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+{"text":"\\section{Introduction}\n\nThe Hanbury Brown and Twiss (HBT) effect can be thought of as Young's double slit experiment using narrowband but incoherent light. Since the source is incoherent the phase is fluctuating \\citep[see for example Figure 7.1 in][]{2006iai..book.....L} and there is no consistent phase difference between the two waves emitted at the slits and thus there will be no interference patterns produced by constructive and destructive superposition visible on the screen, the time averaged intensities (the complex conjugate products of the incoming electric fields) coming from both slits will just add up. But due to the random fluctuations of the phases the intensities will fluctuate. Counting individual photons at photons at two different points on the screen will result in a coincidence rate slightly higher than the Poisson coincidence rate, the intensity fluctuations are correlated. This additional term is what is called the HBT effect.\n\nIn astronomy we need to replace the slits with a two-dimensional superposition of sub-sources. Integrating the amplitude contributions over the new, extended source amounts to taking the Fourier transform of the intensity or brightness distribution on the sky (Van Cittert Zernike theorem). The normalized spatial Fourier transform is called the complex visibility $V$ and can, in principle, be measured at different spatial distances or baselines.  But measuring visibility directly is very difficult for astronomical sources because it requires maintaining optical quality mirrors over long baselines to sub wavelength accuracy. Alternatively, information on the visibility can be obtained by considering the normalized intensity correlation (the probability of coincident detection at two detectors) for two detectors ($1$ and $2$):\n\\begin{equation}\nC_{12} = 1 + |V_{12}|^2\n\\label{eq:corr}\n\\end{equation}\nThis equation is valid over a coherence time which is the time scale\n\\begin{equation}\n\\Delta\\tau \\sim 1\/\\Delta\\nu = \\frac{\\lambda^2}{c \\,\\Delta\\lambda}\n\\end{equation}\non which the phases of incoherent light fluctuate. The phase information of the electrical field is lost when measuring the amplitude squared.  The time resolution $\\Delta t$ for measuring intensity correlations or photon coincidences should be as quick as possible.  If $\\Delta t$ is shorter than $\\Delta\\tau$, the full benefit of the signal in equation \\eqref{eq:corr} is obtained.  If $\\Delta t\\gg\\Delta\\tau$ the signal to noise per measurement interval ${\\rm SNR}(\\Delta t)\\ll1$.  Nevertheless, sufficient SNR can be built up by collecting data over many many $\\Delta t$.  The great advantage of HBT interferometry is that optical parts need to be accurate only to $\\ll c\\Delta t$.  For the $\\Delta t$ achievable nowadays, optical-path tolerances of a millimetre would be adequate.  In contrast, standard interferometry requires optical paths to be precise to better than a wavelength.\n\nIn \\cite{1958RSPSA.248..222B} famously implemented these ideas in the first stellar intensity interferometer, which measured the diameter of Sirius.  In the 1960s the Narrabri intensity interferometer was built to measure more stellar diameters \\citep{1974MNRAS.167..121H}. At that time intensity interferometry was limited to blue-sensitive counting equipment, and after the Narrabri instrument had observed all the hot stars it could, the technique was abandoned for a long time. The possibilities of new, faster counters in red and infrared ranges have recently brought intensity interferometry back into focus and arrays of telescopes such as the proposed Cherenkov Telescope Array (CTA) will allow us to exploit the effect to a greater extent in the years to come \\citep[see for example][]{2013APh....43..331D}. Some illustrative examples are shown in Figures \\ref{fig:simone} and \\ref{fig:fftld}.\n\nThe loss of phase information, as evident from the absolute value in equation (\\ref{eq:corr}) is the major shortcoming of intensity interferometry. But it is not a fundamental shortcoming, because from the theory of quantum optics developed in the 1960s \\citep[see][for a review]{glauber2006}, phase information can be obtained by using three telescopes in concert.\n\nThe normalized three point correlation is:\n\\begin{equation}\nC_{123} = 1 + |V_{12}|^2 + |V_{23}|^2 + |V_{31}|^2 + 2{\\rm Re}[V_{12}V_{23}V_{31}] .\n\\label{eq:3corr}\n\\end{equation}\nThe product in the last term is the real part of the spatial bispectrum of the source.  Its phase is well known in radio astronomy as the closure phase, and useful for eliminating local atmospheric and other influences on phase measurements for individual detectors in a three-antenna setup.  The term has also long been studied in the context of diverse laboratory experiments: \\cite{1963JAP....34..875G} already used photon triple-coincidence as a tool for spectral measurement; \\cite{1978ApOpt..17.2047S} constructed an imaging system using three-point HBT; \\cite{1983JAP....54..473F} performed some remarkable experiments showing three- and four-point acoustic HBT.  The analyses in these works is in terms of classical waves and intensities, so it does not necessarily apply to photon counting, but fortunately it turns out \\citep[see for example][]{1964PPS....84..435M} that for ordinary light, quantum fields can be replaced with classical and intensities interpreted as photon-detection probabilities.\n\n\\cite{2014MNRAS.437..798M} recently reviewed the theory of higher order correlations and provided a simple way of estimating the achievable SNR. In the present work, we will study the feasibility of detecting three-point HBT for astronomical sources. In the following sections we will simulate signals for two and three point correlation measurements of different, simple sources and make some estimates about possible SNR. We will not consider the problem of reconstruction of the actual phase but algorithms for related problems are known \\citep[see, for example][]{Kang:1991:529}.\n\nLet us now briefly preview our results.\n\nIn the next Section we show simulations of the complex visibilities and resulting correlation signals. For example, Figure~\\ref{fig:simone} shows a simulation of a structured disc, inspired by reconstructions of Betelgeuse. Comparing that to the other figures in Section~\\ref{sec:cv} we can immediately see that adequate $(u,v)$ coverage allows to see differences in shape and structure. We also plot the 3-point correlation signals (specifically the bispectra) for a given baseline for different sources.\n\nIn Section~\\ref{snrchap} we rederive the well-known but counterintuitive result that for two-point correlation, the SNR is independent of bandwidth, meaning that decreasing the bandwidth and hence decreasing the count rates will not change the SNR for HBT detection.  A related feature is that the observation time needed goes as the inverse square of the collection area. With larger telescopes such as H.E.S.S. (12~m diameter dishes) and the planned CTA (7m, 12m, and 23m are proposed) this will allow for significant measurements on very short timescales. We also want to emphasize again the possibilities in using arrays with many telescopes increasing the possible $(u,v)$ \\citep[see for example][]{2013APh....43..331D}. For the brightest stars, size measurements would be possible with even a 10~cm diameter mirror with modern off-the-shelf single-photon correlators.  With a 10 cm diameter telescope and a counting system at 50\\% efficiency one could even achieve an SNR of 1 in 36 seconds integration time for Capella b for example. The small baselines possible with such a setup should allow the detection of the 2 point correlation for this binary system, that would require a about a 2 meter aperture to be resolved by a single telescope.\n\nWhile we have considered nearby stars as examples, the results can be trivially scaled.  For example, scaling Sirius up in size and distance by $10^4$ and scaling in luminosity by $10^8$ (keeping apparent size, luminosity and effective temperature the same) would be not unlike a supernova in the LMC.  This suggests that supernova shells in nearby galaxies could be rather easily resolved --- the equivalent of SN~1987a with very modest equipment and the equivalent of SN~2011fe in M101 with the sort of instruments now being proposed.\n\nSection~\\ref{snrchap} also shows the necessary observation times to reach a signal to noise ratio of one for the three-point correlation signal. Skipping ahead to Figure \\ref{fig:qtobs3} we can immediately see that measuring the bispectrum in the visible range using off the shelf photon counters is easily achievable for larger telescopes as the observation time goes with the inverse cubed of the collection area. To emphasize this point: measuring the three-point HBT signal will be feasible and informative for CTA type mirrors.\n\n\n\\label{sec:cv}\n\\begin{figure}\n\\begin{minipage}[b]{0.5\\linewidth}\n\\centering\n\\includegraphics[scale=.45]{structured11.png}\n\\end{minipage}\n\\hspace{0.5cm}\n\\begin{minipage}[b]{0.5\\linewidth}\n\\centering\n\\includegraphics[scale=.45]{structured12.png}\n\\end{minipage}\n\\begin{minipage}[b]{0.5\\linewidth}\n\\centering\n\\includegraphics[scale=.45]{structured13.png}\n\\end{minipage}\n\\hspace{0.5cm}\n\\begin{minipage}[b]{0.5\\linewidth}\n\\centering\n\\includegraphics[scale=.45]{structured14.png}\n\\end{minipage}\n\\caption{Simulation of a structured, crudely limb-darkened disc of 1~mas radius (inspired by Betelgeuse): Top left shows the simulated source brightness distribution, top right shows the absolute value of the visibility squared, the 2 point correlation term in equation (\\ref{eq:corr}) as simulated using a 2D FFT algorithm. The axes of this and the bottom panels are baselines in meters, for $\\lambda=500\\rm\\,nm$. For completeness the bottom panels show the real and imaginary parts of the complex visibility. Note that the imaginary part goes to zero for point symmetric sources.}\\label{fit}  \n\\label{fig:simone}\n\\end{figure}\n\n\\section{Complex visibility vs HBT observables}\n\nThe Van Cittert Zernike theorem allows us to simulate complex visibilities using a 2D FFT. Figure~\\ref{fig:simone} shows a simulation of a structured, crudely limbdarkened disc of 1~mas radius.\\footnote{The baseline scales are always in metres: units of frequency space $(u,v)$ multiplied by an assumed measurement wavelength $\\lambda=500\\rm\\,nm$.} It features three spots that are 25\\% brighter than the rest of the disc. This mimics the features observed on Betelgeuse using standard optical interferometry \\citep{2000MNRAS.315..635Y,2009A&A...508..923H}. Figure \\ref{fig:fftld} shows a similar crudely limb darkened disc without the spots. The power spectrum shows extremely subtle differences, though, looking at real and imaginary parts it is clear one brightness distribution is symmetrical and the other one is not. Clearly it is desirable to have information about the phase of the complex visibility.\n\n\n\\begin{figure}\n\\begin{minipage}[b]{0.5\\linewidth}\n\\centering\n\\includegraphics[scale=.45]{limbdark11.png}\n\\end{minipage}\n\\hspace{0.5cm}\n\\begin{minipage}[b]{0.5\\linewidth}\n\\centering\n\\includegraphics[scale=.45]{limbdark12.png}\n\\end{minipage}\n\\begin{minipage}[b]{0.5\\linewidth}\n\\centering\n\\includegraphics[scale=.45]{limbdark13.png}\n\\end{minipage}\n\\hspace{0.5cm}\n\\begin{minipage}[b]{0.5\\linewidth}\n\\centering\n\\includegraphics[scale=.45]{limbdark14.png}\n\\end{minipage}\n\\begin{minipage}[b]{0.5\\linewidth}\n\\centering\n\\includegraphics[scale=.45]{examplelimbdark26ab.png}\n\\end{minipage}\n\\hspace{0.5cm}\n\\begin{minipage}[b]{0.5\\linewidth}\n\\centering\n\\includegraphics[scale=.45]{examplelimbdark26ph.png}\n\\end{minipage}\n\\caption{Simulation of a roughly limb darkened disc of $1\\rm\\,mas$ radius: left the simulated source, on the right the squared visibility. As we can see the imaginary part of the complex visibility is zero except for roundoff errors. The two lower panels show absolute value and cosine of the phase of the bispectrum at the position of a third detector with respect to a fixed 107~m long baseline of two detectors as indicated by the black line. Note that these quantities cannot be measured directly but their product can.}\n\\label{fig:fftld}\n\\end{figure}\n\nThe bispectrum is a function of two baselines, so it cannot be shown in a single figure. In Figure \\ref{fig:diffat} we show an example with two detectors (hence one baseline) fixed. \\footnote{A program to simulate other baselines as well as different sources, such as a binary, is provided in the supplementary material.}\n\nFeatures of 25\\% difference in brightness appear at the level of $10^{-3}$ in the bispectrum. To assess the feasibility of measuring this one has to calculate the signal to noise ratio.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.\\linewidth]{compare26.png}\n\\caption{Bispectra $2{\\rm Re}[V_{12}V_{23}V_{31}]$ as the position of one detector is varied while the other two detectors are fixed (at opposite ends of the black line indicating a baseline of 107~m as before). The left panel corresponds to the source in figure \\ref{fig:fftld} while the right panel to the disc in figure \\ref{fig:simone}. The center panel shows the difference.}\n\\label{fig:diffat}\n\\end{figure}\n\n\\section{Count rates and Signal to Noise}\n\\label{snrchap}\n\n\\subsection{count rates}\n\n\\begin{figure}\n\\begin{minipage}[b]{0.5\\linewidth}\n\\centering\n\\includegraphics[scale=.45]{qeffpdm-eps-converted-to.pdf}\n\\end{minipage}\n\\hspace{0.5cm}\n\\begin{minipage}[b]{0.5\\linewidth}\n\\centering\n\\includegraphics[scale=.45]{qeffidq-eps-converted-to.pdf}\n\\end{minipage}\n\\caption{Quantum efficiency 7th order polynomial fits for PicoQuant PDM series$^{1}$ (left) and the IDQuantique ID100 series$^{2}$ from figures on the manufacturers' data sheets using the WebPlotDigitizer$^{3}$ by Ankit Rohatgi.}\n\\label{fig:qefffit}  \n\\end{figure}\nFor purposes of estimating count rates we approximate stars by black body discs. In particular we approximate Sirius (temperature 9940~K, diameter $0.006''$), Betelgeuse (3500~K, $0.04''$), and Capella~a (5700~K, $0.003''$). Let $r$ be the average count rate for one detector, the number of counts per coherence time $\\Delta\\tau$ then has the simple form \\citep[][]{2014MNRAS.437..798M}:\n\\begin{equation}\n\\gamma r\\Delta\\tau = \\frac{\\gamma A\\Omega} {\\lambda^2 (\\exp{[h c\/\\lambda k_B T]} - 1)}\n\\label{eq:snrarea2det}\n\\end{equation}\nwhere $\\Omega$ is the solid angle of the source, $A$ the collection area of the detectors, $\\lambda$ the measurement wavelength, $T$ the surface temperature of the star, and $\\gamma$ the quantum efficiency of the detector. Today's possibilities allow for reasonable quantum efficiencies at high time resolutions in the middle of the visible spectrum. \nOff the shelf photon counters such as the Pico Quant PDM series$^{[1]}$ or the ID100 series of IDQuantique$^{[2]}$ can help to get a feeling for achievable count rates (see best case parameters in Table \\ref{detectors} and fits to quantum efficiency measurements by the manufacturers in Figure \\ref{fig:qefffit}). \nZooming in on the wavelength region that is covered by these detectors, Figure \\ref{fig:qrate} shows the expected count rates for different stars using narrowband filters with $\\Delta\\lambda = 1\\rm\\,nm$.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{l*{6}{c}r}\ndetector & max efficiency & wavelength of max efficiency & time resolution $\\Delta t$& dead time \\\\\n\\hline\nPicoquant & 49 \\% & 550 nm & 50 ps & 80 ns \\\\\nPDM series$^{1}$ &&&& \\\\\n\\hline\nIDQuantique & 35 \\% & 500 nm & 40 ps & 45 ns \\\\\nID100 series$^{2}$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Some best case specifications of the two mentioned single photon counters.}\n\\label{detectors}\n\\end{table}\n\\footnotetext[1]{http:\/\/www.picoquant.com\/images\/uploads\/downloads\/pdm\\_series.pdf} \n\\footnotetext[2]{http:\/\/www.idquantique.com\/images\/stories\/PDF\/id100-single-photon-detector\/id100-specs.pdf}\n\\footnotetext[3]{http:\/\/arohatgi.info\/WebPlotDigitizer\/app\/}\n\n\n\\begin{figure}\n\\begin{minipage}[b]{\\linewidth}\n\\centering\n\\includegraphics[scale=.8]{qeffratepico.png} \n\\end{minipage}\n\\begin{minipage}[b]{\\linewidth}\n\\centering\n\\includegraphics[scale=.8]{qeffrateidq.png} \n\\end{minipage}\n\\caption{Upper panel: expected count rates per coherence time $\\Delta\\tau$ using two Picoquant PDM series photon counters after filtering to a bandwidth of $\\Delta\\lambda = 1\\rm\\,nm$ for different stars using the quantum efficiency estimates given in the left panel of figure \\ref{fig:qefffit}. The plot shows equation (\\ref{eq:snrarea2det}) evaluated for Sirius (blue), Betelqeuse (magenta), and Capella a (cyan) at the approximate parameters given in table \\ref{stars}. The lower panel shows the expected count rates per coherence time for measurements using the IDQuantique ID100 series.}\n\\label{fig:qrate}\n\\end{figure}\n\n\n\n\n\\subsection{Signal to Noise Ratio for 2 Detectors}\n\nOver a coherence time $\\Delta \\tau$ there will be $|V_{12}|^2(\\gamma r\\Delta \\tau)^2$ coincidences. \nThe time resolution, or counting time of a detector $\\Delta t$ will be much larger than the coherence time, $\\Delta t \\gg \\Delta\\tau$, so in one counting interval there will be $\\Delta t\/\\Delta\\tau$ such contributions, giving $|V_{12}|^2 (\\gamma r)^2\\Delta\\tau\\Delta t$. Meanwhile there will be $(\\gamma r\\Delta t)^2$ random coincidences corresponding to the first term of equation (\\ref{eq:3corr}),  giving $\\gamma r\\Delta t$ of noise, so the signal to noise ratio in $\\Delta t$ becomes:\n\\begin{equation}\nSNR(\\Delta t) =  |V_{12}|^2 \\gamma r \\Delta\\tau ,\n\\end{equation}\nNote that the signal to noise ratio is independent on the bandwidth: decreasing bandwidth results in lower rates but will not change the signal to noise ratio.\n\nEvaluating the signal to noise ratio over an observation time $T_{\\rm obs}$, we assume that the intensity fluctuations measured in one counting time $\\Delta t$ are not correlated to the ones measured in the next interval, which is a reasonable assumption, since at relevant wavelengths $\\Delta t\\gg\\Delta\\tau\\: (\\sim 10^{-12}$). Thus, uncorrelated, Gaussian noise dominates and the SNR adds in quadrature:\n\\begin{equation}\nSNR = SNR(\\Delta t) \\sqrt{T_{\\rm obs}\/\\Delta t}\n\\end{equation}\nWe can now find the required observation time for a required signal to noise ratio using equation (\\ref{eq:snrarea2det}):\n\\begin{equation}\nT_{\\rm obs}(SNR) = \\frac{SNR^2}{|V_{12}|^4}\\Delta t  \\Big[\\frac{\\gamma A \\Omega} {\\lambda^2 (\\exp{[h c\/\\lambda k_B T]} - 1)}\\Big]^{-2}\n\\label{eq:TOBS}\n\\end{equation}\nFigure \\ref{fig:qtobs2} shows the the observation times necessary for some example cases to reach an SNR of $1$ for $|V_{12}|^2 = 1$. Clearly detector technology has improved greatly in the past half century, observing Sirius Hanbury Brown and Twiss needed a few minutes to reach a signal to noise ratio of 1 using a $5$-foot searchlight mirrors. Now the same could be done using $10\\rm\\,cm$ diameter mirrors.\n\n\\begin{figure}\n\\begin{minipage}[b]{\\linewidth}\n\\centering\n\\includegraphics[scale=.8]{2pthbtpicoq.png} \n\\end{minipage}\n\\vskip -1.4cm\n\\begin{minipage}[b]{\\linewidth}\n\\centering\n\\includegraphics[scale=.8]{2pthbtidq.png} \n\\end{minipage}\n\\caption{Upper panel: observation time estimates in seconds to reach an HBT SNR of $1$ for dishes with diameters of $.1\\rm\\,m$ (scales on the left) and $10\\rm\\,m$ (scales on the right) using the Pico Quant PDM series single photon counters, see table \\ref{stars} for color code. The lower panel shows observation time estimates using ID100 detectors by IDQuantique.}\n\\label{fig:qtobs2}\n\\end{figure}\n\n\n\n\n\\subsection{Signal to Noise Ratio for 3 Detectors}\n\\begin{figure}\n\\begin{minipage}[b]{\\linewidth}\n\\centering\n\\includegraphics[scale=.8]{3pthbtpico.png} \n\\end{minipage}\n\\vskip -1.4cm\n\\begin{minipage}[b]{\\linewidth}\n\\centering\n\\includegraphics[scale=.8]{3pthbtidq.png} \n\\end{minipage}\n\\caption{Upper panel: observation time estimates to reach an SNR of $1$ for $|V_{123}| = 10^{-3}$ for dishes with diameters of $.1\\rm\\,m$ (scales on the left) and $10\\rm\\,m$ (scales on the right) using Pico Quant PDM series single photon counters vs wavelength, see table \\ref{stars} for color code. The lower panel shows observation time estimates using ID100.}\n\\label{fig:qtobs3}\n\\end{figure}\n\nOver a coherence time $\\Delta \\tau$ there will be $V_{123}(\\gamma r\\Delta \\tau)^3$ excess coincidences, where $V_{123}$ is the last term in equation (\\ref{eq:3corr}). So over a time $\\Delta t$ there will be $V_{123}(\\gamma r)^3 \\Delta \\tau^2\\Delta t$ signal coincidences. Meanwhile there will be $(\\gamma r \\Delta t)^{3\/2}$ of noise, hence the signal to noise ratio is:\n\\begin{equation}\nSNR_{3}(\\Delta t) \\sim V_{123}(\\gamma r\\Delta\\tau)^{3\/2} (\\Delta\\tau\/\\Delta t)^{1\/2} ,\n\\label{3detSNR}\n\\end{equation}\nagain understanding $r\\Delta \\tau$ as the count rate at one detector over a coherence time. We can now rewrite and expand as before to find the observation time for a specific signal to noise ratio:\n\\begin{equation}\nT_{\\rm obs}(SNR) = \\frac{SNR^2}{V_{123}^2}\\frac{\\Delta t^2}{\\Delta\\tau}  \\Big[\\frac{\\gamma A \\Omega} {\\lambda^2 (\\exp{[h c\/\\lambda k_B T]} - 1)}\\Big]^{-3}\n\\label{eq:SNRT3}\n\\end{equation}\nComparing with equation (\\ref{eq:TOBS}) we note the steeper dependence not only on the quantum efficiency $\\gamma$ but also on counting time resolution $\\Delta t$ and detector area $A$.\nAs we saw from Figures \\ref{fig:simone} and \\ref{fig:diffat}, features with brightness differences of around 25\\% show up in $V_{123}$ at the level of $10^{-3}$. Figure \\ref{fig:qtobs3} shows observation time estimates for our example cases for an $SNR$ of $1$ and $V_{123} = 10^{-3}$. We can immediately see that attempting three-point HBT with smaller sized telescopes is not an option but increasing the telescope diameters dramatically reduces observation times as the cube of the area thus making it possible to measure the bispectrum. With the equivalent of  $10\\rm\\,m$ diameter mirrors these should be easily detectable in the case of Betelgeuse or Sirius.\n\n\\section{Outlook}\nThe results shown in Figures \\ref{fig:qtobs2} and \\ref{fig:qtobs3} suggest that the recovery of HBT phase is feasible with present day detector technology and may lead to major advances in stellar imaging. Many technical problems will need to be solved first. The most important of these are the following.\n\\begin{itemize}\n\\item Designing suitable configurations for three detectors is essential. In particular, for Betelgeuse $10\\rm\\,m$ diameter mirrors would wash out the interference patterns. Some combination of small mirrors to resolve the large scale and large mirrors to resolve small scale structures is desirable.\n\\item An image reconstruction algorithm using two- and three-point HBT signal is needed.\n\\item Large mirrors such as proposed by CTA introduce non-isochronicity of a nanosecond or more \\citep{2006ApJ...649..399L}.  In order to benefit from fast photon counting, the light paths from different parts of the mirror need to be equalized to the sub $\\Delta t$ level (submillimeter) level.  Since only small fields of view are involved, it is plausible that a simple spherical-aberration compensators could do the job.\n\\end{itemize}\n\n\\section*{Acknowledgments}\n\nWe thank the referee, Paul Nu\\~nez, for many helpful comments.\n\n\\clearpage\n\\bibliographystyle{astron}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nUniaxially modulated structures are observed in very different\nclasses of magnetic and ferroelectric substances. In many cases they exhibit\nrather complex phase diagrams with large varieties of phases. Phase \ntransitions from a high temperature paramagnetic or paraelectric phase \n(paraphase) to commensurately and incommensurately modulated phases occur, as \nexternal control parameters like temperature \nand elastic stresses are varied. Microscopic models are successfully\nused for the description of these modulated systems. \nThey were reviewed e.g.\\ in \\cite{Sel88}. An interesting example is \nthe $p$-state chiral clock model \\cite{Yeo82}, whose Hamiltonian is\n\\begin{equation}\n\\fl H = - J_0 \\sum\\limits_{\\alpha} \\sum\\limits_{<ij>} \\cos \\left[ \\frac{2 \\pi}{p}\n\\left( n_{i, \\alpha} - n_{j, \\alpha} \\right) \\right] \n- J \\sum\\limits_i \\sum\\limits_{\\alpha} \\cos \\left[ \\frac{2 \\pi}{p}\n\\left( n_{i, \\alpha} - n_{i, \\alpha+1} + \\Delta\n\\right) \\right].\n\\label{hamiltonian}\n\\end{equation}\n$\\alpha$ labels the layers perpendicular to the direction of the modulation\n(chiral direction) and $i$, $j$ the crystal units in these layers.\n$\\left< ij \\right>$ runs over neighbouring pairs in the layers.\nThe integer variables $n_{i, \\alpha}$ describe the state of the unit\n$(i,\\alpha)$. They assume one of the values from\n0 to $p-1$. Below they are called spins.\nThe two terms in equation (\\ref{hamiltonian}) describe couplings ($J_0 > 0$,\n$J > 0$)\nbetween nearest neighbours in the same and in\nadjacent layers, respectively.\\\\\nIn the ground state every layer is ferromagnetically ordered.\nDepending on the value of $\\Delta$, various ordering patterns of the different layers are \nrealized. For $0 \\leq \\Delta < \\frac{1}{2}$ nearest neighbours in the chiral direction\ncouple ferromagnetically (ferromagnetic bond),\nthus leading to a ferromagnetic ground state\nwhere all spins are equal. For $\\frac{1}{2} \n< \\Delta \\leq 1$ \nthe spin increases by one for successive layers (chiral bond),\nthus yielding the right-handed chiral pattern\n\\begin{displaymath}\n\\ldots ~ 0 ~ 1 ~ 2 ~ \\ldots ~ (p-1) ~ 0 ~ 1 ~ \\ldots\n\\end{displaymath}\n$\\Delta = \\frac{1}{2}$ is a multiphase point at which an infinity of different phases are\ndegenerate since ferromagnetic and chiral bonds have the same energy.\\\\\nWhereas the three-state model ($p = 3$) \\cite{Ost81,Hus81}\nhas been very thoroughly investigated, \nonly few results are known for the general case $p \\geq 4$. \nThere are derivations of the low\ntemperature phase diagram of the general $p$-state model by an expansion of the free\nenergy in the vicinity of the multiphase point \\cite{Yeo82} as well as by a low\ntemperature mean-field theory \\cite{Yeo84}, in which it was claimed that, for\nthe four-state model ($p = 4$), only the phases $\\left< 1 2^k \\right>$, $\\left< 1 2^k 1 2^{k+1}\n\\right>$, $\\left< 2^k 3 \\right>$, $\\left< 2^k 3 2^{k+1} 3 \\right>$, $\\left< 4\n\\right>$, and $\\left< \\infty \\right>$ ($k = 0, 1, 2, \\ldots$) are stable at low temperatures.\n$\\left<  u_1 \\ldots u_r \\right>$ is a shorthand notation for the phase with a period\nconsisting of $r$ bands with $u_1, u_2, \\ldots , u_r$ \nlayers with spins $n$, $n+1, \\ldots, n+r$ (all modulo $p$) respectively.\nThe phase $\\left< 12 \\right>$, for example, is given by the layer sequence\n\\begin{displaymath}\n\\ldots ~ 0 ~ 1 ~ 1 ~ 2 ~ 3 ~ 3 ~ 0 ~ 1 ~ 1 ~ \\ldots\n\\end{displaymath}\nThe ferromagnetic and chiral ground states are denoted by $\\left< \\infty \\right>$ \nand $\\left< 1 \\right>$ respectively.\\\\\nMcCullough \\cite{Mcc92} investigated the phase diagram for\n$p = 3$, 4, and 5 using the mean-field transfer-matrix (MFTM) method.\nFrom the numerical \nextrapolation of the data it was concluded that the low temperature phase diagrams\nfor $p = 3$ and $p = 4$ were consistent \nwith the results of the low-temperature\nseries expansion \\cite{Yeo82,Yeo81}. It is interesting, that, for $p = 5$,\nnew phases not predicted by the\nlow-temperature series expansion \\cite{Yeo82} were found to be stable at low\ntemperatures.\\\\\nScholten and King \\cite{Sch96} presented Monte Carlo simulations of\nthe four- and the six-state models. They investigated especially the transition\nfrom the modulated phases to the ferromagnetic phase (i.e. $\\Delta < \\frac{1}{2}$).\nAs it was not possible to resolve particular phases, they determined the \"interface\nspacing\" as the average number of layers in a band for a given phase.\nThey claimed that, for $\\Delta = 0.45$, the results were not\ninconsistent with the predictions of Yeomans. In the case $p =4$ and $\\Delta = 0.2$\nnew phases with an interface spacing larger than the interface spacings of the phases\npredicted in \\cite{Yeo82} were observed \nclose to the transition to the ferromagnetic phase.\\\\\nRecently the four-state chiral clock model was shown \\cite{Ple97} to be \na special case of the Double Ising Spin (DIS) model \\cite{Ple94,Neu94,Ple96},\nwhich was introduced to describe uniaxially modulated ferroelectrics.\\\\\nIn the following new results for the four-state chiral clock model are presented. \nIn section \\ref{sec2} and \\ref{sec3}\nwe will reexamine the low temperature phase diagram and \ndiscuss discrepancies with previous results.\nIn section \\ref{sec4} it is shown that the \ntransition from the modulated phases to the paramagnetic phase belongs to the\nuniversality class of the 3d-$XY$ model, and in section 5\nshort conclusions are given.\n\n\\section{The low temperature series expansion\\label{sec2}}\nThe present series expansion technique for the four-state chiral clock ($CC_4$) model\nis similar to the method\ndeveloped by Fisher and Selke \\cite{Fis80} for the Axial Next Nearest \nNeighbour Ising (ANNNI) model. At low temperatures the reduced free energy\nper spin $f = \\frac{F}{Nk_BT}$ ($N$ is the total number of spins)\nmay be expanded in the form \\cite{Fis80}\n\\begin{equation}\nf = \\frac{E_0}{k_BT} - \\frac{1}{N} \\sum\\limits_{n \\geq 1} \\Delta Z_N^{(n)}.\n\\label{free_energy}\n\\end{equation}\n$\\Delta Z_N^{(n)}$ is the total contribution \nto the partition function from configurations in which \n$n$ spins have flipped (as compared to the ground state). $E_0$,\nthe ground state energy per spin, can be expressed \\cite{Yeo82,Yeo81} in terms\nof the structural variables \\cite{Fis80}\n$l_k = L_k\/L$ ($L_k$: number of $k$-layer bands; $L$: total number of layers):\n\\begin{displaymath}\nE_0 \\left( \\left\\{ l_k \\right\\} \\right) = - \\frac{1}{2} q_\\perp J_0 - J_1 \n-J_1 \\, \\delta \\, \\sum\\limits_{k \\geq 1} l_k\n\\end{displaymath}\nwith $J_1 = J \\cos \\left( \\frac{\\pi}{2} \\Delta \\right)$ and $\\delta = \\tan \\left( \n\\frac{\\pi}{2} \\Delta \\right) - 1$. The number of nearest neighbours in the \nlayers is $q_\\perp$; it is 4 for the primitive cubic lattice.\\\\\nThe contributions $\\Delta Z_N^{(n)}$ are expressed\nin terms of the elementary Boltzmann factors\n\\begin{displaymath}\n\\fl w = \\exp \\left( - K_0 \\right), ~ x = \\exp \\left( - 2 K \\, \\cos \\left( \n\\frac{\\pi}{2} \\Delta \\right) \\right)\n~~ \\mbox{and} ~~ y = x^{1+ \\delta} = \\exp \\left( - 2 K \\, \\sin \n\\left( \\frac{\\pi}{2} \\Delta \\right) \\right)\n\\end{displaymath}\nwith $K_0 = J_0\/(k_BT)$ and $K = J\/(k_BT)$. \nThe reduced free energy per spin can be expanded in a convergent power series\nof $w$, provided that $x \\gg w$, i.e.\\ if $J_0$ is large compared to $J$ (which\nis assumed throughout this paper). \nThe weight $w$ results from changing\nan in-layer bond between spins with equal values to a bond between spins\nwith values differing by 1. \nThe lowest orders involved are $w^{q_\\perp}$ (overturning one spin),\n$w^{2 q_\\perp-2}$ (overturning two neighbouring spins in one layer) and\n$w^{2 q_\\perp}$ (overturning two spins not being in-layer nearest neighbours).\\\\\nThere are\nthree possible environments of a given spin (the numbers in\nparentheses are the values of the spins in three consecutive layers where the\nconsidered spin belongs to the middle layer): (a) spins with\ntwo ferromagnetic bonds in the chiral direction (e.g.\\ $0\\hat{0}0$), (b) spins\nwith one ferromagnetic and one chiral bond (e.g.\\ $0\\hat{1}1$), and (c) spins \nwith two chiral bonds (e.g.\\ $0\\hat{1}2$).\\\\\nLet us discuss, as an example, the contribution\nto $\\Delta Z_N^{(1)}$ [first-order term in equation (\\ref{free_energy})] for case (a). \nBy overturning one spin, three different final states can be obtained\n($m$ being the initial state): $(m+1) ~ \\mbox{mod} ~ 4$, $(m+2) ~ \\mbox{mod} ~ 4$, and \n$(m+3) ~ \\mbox{mod} ~ 4$. This leads to the Boltzmann factor\n\\begin{eqnarray}\n\\fl \\sum\\limits_{n = 1}^3 \\exp \\left[ - \\left( E_f(n) - E_i \\right)\/\\left( k_BT \\right) \\right]\n\\nonumber \\\\\n\\lo=  \\sum\\limits_{n = 1}^3 \\left( \\exp \\left\\{ 2K \\, \\cos \\left( \\frac{\\pi}{2} \\Delta \\right)\n\\, \\left[ \\cos \\left( \\frac{\\pi}{2} n \\right) -1 \\right] \\right\\} \\right. \\nonumber \\\\\n\\left. \\times\n\\exp \\left\\{ q_\\perp \\, K_0 \\left[ \\cos \\left( \\frac{\\pi}{2} n \\right) -1 \\right] \\right\\} \\right)\n\\nonumber \\\\\n\\lo= x \\, w^{q_\\perp} + x^2 \\, w^{2q_\\perp} + x \\, w^{q_\\perp}.\n\\label{boltz}\n\\end{eqnarray}\nIt is obvious from\nequation (\\ref{boltz}) that the process $m \\longrightarrow (m+2) ~ \\mbox{mod} ~ 4$ does not contribute\nto the lowest order term in the expansion, as it has the same in-layer Boltzmann factor\n$w^{2q_\\perp}$ as the higher order process by which the values of two uncoupled\nspins change by 1. In fact, this process of the order $w^{2q_\\perp}$\ndoes not even contribute to the \nlowest order correction term, which is of the order $w^{2q_\\perp-2}$ (flipping of two\nneighbouring spins in one layer \\cite{Fis80}).\\\\\nIn reference \\cite{Yeo82} the following contribution to $\\Delta Z_N^{(1)}$ for the case (a) is\ngiven:\n\\begin{eqnarray}\n\\fl \\sum\\limits_{n = 1}^3 \\exp \\left[ - \\left( E_f(n) - E_i \\right)\/\\left( k_BT \\right) \\right]\n\\nonumber \\\\\n\\lo= \\sum\\limits_{n = 1}^3 \\exp \\left\\{ 2K \\, \\cos \\left( \\frac{\\pi}{2} \\Delta\n\\right) \\, \\left[ \\cos \\left( \\frac{\\pi}{2} n \\right) -1 \\right] \\right\\} \\omega^{q_\\perp}\n\\nonumber \\\\\n\\lo= \\left( x + x^2 + x \\right) \\omega^{q_\\perp}\n\\label{boltzyeo}\n\\end{eqnarray}\nwith\n\\begin{equation}\n\\omega = \\sum\\limits_{n = 1}^3 \\exp \\left\\{ K_0 \\left[ \\cos \\left( \\frac{\\pi}{2} n \\right)\n-1 \\right] \\right\\} \n= w + w^2 + w .\n\\label{boltzyeo2}\n\\end{equation}\nA comparison of equations (\\ref{boltzyeo}) and (\\ref{boltzyeo2}) \nwith equation (\\ref{boltz}) reveals that\nthe treatment of the in-layer bonds is erroneous in reference \\cite{Yeo82}. \nThe free energy is written in reference \\cite{Yeo82} as an expansion in terms of the (erroneous)\nBoltzmann factor $\\omega$.\nAs a consequence, contributions from different\norders of the expansion are treated in reference \\cite{Yeo82} as \nif they were of the same order. Thus, in our example,\nthe term $x^2$, resulting from the process $m \\longrightarrow (m+2) ~ \\mbox{mod}\n~ 4$ and contributing to a higher order correction in the polynominal\nexpansion in $w$ [see equation (\\ref{boltz})], contributes \nto the lowest order in \\cite{Yeo82} [see equation (\\ref{boltzyeo})]. This error\nis repeated for all considered spin configurations and for all considered $p$-state\nmodels ($p \\geq 4$), thus\nleading to a wrong low temperature phase diagram not only for the $CC_4$ model, but\nalso for the generalised $p$-state chiral clock model with $p \\geq 4$. One should\nemphasise that the treatment\nof the in-layer bonds is correct in the analyses of the $CC_3$ model \\cite{Yeo81}.\\\\\nWith the correct contributions, the reduced free energy (2) in first order is given by\n\\begin{equation}\n\\fl f = - \\frac{1}{2} q_\\perp K_0 - K_1 - \\frac{1}{2} K_1 \\delta - \\left( 1 + x \\, y \\right) \nw^{q_\\perp} + a_1( \\delta ) \\, l_1 + \\sum\\limits_{k \\geq 3} a_k( \\delta ) \\, l_k\n+ O(w^{2 q_\\perp -2})\n\\label{f_firstorder}\n\\end{equation}\nwith \n\\begin{displaymath}\na_1( \\delta ) = - \\frac{1}{2} K_1 \\delta - \\left( 2 y - x y -1 \\right) w^{q_\\perp}\n\\end{displaymath}\nand\n\\begin{displaymath} \na_k( \\delta ) = \\left( k - 2  \\right) \\left[ \\frac{1}{2} K_1 \\delta - \\left( 2 x - x y -1   \n\\right) w^{q_\\perp} \\right].\n\\end{displaymath}\nThe set of structural variables $l_k$ minimizing\n$f$ for given values of $\\delta$ and $T$ determine the stable phases occurring in\nfirst order (see figure 1): the $\\left< \\infty \\right>$-, the $\\left< 1 \\right>$-,\nand the $\\left< 2 \\right>$-phase.\nPhases $\\left< \\infty \\right>$ and $\\left< 2 \\right>$ are, in this order\nof the expansion, separated by a boundary, at which all phases that are degenerate at\nthe multiphase point and that do not contain 1-layer bands have the same free energy. Likewise,\nphases containing only 1- and 2-layer bands are still degenerate on the boundary\nbetween the $\\left< 1 \\right>$- and the $\\left< 2 \\right>$-phase.\\\\\nOne could now proceed in considering processes involving two spins, then three\nspins and so on. This is very cumbersome and only feasible for processes involving\nfew spins. \nIn the next section the phases stable in general order in the series expansion\nwill be determined using a transfer-matrix method.\n\n\\section{Transfer-matrix method\\label{sec3}}\n\\subsection{Introductory remarks}\nOne should first note that the Hamiltonian (\\ref{hamiltonian}) is left invariant by\nthe transformation\n\\begin{eqnarray}\n&& \\Delta \\longrightarrow \\Delta ' = 1 - \\Delta \\nonumber \\\\\n&& n_{i,\\alpha} \\longrightarrow n_{i,\\alpha}' = \\left( - n_{i,\\alpha}\n + \\alpha \\right) ~ \\mbox{mod} ~ 4 .\n\\label{trafo}\n\\end{eqnarray}\nTherefore, the phase diagram of the $CC_4$ model is invariant under a reflection \nin the line $\\Delta = \\frac{1}{2}$. In the following we will discuss the low\ntemperature phase diagram for the case $\\Delta > \\frac{1}{2}$, i.e.\\ we will analyse\nin detail the stability of the boundary line between the $\\left< 1 \\right>$- and\nthe $\\left< 2 \\right>$-phase, the phase diagram for $\\Delta < \\frac{1}{2}$ being\ninferred by the transformation (\\ref{trafo}).\\\\\nIn the ground state and in the low temperature expansion every phase $\\left< \\nu\n\\right>$ consists of a periodic arrangement of a sequence of $n(\\nu)$ layers called\n$\\nu$-sequences [$n(\\nu)$ is the period of the phase]. Suppose now that in a certain\norder of the series expansion two stable phases, $\\left< \\nu_1 \\right>$ and \n$\\left< \\nu_2 \\right>$, are separated by a boundary at which the phases produced by \n$\\nu_1$- and $\\nu_2$-sequences are degenerate (see figure \\ref{fig2}).\nIn first order the boundary under\nconsideration separates the phases $\\left< 1 \\right>$ and $\\left< 2 \\right>$.\nAt higher order a new phase $\\left< \\nu \\right> = \\left< \\nu_1 \\nu_2 \\right>$ \nconsisting of a structure\nwith alternating $\\nu_1$- and $\\nu_2$-sequences might be stable in the vicinity\nof the boundary. If \n\\begin{equation}\na_\\nu = f_{\\left< \\nu \\right> } - \\frac{n(\\nu_1)}{n(\\nu_1)+n(\\nu_2)} \nf_{\\left< \\nu_1 \\right> } - \\frac{n(\\nu_2)}{n(\\nu_1)+n(\\nu_2)}\nf_{\\left< \\nu_2 \\right> }\n\\label{anu}\n\\end{equation}\nis negative, the new phase has a lower free energy than the phases $\\left< \\nu_1 \\right>$ and \n$\\left< \\nu_2 \\right>$ \\cite{Fis80,Sen93} and it will be stabilized in the vicinity of the\n$\\left< \\nu_1 \\right>  :  \\left< \\nu_2 \\right>$ boundary (see figure 2a).\nThe stability of the boundaries between the phases $\\left< \\nu_1 \\right>$ and \n$\\left< \\nu_1 \\nu_2 \\right>$ and the phases $\\left< \\nu_1 \\nu_2 \\right>$ and\n$\\left< \\nu_2 \\right>$ must then be examined at higher orders.\nIf, on the other hand, $a_\\nu$ is positive,\nthe phase $\\left< \\nu_1 \\nu_2 \\right>$ (and therefore every phase consisting of\n$\\nu_1$- and $\\nu_2$-sequences) has a higher free energy than either $\\left< \\nu_1 \\right>$\nor $\\left< \\nu_2 \\right>$. The boundary is a true phase boundary which remains stable\nin all orders of the low temperature series expansion (see figure 2b).\\\\\nThe reader is referred to references \\cite{Fis80} and \\cite{Yeo81}\nfor details concerning\nthe construction of the series expansion to general order.\\\\\n\n\\subsection{Formulation in terms of transfer matrices and vectors\\label{sec3b}} \nThe sign of $a_\\nu$, and therefore the stability of the phase $\\left< \\nu \\right>$,\nis determined by the leading term in its expansion in terms of $w$.\nThis term is obtained by considering all flipping processes involving\na spin chain of $n(\\nu)-1$ spins in $n(\\nu)-1$ different layers\n\\cite{Yeo81}. Besides the linear configuration with all $n(\\nu)-1$ spins \nconnected, the various decompositions of this configuration into 2, 3, $\\ldots$, $n(\\nu)-1$\ndifferent parts must be taken into account.\nThe contributions from these processes can be written\nas a product of transfer matrices and vectors. \nThe matrices describe a bond between two flipping spins, the vectors \nan initial or a final bond preceding the first or following the last flipped spin\nrespectively.\\\\\nEvery spin can flip to three different values and hence\n$3 \\times 3$ matrices occur. As we are only interested in the\nsign of the $a_\\nu$ we can restrict ourselves to the two processes contributing\nin lowest order, thus excluding the process $m \\longrightarrow (m+2) ~ \\mbox{mod}\n~ 4$ only relevant for the correction term. Of course,\nif one considers all possible processes (i.e.\\ $3 \\times 3$ matrices),\nthe leading term is identical to the term obtained by the $2 \\times 2$ matrices.\nThis has already been noticed in the low temperature analyses of a six-state\nclock model with competing axial nearest and next-nearest neighbour couplings \\cite{Sen93},\nwhere the corresponding $2 \\times 2$ matrices have been considered instead\nof the general $5 \\times 5$ matrices.\\\\\nAs two axial next nearest neighbours\nare either coupled by a ferromagnetic or by a chiral bond, only two different matrices\nare to be constructed. For a ferromagnetic or a chiral bond between two spins\nin the layers $\\alpha$ and $\\alpha$+1 one obtains, respectively, the transfer matrices\n\\begin{equation}\n\\altmat{F}_{\\alpha, \\alpha+1} =\n\\left( \\begin{array}{ll} \n1-x ~ & ~ x(1-y) \\\\\nx(1-y^{-1}) ~ & ~ 1-x\n\\end{array} \\right) w^{q_\\perp}\n\\label{matrix_ferro}\n\\end{equation}\nand\n\\begin{equation}\n\\altmat{C}_{\\alpha, \\alpha+1} =\n\\left( \\begin{array}{ll}\n1-y ~ & ~ y(1-x^{-1}) \\\\\ny(1-x) ~ & ~ 1-y\n\\end{array} \\right) w^{q_\\perp}.\n\\label{matrix_chiral}\n\\end{equation}\nThe matrix elements are the Boltzmann factors\nfor a simultaneous change of the values of the two\nspins. The first (second) row corresponds to a\nchange $\\Delta n_{i,\\alpha} = +1 (-1)$ and\nthe first (second) column to $\\Delta n_{i,\\alpha+1} = +1 (-1)$.\nEvery element of the matrices \\altmat{F} and \\altmat{C} is a sum of two terms,\nthe first term resulting from changing the values of two axially coupled \nspins. As already mentioned, disconnected pairs of spins\n(i.e.\\ two spins that are not neighbour to each other but neighbour\nto an unchanged spin) also contribute to the partition sum. Since every\ndisconnected pair must be associated with a minus sign \\cite{Fis80}, the\ncorresponding Boltzmann factors enter the different matrices\nwith a negative sign.\\\\\nThe factor $w^{q_\\perp}$ resulting from changing \nthe in-layer bonds in layer $\\alpha$+1\nis common to all elements of the matrices \\altmat{F} and \\altmat{C}.\nThis is a direct consequence of the fact that only flipping processes\n$m \\longrightarrow (m \\pm 1) ~ \\mbox{mod} ~ 4 $ are to be considered for\nobtaining the leading order in the expansion of $a_\\nu$.\nFor the full $3 \\times 3$ matrices this is not\nthe case as the flipping process $m \\longrightarrow (m+2) ~ \\mbox{mod} ~ 4 $ has the in-layer\nBoltzmann factor $w^{2 q_\\perp}$.\nIn reference \\cite{Yeo82} the phase diagram has been determined to general order using $3 \\times 3$\ntransfer matrices. Due to the erroneous treatment of the in-layer interactions \n(see section \\ref{sec2}) the \"common term\" $\\omega^{q_\\perp}$ has been\nfactorized, thus leading, again,\nto the treatment of terms belonging to different orders as being of the same order.\\\\\nA spin at the end of the spin chain is neighbour of an unchanged spin. To determine\nthe contributions of these spins, four different cases are to be distinguished:\n(a) the considered spin is the first spin of the chain and its bond to the left (i.e.\\\nto an unchanged spin) is a ferromagnetic or a chiral bond (subscripts $f$ and $c$\nrespectively) or (b) it\nis the last spin of the chain and its bond to the right is \na ferromagnetic or a chiral bond. The Boltzmann factors for the\nflipping of these single spins are written as vectors:\n\\begin{eqnarray}\n\\altvec{a}_f & = & \\left(\n\\begin{array}{l}\ny^{- \\frac{1}{2}} \\\\\ny^{\\frac{1}{2}}\n\\end{array} \\right) \\, x^{\\frac{1}{2}} \\, w^{q_\\perp} \n\\label{af} \\\\\n\\altvec{a}_c & = & \\left(\n\\begin{array}{l}\nx^{\\frac{1}{2}} \\\\\nx^{- \\frac{1}{2}}\n\\end{array} \\right) \\, y^{\\frac{1}{2}} \\, w^{q_\\perp} \n\\label{ac} \\\\\n\\altvec{b}_f & = & \\left(\n\\begin{array}{l}\ny^{\\frac{1}{2}} \\\\\ny^{- \\frac{1}{2}}\n\\end{array} \\right) \\, x^{\\frac{1}{2}} \n\\label{bf} \\\\\n\\altvec{b}_c & = & \\left(\n\\begin{array}{l}\nx^{- \\frac{1}{2}} \\\\\nx^{\\frac{1}{2}}\n\\end{array} \\right) \\, y^{\\frac{1}{2}}\n\\label{bc}\n\\end{eqnarray}\nThe vectors (\\ref{bf}) and (\\ref{bc}) do not include the Boltzmann factor resulting from \nthe change of the in-layer bonds. This factor has already been included in the matrix\ndescribing the overturning of the two last spins in the spin chain.\n\n\\subsection{Derivation of the low temperature phase diagram}\nWith the matrices (\\ref{matrix_ferro}) and (\\ref{matrix_chiral}) and the vectors\n(\\ref{af})-(\\ref{bc}) it is now possible to compute the leading order term $b_\\nu$\nof the quantities $a_\\nu$ (and, thus, to determine the sign of $a_\\nu$) for all phases\ndegenerate at the multiphase point and containing only 1- and 2-layer bands.\nAll considered phases can be viewed as periodic arrangements\nof spin sequences with a 1-layer band as the first and a 2-layer band as the last band\nin the sequence \n\\cite{Yeo81}. The sequence $\\tilde{\\nu}$ obtained by stripping\nthe original sequence $\\nu$ by its last and first band is called core. All sequences\nbased on the same core $\\tilde{\\nu}$ enter in the computation of the $b_\\nu$: The\nsequences $ 1 \\tilde{\\nu} 2$ and $ 2 \\tilde{\\nu} 1$ contribute negatively, the\nsequences $ 1 \\tilde{\\nu} 1$ and $ 2 \\tilde{\\nu} 2$ contribute positively \\cite{Fis80}.\nThe expressions $b_\\nu$ for different families of phases \nare summarised in table 1.\n\n\\subsubsection{Stability of some series of phases}\nFor the series of phases $\\left< 1 2^k \\right>$ the expression\n(see table 1)\n\\begin{equation}\nb_{12^k} = - \\left( \\altvec{a}_c^T - \\altvec{a}_f^T \\right) \\left( \\altmat{C} \\, \n\\altmat{F} \\right)^{k-1} \\altmat{C} \\left( \\altvec{b}_f - \\altvec{b}_c \\right)\n\\label{b12k}\n\\end{equation}\ngives the leading contribution to $a_{12^k}$. \nThe four different sequences based on the core $\\tilde{\\nu}= 2^{k-1}$ yield\nthe four different contributions to $b_{12^k}$.\\\\\nThe eigenvalues $\\exp \\left( - \\Gamma_\\pm \\right)$ of the \nmatrix $\\altmat{C} \\, \\altmat{F}$ are \nreal and positive. Expression (\\ref{b12k}) can be written in the form\n\\begin{displaymath}\nb_{12^k} = A_+ \\, \\exp \\left( - \\frac{k}{2} \\Gamma_+ \\right) + A_- \\, \\exp \\left( - \\frac{k}{2}\n\\Gamma_- \\right)\n\\end{displaymath}\nwith $\\Gamma_+ < \\Gamma_-$. A close\nexamination reveals that for finite temperatures $A_+ < 0$, $A_- > 0$ and $A_+ + A_- < 0$.\nThus, $b_{12^k}$ is negative for all $k$, i.e.\\ all phases of the form\n$\\left< 1 2^k \\right>$ spring from the multiphase point and have a finite stability \nrange at temperatures above zero.\\\\\nThe leading order contribution for the phases $\\left< 1^k 2 \\right>$ is\n\\begin{equation}\nb_{1^k2} = - \\left( \\altvec{a}_c^T - \\altvec{a}_f^T \\right) \\altmat{C}^k \n\\left( \\altvec{b}_f - \\altvec{b}_c \\right).\n\\label{b1k2}\n\\end{equation}\nThe eigenvalues of the matrix $\\altmat{C}$ are complex conjugate. They are\nwritten in the form\n\\begin{equation}\n\\xi_1 \\pm \\i \\xi_2 = \\exp \\left( - \\Gamma_0 \\pm \\i \\, \\Omega \\right)\n\\label{eigen}\n\\end{equation}\nwith $\\Gamma_0 = - \\frac{1}{2} \\ln \\left( \\xi_1^2 + \\xi_2^2 \\right) > 0$, \n$\\Omega = \\arctan \\left( \\frac{\\xi_2}{\\xi_1} \\right)$ and $\\xi_1 = 1 -x^{1+\\delta}$, $\\xi_2 =\n\\left( 1 - x \\right) x^{\\frac{1}{2} + \\delta}$. We then obtain the expression\n\\begin{displaymath}\n\\fl\nb_{1^k2} = - (A_1 + \\i A_2) (\\xi_1 + \\i \\xi_2 ) + (A_1 - \\i A_2) (\\xi_1 - \\i \\xi_2 )\n= - \\left| \\Delta \\right| \\exp \\left( - k \\, \\Gamma_0 \\right) \\cos \\left(\nk \\, \\Omega + \\phi \\right)\n\\end{displaymath}\nwith $\\left| \\Delta \\right| \\exp \\i \\phi = A_1 + \\i A_2$.\nThe temperature-dependent quantities $\\left| \\Delta \\right|$, $\\phi$, \n$\\Gamma_0$, and $\\Omega$ do not depend on $k$.\\\\ \n$b_{1^k2}$ is negative for small\nvalues of $k$. If $k$ exceeds the value $k_{max} = \\frac{1}{\\Omega} \\left( \\frac{\\pi}{2} -\n\\phi \\right)$, then $b_{1^k2}$ becomes positive and, thus, all phases with\n$k > k_{max}$ are unstable at the considered point of the phase diagram. \nSince $k_{max} \\longrightarrow \\infty$ for $T \\longrightarrow 0$, there is, \nfor every $k$, a temperature\nbelow which the phase $\\left< 1^k 2 \\right>$ is stable. Thus, all phases \n$\\left< 1^k 2 \\right>$ spring from the multiphase point, but the higher\ncommensurate phases disappear at higher temperatures. Such a cut-off of the\nhigh commensurate phases at finite temperatures is also observed in the ANNNI \nmodel \\cite{Fis87}.\\\\\nFollowing the general line we\nalso examined the series of phases $\\left< 1 2^k 1 2^{k+1} \\right>$ and\n$\\left< 1^k 2 1^{k-1} 2 \\right>$.\nFor the case $\\left< 1 2^k 1 2^{k+1} \\right>$ we find that all these phases are\nstable at finite temperatures in the vicinity of the multiphase point with no\ncut-off for the phases with a large value of $k$, i.e.\\ the results for the series\n$\\left< 1 2^k 1 2^{k+1} \\right>$  resemble the results for the series $\\left< 1 2^k \\right>$.\nAnalysing the leading contribution for the phases $\\left< 1^k 2 1^{k-1} 2 \\right>$\nwe find a behaviour similar to the behaviour of the phases $\\left< 1^k 2 \\right>$,\ni.e.\\ all phases with $k < k_{max}$ (the value of $k_{max}$ being series-dependent)\nare stable and $k_{max} \\longrightarrow \\infty$ as $T \\longrightarrow 0$.\n\n\\subsubsection{Phases containing general sequences of 1- and 2-layer bands}\nIn the following we will show\nthat all phases consisting only of 1- and 2-layer bands and obeying the rules\nof the structure combination\nspring from the multiphase point, the higher commensurate phases of some series\nbecoming unstable at higher temperatures.\nThe leading contribution to $a_\\nu$ for all these phases is\nof the form (see table 1)\n\\begin{equation}\nb_\\nu = - \\left( \\altvec{a}_c^T - \\altvec{a}_f^T \\right) \\altmat{D} \\, \n\\altmat{C} \\left( \\altvec{b}_f - \\altvec{b}_c \\right)\n\\label{bnu}\n\\end{equation}\nwhere $\\altmat{D}$ is a product of powers of matrices $\\altmat{C}$ \nand $\\left( \\altmat{C} \\, \\altmat{F} \\right)$. The contributions of the\nfirst and last band are given by $\\left( \\altvec{a}_c^T - \\altvec{a}_f^T \\right)$\nand $\\altmat{C}\n\\left( \\altvec{b}_f - \\altvec{b}_c \\right)$ respectively.\nA 1-layer band in the core contributes a\nmatrix $\\altmat{C}$, whereas a 2-layer band yields the matrix product\n$\\left( \\altmat{C} \\, \\altmat{F} \\right)$. The product over all bands\nin the core yields the matrix $\\altmat{D}$ [see equation (\\ref{bnu})].\\\\\nThe diagonal elements of the matrix\n\\begin{equation}\n\\fl \n\\altmat{C} \\, \\altmat{F} = \\left( \\begin{array}{ll}\n2 \\left( 1 -y \\right) \\left( 1 - x \\right) & x \\left( 1 - y \\right)^2 - x^{-1} y\n\\left( 1 - x \\right)^2 \\\\\ny \\left( 1 - x \\right)^2 - x y^{-1} \\left( 1 - y \\right)^2 & \\left( 1 \n+ x y \\right) \\left( 1 -y \\right) \\left( 1 - x \\right)\n\\end{array} \\right) \\, w^{2 q_\\perp}\n\\nonumber\n\\end{equation}\nare positive whereas the non-diagonal elements are negative, since\n$y = x^{1 + \\delta}$ with $x \\ll 1$. We now follow reference\n\\cite{Sen93} and introduce the unitary matrix\n\\begin{displaymath} \n\\altmat{U} = \\left( \\begin{array}{ll}\n-1 &  ~0 \\\\\n~0 & ~1 \n\\end{array} \\right) = \\altmat{U}^{-1}.\n\\end{displaymath} \nAll elements of $\\altmat{U} \\, \\left( \\altmat{C} \\, \\altmat{F} \\right) \\, \\altmat{U}$,\nand therefore of\n$\\altmat{U} \\, \\left( \\altmat{C} \\, \\altmat{F} \\right)^k \\, \\altmat{U}$, are positive.\nThis is also the case for\nthe two vectors $\\left( \\altvec{a}^T_c - \\altvec{a}^T_f \\right) \\, \\altmat{U}$\nand $\\altmat{U} \\, \\altmat{C} \\, \\left( \\altvec{b}_f - \\altvec{b}_c \\right)$,\nthus [see equation (\\ref{b12k}) and table 1]\n\\begin{displaymath}\n\\fl \\left( \\altvec{a}_c^T - \\altvec{a}_f^T \\right) \\left( \\altmat{C} \\, \n\\altmat{F} \\right)^{k-1} \\altmat{C} \\left( \\altvec{b}_f - \\altvec{b}_c \\right)\n= \\left( \\altvec{a}_c^T - \\altvec{a}_f^T \\right) \\, \\altmat{U} \\, \\altmat{U} \\,\n\\left( \\altmat{C} \\, \\altmat{F} \\right)^{k-1} \\, \\altmat{U} \\, \\altmat{U} \\,\n\\altmat{C} \\left( \\altvec{b}_f - \\altvec{b}_c \\right)\n\\end{displaymath}\nis positive, i.e.\\ $b_{12^k} < 0$, in agreement\nwith the aforementioned calculations.\\\\\nPhases of the series $\\left< 1 2^k 1 2^{k+1} \\right>$ contain a single\n1-layer-band in the core yielding the matrix product $\\left( \\altmat{C} \\, \\altmat{F} \n\\right) \\altmat{C} \\left( \\altmat{C} \\, \\altmat{F} \\right)$ with positive diagonal\nand negative non-diagonal elements for small $x$. Hence, the product (see table 1)\n\\begin{eqnarray}\n\\fl \\left( \\altvec{a}_c^T - \\altvec{a}_f^T \\right) \\left( \\altmat{C} \\,\n\\altmat{F} \\right)^k \\altmat{C} \\left( \\altmat{C} \\,\n\\altmat{F} \\right)^k \\altmat{C} \\left( \\altvec{b}_f - \\altvec{b}_c \\right)\n\\nonumber \\\\\n\\lo= \\left( \\altvec{a}_c^T - \\altvec{a}_f^T \\right) \\, \\altmat{U} \\, \\altmat{U} \\,\n\\left( \\altmat{C} \\, \\altmat{F} \\right)^{k-1} \\, \\altmat{U} \\, \\altmat{U} \\,\n\\left( \\altmat{C} \\, \\altmat{F} \\right) \n\\altmat{C} \\left( \\altmat{C} \\, \\altmat{F} \\right) \\, \\altmat{U} \\nonumber \\\\\n\\times \\altmat{U} \\,\n\\left( \\altmat{C} \\, \\altmat{F} \\right)^{k-1} \\altmat{U} \\, \\altmat{U} \\,\n\\altmat{C} \\left( \\altvec{b}_f - \\altvec{b}_c \\right)\n\\nonumber\n\\end{eqnarray}\nis positive, showing the stability of the phases $\\left< 1 2^k 1 2^{k+1} \\right>$.\nFollowing this line of thought one easily shows that all phases appearing between \nthe phases $\\left< 2 \\right>$ and $\\left< 12 \\right>$ (i.e. phases with only isolated\n1-layer-bands in the core) are stable in the vicinity of the multiphase point.\nIndeed, as no new matrix products show up in the computation of \nthe different $b_\\nu$, all these\nexpressions can be written, using the matrix $\\altmat{U}$,\nas a product of vectors and matrices having only positive elements.\\\\\nFor phases containing consecutive 1-layer-bands in the core\nthe following additional vectors and matrices may contribute \nto the $b_\\nu$ as can be seen\nfrom table 1: $\\altmat{U} \\, \\altmat{C}^k \\, \\left( \\altvec{b}_f - \\altvec{b}_c \\right)$,\n$\\left( \\altvec{a}_c^T - \\altvec{a}_f^T \\right) \\, \\altmat{C}^k \\,\n\\altmat{U}$, and $\\altmat{U} \\, \\left( \\altmat{C} \\, \\altmat{F} \\right) \\,\n\\altmat{C}^k \\, \\altmat{U}$ with $k \\geq 2$. Introducing\nthe eigenvalues of the matrix $\\altmat{C}$ [see equation (\\ref{eigen})], we obtain\n\\begin{eqnarray}\n\\mbox{v}_1 & = & \\exp \\left( -k \\Gamma_0 \\right) \\left[ x^\\frac{\\delta}{2} \\left( 1 - x \\right)\n\\cos k \\Omega + x^{-\\frac{1+\\delta}{2}} \\left( 1 - x^{1+\\delta} \\right)\n\\sin k \\Omega \\right] \\, w^{k q_\\perp} \\nonumber \\\\\n\\mbox{v}_2 & = & \\exp \\left( -k \\Gamma_0 \\right) \\left[ x^{-\\frac{\\delta}{2}} \n\\left( 1 - x^{1+\\delta} \\right) \\cos k \\Omega - x^{1+\\delta} \\left( 1 - x \\right)\n\\sin k \\Omega \\right] \\, w^{k q_\\perp}  \\nonumber\n\\end{eqnarray}\nfor the components of the vector $\\altvec{v}= \\altmat{U} \\, \\altmat{C}^k \\,\n\\left( \\altvec{b}_f - \\altvec{b}_c \\right)$.\nWhereas $\\mbox{v}_1$ is always positive, $\\mbox{v}_2 > 0$ only if the\ninequality \n\\begin{displaymath}\n\\tan k \\Omega < \\frac{x^{-\\frac{\\delta}{2}} \\, \\left( 1 - x^{1+ \\delta } \\right)}\n{x^{1+ \\delta } \\, \\left( 1 - x \\right)}\n\\end{displaymath}\nholds. This is the case for temperatures smaller than an upper limit which\ndepends on $\\delta$ and $k$.\nIn a similar way one shows that for temperatures smaller than some $k$-dependent\ntemperature all the components of the vector \n$\\left( \\altvec{a}_c^T - \\altvec{a}_f^T \\right) \\, \\altmat{C}^k \\,\n\\altmat{U}$ and of the matrix $\\altmat{U} \\, \\left( \\altmat{C} \\, \\altmat{F} \\right) \\,\n\\altmat{C}^k \\, \\altmat{U}$ are positive.\nThe free energy differences $a_\\nu$ for all phases \ncontaining consecutive 1-layer-bands in the core are therefore negative below\na certain temperature, \ni.e.\\ these phases possess a stability region below this\ntemperature.\n\n\\subsubsection{Conclusion}\nThe results obtained so far can be summarised as follows: All\nphases consisting only of 1- and 2-layer-bands, that can be formed\nby means of the aforementioned structure combination rules, spring from\nthe multiphase point, where they are degenerate. The higher commensurate\nphases of some series, i.e.\\ those phases formed in higher orders of the combination\nprocess, disappear again at temperatures individually depending\non the series under consideration.\\\\ \nFrom these results the complete low temperature phase diagram\nof the $CC_4$ model is deduced by applying the transformation (\\ref{trafo}).\nAt non-zero temperatures all phases appearing between\nthe phases $\\left< 12 \\right>$ and $\\left< 3 \\right>$ are stable since the \ntransformation (\\ref{trafo}) transforms phase $\\left< 12 \\right>$ into\n$\\left< 3 \\right>$ and leaves the phase $\\left< 2 \\right>$ invariant. \nSome of the long commensurate phases\nappearing between the phases $\\left< 1 \\right>$ and $\\left< 12 \\right>$\nfor $\\Delta > \\frac{1}{2}$ \nand between the phases $\\left< \\infty \\right>$ and $\\left< 3 \\right>$\nfor $\\Delta < \\frac{1}{2}$ are unstable at a given temperature.\nUpon reducing the temperature,\nmore and more of these phases become stable, and in the limit $T \\longrightarrow 0$ all phases\nobeying the rules of the structure combination\nare stable. Therefore, the $CC_4$ model\nexhibits a complete devil's staircase in the low-temperature limit.\n\n\\subsection{Comparison with other work}\nThe low-temperature behaviour of the general $p$-state chiral clock model was\nanalysed in reference \\cite{Yeo82} using a series expansion technique similar \nto the one presented here. Due to the incorrect expansion (see section 2)\nonly some specific families of phases were shown to \npossess a finite stability region at small temperatures. Especially, it was\nclaimed that the phases $\\left< 1^k 2 \\right>$ with $k > 2$ are not stable\nat low temperatures, implying, due to the transformation (\\ref{trafo}), that\nfor $\\Delta < \\frac{1}{2}$ a direct transition from \nthe ferromagnetic phase to the $\\left< 4 \\right>$-phase exists.\nIn order to corroborate these calculations a low temperature mean-field analyses \nof the $CC_p$ model was presented in \\cite{Yeo84} were it was claimed that\nin the vicinity of the multiphase point the mean-field approximation\nyields the same stable phases as reference \\cite{Yeo82}. In that work the model in mean-field\napproximation was mapped onto an one-dimensional array of interacting domain walls. This\nmapping was derived under the approximation that \nthe mean-field average spin\n$( \\langle \\cos \\frac{\\pi}{2} n_{i,\\alpha} \\rangle_{MF},\n\\langle \\sin \\frac{\\pi}{2} n_{i,\\alpha} \\rangle_{MF} )$ \nin each layer (layer spin)\ndoes only deviate from\nthe $T = 0$ value in amplitude but not in phase.\nIn a detailed analyses of the mean-field phase diagram of the $CC_3$ model \nSiegert and Everts \\cite{Sie85} showed that\nthis approximation leads to a wrong phase diagram at low temperatures, thus concluding\nthat the layer spin must also be allowed to deviate in phase from its ground-state\nvalue. This should not only be the case for the three-state but also for the\ngeneral $p$-state model. The results of reference \\cite{Yeo84} for the \nmean-field low temperature\nbehaviour of the $CC_p$ model must therefore be considered with care.\\\\\nAs we have shown in the preceding sections the results of the series\nexpansion in reference \\cite{Yeo82} are erroneous due\nto wrong Boltzmann factors for the in-layer bonds. In fact, the four-state\nmodel exhibits in the low-temperature limit a complete devil's staircase.\nFurthermore, it results from our calculations that no direct\ntransition from the ferromagnetic to the $\\left< 4 \\right>$-phase exists\nas phases with longer periods are stable between these two phases.\\\\\nIn the Monte Carlo simulation of the $CC_4$\nmodel \\cite{Sch96} long-period spin patterns were observed\nwhen going from the ferromagnetic phase to the modulated phases\nat rather high temperatures.\nIn view of the present work one must interpret\nthese patterns as reflecting the existence of phases\nspringing from the multiphase point and intercalating\nbetween the ferromagnetic\nand the $\\left< 4 \\right>$-phases.\n\n\\section{The critical behaviour\\label{sec4}}\nThe critical behaviour of the general $p$-state chiral clock \nmodel at the transition to the paraphase is an interesting topic since \nfor $p=2$ the chiral clock model reduces to the anisotropic Ising model, for\n$p = \\infty$ it corresponds to the classical 3d-$XY$ model. Siegert and Everts \\cite{Sie89}\nshowed that the $CC_3$ model belongs to the universality class of the 3d-$XY$ model.\nOn the basis of his contradictory MFTM results, McCullough \\cite{Mcc92} speculated about\na change in the universality class from 3d-Ising behaviour to 3d-$XY$ behaviour for\n$p$ close to 5.\nIn the following we will show that for $p=4$ an\neffective Ginzburg-Landau-Wilson Hamiltonian can be derived which can be\ntransformed to the effective Hamiltonian of the 3d $XY-$model.\\\\\nFor the case $p=4$\nthe Hamiltonian (\\ref{hamiltonian}) can be rewritten\nin the form\n\\begin{displaymath}\nH = -J \\sum\\limits_{i} \\sum\\limits_{\\alpha} \\altvec{S}_{i,\\alpha} \\, \\altmat{R} \\left( \n\\Delta \\right) \\,\n\\altvec{S}_{i,\\alpha+1} - J_0 \\sum\\limits_{\\alpha} \\sum\\limits_{\\left< ij \\right>} \n\\altvec{S}_{i,\\alpha} \\, \\altvec{S}_{j,\\alpha}\n\\end{displaymath}\nwhere we introduced the spin vector $\\altvec{S}_{i,\\alpha} = \\left( \n\\cos \\frac{\\pi}{2} n_{i,\\alpha},\n\\sin \\frac{\\pi}{2} n_{i,\\alpha} \\right)$ and the rotation matrix\n\\begin{displaymath}\n\\altmat{R} \\left( \\Delta \\right) = \\left( \\begin{array}{ll}\n~ \\cos \\frac{\\pi}{2} \\Delta & \\sin \\frac{\\pi}{2} \\Delta  \\\\\n- \\sin \\frac{\\pi}{2} \\Delta & \\cos \\frac{\\pi}{2} \\Delta \n\\end{array}\n\\right).\n\\end{displaymath}\nRotating all spins in layer $\\alpha$ by the angle $\\frac{\\pi}{2} \\alpha \\Delta$, i.e.\\\nintroducing new vectors $\\altvec{\\sigma}_{i,\\alpha}= \\altmat{R} \\left( \\alpha \\Delta \\right) \\,\n\\altvec{S}_{i,\\alpha}$, leads to the expression\n\\begin{equation}\nZ=\\sum\\limits_{\\left\\{ \\altvec{\\sigma} \\right\\} } \\exp \\left[- \\frac{1}{2} \\sum\\limits_{ij}\n\\sum\\limits_{\\alpha \\, \\beta} \\sum\\limits_{\\kappa=1}^{2} \\sigma_{i,\\alpha}^\\kappa \\,\nK_{i \\, \\alpha, j \\, \\beta} \\, \\sigma_{j, \\beta}^\\kappa \\right]\n\\label{partition}\n\\end{equation} \nfor the partition function, $\\kappa$ labelling the two spin components. The elements \n$K_{i \\, \\alpha, j \\, \\beta}$ of the coupling matrix are zero unless the lattice sites\n$\\left( i,\\alpha \\right)$ and $\\left( j , \\beta \\right)$ are nearest neighbours.\\\\\nExpression (\\ref{partition}) may be transformed \\cite{Bak62,Hub72} to\n\\begin{displaymath}\n\\fl\nZ = C \\sum\\limits_{\\left\\{ \\altvec{\\sigma} \\right\\}} \\left( \\prod_{\\rho=1}^2 \\prod_{k \\, \\gamma} \n\\int dh_{k, \\gamma}^\\rho \\right) \\exp \\left[ - \\frac{1}{2} \\sum\\limits_{ij}\n\\sum\\limits_{\\alpha \\, \\beta} \\sum\\limits_{\\kappa} h_{i,\\alpha}^\\kappa \\,\nL_{i \\, \\alpha, j \\, \\beta}^{-1} \\, h_{j, \\beta}^\\kappa + \n\\sum\\limits_{i} \\sum\\limits_{\\alpha} \\sum\\limits_{\\kappa} h_{i,\\alpha}^\\kappa \\,\n\\sigma_{i,\\alpha}^\\kappa \\right].\n\\end{displaymath}\nHere $C$ is a numerical constant, $N$ is the total number of lattice sites and $\\altmat{I}$ is \nthe $N \\times N$ identity matrix. The matrix $\\altmat{L}$ is given by\n$\\altmat{L}= \\mu \\, \\altmat{I}- \\altmat{K}$ where the positive number $\\mu$ is\nchosen large enough to ensure that all the eigenvalues of $\\altmat{L}$ are positive.\\\\\nThe sum over all states can be easily computed:\n\\begin{eqnarray} \n\\fl \n\\sum\\limits_{\\left\\{ \\altvec{\\sigma} \\right\\}}  \\exp \\left( \n\\sum\\limits_{i} \\sum\\limits_{\\alpha} \\sum\\limits_{\\kappa} h_{i,\\alpha}^\\kappa \\,\n\\sigma_{i,\\alpha}^\\kappa \\right) \\nonumber \\\\\n\\lo= 2^N \\prod_{i \\, \\alpha} \\left[ \\cosh \\left(\nh^1_{i, \\alpha} \\, c_\\alpha - h^2_{i, \\alpha} \\, s_\\alpha \\right) + \\cosh \\left(\nh^1_{i, \\alpha} \\, s_\\alpha + h^2_{i, \\alpha} \\, c_\\alpha \\right) \\right],\n\\label{partcosh}\n\\end{eqnarray}\nwith $c_\\alpha = \\cos \\left( \\frac{\\pi}{2}\n\\alpha \\Delta \\right)$ and $s_\\alpha = \\sin \\left( \\frac{\\pi}{2}\n\\alpha \\Delta \\right)$. Using the expansion \n\\begin{displaymath}\n\\ln \\cosh x = - \\sum\\limits_n \\left( - 1 \\right)^n\n\\frac{2^{2n-1} \\left( 2^{2n} -1 \\right) B_n}{n \\left( 2 n \\right) !} x^{2n}\n\\end{displaymath}\n($B_n$: Bernoulli number) the expression on the righthand side of (\\ref{partcosh})\ncan be written as the exponential of a sum of powers of $h_{i,\\alpha}^1$ and\n$h_{i,\\alpha}^2$:\n\\begin{equation}\n\\fl\nC_1 \\, \\exp \\left[ \\sum\\limits_n \\sum\\limits_{k=0}^{2n} \\sum\\limits_{l=0}^k \\sum\\limits_{l'=0}^{2n-k}\nc(n,k,l,l') \\sum\\limits_i \\sum\\limits_\\alpha c_\\alpha^{l+l'} \\, s_\\alpha^{2n-l-l'}\n\\, \\left( h^1_{i, \\alpha} \\right)^k \\, \\left( h^2_{i, \\alpha} \\right)^{2n-k}\n\\right]\n\\label{h1h2exp}\n\\end{equation}\nwhere $c(n,k,l,l')$ is a number depending on $n$, $k$, $l$ and $l'$.\\\\\nIntroducing new variables $\\phi^\\kappa_{i,\\alpha} = \nL_{i \\, \\alpha, j \\, \\beta}^{-1} \\, h_{j, \\beta}^\\kappa$, taking the continuum limit\nand turning to the wavenumber representation leads to the \npartition function\n\\begin{displaymath}\nZ \\propto  \\left( \\prod\\limits_{\\altvec{q}} \\int d\\altvec{\\tau}_{\\altvec{q}} \\right) \\exp \\left[ - \\bar{H} \\right]\n\\end{displaymath}\nwith the effective Hamiltonian\n\\begin{eqnarray}\n\\fl \\bar{H} = - \\frac{1}{2} \\int_{BZ} \\frac{d^3q}{( 2 \\pi )^3} \\left(\nr + q^2_{\\perp} + \\frac{\\Upsilon}{\\Upsilon_0} q^2_{\\parallel} \\right) \\left[\n\\underline{\\tau} ( \\underline{q} ) ~ \\underline{\\tau} ( - \\underline{q} )\n\\right] \\nonumber \\\\[2mm]\n- u \\int_{BZ} \\frac{d^3q \\, d^3q' \\, d^3q''}{( 2 \\pi )^9}\n\\left[ \\underline{\\tau} ( \\underline{q} ) ~ \\underline{\\tau} (\n\\underline{q}' ) \\right] \\, \\left[ \\underline{\\tau} ( \\underline{q}'' )\n~ \\underline{\\tau} (- \\underline{q} - \\underline{q}' - \\underline{q}'' )\n\\right]\n\\label{effham}\n\\end{eqnarray}\nwith $\\altvec{\\tau}_{\\altvec{q}}=\\left( \\tau_{\\altvec{q}}^1 , \\tau_{\\altvec{q}}^2\n\\right)$ \nand $\\phi^{\\kappa} \\left( \\altvec{r} \\right) = \\int_{BZ} \\frac{d^3q}{( 2 \\pi )^3} \\,\n\\exp ( \\i \\altvec{q} \\cdot \\altvec{r} ) \\, \\tau_{\\altvec{q}}^\\kappa$.\nThe integration is over the first Brillouin zone with $\\altvec{q} = \\left( \n\\altvec{q}_{\\perp}, \\altvec{q}_{\\parallel} \\right)$, \nits components $\\altvec{q}_{\\perp}$ and $\\altvec{q}_{\\parallel}$ being perpendicular\nand parallel to the direction of the modulation respectively.\n$r = \\frac{1}{\\Upsilon_0} \\left( 1 - 2 \\Upsilon_0 - \\Upsilon \\right)$ with\n$\\Upsilon = \\frac{J}{k_BT}$ and $\\Upsilon_0 = \\frac{J_0}{k_BT}$ \nvaries linearly with temperature.\\\\\nIn deriving equation (\\ref{effham}) we neglected\nfourth and higher harmonics, i.e.\\ fast oscillating terms containing\n$\\exp \\left( \\mbox{i} n \\frac{\\pi}{2} \\alpha \\Delta \\right)$ with \n$n \\geq 4$. Furthermore\nwe did not include terms of higher than fourth order in $\\tau$. If we rescale\n$q_\\parallel$ in the effective Hamiltonian \\cite{Nel75} we end with the\neffective Ginzburg-Landau-Wilson Hamiltonian of the 3d-$XY$ model. \n\n\\section{Conclusions\\label{sec5}}\nA low temperature series expansion technique is suitable to obtain exact\nresults on the low \ntemperature behaviour of the four-state chiral clock model. \nAll phases degenerate at the multiphase point ($T=0$, $\\Delta=\\frac{1}{2}$)\nand obeying the structure combination rules spring from\nthe multiphase point. Some of these phases disappear at higher temperatures.\nIn the low-temperature limit the $CC_4$ model exhibits\na complete devil's staircase.\nDifferences in the low temperature phase diagrams derived in the present and in\na previous publication can be traced back to an inconsistency in the series\nexpansion of the latter.\nLong-period spin patterns derived in the present paper as \nstable phases between the ferromagnetic and\nthe $\\left< 4 \\right>$-phase and not occurring in the analyses presented \nin \\cite{Yeo82}, were recently\nseen in Monte Carlo\nsimulations just above the boundary of the ferromagnetic phase.\\\\\nFurthermore,\nthe critical behaviour at the boundary between the paraphase and \nthe modulated structures follows from the derivation of\nan effective Ginzburg-Landau-Wilson\nHamiltonian. It is shown that the latter can be \ntransformed to the effective Hamiltonian of the\n3d-$XY$ model. The four-state model thus belongs to the universality class\nof the $XY$ model.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLast year, an interesting enhancement was reported at 750 GeV by the ATLAS \\cite{ATLAS} and CMS \\cite{CMS} groups at LHC.\\footnote{At  ICHEP 2016 (August 3-10), this enhancement was reported to disappear after adding the new data collected by ATLAS and CMS in 2016.  Even if this paper is triggered by the enhancement, we study the monopolium in general, so that the contents are not affected by whether it may exist or not.}\nThe enhancement $X$ was observed in the $\\gamma \\gamma$ channel and was not observed in the other di-boson channels such as ${ZZ}$, ${W^+} $ and ${W^-}$: \n\\begin{eqnarray}\n\\sigma(p+p \\to X \\to \\gamma \\gamma)=10 \\pm 3~\\mbox{fb}~\\mbox{(ATLAS)} ~\\mbox{and}~ 6 \\pm 3 ~\\mbox{fb}~\\mbox{(CMS)}, \n\\end{eqnarray}\nat 13 TeV (Run2) experiment, but no such enhancement was reported at 8~TeV (Run1). Upon combining the ATLAS and CMS data and including 8~TeV data the following cross-section estimate is obtained \\cite{Buttazzo:2015txu}:\n\\begin{equation}\n\\sigma (pp\\to X \\to \\gamma\\gamma) \\approx  (4.6\\pm1.2)~ {\\rm fb}.\n\\label{combined}\n\\end{equation}\nThe total width $\\Gamma_\\mathrm{tot}$ has not been well determined, but appears to be large $ \\sim45 $ GeV.  Various models have been proposed for $X$ \\cite{750 GeV models}, but if $X$ couples exclusively to $\\gamma \\gamma$ with a large coupling constant, then it may be possible to consider the enhancement as a monopolium, or a monopole-antimonopole dumbbell as described by Nambu \\cite{Nambu}.  A recent paper by M. Yamada {\\it  et al.}  \\cite{Yamada et al.} considers a similar idea, but they identify the enhancement as a magnetic Higgs particle where it is dual to the electric Higgs particle in a hidden $\\mbox{U(1)}_H$ gauge theory.  Ours is more directly a bound state system of a monopole and an antimonopole.  \n\nThe detection of monopolium has already been studied by L. I. Epele {\\it et al.}, aiming for observation at Tevatron and LHC \\cite{Epele1, Epele2}. The main difference between their paper and the following analysis is that we adopt a strong coupling expansion in lattice gauge theory and accordingly a linear term is included in the monopole-antimonopole potential. Therefore, the large binding energy case can be studied. \n  \nThis work is motivated by papers ~\\cite{Neil1, Neil2} written by one of the authors (N. D. B.) on explaining the diphoton excess via photon fusion production of $ X $, in particular where $ X $ is identified as a leptonium of highly charged leptons having a charge $Q=(5-7)~e$.  Higher charge can be naturally understood in the context of monopoles due to the large magnetic charge, which is opposite to the small electric charge. Also, the stability of the monopole is less crucial than the lepton, since our monopole is confined and can't exist individually.\n  \nThe difficulty of highly charged particles is the estimation of the potential between them.  One photon exchange is not enough, and so we will use lattice gauge theory with a finite lattice constant $a$, in which a strong coupling expansion is possible \\cite{lattice gauge theory1, lattice gauge theory2, lattice gauge theory3}.  We will apply this to the magnetic $ U(1) $ part of the manifestly electromagnetic dual formulation of Zwanziger \\cite{Zwanziger1, Zwanziger2}.  In lattice gauge theory, however, there exists another difficulty, that is, $ U(1) $ gauge theories are not well defined; a first-order phase transition exists between the weak coupling perturbative region and the strong coupling confinement region, and so the continuum limit of $a \\to 0$ can't be taken properly \\cite{Creutz-Jacobs-Rebbi}.\nOn the other hand, the continuum limit is properly taken for $ SU(2) $ \\cite{Creutz} and other asymptotically free gauge theories.  So, we consider that the welcoming non-Abelian structure reveals when we go inside the finite sized $ U(1) $ monopole, that is, at some point of taking $a \\to 0$, the gauge group is expected to be enhanced from $ U(1) $ to $ SU(2) $ and the 't Hooft-Polyakov~\\cite{monopole1, monopole2, monopole3} like inside structure will appear.  Even in the 't Hooft-Polyakov monopole,  the $ U(1) $ magnetic charge is unfortunately located at the origin as a point-like singularity.  To relax this situation, we look for a solution in which the $ U(1) $ magnetic charge is distributed nonlocally.  Fortunately, there exists a solution in which the magnetic charge is distributed uniformly on the surface of a sphere with radius $R$.  Inside the sphere ($r<R$) there is no magnetic force, and so the potential becomes flat, while the potential between the monopole and antimonopole becomes linear plus Coulomb outside the sphere. \n\nWe estimate the monopolium masses and wave functions using this potential.  In this model, the monopole and antimonopole are confined like quark and antiquark, so that they are not observed individually.\n  \nIf the monopolium is found and the world of monopoles is opened in reality, then various interesting phenomena are expected. One of these is that the neutrino mass as well as possible lepton and baryon number violation effects may be attributed to the monopolium or the monopole-antimonopole dumb bell system \\cite{sugamoto}. \n\n\\section{Zwanziger's manifestly dual formulation of gauge theory} \nThe manifestly dual formulation of $ U(1) $ gauge theory by Zwanziger \\cite{Zwanziger1, Zwanziger2} is  given by the following action,   \n\\begin{eqnarray}\nS^{\\rm{ZW}}&=&\\int d^4 x \\left[ \\left(\\frac{1}{2} \\eta^{\\mu} \\eta^{\\lambda}\n               \\sqrt{-g} g^{\\nu\\rho}\\right) \\nonumber \\right. \\\\\n&\\times& \\left(F_{\\mu\\nu} F_{\\lambda\\rho} + G_{\\mu\\nu} G_{\\lambda\\rho}+ F_{\\mu\\nu} \\tilde{G}_{\\lambda\\rho} - G_{\\mu\\nu}\\tilde{F}_{\\lambda\\rho} \\right)  \\nonumber \\\\\n&+& \\left. \\sum_i \\overline{\\psi_i} \\gamma^{\\mu}( iD_{\\mu}-m_i) \\psi_i \\right]\n   \\end{eqnarray}\nHere, a constant unit vector $\\eta^{\\mu}$, denoting the direction of Dirac strings \\cite{Dirac1, Dirac2, Dirac3} is displayed in parallel along the space-like direction, and\n \\begin{eqnarray}\n              iD_{\\mu}&=&i \\partial_{\\mu}- e_i A_{\\mu} -g_i B_{\\mu}, \\\\\n            F_{\\mu\\nu}&=&\\partial_{\\mu}A_{\\nu}-\\partial_{\\nu}A_{\\mu}, ~~\n           G_{\\mu\\nu}=\\partial_{\\mu}B_{\\nu}-\\partial_{\\nu}B_{\\mu}, \\\\\n           \\tilde{F}_{\\mu\\nu}&=&\\frac{1}{2} \\varepsilon_{\\mu\\nu\\lambda\\rho}\n                               F^{\\lambda\\rho}, ~~\\tilde{G}_{\\mu\\nu}=\\frac{1}{2} \\varepsilon_{\\mu\\nu\\lambda\\rho}\n                               G^{\\lambda\\rho}, \n\\end{eqnarray}\nwhere fermions with electric charge $e_i$ and magnetic charge $g_i$ are introduced, and $\\varepsilon^{0123}=1$.  We can add scalars if necessary.  From the consistency condition of the finite Lorentz transformation, the following Dirac (or correctly Schwinger type) quantization condition \\cite{Dirac1, Dirac2, Dirac3} appears\n\\begin{eqnarray}\ne_ig_j-g_ie_j= 4\\pi N_{ij},\n\\end{eqnarray}\nwhere $N_{ij}$ is an integer.\nHere, we consider a flat space-time $g_{\\mu\\nu}=\\eta_{\\mu\\nu}=(1, -1, -1, -1)$ and $\\varepsilon^{0123}=-\\varepsilon_{0123}=1$.  If we want to obtain the dual gravity theory, then consider that the metric $g_{\\mu\\nu}(x)$ is formed as a collective excitation of a pair of gauge bosons, $(B_{\\mu}(x)B_{\\nu}(x))$ or $(A_{\\mu}(x)A_{\\nu}(x))$ \\cite{Noguchi-Sugamoto}.  In the Zwanziger formulation, the degrees of freedom are doubled by the introduction of electric and magnetic vector potentials, but are halved by the projection to the $\\eta^{\\mu}$ direction.  Such a formulation in a non-Abelian gauge theory is more complicated but is possible~\\cite{Zwanziger3, Zwanziger4, Zwanziger5}.\n\nIf the axial gauge is taken,\n\\begin{eqnarray}\n\\eta^{\\mu}A_{\\mu}(x)=\\eta^{\\mu}B_{\\mu}(x)=0,\n\\end{eqnarray}\nno ghost fields appear, and the Feynman rules are obtained as follows~\n\\begin{eqnarray}\n\\langle A_{\\mu} A_{\\nu} \\rangle (k)&=& \\frac{-i}{k^2 + i \\varepsilon} \\left( g_{\\mu\\nu} - \\frac{k_{\\mu}\\eta_{\\nu}+k_{\\nu}\\eta_{\\mu}}{k \\cdot \\eta}+\\eta^2 \\frac{k_{\\mu}k_{\\nu}}{(k \\cdot \\eta)^2} \\right), \\\\\n\\langle B_{\\mu} B_{\\nu} \\rangle (k)&=& \\frac{-i}{k^2 + i \\varepsilon} \\left( g_{\\mu\\nu} - \\frac{k_{\\mu}\\eta_{\\nu}+k_{\\nu}\\eta_{\\mu}}{k \\cdot \\eta} +\\eta^2 \\frac{k_{\\mu}k_{\\nu}}{(k \\cdot \\eta)^2}\\right), \\\\\n\\langle A_{\\mu} B_{\\nu} \\rangle (k) &=&-\\langle B_{\\mu} A_{\\nu} \\rangle  (k)= \\frac{-i}{k^2 + i \\varepsilon} \\varepsilon_{\\mu\\nu\\rho\\sigma} \\frac{\\eta^{\\rho}k^{\\sigma}}{k \\cdot \\eta}.\n\\end{eqnarray}\nThe kinetic terms of the gauge field are complicated and depend on $\\eta^{\\mu}$, but  the $2 \\times 2$ matrix form of the propagators is simple and satisfies\n\\begin{eqnarray}\n\\hat{D}^{ab\\mu\\nu} \\langle V_{b\\nu} V_{c\\lambda} \\rangle = \\delta^a_c \\eta^{\\mu}_{\\lambda},\n\\end{eqnarray} \nwhere $V^1_{\\mu}=A_{\\mu}$, $V^2_{\\mu}=B_{\\mu}$, and $\\hat{D}^{ab\\mu\\nu}$ is the differential operator for the gauge fields in the action.\\footnote{Propagators that Zwanziger derived have no $\\eta^2 \\frac{k_{\\mu}k_{\\nu}}{(k \\cdot \\eta)^2}$ term in the numerator \\cite{Zwanziger2}. These propagators are for a gauge-fixing Lagrangian ${\\cal L}_G=-\\frac{1}{2} \\left[ (\\partial_{\\mu}(\\eta \\cdot  A) )^2+ (\\partial_{\\mu}(\\eta \\cdot B) )^2 \\right]$.  Two kinds of propagators with or without $\\eta^2 \\frac{k_{\\mu}k_{\\nu}}{(k \\cdot \\eta)^2}$ term are, of course, equivalent for the on-shell amplitudes, if the Ward identity is taken into account.}\n\n\\section{Production and decay of monopolium at LHC}\nWe assume that there is only one kind of spin 1\/2 monopole $M$ in our world, that is electrically neutral, having magnetic charge $g$.  We have, of course, a number of magnetically neutral spin 1\/2 fields having electric charge $Q=\\pm e$  ($e$ is the unit of the electron charge) and the monopole satisfies the Schwinger quantization condition with these charged particles.  This gives\n\\begin{eqnarray}\ng=\\frac{4\\pi}{e} N ~~(N=\\mbox{integer}).  \n\\end{eqnarray}\nWe denote the mass of the monopole simply as $m$, and assume that the monopole $M$ and its antimonopole $\\overline{M}$ form\na $J=0$ bound state $X_0=(M\\overline{M})$ ($^1S_0$, para-monopolium) and a $J=1$ bound state of $X_1=(M\\overline{M})$ ($^3S_1$, ortho-monopolium) via the exchange of magnetic photons, which we will denote as $\\gamma_M$ and are described by the $B_{\\mu}$ fields. The mass of the monopolium so obtained will be denoted generally as $m_X$, or specifically $m_{X_0}$ and $m_{X_1}$.\n\nAfter obtaining the energy and wave function of the monopolium, we can estimate the various partial decay widths $\\Gamma(X_0 \\to f)$ for $X_0$ or $\\Gamma(X_1 \\to f)$ for $X_1$, and the total decay widths $\\Gamma_\\mathrm{tot} (X_0 \\to all~f's)$ or $\\Gamma_\\mathrm{tot} (X_1 \\to all~f's)$, where $f$ denotes a final state.\n\nIn this section we use perturbation theory even for the strong coupling monopole.  Our strategy is to store the  effects into the bound state problem or the estimation of the monopole and antimonopole potential, and study the multi-photon emission processes perturbatively.  This is because, if we fix the number of emitted photons to $n$, then even if photons convert to magnetic photons and couple to the monopolium strongly, the order of coupling is $O(g^n)$, so that perturbative calculation at this order can be applied. This is valid, however, under the restriction that radiative corrections due to monopole loops are ignored. \n\n\\subsection{Production cross section}\nAt LHC, to produce $X$ we consider that a parton $a$ inside one proton collides with parton $b$ inside another proton to produce $X$ as a resonance scattering, where the parton can be anything produced from the proton, namely, quark, gluon, or photon.  To do this we introduce the parton distribution function inside the proton $f_{a\/p}(x)$, where $x$ is the momentum fraction. Then the production cross section reads\n\\begin{eqnarray}\n&&\\sigma(pp \\to a+b \\to X \\to f)  \\nonumber \\\\\n&&=\\int_0^1 dx_1 \\int_0^1 dx_2 f_{a\/p}(x_1) f_{b\/p}(x_2) \\hat{\\sigma}(\\hat{s}) (a+b \\to X \\to f) \\times (2, \\mbox{if}~ a \\ne b), \\\\\n&&=\\int_0^1 d\\tau \\int_{\\tau}^1 \\frac{dx_1}{x_1} f_{a\/p}(x_1) f_{b\/p}(\\tau\/x_1) \\hat{\\sigma}(\\hat{s}) (a+b \\to X \\to f) \\times (2, \\mbox{if}~ a \\ne b).\n\\end{eqnarray}\n Here, the patrons are labeled $a$ and $b$, and $\\hat{\\sigma}$ is the cross section of the two colliding patrons, $a$ and $b$, with the center of mass (CM) energy $\\sqrt{\\hat{s}}$, $\\tau\\equiv x_1x_2$, and $\\hat{s}=\\tau s$, where $\\sqrt{s}$ is the CM energy of the colliding protons. \n\nIf we introduce the ``parton luminosity\" for $a$ and $b$ as $d{\\cal L}_{ab}(\\tau)\/d\\tau$, and define it by \\cite{Barger-Phillips}\n\\begin{eqnarray}\n\\frac{d {\\cal L}_{ab}(\\tau)}{d\\tau} \\equiv \\frac{1}{c_ac_b}\\int_{\\tau}^1 \\frac{dx_1}{x_1} ~f_{a\/p}(x_1) f_{b\/p}(\\tau\/x_1)\\times (2, ~\\mbox{if}~ a \\ne b),\n\\end{eqnarray}\nwhere $c_a$ and $c_b$ are the color degrees of freedom for $a$ and $b$~ $(c_a= 3$ and $8$ for quark and gluon, respectively), then, the cross section becomes\n\\begin{eqnarray}\n&&\\sigma(pp \\to a+b \\to X \\to f) \\nonumber \\\\\n&&=\\int_0^1 d\\tau ~\\frac{d {\\cal L}_{ab}(\\tau)}{d\\tau} ~ c_ac_b ~\\hat{\\sigma}(\\hat{s}) (a+b \\to X \\to f).\n\\end{eqnarray}\nIf we replace $\\hat{\\sigma}$ by the resonance scattering formula, or its small width approximation \\footnote{The spin degree of freedom $2s+1$ is $2s=2$ for photon and gluon.} ,\n\\begin{eqnarray}\n\\hat{\\sigma}(\\hat{s}) (a+b \\to X \\to f)= 32 \\pi  \\frac{(2J_X+1)F}{c_ac_b(2s_a+1)(2s_b+1)}\\frac{\\Gamma(X \\to a+b) \\Gamma(X \\to f)}{(\\hat{s}-m^2_X)^2 + m_X^2 \\Gamma_\\mathrm{tot}^2} \\\\\n\\approx 32 \\pi  \\frac{(2J_X+1)F}{c_ac_b(2s_a+1)(2s_b+1)}\\frac{\\Gamma(X \\to a+b) \\Gamma(X \\to f)}{m_X\\Gamma_\\mathrm{tot}} \\pi \\delta(\\hat{s}-m_X^2).\n\\end{eqnarray}\nFrom these, we have\n\\begin{eqnarray}\n&&\\sigma(pp \\to a+b \\to X \\to f)  \\nonumber \\\\\n&&=\\left[ \\frac{1}{s} \\frac{d {\\cal L}_{ab}(\\tau)}{d\\tau}\\right]_{\\tau=\\frac{m_X^2}{s}} \\times \\left[  \\frac{32 \\pi^2 (2J_X+1)F}{(2s_a+1)(2s_b+1)} \\frac{\\Gamma(X \\to a+b) \\Gamma(X \\to f)}{m_X\\Gamma_\\mathrm{tot}}\\right], \n\\end{eqnarray}\nwhere the factor $F= 1$ for $a=b$, but is $\\frac{1}{2}$ for $a \\ne b$, since $a+b$ is originally the initial state, but it is converted to the final state in the formula. \n\nNow the production cross section is expressed as the product of two factors, the first is given by the parton luminosity function and $s^{-1}$, while the second one is a model-dependent number determined by the decay widths and mass of the resonance $X$.  If we understand that $s^{-1}=(13~\\mbox{TeV})^{-2}$ and $(8 ~\\mbox{TeV})^{-2}$ correspond to $2.4 ~\\mbox{pb}$ and $6.2 ~\\mbox{pb}$, respectively, in order to obtain 3 - 6 fb (to be consistent with Eq. (\\ref{combined})), we have to find the suppression factor of $10^{-3}$ from the product of $\\frac{d {\\cal L}_{ab}(\\tau)}{d\\tau},  \\Gamma(X \\to a+b)\/m_X$, and $\\mathrm{Br}(X \\to f)$.  This is one of the reasons why it is difficult to understand the 750 GeV enhancement.\n\\subsection{Decay rate}\nThe decay rate of $X_0 \\to 2 \\gamma$ can be estimated from $\\sigma_0(M+\\overline{M} \\to 2\\gamma)$ and the value of the wave function of relative motion of the monopole and antimonopole at the distance $ R $, $\\psi_{^1S_0}(r=R)$, that is\n\\begin{eqnarray}\n\\Gamma(X_0 \\to \\gamma\\gamma)=4 \\vert \\psi_{^1S_0}(R) \\vert^2 \\sigma_0(M+\\overline{M} \\to \\gamma + \\gamma) v_{rel},\n\\end{eqnarray}\nsince our model of the monopole has a finite size $R$, $\\vert \\psi_{^1S_0}(R) \\vert^2$ gives the probability of finding the monopole and antimonopole at a relative distance $ R $ in the bound state, where the collision occurs.  The factor 4 $\\sigma_0 v_\\mathrm{rel}$ gives the reaction rate (the factor 4 comes from the possible combination of monopole and antimonopole spin states). \n\nSimilarly, the decay rate of $X_1 \\to 3 \\gamma$ is estimated from $\\sigma_1(M+\\overline{M} \\to 3\\gamma)$ and the value at $r=R$ of the wave function of the ortho-monopolium $\\psi_{^3S_1}(r=R)$, that is \n\\begin{eqnarray}\n\\Gamma(X_1 \\to 3\\gamma)=\\frac{4}{3} \\vert \\psi_{^3S_1}(R) \\vert^2 \\sigma_1(M+\\overline{M} \\to 3\\gamma) v_\\mathrm{rel}.\n\\end{eqnarray}\n\nTo estimate $\\sigma_0(M+\\overline{M} \\to 2\\gamma)$ or $\\sigma_1(M+\\overline{M} \\to 3\\gamma)$, we first consider the process of emission of two or three magnetic photons, and convert the emitted magnetic photons to the electric photons.  The estimation is done by using the perturbation theory, since the final number of emitted photons is restricted to two or three.  \n\nIn the calculation, the polarization vector of the magnetic photon $\\epsilon_M(k)^{\\mu}$, has to be converted to the usual electric photon's polarization $\\epsilon_{\\gamma}(k)^{\\nu}$, by using the off-diagonal propagator $\\langle B_{\\mu} A_{\\mu} \\rangle (k)$, that is \n\\begin{eqnarray}\n\\epsilon_M(k)^{\\mu}=\\varepsilon^{\\mu\\nu\\rho\\sigma} \\epsilon_{\\gamma}(k)_{\\nu} \\frac{\\eta_{\\rho} k_{\\sigma}}{(k \\cdot \\eta)}. \\label{replacement of polarizations}\n\\end{eqnarray}\nThe polarization sum of photons, $\\sum_{pol}\\epsilon_{\\gamma}(k)^{*\\lambda}\\epsilon_{\\gamma}(k)^{\\rho}=-g^{\\lambda\\rho}$, is performed, then we have\n\\begin{eqnarray}\n\\sum_{\\gamma's~\\mathrm{pol.}}\\epsilon_M(k)^{*\\mu}\\epsilon_M(k)^{\\nu}\n=g^{\\mu\\nu}\\left(-1+\\frac{\\eta^2k^2}{(k \\cdot \\eta)^2}\\right)+\\frac{k^{\\mu}\\eta^{\\nu}+k^{\\nu}\\eta^{\\mu}}{(k \\cdot \\eta)}-\\frac{k^{\\mu}k^{\\nu}\\eta^2+\\eta^{\\mu}\\eta^{\\nu}k^2}{(k \\cdot \\eta)^2}.\n\\end{eqnarray}\nif the photon is on the mass shell, $k^2=0$, and the Ward identity is used, $k^{\\mu}$ and $k^{\\nu}$ terms vanish in the amplitude of monopoles on the mass shell, and hence $\\sum_{\\gamma's~pol.}\\epsilon_{M}(k)^{*\\mu}\\epsilon_M (k)^{\\nu}=-g^{\\mu\\nu}$.\nTherefore, the expression of $\\sigma(M+\\overline{M} \\to 2 ~\\mbox{or}~ 3\\gamma's)$ is obtained from $\\sigma(e^{+}+e^{-} \\to 2 \\gamma ~\\mbox{or}~ 3\\gamma)$, by replacing the mass $m_e$ by $m$ and the coupling constant $e$ by $g$. So, $(g\\beta)$ depending on the velocity $\\beta$ does not appear in our formulation.  On this point, refer to Ref.~\\cite{Epele1, Epele2}.\n\nNow we can obtain the decay rates of the para- and ortho-monopoliums as \n\\begin{eqnarray}\n\\Gamma(X_0 \\to 2\\gamma)&=&4\\pi  \\left(\\frac{g^2}{4\\pi}\\right)^2 \\frac{1}{m^2}\\vert \\psi_{^1S_0}(R)\\vert^2 , \\\\\n\\Gamma(X_1 \\to 3\\gamma)&=&\\frac{16(\\pi^2-9)}{9}\\left(\\frac{g^2}{4\\pi}\\right)^3\\frac{1}{m^2}\\vert \\psi_{^3S_1}(R)\\vert^2, \n\\end{eqnarray}\nwhere $m$ is the monopole mass and is not the monopolium mass $m_X$. These are the formulae valid at low energy, in the case of $\\sqrt{\\hat{s}}=m_X  \\approx 2m $.  When $\\sqrt{\\hat{s}}=m_X  \\gg 2m $ holds, and the binding energy is large, the following high energy limit formulae should be used:\n\\begin{eqnarray}\n\\Gamma(X_0 \\to 2\\gamma)&=&16\\pi \\left(\\frac{g^2}{4\\pi}\\right)^2 \\frac{1}{m_X^2} (\\rho-1) \\vert \\psi_{^1S_0}(R)\\vert^2, \\\\\n\\Gamma(X_1 \\to 3\\gamma)&=&\\frac{16}{3} \\left(\\frac{g^2}{4\\pi}\\right)^3 \\frac{1}{m_X^2} \\left[\\frac{1}{6} \\rho^3-\\frac{1}{4}\\rho^2+\\rho\\left(\\frac{1}{3}\\pi^2-1\\right) +3\\zeta(3)-\\frac{1}{3}\\pi^2+3\\right] \\nonumber \\\\\n&\\times& \\vert \\psi_{^3S_1}(R)\\vert^2,\n\\end{eqnarray}\nwhere $\\rho=\\ln(\\hat{s}\/m^2)=\\ln(m_X^2\/m^2)$, and $\\zeta(3)=\\sum\\limits_{n=1}^{\\infty} n^{-3}$=1.202 \\cite{Eidelman}. \n\nFor the ortho-monopolium $X_1$, it can decay to quark-antiquark, lepton, and weak boson pairs.  The corresponding decay rates are\n\\begin{eqnarray}\n\\Gamma(X_1 \\to q\\bar{q})&=&3 \\cdot \\frac{32\\pi}{9}\\left(\\frac{eg}{4\\pi}\\right)^2 \\frac{Q_q^2}{m_X^2}~\\vert \\psi_{^3S_1}(R)\\vert^2, \\\\\n\\Gamma(X_1 \\to \\ell^+\\ell^-)&=&\\frac{32\\pi}{9}\\left(\\frac{eg}{4\\pi}\\right)^2 \\frac{Q_{\\ell}^2}{m_X^2}~\\vert \\psi_{^3S_1}(R)\\vert^2, \\\\\n\\Gamma(X_1 \\to W^+W^-)&=&\\frac{2\\pi}{9} \\left(\\frac{eg}{4\\pi}\\right)^2\\frac{m_X^2}{m_W^4}~\\vert \\psi_{^3S_1}(R)\\vert^2,\n\\end{eqnarray}\nwhere the electric charge of the quark and lepton is $Q_{q, \\ell} ~e$. These formulae are derived for massless initial fermions. So, they have to be applied for the large binding energy case, or $m \\ll m_X$ and $m_W  \\ll m_X$.  \nIt is noted that a large number $eg\/4\\pi$ appears here, but it is not serious, since quarks, leptons and weak bosons don't carry magnetic charge.\n\n\n\nThe serious contribution to the decay rates comes from the emission of multi-photons, since the (electric or usual) photon can be converted to the magnetic photon and it couples to the monopole with a strong coupling $g$.\nTherefore, the total decay rates become\n\\begin{eqnarray}\n\\Gamma_\\mathrm{tot}(X_0)&=&\\sum_{n=\\mathrm{even}}\\Gamma(X_0 \\to n \\gamma),  \\\\\n\\Gamma_\\mathrm{tot}(X_1)&=&\\sum_{i} \\Gamma(X_1 \\to q_i\\bar{q}_i)+ \\sum_{i}\\Gamma(X_1 \\to \\ell^+_i\\ell^-_i)+\\Gamma(X_1 \\to W^+W^-) \\nonumber \\\\\n&+&\\sum_{n=\\mathrm{odd}}\\Gamma(X_1 \\to n \\gamma).\n\\end{eqnarray}\n\nIt is difficult to estimate the multi-photon emission rate.  However, the contribution of multi-photon emission can reduce the branching ratios $\\mathrm{Br}(X_0 \\to 2\\gamma)$ and $\\mathrm{Br}(X_1 \\to 3\\gamma)$ enormously, so we have to take them into account.  One way is to consider the emitted photons, other than two or three hard photons, to be all soft, namely its energy $E^\\mathrm{soft}_{\\gamma}$ is less than the energy cut $E^\\mathrm{cut}_{\\gamma}$.  The energy resolution $\\Delta E$ of photons at ATLAS is \n\\begin{eqnarray}\n\\Delta E\/E &=& 0.01-0.02 ~~(\\mbox{for}~ E > 100~ \\mbox{GeV}), \\\\\n&=& (0.1-0.2) \/\\sqrt{E\/\\mbox{GeV}} ~~(\\mbox{for}~ E < 100~ \\mbox{GeV}).\n\\end{eqnarray}\n\nHowever, to analyze the data of two photon events, low energy photons are discarded by the energy cutoff  $E^\\mathrm{cut}_{\\gamma}$, \n\\begin{eqnarray}\nE^\\mathrm{cut}_{\\gamma}=25-35~\\mbox{GeV},\n\\end{eqnarray}\nwhich is much larger than the energy resolution \\cite{Energy cut1, Energy cut2}.  We choose the energy cut as 35 GeV.  The strategy is, by using the infrared technique, to sum up the emitted but undetectable photons with energy \n\\begin{eqnarray}\n0 < E^\\mathrm{soft}_{\\gamma} <  E^\\mathrm{cut}_{\\gamma}= 35~\\mbox{GeV} \\ll \\frac{m_X}{2}.\n\\end{eqnarray}\n\nThere is another cut of eliminating the photons emitted forward or backward, which is the cut on the pseudo-rapidity $\\eta=-\\ln \\tan(\\theta\/2)$ ($\\theta$ is the scattering angle), namely,\n\\begin{eqnarray}\n\\vert \\eta \\vert ~<~\\eta^\\mathrm{cut}=2.37 .\n\\end{eqnarray}\nIt is rather technical, but we can convert the energy cut and the rapidity cut into nonvanishing photon masses $\\lambda$, since the introduction of the photon mass prevents the infrared singularity which occurs at the zero photon energy, as well as the collinear (mass) singularity that occurs when the photon is emitted parallel to the emitting fermion (monopole in this case)\n\\footnote{The lower bound of the massless photon propagator is identified with the lower bound of massive photon propagator, namely $\\left. E^\\mathrm{cut}_{\\gamma}=\\sqrt{\\vert \\bm{k} \\vert^2 + \\lambda^2}\\right\\vert_{\\vert \\bm{k}\\vert=0}$, which gives Eq. (\\ref{energy cut}).  Similarly the denominator of the fermion propagator $p\\cdot k$ is compared between the case of a massless photon with an angle cut and that of a massive photon without an angle cut. This gives a massless fermion, $1-\\cos \\theta^\\mathrm{cut}=\\sqrt{1+(\\lambda \/ \\vert \\bm{k} \\vert)^2}-1$, from which Eq. (\\ref{rapidity cut}) is obtained.}.\nThe conversion is done by the following dictionary:\n\\begin{eqnarray}\nE^\\mathrm{cut}_{\\gamma} &\\Leftrightarrow& \\lambda=E^\\mathrm{cut}_{\\gamma}=35 ~\\mbox{GeV}, \\label{energy cut} ~\\mbox{and}  \\\\\n\\eta^\\mathrm{cut} &\\Leftrightarrow& \\lambda=2 E_{\\gamma} e^{-\\eta^\\mathrm{cut}}=0.19 ~E_{\\gamma}< 0.19 \\left(\\frac{m_X}{2}\\right). \\label{rapidity cut}\n\\end{eqnarray} \nUpon combining the two cuts, in the case of $m_X \\ge $ 750 GeV, the angle cut is stronger, so that the soft photons emitted at LHC satisfy \n\\begin{eqnarray}\n0< E^\\mathrm{soft}_{\\gamma} \\le 0.19 \\left(\\frac{m_X}{2}\\right). \\label{real emission photon}\n\\end{eqnarray}\nThe discussion so far has pertained to the emitted photons.\n\nThe technique of summing infrared photons is as follows: The contribution of one soft photon emission to the decay width (or the cross section) has an infrared divergence, but the divergence is canceled by the other infrared divergence coming from the radiative corrections of the one soft photon exchange.  As a result the decay width is multiplied by an infrared free factor $\\Phi$, called the eikonal factor. This eikonal factor can be easily summed to form an exponential factor, by including multi-photons. Virtual photons should also be soft, since the following approximation is taken in the numerator of the fermion propagator, \n\\begin{eqnarray}\n\\gamma_{\\mu} (p+k)^{\\mu}+m \\approx  \\gamma_{\\mu} p^{\\mu}+m,\n\\end{eqnarray}\nwhere $p$ is the fermion's (monopole's) momentum and $k$ is the photon's momentum.\nThis means\n\\begin{eqnarray}\n\\vert \\bm{k} \\vert \\ll \\{m~ \\mbox{and}~ \\vert \\bm{p} \\vert\\} \\le \\frac{1}{2} \\sqrt{(2m)^2+ (2\\bm{p})^2} \\approx \\frac{1}{2} \\left(2m+ \\frac{\\bm{p}^2}{m} \\right) \\approx \\frac{1}{2} m_X.\n\\end{eqnarray}\nTherefore, for the virtual soft photon corrections, we have\n\\begin{eqnarray}\n0< E^\\mathrm{soft}_{\\gamma} \\le \\frac{1}{2} m_X.  \\label{virtual soft photon}\n\\end{eqnarray}\nThen, the cancelation between real emissions and virtual corrections leaves the soft photons in the following energy interval:\n\\begin{eqnarray}\n0.19 \\times \\frac{1}{2} m_X \\le E^\\mathrm{soft}_{\\gamma} \\le \\frac{1}{2} m_X.\n\\end{eqnarray}\n\nIf we assume that the monopole has four-momentum $p_1$, the antimonopole has $p_2$ inside the monopolium, and $t=(p_1-p_2)^2=4p_{rel}^2=-4\\bm{p}^2_{rel}<0$, then\n\\begin{eqnarray}\n\\Phi= -\\frac{g^2}{2} \\int_{0.19 \\times \\frac{1}{2} m_X}^{\\frac{1}{2} m_X} \\frac{d^4k}{(2\\pi)^4}\\frac{-i}{k^2+i \\varepsilon} \\left(\\frac{p_1^{\\mu}}{p_1\\cdot k} - \\frac{p_2^{\\mu}}{p_2\\cdot k}\\right)^2,\n\\end{eqnarray}\nWe can rewrite $\\Phi$ using the above dictionary as follows:\n\\begin{eqnarray}\n\\Phi &=& \\frac{g^2}{2} \\left[ \\int_{0}^{\\infty} \\frac{d^4k}{(2\\pi)^4}\\frac{-i}{k^2-\\lambda^2+i \\varepsilon} \\left(\\frac{p_1^{\\mu}}{p_1\\cdot k} - \\frac{p_2^{\\mu}}{p_2\\cdot k} \\right)^2 \\right]_{\\lambda=0.19 \\times \\frac{1}{2} m_X}^{\\lambda=\\frac{1}{2}m_X}\\\\\n&=&\\left(\\frac{g}{4\\pi}\\right)^2 \\left[-\\int_0^1 dy~\\frac{y}{y^2+(1-y) (\\lambda\/m)^2} \\right. \\nonumber \\\\ \n& & ~~~~~~~~~~+\\left. 4 \\left(1-2m^2\/t \\right)\\times \\int_0^1 dy~\\frac{1}{\\sqrt C} \\ln \\left\\vert \\frac{\\sqrt{C}+y}{\\sqrt{C}-y} \\right\\vert \\right] _{\\lambda=\\lambda=0.19 \\times \\frac{1}{2} m_X}^{\\lambda=\\frac{1}{2} m_X}\\label{Eikonal} \\\\\n&\\approx& \\pi^2 \\left(\\frac{g^2}{4 \\pi} \\right) \\left[\\ln^2 \\frac{\\vert t \\vert}{m^2}+4 \\ln \\frac{\\vert t \\vert}{m^2} \\ln \\frac{m}{\\lambda} \\right]_{\\lambda=0.19 \\times \\frac{1}{2} m_X}^{\\lambda=\\frac{1}{2} m_X},\\label{high energy Eikonal}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nC=C\\left(y, \\frac{m^2}{t}, \\frac{\\lambda^2}{t}\\right) =y^2 \\left(1-\\frac{4m^2}{t}\\right) - 4 (1-y) \\frac{\\lambda^2}{t}.\n\\end{eqnarray}\nThe last approximate formula, in Eq. (\\ref{high energy Eikonal}), being valid in the high energy limit, is written as a reference, but is useful to estimate $\\Phi$ roughly (see for example Ref.~\\cite{Landau-Lifshitz-QED}).  We will use Eq. (\\ref{Eikonal}), since the high energy limit $\\vert t \\vert \\gg m^2$ does not necessarily work in our case.  This double logarithmic approximation in the high energy limit was used to estimate the total cross section $\\sigma(e^{+}e^{-} \\to \\mbox{multi-photons})$.  Please refer to Ref.~\\cite{Eidelman} and references theirein.\n\nUsing this eikonal factor, the total decay widths can be written as,\n\\begin{eqnarray}\n\\Gamma_\\mathrm{tot}(X_0)&=& \\Gamma(X_0 \\to 2 \\gamma) \\cosh \\Phi, \\\\\n\\Gamma_\\mathrm{tot}(X_1)&=&\\Gamma(X_1 \\to 3 \\gamma) \\cosh \\Phi \\nonumber \\\\\n&+&\\sum_{i} \\Gamma(X_1 \\to q_i\\bar{q}_i)+ \\sum_{i}\\Gamma(X_1 \\to \\ell^+_i\\ell^-_i)+\\Gamma(X_1 \\to W^+W^-),\n\\end{eqnarray}\nsince we have to add an even number of soft photons to the $2\\gamma$ decay or $3\\gamma$ decay, the $\\cosh \\Phi$ factor appears. Now we understand that the branching ratio is roughly given by,\n\\begin{eqnarray}\n\\mathrm{Br}(X_0) \\approx \\mathrm{Br}(X_1) \\approx \\frac{1}{\\cosh \\Phi \\left(g, t\/m^2, m_X\/m \\right)}. \\label{Br}\n\\end{eqnarray}\nDerivation of this suppression factor in the branching ratios is very naive, but is indispensable for the strong coupling dynamics of the monopolium.\n\\section{A model of a monopole and a monopolium based on the Zwanziger model and lattice gauge theory}\nThe bound state of the monopoles is formed by the exchange of magnetic photons $\\gamma_M$.  However, the coupling of the monopole to the magnetic photon is very strong, so that we adopt a lattice gauge theory approach with a lattice constant $a$ as a UV cutoff \\cite{lattice gauge theory1, lattice gauge theory2, lattice gauge theory3}.\nThe space-time is considered to be a square lattice $n=(n_0, n_1, n_2, n_3)$ ($n_{\\mu}$ is an integer) with a lattice constant $a$. The link variables $U^{(A)}_{n\\hat{\\mu}}$ and $U^{(B)}_{n\\hat{\\mu}}$ are introduced as usual for the electric and magnetic photons $A_{\\mu}(x)$ and $B_{\\mu}(x)$, respectively.\n\\begin{eqnarray}\nU^{(A)}_{n\\hat{\\mu}}=e^{i \\{e a A_{\\mu}(na)\\}}, ~~U^{(B)}_{n\\hat{\\mu}}=e^{i \\{g a B_{\\mu}(na)\\}},\n\\end{eqnarray}\nand the Wilson loops $W^{(A)}[C]$ and $W^{(B)}[C]$ are defined as the product of the link variables along the loop $C$:\n\n\\begin{eqnarray}\nW^{(A)}[C]= \\prod_{n \\in C, ~\\mu \\parallel C} U^{(A)}_{n\\hat{\\mu}}, ~~W^{(B)}[C]= \\prod_{n \\in C, ~\\mu \\parallel C} U^{(B)}_{n\\hat{\\mu}}.\n\\end{eqnarray}\nThe minimum Wilson loop is given for the boundary curve $C_{n\\mu\\nu}\\equiv \\partial P_{n\\mu\\nu}$ of the minimum plaquette $P_{n\\mu\\nu}=\\mbox{rectangular}~(n, n+\\hat{\\mu}, n+\\hat{\\mu}+\\hat{\\nu}, n+\\hat{\\nu}, n)$. Then, the Zwanziger action can be written as the lattice gauge theory action in the Euclidean metric:\n\\begin{eqnarray}\nS^{\\rm{ZW}}_\\mathrm{lattice}&=&-\\sum_{n, \\nu} \\left( \\frac{1}{2e^2} W^{(A)} [C_{n\\eta\\nu}] +\\frac{1}{2g^2} W^{(B)} [C_{n\\eta\\nu}]+ (h.c.) \\right) \\nonumber \\\\\n&-&\\sum_{n, \\nu} \\frac{1}{2eg}  \\left(W^{(A)} [C_{n\\eta\\nu}]\\tilde{W}^{(B)} [C_{n\\eta\\nu}] - W^{(B)} [C_{n\\eta\\nu}]\\tilde{W}^{(A)} [C_{n\\eta\\nu}] \\right) \\nonumber \\\\\n&+&\\frac{a^3}{2} \\sum_{i, n\\nu} \\left( \\overline{\\psi}_{i, n}\\gamma_{\\nu} \\left(U^{(A)}_{n\\nu}+U^{(B)}_{n\\nu}\\right) \\psi_{i, n+\\nu}-\\overline{\\psi}_{i, n}\\gamma_{\\nu} \\left(U^{(A)}_{n\\nu}+U^{(B)}_{n\\nu}\\right)^{\\dagger} \\psi_{i, n-\\nu} \\right) \\nonumber \\\\\n&-& a^4 \\sum_{i, n} m_i ~\\overline{\\psi}_{i, n} \\psi_{i, n} ,\n\\end{eqnarray}\nwhere the dual Wilson loop reads\n\\begin{eqnarray}\n\\tilde{W}^{(A,B)} [C_{n\\eta\\nu}] =-\\frac{i}{2} \\epsilon_{\\eta\\nu\\lambda\\rho}W^{(A,B)} [C_{n\\lambda\\rho}] ,\n\\end{eqnarray}\nwhere $\\epsilon_{1234}=1$.\n\nIn the action, we assume that the magnetic coupling $g$ is strong and , while the electric coupling $e$ is weak and perturbative.  So, in estimating the expectation value of the large Wilson loop $W^{(B)}[C]$, the strong coupling expansion is used \\cite{lattice gauge theory1, lattice gauge theory2, lattice gauge theory3}.  We choose $C$ to be a rectangle of length $T$ in time and length $r$ in the space-like direction $\\nu$; then we have \n\\begin{eqnarray}\n\\langle W^{(B)}[C] \\rangle &=& \\frac{\\int dU^{(B)}_{n\\nu}~ W^{(B)}[C] ~e^{-S^{\\rm{ZW}}_\\mathrm{lattice}}}{\\int dU^{(B)}_{n\\nu} ~e^{-S^{\\rm{ZW}}_\\mathrm{lattice} } }\\\\\n&=& \\delta_{\\mu\\eta} \\exp\\left(-\\ln(2g^2) \\frac{T \\cdot r}{a^2} \\right) \\left(1+ \\cdots \\right),\n\\end{eqnarray}\nwhere $T \\cdot r$ denotes the minimum area of the rectangle $C$, so that the Wilson's area law is realized only if $\\mu$ is in the $\\eta$-direction.  This point is very important.  Define the potential between a heavy monopole and its antimonopole separated by a distance $r$ to be $V_{M\\overline{M}}(r)$.  Then, the potential has a linear term in $r$, if the monopole and antimonopole are separated in the $\\eta$-direction;\n\\begin{eqnarray}\nV_{M\\overline{M}}(r)= \\delta_{r \\parallel \\eta} \\frac{\\ln(2g^2)}{a^2} r + \\cdots.\n\\end{eqnarray}\nThis clarifies the meaning of the special direction $\\eta$ that appears in the Zwanziger formulation.  It gives the direction of the Dirac string starting from the monopole.  Therefore, the monopole and antimonopole are connected by the string, starting from the monopole and ending at the antimonopole, which contributes to the linear potential between them.\n\nIn addition to this strong coupling contribution, we will add the usual perturbative weak coupling contribution, and so the potential at this stage is\n\\begin{eqnarray}\nV_{M\\overline{M}}(r)= -\\frac{g^2}{4\\pi r} + \\frac{\\ln(2g^2)}{a^2} r.\n\\end{eqnarray}\nThis matches with high precision QCD calculations in which the potential between a quark and antiquark pair is well approximated by the linear plus Coulomb potential \\cite{Bali1, Bali2, Bali3}; this potential is good for point-like quarks.  The monopole, however, may not be point-like, and may have an internal structure.  The $ U(1) $ lattice gauge theory is usually considered not to be well defined, since there exists a first-order phase transition between the confinement phase and the perturbative phase \\cite{Creutz-Jacobs-Rebbi}, and it obstructs the continuum limit of $a \\to 0$.  One way out from this difficulty is to lift the $ U(1) $ theory to $ SU(2) $ theory, or other asymptotic free theory, when we approach to the short distance region.  As is shown by 't Hooft and Polyakov \\cite{monopole1, monopole2, monopole3}, the $ U(1) $ monopole was given as a classical solution of $ SU(2) $ gauge theory, which is broken to $ U(1) $ with a triplet Higgs field $\\phi^a(x)~(a=1-3)$. Therefore, if we go inside the monopole, the non-Abelian gauge theory may appear.  If this happens we may take the continuum limit properly.  \n\nThe 't Hooft-Polyakov monopole is a classical solution of the $ SU(2) $ gauge theory with a triplet Higgs, based on \n\\begin{eqnarray}\n{\\cal L}_{SU(2)\\, \\mathrm{monopole}}=-\\frac{1}{4} (F^a_{\\mu\\nu})^2+ (D_{\\mu}\\phi^a)^2-\\lambda(\\vert\\phi \\vert^2-v)^2,\n\\end{eqnarray}\nand the following ansatz:\n\\begin{eqnarray}\nA^a_i(x)=v~\\epsilon^{aij}\\hat{r}^j \\frac{1-K(\\xi)}{\\xi}, ~\\phi^a(x)=v~\\hat{r}^a \\frac{H(\\xi)}{\\xi},\n\\end{eqnarray}\nwhere we define a dimensionless parameter $\\xi=evr$.\nIf we take the limit $\\lambda \\to 0$ while keeping $v \\ne 0$, the solution, called the BPS solution, is given analytically \\cite{BPS1, BPS2}.  Then, the equations of motion become the first order:\n\\begin{eqnarray}\n\\xi \\frac{dK}{d\\xi}=-KH, ~~\\xi \\frac{dH}{d\\xi}=H-K^2+1,\n\\end{eqnarray}\nfrom which the solutions read\n\\begin{eqnarray}\nK(\\xi)=\\frac{\\xi}{\\sinh \\xi}, ~~H(\\xi)=\\xi \\coth\\xi-1.\n\\end{eqnarray}\nWe want to know the distribution of the magnetic charge inside the monopole, $\\xi <1$ or $r< 1\/ev$.  The $ SU(2) $ gauge potential and the Higgs field are properly reduced by factors of $1-K(\\xi)$ and $H(\\xi)$, but the monopole charge is unfortunately not smeared even inside the monopole.  This can be understood from the $ U(1) $ field strength proposed by 't Hooft.  This gauge invariant expression can be rewritten as follows \\cite{Arafune-Freund-Goebel}:\n\\begin{eqnarray}\nF_{\\mu\\nu}=\\partial_{\\mu} (\\hat{\\phi}^a A^a_{\\nu})-\\partial_{\\nu} (\\hat{\\phi}^a A^a_{\\mu})-\\frac{1}{e}\\epsilon^{abc}\\hat{\\phi}^a \\partial_{\\mu} \\hat{\\phi}^b \\partial_{\\nu} \\hat{\\phi}^c,\n\\end{eqnarray}\nwhere $\\hat{\\phi}^a=\\phi^a\/\\vert \\phi \\vert$.  From this expression the magnetic charge is found to be a topological number:\n\\begin{eqnarray}\n\\Phi_m=\\int \\bm{B}d\\bm{S}=\\frac{4\\pi}{e} \\int d^3x \\frac{\\partial(\\phi^1, \\phi^2, \\phi^3)}{\\partial(x^1, x^2, x^3)}=\\frac{4\\pi}{e} N,\n\\end{eqnarray}\nwhere $\\Phi_m$ is the magnetic flux, and $N$ is the winding (wrapping) number of the sphere of the Higgs fields ($\\bm{\\phi}^2=v^2)$ by the  sphere in space ($\\bm{x}^2=1$).\n\nWe can treat the charged sphere as follows: If the sphere of radius $R$ is uniformly charged, it can be viewed as a point charge $Q(\\infty)$ from a long distance, but inside the sphere $(r<R)$ the charge is reduced to $Q(r)=Q(\\infty)(r\/R)^3$. If we can apply this kind of treatment for the magnetic charge, the Coulomb potential can be relaxed at short distances inside the monopole.  It is not clear, however, whether the magnetic charge can be distributed uniformly inside the sphere-like monopole.\n\nHere, we will consider another possibility in which the magnetic charge is distributed uniformly on the surface of the sphere,  fortunately such a solution exists.  Let's take the following modified gauge and Higgs fields in which the contribution from $r<R$ is cutoff:\n\\begin{eqnarray}\n\\tilde{A}^a_i(x)=\\theta(r-R) A^a_i(x), ~\\tilde{\\phi}^a(x)=\\theta(r-R)\\phi^a(x),\n\\end{eqnarray}\nwhere $\\theta(r-R)$ is a step function $0$ for $r<R$, and $1$ for $r>R$.  Accordingly, $K$ and $H$ are modified:\n\\begin{eqnarray}\n1-\\tilde{K}(r)=\\theta(r-R) (1-K(r)), ~~\\tilde{H}(r)=\\theta(r-R)H(r).\n\\end{eqnarray}\nThis modification inside the sphere ($r<R$) is allowed, since $K(r)=1$ and $H(r)=0$ is the solution.\n\nThe distribution of the $ U(1) $ magnetic charge is easily understood. Since $\\phi^a=0$ inside $r<R$, the magnetic flux on the surface of the radius $r$ sphere reads\n\\begin{eqnarray}\n\\Phi_m(r)= \\frac{4\\pi}{e} \\theta(r-R),\n\\end{eqnarray}\nwhich shows that the magnetic charge $4\\pi\/e$ is distributed uniformly on the surface of the monopole sphere with radius $R$.  The solution satisfies the following equations:\n\\begin{eqnarray}\n\\xi \\frac{d\\tilde{K}}{d\\xi}&=&-\\tilde{K}\\tilde{H} + evR\\left(1-K(evR)\\right)\\delta(\\xi-evR), \\\\\n\\xi \\frac{d\\tilde{H}}{d\\xi}&=&\\tilde{H}-\\tilde{K}^2+1+evRH(evR)\\delta(\\xi-evR),\n\\end{eqnarray}\nwhich are modified only on the surface, where the magnetic charge is distributed.\n\nTherefore, the magnetic force between the monopole and antimonopole vanishes when one goes inside either respective sphere $(r<R)$.  So, one possibility is to use the following monopole-antimonopole potential $V^{(a)}_{M\\overline{M}}(r)$:\n\\begin{eqnarray}\nV^{(a)}_{M\\overline{M}}(r)&=& - \\frac{g^2}{4\\pi r} + \\frac{\\ln(2g^2)}{a^2} r ~~(\\mbox{for}~ r>R), \\\\\n&=&\\mbox{const} = - \\frac{g^2}{4\\pi R} + \\frac{\\ln(2g^2)}{a^2} R~~ (\\mbox{for}~ r<R),\n\\end{eqnarray}\nwhich is defined by three parameters $g$, $a$ and $R$.  The idea of relaxing the Coulomb potential near the monopole is used by Epele {\\it et al.}~\\cite{Epele1, Epele2}. They follow Schiff and Goebel and relax the potential through the existence of Dirac strings \\cite{Schiff-Goebel1, Schiff-Goebel2}.  Their choice of the potential is \n\\begin{eqnarray}\nV^{SG}_ {M\\overline{M}}(r)&=& - \\left(1-e^{-r\/R}\\right)\\frac{g^2}{4\\pi r},\n\\end{eqnarray}\nwithout the linear term.  \n\nAnother possibility is to consider the linear potential to be the dominant contribution and the Coulomb potential just playing the role of lowering the potential.  Then, the corresponding potential becomes\n\\begin{eqnarray}\nV^{(b)}_ {M\\overline{M}}(r) &=& - \\frac{g^2}{4\\pi R} + \\frac{\\ln(2g^2)}{a^2} r ~~(\\mbox{for}~ r>R), \\\\\n&=&\\mbox{const} = - \\frac{g^2}{4\\pi R} + \\frac{\\ln(2g^2)}{a^2} R~~ (\\mbox{for}~ r<R).\n\\end{eqnarray}\nThe difference between the two potentials is\n\\begin{eqnarray}\nV^{(a)}_ {M\\overline{M}}(r)-V^{(b)}_ {M\\overline{M}}(r) &=& +\\frac{g^2}{4\\pi R} -\\frac{g^2}{4\\pi r}~~(\\mbox{for}~ r>R), \\\\\n&=&0~~ (\\mbox{for}~ r<R).\n\\end{eqnarray}\n\n\nWe can also include the spin-dependent hyperfine interaction.  Then, the Hamiltonian $\\hat{H}$, which we will use to study the energy and wave function of the lowest s-wave states ($^1S_0$ and $^3S_1$) of the monopolium, is\n\\begin{eqnarray}\n\\hat{H}=2m+\\hat{H}_\\mathrm{NR}+\\hat{H}_\\mathrm{HF},\n\\end{eqnarray}\nwhere the nonrelativistic Hamiltonian $\\hat{H}_\\mathrm{NR}$ and the hyperfine interaction $\\hat{H}_\\mathrm{HF}$ are defined by\n\\begin{eqnarray}\n\\hat{H}_{NR}&=& - \\frac{1}{m} \\left(\\frac{1}{r^2} \\frac{d}{dr} r^2 \\frac{d}{dr}\\right) + V_{M\\overline M}(r) , \\\\\n\\hat{H}_{HF}&=&\\frac{2\\pi}{3 m^2}~(\\bm{\\sigma}_1\\cdot \\bm{\\sigma}_2) ~\\bm{\\nabla}^2 V_{M\\overline M}(r),\n\\end{eqnarray}\nIn the case of the Coulomb potential, $g^2 \\delta^{(3)}(\\bm{r})$ appears from $\\bm{\\nabla}^2 V(r)$ and the wave function at the origin contributes to the hyperfine interaction.  If we use the potential  $V^{(a)}_ {M\\overline{M}}(r)$ or $V^{(b)}_ {M\\overline{M}}(r)$, the hyperfine interaction becomes\n\\begin{eqnarray}\nH_{HF}^{(a)}&=&\\frac{2\\pi}{3m^2}(\\bm{\\sigma}_1\\cdot \\bm{\\sigma}_2) \\left[ \\left(\\frac{g^2}{4\\pi R^2}+\\kappa \\right) \\delta(r-R) + \\frac{2\\kappa }{r} \\theta (r-R) \\right], \\\\\nH_{HF}^{(b)}&=&\\frac{2\\pi}{3 m^2}(\\bm{\\sigma}_1\\cdot \\bm{\\sigma}_2) \\left[ \\kappa \\delta(r-R) + \\frac{2\\kappa }{r} \\theta (r-R) \\right],\n\\end{eqnarray}\nwhere the string tension is denoted by $\\kappa= \\ln(2g^2)\/a^2$, and $(\\bm{\\sigma}_1\\cdot \\bm{\\sigma}_2)=-1$ for $X_0$ and $+3$ for $X_1$.  We don't include the relativistic corrections, such as the other Breit terms and the annihilation effect for the hyperfine interaction, since we intend to solve the problem semiclassically.\n\nThe mass $m_X$ of the monopole-antimonopole bound state is obtained as the eigenvalue of the total Hamiltonian $\\hat{H}$.\n\n\n\n\\section{Mass and wave function of monopolium} \nIn this section we use the potential $V^{(b)}_ {M\\overline{M}}(r)$, and obtain the mass and wave function of the monopolium semiclassically.  Schr\\\"{o}dinger's equation with the linear potential can be solved in terms of the Airy function, but the solution is essentially equal to the semiclassical treatment \\cite{Landau-Lifshitz}, so we use the semiclassical method.\n\nFor a given nonrelativistic energy $\\tilde{E}$, that is an eigenvalue $E_\\mathrm{NR}$ of the nonrelativistic Hamiltonian $\\hat{H}_\\mathrm{NR}$, classical momentum $p_r$ and $\\vert p_r \\vert$ are given by\n\\begin{eqnarray}\np_r=\\sqrt{m(\\tilde{E}-V(r))}, ~~\\mbox{and}~~\\vert p_r \\vert=\\sqrt{m(V(r)-\\tilde{E})}.\n\\end{eqnarray}\nThe turning point $r=r_{*}$ is the point where $p_{r_{*}}=0$.\n\nIf we define the wave function by $\\psi(r)= \\chi(r)\/r$, then $\\chi(r)$ satisfies the one-dimensional Schr\\\"{o}dinger equation, and the corresponding semiclassical (WKB) wave function reads\n\\begin{eqnarray}\n\\chi(r)_I&=& \\frac{C}{2\\sqrt{\\vert p_r \\vert}} e^{-\\int^r_{r_{*}} \\vert p_r \\vert dr}~~\\mbox{for}~~r>r_{*}, \\\\\n\\chi(r)_{II}&=& \\frac{C}{\\sqrt{p_r} } \\sin \\left(-\\int_r^{r_{*}} p_r dr + \\frac{\\pi}{4} \\right)~~\\mbox{for}~~0<r<r_{*}, \n\\end{eqnarray}\nand the Bohr-Sommerfeld quantization condition is given by\n\\begin{eqnarray}\n2 \\int_0^{r_{*}} p_r ~dr= 2\\pi \\left(n-\\frac{1}{2}\\right), \\label{BS}\n\\end{eqnarray}\nwhere $n~(=1, 2, \\cdots)$ denotes the radial excitation number, or the principal quantum number.\n\nIf we apply this for the potential $V^{(b)}(r)$, then the wave function is analytically given by, \n\\begin{eqnarray}\n\\int^r_{r_{*}} \\vert p_r \\vert dr &=&\\frac{2\\sqrt{m}}{3\\kappa} \\left( \\kappa r-(\\tilde{E}+g^2\/4\\pi R) \\right)^{3\/2}-(r=r_{*})~~\\mbox{for}~~r>r_{*}, \\\\\n\\int_r^{r_{*}} p_r dr&=& \\frac{2\\sqrt{m}}{3\\kappa} \\left((\\tilde{E}+g^2\/4\\pi R)-\\kappa r \\right)^{3\/2}-(r=r_{*})~~\\mbox{for}~~R<r<r_{*}, \\\\\n\\int_r^{r_{*}} p_r dr&=& \\frac{2\\sqrt{m}}{3\\kappa} \\left((\\tilde{E}+g^2\/4\\pi R)-\\kappa r \\right)^{3\/2}(r=R)-(r=r_{*}) \\nonumber \\\\\n&+& \\sqrt{m} \\left((\\tilde{E}+g^2\/4 \\pi R)-\\kappa R\\right)^{1\/2} (R-r)~~\\mbox{for}~~0<r<R, \n\\end{eqnarray} \nwhere we use the notation of the string tension as $\\kappa=\\ln(2g^2)\/a^2$.\n\nThe Bohr-Sommerfeld quantization condition becomes a third order algebraic equation;\n\\begin{eqnarray}\n\\frac{2}{3m\\sigma} (p_R)^3 +R (p_R)=\\pi (n-1\/2),\n\\end{eqnarray}\nfor \n\\begin{eqnarray}\np_R=\\sqrt{m(\\tilde{E}+ \\frac{g^2}{4\\pi R}-\\kappa R)}.\n\\end{eqnarray}\nThis is the case of one real root and two mutually complex conjugate roots, and so the real root is given by the Cardano formula as\n\\begin{eqnarray}\n\\tilde{E}_n=-\\frac{g^2}{4\\pi R}+\\kappa R + \\frac{1}{m} \\left\\{\\sqrt[3]{-\\frac{q}{2}+\\sqrt{\\left(\\frac{q}{2}\\right)^2+\\left(\\frac{p}{3}\\right)^3}} + \\sqrt[3]{-\\frac{q}{2} - \\sqrt{\\left(\\frac{q}{2}\\right)^2+\\left(\\frac{p}{3}\\right)^3}} \\right\\}^2,\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\np=\\frac{3m}{2} \\kappa R,~~\\mbox{and}~~q=-\\frac{3m}{2} \\kappa \\pi(n-1\/2).\n\\end{eqnarray}\n\nHere we consider two cases: (Case 1) is that the monopole mass is small compared to the scale of the string tension, and also $R$ is about the same order as the string scale;\n\\begin{eqnarray}\nm \\ll \\sqrt{\\kappa} = \\frac{\\sqrt{\\ln (2g^2)}}{a}, ~~\\mbox{and}~~\nR = O( 1\/\\sqrt{\\kappa})=O(a),\n\\end{eqnarray}\nUnder this condition, $(q\/2)^2 \\gg (p\/3)^3$ , or {\\bf [$m\\kappa R^3 \\ll 9 \\pi^2(2n-1)^2\/8$]} holds, and we obtain\n\\begin{eqnarray}\n\\tilde{E}^{(1)}_n&=&-\\frac{g^2}{4\\pi R}+\\kappa R+\\frac{1}{m} (\\sqrt[3]{q})^2 \\\\\n&=&-\\frac{g^2}{4\\pi R}+\\kappa R+ \\left(\\frac{3\\pi}{4} \\kappa \\right)^{2\/3} m^{-1\/3} (2n-1)^{2\/3}.\n\\end{eqnarray}\nThe dependence of the energy on the mass $m$ is $(m)^1$ for the Coulomb potential, but is $m^{-1\/3}\\sigma^{2\/3}$ in the linear potential case.\\footnote{This dependence $m^{-1\/3}$ is utilized in the study of pentaquark mass.  See Eq. (104) in Ref.~\\cite{Bando}.}\n\nThe other case (Case 2) is that the monopole mass is heavier than the string scale, that is, the case where $(q\/2)^2 \\ll (p\/3)^3$, or {\\bf [$m\\kappa R^3 \\gg 9 \\pi^2(2n-1)^2\/8$]} holds.  In this case, the energy levels depend on $m^{-1}R^{-2}$:\n\\begin{eqnarray}\n\\tilde{E}^{(2)}_n= -\\frac{g^2}{4\\pi R} +\\kappa R +\\frac{1}{m} \\left(\\frac{\\pi}{2R} \\right)^2 (2n-1)^2.\n\\end{eqnarray}\n\nThe normalization constant $C$ is fixed numerically so as to satisfy the wave function's normalization,\n\\begin{eqnarray}\n4 \\pi \\int_0^{\\infty}  \\vert \\chi (r) \\vert^2 dr = 1,\n\\end{eqnarray}\nand the value of the function at $r=R$ is determined with $C$ or $C_n$ for the $n$th state as \n\\begin{eqnarray}\n\\vert \\psi_n(r=R) \\vert = \\frac{C_n}{R \\sqrt{p_R}} \\left\\vert \\sin (p_R \\cdot R + \\pi\/4)\\right\\vert, \n\\end{eqnarray}\nwhere we consider that the monopole and antimonopole collide when the distance $r=R$ in our model, so that the wave function at $r=R$ is used instead of the usual value at $r=0$.\n\n\\section{Comparison with experiments, LHC and others} \nIn this section, on the basis of the formulae obtained so far, we will compare the numerical estimation of the mass and the decay width of monopoliums ($^1S_0$ and $^3S_1$) , and the production rate with experiments such as LHC and others.   \n\nThere are typically two kinds of Solutions, 1 and 2: \n\nSolution 1 has a large monopole mass about half of the monopolium mass, namely $2m \\approx m_X$.  In this case the nonrelativistic energy $\\tilde{E}$ is one order of magnitude smaller than the monopole mass, and the hyperfine interaction is also negligibly small. This corresponds to (Case 2) in the previous section. See Table~\\ref{tab:monopolium10} in which a few examples of Solution 1 are listed, where $J=0$ and $J=1$ monopoliums are degenerate with a mass of a several TeV. Estimation of the total width $\\Gamma_\\mathrm{tot}(X) \\approx \\mathrm{\\Gamma}(X \\to 2\\gamma ~\\mbox{or} ~3\\gamma) \\cosh \\Phi$ is performed using formulae (\\ref{Eikonal}) and (\\ref{Br}) with $\\Gamma(X_0 \\to 2\\gamma)$ (28) and $\\Gamma(X_1 \\to 3\\gamma)$ (29).  However, the very large total widths and the very small lifetimes suggest that our approximation is too naive, and a more reasonable estimation should be developed.  We can say here that the estimation of the total widths and branching ratios are important for the strong coupling monopole dynamics.\n\\begin{table}[t]\n\\caption{The derived properties of the monopolium spectrum for masses of $ \\mathcal{O}(1-10) $ TeV. \nWe use $g=\\sqrt{4\\pi\\cdot137}$, and $R=a$ is taken, while $ a $ and $ m $ are varied.\n\\label{tab:monopolium10}\n\\centering\n\t\\begin{tabular}{| c || c | c || c | c || c | c |} \\hline\nSpin & 0 & 1 & 0 & 1 & 0 & 1\\\\ \\hline \\hline\nMass & \\multicolumn{2}{|c||}{$5~\\mathrm{TeV}$} & \\multicolumn{2}{|c||}{$6~\\mathrm{TeV}$} & \\multicolumn{2}{|c|}{$7~\\mathrm{TeV}$}\\\\ \\hline\n$m$~[GeV] & \\multicolumn{2}{|c||}{$2624$} & \\multicolumn{2}{|c||}{$3149$}  & \\multicolumn{2}{|c|}{$3674$} \\\\ \\hline\n$\\tilde{E}=E_\\mathrm{NR}$~[GeV] & \\multicolumn{2}{|c||}{$-248$} & \\multicolumn{2}{|c||}{$-298$} & \\multicolumn{2}{|c|}{$-347$} \\\\ \\hline\n$E_\\mathrm{HF}$~[GeV] & $-6\\cdot10^{-6}$ & $2\\cdot10^{-5}$ & $-7\\cdot10^{-6}$ & $2\\cdot10^{-5}$ & $-8\\cdot10^{-6}$ & $2\\cdot10^{-5}$\\\\ \\hline\n$\\Gamma_\\mathrm{tot}$~[GeV] & $2 \\cdot 10^{15}$ &  $4 \\cdot 10^{16}$ & $3 \\cdot 10^{15}$ &  $5 \\cdot 10^{16}$\n& $3 \\cdot 10^{15}$ &  $6 \\cdot 10^{16}$ \\\\ \\hline\n$\\tau$~[s] & $3\\cdot10^{-40}$ &  $2\\cdot10^{-41}$ & $2\\cdot10^{-40}$ &  $1\\cdot10^{-41}$\n& $2 \\cdot 10^{-40}$ & $1 \\cdot 10^{-41}$ \\\\ \\hline\n$a$ [$\\mathrm{GeV}^{-1}$] & \\multicolumn{2}{|c||}{$1\/0.52$} & \\multicolumn{2}{|c||}{$1\/0.43$} & \\multicolumn{2}{|c|}{$1\/0.37$}\\\\ \\hline\n\\end{tabular}\n\\end{table}\nAt this stage, the states considered are difficult to observe as a resonance peak at the LHC due to their high masses, but multi-photon emission decays may be observable at future 100 TeV colliders.  Other than collider experiments, they could possibly be observed in astrophysical observations, through an explosion or burst of $\\gamma$ rays from the monopoliums.  The number distribution function $f_{\\gamma}(n)$ for photons can be very roughly estimated using the eikonal factor as\n\\begin{eqnarray}\nf_{\\gamma}(n)=\\frac{\\Phi^n\/n!}{\\cosh \\Phi} ~~~(n=\\mbox{even~for}~ X_0, ~\\mbox{odd for~} X_1).\n\\end{eqnarray}\n\n\nHowever, in order to give a definite statement for the number distribution function of photons, a more reasonable estimation of $\\Phi$ is necessary.\n\nThe second case, Solution 2, is obtained when considering very light monopole masses, for example 870 MeV.  It is found that the large  energy $\\tilde{E}$ can lead to a bound state mass of 750 GeV and 1410 GeV for the $ X_0 $ and $ X_1 $ states respectively.  See Table~\\ref{tab:monopolium}.\n\n\\begin{table}[t]\n\\caption{Properties of the monopolium and it's decay at a 13 TeV collider.\nThe parameter choices considered are $g=\\sqrt{4\\pi\\cdot137}$, $a=64\/\\mathrm{MeV}$, $R=570\/\\mathrm{MeV}$ \nand monopole mass $m = 870$~MeV, for consistency with the relation in Case 2.\n\\label{tab:monopolium}\n\\centering\n\\begin{tabular}{| c || c | c |} \\hline\nSpin & 0 & 1 \\\\ \\hline \\hline\nMass & $750~\\mathrm{GeV}$ & $1410~\\mathrm{GeV}$ \\\\ \\hline \\hline\n$\\sigma$ ($\\sqrt{s} = 13~\\mathrm{TeV}$) & $0.68$~fb (to 2 $\\gamma$) & $0.0025$~ab (to 3 $\\gamma$) \\\\ \\hline\nFraction ($\\Gamma_i\/\\Gamma_\\mathrm{tot}$) & $0.78$ (2 $\\gamma$ decay) & $1$ (3 $\\gamma$ decay) \\\\ \\hline\n$\\tilde{E}=E_\\mathrm{NR}$ & $914$~GeV & $914$~GeV \\\\ \\hline\n$E_\\mathrm{HF}$ & $-164$~GeV & $496$~GeV \\\\ \\hline\n$\\Gamma_\\mathrm{tot}$ & $5$~MeV & $6$~MeV  \\\\ \\hline\n\\end{tabular}\n\\end{table}\nIn the estimation of the production cross section for $X_0$ we have used the formula of Csaki {\\it et al.} \\cite{Csaki}, \n\\begin{eqnarray}\n\\sigma_{13~\\mbox{\\scriptsize{TeV}}}=10.8~\\mbox{pb} \\left(\\frac{\\Gamma_\\mathrm{tot}}{45~ \\mbox{\\small{GeV}}}\\right) \\mathrm{Br}^2(X_0 \\to \\gamma\\gamma).\n\\end{eqnarray}\nTo attempt to replicate the 750 GeV diphoton excess we make an appropriate choice of parameters to obtain an $ X_0 $ mass of $m_{X_0}=750$ GeV, and an $ X_1 $ mass of $m_{X_1}=1410$ GeV. The production cross section in the 2$\\gamma$ channel at 13 TeV LHC is, however, at most 0.68 fb and is one order of magnitude smaller than the recent announcements.  In this study the high energy limit of the decay rates $\\Gamma(X_0 \\to 2\\gamma)$ (30), $\\Gamma(X_1 \\to 3\\gamma)$ (31) are used, and the total widths are found to be 5 MeV, 6 MeV respectively by using Eq. (56).  It is also found that the hyperfine interaction is large because of the small monopole mass, such that the $X_0$ and $X_1$ state masses are strongly separated.   \n\n For the estimation of the cross section for $X_1$\n we have used the decay width $\\Gamma(X_1 \\to q\\bar{q})$, \na branching ratio of $X_1$ to $3 \\gamma$, $\\mathrm{Br} (X_1 \\to 3 \\gamma) \\sim 1$, and the parton luminosity which is derived from graphs by W.J.~Stirling~(MSTW2008NLO PDF)\\cite{sterling},\n \\begin{eqnarray}\n \\left[\\frac{1}{s} \\frac{d {\\cal L}_{q\\overline{q}}(\\tau)}{d\\tau}\\right]_{\\tau=\\frac{m^2_X}{s}} \\sim 67~\\mathrm{pb}\n \\end{eqnarray} \nAs for the production cross section of $X_1$, the usual fusion mechanism of quark and antiquark inside a proton is used. \nThe value of the cross section 0.0025 ab is rather small.\nThe parton luminosity $\\left[\\frac{1}{s} \\frac{d {\\cal L}_{q\\overline{q}}(\\tau)}{d\\tau}\\right]_{\\tau=\\frac{m_X^2}{s}} $, however,\nincreases when $s$ becomes larger, so that the production cross section can be several times larger for 33 TeV.\n The integrated luminosity of the order of hundred $(\\mathrm{ab})^{-1}$ is necessary at 33 TeV LHC or the future 100 TeV colliders in order to detect 1.4 TeV $X_1$.\n\\section{Conclusion}\nBeing inspired by the announcement of the 750 GeV excess in the two photon channel at LHC,  we have considered seriously a manageable model of monopolium, without missing the essence of it.  As a result, we have a number of small but interesting findings which are summarized as follows:\n\n(1) Zwanziger's electric and magnetic dual formulation works properly.  Especially, the meaning of a special direction ($n^{\\mu}$ in Zwanziger and $\\eta^{\\mu}$ in this paper) becomes a little more manifest, since in the lattice formulation, the linear potential appears only in this direction.  This supports that $\\eta^{\\mu}$ is in the direction of  the string going from the monopole and terminating at the antimonopole.\\footnote{We should comment on the special direction $\\eta ^{\\mu}$ a little more: At the classical level, the classical solution gives a singularity in the direction $\\eta ^{\\mu}$ along the Dirac string\\cite{Zwanziger2}.  At the perturbative level with Feynman rules Eq. (10)-(12), $\\eta ^{\\mu}$ gives the gauge condition, and hence the amplitudes on the mass shell do not depend on it.  At the non-perturbative level, the potential does depend on $\\eta ^{\\mu}$ and the energy is stored along its direction connecting monopole and antimonopole, that is, the tensionless Dirac string at the classical level becomes a tension-full string.  So, in order to match the perturbative calculation with the classical and non-perturbative treatments, we have to fix $\\eta ^{\\mu}$ after the classical solution is obtained. From this observation, a natural way of treating Zwanziger formalism is to relax the frozen property of $\\eta ^{\\mu}$, by replacing with an external string variable $X^{\\mu}(t, \\sigma)$, namely, $\\eta ^{\\mu} \\rightarrow \\partial X^{\\mu}(t, \\sigma)\/\\partial \\sigma$.  Here $t$ is a time and $\\sigma$ is a parameter representing the extension of the string. Then, the non-perturbative effects may lift the external field of $X^{\\mu}(t, \\sigma)$ to a dynamical variable with a tension-ful string action.}\n\n(2) For the strong coupling, the lattice version of Zwanziger gives a linear potential.\n\n(3) As a finite sized monopole, the BPS equation for the 't Hooft-Polyakov monopole has another solution in which the $ U(1) $ magnetic charge is distributed uniformly on the surface of a sphere. So, the potential becomes flat inside the monopole.\n\n(4) Now, the monopole-antimonopole potential $V_ {M\\overline{M}}(r)$ is linear plus Coulomb, but is cutoff inside, that is,\n\\begin{eqnarray}\nV^{(b)}_ {M\\overline{M}}(r) &=& - \\frac{g^2}{4\\pi R} + \\frac{\\ln(2g^2)}{a^2} r ~~(\\mbox{for}~ r>R), \\nonumber \\\\\n&=&\\mbox{const} = - \\frac{g^2}{4\\pi R} + \\frac{\\ln(2g^2)}{a^2} R~~ (\\mbox{for}~ r<R). \\nonumber\n\\end{eqnarray}\n\nThe hyperfine interaction is modified depending on the potential which includes the non-perturbative effects.\n\n(5) The mass and wave function of the monopolium ($n ^1S_0$ and $n ^3S_1$) can be given in a tractable manner, by applying the semiclassical method to the potential.  \n\n(6) The branching ratio can be estimated very naively, by using the technique of infrared divergences, leading to $\\mathrm{Br}(n ^1S_0 \\to 2\\gamma)\\approx \\mathrm{Br}(n ^3S_1 \\to 3\\gamma)\\approx 1\/\\cosh \\Phi$, where $\\Phi$ is the calculable eikonal factor that depends on the detection cut for the photon energy and angle.  However, the total width calculated in this way is very big in Table~\\ref{tab:monopolium10} (not big in Table~\\ref{tab:monopolium}), so that we have to find a more reasonable way to estimate it before a definite statement can be given.\n\nNumerical estimation based on our model of monopolium shows that there are typically two kinds of solutions:\n\n(7) One kind of solution (Solution 1) is the large monopole mass case in which the binding energy and the hyperfine splitting are very small.  So, $X_0$ and $X_1$ are degenerate. A few examples considered give masses of monopoliums to be 5, 6, and 7 TeV.  The signals of the multi-photon explosion or burst may be detected by 100 TeV colliders or by astronomical observations.\n\n(8)  The other kind of solution  (Solution 2) gives an 870 MeV monopole, and the monopole and its antimonopole form a 750 GeV $J=0$ state and a 1.4 TeV $J=1$ state.\nThe solution can be understood similarly as in QCD, where the so-called small current mass of the monopole is heavily dressed with non-perturbative coats and makes the monopoliums heavy. Unfortunately this solution can't explain the 750 GeV excess at LHC, because the predicted diphoton production rate is one order of magnitude smaller than that of the observed excess. The 1.4 TeV $J=1$ state may provide a target at LHC or future 100 TeV colliders.\n\nThis paper has studied how to take into account strong coupling effects in monopole dynamics, especially in the dynamics of monopolium.  We provide a method, in which the bound state is formed non-perturbatively using a strong coupling expansion in lattice gauge theory, while the decay rate is estimated perturbatively, since the final number of emitted photons is fixed.  However, the strong magnetic coupling may give large branching ratios for the multi-photon emission processes, even if they are not intended to be detected.  Overcoming this point is important and is an issue requiring future work.\n\n\\section*{Acknowledgments}\nThe authors give their thanks to Professor Masanori Okawa and Professor Takanori Kono for valuable discussions.\n\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nActive region formation in the Sun is traditionally thought to be a\ndeeply rooted phenomenon, because their size ($\\sim100\\Mm$) is much\nlarger than the naturally occurring scales in the surface layers of\nthe convection zone ($\\sim1$--$10\\Mm$); see \\cite{Golub}.\nThey are also long-lived\n($\\sim1\/2$ year), which seems unnaturally long if associated with\nthe near-surface layers ($40\\Mm$ depth) where typical time scales\nare about a day. On the other hand, a deeply rooted formation\nscenario for active regions has the problem that the azimuthal pattern\nspeed of active regions does not match the\nangular velocity at great depth.\nOther difficulties concern the strong field strength inferred for the\ntachocline to explain the observed tilt angles and the fact that\nmagnetic structures expand tremendously during their ascent.\nThese and several other arguments have led to the consideration of solar\nactivity as a shallow phenomenon; see \\cite{B05} for details.\nAs a possible mechanism for producing magnetic flux concentrations of the\nform of active regions, the negative effective magnetic pressure instability\nhas been discussed \\citep{KRR89,KRR90,KR94,KMR96,RK07,BKR10,BKKMR}.\nOf course a magnetic field $\\BB$ always gives rise to a positive\nmagnetic pressure, $\\BB^2\/2\\mu_0$,\nwhere $\\mu_0$ is the vacuum permeability.  In a turbulent medium,\nhowever, magnetic fields also suppress the turbulence and thus decrease\nthe turbulent pressure $\\rho\\uu^2\/3$, and modify the pressure caused\nby magnetic fluctuations $\\bb^2\/6\\mu_0$.\nHere, $\\uu$ and $\\bb$ are velocity and magnetic fluctuations,\n$\\rho$ is the density, $\\mu_0$ is the vacuum permeability,\nand the coefficients in the turbulent fluid and magnetic pressure are\ngiven for isotropic turbulence.\nMagnetic fluctuations can be due to both small-scale dynamo action\nas well as tangling of a large-scale field, $\\meanBB$.\nThe total field is thus $\\BB=\\meanBB+\\bb$.\nThe sum of both effects,\n$p_{\\rm turb}=\\overline{\\rho\\uu^2}\/3+\\bra{\\bb^2}\/6\\mu_0$,\nis positive definite, but it depends on $\\meanBB$, and $p_{\\rm turb}$\ntends to decline as $\\meanB\\equiv|\\meanBB|$ increases.\nIndeed, $p_{\\rm turb}=2E_K\/3-\\bra{\\bb^2}\/6\\mu_0$, where $E_K=\\overline{\\rho\\uu^2}\/2\n+\\bra{\\bb^2}\/2\\mu_0 \\approx$ const, so that the change of the turbulent\npressure is negative $(\\delta p_{\\rm turb}<0)$ when the magnetic fluctuations\nare generated by tangling of the mean magnetic field by the velocity fluctuations\nat the expanse of turbulent kinetic energy \\citep{KRR90,BKKMR}.\nThus, we write\n\\EQ\np_{\\rm turb}(\\meanB)=p_{\\rm turb}(0)-\\qp(\\meanB)\\meanB^2\/2\\mu_0,\n\\EN\nwhere $p_{\\rm turb}(0)$ is the turbulent pressure at zero mean field.\nThe pressure $p_{\\rm turb}(0)$\nonly includes those contributions from $\\bb^2$ that are associated with\nsmall-scale dynamo action, but not those resulting from the mean field.\nThe relevant magnetic pressure in the evolution equation for the mean flow,\n$\\meanUU$, is then not just $\\meanB^2\/2\\mu_0$, but it is\naffected by the $\\meanB$ dependence of $p_{\\rm turb}$, i.e.,\nit depends on\n\\EQ\np_{\\rm turb}(\\meanB)+\\meanB^2\/2\\mu_0=\np_{\\rm turb}(0)+[1-\\qp(\\meanB)]\\meanB^2\/2\\mu_0,\n\\EN\nwhich is also still positive, but $1-\\qp(\\meanB)$\nmay well become negative, which is what we call a\n{\\em negative effective magnetic pressure}.\nConsequently, the expression\n\\EQ\np_{\\rm eff}=(1-\\qp)\\meanBB^2\/2\\mu_0\n\\EN\nis referred to as the effective magnetic pressure.\nIn addition, there is also the gas pressure $p_{\\rm gas}$.\nOnce the effective magnetic pressure drives a mean flow, the\ngas density changes, and as a consequence the gas pressure,\nso as to re-establish approximate total pressure balance.\nTherefore, $p_{\\rm gas}$ and $\\rho$ will also depend on $\\meanB$.\n\nIn the presence of gravity, the properties of magnetic buoyancy\nare drastically altered by a negative effective magnetic pressure.\nIn the following we illustrate how this can lead to an instability.\nSince the flow velocities are highly subsonic, we can make\nthe anelastic approximation for low Mach numbers,\ni.e., $\\nab\\cdot\\meanrho\\meanUU=0$.\nThis leads to $\\nab\\cdot\\meanUU+\\meanUU\\cdot\\nab\\ln\\meanrho=0$, or\n\\EQ\n\\nab\\cdot\\meanUU={\\meanU_z\\over H_\\rho},\n\\label{divu}\n\\EN\nwhere we have used the density scale height $H_\\rho$,\nso that $\\nab\\ln\\meanrho=(0,0,-1\/H_\\rho)$.\nThis equation shows that a downward motion $\\meanU_z<0$ leads to\na compression, $\\nab\\cdot\\meanUU<0$.\nThis enhances an applied field locally.\nWe consider an applied equilibrium magnetic field of the form $(0,B_0,0)$\nand the mean field has only a $y$ component,\ni.e., $\\meanBB=(0,\\meanB_y(x,z),0)$, so we have\n\\EQ\n{\\DD\\meanB_y\\over\\DD t}=-\\meanB_y\\nab\\cdot\\meanUU,\n\\label{DByDt}\n\\EN\nwhere $\\DD\/\\DD t=\\partial\/\\partial+\\meanUU\\cdot\\nab$\nis the advective derivative.\nNote that for a magnetic field with only a $y$ component, but\n$\\partial\/\\partial y=0$, there is no stretching term, so\nthere is no term of the form $\\meanBB\\cdot\\nab\\meanUU$.\nUsing \\Eq{divu}, and linearizing \\Eq{DByDt} around $\\meanUU=\\bm{0}$\nand $\\meanBB=\\BB_0$, we have\n\\EQ\n{\\partial\\meanB_{1y}\\over\\partial t}=-B_0{\\meanU_{1z}\\over H_\\rho},\n\\label{induct-eq}\n\\EN\nwhere subscripts 1 denote linearized quantities.\nThe vertical velocity perturbation $\\meanU_{1z}$ is caused by\nmagnetic buoyancy.\nAssuming total pressure equilibrium, $p_{\\rm gas}+p_{\\rm eff}=\\const$,\nwe see that an increase in the effective magnetic pressure causes\na {\\it decrease} is the gas pressure, i.e.,\n$\\delta p_{\\rm gas}=-\\delta p_{\\rm eff}$, just like in the regular\nmagnetic buoyancy instability.\nTherefore, the Archimedian buoyancy force is\n\\EQ\n-{\\delta\\rho\\over\\rho}g=-{\\delta p_{\\rm gas}\\over p_{\\rm gas}}g\n={\\delta p_{\\rm eff}\\over\\rho\\cs^2}g\n={\\dd p_{\\rm eff}\\over\\dd\\meanB^2}{\\delta\\meanB^2\\over\\rho\\cs^2}g,\n\\EN\nwhere we have used $p_{\\rm gas}=\\rho\\cs^2$ for an isothermal gas.\nIn the regular magnetic buoyancy instability, without turbulence effects,\nwe have $2\\mu_0\\dd p_{\\rm eff}\/\\dd{\\meanB}{}^2=1$.\nIn the domain where the negative effective magnetic pressure effect causes\n$\\dd p_{\\rm eff}\/\\dd{\\meanB}{}^2$\nto be negative, a magnetic field enhancement leads to\na further reduction of the local pressure, which is compensated by horizontal\ninflows, increasing density (and field strength),\nmaking this fluid parcel heavier, causing it to sink.\nInversely, a local field reduction causes outflows and rises until\nit reaches the region where this feedback reverses.\nThus, the instability loop is closed by considering the momentum equation\nin its linearized form\n\\EQ\n{\\partial\\meanU_{1z}\\over\\partial t}={\\dd p_{\\rm eff}\\over\\dd\\meanB^2}\\,\n{2B_0\\meanB_{1y}\\over \\rho\\cs^2} \\,g.\n\\EN\nUsing $\\cs^2\/g=H_\\rho$ for an isothermal atmosphere, we then find\nthe dispersion relation for the growth rate $\\lambda$\nof the resulting instability\n\\EQ\n\\lambda={\\vA\\over H_\\rho}\\sqrt{-2 \\mu_0\\, p_{\\rm eff}'}-\\etat k^2,\n\\label{disper}\n\\EN\nwhere $\\vA=B_0\/\\sqrt{\\mu_0\\rho}$ is the Alfv\\'en speed and\n\\EQ\n2\\mu_0 \\, p_{\\rm eff}'=2\\mu_0 \\, \\dd p_{\\rm eff}\/\\dd{\\meanB}^2=1-\\qp-\\dd\\qp\/\\dd\\ln\\meanB^2\n\\EN\nis twice the derivative of the effective magnetic pressure.\nWe have here also included the effects of turbulent magnetic diffusivity\n$\\etat$ and turbulent magnetic viscosity $\\nut$, assuming $\\nut\/\\etat=1$.\nHere, $k$ is the effective wavenumber.\nA proper derivation of the growth rate of the instability,\nbut again without including turbulent magnetic diffusivity\nand turbulent magnetic viscosity, is given in \\App{LinTheo}.\n\nThe negative contribution of turbulence to the mean magnetic pressure\nand the resulting large-scale instability has been predicted long\nago \\citep{KRR90,KMR96,KR94}. However, this instability\nhas been detected in DNS only recently \\citep{BKKMR,KBKMR12}.\nThis large-scale instability is called the negative effective magnetic\npressure instability, or NEMPI, for short.\n\n\\Eq{disper} demonstrates that stronger stratification and thus a\nsmaller scale height leads to an increased growth rate\nof the instability.\nThis was qualitatively confirmed by \\cite{KBKR12}.\nUsing numerical solutions of the full mean-field equations,\nthey found furthermore that the\nmaximum growth rate of the instability\nis actually independent of $\\vA$.\nThis seems to be at odds with the \\Eq{disper}.\nTo understand this, we use the following fit formula for $\\qp$:\n\\EQ\n\\qp(\\beta)={\\betastar^2\\over\\betap^2+\\beta^2},\n\\EN\nwhere $\\beta=\\meanB\/\\Beq$ and $\\Beq=\\sqrt{\\mu_0\\rho}\\urms$\nis the equipartition field strength.\nThus, for $\\betastar\\gg\\beta\\gg\\betap$, we have\n\\EQ\n\\lambda\\approx\\betastar{\\urms\\over H_\\rho}-\\etat k^2,\n\\label{E1}\n\\EN\nso the growth rate is indeed independent of the imposed field strength.\n\nIn a mean-field model, $\\urms$ is normally expressed in terms of\n$\\etat=\\urms\/3\\kf$, so \\Eq{E1} turns into\n\\EQ\n{\\lambda\\over\\etat k^2}\\approx3\\betastar{\\kf\/k\\over k H_\\rho}-1,\n\\EN\nwhich illustrates immediately the importance of large enough\nscale separation, i.e., large enough values of $\\kf\/k$.\n\nThe purpose of this paper is to show that\nNEMPI can work over a range of different field strength.\nSuch a result was recently predicted using\nthe mean-field simulations (MFS) by \\cite{KBKR12}.\nWe shall also investigate the close connection between mean\nfield and the resulting effective magnetic pressure in the plane\nperpendicular to the mean field.\nHere we focus on a series of simulations with different field strengths,\nbut for a fixed value of the magnetic Reynolds number and fixed value\nof the scale separation ratio.\nFor a study of the dependence on magnetic Reynolds number and\non scale separation ratio, but fixed field strength, we refer\nto the recent work of \\cite{KBKMR12}.\nIn the following, we discuss first direct numerical simulations (DNS)\nof NEMPI and turn then to proper mean-field simulations (MFS).\nWe begin with a simplistic illustration of the nature of NEMPI.\n\n\\section{Vertical profile of effective magnetic pressure}\n\nThe first successful DNS of NEMPI have been possible under the\nassumption of an isothermally stratified layer with an isothermal\nequation of state \\citep{BKKMR}.\nMuch of the same physics is also possible in adiabatically stratified\nlayers, but NEMPI was found in this case only\nin mean-field models \\citep{BKR10,KBKMR11}.\nThe isothermal case has conceptual advantages that help us understanding\nbetter the underlying physics of this instability.\nWe make use of this advantage in the present paper, too.\n\nIn most of the isothermal setups studied so far, the rms velocity\nis only weakly dependent on\nheight, so the $z$ variation of $\\Beq$ was only caused by that of\n$\\rho=\\rho_0\\exp(-z\/H_\\rho)$.\nThis allows us then to plot the effective magnetic pressure.\nIn the following, we introduce the quantity\n\\EQ\n\\Peff(\\beta)=\\half\\left[1-\\qp(\\beta)\\right]\\beta^2,\n\\EN\nwhich is the effective magnetic pressure normalized by the\nlocal equipartition field strength $\\Beq$,\ni.e., $\\Peff=p_{\\rm eff}\/\\Beq^2$.\nSince $\\beta=\\beta(z)=\\beta_0\\exp(z\/2H_\\rho)$ increases with $z$,\n$\\Peff(z)$ is small at large depths, reaches a negative minimum\nat some depth, and then becomes positive and equal to $\\beta^2$.\nIn \\Fig{MF_init} we show vertical profiles of $\\Peff$,\n$\\dd\\Peff\/\\dd\\beta^2$, and $\\beta(-2\\dd\\Peff\/\\dd\\beta^2)^{1\/2}$\nfor the fit parameters $\\qpz=20$ and $\\betap=0.167$ derived later\nin this paper,\nand the three field strengths $\\beta_0\\equiv B_0\/\\Beqz=0.05$, 0.1, and 0.2\nwithin the $z$ range from $-\\pi$ to $\\pi$, which is also consistent\nwith the DNS and some of the MFS discussed below.\nHere, $\\Beqz=\\Beq(0)$.\n\n\\begin{figure}[t!]\\begin{center}\n\\includegraphics[width=\\columnwidth]{MF_init}\n\\end{center}\\caption[]{\nProfiles of $\\Peff$, $\\dd\\Peff\/\\dd\\beta^2$, and\n$\\beta(-2\\dd\\Peff\/\\dd\\beta^2)^{1\/2}$\nfor fit parameters $\\qpz=20$ and $\\betap=0.167$, and the three\nfield strengths $B_0\/\\Beqz=0.05$, 0.1, and 0.2 within the $z$ range\nfrom $-\\pi$ to $\\pi$, which is consistent with some of the models\ndiscussed below.\nThe vertical lines of similar line types give the location\nwhere the unstable eigenmodes reaches its peak.\n}\\label{MF_init}\\end{figure}\n\nNotice first of all that all three curves of $\\Peff$ have minima with\nleft flanks (negative slopes) within the domain.\nAs the imposed field is increased, these curves shift downward\n(smaller values of $z$).\nThus, we should expect the peak of the instable eigenmode\nto appear somewhere along the left flanks of these curves and\nthat these peaks move further down as the imposed field\nis increased.\nThis is qualitatively reproduced by the DNS and MFS discussed\nbelow, except that the location is consistently a certain\ndistance below the position where the left flanks have their\nsteepest gradient.\nOn the other hand, as is evident from the middle panel of\n\\Fig{MF_init}, the largest value of $\\dd\\Peff\/\\dd\\beta^2$\nis always achieved at the bottom of the domain.\nHowever, the growth rate of NEMPI\nhas still a factor proportional to\n$\\vA=\\urms\\beta$ in front of it; see \\Eq{disper}.\nThis then confines the instability to a narrow strip within the domain.\nIn the third panel of \\Fig{MF_init} we plot therefore also\n$\\beta(-2\\dd\\Peff\/\\dd\\beta^2)^{1\/2}$, and their extrema are\nnow only slightly above the location where DNS and MFS\nshow a peak in the eigenfunction.\nThe reason for the remaining discrepancy is not well understood at present.\n\n\\section{Onset and saturation of NEMPI in DNS}\n\\label{DNS}\n\n\\subsection{Isothermal setup in DNS}\n\\label{model}\n\nFollowing the earlier work of \\cite{BKKMR} and \\cite{KBKMR12},\nwe solve the equations for the velocity $\\UU$,\nthe magnetic vector potential $\\AAA$, and the density $\\rho$,\n\\begin{equation}\n\\rho{\\DD\\UU\\over\\DD t}=-\\cs^2\\nab\\rho+\\JJ\\times\\BB+\\rho(\\ff+\\grav)\n+\\nab\\cdot(2\\nu\\rho\\SSSS),\n\\end{equation}\n\\begin{equation}\n{\\partial\\AAA\\over\\partial t}=\\UU\\times\\BB+\\eta\\nabla^2\\AAA,\n\\end{equation}\n\\begin{equation}\n{\\partial\\rho\\over\\partial t}=-\\nab\\cdot\\rho\\UU,\n\\end{equation}\nwhere $\\nu$ is the kinematic viscosity, $\\eta$ is the magnetic diffusivity\ndue to Spitzer conductivity of the plasma,\n$\\BB=\\BB_0+\\nab\\times\\AAA$ is the magnetic field,\n$\\BB_0=(0,B_0,0)$ is the imposed uniform field,\n$\\JJ=\\nab\\times\\BB\/\\mu_0$ is the current density,\n$\\mu_0$ is the vacuum permeability,\n${\\sf S}_{ij}=\\half(U_{i,j}+U_{j,i})-\\onethird\\delta_{ij}\\nab\\cdot\\UU$\nis the traceless rate of strain tensor, and commas denote\npartial differentiation.\nThe forcing function $\\ff$ consists of random, white-in-time,\nplane non-polarized waves with a certain average wavenumber.\nThe turbulent rms velocity is approximately\nindependent of $z$ with $\\urms=\\bra{\\uu^2}^{1\/2}\\approx0.1\\,\\cs$.\nThe gravitational acceleration $\\grav=(0,0,-g)$ is chosen such that\n$k_1 H_\\rho=1$, so the density contrast between\nbottom and top is $\\exp(2\\pi)\\approx535$.\nHere, $H_\\rho=\\cs^2\/g$ is the density scale height.\nWe consider a domain of size $L_x\\times L_y\\times L_z$ in\nCartesian coordinates $(x,y,z)$, with periodic boundary conditions in\nthe $x$ and $y$ directions and stress-free perfectly conducting\nboundaries at top and bottom ($z=\\pm L_z\/2$).\nIn all cases, we use a scale separation ratio $\\kf\/k_1$ of 30,\na fluid Reynolds number $\\Rey\\equiv\\urms\/\\nu\\kf$ of 18,\nand a magnetic Prandtl number $\\Pm=\\nu\/\\eta $ of 0.5.\nIn our units, $\\mu_0=1$ and $\\cs=1$.\nThe value of $B_0$ is specified in units of the volume\naveraged value, $\\Beqz=\\sqrt{\\mu_0\\rho_0} \\, \\urms$,\nwhere $\\rho_0=\\bra{\\rho}$ is the volume-averaged density,\nwhich is constant in time.\nIn addition to visualizations of the actual magnetic field,\nwe also monitor $\\meanB_y$, which is an\naverage over $y$ and a certain time interval $\\Delta t$.\nSince the simulations are periodic in the $x$ and $y$ directions,\nwe sometimes shift the images such that the peak field strength\nof NEMPI appears in the middle of the frame.\n\nThe simulations are performed with the {\\sc Pencil Code},%\n\\footnote{{\\tt http:\/\/pencil-code.googlecode.com}}\nwhich uses sixth-order explicit finite differences in space and a\nthird-order accurate time stepping method.\nWe use a numerical resolution of $256^3$ mesh points.\n\n\\subsection{Results}\n\\label{DNSResults}\n\n\\begin{figure}[t!]\\begin{center}\n\\includegraphics[width=\\columnwidth]{DNS_B}\n\\end{center}\\caption[]{\n$\\meanB_y$ from  DNS for three values of the imposed field strength at\nthe end of the linear growth phase of NEMPI for $\\Rm=18$ and $\\Pm=0.5$.\n}\\label{DNS_B}\\end{figure}\n\nIn \\Fig{DNS_B} we demonstrate that NEMPI can work over\na range of field strengths.\nAs we increase the strength of the imposed field, NEMPI develops\nat progressively greater depth.\nThis result was recently obtained for MFS, but is now for the first\ntime demonstrated in DNS.\n\\FFig{DNS_grr} shows that the growth of the large-scale field $\\meanB_1$\nof the magnetic structure is similar for three different field strengths.\nHere, $\\meanB_1$ has been determined by taking the maximum value of\nthe mean field in the neighborhood of the position where the flux\nconcentration later develops.\nNote that there is a range over which $\\meanB_1$ grows approximately\nexponentially, independent of the value of $B_0$.\n\n\\begin{figure}[t!]\\begin{center}\n\\includegraphics[width=\\columnwidth]{DNS_grr}\n\\end{center}\\caption[]{\nGrowth of the large-scale field strength $\\meanB_1$ at the center of\nthe magnetic structure for three field strengths.\n}\\label{DNS_grr}\\end{figure}\n\n\\begin{figure}[t!]\\begin{center}\n\\includegraphics[width=.8\\columnwidth]{DNS_P}\n\\end{center}\\caption[]{\n$\\meanB_y$ and $\\Peff$ from DNS at three times showing the descent\nof the potato sack feature for $\\Rm=18$ and $\\Pm=0.5$.\n}\\label{DNS_P}\\end{figure}\n\nIn \\Fig{DNS_P} we show $\\meanB_y$ at early, intermediate, and late stages\nof the saturation process (left), and compare with visualizations of\n$\\Peff$ at the same times.\nHere, $\\Peff=\\half(1-\\qp)\\beta^2$, where $\\qp(\\beta)$\nwith $\\beta=\\meanB\/\\Beq$ is evaluated from\n\\EQ\n\\qp=-2\\Delta\\overline\\Pi_{xx}^{\\rm f}\/\\meanBB^2,\n\\EN\nfor $\\BB_0=(0,B_0,0)$, and\n\\EQ\n\\Delta\\overline\\Pi_{ii}^{\\rm f}\n=\\meanrho\\,(\\overline{u_i^2}-\\overline{u_{0i}^2})\n+\\half(\\overline{\\bb^2}-\\overline{\\bb_0^2})\n-(\\overline{b_i^2}-\\overline{b_{0i}^2}),\n\\label{Pi_ii}\n\\EN\nis applied to the $xx$ component of the total stress from the fluctuating\nvelocity and magnetic fields.\nIn \\Eq{Pi_ii} no summation over the repeated index $i$ is assumed.\n\nIn \\Fig{DNS_P}, blue shades correspond to low values of $\\Peff$ and occur\naround the minimum line (marked in white) where $\\Peff=\\Pmin$.\nAs time progresses, low values of $\\Peff$ are also found at\ngreater depth as the magnetic flux concentration descends.\nThe fact that there is a clear spatial correlation between $\\meanB_y$ and\n$\\Peff$ provides strong evidence that the interpretation of the formation\nof structures in the stratified turbulence simulations in terms of\nNEMPI is indeed the correct one.\n\nThe descending structures have previously been referred to as potato\nsack structures \\citep{BKKMR}, because of their widening cross-section\nwith greater depth.\nWhen such structures were first seen in MFS \\citep{BKR10}, they were\noriginally thought to be artifacts of the model that one would not\nexpect to see in the Sun.\nHowever, such structures were later also found in DNS \\citep{BKKMR},\nhighlighting therefore the strong predictive power of MFS.\n\n\\begin{figure}[t!]\\begin{center}\n\\includegraphics[width=\\columnwidth]{DNS_U}\n\\end{center}\\caption[]{\nVectors of $\\meanUU$ together with a color\/grey scale representation of\n$\\meanB_y$ from DNS at a late time for $\\Rm=18$ and $\\Pm=0.5$.\n}\\label{DNS_U}\\end{figure}\n\nVisualizations of the resulting mean flow $\\meanUU$ are shown in \\Fig{DNS_U}\nas vectors.\nThe flow shows a convergent shape toward the magnetic structures.\nIt is interesting to note that such convergent flow structures are now also\nseen in local helioseismic flow measurements around active regions\n\\citep{Hindman}.\nIn this connection it is instructive to discuss the somewhat peculiar shape\nof such a structure that widens as it descends.\nNormally, in a strongly stratified atmosphere, descending structures\nget compressed and become narrower, but this is not seen in the present\nvisualizations.\nAs already argued in \\cite{BKKR12}, this is because the boundaries of these\nstructures do not coincide with material lines, so the mass is not\nconserved inside them and can leak through the boundaries.\nIndeed, these structures grow as they descend, and may become amenable\nto helioseismic detection; cf.\\ \\cite{Ilonidis}.\nThis phenomenon is well known in the description of turbulent plumes\nas a model of turbulent downdrafts in convection \\citep{RZ95}.\nSuch structures are known to widen as a result of entrainment.\nThe sinking behavior of these apparently disconnected flow structures\ncan be explained as follows:\nwhile inflows dominate downflows throughout the whole lifespan of the field concentration,\nin the initial stage the former can drag in a large fraction of the surrounding magnetic field,\novercompensating the losses by downflows.\nHowever as the environment gets depleted,\nthis dynamical balance shifts and the structures start moving downwards.\n\n\\subsection{Mean-field coefficients from DNS}\n\nIn earlier work by \\cite{KBKR12}, the parameters $\\qpz=40$ and\n$\\betap=0.05$, corresponding to $\\betastar=0.32$, were used.\nThose values are compatible with work by \\cite{BKKR12} and \\cite{KBKMR12}.\nHowever, in the present case we have a larger scale separation ratio,\n$\\kf\/k_1=30$, for which these parameters have not yet been determined.\nIn the \\Fig{DNS_qpc} we show the functional form of $\\Peff(\\beta)$\nfor the present case with $\\kf\/k_1=30$, $\\Rm=18$, and $\\Pm=0.5$.\nHere we have followed the method described in \\cite{BKKR12};\nsee their Eq.~(17).\nFor the present model we find as fit parameters $\\qpz=20$ and $\\betap=0.167$,\nwhich corresponds to $\\betastar=0.75$.\n\n\\begin{figure}[t!]\\begin{center}\n\\includegraphics[width=\\columnwidth]{DNS_qpc}\n\\end{center}\\caption[]{\n$\\Peff(\\beta)$ for the DNS used in this paper with $\\kf\/k_1=30$,\n$\\Rm=18$, and $\\Pm=0.5$.\n}\\label{DNS_qpc}\\end{figure}\n\n\\section{Comparison with MFS}\n\nRecently, many aspects of NEMPI seen in the DNS have also been detected in MFS.\nEstablishing the usefulness and limitations of MFS is important, because\nsuch models are easier to solve and allow one to explore parameters in\nregimes where DNS are harder to apply or have not yet been applied in\nthe limited time span since the close correspondence between DNS and\nMFS was first noted.\n\nIn the following we consider two-dimensional mean-field models, in which\nthe presence of $\\qs$ has no effect on the solutions \\citep{KBKR12}.\nFurthermore, we ignore other effects connected with the anisotropy\nof the turbulence.\nThese effects have previously been found to be weak \\citep{BKKR12,KBKMR11}.\n\n\\subsection{Isothermal setup in MFS}\n\\label{modelMFS}\n\nIn this section we solve the evolution equations for mean velocity $\\meanUU$,\nmean density $\\meanrho$, and mean vector potential $\\meanAA$, in the form\n\\EQ\n{\\partial\\meanUU\\over\\partial t}=-\\meanUU\\cdot\\nab\\meanUU\n-\\cs^2\\nab\\ln\\meanrho+\\grav+\\meanFFFF_{\\rm M}+\\meanFFFF_{\\rm K},\n\\EN\n\\EQ\n{\\partial\\meanrho\\over\\partial t}=-\\meanUU\\cdot\\nab\\meanrho\n-\\meanrho\\nab\\cdot\\meanUU,\n\\EN\n\\EQ\n{\\partial\\meanAA\\over\\partial t}=\\meanUU\\times\\meanBB-(\\etat+\\eta)\\meanJJ,\n\\EN\nwhere $\\meanFFFF_{\\rm M}$ is given by\n\\EQ\n\\meanrho \\, \\meanFFFF_{\\rm M} = -\\half\\nab[(1-q_{\\rm p})\\meanBB^2]\n+ \\meanBB \\cdot \\nab\\left[(1-q_{\\rm s})\\meanBB\\right]\\!,\n\\label{efforce}\n\\EN\nand\n\\EQ\n\\meanFFFF_{\\rm K}=(\\nut+\\nu)\\left(\\nabla^2\\meanUU+\\onethird\\nab\\nab\\cdot\\meanUU\n+2\\meanSSSS\\nab\\ln\\meanrho\\right)\n\\EN\nis the total (turbulent plus microscopic) viscous force.\nHere, $\\meanSSS_{ij}=\\half(\\meanU_{i,j}+\\meanU_{j,i})\n-\\onethird\\delta_{ij}\\nab\\cdot\\meanUU$\nis the traceless rate of strain tensor of the mean flow.\n\nWe approximate $\\qp$ by a simple profile that is only a function\nof the ratio $\\beta\\equiv|\\meanBB|\/\\Beq$.\nWe use an algebraic fit of the form\n\\begin{equation}\n\\qp(\\beta)={\\qpz\\over1+\\beta^2\/\\betap^2}.\n\\label{qpbeta2}\n\\end{equation}\nThe function $\\qp$ quantifies the impact of the mean magnetic field\non the effective pressure force.\n\n\\subsection{Aspects of the MFS}\n\\label{MFSResults}\n\nWe begin by showing $\\meanB_y$ for three values of the imposed field strength\nat the end of the linear growth phase of NEMPI.\nThe results are shown in \\Figs{MF_B_stanb}{MF_B} for two different setups.\nIn the former we use $\\qpz=20$ and $\\betap=0.167$ for the same $z$ range\n($-\\pi\\leq z\/H_\\rho\\leq\\pi$) as in the DNS, while in the latter we use\n$\\qpz=40$ and $\\betap=0.05$ for somewhat stronger fields and a deeper\n$z$ range ($0\\leq z\/H_\\rho\\leq2\\pi$), which is also the fiducial model used\nby \\cite{KBKR12}.\nIn the former case the growth rate is $\\approx11 H_\\rho^2\/\\eta_t$\nwhile in the latter it is $\\approx5.0 H_\\rho^2\/\\eta_t$.\n\n\\begin{figure}[t!]\\begin{center}\n\\includegraphics[width=\\columnwidth]{MF_B_stanb}\n\\end{center}\\caption[]{\n$\\meanB_y$ from mean-field models for three values of the imposed field strength\nat the end of the linear growth phase of NEMPI.\nHere, $\\qpz=20$ and $\\betap=0.167$, which corresponds to $\\betastar=0.75$.\n}\\label{MF_B_stanb}\\end{figure}\n\nUnlike the DNS, the MFS show that in the former series of models with\n$\\qpz=20$ and $\\betap=0.167$ the $x$ extend is slightly larger than the\noptimal horizontal wavelength of the instability, because one sees\nthat some of the structures begin to split into two (\\Fig{MF_B_stanb}).\nThis is not the case for the second model with $\\qpz=40$ and $\\betap=0.05$\n(\\Fig{MF_B}).\n\n\\begin{figure}[t!]\\begin{center}\n\\includegraphics[width=\\columnwidth]{MF_B}\n\\end{center}\\caption[]{\nSimilar to \\Fig{MF_B_stanb}, but for larger field strengths and vertical\ndomain boundaries that a deeper down, so the magnetic field maxima of the\ninstability fit better into the domain.\nHere, $\\qpz=40$ and $\\betap=0.05$, corresponding to $\\betastar=0.32$.\n}\\label{MF_B}\\end{figure}\n\nNext, we compare $\\meanB_y$ with $\\Peff=\\half(1-\\qp)\\beta^2$.\nAgain, there is a close correspondence between the $\\meanB_y$ field\nand the resulting distribution of $\\Peff$; see \\Fig{MF_P}.\nHere, $\\qp(\\beta)$ is evaluated from the assumed fit formula given\nby \\Eq{qpbeta2}.\nFurthermore, there is a close correspondence between regions of\nenhanced magnetic field and enhanced density; see \\Fig{MF_R}.\n\n\\begin{figure}[t!]\\begin{center}\n\\includegraphics[width=.8\\columnwidth]{MF_P}\n\\end{center}\\caption[]{\n$\\meanB_y$ and $\\Peff$ from mean-field models at three times.\n}\\label{MF_P}\\end{figure}\n\n\\begin{figure}[t!]\\begin{center}\n\\includegraphics[width=.8\\columnwidth]{MF_R}\n\\end{center}\\caption[]{\n$\\meanB_y$ compared with $\\meanrho$ for three different times.\n}\\label{MF_R}\\end{figure}\n\n\\section{Conclusions}\n\nThe present work has demonstrated that NEMPI is able to concentrate\nthe magnetic field into large patches encompassing the size of many\nturbulent eddies.\nThe physics of this instability is a straightforward extension\nof the usual magnetic buoyancy instabilities\n\\citep{P66,P79,HP88,Catt88,Wis00,Isobe05,Kers07}, except that the sign\nof the buoyancy force is reversed in a regime of intermediate\nfield strength.\nWe have here re-examined the simple case of an isothermal layer\nin which NEMPI can in principle occur at any depth whose value\nis determined by the strength of the imposed field.\nOur new DNS have verified that the growth rate is indeed independent\nof the strength of the imposed field, provided the peak of the instability\nfits still comfortably within the domain.\nDuring the subsequent nonlinear evolution of the instability, the\noverall density stratification readjusts, allowing the magnetic field\nconcentrations to move further down.\nIt is important to realize that the resulting structures are subject\nto significant turbulent entrainment \\citep{RZ95},\nso their boundaries are not closed.\nThe agreement with corresponding mean field models is astounding\nand much more convincing than what has so been possible to demonstrate\nin mean-field dynamo theory.\nMean-field models provide therefore a strong source of guidance\nwhen designing new setups for DNS.\n\nWhile NEMPI now begins to be fairly well understood for isothermal models,\nmore work is required for non-isothermal ones.\nIn that case, the density scale height is no longer constant and the\ndegree of stratification is much stronger at the top than in deeper layers.\nThe simple result that the instability can occur at any height, depending\njust on the strength of the imposed field, is then no longer valid.\nAt the same time, there is another perhaps more important aspect,\nnamely the possibility of other instabilities.\nOne of them is connected with the suppression of turbulent convective\nenergy flux by the mean magnetic field.\nAs shown by \\cite{KM00}, this effect can also lead to magnetic flux\nconcentrations and it may be sustained for much stronger magnetic\nfield strengths, allowing thus the formation of structures in which the\nmagnetic pressure becomes comparable to the ambient gas pressure.\nThere may also be a connection with flux segregation events seen in\nsimulations of magnetoconvection at large aspect ratios \\citep{Tao,KKWM10},\nwhich have already been shown to produce bipolar regions in simulations\nwith radiation transfer \\citep{Stein}.\nThe study of the possibility of producing sunspots similar to those\nof \\cite{Rem11a,Rem11b}, but without initial flux structures,\nis now of high priority in the quest for\nsolving the solar dynamo problem in terms of distributed dynamo models\nin which magnetic activity is explained as a surface phenomenon.\n\n\\begin{acks}\nComputing resources provided by the Swedish National Allocations Committee\nat the Center for Parallel Computers at the Royal Institute of Technology in\nStockholm and the High Performance Computing Center North in Ume{\\aa}.\nThis work was supported in part by the European Research Council\nunder the AstroDyn Research Project No.\\ 227952\nand the Swedish Research Council under the project grant 621-2011-5076.\n\\end{acks}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\IEEEPARstart{W}{ith} the development of computer vision techniques,\nmany learning, detection, matching, and tracking algorithms that are based on small features of images have appeared recently. However, many of these algorithms are quite sensitive to weather conditions under which the image is taken. In this work, we consider the image recovering with good visual quality from a single color image that is spotted by rain during the capturing.\n\nWeather conditions can be classified into steady and dynamic according to the constituent particles \\cite{Garg_2004_CVPR}. The former one contains small particles (e.g., fog) and the later one includes large particles (e.g., rain and snow). In the steady condition, small particles cannot be captured by cameras; while in the dynamic condition, droplets of rain and snow can be clearly filmed. He \\emph{et al.} proposed a de-haze approach that is based on dark channel priors and has achieved excellent results on various challenging examples \\cite{He_2011_PAMI}. Another fast image dehazing work which based on linear transformation are proposed by Wang \\emph{et al.} \\cite{Wang_2017_TMM}. However, in the case of dynamic weather, the existing rain removal methods still need to be improved. The difficulty lies in two aspects: rain droplets appear in an image randomly and large droplets interfere original image contents.\n\nThe earliest work on rain dates back to the study of statistical characteristics of rain in the atmospheric science in 1948 \\cite{Marshall_1948_JM}. According to these characteristics,\nrain appears in a picture looks quite random and is of different shapes,  which makes it difficult to detect and remove rain streaks from a single image. Therefore, most works pay attentions to rain removal in videos\n\\cite{Garg_2004_CVPR,Zhang_2006_ICME,Brewer_2008_CS,Bossu_2011_CV,Barnum_2007_PACV,Barnum_2010_CV}, where the rain detection is relatively easier. For example, Barnum \\emph{et al.} detected and removed rain streaks for videos in the frequency domain \\cite{Barnum_2007_PACV,Barnum_2010_CV}. To the best of our knowledge, dealing with rain removing in a single image started in 2009 when Roser \\emph{et al.} detected rain streaks in single image \\cite{Roser_2009_CV}. Later on, several other rain removal works based on a single image were proposed, e.g., \\cite{Fu_2011_ASSP,Kang_2012_TIP,Chen_2014_CSVT,Xu_2012_CIS,Kim_2013_ICIP,Ding_2015_MTA,Huang_2014_TMM}.\n\nThis paper aims at removing rain from a single image. Although the detection and removal of rain in a single image is more challenging as compared with videos, there are still some observations that can be utilized for the rain identification. On one hand, rain streaks are more reflective than other parts of images, leading to higher pixel intensities compared with non-rain pixels. On the other hand, rain streaks usually do not occlude objects completely due to their semi-transparency property. The former one can be utilized for the rain detection while the latter one facilitates the reconstruction in the gradient domain after computing image gradients on all non-rain locations.\n\nThere are two main challenges: accurate rain streaks identification and high quality rain-removed image recovery. The detection of rain streaks will not be accurate if only pixel intensities are involved, because other objects with similar or even higher pixel intensities would be mis-classified as rain pixels. After the rain pixels are identified, the final result needs to be reconstructed without introducing noticeable artifacts (e.g., blurring of image contents). Methods such as the weighted mean of neighbouring non-rain pixels and image inpanting~\\cite{Bertalmio_2000_CGIT} are all possible candidates that have been considered previously.\n\nIn this paper, in order to deal with the first challenge, we over-detect rain pixels in single rain image by the\nmethod in \\cite{Wang_2016_ICIP} firstly. To improve the accuracy, we further employ a morphological processing technique\\cite{Gonzalez_2002_PUSR} to refine all detected rain pixels. For the second challenge, we decompose the input image into a rain-layer and a non-rain layer in the gradient domain after the rain pixels are identified. We reconstruct the final result by using the non-rain layer under the image quasi-sparsity priors.\n\nThe contributions of our work are the following aspects. (1) We propose a quite unique detection method of rain streaks. (2) We simplify the sparsity \\cite{Levin_2007_PAMI} to quasi-sparsity and combine it with the detection of rain to complete rain-removal tasks. After simplifying the sparsity to quasi-sparsity, the loss function is derived into a $L_{1}$-norm minimization problem.  (3) An additional constraint is added to solve the color shift problem that often appears in \\cite{Levin_2007_PAMI} successfully. An example of our rain-removed results is shown in Fig. \\ref{fig:rain_derain}.\n\nThe rest of the paper is organized as follows. We briefly review some related works in Section \\ref{sec:RelatedWorks}. We propose the rain streaks detection algorithm in Section \\ref{sec:RainStreaksDetection}. The reconstruction of rain-removed images is presented in Section \\ref{sec:ImageReconstruction}. The results and comparisons are presented and discussed in Section \\ref{sec:ExperimentalResults}. Finally, some conclusions are drawn in Section \\ref{sec:Conclusion}.\n\n\n\n\\section{Related Works}\n\\label{sec:RelatedWorks}\n\nRain removal can be performed in the spatial domain or the frequency domain, and some are focused on the single-image scenario. A brief review of the existing algorithms is presented in the following.\n\n\\textbf{Rain removal from videos in the spatial domain:} Garg and Nayar analyzed the visual effect of rain streaks comprehensively \\cite{Garg_2004_CVPR} by developing a correlation model to describe rain's dynamics and a motion blur model to explain the photometry of rain. Through these two models, rain streaks can be detected efficiently and then removed in videos. To make the study more complete, Garg and Nayar further built a rain appearance model based on a rain oscillation model that was developed in the atmospheric science in \\cite{Garg_2006_TG}. They also developed an image-based rain-rendering algorithm by creating a database to describe different kinds of rain appearances under various lighting and viewing directions. In \\cite{Garg_2007_CV}, Garg and Nayar analyzed various factors that influence the visual effect of rain. Based on these analyses, an efficient algorithm was developed to control rain. Besides, by modeling the distortion of raindrop, they accomplished photorealistic rain-rendering.\n\nAnother rain removal algorithm that is based on both temporal and chromatic characteristics of rain streaks in video was proposed by Zhang \\emph{et al.} \\cite{Zhang_2006_ICME}. This work shows that a certain area is not always infected by rain streaks. On the other hand, when indeed affected by rain, the intensity changes of chromatic components (namely, R, G, B) of a pixel approximately equal to each other. These two properties have been utilized to detect and then remove rain streaks in videos. However, constrained by temporal properties, this method can only deal with the videos that are obtained by using a stationary camera.\n\nIn \\cite{Brewer_2008_CS}, Brewer and Liu suggested that (1) a region with instantaneous intensity spike be probably affected by rain streaks and (2) streak-like objects in a region with a nearly consistent range of aspect ratios be considered as rain streaks. Once detected, rain streaks can be removed by calculating the mean value of two neighbouring frames. A rain streaks detection method that uses a histogram of orientation of streaks (HOS) was introduced by Bossu \\emph{et al.} in \\cite{Bossu_2011_CV}. This method proposes to decompose an image sequence into foreground and background, while potential rain streaks are detected in foreground. Then, HOS is calculated, which follows a model of\nGaussian-uniform mixture. Finally, the Gaussian distribution whose amplitude stands for rain presence and the uniform distribution standing for noise are separated by an algorithm of expectation maximization.\n\n\\textbf{Rain removal from videos in the frequency domain:}\nIn \\cite{Barnum_2007_PACV}, Barnum \\emph{et al.} combined\na physical model of rain streaks (for determining the general shape and brightness of rain) and some statistical properties of rain streaks to show the influence of rain on image sequences in the frequency domain.\nOnce detected, the spectrum of rain streaks can be suppressed to obtain rain-removed image sequences. Later on, they combined a shape model\nwith statistical properties of rain streaks to detect and remove rain\nstreaks, also in the frequency domain, and demonstrated a better accuracy  \\cite{Barnum_2010_CV}.\n\n\\begin{figure}[t]\n\\begin{minipage}{0.48\\linewidth}\n\\centering{\\includegraphics[width=1\\linewidth]{images\/test25}}\n\\centerline{(a)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.48\\linewidth}\n\\centering{\\includegraphics[width=1\\linewidth]{images\/test25_I_nd}}\n\\centerline{(b)}\n\\end{minipage}\n\\caption{(a) Original rain image. (b) Rain-removed image by our method.}\n\\label{fig:rain_derain}\n\\end{figure}\n\n\\begin{figure*}\n\\centering\n\\begin{minipage}{1\\linewidth}\n\\centering{\\includegraphics[width=.9\\linewidth]{images\/pipeline}}\n\\centerline{}\n\\end{minipage}\n\\caption{Pipeline of our method. We first identify rain locations from the input rain image (a) to generate an initial rain map (b). Then, a morphological processing is used to refine these initial locations, producing a new non-rain map (c) and rain locations (d) (white pixels). The final result (e) and optionally a rain layer (f) can be reconstructed based on (c) and (d), respectively.}\n\\label{fig:pipeline}\n\\end{figure*}\n\n\\textbf{Single image rain removal:} Roser \\emph{et al.} detected rain streaks in a single image monocularly, based on a photometric raindrop model \\cite{Roser_2009_CV}. Meanwhile, Halimeh \\emph{et al.} detected raindrops on car windshield by utilizing a model that describes the shape of raindrop and a relationship between raindrops and the environment \\cite{Halimeh_2009_VS}.\n\nFor the first time, Fu \\emph{et al.} accomplished the rain-removal task for a single image by utilizing morphological component analysis (MCA) \\cite{Fu_2011_ASSP}. Some improved or extended versions have been proposed by Kang \\emph{et al.} \\cite{Kang_2012_TIP} ,Chen \\emph{et al.} \\cite{Chen_2014_CSVT}, Wang \\emph{et al.} \\cite{Wang_2016_ICIP} and Wang \\emph{et al.}\n\\cite{Wang_2017_TIP}. In particular, Kang \\emph{et al.} used the histogram of oriented gradients (HOG) \\cite{Dalal_2005_CVPR} to separate rain and non-rain dictionary atoms, while Chen \\emph{et al.} extended rain removal task to a single color image.\nThe denosing paper \\cite{Huang_2014_TMM} on TMM which treats rain as a kind of noise removes rain streak by a self-learning based image decomposition method.\nIn \\cite{Wang_2016_ICIP,Wang_2017_TIP}, Wang \\emph{et al.} developed a rough detection method of rain to remove bright rain streaks.\n\nMore recently, Luo \\emph{et al.} proposed that a rain image be decomposed into the rain layer and non-rain layer by a highly discriminative code on a learned dictionary that is based on a screen blend model \\cite{Luo_2015_ICCV}. On the other hand, a novel rain removal method based on the guided filter was proposed by Xu \\emph{et al.} \\cite{Xu_2012_CIS}, in which a rain-free guidance image is constructed and a guided filter \\cite{He_2013_PAMI} is used to remove rain in a single image. In \\cite{Kim_2013_ICIP}, Kim \\emph{et al.} assumed that rain streaks have an elliptical shape. Then, a kernel regression method \\cite{Takeda_2007_TIP} is used to extract elliptical components in the image to detect rain streaks. Once detected, rain streaks are removed by non-local mean filter \\cite{Buades_2005_CVPR}. In the meantime, Chen \\emph{et al.} proposed a low-rank model of rain streaks to capture the spatio-temporally correlated rain streaks \\cite{Chen_2013_ICCV}.\n\nLately, a rain removal method based on the $L_0$ gradient minimization was proposed by Ding \\emph{et al.} \\cite{Ding_2015_MTA}. By this method, majority of rain streaks can be restrained, but a lot of image details also vanish with the rain streaks removal. Another novel rain removal method was developed by Li \\emph{et al.}  \\cite{Li_2016_CVPR}, in which some patch-based priors for both the background layer and rain layer are used to accomplish the rain removal task. Because these priors are based on Gaussian mixture models and can accommodate the rain streaks with multiple orientations and scales, this method obtains the state-of-the-art effectiveness.\n\n\\textbf{Deep learning based methods:} In recent years, deep learning is utilized in many computer vision tasks, including rain\nremoval. In \\cite{Fu_2017_TIP}, Fu \\emph{et al.} designed a DerainNet to learn the mapping relationship between rain and clean images. They also proposed a deep detail network which\ncan directly reduces the mapping range to simplify the learning\nprocess, then remove rain streaks in single color images \\cite{Fu_2017_CVPR}. Yang \\emph{et al.} built a new model for rain images and designed a multi-task deep learning architecture\nto remove rain streaks in single images \\cite{Yang_2017_CVPR}.\nA DID-MDN net tried to estimate the density of rain\nfirst, then remove rain streaks \\cite{Zhang_2018_CVPR}.\n\n\\section{Rain Streaks Detection}\n\\label{sec:RainStreaksDetection}\n\nFig. \\ref{fig:pipeline} shows the pipeline of our method.\nGiven an input rain image (Fig. \\ref{fig:pipeline}(a)), we first detect the rain\nlocations according to pixel intensities. Since the initial\nlocations (Fig. \\ref{fig:pipeline}(b)) are usually inaccurate,\nthey will be refined using the proposed morphology approach, generating a\nrefined rain location map (Fig. \\ref{fig:pipeline}(d)) as well as a non-rain location map (Fig. \\ref{fig:pipeline}(c)).\nThe final result (Fig. \\ref{fig:pipeline}(e)) is reconstructed from the image gradients\non all non-rain locations.\nOptionally, we can also reconstruct an image that contains rain only (Fig. \\ref{fig:pipeline}(f)).\n\nIn this section, we utilize the rain image in Fig. \\ref{fig:pipeline}(a)\nas an example to present the details of rain streaks detection.\nFirstly, an initial rain location is obtained by the method in \\cite{Wang_2016_ICIP}.\nThen, the mis-detections are refined by a morphological processing and the principal component analysis (PCA).\n\n\\subsection{Initial detection of rain streaks}\n\nRain pixels often possess higher values than\ntheir neighbouring non-rain pixels. Therefore, Wang \\emph{et al.} \\cite{Wang_2016_ICIP}\nover-detected rain location by this characteristic.\n\nFor each pixel $I(i,j)$ in a given rain image $I$,\nWang \\emph{et al.} calculate 5 mean values $\\bar{I}^{(k)}$ $(k=1,2,3,4,5)$\nin the windows $w^{(k)}$ with pixel $I(i,j)$ located in the center,\ntop-left, top-right, bottom-left, and bottom-right of the window, respectively.\nIf the following inequalities\n\\begin{equation}\\label{eq:detect_condition}\n I(i, j)>\\bar{I}^{(k)}=\\frac{\\sum_{\\{m,n\\}\\in w^{(k)}}I(m,n)}{|w^{(k)}|}, k\\!=\\!\\{1,2,3,4,5\\},\n\\end{equation}\nwhere $|w^{(k)}|$ stands for the window size,\nare satisfied for all color channels, $I(i,j)$ is recognized as a rain pixel\nand the corresponding term $S_R(i,j)$ in the so-called binary location map $S_R$ is set to be 1;\notherwise $S_R(i,j)$ is assigned as 0.\nDetection result $S_{R}$ is a binary image as shown in Fig. \\ref{fig:initial_location}.\n\n\\begin{figure}[t]\n\\centering\n\\begin{minipage}{0.48\\linewidth}\n\\centering{\\includegraphics[width=.9\\linewidth]{images\/rain18_loca}}\n\\centerline{(a)}\n\\end{minipage}\n\\caption{Initial rain's location map.}\n\\label{fig:initial_location}\n\\end{figure}\n\n\\subsection{An analysis of mis-detections}\n\nIt can be seen from Fig. \\ref{fig:initial_location}\nthat not only rain streaks but also some non-rain components\nappear in rain detection result. How to recognize those\nnon-rain components and eliminate their influence is\nthus very critical - a lot of image details and useful\ninformation would otherwise get lost after the removal of rain streaks.\n\nIn order to separate rain from non-rain objects,\nsome characteristics of rain can be useful.\nWe describe them as follows:\n\\begin{itemize}\n\\item rain streaks usually do not have too large size in width,\n\\item the directions of all rain streaks in a scene are nearly consistent,\n\\item the color of a rain streak is usually shallow white, and\n\\item the length of a rain streak is usually larger than its width.\n\\end{itemize}\nThese characteristics are very robust to describe rain, and\nsome of them have been utilized in some existing rain-removal\nworks, such as \\cite{Chen_2014_CSVT}, \\cite{Kim_2013_ICIP} and so on.\nLater on, we will see that when these characteristics are\ncombined with our proposed morphological processing,\nthe error detection will be reduced largely.\n\n\\subsection{Refining of initial locations of rain streaks}\n\n\\noindent \\textbf{First}, all connected components shown in Fig. \\ref{fig:initial_location}\nare extracted by the morphology method and the details can be referred to \\cite{Gonzalez_2002_PUSR}.\n\n\\begin{figure}[t]\n\\begin{minipage}{0.48\\linewidth}\n\\centering{\\includegraphics[width=.9\\linewidth]{images\/rain18_L_single_streak}}\n\\centerline{(a)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.48\\linewidth}\n\\centering{\\includegraphics[width=.9\\linewidth]{images\/rain18_L_width}}\n\\centerline{(b)}\n\\end{minipage}\n\\caption{(a) An example of PCA description. (b) Refined result by connected component width.}\n\\label{fig:single_streak}\n\\end{figure}\n\n\\noindent \\textbf{Second}, PCA is used to describe the shape of every connected component.\nIn order to describe this step more visually, we select one connected component\nfrom Fig. \\ref{fig:initial_location} as an example\nto show the refining process, the selected component is in Fig. \\ref{fig:single_streak}(a).\nBecause some colors can not be seen clearly on a black background,\nwe have changed the selected streak to black and the background to white in Fig. \\ref{fig:single_streak}(a).\n\n\\begin{figure*}[t]\n\\begin{minipage}{0.24\\linewidth}\n\\centering{\\includegraphics[width=.9\\linewidth]{images\/rain18_L_angle}}\n\\centerline{(a)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.24\\linewidth}\n\\centering{\\includegraphics[width=.9\\linewidth]{images\/rain18_L_color}}\n\\centerline{(b)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.24\\linewidth}\n\\centering{\\includegraphics[width=.9\\linewidth]{images\/rain18_L_aspect_ratio}}\n\\centerline{(c)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.24\\linewidth}\n\\centering{\\includegraphics[width=.9\\linewidth]{images\/rain18_L}}\n\\centerline{(d)}\n\\end{minipage}\n\\caption{(a) Refined result by the connected component angle. (b) Refined result by the connected component color. (c) Refined result by the connected component aspect ratio. (d) Dilation of rain streaks.}\n\\label{fig:revision}\n\\end{figure*}\n\nFor $p^{th}( p=1, 2, ..., P)$ connected component, we calculate the covariance matrix of location vectors of all pixels in it.\nSuppose that there are $N$ pixels in $p^{th}$ connected component.\nHence, there are $N$ sample vectors of pixel locations so that the mean location\nvector $\\bm{m}_{\\bm{z}}$ and covariance matrix $\\bm{C}_{\\bm{z}}$ can be calculated as\n\\begin{equation} \\label{eq:mean_approximation}\n\\bm{m}_{\\bm{z}} = \\frac {1}{N} \\sum^{N}_{n=1} \\bm{z}_{n}\n\\end{equation}\n\\begin{equation} \\label{eq:covariance_matrix2}\n\\bm{C}_{\\bm{z}} = \\frac {1}{N} \\sum^{N}_{n=1} \\bm{z}_{n}\\bm{z}_{n}^{T} - \\bm{m}_{\\bm{z}}\\bm{m}_{\\bm{z}}^{T}\n\\end{equation}\nwhere $\\bm{z}_{n}= [x_{n}, y_{n}]^{T}$, and $x_n$ and $y_n$ are respectively\nthe corresponding coordinates of the $n^{th}$ pixel ($n=1, 2, \\cdots, N$).\n\nAfter the covariance matrix $\\bm{C}_{\\bm{z}}$ of $p^{th}$ connected component is obtained,\nwe perform the eigenvalue decomposition of $\\bm{C}_{\\bm{z}}$\nand obtain the eigenvalues $\\lambda_{1}$, $\\lambda_{2}$ and\ntheir corresponding eigenvector $\\bm{e}_{1}$, $\\bm{e}_{2}$ ($\\lambda_{1}$ is the larger eigenvalue).\nThe description of PCA to the shape of connected\ncomponents are shown in Fig. \\ref{fig:single_streak}(a).\nThe red arrows stand for two eigenvectors, while two yellow arrows denote\nthe coordinate axes. Here, $\\theta$ is the angle between $x$-axis and eigenvector\n$\\bm{e}_{1}$ and it can be calculated as $\\theta=\\arctan(\\frac {\\bm{e}_{1}(2)}{\\bm{e}_{1}(1)})$.\nNotice that in order to avoid the red direction arrow from occluding the connected component,\nthe origin of the coordinate system is not placed on the connected component.\n\nFrom Fig. \\ref{fig:single_streak}(a), we learn that $\\bm{e}_{1}$\n(corresponding to the larger eigenvalue $\\lambda_{1}$) points to the direction\nwhere the location variance has the maximum value; whereas $\\bm{e}_{2}$\n(corresponding to the smaller eigenvalue $\\lambda_{2}$) is perpendicular to the maximum variance direction.\n\nAccordingly, we define the length of a connected component as\n\\begin{equation} \\label{eq:length}\nL=c\\lambda_{1}\n\\end{equation}\nand its width as\n\\begin{equation} \\label{eq:width}\nW=c\\lambda_{2}\n\\end{equation}\nwhere $c$ is a proportional parameter. We assume that $c$ is a constant in an image.\nThe specific value of $c$ is not important, because it does not affect the ratio of the\nlength and width of a connected component.\nThe more important quantity is the direction angle of a connected components,\nwhich is denoted as $\\theta$ in Fig. \\ref{fig:single_streak}(a), but is now re-defined as\n\\begin{equation} \\label{eq:angle}\nD=\\theta\n\\end{equation}\nand name $D$ as the direction of a connected component.\n\nIn our experiment, the values $\\lambda_{1}$, $\\lambda_{2}$, $\\bm{e}_{1}$,\n$\\bm{e}_{2}$ and $D$ of all $p^{th}(p=1, 2, ..., P)$ connected components are calculated,\n$P$ is the number of connected components.\nAs an example, these values of the given connected components\nin Fig. \\ref{fig:single_streak}(a) are\n$\\lambda_{1}=172.8949$, $\\lambda_{2}=0.5852$,\n$\\bm{e}_{1}=(0.9309, 0.3653)^{T}$,\n$\\bm{e}_{2}=(-0.3653, 0.9309)^{T}$ and $D=21.4286^{\\circ}$ respectively.\n\n\\noindent \\textbf{Third}, after obtaining the quantified characteristics of all connected components,\nwe recognise non-rain connected components as follows.\n\\begin{itemize}\n\n\\item As we said above, rain streaks usually do not have large width\nas compared to some non-rain objects.\nHence, the $K$-means is used here to classify the connected components by their width $W$.\nThe connected components with larger width are mis-detected non-rain components and\nwe set their corresponding values in location map $S_{R}$ as $0$.\nThe refined result in this way is shown in Fig. \\ref{fig:single_streak}(b).\n\nThere are not so many wide non-rain objects in this given image, hence\nthe refinement by width is not too apparent.\nWe can see that some non-rain components at right\nbottom corner disappear in Fig. \\ref{fig:single_streak}(b).\nThis is because when textures of an image are complex,\nsome non-rain streaks combine together and form\na larger connected component so that the width becomes large.\n\n\\item An apparent characteristic of rain streaks is that they follow\nnearly the same falling direction and the angle will not be too large generally.\nIf we use the direction angle $D$ of connected component defined in Equation\n(\\ref{eq:angle}) to describe this characteristic, $\\vert D \\vert$\nof rain components must be less than a threshold $T1$ ($\\vert D \\vert$ is\nthe absolute value of $D$).\nHence, by the threshold $T1$, we can recognize the mis-detected non-rain connected components\nin Fig. \\ref{fig:single_streak} (b). Then the non-rain connected\ncomponents are set to be 0, and the refined result is shown in Fig. \\ref{fig:revision}(a).\n\n\\item After refining by the width and direction constraints,\nmajority of non-rain components are recognized. However, some non-rain components that\nare similar in shape to the rain streaks still remain.\nRain streaks usually possess neutral color.\nAccording to this feature, Chen \\emph{et al.} \\cite{Chen_2014_CSVT} proposed to\nidentify rain dictionary atoms by the eigen color feature \\cite{Tsai_2008_IET_CV}.\nIn our work, we utilize the color characteristics of\nrain to revise the mis-detected non-rain connected components.\n\nFor $p^{th}$ connected component in Fig. \\ref{fig:revision}(a),\nwe calculate the mean color vector of all pixels in it,\nand denote as $[\\bar{R}, \\bar{G}, \\bar{B}]$.\nThen we transform this 3-D RGB color vector into a 2-D vector as follows:\n\\begin{small}\n\\begin{equation}\\label{eq:color_transform}\n\\begin{split}\n u & =\\frac{2\\Phi-\\bar{G}-\\bar{B}} {\\Phi} \\\\\n v & =max \\left \\{\n               \\frac{\\Phi-\\bar{G}}{\\Phi}, \\frac{\\Phi-\\bar{B}}{\\Phi}\n            \\right \\}\n\\end{split}\n\\end{equation}\n\\end{small}\nwhere $\\Phi=\\frac{1}{3}(\\bar{R}+\\bar{G}+\\bar{B})$.\nIt is clear from (\\ref{eq:color_transform}) that,\nafter the transform any connected components having neutral color will\nbe clustered around $(0, 0)$ in the $u$-$v$ space.\nHence, we calculate the magnitude of this 2-D vector $(u, v)$\n(i.e.,the Euclidean distance to the origin of the $u$-$v$ space),\nif the magnitude is larger than a pre-set value $T2$,\nthe $p^{th}$ connected component is recognized as a mis-detected non-rain connected component.\nFor all remaining connected components in Fig. \\ref{fig:revision}(a),\nwe repeat this process and revise the mis-detected non-rain connected components.\nThe refined result is shown in Fig. \\ref{fig:revision}(b).\n\n\\item According to \\cite{Kim_2013_ICIP}, a rain streak has a larger size\nin length than in width. Hence, we classify the connected components\nwhose aspect ratio are less than $\\mu$ as non-rain components.\nBy excluding the connected components that have small aspect ratios,\nthe refined result is shown in Fig. \\ref{fig:revision}(c).\n\n\\item Finally, in order to avoid some slim rain edges\nfrom remaining in our final rain-removed result,\nwe dilate the connected components in\nFig. \\ref{fig:revision}(c) by a $3\\times 3$ 'disk' mask,\nto obtain the final result for rain streaks detection,\nas shown in Fig. \\ref{fig:revision}(d).\n\\end{itemize}\n\nOur rain detection is a stepwise revision method.\nBy utilizing morphology and PCA, we quantify the rain's\ncharacteristics and detect rain streaks relatively accurately.\n\n\\section{Image Reconstruction}\n\\label{sec:ImageReconstruction}\nIn this section, we try to verify the sparsity\nof natural rain images and utilize one Laplacian distribution\nto approximate the sparsity prior of natural rain image,\nwe name the approximate prior as \\emph{quasi-sparsity prior}.\nThen, based on the quasi-sparsity and several constraints,\nthe rain-removed result is obtained by\nseparating a rain image into rain layer and background layer.\n\n\\subsection{Quasi-sparsity of rain images}\n\nIn \\cite{Levin_2007_PAMI}, Levin and Weiss tried to separate\nthe background and reflection from an image by sparsity prior of natural images.\nWe also utilize image sparsity in our rain removal task.\nThe sparsity of an image mentioned in \\cite{Levin_2007_PAMI} can\nbe depicted as: when a derivative filter is applied on an image,\nthe logarithm of the histogram of the obtained gradient image reaches\npeak value at zero and falls off much faster than a Gaussian.\nLevin \\emph{et al.} demonstrated that sparse distributions\nwill lead to a better image decomposition \\cite{Levin_2002_NIPS}.\nHence, the sparsity of a natural\nimage is crucial to its decomposition into several layers.\n\n\\begin{figure}[t]\n\\centering\n\\begin{minipage}{0.48\\linewidth}\n\\centering{\\includegraphics[width=.9\\linewidth]{images\/rain18_distribution_compare}}\n\\centerline{(a)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.48\\linewidth}\n\\centering{\\includegraphics[width=.9\\linewidth]{images\/rain18_sparsity_prior}}\n\\centerline{(b)}\n\\end{minipage}\n\\caption{(a) Log-probability of several distributions. (b) Sparsity verification on one rain image.}\n\\label{fig:sparsity}\n\\end{figure}\n\n\\begin{figure*}[t]\n\\centering\n\\begin{minipage}{0.24\\linewidth}\n\\centering{\\includegraphics[width=.9\\linewidth]{images\/rain18}}\n\\centerline{(a)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{0.24\\linewidth}\n\\centering{\\includegraphics[width=.9\\linewidth]{images\/rain18_I_nd}}\n\\centerline{(b)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.24\\linewidth}\n\\centering{\\includegraphics[width=.9\\linewidth]{images\/rain18_removed_rain}}\n\\centerline{(c)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.24\\linewidth}\n\\centering{\\includegraphics[width=.9\\linewidth]{images\/rain18_L_nonrain}}\n\\centerline{(d)}\n\\end{minipage}\n\\caption{(a) Original rain image. (b) Rain-removed image.\n(c) Rain component removed from (a).\n(d) Non-rain location $S_{NR}$ that is obtained by $S_{NR}=1-S_{R}$ (the white area).}\n\\label{fig:results}\n\\end{figure*}\n\n\\begin{figure*}[t]\n\\centering\n\\begin{minipage}{0.195\\linewidth}\n\\centering{\\includegraphics[width=.9\\linewidth]{images\/test75}}\n\\centerline{(a)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{0.195\\linewidth}\n\\centering{\\includegraphics[width=.9\\linewidth]{images\/test75_I_nd}}\n\\centerline{(b)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.195\\linewidth}\n\\centering{\\includegraphics[width=.9\\linewidth]{images\/test75_I_nd_rain}}\n\\centerline{(c)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.195\\linewidth}\n\\centering{\\includegraphics[width=.9\\linewidth]{images\/test75_I_nd_revised}}\n\\centerline{(d)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.195\\linewidth}\n\\centering{\\includegraphics[width=.9\\linewidth]{images\/test75_I_nd_rain_revised}}\n\\centerline{(e)}\n\\end{minipage}\n\\caption{(a) One rain image; (b) background layer without the third constraint;\n(c) rain layer without the third constraint;\n(d) background layer with the third constraint; (e) rain layer with the third constraint.}\n\\label{fig:correct_separate}\n\\end{figure*}\n\nFig. \\ref{fig:sparsity}(a) illustrates the logarithm probabilities of\nseveral distributions. Laplacian distribution exactly results in a straight line and\nconnects the maximum and minimum values.\nWe can see that Gaussian distribution falls off the slowest and is above the\nstraight line so that it is viewed as non-sparse. The other two distributions\nbelow the straight line are classified as sparse according to \\cite{Levin_2007_PAMI}.\nLaplacian distribution is on the border of\nsparsity and non-sparsity.\n\nIn order to verify the sparsity of rain images,\nwe conduct an experiment on nearly 200 rain images and part of these\nimages are also used in the experiment section. Here, we use the image\nin Fig. \\ref{fig:results}(a) as an example to illustrate the sparsity of rain images.\nFig. \\ref{fig:sparsity}(b) shows the logarithm curve (the blue curve)\nof histogram after applying an horizontal derivative filter on it.\nObviously, the result reveals that the rain image satisfies the sparsity requirement.\n\nHowever, decomposing a rain image $I$ into the rain layer $I_{R}$ and background layer $I_{NR}$ as\n\\begin{equation} \\label{eq:image_decomposition}\nI=I_{R}+I_{NR}\n\\end{equation}\nis a massively ill-posed problem. To simplify this kind of problem,\nLevin \\emph{et al.} proposed that users can label some edges or areas that belong\nto $I_{R}$ and some other edge or areas that belong to $I_{NR}$\nto increase the constraint for this kind of problem \\cite{Levin_2007_PAMI}.\n\nSparsity ensures that an edge of unit contrast will not be split,\nand will appear in one layer \\cite{Levin_2007_PAMI}.\nIn our task, we have detected nearly all rain locations and\nthe remaining region is labelled as the non-rain area.\nOur detection offers better constraints to this ill-posed problem\nthan the manually-labeled operation in \\cite{Levin_2007_PAMI},\nand also realize the role of sparsity in certain degree.\nUnlike in \\cite{Levin_2007_PAMI}, we relax the probability constraint and utilize single\nLaplacian function to approximate the sparsity of rain images\nand named as \\emph{quasi-sparse distribution}:\n\\begin{equation} \\label{eq:histogram_approximation}\nP(x)=e^{- \\vert x \\vert}\n\\end{equation}\nHence, the quasi-sparsity prior over the whole image $I$ is as follows:\n\\begin{equation}\\label{eq:laplacian_approximation}\nP(I)=\\prod_{i, k}e^{- \\vert \\omega_{i, k} \\cdot I \\vert}\n\\end{equation}\nwhere $\\omega_{i,k}$ is the $k^{th}$ filter which centered\nat $i^{th}$ pixel.  The filters\nwith two orientations (horizontal and vertical) and two degrees (the\nfirst derivative and the second derivative) are used here.\n\n\\subsection{Optimization}\n\nFor an given rain image $I$, $S_{R}$ is the detected rain location,\nand the non-rain location can be obtained by $S_{NR}=1-S_{R}$.\nThe following constraints are satisfied to separate an rain image\ninto rain layer $I_{R}$ and background (non-rain) layer $I_{NR}$:\n\\begin{enumerate}\n\\item $I=I_{R}+I_{NR}$;\n\\item the gradients of $I_{R}$ and $I_{NR}$ at their corresponding locations\nin $S_{R}$ and $S_{NR}$ respectively agree with the gradient of image $I$;\n\\item the values of $I_{NR}$ at location $S_{NR}$ are close to the value of $I$.\n\\end{enumerate}\nThe first two constraints are also utilized in \\cite{Levin_2007_PAMI}.\nAs shown later, this will lead some non-normal separation for some specific images.\nTo improve the separation, we add the third constraint and it plays a role\nas boundary condition.\n\nAs the work in \\cite{Weiss_2001_ICCV}, we assume that\nderivative filters are independent over space and orientation;\nrain layer $I_{R}$ and background layer $I_{NR}$ are independent.\nThen the quasi-sparsity prior can be written as follows according\nto the first constraint:\n\\begin{equation} \\label{eq:prior_define}\nP(I)=P(I_{R})P(I_{NR})=\\prod_{i, k}e^{- ( \\vert \\omega_{i, k} \\cdot I_{R} \\vert +  \\vert \\omega_{i, k} \\cdot I_{NR} \\vert)}\n\\end{equation}\nWe would like to obtain $I_{R}$ and $I_{NR}$ which maximize the\nabove likelihood function. It is equal to minimize the following loss function:\n\\begin{equation}\\label{eq:loss_function}\nJ(I_{R}, I_{NR})=\\sum_{i, k} \\vert \\omega_{i, k} \\cdot I_{R} \\vert + \\vert \\omega_{i, k} \\cdot I_{NR} \\vert\n\\end{equation}\nCombined with the second and third constraints,\nwe rewrite Equation (\\ref{eq:loss_function}) as\n\\begin{equation}\\label{eq:loss_function1}\n\\begin{split}\n&J_{1}(I_{R})=\\sum_{i, k} \\vert \\omega_{i, k} \\cdot I_{R} \\vert + \\vert \\omega_{i, k} \\cdot (I-I_{R}) \\vert \\\\\n& \\qquad \\quad + \\lambda \\sum_{i \\in S_{R}, k} \\vert \\omega_{i,k} \\cdot I_{R} - \\omega_{i, k} \\cdot I \\vert \\\\\n& \\qquad \\quad +\\lambda \\sum_{i \\in S_{NR}, k} \\vert \\omega_{i, k} \\cdot I_{R} \\vert  \\\\\n& \\qquad \\quad +\\eta \\sum_{i \\in S_{NR}} \\vert I_{R} \\vert\n\\end{split}\n\\end{equation}\nwhere $\\lambda$ and $\\eta$ are regularization parameters.\n\nIf $v$ is defined as the vectorized version of image $I_{R}$,\nEquation (\\ref{eq:loss_function1}) becomes\n\\begin{equation}\\label{eq:loss_function2}\nJ_{2}(v)= \\Vert Av-b \\Vert_{1}\n\\end{equation}\nwhere $\\Vert \\cdot \\Vert_{1}$ is the $L_{1}$ norm, $A$\nis relative to the derivative filters, $\\lambda$ and $\\eta$, and $b$ is relative to the image derivative,\nthe values of $I$ at location $S_{NR}$, zero and $\\lambda$, $\\eta$.\n\nThis is a $L_{1}$-norm optimization problem,\nand it can be solved by iterative reweighted least squares (IRLS) \\cite{Burrus_2009_CPAM}.\nWe summarize the process in Algorithm \\ref{alg:whole_algorithm}.\nOnce $v$ is obtained, we resize it back to the rain-layer image $I_{R}$.\nThen, the rain-removed image $I_{NR}$ can be obtained as\n\\begin{equation}\\label{eq:rain_remove}\nI_{NR}=I-I_{R}\n\\end{equation}\nOne example of the rain layer and rain-removed image is shown in Fig. \\ref{fig:results}(a) and (b), respectively.\nIn Fig. \\ref{fig:results}(c)(d), we show the constructed rain layer and the non-rain location $S_{NR}$.\n\nAs mentioned above, the third constraint plays an important role\nin the correct separation of rain images. Here, we show\nan example in Fig. \\ref{fig:correct_separate} to suggest the role of this constraint.\nIn Fig. \\ref{fig:correct_separate}(b)(c), we can see that serious\ncolor shift (means that the colors of non-rain details in (b) are abnormal) will\nappear without the third constraint.\nThe reason is that some colors go to the rain layer (c) in under-determined conditions.\nBy adding the third constraint, the separation quality\ncan be improved and we can obtain a natural rain-removed image.\n\n\\begin{algorithm}\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\\caption{IRLS}\n\\label{alg:whole_algorithm}\n\\begin{algorithmic}\n\\REQUIRE $A$, $b$, $Iter$\n\\renewcommand{\\algorithmicrequire}{Initialization}\n\\REQUIRE $v=[ A^{T}A ]^{-1}Ab$\n\\FOR{$t$=1 to $Iter$ }\n\\STATE $e=abs(Av-b)$\n\\STATE $z(i) = e(i)^{-0.5}, i=1, 2, ...$\n\\STATE $\\Omega = diag(z)$\n\\STATE $v =[ A^{T}\\Omega^{T}\\Omega A ]^{-1}A^{T}\\Omega^{T}\\Omega b$\n\\ENDFOR\n\\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n\\ENSURE $v$\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\section{Experimental Results}\n\\label{sec:ExperimentalResults}\n\n\\begin{table*} [t]\n\\centering\n\\caption{The Average Time Consumed by Selected Methods on $256 \\times 256$ Images.}\n\\begin{tabular}{lccccccc}\n\\hline\nMethod     &  \\cite{Ding_2015_MTA}  & \\cite{Chen_2014_CSVT}   & \\cite{Luo_2015_ICCV}  & \\cite{Li_2016_CVPR}  & \\cite{Fu_2017_CVPR}  & \\cite{Zhang_2018_CVPR}  & Ours   \\\\\nTime(s) &  1.25s & 97.15s & 69.69s & 1260.40s & 5.30s   & 0.20s   & 28.01s   \\\\\n\\hline\n\\end{tabular}\n\\label{tab:time}\n\\end{table*}\n\n\\begin{table*} [t]\n\\small\n\\newcommand{\\tabincell}[2]{\\begin{tabular}{@{}#1@{}}#2\\end{tabular}}\n\\centering\n\\caption{Image Performances (Top: \\textbf{PSNR}, Bottom: \\textbf{SSIM}) of Different Methods (Rows) on $11$ Synthesized Rain Images (Columns) against Ground-truth.}\n\\begin{tabular}{l|c|c|c|c|c|c|c|c|c|c|c}\n\\hline\n        & Image 1                       & Image 2                       & Image 3                       & Image 4                       & Image 5                       & Image 6                       & Image 7                       & Image 8                       & Image 9                       & Image 10                      & Image 11                      \\\\\n\\hline\n\\cite{Ding_2015_MTA} & \\tabincell{c}{34.65 \\\\ 0.867} & \\tabincell{c}{33.70 \\\\ 0.889} & \\tabincell{c}{33.89 \\\\ 0.802} & \\tabincell{c}{34.17 \\\\ 0.805} & \\tabincell{c}{35.16 \\\\ 0.861} & \\tabincell{c}{35.93 \\\\ 0.835} & \\tabincell{c}{41.29 \\\\ 0.796} & \\tabincell{c}{31.77 \\\\ 0.811} & \\tabincell{c}{32.50 \\\\ 0.874} & \\tabincell{c}{34.58 \\\\ 0.907} & \\tabincell{c}{33.22 \\\\0.832 } \\\\\n\\hline\n\\cite{Chen_2014_CSVT}   & \\tabincell{c}{34.31 \\\\ 0.803} & \\tabincell{c}{32.36 \\\\ 0.759} & \\tabincell{c}{34.92 \\\\ 0.750} & \\tabincell{c}{34.68 \\\\ 0.738} & \\tabincell{c}{34.95 \\\\ 0.774} & \\tabincell{c}{32.55 \\\\ 0.824} & \\tabincell{c}{38.58 \\\\ 0.775} & \\tabincell{c}{31.84 \\\\ 0.602} & \\tabincell{c}{32.11 \\\\ 0.704} & \\tabincell{c}{34.59 \\\\ 0.854} & \\tabincell{c}{34.15 \\\\ 0.784} \\\\\n\\hline\n\\cite{Luo_2015_ICCV} & \\tabincell{c}{32.69 \\\\ 0.767} & \\tabincell{c}{30.23 \\\\ 0.703} & \\tabincell{c}{31.53 \\\\ 0.748} & \\tabincell{c}{32.43 \\\\ 0.820} & \\tabincell{c}{33.73 \\\\ 0.888} & \\tabincell{c}{29.45 \\\\ 0.841} & \\tabincell{c}{35.95 \\\\ 0.784} & \\tabincell{c}{29.45 \\\\ 0.790} & \\tabincell{c}{30.43 \\\\ 0.879} & \\tabincell{c}{31.63 \\\\ 0.864} & \\tabincell{c}{32.99 \\\\ 0.843} \\\\\n\\hline\n\\cite{Li_2016_CVPR} & \\tabincell{c}{31.55 \\\\ 0.701} & \\tabincell{c}{30.45 \\\\ 0.686} & \\tabincell{c}{31.23 \\\\ 0.789} & \\tabincell{c}{32.27 \\\\ 0.691} & \\tabincell{c}{33.34 \\\\ 0.748} & \\tabincell{c}{31.13 \\\\ 0.754} & \\tabincell{c}{36.39 \\\\ 0.681} & \\tabincell{c}{29.54 \\\\ 0.570} & \\tabincell{c}{30.32 \\\\ 0.686} & \\tabincell{c}{32.35 \\\\ 0.786} & \\tabincell{c}{32.42 \\\\ 0.749} \\\\\n\\hline\nOurs   & \\tabincell{c}{\\textbf{35.46} \\\\ \\textbf{0.886}} & \\tabincell{c}{\\textbf{35.30} \\\\ \\textbf{0.901}} & \\tabincell{c}{\\textbf{35.04} \\\\ \\textbf{0.827}} & \\tabincell{c}{\\textbf{34.86} \\\\ \\textbf{0.832}} & \\tabincell{c}{\\textbf{35.38} \\\\ \\textbf{0.897}} & \\tabincell{c}{\\textbf{36.03} \\\\ \\textbf{0.842}} & \\tabincell{c}{\\textbf{41.31} \\\\ \\textbf{0.846}} & \\tabincell{c}{\\textbf{31.94} \\\\ \\textbf{0.854}} & \\tabincell{c}{\\textbf{33.42} \\\\ \\textbf{0.883}} & \\tabincell{c}{\\textbf{34.91} \\\\ \\textbf{0.916}} & \\tabincell{c}{\\textbf{34.53} \\\\ \\textbf{0.866}} \\\\\n\\hline\n\\end{tabular}\n\\label{tab:psnrssim}\n\\end{table*}\n\n\\begin{figure*}[t]\n\\centering\n\\begin{minipage}{0.1\\linewidth}\n\\centering{\\includegraphics[width=.99\\linewidth]{images\/001_GT}}\n\\end{minipage}\n\\begin{minipage}{0.1\\linewidth}\n\\centering{\\includegraphics[width=.99\\linewidth]{images\/RR1}}\n\\end{minipage}\n\\begin{minipage}{.1\\linewidth}\n\\centering{\\includegraphics[width=.99\\linewidth]{images\/RR1_I_nd_by_Ding}}\n\\end{minipage}\n\\begin{minipage}{.1\\linewidth}\n\\centering{\\includegraphics[width=.99\\linewidth]{images\/RR1_I_nd_by_Kang}}\n\\end{minipage}\n\\begin{minipage}{.1\\linewidth}\n\\centering{\\includegraphics[width=.99\\linewidth]{images\/RR1_I_nd_by_luo}}\n\\end{minipage}\n\\begin{minipage}{.1\\linewidth}\n\\centering{\\includegraphics[width=.99\\linewidth]{images\/RR1_I_nd_by_li}}\n\\end{minipage}\n\\begin{minipage}{.1\\linewidth}\n\\centering{\\includegraphics[width=.99\\linewidth]{images\/RR1_I_nd_by_Fu_label}}\n\\end{minipage}\n\\begin{minipage}{.1\\linewidth}\n\\centering{\\includegraphics[width=.99\\linewidth]{images\/RR1_I_nd_by_Zhang_label}}\n\\end{minipage}\n\\begin{minipage}{.1\\linewidth}\n\\centering{\\includegraphics[width=.99\\linewidth]{images\/RR1_I_nd}}\n\\end{minipage} \\\\\n\\vspace{0.5mm}\n\\begin{minipage}{0.1\\linewidth}\n\\centering{\\includegraphics[width=.99\\linewidth]{images\/005_GT}}\n\\centerline{(a)}\n\\end{minipage}\n\\begin{minipage}{0.1\\linewidth}\n\\centering{\\includegraphics[width=.99\\linewidth]{images\/RR5}}\n\\centerline{(b)}\n\\end{minipage}\n\\begin{minipage}{.1\\linewidth}\n\\centering{\\includegraphics[width=.99\\linewidth]{images\/RR5_I_nd_by_Ding}}\n\\centerline{(c)}\n\\end{minipage}\n\\begin{minipage}{.1\\linewidth}\n\\centering{\\includegraphics[width=.99\\linewidth]{images\/RR5_I_nd_by_Kang}}\n\\centerline{(d)}\n\\end{minipage}\n\\begin{minipage}{.1\\linewidth}\n\\centering{\\includegraphics[width=.99\\linewidth]{images\/RR5_I_nd_by_luo}}\n\\centerline{(e)}\n\\end{minipage}\n\\begin{minipage}{.1\\linewidth}\n\\centering{\\includegraphics[width=.99\\linewidth]{images\/RR5_I_nd_by_li}}\n\\centerline{(f)}\n\\end{minipage}\n\\begin{minipage}{.1\\linewidth}\n\\centering{\\includegraphics[width=.99\\linewidth]{images\/RR5_I_nd_by_Fu_label}}\n\\centerline{(g)}\n\\end{minipage}\n\\begin{minipage}{.1\\linewidth}\n\\centering{\\includegraphics[width=.99\\linewidth]{images\/RR5_I_nd_by_Zhang_label}}\n\\centerline{(h)}\n\\end{minipage}\n\\begin{minipage}{.1\\linewidth}\n\\centering{\\includegraphics[width=.99\\linewidth]{images\/RR5_I_nd}}\n\\centerline{(i)}\n\\end{minipage}\n\\caption{(a) Groudtruth. (b) Original synthesized rain images. (c) Results by Ding \\emph{et al.} in \\cite{Ding_2015_MTA}.\n(d) Results by Chen \\emph{et al.} in \\cite{Chen_2014_CSVT}. (e) Results by Luo \\emph{et al.} in \\cite{Luo_2015_ICCV}.\n(f) Results by Li \\emph{et al.} in \\cite{Li_2016_CVPR}. (g) Results by Fu \\emph{et al.} in \\cite{Fu_2017_CVPR}.\n(h) Results by Zhang \\emph{et al.} in \\cite{Zhang_2018_CVPR}. (i) Results by our method.}\n\\label{fig:result_render_compare}\n\\end{figure*}\n\n\\begin{figure}[t]\n\\begin{minipage}{0.48\\linewidth}\n\\centering{\\includegraphics[width=1\\linewidth]{images\/PSNR_Comparison}}\n\\centerline{(a)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.48\\linewidth}\n\\centering{\\includegraphics[width=1\\linewidth]{images\/SSIM_Comparison}}\n\\centerline{(b)}\n\\end{minipage}\n\\caption{Objective comparisons with two state-of-the-art deep learning works: (a) PSNR comparison. (b) SSIM comparison.}\n\\label{fig:PSNR_SSIM}\n\\end{figure}\n\nIn order to verify the effectiveness of our method, several state-of-the-art traditional and deep learning based rain removal works are selected for comparisons. The method by Ding \\emph{et al.} \\cite{Ding_2015_MTA} removes rain streaks in a single image by developing an $L_0$ smoothing filter that is derived from the guided filter by He \\emph{et al.} \\cite{He_2013_PAMI}. This work produces excellent rain removal results for some kinds of images and keeps good visual quality. Meanwhile, several rain removal works that are based on dictionary learning have appeared in recent years \\cite{Fu_2011_ASSP,Kang_2012_TIP,Chen_2014_CSVT}. Among them, the work by Chen \\emph{et al.} \\cite{Chen_2014_CSVT} produces the best rain removal effect. In addition, two most recent works, by Luo\n\\emph{et al.} \\cite{Luo_2015_ICCV} and Li \\emph{et al.} \\cite{Li_2016_CVPR}, respectively, are also selected in our comparisons.\nFor deep learning based rain-removed methods, we select the most recent two works \\cite{Fu_2017_CVPR} and \\cite{Zhang_2018_CVPR}.\nCompared with other deep learning based works, these two works are more robust and can obtain better rain-removed visual quality.\n\nWe implement our rain removal algorithm using MATLAB on\nan Intel (R) Xeon (R) CPU E5-2643 v2 @ 3.5 GHz 3.5 GHz (2 processors) with 64G RAM.\nSome parameters used in our work are: the size of the window in\nEquation (\\ref{eq:detect_condition}) is $7 \\times 7$;\nthe iteration time of $K$-means in Section \\ref{sec:RainStreaksDetection} is 100;\nthe thresholds $T1$, $T2$ and $\\mu$ are 10, 0.08, 2 respectively;\nregularization parameter $\\lambda$ and $\\eta$ in loss function (\\ref{eq:loss_function1}) is $0.25$ and $0.1$,\nand the iteration time in IRLS is $3$.\nThe parameters here are robust in our experiments.\nWhile for the parameter $T1$, it can be slightly changed for different images.\nBecause, rain's direction is downward\nand its value $D$ is close to $0$ in most image,\nwe set the threshold $T1$ as $10$ in our paper.\nRain's direction $D$ can be approximately evaluated by user easily.\nFor some rain which has large falling direction (e.g. the sixth row in Fig. \\ref{fig:result_compare}),\nthe threshold $T1$ can be changed to a larger value.\n\nWe first test the run-time consumed by the selected methods on images with size $256 \\times 256$. Our method takes $28.01$ seconds. Specifically, the initial detection of rain streaks takes $5.30$ seconds; the rain streaks refining by morphology needs $2.19$ seconds; and the majority of time is spent on rain and non-rain layer separation by using quasi-sparsity prior, which is $20.02$ seconds. Upon the same image, the time consumed by other selected methods are listed in Table \\ref{tab:time}. By comparison, our algorithm is the fourth fastest one in selected methods.\n\nBecause the task in this work is to remove rain streaks in single images,\nwe need to evaluate the effectiveness of our algorithm subjectively and objectively. For the purpose of objective evaluations, we synthesize rain images from clean images. Two such ground-truth images and synthesized rain images are shown in Fig. \\ref{fig:result_render_compare}(a) and (b), respectively, and the other columns are the corresponding rain-removed results by different state-of-the-art algorithms and our method.\n\nWe also collect many real rain images and present the corresponding rain removal results as shown in Figs. \\ref{fig:result_compare} and \\ref{fig:result_compare1} for subjective assessments.\n\n\\begin{table*} [t]\n\\centering\n\\caption{User Study Result. The Numbers Are The Percentages of Votes Which Are Obtained by Each Method.}\n\\begin{tabular}{lccccccc}\n\\hline\nMethod     &  \\cite{Ding_2015_MTA}  & \\cite{Chen_2014_CSVT}   & \\cite{Luo_2015_ICCV}  & \\cite{Li_2016_CVPR} & \\cite{Fu_2017_CVPR}  & \\cite{Zhang_2018_CVPR} & Ours   \\\\\nPercentage &  5.50\\% & 1.25\\% & 2.50\\% & 3.75\\% & 21.00\\% & 9.50\\%  & 56.50\\%   \\\\\n\\hline\n\\end{tabular}\n\\label{tab:statistics}\n\\end{table*}\n\n\\subsection{Objective assessment}\n\nIn order to evaluate the performances of different methods more completely\nand accurately, we synthesize rain images by the method in \\cite{Luo_2015_ICCV} and implement different rain removal algorithms on these synthesized images. Then, we calculate the PSNR and SSIM \\cite{Wang_2004_TIP} values between the rain-removed images and the ground-truth images.\n\nFig. \\ref{fig:result_render_compare} shows two examples where each row presents a ground-truth image, the rain image (obtained by synthesis),\nand the rain-removed results by different methods. Note that we show the corresponding PSNR\/SSIM values at the top-left corner of each rain-removed image. The PSNR\/SSIM values of more examples by selected traditional methods are shown in Table \\ref{tab:psnrssim}. The comparisons of PSNR\/SSIM with deep learning methods are shown in Fig. \\ref{fig:PSNR_SSIM}.\n\nAccording to the PSNR\/SSIM values, the method by Ding \\emph{et al.} \\cite{Ding_2015_MTA} produces very good results compared with the other traditional methods. Because of the use of an $L_0$ threshold, objects with large structures in the image will usually be preserved well, thus leading to higher SSIM values. In the meantime, rain streaks below the $L_0$ threshold will be removed, leading to higher PSNR values.\n\n\\begin{figure*}[!htb]\n\\centering\n\\begin{minipage}{0.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain77}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain77_I_nd_by_Ding}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain77_I_nd_by_Kang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain77_I_nd_by_luo}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain77_I_nd_by_li}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain77_I_nd_by_Fu}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain77_I_nd_by_Zhang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain77_I_nd}}\n\\end{minipage}\n\\vspace{0.5mm}\n\\vfill\n\\begin{minipage}{0.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain77_part}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain77_I_nd_by_Ding_part}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain77_I_nd_by_Kang_part}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain77_I_nd_by_luo_part}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain77_I_nd_by_li_part}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain77_I_nd_by_Fu_part}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain77_I_nd_by_Zhang_part}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain77_I_nd_part}}\n\\end{minipage}\n\\vspace{0.5mm}\n\\vfill\n\\begin{minipage}{0.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain74}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain74_I_nd_by_Ding}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain74_I_nd_by_Kang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain74_I_nd_by_luo}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain74_I_nd_by_li}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain74_I_nd_by_Fu}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain74_I_nd_by_Zhang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain74_I_nd}}\n\\end{minipage}\n\\vspace{0.5mm}\n\\vfill\n\\begin{minipage}{0.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test78}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test78_I_nd_by_Ding}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test78_I_nd_by_Kang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test78_I_nd_by_luo}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test78_I_nd_by_li}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test78_I_nd_by_Fu}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test78_I_nd_by_Zhang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test78_I_nd}}\n\\end{minipage}\n\\vspace{0.5mm}\n\\vfill\n\\begin{minipage}{0.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test9}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test9_I_nd_by_Ding}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test9_I_nd_by_Kang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test9_I_nd_by_luo}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test9_I_nd_by_li}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test9_I_nd_by_Fu}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test9_I_nd_by_Zhang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test9_I_nd}}\n\\end{minipage}\n\\vspace{0.5mm}\n\\vfill\n\\begin{minipage}{0.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test113}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test113_I_nd_by_Ding}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test113_I_nd_by_Kang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test113_I_nd_by_luo}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test113_I_nd_by_li}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test113_I_nd_by_Fu}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test113_I_nd_by_Zhang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test113_I_nd}}\n\\end{minipage}\n\\vspace{0.5mm}\n\\vfill\n\\begin{minipage}{0.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test25}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test25_I_nd_by_Ding}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test25_I_nd_by_Kang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test25_I_nd_by_luo}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test25_I_nd_by_li}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test25_I_nd_by_Fu}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test25_I_nd_by_Zhang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test25_I_nd}}\n\\end{minipage}\n\\vspace{0.5mm}\n\\vfill\n\\begin{minipage}{0.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test105}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test105_I_nd_by_Ding}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test105_I_nd_by_Kang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test105_I_nd_by_luo}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test105_I_nd_by_li}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test105_I_nd_by_Fu}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test105_I_nd_by_Zhang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test105_I_nd}}\n\\end{minipage}\n\\vspace{0.5mm}\n\\vfill\n\\begin{minipage}{0.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test154}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test154_I_nd_by_Ding}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test154_I_nd_by_Kang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test154_I_nd_by_luo}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test154_I_nd_by_li}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test154_I_nd_by_Fu}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test154_I_nd_by_Zhang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test154_I_nd}}\n\\end{minipage}\n\\vspace{0.5mm}\n\\vfill\n\\begin{minipage}{0.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test26}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test26_I_nd_by_Ding}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test26_I_nd_by_Kang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test26_I_nd_by_luo}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test26_I_nd_by_li}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test26_I_nd_by_Fu}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test26_I_nd_by_Zhang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test26_I_nd}}\n\\end{minipage}\n\\vspace{0.5mm}\n\\vfill\n\\begin{minipage}{0.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test110}}\n\\centerline{(a)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test110_I_nd_by_Ding}}\n\\centerline{(b)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test110_I_nd_by_Kang}}\n\\centerline{(c)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test110_I_nd_by_luo}}\n\\centerline{(d)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test110_I_nd_by_li}}\n\\centerline{(e)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test110_I_nd_by_Fu}}\n\\centerline{(f)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test110_I_nd_by_Zhang}}\n\\centerline{(g)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test110_I_nd}}\n\\centerline{(h)}\n\\end{minipage}\n\\vspace{0.5mm}\n\\caption{(a) Original rain images. (b) Results by Ding \\emph{et al.} in \\cite{Ding_2015_MTA}.\n(c) Results by Chen \\emph{et al.} in \\cite{Chen_2014_CSVT}. (d) Results by Luo \\emph{et al.} in \\cite{Luo_2015_ICCV}.\n(e) Results by Li \\emph{et al.} in \\cite{Li_2016_CVPR}. (f) Results by Fu \\emph{et al.} in \\cite{Fu_2017_CVPR}.\n(g) Results by Zhang \\emph{et al.} in \\cite{Zhang_2018_CVPR}. (h) Results by our proposed method.}\n\\label{fig:result_compare}\n\\end{figure*}\n\nThe method by Chen \\emph{et al.} \\cite{Chen_2014_CSVT} can remove the rain steaks that possess lower pixel intensities but the rain streaks with higher intensities will remain (the reason will be described latter).\nFurthermore, because the HOG descriptor used in this method can not identify rain streaks from tenuous details well, it would lose many details (the second image in Fig. \\ref{fig:result_render_compare}).\nFor the above two reasons, the PSNR\/SSIM values are relatively lower than the method by Ding \\emph{et al.}\n\nThe work by Luo \\emph{et al.} \\cite{Luo_2015_ICCV} can not remove rain streaks well. It makes rain streaks more tenuous in size and weaker in intensity. We show the results of Li \\emph{et al.} \\cite{Li_2016_CVPR} in the sixth column of Fig. \\ref{fig:result_render_compare}. This method removes rain streaks quite well. However, a lot of image details have also been removed at the same time. It can be seen from Table \\ref{tab:psnrssim} that these two methods produce lower PSNR and SSIM values.\n\nFinally, it is seen from Table \\ref{tab:psnrssim} that our proposed method produces the best PSNR\/SSIM results consistently for all 11 test images compared with the selected traditional methods. For some test images (5 out of 11), the PSNR value of our method is about 1 dB higher than the second best method (i.e., Ding's method).\n\nWe can see from Fig. \\ref{fig:PSNR_SSIM} that our method produces comparable PSNR\/SSIM values compared with the state-of-the-art\ndeep learning methods. Only for the two rendering rain images shown in Fig. \\ref{fig:result_render_compare}, the work by Fu \\emph{et al.} \\cite{Fu_2017_CVPR} removes nearly all rain streaks and keeps image details relatively well. But its PSNR\/SSIM values are slightly low compared with ours. The reason is that our method only remove rain streaks on\nthe detected rain pixels and the non-rain pixels nearly unchanged.\nThough a few of light rain streaks remain in our results in these selected two images, our method keeps good image details in majority part of images. The work by Zhang \\emph{et al.} removes rain streaks well, but the image details loss seriously.\nThat is why this method has high PSNR values, while the SSIM values are low.\nBesides, these two methods also can not remove all rain streaks in some rendering rain images, but our method can achieve\ngood results, especially, for practical images which will be shown later. As we know, deep learning methods\nare very good at dealing with rendering rain images, because they are trained from them. Real-world images are real challenging\nfor them.\n\n\\subsection{User study}\n\nTo conduct a visual (subjective) evaluation on the performances of selected methods, we invited 20 viewers (14 males and 6 females, they all are undergraduate, master or Ph.D students in computer vision field) to evaluate the visual quality of different methods in terms of the following three aspects:\n\\begin{itemize}\n\\item less rain residual,\n\\item the maintenance of the image details, and\n\\item overall perception.\n\\end{itemize}\nIn the evaluation, $20$ groups of results are selected and every group involves the results by Ding \\emph{et al.}, Chen \\emph{et al.}, Luo \\emph{et al.}, Li \\emph{et al.}, Fu \\emph{et al.}, Zhang \\emph{et al.} and our method. To ensure fairness, the results in each group are arranged randomly. For each group, the viewers are asked to select only one result which they like most by considering the three criterions together.\n\nThe evaluation result is shown in Table \\ref{tab:statistics}. It is clear that our rain removal results are favored by a majority of viewers (56.50\\%).\n\n\\begin{figure*}[!htb]\n\\centering\n\\begin{minipage}{0.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test57}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test57_I_nd_by_Ding}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test57_I_nd_by_Kang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test57_I_nd_by_luo}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test57_I_nd_by_li}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test57_I_nd_by_Fu}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test57_I_nd_by_Zhang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test57_I_nd}}\n\\end{minipage}\n\\vspace{0.5mm}\n\\vfill\n\\begin{minipage}{0.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain9}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain9_I_nd_by_Ding}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain9_I_nd_by_Kang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain9_I_nd_by_luo}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain9_I_nd_by_li}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain9_I_nd_by_Fu}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain9_I_nd_by_Zhang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain9_I_nd}}\n\\end{minipage}\n\\vspace{0.5mm}\n\\vfill\n\\begin{minipage}{0.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain17}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain17_I_nd_by_Ding}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain17_I_nd_by_Kang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain17_I_nd_by_luo}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain17_I_nd_by_li}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain17_I_nd_by_Fu}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain17_I_nd_by_Zhang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain17_I_nd}}\n\\end{minipage}\n\\vspace{0.5mm}\n\\vfill\n\\begin{minipage}{0.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain23}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain23_I_nd_by_Ding}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain23_I_nd_by_Kang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain23_I_nd_by_luo}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain23_I_nd_by_li}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain23_I_nd_by_Fu}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain23_I_nd_by_Zhang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain23_I_nd}}\n\\end{minipage}\n\\vspace{0.5mm}\n\\vfill\n\\begin{minipage}{0.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain36}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain36_I_nd_by_Ding}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain36_I_nd_by_Kang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain36_I_nd_by_luo}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain36_I_nd_by_li}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain36_I_nd_by_Fu}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain36_I_nd_by_Zhang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain36_I_nd}}\n\\end{minipage}\n\\vspace{0.5mm}\n\\vfill\n\\begin{minipage}{0.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain53}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain53_I_nd_by_Ding}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain53_I_nd_by_Kang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain53_I_nd_by_luo}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain53_I_nd_by_li}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain53_I_nd_by_Fu}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain53_I_nd_by_Zhang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain53_I_nd}}\n\\end{minipage}\n\\vspace{0.5mm}\n\\vfill\n\\begin{minipage}{0.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain73}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain73_I_nd_by_Ding}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain73_I_nd_by_Kang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain73_I_nd_by_luo}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain73_I_nd_by_li}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain73_I_nd_by_Fu}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain73_I_nd_by_Zhang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/rain73_I_nd}}\n\\end{minipage}\n\\vspace{0.5mm}\n\\vfill\n\\begin{minipage}{0.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test8}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test8_I_nd_by_Ding}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test8_I_nd_by_Kang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test8_I_nd_by_luo}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test8_I_nd_by_li}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test8_I_nd_by_Fu}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test8_I_nd_by_Zhang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test8_I_nd}}\n\\end{minipage}\n\\vspace{0.5mm}\n\\vfill\n\\begin{minipage}{0.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test58}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test58_I_nd_by_Ding}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test58_I_nd_by_Kang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test58_I_nd_by_luo}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test58_I_nd_by_li}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test58_I_nd_by_Fu}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test58_I_nd_by_Zhang}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test58_I_nd}}\n\\end{minipage}\n\\vspace{0.5mm}\n\\vfill\n\\begin{minipage}{0.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test67}}\n\\centerline{(a)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test67_I_nd_by_Ding}}\n\\centerline{(b)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test67_I_nd_by_Kang}}\n\\centerline{(c)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test67_I_nd_by_luo}}\n\\centerline{(d)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test67_I_nd_by_li}}\n\\centerline{(e)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test67_I_nd_by_Fu}}\n\\centerline{(f)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test67_I_nd_by_Zhang}}\n\\centerline{(g)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{.115\\linewidth}\n\\centering{\\includegraphics[width=.995\\linewidth]{images\/test67_I_nd}}\n\\centerline{(h)}\n\\end{minipage}\n\\vspace{0.5mm}\n\\caption{(a) Original rain images. (b) Results by Ding \\emph{et al.} in \\cite{Ding_2015_MTA}.\n(c) Results by Chen \\emph{et al.} in \\cite{Chen_2014_CSVT}. (d) Results by Luo \\emph{et al.} in \\cite{Luo_2015_ICCV}.\n(e) Results by Li \\emph{et al.} in \\cite{Li_2016_CVPR}.\n(f) Results by Fu \\emph{et al.} in \\cite{Fu_2017_CVPR}. (g) Results by Zhang \\emph{et al.} in \\cite{Zhang_2018_CVPR}.\n(h) Results by our proposed method.}\n\\label{fig:result_compare1}\n\\end{figure*}\n\n\\subsection{Results analysis}\n\nIn this subsection, we try to analyze the rain removal effectiveness\nof different methods. The advantages or disadvantages of different\nmethods are discussed according to the rain-removed results that\nis obtained by applying the selected methods on practical rain images. Notice that some images employed in these experiments have a large size so that the rain streaks look tenuous.\n\n\\textbf{Method by Ding \\emph{et al.}:} The first row of Fig. \\ref{fig:result_compare} shows a rain image with slight rain streaks. The result by Ding \\emph{et al.}, as shown in the second column, seems to have removed the rain streaks quite well at the first glance. However, when the picture is zoomed in, it is found that a lot of non-rain details are lost. To verify this point more clearly, a small part of the rain picture and its corresponding rain-removed results by the selected methods are shown in the second row of Fig. \\ref{fig:result_compare}. Now, it becomes obvious that some details of tree leaves have been removed together with the rain streaks. This is due to the threshold of the $L_0$ filters used in \\cite{Ding_2015_MTA}: some non-rain objects whose size is relatively small would be mistreated as rain streaks and get removed.\n\nThe third row is still a slight rain image, but the rain streaks are denser. When zoomed in, the details-losing becomes more apparent. For heavy rain streaks in images shown in the sixth, seventh, eighth rows of Fig. \\ref{fig:result_compare}, they can not be removed by the method of Ding \\emph{et al.}. This is because the size of rain streaks in these images is beyond the preset threshold of $L_0$ filters. If we set the threshold larger, the rain streaks with wide size will be removed. However more details in the images will also be removed at the same time. For the light rain images which have less tenuous details (the third, forth and sixth rows in Fig. \\ref{fig:result_compare1}), this method has satisfactory rain removal effectiveness.\n\n\\textbf{Method by Chen \\emph{et al.}:} The results by Chen \\emph{et al.} are shown in the third column. For the light rain images that have less subtle details (such as the image in the fifth row of Fig. \\ref{fig:result_compare}, the third, forth and sixth rows in Fig. \\ref{fig:result_compare1}), this method can obtain good rain removal results. However, if the rain images possess subtle details (such as the first, third and forth rows of Fig. \\ref{fig:result_compare}), the detail-losing and image-blurring are inevitable. The reason is that the HOG descriptor used here cannot separate rain streaks and subtle details well.\nThe lost details can be seen clearly in third image of the second row of Fig. \\ref{fig:result_compare}, which is obtained by zooming in a part of the image in the first row. Moreover, low-pass filter cannot filter bright\nrain streaks completely. Consequently, the method by Chen \\emph{et al.} can not deal with heavy rain images (such as the images in the sixth and seventh rows of Fig. \\ref{fig:result_compare}).\n\n\\textbf{Method by Luo \\emph{et al.}:} The results by Luo \\emph{et al.} are in the forth column of Fig. \\ref{fig:result_compare} and \\ref{fig:result_compare1}. Obviously, this method can not remove rain streaks well. This is due to the discrimination of the sparse code used in this work, which is not good to separate a rain image into the rain layer\nand non-rain layer. However, this method can make the intensity of rain streaks a little weaker. Hence, for tenuous rain streaks considered in their work, their method seems to have removed rain well. When rain steaks become brighter or wider, they can not be removed well.\n\n\\textbf{Method by Li \\emph{et al.}:} Li \\emph{et al.} used priors for both background and rain layer (which are based on Gaussian mixture models) to remove rain streaks. We show the results by this method in the fifth column. For the images that have little subtle details (the fifth, seventh, ninth, and tenth image in Fig. \\ref{fig:result_compare}, as well as\nthe third, forth, and sixth images in Fig. \\ref{fig:result_compare1}),\nthis method can obtain good rain-removal effectiveness. However, for rain images that have subtle details (e.g., the first, third and forth of Fig. \\ref{fig:result_compare}), many subtle details are lost. This point can be seen clearly in the fifth image of the second row of Fig. \\ref{fig:result_compare}.\nAs mentioned above, this image is part of the image in the first row that is zoomed in to see the details more clearly.\n\n\\textbf{Method by Fu \\emph{et al.}:} For the majority of selected practical images, this work can achieve good results.\nBut there are still some defects. The first apparent one is that this method can cause slight blur for some rainy images,\nsuch as the second and eighth images in Fig. \\ref{fig:result_compare}. That is also the reason that this method has\nlower PNSR\/SSIM values than ours for the images in Fig. \\ref{fig:result_render_compare}. The second is the generalization.\nThis method can not handle some rain images. For example, the seventh and eighth images in Fig. \\ref{fig:result_compare}, the rain streaks are left in the results.\n\n\\textbf{Method by Zhang \\emph{et al.}:} The work by Zhang \\emph{et al.} is the most recent work\npublished on CVPR. We can see that this method faces the similar problems as the work by Fu \\emph{et al.} \\cite{Fu_2017_CVPR}.\nThe details lose seriously for some practical images, especially, the images with slim details (the last one in Fig.\n\\ref{fig:result_compare} and \\ref{fig:result_compare1} separately, you can enlarge the images in this paper to see clear).\nThis method also can not deal with some rainy images, and some apparent rain streaks are left in some rain-removed results.\n\n\\textbf{Our work:} The results by our proposed method are shown in the eighth column. Compared with other traditional rain removal works, our proposed approach achieves better rain removal results. When compared with deep learning based works,\nour method produces comparable results for majority of rain images. But for some other rain images, the selected deep learning\nbased methods can not handle well and better rain-removed result are obtained by our method.\nBecause our method acquires relatively more accurate locations, the remaining image details can be preserved well. Besides, the image quasi-sparsity prior offers a robust tool to the image recovery. Hence, better PSNR\/SSIM values and good visual quality have been achieved in our proposed method.\n\n\\subsection{Limitations}\n\nBy experiments, our method can deal with majority of rain images.\nHowever, every algorithm has its drawbacks, so does our method. For some images with non-rain objects that are very similar to the shape and color of rain streaks, some mis-detections are inevitable. This will result in the loss of some useful information. Besides, when the rain is very heavy, the rain streak will be combined to produce fog. A shallow thought for this situation is that we can remove the rain streaks by our method first, and a dehaze method can be used to remove haze which is caused by heavy rain. We note that this situation has been discussed in a very recent work in \\cite{Li_2016_CVPR}. We will continue to work on this situation in our future work. Another future work is to further improve the rain detection.\n\n\\section{Conclusions}\n\\label{sec:Conclusion}\nIn this paper, we have proposed a new rain streaks detection and removal method from a single color image. Our results suggest that using a morphological image processing to extract connected components and quantifying the characteristics of extracted connected components by\nthe principal component analysis (PCA) are effective in detecting rain streaks. Once rain streaks are detected, we employ an image sparsity prior to accurately decompose a rain image into the rain layer and non-rain layer, which has also been proven to be effective. In addition, quantitative (objective) evaluations and an user study (subjective) validate the overall rain removal effectiveness of our method, which outperforms four selected traditional methods and is comparable to the most recent deep learning based works that are all proposed very recently and widely regarded as the state-of-the-art.\n\n\n\n\n\n\n\n\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{introduction}\n\n\nIn general relativity and cosmology, our knowledge about spatially\nhomogeneous cosmological models has increased substantially over the\nyears, and we are able to say that, for a large number of models,\nthe qualitative behaviour of solutions is now well understood;\nsee~\\cite{waiell97} for an overview. The majority of results,\nhowever, concerns solutions of the Einstein equations coupled to a\nperfect fluid (usually with a linear equation of state). It is thus\nimportant to note that these results are in general not robust,\ni.e., not structurally stable, under a change of the matter model;\nsignificant changes of the qualitative behaviour of solutions occur,\nfor instance, for collisionless matter.\n\nSeveral fundamental results on spatially homogeneous diagonal models of\nBianchi type~I with collisionless matter have been obtained in~\\cite{ren96}.\nDiagonal locally rotationally symmetric (LRS) models have been investigated\nsuccessfully by using dynamical systems methods, see~\\cite{rentod99} for the\ncase of massless particles, and~\\cite{renugg00,ren02} for the massive case.\nIn particular, solutions have been found whose qualitative behaviour is\ndifferent from that of any perfect fluid model of the same Bianchi type.\n\nThe purpose of this article is to re-investigate the diagonal (non-LRS)\nBianchi type~I models with collisionless matter. Our analysis is based on\ndynamical systems techniques, which enables us to obtain a more detailed\npicture of the global dynamics than the one previously given.\nIn particular, we show that the dynamical behaviour toward\nthe past singularity of the collisionless matter model\ndiffers considerably from the Bianchi type~I perfect fluid model.\n\nThe outline of the paper is as follows. In Section~\\ref{einsteinvlasov}\nwe reformulate Einstein's field equations for the diagonal Bianchi\ntype~I case with collisionless matter as a reduced \ndimensionless dynamical system on a compact state space. In\nSection~\\ref{invfixed} we give the fixed points of the system and list\nand discuss a hierarchy of invariant subsets of the state space,\nwhich is associated with a hierarchy of monotone functions. In\nSection~\\ref{locglo} we first present the results of the local\ndynamical systems analysis; subsequently, we focus on\nthe analysis of the global dynamics: we establish two theorems that\nformulate the future and past asymptotic behaviour of solutions,\nrespectively. As regards the future asymptotics we show that all\nmodels isotropize asymptotically toward the future. The past\nasymptotic behaviour is more complicated since there exists several\ntypes of past asymptotic behaviour; in particular we establish that\nthe past attractive set resides on a set that contains a so-called\nheteroclinic network. The proof of the theorems is based on methods\nfrom global dynamical systems analysis; in particular, we exploit\nthe hierarchy of monotone functions in conjunction with the\nmonotonicity principle. Finally, in Section~\\ref{conc} we conclude with\nsome remarks about our results and their implications.\nAppendix~\\ref{dynsys} provides a brief introduction to relevant\nbackground material from the theory of dynamical systems; in\nparticular we cite the monotonicity principle. The proofs of some of\nthe statements in the main text are given in Appendix~\\ref{FRWLRS}\nand~\\ref{futureproof}. In Appendix~\\ref{Si0space} we discuss the\nphysical interpretation of one of the most important boundaries of\nour state space formulation.\n\n\n\\section{The reflection-symmetric Bianchi type~I Einstein-Vlasov system}\n\\label{einsteinvlasov}\n\nIn a spacetime with Bianchi type~I symmetry the spacetime metric can\nbe written as\n\\begin{equation}\nd s^2 = -d t^2 + g_{i j}(t) d x^i d x^j\\:,\\quad i,j=1,2,3\\:,\n\\end{equation}\nwhere $g_{i j}$ is the induced Riemannian metric on the spatially\nhomogeneous surfaces $t=\\mathrm{const}$. Since the metric is\nconstant on $t=\\mathrm{const}$, it follows that the Ricci tensor of\n$g_{i j}$ vanishes. Einstein's equations, in units $G=1=c$,\ndecompose into the evolution equations,\n\\begin{subequations}\\label{einsteinvlasovsystem}\n\\begin{equation}\\label{evolution}\n\\partial_t g_{i j} = -2 k_{i j} \\:,\\quad\n\\partial_t k^i_{\\:\\, j} = \\mathrm{tr}\\hspace{0.15ex} k\\: k^i_{\\:\\, j} - 8 \\pi T^i_{\\:\\, j} +\n4 \\pi \\delta^i_{\\:\\, j} ( T^k_{\\:\\, k} -\\rho) - \\Lambda \\delta^i_{\\:\\, j}\\:,\n\\end{equation}\nand the Hamiltonian and momentum constraint\n\\begin{equation}\\label{constraints} (\\mathrm{tr}\\hspace{0.15ex} k)^2\n- k^i_{\\:\\, j} k^j_{\\:\\, i} - 16 \\pi \\rho -2 \\Lambda = 0\\:,\\qquad\nj_k = 0\\:.\n\\end{equation}\nHere, $k_{i j}$ denotes the second fundamental form of the surfaces $t=\\mathrm{const}$.\nThe matter variables are defined as components of the energy-momentum\ntensor $T_{\\mu\\nu}$ ($\\mu=0,1,2,3$), according to $\\rho = T_{00}$,\n$j_k= T_{0k}$; $T_{i j}$ denotes the spatial components.\nThe cosmological constant $\\Lambda$ is set to zero in the following;\nthe treatment of the case $\\Lambda>0$ is straightforward once the case $\\Lambda=0$ has been solved,\ncf.~the remarks in the conclusions.\n\nIn this paper we consider collisionless matter (Vlasov matter),\ni.e., an ensemble of freely moving particles described by a\nnon-negative distribution function $f$ defined on the mass shell bundle\n$PM\\subseteq TM$ of the spacetime; for simplicity we consider\nparticles with equal mass $m$.\nThe spacetime coordinates $(t,x^i)$\nand the spatial components $v^i$ of the four-momentum $v^\\mu$\n(measured w.r.t.\\ $\\partial\/\\partial x^\\mu$) provide local\ncoordinates on $PM$ so that $f = f(t,x^i,v^j)$. Compatibility with\nBianchi type~I symmetry forces the distribution function $f$ to\nbe homogeneous, i.e., $f = f(t, v^j)$. The evolution equation for\n$f$ is the Vlasov equation (the Liouville equation)\n\\begin{equation}\\label{Vlasovequation}\n\\partial_t f + \\frac{v^j}{v^0} \\partial_{x^j} f -\n\\frac{1}{v^0} \\Gamma^j_{\\mu\\nu} v^\\mu v^\\nu \\partial_{v^j} f  =\n\\partial_t f + 2 k^j_{\\:\\, l} v^l \\partial_{v^j} f = 0 \\:.\n\\end{equation}\nThe energy-momentum tensor associated with the distribution\nfunction $f$ is given by\n\\[\nT^{\\mu\\nu} = \\int f v^\\mu v^\\nu \\mathrm{vol}_{PM}\\:,\n\\]\nwhere $\\mathrm{vol}_{PM} =  (\\det g)^{1\/2} v_0^{-1} d v^1 d v^2 d\nv^3$ is the induced volume form on the mass shell; $v_0$ is\nunderstood as a function of the spatial components, i.e., $v_0^2 =\nm^2 + g_{i j} v^i v^j$. The components $\\rho$, $j_k$, and $T_{i j}$,\nwhich enter in~\\eqref{evolution} and~\\eqref{constraints} can thus be\nwritten as\n\\begin{align}\n\\label{matterrho}\n\\rho & = \\int f \\left(m^2 + g^{i j} v_i v_j\\right)^{1\/2}\n(\\det g)^{-1\/2} d v_1 d v_2 d v_3\\:, \\\\\n\\label{matterj}\nj_k & = \\int  f v_k (\\det g)^{-1\/2} d v_1 d v_2 d v_3 \\:,\\\\\n\\label{matterT}\nT_{i j} & = \\int f v_i v_j\n\\left(m^2 + g^{k l} v_k v_l\\right)^{-1\/2}\n(\\det g)^{-1\/2} d v_1 d v_2 d v_3 \\:.\n\\end{align}\n\\end{subequations}\nThe Einstein-Vlasov system~\\eqref{einsteinvlasovsystem} is usually\nconsidered for particles of mass $m>0$, however, the system also\ndescribes massless particles if we set $m=0$. (For a detailed\nintroduction to the Einstein-Vlasov system we refer to~\\cite{and05}\nand~\\cite{ren04}.)\n\nThe general spatially homogeneous solution\nof the Vlasov equation~(\\ref{Vlasovequation}) in\nBianchi type~I is\n\\begin{equation}\\label{fisf0}\nf(t,v^i) = f_0(v_i)\\:,\n\\end{equation}\nwhere the $v_i$ are the covariant components of the momenta and\n$f_0$ is an arbitrary non-negative function, see~\\cite{maamah90}.\n(By inserting~(\\ref{fisf0})\ninto~(\\ref{Vlasovequation}) and using that $v_i = g_{i j}(t) v^j$\nit is easy to check that $f_0(v_i)$ is a solution.)\nThe momentum constraint in~(\\ref{constraints}) then reads\n\\begin{equation}\\label{momcons}\n\\int f_0(v_i) v_k  d v_1 d v_2 d v_3 = 0\\:.\n\\end{equation}\nHenceforth, for simplicity, $f_0$ is assumed to be compactly supported.\n\nThere exists a subclass of Bianchi type~I Einstein-Vlasov models\nthat is naturally associated with the constraint~\\eqref{momcons}:\nthe class of ``reflection-symmetric'' (or ``diagonal'') models.\nThe following symmetry conditions are imposed on the initial data:\n\\begin{subequations}\\label{reflsymm}\n\\begin{equation}\\label{reflsymmf0}\nf_0(v_1, v_2,v_3) = f_0(-v_1,-v_2,v_3) = f_0(-v_1,v_2,-v_3) = f_0(v_1,-v_2,-v_3)\\:,\n\\end{equation}\n\\begin{equation}\\label{reflsymmgk}\ng_{i j}(t_0)\\:, k_{i j}(t_0) \\quad\\text{diagonal}\\:.\n\\end{equation}\n\\end{subequations}\nThese conditions ensure that $T_{i j}(t_0)$ is diagonal, hence $g_{i\nj}$, $k_{i j}$, and $T_{i j}$ are diagonal for all times by~(\\ref{einsteinvlasovsystem}). \nIn the present paper, we will be\nconcerned with this class of reflection-symmetric models.\n\nThe Einstein-Vlasov system~\\eqref{einsteinvlasovsystem} thus reduces\nto a system for six unknowns, the diagonal components of the metric\n$g_{i i}(t)$ and the second fundamental form $k^i_{\\:\\, i}(t)$ (no\nsummation). The equations are~\\eqref{evolution} and the Hamiltonian\nconstraint in~\\eqref{constraints}. The initial data consists of\n$g_{i i}(t_0)$, $k^i_{\\:\\, i}(t_0)$; in addition we prescribe a\ndistribution function $f_0(v_i)$ that provides the source terms in\nthe equations via~\\eqref{matterrho} and~\\eqref{matterT}.\n\nIn the following we reformulate the Einstein-Vlasov system as a\ndimensionless system on a compact state space.\nTo that end we introduce new variables and modified matter quantities. Let\n\\begin{equation}\\label{hx}\nH := -\\frac{\\mathrm{tr}\\hspace{0.15ex} k}{3}\\:, \\quad\\qquad x := \\sum_i g^{i i}\\:,\n\\end{equation}\nand define the dimensionless variables\n\\begin{subequations}\n\\begin{align}\n\\label{defdimless}\n& s_i := \\frac{g^{i i}}{x}\\; , & & \\Sigma_i := -\\frac{k^i_{\\:\\, i}}{H} - 1\\; ,\n& z & := \\frac{m^2}{m^2 + x}\\:, \\\\[1ex]\n\\text{where} \\qquad & \\!\\sum_i s_i =1\\; , & & \\sum_i\\Sigma_i = 0\\:.\n\\end{align}\n\\end{subequations}\nThe transformation from the variables $(g_{ii}, k^i_{\\:\\, i})$ to\n$(s_i,\\Sigma_i, x, H)$, where $(s_i,\\Sigma_i)$ are subject to the\nabove constraints, is one-to-one. (Note that $x$ can be obtained\nfrom $z$ when $m>0$.) By distinguishing one direction ($1$, $2$, or\n$3$), one can decompose the $s_i$ and simultaneously introduce a\ntrace-free adaption of the shear to new $\\Sigma_\\pm$ variables as is\ndone in, e.g.,~\\cite{waiell97}; however, since Bianchi type~I does\nnot have a preferred direction we will refrain from doing so here.\n\nNext we replace the matter quantities $\\rho$, $T^i_{\\:\\, i}$ (no\nsummation) by the dimensionless quantities\n\\begin{equation}\n\\Omega :=\\frac{8\\pi\\rho}{3 H^2}\\,,\\qquad w_i := \\frac{T^i_{\\:\\,\ni}}{\\rho}\\,,\\qquad w := \\frac{1}{3} \\sum_i w_i = \\frac{1}{3}\n\\frac{\\sum_i T^i_{\\:\\, i}}{\\rho}\\,.\n\\end{equation}\nExpressed in the new variables, $w_i$ can be written as\n\\begin{equation}\\label{omegai}\nw_i = \\frac{(1-z) s_i {\\displaystyle\\int} f_0\\, v_i^2\n\\left[z+(1-z) \\sum_k s_k \\, v_k^2\\right]^{-1\/2} d v_1 d v_2 d v_3}%\n{{\\displaystyle\\int} f_0 \\left[z+(1-z) \\sum_k s_k \\,\nv_k^2\\right]^{1\/2} d v_1 d v_2 d v_3}\\:.\n\\end{equation}\n\nFinally, let us introduce a new dimensionless time variable $\\tau$ defined by\n\\begin{equation}\n\\partial_\\tau = H^{-1}\\partial_t\\:;\n\\end{equation}\nhenceforth a prime denotes differentiation w.r.t.\\ $\\tau$.\n\nWe now rewrite the Einstein-Vlasov equations as a set of dimensional\nequations that decouple for dimensional reasons and a reduced system\nof dimensionless coupled equations on a compact state space. The\ndecoupled dimensional equations are\n\\begin{subequations}\n\\begin{align}\n\\label{Heq}\nH^\\prime &= -3 H \\left[1 -\\frac{\\Omega}{2} (1-w)\\right]\\\\\n\\label{xeq}\nx^\\prime &= -2 x \\left(1 + \\sum_k \\Sigma_k s_k \\right)\\:.\n\\end{align}\n\\end{subequations}\nThe reduced dimensionless system consists of the Hamiltonian\nconstraint, cf.~\\eqref{constraints},\n\\begin{equation}\\label{omega}\n1- \\Sigma^2-\\Omega = 0\\:, \\qquad\\text{where}\\quad \\Sigma^2 :=\n\\sfrac{1}{6} \\sum_k \\Sigma_k^2\\:,\n\\end{equation}\nwhich we use to solve for $\\Omega$,\nand a coupled system of evolution equations\n\\begin{subequations}\\label{eq}\n\\begin{align}\n\\label{Sigeq}\n\\Sigma_i^\\prime & = -3 \\Omega \\left[ \\frac{1}{2} (1-w) \\Sigma_i -(w_i - w)\\right]\\\\\n\\label{seq}\ns_i^\\prime & = -2 s_i \\left[\\Sigma_i - \\sum_k \\Sigma_k s_k \\right] \\\\\n\\label{zeq}\nz^\\prime & = 2 z\\,(1 - z)\\left(\\,1 + \\sum_k s_k \\, \\Sigma_k\\,\\right)\\:.\n\\end{align}\n\\end{subequations}\nIn the massive case $m>0$ the decoupled equation for $x$ is\nredundant since the equation for $z$ is equivalent. In the massless\ncase $m=0$ we have $z=0$; hence, although the equation for $x$ does\nnot contribute to the dynamics, $x$ is needed in order to\nreconstruct the spatial metric from the new variables.\n\nThe dimensionless dynamical system~(\\ref{eq}) together with the\nconstraint~\\eqref{omega} describes the full dynamics of the\nEinstein-Vlasov system of Bianchi type~I. In the massive case the state space\nassociated with this system is the space of the variables\n$\\{(\\Sigma_i, s_i, z)\\}$, i.e.,\n\\begin{equation}\\label{statespace}\n\\mathcal{X} :=\n\\left\\{(\\Sigma_i, s_i,z)\\:\\big|\\: \\left(\\Sigma^2 < 1\\right)\n\\wedge \\left(s_i > 0\\right) \\wedge \\left(0< z < 1\\right)\\right\\}\\:,\n\\end{equation}\nwhere the $s_i$ and $\\Sigma_i$ are subject to the constraints\n$\\sum_k\\, s_k=1\\:,\\,\\sum_k\\,\\Sigma_k = 0$. (The inequalities for\n$s_i$ and $\\Sigma_i$ follow from the definition~\\eqref{defdimless}\nand the constraint~\\eqref{omega}, respectively.) The state space\n$\\mathcal{X}$ is thus five-dimensional.\n\nIt will turn out eventually that all solutions asymptotically\napproach the boundaries of $\\mathcal{X}$: $z=0$, $z=1$, $s_i=0$,\n$\\Omega = 0$ ($\\Leftrightarrow \\Sigma^2 =1$). This suggests to\ninclude these sets in the analysis, whereby we obtain a compact\nstate space $\\bar{\\mathcal{X}}$.\n\nThe equations on the invariant subset $z=0$ of $\\bar{\\mathcal{X}}$\nare identical to the coupled dimensionless system in the case of\nmassless particles $m=0$. We will therefore refer to the subset\n$z=0$ as the massless subset; it represents the four-dimensional\nstate space for the massless case.\n\nWe conclude this section by looking at some variables in more\ndetail. The inequality $\\Sigma^2 \\leq 1$ together with the\nconstraint $\\sum_k \\Sigma_k =0$ results in $|\\Sigma_i| \\leq 2$ for\nall $i$. Note that equality is achieved when\n$(\\Sigma_1,\\Sigma_2,\\Sigma_3) = (\\pm 2 ,\\mp 1, \\mp 1)$ and\npermutations thereof, cf.~Figure~\\ref{kasneri}. The matter quantities\nsatisfy\n\\begin{equation}\\label{wrelations}\n0 \\leq w \\leq \\textfrac{1}{3} \\:, \\qquad 0 \\leq w_i \\leq 3 w \\leq 1\\:.\n\\end{equation}\nThe equalities hold at the boundaries of the state space: $w =0$ iff\n$z=1$, and $w = \\textfrac{1}{3}$ iff $z=0$; $w_i = 0$ iff $s_i = 0$,\nand $w_i = 3 w$ iff $s_i =1$ (provided that $z<1$; for $z=1$, $0 =\nw_i = 3 w = 0$).\n\nThere exists a number of useful auxiliary equations that complement the system~\\eqref{eq}:\n\\begin{align}\n\\label{omegaeq}\n\\Omega^\\prime & =\n\\Omega\\, \\left[\\,3(1-w)\\Sigma^2 - \\sum_k w_k\\,\\Sigma_k\\,\\right] \\:,\\\\\n\\label{rhoeq} \\rho' & =  -\\rho\\, [\\,3(1+w) + \\sum_k w_k\\,\\Sigma_k\\,]\n\\leq -2\\rho\\:.\n\\end{align}\nThe inequality in~\\eqref{rhoeq} follows by using $\\Sigma_i \\geq -2$\n$\\forall i$ and~\\eqref{wrelations}. This shows that $\\rho$ increases\nmonotonically toward the past which yields a matter singularity,\ni.e., $\\rho\\rightarrow\\infty$ for $\\tau\\rightarrow -\\infty$. It is\noften beneficial to consider the equations of the original variables\nas auxiliary equations, e.g., $(g^{ii})^\\prime=  -2g^{ii}\\,(1 +\n\\Sigma_i)$.\n\n\n\n\\section{Fixed points, invariant subsets, and monotone functions}\n\\label{invfixed}\n\n\\subsection{Fixed points}\n\\label{fixed}\n\nThe dynamical system~\\eqref{eq} possesses a number of fixed points,\nall residing on the boundaries of the state space;\nsee~Table~\\ref{fixtab}.\n\n\\begin{table}[ht]\n\\begin{center}\n\\begin{tabular}{|c|c|}\\hline\nFixed point set & defined by \\\\ \\hline\nFS$^1$ & $z=1$, $\\Sigma_j = 0 \\:\\,\\forall j$ \\\\\nKC$_i^1$ & $z=1$, $\\Sigma^2 =1$, $s_i=1$, $s_j = 0\\:\\,\\forall j \\neq\ni$\n\\\\ \\hline\nTS$_i$ & $0\\leq z\\leq 1$, $\\Sigma_i = 2$, $\\Sigma_j =-1 \\:\\,\\forall\nj \\neq i$, $s_i=0$  \\\\ \\hline\nF$^0$ & $z=0$, $\\Sigma_j = 0 \\:\\,\\forall j$, $w_j = 1\/3 \\:\\,\\forall j$  \\\\\nD$_i^0$ & $z=0$, $s_i = 0$, $\\Sigma_i =-1$, $\\Sigma_{j}=1\/2 =w_j\\:\\,\n\\forall j\\neq i$  \\\\\nQL$_i^0$ & $z=0$, $\\Sigma_i = -2$, $\\Sigma_j = 1 \\:\\,\\forall j \\neq\ni$,\n$s_i=0$  \\\\\nKC$_i^0$ & $z=0$, $\\Sigma^2 =1$, $s_i=1$, $s_j = 0 \\:\\,\\forall j\n\\neq i$  \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{The fixed point sets. The range of the index $i$ is always\n$i=1\\ldots 3$. The superscript denotes the value of $z$; the first\nkernel letter describes the type of fixed point set; if there is no\nsecond kernel letter the fixed point set is just a point; if there\nis a second kernel letter this letter denotes the dimensionality and\ncharacter of the set --- S refers to surface, L stands for line, and\nC for circle.} \\label{fixtab}\n\\end{table}\n\n\\begin{figure}[Ht]\n\\begin{center}\n\\psfrag{Ti1}[cc][cc][1][0]{$\\text{T}^0_{i1}$}\n\\psfrag{Ti2}[cc][cc][1][0]{$\\text{T}^0_{i2}$}\n\\psfrag{Ti3}[cc][cc][1][0]{$\\text{T}^0_{i3}$}\n\\psfrag{Qi1}[cc][cc][1][0]{$\\text{Q}^0_{i1}$}\n\\psfrag{Qi2}[cc][cc][1][0]{$\\text{Q}^0_{i2}$}\n\\psfrag{Qi3}[cc][cc][1][0]{$\\text{Q}^0_{i3}$}\n\\psfrag{S1}[cc][cc][1.2][0]{$\\Sigma_1$}\n\\psfrag{S2}[cc][cc][1.2][0]{$\\Sigma_2$}\n\\psfrag{S3}[cc][cc][1.2][0]{$\\Sigma_3$}\n\\psfrag{S1m}[cc][cc][1][-90]{$\\Sigma_1=-1$}\n\\psfrag{S2m}[cc][cc][1][30]{$\\Sigma_2=-1$}\n\\psfrag{S3m}[cc][cc][1][-30]{$\\Sigma_3=-1$}\n\\psfrag{S1p}[cc][cc][0.6][90]{$\\Sigma_1=1$}\n\\psfrag{S2p}[cc][cc][0.6][30]{$\\Sigma_2=1$}\n\\psfrag{S3p}[cc][cc][0.6][-30]{$\\Sigma_3=1$}\n\\psfrag{s1}[cc][cc][1.0][90]{$\\longleftarrow\\: s_1 =0\\:\n\\longrightarrow$} \\psfrag{s2}[cc][cc][1.0][35]{$\\longleftarrow \\:s_2\n=0 \\:\\longrightarrow$} \\psfrag{s3}[cc][cc][1.0][-35]{$\\longleftarrow\n\\:s_3 =0 \\:\\longrightarrow$}\n\\includegraphics[width=0.6\\textwidth]{kasnercircle.eps}\n\\caption{The disc $\\Sigma^2\\leq 1$ and the Kasner circle $\\text{KC}_i^0$.} \\label{kasneri}\n\\end{center}\n\\end{figure}\n\n$\\text{FS}^1$ is a surface of fixed points that correspond to the\nflat isotropic dust solution. The circles $\\text{KC}_i^{1,0}$\nconsist of fixed points with constant $\\Sigma_ i$, satisfying\n$\\Sigma^2=1,\\sum_k\\,\\Sigma_k=0$, see Figure~\\ref{kasneri}; these fixed\npoints correspond to Kasner solutions. The fixed points on\n$\\text{TS}_i$ are associated with the Taub representation of the\nflat Minkowski spacetime. The intersection of $\\text{TS}_i$ with\n$(z=0)$ yields a line of fixed points which we denote\n$\\text{TL}_i^0$. The fixed points on $\\text{QL}_i^0$ correspond to\nthe non-flat LRS Kasner solutions.\n$\\text{F}^0$ is a fixed point that corresponds to the flat isotropic\nradiation solution. In Appendix~\\ref{FRWLRS} we prove that $\\text{F}^0$\nis well-defined through the equations $w_1 = w_2= w_3 = 1\/3$ (which\nare to be solved for $(s_1,s_2,s_3)\\,$). The location of\n$\\text{F}^0$ depends on the chosen distribution function, since the\nequations $w_1 = w_2= w_3 = 1\/3$ involve $f_0$.\nAnalogously, the equations $w_j = 1\/2$ ($\\forall \\: j\\neq i$) yield\na unique solution, the fixed point D$_i^0$; the location of the\npoint D$_i^0$ also depends on $f_0$. The fixed points D$_i^0$ are\nassociated with a scale-invariant LRS solution (related to a\ndistributional $f_0$; see Appendix~\\ref{Si0space} for details).\n\nThe LRS points on $\\text{KC}_i^0$ play a particularly important role\nin the following, which motivates that they are given individual\nnames. We denote the three Taub points on KC$_i^0$ defined by\n$\\Sigma_j = 2$ (and thus $\\Sigma_l = -1$ $\\forall l\\neq j$) by\n$\\text{T}_{ij}^0$, while we denote the three non-flat LRS point on\nKC$_i^0$ given by $\\Sigma_j = -2$ (and thus $\\Sigma_l = 1$ $\\forall\nl\\neq j$) by Q$_{ij}^0$. The Kasner circles $\\text{KC}_j^0$ and\n$\\text{KC}_k^0$ are connected by the lines $\\text{TL}_i^0$ and\n$\\text{QL}_i^0$; the end points of the line $\\text{TL}_i^0$ are the\nTaub points $\\text{T}_{ji}^0$ and $\\text{T}_{ki}^0$; analogously,\nthe end points of $\\text{QL}_i^0$ are the points $\\text{Q}_{ji}^0$\nand $\\text{Q}_{ki}^0$. (Here, $(i,j,k)$ is an arbitrary permutation\nof $(1,2,3)$.) The remaining points $\\text{T}_{ll}^0$ and\n$\\text{Q}_{ll}^0$ ($l=1\\ldots 3$) do not lie on any of the fixed\npoint sets $\\text{TL}^0_i$ or $\\text{QL}^0_i$. This fixed point\nstructure is depicted in Figure~\\ref{fixedp}.\n\n\\begin{figure}[Ht]\n\\begin{center}\n\\psfrag{TL1}[cc][cc][0.8][0]{$\\text{TL}^0_1$}\n\\psfrag{TL2}[cc][cc][0.8][0]{$\\text{TL}^0_2$}\n\\psfrag{TL3}[cc][cc][0.8][0]{$\\text{TL}^0_3$}\n\\psfrag{QL1}[cc][cc][0.8][0]{$\\text{QL}^0_1$}\n\\psfrag{QL2}[cc][cc][0.8][0]{$\\text{QL}^0_2$}\n\\psfrag{QL3}[cc][cc][0.8][0]{$\\text{QL}^0_3$}\n\\psfrag{T21}[cc][cc][0.6][0]{$\\text{T}^0_{21}$}\n\\psfrag{T22}[cc][cc][0.6][0]{$\\text{T}^0_{22}$}\n\\psfrag{T23}[cc][cc][0.6][0]{$\\text{T}^0_{23}$}\n\\psfrag{Q21}[cc][cc][0.6][0]{$\\text{Q}^0_{21}$}\n\\psfrag{Q22}[cc][cc][0.6][0]{$\\text{Q}^0_{22}$}\n\\psfrag{Q23}[cc][cc][0.6][0]{$\\text{Q}^0_{23}$}\n\\psfrag{T11}[cc][cc][0.6][0]{$\\text{T}^0_{11}$}\n\\psfrag{T12}[cc][cc][0.6][0]{$\\text{T}^0_{12}$}\n\\psfrag{T13}[cc][cc][0.6][0]{$\\text{T}^0_{13}$}\n\\psfrag{Q11}[cc][cc][0.6][0]{$\\text{Q}^0_{11}$}\n\\psfrag{Q12}[cc][cc][0.6][0]{$\\text{Q}^0_{12}$}\n\\psfrag{Q13}[cc][cc][0.6][0]{$\\text{Q}^0_{13}$}\n\\psfrag{T31}[cc][cc][0.6][0]{$\\text{T}^0_{31}$}\n\\psfrag{T32}[cc][cc][0.6][0]{$\\text{T}^0_{32}$}\n\\psfrag{T33}[cc][cc][0.6][0]{$\\text{T}^0_{33}$}\n\\psfrag{Q31}[cc][cc][0.6][0]{$\\text{Q}^0_{31}$}\n\\psfrag{Q32}[cc][cc][0.6][0]{$\\text{Q}^0_{32}$}\n\\psfrag{Q33}[cc][cc][0.6][0]{$\\text{Q}^0_{33}$}\n\\psfrag{KC1}[cc][cc][1.2][0]{$\\text{KC}^0_1$}\n\\psfrag{KC2}[cc][cc][1.2][0]{$\\text{KC}^0_2$}\n\\psfrag{KC3}[cc][cc][1.2][0]{$\\text{KC}^0_3$}\n\\psfrag{s1}[cc][cc][1.0][90]{$\\longleftarrow\\: s_1 =0\\:\n\\longrightarrow$} \\psfrag{s2}[cc][cc][1.0][35]{$\\longleftarrow \\:s_2\n=0 \\:\\longrightarrow$} \\psfrag{s3}[cc][cc][1.0][-35]{$\\longleftarrow\n\\:s_3 =0 \\:\\longrightarrow$}\n\\includegraphics[width=0.8\\textwidth]{fixedpoints9.eps}\n\\caption{A schematic depiction of the fixed points on $z=0$.\nThe underlying structure is the three sides of the $s_i$-triangle $s_1+s_2+s_3 =1$:\neach point represents a disc $\\Sigma^2\\leq 1$; the vertices contain the Kasner circles $\\text{KC}_i^0$.\nBold lines denote the lines of fixed points $\\text{TL}_i^0$, $\\text{QL}_i^0$,\nand $\\mathrm{KC}_i^0$.} \\label{fixedp}\n\\end{center}\n\\end{figure}\n\n\n\n\\subsection{Invariant subsets and monotone functions}\n\\label{inv}\n\nThe dynamical system~\\eqref{eq} possesses a hierarchy of invariant\nsubsets and monotone functions. Since this feature of the dynamical\nsystem will turn out to be of crucial importance in the analysis of\nthe global dynamics, we give a detailed discussion.\n\n$\\mathcal{X}$: On the full (interior) state space $\\mathcal{X}$ define\n\\begin{subequations}\\label{m1eqs}\n\\begin{equation}\\label{m1}\nM_{(1)} = (s_1 s_2 s_3)^{-1\/3} \\frac{z}{1-z} \\:.\n\\end{equation}\nA straightforward computation shows\n\\begin{equation}\nM_{(1)}^\\prime = 2 M_{(1)}\\:,\n\\end{equation}\n\\end{subequations}\ni.e., $M_{(1)}$ is strictly monotonically increasing along orbits in\n$\\mathcal{X}$. Note that $M_{(1)}$ is intimately related to the\nspatial volume density since $M_{(1)} = m^2 \\det(g_{ij})^{1\/3}$.\n\n$\\mathcal{Z}^1$: This subset is characterized by $z=1$. Since $w_i = w =0$,\nthe equations for $s_i$ decouple, and the essential dynamics is\ndescribed by the equations $\\Sigma_i^\\prime = -(3\/2)(1-\\Sigma^2)\n\\Sigma_i$. (Note that these equations are identical to the Bianchi\ntype~I equations for dust --- it is therefore natural to refer to\n$\\mathcal{Z}^1$ as the dust subset.) Explicit solutions for these equations\ncan be obtained by noting that\n$\\Sigma_1\\propto\\Sigma_2\\propto\\Sigma_3$ for all solutions, or by\nusing that $\\Omega^\\prime = 3\\Sigma^2\\,\\Omega$.\n\n$\\mathcal{Z}^0$: This subset is the massless boundary set $z=0$.\nSince $w =1\/3$, the dynamical system~\\eqref{eq} reduces to\n\\begin{equation}\\label{z0eq}\n\\Sigma_i^\\prime = - \\Omega\\,[\\,1+\\Sigma_i -3 w_i\\,]\\:,\\qquad\ns_i^\\prime = -2 s_i \\,[\\,\\Sigma_i - \\sum\\nolimits_k s_k\\, \\Sigma_k \\,]\\:.\n\\end{equation}\nConsider the function\n\\begin{subequations}\\label{m2eqs}\n\\begin{equation}\\label{m2}\nM_{(2)} =\\left(1 -\\Sigma^2\\right)^{-1} (s_1 s_2 s_3)^{-1\/6} \\int f_0\n\\left[ \\sum\\nolimits_k s_k v_k^2\\right]^{1\/2} d v_1 d v_2 d v_3\\:.\n\\end{equation}\nThe derivative is\n\\begin{equation}\nM_{(2)}^\\prime = -2 \\Sigma^2 M_{(2)}\\:,\n\\end{equation}\nwhich yields monotonicity when $\\Sigma^2 \\neq 0$. If $\\Sigma^2 = 0$, then\n\\begin{equation}\nM_{(2)}^\\prime = 0 \\:, \\quad M_{(2)}^{\\prime\\prime} = 0 \\:, \\quad\nM_{(2)}^{\\prime\\prime\\prime} = -6 M_{(2)} \\sum_i \\left(w_i\n-\\textfrac{1}{3}\\right)^2 \\:.\n\\end{equation}\n\\end{subequations}\nHence, $M_{(2)}$ is strictly monotonically decreasing everywhere on\n$z=0$, except at the fixed point $\\text{F}^0$ (for which\n$\\Sigma^2=0$ and $w_1 = w_2 = w_3 = 1\/3$), where $M_{(2)}$ attains a\n(positive) minimum. The latter follows from the fact that\n$(1-\\Sigma^2)^{-1}$ is minimal at the point $\\Sigma_i = 0\n\\:\\,\\forall i$ and that $\\partial M_{(2)}\/\\partial s_i = (2\ns_i)^{-1} [w_i - 1\/3] M_{(2)}$.\n\n$\\mathcal{S}_i$ ($i=1,2,3$): These invariant boundary subsets are defined by\n$s_i=0$ (which yields $w_i=0$). There exists a monotone function on\n$\\mathcal{S}_1$,\n\\begin{equation}\\label{m3}\nM_{(3)}= (s_2 s_3)^{-1\/2}\\,\\frac{z}{1-z}\\, ,\\qquad M_{(3)}^\\prime =\n(2-\\Sigma_1)\\,M_{(3)}\\, ;\n\\end{equation}\nanalogous functions can be obtained on $\\mathcal{S}_2$ and $\\mathcal{S}_3$ through\npermutations.\n\n$\\mathcal{K}$: This boundary subset is the vacuum subset defined by\n$\\Omega=0$ (or equivalently $\\Sigma^2=1$). The $\\Sigma_i$ are\nconstant on this subset, which completely determines the dynamics of\nthe $s_i$ variables (via~\\eqref{seq} or via the auxiliary equation\nfor $g^{ii}$). The Bianchi type~I vacuum solution is the familiar\nKasner solution and we thus refer to $\\mathcal{K}$ as the Kasner subset.\n\nIntersections of the above boundary subsets yield boundary subsets\nof lower dimensions; those that are relevant for the global dynamics\nare discussed in the following.\n\n$\\mathcal{S}_i^0$ and $\\mathcal{S}_i^1$: The intersection between the subset $\\mathcal{S}_i$\nand $\\mathcal{Z}^0$ and $\\mathcal{Z}^1$ yields three-dimensional invariant subsets\n$(s_i=0) \\cap (z=0)$ and $(s_i=0) \\cap (z=1)$, respectively. On\n$\\mathcal{S}_i^0$ there exists a monotonically decreasing function:\n\\begin{equation}\\label{m4}\nM_{(4)} = (1+\\Sigma_i)^2\\:, \\qquad M_{(4)}^\\prime = - 2 \\Omega M_{(4)} \\:.\n\\end{equation}\n\n$\\mathcal{S}_{ij}$: These subsets are defined by setting $s_i =0$ and $s_j\n=0$ ($j\\neq i$), i.e., $\\mathcal{S}_{ij} = \\mathcal{S}_i \\cap \\mathcal{S}_j$. On $\\mathcal{S}_{ij}$,\nwe have $s_k =1$ ($k \\neq i,j$) and $w_k = 3 w$, because $w_i = w_j\n=0$.\n\n$\\mathcal{D}_i^0$: The subsets $\\mathcal{S}_i^0$ admit two-dimensional invariant\nsubsets $\\mathcal{D}_i^0$ characterized by $(z=0) \\cap (s_i=0) \\cap (\\Sigma_i =-1)$.\nOn $\\mathcal{D}_1^0$ consider the function\n\\begin{subequations}\\label{m5eqs}\n\\begin{equation}\nM_{(5)} = \\left(2 + \\Sigma_2 \\Sigma_3 \\right)^{-1} (s_2 s_3)^{-1\/4}\n\\int f_0 \\left[ s_2 v_2^2 + s_3 v_3^2\\right]^{1\/2} d v_1 d v_2 d\nv_3\\:;\n\\end{equation}\nanalogous functions can be defined on $\\mathcal{D}_2^0$ and $\\mathcal{D}_3^0$.\nEqs.~\\eqref{z0eq} imply\n\\begin{equation}\\label{m5d}\nM_{(5)}^\\prime  = -\\textfrac{1}{12}\\, M_{(5)} \\left[ \\left(1 -\n2\\Sigma_2\\right)^2 + \\left(1 - 2\\Sigma_3\\right)^2\\right] \\,,\n\\end{equation}\ni.e., $M_{(5)}$ is strictly monotonically decreasing unless\n$\\Sigma_2 = 1\/2 = \\Sigma_3$. In the special case $\\Sigma_2 = 1\/2 =\n\\Sigma_3$ we obtain\n\\begin{equation}\nM_{(5)}' = 0 \\:, \\quad M_{(5)}^{\\prime\\prime} = 0 \\:, \\quad\nM_{(5)}^{\\prime\\prime\\prime} =\n-\\textfrac{27}{8} M_{(5)} \\left[ \\left(w_2 -\n\\textfrac{1}{2}\\right)^2 +\\left(w_3 -\n\\textfrac{1}{2}\\right)^2\\right] \\:.\n\\end{equation}\n\\end{subequations}\nHence, $M_{(5)}$ is strictly monotonically decreasing everywhere on\n$\\mathcal{D}_1^0$ except for at the fixed point $\\text{D}_1$, for which\n$\\Sigma_2=\\Sigma_3=w_2=w_3=\\frac{1}{2}$, cf.~Table~\\ref{fixtab}. The\nfunction $M_{(5)}$ possesses a positive minimum at $\\text{D}_1$.\nThis is because $(2+\\Sigma_2 \\Sigma_3)^{-1}$ is minimal at the point\n$\\Sigma_2 = \\Sigma_3 = 1\/2$ and $\\partial M_{(5)}\/\\partial s_i = (2\ns_i)^{-1} [w_i - 1\/2] M_{(5)}$ for $i=2,3$.\n\n$\\mathcal{K}^0$: The intersection of the Kasner subset $\\mathcal{K} = (\\Sigma^2 =1)$\nwith the $z=0$ subset yields a 3-dimensional subset, $\\mathcal{K}^0$. This\nsubset will play a prominent role in the analysis of the past\nasymptotic behaviour of solution.\n\nThe remaining subsets we consider are not located at the boundaries\nof the state space, but in the interior; these subsets are invariant\nunder the flow of the dynamical system, if the distribution function\n$f_0$ satisfies certain symmetry conditions.\n\n$\\text{LRS}_i$: We define the subset $\\text{LRS}_1$ of $\\mathcal{X}$\nthrough the equations $\\Sigma_2=\\Sigma_3$, $w_2=w_3$;\n$\\text{LRS}_{2,3}$ are defined analogously. In order for these sets\nto be invariant under the flow of the dynamical system, the\ndistribution function $f_0$ must satisfy conditions that ensure\ncompatibility with the LRS symmetry, see Appendix~\\ref{FRWLRS} for\ndetails. For an orbit lying on $\\text{LRS}_1$, Equation~(\\ref{seq})\nentails that $s_2(\\tau) \\propto s_3(\\tau)$ (where the\nproportionality constant exhibits a dependence on $f_0$, which\nenters through the equation $w_2 =w_3$), and hence $g_{22}\\propto\ng_{33}$; by rescaling the coordinates one can achieve\n$g_{22}=g_{33}$, i.e., a line element in an explicit LRS form.\nHence, the $\\text{LRS}_i$ subsets, if present as invariant subsets,\ncomprise the solutions with LRS geometry.\n\nFRW: If $f_0$ is compatible with an isotropic geometry, see\nAppendix~\\ref{FRWLRS} for details, the one-dimensional subset\ncharacterized by the equations $\\Sigma_i=0$ $\\forall i$ and $w_1\n=w_2=w_3 =w$ is an invariant subset (in fact: orbit), the FRW\nsubset. The equations $\\Sigma_i=0$ yield $s_{i}=\\mathrm{const}$,\nwhereby we obtain a Friedmann-Robertson-Walker (FRW) geometry, since\nthe spatial coordinates can be rescaled so that\n$g_{ij}\\propto\\delta_{ij}$. Note that the location in $s_i$ of the\nFRW subset depends on $f_0$, since the equations $w_1 = w_2= w_3$,\nwhich are to be solved for $(s_1,s_2,s_3)$, involve $f_0$,\ncf.~Appendix~\\ref{FRWLRS}. Remarkably, in the massless case the\nexistence of a FRW solution (which corresponds to the fixed point\n$\\text{F}^0$) does not require any symmetry conditions on $f_0$.\n\n\n\\section{Local and global dynamics}\n\\label{locglo}\n\n\\subsection{Local dynamics}\n\nLet us consider smooth reflection-symmetric\nBianchi type~I Vlasov solutions that approach fixed point sets when\n$\\tau\\rightarrow -\\infty$.\n\n\\begin{theorem}\\label{locthm}\nIn the massive (massless) case there exists\n\\begin{itemize}\n\\item[(a)] a single orbit that approaches (corresponds\nto) $\\mathrm{F}^0$,\n\\item[(b)] three equivalent one-parameter sets of orbits (three single orbits)\nthat approach $\\mathrm{D}_i^0$, $i=1 \\ldots 3$,\n\\item[(c)] one three-parameter\n(two-parameter) set of orbits that approaches $\\mathrm{QL}_1^0$;\n$\\mathrm{QL}_2^0$ and $\\mathrm{QL}_3^0$ yield equivalent sets,\n\\item[(d)] one four-parameter (three-parameter) set of orbits that\napproaches the part of {\\rm KC}$_1^0$ defined by $1<\\Sigma_1<2$;\n{\\rm KC}$_2^0$ and {\\rm KC}$_3^0$ yield equivalent sets.\n\\end{itemize}\n\\end{theorem}\n\n\\proof The statements of the theorem follow from the local stability\nanalysis of the fixed point sets F$^0$, D$_i^0$, $\\text{QL}_i^0$,\n$\\text{KC}_i^0$, when combined with the Hartman-Grobman and the\nreduction theorem, since the fixed points F$^0$, D$_i^0$ are\nhyperbolic and $\\text{QL}_i^0$, $\\text{KC}_i^0$ are transversally\nhyperbolic. This requires the dynamical system to be $\\mathcal{C}^1$\nand this leads to some restrictions on $f_0$. However, it is\npossible to obtain an alternative proof that does not require such\nrestrictions. Such a proof can be obtained from the hierarchical\nstructure of invariant sets; we will refrain from making the details\nexplicit here, since our analysis of the global dynamics below\ncontains all essential ingredients implicitly.\n\n\\textit{Interpretation of Theorem~\\ref{locthm} (massive case)}:\nA three-parameter set of solutions \nconverges to every non-LRS Kasner solution as $t\\rightarrow 0$.\n(In the state space description three equivalent sets of orbits approach three equivalent\ntransversally stable Kasner arcs that cover all non-LRS Kasner\nsolutions; the \nequivalence reflects the freedom of permuting the coordinates.)\nFurthermore, a three-parameter set of\nsolutions approaches the non-flat LRS Kasner solution. \nHence, in total, a four-parameter set of solutions asymptotically approaches \nnon-flat Kasner states. \nThere exist special solutions with\nnon-Kasner behaviour toward the singularity: one solution\nisotropizes toward the singularity and a one-parameter set of solutions\napproaches a non-Kasner LRS solution of the type~\\eqref{Disol} \n(three equivalent one-parameter\nsets of orbits approach three equivalent non-Kasner LRS fixed points\nassociated with this solution).\nFor the latter solutions \n$\\Omega=3\/4$; these solutions cannot be interpreted as a perfect\nfluid solutions since they possess anisotropic pressures.\n\nIn the following we show that the list of Theorem~\\ref{locthm} is\nalmost complete: there exist no other attractors toward the singularity\nwith one exception, a heteroclinic network that connects the flat LRS-Kasner\npoints.\n\n\n\\subsection{Global dynamics}\n\\label{globaldynamics}\n\n\\begin{theorem}\\label{futurethm}\nAll orbits in the interior of the state space $\\mathcal{X}$ of\nmassive particles [state space $\\mathcal{Z}^0$ of massless\nparticles] converge to {\\rm FS}$^1$ [{\\rm F}$^0$] when\n$\\tau\\rightarrow +\\infty$; i.e., all smooth reflection-symmetric\nBianchi type~I Vlasov solutions isotropize toward the future.\n\\end{theorem}\n\nA proof of this theorem has been given in~\\cite{ren96}.\nIn Appendix~\\ref{futureproof} we present an alternative proof based on dynamical\nsystems techniques.\n\n\\begin{theorem}\\label{alphathm}\nThe $\\alpha$-limit set of an orbit in the interior of the state\nspace is one of the fixed points $\\mathrm{F}^0$, $\\mathrm{D}_i^0$,\n$\\mathrm{QL}_i^0$, $\\mathrm{KC}_i^0$, see Theorem~\\ref{locthm}, or\nit is the heteroclinic network $\\mathcal{H}^0$. The $\\alpha$-limit set of a\ngeneric orbit resides on the union of the fixed point sets {\\rm\nKC}$_i^0$ and possibly the heteroclinic network $\\mathcal{H}^0$.\n\\end{theorem}\n\n\\begin{remark}\nA heteroclinic network is defined as a compact connected\nflow-invariant set that is indecomposable (all points are connected\nby pseudo-orbits), has a finite nodal set (the set of recurrent\npoints is a finite union of disjoint compact connected\nflow-invariant subsets), and has finite depth (taking the\n$\\alpha$\/$\\omega$-limit iteratively yields the set of recurrent\npoints after a finite number of iterative steps); for details\nsee~\\cite{Ashwin\/Field:1999} and references therein. A simple case\nis a heteroclinic network of depth $1$ whose nodes are equilibrium\npoints: it can be regarded as a collection of entangled heteroclinic\ncycles. The heteroclinic network $\\mathcal{H}^0$ is of the latter type; it\nwill be introduced in the proof of the theorem.\n\\end{remark}\n\n\nThe remainder of this section is concerned with the proof of Theorem~\\ref{alphathm}.\nThe first step in the proof is to gain a detailed understanding of the dynamics\non the relevant invariant subspaces of the dynamical system.\n\n\\subsubsection*{Dynamics on $\\mathcal{S}_i^0$}\n\n\\begin{lemma}\\label{lemmaSi0}\nConsider an orbit in the interior of $\\mathcal{S}_i^0$. Its $\\alpha$-limit\nset is either a fixed point on {\\rm KC}$_j^0$ or {\\rm KC}$_k^0$\n($i\\neq j \\neq k$),  {\\rm QL}$_i^0$ or {\\rm TL}$_i^0$, or it is the\nheteroclinic cycle $\\mathcal{H}_i^0$. The $\\omega$-limit set consists of the\nfixed point {\\rm D}$_i^0$.\n\\end{lemma}\n\n\\begin{remark}\nThe heteroclinic cycle $\\mathcal{H}_i^0$ will be defined\nin~\\eqref{heterocycle}.\n\\end{remark}\n\n\\proof Without loss of generality we consider $\\mathcal{S}_1^0$, which can\nbe described by the variables\n\\begin{equation}\n0< s_2 < 1 \\;\\, (s_3 = 1- s_2)\n\\quad\\text{ and }\\quad \\Sigma_1, \\Sigma_2,\n\\Sigma_3 \\quad\\left( \\sum\\nolimits_i \\Sigma_i = 0,\n\\Sigma^2 < 1\\right) \\:;\n\\end{equation}\nhence $\\mathcal{S}_1^0$ is represented by the interior of a cylinder, cf.~Figure~\\ref{cylinder}.\nThe boundary of $\\mathcal{S}_1^0$ consists of the lateral boundary $\\mathcal{S}_1^0 \\cap \\mathcal{K}^0$,\nthe base $\\mathcal{S}_{12}^0$, and the top surface $\\mathcal{S}_{13}^0$.\n\nSince $\\mathcal{S}_1^0 \\cap \\mathcal{K}^0$ is part of $\\mathcal{K}$, it follows that\n$\\Sigma_i \\equiv \\mathrm{const}$ for all orbits on $\\mathcal{S}_1^0 \\cap \\mathcal{K}^0$.\nWe observe that $s_2$ is monotonically increasing\n(decreasing) when $\\Sigma_2 < \\Sigma_3$ ($\\Sigma_2 > \\Sigma_3$), since\n$s_2^\\prime = -2 s_2 (1-s_2)(\\Sigma_2 -\\Sigma_3)$;\nthe two domains are separated by the lines of fixed points\nTL$_1^0$ and QL$_1^0$, see Figure~\\ref{cylinder}.\n\nThe key equations to understand the flow on $\\mathcal{S}_{12}^0$ are\n\\begin{equation}\n\\Omega^\\prime = \\Omega (2 \\Sigma^2 - \\Sigma_3)\\quad\\text{ and } \\quad\n\\Sigma_3^\\prime = \\Omega (2-\\Sigma_3)\\, .\n\\end{equation}\nFrom the first equation it follows that all points on\nKC$_3^0$ are transversally hyperbolic repelling fixed points\nexcept for T$_{33}^0$; from the second\nequation we infer that T$_{33}^0$ is the attractor of the entire\ninterior of $\\mathcal{S}_{12}^0$.\nSimilarly, T$_{22}^0$ is the attractor on $\\mathcal{S}_{13}^0$, see Figure~\\ref{cylinder}.\n\n\\begin{figure}[Ht]\n\\begin{center}\n\\psfrag{A}[cc][cc][1.2][0]{$\\text{TL}^0_1$}\n\\psfrag{B}[cc][rc][1.2][0]{$\\text{QL}^0_1$}\n\\psfrag{K2}[cc][cc][1.2][0]{$\\text{KC}^0_2$}\n\\psfrag{K3}[cc][cc][1.2][0]{$\\text{KC}^0_3$}\n\\psfrag{T2}[cc][cc][0.9][0]{$\\text{T}_{32}^0$}\n\\psfrag{T3}[cc][cc][0.9][0]{$\\text{T}_{23}^0$}\n\\psfrag{P2}[cc][cc][0.9][0]{$\\text{T}_{22}^0$}\n\\psfrag{P3}[cc][cc][0.9][0]{$\\text{T}_{33}^0$}\n\\psfrag{S1}[cc][cc][0.8][0]{$\\Sigma_1$}\n\\psfrag{S2}[cc][cc][0.8][0]{$\\Sigma_2$}\n\\psfrag{S3}[cc][cc][0.8][0]{$\\Sigma_3$}\n\\psfrag{Si}[cc][cc][0.8][0]{$\\Sigma_i$}\n\\psfrag{s2}[cc][cc][0.8][0]{$s_2$}\n\\includegraphics[width=0.9\\textwidth]{cylinder2.eps}\n\\caption{Flow on the boundaries and on the invariant subset\n$\\Sigma_1 = -1$ on $\\mathcal{S}_1^0$. The fixed point on $\\Sigma_1=-1$ is\nthe point D$_1^0$; the heteroclinic cycle $\\mathcal{H}_1^0$ consists of the\nfixed points $\\mathrm{T}_{22}^0$, $\\mathrm{T}_{32}^0$,\n$\\mathrm{T}_{33}^0$, $\\mathrm{T}_{23}^0$, and the connecting\norbits.}\n\\label{cylinder}\n\\end{center}\n\\end{figure}\n\nThe plane $\\mathcal{D}_1^0$, defined by $\\Sigma_1 = -1$, is an invariant subset in $\\mathcal{S}_1^0$.\nIn the interior of the plane we find the fixed point\nD$_1^0$; the boundary consists of a heteroclinic cycle $\\mathcal{H}_1^0$,\n\\begin{equation}\\label{heterocycle}\n\\mathcal{H}_1^0: \\mathrm{T}_{22}^0 \\rightarrow \\mathrm{T}_{32}^0 \\rightarrow\n\\mathrm{T}_{33}^0 \\rightarrow \\mathrm{T}_{23}^0 \\rightarrow\n\\mathrm{T}_{22}^0\\, .\n\\end{equation}\n(Note that analogous cycles $\\mathcal{H}_2^0$ and $\\mathcal{H}_3^0$ exist\non $\\mathcal{S}_2^0$ and $\\mathcal{S}_3^0$, respectively.)\nThe function $M_{(5)}$ is monotonically decreasing on $\\mathcal{D}_1^0$, cf.~\\eqref{m5eqs}.\nApplication of the monotonicity principle, see Appendix~\\ref{dynsys},\nyields that $\\text{D}_1^0$ is the $\\omega$-limit and that $\\mathcal{H}_1^0$ is the\n$\\alpha$-limit for all orbits on $\\mathcal{D}_1^0$, cf.~Figure~\\ref{cylinder}.\n\nConsider now an orbit in $\\mathcal{S}_1^0$ with $\\Sigma_1 \\neq -1$.\nThe function $M_{(4)} = (1+\\Sigma_1)^2$ is monotonically decreasing\non $\\mathcal{S}_1^0$, cf.~\\eqref{m4}. The monotonicity\nprinciple implies that the $\\omega$-limit lies on\n$\\Sigma_1 = -1$ or $\\Sigma^2 =1$ (but $\\Sigma_1 \\neq \\pm 2$).\nSince the logarithmic derivative of $\\Omega$ is positive everywhere on $\\mathcal{S}_1^0 \\cap \\mathcal{K}^0$\n(except at $\\text{T}^0_{22}$ and $\\text{T}^0_{33}$), i.e.,\n$\\Omega^{-1}\\Omega^\\prime|_{\\Omega =0} = 2  -\\sum_k w_k \\Sigma_k> 0$,\nit follows that the ``wall'' $\\mathcal{S}_1^0 \\cap \\mathcal{K}^0$ of the cylinder\nis repelling everywhere away from $\\Sigma_1 = -1$.\nConsequently, the\n$\\omega$-limit of the orbit cannot lie on $\\Sigma^2 =1$\nbut is contained in $\\Sigma_1 = -1$.\nThe fixed point $\\text{D}_1^0$ is a hyperbolic sink, as we conclude\nfrom the dynamics on $\\Sigma_1 = -1$ and from\n$(1+\\Sigma_1)^{-1} (1+ \\Sigma_1)^\\prime|_{\\text{D}_1^0} = -3\/4$.\nTherefore, the a priori possible $\\omega$-limit sets on $\\Sigma_1 = -1$\nare $\\text{D}_1^0$ and the heteroclinic cycle $\\mathcal{H}_1^0$.\n\nTo prove that $\\text{D}_1^0$ is the actual $\\omega$-limit we again\nconsider the function $M_{(5)}$. However, we no longer restrict its\ndomain of definition to $\\mathcal{D}_1^0$, but view it as a function\non $\\mathcal{S}_1^0$; we obtain\n\\begin{equation}\\label{M52}\n12 M_{(5)}^\\prime = -M_{(5)} \\left[(\\Sigma_1 + 2 \\Sigma_2)^2 +\n(\\Sigma_1+2 \\Sigma_3)^2 + 6 (\\Sigma_1+1)^2 - 6(\\Sigma_1+1)\\right]\\:.\n\\end{equation}\nThe bracket is positive when $\\Sigma_1 < -1$; hence $M_{(5)}$ is\ndecreasing when $\\Sigma_1 < -1$. This prevents orbits with $\\Sigma_1\n< -1$ from approaching $\\mathcal{H}_1^0$, since the cycle is characterized\nby $M_{(5)} = \\infty$. Now suppose that there exist an orbit in\n$\\Sigma_1 > -1$, whose $\\omega$-limit is $\\mathcal{H}_1^0$. At late times\nthe trajectory shadows the cycle; hence, for late times, the bracket\nin~\\eqref{M52} is almost always positive along the trajectory --\nonly when the trajectory passes through a small neighbourhood of\n$(\\Sigma_1,\\Sigma_2,\\Sigma_3) = (-1,1\/2,1\/2)$ the bracket is\nmarginally negative. Since the trajectory spends large amounts of\ntime near the fixed points and the passages from one fixed point to\nanother become shorter and shorter in proportion,\nit follows that at late times $M_{(5)}$ is decreasing along the\norbit (with ever shorter periods of limited increase). This is a\ncontradiction to the assumption that the orbit is attracted by the\nheteroclinic cycle. We therefore draw the conclusion that\n$\\text{D}_1^0$ is the global sink on $\\mathcal{S}_1^0$.\n\nConsider again an orbit in $\\mathcal{S}_1^0$ with $\\Sigma_1 \\neq -1$.\nInvoking the monotonicity principle with the function $M_{(4)}$ we\nfind that the $\\alpha$-limit of the orbit must be located on\n$\\Sigma^2 =1$, $\\Sigma_1 \\neq -1$. From the analysis of the flow on\nthe boundaries of the cylinder we obtain that all fixed points on\n$\\Sigma^2=1$ except for T$_{22}^0$ and T$_{33}^0$ are transversally\nhyperbolic. The fixed points on KC$_2^0$ with $\\Sigma_2 < \\Sigma_3$\nand the points on KC$_3^0$ with $\\Sigma_2\n> \\Sigma_3$ are saddles; the fixed points on KC$_2^0$ with\n$\\Sigma_2 > \\Sigma_3$ and those on KC$_3^0$ with $\\Sigma_2 <\n\\Sigma_3$ are transversally hyperbolic sources (except for\nT$_{22}^0$, T$_{33}^0$): every point attracts a one-parameter set of\norbits from $\\mathcal{S}_1^0$ as $\\tau\\rightarrow -\\infty$. In contrast,\neach fixed point on TL$_1^0$ and QL$_1^0$ is a source for exactly\none orbit. The structure of the flow on $\\Sigma^2=1$ implies that\nthe $\\alpha$-limit of the orbit in $\\mathcal{S}_1^0$ with $\\Sigma_1 \\neq -1$\nmust be one of the transversally hyperbolic sources.\n\nThis establishes Lemma~\\ref{lemmaSi0}.\n\n\n\\subsubsection*{Dynamics on $\\mathcal{K}^0$}\n\nThe invariant subset $\\mathcal{K}^0$ is defined by setting $z=0$ and $\\Sigma^2=1$;\nit can be represented by the Cartesian\nproduct of the circle ($\\Sigma^2=1$) in the $\\Sigma_i$-space times\nthe $s_i$-triangle given by $\\{0<s_1,s_2,s_3<1$, $\\sum_k s_k = 1\\}$. The flow\non this space possesses a simple structure: since $\\Sigma_i^\\prime\n\\equiv \\mathrm{const}$ for all orbits, the dynamical freedom resides\nin the $s_i$-spaces.\n\nA schematic depiction of the flow on $\\mathcal{K}^0$ is given in\nFigure~\\ref{sigmaquadgleich1}. All fixed points are located on the\nboundaries of $\\mathcal{K}^0$, i.e., on $s_1=0$, $s_2=0$, or $s_3=0$. The\nvertices of the $s_i$-triangle are the Kasner circles KC$_i^0$. If\n$(\\Sigma_1,\\Sigma_2,\\Sigma_3) \\in (\\Sigma^2 =1)$ is such that\n$\\Sigma_k = 2$ (respectively $\\Sigma_k = -2$) for some $k$, then the\nside $s_k = 0$ of the triangle is a line of fixed points, TL$_k^0$\n(respectively QL$_k^0$). Note that all fixed points are\ntransversally hyperbolic on $\\mathcal{K}^0$, and that they constitute the\n$\\alpha$- and $\\omega$-limit sets for all orbits on $\\mathcal{K}^0$. The\ncharacter of the fixed points, i.e., whether they are (transversal)\nattractors or repellors, depends on the sector of the circle\n($\\Sigma^2=1$), see~Figure~\\ref{sigmaquadgleich1}.\n\n\\begin{figure}[Ht]\n\\begin{center}\n\\psfrag{S1}[cc][cc][0.9][0]{$\\Sigma_1$}\n\\psfrag{S2}[cc][cc][0.9][0]{$\\Sigma_2$}\n\\psfrag{S3}[cc][cc][0.9][0]{$\\Sigma_3$}\n\\psfrag{s1}[cc][cc][0.9][0]{$s_1$}\n\\psfrag{s2}[cc][cc][0.9][0]{$s_2$}\n\\psfrag{s3}[cc][cc][0.9][0]{$s_3$}\n\\psfrag{s10}[cc][cc][0.7][55]{$s_1=0$}\n\\psfrag{s20}[cc][cc][0.7][0]{$s_2=0$}\n\\psfrag{s30}[cc][cc][0.7][-60]{$s_3=0$}\n\\psfrag{K1}[lc][lc][0.9][0]{$\\text{KC}^0_1$}\n\\psfrag{K2}[cc][cc][0.9][0]{$\\text{KC}^0_2$}\n\\psfrag{K3}[rc][rc][0.9][0]{$\\text{KC}^0_3$}\n\\psfrag{P}[cc][cc][0.7][0]{$\\mathrm{P}$}\n\\includegraphics[width=0.9\\textwidth]{sigmaquadgleich1_and.eps}\n\\caption{Schematic representation of the flow of the dynamical\nsystem on the invariant set $\\mathcal{K}^0 =(z=0) \\cap (\\Sigma^2 =1)$, which\nis the Cartesian product of the $(\\Sigma^2=1)$-circle and the\n$s_i$-triangle. The depicted fixed points are the Kasner fixed\npoints and the lines of fixed points $\\text{TL}_i^0$\n($\\leftrightarrow \\Sigma_i =2$) and $\\text{QL}_i^0$\n($\\leftrightarrow\\Sigma_i = -2$). All orbits are heteroclinic, like\nthe orbit through the arbitrary point $\\mathrm{P}$ that connects\n$\\text{KC}_2^0$ with $\\text{KC}_3^0$.} \\label{sigmaquadgleich1}\n\\end{center}\n\\end{figure}\n\n\nThe results about the global dynamics on $\\mathcal{S}^0_i$ and $\\mathcal{K}^0$\nwill turn out to be an integral part in the proof of Theorem~\\ref{alphathm},\nwhich we will address next.\nFirst, we will prove the massless case of the theorem.\n\n\n\\subsubsection*{Dynamics on $\\mathcal{Z}^0$}\n\nLet $\\gamma$ be an arbitrary orbit in the interior of $\\mathcal{Z}^0$,\n$\\gamma \\neq \\{ \\text{F}^0 \\}$. The function $M_{(2)}$ is strictly\nmonotonically decreasing on $\\mathcal{Z}^0$ (except at $\\text{F}^0$, where\nit has a minimum), cf.~\\eqref{m2eqs}ff.; hence we can use the\nmonotonicity principle: the $\\alpha$-limit set $\\alpha(\\gamma)$ of\n$\\gamma$ must be located on the boundaries of $\\mathcal{Z}^0$, i.e., on\n$\\mathcal{S}_i^0$ or $\\mathcal{K}^0$. The first step in our analysis is to prove\nthat the interior of the subsets $\\mathcal{S}_i^0$ and $\\mathcal{K}^0$ cannot belong\nto $\\alpha(\\gamma)$, unless $\\gamma$ is one of three special orbits.\n\nRecall from our analysis of $\\mathcal{S}_i^0$ that the fixed point\n$\\text{D}_i^0 \\in \\mathcal{S}_i^0$ is a hyperbolic sink on $\\mathcal{S}_i^0$. In the\northogonal direction, however, we obtain $s_i^{-1}s_i^\\prime\n|_{\\text{D}_i^0} = 3$. It follows that $\\text{D}_i^0$ is a\nhyperbolic saddle in the state space $\\mathcal{Z}^0$ and that there exists\nexactly one orbit $\\delta^0_i$ that emanates from $\\text{D}_i^0$\ninto the interior of $\\mathcal{Z}^0$. (Theorem~\\ref{futurethm} implies that\n$\\delta^0_i$ converges to the global sink $\\text{F}^0$ as\n$\\tau\\rightarrow \\infty$.)\n\nHenceforth, let $\\gamma$ be different from $\\delta^0_i$. In order to\nshow that $\\alpha(\\gamma)$ does not contain any point of the\ninterior of $\\mathcal{S}^0_i$, we perform a proof by contradiction: assume\nthat $\\alpha(\\gamma)$ contains a point $\\mathrm{P}$ of the interior\nof $\\mathcal{S}_i^0$; then the whole orbit through $\\mathrm{P}$ and the\n$\\omega$-limit $\\omega(\\mathrm{P})$ (as well as the $\\alpha$-limit)\nof that orbit must be contained in $\\alpha(\\gamma)$. As already\nshown, $\\text{D}_i^0$ is the global attractor on $\\mathcal{S}_i^0$, hence\n$\\alpha(\\gamma) \\ni \\omega(\\mathrm{P}) = \\text{D}_i^0$. Since the\nsaddle $\\text{D}_i^0$ is in $\\alpha(\\gamma)$, the unique orbit\n$\\delta^0_i$ emanating from it is contained in $\\alpha(\\gamma)$ as\nwell. Thus, ultimately, $\\omega(\\delta^0_i)$, i.e., the point\n$\\text{F}^0$, must be contained in $\\alpha(\\gamma)$; this is a\ncontradiction, since $\\text{F}^0$ is a sink. Therefore, $\\gamma$\ncannot contain any $\\alpha$-limit point in the interior of\n$\\mathcal{S}_i^0$. We will now use similar reasoning repeatedly.\n\nAssume next that $\\alpha(\\gamma)$ contains a point $\\mathrm{P}$ in\nthe interior of $\\mathcal{K}^0$, i.e., a point with $\\Sigma^2 =1$, $0< s_i\n<1\\:\\forall i$. Suppose first that $\\Sigma_k\\neq \\pm 2$ for all $k$.\nSince $\\mathrm{P}$ is an element of $\\alpha(\\gamma)$, the whole\norbit through $\\mathrm{P}$ and the $\\alpha$-limit\n$\\alpha(\\mathrm{P})$ of that orbit must be contained in\n$\\alpha(\\gamma)$. From the dynamics on $\\mathcal{K}^0$,\ncf.~Figure~\\ref{sigmaquadgleich1}, it follows that\n$\\alpha(\\mathrm{P})$ is one of the Kasner fixed points on\n$\\text{KC}_i^0$, where $i$ corresponds to the direction determined\nby $\\Sigma_i = \\max_k \\Sigma_k$; we hence denote\n$\\alpha(\\mathrm{P})$ as $\\mathrm{K}_\\mathrm{P}$. Since $\\Sigma_k\\neq\n\\pm 2$, it follows from the previous analysis that\n$\\mathrm{K}_\\mathrm{P}$ is a transversally hyperbolic source on the\nsubspace $\\mathcal{K}^0$; $\\Omega^{-1} \\Omega^\\prime|_{\\Sigma^2=1} = 2 -\n\\sum_k w_k \\Sigma_k > 0$ yields that $\\mathrm{K}_\\mathrm{P}$ is a\ntransversally hyperbolic source on the whole space $\\mathcal{Z}^0$. Since\n$\\alpha(\\gamma)$ contains the transversally hyperbolic source\n$\\mathrm{K}_\\mathrm{P}$, that fixed point necessarily constitutes\nthe entire $\\alpha$-limit set, i.e., $\\alpha(\\gamma)\n=\\mathrm{K}_\\mathrm{P}$. This is in contradiction to our assumption\n$\\alpha(\\gamma) \\ni \\mathrm{P}$. The omitted cases $\\Sigma_i = \\pm\n2$ for some $i$ will be dealt with next.\n\nSuppose that $\\Sigma_i = -2$ for one index $i$. Assume that\n$\\mathrm{P}$ lies in $\\alpha(\\gamma)$, therefore\n$\\alpha(\\mathrm{P})$ is contained in $\\alpha(\\gamma)$ as well. The\ndynamics on $\\mathcal{K}^0$ implies that $\\alpha(\\mathrm{P})$ is a fixed\npoint $\\mathrm{Q}_{\\mathrm{P}}$ on QL$_i^0$,\ncf.~Figure~\\ref{sigmaquadgleich1}. This point is a transversally\nhyperbolic source; $\\Omega^{-1} \\Omega^\\prime|_{\\text{QL}_i^0} = 1$\nin this case. By the same argument as above we obtain a\ncontradiction to the assumption $\\alpha(\\gamma) \\ni \\mathrm{P}$.\n\nFinally suppose that $\\Sigma_i = 2$ for one index $i$. When we\nassume that $\\mathrm{P}$ is in $\\alpha(\\gamma)$, then the\n$\\omega$-limit $\\omega(\\mathrm{P})$ is contained in\n$\\alpha(\\gamma)$. From Figure~\\ref{sigmaquadgleich1} we see that\n$\\omega(\\mathrm{P})$ is a fixed point $\\mathrm{T}_{\\mathrm{P}}$ on\nTL$_i^0$. The point $\\mathrm{T}_{\\mathrm{P}}$ is a transversally\nhyperbolic saddle, since $\\Omega^{-1} \\Omega^\\prime|_{\\text{TL}_i^0}\n= 3$, and there exists exactly one orbit that emanates from it,\nnamely the orbit that connects $\\mathrm{T}_{\\mathrm{P}}$ with\n$\\text{D}_i$ in $\\mathcal{S}_i^0$. Since $\\mathrm{T}_{\\mathrm{P}} \\in\n\\alpha(\\gamma)$, that orbit must also be contained in\n$\\alpha(\\gamma)$. This is in contradiction to the previous result:\n$\\alpha(\\gamma)$ cannot contain interior points of $\\mathcal{S}_i^0$. Hence\nour assumption $\\alpha(\\gamma) \\ni \\mathrm{P}$ was false: the\n$\\alpha$-limit of $\\gamma$ cannot contain any interior point of\n$\\mathcal{K}^0$.\n\nOur analysis results in the following statement: There exist four\nspecial orbits, one trivial orbit corresponding to the fixed point\n$\\text{F}^0$ and three orbits, the orbits $\\delta^0_i$, that\nconverge to the fixed points $\\text{D}^0_i \\in \\mathcal{D}^0_i$. The\n$\\alpha$-limit set of every orbit $\\gamma$ in $\\mathcal{Z}^0$ different from\n$\\text{F}^0$ and $\\delta^0_i$ must be located on the boundaries of\nthe spaces $\\mathcal{S}_i^0$ and $\\mathcal{K}^0$, i.e., on the union of the\nboundaries of the cylinders $\\mathcal{S}_i^0$, which we denote by\n$\\partial\\mathcal{S}^0=\\partial \\mathcal{S}_1^0 \\cup\n\\partial \\mathcal{S}_2^0 \\cup\n\\partial \\mathcal{S}_3^0$.\nThe set $\\partial\\mathcal{S}^0$ is depicted in Figure~\\ref{fixedp}: it\ncomprises the lateral surfaces of the cylinders and the base\/top\nsurfaces.\n\nAll fixed points on $\\partial\\mathcal{S}^0$ are transversally hyperbolic except\nfor the points $\\text{T}_{ii}^0$:\n$\\text{TL}_i^0$ consists of transversally hyperbolic saddles; in\ncontrast, the fixed points on $\\text{QL}_i^0$ are transversally\nhyperbolic sources; points on $\\text{KC}_i^0$ with $\\Sigma_i\n> 1$, $\\Sigma_i \\neq 2$ are sources while those with $\\Sigma_i < 1$\nare saddles.\nCombining the analysis of the preceding sections,\nsee~Figs.~\\ref{cylinder} and~\\ref{sigmaquadgleich1}, we obtain, more\nspecifically: each point on $\\text{QL}_i^0$ is a source for a\none-parameter family of orbits that emanate into the interior of\n$\\mathcal{Z}^0$, and each point on $\\text{KC}_i^0$ with $\\Sigma_i >\n1$ ($\\Sigma_i \\neq 2$) is the source for a two-parameter family.\n(The points with $\\Sigma_i =1$ on $\\text{KC}_i^0$ are the two points\n$\\text{Q}^0_{ij} \\in \\text{QL}_j^0$ and $\\text{Q}^0_{ik}\\in\n\\text{QL}_k^0$. Each of these two points is a transversally\nhyperbolic source for a one-parameter family of orbits, however,\nthose orbits are not interior orbits, but remain on the boundary of\n$\\mathcal{Z}^0$.)\n\nThe non-transversally hyperbolic fixed points $\\text{T}_{ii}^0$ are\npart of a special structure that is present on $\\partial\\mathcal{S}^0$: the\nset $\\partial\\mathcal{S}^0$ exhibits a robust heteroclinic network $\\mathcal{H}^0$\n(of depth $1$), see, e.g.,~\\cite{Ashwin\/Field:1999} for a discussion\nof heteroclinic networks; the network $\\mathcal{H}^0$ is depicted in\nFigure~\\ref{network}; a schematic depiction is given in\nFigure~\\ref{networkschematic}. In particular we observe that the\nheteroclinic cycles $\\mathcal{H}_i^0$ of the spaces $\\mathcal{S}_i^0$ appear as\nheteroclinic subcycles of the network.\n\n\n\\begin{figure}[Ht]\n\\begin{center}\n\\psfrag{TL1}[cc][cc][0.8][0]{$\\text{TL}^0_1$}\n\\psfrag{TL2}[cc][cc][0.8][0]{$\\text{TL}^0_2$}\n\\psfrag{TL3}[cc][cc][0.8][0]{$\\text{TL}^0_3$}\n\\psfrag{QL1}[cc][cc][0.8][0]{$\\text{QL}^0_1$}\n\\psfrag{QL2}[cc][cc][0.8][0]{$\\text{QL}^0_2$}\n\\psfrag{QL3}[cc][cc][0.8][0]{$\\text{QL}^0_3$}\n\\psfrag{T21}[cc][cc][0.6][0]{$\\text{T}^0_{21}$}\n\\psfrag{T22}[cc][cc][0.6][0]{$\\text{T}^0_{22}$}\n\\psfrag{T23}[cc][cc][0.6][0]{$\\text{T}^0_{23}$}\n\\psfrag{Q21}[cc][cc][0.6][0]{$\\text{Q}^0_{21}$}\n\\psfrag{Q22}[cc][cc][0.6][0]{$\\text{Q}^0_{22}$}\n\\psfrag{Q23}[cc][cc][0.6][0]{$\\text{Q}^0_{23}$}\n\\psfrag{T11}[cc][cc][0.6][0]{$\\text{T}^0_{11}$}\n\\psfrag{T12}[cc][cc][0.6][0]{$\\text{T}^0_{12}$}\n\\psfrag{T13}[cc][cc][0.6][0]{$\\text{T}^0_{13}$}\n\\psfrag{Q11}[cc][cc][0.6][0]{$\\text{Q}^0_{11}$}\n\\psfrag{Q12}[cc][cc][0.6][0]{$\\text{Q}^0_{12}$}\n\\psfrag{Q13}[cc][cc][0.6][0]{$\\text{Q}^0_{13}$}\n\\psfrag{T31}[cc][cc][0.6][0]{$\\text{T}^0_{31}$}\n\\psfrag{T32}[cc][cc][0.6][0]{$\\text{T}^0_{32}$}\n\\psfrag{T33}[cc][cc][0.6][0]{$\\text{T}^0_{33}$}\n\\psfrag{Q31}[cc][cc][0.6][0]{$\\text{Q}^0_{31}$}\n\\psfrag{Q32}[cc][cc][0.6][0]{$\\text{Q}^0_{32}$}\n\\psfrag{Q33}[cc][cc][0.6][0]{$\\text{Q}^0_{33}$}\n\\psfrag{KC1}[cc][cc][1.2][0]{$\\text{KC}^0_1$}\n\\psfrag{KC2}[cc][cc][1.2][0]{$\\text{KC}^0_2$}\n\\psfrag{KC3}[cc][cc][1.2][0]{$\\text{KC}^0_3$}\n\\psfrag{s1}[cc][cc][1.0][90]{$\\longleftarrow\\: s_1 =0\\: \\longrightarrow$}\n\\psfrag{s2}[cc][cc][1.0][35]{$\\longleftarrow \\:s_2 =0 \\:\\longrightarrow$}\n\\psfrag{s3}[cc][cc][1.0][-35]{$\\longleftarrow \\:s_3 =0 \\:\\longrightarrow$}\n\\includegraphics[width=0.8\\textwidth]{network3D3.eps}\n\\caption{The heteroclinic network $\\mathcal{H}^0$ that exists on the set\n$\\partial\\mathcal{S}^0$. Its building blocks are the heteroclinic cycles\n$\\mathcal{H}^0_1$, $\\mathcal{H}^0_2$, $\\mathcal{H}^0_3$.} \\label{network}\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{figure}[Ht]\n\\begin{center}\n\\psfrag{T11}[cc][cc][0.9][0]{$\\text{T}_{11}^0$}\n\\psfrag{T22}[cc][cc][0.9][0]{$\\text{T}_{22}^0$}\n\\psfrag{T33}[cc][cc][0.9][0]{$\\text{T}_{33}^0$}\n\\psfrag{T12}[cc][cc][0.9][0]{$\\text{T}_{12}^0$}\n\\psfrag{T21}[cc][cc][0.9][0]{$\\text{T}_{21}^0$}\n\\psfrag{T13}[cc][cc][0.9][0]{$\\text{T}_{13}^0$}\n\\psfrag{T31}[cc][cc][0.9][0]{$\\text{T}_{31}^0$}\n\\psfrag{T23}[cc][cc][0.9][0]{$\\text{T}_{23}^0$}\n\\psfrag{T32}[cc][cc][0.9][0]{$\\text{T}_{32}^0$}\n\\includegraphics[width=0.7\\textwidth]{network.eps}\n\\caption{Schematic representation of $\\mathcal{H}^0$.}\n\\label{networkschematic}\n\\end{center}\n\\end{figure}\n\nA straightforward analysis of the flow on $\\partial \\mathcal{S}^0$ using the\nsame type of reasoning as above leads to the result that there exist\nno other structures on $\\partial \\mathcal{S}^0$ that could serve as\n$\\alpha$-limits for an interior $\\mathcal{Z}^0$-orbit $\\gamma$. We\nhave thus proved the following statement: The $\\alpha$-limit of\n$\\gamma$ is one of the transversally hyperbolic sources listed\nabove, or it is the heteroclinic network (or a heteroclinic subcycle\nthereof). This concludes the proof of the massless case of\nTheorem~\\ref{alphathm}.\n\n\n\\subsection*{Global dynamics in the massive case}\n\nLet $\\gamma$ be an arbitrary orbit in the interior of the state\nspace $\\mathcal{X}$. The function $M_{(1)}$ is strictly\nmonotonically increasing on $\\mathcal{X}$ (and on $\\mathcal{K}$),\ncf.~\\eqref{m1eqs}ff.; moreover, $M_{(1)}$ vanishes for $z\\rightarrow\n1$ and $s_i \\rightarrow 0$ (unless $z\\rightarrow 0$ simultaneously).\nHence, by applying the monotonicity principle we obtain that the\n$\\alpha$-limit set $\\alpha(\\gamma)$ of $\\gamma$ must be located on\n$\\mathcal{Z}^0$ including its boundaries.\n\nConsider the fixed point $\\text{F}^0 \\in \\mathcal{Z}^0$. By\nTheorem~\\ref{futurethm} this fixed point is a global sink on\n$\\mathcal{Z}^0$. In the orthogonal direction, however, we have $z^{-1}\nz^\\prime |_{\\text{F}^0} = 2$. It follows that $\\text{F}^0$ is a\nhyperbolic saddle in the space $\\mathcal{X}$ and that there exists exactly\none orbit $\\phi$ that emanates from $\\text{F}^0$ into the interior\nof $\\mathcal{X}$. (Theorem~\\ref{futurethm} implies that $\\phi$ converges to\n$\\text{FS}^1$ as $\\tau\\rightarrow \\infty$; thus, $\\phi$ represents\nthe unique solution of the Einstein-Vlasov equations that\nisotropizes toward the past and the future.)\n\nLet $\\gamma$ be different from $\\phi$. Assume that $\\alpha(\\gamma)$\ncontains a point $\\mathrm{P}$ of the interior of $\\mathcal{Z}^0$; then the\nwhole orbit through $\\mathrm{P}$ and the $\\omega$-limit\n$\\omega(\\mathrm{P})$ must be contained in $\\alpha(\\gamma)$.\nTheorem~\\ref{futurethm} implies $\\omega(\\mathrm{P}) = \\text{F}^0$,\nhence $\\text{F}^0 \\in \\alpha(\\gamma)$. Since the saddle $\\text{F}^0$\nis in $\\alpha(\\gamma)$, the unique orbit $\\phi$ emanating from it is\ncontained in $\\alpha(\\gamma)$ as well. Thus, ultimately,\n$\\omega(\\phi)$, i.e., a point on $\\text{FS}^1$, must be contained in\n$\\alpha(\\gamma)$; this is a contradiction, since $\\text{FS}^1$\nconsists of transversally hyperbolic sinks. We conclude that\n$\\gamma$ cannot contain any $\\alpha$-limit point in the interior of\n$\\mathcal{Z}^0$.\n\nSince $\\alpha(\\gamma)$ must be located on the boundary on $\\mathcal{Z}^0$,\ni.e., on $\\mathcal{S}_i^0$ or $\\mathcal{K}^0$, the proof can be completed in close\nanalogy to the proof in the massless case. We thus restrict\nourselves here to giving some relations that establish that the\nsources on $\\mathcal{Z}^0$ generalize to sources on $\\mathcal{X}$: on\n$\\text{KC}_i^0$ we have $z^{-1} z^\\prime |_{\\text{KC}_i^0} = 2 ( 1+\n\\Sigma_i)$, which is positive for all $\\Sigma_i> -1$ and thus for\n$\\Sigma_i> 1$ in particular; for $\\text{QL}_i^0$ we obtain $z^{-1}\nz^\\prime |_{\\text{QL}_i^0} = 4$. We note further that $z^{-1}\nz^\\prime |_{\\text{D}_i^0} = 3$; thus $\\text{D}_i^0$ possesses a\ntwo-dimensional unstable manifold. (Orbits in that manifold converge\nto $\\text{FS}^1$.) Finally, note that along the heteroclinic cycle\n$\\mathcal{H}_1^0: \\mathrm{T}_{22}^0 \\rightarrow \\mathrm{T}_{32}^0\n\\rightarrow \\mathrm{T}_{33}^0 \\rightarrow \\mathrm{T}_{23}^0\n\\rightarrow \\mathrm{T}_{22}^0$, we obtain that $z^{-1} z^\\prime$\nequals $6 s_2$, $2(1+\\Sigma_3)$, $2(1+\\Sigma_2)$, $6(1-s_2)$,\nrespectively; hence $z^{-1} z^\\prime$ is non-negative along the\nheteroclinic network $\\mathcal{H}$.\n\nThis concludes the proof of Theorem~\\ref{alphathm}.\n\n\n\n\n\\section{Concluding remarks}\n\\label{conc}\n\nIn this article we have analyzed the asymptotic behaviour of\nsolutions of the Einstein-Vlasov equations with Bianchi type I\nsymmetry. To that end we have reformulated the equations as a system\nof autonomous differential equations on a compact state space, which\nenabled us to employ powerful techniques from dynamical systems\ntheory.\n\nBased on the global dynamical systems analysis we have identified\nall possible past attractors of orbits\n--- both in the massless and massive case.\nWe have found that an open set of solutions converges to the Kasner\ncircle(s); in particular, for these solutions, the rescaled matter\nquantity $\\Omega$ satisfies $\\Omega\\rightarrow 0$ toward the\nsingularity, so that ``matter does not matter.'' However, we have\nseen that there exists an interesting structure that might\ncomplicate matters: there exists a heteroclinic network $\\mathcal{H}^0$ that\nmight be part of the past attractor set. For solutions that converge\nto $\\mathcal{H}^0$, $\\Omega$ has no limit toward the singularity, since\n$\\Omega \\neq 0$ along parts of the network $\\mathcal{H}^0$, i.e., matter\ndoes matter for such solutions. It is not clear at the present stage\nwhether the set of solutions converging to $\\mathcal{H}^0$ is empty, or, if\nnon-empty, of measure zero or not (the flow on the boundary subsets\ngives a hint that it might be a three-parameter set, i.e., a set of\nmeasure zero). In any case $\\mathcal{H}^0$ will be important for the\nintermediate dynamical behaviour of some models, and thus there are\nsignificant differences between Bianchi type I perfect fluid models\nand models with Vlasov matter.\n\nIf a generic set of orbits converges to $\\mathcal{H}^0$, this will have\nconsiderable consequences. Bianchi type I perfect fluid models play\na central role in understanding the singularity of more general\nspatially homogeneous models~\\cite{waiell97}, as well as general\ninhomogeneous models~\\cite{uggetal03}. The importance of Bianchi\ntype I perfect fluid models is due to Lie and source\n``contractions'' in spatially homogeneous cosmology and the\nassociated invariant subset and monotone function structure (which\nis quite similar to the hierarchical structure we have encountered\nin the present paper), and asymptotic silence in general\ninhomogeneous models~\\cite{uggetal03}. Similarly, we expect that\nBianchi type I Einstein-Vlasov models hold an equally prominent\nplace as regards singularities in more general\n--- spatially homogeneous and inhomogeneous --- Einstein-Vlasov\nmodels. Hence the resolution of the problem of whether the\nheteroclinic network attracts generic solutions determines if\nEinstein-Vlasov models are generically different from general\nrelativistic perfect fluid models in the vicinity of a generic\nsingularity, or not.\n\nIn this article we have not considered a cosmological constant,\n$\\Lambda$. The effects of a positive cosmological constant can be\noutlined as follows: since $\\rho \\rightarrow \\infty$ toward the\nsingularity, it follows that $\\Lambda$ can be asymptotically\nneglected and hence that the singularity structure is qualitatively\nthe same as for $\\Lambda =0$. However, toward the future $\\Lambda$\ndestabilizes FS$^1$, which becomes a saddle, and instead solutions\nisotropize and asymptotically reach a de Sitter state.\n\nWe conclude with some remarks on different formulations. We have\nseen that the variables we have used to reformulate the equations as\na dynamical system yielded multiple representations of some\nstructures, e.g., the Kasner circle. Replacing $s_i$ with\n$E_i=\\sqrt{g^{ii}}\/H$, i.e., the Hubble-normalized spatial frame\nvariables of~\\cite{uggetal03,rohugg05}, and using $y=m^2H^{-2}$\ninstead of $z$, yields a single Kasner circle. The latter variables,\nhowever, are not bounded; indeed, they blow up toward the future in\nthe present case. It is possible to replace the variables by bounded\nvariables, however, variables of this type lead to differentiability\ndifficulties toward the singularity. Issues like these made the\nvariables we employed in this article more suitable for the kind of\nanalysis we have performed. However, $E_i$-variables, or\n``$E_i$-based'' variables would have been more suitable to relate\nthe present results to a larger context; but it is not difficult to\ntranslate our results to the $E_i$-variables variables used\nin~\\cite{uggetal03,rohugg05}, where the relationship between the\ndynamics of inhomogeneous and spatially homogeneous models was\ninvestigated and exploited.\n\n\\\n\n\\noindent\n{\\bf Acknowledgement}\n\n\\noindent We gratefully acknowledge the hospitality and the support\nof the Isaac Newton Institute for Mathematical Sciences in\nCambridge, England, where part of this work was done. We also thank\nAlan Rendall for useful questions and comments. CU is supported by\nthe Swedish Research Council.\n\n\\\n\n\n\\begin{appendix}\n\n\\section{Dynamical systems}\n\\label{dynsys}\n\n\nIn this appendix we briefly recall some concepts from the theory of\ndynamical systems which we use in the article.\n\nConsider a dynamical system defined on an invariant set $X\\subseteq\n\\mathbb{R}^m$. The $\\omega$-limit set $\\omega(x)$ [$\\alpha$-limit\nset $\\alpha(x)$] of a point $x\\in X$ is defined as the set of all\naccumulation points of the future [past] orbit of $x$. The simplest\nexamples are fixed points and periodic orbits.\n\nThe monotonicity principle~\\cite{waiell97} gives\ninformation about the global asymptotic behaviour of the dynamical\nsystem. If $M: X\\rightarrow \\mathbb{R}$ is a ${\\mathcal C}^1$\nfunction which is strictly decreasing along orbits in $X$, then\n\\begin{subequations}\\label{omegalimitmon}\n\\begin{align}\n\\omega(x) &\\subseteq\n\\{\\xi \\in \\bar{X}\\backslash X\\:|\\: \\lim\\limits_{\\zeta\\rightarrow \\xi} M(\\zeta) \\neq\n\\sup\\limits_{X} M\\} \\\\\n\\alpha(x) &\\subseteq\n\\{\\xi \\in \\bar{X}\\backslash X\\:|\\:\\lim\\limits_{\\zeta\\rightarrow \\xi} M(\\zeta) \\neq\n\\inf\\limits_{X} M\\}\n\\end{align}\n\\end{subequations}\nfor all $x\\in X$.\n\nLocally in the neighbourhood of a fixed point, the flow of the\ndynamical system is determined by the stability features of the\nfixed point. If the fixed point is hyperbolic, i.e., if the\nlinearization of the system at the fixed point is a matrix\npossessing eigenvalues with non-vanishing real parts, then the\nHartman-Grobman theorem applies: in a neighbourhood of a hyperbolic\nfixed point the full nonlinear dynamical system and the linearized\nsystem are topologically equivalent. Non-hyperbolic fixed points are\ntreated in centre manifold theory: the reduction theorem generalizes\nthe Hartman-Grobman theorem; for further details see,\ne.g.,~\\cite{cra91}. If a fixed point is an element of a connected\nfixed point set (line, surface,\\nolinebreak \\ldots) and the number\nof eigenvalues with zero real parts is equal to the dimension of the\nfixed point set, then the fixed point is called transversally\nhyperbolic. Application of the centre manifold reduction theorem is\nparticularly simple in this case. (The situation is analogous in the\nmore general case when the fixed point is an element of an a priori\nknown invariant set that coincides with the centre manifold of the\nfixed point.)\n\n\\section{FRW and LRS$_i$ symmetry}\n\\label{FRWLRS}\n\nIn this section we discuss in detail the sets $\\text{FRW}$ and $\\text{LRS}_i$,\nconnected with solutions exhibiting FRW or LRS geometry.\n\nTo begin with, we prove that the fixed point $\\text{F}^0$ on $z=0$\nis well-defined. Since the defining equations for $\\text{F}^0$ are\n$w_1 = w_2= w_3 = 1\/3$, we must show that these equations indeed\npossess a unique solution $(s_1,s_2,s_3)$ for all distribution\nfunctions $f_0$. Setting $z=0$ in~\\eqref{omegai} implies that \nequations $w_1 = w_2= w_3 = 1\/3$ are equivalent to the system\n\\begin{equation}\\label{udef}\nu := \\int f_0 \\, \\left[s_1 v_1^2 -s_2 v_2^2\\right] \\left(\\sum\\nolimits_k s_k v_k^2 \\right)^{-1\/2} d^3 v =0\n\\end{equation}\nand $v = 0$, where $v$ is defined by replacing $[s_1 v_1^2 -s_2\nv_2^2]$ by $[s_1 v_1^2 -s_3 v_3^2]$ in~\\eqref{udef}. On the three\nboundaries of the space $\\{(s_1,s_2,s_3)\\:|\\: s_i \\geq 0, \\sum_k s_k\n=1\\}$ the functions $u$ and $v$ are monotonic; their signs are given\nin Figure~\\ref{dreiecken}. The derivative $\\partial u\/\\partial s_1$ is\nmanifestly positive, $\\partial u\/\\partial s_2$ is negative, hence\n$\\mathrm{grad}\\, u$ is linearly independent of the surface normal\n$(1,1,1)$, and it follows that\n$u=\\mathrm{const}$ describes a curve\nfor all $\\mathrm{const} \\in \\mathbb{R}$. The same argument applies\nto $v$, since $\\partial v\/\\partial s_1>0$ and $\\partial v\/\\partial\ns_3 <0$. Figure~\\ref{dreiecken} reveals that $u=0$ ($v=0$) connects\nthe upper (right) vertex of the $(s_1,s_2,s_3)$-space with the\nopposite side. Investigating $(\\mathrm{grad} \\,u - \\lambda\\,\n\\mathrm{grad} \\,v)$ we find that the first component is manifestly\npositive when $\\lambda \\leq 2\/3$ and negative when $\\lambda\\geq\n3\/2$, the second component is negative when $\\lambda \\leq 3$, and\nthe third component is positive when $\\lambda \\geq 1\/3$, which\nimplies that $(\\mathrm{grad}\\, u - \\lambda\\, \\mathrm{grad}\\, v)$ is\nlinearly independent of the surface normal $(1,1,1)$ for all\n$\\lambda$. It follows that all equipotential curves of the functions\n$u$ and $v$ intersect transversally; hence $u=0$ and $v=0$ possess a\nunique point of intersection, which proves the claim.\n\n\\begin{figure}[Ht]\n\\begin{center}\n\\psfrag{s1}[cc][cc][0.8][0]{$s_1$}\n\\psfrag{s2}[cc][cc][0.8][0]{$s_2$}\n\\psfrag{s3}[cc][cc][0.8][0]{$s_3$}\n\\psfrag{a}[cc][cc][0.7][0]{$v>0$}\n\\psfrag{b}[cc][cc][0.7][0]{$\\begin{array}{cc} u<0 \\\\ v<0 \\end{array}$}\n\\psfrag{c}[cc][cc][0.7][0]{$u>0$}\n\\psfrag{er}[cc][cc][0.7][0]{$\\begin{array}{cc} u<0 \\\\ v=0 \\end{array}$}\n\\psfrag{el}[cc][cc][0.7][0]{$\\begin{array}{cc} u>0 \\\\ v>0 \\end{array}$}\n\\psfrag{eo}[cc][cc][0.7][0]{$\\begin{array}{cc} u=0 \\\\ v<0 \\end{array}$}\n\\includegraphics[width=0.5\\textwidth]{dreiecken.eps}\n\\caption{The functions $u$ and $v$ are monotonic along the boundaries of the space\n$\\{(s_1,s_2,s_3)\\:|\\: s_i \\geq 0, \\sum_k s_k =1\\}$.}\n\\label{dreiecken}\n\\end{center}\n\\end{figure}\n\nBy establishing existence and uniqueness of the fixed point $\\text{F}^0$ for all $f_0$,\nwe have shown that for all distribution functions\nthere exists a unique FRW solution of the massless Einstein-Vlasov\nequations.\n\nThe situation is different in the massive case.\nA FRW solution is characterized by the equations\n$\\Sigma_i=0$ $\\forall i$, $w_1 =w_2=w_3 =w$, since\nthis yields $s_{i}=\\mathrm{const}$\n(and a rescaling of the spatial coordinates then results in $g_{ij}\\propto\\delta_{ij}$.)\nHowever, for a general distribution function $f_0$, these equations\nare incompatible with\nthe Einstein-Vlasov equations; in other words, the straight line\n$\\Sigma_i=0$ $\\forall i$, $w_1 =w_2=w_3 =w$ is not an orbit of the\ndynamical system. Hence, in the massive case, the Einstein-Vlasov\nequations do not admit a FRW solution for arbitrary $f_0$; the\ndistribution function $f_0$ is required to satisfy FRW compatibility\nconditions, see below, in order for a FRW solution to exist.\n\nNote, however, that for each $f_0$, there exists exactly one orbit\nthat originates from $\\text{F}^0$ and ends on $\\text{FS}^1$, see\nSection~\\ref{locglo}, i.e., there exists a unique solution of\nthe Einstein-Vlasov equations that isotropizes toward the past and\ntoward the future. This anisotropic solution can be regarded as a\ngeneralized FRW solution; if $f_0$ is compatible with the FRW\ngeometry, then the generalized FRW solution reduces to an ordinary\nFRW solution.\n\nThe treatment of the LRS case is analogous: the subset\n$\\text{LRS}_1$ (and, analogously, $\\text{LRS}_{2,3}$), defined\nthrough the equations $\\Sigma_2=\\Sigma_3$, $w_2=w_3$, describes\nsolutions exhibiting LRS geometry. (For a solution on\n$\\text{LRS}_1$, Equation~(\\ref{seq}) entails $s_2(\\tau) \\propto\ns_3(\\tau)$; by rescaling the coordinates one can achieve\n$g_{22}=g_{33}$, i.e., a line element in an explicit LRS form.)\nHowever, for general $f_0$, the set $\\text{LRS}_1$ is not invariant\nunder the flow of the dynamical system. Consequently, for general\n$f_0$, the Einstein-Vlasov equations do not admit solutions with LRS\ngeometry.\n\nMore specifically, consider\n\\begin{equation}\n\\left(\\Sigma_2 - \\Sigma_3\\right)^\\prime =\n- 3 \\Omega \\left[ \\textfrac{1}{2} (1-w) (\\Sigma_2 -\\Sigma_3) - (w_2-w_3)\\right]\\:.\n\\end{equation}\nHence, $(\\Sigma_2-\\Sigma_3)^\\prime$ vanishes when $\\Sigma_2 = \\Sigma_3$ and $w_2 = w_3$.\nFrom~\\eqref{seq} and~\\eqref{zeq} we obtain an equation for $w_i^\\prime$,\n\\begin{equation}\\label{omegaieq}\nw_i^\\prime = -2 w_i \\left[ \\Sigma_i -\n\\sum_k \\Sigma_k \\left(\\frac{1}{2}w_k + \\frac{1}{2} w_i^{-1} \\beta_{i k}^{\\text{\\tiny (0)}}\\right)\n+\\frac{z}{2} \\left(w_i^{-1} \\beta_{i}^{\\text{\\tiny (1)}}  + \\beta^{\\text{\\tiny (1)}} \\right) \\right]\\:,\n\\end{equation}\nwhere we have defined\n\\begin{equation}\n\\beta_{i_1\\ldots i_k}^{\\text{\\tiny (m)}} =\n\\frac{(1-z)^k {\\displaystyle\\int} f_0\\,\n\\left(\\Pi_{n=1}^{k} s_{i_n} v_{i_n}^2\\right) \\left[z+(1-z) \\sum_k s_k v_k^2\\right]^{1\/2-k-m} d v_1 d v_2 d v_3}%\n{{\\displaystyle\\int} f_0 \\left[z+(1-z) \\sum_k s_k v_k^2\\right]^{1\/2} d v_1 d v_2 d v_3}\\:;\n\\end{equation}\nnote that $w_i = \\beta_{i}^{\\text{\\tiny (0)}}$. Equation~\\eqref{omegaieq} implies\n\\begin{equation}\n(w_2-w_3)^\\prime = -\\left(\\Sigma_1 -\\Sigma_2\\right)\n\\left(\\beta_{22}^{\\text{\\tiny (0)}} - \\beta_{33}^{\\text{\\tiny (0)}}\\right) -\nz \\left(\\Sigma_1 +1\\right) \\left(\\beta_{2}^{\\text{\\tiny (1)}}- \\beta_{3}^{\\text{\\tiny (1)}}\\right)\\:,\n\\end{equation}\nwhen $\\Sigma_2 = \\Sigma_3$ and $w_2 = w_3$. We conclude that the set\n$\\Sigma_2 \\equiv \\Sigma_3$, $w_2 \\equiv w_3$ is an invariant set of\nthe dynamical system, iff $w_2 = w_3$ implies\n$\\beta_{22}^{\\text{\\tiny (0)}} = \\beta_{33}^{\\text{\\tiny (0)}}$ and\n$\\beta_{2}^{\\text{\\tiny (1)}} = \\beta_{3}^{\\text{\\tiny (1)}}$. (In\nthe massless case, only the first condition is required.) These\nconditions are violated for general distribution functions; for the\ncondition to hold $f_0$ must be of a certain type that ensures\ncompatibility with the LRS symmetry. This is the case, for instance,\nwhen there exist constants $a_2>0$, $a_3>0$, such that $f_0$ is\ninvariant under the transformation $v_2 \\rightarrow (a_3\/a_2)\\, v_3$,\n$v_3\\rightarrow (a_2\/a_3)\\, v_2$; e.g., $f_0 = \\tilde{f}_0(v_1,\nv_2^2 v_3^2)$, or $f_0 = \\tilde{f}_0(v_1, a_2^2 v_2^2 + a_3^2\nv_3^2)$; in the latter case $w_2(\\tau) \\equiv w_3(\\tau)$ implies\n$a_3^2 s_2(\\tau) \\equiv a_2^2 s_3(\\tau)$.\n\nNote finally that a distribution function $f_0$ is compatible with a\nFRW geometry, if it is compatible with all LRS symmetries. This\nmeans, that for instance $f_0 = \\tilde{f}_0(a_1^2 v_1^2 +a_2^2 v_2^2\n+a_3^2 v_3^2)$ is compatible with the FRW symmetry and thus admits a\nunique FRW solution of the Einstein-Vlasov equations.\n\n\n\n\\section{Future asymptotics}\n\\label{futureproof}\n\nIn this section we give the proof of Theorem~\\ref{futurethm}:\n\n\\noindent\n\\textbf{Theorem~\\ref{futurethm}.}\n\\textit{The $\\omega$-limit of every orbit in the interior of the\nmassive state space $\\mathcal{X}$ [massless state space $\\mathcal{Z}^0$]\nis one the fixed points $\\,\\mathrm{FS}^1$ [the fixed point $\\,\\mathrm{F}^0$].}\n\n\n\\proof Consider first the state space $\\mathcal{Z}^0$ of massless\nparticles and the associated system~\\eqref{z0eq}. The function\n$M_{(2)}$, cf.~\\eqref{m2eqs}ff., is well-defined and monotonically\ndecreasing everywhere except for at the fixed point $\\text{F}^0$,\nwhere it has a global minimum. On the boundaries $\\mathcal{S}_i^0$\n(given by $s_i=0$) and $\\mathcal{K}^0$ ($\\Sigma^2 =1$) of the state\nspace $\\mathcal{Z}^0$, the function $M_{(2)}$ is infinite.\nTherefore, application of the monotonicity principle yields that the\n$\\omega$-limit of every orbit must be the fixed point $\\text{F}^0$.\n\nIn the massive case consider~\\eqref{Sigeq} in the form\n\\begin{equation}\\label{sigmaiprime}\n\\Sigma_i^\\prime = -3 \\Omega \\left[ \\frac{1}{2} (1-w) (1 +\\Sigma_i) - \\frac{1}{2} (1- 3w) - w_i \\right]\\:.\n\\end{equation}\nThe r.h.s.\\ is positive when $\\Sigma_i \\leq -1$ and $z>0$ ($w<1\/3$).\nThis implies that the hyperplanes $\\Sigma_i = -1$ constitute semipermeable membranes in the state space $\\mathcal{X}$,\nwhereby the ``triangle'' $(\\Sigma_1 > -1) \\cap (\\Sigma_2 > -1) \\cap (\\Sigma_3 > -1)$ becomes a future invariant\nsubset of the flow~\\eqref{eq}.\n\nThe first part of the proof is to show that every orbit enters the\ntriangle at some time $\\tau_e$ (and consequently remains inside for\nall later times).\n\nAssume that there exists an orbit with $\\Sigma_i(\\tau) \\leq -1$ for all $\\tau$ (for some $i$).\nFrom~\\eqref{seq} we infer that\n\\begin{equation}\ns_i^\\prime = -2 s_i \\left[ s_j (\\Sigma_i - \\Sigma_j) + s_k (\\Sigma_i - \\Sigma_k) \\right] > 0\n\\end{equation}\nif $\\Sigma_i< -1$, and that $s_i^\\prime \\geq 0$ if $\\Sigma_i =-1$;\nhence $s_i(\\tau) \\geq s_i(\\tau_0)=\\mathrm{const} > 0$ for all $\\tau \\in [\\tau_0,\\infty)$.\nFrom~\\eqref{omegaeq} we obtain\n\\begin{equation}\n\\begin{split}\n\\frac{1}{3} \\Omega^{-1} \\Omega^\\prime \\,\\Big|_{\\Omega=0} & =\n1 - \\frac{1}{3} w_i (1+\\Sigma_i) - \\frac{1}{3} w_j (1+\\Sigma_j) - \\frac{1}{3} w_k (1+\\Sigma_k) \\,\\geq \\\\\n& \\geq 1 -w_j - w_k = (1-3 w) + w_i \\geq \\mathrm{const} >0 \\:,\n\\end{split}\n\\end{equation}\nsince $s_i \\geq \\mathrm{const} > 0$. Consequently, $\\Omega(\\tau)\n\\geq \\mathrm{const} > 0$ for all $\\tau \\in [\\tau_0,\\infty)$. It\nfollows from~\\eqref{sigmaiprime} that\n\\begin{equation}\n\\Sigma_i^\\prime \\geq \\mathrm{const} > 0\n\\end{equation}\nfor all $\\tau\\in [\\tau_0,\\infty)$ by the same argument.\nThis is in contradiction to the assumption $\\Sigma_i \\leq -1$ for all $\\tau$.\n\nThus, in the second part of the proof, we can consider an arbitrary\norbit $\\gamma$ and assume, without loss of generality, that\n$\\gamma(\\tau)$ lies in the $\\Sigma$-triangle for all $\\tau \\in\n[\\tau_e,\\infty)$. Equation~\\eqref{zeq} leads to\n\\begin{equation}\nz^\\prime = 2 z (1-z) \\sum\\limits_n s_n (1+\\Sigma_n) \\geq 0\n\\end{equation}\nfor all $\\tau \\in [\\tau_e,\\infty)$, hence\n$z(\\tau) \\geq z(\\tau_e) > 0$ for all $\\tau \\in [\\tau_e,\\infty)$.\n\nWe define the function $N$ by\n\\begin{equation}\nN = (1+\\Sigma_1) (1+ \\Sigma_2) (1+\\Sigma_3)\\:.\n\\end{equation}\nThe derivative can be estimated by\n\\begin{equation}\nN^\\prime \\geq 3 \\Omega N \\left[ -\\frac{3}{2} (1-w) + \\frac{1}{2} \\sum_n \\frac{1-3 w}{1+\\Sigma_n}\\right]\\:.\n\\end{equation}\nSince $w(\\tau) \\leq \\mathrm{const} < 1\/3$ (because $z(\\tau) \\geq\n\\mathrm{const} > 0$), $N^\\prime$ is positive when at least one of\nthe $\\Sigma_i$ is sufficiently small, i.e., when $N$ itself is small\n(a detailed analysis shows that $N^\\prime \\geq 3 \\Omega N [-(3\/2)\n(1-w) + \\sqrt{3} (1-3 w) N^{-1\/2}]$). We conclude that there exists\na positive constant $N_0$ such that $N(\\tau) \\geq N_0$ for all $\\tau\n\\in [\\tau_e,\\infty)$. This in turn implies that there exists $\\nu>0$\nsuch that $\\Sigma_i(\\tau) \\geq -1 + \\nu$ for all $i$ for all $\\tau\n\\in [\\tau_e,\\infty)$, whereby $z^\\prime \\geq 2 z (1-z) \\nu$ for all\n$\\tau \\in [\\tau_e,\\infty)$.\n\nIt follows that the $\\omega$-limit of $\\gamma$ must lie on $z=1$,\ni.e., on $\\mathcal{Z}^1$. Taking into account the simple structure\nof the flow on $\\mathcal{Z}^1$, characterized by $\\Omega^\\prime = 3\n(1-\\Omega) \\Omega$, we conclude that the fixed points $\\text{FS}^1$\ngiven by $\\Sigma_1 =\\Sigma_2 =\\Sigma_3 = 0$ are the only possible\n$\\omega$-limits. \\hspace*{\\fill}\\rule{0.2cm}{0.2cm}\n\n\\begin{remark}\nIn order to demonstrate the versatility of the dynamical systems\nmethods, we have chosen here to prove Theorem~\\ref{futurethm} by\nusing techniques that are slightly different from those employed in\nSection~\\ref{locglo} (which exploit the monotonicity principle).\nHowever, it is straightforward (in fact, even simpler) to give a\nproof by making use of the hierarchy of monotone functions. Indeed,\nthe function $M_{(1)}$ ensures that the $\\omega$-limit of every\norbit lies on $\\mathcal{Z}^1$ or $\\mathcal{S}_i$; modulo some\nsubtleties, we can exclude that $\\mathcal{S}_i$ is attractive by\nusing the monotone function $M_{(3)}$ and the local properties of\nthe fixed points.\n\\end{remark}\n\n\n\n\\section{The spaces $\\mathcal{S}^0_i$ -- interpretation of solutions}\n\\label{Si0space}\n\nThe flow on the boundary subsets $\\mathcal{S}^0_i$\nis of fundamental importance in the analysis of the global dynamics of the state space,\nsee Section~\\ref{globaldynamics}. Note that except for $\\text{F}^0$ all attractors\n($\\mathrm{D}_i^0$, $\\mathrm{QL}_i^0$, $\\mathrm{KC}_i^0$, and the heteroclinic network)\nlie on $\\mathcal{S}^0_i$. For a depiction of the flow on $\\mathcal{S}^0_1$ see Figure~\\ref{cylinder}.\nIn the following we show that orbits on $\\mathcal{S}^0_1$ represent solutions\nof the Einstein-Vlasov system that are associated with a special class of\ndistribution functions. Furthermore, we investigate in detail solutions that\nconverge to the subcycle $\\mathcal{H}^0_1$ of the heteroclinic network.\n\nConsider a distribution function $f_0$ of the form\n\\begin{equation}\\label{disdisfct}\nf_0(v_1,v_2,v_3) = \\delta(v_1) f_0^{\\mathrm{red}}(v_2,v_3)\\:,\n\\end{equation}\nwhere $f_0^{\\mathrm{red}}(v_2,v_3)$ is even in $v_2$ and $v_3$.\nIn the case of massless particles, $m=0$ (and $z=0$ respectively),\nwe obtain\n\\begin{equation}\nw_1 = 0 \\:,\\qquad w_j = \\frac{g^{jj} {\\displaystyle\\int} f_0^{\\mathrm{red}} \\, v_j^2\n\\left[g^{22} v_2^2 + g^{33} v_3^2 \\right]^{-1\/2} d v_1 d v_2 d v_3}%\n{{\\displaystyle\\int} f_0^{\\mathrm{red}}\n\\left[g^{22} v_2^2 + g^{33} v_3^2 \\right]^{1\/2} d v_1 d v_2 d v_3}\\:\\,(j=2,3)\\:,\n\\end{equation}\nwhere $g^{22}$ and $g^{33}$ can be replaced by $s_2$ and $s_3$, if desired.\nIn the unbounded variables $g^{ii}$ the equations read\n\\begin{subequations}\\label{unboundedvarsz0sys}\n\\begin{align}\n\\Sigma_1^ \\prime & = - \\Omega [1+\\Sigma_1] \\:,  & (g^{11})^\\prime & = -2 g^{11} (1+\\Sigma_1)  \\\\\n\\Sigma_j^\\prime &= - \\Omega [1+\\Sigma_j -3 w_j] \\:, & (g^{jj})^\\prime &= -2 g^{jj} (1+\\Sigma_j) \\quad\\qquad(j=2,3)\\:,\n\\end{align}\n\\end{subequations}\ncf.~the remark at the end of Section~\\ref{einsteinvlasov}.\nIn particular we note that the equation for $g^{11}$ decouples; hence the full dynamics is\nrepresented by a reduced system in the variables $(\\Sigma_1, \\Sigma_2, \\Sigma_3, g^{22}, g^{33})$,\nwhich coincides with the system~\\eqref{unboundedvarsz0sys} on the invariant subset $g^{11} = 0$.\nIn analogy to the definitions~\\eqref{defdimless} we set\n\\begin{equation}\ns_1 =0 \\:, \\qquad s_2 = \\frac{g^{22}}{g^{22}+g^{33}} \\:,\\qquad s_3 = \\frac{g^{33}}{g^{22}+g^{33}}\\:,\n\\end{equation}\nso that $s_2 + s_3 =1$. This results in the dynamical system\n\\begin{subequations}\\label{boundedvarsz0sys}\n\\begin{align}\n\\Sigma_1^ \\prime & = - \\Omega [1+\\Sigma_1] \\:,  & s_1 &  \\equiv 0 \\\\\n\\Sigma_j^\\prime &= - \\Omega [1+\\Sigma_j -3 w_j] \\:,\n& s_j^\\prime &= -2 s_j [\\Sigma_j - (s_2 \\Sigma_2 + s_3 \\Sigma_3)] \\quad\\qquad(j=2,3)\\:.\n\\end{align}\n\\end{subequations}\nThis system~\\eqref{boundedvarsz0sys} coincides with the dynamical system~\\eqref{eq} induced on $\\mathcal{S}^0_1$\n(which is obtained by setting $z=0$, thus $w=1\/3$, and $s_1 = 0$ in~\\eqref{eq}).\n\nOur considerations show that the flow on $\\mathcal{S}^0_1$ possesses a direct physical interpretation:\norbits on $\\mathcal{S}^0_1$ represent solutions of the massless Einstein-Vlasov system\nof Bianchi type~I with a ``distributional'' distribution function of the\ntype~\\eqref{disdisfct}.\nNote that the system~\\eqref{boundedvarsz0sys} on $\\mathcal{S}^0_1$ must be supplemented\nby the decoupled equations~\\eqref{xeq} and $(g^{11})^\\prime  = -2 g^{11} (1+\\Sigma_1)$\nin order to construct the actual solution from an orbit in $\\mathcal{S}^0_1$.\n\nTwo structures in $\\mathcal{S}^0_1$ are of particular interest: the fixed point $\\mathrm{D}_1^0$ and\nthe heteroclinic cycle $\\mathcal{H}^0_1$, see Figure~\\ref{cylinder}.\nThe fixed point $\\mathrm{D}_1^0$ represents an LRS solution (associated with a distributional\n$f_0$); it is straightforward to show that the metric is of the form\n\\begin{equation}\\label{Disol}\ng_{11} = \\mathrm{const} \\:,\\qquad\ng_{22} \\propto t^{4\/3} \\:,\\qquad\ng_{33} \\propto t^{4\/3}\\:,\n\\end{equation}\nand $H = (4\/9) t^{-1}$.\n\nThe orbit $\\mathrm{T}^0_{22} \\rightarrow \\mathrm{T}^0_{32}$, which is part of $\\mathcal{H}^0_1$, corresponds to a solution\n\\begin{equation}\\label{teil1}\ng_{11} = g_{11}^0 \\:, \\qquad\ng_{22} = g_{22}^0 ( 3 H_0 t)^2 \\:, \\qquad\ng_{33} = g_{33}^0\\:;\n\\end{equation}\nhere, $H = (3 t)^{-1}$; $H_0$ is a characteristic value of $H$.\nFor the orbit $\\mathrm{T}^0_{33} \\rightarrow \\mathrm{T}^0_{23}$\nthe result is analogous with $g_{22}$ and $g_{33}$ interchanged.\nA more extensive computation shows that the orbit $\\mathrm{T}^0_{32} \\rightarrow \\mathrm{T}^0_{33}$\nleads to\n\\begin{equation}\\label{teil2}\ng_{11} = g_{11}^0 \\:, \\qquad\ng_{22} = g_{22}^0 \\left[ \\log(1+3 H_0 t) \\right]^2 \\:,\\qquad\ng_{33} = g_{33}^0 (1 +3 H_0 t)^2\\:,\n\\end{equation}\ntogether with $H= H_0 (1 +3 H_0 t)^{-1} (1 + [\\log (1+3 H_0 t)]^{-1})$.\n(Note that $3 H t$ is always close to $1$ and approaches $1$ for $t \\rightarrow 0$\nand $t\\rightarrow \\infty$.)\nThe result for the orbit $\\mathrm{T}^0_{23} \\rightarrow \\mathrm{T}^0_{22}$ is analogous with\n$g_{22}$ and $g_{33}$ interchanged.\n\nNow consider an orbit converging to the heteroclinic cycle as $\\tau \\rightarrow -\\infty$, i.e., $t \\searrow 0$.\nSince the orbit alternates between episodes where it is close to one of the four heteroclinic orbits,\nwe obtain a solution with alternating episodes of characteristic behaviour of\nthe type~\\eqref{teil1} and~\\eqref{teil2}; transitions between the episodes\ncorrespond to the orbit being close to the fixed points.\n\nLet $t^{(n)}$ denote a monotone sequence of times such that\nthe solution is in episode $(n)$ at time $t^{(n)}$\n(i.e., the orbit is close to one of the four heteroclinic orbits and\nfar from the fixed points); $t^{(n)}\\searrow 0$ as $n\\rightarrow \\infty$.\nSince $3 H t \\approx 1$ as $t\\searrow 0$, the sequence $t^{(n)}$ gives rise\nto a sequence $H^{(n)}$ defined by $3 H^{(n)} t^{(n)} = 1$.\nDuring episode $(n)$ the solution exhibits a characteristic behaviour of\nthe type~\\eqref{teil1} or~\\eqref{teil2} with $H_0 =H^{(n)}$\n(and $g_{22}^0 = g_{22}^{(n)}$, $g_{33}^0 = g_{33}^{(n)}$).\nA transition from one episode to another involves\na matching of the constants.\n\n\\textit{Example}.\nSuppose that the orbit is close to the heteroclinic orbit\n$\\mathrm{T}^0_{32} \\rightarrow \\mathrm{T}^0_{33}$ in episode $(n)$.\nWe obtain a behaviour of the type~\\eqref{teil2} with $H_0=H^{(n)}$.\nAs $H^{(n)} t$ gets small we see that $g_{22} \\approx g_{22}^{(n)} (3 H^{(n)} t)^2$,\n$g_{33} \\approx g_{33}^{(n)}$.\nThe next (as $t\\searrow 0$) episode corresponds to the orbit running close\nto $\\mathrm{T}^0_{22} \\rightarrow \\mathrm{T}^0_{32}$;\nthe behaviour of the solution is~\\eqref{teil1} with $g_{22}^{(n+1)}$, $g_{33}^{(n+1)}$,\nand $H_0=H^{(n+1)}$.\nThe transition between the episodes $(n)$ and $(n+1)$ is thus straightforward:\n$g_{22}^{(n+1)} (H^{(n+1)})^2 = g_{22}^{(n)}  (H^{(n)})^2$ and\n$g_{33}^{(n+1)} = g_{33}^{(n)}$.\nMatching episodes $(n+1)$ and $(n+2)$ is slightly more involved.\nThe orbit is close to the heteroclinic orbit $\\mathrm{T}^0_{23} \\rightarrow \\mathrm{T}^0_{22}$\nin episode $(n+2)$, where\n\\begin{equation}\\label{teil4}\ng_{11} = g_{11}^0 \\:, \\qquad\ng_{22} = g_{22}^{(n+2)} (1 +3 H^{(n+2)} t)^2 \\:,\\qquad\ng_{33} = g_{33}^{(n+2)} \\left[ \\log(1+3 H^{(n+2)} t) \\right]^2 \\:.\n\\end{equation}\nClose to transition time, when $H^{(n+2)} t$ is large,\nwe get $g_{22} = g_{22}^{(n+2)} (3 H^{(n+2)} t)^2$ and $g_{33} = g_{33}^{(n+2)} (\\log 3 H^{(n+2)} t)^2$.\nThe transition between episode $(n+1)$ and $(n+2)$ thus involves\nthat $g_{33}$ begins to decay logarithmically from having been approximately\nconstant.\n\n\n\n\n\n\n\n\\end{appendix}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n\n\n\\label{intro}\n\n\\textit{Cloud computing} is inexorably becoming the technology of choice among big and small businesses to deploy and manage their IT infrastructures and applications \\cite{wang2016spatial}. Infrastructure-as-a-Service (\\textit{IaaS}) is a key cloud service delivery model. Large companies such as Amazon, Google, Microsoft, and IBM provide IaaS solutions to their consumers. Examples of IaaS consumers include Software-as-a-Service (SaaS) providers and organizations such as governments, universities, and research centers \\cite{fattah2020long}. The computing resources or Virtual Machine (VM) instances are the most common IaaS services \\cite{hwang2015cloud}. The \\textit{functional} properties of a VM instance or an IaaS service include computing resources such as CPU units, memory units, storage, and network bandwidths. Examples of \\textit{non-functional} properties or Quality of Service (QoS) attributes of IaaS services are availability, price, response time, throughput, and energy efficiency \\cite{fattah2020event,fattah2020signature}.\n\n\n\nThe \\textit{market-driven} cloud service provisioning is a topical research area \\cite{varghese2018next}. There exist several key service \\textit{provisioning} models in the cloud market such as on-demand, reservation \\cite{chaisiri2011optimization, zheng2016probabilistic}, and economic models \\cite{sharma2012pricing,Thanakornworakij,PAL2013113}. We propose an \\textit{alternative strategy to the on-demand model} which would be used in conjunction. The premise is that the on-demand or reservation model makes it difficult to accurately predict service demand, thus potentially leading to either under-provisioning or over-provisioning \\cite{dustdar2011principles,jiang2011asap, islam2012empirical}.\nThe alternative model focuses on the economic model-based cloud service selection and provision for long-term IaaS composition. In that regard, the economic model-based service provisioning approach is fundamentally different from the typical on-demand and reservation models. According to the \\textit{on-demand}, the provider has a fixed set of VM instances associated with QoS and price \\cite{Hong}. The consumer may acquire and release on-demand VM instances anytime and only pay for the usage by per hour or per second. \nThe provider usually sets a discounted flat rate for the reserved instances in the \\textit{reservation} model \\cite{fattah2019long}. The consumers reserve the VM instance for a fixed time period and pay for it regardless of usage.\nAccording to \\textit{economic model} based service provisioning approaches, there exists \\textit{market competitions} among providers to set the price and QoS of their services \\cite{PAL2013113}. The market competition forms \\textit{non-cooperative games} among competitive providers and consumers \\cite{Thanakornworakij}.\n\n\n\n\\textit{We focus on the economic model-based cloud service selection and provision for a long-term period}. Economic expectations are \\textit{formally} expressed in terms of economic models \\cite{ye2014economic,sajibtsc2015,goiri2012economic}. According to the economic model-based service selection approaches \\cite{ye2014economic,yu2007efficient,kholidy2014qos}, a consumer's requests include custom VM instances, QoS parameters and the price the consumer is willing to pay for the instance (usually determined by their market research or the \\textit{consumer economic models}). Similarly, the providers follow their own economic model to \\textit{accept} or \\textit{reject} the requests from the consumers \\cite{sajibtsc2015}.\nThe economic model-based service selection and provisioning approaches are applied in different cloud markets such as spot market, SLA negotiation, and auction-based reservation models \\cite{zaman2013combinatorial}.     \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nWe assume that consumers create their \\textit{IaaS requests} following their own economic models. An IaaS provider receives a set of IaaS requests from \\textit{different} consumers. \\textit{The IaaS composition from the provider's perspective is defined as the selection of an optimal set of consumer requests \\cite{sajibtsc2015}}. The IaaS composition is a \\textit{decision-making} problem where the provider decides which requests it should \\textit{accept} or \\textit{reject}. An effective IaaS composition \\textit{maximizes} the provider's long-term economic expectations, such as profit, reputation, and consumer growth \\cite{fattah2018cp}. The economic model is the \\textit{formal tool} to select the optimal set of consumer requests to meet the provider's expectations \\cite{dash2009economic}.\n\n\n\nOur objective is to design a \\textit{qualitative economic model} for the long-term IaaS composition \\cite{sajibicsoc2016}. The qualitative economic models provide an \\textit{effective} way to select consumer requests where there exists \\textit{uncertainties} or \\textit{incomplete} information. The consumer requirements are typically uncertain and probabilistic \\cite{fattah2018cp} for the long-term period. \\textit{The provider's long-term economic expectations are also dynamic} \\cite{mistry2018long}. The qualitative economic model specifies the provider's \\textit{temporal business strategies} such as reputation building, risk management, revenue, and profit maximization \\cite{sajibicsoc2016}. These business strategies determine the service provisioning \\textit{preferences}. For example, the provider may observe very high demand for \\textit{Network-intensive services} (i.e., VM instances designed for Network-intensive applications, e.g., C5n instance type in Amazon EC2\\footnote{https:\/\/aws.amazon.com\/ec2\/instance-types\/c5\/}) in the Christmas or holiday period. The provider may prefer to provision Network-intensive services than CPU-intensive services (i.e., VM instances designed for CPU intensive applications, e.g., P3 instance in Amazon EC2\\footnote{https:\/\/aws.amazon.com\/ec2\/instance-types\/p3\/}) to increase its revenue.\n\n\n\nWe assume that an IaaS provider has its long-term qualitative economic model, i.e., the temporal service provisioning \\textit{preferences} \\cite{sajibicsoc2016}. The provider receives long-term IaaS requests from different consumers which are represented in \\textit{time series} and associated with QoS parameters and price. The \\textit{qualitative IaaS composition} is defined as the \\textit{selection} or \\textit{acceptance} of an optimal set of IaaS requests using the qualitative economic model of the provider. \\textit{We aim to provide a comprehensive framework for long-term qualitative IaaS composition}. To the best of our knowledge, apart from our previous work \\cite{sajibicsoc2016,mistry2018long}, existing research mainly focus on the \\textit{quantitative IaaS composition}. The target of the quantitative composition is to \\textit{maximize revenue and profit} of the provider for a \\textit{short-term} period without any long-term business strategies or economic model \\cite{ye2013qos,chaisiri2012optimization,zhu2010resource}. In contrast, the target of the qualitative composition is to \\textit{maximize the similarity measure} between a given set of consumer requests and the provider's qualitative economic model.\n\n\n\nWe represent the provider's long-term qualitative economic model using \\textit{Temporal Conditional Preference Networks} (TempCP-nets) \\cite{mistry2017probabilistic}. The TempCP-net \\textit{ranks} the short-term and long-term consumer requests using k-d tree according to provider's preferences \\cite{sajibtsc2014}. The qualitative composition is \\textit{transformed} into a \\textit{combinatorial optimization problem} where the objective is to select the consumer requests that maximizes the preference rankings. We explore two composition approaches: a) \\textit{global composition}, and b) \\textit{local composition} \\cite{alrifai2009combining, yu2008framework}. The global composition approach considers all the consumer requests within the composition interval which is computationally expensive \\cite{sajibicsoc2016}. The local composition approach \\textit{divides} the composition interval into several time segments and optimizes a \\textit{partial set} of requests (acceptance or rejection) in each time segment. It may \\textit{significantly improve} the runtime efficiency as we do not need to consider the whole set of requests during the entire composition period. However, the local composition approach is a \\textit{greedy approach} of sequential optimization \\cite{gnanlet2009sequential,pednault2002sequential} and may not produce the optimal result as the request selection is \\textit{temporal-dependent} on the previous acceptance or rejection decisions in other time segments. For example, when we optimize the requests from left to right time segments (i.e., January, February, March), the composition result may be different than the optimization from right to left time segments (i.e., March, February, and January). A reinforcement learning based approach called 3d Q-learning \\cite{mistry2018long} is proposed to \\textit{find the optimal sequence} of temporal selections. The proposed 3d Q-learning based composition approach is considered \\textit{off-policy} as the \\textit{learning approach has no restrictions over exploration}. The proposed approach does not consider the \\textit{temporal distribution} and \\textit{correlations} of the \\textit{historical request sets} to compose a new set of requests using \\textit{policy reuse}.  \n\n\nWe propose a novel \\textit{on-policy} based 3d Q-learning approach that effectively utilizes historical information to find the optimal selection of requests. First, the proposed learning approach reduces the run-time by removing redundant state transitions in the off-policy based 3d Q-learning approaches. Next, a novel request annotation approach based on agglomerative clustering techniques \\cite{fernandez2008solving,bouguettaya2015efficient} is proposed to capture the \\textit{intrinsic characteristics} of the historical requests such as the temporal distribution and the global preference ranking. A \\textit{novel policy reuse approach} is proposed to compose a new set of requests that effectively utilizes previous policies which are learned from historical information. The key contributions of this work are as follows:\n\n\\begin{itemize}\n    \\item A comprehensive framework to compose long-term and short-term IaaS requests based on the provider's qualitative economic model.\n    \\item An on-policy based 3d Q-learning approach to finding the optimal request selection sequence.\n    \\item A novel request annotation approach to capture the intrinsic characteristics of historical request sets.\n    \\item A novel policy reuse approach to enable effective utilization of historical information in the proposed 3d Q-learning.  \n\\end{itemize}\n\nThe rest of the paper is structured as follows. In section 2, we introduce a set of terminologies and concepts that are used to formulate the qualitative composition problem. Section 3 illustrates a motivation scenario to explain the need for sequential learning in IaaS composition. Section 4 provides a general overview of the proposed qualitative IaaS composition framework. In section 5, we describe the proposed IaaS composition approach for a new set of requests. Section 6 describes the proposed long-term qualitative composition with the previous learning experience. Section 7 presents our experiments to evaluate the proposed approaches. Section 8 summarizes the related work on economic model based cloud service composition and sequential learning approaches. Finally, Section 9 concludes the paper and discusses the limitation and future work of this paper.  \n\n\\section{Preliminaries}\n\nIn this section, We introduce a set of terminologies and concepts that are used to formulate the qualitative IaaS composition problem in this paper. We will use these terminologies throughout the paper to describe the problem and proposed solution.\n\n\n\\begin{itemize}\n    \\item \\textit{Providers and Services:} IaaS providers are referred as the provider. We consider the composition problem from the perspective of a single provider. IaaS services usually include a wide selection of VM instance types optimized to fit different use cases such as general purpose, compute optimized, memory optimized, and storage optimized \\cite{hwang2015cloud}. The VM Instance types comprise varying combinations of CPU, memory, storage, and networking capacity and give consumers the flexibility to choose the appropriate mix of resources for their applications. We consider VM instances as services. The VM instances designed for Memory-intensive applications are termed as Memory-intensive services. For example, The R5 instance in Amazon EC2 is an example of Memory-intensive services\\footnote{https:\/\/aws.amazon.com\/ec2\/instance-types\/r5\/}. Similarly, VM instances designed for CPU-intensive applications are termed as CPU-intensive services. For example, the P3 instance in Amazon EC2 is an example of Memory-intensive services\\footnote{ https:\/\/aws.amazon.com\/ec2\/instance-types\/p3\/}. \n    \n    \\item \\textit{Resources:} The resources are the capacity of the physical machines that are used to offer the VM services such as: CPU cores, memory unit, and network bandwidths \\cite{hwang2015cloud}. \\textit{We assume that provider has a fixed set of resources.} Here, the number of fixed set of resources refers to the maximum number of resource the provider may have at a certain point of time. The maximum number of resource however may change over period of time. In such a case, the number of fixed of resources is required to be updated according to the new maximum number of resources. The proposed approach can be considered as a proactive approach, where provider anticipates the maximum number of resources it can have.  Our aim to utilize these resources based on the provider's economic model.\n    \n    \\item \\textit{Consumers}: The targeted consumers are mainly the medium to large business organizations such as SaaS providers, governments, universities, and research institutes. These organizations may require services for a long-term period (e.g., 1 to 3 years).  \n    \n    \\item \\textit{IaaS Requests}: IaaS requests refer to the configuration of \\textit{functional} and \\textit{non-functional} requirements of the VM over a period of time. A consumer may need a VM with 2 vCPU, 2gb memory, and 99\\% availability in first six months of a year. The IaaS requests for that period is represented as (2 vCPU, 2 gb memory, 99\\% availability). \\textit{We assume the deterministic IaaS requests, i.e., the provider has knowledge about the long-term requests prior to the composition}. We compose these requests based on the provider's economic models. The provider defines the requests as either \\textit{short-term} or \\textit{long-term}. A request is considered short-term if only one business strategy is applicable for the life time of the requests. A request is considered long-term if more than one business strategies are applicable to the request. For instance, if the provider changes its business strategies or economic models quarterly, a request that needs reservation for a VM for 1 year is considered as a long-term request. In this circumstances, a request that reserves the VM for a one month is considered as short-term requests. Note that, we focus on the economic model-based service provisioning where VMs are reserved for a certain time period. \\textit{The burstable on-demand resources are outside the focus of this paper}. \n    \n    \\item \\textit{Conditional Preference Networks (CP-nets)}: The CP-net is a widely used tool that captures a user's conditional preferences qualitatively \\cite{cp1}. CP-Nets \\cite{boutilier2004cp} is a compact and intuitive formalism for representing and reasoning with conditional preferences under the ceteris paribus (``all else being equal\") semantics. The dynamic semantics of the preferences are indicated using a Conditional Preference Table (CPT) \\cite{boutilier2004cp}. One CP-net can only represent one business strategy at a time \\cite{sajibicsoc2016}. For example, if the business strategy is to build reputation for the first three months, a CP-net could be constructed that can graphically express the preference on higher QoS in a service than higher prices. \n    \\item \\textit{Temporal Conditional Preference Networks (TempCP-nets)}: The TempCP-net is a set of CP-nets that represents a provider's economic expectations over the long-term period \\cite{sajibicsoc2016}. If there are three business strategies for a year, the TempCP-net could be constructed with a set of three CP-nets.\n    \n    \\item \\textit{k-d Tree}: The induced graph in a CP-net may contain nodes with multi-dimensional tuples and the annotated ranking of preferences \\cite{boutilier2004cp}. The \\textit{k-d tree} is a graph indexing technique in which every node is a \\textit{k}-dimensional point \\cite{andoni2006near}. Every non-leaf node in a k-d tree can be thought of as implicitly generating a splitting hyperplane that divides the space into two parts, known as half-spaces. Points on the left and right sides of this hyperplane are represented by the left and right subtree of that node respectively. We apply the k-d tree to index the service preference rankings based on the provider's TempCP-net.\n    \n    \\item \\textit{Local IaaS Ranking}: The rank of an IaaS request is determined by its k-d tree index. The ranking of a request or a set of request considering only one period is called local IaaS ranking. For example, if an IaaS request expands from January to December, then its local IaaS ranking of January is computed considering its preference rankings on January.\n    \n    \\item \\textit{Global IaaS Ranking}: The ranking of a long-term request or set of requests considering each period is called its global IaaS ranking. For example, if an IaaS request expands from January to December, then its global ranking considers is the aggregated local ranking of each month from January to December. \n\n    \\item \\textit{Sequential Optimization}: The sequential optimization approach is a series of local optimization where the initial problem is divided into sub-problems \\cite{gnanlet2009sequential}. The local optimizations have cascading effect as the decisions in each local optimization affects the sequential decision making in successive local optimizations. The key benefit of the sequential optimization is that it reduces the search space of the global optimization significantly \\cite{pednault2002sequential}.    \n    \n    \\item \\textit{Q-learning}: Q-learning \\cite{watkins1992q} is a model-free reinforcement learning approach. The goal of Q-learning is to learn a policy, which tells an agent what action to take under what circumstances. It does not require a model of the environment, and it can handle problems with stochastic transitions and rewards, without requiring adaptations \\cite{watkins1992q}. In the context of IaaS compositions, the Q-learning is used to learn the optimal sequences of request selection. The two-dimensional Q-learning has no start and terminal states as it accepts only model-free state transitions \\cite{shani2005mdp}. A Q-learning process is termed as \\textit{off-policy} if the learning approach has no restrictions over exploration \\cite{munos2016safe}. In contrast, the \\textit{on-policy} based Q-learning process is \\textit{smart} and removes redundant state transitions considering historical information \\cite{van2016deep}.\n\\end{itemize}\n\n\n\n\n\\section{Motivation Scenario}\n\nIn this section, We illustrate an example scenario to describe the need of sequential learning in IaaS composition.  Let us consider an IaaS provider offers virtual CPU services. The provider offers 100 CPU units with 100\\% availability. We are only considering availability as a QoS parameter for simplicity. We defined three semantic levels - high, moderate, and low to express qualitative preferences of the provider for services attributes  as shown in Figure \\ref{fig:semanticTable}. We assume the provider may change the interpretation of the semantic levels based on the cloud market condition. The provider considers more than \\$1000 as high price according to Figure \\ref{fig:semanticTable} in the first year. The provider considers more than \\$1300 as high price due to predicted inflation in the second and third years. Three different preference rankings are set based on the provider's annual goal in three years.\n\n\\begin{figure}\n    \\centering\n       \n      \\includegraphics[width = .8\\textwidth]{semanticTable.pdf}\n    \\caption{Semantic preference table}\n       \\label{fig:semanticTable}\n       \\vspace{-4mm}\n\\end{figure}\n\nWe adopt the economic models of the provider as described in \\cite{mistry2018economic} to continue in this example. Figure \\ref{fig:cp} shows the provider's three economic models for three different years. In the first year, the provider wants to offer high-quality service in with a lower price to create its reputation. The most important attribute in the first year is the ``availability'' of a service followed by the ``CPU'' and the ``price''. The provider decides to maximize its profit by offering services at a higher price for lower resources and QoS in the second year. The ``price'' therefore sets the ``CPU'' and the ``availability'' in the second year. The provider's preference for the third year is to provision lower CPU-intensive services. Let us assume the provider wants to receive requests that are long-term. Therefore, the provider offers discounts on long-term service requests. A decision variable labeled $N$ is used to distinguish the type of requests. The value of $N$ is set to true ($T$) when a request is long-term. A request is considered long-term if it spans over the next period. Otherwise, the value of $N$ is set to false ($F$) to indicate the request as a short-term request. In figure \\ref{fig:cp}, $N$ is associated with ``price'' ($P$) for the first two years. In these periods ($CP1$ and $CP2$), the provider considers the high and moderate ``price'' level indifferently for long-term requests. $N$ is associated with ``availability'' ($A$) in the third year. According to $CP3$, short-term requests are provided with relatively lower ``availability\" at the same moderate price. More details about how to represent these economic models using CP-nets can be found in \\cite{sajibicsoc2016}.\n\n\n\n\n\n\n\\begin{figure}\n    \\centering\n      \\includegraphics[width = 0.9\\textwidth]{CPNets} \n    \\caption{CP-nets of a provider}\n    \\label{fig:cp}\n    \\vspace{-4mm}\n\\end{figure}\n\n\n\n\n\n\nLet us assume a set of requests is represented by $A$ in Figure \\ref{fig:req}(a). $A$ has four requests i.e.,  $\\{R1\\}$,$\\{R2\\}$,$\\{R3\\}$, and  $\\{R4\\}$ as shown in Figure \\ref{fig:req}(a). Each of these requests arrives at the beginning of the composition. A request is represented in annual segments for simplicity. For instance, $(C: High, A: low, P: moderate)$ represents a request segment of $\\{R1\\}$ in the first year. Similarly, Figure \\ref{fig:req}(a) shows the annual requirements of other consumer requests $\\{R2\\}, \\{R3\\} $ and $ \\{R4\\}$(a) for three years. The provider can select these four requests from $2^{4} = 16$ possible combinations to find the optimal composition in a brute force manner. \n\n\n\n\nThe number of possible ways to select the requests grows exponentially with the number of requests. A sequential optimization process may be applied to reduce the total number of comparisons to find the global optimal composition. Let us consider a sequential optimization approach for the request set $A$ where requests are selected from the right to left year i.e., $3^{\\text{rd}}$,$2^{\\text{nd}}$ and $1^{\\text{st}}$. There are $2^3 = 8$ comparisons are required in the third year to select the highest ranked $R3$ according to the $CP3$. Note that, in $CP3$ the highest preference order is low CPU, high price, and low availability. In $R3$, the consumer's preference order is low CPU, moderate availability, and low price. Therefore, $R3$ is the closest match request according to $CP3$. Details of the ranking technique using CP-nets can be found in \\cite{sajibicsoc2016}. Once we only accept  $R3$ in the third year, $R1$ and $R2$ are rejected in the subsequent years. In following years, the local optimization accepts $R4$ and update the solution. The optimal solution $\\{R3, R4\\}$ is calculated in ten comparisons. If we change the sequence of optimization process e.g., left to right i.e., $1^{\\text{st}}$, $2^{\\text{nd}}$ and $3^{\\text{rd}}$, the total number of comparison in the first year becomes $2^4 = 16$ to select the highest ranked $R1$. The request $R1$ has ``N\/A\" ranking in the following years. As a result, the left to right sequence produces an unacceptable solution. The right to left sequence generates optimal result when sequential optimization is applied on $A$. The same sequence may not work or give a good solution for a different set of requests. Let us consider the request set $B$ in Figure \\ref{fig:req}(c). The distribution of requests in $B$ is different from $A$. The number of comparisons becomes $2^5 = 32$  to select the highest ranked $R3$ in the third year (Figure \\ref{fig:req}(d)). As $R3$ has ``N\/A\" ranking in the second year, it can not be selected. As a result, the right to left sequence does not work for $B$. We propose a model-free learning approach to generate the optimal sequence of local optimizations.\n\n\\begin{figure}\n   \n    \\centering\n      \\includegraphics[width= 0.8\\textwidth]{images\/motive2.pdf}  \n    \\caption{ Sets of requests (a) request set $A$  (c) request set $B$}\n    \\label{fig:req}\n    \\vspace{-5mm}\n\\end{figure}\n\n\n\n\\section{A Qualitative IaaS Composition Framework}\n\nIn this section, we provide a general overview of the proposed qualitative IaaS composition framework. We also describe some common concepts in qualitative IaaS composition such as long-term IaaS request representation, long-term qualitative preferences representation, combinatorial optimization in qualitative IaaS composition.\n\nA qualitative composition framework is proposed that learns from historical information of the past request sets as shown in Figure \\ref{fig:propframe}. We assume that a set of long-term requests of consumers and the provider's qualitative preferences are available at the beginning of the composition. Our target is to find the optimal composition using a learning based approach that utilizes information of the past consumer requests. For each set of new requests, we learn the sequence of the optimal composition and save it for future request sets.\n\n\\begin{figure}\n\n\t\\centering\n  \t\\includegraphics[width= .6\\textwidth]{propframe.pdf}   \n    \\caption{Long-term qualitative composition framework}\n  \t\\label{fig:propframe}\n   \n\\end{figure}\n\nWe use TempCP-nets to represent the qualitative preferences which enable qualitative composition of the requests. The proposed framework performs indexing and ranking of the requests configuration of the TempCP-net. The indexing of the TempCP-net is built using \\textit{k}-d tree indexing that enables efficient searching of the rank of the consumer requests. The ranking of the requests configurations is computed using the provider's TempCP-nets. The ranked requests are taken as an input of a learning module. The learning module applies a reinforcement learning based method that utilizes the historical information of the past requests to find out optimal composition efficiently.  \n\n\n\n\n\n\n\n\\subsection{\\textbf{Long-term IaaS Request Representation}}\n\nThe long-term requests of the consumers are represented as time series group (TSG) of the service attributes. We denote \\(T\\) as the total service usage time. The TSG of the consumer requests is defined as \\(R_c = \\{s_{c1},s_{c2},...,s_{cn}\\} \\), where $s_{cn}$ represents a service attribute and \\(cn\\) is the number of service attributes in \\(R_c\\). We represent the service attribute time series as \\( s_{cn} = \\{(x_n,t_n)|n=1,2,3,...., T\\} \\), where \\(x_n\\) is the value of \\(s_{cn}\\) at the time interval \\(t_n\\). Figure \\ref{fig:req} shows two sets of requests where each request in a set has 3 service attributes ($cn=3$) i.e., CPU, availability, and price. Each requests has three year intervals ($T =3$). Each service attribute may have different types of value during these intervals.\n\n\n\n\n\\subsection{\\textbf{Long-term Qualitative Preferences based on TempCP-nets}}\n\\label{sec:cp}\n\n\n\n\nWe need an efficient tool to represent the provider's qualitative preferences. We define a set of attributes $V = \\{X_{1},..., X_{n}\\}$ which is defined over the finite, discrete domain $D(X_n)$ and semantic domains $S(X_n)$. The attributes are either functional or non-functional. Examples of functional attributes are CPU ($C$), Memory ($M$), and so on. Availability ($A$), Price ($P$), Latency ($LT$) are examples of QoS attributes. A mapping table $Sem\\_Table(X_{n}, x_{n})$ is used to map $x_{n}$ in $D(X_{n})$ into $s_{n}$ in  $S(X_{n})$ where $s_{n} = Sem\\_Table(X_{n}, x_{n})$. An example of a semantic table is shown in Figure \\ref{fig:semanticTable}. We assume the preferences order and semantics of $V$ is static within an interval in a long-term composition period. However, they may vary within different intervals. We consider a set of decision variables $DN = \\{N_{1}, N_{2},....,N_{d}\\}$. A decision variable may represent request type, requests duration, and etc. We assume that the decision variable is a binary variable. Therefore, it takes true of false $\\{T, F\\}$ values. For instance, the decision variable is set true for a request if it spans to the next interval. \n\n\\begin{figure}\n\n\t\\centering\n  \t\\includegraphics[width= .8\\textwidth]{induced.pdf}   \n    \\caption{Induced preference graph with decision variable for CP1}\n  \t\n\t\\label{fig:f3}\n\t\\vspace{-3mm}\n\\end{figure}\n\nThe service provisioning time $T$ is considered into $m$ intervals and represented as  $T=\\sum_{k=1}^{m}I_{k}$, where $I_k$ is an interval in $T$. We assume that the provider sets a preference ranking of each service configuration at each interval $I_k$. The preference rankings of service configurations are expressed over the complete assignments on $V$ and $DN$ with the semantic domain $Sem\\_D^{I_{k}}(V)$. $O^{I_{k}}$ is denoted as the set of service configurations for an interval $I_{k}$. A total order $(\\succeq)$ of the service configuration set represents a preference ranking for an interval. For example, $o_{1} \\succeq o_{2}$ denotes that a service configuration $o_{1}$ is equally or more preferred over $o_{2}$. The preference relation $o_{1} \\succ o_{2}$ denotes that the service configuration $o_{1}$ is preferred over $o_{2}$. If the preferences are indifferent or non-comparable, we denote the relation using $o_{1} \\sim o_{2}$. $T \\sim F$ means that the provider does not care about the true and false values of a decision variable. \n\nThe size of the service configuration set $O^{I_{k}}$ grows exponentially with the number of intervals. Therefore, direct assignment of all possible preferences over the long-term period is always not feasible. We represent the provider's long-term preferences on service configurations using a TempCP-net. A TempCP-net is represented as set of CP-nets with semantic preference tables for each interval of the composition period. We denote a TempCP-net as $\\text{TempCP-Net} = \\{(CP^{I_{k}}, Sem\\_Table^{I_{k}}, I_{k})\\;|\\;\\forall k \\in [1,m]\\}$. A CP-net can be considered as a graphical model that formally represents qualitative preferences and reasons about them. A CP-net $CP^{I_{k}}$ in the interval $I_{k}$ consists a direct graph $G$ which is defined using $V$ and $DN$. Each node in $G$ represents an attribute $X_i \\in V$. The nodes of $DN$ are represented by a dashed circle. A node of $V$ is represented by a solid circle. In this work, we only consider acyclic CP-nets to represent the provider's qualitative preferences. The CPT of each node is denoted by  $CPT(X_{i})$ which contains a total order $\\succ^{i}_{u}$ with each instantiation $u$ of $X_{i}$'s parents $Pa(X_{i}) = U$ \\cite{cp1}. For example, $Pa(P)=C$ and $CPT(C)$ contains $\\{C1, C2\\}$ in $CP3$ while preferences are made over $\\{P1, P2, P3\\}$ (Figure \\ref{fig:cp}). A preference outcome $o$ of a CP-net is obtained by sweeping through the CP-net from top to bottom setting each variable to its preferred value given the instantiation of its parents \\cite{wang2012wcp}. A preference order $o \\succ \\acute{o}$ is called a consequence of a CP-net, if $o \\succ \\acute{o}$ can be obtained directly from one of the CPTs in the CP-net. For example, the fact that $(A2, C2, P2)$ is preferred to $(A2, C1, p2)$ is a direct consequences of the semantics of $CPT(C)$ in $CP1$ for the long-term requests (Figure \\ref{fig:cp}). The set of consequences $o\\succ \\acute{o}$ creates a partial order over all possible service configurations of an acyclic CP-net.\n\n\n\nFigure \\ref{fig:f3} shows the induced preference graph generated by $CP1$ which is a directed acyclic graph (DAG). There are two induced graph generated based on the value of the decision variable $N$ (i.e., true and false). The true value of $N$ represents the induced preference graph for long-term requests. The false value of $N$ represents short-term requests. $(A1, C1, P1)$ is considered as the most preferred request for the short-term requests. There is no outgoing edge from $(A1, C1, P1)$. Similarly, $(A2, C1, P3)$ has no incoming edge because it is least preferred request configuration. There is an edge between $(A2, C1, P1)$ and $(A1, C1, P1)$. According the preference statement $A1 \\succ A2$ in the $CPT(CP1)$, $(A1, C1, P1) \\succ (A2, C1, P1)$. $(A1, C1, P1)$ and $(A1, C1,\\\\ P2)$ does not have any outgoing edge because they are the highest preferred configuration (Figure \\ref{fig:f3}). The induce preference graph of a CP-net is constructed by pairwise comparison of all service configurations. The complexity for ordering queries for a TempCP-net in an interval is $O(ndq^2)$ where $n$ and $d$ is the number of attributes and decision variables respectively and the number of output configurations is $q$.\n\n\n\n\n\\subsection{\\textbf{Combinatorial Optimization in Qualitative IaaS Composition}}\n\n\\begin{figure}[t!]\n\n\t\\centering\n  \t\\includegraphics[width= 0.8\\textwidth]{kd.pdf}   \n    \\caption{\\textit{k}-d tree indexing of the induced preference graphs}\n    \\vspace{-3mm}\n\t\\label{fig:f4}\n\\end{figure}\n\nGiven a provider's TempCP-net and a set of long-term requests $R$, the IaaS composition is defined as the selection of an optimal set $\\bar{r} \\subseteq \\bar{R}$ that produces the best similarity measure with the TempCP-Net. We consider qualitative preference rankings as the foundation of the similarity measure. First, we index preference rankings from TempCP-Net. We then perform a similarity search on the indexed TempCP-Net which is denoted as $Pref(\\text{TempCP-Net}, \\bar{r})$. Hence, the objective of the IaaS composition is to minimize the ranking output $Pref(\\text{TempCP-Net}, \\bar{r})$. \n\n\\subsubsection{\\textbf{Indexing Preference ranks}} \n\\label{sec:kd}\n\n\n\nThe preference rank of a request configuration is denoted as $Sem\\_Req$  $=(s_{1}, ...,s_{n}) \\;|\\; \\text{where } s_{i} \\in S(X_{i}),  \\text{and }X_{i} \\in V$. It is found by a pre-order traversal of the induced graph. The time complexity of searching the preference rank over the induced graph is $O(n)$. A request configuration $(s_{1}, ...,s_{n})$ is considered as a multidimensional vector. We use a \\textit{k}-d tree \\cite{jia2010optimizing} to improve the searching process. There exist different multi-dimensional indexing structures such as B-tree, B+-tree, k-d Trees, Point Quadtrees, R, R*, R+ Trees \\cite{sellis1997multidimensional,robinson1981kdb}. The k-d tree is a \\textit{congruent} choice in IaaS composition as the IaaS preference ranking requires the multi-dimensional value queries \\cite{sajibicsoc2016,nam2004comparative}. \\textit{Note that, finding the optimal multi-dimensional indexing structures for IaaS composition is outside the focus of this paper}.\n\nThe \\textit{k}-d tree is a binary tree that is used for indexing some nodes in a space with k dimensions. We represent each service configuration $o$ (i.e., a node in the induced graph) as a k-dimensional point in the \\textit{k}-d tree. Each node in each level splits its all children along a specific dimension into two subspace, known as half-spaces. Each subspace is represented by either a left or a right sub-tree of that node. A canonical method is used to build the \\textit{k}-d tree \\cite{jia2010optimizing}. The construction algorithm follows a cycle during the selection of splitting planes. For example, at the root, all children are split based on ``availability\" plane in Figure \\ref{fig:f4}. The children of the root split their children along ``CPU\" plane. The grandchildren of the root have ``price-aligned\" planes. Finally, the great-grandchildren have again planes aligned with availability. \n\nLet us assume there are $n$ points in an induced preference graph. We place the median point found in one dimension at the root of the \\textit{k}-d tree. Every other point smaller and larger than the root in the same dimension are placed into right and left sub-tree respectively. This process creates a balanced \\textit{k}-d tree where the runtime is $O(n \\;log(n))$ \\cite{jia2010optimizing}. We annotate each node of the \\textit{k}-d tree with its respective preference ranking obtained from the induced graph. For instance, the root node $(A2,C2,P2)$ in the \\textit{k}-d tree in Figure \\ref{fig:f4} has the preference ranking 6 that is obtained from its induced graph. We construct the \\textit{k}-d tree indexing for each value of the decision variable $N$. For example, two different \\textit{k}-d tree indexing are shown in Figure \\ref{fig:f4} to represent short-term and long-term service configurations. The service configurations with indifferent preference have the same ranking value. For example, the provider's preferences on $(A2,C1,P2)$ and $(A1,C2, P1)$ are indifferent for long-term requests. Both service configurations are annotated with preference ranking 4 in Figure \\ref{fig:f4}.\n\n\\subsubsection{\\textbf{Local and Global Preference Ranking}}\n\n\n\nA request may not be inclusive, i.e., exactly fit within an interval of TemCP-net. It may overlap two or more intervals of TempCP-nets. An overlapping request $R$ (interval $[T_{0}, T_{m}]$) is divided into smaller inclusive segments where each segment fits within an interval of the TempCP-net. The attributes that have \\textit{temporal semantics} require such segmentation. For instance, ``Price\" is considered as an attribute with \\textit{temporal semantics} in a consumer request. If a request requires 20 units of CPU for 12 months with total \\$120, a monthly segmentation interprets the provisioning of 20 CPU units for \\$10. Let us consider an attribute $X$ in $R$ that has \\textit{temporal semantics}. If the segmentation is applied in $[T_{j}, T_{k}]$, the new segments are calculated using the following equation according to \\cite{sajibicsoc2016}: \n\\vspace{-3mm}\n\n\\begin{equation}\n\\label{eqn:rank}\nx_{i}^{[T_{j},T_{k}]} =  x_{i}^{[T_{0},T_{m}]} \\times \\frac{|T_{k}-T_{j}|}{|T_{m}-T_{0}|}\n\\end{equation} \n\n The requests are ready to be composed after the temporal segmentation. We define a set of $N$ requests as $\\bar{R} = \\sum_{i=1}^{N} R_i$. We use the following composition rules to combine the requests in a set $\\bar{R}$ \\cite{sajibtsc2015}:\n\n\\begin{itemize}\n    \\item The rule of summation: $\\bar{x_{i}} = \\sum_{i=1}^{N} x_{i}$, where $X_{i} \\in \\{C, M, NB, RT, P\\}$.\n    \\item The rule of maximization: $\\bar{y_{i}} = max(y_{i}), \\forall \\; i \\in [1,N]$, where $Y_{i} \\in \\{A, TP\\}$.\n\\end{itemize}\n\nThe preference ranking function of a set of requests is $Pref(\\text{TempCP-Net}, \\bar{R}): V \\rightarrow [1,n]$, which outputs the order of $\\bar{R}$ according to the preferences (\\textit{k}-d tree of the TempCP-net). $\\bar{R}$ is transformed  $\\acute{\\bar{R}} = \\{(s_{i}, I_{j})\\;|\\;s_{i} \\in S(X_{i}),  X_{i} \\in V, \\text{and } T = \\sum_{j=1}^m I_{j}\\}$ based on $Sem\\_Table$ of the TempCP-net. First, we define the local similarity measure, i.e., preference rankings for a time segment and then we define the global objective function for the entire composition period.\n\n\\begin{itemize}\n    \\item \\textit{Local preference ranking}: Let us consider $M^{i}(s_{i})$ in the interval $i$ is the function that outputs the preference ranking by temporal matching of  $\\acute{\\bar{R}}$ segments with the \\textit{k}-d tree. The temporal matching process or the searching algorithm starts from the root node and traverses the tree recursively. The search algorithm returns the preference ranking of a node if that matches with a request configuration. For instance, the algorithm returns ranking 10  by performing 10 comparisons for the short-term request $(A2, C1, P3)$ in Figure \\ref{fig:f4}. If a query search point is not found in the \\textit{k}-d tree, it is discarded in the composition. The complexity of performing a query in a \\textit{k}-d tree is on average $O(log(n))$ for each service.\n    \n    \\item \\textit{Global Preference Ranking}:  As the long-term requests are divided into local segments, we aggregate the local preference rankings to generate the global preference ranking as follows:\n    \\begin{equation}\n    \\label{eq:ranking}\n        Pref(\\text{TempCP-Net}, \\bar{R}) = \\sum_{i=1}^{m} M^{i}(s_{i})\n    \\end{equation}\n\\end{itemize}\n\n\\section{IaaS Composition for a New Set of Requests}\n\nIn this section, we illustrate the proposed IaaS composition approach for a new set of requests, i.e., IaaS composition without any prior knowledge of incoming requests. We introduce a sequential IaaS composition approach that leverages reinforcement learning techniques to compose incoming requests.\n\nWe assume that initially the IaaS provider does not store a history of incoming requests. Each set of incoming requests are considered as a new set of requests and the composition is performed from scratch for the new set of requests. We identify three approaches to compose a new set of requests:\n\\begin{itemize}\n    \\item \\textit{Brute-force approach}: This approach generates all the combinations of requests over the total composition period. The preference ranking of each combination is computed using the global preference ranking Equation \\ref{eq:ranking} and pairwise compared. The combination of requests that generates the minimum global ranking is returned as the optimal composition. If the number of requests is $N$, the time complexity of this approach is exponential ($2^N$).\n    \\item \\textit{Global optimization approach}: The target of the global optimization is to improve the runtime efficiency from the brute-force approach. We apply Dynamic Programming (DP) \\cite{sajibicsoc2016} to reduce the re-computation of similarity measure of the same combinations of requests. The DP is designed to compute the similarity measure of a large combination of request sets by breaking it into smaller combinations (overlapping subproblem) structure. The results of the subproblems are stored in a temporary array which enables avoiding repeated computation. We denote $\\bar{R}(N)$ as a set of $N$ requests and $i \\in [1,N]$ as the $i_{th}$ request. $\\tau(\\bar{R}(N), k)$ denotes the subset of requests of size $k$ which generates the maximum preference rankings among all requests of size $k$. We start with base case $k=1$, i.e., a set consists of only one requests. The highest ranked request $i$ is computed by pairwise comparison of preference rankings:\n    \\vspace{-5mm}\n    \n    \\begin{align*}\n    &\\text{Base case, } \\tau(\\bar{R}(N), k=1) = R_{i}\\\\ \\notag &\\text{where } Pref(\\text{TempCP-Net}, R_{i}) \\text{ is minimum.} \\notag\n    \\end{align*}\n\n    \n     For $k>1$, it either accepts the $N_{th}$ request (the $k_{th}$ place is already filled) or rejects it (reduces $\\bar{R}(N)$ to $\\bar{R}(N-1)$). We have two optimal substructures:\n     \n     \\vspace{-6mm}\n    \n    \\begin{align*}\n        \\bar{R_{i}} &= \\{N \\cup \\tau(\\bar{R}(N-1), k-1)\\} \\\\\n        \\bar{R_{j}} &= \\tau(\\bar{R}(N-1), k) \\notag\n    \\end{align*}\n     \n     \n     The $\\bar{R_{i}}$ and $\\bar{R_{j}}$ are computed separately. The re-computation of overlapping substructures is avoided by building a temporary array in a bottom-up manner \\cite{kimes2004restaurant}. if $\\bar{R_{i}}$ returns the minimum preference ranking, it should be returned as the optimal composition,  $\\tau(\\bar{R}(N), k) =\\bar{R_{i}}$ if $Pref(\\text{TempCP-Net},\\bar{R_{i}})$ $<  Pref(\\text{TempCP-Net},  \\bar{R_{j}})$. Otherwise, the request is removed from the composition, i.e., $\\tau(\\bar{R}(N), k) =\\bar{R_{j}},\\text{if } Pref(\\text{TempCP-Net},  \\bar{R_{i}}) \\geq  Pref(\\text{TempCP-Net}, \\bar{R_{j}})$. The complexity of finding $C(\\bar{R}(N), k)$ is $O(N^{k})$. As there are at most $N$ requests to be considered in a set, we solve the DP from bottom up manner in the following sequence:  $\\tau(\\bar{R}(N), \\;1)$,  $\\tau(\\bar{R}(N), \\;2), \\cdots, \\tau(\\bar{R}(N), N)$.  The final complexity of the DP based solution is $O(N^{O(N)})$.  \n\n    \\item \\textit{Local sequential optimization approach:} This approach optimizes requests in each time segment. The key reason is that we do not need to consider the whole set of requests during the entire composition period. We only consider a partial set which is applicable in a specific temporal segment. It should reduce the runtime complexity significantly. In Fig \\ref{fig:newExample1}, the local optimization could be divided into two segments: optimization with ${A,B}$ in the first year ($OP_{i}$) and optimization with ${A, B, C}$ ($OP_{j}$) in the second year. As the request sets are deterministic, we can perform the optimization sequences in different orders, i.e., $<OP_{i},OP_{j}>$ or $<OP_{j},OP_{i}>$. Local optimizations are dependent on the accepted or rejected requests during previous optimizations in a sequence. For example, if the sequence is $<$1st year, second year$>$ in Fig \\ref{fig:newExample1} and we reject request $B$ in the first year, the candidate request set for local optimization is the second year reduces to $<A, C>$.\n\\end{itemize}\n\n\\begin{figure}[t!]\n\t\\centering\n  \t\\includegraphics[width=.5\\textwidth]{Example-Overlap.pdf}  \n    \\caption{Overlapping requests in different temporal segments}\n    \\vspace{-4mm}\n\t\\label{fig:newExample1}\n\\end{figure}\n\nWe propose a heuristic based sequential optimization approach for an IaaS composition in \\cite{sajibicsoc2016}. To improve the quality of the solution, we develop a reinforcement learning based approach to find the best local service provision policy, i.e., the best selection of requests in the optimal temporal sequence. \n\n\n\\subsection{\\textbf{Sequential IaaS Composition using Reinforcement Learning}}\n\nWe formulate the long-term IaaS composition problem, i.e., the selection of requests, as a sequential decision process. We begin with a time segment in the TempCP-net and select a request for the composition. The selection of the requests for the next segment depends on the previous selections of requests as accepted overlapping requests are already committed for both the segments. \n\n\nA sequential decision process is modeled in different approaches such as Multi-Armed Bandit (MAB), Markov Decision Process (MDP), or Partially Observable Markov Decision Process (POMDP) \\cite{kaelbling1996reinforcement}. We observe that the state-action-reward situation in the IaaS composition is similar to the MDP. We may start the selection of requests (actions) in a time segment and can compute the local preference ranking of the selected requests by matching the temporal segment in the selected requests with the corresponding temporal \\textit{k}-d tree of the given TempCP-net. The local preference ranking may be considered as the reward function. After the selection of requests in a segment, the composition approach may transit to any segment and make new selections. The process may continue until total reward cannot be further maximized (convergence) (as we consider ranking values as the reward, maximizing total rewards refer to minimizing global ranking). MAB or POMDP is also applicable in our context as they are special cases of MDP. For example, MAB is a special case of MDP that has only one state. We select MDP as the general sequential decision process.\n\n\n\\begin{figure}[t!]\n\n\t\\centering\n  \t\\includegraphics[width=.5\\textwidth]{newExample2.pdf}  \n    \\caption{State-action transitions for sequential composition}\n   \n\t\\label{fig:newExample2}\n\t\\vspace{-4mm}\n\\end{figure}\n\nFigure \\ref{fig:newExample2} shows the all possible state transitions for the request sets in the Figure \\ref{fig:newExample1}. The state-action is represented by the pair [interval, Request]. In Fig \\ref{fig:newExample2}, [1,A] refers that request A in Fig \\ref{fig:newExample2} is selected in the interval 1. We consider bi-directional transition edges. The key reason is it does not specify the transition sequence. For example, if request C is selected in the second interval first, i.e. [2,C], the next transition may happen to any of the [1,A], [1,B] or [1,AB]. [1,AB] represents that both requests A and B are selected in the interval 1.  \n\n\n\nAs the composition environment is dynamic, model-free learning, e.g., reinforcement learning (RL) is usually more applicable than the model-based learning algorithms to implement the MDP. To solve the composition problem using a reinforcement learning (RL) approach, we treat each new request sets as a new environment and learn the optimal selection of requests through multiple interactions with the environment. We focus on the Q-learning approach as a reinforcement learning (RL) approach. Note that other deep learning approaches could be implemented in our context. However, we are not focused on providing a comparative study of machine learning approaches in this paper; instead, we use what we think is a sound approach in our context. Our primary target is to apply an unsupervised machine learning approach in the long-term IaaS composition. In that respect, our primary target is to evaluate the effectiveness of general reinforcement learning, i.e., Q-learning and its proposed variations in our context. In the future work, we will compare the performance of the proposed approach with other deep reinforcement learning approaches.\n\n\\subsubsection{\\textbf{Q-learning based Approach in IaaS Composition}}\n\nA widely used reinforcement learning is Q-learning. In a Q-learning based method, past interactions in the same environment are utilized to learn the optimal policy \\cite{watkins1992q}. The sequences of request selection i.e., \\textit{experience} over different intervals can be treated as past interactions in the context of IaaS composition. We define experience as a tuple  $<s, a ,r, \\acute{s}>$ where $s$ is the current interval, $a$ is the selected request in $s$, $r$ is the reward for selecting $a$, and $\\acute{s}$ is the next interval. We represent the history of interactions as $<s_{0}, a_{0}, r_{1}, s_{1}, a_{1}, r_{2},....>$. We formally define the Q-learning environment in the context of qualitative composition as follows:\n \n\\begin{itemize}\n\\item \\textit{Environment}: The environment consists of consumers and the provider. The consumer requests are represented into time series groups and the provider's long-term qualitative preferences are represented in TempCP-net. The environment is deterministic, i.e., incoming requests and TempCP-net is given prior to the composition.\n\n\\item \\textit{State ($s$)}: The tempCP-net is usually consists of several temporal CP-nets for different time intervals or segments. Each time interval or segment is treated as a state. The number of states is finite.  \n\\item \\textit{Action ($a$)}: The selection or rejection of a request is treated as an action in our context. In Figure \\ref{fig:newExample1}, the second segment (2nd year) has 3 available requests $\\{A,B, \\text{ and} \\;C\\}$. Hence, the possible set of acceptance actions is $\\{A,B,C,AB,AC,BC,ABC\\}$.  \n\n\\item \\textit{Policy ($\\pi$)}: It is a function that determines which action to perform, i.e., selection of requests in a given state. If $S$ is the set of all states and $A$ is the set of all possible actions, the policy ($\\pi$) is represented as $\\pi(s): S \\longrightarrow A$.\n\n\\item \\textit{Reward function ($RWD$)}: We match the action, i.e., selected request segments with the corresponding segment in TemCP-net. We consider the local preference ranking as the reward. The reward function is defined based on the equation \\ref{eq:ranking} as $RWD(s,a)=Pref(\\text{TempCP-net(s)},a)$ at a state $s$ and actions $a$.\n \n\\item \\textit{Value function ($V$)}: $V^{\\pi}(s)$ is the state-value function in the sequential decision process. It is the expected cumulative preference ranking starting from state $s$ following a policy $\\pi$.\n\\end{itemize}\n\nWe apply the basic Q-learning approach to the IaaS composition. We propose a modified Q-learning approach for the long-term IaaS composition.\n\n\n\\subsubsection{\\textbf{IaaS Composition using 2d Q-learning}}\n\nA value function $V^{\\pi}(s)$ represents how good is the temporal sequence of a segment for the composer to select requests. The value function depends on the policy by which the composer chooses its actions. Among all possible value-functions, there exists an optimal value function that has a higher value than other state functions:\n\\vspace{-5mm}\n\n\\begin{equation}\n    V^{*}(s) = max_{\\pi} V^{\\pi}(s) \\;\\;\\forall s \\in S \n\\end{equation}\n\nThe optimal policy $\\pi^{*}$ corresponds to the optimal value function: \\vspace{-3mm}\n\n\\begin{equation}\n    \\pi^{*}= arg\\;max_{\\pi} V^{\\pi}(s) \\;\\;\\forall s \\in S \n\\end{equation}\n\nA recursive function called $Q$ is usually used in a Q-learning process and represented as $Q^{\\pi}(s,a)$ \\cite{watkins1992q}. It is used to calculate the cumulative reward using the policy $\\pi$. We map $Q^{\\pi}(s,a)$ with the expected global preference ranking of choosing the request $a$ in the interval $s$ for the policy $\\pi$. The probability of moving from interval $s$ to interval $\\acute(s)$ is denoted by $P(\\acute{s}| s,a)$ in Equation \\ref{eq:q3} where $a$ is the selected requests. The current preference ranking of $a$ in $s$ is denoted by $R(s,a,\\acute{s})$ where $\\acute{s}$ is the next interval. The future global ranking is denoted by $V^{\\pi}(\\acute{s})$ in Equation \\ref{eq:q3}.\n\n\n\n\\vspace{-3mm}\n\n\\begin{equation}\n\\label{eq:q3}\nQ^{\\pi}(s,a) = \\sum_{\\acute{s}} P(\\acute{s}| s,a)(R(s,a,\\acute{s})+\\gamma V^{\\pi}(\\acute{s}))\n\\end{equation}\n\n\n\n\nIn a Q-learning process, a table $Q[S,A]$ is maintained where $S$ denotes the set of states and $A$ denotes the set actions \\cite{watkins1992q}. $Q[S,A]$ is used to store the current value of $Q^{\\pi}(S,A)$. The value of $Q^{\\pi}(S,A)$ in the context of long-term IaaS composition is computed using temporal differences. Therefore, we create a table $Q[S,A]$ where $s$ denotes the set of interval or segment and the set of action is denoted by $a$. We set the initial $Q[s,a]$ to 0 for each $(s,a)$. we start the process from an random state ($s$) and executes a random action ($a$) for a reward $r$. The next interval is also selected randomly. We use $\\epsilon$-greedy policy to restrict the randomness over the time. The idea is the composer should explore the state-action sequences randomly, in the beginning, to find better future discounted preference rankings, later the randomness should be reduced. Here, $\\epsilon$ is defined as the probability of exploration. The exploration is equivalent to picking a random action in action space. If $\\epsilon$ = 1, the composer will always explore, and never act greedily concerning the action-value function. Therefore, $\\epsilon < 1$ in practice, so that there is a good balance between exploration and exploitation. The higher value of alpha usually assigns higher weights to the current estimate than the previous estimate. The learning process terminates when there are no further updates on Q-values. It is also known as convergence values. Once the Q-learning reaches the convergence state, the optimal policy is found using $Q[S,A]$ in Equation \\ref{eq:q5}. At each segment the action with highest reward is selected. An example of 2d $Q[S,A]$ is shown in Figure \\ref{fig:f7} where the number of segments is 5 and the number of actions is 10. The best action in segment 5 is $A10$ or $A5$.\n\n\n\\begin{equation}\n\\label{eq:q4}\nQ[s,a] = (1-\\alpha)Q[s,a] + \\alpha (r+\\gamma max_{\\acute{a}}Q[\\acute{s},\\acute{a}]) \n\\end{equation} \n\n\n\\begin{equation}\n\\label{eq:q5}\n\\pi^{*}(s) = argmax_{a}Q[s,a] \\;|\\; \\forall a \\in A(s)\n\\end{equation}\n\n\\subsubsection{\\textbf{IaaS Composition using 3d Q-learning}}\n\n\\begin{figure}[t!]\n\n  \t\\centering\n  \t\\includegraphics[width=0.8\\textwidth]{fig7_qvalues.pdf}   \n    \\caption{(a) Q-values in a 2d $Q[S,A]$ (b) Q-values in a 3d $Q[S,A, O]$}\n    \\vspace{-3mm}\n\t\\label{fig:f7}\n\\end{figure} \n\n\nThe 2d Q-learning has no start and terminal states as it accepts only model-free state transitions \\cite{shani2005mdp}. The long-term effect of sequential order is indicated by $Q[S,A]$. However, it is not possible to keep track of the execution order in the 2d Q-learning process. For example, there are several possible state transitions can take places in Figure \\ref{fig:newExample2} such as $\\{[1,A]\\rightarrow [2,B]\\}$ and  $\\{[2,B]\\rightarrow [1,A]\\}$. Here, $Q[1,A]$ represents the expected global preference rankings of selecting action $A$ in the first year irrespective of the sequence of selection (first or second). A similar explanation is also applied to $Q[1,A]$. Besides, the existing Q-learning approaches for the composition allow multiple execution order. Such order in Figure \\ref{fig:newExample2} is  $\\{[1,A] \\rightarrow [2,BC] \\rightarrow [1,B]\\}$. Note, the selection of request $A$ and $B$ in the first year is taken at two different positions in a sequence.\n\n\n\n\nWe introduce a three-dimensional Q-learning in \\cite{mistry2018long} using a 3d table $Q[S,A,O]$ to store the Q values. $O$ represents the set of execution orders. Therefore, a particular state-action pair $(s,a)$ may have different values depending on the execution order $o$. If a set of requests is selected from the first segment at the first step of the composition, it may have a different preference ranking than if it is selected at the last step of the composition. Figure \\ref{fig:f7} illustrates an example of 3d $Q[S,A,O]$. Here, the ranking of $A10$ is 1 in segment 5 when it is performed as the starting state. However, the preference ranking of $A10$ changes into 9 when it is performed in the third step. An extension of Equation \\ref{eq:q4} is shown in Equation \\ref{eq:q6} for a 3d Q-learning process. The $\\acute{o}$ denotes the next execution order after $o$ and $\\alpha$ denotes the learning rate. The 3d Q-learning selects the start state ($s$), perform an action ($a$) with reward $r$ arbitrarily. From the start state, it observes all the possible states in different orders. \n\n\n\n\\vspace{-4mm}\n\n\\begin{align}\n\\label{eq:q6}\nQ[s,a,o] = &(1-\\alpha)Q[s,a,o] \\\\ \\notag\n&+ \\alpha (r+\\gamma max_{(\\acute{a},\\acute{o})}Q[\\acute{s},\\acute{a},\\acute{o}]) \n\\end{align} \n\n\n\n\n\\subsubsection{\\textbf{On-policy based 3d Q-learning}}\n\\begin{figure}\n\n  \t\\centering\n  \t\\includegraphics[width= .8\\textwidth]{redundant-states.pdf}   \n    \\caption{On-policy state-action transitions in 3d Q-learning (Green colored are allowed from [1,A,1] and [1,B,2]}\n    \\vspace{-4mm}\n\t\\label{fig:newexample3}\n\\end{figure}\n\nThe 3d Q-learning increases the number of explorations in the learning process than the 2d Q-learning. The $n\\times m$ Q-matrix from two dimensions is extended to $n\\times m \\times p$ Q-matrix. The number of states is denoted by $n$, the number of actions is denoted by $m$ and $p$ is the number of segments. As the exploration space increases, 3d Q-learning requires more time to learn compared to the 2d Q-learning. The 3d Q-learning based composition approach is considered as off-policy, as the composer has no restrictions over exploration. We transform the off-policy approach into an on-policy learning approach by intelligent reduction of state transitions as follows:\n\n\n\\begin{itemize}\n    \\item \\textit{State sequence policy}: Once a request is rejected, it is not considered anymore in following local optimization. Therefore, we do not need to consider the same states multiple times. We introduce state sequence policy where each state is visited only once in a policy $\\pi$. \n    \\item \\textit{Removing redundant state transitions}: The rejected request in a local segment may appear in another segment if the request overlaps the two segments. All the state-action pairs that contain the rejected request should not be considered as the next state transitions. For example, if we accept request $A$ in the first year, request $B$ is rejected in Figure \\ref{fig:newExample1}. However, request $B$ overlaps into the second year. Hence it should be removed from the next candidate transitions. Figure \\ref{fig:newexample3} depicts the on-policy state transitions when the request $A$ in the first year is selected at first in the sequence.        \n\\end{itemize}\n\nWe represent the on-policy 3d Q-learning process for IaaS composition in Algorithm \\ref{alg:qlearning}. Algorithm \\ref{alg:qlearning} runs multiple episodes or round to perform the 3d Q-learning process. The algorithm uses Equation \\ref{eq:q6} to update the Q-values in each episode. A $\\epsilon$-greedy policy is used by algorithm \\ref{alg:qlearning} where the optimal action is selected for execution. The optimal action is denoted by $arg\\;max_{(a,o)}Q(s, a,o))$ where the probability of the optimal action is $(1 - \\epsilon)$. The greedy policy optimizes $Q(s, a,o)$ continuously. It incorporates the unique state sequence policy. When a request is rejected, we remove it from the candidate action sets. The learning process continues in a loop up to a maximum number $k$ or Q-values converge to the optimal value. Once the Q-values reach convergence state, we use Equation \\ref{eq:q7} to compute the optimal policy which is similar to 2d Q-learning process \\cite{wang2010adaptive}.  \n\n\\vspace{-4mm}\n\\begin{align}\n\\label{eq:q7}\n\\pi^{*}(s) = &argmax_{(a,o)}Q[s,a,o] \\\\ \\notag\n&\\text{where } \\forall a \\in A(s) \\text{ and } o \\in [1, |A(s)|]\n\\end{align}  \n \n\n\\begin{algorithm}\n\\fontsize{8pt}{8pt}\\selectfont\n    \\caption{The on-policy 3d Q-learning process to compose IaaS requests}\n    \\label{alg:qlearning}\n    \\begin{algorithmic}[1]\n  \n   \n    \\STATE Initialize Q(s,a,o) to 0\n    \\FOR {each episode to $K$}\n    \\STATE $s \\gets s_{0}$\n    \\STATE execution order, $o\\gets 1$\n    \t\\WHILE{$o \\neq$ total number of segments}\n    \t \t\\STATE Choose action $a$ from $s$ in $o$ using $\\epsilon$-greedy policy.\n    \t \t\\STATE  Execute $a$, observe reward $r$ and next state $\\acute{s}$\n    \t \t\\STATE $Q[s,a,o] \\gets (1-\\alpha)Q[s,a,o] + \\alpha (r+\\gamma max_{(\\acute{a},\\acute{o})}Q[\\acute{s},\\acute{a},\\acute{o}])$\n    \t \t\\STATE create candidate $[s,a]$ based on redundant state transitions\n    \t \t\\STATE $s \\gets \\acute{s}$ using $\\epsilon$-greedy policy. \n    \t \t\\STATE  $o \\gets o+1$ \n\t\t\\ENDWHILE\n    \\ENDFOR\n    \\end{algorithmic}\n    \n\\end{algorithm} \n\n\n\\section{Long-term Qualitative Composition with Previous Learning Experiences}\n\nWe aim to utilize the knowledge of composing past requests to compose new incoming requests efficiently. In this section, we describe the proposed long-term qualitative composition approach that leverages past learning experience. We analyze different types of requests patterns and illustrate how to identify similar request sets. Once we identify a similar requests set from the past incoming requests, we reuse previously learned policy to compose new incoming requests. \n\nThe optimal sequence of state-action may vary depending on the distribution of the requests over the time and their rankings. For example, Figure \\ref{fig:pattern} shows four types of request patterns, i.e., almost sparse pattern, almost dense pattern, chain pattern, and mixed pattern. \n\n\n\n\\begin{itemize}\n    \\item \\textit{Almost Sparse Pattern}: Figure \\ref{fig:pattern}(a) shows a set of \\textit{almost sparse  request pattern}. Most requests are short and disjoint, i.e., do not overlap between two intervals. The composition may be performed in parallel by taking the short-term requests.\n    \\item\\textit{Almost dense Pattern}: Figure \\ref{fig:pattern}(b) shows a set of requests where the requests are mostly \\textit{overlapped} between intervals. An overlapping request spans for the next interval. Most requests are accepted in the first step of the selection.\n    \\item\\textit{Chain Pattern}: Figure \\ref{fig:pattern}(c) shows a chain pattern where the requests are mostly short-term and overlapped between intervals. The requests are also evenly distributed between the intervals in a chain pattern.\n    \\item \\textit{Mixed Pattern}: Figure \\ref{fig:pattern}(d) shows a mixed pattern where both long-term and short-term requests are overlapped and evenly distributed.\n\\end{itemize}\n\n\\begin{figure}\n    \\centering\n      \\includegraphics[width=.9\\textwidth]{requests_pattern}   \n    \\caption{Different requests patterns: a) almost sparse pattern, b) almost dense pattern, c) chain, (d) mixed patterns \\cite{sajibicsoc2016}}\n    \\vspace{-4mm}\n    \\label{fig:pattern}\n\\end{figure} \n\n\nApplying the Q-learning method each time when a new set of requests arrives can be expensive. Instead of learning a new set of incoming request every time, we apply the experience from the previously learned request sets. We propose a qualitative composition framework as shown in Figure \\ref{fig:flowchart}. The proposed framework takes a set of incoming requests and the TempCP-net of the provider. First, the given request set is annotated with its global preference ranking and overlapping ratio and matched with the existing sets of requests. Initially, there is no existing request set. A Q-learning algorithm is applied to the request set using the TempCP-net. The output of the Q-learning algorithm is a matrix called Q-value. The learned Q-value matrix is stored with the corresponding request set for future. \n\n \\begin{figure}\n   \n    \\centering\n      \\includegraphics[width=.6\\textwidth]{flowcharts.pdf}   \n    \\caption{A qualitative composition framework using policy library from Q-learning}\n      \\vspace{-4mm}\n    \\label{fig:flowchart}\n\\end{figure}\n\nEach time a new set of requests arrives the proposed framework find the similarity with existing request sets. The similarity is measured through a hierarchical clustering method called \\textit{agglomerative clustering method} \\cite{Growendt2017,fernandez2008solving,bouguettaya2015efficient}. For each set of requests, we apply the\\textit{ agglomerative clustering method} to build a corresponding clustering tree. To measure the similarity between two request sets, the correlation coefficient of their corresponding clustering trees is computed. If the similarity is greater than a predefined variable $S$, then the Q-value matrix of the corresponding matched request set is applied to compose the new request set. We set the value of $S$ based on the trial and error in the experiment. The existing Q-value matrix may not be fully applicable to a new set of requests. In such a case, the proposed framework applies the Q-value matrix partially (a policy reuse approach) and learns the rest of the sequence. \n\n\n\n\\label{sec:dq}\n\n\nIn our previous work, we calculate similarity between different types of requests concerning the statistical distribution of their resources attributes such as normal, left-skewed and right-skewed distributions of CPU, memory, and network bandwidth \\cite{mistry2018long}. The proposed approach however is unable to capture intrinsic characteristics of different request patterns such as temporal distribution or global ranking. Therefore, it may not correctly utilize the historical information of the previous consumers' request sets.\n\n\n\n\\subsection{\\textbf{Clustering Methods to Find Similar Request Sets}}\n\nWe use clustering techniques to compute the similarity between a new request set and past request sets. Clustering is a well-known data analytic techniques to capture the intrinsic features of a set of data and group them on different sets based on similarity. Clustering is suitable where manual tagging of data is expensive. Moreover, the prior knowledge for manual tagging may not be available or insufficient. In such a case, clustering is a preferred option over supervised learning approaches such as classification and regression \\cite{Growendt2017,fernandez2008solving,bouguettaya2015efficient}. There are many clustering techniques in the existing literature. We focus on partitional and hierarchical clustering approaches in the IaaS composition. \n\n\\subsubsection{\\textbf{Partitional Clustering}}\n\nPartitional clustering methods produces a flat partition of that optimizes an objective function. The objective function needs to be predefined. The most well-known partitional algorithm is \\textit{K-means} clustering. The main steps of K-means clustering of a request set are as follows:\n\\begin{enumerate}\n    \\item Create randomly few centroid points for the requests. \n    \\item Assign each request to the closest centroid. \n    \\item Calculate the central point of the newly created clusters and update the centroid accordingly\n    \\item Repeat the previous last two steps until there is an object left to resign to another cluster. \n\\end{enumerate}\n\nThe computational complexity of the K-means clustering is $O(NK)$ where $N$ is the number of requests and $K$ is the number of the clusters. The K-means clustering is efficient concerning computational clustering as it requires linear computational time. However, the performance of K-means depends on how the value of $K$ is chosen. It is difficult to determine the optimal value of $K$ when prior knowledge is inadequate or absent. \n\n\n\\subsubsection{\\textbf{Hierarchical Clustering}} \n\n\nThe hierarchical clustering method is a mainstream clustering method because it can be applied to the most types of data. Although hierarchical clustering method has a higher complexity compare to the K-means, it does not need predefined parameters. Therefore, hierarchical clustering are more suitable for handling real-world data. There are two main approaches for hierarchical clustering bottom up and top down. The bottom-up approach aggregate individual data points to the most high-level cluster. The complexity is usually $O(N^2)$ for the bottom-up approach. However, it may go up to $O(N^2logN)$. The complexity of the top-down approach is $O(2^N)$. The top-down approach is usually more expensive than a bottom-up approach. The bottom-up approach is generally known as \\textit{agglomerative hierarchical clustering} \\cite{murtagh2012algorithms,bouguettaya1996line} . \n\n\\subsubsection{\\textbf{Agglomerative Clustering based Similarity Measure}} \n\nWe use an agglomerative clustering based approach to reuse the existing policies for a new incoming request set based on the history of the past request sets. The clustering approach is applied to a set of requests to construct a clustering tree. The clustering tree captures the intrinsic features of the requests and group the requests based on their similarities. When a set of requests arrives, we build a clustering tree and compare it with the existing clustering trees to find the most similar clustering tree using the correlation coefficient.\n\n\nWe annotate each request in a set of requests with its global rank and overlapping ratio to construct a clustering tree to capture the temporal aspects and global ranking of a request set. The global rank is computed by the equation \\ref{eq:ranking}. The overlapping ratio of a request is the ratio of the number of operation interval and the number of total intervals of the composition. The overlapping ratio of a request $R_i$ is computed by the following equation: \n\n\\begin{equation}\n    O(R_i) = \\frac{\\text{Number of intervals of } R_i}{\\text{Total Number of Intervals}}\n\\end{equation}\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=.7\\textwidth]{plot.pdf}\n    \\caption{Annotation of a request set using global preference ranking and overlapping ratio (a) annotation table (b) annotation plot}\n    \\label{fig:plot}\n    \\vspace{-2mm}\n\\end{figure}\n\nThe overlapping ratio for the request $R3$ in the request set $A$ (Figure \\ref{fig:req}) is $2\/3=.667$. Let us assume a request set is annotated with the global preference ranking and the overlapping ratio of each request as shown in Figure \\ref{fig:plot}. Now, we construct the clustering tree based on the following steps:\n\n\n\\begin{enumerate}\n\n    \\item Each request is considered as a cluster. If there are $N$ requests in a set of requests, then the number of clusters is $N$.\n    \\item A $N X N$ distance matrix is constructed based on the \\textit{Euclidean Distance} (Equation \\ref{eqn:euclid}) of each pair of requests according to their global preference ranking $GPR$ and overlapping ratio $OR$. \n    \\item The closest pair of clusters is selected and merge them into a single cluster.\n    \\item The distances between the new cluster and each of the old clusters are computed.\n    \\item Steps 3 and 4 are repeated until there exists only a single cluster. \n\\end{enumerate}\n\n\\vspace{-3mm}\n\n\\begin{equation}\n    E.D = \\sqrt{\\sum{GPR(R1-R2)}^2+OR(R1-R2)^2}\n    \\label{eqn:euclid}\n\\end{equation}\n\nWe can perform step 4 in different ways based on different hierarchical clustering approaches \\cite{murtagh2012algorithms,fernandez2008solving,Growendt2017,bouguettaya1996line}. We consider different conventional approaches to measure the distance between two clusters which are as follows:\n\n\\begin{enumerate}\n    \\item SLINK: SLINK stands for single linkage clustering method. In this method, two clusters are joined based on the distance of their nearest pair of elements. Only one member of each cluster is considered to compute the distance \\cite{murtagh2012algorithms}. It is also known as \\textit{nearest neighbour}(NN) clustering method.  \n    \\item CLINK: CLINK is short for complete linkage clustering method. Two clusters are joined based on the distance of the farthest pair of elements \\cite{Growendt2017}. It is also known as \\textit{farthest neighbour}(FN) clustering method.  \n    \\item UPGMA: This unweighted pair-group method using arithmetic average is also known as the average linkage clustering method \\cite{fernandez2008solving}. We compute the distance between two clusters based on the average distance of each pair of elements from two clusters.\n\\end{enumerate}\n\n\\begin{figure}[t!]\n    \\centering\n   \n    \\includegraphics[width=0.9\\textwidth]{clusters}\n    \\caption{Hierarchical clustering steps}\n    \\label{fig:clusters}\n     \\vspace{-3mm}\n\\end{figure}\n   \n\nAny of the above hierarchical clustering approaches, i.e., SLINK, CLINK, UPGMA could be used for the IaaS composition of new requests. We use the SLINK or nearest neighbor approach as it is a widely used clustering approach and effective in time-series data clustering \\cite{berkhin2006survey}. \\textit{Note that, finding the optimal clustering approach for IaaS composition is out of the focus of this paper}.\n\nA clustering construction process is shown in Figure \\ref{fig:clusters}. Initially, we perform the annotation as shown in Figure \\ref{fig:plot}. In Figure \\ref{fig:clusters}(a), \\{R4\\} and \\{R5\\} are the nearest to each other. \\{R4\\} and \\{R5\\} are put in the same cluster. Similarly, \\{R2\\} and \\{R3\\} are the nearest to each other and joined in the same cluster. In Figure \\ref{fig:clusters}(b), \\{R6\\} is the nearest to the \\{R4\\}. Therefore, \\{R6\\} is joined with \\{\\{R4\\},\\{R5\\}\\}. In the next step, \\{R1\\} is the nearest to \\{R3\\}. \\{R1\\} is joined with \\{\\{R2\\},\\{R3\\}\\}. In Figure \\ref{fig:clusters}(c), there are only two clusters which are \\{\\{R4\\},\\{R5\\},\\{R6\\}\\} and \\{\\{R1\\},\\{R2\\},\\{R3\\}\\}. These two clusters are joined based on \\{\\{R5\\} and \\{R2\\} who are the nearest request to each other from two clusters. Finally, there is only one cluster left. \n\n\nOnce we build clustering trees for different sets of requests, we need to compute the coefficient of correlation between different clustering trees. We use \\textit{Cophenetic correlation coefficient} to compare two clustering trees \\cite{sokal1962comparison}. A cophenetic correlation coefficient determines how well a clustering tree preserves the pairwise distance between the original requests before clustering. \\textit{The cophenetic distance between two requests in a clustering tree is the height of the clustering tree where the two branches that include the two requests merge into a single branch.} We compute the cophenetic correlation coefficient for each clustering tree and use them to measure similarities between two clustering trees. Given a set of requests R, its corresponding clustering tree T, R(i,j) is the Euclidean distance between $i$th and $j$th requests, and T(i,j) is the cophenetic distance where two requests merged first time in the clustering tree, we compute the cophenetic correlation coefficient using the following equation \\cite{sokal1962comparison}: \n\\begin{equation}\n    c = \\frac{\\sum_{i<j}(R(i,j)-\\bar{R})(T(i,j)-\\bar{T})}{\\sqrt{[\\sum_{i<j}(R(i,j)-\\bar{R})^2][\\sum_{i<j}(T(i,j)-\\bar{T})^2]}}\n    \\label{eqn:cop}\n\\end{equation}\n\n\\subsection{\\textbf{Q-learned Policy Reuse in IaaS Composition}}\nLet us assume we are given a library of Q-matrices where the corresponding incoming requests set has a cophenetic distance to a new set of requests, $c$ below a certain threshold $ThC$. Our target is to reuse the Q-matrices to find the optimal composition of the new set of requests. We reuse existing policy library in two ways: a) greedy policy reuse approach, b) policy reuse in on-policy 3d Q-learning approach:\n\n\\subsubsection{\\textbf{Greedy Policy Reuse Approach}}\n\nThe best policy reuse approach should not involve Q-learning approach for a new set of requests at all. However, it is challenging to apply an existing policy directly to a new set of requests due to the new request sets, e.g., new states and actions in the environment. However, the number of temporal segments remain the same as the provider's long-term TempCP-net does not change. As a greedy approach we can partially reuse the policy, i.e., only the temporal sequences from the previous history could be applied to perform local optimization. Note that, the greedy policy reuse approach does not involve any reinforcement learning to compose a new set of requests. The steps for greedy policy reuse approach are described below:\n\n\\begin{enumerate}\n    \\item Find the similar Q matrices where cophenetic distance $c$ is below a certain threshold $ThC$ using the Agglomerative Clustering approach.\n    \\item Select the Q matrix that is learned for the request set of min($c$).\n    \\item Generate the optimal policy ($\\pi^{*}$) from the Q matrix.\n    \\item Let us assume  $D$ is the sequence of state transitions in ($\\pi^{*}$).\n    \\item Following the sequence $D$, perform local optimization using Dynamic programming in each segment. Accept the requests from the output of the optimization. Rejected requests are removed from the candidate list for the next available sequence.\n    \\item Return the set of accepted requests when local optimization is performed in all segments in $D$. \n\\end{enumerate}\n\n\n\n\n\\subsubsection{\\textbf{Policy Reuse in the On-policy 3d Q-learning Approach}}\n\nWe omit the reuse of learned actions, i.e., selection of requests in the greedy approach. Here, we focus on policy reuse in the exploration process in the proposed Q-learning approach. We adopt the policy reuse approach for two-dimensional Q-learning approach \\cite{fernandez2006probabilistic} for the proposed on-policy 3d Q-learning approach. The key idea in policy reuse is to integrate the past policy as a probabilistic bias in the exploration strategy of the new learning process for composition. \n\nWe define $\\mu$ as the policy exploitation probability that defines the exploitation of past policy, the new policy or a random exploration. Hence, the $\\epsilon$-greedy state transition is replaced with $\\mu$-$\\epsilon$-greedy state transition as follows:\n\\begin{enumerate}\n    \\item Follow the past policy $\\pi_{past}(s)$ to select the next state-action with the probability $\\mu$\n    \\item Exploit the current policy $\\pi_{past}(s)$ to select the next state-action with the probability $(1-\\mu)\\epsilon$.\n    \\item Select the next state-action randomly with the probability $(1-\\mu)(1-\\epsilon)$.\n\\end{enumerate}\n\nOne issue in the policy reuse is to find the corresponding actions as the partial request, i.e., request in an interval may not be an exact match with the new request. We perform time-series similarity measure \\cite{sajibtsc2015} between the request sets. If the similarity index is higher than a threshold $(TMS)$, we consider a one to one mapping between the actions and apply the past policy to the new request sets. If we have a library of $N$ Q matrices, we modify the on-policy 3d Q-learning approach with policy reuse strategy in the Algorithm \\ref{alg:qlearningp}.\n\n\n\\begin{algorithm}[t!]\n\\fontsize{8pt}{8pt}\\selectfont\n    \\caption{The on-policy 3d Q-learning process with policy reuse}\n    \\label{alg:qlearningp}\n    \\begin{algorithmic}[1]\n  \n   \n    \\REQUIRE $L$ is library of $N$ Q matrices shorted on cophenetic distance $c$\n    \\REQUIRE $H$ is library of $N$ past optimal policy  $\\pi_{past}(s)$ from the Q matrices.\n    \\REQUIRE $E$ is the extra episodes for random exploration\n    \\REQUIRE the policy exploitation probability $\\mu$\n    \\STATE Initialize Q(s,a,o) to 0\n    \\FOR {each episode to $N+E$}\n    \\STATE $s \\gets s_{0}$\n    \\STATE execution order, $o\\gets 1$\n        \\WHILE{$o \\neq$ total number of segments}\n             \\STATE Choose action $a$ from $s$ in $o$ using $\\mu$-$\\epsilon$-greedy policy.\n             \\STATE  Execute $a$, observe reward $r$ and next state $\\acute{s}$\n             \\STATE $Q[s,a,o] \\gets (1-\\alpha)Q[s,a,o] + \\alpha (r+\\gamma max_{(\\acute{a},\\acute{o})}Q[\\acute{s},\\acute{a},\\acute{o}])$\n             \\STATE create candidate $[s,a]$ based on redundant state transitions\n             \\STATE $s \\gets \\acute{s}$ using $\\mu$-$\\epsilon$-greedy policy. \n             \\STATE  $o \\gets o+1$ \n        \\ENDWHILE\n    \\ENDFOR\n    \\end{algorithmic}\n        \n\\end{algorithm} \n\n\n\n\n\n\n\n\n\n\n\n\\section{Experiments}\n\\label{exper}\n\nIn this section, we discuss our experiments to evaluate the proposed approach. First, we discuss different techniques to evaluate the proposed approach. Next, we discuss the experiment setup with real world datasets. Finally, the experiment results and their evaluation are discuss.\n\nWe conduct a set of experiments to evaluate the proposed long-term qualitative composition framework in terms of accuracy and runtime efficiency. First, we analyze the efficiency of the proposed long-term qualitative composition framework for a new set of requests (without history). We compare the proposed approaches with four state-of-the-art techniques: a) Global Dynamic Programming \\cite{zou2012qos}, b) 2-d Q-learning \\cite{moustafa2013multi}, c) Heuristics based optimization \\cite{sajibicsoc2016}, and d) On-policy SARSA learning approach \\cite{wang2017integrating}. \n\n\n\n\\begin{itemize}\n    \\item \\textit{Global Dynamic Programming}: A QoS-aware dynamic service composition approach is proposed in \\cite{zou2012qos} that utilizes the concept of global dynamic programming (DP). The global DP approach finds the all possible solution through developing sub-optimal solutions considering the provider's economic model. The global DP approach is able to find the best solution if exists. However, it poses scalability issue during runtime due to comparisons among a large number of candidates. \n    \n    \\item \\textit{2-D Q-learning}: A multi-objective service composition approach is proposed in \\cite{moustafa2013multi} that leverages reinforcement learning technique to deal with conflicting objectives and various QoS constraints. The proposed approach uses reinforcement learning to deal with the uncertainty characteristic inherent in open and decentralized environments.  \n    \n    \\item \\textit{Heuristic based optimization}: A heuristic based approach is proposed in \\cite{sajibicsoc2016} that utilizes sequential optimization. The proposed approach considers several sequences of local optimizations, i.e., left to right, right to left and random. During each iteration, higher utility requests (i.e., trade-off ratio of higher rankings with smaller request length) are selected over higher utility requests. Finally, long-term requests are accepted through collaborative decisions of local optimizations, i.e., based on the global utility score.\n    \\item \\textit{On-policy SARSA learning approach}: A modified SARSA (State-Action-Reward-State-Action) algorithm is proposed as the on-policy reinforcement learning approach for adaptive service composition \\cite{wang2017integrating}. The difference between SARSA and Q-learning is that SARSA selects requests (action) following the same current policy and updates its Q-values. However, Q-learning applies the greedy policy, i.e.,  updates its Q-values based on the maximum preference rankings of available actions.\n\\end{itemize}\n\nThe efficiency of the proposed long-term qualitative composition framework is analyzed for a given set of historical request sets. \\textit{We consider the outputs of the brute-force approach as the baseline or validation set}. In the brute-force approach, the minimum preference ranking of all the combinations of requests over the total composition period are computed using the global preference ranking Equation \\ref{eq:ranking} and pairwise comparisons. As the brute-force approach has exponential runtime, it is not applicable in real world applications. We compare the performance of proposed the greedy policy reuse and the 3d Q-learning with policy reuse approach when the Agglomerative clustering is used for similarity measure. We compare the proposed approach with a Kolmogorov-Smirnov (K-S) test based composition approach \\cite{mistry2018long} which is a well-known technique to find correlations between incoming requests and the set of historical requests based on statistical distribution. The historical learned Q-values are used as the initial values (rather than 0) in the K-S test based Q-learning approach \\cite{mistry2018long}.\n\n\n\\subsection{\\textbf{Simulation Setup with Real World Datasets}}\n\\label{ex:sim}\n\nLong-term consumer requests are generated from publicly available datasets. Google cluster trace \\cite{clusterdata} is used to generate functional requirements of the consumer requests. Google cluster traces provide CPU and Memory workloads of 70 jobs. It does not include the QoS requirements of IaaS consumers. WS-dream dataset \\cite{JiangICWS2012} is a real-world cloud QoS performance datasets consist response time and throughput information of one hundred cloud services. As we consider both the functional and non-functional (QoS) attributes in IaaS requests, we generate 70 long-term IaaS requests by creating a random one-to-one mapping between Google cluster and WS-dream dataset. Note that WS-dream dataset does not include IaaS requirements on two QoS attributes, i.e., availability and price which are important for the proposed economic model-based IaaS composition framework. Hence, we randomly generated availability and price values for each IaaS requests.\n\n\n\\begin{table}[t!]\n\\centering\n\\caption{Requests distribution description}\n\\label{tab:request}\n\\scalebox{1}{\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}Distribution \\\\ type\\end{tabular}} & \\multicolumn{3}{c|}{\\begin{tabular}[c]{@{}c@{}}Request length \\\\ (in months)\\end{tabular}} \\\\ \\cline{2-4} \n & 1-3 & 4-8 & 9-12 \\\\ \\hline\nNormal & 20\\% & 60\\% & 20\\% \\\\ \\hline\nRight-skewed & 20\\% & 20\\% & 60\\% \\\\ \\hline\nLeft-skewed & 60\\% & 20\\% & 20\\% \\\\ \\hline\nRandom & random & random & random \\\\ \\hline\n\\end{tabular}\n}\n\\end{table}\n\nFinding actual IaaS provider's business strategy is difficult in the real world. Therefore, Ten tempCP-nets are synthesized where each CP-net has twelve intervals. We consider six attributes namely CPU, memory, response time, availability, price, throughput, and request length. A decision variable is used to represent request length. We generate the dependency randomly to build different types of business strategy. We define ten levels of semantic to generate semantic preference tables. The preference statements for the CPT of each attribute is created randomly. The requests are categorized into four different distributions which are normal, right-skewed, left-skewed, and random. When most of the consumers request the median length of services, we consider the distribution as a normal distribution. Similarly, temporal higher overlapping requirements generate right-skewed distribution and temporal lower overlapping requirements generates a left-skewed distribution \\cite{mistry2018long}. For each category of the distribution, we generate five request sets where each set has ten to thirty requests. Table \\ref{tab:request} shows the description of each distribution. To create a normal distribution of requests, sixty percent of the total requests are segmented into 4 to 8 month intervals. Rest of the requests are segmented into 1 to 3 month interval and 9 to 12 month intervals. We create other distributions in a similar manner. The starting time of the requests in distribution is generated using poison distribution. \n\n\n\n\n\n\n\n\\subsection{\\textbf{Efficiency of the Proposed Framework for a New Set of Requests}}\n\n\\subsubsection{\\textbf{Evaluating Accuracy}}\n\\label{ex:s1}\n\n\n\n\nIn the first experiment, we do not consider the history of Q-learned values. Each request set is composed individually against 10 different TempCP-nets. Initially, moderate learning rate ($\\alpha =0.5$) is used for both off-policy and on-policy Q-learning based composition approaches. The outputs are normalized, averaged and compared against the brute force approach. We define the accuracy of a composition as the normalized preference ($NP$):\n\n\\begin{equation*}\n    NP = \\frac{\\text{ranking from a composition approach}}{\\text{ranking from the brute-force approach}} \n    \\label{eqn:cop1}\n\\end{equation*}\n\n\n\n\n\n\n  \n  \n\n\n\nA higher normalized preference value means higher accuracy and a lower value means lower accuracy. The Figure \\ref{fig:exp1} depicts the performances of the proposed approach in different request distributions. In all the distributions (Figure \\ref{fig:exp1}), both the off-policy and on-policy 3D Q-learning approach have similar accuracy to the DP based approach. Both approaches have significant higher accuracy than the 2d Q-learning, heuristic-based approaches, and on-policy SARSA approach. The accuracy in the proposed off-policy 3d Q-learning approach is inversely correlated with the number of requests. An increase in the number of requests is followed by a decrease in accuracy for off-policy 3d Q-learning approach irrespective of the type of distribution. It is mainly because an increase in the number of requests results in an increase of transition probabilities in state-action sequences of Q-learning which may cause local optima. There is no significant decrease in the on-policy Q-learning approach for a higher number of request sets except for left-skewed distribution. The possible explanation is that the on-policy Q-learning approach removes redundant state transition probabilities. As there is less number of overlapping requests in left-skewed distribution, the on-policy Q-learning approach behaves similarly to the off-policy 3d Q-learning approach. The accuracy in the heuristic based approach is increased when the number of requests is increased. The accuracy in SARSA is almost similar to the heuristic-based approach. \n\nThe proposed off-policy and on-policy 3D Q-learning approaches demonstrate 35\\% higher accuracy than the existing approaches, i.e., SARSA, Q-learning, and Heuristic based approaches in normal, right-skewed, and random distribution. In the case of left-skewed distribution, the accuracy reduces slightly (especially for large number of request sets), but still 15\\% higher than the existing approaches. A large portion (around 60\\% to 80\\%) IaaS requests in the left-skewed distribution is short-term (0-3 months). However, the focus of the paper is long-term compositions. We will focus on achieving the consistent higher accuracy in different distributions in the future work.\n\n\n\n\\begin{figure}[t!]\n    \\vspace{-8mm}\n      \\subfloat[]{\\includegraphics[width=0.50\\textwidth, height=0.5\\textwidth]{normal.pdf}} \\hspace*{-.01em}\n      \\subfloat[]{\\includegraphics[width=0.50\\textwidth, height=0.5\\textwidth]{Right.pdf}}  \\\\ [-3ex]\n      \\subfloat[]{\\includegraphics[width=0.50\\textwidth, height=0.5\\textwidth]{Left.pdf}} \\hspace*{-.01em}\n      \\subfloat[]{\\includegraphics[width=0.50\\textwidth, height=0.5\\textwidth]{random.pdf}}  \\\\     \n    \\caption{\\small Accuracy of the DP, on-policy and off-policy Q-learning, SARSA and heuristic approaches in different distributions of requests(a) normal, (b) right-skewed, (c) left-skewed, and  (d) random. \\normalsize}\n    \\label{fig:exp1}\n    \\vspace{-3mm}\n\\end{figure}\n\n\nNext, we analyze the effect of learning rate $(\\alpha)$ on the accuracy of the proposed framework. We use 3 different learning rates: a) low ($\\alpha =0.2$), b) moderate ($\\alpha =0.5$), and c) high ($\\alpha =0.8$) in the proposed on-policy and off-policy Q-learning approaches in the random distribution set. We randomly choose 10 request sets and the proposed framework is applied to each set separately with the three different learning rates. The averaged accuracy in different learning rate is shown in Fig \\ref{fig:learning-rate}. We find that the low and moderate learning rate generates higher and acceptable accuracy for both the on-policy and off-policy 3D Q-learning approaches. However, the higher learning rate reduces the accuracy by around 30\\% for the proposed 3d and 2d Q-learning based composition approaches. The key reason is that the higher learning rate overestimates the expected future reward or preference ranking and reduces the state-action exploration process. However, the learning rate has a lower influence on the accuracy of SARSA. As SARSA applies in-policy Q-updates with reduced exploration, different learning rate creates an almost similar number of random explorations which can bring any random update (increase, decrease or remain the same) on the output.  \n\n\n\\begin{figure}[ht]\n\\vspace{-8mm}\n    \\centering\n      \\includegraphics[width=0.5\\textwidth]{learning-rate.pdf} \n      \\vspace{-5mm}\n    \\caption{Preference rankings in different learning rates}\n    \\vspace{-4mm}\n    \\label{fig:learning-rate}\n\\end{figure} \n\n\n\\subsubsection{\\textbf{Evaluating Run-time Efficiency in the Composition without History}}\n\n\\begin{figure}[ht]\n\\vspace{-8mm}\n    \\centering\n      \\includegraphics[width=0.5\\textwidth]{Runtime-historyless.pdf}   \n    \\caption{Runtime efficiency in composition approaches}\n   \n    \\label{fig:runtimehistoryless}\n\\end{figure}\n\nWe calculate the execution time of each composition for different numbers of incoming requests. We randomly pick 5 requests sets from all the distributions with the same number of request sets. Total 25 compositions are performed on request sets with 10, 15, 20, 25 and 30 incoming requests. The average execution time is described in Figure \\ref{fig:runtimehistoryless}. All the composition approaches have similar execution time for a lower number of incoming requests, i.e., sets with 10 and 15 requests. However, for a higher number of requests, the best runtime is generated by the sequential approach and the global DP approach generates the worst case. It defines the applicability of sequential local composition approach over global composition approach for a higher number of requests. We find that the SARSA approach is the quickest among the reinforcement learning approaches due to its on-policy nature. Similarly, the proposed on-policy 3d learning approach runs around 35\\% faster than the off-policy 3d q-learning approach and around 65\\% faster than the 2d q-learning composition approach. The key reason is the on-policy 3d q-learning approach removes redundant overlapping requests in each episode of its convergence for a composition.\n\nThe existing heuristic-based sequential composition approach aggressively reduces the solution space and computation time \\cite{sajibicsoc2016}. As a result it is highly effective in real world scenarios where runtime efficiency is more important than accuracy. The proposed approach achieves 35\\% higher accuracy using only 5\\% increase in computation time than the heuristic-based approach. The impact of this slight increase is negligible in runtime. Moreover, Figure 17 depicts that the proposed approach is scalable in runtime with a higher number of requests set. We conclude that the proposed approach has a higher time efficiency in real world scenarios to achieve the high accuracy.\n\n\n\n\\subsection{\\textbf{Efficiency of the Proposed Policy Reuse Approach with History}}\n\nIn the second experiment, we consider that a history of learning experiences are available as a database of learned Q-matrices during a composition. The proposed policy reuse approaches apply the learned policies from a similar set of requests based on the agglomerative clustering approach. All 70 requests are clustered into three sets based on the cophenetic correlation coefficient:\n\n\\begin{itemize}\n    \\item \\textit{Set A}: It consists of a highly similar request sets with normalized cophenetic correlation coefficient ranging from (.8 to 1).\n    \\item \\textit{Set B}: It consists of an average similar request sets with normalized cophenetic correlation coefficient ranging from (.6 to .8).\n    \\item \\textit{Set C}: It consists of a lowly similar request sets with normalized cophenetic correlation coefficient ranging from (.3 to .6).\n\\end{itemize}\n\n\\begin{figure}[t!]\n \\vspace{-8mm}\n      \\subfloat[]{\\includegraphics[width=0.50\\textwidth, height=0.5\\textwidth]{History-high-similarity.pdf}} \\hspace*{-.01em}\n      \\subfloat[]{\\includegraphics[width=0.50\\textwidth, height=0.5\\textwidth]{History-med-similarity.pdf}}  \\\\ [-3ex]\n      \\subfloat[]{\\includegraphics[width=0.50\\textwidth, height=0.5\\textwidth]{History-low-similarity.pdf}} \\hspace*{-.01em}\n      \\subfloat[]{\\includegraphics[width=0.50\\textwidth, height=0.5\\textwidth]{Runtime-with-history.pdf}}  \\\\     \n    \\caption{\\small Accuracy of policy reuse approaches in different distributions, (a) highly similar (Set A), (b) average similar (Set B), (c) lowly similar (Set C), and (d) Average Runtime efficiency in the policy reuse approaches \\normalsize}\n    \\label{fig:exp2}\n    \\vspace{-3mm}\n\\end{figure}\n\nNote that, we consider a request set as new if its cophenetic correlation coefficient ranges from 0 to 0.3 when compared with existing request sets. All the requests in set A, B and C are composed individually with the off-policy Q-learning approach and the resulted Q-matrices and policies are stored in a database. Figure \\ref{fig:exp2}(a), (b) and (c) depicts the accuracy of the proposed approaches in Set A, B and C. We find that the greedy policy reuse approach performs poorly in every distribution (average accuracy around 35\\%). The key reason is that it ignores critical information of request overlapping and the future expected to rank from the current selection. We consider the output from the individual composition without history as a baseline. The efficiency of the policy reuse approach is calculated based on how close output it can produce in comparison to the individual composition without history. According to Figure \\ref{fig:exp2}(a), the proposed Q-learned based policy reuse approach performs closely to the output which is generated without history (around 80\\% accuracy). However, the accuracy drops to about 60\\% in average and lower similar sets (Set B and C) which is not acceptable (Figure \\ref{fig:exp2}(b), (c)). The K-S test based approach also has around 60\\% accuracy in all the three sets. The K-S test based approach is about 30\\% and 5\\% less accurate than the proposed Q-learned policy reuse approach in Set A and B respectively (Figure \\ref{fig:exp2}(a) and (b)). However, it is 5\\% more accurate than the policy based approaches in Set C (Figure \\ref{fig:exp2}(c)). The proposed policy reuse approach should only be applied when highly similar request sets are found in the history.\n\nAccording to existing literature \\cite{koenig1993complexity}, the time complexity of 2d Q-learning approach is $O(ep)$, where $e$ is the total number of actions, $e = \\sum_{s \\epsilon S} a(s)$, and $p$ is the total number of states, $p = |S|$. In our context, the state is dependent on the number of time intervals, i.e., $I$ and the state transitions are laid out on the two dimensional state-space $[I \\times I]$. Similarly, the pairwise comparison is performed in each action, i.e., selecting a request set in a composition. Hence, if the number of request set is $N$, the time complexity of 2d Q-learning approach for the IaaS composition is $O(N^{2}I^{2})$. We modified the 2d Q-learning approach for IaaS composition into 3d On-policy learning process in Algorithm 1. As one more dimension is added in the 3d learning, the time complexity of Algorithm 1 is $O(N^{3}I^{3})$. Algorithm 2 applies policy reuse to reduce the state-action transitions by removing the redundant states. As the complexity of the clustering based similarity in Algorithm 2 is $O(N^{2})$, the time complexity of the proposed policy reuse approach is $O(N^{2})$ + $O(N^{2}I^{2})$. Figure \\ref{fig:exp2}(d) depicts the average runtime efficiency of the proposed policy reuse approach in the three sets. For a smaller number of sets (10 and 15) the policy reuse approach could not improve the runtime efficiency significantly than the K-S test based approach or composition without history approach. However, the policy reuse approach improves the runtime efficiency significantly (around 40\\%) than the Q-learning approach without history for a higher number of requests composition. The runtime of the greedy policy reuse approach is similar to the heuristic based sequential optimization approach.\n\nWe analyze the effect of the library size, i.e., the number of learned Q-matrices for the proposed policy reuse approach. According to Figure \\ref{fig:librarysize}, although a lower library size (i.e., 5) enables a quick convergence for the Q-leaning based policy reuse approach, the resulted accuracy also stays low. The key reason is the exploration method in the learning approach may not receive enough policies from the library of similar policies. Figure \\ref{fig:librarysize} depicts that a bigger library size improves the accuracy of the proposed approach. However, the accuracy is not linearly related to the library size. Once, the library size reaches a certain threshold, adding newly learned matrices may not necessarily improve the result. The proposed approach performs a satisfactory level of exploration with the threshold library size and newly added policies may only reiterate over the same explored states. N=15 is the threshold value of library size in Figure \\ref{fig:librarysize}. On the other side, a bigger library size also increases the runtime. In Figure \\ref{fig:librarysize} the runtime increases at the rate of 20\\% if library size is increased by a factor of 5. The threshold library size is dependent on the TempCP-net, the number of intervals and the distribution of the learned Q-matrices. In the future, we will investigate determining the optimal library size of the Q-learned database for the composition framework.\n\n\\begin{figure}[t!]\n    \\centering\n      \\includegraphics[width=0.5\\textwidth]{Effect-of-library-size-new.pdf}  \n      \\vspace{-6mm}\n    \\caption{Effects of different library sizes of Q-matrices in the policy reuse approach}\n    \\label{fig:librarysize}\n\\end{figure}    \n\nOne of our assumption is that the semantic levels of the provider's qualitative preferences are determined by the providers prior to the composition. The proposed approach determines the optimal qualitative IaaS composition for the provider based on the provided semantic preference table. Note, determining the optimal semantic levels, i.e., value ranges are out of the focus of this paper. We have added a new experiment to depict the effect of different semantic levels $(Slv = [2,3,5,10,20])$ on the accuracy of the proposed approach. The value ranges are uniformly distributed in the experiment. For example, the value range [66,100] is considered ``High'' in 3 semantic levels where the value range [80,100] is considered ``High'' in 5 semantic levels. Figure \\ref{fig:levelsize} depicts that a higher semantic levels improves the accuracy of the proposed approach slightly. As the lower semantic levels are coarse-grained, the number of candidate solutions are not effectively filtered in contrast to higher semantic levels. However, after the semantic levels reaches a certain threshold, adding new levels may not necessarily improve the result. In the future, we will investigate determining the optimal semantic levels for the composition framework. \n\n\\begin{figure}\n    \\centering\n      \\includegraphics[width=0.5\\textwidth]{images\/Effect-of-semantic-levels.pdf}  \n     \n    \\caption{Effects of different semantic levels in the proposed approach}\n    \\vspace{-6mm}\n    \\label{fig:levelsize}\n\\end{figure}    \n\n\n\n\\section{Related Work}\n\nCloud service composition has been studied extensively in the existing literature \\cite{mistry2018economic}. We identify two types of economic models for cloud service composition - a) quantitative , and b) qualitative composition \\cite{mistry2018economic, gavvala2019qos,souri2020hybrid}. We find that most quantitative approaches are mainly considered from a provider's perspective where the objective is to select the optimal set of service requests based on the provider's economic goal such as maximizing profit, revenue, or resource utilization, We also find that qualitative approaches are mainly considered from a consumer's perspective. Composition from a consumer's perspective selects the best cloud service from a set of cloud services based on the consumer's service requirements.\n\n\n\n\\begin{itemize}\n    \\item \\textit{Quantitative composition from a provider's perspective}: The objective of the quantitative composition is to maximize a provider's economic expectations such as revenue, profit, and reputation by optimizing available resources \\cite{cong2020survey}. For example, let us assume an IaaS provider receives requests from two perspective consumers, a bank and a university. The bank requests a VM instance with 8 cores CPU, 1TB memory, 1GB bandwidth, 10ms latency for 1 year and is willing to pay \\$500 upfront. The university requests a different VM instance with 16 cores CPU, 1972 CUDA cores GPU, 2TB memory, 500MB bandwidth, 5ms latency for 1 year and is willing to pay \\$800 upfront. The provider has four options: a) accept both the requests, b) accept the bank's request, c) accept the university's request, and d) reject both the requests. The economic model of the provider will guide the provider to take the appropriate decision. For example, the economic model of the provider may \\textit{suggest} that GPU based services have larger operation costs (due to the higher demand for electricity). As a result, the bank's IaaS request may be \\textit{preferred} and \\textit{accepted} over the university's IaaS request to \\textit{maximize profit}. Note that, if the provider has a \\textit{different economic model}, i.e., revenue maximization and reputation building for services with higher throughput, the university's IaaS request may be \\textit{preferred} and \\textit{accepted} over the bank's IaaS request (the university is paying more upfront that \\textit{maximizes revenue}). An economic model is proposed in \\cite{goiri2012economic} to maximize resource utilization. The proposed approach assesses the cost of resource utilization from a cloud federation perspective and develops a resource management core for the \\textit{short-term} profit maximization. An economic model is proposed to calculate the short-term operation cost of VM provisioning using power ratings of the physical servers in \\cite{Thanakornworakij}. The proposed economic model aims at maximizing profit based on right pricing and rightsizing in the Cloud data centre. However, short-term economic benefits may negatively affect the long-term economic gain. A meta-heuristic optimization approach is proposed in \\cite{sajibtsc2015} to compose stochastic service requests considering long-term economic gain. However, these proposed quantitative models could not be applied in qualitative composition context as these models do not consider the provider's business strategies or long-term qualitative preferences to accept or reject requests in the composition.\n    \n    \\item \\textit{Qualitative composition from a consumer's perspective}: Qualitative economic models are often used to express the preferences of a consumer where there exists incomplete or uncertain information. Qualitative models facilitate the selection and composition of multiple providers when quantitative approaches are not applicable. A qualitative economic model is proposed in \\cite{fattah2018cp} to represent a consumer's service preferences. The qualitative model is built using a dependency graph and conditional preference tables. The dependency graph and the conditional preference tables form a Conditional Preference Network (CP-net). A CP-net represents the consumer's preferences qualitatively in a natural and intuitive way. The conditional preference statements for each QoS attribute are expressed using a CPT (Conditional Preference Table). A CP-net based qualitative economic model is presented in \\cite{cp1} to select and compose multiple providers based on qualitative preferences of a consumer. The semantic consistency of the CP-net decides the quality of the composition. There are several variations of CP-nets in the existing literature. Consumers are allowed to express the weight or relative importance among attributes using the weighted CP-net (WCP-Net) \\cite{wang2012wcp}. The preferences can also be incomplete or inconsistent. A web service selection model using CP-net is proposed in \\cite{wang2009web} to select with incomplete or inconsistent user preferences. Probabilities are attached to the CPTs when there exist uncertainties in the qualitative preferences \\cite{cornelio2013updates}. A CP-net based user similarity search and conflict removal technique are proposed to remove inconsistency or incompleteness in the user preferences \\cite{bohanec2008qualitative}. The qualitative composition approaches from the consumer's perspective could not be applied directly to a provider as multiple long-term requests are composed at the same time according to the qualitative preferences of the provider.   \n\\end{itemize}\n\nAn IaaS provider's preferences are expressed using qualitative economic models. The ordering of the preference outcomes is calculated using an economic variable. Examples of economic variables are cost, reputation, revenue, and so on. The composition depends on the ordering of the preference outcomes. A qualitative economic model usually depends on multiple domain-specific properties \\cite{chung2012general}. Several existing studies consider qualitative economic models from the perspective of a provider. A CP-net based composition is proposed to select an optimal set of consumers from a provider's perspective \\cite{fattah2018cp}. The proposed approach defines a composability model to determine the inconsistency of the preference composition of the consumers from a provider's perspective. A qualitative economic model for long-term IaaS composition is proposed to capture provider's long-term business strategies using temporal CP-nets \\cite{sajibicsoc2016}. A novel sequential optimization method is proposed to select the optimal set of long-term consumer requests. The proposed approach is compared with global dynamic approach and greedy approach which shows that sequential optimization can achieve better time efficiency with acceptable accuracy. However, the proposed method does not include decision variables and business strategies to capture the types of consumers. The decision variables distinguish different types of consumers and enable a natural way to represent qualitative preferences. Decision variables are introduced in \\cite{mistry2018long} to capture the provider's qualitative preferences for short-term and long-term requests. The optimal composition is performed using sequential optimization through reinforcement learning following the long-term qualitative economic model.\n\n\nThe sequential local optimization method is a well-known approach in many areas such as operational research, image processing, pattern recognition and so on. Sequential local optimization is usually applicable when the consequences of any given action regarding high-level criteria need to be determined \\cite{kimes2004restaurant,chaefer2005modeling}. Many existing approaches use machine learning techniques in a sequential optimization to determine the parameters in state-action sequences \\cite{watkins1992q}. As the IaaS composition is dynamic in nature, a model-free learning platform might be more suitable than a model-based learning platform \\cite{dorigo2016ant}. Adaptive service composition utilizes reinforcement learning based methods to compose services from a consumer's perspective \\cite{wang2020integrating}. Markov Decision Process or MDP is often used to model service composition problem \\cite{alizadeh2020reinforcement}. A Q-learning based composition approach is proposed in \\cite{lewis2012reinforcement}. The proposed approach models adaptive controllers through a sequential decision process. The optimal policy is constructed using a reward function to approximate the optimal function. A heuristic based optimization approach is proposed without considering historical requests patterns  \\cite{sajibicsoc2016}. It creates random sequences of local optimizations. The heuristics treat different types of consumers in a similar manner which may output a non-optimal composition. A reinforcement learning (Q-learning) approach is proposed in \\cite{mistry2018long} that learns service provision strategies for different types of consumers without historical information of the past requests. It invalidates the necessity of learning if every new composition needs to be learned from scratch. In this paper, we remove the limitation of the existing approach in \\cite{mistry2018long}. We propose a new on-policy 3d Q-learning based IaaS composition framework that has the ability o reuse historical learning experiences or policies using agglomerative clustering.\n\n\\section{Conclusion}\n\nWe propose a qualitative composition approach for deterministic consumer requests according to the provider's long-term qualitative preferences represented in TempCP-nets. We perform the selection of consumer requests in each temporal segments sequentially. We propose Q-leaning based reinforcement learning approaches to find the best sequence of local optimization or selection of requests. We design an on-policy based three-dimensional Q-learning approach that intuitively removes redundant computations, i.e., state-action transitions using the length of overlapping requests. We propose different policy reuse approaches to perform the composition based on the learned compositions. The similar policy is selected using the cophenetic distance from agglomerative clustering of requests. Experimental results show that the proposed on-policy based 3d Q-learning approach is effective in real world situations. The proposed approach generates more accurate results than other approaches, i.e., 2d Q-learning, SARSA and heuristic based sequential approaches. The accuracy of the proposed on-policy based learning approach closely approximates to the accuracy of the off-policy 3d Q-learning composition approach in significantly reduced runtime. Experimental results also suggest that higher learning rates should not be used in the proposed approach. The proposed on-policy reuse approach produces better accuracy with agglomerative clustering than the greedy and K-S test based reuse approaches in an acceptable runtime. \n\nOne of the key limitations of the proposed work is to determine the optimal library size of Q-matrices which will be addressed in the future work. The proposed framework also does not consider the stochastic arrival of incoming requests and the probabilistic qualitative preferences. We will explore the dynamic qualitative IaaS composition in the future work.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Acknowledgement}\n\nThis research was partly made possible by DP160103595 and LE180100158 grants from the Australian Research Council. The statements made herein are solely the responsibility of the authors.\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Unit Interval Mappings}\n\n\\subsection{The $n$-th Harmonic Sawtooth Function $w_n (x)$}\n\n\\subsubsection{Infinite Sum Decomposition}\n\nLet the $n$-th component$w_n (x) \\in [0, 1] \\forall x \\in [0, 1]$ of the\nharmonic sawtooth function $w (x$) {\\cite[2.3]{two-new-zeta-constants}} be\ndefined as\n\\begin{equation}\n  \\begin{array}{ll}\n    w_n (x) & = n (x n + x - 1) \\chi_{} \\left( x, I^H_n \\right)\n  \\end{array} \\label{wn}\n\\end{equation}\nwhere\n\\begin{equation}\n  \\text{$\\chi_{} \\left( x, I^H_n \\right) = \\left\\{ \\begin{array}{ll}\n    1 & \\frac{1}{n + 1} < x \\leqslant \\frac{1}{n}\\\\\n    0 & \\tmop{otherwise}\n  \\end{array} \\right.$}\n\\end{equation}\nis the characteristic function of the $n$-th harmonic interval(\\ref{hii}). By\nsetting $n = \\left\\lfloor x^{- 1} \\right\\rfloor$ as in ($\\ref{hieye}$) we get\nthe unit interval mapping\n\\begin{equation}\n  \\begin{array}{ll}\n    w (x) & = w_{\\left\\lfloor x^{- 1} \\right\\rfloor} (x)\\\\\n    & = \\left\\lfloor x^{- 1} \\right\\rfloor (x \\left\\lfloor x^{- 1}\n    \\right\\rfloor + x - 1) \\chi_{} \\left( x, I^H_{\\left\\lfloor x^{- 1}\n    \\right\\rfloor} \\right)\\\\\n    & = \\sum_{n = 1}^{\\infty} w_n (x)\\\\\n    & = \\sum_{n = 1}^{\\infty} n (x n + x - 1) \\chi_{} \\left( x, I^H_n\n    \\right)\\\\\n    & = \\left\\lfloor x^{- 1} \\right\\rfloor \\left( x \\left\\lfloor x^{- 1}\n    \\right\\rfloor + x - 1 \\right)\\\\\n    & = x \\left\\lfloor x^{- 1} \\right\\rfloor^2 + x \\left\\lfloor x^{- 1}\n    \\right\\rfloor - \\left\\lfloor x^{- 1} \\right\\rfloor\n  \\end{array} \\label{w}\n\\end{equation}\n\\begin{figure}[h]\n  \\resizebox{12cm}{12cm}{\\includegraphics{harmonicsaw.eps}}\n  \\caption{\\label{wfig}The Harmonic Sawtooth map}\n\\end{figure}\n\nAs can be seen in Figure \\ref{wfig}, $w (x)$ is discontinuous at a countably\ninfinite set of points of Lebesgue measure zero\n\\begin{equation}\n  \\begin{array}{ll}\n    \\text{$\\mathbbm{H}$} & = \\left\\{ y : \\lim_{x \\rightarrow y^-} w (x) \\neq\n    \\lim_{x \\rightarrow y^+} w (x) \\right\\}\\\\\n    & = \\left\\{ 0, \\frac{1}{n} : n \\in \\mathbbm{Z} \\right\\}\n  \\end{array}\n\\end{equation}\nThe left and right limits at the discontinuous points are\n\\begin{equation}\n  \\begin{array}{lll}\n    \\lim_{x \\rightarrow \\in \\mathbbm{H}^-} w (x) & = 1 & \\\\\n    \\lim_{x \\rightarrow \\in \\mathbbm{H}^{+_{}}} w (x) & = 0 & \n  \\end{array}\n\\end{equation}\n\n\\subsubsection{The Fixed Points of $w (x)$ as an Iterated Function System}\n\nThe iterates of the map\n\\begin{equation}\n  \\text{ $[w (x), w (w (x)), w (w (w (x))), w (w (w (w (x)))), \\ldots] =\n  \\left[ \\overset{}{w^1} (x), \\overset{}{w^2} (x), \\overset{}{w^3} (x),\n  \\overset{}{w^4} (x), \\ldots \\right]$}\n\\end{equation}\nhave the form\n\\begin{equation}\n  \\begin{array}{ll}\n    w^r (x) & = a_r - (x - c) b_r \n  \\end{array} \\label{wform}\n\\end{equation}\nwhere $^{} \\{a_r, b_r \\in \\mathbbm{Z}: r \\in \\mathbbm{N}\\}$ is a pair of\ninteger sequences and $c \\in \\mathbbm{R}$ is some constant. The sequence of\nquotients $\\frac{a_r}{b_r}$ converges rather quickly to the fixed value\n\\begin{equation}\n  \\begin{array}{l}\n    \\lim_{r \\rightarrow \\infty} \\frac{a_r}{b_r} = x - c\n  \\end{array} \\forall x \\in [0, 1]\n\\end{equation}\nThe explicit equation(sometimes called Schr\\\"oder's equation) for the fixed\npoints of $w (x)$, is\n\\begin{equation}\n  \\begin{array}{ll}\n    \\tmop{Fix}_w^n & =\\{x : w_n (x)_{} = x\\}\\\\\n    & =\\{x : n (x n + x - 1) \\chi_{} (x, I^H_n)_{} = x\\}\\\\\n    & =\\{x : n (x n + x - 1) = x\\}\\\\\n    & = \\lim_{x \\rightarrow 0} \\frac{\\frac{\\mathrm{d}}{\\mathrm{d} x^n} \\tmop{Fix}_w\n    (x)}{n!}\\\\\n    & = \\lim_{x \\rightarrow 0} \\frac{x^{1 - n}}{n!} G_{4, 4}^{1, 4} \\left( -\n    x | \\begin{array}{llll}\n      - 1 & - 1 & - \\phi & \\phi - 1\\\\\n      0 & n - 1 & 2 - \\phi & - 1 - \\phi\n    \\end{array} \\right)\\\\\n    & = \\frac{n}{n^2 + n - 1}\n  \\end{array}\n\\end{equation}\nwhere $G$ is the Meijer-G function and $\\tmop{Fix}_w (x)$ is the\ngeneration function\n\\begin{equation}\n  \\begin{array}{ll}\n    \\tmop{Fix}_w (x) & = \\begin{array}{l}\n      \\sum_{n = 1}^{\\infty}\n    \\end{array} \\tmop{Fix}^n_w x^n\\\\\n    & = \\sum_{n = 1}^{\\infty} \\frac{n x^n}{n^2 + n - 1}\\\\\n    & = \\frac{1}{10} \\left( (5 - \\sqrt{5}) \\Phi \\left( x, 1, 1 - \\phi \\right)\n    + (5 - \\sqrt{5}) \\Phi \\left( x, 1, \\phi \\right) \\right)\n  \\end{array}\n\\end{equation}\nwhere $\\Phi (z, a, v)$ is the Lerch Transcendent (\\ref{lerch}), and $\\phi$ is\nthe Golden Ratio, which is the ratio of two numbers having the property that\nthe ratio of the sum to the larger equals the ratio of the larger to the\nsmaller.\n{\\cite[Ch.XX]{col}}{\\cite[I.7]{cf}}{\\cite[p.50]{cet}}{\\cite{goldenRatio}} The\nnumber $\\phi$ can be called the ``most irrational number'' because its\ncontinued fraction expansion, given by iterations of the Gauss map (\\ref{h}),\nconverges more slowly than any other number. The constant $\\phi$ satisfies the\nsimple identities\n\\begin{equation}\n  \\phi = \\frac{1}{\\phi - 1} = \\frac{\\phi}{\\phi + 1}\n\\end{equation}\n\\begin{equation}\n  \\phi^2 - \\phi - 1 = 0\n\\end{equation}\nAn interesting fact is that the density of a motif in a certain noncommutative\nspace described in {\\cite[5.1]{fractalzetastrings}} must necessarily be an\nelement of the group $\\mathbbm{Z}+ \\phi \\mathbbm{Z}$.\n\n\\subsection{Integrals Transforms of w(x)}\n\n\\subsubsection{$\\tmop{Dirichlet} \\tmop{Polynomial} \\tmop{Series}$ and the\nMellin Transform of $w (x)$}\n\nThe Mellin transform of the harmonic saw map $w (x)$, multiplied by\n\\begin{equation}\n  \\tau (s) = s \\frac{s + 1}{s - 1} \\label{tau}\n\\end{equation}\nis an analytic continuation of the Riemann zeta function $\\zeta (s) \\forall -\n\\tmop{Re} (s) \\not\\in \\mathbbm{N}$. This form of the zeta function, denoted by\n$\\zeta_w (s)$, is the infinite sum of the Mellin transformations of the\ncomponent functions.\n\\begin{equation}\n  \\begin{array}{ll}\n    \\left. M_{} [w_n (x) ; x \\rightarrow s \\right] & = \\int_{\\frac{1}{n +\n    1}}^{\\frac{1}{n}} w_n (x) x^{s - 1} \\mathrm{d} x\\\\\n    & = \\int_n^{n + 1} w_n (x^{- 1}) x^{- s - 1} \\mathrm{d} x\\\\\n    & = \\int_0^1 n (x n + x - 1) \\chi_{} (x, I^H_n) x^{s - 1} \\mathrm{d} x\\\\\n    & = \\int_{\\frac{1}{n + 1}}^{\\frac{1}{n}} n (x n + x - 1) x^{s - 1} \\mathrm{d}\n    x\\\\\n    & = - \\frac{e^{\\ln (n) (1 - s)} - n e^{s \\ln \\left( \\frac{1}{n + 1}\n    \\right)} - s e^{s \\ln \\left( \\frac{1}{n} \\right)}}{s^2 + s}\\\\\n    & = \\frac{n^s n + s (n + 1)^s - (n + 1)^s n}{n^s s (n + 1)^s + n^s s^2 (n\n    + 1)^s}\\\\\n    & = - \\frac{n^{1 - s} - n (n + 1)^{- s} - s n^{- s}}{s^2 + s}\n  \\end{array} \\label{mwn}\n\\end{equation}\nThere is a conjugate pair of inverse branches of $\\tau (s)$ found by solving\n\\begin{equation}\n  \\begin{array}{ll}\n    \\tau_{\\pm}^{- 1} (t) & =\\{s : \\tau (s) = t\\}\\\\\n    & = \\left\\{ s : s \\frac{s^{} + 1}{s - 1} = t \\right\\}\\\\\n    & = \\frac{t}{2} - \\frac{1}{2} \\pm \\frac{\\sqrt{1 - 6 t + t^2}}{2}\n  \\end{array}\n\\end{equation}\nwhere $\\tau_+^{- 1} (t)$ and $\\tau_-^{- 1} (t)$ denote the positive and\nnegative solutions respectively. The coefficients in the series expansions are\nintegers enumerating the {\\tmem{large Schr\\\"oder numbers}} $S_n$ which count\nthe number of perfect matchings in a triangular grid of $n$ squares, named\nafter Ernst\nSchr\\\"oder(1841-1902).{\\cite[A006318]{oeis}}{\\cite[p.340]{combPerm}}{\\cite{cms}}\n\\begin{equation}\n  \\begin{array}{ll}\n    S_n & = {_2 F_1} \\left( \\begin{array}{ll}\n      n + 1 & 2 - n\\\\\n      2 & \n    \\end{array} | - 1 \\right) 2\n  \\end{array}\n\\end{equation}\nwhere $_p F_q$ is a hypergeometric function. We have\n\\begin{equation}\n  \\begin{array}{ll}\n    \\lim_{t \\rightarrow 0} \\frac{\\frac{\\mathrm{d}^n}{\\mathrm{d} t^n} \\text{$\\tau_+^{-\n    1} (t)$}}{n!} & = \\left\\{ \\begin{array}{ll}\n      0 & n = 0\\\\\n      - 1 & n = 1\\\\\n      - S_n & n \\geqslant 2\n    \\end{array} \\right.\\\\\n    \\lim_{t \\rightarrow 0} \\frac{\\frac{\\mathrm{d}^n}{\\mathrm{d} t^n} \\text{$\\tau_-^{-\n    1} (t)$}}{n!} & = \\left\\{ \\begin{array}{ll}\n      - 1 & n = 0\\\\\n      2 & n = 1\\\\\n      S_n & n \\geqslant 2\n    \\end{array} \\right.\n  \\end{array}\n\\end{equation}\n\\begin{equation}\n  \\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n    \\hline\n    n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14\\\\\n    \\hline\n    S_n & 1 & 1 & 2 & 6 & 22 & 90 & 394 & 1806 & 8558 & 41586 & 206098 &\n    1037718 & 5293446 & 27297738 & 142078746\\\\\n    \\hline\n  \\end{array}\n\\end{equation}\nThe residue at the singular point $s = 1$ of $\\tau (s)$ is\n\\begin{equation}\n  \\begin{array}{ll}\n    \\underset{}{\\underset{s = 1}{\\tmop{Res}} \\left( \\tau (s) M [w_n (x) ; x\n    \\rightarrow s] \\right)} & = \\underset{}{\\underset{s = 1}{\\tmop{Res}}\n    \\left( \\frac{s^2 + s}{s - 1} \\left( - \\frac{n^{1 - s} - n (n + 1)^{- s} -\n    s n^{- s}}{s^2 + s} \\right) \\right)}\\\\\n    & = \\underset{}{\\underset{s = 1}{\\tmop{Res}} \\left( \\frac{n (n + 1)^{- s}\n    - n^{1 - s} + s n^{- s}}{s - 1} \\right)}\\\\\n    & = 2 \\int_{\\frac{1}{n + 1}}^{\\frac{1}{n}} w_n (x) \\mathrm{d} x\\\\\n    & = 2 \\int_{\\frac{1}{n + 1}}^{\\frac{1}{n}} n (x n + x - 1) \\mathrm{d} x\\\\\n    & = \\frac{1}{n^2 + n}\n  \\end{array} \\label{wm1r}\n\\end{equation}\nThe infinite sum of the Mellin transforms multiplied by $\\tau (s)$\nanalytically continues $\\zeta (s) \\text{$\\forall (- \\tmop{Re} (s)) \\not\\in\n\\mathbbm{N}_{}$}$\n\\begin{equation}\n  \\begin{array}{lll}\n    \\zeta_w (s) & = \\zeta (s) & \\\\\n    & = \\tau (s) M_{} [w_{} (x) ; x \\rightarrow s] & \\\\\n    & = \\tau (s) \\int_0^1 w (x) x^{s - 1} \\mathrm{d} x & \\\\\n    & = s \\frac{s^{} + 1}{s - 1} \\int_0^1 \\left\\lfloor x^{- 1} \\right\\rfloor\n    \\left( x \\left\\lfloor x^{- 1} \\right\\rfloor + x - 1 \\right) x^{s - 1}\n    \\mathrm{d} x & \\\\\n    & = s \\frac{s^{} + 1}{s - 1} \\sum_{n = 1}^{\\infty} M [w_n (x) ; x\n    \\rightarrow s] & \\\\\n    & = s \\frac{s^{} + 1}{s - 1} \\sum_{n = 1}^{\\infty} \\int_{\\frac{1}{n +\n    1}}^{\\frac{1}{n}} n (x n + x - 1) x^{s - 1} \\mathrm{d} x & \\text{ }\\\\\n    & = \\sum_{n = 1}^{\\infty} s \\frac{s^{} + 1}{s - 1} \\left( - \\frac{n^{1 -\n    s} - n (n + 1)^{- s} - s n^{- s}}{s \\left( s + 1 \\right)} \\right) & \\\\\n    & = \\sum_{n = 1}^{\\infty} \\frac{n (n + 1)^{- s} - n^{1 - s} + s n^{-\n    s}}{s - 1} & \\\\\n    & = \\frac{1}{s - 1} \\sum_{n = 1}^{\\infty} n (n + 1)^{- s} - n^{1 - s} + s\n    n^{- s} & \n  \\end{array} \\label{sawzeta}\n\\end{equation}\nUnlike the Mellin transform of the Gauss map (\\ref{h}), which must be\nmultiplied by the factor $s$ then subtracted from $\\frac{s}{s - 1}$ before it\nequals $\\zeta (s$), the ``harmonic sawtooth continuation'' $\\zeta_w (s)$ of\nthe zeta function $\\zeta (s$) has the fortuitous property that it equals\n$\\zeta (s$) after multiplying $M^{} [w_{} (x) ; x \\rightarrow s]$ by $\\tau\n(s$). \\ This property of $w (x$) allows us to put $\\tau (s$) inside the sum to\nget an expression, denoted by $\\zeta_w (N ; s)$, involving the difference of\ntwo Dirichlet polynomials, one of which is scaled by $s$. The substition\n$\\infty \\rightarrow N$ is made in the infinite sum appearing the expression\nfor $\\zeta_w (s$) to get\n\\begin{equation}\n  \\begin{array}{ll}\n    \\zeta_{w_{}} (N ; s) & = \\tau (s) \\sum_{n = 1}^N M_{}^{} [w_n (x) ; x\n    \\rightarrow s]\\\\\n    & = \\frac{1}{s - 1} \\sum_{n = 1}^N n (n + 1)^{- s} - n^{1 - s} + s n^{-\n    s}\\\\\n    & = \\frac{1}{s - 1} \\left( s + (N + 1)^{1 - s} - 1 + s \\sum_{n = 2}^N\n    n^{- s} - \\sum_{n = 2}^{N + 1} n^{- s} \\right)\\\\\n    & = \\frac{N}{\\left( s - 1 \\right) \\left( N + 1 \\right)^s} - \\frac{\\cos\n    \\left( \\pi s \\right) \\Psi \\left( s - 1, N + 1 \\right)}{\\Gamma \\left( s\n    \\right)} + \\zeta \\left( s \\right) \\forall s \\in \\mathbbm{N}^{\\ast}\n  \\end{array} \\label{zwpoly}\n\\end{equation}\nwith equality in the limit except at the negative integers\n\\begin{equation}\n  \\begin{array}{ll}\n    \\lim_{N \\rightarrow \\infty} \\zeta_{w_{}} (N ; s) & = \\zeta (s) \\forall - s\n    \\not\\in \\mathbbm{N}^{\\ast}\n  \\end{array}\n\\end{equation}\nThe functions $\\zeta_w \\left( N ; s \\right)$ have real roots at $s = 0$ and $s\n= - 1$. That is\n\\begin{equation}\n  \\lim_{s \\rightarrow 0} \\zeta_w \\left( N ; s \\right) = \\lim_{s \\rightarrow -\n  1} \\zeta_w \\left( N ; s \\right) = 0\n\\end{equation}\nThe residue of $\\zeta_{w_{}} (N ; s)$ at $s = 1$ is given by\n\\begin{equation}\n  \\begin{array}{ll}\n    \\underset{}{\\underset{s = 1}{\\tmop{Res}} (\\zeta_{w_{}} (N ; s))} & =\n    \\underset{}{\\underset{s = 1}{\\tmop{Res}} \\left( \\tau (s) \\sum_{n = 1}^N\n    M_{}^{} [w_n (x) ; x \\rightarrow s] \\right)}\\\\\n    & = \\sum_{n = 1}^N \\underset{s = 1}{\\tmop{Res}} \\left( \n    \\underset{}{\\underset{}{} \\tau (s)} M_{}^{} [w_n (x) ; x \\rightarrow s]\n    \\right)\\\\\n    & = \\sum_{n = 1}^N \\frac{1}{n^2 + n}\\\\\n    & = \\frac{N}{N + 1}\n  \\end{array}\n\\end{equation}\nThus, as required\n\\begin{equation}\n  \\begin{array}{ll}\n    \\lim_{N \\rightarrow \\infty} \\underset{}{\\underset{s = 1}{\\tmop{Res}}\n    (\\zeta_{w_{}} (N ; s))} & = \\lim_{N \\rightarrow \\infty} \\sum_{n = 1}^N\n    \\frac{1}{n^2 + n}\\\\\n    & = \\lim_{N \\rightarrow \\infty} \\frac{N}{N + 1}\\\\\n    & = 1\n  \\end{array}\n\\end{equation}\nThe function $\\tau (s)$ has zeros at $- 1$ and $0$ and a simple pole at $s =\n1$ with residue\n\\begin{equation}\n  \\begin{array}{ll}\n    \\underset{s = 1}{\\tmop{Res}} \\left( \\tau (s) \\right) & =\n    \\underset{}{\\underset{s = 1}{\\tmop{Res}} \\left( s \\frac{s + 1}{s - 1}\n    \\right)} = 2\n  \\end{array} \\label{wtaures}\n\\end{equation}\nThe Mellin transform of $\\tau (s)$ has an interesting Laurent series,\nconvergent on the unit disc, given by\n\\begin{equation}\n  \\begin{array}{lll}\n    M [\\tau (s) ; s \\rightarrow t] & = \\int_0^{\\infty} \\tau (s) s^{t - 1}\n    \\mathrm{d} s & \\\\\n    & = \\int_0^{\\infty} s \\frac{s^{} + 1}{s - 1} s^{t - 1} \\mathrm{d} s & \\\\\n    & = \\sum_{n = 1}^{\\infty} 4 \\zeta (2 n - 2) t^{2 n - 3} & \\text{$\\forall\n    |t| < 1$}\\\\\n    & = - 4 i \\pi \\frac{\\left( \\frac{1}{2} e^{i \\pi t} + \\frac{1}{2} e^{- i\n    \\pi t} \\right)}{e^{i \\pi t} - e^{- i \\pi t}} & \\\\\n    & = - 2 \\pi \\frac{\\cos (\\pi t)}{\\sin (\\pi t)} & \n  \\end{array}\n\\end{equation}\nThe transformations $M [w_n (x) ; x \\rightarrow s]$ have removable\nsingularities at $- 1$ and $0$ where the limits are given by\n\\begin{equation}\n  \\begin{array}{ll}\n    \\underset{s \\rightarrow - 1}{\\lim} \\underset{}{M [w_n (x) ; x \\rightarrow\n    s]}_{} & = n^2 \\ln \\left( \\frac{n + 1}{n} \\right) + n \\ln \\left( \\frac{n +\n    1}{n} \\right) - n\\\\\n    \\underset{s \\rightarrow 0}{\\lim} \\underset{}{M^{} [w_n (x) ; x \\rightarrow\n    s]}_{} & = 1 + n \\ln \\left( \\frac{n + 1}{n} \\right)\n  \\end{array}\n\\end{equation}\n\\begin{equation}\n  \\begin{array}{ll}\n    \\lim_{s \\rightarrow 0} e^{- M [w_n (x) ; x \\rightarrow s]} & = (n + 1)^n\n    n^{- n} e^{- 1}\\\\\n    \\begin{array}{l}\n      \\lim_{s \\rightarrow - 1} e^{- M [w_n (x) ; x \\rightarrow s]}\n    \\end{array} & = n^{(n^2 + n)} (n + 1)^{(- n^2 - n)} e^n\n  \\end{array}\n\\end{equation}\nSo,  (\\ref{mwn}) can be rewritten as\n\\begin{equation}\n  \\begin{array}{ll}\n    M^{} \\left[ w_n (x) ; x \\rightarrow s \\right] & = \\left\\{\n    \\begin{array}{ll}\n      n^2 \\ln \\left( \\frac{n + 1}{n} \\right) + n \\ln \\left( \\frac{n + 1}{n}\n      \\right) - n & s = - 1\\\\\n      1 + n \\ln \\left( \\frac{n + 1}{n} \\right) & s = 0\\\\\n      \\begin{array}{l}\n        - \\frac{n^{1 - s} - n (n + 1)^{- s} - s n^{- s}}{s^2 + s}\n      \\end{array} & \\tmop{otherwise}\n    \\end{array} \\right.\n  \\end{array}\n\\end{equation}\nFurthermore, we have the limits\n\\begin{equation}\n  \\begin{array}{ll}\n    \\lim_{n \\rightarrow \\infty} \\underset{s \\rightarrow - 1}{\\lim} M^{} \\left[\n    w_n (x) ; x \\rightarrow s \\right]  & = \\lim_{n \\rightarrow \\infty} \\lim_{s\n    \\rightarrow - 1} - \\frac{n^{1 - s} - n (n + 1)^{- s} - s n^{- s}}{s^2 +\n    s}\\\\\n    & = \\lim_{n \\rightarrow \\infty} n^2 \\ln \\left( \\frac{n + 1}{n} \\right) +\n    n \\ln \\left( \\frac{n + 1}{n} \\right) - n\\\\\n    & = \\frac{1}{2}\n  \\end{array}\n\\end{equation}\nand\n\\begin{equation}\n  \\begin{array}{ll}\n    \\lim_{n \\rightarrow \\infty} \\underset{s \\rightarrow 0}{\\lim} M^{} \\left[\n    w_n (x) ; x \\rightarrow s \\right] & = \\lim_{n \\rightarrow \\infty} \\lim_{s\n    \\rightarrow 0} - \\frac{n^{1 - s} - n (n + 1)^{- s} - s n^{- s}}{s^2 + s}\\\\\n    & = \\lim_{n \\rightarrow \\infty} 1 + n \\ln \\left( \\frac{n + 1}{n}\n    \\right)\\\\\n    & = 0\n  \\end{array}\n\\end{equation}\nThe sum of limits over $n$ at $s = - 1$ is Euler's constant.\n{\\cite{gamma}}{\\cite[1.1]{htf1}}\n\\begin{equation}\n  \\begin{array}{ll}\n    \\sum_{n = 1}^{\\infty} \\frac{M^{} \\left[ w_n (x) ; x \\rightarrow - 1\n    \\right]}{n} & = \\sum_{n = 1}^{\\infty} \\frac{1 + n \\ln \\left( \\frac{n +\n    1}{n} \\right)}{n}\\\\\n    & = \\lim_{s \\mathop{\\rightarrow}\\limits 1} \\zeta (s) - \\frac{1}{s - 1}\\\\\n    & = \\lim_{n \\rightarrow \\infty} \\sum_{k = 1}^n \\frac{1}{k} - \\ln (n)\\\\\n    & = \\gamma\\\\\n    & \\cong 0.577215664901533 \\ldots\n  \\end{array} \\label{gamma}\n\\end{equation}\n\n\\subsubsection{The Reflection Formula for $\\zeta_w \\left( N ; s \\right)$}\n\nThere is a reflection equation for the finite-sum approximation $\\zeta_{w_{}}\n(N ; s)$ which is similiar to the well-known formula $\\zeta \\left( 1 - s\n\\right) = \\chi \\left( s \\right) \\zeta \\left( s \\right)$ with $\\chi \\left( s\n\\right) = 2 \\left( 2 \\pi \\right)^{- s} \\cos \\left( \\frac{\\pi s}{2} \\right)\n\\Gamma \\left( s \\right)$. The solution to\n\\begin{equation}\n  \\zeta_w \\left( N ; 1 - s \\right) = \\chi \\left( N ; s \\right) \\zeta_w \\left(\n  N ; s \\right)\n\\end{equation}\nis given by the expression\n\\begin{equation}\n \\begin{array}{ll}\n     \\chi \\left( N ; s \\right) & = \\frac{\\zeta_w \\left( N ; 1 - s\n     \\right)}{\\zeta_w \\left( N ; s \\right)}\\\\\n     & = \\frac{\\sum_{n = 1}^N - \\frac{- n^s + \\left( n + 1 \\right)^{s - 1} n\n     + n^{s - 1} - n^{s - 1} s}{s}}{\\sum_{n = 1}^N \\frac{- n^{1 - s} + \\left(\n     n + 1 \\right)^{- s} n + n^{- s} s}{s - 1}}\\\\\n     & = - \\frac{\\left( s - 1 \\right) \\sum_{n = 1}^N - n^s + \\left( n + 1\n     \\right)^{s - 1} n + n^{s - 1} - n^{s - 1} s}{s \\sum_{n = 1}^N - n^{1 - s}\n     + \\left( n + 1 \\right)^{- s} n + n^{- s} s}\n   \\end{array} \n\\end{equation}  \nwhich satisfies\n\\begin{equation}\n  \\chi \\left( N ; 1 - s \\right) = \\chi \\left( N ; s \\right)^{- 1}\n\\end{equation}\nThe functions $\\chi \\left( N ; s \\right)$, indexed by $N$, have singularities\nat $s = 0$. Let\n\\begin{equation}\n  \\begin{array}{ll}\n    a \\left( N \\right) & = \\sum_{n = 1}^N n \\left( \\ln \\left( n + 1 \\right) -\n    \\ln \\left( n \\right) \\right)\\\\\n    b \\left( N \\right) & = \\sum_{n = 1}^N \\frac{\\ln \\left( n \\right) n^2 - \\ln\n    \\left( n + 1 \\right) n^2 - \\ln \\left( n \\right)}{n \\left( n + 1 \\right)}\\\\\n    c \\left( N \\right) & = \\frac{1}{2} \\sum_{n = 1}^N n \\left( \\ln \\left( n +\n    1 \\right)^2 - \\ln \\left( n \\right)^2 \\right)\n  \\end{array}\n\\end{equation}\nthen the residue at the singular point $s = 0$ is given by the expression\n\\begin{equation}\n  \\begin{array}{ll}\n    \\underset{s = 0}{\\tmop{Res}} (\\chi_{_{}} (N ; s)) & = - \\underset{s =\n    1}{\\tmop{Res}} (\\chi_{_{}} (N ; s)^{- 1})\\\\\n    & = \\frac{1 + \\gamma + \\Psi \\left( n + 2 \\right) - \\frac{2}{N + 1} + b\n    \\left( N \\right) - \\frac{N \\left( \\ln \\left( \\Gamma \\left( N + 1 \\right)\n    \\right) - c \\left( N \\right) \\right)}{\\left( N - a \\left( N \\right)\n    \\right) \\left( N + 1 \\right)}}{a \\left( N \\right) - N}\\\\\n    & = \\frac{1 + \\gamma + \\Psi \\left( n + 2 \\right) - \\frac{2}{N + 1} +\n    \\sum_{n = 1}^N \\frac{\\ln \\left( n \\right) n^2 - \\ln \\left( n + 1 \\right)\n    n^2 - \\ln \\left( n \\right)}{n \\left( n + 1 \\right)} - \\frac{N \\left( \\ln\n    \\left( \\Gamma \\left( N + 1 \\right) \\right) - \\frac{1}{2} \\sum_{n = 1}^N n\n    \\left( \\ln \\left( n + 1 \\right)^2 - \\ln \\left( n \\right)^2 \\right)\n    \\right)}{\\left( N - \\sum_{n = 1}^N n \\left( \\ln \\left( n + 1 \\right) - \\ln\n    \\left( n \\right) \\right) \\right) \\left( N + 1 \\right)}}{\\left( \\sum_{n =\n    1}^N n \\left( \\ln \\left( n + 1 \\right) - \\ln \\left( n \\right) \\right)\n    \\right) - N}\n  \\end{array}\n\\end{equation}\nwhich has the limit\n\\begin{equation}\n  \\lim_{N \\rightarrow \\infty} \\underset{s = 0}{\\tmop{Res}} (\\chi_{_{}} (N ;\n  s)) = 1\n\\end{equation}\nWe also have the residue of the reciprocal at $s = 2$\n\\begin{equation}\n  \\begin{array}{ll}\n    \\underset{s = 2}{\\tmop{Res}} (\\chi_{_{}} (N ; s)^{- 1}) & = \\frac{\\frac{2\n    N}{\\left( N + 1 \\right)^2} - 2 \\Psi \\left( 1, N + 1 \\right) + 2 \\zeta\n    \\left( 2 \\right)}{\\frac{\\left( N + 1 \\right)^2}{2} - \\frac{N}{2} -\n    \\frac{1}{2} - \\sum_{n = 1}^N n \\left( \\ln \\left( n + 1 \\right) + \\ln\n    \\left( n + 1 \\right) n - \\ln \\left( n \\right) - n \\ln \\left( n \\right)\n    \\right)}\n  \\end{array}\n\\end{equation}\nwhich vanishes as $N$ tends to infinity\n\\begin{equation}\n  \\lim_{N \\rightarrow \\infty} \\underset{s = 2}{\\tmop{Res}} (\\chi_{_{}} (N ;\n  s)^{- 1}) = 0\n\\end{equation}\nAs can be seen in the figures below, the residue at $s = 0$ changes sign from\nnegative to positive between the values of $N = 176$ and $N = 177$.\n\n\\begin{figure}[h]\n  \\resizebox{5in}{3in}{\\includegraphics{wmzres.eps}}\n  \\caption{$\\left\\{ \\underset{s = 0}{\\tmop{Res}} (\\chi_{_{}} (N ; s)) : N = 1\n  \\ldots 250 \\right\\}$}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\resizebox{5in}{3in}{\\includegraphics{wmzresinv.eps}}\n  \\caption{$\\left\\{ \\underset{s = 0}{\\tmop{Res}} (\\chi_{_{}} (N ; s))^{- 1} :\n  N = 1 \\ldots 250 \\right\\}$}\n\\end{figure}\n\nFor any positive integer N, we have the limits\n\\begin{equation}\n  \\begin{array}{ll}\n    \\lim_{s \\rightarrow 0} \\chi \\left( N ; s \\right) & = \\infty\\\\\n    \\lim_{s \\rightarrow 0} \\frac{\\mathrm{d}^n}{\\mathrm{d} s^n} \\chi \\left( N ; s\n    \\right) & = \\infty\\\\\n    \\lim_{s \\rightarrow \\frac{1}{2}} \\chi \\left( N ; s \\right) & = 1\\\\\n    \\lim_{s \\rightarrow 1} \\chi \\left( N ; s \\right) & = 0\\\\\n    \\lim_{s \\rightarrow 2} \\chi \\left( N ; s \\right) & = 0\\\\\n    \\lim_{s \\rightarrow 1} \\frac{\\mathrm{d}}{\\mathrm{d} s} \\chi \\left( N ; s \\right) &\n    = 0\n  \\end{array}\n\\end{equation}\nThe line $\\tmop{Re} \\left( s \\right) = \\frac{1}{2}$ has a constant modulus\n\\begin{equation}\n  \\left| \\chi \\left( N ; \\frac{1}{2} + i s \\right) \\right| = 1\n\\end{equation}\nThere is also the complex conjugate symmetry\n\\begin{equation}\n  \\chi \\left( N ; x + i y \\right) = \\overline{\\chi \\left( N ; x - i y \\right)}\n\\end{equation}\nIf $s = n \\in \\mathbbm{N}^{\\ast}$ is a positive integer then $\\chi \\left( N ;\nn \\right)$ can be written as\n\\begin{equation}\n  \\begin{array}{cc}\n    \\chi \\left( N ; n \\right) & = \\frac{\\zeta_w \\left( N ; 1 - n\n    \\right)}{\\zeta_w \\left( N ; n \\right)}\\\\\n    & = \\frac{\\sum_{m = 1}^N - \\sum_{k = 1}^{n - 2} \\frac{m^k}{n} \\binom{n -\n    1}{k - 1}}{\\frac{N}{\\left( n - 1 \\right) \\left( N + 1 \\right)^n} -\n    \\frac{\\cos \\left( \\pi n \\right) \\Psi \\left( n - 1, N + 1 \\right)}{\\Gamma\n    \\left( n \\right)} + \\zeta \\left( n \\right)}\\\\\n    & = \\frac{- \\sum_{m = 1}^N \\frac{1}{n} \\left( \\left( n - 1 \\right) m^{n -\n    1} + m^n - \\left( m + 1 \\right)^{n - 1} m \\right)}{\\frac{N}{\\left( n - 1\n    \\right) \\left( N + 1 \\right)^n} - \\frac{\\cos \\left( \\pi n \\right) \\Psi\n    \\left( n - 1, N + 1 \\right)}{\\Gamma \\left( n \\right)} + \\zeta \\left( n\n    \\right)}\n  \\end{array}\n\\end{equation}\nThe Bernoulli numbers{\\cite{bernoilli-euler-maclaurin}} make an appearance\nsince\n\\begin{equation}\n  \\begin{array}{ll}\n    \\chi \\left( N ; 2 n \\right) \\zeta_{w_{}} (N ; 2 n) & = B_{2 n}  \\left( N +\n    1 \\right)^2 \\frac{\\left( 2 n + 1 \\right)}{2} + \\ldots\n  \\end{array}\n\\end{equation}\nThe denominator of $\\chi \\left( N ; n \\right)$ has the limits\n\\begin{equation}\n  \\begin{array}{cl}\n    \\lim_{N \\rightarrow \\infty} \\zeta_w \\left( N ; n \\right) & = \\zeta \\left(\n    n \\right)\\\\\n    \\lim_{n \\rightarrow \\infty} \\zeta_w \\left( N ; n \\right) & = 1\n  \\end{array}\n\\end{equation}\nAnother interesting formula gives the limit at $s = 1$ of the quotient of\nsuccessive functions\n\\begin{equation}\n  \\begin{array}{cl}\n    \\lim_{s = 1} \\frac{\\chi \\left( N + 1 ; s \\right)}{\\chi \\left( N ; s\n    \\right)} & = \\frac{\\left( N + 2 \\right) N \\left( N + 1 - a \\left( N + 1\n    \\right) \\right) }{\\left( N + 1 \\right)^2  \\left( N - a \\left( N \\right)\n    \\right)}\\\\\n    & = \\frac{\\left( N + 2 \\right) N \\left( N + 1 - \\sum_{n = 1}^{N + 1} n\n    \\left( \\ln \\left( n + 1 \\right) - \\ln \\left( n \\right) \\right) \\right)\n    }{\\left( N + 1 \\right)^2  \\left( N - \\sum_{n = 1}^N n \\left( \\ln \\left( n\n    + 1 \\right) - \\ln \\left( n \\right) \\right) \\right)}\n  \\end{array}\n\\end{equation}\n\n\n\\begin{figure}[h]\n  \\resizebox{5in}{3.25in}{\\includegraphics{chi1-25.eps}}\n  \n  \n  \\caption{$\\left\\{ \\chi \\left( N ; s \\right) : s = 1 \\ldots 2, N = 1 \\ldots\n  25 \\right\\}$}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\resizebox{5in}{3.25in}{\\includegraphics{chi1-100.eps}}\n  \n  \n  \\caption{$\\left\\{ \\chi \\left( N ; s \\right) : s = \\frac{1}{2} \\ldots 2, N =\n  1 \\ldots 100 \\right\\}$}\n\\end{figure}\n\nLet\n\\begin{equation}\n  \\nu \\left( s \\right) = \\chi \\left( \\infty ; s \\right) = \\frac{\\zeta \\left( 1\n  - s \\right)}{\\zeta \\left( s \\right)}\n\\end{equation}\nThen the residue at the even negative integers \\ is\n\\begin{equation}\n  \\underset{s = - n}{\\tmop{Res}} (\\nu_{_{}} (s)^{}) = \\left\\{\n  \\begin{array}{ll}\n    \\frac{\\zeta \\left( 1 - n \\right)}{\\frac{\\mathrm{d}}{\\mathrm{d} s} \\zeta \\left( s\n    \\right)_{} |_{s = - n}} & n \\tmop{even}\\\\\n    0 & n \\tmop{odd}\n  \\end{array} \\right.\n\\end{equation}\n\n\\subsubsection{The Laplace Transforms $L [w_n (x) ; x \\rightarrow s]$}\n\nThe Laplace transform $L^{} [w_n (x) ; x \\rightarrow s]$ and its roots are\ncalculated to shed light on the behaviour roots of the Mellin transforms $M\n[w_n (x) ; x \\rightarrow s]$ but it is unclear whether this is accomplished. \\\nThe Laplace transform (\\ref{laplace}) of the $n$-th component function is\ngiven by\n\\begin{equation}\n  \\begin{array}{ll}\n    L [w_n (x) ; x \\rightarrow s] & = \\int_0^1 w_n (x) e^{- x s} \\mathrm{d} x\\\\\n    & = \\int_0^1 n (x n + x - 1) \\chi_{} (x, I^H_n) e^{- x s} \\mathrm{d} x\\\\\n    & = \\int_{\\frac{1}{n + 1}}^{\\frac{1}{n}} n (x n + x - 1) e^{- x s} \\mathrm{d}\n    x\\\\\n    & = \\frac{n (n + 1) e^{- \\frac{s}{n + 1}} - (n^2 + n + s) e^{-\n    \\frac{s}{n}}}{s^2}\n  \\end{array}\n\\end{equation}\nThere is a removable singularity at $s = 0$ which has the limit\n\\begin{equation}\n  \\begin{array}{ll}\n    L [w_n (x) ; x \\rightarrow 0] & = \\lim_{s \\rightarrow 0} L [w_n (x) ; x\n    \\rightarrow s]\\\\\n    & = \\lim_{s \\rightarrow 0} \\frac{n (n + 1) e^{- \\frac{s}{n + 1}} - (n^2 +\n    n + s) e^{- \\frac{s}{n}}}{s^2}\\\\\n    & = \\frac{1}{2}  \\underset{s = 1}{\\tmop{Res}} \\left( \\tau (s)_{} M [w_n\n    (x) ; x \\rightarrow s] \\right)\\\\\n    & = \\frac{1}{2 n \\left( n + 1 \\right)}\n  \\end{array}\n\\end{equation}\nAdditionally,\n\\begin{equation}\n  \\begin{array}{ll}\n    \\sum_{n = 1}^{\\infty} L [w_n (x) ; x \\rightarrow 0] & = \\sum_{n =\n    1}^{\\infty} \\lim_{s \\rightarrow 0} \\frac{n (n + 1) e^{- \\frac{s}{n + 1}} -\n    (n^2 + n + s) e^{- \\frac{s}{n}}}{s^2}\\\\\n    & = \\sum_{n = 1}^{\\infty} \\frac{1}{2 n \\left( n + 1 \\right)}\\\\\n    & = \\lim_{n \\rightarrow \\infty} \\underset{s \\rightarrow - 1}{\\lim} M_{}\n    [w_n (x) ; x \\rightarrow s]\\\\\n    & = \\lim_{n \\rightarrow \\infty} \\lim_{s \\rightarrow - 1} - \\frac{n^{1 -\n    s} - n (n + 1)^{- s} - s n^{- s}}{s^2 + s}\\\\\n    & = \\frac{1}{2}\n  \\end{array}\n\\end{equation}\nThe roots of $L [w_n (x) ; x \\rightarrow s]$ are enumerated by\n\\begin{equation}\n  \\begin{array}{ll}\n    \\rho^L_{w_n} (m) & \\text{$= \\left\\{ s : \\text{$L [w_n (x) ; x \\rightarrow\n    s]$} = 0 \\right\\}$} = - n (n + 1) W (m, - e^{- 1}) - n (n + 1)\n  \\end{array}\n\\end{equation}\nwhere $W (m, x)$ is the Lambert W function (\\ref{lambertw}) and $m \\in\n\\mathbbm{Z}, \\text{$n \\in \\mathbbm{N}$}$. It can be verified that\n\\begin{equation}\n  \\begin{array}{ll}\n    L [w_n (z) ; x \\rightarrow \\rho^L_{w_n} (m)] & = \\frac{e^{(n + 1) (1 + W\n    (m, - e^{- 1}))} W (m, - e^{- 1}) + e^{n (1 + W (m, - e^{- 1}))}}{n (1 + W\n    (m, - e^{- 1}))^2 (n + 1)} = 0\n  \\end{array}\n\\end{equation}\nwhere $- e^{- 1}$ is expressed as the continued fraction via its quotient\nsequence\n\\begin{equation}\n  - e^{- 1} = [- 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12,\n  1, 1, 14, 1, 1, 16, 1, 1, 18, 1, 1, 20, ...]\n\\end{equation}\nThe roots $\\rho^L_{w_n} (m)$ satisfy a functional reflection equation with\nrespect to $m$\n\\begin{equation}\n  \\begin{array}{ll}\n    \\rho^L_{w_n} (m) & = \\overline{\\rho^L_{w_n} (- m - 1)}\\\\\n    - n (n + 1) W (m, - e^{- 1}) - n (n + 1) & = - n (n + 1) \\overline{W (- m\n    - 1, - e^{- 1})} - n (n + 1)\n  \\end{array}\n\\end{equation}\nwhere $\\bar{x} = \\tmop{Re} (x) - \\tmop{Im} (x)$ denotes complex conjugation.\nThe quotients of the roots of consecutive transforms is\n\\begin{equation}\n  \\begin{array}{ll}\n    \\frac{\\rho^L_{w_n} (m)}{\\rho^L_{w_{n - 1}} (m)} & = \\frac{n + 1}{n - 1}\n  \\end{array}\n\\end{equation}\nThus\n\\begin{equation}\n  \\begin{array}{ll}\n    \\lim_{n \\rightarrow \\infty} \\frac{\\rho^L_{w_n} (m)}{\\rho^L_{w_{n - 1}}\n    (m)} & = \\lim_{n \\rightarrow \\infty} \\frac{n + 1}{n - 1} = 1\n  \\end{array}\n\\end{equation}\n\\begin{figure}[h]\n  \\resizebox{13cm}{10cm}{\\includegraphics{wlroots1-9.eps}}\n  \\caption{The Roots $\\rho^L_{w_n} (m)$ of $\\left\\{ L [w_n (x) ; x \\rightarrow\n  s] \\right. \\left. : n \\in 1 \\ldots 9 \\right\\}$}\n\\end{figure}\n\n\\subsubsection{The Roots $\\rho^M_{w_n} \\left( m \\right)$ of $M [w_n (x) ; x\n\\rightarrow s]$}\n\nDefine\n\\begin{equation}\n  \\begin{array}{ll}\n    M^{}_{\\tau w} (s, n) & = \\text{$\\tau (s) \\text{$M [_{} w_n (x) ; x\n    \\rightarrow s]$}$}\\\\\n    & = \\frac{n (n + 1)^{- s} - n^{1 - s} + s n^{- s}}{s - 1}\n  \\end{array}\n\\end{equation}\nas in (\\ref{zwpoly}) and its infinte number of inverse branches (which are\ncurrently lacking closed-form expression if such a thing is possible except\nwhen $n = 1$), where the branches are indexed by $m$\n\\begin{equation}\n  \\begin{array}{ll}\n    M^{- 1}_{\\tau w} (z, n, m) & =\\{s : \\tau (s) \\text{$M [_{} w_n (x) ; x\n    \\rightarrow s]$} = z\\}\\\\\n    & = \\left\\{ s : \\frac{n (n + 1)^{- s} - n^{1 - s} + s n^{- s}}{s - 1} = z\n    \\right\\}\n  \\end{array}\n\\end{equation}\nthen we see that the first function where $n = 1$, $M^{- 1}_{\\tau w} (z, 1,\nm)$, has the closed-form\n\\begin{equation}\n  \\begin{array}{ll}\n    \\text{$M^{- 1}_{\\tau w} (z, 1, m)$} & = \\frac{W \\left( m, \\frac{\\ln (\n    \\sqrt{2})}{z - 1} \\right) + \\ln (2)}{\\ln (2)}\n  \\end{array}\n\\end{equation}\nwhere $W$ is the Lambert W function (\\ref{lambertw}). It is verified that\n\\begin{equation}\n  \\begin{array}{ll}\n    \\text{$M_{\\tau w} (M^{- 1}_{\\tau w} (z, 1, m), 1, m)$} & = - \\frac{1 -\n    2^{- \\frac{W \\left( m, \\frac{\\ln ( \\sqrt{2})}{z - 1} \\right) + \\ln\n    (2)}{\\ln (2)}} - \\frac{W \\left( m, \\frac{\\ln ( \\sqrt{2})}{z - 1} \\right) +\n    \\ln (2)}{\\ln (2)}}{\\frac{W \\left( m, \\frac{\\ln ( \\sqrt{2})}{z - 1} \\right)\n    + \\ln (2)}{\\ln (2)} - 1} = z\n  \\end{array}\n\\end{equation}\nFurthermore, let $\\rho^M_{w_n} \\left( m \\right)$ denote the $m$-th root of\n$\\text{$M [_{} w_n (x) ; x \\rightarrow s]$}$\n\\begin{equation}\n  \\begin{array}{ll}\n    \\rho_{w_n}^M (m) & =\\{s : \\text{$M [_{} w_n (x) ; x \\rightarrow s]$} =\n    0\\}\\\\\n    & = M^{- 1}_w (0, n, m)\n  \\end{array} \\label{wmroots}\n\\end{equation}\nwhich satisfies\n\\begin{equation}\n  \\begin{array}{l}\n    \\tmop{Im} \\left( \\rho^M_{w_n} \\left( m \\right) \\right) > \\tmop{Im} \\left(\n    \\rho^M_{w_{n - 1}} \\left( m \\right) \\right)\n  \\end{array}\n\\end{equation}\n\\begin{equation}\n  \\begin{array}{ll}\n    \\lim_{m \\rightarrow \\pm \\infty} \\tmop{Re} (\\rho^M_{w_n} \\left( m \\right))\n    & = 0\n  \\end{array}\n\\end{equation}\nThus,\n\\begin{equation}\n  \\begin{array}{ll}\n    \\lim_{m \\rightarrow \\pm \\infty} \\arg \\left( \\rho^M_{w_n} \\left( m \\right)\n    \\right) & = \\frac{\\pi}{2}\n  \\end{array}\n\\end{equation}\n\n\\subsubsection{Quotients and Differences of $\\rho^M_{w_n} \\left( m \\right)$}\n\nLet\n\\begin{equation}\n  \\begin{array}{ll}\n    \\Delta \\rho^M_{w_n} (m) & = \\rho^M_{w_n} (m + 1) - \\rho^M_{w_n} \\left( m\n    \\right)\n  \\end{array}\n\\end{equation}\nbe the forward difference of consecutive roots of $M [w_n (x) ; x \\rightarrow\ns]$. The limiting difference between consecutive roots is the countably\ninfinite set of solutions to the equation $n^{\\frac{s}{2}} +_{} (n +\n1)^{\\frac{s}{2}} = 0$ given by\n\\begin{equation}\n  \\begin{array}{ll}\n    \\Delta \\rho^M_{w_n} (\\pm \\infty) & = \\lim_{m \\rightarrow \\pm \\infty}\n    \\overset{}{\\Delta^{}} \\rho^M_{w_n} (m)\\\\\n    & = \\left\\{ s : n^{\\frac{s}{2}} +_{} (n + 1)^{\\frac{s}{2}} = 0 \\right\\}\\\\\n    & = \\lim_{m \\rightarrow \\pm \\infty} \\left( \\rho^M_{w_n} (m + 1) -\n    \\rho^M_{w_n} (m) \\right)\\\\\n    & = \\lim_{m \\rightarrow \\pm \\infty} \\left( \\frac{\\rho^M_{w_n} (m)}{m}\n    \\right)\\\\\n    & = \\frac{2 \\pi i}{M [\\chi \\left( x, I^H_n \\right) ; x \\rightarrow 0]}\\\\\n    & = \\frac{2 \\pi i}{\\lim_{s \\rightarrow 0} \\left( \\frac{n^{- s} - (n +\n    1)^{- s}}{s} \\right)}\\\\\n    & = \\frac{2 \\pi i}{\\ln (n + 1) - \\ln (n)}\n  \\end{array} \\text{\\label{wmrad}}\n\\end{equation}\nLet $\\mathcal{Q}^{\\infty}_{\\rho^M_{w_n}}$ denote the limit\n\\begin{equation}\n  \\begin{array}{ll}\n    \\mathcal{Q}^{\\infty}_{\\rho^M_{w_n}} & = \\lim_{m \\rightarrow \\pm \\infty}\n    \\frac{\\rho^M_{w_n} (m)}{\\rho^M_{w_{n - 1}} (m)}\\\\\n    & = \\frac{\\Delta \\rho^M_{w_n} (\\pm \\infty)}{\\Delta \\rho^M_{w_{n - 1}}\n    (\\pm \\infty)}\\\\\n    & = \\frac{\\overset{}{M} \\left[ \\chi \\left( x, \\left( \\frac{1}{n + 1},\n    \\frac{1}{n} \\right) \\right) ; x \\rightarrow 0 \\right]}{\\overset{}{M}\n    \\left[ \\chi \\left( x, \\left( \\frac{1}{n}, \\frac{1}{n - 1} \\right) \\right)\n    ; x \\rightarrow 0 \\right]}\\\\\n    & = \\frac{\\ln (n) - \\ln (n - 1)}{\\ln (n + 1) - \\ln (n)}\n  \\end{array}\n\\end{equation}\nthen we also have the limit of the limits\n$\\mathcal{Q}^{\\infty}_{\\rho^M_{w_n}}$ \\ as $n \\rightarrow \\infty$ given by\n\\begin{equation}\n  \\begin{array}{ll}\n    \\lim_{n \\rightarrow \\pm \\infty}\n    \\text{$\\mathcal{Q}^{\\infty}_{\\rho^M_{w_n}}$ } & = \\lim_{n \\rightarrow \\pm\n    \\infty} \\left( \\frac{\\Delta \\rho^M_{w_n} (\\pm \\infty)}{\\Delta \\rho^M_{w_{n\n    - 1}} (\\pm \\infty)} \\right)\\\\\n    & = \\lim_{n \\rightarrow \\pm \\infty} \\left( \\frac{\\frac{\\ln (n) - \\ln (n -\n    1)}{\\ln (n + 1) - \\ln (n)}}{\\frac{\\ln (n - 1) - \\ln (n - 2)}{\\ln (n) - \\ln\n    (n - 2)}} \\right)\\\\\n    & = \\lim_{n \\rightarrow \\pm \\infty} \\left( \\frac{\\ln (n) - \\ln (n -\n    1)}{\\ln (n + 1) - \\ln (n)} \\right)\\\\\n    & = 1\n  \\end{array}\n\\end{equation}\n\\begin{figure}[h]\n  \\resizebox{13cm}{9cm}{\\includegraphics{wmroots.eps}}\n  \\caption{$\\{\\rho^M_{w_n} (m) : m = 1 \\ldots 5\\}$}\n\\end{figure}\n\nThe limiting quotients $\\frac{e^{- \\rho^M_{w_n} (m + 1)} }{e^{- \\rho^M_{w_n}\n(m)}} = e^{\\rho^M_{w_n} (m) - \\rho^M_{w_n} (m + 1)}$ as $m \\rightarrow \\infty$\nare given by\n\\begin{equation}\n  \\begin{array}{ll}\n    \\begin{array}{l}\n      \\lim_{m \\rightarrow \\infty} e^{\\rho^M_{w_n} (m) - \\rho^M_{w_n} (m + 1)}\n    \\end{array} & = 1 - 2 \\sin \\left( \\frac{\\pi}{\\ln (n + 1) - \\ln (n)}\n    \\right)^2 - 2 i \\cos \\left( \\frac{\\pi}{\\ln (n + 1) - \\ln (n)} \\right) \\sin\n    \\left( \\frac{\\pi}{\\ln (n + 1) - \\ln (n)} \\right) \\\\\n    & = e^{- \\frac{2 \\pi i}{\\ln (n + 1) - \\ln (n)}}\n  \\end{array}\n\\end{equation}\nwhere we have\n\\begin{equation}\n  \\begin{array}{lll}\n    | \\lim_{m \\rightarrow \\infty} e^{\\rho^M_{w_n} (m) - \\rho^M_{w_n} (m + 1)}\n    | & = \\lim_{m \\rightarrow \\infty} | e^{\\rho^M_{w_n} (m) - \\rho^M_{w_n} (m\n    + 1)} | & \\\\\n    & = \\sqrt{e^{- \\frac{2 \\pi i}{\\ln (n + 1) - \\ln (n)}}} & \\\\\n    & = 1 & \n  \\end{array}\n\\end{equation}\nand\n\\begin{equation}\n  \\begin{array}{l}\n    \\lim_{n \\rightarrow \\infty} \\lim_{m \\rightarrow \\infty} e^{\\rho^M_{w_n}\n    (m) - \\rho^M_{w_n} (m + 1)} = - 1\n  \\end{array}\n\\end{equation}\n\n\\subsubsection{The Laplace Transform $L \\left[ (s - 1) M [w_n (x) ; x\n\\rightarrow s \\right] ; s \\rightarrow t]$}\n\nThe Laplace transform of $(s - 1) M [w_n (x) ; x \\rightarrow s]$ defined by\n\\begin{equation}\n  \\begin{array}{lll}\n    \\left. L [(s - 1) M [w_n (x) ; x \\rightarrow s \\right] ; s \\rightarrow t]\n    &  & = L [n (n + 1)^{- s} + sn^{- s} - n^{1 - s} ; s \\rightarrow t]\\\\\n    &  & = \\int_0^{\\infty} n (n + 1)^{- s} + sn^{- s} - n^{1 - s} e^{- s t}\n    \\mathrm{d} s\\\\\n    &  & = \\frac{t + \\ln \\left( \\frac{n^n}{(n + 1)^n} \\right) t + \\ln (n + 1)\n    + n \\ln (n)^2 - n \\ln (n) \\ln (n + 1)}{(\\ln (n) + t)^2 (\\ln (n + 1) + t)}\n  \\end{array}\n\\end{equation}\nhas poles at $- \\ln (n)$ and $- \\ln (n + 1)$ with residues\n\\begin{equation}\n  \\begin{array}{ll}\n    \\underset{}{\\underset{t = - \\ln (n)}{\\tmop{Res}} \\left( \\underset{}{L} [(s\n    - 1) M_{}^{} [w_n (x) ; x \\rightarrow s] ; s \\rightarrow t] \\right)} & = -\n    n\\\\\n    \\underset{}{\\underset{t = - \\ln (n + 1)}{\\tmop{Res}} \\left( \\underset{}{L}\n    [(s - 1) M_{}^{} [w_n (x) ; x \\rightarrow s] ; s \\rightarrow t] \\right)} &\n    = n\n  \\end{array}\n\\end{equation}\n\n\\subsection{The Gauss Map $h (x)$}\n\n\\subsubsection{$\\tmop{Continued} \\tmop{Fractions}$}\n\nThe Gauss map $h (x)$, also known as the Gauss function or Gauss\ntransformation, maps unit intervals onto unit intervals and by iteration gives\nthe continued fraction expansion of a real number\n{\\cite[A.1.7]{Ctsa}}{\\cite[I.1]{cf}}{\\cite[X]{itn}} The $n$-th component\nfunction $h_n (x)$ of the map $h (x)$ is given by\n\\begin{equation}\n  \\begin{array}{ll}\n    h_n (x) & = \\frac{1 - x n}{x} \\chi_{} \\left( x, I^H_n \\right)\n  \\end{array}\n\\end{equation}\nThe infinite sum of the component functions is the Gauss map\n\\begin{equation}\n  \\begin{array}{ll}\n    h (x) & = \\sum_{n = 1}^{\\infty} h_n (x)\\\\\n    & = \\sum_{n = 1}^{\\infty} \\frac{1 - x n}{x} \\chi_{} \\left( x, I^H_n\n    \\right)\\\\\n    & = x^{- 1} - \\left\\lfloor x^{- 1} \\right\\rfloor\\\\\n    & =\\{x^{- 1} \\}\n  \\end{array} \\label{h}\n\\end{equation}\n\\begin{figure}[h]\n  \\resizebox{12cm}{12cm}{\\includegraphics{h.eps}}\n  \\caption{\\label{gaussmapfig}The Gauss Map}\n\\end{figure}\n\nThe fixed points of $h (x)$ are the (positive) solutions to the equation $h_n\n(x) = x$ given by\n\\begin{equation}\n  \\begin{array}{ll}\n    \\tmop{Fix}_h^n & =\\{x : h_n (x)_{} = x\\}\\\\\n    & = \\left\\{ x : \\frac{1 - x n}{x} \\chi_{} \\left( x, I^H_n \\right)_{} = x\n    \\right\\}\\\\\n    & = \\left\\{ x : \\frac{1 - x n}{x} = x \\right\\}\\\\\n    & = \\frac{\\sqrt{n^2 + 4}}{2} - \\frac{n}{2}\n  \\end{array}\n\\end{equation}\n\n\\subsubsection{The Mellin Transform of $h (x)$}\n\nThe Mellin transform (\\ref{mellin}) of the Gauss map $h (x$) over the unit\ninterval, scaled by $s$ then subtracted from $\\frac{s}{s - 1}$, is an analytic\ncontinuation of $\\zeta (s$), denoted by $\\zeta_h (s$), valid for all$-\n\\tmop{Re} (s)) \\not\\in \\mathbbm{N}_{}$. The transfer operator and thermodynamic\naspects of the Gauss map are discussed in\n{\\cite{gkwoperator}}{\\cite{newtonzeta}}{\\cite{gkwzeta}}{\\cite{yarh}}{\\cite{dzf}}.\nThe Mellin transform of the $n$-th component function $w_n (x$) is given by\n\\begin{equation}\n  \\begin{array}{ll}\n    M [h_n (x) ; x \\rightarrow s] & = \\int_0^1 h_n (x) x^{s - 1} \\mathrm{d} x\\\\\n    & = \\int_0^1 \\frac{1 - x n}{x} \\chi_{} \\left( x, I^H_n \\right) x^{s - 1}\n    \\mathrm{d} x\\\\\n    & = \\int_{\\frac{1}{n + 1}}^{\\frac{1}{n}} \\left( x^{- 1} - \\left\\lfloor\n    x^{- 1} \\right\\rfloor \\right) x^{s - 1} \\mathrm{d} x\\\\\n    & = \\int_{\\frac{1}{n + 1}}^{\\frac{1}{n}} \\frac{1 - x n}{x} x^{s - 1}\n    \\mathrm{d} x\\\\\n    & = - \\frac{n (n + 1)^{- s} + s (n + 1)^{- s} - n^{1 - s}}{s^2 - s}\n  \\end{array}\n\\end{equation}\nwhich provides an analytic continuation $\\zeta_h (s) = \\zeta (s) \\forall (-\n\\tmop{Re} (s)) \\not\\in \\mathbbm{N}_{}$\n\\begin{equation}\n  \\begin{array}{ll}\n    \\zeta_h (s) & = \\frac{s}{s - 1} - s M_{} [h (x) ; x \\rightarrow s]\\\\\n    & = \\frac{s}{s - 1} - s \\int_0^1 h (x) x^{s - 1} \\mathrm{d} x\\\\\n    & = \\frac{s}{s - 1} - s \\int_0^1 \\left( x^{- 1} - \\left\\lfloor x^{- 1}\n    \\right\\rfloor \\right) x^{s - 1} \\mathrm{d} x\\\\\n    & = \\frac{s}{s - 1} - s \\sum_{n = 1}^{\\infty} M_{} [h_n (x) ; x\n    \\rightarrow s]\\\\\n    & = \\frac{s}{s - 1} - \\frac{1}{s - 1} \\sum_{n = 1}^{\\infty} - (n (n +\n    1)^{- s} + s (n + 1)^{- s} - n^{1 - s})\\\\\n    & = \\frac{s}{s - 1} -  \\frac{1 }{s - 1} \\sum_{n = 1}^{\\infty} n^{1 - s} -\n    n (n + 1)^{- s} - s (n + 1)^{- s}\n  \\end{array} \\label{gaussmap}\n\\end{equation}\n\n\\subsection{The Harmonic Sawtooth Map w(x) as an Ordinary Fractal String}\n\n\\subsubsection{Definition and Length}\n\nLet\n\\begin{equation}\n  _{} \\begin{array}{ll}\n    I^H_n & = \\left( \\frac{1}{n + 1}, \\frac{1}{n} \\right)\n  \\end{array} \\label{hi}\n\\end{equation}\nbe the $n$-th harmonic interval, then $\\{w (x) \\in \\mathcal{L}_w : x \\in\n\\Omega\\}$ is the piecewise monotone mapping of the unit interval onto itself.\nThe fractal string $\\mathcal{L}_w$ associated with $w (x)$ is the set of\nconnected component functions $w_n (x) \\subset w (x)$ \\ where each $w_n (x)$\nmaps$I_n^H$ onto $(0, 1)$ and vanishes when $x \\not\\in I_n^H$. Thus, the\ndisjoint union of the connected components of $\\mathcal{L}_w$ is the infinite\nsum $w (x) = \\sum_{n = 1}^{\\infty} w_n (x)$ where only 1 of the $w_n (x)$ is\nnonzero for each $x$, thus $w (x)$ maps entire unit interval onto itself\nuniqely except except for the points of discontinuity on the boundary\n$\\partial \\mathcal{L}_w =\\{0, \\frac{1}{n} : n \\in \\mathbbm{N}^{\\ast} \\}$ where\na choice is to be made between 0 and 1 depending on the direction in which the\nlimit is approached. Let\n\\begin{equation}\n  \\begin{array}{ll}\n    w_n (x) & = n (x n + x - 1) \\chi_{} (x, I^H_n)\n  \\end{array} \\label{wn}\n\\end{equation}\nwhere $\\chi_{} (x, I^H_n)$ is the $n$-th harmonic interval indicator\n(\\ref{ii})\n\\begin{equation}\n  \\begin{array}{ll}\n    _{} \\chi_{} (x, I_n^H) & = \\theta \\left( \\frac{x n + x - 1}{n + 1} \\right)\n    - \\theta \\left( \\frac{x n - 1}{n} \\right)\n  \\end{array} \\label{hii}\n\\end{equation}\nThe substitution $n \\rightarrow \\left\\lfloor \\frac{1}{x} \\right\\rfloor$ can be\nmade in (\\ref{hii}) where it is seen that\n\\begin{equation}\n  \\begin{array}{ll}\n    \\text{$\\chi_{} \\left( x, I^H_{\\left\\lfloor x^{- 1} \\right\\rfloor}\n    \\right)$} & = \\theta \\left( \\frac{x \\text{$\\left\\lfloor x^{- 1}\n    \\right\\rfloor$} + x - 1}{\\text{$\\left\\lfloor x^{- 1} \\right\\rfloor$} + 1}\n    \\right) - \\theta \\left( \\frac{x \\left\\lfloor x^{- 1} \\right\\rfloor \\text{}\n    - 1}{\\left\\lfloor x^{- 1} \\right\\rfloor} \\right) = 1 \\label{hieye}\n  \\end{array}\n\\end{equation}\nand so making the same substitution in (\\ref{wn}) gives\n\\begin{equation}\n  \\begin{array}{ll}\n    w (x) & = \\sum_{n = 1}^{\\infty} w_n (x)\\\\\n    & = \\sum_{n = 1}^{\\infty} n (x n + x - 1) \\chi_{} (x, I^H_n)\\\\\n    & = \\left\\lfloor x^{- 1} \\right\\rfloor \\text{} \\left( x \\left\\lfloor x^{-\n    1} \\right\\rfloor \\text{} + x - 1 \\right)\\\\\n    & = x \\left\\lfloor x^{- 1} \\right]^2 + x \\left\\lfloor x^{- 1}\n    \\right\\rfloor \\text{} - \\left\\lfloor x^{- 1} \\right\\rfloor \\text{}\n  \\end{array} \\label{w}\n\\end{equation}\n\\begin{figure}[h]\n  \\resizebox{12cm}{12cm}{\\includegraphics{harmonicsaw.eps}}\n  \\caption{The Harmonic Sawtooth Map}\n\\end{figure}\n\nThe intervals $_{} I_n^w$ will be defined such that $\\ell w_n = \\left| I_n^w\n\\right| = \\left| w_n (x) \\right|$. Let\n\\begin{equation}\n  \\begin{array}{ll}\n    \\mathfrak{h}_n & = \\int^{}_{I^H_n} x (n + 1) n \\mathrm{d} x\\\\\n    & = \\frac{\\left( \\frac{1}{n + 1} + \\frac{1}{n} \\right)}{2}\\\\\n    & = \\frac{2 n + 1}{2 n (n + 1)}\n  \\end{array}\n\\end{equation}\nbe the midpoint of $I^H_n$ then\n\\begin{equation}\n  \\begin{array}{ll}\n    I_n^w & = \\left( \\mathfrak{h}_n - \\frac{\\left| w_n (x) \\right|}{2},\n    \\mathfrak{h}_n + \\frac{\\left| w_n (x) \\right|}{2} \\right)\\\\\n    & = \\left( \\frac{4 n + 1}{4 n (n + 1)}, \\frac{4 n + 3}{4 n (n + 1)}\n    \\right)\n  \\end{array}\n\\end{equation}\nso that\n\\begin{equation}\n  \\begin{array}{ll}\n    \\ell w_n & = \\left| w_n (x) \\right|\\\\\n    & = \\int_0^1 n (x n + x - 1) \\chi_{} (x, I^H_n) \\mathrm{d} x\\\\\n    & = \\int^{\\frac{1}{n}}_{\\frac{1}{n + 1}} w (x) \\mathrm{d} x\\\\\n    & = \\left| I_n^w \\right|\\\\\n    & = \\int_0^1 \\chi (x, I_n^w) \\mathrm{d} x\\\\\n    & = \\frac{4 n + 3}{4 n (n + 1)} - \\frac{4 n + 1}{4 n (n + 1)}\\\\\n    & = \\frac{1}{2 n (n + 1)}\n  \\end{array} \\label{wlen}\n\\end{equation}\n\\begin{figure}[h]\n  \\resizebox{10cm}{8cm}{\\includegraphics{reciplen.eps}}\n  \\caption{Reciprocal lengths $\\ell w_n^{- 1}$}\n\\end{figure}\n\nThe total length of $\\mathcal{L}_w$ is\n\\begin{equation}\n  \\begin{array}{ll}\n    |\\mathcal{L}_w | & = \\int_0^1 w (x) \\mathrm{d} x\\\\\n    & = \\sum_{n = 1}^{\\infty} \\ell w_n\\\\\n    & = \\sum_{n = 1}^{\\infty} \\frac{1}{2 n (n + 1)}\\\\\n    & = \\frac{1}{2}\n  \\end{array}\n\\end{equation}\n\\begin{figure}[h]\n  \\resizebox{10cm}{10cm}{\\includegraphics{sawfi.eps}}\n  \\caption{$\\chi (x, I_n^w)$ green and $w_n (x)$ blue for $n = 1 \\ldots 3$ and\n  $x = \\frac{1}{4} ..1$}\n\\end{figure}\n\n\\subsubsection{Geometry and Volume of the Inner Tubular Neighborhood}\n\nThe geometric counting function (\\ref{geocount}) of $\\mathcal{L}_w$ is\n\\begin{equation}\n  \\begin{array}{ll}\n    N_{\\mathcal{L}_w} (x) & =\\#\\{n \\geqslant 1 : \\ell w_n^{- 1} \\leqslant\n    x\\}\\\\\n    & =\\#\\{n \\geqslant 1 : 2 (n + 1) n \\leqslant x\\}\\\\\n    & = \\left\\lfloor \\frac{\\sqrt{2 x + 1}}{2} - \\frac{1}{2} \\right\\rfloor\n  \\end{array} \\label{wgeocount}\n\\end{equation}\nwhich is used to calculate the limiting constant (\\ref{mconst}) $C_w$\nappearing in the equation for the Minkowski content \\\n\\begin{equation}\n  \\begin{array}{ll}\n    C_w & = \\lim_{x \\rightarrow \\infty} \\frac{N_{\\mathcal{L}_w}\n    (x)}{x^{D_{\\mathcal{L}_w}}}\\\\\n    & = \\lim_{x \\rightarrow \\infty} \\frac{\\frac{\\sqrt{2 x + 1}}{2} -\n    \\frac{1}{2}}{\\sqrt{x}}\\\\\n    & = \\frac{\\sqrt{2}}{2}\n  \\end{array} \\label{wmconst}\n\\end{equation}\nThe function $N_{\\mathcal{L}_w} (x)$ happens to coincide with\n{\\cite[A095861]{oeis}}, which is the number of primitive Pythagorean triangles\nof the form $\\{(a, b, b + 1) : (b + 1) \\leqslant n\\}$.\n{\\cite[171-176]{bon}}{\\cite[10.1]{pt}}{\\cite[11.2-11.5]{ent}} Let\n\\begin{equation}\n  \\begin{array}{lll}\n    v_{} (\\varepsilon) & = \\min (j : \\ell w_j < 2 \\varepsilon) & =\n    \\left\\lfloor \\frac{\\varepsilon + \\sqrt{\\varepsilon^2 + \\varepsilon}}{2\n    \\varepsilon} \\right\\rfloor\n  \\end{array}\n\\end{equation}\nwhich is the floor of the solution to the inverse length equation\n\\begin{equation}\n  \\begin{array}{ll}\n    \\frac{\\varepsilon + \\sqrt{\\varepsilon^2 + \\varepsilon}}{2 \\varepsilon} =\n    \\left\\{ n : \\ell w_{n - 1} = 2 \\varepsilon \\right\\} = \\left\\{ n :\n    \\frac{1}{2 n (n - 1)} = 2 \\varepsilon \\right\\} & \\\\\n    \\frac{\\ell w_{\\frac{\\varepsilon + \\sqrt{\\varepsilon^2 + \\varepsilon}}{2\n    \\varepsilon} - 1}}{2} = \\varepsilon & \n  \\end{array}\n\\end{equation}\nThen the volume of the inner tubular neighborhood of $\\partial \\mathcal{L}_w$\nwith radius $\\varepsilon$ (\\ref{tnv}) is\n\\begin{equation}\n  \\begin{array}{ll}\n    V_{\\mathcal{L}_w} (\\varepsilon) & = 2 \\varepsilon N_{\\mathcal{L}_w} \\left(\n    \\frac{1}{2 \\varepsilon^{}} \\right) + \\sum_j^{\\ell w_j < 2 \\varepsilon}\n    \\ell w_j\\\\\n    & = 2 \\varepsilon N_{\\mathcal{L}_w} \\left( \\frac{1}{2 \\varepsilon^{}}\n    \\right) + \\sum_{\\tmscript{\\begin{array}{l}\n      n = v (\\varepsilon)\n    \\end{array}}}^{\\infty} \\frac{1}{2 (n + 1) n}\\\\\n    & = 2 \\varepsilon N_{\\mathcal{L}_w} \\left( \\frac{1}{2 \\varepsilon^{}}\n    \\right) + \\frac{1}{2 v (\\varepsilon)^{}}\\\\\n    & = 2 \\varepsilon \\left\\lfloor \\frac{\\sqrt{\\frac{1}{\\varepsilon} + 1}}{2}\n    - \\frac{1}{2} \\right\\rfloor + \\frac{1}{2 v (\\varepsilon)^{}}\\\\\n    & = \\frac{4 \\varepsilon v (\\varepsilon)^2 - 4 \\varepsilon v (\\varepsilon)\n    + 1}{2 v (\\varepsilon)}\n  \\end{array} \\label{wvol}\n\\end{equation}\nsince\n\\begin{equation}\n  \\begin{array}{ll}\n    \\sum_{n = m}^{\\infty} \\frac{1}{2 n (n + 1)} & = \\frac{1}{2 m}\n  \\end{array}\n\\end{equation}\nand by defintion we have\n\\begin{equation}\n  \\begin{array}{ll}\n    \\lim_{\\varepsilon \\rightarrow 0^+} V_{\\mathcal{L}_w} (\\varepsilon) & = 0\\\\\n    \\lim_{\\varepsilon \\rightarrow \\infty} V_{\\mathcal{L}_w} (\\varepsilon) & =\n    |\\mathcal{L}_w | = \\frac{1}{2}\n  \\end{array}\n\\end{equation}\nThus, using (\\ref{wmconst}) and (\\ref{wvol}), the Minkowski content\n(\\ref{mcontent}) of $\\mathcal{L}_w$ is\n\\begin{equation}\n  \\begin{array}{lll}\n    \\mathcal{M}_{\\mathcal{L}_w} & = \\lim_{e \\rightarrow 0^+}\n    \\frac{V_{\\mathcal{L}w} (\\varepsilon)}{\\varepsilon^{1 - D_{\\mathcal{L}_w}}}\n    & \\\\\n    & = \\lim_{e \\rightarrow 0^+} \\frac{1}{\\sqrt{\\varepsilon}}  \\left( 2\n    \\varepsilon \\left\\lfloor \\frac{\\sqrt{\\frac{1}{\\varepsilon} + 1}}{2} -\n    \\frac{1}{2} \\right\\rfloor + \\frac{1}{2} \\left\\lfloor \\frac{\\varepsilon +\n    \\sqrt{\\varepsilon^2 + \\varepsilon}}{2 \\varepsilon} \\right\\rfloor^{- 1}\n    \\right) & \\\\\n    & = \\frac{C_w 2^{1 - D_{\\mathcal{L}_w}}}{1 - D_{\\mathcal{L}_w}} & \\\\\n    & = \\frac{\\frac{\\sqrt{2}}{2} 2^{1 - \\frac{1}{2}}}{1 - \\frac{1}{2}} & \\\\\n    & = 2 & \n  \\end{array} \\label{wmcontent}\n\\end{equation}\n\\begin{figure}[h]\n  \\resizebox{12cm}{10cm}{\\includegraphics{sawgeocount.eps}}\n  \\caption{Geometric Counting Function $N_{\\mathcal{L}_w} (x)$ of $w (x)$}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\resizebox{12cm}{10cm}{\\includegraphics{minleps.eps}}\n  \\caption{$\\{v (\\varepsilon) = \\min (n : \\ell w_n < 2 \\varepsilon) :\n  \\varepsilon = \\frac{1}{1000} \\ldots \\frac{1}{4} \\}$}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\resizebox{12cm}{10cm}{\\includegraphics{Vw.eps}}\n  \\caption{Volume of the inner tubular neighborhood of $\\partial\n  \\mathcal{L}_w$ with radius $\\varepsilon$ $\\left\\{ V_{\\mathcal{L}_w}\n  (\\varepsilon) : \\varepsilon = 0 \\ldots \\frac{1}{8} \\right\\}$}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\resizebox{12cm}{10cm}{\\includegraphics{vse.eps}}\n  \\caption{$\\left\\{ \\frac{V\\mathcal{L}_w (\\varepsilon)}{\\sqrt{\\varepsilon}} :\n  \\varepsilon = 0 \\ldots \\frac{1}{8} \\right\\}$ and $\\frac{V\\mathcal{L}_w (8^{-\n  1})}{\\sqrt{8^{- 1}}} = \\sqrt{2}$}\n\\end{figure}\n\n\\subsubsection{The Geometric Zeta Function $\\zeta_{\\mathcal{L}_w} (s)$}\n\nThe geometric zeta function (\\ref{geozeta}) of $w (x$) is the Dirichlet series\nof the lengths $\\ell w_n$ (\\ref{wlen}) and also an integral over the geometric\nlength counting function (\\ref{wgeocount}) $N_{\\mathcal{L}_w} (x)$\n\\begin{equation}\n  \\begin{array}{ll}\n    \\zeta_{\\mathcal{L}_w} (s) & = \\sum_{n = 1}^{\\infty} \\ell w_j^s\\\\\n    & = \\sum_{n = 1}^{\\infty} \\left( \\frac{1}{2 n (n + 1)} \\right)^s\\\\\n    & = \\sum_{n = 1}^{\\infty} 2^{- s} (n + 1)^{- s} n^{- s}\\\\\n    & = s \\int_0^{\\infty} N_{\\mathcal{L}_w} (x) x^{- s - 1} \\mathrm{d} x\\\\\n    & = s \\int_0^{\\infty} \\left\\lfloor \\frac{\\sqrt{2 x + 1}}{2} - \\frac{1}{2}\n    \\right\\rfloor x^{- s - 1} \\mathrm{d} x\n  \\end{array} \\label{wgeozeta}\n\\end{equation}\nThe residue (\\ref{geozetares}) of $\\zeta_{\\mathcal{L}_w} (s$) at\n$D_{\\mathcal{L}_w}$ is\n\\begin{equation}\n  \\begin{array}{ll}\n    \\underset{s = D_{\\mathcal{L}}}{\\tmop{Res} (\\zeta_{\\mathcal{L}} (s))} & =\n    \\lim_{s \\rightarrow D_{\\mathcal{L}_w}^+} (s - D_{\\mathcal{L}})\n    \\zeta_{\\mathcal{L}} (s)\\\\\n    & = \\lim_{s \\rightarrow \\frac{1}{2}^+} \\left( s - \\frac{1}{2} \\right)\n    \\zeta_{\\mathcal{L}} (s)\\\\\n    & = \\lim_{s \\rightarrow \\frac{1}{2}^+} \\left( s - \\frac{1}{2} \\right)\n    \\sum_{n = 1}^{\\infty} 2^{- s} (n + 1)^{- s} n^{- s}\\\\\n    & = \\lim_{s \\rightarrow \\frac{1}{2}^+} \\left( s - \\frac{1}{2} \\right) s\n    \\int_0^{\\infty} \\left\\lfloor \\frac{\\sqrt{2 x + 1}}{2} - \\frac{1}{2}\n    \\right\\rfloor x^{- s - 1} \\mathrm{d} x\\\\\n    & = 0\n  \\end{array}\n\\end{equation}\nThe values of $\\zeta_{\\mathcal{L}_w} (n)$ at positive integer values $n \\in\n\\mathbbm{N}^{\\ast}$ are given explicitly by a rather unwieldy sum of binomial\ncoefficients and the Riemann zeta function $\\zeta (n)$ at even integer values.\nFirst, define\n\\begin{equation}\n  \\begin{array}{ll}\n    a_n & = \\frac{\\left( n - 1 \\right) \\left( 1 - \\left( - 1 \\right)^{n + 1}\n    \\right)}{2}\\\\\n    b_n & = \\frac{\\left( - 1 \\right)^{n + 1} \\left( n - 1 \\right)}{2} + n -\n    \\frac{7}{4} + \\frac{(- 1)^n}{4}\\\\\n    c_n & = (- 1)^n (n - 1)\\\\\n    d_n & = \\frac{(- 1)^n}{2}\n  \\end{array}\n\\end{equation}\nthen\n\n\\begin{equation}\n  \\begin{array}{ll}\n    \\zeta_{\\mathcal{L}_w} (n) & = \\frac{(  - 1)^n  \\binom{2 n - 1}{n - 1}}{\n    2^n} + \\sum_{m = a_n}^{b_n} \\frac{2 (-  1)^n \\binom{2 m + c_n - d_n +\n    \\frac{1}{2}}{n - 1} \\zeta \\left( d_n + 2 n -  \\frac{3}{2} -  2 m -  c_n\n    \\right)}{ 2^n}\n  \\end{array}\n\\end{equation}\n\nThe terms of $\\zeta_{\\mathcal{L}_w} (n)$ from $n = 1$ to $10$ are shown below\nin Table \\ref{wgztable}.\n\n\\begin{table}[h]\n  $\\left( \\begin{array}{llllll}\n    + \\frac{1}{2} &  &  &  &  & \\\\\n    - \\frac{3}{4} & + \\frac{1}{2} \\hspace{0.25em} \\zeta \\left( 2 \\right) &  & \n    &  & \\\\\n    + \\frac{5}{3} & - \\frac{3}{4} \\hspace{0.25em} \\zeta \\left( 2 \\right) &  & \n    &  & \\\\\n    - \\frac{35}{16} & + \\frac{5}{4} \\hspace{0.25em} \\zeta \\left( 2 \\right) & +\n    \\frac{1}{8} \\hspace{0.25em} \\zeta \\left( 4 \\right) &  &  & \\\\\n    + \\frac{63}{16} & - \\frac{35}{16} \\hspace{0.25em} \\zeta \\left( 2 \\right) &\n    - \\frac{5}{16} \\hspace{0.25em} \\zeta \\left( 4 \\right) &  &  & \\\\\n    - \\frac{231}{32} & + \\frac{63}{16} \\hspace{0.25em} \\zeta \\left( 2 \\right)\n    & + \\frac{21}{32} \\hspace{0.25em} \\zeta \\left( 4 \\right) & + \\frac{1}{32}\n    \\zeta \\left( 6 \\right) &  & \\\\\n    + \\frac{429}{32} & - \\frac{231}{32} \\hspace{0.25em} \\zeta \\left( 2 \\right)\n    & - \\frac{21}{16} \\hspace{0.25em} \\zeta \\left( 4 \\right) & - \\frac{7}{64}\n    \\hspace{0.25em} \\zeta \\left( 6 \\right) &  & \\\\\n    - \\frac{6435}{256} & + \\frac{429}{32} \\hspace{0.25em} \\zeta \\left( 2\n    \\right) & + \\frac{165}{64} \\hspace{0.25em} \\zeta \\left( 4 \\right) & +\n    \\frac{9}{32} \\hspace{0.25em} \\zeta \\left( 6 \\right) & + \\frac{1}{128}\n    \\hspace{0.25em} \\zeta \\left( 8 \\right) & \\\\\n    + \\frac{12155}{256} & - \\frac{6435}{256} \\hspace{0.25em} \\zeta \\left( 2\n    \\right) & - \\frac{1287}{256} \\hspace{0.25em} \\zeta \\left( 4 \\right) & -\n    \\frac{165}{256} \\hspace{0.25em} \\zeta \\left( 6 \\right) & - \\frac{9}{256}\n    \\hspace{0.25em} \\zeta \\left( 8 \\right) & \\\\\n    - \\frac{46189}{512} & + \\frac{12155}{256} \\hspace{0.25em} \\zeta \\left( 2\n    \\right) & + \\frac{5005}{512} \\hspace{0.25em} \\zeta \\left( 4 \\right) & +\n    \\frac{715}{512} \\hspace{0.25em} \\zeta \\left( 6 \\right) & + \\frac{55}{512}\n    \\hspace{0.25em} \\zeta \\left( 8 \\right) + &  \\frac{1 }{ 512 \\mathfrak{}}\n    \\hspace{0.25em} \\zeta \\mathbb{} \\left( \\mathfrak{} 1  0 \\mathsf{}  \\right)\n  \\end{array} \\right)$\n  \n  \n  \\caption{$\\{ \\text{$\\zeta_{\\mathcal{L}_w} (n) = \\Sigma_{}\n  \\tmop{row}_n$:n=1..10\\}}$\\label{wgztable}}\n\\end{table}\n\n\\section{Fractal Strings and Dynamical Zeta Functions}\n\n\\subsection{Fractal Strings}\n\nA a fractal string $\\mathcal{L}$ is defined as a nonempty bounded open subset\nof the real line $\\mathcal{L} \\subseteq \\mathbbm{R}$ consisting of a countable\ndisjoint union of open intervals $I_j$\n\\begin{equation}\n  \\mathcal{L}= \\bigcup_{j = 1}^{\\infty} I_j\n\\end{equation}\nThe length of the $j$-th interval $I_j$ is denoted by\n\\begin{equation}\n  \\text{$\\ell_j = \\left| I_j \\right|_{}$}\n\\end{equation}\nwhere $\\left. \\left. \\right| \\cdot \\right|_{}$ is the $1$-dimensional Lebesgue\nmeasure. The lengths $\\ell_j$ must form a nonnegative monotically decreasing\nsequence and the total length must be finite, that is\n\\begin{equation}\n  \\begin{array}{l}\n    |\\mathcal{L}|_1 = \\sum_{j = 1}^{\\infty} \\ell_j < \\infty\\\\\n    \\ell_1 \\geqslant \\ell_2 \\geqslant \\ldots \\geqslant \\text{$\\ell_j \\geqslant\n    \\ell_{j + 1} \\geqslant \\cdots \\geqslant 0$}\n  \\end{array}\n\\end{equation}\nThe case when $\\ell_j = 0$ for any $j$ will be excluded here since $\\ell_j$ is\na finite sequence. The fractal string is defined completely by its sequence of\nlengths so it can be denoted\n\\begin{equation}\n  \\begin{array}{ll}\n    \\mathcal{L} & =\\{\\ell_j \\}_{j = 1}^{\\infty}\n  \\end{array}\n\\end{equation}\nThe boundary of $\\mathcal{L}$ in $\\mathbbm{R}$, denoted by $\\partial\n\\mathcal{L} \\subset \\Omega$, is a totally disconnected bounded perfect subset\nwhich can be represented as a string of finite length, and generally any\ncompact subset of $\\mathbbm{R}$ also has this property. The boundary $\\partial\n\\mathcal{L}$ is said to be perfect since it is closed and each of its points\nis a limit point. Since the Cantor-Bendixon lemma states that there exists a\nperfect set $P \\subset \\partial \\mathcal{L}$ such that $\\partial \\mathcal{L}-\nP$ is a most countable, we can define $\\mathcal{L}$ as the complenent of\n$\\partial \\mathcal{L}$ in its closed convex hull. The connected components of\nthe bounded open set $\\mathcal{L} \\backslash \\partial \\mathcal{L}$ are the\nintervals $I_j$. {\\cite[1.2]{gsfs}}{\\cite[2.2\nEx17]{mti}}{\\cite[3.1]{fractalzetastrings}}{\\cite{weylberry}}{\\cite{fdisp}}{\\cite{hearfractaldrumshape}}{\\cite{gmc}}{\\cite{ncfg}}{\\cite{zflf}}{\\cite{possf}}{\\cite{cdssfs}}\n\n\\subsubsection{The Minkowski Dimension $D_{\\mathcal{L}}$ and Content\n$\\mathcal{M}_{\\mathcal{L}}$}\n\nThe Minkowski dimension $D_{\\mathcal{L}} \\in [0, 1]$, also known as the box\ndimension, is maximum value of $V (\\varepsilon)$\n\\begin{equation}\n  \\begin{array}{ll}\n    D_{\\mathcal{L}} & = \\inf \\{\\alpha \\geqslant 0 : V (\\varepsilon) = O\n    (\\varepsilon^{1 - \\alpha}) \\tmop{as} \\varepsilon \\rightarrow 0^+ \\}=\n    \\zeta_{\\mathcal{L}} (1) = \\sum_{n = 1}^{\\infty} \\ell_j^{}\n  \\end{array}\n\\end{equation}\nwhere $V (\\varepsilon)$ is the volume of the inner tubular neighborhoods of\n$\\partial \\mathcal{L}$ with radius $\\varepsilon$\n\\begin{equation}\n  \\text{$\\begin{array}{ll}\n    V_{\\mathcal{L}} (\\varepsilon) & \\left. = | x \\in \\mathcal{L}: d (x,\n    \\partial \\mathcal{L}) < \\varepsilon \\right|_{}\\\\\n    & = \\sum_j^{\\ell_j \\geqslant 2 \\varepsilon} 2 \\varepsilon +\n    \\sum_j^{\\ell_j < 2 \\varepsilon} \\ell_j\\\\\n    & = 2 \\varepsilon N_{\\mathcal{L}} \\left( \\frac{1}{2 \\varepsilon^{}}\n    \\right) + \\sum_j^{\\ell_j < 2 \\varepsilon} \\ell_j\n  \\end{array}$} \\label{tnv}\n\\end{equation}\nand $N_{\\mathcal{L}} (x)$ is the geometric counting function which is the\nnumber of components with their reciprocal length being less than or equal to\n$x$.\n\\begin{equation}\n  \\begin{array}{ll}\n    N_{\\mathcal{L}} (x) & =\\#\\{j \\geqslant 1 : \\ell_j^{- 1} \\leqslant x\\}\\\\\n    & = \\sum^{\\ell_j^{- 1} \\leqslant x}_{\\tmscript{\\begin{array}{l}\n      j \\geqslant 1\n    \\end{array}}} 1\n  \\end{array} \\label{geocount}\n\\end{equation}\nThe Minkowski content of $\\mathcal{L}$ is then defined as\n\\begin{equation}\n  \\begin{array}{ll}\n    \\mathcal{M}_{\\mathcal{L}} & = \\lim_{e \\rightarrow 0^+}\n    \\frac{V_{\\mathcal{L}} (\\varepsilon)}{\\varepsilon^{1 - D_{\\mathcal{L}}}} \\\\\n    & = \\frac{C_{\\mathcal{L}} 2^{1 - D_{\\mathcal{L}}}}{1 - D_{\\mathcal{L}}}\\\\\n    & = \\frac{\\tmop{Res} (\\zeta_{\\mathcal{L}} (s) ; D_{\\mathcal{L}}) 2^{1 -\n    D_{\\mathcal{L}}}}{D_{\\mathcal{L}} (1 - D_{\\mathcal{L}})} \n  \\end{array} \\label{mcontent}\n\\end{equation}\nwhere $C_{\\mathcal{L}}$ is the constant\n\\begin{equation}\n  \\begin{array}{ll}\n    C_{\\mathcal{L}} = & \\lim_{x \\rightarrow \\infty} \\frac{N_{\\mathcal{L}_{}}\n    (x)}{x^{D_{\\mathcal{L}_{}}}}\n  \\end{array} \\label{mconst}\n\\end{equation}\nIf $\\mathcal{M}_{\\mathcal{L}} \\in (0, \\infty)$ exists then $\\mathcal{L}$ is\nsaid to be Minkowski measurable which necessarily means that the geometry of\n$\\mathcal{L}$ does not oscillate and vice versa. {\\cite[1]{cdfs}}\n{\\cite{cizlm}}{\\cite{fsscd}}{\\cite[6.2]{fgnt}}\n\n\\subsubsection{The Geometric Zeta Function $\\zeta_{\\mathcal{L}} (s)$}\n\nThe geometric Zeta function $\\zeta_{\\mathcal{L}} (s)$ of\n\\tmtexttt{$\\mathcal{L}$} is the Dirichlet series\n\\begin{equation}\n  \\begin{array}{ll}\n    \\zeta_{\\mathcal{L}} (s) & = \\sum_{n = 1}^{\\infty} \\ell_j^s\\\\\n    & = s \\int_0^{\\infty} N_{\\mathcal{L}_w} (x) x^{- s - 1} \\mathrm{d} x\n  \\end{array} \\label{geozeta}\n\\end{equation}\nwhich is holomorphic for $\\tmop{Re} (s) > D_{\\mathcal{L}}$. If\n$\\mathcal{L}_{}$ is Minkowski measurable then $0 < D_{\\mathcal{L}} < 1$ is the\nsimple unique pole of $\\zeta_{\\mathcal{L}} (s)$ on the vertical line\n$\\tmop{Re} (s) = D_{\\mathcal{L}}$. Assuming $\\zeta_{\\mathcal{L}} (s)$ has a\nmeromorphic extension to a neighboorhood of $D_{\\mathcal{L}}$ then\n$\\zeta_{\\mathcal{L}} (s)$ has a simple pole at $\\zeta_{\\mathcal{L}}\n(D_{\\mathcal{L}})$ if\n\\begin{equation}\n  N_{\\mathcal{L}} (s) = O (s^{D_{\\mathcal{L}}}) \\tmop{as} s \\rightarrow \\infty\n\\end{equation}\nor if the volume of the tubular neighborhoods satisfies\n\\begin{equation}\n  V_{\\mathcal{L}} (\\varepsilon) = O (\\varepsilon^{1 - D_{\\mathcal{L}}})\n  \\tmop{as} \\varepsilon \\rightarrow 0^+\n\\end{equation}\nIt can be possible that the residue of $\\zeta_{\\mathcal{L}} (s)_{}$ at $s =\nD_{\\mathcal{L}}$ is positive and finite\n\\begin{equation}\n  0 < \\lim_{s \\rightarrow D_{\\mathcal{L}}} (s - D_{\\mathcal{L}})\n  \\zeta_{\\mathcal{L}} (s) < \\infty \\label{geozetares}\n\\end{equation}\neven if $N_{\\mathcal{L}} (s)$ is not of order $s^{D_{\\mathcal{L}}}$ as $s\n\\rightarrow \\infty$ and $V_{\\mathcal{L}} (\\varepsilon)$ is not of order\n$\\varepsilon^{1 - D_{\\mathcal{L}}}$, however this does not contradict the\nMinkowski measurability of $\\mathcal{L}_{}$.\n\n\\subsubsection{Complex Dimensions, Screens and Windows}\n\nThe set of visible complex dimensions of $\\mathcal{L}$, denoted by\n$\\mathcal{D}_{\\mathcal{L}} (W)$, is a discrete subset of $\\mathbbm{C}$\nconsisting of the poles of $\\{\\zeta_{\\mathcal{L}} (s) : s \\in W\\}$.\n\\begin{equation}\n  \\begin{array}{ll}\n    \\mathcal{D}_{\\mathcal{L}} (W) & =\\{w \\in W : \\zeta_{\\mathcal{L}} (w)\n    \\tmop{is} a \\tmop{pole}\\}\n  \\end{array}\n\\end{equation}\nWhen $W$ is the entire complex plane then the set $\\mathcal{D}_{\\mathcal{L}}\n(\\mathbbm{C}) =\\mathcal{D}_{\\mathcal{L}}$ is simply called the set of complex\ndimensions of $\\mathcal{L}$. The presence of oscillations in $V (\\varepsilon)$\nimplies the presence of imaginary complex dimensions with $\\tmop{Re} (\\cdot) =\nD_{\\mathcal{L}}$ and vice versa. More generally, the complex dimensions of a\nfractal string $\\mathcal{L}$ describe its geometric and spectral oscillations.\n\n\\subsubsection{Frequencies of Fractal Strings and Spectral Zeta Functions}\n\nThe eigenvalues $\\lambda_n$ of the Dirichlet Laplacian $\\Delta u (x) = -\n\\frac{\\mathrm{d}^2}{\\mathrm{d} x^2} u (x)$ on a bounded open set $\\Omega \\subset\n\\mathbbm{R}$ correspond to the normalized frequencies $f_n =\n\\frac{\\sqrt{\\lambda_n}}{\\pi}$ of a fractal string. The frequencies of the unit\ninterval are the natural numbers $n \\in \\mathbbm{N}^{\\ast}$ and the\nfrequencies of an interval of length $\\ell$ are $n \\ell^{- 1}$. The\nfrequencies of $\\mathcal{L}$ are the numbers\n\\begin{equation}\n  \\begin{array}{l}\n    f_{k, j} = k \\ell_j^{- 1} \\forall \\text{ $k, j \\in \\mathbbm{N}^{\\ast}$}\n  \\end{array}\n\\end{equation}\nThe spectral counting function $N_{v\\mathcal{L}} (x)$ counts the frequencies\nof $\\mathcal{L}$ with multiplicity\n\\begin{equation}\n  \\begin{array}{ll}\n    N_{v\\mathcal{L}} (x) & = \\sum_{j = 1}^{\\infty} N_{\\mathcal{L}} \\left(\n    \\frac{x}{j} \\right)\\\\\n    & = \\sum_{j = 1}^{\\infty} \\left\\lfloor x \\ell_j \\right\\rfloor\n  \\end{array} \\label{spectralcount}\n\\end{equation}\nThe spectral zeta function $\\zeta_{\\upsilon \\mathcal{L}} (s)$ of $\\mathcal{L}$\nis connected to the Riemann zeta function (\\ref{zeta}) by\n\\begin{equation}\n  \\begin{array}{ll}\n    \\zeta_{\\upsilon \\mathcal{L}} (s) & = \\sum_{k = 1}^{\\infty} \\sum_{j =\n    1}^{\\infty} k^{- s} \\ell^s_j\\\\\n    & = \\zeta (s) \\sum_{n = 1}^{\\infty} \\ell_j^s\\\\\n    & = \\zeta (s) \\zeta_{\\mathcal{L}} (s)\n  \\end{array} \\label{spectralzeta}\n\\end{equation}\n{\\cite[1.1]{cdfs}}{\\cite[1.2.1]{gsfs}}\n\n\\subsubsection{Generalized Fractal Strings and Dirichlet Integrals}\n\nA generalized fractal string is a positive or complex or local measure $\\eta\n(x)$ on $(0, \\infty)$ such that\n\\begin{equation}\n  \\int_0^{x_0} \\eta (x) \\mathrm{d} x = 0 \\label{gfs}\n\\end{equation}\nfor some $x_0 > 0$. A local positive measure is a standard positive Borel\nmeasure $\\eta (J)$ on $(0, \\infty)$ where $J$ is the set of all bounded\nsubintervals of $(0, \\infty)$ in which case $\\eta (x) = | \\eta (x) |$. More\ngenerally, a meausre $\\eta (x)$ is a local complex measure if $\\eta (A)$ is\nwell-defined for any subset $A \\subset [a, b]$ where $[a, b] \\subset [0,\n\\infty]$ is a bounded subset of the positive half-line $(0, \\infty)$ and the\nrestriction of $\\eta$ to the Borel subsets of $[a, b]$ is a complex measure on\n$[a, b]$ in the traditional sense. The geometric counting function of $\\eta\n(x)$ is defined as\n\\begin{equation}\n  \\begin{array}{ll}\n    N_{\\eta} (x) & = \\int_0^x \\eta (x) \\mathrm{d} x\n  \\end{array} \\label{ggcf}\n\\end{equation}\nThe dimension $D_{\\eta}$ is the abscissa of conergence of the Dirichlet\nintegral \\\n\\begin{equation}\n  \\zeta_{| \\eta |} (\\sigma) = \\int_0^{\\infty} x^{- \\sigma} | \\eta (x) | \\mathrm{d}\n  x \\label{gd}\n\\end{equation}\nIn other terms, it is the smallest real positive $\\sigma$ such that the\nimproper Riemann-Lebesgue converges to a finite value. The geometric zeta\nfunction is defined as the Mellin transform\n\\begin{equation}\n  \\begin{array}{ll}\n    \\zeta_{\\eta} (s) & =\n  \\end{array} \\int_0^{\\infty} x^{- s} \\eta (x) \\mathrm{d} x \\label{ggz}\n\\end{equation}\nwhere $\\tmop{Re} (s) > D_{\\eta}$.\n\n\\subsection{Fractal Membranes and Spectral Partitions}\n\n\\subsubsection{Complex Dimensions of Dynamical Zeta Functions}\n\nThe fractal membrane $\\mathcal{T}_L$ associated with $\\mathcal{L}$ is the\nadelic product\n\\begin{equation}\n  \\mathcal{T}_L = \\coprod_{j = 1}^{\\infty} \\mathcal{T}_j\n\\end{equation}\nwhere each $\\mathcal{T}_j$ is an interval $I_j$ of length $\\log (\\ell_j^{-\n1})^{- 1}$. To each $\\mathcal{T}_j$ is associated a Hilbert space\n$\\mathcal{H}_j = L^2 (I_j)$ of square integrable functions on $I_j$. The\nspectral partition function $Z_{\\mathcal{L}} (s)$ of $\\mathcal{L}$ is a Euler\nproduct expansion which has no zeros or poles in $\\tmop{Re} (s) > D_M\n(\\mathcal{L})$.\n\\begin{equation}\n  \\begin{array}{ll}\n    Z_{\\mathcal{L}} (s) & = \\prod_{j = 1}^{\\infty} \\frac{1}{1 - \\ell_j^s}\\\\\n    & = \\prod_{j = 1}^{\\infty} Z_{\\mathcal{L}_j} (s)\n  \\end{array}\n\\end{equation}\nwhere $D_M (\\mathcal{L})$ is the Minkowski dimension of $\\mathcal{L}$ and\n$Z_{\\mathcal{L}_j} (s) = \\frac{1}{1 - \\ell_j^s}$ is the $j$-th Euler factor,\nthe partition function of the $j$-th component of the fractal membrane.\n{\\cite[3.2.2]{fractalzetastrings}}\n\n\\subsubsection{Dynamical Zeta Functions of Fractal Membranes}\n\nThe dynamical zeta function of a fractal membrane $\\mathcal{L}$ is the\nnegative of the logarithmic derivative of the Zeta function associated with\n$\\mathcal{L}$.\n\\begin{equation}\n  \\begin{array}{ll}\n    Z_{\\mathcal{L}} (s) & = - \\frac{\\mathrm{d}}{\\mathrm{d} s} \\ln (\\zeta_{\\mathcal{L}}\n    (s))\\\\\n    & = - \\frac{\\frac{\\mathrm{d}}{\\mathrm{d} s} \\zeta_{\\mathcal{L}}\n    (s)}{\\zeta_{\\mathcal{L}} (s)}\n  \\end{array} \\label{dzfm}\n\\end{equation}\n\n\\section{Special Functions, Definitions, and Conventions}\n\n\\subsection{Special Functions}\n\n\\subsubsection{The Interval Indicator (Characteristic) Function $\\chi (x, I)$}\n\nThe (left-open, right-closed) interval indicator function is $\\chi (x, I)$\nwhere $I = (a, b]$\n\\begin{equation}\n  \\begin{array}{ll}\n    \\chi (x, I) & = \\left\\{ \\begin{array}{ll}\n      1 & x \\in I\\\\\n      0 & x \\not\\in I\n    \\end{array} \\right.\\\\\n    & = \\left\\{ \\begin{array}{ll}\n      1 & a < x \\leqslant b\\\\\n      0 & \\tmop{otherwise}\n    \\end{array} \\right.\\\\\n    & = \\theta (x - a) - \\theta (x - a) \\theta (x - b)\n  \\end{array} \\text{} \\label{ii}\n\\end{equation}\nand $\\theta$ is the Heaviside unit step function, the derivative of which is\nthe Dirac delta function $\\delta$\n\\begin{equation}\n  \\begin{array}{ll}\n    \\text{$\\int \\delta (x) \\mathrm{d} x$} & = \\theta (x)\\\\\n    \\theta (x) & = \\left\\{ \\begin{array}{ll}\n      0 & x < 0\\\\\n      1 & x \\geqslant 0\n    \\end{array} \\right.\n  \\end{array} \\label{deltastep}\n\\end{equation}\nThe discontinous point of $\\theta (x)$ has the limiting values\n\\begin{equation}\n  \\begin{array}{ll}\n    \\lim_{x \\rightarrow 0^-} \\theta (x) & = 0\\\\\n    \\lim_{x \\rightarrow 0^+} \\theta (x) & = 1\n  \\end{array}\n\\end{equation}\nthus the values of $\\chi (x, (a, b))$ on the boundary can be chosen according\nto which side the limit is regarded as being approached from.\n\\begin{equation}\n  \\begin{array}{ll}\n    \\lim_{x \\rightarrow a^-} \\text{$\\chi (x, (a, b])$} & = 0\\\\\n    \\lim_{x \\rightarrow a^+} \\text{$\\chi (x, (a, b])$} & = 1 - \\theta (a -\n    b)\\\\\n    \\lim_{x \\rightarrow b^-} \\text{$\\chi (x, (a, b])$} & = \\theta (b - a)\\\\\n    \\lim_{x \\rightarrow b^+} \\text{$\\chi (x, (a, b])$} & = 0\n  \\end{array}\n\\end{equation}\n\n\\subsubsection{``Harmonic'' Intervals}\n\nLet the $n$-th harmonic (left-open, right-closed) interval be defined as\n\\begin{equation}\n  _{} \\begin{array}{ll}\n    I^H_n & = \\left( \\frac{1}{n + 1}, \\frac{1}{n} \\right]\n  \\end{array} \\label{hi}\n\\end{equation}\nthen its characteristic function is\n\\begin{equation}\n  \\begin{array}{lll}\n    _{} \\chi_{} (x, I_n^H) &  & = \\theta \\left( x - \\frac{1}{n + 1} \\right) -\n    \\theta \\left( x - \\frac{1}{n} \\right)\\\\\n    &  & = \\theta \\left( x - \\frac{n}{n + 1} \\right) - \\theta \\left( x - n\n    \\right)\\\\\n    &  & = \\left\\{ \\begin{array}{ll}\n      1 & \\frac{1}{n + 1} < x \\leqslant \\frac{1}{n}\\\\\n      0 & \\tmop{otherwise}\n    \\end{array} \\right.\n  \\end{array} \\label{hii}\n\\end{equation}\nAs can be seen\n\\begin{equation}\n  \\begin{array}{llll}\n    \\bigcup_{n = 1}^{\\infty} I_n^H & = & \\bigcup_{n = 1}^{\\infty} \\left(\n    \\frac{1}{n + 1}, \\frac{1}{n} \\right] & = (0, 1]\\\\\n    \\sum_{n = 1}^{\\infty} \\chi (x, I_n^H) & = & \\sum_{n = 1}^{\\infty}\n    \\chi_{} \\left( x, \\left( \\frac{1}{n + 1}, \\frac{1}{n} \\right] \\right) & =\n    \\chi (x, (0, 1])\n  \\end{array}\n\\end{equation}\nThe substitution $n \\rightarrow \\left\\lfloor \\frac{1}{x} \\right\\rfloor$ can be\nmade in (\\ref{hii}) where it is seen that\n\\begin{equation}\n  \\begin{array}{lll}\n    \\text{$\\chi_{} \\left( x, I^H_{\\left\\lfloor x^{- 1} \\right\\rfloor}\n    \\right)$} & = \\theta \\left( \\frac{x \\text{$\\left\\lfloor x^{- 1}\n    \\right\\rfloor$} + x - 1}{\\text{$\\left\\lfloor x^{- 1} \\right\\rfloor$}\n    \\text{} + 1} \\right) - \\theta \\left( \\frac{x \\text{$\\left\\lfloor x^{- 1}\n    \\right\\rfloor$} - 1}{\\left\\lfloor x^{- 1} \\right\\rfloor} \\right) = 1 &\n    \\forall x \\in [- 1, + 1] \\label{hieye}\n  \\end{array}\n\\end{equation}\n\n\\subsubsection{The Laplace Transform $L_a^b [f (x) ; x \\rightarrow s]$}\n\nThe Laplace transform {\\cite[1.5]{tlt}} is defined as\n\\begin{equation}\n  \\begin{array}{ll}\n    \\text{} \\text{$L_a^b [f (x) ; x \\rightarrow s]$} & = \\int_a^b f (x) e^{- x\n    s} \\mathrm{d} x\n  \\end{array} \\label{laplace}\n\\end{equation}\nwhere the unilateral Laplace transform is over the interval ($a, b) = (0,\n\\infty$) and the bilateral transform is over ($a, b) = (- \\infty, \\infty$).\nWhen ($a, b$) is not specified, it is assumed to range over the support of $f\n(x$) if the support is an interval. If the support of $f (x$) is not an\ninterval then ($a, b$) must be specified. Applying $L$ to the interval\nindicator function (\\ref{ii}) gives\n\\begin{equation}\n  \\begin{array}{lll}\n    \\text{$L_a^b [\\chi (x, (a, b)) ; x \\rightarrow s]$} & = \\int_a^b \\chi (x,\n    (a, b)) e^{- x s} \\mathrm{d} x & \\\\\n    & = \\int_a^b (\\theta (x - a) - \\theta (x - b) \\theta (x - a)) e^{- x s}\n    \\mathrm{d} x & \\\\\n    & = \\frac{e^{b s} - e^{\\tmop{as}}}{s e^{b s} e^{a s}} & \\\\\n    & = - \\frac{(e^{a s} - e^{b s}) e^{- s (b + a)}}{s} & \\\\\n    &  & \n  \\end{array} \\label{iilt}\n\\end{equation}\nThe limit at the singular point $s = 0$ is\n\\begin{equation}\n  \\begin{array}{ll}\n    \\lim_{s \\rightarrow 0} \\text{$L_a^b [\\chi (x, (a, b)) ; x \\rightarrow s]$}\n    & = \\lim_{s \\rightarrow 0} - \\frac{(e^{a s} - e^{b s}) e^{- s (b +\n    a)}}{s}\\\\\n    & = b - a\n  \\end{array}\n\\end{equation}\n\n\\subsubsection{The Mellin Transform $M_a^b [f (x) ; x \\rightarrow s]$}\n\nThe Mellin transform\n{\\cite[3.2]{ambi}}{\\cite[II.10.8]{mmp}}{\\cite[3.6]{piagm}} is defined as\n\\begin{equation}\n  \\begin{array}{ll}\n    \\text{$M_a^b [f (x) ; x \\rightarrow s]$} & = \\int_a^b f (x) x^{s - 1}\n    \\mathrm{d} x\n  \\end{array} \\label{mellin}\n\\end{equation}\nwhere the standard Mellin transform is over the interval ($a, b) = (0,\n\\infty$). Again, as with the notation for Laplace transform, the integral is\nover the support of $f (x$) if the support is an interval and ($a, b$) is not\nspecified, otherwise ($a, b$) must be specified. Applying $M$ to the interval\nindicator function (\\ref{ii}) gives\n\\begin{equation}\n  \\begin{array}{lll}\n    & M_a^b [\\chi (x, (a, b)) ; x \\rightarrow s] & = \\int_a^b \\chi (x, (a,\n    b)) x^{s - 1} \\mathrm{d} x\\\\\n    &  & = \\int_a^b (\\theta (x - a) - \\theta (x - b) \\theta (x - a)) x^{s -\n    1} \\mathrm{d} x\\\\\n    &  & = \\frac{b^s - a^s}{s}\n  \\end{array} \\label{iimt}\n\\end{equation}\nThe limit at the singular point $s = 0$ is\n\\begin{equation}\n  \\begin{array}{ll}\n    M_a^b \\left[ \\chi (x, (a, b)) ; x \\rightarrow 0 \\right] & = M \\left[ \\chi\n    (x, (a, b)) ; x \\rightarrow 0 \\right]\\\\\n    & = \\lim_{s \\rightarrow 0} M_a^b [\\chi (x, (a, b)) ; x \\rightarrow s]\\\\\n    & = \\lim_{s \\rightarrow 0} \\frac{b^s - a^s}{s}\\\\\n    & = \\ln (b) - \\ln (a)\n  \\end{array}\n\\end{equation}\nThe Mellin transform has several identities {\\cite[3.1.2]{ambi}}, including\nbut not limited to\n\\begin{equation}\n  \\begin{array}{ll}\n    M [f (\\alpha x) ; x \\rightarrow s] & = \\alpha^{- s} M [f (x) ; x\n    \\rightarrow s]\\\\\n    M [x^{\\alpha} f (x) ; x \\rightarrow s] & = M [f (x) ; x \\rightarrow s +\n    \\alpha]\\\\\n    M [f (x^{\\alpha}) ; x \\rightarrow s] & = \\frac{1}{a} M \\left[ f (x) ; x\n    \\rightarrow \\frac{s}{\\alpha} \\right]\\\\\n    M [f (x^{- \\alpha}) ; x \\rightarrow s] & = \\frac{1}{a} M \\left[ f (x) ; x\n    \\rightarrow - \\frac{s}{\\alpha} \\right]\\\\\n    M [x^{\\alpha} f (x^{\\mu}) ; x \\rightarrow s] & = \\frac{1}{\\mu} M \\left[ f\n    (x) ; x \\rightarrow \\frac{s + \\alpha}{\\mu} \\right]\\\\\n    M [x^{\\alpha} f (x^{- \\mu}) ; x \\rightarrow s] & = \\frac{1}{\\mu} M \\left[\n    f (x) ; x \\rightarrow - \\frac{s + \\alpha}{\\mu} \\right]\\\\\n    M [\\ln (x)^n f (x) ; x \\rightarrow s] & = \\frac{\\mathrm{d}^n}{\\mathrm{d} s^n} M\n    \\left[ f (x) ; x \\rightarrow s \\right]\n  \\end{array}\n\\end{equation}\nwhere $\\alpha > 0, \\mu > 0 $, and $n \\in \\mathbbm{N} $. The Mellin transform\nof the harmonic interval indicator function (\\ref{hii}) is\n\\begin{equation}\n  \\begin{array}{lll}\n    M \\left[ \\chi \\left( x, I^H_n \\right) ; x \\rightarrow 0 \\right] &  & =\n    \\int_{\\frac{1}{n + 1}}^{\\frac{1}{n}} \\chi \\left( x, \\left( \\frac{1}{n +\n    1}, \\frac{1}{n} \\right) \\right) x^{s - 1} \\mathrm{d} x\\\\\n    &  & = \\int_{\\frac{1}{n + 1}}^{\\frac{1}{n}}  \\left( \\theta \\left( t -\n    \\frac{1}{n + 1} \\right) - \\theta \\left( t - \\frac{1}{n} \\right) \\right)\n    x^{s - 1} \\mathrm{d} x\\\\\n    &  & = \\frac{n^{- s} - (n + 1)^{- s}}{s}\n  \\end{array} \\label{himt}\n\\end{equation}\nwhich has the limit\n\\begin{equation}\n  \\begin{array}{ll}\n    M \\left[ \\chi \\left( x, I^H_n \\right) ; x \\rightarrow 0 \\right] & =\n    \\lim_{s \\rightarrow 0} M \\left[ \\chi \\left( x, I^H_n \\right) ; x\n    \\rightarrow s \\right]\\\\\n    & = \\lim_{s \\rightarrow 0} \\frac{n^{- s} - (n + 1)^{- s}}{s}\\\\\n    & = \\ln (n + 1) - \\ln (n)\n  \\end{array} \\label{himtl}\n\\end{equation}\nThe Mellin and bilaterial Laplace transforms are related by the change of\nvariables $x \\rightarrow - \\ln (y)$ resulting in the identity\n{\\cite[3.1.1]{ambi}}\n\\begin{equation}\n  \\begin{array}{lll}\n    M_0^{\\infty} [f (- \\ln (x)) ; x \\rightarrow s] & = L_{- \\infty}^{+ \\infty}\n    [f (y) ; y \\rightarrow s] & \\\\\n    \\int_0^{\\infty} f (- \\ln (x)) x^{s - 1} \\mathrm{d} x & = \\int_{- \\infty}^{-\n    \\infty} f (y) e^{- y s} \\mathrm{d} y & \n  \\end{array}\n\\end{equation}\n\n\\subsubsection{The Lambert W Function $W (k, x)$}\n\nThe Lambert W function {\\cite{lambertw}}{\\cite{lambertwss}} is the inverse of\n$x e^x$ given by\n\\begin{equation}\n  \\begin{array}{ll}\n    W (z) & =\\{x : x e^x = z\\}\\\\\n    & = W (0, z)\\\\\n    & = 1 + (\\ln (z) - 1) \\exp \\left( \\frac{i}{2 \\pi} \\int_0^{\\infty}\n    \\frac{1}{x + 1} \\ln \\left( \\frac{x - i \\pi - \\ln (x) + \\ln (z)}{x + i \\pi\n    - \\ln (x) + \\ln (z)} \\right) \\hspace{-0.25em} \\mathrm{d} x \\right)\\\\\n    & = \\sum_{k = 1}^{\\infty} \\frac{(- k)^{k - 1} z^k}{k!}\n  \\end{array}  \\label{linv}\n\\end{equation}\nwhere $W (a, z) \\forall a \\in \\mathbbm{Z}, z \\not\\in \\{0, - e^{- 1} \\}$ is\n\\begin{equation}\n  \\begin{array}{ll}\n    W (a, z) & = 1 + (2 i \\pi a + \\ln (z) - 1) \\exp \\left( \\frac{i}{2 \\pi}\n    \\int_0^{\\infty} \\frac{1}{x + 1} \\ln \\left( \\frac{x + \\left( 2\n    \\hspace{0.25em} a - 1 \\right) i \\pi - \\ln \\left( x \\right) + \\ln \\left( z\n    \\right)}{x + \\left( 2 \\hspace{0.25em} a + 1 \\right) i \\pi - \\ln \\left( x\n    \\right) + \\ln \\left( z \\right)} \\right) \\hspace{-0.25em} \\mathrm{d} x \\right)\n  \\end{array}  \\label{lambertw}\n\\end{equation}\nA generaliztion of (\\ref{linv}) is solved by\n\\begin{equation}\n  \\begin{array}{ll}\n    \\{x : x b^x = z\\} & = \\frac{W (\\ln (b) z)}{\\ln (b)}\n  \\end{array}\n\\end{equation}\nThe W function satisifes several identities\n\\begin{equation}\n  \\begin{array}{lll}\n    W (z) e^{W (z)} & = z & \\\\\n    W (z \\ln (z)) & = \\ln (z) & \\forall z < 1\\\\\n    |W (z) | & = W (|z|) & \\\\\n    e^{n W (z)} & = z^n W (z)^{- n}  & \\\\\n    \\ln (W (n, z)) & = \\ln (z) - W (n, z) + 2 i \\pi n & \\\\\n    W \\left( - \\frac{\\ln (z)}{z} \\right) & = - \\ln (z) & \\forall z \\in [0,\n    e]\\\\\n    \\frac{W (- \\ln (z))}{- \\ln (z)} & = z^{z^{z^{z^{.^{.^.}}}}} & \n  \\end{array}\n\\end{equation}\nwhere $n \\in \\mathbbm{Z}$. Some special values are\n\\begin{equation}\n  \\begin{array}{lll}\n    W \\left( - 1, - e^{- 1} \\right) & = - 1 & \\\\\n    W (- e^{- 1}) & = - 1 & \\\\\n    W (e) & = 1 & \\\\\n    W (0) & = 0 & \\\\\n    W (\\infty) & = \\infty & \\\\\n    W (- \\infty) & = \\infty + i \\pi & \\\\\n    W \\left( - \\frac{\\pi}{2} \\right) & = \\frac{i \\pi}{2} & \\\\\n    W \\left( - \\ln ( \\sqrt{2}) \\right) & = - \\ln (2) & \\\\\n    W \\left( - 1, - \\ln ( \\sqrt{2}) \\right) & = - 2 \\ln (2) & \n  \\end{array}\n\\end{equation}\nWe also have the limit\n\\begin{equation}\n  \\begin{array}{ll}\n    \\lim_{a \\rightarrow \\pm \\infty} \\frac{W (a, x)}{a} & = 2 \\pi i\n  \\end{array}\n\\end{equation}\nand differential\n\\begin{equation}\n  \\begin{array}{lll}\n    \\frac{\\mathrm{d}}{\\mathrm{d} z} W (a, f (z)) &  & = \\frac{W (a, f (z))\n    \\frac{\\mathrm{d}}{\\mathrm{d} z} f (z)}{f (z) (1 + W (a, f (z)))}\n  \\end{array}\n\\end{equation}\nas well as the obvious integral\n\\begin{equation}\n  \\begin{array}{ll}\n    \\int_0^1 W \\left( - \\frac{\\ln (x)}{x} \\right) \\mathrm{d} x & = \\int_0^1 - \\ln\n    (x) \\mathrm{d} x = 1\n  \\end{array}\n\\end{equation}\nLet us define, for the sake of brevity, the function\n\n\\begin{equation}\n  \\begin{array}{ll}\n    W_{\\ln} (z) & = W \\left( - 1, - \\frac{\\ln (z)}{z} \\right)\\\\\n    & = 1 + \\left( \\ln \\left( - \\frac{\\ln (z)}{z} \\right) - 1 - 2 \\pi i\n    \\right) \\exp \\left( \\frac{i}{2 \\pi} \\int_0^{\\infty} \\frac{1}{x + 1} \\ln\n    \\left( \\frac{x - 3 i \\pi - \\ln \\left( x \\right) + \\ln \\left( - \\frac{\\ln\n    (z)}{z} \\right)}{x - i \\pi - \\ln \\left( x \\right) + \\ln \\left( - \\frac{\\ln\n    (z)}{z} \\right)} \\right) \\hspace{-0.25em} \\mathrm{d} x \\right)\n  \\end{array}\n\\end{equation}\n\n\\begin{figure}[h]\n  \\resizebox{15cm}{12cm}{\\includegraphics{lw1.eps}}\n  \\caption{$W \\left( - \\frac{\\ln (x)}{x} \\right) = - \\ln (x)$ and $W_{\\ln} (x)\n  = W \\left( - 1, - \\frac{\\ln (x)}{x} \\right)$}\n\\end{figure}\n\nThen we have the limits\n\\begin{equation}\n  \\begin{array}{ll}\n    \\lim_{x \\rightarrow - \\infty} W_{\\ln} (x) & = 0\\\\\n    \\lim_{x \\rightarrow + \\infty} W_{\\ln} (x) & = - \\infty\n  \\end{array}\n\\end{equation}\nand\n\\begin{equation}\n  \\tmop{Im} \\left( W_{\\ln} (x) \\right) = \\left\\{ \\begin{array}{ll}\n    - \\pi & - \\infty < x < 0\\\\\n    \\ldots & 0 \\leqslant x \\leqslant 1\\\\\n    0 & 1 < x < \\infty\n  \\end{array} \\right.\n\\end{equation}\n\\begin{equation}\n  \\begin{array}{lll}\n    W_{\\ln} (x) & = - \\ln (x) & \\forall x \\not\\in [0, e]\n  \\end{array}\n\\end{equation}\nThe root of $\\tmop{Re} \\left( W_{\\ln} (x) \\right)$ is given by\n\\begin{equation}\n  \\begin{array}{ll}\n    \\left\\{ x : \\tmop{Re} \\left( W_{\\ln} (x)) = 0 \\right\\} \\right. & =\\{- x :\n    \\left( x^2 \\right)^{\\frac{1}{x}} = e^{3 \\pi} \\}\\\\\n    & = \\frac{2}{3} \\pi W \\left( \\frac{3}{2} \\pi \\right)\\\\\n    & \\cong 0.27441063190284810044 \\ldots\n  \\end{array} \\label{wlrr}\n\\end{equation}\nwhere the imaginary part of the value at the root of the real part of $W_{\\ln}\n(z)$ is\n\\begin{equation}\n  \\begin{array}{ll}\n    W_{\\ln} \\left( \\frac{2}{3} \\pi W \\left( \\frac{3}{2} \\pi \\right) \\right) &\n    = W \\left( - 1, - \\frac{\\ln \\left( \\frac{2}{3} \\pi W \\left( \\frac{3}{2}\n    \\pi \\right) \\right)}{\\frac{2}{3} \\pi W \\left( \\frac{3}{2} \\pi \\right)}\n    \\right)\\\\\n    & = W \\left( - 1, \\frac{3 \\pi}{2} \\right)\\\\\n    & = \\frac{3 \\pi i}{2}\\\\\n    & \\cong i 4.712388980384689857 \\ldots\n  \\end{array}\n\\end{equation}\n\n\\subsubsection{The Lerch Transcendent $\\Phi (z, a, v)$}\n\nThe Lerch Transcendent {\\cite[1.11]{htf1}} is defined by\n\\begin{equation}\n  \\begin{array}{lll}\n    \\Phi (z, a, v) & = \\sum_{n = 0}^{\\infty} \\frac{z^n}{(v + n)^a} & \\forall\n    \\{|z| < 1\\} \\tmop{or} \\{|z| = 1 \\tmop{and} \\tmop{Re} (a) > 1\\}\n    \\label{lerch}\n  \\end{array}\n\\end{equation}\nThe Riemann zeta function is the special case\n\\begin{equation}\n  \\begin{array}{ll}\n    \\zeta (s) & = \\Phi (1, s, 1) = \\sum_{n = 0}^{\\infty} \\frac{1}{(1 + n)^s}\n  \\end{array}\n\\end{equation}\n\n\\subsection{Applications of w(x)}\n\n\\subsubsection{Expansion of $\\gamma$}\n\nConsider Euler's constant $\\gamma = 0.577215664901533 \\ldots$ (\\ref{gamma})\n\\begin{equation}\n  \\begin{array}{ll}\n    w^n (\\gamma) & = a_n - b_n \\gamma\n  \\end{array}\n\\end{equation}\nwhereupon iteration we see that\n\\begin{equation}\n  \\begin{array}{ll}\n    \\left(\\begin{array}{c}\n      n\\\\\n      - a_n\\\\\n      - b_n\n    \\end{array}\\right) & \\text{$= \\left(\\begin{array}{cccccccccccc}\n      0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & \\ldots\\\\\n      0 & 1 & 48 & 290 & 581 & 1163 & 2327 & 13964 & 7492468716 & 14984937433\n      & 1078915495184 & \\ldots\\\\\n      1 & 2 & 84 & 504 & 1008 & 2016 & 4032 & 24192 & 12980362752 &\n      25960725504 & 1869172236288 & \\ldots\n    \\end{array}\\right)$}\n  \\end{array}\n\\end{equation}\n\n\\subsection{Conventions and Symbols}\n\nMany of these symbols are from {\\cite[p491]{fractalzetastrings}}.\n\\begin{equation}\n  \\begin{array}{ll}\n    i & \\sqrt{- 1}\\\\\n    \\mathbbm{R} & \\{x : - \\infty < x < \\infty\\}\\\\\n    \\bar{\\mathbbm{R}} & \\{x : - \\infty \\leqslant x \\leqslant \\infty\\}\\\\\n    \\mathbbm{R}^+ & \\{x : 0 \\leqslant x < \\infty\\}\\\\\n    \\mathbbm{R}^d & \\{x_1 \\ldots x_d : - \\infty < x_i < \\infty\\}\\\\\n    \\mathbbm{C} & \\{x + i y : x, y \\in \\mathbbm{R}\\}\\\\\n    \\mathbbm{Z} & \\{\\ldots, - 2, - 1, 0, 1, 2, \\ldots\\}\\\\\n    \\mathbbm{N} & \\{0, 1, 2, 3, \\ldots .\\}\\\\\n    \\mathbbm{N}^{\\ast} & \\{1, 2, 3, \\ldots .\\}\\\\\n    \\text{$\\mathbbm{H}$} & \\left\\{ 0, \\frac{1}{n} : n \\in \\mathbbm{Z}\n    \\right\\}\\\\\n    f (x) = O (g (x)) & \\frac{f (x)}{g (x)} < \\infty\\\\\n    f (x) = o (g (x)) & \\lim_{x \\rightarrow \\infty} \\frac{f (x)}{g (x)} = 0\\\\\n    f (x) \\asymp g (x) & \\left\\{ a \\leqslant \\frac{f (x)}{g (x)} \\leqslant b :\n    \\{a, b\\}> 0 \\right\\}\\\\\n    \\#A & \\tmop{numbers} \\tmop{of} \\tmop{elements} \\tmop{in} \\tmop{the}\n    \\tmop{finite} \\tmop{set} A\\\\\n    |A |_d & d \\text{-dimensional} \\tmop{Lebesgue} \\tmop{measure}\n    (\\tmop{volume}) \\tmop{of} A \\subseteq \\mathbbm{R}^d\\\\\n    \\text{$d (x, A)$} & \\{\\min (|x - y|) : y \\in A\\} \\tmop{Euclidean}\n    \\tmop{distance} \\tmop{between} x \\tmop{and} \\tmop{the} \\tmop{nearest}\n    \\tmop{point} \\tmop{of} A\\\\\n    \\exp (x) & \\tmop{exponential} e^x = \\sum_{n = 0}^{\\infty} \\frac{x^n}{n!}\\\\\n    \\underset{}{\\underset{x = y}{\\tmop{Res}} (f (x))} & \\tmop{complex}\n    \\text{residue of $f (x)$ at $x = y$}\\\\\n    \\left\\lfloor x \\right\\rfloor & \\tmop{floor}, \\tmop{the} \\tmop{greatest}\n    \\tmop{integer} \\leqslant x\\\\\n    \\{x\\} & x - \\left\\lfloor x \\right\\rfloor, \\tmop{the} \\tmop{fractional}\n    \\tmop{part} \\tmop{of} x\\\\\n    \\text{$\\bar{x}$} & \\tmop{complex} \\tmop{conjugate}, \\tmop{Re} (x) -\n    \\tmop{Im} (x)\\\\\n    \\tmop{Fix}_f^n & \\text{$n$-th} \\tmop{fixed} \\tmop{point} \\tmop{of}\n    \\tmop{the} \\tmop{map} f (x), \\text{$n$-th} \\tmop{solution} \\tmop{to} f (x)\n    = x\\\\\n    p_k & \\text{$k$-th} \\tmop{prime} \\tmop{number}\\\\\n    \\ln_b (a) & \\frac{\\ln (a)}{\\ln (b)}\n  \\end{array} \\label{notation}\n\\end{equation}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec-introduction}\n\nPooling samples in biomedical studies has now become a frequent\npractice among many researchers. For example, more than $15\\%$ of\nthe data sets deposited in the Gene Expression Omnibus Database\ninvolve pooled RNA samples\\citep{Kendziorski-etal-PNAS2005}. The\npractice of pooling biological samples though is not a new\nphenomenon, as it can be traced back at least to 1940s\n\\citep{Dorfman1943} and has been used in different application\nareas \\citep{Gastwirth2000}, e.g., for the detection of certain\nmedical conditions and estimation of prevalence in a population.\nIn the context of detecting differential gene expressions using\nmicroarrays, divergent views on the wisdom of pooling samples can\nbe found in the literature\n\\citep{Agrawal-etal-JNatCancerInst2002,Affymetrix2004,\nShih-etal-Bioinformatics2004, Churchill&Oliver-NatureGenetics2001,\nPeng-etal-BMCBioinformatics2003,Jolly-etal-PhysiolGenomics2005}.\nOne of the arguments supporting the practice of pooling biological\nsamples is that biological variation can be reduced by pooling RNA\nsamples in microarray\nexperiments\\citep{Churchill&Oliver-NatureGenetics2001}. As more\ncarefully described by Kendziorski {\\em et al}\n\\citep{Kendziorski-etal-PNAS2005}, pooling can reduce the effects\nof biological variation, but not the biological variation itself.\nAnother argument in support of pooling samples in microarray\nexperiments is that it reduces financial cost. However, cost\nreduction is meaningful only if statistical equivalence between\nthe pooled and the non-pooled experimental setups is maintained.\nHere we address this issue and present formulas to determine the\nconditions under which pooled and non-pooled designs are\nstatistically equivalent.\n\nTo compare experimental designs with and without sample pooling\nthe two designs must have something in common that can be\nmeasured, e.g., using the same or equivalent amount of resources,\nor, yielding the same level of detection power. Kendziorski {\\em\net al}\\citep{Kendziorski-etal-Biostatistics2003} used the width of\nthe 95\\% confidence interval for gene expression to compare\ndifferent experimental designs with and without sample pooling.\nThe criterion was that the narrower the confidence interval, the\nmore accurate the results from the experimental design. In a\ncomparative study where two groups of biological subjects are\ncompared the common goal of the different experimental designs is\nto detect a change between the two groups with a given power at a\ngiven false positive rate, as adopted in\n\\citep{Shih-etal-Bioinformatics2004}. We shall use the latter\nmethod to compare different designs. So in this work statistical\nequivalence means that the designs have the same statistical power\nat the same level of significance. Therefore the more appropriate\nexperimental design will be the one which uses less resources to\nachieve this statistical equivalence.\n\nThe basic assumption underlying sample pooling is biological\naveraging; that the measure of interest taken on the pool of\nsamples is equal to the average of the same measure taken on each\nof the individual samples which contributed to the pool. For\nexample in the situation of a microarray experiment, if $r$\nindividual samples contribute equally to a pool, and the\nconcentrations of a gene's mRNA transcripts for the $r$ samples\nare denoted by $T_i$ with $i=1, 2, \\cdots, r$ indexing the\nindividual samples, the assumption of biological averaging says\nthat the concentration of this gene's mRNA transcripts in the pool\nis $T=1\/r\\sum_{i=1}^rT_i$. However, for microarray experiment\nthere is some debate on whether the basic assumption of pooling\nholds. Kendziorski {\\em et al}\n\\citep{Kendziorski-etal-Biostatistics2003,Kendziorski-etal-PNAS2005}\nargue that there is limited support for this assumption. Here we\ndo not seek to enter into this debate but rather take the\nassumption of biological averaging as valid, or at least\napproximately so, so that we are in a position to determine\nwhether pooling samples is financially beneficial or not. The\nvalidity of biological averaging makes it possible (or easier) to\nderive a neat theoretical formulation. On a practical level\nthough, the requirement for the validity of this assumption may\nnot be as stringent as a theoretical formulation does. For\ninstance, in \\citep{Kendziorski-etal-PNAS2005} it was shown that\neven when biological averaging does not hold, pooling can be\nuseful and inferences regarding differential gene expression are\nnot adversely affected by pooling.\n\n\nOne situation where there is little alternative but to pool\nbiological samples is where there is insufficient amount of RNA\nfrom each individual biological subject to perform single\nmicroarray hybridization. RNA amplification may be a possible way\nof obtaining more RNA, but may not be practically feasible when\nmany individual biological subjects are involved as in the case of\n\\citep{Jin-etal2001}. In such a circumstance, pooling samples is\njustified by the lack of alternative and will not be considered\nfurther here. Similarly we will not consider here the case where\nall the biological samples of the same group were pooled together,\nand multiple technical replicate measurements were carried out on\nthe sample pool. This is sometimes seen in the literature\n\\citep{Muckenthaler-etal2003}, but such an experimental design\nleaves no degree of freedom to estimate the biological variance.\nThus valid inferences about the differences between the two\npopulations of biological subjects under study cannot be made.\nHere we only consider situations other than the above two and\nwhere pooling may reduce the overall costs of the experiments.\n\n\n\\section{A general formalism}\n\\label{sec-formalism}\n\nFor every comparative study, there is at least one measurable\nquantity which is the quantity of interest. The goal of the study\nis to deduce from the data collected if there is any difference\nbetween the means of the two populations. As measuring all the\nbiological subjects in two populations is rarely possible in most\nsituations representatives from a population are randomly selected\nand measurements made on these. These are then taken to infer the\nproperties of the population.\n\nLet $X$ be the measurable quantity that is being determined in the\nthe experiment, e.g., the expression level of a gene. In the case\nof one-channel microarray, $X$ could denote the logarithm (most\ncommonly base 2 is used) of fluorescence intensity; or the\nlogarithm of the fluorescence ratio in the case of two-channel\nmicroarray. Let $x^c_i$ denote the value of $X$ for an individual\nsubject $i$ in the control population (c), and $x^t_j$ that of the\nindividual subject $j$ in the treatment population (t). We assume\nthat $x^c_i$s for all individuals in the control population are\nindependent normally distributed with a mean $\\mu_c$ and a\nvariance $\\sigma _c^2$, denoted by $x^c_i \\sim N(\\mu_c,\\sigma\n_c^2)$ for all $i$. Similarly, $x^t_j \\sim N(\\mu_t,\\sigma _t^2)$\nfor all $j$.\n\n\n\\subsection{A general experimental setup}\nFor a general experimental setup individual subjects from both\npopulations are randomly selected and tissue samples collected\nfrom each. Tissue sample pools are made by pooling a given number\n$r$ of randomly selected tissue samples (of the same population)\ntogether. Note that to make $n$ pools we need to have selected\n$nr$ individual subjects from the population. $m$ measurements are\nthen made on each pool of tissue samples. So $m$ is the number of\ntechnical replications of measurement on each pool. Notice that by\nintroducing two parameters $r$ and $m$ a general and flexible\nexperimental setup has been created. For instance, if we set\n$r=1$, the experiment would be equivalent to no pooling of tissue\nsamples. And if we set $m=1$ there is no technical replication.\nUnder the basic assumption of biological averaging, the result of\npooling $r$ tissue samples in equal proportions together is that\nthe value of $X$ for the pool is the average of those subjects\nwhich formed this pool,\n\\begin{equation}\n\\tilde{x}=\\frac{1}{r}\\sum _{i=1}^{r}x_i.\n\\end{equation}\nIt follows that $\\tilde{x} \\sim N(\\mu_c, \\sigma_c ^2\/r)$ for a\npool from the control population, or $\\tilde{x} \\sim N(\\mu_t,\n\\sigma_t ^2\/r)$ for a pool from the treated population. Note that\nin this paper we shall only discuss pooling samples with equal\nindividual contributions. While pools formed by un-equal\ncontributions from individual samples are possible, such pooled\nexperimental design is generally less effective than the equal\npooling, as already shown by Peng {\\em et al}\n\\citep{Peng-etal-BMCBioinformatics2003} with their simulated\nresults.\n\nWhen we take a measurement on a pool $p$, the measured value is\n\\begin{equation}\ny_{p,k}=\\tilde{x}_p+\\epsilon _k,\n\\end{equation}\nwhere $p$ indexes pools, $k$ indexes measurements, and $\\epsilon\n_k$ is a random error term assumed to be independently and\nnormally distributed as $\\epsilon _k \\sim\nN(0,\\sigma^2_{\\epsilon})$. Hereafter $\\sigma^2_{\\epsilon}$ will be\nreferred to as the technical variance, $\\sigma^2_c$ the biological\nvariance for the control population, and $\\sigma^2_t$ the\nbiological variance for the treatment population.\n\nThe output of the experiment are the measurements on the two sets\nof pools. For the control group, we have $y^c_{p,k}$ for\n$p=1,\\dots, n_{c}$ and $k=1,\\dots, m$. And for the treatment\ngroup, we have $y^t_{p,k}$ for $p=1,\\dots, n_{t}$ and $k=1,\\dots,\nm$. Here $n_{c}$ and $n_{t}$ are the numbers of pools prepared for\nthe control and treatment population respectively. Our task is to\ninfer population properties from these measured data. In\nparticular, we want to know whether there is any difference\nbetween the two populations means $\\mu_c$ and $\\mu_t$. It can be\nshown that\n\\begin{equation}\n\\overline{Y}^c=\\frac{1}{mn_{c}}\\sum_{p=1}^{n_c}\\sum_{k=1}^my^c_{p,k}\n\\label{eq-Y^c}\n\\end{equation}\nis an unbiased estimator of $\\mu_c$, with a variance\n\\begin{equation}\n\\frac{1}{n_c}\\left(\\frac{\\sigma^2_c}{r}+\\frac{\\sigma^2_{\\epsilon}}{m}\\right),\n\\end{equation}\nand similarly,\n\\begin{equation}\n\\overline{Y}^t=\\frac{1}{mn_{t}}\\sum_{p=1}^{n_t}\\sum_{k=1}^my^t_{p,k}\n\\label{eq-Y^t}\n\\end{equation}\nis an unbiased estimator of $\\mu_t$, with a variance\n\\begin{equation}\n\\frac{1}{n_t}\\left(\\frac{\\sigma^2_t}{r}+\\frac{\\sigma^2_{\\epsilon}}{m}\\right).\n\\end{equation}\nIf we make an additional assumption that the variances for the two\npopulations of biological subjects are the same, i.e.,\n$\\sigma^2_c=\\sigma^2_t=\\sigma^2$, then the difference between\nEqs.(\\ref{eq-Y^t}) and  (\\ref{eq-Y^c}),\n$D=\\overline{Y}^t-\\overline{Y}^c$, is an unbiased estimator of\n$\\mu=\\mu_t-\\mu_c$ with a variance\n\\begin{equation}\n\\sigma^2_D=\\left(\\frac{1}{n_c}+\\frac{1}{n_t}\\right)\n\\left(\\frac{\\sigma^2}{r}+\\frac{\\sigma^2_{\\epsilon}}{m}\\right).\n\\label{eq-var(D)}\n\\end{equation}\nThe factor $(\\sigma^2\/r+\\sigma^2_{\\epsilon}\/m)$ in\nEq.(\\ref{eq-var(D)}) can be estimated without bias by\n\n\n\\begin{eqnarray}\ns^2_p=\\frac{1}{n_c+n_t-2}\n\\sum_{p=1}^{n_c}\\left(\\frac{1}{m}\\sum_{k=1}^my^c_{p,k}-\\overline{Y}^c\\right)^2 \\nonumber \\\\\n+\\frac{1}{n_c+n_t-2}\n\\sum_{p=1}^{n_t}\\left(\\frac{1}{m}\\sum_{k=1}^my^t_{p,k}-\\overline{Y}^t\\right)^2.\n\\end{eqnarray}\nIt is then clear that\n\\begin{equation}\nt=\\frac{(\\overline{Y}^t-\\overline{Y}^c)-(\\mu_t-\\mu_c)}{s_p\\sqrt{1\/n_c+1\/n_t}}\n\\end{equation}\nfollows the Student's t distribution with $n_c+n_t-2$ degrees of\nfreedom. In detecting a differential gene expression, we want to\ntest the null hypothesis $\\mu_c=\\mu_t$ against an alternative\nhypothesis $\\mu_c\\neq \\mu_t$. So our test statistic is\n\\begin{equation}\nt_0=\\frac{(\\overline{Y}^t-\\overline{Y}^c)}{s_p\\sqrt{1\/n_c+1\/n_t}},\n\\label{eq-t_0}\n\\end{equation}\nand there are no unknowns in Eq.(\\ref{eq-t_0}). Note that $t_0$\ncan be seen as a generalized two-sample-t-test statistic, which\nreduces to the statistic of the traditional two-sample t test with\nequal variance when we set the parameters $r=1$ (no pooling of\ntissue samples) and $m=1$ (no technical replication of\nmeasurements). In Ref. \\citep{Shih-etal-Bioinformatics2004},\n\\citeauthor{Shih-etal-Bioinformatics2004} arrived at two separate\nstatistics, one for non-pooled design, the other for pooled\ndesign. The $t_0$ defined by Eq.(\\ref{eq-t_0}) is in more general\nform, setting $r=1$ and $m=1$ in Eq.(\\ref{eq-t_0}) recovers Shih\n{\\em et al}'s statistic for non-pooled design; while setting $r>1$\nand $m=1$ recovers Shih {\\em et al}'s statistic for pooled design.\nNote that $m$ does not need to equal 1. Here by incorporating two\nadditional parameters $r$ and $m$, the statistic $t_0$ can deal\nwith situations where there are pooled tissue samples and multiple\ntechnical replications.\n\n\\subsection{Criteria of significance}\nAs with any statistical test we need to specify a threshold\np-value $P_{th}$ to claim significant results in the test. When\nall the other parameters are given, setting $P_{th}$ is equivalent\nto setting a threshold, say $|\\xi|$, for the statistics $t_0$\ndefined in Eq.(\\ref{eq-t_0}). With this threshold t-value, our\ncriteria for claiming a significant test is as follows: If\n$t_0>|\\xi|$, we declare that $\\mu_t-\\mu_c>0$; if $t_0<-|\\xi|$, it\nis claimed as $\\mu_t-\\mu_c<0$. So the rate at which false positive\nclaims are made is\n\\begin{eqnarray}\nP_{th}=\\int _{-\\infty}^{-|\\xi|}\\rho _{n_c+n_t-2}(t_0)dt_0\n+\\int_{|\\xi|}^{\\infty}\\rho _{n_c+n_t-2}(t_0)dt_0 \\nonumber \\\\\n=2\\int _{-\\infty}^{-|\\xi|}\\rho _{n_c+n_t-2}(t_0)dt_0\n=2T_{n_c+n_t-2}(-|\\xi|), \\label{eq-P_th}\n\\end{eqnarray}\nwhere $\\rho_{n_c+n_t-2}(.)$ is the probability density function\n(PDF) of the Student's t distribution with $n_c+n_t-2$ degrees of\nfreedom, and $T_{n_c+n_t-2}(.)$ is the corresponding cumulative\nprobability distribution function (CDF). It is therefore apparent\nthat the threshold t-value $|\\xi|$ can be obtained by solving the\nequation $2T_{n_c+n_t-2}(-|\\xi|)=P_{th}$ with a given false\npositive rate $P_{th}$.\n\n\n\n\\section{Power function}\n\\label{sec-power-function} In\n\\citep{Zhang&Gant-Bioinformatics2004} we presented a power\nfunction for a new statistical t test (hereafter referred to as\n\"two-labelling t test\") in the context of using two-color\nmicroarrays to detect differential gene expression. Following\nsimilar steps we can derive the power function for the generalized\ntwo-sample t test presented in this paper, which reads\n\n\n\\begin{eqnarray}\nS=\\int _0^{\\infty}p_{n_c+n_t-2}(Y)\\Phi\\left[{-|\\xi|\\sqrt{Y}\\over\n\\sqrt{n_c+n_t-2}} +\\frac{|\\mu|}{\\sigma_D} \\right]dY,\n \\label{eq-S}\n\\end{eqnarray}\nwhere $p_{n_c+n_t-2}(Y)$ is the PDF for the $\\chi ^2$ distribution\nwith $n_c+n_t-2$ degrees of freedom, and $\\Phi(.)$ is the CDF for\nthe standard normal distribution. The rate $S$ at which a true\ndifference between $\\mu_t$ and $\\mu_c$ can be successfully\ndetected is a function of $n_c$, $n_t$, $|\\mu|\/\\sigma _D$, and\n$|\\xi|$. With $\\sigma_D$ given by the square root of\nEq.(\\ref{eq-var(D)}), and $|\\xi|$ determined by solving\nEq.(\\ref{eq-P_th}) at a given false positive rate $P_{th}$, $S$\nis, eventually, a function of $P_{th}$, $n_c$, $n_t$, and\n$|\\mu|\/\\sigma _D$.\n\n\nA few points are worth noting here.\n\n1. The two-labelling t test presented in\n\\citep{Zhang&Gant-Bioinformatics2004} was designed to deal with\nsystematic labelling biases generated during microarray\nexperimentation. The t test presented in this paper, however,\nassumes no systematic data biases. In the case of two-color\nmicroarrays this requires a common reference design. In such an\nexperimental design the labelling biases cancel themselves out in\nthe calculation of the test statistic.\n\n2. In \\citep{Zhang&Gant-Bioinformatics2004}, the biological\nvariances of the two populations under comparison do not have to\nbe the same, that is, we did not assume $\\sigma^2_c=\\sigma^2_t$.\nFor the t test in this paper, we have made an additional\nassumption that $\\sigma^2_c=\\sigma^2_t$. Relaxing this requirement\nwas possible, as in the case of the traditional two-sample t test\nwith unequal variance \\citep{Brownlee1965}, but an exact power\nfunction could not be readily obtained.\n\n3. The exact power function obtained in this paper allows\nevaluation of the effects of pooling biological samples and the\neffects of taking multiple technical measurements, thus giving\nresearchers quantitative guidance on the practice of pooling\nsamples.\n\n4. By setting the parameters $r=1$ and $m=1$, an exact power\nfunction is provided for the traditional two-sample t test with\nequal variance.\n\n\n\\section{Results}\n\nWe have implemented the computation of the power function $S$ of\nEq.(\\ref{eq-S}) as a Java application, which can be accessed at\nthe URL given in the abstract. Here we apply this to microarray\ncomparative studies for finding differentially expressed genes,\nand investigate the effect of pooling RNA samples in the\nexperiments. We also compare our exact results with some\napproximate results presented by other authors\n\\citep{Shih-etal-Bioinformatics2004} to demonstrate why an exact\nformula is desirable.\n\n\\subsection{Comparison with approximate results}\nBased on their approximate formulas,\n\\citeauthor{Shih-etal-Bioinformatics2004} considered two scenarios\nto compare the number of biological subjects and number of\nmicroarrays in the non-pooled and pooled\ndesigns\\citep{Shih-etal-Bioinformatics2004}. Here we give exact\nresults for the two scenarios to show the difference to the\napproximate results. In the first scenario, we consider that the\ncommon biological variance of the two populations is $\\sigma\n^2=0.05$, and the technical variance $\\sigma_{\\epsilon}^2=0.0125$,\nwhich gives the biological-to-technical variance ratio $\\lambda\n=\\sigma^2\/\\sigma_{\\epsilon}^2=4$. The preset target of the\nexperiment in this scenario is that the false positive rate being\ncontrolled at $P_{th}=0.001$ and the power being no less than\n$S=0.95$ to detect a two-fold differential gene expression, which\ncorresponds to $\\mu=1$ with base 2 logarithm\n\\citep{Shih-etal-Bioinformatics2004}. In Table\n\\ref{tab-scenario1}, we present results for different pooling\nparameter $r$. It can be seen from the first panel of this Table\nthat in order to hit the preset target, the non-pooled design\n($r=1$) requires at least $12$ biological subjects divided evenly\nto the two populations, i.e., $6$ from each of the two\npopulations. Having $7$ subjects from one population and $5$\nsubjects from the other is insufficient to achieve the target of\n$95\\%$ detection power. The effects of other levels of pooling on\nthe detection power are also shown in Table \\ref{tab-scenario1}.\nThe minimum number of biological subjects ($N_s$) and microarrays\n($N_m$) that meet the preset targets are highlighted with bold\nfonts. It is clear that as the level of pooling is increased (with\nincreasing $r$), the number of microarrays $N_m$ can be reduced,\nbut the number of biological subjects $N_s$ has to be increased.\nFor example, in order to reduce the number of arrays from $12$\n(Table \\ref{tab-scenario1}, first panel) to $8$ (Table\n\\ref{tab-scenario1}, fourth panel), the number of biological\nsubjects to form the pools must be increased from $12$ to $40$.\n\n\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=8.5cm,height=6.0cm,angle=0]{Senario2-Scurve.eps}}\n\\caption{The power $S$ as a function of the total number of pools\n$n_c+n_t$. The parameters used are for the second scenario $\\sigma\n^2=0.2$, $\\sigma_{\\epsilon}^2=0.05$, $\\lambda\n=\\sigma^2\/\\sigma_{\\epsilon}^2=4$, $P_{th}=0.001$, $\\mu=1$, and\n$m=1$. The five solid curves correspond to different levels of\npooling, from right to left, $r=1$, $r=2$, $r=4$, $r=6$, and\n$r=15$ respectively. The dashed line indicates the $95\\%$ power,\nthe intersections of which with the power curves specify the total\nnumbers of pools (assuming $n_c=n_t$) needed to achieve the target\npower. The total number of biological subjects and the total\nnumber of arrays can then be calculated simply by\n$N_s=r(n_c+n_t)$, and $N_m=m(n_c+n_t)$ respectively.}\n\\label{fig-Scurve}\n\\end{figure}\n\n\n\n\nFor the second scenario we consider the case $\\sigma ^2=0.2$,\n$\\sigma_{\\epsilon}^2=0.05$, which gives $\\lambda\n=\\sigma^2\/\\sigma_{\\epsilon}^2=4$. Again the preset targets are to\ndetect a true differential expression $\\mu=1$ with no less than\n$95\\%$ power while the false positive rate is set at\n$P_{th}=0.001$. Using these parameters, the power $S$ as a\nfunction of $n_c+n_t$ is plotted in Fig.\\ref{fig-Scurve} for\ndifferent levels of sample-pooling. For the non-pooled design\n($r=1$), $N_s=30$ total biological subjects and $N_m=30$ arrays\nare required to hit the preset targets. Similar to the first\nscenario, as the level of pooling is increased, the number of\narrays $N_m$ is reduced while the number of subjects increased to\nmeet the preset targets.\n\n\nIn Table \\ref{tab-exact-vs-approx}, we summarize our exact results\nand the approximate results of\n\\citep{Shih-etal-Bioinformatics2004}. It can be seen that the\ndifference between the two can be very large, indicating the need\nfor exact results.  For example, in the first scenario when\n$N_m=8$ the approximate result of\n\\citep{Shih-etal-Bioinformatics2004} predicts that a minimum of\n$21$ biological subjects are required. In practice $24$ subjects\nare required as $24$ is the minimum number larger than $21$ and\ndivisible by $8$. However this experiment setup ($24$ subjects\nforming $8$ pools, $8$ microarrays) will only give a detection\npower of $90\\%$. To meet the target power of $95\\%$, $40$\nbiological subjects are actually required by our exact result. If\nan experiment with $N_m=7$ microarrays is planned,\n\\citeauthor{Shih-etal-Bioinformatics2004} predicts that $37$\nsubjects are required\\citep{Shih-etal-Bioinformatics2004}, but in\nfact $126$ subjects must be used to achieve the target. Generally,\nthe approximate formulas of \\citep{Shih-etal-Bioinformatics2004}\nare too optimistic in assessing the benefits of pooling samples\nand reducing the number of microarrays, because they underestimate\nthe number of biological subjects required.\n\n\\subsection{Cost analysis}\nDepending on the material costs involved in the biological\nsubjects and microarrays, the conditions where pooling samples\nbecomes beneficial may be different from lab to lab. Here we show\nexamples to determine these conditions. Denoting the cost\nassociated with each biological subject as $C_s$ (including\nmaterials and labor etc) and the cost associated with a microarray\nas $C_m$, then the total costs for an experiment in microarray\ncomparative study is $C_T=N_sC_s+N_mC_m$. Taking the first\nscenario as an example, the total cost of a non-pooled design to\nachieve our preset targets is\n$$C_T(r=1)=12C_s+12C_m,$$\nand the total cost for pooled design with $r=2$ is\n$$C_T(r=2)=20C_s+10C_m.$$\nTherefore in order that the pooled design with $r=2$ is beneficial\nwe must have\n\n\\begin{equation}\nC_T(r=2) \\le C_T(r=1),\n\\end{equation}\nwhich requires that $C_m \\ge 4C_s$. Put another way, only when the\ncost associated with one microarray $C_m$ is more than $4$ times\nthe cost of a subject $C_s$, does the pooling design with $r=2$\nbecome preferable to the non-pooled design. Similarly a higher\nlevel of pooling with $r=3$ becomes  preferable to $r=2$ only when\n$C_m \\ge 7C_s$. Furthermore the conditions for increasing the\nlevel of pooling from $r=3$ to $r=5$ are $C_m \\ge 13C_s$, and so\non. Table \\ref{tab-exact-vs-approx} gives these conditions for\nfurther levels of pooling.\n\nFor the first scenario using the actual cost figures given in\n\\citep{Shih-etal-Bioinformatics2004} where $C_s=\\$230$ and\n$C_m=\\$300$, it can be seen that none of the pooling conditions is\nmet. Therefore for this laboratory pooling samples is not\nrecommended. However, if we use the cost figures of\n\\citep{Kendziorski-etal-Biostatistics2003} where $C_s=\\$50$ and\n$C_m=\\$700$, an optimal design is a pooled design with $r=5$.\n\n\nFor the second scenario, it is a similar story. The cost figures\nof Ref. \\citep{Shih-etal-Bioinformatics2004} ($C_s=\\$230$ and\n$C_m=\\$300$) gives $C_m=1.30C_s$, which does not satisfy any of\nthe pooling conditions. So again the non-pooled design with\n$N_m=30$ and $N_s=30$ is recommended. On the other hand, the cost\nfigures of \\citep{Kendziorski-etal-Biostatistics2003} ($C_s=\\$50$\nand $C_m=\\$700$) give $C_m=14C_s$ which satisfies all the pooling\nconditions in the lower panel of Table \\ref{tab-exact-vs-approx}\nexcept the last row. So in Kendziorski et al's lab, the pooled\ndesign with $N_m=14$ and $N_s=84$ would be recommended.\n\n\n\\section{Discussion} \\label{sec-discussion} We have in this paper\npresented exact formulas for calculating the power of microarray\nexperimental design with different levels of pooling. These\nformulas can be used to determine the conditions of statistical\nequivalence between different pooling setups. As in\n\\citep{Kendziorski-etal-Biostatistics2003} and\n\\citep{Shih-etal-Bioinformatics2004}, the calculations presented\nin this paper are for an individual gene, so the statistical\nequivalence for different designs of pooling can be determined\nwith regard to one particular gene. However, microarray monitors\nthousands of genes simultaneously, and the biological and\ntechnical variances vary from gene to gene, therefore no single\nresult of statistical equivalence between pooled and non-pooled\ndesigns applies equally to all genes on the array. So in practice\nhow would the formulations in this work be used? One possible way,\nas suggested by Kendziorski {\\em et al}.\n\\citep{Kendziorski-etal-Biostatistics2003}, is to specify the\ndistributions of $\\sigma ^2$ and $\\sigma_{\\epsilon}$ and calculate\nthe total number of subjects and arrays that maximize the average\npower across the the array. In theory, if the biological variances\nand technical variances were known for all genes on the array, an\nequivalence condition between pooled and non-pooled designs could\nbe determined for each gene individually. The overall (or say,\naverage) equivalence condition between pooled and non-pooled\ndesigns could be obtained, for example, by some form of averaging\noperation over all genes. An alternative and probably more\npractical way is to use representative values of $\\sigma ^2$ and\n$\\sigma_{\\epsilon}$. We therefore propose that parameters for\n\"typical gene\" be used as inputs for the power and sample size\ncalculations. A typical gene is a gene whose biological and\ntechnical variance have the most probable values among the genes,\ni.e., the mode of the distribution for biological and technical\nvariance of genes. Alternatively, the median or mean variances\nacross genes could be used as representative\nvalues\\citep{Shih-etal-Bioinformatics2004}.\n\n\nAn issue associated with microarray experiments is the problem of\nmultiple inferences, where a separate null hypothesis is being\ntested for each gene. Given thousands of null hypotheses being\ntested simultaneously, the customary significance level $\\alpha\n=0.05$ for declaring positive tests will surely give too many\nfalse positives. For example, if among a total number $N=10000$ of\ngenes being tested, $N_0=4000$ are truly null genes (genes that\nare non-differentially expressed between the two classes), the\nexpected number of false positive results would be $4000\\times\n0.05=200$, which may be too many to be acceptable. Thus a smaller\nthreshold p-value for declaring differentially expressed genes\nshould be used. Effectively controlling false positives in a\nmultiple testing situation such as microarray experiments is an\narea which has drawn much attention in recent years due to the\nwider application of microarray technology. As discussed in our\nprevious work in \\citep{Zhang&Gant-Bioinformatics2004}, generally\nspeaking, all different multiple-testing adjustment methods\neventually amount to effectively setting a threshold p-value, and\nthen rejecting all the null hypotheses with p-value below this\nthreshold. The classical Bonferroni multiple-testing procedure,\nwhich controls family-wise error rate at $\\alpha$ by setting the\nthreshold $P_{th}=\\alpha \/N$, is generally regarded as being too\nconservative in the microarray context. The FDR (False Discovery\nRate) idea, initially due to \\citep{Benjamini&Hochberg1995} in\ndealing with the multiple testing problem, has now been widely\naccepted as appropriate to the microarray situation. Recently,\nEfron \\citep{Efron2004} extended the FDR idea by defining fdr, a\nlocal version FDR (local false discovery rate). When planning\nmicroarray experiments in terms of power and sample size\ncalculation, the FDR of \\citep{Benjamini&Hochberg1995} is more\nappropriate and convenient to use. There are now in the literature\na few slightly different variants of the definition of FDR\n\\citep{Benjamini&Hochberg1995,Storey&Tibshrirani-pnas2003,Grant-etal-Bioinformatics2005},\nbut in essence it is defined as the proportion of false positives\namong all positive tests declared. To provide an interface between\nFDR and the formulation in the previous sections, here we show\nthat there is a simple correspondence between controlling FDR and\nspecifying the traditional type I error rate and power. Suppose\nthat there are a total number $N$ of genes being monitored by\nmicroarray, so there will be $N$ hypotheses being tested, one for\neach gene. Suppose that a fraction $\\pi_0$ of the $N$ genes are\ntrue null genes, i.e., genes that are non-differentially expressed\nbetween the two classes. Given the type I error rate $P_{th}$, the\nexpected number of false positive tests is $P_{th}N\\pi_0$; Given\nthe power $S$, the expected number of non-null genes (truly\ndifferentially expressed genes) that are declared positive is\n$SN(1-\\pi_0)$. So the FDR achieved by this setting is\n\\begin{equation}\n\\mbox{FDR}=\\frac{P_{th}N\\pi_0}{P_{th}N\\pi_0+SN(1-\\pi_0)}\n=\\frac{P_{th}\\pi_0}{P_{th}\\pi_0+S(1-\\pi_0)}. \\label{eq-fdr}\n\\end{equation}\nHere $\\pi_0$ is an important parameter in controlling FDR, for\nwhich  several different methods of estimating this parameter have\nbeen proposed\n\\citep{Pounds&Morris2003,Storey&Tibshrirani-pnas2003,Zhang&Gant-Bioinformatics2004}.\nEspecially the method we presented in\n\\citep{Zhang&Gant-Bioinformatics2004} is an accurate yet\ncomputationally much simpler algorithm than the one proposed by\nStorey and Tibshrirani in \\citep{Storey&Tibshrirani-pnas2003}.\nWith the interface Eq.(\\ref{eq-fdr}), FDR can be readily presented\nand incorporated into the calculations.\n\n\n\n\\vspace{0.2cm}\\noindent {\\Large\\bf Acknowledgments}\n\n\\noindent We wish to acknowledge the support of the microarray\nteam of the MRC Toxicology Unit particularly Reginald Davies,\nJinLi Luo and Joan Riley. We also thank two anonymous reviewers\nfor their helpful and constructive comments.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMode-coupling theory (MCT) is considered by many the most\ncomprehensive first-principle approach to the dynamics of supercooled\nliquids~\\cite{Gotze}.  Nevertheless, its status is rather problematic\nfrom a fundamental point of view, as the physical nature of the glass\nstate and the microscopic interpretation of structural arrest are not\nyet fully elucidated. This is all the more so when we look at the\nhigher-order glass singularities in structured and complex liquids.\nIn this Rapid Communication, I show that multiple glassy states and\nglass-glass transition in MCT can be understood in terms of a\ngeneralisation of the notion of dynamic facilitation~\\cite{FrAn,RiSo}\nand bootstrap percolation~\\cite{ChLeRe,BP_rev}. The latter is known to\nemerge in a variety of contexts including jamming of granular\nmaterials~\\cite{Liu}, NP-hard combinatorial optimization\nproblems~\\cite{MoZe}, neural and immune networks~\\cite{Tlusty,\n  Stauffer}, and evolutionary modeling~\\cite{Klimek}.\n\nThe formal structure of glass singularities predicted by MCT is\nencoded in the self-consistent equation\n\\begin{equation}\n\\label{eq.Phi}\n\\Phi = (1-\\Phi) \\ {\\mathsf M} (\\Phi),\n\\end{equation}\nwhere $\\Phi$ is the asymptotic value of the correlator, and ${\\mathsf\n  M}$ is the memory kernel describing the retarded friction effect\ncaused by particle caging, a physical feature associated with the de\nGennes narrowing.  We shall be concerned in the following with\none-component schematic models in which the wavevector dependence of\n$\\Phi$ is disregarded and ${\\mathsf M}$ is a low order polynomial.\nEquation~(\\ref{eq.Phi})--derived by taking the long-time of the\nintegro-differential equation describing the evolution of the\ncorrelator of particle density fluctuations--generates a hierarchy of\ntopologically stable glass singularities, which can be classified in\nterms of bifurcations exhibited by the roots of the real polynomial\n\\begin{equation}\n{\\mathcal Q}(\\Phi) = \\Phi - (1-\\Phi) \\ {\\mathsf M} (\\Phi) .\n\\end{equation}\nFollowing Arnol'd notation, adopted in~\\cite{Gotze}, an ${\\mathsf\n  A}_{\\ell}$ glass singularity occurs when the corresponding maximum\nroot of ${\\mathcal Q}$ has a degeneracy $\\ell$, $\\ell \\ge 2$, and is\ndefined by\n\\begin{equation}\n\\frac{d^n {\\mathcal Q}}{d\\Phi^n} = 0 \\,, \\qquad n=0, \\cdots, \\ell-1,\n\\label{eq.dQ}\n\\end{equation}\nwhile the $\\ell$th derivative is nonzero. The polynomial ${\\mathcal\n  Q}$ has always the trivial root $\\Phi=0$, corresponding to a liquid\nergodic state, whereas nonzero values of $\\Phi$ correspond to a system\nthat is unable to fully relax and hence can be identified with a glass\nnonergodic state.\n\n\nFor two-parameter systems there are two basic singularities, ${\\mathsf\n  A}_2$ and ${\\mathsf A}_3$, also known as {\\em fold} and {\\em cusp}\nbifurcations. They have been extensively studied by using memory\nkernels given by a superposition of linear and nonlinear terms.  In\nthe ${\\mathsf F}_{12}$ schematic model the memory kernel is ${\\mathsf\n  M}(\\Phi) = v_1 \\Phi + v_2 \\Phi^2$ while the ${\\mathsf F}_{13}$ model\nis defined by ${\\mathsf M}(\\Phi) = v_1 \\Phi + v_3 \\Phi^3$.  The\ncompetition between the two terms produces a variety of nonergodic\nbehaviors: the linear term gives rise to a continuous liquid-glass\ntransitions at which $\\Phi \\sim \\epsilon$, where $\\epsilon$ is the\ndistance from the critical point (e.g., $\\epsilon=T-T_{\\scriptstyle \\rm c}$), while the\nnonlinear term induces a discontinuous liquid-glass transition, with\nthe well known square-root anomaly $\\Phi-\\Phi_{\\scriptstyle \\rm c} \\sim\n\\epsilon^{1\/2}$. In the ${\\mathsf F}_{12}$ scenario the discontinuous\nline joins smoothly the continuous one at a tricritical point. In the\n${\\mathsf F}_{13}$ scenario, the discontinuous transition line\nterminates at an ${\\mathsf A}_3$ singularity inside the glass phase\ngenerated by the continuous liquid-glass transition, and therefore\ninducing a glass-glass transition (see Fig.~1 for a representative\nphase diagram). The scaling form of the order parameter near the\n${\\mathsf A}_3$ endpoint is $\\Phi-\\Phi_{\\scriptstyle \\rm c} \\sim \\epsilon^{1\/3}$, and more\ngenerally $\\Phi-\\Phi_{\\scriptstyle \\rm c} \\sim \\epsilon^{1\/\\ell}$ for an ${\\mathsf\n  A}_{\\ell}$ singularity, as implied by the Taylor expansion of\n${\\mathcal Q}$ near the critical surface and Eqs.~(\\ref{eq.dQ}).  Thus\none can observe a rich variety of nonergodic behaviors whose\ncomplexity is comparable to that of multicritical points in phase\nequilibria~\\cite{Gilmore}.  It is a nontrivial result that only\nbifurcation singularities of type ${\\mathsf A}_{\\ell}$ can occur in\nMCT~\\cite{Gotze}.\n\n\nThe ${\\mathsf F}_{12}$ and ${\\mathsf F}_{13}$ scenarios were first\nintroduced with the mere intention of demonstrating the existence of\nhigher-order singularities and glass-glass transition, and then were\nsubsequently observed in a number of experiments and numerical\nsimulations of realistic model\nsystems~\\cite{Dawson,Pham,Eckert,Chong,Krako,Kurz,Kim,Sperl,Voigtmann}.\nIt is important to emphasize that the parameters $v_i$ entering the\nmemory kernel are smooth functions of the thermodynamic variables,\ne.g.  temperature and density, therefore the nature of nonergodic\nbehaviors predicted by MCT is purely dynamic. This is rather puzzling\nfrom the statistical mechanics perspective of critical phenomena where\ndiverging relaxation time-scales are closely tied to thermodynamic\nsingularity.  It has been argued that this unusual situation stems\nfrom uncontrolled approximations.  For example, the intimate\nconnection of some spin-glass models with MCT has brought to the fore\nthe existence of a genuine {\\em thermodynamic} glass phase at a\nKauzmann temperature $T_{\\scriptscriptstyle \\rm K}$ below the putative dynamic glass transition\npredicted by MCT~\\cite{KiWo,KiTh}. A non-trivial Gibbs measure,\ninduced by a replica-symmetry breaking, would therefore be actually\nresponsible for the observed glassy behavior~\\cite{MePa}.  For this\nreason, the nature of the MCT has been much debated since its first\nappearance and several approaches have been attempted to clarify its\nstatus~\\cite{KiWo,KiTh,MePa,BoCuKuMe,Dave,Andrea,Ikeda,Schmid,Szamel,Silvio}.\nI will show here that the idea of dynamic facilitation~\\cite{RiSo},\nfirst introduced by Fredrickson and Andersen~\\cite{FrAn}, offers some\nclues in this direction for its relation with bootstrap percolation\nprovides a transparent microscopic mechanism of structural\narrest~\\cite{ChLeRe,Branco}.  In the dynamic facilitation approach the\ncoarse-grained structure of a supercooled liquid is represented by an\nassembly of higher\/lower density mesoscopic \\emph{cells}. In the\nsimplest version a binary spin variable, $s_i=\\pm 1$, is assigned to\nevery cell $i$ depending on its solid or liquid like structure and no\nenergetic interaction among cells is assumed, ${\\mathcal H} = -h\n\\sum_i s_i$.  The crucial assumption is that the supercooled liquid\ndynamics is essentially dominated by the cage effect: fluctuations in\nthe cells structure occur if and only if there is a certain number,\nsay $f$, of nearby liquid-like cells. $f$ is called the facilitation\nparameter and can take values in the range $0 \\le f \\le z$, where $z$\nis the lattice coordination: cooperative facilitation imposes $f \\ge\n2$, while non-cooperative dynamics only requires $f=1$.  This very\nschematic representation of the cage effect gives rise to a large\nvariety of remarkable glassy behaviors, and it has long been noticed\nthat they are surprisingly similar to those found in the dynamic of\nmean-field disordered systems~\\cite{KuPeSe,Se,SeBiTo}.  It has been\nrecently observed that in a special case, an exact mapping between\nfacilitated and disordered models with $T_{\\scriptscriptstyle \\rm K}=0$,\nexists~\\cite{FoKrZa}. Since such models are so utterly different in\ntheir premises, it is by no means obvious that such a correspondence\nis not accidental and can be extended to systems with higher-order\nglass singularities.  To clarify this issue, I will consider a\ngeneralization of the facilitation approach~\\cite{SeDeCaAr} in which\nevery cell $i$ is allowed to have its own facilitation parameter $f_i$\n(or, equivalently, an inhomogeneous local lattice connectivity).\nPhysically, this situation may arise from the coexistence of different\nlengthscales in the system, e.g., mixtures of more or less mobile\nmolecules or polymers with small and large size, (or from a\ngeometrically disordered environment, e.g., a porous matrix).  In such\nfacilitated spin mixtures the facilitation strength can be tuned\nsmoothly and is generally described by the probability distribution\n\\begin{eqnarray}\n  \\pi(f_i) & = & \\sum_{\\zeta=0}^z \\ w_{\\zeta} \\ \\delta_{f_i,\\zeta} ,\n\\label{eq.distf.general}\n\\end{eqnarray}\nwhere the weights $\\{w_{\\zeta}\\}$ controlling the facilitation strength\nsatisfy the conditions\n\\begin{eqnarray}\n\\sum_{\\zeta=0}^z w_{\\zeta} = 1 ,\\qquad 0 \\le w_{\\zeta} \\le 1.\n\\end{eqnarray}\nBy tuning the weights one can thus explore a variety of different\nsituations.  Generally, one observes that when the fraction of spins\nwith facilitation $f=z-1,z$ is larger than that with $2 \\le f \\le\nz-2$, the glass transition is continuous while in the opposite case it\nis discontinuous.  One advantage of the facilitation approach is that\nwhen the lattice topology has a local tree-like structure, one can\ncompute exactly some key quantities, such as the critical temperature\nand the arrested part of correlation and its scaling properties near\ncriticality. This can be done by exploiting the analogy with bootstrap\npercolation.  Let $p$ be the density of up spins in thermal\nequilibrium,\n\\begin{eqnarray}\n  p &=& \\frac{1}{ 1 + {\\rm e}^{-h\/k_{\\scriptscriptstyle \\rm B} T} },\n\\end{eqnarray}\nfor a generic spin mixture on a Bethe lattice with branching ratio\n$k=z-1$.  As usual, one arranges the lattice as a tree with $k$ branches\ngoing up from each node and one going down, and then proceeds\ndownwards. In analogy with the heterogeneous bootstrap percolation\nproblem, the probability $B$ that a cell is in, or can be brought into, the\nliquid-like state by only rearranging the state of $k$ cells above\nit~\\cite{ChLeRe,Branco,SeBiTo,SeDeCaAr}, can be cast in the form\n\\begin{eqnarray}\n  1-B &=& B \\ p \\ \\left\\langle \\sum_{i=k-f+1}^k {k \\choose i}\n  B^{k-i-1} (1-B)^i \\right\\rangle_f,\n\\label{eq.B}\n\\end{eqnarray}\nwhere $\\left\\langle \\cdots \\right\\rangle_f$ represents the average\nover the probability distribution Eq.~(\\ref{eq.distf.general}). The\nright-hand side of Eq.~(\\ref{eq.B}) is a polynomial of $1-B$, and\nhence the formal structure of Eq.~(\\ref{eq.B}) is similar to that of\nschematic MCT (once $1-B$ is formally identified with $\\Phi$).\nSingularities can therefore be classified according to the criteria\nalready mentioned in the introduction.  Nevertheless, it should be\nnoticed that what would be the anolog of the MCT kernel in\nEq.~(\\ref{eq.B}) can also have negative coefficients (besides\ncontaining an extra term of the form $(1-B)^k\/B$), while the\npolynomial coefficients of the MCT memory kernel are restricted to\nnon-negative ones.  In fact, the sets of critical states which specify\nsome ${\\mathsf A}_{\\ell}$ glass-transition singularity are not\nidentical to those describing the full bifurcation scenario of real\npolynomials of degree $\\ell$, because the coefficients of the\nadmissible polynomials ${\\mathcal Q}$ form only a subset of all real\ncoefficients. This observation means that the correspondence between\nMCT and the heterogeneous facilitation approach is not an identity,\nbut this still leaves enough room for building up models with MCT\nfeatures, although some ingenuity may be required. It has already been\nshown, for example, that the ${\\mathsf F}_{12}$ scenario is faithfully\nreproduced in this framework~\\cite{SeDeCaAr,ArSe}. To substantiate the\nabove observation, I now will focus on the next higher-order glass\nsingularity, which is the ${\\mathsf F}_{13}$ scenario.\n\\begin{figure}\n\\includegraphics[width=8.5cm]{phasediagram}\n\\caption{Phase diagram for a Bethe lattice with $z=5$ and facilitation\n  as in Eq.~(\\ref{eq.distf}).  The liquid-glass 1 transition is\n  continuous while the liquid-glass 2 and the glass 1-glass 2 are\n  discontinuous. The light dashed line is the unstable branch of the\n  phase diagram and shows the cuspoid structure of the terminal\n  endpoint.}\n\\label{fig.diag}\n\\end{figure}\nLet us consider, for simplicity, a binary mixture on a Bethe lattice with\n$z=5$ and\n\\begin{equation}\n\\pi(f_i) = (1-q) \\delta_{f_i,2} + q \\delta_{f_i,4}.\n\\label{eq.distf}\n\\end{equation}\nFor such a mixture, denoted here as (2,4), the probability $B$ obeys\nthe fixed-point equation:\n\\begin{equation}\n  1-B = p \\left[ q (1-B^4) + (1-q) (1-B)^3 (1+3B)\\right].\n\\label{eq.P245}\n\\end{equation}\nThis equation is always satisfied by $1-B=0$, while an additional\nsolution with $1-B>0$ is found by solving\n\\begin{equation}\n  p^{-1} = 1+B-5B^2+3B^3 + 2 q B^2 (3-B).\n\\label{eq.P245_1}\n\\end{equation}\nA continuous glass transition is obtained by setting $B=1$ in the\nprevious equation, giving: $p_{\\scriptstyle \\rm c} = 1\/4q$. Using the relation between $T$\nand $p$ (and setting $h\/k_{\\scriptscriptstyle \\rm B}=1$), one gets $T_{\\scriptstyle \\rm c}(q) = -1\/\\ln(4q-1)$,\nimplying that the continuous transition exists in the range $1\/2 \\ge q\n\\ge 1\/4$.  The discontinuous transition instead occurs when\nEq.~(\\ref{eq.P245_1}) is satisfied and its first derivative with\nrespect to $B$ vanishes. The latter condition implies\n\\begin{equation}\n  q = \\frac{ (9B-1)(1-B)}{6B(2-B)} ,\n\\label{eq.P245_2}\n\\end{equation}\nand naturally leads to the square-root scaling near the discontinuous\ntransition line. Thus the discontinuous transition can be graphically\nrepresented by expressing Eqs.~(\\ref{eq.P245_1}) and (\\ref{eq.P245_2})\nin parametric form in terms of $B$. The phase diagram in the plane\n$(T,q)$ is shown in the Fig.~\\ref{fig.diag}.\nIt exhibits two crossing glass transition lines, with continuous and\ndiscontinuous nature, corresponding to a degenerate and generic\n${\\mathsf A}_2$ singularities. The discontinuous branch extends into\nthe glass region below the continuous line up to a terminal endpoint\nwhich corresponds to an ${\\mathsf A}_3$ singularity. The location of\nthe endpoint is found by simultaneously solving equation\n\\begin{equation}\n  B = \\frac{5-6q}{9-6q} ,\n\\label{eq.P245_3}\n\\end{equation}\n(which is obtained by setting the second derivative of\nEq.~(\\ref{eq.P245_1}) to zero), along with Eqs.(\\ref{eq.P245_1}) and\n(\\ref{eq.P245_2}).  The discontinuous branch located between the\ncrossing point and the endpoint corresponds to a transition between\ntwo distinct glass states, called here glass 1 and glass 2.  They are\nrespectively characterized by a fractal and compact structure of the\nspanning cluster of frozen particles. The passage from one glass to\nthe other can take place either discontinuously or without meeting any\nsingularity, i.e. by circling around the endpoint (in a way much\nsimilar to liquid-gas transformation).  The existence of two\ntransitions in bootstrap percolation was first discovered by Fontes\nand Schonmann~\\cite{FoSc} in homogeneous trees and then found in\nErd\\H{o}s-R\\'enyi graphs and complex networks in~\\cite{Porto,Cellai}.\nHowever, its relation with glass-glass transition and MCT went\nunnoticed. In fact, the correspondence between Eqs.~(\\ref{eq.Phi}) and\n(\\ref{eq.B}) naturally suggests the existence of further singularities\nin bootstrap percolation and cooperative facilitated models.\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{orderparameter}\n\\caption{ Fraction of permanently frozen spins, $\\Phi$, vs temperature\n  ratio, $T\/T_{\\scriptstyle \\rm c}(q)$, with values of $q$ larger than the crossing\n  point.}\n\\label{fig.Phi}\n\\end{figure}\n\nFig.~\\ref{fig.Phi} reports the behavior of the fraction of frozen\nspins, which is the analog of the nonergodicity parameter in the\nfacilitation approach, when the temperature crosses the liquid-glass\ncontinuous transition and the glass-glass transition.  This quantity\ncan be exactly computed from $B$~\\cite{SeDeCaAr,ArSe}, and its\nexpression is not reported here--we only notice that its general\nfeatures, and in particular the scaling properties near the critical\nstates, are similar to those of $B$.  We observe that the fraction of\nfrozen spins first increases smoothly at the liquid-glass continuous\ntransition and then suddenly jumps at the glass-glass transition. The\njump decreases when $q$ approaches the endpoint and eventually\ndisappears. At this special point, the additional condition that the\nsecond-order derivative of Eq.~(\\ref{eq.P245_1}) with respect to $B$\nvanishes, implies a cube-root scaling near the endpoint. These scaling\nfeatures are exactly those expected from the ${\\mathsf F}_{13}$\nscenario, and we obtain similar results for the mixtures (3,5) on a\nBethe lattice with $z=6$.\n\nTo summarise, a close relationship exists between the structure\nof glass singularities in MCT and that of heterogeneous bootstrap\npercolation.  This allows the construction of microscopic realizations\nof MCT scenarios based on the heterogeneous cooperative facilitation\napproach and provides further insights into the degree of universality\nof MCT. The role of the linear and nonlinear terms in the MCT memory\nkernel is played in facilitated spin mixtures by the fraction of spins\nwith facilitation $f=k, k+1$ and $k-1 \\ge f\\ge 2$, respectively. Their\ncompetition generates continuous and discontinuous liquid-glass\ntransitions, while the order of singularity is primarily controlled by\nthe lattice connectivity. This leads to multiple glassy states,\nglass-glass transition and more complex glassy behaviors.  In this\nframework, the mechanism of structural arrest can be geometrically\ninterpreted in terms of the formation of a spanning cluster of frozen\nspin having fractal or compact structure depending on the continuous\nor discontinuous nature of the glass transition.  Finally, from the\nrelation between MCT and mean-field disordered\nsystems~\\cite{KiWo,KiTh} it follows that quenched disorder and\ncooperative facilitation are two complementary, rather than\nalternative, descriptions of glassy matter, and this contributes to\nthe long sought unifying approach to glass physics.\n\n\n\\bibliographystyle{apsrev}   \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nSmartphones, tablets, and similar smart mobile devices have been greatly successful as personal communication and computing devices owing to their always-on connectivity, mobility, usability, and operating systems that enable a vast market of applications tailored for the needs of mobile users. Smart devices are undeniably taking over the mobile device market, and this trend is not expected to slow down in the near future. Table \\ref{tab:deviceShipments} gives the number of mobile devices shipped worldwide in 2012 based on data from Canalys~\\cite{bib:canalysMobileMarket13}. The data shows that smartphones and tablets represented $42\\%$ of all mobile devices shipped in 2012, and their combined market share is expected to grow to $66\\%$ by 2016. The significance of these numbers is highlighted by the fact that among all consumer devices, only the television has had such a fast market penetration rate in the USA~\\cite{bib:degustaTech12}. Furthermore, while smartphones represented only $18\\%$ of total global mobile phones in use in 2012, they were responsible for $92\\%$ of total global mobile phone traffic~\\cite{bib:ciscoMobileForecast13}. With both the device market share and the amount of data traffic due to smart devices expected to continue to grow significantly, their importance is evident in the current and future mobile landscape.\n\n\\setlength{\\tabcolsep}{8pt}\n\n\\begin{table}[tbp]\n\n\t\\caption{The number of mobile devices shipped worldwide in 2012 and forecast for 2016, in millions of units, according to Canalys (Feb. 2013) \\cite{bib:canalysMobileMarket13}.}\n\t\\label{tab:deviceShipments}\n\t\\centering\n\t\\begin{tabular}{r l l l}\n\t\t\\toprule\n\t\tDevice type\t\t& 2012 shipments & 2016 shipments & 2012--2016 CAGR\\\\\n\t\t\\midrule\n\t\tBasic phone\t\t& $122.0$ & $58.0$ & $-17.0\\%$\\\\\n\t\tFeature phone\t& $770.8$ & $660.9$ & $-3.8\\%$\\\\\n\t\tSmartphone\t\t& $694.8$ & $1,342.5$ & $17.9\\%$\\\\\n\t\tTablet\t\t\t& $114.6$ & $383.5$ & $35.3\\%$\\\\\n\t\tNotebook\t\t\t& $215.7$ & $169.1$ & $-5.9\\%$\\\\\n\t\tNetbook\t\t\t& $18.3$ & $0.3$ & $-64.2\\%$\\\\\n\t\t\\midrule\n\t\tTotal \t\t\t& $1,936.2$ & $2,614.2$ & $7.8\\%$\\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\\end{table}\n\n\\begin{figure*}[tbp]\n\t\\centering\n\t\\begin{subfigure}[b]{0.45\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth, trim = 0cm -1.6cm 0cm 0cm]{.\/figures\/malwareNumbers.eps}\n\t\t\\caption{The number of detected Android malware in 2012. The data shows a rapid increase in malware, especially in Q3-Q4 2012, which was due to aggressive adware.}\n\t\t\\label{fig:androidMalware}\n\t\\end{subfigure} \\hspace{0.5cm}\n\t\\begin{subfigure}[b]{0.45\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{.\/figures\/malwareTypes.eps}\n\t\t\\caption{The distribution of malware types for the top ten Android malware families in 2012}\n\t\t\\label{fig:malwareTypes}\n\t\\end{subfigure}\n\t\\caption{Android malware in 2012, based on data provided by TrendLabs~\\cite{bib:trendmicroMobileThreats12}}\n\t\\label{fig:malware}\n\\end{figure*}\n\nSmart devices increasingly store and provide access to a range of personal, corporate, financial and security-related data. While the availability and ease of access to such data via different types of network connections (e.g.~\\mbox{Wi-Fi}, cellular), the ability to download, install and use mobile applications, and the practicality of using networked services while on the go make the smart mobile device an inseparable companion to the modern man, they also provide the perfect breeding ground for malicious software. Cyber-criminals have not had continuous access to such varied sources of personal and financial data before the advent of the smartphone. Increasing use of mobile payments presents a new area to exploit, and attackers have already devised covert ways to make direct financial gain from the mobile users, e.g. via premium calls and SMS~\\cite{bib:feltSurveyMobileMalware11}. Furthermore, the use of multiple communication technologies in smart devices allows attackers to cross service boundaries. For example, an attack carried out over \\mbox{Wi-Fi} would allow the attacker to launch attacks over the mobile phone network. In addition to exposing the mobile user to heterogeneous attack vectors, the increasing use of complementary access to the mobile core network, via \\mbox{Wi-Fi} and femtocells, introduces new vulnerabilities to the core network~\\cite{bib:goldeFemtocell12}. The amount of data carried over complementary access should not be underestimated; according to Cisco~\\cite{bib:ciscoMobileForecast13}, $429$ petabytes per month of global mobile data was offloaded onto the fixed network through \\mbox{Wi-Fi} or femtocell radio access, which represents $33\\%$ of total mobile traffic in 2012. Cisco also estimates that mobile data offload will reach $9.6$ exabytes per month by 2017, which will be $46\\%$ of total mobile traffic.\n\nSmart mobile devices are also increasingly at the center of security systems for managing small or large emergencies in built environments, or during sports or entertainment events~\\cite{bib:gelenbeICCCN12,bib:gelenbeWuSimEvac12}, and they are also used increasingly for online search of difficult-to-get sensitive information~\\cite{bib:gelenbeSearch10,bib:abdelrahmanSearch13}. Thus such technologies will necessarily be targeted and may be breached in conjunction with other physical or cyber attacks, as a means of disrupting safety and confidentiality of individuals and emergency responders~\\cite{bib:gorbilANT11,bib:gorbilISCIS11,bib:gorbilPernem13}.\n\nIn order to address the growing mobile threat, there is an urgent need to detect, analyze and understand the new vulnerabilities and threats in the smart mobile ecosystem, which are a result of the evolution of mobile networks and smart devices, the changing way users interact with technology, the popularity of smart devices, and the heterogeneity of the wireless interfaces, supported platforms and offered services. In order to advance in the fast moving field of cyber-security and to counter existing and potential mobile threats, we need to be proactive and work on predicting threats and vulnerabilities to build our defenses before threats materialize. To this purpose, the EU FP7 NEMESYS project\\footnote{\\url{http:\/\/www.nemesys-project.eu\/nemesys\/}} will develop a novel security framework to gather and analyze information about the nature of cyber-attacks targeting mobile devices and the mobile core network, as well as identify and predict abnormal behaviour observed on smart mobile devices and the mobile network so that appropriate countermeasures can be taken. We aim to understand the modus operandi of cyber-criminals, and to identify and reveal the possible shift in the way they launch attacks against mobile devices through root cause analysis and correlation of new findings with known patterns of attacks on wireline networks.\n\n\\section{The Current Mobile Threat Landscape}\n\\label{sec:landscape}\n\nSmart devices are open to both traditional and mobile-specific threats due to the multiple roles smart mobile devices play and the heterogeneity of mobile communication technologies and networked services~\\cite{bib:becherMobileSecurity11}. Among the traditional threats that smart mobile devices face, we include physical attacks that require physical access to the device, device-independent attacks such as eavesdropping on the wireless medium or man-in-the-middle attacks, e-mail-based spam and phishing, and IP-based attacks. Current IP-based attacks encountered on mobile devices~\\cite{bib:wahlischMobileHoneypot13} have been found to be largely similar to those on non-mobile devices~\\cite{bib:gelenbeLoukasDoS07,bib:gelenbeSelfAware09}, but we are more interested in the traits of attacks that are tailored specifically for mobile devices. With the growing popularity of smart devices, mobile-specific threats have evolved from SMS\/MMS-based denial-of-service (DoS) attacks~\\cite{bib:traynorSMSAttack09,bib:mullinerSMSAttack11} to more sophisticated attacks that usually come in the form of malware and target both the core network and the mobile users. The ability of smart devices to install and run applications not only from official markets but also from unknown sources exposes them to malware~\\cite{bib:feltSurveyMobileMalware11,bib:zhouAndroidMalware12}, and while the mobile malware threat is not new~\\cite{bib:dagonMobileVirus04}, it is clearly evolving and growing as attackers experiment with new business models by targeting mobile users~\\cite{bib:lookoutMobileSecurity12,bib:kasperskyMalwareEvolution12}. We next provide a taxonomy of mobile malware based on behavioral classification, and present an overview of malware detection techniques.\n\n\\subsection{Mobile Malware}\n\nAndroid has been the most targeted mobile platform in 2012, with almost $99\\%$ of all encountered mobile malware being designed for Android~\\cite{bib:kasperskyStatistics12}. The number of malicious Android applications detected by Kaspersky Lab in 2012 was more than $35,000$, which reflects a six-fold increase from 2011~\\cite{bib:kasperskyMalwareEvolution12}. Figure \\ref{fig:androidMalware} shows the rapid growth of Android malware in 2012 based on data from TrendLabs~\\cite{bib:trendmicroMobileThreats12}, which shows the significance of the growing mobile malware threat. 2012 has also seen the emergence of the first mobile botnets~\\cite{bib:kasperskyStatistics12}. A botnet is a collection of Internet-connected devices acting together to perform tasks, often under the control of a command and control server. In wireline networks, malicious botnets are used to generate various forms of spam, phishing, and distributed denial-of-service (DDoS) attacks. Mobile botnets extend such capability to cellular networks, give cyber-criminals the advantages of control and adaptability, and pose a significant threat to the mobile core network as they could be used to launch debilitating signaling-based DDoS attacks~\\cite{bib:leeDetectionDoS3G09,bib:traynorCellularBotnet09,bib:mullineriBot10,bib:mullinerSignaling12}.\n\nMobile malware uses various infection vectors in order to gain access to the device; the top two main categories of attacks based on the vulnerabilities they use are:\n\\begin{itemize}\n  \\item \\textit{Exploiting} hardware or software vulnerabilities of devices to completely bypass the user and instal malware. Some of the exploits used to  attack smart devices are near field communication (NFC) technology, third-party kernel drivers, Android firmware vulnerabilities, and mobile web browsers. For example, some malicious websites use mobile browser exploits to install malware on the device without any user interaction other than visiting the site. Android firmware and third-party driver exploits have been used by malware to elevate their privileges and thus gain root access to the device, allowing them to practically do anything they want without the user's knowledge~\\cite{bib:liebergeldAndroidSecurity13}.\n  \n  \\item \\textit{Social engineering} is by far the most common method used to infect smart mobile devices, where users are ``tricked'' into installing the malware themselves. Social engineering includes all techniques that exploit the human user, such as phishing, application repackaging, etc., in order to infect the device. Social engineering is popular since it does not require any technical investment by the attacker, i.e.~the identification of a new exploit and the development of a delivery system that uses it. Upcoming malware will continue to employ social engineering in new ways; for example, we have already witnessed the first malicious QR codes, which need to be scanned by the user for their activation.\n\\end{itemize}\nIndependently of how it infects the device, once the malware is installed, it performs one or more malicious activities, which are classified in Table \\ref{tab:malwareBehaviour} according to their behavior. Figure \\ref{fig:malwareTypes} shows the distribution of malware types for the top ten Android malware families in 2012, based on data from TrendLabs~\\cite{bib:trendmicroMobileThreats12}. The data shows that some malware families exhibit multiple malicious activities (e.g. both premimum service abuse and data stealing), and that premium service abuse has been the favorite malware type, most probably because it is simple to create and allows a direct source of revenue for the cyber-criminal. It is closely followed by adware, which generates indirect profit through advertisement fraud. Malicious downloaders appear to be popular malware delivery methods since they can evade detection by malware detectors and do not alarm the user at installation time by requesting many high-level privileges.\n\n\\setlength{\\tabcolsep}{8pt}\n\\renewcommand{\\arraystretch}{1.5}\n\n\\begin{table}[tbp]\n\n\t\\caption{Malware behavioral classification.}\n\t\\label{tab:malwareBehaviour}\n\t\\centering\n\t\\begin{tabular}{l p{5cm}}\n\t\t\\toprule\n\t\tActivity & Description\\\\\n\t\t\\midrule\n\t\tStealing user information\t& Steals user information and credentials. The most commonly targeted data are the contact list, IMEI and IMSI numbers, API authentication keys, banking credentials, user's location, network operator, phone ID and model, phone number, and text messages.\\\\\n\t\tMonitoring\t\t\t\t\t& Tracks user's location, records conversations, takes photos. Spyware is considered to exhibit this type of behavior.\\\\\n\t\tAdware\t\t\t\t\t\t& Presents unwanted advertisements to the user. Most mobile adware have evolved to incorporate other types of behavior, such as monitoring user activity, especially browsing behavior, and stealing user information. Malicious advertisement networks are increasingly finding their way into legitimate applications and being used as infection vectors.\\\\\n\t\tPremimum service abuse\t\t& Sends premium SMS\/MMS, makes premium calls, subscribes the user to premium services without the user's knowledge. Cyber-criminals often make direct financial gain from such premium service abuse.\\\\\n\t\tClick fraud\t\t\t\t\t& Generates ``clicks'' on ads shown on websites and applications, generating indirect financial gain to the cyber-criminal through payment from the advertisement networks.\\\\\n\t\tSearch engine optimization\t& Manipulates the search results shown on the mobile browser and other applications to improve website rankings in search engine results.\\\\\n\t\tSMS and e-mail spam\t\t\t& Sends spam SMS and e-mail either to the user's contacts or to a specified list of people. Could be used for phishing attacks.\\\\\n\t\tMalicious downloading\t\t& Downloads malicious content onto the device. Mainly used to evade detection by malware detectors and the user.\\\\\n\t\tBotclient\t\t\t\t\t& Turns the device into a botclient that receives commands from a remote command and control server. Once part of a mobile botnet, the device can be used to launch a variety of attacks, ranging from spam to DDoS attacks on the network.\\\\\n\t\tRooting\t\t\t\t\t\t& Roots the device to allow execution of otherwise restricted commands and programs. Malware that has root access potentially has full control of the device.\\\\\n\t\tRansom\t\t\t\t\t\t& Locks the device and demands a ransom to be paid in order to unlock the device.\\\\\n\t\tDestruction\t\t\t\t\t& Causes physical damage by deleting important files or personal information.\\\\\n\t\tDenial-of-service\t\t\t& Launches a denial-of-service attack either on the mobile device itself or on the core network. The mobile device may be attacked by repeatedly switching the device off or depleting the battery.\\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\\end{table}\n\n\\subsection{Malware Detection Techniques}\n\nThe traditional approach to malware detection is \\textit{signature-based}, where signatures for identified malware are extracted and used to detect new infections of the same malware on other devices. While this can be very effective for controlling an outbreak, it cannot defend against malware unless samples have already been obtained and analyzed. Therefore, new and unknown malware cannot be detected via signature-based approaches. In order to detect previously unknown malware, \\textit{behavioral methods} are useful in which the activities of an application are analyzed via either static or dynamic analysis. \\textit{Static analysis} provides a quick and efficient way to detect malware without executing them, but they are ultimately limited in their effectiveness unless the source code for the application is available. Unlike static analysis, \\textit{dynamic analysis methods} execute the application code in an isolated environment, for example a virtual machine or a sandbox, so its behavior can be directly observed. Dynamic analysis techniques include function call monitoring, function parameter analysis, information flow tracking, and instruction tracing. An overview of mobile malware detection methods is presented in~\\cite{bib:chandraMobileMalware12}, and a more comprehensive survey is given by~\\cite{bib:egeleMalwareAnalysis12}. In the rest of this section, we will only consider behavior-based detection methods.\n\nBased on where the detection is performed, we can classify malware detection methods into three categories. \\textit{Client-side detectors} reside in the mobile device, but are constrained by its limited physical resources, especially battery. \\textit{Network-side detection} methods analyze mobile network traffic from many users and offer a complementary means for detecting attacks targeting mobile users. They can be used in conjunction with client-side methods to improve detection rates, and provide a broad view of malicious activities within a carrier's network, enabling detection of anomalous behavior that would not be visible on a per user basis. However, network-based methods are limited in that they can only monitor and analyze mobile traffic that goes through the cellular network, and therefore may not be suitable for certain malware types, such as malware that exclusively uses \\mbox{Wi-Fi} for communication. \\textit{Cloud-based detection} offers a trade-off between network-level analysis and on-device security by offloading intensive security analysis and computations to the cloud while monitoring internal mobile device events as well as different types of wireless communications from many users. However, cloud-based solutions can only protect those users that install the application and require a large number of subscribers in order to identify large-scale events, while network-based detection does not require the user to do anything as all detection is performed using data available to the network operator.\n\n\\section{The NEMESYS Approach}\n\\label{sec:nemesys}\n\nDespite recent advances in mobile security, large gaps remain in our understanding of the new and future threats and vulnerabilities in the smart mobile ecosystem. Our initial research has identified the following open issues with respect to the general problem of cyber threats against smart mobile devices:\n\n\\textit{- Missing infrastructure for collecting attack traces.} Without an infrastructure to collect attack traces against mobile devices, we will not be able to detect, analyze and understand the evolving attack strategies employed by cyber-criminals. These are crucial in order to develop effective mitigation strategies and to provide seamless services in the smart mobile ecosystem.\n\n\\textit{- Virtualization.} By leveraging advances in mobile virtualization technology, we can protect mobile users from the consequences of malware by appropriately restricting access to device functionality. Virtualization would also allow an easy way to reset mobile devices to a state before infection.\n\n\\textit{- Mobile honeypot development.} Honeypots are successfully used in wired networks in order to study the strategies of attackers and to protect production systems from attacks. However, the development of mobile honeypots is still at an early stage. For example, the mobile honeypot presented in~\\cite{bib:songMobileHoneypot12} can connect to the Internet only over Wi-Fi or a wired connection. Although the mobile honeypot presented in~\\cite{bib:wahlischMobileHoneypot13} can connect to a UMTS network via a USB dongle, the honeypot uses desktop computers to emulate mobile devices. We aim to address the deficiencies of these early efforts by developing a high-interaction mobile honeypot using real mobile devices.\n\n\\textit{- New potential for exploiting security vulnerabilities through mobile botnets.} While botnets are a well-known phenomenon in the wired Internet, we have just witnessed the first mobile botnets in 2012. Mobile botnets pose interesting questions as to their capabilities and uses since smart mobile devices possess many abilities not present on a desktop computer. Such mobile botnets could be used to attack mobile users (e.g. SMS spam) and the core network (e.g. DDoS attack); the ability of a large number of mobile devices to effectively attack the core network has been demonstrated in recent cases of legitimate but poorly-designed mobile applications. We need to explore the new threats posed by the emergence of mobile botnets in more detail.\n\n\\textit{- Adaptability of cyber-criminals and rapidly changing cyber-crime tactics.} Cyber-criminals have become adept at modifying their strategies and tactics as methods are developed to counter their activities. Thus, security solutions should rely less on signatures and instead adopt other forms of detection.\n\n\\textit{- New types of attacks that cross service, platform and network boundaries.} Identification of anomalous behavior within a large set of heterogeneous data is difficult and time-consuming, particularly across layers. Statistical analysis is further challenged by rare anomalous events in massive amounts of data.\n\n\\textit{- Attack attribution and understanding of the cyber-criminals' modus operandi.} The selection of the best mitigation strategy requires understanding of new phenomena and recognizing changes in how the malicious actors operate. This requires that attacks are analyzed in a detailed way, in order to ``attribute'' responsibility to an exact attacker or to protect the true targets.\n\nIn the EU FP7 NEMESYS project, we are developing a data collection, visualization and analysis infrastructure (Fig.~\\ref{fig:architecture}) in order to address these open issues. The core of this architecture consists of a data collection infrastructure that incorporates high-interaction virtualized mobile honeypots and honeyclients in order to gather data regarding mobile attacks. The collected data is enriched by the data collection infrastructure through interaction with external sources and made available to anomaly detection, visualization and analysis modules running on the mobile network operator's site. The purpose of the anomaly detection mechanisms is to detect deviations from normal behaviour of mobile users and the core network in real-time. These mechanisms will utilize charging data records (CDRs) for the users and control-plane protocol data, combined with enriched mobile attack traces made available by the data collection infrastructure. In addition to monitoring abnormal behaviour of users connected to the core network through the radio access network, the architecture contains a module that performs anomaly detection within the femtocell complementary access system.\n\n\\begin{figure*}[tbp]\n\t\\centering\n\t\\includegraphics[width=0.8\\linewidth]{.\/figures\/architecture2.eps}\n\t\\caption{The NEMESYS architecture}\n\t\\label{fig:architecture}\n\\end{figure*}\n\nEnriched attack traces and normal behaviour statistics from the data collection infrastructure and the output of the anomaly detection module are fed into the visualization and analysis module. The visualization and analysis module will aid in the detection of existing and emerging threats in the mobile ecosystem through attack attribution, root cause identification, and correlation of observed mobile attacks with known attack patterns. It will present large sets of data related to the observed attacks from heterogeneous sources and facilitate the role of the security analyst in reasoning, hypothesis testing and decision making. We describe the components of the NEMESYS architecture in the following sections in more detail.\n\n\\subsection{Virtualized Mobile Client Honeypot}\n\\label{sec:honeypot}\n\nHoneypots are networked computer system elements that are designed to be attacked and compromised so we can learn about the methods employed by the attackers~\\cite{bib:provosHoneypot07}. Traditional honeypots are servers that passively wait to be attacked, whereas client honeypots are security devices that actively search for malware, compromised websites and other forms of attacks. High-interaction client honeypots are fully functional, realistic client systems and they generally do not impose any limitations on the attacker other than those required for containing the attack within the compromised system. Despite their complexity and maintenance difficulty, high-interaction client honeypots are effective at detecting unknown attacks and they are harder to detect by the attacker~\\cite{bib:provosHoneypot07}. They also enable in-depth analysis of the attacks during and after the attack has taken place.\n\nAs part of NEMESYS, we are developing a high-interaction virtualized client honeypot for the Android mobile platform in order to attract and collect mobile attack traces. We have chosen Android considering its popularity among mobile users and the extremely high ratio of malware targeting Android. Our virtualization technology logically partitions the physical device into two virtual machines (VMs): the \\textit{honeypot VM} and the \\textit{infrastructure VM}. The honeypot VM will host the largely unmodified mobile device operating system, and it will not have direct access to the device's communication hardware. The infrastructure VM will mediate all access to the communication hardware, and employ sensors to wiretap any communication and detect suspicious behaviour. It will also provide the event monitoring, logging and filesystem snapshot facilities, as well as transmit threat information to the NEMESYS data collection infrastructure. It will host a lightweight malware detection module in order to identify malicious applications running on the honeypot VM. For this purpose, both signature-based and behaviour-based approaches will be considered. In order to improve the efficiency of malware detection, we will identify and prioritize the most important attributes in the system state space to monitor.\nOur virtualization technology will ensure that an attack is confined within the compromised device and that it will not put other devices in the network at risk. Furthermore, through this approach, we will be able to stop malware from abusing premium services and from subscribing the user to services without her knowledge. Thus, the user will be spared from any financial distress that may arise as a result of using the mobile honeypot. The virtualization solution also enables taking full snapshots of the honeypot VM filesystem for further forensic analysis of an attack, as well as improving honeypot maintenance since a compromised honeypot could be restored more quickly.\n\nOur initial research has shown that the infection vector of most mobile malware is social engineering, where users are ``tricked'' into installing the malware themselves. This observation has led us to the conclusion that the user should not be ignored in the construction of an effective mobile honeypot. To this end, we introduce the \\textit{nomadic honeypot} concept, which utilizes real smartphone hardware with the virtualization solution that will be developed within NEMESYS~\\cite{bib:liebergeldHoneypot13}. We plan to deploy nomadic honeypots by handing them out to a chosen group of volunteers, who will use the honeypot as their primary mobile device. It will be up to these human users to get the honeypot infected by visiting malicious sites, installing dubious applications, and so forth. Traces from malware and other types of mobile attacks collected and identified through the nomadic honeypots will be provided to the data collection infrastructure, which is described next.\n\n\\subsection{Data Collection Infrastructure}\n\\label{sec:dataCollection}\n\nThe data collection infrastructure will gather and store mobile attack traces that will be provided by the virtualized mobile client honeypot and the honeyclient, and combine them with data from the mobile core network and external sources for enrichment, correlation analysis, and visualization. As an initial step in the design of this infrastructure, we are identifying available external data sources relating to wireline network attacks which will enable correlation of data from multiple heterogeneous sources. Examples of such data sources are the SGNET~\\cite{bib:leitaSgnet08}, HARMUR~\\cite{bib:leitaHarmur11}, and VirusTotal databases. A source aggregator is being designed and developed to harvest and consolidate data from these sources and the NEMESYS mobile honeypot in a scalable database. Scalable design of the database is important in order to be able to efficiently store and handle large heterogeneous data sets.\nOnce data from multiple sources have been consolidated, they will be enriched by analyzing the data itself or accessing external sources. For example, TCP\/IP stack fingerprinting in order to identify the remote machine's operating system, and clustering of the traces are passive methods of data enrichment. On the other hand, DNS reverse name lookup, route tracing, autonomous system identification, and geo-localization are methods to improve characterization of remote servers but these functions may require access to external sources, possibly in real time. As a final step, the data collection infrastructure will help in the definition of the appropriate inputs representing normal and malicious network activity, which will then be used as the fundamental representation of information in the visualization and analysis module.\n\nThe \\textit{honeyclient}~\\cite{bib:delosieresMalwareDetection13} being developed as part of the data collection infrastructure is similar in concept to the virtualized mobile honeypot, but instead of using real hardware and being driven by real users, the honeyclient uses an Android emulator driven by artificially generated user input to automate interaction with web sites, application markets and applications in order to collect mobile attack traces. The honeyclient consists of a crawler, client, and detector components. The crawler discovers web sites, application markets and applications of interest and generates a list of web pages to visit and applications to download. The client runs the Android emulator (e.g. on a desktop computer) and processes the list generated by the crawler, visiting the web sites using a mobile browser and downloading, installing and executing applications. The behavior of the applications, e.g. function calls, and changes to the system as a result of executing the applications are recorded by the client, which are used by the malware detector to identify malware. The honeyclient provides data relating to identified malware and malicious web sites to the data collection infrastructure.\n\n\\subsection{Anomaly Detection Using Control Plane and Billing Data}\n\\label{sec:anomalyDetection}\n\nThe anomaly detection module that operates at the mobile network operator's site is used for the identification and prediction of abnormal behavior observed on smart mobile devices and the mobile network. In addition to user-oriented attacks, mobile networks are vulnerable to a novel DoS attack called the signaling attack~\\cite{bib:leeDetectionDoS3G09}. Signaling attacks seek to overload the control plane of the mobile network using low-rate, low-volume attack traffic, based on the structure and characteristics of mobile networks. Unlike conventional DoS attacks that focus on the data plane, the signaling attack creates havoc in the control plane of a mobile network by repeatedly triggering radio channel allocations and revocations. In order to identify such DoS attacks against the mobile network and attacks against the mobile users in real time, we will use signaling data from control-plane protocols and sanitized (anonymized) CDRs from mobile users, respectively. For this purpose, we will use normal user behavior statistics, as well as synthetic ``typical'' user playbacks, to create traces of signaling events and billing data so as to characterize and extract their principal statistics such as frequencies, correlations, times between events, and possible temporal tendencies over short (milliseconds to seconds) and long (hours to days) intervals. We will then employ Bayesian techniques such as maximum likelihood detection, neuronal techniques based on learning, and a combination of these two in order to design and develop robust and accurate change detection algorithms to detect the presence of an attack, and classification algorithms to identify the type of attack when it is detected with high confidence.\n\nNovel femtocell architectures provide a specific opportunity for user-end observation of network usage, while they also have specifics for attacks within the femtocells~\\cite{bib:borgaonkarSecurityFemtocell11}. To address attacks specific to femtocells, we will conduct a survey and evaluation of how users may be monitored and attacks detected within a femtocell, and how these are linked to overall mobile network events.\n\nA number of novel ideas are also being investigated~\\cite{bib:abdelrahmanAnomalyDetection13} such as modeling the signaling and billing network as a queueing network~\\cite{bib:gelenbeMuntzProb76,bib:GelenbeActa} to capture the main events that involve hundreds of thousands of mobile calls and interactions, while only a few may be subject to an intrusion or attack at any given time. Detection of anomalies is studied using learning with neural networks~\\cite{bib:gelenbeRNN99,bib:gelenbeNatural12} that provide fast low-order polynomial detection complexity required for massive real-time data, and the need to detect and respond to threats in real-time. Such techniques can also benefit from distributed task decomposition and execution for greater efficiency~\\cite{bib:aguilarTask97}.\nOur analytical models and anomaly detection algorithms will be augmented and validated with simulation tools. As an initial step, we are developing realistic simulations of UMTS and LTE networks using the OPNET simulator in order to extract data regarding control-plane events that take place during normal mobile communications. Characteristics of these control events will be used to drive the development of our analytical models. We will later conduct large-scale mobile network simulations to validate our mathematical results. Another set of simulations will focus on user-level events, such as voice calls and packet communications, and include charging system components to monitor the use of internal and external network resources. Such simulations will be used to test the performance of our real-time and offline anomaly detection methods.\n\n\\subsection{Root Cause Analysis, Correlation and Visualization}\n\\label{sec:analysisVisualization}\n\nThe role of the visualization and analysis module is to process the data obtained from the data collection infrastructure and the anomaly detection module in order to identify and reveal correlations between network events, and to provide a visual analytics framework for the security analyst to perform hypothesis formulation and testing. The data provided to this module represents a large and heterogeneous data set that needs to be presented in a meaningful way to the operator without overwhelming her or restricting available views and actions on the data. In addition to mere representation of data, the visualization and analysis module aims to provide visual analytics tools to the operator. This task is compounded by different uses of visualization by the operator: (i) real-time monitoring of the status of users and the mobile network, and (ii) exploratory data analysis. For real-time monitoring, the security status of a large set of mobile users and more importantly the mobile network need to be presented to the operator. This includes providing early alerts for abnormal behaviour, DoS attacks, malware spreading among the users of the mobile network, etc. The analytics module must also provide visual analytics tools so the analyst can perform attack attribution and correlation analysis with the help of algorithms running in the background.\n\nIn order to effectively visualize and explore large sets of heterogeneous, dynamic, complex data, it is necessary to create multiple coordinated views of the data that allow a multi-faceted perception and the discovery of any hidden attributes. The analysis methods also need to be scalable for early network alerts and fast access to the underlying data. We will therefore focus on enabling a real-time analysis framework by means of incremental analysis and visualization methods~\\cite{bib:papaVisualNetwork13}, such as multi-level hierarchical screen visualizations that update smoothly rather than showing abrupt changes.\n\n\\subsection{Integration and Validation}\n\nIn order to evaluate and validate the technologies that are being developed and to demonstrate their impact to interested parties, NEMESYS will construct a virtual testing environment based on guidelines provided by our industrial partners that is as close to a real mobile network as possible within feasibility limitations. The different modules being developed by various partners will be integrated in the virtual testing environment, and validation tests will be conducted based on realistic use-cases. We aim to use the OPNET simulator as part of the virtual testing environment in order to conduct simulations of different types of mobile networks, e.g. UMTS and LTE, and to drive the large-scale networking experiments.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nThe evolving and growing nature of the mobile threat in the smart mobile ecosystem is evident from the increasing marketshare of smartphones and tablets, the large amount of data due to smart devices, and the number of detected mobile malware. We must therefore address the mobile threat and understand the new and potential vulnerabilities, threats, and operating methods of cyber-criminals. NEMESYS will provide new insight into the nature of next generation network security in the smart mobile ecosystem. The main innovation of NEMESYS is the research and development of novel security technologies for the identification and prediction of abnormal behavior observed on smart mobile devices, as well as for gathering and analyzing information about the nature of cyber-attacks targeting mobile devices, so that appropriate countermeasures can be taken to combat them. It will involve the development of virtualized honeypots for mobile devices, a data collection infrastructure, and the introduction of novel attack attribution and visual analytics technologies for the mining, presentation and representation of large amounts of heterogeneous data that are related to the smart mobile ecosystem.\n\n\\section*{Acknowledgments}\n\nThe work presented in this paper was supported by the EU FP7 collaborative research project NEMESYS (Enhanced Network Security for Seamless Service Provisioning in the Smart Mobile Ecosystem), under grant agreement no. 317888 within the FP7-ICT-2011.1.4 Trustworthy ICT domain.\n\n\n\\IEEEtriggeratref{27}\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Intro}\nWe extend the stability theorem of the depolarizing channel \nto the output quantum $p$-R\\'{e}nyi entropy \nfor $p \\ge 2$ or $p=1$. \nThe original stability theorem with the output purity is essentially equivalent to our stability theorem for the case $p=2$ \nand was used in proving the equality \n$\\mathrm{QMA}(k)=\\mathrm{QMA}(2)$ for all $k \\ge 2$~\\cite{HM13}. \nWe generalize it to the output quantum $p$-R\\'{e}nyi entropy \nto create a more powerful tool,\nand we apply it to a type of polygraph test as discussed below.\nGeneralization is accomplished by defining the notion of stability of a quantum channel with respect to any real valued continuous function.\nThat is, if a state is close to achieving \nthe minimal\/maximal output value of a particular quantity (entropy function) through the channel,\nthen it must be close to an input state giving the minimal\/maximal value.\nIn particular, we show that the depolarizing channel is stable \nwith respect to the output quantum R\\'{e}nyi entropy.\n\nOur theorem is constructed by generalizing the Taylor expansion of von~Neumann entropy~\\cite{GF13} to the quantum R\\'{e}nyi entropy.\nWhereas the original work employed the output purity~\\cite{HM13},\nwhich is a relatively simpler function,\nwe use a more general (and complicated) function,\nnamely the output quantum R\\'{e}nyi entropy.\nThe Taylor expansion of the output quantum R\\'{e}nyi entropy is the technique we use\nto prove the stability theorem for a depolarizing channel with respect to the the output quantum R\\'{e}nyi entropy.\nThe protocol is described in \\S\\ref{sec:Poly} \nand provides us meaning and intuition for \nthe stability theorem of the depolarizing channel.\nFurthermore, the protocol shows that our stability theorem has a benefit \nas our protocol has a smaller undecidable gap than the original case. \n\nWe organize our paper as follows. \nIn \\S\\ref{sec:Stable}, we provide some notions to define a stable channel clearly.\nOur main result appears in \\S\\ref{sec:Stability}\nwhere we generalize the Taylor expansion of the von~Neumann entropy \nto calculate the Taylor expansions of the quantum R\\'{e}nyi entropies.\nWe use this result to show that the depolarizing channel is stable with respect to the output quantum $p$-R\\'{e}nyi entropies for $p \\ge 2$ or $p=1$. \nIn \\S\\ref{sec:Poly}, \nwe introduce a polygraph test as an application of our stability theorem. \nFinally, in \\S\\ref{sec:Conclusions}, we conclude with discussion on our results.\n\n\n\\section{Stable channels}\n\\label{sec:Stable}\nIn this section, we define the notions of \na {\\em quantity}, an {\\em extremal} state, an {\\em~$\\epsilon$-almost extremal} state, \nan {\\em~$\\epsilon$-stable} channel, and a {\\em stable} channel. \nExcept for this section, in the rest of the paper, we will be using these notions for only \nthe depolarizing channel and the quantum R\\'{e}nyi entropy as a quantity.\n\n\\begin{definition}\nLet \n$\\mathcal{E}: \\mathcal{B}(\\mathcal{H}_{\\text i}) \\rightarrow \\mathcal{B}(\\mathcal{H}_{\\text o})$ \nbe a quantum channel (i.e., a trace preserving completely positive map) and let $Q$ be a real-valued continuous function on $\\mathcal{B}(\\mathcal{H}_{\\text o})$, \nwhere $\\mathcal{B}(\\mathcal{H}_{\\text i})$ and $\\mathcal{B}(\\mathcal{H}_{\\text o})$ are the sets of all states in the input space $\\mathcal{H}_{\\text i}$ and the output space $\\mathcal{H}_{\\text o}$, respectively.\n\nFor any $\\epsilon>0$, \na state $\\sigma \\in \\mathcal{B}(\\mathcal{H}_{\\text i})$ is {\\em~$\\epsilon$-almost extremal}\nwith respect to the function $Q$ and the channel $\\mathcal{E}$ \nif\n\\begin{equation}\n\t\\left| Q\\left(\\mathcal{E}(\\sigma)\\right)-\\operatorname{ext}_\\rho Q\\left( \\mathcal{E} \\left(\\rho \\right) \\right) \\right|\n\t\t\\in O(\\epsilon),\n\\end{equation} \nwhere the extremal value, $\\operatorname{ext}_\\rho$, refers to \neither the maximal value or the minimal value of $Q$ over all states $\\rho$ in $\\mathcal{B}(\\mathcal{H}_{\\text i})$ \naccording to a given quantity.\nA state $\\sigma_0$ is said to be {\\em extremal} \nwith respect to $Q$ and $\\mathcal{E}$ if \n\\begin{equation}\n\tQ\\left(\\mathcal{E}(\\sigma_0)\\right)=\\operatorname{ext}_\\rho Q \\left( \\mathcal{E} \\left(\\rho\\right) \\right).\n\\end{equation}\n\\end{definition}\nNow we define a stable channel with respect to the function $Q$.\n\\begin{definition}\nFor a given $\\epsilon>0$, a channel $\\mathcal{E}$ is {\\em~$\\epsilon$-stable} \nwith respect to a quantity $Q$\nif, for all $\\sigma$~$\\epsilon$-almost extremal, \nan extremal state $\\sigma_0$ exists with respect to $Q$ and $\\mathcal{E}$ such that\n\\begin{equation}\n\t\\left\\| \\sigma- \\sigma_0 \\right\\|^2_1 \\in O(\\epsilon),\n\\end{equation}\nwhere $\\left\\| \\cdot \\right\\|_1$ denotes the trace norm.\nA channel $\\mathcal{E}$ is {\\em stable} with respect to a quantity $Q$\nif it is~$\\epsilon$-stable with respect to the quantity $Q$ \nfor all $\\epsilon>0$.\n\\end{definition}\n\nWe have provided some generalized definitions to establish the notion of a stable channel.\nIn the next section, as our main result, we present the stability theorem of the depolarizing channel for the output quantum R\\'{e}nyi entropy and prove that the depolarizing channel is stable with respect to the quantum R\\'{e}nyi entropy.\n \n\n\\section{Stability of the depolarizing channel}\n\\label{sec:Stability}\nIn this subsection, we present and prove the stability theorem of the depolarizing channel with respect to the output quantum R\\'{e}nyi entropy. This section consists of two subsections. In the first subsection, we evaluate the Taylor expansion of the quantum R\\'{e}nyi entropy which is crucial to prove our main theorem in the second subsection.\n\n\n\\subsection{The Taylor expansion of the quantum $p$-R\\'{e}nyi entropy}\n\\label{subsec:Taylor}\nIn this subsection, \nwe the Taylor expansion technique for the von~Neumann entropy~\\cite{GF13}\nto calculate the Taylor expansion of the quantum $p$-R\\'{e}nyi entropy.\nThis technique is key to prove the stability theorem of the depolarizing channel for the output quantum $p$-R\\'{e}nyi entropy.\n\nFor $p > 0$ ($p\\neq1$),\nthe quantum $p$-R\\'{e}nyi entropy~\\cite{H3} of a state~$\\rho$ is\n\\begin{equation}\n\tS_p\\left(\\rho\\right):=\\frac{1}{1-p} \\log\\operatorname{Tr}\\rho^{p}.\n\\label{eq:Renyi}\n\\end{equation}\nThe minimal output quantum $p$-R\\'{e}nyi entropy of a quantum channel $\\mathcal{E}$ \nis defined as\n\\begin{equation}\n\tS_p^{\\min}\\left(\\mathcal{E}\\right):=\\min_\\rho S_p\\left(\\mathcal{E}(\\rho)\\right),\n\\label{eq:minRenyi}\n\\end{equation}\nwhere the minimum is taken over all input states $\\rho$ of $\\mathcal{E}$.\nThe quantum $p$-R\\'{e}nyi entropy converges to the von~Neumann entropy as $p$ tends to one, \nand we can thus consider the quantum R\\'{e}nyi entropy \nas a generalization of the von~Neumann entropy~\\cite{H3}. \n\nIn order to obtain the Taylor expansion of the quantum R\\'{e}nyi entropy, \nwe exploit the following lemma.\n\\begin{lemma}[Gour and Friedland~\\cite{GF13}]\n\\label{lem:DividedDifference}\n\tLet $A=diag\\left(p_1,\\cdots,p_m\\right) \\in \\mathbb{C}^{m \\times m}$ be a diagonal square matrix, and $B=\\left[b_{ij} \\right] \\in \\mathbb{C}^{m \\times m}$ be a complex square matrix. \nLet $f$ be a $C^2$ function defined on a real open interval $\\left(a,b\\right)$.\nThen\n\\begin{equation}\n\tf\\left(A+tB\\right)\n\t\t=f\\left(A\\right)+t L_A\\left(B\\right)+t^2 Q_A\\left(B\\right)+O\\left(t^3\\right)\n\\label{eq:Lemma_GF}\n\\end{equation}\nfor $L_A : \\mathbb{C}^{m \\times m} \\rightarrow \\mathbb{C}^{m \\times m}$ a linear operator\nand $Q_A:\\mathbb{C}^{m \\times m} \\rightarrow \\mathbb{C}^{m \\times m} $ \na quadratic homogeneous non-commutative polynomial in $B$. \nFor $i,j=1,\\cdots,m$, we have\n\\begin{align}\n&\\left[L_A\\left(B\\right)\\right]_{ij}\n=\\Delta f\\left(p_i, p_j\\right)b_{ij}\n=\\frac{f\\left(p_i\\right)-f\\left(p_j\\right)}{p_i-p_j}b_{ij}, \n\\nonumber \\\\\n&\\left[Q_A\\left(B\\right)\\right]_{ij}\n=\\sum_{k=1}^m\\Delta^2 f\\left(p_i, p_k, p_j\\right)b_{ik}b_{kj}.\n\\label{eq:lemma_GF1}\n\\end{align}\nIn particular,\n\\begin{align}\n&\\operatorname{Tr}\\left(L_A\\left(B\\right)\\right)\n=\\sum_{j=1}^m f^\\prime\\left(p_j\\right)b_{jj}, \n\\nonumber \\\\\n&\\operatorname{Tr}\\left(Q_A\\left(B\\right)\\right)\n=\\sum_{i,j=1}^m \\frac{f^\\prime\\left(p_i\\right)-f^\\prime\\left(p_j\\right)}\n{2\\left(p_i-p_j\\right)}b_{ij}b_{ji}.\n\\label{eq:lemma_GF2}\n\\end{align}\n\\end{lemma}\nNow we use Lemma~\\ref{lem:DividedDifference} \nto calculate the Taylor expansion of $S_p\\left(\\rho\\left(t\\right)\\right)$.\n\\begin{theorem}\n\\label{thm:TER}\nA nonsingular density matrix\n\\begin{equation}\n\t\\rho\\left(t\\right)=\\rho+ t \\gamma_0+t^2 \\gamma_1+ O(t^3),\n\\end{equation}\nwith~$\\rho$ diagonal, $\\gamma_0$ all zeroes along the diagonal \nand~$\\gamma_1$ having zero trace,\nhas quantum $p$-R\\'{e}nyi entropy\n\\begin{align}\n\tS_p\\left(\\rho\\left(t\\right)\\right)\n=&S_p\\left(\\rho\\right) +\\frac{1}{1-p}t^2\\left(p\\frac{\\operatorname{Tr}\\left(\\rho^{p-1}\\gamma_1\\right)}{\\operatorname{Tr}\\left(\\rho^{p}\\right)}+\\frac{\\operatorname{Tr}\\left(Q_{\\rho}\\left(\\gamma_0\\right)\\right)}{\\operatorname{Tr}\\left(\\rho^{p}\\right)}\n\\right)+O\\left(t^3\\right).\n\\label{eq:Salphat}\n\\end{align}\n\\end{theorem}\n\\begin{remark}\nAs $p$ tends to one, \nTheorem~\\ref{thm:TER} \nimplies the Taylor expansion of the von~Neumann entropy, \nhence generalizes the von~Neumann entropy.\n\\end{remark}\n\\begin{proof}\nAs $\\rho$ is nonsingular, $\\mathds{1}-\\rho\\left(t\\right) < \\mathds{1}$ for small~$t$. \nThus, we can employ the Taylor expansion with respect to~$t$. \nFrom the following Taylor expansion\n\\begin{equation}\n\t\\rho^{p}\\left(t\\right)\n\t\t=\\left[\\mathds{1}-\\left(\\mathds{1}-\\rho\\left(t\\right)\\right)\\right]^{p}\n=\\sum_{n=0}^{\\infty} \\binom{p}{n}\\left(-1\\right)^n\\left(\\mathds{1}-\\rho\\left(t\\right)\\right)^n,\n\\label{eq:TER1}\n\\end{equation}\nwe obtain\n\\begin{equation}\n\t\\operatorname{Tr}\\left(\\rho^{p}\\left(t\\right)\\right)\n\t\t=\\sum_{n=0}^{\\infty} \\binom{p}{n}\\left(-1\\right)^n\n\\operatorname{Tr}\\left[\\left(\\mathds{1}-\\rho\\left(t\\right)\\right)^n\\right].\n\\label{eq:TER2}\n\\end{equation}\nExpanding the trace term in the right-hand side of Eq.~(\\ref{eq:TER2}) \nup to second order in~$t$ yields\n\\begin{equation}\n\t\\operatorname{Tr}\\left[\\left(\\mathds{1}-\\rho\\left(t\\right)\\right)^n\\right]\n\t\t=\\operatorname{Tr}\\left[\\left(\\mathds{1}-\\sigma\\left(t\\right)\\right)^n\\right]-t^2 n \n\\operatorname{Tr}\\left[\\left(\\mathds{1}-\\rho\\right)^{n-1}\\gamma_1\\right]+O\\left(t^3\\right),\n\\label{eq:TER3}\n\\end{equation}\nwhere $\\sigma\\left(t\\right)=\\rho+t\\gamma_0$.\nFrom Eq.~(\\ref{eq:TER2}) and Eq.~(\\ref{eq:TER3}),\n\\begin{equation}\n\\operatorname{Tr}\\left(\\rho^{p}\\left(t\\right)\\right)\n=\\operatorname{Tr}\\left(\\sigma^{p}\\left(t\\right)\\right)\n+p t^2 \\operatorname{Tr}\\left( \\rho^{p-1}\\gamma_1\\right)\n+O\\left( t^3\\right).\n\\label{eq:TER4}\n\\end{equation}\nAs Lemma~\\ref{lem:DividedDifference} yields the equality\n\\begin{equation}\n\t\\operatorname{Tr}\\left(\\sigma^{p}\\left(t\\right)\\right)\n\t\t=\\operatorname{Tr}\\left( \\rho^{p}\\right)+t p \\operatorname{Tr}\\left(\\rho^{p-1}\\gamma_0\\right)\n\t\t\t+t^2 \\operatorname{Tr}\\left( Q_{\\rho}\\left( \\gamma_0\\right)\\right)+O\\left( t^3\\right),\n\\label{eq:TER5}\n\\end{equation}\nand $\\gamma_0$ is zero along the diagonal, \nwe obtain\n\\begin{equation}\n\t\\operatorname{Tr}\\left(\\rho^{p}\\left(t\\right)\\right)\n\t\t=\\operatorname{Tr}\\left(\\rho^{p}\\right)+t^2\\left( p \\operatorname{Tr}\\left(\\rho^{p-1}\\gamma_1\\right)\n\t\t\t+\\operatorname{Tr}\\left( Q_{\\rho}\\left( \\gamma_0\\right)\\right)\\right)+O\\left( t^3\\right).\n\\label{eq:TER6}\n\\end{equation}\nUsing the Taylor expansion of the logarithm function,\n\\begin{equation}\n\\log\\left(1+x\\right)=\\sum_{n=1}^{\\infty} (-1)^{n+1}\\frac{x^n}{n},\n\\label{eq:TER7}\n\\end{equation}\nwe obtain\n\\begin{align}\n\t\\log\\operatorname{Tr}\\left(\\rho^{p}\\left(t\\right)\\right)\n\t\t=&\\log\\left[\\operatorname{Tr}\\left(\\rho^{p}\\right)\\left( 1+ t^2\\left( p \\frac{\\operatorname{Tr}\n\t\t\t\\left(\\rho^{p-1}\\gamma_1\\right)}{\\operatorname{Tr}\\left(\\rho^{p}\\right)}\n\t\t\t\t+\\frac{\\operatorname{Tr}\\left(Q_{\\rho}\\left(\\gamma_0\\right)\\right)}\n\t\t\t\t\t{\\operatorname{Tr}\\left(\\rho^{p}\\right)}\\right) +O\\left( t^3\\right) \\right) \\right]\n\t\t\t\t\t\t\\nonumber\\\\\n\t\t=&\\log\\operatorname{Tr}\\left(\\rho^{p}\\right)+ t^2\\left( p \\frac{\\operatorname{Tr}\\left(\\rho^{p-1}\n\t\t\t\\gamma_1\\right)}{\\operatorname{Tr}\\left(\\rho^{p}\\right)}\n\t\t\t\t+\\frac{\\operatorname{Tr}\\left(Q_{\\rho}\\left(\\gamma_0\\right)\\right)}{\\operatorname{Tr}\\left(\\rho^{p}\\right)}\\right)\n\t\t\t\t\t+O\\left( t^3\\right).\n\\label{eq:TER8}\t\t\t\n\\end{align}\nTherefore, by definition of the quantum $p$-R\\'{e}nyi entropy in Eq.~(\\ref{eq:Renyi}), \nthe equality~(\\ref{eq:Salphat}) can be readily obtained from Eq.~(\\ref{eq:TER8}).\nThis completes the proof.\n\\end{proof}\nWe have evaluated the Taylor expansion of the quantum R\\'{e}nyi entropy.\nIn the next subsection, we use this result to prove the stability theorem of the depolarizing channel for the output quantum R\\'{e}nyi entropy.\n\n\\subsection{The stability theorem of the depolarizing channel for the output quantum R\\'{e}nyi entropy}\n\\label{subsec:Stability}\n\nIn this subsection, we prove our main theorem, \nnamely the stability theorem of the output quantum $p$-R\\'{e}nyi entropy for the depolarizing channel \nfor $p \\ge 2$. \nFirst, we present the following lemma, \nwhich is crucial to prove the theorem. \n\\begin{lemma}\n\\label{lem:f_p(x)}\nFor $p \\ge 2$, $r> 1$ and $d \\ge 2$, \n\\begin{equation}\n\t f_p(x)\n\t \t:= \\frac{p}{1-p}\\left[\\left(\\frac{({r}^{x})^{p-1}-1}{r^x-1}\\right)\n\t\t\t\\left(\\frac{(r-1)^2}{(d+r-1)^2(r^p+(d-1))}\\right)^{x}\n+\\left(\\frac{r^{p-1}+r+(d-2)}{r^p+(d-1)}\\right)^{x} -1 \\right]\n\\label{eq:falphax}\n\\end{equation}\nis monotonically increasing on $[2,\\infty)$.\n\\end{lemma}\n\\begin{remark}\nLet  $\\ket{\\psi}$ be an $n$-qudit pure state satisfying $\\left| \\inn{\\psi}{\\phi}\\right|^{2}=1-t^2$\nfor an $n$-qudit product state $\\ket{\\phi}$. Then the function $f_p$~(\\ref{eq:falphax}) is the coefficient of the second order term \nin the Taylor expansion of $S_p\\left(\\mathcal{D}_{\\lambda}^{\\otimes n}\\ket{\\psi}\\bra{\\psi}\\right)$.\n\n\\end{remark}\n\\begin{proof}\n\\label{proof:f_p(x)}\nObserve that\n\\begin{equation}\nf'_p(x)=A_p\\left[ B_p \\log\\left(\\frac{r^p+d-1}{r^{p-1}(r-1)^2}\\right)+C_p\\log r \\right],\n\\label{eq:f_p0}\n\\end{equation}\nwhere\n\\begin{align}\nA_p=&\\frac{p}{1-p}\\frac{(r-1)^2}{(r^x-1)^2(r^p+d-1)}, \\nonumber \\\\\nB_p=&(r^x-1)(1-(r^x)^{p-1}), \\nonumber \\\\\nC_p=&-(r^x)^p+p(r^x-1)+1. \n\\label{eq:f_p1}\n\\end{align}\nAs $r>1$, straightforward calculations yield $A_p \\leq 0$, $C_p \\leq B_p \\leq 0$ for $p \\geq 2$.\nThus, we obtain the inequality \n\\begin{align}\n\tf'_p(x) \n\t\t\\ge& A_p \\left[ B_p\\log\\left(\\frac{r^p+d-1}{r^{p-1}(r-1)^2}\\right)+B_p \\log r \\right] \n\t\t\\nonumber\\\\\n\t\t=& A_p B_p \\log\\left(\\frac{r^{p+1}+(d-1)r}{r^{p-1}(r-1)^2}\\right).\t\n\\label{eq:f_p2}\n\\end{align}\nHere the right-hand side of the inequality (\\ref{eq:f_p2}) is clearly nonnegative \nas the inequality\n\\begin{equation}\n\t\\log\\left(\\frac{r^{p+1}+(d-1)r}{r^{p-1}(r-1)^2}\\right) >0\n\t\\label{eq:f_p3}\n\\end{equation}\n\t\tcan be easily proved due to the inequality\n\\begin{equation}\n\t2r^{p-1}-r^{p-2}+(d-1) >0. \n\t\\label{eq:f_p4}\n\\end{equation}\nTherefore, the function $f_p(x)$ is monotonically increasing.\n\\end{proof}\n\nWe now present one more lemma,\nwhich tells us that the minimal output quantum $p$-R\\'{e}nyi entropy of the depolarizing channel is achieved for product state inputs. \nThe lemma can be readily obtained \nby the additivity of the minimal output quantum $p$-R\\'{e}nyi entropy~\\cite{Kin03}.\n\n\\begin{lemma}\n\\label{lem:minRenyi}\nFor the $n$-partite product depolarizing channel~$\\mathcal{D}_{\\lambda}^{\\otimes n}$,\nthe quantum $p$-R\\'{e}nyi entropy of the output state is minimized for product state inputs\nand furthermore has the same value for all product state inputs;\nthat is, \n\\begin{equation}\n\tS_p^{\\min}\\left( \\mathcal{D}_{\\lambda}^{\\otimes n}\\right)\n\t\t=S_p\\left(\\mathcal{D}_{\\lambda}^{\\otimes n}\\ket{\\phi}\\bra{\\phi}\\right),\n\t\t\\label{eq:lemma_product}\n\\end{equation}\t\nfor any $n$-partite pure product state $\\ket{\\phi}$.\n\\end{lemma}\n\nWe now use Theorem~\\ref{thm:TER} and the above lemmas \nto obtain the stability theorem of the output quantum $p$-R\\'{e}nyi entropy \nfor the depolarizing channel.\n\\begin{lemma}\n\\label{lemma:STRenyi}\nLet $p \\ge 2$, $\\epsilon>0$ and $|\\psi\\rangle\\in(\\mathbb{C}^d)^{\\otimes n}$ be a state. Then\n\\begin{equation}\n\tS_p\\left(\\mathcal{D}_{\\lambda}^{\\otimes n}\\ket{\\psi}\\bra{\\psi}\\right)\n\t\t< S_p^{\\min}\\left( \\mathcal{D}_{\\lambda}^{\\otimes n}\\right)\n\t\t\t+2\\epsilon\\frac{p}{p-1}\n\t\t\t\t\\frac{r-1}{r+1}\n\t\t\t\t\t\\frac{(r^{p-1}-1)(2r^{p}+dr+d-2)}{(r^{p}+d-1)^2}\n\t\t\t\t\t+ O(\\epsilon^{3\/2})\n\\label{eq:True}\n\\end{equation}\nholds only if a pure product state~$\\ket{\\phi}$ exists such that~$\\ket{\\psi}$ satisfies\n\\begin{equation}\n\\label{eq:close}\n\t\\left|\\left\\langle\\psi\\right|\\phi\\rangle\\right|^{2} \\ge 1-\\epsilon.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nWe prove the contrapositive of the theorem. \nLet\n\\begin{equation}\n\\epsilon_0 \n= 1-\\max\\left\\{ \\left| \\left\\langle\\psi\\right| \\phi_{1},\\cdots,\\phi _{n}\\rangle\\right|^{2}:\n\t\t\t\\left| \\phi_{i}\\right\\rangle \\in \\mathbb{C}^d\\right\\}>\\epsilon.\n\\label{eq:MaxFidelity}\n\\end{equation} \nWithout loss of generality, \nwe may assume that \none of the states achieving the maximum in Eq.~(\\ref{eq:MaxFidelity}) is $\\ket{0^n}=\\ket{0}$. \nWe then have\n\\begin{equation}\n\t\\ket{\\psi}=\\sqrt{1-\\epsilon_0}\\ket{0}+\\sqrt{\\epsilon_0}\\ket{\\phi}\n\t\\label{eq:main_Thm1}\n\\end{equation}\nfor some state $\\ket{\\phi}$ such that $\\inn{0}{\\phi}=0$;\nthat is, $\\ket{\\phi}=\\sum_{x\\neq0}\\alpha_x\\ket{x}$ \nfor some $\\alpha_x $ such that $\\sum_{x\\neq0}\\left|\\alpha_x\\right|^2=1$.\nWe can write explicitly\n\\begin{equation}\n\\ket{\\psi}\\bra{\\psi}\n=\\left(1-\\epsilon_0\\right)\\ket{0}\\bra{0}+\\sqrt{\\epsilon_0\\left(1-\\epsilon_0\\right)}\n\\left( \\ket{0}\\bra{\\phi}+\\ket{\\phi}\\bra{0}\\right)+\\epsilon_0\\ket{\\phi}\\bra{\\phi}.\n\\label{eq:main_Thm2}\n\\end{equation}\nTherefore, we have\n\\begin{align}\n\t\\mathcal{D}_{\\lambda}^{\\otimes n}\\ket{\\psi}\\bra{\\psi}\n\t\t=& \\mathcal{D}_{\\lambda}^{\\otimes n}\n\t\t\t\\ket{0}\\bra{0}+\\sqrt{\\epsilon_0}\\sqrt{1-\\epsilon_0}~\\mathcal{D}_{\\lambda}^{\\otimes n}\n\t\t\t\t\\left(\\ket{0}\\bra{\\phi}+\\ket{\\phi}\\bra{0}\\right)\n\t\t\t\t\t+\\epsilon_0~\\mathcal{D}_{\\lambda}^{\\otimes n}\\left(\\ket{\\phi}\\bra{\\phi}\n\t\t\t\t\t\t-\\ket{0}\\bra{0}\\right) \\nonumber\\\\\n\t\t=& \\mathcal{D}_{\\lambda}^{\\otimes n}\n\t\t\t\\ket{0}\\bra{0}+\\sqrt{\\epsilon_0}~\\mathcal{D}_{\\lambda}^{\\otimes n}\n\t\t\t\t\\left(\\ket{0}\\bra{\\phi}+\\ket{\\phi}\\bra{0}\\right)\n\t\t\t\t\t+\\epsilon_0~\\mathcal{D}_{\\lambda}^{\\otimes n}\\left(\\ket{\\phi}\\bra{\\phi}\n\t\t\t\t\t\t-\\ket{0}\\bra{0}\\right)+O( \\epsilon_0^{3\/2} ).\n\\label{eq:main_Thm3}\n\\end{align}\nFor the last equality in Eq.~(\\ref{eq:main_Thm3}), \nwe use the Taylor expansion\n\\begin{equation}\n\t\\sqrt{1-x}=1-\\frac{1}{2}x+O\\left( x^2\\right)\n\\end{equation}\nfor all $|x|<1$.\nNow we use \n\\begin{align}\n\\rho &= \\mathcal{D}_{\\lambda}^{\\otimes n}\\ket{0}\\bra{0}, \\nonumber \\\\\n\\gamma_0 &= \\mathcal{D}_{\\lambda}^{\\otimes n}\\left(\\ket{0}\\bra{\\phi}+\\ket{\\phi}\\bra{0}\\right), \\nonumber \\\\\n\\gamma_1&= \\mathcal{D}_{\\lambda}^{\\otimes n}\\left(\\ket{\\phi}\\bra{\\phi} -\\ket{0}\\bra{0}\\right), \\nonumber \\\\\nt&=\\sqrt{\\epsilon_0},  \\nonumber \\\\\n\\rho\\left(t\\right) &= \\rho+t \\gamma_0+t^2 \\gamma_1+O\\left(t^3\\right). \\nonumber \\\\\n\\end{align}\nThen Theorem~\\ref{thm:TER} implies that\n\\begin{align}\n\tS_p\\left(\\mathcal{D}_{\\lambda}^{\\otimes n}\\ket{\\psi}\\bra{\\psi}\\right) \n\t\t=&S_p\\left( \\rho\\left(t\\right)\\right)\n\t\t\t\t\t\t\\nonumber\\\\\n\t\t=&S_p\\left(\\rho\\right)+\\frac{1}{1-p}t^2\n\t\t\t\\left(p\\frac{\\operatorname{Tr}\\left(\\rho^{p-1}\\gamma_1\\right)}\n\t\t\t\t{\\operatorname{Tr}\\left(\\rho^{p}\\right)}+\\frac{\\operatorname{Tr}\\left(Q_{\\rho}\\left(\\gamma_0\\right)\\right)}\n\t\t\t\t\t{\\operatorname{Tr}\\left(\\rho^{p}\\right)}\\right)\n\t\t\t\t\t\t+O\\left(t^3\\right).\n\\label{eq:RentropyPerturbation}\n\\end{align}\nFor convenience, we let\n\\begin{equation}\n\ta=\\left(1+(d-1)\\lambda\\right)\/d,\\;\n\tb=(1-\\lambda)\/d.\n\\end{equation}\nThen we obtain the following three facts.\n\\begin{enumerate}\n\t\\item\tAs $\\rho=\\mathcal{D}_{\\lambda}^{\\otimes n}\\ket{0}\\bra{0}$ can be rewritten as\n$\\sum_{y} a^{n-|y}b^{|y|}\\ket{y}\\bra{y}$, \n\t\\begin{equation}\n\t\t\\operatorname{Tr}\\left( \\rho^{p}\\right)\n\t\t\t=\\operatorname{Tr}\\left(\\sum_{y}\\left(a^{p}\\right)^{n-|y|}\\left(b^{p}\\right)^{|y|}\\ket{y}\\bra{y}\\right)\n\t\t\t=\\left( a^p+(d-1)b^p\\right)^n,\n\t\\end{equation}\n\tfor~$|y|$ denoting Hamming weight of an $n$-bit string~$y$. \n\t\\item We can be evaluate\n\t\\begin{align}\n\t\t\\operatorname{Tr}\\left(\\gamma_1\\rho^{p-1}\\right) \n\t\t=& \\operatorname{Tr}\\left(\\mathcal{D}_{\\lambda}^{\\otimes n}\n\t\t\t\\ket{\\phi}\\bra{\\phi}\\right)\\left( \\mathcal{D}_{\\lambda}^{\\otimes n}\n\t\t\t\t\\ket{0}\\bra{0}\\right)^{p-1} -\\operatorname{Tr}\\left( \\rho^p\\right) \\nonumber\\\\\n\t\t=& \\sum_{x,x' \\neq 0,y } \\alpha_x {\\alpha}_{x'}^*\n\t\t\t\\left(a^{p-1}\\right)^{n-|y|}\\left( b^{p-1}\\right)^{|y|}\n\t\t\t\t\\operatorname{Tr}\\left( \\left(\\mathcal{D}_{\\lambda}^{\\otimes n}\n\t\t\t\t\t\\ket{x}\\bra{x'}\\right)\\ket{y}\\bra{y}\\right) -\\operatorname{Tr}\\left( \\rho^p\\right)\\nonumber\\\\\n\t\t=& \\sum_{x \\neq 0,y} |\\alpha_x|^2\\left(a^{p-1}\\right)^{n-|y|}\\left( b^{p-1}\\right)^{|y|} \\prod_{i=1}^n \\frac{1-\\lambda(1-d)^{\\delta_{x_i y_i}}}{d}-\\operatorname{Tr}\\left( \\rho^p\\right) \\nonumber\\\\\n\t\t=& \\sum_{x \\neq 0} |\\alpha_x|^2\\left( a^p+(d-1)b^p\\right)^n\n\t\t\\left(\\frac{ a^{p-1}b+ab^{p-1}+(d-2)b^p}{a^p+(d-1)b^p}\\right)^{|x|}\n\t\t-\\operatorname{Tr}\\left( \\rho^p\\right).\n\t\t\\label{eq:main_Thm4}\n\\end{align}\n\t\\item For $n$-bit strings $j$ and $k$, let \n\t\\begin{equation}\n\t\tg_{jk}\n\t\t\t:= \\frac{\\left(a^{n-|j|}b^{|j|}\\right)^{p-1}-\\left(a^{n-|k|}b^{|k|}\\right)^{p-1}}{2\\left(a^{n-|j|}b^{|j|}-a^{n-|k|}b^{|k|}\\right)}.\n\t\\label{eq:main_Thm5}\n\\end{equation}\nThen we write\n\\begin{align}\n\t\\operatorname{Tr}\\left( Q_\\rho\\left( \\gamma_0\\right)\\right) \n\t\t=& p\\left( \\sum_{jk} g_{jk}\\right) \\left|\\left( \\mathcal{D}_{\\lambda}^{\\otimes n}\n\t\t\t\\ket{0}\\bra{\\phi}\\right)_{jk}\\right|^2 \n\t\t\t\\label{eq:main_Thm6} \\\\\n\t\t=& p \\sum_{x \\neq 0} |{\\alpha}_x|^2 \n\t\t\t\\left(\\frac{(a^{|x|})^{p-1}-(b^{|x|})^{p-1}}{a^{|x|}-b^{|x|}}\\right)\n\t\t\t\t(a^p+(d-1)b^p)^{n-|x|}.\n\t\t\t\t\\label{eq:main_Thm7}\n\\end{align}\n\\end{enumerate}\nHere $Q_\\rho$~(\\ref{eq:main_Thm6}) is a polynomial defined in Eq.~(\\ref{eq:lemma_GF1}), and \nall equalities can be proved by tedious but straightforward calculations \nexcept the last equality in Eq.~(\\ref{eq:main_Thm4}), \nwhich can be shown by mathematical induction on $n$.\n\nCombining the above facts, we have\n\\begin{equation}\n\tS_p\\left(\\rho\\left(t\\right)\\right)-S_p\\left(\\rho\\right)\n\t\t=  t^2 \\sum_{x \\neq 0} |{\\alpha_x}|^2 f_p(|x|)+O(t^3)\n\t\t\\label{eq:main_Thm8}\n\\end{equation}\nwith \n\\begin{align}\n\tf_p(|x|)\n\t\t=&\\frac{p}{1-p}\n\t\t\t\\left(\\frac{(a^{|x|})^{p-1}-(b^{|x|})^{p-1}}{a^{|x|}-b^{|x|}}\\right)\n\t\t\t\t\\left(\\frac{\\lambda^2}{a^p+(d-1)b^p}\\right)^{|x|}\n\t\t\t\t\t\t\t\\nonumber\\\\\n\t\t\t\t\t&+\\frac{p}{1-p}\\left[\\left(\\frac{a^{p-1}b\n\t\t\t\t\t\t+ab^{p-1}+(d-2)b^p}{a^p+(d-1)b^p}\\right)^{|x|} -1\\right].\n\t\t\t\t\t\t\\label{eq:main_Thm9}\n\\end{align}\nHere it can be shown that \nthe function $f_p$~(\\ref{eq:main_Thm9}) is equal to \nthe function $f_p$ defined in Eq.~(\\ref{eq:falphax})\nby taking $r=a\/b$.  \nHence, Lemma~\\ref{lem:f_p(x)} implies that \n$f_p(|x|)$ is monotonically increasing on $|x|$ for all $p \\geq 2$.\nAs $\\ket{\\phi}$ does not have any weight-one components~\\cite{HM13};\nthat is, $\\alpha_x=0$ for $|x|< 2$, from Theorem \\ref{thm:TER} and Lemma \\ref{lem:minRenyi},\nwe can finally obtain the following inequality, \nand thereby complete the proof.\n\\begin{align}\n\tS_p\\left(\\mathcal{D}_{\\lambda}^{\\otimes n}\\ket{\\psi}\\bra{\\psi}\\right)-\n\t\tS_p^{\\min}\\left(\\mathcal{D}_{\\lambda}^{\\otimes n}\\right)\n\t\t\t=&S_p\\left(\\rho\\left(t\\right)\\right)-S_p\\left(\\rho\\right) \n\t\t\t\t\t\t\t\\nonumber\\\\\n\t\t\t\\ge& \\epsilon_0 f_p(2)+O(\\epsilon_0^{3\/2})\n\\label{eq:ineq}\n\t\t\t\t\t\t\t\\nonumber\\\\\n\t\t\t>&\\epsilon \\frac{p}{p-1}\n\t\t\t\t\\left[\\frac{2\\lambda(1-\\lambda)(a^{p-1}-b^{p-1})\n\t\t\t\t\t(2(a^{p}-b^{p})+db^{p-1}(a+b))}\n\t\t\t\t\t\t{(2+(d-2)\\lambda)(a^{p}+(d-1)b^p )^2}\\right] +O(\\epsilon^{3\/2})\n\t\t\t\t\t\t\t\\nonumber\\\\\n\t\t=& 2\\epsilon\\frac{p}{p-1}\n\t\t\t\\frac{(r-1)(r^{p-1}-1)(2r^{p}+dr+d-2)}{(r+1)(r^{p}+d-1)^2}\n\t\t\t\t+O(\\epsilon^{3\/2}).\n\\end{align}\n\\end{proof}\n\\begin{remark}\nAlthough we have not yet established the stability theorem for $1<p<2$,\nwe can show that it still holds for $p=1$;\nthat is, the stability theorem for the von~Neumann entropy of the depolarizing channel holds\nby using a similar method to what we have used.\n(We have numerically checked that the same result holds \nfor the several cases of $p$ with $1<p<2$.)\nLet us see the proof for the case of $p=1$.\n\nWe calculate the Taylor expansion of the von~Neumann entropy \nto get the difference between the output von~Neumann entropy and its minimum value as follows.\n\\begin{equation}\nS\\left(\\mathcal{D}_{\\lambda}^{\\otimes n}\\ket{\\psi}\\bra{\\psi}\\right)\n-S^{\\min}\\left(\\mathcal{D}_{\\lambda}^{\\otimes n}\\right)\n=\\epsilon\\sum_{x \\neq 0} |{\\alpha_x}|^2 f(|x|)+O(\\epsilon^{3\/2}),\n\\label{eq:output_von_Neumann_entropy}\n\\end{equation}\nwhere the function $f$ is defined as\n\\begin{equation}\nf(|x|)= |x|(a-b) \\log \\frac{a}{b}-\\left(a-b\\right)^{2|x|}\\left(\\frac{\\log a^{|x|}-\\log b^{|x|}}{a^{|x|}-b^{|x|}}\\right),\n\\end{equation}\nwhich is the limit of $f_p(|x|)$ for $p$ tends to one. \nAs it is easier than Lemma~\\ref{lem:f_p(x)} \nto prove that $f(|x|)$ is monotonically increasing, \nwe can easily obtain the almost same result as \nour stability theorem.\n\\end{remark}\n\n\\begin{theorem}\n\\label{thm:stability_new}\nThe depolarizing channel is stable with respect to the output quantum $p$-R\\'{e}nyi entropy \nfor $p \\ge 2$ or $p=1$.\n\\end{theorem}\n\\begin{proof}\nLet $\\epsilon>0$ be given,\nand let the quantity $Q$ be~$S_p$.\nLet $\\ket{\\psi}\\bra{\\psi}$ be\nan~$\\epsilon$-almost extremal state with respect to~$S_p$ and $\\mathcal{D}_{\\lambda}^{\\otimes n}$.\nThen\n\\begin{equation}\n\t\\left|S_p\\left(\\mathcal{D}_{\\lambda}^{\\otimes n}\\ket{\\psi}\\bra{\\psi}\\right)\n\t\t-S_p^{\\min}\\left(\\mathcal{D}_{\\lambda}^{\\otimes n}\\right)\\right|\n\t\t\t\\in O(\\epsilon),\n\\end{equation}\nby Lemma~\\ref{lemma:STRenyi}, \nthere exists some extremal state $\\ket{\\phi}\\bra{\\phi}$ \nwith respect to~$S_p$ and $\\mathcal{D}_{\\lambda}^{\\otimes n}$ such that\n\\begin{equation}\n\\left\\| \\ket{\\psi}\\bra{\\psi}-\\ket{\\phi}\\bra{\\phi} \\right\\|^2_1\n\\le \\epsilon.\n\\end{equation}\nThus, the depolarizing channel $\\mathcal{D}_{\\lambda}^{\\otimes n}$ is~$\\epsilon$-stable with respect to the quantum $p$-R\\'{e}nyi entropy~$S_p$\nfor any $\\epsilon>0$, \nand hence it is stable with respect to~$S_p$.\n\\end{proof}\n\n\\begin{remark}\n\tWe obtain the stability theorem of the depolarizing channel with respect to the output purity~\\cite{HM13}\n\tas a corollary of Theorem~\\ref{thm:stability_new}.\n\\end{remark}\n\nFurthermore, we can similarly show that if $n$-qudit pure state is close to be product \nthen its output quantum $p$-R\\'{e}nyi entropy is close to \nthe minimal output quantum $p$-R\\'{e}nyi entropy with a specific precision as follows.  \n\\begin{theorem}\n\\label{theorem:STRenyi2}\nLet $p \\ge 2$, $\\epsilon>0$ and $|\\psi\\rangle\\in(\\mathbb{C}^d)^{\\otimes n}$ be a state. Then\n\\begin{equation}\n\\label{eq:False}\n\tS_p\\left(\\mathcal{D}_{\\lambda}^{\\otimes n}\\ket{\\psi}\\bra{\\psi}\\right)\n\t\t\\geq S_p^{\\min}\\left( \\mathcal{D}_{\\lambda}^{\\otimes n}\\right)\n\t\t\t+\\epsilon\\frac{p}{p-1}+ O(\\epsilon^{3\/2})\n\\end{equation}\nimplies \n\\begin{equation}\n\\left| \\inn{\\psi}{\\phi}\\right|^{2}<1-\\epsilon\n\\label{eq:False2}\n\\end{equation}\n for any product state $|\\phi\\rangle$.\n\\end{theorem}\n\\begin{proof}\nSuppose that\n\\begin{equation}\n\\label{eq:tau}\n1-\\epsilon_1=\\max\\left\\{\\left| \\left\\langle\\psi\\right|\\phi_{1},\\cdots,\\phi _{n}\\rangle\\right|^{2}:\n\t\t\t\\ket{\\phi_{i}} \\in \\mathbb{C}^d\\right\\} \\ge 1-\\epsilon.\n\\end{equation}\nFrom the same arguments in Theorem~\\ref{lemma:STRenyi}, \nwe obtain the same equality as in Eq.~(\\ref{eq:main_Thm8}).\nThen\n\\begin{align}\n\tS_p\\left(\\mathcal{D}_{\\lambda}^{\\otimes n}\\ket{\\psi}\\bra{\\psi}\\right)-\n\t\tS_p^{\\min}\\left(\\mathcal{D}_{\\lambda}^{\\otimes n}\\right)\n\t\t\t=&  \\epsilon_1 \\sum_{x \\neq 0} |{\\alpha_x}|^2 f_p(|x|)+O(\\epsilon_1^{2\/3})\n\t\t\t\t\t\t\t\\nonumber\\\\\n\t\t\t<& \\epsilon_1\\frac{p}{1-p}+O(\\epsilon_1^{3\/2})\n\t\t\t\t\t\t\t\\nonumber\\\\\n\t\t\t\\le& \\epsilon \\frac{p}{1-p}+O(\\epsilon^{3\/2}),\n\\label{eq:epsilon}\t\n\\end{align}\nwhere the first inequality is obtained due to the monotonicity of $f_p(|x|)$\nand the second inequality results from Eq.~(\\ref{eq:tau}). \n\\end{proof}\n\n\\begin{remark}\nThe coefficient of~$\\epsilon$ in Eq.~(\\ref{eq:True}) is smaller than \nthe coefficient of~$\\epsilon$ in Eq.~(\\ref{eq:False}),\nwhich means that some gap exists between them \neven though it is close to zero for sufficiently small~$\\epsilon$. \nFurthermore, for a sufficiently large $p$, \nthe gap can be smaller than the gap for the case of $p=2$ \nas we will see in \\S\\ref{sec:Poly}.\n\\end{remark}\n\n\\section{An application: A Polygraph Test}\n\\label{sec:Poly}\nIn this section, we introduce a polygraph test \nas an application of our main results,\nnamely Theorems~\\ref{lemma:STRenyi} and~\\ref{theorem:STRenyi2}.\nLet us consider the following protocol \nwherein sender Alice transmits multiple copies of an $n$-qudit state \nthrough depolarizing channels to receiver Bob.\n\\renewcommand{\\labelenumi}{\\arabic{enumi}.}\n\\begin{enumerate} \n\t\\item Bob informs Alice of a small enough $\\epsilon>0$ chosen as an error bound.\n\t\\item Alice prepares an $n$-qudit pure state \nthat is close to a product state with fidelity at least $1-\\epsilon$ as in Eq.~(\\ref{eq:close})\nand sends multiple copies to Bob through depolarizing channels.\n\t\\item Bob estimates its output quantum R\\'{e}nyi entropy.\n\t\\item Bob determines whether Alice's preparation satisfies the requirement or not,\n\t\tand our results help Bob make the correct decision as discussed below.\n\t\t\\begin{itemize}\n\t\t\t\\item[Accept:]\n\t\t\t\tIf Bob's estimate of the output quantum R\\'{e}nyi entropy \n\t\t\t\tsatisfies Inequality~(\\ref{eq:True}) then \n\t\t\t\tAlice definitely prepared a correct state according to\n\t\t\t\tTheorem~\\ref{lemma:STRenyi}. \n\t\t\t\\item[Reject:]\n\t\t\t\tIf Bob's estimate of the output quantum R\\'{e}nyi entropy \n\t\t\t\tsatisfies Inequality~(\\ref{eq:False}) then\n\t\t\t\tTheorem~\\ref{theorem:STRenyi2} guarantees\n\t\t\t\tthat Alice's preparation fails the requirement.\n\t\t\\end{itemize}\n\\end{enumerate}\n\\begin{remark}\n\tSome gap exists between the coefficients of~$\\epsilon$ in Eq.~(\\ref{eq:True}) \n\tand of~$\\epsilon$ in Eq.~(\\ref{eq:False}), \n\twhich means that Bob cannot detect Alice's lie \n\twhen neither Eqs.~(\\ref{eq:True}) nor~(\\ref{eq:False}) holds\n\tfor the output quantum R\\'{e}nyi entropy of the state Alice sent. \n\tHowever, the probability that Alice cheats Bob can be forced to be close to zero \n\tif Bob chooses small enough~$\\epsilon$. \n\tThus, the gap problem can be resolved in this way.\n\\end{remark}\n\nWe note that the above polygraph test can be also realized \nby the original stability theorem~\\cite{HM13} in the same way as ours,\nas the original stability theorem is essentially equivalent to \nthe case of $p=2$ in ours.\nHowever, we show that \nif $p$ is sufficiently large then \nour gap can be smaller than the gap from the original stability as follows.\nLet a nonzero~$\\epsilon$ be fixed, \nand define the gap function\n\\begin{equation}\n\t\\operatorname{gap}(p)\n\t\t:=\\frac{p}{p-1}\\left(1-\\frac{2(r-1)(r^{p-1}-1)(2r^p+dr+(d-2))}{(r+1)(r^p+(d-1))^2}\\right),\n\\end{equation}\nwhich is the gap between the coefficients of~$\\epsilon$ \nin Eqs.~(\\ref{eq:True}) and~(\\ref{eq:False}). \nWe claim \n\\begin{equation}\n\t\\operatorname{gap}(2)>\\lim_{p \\rightarrow \\infty}\\operatorname{gap}(p), \n\\label{eq:gap}\n\\end{equation}\nwhich is equivalent to the inequality\n\\begin{equation}\n\t (r^2+(d-1))^2(r^2+5r-4) > 4r(r-1)^2(2r^2+dr+(d-2)).\n \\label{eq:gap2}\n\\end{equation}\nMoreover, in order to prove the inequality (\\ref{eq:gap2})\nit is enough to show that\n\\begin{equation}\n\th(r)\n\t\t:=r^3-3r^2+(-2d+10)r+(14d-10)\n\\label{eq:gap3}\n\\end{equation}\nis positive for all $r>1$.\nAs its derivative is \n\\begin{equation}\nh'(r)=3r^2-6r+(-2d+10),\n\\label{eq:h2}\n\\end{equation}\n$h(r)$ is evidently positive if $d$ is 2 or 3. \nIf $d\\ge4$ then it can be readily shown that\n\\begin{equation}\nh(r)\\ge h\\left(1+\\frac{\\sqrt{6d-21}}{3}\\right)=\\frac{2\\sqrt{6d-21}}{3}-2+12d>0\n\\label{eq:gap4}\n\\end{equation} \nfor all $r>1$.\n\nWe have introduced the polygraph test as an application of out main theorem,\nand we have proved that the protocol for our stability theorem has a smaller undecidable gap than for the protocol in the original stability theorem.\n\n\\section{Conclusions}\n\\label{sec:Conclusions}\nWe have shown that the stability theorem of the depolarizing channel holds \nfor the output quantum $p$-R\\'{e}nyi entropy for $p \\ge 2$, \nwhich was one of the open questions in Ref.~\\cite{HM13}. \n Furthermore, we have also proved that the stability theorem of the depolarizing channel holds for the output von~Neumann entropy ($p=1$), \nand have numerically checked that \nthe stability theorem \nholds \nfor the several cases $p$ with $1<p<2$.\nTherefore, we expect that the stability theorem \nholds \nfor all $p \\ge 1$, and \nleave this for future work.\n\nAs an application of our main results, we have introduced a polygraph test\nand have presented its protocol. \nThe original stability theorem can be also applied to the polygraph test, \nas the original one is essentially equivalent to our stability theorem\nwhen $p=2$. \nIn the protocol, Bob determines \nwhether Alice prepares a pure quantum state close to a product state.\nHowever, Bob cannot perfectly decide whether her preparation is proper, \nthat is, there is an undecidable gap in which \nhe can decide nothing. \nWe have shown that the undecidable gap of our protocol \ncan be smaller than the original case.\nTherefore, our results improve the original stability theorem as well as generalize it.\n\n\n\n\n\\acknowledgements{%\nWe acknowledge discussions with A. W. Harrow and A. Montanaro.\nThis research was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (NRF-2012R1A1A2003441).\nE. B.\\ and B. C. S.\\ acknowledge financial support from AITF and NSERC,\nand B.C.S.\\ acknowledges support from China's 1000 Talent Plan.\nG.G.\\ acknowledges financial support from NSERC.%\n}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nBeyond routing their vehicles at the least cost, goods carriers often want to\nknow what is the best fleet to use. For example, some vehicle types are more\nexpensive, in terms of maintenance and usage than others. Our work is based on a\ntender for grocery delivery in Queensland, Australia. A goods carrier, handling\ndelivery of chilled and ambient temperature products, provided us with a year of\ndemand data. We want to determine what is the best fleet and the cheapest\nrouting to satisfy customers.\n\nOur main contribution is to compare different modelling approaches, and provide\nextensive computational experiments based on real-world instances. The ultimate\ngoal of this project is to find good modelling techniques for a more general\nmulti-day setting. In this work we focus on the single day setting as a\nfirst step.\n\nThe Vehicle Routing Problem \\citep[VRP,][]{Dantzig59,clarke1964scheduling}\nconsists in routing a fleet of vehicles to visit a set of customers. It was\nfirst defined as an extension of the Travelling Salesman Problem (TSP) with\nmultiple salespeople. As such, it belongs to the class of \\textsc{np}-hard problems\n\\citep{Lenstra81}. Its most common extension is the Capacitated VRP\n\\citep[CVRP,][]{Toth01}. In this case, customers have an associated demand and\nvehicles have a limited capacity.\n\nAnother common extension is the Split Delivery VRP \\citep[SDVRP,][]{Dror89}. In\nthe classic VRP, customers are visited exactly once. The SDVRP relaxes this\nconstraint, allowing more than one visit per customer. It is especially useful\nwhen demands exceed the vehicle capacity. Although it is a relaxation, it has\nbeen proven to be harder because because of the additional degree of freedom in\nplanning the deliveries. Despite its complexity, there are good reasons for\nstudying the SDVRP as it can lead to significant savings.\n\nIn recent years, driven by industrial needs, several realistic variants of the\nVRP appeared. First, models were extended to handle multiple commodities (types\nof goods). Usually such extension comes with compatibility issues, as vehicles\ncannot transport all commodities. A more ambitious extension is the Fleet Size\nand Mix VRP (FSMVRP) that considers finding a good fleet size and composition.\n\nWe call Rich VRP a problem merging many such extensions, see \\citet{Caceres14}\nfor a survey. In this paper we consider the fleet size and mix, multi-commodity,\ncapacitated, and split delivery VRP.\n\nOur current work tries to address issues raised in the experimental analysis of\n\\citet{Kilby16}. We propose a stronger MIP scheme to compare with their CP\napproach.\n\nTo solve this problem, we focus on the two distinct, but interdependent parts of\nthe problem. First is the fleet design component, and second is the routing\ncomponent. Bigger vehicles with better compatibility are more expensive to\nrun. We propose two different MIP models for each component, amounting to four\ncomplete MIPs.\n\nFinally, we propose a lightweight integration of a CP model into our solving\nprocess. The main strength of the CP model is to be able to find good solution\nfast. Thus, we use it to prime our MIP search. Instead of letting the solver\nfind a starting solution, we use the CP model to find one.\n\n\\section{Related Work}\n\\label{sec:literature}\n\nIn this section we review some of the relevant literature on SDVRP, FSMVRP, and\nRVRP formulations, and we consider some works on valid inequalities, some of\nwhich we adapt to work with our models.\n\nIn \\citet{Dror89} the authors present SDVRP together with an analysis of the\nproblem's complexity and the potential savings that can be attained by using it\nas a delivery scheme. The conclusion of this work is that, despite the SDVRP\nbeing a relaxation of the VRP, it is harder to solve. However, the additional\ncomplexity pays off, since allowing split deliveries can lead to substantial\ngains in terms of distances and number of vehicles needed, especially when the\ndemand of the customers is slightly less than a full truckload. Our work extends\nthe above, in that we consider additional compatibility constraints and a\nmulti-commodity scenario.\n\nRegarding RVRP formulations we refer the interested reader to a recent,\nexcellent survey by \\citet{Caceres14}.\n\nConcerning exact algorithms to solve the VRP under capacity or time window\nconstraints, the interested reader can refer to \\citet{Baldacci12}. The authors\npresent a comparison of different exact formulations. Two main class of\nalgorithms emerge: branch-and-cut algorithms and set covering algorithms. The\nformer are based on the idea of adding valid cuts during a branch-and-bound\nsearch. The latter employ an exponential number of variables, and are therefore\nsolved using column generation. Our work can be seen as a preliminary study to\ncompare MIP models to be used in a decomposition scheme.\n\nRegarding the research on valid inequalities for VRP, the literature is quite\nrich. In an extension \\citep{Dror94} of \\citet{Dror89}, the authors focus on\nimproving the solution process by devising valid cuts for their previous model.\nBoth works are based on the traditional CVRP, where a single commodity is\nconsidered, and the fleet is fixed.\n\nIn \\citet{Letchford06}, a study on the relative strengths of different classes\nof inequalities is presented, this time for the CVRP. In a follow-up to this\nwork \\citep{Letchford15} the authors add two new multi-commodity formulations,\nand study additional families of inequalities. In \\citet{Baldacci12} a number of\nvalid cuts are also discussed; often these are not trivial as few of them have\npolynomial algorithms available. Our approach adopts several of the valid\ninequalities presented in the above works.\n\nThe FSMVRP is a variant of the Heterogeneous VRP (HVRP) where the number of\nvehicles is not fixed. The HVRP has been first defined in the work of\n\\citet{Golden1984} as a relaxation of the CVRP where the vehicles do not have a\nfixed type. The authors discuss a number of heuristic algorithms, some based on\nthe Clarke and Wright Savings algorithm, some based on the concept of\n\\emph{opportunity cost}, a way to evaluate the potential gain by using a vehicle\nof a different type.\n\nMost of the works on FSMVRP are limited to single-commodity and single-day\nscenarios, furthermore very few exact algorithms exist in the literature, the\nmain focus being on improving the quality of linear relaxations in order to\nstrengthen bounding procedures -- e.g. \\citet{Jabali2012}.\n\n\n\\citet{Vidal14} use a component-based framework to solve a variety real-world\nVRP variants, including fleet size and mix vehicle routing, is proposed. The\napproach, however, does not handle split deliveries or multiple commodities.\n\nAn exact algorithm was proposed by \\citet{Baldacci2009} for the HVRP. Their\napproach is based on a set partitioning formulation, which they solve using a\nreduced model in combination with a bounding procedure. Although this approach\nencompasses a wide variety of possible variations of the HVRP, it does not\nhandle split deliveries or multiple commodities.\n\n\\citet{Yaman2006} offers a work similar to ours in the sense that they design\nand compare six different formulations for the HVRP: four based on the\nMiller-Tucker-Zemlin constraints and two based on a flow formulation. They\ndesign a number of valid inequalities for the different models, then report\ncomputational experiments using standard benchmarks from the literature.\nHowever, they only provide the time and value of the linear relaxations, no\nexact result is provided.\n\nFor a recent review on different FSMVRP algorithms and problem classification,\nwe refer the interested read to a survey by \\citet{kocc2016thirty}. Following\nthe classification proposed in another review by \\citet{Baldacci2008a} our\nproblem is a FSMFD, with unlimited fleet size, fixed costs, and routing costs.\n\nTo the best of our knowledge the work of \\citet{Kilby16} is the only combining\nFSMVRP with split deliveries and multiple commodities. The authors present a CP\nand Large Neighbourhood Search (LNS) approach to solve the same problem we\naddress here. The experimental analysis revealed the strength of MIP on some\nparticular problem configurations, and therefore motivated the present\nresearch. Moreover, we employ the CP model presented in that work to prime our\nMIP formulation, so as to increase the performance of our approach.\n\n\\section{Problem}\n\\label{sec:problem}\n\\newcommand{\\dep}{\\mathbf{dep}}%\n\\newcommand{\\dem}{\\mathbf{dem}}%\n\\newcommand{\\cost}{\\mathbf{cost}}%\n\\newcommand{\\dist}{\\mathbf{dist}}%\n\\newcommand{\\capa}{\\mathbf{cap}}%\n\\newcommand{\\comp}{\\mathbf{comp}}%\n\\newcommand{\\comms}{\\mathbf{K}}%\n\\newcommand{\\comm}{k}%\n\\newcommand{\\vtypes}{\\mathbf{T}}%\n\\newcommand{\\vtype}{t}%\n\\newcommand{\\custs}{\\mathbf{C}}%\n\\newcommand{\\cust}{i}%\n\\newcommand{\\custi}{\\mathbf{C^\\prime}}%\n\\newcommand{\\edges}{\\mathbf{E}}%\n\\newcommand{\\edge}{e}%\n\\newcommand{\\veh}{v}%\n\\newcommand{\\vehs}{\\mathbf{V}}%\n\\newcommand{\\use}{u}%\nIn this section we provide a description of the problem components: entities,\nconstraints, and objective. We defer the description of routing and fleet\nconstraints to \\Cref{sec:approach} as they are specific to each model. We\ndecided to limit the use of equations in this section because they are often\nvery similar. \\Cref{ap:models} presents the complete mathematical formulation of\nthe resulting models which combine: the objective function, the core\nconstraints, one fleet model, and one routing model. \\Cref{sec:complete}\npresents an example augmented with the constraints.\n\n\\subsection{Entities}\n\n\\begin{table}[htb]\n  \\begin{tabularx}{\\textwidth}{r X}\n    $\\dep$             & the depot where the vehicles start and end their routes; \\\\\n    $\\comms$           & the set of product types (or \\emph{commodities}) that %\n                         must be dispatched;  \\\\\n    $\\vtypes$          & the set of vehicle types that can be used to deliver the %\n                         demand, which differ in capacity and cost;  \\\\\n                         $\\capa_\\vtype$     & the total capacity of a type of vehicle (across all the commodities); \\\\\n    $\\custs$           & the set of customers to be served. We also denote by %\n                         $\\custi = \\custs \\cup \\{\\dep\\}$ the set of all locations %\n                         in the problem;  \\\\\n    $\\dem_{i,\\comm}$   & the demand of commodity $\\comm \\in \\comms$ by customer %\n                         $i \\in \\custs$, we also denote by $\\dem_i$ the total %\n                         demand of customer $i$ (i.e. %\n                         $\\dem_i = \\sum_{\\comm \\in \\comms} \\dem_{i, \\comm}$);  \\\\\n    $\\comp_{t,\\comm}$  & the compatibility between vehicle type %\n                         $\\vtype \\in \\vtypes$ and commodity $\\comm \\in \\comms$\n    \\begin{equation*}\n      \\comp_{t,\\comm} = \\begin{cases}\n        1 & \\text{if vehicle type}~\\vtype~\\text{can transport commodity}~\\comm,\\\\\n        0 & \\text{otherwise;}\n      \\end{cases}\n    \\end{equation*}  \\\\\n    $\\edges$           & the connections $\\edge_{i,j},\\;\\forall i, j \\in \\custi$ %\n                         between the locations; and \\\\\n    $\\cost_{t,i,j}$    & the cost of travelling on a connection $\\edge_{i,j}$ with %\n                         a vehicle of type $t$. \\\\\n    $\\dist_{i,j}$      & the the length of edge $\\edge_{i, j}$. \\\\\n  \\end{tabularx}\n  \\label{tab:entities}\n\\end{table}\n\nWe consider the distribution network to be a graph $\\mathbf{G} = \\langle \\custi,\n\\edges\\rangle$ where the locations $\\custi$ represent nodes and the connections\n$\\edges$ represent the edges. Moreover, all the models will refer to a set of\nvehicles $\\veh \\in \\vehs$ that can be either fixed or to be decided. We list the\ndifferent entities used to model the problem in \\Cref{tab:entities}.\n\n\n\\subsection{Objective}\n\nThe problem we address is the daily component of an annual fleet size-and-mix\nproblem. The long term costs of running vehicles can be amortised to a daily\nfixed cost, but we decided not to take them into account as:\n\\begin{inparaenum}[\\itshape a)]\n\\item our partenr was not able to provide those costs;\n\\item they make little sense in the daily setting, although they are an easy\n  extension.\n\\end{inparaenum}\n\nTherefore the objective in our models is only to minimise the routing costs.\n\n\\subsection{Constraints}\n\nWe distinguish four types of \\emph{core} constraints that will be present in\nevery problem variant.\n\\begin{description}\n\\item[\\textbf{Demand satisfaction}]\n  the demands of all customers must be satisfied, that is the sum of all\n  deliveries of a given commodity to a customer must equal its demand.\n\\item[\\textbf{Capacity constraints}]\n  the capacities of vehicles must be respected, that is the sum of all\n  deliveries by a single vehicle must not exceed its capacity.\n\\item[\\textbf{Used vehicles}]\n  unused vehicles stay at the depot, and therefore cannot travel on any\n  edge or serve any customers.\n\\item[\\textbf{Visited customers}]\n  vehicles can only deliver customers that they visited.\n\\end{description}\n\n\\section{Our Approach}\n\\label{sec:approach}\n\nIn this section we present our approach, whose modularity emerges from the\nnatural decomposition of the addressed problem into fleet design and routing\ncomponents. In particular, we describe two models for each component, which can\nbe combined to assemble four different complete models. Moreover, we present a\nnumber of valid cuts and symmetry breaking constraints that can be applied to\nthe models.\n\n\\subsection{Common Variables}\n\nOur models share three sets of variables. The routing decisions are modelled\nusing the Boolean variables $x$ indicating whether a vehicle travels on a given\nedge in the solution. The amount of each commodity delivered by a vehicle at a\ncustomer is modelled by the continuous variables $y$. Finally, the Boolean\nvariables $\\use$ represent the \\emph{unused} vehicles, i.e., if $\\use$ is equal\nto one, it means that the vehicle is not used.\n\nThe usage variable comes from the yearly problem and tells whether a vehicle is\nused on a given day. We decided to keep it in the daily setting as we found it\nlead to stronger relaxations when deciding the fleet composition, it is also\nused to formulate some valid cuts (e.g., \\eqref{eq:max_vehs}).\n\n\\subsection{Fleet Models}\n\nIn this section we describe two different models to deal with the fleet design\ncomponent of the problem. In the \\emph{Flexible} fleet, we have a fixed number\nof trucks and use a Boolean variable $z$ to determine the type of each\nvehicle. We can easily compute the maximum size of the fleet by considering only\nthe smallest trucks, then the question of the fleet composition remains. In the\n\\emph{Stable} fleet, we assign each vehicle a type beforehand and size the fleet\naccording to their respective characteristics.\n\n\\subsubsection{Flexible}\n\nIn addition to the core variables this model uses an additional set of Boolean\nvariables $z_{v,t}$ to indicate if vehicle $\\veh \\in \\vehs$ is of type $\\vtype\n\\in \\vtypes$. In order to keep a linear model, as different vehicle types have\ndifferent costs, we have to decompose the routing decision on the type of the\nvehicle and therefore use variables $x_{\\veh, \\vtype, i, j}$ to indicate if a\nvehicle $\\veh \\in \\vehs$ of type $\\vtype \\in \\vtypes$ travels on edge\n$\\edge_{i,j} \\in \\edges$. As such, vehicles have access to networks of different\ncosts, but can only use the network of their chosen type. Finally, we need to\nenforce compatibility because the type of each vehicle is not known in advance.\n\\model{%\n    \\sum_{i \\in \\custs} y_{v, i, k} & \\leq \\sum_{t \\in \\vtypes} z_{v, t} \\cdot \\capa_t \\cdot \\comp_{t, k} & \\forall \\veh \\in \\vehs, k \\in \\comms \\\\\n    \\sum_{t \\in \\vtypes} z_{v, t} &= 1 & \\forall \\veh \\in \\vehs \\\\\n    \\sum_{\\edge_{i, j} \\in \\edges} x_{v, t, i, j} & \\leq |\\edges| \\cdot z_{v, t} & \\forall \\veh \\in \\vehs, t \\in \\vtypes\n}{%\n    \\item Compatibility constraint.\n    \\item All vehicles must have exactly one type.\n    \\item Vehicles can only use edges associated with their chosen type.\n}\n\n\\subsubsection{Stable}\n\nInstead of determining the composition of the fleet during the optimisation\nanother approach is to have vehicles with fixed types from the start. We obtain\na model with \\emph{pools} of vehicles of a given type. We then need to size each\nof these pools so that we are able to satisfy the demands.\n\nPresetting the type of the vehicles allows us to use a three-index routing\nvariable, $x_{v, i, j}$, thus reducing greatly the number of variables in the\nmodel.\n\nFurthermore, we reduce the number of $y$ variables as compatibility becomes an\nimplicit characteristic. As it also eliminates infeasible deliveries, the\ncompatibility constraints can be removed. In the following we denote $\\comms^v$\nthe set of compatible commodities for vehicle $v$. Finally, the $y$ variables\nare bounded by the size of the vehicle instead of the maximum size in the fleet.\n\n\\subsection{Routing Models}\n\nIn this section we present two different models to deal with the routing aspects\nof the problem. Both are based on classic formulations in the literature. The\nfirst model is called \\emph{Vehicle Flow}, reflecting the fact that the main\nconstraint models the balance in the in-flow and out-flow from a node.  The\nsecond model is called \\emph{Commodity Flow}, since it employs an additional\nvariable tracking the amount of goods carried on the edges.\n\n\\subsubsection{Vehicle Flow}\n\nThis model is very close to the standard 2-index model from the literature\n\\citet{Laporte83}, and requires a rather small number of variables. The\ndecision variables are the amount of flow on each vehicle between each pair of\ncustomers.\n\\begin{align}\n  \\sum_{j \\in \\custi} x_{\\veh,i,j} &= \\sum_{j \\in \\custi} x_{\\veh,j,i} & \\forall \\veh \\in \\vehs, i \\in \\custi\n\\end{align}\n\nThis formulation does not automatically eliminate sub-tours, and hence sub-tour\nelimination constraints must be included. Because these are exponential in the\nnumber of nodes, we use a classic sub-tour elimination procedure, which fires\nwhenever an integer solution is found during the branch-and-bound search. If we\nfind any sub-tour $S$ not including the depot, the following constraint is\nadded, with $\\edges^S$ being the set of edges in S:\n\\begin{align}\n  \\sum_{\\edge_{i,j} \\in \\edges^S} x_{\\veh,i,j} & \\leq |S|-1 & \\forall S \\subseteq \\custs, \\veh \\in \\vehs \\label{eq:subtours}\n\\end{align}\n\n\\subsubsection{Commodity Flow}\n\nThis model uses a set of variables $f_{v, k, i, j}$ representing the amount of\ncommodity $\\comm$ transported by vehicle $\\veh$ on $\\edge_{i,j}$\n\\citep[see][]{Gavish78}. As a consequence, the Commodity flow formulation has more\nvariables than the Vehicle flow formulation but results in a polynomial-sized model\nsince no sub-tour elimination constraints are required.\nIn order to guarantee flow consistency, we need to add the following set of\nconstraints:\n\\model{%\n    \\sum_{j \\in \\custi} \\left( f_{\\veh,\\comm, j, i} - f_{\\veh, \\comm, i, j} \\right) & = y_{\\veh, i, \\comm} & \\forall \\veh \\in \\vehs, i \\in \\custs, \\comm \\in \\comms \\\\\n    \\sum_{\\comm \\in \\comms} f_{\\veh, \\comm, i, j} & \\leq x_{\\veh, i, j} \\cdot \\capa_\\veh & \\forall \\veh \\in \\vehs, \\edge_{i,j} \\in \\edges\n}{%\n    \\item The amount deposited at $i$ by vehicle $v$ is the difference between\n      the amount carried before the visit and after the visit.\n    \\item A vehicle can only carry a load on the edges it uses.\n}\n\n\\subsection{Valid Cuts}\n\n\\emph{Valid cuts} (or \\emph{inequalities}) are an extremely effective technique\nto solve MIP problems efficiently. Similarly to \\emph{redundant constraints} in\nCP, they are used to improve the strength of currently existing constraints\nwithout removing optimal solutions. In this section we will present a set of\nvalid cuts derived from the literature and adapted to work in our rich VRP\ncontext.\n\nWe do not present all variants of the constraints, only their most generic\nexpression. If a constraint differs considerably between two models, we present\nthe two formulations; otherwise, assume no changes apart from the indices.\n\n\\subsubsection{Minimum Visits}\n\nThis cut \\citep{Dror94} provides a lower bound on the number of vehicles\nrequired to visit a given customer, it is obtained by dividing its total demand\nby the maximum capacity in our fleet.\n\\begin{align}\n  \\sum_{\\veh \\in \\vehs} \\sum_{j \\in \\custi} x_{\\veh,j,i} & \\geq \\left\\lceil\\frac{\\dem_{i}}{\\capa_{max}}\\right\\rceil & \\forall i \\in \\custs\n\\end{align}\n\n\\subsubsection{Minimum Vehicles}\n\nThis constraint, together with the branching strategy, was the most successful\nimprovement in the search for feasible solutions with a flexible fleet in\n\\citet{Kilby16}.\n\nIn the Vehicle model a similar cut can be implemented by ensuring that we will\nhave enough vehicle capacity to serve all demands.\n\\begin{align}\n  \\sum_{\\veh \\in \\vehs} \\sum_{\\vtype \\in \\vtypes} z_{\\veh, \\vtype} \\cdot \\capa_\\vtype & \\geq \\sum_{i \\in \\custs} \\dem_{i,\\comm} & \\forall \\comm \\in \\comms \\label{eq:min_veh}\n\\end{align}\n\nIn the Stable model, since capacities are already fixed, we simply ensure that\nenough vehicles leave the depot by using the $\\use$ (usage) variables. This\nprovides an earlier cut in the branch-and-bound search since $\\use$s have high\nbranching priority. Let $F_\\vtype$ be the total number of vehicle of type\n$\\vtype$.\n\\begin{align}\n  \\sum_{\\veh \\in \\vehs} \\use_\\veh & \\leq \\sum_{\\vtype \\in \\vtypes} F_\\vtype - \\sum_{i \\in \\custs} \\left\\lceil\\frac{\\dem_{i}}{\\capa_{max}}\\right\\rceil \\tag{\\ref{eq:min_veh}$^\\prime$}\n\\end{align}\n\n\\subsubsection{Maximum Vehicles}\n\nConversely, when using the Stable fleet model, we can limit the number of\nvehicle in operation by imposing a lower bound on the number $V^{\\dep}$ of\nvehicles staying at the depot.\n\\begin{align}\n  \\sum_{\\veh \\in \\vehs} \\use_\\veh & \\geq V^{\\dep} \\label{eq:max_vehs}\n\\end{align}\n\n\\subsubsection{Fractional Sub-Tours}\n\nThis cut \\citep{Dror94} requires that if a vehicle visits a customer it has to\ntravel somewhere else afterwards.\n\\begin{align}\n  x_{\\veh, i, j} & \\leq \\sum_{\\mathclap{l \\in \\custi, l \\neq i}} x_{\\veh, j, l} & %\n  \\forall \\veh \\in \\vehs, \\edge_{i, j} \\in \\edges\n\\end{align}\n\n\\subsubsection{Depot Outgoing Degree}\n\nIn its general form, this cut \\citep{Dror94} states that all vehicles have to\nexit the depot, however we amend it to allow unused vehicles to stay at the\ndepot.\n\\begin{align}\n  \\sum_{j \\in \\custs} x_{\\veh, \\dep, j} + \\use_\\veh & = 1 & \\forall \\veh \\in \\vehs\n\\end{align}\n\n\\subsubsection{Single visit.}\nThis cut enforces the fact that a vehicle can visit a customer at most once,\nusing the following symmetric constraints:\n\\begin{align}\n  \\sum_{j \\in \\custi} x_{\\veh,i,j} & \\leq 1 & \\forall \\veh \\in \\vehs, i \\in \\custs \\\\\n  \\sum_{j \\in \\custi} x_{\\veh,j,i} & \\leq 1 & \\forall \\veh \\in \\vehs, i \\in \\custs\n\\end{align}\n\n\\subsection{Symmetry Breaking Constraints}\n\n\\emph{Symmetry breaking} constraints try to alleviate the degeneracies of a\nmodel by removing equivalent solutions. In this section we assume that the input\ndata, such as vehicles or customers, is ordered as integer sequences. In\nparticular, the depot would be customer $0$.\n\n\\subsubsection{Usage}\n\nIf a vehicle is unused, the \\emph{next} vehicle must be unused too so as to\navoid recombination to determine the number of used vehicles.\n\\begin{align}\n  \\use_{\\veh-1} & \\geq \\use_{\\veh} & \\forall \\veh \\in \\vehs\n\\end{align}\n\n\\subsubsection{Visit Order}\n\nWe impose that a vehicle $\\veh$ can visit customer $i$ only if vehicle $\\veh-1$\nhas visited any location in $[1..i]$.\n\n\\begin{align}\n  \\sum_{h \\in \\custi} \\sum_{l = 1}^j x_{\\veh - 1, h, l} & \\geq x_{\\veh, i, j} & \\forall \\veh \\in \\vehs, i \\in \\custs, j \\in \\custi \\label{eq:visit_order}\n\\end{align}\n\nIn the Flexible model this must be restricted to vehicles of the same type.\n\\begin{align}\n  \\sum_{u = 0}^t z_{\\veh-1, u} + \\sum_{h \\in \\custs} \\sum_{l = 1}^{j} x_{\\veh - 1, \\vtype, h, l} & \\geq x_{\\veh, \\vtype, i, j} & \\forall \\veh \\in \\vehs, \\vtype \\in \\vtypes, i \\in \\custs, j \\in \\custi \\tag{\\ref{eq:visit_order}$'$}\n\\end{align}\n\n\\subsubsection{Fleet Order}\n\nWhen using the Flexible model, we force the vehicle types to be ordered, that is\nif vehicle $\\veh$ is of type $\\vtype$, then vehicle $\\veh + 1$ can only take\ntypes $[t .. T]$.\n\\begin{align}\n  \\sum_{\\tau = 0}^{\\vtype} z_{\\veh - 1, \\tau} & \\geq z_{\\veh, \\vtype} & \\forall \\veh \\in \\vehs, \\vtype \\in \\vtypes\n\\end{align}\n\n\\subsubsection{Customer Assignment}\n\nInspired by \\citet{Dror94}, where the authors design a cut assigning the first\nvehicle to the farthest customer, we re-order the input data so that the farther\ncustomers (from the depot) are assigned smaller indices.\n\nUsing a Flexible fleet, with $\\hat{\\jmath}$ being the farthest customer, we\nhave:\n\\begin{align}\n  \\sum_{i \\in \\custs} \\sum_{t \\in \\vtypes} x_{0, t, i, \\hat{\\jmath}} = 1 \\label{eq:farthest_customer}\n\\end{align}\n\nWith a Stable fleet we can only have a weaker constraint where we force at least\none of the first vehicles of each type to visit the farthest customer. With\n$\\Pi$ being the sets of such vehicles.\n\\begin{align}\n  \\sum_{\\veh \\in \\Pi} \\sum_{i \\in \\custs} x_{\\veh, i, \\hat{\\jmath}} & \\geq 1 \\tag{\\ref{eq:farthest_customer}$^\\prime$}\n\\end{align}\n\n\\subsubsection{Total Load}\n\nIn the Commodity flow model we also enforce that all vehicles have to leave the\ndepot fully loaded. However, we do not constrain their final load as this can be\nadjusted easily.\n\\begin{align}\n\\sum_{\\comm \\in \\comms} \\sum_{j \\in \\cust} f_{\\veh, \\comm, \\dep, j} & = \\capa_\\veh & \\forall \\veh \\in \\vehs\n\\end{align}\n\n\\subsection{Complete Example}\n\\label{sec:complete}\n\nWe present here a complete mathematical formulation using: Commodity routing\nmodel (\\Cref{eq:balance,eq:carry}) and Stable fleet model (e.g. implicit\ncompatibility with $\\comms^\\veh$). The model is composed of: the objective\nfunction, core constraints (\\Cref{eq:demand,eq:capacity,eq:usage,eq:visited}),\nvalid cuts (\\Cref{eq:nb_visits,eq:min_vehs,eq:max_vehs,eq:frac_subtour,%\n  eq:depot_exit,eq:depot_load,eq:single_entry,eq:single_exit}), symmetry\nbreaking constraints (\\Cref{eq:use_order,eq:cust_order}), and ordering\nconstraints (\\Cref{eq:farthest_visit}).\n\n\\named{mip}%\n      {MIP}%\n      {\\sum_{\\veh \\in \\vehs} \\sum_{\\edge_{i, j} \\in \\edges} x_{\\veh, i, j} \\cdot \\cost_\\veh \\cdot \\dist_{i, j} \\label{eq:objective}}%\n      {\\sum_{\\veh \\in \\vehs} y_{\\veh, \\cust, \\comm} & = \\dem_{\\cust, \\comm} & \\forall \\cust \\in \\custs, \\comm \\in \\comms \\label{eq:demand} \\\\\n        && \\sum_{\\cust \\in \\custs} \\sum_{\\comm \\in \\comms^\\veh} y_{\\veh, \\cust, \\comm} & \\leq \\capa_{\\veh} & \\forall \\veh \\in \\vehs  \\label{eq:capacity} \\\\\n        && \\sum_{\\edge_{i,j} \\in \\edges} x_{\\veh, i, j} & \\leq u_{\\veh} & \\forall \\veh \\in \\vehs  \\label{eq:usage} \\\\\n        && \\sum_{j \\in \\custs} x_{\\veh, j, \\cust} \\cdot \\dem_{\\cust, \\comm} & \\geq y_{\\veh, \\cust, \\comm} & \\forall \\veh \\in \\vehs, i \\in \\custs, \\comm \\in \\comms^\\veh \\label{eq:visited} \\\\\n        && \\sum_{j \\in \\custi} (f_{\\veh,\\comm, j, i} - f_{\\veh, \\comm, i, j}) & =  y_{\\veh, \\cust, \\comm}  & \\forall \\veh \\in \\vehs, i \\in \\custs,  \\comm \\in \\comms ^\\veh \\label{eq:balance} \\\\\n        && \\sum_{\\comm \\in \\comms^\\veh} f_{\\veh,\\comm, i, j} & \\leq x_{\\veh, i, j} \\cdot \\capa_\\veh & \\forall \\veh \\in \\vehs, \\edge_{i,j} \\in \\edges  \\label{eq:carry} \\\\\n        && \\sum_{\\veh \\in \\vehs} \\sum_{j \\in \\custi} x_{\\veh,j,i} & \\geq \\left\\lceil \\frac{\\dem_{i}}{\\capa_{max}} \\right\\rceil & \\forall i \\in \\custs  \\label{eq:nb_visits} \\\\\n        && \\sum_{\\veh \\in \\vehs} \\use_\\veh & \\leq \\sum_{\\vtype \\in \\vtypes} F_\\vtype - \\sum_{i \\in \\custs} \\left\\lceil \\frac{\\dem_{i}}{\\capa_{max}} \\right\\rceil \\span  \\label{eq:min_vehs} \\\\\n        && \\sum_{\\veh \\in \\vehs} \\use_\\veh & \\geq V^{\\dep}  \\label{eq:max_vehs} \\\\\n        && x_{\\veh, i, j} & \\leq \\sum_{\\mathclap{l \\in \\custi, l \\neq i}} x_{\\veh, j, l} & \\forall \\veh \\in \\vehs, \\edge_{i, j} \\in \\edges  \\label{eq:frac_subtour} \\\\\n        && \\sum_{j \\in \\custs} x_{\\veh, \\dep, j} + \\use_\\veh & = 1 & \\forall \\veh \\in \\vehs  \\label{eq:depot_exit} \\\\\n        && \\sum_{j \\in \\custs} \\sum_{\\comm \\in \\comms^\\veh} f_{\\veh, \\comm, \\dep, j} &  = \\capa_\\veh & \\forall \\veh \\in \\vehs  \\label{eq:depot_load} \\\\\n        && \\sum_{j \\in \\custi} x_{\\veh,i,j} & \\leq 1 & \\forall \\veh \\in \\vehs, i \\in \\custs  \\label{eq:single_entry} \\\\\n        && \\sum_{j \\in \\custi} x_{\\veh,j,i} & \\leq 1 & \\forall \\veh \\in \\vehs, i \\in \\custs  \\label{eq:single_exit} \\\\\n        && \\use_{\\veh-1} & \\geq \\use_{\\veh} & \\forall \\veh \\in \\vehs  \\label{eq:use_order} \\\\\n        && \\sum_{h \\in \\custi} \\sum_{l = 1}^j x_{\\veh - 1, h, l} & \\geq x_{\\veh, i, j} & \\forall \\veh \\in \\vehs, i \\in \\custs, j \\in \\custi  \\label{eq:cust_order} \\\\\n        && \\sum_{\\veh \\in \\Pi} \\sum_{i \\in \\custs} x_{\\veh, i, \\hat{\\jmath}} & \\geq 1  \\label{eq:farthest_visit}\n      }%\n      {x \\in \\mathbb{B}, u \\in \\mathbb{B}, y \\geq 0, f \\geq 0}\n\n\\section{Experimental Comparison} \\label{sec:expes}\n\nIn this section we compare several variants of the presented models. Each\nexperiment is run for $15$ minutes on an AMD Opteron $4334$ machine running at\n$3.1$ GHz and with $64$ GB of RAM. The models were implemented in \\textsc{Gurobi}~$6.5$\\xspace\n\\citep{gurobi}, and run using the default parameters except a higher branching\npriority for the usage variable for all models, and then for the truck type in\nthe Flexible variant.\n\nWe have three groups of instances: small, medium, and large. The latter\nrepresent real demand data, whereas in medium and small a different degree of\nclustering is used to aggregate the demands of nearby customers. On average,\nthere are $5$ customer nodes in a small instance, $15$ in a medium, and $18$ in\na large.\n\nTo size the fleet, we rely on a characteristic of our case study: refrigerated\nvehicles are more expensive to run than regular vehicles. Hence, we only\nconsider chilled demand for determining the number of refrigerated vehicles\nrequired, and ambient demand for regular vehicles.\n\nIn our results we use the following abbreviations: the first letter is for the\nfleet model (\\emph{s} for Stable and \\emph{f} for Flexible); the second is for\nthe routing model (\\emph{c} for Commodity and \\emph{v} for Vehicle).\n\n\\subsection{MIP Comparison}\n\\label{sec:mip_results}\n\nThis first set of experiments reports the results obtained over nine small\ninstances by the four MIP models, augmented with valid cuts, symmetry\nconstraints, and ordering constraints. Each instance was run for a maximum of\nfifteen minutes. We report the best objective value found across all variants;\nthen the run time in seconds (if the time limit was not exceeded) or the final\nintegrality gap (if the time limit was hit) for each of the variants; and\nfinally the number of sub-tour elimination constraints \\eqref{eq:subtours} added\nin the two models based on Vehicle flow. The last line recaps the average gap\nfound on instances not solved to optimality. We highlight rows where optimality\nwas not reached by any model as well as the best time or gap per row.\n\nThe tables in \\Cref{ap:results} provide the results for the complete set of\nexperiments, starting with the standard MIP formulation as presented in\n\\citet{Kilby16}. \\Cref{tab:order} presents the results of the most complete\nmodel which consists of: valid cuts, symmetry breaking constraints, and\nordering.\n\nIn \\Cref{tab:relaxation} we provide the time taken in seconds and the gap\nbetween the optimal solution, or best known solution, and the linear relaxation;\nand respectively with the first heuristic solution in \\Cref{tab:heuristic}.\n\n\\ctab{Results of using different models. For each instance we report the best value found, time taken or integrality gap at $15$ min, and number of sub-tour elimination constraints. The best results are in bold. Instances where the optimum is not reached are greyed.}\n     {order}\n     {*{8}{C}}\n     {\n     \\multirow{2}{*}{Instance} & \\multirow{2}{*}{Value} & \\multicolumn{4}{c}{Time (s) or gap at 900s} & \\multicolumn{2}{c}{\\# subtours \\eqref{eq:subtours}} \\\\\n     \\cmidrule(lr){3-6} \\cmidrule(lr){7-8}\n     & & sc & sv & fc & fv & sv & fv \\\\\n     \\midrule\n     1 & 31358.95 & \\bfseries 15.19 & 26.02 & 32.63 & 33.81 & 216 & 250 \\\\\n     2 & 45538.02 & \\bfseries 723.32 & 758.51 & (0.20\\%) & (1.13\\%) & 196 & 325 \\\\\n     3 & 54150.16 & \\bfseries 70.89 & 343.70 & 503.72 & (0.62\\%) & 456 & 1020 \\\\\n     4 & 45435.52 & \\bfseries 326.36 & 680.91 & 505.64 & (1.30\\%) & 341 & 490 \\\\\n     5 & 53989.22 & \\bfseries 817.45 & (1.02\\%) & (1.27\\%) & (1.79\\%) & 117 & 255 \\\\\n     \\rowcolor{Gray}\n     6 & 60943.93 & \\bfseries (0.11\\%) & (1.55\\%) & (0.83\\%) & (3.13\\%) & 765 & 1700 \\\\\n     \\rowcolor{Gray}\n     7 & 62864.21 & \\bfseries (0.62\\%) & (1.81\\%) & (1.12\\%) & (3.77\\%) & 430 & 1700 \\\\\n     8 & 23219.89 & \\bfseries 2.25 & 2.35 & 25.46 & 22.99 & 0 & 0 \\\\\n     9 & 44738.91 & \\bfseries 125.65 & 259.11 & 281.49 & 550.84 & 203 & 140 \\\\\n     \\cmidrule{2-6}\n     \\multicolumn{2}{r}{Avg. gap} & \\bfseries 0.36\\% & 1.46\\% & 0.85\\% & 1.96\\% \\\\\n     }\n\n\\ctab{Root node relaxation results. The Time columns give the time taken to solve the relaxed problem; the Gap columns give the percentage difference with the best known value. The smallest gaps are in bold. Instances where optimum is not reached are greyed.}\n     {relaxation}\n     {c *{8}{C}}\n     {\n       \\multirow{2}{*}{Instance} & \\multicolumn{2}{c}{sc} & %\n       \\multicolumn{2}{c}{sv} & \\multicolumn{2}{c}{fc} & \\multicolumn{2}{c}{fv} \\\\\n       \\cmidrule(lr){2-3} \\cmidrule(lr){4-5} \\cmidrule(lr){6-7} \\cmidrule(lr){8-9}\n       & Time (s) & Gap (\\%) & Time (s) & Gap (\\%) & Time (s) & Gap (\\%) %\n       & Time (s) & Gap (\\%)  \\\\\n       \\midrule\n       1 & 0.44 & \\bfseries 6.61 & 0.35 & 88.64 & 2.04 & 8.53 & 1.18 & 86.36 \\\\\n       2 & 0.52 & \\bfseries 8.41 & 0.29 & 81.05 & 3.07 & 8.92 & 1.23 & 99.98 \\\\\n       3 & 1.09 & \\bfseries 3.35 & 0.48 & 112.97 & 4.99 & 4.13 & 3.02 & 136.74 \\\\\n       4 & 0.73 & \\bfseries 3.43 & 0.45 & 112.15 & 3.58 & 3.56 & 1.92 & 138.54 \\\\\n       5 & 1.05 & 4.38 & 0.42 & 83.89 & 4.68 & \\bfseries 4.12 & 2.16 & 128.39 \\\\\n       \\rowcolor{Gray}\n       6 & 1.10 & \\bfseries 4.47 & 0.61 & 111.09 & 6.86 & 4.57 & 3.48 & 131.70 \\\\\n       \\rowcolor{Gray}\n       7 & 1.47 & \\bfseries 4.55 & 0.50 & 111.39 & 6.11 & 4.88 & 4.12 & 153.74 \\\\\n       8 & 0.10 & 13.99 & 0.07 & 14.36 & 0.67 & \\bfseries 13.22 & 0.40 & 77.54 \\\\\n       9 & 0.66 & \\bfseries 8.05 & 0.33 & 79.06 & 2.73 & 8.86 & 1.59 & 95.60 \\\\\n       \\midrule\n       \\multicolumn{2}{l}{Avg. gap (\\%)} & \\bfseries 6.89 && 81.73 && 7.33 && 109.02 \\\\\n     }\n\n\\ctab{First integer solution results. The Time columns give the time taken to find the first integer solution; the Gap columns the percentage difference with the best known value. The smallest gaps are in bold. Instances where optimum is not reached are greyed.}\n     {heuristic}\n     {*{9}{C}}\n     {\n       \\multirow{2}{*}{Instance} & \\multicolumn{2}{c}{sc} & %\n       \\multicolumn{2}{c}{sv} & \\multicolumn{2}{c}{fc} & \\multicolumn{2}{c}{fv} \\\\\n       \\cmidrule(lr){2-3} \\cmidrule(lr){4-5} \\cmidrule(lr){6-7} \\cmidrule(lr){8-9}\n       & Time (s) & Gap (\\%) & Time (s) & Gap (\\%) & Time (s) & Gap (\\%) %\n       & Time (s) & Gap (\\%)  \\\\\n       \\midrule\n       1 & 0.15 & 56.16 & 0.08 & 54.20 & 0.46 & 55.85 & 0.44 & \\bfseries 42.76 \\\\\n       2 & 0.10 & 54.63 & 0.01 & 50.24 & 9.27 & 13.84 & 8.99 & \\bfseries 13.00 \\\\\n       3 & 0.15 & 56.68 & 0.38 & 37.90 & 7.13 & \\bfseries 34.25 & 2.53 & 38.10 \\\\\n       4 & 0.10 & 55.43 & 0.03 & 45.64 & 16.71 & \\bfseries 21.59 & 1.69 & 27.01 \\\\\n       5 & 0.16 & 56.08 & 0.11 & 32.80 & 8.00 & 32.46 & 1.89 & \\bfseries 31.66 \\\\\n       \\rowcolor{Gray}\n       6 & 0.16 & 57.59 & 0.01 & 49.69 & 29.16 & \\bfseries 10.33 & 2.99 & 32.88 \\\\\n       \\rowcolor{Gray}\n       7 & 0.18 & 57.31 & 0.39 & 39.54 & 64.35 & \\bfseries 19.13 & 3.62 & 38.09 \\\\\n       8 & 0.01 & 44.76 & 0.00 & \\bfseries 36.79 & 0.19 & 47.12 & 0.14 & 44.07 \\\\\n       9 & 0.14 & 58.67 & 0.01 & 53.48 & 23.59 & \\bfseries 17.76 & 1.24 & 29.88 \\\\\n       \\midrule\n       \\multicolumn{2}{l}{Avg. gap (\\%)} & 54.63 && 44.44 && \\bfseries 31.84 && 32.35 \\\\\n     }\n\nFrom these results we can clearly see that a model dominates the others: the\nStable fleet with Commodity routing, Model SC, presented in its augmented\nversion as Model \\eqref{mod:mip}. First of all, predefining the types of each\nvehicle instead of having to determine their type during the optimization is the\nbiggest improvement. While it increases the number of vehicles to route it\ndecreases the number of decision variables.\nSecondly, having a polynomial-sized formulation by using a flow-based\nformulation for the routing gives far better results than relying on sub-tour\nelimination. This is explained by the fact that such a formulation has a better\nlinear relaxation (see \\Cref{tab:relaxation}), and therefore a smaller search\ntree. Conversely, the reason why the first integer solution for flow based\nmodels is better is because they need to add sub-tour elimination constraints to\nreach feasibility, and it is reflected in the time they take to find that first\nsolution.\n\nThis result concurs with those of \\citet{Yaman2006} whose best performing model\nis based on a flow model and fleet is always predefined.\n\n\\subsection{Warm-Starting the MIP Using CP}\n\nIn this second set of experiments we attempt to integrate CP to MIP. We devised\na rather lightweight integration between the CP solver in \\citet{Kilby16} and\nour MIP models. Since the CP-based solver returns a reasonably good solution\nquickly, the idea is: first, find a solution with CP; then use that solution as\na starting solution for the MIP; and, from there, let the MIP search proceed\nnormally. We call this: \\emph{warm starting} the MIP.\n\nIn the following experiments, we keep the same $15$-minutes time limit as\nbefore, but we allow a variable amount of time (ranging between $0$ and $15$\nminutes) to the CP solver to find an initial solution. In order to keep the\nlength of the experiments constant, the longer the CP search runs, the shorter\nthe time limit for the MIP search.\n\nThe CP solver functions by finding a first feasible solution and then proceeding\nto improve it by means of an LNS procedure. Therefore, in our experiments,\nallocating a time of $0$ seconds to the CP solver means we use the first\nfeasible solution it finds instead of letting the CP solver improve it.\n\n\\Cref{fig:gap} presents the evolution of the final MIP gap on a selected\ninstance, day 6, for all instance sizes and model variations. We report the\nfinal integrality gap function of the time allocated to CP.\n\n\\eps{gap}{Evolution of final gap when using CP solution, based on a single\n  instance, day 6, for the four models. The $x$ axis is the time allotted to the\n  CP model, then the MIP has the remaining of the fifteen minutes. A dot means\n  the MIP finished. A missing curve means that the LP relaxation at the root\n  node did not finish under the remaining time.}\n\nThe slight dip early in \\Cref{fig:gap} shows that the first solution found by CP\nis not the most efficient for warm starting the MIP search, and letting CP\nimprove it for about one minute yields better results. Allowing longer CP\nrun-times decreases the overall solution quality. The reason behind this is that\nCP finds a feasible solution quickly, then has to spend more and more time\nimproving it as the solutions get better. At a given point, letting MIP improve\nthe solution proves to be a better strategy than letting CP do it. For the\nflexible fleet with commodity flow, it is even impossible to find even the\nsolution to the linear relaxation at the root node under fifteen minutes on\nlarge instances.\n\n\\Cref{tab:prime} reports the results of warm-starting the MIP using a solution\nfound by the CP model in 1 minute; it presents the results in a similar fashion\nto those in \\Cref{sec:mip_results}. Do note that the time reported is for the\nMIP only, not taking the CP runtime into account.\n\n\\ctab{Warm-starting MIP with a CP solution found in 1 minute}%\n     {prime}%\n     {*{8}{C}}\n     {\n     \\multirow{2}{*}{Instance} & \\multirow{2}{*}{Value} & \\multicolumn{4}{c}{Time (s) or gap at 900s} & \\multicolumn{2}{c}{\\# subtours \\eqref{eq:subtours}} \\\\\n     \\cmidrule(lr){3-6} \\cmidrule(lr){7-8}\n     & & sc & sf & fc & ff & sf & ff \\\\\n     \\midrule\n     1 & 31358.95 & 29.31 & 27.82 & \\bfseries 19.83 & 58.84 & 120 & 300 \\\\\n     2 & 45538.02 & \\bfseries 522.84 & 806.01 & 598.22 & (0.36\\%) & 196 & 260 \\\\\n     3 & 54150.16 & \\bfseries 72.99 & 664.07 & 506.86 & (0.26\\%) & 342 & 340 \\\\\n     4 & 45435.52 & 222.91 & \\bfseries 207.41 & 438.87 & 517.95 & 62 & 770 \\\\\n     5 & 53989.22 & \\bfseries 447.78 & (1.48\\%) & (1.37\\%) & (1.65\\%) & 78 & 425 \\\\\n     6 & 60943.93 & \\bfseries 664.57 & (2.15\\%) & (1.10\\%) & (3.51\\%) & 270 & 800 \\\\\n     \\rowcolor{Gray}\n     7 & 62864.21 & \\bfseries (0.45\\%) & (1.09\\%) & (1.53\\%) & (3.16\\%) & 301 & 1200 \\\\\n     8 & 23219.89 & \\bfseries 4.58 & 4.61 & 27.21 & 18.78 & 36 & 40 \\\\\n     9 & 44738.91 & \\bfseries 160.73 & 183.37 & 161.06 & 438.17 & 261 & 770 \\\\\n     \\cmidrule{2-6}\n     \\multicolumn{2}{r}{Avg. gap} & \\bfseries 0.45\\% & 1.58\\% & 1.34\\% & 1.79\\% \\\\\n     }\n\nWarm starting the MIP search with a CP solution gives satisfying results: one\nmore instance is now solved to optimality under the 15 minutes constraint;\noverall solving times are reduced; and, although the average gap reported on the\nlast line is higher for some models -- e.g. SC -- this comes from having less\nunsolved days -- and then the gap is lower, $0.62\\%$ vs. $0.45\\%$.\n\nOn the other hand, we can observe a slight decrease in performance in some\ncases, with higher time or final gaps. This comes from the \\emph{starting\n  location} for the MIP; in other words, imposing a starting solution to the MIP\nmodel makes the search start at a given position in the tree, with no warranty\nthat the optimal solution is close. In that case, letting the MIP model do the\nexploration and pruning may lead to a smaller search tree as the region of the\ninitial solution will never be explored.\n\nWe can observe that most model find the optimal solution early on but fail to\nprove its optimality, this is shown by the best value reported for unsolved\ninstances being equal to the optimal value. Using the CP to warm start the\nsearch helps reducing the size of the search tree, but not always. A way to\nimprove the solving process would be to determine which solutions are\ninteresting, or find a way to close the integrality gap faster.\n\n\\subsection{Results Over a Year}\n\nIn the final experiment from \\citet[Table 2]{Kilby16} the authors compared the\nresults of two CP models, one with a standard B\\&B search and one with the\ncustom LNS search, and their MIP model using one year of real world demand\nprovided by the client.\n\nWe ran the same experiment using the MIP model \\eqref{mod:mip}. \\Cref{tab:year}\nsummarizes the results on all instances, grouped by quarter, of: the CP model\nusing LNS search, our MIP, and our MIP warm-started with a CP solution obtained\nin one minute.\n\nFor each quarter, we report the average value found across all the instances,\nhow often the model produced the best known solution, the percentage difference\nwith the best found value for each instance, and the final gap for the MIP and\nthe combination CP+MIP.\n\n\\ctab{Summary of results using one year of data, grouped by quarter, for each instance size. For each approach we have the average value over all instances, the percentage of instances where the approach found the best known value, and the percentage difference with the best known value. We also report the average integrality gap for the MIP based approaches.}\n     {year}\n    {p{0.5em} p{0.25em} c C C *{2}{c C C C}}\n     {\n       && \\multicolumn{3}{c}{CP} & \\multicolumn{4}{c}{MIP} & \\multicolumn{4}{c}{CP (1min) + MIP} \\\\\n       \\cmidrule(lr){3-5} \\cmidrule(lr){6-9} \\cmidrule(lr){10-13}\n       && \\multirow{2}{*}{Value} & Best & Diff. %\n       & \\multirow{2}{*}{Value} & Best & Diff. & Gap %\n       & \\multirow{2}{*}{Value} & Best & Diff. & Gap  \\\\\n       && & (\\%) & (\\%) && (\\%) & (\\%) & (\\%) && (\\%) & (\\%) & (\\%) \\\\\n       \\midrule\n       \\multirow{3}{*}{Q1} & S & 48959 & 0.00 & 0.78 & 48582 & 7.78 & \\bfseries 0.00 & \\bfseries 0.81 & \\bfseries 48582 & \\bfseries 92.22 & 0.02 & 0.82 \\\\\n       & M & 51353 & 7.78 & 3.59 & 51320 & 40.00 & 3.50 & 7.39 & \\bfseries 50068 & \\bfseries 52.22 & \\bfseries 0.98 & \\bfseries 6.03 \\\\\n       & L & 52406 & 6.67 & 4.96 & 52455 & 27.78 & 5.00 & 10.17 & \\bfseries 50456 & \\bfseries 65.56 & \\bfseries 1.33 & \\bfseries 7.46 \\\\\n       \\midrule\n       \\multirow{3}{*}{Q2} & S & 47511 & 1.11 & 1.06 & 47030 & 7.78 & 0.01 & \\bfseries 0.68 & \\bfseries 47026 & \\bfseries 91.11 & \\bfseries 0.01 & 0.76 \\\\\n       & M & 49896 & 5.56 & 4.06 & 48954 & \\bfseries 47.78 & 2.42 & 6.50 & \\bfseries 48634 & 46.67 & \\bfseries 1.59 & \\bfseries 6.27 \\\\\n       & L & 50956 & 6.67 & 5.45 & 50602 & 27.78 & 4.43 & 9.89 & \\bfseries 48921 & \\bfseries 65.56 & \\bfseries 1.32 & \\bfseries 7.45 \\\\\n       \\midrule\n       \\multirow{3}{*}{Q3} & S & 49043 & 0.00 & 1.04 & 48534 & 7.69 & \\bfseries 0.00 & \\bfseries 0.74 & \\bfseries 48532 & \\bfseries 92.31 & 0.01 & 0.80 \\\\\n       & M & 51359 & 5.49 & 3.72 & 50551 & 37.36 & 2.37 & 6.88 & \\bfseries 50062 & \\bfseries 57.14 & \\bfseries 1.61 & \\bfseries 6.20 \\\\\n       & L & 52532 & 8.79 & 5.19 & 52090 & 26.37 & 4.07 & 9.99 & \\bfseries 50652 & \\bfseries 64.84 & \\bfseries 1.75 & \\bfseries 8.02 \\\\\n       \\midrule\n       \\multirow{3}{*}{Q4} & S & 51700 & 2.20 & 0.93 & 51261 & 12.09 & \\bfseries 0.00 & \\bfseries 0.73 & \\bfseries 51259 & \\bfseries 85.71 & 0.01 & 0.77 \\\\\n       & M & 54182 & 12.09 & 3.94 & 53972 & 30.77 & 2.88 & 7.12 & \\bfseries 53022 & \\bfseries 57.14 & \\bfseries 1.65 & \\bfseries 6.00 \\\\\n       & L & 54179 & 8.79 & 4.72 & 53654 & 23.08 & 3.17 & 9.34 & \\bfseries 52557 & \\bfseries 67.03 & \\bfseries 1.85 & \\bfseries 7.81 \\\\\n       \\midrule\n       \\multirow{3}{*}{Y} & S & 49303 & 0.83 & 0.95 & 48852 & 8.83 & \\bfseries 0.00 & \\bfseries 0.74 & \\bfseries 48850 & \\bfseries 90.34 & 0.01 & 0.79 \\\\\n       & M & 51698 & 7.73 & 3.83 & 51199 & 38.98 & 2.79 & 6.97 & \\bfseries 50446 & \\bfseries 53.29 & \\bfseries 1.46 & \\bfseries 6.12 \\\\\n       & L & 52518 & 7.73 & 5.08 & 52200 & 26.25 & 4.17 & 9.85 & \\bfseries 50646 & \\bfseries 65.74 & \\bfseries 1.56 & \\bfseries 7.68 \\\\\n     }\n\nThe combination CP+MIP consistently dominates its individual components. Even\nthough the MIP sometimes reaches a better value in the end, especially on medium\nsize instances, overall the combination CP+MIP has a lower value overall, as\nshown by the lower percentage difference overall. The combined models also have\na lower integrality gap after 15 minutes. Furthermore, the combined CP+MIP has a\ngreater success rate in finding the optimal under the time limit ($64\\%$)\ncompared to the simple MIP ($48\\%$).\n\nInstances \\emph{better} solved using the MIP without warm-start are symptomatic\nof those cases where the CP solution, albeit of good quality, is \\emph{far} from\nthe actual optimal and the MIP spends a lot of time backtracking in the search\ntree. In \\emph{small} instances, the MIP has a lower percentage difference and\nbetter gap because it solves less instances to optimality, therefore the\naverages are lower.\n\n\\section{Conclusions}\n\nWe presented various MIP formulations for the fleet size and mix,\nmulti\\hyp{}commodity, split-delivery vehicle routing problem with compatibility\nconstraints. Our formulation is modular, in that we split the fleet design and\nthe routing components of the problem and we develop two models for each\ncomponent.\n\nWe carried out an extensive experimental analysis to compare the MIP models, and\nto measure their performance with respect to the CP-based LNS approach presented\nin \\citet{Kilby16}. Overall, the Stable fleet model and Commodity routing model\ncombination seem to attain the best performance, outperforming the LNS approach\non most instances, and represents the most promising candidate for a further\nexploration of decomposition models for the multi-day extension of the problem.\nWithin our experimental analysis, we have also explored a lightweight\nintegration of CP and MIP, which consists in priming the MIP model with a\nsolution found by CP, replacing the heuristic solution found by the standard MIP\nsolver (in our case \\textsc{Gurobi}~$6.5$\\xspace). The effect on the integrality gap is positive,\nhowever enough time must be allotted to the MIP search procedure to find a\nbetter solution.\n\n\n\\bibliographystyle{spbasic}     \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Wright-Fisher model \\citep{fisher1930,wright1931} is the most widely used model for the evolution of genetic variation at a locus over time. In its most basic form, the Wright-Fisher model assumes a population of $N$ individuals, each with two possible genetic variants (called alleles) at the locus of interest, for a total of $2N$ alleles in the population.  Of those $2N$ alleles,  $X$ are assumed to be of type $A$, and the remaining $2N-X$ alleles are assumed to be of type $a$. The Wright-Fisher model traces the proportion of $A$ alleles across generations under the assumption of discrete, non-overlapping generations of constant size, in which the next generation of $2N$ alleles is produced by sampling from the current  generation at random with replacement.  Under this model, it is easy to see that the number of $A$ alleles at time $t$, denoted $X_t$, is a discrete-time Markov process with transition probabilities given by the Binomial distribution with $2N$ trials and probability  $q_{t-1} = X_{t-1}\/(2N)$ of success.  In this model, states $0$ and $2N$ are absorbing states.\n\nApplication of the basic Wright-Fisher model is limited, however, in several ways. For example, it deals only with variation at a single locus in a single population, and it neglects other important aspects of the evolutionary process, such as selection for advantageous or against deleterious mutations, changes in population size over time, and the generation of new allelic variants through spontaneous mutation. While these processes may be straight-forwardly modeled, inference in the discrete-time framework in these more complex situations is generally not computationally tractable. \n\nFor this reason, two common continuous-time approximations are typically employed. The first is generally referred to as Kingman's coalescent process \\citep{kingman1982a, kingman1982b,kingman1982c}, and arises by examining the limiting distribution of the time at which a collection of sampled alleles last shared a common ancestor as the population size becomes large relative to the sample size. The coalescent approach has proven extremely useful for species-level phylogenetic inference for moderate numbers of species or populations (generally, three to 20 species\/populations) under the basic Wright-Fisher model in which the effects of selection are negligible and in which no gene flow between distinct species occurs \\citep{edwards2009,knowleskubatko2010}.  However, inference in the presence of selection and\/or migration between species or populations is much more difficult using Kingman's coalescent  \\citep{wakeleyhey1997,nielsenwakeley2001,heynielsen2007,wakeley2009}, and both simulation and empirical studies indicate that the inferences obtained under models that ignore these processes can be biased in important ways \\citep{eckertcarstens2008,leacheetal2014,burbrinkguiher2015}.\n\nThe second continuous-time approximation to the Wright-Fisher model is based on modeling $Q_t$, the proportion of $A$ alleles in a population of size $N$ at time $t$,  as a diffusion process. This process is defined through the following stochastic differential equation (SDE) \n\\begin{equation} \n\\label{eq:WFgeneric}\n\\left\\{\n\\begin{array}{l}\ndQ_t = \\beta(Q_t)dt + \\sqrt{Q_t(1-Q_t)} dB_t\\\\\nQ_0 = q_0, \\quad 0\\le t\\le T\n\\end{array}\n\\right.\n\\end{equation}\nwhich will be referred to as the Wright-Fisher (WF) diffusion. In \\eqref{eq:WFgeneric}, the initial condition is $q_0\\in(0,1)$, $\\{B_t, t\\ge 0\\}$ is a standard Brownian motion process and the coefficient $\\beta: [0,1] \\rightarrow \\mathbb{R}$ is assumed to satisfy sufficient smoothness conditions such that \\eqref{eq:WFgeneric} admits a unique weak solution. Such conditions can be found in Section 5.3 of \\citet{karatzas2012brownian}. In applications, the drift function $\\beta(\\cdot)$ is selected to incorporate complex evolutionary processes such as selection and mutation.   Because such processes are easily specified in this approach, the WF diffusion has been increasingly applied to settings in which the number of populations is small (typically on the order of one to three populations) and primary interest is on inferring the contribution of these evolutionary processes to the current genetic configuration of the population(s).  \n\nPractical use of (\\ref{eq:WFgeneric}) is complicated by the fact that the transition density for the WF diffusion does not have a closed form.  Current approaches using this model are therefore based on approximations of this density.  A typical solution is to approximate the transition density using a numerical scheme for the SDE \\citep[see][for a classical overview]{kloedenplaten}.  Other recent approaches to estimating the transition density of an SDE include methods based on importance sampling \\citep[see][and the references therein for a review]{Pedersen1995, elerian2001likelihood, brandt2002simulated, DurhamGallant2002, lin2010generating}, closed-form approximations based on a Hermite polynomials expansion \\citep{AitSahalia,ait2008closed}, a strategy based on approximating the solution of the Kolmogorov forward equation \\citep{lo1988maximum}, and exact sampling \\citep{EAEstimate}.  Another approach, specific to approximating the transition density of the WF diffusion, involves truncation of a spectral representation of the target density \\citep{songetal2012,steinruckenetal2013}. \n\nGiven the wealth of genetic data currently available as a result of ever-improving sequencing technologies, an additional challenge for the diffusion-based inferential framework is the desire to use data from many loci simultaneously for the inference of evolutionary process parameters. This is typically done by making the assumption that the loci included in the study are unlinked, and thus independent, as would typically be the case for single nucleotide polymorphism (SNP) data collected across the genome.   Under this assumption, the likelihood  is computed by multiplying single-locus likelihoods computed under the model across loci.  When this assumption is violated and loci are actually linked, the likelihood is viewed as a composite likelihood, and inference proceeds as usual.\n\nThis framework has been considered in a variety of contexts. \\citet{williamsonetal2005} considered specification of the drift coefficient to include the effect of selection, and fit models that allowed for changes in population size throughout time.  To carry out the computations required by the WF diffusion, the Crank-Nicolson approximation was used to solve the relevant SDE in order to compute the likelihood. Estimation of the selection coefficient and population size parameters was then carried out in a likelihood framework. These authors later extended the method to include multiple populations with the possibility of gene flow between populations, again using a finite differencing method to provide a numerical solution to the SDE \\citep{gutenkunstetal2009}.  As before, inference was carried out in a likelihood framework, with variance estimates and significance values for hypothesis testing obtained using bootstrapping.  These methods were implemented in the program $\\partial a \\partial i$ for up to three populations \\citep{gutenkunstetal2010}, and  have been applied to a variety of organisms, including humans, cattle, rice, and bees \\citep{robinsonetal2014}.  In addition, recent attention has been given to methodology for efficient computation of estimates of uncertainty in parameter estimates, with applications to model selection \\citep{coffmanetal2015, robinsonetal2014}.\n\nDespite a great deal of previous work in this area, inference in a Bayesian framework has been limited.  \\citet{schraiberetal2016} and \\citet{ferreradmetllaetal2016} applied Bayesian techniques to infer selection in allele frequency time series data.  Their methods require approximation of the transition density of the WF diffusion.  \\citet{jenkinsspano2015} recently developed an algorithm for obtaining exact draws from the WF diffusion with general drift function, using draws from the neutral WF diffusion with mutation.  The work of \\citet{jenkinsspano2015} thus enables the development of Markov chain Monte Carlo (MCMC) approaches to inference of evolutionary process parameters from genome-wide SNP data based on the composite likelihood, but without the need for numerical approximation of the solution to the SDE.\n\nHere, we implement one such method for a single-population model in which interest is focused on estimation of the selection parameter.  Unlike \\citet{schraiberetal2016}, \\citet{ferreradmetllaetal2016}, \\citet{folletal2015}, and \\citet{malaspinasetal2012}, who considered time series data, we assume that allele frequency data are only available at a single time point.  We note that previous work in this single time point setting has focused primarily on likelihood- or summary statistic-based inference, rather than on Bayesian approaches. In particular, \\citet{nielsenetal2005} proposed a composite likelihood statistic for the detection of selective sweeps that allowed estimation of the magnitude of the selection coefficient as well as the location of the sweep.  This method, called SweepFinder, was later made more computationally efficient in an implementation called SweeD \\citep{pavlidisetal2013}.  Similarly, the linkage disequilibrium-based method of \\citet{kimnielsen2004} has recently been implemented with improved computational efficiency in the software OmegaPlus \\citep{alachiotisetal2012}.  Finally, the methodology proposed in \\citet{zivkovicetal2015} could also be applied to data sampled at a single time point.  All of these approaches, however, involve computation of statistics (e.g., composite likelihoods) at various positions in the genome, with no naturally associated measure of variance, aside from those obtained via bootstrap procedures, as in \\citet{zivkovicetal2015}.  We thus propose a Bayesian approach to the problem of data sampled at a single time point. \n\nIn the next section, we describe the details of our proposed model, as well as the computational approach we take to implement inference in an MCMC-based Bayesian framework.   We apply the method to simulated data to study its performance, which provides insight into the role of the prior distribution in inference under the model. We then apply the method to an empirical data set \\citep{zhouetal2011} consisting of SNP data on flies that were subjected to either a hypoxic environment or a control (e.g., normal oxygen) environment for 200 generations, with the goal of estimating the extent of selection for the flies that were oxygen-deprived. We compare our results with those of \\citet{ronenetal2013}, who analyzed the same data. Finally, we discuss potential extensions of our results to more general settings, as well as the advantages of inference in the Bayesian setting for these problems.\n\n\\section{Methods}\n\\label{sec:approach}\n\nSuppose that a total of $n$ individuals are sampled, and that the type of allele ($A$ or $a$) is recorded for each individual at each of $K$ loci.  At locus $k$, the number of individuals with allele $A$ is denoted $y_k$.  Let ${\\mathbf y} = (y_1, y_2, \\ldots , y_K)$ be the complete vector of data across all $K$ loci. For each locus, $y_k$ is assumed to be an observation from a Binomial distribution with $n$ trials and probability $q_k$ of success.  Note that $q_k = q_{T,k}$ represents the proportion of $A$ alleles in the population at locus $k$ at time $T$ -- the time at which the sample is collected -- however, we will suppress the dependence on $T$ throughout the manuscript, in order to simplify notation.  Conditional on the vector $\\mathbf{q} = (q_1, q_2, \\ldots , q_K)$, the likelihood is given by \n\\begin{equation} \\label{eq:lik}\ng(\\mathbf{y} | \\mathbf{q}) = \\prod_{k=1}^{K} \\binom{n}{y_k} q_k^{y_k} (1-q_k)^{n-y_k} .\n\\end{equation}\nNote that this assumes conditional independence of the data across loci; if the loci are not independent given the allele frequencies, then the likelihood in (\\ref{eq:lik}) can be viewed as a pseudo-likelihood. This is the same likelihood function used in other applications \\citep[e.g.,][]{williamsonetal2005}.\n\nThe allele frequencies $q_k$, $k=1,\\dots, K$ are assumed to be realizations (at time $T$) from WF diffusion processes with  starting values $q_{0,k}$,\\ $k=1,\\dots, K$. Often, $T$ is a parameter of interest in the model, however, in this case we will assume that $T$ is fixed and known.  The drift coefficient $\\beta(\\cdot)$ is specified to incorporate the evolutionary processes of interest and associated model parameters.   The goal of the inference procedure is to estimate the evolutionary model parameters given the data $\\mathbf{y}$ and carry out standard inference (e.g., hypothesis testing).\n\nHere we consider a model in which selection and mutation operate within a single population.  Selection refers to the evolutionary mechanism in which one allele is preferred over other potential alleles to survive to the next generation, while mutation refers to the origination of new types of alleles within the population.  Over time, both selection and mutation can act to alter the frequency of allele $A$ at a particular locus in the population beyond what is expected under the basic Wright-Fisher model that does not include these processes.   The WF diffusion corresponding to the model with mutation and selection is given by \n\\begin{equation}\n\\left\\{\n\\begin{array}{l}\n\\label{eq:WFmutsel}\ndQ_t = [sQ_t(1-Q_t) + \\frac{1}{2}(\\theta(1-2Q_t))]  dt \\\\\n\\hspace{2.9cm}  + \\sqrt{Q_t(1-Q_t)} dB_t,\\\\\nQ_0 = q_0, \\quad 0\\le t\\le T, \n\\end{array}\n\\right.\n\\end{equation}\nwhere $s$ is the population-scaled selection parameter and $\\theta$ is the population-scaled mutation parameter \\citep[see][Chapter 5]{etheridge2011some}.  For each locus $k$, define the density of the allele frequency at time $T$ under this WF diffusion by $f(q_k | s, \\theta, q_{0,k}, T)$, where $q_{0,k}$ is the initial allele frequency at locus $k$.\n\nAssuming that $\\theta$, $T$, and $\\mathbf{q}_0 = (q_{0,1}, q_{0,2}, \\ldots , q_{0,K})$ are known, we focus on estimating the extent of selection via inference of the parameter $s$. We take a Bayesian viewpoint and base our statistical inference on the posterior density function\n\\begin{equation}\n\\label{eq:posterior}\n\\pi(s, \\mathbf{q} \\giv \\mathbf{y}) \\propto g(\\mathbf{y}\\giv\\mathbf{q}){\\mathbf f}(\\mathbf{q}\\giv s, \\theta,\\mathbf{q}_0,T)\\pi_s(s),\n\\end{equation}\nwhere we use the notation\n$$\n{\\mathbf f}(\\mathbf{q}\\giv s, \\theta,\\mathbf{q}_0,T) = \\prod_{k=1}^K f(q_k | s, \\theta, q_{0,k}, T)\n$$\nand $\\pi_s(s)$ is a specified prior distribution on the selection parameter $s$. Subsequently, we use an MCMC algorithm to explore the distribution given by \\eqref{eq:posterior}.\n\nTo that end, note that a typical Metropolis within Gibbs (MwG) sampler would alternate between updating $s$ and $\\bfq$, conditional on everything else, using some proposal-acceptance mechanism. However, a direct implementation of this procedure suffers from several issues. For example, during the ``update $\\mathbf q$ given $s$'' step, under the assumption that the proposal distribution for $\\bfq$ is symmetric, the MwG accept-reject mechanism requires evaluating the acceptance ratio\n\\begin{equation*}\n\\alpha = \\frac{g(\\bfy\\giv\\bfq'){\\mathbf f}(\\bfq'\\giv s,\\theta,\\bfq_0,T)}{g(\\bfy\\giv\\bfq_n){\\mathbf f}(\\bfq_n\\giv s,\\theta,\\bfq_0,T)},\n\\end{equation*}\nwhere $\\bfq_n$ and $\\bfq'$ are the current and proposed states respectively. However, since the transition density for the WF diffusion $f(q_k\\giv s,\\theta,q_{0,k},T)$ does not have a closed form, the ratio above cannot be evaluated exactly in practice. \n\n\nMost of the methods described in Section~1 for estimating $f(q_k\\giv s,\\theta,q_{0,k},T)$ come with a significant computational cost.  Such methods are typically designed to work in a maximum likelihood estimation setting, where one would have a relatively small number of transition density evaluations. In a Bayesian setting however, simulating a Markov chain would require a very large number of transition density evaluations and thus many of these methods become impracticable.  \n\nIn this manuscript we suggest approaches based on exact sampling. Our goal is to use the transition density $f(\\cdot)$ in an MCMC setting and thus we require computational efficiency. We propose two sampling strategies: (i) an exact sampler which avoids evaluating the transition density $f(\\cdot)$ entirely but suffers from some computational overhead and (ii) an efficient approximate sampler based on exact simulations of the diffusion process \\eqref{eq:WFmutsel}. In both cases, our method relies on the ability to obtain exact draws from the density $f(\\cdot \\giv s,\\theta,\\bfq_0,T)$. We explain how this is done in detail in the Appendix. Henceforth, we assume that we can simulate variates\n$$\n{\\mathsf Q}_k\\sim f(\\cdot \\giv s,\\theta,q_{0,k},T) \\quad \\mbox{for} \\quad k=1,\\dots, K.\n$$\nAlthough such simulation is possible, it can be extremely time-consuming.\n\n\\vspace{0.5cm}\n\\noindent\n\\textbf{Exact Sampler}. Our exact MCMC sampler uses a joint proposal mechanism as follows. Let $(s_n, \\bfq_n)$ be the current state of the Markov chain exploring the distribution \\eqref{eq:posterior} and consider simulating a new proposed state $(s', \\bfq')$ from the (joint) prior distribution as follows:\n\\begin{algorithm}\n\\caption{Joint simulation of $(s',\\bfq')$ from the prior model}\n\\label{alg:prior}\n\\begin{algorithmic}[1]\n\\State Draw ${\\mathsf S}'\\sim \\pi_s(\\cdot)$;\n\\State Given ${\\mathsf S}' = s'$, draw ${\\mathsf Q}'_{k} \\sim f(\\cdot\\giv s', \\theta,q_{0,k},T)$ for $k=1,\\dots,K$;\n\\State Set $\\bfq'=({\\mathsf Q}_{1}', \\dots {\\mathsf Q}_{K}')$;\n\\State \\textbf{Return} $(s', \\bfq')$.\n\\end{algorithmic}\n\\end{algorithm}\n\nIn this case one can derive the acceptance ratio for $(s', \\bfq')$ to be\n$$\n\\alpha = \\frac{g(\\bfy\\giv\\bfq')}{g(\\bfy\\giv\\bfq_n)},\n$$\nwhich does not depend on $f(\\cdot \\giv s,\\theta,q_{0,k},T)$ and can therefore be easily evaluated. Exact draws from $f(\\cdot \\giv s,\\theta,q_{0,k},T)$ are used to generate proposals, but it is not necessary to evaluate $f(\\cdot \\giv s,\\theta,q_{0,k},T)$ to compute the acceptance ratio.  Thus, unlike existing methods, this strategy does not require one to approximate the transition density.  \n\nThere are, however, a number of drawbacks that one should consider. Given that this algorithm is essentially an independent Metropolis-Hastings sampler, it is well known that it may mix very poorly. Further, the second sampling step requires one to simulate the SDE \\eqref{eq:WFmutsel}. As mentioned above and discussed in greater detail in Section~\\ref{sec:results}, such simulation can be extremely time-consuming.  Since each update in the MCMC procedure requires this sampling step, simulation of the SDE becomes a bottleneck in the algorithm. Due to the combination of poor mixing and a slow sampling step, we found this algorithm to be impracticable computationally, especially when studying a large number of loci simultaneously. Nonetheless, given sufficient computing resources, we believe this procedure would perform adequately.\n\n\\vspace{0.5cm}\n\\noindent\n\\textbf{Approximate MwG}. A natural solution to the issues described above is to find an analytical expression for the transition density $f(\\cdot\\giv s,\\theta,\\mathbf{q}_0,T)$,  which would allow one to implement a standard MwG algorithm.  As discussed earlier, this is a difficult task.  To take advantage of our ability to sample exactly from $f(\\cdot\\giv s,\\theta,\\mathbf{q}_0,T)$, we suggest approximating the transition density using a kernel density estimate   \n\\begin{equation}\n\\label{eq:gaussiankde}\n\\tilde{f}(q \\giv s,\\theta,q_{0,k},T)=\\frac{1}{M} \\sum_{i=1}^M \\frac{1}{h}\\phi\\bigg{(}\\frac{q-{\\mathsf Q_i}}{h}\\bigg{)},\n\\end{equation}\nwhere $\\phi(\\cdot)$ is the density of the standard Gaussian distribution, $h$ is a bandwidth parameter, and ${\\mathsf Q}_1, \\dots, {\\mathsf Q}_M$ are independent and identically distributed draws from the density $f(\\cdot\\giv s,\\theta,q_{0,k},T)$.  We choose $h\\approx M^{-1\/5}$ according to Scott's Rule \\citep{scott1992} and in our simulations we use $M = 10,000$, see Section \\ref{sec:results}. It is known that the mean integrated squared error in this case is \n$$E\\left( \\int_0^1(f(q) - \\tilde f(q))^2\\ dq \\right)  = {\\mathcal O}(M^{-4\/5})\\ ,$$\nwhere the expected value is taken with respect to the Monte Carlo draws. Once the transition density has been approximated, the posterior distribution \\eqref{eq:posterior} can be replaced by\n\\begin{equation}\n\\label{eq:post}\n\\tilde \\pi(s, \\mathbf{q} \\giv \\mathbf{y}) \\propto g(\\mathbf{y}\\giv\\mathbf{q})\\tilde {\\mathbf f}(\\mathbf{q}\\giv s, \\theta,\\mathbf{q}_0,T)\\pi_s(s),\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eq:densityest}\n\\tilde{\\mathbf f}(\\mathbf{q}\\giv s, \\theta,\\mathbf{q}_0,T) = \\prod_{k=1}^K \\tilde f(q_k | s, \\theta, q_{0,k}, T)\\ .\n\\end{equation}\nThe distribution \\eqref{eq:post} can then be explored directly using a standard MwG algorithm.  \n\nThis approximate MwG algorithm requires a kernel density estimate of $f(q_k \\giv s,\\theta,q_{0,k},T)$ for each potential set of proposed values for $s$, $\\theta$, $q_{0,k}$, and $T$.  For this to be feasible we must restrict ourselves to a finite set of plausible values for these parameters.  We choose to treat both $\\theta$ and $T$ as fixed and known, and to assume that $s$ and $q_{0,k}$ are contained in the finite sets $\\mathcal{S}$ and $\\mathcal{Q}_0$, respectively.  The initial allele frequencies are generally not known in practice, so it is not reasonable to treat $\\mathbf{q}_0$ as fixed and known.  Instead, we view each $q_{0,k}$ as an independent draw from a discrete distribution $\\pi_q(\\cdot)$ that puts positive mass on $\\mathcal{Q}_0$.  This allows us to account for uncertainty in $\\mathbf{q}_0$, but presents a challenge in that we cannot estimate $s$ and $\\mathbf{q}_0$ simultaneously.  We therefore ``integrate out\" $q_{0,k}$ to leave $s$ as the only parameter to be estimated.  \n\nUltimately, for each $s \\in \\mathcal{S}$ and $q_{0,k} \\in \\mathcal{Q}_0$, we use (\\ref{eq:gaussiankde}) to produce a kernel density estimate $\\tilde{f}(q \\giv s,\\theta,q_{0,k},T)$ of the transition density and then sum over $\\mathcal{Q}_0$ to compute an alternative estimate $\\hat{f}(q \\giv s,\\theta,T)$ that does not depend on $q_{0,k}$.  Specifically, we compute \n\\begin{equation} \\label{eq:sumoverq0k}\n\\hat{f}(q_k|s,\\theta,T)=\\sum_{q_{0,k} \\in \\mathcal{Q}_0} \\tilde{f}(q_k \\giv s,\\theta,q_{0,k},T) \\pi_q(q_{0,k}).\n\\end{equation} \nThe density estimate $\\hat{f}(q_k \\giv s,\\theta,T)$ replaces $\\tilde{f}(q_k \\giv s,\\theta,q_{0,k},T)$ in (\\ref{eq:densityest}) and the algorithm proceeds as if $\\mathbf{q}_0$ were known.\n\nThe upfront computational cost of this approach is substantial as one must simulate many draws from many transition densities in order to generate a sufficient number of accurate kernel density estimates with which to effectively explore the parameter space.  However, once these density estimates have been produced, the MCMC algorithm is able to run efficiently.  This is in contrast to the exact sampler, which does not have the same upfront costs but performs each update inefficiently.  Further, by updating the parameters individually, conditional on all other parameters in the model, the MwG sampler mixes better than the exact sampler and therefore requires far fewer updates to obtain a sample of approximately independent draws from the target posterior.\n\n\n\\section{Results}\n\\label{sec:results}\n\n\\noindent\n\\textbf{Simulated Data.} In order to assess the viability of our approach we first applied it to simulated data with known selection parameter.  Specifically, for each of three different selection parameters ($s=0.0$, $s=5.5$, and $s=11.0$) we simulated 500 data sets with $n=200$ individuals and $K=324$ loci.  This was accomplished by using the algorithm described in the Appendix to simulate each $q_k$ from the SDE given in (\\ref{eq:WFmutsel}) and drawing each corresponding $y_k$ from the appropriate Binomial distribution.  Model parameters were given values matching those used with the empirical fly data in the example described below.  Specifically, for each data set we set $\\theta=0.00014$ and $T=0.1$, and let $\\mathbf{q}_0$ follow the ending distribution of allele frequencies for the neutral fly population in the example. \n\nWe conducted two sets of simulations.  In both cases we defined the set of plausible values for $s$ as $\\mathcal{S}=\\{-12.0,-11.5,\\dots,17.0\\}$ and assumed a distribution $\\pi_q(\\cdot)$ for the initial allele frequencies that put mass on the set $\\mathcal{Q}_0=\\{0.01,0.02,\\dots,0.99\\}$.  As such, we computed $59 \\times 99 = 5,841$ kernel density estimates $\\tilde{f}(q_k \\giv s,\\theta,q_{0,k},T)$ with one corresponding to each combination of $s \\in \\mathcal{S}$ and $q_{0,k} \\in \\mathcal{Q}_0$.  Each of these density estimates was based on 10,000 draws from the true transition density, produced via exact simulation of the SDE in (\\ref{eq:WFmutsel}). We then summed over $\\mathcal{Q}_0$ as in (\\ref{eq:sumoverq0k}) to obtain an estimate $\\hat{f}(q_k \\giv s,\\theta,T)$ for each of the $59$ values of $s$ in $\\mathcal{S}$.  \n\nGenerating the samples required to compute these density estimates took nearly 1,600 hours on a NVIDIA Tesla C2050 GPU computing card.  However, a disproportionate amount of this time was required to generate samples for the largest values of $s$.  It took about 607 hours for $s=17.0$ alone, about 384 hours for $s=16.5$, and about 215 hours for $s=16.0$.  In contrast, generating the required samples for all values of $s$ in the set $\\{-8.0,-7.5,\\dots,8.0\\}$ took a total of just 32 minutes.  Large values of $s$ require more time because the proposals in the acceptance-rejection algorithm are generated from a process with $s=0.0$.  Thus, as $s$ increases in magnitude there is a bigger discrepancy between the process used for proposals and the target process, so far fewer proposals are accepted.   \n\nFor the first set of simulations we defined $\\pi_q(\\cdot)$ as a discretized version of the distribution of $\\mathbf{q}_0$ that was used to generate the data whereas in the second set of simulations we defined $\\pi_q(\\cdot)$ as a discrete Uniform distribution over the set $\\mathcal{Q}_0$.  Altering the assumed distribution for $\\mathbf{q}_0$ allows us to assess the importance of this choice and its impact on inference for $s$.\n\nFor all simulations we placed a discrete Uniform prior on $s$ so that $\\pi_s(s)=\\frac{1}{59}$ for each $s \\in \\mathcal{S}$.  Our proposal density for $s$ was a discrete Uniform centered on the current value of $s$ that put positive mass on the five grid values on either side of the current value \\big{(}$p(s|s_n)=\\frac{1}{11}$ for $s \\in \\{s_n-2.5,s_n-2.0,\\dots,s_n+2.5\\}$ and $p(s|s_n)=0$ otherwise\\big{)}. Proposals for each $q_k$ came from a $\\mathrm{N}(q_{n,k},\\sigma_q^2)$ distribution where $q_{n,k}$ was the current value for $q_k$ and $\\sigma_q^2=0.05^2$ was the proposal variance.  Since these proposal densities are symmetric they do not play a role in the acceptance ratios of the MwG algorithm.  For each simulated data set, a Markov chain was run for 10,000 steps.  The first 1,000 steps were discarded as burn-in and every tenth step thereafter was retained to form a sample of size 900 from the posterior distribution.  Trace plots and autocorrelation plots for a random subsample of the replicates (not shown) suggest that the Markov chains converge to a stationary distribution within 1,000 steps and that retaining only every tenth step eliminates virtually all of the autocorrelation among values in the posterior sample.  \n\nFor each of the three simulated values of $s$ (0.0, 5.5, and 11.0) and each choice of $\\pi_q(\\cdot)$ we generated a posterior sample for each of the 500 simulated data sets.  The posterior mean of $s$ was then computed for each of these samples, yielding 500 estimates of $s$ for each simulated value of $s$ and choice of $\\pi_q(\\cdot)$.  Histograms of these posterior estimates when $\\pi_q(\\cdot)$ was chosen to approximate the initial allele frequency used to generate the data can be seen in Figure~\\ref{fig:simresults}.  For all three values of $s$ the empirical distribution of the estimator covers the true value of $s$.  However, the empirical distribution of the estimator is not always centered at the true value of $s$.  Thus, this approach appears to be able to successfully distinguish among weak, strong, and moderate selection, but it may not yield an unbiased estimator of $s$. \n\nAdditionally, a 99\\% credible interval for $s$ was computed from the posterior sample of each replicate.  When $s=0.0$, 97.2\\% of these intervals included the true value of $s$, whereas for $s=5.5$ and $s=11.0$ the coverage was 99.2\\% and 84.4\\%, respectively.  In all three cases the average length of the credible intervals was about three units.  Thus, the lower coverage for $s=11.0$ is presumably due not to narrower intervals, but rather to the fact that those intervals tend to lie above $s=11.0$.  Ultimately, our approach may lead to slightly biased estimates, but we feel that we effectively quantify the uncertainty of the estimates.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=3.2in]{figure1}\n\\caption{\\label{fig:simresults} Histograms showing estimates of the selection parameter $s$ from 500 simulations of data with true values of $s=0.0$ (white), $s=5.5$ (light gray), and $s=11.0$ (dark gray) when $\\pi_q(\\cdot)$ is chosen to reflect the distribution of initial allele frequencies used to generate the data.  The vertical lines indicate the true values of $s$.} \n\\end{center}\n\\end{figure} \n\nHistograms of the posterior estimates for the cases of $s=5.5$ and $s=11.0$ when $\\pi_q(\\cdot)$ was chosen to be a discrete Uniform can be seen in Figure~\\ref{fig:unifsimresults}.  The plot for $s=0.0$ is excluded because all of the Markov chains became trapped in a state with $s=-12.0$, which is the lower boundary of our set $\\mathcal{S}$ of plausible values for $s$.  This suggests that our posterior estimate for $s$ would have been less than $-12.0$ if the algorithm had been able to search for lesser values of $s$.  Further, none of the 500 replicates for any of the simulated values of $s$ yields a single 99\\% credible interval that captures the true value of $s$.  It is evident from these results that our approach relies on an accurate assumption for the distribution of the initial allele frequencies.  Assuming a discrete Uniform distribution for $q_{0,k}$ puts more mass on higher values of $q_{0,k}$ than the distribution used to generate the data did.  Consequently, we estimate values for $s$ that are too small when we assume such a distribution for $q_{0,k}$. We address the dependence of our method on an accurate choice for $\\pi_q(\\cdot)$ in Section~\\ref{sec:discussion}.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=3.2in]{figure2}\n\\caption{\\label{fig:unifsimresults} Histograms showing estimates of the selection parameter $s$ from 500 simulations of data with true values of $s=5.5$ (light gray) and $s=11.0$ (dark gray) when $\\pi_q(\\cdot)$ is chosen to be Uniform over $\\mathcal{Q}_0$.  The vertical lines indicate the true values of $s$.} \n\\end{center}\n\\end{figure}   \n\n\\noindent \\textbf{Example Data: Hypoxia in Flies.} An example to which this methodology can be applied is a study conducted by \\citet{zhouetal2011} in which flies (\\textit{Drosophila melanogaster}) were subjected to a hypoxic environment or a normal oxygen environment for $200$ generations.  Flies in the hypoxic environment were under selective pressure to adapt to the lack of oxygen in their environment.  The data consist of allele frequencies at the end of the study among approximately 200 surviving flies in each population.  These frequencies were measured at 75,999 loci along the 2L chromosome for the population under selective pressure and 71,335 loci along the 2L chromosome for the neutral population.  A subsequent analysis of these data conducted by \\citet{ronenetal2013} identified a region of the 2L chromosome showing strong evidence of selection.  Within this region we have measurements at 459 loci for the population under selective pressure and measurements at 324 loci for the neutral population.  \n\nWe applied our proposed methodology to the region of the 2L chromosome of the fly data of \\citet{zhouetal2011} that \\citet{ronenetal2013} identified as being subject to selection.  We used the same settings as in the first set of simulations including choosing $\\pi_q(\\cdot)$ to be a discretized version of the distribution of allele frequencies at the end of the study for the neutral population.  Thus, we assumed that both the neutral population and the population under selective pressure began the study with a distribution of allele frequencies similar to that of the neutral population at the end of the study.  As such, we should estimate a selection parameter near zero for the neutral population.  A Markov chain was run to obtain a posterior sample of size 900 for each of the two populations of flies.  \n\nFigure~\\ref{fig:flyresults} shows a histogram of the marginal posterior sample of $s$ for each population of flies.  These posterior samples correspond to the region of the 2L chromosome identified by \\citet{ronenetal2013} as being subject to selection.  The white bars show the posterior sample for the neutral population whereas the gray bars  show the posterior sample for the population subjected to the hypoxic environment.  For the neutral population the posterior mean for $s$ is $-1.13$ and a corresponding 99\\% credible interval contains zero.  Thus, as expected, there is no evidence of selection.  For the population subjected to the hypoxic environment the posterior mean for $s$ is $16.10$ and the entire sample from the posterior density of $s$ lies above zero.  This indicates strong positive selection for this population.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=3.2in]{figure3}\n\\caption{\\label{fig:flyresults} Histograms of the posterior sample of $s$ from fitting our model to the neutral fly population (white) and the fly population subjected to a hypoxic environment (gray).} \n\\end{center}\n\\end{figure} \n\nFinally, to determine if any other regions of the 2L chromosome display evidence of selection we applied our proposed methodolgy to sliding windows of $K=500$ loci with each window shifted 250 loci from the preceding window.  This was done for the population subjected to the hypoxic environment. Note that this procedure assumes that all loci within a sliding window have identical values for the selection coefficient, a condition that is unlikely to be strictly true.  However, we view this as a reasonable approximation given that the relatively small window size for this fairly dense SNP sample means that SNPs within a single window are likely to be linked.  In this situation, the selection coefficient estimated in our model can be viewed as an ``average'' value for the strength of selection across the loci included in the window. Thus, our method can be  used to identify regions of the genome that are likely to be under selection, but does not necessarily pinpoint individual SNPs. For each window, a Markov chain was run for 10,000 steps and every tenth step after the first 1,000 was retained to form a posterior sample of size 900.  These posterior samples were used to produce 99\\% credible intervals for $s$.  \n\nFigure~\\ref{fig:windowresults} displays 99\\% credible intervals for $s$ for the sliding windows along the 2L chromosome for the population subjected to the hypoxic environment.  Intervals displayed in gray contain zero whereas those shaded black do not.  We find evidence of selection in more regions than \\citet{ronenetal2013} did, including several regions under negative selection.  However, we detect the strongest signal in the same region that \\citet{ronenetal2013} identified. In fact, the credible interval that lies furthest above zero falls directly in the relatively narrow region (highlighted by a gray vertical bar in Figure~\\ref{fig:windowresults}) that \\citet{ronenetal2013} identified using the XP-SFselect method.  To further compare our method to existing approaches, we used the SweeD software \\citep{pavlidisetal2013} with grid size 30,000 to compute the composite likelihood ratio (CLR) at various points along the chromosomal region (shown in medium-gray at the bottom of Figure~\\ref{fig:windowresults} with scaling given by the right-handed y-axis).  In general, the magnitude of the CLR statistic corresponds to the signal indicated by our credible intervals.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=3.2in]{figure4_new}\n\\caption{\\label{fig:windowresults}  Results of our analysis of selection for the 2L chromosome of the fly population subjected to the hypoxic environment. Location (x-axis) is displayed in units of $10^6$. The left-handed y-axis gives units for \n99\\% credible intervals for $s$ for sliding windows along the chromosome.  Intervals containing zero are gray whereas those not containing zero are black.  The right-handed y-axis gives units for the composite likelihood ratio statistic computed by the SweeD software \\citep{pavlidisetal2013}, shown in medium-gray at the bottom of the plot. The vertical gray bar indicates a region previously identified by \\citet{ronenetal2013} as being under positive selection, which corresponds to the strongest signal of selection detected by our analysis. }\n\\end{center}\n\\end{figure} \n\n\n\\section{Discussion}\n\\label{sec:discussion}\nHere we propose a Bayesian method for inference of population genetics parameters under the Wright-Fisher diffusion model. In particular, we consider estimation of the selection coefficient from allele frequency data sampled at a single time point by constructing an MCMC algorithm that allows one to draw from the joint posterior distribution of the selection coefficient and the allele frequencies. We show that  when assumptions about the initial allele frequencies are accurate our method performs well in simulations as well as in the analysis of an empirical data set on hypoxia in flies. \n\nExisting methods for inference in similar settings have used a maximum likelihood framework and have relied on numerical approximation of the solution of the SDE corresponding to the WF diffusion.  In addition to requiring significant computation, a drawback of such methods is that they do not  provide a natural way to assess variability in the estimates. Recent work in this area has involved the development of bootstrap procedures for estimation of error variances \\citep{coffmanetal2015,robinsonetal2014}, requiring significant additional computation time. An advantage of a Bayesian approach is that MCMC can be used to approximate the entire posterior distribution, providing an automatic assessment of uncertainty.  \n\nBayesian approaches to this problem  require an efficient method for sampling from the WF diffusion or for accurately approximating its transition density.  The important work of \\citet{jenkinsspano2015} provides a method to draw exact samples from WF diffusions that incorporate features of the evolutionary process such as mutation and selection.  Their work will enable much more general developments in the application of Bayesian methods to population-level diffusion models. Our example on hypoxia in flies demonstrates the potential utility of these approaches; for example, we compute 99\\% credible intervals along a chromosomal region using a sliding window approach, and recover evidence for strong positive selection in a region identified in previous studies.\n\nAlthough the recent work of \\citet{jenkinsspano2015} enables MCMC-based Bayesian inference in this setting, the method we have developed based on this work is also fairly computationally intensive.  This is because of the difficulty of obtaining exact draws from the general WF diffusion model.  To deal with this, we chose to use kernel density estimates of the transition density, and most of our computational effort is devoted to computing these kernel density estimates.  The computational effort needed for this step will vary depending on the true underlying parameters that generated the data. For our empirical data set for which $s \\approx 16$, the method required a great deal of effort because $s$ is large compared to the neutral Wright-Fisher model, and sampling from the WF diffusion is inefficient. When the true value of $s$ is closer to zero, transition density estimates for larger values of $s$ will not be needed, and the overall computation time will be reduced.  Also, although we have exploited GPU computing in order to make this approach computationally feasible, it is possible that more efficient algorithms could be developed to speed computation further.  It may also be possible to use alternative approaches to the kernel density estimates we have used that might provide similar accuracy with reduced computational expense.\n\nOur simulation studies clearly demonstrate that the choice of prior distribution for the initial allele frequencies across loci will have a strong effect on the posterior distribution. This is intuitive in the case of selection:  a large initial allele frequency and strong negative selection can yield the same ending allele frequency distribution as a small initial allele frequency with strong positive selection. Thus, two markedly different processes can result in very similar data, and it is not possible to disentangle the effects of these two parameters. This means that a requirement for effective use of this method is either prior knowledge of the approximate allele frequency distribution at neutrality, or a thorough exploration of the sensitivity of the results to the choice of prior distribution. Fortunately, many studies in this area do devote effort to the assessment of neutral processes in the populations under consideration and thus there will often be prior information about the distribution of initial allele frequencies available to guide choice of the prior distribution \\citep[see, e.g.,][]{listephan2006, williamsonetal2005}. \n\nThe method presented here addresses perhaps the simplest scenario one might consider in this setting: a single population and a single parameter of interest (the selection coefficient).  In practice, however, data may be collected for multiple populations or species at multiple time points, and inference for all relevant parameters (e.g., divergence times, migration rates, and selection coefficients) may be desired. While diffusion models such as this are now commonly used to model up to three populations and to infer multiple parameters simultaneously, they are generally computationally demanding and, as mentioned above, the current likelihood-based approaches require significant additional computation in order to carry out complete statistical inference that includes an assessment of uncertainty. As genome-scale data become more widely available and sampling of individuals within populations allows more precise estimates of population allele frequencies throughout the genome, methods based on allele frequencies provide a unique opportunity for efficient population-level evolutionary inference.  Methods that utilize a Bayesian framework for these problems have the potential to contribute in important ways to these developments, and our Bayesian framework provides a useful starting point for these future developments. \n\n\\section{Acknowledgement}\nThis work was supported by National Science Foundation grants DMS-1209142 and DMS-1407604 (R.H.) and DMS-1106706 (L.S.K.). \n\n\\section*{Appendix}\n\\label{sec:appendix}\n\n\\subsection*{Exact simulation of the neutral Wright-Fisher diffusion}\n\nConsider a neutral WF diffusion process $Q_t$\n\\begin{equation}\n\\label{eq:wfneutral}\n\\left\\{\n\\begin{array}{l}\ndQ_t = \\beta_0(Q_t)dt + \\sqrt{Q_t (1-Q_t)} dB_t\\ , \\\\\nQ_0 = q_0, \\quad 0\\le t\\le T\n\\end{array}\n\\right.\n\\end{equation}\nwhere the drift function is $\\beta_0(x) = (1\/2)(\\theta_1(1-x) - \\theta_2 x)$, for $0\\le x\\le 1$ and $\\theta_1, \\theta_2 >0$ (note the absence of the selection parameter). The necessary details for simulating a variate $Q_T$ from the model \\eqref{eq:wfneutral} for a fixed $T>0$, given that $Q_0 = q_0\\in(0,1)$ are given in \\citet{jenkinsspano2015} and we briefly review the main ideas. It is well-known \\citep{griffiths1979,tavare1984,ethiergriffiths1993,griffithsspano2013} that, conditional on $Q_0  = q_0$, the variate $Q_T$ has a probability density function given by\n\\begin{equation}\n\\label{eq:pdfWF0}\nf(x\\giv q_0) = \\sum_{m=0}^\\infty q_m^\\theta(T) \\sum_{l=0}^m {\\mathcal Bin}(l;m,q_0){\\mathcal Beta}(x;\\theta_1+l, \\theta_2+m-l)\n\\end{equation}\nwhere $\\theta = \\theta_1 + \\theta_2$, and ${\\mathcal Bin}(\\cdot)$ and ${\\mathcal Beta}(\\cdot)$ are the Binomial and Beta density functions, given by\n\\begin{align*}\n{\\mathcal Bin}(l;m,q_0) &= {m\\choose l} q_0^l (1-q_0)^{m-l} \\\\\n{\\mathcal Beta}(x;\\theta_1, \\theta_2)&\\propto x^{\\theta_1 - 1}(1-x)^{\\theta_2 -1}\\ .\n\\end{align*}\nThe mixture weights $q_m^\\theta(T)$ in \\eqref{eq:pdfWF0} have an alternating series expansion\n\\begin{equation}\n\\label{eq:q0}\nq_m^\\theta(T) = \\sum_{k=m}^\\infty (-1)^{k-m}a_{km}^\\theta e^{-k(k+\\theta-1)T\/2}\n\\end{equation}\nwhere\n\\begin{equation*}\na_{km}^\\theta = \\frac{(\\theta+2k-1)(\\theta+m)_{(k-1)}}{m!(k-m)!}\n\\end{equation*}\nwith the convention that $a_{(x)} \\equiv \\Gamma(a+x)\/\\Gamma(a)$ for $a>0$ and $x\\ge-1$. Sampling from the density \\eqref{eq:pdfWF0} is achieved via Agorithm \\ref{alg:WF0} described below \\citep[see][]{jenkinsspano2015}.\n\n\\begin{algorithm}\n\\caption{Exact Sampling for the neutral WF diffusion:}\n\\label{alg:WF0}\n\\begin{algorithmic}[1]\n\\State Draw $\\sM$ from the discrete distribution $\\{q_m^\\theta(T), m=0,1,\\dots\\}$;\n\\State Draw $\\sL \\sim {\\mathcal Bin}(\\sM, q_0)$;\n\\State Draw $\\sQ \\sim {\\mathcal Beta}(\\theta_1+\\sL, \\theta_2+\\sM-\\sL)$;\n\\State \\textbf{Return} $\\sQ$.\n\\end{algorithmic}\n\\end{algorithm}\n\nThe computational burden in Algorithm \\ref{alg:WF0} is in Step 1, since the weights $q_m^\\theta(T)$ do not have a closed form expression and \\eqref{eq:q0} suggests an infinite amount of computation. However, given the alternating series representation in \\eqref{eq:q0}, one can use ideas described in Chapter 5 of  \\citet{devroye1986} to devise an exact procedure for simulating the variate $\\sM$, which only requires finite computing time.   Details of such a procedure are provided in Section~3.2 of \\citet{jenkinsspano2015}.   \n\n\\subsection*{Exact simulation of general WF-diffusion models}\nAs we mention above, our approach relies on being able to obtain exact draws from a general Wright-Fisher diffusion \\eqref{eq:WFgeneric}. We briefly review the general setup for sampling an SDE. Assume that $\\{X_t,\\ 0\\le t\\le T\\}$ is a stochastic process described via \n\\begin{equation}\n\\label{eq:sde}\ndX_t = \\beta(X_t)dt + \\sigma(X_t)dB_t, \\quad\\quad X(0) = x_0\\ , \\quad 0\\le t\\le T\n\\end{equation}\nwhere the functions $\\beta(\\cdot)$  and $\\sigma(\\cdot)$ are assumed to be smooth enough such that \\eqref{eq:sde} has a unique weak solution. Such conditions can be found in \\citet{karatzas2012brownian}, for example. Our goal is to simulate exact draws from the distribution of $X_t$, for some fixed $t>0$, conditional on $X_0=x_0$. In general, the transition density for the process $X_t$ is not available in closed form, not even in an infinite series representation as is the case with the neutral Wright-Fisher diffusion. Recently, in a series of papers \\citep{EA1,EA2,EA3}, the authors have developed a rejection sampling approach, which yields an exact (distribution-wise) skeleton of the full path $(X_t, 0\\le t\\le T)$ for a very large class of diffusions. Briefly, let $\\Omega = {\\mathcal C}([0,T])$ and let $\\omega$ be a typical element of $\\Omega$. Let $\\QQ$ be the probability measure induced by $\\{X_t,\\ 0\\le t\\le T\\}$ onto $\\Omega$ and $\\ZZ$ be another probability measure on $\\Omega$ which is user-selected in an appropriate way. Under regularity conditions \\citep[see][]{EA1} one can use the Girsanov formula to derive and expression for the Radon-Nikodym derivative\n$$\n\\frac{d\\QQ}{d\\ZZ}(\\omega) \\propto {\\mathsf G}(\\omega)\n$$\nwhere, in principle, one can arrange that ${\\mathsf G}(\\cdot) \\in (0,1)$. Thus, a rejection sampling strategy will be appropriate:\n\n\\begin{algorithm}\n\\caption{Exact sampling for general diffusions}\n\\label{alg:rejection}\n\\begin{algorithmic}[1]\n\\State Sample $\\omega\\sim \\ZZ$;\n\\State Accept $\\omega$ as a draw from $\\QQ$ with probability $\\sG(\\omega)\\in (0,1)$.\n\\end{algorithmic}\n\\end{algorithm}\n\nThe proposal distribution $\\ZZ$ has to be selected in a way that Step 1 of Algorithm \\ref{alg:rejection} can be done efficiently, i.e. a biased Brownian Bridge measure. We refer the reader to \\citet{EA1} for a detailed description of all of the conditions and the general setup. \n\n\\vspace{0.5cm}\n\\noindent\n\\textit{General Wright-Fisher diffusions}. If the diffusion coefficient in \\eqref{eq:sde} takes the form\n$$\n\\sigma(x) = \\sqrt{x(1-x)},\n$$\nthen the SDE \\eqref{eq:sde} is a general WF diffusion. As \\citet{jenkinsspano2015} suggest, in this case, it is efficient to select the proposal measure $\\ZZ$ to be the law induced by the neutral WF diffusion \\eqref{eq:wfneutral}. \n\n\n\\bibliographystyle{apalike}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nHigh-speed Monte Carlo simulations are used for a large spectrum of\napplications from mathematics to economy. As input for such simulations, the\nprobability distribution are usually generated by pseudo-random number\nsambling, a method going back to a work of John von Neumann from\n1951~\\cite{vonNeumann:1951}. In the era of ``big data'' such methods have to\nbe fast and reliable, a sign of such neccessity being the release of the very\nfirst randomness Quside processing unit in 2023~\\cite{111}. Still, these\nsamblings need to be cross-checked by exact methods, and for these the\nknowledge of analytical functions to describe the stochastic processes, among\nthose the error function, are of tremendous importance.\n\nBy definition, a function is called analytic if it locally given by a\nconverging Taylor series expansion. Even if a function itself turns out not\nto be analytic, its inverse can be analytic. The error function can be given\nanalytically, one of these analytic expressions is the integral representation\ngiven by Craig in 1991~\\cite{Craig:1991}. Craig mentioned this representation\nonly in passing and did not give a derivation of it. In the following, there\nhave been a couple of derivations of this formula~\\cite{Lever:1998,%\nTellambura:1999,Stewart:2017}. In Sec.~2 we add a further one which is based\non the same geometric considerations as employed in Ref.~\\cite{Martila:2022}.\nIn Sec.~3 we give the series expansion for Craig's integral representation and\nshow the fast convergence of this series.\n\nFor the inverse error function, handbooks for special functions (cf.\\ e.g.\\\nRef.~\\cite{book}) do not unveil such an analytic property. Instead, this\nfunction have to be approximated. Known approximations are dating back to the\nlate 1960s and early 1970s~\\cite{Strecok:1968,Blair:1976}) and reach up to\nsemi-analytical approximations by asymptotic expansion (cf., e.g.,\nRefs.~\\cite{Bergsma:2006,Dominici:2007,Dominici:2008,Winitzki:2008,Giles:2011,%\nSoranzo:2012}. Using the same geometric considerations, in Sec.~4 we develop a\ncouple of handy approximations which can easily be implemented in different\ncomputer languages, indicating the deviations from an exact treatment. In\nSec.~5 we test the CPU time and give our conclusions.\n\n\\section{Derivation of Craig's integral representation}\nRef.~\\cite{Martila:2022} provides an approximation for the Gaussian normal\ndistribution obtained by geometric considerations. The same considerations\napply to the error function $\\mathop{\\rm erf}\\nolimits(t)$ which is given by the normal\ndistribution $P(t)$ via\n\\begin{equation}\\label{erf}\n\\mathop{\\rm erf}\\nolimits(t)=\\frac1{\\sqrt\\pi}\\int_{-t}^te^{-x^2}dx\n  =\\frac1{\\sqrt{2\\pi}}\\int_{-\\sqrt2t}^{\\sqrt2t}e^{-x^2\/2}dx=P(\\sqrt2t).\n\\end{equation}\nTranslating the results of Ref.~\\cite{Martila:2022} to the error function, one\nobtains the approximation of order $p$ to be\n\\begin{equation}\\label{Eqf2}\n\\mathop{\\rm erf}\\nolimits_p(t)^2=1-\\frac1N\\,\\sum_{n=1}^Ne^{-k_n^2t^2},\n\\end{equation}\nwhere the $N=2^p$ values $k_n$ ($n=1,2,\\ldots,N$) are found in the intervals\nbetween $1\/\\cos(\\pi(n-1)\/(4N))$ and $1\/\\cos(\\pi n\/(4N))$ and can be optimised.\nIn Ref.~\\cite{Martila:2022} it is shown that\n\\begin{equation}\\label{rteq}\n\\Big|\\mathop{\\rm erf}\\nolimits(t)-\\sqrt{1-e^{-k_{0,1}^2t^2}}\\Big|<0.0033\n\\end{equation}\nfor $k_{0,1}=1.116$, and with $14\\approx 0.0033\/0.00024$ times larger\nprecision\n\\begin{equation}\\label{rteq2}\n\\Big|\\mathop{\\rm erf}\\nolimits(t)-\\sqrt{1-\\frac12(e^{-k_1^2t^2}+e^{-k_2^2t^2})}\\Big|<0.00024,\n\\end{equation}\nwhere $k_{1,1}=1.01$, $k_{1,2}=1.23345$. For the parameters $k_n$ taking the\nvalues $k_n=1\/\\cos(\\pi n\/(4N))$ of the upper limits of those intervals, it can\nbe shown that the deviation is given by\n\\begin{equation}\\label{Eqf299}\n|\\mathop{\\rm erf}\\nolimits(t)-\\mathop{\\rm erf}\\nolimits_p(t)|<\\frac{\\exp(-t^2)}{2N}\\,\\sqrt{1-\\exp(-t^2)}\\,.\n\\end{equation}\nGiven the values $k_n=1\/\\cos\\phi(n)$ with $\\phi(n)=\\pi n\/(4N)$, in the limit\n$N\\to\\infty$ the sum over $n$ in Eq.~(\\ref{Eqf2}) can be replaced by an\nintegral with measure $dn=(4N\/\\pi)d\\phi(n)$ to obtain\n\\begin{equation}\\label{A6}\n\\mathop{\\rm erf}\\nolimits(t)^2=1-\\frac4\\pi\\int_0^{\\pi\/4}\\exp\\left(\\frac{-t^2}{\\cos^2\\phi}\\right)\n  \\,d\\phi.\n\\end{equation}\n\n\\section{Power series expansion}\nThe integral in Eq.~(\\ref{A6}) can be expanded into a power series in $t^2$,\n\\begin{equation}\\label{A7}\n\\mathop{\\rm erf}\\nolimits(t)^2=1-\\frac4\\pi\\sum_{n=0}^\\infty c_n\\frac{(-1)^n}{n!}(t^2)^n\n\\end{equation}\nwith\n\\begin{eqnarray}\\label{cn}\nc_n&=&\\int_0^{\\pi\/4}\\frac{d\\phi}{\\cos^{2n}\\phi}\\ =\\ \\int_0^{\\pi\/4}\n  (1+\\tan^2\\phi)^nd\\phi\\ =\\ \\int_0^1(1+y^2)^{n-1}dy\\nonumber\\\\\n  &=&\\sum_{k=0}^{n-1}\\begin{pmatrix}n-1\\\\k\\\\\\end{pmatrix}\\int_0^1y^{2k}dy\n  \\ =\\ \\sum_{k=0}^{n-1}\\frac{1}{2k+1}\\begin{pmatrix}n-1\\\\ k\\\\\\end{pmatrix},\n\\end{eqnarray}\nwhere $y=\\tan\\phi$. The coefficients $c_n$ can be expressed by the\nhypergeometric function, $c_n={}_2F_1(1\/2,1-n;3\/2;-1)$, also known as Barnes'\nextended hypergeometric function. On the other hand, we can derive a\nconstraint for the explicit finite series expression for $c_n$ that renders\nthe series in Eq.~(\\ref{A7}) to be convergent for all values of $t$. In order\nto be self-contained, intermediate steps to derive this constraint and to show\nthe convergence are shown in the following. Necessary is Pascal's rule\n\\begin{eqnarray}\n\\lefteqn{\\begin{pmatrix}n\\\\k\\end{pmatrix}+\\begin{pmatrix}n\\\\k-1\\end{pmatrix}\n  \\ =\\ \\frac{n!}{k!(n-k)!}+\\frac{n!}{(k-1)!(n-k+1)!}}\\nonumber\\\\\n  &=&\\frac{n!(n-k+1+k)}{k!(n-k+1)!}\\ =\\ \\frac{(n+1)!}{k!(n+1-k)!}\n  \\ =\\ \\begin{pmatrix}n+1\\\\k\\end{pmatrix}\n\\end{eqnarray}\nand the sum over the rows of Pascal's triangle,\n\\begin{equation}\\label{row}\n\\sum_{k=0}^n\\begin{pmatrix}n\\\\k\\end{pmatrix}=2^n\n\\end{equation}\nwhich can be shown by mathematical induction. The base case $n=0$ is obvious,\nas $(\\begin{smallmatrix}0\\\\0\\end{smallmatrix})=1=2^0$. For the induction step\nfrom $n$ to $n+1$ we write the first and last elements\n$(\\begin{smallmatrix}n+1\\\\0\\end{smallmatrix})=1$ and\n$(\\begin{smallmatrix}n+1\\\\n+1\\end{smallmatrix})=1$ separately and use\nPascal's rule to obtain\n\\begin{eqnarray}\n\\lefteqn{\\sum_{k=0}^{n+1}\\begin{pmatrix}n+1\\\\k\\end{pmatrix}\n  \\ =\\ 1+\\sum_{k=1}^n\\begin{pmatrix}n+1\\\\k\\end{pmatrix}\n  +1\\ =}\\nonumber\\\\\n  &=&1+\\sum_{k=1}^n\\begin{pmatrix}n\\\\k\\end{pmatrix}\n  +\\sum_{k=1}^n\\begin{pmatrix}n\\\\k-1\\end{pmatrix}+1\n  \\ =\\ 2\\sum_{k=0}^n\\begin{pmatrix}n\\\\k\\end{pmatrix}\\ =\\ 2^{n+1}.\n\\end{eqnarray}\nThis proves Eq.~(\\ref{row}). Returning to Eq.~(\\ref{cn}), one has\n$0\\le k\\le n-1$ and, therefore,\n\\begin{equation}\n\\frac1{2n-1}\\le\\frac1{2k+1}\\le 1.\n\\end{equation}\nFor the result in Eq.~(\\ref{cn}) this means that\n\\begin{equation}\n\\frac1{2n-1}\\sum_{k=0}^{n-1}\\begin{pmatrix}n-1\\\\k\\end{pmatrix}\\le c_n\\le\n\\sum_{k=0}^{n-1}\\begin{pmatrix}n-1\\\\k\\end{pmatrix}=2^{n-1},\n\\end{equation}\ni.e., the existence of a real number $c_n^*$ between $1\/(2n-1)$ and $1$ such\nthat $c_n=c_n^*2^{n-1}$. One has\n\\begin{equation}\n\\mathop{\\rm erf}\\nolimits_p(t)^2=1-\\frac4\\pi\\sum_{n=0}^Nc_n\\frac{(-1)^n}{n!}(t^2)^n\n  =1-\\frac2\\pi\\sum_{n=0}^Nc_n^*\\frac{(-2t^2)^n}{n!},\n\\end{equation}\nand because of $0\\le c_n^*\\le 1$ there is again a real number $c_N^{**}$ in\nthe corresponding open interval so that\n\\begin{equation}\n\\frac2\\pi\\sum_{n=0}^Nc_n^*\\frac{(-2t^2)^n}{n!}=c_N^{**}\\frac2\\pi\\sum_{n=0}^N\n\\frac{(-2t^2)^n}{n!}<\\frac2\\pi\\sum_{n=0}^N\\frac{(-2t^2)^n}{n!}.\n\\end{equation}\nAs the latter is the power series expansion of $(2\/\\pi)e^{-2t^2}$ which is\nconvergent for all values of $t$, also the original series is convergent and,\ntherefore, $\\mathop{\\rm erf}\\nolimits_p(t)^2$ with the limiting value shown in Eq.~(\\ref{A7}). A\nmore compact form of the power series expansion is given by\n\\begin{equation}\n\\mathop{\\rm erf}\\nolimits(t)^2=\\sum_{n=1}^\\infty c_n\\frac{(-1)^{n-1}}{n!}(t^2)^n,\\qquad\nc_n=\\sum_{k=0}^{n-1}\\frac{1}{2k+1}\\begin{pmatrix}n-1\\\\ k\\\\\\end{pmatrix}.\n\\end{equation}\n\n\\section{Approximations for the inverse error function}\nBased on the geometric approach from Ref.~\\cite{Martila:2022}, in the\nfollowing we describe how to find simple, handy formulas that, guided by\nhigher and higher orders of the approximation~(\\ref{Eqf2}) for the error\nfunction lead to more and more advanced approximation of the inverse error\nfunction. Starting point is the degree $p=0$, i.e., the approximation in\nEq.~(\\ref{rteq}). Inverting $E=\\mathop{\\rm erf}\\nolimits_0(t)=(1-e^{-k_{0,1}^2t^2})^{1\/2}$ leads\nto $t^2=-\\ln(1-E^2)\/k_{0,1}^2$, and using the parameter $k_{0,1}=1.116$ from\nEq.~(\\ref{rteq}) gives\n\\begin{equation}\nT_0=\\sqrt{-\\ln(1-E^2)}\/k_{0,1}^2.\n\\label{dft2}\n\\end{equation}\nFor $0\\le E \\le 0.92$ the relative deviation $(T_{(0)}-t)\/t$ from the exact\nvalue $t$ is less than $1.11\\%$, for $0\\le E<1$ the deviation is less than\n$10\\%$. Therefore, for $E>0.92$ a more precise formula has to be used. As such\nhigher values for $E$ appear only in $8\\%$ of the cases, this will not\nessentially influence the CPU time.\n\nContinuing with $p=1$, we insert $T_0=\\sqrt{-\\ln(1-E^2)}\/k_{0,1}^2$ into\nEq.~(\\ref{Eqf2}) to obtain\n\\begin{equation}\n\\mathop{\\rm erf}\\nolimits_1(T_0)=\\sqrt{1-\\frac12(e^{-k_{1,1}^2T_0^2}+e^{-k_{1,2}^2T_0^2})},\n\\end{equation}\nwhere $k_{1,1}=1.01$ and $k_{1,2}=1.23345$ are the same as for\nEq.~(\\ref{rteq2}). Taking the derivative of Eq.~(\\ref{erf}) and approximating\nthis by the difference quotient, one obtains\n\\begin{equation}\n\\frac{\\mathop{\\rm erf}\\nolimits(t)-\\mathop{\\rm erf}\\nolimits(T_0)}{t-T_0}=\\frac{\\Delta\\mathop{\\rm erf}\\nolimits(t)}{\\Delta t}\\Big|_{t=T_0}\n  \\approx\\frac{d\\mathop{\\rm erf}\\nolimits(t)}{dt}\\Big|_{t=T_0}=\\frac2{\\sqrt\\pi}e^{-T_0^2},\n\\end{equation}\nleading to $t\\approx T_1=T_0+\\frac12\\sqrt\\pi e^{T_0^2}(E-\\mathop{\\rm erf}\\nolimits_1(T_0))$. In\nthis case, for in the larger interval $0\\le E\\le 0.995$ the relative deviation\n$(T_1-t)\/t$ is less than $0.1\\%$. Using $\\mathop{\\rm erf}\\nolimits_2(t)$ instead of $\\mathop{\\rm erf}\\nolimits_1(t)$ and\ninserting $T_1$ instead of $T_0$ one obtains $T_2$ with a relative deviation\nof maximally $0.01\\%$ for the same interval. The results are shown in\nFig.~\\ref{rval012}.\n\n\\begin{figure}[ht]\\begin{center}\n\\epsfig{figure=rval012.eps, scale=0.7}\n\\caption{\\label{rval012}Relative deviations for the statical approximations}\n\\end{center}\\end{figure}\n\nThe method to can be optimised by a method similar to the shooting method in\nboundary problems, giving dynamics to the calculation. Suppose that following\none of the previous methods, for a particular argument $E$ we have found an\napproximation $t_0$ for the value of the inverse error function at this\nargument. Using $t_1=1.01t_0$, one can adjust the improved result\n\\begin{equation}\nt=t_0+A(E-\\mathop{\\rm erf}\\nolimits(t_0))\n\\end{equation}\nby inserting $E=\\mathop{\\rm erf}\\nolimits(t)$ and and calculating $A$ for $t=t_1$. In general, this\nprocedure gives a vanishing deviation close to $E=0$. In this case and for\n$t_0=T_1$, in the interval $0\\le E\\le 0.7$ the maximal deviation is slightly\nlarger than $10^{-6}=0.0001\\%$ while up to $E=0.92$ the deviation is\nrestricted to $10^{-5}=0.001\\%$. A more general ansatz\n\\begin{equation}\nt=t_0+A(E-\\mathop{\\rm erf}\\nolimits(t_0))+B(E-\\mathop{\\rm erf}\\nolimits(t_0))^2\n\\end{equation}\ncan be adjusted by inserting $E=\\mathop{\\rm erf}\\nolimits(t)$ for $t=1.01t_0$ and $t=1.02t_0$, and\nthe system of equations\n\\begin{equation}\n\\Delta t=A\\Delta E_1+B\\Delta E_1^2,\\qquad\n2\\Delta t=A\\Delta E_2+B\\Delta E_2^2\n\\end{equation}\nwith $\\Delta t=0.01t_0$, $\\Delta E_i=\\mathop{\\rm erf}\\nolimits(t_i)-\\mathop{\\rm erf}\\nolimits(t_0)$ can be solved for\n$A$ and $B$ to obtain\n\\begin{equation}\nA=-\\frac{(2\\Delta E_1^2-\\Delta E_2^2)\\Delta t}{\\Delta E_1\\Delta E_2\n  (\\Delta E_1-\\Delta E_2)},\\quad\nB=\\frac{(-2\\Delta E_1+\\Delta E_2)\\Delta t}{\\Delta E_1\\Delta E_2\n  (\\Delta E_1-\\Delta E_2)}.\n\\end{equation}\nFor $0\\le E\\le 0.70$ one obtains a relative deviation of $1.5\\cdot 10^{-8}$,\nfor $0\\le E\\le 0.92$ the maximal deviation is $5\\cdot 10^{-7}$. Finally, the\nadjustment of\n\\begin{equation}\nt=t_0+A(E-\\mathop{\\rm erf}\\nolimits(t_0))+B(E-\\mathop{\\rm erf}\\nolimits(t_0))^2+C(E-\\mathop{\\rm erf}\\nolimits(t_0))^3\n\\end{equation}\nleads to\n\\begin{eqnarray}\nA&=&(3\\Delta E_1^2\\Delta E_2^2(\\Delta E_1-\\Delta E_2)\n  -2\\Delta E_1^2\\Delta E_3^2(\\Delta E_1-\\Delta E_3)\\strut\\nonumber\\\\&&\\strut\n  +\\Delta E_2^2\\Delta E_3^2(\\Delta E_2-\\Delta E_3))\\Delta t\/D,\\nonumber\\\\\nB&=&(-3\\Delta E_1\\Delta E_2(\\Delta E_1^2-\\Delta E_2^2)\n  +2\\Delta E_1\\Delta E_3(\\Delta E_1^2-\\Delta E_3^2)\\strut\\nonumber\\\\&&\\strut\n  -\\Delta E_2\\Delta E_3(\\Delta E_2^2-\\Delta E_3^2))\\Delta t\/D,\\nonumber\\\\\nC&=&(3\\Delta E_1\\Delta E_2(\\Delta E_1-\\Delta E_2)\n  -2\\Delta E_1\\Delta E_3(\\Delta E_1-\\Delta E_3)\\strut\\nonumber\\\\&&\\strut\n  +\\Delta E_2\\Delta E_3(\\Delta E_2-\\Delta E_3))\\Delta t\/D,\n\\end{eqnarray}\nwhere $D=\\Delta E_1\\Delta E_2\\Delta E_3(\\Delta E_1-\\Delta E_2)\n(\\Delta E_1-\\Delta E_3)(\\Delta E_2-\\Delta E_3)$. For $0\\le E\\le 0.70$ the\nrelative deviation is restricted to $5\\cdot 10^{-10}$ while up to $E=0.92$\nthe maximal relative deviation is $4\\cdot 10^{-8}$. The results for the\ndeviations of $T_{(n)}$ ($n=1,2,3$) for linear, quadratic and cubic dynamical\napproximation are shown in Fig.~\\ref{rval456}.\n\n\\begin{figure}[ht]\\begin{center}\n\\epsfig{figure=rval456.eps, scale=0.7}\n\\vspace{7pt}\n\\epsfig{figure=rval4567.eps, scale=0.7}\n\\caption{\\label{rval456}Relative deviation for the dynamical approximations\n  (the degree is chosen to be $p=1$)}\n\\end{center}\\end{figure}\n\n\\section{Conclusions}\nIn order to test the feasibility and speed, we have coded our algorithm in the\ncomputer language C under {\\tt Slackware~15.0} on an ordinary hp laptop. The\ndependence of the CPU time for the calculation is estimated by calculating the\nvalue $10^6$ times in sequence. The speed of the calculation of course turns\ndoes not depend on the value for $E$, as the precision is not optimised. This\nwould have to be neccessary for a practical application. For an arbitrary\nstarting value $E=0.8$ we perform this test, and the results are given in\nTable~\\ref{tab1}. An analysis of the table shows that a further step in the\ndegree $p$ doubles the run time while the dynamics for increasing $n$ adds a\nconstant value of approximately $0.06$ seconds to the result. Despite the fact\nthe increase of the dynamics needs the solution of a linear system of\nequations and the coding of the result, this endeavour is justified, as by\nusing the dynamics one can increase the precision of the result without\nloosing calculational speed. \n\n\\begin{table}[ht]\\begin{center}\n\\caption{\\label{tab1}Run time experiment for our algorithm under C for\n$E=0.8$ and different values of $n$ and $p$ (CPU time in seconds). As\nindicated, the errors are in the last displayed digit, i.e., $\\pm 0.01$\nseconds.}\n\\begin{tabular}{|r||c|c|c|c|}\\hline\n&$n=0$&$n=1$&$n=2$&$n=3$\\\\\\hline\\hline\n$p=0$&$0.06(1)$&$0.12(1)$&$0.15(1)$&$0.18(1)$\\\\\\hline\n$p=1$&$0.13(1)$&$0.19(1)$&$0.22(1)$&$0.25(1)$\\\\\\hline\n$p=2$&$0.24(1)$&$0.29(1)$&$0.33(1)$&$0.36(1)$\\\\\\hline\n\\end{tabular}\\end{center}\n\\end{table}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \\label{sec:introduction}\nRecent semantic segmentation algorithms \\cite{DeeplabV3+:2018Chen,Wider:2019Wu} have successfully solved challenging benchmark datasets for semantic segmentation tasks like PASCAL VOC~\\citep{VOC:Everingham2015} and MS COCO~\\citep{COCO:Lin2014}.\nTo do so, however, they use a large number of pixel-level annotations,\nwhich require extensive human labor to obtain.\nGreat attention in computer vision research has thus turned to \\emph{weakly-supervised} learning \\citep{OAA:2019Jiang,SEAM:2020Wang,MCIS:2020Sun}.\nWeakly-supervised semantic segmentation aims to classify each pixel of test images, trained only on image-level labels\n(whether a class is present in the image, but not its location).\nAlthough weakly-supervised approaches have seen success in both semantic segmentation and object localization tasks, \\citet{EvalWSOL:Choe2020} cast significant doubt on their validity and practicality.\nThey argue that although weakly-supervised learning algorithms are designed to be trained only on image-level labels, they inevitably use explicit pixel-level labels (or, equivalently, manual judgement of outputs) in hyperparameter tuning.\nSince at least \\emph{some} fully-supervised inputs are necessary,\n\\citeauthor{EvalWSOL:Choe2020} point out that simply using a small number of these,\n{e.g}\\onedot} \\def\\Eg{{E.g}\\onedot five per class,\nto train a fully-supervised localization model\nsubstantially outperforms a weakly-supervised counterpart.\nTo still take advantage of less-expensive weakly-supervised data points, though,\nperhaps the most natural change is\nto the semi-weakly supervised semantic segmentation (SWSSS) task:\nhere only a small number of pixel-level labels are provided, as well as a large number of image-level labels.\n\nSegmentation networks trained on a small number of pixel-level labels often confuse similar classes, {e.g}\\onedot} \\def\\Eg{{E.g}\\onedot \\labelname{cat} and \\labelname{horse}, as they architecturally tend to focus on local features rather than farther-away distinguishing areas.\nThus, the additional supervision from image-level labels can be potentially quite helpful.\nMost existing SWSSS methods generate pseudo-labels from a classifier using class activation maps (CAMs) \\cite{CAM:Zhou2015},\nthen train a segmentation network using both pseudo-labels and true pixel labels.\nThese pseudo-labels, however, are difficult to extract:\nthey tend to focus on small discriminative regions of objects,\nignoring less-distinctive bulks of objects and often including nearby pixels that are not part of objects.\nOur analysis shows that as the baseline segmentation model improves with more training data,\nthe pseudo-labels quickly provide more incorrect than correct supervision to what the model already would have predicted.\nPrevious methods have thus employed additional information, such as saliency maps, or additional processing methods, adding complexity and many more hyperparameters to tune on a fully-supervised validation set.\nWe use weak supervision data differently,\nwithout requiring any side information and introducing far fewer new hyperparameters.\n\n\n\nTo motivate our method, consider \\cref{fig:gt_filtering}.\nBaseline models with small training sets predict many classes which are not present in the image.\nIf we ignore the predictions for the class of any pixel which are not present in the image at all, which we term \\textit{oracle filtering}, then the segmentation performance improves dramatically.\nInspired by this, we propose a simple algorithm we call \\textit{prediction filtering}.\nPrediction filtering uses a multi-label classifier trained on only image-level labels\nto filter out segmentation predictions deemed very unlikely by the classifier, replacing predictions for those pixels with the next-most-likely class allowed through the filter.\nIt is compatible with any segmentation model, and\nthe threshold for ``very unlikely'' is the only new hyperparameter introduced.\n\nAlthough the classifier is not perfect, because it is trained on a large weakly-supervised set, its predictions tend to be quite accurate.\nMoreover, it is trying to solve an easier problem than the segmentation network, using a different architecture.\nAs we will see in the experiments, even without any additional weakly-supervised data, prediction filtering tends to improve the segmentation performance.\nWhen applied to baseline segmentation models, the performance significantly improves; adding it to the baseline model variant from \\citet{Dual:2020Luo} achieves (to our knowledge) the new highest performance on PASCAL VOC in the SWSSS regime.\nAs prediction filtering is so general, it can even be easily applied to models which already exploit weakly-supervised data via pseudo-labels;\ndoing so on the state-of-the-art SWSSS algorithms \\citep{CCT:2020Ouali,Dual:2020Luo} yields a new model with significantly higher performance,\nwith more improvement for models trained on fewer fully-labeled images.\n\n\n\n\n\n\n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{figures\/gt_filtering_smaller.pdf}\n  \\caption{Filtering this model's predicted classes drastically improves segmentation quality.}\n  \\label{fig:gt_filtering}\n  \\vspace*{-3.5mm}\n\\end{figure}\n\n\n\\section{Approaches to Semantic Segmentation}\n\\label{sec:related_work}\n\n\\paragraph{Fully-supervised semantic segmentation.}\nIn this task, we have a training set of images with pixel-level class labels:\n$\\mathcal{D}_\\mathit{pixel} = \\{(x_i, y_i)\\}_{i=1}^M$, where\n$x_i \\in \\mathbb{R}^{3 \\times H_i\\times W_i}$ are images\nand $y_i \\in \\{0,1\\}^{K\\times H_i\\times W_i}$ are pixel labels,\nwith $K$ the number of classes.\nOur goal is to find a model that can predict pixel labels $y$ given a new image $x$.\n\nCurrent approaches are mostly based on convolutional networks.\nOne important factor to their success is using larger receptive fields via dilated convolutional layers \\citep{Dilated:2015Yu,DeeplabV3:2017Chen}.\nEven so, state-of-the-art algorithms still misclassify many pixels\nwhen multi-label classifiers on the same data obtain near-perfect accuracy.\nWe conjecture this is because segmentation models still miss global structure of the image when looking at an individual pixel.\nWe will exploit that a classifier ``looks at images differently.''\n\n\n\\paragraph{Weakly-supervised semantic segmentation.}\nTo avoid extensive human labor for pixel-level annotation, there have been many attempts to replace pixel-level labels with image-level labels:\n$\\mathcal{D}_\\mathit{image} = \\{(x_i, z_i)\\}_{i=1}^N$,\nwith $z_i \\in \\{0,1\\}^{K}$ a ``logical or'' of each channel of the unknown $y_i$.\nWe still want a model to produce $y$.\nThe most common pipeline for weakly-supervised semantic segmentation is to generate a class activation map (CAM) \\citep{CAM:Zhou2015}, refine it with various post-processing methods, then use it as a pseudo-label to train a semantic segmentation network.\nHowever, CAMs tend to focus only on discriminative regions of an object.\nPrior work has attempted to expand the CAM to entire objects by masking out parts of an image \\citep{HaS:Singh2017} or intermediate feature \\citep{ACoL:Zhang2018,ADL:Choe2019};\nthese methods do indeed expand the resulting CAM, but often too much, in very unstable ways.\nAnother popular approach to grow the CAM regions, via methods proposed by \\citet{CRF:Krahenbuhl2011}, \\citet{DSRG:2018Huang}, or \\citet{PSA:2018Ahn}, until it converges to the object region \\cite{Boundary:2020Chen}.\n\n\n\n\n\\paragraph{Semi-weakly supervised semantic segmentation.}\n\\citet{EvalWSOL:Choe2020} point out fundamental problems with weakly-supervised learning, as discussed in \\cref{sec:introduction}.\nWe thus consider combining a small number of pixel-annotated images $\\mathcal{D}_\\mathit{pixel}$ with many weakly-supervised images $\\mathcal{D}_\\mathit{image}$, which we refer to as semi-weakly supervised semantic segmentation (SWSSS).\nAlthough they have not used exactly this name, many papers have already addressed this setting.\nBroadly speaking, the most common approach is to generate pseudo-labels from a CAM, and use these in combination with true labels to train a segmentation network\n\\citep{WSSL:2015Papandreou,MDC:2018Wei,GAIN:2018Li,CCT:2020Ouali,Dual:2020Luo,AdvCAM:2021Lee,vessel2022dang,semi_context_const2021lai}.\n(For lack of space, we unfortunately elide all details of these approaches.)\nBecause image-level labels have no spatial information, however, it is fundamentally difficult to make accurate pseudo-labels.\nAs we will now argue, as the number of pixel labels increases and base models improve, the benefit of pseudo-labels drastically diminishes.\n\n\n\n\n\n\n\n\n\n\n\\label{sec:problem}\n\n\n\n\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{figures\/problems_extended3.pdf}\n    \\caption{A segmentation network trained on few pixel-level labels confuses similar classes, as it does not capture the global features (panel a). To alleviate this issue, pseudo-labels have been widely used; this approach is sensitive to the values of additional hyperparameters, and the quality of its supervision drastically diminishes as the number of pixel labels increases (panel c).}\n    \\vspace*{-2mm}\n    \\label{fig:problems}\n\\end{figure*}\n\n\n\nTo demonstrate this, we train a DeeplabV1 segmentation network \\citep{DeeplabV1:2015Chen} on $\\mathcal{D}_\\mathit{pixel}$ consisting of $1{,}464$ images from PASCAL VOC training set, and a VGG16 classifier on $\\mathcal{D}_\\mathit{image}$ containing all $10{,}582$ images in the full training set.\nTo examine what part of the image causes this prediction, we extract CAMs for \\labelname{cat} and \\labelname{horse} classes using GradCAM~\\cite{Gradcam:Selvaraju2016}.\nAlthough the classifier confidently predicts that only the \\labelname{cat} class is present in the given image ($\\Pr(\\labelname{cat}) = 0.97$, $\\Pr(\\labelname{horse}) = 0.03$), \\cref{fig:problems}(a) shows that the segmentation model predicts most of the cat's body as \\labelname{horse} (pink region).\nThe CAM shows that the classifier makes this decision based on the most discriminative region of the object, {i.e}\\onedot} \\def\\Ie{{I.e}\\onedot the cat's head.\nThe segmentation model does the same at the green (top) location, correctly predicted as \\labelname{cat};\nat the yellow (middle) location, however, the \\labelname{horse} prediction is based mostly on the more ambiguous body area.\nAs $\\mathcal{D}_\\mathit{pixel}$ grows, this phenomenon is largely alleviated; it seems $1{,}464$ images were not enough for the segmentation model to learn ``where to look.''\n\n\nSupervision added by previous models from classifier CAMs, then,\nwill also tend to focus on discriminative regions of an object,\nand can therefore be misleading.\nTo estimate the effectiveness of the pseudo-labels, we define a measure called \\textit{mNet}, the mean over classes $c$ of\n\\[\n    \\mathit{net}_c =\n   \n    \\frac{\\sum_{i=1}^N \\!\\!\\splitfrac{%\n        \\textcolor{myBlue}{\\operatorname{area}\\left(\n            \\left( \\mathit{Pseudo}_{i,c} \\setminus \\mathit{Pred}_{i,c} \\right)\n            \\cap \\mathit{GT}_{i,c}\n        \\right)} }{%\n        - \\textcolor{myRed}{\\operatorname{area}\\left(\n            \\left( \\mathit{Pseudo}_{i,c} \\setminus \\mathit{Pred}_{i,c} \\right)\n            \\setminus \\mathit{GT}_{i,c}\n        \\right))} }\n    }{\n        \\sum_{i=1}^N \\operatorname{area}(\\mathit{Pseudo}_{i,c} \\setminus \\mathit{Pred}_{i,c})\n    }\n    \\label{eq:net}\n.\\]\nHere\nthe subscript $\\cdot_{i,c}$ refers to the set of pixels of $i$-th training image whose label is $c$;\n$\\mathit{GT}$ refers to the ground truth labels $y$,\n$\\mathit{Pred}$ to the predicted labels from a baseline segmentation model,\nand $\\mathit{Pseudo}$ to the CAM-based pseudo-labels.\nThe first (blue) term in the numerator measures how much correct supervision the pseudo-label adds (see \\cref{fig:problems}(b));\nthe second (red) term, how much incorrect supervision is added.\nThe denominator combines these two regions.\n$\\mathit{mNet}$ does not exactly measure how much the pseudo-labels help predictions, but it gives a rough sense of how correct their information is.\n\n\\Cref{fig:problems}(c) shows that ``plain'' CAM (in green, lowest) indeed helps when $\\mathcal{D}_\\mathit{pixel}$ is very small, but as it grows, $\\mathit{mNet}$ even becomes negative.\nIt is possible to improve these predictions by, for instance, post-processing with a CRF (blue line, top).\nThis, however, requires a complicated structure with several additional hyperparameters to tune on a fully-supervised validation set;\nsetting these parameters correctly significantly affects performance,\nas shown {e.g}\\onedot} \\def\\Eg{{E.g}\\onedot by the substantially worse $\\mathit{mNet}$ when changing the threshold for a foreground region $\\theta$ from $0.3$ to $0.2$ (orange line, middle).\n\n\n\n\n\n\n\n   \n   \n   \n   \n   \n\n\\section{Prediction Filtering}\n\\label{sec:prediction_filtering}\n\\paragraph{Motivation.}\nGiven a segmentation network $f$,\nwhose output on the $i$-th image is $f(x_i) \\in \\mathbb{R}^{K \\times H\\times W}$,\nthe final prediction at each pixel $(h, w)$ is normally\n\\begin{align}\n    \\hat{y}_{h,w} = \\argmax_{c \\in \\mathcal K} {f(x_i)_{c, h, w}}\n    \\label{eq:normal}\n,\\end{align}\nwhere $\\mathcal K = \\{1, 2, \\dots, K\\}$.\n\\emph{Oracle filtering} (\\cref{fig:gt_filtering}) instead only considers classes actually present in the image, maximizing over $\\mathcal{\\tilde K}_i = \\{ c : z_{i,c} = 1 \\}$.\nThis improves segmentation performance substantially;\nthe mIoU (mean intersection over union, the standard segmentation performance metric) \nof a DeeplabV1 segmentation network trained on $\\mathcal{D}_\\mathit{pixel}$ with $1{,}464$ images improves from $61.8$ to $70.6$.\nWe conjecture this is because the segmentation network has not learned to appropriately consider global features when predicting at each pixel of the image,\nwhile the classifier, solving an easier problem with more data, can immediately identify relevant areas.\n\n\n\n\n\n\n\\begin{table*}[t!]\n\\centering\n\\fontsize{9.0}{12.0}\\selectfont\n\\begin{tabular}{c|c|l|c|c|c|l}\n\\hline\nBackbone & Add. 9.1K Images & \\multicolumn{1}{c|}{Method} & Bkg. Cues & CRF & Pred. Filter & $\\!$mIoU \\\\\n\\hline\\hline\n\\multirow{8}{*}{VGG} \n  & -- & DeeplabV1~\\citep{DeeplabV1:2015Chen}  & -- & \\checkmark & -- &  61.8 \\\\\n  \\hhline{*{1}{~}*{6}{|-}}\n  & \\multirow{6}{*}{Image-level} & \\cellcolor{myGray}DeeplabV1~\\citep{DeeplabV1:2015Chen}  & \\cellcolor{myGray}-- & \\cellcolor{myGray}\\checkmark & \\cellcolor{myGray}\\checkmark & \\cellcolor{myGray}\\ub{67.4} \\\\\n  & & WSSL~\\citep{WSSL:2015Papandreou}  & --  & \\checkmark & -- & 64.6 \\\\\n  & & \\cellcolor{myGray}WSSL~\\citep{WSSL:2015Papandreou} & \\cellcolor{myGray}-- & \\cellcolor{myGray}\\checkmark & \\cellcolor{myGray}\\checkmark & \\cellcolor{myGray}67.1 \\\\\n  & & \\citet{SemiGAN2017Souly}  & -- & -- & -- & 65.8$^*$ \\\\\n  & & MDC~\\citep{MDC:2018Wei}  & \\checkmark   & \\checkmark & -- & 65.7$^*$  \\\\\n  & & FickleNet~\\citep{FickleNet:2019Lee} & \\checkmark  & \\checkmark & -- & 65.8$^*$ \\\\ \n  \\hhline{*{1}{~}*{6}{|-}}\n  & Pixel-level & DeeplabV1~\\citep{DeeplabV1:2015Chen}  & -- & \\checkmark & -- & 69.0  \\\\\n\\Xhline{2\\arrayrulewidth}\n\\multirow{4}{*}{VGG-W}\n  & -- & DeeplabV1-W~\\citep{Dual:2020Luo}  & -- & \\checkmark & -- & 69.2 \\\\\n  \\hhline{*{1}{~}*{6}{|-}}\n  & \\multirow{3}{*}{Image-level} & \\cellcolor{myGray}DeeplabV1-W~\\citep{Dual:2020Luo}  & \\cellcolor{myGray}-- & \\cellcolor{myGray}\\checkmark & \\cellcolor{myGray}\\checkmark & \\cellcolor{myGray}73.8 \\\\\n  & & DualNet~\\citep{Dual:2020Luo} & \\checkmark  & \\checkmark & -- & 73.9 \\\\\n  & & \\cellcolor{myGray}DualNet~\\citep{Dual:2020Luo} & \\cellcolor{myGray}\\checkmark & \\cellcolor{myGray}\\checkmark & \\cellcolor{myGray}\\checkmark & \\cellcolor{myGray}\\ub{75.1} \\\\ \\Xhline{2\\arrayrulewidth}\n\\multirow{9}{*}{ResNet}\n  & \\multirow{2}{*}{--} & DeeplabV3~\\citep{DeeplabV3:2017Chen} & -- & \\checkmark & -- & 72.4 \\\\\n  & & \\citet{semi_context_const2021lai} & -- & -- & -- & 74.5 \\\\\n  \\hhline{*{1}{~}*{6}{|-}}\n  & \\multirow{6}{*}{Image-level} & \\cellcolor{myGray}DeeplabV3~\\citep{DeeplabV3:2017Chen} & \\cellcolor{myGray}-- & \\cellcolor{myGray}\\checkmark & \\cellcolor{myGray}\\checkmark & \\cellcolor{myGray}75.3 \\\\ \n  & & \\citet{semi_context_const2021lai} & -- & -- & -- & 76.1 \\\\\n  & & CCT~\\citep{CCT:2020Ouali} & -- & \\checkmark & -- & 74.7 \\\\\n  & & \\cellcolor{myGray}CCT~\\citep{CCT:2020Ouali} & \\cellcolor{myGray}-- & \\cellcolor{myGray}\\checkmark & \\cellcolor{myGray}\\checkmark & \\cellcolor{myGray}76.0 \\\\\n  & & AdvCAM~\\citep{AdvCAM:2021Lee} & -- & \\checkmark & -- & 76.1 \\\\\n  & & \\cellcolor{myGray}AdvCAM~\\citep{AdvCAM:2021Lee} & \\cellcolor{myGray}-- & \\cellcolor{myGray}\\checkmark & \\cellcolor{myGray}\\checkmark & \\cellcolor{myGray}\\ub{77.1} \\\\ \n  \\hhline{*{1}{~}*{6}{|-}}\n  & Pixel-level & DeeplabV3~\\citep{DeeplabV3:2017Chen} & -- & \\checkmark &-- & 77.4 \\\\ \n\\Xhline{2\\arrayrulewidth}\n\\multirow{4}{*}{ResNet-W}\n  & -- & DeeplabV3-W~\\citep{Dual:2020Luo} & -- & \\checkmark & -- & 76.2 \\\\ \n  \\hhline{*{1}{~}*{6}{|-}}\n  & \\multirow{3}{*}{Image-level} & \\cellcolor{myGray}DeeplabV3-W~\\citep{Dual:2020Luo} & \\cellcolor{myGray}-- & \\cellcolor{myGray}\\checkmark & \\cellcolor{myGray}\\checkmark & \\cellcolor{myGray}\\ub{77.5} \\\\ \n  & & DualNet~\\citep{Dual:2020Luo} & \\checkmark  & \\checkmark & -- & 76.7 \\\\\n  & & \\cellcolor{myGray}DualNet~\\citep{Dual:2020Luo} & \\cellcolor{myGray}\\checkmark  & \\cellcolor{myGray}\\checkmark & \\cellcolor{myGray}\\checkmark & \\cellcolor{myGray}77.3 \\\\\n\\Xhline{2\\arrayrulewidth}\n\\multirow{3}{*}{HRNetV2-W48}\n  & -- & OCRNet~\\citep{OCRNet:2021Strudel} & -- & -- & -- & 74.0 \\\\ \n  \\hhline{*{1}{~}*{6}{|-}}\n  & Image-level & \\cellcolor{myGray}OCRNet~\\citep{OCRNet:2021Strudel}& \\cellcolor{myGray}-- & \\cellcolor{myGray}-- & \\cellcolor{myGray}\\checkmark & \\cellcolor{myGray}\\ub{75.8} \\\\ \n  \\hhline{*{1}{~}*{6}{|-}}\n  & Pixel-level & OCRNet~\\citep{OCRNet:2021Strudel}& -- & -- & -- & 77.7 \\\\ \n\\Xhline{2\\arrayrulewidth}\n\\end{tabular}\n\\vspace*{-1.0mm}\n\\caption{Comparison of the state-of-the-art methods on $1{,}464$ images with pixel labels. The ``Add. 9.1K Images'' column gives which type of supervision is used for the 9.1K additional images (augmented dataset). Numbers marked with $*$ are as reported by the corresponding paper.}\n\\label{tbl:sota}\n\\vspace*{-1.0mm}\n\\end{table*}\n\n\\newcolumntype{C}{>{\\centering\\arraybackslash}p{3.2em}}\n\\newcolumntype{D}{>{\\centering\\arraybackslash}p{6.1em}}\n\\begin{table*}[t!]\n\\setlength{\\tabcolsep}{0.7pt}\n\\centering\n\\fontsize{6.5}{9.0}\\selectfont\n\\begin{tabu}{D|CCCCCCCCCCCCCCCCCCCC|C}\n\\hline\nMethod & plane & bike & bird & boat & bottle & bus & car & cat & chair & cow & table & dog & horse & mbike & person & plant & sheep & sofa & train & tv & mIoU \\\\\n\\hline\\hline\nDeeplabV1 & 71.1 & 37.1 & 78.5 & 52.7 & 58.3 & 79.4 & 72.0 & 73.2 & 20.6 & 58.0 & 56.1 & 66.5 & 55.9 & 76.1 & 76.4 & 40.6 & 65.4 & 42.8 & 65.2 & 53.3 & 61.4 \\\\\n+ Pred.\\ Filter    & \\ub{76.2} & \\ub{38.9} & \\ub{82.2} & \\ub{58.2} & \\ub{61.2} & \\ub{85.1} & \\ub{76.8} & \\ub{84.4} & \\ub{22.6} & \\ub{73.0} & \\ub{56.2} & \\ub{79.3} & \\ub{76.2} & \\ub{82.4} & \\ub{78.2} & \\ub{46.0} & \\ub{80.4} & \\ub{43.6} & \\ub{71.5} & \\ub{55.2} & \\ub{67.6} \\\\\n\\hline\nDeeplabV3-W & \\ub{89.3} & 60.2 & 80.5 & 56.4 & 73.7 & 92.5 & 83.8 & \\ub{92.4} & 31.1 & 83.6 & \\ub{69.9} & \\ub{85.3} & 81.9 & 84.8 & \\ub{85.4} & 63.2 & 84.2 & 52.7 & \\ub{83.9} & \\ub{69.8} & 76.1 \\\\\n+ Pred.\\ Filter  & \\ub{89.3} & \\ub{61.7} & \\ub{81.5} & \\ub{58.0} & \\ub{73.8} & \\ub{92.6} & \\ub{84.3} & 91.3 & \\ub{34.4} & \\ub{84.9} & 69.8 & 85.1 & \\ub{89.4} & \\ub{86.2} & 85.1 & \\ub{64.3} & \\ub{89.0} & \\ub{54.3} & 83.2 & 68.1 & \\ub{77.2} \\\\\n\\Xhline{2\\arrayrulewidth}\n\\end{tabu}\n\\vspace*{-1.0mm}\n\\caption{Evaluation results of DeeplabV1 (VGG-based) and DeeplabV3-W (ResNet-based) models on the test set.}\n\\vspace*{-4.0mm}\n\\label{tbl:test}\n\\end{table*}\n\n\\paragraph{Prediction filtering.}\nInspired by this phenomenon, we propose a simple post-processing method, \\emph{prediction filtering}.\nGiven $D_{image}$ with a large number of images, we can train a highly accurate multi-label classifier $g$;\na ResNet50 achieves $99\\%$ accuracy and $97.5\\%$ average precision on PASCAL VOC.\nHence, we constrain predictions to come from the classifier's predictions instead of ground truth classes,\n$\\mathcal{\\hat K}_i = \\{ c : g(x_i)_c > \\tau \\}$,\nwhere $g(x_i)_{c}$ is the output logit of $g$ for class $c$,\nand $\\tau$ is a threshold to determine the presence of a class in an image.\nWe provide full pseudocode in the appendix.\n\n\n\n\nCompared to other SWSSS algorithms, prediction filtering has several advantages.\nFirst, the architecture is simple. It only requires an additional classifier, which can be trained in parallel with the segmentation network; most existing methods require training a classifier first.\nSecond, it requires only a single additional hyperparameter, the threshold $\\tau$, far fewer than required by other SWSSS algorithms.\nFor instance, MDC~\\cite{MDC:2018Wei} requires two thresholds to determine the foreground and background regions, in addition to selecting the number of dilation layers with different rate for each layer.\n(We provide a comprehensive comparison of hyperparameter counts in the appendix).\nPrediction filtering thus minimizes the requirements on the additional fully-supervised validation set.\nThird, it can be independently added to any segmentation algorithm, including existing SWSSS algorithms; we do so in our experiments. \n\n\n\n\n\n\n\n\n\n\\paragraph{Effect on performance.}\n\nPrediction filtering helps performance when an incorrect prediction is filtered out and the ``backup'' prediction is correct;\nit hurts when a correctly-predicted object is incorrectly filtered.\nIt can also change an incorrect prediction to a different incorrect prediction;\nthis can in fact either increase or decrease the mIoU score.\n\nFor a non-perfect classifier and reasonable setting of $\\tau$, it is conceivable for prediction filtering to hurt segmentation performance -- although we did not observe this in our experiments.\nThere is always a value of the threshold $\\tau$, however, for which prediction filtering at least does not hurt:\njust take $\\tau \\to -\\infty$, in which case no predictions are changed.\nAs $\\tau$ increases, it likely (though not certainly) removes incorrect predictions before it begins removing any correct predictions.\nIn another extreme, for a perfect classifier, prediction filtering approaches oracle filtering;\nclearly, oracle filtering may not achieve perfect performance, but it can only help.\n\n\n\n\n\n\\begin{figure*}[t!]\n    \\centering\n    \\includegraphics[width=0.95\\textwidth]{figures\/various_supervision_resnet_6_v4.pdf}\n    \\caption{Performance of the prediction filtering on various models and levels of supervision (M).}\n    \\label{fig:various_supervision}\n\\end{figure*}\n\n\n\\newcolumntype{F}{>{\\centering\\arraybackslash}p{2.0em}}\n\\newcolumntype{E}{>{\\centering\\arraybackslash}p{4.0em}}\n\\begin{table*}[t!]\n\\begin{minipage}{.53\\textwidth}\n\\centering\n\\fontsize{9.0}{12.0}\\selectfont\n\\begin{tabu}{cc|EE|EE}\n\\hline\n\\multirow{2}{*}{Filtering} &\n\\multirow{2}{*}{CRF} &\n\\multicolumn{2}{c|}{DeeplabV1} &\n\\multicolumn{2}{c}{DeeplabV3}\\\\\n\\cline{3-6}\n& & $M$=$500$ & $M$=$1{,}464$ & $M$=$500$ & $M$=$1{,}464$ \\\\\n\\hline\n\\hline\n & & 49.6 & 57.2 & 57.0 & 70.6 \\\\\n \\checkmark & & 61.4 & 64.6 & 64.8 & 74.0  \\\\\n & \\checkmark & 53.9 & 61.8 & 58.4 & 72.4 \\\\\n \\checkmark & \\checkmark & \\ub{63.6} & \\ub{67.4} & \\ub{65.9} & \\ub{75.3} \\\\\n\\Xhline{2\\arrayrulewidth}\n\\end{tabu}\n\\end{minipage}%\n\\begin{minipage}{.52\\textwidth}\n\\centering\n\\fontsize{9.0}{12.0}\\selectfont\n\\begin{tabu}{cc|FFFF}\n\\hline\n\\multirow{2}{*}{Filtering} & \\multirow{2}{*}{CRF} & \\multicolumn{4}{c}{Pixel-level labels ($M$)} \\\\\n\\cline{3-6}\n & & 200 & 500 & 800 & 1,464 \\\\\n\\hline\n\\hline\n & & 43.3 & 49.9 & 52.7 & 57.2 \\\\\n \\checkmark & & 46.1 & 54.0 & 57.1 & 62.0\\\\\n & \\checkmark & 46.3 & 53.9 & 56.8 & 61.8 \\\\\n \\checkmark & \\checkmark & \\ub{48.2} & \\ub{56.5} & \\ub{59.8} & \\ub{64.9} \\\\\n\\Xhline{2\\arrayrulewidth}\n\\end{tabu}\n\\end{minipage}%\n\\caption{{Left}: Performance of model variants with $\\lvert\\mathcal{D}_\\mathit{pixel}\\rvert = M$ and $\\lvert\\mathcal{D}_\\mathit{image}\\rvert = 10,582$. {Right}: the same, for a DeeplabV1 baseline, but with the classifier trained only on the same $M$ images in $D_{pixel}$.}\n\\vspace*{-1.0mm}\n\\label{tbl:crf_and_partial}\n\\end{table*}\n\n\n\\section{Experimental Evaluation}\n\\label{sec:experiment}\n\n\n\\paragraph{Dataset.}\n\nWe evaluate prediction filtering on PASCAL VOC 2012 \\citep{VOC:Everingham2015} \nwhich contains $10{,}582$ training, $1{,}449$ validation, and $1{,}456$ test images.\nFor SWSSS, we follow the training splits of \\citet{CCT:2020Ouali}, where $1{,}464$ images are used for $\\mathcal{D}_\\mathit{pixel}$.\nAs with previous work, we evaluate segmentation performance by mean Intersection over Union (mIoU),\ngenerally on the validation set.\nTest set performance is obtained from the PASCAL VOC evaluation server, without any tricks such as multi-scale or flipping.\n\n\\paragraph{Implementation.}\nTo verify the robustness of our method, we experiment with five semantic segmentation baselines: DeeplabV1 (based on VGG16), DeeplabV3 (based on ResNet-101), their deeper variants with wider receptive fields used by \\citet{Dual:2020Luo} which we call DeeplabV1-W and DeeplabV3-W, and a Transformer model called OCRNet~\\cite{OCRNet:2021Strudel} (based on HRNetV2-W48~\\cite{HRNet2020Wang}).\nFor prediction filtering, we use a ResNet50 classifier.\nWe also apply prediction filtering to several existing SWSSS models.\nAlthough CRF post-processing is no longer commonly used in semantic segmentation tasks,\nin our experiments it still significantly improves the performance of models trained on a small number of pixel-level labels.\nWe thus apply CRFs as default except when otherwise specified.\n\n\n\n\n\n\n\n\n\\begin{figure*}[t!]\n    \\centering\n    \\begin{subfigure}[b]{0.47\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/orig_resnet.png}\n        \\caption{Baseline model}\n        \\label{fig:confusion-nofilter}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.47\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/updated_resnet.png}\n        \\caption{With prediction filtering}\n        \\label{fig:confusion-filtered}\n    \\end{subfigure}\n    \\vspace*{-0.8mm}\n    \\caption{Pixel-level confusion matrices for DeeplabV1 models trained on $1{,}464$ pixel-level labels. Each entry shows the number of pixels (on a logarithmic scale; $0$ values are plotted as if they were $1$) whose true label is given according to the row, and whose predicted label is that in the column. Labels are sorted into rough categories to show block structure.}\n     \\vspace*{-1.0mm}\n    \\label{fig:confusion-matrices}\n\\end{figure*}\n\n\\begin{figure*}[h!]\n   \n    \\centering\n    \\includegraphics[width=0.92\\textwidth]{figures\/qualitative4.pdf}\n    \\vspace*{-1.5mm}\n    \\caption{Qualitative results, the top row of a successful case and bottom a failure\n   \n    using DeeplabV1 trained on $1{,}464$ pixel-labeled images.\n   \n   \n    }\n     \\vspace*{-3.0mm}\n    \\label{fig:qualitative}\n\\end{figure*}\n\n\n\n\n\\paragraph{Comparison with state-of-the-art.}\nIn \\cref{tbl:sota}, we compare the performance of prediction filtering to existing methods when $\\lvert\\mathcal{D}_\\mathit{pixel}\\rvert$ is $1{,}464$.\nWe reproduce results for Deeplab, OCRNet, WSSL, CCT, DualNet, and AdvCAM.\\footnote{%\nOur AdvCAM result is substantially worse than reported by \\citet{AdvCAM:2021Lee}, because we did not apply tricks such as multi-scale and flipping in inference time for fair comparison with the other state-of-the-art methods.}\nAmong VGG-based methods, DeeplabV1 with prediction filtering outperforms the other methods without using any additional information.\nSimilarly, filtering also improves models with stronger backbones, though the margin of improvement is less dramatic since the baseline is better.\nPrediction filtering can help even when it does not involve adding any new training data:\nit significantly helps WSSL, CCT, AdvCAM, and DualNet although they already use weakly-labeled data.\nIt is worth highlighting that simply adding prediction filtering to the DeeplabV3-W baseline achieves the new highest performance, slightly higher than DualNet (with a ResNet-W backbone) with prediction filtering, both of which are notably better than the previous state-of-the-art (DualNet with a ResNet-W backbone without prediction filtering).\nPrediction filtering on top of DualNet also sets the new state-of-the-art for VGG-based models.\n\n\n\n\n\n\n\n\n\n\\paragraph{Results on the test set.}\nWe further evaluate prediction filtering on the test set of VOC2012.\nIn \\cref{tbl:test}, we provide the performance of DeeplabV1 and DeeplabV3-W on the test set, as well as with prediction filtering applied.\nWe can observe that prediction filtering improves the intersection-over-union (IoU) scores for most of the $21$ classes,\nleading to significant improvements in terms of mIoU (as on the validation set).\n\n\n\n\n\n\n\n\n\n\\paragraph{Various levels of supervision.}\n\\Cref{fig:various_supervision} shows the segmentation performance of DeeplabV1, DualNet with a VGG backbone, and DeeplabV3-W trained on $200$, $500$, $800$ and $1{,}464$ images with pixel labels, with and without prediction filtering.\nThe blue (bottom) line shows performance of the base model;\norange (middle) shows with prediction filtering;\ngreen (top) is with oracle filtering, which upper-bounds the possibility of improvement of prediction filtering with a better classifier.\nFor smaller numbers of pixel labels, the performance gain from prediction filtering is drastically larger.\nFor example, at $800$ images with pixel labels,\nDeeplabV1 goes from $56.8 \\rightarrow 64.4$,\nDeeplabV3-W from $72.3 \\rightarrow 74.3$.\nThe improvements for $200$ pixel labels are  $12.4$, $1.8$, and $4.8$.\n\n\n\n\n\\paragraph{Relationship with CRF.}\nCRFs adjust the prediction for each pixel by encouraging nearby, similar-colored pixels to have the same label. This usually improves the segmentation performance by refining detailed boundaries and making class predictions across an object more consistent.\nThe latter role overlaps to some extent with prediction filtering.\nIf the size of the wrong prediction is large, though, a CRF might expand the area of the wrong prediction, rather than remove it.\n\\Cref{tbl:crf_and_partial}, as well as the qualitative results to come shortly, shows that the methods complement one another.\n\n\n\\paragraph{Without image-level labels.}\nAlthough image-level labels are easier to obtain than pixel-level labels, annotation effort is still nontrivial.\nOur hypothesis about why prediction filtering works, however, is largely that classifiers ``looks at images differently'' than segmentation networks do.\nIt thus might help even if it does not use any additional data: $\\mathcal{D}_\\mathit{pixel}$ and $\\mathcal{D}_\\mathit{image}$ contain the same images.\n\\cref{tbl:crf_and_partial} (right) shows this is the case.\nEven without introducing any actual new information, prediction filtering still improves mIoU significantly.\n\n\n\n\n\n\n\n\n\n\n\\paragraph{Changes between classes.}\nIn \\cref{sec:problem}, we showed some qualitative evidence that the segmentation network with low pixel labels tends to be confused between similar classes,\nand that prediction filtering can help to compensate it by looking at other parts of the image.\nTo further demonstrate this, \\cref{fig:confusion-matrices} shows the pixel-level confusion matrix for a DeeplabV1 model with CRF before and after prediction filtering.\n\\cref{fig:confusion-matrices}(a) shows a strong block structure where pixels from one animal class are often confused for another animal class, or vehicles for vehicles.\nIn \\cref{fig:confusion-matrices}(b), prediction filtering has dramatically reduced these types of mistakes.\n\n\n\n\n\n\n\n\n\n\n\\paragraph{Qualitative results.}\n\\Cref{fig:qualitative} shows a success (top) and failure (bottom) for prediction filtering.\nAt top, an object is mostly incorrectly predicted -- \\labelname{bike} as \\labelname{motor bike} -- and CRF only makes this worse.\nThe classifier in filtering, however, correctly identifies there is no \\labelname{motor bike}, and the model's ``second choice'' prediction is largely correct.\nIn the failure case, an object (\\labelname{table}) is mostly occluded;\nthe segmentation model still identifies it, but the classifier misses it.\n\n\\paragraph{Additional ablation studies.}\nThe appendix provides further experiments, including alternative approaches to prediction filtering, performance with model variants, and a runtime comparison between prediction filtering and CRFs. \n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nMost existing semi-weakly supervised semantic segmentation algorithms exploit the pseudo-labels extracted from a classifier.\nDoing so, however, requires a complicated architecture and extensive hyperparameter tuning on fully-supervised validation sets.\nWe propose \\textit{prediction filtering}, a simple post-processing method that only considers the classes for segmentation a classifier is confident are present.\nOur experiments demonstrated adding this method to baselines achieves the new highest performance on PASCAL VOC in SWSSS regimes, and adding it to existing SWSSS algorithms uniformly improves their performance.\nWe expect prediction filtering can become a standard post-processing method for segmentation, along with CRFs, at least when a relatively large number of weakly-labeled images are available and the portion of class labels present in most images is low.\n\n\\bibliographystyle{named}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nIn the last few years there has been a renewed interest in \nthe study of the superconformal index of 4d~$\\mathcal{N}=1$ superconformal field theories (SCFTs) and,\nin particular, $\\mathcal{N}=4$ super Yang-Mills (SYM). \nThe index in question is the supersymmetric partition function of the SCFT on~$S^3 \\times S^1$\nwhich receives contributions from BPS states that preserve two supercharges~$(\\mathcal Q, \\overline{\\mathcal{Q}})$. \nIn the large-$N$ limit, the expectation from AdS\/CFT \nis that the index should account for the entropy of the BPS black holes (BH) \nthat preserve the same two supercharges in the dual supergravity on~AdS$_5$. \nThis question was introduced in~\\cite{Sundborg:1999ue, Aharony:2003sx, Kinney:2005ej}, \nand the work of the last few years has shown that the index indeed captures the BH entropy\nin different asymptotic \nlimits~\\cite{Hosseini:2017mds,Cabo-Bizet:2018ehj, Choi:2018hmj,Benini:2018ywd,\nChoi:2018vbz, Honda:2019cio,ArabiArdehali:2019tdm,Kim:2019yrz,Cabo-Bizet:2019osg,\nCabo-Bizet:2019eaf, Benini:2020gjh, Amariti:2019mgp,Lezcano:2019pae,Lanir:2019abx,David:2020ems, \nCabo-Bizet:2020nkr,Murthy:2020rbd,Agarwal:2020zwm,Cabo-Bizet:2020ewf,Copetti:2020dil,Goldstein:2020yvj}. \n\nThe focus of the present paper is the \\emph{Cardy-like limit} in which the BH entropy \nbecomes very large. In the canonical ensemble, this translates to the study of the exponential growth of the \nindex as~$\\t \\to 0$, where the parameter~$\\t$ is the chemical potential dual to the charge.\nAs pointed out in~\\cite{Cabo-Bizet:2019eaf}, the $\\t \\to 0$ limit is in fact one of an infinite number of \ninequivalent Cardy-like limits in which the index is expected to grow exponentially. \nThese limits correspond to~$\\t$ approaching a rational number or, equivalently,~$q = e^{2\\pi \\i \\t}$ \napproaching a root of unity. \nIn this paper we analyze the 4d superconformal index near a general root of unity, \nand find interesting relations to three-dimensional Chern-Simons (CS) theory.\nThe main statement is that  the asymptotics of the index near a rational point~$-n\/m$ is equal \n(to all orders in perturbation theory in deviations~$\\widetilde \\t = m\\t +n$ from the rational point) to \nthe partition function of a certain 3d $\\mathcal{N}=2$ gauge theory with \nChern-Simons couplings that involve background as well as dynamical fields on \nan~$S^3\/\\mathbb{Z}_m$ orbifold. The background couplings give rise to singular terms \nat~$\\text{O}(1\/\\widetilde \\t^2)$ and~$\\text{O}(1\/\\widetilde \\t)$ that govern the growth of the index, while the \nconstant~$\\text{O}(1)$ term receives contributions from both background fields and \nthe dynamical Chern-Simons theory.  \n\nWe demonstrate this statement from two points of view---by direct asymptotic analysis of \nthe index near rational points, and from an analysis of the reduced three-dimensional theory and calculating the\nvarious couplings using high-temperature effective-field theory (EFT) techniques. \nThe latter method, based on~\\cite{DiPietro:2014bca, DiPietro:2016ond}, \nrelates the high-temperature asymptotics \nof the index to a low-energy effective \nfield theory, in the spirit of the Cardy formula.\\footnote{In the high-temperature picture \nthe (Euclidean) time direction is taken along the $S^1$, while in the low-energy picture \ntime is a fiber inside the $S^3$. Relating the two pictures involves swapping time and \nspace as in the derivation of the 2d Cardy formula~\\cite{Cardy:1986ie}. Unlike in the \ntwo-dimensional context where one uses~$SL(2,\\mathbb{Z})$ automorphy to relate the \nswapped problem to the original one, here we do not have an a priori understanding \nof the automorphic properties of the 4d index~$\\mathcal I(\\tau)$. Aspects of this question are being \naddressed in~\\cite{GMZ}. See also~\\cite{Shaghoulian:2016gol,Gadde:2020bov} for \nrelated work on modular-type transformation properties relating different indices, \nand~\\cite{Razamat:2012uv} for a discussion of the automorphic behavior of a different \nindex in $\\mathcal{N}=2$ SCFTs.}\n\n\n\n\\vskip 0.4cm\n\n\\noindent {\\bf The four-dimensional superconformal index and its asymptotic growth}\n\n\\vskip 0.1cm\n\nIn this paper we study~$\\mathcal{N}=1$ gauge theories with a Lagrangian description and a $U(1)_R$ symmetry, \nwith a focus on~$\\mathcal{N}=4$ SYM which we use to illustrate some statements in detail. \nThe symmetry algebra of $\\mathcal{N}=1$ SCFT on $S^1\\times S^3$ is~$SU(2,2|1)$,\nwhich includes the energy~$E$ which generates translations around~$S^1$, \nthe angular momenta~$J_1$, $J_2$ on~$S^3$, \nand the $U(1)$ R-charge~$Q$. \nOne can pick a complex supercharge obeying the following algebra, \n\\be \\label{QQbaralg}\n\\{\\mathcal Q, \\overline{\\mathcal{Q}} \\} \\= E-J_1-J_2-\\tfrac{3}{2}\\,Q \\,.\n\\ee\nThe most general index built out of the~$\\mathcal{N}=1$ superconformal algebra is an extension of the \nWitten index of~$\\mathcal Q$ and is defined as the following trace over the physical Hilbert space,\n\\be \\label{defindex}\n\\mathcal I(\\sigma, \\t) \n\\=  \\, {\\rm Tr}_{\\mathcal{H}}\\,  (-1)^F {\\rm e}^{- \\gamma \\{\\mathcal Q, \\overline{\\mathcal{Q}} \\}   \n+2 \\pi \\i \\sigma (J_1+\\frac{1}{2}Q)+2 \\pi \\i \\t (J_2+\\frac{1}{2}Q)} \\,.\n\\ee\nThe trace~\\eqref{defindex} only receives contributions from states annihilated by the supercharges \n($\\frac14$-BPS states) so that the right-hand side of~\\eqref{QQbaralg} vanishes for these states.  \nThis index~$\\mathcal I(\\sigma,\\t)$ can be calculated from either Hamiltonian or functional integral methods \nand reduces to a unitary matrix integral~\\cite{Romelsberger:2005eg,Kinney:2005ej,Nawata:2011un,Assel:2014paa}, \nwhich can be written as an  integral over the space of gauge holonomies around the~$S^1$\nof certain infinite products, as written in Equation~\\eqref{eq:pqIndex}.\n\n\nOur focus in this paper is the analog, in the present context, of the \nhigh-temperature Cardy limit of 2d CFT. This means fixing the rank and taking the \ncharges~$(J_i,Q)$ to be larger than any other scale in the theory.\nIn the canonical ensemble this translates to taking~$\\Im \\, \\sigma, \\, \\Im \\, \\t \\to 0$ at fixed rank. \nIn order to calculate the asymptotic growth of states along a certain direction in the charge lattice, \none needs to fix the relation between~$\\sigma$ and~$\\t$.\nWe study\\footnote{Our methods can be generalized to study the case where~$\\sigma$ and~$\\t$ are linearly\ndependent over the rationals, but we shall not develop this in the present paper.\n} \nthe slice~$\\sigma=\\t-n_0$ with~$n_0$ an integer, \nas in~\\cite{Cabo-Bizet:2019osg, Cabo-Bizet:2019eaf, Cabo-Bizet:2020nkr}.\nSetting~$2J =J_1+J_2$, the resulting canonical index~$\\mathcal I$ is given by\n\\be \\label{eq:n0Index}\n\\mathcal I(\\t;n_0) \\=  \\, {\\rm Tr}_{\\mathcal{H}}\\,  (-1)^F {\\rm e}^{-\\gamma \\{\\mathcal Q, \\overline{\\mathcal{Q}} \\}   \n-2 \\pi \\i n_0 (J_1+\\frac{1}{2}Q)+2 \\pi \\i \\t (2J+Q)} \\,.\n\\ee\nThe large-charge asymptotics then implies~$\\Im\\,\\t \\to 0$, \nwhile~$\\Re\\,\\t$ is not fixed a priori by the limit. \nWe consider asymptotic limits as~$\\t$ approaches a rational \nnumber~$\\t \\to -n\/m$ with~$\\text{gcd}(m,n)=1$,\nintroduced in the present context as new Cardy-like limits in~\\cite{Cabo-Bizet:2019eaf}. \nThe index $\\mathcal{I}$ clearly depends on the gauge group $G$. We generally suppress \nit in our notation, but sometimes use the notation $\\mathcal{I}_N$ to emphasize the \ndependence on $N$ for $U(N)$ or $SU(N)$ $\\mathcal{N}=4$ SYM theory (which \nshould be clear from the context). \n\nOur motivation to consider these rational points comes from the study\nof the index~$\\mathcal I_N(\\t)$ of $\\mathcal{N}=4$ SYM in the large-$N$ \nlimit.\\footnote{Another motivation comes from the mathematical literature on $q$-series, where \nit is also natural to consider expansions around roots of unity. We thank D.~Zagier for emphasizing this point to us.} \nIn this limit one considers charges scaling as~$N^2$ as~$N \\to \\infty$, \nwhich translates to~$N \\to \\infty$ at fixed~$\\t$ in the canonical ensemble~\\cite{Murthy:2020rbd}. \nIn this large-$N$ limit one expects the field theory index~$\\mathcal I_N(\\t)$ to be written as a \nsum over saddles. This picture has been partially realized in the last few years using two different approaches---the \nBethe-ansatz-like approach developed in~\\cite{Closset:2017bse,Benini:2018mlo, Benini:2018ywd}, \nand the direct study of large-$N$ saddle points using an elliptic extension of the \naction~\\cite{Cabo-Bizet:2019eaf, Cabo-Bizet:2020nkr}. \nIn particular, the large-$N$ approach in~\\cite{Cabo-Bizet:2019eaf} found a class of  \nsaddles labelled by rational numbers~$-n\/m$, where the perturbation expansion \naround each saddle is given by the asymptotic limit~$\\t \\to -n\/m$.\\footnote{These saddles \nmap to residues of the Bethe-ansatz type approach---see~\\cite{Cabo-Bizet:2020ewf} \nfor a recent discussion of the connections between the two approaches. \nA larger set of saddles have been classified in~\\cite{Cabo-Bizet:2020nkr}, but \nthe full set of important\/contributing saddles is not understood in either approach.  \nIn particular, interesting continuum configurations of the Bethe-ansatz equations have been \nrecently discovered in \\cite{ArabiArdehali:2019orz,Benini:2021ano,Lezcano:2021qbj} \nwhose role in the large-$N$ limit is not fully understood.}\nSetting~$n_0=-1$, we have \n\\be \\label{mnasymp}\n\\log \\mathcal I_N(\\t) \\; \\sim \\; -S_\\text{eff}(m,n; \\tau) \\,, \\qquad \\t \\to -n\/m \\,, \\\\\n\\ee\nwhere the effective action at each saddle is given by\n\\be\\label{actionEll}\nS_\\text{eff}(m,n; \\tau) \\= \\frac{N^2 \\pi \\i }{27\\,m} \\,\\frac{  \\bigl(2 \\widetilde \\t  +  \\chi_1(m+n) \\bigr)^3}{{\\widetilde \\t}^2}  \\,, \n\\qquad \\widetilde \\t \\; \\coloneqq \\;} % This only works with \\usepackage{mathtools m\\t +n \\,.\n\\ee\nwhere~$\\chi_1(n)$ is the Dirichlet character equal to~$0, \\pm 1$ when~$n \\equiv 0, \\pm 1$ (mod~3), respectively.\nThere was one caveat in the above result, which was stressed in~\\cite{Cabo-Bizet:2019eaf, Cabo-Bizet:2020nkr}, \nnamely that the pure-imaginary~$\\widetilde \\t$-independent term could not be fixed by the methods used in those papers.\nThe constant term in the effective action~\\eqref{mnasymp}, therefore, was a convenient choice made using \ninputs coming from outside the field-theory analysis.  \n\nAlthough we do not have a rigorous notion of the sum over saddles yet,\nit should be clear that if the effective action of the~$(m,n)$ saddle has negative real part it dominates \nover the others as~$m\\tau +n \\to 0$. \nIt is also clear from~\\eqref{actionEll} that the fastest growth among these saddles comes from~$(m,n)=(1,0)$. \nThe~$(1,0)$ saddle \nin the SYM theory is identified as a fully deconfined phase whose \nentropy scales as~$N^2$, while the other $(m,n)$ saddles have entropies that are suppressed by a factor of~$m$.\nFor this reason they can be called partially deconfined saddles (in the sense of asymptotic growth, \nbut not in the sense of center symmetry breaking---cf.~\\cite{ArabiArdehali:2019orz}). \nOn the gravitational side, the action~$S_\\text{eff}(1,0; \\tau)$ agrees precisely with the canonical on-shell action of the \nblack hole solution in the dual AdS$_5$  supergravity~\\cite{Cabo-Bizet:2018ehj},\nwhich leads to the identification of the AdS$_5$ BH as the saddle~$(1,0)$.\nThe~$(m,n)$ solutions have been identified with orbifolds of the Euclidean AdS$_5$ BH~\\cite{AhaSBTalk}.\n\nBecause of the dominance of the~$(1,0)$ saddle near~$\\t \\to 0$, one can capture it directly\nin an asymptotic expansion---even for finite~$N$.\nIn this calculation, one writes the index~\\eqref{eq:n0Index} as an integral over gauge \nholonomies~$u_i$ (see~\\eqref{eq:pqIndex} below), estimates the integrand in the \nCardy-like limit~$\\t \\to 0$, and then performs the integrals. \nThe initial studies~\\cite{Choi:2018vbz, ArabiArdehali:2019tdm,Honda:2019cio,Kim:2019yrz,Cabo-Bizet:2019osg} \nsuccessfully reproduced the singular parts of the action as~$\\t \\to 0$, \ni.e.~the~$1\/\\t^2$ and the~$1\/\\t$ terms with the correct coefficients.\nMore recently, the complete action~\\eqref{actionEll} for~$(m,n) = (1,0)$ was obtained \nin~\\cite{GonzalezLezcano:2020yeb} by a direct method,  \ninvolving a careful analysis of all perturbative terms in the Cardy-like limit. \n(See \\cite{Amariti:2020jyx,Amariti:2021ubd} for more recent related work.)\n\nOur first goal in this paper is to obtain the complete perturbative action at all the~$(m,n)$ saddles by a \ndirect asymptotic analysis of the index as~$\\t \\to -n\/m$. \nThis analysis is described in Section~\\ref{sec:SCI}, the result of which is a \nperfect agreement with the action~\\eqref{actionEll}, up to the constant terms as mentioned above.\nThe asymptotic analysis requires developing the asymptotics of the elliptic gamma function \n\\cite{Ruijsenaars:1997,felder2000elliptic} near rational points. The~$\\tau\\to0$ asymptotic \nestimates were available in previous literature~\\cite{Rains:2006dfy}. Here we develop the \nanalysis for~$\\tau$ approaching rational numbers. The analysis is presented in \nAppendix~\\ref{app:Estimates}. (See also~\\cite{Kels:2017vbc} for related work motivated \nby integrable-systems considerations.)\n\nFurthermore, we note that for given~$m,n$, depending on the sign of~$\\mathrm{arg}\\widetilde\\t-\\pi\/2$ \nthe action in~\\eqref{actionEll} can have negative or positive real part, which respectively yields \na growing or decaying contribution to the index. Therefore in essentially half of the parameter \nspace the saddles in~\\eqref{actionEll} do not capture any growth in the index. As demonstrated \nin Section~\\ref{sec:Ccenter}, when the $(m,n)$ saddle in~\\eqref{actionEll} gives a decaying \ncontribution to the index as~$\\widetilde\\t\\to0$, a ``2-center saddle'' takes over which yields exponential \ngrowth again. In other words, in half of the parameter space the growth of the \nindex~$\\mathcal{I}_N(\\tau)$ is captured by 2-center saddles. (These turn out to be partially \ndeconfined saddles both in the sense of asymptotic growth and in the sense of center \nsymmetry breaking---cf.~\\cite{ArabiArdehali:2019orz}.)\n\n\\vskip 0.4cm\n\n\\noindent {\\bf Chern-Simons theory from the asymptotics of the 4d index}\n\n\\vskip 0.1cm\n\n\n\nThe second goal of the paper is partly inspired by an interesting pattern \nappearing in the asymptotic calculations. \nAs emphasized in the context of SU($N$)~$\\mathcal{N}=4$ SYM in~\\cite{GonzalezLezcano:2020yeb}, \nin the part of the parameter space where the index is dominated by isolated, 1-center saddles, \nthe complete asymptotic expansion in~$\\t$\nterminates at~$O(\\t)$---i.e.~the perturbation theory only contains~$1\/\\t^2$, $1\/\\t$, $\\t^0$ and~$\\t$ \nup to exponentially suppressed corrections. (This is, in fact, more generally true when the index is \ndominated by isolated saddles, and not true when there are flat directions; see~\\cite{Ardehali:2015bla}.) \nInterestingly, it was found in~\\cite{GonzalezLezcano:2020yeb} that the constant term in the \nexpansion contains the partition function of SU($N$) pure Chern-Simons theory on~$S^3$ at level~$\\pm N$.\n\nIn this paper we find that the same structure persists at all rational points. We see that \nthe constant term in the expansion as~$\\t \\to -n\/m$ involves Chern-Simons theory \nwhose action is~$1\/m$ times the action as~$\\t \\to 0$. We present evidence that this corresponds \nto CS theory on an orbifold space~$S^3\/\\mathbb{Z}_m$ (with the action of~$\\mathbb{Z}_m$ depending on~$n$ \nsuch that the orbifold coincides with the lens space~$L(m,-1)$ when~$n=1$) at \nlevel~$\\pm N$~\\cite{Garoufalidis:2006ew,Gang:2019juz}. In other words, the~4d SYM index appears to play the \nrole of a master index which governs the partition function of three-dimensional CS theory on an infinite family of~$S^3$ orbifolds.\n\nThe appearance of 3d Chern-Simons theory from the 4d superconformal index is intriguing, and gives rise to \ntwo related questions:\\\\\n(a) is there a direct three-dimensional physics explanation of the appearance of Chern-Simons theory?\\\\\n(b) can we also understand the singular terms in the asymptotic expansions around \nrational points as being related to 3d Chern-Simons theory?\\\\\nThe answers to both these questions are positive, as we now explain. \n\n\n\\vskip 0.4cm\n\n\\noindent {\\bf The asymptotics of the 4d index from supersymmetric Chern-Simons theory}\n\n\\vskip 0.1cm\n\nThe natural idea is that the reduction of the four-dimensional theory on~$S^1$ gives rise to a three-dimensional \ntheory on~$S^3$ in a ``high-temperature\" expansion in powers of the circumference~$\\beta$ of the shrinking \ncircle. \nIf we calculate the functional integral of the three-dimensional theory, we should \nrecover the four-dimensional functional integral as~$\\beta \\to 0$.   \nThe three-dimensional effective field theory\nis known to have a derivative expansion, where the most relevant terms are  \nChern-Simons terms~\\cite{Banerjee:2012iz,Jensen:2012jh}.\nThis EFT approach was developed in the supersymmetric context \nin~\\cite{DiPietro:2014bca, DiPietro:2016ond} who presented \\emph{supersymmetrized} CS actions \ninvolving the dynamical as well as background fields, which are \nnecessary for preserving supersymmetry on~$S^3\\times S^1$. In particular, \nthe~$1\/\\beta^2$ and~$1\/\\beta$ effective actions derived this way in~\\cite{DiPietro:2016ond} \nreproduced the asymptotics of the the index as found in~\\cite{Ardehali:2015bla} for $n_0=0$ and $\\mathrm{arg}(\\t)=\\pi\/2$.\n(Note that when the metric on $S^3\\times S^1$ has a direct product form with $S^3$ the unit round three-sphere, \na real value of $\\beta$ determines a purely imaginary~$\\t=\\frac{\\i\\beta}{2\\pi}$.)\nThe coefficient of the leading~$1\/\\beta^2$ term in these works is pure imaginary, and \nalso does not grow as $N^2$ (it is in fact zero for non-chiral theories), therefore the \nexponential growth of states corresponding to the BH is not captured there.\n\n\n\nOne of the motivations for the current paper is to explain the exponential growth associated to the bulk black holes\nfrom the three-dimensional point of view, which \nrequires  $\\arg(\\tau) \\neq \\pi\/2$ and $n_0\\neq0$.\\footnote{The leading order~$1\/\\tau^2$ behavior was found \nfrom similar considerations in~\\cite{Choi:2018hmj}\nfor~$\\mathcal{N}=4$ SYM with flavor chemical potentials, and in \\cite{Kim:2019yrz} \nfor more general gauge theories in a setting similar to ours. In this paper we follow a \nsystematic, manifestly supersymmetric approach developed in~\\cite{DiPietro:2014bca,DiPietro:2016ond}, \nwhich allows us to obtain all-order results for general gauge theories around generic rational points.}\n(Note, in particular, that the $(m,n)=(1,0)$ saddle in \\eqref{actionEll} given for $n_0=-1$, would \nhave its leading piece a pure phase if $\\mathrm{arg}(\\t)=\\pi\/2$.) For this purpose we consider, as \nin~\\cite{Cabo-Bizet:2018ehj}, a background geometry of the \nform~$S^3\\times_\\Omega S^1$, with $S^3$ the unit round three-sphere, \n$\\gamma$ the circumference of the circle, and~$\\Omega$ \na twist parameter\\footnote{Similar twists had been described in slightly different contexts \nin~\\cite{Kim:2009wb,Nawata:2011un}.} \ncontrolling the deviation of the metric from a direct product form (Equation~\\eqref{4dbackgnd}).\nThe imaginary part of the twist parameter determines a non-zero real part \nof~$\\tau$ via (Equation~\\eqref{Omomrel})\n\\be\n\\t \\= \\frac{\\i\\gamma}{2\\pi} (1-\\O) \\,.\n\\label{eq:tauAndOmega}\n\\ee\nAs shown in~\\cite{Cabo-Bizet:2018ehj}, \nthe integer~$n_0$ in~\\eqref{eq:n0Index} controls the \nperiodicity of the fermions in this background, and~$n_0=\\pm 1$ (which is naturally dual to the BH) corresponds to \nanti-periodic fermions, i.e.,~as in a Scherck-Schwarz reduction. \nIn the present context we insist on supersymmetry being preserved---and that necessitates the \nturning on of other background fields under which the fermions are charged. \nIn the three-dimensional background supergravity, we have a non-zero graviphoton from the fibration  \nas well as non-zero auxiliary background gauge and scalar fields. As we explain in Section~\\ref{sec:4dto3d}, \nthe resulting configuration is effectively described by a circle of radius~$R$, which in the limit~$\\gamma \\to 0$, $\\Omega\\to\\infty$ with~$\\t$ fixed obeys~$R \\to \\t$.\n\n\nNow, what is the actual calculation? \nThere are two types of fields in the three-dimensional functional integral---background fields \nwhich take constant values, and dynamical modes which fluctuate in the integral. The latter is further \nmade up of light modes (with zero momentum around~$S^1$) and heavy (Kaluza-Klein) modes. \nThe first step is to integrate out the heavy modes in order to obtain an effective action for the light modes. \nThe integration over heavy modes also generates corrections to the coefficients of the supersymmetric\nChern-Simons terms of the non-zero background fields, see e.g.~\\cite{Intriligator:2013lca,DiPietro:2014bca}.\nIn these calculations, we need to include, in addition to the couplings discussed \nin~\\cite{DiPietro:2016ond}, the supersymmetrized RR and gravitational CS actions which were \ndiscussed in~\\cite{Closset:2018ghr}. The effective actions of the background gauge fields turn out to \nproduce precisely the singular pieces~$1\/\\t^2$ and~$1\/\\t$ in the asymptotic expansion of the index, \nas well as a constant piece. The remaining functional integral is described by an~$\\mathcal{N}=2$\nSYM theory with a certain one-loop induced CS coupling on~$S^3$, whose partition function is \nknown to agree, up to a sign, with that of pure Chern-Simons theory~\\cite{Kapustin:2009kz}. \nThis explains the appearance of the dynamical Chern-Simons theory in the constant term\nof the asymptotic expansion.\n\n\n\nTwo technical remarks are in order. \nFirstly, recall that supersymmetry implies that the 4d superconformal index should not depend on~$\\gamma$\nand $\\Omega$ separately, but only on their combination~$\\tau$ as in~\\eqref{eq:tauAndOmega}. \nIn~\\cite{Cabo-Bizet:2018ehj} this was shown to be true in 5d gravitational variables, as well as \nthrough a localization computation in 4d field theory. In this paper we verify this also in 3d  effective field theory.\nSecondly, the order of limits is important to have a smooth calculational set up. \nWe first send~$\\gamma \\to0$ keeping~$\\Omega$ fixed, so that the three-dimensional geometry is smooth and finite. \nThen we take~$\\Omega \\to \\infty$ at fixed~$\\t$ and express the \nresult in terms of~$\\tau$ using~\\eqref{eq:tauAndOmega}. We find there are no singularities generated in \nthe latter step and thus the limiting procedure is perfectly smooth.\n\n\n\n\n\nFinally, we repeat the same analysis as~$\\t$ approaches rational points. The dimensional reduction in this\ncase naturally leads us to considering orbifolds of~$S^3\\times S^1$, \nwhich, as far as we understand,\nare related to the orbifolds discussed in~\\cite{AhaSBTalk}. The three-dimensional calculation \nthen leads to the~$1\/\\widetilde \\t^2$ and~$1\/\\widetilde \\t$ terms as well as a constant piece from the background \nfields, and we provide evidence that the remaining dynamical piece is the partition function \nof~$\\mathcal{N}=2$ SYM with a one-loop induced CS coupling on~$S^3\/\\mathbb{Z}_m$. \n\n\n\n\n\\vskip 0.4cm\n\n\n\\noindent {\\bf Notation}. We have~$\\sigma,\\t \\in \\mathbb{H}$ and~$z, u \\in \\mathbb{C}$, and \nwe set~$p={\\rm e}^{2\\pi i\\sigma}$, $q={\\rm e}^{2 \\pi \\i \\t}$, $\\zeta={\\rm e}^{2\\pi\\i z}$. \\\\\nWe use~$\\simeq$ to mean an all-order asymptotic equality of the logarithms of the two sides. \n\n\n\n\n\n\n\n\\section{The 4d superconformal index and its asymptotic expansion \\label{sec:SCI}}\n\n\n\n\nWe consider a four-dimensional $\\mathcal{N}=1$ gauge theory which flows to a superconformal fixed point. \nThe theory has gauge group~$G$ (which we take to be semi-simple, and separately comment \non the $U(N)$ case), and a number of chiral multiplets labelled by~$I$  \nwith R-charge $r_I$ and in the representation $\\mathcal{R}_I$ of the gauge group. \nWe assume $0<r_I<2$ for all chiral multiplets. \nThe superconformal index for these theories on~$S^3\\times S^1$\nhas been calculated in the Hamiltonian as well as functional integral \nformalism~\\cite{Romelsberger:2005eg,Kinney:2005ej,Nawata:2011un,Assel:2014paa}, \nand the answer is expressed as an integral over the Cartan torus which we parameterize by the \nvector of gauge holonomies~$\\underline{u} = (u_1, \\dots, u_{\\text{rk}(G)})$, with~$u_i \\in \\mathbb{R}\/\\mathbb{Z}$.\nThe index is given by the following expression~\\cite{Dolan:2008qi,Spiridonov:2009za,Spiridonov:2011hf}\n\\be\\label{eq:pqIndex}\n\\mathcal I(\\sigma,\\tau) \n\\= \\int [D\\underline{u}]   \\, \\mathcal{Z}_{\\rm vec}  (\\underline{u};\\sigma,\\t) \\, \\mathcal{Z}_{\\rm chi}  (\\underline{u};\\sigma,\\t) \\, .\n\\ee\nHere we have used the measure~$D\\underline{u} = \\frac{1}{|\\mathcal{W}|} \\prod_{i=1}^{\\text{rk}(G)} \\mathrm{d} u_i$  \nwith $|\\mathcal{W}|$ the order of the Weyl group of $G$. \nFor~$U(N)$ we have~$D\\underline{u} = \\frac{1}{N!} \\prod_{i=1}^{N} \\mathrm{d} u_i$, while \nfor~$SU(N)$ one can work with~$u_1, \\dots , u_N$ subject to~$\\sum_i u_i  \\in \\mathbb{Z}$ \nand~$D\\underline{u} = \\frac{1}{N!} \\prod_{i=1}^{N-1} \\mathrm{d} u_i$.\nThe factors~$\\mathcal{Z}_{\\rm vec}$, $\\mathcal{Z}_{\\rm chi}$ denote the vector multiplet and\nchiral multiplet contribution respectively given by\n\\be\n\\begin{split}\n\\mathcal{Z}_{\\rm vec} (\\underline{u};\\sigma,\\t) &\\= (p;p)^{{\\rm rk}(G)}(q;q)^{{\\rm rk}(G)} \\prod_{ \\alpha_+}   \n\\Gamma_\\text{e} \\bigl(\\alpha_+\\cdot u + \\sigma + \\tau ; \\sigma , \\tau \\bigr)  \\, \n\\Gamma_\\text{e} \\bigl( - \\alpha_+\\cdot u + \\sigma + \\tau ; \\sigma , \\tau \\bigr) \\, ,\\\\\n\\mathcal{Z}_{\\rm chi} (\\underline{u};\\sigma,\\t) &\\= \\prod_{I} \\prod_{ \\rho \\in \\mathcal{R}_I}\n\\Gamma_\\text{e} \\Bigl(\\rho\\cdot u+\\frac{r_I}{2} (\\sigma+\\tau); \\sigma,\\tau \\Bigr) \\,.\n\\end{split}\n\\label{ZvecZchi}\n\\ee\nHere $\\a_+$ runs over the set of positive roots of the gauge group $G$, \n$I$ runs over all the chiral multiplets of the theory, \nand $\\rho$ is the weight of the gauge representation $\\mathcal{R}_I$.  \nThe \\emph{Pochhammer symbol} is defined by \n\\be\n    (\\zeta;q) \\= \\prod_{k=0}^{\\infty}(1- \\zeta \\, q^k) \\,, \\label{eq:PochDef}\n\\ee\nand\nthe \\emph{elliptic gamma function}~\\cite{Ruijsenaars:1997,Felder}\nis defined by the infinite product formula \n\\be\n\\Gamma_\\text{e}(z;\\sigma,\\t) \\= \n\\prod_{j,k\\ge 0}\\frac{1-\\,p^{j+1} \\,q^{k+1} \\, \\zeta^{-1}}{1-p^{j} \\,q^{k}\\, \\zeta } \\,. \n\\label{eq:GammaDef}\n\\ee\n\nFrom now on in this section we set~$\\sigma=\\t-n_0$, and use the notation~$\\Gamma_\\text{e}(z) = \\Gamma_\\text{e}(z;\\t,\\t)$.\nWe have \n\\be \n\\mathcal I(\\t;n_0)  \\=   \\int [D \\underline{u}] \\, \\exp \\bigl( -S_\\text{ind}(\\underline{u};\\t)  \\bigr) \\,, \n\\label{eq:SYMIndex} \n\\ee\nwhere the action~$S_\\text{ind}(\\underline{u}) = S_\\text{ind}(\\underline{u};\\t)$ is given by\n\\be\n\\begin{split}\n -S_\\text{ind}(\\underline{u})  & \\=  2 \\, {\\rm rk}(G) \\log (q;q) +  \n \\sum_{ \\alpha_+}  \\log \\bigl( \\Gamma_\\text{e} \\bigl(\\alpha_+\\cdot u +  2\\tau \\bigr) \n \\Gamma_\\text{e} \\bigl( - \\alpha_+\\cdot u + 2\\tau \\bigr) \\bigr) \\\\\n&  \\qquad  + \\sum_{I} \\sum_{ \\rho \\in \\text{R}_I}\n\\log \\Gamma_\\text{e} \\bigl(\\rho\\cdot u+ r_I (\\tau - \\tfrac12 n_0) \\bigr) \\,.\n\\label{defVmicro}\n\\end{split}\n\\ee\nOur goal now is to calculate the asymptotics of the function~$\\mathcal I(\\t,n_0)$ as~$\\t$ \napproaches a rational number or, equivalently,~$q=e^{2 \\pi \\i \\tau}$ approaches a root of unity.\nFor~$\\mathcal{N}=4$ SYM we have \n\\be \n\\begin{split}\n -S_\\text{ind}^{\\mathcal{N}=4}(\\underline{u})  \\= & 2N \\log (q;q) +  3N \\log \\Gamma_\\text{e} \\bigl(\\,\\tfrac13 (2\\t - n_0)\\bigr)   \\\\\n& \\quad + \\sum_{i\\neq j}  \\log \\Gamma_\\text{e} \\bigl(u_{ij} +2\\t\\bigr) + \n3 \\sum_{i\\neq j} \\log  \\Gamma_\\text{e} \\bigl(u_{ij} + \\tfrac13 (2\\t - n_0) \\bigr)\n\\label{eq:N4Index} \n\\end{split}\n\\ee\nfor~$U(N)$ and a similar expression for~$SU(N)$ with~$N$ replaced by~$N-1$.\nUsing the product expression~\\eqref{eq:GammaDef} we see \nthat for~$\\mathcal{N}=4$ SYM the index $\\mathcal I_N(\\t;n_0)$ has the symmetry~$\\t \\mapsto \\t+3$ for fixed~$n_0$, \nso that we can restrict our attention to, say,~$\\t \\in [0,3]$. Relatedly, the independent values of~$n_0$\nare~$0$, $\\pm 1$.  \nMore generally, the periodicity of~$\\t$ depends on the quantization of R-charge in the theory.\n\nBefore analyzing these asymptotic limits \nwe briefly discuss a slightly independent motivation to study these new limits and, relatedly, the \norigin of the number~$n_0$ in~\\eqref{eq:SYMIndex}, \\eqref{eq:N4Index}. \nOne of the motivations in the recent developments in this subject has been to ``find the dual black hole\"\nin the superconformal index. In terms of the microcanonical Fourier coefficients\n\\be\n\\mathcal I_N (\\t;n_0) \\= \\sum_n d_N(n;n_0) \\, {\\rm e}^{2 \\pi \\i n \\t} \\,,\n\\ee \nthe problem in the context of the Cardy-like limit is to check if~$|d_N(n;n_0)|  \\sim N^2 \\, s(n\/N^2)$  \nas~$n \\to \\infty$~\\cite{Murthy:2020rbd}.\nIn the canonical setting this is reflected by a corresponding asymptotic growth of the function~$\\mathcal I(\\t)$ \nas~$\\t$ approaches the real axis or, equivalently, as~$q = e^{2 \\pi \\i \\t}$ approaches the unit circle. \nThe leading asymptotics of the growth of microcanonical degeneracies is governed by the dominant \nsingularity of~$\\mathcal I$. \nAs it turns out, the index~$\\mathcal I_N(\\t;0)$ of~$\\mathcal{N}=4$ SYM does not have any exponential growth \nas~$\\t \\to 0$ (the growth is power-law \\cite{Ardehali:2015bla}). \nIt is the asymptotic growth of~$\\log \\mathcal I_N(\\t;0)$ as~$\\t \\to \\pm 1$ instead that matches the \non-shell action of the AdS$_5$ BH (the two points giving growth of equal magnitude and opposite phases). \nFrom a numerical study of the microcanonical degeneracies one can deduce that this is, in fact, the \nleading growth of the index~\\cite{Murthy:2020rbd}. \nIn this case, noting that~$\\mathcal{I}_{N}(\\tau\\pm1; n_0) =\\mathcal{I}_{N}(\\tau;n_0\\mp1)$, \nwe see that the leading growth can be stated as coming from the growth of the function~$\\mathcal I_N(\\t,\\mp1)$\nas~$\\t \\to 0$. Actually, one finds that the growth of states at~$n_0=\\pm 1$ matches the BH growth of states \nfor very large classes of~$\\mathcal{N}=1$ SCFTs~\\cite{Cabo-Bizet:2019osg,Kim:2019yrz}. \nOnce we understand that the growth can come from a region with~$\\Im \\, \\t \\to 0$ but~$\\Re \\, \\t \\neq 0$,\nit is perhaps more natural to set~$n_0=0$ (for which the two regions of leading growth have a symmetric\nplacement around~$\\t=0$).\nWe will, nevertheless, keep~$n_0$ as an explicit parameter in the following to make contact with related literature. \n\nIt is clear from the above discussion that one should equally well explore other points on the unit circle \nin~$q$.\\footnote{The superconformal index as a function of $q$ is defined on a branched  \ncover of the complex plane and one should explore the full covering space. \nFor~$\\mathcal{N}=4$ SYM one has a three-sheeted cover and the leading \ngrowth occurs on two of the three sheets~\\cite{Cabo-Bizet:2019osg,Kim:2019yrz,ArabiArdehali:2019orz}.}\nAs it turns out there is exponential growth near any root of unity consistent with~\\eqref{mnasymp}, \n\\eqref{actionEll}, i.e.~partial deconfinement in the sense of asymptotic growth \\cite{ArabiArdehali:2019orz}. \nIn the following subsections we proceed to analyze the asymptotic\nbehavior of the index as~$\\t \\to 0$ and then as~$\\t$ approaches any rational number. \n\n\n\n\n\\subsection{Asymptotics of the index as $\\tau\\to0$}\n\n\nIn this subsection we perform an all-order asymptotic analysis of the integral~\\eqref{eq:SYMIndex} \nas~$\\t \\to 0$. This calculation was done for~$\\mathcal{N}=4$ SYM recently in~\\cite{GonzalezLezcano:2020yeb} \nusing a saddle-point  analysis. \nHere we find the asymptotics for the general class of theories discussed in the introduction, \nusing the rigorous method of~\\cite{Rains:2006dfy,Ardehali:2015bla} (see in particular Section~3.1 \nof \\cite{Ardehali:2015bla}). \nThe application in~\\cite{Ardehali:2015bla} was restricted to real~$\\t$ and~$n_0=0$, but the method is \nmore general and we apply it to the case of complex~$\\t$ and general~$n_0$. \n\n\nWe first calculate the all-order asymptotic expansion of the integrand. \nIn order to do this we need the the asymptotic behavior of the elements in~\\eqref{defVmicro}, \nnamely the Pocchammer symbol and the elliptic gamma function, which \nwe review in Equations~(\\ref{eq:PochEst}), (\\ref{eq:numEst}), \\eqref{eq:denomEst}.\nUsing these estimates we find that in the range~$\\a_+ \\cdot \\underline{u} \\in (-1+\\delta,1-\\delta)$ (for fixed small~$\\delta$)\nthe integrand of~\\eqref{eq:SYMIndex} can be written, up to exponentially suppressed corrections, as \n\\be\n \\exp \\bigl( -S_\\text{ind}(\\underline{u};\\t)  \\bigr) \\; \\simeq \\; \\frac{1}{(-\\i\\, \\tau)^{\\mathrm{rk}(G)}}\\, \n\\prod_{\\alpha_+}4\\sinh^2\\Bigl(\\frac{\\pi \\alpha_+ \\cdot \\underline{u}}{-\\i\\, \\tau}\\Bigr) \\,  \n\\exp \\bigl( -2\\pi \\i\\tau E_{\\mathrm{susy}} -V (\\underline{u}) \\bigr)\\, .\n\\label{eq:In0semi-simpleAsy0}\n\\ee\nThe all-order effective potential as~$\\t\\to 0$  is given by\n\\be\nV(\\underline{u}) \\= \\frac{1}{\\t^2} V_2(\\underline{u})+ \\frac{1}{\\t}  V_1(\\underline{u})+V_0(\\underline{u}) \\,,\n\\label{defVsum}\n\\ee\nwith\n\\be\n\\begin{split}\n    V_2(\\underline{u})&\\=\\sum_{I, \\,\\rho_I}\\, \\frac{\\i\\pi}{3} \\, \\overline{B}_3 \\bigl(\\rho_I \\cdot \\underline{u}  - \\tfrac12 r_I n_0 \\bigr) \\,,\\\\\n    V_1(\\underline{u})&\\=\\sum_{I,\\,\\rho_I} \\, \\i\\pi(r_I-1) \\, \n    \\overline{B}_2 \\bigl(\\rho_I \\cdot \\underline{u}  - \\tfrac12 r_I n_0\\bigr)+\n    \\sum_{\\alpha} \\i\\pi \\, \\Bigl((\\alpha \\cdot \\underline{u})^2  + \\frac{1}{6}\\Bigr) \\,,\\\\\n    V_0(\\underline{u})&\\=\\sum_{I,\\,\\rho_I} \\, \\i\\pi \\Bigl( (r_I-1)^2-\\frac{1}{6} \\Bigr) \\, \n    \\overline{B}_1  \\bigl( \\rho_I \\cdot \\underline{u}  - \\tfrac12 r_I n_0\\bigr) \\,,\n\\label{eq:defVs}\n\\end{split}\n\\ee\nwhere~$\\a$ runs over all the roots of $G$ including the rk$(G)$ zero roots,~$I$ runs over all the \nchiral multiplets, and~$\\rho_I$ runs over all the weights of the representation~$\\mathcal{R}_I$.\nNote that in~\\eqref{eq:In0semi-simpleAsy0} we have separated\nthe supersymmetric Casimir energy given by~\\cite{Assel:2015nca}\n\\be\nE_{\\mathrm{susy}} \\= \\frac{1}{6} \\, \\mathrm{Tr}R^3 -\\frac{1}{12} \\, \\mathrm{Tr}R\\,.\n\\label{Esusy}\n\\ee\n\nWe make a brief comparison to~\\cite{Cabo-Bizet:2019osg} in which the singular pieces were studied. \nThe potential~$V_2$ in~\\eqref{eq:defVs} coincides (up a multiplicative~$-\\i\\pi\/6$ factor) with \nthe~$V_2$ studied in~\\cite{Cabo-Bizet:2019osg}. \nAt finite $\\underline{u}$, the sinh$^2(\\frac{\\pi\\alpha_+\\cdot\\underline{u}}{-\\i\\tau})$ factors in~\\eqref{eq:In0semi-simpleAsy0} \nalso contribute to~$\\text{O}(1\/\\t)$. Including this piece in~$V_1$ renormalizes it to~$V^r_1$ \nas\\footnote{Instead of using~\\eqref{eq:denomEst} for vector multiplet gammas and then \nsimplifying the sinh term for finite~$\\underline{u}$ as above, we could alternatively use~\\eqref{eq:numEst} \nfor vector multiplet gammas (assuming~$\\alpha_+\\cdot\\underline{u}\\notin\\mathbb{Z}$) and get~$V_1^r$ \ndirectly as in~\\cite{Cabo-Bizet:2019osg}.}\n\\begin{equation}\n    V_1(\\underline{u})\\to V^r_1(\\underline{u}) \\=\\sum_{I,\\,\\rho_I} \\, \\i\\pi(r_I-1) \\, \n    \\overline{B}_2 \\bigl(\\rho_I \\cdot \\underline{u}  - \\tfrac12 r_I n_0\\bigr)+\n    \\sum_{\\alpha} \\i\\pi \\, \\overline{B}_2 \\bigl(\\alpha \\cdot \\underline{u} \\bigr) \\,.\\label{eq:V1ren}\n\\end{equation}\nThe potential~$V^r_1$ coincides (up to a multiplicative $-\\i\\pi$ factor) with the~$V_1$ studied in~\\cite{Cabo-Bizet:2019osg}. \nIn our treatment below we keep the sinh$^2$ factors separate and place them in the ``dynamical measure\" \n\\be\n\\frac{D\\underline{u}}{(-\\i\\tau)^{\\mathrm{rk}(G)}}\\, \n\\prod_{\\alpha_+}4\\sinh^2\\Bigl(\\frac{\\pi \\alpha_+ \\cdot \\underline{u}}{-\\i\\tau}\\Bigr) \\,. \n\\ee\nCompared to~\\cite{Cabo-Bizet:2019osg}, here we also include the $\\text{O}(\\tau^0)$ piece \ncorresponding to $\\exp(-V_0)$. \nFinally, the $\\text{O}(\\tau)$ piece of the exponent is determined by the \nsupersymmetric Casimir energy and, notably, there are no $\\text{O}(\\tau^2)$ or higher \ncorrections in the perturbative effective potential.\n\nWe now investigate the local behavior of the potential near~$\\underline{u} =0$. \nThe potentials~$V_{2}, V_1, V_0$ are piecewise polynomials,\nand using~$\\overline{B}'_j=j \\overline{B}_{j-1}$ we obtain their Taylor expansion near~$\\underline{u}=0$ as \n\\be\n\\begin{split}\n    V_2(\\underline{u}) &\\=\\sum_{I,\\,\\rho_I} \\Bigl(\\frac{\\i\\pi}{3} \\, \\overline{B}_3 \\bigl(- \\tfrac12 r_I \\, n_0 \\bigr) \\+\n    \\i\\pi \\, \\overline{B}_2 \\bigl(-\\tfrac12 r_I \\, n_0 \\bigr)\\rho_I\\cdot\\underline{u}\n    \\+2\\pi \\i \\, \\overline{B}_1\\bigl(-\\tfrac12 r_I \\, n_0 \\bigr)\\frac{(\\rho_I\\cdot\\underline{u})^2}{2} \\Bigr)\\\\\n    &\\qquad \\+\\sum_{I,\\,\\rho_I} \\, 2\\pi\\i \\, \\frac{(\\rho_I\\cdot\\underline{u})^3}{3!}\\, ,\\\\\n    V_1(\\underline{u}) &\\=\\sum_{I,\\,\\rho_I} \\, \\i\\pi(r_I-1) \\, \n    \\overline{B}_2\\bigl(-\\tfrac12 r_I \\, n_0\\bigr)\\+\\frac{\\i\\pi}{6}\\text{dim}G\\+\n    \\sum_{\\rho_I}2\\pi\\i \\, (r_I-1) \\, \\overline{B}_1(-\\tfrac12 r_I \\, n_0)\\, \\rho_I\\cdot\\underline{u}\\\\ \n    &\\qquad \\+\\sum_{I,\\,\\rho_I} \\, 2\\pi\\i \\,(r_I-1) \\, \\frac{(\\rho_I\\cdot\\underline{u})^2}{2}\n    \\+\\sum_{\\alpha} \\i\\pi \\, (\\alpha \\cdot \\underline{u})^2,\\\\\n    V_0(\\underline{u}) &\\=\\sum_{I,\\,\\rho_I} \\, \\i\\pi \\Bigl((r_I-1)^2-\\frac{1}{6} \\Bigr)\n    \\overline{B}_1\\bigl(-\\tfrac12 r_I \\, n_0\\bigr)\\\\\n    &\\qquad \\+\\sum_{I,\\,\\rho_I} \\, \\i\\pi \\Bigl((r_I-1)^2-\\frac{1}{6} \\Bigl)\\rho_I\\cdot\\underline{u}.\n\\end{split}\n\\ee\nImportantly, the second lines of $V_{2}$, $V_1$, $V_0$ above vanish due to gauge$^3$, \n$U(1)_R$-gauge$^2$, and $U(1)_R^2$-gauge and gravity$^2$-gauge anomaly cancellations. \nTherefore $V_2$ is actually piecewise quadratic in $\\underline{u}$, while $V_1$ is piecewise linear \nand $V_0$ is piecewise constant. This is similar to Section~3.1 of~\\cite{Ardehali:2015bla}.\n\nThe leading asymptotic behavior of~$V$~as $\\tau\\to0$ is determined by~$V_2$.  \nIn order to obtain a local minimum of~$\\text{Re}(V)$ at $\\underline{u}=0$, we want \n(i) the linear term in $V_2$ to vanish, and (ii) the quadratic term to be on the negative (respectively positive) \nimaginary axis for~$\\mathrm{arg}(\\tau)-\\frac{\\pi}{2}>0$ (respectively~$\\mathrm{arg}(\\tau)-\\frac{\\pi}{2}<0$). \nAs found in~\\cite{Cabo-Bizet:2019osg}, we can achieve both \nof these requirements in any theory in which~$0<r_I<2$ by specializing to~$n_0=\\pm1$. \nExplicitly, for $n_0=\\pm1$ we can use the fact that for~$|x|<1$ we \nhave~$\\overline{B}_2(x)=x^2-|x|+\\frac{1}{6}$ to \ndeduce that the linear term in~$V_2$ is equal to\n\\be\n\\sum_{I,\\,\\rho_I}\\, \\frac{\\i\\pi}{4} \\, \\Bigl( (r_I-1)^2-\\frac{1}{3} \\Bigl) \\, \\rho_I\\cdot\\underline{u} \\,,\n\\ee\nwhich vanishes thanks to the~$U(1)_R^2$-gauge and gravity$^2$-gauge anomaly cancellations. \nSimilarly we can use the fact that for~$0<|x|<1$ we have~$\\overline{B}_1(x)=x-\\frac{\\mathrm{sign}(x)}{2}$ \nto deduce that the quadratic term in $V_2$ is equal to\n\\be\n-\\i\\pi n_0 \\, \\sum_{I,\\,\\rho_I} \\, (r_I-1)\\frac{(\\rho_I\\cdot\\underline{u})^2}{2}\\=\\i\\pi n_0 \\, \n\\sum_{\\alpha} \\, \\frac{(\\alpha\\cdot\\underline{u})^2}{2} \\,,\n\\ee\nwhere the equality follows from~$U(1)_R$-gauge$^2$ anomaly cancellation, and we have \nused that~$\\mathrm{sign}(n_0)=n_0$ for~$n_0=\\pm1$.\nThis quadratic piece is on the positive (respectively negative) imaginary axis for $n_0=+1$ \n(respectively $n_0=-1$). In this manner we see that~$\\underline{u}=0$ is a local minimum of~$\\text{Re}(V)$. \nTherefore in the rest of this subsection \n{we focus on~$n_0=\\pm1$, and take~$\\mathrm{arg}(\\tau)-\\frac{\\pi}{2}$ to have the \nopposite sign to~$n_0$}, i.e.~$n_0\\, (\\mathrm{arg}(\\tau)-\\frac{\\pi}{2})<0$.\n\nUsing the explicit expressions of~$\\overline{B}_{1,2,3}(x)$ in the range~$0<|x|<1$, and using the \nanomaly cancellation conditions, the potentials~$V_2$,~$V_1$,~$V_0$ simplify, for~$n_0=\\pm1$, to\n\\be\n\\begin{split}\n    V_2(\\underline{u})&\\=-\\frac{\\i\\pi n_0}{24}\\, \\bigl(\\mathrm{Tr}R^3-\\mathrm{Tr}R\\bigr) + \\i\\pi n_0 \\, \n    \\sum_{\\alpha}\\frac{(\\alpha\\cdot\\underline{u})^2}{2} \\,,\\\\\n    V_1(\\underline{u})&\\=\\frac{\\i\\pi}{12} \\, \\bigl(3\\mathrm{Tr}R^3-\\mathrm{Tr}R\\bigr) \\,,\\\\\n    V_0(\\underline{u})&\\=\\sum_{I,\\,\\rho_I}\\, \\i\\pi \\, \\Bigl(\\frac{r_I-1}{6} - (r_I-1)^3\\Bigr)\n    \\frac{n_0}{2} \\,.\n\\label{eq:simplifiedVs}\n\\end{split}\n\\ee\nNote that, as a bonus, $V_1$ also becomes independent of $\\underline{u}$ for $n_0=\\pm1$ and \nsmall enough~$\\underline{u}$.\\footnote{We will shortly interpret the quadratic term in $V_2$ as inducing a \nChern-Simons type coupling in the integrand. If the linear term in $V_1$ were present, \nit would similarly induce an FI parameter in the integrand. While this is impossible for semi-simple \ngauge theories near~$\\underline{u}=0$ which we are focussing on in this section, there are cases of \nsemi-simple gauge theories in which one must expand \naround~$\\underline{u}\\neq0$ and as a result one finds the measure of an abelian gauge theory in the integrand, \nwhere such induced FI parameters do arise. See Section~3.3.1 of~\\cite{Ardehali:2015bla} where \nthe ISS model displaying an SU(2)$\\to U(1)$ breaking pattern with an induced FI parameter in \nthe~$\\tau\\to0$ limit is discussed, and see Appendix~A of~\\cite{ArabiArdehali:2019zac} where that \ninduced FI parameter is given an effective field theory explanation.}\n\n\nWe now consider a small neighborhood $\\mathfrak{h}_{cl}^\\epsilon$ around $\\underline{u}=0$, \ndefined by the cutoff $|u_j|<\\epsilon$, whose contribution to the index is \n\\be\n\\mathcal{I}(\\tau;n_0)_{|u_j|<\\epsilon} \\; \\simeq \\;\n{\\rm e}^{-2\\pi \\i\\tau E_{\\mathrm{susy}}}\\int_{\\mathfrak{h}_{cl}^\\epsilon} \\frac{D\\underline{u}}{(-\\i\\tau)^{\\mathrm{rk}(G)}}\\, \n\\prod_{\\alpha_+}4\\sinh^2\\Bigl(\\frac{\\pi \\alpha_+ \\cdot \\underline{u}}{-\\i\\tau}\\Bigr) \\,  \n\\exp \\bigl( -V(\\underline{u}) \\bigr)\\, .\n\\label{eq:In0semi-simpleAsy1}\n\\ee\nFrom the above discussion we have that \n\\be\n\\mathcal{I}(\\tau;n_0=\\pm1)_{|u_j|<\\epsilon} \\; \\simeq \\; {\\rm e}^{-2\\pi \\i\\tau E_{\\mathrm{susy}}} \\, \nZ^{\\text{bgnd}}(\\tau;n_0) \\  Z^{\\text{dyn}}_\\epsilon(\\tau;n_0)\\,,\n\\label{eq:In0semi-simpleAsy2}\n\\ee\nwhere the \\emph{background piece} is \n\\be\n\\begin{split}\n& Z^{\\text{bgnd}}(\\tau;n_0)  \\= \\\\ \n& \\quad \\exp \\Bigl( \\frac{\\i\\pi\\, n_0}{24\\tau^2}(\\mathrm{Tr}R^3-\\mathrm{Tr}R)\n-\\frac{\\i\\pi}{12\\tau}(3\\mathrm{Tr}R^3-\\mathrm{Tr}R) + \n\\sum_{\\rho_I}\\frac{\\i\\pi\\, n_0}{2} \\bigl((r_I-1)^3-\\tfrac16(r_I-1) \\bigr) \\Bigr)\n\\end{split}\n\\ee\nand the \\emph{dynamical piece} is\n\\be\nZ^{\\text{dyn}}_\\epsilon(\\tau;n_0)=\\int_{\\mathfrak{h}_{cl}^\\epsilon} \\frac{D\\underline{u}}{(-\\i\\, \\tau)^{\\mathrm{rk}(G)}}\\,\n\\prod_{\\alpha_+}4\\sinh^2\\Bigl(\\frac{\\pi \\alpha_+ \\cdot \\underline{u}}{-\\i\\,\\tau}\\Bigr) \\,  \n\\exp \\Bigl(\\frac{\\i\\pi n_0}{2}\\sum_\\alpha \\bigl(\\frac{\\alpha\\cdot\\underline{u}}{-\\i\\,\\tau} \\bigr)^2 \\Bigr) \\,.\\label{eq:ZdynComplex}\n\\ee\nHere we suppress the dependence of these functions on the gauge group and the matter content.\n\nTo simplify $Z_\\epsilon^{\\text{dyn}}$ further, we first define $x_j=\\frac{u_j}{-\\i\\tau}$, \nso that the integral becomes along straight contours from \n$x_j=-\\frac{\\epsilon}{-\\i\\tau}$ to $x_j=+\\frac{\\epsilon}{-\\i\\tau}$. \nWith our choice of~$n_0$ and~arg$(\\t)$, the integrand is \nlocally exponentially suppressed away from $\\underline{u}=0$, so we can complete the tails of the contours \nalong straight lines to infinity (i.e.~send~$\\epsilon\\to+\\infty$) introducing only exponentially small error. \nThe contours make an angle~$\\frac{\\pi}{2}-\\mathrm{arg}(\\tau)$ with the positive real axis. \nHowever, observing that (i)~the integrand is exponentially suppressed as $|x_j|\\to\\infty$, and (ii) \nthere are no poles between the contour of $x_j$ and the real axis, we can deform the contours \nback to the real axis. We thus obtain, with~$ \\boldsymbol{x} = (x_1,\\dots,x_n)$ \n\\be\nZ^{\\text{dyn}}_\\epsilon(\\tau;n_0) \\;\\simeq \\; \\int_{-\\infty}^\\infty D \\boldsymbol{x} \\,\n\\prod_{\\alpha_+}4\\sinh^2(\\pi \\alpha_+ \\cdot \\boldsymbol{x}) \\,  \n\\exp \\Bigl( \\frac{\\i\\pi n_0}{2}\\sum_\\alpha (\\alpha\\cdot \\boldsymbol{x} )^2 \\Bigr) \\; =: \\; Z^{\\text{dyn}}(n_0) \\,.\\label{eq:ZdynS3}\n\\ee\nAs noted in~\\cite{GonzalezLezcano:2020yeb} for~$\\mathcal{N}=4$ SYM,\nand in \\cite{Amariti:2020jyx,Amariti:2021ubd} for more general groups, \n$Z^{\\text{dyn}}$ is related to the\npartition function of pure Chern-Simons theory~\\cite{Witten:1988hf} on~$S^3$ as\n\\be\nZ^{\\text{dyn}} ( n_0)\\=(-1)^{(\\text{dim}G-\\text{rk}G)\/2} \\, Z^{\\text{CS}}(k_{ij}) \\,,\n\\label{dynCSrel}\n\\ee\nwith the gauge group implicit and the same on both sides, and with Chern-Simons coupling given by \n\\be\nk_{ij}\\=-\\frac{n_0}{2} \\,\\sum_\\alpha \\alpha_i \\, \\alpha_j \\,.\n\\ee\nNotice that~$Z^{\\text{dyn}} ( n_0)$ is independent of~$\\t$. The tails completion \n(i.e.~sending~$\\epsilon\\to\\infty$) and contour deformation mentioned above \nremoved the~$\\tau$-dependence of~$Z_\\epsilon^{\\mathrm{dyn}}$ at the cost of an exponentially small error.\n\nThe considerations of three-dimensional effective field theory in the next section show \nthat~$Z^{\\text{dyn}}$ arises naturally in fact as the partition function of three-dimensional~$\\mathcal{N}=2$ \ngauge theory on~$S^3$ with the same gauge group and the same CS coupling. (The latter is \nwell-known to be related  to~$Z^{\\text{CS}}$ exactly as in~\\eqref{dynCSrel}.)\n\n\n\n\n\\vskip 0.4cm\n\nThe above analysis was local around~$\\underline{u}=0$. \nWe now focus on SU($N$) $\\mathcal{N}=4$ SYM for which we know that~$\\underline{u}=0$ is a global minimum \nof the leading potential~$V_2\/\\t^2$ for $n_0=\\pm1$ and \n$n_0(\\mathrm{arg}(\\t)-\\frac{\\pi}{2})<0$.\\footnote{This was shown \nin~\\cite{Cabo-Bizet:2019osg,Kim:2019yrz} where experimental evidence that this is true for a large class \nof theories was also discussed.} \nHowever, this is not the only global minimum---there are~$N$ isolated global mimima \nlabelled by~$k = 1, \\dots ,N$ \nwhich are related to~$u_j=0$ by center symmetry, namely the points~$u_j=(k-1)\/N$, \n$k=1, \\dots, N$ \\cite{ArabiArdehali:2019tdm}. \nUpon summing over these minima we obtain\n\\be \n \\mathcal I_N(\\t;n_0) \\; \\simeq \\;  N {\\rm e}^{-2\\pi\\i\\tau E_{\\text{susy}}}  \\, Z^\\text{bgnd}(\\t;n_0) \\,  Z^\\text{dyn}(n_0) \\,. \n\\label{eq:almostThere}\n\\ee\nThe factor of $N$ arises from the sum over~$N$ minima as explained above. \nThe other three factors can be calculated by \nspecializing our general discussion to this case: \n\\bea\nE_{\\mathrm{susy}}& \\= & \\frac{4}{27}\\big(N^2-1\\big) \\,, \\\\\nZ^\\text{bgnd}(\\t;n_0) & \\= &   \\exp\\left(-\\frac{\\i\\pi}{\\tau^2}(N^2-1)\n    \\biggl(\\Bigl( \\frac{-n_0+2\\tau}{3} \\Bigr)^3 + \n    \\frac{5n_0\\tau^2}{12} \\biggr)+2\\pi\\i\\tau\\cdot\\frac{4}{27}\\big(N^2-1\\big)\\right) \\,, \n    \\qquad\\qquad \n\\label{eq:Zback} \\\\\nZ^\\text{dyn}(n_0)  & \\=  &\n \\int_{-\\infty}^\\infty D \\boldsymbol{x} \\,    \\prod_{i<j} 4 \\sinh^2 \\bigl(\\pi x_{ij} \\bigr) \\,\n    \\exp \\Bigl(  \\i\\pi n_0 N \\sum_{j=1}^N x_j^2 \\, \\Bigr)  \\,.\n    \\label{Zdyn}\n\\eea\nIn this case the matrix of Chern-Simons couplings reduces to a single level ($k_{ij} = k \\delta_{ij}$), \nand we have\n\\be\nZ^\\text{dyn}(n_0) \\= (-1)^{N(N-1)\/2} \\, Z^{\\text{CS}}(k) \\,, \\qquad k \\= -n_0 N \\,.\n\\label{ZdynCS}\n\\ee\n\n\nFor $n_0=\\pm1$ the SU($N$) Chern-Simons partition was found in \\cite{GonzalezLezcano:2020yeb} to simplify to\n\\begin{equation}\nZ^\\text{CS}(-n_0 N) \\=(-1)^{N(N-1)\/2}\\exp \\bigl(5 \\i\\pi \\,n_0 \\ (N^2-1)\/12 \\bigr) \\,.    \n\\end{equation}\nUpon combining this equation with~\\eqref{ZdynCS} and \\eqref{eq:almostThere}, \nwe obtain\\footnote{For comparison with~\\cite{Lezcano:2019pae}, we set~$\\xi_a^{\\text{there}}=-n_0^{\\text{here}}\/3$.\nThe result in that paper contains the number $\\eta\\in\\{-1,+1\\}$. This is related to our~$n_0$ as \n$\\eta=6\\overline{B}_1(-\\frac{n_0}{3})$. For~$n_0 = \\pm 1$, a simple calculation shows that $\\eta=n_0$.}\n\\begin{equation}\n\\begin{split}\n   \\mathcal I_N(\\t;n_0) \\; \\simeq \\; \n    N \\, \\exp \\biggl(-\\frac{\\i\\pi}{\\tau^2} \\, (N^2-1) \\Bigl(\\frac{-n_0+2\\tau}{3} \\Bigr)^3 \\biggr) \\,.\n    \\label{eq:N=4indexAsyFinal}\n    \\end{split}\n\\end{equation}\nThe analogous result for the case with $U$($N$) gauge group is obtained by adding the \ncontribution of a decoupled~$U$(1) $\\mathcal{N}=4$ multiplet to that of the SU($N$) theory:\n\\begin{equation}\n\\begin{split}\n    \\mathcal{I}^{U(N)}(\\t;n_0) \\; \\simeq \\;  \n    N \\, \\frac{1}{-\\i\\tau} \\, \\exp \\biggl( -\\frac{\\i\\pi}{\\tau^2} \\, N^2 \\Bigl(\\frac{-n_0+2\\tau}{3} \\Bigr)^3\n    -\\frac{5\\pi \\i \\, n_0}{12} \\biggr) \\,.\n    \\label{eq:N=4indexAsyFinalU(N)}\n    \\end{split}\n\\end{equation}\n\n\nThis finishes the discussion of our methods illustrated in the special case~$\\tau\\to0$. \nBefore moving on to the more general case of rational points, we make a few technical remarks.\n\nFirstly, since we are analyzing the index by estimating its integrand, we need uniform estimates. \nFor~$n_0=\\pm1$, the estimate~\\eqref{eq:numEst} when applied to the chiral multiplet gamma functions \ngives uniformly valid asymptotics near~$\\underline{u}=0$, because the~$-n_0 \\, r_I\/2$ shift in the argument takes \nus safely into the domain of validity \nof~\\eqref{eq:numEst}. For the vector multiplet gamma functions, however, there is no finite shift in the \nargument, so~\\eqref{eq:numEst} does not apply uniformly around~$\\underline{u}=0$. We had to use \ninstead~\\eqref{eq:denomEst} to obtain uniform asymptotics near~$\\underline{u}=0$ for the vector multiplet gammas.\n\nSecondly, we emphasize that our asymptotic analysis is essentially real-analytic \n(as in~\\cite{Rains:2006dfy,Ardehali:2015bla}). We only appeal to complex-analytic tools \n(specifically, contour deformation), after having done the asymptotic analysis, \nto simplify the final answer for~$Z_\\epsilon^{\\mathrm{dyn}}$ \nin~\\eqref{eq:ZdynComplex} to the more familiar form~\\eqref{eq:ZdynS3}. \n\nThirdly, we note that when actually doing the saddle-point analysis, one finds that the dominant \nholonomy configurations spread into the complex plane, as in the analysis \nof~\\cite{Cabo-Bizet:2019eaf,Benini:2018ywd,GonzalezLezcano:2020yeb}. \nUpon taking the~$\\tau\\to0$ limit the spreading shrinks, and the answers from those approaches \nindeed agree with our results.\n\n\\subsection{Asymptotics of the index as $\\tau\\to\\mathbb{Q}$}\n\nWe now study the index~\\eqref{eq:SYMIndex} in the limit \n\\begin{equation}\n     \\widetilde \\tau \\; \\equiv \\; m\\tau+n \\to 0 \\,, \n\\label{eq:rationalLim}\n\\end{equation}\nwith $m,n$ relatively prime, keeping~$\\mathrm{arg}(\\widetilde \\tau)$ away from integer multiples \nof~$\\pi\/2$. \n\nAs in the previous subsection we first calculate the all-order asymptotics of the integrand. \nThe required small-$\\widetilde\\t$ estimates for the Pocchammer symbol and the elliptic gamma \nfunction are given in Equations~(\\ref{eq:PochRationalEst}), (\\ref{eq:gammaRationalEstWithR}), \n\\eqref{eq:denomEstRational}.\nUsing these estimates we find that in the \nrange~$\\a_+ \\cdot \\underline{u} \\in (-\\frac{1}{m}+\\delta,\\frac{1}{m}-\\delta)$, for some fixed small~$\\delta$,\nthe integrand of~\\eqref{eq:SYMIndex} can be written up to exponentially suppressed corrections as \n\\be\n \\exp \\bigl( -S_\\text{ind}(\\underline{u};\\t)  \\bigr) \\; \\simeq \\; \\frac{1}{(-\\i\\, \\widetilde\\tau)^{\\mathrm{rk}(G)}}\\, \n\\prod_{\\alpha_+}4\\sinh^2\\Bigl(\\frac{\\pi \\alpha_+ \\cdot \\underline{u}}{-\\i\\, \\widetilde\\tau}\\Bigr) \\,  \n\\exp \\bigl( -2\\pi \\i \\, \\widetilde\\tau \\, \\frac{E_{\\mathrm{susy}}}{m} -\\frac{\\widetilde V (\\underline{u})}{m} \\bigr)\\, .\n\\label{eq:In0semi-simpleAsyRat0}\n\\ee\nThe all-order effective potential as~$\\widetilde\\t\\to 0$  is given by\n\\be\n\\widetilde V(\\underline{u}) \\= \\frac{1}{\\widetilde\\t^2} \\widetilde V_2(\\underline{u})+ \\frac{1}{\\widetilde\\t}  \\widetilde V_1(\\underline{u})+\\widetilde V_0(\\underline{u}) \\,,\n\\label{defVsumRat}\n\\ee\nwith\n\\be\n\\begin{split}\n    \\widetilde V_2(\\underline{u})&\\=\\sum_{I, \\,\\rho_I}\\, \\frac{\\i\\pi}{3} \\, \\overline{B}_3 \\bigl(m\\rho_I \\cdot \\underline{u} +m\\xi_I \\bigr) \\,,\\\\\n    \\widetilde V_1(\\underline{u})&\\=\\sum_{I,\\,\\rho_I} \\, \\i\\pi(r_I-1) \\, \n    \\overline{B}_2 \\bigl(m\\rho_I \\cdot \\underline{u}  +m\\xi_I\\bigr)+\n    \\sum_{\\alpha} \\i\\pi \\, \\Bigl((m\\alpha \\cdot \\underline{u})^2  + \\frac{1}{6}\\Bigr) \\,,\\\\\n    \\widetilde V_0(\\underline{u})&\\=-2\\pi\\i\\cdot\\mathrm{dim}(G)\\, s(n,m)+\\sum_{I,\\,\\rho_I} \\,2\\pi\\i\\, C(m,n,\\rho_I\\cdot\\underline{u}-\\frac{n_0}{2}r_I,r_I) \\,,\n\\label{eq:defVsRat}\n\\end{split}\n\\ee\nwhere~$\\a$ runs over all the roots of $G$ including the rk$(G)$ zero roots,~$I$ runs over all the \nchiral multiplets, and~$\\rho_I$ runs over all the weights of the representation~$\\mathcal{R}_I$. \nHere~$s(n,m)$ is the Dedekind sum defined in~\\eqref{eq:ourDedekind Sum} and the function~$C(m,n,r,z)$ is defined in~\\eqref{eq:C(m,n,r,z)}. \nWe have defined\n\\begin{equation}\n    \\xi_I\\; \\coloneqq \\;} % This only works with \\usepackage{mathtools-\\frac{r_I}{2}\\big(n_0+\\frac{2n}{m}\\big),\\label{eq:defXiGen}\n\\end{equation}\nto emphasize an analogy with the analysis in~\\cite{ArabiArdehali:2019orz,ArabiArdehali:2019tdm} \nof the index with flavor chemical potential~$\\xi$, although we do not have flavor fugacities \nin our problem. Note also that we have separated the supersymmetric Casimir energy \nin~\\eqref{eq:In0semi-simpleAsyRat0} as in the~$\\t \\to 0$ case.\n\nNext, as in the previous subsection we expand the potentials near~$\\underline{u}=0$. \nAnomaly cancellations again lead to simplifications, \nbut here we further assume the theory is non-chiral (i.e. that~$\\rho_I$ come \nin pairs of opposite sign) so that the answer takes a particularly simple form. \nAnalogously to~\\eqref{eq:simplifiedVs} we obtain\n\\be\n\\begin{split}\n    \\widetilde V_2(\\underline{u}) &\\=\\sum_{I,\\,\\rho_I} \\Bigl(\\frac{\\i\\pi}{3} \\, \\overline{B}_3 \\bigl(m\\xi_I \\bigr) \n    \\+2\\pi \\i \\, \\overline{B}_1\\bigl(m\\xi_I \\bigr)\\frac{(m\\rho_I\\cdot\\underline{u})^2}{2} \\Bigr) \\,,\\\\\n    \\widetilde V_1(\\underline{u}) &\\=\\sum_{I,\\,\\rho_I} \\, \\i\\pi(r_I-1) \\, \n    \\overline{B}_2\\bigl(m\\xi_I\\bigr)\\+\\frac{\\i\\pi}{6}\\text{dim}(G)\\,,\\\\\n    \\widetilde V_0(\\underline{u}) &\\=\\widetilde V_0(0)\\= -2\\pi\\i \\, \\mathrm{dim}(G)\\, s(n,m)\n    +\\sum_{I,\\,\\rho_I} \\,2\\pi\\i\\, C(m,n,-\\frac{n_0}{2}r_I,r_I) \\,.\n\\label{eq:simplifiedVtildes}\n\\end{split}\n\\ee\n\nNext we focus on a small neighborhood~$\\mathfrak{h}_{cl}^\\epsilon$ around~$\\underline{u}=0$, \ndefined by the cutoff~$|u_j|<\\epsilon$, whose contribution to the index as~$\\widetilde\\t\\to0$ is \n\\be\n\\mathcal{I}(\\tau;n_0)_{|u_j|<\\epsilon} \\; \\simeq \\;\n{\\rm e}^{-2\\pi \\i\\widetilde\\tau \\frac{E_{\\mathrm{susy}}}{m}}\n\\int_{\\mathfrak{h}_{cl}^\\epsilon} \\frac{D\\underline{u}}{(-\\i\\widetilde\\tau)^{\\mathrm{rk}(G)}}\\, \n\\prod_{\\alpha_+}4\\sinh^2\\Bigl(\\frac{\\pi \\alpha_+ \\cdot \\underline{u}}{-\\i\\widetilde\\tau}\\Bigr) \\,  \n\\exp \\bigl( -\\frac{\\widetilde V(\\underline{u})}{m} \\bigr)\\, .\n\\label{eq:In0semi-simpleAsyRat1}\n\\ee\nUpon putting the above discussion together we obtain\n\\be\n\\mathcal{I}(\\tau;n_0)_{|u_j|<\\epsilon} \\; \\simeq \\; {\\rm e}^{-2\\pi \\i\\widetilde\\tau \\frac{E_{\\mathrm{susy}}}{m}} \\, \nZ^{\\text{bgnd}}(\\tau;m,n,n_0) \\  Z_\\epsilon^{\\text{dyn}}(\\tau;m,n,n_0)\\,,\n\\label{eq:In0semi-simpleAsyRat2}\n\\ee\nwhere the \\emph{background piece} is \n\\be\n\\begin{split}\n& Z^{\\text{bgnd}}(\\tau;m,n,n_0)  \\= \\\\ \n& \\quad \\exp \\Bigl( -\\frac{1}{m\\widetilde\\t^2}\\sum_{I,\\,\\rho_I} \\frac{\\i\\pi}{3} \\, \\overline{B}_3 \\bigl(m\\xi_I \\bigr)\n-\\frac{1}{m\\widetilde\\t}\\sum_{I,\\,\\rho_I} \\, \\i\\pi(r_I-1) \\, \n    \\overline{B}_2\\bigl(m\\xi_I\\bigr)\\+\\frac{\\i\\pi}{6}\\text{dim}(G) + \n\\widetilde V_0(0) \\Bigr)\\,,\n\\end{split}\n\\ee\nand the \\emph{dynamical piece} is\n\\be\nZ_\\epsilon^{\\text{dyn}}(\\tau;m,n,n_0)=\\int_{\\mathfrak{h}_{cl}^\\epsilon} \\frac{D\\underline{u}}{(-\\i\\, \\widetilde\\tau)^{\\mathrm{rk}(G)}}\\,\n\\prod_{\\alpha_+}4\\sinh^2\\Bigl(\\frac{\\pi \\alpha_+ \\cdot \\underline{u}}{-\\i\\,\\widetilde\\tau}\\Bigr) \\,  \n\\exp \\Bigl(+\\frac{\\i\\pi}{m} \\sum_{I,\\,\\rho_I}  \\overline{B}_1 \\bigl(m\\xi_I \\bigr)\\bigl(\\frac{m\\rho_I\\cdot\\underline{u}}{-\\i\\,\\tau} \\bigr)^2 \\Bigr) \\,.\n\\ee\nUpon defining the rescaled variable~$x_j=\\frac{u_j}{-\\i\\widetilde\\tau}$, \nwe recognize~$Z_\\epsilon^{\\text{dyn}}(\\tau;m,n,n_0)$ as the CS partition function \non~$S^3$ with gauge group~$G$  and \nlevel~$k^{ij}\\=-\\frac{1}{m}\\sum_{I,\\rho_I}\\overline{B}_1 \\bigl(m\\xi_I \\bigr) \\rho_I^i \\rho_I^j$. \nWe will see momentarily that it is more natural to define the rescaled variable \nas~$x_j=\\frac{mu_j}{-\\i\\widetilde\\tau}$. Upon tails completion and deforming the integration \ncontour we obtain\n\\be\n\\begin{split}\nZ_\\epsilon^{\\text{dyn}}(\\tau;m,n,n_0)\\; &\\simeq \\; m^{-\\mathrm{rk}(G)}\\int_{-\\infty}^{\\infty} D\\boldsymbol{x}\\,\n\\prod_{\\alpha_+}4\\sinh^2\\Bigl(\\frac{\\pi \\alpha_+ \\cdot \\boldsymbol{x}}{m}\\Bigr) \\,  \n\\exp \\Bigl(+\\frac{\\i\\pi}{m} \\sum_{I,\\,\\rho_I}  \n\\overline{B}_1 \\bigl(m\\xi_I \\bigr)\\bigl(\\rho_I\\cdot\\boldsymbol{x} \\bigr)^2 \\Bigr)\\\\\n&=:\\; m^{-\\mathrm{rk}(G)}\\, Z_{\\underline{0}}^{\\text{dyn}}(m,n,n_0)\\,.\\label{eq:Z0Rational}\n\\end{split}\n\\ee\nUp to the overall $m^{-\\mathrm{rk}(G)}$ factor, this coincides~\\cite{Gang:2019juz} with the topologically \ntrivial sector of the~$S^3\/\\mathbb{Z}_m$ partition function of $\\mathcal{N}=2$ SYM with Chern-Simons coupling\n\\be\nk^{ij}\\=-\\sum_{I,\\rho_I}\\overline{B}_1 \\bigl(m\\xi_I \\bigr) \\,  \\rho_I^i \\, \\rho_I^j \\,.\n\\label{eq:CScouplingRat}\n\\ee\n\nWhile the explicit expression for the dominant potential~$\\widetilde V_2$ in~\\eqref{eq:defVsRat} \nwas derived in a neighborhood $(-\\frac{1}{m}+\\delta,\\frac{1}{m}-\\delta)$ of~$\\underline{u}=0$, it is \nactually correct more generally, because it follows from~\\eqref{eq:gammaRationalEstWithR} which \nwe can use as long as $m\\rho_I \\cdot \\underline{u} +m\\xi_I\\notin\\mathbb{Z}$. Moreover, since~$\\rho^j_I$ \nare integers and~$u_j$ appears in~$\\widetilde V_2$ in the combination~$m\\,\\rho_I\\cdot\\underline{u}$, the \n1-periodicity of~$\\widetilde V_2$  implies that any holonomy configuration with~$u_{j}$ a multiple \nof~$1\/m$ gives the same leading asymptotics as the~$\\underline{u}=0$ configuration. \nIn the SU($N$) case these non-trivial holonomy configurations correspond to\n\\be\n\\underline{u}=\\frac{1}{m} \\, \\underline{m} \\= \\bigl(\\frac{m_1}{m} \\,,\\dots,\\frac{m_N}{m} \\bigr) \\, ,\n\\label{eq:nontrivialSectors}\n\\ee\nwith~$m_j\\in\\mathbb{Z}\/m\\mathbb{Z}$, and~$\\sum_{j=1}^N m_j=0$ (mod $m$).\n\nFor~$n=1$, we can use the estimate~\\eqref{eq:denomEstRational0.75} for the vector multiplet \ngamma functions to compute the contribution of the saddles~\\eqref{eq:nontrivialSectors}. \nThe result is similar to~\\eqref{eq:In0semi-simpleAsyRat1}, with the same~$\\widetilde V_{2,1}$ and \nthe same SUSY Casimir piece, but with the dynamical piece modified to (modulo an overall constant factor)\n\\be\n\\begin{split}\nZ_{\\epsilon_{\\underline{m}}}^{\\text{dyn}}(\\tau;m,n,n_0)\\; &\\simeq \\; \\int_{-\\infty}^{\\infty} D\\boldsymbol{x}'\\,\n\\prod_{\\alpha_+}4\\sinh^2\\Bigl(\\frac{\\pi \\alpha_+ \\cdot (\\boldsymbol{x}'+\\i\\underline{m})}{m}\\Bigr) \\,  \n\\exp \\Bigl(+\\frac{\\i\\pi}{m} \\sum_{I,\\,\\rho_I}  \n\\overline{B}_1 \\bigl(m\\xi_I \\bigr)\\bigl(\\rho_I\\cdot\\boldsymbol{x}' \\bigr)^2 \\Bigr)\\\\\n&=:\\; Z_{\\underline{m}}^{\\text{dyn}}(m,n,n_0)\\,,\\label{eq:Z0RationalNonTrivialSect}\n\\end{split}\n\\ee\nwhere~$\\epsilon_{\\underline{m}}$ indicates that we are considering the contribution from a \nneighborhood~$|u_j-m_j\/m|<\\epsilon$, and the re-scaled integration variable arises \nas~$x_j'=\\frac{m(u_j-m_j\/m)}{-\\i\\widetilde\\t}$. This coincides (again up to an overall constant factor) \nwith the topologically non-trivial sector of the partition function of~SU($N$) Chern-Simons theory \nwith coupling~\\eqref{eq:CScouplingRat} on the lens space~$L(m,-1)$~\\cite{Gang:2019juz}.\n\nWe expect that similarly for general~$n$, including the contribution of the non-trivial \nsaddles~\\eqref{eq:nontrivialSectors} to the index would complete~$Z_{\\underline{0}}^{\\text{dyn}}(m,n,n_0)$ \nto the full~$S^3\/\\mathbb{Z}_m$ partition function, including all the topologically \nnon-trivial sectors. We motivate this expectation \nfurther from an EFT perspective in the next section where we also present the ($n$-dependent) \naction of~$\\mathbb{Z}_m$ on the~$S^3$. The explicit demonstration, which we leave to future work, \nrequires generalizing the estimate~\\eqref{eq:denomEstRational0.75} to arbitrary~$n$, and improving \nit to include the overall constant.\n\nThe above analysis was local in nature: we considered the contribution to the index from only a \nsmall neighborhood of~$\\underline{u}=0$. We now study the specific case of SU($N$)~$\\mathcal{N}=4$ \nSYM for which we present a global picture of the dominant holonomy configurations. \nNote that in the previous subsection rather than performing the global analysis from scratch \nwe had borrowed the result of~\\cite{Kim:2019yrz,Cabo-Bizet:2019osg} that in a certain \ndomain of parameters ($n_0(\\mathrm{arg}(\\t)-\\pi\/2)<0$) the~$\\underline{u}=0$ configuration is \nglobally dominant (see Section~\\ref{sec:Ccenter} for the complementary domain).\n\n\\subsubsection*{The global structure of the leading potential for $\\mathcal{N}=4$ SYM}\n\nFor SU($N$) $\\mathcal{N}=4$ theory the potential~$\\widetilde V_2$ reads\n\\begin{equation}\n    \\begin{split}\n    \\widetilde V_2 (\\underline{u};\\xi) \\= \n    \\frac{\\i\\pi}{3} \\times 3 \\; \\Bigl((N-1)\\overline{B}_3(m\\xi)+\n    \\sum_{i<j} \\bigl( \\overline{B}_3(m\\xi+m u_{ij})+ \\overline{B}_3 (m\\xi + m u_{ji}) \\bigr)\\Bigr) \\,,\n    \\label{eq:QhDef}\n    \\end{split}\n\\end{equation}\nwhere the factor of 3 comes from the sum over three chiral multiplets, and \nwith\n\\be\n\\xi \\= - \\tfrac13 \\bigl(n_0 + \\tfrac{2n}{m} \\bigr) \\,.\n\\label{defxi}\n\\ee \nAs mentioned below \\eqref{eq:CScouplingRat} the expression \\eqref{eq:QhDef} \napplies as long as $u_{ij}+\\xi_I$ avoid $\\frac{\\mathbb{Z}}{m}$.\n\n\n\nWe now have to minimize the real part of~$\\widetilde V_2(\\underline{u})\/\\widetilde \\t^{\\,2}$ as~$|\\widetilde \\t| \\to 0$.\nSince the~$u_{ij}$-independent piece and the real positive overall multiplicative constants \nare irrelevant in finding the dominant holonomy configurations, our problem boils down \nto minimizing the potential \n\\begin{equation}\n    \\begin{split}\n    V^Q(u_{ij};\\arg\\widetilde{\\tau},\\xi)\\=-\\mathrm{sign}(\\arg\\widetilde{\\tau}-\\frac{\\pi}{2})\\,\n    \\left(\\overline{B}_3(m\\xi+ m u_{ij})+\\overline{B}_3(m\\xi- m u_{ij})\\right) \\,,\n    \\label{eq:QhPot}\n    \\end{split}\n\\end{equation}\nwhich is analogous to the \\emph{pairwise holonomy potential} in~\\cite{ArabiArdehali:2019tdm}. \nAs in that work, we first consider the \\emph{qualitative} behavior of $V^Q$. \nWe assume $\\mathrm{arg}(\\widetilde{\\tau})-\\frac{\\pi}{2}>0$, and comment below on what \nhappens for the opposite sign.\nWe find that the potential is (see Figure~\\ref{fig:MW}) \n\\begin{equation*}\n\\begin{split}\n    \\text{M-shaped for }&0<\\{m\\xi\\}<1\/2 \\,,\\\\\n\\text{W-shaped for }&1\/2<\\{m\\xi\\}<1 \\,.\n\\end{split}\n\\end{equation*}\nWe also see from Equation~(\\ref{defxi}) that we have~$\\{m\\xi\\}\\in\\{0,\\frac{1}{3},\\frac{2}{3}\\}$.\n\n\n\\begin{figure}[h]\n\\centering\n    \\includegraphics[scale=.6]{MWlineF}\n\\caption{The catastrophic behavior of $V^Q(u_{ij})$, drawn over the \nrange~$mu_{ij}\\in(-1,1)$, for $\\mathrm{arg}\\widetilde{\\tau}>\\frac{\\pi}{2}$. \nThe control parameter $m\\xi$ determines the M or W type behavior. \\label{fig:MW}}\n\\end{figure}\n\n\n\n\n\\subsubsection*{The $\\text{O}(1\/\\widetilde \\t^2)$ exponent}\n\nLet us now assume $m,n$ are chosen such that~$\\{m\\xi\\}=\\{\\frac{-m n_0-2n}{3}\\}=\\frac{1}{3}$, \nso we are in the M-region with the dominant holonomy configurations corresponding to~$\\{m u_{ij}\\}=0$. \nAlthough this is analogous to the~\\emph{1-center phase} in~\\cite{ArabiArdehali:2019orz}, \nas mentioned around~\\eqref{eq:nontrivialSectors} \nhere in fact $u_{ij}$ can be any integer multiple of~$\\frac{1}{m}$. All these saddles contribute \nequally to the~$\\text{O}(1\/\\widetilde\\t^2)$ exponent though, and hence the preceding analysis \naround~$\\underline{u}=0$ gives the correct leading asymptotics of the index, which up to~$\\text{O}(1\/\\widetilde \\t)$ \ncorrections in the exponent reads\n\\begin{equation}\n\\begin{split}\n     &\\exp \\Bigl(-\\frac{\\pi \\i}{m \\, \\widetilde \\t^{\\, 2}}(N^2-1) \\overline{B}_3(m\\xi) \\Bigr) \\=\n        \\exp\\Bigl(-\\frac{\\i\\pi }{27 m \\, \\widetilde \\t^{\\, 2}}(N^2-1)  \\Bigr) \\,,        \n\\label{eq:n0IndexRationalAsy1} \\\\ \n&\\qquad \\text{for} \\quad \\mathrm{arg} (\\widetilde \\t)>\\frac{\\pi}{2} \\,, \\quad \\{m\\xi\\}\\=\\{\\frac{-m n_0-2n}{3}\\} \\=\\frac{1}{3} \\,. \n\\end{split}\n\\end{equation}\nFor $\\mathrm{arg} (\\widetilde \\t)-\\frac{\\pi}{2}<0$, the M- and W-regions switch places. \nSo in order to have $u_{ij}=0$ as the dominant saddle we must assume $m,n$ are such \nthat $ \\{m\\xi\\}=\\{\\frac{-m n_0-2n}{3}\\}=\\frac{2}{3}$. In this case we \nhave~$\\overline{B}_3(2\/3) = - \\overline{B}_3(1\/3) =1\/27$, which leads to \n\\begin{equation}\n\\begin{split}\n     &\\exp \\Bigl(-\\frac{\\pi \\i}{m \\, \\widetilde \\t^{\\, 2}}(N^2-1) \\overline{B}_3(m\\xi) \\Bigr) \\=\n        \\exp\\Bigl(\\frac{\\i\\pi }{27 m \\, \\widetilde \\t^{\\, 2}}(N^2-1)  \\Bigr) \\,,        \n\\label{eq:n0IndexRationalAsy2} \\\\ \n&\\qquad \\text{for} \\quad \\mathrm{arg} (\\widetilde \\t)<\\frac{\\pi}{2} \\,, \\quad \\{m\\xi\\}\\=\\{\\frac{-m n_0-2n}{3}\\} \\=\\frac{2}{3} \\,. \n\\end{split}\n\\end{equation}\n\nIn the remaining case where~$\\{m\\xi\\}=\\{\\frac{-m n_0-2n}{3}\\}=0$, we have~$\\widetilde V_2 (\\underline{u};\\xi)=0$ \nand hence no~$\\text{O}(\\frac{1}{\\widetilde \\tau^2})$ exponent. As we discuss momentarily there is \nno~$\\text{O}(\\frac{1}{\\widetilde \\tau})$ exponent in this case either. There are thus~$\\mathrm{rk}(G)$ \nflat directions in the moduli space, leading to a~$(1\/\\widetilde{\\t})^{\\mathrm{rk}(G)}$ growth for the index, \nas in the~$n_0=0$ and~$\\tau$ pure imaginary case studied in~\\cite{Ardehali:2015bla}.\n\n\\subsubsection*{The $\\text{O}(1\/\\widetilde \\t)$ exponent}\n\nThe~$\\text{O}(1\/\\widetilde \\t)$ exponent comes from $\\widetilde V_1\/m\\widetilde\\t$. Although the expression \nfor~$\\widetilde V_1$ in~\\eqref{eq:simplifiedVtildes} was obtained near~$\\underline{u}=0$, the~$\\text{O}(1\/\\widetilde \\t)$ \nexponent is correctly captured by~\\eqref{eq:gammaRationalEstWithR}, which implies \nthat~\\eqref{eq:simplifiedVtildes} remains correct near the nontrivial \nsaddles with~$u_{ij}\\in\\frac{1}{m}\\mathbb{Z}$ as well.\nSo we can specialize~$\\widetilde V_1$ in~\\eqref{eq:simplifiedVtildes} to the SU($N$) $\\mathcal{N}=4$ theory\nand obtain \n\\begin{equation}\n    \\exp \\Bigl(- \\frac{\\pi \\i}{m \\widetilde \\t} (N^2-1) \\, \\bigl(-\\overline{B}_2(m\\xi)+\\tfrac{1}{6} \\bigr) \\Bigr) \\,.\\label{eq:N=4at1\/tau}\n\\end{equation}\nIn this case we have that~$\\overline{B}_2(2\/3) = +\\overline{B}_2(1\/3) = - 1\/18$, which leads to  \n\\begin{equation}\n    \\exp \\biggl(-\\frac{2\\pi \\i \\,}{9}\\frac{(N^2-1)}{m \\widetilde \\t}\\biggr) \\,,\n\\end{equation}\nfor~$\\mathrm{arg} (\\widetilde \\t)>\\frac{\\pi}{2}$  as well as~$\\mathrm{arg} (\\widetilde \\t)<\\frac{\\pi}{2}$. \nNote that since~$\\overline{B}_2(0) = 1\/6$, we see from~\\eqref{eq:N=4at1\/tau} that there is\nno~$\\text{O}(1\/\\widetilde\\t)$ exponent for~$\\{m\\xi\\}=0$, as alluded to above.\n\n\\subsubsection*{The Chern-Simons coupling}\n\nSpecializing the Chern-Simons coupling \\eqref{eq:CScouplingRat} to SU($N$) $\\mathcal{N}=4$ theory we find\n\\be\nk_{ij}=-\\widetilde\\eta \\, N \\, \\delta_{ij},\n\\ee\nwith\n\\be\n\\widetilde\\eta\\; \\coloneqq \\;} % This only works with \\usepackage{mathtools 6\\overline{B}_1(m\\xi)\\= 6\\overline{B}_1\\bigl(\\frac{-mn_0-2n}{3}\\bigr) \\,.\\\\\n\\ee\n\\vspace{.5cm}\n\nWe emphasize that all the topologically nontrivial sectors necessary for agreement with \nan~$S^3\/\\mathbb{Z}_m$ partition function are present in our analysis, but we leave the \ninvestigation of their explicit contributions to future work. \n\n\n\\subsubsection*{The $\\text{O}(\\widetilde \\t)$ exponent}\n\nThe linear (in $\\widetilde{\\tau}$) exponent can be read from~\\eqref{eq:In0semi-simpleAsyRat1} to \nbe~$-2\\pi i\\widetilde\\tau E_{\\mathrm{susy}}\/m$. Note again that while~\\eqref{eq:In0semi-simpleAsyRat1} \nwas derived near~$\\underline{u}=0$, as the estimate~\\eqref{eq:denomEstRational0.75} shows the~$\\text{O}(\\widetilde \\t)$ \nexponent remains valid near~$\\underline{u}\\in\\frac{\\mathbb{Z}}{m}$ as well (at least for~$n=1$, and \nwe expect more generally as well). Since for SU($N$) \n$\\mathcal{N}=4$ theory $E_{\\mathrm{susy}}= \\frac{4}{27}\\big(N^2-1\\big)$, we have the $\\mathcal{O}(\\widetilde{\\tau})$ \nexponent as in\n\\begin{equation}\n    \\exp \\biggl(-\\frac{8\\pi \\i}{27m}(N^2-1) \\widetilde{\\tau} \\biggr) \\,.\n\\end{equation}\n\n\n\\subsection*{Summary: the small-$\\widetilde\\t$ asymptotics for $\\mathcal{N}=4$ SYM}\n\nWe can summarize the asymptotics of the SU($N$) $\\mathcal{N}=4$ SYM index analyzed above as follows\n\\begin{equation}\n    \\mathcal I_N(\\t;n_0)  \\; \\simeq \\; N \\, \\widetilde {C}_N(n_0,m,n)\\,  \n    \\exp \\biggl(-\\frac{\\i\\pi}{m \\,\\widetilde\\tau^2} \\, (N^2-1) \\Bigl(\\frac{-\\widetilde \\eta+2\\widetilde\\tau}{3} \\Bigr)^3 \\biggr) \\;\nZ^{\\text{CS}}_{S^3\/\\mathbb{Z}_m}(k)  \\,,\n  \\label{eq:N=4indexAsyRational}\n\\end{equation}\nfor $\\tau$ near any rational point~$-n\/m$, with  \n\\be\n\\widetilde \\t \\= m\\t+n \\,, \\qquad \\widetilde \\eta \\= 6\\overline{B}_1(\\frac{-mn_0-2n}{3}) \\= -\\mathrm{sign}(\\mathrm{arg}(\\widetilde \\t)-\\tfrac{\\pi}{2}) \\,,\n\\qquad k \\=- \\widetilde \\eta \\, N  \\,,\n\\ee\nand with~$\\widetilde {C}_N(n_0,m,n)$ an overall constant. Note that we have used~$\\widetilde{\\eta}^3=\\widetilde{\\eta}=\\pm1$ \nto simplify the final expression. \nAlso, by completing the cube inside the exponent we have introduced an~$\\mathcal{O}(\\widetilde{\\tau}^0)$ \nfactor at the cost of redefining~$\\widetilde{C}_N(n_0,m,n)$.\n\nWe have only demonstrated that there is a contribution to~$Z^{\\text{CS}}_{S^3\/\\mathbb{Z}_m}(k)$ from \nnear~$\\underline{u}=0$ that coincides with the topologically trivial sector of the~$S^3\/\\mathbb{Z}_m$ \npartition function of Chern-Simons theory with coupling~$k$. As mentioned below~\\eqref{eq:nontrivialSectors} \nwe expect that summing over the contributions from neighborhoods of the non-trivial \nconfigurations~$u_j=m_j\/m$ would lead to the complete orbifold partition function.\n\nWe can include the contribution of a decoupled $U$(1) $\\mathcal{N}=4$ multiplet in a straightforward manner. \nThis effectively changes the dimension of the group in the exponent to~$N^2$, introduces a prefactor~$1\/\\widetilde \\t$,\nand change the constant from~$\\widetilde{C}_N(n_0,m,n)$ to a new constant~$\\widetilde{C'}_N(n_0,m,n)$, so that we have\n\\begin{equation}\n    \\mathcal I^{U(N)}(\\t;n_0)  \\; \\simeq \\; \\frac{N}{\\i \\widetilde \\t} \\, \\widetilde{C'}_N(n_0,m,n) \\, \n    \\exp \\biggl(-\\frac{\\i\\pi}{m \\,\\widetilde\\tau^2} \\, N^2 \\Bigl(\\frac{-\\widetilde \\eta+2\\widetilde\\tau}{3} \\Bigr)^3 \\biggr) \\;\nZ^{\\text{CS}}_{S^3\/\\mathbb{Z}_m}(k)  \\,.\n  \\label{eq:N=4indexAsyRationalU(N)}\n\\end{equation}\nWe see that the background (and the SUSY Casimir) piece in~\\eqref{eq:N=4indexAsyRationalU(N)} matches the effective \naction~\\eqref{actionEll} and, in addition, we have a dynamical Chern-Simons term.\nIn the following section we explain both these pieces from the point of view of 3d $\\mathcal{N}=2$ field theory.\n\n\n\\subsection{$C$-center phases}\\label{sec:Ccenter}\n\nFocussing on SU($N$) $\\mathcal{N}=4$ theory, we now move on to studying the $\\widetilde\\t\\to0$ limit \nof the index in the \\emph{W region}, which as shown in Figure~\\ref{fig:MW} for \n$\\mathrm{arg}\\widetilde{\\tau}>\\pi\/2$ corresponds to $1\/2<\\{m\\xi\\}<1$. As before we assume \n$\\mathrm{arg}\\widetilde\\t$ is in compact domains avoiding integer multiples of $\\pi\/2$ as $|\\widetilde\\t|\\to0$.\n\nRecall from (\\ref{defxi}) that only the values $\\{m\\xi\\}=0,1\/3,2\/3$ are realized in our problem. \nBut to highlight the parallels with the analysis of partially-deconfined phases in the W regions \nof the (flavored) 4d $\\mathcal{N}=4$ index in \\cite{ArabiArdehali:2019orz}, we will study the \nphase structure for arbitrary $\\{m\\xi\\}\\in(\\frac{1}{2},1)$ below, and only at the end specialize \nour result to the single ``physical'' point $\\{m\\xi\\}=2\/3$ in that interval.\n\n\nAsymptotic analysis of the index for arbitrary $\\{m\\xi\\}\\in(\\frac{1}{2},1)$ is difficult for general $N$, \nbecause finding the dominant holonomy configurations is not possible analytically in the W regions. \nAnalogously to \\cite{ArabiArdehali:2019orz} we consider now the large-$N$ limit (on top of \nthe $\\widetilde\\t\\to0$ limit), and conjecture that the $C$-center phases suffice for extremizing the \npotential in the W region. Also, similarly to \\cite{ArabiArdehali:2019orz} we consider only the \nleading (here $\\text{O}(1\/\\widetilde\\t^2)$) exponent of the index in the W region.\n\nA $C$-center holonomy configuration consists of $C$ packs of $N\/C$ holonomies uniformly \ndistributed on the circle such that the SU($N$) constraint is satisfied. While at finite $N$ it is \npossible to have such configurations only for $C$ a divisor of $N$, in the large-$N$ limit any \ninteger $C\\ge1$ provides an acceptable $C$-center configuration \\cite{ArabiArdehali:2019orz}. \nFor such a distribution the ``on-shell'' value of the potential $\\widetilde V_2$ in (\\ref{eq:QhDef}) becomes\n\\begin{equation}\n   \\widetilde V_2^{(C)}=\\i\\pi\\Bigl((N-1)\\overline{B}_3(m\\xi)+\\frac{N}{d}\\frac{d(d-1)}{2}\\,\n   2\\overline{B}_3(m\\xi)+d^2 \\sum_{J=1}^{C-1}J\\big(\\overline{B}_3(m\\xi+m\\frac{J}{C})\n   +\\overline{B}_3(m\\xi-m\\frac{J}{C})\\big)\\Bigr),\n\\end{equation}\nwhere $d:=N\/C$. The second term above is the contribution from pairs in the same pack, and the \nthird term is from pairs with each end on a different pack. To simplify the above expression further, \nwe use the following identity which can be proven from (\\ref{eq:Raabe}) and (\\ref{eq:remarkableId}):\n\\begin{equation}\n    \\sum_{J=1}^{C-1}J\\big(\\overline{B}_3(\\Delta+m\\frac{J}{C})+\\overline{B}_3(\\Delta-m\\frac{J}{C})\\big)\n    =\\frac{g^2\\overline{B}_3(C'\\Delta)}{C'}-C\\overline{B}_3(\\Delta),\n\\end{equation}\nwhere $g:=\\mathrm{gcd}(m,C)$ and $C':=C\/g$. Keeping only the $O(N^2)$ terms we hence end up with\n\\begin{equation}\n    \\widetilde V_2^{(C)}=\\i\\pi N^2\\,\\frac{\\overline{B}_3(C'm\\xi)}{C'^3}.\n\\end{equation}\nSince the leading asymptotics of the index is given as $\\exp(-\\widetilde V_2\/\\widetilde\\t^2)$, we then find the \nanalog of the main result of \\cite{ArabiArdehali:2019orz} (Equation~(3.19) of that work) for our case to be\n\\begin{equation}\n    \\mathcal{I}_{N\\to\\infty}\\xrightarrow{\\widetilde\\t\\to0}\\sum_{C=1}^{\\infty}\n    \\exp\\left(-\\frac{i\\pi N^2}{m\\widetilde{\\tau}^{\\,2}}\\,\\frac{\\overline{B}_3(C'm\\xi)}{C'^3}\\right),\n    \\label{eq:doubleScalingConjecture}\n\\end{equation}\nwith $m\\xi=-\\frac{mn_0+2n}{3}$ as before.\n\nThe competition between various terms in (\\ref{eq:doubleScalingConjecture}) can be visualized by \ncomparing the exponents as in Figure~\\ref{fig:singleDelta}, which shows the range of $\\Delta:=\\{m\\xi\\}$ \nfor which a given phase dominates when $\\mathrm{arg}\\widetilde\\t-\\pi\/2>0$. The figure implies that for the \n``physical'' values $\\{m\\xi\\}=1\/3,2\/3$, the index is respectively in the 1-center, and 2-center phase \nwhen $\\mathrm{arg}\\widetilde\\t-\\pi\/2>0$, and vice versa for $\\mathrm{arg}\\widetilde\\t-\\pi\/2<0$. As mentioned above, \nfor $\\{m\\xi\\}=0$ the index is in a confined phase and does not yield exponential $O(N^2)$ growth. \nTherefore up to an $o(N^2\/\\widetilde{\\tau}^{2})$ error in the exponents we have the following simplification of \n(\\ref{eq:doubleScalingConjecture}) by restricting to $C'=1,2$:\n\\begin{equation}\n    \\mathcal{I}_{N\\to\\infty}\\xrightarrow{\\widetilde\\t\\to0}e^{-\\frac{i\\pi N^2}{m\\widetilde{\\tau}^2}\\,\n    \\overline{B}_3(m\\xi)}+e^{-\\frac{i\\pi N^2}{m\\widetilde{\\tau}^2}\\,\\frac{\\overline{B}_3(2m\\xi)}{8}}.\n    \\label{eq:doubleScalingConjecture2}\n\\end{equation}\nThis is the analog of Conjecture~1 in \\cite{ArabiArdehali:2019orz}.\n\nSince $\\overline{B}_3(2\/3)=-\\overline{B}_3(1\/3)$, we see from~\\eqref{eq:doubleScalingConjecture2} \nthat the action of the 2-center saddle has the opposite sign and is smaller in absolute value by a \nfactor of 8 compared to that of the 1-center saddle.\n\n\n\\begin{figure}[h]\n\\centering\n    \\includegraphics[scale=.45]{singleDelta}\n\\caption{The functions $C'^{-3}\\overline{B}_3(C'\\Delta)$ with\n$C'=1,\\cdots,13$, for $0\\le\\Delta\\le1$. For $0<\\Delta<1\/2$ the blue curve corresponding to the \nfully-deconfined phase takes over. The take-over of the\norange curve signifies the partially-deconfined 2-center phase in the corresponding \nregion ($1\/2<\\Delta\\lesssim .72$),\nand so on. \\label{fig:singleDelta}}\n\\end{figure}\n\n\n\n\n\n\n\n\\section{Asymptotics of the 4d index from 3d field theory \\label{sec:4dto3d}}\n\nIn this section we consider the dimensional reduction of the four-dimensional~$\\mathcal{N}=1$ \ngauge theory on a Hopf surface. This surface is topologically~$S^3 \\times S^1$ and  \nwe reduce along the~$S^1$ fiber.  \nThe dimensionally reduced theory describes a three-dimensional dynamical gauge \nsupermultiplet coupled to background three-dimensional supergravity on~$S^3$. \nThe Wilsonian effective action of the gauge multiplet can be calculated by integrating \nout the tower of massive Kaluza-Klein modes, and the resulting theory is described by a \nfunctional integral over the gauge multiplet fields with this effective action. \nWe find that the functional integral of the three-dimensional theory \ncan be written as a perturbative expansion in~$\\t$. \nThe singular terms in the expansion behave as~$\\text{O}(1\/\\t^2)$ and~$\\text{O}(1\/\\t)$, and are captured \nby three-dimensional effective field theory. \nIn particular, these terms are independent of the dynamical fields, and are completely \naccounted by the (supersymmetrized) Chern-Simons couplings of the background supergravity. \nThe result agrees with the corresponding singular terms in the microscopic \nexpansion~\\eqref{eq:almostThere},~\\eqref{eq:Zback}.\n\nThe all-order asymptotic formula from the microscopic index includes, in addition to these singular terms, \nconstant and linear terms in~$\\tau$. Using a localization argument we show that \nthe constant term in~$\\t$, besides a background part, has a dynamical piece captured by the \nintegral over the fluctuations of the dynamical fields in three-dimensional path integral, which is \nessentially the partition function of~$\\mathcal{N}=2$ supersymmetric CS theory at level~$\\pm N$. \nFinally, the linear term in the microscopic formula is precisely the supersymmetric Casimir energy \nwhich is needed to translate between the microscopic Hamiltonian index and the macroscopic \nfunctional integral.\\footnote{The supersymmetric Casimir energy that appears in our asymptotic \nformulas is the one given in~\\cite{Assel:2015nca}. Note in particular that (unlike \nin~\\cite{Cabo-Bizet:2018ehj}) this is independent of~$n_0$. \nWe can understand this in the path-integral picture by appealing to the result in Section~4 of~\\cite{ArabiArdehali:2019tdm} \n(based on the regularization method of \\cite{Ardehali:2015hya}) which demonstrated that the \nsupersymmetric Casimir energy is independent of flavor fugacities when they are on the unit circle, \nand by noting that~${\\rm e}^{2\\pi \\i(-n_0 r_I\/2)}$ is effectively a flavor fugacity in our problem.}\nIn this manner the full asymptotic formula for the four-dimensional index is explained \nby three-dimensional physics. \nThe fact that the asymptotic formula does not contain any higher order terms in~$\\t$ \nimplies a non-renormalization theorem, namely \nthat there are no corrections to the three-dimensional effective action at any polynomial order in~$\\t$. \nWe leave the explanation of this interesting point to future work.  \nFinally, we show that corresponding statements also hold near rational points \nwhen~$\\t \\to -n\/m$. Here we present evidence that the relevant three-dimensional manifold is a~$\\mathbb{Z}_m$ orbifold of~$S^3$ and the results agree with the microscopic asymptotic expansion given \nin~\\eqref{eq:N=4indexAsyRational}.\n\n\\vskip 0.4cm\n\n\nWe begin by recalling the functional integral definition of the~$\\mathcal{N}=1$ superconformal index on~$S^3 \\times S^1$. \nIn the Hamiltonian trace definition~\\eqref{defindex} we have two chemical potentials that couple to linear \ncombinations of the two angular momenta~$J_1, J_2$ on~$S^3$ and the~$U(1)$ R-charge~$Q$. \nThis is equal to the supersymmetric functional integral of the theory on~$S^3 \\times S^1$ with twisted boundary \nconditions on the fields as we go around the~$S^1$.\nEquivalently, one can explicitly introduce a background gauge field (for the R charge) and background \noff-diagonal terms in the metric (for the angular momenta) in a manner, so as to preserve supersymmetry.  \nAs explained in \\cite{Festuccia:2011ws}, such background configurations can be obtained as solutions to \nthe condition of vanishing gravitino variations of off-shell supergravity (and then taking a rigid limit so as to decouple the \nfluctuations of gravity). \n\n\nThe relevant background configuration for the calculation of the 4d superconformal index for \ncomplex~$\\tau$ and nonzero $n_0$ was studied\nin~\\cite{Cabo-Bizet:2018ehj} in the context of 4d new minimal supergravity~\\cite{Sohnius:1981tp, Sohnius:1982fw}.\nRecall that the bosonic fields of new minimal supergravity are the metric, a gauge field~$A^\\text{nm}$, \nand another vector field~$V^\\text{nm}$ which is covariantly conserved. \nThe background configuration~\\cite{Cabo-Bizet:2018ehj} preserving the supercharges~$(\\mathcal Q, \\overline{\\mathcal{Q}})$ \nis\\footnote{A real metric corresponds to pure imaginary~$\\Omega_i$. \nGeneral complex $\\Omega_i$ correspond to analytic continuation in the background metric.} \n\\be \\label{4dbackgnd}\n\\begin{split}\n\\mathrm{d} s_4^2 & \\=  \\mathrm{d} t_E^2 + \\mathrm{d} \\theta^2 + \\sin^2 \\theta \\, \n\\bigl(\\mathrm{d} \\phi_1 -\\i \\,\\Omega_1 \\, \\mathrm{d} t_E \\bigr)^2 + \\cos^2 \\theta \\, \\bigl(\\mathrm{d} \\phi_2 -\\i \\, \\Omega_2 \\, \\mathrm{d} t_E \\bigr)^2  \\,,\\\\\nA^\\text{nm} & \\= \\i\\, \\Bigl(\\Phi - \\frac{3}{2} \\Bigl) \\mathrm{d} t_E \\,, \\qquad V^\\text{nm}  \\=  -\\i \\,  \\mathrm{d} t_E \\, .\n\\end{split}\n\\ee\nHere~$\\theta \\in [0, \\pi\/2]$, the angles~$\\phi_1$, $\\phi_2$ are $2\\pi$-periodic, and \nthe Euclidean time coordinate has the independent periodicity \ncondition\\footnote{In~\\cite{Cabo-Bizet:2018ehj} the parameter~$\\gamma$ was called~$\\b$.} \n\\be\nt_E \\sim t_E + \\gamma  \\,.\n\\ee\nThis configuration admits the following Killing spinor which is identified with~$\\mathcal Q$, \n\\be \\label{KSsol}\n\\varepsilon \\=\n\\left(\n\\begin{array}{c}\n{\\rm e}^{\\i \\, z \\,t_E } \\\\\n0 \\\\\n0 \\\\\n{\\rm e}^{- \\i \\, z \\, t_E } \\\\\n\\end{array}\n\\right) \\,, \\qquad z \\= \\frac{\\pi n_0}{\\gamma}  \\,.\n\\ee\n\n\nThe twist parameters~$\\Omega_i$, $\\Phi$ are related to the chemical potentials~$\\sigma$,~$\\t$ in the index as \nfollows\\footnote{Here~$(\\Omega_1^*,\\Omega_2^*, \\Phi^*) = (1,1,\\frac32)$ are the values of the potentials\non the supersymmetric BH solution.}\n\\be\n\\Omega_i \\= 1 + \\frac{\\omega_i}{\\gamma} \\,, \\qquad \\Phi \\= \\frac32 +  \n\\frac{1}{\\gamma} \\Bigl(\\frac{\\omega_1+\\omega_2}{2} - \\pi \\i \\, n_0 \\Bigr) \\,,\n\\label{Omomrel}\n\\ee \nwith \n\\be\n\\omega_1 \\= 2 \\pi \\i \\,\\sigma \\,, \\qquad \\omega_2 \\= 2 \\pi \\i \\, \\t \\,.\n\\ee\nIn this section for ease of presentation we focus on the case with~$\\Omega_1=\\Omega_2=\\Omega,$ \nwhich implies~$\\sigma=\\tau=\\frac{\\i\\gamma}{2\\pi}(1-\\Omega)$. \nThe partition function on the above background is related to the index~$\\mathcal I(\\sigma-n_0,\\tau)$, \nwhich for $\\sigma=\\tau$ coincides with the index~$\\mathcal I(\\tau;n_0)$ in~\\eqref{eq:n0Index}. \nIn Appendix~\\ref{sec:diffomegas} we comment on the more general case \nwith~$\\O_1\\neq\\O_2$ and hence $\\sigma\\neq\\tau$.\n\n\nThe four-dimensional supersymmetric partition function of the theory corresponding to the Hamiltonian \nindex~\\eqref{defindex} can then be expressed as a functional integral \nof the gauge theory with 4d~$\\mathcal{N}=1$ chiral and vector multiplets on the  \nbackground~\\eqref{4dbackgnd}.\\footnote{More precisely the Hamiltonian index equals the functional \nintegral for the supersymmetric partition function up to the supersymmetric Casimir energy \nfactor~\\cite{Ardehali:2015hya,Assel:2015nca}.}\nAs discussed in~\\cite{Cabo-Bizet:2018ehj}, this functional integral \nlocalizes to an integral over flat connections of the gauge field on the KK circle, \n\\be \\label{AYval}\n\\oint A^i \\= 2 \\pi \\, u_i \\,.\n\\ee\nThe Wilson loop~\\eqref{AYval} maps to the scalar in the three-dimensional vector multiplet \nin the KK reduction. \nWe now proceed to derive an expression for the supersymmetric partition function of \nthe three-dimensional gauge theory. \n\n\n\n\\subsection{Dimensional reduction to three dimensions \\label{sec:dimred}}\n\nWe first consider the reduction of the above four-dimensional background as a \nconfiguration in three-dimensional supergravity. \nIn three dimensions we use the off-shell supergravity \nformalism~\\cite{Kuzenko:2011rd, Kuzenko:2011xg,  Kuzenko:2012bc, Kuzenko:2013uya}, \nand follow the treatment~\\cite{Closset:2012vp, Closset:2012vg, Closset:2012ru, Assel:2014paa} \nfor the reduction from four to three dimensions.  The bosonic fields in the off-shell three-dimensional \nsupergravity are the metric, the KK gauge field (the graviphoton) written as a one-form~$c$, a \ntwo-form~$B$, and the R-symmetry gauge field one-form~$\\mathcal{A}^R$. \nThe equations are often presented in terms of the dual one-form~$v=-\\i *\\mathrm{d} c$\nand the dual scalar~$H = \\i * \\mathrm{d} B$. \n\n\nWe begin by writing the background in~\\eqref{4dbackgnd} as a Kaluza-Klein (KK) compactification \nto three dimensions, i.e.~a circle fibration on a 3-manifold~$\\mathcal{M}_3$.\nWe define the rescaled~$S^1$ coordinate \n\\be \\label{X4period}\nY \\= \\sqrt{1 - \\Omega^2} \\; t_E \\,,\n\\ee\nwhich obeys the periodicity condition \n\\be\nY \\sim Y + 2 \\pi R\\,, \\qquad  R \\= \\frac{\\gamma}{2 \\pi}  \\sqrt{1-\\O^2} \\,.\n\\label{Rgamrel}\n\\ee\n\nWriting the metric~\\eqref{4dbackgnd} in the KK form,\n\\be  \\label{4dmetricKKform}\n\\mathrm{d}  s_4^2  \\=  \\mathrm{d} s_3^2 + (\\mathrm{d} Y + c)^2 \\,,\n\\ee \nwe find that the graviphoton field is \n\\be \\label{3dKKc}\nc \\= c_\\mu \\, \\mathrm{d} x^\\mu   \n \\= -\\i \\frac{\\O}{\\sqrt{1-\\O^2}} \\, \\bigl( \\sin^2 \\theta \\,\\mathrm{d} \\phi_1  + \\cos^2 \\theta \\,\\mathrm{d} \\phi_2 \\bigr) \\,,\n\\ee\nand the metric on the 3-manifold~$\\mathcal{M}_3$ is \n\\be \\label{3dmetric1}\n\\mathrm{d} s_3^2  \\= g_{\\mu\\nu} \\, \\mathrm{d} x^\\mu \\, \\mathrm{d} x^\\nu \n\\=  \\mathrm{d} \\theta^2 + \\sin^2 \\theta \\,  \\mathrm{d} \\phi_1^2 + \\cos^2 \\theta \\, \\mathrm{d} \\phi_2^2  - c^2 \\,.\n\\ee\nThe three-dimensional metric obeys \n\\be \\label{sqrtg}\n\\sqrt{g} \\= \\frac{\\sin 2 \\theta}{2 \\sqrt{1-\\O^2}} \\,.\n\\ee\nWe see that we effectively have a KK reduction on a circle of radius~$R$. \n\n\\vskip 0.4cm\n\nIn order to study the effective theory in three dimensions, we consider the limit~$R \\to 0$.\nFrom the relation~\\eqref{Rgamrel} we see that this is implemented by taking the original circle size~$\\gamma \\to 0$. \nOur eventual interest is in the limit~$\\t \\to 0$. The question is how to correlate these two limits of~$\\gamma$ and~$\\t$.  \nIf we take~$\\gamma \\to 0$ first, then we see from the relation~\\eqref{Omomrel} that~$\\Omega \\to \\infty$ \nand from~\\eqref{sqrtg} that~$\\mathcal{M}_3$ shrinks to zero size. \nAlthough the local Lagrangian involves background fields and terms such as the Ricci scalar which diverge in this limit, \nthe three-dimensional effective action turns out to be finite.  \nWe can understand this in a cleaner manner as follows. \nWe first scale~$\\t$ and~$\\gamma$ to zero at the same rate keeping~$\\O$ finite and fixed, \ni.e.~take~$\\gamma = \\varepsilon \\t$ with fixed~$\\varepsilon = 2 \\pi \\i \/(\\O -1)$, and only take~$\\varepsilon\\to0$ at the end of all calculations. \nIn particular, the three-dimensional calculations\nare all performed at finite~$\\varepsilon$, i.e.~on smooth backgrounds. \nThe action turns out to have two pieces, one of which stays finite and the other vanishing in the limit~$\\varepsilon \\to 0$,\nand, in particular, there are no diverging terms in this limit. \nThus we can safely take the limit~$\\O \\to \\infty$ at the end of calculations. \nIn this limit we have that~$R \\to \\t$, so that the effective field theory answers are effectively written as \na perturbative series in~$\\t$. \n\n\n\n\\vskip 0.4cm\n\nIn the treatment of three-dimensional background supergravity we need the Hodge dual of the graviphoton,\n\\be \\label{defv}\nv \\= v_\\mu \\, \\mathrm{d} x^\\mu \\= - \\i * \\mathrm{d} c \\,,\n\\ee\nwhose value in the above background is\n\\be\nv \\= \\frac{2 \\, \\O}{1-\\O^2} \\, \\bigl( \\sin^2 \\theta \\,\\mathrm{d} \\phi_1  + \\cos^2 \\theta \\,\\mathrm{d} \\phi_2 \\bigr) \\,,\n\\ee\nso that~$v^\\mu = 2 \\,\\O(1,1,0)$. \nThe associated Chern-Simons action is \n\\be\nS^\\text{CS} (c)  \\= \\int_{\\mathcal{M}_3} \\, c \\wedge \\mathrm{d} c  \\= \\i \\int_{\\mathcal{M}_3} \\mathrm{d}^3 x \\,  \\sqrt{g} \\, v^\\mu \\, c_\\mu\\,. \n\\ee\n\n\n\\vskip 0.4cm\n\n\nThe identification between the four-dimensional and the three-dimensional gauge fields \nis made by comparing the respective Killing spinor equations. \nAs shown in~\\cite{Closset:2012ru, Assel:2014paa}, one has \n\\be\n\\frac12 v_\\mu \\= V^\\text{nm}_\\mu - V^\\text{nm}_Y c_\\mu \\,, \\qquad H \\= V^\\text{nm}_Y \\,,\n\\qquad \\mathcal{A}^R_\\mu \\= A^\\text{nm}_\\mu - A^\\text{nm}_Y c_\\mu  + \\frac12 v_\\mu \\,.\n\\ee\nThe background gauge fields in~\\eqref{4dbackgnd} are given by\n\\be \\label{AYVYval}\nA^\\text{nm}\n \\=  \\Bigl( -\\frac{\\t}{R}+ \\frac{n_0}{2R} \\Bigr) \\, \\mathrm{d} Y \\,, \n\\qquad V^\\text{nm}  \\=  -\\frac{\\i}{\\sqrt{1-\\O^2}} \\,  \\mathrm{d} Y \\,,\n\\ee \nso that the auxiliary fields in the background supergravity multiplet are \n\\be \\label{vHvalues}\nv_\\mu \\=  - 2 \\, V^\\text{nm}_Y \\, c_\\mu  \\,, \\qquad  \nH \\= V^\\text{nm}_Y \\,, \\qquad \n\\mathcal{A}^R_\\mu \\= - (A^\\text{nm}_Y + V^\\text{nm}_Y) \\, c_\\mu \\,.\n\\ee\n(The above equation for~$v_\\mu$ is consistent with Equations~\\eqref{3dKKc},~\\eqref{defv}.)\n\n\n\\vskip 0.4cm\n\n\nWe now discuss the Kaluza-Klein reduction of the dynamical gauge multiplet. \nThe~$\\mathcal{N}=1$ gauge multiplet in four dimensions reduces to an~$\\mathcal{N}=2$\ngauge multiplet in three dimensions, whose bosonic field content is a vector~$\\mathcal{A}_\\mu$, \na scalar~$\\sigma$, and  the auxiliary~$\\mathcal{D}$ field. \nThese are related to the four-dimensional fields as follows,\n\\be\\label{eq:susy3dvec}\n\\sigma^i \\= A^i_Y \\,, \\qquad \\mathcal{A}^i_\\mu \\= A^i_\\mu -  A^i_Y \\, c_\\mu \\,, \\qquad \\mathcal{D}^i \\= D^i -  A^i_Y \\, H \\,,\n\\ee\nand the three-dimensional fermions are the reduction of the corresponding four-dimensional fermions.\nAs discussed above, the theory localizes on the BPS configurations given by \n\\be \nA^i \\= \\frac{u_i}{R}  \\, dY \\,, \\qquad D^i \\= 0 \\,, \n\\label{AYval}\n\\ee\nwith vanishing values of all other fields in the off-shell gauge and chiral multiplets. \nIn the three-dimensional theory the non-zero fields on the BPS locus are \n\\be\n\\sigma^i \\= \\frac{u_i}{R} \\,, \\qquad \n\\mathcal{A}^i_\\mu \\=   - \\frac{u_i}{R} \\, c_\\mu \\,, \\qquad \\mathcal{D}^i \\=  -  \\frac{u_i}{R}  \\, H \\,.\n\\label{AYval3d}\n\\ee\n\n\n\n\\subsection{Effective action and functional integral of the three-dimensional theory \\label{sec:3deffact}}\n\n\nWe now turn to the calculation of the partition function of the three-dimensional supersymmetric theory\nthat we just discussed. \nOur strategy is to first calculate the three-dimensional Wilsonian effective action of~$u_i$, and then use \nthis to calculate the three-dimensional partition function. \nThe tree-level action (coming from a mode expansion of the four-dimensional theory) \nconsists of matter-coupled super Yang-Mills theory.\nThe full quantum effective action of the three-dimensional theory is obtained by integrating out the \ntower of massive KK modes on the circle. In order to calculate this action, we draw from known results \nin the effective field theory in three dimensions. \n\nThe effective field theory on backgrounds of the type~$\\mathcal{M}_3 \\times S^1_R$ was studied in a  \ngeneral context in~\\cite{Banerjee:2012iz,Jensen:2012jh},\nand in the special context of supersymmetry in~\\cite{DiPietro:2014bca, DiPietro:2016ond}.  \nThe resulting three-dimensional action begins with a term proportional to~$1\/R^2$,\nand continues as a perturbation expansion as the radius~$R \\to 0$. \nAt each order in~$R$ one has \na combination of three-dimensional actions of the background and the dynamical fields, \nwhich are all related by supersymmetry to a certain Chern-Simons term.\nThe Chern-Simons terms are of the form~$\\int_{\\mathcal{M}_3} A_x \\wedge \\mathrm{d}A_y$, where~$A_x$\nand~$A_y$ represent the various gauge fields. As discussed in the previous subsection, \nthese are the dynamical gauge field, the background graviphoton, the background R-gauge field, \nand the spin connection. We follow, and review in Appendix~\\ref{sec:CSactions}, \nthe treatment of~\\cite{DiPietro:2016ond} for the supersymmetrized  Chern-Simons action of all \nthe background and the dynamical gauge fields up to~$O(R^0)$. The full effective action also \nincludes RR and gravitational supersymmetrized CS terms discussed in \\cite{Closset:2018ghr}, \nwhich turn out to be crucial for our purposes.\n\nIt follows from the above discussion that the overall coefficient at each order in~$R$ \ncan be fixed by calculating the coefficient of the Chern-Simons terms themselves. \nThese coefficients, in turn, can be obtained by integrating out all the fermions coupling \nto the corresponding gauge fields. The resulting induced Chern-Simons coefficient is one-loop exact. \nThus the strategy is to integrate out the fermions in each KK mode, write the \nresulting Chern-Simons action, and sum over all the fermions in the theory. \nThe KK momenta of the fermions take the values ~$p_Y = k_Y\/R$, with $k_Y =n + \\frac{n_0}{2}$, $n \\in \\mathbb{Z}$.\nThe shift~$n_0\/2$ appears because of the gauge fields in the background~\\eqref{4dbackgnd}. \n(Recall, for example, that the four-dimensional Killing spinor~\\eqref{KSsol} has momentum~$n_0\/2$.) \n\n\nThe result for the complete action obtained by integrating out a fermion~$\\tf$ of \nR-charge~$r_\\tf$ and transforming in a representation of weight~$\\rho_\\tf$ under the gauge group \nis given in Appendix~\\ref{sec:CSactions} and take the following form,\n\\be\n\\delta S^\\tf_\\text{1-loop} \\= \\widetilde{S}^\\tf_\\text{g-g} + 2 \\, \\widetilde{S}^\\tf_\\text{g-R} + S^\\tf_\\text{R-R} + S^\\tf_\\text{grav} \\,.\n\\label{Sftot1}\n\\ee\nThe terms in~\\eqref{Sftot1} depend on the real mass~$m_\\tf$ (related to the central charge appearing \nin the three-dimensional algebra). \nThe first two terms depend on the dynamical gauge field. \nOn the configuration~\\eqref{AYval3d} they take the following values, \n\\be\n\\begin{split}\n\\widetilde{S}^\\tf_\\text{g-g} & \\=-\\i \\pi \\, \\frac{ \\mathrm{sgn}(m_\\tf)}{8R^2} \\,  \n\\bigl(\\rho_{\\tf}\\cdot \\underline{u} - k_Y \\bigr)^2\\, A_{\\mathcal{M}_3} \\,, \\\\\n2 \\, \\widetilde{S}^\\tf_\\text{g-R} & \\=-\\i \\pi \\,  \\frac{\\mathrm{sgn}(m_\\tf)}{8R} \\, \n2 \\, r_\\tf  \\, \\bigl(\\rho_{\\tf}\\cdot \\underline{u} - k_Y \\bigr) \\; L_{\\mathcal{M}_3} \\,,\n\\end{split}\n\\label{S1S2fin}\n\\ee\nwhere~$A_{\\mathcal{M}_3}$ and $L_{\\mathcal{M}_3}$ are functions\nof the three-dimensional background given in~\\eqref{defALl}.\nThe last two terms in~\\eqref{Sftot1} do not depend on the dynamical gauge field,\nand given by \n\\be\n\\begin{split}\n    S^\\tf_\\text{R-R}&\\=-\\i \\pi \\, \\frac{\\mathrm{sgn}(m_\\tf)}{8} \\, \\bigl(r_\\tf^2-\\frac{1}{6}\\bigr) \\, R_{\\mathcal{M}_3}  \\,, \\\\\n  S^\\tf_\\text{grav}&\\=-\\i \\pi \\, \\frac{\\mathrm{sgn}(m_\\tf)}{192} \\, G_{\\mathcal{M}_3} \\,, \n\\label{S:susy:explicit3}%\n\\end{split}\n\\ee\nwhere~$R_{\\mathcal{M}_3}$ and~$G_{\\mathcal{M}_3}$ are functions\nof the three-dimensional background given in~\\eqref{eq:CSactions2}.\n\n\n\nIn Appendix~\\ref{sec:CSactionvals} we calculate the values of these background  \nactions.\\footnote{We note that there is a subtlety with the gravitational CS term in~\\eqref{eq:CSactions2},\nconcerning the dependence of the term on the frame~\\cite{Witten:1988hf}. \nThere should be a choice of frame which is consistent with the supersymmetry and the 4d to 3d reduction.\nWe do not work out the details of this issue in this paper, and instead rely on consistency \nwith~\\cite{Closset:2018ghr} where this term is obtained indirectly by considering integrating out chiral multiplets.\nWe thank Cyril Closset for a discussion on this point.}\nAs explained above, we perform the calculations keeping~$R$, $\\O$ finite so that \nthe three-dimensional physics is manifestly smooth. The result is that there is a smooth \nlimit as~$\\gamma \\to 0$ keeping fixed~$\\t$. \nThe limiting values of the actions are as follows,\n\\be\n\\begin{split}\n&A_{\\mathcal{M}_3}  \\= - 4 \\,, \\qquad \n L_{\\mathcal{M}_3} \\= - 4  \\Bigl( 1 - \\frac{n_0}{2R} \\Bigr) \\,, \\\\ \n& R_{\\mathcal{M}_3}  \\= - 4 \\, \\Bigl( 1 -\\frac{n_0}{2R} \\Bigr)^2 \\,, \\qquad \nG_{\\mathcal{M}_3} \\= - 16 + 4 \\, R_{\\mathcal{M}_3} \\,.\n\\end{split}\n\\label{ALvals}\n\\ee\nUsing these values, we obtain the total effective action of the fermion~$\\tf$  to be\n\\be\n\\begin{split}\n\\delta S^\\tf_\\text{1-loop} \n& \\= \\i \\pi \\,  \\frac{\\mathrm{sgn}(m_\\tf)}{2R^2}  \\, \n\\bigl(\\rho_{\\tf}\\cdot \\underline{u} - k_Y - \\tfrac{1}{2} n_0 \\, r_\\tf \\bigr)^2 \n+ \\i \\pi \\,  \\frac{\\mathrm{sgn}(m_\\tf)}{R}  \\, r_\\tf \\, \\bigl(\\rho_{\\tf}\\cdot \\underline{u} - k_Y - \\tfrac{1}{2} n_0 \\, r_\\tf \\bigr) \\\\\n& \\qquad  + \\i \\pi \\,  \\frac{\\mathrm{sgn}(m_\\tf)}{2}  \\, r_\\tf^2 \n\\; - \\; \\i \\pi \\,  \\frac{\\mathrm{sgn}(m_\\tf)}{12} \\,.\n\\end{split}\n\\label{defSf}\n\\ee\n\n\\vskip 0.4cm\n\nNow we turn to the sum over all the fermions in the theory. \nThe value of the real mass is given in~\\eqref{realmass} to be, \nas~$R \\to 0$,\\footnote{In fact the first three terms in~\\eqref{defSf} sum up to\n\\be \\i \\frac{\\pi}{2} \\, \\mathrm{sgn}(m_\\tf)  \n\\Bigl( \\frac{ \\rho_{\\tf}\\cdot \\underline{u} - n - \\tfrac{1}{2} n_0 \\, r_I  + (r_I-1) R }{R}  \\Bigr)^2 \\,,\n\\ee\nand, using~\\eqref{eq:KKsumBbar} with~$x=  \\rho_{\\tf}\\cdot \\underline{u} - n - \\tfrac{1}{2} n_0 \\, r_I  + (r_I-1) R$ \nto perform the sum over the KK modes, we obtain an effective potential which \nreproduces the chiral multiplet contributions in~\\eqref{Veffans3d}. \nEssentially the same comment can be made in the microscopic analysis of Section~\\ref{sec:SCI}.\n} \n\\be\nm_{\\tf,n} \\= -\\frac{1}{R} \\bigl(\\rho_\\tf\\cdot \\underline{u} - n - \\tfrac12 n_0  \\, (r_\\tf+1)   \\bigr) \\,,\n\\ee\nIn order to obtain the full effective action we now have to sum over all the fermions.\nFor the chiral multiplets, this implies summing over all the weights in \nrepresentations~$\\rho_\\tf\\in\\mathcal{R}_\\tf$, as well as over all momenta labelled by $n\\in\\mathbb{Z}$. \nThe summation over KK modes can be evaluated using\n\\begin{equation}\n    \\sum_{n\\in\\mathbb{Z}}\\mathrm{sgn}(n+x)(n+x)^{j-1} \\= -\\frac{2}{j} \\, \\overline{B}_{j}(x) \\,,\n\\label{eq:KKsumBbar}\n\\end{equation}\nwith~$x=\\rho_\\tf\\cdot \\underline{u} - \\tfrac12  n_0  \\, r_I $, for $j=1,2,3$ (cf.~Section~4 \nof~\\cite{ArabiArdehali:2019tdm}).\\footnote{Here, the~$\\mbox{sgn}$ function is interpreted as applying \nto~$m_\\tf$ with~$R$ a real positive number (which therefore scales out of the formula so \nas to give~$\\mbox{sgn}(\\rho_\\tf\\cdot \\underline{u} - \\tfrac12  n_0  \\, r_I)$). \nNote that in the subsequent formulas $R$ is taken to be complex.}\nHere we have used the relation~$r_\\tf = r_I-1$ between the R-charge of the fermion and that of the \nbottom component of the multiplet~$I$ to which the fermion belongs. \n\n\\vskip 0.4cm\n\nFor the vector multiplet contribution the analysis is quite similar: there is a tower of massive KK gaugino \nmodes that are integrated out. These generate CS actions whose supersymmetrization yields the \nvector multiplet contribution to $\\delta S_{\\text{1-loop}}$. In the present context there is an important difference \nwith the chiral multiplet analysis however. Near $\\underline{u}=0$ there is a single gaugino mode in the tower that \nhas real mass of order $\\alpha\\cdot\\underline{u}\/R$, and is therefore considered a ``light'' mode \nfor small enough~$|\\alpha\\cdot\\underline{u}|$. \nTherefore we do not integrate out this mode and, instead, keep it as a dynamical mode in \nthe path integral of the three-dimensional theory.\n\nMore precisely, recall that the $n$th KK gaugino mode associated to a root~$\\alpha$ of \nthe gauge group has~$p_Y=(n+n_0)\/R$ and hence a real mass~$(\\alpha\\cdot\\underline{u} -n-n_0)\/R$. \nTherefore the mode corresponding to~$n=-n_0$ is light near~$\\alpha\\cdot\\underline{u}=0$. \nWe now describe how removing this term from the sum over the KK tower modifies the result \ncompared to the chiral multiplet computation. \nThe vector multiplet contributions is a sum over roots~$\\alpha$ that come in pairs~$\\pm\\alpha_+$, \nas a result of which they give vanishing contributions to the quadratic and constant terms in~$\\underline{u}$\nin the action of a single KK mode.\nWe therefore focus on the contribution to the linear term in~$\\underline{u} \\,,$ which is proportional to~$1\/R$. \nThe calculation is similar to the corresponding chiral multiplet calculation.  \nUpon summing over all the KK modes, we obtain \nthe vector multiplet contribution \nfrom a root~$\\alpha$ to be \\\\\n\\be\n-\\frac{\\pi \\i}{R} \\, \\sum_{n\\in\\mathbb{Z}}{}^{'}\\mathrm{sgn} \\, (\\alpha\\cdot\\underline{u} -n-n_0)\\, \n\\bigl(\\alpha\\cdot\\underline{u} -n-n_0\\bigr)\\label{eq:sumPrimeVec} \\,,\n\\ee\nwhere the prime indicates that we are not including the light mode corresponding to $n=-n_0$. \nUpon adding and subtracting the $n=-n_0$ contribution, we obtain, using~\\eqref{eq:KKsumBbar},  \n\\be\n\\frac{\\i\\pi}{R} \\, \\Bigl( \\overline{B}_2(\\alpha\\cdot\\underline{u})+ |\\alpha\\cdot\\underline{u}|  \\Bigr)\\,.\n\\ee\nNow, since we are interested in the proximity of~$\\underline{u}=0$, we use the fact that for~$|x|<1$ \nwe have~$\\overline{B}_2(x)=x^2-|x|+\\frac{1}{6}$, to simplify the result to\n\\be\n\\frac{\\i\\pi}{R} \\, \\Bigl((\\alpha \\cdot \\underline{u})^2  + \\frac{1}{6}\\Bigr) \\,.\n\\ee\n\n\nUpon putting all the pieces together, we obtain the total one-loop correction to the \nWilsonian action of the three-dimensional theory, \nwhich we call~$V_\\text{eff}(\\underline{u})$ (we justify this name below). We have\n\\be\n\\begin{split}\n  V_\\text{eff}(\\underline{u}) \n \\= & \\sum_\\tf\\sum_{\\rho_\\tf\\in\\mathcal{R}_\\tf} \\delta S^\\tf_\\text{1-loop} \\, \\\\\n     \\= & \\i\\pi  \\sum_{I, \\, \\rho_I} \\,\n    \\Bigl(\\frac{1}{3R^2} \\, \\overline{B}_3 \\bigl(\\rho_\\tf\\cdot \\underline{u} - \\tfrac{1}{2} n_0 \\, r_I \\bigr)\n    + \\frac{r_I-1}{R} \\, \\overline{B}_2 \\bigl(\\rho_\\tf\\cdot \\underline{u} -  \\tfrac{1}{2} n_0 \\, r_I \\bigr) \\\\\n& \\qquad\\qquad\\qquad +  \\frac{1}{R}  \\sum_{\\alpha}  \\, \\bigl((\\alpha \\cdot \\underline{u})^2  + \\tfrac{1}{6} \\bigr) \n    +   \\bigl( (r_I-1)^2 \\, - \\tfrac{1}{6} \\bigr) \\overline{B}_1 \n    \\bigl(\\rho_\\tf\\cdot \\underline{u} -  \\tfrac{1}{2} n_0 \\, r_I \\bigr) \\Bigr) \\,.\n\\end{split}\n\\label{Veffans3d}\n\\ee\n\nWe now localize the path integral of the light gauge multiplet mode that was excluded from \nthe sum~\\eqref{eq:sumPrimeVec}, using its Wilsonian effective action, which consists of the \ntree-level action coming from the light~$n=-n_0$ mode in 4d, as well as the one-loop \naction~$\\delta S_{\\text{1-loop}}$ derived above (in the bosonic sector, which is relevant \nfor the localization calculation) from integrating out the heavy modes.\nIt is useful to keep in mind the different but related problem of calculating the partition function of \nsuperconformal CS theory coupled to matter on~$\\mathcal{M}_3$ \\cite{Kapustin:2009kz}, \\cite{Willett:2016adv}. \nIn that case the theory localizes onto \narbitrary constant values of the scalar~$\\sigma$ and is supported by the auxiliary scalar~$H$. \nThe measure including the one-loop determinant of the localizing action in the non-BPS \ndirections is\\footnote{Compare with Section~5 of~\\cite{Aharony:2013dha}, noting that for \nsquashed~$S^3$ with squashing parameter~$b$ one has~$\\omega_{1}^{\\text{thf}}=\\i b$, \n$\\omega_{2}^{\\text{thf}}=\\i b^{-1}$. We leave the derivation of~\\eqref{eq:thfModuli} from the \nmetric~\\eqref{3dmetric1} to future work.} \n\\be\n\\int\n\\frac{D \\underline{\\sigma}}{\\sqrt{- \\omega_1^{\\text{thf}}\\, \\omega_2^{\\text{thf}}}}\\, \n\\prod_{\\alpha_+}4\\sinh\\bigl( \\frac{\\pi \\alpha_+ \\cdot \\underline{\\sigma}}{-\\i\\, \\omega_1^{\\text{thf}}} \\bigr)\n\\sinh\\bigl( \\frac{\\pi \\alpha_+ \\cdot \\underline{\\sigma}}{-\\i\\,  \\omega_2^{\\text{thf}}} \\bigr) \\,,\n\\label{1loopM3}\n\\ee\nwith~$\\omega_{1,2}^{\\text{thf}}$ the moduli of the transversely holomorphic foliation \n(THF)~\\cite{Closset:2013vra} of~$\\mathcal{M}_3$, which we expect to be\n\\be\n \\omega_{1}^{\\text{thf}}\\=\\omega_{2}^{\\text{thf}}\\=\\i\\sqrt{\\frac{1-\\Omega}{1+\\Omega}}.\\label{eq:thfModuli}\n\\ee\n\n\nRecalling from~\\eqref{AYval3d} that~$\\sigma^i \\= u_i\/R$, and adding the contribution \nfrom~$\\delta S_{\\text{1-loop}}$ in~(\\ref{Veffans3d}) (which although arises at one-loop \nin high-temperature EFT, contributes as a ``classical'' piece in the localization computation), \nwe obtain the final result for the \nthree-dimensional partition function \n\\be\nZ(\\t) \\= \\int \\frac{D\\underline{u}}{(-\\i\\, \\tau)^{\\mathrm{rk}(G)}}\\, \n\\prod_{\\alpha_+}4\\sinh^2\\Bigl(\\frac{\\pi \\alpha_+ \\cdot \\underline{u}}{-\\i\\, \\tau}\\Bigr)  \\exp \\bigl( - V_\\text{eff}(\\underline{u}) \\bigr) \\,.\n\\label{Z3d}\n\\ee\nNoting that the supersymmetric partition function and the Hamiltonian\nindex are related as \\cite{Ardehali:2015hya,Assel:2015nca}\n\\be\nZ(\\t) \\= e^{2\\pi\\i\\tau E_\\text{susy}} \\, \\mathcal I (\\t) \\,,\n\\ee\nwe see that the result~\\eqref{Z3d} agrees precisely with the microscopic\nresult~\\eqref{eq:In0semi-simpleAsy0}--\\eqref{Esusy}.\n\nWe emphasize that while the above derivation of~$V_{\\text{eff}}$ in~\\eqref{Veffans3d} \napplies to~$\\underline{u}$ near 0, it can be easily extended to generic finite~$\\underline{u}$ by modifying the \nvector multiplet discussion. For generic~$\\underline{u}$, the non-Cartan components of the~$n=-n_0$ \nmode of the vector multiplet are also heavy, and ought to be integrated out. Consequently the \nsum in~\\eqref{eq:sumPrimeVec} would no longer have a prime, and we end up with~$V^r_1$ \nas in~\\eqref{eq:V1ren} rather than~$V_1$ in~\\eqref{Veffans3d}. This is the EFT derivation of \nthe finite-$\\underline{u}$ potentials~$V_{1,2}$ found microscopically in~\\cite{Cabo-Bizet:2019osg}.\n\nOn the other hand, when $n_0=0$, the small-$\\underline{u}$ discussion leading up to~\\eqref{Veffans3d} \nneeds to be modified because now the chiral multiplets have light modes (corresponding to $n=0$). \nAs in the discussion around~\\eqref{eq:sumPrimeVec} the light mode should be removed from the \nKK sum and instead be included in the dynamical part (to be localized). Indeed, it is well-known that \nfor~$n_0=0$ the constant piece of the small-$\\tau$ expansion coming from the~$\\underline{u}=0$ saddle \ncontains the (localized)~$S^3$ partition function of the dimensionally reduced chiral as well as \nvector multiplets~\\cite{Ardehali:2015bla} \n(see~\\cite{Dolan:2011rp,Spiridonov:2012ww,Niarchos:2012ah,Gadde:2011ia,Imamura:2011uw} \nfor earlier work on the connection between 4d indices and $S^3$ partition functions). \n\nFinally, as  discussed in Appendix~\\ref{sec:diffomegas}, we find that the \neffective potential for~$\\t$ and~$\\sigma$ not necessarily equal is given by \nmaking the replacement \n\\be\n\\frac{1}{R^2} \\to \\frac{1}{\\t \\, \\sigma} \\,, \\qquad \\frac{1}{R} \\to \\frac{\\t + \\sigma}{2\\t \\, \\sigma}\n\\ee\nin the effective potential~\\eqref{Veffans3d}. The singular pieces are indeed in agreement \nwith the microscopic calculations reported \nin~\\cite{Kim:2019yrz,Cabo-Bizet:2019osg}.\n\n\n\n\\subsection{Rational points}\n\nWe now turn our attention to the limit of~$\\t$ approaching a rational point. \nIn the discussion of the previous subsection we used the fact that the radius of the circle~$R$ \nequals~$\\t$ which becomes small in the limit, so that we could use an effective three-dimensional description. \nNow we are interested in~$\\widetilde \\t = m\\t +n \\to 0$, with~$n, m \\in \\mathbb{Z}$ (with no common factor) \nas in~\\cite{Cabo-Bizet:2019eaf}. \nIn terms of the variable~$\\widetilde \\t$ we have that~$\\omega = 2 \\pi \\i \\t = 2 \\pi \\i (\\widetilde \\t -n)\/m$ so that\n\\be\n\\O \\= 1+ \\frac{\\omega}{\\gamma} \\= 1- \\frac{2 \\pi \\i n}{m\\gamma} + \\frac{2 \\pi \\i \\widetilde \\t}{m\\gamma} \\,,\n\\ee\nand the four-dimensional metric background~\\eqref{4dbackgnd} is now\n\\be \\label{4dbackgndrat}\n\\begin{split}\n\\mathrm{d} s_4^2 \\= & \\mathrm{d} t_E^2 + \\mathrm{d} \\theta^2 \n+ \\sin^2 \\theta \\, \n\\Bigl(\\mathrm{d} \\phi_1 - \\frac{2 \\pi n}{m\\gamma} \\, \\mathrm{d} t_E -\\i \\, \\bigl(1 + \\frac{2 \\pi \\i \\widetilde \\t}{m\\gamma}\\bigr) \\, \n\\mathrm{d} t_E \\Bigr)^2 \\\\\n& \\qquad \\qquad \\qquad + \\cos^2 \\theta \\, \n\\Bigl(\\mathrm{d} \\phi_2 - \\frac{2 \\pi n}{m\\gamma} \\, \\mathrm{d} t_E -\\i \\, \\bigl(1+ \\frac{2 \\pi \\i \\widetilde \\t}{m\\gamma}\\bigr) \\, \n\\mathrm{d} t_E \\Bigr)^2  \\,.\n\\end{split}\n\\ee\nIn terms of the following new coordinates and new parameters,\n\\be\n\\widetilde \\gamma \\= m \\gamma \\,, \\qquad \n\\widetilde \\O \\= 1+ \\frac{2 \\pi \\i \\widetilde \\t}{\\widetilde \\gamma}  \\,,  \\qquad \n\\widetilde \\phi_i \\= \\phi_i - \\dfrac{2 \\pi n}{\\widetilde \\gamma} t_E \\,,\n\\ee\nthe above metric is  \n\\be \\label{4dbackgndrat1}\n\\mathrm{d} s_4^2 \\=  \\mathrm{d} t_E^{2} + \\mathrm{d} \\theta^2 \n+ \\sin^2 \\theta \\, \n\\bigl(\\mathrm{d} \\widetilde \\phi_1  -\\i \\, \\widetilde \\O  \\, \\mathrm{d} t_E \\bigr)^2 + \\cos^2 \\theta\\,  \\bigl(\\mathrm{d} \\widetilde \\phi_2  -\\i \\, \\widetilde \\O  \\,\\mathrm{d} t_E \\bigr)^2  \\,,\n\\ee\nwith~$\\widetilde \\phi_1$, $\\widetilde \\phi_2$ being $2\\pi$-periodic as before, \nand the periodic identification going around the time circle is \n\\be \\label{newident}\n\\bigl(t_E \\,, \\;  \\widetilde \\phi_1 \\,, \\;  \\widetilde \\phi_2 \\bigr) \\; \\sim \\;\n\\Bigl( t_E + \\frac{\\widetilde \\gamma}{m} \\,, \\;  \\widetilde \\phi_1 - \\frac{2 \\pi n}{m} \\,, \\;  \\widetilde \\phi_2 - \\frac{2 \\pi n}{m} \\Bigr) \\,.\n\\ee\nThe metric configuration~\\eqref{4dbackgndrat1} with the identifications~\\eqref{newident} is \nsimply a global identification, or orbifold, of the configuration considered in the previous subsection with \nthe new parameters~$(\\widetilde \\gamma, \\widetilde\\t, \\widetilde \\O)$ replacing $(\\gamma,\\t, \\O)$.~\\footnote{We learned about \nthese orbifolds from a talk by O.~Aharony at the Stony Brook seminar series in November \n2020~\\cite{AhaSBTalk}.}\n\n\nOn the covering space, \ngoing around the time circle shifts~$\\widetilde t_E \\to \\widetilde t_E + \\widetilde \\gamma$ and~$\\widetilde \\phi_i \\to \\widetilde \\phi_i + 2 \\pi n$.\nThe latter identification can be trivialized by using the independent~$2\\pi$-periodicity of~$\\widetilde \\phi_i$, so that \nwe have the identification~$\\bigl(\\,\\widetilde t_E \\,,  \\widetilde \\phi_1 \\,,  \\widetilde \\phi_2 \\, \\bigr) \\sim \n\\bigl(\\, \\widetilde t_E + \\widetilde \\gamma \\,,   \\widetilde \\phi_1  \\,,   \\widetilde \\phi_2 \\, \\bigr)$. \nOn this configuration we can perform the dimensional reduction to three dimensions.\nThe relevant considerations of the previous subsection go through exactly as before \nwith the replacement~$(\\gamma,\\t, \\O) \\mapsto (\\widetilde \\gamma, \\widetilde\\t, \\widetilde \\O)$. Actually, because the gauge \nholonomies on the cover wrap a circle~$m$ times larger than the original~$S^1$, we also get \na replacement~$u_j\\to mu_j$. Moreover, since~$\\xi_I$ (which equals~$-n_0r_I\/2$ for~$(m,n)=(1,0)$) \neffectively plays the role of a flavor chemical potential in our problem as mentioned around~\\eqref{eq:defXiGen}, \nwe expect a similar replacement $-n_0r_I\/2\\to m\\xi_I$ (we postpone a direct 3d explanation of this \nreplacement to future work). With the preceding substitutions in the results of the previous \nsubsection, we thus arrive at the potentials~$\\widetilde V_{2,1}$ in~\\eqref{eq:defVsRat}. \nWe then take the~$\\mathbb{Z}_m$ quotient which has two effects as usual. \nFirstly it reduces the volume of the three-dimensional space, and \nsecondly it introduces new topologically non-trivial sectors in the path integral over the \ngauge-field configurations.\nThe change in calculations involving local gauge-invariant Lagrangians will therefore \nbe only a reduction in the action by a factor of~$m$. \nThis explains the reduction of the effective potential by a factor of~$m$ as in~\\eqref{eq:In0semi-simpleAsyRat0}.\n\nFinally we discuss the constant terms (in~$\\widetilde \\t$) arising from the functional integral over the \ndynamical gauge multiplet. \nThere are a few subtleties. Firstly the actions like the gravitational CS action will depend on the \nglobal properties of the orbifold. \nThen we need to calculate the partition function of the orbifold space with a background graviphoton.\nAssuming as in the previous subsection that the expected THF moduli arise, and that by \nre-scaling and contour deformation (as discussed around~\\eqref{eq:Z0Rational}) the THF\nmoduli can be replaced with those of round~$S^3$, the calculation presumably reduces to \nan~$S^3\/\\mathbb{Z}_m$ partition function as in~\\cite{Benini:2011nc,Alday:2012au,Gang:2019juz,Willett:2016adv}, \nwith the~$\\mathbb{Z}_m$ action following from~\\eqref{newident} to be\n\\be\n(\\widetilde\\phi_1,\\widetilde\\phi_2) \\; \\sim \\; (\\widetilde\\phi_1-\\frac{2\\pi n}{m},\\widetilde\\phi_2-\\frac{2\\pi n}{m}),\n\\ee\nwhich for~$n=1$ coincides with that of the lens space~$L(m,-1)$.\nHere one has to be careful about how the measure on the space \nof constant scalars~$\\sigma_i$ is affected by the  four-dimensional orbifold~\\eqref{newident}. \nWe leave these interesting questions to future work, noting that \nthe result of these considerations indeed agrees with the microscopic \nanswer~\\eqref{eq:N=4indexAsyRational}, with the~$\\text{O}(\\widetilde \\t)$ piece explained by the \nsupersymmetric Casimir energy factor as before.\n\n\\vspace{0.4cm}\n\n\\noindent \\textbf{Note Added.} The paper~\\cite{Cassani:2021fyv}, which appeared on the arXiv \nthe same day as the first version of this paper, has some overlap with our section~\\ref{sec:4dto3d}. \nThe paper~\\cite{Jejjala:2021hlt}, which appeared on the arXiv soon after, has some overlap \nwith our section~\\ref{sec:SCI}.\n\n\n\n\n\n\\section*{Acknowledgements}\n\nWe would like to thank Alejandro Cabo-Bizet, Daniel Butter, Davide Cassani, Cyril Closset, Zohar Komargodski, Neil Lambert, Stavros Garoufalidis,\nBernard de Wit, and Don Zagier for useful discussions and comments.\nThis work is supported by the ERC Consolidator Grant N.~681908, ``Quantum black holes: A macroscopic \nwindow into the microstructure of gravity'', and by the STFC grant ST\/P000258\/1. \nAAA would like to especially thank Junho~Hong for several helpful discussions and \ncollaboration on a related project.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n Geometry has been a testbed for automated deduction almost\nas long as computers have existed; the first experiments were done\nin the 1950s.  In the nineteenth century, geometry was the testbed for the development \nof the axiomatic method in mathematics, spurred by the efforts to prove Euclid's\nparallel postulate from his other postulates and ultimately the development of\nnon-Euclidean geometry.  This effort culminated in Hilbert's seminal 1899 book \\cite{hilbert1899}.\n In the period 1927--1965, Tarski developed his \nsimple and short axiom system (described in \\S\\ref{section:axioms}). \nSome 35 years ago, Wos experimented with finding proofs\nfrom Tarski's axioms, reporting success with simple theorems, but leaving several\nunsolved challenge problems.   The subject was revisited\nby Art Quaife, who in his 1992 book \\cite{quaife1992} reported on the successful \nsolution of some of those challenge problems using an early version of McCune's theorem prover, \\Ott.  \nBut  several theorems remained that Quaife was not able to get {\\textsc OTTER\\ } to prove,\nand he stated them as ``challenge problems'' in his book.\nAs far as we know, nobody took up the subject again until 2012, when we \nset out to see whether automated reasoning techniques, and\/or computer\nhardware,  had improved enough to let us progress beyond Quaife's achievements.%\n\\footnote{There is also a long tradition, going back to Descartes, of reducing geometric\nproblems to algebra calculations by introducing coordinates.  Algorithms for carrying out \nsuch calculations by computer have been extensively studied, including special  methods intended\nfor geometry and \nTarski's general decision  procedure for real closed fields.  We mention these only to \nemphasize that such methods are irrelevant to this paper, which is concerned with proofs\nin an axiomatic system for geometry.}\n\nThe immediate stimulus leading to\n our 2012 work was the existence of the almost-formal development of \nmany theorems in Tarskian geometry in Part I of \\cite{schwabhauser}.  This Part I \nis essentially the manuscript developed by Wanda Szmielew for her 1965 Berkeley lectures\non the foundations of geometry, with ``inessential modifications'' by Schw\\\"abhauser.\nThere are 16 chapters.  Quaife's challenge problems (listed in \\S\\ref{section:challenges})\n occur in the first nine chapters.\nThe rest contain other important geometrical theorems (described in \\S\\ref{section:results} \nand \\S\\ref{section:hardtheorems}).\nWe set ourselves the goal to find {\\textsc OTTER\\ } proofs of each of the theorems in\nSzmielew's 16 chapters; we completed twelve of them, which is enough to make our points.\nOur methodology focuses on separate problems, each targeting one theorem.  For\neach problem, we supplied the axioms and the previously-proved \ntheorems, as well as the (negated) goal expressing the theorem to be proved.\nWe were often forced to supply {\\textsc OTTER\\ } with more information than that,\nas we describe in \\S\\ref{section:diagrams} and \\S\\ref{section:hints}.\n\n\nWe do not know of another \nbody of mathematics of this size that has been formalized by using a theorem prover. \nNormally proof checkers are used to formalize a whole theory, and theorem provers\nare used for individual theorems.  Our first report on this work \\cite{beeson2014-wos}\nemphasized the solution of two hundred individual problems.  We prepared two \nhundred {\\textsc OTTER\\ } input files by hand.   Since the time of that report, we developed \na methodology for the automated preparation and checking of those input files, to \nensure that no human error has corrupted the formal development of an entire theory\nas embodied in two hundred input files and proofs.   This methodology makes it \npossible to easily conduct experiments involving many files.  For example,  \none can easily generate input files for the theorem prover of one's choice, \ninstead of for the prover we used (\\Ott).  \nOn the web site for this project \\cite{tarski-archive}, we have made available\nour input files, the resulting proofs, and various tools for manipulating theorems\nand files,  which we hope will be of use to others wishing to work with these problems.\n\nWe distinguish between proofs that were found completely mechanically (without \nreference to the steps of a book proof)  and proofs that were constructed by \nsome technique that involved a human knowing the steps of a book proof.  Roughly \nspeaking, we were able to derive mechanically most of the theorems that have \n``short'' proofs (40 or fewer deduction steps).     \nProofs of length 40--100, roughly speaking, are difficult\nexercises for a human,  and proofs of 100-250 steps belong in a Ph.D. thesis or\npublication.  At first (that is, in \\cite{beeson2014-wos}),  we could not obtain\nsuch long proofs completely mechanically.   Our main goal at that time (2013)\nwas to obtain formal proofs by any means possible.  That meant starting with \na book proof  and using techniques explained in \\S\\ref{section:hints} (lemma adjunction and hint injection)\nto eventually obtain a formal proof.  We did obtain proofs of all two hundred theorems.  \nBut that left an obvious challenge:  Can one develop methods that \nenable a theorem prover to prove these theorems without reference to the steps of a book proof?\nWe were able to prove some (but not all) \nquite long theorems (of Ph.D. difficulty) \ncompletely mechanically,  using a technique called the ``subformula strategy''\n\\cite{wos-notebook2008}.\nThe challenge of developing strategies to reliably find proofs \nof such difficult theorems (by methods independent of knowledge of a book proof) still stands.  \n\nIn this paper, we give Tarski's axioms, explain the challenge problems of Quaife\nand some of the axioms of Hilbert, discuss the difficulties of finding {\\textsc OTTER\\ } \nproofs of these theorems, and explain what techniques we used to find those proofs. \nThen we explain our automated file-generation\nand checking methodologies and the extent to which those methods ensure the reliability\nof the whole theory. \n\n \n\\section{The challenge problems} \\label{section:challenges}\nQuaife's four challenge problems were as follows: every line segment has a midpoint;\nevery segment is the base of some isosceles triangle; the outer Pasch axiom (assuming \ninner Pasch as an axiom); and the first outer connectivity property of betweenness.\nThese are to be proved without any parallel axiom and without even line-circle continuity.\nThe first proofs of these theorems were the heart of Gupta's \nPh.D. thesis \\cite{gupta1965}  under Tarski.  {\\textsc OTTER\\ } proved them all in 2013.   \nAll Quaife's challenge problems occur in Szmielew; the last of them is Satz 9.6,\nso solving these challenge problems is a consequence of formalizing Chapters 2-11 of \nSzmielew.   \n\nThe theory of Hilbert (1899) can be translated into Tarski's language, interpreting\nlines as pairs of distinct points and interpreting angles as ordered triples of non-collinear points.\nUnder this interpretation, the axioms of Hilbert either occur among or are easily \ndeduced from theorems in the first 11 (of 16) chapters of Szmielew.  We have found\n{\\textsc OTTER\\ } proofs of all of Hilbert's axioms from Tarski's axioms (through Satz 11.49 \nof Szmielew, plus Satz 12.11). \n\n \n\\section{Related work}\nWe know of several other projects involving formal proofs in Tarskian geometry.\nBraun and Narboux \\cite{narboux2012}, \\cite{narboux2015} have  checked many theorems of Tarskian geometry in Coq; they have now gotten as far as Pappus's theorem in Chapter 15. \n Richert has checked some in HOL Light (unpublished, but distributed with HOL Light).  \n Urban and Veroff are also checking some in HOL-Light.  \n Durdevic {\\em et.al.} \\cite{narboux2015b} have used Vampire, E, and {\\textsc SPASS}\n to check some of these theorems.   \n \nWos has also conducted many experiments aimed at shortening some of \nthese proofs, and other proofs in Tarski's older axiom system,  or finding forward \nproofs instead of backwards or bidirectional proofs;  some of these are\ndescribed in \\cite{wos-notebookTarski}. \n\n\n  \n \n\\section{Tarski's axioms} \\label{section:axioms}\n In about 1927,  Tarski first lectured on his axiom system for geometry, which \nwas an improvement on Hilbert's 1899 axioms in several ways: First, the language\nhad only one sort of variables (for points), instead of having three primitive\nnotions (point, line, and angle).   Second, it was a first-order theory (Hilbert's\naxioms mentioned sets, though not in an essential way).   Third, the axioms were short,\nelegant, and few in number (only twenty in 1927, decreasing to twelve in 1957 \\cite{tarski-givant}).  They could be expressed comprehensibly\nin the  primitive syntax, without abbreviations.   \n\n\\subsection{History}\n The development of Tarski's theory, started in 1927 or before, was delayed, first \n by Tarski's involvement in other projects, and then by  \nWorld War II (galley proofs of Tarski's article about it were destroyed by bombs).\nThe first publication of Tarski's axioms came in 1948 \\cite{tarski1951} and contained little more \nthan a list of the axioms and a statement of the important metamathematical theorems \nabout the theory (completeness, representation of models as ${\\mathbb F}^2$ for ${\\mathbb F}$ a \nreal-closed field, quantifier-elimination, and decidability).  Tarksi\nthen lectured on the subject at Berkeley in 1956 to 1958, and published a reduced\nset of axioms in 1959 \\cite{tarski1959}.  In the 1960s, \nTarski, Szmielew, Gupta, and Schw\\\"abhauser (and some students) \nreduced the number of axioms still further.  \nThe manuscript that Szmielew prepared for her 1965 course became Part I of \\cite{schwabhauser}. \nMore details of the history of these axioms can be found in \\cite{tarski-givant} (our main source) \nand \nthe foreword to (the Ishi Press edition of) \\cite{schwabhauser}. \nFor our present purposes, the relevance of the\nhistory is mainly that there are three versions of Tarski's theory: the 1948 version, \nthe 1959 version, and the 1965 version (published in 1983).\nThe earlier experiments of Wos used the 1959 axioms, but Quaife used the 1965 version,\nas we do.   The exact differences are explained in \\S\\ref{section:betweenness} and \n\\S\\ref{section:pasch}. \n\n\\subsection{Syntax}\nThe fundamental relations in the \ntheory (first introduced by Pasch in 1852) are ``betweenness'',  which we here write\n${\\bf T}(a,b,c)$,  and ``equidistance'', or ``segment congruence'',  which is officially \nwritten $E(a,b,c,d)$  and unofficially as $ab = cd$, segment $ab$ is congruent to segment\n$cd$.  The intuitive meaning of ${\\bf T}(a,b,c)$ is that $b$ lies between $a$ and $c$ on\nthe line connecting $a$ and $c$;  Tarski used non-strict betweenness, so we do \nhave ${\\bf T}(a,a,c)$ and ${\\bf T}(a,c,c)$ and even ${\\bf T}(a,a,a)$. Hilbert used strict betweenness.\nBoth of them  wrote ${\\bf B}(a,b,c)$, which is a potential source of confusion.  We therefore\nreserve ${\\bf B}$ for strict betweenness and use ${\\bf T}$ for Tarski's non-strict betweenness.\nThe fact that Tarski's initial is `T' should serve as a mnemonic device.  Of course\nthe equality relation between points is also part of the language.  \n\n\\subsection{Betweenness and congruence axioms}\\label{section:betweenness}\nWe sometimes write $ab=cd$ instead of $E(a,b,c,d)$ to enhance human readability. In {\\textsc OTTER\\ } files\nof course we use $E(a,b,c,d)$.  The following are five axioms from the 1965 system.\n\\smallskip\n\n\n\\begin{tabular*} {0.7\\textwidth}{@{\\extracolsep{\\fill}}ll}\n$ab = ba$& (A1)  reflexivity for equidistance\\\\\n$ab=pq \\land ab = rs \\ \\rightarrow\\  pq = rs$& (A2) transitivity for equidistance \\\\\n$ab = cc \\ \\rightarrow\\  a=b$& (A3) identity for equidistance \\\\\n$\\exists x\\,({\\bf T}(q,a,x) \\land ax = bc)$& (A4) segment extension\\\\\n${\\bf T}(a,b,a) \\ \\rightarrow\\  a=b$& (A6) identity for betweenness\n\\end{tabular*}\\smallskip\n\nWhen using (A4) in \\Ott, we Skolemize it:\n\\smallskip\n\n\\begin{tabular*} {0.7\\textwidth}{@{\\extracolsep{\\fill}}ll}\n$ {\\bf T}(q,a,ext(q,a,b,c)) \\land E(a,ext(q,a,b,c),b,c)$& (A4) Skolemized\\\\\n\\end{tabular*}\\smallskip\n\nThe original (1948) theory had the following additional fundamental properties\nof betweenness listed as axioms. (We follow the numbering of \\cite{tarski-givant}.)\n\\smallskip\n\n\\hskip-0.5cm\n\\begin{tabular*} {0.7\\textwidth}{@{\\extracolsep{\\fill}}ll}\n${\\bf T}(a,b,b)$&  (A12) Reflexivity for ${\\bf T}$ \\\\\n${\\bf T}(a,b,c) \\ \\rightarrow\\  {\\bf T}(c,b,a)$&  (A14) Symmetry for ${\\bf T}$   \\\\\n${\\bf T}(a,b,d) \\land {\\bf T}(b,c,d) \\ \\rightarrow\\  {\\bf T}(a,b,c)$& (A15) Inner transitivity \\\\\n${\\bf T}(a,b,c) \\land {\\bf T}(b,c,d) \\land b \\neq c \\ \\rightarrow\\  {\\bf T}(a,b,d)$&(A16) Outer transitivity \\\\\n${\\bf T}(a,b,d) \\land {\\bf T}(a,c,d) \\ \\rightarrow\\  {\\bf T}(a,b,c) \\lor {\\bf T}(a,c,b)$ & (A17) Inner connectivity \\\\\n${\\bf T}(a,b,c) \\land {\\bf T}(a,b,d) \\land a\\neq b$& (A18) Outer connectivity \\\\\n$\\qquad \\ \\rightarrow\\  {\\bf T}(a,c,d) \\lor {\\bf T}(a,d,c)$& \n\\end{tabular*}\\smallskip\n\\smallskip\n\n\\noindent\nOf these only (A15) and (A18) appear in the 1959 version, because in 1956--57 Tarski and \nhis students Kallin and Taylor showed that the other four are dependent (derivable from the \nremaining axioms).  H.~N.~Gupta\nshowed in his 1965 Ph.D. thesis \\cite{gupta1965} that (A18) is also dependent.  The \nproof of (A18) is one of Quaife's challenge problems.\nGupta also showed that (A15) implies (A6) using the other axioms of the 1959 system.\nThen one could have dropped (A6) as an axiom; but instead, Szmielew\ndropped (A15), keeping (A6) instead; then (A15) becomes a \ntheorem.   \n\nAll six of these axioms  occur\nas theorems in \\cite{schwabhauser}:  (A12) is Satz~3.1, (A14) is Satz~3.2, (A15) is Satz~3.5,\n(A16) is Satz~3.7, (A18) is Satz~5.1, and (A17) is Satz~5.3.  \nHence, our research program of proving all the \ntheorems in Szmielew's development using {\\textsc OTTER\\ }  systematically captured these results as \nsoon as we reached Satz~5.3. \n\n\\subsection{The five-segment axiom}\nHilbert \\cite{hilbert1899} treated angles as primitive objects and angle congruence\nas a primitive relation, and he took SAS (the side-angle-side triangle congruence principle) \nas an axiom. In Tarski's theory, angles are\n  treated as  ordered triples \nof points, and angle congruence is a defined notion,\nso a points-only formulation of the SAS principle is required.   \n The key idea is Tarski's  ``five-segment axiom'' (A5), shown in Fig.~\\ref{figure:5segment}.\n\n \n \\begin{figure}[ht]\n\\TarskiFiveSegmentFigure\n\\caption{The five-segment axiom (A5)}\n\\label{figure:5segment}\n\\end{figure}\n\n\nIf the four solid segments in Fig.~\\ref{figure:5segment} are pairwise congruent, then the fifth (dotted) segments are congruent too.\nThis is essentially SAS for triangles $dbc$ and $DBC$.  The triangles $abd$ \nand $ABD$ are surrogates, used to express the congruence of angles $dbc$ and $DBC$.\n By using Axiom A5, we \ncan avoid all mention of angles.\n\n\\subsection{Pasch's axiom} \\label{section:pasch}\nMoritz Pasch \\cite{pasch1882} (see also \\cite{pasch1926}, with an historical appendix by Max Dehn)\n supplied (in 1882) an axiom that repaired many of the defects that \nnineteenth-century rigor found in Euclid.  Roughly, a line that enters a \ntriangle must exit that triangle.\nAs Pasch formulated it,  it is not in $\\forall\\exists$ form.  There are two $\\forall\\exists$  versions,\nillustrated in Fig.~\\ref{figure:InnerOuterPaschFigure}.  These formulations of Pasch's axiom \ngo back to Veblen \\cite{veblen1904}, who proved outer Pasch implies inner Pasch. \nTarski took outer Pasch as an axiom in \\cite{tarski1959}.\n \n \n\\begin{figure}[ht]   \n\\hskip 0.7cm\n\\InnerOuterPaschFigure\n\\caption{ Inner Pasch (left) and Outer Pasch (right).  Line $pb$ meets triangle $acq$ in one side.\n The open circles show the points asserted to exist on the other side. }\n\\label{figure:InnerOuterPaschFigure}\n\\end{figure}\n\\smallskip\n\n\\begin{tabular*} {0.7\\textwidth}{@{\\extracolsep{\\fill}}ll}\n${\\bf T}(a,p,c) \\land {\\bf T}(b,q,c) \\ \\rightarrow\\  \\exists x\\,({\\bf T}(p,x,b) \\land {\\bf T}(q,x,a))$& (A7) inner Pasch \\\\\n${\\bf T}(a,p,c)\\land {\\bf T}(q,c,b) \\ \\rightarrow\\  \\exists x\\,({\\bf T}(a,x,q) \\land {\\bf T}(b,p,x))$&  outer Pasch \n\\end{tabular*}\\smallskip\n\\smallskip\n\nFor use in \\Ott, we introduce Skolem symbols \n$$ip(a,p,c,b,q) \\qquad \\mbox{ and} \\qquad   op(p,a,b,c,q)$$ \nfor the point $x$ asserted to exist.  The use of these symbols makes these axioms\nquantifier free. \n\n \n\nTarski originally took outer Pasch as an axiom.  In \\cite{gupta1965}, Gupta proved\nboth that inner Pasch implies outer Pasch and that outer Pasch implies inner Pasch,\nusing the other axioms of the 1959 system.   \nThe proof of outer Pasch from inner Pasch\nis one of Quaife's four challenge problems.\n \n \\subsection{Dimension axioms}\nWith no dimension axioms, Tarski's geometry axiomatizes theorems that are true in $n$-dimensional\ngeometries for all $n$.  For each positive integer $n$, we can specify that the dimension of \nspace is at least $n$ (with a lower-dimension axiom A8${}^{(n)}$), or at most $n$ (with an \nupper-dimension axiom A9${}^{(n)}$).  The uppe--dimension axiom says (in a first-order way) that the set of points\nequidistant from $n$ given points is at most a line.   The lowe--dimension axiom for $n$ is the \nnegation of the upper-dimension axiom for $n-1$.  For the exact statements of these axioms \nsee \\cite{tarski-givant}.\n\nInner and outer Pasch have another advantage over Pasch's original version, besides \nlogical simplicity.  Namely, they hold even in 3-space, where Pasch's original version fails.\nThat is, they can be used without an ``upper-dimension'' axiom, whose use is thereby postponed\n a long time in Szmielew.\n\n \n\\subsection{Tarski's parallel axiom (A10)}\nIn  Fig.~\\ref{figure:parallelaxiom}, open circles indicate points asserted to exist.\n \\vskip-0.5cm\n  \\begin{figure}[ht]\n \\hskip 2cm\n \\TarskiParallelFigure \n \\caption{Tarski's parallel axiom}\n \\label{figure:parallelaxiom}\n \\vskip -1.2cm\n \\end{figure}\n\n\\begin{eqnarray*}\n& {\\bf T}(a,d,t) \\land {\\bf T}(b,d,c) \\land a \\neq d \\ \\rightarrow\\   \\\\\n& \\qquad \\exists x \\exists y\\,({\\bf T}(a,b,x) \\land {\\bf T}(a,c,y) \\land {\\bf T}(x,t,y)) \\qquad ({\\rm A}10)  \n \\end{eqnarray*}\n\n \n The hypothesis says that $t$ lies in the interior of angle $a$, as witnessed by $b$, $c$, and $d$.\n The conclusion says that some line through $t$ meets both sides of the angle.  Of course this fails\n in non-Euclidean geometry when both $ab$ and $ac$ are parallel to some line $L$ through $t$.\n \n  According to \\cite{tarski-givant}, Szmielew preferred to use the ``triangle circumscription \n  principle '' (A10${}_2$) as the parallel axiom.  Substituting (A10${}_2$) was apparently one of the \n``inessential changes'' made by Schwabh\\\"auser.  This principle says that if $a$, $b$, and\n $c$ are not collinear, then there exists a point equidistant from all three. (That point is the \n center of the circumscribed triangle, hence the name.)  See \\cite{tarski-givant}\n for a formal statement.\n \n \\subsection{Continuity axioms}\n Axiom schema (A11) is not a single axiom but an axiom schema, essentially asserting that \n first-order Dedekind cuts are filled.  See \\cite{tarski-givant}\n for a formal statement.\n Models of A1-A11 are all isomorphic to planes ${\\mathbb F}^2$\n where ${\\mathbb F}$ is a real-closed field.  One can also consider instead of (A11) the axioms of \n line-circle continuity and\/or circle-circle continuity,  which assert the existence of intersection \n points of lines and circles, or circles and circles, under appropriate hypotheses.  None of \n the continuity axioms are used in the work reported in this paper.  Szmielew's development \n proceeds strictly on the basis of A1-A10.\n \n \n\n\n\\section{Completely mechanical methods for finding proofs}\nIn this section we describe, with illustrative examples, the techniques we used \nthat do not involve knowing the steps of a book proof (not \neven the diagram accompanying the proof).\n \n\\subsection{How {\\textsc OTTER\\ } works}\nReaders familiar with {\\textsc OTTER\\ } can skip this subsection.  It is not an introduction to {\\textsc OTTER\\ } \nbut an attempt to make the subsequent information about our methodology comprehensible to \nthose who do not have expertise with \\Ott; at least, it should enable such readers to \nmake sense of the input files and proofs we exhibit on the project's website\n\\cite{tarski-archive}.\nFor more information about \\Ott, see \\cite{wos-fascinating}.\n\n{\\textsc OTTER\\ } is a clause-based resolution theorem prover.  One writes {\\tt -A}  for the \nnegation of $A$.   One writes {\\tt A | B} for disjunction (``or'').  One does not \nwrite ``and'' at all, but instead one enters the two clauses separately.  One writes\n$A \\ \\rightarrow\\  B$  as {\\tt -A | B}.   Similarly one writes $P \\land Q \\ \\rightarrow\\  R$ as \n{\\tt -P | -Q | R}. \n\nVariables begin with {\\tt x,y,z,w,u,v}.  Names beginning with any other letter are constants.\nA resolution theorem prover requires the goal to be negated and entered as clauses.\nFor example, to prove $A(x) \\ \\rightarrow\\  \\exists y\\, B(x,y)$,  we would enter the following clauses:\n\n\\begin{verbatim}\nA(c).\n-B(c,y).\n\\end{verbatim}\nAfter proving this theorem, if we want to use it to prove the next theorem, we invent \na new Skolem symbol {\\tt f} and enter the theorem as\n{\\tt -A(x) | B(x,f(x))}.\n \n \nThe input to {\\textsc OTTER\\ } is contained in two lists,  the ``set of support'' (sos) and \n``usable''.  The fundamental run-time loop of {\\textsc OTTER\\ } moves a clause from sos to usable,\nand then tries to use one of the specified inference rules to generate new clauses from \nthat clause and other clauses in usable.  If conclusions are generated, {\\textsc OTTER\\ } has to decide\nwhether to keep them.  If it decides to keep them, they are placed on sos, where they\ncan eventually be used to generate yet more new clauses.   If the empty clause is generated,\nthat means a proof has been found, and it will be output.  \n\nThe fundamental problem of automated deduction is to avoid drowning in a sea of useless conclusions\nbefore finding the desired proof.  One tries to get control over this by assigning ``weights''\nto clauses, adjusting those weights in various ways, and using them to control both which \nclauses are kept and which clause is selected from sos for the next iteration of the loop.\nBy default: the weight of a clause is the number of its symbols; the next clause selected\nis the lightest one in sos; and clauses are kept if their weight does not exceed a \nparameter {\\tt max\\_weight}.   More sophisticated ways of setting the weights have been developed\nover the past decades and are discussed in \\S\\ref{section:hints}.\n  The idea is to get the weights of the \nimportant clauses to be small, and then to squeeze down {\\tt max\\_weight} to prevent\ndrowning.\n\nIn addition to techniques involving weighting, there are other ways to control \\Ott's search:\n\\begin{itemize}\n\\item Use a propitious combination of rules of inference.  For an introduction to these rules please\nrefer to \\cite{wos-fascinating}. \n\\item One can exert some control over which clause will\nbe selected from sos at the next iteration by using \\Ott's {\\tt pick\\_given\\_ratio}.  \n\\item One can exert some control over how the search starts and what kind of proof to seek\n(forward, backward, or bi-directional) by choosing which clauses to put in sos and which \nto put in usable.\n\\end{itemize}\n\n\n\\subsection{Hints}  \nPutting a clause \ninto {\\tt list(hints)} causes {\\textsc OTTER\\ } to give that clause, if deduced, a low weight, causing \nit to be retained, even if its default weight would have been so large as to cause it to be \ndiscarded.   One has options (specified at the top of an {\\textsc OTTER\\ } file) to cause this weight \nadjustment to apply to clauses that match the hints, or subsume the hints, or are \nsubsumed by the hints.  We use hints in the implementation of several strategies, \nincluding the subformula strategy and lemma adjunction, which are described \nin \\S\\ref{section:subformulastrategy} and \\S\\ref{section:hints}.\nThe technique of hints was invented by Veroff \\cite{veroff1996} and later incorporated\ninto \\Ott.  As a technical note: when using hints, one should always include these lines,\nwithout which the hints will not have the desired effect.\n\n\\begin{verbatim}\nassign(bsub_hint_wt,-1).\nset(keep_hint_subsumers).\n\\end{verbatim}\n\nThere is also a technical difference in {\\textsc OTTER\\ } between {\\tt list(hints)} and {\\tt list(hints2)};\n researchers should consult the manual before using either.  \nAnother similar technique is known as {\\em resonators}.  This is more useful when one has\na proof in hand and wishes to find a shorter proof.  For the exact differences between \nhints and resonators, see \\cite{wos2003}, p.~259.\n\n\\subsection{Choice of inference rules and settings}\nWe mentioned that one of the ways {\\textsc OTTER\\ } can be controlled is through a \npropitious choice of inference rules and settings.  We tinkered with our choices often,\nin the hope that a different choice would be better.  We did not find that one \nchoice was always best, but a carefully chosen default choice served us well \nin most cases.  Our default choice was to use all three of \nhyperresolution, binary resolution, and unit resolution.  In addition, \nwe always used paramodulation (for equality reasoning). \n   We {\\em often} \nchanged the values of the parameters {\\tt max\\_weight} and {\\tt max\\_proofs}.\nIn particular, when using lemma adjunction, we needed {\\tt max\\_proofs} large enough \nto accommodate many intermediate goals (on the passive list); when using many hints,\nwe chose {\\tt max\\_weight} to be 8 or even smaller;  but when searching a ``raw''\nsearch space, we often used 16, 20, or 24, and occasionally even higher. \nIf {\\tt max\\_weight} is too high, then one may be swamped with too many conclusions,\nbut if {\\tt max\\_weight} is too low, one may throw away a clause that was needed.  Novices\nmay fail to realize another danger:  one might derive a useful clause,  and keep it, \nbut if, say, its weight is 10 and there are many clauses of smaller weight, your \nuseful clause may ``never'' become the given clause and give rise to the further \nconclusions that are needed to complete the proof.\n\nOccasionally we changed the values of \n{\\tt max\\_distinct\\_vars},  but usually without much effect.  Rarely do the \nproofs we are seeking  contain more than one variable per clause.  We think \nthis is because Szmielew has already organized the theorems in such a way that we \ndo not need to prove many extra lemmas;  a derived clause with variables amounts to a lemma.\nWe performed experiments in which {\\tt max\\_distinct\\_vars} was given increasing values,\nholding other settings fixed.  There were only about four theorems that required \nit to have the value 4,  but setting it lower did not speed up finding the other proofs\nnoticeably.  \n\n\nWe used paramodulation for equality reasoning.  Wos believes that\nfailing to use paramodulation was an important reason for the limited success\nof the experiments he made in the 1980s in Tarskian geometry; but we also note\nthat Quaife writes that paramodulation was available for use but seldom actually used in his proofs.\n\n\\paragraph{Demodulation.}  A {\\em demodulator} is an equation, used left-to-right \nto rewrite terms to ``simpler forms.''  We used demodulators in connection with \ndiagram equations, but usually we could also obtain proofs without those demodulators.\nIn other subjects, Wos has made use of the technique of ``demodulating to junk'',\nin which unwanted classes of term or types of conclusions are demodulated to an atom {\\tt junk},\nwhich is eliminated by giving it a high weight.  But this technique was not useful in \nour work on geometry.  Only one of the two hundred theorems of Tarskian geometry \nis an equation.  That theorem says that reflection in a point is an involution; that is, \nthe reflection of the reflection of a point is the original point.  We write $s(p,x)$ for \nthe reflection of $x$ in $p$;  then the equation is $s(p,x(p,x)) = x$.%\n\\footnote{There is a similar theorem, Satz 10.5, about reflection in a line, but that\ntheorem is not quite equational, because it requires that the two points determining the \nline be distinct.}     It was \n useful to make this equation a demodulator;  but since there is  only {\\em one}\nequational theorem, demodulation is not an important technique in the development\nof Tarskian geometry based on Szmielew's manuscript.\n\nWe did, however, make some use of demodulators in the process of lemma adjunction.\nFor example, $midpoint(x,y) = midpoint(y,x)$ is a useful equation (which does not occur\nin Szmielew).   Also, in several situations   it is natural to rewrite \n(not terms but) formulas, for example, we would naturally regard all the following \nas equivalent ways to express the congruence of two segments:  $ab = cd$, $ab=dc$,\n$ba = cd$, $ba = dc$,  $cd = ab$, $dc=ab$, $cd = ba$, and $dc = ba$.  Similarly \nfor the 4-ary predicate $ab \\perp cd$.   {\\textsc OTTER\\ } is able to use demodulators for \nformulas, and setting the {\\tt lrpo} flag tells it to use them to demodulate terms \nto what amounts to alphabetical order; thus we would keep $midpoint(a,b)$ unchanged\nand demodulate $midpoint(b,a)$ to $midpoint(a,b)$.  Technically, however,  the use of \ndemodulators at a formula level is not permitted in a first-order development, so \nin the end we tried (successfully) to eliminate their use, mostly by hand-editing the \nhints resulting from proof steps using demodulators.  Like Gauss, we removed our \nscaffolding,  so this paragraph is the only trace of that technique. \n\n \n\\paragraph{  Hot list.}  The hot list is a feature of {\\textsc OTTER\\ }\\ that works as \nfollows:  Whatever clauses are on the hot list are applied immediately to newly deduced\nformulas.  This is useful in the following situation:  suppose you think that clause\n$P$ should be derived, and then clause $Q$ should be derived from $P$ by axiom 7. \nYou see that $P$ is derived, but then $P$ never becomes the given clause,  because \nit has a fairly high weight and the sea of conclusions contains many clauses of \nlower weight.   If you put Axiom 7 on the hot list, then $Q$ will be deduced immediately\nwhen $P$ is deduced, without having to wait until $P$ is the given clause.\nWe used this technique in perhaps fifty out of two hundred hand-crafted input \nfiles.   Perhaps it wasn't really necessary.  In the mechanically generated\nfiles described in \\S\\ref{section:mechanicalinputfiles}, the hot list disappears, but its influence lingers, because\nthe steps of the proof found originally using a hot list remain as hints. \n \n\\paragraph{Set of support.} Normally, we placed the axioms and previously proved theorems\non the list of ``usable'' clauses, and the negated form of the current theorem on the \nset of support.  One of the things that can be tried is putting \neverything  on the set of support.  This goes against the basic idea behind the set of support (to\nprevent deducing many consequences of the axioms that are irrelevant to the current goal),\nbut it has worked for Wos in some situations.  In geometry, however, it \nworked on exactly one theorem out of two hundred, namely Satz 3.1.\n \n\n\\paragraph{ Level saturation.}  ``Level saturation''\n refers to a gradual increase in allowed complexity.\nThe program starts\nby allowing the simplest next conclusions, for us of weight 1, then 2, 3, $\\ldots$,\n and continues, run after run.   This will be useful only if there is a very short proof\n that for obscure reasons does not come out with more normal settings.   Once or twice\n we found a short proof this way, but we always found a proof later with more normal settings,\nso this was not actually a useful technique.\n \n \n\\paragraph{Ratio strategy.}  The pick-given ratio in {\\textsc OTTER\\ } alters the basic strategy \nof selecting the lowest-weight clause from the set of support.  For example, if this \nratio is set to 4 (our default), then after four clauses have been selected by weight,\nthe oldest one on the set of the support is chosen, regardless of weight.  We sometimes,\nbut rarely, had success by setting this ratio to 2.\n\n\\section{Using the diagram} \\label{section:diagrams}\nIn this section, we consider the method of ``diagrams'',  which involves looking at the diagram \nin the book accompanying the proof,  but not the steps of the proof itself.\n\nWe learned this technique from \nQuaife \\cite{quaife1992}.  Suppose we are trying to find an $x$ such that \n$A(a,b,x)$.   If we let a theorem prover search, without making any attempt to guide the search, \nessentially it will generate all the points that can be constructed from $a$ and $b$ (and \nother points in the problem or previously constructed in clauses that were kept)\nby the Skolem functions for segment extension and inner Pasch.  Imagine trying to prove \na geometry theorem that way:  just take your ruler, draw all the lines you can between \nthe points of the diagram,  label all the new points formed by their intersections (that is\nby using \nPasch), and construct every point that can be constructed by extending a segment you have by \nanother segment you have.  See if any of the new points is the point \nyou are trying to construct. If not,  repeat.  You will generate a sea of useless points,\neven if you discard those with too many construction steps.  \n\nTo guide {\\textsc OTTER\\ } (or any other theorem prover)\n to construct the right points, we ``give it the diagram'' by \ndefining the points to be constructed, using  Skolem functions.  For example, \nconsider the ``crossbar theorem'' (Satz 3.17).  See Figure~\\ref{figure:crossbar},\n shows the diagram and the input (not shown are the axioms A1-A6 \nand the previous theorems, which are placed in {\\tt list(usable)}).\nThe two lines defining $r$ and $q$ are what we call ``giving {\\textsc OTTER\\ } the diagram.''\nWith those lines present, {\\textsc OTTER\\ } finds a proof instantly.  Remove them, \nand {\\textsc OTTER\\ } does not find a proof (at least, not in ten minutes).\n\n\\begin{figure}[ht]\n\\hskip 0.3in\n\\begin{verbatim}\n     list(sos).\n     T(a,s,c).\n     T(b,t,c).\n     T(a,p,b).\n     -T(p,x,c)|-T(s,x,t).\n     r = ip(c,t,b,a,p).\n     q = ip(c,s,a,t,r).\n     end_of_list.\n\\end{verbatim}\n\\vskip-1.6in\n\\CrossBarFigure\n\\caption{The crossbar theorem asserts that $q$ exists, given the other points\nexcept $r$.\nTo prove it, we construct first $r$ and then $q$, using inner Pasch twice. Here {\\tt ip}\nis the Skolem function for the inner Pasch axiom.}\n\\label{figure:crossbar}\n \n\\end{figure}\n \nThe reason this works is that $q$, being a single letter, has weight 1,  so \nclauses involving $q$ have much lower weight than would the same clauses with \n$q$ replaced by $ip(c,s,a,t,r)$, which has weight 6.   A clause involving the \nterms defining both $q$ and $r$ would have weight at least 14,  so it would \nnot be considered very soon, and meantime many other points will be constructed.\n\nIn this simple example, if one puts all clauses into sos and nothing in usable, \n{\\textsc OTTER\\ } does eventually find a proof without being given the diagram; but it is certainly \nmuch slower than with the diagram.  In more complex examples, we  \nthink this technique is essential.  Here, for example, are the lines \nfrom the input file for Satz~5.1, the inner connectivity of betweenness, \ndescribing the two  complicated diagrams on pp.~39--40 of \\cite{schwabhauser}:\n\n \n\\begin{verbatim}\n         c1=ext(a,d,c,d).  \n         d1=ext(a,c,c,d).\n         p=ext(c1,c,c,d).\n         r=ext(d1,c,c,e).\n         q=ext(p,r,r,p).\n         b1 = ext(a,c1,c,b).\n         b2 = ext(a,d1,d,b).\n         e = ip(c1,d,b,d1,c).\n\\end{verbatim}\n \n\\section{The subformula strategy} \\label{section:subformulastrategy}\nWos invented the subformula strategy sometime in 2007--2008 and \nused it to study single axioms in relevance logics.  He wrote up this work \nin \\cite{wos-notebook2008}.  \n  The subformula strategy consists of this:\n\n\\begin{quote}\n{\\em For every literal occurring in a clause in the set of support,\nput both the literal and its negation into the hints list.}\n\\end{quote}\n\n\\noindent\nSince clauses that subsume a hint are kept and given low weight, the result is that \nwhen any subformula of a clause in the set of support is deduced, it is kept, and \nmoreover,  it will soon be used as the ``given clause'' to deduce more conclusions.\nThe idea is that the proof we are seeking is likely to involve such clauses, so \nwhen we deduce them, we are getting closer to a good result.\n\nThis strategy is particular easy to implement in a modern programming language \nwith a string manipulation function like PHP's {\\tt explode} or Python's {\\tt split}.\nNo parsing is required.  Just explode each clause in the set of support on the \nsymbol for disjunction, resulting in an array of literals. Negating a literal is \nalso a simple string manipulation: for example, in PHP one can use {\\tt str\\_replace} to \nreplace {\\tt !=} by {\\tt =} and vice-versa.  To compute the required hints and write \nthem into an already-generated input file takes less than 30 lines of code.\n\nThis strategy is completely general;  it can be applied to any problem and \nrequires no advance knowledge of a proof.  Perhaps not even its creator \nnoticed its generality at the time.  \n We failed to use\nthe subformula strategy in 2012--2014,  and at that time we found no proofs of length \ngreater than 30 in a completely mechanical fashion, that is, without any reference to \na book proof  (although we did find proofs of all the Tarski theorems we tried,\nby methods discussed in \\S\\ref{section:hints} that did start by reference to a book proof).\n\nWhen the light bulb came on over our heads, we used the subformula strategy  \nand found several amazingly long proofs completely mechanically,  as described in \n\\S\\ref{section:results} and \\S\\ref{section:hardtheorems}.  We kept {\\tt max\\_weight}\nas low as possible,  typically 8. \n\n\n\\section{Methods that require using steps of the book proof} \\label{section:hints}\nIn this section we describe a method, or perhaps combination of methods, \nthat starts with the steps of a book proof (thus, human steps rather than \nmachine steps)   and ends with an {\\textsc OTTER\\ } proof.\n\n\n\nOur methodology was as follows:  when trying to prove a theorem, we prepared an \ninput file with the negated goal in {\\tt list(sos)}  and the axioms and previously proved theorems \nin list(usable), and a default choice of inference rules.  If this \ndid not find a proof, and the proof in Szmielew had a diagram, we supplied the diagram.\nThis section concerns what we did with theorems that could not be proved \nfrom these input files alone.  Namely, we supplied some hints based on the book proof.\nHere are the details about how this idea led to proofs.\n\n\n\\paragraph{From the passive list to hints.}\nWe tried supplying some intermediate goals, namely, \nsome or even all of the steps of the book proof.\n\nThese steps were supplied in two places:  we put them in the hints list, \nand we also put their negated forms in {\\tt list(passive)},\nwith answer literals having numbers for those goals.  The passive list is used\nas follows:  When a clause is generated that conflicts with an element of the passive\nlist, a proof is generated.  After that, proof search continues.\nHence, when this file is run,\none sees which of the intermediate goals are being proved.  Among other things, this \nshowed us where {\\textsc OTTER\\ } was ``getting stuck.''   But even though we found no proof of the \nmain goal, we had proofs of some intermediate goals.  We converted these proof steps\nto hints  and ran {\\textsc OTTER\\ } again.   Sometimes more intermediate goals would be reached. \nOne can tinker with {\\tt max\\_weight}: sometimes a smaller {\\tt max\\_weight} may keep \none from drowning, and sometimes a larger {\\tt max\\_weight} may allow a vital clause \nto be kept.  \nAfter adding the new hints, the previous proofs were found immediately, and perhaps\nmore intermediate goals could be found,  resulting in still more hints.\n\nWith luck this process would converge to a proof.\n\n\\paragraph{Lemma adjunction and hint injection.}  Often, especially with more difficult theorems,\nthe process described above would lead not to a proof but to a situation in which \nsome of the book steps would not be proved, even though others would be.  In that\ncase, our next move was to place one or more of these steps,  or their negations, \nin the set of support.  If this generated a proof (of the original goal, or of \nmore intermediate steps), then again we put those steps in as hints.\nWith those additional hints,  then sometimes we could remove (one, some, or even all)\nof the extra clauses\nwe added to the set of support and still get a proof.  If we did, we then \nreplaced the hints that came from the proof obtained by using those additional clauses,\nenabling that same proof to be found more easily.  \n\n\\paragraph{Divide and conquer.} \nIf all the extra steps can be removed from the set of support, then we have the \nproof we were seeking.  But if not, then we wind up, for example, with a \nproof of the theorem from an extra hypothesis $P$.   At that point, we replace\nthe extra clause $P$ with its negation $-P$ in the set of support  and try \nagain to find a proof, using the techniques described above, accumulating the steps\nof any (partial) proofs found in the hints list.  Eventually (in practice) we \nalways can succeed to find a proof from the extra hypothesis $-P$.  Now we have\n two proofs,  one from $P$ and one from $-P$.   If there had \nbeen two extra hypotheses $P$ and $Q$,  we might end up with four proofs, \none from $P,Q$, one from $P,-Q$,  one from $-P,Q$, and one from $-P,-Q$.   \nAt that point we have ``reduced the problem to combining the cases.''\n\n\n\n\\section{Proof by cases}  \\label{section:cases}\nProof by cases turned out to be difficult.  A number of proofs\nin the book proceed by cases, for example according to whether $p$ lies on line $L$\nor does not lie on line $L$.   Moreover, the lemma adjunction method \noften resulted in uncovering an intermediate step $A$ such that if we added\n$A$ as an additional hypothesis, we could find a proof,  and we could also \nprove that step, by adding $-A$ as an additional hypothesis.  That amounts\nto a proof by cases according as $A$ or $-A$.  It may seem that we must be \nnearly finished at that point.  But we found\nthat in such situations it was often not trivial, and sometimes\nnot possible,  to combine the two proofs into a single proof with no \nadditional assumptions.  In this section, we discuss this difficulty.\n\n Let us suppose that \nwe have a proof of the original goal from $A$  and another from $-A$.\nWhat do we do?\nWe could of course just formulate two lemmas: $P$ implies the goal,\nand $-P$ implies the goal. Then (in a third file), we could prove the goal from those two \nlemmas.  (Possibly just one of those lemmas would be enough.)\n Sometimes in a mathematics book, that is what is really done, even though \nthe theorem is stated as a single theorem;  particularly when there is a certain \nsymmetry to the theorem, instead of formulating a lemma explicitly, one simply says,\n``without loss of generality we can assume.''  Still, we tried to eliminate the \ncases and give a proof in a single file, when we could do so.  We now \ndescribe our technique for that. \n\n\\subsection{Tautology adjunction} \n   We could sometimes succeed by simply\nadding a clause {\\tt A | -A}, where the proof needs to proceed by cases on $A$.\n{\\textsc OTTER\\ } seems to prefer constructive proofs!    This technique is \ncalled ``tautology adjunction'' by Wos, who used it \ndecades ago in proving that subgroups of index 2 are normal.  We used this \nin many input files.  Here we discuss just one example. The inner connectivity\nof betweenness (A17 above, Satz~5.3 in Szmielew) is derived as an \neasy corollary of Satz~5.1, which is\n$$ a \\neq b \\land {\\bf T}(a,b,c) \\land {\\bf T}(a,b,d) \\ \\rightarrow\\  {\\bf T}(a,c,d) \\lor {\\bf T}(a,d,c).$$\n The natural \nway to formulate {\\tt list(sos)} for this problem would be\n\n\\begin{verbatim}\n     a != b.\n     T(a,b,c).\n     T(a,b,d).\n     -T(a,c,d).\n     -T(a,d,c).\n\\end{verbatim}\n\n\\noindent\nOf course, that does not suffice to find a proof.  So, we added the \ndescription of the diagram, as given in \\S\\ref{section:diagrams}.  Unfortunately\n{\\textsc OTTER\\ } could still not find a proof, even with hints.  \n\nThe proof in Szmielew proceeds by cases.   The methodology we followed \nin such cases was this:\n\n\\begin{itemize}\n\\item Add one case to sos, e.g. {\\tt c=c1}.  ({\\tt c1} is a constant from the diagram, above.)  \n\\item If we find a proof,  add the steps of that proof as hints.\n\\item Remove that case, and add its negation, e.g. {\\tt c != c1}\n\\item If we find a proof,  add its steps also as hints.\n\\item Now remove both cases, and add their disjunction: {\\tt c = c1 | c != c1}.\n\\end{itemize}\n\nOften we could find a proof.   The example at hand required  two divisions into cases \n(so tautology disjunction was applied recursively).\n The first is the case whether {\\tt c=c1} or not, \nand the second  whether {\\tt d1=e} or not.  Here \none can see in the two commented lines in {\\tt list(sos)}  the \ntraces of completing the last argument by cases. \n\n\\begin{verbatim}\n    \n    \n     d1=e | d1!=e.\n     c = c1 | c != c1.\n\\end{verbatim}\n\nWe do not mean to imply that this was all there was to proving Satz~5.1.  This was just \nthe last difficulty.  By that time, the input file already contained a long list of hints\nobtained by methods described above.  The final proof had 127 steps.\n\n\\subsection{Remarks on tautology adjunction}\nSuppose we have a resolution proof of contradiction from assumptions $L$ \nand the additional assumption $P$, and another proof \nfrom $L$ together with $-P$.  Then if we form the list of all clauses $Q | -P$,  where $Q$\nis a clause in the proof from $P$,  we arrive at the conclusion $-P$, where we formerly\narrived at a contradiction.  Then we can continue with the proof of contradiction from $-P$.\nThis method, however, results in a proof only if the original proof from $P$ proceeded\nby binary resolution.   If instead it used hyperresolution, then the resulting clauses \n$Q | -P$  do not form a proof, as the steps originally made by hyperresolution \nare no longer legal hyperresolution steps.  Steps by paramodulation might also be \nspoiled if the terms involved in the paramodulation occur in $P$.  These observations \nexplain why it can be difficult to combine cases.  \n\nAt least we would like clauses of the form $Q | -P$ to be kept if they are deduced.\nTo achieve that when the $Q$ are in the hints list, we need (in \\Ott)  to use {\\tt hints}\ninstead of {\\tt hints2}, and {\\tt assign(fsub\\_hint\\_wt,-1)} to get clauses kept that are \nsubsumed by hints.  Then we set {\\tt max\\_weight} very low, perhaps 6 or 8, which might be \nenough to allow the clauses $Q | -P$ to be deduced (in several steps) by binary resolution.\n Sometimes this approach worked, but {\\tt hints} is a {\\em lot} slower than \n{\\tt hints2}.  (In Prover9,  {\\tt hints} does not exist; what is called {\\tt hints} is \nsimilar to {\\tt hints2} in \\Ott.)  Moreover, it did not always work (e.g., on Satz~12.1),\nprobably because some of the required inferences by binary resolution contain clauses that \nare not in the hints list, and so their weight is too large.  In principle there must be \na setting of {\\tt max\\_weight} and a running time that produces the desired proof, but \nit sometimes eluded us.\n\nIn principle it should be possible to implement an algorithm for combining two proofs\n(one from $P$ and one from $-P$)  and producing a valid proof by binary resolution and paramodulation.\nThen the steps of that proof could be injected as hints.  The resulting proof \nwould not rely for its correctness on the algorithm for combining proofs,  since if the \nhints work, it doesn't matter where they came from.   But we have not \ntried to implement such an algorithm.  We note, however, that such an algorithm could \nbe implemented ``on top of'' existing theorem provers, without modifying the theorem prover,\nusing PHP scripts (or scripts written in some other language). \n\n\\section{The challenge problems} \\label{section:results}\n\nAll the input files and resulting {\\textsc OTTER\\ } proofs that we found are posted \non the web at \\cite{tarski-archive}.     Here\nwe list some of the more difficult proofs we found, as well as some of interest\nfor reasons other than difficulty.\n\n\\subsection{Properties of betweenness}\nIn \\S\\ref{section:betweenness}, we listed six difficult theorems (A12-A18 except A13),\neach of which Tarski originally took as an axiom.  They must have been fairly\ndifficult if Tarski did not notice that they could be eliminated as axioms.\n(A13 disappears when equality axioms are included; in 1926 equality axioms were not\nincluded.)\nAll six occur  as theorems\nin \\cite{schwabhauser}. \nWe found {\\textsc OTTER\\ } proofs of all those theorems, several of them  short in spite\nof the perceived difficulty of the theorems. \nTable~\\ref{table:1} gives the\nlength of the proofs we found, which is perhaps some indication of the relative \ndifficulty of finding them. We found the short proofs completely mechanically.\nSatz~5.1, with 121 steps, is a theorem from Gupta's Ph.D. thesis \\cite{gupta1965},\nand it is the first theorem in Szmielew's development that we could not prove \ncompletely mechanically.  We had to use lemma adjunction and hint injection.\n\n\n\n\\begin{table}\n\\caption{Proofs found for some of Tarski's original axioms}\n\\label{table:1}\n\\center{\n\\begin{tabular}{l l l r l}\nA12 \\ \\ & Satz 3.1 \\ \\ & Reflexivity for betweenness & 4 steps \\\\   \nA14 & Satz 3.2 & Symmetry for betweenness &  4 steps \\\\\nA15 & Satz 3.5 &  Inner transitivity & 4 steps\\\\\nA16 & Satz 3.7 & Outer transitivity & 16 steps\\\\\nA17 & Satz 5.3 & Inner connectivity & 15 steps\\\\\nA18 & Satz 5.1 & Outer connectivity & 121 steps\n\\end{tabular} }\n\\vskip -0.3cm \n\\end{table}\n\nAnother theorem about betweenness   occurs as a \nchallenge problem in \\cite{wos1988}, namely, the ``five-point theorem'':\n$$ {\\bf T}(z,w,v) \\land {\\bf T}(z,y,v) \\land {\\bf T}(w,x,y) \\ \\rightarrow\\  {\\bf T}(z,x,v).$$\nThis  theorem does not occur in Szmielew's manuscript, \nbut it has a proof with 6 steps using the theorems mentioned above,\nfound completely mechanically. \n \n\n\\subsection{Midpoints, perpendiculars, and isosceles triangles}\nThe ``midpoint theorem'' asserts that every segment has a midpoint.  The traditional Euclidean \nconstruction involves the intersection points of two circles, but we are required to \nprove the theorem from A1-A9.  (Not even the parallel axiom A10 is to be used).  \nThis is a difficult problem, and was apparently not solved until Gupta's 1965 thesis\n\\cite{gupta1965}.  Two  important preliminary steps are the erection of a perpendicular\nto a line at a given point, and the ``Lotsatz'', which says we can drop a perpendicular\nto a line from a point not on the line.  Remember this must be done without circles!\n A clever observation of Gupta was \nthat one can fairly easily construct the midpoint of $ab$ if $ab$ is the base of an isosceles\ntriangle (only two applications of inner Pasch are needed).  This plays a key role in \nthe proof of the Lotsatz.   The two theorems on perpendiculars are used to construct\nthe midpoint.  Of course, once we have midpoints and perpendiculars, it is trivial to \nshow that every segment is the base of an isosceles triangle; that theorem does not \neven occur explicitly in \\cite{schwabhauser}.  An important lemma used in the proofs\nof these theorems is the  ``Krippenlemma'', a key result in Gupta's thesis.  For a diagram \nand formal statement, see Lemma~7.22 of \\cite{schwabhauser}, pp.~53--54.  \nTable~\\ref{table:2} shows the lengths of our proofs of these theorems, obtained using lemma adjunction and hint injection.\n \n\\begin{table}[ht]\n\\caption{Proofs of Gupta's theorems found using lemma adjunction and hint injection}\n\\label{table:2}\n\\center{\n\\begin{tabular}{l l r l}\n Satz 7.22 \\ \\ & Krippenlemma & 96 steps\\\\\n Satz 7.25 & Base of isosceles triangle has a midpoint &  113 steps\\\\\n Satz 8.18 & Lotsatz: there is a perpendicular to a & \\\\\n & line from a point not on the line & 227 steps \\\\\n Satz 8.21a & There is a perpendicular to a line through  \\\\\n &  a point on the line on the opposite side from & \\\\\n &  a given point not on the line. & 106 steps \\\\\nSatz 8.24b & Given segment $ab$ and perpendiculars $ap$ and $qb$,    \\\\\n& and point $t$ on line $ab$ between $p$ and $q$, \\\\\n&with $ap\\le qb$, then  segment $ab$ has a  midpoint. & 201 steps \\\\\nSatz 8.22 & Every segment has a midpoint & 22 steps\n\\end{tabular} }\n\\vskip-0.3cm\n\\end{table}\n\\medskip\n\nWe obtained a 108-step proof of the Krippenlemma using the subformula strategy, which \nworks without reference to the steps of the book proof.  It took 3,189 seconds to find.\nWe were surprised to find such a long proof completely mechanically!   \nNone of the other theorems listed here could be \nproved by the subformula strategy in one hour, although, \nas described in \\S\\ref{section:hardtheorems}, we did find several more proofs of \nother difficult theorems.\n\n\n\n\\subsection{The diagonals of a rhomboid bisect each other}\nA rhomboid is a quadrilateral whose opposite sides are equal.\nOne of the challenges in \\cite{wos1988} (see p.~214) solved by Quaife,\nwas to prove that the diagonals of a rectangle bisect each other. \nA more general problem is found in Satz~7.21, which asserts that \nif the diagonals of a rhomboid meet, then they bisect each other.  Quaife\nalso proved this theorem with \\Ott. \nOur proof of this theorem has 26 steps.  \nNote that no upper-dimension axiom is used, so it is necessary to assume the diagonals \nmeet, since otherwise the theorem fails in 3-space.  Even though 26 steps is fairly short,\nwe could not find this proof completely mechanically, even with the subformula strategy. \n\n\\subsection{Inner and outer Pasch}\nThe proof that inner Pasch implies outer Pasch (using A1-A6 and A8) was one of the \nmajor results of Gupta's thesis, and enabled Szmielew to replace outer Pasch by \ninner Pasch as an axiom.  This theorem was one of Quaife's four challenges.  It\nis Satz~9.6 in \\cite{schwabhauser}.   The proof posted on our archive is 111 steps,\npreceded by proofs Satz~9.4 and Satz~9.5 of 57 and 45 steps, respectively. Satz 9.5 is the ``plane\nseparation theorem'',  important in its own right.  \n\n\\subsection{Hilbert's axioms}  \nHilbert's theory can be interpreted in Tarski's, using pairs of \npoints for lines and ordered triples of points for angles and planes.   His \naxioms (so interpreted)\n all turn out to be either axioms, theorems proved in \\cite{schwabhauser}, or \nextremely elementary consequences of theorems proved in \\cite{schwabhauser}.   \n  The theorems of \\cite{schwabhauser}\nneeded are 2.3,2.4,2.5,2.8; 3.2,3.13;6.16, 6.18; 8.21, 8.22; 9.8, 9.25, 9.26, 11.15, 11.49;\nand   Hilbert's parallel axiom is Satz 12.11.  We have posted {\\textsc OTTER\\ } proofs of all these\ntheorems.  Narboux and Braun have proof-checked Hilbert's axioms in Tarskian geometry, using Coq \\cite{narboux2012}.\n\n\\section{1992 {\\em vs} 2015: improved strategies or faster computers?}\n  \nThe research reported here shows how much progress has occurred in automated reasoning in that \ntime period.  Indeed, approximately thirty years ago, almost all of the theorems cited in this article were out of reach.\nThe question arises whether this advance might be due simply to the increased memory capacity and \nspeed of modern computers.   Perhaps Quaife, equipped with one of our computers, would have found \nthese proofs?  Perhaps we, constrained to run on a computer from 1990, might not have found them?\nWe argue that this is not the case:  the improvements are due not to faster hardware but to \nthe techniques described above, namely generating partial proofs (of intermediate steps) and using \ntheir steps as hints; using the right combination of inference rules and settings; using \ntautology adjunction to help with proofs by cases; divide-and-conquer; and finally,\nthe spectacular success of the subformula strategy.    We note that Quaife\ndid have (in fact, invented) the technique of giving {\\textsc OTTER\\ } the diagram.  We did not actually try \nto run on a 1990 computer, and we do not doubt that it would have been painful and discouraging;\nbut we think the main credit should go to Veroff's invention of hints, and the uses of hints\ndeveloped by Wos and applied here.  Proofs of length more \nthan 40 were out of reach in 2014 for completely mechanical derivations, and could then only \nbe found by lemma adjunction and hint injection.  But in 2015 several proofs of length more \nthan 90 were found completely mechanically by the subformula strategy,  using the same \ncomputer we used in 2014.  \n\n\n\\section{Proof checking {\\em vs.} proof finding}\n``Proof checking'' refers to obtaining computer-verified proofs, starting with human-written proofs.\n``Proof finding'' refers to the traditional task of automated deduction, finding a proof by searching a large space of possible proofs,  either without possessing a proof or without making use of a known proof.\nOur work shows that this distinction is not as clearcut as it might seem. \n  If we have a proof in hand (whether generated by \nhuman or machine), and we enter its steps as hints, with a low {\\tt max\\_weight}, we \nforce {\\textsc OTTER\\ } to find a proof containing mostly the same formulas as the proof in the hints.  \n(The order of deductions might be different.)   This happens immediately if the steps\nwere machine steps, and we have explained above the techniques we use if the original steps\nare human proof steps.   By those methods, we can almost always ensure that {\\textsc OTTER\\ } finds \na proof, if we have a proof in hand.  One could plausibly claim that this is proof checking,\nnot proof finding. \n\nOn the other hand, today's proof checkers are increasingly incorporating (calls to) external\ntheorem provers to help reduce the number of steps a human must supply.   If a theorem prover\ncan get from the existing proof to the next goal,  then the returned proof can often \nbe translated into the proof checker's language and checked.  For example, the Sledgehammer\nprogram \\cite{paulson2010} is used with Isabelle, and HOLY-Hammer \\cite{holyhammer}\n is used with HOL Light. \nIn this way, proof checking is incorporating elements of automated theorem proving. \nSee \\cite{blanchette2015} for a survey of this research and further references.\n\n\n\\section{Developing an entire theory using a theorem prover}\nWhen we made our first report on this work \\cite{beeson2014-wos}, we \nhad two hundred hand-crafted input files,  one for each theorem.   These files \nhad been made by cutting and pasting lists of axioms,  adding the positive form \n(Skolemization) of the last theorem by hand,  and often had been heavily edited\nduring the process of applying our proof-finding methods.  There was plenty\nof room for human error in the preparation of these files; and as it turned out \nin 2015,  they contained numerous errors;  none irreparable, but one wishes \nfor a standard of perfection in computer-generated proofs.  \n\nThinking about the sources of error and the means to remedy them,  we realized\nthat a major source of error could be the hand-translation from the negated\nform of a theorem (used in the set of support to prove it) to the ``positive form''\nor Skolemization,  in which the theorem is entered into subsequent input files \nfor use in proving further theorems.   We therefore prepared a ``master list''\ncontaining, for each theorem,  the positive and negative forms and, in addition,\nthe ``diagram equations'' defining the diagram for that theorem, if any.\nThe master list also contains the names of Skolem functions to be used in \nSkolemizing that theorem,  if the theorem contains (existential) variables.\nWe also optionally included a list of tautologies that were needed to prove \nthat theorem,  if we could not eliminate the tautologies.\n\nThe plan was that the master list could then be used to mechanically generate \ninput files.   The correctness issues would then have been conveniently divided:\n\n\\begin{itemize}\n\\item  Is the positive form of each theorem in the master list really the correct\nSkolemization of the negative form?\n\\item Does the diagram entry in the master list have the form of an equation with a new \nconstant on the left?  And on the right, does it mention only previously-introduced Skolem functions?\n\\item Is each Skolem function introduced in exactly one theorem? \n\\item  Does the master list actually correspond to the statements of the theorems in the book?\n\\item  Are the input files being correctly generated?\n\\item  Do they all produce proofs?\n\\item  And of course:  are the proofs produced by {\\textsc OTTER\\ } correct? (This is not a novel\nissue and will not be discussed.)\n\\end{itemize}\n\nThe question whether the diagram in the master list corresponds to the book's\ndiagram is not relevant.  If one gets a proof with any diagram whatsoever (meeting the \nabove conditions),  it is fine.\nAdding equations defining a new constant is conservative; that is,  no new theorems can be \nproved by adding such axioms.  However, the meaning of ``new constant'' is not quite\nas obvious as one might think.  Consider, for example, the diagram for Satz 3.17.\nThere are two equations:\n$$ e = ip(c,b1,a1,a,p)$$\n$$ d = ip(c,b,a,b1,e)$$\nThe new constants are $d$ and $e$.  The equation for $d$ has $e$ on the right, but \nthat is okay.  What would be wrong is if the equation for $d$ had $d$ on the right,\nor if the equation for $e$ had $d$ on the right and the equation for $d$ had $e$ on the right.\nWhat we mean by ``new constant'' is  ``constant not occurring in the theorem or the \nright sides of the previous diagram equations.''\n\nSimilarly, it does not matter what one puts in the hints list.  If one gets a proof,\nit is a proof, regardless of what was in the hints list.  In our mechanical generation \nof input files,  we make use of the (possibly unreliable) proofs we found in 2012.\nIf we cannot find a proof without hints, then we use the steps of an existing (alleged)\nproof as hints.   Thus, the sources for generating input files are \n\n\\begin{itemize}\n\\item The master list\n\\item The existing (alleged) proofs\n\\item The program that produces the input files from the master list and proofs\n\\end{itemize}\n \n\n\\subsection{The master list}\nThe master list is a text file, a piece of code in the PHP programming language.\nThat file begins with class definitions of classes called {\\em Theorem}, {\\em Definition},\nand {\\em Axiom}.  The class {\\em Theorem}, for example, looks like this (but \nfor simplicity we here omit the constructor function).  (Those unfamiliar with PHP\nshould just ignore {\\em var} and the dollar signs.)\n\n\\begin{verbatim}\nclass Theorem\n{ var $name;\n  var $PositiveForm;     \/\/ array of clauses \n  var $NegatedForm;      \/\/ array of clauses\n  var $Diagram;          \/\/ array of diagram equations\n  var $SkolemSymbols = \"\"; \/\/ array of Skolem symbols introduced\n  var $Cases = \"\"; \/\/ cases we used to split the problem, if any.\n}\n\\end{verbatim}\n\n\\hyphenation{Tarski-Theorems}\n\\noindent\nAfter that come arrays defining {\\em TarskiAxioms}, \n{\\em TarskiDefinitions}, and  {\\em TarskiTheorems}.  \nThe {\\em TarskiTheorems} array is the heart of the master list.\nIt starts off like this:\n\n\\begin{verbatim}\n$TarskiTheorems = array(\n\tnew Theorem( \"Satz2.1\", \n\t\tarray(\"E(xa,xb,xa,xb)\"), \n\t\tarray(\"-E(a,b,a,b)\")\n\t),  \n\tnew Theorem( \"Satz2.2\", \n\t\tarray(\"-E(xa,xb,xc,xd) | E(xc,xd,xa,xb)\"),\n\t\tarray(\"E(a,b,c,d)\",\"-E(c,d,a,b)\")\n\t), \n\\end{verbatim}\n\n\\noindent\nAs one can see, the master list is human readable, not much encumbered\nby the occurrences of {\\tt new} and {\\tt array} that make it into PHP code.\nThe master list was prepared by hand, by the following method:\nFirst, we entered the negated form of a theorem, by copying \nit from {\\tt list(sos)} of the existing hand-crafted input file.\nIt was, we realized, important that the constants in the master list\nhave the same names as the constants in the corresponding \nexisting proofs,  to avoid incompatibilities when those proofs are\nused as hints. (Of course, that  matters only for the theorems that\ncannot be proved without hints.)%\n\\footnote{Therefore we could not satisfy a request that the constants\nhave names unique across all files.  Researchers with that wish can \nwrite PHP code to append a number to each constant.  Then, however, they \nwon't be able to use hints extracted from our proofs.\n}\n\nThen, we computed (by hand) the positive form. \nHere constants such as $a$ were changed to variables such as $xa$.\nIn some cases we had constants $cu$ or $cx$, where the book proof \nhad $u$ or $x$;  in computing the positive form these become $u$\nor $x$ again, rather than $xcx$ or $xcu$.  Although this was\noriginally done by hand, we checked it (and other things about the \nmaster list) with a PHP program {\\em TestMasterList.php}.  This program\ncarries out Skolemization  on a text-only basis, without parsing\nterms or clauses.  (Any parse errors will turn up when we run the resulting input files.)\nThe program copes correctly when the negative form has one clause that is a disjunction.\nIn the entire Tarski corpus, there are only two theorems whose negative form contains\ntwo disjunctions.  These were Skolemized by hand and checked carefully.\nThat the resulting input files do result in proofs is another positive indication.\nThis mechanical check of correct Skolemization is our answer to the first\nitemized correctness concern above.\n\nOne issue was the order of arguments to Skolem functions.  In 2015 we could \nnot change the conventions that we adopted in 2013, since that would have rendered the \nhints obtained from our existing proofs useless.  That necessitated some \nnot very beautiful code that specifies the order per Skolem function.  The correctness\nof this code is evidenced by the fact that it did work with hints obtained from \nexisting proofs.  Luckily, in 2013 when we worked on one file at a time, we were consistent\nabout the order of arguments of the same Skolem function in different files.\n\nThe second concern was whether each diagram clause has the form of an \nequation with a new constant on the left, in the precise sense defined above.\nOur program {\\tt TestMasterList.php} checks this condition mechanically.  It \nalso checks that each Skolem function is introduced by exactly one theorem and \nis not used (even in a diagram) before it is introduced. \n\n\nThe third concern was whether each theorem in the master list corresponds\nto the version in the printed book.  Several considerations arise here:\nFirst, Szmielew treated lines as sets of points, so her treatment is not \nstrictly first order. We translated ``line $L$'' to two points $p,q$ and \nthe literal $p \\neq q$, and $x \\in L$  to $Col(p,q,x)$,  where $Col$ means\n``collinear'' and is defined in terms of betweenness.  The correctness issue\nhas to be understood modulo this translation.  Second, as a result of that\ntreatment of lines,  some simple theorems had to be formulated in our development\nthat are not in the book.  For example, perpendicularity of lines becomes\na 4-ary relation $perp(a,b,c,d)$, or $ab \\perp cd$,  and we have to prove that, for example, \n$a$ and $b$ can be interchanged and $c$ and $d$ can be interchanged, or both.\nThis gives rise to some theorems that do not occur in the book.  But for those\ntheorems that do occur in the book, one can check by hand that they do correspond \nto the book.\n Since the Skolemization has been checked by machine,\nit is only necessary to check that the negative form corresponds to the book.\nThere are only a few cases where this is at all problematic.  Lemma~9.4, p.~68\nis an example.  The question whether the formal theorem corresponds to \nthe intended theorem arises also in using a proof checker. One\ncannot check by means of a computer program that the master list\n corresponds to the book.  Mathematics books\nwritten in the twentieth century do not contain machine-readable formulas,\nand even if they did,  we would still need to know if those really represented\nSzmielew's intent.  (There are a few typographical errors in Szmielew.)\n\n\\subsection{Size and availability of the master list}\nWe briefly contemplated including the master list as an appendix to this paper.\nWe discovered that, when printed, it is 43 pages long.  Moreover, in order\nto be useful, it must be in electronic form.  It could well turn out that \nthe master list is the most useful product of this research, since it can be \nused immediately by anyone else wishing to study the Tarski axioms and Szmielew's\naxiomatic development from them.  We therefore plan to post a separate\ndocument to the ArXiv containing the master list in the \\TeX\\ source.  Then,\nas long as the ArXiv lives, the master list will be available. Of course\nthe master list, as well as the PHP programs used for this research, will be\navailable at \\cite{tarski-archive} for at least the next few years.\n\n\\subsection{Generating input files mechanically} \\label{section:mechanicalinputfiles}\nOnce the master list is correct, it is a routine programming exercise\ninvolving only textual (string) manipulations to generate input files.\nThe algorithm is simple.  Consider, for example, the function\nwith the specification given in the following comments.\n\n\\begin{verbatim}\nfunction InputFileAsArray($t, $diagrams, $sos, $settings, $cases)\n\/\/ $t should be a member of $TarskiTheorems\n\/\/ Generate the lines of an input file for proving $t\n\/\/ and return an array of strings containing those lines.\n\/\/ Make sure no line exceeds 80 characters.\n\/\/ include the diagram if $diagrams is not false.\n\/\/ include the members of $t->Cases if it is an array\n\\end{verbatim}\n\nHere {\\tt \\$settings} is an array of the {\\textsc OTTER\\ } settings to be used\nin this file; the strings in the arrays\n {\\tt \\$t->Diagrams}, {\\tt \\$t->NegatedForm }, and {\\tt \\$t->Cases}\nare ready to be copied out of {\\tt \\$t} into the appropriate places in a file\ntemplate.   All the function has to do is \ncreate an array {\\tt \\$ans} and add these things to that array in the right\norder,  interspersing template lines like ``{\\tt list(sos).}''  at the appropriate places.\n\nThis function, however, doesn't put any hints into the file it is creating.\nThat is done by the following code.\n\n\\begin{verbatim}\nfunction InsertHintsIntoInputFile($theorem_name, $outdir, $indir)\n\/\/ Get hints from the proof, i.e. from the .prf file  \n\/\/ found in directory $outdir, and insert them in \n\/\/ list(hints) in the input file.\n\\end{verbatim}\n\nThe heart of this program is {\\tt GetHintsFromProof},  which we have been \nusing since 2012 to extract hints by hand from a proof; we have confidence \nin the correctness of that program based on long use and on many comparisons\nof the output with the input.  Again, the correctness of that program doesn't\nreally matter:  whatever we put into the hints list, if we get a proof,\nthe proof is right.   In order to ensure correctness of any proofs obtained, \nthe only thing that really matters about {\\tt InsertHintsIntoInputFile}\nis that all it does to the input file is add a hints list.  (Of course,\nto actually obtain some proofs, it had better be correct in other ways.)\n\n\\subsection{Experimenting with many files at a time}\n\nWith the aid of the PHP programs discussed above, we wrote short\nPHP programs that test a given set of files (such as, for example, those belonging to Chapter 8),\nwith a given set of settings for the theorem prover, and given values of variables\ncontrolling whether we will use hints extracted from a proof or not, whether we will \nuse diagrams, whether we will use the case listed in the master list,\nwhether we will try to prove theorems that have already been proved or only ones\nthat have not been proved yet, and so forth.  We also could have the settings used on each \ninput file depend on previously-obtained results, the record of which was kept in \na machine-readable (but handwritten) file.   The PHP tools that we used are\n available, with descriptions, at \\cite{tarski-archive}. \n\n\\subsection{Summary of the correctness discussion}\nWe were able to replace the hand-crafted input files we posted in 2013--2014\nby mechanically generated input files.  In the case of difficult theorems that \nwe did not prove completely mechanically, our mechanically generated files used\n hint injection,  with hints extracted\nfrom the old proofs.   Sometimes, if the proof was shorter, we iterated,  injecting\nhints obtained from the new proof.   In most cases the resulting proofs were\nshorter than the ones we had before.  As discussed above, if our PHP programs for \ngenerating those files are correct and the Skolemizations in the master list are correct,\n then each theorem has been deduced from the \nones preceding it in the master list.  In other words, it is a conclusion, not a \nhypothesis, that the theorems are ordered correctly in the master list.   \nThe correctness of the Skolemizations in the master list has been machine-checked\n(except for the one theorem with two disjunctions, which was checked manually).  \nThat the master list correctly corresponds \nto the book has been checked manually (the only possible way to check it).\n\n\n\\section{Easy and hard theorems and their proofs} \\label{section:hardtheorems}\nWe divide the Tarski theorems into ``easy'' and ``hard'' theorems, according to the somewhat\narbitrary rule that a theorem is ``easy'' if it has a proof of 40 or fewer steps. The idea\nbehind the choice of the number 40 is\n that we were able to prove many of the easy theorems completely automatically,\nwhile to prove the hard theorems,  we had to use hints.   There are 214 \ntheorems all together, of which 29 are ``hard'' by this definition.  We proved about \ntwo-thirds of the \neasy theorems mechanically, and we proved four very hard theorems completely mechanically\ntoo, using the subformula strategy,  leaving 25 theorems that required hints.  \nHere ``mechanical'' refers to proofs\n obtained with no \nreference to the book proof. An additional 14 proofs were found by ``giving the prover\nthe diagram'',  without additional use of the book proof.   That raises ``about two-thirds''\nto ``about three-quarters''.\n \nThe rest of the easy proofs, and all the hard proofs, \nwere proved using ``hints''.  These hints came from using some steps from the \nbook proofs, and from  proofs of related theorems obtained by the \nmethods described above (lemma adjunction and eliminating cases).  \n\nIf we had to measure the strength of \\Ott\\ on this problem set by a single number, we would say \nthat 40 is the answer:  we expect that \\Ott\\  has a good chance to automatically find \nproofs for theorems that have proofs of length less than or equal to 40,  and\n only occasionally to find \nproofs for theorems much longer than 40.   \n \n \nThe book starts   with easy theorems.  Chapters 2 through 6 contain 81 \ntheorems.  Of those only two are hard:  Satz 4.2 (44 steps) and Satz 5.1 (127 steps).\n  Satz~5.1 expresses in Tarski's language that \nif we fix $a$ and define $x \\le y$ by ${\\bf T}(a,x,y)$, then $x \\le y \\lor y \\le x$.  \nThis was first proved from Tarski's axioms in Gupta's thesis \\cite{gupta1965}.\nWe had to use diagrams on six of the remaining theorems.  We \n used hints on two ``easy'' theorems:   Satz~4.16 (23 steps) \nand Satz~5.12b (16 steps).  We could prove Satz~4.16 without hints,\nbut we had to wait 95696.25 seconds (more than 26 hours).\nSatz 4.5 required the subformula strategy in order to find a proof automatically.\nSatz 6.16b appeared to be hard at first, but the \nsubformula strategy found a short proof.   Of the remaining 69 theorems, we proved 67 \ncompletely mechanically, without hints or diagrams, using our default settings.   The \nother two are Satz 3.1 and Satz 4.6.  The former we could prove mechanically\nwith a diagram,  but we could also prove it without a diagram if we put everything\nin the set of support, instead of putting the axioms in list(usable).  Satz~4.6,\nstrangely enough,  could be proved immediately after adding two instances of \nequality axioms explicitly to the set of support,  but we could not get a proof \nwithout adding those equality axioms.  Of course, once we found the proof by adding\nequality axioms, we could put the steps of the proof in as hints and find a proof\nwithout explicit equality axioms. \n\nIn Chapters 7 and 8, there are 46 more theorems that we proved completely mechanically,\n8 more hard theorems, 3 short theorems on which we had to use hints (i.e., could \nnot prove mechanically even with a diagram),  and five theorems that we could \nprove mechanically with a diagram, but not without.   Of those eight hard theorems,\ntwo were proved mechanically using the subformula strategy: Satz 7.13 and \nSatz 7.22a.  The latter of these is the Krippenlemma,  one of the important theorems\nof Gupta's thesis.  That a 108-step proof of a major result could be found \ncompletely automatically was completely unexpected. \n\nChapter 9 gets harder.  We could prove only four theorems  completely mechanically;\ndiagrams were not needed and did not help with the other theorems.  There are five hard theorems,\nwhich we proved using hints. \nWe were also able to prove two of those hard theorems with the subformula strategy,\nincluding the gem of Gupta's thesis, that outer Pasch implies inner Pasch.  \nThere are also five short theorems that we could prove only by using hints, although\nfour of them have proofs between 35 and 40 steps, so they are ``borderline short.''\n\nChapters 10 gets easier again;  it is about reflection in a line\nand right angles, and the high point is the proof of Euclid's fourth postulate, that \nall right angles are congruent, in the form that two right triangles with congruent legs\nalso have congruent hypotenuses.  Although five proofs in Chapter 10 are longer\nthan 45 steps, only two are longer than 75 steps.  The chapter ends with a proof \nof Hilbert's triangle construction axiom.\n\nChapter 11 deals with the definition and properties\nof angles (as triples of points).  Only two proofs in Chapter 11 are longer than 40 steps.\n\nTable~\\ref{table:3} summarizes the results just described.%\n\\footnote{There are 214 theorems in our Master List; but the last three are all the \nHilbert parallel axiom, in two cases and in a combined statement. So really, there are 212\ntheorems, of which 29 are hard and 183 are easy.}\n\n\\begin{table}\n\\caption{One hundred eighty three theorems with proofs of length $\\le 40$}\n\\label{table:3}\n\\center{\n\\begin{tabular} { l r r r r}\n{\\bf Chapter} & {\\bf no diagram} & {\\bf diagram} & {\\bf Subformula strategy} & {\\bf  hints}   \\\\\n2 to 6 & 72 & 5 & 3 & 2\\\\\n7 & 20 & 0 & 1 & 1  \\\\\n8 & 16 & 6 & 2 & 2  \\\\\n9 & 5 & 0 & 3 & 2  \\\\\n8A & 6 & 1 & 0 & 1  \\\\\n9A & 6 & 0 & 0 & 4  \\\\\n10 & 6 & 1 & 0 & 3  \\\\\n11 & 3 & 1 & 0 & 6  \\\\\n12 & 0 & 0 & 0 & 5  \\\\\nTotal & 134 & 14 & 9 & 26\n\\end{tabular}\n}\n\\end{table}\n\n\\begin{table}\n\\caption{Twenty-nine hard theorems and their proofs}\n\\label{table:4}\n\\center{\n \n\\begin{tabular} {l l r r r }\n  {\\bf Theorem} & {\\bf description} & {\\bf sub. str.} & {\\bf hints}  & {\\bf book }\\\\\n  Satz 4.2  &  inner 5-segment theorem && 44  & 13\\\\   \n Satz 5.1 & trichotomy &&  127 &  70 \\\\\n  Satz 7.13 & refl. in pt. preserves congruence  & $(99, 921)$ & 72  & 28\\\\\n Satz 7.22a & Krippenlemma & $(108,3189)$& 96 & 27 \\\\\n Satz 7.25 & isosceles triangle has midpoint && 113 &34\\\\\n Satz 8.18 & Lotsatz (dropped perpendicular)  && 227 & 32\\\\\n Satz 8.21a & uniqueness of dropped perp   && 106& 5\\\\\n  Satz 8.22b & existence of midpoint   &&201 & 42\\\\\n   Satz 8.24a &  midpoint   details      && 171& 42\\\\\n    Satz 8.24b &  midpoint details     && 163& 42\\\\\n  Satz 9.3 & opposite-side ray theorem   && 52 & 14\\\\\n  Satz 9.4b & $r=s$ case of 9.4       &&44 &7\\\\\n    Satz 9.4c & second half of Lemma 9.4 in book       &&41 & ?\\\\\n   Satz 9.6 & inner Pasch implies outer Pasch  &$(98,186)$ &91& 27 \\\\\n    Satz 9.8 & opposite, same implies opposite  &$(63,1646)$ &48 &21\\\\\n    Satz 9.13 & transitivity of samesideline   && 71& 1?\\\\\n    Lemma 9.13f & (case 2)                     && 42 & ? \\\\\n     Satz 10.2a & existence of reflection in line  && 45 & 23\\\\\n     Satz 10.2b & uniqueness of reflection in line  && 49& 23\\\\\n     Satz 10.10 &  refl. in line preserves congruence                    && 72 & 26\\\\\n     Satz 10.12a & \tright triangles with common vertex & & \\\\\n     & $\\ldots$ have congruent hypotenuses & & 60 & 6\\\\\n      Satz 10.12b &  right triangle theorem (Euclid 4)  && 91 &14\\\\\n      Sata 10.15 &  perpendicular on given side of line && 74 & ? \\\\\n      Satz 10.16a & triangle construction  && 57 & 24?\\\\\n      Satz 10.16b & triangle uniqueness   && 60&22\\\\\n      Satz 11.3a & angle congruence theorem  && 43& 18\\\\\n      Satz 11.15a &   angle transport(existence)  && 78 & 1? \\\\\n      Satz 11.15b &   angle transport (uniqueness)  && 78 & 1? \\\\\n      Satz 12.11  &  Hilbert's parallel axiom && 105 & 18?\n\\end{tabular} }\n\\end{table}\n\n\\FloatBarrier \n\n\nTable~\\ref{table:4} shows the hard theorems from the entire book,  with the \nnumber of steps in the proof(s) that we obtained.  A paired number like $(99,921)$ \nindicates  \na proof found mechanically using the subformula strategy, of length 99 and found in 921 seconds.\nThe last column is the number of steps in the book proof.  A question mark in this column indicates\nthat the book proof contains non-trivial gaps or is omitted. \nNot counting the lines with a question mark, we have 2032 {\\textsc OTTER\\ } steps\ncorresponding to 516 book steps, giving a de Bruijn factor (the ratio of the two) \nof 4. \n\nOf the 212 theorems we tried to prove, we proved 147  completely automatically \n(without any use of the book proof or diagram);   we needed the \nsubformula strategy for 13 of those proofs.   We were able to prove 15 more theorems\nby using the diagram, without using the steps of the book proof.  And when we allowed\nourselves to use some of the steps of the book proof as hints,  we achieved a 100\\% success rate.\n \n\n \n \n\\section{What about Prover9? Or E, {\\textsc SPASS}, Vampire?}\n \n \n\nNearly every person with whom \nwe have discussed this work, including the referees, has suggested that another prover\nor provers might do better in some respect than \\Ott.  It was not our purpose \nto evaluate the performance of any theorem prover or to compare the performances of \ndifferent theorem provers.  Our purpose was to show that a combination of strategies \nin {\\em some} theorem prover could prove all the approximately 200 theorems in Tarski\ngeometry.  \n\n\n In \\cite{narboux2015b},  Durdevic {\\em et.al.} tried E, {\\textsc SPASS}, and Vampire\non the theorems from Szmielew, with a 37\\% success rate.   The corresponding \npercentage in our work, as discussed in the previous section, was 62\\% without the subformula strategy, and \n76\\% with the subformula strategy.\n These numbers may not be exactly comparable, since the set of theorems may not have been identical.\nThey used a very low time\nlimit,  but most of the theorems we proved mechanically took less than 20 seconds, so \nthe few that took longer did not change our percentage much.   We attribute our higher \nsuccess rate (without any strategy at all)  \nto a good choice of settings for \\Ott.   The fact that we did achieve a 100\\% success\nrate with the use of lemma adjunction and hint injection  is another reason for not using\nanother prover:  we could not improve on that success rate. \n\n\nOur master list of the Tarski theorems is now available for others to use; also our \nPHP programs for manipulating the master list and files are available.  Conducting experiments\nwith other theorem provers then involves only modifying the template for an input file\nthat is used by {\\em InputFileAsArray}.  \n  We have not conducted such experiments, but others are doing so as this article goes\nto press,  and will report on their experiments in due course.  We note that in order\nto duplicate our result with \\Ott (134 out of 212 proved mechanically),\n it was necessary to use the default settings we used \nfor \\Ott, which can be found in the PHP code supplied on our website.   \nTentative results for Vampire and E are 154 and 142 out of 212.\n\n\\section{Summary}\n\\vskip-0.3cm\nWe used {\\textsc OTTER\\ } to find proofs of the theorems in Tarskian geometry in the first nine \nchapters of Szmielew's development in Part I of \\cite{schwabhauser}.  Those theorems \ninclude the four unsolved challenge problems from Quaife's book\\cite{quaife1992}, and \nthe verification of Hilbert's axioms.  Of the 214 theorems, we proved\n62\\% mechanically and without strategies (except for a good choice of settings), \nand by the use of the subformula strategy we increased that to 76\\%.  By using \nlemma adjunction and hint injection, which depend on starting with a book proof, \nwe achieved a 100\\% success rate.\n\n\nMechanically generated input files and the resulting proofs for all the theorems that we have proved  \nare archived at \\cite{tarski-archive}. \n\n \n\n \n\n \n\n \n \n \n\n\n \n\n\n\\begin{acknowledgements}\nThis material was based in part on work supported by the U.S. Department of Energy,\nOffice of Science, under contract DE-ACO2-06CH11357.\n\\end{acknowledgements}\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{#1}\\setcounter{equation}{0}}\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n\n\\newdimen\\tableauside\\tableauside=1.0ex\n\\newdimen\\tableaurule\\tableaurule=0.4pt\n\\newdimen\\tableaustep\n\\def\\phantomhrule#1{\\hbox{\\vbox to0pt{\\hrule height\\tableaurule width#1\\vss}}}\n\\def\\phantomvrule#1{\\vbox{\\hbox to0pt{\\vrule width\\tableaurule height#1\\hss}}}\n\\def\\sqr{\\vbox{%\n  \\phantomhrule\\tableaustep\n  \\hbox{\\phantomvrule\\tableaustep\\kern\\tableaustep\\phantomvrule\\tableaustep}%\n  \\hbox{\\vbox{\\phantomhrule\\tableauside}\\kern-\\tableaurule}}}\n\\def\\squares#1{\\hbox{\\count0=#1\\noindent\\loop\\sqr\n  \\advance\\count0 by-1 \\ifnum\\count0>0\\repeat}}\n\\def\\tableau#1{\\vcenter{\\offinterlineskip\n  \\tableaustep=\\tableauside\\advance\\tableaustep by-\\tableaurule\n  \\kern\\normallineskip\\hbox\n    {\\kern\\normallineskip\\vbox\n      {\\gettableau#1 0 }%\n     \\kern\\normallineskip\\kern\\tableaurule}%\n  \\kern\\normallineskip\\kern\\tableaurule}}\n\\def\\gettableau#1{\\ifnum#1=0\\let\\next=\\null\\else\n\\squares{#1}\\let\\next=\\gettableau\\fi\\next}\n\n\\tableauside=1.0ex\n\\tableaurule=0.4pt\n\\def\\mathbb{E}{\\mathbb{E}}\n\\def{\\bf{r}}{{\\bf{r}}}\n\\def\\MM#1{{ $\\ll\\!\\!\\ll\\!\\!\\ll$ \\bf MM: {\\it #1} $\\gg\\!\\!\\gg\\!\\!\\gg $}}\n\\def\\PP#1{{ $\\ll\\!\\!\\ll\\!\\!\\ll$ \\bf PP: {\\it #1} $\\gg\\!\\!\\gg\\!\\!\\gg $}}\n\\def\\ND#1{{ $\\ll\\!\\!\\ll\\!\\!\\ll$ \\bf ND: {\\it #1} $\\gg\\!\\!\\gg\\!\\!\\gg$}}\n\n\\newcommand{\\figref}[1]{Fig.~\\protect\\ref{#1}}\n\n\n\n\\title{\\boldmath New results in ${\\cal N} =2$ theories from non-perturbative string}\n\n\n\\author{ Giulio Bonelli$^a$, Alba Grassi$^b$ and Alessandro Tanzini$^a$}\n\n\n\\affiliation{\n$^a$ International School of Advanced Studies (SISSA), \\\\\nvia Bonomea 265, 34136 Trieste, Italy and INFN, Sezione di Trieste\\\\\n\\\\\n$^b$International Center for Theoretical Physics,\\\\\n ICTP, Strada Costiera 11, Trieste 34151, Italy \n and INFN, Sezione di Trieste\\\\}\n\n\n\n\n\\emailAdd{bonelli@sissa.it, agrassi@ictp.it, tanzini@sissa.it}\n\\preprint{\n\\begin{flushright}\nSISSA  14\/2017\/FISI-MATE \\\\\n\\end{flushright}\n}\n\n\n \n\n\\abstract{We describe the magnetic phase of  $SU(N)$ ${\\cal N}=2$ Super Yang-Mills theories in the self-dual $\\Omega$ background in terms of a new class of multi-cut matrix models.  These arise from \na non-perturbative completion of topological strings in the dual four dimensional limit which engineers the gauge theory in the strongly coupled magnetic frame.  \nThe corresponding spectral determinants provide natural candidates for the $\\tau$-functions of isomonodromy problems\nfor flat spectral connections associated to the Seiberg-Witten geometry.\n  }\n\n\n\\begin{document}\n\\maketitle\n\n\\flushbottom\n\n\n\\section{Introduction}\n\nSince the seminal work of Seiberg and Witten (SW) \\cite{sw2} a lot of progress has been made to understand ${\\cal N}=2$ gauge theories in four dimensions. \nA crucial progress has been obtained by applying equivariant localisation methods to the supersymmetric path integral coupled to the so-called $\\Omega$-background \\cite{n, Flume:2002az, Bruzzo:2002xf}. This reduces it to a combinatorial expression which has been subsequently linked to quantum integrable systems \\cite{ns} and to two-dimensional Conformal Field Theory \\cite{agt}.\nLocalisation methods have been applied so far in the weak coupling limit of the gauge theory rebuilding the low-energy effective field theory of SW in the electric\npolarization of the special K\\\"aehler manifold of Coulomb vacua. More recently \\cite{gil1,ilt1,blmst}  it has been realized that the $SU(2)$  Nekrasov and Okounkov (NO) \\cite{no2} partition functions in the self--dual  $\\Omega$-background are $\\tau$-functions of isomonodromy problems related to the corresponding SW geometry, realized as the spectral curve of Hitchin's integrable system.\nIn this case one can use the well-known relation between isomonodromic deformation problems and Painlev\\'e equations to reduce\nthe evaluation of the NO partition function of gauge theories at strong coupling to the calculation of the relevant $\\tau$-function in the long-distance expansion \\cite{blmst}. \n\nOn the other hand, four dimensional gauge theories can be engineered by using topological string theory \\cite{kkv}. Actually, the latter is richer, in particular at the non-perturbative level there are new effects arising from the embedding of the gauge theory in string theory. Along this line of research a lot of work has been done during the last decade to understand topological string beyond perturbation theory starting with the seminal works \\cite{mmopen, Marino:2007te, mmnp}.\n In particular in \\cite{ghm},  in the spirit of large $N$ dualities,  a non--perturbative formulation of topological string on toric Calabi-Yau (CY) has been proposed. This formulation has proved to be extremely rich and constructive leading to several new results and applications in various related fields such as integrable systems \\cite{hm,Franco:2015rnr,Marino:2016rsq}, supersymmetric gauge theories \\cite{bgt, hel} and condensed matter \\cite{Hatsuda:2017zwn, Hatsuda:2016mdw}. \nThe non-perturbative  proposal of \\cite{ghm}  was originally formulated only for CYs whose mirror curve has genus one but it has been  extended to higher genus mirror curves in \\cite{cgm2}.\n\nIn \\cite{bgt} a link between this non-perturbative completion of topological string and isomonodromy problems arising from four dimensional gauge theories has been found in the special case of $SU(2)$ Super Yang-Mills (SYM). The relevant isomonodromy problem in this case is the one associated to Painlev\\'e $\\rm III_3$ (also called \n$ \\rm III^{D_8}$) equation whose $\\tau$-function is known since a long time \\cite{zamo} to admit a Fredholm determinant description.\nUpon a suitable four dimensional scaling limit, the non-perturbative completion of topological string has been shown to be directly related to the Fredholm determinant above. This produces a matrix model presentation of the gauge theory partition function in the strongly coupled magnetic frame  and provides an operator theory interpretation of the self--dual $\\Omega$ background  ($\\epsilon_1=-\\epsilon_2=\\epsilon$). \nMoreover, it also provides exact S-duality transformation formula for the Nekrasov partition function with self-dual $\\Omega$-background including non-perturbative corrections\nin $\\epsilon$.\n\nThe purpose of this paper is to extend these results to $SU(N)$ gauge theories. More precisely, in section \\ref{su2rev} we review the  consequences of the genus one proposal for $SU(2)$ theories  and we provide the exact S-duality transformation for the corresponding gauge theory partition function. In section \\ref{nps}, by following the general prescription of \\cite{cgm2},  we derive the matrix models computing the topological string partition function on the $Y^{N,0}$ geometries.  The result is given by the $N-1$ cut matrix model shown in \\eqref{mmsu35d}. Then, in section \\ref{sun4}, we perform  the so-called {\\it{dual}} four dimensional limit \\cite{bgt} on these models and we make contact with ${\\cal N}=2$ $SU(N)$ SYM in the four dimensional self-dual  $\\Omega $ background \\cite{no2}. More precisely we find that the partition function in the magnetic frame is given by \n\\begin{equation} \\label{su3mmintro} {\\begin{aligned} Z_{\\rm N}^{\\rm 4d}(M_1,\\cdots,M_{N-1})=&  {1 \\over M_1!  \\cdots M_{N-1}!} \\int {{\\rm d} ^M x\\over (2\\pi)^M} \\prod_{j=1}^{N-1} \\prod_{i_j\\in I_j}{\\rm e}^{-  \\frac{N \\Lambda }{ \\pi ^2 \\epsilon}{ \\sin \\left(\\frac{\\pi  j }{N}\\right)} \\cosh(x_{i_j})} \\\\\n&   \\times {\\prod_{ 1\\leq i<j \\leq M}2\\sinh\\left({x_i-x_j\\over 2}+{1\\over 2} (d_i-d_j) \\right)2\\sinh\\left({ x_i-x_j \\over 2}+{1\\over 2}(f_i-f_j)\\right) \\over \\prod_{i,j=1}^M2\\cosh\\left( {x_i-x_j \\over 2} +{1\\over 2} (d_i-f_j) \\right)},\n\\end{aligned} }\\end{equation}\nwhere $\\Lambda$ denotes the instanton counting parameter in gauge theory. The shifts\n $f_i, d_i$  are given in  \\eqref{mmsu35d} and they depend on the rank $N$ of the gauge group. We also\n used \\begin{equation} I_j= \\left[{\\sum_{s=0}^{j-1} M_{s}},\\sum_{s=1}^{j} M_{s}  \\right] \\cap {\\mathbb N} , \\quad M_0=1, \\quad M=\\sum_{i=1}^{N-1}M_i . \\end{equation}\nAs a consequence we have a spectral determinant representation for the four dimensional Nekrasov-Okounkov  partition function associated to these $SU(N)$ theories as shown in section \\ref{opno}. We  expect this to be the $\\tau$-function of the isomonodromy problem associated to the Hitchin's system describing the relevant\nSW curve, see the end of section \\ref{opno}. The spectral determinant presentation also allows to compute the exact S-duality transformation of the $SU(N)$ Nekrasov partition function. \n\n\n\\section{Reviewing the $SU(2)$ case} \\label{su2rev}\n\nThe TS\/ST duality \\cite{ghm} has led to various exact results in topological string and in spectral theory which  allows us to explore all range of the couplings in both side of the duality. We denote by $g_s$ the coupling constant of string theory and by $\\hbar\\sim g_s^{-1}$ the Planck constant appearing in the spectral theory side of the correspondence. \n\nIt was pointed out in \\cite{bgt} that there are two limits in which the TS\/ST duality makes contact with gauge theory.  In one limit,  which we refer to as  the {\\it standard} four dimensional limit, one takes $\\hbar\\to 0$. It was shown in \\cite{hm} that in this limit the TS\/ST correspondence  reproduces  the well known Nekrasov--Shatashvili  (NS) conjecture in four dimensions \\cite{ns}. In the second limit instead,  which we refer to as  the {\\it dual} four dimensional limit,  one takes $\\hbar\\to \\infty$ and we obtain some new results in ${\\cal N}=2$ gauge theories. \nIn this section we review these two limits and their consequences for  $SU(2)$ gauge theories.\n Even tough we mainly focus on the  pure $SU(2)$ theory this procedure can in principle be extended as well to other  geometries which engineer $SU(2)$ gauge theories with $N_f$ matter multiplets.\n\n\n\\subsection{Operator theory and the self-dual $\\Omega$ background}\nIt is  known \\cite{adkmv,acdkv, ghm, kama,Laptev:2015loa} that quantization of mirror curves to toric CY leads to well defined operators with a positive and discrete spectrum.\nOne of the simplest examples is the canonical bundle over ${\\mathbb P}^1\\times {\\mathbb P}^1$. The quantization of its mirror curve, namely\n\\begin{equation}   \\label{mirrorp1}  \\xi^{1\/2} \\left({\\rm e}^x +{\\rm e}^{-x} \\right)+{\\rm e}^p +{\\rm e}^{-p}=\\kappa, \\quad x, p \\in {\\mathbb C}, \\end{equation}\n leads to the following operator which corresponds to the Hamiltonian of $SU(2)$ relativistic Toda lattice \n\\begin{equation} \\label{rt} {\\rm O}= \\xi ^{1\/2} \\left({\\rm e}^{\\mathsf x}+{\\rm e}^{-\\mathsf x}\\right) +{\\rm e}^{\\mathsf p}+{\\rm e}^{-\\mathsf p}, \\qquad [\\mathsf x, \\mathsf p]={\\rm i} \\hbar. \\end{equation}\nFor the propose of this paper we take \\begin{equation} \\xi, \\hbar \\in {\\mathbb R}_+. \\end{equation} \n The spectral properties of the operator ${\\rm O}$ are determined by a special combination of the NS and  the unrefined (or GV) limit of topological string \\cite{nek5,acdkv,Mironov:2009uv,ghm}. \n More precisely the perturbative WKB part of the spectrum is closely related to the NS limit  as pointed out in \\cite{nek5,Mironov:2009uv,Bonelli:2011na,acdkv}. However there are additional non--perturbative corrections which are encoded in the  GV limit of topological string \\cite{km,ghm}  \\footnote{In this paper we are  interested in the spectral determinant  formulation of the TS\/ST duality in which the GV part plays a crucial role. However if one focus exclusively on the quantization condition, by using blowup equations it is possible to rewrite the GV part of the quantum volume by using the NS free energies  in a way which features S-duality \\cite{wzh,hm,Franco:2015rnr,huang1606,Grassi:2016nnt}. Recently  this relation  has lead to an extension of the blowup formalism \\cite{Gu:2017ccq}.}.\n %\n   In the following we will describe two different limits of the operator  \\eqref{rt} dominated respectively by one of the two contributions. \n \n\n\n\\subsubsection{The standard four dimensional limit}\n \n In the standard four dimensional limit  \\cite{nek5,kkv,ikp3,hm}\n one  scales the parameters  of  the operator \\eqref{rt} as\n { \\begin{equation} \\label{s4} \\xi= \\beta^{-4}, \\quad \\hbar=\\beta \\hbar_{\\rm  T}, \\quad \\mathsf x = \\mathsf  {\\overline x  }\\beta \\qquad \\beta \\to 0.\\end{equation}}\nThis leads to\n\\begin{equation} {\\rm O} \\quad \\xrightarrow{\\beta \\to 0} \\quad {\\rm H_{\\rm 4D}}\\end{equation}\nwhere ${\\rm H}_{\\rm 4D}$ is the Hamiltonian of quantum Toda namely\n{ \\begin{equation} {\\rm H}_{\\rm 4D}={\\rm e} ^{\\mathsf p}+{\\rm e}^{- \\mathsf p}+\\mathsf{ {\\overline x  }} ^2 \\quad [\\mathsf {\\overline x}, \\mathsf  { p  }]={\\rm i}\\hbar_{\\rm T}.\\end{equation}}\nThe eigenvalues of this Hamiltonian are  \\begin{equation} E_{\\rm 4D}^{n}=\\lim_{\\beta \\to 0}\\left({\\rm e}^{E_n}-2\\beta^{-2}\\right),\\end{equation}\nwhere ${\\rm e}^{E_n}$ denotes the energies of \\eqref{rt}.\nAs conjectured in \\cite{ns}, the spectrum of  $\\rm H_{\\rm 4D}$ is exactly determined by the NS limit of ${\\cal N}=2$ $SU(2)$  SYM  theory. Indeed in this standard 4d limit  the additional non-perturbative corrections appearing in the relativistic version of the operator, and computed in \\cite{ghm}, are not present as pointed out in \\cite{hm}. \n\nAt the level of  flat coordinates this limit can be implemented  by taking the so--called geometric engineering limit \\cite{kkv}\n{ \\begin{equation} \\begin{aligned}  T_b=- \\log \\left(\\beta ^4\\right)+\\beta L, \\quad\nT_f=\\beta L, \\quad\ng_{st}=4 \\pi^2 \\beta^{-1} \\hbar_{\\rm T}^{-1}, \\quad \\beta \\to 0 \\end{aligned}\\end{equation}}\nwhere $T_b , T_f$ are the Kahler parameters of the canonical bundle over ${\\mathbb P}^1\\times {\\mathbb P}^1$ .\n\n\n \\subsubsection{The dual four dimensional  limit} \\label{dus}\nAs discussed above in the standard four dimensional limit \\eqref{s4} we take \n$ \\hbar \\to 0$  and\nwe make contact with ${\\cal N}=2$ SYM in the four dimensional NS phase of the $\\Omega$ background.\n  It is therefore natural to ask  whether there is a different limit of the relativistic operator taking\n\\begin{equation} \\hbar \\to \\infty  \\end{equation}\nand still making contact with a different phase of  four dimensional ${\\cal N}=2$ theories. In \\cite{bgt} it was pointed out that this limit  indeed exists,   we refer to it as {\\it dual four dimensional limit}. In the case of local ${\\mathbb P}^1 \\times {\\mathbb P}^1$ this consists in taking \n\\begin{equation} \\label{dl} \\begin{aligned} &  \\hbar= { 1 \\over \\epsilon \\beta}, \\quad \\log \\xi =   {a\\over 2 \\pi} -\\frac{  \\log \\left(\\beta ^4 \\Lambda^4 \\right)}{ 2 \\pi \\beta  \\epsilon }, \\\\\n& \\log \\kappa=- {\\hbar \\over 4 \\pi} \\log \\left(\\beta^{4} \\Lambda^{4}\\right)+\\log  \\left( 1+{\\rm e}^{  {a\\over 2 \\pi} }\\right)+ \\mathcal{O}({\\rm e}^{\\beta^{-1}\\log \\beta^4}), \\qquad { \\beta \\to 0^+} , \\end{aligned} \\end{equation}\n where $\\kappa$ is the complex modulus of  local ${\\mathbb P}^1\\times {\\mathbb P}^1$ and, after quantization, it is related to the  eigenvalues of the operator \\eqref{rt}. From the four dimensional perspective the parameter $a$\nis related to the A period of the $SU(2)$ SW curve \\footnote{ The parameter $a$ is related to the SW variable $\\tt a$ as $a \\sim {\\tt a \/ \\epsilon}$.},   $\\Lambda$ is the instanton counting parameter and  $\\epsilon$ is the twisting parameter of the self-dual $\\Omega$ background. \n In general, to implement the dual limit on the operator \\eqref{rt} it is more laborious  than the standard four dimensional limit but it can be done  as explained in \\cite{bgt}.  In particular one has to use unitary transformations to write the operator in  a suitable form.\nIn the case  of local ${\\mathbb P}^1 \\times {\\mathbb P}^1$ one has to rescale $\\mathsf x, \\mathsf p$ as\n \\begin{equation} \\mathsf x=\\mathsf v {1\\over 2}+\\mathsf u {\\hbar \\over 2 \\pi }+{1\\over 4 } \\log \\xi , \\quad  \\mathsf p=\\mathsf v{1\\over 2}-\\mathsf u {\\hbar \\over 2 \\pi }+{1\\over 4 } \\log \\xi , \\quad  [\\mathsf u , \\mathsf v ]= 2 \\pi {\\rm i} .\\end{equation}\n By using unitary transformations, one can write \\eqref{rt} as \\cite{kmz}\n\\begin{equation} \\label{uvt} \\xi ^{1\/4} | g\\left(b{\\mathsf{u}\\over 2 \\pi }\\right) | { 2\\cosh(\\mathsf{v}\/2)} | g\\left(b{\\mathsf{u}\\over 2 \\pi }\\right)|,\\end{equation}\nwhere \\begin{equation}  g (\\mathsf{s}): ={\\rm e}^{-\\pi b \\mathsf{s}\/2}{  \\operatorname{\\Phi}_{\\mathsf{b}}(\\mathsf{s}+\\log \\xi \/(4 \\pi b)- {\\rm i} b\/4)\\over \\operatorname{\\Phi}_{\\mathsf{b}}(\\mathsf{s}-\\log \\xi \/(4 \\pi b)+ {\\rm i} b\/4) }, \\quad  b^2={\\hbar \\over \\pi },\n\\end{equation}\nand  $\\operatorname{\\Phi}_{\\mathsf{b}} $ denotes the Faddeev's quantum dilogarithm \\cite{faddeev,fk}.\nOnce the operator  is written in  the form \\eqref{uvt} it is easy to implement the dual limit \\eqref{dl}. As explained in  \\cite{bgt}  one obtains, after a suitable normalisation,  the following operator \n\\begin{equation} \\label{rho4d} {\\rm O}_{\\rm 4D}= {\\rm e}^{{ \\pi^{-2}  \\Lambda \\epsilon^{-1}  } \\cosh(\\mathsf u)}{ \\left({\\rm e}^{\\mathsf v\/2}+ {\\rm e}^{-\\mathsf v\/2} \\right)}{\\rm e}^{{\\pi^{-2}  \\Lambda \\epsilon^{-1} } \\cosh(\\mathsf u)}, \\qquad [\\mathsf u, \\mathsf v]= 2 \\pi {\\rm i}  .\\end{equation} \nThe spectrum of  this operator is exactly computed by the  ${\\cal N}=2$ $SU(2)$  SYM  in the self--dual $\\Omega$ background.  Therefore the dual limit provides an operator interpretation of the self-dual $\\Omega$ background.\n\nAt the level of  flat coordinates  this limit consists in taking\n\\begin{equation} \\begin{aligned}  T_b =-{1\\over 2 \\pi \\beta \\epsilon}\\log \\left(\\beta^4 \\Lambda^4\\right) \\quad\nT_f =-{ a\\over 2\\pi}, \\quad\ng_{st}=4 \\pi^2 \\epsilon \\beta, \\quad \\beta \\to 0 \\end{aligned}\\end{equation}\nwhere $T_b , T_f$ are the Kahler parameters of the canonical bundle over ${\\mathbb P}^1\\times {\\mathbb P}^1$. This can be viewed as a sort of dual (or rescaled) geometric engineering limit where we replaced  \\begin{equation}  T_b,\\quad T_f, \\quad  g_{st}\\quad \\to  \\quad g_{st}T_b,\\quad g_{st}T_f, \\quad g_{st}^{-1} .\\end{equation}\n\n \n  \\subsection{The TS\/ST duality and Painlev\\'e equations}\n  \n  The implementation of the dual limit has also an additional interesting feature: it connects the TS\/ST duality to the theory of Painlev\\'e equations \\cite{bgt}. This goes as follows.\n\n\nIn a series of papers \\cite{ilt1,ilt,gil1,ilte, bes,gil,blmst} it was found that the  $\\tau$ functions of Painlev\\'e equations are computed by four dimensional ${\\cal N}=2$, $SU(2)$ gauge theories in the self--dual $\\Omega$ background. Different types of Painlev\\'e equations correspond to different matter content in the gauge theory. For instance the pure $SU(2)$  theory computes the $\\tau$ function of Painlev\\'e $\\rm III_3$ (also known as $ \\rm III^{D_8}$) while the Painlev\\'e $\\rm V$ is related to  $SU(2)$ gauge theory with $N_f=3$ fundamentals multiplets. \n\n\nFrom the string theory viewpoint  we expect that  these results can be obtained by implementing the dual four dimensional limit to the TS\/ST duality  on some specific CY  $X$ with genus one mirror curve \\cite{bgt}.\n In this duality one associates  a trace class operator $\\rho_X$ to any of these geometries and gives an exact expression for the corresponding spectral determinant.  Schematically we have \\cite{ghm}\n\\begin{equation} \\label{det5d}\\det (1+\\kappa \\rho_X)=\\sum_{n\\in {\\mathbb Z}} {\\rm e}^{J_X(\\mu+2\\pi {\\rm i} n, \\hbar)}, \\quad \\kappa={\\rm e}^{\\mu} \\end{equation}  \nwhere  $J_X$ is the topological string grand potential studied in \\cite{mp,hmo2,cm,hmo3,hmmo,ghm}. If $X$ is the  canonical bundle  over ${\\mathbb P}^1\\times {\\mathbb P}^1$ the operator $\\rho_{\\rm {\\mathbb P}^1\\times {\\mathbb P}^1}$ is the inverse of \\eqref{rt}.\nWhen the underling geometry $X$ can be used to engineer gauge theories \\cite{kkv,ta,ikp3,Eguchi:2000fv} we can implement the dual four dimensional limit described above. After performing this limit  \\eqref{det5d} takes the following form\n\\begin{equation} \\label{det4d} \\begin{aligned} & \\det (1+\\kappa \\rho_{X}^{\\rm 4D})=\\sum_{n\\in {\\mathbb Z}} {\\cal Z}_X^{\\rm 4D}{(\\sigma+ n, t)},\\\\\n& \\quad   t\\sim \\Lambda^4\/\\epsilon^4, \\qquad  \\sigma \\sim a , \\quad \\kappa \\sim \\cos( 2  \\pi\\sigma)\\end{aligned}\\end{equation}\nwhere $a, \\epsilon, \\Lambda$ are  the gauge theory parameters introduced in section \\ref{dus}.   In the  context of Painlev\\'e equations $t$ is the time and the variable $\\sigma$ is related to the asymptotic conditions. We  denote  by $\\rho_{X}^{\\rm 4D}$ the operator obtained after performing the dual limit on $\\rho_{X}$. \nMoreover, up to an overall factor, $ {\\cal Z}_X^{\\rm 4D}$ is the partition function of  a corresponding ${\\cal N}=2$ $SU(2)$ SYM theory in the four dimensional self--dual background. The specific matter content of the gauge theory depends on the starting geometry $X$ according to the geometric engineering construction \\cite{kkv,ta,ikp3,Eguchi:2000fv}.  \nIn particular on the r.h.s of \\eqref{det4d} we recover, up to an overall factor, the $\\tau$ function of a corresponding Painlev\\'e  equation as computed in \\cite{ilt1,ilt,gil1,gil,blmst} \\footnote{\\label{fn3}The general form for the $\\tau$ function of Painlev\\'e equations  depends in general on two variables $\\sigma, \\eta$ which are related to the monodromy data of the corresponding Fuchsian system. To make contact with TS\/ST we set $\\eta=0$ \\cite{bgt}. As explained in \\cite{mmnp,fki3} fixing $\\eta$   corresponds to fix the non perturbative ambiguity. Hence the TS\/ST duality  corresponds to one particular non-perturbative choice. }.   On the l.h.s.~instead we obtain the spectral determinant representation of the same $\\tau $ function.\nHence from this perspective  different Painlev\\'e equations correspond  to different background geometry in string theory as illustrated on \\figref{fig1}.\n\\begin{figure}[h!] \\begin{center}\n {\\includegraphics[scale=0.35]{p1.pdf}}\n\\caption{In the upper line we have a list of  polyhedra representing del Pezzo surfaces $S$  \\cite{Bat,ckyz} connected to the coalescence diagram of Painlev\\'e equations through an arrow. The total space of the canonical bundle  over $S$ is a CY manifold and we can put topological string theory on it. In particular  we can consider the TS\/ST duality for these manifolds. Once we implement the dual 4d limit on the determinants appearing in this duality  we recover the tau function of a corresponding Painlev\\'e equation.  Also notice that there may be different del Pezzo's which engineers  the same Painlev\\'e equation. In the figure we have chosen one of them \\cite{Eguchi:2000fv}. } \\label{fig1}\n\\end{center}\n\\end{figure}\nTherefore  this construction gives a geometrical meaning to the operators whose spectral determinant compute the $\\tau$ function of Painlev\\'e equations: these arise by quantizing mirror curves to  CY geometries and performing the dual limit.\n The details of this limit have been worked out for pure $SU(2)$ in \\cite{bgt} which makes contact with the local ${\\mathbb P}^1\\times  {\\mathbb P}^1$ geometry and the Painlev\\'e $\\rm III_3$ equation.   In this case the equation \\eqref{det4d} reads\n \\begin{equation}  \\label{4dstat}\\det (1+\\kappa \\rho_{{\\mathbb P}^1 \\times {\\mathbb P}^1}^{\\rm 4D})={\\rm e}^{\\frac{\\log (2)}{12} +3\\zeta'(-1)}t^{-1\/16}{\\rm e}^{4 \\sqrt{t}} \\sum_{n \\in {\\mathbb Z}} Z^{\\rm Nek}(\\sigma+n, t),   \\end{equation}\n where we used \\begin{equation} \\label{dic4}  \\kappa={\\cos \\left(2 \\pi \\sigma\\right)\\over 2 \\pi} , \\quad  t= \\left(\\frac{\\Lambda }{4 \\pi ^2 \\epsilon }\\right)^4\\end{equation}\n  and $\\rho_{{\\mathbb P}^1 \\times {\\mathbb P}^1}^{\\rm 4D}$ is the inverse of \\eqref{rho4d}. More precisely\n \\begin{equation}  \\rho_{{\\mathbb P}^1 \\times {\\mathbb P}^1}^{\\rm 4D}= {\\rm e}^{-{4 t^{1\/4} } \\cosh(\\mathsf u )}{4 \\pi  \\over \\left({\\rm e}^{\\mathsf v\/2}+ {\\rm e}^{-\\mathsf v \/2} \\right)}{\\rm e}^{-{4 t^{1\/4} } \\cosh( \\mathsf u)}, \\qquad [\\mathsf u,  \\mathsf v]= 2 \\pi {\\rm i} .\\end{equation}\nMoreover we denoted by $Z^{\\rm Nek}(\\sigma, t)$ the Nekrasov partition function of pure ${\\cal N}=2$, $SU(2)$ theory, namely  \n  \\begin{equation} \\label{nek4d}\\begin{aligned} Z^{\\rm Nek}(\\sigma, t)={t^{\\sigma^2}  \\over  G(1-2\\sigma)G(1+2\\sigma)} \\left( 1+\\frac{t}{2 \\sigma ^2}+ \\frac{\\left(8 \\sigma ^2+1\\right) t^2}{4 \\sigma ^2 \\left(4 \\sigma ^2-1\\right)^2}+\\mathcal{O}(t^3)\\right). \\end{aligned}\\end{equation}\n  Sometimes we refer to\n  \\begin{equation} Z^{\\rm NO}(\\sigma, \\Lambda, \\epsilon)= \\sum_{n \\in {\\mathbb Z}} Z^{\\rm Nek}(\\sigma+n, t)\\end{equation}\n  as the Nekrasov-Okounkov  partition function.  \n The statement \\eqref{4dstat} has been proved  for   $\\sigma \\neq {\\mathbb Z}\/2$  in \\cite{bgt} by using the relation with the Painlev\\'e $\\rm III_3$ equation.\n \n Even tough the details have been worked out only for the pure $SU(2)$ gauge theory we expect that a straightforward generalisation should be possible for the others geometries illustrated on \\figref{fig1} as well. \n\nWe would like to observe that this approach only holds when some particular initial conditions are imposed on the solutions to Painlev\\'e equations (see footnote \\ref{fn3}). Recently a more general formalism to compute generic  Fredholm determinant representations of isomonodromic tau functions has been developed in  \\cite{lnew}.  It would be interesting to derive the kernels $\\rho_{X}^{\\rm 4D}$ as limiting cases of this general formulation.\n\n\n\\subsection{Dual matrix models in four dimensions }\\label{newsection}\nIn addition the TS\/ST duality can be used to compute the exact expression for the partition function of topological string theory on $X$. Let us review how this goes. By using standard results in Fredholm theory we have\n\\begin{equation} \\det (1+\\kappa \\rho_{X})=\\sum_{N\\geq 0} \\kappa^N Z_X(N), \\end{equation}\nwhere $Z_X(N)$ are the fermionic spectral traces of $\\rho_{X}$, namey\n\\begin{equation} \\label{mm} Z_X(N)={1\\over N!}\\sum_{\\sigma \\in S_N}(-1)^{\\sigma} \\int_{{\\mathbb R}} {\\rm d} ^N x  \\rho_{X}(x_i, x_{\\sigma(i)}).\\end{equation}\nWe denoted by $S_N$  the permutation group of $N$ elements.  The TS\/ST duality states that $Z_X(N)$ is the non-perturbative partition function of topological string on $X$. More precisely, at the level of the fermionic spectral traces the equality \\eqref{det5d} reads \\cite{ghm}\n\\begin{equation} \\label{zj}Z_X(N)={1\\over 2 \\pi {\\rm i}} \\int_{\\mathcal{C}}  {\\rm e}^{J_X(\\mu, \\hbar)- \\mu N}{\\rm d} \\mu ~ ,\\end{equation}\nwhere $\\mathcal{C}$ is the Airy contour.  Furthermore by using the Cauchy identity it is possible to write \\eqref{mm} as a matrix model which computes the partition function of  topological string in the conifold frame \\cite{mz}.    Even tough $Z_X(N)$ is originally defined in \\eqref{mm} for integers $N$, it was pointed out in \\cite{cgm}  that  \\eqref{zj} provides  an analytic continuation of the fermionic spectral traces to an entire function in the full complex plane. \n\nWhen we implement the dual limit at the level of the fermionic spectral traces  \\eqref{mm},  we obtain a matrix model expression for the partition function of  $d=4$, ${\\cal N}=2$, $SU(2)$ gauge theories  in the self-dual $\\Omega$-background in the magnetic frame.  This was done in details for the pure $SU(2)$ theory in \\cite{bgt} where it was found that the matrix model computing its partition function is a well known $O(2)$  model\n\\begin{equation} Z_2^{\\rm 4d}(M)= {1\\over M!} \\int \\prod_{i=1}^M  \\frac{{\\rm d} x_i}{4 \\pi } {\\rm e}^{-{1\\over g_s }\\cosh x_i} \\prod_{i < j} \\tanh\\left({x_i-x_j\\over 2}\\right)^2, \\quad {  g_s^{-1}={2  \\Lambda \\over \\pi^{2}\\epsilon} >0}~.\\end{equation}\nLikewise the counterpart of  \\eqref{zj} in this limit is\n\\begin{equation} \\label{zj4d}  \\begin{aligned} Z_2^{\\rm 4d}(M)=&{\\rm i}^{-1} \\int_{ {\\mathbb R}+\\sigma_0}{{\\rm d} \\sigma}  \\tan \\left(2 \\pi \\sigma\\right) Z^{\\rm Nek}(\\sigma,t){\\rm e}^{- \\log \\left[2 \\cos (2 \\pi  \\sigma )\\right] M}  {\\rm e}^{\\frac{\\log (2)}{12} +3\\zeta'(-1)}t^{-1\/16}{\\rm e}^{4 \\sqrt{t}} , \\\\\n& \\qquad \\sigma_0=(2 \\pi )^{-1}{{\\rm i} \\cosh ^{-1}(2 \\pi )} , \\quad  t= \\left(\\frac{\\Lambda }{4 \\pi ^2 \\epsilon }\\right)^4,\\end{aligned}\\end{equation}\nSince $Z^{\\rm Nek}(\\sigma, t)$ has poles only for $\\sigma \\in {\\mathbb Z}\/2$ one can take a generic $\\sigma_0 \\in {\\rm i} {\\mathbb R}_+$. \nThis equality also follows from \\eqref{4dstat} and \\eqref{dic4}.  Indeed one has\n\\begin{equation} \\label{interm}Z_2^{\\rm 4d}(M)={1\\over 2 \\pi {\\rm i} } \\oint_0\\kappa^{-N-1}\\det (1+\\kappa \\rho^{\\rm 4D}_{{\\mathbb P}^1\\times {\\mathbb P}^1}) {\\rm d} \\kappa.\\end{equation}\nBy using the change of variable \\eqref{dic4} together with the identity  \\eqref{4dstat}  we can write \\eqref{interm} as \\eqref{zj4d}.\n Moreover \\eqref{zj4d} can also be tested numerically in an easy way thanks to the good convergent properties of \\eqref{nek4d}. For instance on the l.h.s. we have\n\\begin{equation} \\label{tt}\\begin{aligned}& Z_2^{\\rm 4d}(0)=1, \\quad Z_2^{\\rm 4d}(1)=\\left(2 \\pi \\right)^{-1}{K_0\\left(8 \\sqrt[4]{t}\\right)}, \\\\\n& Z_2^{\\rm 4d}(2)=\\frac{G_{1,3}^{3,0}\\left(64 \\sqrt{t}|\n\\begin{array}{c}\n \\frac{3}{2} \\\\\n 0,0,0 \\\\\n\\end{array}\n\\right)}{32 \\pi ^{3\/2}},\\end{aligned}\\end{equation}\nwhere $K_0$ denotes the Bessel function and $G$ the Meijer G function. It is easy to check that the numerical integration on the r.h.s of \\eqref{zj4d} reproduces \\eqref{tt}. \n\nEquation \\eqref{zj4d} provides the integral kernel implementing the exact S-duality transformation of pure $SU(2)$ Nekrasov partition in a self-dual $\\Omega$-background parametrized by $\\epsilon$. It is easy to see that in the semiclassical limit  $\\epsilon\\to 0$ it reduces to a simple Fourier transform as it is expected from Seiberg-Witten special geometry relations  \\cite{sw2,abk}. Indeed, by introducing the SW variables $   {\\tt a}\/\\epsilon\\equiv \\sigma $ and $ {\\tt a}_D\/\\Lambda \\propto g_s M$  one has in the $\\epsilon\\to 0$ limit \n\\begin{equation} \n\\log 2 \\cos \\frac{2\\pi {\\tt a}}{\\epsilon} =- \\frac{2\\pi {\\rm i} {\\tt a}}{\\epsilon}+ \\mathcal{O}({\\rm e}^{4 \\pi {\\rm i} {\\tt a} \\over \\epsilon  }) \\  \\  , \\quad \\  \\ \n  {\\rm i}^{-1}\\tan \\frac{2\\pi {\\tt a}}{\\epsilon} =1+  \\mathcal{O}({\\rm e}^{4 \\pi {\\rm i} {\\tt a} \\over \\epsilon  }) \\ \\ .\n\\end{equation}\n Because of the imaginary part in the integration contour we have  ${\\rm Re}(4 \\pi {\\rm i} {\\tt a} ) <0$, hence the corrections $\\mathcal{O}({\\rm e}^{4 \\pi {\\rm i} {\\tt a} \\over \\epsilon })$ are exponentially suppressed and do not appear in the perturbative limit $\\epsilon \\to 0$. Notice also that\nequation \\eqref{zj4d}   agrees with the general philosophy for the change of frame in topological strings  \\cite{abk} and provides the exact S-duality transformation properties of the gauge theory\npartition function with gravitational corrections. \n\n\n\nAll this procedure can be in principle extended to $SU(2)$ theories with matter multiplets as well, however the details still have to be worked out.\nSummarizing the implementation of the dual limit on the TS\/ST duality leads to  the following  results in connection with four dimensional ${\\cal N}=2$ gauge theories: it gives an operator theory interpretation of the self-dual $\\Omega$ background, it gives Fredholm determinant representation for the $\\tau$ functions of  Painlev\\'e equations, it provides a matrix model  for the partition function in the magnetic frame\nand an exact formula for its S-duality transformation.\n\n\n \n\\section{Non-perturbative string on $Y^{N,0}$  geometries } \\label{nps}\n\n\nThe TS\/ST duality \\cite{ghm} has been generalized to higher genus mirror curve in \\cite{cgm2}.   \nAccording to this construction one can associate a set of  $g$ operators \\begin{equation} \\left\\{{\\rm O}_i \\right\\}_{i=1}^g \\end{equation} to any toric CY manifold, $g$ being the genus of its mirror curve.\nOf particular interest for this paper are those CYs from which one can engineer  $SU(N)$ supersymmetric  gauge theories \\cite{kkv,ikp3,ta}.  \nExamples of such  geometries are  the resolution of the cone over the $Y^{N,0}$ singularity studied  for instance in  \\cite{bz}.\n  The corresponding mirror curve has genus $N-1$ and therefore  there are $N-1$ different \"canonical\" forms for this curve which reads \n\\begin{equation} O_i(x_1,x_2, \\xi)+\\kappa_i=0, \\quad i=1, \\cdots , N-1,\\end{equation}\nwhere $\\kappa_i$ denote the complex moduli of the geometry.\nFor instance we have\n\\begin{equation} \\label{cl1}O_1(x_1,x_2, \\xi)+\\kappa_1={\\rm e}^{x_2}+{\\rm e}^{-x_2 +(-N+2)x_1}+\\sum_{i=1}^{N-1} \\kappa_{N-i} {\\rm e}^{(i-N+1)x_1} +\\xi {\\rm e}^{(-N+1)x_1} +{\\rm e}^{ x_1}=0 ,\\end{equation}  \nwhere $\\xi$ is the mass parameter and should be distinguished from the others moduli $\\kappa_i$ as  emphasized for instance in  \\cite{hkp}.\nTherefore, the quantization procedure for  the $Y^{N,0}$ geometry  leads to the following $N-1$ operators \n\\begin{equation} \\begin{aligned} \n&{ {\\rm O_1}+\\kappa_1={\\rm e}^{\\mathsf x_2}+{\\rm e}^{-\\mathsf x_2 +(-N+2)\\mathsf x_1}+\\sum_{i=1}^{N-1} \\kappa_{N-i} {\\rm e}^{(i-N+1) \\mathsf x_1} +\\xi {\\rm e}^{(-N+1) \\mathsf x_1} +{\\rm e}^{\\mathsf  x_1}},\\\\\n&  {\\rm O}_{j}+\\kappa_{j}= {\\rm Q}_j^{-1\/2}\\left( {\\rm O}_1+\\kappa_{1}\\right) {\\rm Q}_j^{-1\/2}, \\quad 1 < j \\leq  N-1,\n\\end{aligned}\\end{equation}\nwhere $[\\mathsf x_1, \\mathsf x_2]={\\rm i} \\hbar$\nand we denote\n\\begin{equation} {\\rm Q}_j={\\rm e}^{ -(j-1) \\mathsf x_1}.\\end{equation}\nLet us define the following two operators \n\\begin{equation}  \\label{rad}\\begin{aligned}&  \\rho_{1, N-2,\\xi}=\\left({\\rm e}^{\\mathsf x_2}+{\\rm e}^{-\\mathsf x_2 +(-N+2)\\mathsf x_1}+\\xi {\\rm e}^{(-N+1)\\mathsf x_1} +{\\rm e}^{\\mathsf  x_1}\\right)^{-1}, \\\\\n& {\\rm A}_j^{\\rm 5D}=\\rho_{1, N-2,\\xi}{\\rm Q}_j.\\end{aligned}\\end{equation} \nThe conjecture \\cite{cgm2} states that the non--perturbative   topological string partition function on  the $ Y^{N,0}$ geometry in the conifold frame is given by\n\\begin{equation}\n\\label{gen-fred-th}\nZ_N({M_1}, \\cdots M_{N-1})= {1\\over M_1! \\cdots M_{N-1}!} \\int  {\\rm det}_{m,n} \\left(R(x_m, x_n)\\right){\\rm d}^M x, \n\\end{equation}\nwhere\n\\begin{equation}\nM=\\sum_{j=1}^{N-1} M_j,\n\\end{equation}\nand \n\\begin{equation}\nR (x_m, x_n) = A_{j}^{\\rm 5D}(x_m , x_n) \\,\\, \\,\\, \\,\\, \\,\\,  \\text{if} \\, \\,  \\,\\, \\,\\, \\,\\,\\sum_{s=0}^{j-1} M_s\\leq m \\le \\sum_{s=1}^j M_s.\n\\end{equation} We use $M_0=1$ and \n\\begin{equation} A_{j}^{\\rm 5D}(x_m , x_n)\\end{equation}\ndenotes the kernel of the operator  ${\\rm A}_{j}^{\\rm 5D}$ defined in \\eqref{rad}.\nThe explicit expression for a  kernel of  the form \\begin{equation}\\rho_{n,m,\\xi }= \\left({\\rm e}^{\\mathsf  x_1}+ {\\rm e}^{\\mathsf  x_2} + {\\rm e}^{-m \\mathsf  x_1-n \\mathsf  x_2} + \\xi {\\rm e}^{-(1+m)\\mathsf  x_1-(n-1)\\mathsf  x_2} \\right)^{-1}\\end{equation}\nwas computed in \\cite{cgum}. Let us review how this goes. We introduce some new variables $\\mathsf q, \\mathsf p$ such that \\cite{kama,cgum} \n \\begin{equation} \\label{newv} \\begin{aligned}  \\mathsf x_1= {2\\pi b \\over m+n+1}\\left({\\mathsf p+n \\mathsf q }\\right)\\quad& \\mathsf x_2=-{2\\pi b \\over m+n+1}\\left({- \\mathsf p+(m+1)\\mathsf q }\\right),   \\end{aligned}\\end{equation} \n where \\begin{equation} \\hbar={2\\pi \\over m+n+1} b^2.\\end{equation}\n In particular we have\n\\begin{equation} [\\mathsf q, \\mathsf p]={{\\rm i} \\over 2 \\pi}.\\end{equation}\n Then, the kernel of $\\rho_{n,m,\\xi }$ in the momentum representation w.r.t.~the new variables \n  reads \\cite{cgum} \n\\begin{equation}  \\label{rhomon} \\rho_{n,m, \\xi}(p, p')= {{\\overline  f_\\zeta(  p)}  f_\\zeta(p')  \\over 2 b \\cosh\\left( \\pi {p-p'+{\\mathsf{i}} h \\over b} \\right)},\\end{equation}\nwhere\n \\begin{equation}\n\\label{fx}\\begin{aligned} \nf_\\zeta(x)=&{\\operatorname{\\Phi}_{\\mathsf{b}}(x-\\zeta + {\\rm i} n { c} ) \\over \\operatorname{\\Phi}_{\\mathsf{b}}(x-{\\mathsf{i}} (\\alpha+c))} {\\rm e}^{ 2 \\pi (\\alpha+ n c) x } {\\rm e}^{ -2 \\pi  c n  \\zeta}, \n\\end{aligned}\\end{equation}\n \\begin{equation}\n\\label{fx}\\begin{aligned} \n\\overline f_\\zeta(x)= &{\\operatorname{\\Phi}_{\\mathsf{b}}(x+{\\mathsf{i}} (\\alpha+c)) \\over \\operatorname{\\Phi}_{\\mathsf{b}}(x-\\zeta - {\\rm i} n c ) } {\\rm e}^{ 2 \\pi (\\alpha+ n c) x } {\\rm e}^{ -2 \\pi  c n  \\zeta}. \\end{aligned}\\end{equation}\nWe  denote by $\\operatorname{\\Phi}_{\\mathsf{b}} $ the Faddeev's quantum dilogarithm \\cite{faddeev,fk} and we use\n\\begin{equation}\n{  \\alpha={b m\\over 2(m+n+1)}}, \\qquad c= {b \\over 2(m+n+1)}, \\qquad h = \\alpha+c-nc, \\qquad  \\zeta={1\\over 2 \\pi b}\\log \\xi.\n\\end{equation}\nSome useful properties of the quantum dilogarithm can be found in Appendix A of \\cite{kama}.\nIn our case we specialize the  above formulae to $n=1$ and $m=N-2$. Therefore, in the particular case of the $Y^{N,0}$ geometries, the partition function  \\eqref{gen-fred-th} reads \n\\begin{equation} \\label{gmat} \\begin{aligned} Z_N(M_1, \\cdots,  M_{N-1})=&{1\\over M_1! \\cdots M_{N-1}! } \\sum_{\\sigma \\in S_{M}}(-1)^\\sigma\\int  {\\rm d} ^M x \\left(\\prod_{i=1}^{M_1} A_1^{\\rm  5D}(x_{\\sigma(i)},x_i) \\right)\\\\\n&\\left(\\prod_{i=1+M_1}^{M_1+M_2} A_2^{\\rm    5D}(x_{\\sigma(i)},x_i)\\right) \\cdots\n  \\left( \\prod_{i=1+\\cdots +M_{N-2}}^{M_1+\\cdots +M_{N-1}} A_{N-1}^{\\rm   5D}(x_{\\sigma(i)},x_i)\\right)\\\\\n\\end{aligned}\\end{equation}\nwhere \n\\begin{equation} \\begin{aligned}\n &A_j^{\\rm   5D}(p,p')={\\rm e}^{-{\\rm i}  \\pi b^2( j-1)^2\/N^2}{\\rm e}^{-2\\pi ( j-1)b p'\/N } \\rho_{1, N-2,\\xi}(p,p'+{{\\rm i} b ( j-1) \\over N}), \\\\\n\\end{aligned}\\end{equation}\nand \\begin{equation}  \\rho_{1, N-2,\\xi}(p,p') \\end{equation} is given by \\eqref{rhomon}.\nBy using the Cauchy identity\n\\begin{equation} \\begin{aligned} &{\\prod_{1\\leq i<j\\leq N}2\\sinh\\left({\\mu_i-\\mu_j\\over 2} \\right)2\\sinh\\left({ \\nu_i-\\nu_j \\over 2}\\right) \\over \\prod_{i,j=1}^N2\\cosh\\left( {\\mu_i-\\nu_j \\over 2}  \\right)}= \\sum_{\\sigma\\in S_N}(-1)^\\sigma \\prod_{i=1}^N{1\\over 2 \\cosh \\left({\\mu_i- \\nu_{\\sigma(i)}\\over 2}\\right)},\\end{aligned} \\end{equation}\n together with some algebraic manipulations, we can write  \\eqref{gmat}  as \n\\begin{equation} \\label{mmsu35d}\\begin{aligned} Z_N(M_1, \\cdots,  M_{N-1})=& { 1\\over M_1!  \\cdots M_{N-1}!} \\int {{\\rm d} ^M x\\over (2\\pi)^M} \\prod_{j=1}^{N-1} \\prod_{i_j\\in I_j}{\\rm e}^{V_j^{\\rm 5D}(x_{i_j})} \\\\\n & \\times {\\prod_{1 \\leq i<j \\leq M}2\\sinh\\left({x_i-x_j\\over 2}+{1\\over 2} (d_i-d_j) \\right)2\\sinh\\left({ x_i-x_j \\over 2}+{1\\over 2}(f_i-f_j)\\right) \\over \\prod_{i,j=1}^M2\\cosh\\left( {x_i-x_j \\over 2} + {1\\over 2}\\left( d_i-f_j\\right) \\right)}.\n\\end{aligned} \\end{equation}\nWe denote \n \\begin{equation} \\label{genn} \\begin{aligned} d_j = &-  {(N-1-k) {\\rm i} \\pi  \\over N}, \\quad  \\text{if}\\quad  \\sum_{s=0}^{k-1}M_s \\leq j \\leq  \\sum_{s=1}^{k}M_s,\\\\\n  f_j=&- {(N-2) {\\rm i} \\pi  \\over N} -d_j,\n  \\end{aligned}\\end{equation}\n  and\n\\begin{equation} \\begin{aligned}\n{\\rm e}^{V_k^{\\rm 5D}(x)}= &{\\rm e}^{- b^2  (k-1)x \\over N}{\\rm e}^{-{(k-1)\\over 2N}\\log \\xi } {f_\\zeta({b\\over 2 \\pi } x+{\\rm i} b  {(k-1)\\over 2 N}+\\zeta\/2)} \\overline{f}_\\zeta({b\\over 2 \\pi } x-{\\rm i} b  {(k-1)\\over 2 N}+\\zeta\/2). \\end{aligned}\\end{equation}\nMoreover\n\\begin{equation} \\label{inter} I_j=\\left[{\\sum_{s=0}^{j-1} M_{s}},\\sum_{s=1}^{j} M_{s}  \\right] \\cap {\\mathbb N} , \\quad M_0=1, \\quad M=\\sum_{i=1}^{N-1}M_i . \\end{equation}\n %\nAccording to  the conjecture \\cite{ghm,cgm2,mz} the above matrix model  computes the topological string partition function of the $Y^{N,0}$ geometry in the conifold frame.  When $N=2$ this conjecture was tested in details in \\cite{ghm,kmz} while for $N=3$ several tests have been performed in \\cite{cgum}.  In the following we will compute the dual 4d limit of these matrix models for genetic $N$ and test the conjecture  by cross--checking  our result with some existing results in the gauge theory literature in particular with the works of  \\cite{klemmlt,dp1,Edelstein:1999fz,Edelstein:2000aj}.\n\n\n\\section{ Four dimensional $SU(N)$ gauge theory} \\label{sun4}\nAs described in section \\ref{su2rev}, given the non-perturbative  topological string formulation of \\cite{ghm, mz, cgm2},\nthere are two four dimensional limits in which we make contact with four dimensional ${\\cal N}=2$ gauge theory. \nIn the {\\it standard} four dimensional limit of the  $Y^{N,0}$ geometry  \n one takes  $\\hbar \\to 0$\n and it was shown in \\cite{hm} that in this limit the TS\/ST duality reduces to the well known four dimensional NS conjecture relating the spectral properties of $SU(N)$ quantum Toda to ${\\cal N}=2$ $SU(N)$ SYM in the NS phase of the $\\Omega$ background \\cite{ns}.   \nIn the {\\it dual} four dimensional limit instead, we take\n\\begin{equation} \\label{dlg} \\begin{aligned} & \\hbar={  1 \\over \\epsilon  \\beta}, \\quad \\log \\xi ={\\hbar \\over 2\\pi } \\left(a \\epsilon \\beta -\\log(\\beta^{2N}\\Lambda^{2N}) \\right),  \\\\\n &\\log \\kappa_i=- {\\hbar \\over 2 \\pi N} \\log \\left(\\beta^{2N} \\Lambda^{2N}\\right)+\\log  \\left(G_i\\right) + \\mathcal{O}({\\rm e}^{\\beta^{-1}\\log \\beta^4}), \\quad \\beta \\to 0^+ . \\end{aligned}\n\\end{equation}\nAfter implementing this limit we make contact with $d=4$, ${\\cal N}=2$, $SU(N)$ SYM in the self--dual  $\\Omega$ background. The parameter $\\epsilon$ is the twisting parameter of the $\\Omega$ background,  $\\Lambda$ is the instanton counting parameter while $a, G_i$ are related to the A  periods of the SW curve underling the  four dimensional theory.\n\n\n\n\n\\subsection{The matrix model }\n\nIn this section we implement the dual limit \\eqref{dlg} on the matrix model  \\eqref{mmsu35d}.\nWe obtain\n\\begin{equation} \\left(\\prod_{i=1}^{N-1}\\kappa_i^{M_i}\\right)Z_N(M_1, \\cdots,  M_{N-1})\\xrightarrow{\\text{ dual 4d} }\\left(\\prod_{i=1}^{N-1}\\left({\\rm e}^{-a\/( 2N \\pi)}G_i\\right)^{M_i}\\right) Z_{\\rm N}^{\\rm 4d}(M_1,\\cdots,M_{N-1})\\end{equation}\nwhere \n\\begin{equation} \\label{su3mm} {\\begin{aligned} Z_{\\rm N}^{\\rm 4d}(M_1,\\cdots,M_{N-1})=&  {1 \\over M_1!  \\cdots M_{N-1}!} \\int {{\\rm d} ^M x\\over (2\\pi)^M} \\prod_{j=1}^{N-1} \\prod_{i_j\\in I_j}{\\rm e}^{{ -{1\\over g_s}}V_j (x_{i_j})}  \\\\\n&   \\times {\\prod_{1\\leq i<j \\leq M}2\\sinh\\left({x_i-x_j\\over 2}+{1\\over 2} (d_i-d_j) \\right)2\\sinh\\left({ x_i-x_j \\over 2}+{1\\over 2}(f_i-f_j)\\right) \\over \\prod_{i,j=1}^M2\\cosh\\left( {x_i-x_j \\over 2} +{1\\over 2} (d_i-f_j), \\right)}.\n\\end{aligned} }\\end{equation}\nThe quantities $f_i, d_i$  are as in  \\eqref{mmsu35d}. Likewise $I_j$ is defined in \\eqref{inter} while the potential  is now given by\n\\begin{equation} \\begin{aligned} V_{k}(x)=& { \\sin \\left(\\frac{\\pi  k }{N}\\right)\\over  \\sin \\left(\\frac{\\pi  }{N}\\right)} \\cosh(x) ,\\end{aligned}\\end{equation} \nand \\begin{equation} \\label{gsdef}  g_s = \\frac{ \\pi ^2 \\epsilon}{N \\Lambda  \\sin \\left(\\frac{\\pi   }{N}\\right) }.\n\\end{equation}\n Hence in the dual 4d limit we recover a $(N-1)$-cut matrix model, whose   't Hooft couplings are\n\\begin{equation}  T_i= { g_s M_i }.\\end{equation}\nAccording to \\cite{bgt} to make contact with gauge theory we identify the dual periods $a_D^{(i)} $ with the 't Hooft couplings, namely  \n\\begin{equation} a_D^{(i)} = T_i .\\end{equation}\nBy combining the   TS\/ST conjecture \\cite{ghm,cgm2,mz}  with the work of \\cite{bgt} we expect that the matrix model  \\eqref{su3mm}  computes the partition function of $d=4$, $SU(N)$ ${\\cal N}=2$ SYM  in the self--dual $\\Omega$ background  in the magnetic frame.  Therefore from our perspective the rank of the gauge group is related to the number of cuts of the matrix model and not to the sizes of the matrices.\n  The above conjecture can be tested  against existing results in the literature.\nIn particular we will test the 't Hooft expansion of \\eqref{su3mm} for small values of the 't Hooft couplings. This expansion is obtained by considering \\begin{equation} M_i, g_s^{-1} \\to \\infty, \\quad T_i =M_i g_s \\quad \\text{fixed}.\\end{equation}\nIn this limit the matrix model \\eqref{su3mm} displays the following perturbative behavior\n\\begin{equation} \\label{thooft1} \\log Z_N^{\\rm 4d}(M_1, \\dots, M_{N-1} )={  \\sum_{g\\geq 0}g_s^{2g-2}F_g^{ D}(T_1,\\cdots,T_{N-1})}, \\end{equation}\nwhere $ F^D_g (T_1,\\cdots,T_{N-1})$ are the genus $g$ free energies.\nOn the other hand, as explained for instance in  \\cite{akmv,gm,cgum}, we can expand  the matrix model around the gaussian point for small values of the $g_s$ and fixed values of $M_i$. One finds  that the   logarithm of the partition function has the following behavior\n\\begin{equation}\\label{weak}\\begin{aligned} &  \\log Z_N^{\\rm 4d}(M_1, \\dots, M_{N-1} )=\\sum_{i=1}^{N-1}{1\\over 2}\\left( M_i^2\\log \\left(g_s M_i\\right)\\right)+\n\\mu_i \\log M_i+ \\zeta\\\\\n&+\\sum_{n\\geq -1}g_s^{n}\\sum_{i_1,\\cdots, i_{N-1}}' {\\cal F}_{(n-i_1-\\cdots-i_{N-1})\/2+1,i_1,\\cdots,i_{N-1}}M_1^{i_1}\\cdots M_{N-1}^{i_{N-1}},\n\\end{aligned}\\end{equation}\nwhere $\\sum_{i_1,\\cdots, i_{N-1}}'$ runs only over $i_1,\\cdots, i_{N-1}$ such that\n\\begin{equation}\\begin{aligned} {i_1,\\cdots, i_{N-1}}\\geq {\\bf {0}}, & \\quad  \\left\\{i_1,\\cdots, i_{N-1}\\right\\}\\neq {\\bf {0}}, \\quad \n\\sum_k i_k=n \\mod 2,\n\\end{aligned}\\end{equation}\nwhile $\\mu_i, \\zeta$ are constants. By performing the gaussian integration for severals fixed values of $M_i$, the coefficients ${\\cal F}$ appearing in \\eqref{weak} can be computed exactly.\nBy combining the two expansions \\eqref{thooft1}, \\eqref{weak} it follows that the small $T_i$ expansion of the genus $g$ free energy\n \\begin{equation} F_g^D (T_1,\\cdots,T_{N-1})\\end{equation} is determined by  ${\\cal F}_{g,i_1,\\cdots,i_{N-1}}$. \nFor instance we have  (see appendix \\ref{testsu3} for more details)\n\\begin{equation} \\label{fogen} \\begin{aligned} F_0^D (T_1, \\cdots T_{N-1})=&\\sum_{i=1}^{N-1} \\left( {1\\over 2}{T_i}^2\\log T_i \\right) +\\sum_{ i_1, \\cdots  i_{N-1}}'{\\cal F}_{0, i_1, \\cdots, i_{N-1}}T_{1}^{i_1}\\cdots T_{N-1}^{i_{N-1}}.\n\\end{aligned}\\end{equation}\nLet us start by applying  these techniques  to compute the genus zero free energy \\eqref{fogen} of the matrix model  \\eqref{su3mm}.  We obtain \n\\begin{equation} \\label{myf0n} \\begin{aligned} F_0^D(T_1,\\cdots, T_{N-1})=&\\sum_{j=1}^{N-1}\\left[{T_j^2\\over 2}\\left(\\log \\left( {T_j \\over c_j}\\right) -{3\\over 2}\\right) - T_j {  \\sin\\left({\\pi j \\over N}\\right) \\over   \\sin\\left({\\pi  \\over N}\\right)}\\right]+\\sum_{i<j}{  T_i T_j \\log \\left(a_{ij}\\right)}  + \\mathcal{O}(T_i^3),\\end{aligned} \\end{equation}\nwhere\n\\begin{equation} \\label{coefac}\\begin{aligned} a_{ij}=&\\frac{\\cos \\left(\\frac{\\pi  (i-j)}{N}\\right)-1}{\\cos \\left(\\frac{\\pi  (i+j)}{N}\\right)-1}, \\\\\nc_j=&4 \\sin\\left(\\frac{\\pi  j}{N}\\right)^3 \/\\sin \\left(\\frac{\\pi }{N}\\right), \\end{aligned}\\end{equation}\nand $\\mathcal{O}(T_i^3)$ are the so called instantons  corrections to the prepotential and are computed  in appendix \\ref{testsu3}. \nThe result in \\eqref{myf0n} and \\eqref{coefac} agrees with the computation of   the $SU(N)$  ${\\cal N}=2$ prepotential  obtained in \\cite{klemmlt,dp1, Edelstein:1999fz,Edelstein:2000aj}\n\\footnote{  The agreement is up to the the  $c_j$ term. Our normalization agrees with the one in \\cite{klemmlt}. }.\nIn particular the off diagonal terms of  the prepotential namely \n\\begin{equation} \\sum_{i<j}{  T_i T_j \\log \\left(a_{ij}\\right)} \\end{equation}\nhave been computed for $SU(N)$ gauge theories in \\cite{Edelstein:1999fz, Edelstein:2000aj}. We checked that our result \\eqref{coefac} agrees with the one of \\cite{Edelstein:1999fz,Edelstein:2000aj} namely\n\\begin{equation} \\log \\left(a_{ij}\\right)=2 { \\pi {\\rm i} } \\tau_{\\rm D}^{ij},\\end{equation}\nwhere $\\tau_{\\rm D}^{ij}$ is defined in \\cite{Edelstein:2000aj}.\n\nLet us now look at the instanton part of the prepotential. \nThe case $SU(2)$ was discussed in details in \\cite{bgt} where it was shown that the matrix model \\eqref{su3mm} matches  the existing results in the gauge theory literature which include higher genus $F_g$ computations as well.\nFor $SU(3)$  several computations have been performed in the 90's on the gauge theory side in particular in \\cite{klemmlt}. As explained  in appendix \\ref{testsu3}  the matrix model \\eqref{su3mm} reproduces all these results. \nFor generic $SU(N)$ theories the first instanton correction to \\eqref{myf0n} was computed in \\cite{dp1}.  We  have checked explicitly  that  the matrix model \\eqref{su3mm} reproduces  the results of  \\cite{dp1} if $N=4$ and $N=5$. \n\nAs an additional observation we briefly comment on other proposals for matrix models describing  $SU(N)$ SYM theories. \nLet us first underline that all of them aim to describe the gauge theory partition function in the weak coupling {\\it electric} frame while the one proposed here computes the gauge theory partition function in the strongly coupled {\\it magnetic} frame. \n\nThe idea of computing partition functions of supersymmetric gauge theories as matrix models goes back to the work of \n Dijkgraaf-Vafa \\cite{dv}. Even though the original formulation applies  to ${\\cal N}=1$ theories, one can obtain some results in ${\\cal N}=2$ theories  as well  (see for instance \\cite{cv}). In the case of  $d=4$, ${\\cal N}=2$, $SU(N)$ theories  it was found in  \\cite{kmt} that \n these matrix models are expected \n to reproduce  only the first few terms in the perturbative expansion of the gauge theory, namely  only $F_0$ and $F_1$. \n Our models instead reproduce the full dual free energies $F^D_g$s of the gauge theory\nin the self-dual $\\Omega$ background. For instance in the $SU(2)$ case we have checked explicitly that the genus $g$ free energies $F^D_g$ of \\eqref{su3mm} agree with those of the gauge theory (computed in \\cite{hk06})  up to genus  $g=4$. \n\n On the other hand a different class of models was also proposed for instance in \\cite{ksp} \\footnote{ A more exhaustive list of references as well as a detailed discussion on the limits of these other models can be found in \\cite{mz}.}. These proposals are very different from our one and are constructed by elaborating from the Nekrasov partition function itself. This  leads to  formal models which are  not well defined beyond perturbation theory.  Our matrix models instead, as well as the ones in \\cite{mz,kmz,cgm2,bgt},  are obtained by quantizing mirror curves and in particular without using  Nekrasov results. Therefore the fact that we reproduce the gauge theory expansion  is highly--non trivial. Moreover our models are  well-defined matrix models also beyond perturbation theory i.e.~at finite $M$ and finite $g_s$ as it is manifest from \\eqref{su3mm} and \\eqref{mmsu35d}. \n\n\\subsection{Operator theory }\\label{opno}\nIn this subsection  we give an explicit expression for the operators behind the dual four dimensional limit of the the $Y^{N,0}$ geometry.\nBy  using the Cauchy identity \nwe can rewrite  the matrix model \\eqref{su3mm}  as \n\\begin{equation} \\label{gmat4d} \\begin{aligned}  Z^{\\rm 4d}_N(M_1, \\cdots,  M_{N-1})=&{1\\over M_1! \\cdots M_{N-1}! } \\sum_{\\sigma \\in S_{M}}(-1)^\\sigma\\int  {\\rm d} ^M x \\left(\\prod_{i=1}^{M_1} A_1(x_i,x_{\\sigma(i)}) \\right)\\\\\n&\\left(\\prod_{i=1+M_1}^{M_1+M_2} A_2(x_i,x_{\\sigma(i)})\\right) \\cdots\n  \\left( \\prod_{i=1+\\cdots +M_{N-2}}^{M_1+\\cdots +M_{N-1}} A_{N-1}(x_i,x_{\\sigma(i)})\\right)\n\\end{aligned}\\end{equation}\nwhere $M=\\sum_{i=1}^{N-1}M_i$ and  \n\\begin{equation} A_{j}(x,y)= {     f_{\\rm 4d}(x+\\frac{{\\rm i} \\pi  (4j-N-2)}{2 N})  f_{\\rm 4d}(y+{\\rm i}\\pi   {N-2\\over2 N})  \\over 4 \\pi \\cosh \\left({x- y\\over 2}+\\frac{i \\pi  (2 j-N)}{2 N}\\right)}, \\quad j=1, \\cdots, { N-1}\n \\end{equation} with\n\\begin{equation}  \\begin{aligned}\n f_{\\rm 4d} (u) = \\exp \\left(-\\frac{1}{ 2 g_s  \\sin \\left(\\frac{\\pi   }{N}\\right) } \\cosh (u)\\right) . \\\\\n\\end{aligned} \\end{equation}\nThese are the kernels (in the momentum representation) of the following operators \n\\begin{equation} \\begin{aligned}    A_{1}&=  {\\rm e}^{ - {N-2\\over 4 N}{ \\mathsf q  } }{ f_{\\rm 4d}(\\mathsf p)}{ 1 \\over 2 \\cosh ( \\mathsf q\/2) } {  f_{\\rm 4d}(\\mathsf p)}  {\\rm e}^{ - {N-2\\over 4 N}{ \\mathsf q  } },  \\\\\n A_{j}&=  P_{j} A_1, \\quad P_j={\\rm e}^{  {(j-1)\\over  N} {\\mathsf q} }, \\qquad j=1, \\cdots, N-1, \\qquad [\\mathsf q, \\mathsf p]= 2 \\pi {\\rm i}\n\\end{aligned} \\end{equation}\nHence\n\\begin{equation} \\label{det4}\\sum_{{ {M_i}}\\geq 0} x_1^{M_1} \\cdots x_{N-1}^{M_{N-1}}  Z^{\\rm 4d}_N(M_1, \\cdots,  M_{N-1})=\\det \\left( 1+x_1 A_1+\\cdots+ x_{N-1}A_{N-1} \\right) \\end{equation}\nwhere, in terms of the dictionary \\eqref{dlg}, we have \\begin{equation} x_i = {\\rm e}^{-a\/( 2N \\pi)}G_i.\\end{equation}\nNotice that the operators $A_j$ are of trace class hence the determinant  \\eqref{det4} is well defined and analytic in the $x_i$'s.\nFurthermore  by combining  \\cite{ghm,cgm2,bgt}  with the above computation, it follows that the  Nekrasov--Okounkov \\cite{no2} partition function $ Z^{\\rm NO}$ for pure  $d=4$, ${\\cal N}=2$, $SU(N)$ SYM  in the self--dual background has a spectral determinant interpretation, namely\\footnote{  In this identification one has to be careful in imposing a suitable  domain of definition on the parameters $ x_i$  in the l.h.s. on the equality. For instance in the $SU(2)$ case we usually consider $\\sigma \\neq {\\mathbb Z}\/2$ otherwise an appropriate regularisation of $Z^{\\rm NO}$ could be required (see \\cite{bgt}). }\n\\begin{equation}  \\label{4dsd}\\begin{aligned} Z^{\\rm NO}(x_1,\\cdots, x_{N-1}, \\Lambda, \\epsilon)={\\cal N}(\\Lambda, \\epsilon)\\det \\left( 1+x_1 A_1+\\cdots+ x_{N-1}A_{N-1} \\right), \\end{aligned}\\end{equation}\nwhere ${\\cal N} (\\Lambda, \\epsilon)$ is a non-vanishing proportionality constant.  From the topological string viewpoint this constant is related to the constant map contribution as well as the classical polynomial part of the topological string free energy.\nFor instance in the case of $SU(2)$ we have \\cite{bgt}\n \\begin{equation} {\\cal N} (\\Lambda, \\epsilon)={\\rm e}^{- \\Lambda^2\/(4 \\pi^4 \\epsilon^2)} \\left({\\Lambda\\over 4 \\pi^2 \\epsilon}\\right)^{1\/4} {\\rm e}^{-{1\\over 12}\\log 2-3 \\zeta' (-1)}.\\end{equation}\nHence the spectrum of the above operators is completely determined by  the ${\\cal N}=2$, $SU(N)$ SYM in the self--dual $\\Omega$ background. \n Moreover, as discussed in section \\ref{newsection} the spectral determinant representations of the NO partition function determines its exact S-duality transformation. In the $SU(2)$ case this is given by \\eqref{zj4d} and  a similar behaviour also holds in the $SU(N)$ case. Indeed we have\n\\begin{equation} \\label{zdet4}Z^{\\rm 4d}_N(M_1, \\cdots,  M_{N-1})= {1 \\over (2 \\pi {\\rm i})^{N-1}}\\oint _0 {{\\rm d} x_1 \\over x_1^{M_1+1}}\\cdots \\oint _0 {{\\rm d} x_{N-1} \\over x_{N-1}^{M_{N-1}+1}}\\det \\left( 1+x_1 A_1+\\cdots+ x_{N-1}A_{N-1} \\right).\\end{equation}\nBy combining  \\eqref{zdet4} with \\eqref{4dsd} one obtains  the exact S-duality transformation which generalises   \\eqref{zj4d} in the higher rank case.\nNotice that, after a suitable parametrization, the classical version of \\begin{equation} 1+\\sum_{i=1}^{N-1}x_i A_i=0\\end{equation}\ncoincides with the classical $SU(N)$ Seiberg-Witten curve, namely \\begin{equation} \\label{si} z+z^{-1}-v^N+\\sum_{i=0}^{N-2}x_{N-i-1} v^i=0,\\end{equation}\nwhere we used \\begin{equation} z+z^{-1}=2\\cosh(q\/2) \\exp \\left[{-{ N-2\\over 2 N}q+{N\\Lambda\\over \\pi^2 \\epsilon} \\cosh p}\\right] +{\\rm e}^{-q}, \\quad v={\\rm e}^{-q\/N}. \\end{equation}\n\n\nAs an additional observation we  recall that in the pure $SU(2)$ case the  determinant \\eqref{4dsd} corresponds to the $\\tau$ function of  the Painlev\\'e $\\rm III_3$ \\cite{ilt,bgt,zamo}. Moreover, by changing the matter content of the $SU(2)$ theory, we  recover the $\\tau$ function for  the other Painlev\\'e equations  \\cite{blmst}.   It is therefore natural to expect that in the generic $SU(N)$ case  \\eqref{4dsd} provides the solution for an isomonodromy problem which generalizes the one related to Painlev\\'e equations. \n\nAccording to \\cite{blmst} the connection of which one has to study the isomonodromy problem is obtained from the Hitchin system describing the relevant Seiberg-Witten geometry. \nIn the forthcoming subsection \\ref{opp} we will describe the isomonodromy problem associated to the one-period phase\nof the matrix model \\eqref{su3mm} which as we will see give rise to the closed Toda chain equations. We remark that in order to get the appropriate connection one has to \nconsider Hitchin's system on a cylinder: this is in line with the findings of 4d\/2d correspondence discussed in \\cite{cv} which naturally connects the isomonodromy problem to $tt^*$ equations \\cite{ttstar}. \n\n\n\n  \n\n\n\n\\subsection{The  one period phase }\n\\label{opp}\n\nLet us consider the particular point in the moduli space of the $SU(N)$ theory where only one dual period is non vanishing.\nWe can think of this point as a 1-period deformation of the  strong coupling singularity described in \\cite{Douglas:1995nw}.\n  %\n  At this singularity the $N-1$ monopoles become massless and the dual periods of the $SU(N)$ Seiberg--Witten curve vanish\n   \\begin{equation} a_D^{(k)}=0, \\quad k=1, \\cdots N-1.\\end{equation}  \nIf we keep only one non--vanishing period, namely\n \\begin{equation} a_D^{(1)} \\neq 0, \\end{equation} the partition function is described by the following one-cut matrix model \n \\begin{equation} \\label{ad0}\\begin{aligned} Z_{\\rm 4d}^{(1)}(M)= &{ 1 \\over M! }  \\int {{\\rm d} ^M x\\over (2\\pi)^M} \\prod_{i=1}^{M } {\\rm e}^{ -\\frac{1}{ g_s }  \\cosh (x_i)} {\\prod_{i<j}4\\sinh\\left({x_i-x_j\\over 2}  \\right)^2\\over \\prod_{i, j}2\\cosh\\left( {x_i-x_j \\over 2} +{\\rm i} \\pi \\beta, \\right)}, \\quad \\beta={2-N\\over 2 N}.\\end{aligned} \\end{equation}\nThis matrix model computes  the fermionic spectral traces of the following \nkernel \n\\begin{equation}\\label{k1} { \\rm K}(x,y)={{\\rm e}^{-\\cosh (x)  \/2 g_s}}{ 1 \\over 4 \\pi \\cosh \\left( {x-y\\over 2}- {\\rm i} \\pi {(N-2) \\over 2N}\\right) } {{\\rm e}^{-\\cosh(y)  \/2 g_s}}, \\quad x,y \\in {\\mathbb R},\\end{equation}\nnamely\n\\begin{equation} Z_{\\rm 4d}^{(1)}(M)= \\sum_{\\sigma\\in S_M}(-1)^{\\sigma} { 1 \\over M! }  \\int {{\\rm d} ^M x}   \\prod_{i=1}^{M }  K(x_i, x_{\\sigma(i)}). \\end{equation}\nHence, it follows from \\cite{wid} that \n \\begin{equation} \\label{qdef}q_k=\\log \\left[\\det(1-\\kappa {\\rm e}^{2(k+1){\\rm i} \\pi\/N }K)\/\\det(1-\\kappa {\\rm e}^{2k{\\rm i} \\pi\/N} K)\\right], \\quad k \\in {\\mathbb Z}\\end{equation}\n fulfills the first equation of Toda hierarchy which reads\n \\begin{equation} \\label{qt} q_{\\ell} ^{''}+{1\\over t}q_{\\ell}^{'}= {\\rm e}^{q_{\\ell}-q_{{\\ell}-1}}-{\\rm e}^{q_{{\\ell}+1}-q_{{\\ell}}}, \\end{equation}\nwhere  \\begin{equation} t=(2 \\sin \\left({\\pi\/N}\\right)g_s)^{-1} .\\end{equation}\n It was found in  \\cite{its1,its2} that \\eqref{qt} arises as compatibility condition for the isomonodromy deformations of an $SL(N,{\\mathbb C})$ connection on a cylinder with regular singularities at zero and at infinity. This agrees with our general expectation (see discussion at the end of the previous subsection) and it gives a concrete relation between the $SU(N)$ matrix models presented above and the tt* equations of \\cite{ttstar}.\nMore precisely, by considering the $N$-covering $z=w^N$ of the SW geometry \\eqref{si} \none gets the \nradial component of the \nHitchin's connection as (see Chapter 5 of \\cite{swbook})\n\\begin{equation}\n{\\cal A}=\\frac{\\partial}{\\partial r} {\\bf q} + w^{-1}\\left(e^{-\\alpha_0 q}E_{-\\alpha_0}+\\sum_{\\alpha\\,\\, {\\rm simple}} e^{\\alpha q} E_\\alpha\\right)\n+ w\\left(e^{-\\alpha_0 q}E_{\\alpha_0}+\\sum_{\\alpha\\,\\, {\\rm simple}} e^{\\alpha q} E_{-\\alpha}\\right)\n\\end{equation}\nwhere ${\\bf q}={\\rm diag}(q_0,\\ldots,q_{N-1})$ is subject to the traceless condition $q_0+\\ldots+q_{N-1}=0$ and $\\alpha$'s are the roots of  $\\hat A_{N-1}$  algebra, $\\alpha_0$ being the one associated to the affinization. The above is precisely the connection whose isomonodromy problem gives rise \nto closed Toda chain equations as explained in \\cite{its1,its2}. Notice that this is describing the $tt^*$ geometry of the  Landau-Ginzburg model  whose potential is associated to the $A_{N-1}$ singularity.\nIn the one-period case one considers isomonodromy with respect to the length of the cylinder which corresponds to the Yang-Mills coupling of gauge theory. We expect the higher times of Toda hierarchy\nto be related with the insertion of local observables giving rise to new terms in the potential of the kind $T_n \\cosh (n x)$, $T_n$ being related to higher times of Toda hierarchy.\nOn the other hand, the full isomonodromy problem considering Whitham deformations of the Hitchin's system \\cite{jimbo2,Krichever:1992qe,Nakatsu:1995bz,Itoyama:1995nv, Gorsky:1995zq,Edelstein:1999xk,Bonelli:2009zp} \nshould give rise to a system of PDEs satisfied by the gran-canonical partition function of the full matrix model \\eqref{su3mm}.   \n\n\n\nIn the particular case of $N=2$ the partition function \\eqref{ad0} becomes the well studied polymer matrix model \\cite{zamo,gm, fadsal,cfiv,ceva} and we have \n\\begin{equation} q_k=q_{k+2}=-q_{k+1} .\\end{equation}\nAs expected  \\eqref{qt} becomes the $\\sinh$ form of the Painlev\\'e $\\rm III_3$ equation\n \\begin{equation} \\label{qt2} q_{0} ^{''}+{1\\over t}q_{0}^{'}= {\\rm e}^{2q_0}-{\\rm e}^{-2 q_{0}}. \\end{equation}\n\nInterestingly  this is not the first time than the matrix model \\eqref{ad0}  appeared in the literature, indeed it is of the same type as the one considered by Kostov  in   \\cite{kos} to describe the 6-vertex model. \nIn \\cite{oz} it was shown that such a matrix model can be evaluated exactly for fixed values of $M$ and $\\hbar$  by using a TBA system which generalizes the one of  \\cite{cfiv,zamo,tw}.\nMoreover in \\cite{Hollowood:2003gr} it was argued that the  matrix model \\eqref{ad0} computes the partition function of $SU(M)$ ${\\cal N}=1^*$ SYM   in 5 dimensions. More precisely to relate \\eqref{ad0} to  \\cite{Hollowood:2003gr} we need the following dictionary  \n \\begin{equation} {2  \\pi \\over N}= R m,\\end{equation}\nwhere in the context of  \\cite{Hollowood:2003gr}  $R$ is the radius of the fifth dimension and $m$ the mass. To our knowledge this is a curious coincidence.  \n Moreover if we  rescale $$x \\to 2 x \\pi \/N$$  and we take   $N \\to \\infty$ then \\eqref{ad0} reduces to the  D-particle matrix model studied in  \\cite{kkn} where  the  authors show that the corresponding spectral determinant is related to the $\\tau$ function of the KP hierarchy.\nWe also observe that in  \\cite{hkksym} a similar type of model, but with a slightly different contour,  was used in   to compute the partition function of the one dimensional reduction of ${\\cal N}=1$ $SU(M)$ SYM on a circle  of radius $\\beta$ with insertion of a source term.\nThe same matrix model was also used in the context of 2d black hole in \\cite{Kazakov:2000pm}. \n It would be interesting to understand if this is just a mathematical coincidence or if there is a deeper physical meaning.\n\n\n\n\n\n\n \n\n\n\n\n\\section{Conclusions}\n\\raggedbottom\nIn this paper we proposed a matrix model computing  the partition function of $d=4$ ${\\cal N}=2$, $SU(N)$ SYM in the magnetic frame, coupled to self-dual $\\Omega $ background. In particular the 't Hooft expansion of this matrix model reproduces the genus expansion of the gauge theory in the magnetic frame.   Our method relies on the formalism developed in \\cite{cgm2}  for the quantization of  higher genus mirror curves and  as a consequence it provides a  spectral determinant representation for the  four dimensional Nekrasov--Okounkov partition function in the case of higher rank gauge theories, generalizing the results of \\cite{bgt}.  Such a spectral determinant representation also  provides  exact S-duality transformation properties for the pure  gauge theory, which we have  spelled out in detail in the $SU(2)$ case, see \\eqref{zj4d}. \n\nAs explained in \\cite{bgt}, when the gauge group is $SU(2)$ the matrix model  \\eqref{su3mm} coincides with a well known $O(2)$ matrix model \\cite{kos39,kos} which is related to the 2d Ising model and the physics of 2d self--avoiding polymers. It would be interesting to see if the  higher rank matrix model \\eqref{su3mm}  can be connected to statistical systems too.\n\n\nIn the one-period phase, the spectral determinants \\eqref{qdef} satisfy Toda lattice equation, see subsection \\ref{opp}. We expect the corresponding hierarchy of differential equations to be satisfied by the spectral determinants associated to the generating function of the local chiral observables  (see for instance \\cite{Fucito:2005wc}). It would be interesting to further explore this aspect.  Also it is worth investigating the insertion of surface operators (see for instance \\cite{Ashok:2017odt}) and their description in the magnetic phase of the gauge theory as eigenfunctions of the quantum operators arising from the dual four-dimensional limit of the non-perturbative open topological strings \\cite{Marino:2016rsq}. \n\n Furthermore the matrix models proposed here are obtained by combining the quantization of mirror curves with the Fermi gas formalism \\cite{mp,ghm,mz,cgm2,bgt}, however it would be important to derive them directly in gauge theory.\n\n\n\nAnother interesting point would be to prove explicitly  that the higher rank NO partition function is proportional to a suitably defined $\\tau$ function for the isomonodromy problem related to the Hitchin's integrable system corresponding to  the $SU(N)$ SW geometry.  Finally, the extension of our results to gauge theories coupled to matter is still to be worked out. We hope to report on this in the near future. \n\n\\section*{Acknowledgements}\nWe would like to thank Marco Bertola, Simone Giacomelli, Jie Gu, Rajesh Gupta, Victor I. Giraldo , Kohei Iwaki,  Vladimir Kazakov, Oleg Lisovyy and Marcos Mari\\~no for valuable  discussions and in particular Jie Gu for a careful reading of the draft. \nA.G. thanks  Durham University, ENS Paris, Kyoto and Osaka Universities for the kind hospitality during the course of this project.\nA.T. thanks IIP, Natal and Universit\\'e Paris VII for kind hospitality during the course of this project.\nThe work of G.B. is supported by the PRIN project \"Non-perturbative Aspects Of Gauge Theories And Strings\".\nG.B. and A.G. acknowledge support by INFN Iniziativa Specifica ST\\&FI.\nA.T. acknowledges support by INFN Iniziativa Specifica GAST.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction: Renormalization and universal low-energy theorem}\n\\vspace{1cm}\n\nThe renormalization with boundary conditions (BC) are best illustrated\nby the effective range expansion where the S-wave phase shift is\ngiven by ($p$ is CM momentum)\n\\begin{eqnarray}\np \\cot \\delta_0(p) = - \\frac1{\\alpha_0} + \\frac12 r_0 p^2 + \\dots \n\\end{eqnarray}\nwhere $r_0$ is the effective range and $\\alpha_0$ the scattering\nlength. Using the superposition principle one obtains a bilinear\nrelation dubbed as {\\it universal low energy\n  theorem}~\\cite{PavonValderrama:2005wv,RuizArriola:2007wm},\n\\begin{eqnarray}\nr_{0} = A + \\frac{B}{\\alpha_{0}}+ \\frac{C}{\\alpha_{0}^2} \\, ,\n\\label{eq:r0-a0}\n\\end{eqnarray}\nwhere the coefficients $A$, $B$ and $C$ depend {\\it only} on the long\ndistance potential. This way $V(r)$ and $\\alpha_0$ are regarded as\n{\\it independent} variables.  We present below a variety of\nsituations.\n\n\nThe case of ultra-could collisions between neutral atoms described by\nthe van der Waals (vdW) potential $V(r) = - \\sum_{n=6}^{\\infty}\n\\frac{C_n}{r^n}$ where $C_n$ are the vdW coefficients was analyzed in\nRefs.~\\cite{Arriola:2009wi,PhysRevA.81.044701}.  The terms $n=6,8,10$\nare usually retained and the singularity appearing in the vdW\npotential led to the use of phenomenological potentials, like the\nLennard-Jones, modeling short-distance physics.\nThe correlation, Eq.~(\\ref{eq:r0-a0}), is plotted in\nFig.~\\ref{fig:r0-a0} (upper left panel) for the term $n=6$ in units of\nthe vdW radius $R_6 = (2\\mu C_6)^{1\/4}$ where the coefficients\nare $A\/R_6 = 1.39473$, $B\/R_6 = -1.33333$ and $C\/R_6 = 0.637318$,\ntogether with the experimental and potential model determinations of\n$\\alpha_0$ and $r_0$ known up to date and compiled in\nRef.~\\cite{Arriola:2009wi,PhysRevA.81.044701}.\nOn the light of this clear universality one realizes that the leading $1\/r^6$\ncontribution suffices to accurately describe low energy atom-atom scattering\nwith just two parameters in a wide energy range.\n\n\nIn our studies of the NN interaction we consider the OBE potential\nwith exchange of $\\pi$, $\\sigma$, $\\rho$, $\\omega$ keeping the\nspin-flavor structure of the large-$N_c$ limit according with\nRefs.~\\cite{Kaplan:1996rk,Kaplan:1995yg} with the parameters fixed to\nthe condition that the $^1S_0$ $np$ phase shift be reproduced,\n\\begin{eqnarray}\nV_{^1S_0}(r) =\n - \\frac{g_{\\pi NN}^2 m_\\pi^2} {16 \\pi M_N^2} \\frac{e^{-m_\\pi r}}{r}\n - \\frac{g_{\\sigma NN}^2}{4 \\pi}\\frac{e^{-m_\\sigma r}}{r}\n+ \\frac{{g_{\\omega NN}}^2}{4 \\pi}\\frac{e^{-m_\\omega r}}{r} \n- \\frac{{f_{\\rho NN}}^2 m_\\rho^2}{8 \\pi M_N^2 } \\frac{e^{-m_\\omega r}}{r} \\, .\n\\label{eq:OBE}\n\\end{eqnarray}\nAs was shown in Ref.~\\cite{Cordon:2009pj} and more recently in\nRef.~\\cite{Cordon:2010sq} the unnaturally large scattering length in this\nchannel triggers a fine-tuning problem in the potential parameters. However\nusing renormalization one disentangles the unknown short-distance physics from\nthe well established long-distance one by fixing from the start low-energy\nparameters (LEPs).\nIn particular, one finds vector mesons to play a marginal role in the\ndescription of NN scatering below pion production\nthrehold~\\cite{Cordon:2009pj}. \nIn addition \none obtains\n$m_{\\sigma} = 501(25)\\,{\\rm MeV}$, $g_{\\sigma NN} = 9(1)$,\n $g_{\\omega NN} \\sim g_{\\omega NN}^{SU(3)} = 3 g_{\\rho NN}$\neven when charge symmetry breaking is implemented by\nconsidering the neutron-proton and charged-neutral pion mass\ndifferences~\\cite{Cordon:2010sq}.\nUsing the more conventional regular boundary condition $u(0) = 0$ two\nvery good but mutually incompatibles fits are obtained, two scenarios\nwhich correspond with the absence or presence of spurious bound\nstates~\\cite{Cordon:2009pj} and in any case with or without large\n$SU(3)$ violation of $g_{\\omega NN} $ respectively. \n\nUsing the previous potential in Ref.~\\cite{CalleCordon:2008cz} we have\nre-interpreted the old nuclear Wigner (spin-isospin) SU(4) symmetry as\na {\\it long distance symmetry}.\nThis symmetry implies equal singlet and triplet potentials $V_{1S0} =\nV_{3S1}$, a fact that agrees with the large-$N_c$ expectations.\nThe renormalization approach allows a breaking of the symmetry at\nshort distances having $\\alpha_{0,1S0} \\neq \\alpha_{0,3S1}$ and $r_{0,1S0}\n\\neq r_{0,3S1}$ even with the same long-distance potential. This can be\nclearly seen in Fig.~\\ref{fig:r0-a0} (upper right panel) where we plot\nEq.~(\\ref{eq:r0-a0}) in the case of OPE and OPE+$\\sigma$. The only\ndifference between $r_{0,1S0}$ and $r_{0,3S1}$ resides in the\nnumerical values of the scattering lengths $\\alpha_{0,1S0}$ and\n$\\alpha_{0,3S1}$ but the coefficients $A$,$B$ and $C$ in\nEq.~(\\ref{eq:r0-a0}) are the same in both channels. The experimental\nvalues fall almost on top of the curve.\n\n\nThe posibility of connecting different NN isospin channels has been\nanalyzed in Ref.~\\cite{Cordon:2010sq} using a {\\it short distance\n  connection}. One is able to obtain a bilinear relation between the\nscattering lengths of different problems $ \\alpha_{0,2} =\n(a\\ \\alpha_{0,1} + b)\/ (c\\ \\alpha_{0,1} + d) $ where the coefficients\n$a$,$b$,$c$ and $d$ depend {\\it only} on the long distance\npotential. Fixing the scattering length in one channel,\ne.g. $\\alpha_{0,np} = -23.74\\,{\\rm fm}$, with the suitable\nincorporation of the Coulomb interaction $V_C(r) = \\alpha\/r$, with\n$\\alpha \\simeq 1\/137$ for the pp(C) case, we obtain the rest of LEPs\ndisplayed in Table~\\ref{tab:ABC-CSB}.\n\\begin{table}\n\\centering\n\\begin{tabular}{c|ccc|cc|cc}\n\\hline\nBB    & A [${\\rm fm}$] & B [${\\rm fm^2}$] &  C [${\\rm fm^3}$]\n      &  $\\alpha_0 [ {\\rm fm} ] $ & $r_0 [ {\\rm fm} ] $ \n      &  $\\alpha_0^{\\rm exp} [ {\\rm fm} ] $ & $r_0^{\\rm exp} [ {\\rm fm} ] $ \n\\\\\\hline\nnp    & 2.437 & -5.345 & 5.277 & input   & 2.672  & -23.74(2) & 2.77(5)\n\\\\\npp(S) & 2.465 & -5.661 & 5.996 & -17.806 & 2.802  & -17.46    & 2.845\\\\\npp(C) & 1.982 & -4.985 & 7.009 &  -7.706 & 2.747  &  -7.8149(29) & 2.769(14) \\\\\nnn    & 2.467 & -5.666 & 6.003 & -19.626 & 2.771  & -18.9(4)  & 2.75(11) \\\\\n\\hline\n\\end{tabular}\n\\caption{Coefficients $A$,$B$ and $C$ of Eq.~(\\ref{eq:r0-a0}) using\n  the OBE potential Eq.~(\\ref{eq:OBE}) implementing CSB and values for\n  the LEPs using the short distance connection. We also list\n  experimental or recommended values.}\n\\label{tab:ABC-CSB}\n\\end{table}\nIn Fig.~\\ref{fig:r0-a0} (lower left panel) we have plotted\nEq.~(\\ref{eq:r0-a0}) for the np system and the predicted LEPs for the\nother channels. This figure indicates that the strongly interacting\ncomponent is close to a unique curve whereas adding the\nCoulomb interaction distortes the long distance potential and hence\nthe pp(C) LEPs move away from the curve. No explicit $\\rho-\\omega$\nmixing seems necessary as it is a short range effect which is\nreparameterized in terms of $\\alpha_0$.\n\nFinally, we can assume systems with strangeness such as\n$\\Lambda\\Lambda$ for example (see e.g.  \\cite{Yamamoto:2010zzi} and\nreferences therein). If $\\Lambda$-hyperons interact exchanging\n$1\\sigma$- and $1\\omega$-mesons and we use SU(3) relations for the\ncouplings $g_{\\sigma \\Lambda\\Lambda} = g_{\\omega NN}$ and $g_{\\omega\n  \\Lambda\\Lambda} = 2\/3 g_{\\omega NN}$ we obtain $A= 1.705 {\\rm fm} $,\n$B= -1.475 {\\rm fm}^2$ and $ C=1.245 {\\rm fm}^3$. The universal curve\nfor such a simple OBE potential is displayed in Fig.~\\ref{fig:r0-a0}\n(lower right panel) and is not far from more sophisticated potential\nmodel calculations~\\cite{Yamamoto:2010zzi}.\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=5cm,width=4cm,angle=270]{solid-atoms-curve-in.eps}\n\\hspace{0.2cm}\n\\includegraphics[height=5cm,width=4cm,angle=270]{r0-a0.Wigner.eps}\\\\\n\\includegraphics[height=5.3cm,width=4cm,angle=270]{r0-a0.CSB.eps}\n\\includegraphics[height=5.3cm,width=4.3cm,angle=270]{r0LL-a0LL.eps}\n\\end{center}\n\\caption{Universal relations: (upper left panel) Atom-Atom vdW correlation in\n  units of $R_6$, (upper right panel) NN $^1S_0$-$^3S_1$ Wigner\n  correlation, (lower left panel) nn,np and pp(S) short-distance\n  connection correlation and (lower right panel) $\\Lambda\\Lambda$ long\n  distance correlation compared to potential models calculations (see\n  e.g.  \\cite{Yamamoto:2010zzi}) .}\n\\label{fig:r0-a0}      \n\\end{figure}\n\n\\begin{acknowledgements}\nWe thank M. Pav\\'on Valderrama for his collaboration on\nRef.~\\cite{Cordon:2010sq}.  This work is supported by the Spanish DGI\nand FEDER funds with grant FIS2008-01143\/FIS, Junta de Andaluc\u00eda grant\nFQM225-05. \n\\end{acknowledgements}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Implementation Details}\n\nThe simulation was designed for infinite time horizons and infinite number of runs. For each run, the unit for the length of simulation is defined as ``Second'' for simplicity. The settings that are consistent for both model-based approaches and model-free approaches are listed below.\n\n\\begin{table}[h!]\n\\centering\n \\begin{tabular}{|c | c|} \n \\hline\n Length of each simulation run & $50, 000$ \\\\\n \\hline\n Maximum number of commuters waiting at each node& $1, 000, 001$ \\\\\n \\hline\nMaximum sample size collected for each run & $512, 000$ \\\\\n\\hline\n\n\\end{tabular}\n\\label{tab:sim_parameters} \n\\end{table}\n\nThe imitation learning used to initialise model-free leaner will update the policy network until the cross entropy loss stop decreasing or reach maximum update steps $30, 000$. The hyperparameters used in training the model-free RL and the hybrid approach are the same for all experiment settings and are listed below.\n\n\\begin{table}[h!]\n\\centering\n \\begin{tabular}{|c | c|} \n \\hline\n Learning rate & $0.01$ \\\\\n \\hline\n Clipping parameter epsilon& $0.1$ \\\\\n \\hline\nIterations & $10$ \\\\\n\\hline\nBatch size &  Input data size\\\\\n\\hline\n\n\\end{tabular}\n\\label{tab:imitation_parameters} \n\\end{table}\n\n\\section{Experimental Details}\n\nThe experimental details for both artificial road network and community-based Singapore road network are listed below. All experiments were conducted on a commodity server with two NVIDIA Quadro P4000 GPUs with $8$ GB GDDR5 memory.\n\n\\begin{table}[h!]\n\\centering\n \\begin{tabular}{|c| c| c | c|} \n \\hline\n Experiment & Road Network Type &Edge Travel Time & Epochs \\\\\n \\hline\n S1 & $10 \\times 10$ Lattice & $X\\sim unif\\{1, 10\\}$ & $1000$  \\\\\n \\hline\n S2& $33 \\times 33$ Lattice& $X\\sim unif\\{1, 10\\}$  & $1000$\\\\\n \\hline\nS3 & Road network ($2591$ nodes) & Travel time (s) & $1000$ \\\\\n\\hline\nS4 & Road network ($2591$ nodes) &Travel time (s)&  $3000$\\\\\n\\hline\n\n\\end{tabular}\n\\label{tab:exp_parameters} \n\\end{table}\n\n\\section{CONCLUSIONS AND OUTLOOKS}\n\n\n\n\n\nThe methodologies presented in this paper could have wider applications beyond taxi systems, for a wide range of empirical and synthetic systems. For future work, we will work on incorporating time-dependent traffic conditions and demand patterns for real-world impact. We will also extend the agent-based simulation and the proposed model-based and model-free approaches to cases with ride-sharing, for the optimal routing and route-matching between shared commuters. Our ultimate goal is to optimise routing of demand-responsive transit system in a hybrid form consisting of real-time booking, street-hailing, as well as ride-sharing. \n\n\n\n\\section{Statement of Contribution\/Potential Impact}\n\nIn this paper, we propose methodologies for optimising vacant vehicle routing in a decentralised way to meet stochastic spatio-temporal demand patterns. The research is motivated by the grand challenge of urban mobility to meet increasing demands for highly efficient yet high-quality transport services with minimal costs. The difficulty of the problem stems from the stochastic spatio-temporal distribution of the commuter demand, on a large road network with complex structures.\nWe design a set of methodologies for intelligent routing of taxi fleet with reinforcement learning primed with a top-down dispatch model, evaluated on both synthetic and empirical data for commuter demand patterns and road networks. The proposed methodologies can potentially benefit everyone in tackling the urban mobility challenges: shorter waiting time for commuters, higher profits for service providers, easy-to-implement and versatile solutions for government agencies, and less emissions to the environment. The methodologies presented in this paper are in time for real-world applications under current technology and in near future when wireless communication and autonomous vehicles enable new ways of taxi fleet operation. They could also have wider applications beyond taxi systems, for on-demand transportation and logistics services such as dynamic reallocation of shared resources and last-mile delivery.\n\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\\input{Introduction}\n\n\\section{Related Works}\n\\label{sec:related works}\n\\input{Review}\n\n\\section{Problem Formulation}\n\\label{sec:problem}\n\\input{Problem}\n\n\\section{System Overview}\n\\label{sec:system}\n\\input{System}\n\n\\section{Methodology}\n\\label{sec:methodology}\n\\input{Methodology}\n\n\\section{Experiments}\n\\label{sec:experiment}\n\\input{Experiment}\n\n\n\\section{Results}\n\\label{sec:results}\n\\input{Results}\n\n\n\\section{Summary and Outlook}\n\\label{sec:discussion}\n\\input{Discussion}\n\n\\section*{Acknowledgements}\nThis research is supported by the National Research Foundation, Singapore, and the Land Transport Authority under its Urban Mobility Grand Challenge Programme (Award No. UMGC-L005) and the National Research Foundation, Singapore under the NRF fellowship award (NRF-NRFF12-2020-005).\nAny opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not reflect the views of National Research Foundation, Singapore and the Land Transport Authority.\n\n\n\n\n\\bibliographystyle{abbrv}\n\n\\subsection{Model-based Routing Policy}\n\\label{sec: MB routing}\n\n\n\n\nFor this target problem, instead of learning a model of the dynamic function of environment (like in most model-based RL), we can construct routing policy from transportation theory using the MDP. In order to find the optimal $\\mathcal P_{ij}$ which allows us to route a vacant taxi at any location in a given road network, the intuition behind our proposed model-based dispatch algorithm is to direct the vacant taxis to a set of ``hotspots\" in the road network, which are identified either statically or dynamically. These hotspots are locations where the probability of finding a waiting commuter is high, so that vacant taxis are dispatched to these hotspots, which can be formally described by a dispatch matrix $\\mathcal D_{ij}$: given a vacant taxi at node $i$, $\\mathcal D_{ij}$ gives the probability of the taxi being dispatched to node $j$ (which potentially can be quite far away). The dispatch probability is also inversely proportional to the distance between $i$ and $j$. There will also be some routing choice from node $i$ to node $j$ (the simplest being the shortest path between the two nodes). Once $\\mathcal D_{ij}$ and the routing choice are specified, the turn-by-turn routing of $\\mathcal P_{ij}$ as model-based routing policy can also be uniquely determined via microscopic simulations and sampling.\nThroughout this paper we assume shortest path routing between two nodes for $\\mathcal D_{ij}$, as the specific choice of routing algorithm is not essential in this particular context. \n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{Dispatch Algorithm for Hotspots}\nOne of the common approaches for vacant taxi manoeuvring is a good algorithm for dispatching them to certain locations on the map where higher probabilities of commuter generation are expected (the hotspots). This is also implicitly the strategy of experienced taxi drivers. Following the problem statements and definitions in Sec.~\\ref{sec:problem}, we first propose models of efficient dispatch algorithm, using $\\mathcal A_{ij},g_i,M_{ij}$ as inputs, to obtain an efficient dispatch matrix $\\mathcal D_{ij}$. Physically, for a vacant taxi appearing at node $i$, $\\mathcal D_{ij}$ gives the probability for this vacant taxi to be dispatched to a (not necessarily adjacent) node $j$, with the constraint $\\sum_j\\mathcal D_{ij}=1$. We also allow $\\mathcal D_{ii}\\neq 0$, implying the vacant taxi stay put at its location waiting for commuters to appear. If node $j$ is far away from node $i$, the vacant taxi will be dispatched to node $j$ via the shortest path obtained from the network adjacency matrix $\\mathcal A_{ij}^{-1}$. Without loss of generality, here we only make $\\mathcal D_{ij}$ dependent on $g_i$ and $\\mathcal A_{ij}$, and not explicitly dependent on $M_{ij}$.\n\nLet $d_{ij}$ be the shortest distance in terms of travel time between node $i$ and $j$, as computed from the adjacency matrix $\\mathcal A_{ij}^{-1}$. For $i=j$ we set $d_{ii}=1$. A rather general dispatch model is given as follows:\n\\begin{eqnarray}\\label{dispatchmodel}\n\\mathcal D_{ij}=f\\left(d_{ij}^{-1},g_j\\right)\n\\end{eqnarray}\nIntuitively, $\\mathcal D_{ij}$ should increase monotonically with $g_j$, while decreasing monotonically with $d_{ij}$. This is because if all other conditions are identical, we should have a higher probability of dispatching vacant taxis to nodes with higher rate of commuter generation, the so-called hotspots. On the other hand, for two nodes with the same rate of commuter generation, there should be a reduced probability to dispatch the vacant taxi to a node that is further away. Optimisation of Eq.(\\ref{dispatchmodel}) can be done by Taylor expanding $f$ with respect to $d_{ij}^{-1}$ and $g_j$, and tuning the expansion coefficients at different orders to maximise the utility functions computed from agent based simulations. Here we use a rather simple model with the lowest order expansion as follows:\n\\begin{eqnarray}\\label{dm1}\n\\mathcal D_{ij}=\\lambda_i \\frac{g_j}{d_{ij}}\n\\end{eqnarray}\nwhere $\\lambda_i=\\left(\\sum_j\\mathcal D_{ij}\\right)^{-1}$ is the normalisation factor. The performance of this model has been satisfactory based on extensive agent-based numerical simulations shown in later section Sec.\\ref{sec:mb result}.\n\n\\subsubsection{Effective Routing Policy from Dispatch Model}\n\nOne major difference between $\\mathcal D_{ij}$ and $\\mathcal P_{ij}$ for vacant taxi manoeuvring is that for the former, nodes $i$ and $j$ can be quite far away. On the other hand, $\\mathcal P_{ij}$ is only non-zero when nodes $i$ and $j$ are neighbours. The degrees of freedom for $\\mathcal D_{ij}$ is thus much larger and scale as the square of the network size. This makes optimising $\\mathcal D_{ij}$ a much more difficult problem, as compared to optimising $\\mathcal P_{ij}$ (whose degrees of freedom scale linearly with network size). On the other hand, when the vacant taxi is dispatched from node $i$ to $j$, the path between nodes $i$ and $j$ is deterministic (and we choose the shortest path in this work). Effectively, given a specific $\\mathcal A_{ij},g_i,M_{ij},\\mathcal D_{ij}$ there is still a statistical distribution for the choice of neighbouring nodes from any node $i$, if we look at a large number of vacant taxis arriving at node i (either as the destination of the dispatch, or as an intermediate node enroute to the dispatch destination), and how these vacant taxis choose the next node. We can thus generate an effective $\\mathcal P_{ij}$ via agent based simulations, and this particular $\\mathcal P_{ij}$ is unique and well-defined in the limit of long time simulations.\n\nFormally, for a simulation time with a total number of discrete time-steps $H$, we denote $n_i^H$ as the number of times when an vacant taxi lands on node $i$, and we do not distinguish between different taxis. Let $p_{ij}^H$ be the number of times the vacant taxi at node $i$ moves towards one of the neighbouring node $j$. The effective turn-by-turn routing policy is defined as follows:\n\\begin{eqnarray}\n\\bar{\\mathcal P}_{ij}=\\lim_{H\\rightarrow\\infty}\\frac{1}{n_i^T}p_{ij}^H\n\\end{eqnarray}\nIn this way, the effective policy can be unambiguously defined for almost any manoeuvring strategies. If the routing strategy for vacant taxis is a turn-by-turn policy $\\mathcal P_{ij}$ to start with, then obviously we will have $\\bar{\\mathcal P}_{ij}=\\mathcal P_{ij}$. If the routing strategy is from a dispatch policy $\\mathcal D_{ij}$, there is a unique one-to-one mapping from $\\mathcal D_{ij}$ to $\\bar{\\mathcal P}_{ij}$, given all other conditions are fixed. We can thus evaluate and compare the performances of $\\mathcal D_{ij}$ and $\\bar{\\mathcal P}_{ij}$ in a meaningful manner. More importantly, $\\bar{\\mathcal P}_{ij}$ can be used as an optimised initial state for reinforcement learning, as we will explain in the rest of this paper.\n\n\\subsection{Model-free Reinforcement Learning with Model-based Initialisation}\n\nTop-down approaches in constructing $\\mathcal P_{ij}$ can be highly efficient using no data samples on the states $s_t$, given a specific $\\mathcal A_{ij},g_i,M_{ij}$, but they generally do not yield optimal turn-by-turn maneuvring policies. Model-free RL, on the other hand, can optimise the policies variationally, starting with any specific initial policy using policy gradient based methods. To combine the benefits of model-based approach and model-free RL, we propose to train a model-free RL from either random policy or the effective policy from dispatch algorithm as our initialisation policy. Thus, we can use RL to obtain high performance routing policies either initialised from a random policy without the information on $\\mathcal A_{ij},g_i,M_{ij}$, or from model-based policies constructed from top-down approaches.\n\n\\subsubsection{Model-free Reinforcement Learning}\n\\label{sec: MF RL}\n\nOur model-free RL is initialised with some specific policies, which are then updated by Actor-Critic Method with linear function approximators and on-policy policy gradient \\cite{sutton2018reinforcement}. To improve the performance and sample efficiency, Proximal Policy Optimisation (PPO) \\cite{schulman2017proximal} is used to optimise a ``surrogate'' objective function using stochastic gradient ascent to update policy $\\mathcal P_{ij}$ based on data sampled from $\\mathcal P_{ij}$. An exploration bonus is further introduced to encourage the learning \\textit{agent} to visit novel states even when the environmental extrinsic reward is sparse. \n\n\\textbf{Policy Optimisation:}\nMore concretely, we employ Advantage Actor-Critic (A2C) \\cite{mnih2016asynchronous} combined with PPO as a RL framework which serves as a well-performed baseline for a lot of discrete and continuous benchmarks, for example Atari, 3D humanoid, and MuJoCo Physics engine. In general, advantage actor-critic has two networks named as \\emph{actor} and \\emph{critic} accordingly. The actor is to learn the optimised policy using a policy network, as defined in Eq.(\\ref{eq:policygradient}), and the critic is to evaluate the policy created by the actor and suggest the adjustment using a value network with calculated advantage, as defined in Eq.(\\ref{eq:qfunction}) and (\\ref{eq:advantage}. PPO is to used to reduce the variance of vanilla policy gradient by reducing reinforcement learning to a numerical optimisation problem on the policy.\n\nIn essence, the policy matrix $\\mathcal P_{ij}(\\theta)$ is parameterised by actor parameter vector $\\theta$, where $\\theta$ are trainable weights to maximise future rewards. An ``surrogate'' objective is maximised subject to the constraint that the distribution of the policy before update $\\mathcal P_{ij}(\\theta_{old})$ is close to the distribution of update policy $\\mathcal P_{ij}(\\theta)$. Hence, the objective can be modified to penalise changes to the policy that moves too far from the $\\mathcal P_{ij}(\\theta_{old})$. Formally, the objective to be optimised is shown as:\n\n\\begin{equation}\nL^{\\text{clip}}(\\theta) = \\mathbb{\\hat E}_t[\\min (r_t(\\theta)\\hat A_t, \\text{clip}(r_t(\\theta)), 1-\\epsilon, 1+\\epsilon)\\hat A_t]\n\\label{eq:ppo}    \n\\end{equation}\n\nwhere $\\epsilon$ is a small hyperparameter to be tuned, $r_t(\\theta)$ is the probability ratio $\\frac{\\mathcal P_{ij}(\\theta)(a_t|s_t)}{\\mathcal P_{ij}(\\theta_{\\text{old}})(a_t|s_t)}$. $\\hat A_t$ is the estimated advantage at time $t$ from the critic, which captures how a particular policy is better than the others at a given state. The expectation in equation \\ref{eq:ppo} is the empirical average of samples from the distribution of ${\\mathcal P_{ij}(\\theta_{\\text{old}})(a_t|s_t)}$, using importance sampling to estimate expectation of $\\mathcal P_{ij}(\\theta)(a_t|s_t)$. \n\nThe processing of policy optimisation is shown in Fig. \\ref{fig:ppo}. The PPO algorithm in A2C style is explained in detail in \\cite{schulman2017proximal}. \n\n\\begin{figure}[tbh]\n\\centering\n\\includegraphics[width=9cm]{diagram\/ppo.png}\n\\caption{An illustration of proximal policy optimiser}\n\\label{fig:ppo}\n\\end{figure}\n\n\\textbf{Combining Intrinsic and Extrinsic Reward:}\nFor this problem of taxi routing, the extrinsic reward is defined by bestowing a unit reward for every time step when a commuter is in taxi, for each taxi operating in the road network, characterising the optimisation objective defined in Sec.\\ref{sec:problem}. There is no punishment for vacant taxis cruising. The extrinsic reward at each time $t$ is formally defined as cumulative future reward from time $t$ to finite horizon $H$, shown in the following Eq.(\\ref{eq:e_reward}). \n\n\n\\begin{align}\n   & R_e(s_t, a_t)  = \\sum_t^{H}\\sum_{v \\in V}(o_v) \\\\\n    & o_v  = \\begin{cases}\n        1, \\text{when vehicle $v$ is occupied}\\\\\n        0, \\text{when vehicle $v$ is vacant}\\\\\n    \\end{cases}\n\\label{eq:e_reward} \n\\end{align}\n\n\n\nWell-performed policy in RL maintain a right balance of maximising extrinsic rewards (exploitation) and trying out new states (exploration), especially when the extrinsic rewards are sparse and the target task is temporally extended. Without proper exploration method, the proposed RL algorithm may converge to sub-optimal solutions without attempting enough novel actions for potentially higher expected rewards. Though finding optimal exploration strategies under large, infinite MDPs (like this problem) is theoretically intractable, many exploration strategies are proposed to practically solve this problem, including optimistic exploration \\cite{burda2018exploration, fu2017ex2, tang2017exploration}, Thompson sampling style algorithms \\cite{osband2016deep}, information gain style algorithms \\cite{houthooft2016vime}, etc. We adopt a random network distillation (RND) approach \\cite{burda2018exploration} to construct our intrinsic rewards, which is a heuristic estimation of state counts via prediction errors as exploration bonus. Generally, the intuition behind this approach is to encourage the agent to visit novel states by telling if the current state is novel or not. The RND approach takes inspirations of count-based optimistic exploration and information gain approach. This approach can also handle large input states where most states are visited at most once, as compared to the other approaches.\n\nSpecifically, the RND includes two networks: a target network and a predictor network. The target network takes input state to an embedding $f^*(s, a)=f_\\psi(s,a)$, where $\\psi$ is a random parameter vector. The predictor network is trained by gradient descent to minimise the Mean Squared Error (MSE) between output prediction $\\hat f_\\phi (s,a)$ and the embedding $f^\\psi(s,a)$. The prediction error $R_i (s,a) = ||\\hat f_\\phi (s,a) - f_\\psi (s,a)||^2$ is set as the intrinsic reward (exploration bonus). \n\nThe final reward function is a linear combination of the extrinsic reward and intrinsic reward. The reward function proposed does not tell the vehicles anything about where the commuters are and how to reach the potential commuters. Further normalisation has been done for the intrinsic reward to divide it by a running estimate of standard deviation, so as to keep the intrinsic reward within a reasonable scale when combined with the extrinsic reward. The mathematical representation of the combined reward is in Eq.(\\ref{eq:reward}), where $R_e$ and $R_i$ are extrinsic reward and intrinsic reward respectively, $n_s$ represents the number of total observed states and the corresponding number of actions, $\\sigma$ is the normalisation parameter, and $\nR$ is the total reward.\n\n\\begin{align}\n    & \\sigma  = \\sqrt{\\frac{n_s\\sum_{j=0}^{n_s} R_{i}(s_j, a_j)^2 - (\\sum_{j=0}^{n_s} R_{i}(s_j, a_j))^2}{n_s(n_s-1)}} \\\\\n    & R(s_t, a_t)  = R_e(s_t, a_t) + \\frac{R_i(s_t, a_t)}{\\sigma}\n    \\label{eq:reward}\n\\end{align}\n\n\\subsubsection{Initialising Model-free Reinforcement Learning}\nIn practice, the model-based routing policies leverage the full knowledge of the environment and the MDP, but the best-performing dispatch algorithm (non-committed normalised dispatch algorithm as shown in Sec.\\ref{sec:mb result}) still in most cases lags behind the pure model-free learners especially when the road network size is small and\/or the normalised entropy of given demand pattern is also relatively small. The reasons may lie in the mismatch between the optimal state-action distribution and the dispatch algorithm guided state-action distribution as a result of open-loop control algorithm. \n\nTo combine the benefits of model-free learners and model-based routing policies, we first use imitation learning to train a model-free RL agent to imitate the model-based routing policy as an initialisation, and then further train the model-free learner to gather on-policy data in order to minimise the mismatch between the gathered data state-action distribution and the learned state-action distribution. The model-based routing policy $\\mathcal{P}_{ij}$ generated from Sec.\\ref{sec: MB routing} is first collected to train a initial policy of our model-free learner described in Sec.\\ref{sec: MF RL}. The policy network parameterised by $\\theta$ is trained to match the trained policy $\\hat{\\mathcal P}_{ij}\\left(\\theta \\right)$ to the target effective policy $\\mathcal{P}_{ij}$. Let $n$ be the number of nodes in road network $G$, and let $J$ represent the number of direct connections from $i$ where $i \\neq j$ and $\\mathcal A_{ij} \\neq 0$, the objective is to minimise cross entropy loss as Eq.(\\ref{eq:ce}) between the model-based target policy and trained policy, using the Adam optimiser.\n\n\\begin{equation}\n    \\operatornamewithlimits{argmin}_\\theta \\left( -\\sum_{i=0}^{n} \\sum_{j=0}^{J}\\mathcal{P}_{ij} \\log{\\hat{\\mathcal P}_{ij}\\left(\\theta \\right)}\\right)\n    \\label{eq:ce}\n\\end{equation}\n\nThe trained policy is used as initial policy of our model-free learner. After initialisation, on-policy data is collected at each optimisation iteration to update learned policy $\\mathcal{P}_{ij} \\left(\\theta\\right)$ parameterised by $\\theta$ using model-free RL proposed. The process is carried out until the optimisation objective based on average hiring time of taxis for each second meets expectation.\n\n\\subsection{Agent-based Simulation}\n\\label{sec:simulation}\n\nWe have designed and implemented a large scale microscopic agent-based simulation platform \\cite{yang2020phase} for evaluation of the performance of $\\mathcal P_{ij}$, using different $\\mathcal A_{ij},g_i,M_{ij}$ as inputs. A collection of instances containing state information on the positions and actions of vacant taxis, together with relevant extrinsic rewards needed for reinforcement learning is generated from the simulation after running the simulation for finite time horizon $H$ (measured in time steps). While different physical time unit can be used to correspond to each time step, in this study, each time step is taken to be one second for generality. The platform simulates the dynamics of two types of agents: the commuters and the taxis, and produces outputs characterising the performance of stochastic policy $\\mathcal P_{ij}$ from different perspectives using various performance metrics. We focus on average extrinsic reward and average waiting time of commuters in this paper, further explained in Sec.\\ref{sec:experiment}.\n\nThe behaviours of commuters are determined by probability distributions $g_i$ and $M_{ij}$, and the initial positions of all taxis are assigned to random nodes at the beginning of each epoch of the simulation. Vacant taxis will only pick up a commuter when the location of a vacant taxi and the origin of a waiting commuter coincides. The movements of the occupied taxis are computed efficiently by shortest path algorithm using our customised Contraction Hierarchy algorithm \\cite{yang2020phase}, while the movements of the vacant taxis will follow the stochastic routing policy $\\mathcal P_{ij}$. Both real-time booking and road-side hailing with three potential status for the taxis: vacant, booked, and occupied, can be implemented. In this paper, we focus on street-hailing instead of real-time booking so taxis will not take bookings. But in scenarios where some of the taxis take bookings, the methods proposed in this paper can still be applied to route those vacant taxis available for both booking and street-hailing by potential commuters. \n\n\n\n\n\n\\subsection{The Markov Decision Process}\n\n\nThe simulation platform can be used to evaluate policies $\\mathcal P_{ij}$ from top-down construction using model-based methods, or from bottom-up learning using reinforcement learning. An illustration of our proposed framework for optimising taxi routing policies is shown in Figure \\ref{fig:framework}.\n\n\\begin{figure}[tbh]\n\\centering\n\\includegraphics[width=9cm]{diagram\/schematics.pdf}\n\\caption{An illustration of framework for optimising vacant taxi routing using model-based and model-free approaches.}\n\\label{fig:framework}\n\\end{figure}\n\nThe sequential decision makings of the taxis in above framework can be modelled as a Multi-agent Markov Decision Process (MMDP) involving a collective of taxis as agents, each with their own set of actions, operating in a common environment with the same shared goal (utility function) to achieve a fully cooperative system \\cite{boutilier1996planning}. \nThe challenge of a MMDP modelling of this problem (formulated in Sec.\\ref{sec:problem}) lies in the set of actions to be represented and the policies to be optimised, as each vacant taxi will perform their own policies and choose actions at every location in the road network accordingly. While all agents are improving their own policies concurrently, the environment faced by each agent may become non-stationary.\nAnd the joint action space increases exponentially with the number of agents \\cite{zhang2019multi}, which may cause scalability issues in our problem when the number of taxis in systems is large.\n\nTo deal with the above challenges and to work for realistic taxi system with a large taxi fleet, we apply a \"mean-field\" approximation and model the given problem as a single-agent Markov Decision Process (MDP), assuming that each taxi will follow the same stochastic policy to be learned.\nInstead of each taxi as an agent, a learner to optimise routing policies $\\mathcal P_{ij}$ is formulated as the \\textit{agent} and the simulation that the learner interacts with is the \\textit{environment}. The resulting stochastic routing policies $\\mathcal P_{ij}$ is used to determine the routing decisions for each vacant taxi in system.\n\nMathematically, the modelled finite MDP is specified by $(\\mathcal S, \\mathcal A, P, r, \\gamma, \\rho)$, where: $\\mathcal S$ is a state space of the environment, $\\mathcal A$ is an action space, $P$ is a transition model $P(s^\\prime|s, a)$ as the probability of transition into state $s^\\prime$ from $s$ by taking action $a$, $r$ is a reward function where $r(s, a)$ is the immediate reward in taking action $a$ in state $s$, $\\gamma$ is a discount factor $\\gamma \\in [0,1]$, $\\rho$ is a starting state distribution over $S$. In this problem, the state $\\mathcal S$ is given as the state of all taxis in system. The action $\\mathcal A$ are decisions made for vehicles at each node on which adjacent node to move to. As a consequence of the actions, the \\textit{agent} receives reward $r=r(s_t, a_t), r \\in \\mathbb{R}$ characterising the optimisation objective defined in Sec.\\ref{sec:problem}, and transit to the next state $s^\\prime$ according to transition model $P$. The construction of the reward function is further explained in Sec.\\ref{sec: MF RL}.\n\nLet $\\pi: \\mathcal S \\rightarrow \\Delta (\\mathcal A)$ be a stochastic policy specifies the decision-making strategy in choosing the probability of actions for each taxi in system based on current state, where $\\Delta (\\mathcal A)$ is the probability simplex over action space $\\mathcal A$, and $a_t \\sim \\pi(\\cdot|s_t)$. $\\pi$ is represented as the policy matrix $\\mathcal P_{ij}$ in this problem. By following $\\pi$, a distribution of a state-action sequence $\\tau = (s_0, a_0, \\cdots , a_{H-1}, s_{H})$ is denoted as Eq.(\\ref{eq: sadistribution}), starting from state $s_0$ drawing from starting state distribution $\\rho$, and for all subsequent $i, i\\in[0, H)$ where $a_i \\sim \\pi(\\cdot|s_i)$ and $s_{i+1} \\sim P(\\cdot|s_i, a_i)$.\n\n\\begin{equation}\n    D^{\\pi}(\\tau) \\coloneqq \\prod_{i=0}^{H-1} \\pi(s_i, a_i) P(s_i, a_i, s_{i+1})\n    \\label{eq: sadistribution}\n\\end{equation}\n\nGiven a discount factor $0\\leqslant \\gamma \\leqslant 1$, the value function for a policy $P_{ij}$ is defined as discounted sum of future rewards from starting state $s$ over a finite horizon $H$, i.e.\n\n\\begin{equation}\n    V^{\\pi} (s) \\coloneqq \\mathbb{E} \\left[\\sum_{t=0}^{H-1} \\gamma^{t-1} r(s_t, a_t)|\\pi, s_0=s \\right]\n\\end{equation}\n\nThe expected value of initial state is further defined as Eq.(\\ref{eq:valuefirststate}).\n\n\\begin{equation}\n    V^{\\pi}(\\rho) \\coloneqq \\mathbb{E}_{s_0\\sim \\rho}[V^{\\pi}(s_0)]\n    \\label{eq:valuefirststate}\n\\end{equation}\n\nFor a given policy $\\pi$, the state-action value function (Q-value function) $Q^\\pi$ and the advantage function $A^\\pi$ are defined as Eq.(\\ref{eq:qfunction}) and (\\ref{eq:advantage}).\n\n\\begin{equation}\n    Q^{\\pi}(s,a) \\coloneqq \\mathbb{E}\\left[\\sum_{t=0}^{H-1} \\gamma^{t-1} r(s_t, a_t)|\\pi, s_0=s, a_0=a \\right]\n    \\label{eq:qfunction}\n\\end{equation}\n\n\\begin{equation}\n    A^{\\pi}(s,a) \\coloneqq Q^{\\pi}(s,a) - V^{\\pi}(s)\n    \\label{eq:advantage}\n\\end{equation}\n\nThe objective of the learning agent is to use ascent methods for the optimisation problem given as Eq.(\\ref{eq:rlobjective}), which is to maximise expected value from the initial state over all policies.\n\n\\begin{equation}\n    \\max_{\\theta \\in \\Theta} V^{\\pi_\\theta}(\\rho)\n    \\label{eq:rlobjective}\n\\end{equation}\n\nwhere $\\pi$ in this study is parameterised by linear function approximator as\n\n\\begin{equation}\n    \\pi_\\theta(a|s) = \\theta_{s,a}\n\\end{equation}\n\nwhere $\\theta$ is a row stochastic matrix, $\\theta_{s,a} \\geqslant 0$ and $\\sum_{a\\in \\mathcal A} \\theta_{s,a} = 1$ for $s\\in \\mathcal S$ and $a \\in \\mathcal A$.\n\nIt has been proved in \\cite{agarwal2019optimality} that $V^{\\pi_\\theta}(s)$ is non-concave in $\\theta$ for the above parameterisation, so the standard solvers for convex optimisation problems are not applicable for this target problem. Throughout this paper, we will use RL approaches to optimise the routing policies based on policy gradient theorem \\cite{sutton2000policy} as follows.\n\nLet $R(\\tau)$ be the total reward of state-action sequence $\\tau$ on following $\\pi$ starting from $s_0$, where in this problem, the reward $R(\\tau)$ is given by\n\n\\begin{equation}\n    R(\\tau) \\coloneqq R(s_0, a_0)\n\\end{equation}\n\nwhere $R(s_0, a_0)$ is defined by Eq. (\\ref{eq:reward}) in Sec.\\ref{sec: MF RL}.\n\nThen,\n\n\\begin{equation}\n    \\nabla_\\theta V^{\\pi_\\theta}(\\rho) = \\mathbb{E}_\\tau \\left[ r(\\tau)\\nabla_\\theta \\log(D^{\\pi_{\\theta}}(\\tau))|\\rho\\right]\n    = \\mathbb{E}_\\tau \\left[ r(\\tau) \\sum_{t=0}^{H-1} \\nabla_\\theta \\log(\\pi_{\\theta}(s_t, a_t))|\\rho\\right]\n    \\label{eq:policygradient}\n\\end{equation}\n\n\nThough the problem of this paper is formally framed as a MDP, the RL method proposed in this paper is based on Monte-Carlo method with undiscounted setting where $\\gamma \\rightarrow 1$, which does not require the system to be Markovian. The details of the model-free reinforcement learning will be further explained in Sec.\\ref{sec:methodology}.\n\n\\section{CONCLUSIONS AND OUTLOOKS}\n\n\n\n\n\nThe methodologies presented in this paper could have wider applications beyond taxi systems, for a wide range of empirical and synthetic systems. For future work, we will work on incorporating time-dependent traffic conditions and demand patterns for real-world impact. We will also extend the agent-based simulation and the proposed model-based and model-free approaches to cases with ride-sharing, for the optimal routing and route-matching between shared commuters. Our ultimate goal is to optimise routing of demand-responsive transit system in a hybrid form consisting of real-time booking, street-hailing, as well as ride-sharing. \n\n\n\\section{Statement of Contribution\/Potential Impact}\n\nIn this paper, we propose methodologies for optimising vacant vehicle routing in a decentralised way to meet stochastic spatio-temporal demand patterns. The research is motivated by the grand challenge of urban mobility to meet increasing demands for highly efficient yet high-quality transport services with minimal costs. The difficulty of the problem stems from the stochastic spatio-temporal distribution of the commuter demand, on a large road network with complex structures.\nWe design a set of methodologies for intelligent routing of taxi fleet with reinforcement learning primed with a top-down dispatch model, evaluated on both synthetic and empirical data for commuter demand patterns and road networks. The proposed methodologies can potentially benefit everyone in tackling the urban mobility challenges: shorter waiting time for commuters, higher profits for service providers, easy-to-implement and versatile solutions for government agencies, and less emissions to the environment. The methodologies presented in this paper are in time for real-world applications under current technology and in near future when wireless communication and autonomous vehicles enable new ways of taxi fleet operation. They could also have wider applications beyond taxi systems, for on-demand transportation and logistics services such as dynamic reallocation of shared resources and last-mile delivery.\n\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\\input{Introduction}\n\n\\section{Related Works}\n\\label{sec:related works}\n\\input{Review}\n\n\\section{Problem Formulation}\n\\label{sec:problem}\n\\input{Problem}\n\n\\section{System Overview}\n\\label{sec:system}\n\\input{System}\n\n\\section{Methodology}\n\\label{sec:methodology}\n\\input{Methodology}\n\n\\section{Experiments}\n\\label{sec:experiment}\n\\input{Experiment}\n\n\n\\section{Results}\n\\label{sec:results}\n\\input{Results}\n\n\n\\section{Summary and Outlook}\n\\label{sec:discussion}\n\\input{Discussion}\n\n\\section*{Acknowledgements}\nThis research is supported by the National Research Foundation, Singapore, and the Land Transport Authority under its Urban Mobility Grand Challenge Programme (Award No. UMGC-L005) and the National Research Foundation, Singapore under the NRF fellowship award (NRF-NRFF12-2020-005).\nAny opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not reflect the views of National Research Foundation, Singapore and the Land Transport Authority.\n\n\n\n\n\\bibliographystyle{abbrv}\n\n\\subsection{Model-based Routing Policy from Dispatch Algorithm}\n\\label{sec:mb result}\n\nMultiple dispatch algorithms have been proposed and compared based on 1) whether the taxis are committed to dispatch to the target node to serve commuters (do not serve other commuters during their travels) and 2) whether normalisation factor $\\lambda_i$ in Equation \\ref{dm1} has been used. Experiments were conducted to evaluate the performance of the proposed dispatch algorithms on artificial network of $100$ nodes (experiment setting S1). The performance of four dispatch algorithms: committed dispatch, committed normalised dispatch, non-committed dispatch, and non-committed normalised dispatch in terms of average waiting time of commuters are shown in Figure \\ref{fig:dispatch}. From the comparison, non-committed normalised dispatch algorithm outperforms the other algorithms and is selected to be used to derive effective policy $\\mathcal{P}_{ij}$ to initialise model-free RL.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=9cm]{diagram\/fig1.pdf}\n\\caption{Performance comparison of dispatch algorithms on artificial lattice network (experiment setting S1)}\n\\label{fig:dispatch}\n\\end{figure}\n\n\\subsection{Design Decisions for Model-free RL}\n\\label{sec:mf result}\n\nThe design decisions for our proposed model-free RL were evaluated by comparing the final reward and average waiting time of commuters from learned policies by vanilla policy gradient, mode-free RL without exploration bonus, and proposed model-free RL algorithm (Figure \\ref{fig:comp 100}). All the other design decisions including epochs, learning rate, iterations, clipping range, simulation time, and number of instances collected have been fine-tuned to the best design choices for each method. The results show that our designed model-free algorithm combining intrinsic and extrinsic rewards outperform vanilla policy gradient and the design without exploration bonus, especially when the average waiting time of commuters is affected sub-linearly with increased number of taxis (discussed in detail later). \n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=12.5cm]{diagram\/intra_compare.pdf} \n\\caption{Evaluation of design decisions for our model-free RL on lattice network of $100$ nodes (experiment setting S1). The number of epochs is $1000$. The final reward and average waiting time presented are average results over the last $10$ epochs. (a) final reward (average hiring time of taxis for each second), (b) final average waiting time of commuters.}\n\\label{fig:comp 100}\n\\end{figure}\n\n\n\\subsection{Evaluation on Artificial Road Network}\n\\label{sec:mfmb result}\n\nThe final performances of our proposed three approaches --- model-based dispatch algorithm, model-free RL and model-free RL with model-based initialisation are evaluated on the same benchmark tasks specified in Sec.\\ref{sec:experiment}. In particular, five different policies are compared, including random policy, learned policy from model-free RL, dispatch policy from model-based approach, effective policy for initialisation, and final learned policy from model-free RL with model-based initialisation. The comparison of final results on the artificial road networks (lattice networks of $100$ nodes and $1089$ nodes) with varied number of taxis from $0$ to $200$ is first presented in Figure \\ref{fig:rl_lattice}. The results show that the final performance (after training $1000$ epochs) in terms of reward and average waiting time of commuters between learned policies from model-free RL, and model-free RL with initialisation, are similar. Both are better than policy derived from non-committed normalised dispatch algorithm, effective policy and random policy. This is especially true when the average waiting time of commuters are relatively long when demand exceeds supply. \n\n\\begin{figure}[bh!]\n\\centering\n\\includegraphics[width=12.5cm]{diagram\/lattice_compare.pdf} \n\\caption{Comparison of performance on artificial road network among random policy, learned policy from model-free RL (MF-RL), dispatch algorithm from model-based approach (MB Approach), effective policy as model-based initialisation (MB Init), and learned policy from model-free RL with model-based initialisation (MF-RL with MB Init). The number of epochs is $1000$. The final reward and average waiting time presented are average results over the last $10$ epochs. (a) and (b) are final reward and average waiting time of commuters experimented on artificial lattice of $100$ nodes (experiment setting S1), (c) and (d) are corresponding results on artificial lattice of $1089$ nodes (experiment setting S2).}\n\\label{fig:rl_lattice}\n\\end{figure}\n\nRegarding the efficiency of learning, both the model-free RL and the hybrid approach can learn a fairly good policy with better performance than random policy and effective policy from model-based approach very quickly (within $200$ epochs), as shown in Figure \\ref{fig:efficiency_lattice}. Even with similar converged performance at the end as in Figure \\ref{fig:rl_lattice}, the hybrid approach with model-based initialisation can be much more sample-efficient in learning compared with randomised initialisation. A reasonable reward can be achieved by significantly less training epochs (less than $100$ epochs of training). Note that each epoch entails a simulation run of $50,000$ seconds (simulation time) with maximum $512,000$ data samples collected, detailed in Appendix. \n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=12.5cm]{diagram\/Efficiency_Evaluation_of_Lattice_Network.pdf} \n\\caption{Efficiency evaluation between proposed model-free RL and hybrid approach on artificial road network. (a) and (b) show reward experimented on lattice network of $100$ nodes using $20$ taxis and $40$ taxis respectively, (c) and (d) are corresponding results on lattice network of $1089$ nodes.}\n\\label{fig:efficiency_lattice}\n\\end{figure}\n\nWe have previously identified a transition between the oversaturated phase when demand exceeds supply, and the undersaturated phase when supply exceeds demand, when the structure of road network is given, as discussed in our previous work on phase transition in taxi dynamics \\cite{yang2020phase}. The phase transition is also shown in Figure \\ref{fig:rl_lattice} that in the oversaturated phase the average waiting time of commuters is affected exponentially with increased number of taxis, while in the undersaturated phase it is affected sub-linearly. In practice, we are more interested in optimising vacant taxi routing in the oversaturated phase to meet the service requirements of commuters efficiently. When the random policy for vacant taxis can already achieve low average waiting time in the undersaturated phase, there is no much room left for optimisation and the redundancy of taxis on roads creates more problems than they save. Specifically for these benchmark tasks, we are mainly evaluating the performance of our proposed algorithm when the number of taxi is less than $50$ for lattice network of $100$ nodes (experiment setting S1) and less than $150$ for lattice network of $1089$ nodes (experiment setting S2).\n\n\\subsection{Evaluation on Singapore Road Network}\n\nThe performance of our proposed approaches is also evaluated on real community-based Singapore road network as described in Sec.\\ref{sec:experiment}. We compare the results using both randomised demands and empirical demands with taxi numbers ranging from $30$ to $200$, as shown in Figure \\ref{fig:rl_real}. Results demonstrate a consistently better performance of proposed hybrid approach, in compare with all the other approaches on a medium-sized real-world road network of $2591$ nodes. Figure \\ref{fig:rl_real} also shows that when the normalised entropy of the empirical demands is larger as compared with random demands (the empirical demands of these areas are more uniformly distributed spatially with less ``hotspots''), the number of taxis needed to operate on the road network can be significantly less, but still with higher average hiring time of taxi drivers (reward) and lower average waiting time of commuters when using efficient routing policies proposed in this work.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=12.5cm]{diagram\/realnetwork_compare.pdf} \n\\caption{Comparison of performance on community-based Singapore road network among random policy, learned policy from model-free RL (MF-RL), dispatch algorithm from model-based approach (MB Approach), effective policy as model-based initialisation (MB Init) for model-free RL, and learned policy from model-free RL with model-based initialisation (MF-RL with MB Init). The number of epochs is $3000$. The final reward and average waiting time presented are average results over the last $10$ epochs. (a) and (b) are final reward and average waiting time of commuters experimented on randomised demands (experiment setting S3), (c) and (d) are corresponding results on empirical demands (experiment setting S4).}\n\\label{fig:rl_real}\n\\end{figure}\n\nBy comparing the efficiency of proposed hybrid approach and pure model-free approach on Singapore road network (shown in Figure \\ref{fig:efficiency_real}), the hybrid approach can achieve much higher final performance and uses significantly less data samples and training epochs, especially when using empirical demand data (for which the normalised entropy is larger compared to the randomised demands). \n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=12.5cm]{diagram\/Efficiency_Evaluation_of_Real_Network.pdf} \n\\caption{Efficiency evaluation between proposed model-free RL and hybrid approach on community-based Singapore road network. (a) and (b) show reward experimented on same network with random demands using $100$ taxis and $130$ taxis respectively, (c) and (d) are corresponding results with empirical demands.}\n\\label{fig:efficiency_real}\n\\end{figure}\n\n\n\n\n\\subsection{Constraints and Objective Function}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Model-based Routing Policy}\n\\label{sec: MB routing}\n\n\n\n\nFor this target problem, instead of learning a model of the dynamic function of environment (like in most model-based RL), we can construct routing policy from transportation theory using the MDP. In order to find the optimal $\\mathcal P_{ij}$ which allows us to route a vacant taxi at any location in a given road network, the intuition behind our proposed model-based dispatch algorithm is to direct the vacant taxis to a set of ``hotspots\" in the road network, which are identified either statically or dynamically. These hotspots are locations where the probability of finding a waiting commuter is high, so that vacant taxis are dispatched to these hotspots, which can be formally described by a dispatch matrix $\\mathcal D_{ij}$: given a vacant taxi at node $i$, $\\mathcal D_{ij}$ gives the probability of the taxi being dispatched to node $j$ (which potentially can be quite far away). The dispatch probability is also inversely proportional to the distance between $i$ and $j$. There will also be some routing choice from node $i$ to node $j$ (the simplest being the shortest path between the two nodes). Once $\\mathcal D_{ij}$ and the routing choice are specified, the turn-by-turn routing of $\\mathcal P_{ij}$ as model-based routing policy can also be uniquely determined via microscopic simulations and sampling.\nThroughout this paper we assume shortest path routing between two nodes for $\\mathcal D_{ij}$, as the specific choice of routing algorithm is not essential in this particular context. \n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{Dispatch Algorithm for Hotspots}\nOne of the common approaches for vacant taxi manoeuvring is a good algorithm for dispatching them to certain locations on the map where higher probabilities of commuter generation are expected (the hotspots). This is also implicitly the strategy of experienced taxi drivers. Following the problem statements and definitions in Sec.~\\ref{sec:problem}, we first propose models of efficient dispatch algorithm, using $\\mathcal A_{ij},g_i,M_{ij}$ as inputs, to obtain an efficient dispatch matrix $\\mathcal D_{ij}$. Physically, for a vacant taxi appearing at node $i$, $\\mathcal D_{ij}$ gives the probability for this vacant taxi to be dispatched to a (not necessarily adjacent) node $j$, with the constraint $\\sum_j\\mathcal D_{ij}=1$. We also allow $\\mathcal D_{ii}\\neq 0$, implying the vacant taxi stay put at its location waiting for commuters to appear. If node $j$ is far away from node $i$, the vacant taxi will be dispatched to node $j$ via the shortest path obtained from the network adjacency matrix $\\mathcal A_{ij}^{-1}$. Without loss of generality, here we only make $\\mathcal D_{ij}$ dependent on $g_i$ and $\\mathcal A_{ij}$, and not explicitly dependent on $M_{ij}$.\n\nLet $d_{ij}$ be the shortest distance in terms of travel time between node $i$ and $j$, as computed from the adjacency matrix $\\mathcal A_{ij}^{-1}$. For $i=j$ we set $d_{ii}=1$. A rather general dispatch model is given as follows:\n\\begin{eqnarray}\\label{dispatchmodel}\n\\mathcal D_{ij}=f\\left(d_{ij}^{-1},g_j\\right)\n\\end{eqnarray}\nIntuitively, $\\mathcal D_{ij}$ should increase monotonically with $g_j$, while decreasing monotonically with $d_{ij}$. This is because if all other conditions are identical, we should have a higher probability of dispatching vacant taxis to nodes with higher rate of commuter generation, the so-called hotspots. On the other hand, for two nodes with the same rate of commuter generation, there should be a reduced probability to dispatch the vacant taxi to a node that is further away. Optimisation of Eq.(\\ref{dispatchmodel}) can be done by Taylor expanding $f$ with respect to $d_{ij}^{-1}$ and $g_j$, and tuning the expansion coefficients at different orders to maximise the utility functions computed from agent based simulations. Here we use a rather simple model with the lowest order expansion as follows:\n\\begin{eqnarray}\\label{dm1}\n\\mathcal D_{ij}=\\lambda_i \\frac{g_j}{d_{ij}}\n\\end{eqnarray}\nwhere $\\lambda_i=\\left(\\sum_j\\mathcal D_{ij}\\right)^{-1}$ is the normalisation factor. The performance of this model has been satisfactory based on extensive agent-based numerical simulations shown in later section Sec.\\ref{sec:mb result}.\n\n\\subsubsection{Effective Routing Policy from Dispatch Model}\n\nOne major difference between $\\mathcal D_{ij}$ and $\\mathcal P_{ij}$ for vacant taxi manoeuvring is that for the former, nodes $i$ and $j$ can be quite far away. On the other hand, $\\mathcal P_{ij}$ is only non-zero when nodes $i$ and $j$ are neighbours. The degrees of freedom for $\\mathcal D_{ij}$ is thus much larger and scale as the square of the network size. This makes optimising $\\mathcal D_{ij}$ a much more difficult problem, as compared to optimising $\\mathcal P_{ij}$ (whose degrees of freedom scale linearly with network size). On the other hand, when the vacant taxi is dispatched from node $i$ to $j$, the path between nodes $i$ and $j$ is deterministic (and we choose the shortest path in this work). Effectively, given a specific $\\mathcal A_{ij},g_i,M_{ij},\\mathcal D_{ij}$ there is still a statistical distribution for the choice of neighbouring nodes from any node $i$, if we look at a large number of vacant taxis arriving at node i (either as the destination of the dispatch, or as an intermediate node enroute to the dispatch destination), and how these vacant taxis choose the next node. We can thus generate an effective $\\mathcal P_{ij}$ via agent based simulations, and this particular $\\mathcal P_{ij}$ is unique and well-defined in the limit of long time simulations.\n\nFormally, for a simulation time with a total number of discrete time-steps $H$, we denote $n_i^H$ as the number of times when an vacant taxi lands on node $i$, and we do not distinguish between different taxis. Let $p_{ij}^H$ be the number of times the vacant taxi at node $i$ moves towards one of the neighbouring node $j$. The effective turn-by-turn routing policy is defined as follows:\n\\begin{eqnarray}\n\\bar{\\mathcal P}_{ij}=\\lim_{H\\rightarrow\\infty}\\frac{1}{n_i^T}p_{ij}^H\n\\end{eqnarray}\nIn this way, the effective policy can be unambiguously defined for almost any manoeuvring strategies. If the routing strategy for vacant taxis is a turn-by-turn policy $\\mathcal P_{ij}$ to start with, then obviously we will have $\\bar{\\mathcal P}_{ij}=\\mathcal P_{ij}$. If the routing strategy is from a dispatch policy $\\mathcal D_{ij}$, there is a unique one-to-one mapping from $\\mathcal D_{ij}$ to $\\bar{\\mathcal P}_{ij}$, given all other conditions are fixed. We can thus evaluate and compare the performances of $\\mathcal D_{ij}$ and $\\bar{\\mathcal P}_{ij}$ in a meaningful manner. More importantly, $\\bar{\\mathcal P}_{ij}$ can be used as an optimised initial state for reinforcement learning, as we will explain in the rest of this paper.\n\n\\subsection{Model-free Reinforcement Learning with Model-based Initialisation}\n\nTop-down approaches in constructing $\\mathcal P_{ij}$ can be highly efficient using no data samples on the states $s_t$, given a specific $\\mathcal A_{ij},g_i,M_{ij}$, but they generally do not yield optimal turn-by-turn maneuvring policies. Model-free RL, on the other hand, can optimise the policies variationally, starting with any specific initial policy using policy gradient based methods. To combine the benefits of model-based approach and model-free RL, we propose to train a model-free RL from either random policy or the effective policy from dispatch algorithm as our initialisation policy. Thus, we can use RL to obtain high performance routing policies either initialised from a random policy without the information on $\\mathcal A_{ij},g_i,M_{ij}$, or from model-based policies constructed from top-down approaches.\n\n\\subsubsection{Model-free Reinforcement Learning}\n\\label{sec: MF RL}\n\nOur model-free RL is initialised with some specific policies, which are then updated by Actor-Critic Method with linear function approximators and on-policy policy gradient \\cite{sutton2018reinforcement}. To improve the performance and sample efficiency, Proximal Policy Optimisation (PPO) \\cite{schulman2017proximal} is used to optimise a ``surrogate'' objective function using stochastic gradient ascent to update policy $\\mathcal P_{ij}$ based on data sampled from $\\mathcal P_{ij}$. An exploration bonus is further introduced to encourage the learning \\textit{agent} to visit novel states even when the environmental extrinsic reward is sparse. \n\n\\textbf{Policy Optimisation:}\nMore concretely, we employ Advantage Actor-Critic (A2C) \\cite{mnih2016asynchronous} combined with PPO as a RL framework which serves as a well-performed baseline for a lot of discrete and continuous benchmarks, for example Atari, 3D humanoid, and MuJoCo Physics engine. In general, advantage actor-critic has two networks named as \\emph{actor} and \\emph{critic} accordingly. The actor is to learn the optimised policy using a policy network, as defined in Eq.(\\ref{eq:policygradient}), and the critic is to evaluate the policy created by the actor and suggest the adjustment using a value network with calculated advantage, as defined in Eq.(\\ref{eq:qfunction}) and (\\ref{eq:advantage}. PPO is to used to reduce the variance of vanilla policy gradient by reducing reinforcement learning to a numerical optimisation problem on the policy.\n\nIn essence, the policy matrix $\\mathcal P_{ij}(\\theta)$ is parameterised by actor parameter vector $\\theta$, where $\\theta$ are trainable weights to maximise future rewards. An ``surrogate'' objective is maximised subject to the constraint that the distribution of the policy before update $\\mathcal P_{ij}(\\theta_{old})$ is close to the distribution of update policy $\\mathcal P_{ij}(\\theta)$. Hence, the objective can be modified to penalise changes to the policy that moves too far from the $\\mathcal P_{ij}(\\theta_{old})$. Formally, the objective to be optimised is shown as:\n\n\\begin{equation}\nL^{\\text{clip}}(\\theta) = \\mathbb{\\hat E}_t[\\min (r_t(\\theta)\\hat A_t, \\text{clip}(r_t(\\theta)), 1-\\epsilon, 1+\\epsilon)\\hat A_t]\n\\label{eq:ppo}    \n\\end{equation}\n\nwhere $\\epsilon$ is a small hyperparameter to be tuned, $r_t(\\theta)$ is the probability ratio $\\frac{\\mathcal P_{ij}(\\theta)(a_t|s_t)}{\\mathcal P_{ij}(\\theta_{\\text{old}})(a_t|s_t)}$. $\\hat A_t$ is the estimated advantage at time $t$ from the critic, which captures how a particular policy is better than the others at a given state. The expectation in equation \\ref{eq:ppo} is the empirical average of samples from the distribution of ${\\mathcal P_{ij}(\\theta_{\\text{old}})(a_t|s_t)}$, using importance sampling to estimate expectation of $\\mathcal P_{ij}(\\theta)(a_t|s_t)$. \n\nThe processing of policy optimisation is shown in Fig. \\ref{fig:ppo}. The PPO algorithm in A2C style is explained in detail in \\cite{schulman2017proximal}. \n\n\\begin{figure}[tbh]\n\\centering\n\\includegraphics[width=9cm]{diagram\/ppo.png}\n\\caption{An illustration of proximal policy optimiser}\n\\label{fig:ppo}\n\\end{figure}\n\n\\textbf{Combining Intrinsic and Extrinsic Reward:}\nFor this problem of taxi routing, the extrinsic reward is defined by bestowing a unit reward for every time step when a commuter is in taxi, for each taxi operating in the road network, characterising the optimisation objective defined in Sec.\\ref{sec:problem}. There is no punishment for vacant taxis cruising. The extrinsic reward at each time $t$ is formally defined as cumulative future reward from time $t$ to finite horizon $H$, shown in the following Eq.(\\ref{eq:e_reward}). \n\n\n\\begin{align}\n   & R_e(s_t, a_t)  = \\sum_t^{H}\\sum_{v \\in V}(o_v) \\\\\n    & o_v  = \\begin{cases}\n        1, \\text{when vehicle $v$ is occupied}\\\\\n        0, \\text{when vehicle $v$ is vacant}\\\\\n    \\end{cases}\n\\label{eq:e_reward} \n\\end{align}\n\n\n\nWell-performed policy in RL maintain a right balance of maximising extrinsic rewards (exploitation) and trying out new states (exploration), especially when the extrinsic rewards are sparse and the target task is temporally extended. Without proper exploration method, the proposed RL algorithm may converge to sub-optimal solutions without attempting enough novel actions for potentially higher expected rewards. Though finding optimal exploration strategies under large, infinite MDPs (like this problem) is theoretically intractable, many exploration strategies are proposed to practically solve this problem, including optimistic exploration \\cite{burda2018exploration, fu2017ex2, tang2017exploration}, Thompson sampling style algorithms \\cite{osband2016deep}, information gain style algorithms \\cite{houthooft2016vime}, etc. We adopt a random network distillation (RND) approach \\cite{burda2018exploration} to construct our intrinsic rewards, which is a heuristic estimation of state counts via prediction errors as exploration bonus. Generally, the intuition behind this approach is to encourage the agent to visit novel states by telling if the current state is novel or not. The RND approach takes inspirations of count-based optimistic exploration and information gain approach. This approach can also handle large input states where most states are visited at most once, as compared to the other approaches.\n\nSpecifically, the RND includes two networks: a target network and a predictor network. The target network takes input state to an embedding $f^*(s, a)=f_\\psi(s,a)$, where $\\psi$ is a random parameter vector. The predictor network is trained by gradient descent to minimise the Mean Squared Error (MSE) between output prediction $\\hat f_\\phi (s,a)$ and the embedding $f^\\psi(s,a)$. The prediction error $R_i (s,a) = ||\\hat f_\\phi (s,a) - f_\\psi (s,a)||^2$ is set as the intrinsic reward (exploration bonus). \n\nThe final reward function is a linear combination of the extrinsic reward and intrinsic reward. The reward function proposed does not tell the vehicles anything about where the commuters are and how to reach the potential commuters. Further normalisation has been done for the intrinsic reward to divide it by a running estimate of standard deviation, so as to keep the intrinsic reward within a reasonable scale when combined with the extrinsic reward. The mathematical representation of the combined reward is in Eq.(\\ref{eq:reward}), where $R_e$ and $R_i$ are extrinsic reward and intrinsic reward respectively, $n_s$ represents the number of total observed states and the corresponding number of actions, $\\sigma$ is the normalisation parameter, and $\nR$ is the total reward.\n\n\\begin{align}\n    & \\sigma  = \\sqrt{\\frac{n_s\\sum_{j=0}^{n_s} R_{i}(s_j, a_j)^2 - (\\sum_{j=0}^{n_s} R_{i}(s_j, a_j))^2}{n_s(n_s-1)}} \\\\\n    & R(s_t, a_t)  = R_e(s_t, a_t) + \\frac{R_i(s_t, a_t)}{\\sigma}\n    \\label{eq:reward}\n\\end{align}\n\n\\subsubsection{Initialising Model-free Reinforcement Learning}\nIn practice, the model-based routing policies leverage the full knowledge of the environment and the MDP, but the best-performing dispatch algorithm (non-committed normalised dispatch algorithm as shown in Sec.\\ref{sec:mb result}) still in most cases lags behind the pure model-free learners especially when the road network size is small and\/or the normalised entropy of given demand pattern is also relatively small. The reasons may lie in the mismatch between the optimal state-action distribution and the dispatch algorithm guided state-action distribution as a result of open-loop control algorithm. \n\nTo combine the benefits of model-free learners and model-based routing policies, we first use imitation learning to train a model-free RL agent to imitate the model-based routing policy as an initialisation, and then further train the model-free learner to gather on-policy data in order to minimise the mismatch between the gathered data state-action distribution and the learned state-action distribution. The model-based routing policy $\\mathcal{P}_{ij}$ generated from Sec.\\ref{sec: MB routing} is first collected to train a initial policy of our model-free learner described in Sec.\\ref{sec: MF RL}. The policy network parameterised by $\\theta$ is trained to match the trained policy $\\hat{\\mathcal P}_{ij}\\left(\\theta \\right)$ to the target effective policy $\\mathcal{P}_{ij}$. Let $n$ be the number of nodes in road network $G$, and let $J$ represent the number of direct connections from $i$ where $i \\neq j$ and $\\mathcal A_{ij} \\neq 0$, the objective is to minimise cross entropy loss as Eq.(\\ref{eq:ce}) between the model-based target policy and trained policy, using the Adam optimiser.\n\n\\begin{equation}\n    \\operatornamewithlimits{argmin}_\\theta \\left( -\\sum_{i=0}^{n} \\sum_{j=0}^{J}\\mathcal{P}_{ij} \\log{\\hat{\\mathcal P}_{ij}\\left(\\theta \\right)}\\right)\n    \\label{eq:ce}\n\\end{equation}\n\nThe trained policy is used as initial policy of our model-free learner. After initialisation, on-policy data is collected at each optimisation iteration to update learned policy $\\mathcal{P}_{ij} \\left(\\theta\\right)$ parameterised by $\\theta$ using model-free RL proposed. The process is carried out until the optimisation objective based on average hiring time of taxis for each second meets expectation.\n\n\\section{Implementation Details}\n\nThe simulation was designed for infinite time horizons and infinite number of runs. For each run, the unit for the length of simulation is defined as ``Second'' for simplicity. The settings that are consistent for both model-based approaches and model-free approaches are listed below.\n\n\\begin{table}[h!]\n\\centering\n \\begin{tabular}{|c | c|} \n \\hline\n Length of each simulation run & $50, 000$ \\\\\n \\hline\n Maximum number of commuters waiting at each node& $1, 000, 001$ \\\\\n \\hline\nMaximum sample size collected for each run & $512, 000$ \\\\\n\\hline\n\n\\end{tabular}\n\\label{tab:sim_parameters} \n\\end{table}\n\nThe imitation learning used to initialise model-free leaner will update the policy network until the cross entropy loss stop decreasing or reach maximum update steps $30, 000$. The hyperparameters used in training the model-free RL and the hybrid approach are the same for all experiment settings and are listed below.\n\n\\begin{table}[h!]\n\\centering\n \\begin{tabular}{|c | c|} \n \\hline\n Learning rate & $0.01$ \\\\\n \\hline\n Clipping parameter epsilon& $0.1$ \\\\\n \\hline\nIterations & $10$ \\\\\n\\hline\nBatch size &  Input data size\\\\\n\\hline\n\n\\end{tabular}\n\\label{tab:imitation_parameters} \n\\end{table}\n\n\\section{Experimental Details}\n\nThe experimental details for both artificial road network and community-based Singapore road network are listed below. All experiments were conducted on a commodity server with two NVIDIA Quadro P4000 GPUs with $8$ GB GDDR5 memory.\n\n\\begin{table}[h!]\n\\centering\n \\begin{tabular}{|c| c| c | c|} \n \\hline\n Experiment & Road Network Type &Edge Travel Time & Epochs \\\\\n \\hline\n S1 & $10 \\times 10$ Lattice & $X\\sim unif\\{1, 10\\}$ & $1000$  \\\\\n \\hline\n S2& $33 \\times 33$ Lattice& $X\\sim unif\\{1, 10\\}$  & $1000$\\\\\n \\hline\nS3 & Road network ($2591$ nodes) & Travel time (s) & $1000$ \\\\\n\\hline\nS4 & Road network ($2591$ nodes) &Travel time (s)&  $3000$\\\\\n\\hline\n\n\\end{tabular}\n\\label{tab:exp_parameters} \n\\end{table}\n\n\\subsection{Constraints and Objective Function}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Model-based Routing Policy from Dispatch Algorithm}\n\\label{sec:mb result}\n\nMultiple dispatch algorithms have been proposed and compared based on 1) whether the taxis are committed to dispatch to the target node to serve commuters (do not serve other commuters during their travels) and 2) whether normalisation factor $\\lambda_i$ in Equation \\ref{dm1} has been used. Experiments were conducted to evaluate the performance of the proposed dispatch algorithms on artificial network of $100$ nodes (experiment setting S1). The performance of four dispatch algorithms: committed dispatch, committed normalised dispatch, non-committed dispatch, and non-committed normalised dispatch in terms of average waiting time of commuters are shown in Figure \\ref{fig:dispatch}. From the comparison, non-committed normalised dispatch algorithm outperforms the other algorithms and is selected to be used to derive effective policy $\\mathcal{P}_{ij}$ to initialise model-free RL.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=9cm]{diagram\/fig1.pdf}\n\\caption{Performance comparison of dispatch algorithms on artificial lattice network (experiment setting S1)}\n\\label{fig:dispatch}\n\\end{figure}\n\n\\subsection{Design Decisions for Model-free RL}\n\\label{sec:mf result}\n\nThe design decisions for our proposed model-free RL were evaluated by comparing the final reward and average waiting time of commuters from learned policies by vanilla policy gradient, mode-free RL without exploration bonus, and proposed model-free RL algorithm (Figure \\ref{fig:comp 100}). All the other design decisions including epochs, learning rate, iterations, clipping range, simulation time, and number of instances collected have been fine-tuned to the best design choices for each method. The results show that our designed model-free algorithm combining intrinsic and extrinsic rewards outperform vanilla policy gradient and the design without exploration bonus, especially when the average waiting time of commuters is affected sub-linearly with increased number of taxis (discussed in detail later). \n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=12.5cm]{diagram\/intra_compare.pdf} \n\\caption{Evaluation of design decisions for our model-free RL on lattice network of $100$ nodes (experiment setting S1). The number of epochs is $1000$. The final reward and average waiting time presented are average results over the last $10$ epochs. (a) final reward (average hiring time of taxis for each second), (b) final average waiting time of commuters.}\n\\label{fig:comp 100}\n\\end{figure}\n\n\n\\subsection{Evaluation on Artificial Road Network}\n\\label{sec:mfmb result}\n\nThe final performances of our proposed three approaches --- model-based dispatch algorithm, model-free RL and model-free RL with model-based initialisation are evaluated on the same benchmark tasks specified in Sec.\\ref{sec:experiment}. In particular, five different policies are compared, including random policy, learned policy from model-free RL, dispatch policy from model-based approach, effective policy for initialisation, and final learned policy from model-free RL with model-based initialisation. The comparison of final results on the artificial road networks (lattice networks of $100$ nodes and $1089$ nodes) with varied number of taxis from $0$ to $200$ is first presented in Figure \\ref{fig:rl_lattice}. The results show that the final performance (after training $1000$ epochs) in terms of reward and average waiting time of commuters between learned policies from model-free RL, and model-free RL with initialisation, are similar. Both are better than policy derived from non-committed normalised dispatch algorithm, effective policy and random policy. This is especially true when the average waiting time of commuters are relatively long when demand exceeds supply. \n\n\\begin{figure}[bh!]\n\\centering\n\\includegraphics[width=12.5cm]{diagram\/lattice_compare.pdf} \n\\caption{Comparison of performance on artificial road network among random policy, learned policy from model-free RL (MF-RL), dispatch algorithm from model-based approach (MB Approach), effective policy as model-based initialisation (MB Init), and learned policy from model-free RL with model-based initialisation (MF-RL with MB Init). The number of epochs is $1000$. The final reward and average waiting time presented are average results over the last $10$ epochs. (a) and (b) are final reward and average waiting time of commuters experimented on artificial lattice of $100$ nodes (experiment setting S1), (c) and (d) are corresponding results on artificial lattice of $1089$ nodes (experiment setting S2).}\n\\label{fig:rl_lattice}\n\\end{figure}\n\nRegarding the efficiency of learning, both the model-free RL and the hybrid approach can learn a fairly good policy with better performance than random policy and effective policy from model-based approach very quickly (within $200$ epochs), as shown in Figure \\ref{fig:efficiency_lattice}. Even with similar converged performance at the end as in Figure \\ref{fig:rl_lattice}, the hybrid approach with model-based initialisation can be much more sample-efficient in learning compared with randomised initialisation. A reasonable reward can be achieved by significantly less training epochs (less than $100$ epochs of training). Note that each epoch entails a simulation run of $50,000$ seconds (simulation time) with maximum $512,000$ data samples collected, detailed in Appendix. \n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=12.5cm]{diagram\/Efficiency_Evaluation_of_Lattice_Network.pdf} \n\\caption{Efficiency evaluation between proposed model-free RL and hybrid approach on artificial road network. (a) and (b) show reward experimented on lattice network of $100$ nodes using $20$ taxis and $40$ taxis respectively, (c) and (d) are corresponding results on lattice network of $1089$ nodes.}\n\\label{fig:efficiency_lattice}\n\\end{figure}\n\nWe have previously identified a transition between the oversaturated phase when demand exceeds supply, and the undersaturated phase when supply exceeds demand, when the structure of road network is given, as discussed in our previous work on phase transition in taxi dynamics \\cite{yang2020phase}. The phase transition is also shown in Figure \\ref{fig:rl_lattice} that in the oversaturated phase the average waiting time of commuters is affected exponentially with increased number of taxis, while in the undersaturated phase it is affected sub-linearly. In practice, we are more interested in optimising vacant taxi routing in the oversaturated phase to meet the service requirements of commuters efficiently. When the random policy for vacant taxis can already achieve low average waiting time in the undersaturated phase, there is no much room left for optimisation and the redundancy of taxis on roads creates more problems than they save. Specifically for these benchmark tasks, we are mainly evaluating the performance of our proposed algorithm when the number of taxi is less than $50$ for lattice network of $100$ nodes (experiment setting S1) and less than $150$ for lattice network of $1089$ nodes (experiment setting S2).\n\n\\subsection{Evaluation on Singapore Road Network}\n\nThe performance of our proposed approaches is also evaluated on real community-based Singapore road network as described in Sec.\\ref{sec:experiment}. We compare the results using both randomised demands and empirical demands with taxi numbers ranging from $30$ to $200$, as shown in Figure \\ref{fig:rl_real}. Results demonstrate a consistently better performance of proposed hybrid approach, in compare with all the other approaches on a medium-sized real-world road network of $2591$ nodes. Figure \\ref{fig:rl_real} also shows that when the normalised entropy of the empirical demands is larger as compared with random demands (the empirical demands of these areas are more uniformly distributed spatially with less ``hotspots''), the number of taxis needed to operate on the road network can be significantly less, but still with higher average hiring time of taxi drivers (reward) and lower average waiting time of commuters when using efficient routing policies proposed in this work.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=12.5cm]{diagram\/realnetwork_compare.pdf} \n\\caption{Comparison of performance on community-based Singapore road network among random policy, learned policy from model-free RL (MF-RL), dispatch algorithm from model-based approach (MB Approach), effective policy as model-based initialisation (MB Init) for model-free RL, and learned policy from model-free RL with model-based initialisation (MF-RL with MB Init). The number of epochs is $3000$. The final reward and average waiting time presented are average results over the last $10$ epochs. (a) and (b) are final reward and average waiting time of commuters experimented on randomised demands (experiment setting S3), (c) and (d) are corresponding results on empirical demands (experiment setting S4).}\n\\label{fig:rl_real}\n\\end{figure}\n\nBy comparing the efficiency of proposed hybrid approach and pure model-free approach on Singapore road network (shown in Figure \\ref{fig:efficiency_real}), the hybrid approach can achieve much higher final performance and uses significantly less data samples and training epochs, especially when using empirical demand data (for which the normalised entropy is larger compared to the randomised demands). \n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=12.5cm]{diagram\/Efficiency_Evaluation_of_Real_Network.pdf} \n\\caption{Efficiency evaluation between proposed model-free RL and hybrid approach on community-based Singapore road network. (a) and (b) show reward experimented on same network with random demands using $100$ taxis and $130$ taxis respectively, (c) and (d) are corresponding results with empirical demands.}\n\\label{fig:efficiency_real}\n\\end{figure}\n\n\n\n\n\\subsection{Agent-based Simulation}\n\\label{sec:simulation}\n\nWe have designed and implemented a large scale microscopic agent-based simulation platform \\cite{yang2020phase} for evaluation of the performance of $\\mathcal P_{ij}$, using different $\\mathcal A_{ij},g_i,M_{ij}$ as inputs. A collection of instances containing state information on the positions and actions of vacant taxis, together with relevant extrinsic rewards needed for reinforcement learning is generated from the simulation after running the simulation for finite time horizon $H$ (measured in time steps). While different physical time unit can be used to correspond to each time step, in this study, each time step is taken to be one second for generality. The platform simulates the dynamics of two types of agents: the commuters and the taxis, and produces outputs characterising the performance of stochastic policy $\\mathcal P_{ij}$ from different perspectives using various performance metrics. We focus on average extrinsic reward and average waiting time of commuters in this paper, further explained in Sec.\\ref{sec:experiment}.\n\nThe behaviours of commuters are determined by probability distributions $g_i$ and $M_{ij}$, and the initial positions of all taxis are assigned to random nodes at the beginning of each epoch of the simulation. Vacant taxis will only pick up a commuter when the location of a vacant taxi and the origin of a waiting commuter coincides. The movements of the occupied taxis are computed efficiently by shortest path algorithm using our customised Contraction Hierarchy algorithm \\cite{yang2020phase}, while the movements of the vacant taxis will follow the stochastic routing policy $\\mathcal P_{ij}$. Both real-time booking and road-side hailing with three potential status for the taxis: vacant, booked, and occupied, can be implemented. In this paper, we focus on street-hailing instead of real-time booking so taxis will not take bookings. But in scenarios where some of the taxis take bookings, the methods proposed in this paper can still be applied to route those vacant taxis available for both booking and street-hailing by potential commuters. \n\n\n\n\n\n\\subsection{The Markov Decision Process}\n\n\nThe simulation platform can be used to evaluate policies $\\mathcal P_{ij}$ from top-down construction using model-based methods, or from bottom-up learning using reinforcement learning. An illustration of our proposed framework for optimising taxi routing policies is shown in Figure \\ref{fig:framework}.\n\n\\begin{figure}[tbh]\n\\centering\n\\includegraphics[width=9cm]{diagram\/schematics.pdf}\n\\caption{An illustration of framework for optimising vacant taxi routing using model-based and model-free approaches.}\n\\label{fig:framework}\n\\end{figure}\n\nThe sequential decision makings of the taxis in above framework can be modelled as a Multi-agent Markov Decision Process (MMDP) involving a collective of taxis as agents, each with their own set of actions, operating in a common environment with the same shared goal (utility function) to achieve a fully cooperative system \\cite{boutilier1996planning}. \nThe challenge of a MMDP modelling of this problem (formulated in Sec.\\ref{sec:problem}) lies in the set of actions to be represented and the policies to be optimised, as each vacant taxi will perform their own policies and choose actions at every location in the road network accordingly. While all agents are improving their own policies concurrently, the environment faced by each agent may become non-stationary.\nAnd the joint action space increases exponentially with the number of agents \\cite{zhang2019multi}, which may cause scalability issues in our problem when the number of taxis in systems is large.\n\nTo deal with the above challenges and to work for realistic taxi system with a large taxi fleet, we apply a \"mean-field\" approximation and model the given problem as a single-agent Markov Decision Process (MDP), assuming that each taxi will follow the same stochastic policy to be learned.\nInstead of each taxi as an agent, a learner to optimise routing policies $\\mathcal P_{ij}$ is formulated as the \\textit{agent} and the simulation that the learner interacts with is the \\textit{environment}. The resulting stochastic routing policies $\\mathcal P_{ij}$ is used to determine the routing decisions for each vacant taxi in system.\n\nMathematically, the modelled finite MDP is specified by $(\\mathcal S, \\mathcal A, P, r, \\gamma, \\rho)$, where: $\\mathcal S$ is a state space of the environment, $\\mathcal A$ is an action space, $P$ is a transition model $P(s^\\prime|s, a)$ as the probability of transition into state $s^\\prime$ from $s$ by taking action $a$, $r$ is a reward function where $r(s, a)$ is the immediate reward in taking action $a$ in state $s$, $\\gamma$ is a discount factor $\\gamma \\in [0,1]$, $\\rho$ is a starting state distribution over $S$. In this problem, the state $\\mathcal S$ is given as the state of all taxis in system. The action $\\mathcal A$ are decisions made for vehicles at each node on which adjacent node to move to. As a consequence of the actions, the \\textit{agent} receives reward $r=r(s_t, a_t), r \\in \\mathbb{R}$ characterising the optimisation objective defined in Sec.\\ref{sec:problem}, and transit to the next state $s^\\prime$ according to transition model $P$. The construction of the reward function is further explained in Sec.\\ref{sec: MF RL}.\n\nLet $\\pi: \\mathcal S \\rightarrow \\Delta (\\mathcal A)$ be a stochastic policy specifies the decision-making strategy in choosing the probability of actions for each taxi in system based on current state, where $\\Delta (\\mathcal A)$ is the probability simplex over action space $\\mathcal A$, and $a_t \\sim \\pi(\\cdot|s_t)$. $\\pi$ is represented as the policy matrix $\\mathcal P_{ij}$ in this problem. By following $\\pi$, a distribution of a state-action sequence $\\tau = (s_0, a_0, \\cdots , a_{H-1}, s_{H})$ is denoted as Eq.(\\ref{eq: sadistribution}), starting from state $s_0$ drawing from starting state distribution $\\rho$, and for all subsequent $i, i\\in[0, H)$ where $a_i \\sim \\pi(\\cdot|s_i)$ and $s_{i+1} \\sim P(\\cdot|s_i, a_i)$.\n\n\\begin{equation}\n    D^{\\pi}(\\tau) \\coloneqq \\prod_{i=0}^{H-1} \\pi(s_i, a_i) P(s_i, a_i, s_{i+1})\n    \\label{eq: sadistribution}\n\\end{equation}\n\nGiven a discount factor $0\\leqslant \\gamma \\leqslant 1$, the value function for a policy $P_{ij}$ is defined as discounted sum of future rewards from starting state $s$ over a finite horizon $H$, i.e.\n\n\\begin{equation}\n    V^{\\pi} (s) \\coloneqq \\mathbb{E} \\left[\\sum_{t=0}^{H-1} \\gamma^{t-1} r(s_t, a_t)|\\pi, s_0=s \\right]\n\\end{equation}\n\nThe expected value of initial state is further defined as Eq.(\\ref{eq:valuefirststate}).\n\n\\begin{equation}\n    V^{\\pi}(\\rho) \\coloneqq \\mathbb{E}_{s_0\\sim \\rho}[V^{\\pi}(s_0)]\n    \\label{eq:valuefirststate}\n\\end{equation}\n\nFor a given policy $\\pi$, the state-action value function (Q-value function) $Q^\\pi$ and the advantage function $A^\\pi$ are defined as Eq.(\\ref{eq:qfunction}) and (\\ref{eq:advantage}).\n\n\\begin{equation}\n    Q^{\\pi}(s,a) \\coloneqq \\mathbb{E}\\left[\\sum_{t=0}^{H-1} \\gamma^{t-1} r(s_t, a_t)|\\pi, s_0=s, a_0=a \\right]\n    \\label{eq:qfunction}\n\\end{equation}\n\n\\begin{equation}\n    A^{\\pi}(s,a) \\coloneqq Q^{\\pi}(s,a) - V^{\\pi}(s)\n    \\label{eq:advantage}\n\\end{equation}\n\nThe objective of the learning agent is to use ascent methods for the optimisation problem given as Eq.(\\ref{eq:rlobjective}), which is to maximise expected value from the initial state over all policies.\n\n\\begin{equation}\n    \\max_{\\theta \\in \\Theta} V^{\\pi_\\theta}(\\rho)\n    \\label{eq:rlobjective}\n\\end{equation}\n\nwhere $\\pi$ in this study is parameterised by linear function approximator as\n\n\\begin{equation}\n    \\pi_\\theta(a|s) = \\theta_{s,a}\n\\end{equation}\n\nwhere $\\theta$ is a row stochastic matrix, $\\theta_{s,a} \\geqslant 0$ and $\\sum_{a\\in \\mathcal A} \\theta_{s,a} = 1$ for $s\\in \\mathcal S$ and $a \\in \\mathcal A$.\n\nIt has been proved in \\cite{agarwal2019optimality} that $V^{\\pi_\\theta}(s)$ is non-concave in $\\theta$ for the above parameterisation, so the standard solvers for convex optimisation problems are not applicable for this target problem. Throughout this paper, we will use RL approaches to optimise the routing policies based on policy gradient theorem \\cite{sutton2000policy} as follows.\n\nLet $R(\\tau)$ be the total reward of state-action sequence $\\tau$ on following $\\pi$ starting from $s_0$, where in this problem, the reward $R(\\tau)$ is given by\n\n\\begin{equation}\n    R(\\tau) \\coloneqq R(s_0, a_0)\n\\end{equation}\n\nwhere $R(s_0, a_0)$ is defined by Eq. (\\ref{eq:reward}) in Sec.\\ref{sec: MF RL}.\n\nThen,\n\n\\begin{equation}\n    \\nabla_\\theta V^{\\pi_\\theta}(\\rho) = \\mathbb{E}_\\tau \\left[ r(\\tau)\\nabla_\\theta \\log(D^{\\pi_{\\theta}}(\\tau))|\\rho\\right]\n    = \\mathbb{E}_\\tau \\left[ r(\\tau) \\sum_{t=0}^{H-1} \\nabla_\\theta \\log(\\pi_{\\theta}(s_t, a_t))|\\rho\\right]\n    \\label{eq:policygradient}\n\\end{equation}\n\n\nThough the problem of this paper is formally framed as a MDP, the RL method proposed in this paper is based on Monte-Carlo method with undiscounted setting where $\\gamma \\rightarrow 1$, which does not require the system to be Markovian. The details of the model-free reinforcement learning will be further explained in Sec.\\ref{sec:methodology}.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Abstract}\nPredicting volume swelling of structural materials in nuclear reactors under high-dose neutron irradiations based on existing low-dose experiments or irradiation data with high-dose-rate energetic particles has been a long-standing challenge for safety evaluation and rapidly screening irradiation-resistant materials in nuclear energy systems. Here, we build an Additional Defect Absorption Model that describes the irradiation-induced swelling effects produced by energetic electrons, heavy-ions, and neutrons by considering additional defect sinks inherent in the irradiation process. Based on this model, we establish quantitative methods to predict high-dose swelling from low-dose behavior and obtain the equivalent irradiation dose for different energetic particles when the dose rates differ by several orders of magnitude. Furthermore, we propose a universal parameter to characterize the swelling resistance of various alloys and predict their radiation tolerances under different radiation conditions. This work provides quantitative prediction methods for evaluating irradiation-induced swelling effects of structural materials, which is critical to the safety and material development for advanced nuclear reactors.\n\n\n\\section*{Introduction}\n\nThe development of advanced nuclear energy systems is one of the most promising ways to address the anticipated global climate change and energy crisis. In nuclear reactors, the economics and safety depend critically on the durability of structural materials in the reactor core, where the materials need to survive extremely harsh environments, including irradiation by energetic neutrons up to 200 dpa (displacement per atom) \\cite{Zinkle2014,Norgett1975}. In addition, recent efforts to extend the lifetime of current nuclear reactors also raise higher demands on irradiation tolerance of structural materials\\cite{Zinkle2013}. Therefore, evaluating and quantitatively predicting the performance of materials under high-dose neutron irradiation and R\\&D of radiation-resistant materials are critical for the realization of advanced nuclear energy systems and the lifetime extension of current operating reactors. \n\nNeutron irradiation can lead to severe property deterioration of structural materials, such as volume swelling, hardening, embrittlement, creep, phase instability, etc\\cite{Zinkle2012}. Among them, volume swelling is a vital material degradation mechanism. The degree of volume swelling has been used as one of the most critical criteria for materials selection in the reactor design\\cite{Mattas1984}, and the swelling phenomena have been widely investigated since the 1960s\\cite{Cawthorne1967, Mattas1984}. Although a linear relationship between the swelling rate and the irradiation dose is expected at high swelling levels \\cite{Was2007}, it is often difficult to evaluate the swelling trend as it is intrinsically a non-linear relationship at low to intermediate doses. Particularly, swelling in materials exhibits several distinct stages, i.e., a nucleation\/incubation stage, a transient stage, and a steady-growth stage. Until now, there is no suitable general model to describe the volume swelling at different doses, which greatly hinders the evaluation of structural materials in nuclear plants and the development of new materials for advanced nuclear reactors. \n\nTraditionally, neutron radiation effects are studied through test reactors. After irradiation, the bulk properties of irradiated materials are tested, and the microstructures are characterized by post-irradiation examination (PIE) techniques. However, for advanced nuclear reactors, it would take a long time (more than ten years) for irradiation with reactor neutrons because of the high dpa level to achieve lifetime-relevant doses, resulting in an enormous cost and testing difficulty associated with potentially high radioactivation of materials. Irradiation-induced volumetric swelling in alloys is a bulk property of materials that can be measured with TEM (transmission electron microscopy) since the dimensional changes are typically well correlated with void formation\\cite{Zinkle1989,Garner2020}. Since the 1970s, the energetic electron-beam from HVEM (high voltage electron microscope) and heavy ions from accelerators have been extensively used for irradiation and swelling investigation based on TEM analysis \\cite{El-Atwani2019, Massee2015, Agarwal2020, Du2018, Wang2019, Was2007}. For example, some experts claimed that the void swelling of thin samples irradiated by HVEM could represent the bulk property, and the fitting result showed a swelling rate of (dose)1.58.\\cite{Garner1972} Experimental results also show that the swelling rates decrease significantly with the increase of dose rates. Typical examples are that much lower swelling rates for heavy-ion and electron irradiations are found than neutron irradiation due to their higher damage rate\\cite{Gigax2015, Mansur1993, Was2015, Jiao2018, Okita2004, Garner2012, Was2007, Zinkle2012}. Indeed, ion- and electron-beam irradiation experiments could achieve high damage levels in a short time and a shallow depth at a low cost with no or little radioactivation of materials\\cite{Was2012, Zinkle2017a,Hosemann2012,  Was2015, Zinkle2012,  Zinkle2018}. However, because of the much higher damage rate (103-104 orders of magnitude higher) and the differences in the primary in-cascade process, the extent to which electron or ion irradiation effects can represent neutron irradiation needs to be scrutinized. Notably, there exist two critical and long-standing unsolved questions for irradiation-induced volume swelling in practice: can volume swelling at high doses be predicted from easily accessible low-dose irradiation experiments, and how to quantitatively predict neutron radiation damage from high-dose-rate heavy-ion irradiation experiments? \n\nVolume swelling originates from the evolution of irradiation-induced defects. It is well established that energetic particles interact with materials by transferring their kinetic energies into the electronic and atomic subsystems of target materials. Irrespective of the dose and dose rate difference, this process can be described by two main simplified processes\\cite{Was2007}: (1) Energetic particles first collide with lattice atoms and produce primary knock-on atoms (PKAs), giving rise to Frenkel defects (i.e., vacancies and self-interstitial atoms) with the same overall number of interstitials and vacancies. If the PKAs have sufficient kinetic energy, they can lead to further atomic cascade displacements and produce more point defects and defect clusters. (2) Defect evolution occurs through either recombination of interstitials and vacancies or diffusion-induced defect absorption\/accumulation at various defect sinks, such as dislocations, defect clusters, grain boundaries, interfaces, and precipitates. The remaining mobile vacancies aggregate for void formation and growth. For energetic heavy-ions and neutrons, their PKAs can induce displacement cascades. In contrast, electron bombardment creates uniform Frenkel defects without cascades. Although these energetic particles have differences in charge states, mass, and damage rate, similar irradiation effects have been observed, especially the microstructural changes in materials under irradiation such as volume swelling\\cite{Was2007,Zinkle2012}.\n\nThere have been sustained efforts to understand and describe the irradiation effects at different doses and dose rates, especially regarding volume swelling. In the early 1970s, Brailsford and Mansur\\cite{Brailsford1972, Mansur1978} developed a model that states that changes of an irradiation variable can be accommodated by a shift in other variables so as to keep the overall defect aggregation properties (such as swelling) invariant\\cite{Mansur1978,Mansur1986, Mansur1978a, Mansur1993, Mansur1994}. Was et al. applied this model successfully in case of irradiation effects induced by protons and neutrons at low doses (less than 5 dpa) and low dose rates (~10-6 dpa\/s)\\cite{Was2014,Was2002}. At high dose and high dose rate, a direct comparison between neutron and ion or electron irradiation effects becomes difficult, and discrepancies have been found\\cite{Sun2015,Gan2001,Hardie2013}. Based on the production bias model (PBM), Golubov and Singh et al. formulated a theoretical description of volume swelling in pure metals and simple alloys, but with a number of complicated parameters, making its application difficult, especially for actual alloys with complex chemical compositions and microstructures \\cite{Singh1997,Golubov2001, Golubov2012,Okita2004}. Furthermore, it has been shown consistently from frequently worldwide-used ion-irradiation experiments that swelling rates are lower at higher-dose-rate irradiation\\cite{Garner2012, Gigax2015, Okita2004, Was2007} Most recently, Ren et. al. proposed a general radiation polarization theory and a sample spinning strategy technique to mitigate the excess polarization artifact and point-defect imbalance which might be crucial for making the simulation of neutron radiation by ion-beam radiation more realistic\\cite{ren2020sample}. Therefore, an application-oriented general model and prediction methods of irradiation swelling in alloys at different irradiation doses and dose rates are highly desired for the developments of advanced nuclear energy systems. \n\nGenerally, volume swelling occurs on a relatively large spatial scale and a long-time scale. Consequently, the details of in-cascade characteristics and the transient defect behavior are not so important when comparing irradiation effects at different dose rates or with different particles, which justifies the application of rate theory models\\cite{Was2007} by neglecting the details of the atomic displacement process. In addition, volume swelling provides a direct link between the observed void distributions at the microscopic level and the material response at the macroscopic scale, which enables us to establish equivalent relations between irradiations with different energetic particles. Taking into consideration the importance of volume swelling in nuclear reactors, in this paper, we develop an additional defect absorption model (ADAM) that separates material-related properties and irradiation conditions and can be easily applied to predict volume swelling of alloys at different doses and different dose rates. We first introduce ADAM in the context of electron irradiation, where the produced defects are in the form of uniformly distributed point defects without cascades. After validating this model with experimental data, we extend ADAM to the case of neutron and heavy-ion irradiation by including the energetic displacement cascade as one of the stages in the whole defect evolution. Overall, our model provides a unified description of volume swelling at different irradiation conditions and further, can be used to make quantitative predictions of the swelling level of actual alloys at different doses and dose rates based on available experimental data.\n\n\n\n\\section*{Results}\n\\subsection*{Swelling induced by energetic electrons}\n\nTo illustrate the radiation damage process produced by energetic electrons, we schematically show the irradiation effects in Figure 1, specifically the void formation, at different dose rates. \n\n\\begin{figure}[!ht]\n\\includegraphics[width=\\textwidth]{figs\/FIG1.jpg}\n\\caption{\\textbf{Schematic illustration of the electron irradiation effects in materials at low and high dose rates}. The energetic electrons generate uniformly distributed point defects. (a) shows the defect process at low dose rates. The defects can either recombine or be absorbed by defect sinks, including dislocations and boundaries. The surviving defects lead to void formation and growth. (b) shows the defect process at high dose rates, where the irradiation induces additional defect sinks can annihilate and absorb mobile defects.}\n\\label{fig1}\n\\end{figure}\n\nElectron irradiation induces uniformly distributed point defects. When the irradiation dose rate is high, there is a high probability that Frenkel defects and a few small defect clusters are produced in a high density in a short time period. If the dose rate is sufficiently high, e.g., $10^{-2}-10^{-3}$ dpa\/s for electrons in HVEM, the newly produced point defects could be annihilated by the near-by and co-existing Frenkel defects or small clusters. The effect of this process is similar to the capture of defects by the intrinsic sinks in the unirradiated alloys. So, we classify the irradiation-generated near-by and co-existing Frenkel defects or small clusters at high dose rates as additional sinks or irradiation-generated defect sinks. These additional sinks enhance vacancy absorption and reduce the vacancy flux contributing to the void nucleation and growth. Meanwhile, the high-dose-rate irradiation may make the lattice atoms vibrate quickly and increase the local temperature (dynamic temperature), increasing the mobility of defects in a small region and enhancing the additional defect absorption locally. To account for these effects of enhanced defect absorption, we introduce an additional term $S'$ into the rate theory to describe the loss of defects by the irradiation-generated defect sinks. The total adsorption rate then becomes $S_i=S_{i0}+S_i'$ and $S_v=S_{v0}+S_v'$, in which $S_{i0}$ and $S_{v0}$ are the absorption rate of the intrinsic defect sinks of the materials toward interstitials and vacancies respectively, and $S_i'$ and $S_v'$ are the absorption rate of irradiation-generated defect sinks for interstitials and vacancies, respectively. \n\nThe additional sinks mainly result from the product of the near-by and co-existing Frenkel defects at high dose rates, which may not be the final stable extended defects (clusters) but can enhance local defect absorption in a locally \"heated\" region associated with high dose rates, leading to point defect being captured locally. Under irradiation, the generation rate of such additional defect sinks $\\chi_{i,v}$ in reducing mobile defects is proportional to the defect generation rate G for interstitials and vacancies respectively, i.e. $\\chi_i=\\delta_i G_i$ and $\\chi_v=\\delta_v G_v$ at relevant temperatures for swelling, where $\\delta_{i,v}$ are the ratio constants. Consequently, the net absorption rate of the additional sinks is proportional to their generation rate, i.e. $S_i'=\\gamma_i \\chi_i=\\gamma_i \\delta_i G_i$ and  $S_v'=\\gamma_v \\chi_v=\\gamma_v \\delta_v G_v$, with $\\gamma_{i,v}$ being the proportional constants. Since the electron-irradiation-induced defects are generated in the form of Frenkel pairs, the generation rate of interstitials $G_i$ and the generation rate of vacancies $G_v$ are the same, $G_i=G_v=G$. The additional sink introduced in our model is dynamic, depending on local temperature and nearby defect cluster concentrations, etc. Since the role of the additional sink is the same as the intrinsic sink in capturing interstitials and vacancies, the characteristics of these two types of sinks are similar.  The concentration and the sink strength of the additional sinks may vary during irradiation, but the sink absorption rate bias may not change significantly. Therefore, we assume that the sink bias towards vacancies and interstitials from all sinks is roughly proportional to that of intrinsic sinks, i.e., $S_i\/S_v =(S_{i0}+S_i')\/(S_{v0}+S_i' )=b, S_{i0}\/S_{v0} =b_0$, and $b=\\beta b_0$. As a result, the total sink absorption rate for interstitials and vacancies are $S_i=S_{i0}+\\gamma_i \\delta_i G_i$, and $S_v=S_{v0}+\\gamma_v \\delta_v G_v$, respectively. Note that additional sinks might be unstable because of the possible irradiation-induced dissociation. In our model, the absorption rate of additional sinks can be taken as the net absorption rate, i.e. $S'=S_{absorption}'-S_{dissociation}'$. Therefore, only those defects that survive from dissociations contribute to defect absorption in our model.\n\nWith the additional defect sinks, we solve the classical rate theory equation for void growth. We consider the sink dominant case since the total defect sinks play majority roles in the considered conditions. The void-induced swelling is derived as:\n\n\\begin{equation}\n    swelling(\\%)=\\alpha \\left[\\frac{\\Delta_{dpa,ele}}{1+k'G_{dpa,ele}}\\right]^{3\/2}-c.\n\\end{equation}\n\nwhere  $\\alpha=\\frac{4\\pi}{3V_0}\\left[2\\Omega \\frac{D_v b-D_i}{b\\eta S_{v0}}\\right]^{3\/2}, k'=\\frac{\\gamma_v \\delta_v}{\\eta S_{v0}}$, and $c=\\frac{4\\pi r_0^3}{3V_0}$. Here $\\eta$ is the ratio of the dose rate ($G_{dpa}, dpa\/s$) and defect generation rate (G), $V_0$ is the original volume without voids, $\\Omega$ is the defect volume, and $D_{v,i}$ is the diffusion coefficient of vacancies\/interstitials. We also introduce $r_0$ to account for the void nucleation stage. Specifically, the value of $r_0$ is the critical size of a void embryo that must be achieved in order for the embryo to grow into a void. Eq. (1) provides the relation between volume swelling and irradiation dose at different dose rates. It indicates that at a fixed dose rate, swelling increases with irradiation dose slowly when the dose is relatively low, but it undergoes a sharp increase at high doses. Furthermore, Eq.(1) predicts lower swelling with increasing dose rate at a given irradiation dose due to extension of the transient regime, which is in accordance with experiments \\cite{Garner2012,Okita2007,Okita2004}.\n\nThe above model provides the quantitative dependence of volume swelling on different parameters at a fixed temperature, including materials properties ($\\alpha$ and c), irradiation dose ($\\Delta_{dpa,ele}$), and dose rate ($G_{dpa,ele}$). For materials properties, $\\alpha$ is governed by intrinsic properties, such as defect diffusion coefficients and the absorption rate of intrinsic defect sinks, including the influence of temperature, pressure, and cold-work conditions on these properties. The parameter c is governed by the void nucleation ability of materials. \n\n\n\\begin{figure}[!ht]\n\\includegraphics[width=\\textwidth]{figs\/FIG2.jpg}\n\\caption{\\textbf{Validation of the model against volume swelling data from electron irradiation experiments}. (a) 316L irradiated by electrons at 530 \u00b0C \\cite{Garner1972}, (b) EN58B irradiated by electrons at 500 \u00b0C \\cite{Norris1971}, (c) nickel irradiated by electrons at 450 \u00b0C \\cite{Norris1971}, (d) 316SS irradiated by electrons at different temperatures \\cite{Hishinuma1977}. The solid lines are our model-fitting results. (Note: some data point with high swelling rate are not included since the saturation may result from surface effects of thin TEM samples)}\n\\label{fig2}\n\\end{figure}\n\nIn Eq. (1), $k'$ is related to the additional sink absorption rate per defect generation rate, scaled by  $\\gamma_{i,v}$, $\\delta_{i,v}$ and the absorption rate of intrinsic sinks. Since this model accounts for the additional defect absorption at high dose rates, it is named as Additional Defect Absorption Model. We validate ADAM based on volume swelling data from electron irradiation with HVEM at different conditions. With the irradiation-generated sinks considered, we are able to fit the experimental data to Eq. (1). The results shown in Figure 2 suggest that ADAM agrees well with experimental data, indicating our model provides a robust description of the dose dependence of volume swelling. \n\nIt is shown in Figure 2 that volume swelling may start to grow continuously only after a threshold irradiation dose. This corresponds to the nucleation process or incubation period for volume swelling that is frequently observed in experiments \\cite{Garner2000,Garner2012}. By extrapolating to the dose where the swelling equals to zero, we obtain the threshold dose $\\Delta_0$ after which pronounced swelling begins (marked with an arrow in Figure 2(a)):\n\n\\begin{equation}\n    \\Delta_0=\\frac{r_0^2}{2\\Omega} \\frac{(1+ \\frac{\\gamma_v \\delta_v}{\\eta S_{v0}} G_{dpa} )b\\eta S_{v0}}{D_v b-D_i}\n\\end{equation}\n\nwhich reasonably relates to $r_0$ and other materials parameters. Besides, $\\Delta_0$ increases linearly with increasing $G_{dpa}$ , a signature of the dose rate effects on the nucleation\/incubation regime. Our model thus provides a general description of the swelling over the whole dose range.\n\nIn Figure 2, all the irradiation experiments were conducted at a fixed dose rate (constant $G_{dpa,ele}$). As a consequence, we cannot isolate the parameter $k'$ but can only take  $k' G_{dpa,ele}$ as a whole in our model fitting. Therefore, there are only two parameters to obtain the relationship between the swelling and dose, i.e., $\\alpha[1\/(1+k' G_{dpa,ele} )]^{3\/2}$ and c. The obtained parameters are provided in the Supplementary materials. It is found that c is small in all cases, suggesting a short incubation period for void swelling in electron irradiation. The prefactor $\\alpha[1\/(1+k' G_{dpa,ele} )]^{3\/2}$ increases with increasing temperature for 316SS, which agrees with the higher swelling observed in experiments at 600 \u2103.\n\n\n\\subsection*{Swelling induced by energetic heavy-ions and neutrons}\n\nIn contrast to the isolated Frenkel pairs produced by electron irradiation, irradiation by heavy-ions and neutrons induces energetic displacement cascades with nearly the same features at the early stage of defect production. The damage process created by heavy-ions and neutrons at different dose rates is schematically illustrated in Figure 3. After the PKAs and then the collision cascades are produced by heavy-ions and neutrons, athermal recombination between vacancies and interstitials dominates the in-cascade process \\cite{Nordlund2018}. We consider that this process is similar to the situation of electron irradiation at a higher beam current, and the same idea of additional sink is used to account for the enhanced defect absorption. Although the defects induced by electron and ion\/neutron irradiations may be different, all these irradiation sources produce defect clusters. Particularly, due to the high dose rate ($10^{-3}-10^{-2}$ dpa\/s) of high voltage electrons, the produced Frenkel defects are densely distributed spatially, forming high-density defect spots. Such high-density defects are highly likely to aggregate to defect clusters in their evolution processes. For ion\/neutron irradiations that create densely populated cascades and subcascades, defect clusters are directly formed. In this sense, defect clusters can be created in both electron and ion\/neutron irradiations, contributing to subsequent defect evolution. The fate of these defect clusters depends on various factors, such as irradiation-induced dissolution. The in-cascade process in the case of neutron or heavy-ion irradiation may affect the number of residual interstitials and vacancies entering into the governing equations \\cite{Golubov2000,Golubov2012}. The afterward process of defect evolution under these conditions is similar to that of electron irradiations in terms of volume swelling. By treating the in-cascade process as one of the stages of whole defect evolution (since the volume swelling occurs both at a large spatial scale and a long-time scale), we introduce the irradiation-generated defect sinks as stated above. Some experimental and simulation results \\cite{Kiritani1994,Pokor2004,Stoller2008,Muroga1990,Phythian1995,Gan2001} showed that the neutron irradiation-induced stable dislocation loop or cluster density increase quickly with dose and then saturate at around 2-3 dpa. So, at the high dose range, these stable and extended defects as sinks to absorb mobile defects could be classified into the intrinsic defect sinks, i.e., $S_{v0}$  and $S_{i0}$. Comparing to neutron irradiation, the dose rate of heavy-ion irradiation is $10^3-10^4$ higher, similar to electron irradiation, and will also increase the local dynamic temperature. As the energy of majority PKAs produced by neutron irradiation in iron is below 10 keV \\cite{gilbert2015energy}, we calculate a single cascade morphology, as shown in Figure 4(a) ,  produced by 10 keV PKA(10 keV Fe ion) through SRIM program \\cite{ziegler2010srim} in \\textit{Full Damage Cascades} mode. And because the mean free path of neutron is very large (about 4 centimeters in iron), neutron-induced PKAs are spatially separated. Also with SRIM calculation, the cascades produced by PKAs along the transport track of incident 1 MeV Fe ion are drawn in Figure 4(b) and Figure 4(c) . Unlike neutron irradiation, a single heavy ion can induce dense PKAs along its transport track. The cascades from these PKAs are produced in picoseconds \\cite{antoshchenkova2015fragmentation} and are close to each other. Thus, during heavy-ion irradiations, the additional sinks from the near-by and co-existing defects, some of which are transitional defects, could also enhance local defect absorption at a high dynamic temperature, leading to the extra capture of defects in a local region that survive the in-cascade process. Recent MD results for high energy PKAs also demonstrated that there are strong interactions between the closed-by subcascades, which would help annihilating pre-created defects.\\cite{de2018model}\n\n\\begin{figure}[!ht]\n\\includegraphics[width=\\textwidth]{figs\/FIG3.jpg}\n\\caption{\\textbf{Schematic illustration of the heavy-ion and neutron irradiation effects in materials at low and high dose rates}. The PKAs from heavy-ion and neutrons produce defects through displacement cascades. (a) shows the defect process at low dose rates, and (b) shows the defect process at high dose rates.}\n\\label{fig3}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\includegraphics[width=\\textwidth]{figs\/FIG4.jpg}\n\\caption{\\textbf{cascade morphology produced by neutron or ion}. (a) a cascade produced by 10 keV PKA from neutron irradiation in iron; (b) spatially close cascades produced by PKAs from 1 MeV $\\textrm{Fe}^{2+}$ irradiation in iron; (c) the partial enlarged view of (b).}\n\\label{fig4}\n\\end{figure}\n\nWith the above consideration, we apply the derived ADAM to the case of heavy-ion and neutron irradiations. The volume swelling becomes\n\n\\begin{equation}\n    swelling(\\%)=\\alpha \\left[\\frac{\\Delta_{dpa}}{1+k'G_{dpa}} \\right]^{3\/2} -c\n\\end{equation}\n\nHere we have omitted the specific denotation in $\\Delta_{dpa}$ and $G_{dpa}$  for different energetic particles, which means this relation is suitable for all these energetic particles. Based on this equation, we fit experimental data collected from experiments for different alloys at varying temperatures and for the same alloy at varying dose rates. As representative cases, Figure 5 shows the fitting results for irradiation data in different material conditions, including varying materials, temperatures, pressure, and helium incorporation (different $\\alpha$ , $k'$, and c characterizing material conditions), while Figure 6 demonstrates the fitting results for the same material irradiated at different dose rates (different $G_{dpa}$  but with same $\\alpha$ , $k'$, and c) \\cite{Garner1983a,Zinkle2013a,Getto2016, Dubuisson1992,Garner2000,Garner2012,Holmes1969,Norris1971,Okita2004,Sun2015,Zinkle2013}. (A more thorough validation of ADAM is provided in Supplement material). The fitting results shown in Figure 5 and Figure 6 demonstrate ADAM is suitable for describing different alloys at neutron and heavy-ion irradiation conditions in terms of volume swelling at the same elevated temperature or in a narrow temperature range. Particularly, our model captures the swelling trend even when He is present in the material, as shown in Figure 5 (e) and (f).\n\n\\begin{figure}[!ht]\n\\includegraphics[width=\\textwidth]{figs\/FIG5.jpg}\n\\caption{\\textbf{Validation of the derived model using available irradiation data on different alloys or alloy conditions (different $\\alpha$, $k'$, and c)}. (a) Fe-20Ni-xCr and Fe-35Ni-xCr irradiated by neutrons at 538 \u2103 \\cite{Garner2000}; (b) AISI316 irradiated by neutrons at different temperatures \\cite{Garner2000}; (c) 304L SS, 316SS, D9, and 9-12Cr FM irradiated by neutrons \\cite{Garner2000,Garner1983a,Zinkle2013}; (d) 316Ti SS irradiated by neutrons at different pressure \\cite{Dubuisson1992}; (e) HT9 and T91 irradiated by 5 MeV Fe++ at 460 \u2103 \\cite{Getto2016}; (f) F82H irradiated by neutrons with different helium injection \\cite{Zinkle2013a}. The solid lines are our model-fitting results.}\n\\label{fig5}\n\\end{figure}\n\n\\subsection*{Prediction methods based on ADAM}\n\nFor fast screening and evaluation of structural materials for advanced reactors and current reactor lifetime extension, we predict volume swelling in alloys from low-dose irradiation experiments to high-dose cases based on ADAM. Note that such high-dose data usually requires at least several years to carry out neutron irradiation experiments and PIE. The fitting results displayed in Figure 2, Figure 5, and Figure 6 provide the growth trend of swelling at high doses beyond those in the experimental data, representing the prediction merit of ADAM. In addition, the fitting results in Figure 6 indicate the quantitative correlation of the swelling rate with irradiation dose at different dose rates ($G_{dpa}$ ), or with different energetic particles, making the equivalence possible between different energetic particles, especially between heavy-ion and neutron irradiations in term of volume swelling in the same material at elevated temperatures.\nWhen we compare two irradiation conditions that lead to different swelling levels in the same material and the same elevated temperature, we have\n\n\\begin{equation}\n    \\frac{swelling_1}{swelling_2}=\\frac{\\alpha \\left[\\frac{\\Delta_{dpa1}}{1+k'G_{dpa1}} \\right]^{3\/2} -c}{\\alpha \\left[\\frac{\\Delta_{dpa2}}{1+k'G_{dpa2}} \\right]^{3\/2} -c}\n\\end{equation}\n\nIn this case, derivation of the irradiation conditions with a given swelling level based on another irradiation experiment would require the knowledge of $\\alpha$ , $k'$, and c. When the swelling level is the same for the identical material, an equivalent relation between the irradiation dose and dose rate can be obtained as:\n\n\\begin{equation}\n    \\frac{\\Delta_{dpa1}}{1+k'G_{dpa1}}=\\frac{\\Delta_{dpa2}}{1+k'G_{dpa2}}\n\\end{equation}\n\nwhich is only governed by the parameter of $k'$.\n\n\\begin{figure}[!ht]\n\\includegraphics[width=\\textwidth]{figs\/FIG6.jpg}\n\\caption{\\textbf{Validation of the derived model using available irradiation data with different dose rates for the same alloy (constant $\\alpha$ , $k'$, and c)}.  (a) 304 SS irradiated with different dose-rate neutrons \\cite{Garner2012}, (b) Fe-15Cr-16Ni irradiated with different dose-rate neutrons \\cite{Okita2004}, (c) Ni irradiated by 1 MeV electrons and neutrons\\cite{Holmes1969,Norris1971}, and (d) 304L SS irradiated by neutrons and heavy ions\\cite{Sun2015}. The solid lines are our model-fitting results.}\n\\label{fig6}\n\\end{figure}\n\nEquations (4-5) serve as the basis for predicting the swelling level for an identical material at the same or similar elevated temperature at given irradiation conditions, as well as the equivalent irradiation conditions at the same swelling level based on available experiments. In Figure 7, we show three examples of such predictions. In Figure 7(a), the parameters $\\alpha$ , $k'$, and c are calculated from the low-dose irradiation data at 435\u2012487 \u00b0C. Then Eq. (5) is used to predict the swelling rate at higher doses. Our prediction is consistent with another set of high-dose irradiation data at 446 \u00b0C. Figure 7(b) and (c) illustrate the derivation of equivalent irradiation conditions with the same swelling level, which only requires $k'$. In Figure 7(b), we calculate $k'$ for Fe-15Cr-16Ni-0.25Ti from three sets of irradiation data (solid black symbols)\\cite{Garner2012}. Based on $k'$, the irradiation dose required to produce the same swelling at $7.8\\times 10^{-7}$ dpa\/s can be deduced from the data at $1.7\\times 10^{-6}$ dpa\/s. The predicted data (empty pentagons) lie within the range of experimental data. In Figure 7(c), the equivalent neutron irradiation dose is predicted from the self-ion (Fe ions) irradiation experiment, with the $k'$ parameter calculated from two sets of irradiation data at different dose rates (solid black symbols)\\cite{Getto2016,VanDenBosch2013}. It can be seen that the prediction agrees reasonably well with available neutron irradiation data. Therefore, these examples suggest that our derived additional defect absorption model is capable of predicting volume swelling at high doses from the low dose data, correlating the irradiation dose with different given dose rates, and quantitatively predicting equivalent dose from high dose-rate experiments (such as heavy-ions) to low dose-rate irradiation (such as neutrons).\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.6\\textwidth]{figs\/FIG7.jpg}\n\\caption{\\textbf{Quantitative prediction of swelling level by ADAM}. (a) Prediction of swelling level in 304L from low doses to high doses. The parameters are calculated from a set of irradiation data (blue triangle symbols) \\cite{Garner1983a}. (b) Prediction for Fe-15Cr-16Ni-0.25Ti from $1.7\\times10^{-6}$ dpa\/s to $7.8\\times10^{-7}$ dpa\/s at the same swelling level. The parameters are calculated from three sets of irradiation data (solid black symbols) \\cite{Garner2012}. (c) Prediction for T91 from self-ion irradiation ($1\\times10^{-3}$ dpa\/s) to neutron irradiation ($3\\times10^{-6}$ dpa\/s) at the same swelling level. The parameters are calculated from two sets of irradiation data with different dose rates (solid black symbols) \\cite{Getto2016,VanDenBosch2013}}\n\\label{fig7}\n\\end{figure}\n\n\\subsection*{Parameters in ADAM governing swelling}\n\nIrradiation effects in materials are complex processes, including defects production and evolution. Previous works have established that volumetric swelling can be influenced by different factors, including properties of materials, e.g., compositions and their atomic weight, crystal structures, grain boundaries and interfaces, atomic bonds, solute segregation and precipitation, the diffusivity of defects, sink strength, etc., at a fixed temperature, and irradiation conditions, i.e., PKA spectrum, irradiation temperature, dose and dose rate\\cite{Okita2007,Okita2004}. Based on ADAM, Eq. (3) shows that irradiation-induced swelling is governed by $\\alpha$ , $k'$, c, and $\\Delta_{dpa}$. The parameter $\\alpha$ contains different intrinsic properties of materials (e.g., diffusivities of vacancies and interstitials at fixed temperatures, various intrinsic sink strength) and determines how fast the swelling grows. So, a higher $\\alpha$ means faster swelling growth. The value of c is also a material property and relates to the nucleation stage or incubation period for void evolution, which is governed by the threshold dose $\\Delta_0$ due to the nucleation\/incubation process for swelling. Therefore, a larger $\\Delta_0$ indicates a longer nucleation\/incubation time, suggesting better swelling resistance. The value of $\\Delta_0$ is very important for the development of nuclear materials since the extension of the incubation period is the only meaningful solution for designing swelling-resistant materials up to high doses. The overall swelling of materials relies on both $\\Delta_0$ and $\\alpha$ since the dose-dependent swelling includes an initial nucleation\/incubation period, followed by a growth stage \\cite{Wolfer1984,Russell1971}. Finally, $k'$ relates to irradiation conditions and characterizes the ability of materials to absorb mobile defects by the irradiation-generated additional sinks in response to the high dose rate irradiations, which is related to $\\gamma_{i,v}$, $\\delta_{i,v}$, and the absorption rate of the intrinsic defect sinks. Since irradiation-induced defects, which are the source of additional defect absorption, strongly depend on crystal structures and microstructures, the parameter of $\\delta_v$ is expected to be relating to these materials conditions. Therefore, $k'$ is related to both intrinsic sinks and additional sink generation processes, indirectly related to crystal structure and microstructures. The purpose of $k'$ in our model is to reconcile the high density of additional sinks generated at high dose rates, which tends to suppress swelling. As such, it is a key parameter to reveal the impact of dose rate on swelling, as encountered in the comparison between ion and neutron irradiations. All these parameters from our model are provided in the Supplement materials. In this work, we did not explicitly consider the temperature. Therefore, our model describes swelling from different irradiation conditions at the same temperature, or in a narrow temperature window. Since material properties change with temperatures, the swelling also depends on temperature besides the above three parameters. The relations of swelling with all these different parameters are provided in Figure 8.\n\nIt is well known that the swelling rate of some alloys exhibits a peak at an intermediate temperature, as shown in the upper panel of Figure 8(a). At low temperatures, defect mobility is low, and a relatively high density of irradiation-induced defect sinks are produced, both of which limits void growth. When the temperature is sufficiently high, swelling is also reduced since the mobility of interstitials and vacancies is high, and the recombination between them increases significantly. Besides, vacancy supersaturation is reduced compared to thermal equilibrium, and the emission of vacancies from cavities increases. The obtained parameters $\\alpha$ and $\\Delta_0$ from our model based on the experimental data agree with this conclusion. In particular, $\\alpha$ exhibits a peak, whereas $\\Delta_0$ shows a dip around the peak swelling temperature. \n\nThe dependence of $\\alpha$ and $\\Delta_0$ on alloy composition is shown in Figure 8(b) for several well-studied alloys. In both Fe-35Ni-xCr and Fe-20Ni-xCr, $\\alpha$ decreases, whereas $\\Delta_0$ increases with decreasing Cr content. Therefore, the results suggest that reducing Cr enhances the swelling resistance for these alloys. The variation of $\\Delta_0$ is more significant than $\\alpha$ , signifying a primary influence of Cr on the duration of the incubation regime \\cite{Garner2000}. This agrees with prior phenomenological experimental assessments of Fe-Ni-Cr austenitic alloys that the chemical changes predominantly affect the incubation dose rather than the steady-state swelling rate \\cite{Garner2012,Okita2005}. To access the parameter of $k'$ for different materials, we need experimental data from different dose rates. Among all the data that we have gathered, the swelling data on Fe-15Cr-16Ni and Fe-15Cr-16Ni-0.25Ti can be used to compare $k'$ for these two materials, as they were irradiated with varying dose and dose rate at the same temperature. Figure 8(c) displays the dependence of $k'$ and $\\Delta_0$ on the dose rate for Fe-15Cr-16Ni and Fe-15Cr-16Ni-0.25Ti. For the same material, it is seen that $k'$ is constant. It's interesting to find that Fe-15Cr-16Ni-0.25Ti, with better swelling resistance than Fe-15Cr-16Ni \\cite{Garner2000}, has a smaller $k'$, which indicates that if an alloy possesses more intrinsic vacancy absorption centers, or higher intrinsic sink strength, i.e., larger $S_{v0}$, the irradiation-induced additional sinks are less important for volume swelling. On the other hand, $\\Delta_0$ increases linearly with the dose rate for both alloys, consistent with Eq. (2). This conclusion is in line with experiments \\cite{Okita2005}.\n\n\n\\begin{figure}[!ht]\n\\includegraphics[width=\\textwidth]{figs\/FIG8.jpg}\n\\caption{\\textbf{Dependence of swelling (characterized by $\\Delta_0$ , $\\alpha$ , and $k'$) on different factors}. (a) Temperature \\cite{Garner2000}, (b) materials \\cite{Garner2000}, and (c) dose rate \\cite{Garner2012,Okita2005}. A summary of the results provided in (d) shows the relations of swelling (characterized by $\\Delta_0\/\\alpha$) with temperature for different materials under neutron irradiation \\cite{zinkle2017,Dubuisson1993,Garner2000,Garner1983a,Holmes1969,Okita2004,Singh1993,Sun2015,Zinkle2013a,Zinkle2013}, where the irradiation dose rates are comparable.}\n\\label{fig8}\n\\end{figure}\n\nBased on the above discussion, a material with enhanced swelling resistance would have a high $\\Delta_0$ but a low $\\alpha$ so that there are a prolonged incubation period and a low growth rate. To quantify this relation, we use the simple parameter ratio of $\\Delta_0\/\\alpha$ as a figure of merit to characterize the relative swelling resistance of materials at comparable dose rates. With this definition, larger values of $\\Delta_0\/\\alpha$ correspond with better swelling resistance in a material. A summary of $\\Delta_0\/\\alpha$ values for different materials under neutron irradiation is demonstrated in Figure 8(d), together with the temperature at which $\\Delta_0\/\\alpha$ is obtained. Among all the materials, Cu exhibits the lowest $\\Delta_0\/\\alpha$ due to its poor swelling resistance. On the other hand, MA957, a type of ODS (Oxide Dispersion Strengthened) steel, possesses the highest $\\Delta_0\/\\alpha$ among the materials examined, suggesting that its excellent ability to suppress swelling is due to the high density of nano-particles that provide numerous defect sinks to absorb the mobile vacancies. For the same material AISI316, the curve of $\\Delta_0\/\\alpha$ with the temperature at reactor-relevant dose rates is in accordance with the experimental swelling peak at about 510\u00b0C. Note that $\\Delta_0\/\\alpha$ also depends on the dose rate, as shown in Figure 8(c); the comparison in Figure 8(d) is made at comparable dose rates. It also can be seen from Figure 8(c) that a smaller $k'$ corresponds to a larger $\\Delta_0\/\\alpha$ at a given dose rate, indicating that an alloy with higher intrinsic sink strength exhibits better swelling resistance. \n\nHelium, which can be produced through transmutation reactions under fast fission and fusion neutron irradiation, is of great concern for nuclear materials. The results shown in Figure 5(e) and (f) suggest that ADAM can be used to describe the effects of He on swelling. As the migration of He to sinks, e.g. grain boundaries and dislocations, will reduce sink capture efficiency \\cite{Dai2020,li2019radiation,farrell1980experimental}, thus reducing sink absorption rate $S_{v0}$, increasing parameter $\\alpha=\\frac{4\\pi}{3V_0}\\left[2\\Omega \\frac{D_v b-D_i}{b\\eta S_{v0}}\\right]^{3\/2}$, reducing threshold dose  $\\Delta_0=\\frac{r_0^2}{2\\Omega} \\frac{(1+ \\frac{\\gamma_v \\delta_v}{\\eta S_{v0}} G_{dpa} )b\\eta S_{v0}}{D_v b-D_i}$, and facilitating swelling in materials (as indicated by the decrease of $\\Delta_0\/\\alpha$ for \"F82H\" to \"F82H+He\" as shown in Figure 8(d)). \n\n\n\\subsection*{Implication of the prediction neutron irradiation from heavy-ion irradiation }\n\nBy introducing the irradiation-generated additional defect sinks, we provide a simple unified model (ADAM) to describe the volume swelling in alloys irradiated by different energetic particles with varying dose rates at a given temperature. These additional sinks come from the near-by and co-existing defects at high dose rates, enhancing local defect absorption, compensating for the effects of high dose rates, and leading to fewer mobile defects that contribute to volume swelling. This is consistent with the general observation that the efficiency of producing swelling is lower for high-dose-rate than low-dose-rate irradiations (shown in Figure 6), which highlights the role of the additional sinks in influencing volume swelling when the dose rates vary by several orders (e.g., heavy-ions vs. neutrons). To provide further validation of our model, we have simulated void swelling in the framework of phase-field modeling, as provided in the Supplement materials. The modeling results are consistent with our ADAM analysis, where swelling grows steadily with increasing doses. Inspiringly, the swelling relation with irradiation dose can be well modeled by our ADAM formulae. Besides, the phase-field results reproduce the suppressed swelling trend with an increasing dose rate, in line with our ADAM prediction and experiments. Therefore, these integrated computational efforts shed light on the swelling mechanism in materials under different irradiation conditions, which serve as important guidelines for designing swelling-resistant structural materials. \n\nAt the same swelling level, our model indicates that by selecting suitable experimental conditions, heavy-ions, especially self-ion irradiation, can be used not only to investigate the irradiation damage mechanism but also to predict equivalent neutron irradiation dose at the same volume swelling level. This prediction from ion irradiation to neutron radiation makes heavy-ion irradiation quite important and useful in the fast screening of advanced nuclear structural materials. However, the high dose rate screening results need to be adjusted by a factor to account for dose rate effects. From Eq. (5), we have the factor  $\\Delta_{ions}\/\\Delta_{neutrons} =(1+k' G_{ions})\/(1+k' G_{neutrons} )>1$, where $\\Delta_{ions}$ and $\\Delta_{neutrons}$  are doses at the same swelling level for ions and neutrons, respectively. As a consequence, the difference between the dose of heavy-ions and neutrons $(\\Delta_{ions}-\\Delta_{neutrons})=\\Delta_{ions} k'  (G_{ions}-G_{neutrons})\/(1+k' G_{ions} ))$ increases with increasing ion dose ($\\Delta_{ions}$). Therefore, the radiation tolerance of alloys determined solely by heavy-ion irradiation results may be quantitatively overestimated by this factor, e.g., 1.38 for 304L in Figure 6(d) and 2.39 for T91 in Figure 7(c), where the differences in factors are due to different material parameters of $k'$. It means that a dose of 300 dpa or 600 dpa of ion irradiation of T91 is only equivalent to 125 dpa or 250 dpa (5\/12) of neutron irradiation in terms of void swelling, respectively. Our model indicates that in the high dose regime, increasing the irradiation dose of heavy-ions only leads to a limited increase in the equivalent neutron dose. \n\nIn our model, we assume that there is no saturation of void swelling in materials under irradiation, agreeing with most experiments. In principle, when voids are large enough, the void surfaces may become the predominant sink for all types of radiation defects. Besides, there is a possibility of the reduction of network dislocation densities at high doses. In these cases, void swelling may saturate at high irradiation doses. However, most experiments show no saturation of void swelling within a given irradiation dose. Particularly, only under certain conditions is found saturation of swelling. For instance, electron irradiation of stainless-steel samples that are too thin leads to an apparent saturation of swelling, which is actually an artifact of the surface impact. The irradiation of bulk samples, on the other hand, gives rise to steady swelling without saturation. To validate our model, we have chosen the experimental data that mostly show no saturation of void swelling under irradiation. Accordingly, the total sink absorption rate bias (k) is constant in our model. Therefore, our model cannot be used to predict swelling in the saturation regime. Overall, experimental evidence for a high-dose swelling saturation regime is very limited, and convincing evidence for swelling saturation did not emerge until ~100 dpa for neutron-irradiated Fe-Cr-Ni alloys\\cite{Garner2000} or ~1000 dpa for ion irradiated T91.\\cite{gigax2016radiation} Our model is proved to be applicable at such high doses.\n\nOur additional defect absorption model is application-oriented and is based on a simplified description of irradiation-induced volume swelling in alloys considering point defect production and evolution, such as those in the electron irradiation. We extended ADAM to heavy-ion and neutron irradiation cases, but the detailed in-cascade atomistic processes are not explicitly considered. We also did not consider the details of sink strength from different types of defect sinks since our focus is on the effects induced by the dose and dose rate difference. Besides, we assume that the strength of intrinsic defect sinks is approximately constant during the irradiation. The details of defect forms, such as different types of defect clusters, are also neglected. Therefore, although our model can be used to describe the whole stage of volume swelling and fit the experimental data well at different irradiation conditions, the quantitative prediction of equivalent irradiation dose at different dose rates and from low-dose to high-dose cases may not be used for all irradiation conditions. In ADAM, we have omitted the contribution of thermal vacancies at the void surface. This approximation is reasonable at intermediate temperatures but may not be suitable at high temperatures (such as $T > 0.5 T_{Melting}$). Finally, it's worth noting that it is impractical to build equivalent relations based on the present model for all the irradiation effects produced by neutrons and heavy-ions, such as irradiation-induced hardness, embrittlement and creep. Hence, there is still a high challenge to study and quantify comprehensive materials' performance under high-dose neutron irradiation by virtue of heavy-ion or self-ion irradiation techniques.\n\nIn summary, considering the main characteristics of defect evolution under energetic particle irradiation, this paper constructs an integrated kinetic rate theory model to correlate the volume swelling in alloys under different irradiation conditions. Specifically, our model introduces an additional defect sink in the defect master equations when considering the differences in dose rates of irradiations. With the irradiation-generated additional defect sink, our model provides a unified description of volume swelling at different doses and dose rates, especially for different energetic particles. Based on ADAM, quantitative methods are established to predict irradiation swelling in alloys for the advanced nuclear energy system, from low doses to high doses, from one dose rate to another dose rate. It can particularly be used for predicting the equivalent dose from heavy-ion irradiation to neutron irradiation experiments. In addition, the parameters of $\\Delta_0$ and $\\alpha$ in ADAM well capture the swelling features, and the combined parameter of $\\Delta_0\/\\alpha$ is proposed to quantitatively characterize the relative swelling resistance of alloys for nuclear plants. Therefore, this work provides an innovative practical solution to evaluate the swelling effects of the structural materials in reactor cores and a fast screening way to assess the performance of newly-developed alloys for advanced nuclear energy systems.\n\n\n\\section*{Materials and Methods}\n\nThe governing equations for interstitials and vacancies respectively under electron irradiation, are\n\n\\begin{equation}\n    \\frac{\\partial c_i}{\\partial \\tau}=G_i-R_{iv}c_i c_v-S_i c_i,\n\\end{equation}\n\n\\begin{equation}\n    \\frac{\\partial c_v}{\\partial \\tau}=G_v-R_{iv}c_i c_v-S_v c_v.\n\\end{equation}\n\nIn these two equations, $c_i$ and $c_v$ are the concentration of interstitials and vacancies respectively, and $R_{iv}$ is the recombination rate of interstitials and vacancies. If local equilibrium is reached ($\\tau$ is sufficiently large), meaning steady-state conditions, and assuming $G_i=G_v=G$, the solutions of Eqs. (6) and (7) give the defect concentrations:\n\n\\begin{equation}\n    c_i=\\frac{S_v}{S_i} c_v, c_v=\\frac{2G}{S_v+\\sqrt{4GR_{iv} \\frac{S_v}{S_i}+S_v^2}}\n\\end{equation}\n\nThe steady-state solutions are used in the following volume swelling model since it is generally considered that defects can achieve their equilibrated states quickly compared to the time scale of volume growth \\cite{Was2007}. The classical rate theory for void growth is \\cite{Was2007}:\n\n\\begin{equation}\n    \\frac{dr}{dt}=\\frac{\\Omega}{r}[D_v c_v-D_i c_i]\n\\end{equation}\n\nwhere $r$ is the void radius, and $\\Omega$ is the defect volume. Here we have omitted the concentrations of thermal vacancies in the void surface. After inserting the solutions of $c_{v,i}$, we have\n\n\\begin{equation}\n    \\frac{dr}{dt}=\\frac{\\Omega}{r}[D_v S_i-D_i S_v]\\frac{-1+\\sqrt{1+\\frac{4GR_{iv}}{S_i S_v}}}{2R_{iv}}\n\\end{equation}\n\nWe consider the sink dominant case, i.e.,$\\frac{4GR_{iv}}{S_i S_v}\\ll 1$. In this case, Eq. (10) can be simplified to:\n\n\\begin{equation}\n    \\frac{dr}{dt}=\\frac{\\Omega}{r}[D_v S_i-D_i S_v]\\frac{G}{S_i S_v}\n\\end{equation}\n\nThe solution of this equation is $r=\\left[t \\frac{2G\\Omega}{S_{v0}+\\gamma_v \\delta_v G} (D_v-\\frac{D_i}{b}) \\right]^{1\/2}$. After inserting the assumption $b=\\beta b_0$, the total void-induced swelling can be written as:\n\n\\begin{equation}\n    swelling(\\%)=\\frac{\\Delta V}{V_0}=\\frac{\\frac{4\\pi r^3}{3}-\\frac{4\\pi r_0^3}{3}}{V_0}=\\frac{4\\pi}{3V_0}\\left[ t \\frac{2G\\Omega}{S_{v0}+\\gamma_v \\delta_v G} \\frac{D_v\\beta S_{i0}-D_i S_{v0}}{\\beta S_{i0} } \\right]^{3\/2}-\\frac{4\\pi r_0^3}{3V_0}\n\\end{equation}\n\nConsidering the relations among dose ($\\Delta_{dpa}$, dpa), dose rate ($G_{dpa}$, dpa\/s), and defect generation rate (G) of electrons, $G_{dpa,ele}=\\eta G$, and $\\Delta_{dpa,ele}=G_{dpa,ele} t$, we can rewrite the swelling to\n\n\\begin{equation}\n    swelling(\\%)\n    =\\frac{4\\pi}{3V_0}\\left[ 2\\Omega \\frac{D_v \\beta S_{i0} -D_i S_{v0} }{\\beta \\eta S_{i0} S_{v0} } \\frac{\\Delta_{dpa}}{1+\\frac{\\gamma_v \\delta_v}{\\eta S_{v0}} G_{dpa}} \\right]^{3\/2}-\\frac{4\\pi r_0^3}{3V_0}\n    =\\alpha \\left[ \\frac{\\Delta_{dpa,ele}}{1+k'G_{dpa,ele}}\\right]^{3\/2}-c.     \n\\end{equation}\n\n\n\n\n\\section*{Data Availability}\nAll data generated or analyzed during this study are included in this published article (and its supplementary information files).\n\n\\section*{Code Availability}\nThe Phase-Field Modeling program in this paper are deposited at a website, https:\/\/github.com\/pku-ion-beam\/PFM.\n\n\n\n\\section*{Acknowledgments}\nThe authors thank Dr. Roger Stoller and Dr. Yong Dai for valuable discussions and suggestions. This work was supported by National Science Foundation of China (Grant No. 11935004 and Grant No. 12192280). S. Zhao was support by City University of Hong Kong (Grant No. 9610425).\n\n\n\\section*{Contributions}\nY. W. and S. J. Z. conceived the research. W.G. and S.Z. performed the theoretical investigation. C. W., H. L., Y.S, and J. X. assisted in data collection and analysis. S. Z., Y.W., W.G., C. W. and S. J. Z. prepared the manuscript. All authors discussed the results, commented on the manuscript, and contributed to the writing of the paper.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nFundamental stability of edge states \\cite{Halperin82, Laughlin81, Hatsugai93-edge} in the quantum Hall\neffect (QHE)\nimplies the phase\nis topological.\nThe bulk is gapped but, with boundaries, there exists localized modes\nin the gap.\nConversely non trivial topological phases are\nnecessarily associated with  edge states as generic localized modes\nnear the boundaries of the system or impurities.\nIt applies to any of the topological phases \nthat include the fractional QH states,\nquantum spin Hall states as the topological insulators, the\nHaldane phase of integer spin chains and so on.\nGenerically emergence of the edge states\nis reduced to the feature of the bulk,\nthat  is the bulk-edge correspondence \\cite{Hatsugai93-bec}.\n\n{\\color{black} The phase is ``topological''\nimplies it is }``hidden''\\cite{Wen89}, in the sense that\none can hardly observe its character in a bulk\\cout{\\cite{Wen89}},\neven though the phase\nis  characterized by a topological invariant\\cite{Thouless82,KM05,Bernevig06}.\nFurther, \\cout{in most cases} \nthey are\n{mostly} not physical observables. Even in the QHE, whether  the observed\nHall conductance in \nexperiments is of the bulk or of the edges is\nstill controversial.\nThe bulk topological character\nis hidden but the edge states are real observables.\nA typical example\nis a surface Dirac cone of the three dimensional topological insulators,\nthat is observed by \nthe angle resolved photoemission spectroscopies.\n  \\cout{clearly observe the Dirac cones\nas the edge states for some of the semiconductors.}\nThe bulk-edge correspondence is thus\n a rigorous\/conceptual bridge between the\n bulk and with boundaries,\n that was, at first, rigorously shown for the QHE \\cite{Hatsugai93-bec} and\n justified today for the\nothers as well \\cite{Kitaev01,Ryu02,KM05,Bernevig06,Qi06,Schulz00,Graf13}. \n\nThis reverse way of thinking is \\cout{crucially} important since one \ncan realize why the edges are there from a\nuniversal point of view.\nSome of them are historically well-known such as\ndangling bonds of semiconductors and some are new as\nthe chiral edge modes of photonic crystals \\cite{Haldane08,Wang08} and\nphoton's \\cite{Hafezi14}.\n\\cout{It is now clear that we do not need to restrict\nourselves to the quantum world. The photonic crystals are surely governed by\nthe classical Maxwell equation. }\nThis analogy even extends to the \nclassical Newton equation\n\\cite{Prodan09,Kane14,Kariyado15}.\nIt implies the universal feature of the bulk-edge correspondence.\n\nUltimate developments of recent\nquantum technology also trigger further studies of \ntopological phases, such as realization of the Hofstadter butterfly\nby a synthetic gauge field\nin a artificial lattice using cold atoms \\cite{Dalibard11}.\nMany of the theoretical ideas for the topological phases and gauge structures\nof matter \nare now directly measured experimentally where\nquantum coherence and structures\nare under the ultimate control and one can handle or\neven synthesize \ndimensions \\cite{Dalibard11,Fallani15}.\nOne of the most clear and fundamental examples is\na topological pump proposed long time ago \\cite{Thouless83},\nwhere the time is used as a synthetic dimension and the topological\neffects in the QHE in 2D is realized in a simple one dimensional system.\nEven though the proposal is quite old as the QHE,\nit took so long time until its experimental realization \n  in cold atoms \\cite{Nakajima15,Lohse15}, \nand intense studies have just began.\nAlthough the bulk-edge correspondence is fundamental in\ntopological phases, it \\cin{has never been} applied to this topological pump.\n\\cin{Here we have first made\n  the role of the edge states clear in the topological pump.}\n{\\color{red}\\sout{\nWe here revisit the problem and clarify\nthe bulk-edge correspondence in the\ntopological pump.\n  Surprisingly it provides conceptually new aspects of the\n  bulk-edge correspondence, that is again closely related to the experiments.}}\n\\cin\n  In this letter, a new expression of the pumped charge\nis derived  \n  by the Berry connection for the system with boundaries. \n  In the adiabatic limit, \n  the contribution of the the bulk and the edge are \n  clearly separated.\n  The pumped charge is carried by the bulk but\n  its quantization is guaranteed by the locality of the\n  edge states. \n\\csoutf{Although the topological pumping is redesciption of the\n  quantum Hall effect in 2D, physical contents of the bulk-edge correspondence\n  is quite different.} Local $U(1)$ gauge symmetry as the charge conservation\n  is crucially important but plays a role \\cinf{different from the QHE. \nNamely, in the QHE, the right- and left-edge states are always paired on the Fermi surface, \nwhereas in the pumping, they can be\nseparately observed at different times.\nPhysically this is a fractionalization of the electron into the massive\nDirac fermions carrying half charge quantum.\n}\n}\n\nLet us consider a many body\ntopological pump by\nthe time dependent one dimensional hamiltonian of\nlattice free fermions of $L_x$ sites\n\\begin{eqnarray*}\n  H(\\theta,t )\n  = \\sum_{j}^{L_x}\\big[ \n  -t_x^\\theta c_{j+1}^\\dagger c_j +h.c.\n  + v_j(t) c_j ^\\dagger c_j \\big], \n\\end{eqnarray*}\n($t_x^\\theta=t_x  e^{-{\\mathrm i}\\theta\/L_x}, t_x,t_y\\in\\mathbb{R}$. \nThe twist $\\theta$ is introduced to\ndefine the current operator as well as the Berry connection\n for a generic many-body state).\nThe time dependent potential $v_j(t)$ can be any {\\color {black} if it is periodic as \n$v_{j+q}(t)=v_j(t)$ where $q$ is a positive integer.}\nFor simplicity, we choose as\n$  v_j(t) =   -2t_y  \\cos (  \\frac {2\\pi t}{T}- 2\\pi \\phi j)  $,\n  $  \\phi =p\/q$\n  with mutually prime $p$ and $q$. This is equivalent to the 2D QHE under\n  a periodic potential\\cite{Thouless82,Hatsugai93-edge,Hatsugai93-bec}.\n  When mapping back to the QHE in 2D, the time as a synthetic dimension\n  corresponds to\n  the momentum in $y$ direction\n  ($\\frac {2\\pi t}{T}\\to k_y  $\\cite{Hatsugai93-edge}).\n  By writing the  one particle state as\n  $|\\psi \\rangle  =\\sum_j\\psi_j c_j ^\\dagger | 0 \\rangle $, the wave function $\\psi_j$\n  satisfies the Harper equation,\n  $-t_x^\\theta \\psi_{j-1\\remove{,\\ell_L}}-2 t_y \\cos (2\\pi\\frac {t}{T}- 2\\pi\\phi j )\\psi_{j\\remove{,\\ell_L}}-(t_x^\\theta)^* \\psi_{j+1\\remove{,\\ell_L}}=E_{\\ell_L}\\psi_{j\\remove{,\\ell_L}}$.\n  The time-reversal (TR) is broken by a finite $\\theta$ \n  but  is \n  recovered by taking the  $\\theta\\to 0$ limit after\n  the calculation. The pumping period, $T$, controls the\n  adiabaticity of the pumping.\n  For simplicity, we take $T=2\\pi$ and control the adiabaticity\n \n  by the energy scale of $t_x$ and $t_y$.\n  Two boundary conditions are discussed.\n  One is the open boundary condition with edges, $\\ed$: $\\psi_0=\\psi_{L_x}=0$ and the other is\n  periodic one  for the bulk $\\bl$: $\\psi_{j+L_x}=\\psi_j$.\n  The many body eigen state of the snap shot hamiltonian $H(t)$ is given\n  by specifying a set of occupied  one particle states $\\{\\ell_L\\}$\n  as\n  $|\\alpha \\rangle =\\prod_{\\ell_L}(\\sum_j\\psi_{j,\\ell_L}c_j ^\\dagger |0 \\rangle $.\n  \n  Using the current operator\n  \\crep\n      {\n        $\n        J\n  = \\frac {1}{L_x} ({\\mathrm i} \\frac {t_x}{\\hbar } e^{-{\\mathrm i} \\theta\/L_x})\\sum_j  c_{j+1} ^\\dagger c_j + h.c\n= \\hbar ^{-1} \\partial_\\theta H(\\theta )\n$,\n  }\n  {\n  \\begin{eqnarray*}\n    J\n&=&  \\frac {1}{L_x} ({\\mathrm i} \\frac {t_x}{\\hbar } e^{-{\\mathrm i} \\theta\/L_x})\\sum_j  c_{j+1} ^\\dagger c_j + h.c\n= \\hbar ^{-1} \\partial_\\theta H(\\theta,t ),\n  \\end{eqnarray*}\n  }\n  the measured current at the time $t$, \n  $  \\delta j = \\langle G(t)|J|G(t) \\rangle - \\langle g(t) | J| g(t) \\rangle $,\n  is evaluated by the adiabatic approximation \\cite{Thouless83}\n  assuming a finite energy gap above the snap shot\n  ground state $|g(t) \\rangle  $,\n  $H(\\theta,t )|g \\rangle = | g \\rangle E$. The state $|G(t) \\rangle  $ is a\n  true many body state \n  that obeys the time dependent Schr\\\"odinger equation, ${\\mathrm i} \\hbar \\partial _t |G(t) \\rangle =H(\\theta,t )|G(t) \\rangle $. It reads\n\n \\delta j\n  =\n \n  -{\\mathrm i} B, \\ B=\n\\partial _\\theta A_t      -\\partial _t A_\\theta \n \n  $\n  where $B$ is a field strength of the Berry connection $A_\\mu = \\langle g|\\partial_\\mu g \\rangle $, $\\mu =\\theta,t $.\n  Note that $ \\langle g(t) | J| g(t) \\rangle| _{\\theta =0}=0$ due to the TR invariance.\n  Since the field strength $B$ is gauge invariant for the gauge transformation $| g ^\\prime \\rangle = |g \\rangle e^{{\\mathrm i} \\chi}$,\n  $  A ^\\prime _\\mu  = \\langle g ^\\prime | \\partial _\\mu g ^\\prime  \\rangle\n  =A_\\mu +{\\mathrm i} \\partial _\\mu \\chi\n  $, let us take a temporal gauge by imposing a gauge condition,\n  $A^{(t)}_t=0$, which is quite useful for the discussion of the pumping.\n\\cin{Then} the field strength is given as\n $B = -\\remove{{\\mathrm i} }\\partial _t A^{(t)}_\\theta $.\nNow we have a pumped charge between the time period $[t_a,t_b]$ as\n\\begin{eqnarray*}\n \\Delta Q_{[t_a,t_b]}\n &=&  \\int_{t_a}^{t_b} dt \\, \\delta j_x   = {\\mathrm i}  \\int_{t_a}^{t_b} dt \\, \\partial _t A^{(t)}_\\theta\n ={\\mathrm i} A^{(t)}_\\theta(t)\\bigg|^{t_b}_{t_a}, \n\\end{eqnarray*}\nwhere the integration over $t$ has been carried out but it\nneeds a special care, as discussed later.\nFor any (regularly) gauge fixed many body state $| g \\rangle   $, \nthe state in the temporal gauge is\ngiven by\n$\n  | g ^{(t)}(\\theta,t)\\rangle = | g (\\theta,t)\\rangle e^{{\\mathrm i} \\chi(\\theta,t)}\n$,\n  with  the phase factor, $\\chi$, that is path dependent and is  explicitly\n  given as\n  \\crep{\n$\n  \\chi (\\theta,t)=- {\\rm Im\\,} \\int_C ds_\\mu \\langle g | \\partial _\\mu g \\rangle   = \n  {\\mathrm i} \\int_0^t d \\tau\\, \\langle g (\\theta,\\tau)\n  | \\partial _\\tau g (\\theta,\\tau) \\rangle\n \n+  {\\mathrm i} \\int_0^\\theta d \\vartheta\\, \\langle g (\\vartheta,0)\n  | \\partial _\\vartheta g (\\vartheta,0) \\rangle\n  $ where the path is  piecewise linear, $C:(0,0)\\to (\\theta,0 )\\to(\\theta,t)$.\n  }\n       {\n$\n  \\chi (\\theta,t)=\n  {\\mathrm i} \\int_0^t d \\tau\\, A_t(\\theta ,\\tau )\n \n+  {\\mathrm i} \\int_0^\\theta d \\vartheta\\, A_\\theta (\\vartheta,0)$.\n         }\nIt surely satisfies the gauge condition, \n\\crep{\n  $  A^{(t)}_t  (\\theta,t)= \\langle g^{(t)}| \\partial _t g^{(t)} \\rangle  =0$\n}\n     {  $  A^{(t)}_t  (\\theta,t)= 0$     }\nand\none has\n\\crep{\n\\begin{eqnarray*}\n  A^{(t)}_\\theta &&(\\theta,t) = \\!\\langle g| \\partial _\\theta  g \\rangle ^{(t)}\n=   \n  \\langle g (\\theta,t)| \\partial _\\theta g  (\\theta,t)\\rangle\n  \\\\  &&\n  -\\partial _\\theta \n \\int_0^t d \\tau\\, \\langle g (\\theta,\\tau)  | \\partial _\\tau g (\\theta,\\tau) \\rangle\n-\\langle g (\\theta,0)  | \\partial _\\theta g (\\theta,0) \\rangle.\n\\end{eqnarray*}\n}\n     {\n\\begin{eqnarray*}\n  A^{(t)}_\\theta (\\theta,t) &=& \nA_\\theta (\\theta,t)\n  -\\partial _\\theta \n \\int_0^t d \\tau\\, A_t (\\theta,\\tau)   \n- A_\\theta (\\theta,0).\n\\end{eqnarray*}\n       }\nThis is\ngauge invariant:  it is directly confirmed but  is clear in a discretized form \\cite{Fukui05}\nas shown in Fig.\\ref{fig:t-gauge}.\nSubstituting this into $\\Delta Q$ above, we have  a  pumped charge  in\na {\\it novel} gauge invariant form.\n\\begin{figure}[h]\n\\includegraphics[width=80mm]{fig1n.eps\n\\caption{\\label{fig:t-gauge} \n  (a) \n  The Berry connection $A^{(t)}$ in the temporal gauge\n  and (b) its lattice analogue \\cite{Fukui05}.}\n\\end{figure}\nThe discussion up to this point is general and applicable  for both \nwith\/without edges.\n\nNow let us consider a system with edges\nby imposing the open boundary condition.\nIn this case, the twist of the hopping $e^{-{\\mathrm i} \\theta\/L_x}$ is gauged out by the many body gauge transformation\n\\begin{eqnarray*}\n  {\\cal U}(\\theta) &=& \\prod_{j=1}  e^{-{\\mathrm i} \\theta n_j (j-j_0)\/L_x},\n  \\quad j_0=L_x\/2,\n\\end{eqnarray*}\nwhich operates for the fermion operator as $  {\\cal U} c_j {\\cal U}  ^\\dagger = e^{{\\mathrm i} \\theta j\/L_x}c_j$ and\nwe have\n$| g(\\theta )\\rangle = {\\cal U}|g_0 \\rangle  $\nwhere  $|g_0 \\rangle $ is a snap shot ground state of the hamiltonian $H(0,t)$.\nThen noting that $A_t=\\langle g| \\partial _t g \\rangle\n=\\langle g_0| \\partial _t g_0 \\rangle $ is $\\theta  $ independent\nand\n$\n  A_\\theta = \n  \\langle g_0|{\\cal U}^\\dagger   \\partial _\\theta {\\cal U} | g_0 \\rangle\n$, \n  the Berry connection {\\it in the temporal gauge }\n  for the system with edges is\n  \\crep\n  {\n\\begin{eqnarray*}\n  A^{(t),\\ed}_\\theta \n  &=& -{\\mathrm i} \\Delta   P(t),\\ \n  P(t)  = \n \\sum_j \\frac {j-j_0}{L_x}  \\langle g_0(t)|  n_j  | g_0 (t)\\rangle,\n  \\label{eq:CM}\n\\end{eqnarray*}\nwhere {$P(t)$} is a \\coutn{normalized} CM of $|g_0(t) \\rangle $\nand $\\Delta \\bar P=\\bar P(t)-\\bar P(0)$\n($-1\/2\\le \\bar P\\le 1\/2$). \n  }\n  {\n\\begin{eqnarray*}\n  A^{(t),\\ed}_\\theta \n  &=& -{\\mathrm i} \\big[  P(t)-P(0)\\big],\\ \n   P(t)  = \\sum\\nolimits_j x_j \\rho_j(t)\n  \\label{eq:CM}\n\\end{eqnarray*}\nwhere\n$  \\rho_j(t) = \\langle g_0(t)|  n_j  | g_0 (t)\\rangle$,\n$ x_j =  \\frac {j-j_0}{L_x}  $  and  $P(t) $ is the CM.\nNow we have the pumped charge as\n\\begin{eqnarray*}\n \\Delta Q_{[t_a,t_b]} &=& P(t_b)-P(t_a).\n\\end{eqnarray*}\nNote that this is only well defined for a system with boundaries.\n  }\n  \\crep{It should be  noted }\n  {We also stress } that the \\coutn{normalized} CM derived here is distinct from the Zak phase for infinite systems.\n   \n  We assume\n  \\crep{$\\bar P(t)$}{$P(t)$} is a CM measured \n      for the {\\it snap shot ground state }\n      {\\it in contact with a particle reservoir},  that is, the system\n        is specified by the chemical potential $\\mu  $ and the\n        temperature is sufficiently low.\n\\cout{        \\footnote\n            {{\\color{blue}\n                We expect\n                the discussion here is justified even for a closed\n                system\n                in the limit $L_x\\to \\infty$,\n                since the total number of particles is\n                conserved after the pumping of one cycle.}\n            }.            }\nIt should be  also \n  distinguished from the CM of {\\it the time dependent wave function\n    \\crep{\n      $\n\\sum_j \\frac {j-j_0}{L_x}  \\langle G(t)|  n_j  | G (t)\\rangle,\n$\n    }\n         {\n      $\n\\sum_j x_j  \\langle G(t)|  n_j  | G (t)\\rangle,\n$         }\nwhich is recently observed in real\nexperiments}\n\\cite{Nakajima15,Lohse15,Wang:2013fk_pump}.\nEven  though the fermi energy is in the bulk gap,\nwhen the one particle energy of the edge state coincides\nto the fermi energy, \n{\\it the many body   gap} of the snap shot hamiltonian necessarily closes\nand the edge state becomes suddenly occupied\/unoccupied\n(See Fig.\\ref{fig:example}).\nThis sudden change of the snap shot ground state\ncauses singularities (discontinuities) in $\\crep{\\bar P}{P}$ since the\nedge state is spatially localized and its contribution to\nthe normalized CM is\n$\\pm 1\/2 $ in the limit $L_x\\to \\infty$.\nThis is inevitable since topologically non trivial ground state\nis associated with the edge states passing through the gap.\n      Then labeling the gap closing time's by $t_i$'s ($t_i<t_{i+1}$, $i=1,2,\\cdots$) and\n        dividing\n      the period into the time intervals without the singularities\n      (patch-working in the the time domain),\n     \n      the total pumped charge in a single period $T$\n      is given as\n      \\crep{\n        \\begin{eqnarray*}\n\\Delta {Q^\\ed} \n&=& \\sum_{i} \\int_{t_i^+}^{t_{i+1}^-} dt\\, \\partial _t \\bar P(t)\n=\\sum_{i}  \\big[\\bar P(t_{i+1}^-)-\\bar P(t_{i}^+)\\big]\n\\\\\n&=&\n-\\sum_{i}  \\big[ \\bar P(t_{i}^+)-\\bar P(t_{i}^-)\\big]\n= -\\sum_i \\Delta \\bar P(t_i),\n        \\end{eqnarray*}\n      }\n      {\n    \\begin{eqnarray*}\n\\Delta {Q^\\ed} \n&=& \\sum_{i} \\int_{t_i^+}^{t_{i+1}^-} dt\\, \\partial _t P(t)\n=\\sum_{i}  \\big[P(t_{i+1}^-)-P(t_{i}^+)\\big]\n\\\\\n&=&\n-\\sum_{i}  \\big[ P(t_{i}^+)-P(t_{i}^-)\\big]\n= -\\sum_i \\Delta P(t_i),\n    \\end{eqnarray*}\n    }\n    where $t_i^\\pm\\equiv t_i\\pm 0$ and \n    $\\Delta {P}(t_i)\\equiv P(t_i^+)-P(t_i^-)$ is a discontinuity of the CM at $t_i$.\n    Also we used the periodicity of the CM in the adiabatic limit\n    $P(t+T)=P(t)$, which follows from the periodicity of the hamiltonian\n    $H(t+T)=H(t)$. \n    This simple expression can be understood as a bulk-edge correspondence\n    in time domain.\n    We put ''$\\ed$'' to mark the expression for the system with edges.\nNote here that {\\it $P$ is only well defined with edges} and can not be\ndefined for a system with periodic boundary condition.\nDepending on the position of the edge state that cuts the fermi energy and the way it does, this discontinuity is determined as\n      \\begin{eqnarray*}\n        \\Delta P(t_i) &=&\n\\left\\{\n\\begin{array}{ccc}\n-  1\/2 & \\text{Right}:& \\text{become\\ unoccupied}\n  \\\\\n+  1\/2 & \\text{Right}:& \\text{become\\ occupied}\n  \\\\\n+  1\/2 & \\text{Left}:& \\text{become\\ unoccupied}\n  \\\\\n-  1\/2 & \\text{Left}:& \\text{become\\ occupied},\n\\end{array}\n\\right.\n      \\end{eqnarray*}\n      where ``Right'' implies the edge state is localized near the boundary $j=L_x$ and the ``Left'' is for the edge states near the boundary $j\\sim 0$.\n\n      Since the pair of the discontinuity coincides to the winding of the\n      corresponding edge state energy around the ``hole'' (that corresponds to\n      the energy gap) on the Riemann surface \\cite{Hatsugai93-edge},\n      total discontinuity is given by the winding number\n      $I_M$ of the edge states\n      \\begin{eqnarray*}\n\\sum_i \\Delta P(t_i) &=& -I_M,\\qquad      \\Delta {Q^\\ed}  = I_M,\n    \\end{eqnarray*}\n      where we assume that the fermi energy is in the $M$-th energy gap\n      from below. This algebraic definition of the winding number is\n      only possible for the Harper equation, that is, only for special\n      form of the $v_j(t)$.\n      However \n      the relation  is generically justified\n      by defining the winding number $I_M $ \n      as the number of paired  edge states \n      with suitable sign depending on the direction of the crossing of\n      the spectral flow with the fermi energy.\n      This is a modified Laughlin argument \\cite{Laughlin81}\n      which is widely used for\n      the\n      topological number of edge states for\n      various topological phases\\cite{Hatsugai93-edge,KM05,Qi06}.\n      \\cin{\n       \n        Since the total number of particles is conserved,\nthe        gapless times \n$t_i$'s that correspond to the\n(dis)appearance of the edge state\n        are paired (irrespective to the position).\n        It guarantees the quantization of the total pumped charge\n        $\\Delta   Q^\\ed$ as an integer since even number of additions\n        of $\\pm 1\/2$ is an integer.\n        It is a consequence of the local\n        $U(1)$ gauge symmetry as the Laughlin argument,\n        \\cinf{but has sharp contrast to the QHE in which\n          \\sout{the discontinuities $\\Delta P$ }\n               \n                 contribution of the edge states to the Hall conductance}\n          are always integers  in the process of a unit flux penetration, since the right\/left edge states are always paired on the Fermi surface.}\n        This $\\pm 1\/2$ contribution can be understood as a {\\em fractionalization}\n        of electrons into massive Dirac fermions\\cite{Niemi,spec}.\n\n      }\n\n      Also\ncounting the topological number \\coutn{by the spectral flow}\n      with discontinuities here should be compared with the counting of the\n       singularities of the $\\eta$-invariant for the Atiyah-Patodi-Singer index\n       theorem \\cite{PSP:2128256,Alvarez85}.\n       \\cin\n         Here we have clarified the close inter-relation between\n         the topological nature of the discontinuities\n         and local $U(1)$ gauge symmetry. This is not just a math but \\sout{is also experimentally proved recently}\n          plays a crucial role in the recent experiments\\cite{Nakajima15,Lohse15}. \\remove{It is a surprise.}}\n\n      As an example,\n      we calculated the \\coutn{normalized} CM, $P(t)$,\n      numerically for\n      $\\phi=2\/7$ and $L_x=1750$ and $L_x=350$.\n      By the clear finite size effects, the discontinuities deviates from\n      $\\pm 1\/2$ but approaches to the quantized values $\\pm 1\/2$ by the limit\n        $L_x\\to\\infty$.\n      In this  case, we have $\\Delta Q^\\ed=-2$.\n     \n     \n      \n    \\noindent\n    \\begin{figure}[h]\n\\includegraphics[width=80mm]{fig2n.eps\n\\caption{\\label{fig:example} \n  (a)  One particle spectrum $E_{\\ell_L}(t)$\n  \\cout{of the hamiltonian (black lines)} and the\n  fermi energy (blue line)\\cout{.\n{:} $t_x=t_y=1$, $\\phi=2\/7$ and $L_x=1750$\n{(350)}.\n  (b) Enlarged spectrum near the fermi energy.\n  (c) The \\coutn{normalized} CM, \\crep{$\\bar P(t)$}{$P(t)$}, with edges by numerical calculation.\n\\cout{Numerically obtained discontinuities are}$\\Delta \\crep{\\bar P}{P}(t_i)=\n-0.499503 (-0.4954)$,\n$0.496542 (0.487094)$,\n$0.498412 (0.494115)$,\n$0.497435 (0.491092)$,\n$0.497435 (0.491092)$,\n$0.498412 (0.494115)$,\n$0.496542 (0.487094)$,\n and \n$-0.499503 (-0.4954)$\n   for $i=1,\\cdots,8$.\n\\cout{The total discontinuity is} $\\sum_i \\Delta \\crep{\\bar P}{P}(t_i)=1.98577(1.9538)$.\n\\cout{The results for the system $L_x=350$ are shown in the parentheses.}\n}\n    \\end{figure}\n\n    Note that although the total  pumped charge is governed by the\n    discontinuity $\\sum_i\\Delta \\crep{\\bar P}{P}(t_i) $\n    due to the edge states,\n    pumped charge is not carried by singularities caused by\n    the edge states.\n    The charge is still carried \n    by the bulk\n    as we explain below.\n    This is the bulk-edge correspondence in the topological pumping. \n    As one can see in Fig.\\ref{fig:example},\n    the  charge is pumped in the intervals between the singularities\n    (red lines),\nwhich\n    is the bulk contribution.\n    Even though the system has boundaries, the effects of the edges\n    are negligible for the bulk state since the one particle state of\n    the bulk is extended and the amplitude near the boundaries is vanishing in the limit $L_x\\to \\infty$.\n\n\n    \n    Although the \\coutn{normalized} CM is ill-defined for bulk (both for\n      a periodic\/infinite system),\n    the pumped charge is well-defined as discussed by Thouless \\cite{Thouless83}.\n    As for an infinite system, the one particle state is given by the Bloch state\n    \\cout{as}\n    $\\psi_{j,\\ell_L}\\propto e^{{\\mathrm i} k_x j}u_j(k_x)\n{.}\\cout{,\n    $u_{j+q}(k_x)=u_{j}(k_x) $, \n    where $u_j(k_x)=u_{j+q}(k_x)$ satisfies a reduced $q$ site problem\n    replacing the twist $\\theta \/L_x$ by $ k_x$.\n    The translational invariance $\\psi_{j+q}(k_x)=e^{{\\mathrm i} k_x q}\\psi_j(k_x)$ implies\n    $k_x\\in[0,\\Delta k]$, $\\Delta k=2\\pi\/q$.\n    The one particle states for the large periodic system (bulk) $L_x=q l$ ($l\\gg 1$) are all reconstructed from  the Bloch state as\n$      \\psi_{j,\\ell}^n \n      =\n      e^{{\\mathrm i} \\frac { n\\Delta k}l \\cdot j}u_{j,\\ell}(k_x)\/\\sqrt{l}\n$ where\n$    k_x\n    = \\frac {\\Delta k}{l}(n+ \\frac {\\theta}{2\\pi})\\in[\\frac {n\\Delta k}l,\\frac {(n+1)\\Delta k}{l}) ]$,\n($    n = 0,1,\\cdots, l-1$) and \n$   \\ell = 1,\\cdots, q$\n is a band index. }\n    \\cout{\n      Then the many body ground state of the snap shot hamiltonian when the fermi energy is in the $M$-th gap  is given by\n$\n  | g \\rangle = \\prod _{\\ell=1}^M\\prod _{n=0}^{l-1}\n( \\sum_{j=1}^{L_x} \\psi_{j,\\ell}^n  c_j ^\\dagger )|0 \\rangle\n  $ and the Berry connection for the $\\theta $ direction is given by that of the $U(1)$ part of\n  the Berry connection of the filled fermi sea.\n    }\n   \n\n        {\n      Now let $|g^{(t)} \\rangle  $ be a many body ground state of the snap shot hamiltonian in the temporal gauge.\n          }\n\\cout{  By using  the temporal gauge of the Bloch state,\n  one}\n{Assuming the fermi energy is in the $M$-th gap,  one} has\n  \\begin{eqnarray*}\n  A^{(t),\\bl}_{\\theta } &=&   \\langle g^{(t)} | \\partial_\\theta  g^{(t)} \\rangle\n  =  \\int_0^{\\Delta k} \\frac {dk_x}{2\\pi}    \\, a^{(t)}_{k_x},\n  \\\\\n  a^{(t)}_{k_x } &=&     {\\rm Tr} _M {\\cal A}^{(t)} _{k_x },\\\n  {\\cal A}^{(t)} _{k_x } = u ^\\dagger \\partial_{k_x } u,\\ u= (\\bm{u} _1,\\cdots,\\bm{u} _M),\n  \\end{eqnarray*}\n  where \n  the limit $L_x\\to\\infty$ is taken and additional gauge condition\n  $a^{(t)}_t=   {\\rm Tr} _M u^\\dagger   \\partial_t  u  =0$ is impose\n{.}\n\\cout{  for the $U(1)$ part of the Bloch states $u$.}\nAlso we put ``$\\bl$'' to specify that it is purely from bulk.\n  By using the gauge invariant form of the temporal gauge,  \n    \\begin{eqnarray*}\n      a^{(t)}_{k_x} (k_x,t)&=& a_{k_x}(k_x,t)- \\partial _{k_x}\\int_0^t d\\tau\\, a_t(k_x,\\tau) - a_{k_x}(k_x,0),\n    \\end{eqnarray*}\n    the time derivative\n    of $A^{(t),\\bl}_\\theta $ is written by\n    the field strength of the Bloch state,\n    $b=\\partial _{k_x} a_t-\\partial _t a_{k_x} =-\\partial _t a^{(t)}_{k_x}\n{.}\n    \\cout{\n, as      \n    \\begin{eqnarray*}\n\\partial _t  A^{(t),\\bl}_{\\theta } =\n  \\int_0^{\\Delta k} \\frac {dk_x}{2\\pi}\n\\partial _t   a^{(t)}_{k_x} (k_x,t)\n= \n-\\int_0^{\\Delta k} \\frac {dk_x}{2\\pi}\nb(k_x,t).\n    \\end{eqnarray*}\n    }\n    Now \n    the bulk contribution of the pumped charge\n    between the time period $[t_a,t_b]$ is written as\\cite{Thouless83}\n\\begin{eqnarray*} \n  \\Delta Q^\\bl _{[t_a,t_b]} &=& \n  {\\mathrm i}\\int_{t_a}^{t_b}dt \\partial _t  A^{(t),\\bl}_{\\theta }\n=\\frac{1}{2\\pi{\\mathrm i}}  \\int_{t_a}^{t_b} dt  \\int_0^{\\Delta k}  {dk_x}  b(k_x,t).\n\\end{eqnarray*}\n\\cout{These equations imply}\n{It implies}\nthat \\cout{the average of the\n  temporal gauge of the Bloch function ${\\mathrm i}\/(2\\pi) \\int_0^{\\Delta k}dk_x  \\,  a^{(t)}_{k_x}(k_x,t)$}\n{${\\mathrm i}\\partial _t  A^{(t),\\bl}_{\\theta } $}\nis an {\\it effective CM} \n\\cout{at\n  the time $t$}\n{of the bulk}, even though the CM itself\nis not well defined\\cout{for the bulk}.\n\n\n\nTo demonstrate\nthe contribution of the pumping with edges\nis from  the bulk,\nlet us define a pumped charge between the time period $[t_a,t_b]$\nby compensating (skipping) the singularities from the edges as\n    \\begin{eqnarray*}\n      \\Delta Q^\\ed  _{[t_a,t_b]} \n      &=&\n     \\crep{ \\bar  P}{P}(t_b)-\\crep{\\bar P}{P}(t_a)\n      -\n        \\int_{t_a}^{t_b} d\\tau\\,\n        \\sum_i \\Delta \\crep{\\bar P}{P}(t_i) \\delta (\\tau-t_i).\n      \\label{eq:pumped-e-period}\n    \\end{eqnarray*}\n      This is a bulk contribution to the \\coutn{normalized} CM for\n      the system with edges.\n    According to the consideration here, \n      we  expect this bulk contribution\n      $\\Delta Q^\\ed  _{[t_a,t_b]} $  asymptotically approaches to\n$\\Delta Q^\\bl _{[t_a,t_b]}$ in the large system.\n    We have numerically   evaluated them \n    for $\\phi=2\/7$  and shown in Fig.\\ref{fig:bulk-edge-pump}. They show clear coincidence within numerical errors, indicating the relation between the bulk and the edges.\n\n\n    \n\\begin{figure}[h]\n\\includegraphics[width=70mm]{fig3n.eps\n\\caption{\n  \\label{fig:bulk-edge-pump} \n  \\cout{Total amount of the pumped}\n{Pumped} charge between the time interval $[0,t]$.\n  The red line is by the bulk (Bloch states)\n  $\\Delta Q^\\bl_{[0,t]}$. \n  The shifted pumped charge $\\Delta {Q^\\ed} _{[0,t]}$'s are shown \\cout{by}\n{for} $L_x=140$ (blue)\n  and $L_x=1750$ (black) lines.\n  The parameters are the same as Fig.\\ref{fig:example}.\n}\n\\end{figure}\n{The}\\cout{Further the\n total} pumped charge by bulk in a cycle\nis given by the Chern number $C_M$ \\cite{Thouless83}\nof the Berry connection $a_\\mu $ \nas $\\Delta Q^\\bl=  \\Delta Q^\\bl _{[0,T]}= C_M $.\n\\cout{\n  If we interpret $b$ as\n$b = \\partial _{k_x} a_{k_y}-\\partial _{k_y}a_{k_x}$ ($\\partial _{k_y}\\equiv \\partial _t$, $a_{k_y}\\equiv a_t$ ),\nit is a natural definition of the Chern number in  QHE,\nwhich corresponds to the Chern number of the filled $M$ bands (Bloch states) in the present pumping.\n}\nNow we have established the physical bulk-edge correspondence\nin the topological pumping\nas\n\\begin{eqnarray*}\n  \\Delta Q^\\ed       &=&     \\Delta Q^\\bl      \n\\end{eqnarray*}\n{that} gives the relation between the two topological invariants as was\nalso shown in the QHE\nas\n\\begin{eqnarray*}\nI_M &=&   C_M .\n\\end{eqnarray*}\nIt guarantees quantization of the Chern number $C_M$, since\n$I_M$ due to the singularities of edge states\nis quantized by definition.\n\n\n\n\n{\n \n  \\cin\n    Generically the bulk-edge correspondence is for the\n    manybody state and the bulk and the edge can not be separated.\n    Without gapped bulk, edge states are not well defined.\n   \n    We put a stress on\n    that in the present pumping of the non interacting system, the manybody state\n    is constructed from the one particle states which are classified into\n    the bulk(unnormalizable) and the edge(localized). Then the contributions to\n    the pumped charge are also clearly separated (Fig.\\ref{fig:bulk-edge-pump}).\n  }\n  \n\n{\n    To describe the topological  pumping by the CM,\n    one necessarily needs to use a system with edges\n    in contact with a particle reservoir.\n      As we have established here,\n      the pumped charge of the system with edges\n      is carried by the bulk not by the edge states\n          and the CM observed in the cold atom experiments\n          \\crep{    $P_{{\\rm exp} }(t)=\\langle G(t)|\\sum_j n_j\\frac {j-j_0}{L_x} |G(t) \\rangle $}\n{$P_{{\\rm exp} }(t)=\\sum_j x_j\\langle G(t)| n_j|G(t) \\rangle $}          \n    is described by the bulk.\n  Still the quantization of the pumped charge is governed by the edge states where\n    the edge states induce discontinuities in the \\coutn{normalized} CM \n    in the adiabatic limit.\n      Even though the \\coutn{normalized} CM shows singularities in the adiabatic limit,\n    {\\it\n      these singularities can not be observed in realistic experiments\n      in cold atoms}, since the appearance of the edge states at the fermi energy\n      necessarily implies the breakdown of the adiabaticity.\n      As in an  realistic experiment with\n      finite speed pumping, \n      the many body wave function\n      approximately remains unchanged passing through the\n      gapless point,\n      that is, the bulk is \n      adiabatic but the edge states\n      is well described by the sudden approximation.\n      It causes excitation across the gap, which eventually results in\n      accumulation\/depletion of the surface charges after one cycle.\n      These two conditions, as the bulk is adiabatic and the edge is sudden,\n      are  conflicting requirements, if one requires both limits rigorously.\n      Then in the  experiment,\n      it may not be easy to observe exact quantization of\n      the pumped charge, especially for cases with high Chern numbers.\n      We have also confirmed consistency of the\n      argument\n      by a direct numerical\n    integration of the  Liouville$-$von Neumann equation\\cout{,\n      ${\\mathrm i} \\hbar    \\partial _t \\rho(t)=\\cmt{H(t)}{\\rho(t)}$,}\n      for\n      the density matri\n{.}\\cout{ $\\rho(t)$.}\n\n   \n   \n    \n{\\color{green}\\sout{\n  To summarize,\n      the topological pumping, proposed by Thouless\n      and is observed by\nmeasuring the CM  of the system with edges, is governed\nby the edge states.\nThe bulk is hidden but carries the pumped charge in the experiments\nwith edges.\nThe discontinuities of the \\coutn{normalized} CM in the adiabatic limit\nassociate with the edge states \ncan  not be measured experimentally\nbut guarantees the quantization of the pumped charge \nas the bulk-edge correspondence in the topological pumping.\n\n}}\n\n\n\nWe thank stimulating discussions on the topological pump with Y. Takahashi, S. Nakajima and M. Lohse.\nThe work is supported in part by\nKAKENHI from JSPS\n(Nos.26247064, 25400388,16K13845).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:intro}\n\n\n\nIn medical imaging, the characteristics purely derived from a disease should reflect the extent to which abnormal findings deviate from the normal features that would have existed. Indeed, physicians often need corresponding normal images without abnormal findings of interest or, conversely, images that contain similar abnormal findings regardless of normal anatomical context. This is called comparative diagnostic reading of medical images, which is essential for a correct diagnosis. To support the comparative diagnostic reading, content-based image retrieval (CBIR) utilizing either normal or abnormal features will be useful.\n\n\nHere, we define this two-tiered nature of normal and abnormal features as the {\\it compositionality} of medical images. Subsequently, we consider a method to decompose a medical image into two low-dimensional representations, where the two latent codes representing normal and abnormal anatomies should be collective for reconstructing the original image (Figure 1). \n\nTo our best knowledge, few studies have focused on the compositionality of medical images. Recently, image-to-image translation techniques mainly derived from CycleGAN \\citep{cyclegan8237506} were exploited to disentangle the domain-specific variations of medical images \\citep{XIA2020101719, ADN8788607, vorontsov2019semisupervised}. For example, Tian et al. successfully transformed an input image with pathology into a normal-appearing image by leveraging several cycle-consistency losses \\citep{XIA2020101719}. However, these approaches did not treat the compositionality in a separable manner, i.e., the latent spaces of these methods were not explicitly designed for down-stream tasks.\n\nIn this paper, we propose an encoder-decoder network to project a medical image into a pair of latent spaces, each of which produces a {\\it normal anatomy code} and an {\\it abnormal anatomy code}. Using these latent codes, our CBIR framework can retrieve images by focusing on either normal or abnormal features, providing excellent performances in qualitative evaluations.\n\n\\begin{figure}[t]\n  \\centering\n  \\label{fig:concept_of_segregation}\n  \\includegraphics[width=\\linewidth]{.\/figures\/concept_of_segregation_v2.png}\n  \\caption{Our concept of the compositionality of medical images, indicating that the entire image should be composed of the representation of normal and abnormal features.}\n\\end{figure}\n\n\n\\begin{figure*}[t]\n  \\centering\n  \\label{fig:model_overview}\n  \\includegraphics[width=0.8\\linewidth]{.\/figures\/model_overview.png}\n  \\caption{Schematic of the proposed encoder-decoder network.}\n\\end{figure*}\n\n\n\\begin{figure*}[t]\n  \\centering\n  \\label{fig:cbir_result}\n  \\includegraphics[width=0.8\\linewidth]{.\/figures\/cbir_result_v2.png}\n  \\caption{Example results of content-based image retrieval based on similarities of decomposed latent codes. Images are shown with normal or abnormal anatomical labels that are corresponded to the semantics utilized in the retrieval. Although the image retrieval employed latent codes instead of label information, the retrieved images accompanied similar labels to those of the query images. Patient identifiers with slice numbers are noted.}\n\\end{figure*}\n\n\\section{Methodology}\n\\label{sec:method}\n\nThe proposed network consists of an encoder and two decoders, a segmentation decoder and an image decoder (Figure 2). A pair of discrete latent spaces exists at the bottom that produces normal and abnormal anatomy codes, separately. See ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe pseudoscalar spectrum {[}$I^{G}(J^{PC})=1^{-}(0^{-+})${]} contains three states below 2 GeV: $\\pi(140)$; $\\pi(1300)$; and $\\pi(1800)$.  The lightest has been much studied.  However, a comprehensive understanding of QCD requires an approach that admits the simultaneous study of the heavier pseudoscalars and, indeed, other systems.  An understanding of the hadron spectrum and its realisation within QCD is necessary in order to unravel the nature of the long-range force between light quarks.  \n\nThe lightest pseudoscalar meson is both a bound state of $u$- and $d$- quarks and the Goldstone mode in QCD associated with the dynamical breaking of chiral symmetry.  This is readily and clearly understood using the Dyson-Schwinger equations (DSEs) \\cite{Maris:1997hd,Maris:1997tm}.  The DSEs have proven a particularly useful device for studying the spectrum and properties of light-quark systems.  Modern applications are reviewed in Refs.\\ \\cite{bastirev,reinhardrev,Maris:2003vk}.\n\nHerein we present results obtained from studies of the inhomogeneous Bethe-Salpeter equations (BSEs) for the scalar and pseudoscalar vertices in QCD.  This is a practical means of mapping out the domain of applicability of the leading order term in a systematic and symmetry preserving DSE truncation scheme \\cite{munczek,bender,mandarvertex}.  Furthermore, we explore the capacity of such studies to complement contemporary numerical simulations of lattice-regularised QCD.\n\n\\section{Bound states from spacelike data}\n\\label{sec:Numerical-Tools}\nThe BSE provides a Poincar\\'e covariant tool with which to calculate the properties of bound states in quantum field theory.  The inhomogeneous equation for a pseudoscalar quark-antiquark vertex is\\footnote{For simplicity we work with two degenerate flavours of quarks.  Hence, Pauli matrices are sufficient to represent the flavour structure.  We employ a Euclidean metric with the conventions described, e.g., in Sec.~2.1 of Ref.\\,\\protect\\cite{bastirev}.}\n\\begin{eqnarray}\n\\nonumber \\lefteqn{\\left[\\Gamma_5^j(k;P)\\right]_{tu}= \nZ_{4}\\gamma_{5}\\,\\frac{\\tau^j}{2}  }\\\\\n&&  +\\int_{q}^{\\Lambda}\\left[\\chi_5^j(q;P)\\right]_{sr}K_{rs}^{tu}(q,k;P)\\,,\n\\label{eq:DSE_inhomogeneous}\n\\end{eqnarray}\nwhere $k$ is the relative and $P$ the total momentum of the constituents;\n$r,\\ldots,u$ represent colour, Dirac and flavour matrix indices;\n\\begin{equation}\n\\chi_5^j(q;P) = S(q_{+}) \\Gamma_5^j(q;P) S(q_{-})\\,,\n\\end{equation}\n$q_{\\pm}=q\\pm P\/2$; and $\\int_{q}^{\\Lambda}$ represents a Poincar\\'{e}\ninvariant regularisation of the integral, with $\\Lambda$ the regularisation\nmass-scale \\cite{Maris:1997hd,Maris:1997tm}.  In Eq.~(\\ref{eq:DSE_inhomogeneous}),\n$K$ is the fully amputated and renormalised dressed-quark-antiquark scattering kernel and $S$ is the renormalised dressed-quark propagator. ($SSK$ is a renormalisation group invariant).  The dressed-quark propagator has the form \n\\begin{eqnarray} \n S(p)^{-1}\n& =& \\frac{1}{Z(p^2,\\zeta^2)}\\left[ i\\gamma\\cdot p + M(p^2)\\right] ,\n\\label{sinvp} \n\\end{eqnarray} \nand is obtained as the solution of QCD's gap equation:\n\\begin{equation}\nS(p)^{-1}=Z_{2}\\left(i\\gamma\\cdot p+m_{\\mathrm{bm}}\\right)+\\Sigma(p)\\,,\\label{eq:quark_prop}\n\\end{equation}\n\\begin{equation}\n\\Sigma(p) = Z_{1} \\int_{q}^{\\Lambda} g^{2} D_{\\mu\\nu}(p-q) \\frac{\\lambda^{a}}{2} \\gamma_{\\mu} S(q) \\Gamma_{\\nu}^{a}(q;p)\\,,\n\\end{equation}\naugmented by the renormalisation condition\n\\begin{equation}\n\\label{renormS} \\left.S(p)^{-1}\\right|_{p^2=\\zeta^2} = i\\gamma\\cdot p +\nm(\\zeta)\\,,\n\\end{equation}\nwhere $m(\\zeta)$ is the running current-quark mass at the renormalisation point $\\zeta$.  These equations involve the quark-gluon-vertex, quark wave function and Lagrangian mass renormalisation constants, $Z_{1,2,4}(\\zeta,\\Lambda)$, each of which depends on the gauge parameter, the renormalisation point and the regularisation mass-scale.\n\nThe solution of Eq.\\,(\\ref{eq:DSE_inhomogeneous}) has the form \n\\begin{eqnarray}\n\\nonumber\n\\lefteqn{i \\Gamma_{5 }^j(k;P) = \\frac{\\tau^j}{2} \\gamma_5\n\\left[ i E_5(k;P) + \\gamma\\cdot P \\, F_5(k;P) \\right.} \\\\\n\\nonumber & &\n\\left.+ \\, \\gamma\\cdot k \\,k\\cdot P\\, G_5(k;P)+\n \\sigma_{\\mu\\nu}\\,k_\\mu P_\\nu \\,H_5(k;P) \\right].\\\\\n\\label{genpvv}\n\\end{eqnarray}\nThis is the minimal and complete form required by Poincar\\'e covariance.  The homogeneous pseudoscalar BSE is obtained from Eq.\\,(\\ref{eq:DSE_inhomogeneous}) merely by omitting the \\emph{driving term}; viz., $Z_{4}\\gamma_{5}\\frac{\\tau^j}{2} $.  The equation thus obtained defines an eigenvalue problem, with the bound state's mass-squared being the eigenvalue and its Bethe-Salpeter amplitude, the eigenvector.  As such, the equation only has solutions at isolated timelike values of $P^2$.  On the other hand, the solution of the inhomogeneous equation, Eq.\\,(\\ref{eq:DSE_inhomogeneous}), exists for all values of $P^2$, timelike and spacelike, with each bound state exhibited as a pole.  This is illustrated by Fig.\\,\\ref{fig:DSE_Inverted}, wherein the solution is seen to evolve smoothly with $P^2$ and the pole associated with the pseudoscalar ground state is abundantly clear. \n\n\\begin{figure}[t]\n\\includegraphics[clip,width=0.45\\textwidth]{bse_data.eps}\\vspace*{-4ex}\n\n\\caption{\\label{fig:DSE_Inverted} Pseudoscalar amplitude $E_5(0;P^2)$ obtained by solving the inhomogeneous BSE, Eq.\\,(\\ref{eq:DSE_inhomogeneous}), using the renormalisation-group-improved rainbow-ladder truncation described in \\protect\\cite{mariscairns}.  The vertical dotted line indicates the position of the ground state $\\pi$ mass pole.  The inset shows $1\/E_5(0;P^2)$.\\vspace*{-4ex}}\n\\end{figure}\n\nA numerical determination of the precise location of the first pole in $E_5(k^2=0;P^2)$ will generally be difficult.  The task becomes harder if one seeks to obtain the positions of excited states in addition.  It is for these reasons that the homogeneous equation is usually used.  However, if one is employing a framework that can only provide information which is equivalent to the form of this vertex at spacelike momenta, then a scheme must be devised that will yield the pole positions.  \n\nOne obvious alternative is to focus on $P_E(P^2)=1\/E(k^2=0;P^2)$ and locate its zeros, and it is plain from Fig.\\,\\ref{fig:DSE_Inverted} that this approach can at least be successful for the ground state.  It is important to determine whether this (or another) approach can also be used in practice to determine some properties of excited states when information is available only for spacelike $P^2$.  \n\nWhile the DSEs can be used to generate such information, and the analysis of that information is what brought us to this point, herein for this purpose we consider a simple model for an inhomogeneous vertex whose analytic structure is known precisely; namely, \n\\begin{equation}\nV(P^2)=b + \\sum_{i=1}^M\\frac{a_{i}}{P^{2}+m_{i}^{2}}\\,, \\label{eq:simple_model}\n\\end{equation}\nwhere: for each $i$, $m_{i}$ is the bound state's mass and $a_{i}$ is the residue of the bound state pole in the vertex, which is related to the state's decay constant; and $b$ is a constant that represents the perturbative background that is necessarily present in the ultraviolet.  The particular parameter values we employ are listed in Table\\,\\ref{tab:Model_Param}.   This \\textit{Ansatz} provides a data sample that captures the essential qualitative features of true DSE solutions for colour-singlet three-point Schwinger functions, such as that depicted in Fig.\\,\\ref{fig:DSE_Inverted}.\n\n\\begin{table}[t]\n\\begin{center}\n\\caption{\\label{tab:Model_Param} Parameters characterising our vertex \\textit{Ansatz}, Eq.\\,(\\protect\\ref{eq:simple_model}).  They were chosen without prejudice, subject to the constraint in quantum field theory that residues of poles in a three-point Schwinger function must alternate in sign \\cite{Holl:2004fr}, and ordered such that $m_i<m_{i+1}$.  We use $b=0.78$.  This is the calculated value of $Z_4(\\zeta = 19\\,{\\rm GeV},\\Lambda=200\\,{\\rm GeV})$ used to obtain the curves in Fig.\\,\\protect\\ref{fig:DSE_Inverted}.}\n\n\\begin{tabular*}{0.45\\textwidth}{\n|c@{\\extracolsep{0ptplus1fil}}|l@{\\extracolsep{0ptplus1fil}}|l@{\\extracolsep{0ptplus1fil}}|}\\hline\n$i$~ & Mass & Residue \\\\\\hline\n1& 0.14& ~4.23\\\\  \n2& 1.06& -5.6 \\\\\n3&1.72& ~3.82\\\\  \n4&2.05 & -3.45 \\\\  \n5&2.2& ~2.8 \\\\\\hline\n\\end{tabular*}\\vspace*{-4ex}\n\\end{center}\n\\end{table}\n\nTo proceed, we employ a diagonal Pad\\'e approximant of order $N$ to analyse the data sample generated by Eq.\\,(\\ref{eq:simple_model}); viz., we use\n\\begin{equation}\nf_{N}(P^{2}) = \\frac{c_{0}+c_{1}P^{2}+\\ldots+c_{N}P^{2N}} {1+c_{N+1}P^{2}+\\ldots+c_{2N}P^{2N}}\n\\label{eq:pade}\n\\end{equation}\nas a means by which to fit $1\/V(P^2)$.  The known ultraviolet behaviour of the vertex requires that we use a diagonal approximant.  NB.\\ A real-world data sample will exhibit logarithmic evolution beyond our renormalisation point.  No simple Pad\\'e approximant can recover that.  However, this is not a problem in practical applications because the approximant is never applied on that domain.\n\nIn a confining theory it is likely that a colour-singlet three-point function exhibits a countable infinity of bound state poles.  Therefore no finite order approximant can be expected to recover all the information contained in that function.  Our vertex model exhibits $M$ bound state poles.  We expect that an approximant of order $N<M$ will at most provide reliable information about the first $N-1$ bound states, with the position and residue associated with the $N^{\\rm th}$ pole providing impure information that represents a mixture of the remaining $M-(N-1)$ signals and the continuum.  We anticipate that this is the pattern of behaviour that will be observed with any rank-$N$ approximation to a true Schwinger function.  To explore this aspect of the procedure we studied the $N$-dependence of the Pad\\'e fit.  \n\nThe domain of spacelike momenta for which information is available may also affect the reliability of bound state parameters extracted via the fitting procedure.  We analysed this possibility by fitting Eq.\\,(\\ref{eq:pade}) to our \\textit{Ansatz} data on a domain $(0,P_{\\mathrm{max}}^{2}]$, and studying the $P_{\\mathrm{max}}^{2}$-dependence of the fit parameters.\n\nIn all cases we found that a Pad\\'e approximant fitted to $1\/V(P^2)$ can accurately recover the pole residues and locations associated with the ground and first excited states.  However, there is no cause for complacency. \n\n\\begin{figure}[t]\n\\includegraphics[clip,width=0.45\\textwidth]{masses-3.eps}\\vspace*{-4ex}\n\n\\caption{\\label{fig:3pade_mass} Pole positions (mass values) obtained through a fit of Eq.\\,(\\protect\\ref{eq:pade}) with $N=3$ to data for $1\/V(P^2)$ generated from Eq.\\,(\\protect\\ref{eq:simple_model}) with the parameters listed in Table \\ref{tab:Model_Param}.  The coordinate $P_{\\mathrm{max}}^{2}$ is described in the text. Horizontal dotted lines indicate the three lightest masses in Table \\ref{tab:Model_Param}.  The ground state mass (solid line) obtained from the Pad\\'e approximant lies exactly on top of the dotted line representing the true value.\\vspace*{-4ex}}\n\\end{figure}\n\nIn Fig.\\,\\ref{fig:3pade_mass} we exhibit the $P_{\\mathrm{max}}^{2}$-dependence of the mass-parameters determined via a $N=3$ Pad\\'e approximant.  Plateaux appear for three isolated zeros, which is the maximum number possible, and the masses these zeros define agree very well with the three lightest values in Table \\ref{tab:Model_Param}.  This appears to suggest that the procedure has performed better than we anticipated.  However, that inference is seen to be false in Fig.\\,\\ref{fig:3pade_residue}, which depicts the $P_{\\mathrm{max}}^{2}$-dependence of the pole residues.  While the results for $a_{1,2}$ are correct, the result inferred from the plateau for $a_3$ is incorrect.  It is important to appreciate that if we had not known the value of $a_3$ \\textit{a priori}, then we would very likely have been misled by the appearance of a plateau and produced an erroneous \\emph{prediction} from the fit to numerical data.  Plainly, a $N=3$ approximant can at most only provide reliable information for the first $M-N=2$ bound states.  \n\nWe explored this further and applied a $N=4$ approximant to the same model.  In this case we could still only extract reliable information for the first two poles.  We subsequently biased the fit procedure by \\emph{hard-wiring} in Eq.\\,(\\ref{eq:pade}) the residue and position of the lightest pole masses.  This did not help.  The $N=4$ approximant could still not provide results that improved upon what we had already learnt with the $N=3$ approximant.  \n \n\\begin{figure}[tb]\n\\includegraphics[clip,width=0.45\\textwidth]{residues-3.eps}\\vspace*{-4ex}\n\n\\caption{\\label{fig:3pade_residue} Pole residues obtained through a fit of Eq.\\,(\\protect\\ref{eq:pade}) with $N=3$ to data for $1\/V(P^2)$ generated from Eq.\\,(\\protect\\ref{eq:simple_model}) with the parameters in Table \\ref{tab:Model_Param}.  Horizontal dotted lines indicate the residues associated with the three lightest masses in Table \\ref{tab:Model_Param}.  The residue associated with the ground state (solid line) lies exactly atop the dotted line representing the true value.  The residue for the second pole exhibits a plateau at the correct (negative) value.  However, the plateau exhibited by the result for the residue of the third pole is wrong.\\vspace*{-4ex}}\n\\end{figure}\n\nThe results described herein, and our continuing analysis and exploration of other methods, including those popular in contemporary simulations of lattice-regularised QCD \\cite{latticecorrelator}, suggest that solely from spacelike data it is only ever possible to extract reliable information about bound states with masses which do not much exceed $1\\,$GeV.  While these results are preliminary, they nevertheless provide sound reasons for caution.\n\n\\begin{figure}[tb]\n\\includegraphics[clip,width=0.45\\textwidth]{mass_0.33.eps}\\vspace*{-4ex}\n\n\\caption{\\label{fig:dse_masses} \nPole positions (mass values) obtained through a fit of Eq.\\,(\\protect\\ref{eq:pade}) with $N=3$ to the DSE result for $1\/E_5(k^2=0,P^2)$ depicted in Fig.\\,\\protect\\ref{fig:DSE_Inverted}.  Horizontal dotted lines indicate the masses obtained for the ground and first excited state via a direct solution of the homogeneous BSE \\protect\\cite{Holl:2004fr}.\\vspace*{-4ex}}\n\\end{figure}\n\nFollowing this background work on the viability \\emph{in principle} of using spacelike data alone to extract bound state information, we applied the method to the true DSE-calculated pseudoscalar vertex that is in depicted Fig.\\,\\ref{fig:DSE_Inverted}.  The decay constants and masses for the ground state pseudoscalar and first radial excitation were obtained from the homogeneous BSE in Ref.\\,\\cite{Holl:2004fr}.  The comparison between these masses and those inferred from the Pad\\'e approximant is presented in Fig.\\,\\ref{fig:dse_masses}.  As suggested by our background work, with perfect (effectively noiseless) spacelike data at hand, reliable information on the first two states in this channel can be obtained.\\footnote{We are currently exploring the impact of Gaussian noise in the data.}  We do not present a plot of the residues but they are accurate.  In particular, the ground state residue is positive and that of the first excited state is negative, and this is obtained without any bias in the fit.\n\n\\section{Sigma terms}\nThe $\\sigma$-term for a state $O$ is given by\n\\begin{equation}\n\\label{sigmasystem}\n\\sigma_{O} = m(\\zeta) \\frac{\\partial m_O}{\\partial  m(\\zeta)} \\,,\n\\end{equation}\nwhere $m_O$ is the mass of the state, and it is a keen probe of the impact of explicit chiral symmetry breaking on a hadron's mass.  The nucleon's $\\sigma$-term, $\\sigma_N$, has been estimated using: chiral effective theory, e.g.\\ Refs.\\,\\cite{Gasser:1991ce,ulf}; lattice-QCD, e.g.\\ Ref.\\,\\cite{Leinweber:2000sa,ulf2}; and QCD-based models, e.g.\\ Ref.\\,\\cite{lyubovitsky}.  Our recent interest in hadron $\\sigma$-terms is motivated by their utility in using observational data to place constraints on the variation of nature's fundamental parameters \\cite{uzan}.  In Table \\ref{sigmaterms} we reproduce results calculated in Ref.\\,\\cite{Flambaum:2005kc} along with new results described in the text.  We note that \n\\begin{equation}\n\\frac{\\delta m_O}{m_O} = \\frac{\\sigma_O}{m_O} \\frac{\\delta m(\\zeta)}{m(\\zeta)} \\,,\n\\end{equation}\nso that the dimensionless quantity tabulated measures the linear relative response of the mass of interest to a fractional change in the current-quark mass, which is a renormalisation group invariant.\n\n\\begin{table}[t]\n\\begin{center}\n\\caption{\\label{sigmaterms} Calculated $\\sigma$-terms.  Those for $\\pi$, $N$ and $\\Delta$ were reported in Ref.\\,\\protect\\cite{Flambaum:2005kc}.  The values for $\\rho$ and $\\omega$ listed herein were obtained via a direct analysis of the $m(\\zeta)$-dependence of the vector meson mass obtained in a solution of the rainbow-ladder truncation of the quark DSE and homogeneous meson BSE.  They improve the values reported in Ref.\\,\\protect\\cite{Flambaum:2005kc}, which were obtained using a simple fit to $m_\\rho(m(\\zeta))$ provided in Ref.\\,\\cite{marisvienna}.  All results are renormalisation-point-independent and were obtained with: $m_{u,d}(\\zeta)= 3.7\\,$MeV, $m_{s}(\\zeta)= 82\\,$MeV, $m_{c}(\\zeta)= 0.97\\,$GeV and $m_{b}(\\zeta)= 4.1\\,$GeV, where $\\zeta=19\\,$GeV.  Perturbative evolution can be used to determine the associated renormalisation-point-independent current-quark-masses.}\n\n\\begin{tabular*}{0.45\\textwidth}{c@{\\extracolsep{0ptplus1fil}}\n|c@{\\extracolsep{0ptplus1fil}}\n c@{\\extracolsep{0ptplus1fil}}\n c@{\\extracolsep{0ptplus1fil}}\n c@{\\extracolsep{0ptplus1fil}}\n}\\hline\nH & $\\pi$ & $\\pi_1$ & $\\sigma$ &   \\\\\n$\\rule{0em}{3.5ex}\\displaystyle\\frac{\\sigma_H}{m_H}$ \n& 0.498 & 0.017 & 0.013 &  \\\\\\hline\nH & $\\rho$ & $\\omega$ & $N$ & $\\Delta$  \\\\\n$\\rule{0em}{3.5ex}\\displaystyle\\frac{\\sigma_H}{m_H}$ \n& 0.021 & 0.034 & 0.064 & 0.041 \\\\\\hline\n$q$ & $u$,$d$ & $s$ & $c$ & $b$  \\\\\n$\\rule{0em}{3.5ex}\\displaystyle\\frac{\\sigma_q}{M^E_q}$ \n& 0.023 & 0.230 & 0.637 & 0.851 \\\\\\hline\n\\end{tabular*}\\vspace*{-4ex}\n\\end{center}\n\\end{table}\n\n\\medskip\n\n\\hspace*{-\\parindent}\\underline{\\textit{Radial Excitation of the Pion}}.\\hspace*{0.5em}\nWe have a framework which enables the calculation of $\\sigma_{\\pi_1}$ using Eq.\\,(\\ref{sigmasystem}), where $\\pi_1$ denotes the first pseudoscalar radial excitation.  To be specific, we can straightforwardly obtain the current-quark-mass-dependence of the $\\pi_1$ in the renormalisation-group-improved rainbow-ladder (RL) DSE truncation described in Ref.\\,\\cite{mariscairns}.  In this truncation \\mbox{$m^{\\rm RL}_{\\pi_1}=1.06\\,$GeV \\cite{Holl:2004fr}} and we find\\footnote{We are currently unable to provide a reliable estimate of meson-loop corrections to the mass and $\\sigma$-term of the $\\pi_1$.}\n\\begin{equation}\n\\sigma_{\\pi_{1}}^{\\rm RL}=0.018\\,\\textrm{GeV}.\n\\end{equation}\nThis value is considerably smaller than that for the ground state pion: $\\sigma_\\pi = 0.069\\,$GeV.  However, that merely serves again to emphasise the particular character of the mass of QCD's Goldstone mode and its amplification by dynamical chiral symmetry breaking.  \n\n\\medskip\n\n\\hspace*{-\\parindent}\\underline{\\textit{Scalar Meson}}.\\hspace*{0.5em}\nWe do not pretend to have a complete understanding of the lowest mass $0^{++}$ state in the hadron spectrum.  In this channel the rainbow-ladder truncation may be unreliable because the cancellations between higher-order terms in the systematic DSE truncation, which are so effective and important in pseudoscalar and vector channels, do not occur \\cite{cdrQC2}.  This is entangled with the phenomenological difficulties encountered in understanding the scalar states below $1.4\\,$GeV (see, e.g., Refs.\\,\\cite{MikeP}).  \n\nNevertheless, the rainbow-ladder DSE truncation provides a light-quark scalar-meson solution.  Combining the results from a number of sources, one finds \\cite{bastirev} $m_\\sigma^{\\rm RL} = 0.64\\pm 0.06\\,$GeV.  With the kernel described in Ref.~\\cite{mariscairns}, which we have used throughout, one obtains\n\\begin{eqnarray}\n\\label{massRL}\nm_{\\sigma}^{\\rm RL} & = & 0.675\\,{\\rm GeV}\\,,\\\\\n\\label{sigmaRL}\n2\\, m_\\sigma^{\\rm RL} \\sigma_{\\sigma}^{{\\rm RL}} & = & (0.184\\,{\\rm GeV})^2 \\\\\n& \\Rightarrow & \\sigma_{\\sigma}^{{\\rm RL}} = 0.025\\,{\\rm GeV}. \\label{sigmaRLR}\n\\end{eqnarray}\n\nThe scalar meson described by the rainbow-ladder truncation has a large coupling to two pions \\cite{Maris:2000ig}.  It is therefore important to consider the effect of a $\\pi \\pi$ loop correction to this state's mass and $\\sigma$-term.  Such meson loop corrections can be estimated \\cite{Leinweber:2000sa,Flambaum:2005kc,Wright:2000gg} and have a modest quantitative impact ($\\lsim 15$\\%) on $\\sigma_\\rho$, $\\sigma_N$ and $\\sigma_\\Delta$ and a larger effect ($\\lsim 30$\\%) on $\\sigma_\\omega$.  Their effect in this case can be analysed in the same way.  \n\nWe consider a single $\\pi\\pi$-loop self-energy \n\\begin{eqnarray}\n\\nonumber \\lefteqn{\n\\Pi^{\\pi\\pi}_{\\sigma}(m_\\sigma^2)}\\\\\n&& = \\frac{-3g^2_{\\sigma\\pi\\pi}}{16\\pi^{2}} \\int_{0}^{\\infty}dk \\frac{k^{2}\\, u_{\\Lambda_\\sigma}(k)^2} {\\omega(k)\\left(\\omega(k)^2-m_{\\sigma}^{2}\/4\\right)}\\,,\n\\label{sigmaloop}\n\\end{eqnarray}\nwhere $\\omega(k)=\\sqrt{m_{\\pi}^{2}+k^{2}}$.  The self energy both corrects $m_\\sigma^{\\rm RL}$ and provides for the $\\sigma\\to \\pi\\pi$ width.  In Eq.\\,(\\ref{sigmaloop}) \n\\begin{equation}\n\\frac{g_{\\sigma\\pi\\pi}}{m_\\sigma^{\\rm RL}}=5.51 \\; \\Rightarrow \\; \n\\Im \\sqrt{s}_{\\rm T} = 0.300 {\\rm GeV}.\n\\end{equation}\nThis is a typical value for the imaginary part of the T-matrix pole \\cite{pdg} and corresponds to \n\\begin{equation}\n\\Gamma_{\\sigma\\pi\\pi} = 0.55\\,{\\rm GeV}\n\\,,\n\\end{equation} \nwhich equates numerically to $0.92\\,(2\\,\\Im \\sqrt{s}_{\\rm T})$.\n\nThe function $u_{\\Lambda_\\sigma}(k^2) = 1\/(1+k^{2}\/\\Lambda_{\\sigma}^{2})^{2}$ in Eq.\\,(\\ref{sigmaloop}) is a form factor, introduced to represent the nonpointlike nature of the $\\pi$ and $\\sigma$ and hence the $\\sigma\\pi\\pi$ vertex.  The analysis of Refs.\\,\\mbox{\\cite{Bloch:1999yk,MarisFBS}} indicates that the intrinsic size of the $\\sigma$ described herein is 84\\% of that of the $\\rho$.  We therefore choose a value of the regularisation mass-scale $\\Lambda_{\\sigma}=\\Lambda_{\\rho\\pi\\pi}\/0.84$.  With $\\Lambda_{\\rho\\pi\\pi}=1.23\\,$GeV at $m_\\pi=0.14\\,$GeV, as determined in Ref.\\,\\cite{Leinweber:2001ac}, $\\Re\\Pi_\\sigma^{\\pi\\pi}((m^{\\rm RL}_\\sigma)^2) = - (0.395\\,{\\rm GeV})^2$.  Hence, the $\\pi\\pi$-loop acts to reduce the mass of the rainbow-ladder $\\sigma$-meson and, from the shifted pole position,\n\\begin{equation}\n\\label{sigma1loop}\nm_\\sigma^{{\\rm RL}+\\pi\\pi} = 0.624\\,{\\rm GeV.}\n\\end{equation}\nThis is an $8$\\% reduction cf.\\ Eq.\\,(\\ref{massRL}).  With all else kept fixed in Eq.\\,(\\ref{sigmaloop}), a variation of $\\Lambda_\\sigma$ by $\\pm 20$\\% alters the result in Eq.\\,(\\ref{sigma1loop}) by $\\mp 6$\\%.  NB.\\ From Ref.\\,\\cite{pdg}, one might consider $0.60 \\pm 0.25\\,$GeV as typifying the mass of the lightest scalar meson.  \n\nWe define the $\\pi\\pi$-loop correction to the scalar $\\sigma$-term via \n\\begin{equation}\n\\left. m_{\\pi}^{2}\\frac{\\partial}{\\partial m_{\\pi}^{2}}\\, \\Re\\Pi_{\\sigma}^{\\pi\\pi}\\right|_{(m_{\\pi}^{2})_{{\\rm expt.}}}\n = -\\,(0.161\\,{\\rm GeV})^{2}\\,,\n\\end{equation}\nand combine this with Eq.\\,(\\ref{sigmaRL}) accordingly:\n\\begin{eqnarray}\n\\nonumber \\lefteqn{m(\\zeta) \\frac{\\partial\\,\\Re m_\\sigma^2 }{\\partial m(\\zeta)}\n=(0.090\\,{\\rm GeV})^2}\\\\\n& &=:  2 \\,m_\\sigma^{\\Re({\\rm 1-loop})} \\, \\sigma_\\sigma^{\\Re({\\rm 1-loop})}\\\\\n&  \\Rightarrow & \\sigma_\\sigma^{\\Re({\\rm 1-loop})} = 0.0073 \\,{\\rm GeV}\\,,\n\\label{sigmaresult}\n\\end{eqnarray}\nsince $m_\\sigma^{\\Re({\\rm 1-loop})}=0.547\\,$GeV.  Equation (\\ref{sigmaresult}) is a $71$\\% reduction cf.\\ Eq.\\,(\\ref{sigmaRLR}).  The effect is large because the scalar-meson is very broad.  The $\\Pi_\\rho^{\\pi\\pi}$ self-energy contributes much less to properties of the rainbow-ladder $\\rho$-meson because the width\/mass ratio is significantly smaller \\cite{Flambaum:2005kc,pichowsky}. \n\nIt is noteworthy that a self-consistent solution of $s-(m_\\sigma^{\\rm RL})^2-\\Pi_\\sigma^{\\pi\\pi}(\\Re s)=0$ gives a pole position \n\\begin{equation}\n\\label{polemass}\n\\surd s_\\sigma = 0.578- i \\, 0.311\\,{\\rm GeV}.  \n\\end{equation}\nThe third iteration is the last to introduce a change $>1$\\%, and all iterations beyond the first reduce the real part in Eq.\\,(\\ref{polemass}) by a total of $< 7$\\%.  The effects are greater than that of the nucleon's $\\pi N$ self energy \\cite{Flambaum:2005kc,NpiN}.  These notes provide a gauge for the accuracy of the one loop analysis.\n\n\\medskip\n\n\\hspace*{-\\parindent}\\underline{\\textit{Constituent quarks}}.\\hspace*{0.5em}\nOne measure of the importance of dynamical chiral symmetry breaking to the dressed-quark mass function is the magnitude, relative to the current-quark mass, of the Euclidean constituent-quark mass; viz., \n\\begin{equation}\n\\label{CQM}\n(M^E)^2 := s  \\ni s = M(s)^2 .\n\\end{equation}\nFor the current-quark-masses in Table \\ref{sigmaterms}\n\\begin{equation}\n\\begin{array}{c|cccc}\nQ & u,d & s & c & b \\\\\\hline\n\\rule{0em}{3ex} M^E_Q\\,({\\rm GeV}) & 0.42 & 0.56 & 1.57 & 4.68\n\\end{array}\n\\end{equation}\n\nA constituent-quark $\\sigma$-term can subsequently be defined \\cite{Flambaum:2005kc}\n\\begin{equation}\n\\label{sMEQ}\n\\sigma_Q := m(\\zeta) \\, \\frac{\\partial  M^E}{\\partial m(\\zeta)}\\,.\n\\end{equation}\nIt is a renormalisation-group-invariant that can be determined from solutions of the gap equation, Eq.\\,(\\ref{eq:quark_prop}).  Our results are presented in Table \\ref{sigmaterms}.\n\n\\begin{figure}[t]\n\\includegraphics[clip,width=0.45\\textwidth]{quark_mass_ratio.eps}\\vspace*{-4ex}\n\n\\caption{\\label{fig:quark_ratio} Ratio $\\sigma_Q\/M^E_Q$ [Eqs.\\,(\\protect\\ref{CQM}) \\& (\\protect\\ref{sMEQ})].\nIt is a measure of the current-quark-mass-dependence of dynamical chiral symmetry breaking.  The vertical dotted lines correspond to the $u=d$, $s$, $c$ and $b$ current-quark masses listed in Table \\protect\\ref{sigmaterms}.\\vspace*{-4ex}}\n\\end{figure}\n\nIn Fig.\\,\\ref{fig:quark_ratio} we depict $\\sigma_{Q}\/M^E_Q$.  It is a measure of the effect on the dressed-quark mass-function of explicit chiral symmetry breaking compared with the sum of the effects of explicit and dynamical chiral symmetry breaking.  One anticipates that for light-quarks this ratio must vanish because the magnitude of their constituent-mass owes primarily to dynamical chiral symmetry breaking, while for heavy-quarks it should approach one.  The figure confirms these expectations.\n\n\\section{Summary and Conclusion}\n\nWe presented arguments which indicate that using solely spacelike information about a Schwinger function, it may only be possible to extract properties of bound states with masses $\\lsim 1.2\\,$GeV.  With perfect spacelike information it is possible to accurately reconstruct the pole contributions from states which satisfy this bound.  However, such reconstruction methods provide no information beyond that which is already available in studies that employ physical (timelike) bound state momenta.  We illustrated this via the Dyson-Schwinger equations.  \n\nIn the context of numerical simulations of lattice-regularised QCD, we speculate that it may be possible to reach higher mass states by employing lattice data to constrain the infrared behaviour of DSE integral equation kernels and subsequently using the DSEs to provide information on Schwinger functions at timelike momenta.\n\nThe $\\sigma$ term is one useful gauge of the impact of explicit chiral symmetry breaking on a hadron's mass.  Such information is important for the interpretation of measurements that indicate a spacial and\/or temporal variation in Nature's fundamental parameters.  We calculated and reported $\\sigma$-terms for a range of hadrons: the ground state pion; the pion's first radial excitation; a light scalar meson; the $\\rho$ and $\\omega$; the nucleon and $\\Delta$; and the $u$, $s$, $c$ and $b$ constituent-quarks.  \n\n\nThe analysis of the scalar channel is interesting.  It is consistent with a picture of the lightest scalar as a bound state of a dressed-quark and \\mbox{-antiquark} combined with a considerable two-pion component, which reduces the quark-core mass by $\\lsim 15$\\% but dramatically alter the current-quark-mass-dependence of the $0^{++}$ pole position.\n\nThe constituent-quark $\\sigma$-terms provide a tool that is useful for assessing the flavour-dependence of dynamical chiral symmetry breaking.\n\n\\medskip\n\n\\hspace*{-\\parindent}\\textbf{Acknowledgments.}\\hspace*{0.5em}\nWe avow discussions with P.\\,O.~Bowman, V.\\,V.~Flambaum, A.~Krass\\-nigg, D.\\,B.~Leinweber, P.\\,C.~Tandy and A.\\,G.~Williams.\nThis work was supported by: \nDept.\\ of Energy, Office of Nucl.\\ Phys., contract nos.\\ DE-FG02-00ER41135 and W-31-109-ENG-38; \nNational Science Foundation grant no.\\ PHY-0301190;\nand the \\textit{A.\\,v.\\ Humboldt-Stiftung} via a \\textit{F.\\,W.\\ Bessel Forschungspreis}.\nIt benefited from the facilities of ANL's Computing Resource Center.\n\n\\begingroup\\raggedright\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Introduction}\n\nIn her ground-breaking work~\\cite{fouresbruhat_thmexistencecertainespde}, Choquet-Bruhat showed that Einstein's field equations can be formulated as an initial value problem, where the initial data is given on a spacelike Cauchy hypersurface. To given initial data, there is a ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ which is unique up to isometry, as was shown by Choquet-Bruhat and Geroch in~\\cite{choquetbruhatgeroch_globalaspectsgr}.\nAs understood in physics, this implies that this spacetime is uniquely determined by the initial data. The question arises whether one can find a larger development than this when dropping the requirement of global hyperbolicity, and whether this larger development is still unique in a meaningful sense. If two inequivalent developments existed, this would imply that determinism breaks, as the data on the initial Cauchy hypersurface does no longer suffice for determining which of the two developments the universe chooses.\nThe Strong Cosmic Censorship conjecture states that this does not happen, at least not generically, as it conjectures that there is no development larger than the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment.\n\\begin{conj}[Strong Cosmic Censorship]\n\\label{conj_scc}\n\tFor generic initial data to Einstein's equations, the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ (MGHD) is inextendible.\n\\end{conj}\nAs the currently preferred spacetime models exhibit singularities at early times, this conjecture is of additional interest: It claims that the occurrence of such a singularity is not a consequence of a particularly bad choice of initial data hypersurface, i.\\,e.\\  of our point of view, but inherent to the four-dimensional spacetime.\n\nIn the form given in Conjecture~\\ref{conj_scc}, the statement will not hold, one has to add some form of boundary conditions such as spatial compactness or asymptotic flatness, or homogeneity.\nTo this date, a full answer in the general case has not been given, and it is therefore of interest to consider subsets of initial data with additional properties.\nThe natural starting point for this discussion are the maximally symmetric de~Sitter, Anti-de~Sitter\\ and Minkowski spacetimes, which in a first step generalise to the spatially homogeneous and isotropic Friedman-Lema\u00eetre-Robertson-Walker spacetimes.\nThese can be further generalised to the spatially homogeneous Bianchi spacetimes which admit a three-dimensional symmetry group. Some of the Bianchi spacetimes in turn appear as special cases of~$G_2$ cosmologies, where the symmetry group is only of dimension~two. In this paper, we discuss the case of Bianchi spacetimes. The expectation is that the results obtained here will in part translate to the more general~$G_2$ cosmologies and pave the way for the fully general setting.\n\n\nThe terms generic and inextendible in the conjecture have to be made precise in order to obtain a meaningful statement.\nWhat we mean by  genericity will become clear below.\nThere are first results on inextendibility in the $C^0$-sense (for Schwarzschild,~\\cite{sbierski_C0inextschwarzschild}, and for certain spherically symmetric spacetimes,~\\cite{christodoulou_instabnakedsinggravcollapsescalfield} and~\\cite{gallowayling_remarksC0inextend}), but here we consider inextendibility in the~$C^2$-sense, as we can then replace Conjecture~\\ref{conj_scc} with the following, stronger conjecture (see~\\cite[Conj.~17.2]{ringstrom_cauchyproblem}).\n\\begin{conj}[Curvature blow up]\n\\label{conj_blowup}\n\tFor generic initial data to Einstein's equations, the Kretsch\\-mann scalar $R_{\\alpha\\beta\\gamma\\delta}R{}^{\\alpha\\beta\\gamma\\delta}$ is unbounded in the incomplete directions of causal geodesics in the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ (MGHD).\n\\end{conj}\n\\begin{rema}\n\\label{rema_blowupwithmatter}\n\tWhenever the spacetime is not vacuum, i.\\,e.\\  the Ricci tensor does not vanish, it may be enough to determine whether the contraction of the Ricci curvature~$R_{\\alpha\\beta}R^{\\alpha\\beta}$ is unbounded in the incomplete directions of causal geodesics in the MGHD. Just like the Kretschmann scalar, this is a four-dimensional quantity invariant under isometries, and its unboundedness contradicts the existence of a~$C^2$ extension.\n\\end{rema}\n\n\n\n\\subsection{Bianchi perfect fluid spacetimes}\nWe restrict our discussion to spacetimes~$(M,g)$ satisfying Einstein's field equations\n\\begin{equation}\n\\label{eqn_einsteineqn}\n\tR_{\\alpha\\beta}-\\frac12Sg_{\\alpha\\beta}=T_{\\alpha\\beta},\n\\end{equation}\nwith~$R_{\\alpha\\beta}$ and~$S$ the Ricci and scalar curvature of the spacetime\nand~$T_{\\alpha\\beta}$ the stress-energy tensor of a perfect fluid\n\\begin{equation}\n\\label{eqn_stressenergyperfectfluid}\n\tT_{\\alpha\\beta}=\\mu u_\\alpha u_\\beta+p(g_{\\alpha\\beta}+u_\\alpha u_\\beta),\n\\end{equation}\nwhere~$u$ is a unit timelike vector field. We assume that the pressure~$p$ and the energy density~$\\mu$ satisfy a linear equation of state\n\\begin{equation}\n\\label{eqn_lineareqnofstate}\n\tp=(\\gamma-1)\\mu\n\\end{equation}\nfor some constant~$\\gamma$.\nThe case of vacuum is included as~$\\mu=0$.\nWe note at this point that Greek indices $\\alpha,\\beta,\\ldots$ range from~$0$ to~$3$, while lower case Latin ones $i,j,\\ldots$ range from~$1$ to~$3$.\n\n\nWe consider\nBianchi perfect fluids, which are perfect fluid spacetimes such that additionally\nthere is a three-dimensional Lie group~$G$ acting on the spacetime~$(M,g)$\n\\begin{equation}\n\tG\\times M\\rightarrow M.\n\\end{equation}\nThe metric~$g$ is invariant under this action, and the unit timelike vector field~$u$ appearing in the stress-energy tensor~\\eqref{eqn_stressenergyperfectfluid} of the perfect fluid is or\\-tho\\-go\\-nal\\ to the orbits of this action. This defines non-tilted perfect fluids.\nThe name Bianchi spacetimes stems from the classification of three-dimensional Lie group s by Bianchi from~1903, see~\\cite{krasinskibehrschueckingestabrookwahlquistellisjantzenkundt_bianchiclass} for a historical overview:\nUnimodular Lie group s are called class~A, non-unimodular ones class~B, and depending on the form of the structure constants of the corresponding Lie al\\-ge\\-bras, they can be further separated into the following types:\n\\begin{itemize}\n\t\\item Bianchi class~A: types~I, II, VI$_0$, VII$_0$, VIII, IX;\n\t\\item Bianchi class~B: types~IV, V, VI$_\\eta$, VII$_\\eta$, where~$\\eta\\in\\maths{R}$ is a parameter.\n\\end{itemize}\nIn this paper, we focus on Bianchi class~B models, but types~I and~II appear as boundary cases.\nThe models of class~A have been the subject of more detailed study in the past, and a number of results have been achieved,\nincluding an affirmative answer to the Strong Cosmic Censorship conjecture. We refer to~\\cite{ringstrom_cauchyproblem} for a detailed exposition, in particular Prop.~22.23 therein, and are going to relate our own findings to a number of other results further down.\n\n\n\n\n\n\nIn terms of initial data, we arrive at the following setting: The initial data mani\\-fold\\ is a three-dimensional Lie group~$G$ with a metric and second fun\\-da\\-men\\-tal form\\ which are left-invariant, meaning that they are invariant under the action of~$G$ on itself via multiplication from the left. Further, a given constant represents the initial matter configuration.\n\\begin{defi}\n\\label{defi_generalinitialdatabianchib}\n\tBianchi perfect fluid initial data consists of a Lie group~$G$, a left-invariant Rie\\-mann\\-ian\\ metric~${h}$ on~$G$, a left-invariant symmetric covariant two-tensor~${k}$ on~$G$, and a constant $\\mu_0\\ge0$, satisfying the constraint equations\n\t\\begin{align}\n\t\t\\overline R-{k}_{ij}{k}^{ij}+(\\operatorname{tr}_{h}{k})^2={}&2\\mu_0,\\\\\n\t\t\\overline\\nabla_i\\operatorname{tr}_{h}{k}-\\overline\\nabla^j{k}_{ij}={}&0.\n\t\\end{align}\n\tIn these equations, $\\overline\\nabla$ is the Levi-Civita\\ connection of~${h}$, and~$\\overline R$ the corresponding scalar curvature. Indices are lowered and raised by~${h}$.\n\\end{defi}\nHere and in the following, we use the Einstein summation convention and sum over indices which occur twice, both as a sub- and as a superindex.\n\nThe symmetry of the metric and second fun\\-da\\-men\\-tal form\\ is preserved under Einstein's equations, and the resulting ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ is isometric to the spatially homogeneous four-dimensional spacetime\n\\begin{equation}\n\\label{eqn_mghdform}\n\t(I\\times G\\,,\\,g={-}dt^2+{}^t\\overline g),\n\\end{equation}\nwith~$I$ an open interval and $\\{{}^t\\overline g\\}_{t\\in I}$ a family of left-invariant Riemannian metrics on~$G\\cong\\{t\\}\\times G$. We give an explicit construction of the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ in Section~\\ref{section_equivalenceinitialdataexpansionnormalised}.\n\n\n\\subsection{Results for orthogonal perfect fluid initial data}\nLet us now state the precise setting we discuss in this paper and give the main results we obtain regarding the Strong Cosmic Censorship conjecture.\n\nWe focus our attention on Bianchi perfect fluid initial data with a Lie group\\ of Bianchi class~B. Type~I and~II appear as boundary cases. In particular, we exclude Lie group s of type~VIII and~IX. The Lie al\\-ge\\-bra~$\\ggg$ associated to the initial data Lie group~$G$ then admits an Abelian subalgebra, see Lemma~\\ref{lemm_abeliansubalgstructconstants}. In case of a Lie group\\ of Bianchi class~B, this subalgebra can be characterised geometrically as the kernel of a certain one-form, see Lemma~\\ref{lemm_abeliansubalginvardefi}. Using this geometric construction, the subalgebra is unique and denoted by~$\\ggg_2$.\n\nWe introduce an or\\-tho\\-nor\\-mal\\ basis $e_1,e_2,e_3$ of~$\\ggg$ such that~$e_2,e_3$ span the Abelian subalgebra~$\\ggg_2$ and~$e_1$ is or\\-tho\\-go\\-nal\\ to it with respect to\\ the initial metric. By the previous argument, this can be done uniquely up to rotation in the~$e_2e_3$-plane and a choice of sign in~$e_1$, provided the Lie group\\ is of class~B.\n\nFor Bianchi perfect fluid initial data with a Lie group\\ of class~B, one realises that the momentum constraint gives an algebraic relation for the components of the initial symmetric two-tensor~$k$ with respect to this basis.\nFor all initial data apart from certain cases where the Lie group\\ is of type~VI$_{{-}1\/9}$, this relation implies that the off-diagonal components~$k_{12}$ and~$k_{13}$ of this tensor vanish, see Lemma~\\ref{lemm_exceptionalbianchiasinitialdata}. In the remaining special cases, the sets of initial data admit an additional degree of freedom and their ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ is a so-called `exceptional` Bianchi cosmological spacetime, denoted Bbii in~\\cite{ellismaccallum_classofhomogcosmmodels}.\nThis term should be understood as 'having exceptional behaviour'.\nWe refer to Remark~\\ref{rema_exceptionalityexplained} for more details on this terminology.\n\nIn this paper, we exclude initial data with a Lie group\\ of Bianchi type~VI$_{{-}1\/9}$ and thereby ensure that the resulting spacetime is a 'non-exceptional' or 'or\\-tho\\-go\\-nal ' Bianchi spacetime.\n\\begin{defi}\n\\label{defi_initialdatabianchib}\n\tOrthogonal perfect fluid Bianchi class~B initial data consists of a Lie group~$G$ of class~B other than type~VI$_{{-}1\/9}$, a left-invariant Rie\\-mann\\-ian\\ metric~${h}$ on~$G$, a left-invariant symmetric covariant two-tensor~${k}$ on~$G$, and a constant $\\mu_0\\ge0$, satisfying the constraint equations\n\t\\begin{align}\n\t\t\\overline R-{k}_{ij}{k}^{ij}+(\\operatorname{tr}_{h}{k})^2={}&2\\mu_0,\\\\\n\t\t\\overline\\nabla_i\\operatorname{tr}_{h}{k}-\\overline\\nabla^j{k}_{ij}={}&0.\n\t\\end{align}\n\n\tOrthogonal perfect fluid Bianchi type~I and~II initial data for Einstein's orthogonal perfect fluid equation s is defined similarly, allowing any type~I or~II Lie group ~$G$.\n\\end{defi}\nThe reason for this restriction is the technique we apply in this paper. We wish to transform initial data sets into a specific set of variables and prove statements in this setting before translating them back. For Lie group s of type~VI$_{{-}1\/9}$ we encounter difficulties as this transformation, described in detail in Section~\\ref{section_equivalenceinitialdataexpansionnormalised}, can only be carried out for spacetimes which are `non-exceptional`.\n\nCertain initial data sets with higher symmetry will be of importance in the discussion of the Strong Cosmic Censorship conjecture. They are defined using a suitably adapted basis of the Lie al\\-ge\\-bra~$\\ggg$:\n\\begin{defi}[Locally rotationally symmetric initial data]\n\\label{defi_lrsinitialdata}\n\tConsider initial data $(G,{h},{k},\\mu_0)$ as in Def.~\\ref{defi_initialdatabianchib}, and denote\n\tby~$\\ggg$ the corresponding Lie al\\-ge\\-bra\\ with two-dimensional Abelian subalgebra~$\\ggg_2$. Let\n\t$e_1,e_2,e_3$ an or\\-tho\\-nor\\-mal\\ basis of~$\\ggg$ such that $e_2,e_3$ span~$\\ggg_2$.\n\tThe initial data is said to be locally rotationally symmetric (LRS)\n\tif the basis can be chosen such that\n\t\\begin{itemize}\n\t\t\\item $e_2$ commutes with~$e_1$ and~$e_3$\n\t\t\\begin{equation}\n\t\t\t\\liebr{e_2}{e_1}=0=\\liebr{e_2}{e_3},\n\t\t\\end{equation}\n\t\t\\item the commutator~$\\liebr{e_1}{e_3}$ is a multiple of~$e_2$,\n\t\t\\item the two-tensor~${k}_{ij}$ is diagonal, with ${k}_{11}={k}_{33}$.\n\t\\end{itemize}\n\\end{defi}\n\\begin{rema}\n\tThe notion locally rotationally symmetric in the previous definition stems from the fact that a rotation in the~$e_1e_3$-plane is a Lie group\\ isomorphism and an isometry of the initial data.\n\n\tConsidering a three-dimensional Lie group~$G$ of class~B with corresponding Lie al\\-ge\\-bra~$\\ggg$, we find that the following holds: In case~$\\liebr\\ggg\\ggg$ is two-dimensional, there is no locally rotationally symmetric initial data on this Lie group. If instead~$\\liebr\\ggg\\ggg$ is one-dimensional, then the vector spanning this set defines a rotation axis contained in~$\\ggg_2$ and leaving the Lie al\\-ge\\-bra\\ invariant. Initial data on the given Lie group\\ is locally rotationally symmetric if and only if\\ the two-tensor~$k$ is invariant under this rotation. For more details, we refer to Subsection~\\ref{constr_highersymmetrysolutions}.\n\\end{rema}\n\\begin{defi}[Plane wave equilibrium initial data]\n\\label{defi_planewaveinitialdata}\n\tConsider initial data $(G,{h},{k},\\mu_0)$ as in Def.~\\ref{defi_initialdatabianchib}, and denote\n\tby~$\\ggg$ the corresponding Lie al\\-ge\\-bra\\ with two-dimensional Abelian subalgebra~$\\ggg_2$. Let\n\t$e_1,e_2,e_3$ be an or\\-tho\\-nor\\-mal\\ basis of~$\\ggg$ such that $e_2,e_3$ span~$\\ggg_2$, and denote by~$\\gamma_{ij}^k$ the structure constants, i.\\,e.\\ \n\t\\begin{equation}\n\t\t\\liebr{e_i}{e_j}=\\gamma_{ij}^ke_k.\n\t\\end{equation}\n\tThe initial data is said to be of plane wave equilibrium type\n\tif the basis can be chosen such that\n\t\\begin{equation}\n\t\t\\gamma_{1A}^B+\\gamma_{1B}^A={-}2{k}_{AB}.\n\t\\end{equation}\n\\end{defi}\nWe note here that upper case Latin indices $A,B,\\ldots$ range from~$2$ to~$3$, where we assume the frame elements~$e_2,e_3$ to span~$\\ggg_2$.\n\\begin{defi}\n\\label{defi_lrsplanewavemghd}\n\tA spacetime which is the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ of locally rotationally symmetric initial data is called a locally rotationally symmetric spacetime.\n\tA spacetime which is the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ of initial data of plane wave equilibrium type is called a plane wave equilibrium spacetime.\n\\end{defi}\nWe show in Section~\\ref{section_equivalenceinitialdataexpansionnormalised} how to construct the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment~\\eqref{eqn_mghdform} to given initial data as in Def.~\\ref{defi_initialdatabianchib}.\n\n\nTo answer the Strong Cosmic Censorship conjecture, we determine whether geometric quantities invariant under isometries of this spacetime remain bounded in the incomplete directions of causal geodesics.\nFor vacuum models we find that a bounded Kretschmann scalar~$R_{\\alpha\\beta\\gamma\\delta}R^{\\alpha\\beta\\gamma\\delta}$ corresponds to a spacetime with local rotational symmetry or of plane wave equilibrium type.\nWhenever matter is present and $\\gamma\\not=0$, we do not have to compute the full Kretschmann scalar, but it is enough to determine whether the contraction of the Ricci curvature~$R_{\\alpha\\beta}R{}^{\\alpha\\beta}$ is unbounded. We show that this is the case for all causal geodesics, there are no exceptions.\nThe arguments for the matter case work similarly as in the Bianchi~A case, which was discussed in~\\cite{ringstrom_asymptbianchiaspacetimes}.\nThe full statement about curvature blow-up in the incomplete directions of causal geodesics is collected in the following two theorems.\n\\begin{theo}\n\\label{theo_curvblowupinitialdatamatter}\n\tConsider orthogonal perfect fluid Bianchi class~B initial data~$(G,{h},{k},\\mu_0)$ for matter~$\\mu_0>0$. Let $(M,g,\\mu)$ be the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ of the data, solving Einstein's equations for a perfect fluid~\\eqref{eqn_stressenergyperfectfluid} with linear equation of state~\\eqref{eqn_lineareqnofstate}, where~$\\gamma>0$.\n\tThen the contraction of the Ricci tensor with itself $R_{\\alpha\\beta}R^{\\alpha\\beta}$\n\tis unbounded in the incomplete directions of causal geodesics.\n\\end{theo}\n\\begin{theo}\n\\label{theo_curvblowupinitialdatavacuum}\n\tConsider orthogonal perfect fluid Bianchi class~B initial data~$(G,{h},{k},\\mu_0)$ for vacuum~$\\mu_0=0$, which is neither locally rotationally symmetric of Bianchi type~VI$_{{-}1}$ nor of plane wave equilibrium type.\n\tLet $(M,g)$ be the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ of the data, solving Einstein's equations for vacuum~$T_{\\alpha\\beta}=0$.\n\tThen the Kretschmann scalar $R_{\\alpha\\beta\\gamma\\delta}R^{\\alpha\\beta\\gamma\\delta}$\n\tis unbounded in the incomplete directions of causal geodesics.\n\\end{theo}\n\\begin{rema}\n\tWe recall that orthogonal perfect fluid Bianchi class~B initial data excludes Lie group s of Bianchi type~VI$_{{-}1\/9}$, see Definition~\\ref{defi_initialdatabianchib}. In case of such initial data, more precisely for such initial data which is 'exceptional', see Remark~\\ref{rema_exceptionalityexplained}, our statements do not apply. In fact, their behaviour is expected to differ significantly: close to the initial singularity, the corresponding spacetimes are expected to show chaotic oscillatory behaviour, much the same as Bianchi type~VIII and~IX spacetime in class~A.\n\t\n\tDue to the additional complications one meets in these models, they have not been studied extensively to this date. An approach similar to the one we choose here is made in~\\cite{hewitthorwoodwainwright_asymptdynamexceptbianchicosm}. In future works, one can hope that their results and conjectures can be used to find an answer to Strong Cosmic Censorship which also applies to the remaining Bianchi type we do not treat here.\n\\end{rema}\n\n\\begin{rema}\n\tInitial data such that both the Kretschmann scalar $R_{\\alpha\\beta\\gamma\\delta}R^{\\alpha\\beta\\gamma\\delta}$ and the contraction of the Ricci tensor with itself $R_{\\alpha\\beta}R^{\\alpha\\beta}$ remain bounded in the incomplete directions of causal geodesics in the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ are\n\tlocally rotationally symmetric or of plane wave equilibrium type.\n\tIn particular, such initial data has additional symmetry and can therefore be considered non-generic. As a consequence, Strong Cosmic Censorship holds in the class of orthogonal perfect fluid Bianchi class~B initial data.\n\t\n\tIt is interesting to note here that we only in the vacuum case have to exclude certain non-generic initial data sets. In the matter case, each initial data set has a corresponding development which is inextendible.\n\tThe presence of matter appears to simplify things, at least in terms of the Strong Cosmic Censorship conjecture.\n\\end{rema}\n\\begin{rema}\n\\label{rema_curvblowupcosmconst}\n\tIn case~$\\mu>0$ and~$\\gamma=0$, the stress-energy tensor~\\eqref{eqn_stressenergyperfectfluid} with linear equation of state~\\eqref{eqn_lineareqnofstate} takes the form\n\t\\begin{equation}\n\t\tp={-}\\mu\n\t\\end{equation}\n\tand can be interpreted as the stress-energy tensor of vacuum with a positive cosmological constant. In this case, we find a statement similar to the previous one: We can show that the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment~$(M,g)$ is of the form~\\eqref{eqn_mghdform}, with $I=(t_-,t_+)$ an interval,\n\tand every timeslice~$\\{t\\}\\times G$ has positive mean curvature\\ with respect to\\ the normal vector~$\\partial_t$. Then the Kretschmann scalar $R_{\\alpha\\beta\\gamma\\delta}R^{\\alpha\\beta\\gamma\\delta}$ or the contraction of the Ricci tensor with itself $R_{\\alpha\\beta}R^{\\alpha\\beta}$\n\tis unbounded in the incomplete directions of causal geodesics, with the following possible exceptions:\n\t\\begin{itemize}\n\t\t\\item local rotationally symmetric initial data of Bianchi type~VI$_{{-}1}$\n\t\t\\item initial data of Bianchi class~B, and in scale free variables the metric and second fun\\-da\\-men\\-tal form\\ of every slice $\\{t\\}\\times G$ converge to initial data of plane wave equilibrium type,\n\t\tas $t\\rightarrow t_-$.\n\t\\end{itemize}\n\tNote that we did not make precise in what sense the initial data converges, and that for initial data on a Lie group\\ of type other than Bianchi~VI$_{{-}1}$, we have not clearly stated any property which can be interpreted as non-generic apart from this convergence. Further, we have used the existence of a specific foliation in order to formulate the statements.\n\tPrecise results are given and proven in scale free variables using a notion of convergence which we introduce in the following. The results, both for the two previous theorems and the special case of positive cosmological constant in vacuum, then follow directly from translating back into the current formulation.\n\\end{rema}\n\n\n\n\\subsection{Expansion-normalised variables and previous results}\nMost of the work in this paper is carried out in a setting different from the one above. Instead of starting with initial data~$(G,{h},{k},\\mu_0)$ to Einstein's equations and proving properties of the resulting ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment, we adopt a different view-point. The information given by initial data $(G,{h},{k},\\mu_0)$ can be translated into the form of expansion-normalised and dimensionless variables\n\\begin{equation}\n\t(\\Sigma_+\\,,\\,\\tilde\\Sigma\\,,\\,\\Delta\\,,\\,\\tilde A\\,,\\,N_+),\n\\end{equation}\nwhich have been introduced in~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}. Einstein's field equations translate into an ordinary differential equation in these variables, defined on a compact subset of~$\\maths{R}^5$ and with a changed time coordinate. It is in these variables that we work and obtain the results on curvature blow-up. Every initial data set corresponds to a point in the compact subset of~$\\maths{R}^5$, and once one has determined the solution curve in expansion-normalised variables, the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment~\\eqref{eqn_mghdform} which solves Einstein's field equations with correct initial data can be determined. The detailed construction is carried out in Section~\\ref{section_equivalenceinitialdataexpansionnormalised}. Via this construction, our results in expansion-normalised variables can be carried over to the setting of geometric initial data.\n\nThe evolution of these expansion-normalised variables is defined in the following way: To start with, one fixes two parameters, ${\\kappa}\\in\\maths{R}$ and~$\\gamma\\in[0,2]$. The evolution equations for the variables~$(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)$ are\n\\begin{equation}\n\\label{eqns_evolutionbianchib}\n\\begin{subaligned}\n\t\\Sigma_+'={}&(q-2)\\Sigma_+-2\\tilde N,\\\\\n\t\\tilde\\Sigma'={}&2(q-2)\\tilde\\Sigma-4\\Sigma_+\\tilde A-4\\Delta N_+,\\\\\n\t\\Delta'={}&2(q+\\Sigma_+-1)\\Delta+2(\\tilde\\Sigma-\\tilde N)N_+,\\\\\n\t\\tilde A'={}&2(q+2\\Sigma_+)\\tilde A,\\\\\n\tN_+'={}&(q+2\\Sigma_+)N_++6\\Delta,\n\\end{subaligned}\n\\end{equation}\nwhere~$'$ denotes differentiation ${d}\/{d\\tau}$ with respect to the dimensionless time~$\\tau$.\nThe deceleration parameter~$q$ appearing in the equations satisfies\n\\begin{equation}\n\\label{eqn_definitionqgeneral}\n\tq=\\frac32(2-\\gamma)(\\Sigma_+^2+\\tilde\\Sigma)+\\frac12(3\\gamma-2)(1-\\tilde A-\\tilde N),\n\\end{equation}\nand one has set\n\\begin{equation}\n\\label{eqn_definitiontilden}\n\t\\tilde N=\\frac13(N_+^2-{\\kappa}\\tilde A).\n\\end{equation}\nThe evolution is constrained to the set of points in~$\\maths{R}^5$ satisfying\n\\begin{equation}\n\\label{eqn_constraintgeneralone}\n\t\\tilde\\Sigma\\tilde N-\\Delta^2-\\Sigma_+^2\\tilde A=0,\n\\end{equation}\nand\n\\begin{equation}\n\\label{eqn_constraintgeneraltwo}\n\t\\tilde\\Sigma\\ge0, \\qquad \\tilde A\\ge0, \\qquad\\tilde N\\ge0, \\qquad \\Sigma_+^2+\\tilde\\Sigma+\\tilde A+\\tilde N\\le 1.\n\\end{equation}\nOne further introduces the density parameter\n\\begin{equation}\n\\label{eqn_omegageneral}\n\t\\Omega=1-\\Sigma_+^2-\\tilde\\Sigma-\\tilde A-\\tilde N,\n\\end{equation}\nwhich satisfies~$\\Omega\\ge0$ and, as a consequence of the differential equations~\\eqref{eqns_evolutionbianchib}, evolves according to\n\\begin{equation}\n\\label{eqn_evolutionomega}\n\t\\Omega'=(2q-(3\\gamma-2))\\Omega.\n\\end{equation}\nNote that the relations~\\eqref{eqn_constraintgeneralone} and~\\eqref{eqn_constraintgeneraltwo} are preserved by the evolution equations~\\eqref{eqns_evolutionbianchib}, see Remark~\\ref{rema_expnormvarconstraintspreserved}.\n\nWe now give a rough explanation as to how the expansion-normalised variables are related to the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ to given initial data~$(G,{h},{k},\\mu_0)$ as in Def.~\\ref{defi_initialdatabianchib}. For this, assume that we are given a spacetime as in~\\eqref{eqn_mghdform}, together with a function~$\\mu$, which solve Einstein's field equations~\\eqref{eqn_einsteineqn} for a perfect fluid~\\eqref{eqn_stressenergyperfectfluid} with linear equation of state~\\eqref{eqn_lineareqnofstate}. Assume that~$\\partial_t$, the vector field in the~$I$ direction which is unit timelike and or\\-tho\\-go\\-nal\\ to every~$\\{t\\}\\times G$, coincides with the vector field~$u$ from the stress-energy tensor. Assume further that the induced metric and second fundamental form on~$\\{0\\}\\times G$ coincide with~${h}$ and~${k}$ from the initial data, i.\\,e.\\  $(I\\times G,g,\\mu)$ is a development of the data.\nOne obtains consistency between the evolution in expansion-normalised variables and the initial data approach if one chooses the parameter~$\\gamma$ from the expansion-normalised variables to coincide with the same-named constant from the linear equation of state~\\eqref{eqn_lineareqnofstate}. Further, if the Lie group~$G$ has a group parameter~${\\eta}$, then the parameter~${\\kappa}$ from the expansion-normalised variables is set to satisfy~${\\kappa}=1\/{\\eta}$. If the Lie group\\ has no group parameter, then~${\\kappa}=0$.\n\nFor every dimensionless time~$\\tau$, the point $(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)(\\tau)$ in expansion-normalised variables describes the geometric and dynamical properties of the timeslice $\\{t(\\tau)\\}\\times G$.\nThe variables~$\\Delta,\\tilde A,N_+$ carry information on the three-dimensional metric of this timeslice, while~$\\Sigma_+,\\tilde\\Sigma$ describe the shear of the vector field~$\\partial_t$.\nA perfect fluid spacetime is described by a curve in~$\\maths{R}^5$ solving the constrained evolution equations~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_constraintgeneraltwo}, and we sometimes call such a solution an orbit.\nTransforming the momentum constraint from the point of view of geometric initial data into the expansion-normalised variables yields the constraint equation~\\eqref{eqn_constraintgeneralone}. Similarly, the Hamiltonian constraint justifies the definition of~$\\Omega$ in~\\eqref{eqn_omegageneral}, which is the expansion-normalised version of the energy density~$\\mu$.\n\nStatements about geometric initial data can be transformed into statements about specific points in the variables~$(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)$. Conversely, we show in Section~\\ref{section_equivalenceinitialdataexpansionnormalised} how one can, given the solution in expansion-normalised variables through this point, construct the corresponding ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment.\nHence, statements about spacetimes solving Einstein's perfect fluid equation on the one hand and orbits solving the evolution equations in expansion-normalised variables~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} on the other hand carry equivalent information.\nIt is for this reason that we can carry out our analysis wholly in the setting of expansion-normalised variables, and only at the end translate the results back to the point of view of geometric initial data. The setting of expansion-normalised variables is a rather approachable one, as we discuss a polynomial ordinary differential equation on a compact subset of~$\\maths{R}^5$. For instance, this allows the use of techniques from the theory of dynamical systems.\n\n\n\\smallskip\n\n\nThe system of differential equations~\\eqref{eqns_evolutionbianchib}-\\eqref{eqn_definitiontilden} with constraints~\\eqref{eqn_constraintgeneralone}-\\eqref{eqn_constraintgeneraltwo} was first introduced and studied in~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}. In this reference, Hewitt and Wainwright identify invariant subsets representing the different Bianchi types, and we recall these subsets in Tables~\\ref{table_bianchibsubsets} and~\\ref{table_bianchiasubsets} below, together with several other invariant subsets representing models with additional symmetry, Table~\\ref{table_bianchihighersymmetry}. Additionally, \\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB} discuss several sets of equilibrium points of the evolution equations. Of importance to our work are the Kasner parabola with the two Taub points and the plane wave equilibrium points, as convergence to points in these sets corresponds to the incomplete direction of causal geodesics in the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment, which is what we need to investigate in light of the Strong Cosmic Censorship conjecture.\n\\begin{defi}\n\\label{defi_kasnerparabola}\n\tThe Kasner parabola is the subset~$\\K$ defined by\n\t\\begin{equation}\n\t\t\\Sigma_+^2+\\tilde\\Sigma=1, \\qquad \\Delta=\\tilde A=N_+=0.\n\t\\end{equation}\n\\end{defi}\n\\begin{figure}[ht]\n\t\\begin{tikzpicture}[xscale=2.5,yscale=2.5]\n\t\\draw [->] (-1.2,0) -- (1.2,0);\n\t\\draw [->] (0,0) -- (0,1.2);\n\t\\draw[gray, thick, domain=-1:1] plot (\\x, {(\\x+1)*(1-\\x)});\n\t\\draw[fill] (-1,0) circle [radius=0.025];\n\t\\node [below] at (0,0) {$0$};\n\t\\draw (1,-0.03) -- (1,0.04);\n\t\\node [below] at (1,0) {$1$};\n\t\\node [below] at (-1,0) {${-}1$};\n\t\\draw (-0.04,1) -- (0.04,1);\n\t\\node [above left] at (0,1) {$1$};\n\t\\node [right] at (1.2,0) {$\\Sigma_+$};\n\t\\node [above] at (0,1.2) {$\\tilde\\Sigma$};\n\t\\node [above left] at (-1,0) {$\\operatorname{T1}$};\n\t\\draw [dashed] (0.5,0) -- (0.5,0.75);\n\t\\node [below] at (0.5,0) {$1\/2$};\n\t\\draw[fill] (0.5,0.75) circle [radius=0.025];\n\t\\node [above right] at (0.5,0.75) {$\\operatorname{T2}$};\n\t\\node [above left] at (-0.5,0.75) {$\\K$};\n\t\\end{tikzpicture}\n\t\\caption{The Kasner parabola~$\\K$, projected to the $\\Sigma_+\\tilde\\Sigma$-plane. The special points Taub~1 and~2 satisfy~$\\Sigma_+={-}1$ and $\\Sigma_+=1\/2$.}\n\t\\label{figure_kasnerparabola}\n\\end{figure}\n\\begin{defi}\n\\label{defi_taubpoints}\n\tOn the Kasner parabola~$\\K$, the points Taub~1 and 2 are defined by\n\t\\begin{itemize}\n\t\t\\item Taub~1: $\\operatorname{T1}\\coloneqq\\left\\{(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)=({-}1,0,0,0,0)\\right\\}$,\n\t\t\\item Taub~2: $\\operatorname{T2}\\coloneqq\\left\\{(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)=\\left(1\/2,3\/4,0,0,0\\right)\\right\\}$.\n\t\\end{itemize}\n\\end{defi}\n\\begin{defi}\n\\label{defi_planewaveexpansionnorm}\n\tThe plane wave equilibrium points are the elements of the set~$\\planewave {\\kappa}$ defined by\n\t\\begin{equation}\n\t\\label{eqn_defiplanewave}\n\t\\begin{split}\n\t\t&\\Sigma_+>-1,\\qquad\n\t\t\\tilde\\Sigma={-}\\Sigma_+(1+\\Sigma_+),\\qquad\n\t\t\\Delta=0,\\qquad\n\t\t\\tilde A=(1+\\Sigma_+)^2,\\\\\n\t\t&N_+^2=(1+\\Sigma_+)({\\kappa}(1+\\Sigma_+)-3\\Sigma_+).\n\t\\end{split}\n\t\\end{equation}\n\\end{defi}\n\\begin{figure}[ht]\n\t\\begin{tikzpicture}[xscale=2.5,yscale=2.5]\n\t\\draw [->] (-1.2,0) -- (1.2,0);\n\t\\draw [->] (0,0) -- (0,1.2);\n\t\\draw[gray, dashed, domain=-1:1] plot (\\x, {(\\x+1)*(1-\\x)});\n\t\\draw[gray, thick, domain=-1:0] plot (\\x, {-\\x*(1+\\x)});\n\t\\node [below] at (0,0) {$0$};\n\t\\draw (1,-0.03) -- (1,0.04);\n\t\\node [below] at (1,0) {$1$};\n\t\\draw (-1,-0.03) -- (-1,0.04);\n\t\\node [below] at (-1,0) {${-}1$};\n\t\\draw (-0.04,1) -- (0.04,1);\n\t\\node [above left] at (0,1) {$1$};\n\t\\node [right] at (1.2,0) {$\\Sigma_+$};\n\t\\node [above] at (0,1.2) {$\\tilde\\Sigma$};\n\t\\node [above] at (-0.6,0.25) {$\\planewave k$};\n\t\\node [above left] at (-0.5,0.75) {$\\K$};\n\t\\end{tikzpicture}\n\t\\caption{The plane wave equilibrium points~$\\planewave {\\kappa}$, projected to the $\\Sigma_+\\tilde\\Sigma$-plane. For reference, the Kasner parabola~$\\K$ is plotted as a dashed line.}\n\t\\label{figure_planewave}\n\\end{figure}\n\nBoth the Kasner parabola~$\\K$ and the plane wave equilibrium points~$\\planewave k$ satisfy~$\\Omega=0$, are contained in the set defined by the constraint equations~\\eqref{eqn_constraintgeneralone}--\\eqref{eqn_constraintgeneraltwo} and consist of equilibrium points, i.\\,e.\\  the right-hand side\\ of the evolution equations~\\eqref{eqns_evolutionbianchib} is zero.\n\nInformation about the local stability can be drawn from the linearised evolution equations in the extended five-dimensional space, by which one means the linear approximation of the evolution equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_definitiontilden}, without restricting to the constraint equations~\\eqref{eqn_constraintgeneralone}--\\eqref{eqn_constraintgeneraltwo}. We give the explicit form of this vector field for points on the Kasner parabola~$\\K$ in Appendix~\\ref{subsect_appendixlinearisedevolutionkasner}.\nThe eigenvalues of this vector field are\n\\begin{equation}\n\\label{eqn_eigenvalueskasner}\n\t0\\qquad\n\t2(1+\\Sigma_+\\pm\\sqrt{3(1-\\Sigma_+^2)})\\qquad\n\t4(1+\\Sigma_+)\\qquad\n\t3(2-\\gamma).\n\\end{equation}\nWe note at this point that there appear to be typos in the eigenvalues in both~\\cite[Sect.~4.4]{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB} and~\\cite[Sect.~7.2.3]{wainwrightellis_dynamsystemsincosm}. We give the corrected eigenvalues and state the corresponding eigenvectors in Appendix~\\ref{subsect_appendixlinearisedevolutionkasner}.\nThe signs of the individual eigenvalues in these two references however are given correctly, and with this information Hewitt--Wainwright are able to identify the points on the Kasner parabola to the right of Taub~2 (with $1\/2<\\Sigma_+\\le1$) as local sources, and the points to the left of Taub~2 (with ${-}1<\\Sigma_+<1\/2$) as saddles. For the two Taub points, the linearisation of the evolution equations alone does not determine the local stability, as two, or even three in case of the point Taub~1, of the eigenvalues vanish in these two points.\n\nSimilarly, one considers the linearised evolution equations in the extended five-dimensional state space for plane wave equilibrium points. The eigenvalues of this vector field are\n\\begin{equation}\n\\label{eqn_eigenvaluesplanewave}\n\t0\\qquad\n\t{-}4(1+\\Sigma_+)\\qquad\n\t{-}4\\Sigma_+-(3\\gamma-2)\\qquad\n\t{-}2(1+\\Sigma_+)\\pm4iN_+,\n\\end{equation}\nwith~$N_+$ as in equation ~\\eqref{eqn_defiplanewave}. See Appendix~\\ref{subsect_appendixlinearisedevolutionplanewave} for more details. Again, the number of positive, negative, and zero eigenvalues can be used to determine the local stability, and~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB} identify points with~$\\Sigma_+>{-}(3\\gamma-2)\/4$ as local sources, and points with~$\\Sigma_+<{-}(3\\gamma-2)\/4$ as saddles. In the point~$\\Sigma_+={-}(3\\gamma-2)\/4$ two eigenvalues vanish, and the local stability cannot be determined this way.\n\nUsing dynamical systems method, Hewitt--Wainwright show that the Kasner parabola~$\\K$ is of central importance for the asymptotic behaviour of solutions to the evolution equation as~$\\tau\\rightarrow{-}\\infty$, as it contains the~$\\alpha$-limit set of (non-constant, generic) solutions. We state and prove a refined version of~\\cite[Prop.~5.1, Prop.~5.2]{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB} in Prop.~\\ref{prop_alphalimitsets_vacuum_inflat}.\n\n\n\n\\subsection{New results: convergence behaviour in expansion-normalised variables, curvature blow-up}\n\nIn the present paper, we obtain more detailed results about the behaviour of solutions to the evolution equations~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} as~$\\tau\\rightarrow{-}\\infty$.\nWe refine a statement on the $\\alpha$-limit sets of solutions by~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}: all~$\\alpha$-limit points are contained in the union of the Kasner parabola~$\\K$, the plane wave equilibrium points~$\\planewave {\\kappa}$ and the point $\\{\\Sigma_+=\\tilde\\Sigma=\\Delta=\\tilde A=N_+=0\\}$, see Prop.~\\ref{prop_alphalimitsets_vacuum_inflat}.\n\nWe further prove that every solution has one unique~$\\alpha$-limit point, i.\\,e.\\  solutions to the evolution equations converge, to a limit point contained the plane wave equilibrium points~$\\planewave k$ or the Kasner parabola~$\\K$, see Prop.~\\ref{prop_convergencetoplanewave} and Prop.~\\ref{prop_convergencetokasner}. Only constant solutions converge to the point $\\{\\Sigma_+=\\tilde\\Sigma=\\Delta=\\tilde A=N_+=0\\}$, see Prop.~\\ref{prop_alphalimitsets_vacuum_inflat}.\n\nDifferent subsets of this set of limit points have very different qualitative properties:\nWhile solutions converging to the plane wave equilibrium points~$\\planewave {\\kappa}$, to the point~$\\{\\Sigma_+=\\tilde\\Sigma=\\Delta=\\tilde A=N_+=0\\}$ and to the subarc of the Kasner parabola~$\\K$ with~${-}1\\le\\Sigma_+\\le1\/2$ are contained in a `small` set, namely a countable union of $C^1$ submani\\-fold s of positive codimension (Prop.~\\ref{prop_alphalimitsets_vacuum_inflat}, Prop.~\\ref{prop_taubone}, Thm~\\ref{theo_taubtwocharacterisationoforbits}, Thm~\\ref{theo_leftoftaubtworesult}, Thm~\\ref{theo_centreunstablegloballyplanewave}),\nthe remaining arc of the Kasner parabola contains the limit points of all remaining solutions.\nConsidering the set of all possible limit points as a whole, this yields the following statement:\n\\begin{theo}\n\\label{theo_fullmeasurelimitset}\n\tAssume either vacuum or inflationary matter, i.\\,e.\\  either $\\Omega=0$ or $\\Omega>0$, $\\gamma\\in\\left[0,2\/3\\right)$. Then the following holds for solutions to the evolution equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega}:\n\t\\begin{itemize}\n\t\t\\item Consider the sets describing Bianchi type~VI$_{\\eta}$ or~VII$_{\\eta}$, i.\\,e.\\  the sets\n\t\t\\begin{align}\n\t\t\tB(VI_{\\eta}) ={}&\\{\\eqref{eqn_constraintgeneralone}-\\eqref{eqn_constraintgeneraltwo}\\text{ hold},\\,{\\kappa}=\\nicefrac1{\\eta}<0,\\, \\tilde A>0\\}, \\\\\n\t\t\tB^\\pm(VII_{\\eta}) ={}&\\{\\eqref{eqn_constraintgeneralone}-\\eqref{eqn_constraintgeneraltwo}\\text{ hold},\\, {\\kappa}=\\nicefrac1{\\eta}>0,\\,\\tilde A>0,\\,N_+>0\\textit{ or }N_+<0\\}.\n\t\t\\end{align}\n\t\tThen the subset of points such that the corresponding solution converges to a point in~$\\K\\cap\\{\\Sigma_+>1\/2\\}$, as~$\\tau\\rightarrow{-}\\infty$, is\n\t\tof full measure and a countable intersection of open and dense sets\n\t\tin B(VI$_{\\eta}$) or B$^\\pm$(VII$_{\\eta}$), respectively.\n\t\t\\item Consider the set describing Bianchi type~IV, i.\\,e.\\  the set\n\t\t\\begin{equation}\n\t\t\tB^\\pm(IV) =\\{\\eqref{eqn_constraintgeneralone}-\\eqref{eqn_constraintgeneraltwo}\\text{ hold},\\,{\\kappa}=0,\\,\\tilde A>0,\\,N_+>0\\textit{ or }N_+<0\\}.\n\t\t\\end{equation}\n\t\tThen the subset of points such that the corresponding solution converges to a point in~$\\K\\cap\\{\\Sigma_+>1\/2\\}$ or to the point~$\\K\\cap\\{\\Sigma_+=0\\}$, as~$\\tau\\rightarrow{-}\\infty$, is\n\t\tof full measure and a countable intersection of open and dense sets\n\t\tin B$^\\pm$(IV).\n\t\t\\item Consider the set describing Bianchi type~V, i.\\,e.\\  the set\n\t\t\\begin{equation}\n\t\t\tB(V)=\\{\\eqref{eqn_constraintgeneralone}-\\eqref{eqn_constraintgeneraltwo}\\text{ hold},\\,{\\kappa}=0,\\,\\tilde A>0,\\,\\Sigma_+=\\Delta=N_+=0\\}.\n\t\t\\end{equation}\n\t\tThen every non-constant solutions converges to the point~$\\K\\cap\\{\\Sigma_+=0\\}$, as~$\\tau\\rightarrow{-}\\infty$.\n\t\\end{itemize}\n\\end{theo}\nFor Kasner points situated to the right of Taub~2, we further find that they govern the behaviour of all solutions starting close to them.\nIn fact, we show in Prop.~\\ref{prop_rightoftaub2source} that for every element of the subarc~$\\K\\cap\\{\\Sigma_+>1\/2\\}$, points contained in a sufficiently small neighborhood also converge to this subarc as~$\\tau\\rightarrow{-}\\infty$, to a limit point close to the original one.\n\nFor the subarc of the Kasner parabola~$\\K$ to the left of the point Taub~2, i.\\,e.\\  satisfying~${-}1<\\Sigma_+<1\/2$, we additionally find the following restriction, compare Prop.~\\ref{prop_main_generalasymptoticproperties}.\n\\begin{prop}\n\\label{prop_specialrelationbparamkslimit}\n\tLet $\\gamma\\in[0,2)$ and consider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} converging to $(\\operatorname s,1-\\operatorname s^2,0,0,0)$ as~$\\tau\\rightarrow{-}\\infty$. If $\\operatorname s\\in\\left[{-}1,1\/2\\right]$, then\n\t\\begin{equation}\n\t\t\\tilde A(3\\operatorname s^2+{\\kappa}(1-\\operatorname s^2))=0\n\t\\end{equation}\n\talong the whole orbit.\n\\end{prop}\nAs a consequence of this result, for a given parameter~$\\kappa$, a solution converging to a point on the Kasner parabola~$\\K$ situated to the left of Taub~2 has to be either a Bianchi class~A ($\\tilde A=0$) solution, or it is a Bianchi class~B ($\\tilde A>0$) solution but can only converge to one of two specific points on~$\\K$, namely one with~$\\Sigma_+=\\pm\\sqrt{\\kappa\/(\\kappa-3)}$. For this to hold, the parameter~$\\kappa$ has to be non-positive. In particular, no Bianchi type~VII$_\\eta$ solution can converge to a Kasner point situated to the left of Taub~2, as in this Bianchi type one has~$\\kappa=1\/\\eta>0$.\n\nThis statement is very different in spirit than the property of a subarc being a local source or saddle and goes far beyond what can be obtained by considering only the local stability. Instead, we discuss the asymptotic convergence behaviour of the individual expansion-normalised variables one by one, taking into account the full non-linear properties of the evolution equations~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_definitiontilden} in combination with the exact form of the constraint equation~\\eqref{eqn_constraintgeneralone}.\n\nUsing the same methods, we also obtain a detailed statement about the Taub points, which due to the multiple zero eigenvalues of the linearised evolution equations in these points cannot be treated by local stability at all. We prove that only the constant orbit converges to the point Taub~1, see Prop.~\\ref{prop_taubone} and Prop.~\\ref{prop_tauboneothermatter}. Solutions converging to the point Taub~2 have to be locally rotationally symmetric, see Definition~\\ref{defi_lrsexpansionnorm} and Thm~\\ref{theo_taubtwocharacterisationoforbits}, and these locally rotationally symmetric solutions appear as possible exceptions to an unbounded Kretschmann scalar below in Thm~\\ref{theo_curvblowupexpansionnormalisedvacuum}.\n\nFor the plane wave equilibrium points with more than one vanishing eigenvalue, we make use of techniques from the theory of dynamical systems and show that solutions are contained in a countable union of submani\\-fold s of dimension at most two and of positive codimension, see Prop.~\\ref{theo_centreunstablegloballyplanewave} and Remark~\\ref{rema_planewaveconvergencebianchiseparated}.\n\nIn addition to the statements about asymptotic behaviour in the limit~$\\tau\\rightarrow{-}\\infty$, we also obtain the following result on the late time behaviour.\n\\begin{rema}\n\tIn case of inflationary matter, i.\\,e.\\  $\\Omega>0$, $\\gamma\\in\\left[0,2\/3\\right)$, all solutions converge to the point $(0,0,0,0,0)$ as~$\\tau\\rightarrow{+}\\infty$, see Prop.~\\ref{prop_omegalimitsetinflationary}. Transforming this statement back to the setting of geometric initial data and its ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment, this implies isotropisation at late times.\n\\end{rema}\n\nThe main objective of the present paper is to prove the Strong Cosmic Censorship conjecture in the setting of or\\-tho\\-go\\-nal\\ Bianchi class~B perfect fluid and vacuum spacetimes. This is done by showing that boundedness of the Kretschmann scalar~$R_{\\alpha\\beta\\gamma\\delta}R^{\\alpha\\beta\\gamma\\delta}$ or the contraction of the Ricci curvature\\ with itself~$R_{\\alpha\\beta}R^{\\alpha\\beta}$ in the incomplete direction of causal geodesics is a non-generic property.\nMaking use of the transformation between geometric initial data with corresponding ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment~\\eqref{eqn_mghdform} on the one hand, and points in the state space of expansion-normalised variables with corresponding solutions to the evolution equations~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_constraintgeneraltwo} on the other hand, we can express these two curvature expressions in terms of the expansion-normalised variables. Both quantities then depend only on the initial mean curvature\\ and energy density, the parameters~$\\gamma$ and~$\\kappa$, the time~$\\tau$ and the point in expansion-normalised variables, as we explain in the beginning of Section~\\ref{section_proofmainthms}. The incomplete direction of causal geodesics corresponds to the limit~$\\tau\\rightarrow{-}\\infty$, see Prop.~\\ref{prop_mghdincompletedirections}.\nThe statements about curvature blow-up in expansion-normalised variables are as follows.\n\\begin{theo}\n\\label{theo_curvblowupexpansionnormalisedmatter}\n\tConsider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} with~$\\Omega>0$ and~$\\gamma>0$. Then the contraction of the Ricci tensor with itself $R_{\\alpha\\beta}R^{\\alpha\\beta}$ is unbounded as $\\tau\\rightarrow{-}\\infty$.\n\\end{theo}\n\\begin{rema}\n\tWe remark at this point that the statement in the matter case is obtained without knowledge on the detailed asymptotic behaviour of the individual variables. In fact, only the evolution equation~\\eqref{eqn_evolutionomega} for~$\\Omega$ is considered to prove this statement, combined with details on how the expansion-normalised variables are obtained from geometric initial data sets. \n\tIn contrast, the vacuum case treated in the next theorem is rather intricate, and necessitates a detailed discussion of all variables.\n\t\n\tIt is interesting to see that in the vacuum case, there are certain exceptions to unboundedness of the curvature, while there are none in matter. In a way, one could therefore consider the matter case as the easy case, while it is the vacuum case where interesting---and ultimately more difficult---behaviour becomes visible.\n\tIn Bianchi class~A models, a similar observation has been made, see~\\cite{ringstrom_asymptbianchiaspacetimes} and~\\cite{ringstrom_cauchyproblem}.\n\\end{rema}\n\\begin{theo}\n\\label{theo_curvblowupexpansionnormalisedvacuum}\n\tConsider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} with~$\\Omega=0$ which is neither\n\tthe constant solution in the point Taub~1, nor\n\ta locally rotationally symmetric Bianchi type~I, II or~VI$_{{-}1}$ solution, nor a plane wave equilibrium solution.\n\tThen the Kretschmann scalar $R_{\\alpha\\beta\\gamma\\delta}R^{\\alpha\\beta\\gamma\\delta}$\n\tis unbounded as $\\tau\\rightarrow{-}\\infty$.\n\\end{theo}\n\\begin{rema}\n\tIn the previous theorem, locally rotationally symmetric solutions are defined by being contained in the set\n\t\\begin{equation}\n\t\t\t\t3\\Sigma_+^2=\\tilde\\Sigma, \\qquad\n\t\t\\Sigma_+N_+=\\Delta, \\qquad\n\t\t({\\kappa}+1)\\tilde A=0,\n\t\\end{equation}\n\tsee Definition~\\ref{defi_lrsexpansionnorm}, and consult Tables~\\ref{table_bianchibsubsets}, \\ref{table_bianchiasubsets}, and~\\ref{table_bianchihighersymmetry} for more details on the separation into the different Bianchi types.\n\n\tIn vacuum~$\\Omega=0$,\n\ta bounded Kretschmann scalar is possible only for solutions converging to one of the Taub points or to the plane wave equilibrium points~$\\planewave k$. Only in case of Taub~2 can such a solution be non-constant, and it necessarily has to be locally rotationally symmetric.\n\\end{rema}\n\n\\begin{theo}\n\\label{theo_curvblowupexpansionnormalisedcosmconst}\n\tConsider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} with~$\\Omega>0$ and~$\\gamma=0$ which is neither\tthe constant solution\n\tin the point $\\Sigma_+=\\tilde\\Sigma=\\Delta=\\tilde A=N_+=0$,\n\tnor a locally rotationally symmetric Bianchi type~I, II or~VI$_{{-}1}$ solution,\n\tnor a solution contained in the set\n\t\\begin{align}\n\t\t\\C\\coloneqq\n\t\t\t&\\{\\text{solutions to equations~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} with }\\Omega>0,\\,\\gamma=0,\\\\\n\t\t\t&\\qquad\\text{ and converging to an element of }\\planewave k\\text{ as } \\tau\\rightarrow{-}\\infty\\}.\n\t\\end{align}\n\tThen the Kretschmann scalar $R_{\\alpha\\beta\\gamma\\delta}R^{\\alpha\\beta\\gamma\\delta}$ is unbounded as $\\tau\\rightarrow{-}\\infty$.\n\n\tFurther, there is a countable family of~$C^1$ submani\\-fold s~$\\{L_m\\}_{m\\in\\maths{N}}$ of dimension at most~two such that\n\t\\begin{equation}\n\t\t\\C\\subset\\bigcup_m L_m.\n\t\\end{equation}\n\tFor certain Bianchi types, or equivalently certain values of the parameter~$\\kappa$, the following additional restrictions hold:\n\t\\begin{itemize}\n\t\t\\item Bianchi type~VI$_{\\eta}$, which implies~${\\kappa}=1\/{\\eta}<0$: Every solution in~$\\C$ converges to an element of the plane wave equilibrium points~$\\planewave {\\kappa}$ with~$\\Sigma_+=\\operatorname s$ satisfying\n\t\t\\begin{equation}\n\t\t\t\\operatorname s\\le\\frac{{\\kappa}}{3-{\\kappa}}<0.\n\t\t\\end{equation}\n\t\t\\item Bianchi type~$V$, which implies~$\\kappa=0$: Every solution in~$\\C$ is contained in~$\\Sigma_+=\\tilde\\Sigma=\\Delta=N_+=0$, and~$\\tilde A$ decreases monotonically from~$1$ to~$0$.\n\t\\end{itemize}\n\\end{theo}\n\\begin{rema}\n\tIn the state space in expansion-normalised variables, which is the subset of~$\\maths{R}^5$ given by the constraint equations~\\ref{eqn_constraintgeneralone}--\\eqref{eqn_constraintgeneraltwo}, the different Bianchi types are represented by different invariant subsets, see Table~\\ref{table_bianchibsubsets}. In case of Bianchi type~VI$_\\eta$, VII$_\\eta$, or~IV, this is a subset of dimension four. Consequently, the sets~$L_m$ defined in the previous theorem are of positive codimension.\n\tThe invariant subset describing Bianchi type~V solutions is of dimension two, and by the additional restriction stated in the theorem the set~$\\C$ is contained in a set of dimension one, which thus is also of positive codimension.\n\\end{rema}\nCompared to the statements in terms of initial data to Einstein's equations, Thm~\\ref{theo_curvblowupinitialdatamatter} and Thm~\\ref{theo_curvblowupinitialdatavacuum}, there is a larger number of exceptions to unboundedness of either geometric scalar in Thm~\\ref{theo_curvblowupexpansionnormalisedmatter}, Thm~\\ref{theo_curvblowupexpansionnormalisedvacuum} and Thm~\\ref{theo_curvblowupexpansionnormalisedcosmconst}. The reason is that the expansion-normalised formulation allows for every initial data to the evolution equations~\\eqref{eqns_evolutionbianchib} which satisfies the constraint equations~\\eqref{eqn_constraintgeneralone} and~\\eqref{eqn_constraintgeneraltwo}. This in particular includes certain points in~$\\maths{R}^5$ which correspond to Bianchi class~A initial data. In Thm~\\ref{theo_curvblowupinitialdatamatter} and Thm~\\ref{theo_curvblowupinitialdatavacuum}, these are excluded by the assumption on the initial data set.\n\n\n\n\n\\subsection*{Acknowledgements}\n\nThe author is indebted to Hans Ringstr\u00f6m who suggested the topic, for stimulating discussions, ongoing supervision, and his numerous comments on the article.\nThanks is owed to Christopher Nerz for proof-reading parts of an earlier version of this paper.\nFurther thanks goes to Jaap Eldering for his hints on existing literature via MathOverflow that lead to App~\\ref{section_appendixdynamsystheo}, as well as Ingemar Bengtsson for his insightful comments concerning previous work on the de~Sitter\\ spacetime.\n\nThe author would like to acknowledge the support of the G\u00f6ran Gustafsson Foundation for Research in Natural Sciences and Medicine.\n\nPart of this material is based upon work supported by the National Science Foundation under Grant No.~0932078~000, while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during\nthe winter semester of~2013.\n\n\n\\section{Structure of the paper}\n\nAs the scope of this article is rather large, touching a number of different areas and techniques, we wish to give the reader a guide on how to read it, depending on the interests and prerequisites. We also shortly present the main ideas of the proofs.\n\nWe have given, in equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega}, the evolution equation s which correspond to Einstein's equation s for or\\-tho\\-go\\-nal\\ perfect fluid Bianchi~B initial data, and stated our main theorems in the introduction.\nIn Sections~\\ref{section_basicpropertiesexpnormevolution}-\\ref{section_asymptoticsplanewave}, we discuss various properties of this evolution problem. We explain their content in more detail a bit further down.\n\nIn Section~\\ref{section_equivalenceinitialdataexpansionnormalised}, we move our focus away from a detailed discussion of the evolution equations and towards the relation of these equations to spacetimes solving Einstein's equations. It is here that we\nprove equivalence between the evolution problem in expansion-normalised variables and the problem of finding, to given geometric initial data, the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment.\nWe recall the classification of three-dimensional Lie group s and how it relates to Lie group s as initial data sets.\nThis leads, in and after Lemma~\\ref{lemm_exceptionalbianchiasinitialdata}, to an explanation of why we excluded Bianchi~VI$_{{-}1\/9}$ Lie group s in the definition of initial data.\nWe recall the construction of expansion-normalised variables for or\\-tho\\-go\\-nal\\ Bianchi class~B cosmological models, proposed by~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}, and show how this can be used to construct a spacetime which is shown to by the maximal globally hyperbolic development.\nTo the author's knowledge, these constructions and proofs have not been given before.\n\nIn the final section, we give the proofs of the main theorems, which are stated in the introduction. Those which are formulated in expansion-normalised variables, Thm~\\ref{theo_fullmeasurelimitset} Thm~\\ref{theo_curvblowupexpansionnormalisedmatter}, Thm~\\ref{theo_curvblowupexpansionnormalisedvacuum}, and Thm~\\ref{theo_curvblowupexpansionnormalisedcosmconst} are treated first.\nApart from computations of a number of curvature\\ quantities in expansion-normalised variables which we carry out there, most of the work has been done in the sections before, and the proofs of these theorems merely collect the necessary results from Sections~\\ref{section_basicpropertiesexpnormevolution}-\\ref{section_asymptoticsplanewave}.\nIn order to then prove the remaining main theorems, Thm~\\ref{theo_curvblowupinitialdatamatter} and Thm~\\ref{theo_curvblowupinitialdatavacuum}, we use the equivalence between the setting of geometric initial data to Einstein's equation and that of the evolution equations in expansion-normalised variables. With this information at hand, proving the remaining main theorem is equivalent to translating the results from Thm~\\ref{theo_curvblowupexpansionnormalisedmatter} and Thm~\\ref{theo_curvblowupexpansionnormalisedvacuum} back to the setting of geometric initial data.\n\n\n\\smallskip\n\nLet us now give a more detailed description of the content of Sections~\\ref{section_basicpropertiesexpnormevolution}-\\ref{section_equivalenceinitialdataexpansionnormalised}, which is where the main part of the work is carried out. These sections all discuss more and more detailed properties of the evolution in expansion-normalised variables.\nIn the end, we wish to make a statement about Strong Cosmic Censorship in the $C^2$-sense, and one way of doing so is to determine for which initial data both the Kretschmann scalar $R_{\\alpha\\beta\\gamma\\delta}R^{\\alpha\\beta\\gamma\\delta}$ and the contraction of the Ricci tensor with itself $R_{\\alpha\\beta}R^{\\alpha\\beta}$ remain bounded in the incomplete direction of causal geodesics in the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment, see Conjecture~\\ref{conj_scc}, Conjecture~\\ref{conj_blowup} and Remark~\\ref{rema_blowupwithmatter}.\nFor spacetimes corresponding to solutions to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega}, this direction corresponds to $\\tau\\rightarrow{-}\\infty$. Consequently, we have to understand the asymptotic behaviour of solutions as $\\tau\\rightarrow{-}\\infty$, and determine boundedness of the Kretschmann scalar and the contraction of the Ricci tensor with itself along such solutions. If we can show that the set of solutions along which both quantities are bounded at negative times is suitably small and can be considered non-generic, we have proven Strong Cosmic Censorship for this class of initial data.\n\nSection~\\ref{section_basicpropertiesexpnormevolution} starts with a general discussion of basic properties of equation s~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega}, such as compactness of the state space, the smoothness of the constraint surface and its division into different invariant subsets corresponding to the different Bianchi types. In Section~\\ref{section_kasnerparabolaplanewave}, we examine the so-called $\\alpha$-limit set, which is the set of points where solutions accumulate as $\\tau\\rightarrow{-}\\infty$.\nIn the case of vacuum and inflationary matter, the $\\alpha$-limit set is contained in the Kasner parabola~$\\K$ and the plane wave equilibrium points~$\\planewave {\\kappa}$, together with one additional point in the case of inflationary matter, see Prop.~\\ref{prop_alphalimitsets_vacuum_inflat}.\nThis result about accumulation points is subsequently strengthened to convergence, as $\\tau\\rightarrow{-}\\infty$, in Prop.~\\ref{prop_convergencetoplanewave} and Prop.~\\ref{prop_convergencetokasner}, where the latter requires additional assumptions.\nIn particular, one needs to argue that the only solution with the point Taub~1 as an $\\alpha$-limit point is the constant orbit, which is the main result we achieve in Section~\\ref{section_taubone}.\nWe additionally prove a statement about isotropisation at late times: In case of inflationary matter, all solutions converge to the same point as~$\\tau\\rightarrow{+}\\infty$, see Prop.~\\ref{prop_omegalimitsetinflationary}.\n\n\\smallskip\n\nBefore we continue with a summary of the results of Sections~\\ref{section_asymptoticdecaykasner}-\\ref{section_equivalenceinitialdataexpansionnormalised}, let us briefly describe the main techniques.\nIn the end, we wish to understand exactly which solutions or sets of solutions converge to specific points, for example to those points which are of interest for the Strong Cosmic Censorship conjecture. To do so, we determine exponential decay or convergence properties of the individual variables using the evolution equation s with constraints~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega}, for example the variable~$\\tilde A$:\nFor a solution converging to a point $(\\operatorname s,1-\\operatorname s^2,0,0,0)$, $\\operatorname s\\in[{-}1,1]$, on the Kasner parabola~$\\K$, as $\\tau\\rightarrow{-}\\infty$,\nthe evolution equation\n\\begin{equation}\n\t\\tilde A'=2(q+2\\Sigma_+)\\tilde A\n\\end{equation}\nimmediately implies that either $\\tilde A\\equiv0$ or $\\tilde A=\\O(e^{(4+4\\operatorname s-\\varepsilon)\\tau})$.\nThis is a consequence of Lemma~\\ref{lemm_decaylemmageneral}, and in this particular case stated as Lemma~\\ref{lemm_decaytildea}.\nFurther in our discussion, we can improve this exponential decay estimate to not only include the slowest, but also the second-slowest exponential term, see also Prop.~\\ref{prop_leftoftaub2additionaldecay}.\n\n\nSimilar convergence properties are obtained for the remaining variables upon convergence to the Kasner parabola.\nWe find that the five variables split into two groups, with identical decay rates in each group.\nThe variables $\\Sigma_+$, $\\tilde\\Sigma$ and $\\tilde A$, converge to their respective limit value exponentially to order~$4+4\\operatorname s-\\varepsilon$, while~$\\Delta$ and~$N_+$ decay to zero exponentially to order~$2+2\\operatorname s\\pm2\\sqrt{3(1-\\operatorname s^2)}-\\varepsilon$, where the sign depends on the sign of $\\Delta N_+$ at sufficiently negative times. In some cases, this sign is determined by where the limit point on the Kasner parabola is located with respect to\\ the Taub point~2.\nThis splitting into two different decay rates leads to a tension which we exploit on several occasion: The constraint equation~\\eqref{eqn_constraintgeneralone} written in the form\n\\begin{equation}\n\t\\tilde\\Sigma N_+^2-3\\Delta^2=(3\\Sigma_+^2+{\\kappa}\\tilde\\Sigma)\\tilde A\n\\end{equation}\nseparates the variables according to their decay rate. The fast decay of the left-hand side\\ can only be achieved if the other side vanishes altogether, which is what we use in Lemma~\\ref{lemm_deltanplussamesign} to show the relation stated in Prop.~\\ref{prop_specialrelationbparamkslimit} between the parameter~$\\kappa$ and the limit value~$\\operatorname s$ in Bianchi~B solutions. We further use this splitting to obtain lower bounds on the decay rates or even exclude certain subsets of the state space altogether upon convergence to specific subarcs of the Kasner parabola, see the proofs of Lemmata~\\ref{lemm_deltanplussamesign},~\\ref{lemm_deltanplusoppositesign}, and~\\ref{lemm_deltanplusrightoftaub2oppositesign}.\nOnce these decay and convergence rates are found, they are used to determine the subsets containing all solutions converging to the point Taub~2, or the subarcs of the Kasner parabola to the left and right of it, respectively.\n\n\n\n\n\\smallskip\n\nThe most important of the convergence statements achieved in Section~\\ref{section_asymptoticdecaykasner} are the following: For all variables, the lowest order exponential terms upon convergence to a point on the Kasner parabola $(\\operatorname s,1-\\operatorname s^2,0,0,0)$ with $\\operatorname s\\in[{-}1,1\/2]$ are obtained in Prop.~\\ref{prop_main_generalasymptoticproperties}.\nFor specific cases, this statement is refined in Prop.~\\ref{prop_leftoftaub2additionaldecay}, where we determine the lowest order exponential term more precisely and find the second-lowest, again for all variables. Convergence to a point on the Kasner parabola which is situated to the left of the point Taub~2 requires that $\\Delta N_+\\ge0$, and we have to treat the case of convergence to the other subarc of the Kasner parabola while $\\Delta N_+<0$ separately. In Lemma~\\ref{lemm_deltanplusrightoftaub2oppositesign}, we determine the lowest order exponential terms for all variables for this situation. This section is the foundation of most of the results obtained later.\n\nIn Section~\\ref{section_taubtwo}, we determine exactly which solutions converge to the point Taub~2. We rely on the results from the previous section and find that all such solutions have a local rotational symmetry.\nThis result is directly linked to Strong Cosmic Censorship: The only non-constant solutions converging to a point on the Kasner parabola for which both the Kretschmann scalar $R_{\\alpha\\beta\\gamma\\delta}R^{\\alpha\\beta\\gamma\\delta}$ and the contraction of the Ricci tensor with itself $R_{\\alpha\\beta}R^{\\alpha\\beta}$ possibly remain bounded as $\\tau\\rightarrow{-}\\infty$ are those which converge to the point Taub~2. As all such solutions are locally rotationally symmetric, convergence to the point Taub~2 and thereby boundedness of these two curvature invariants can be considered non-generic.\n\nSections~\\ref{section_leftoftaubtwo} and~\\ref{section_asymptoticsplanewave} treat the equivalent question for the arc of the Kasner parabola~$\\K$ to the left of Taub~2 and the plane wave equilibrium points~$\\planewave {\\kappa}$. We give qualitative results on the set of solutions with this convergence behaviour, using a theorem from dynamical systems theory which we recall in Appendix~\\ref{theo_centremftheory}. The statements we prove using dynamical systems theory are Thm~\\ref{theo_leftoftaubtworesult} and Thm~\\ref{theo_centreunstablegloballyplanewave}, which build upon where the eigenvalues to the linearised evolution equations are located in the complex plane.\n\nConvergence towards one of the remaining points on the Kasner parabola~$\\K$, located to the right of the point Taub~2, is treated in Section~\\ref{section_rightoftaubtwo}. In~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}, these points have been identified is local sources, and we make this statement more precise. For such perfect fluids where the convergence behaviour at early times is sufficiently well understood, we show that given a solution converging to a Kasner point to the right of Taub~2, all solutions intersecting a sufficiently small neighborhood of its limit point converge to a Kasner point close to this limit point.\n\n\n\n\\smallskip\n\n\nIn Appendix~\\ref{section_appendixbianchivariables}, we discuss the linear approximation of the evolution equations in the extended state space, which we investigate both on the Kasner parabola~$\\K$ and on the plane wave equilibrium points~$\\planewave {\\kappa}$.\n\nIn the second appendix, Appendix~\\ref{section_appendixdynamsystheo}, we present a result from the theory of dynamical systems.\nThis statement is an important prerequisite for Sections~\\ref{section_leftoftaubtwo} and~\\ref{section_asymptoticsplanewave}, where we discuss the properties of orbits converging to the Kasner parabola~$\\K$ to the left of the point Taub~2 and to the plane wave equilibrium points~$\\planewave {\\kappa}$.\n\n\n\n\n\\section{Basic properties of the expansion-normalised evolution}\n\\label{section_basicpropertiesexpnormevolution}\n\nIn this section, we start discussing properties of solutions to the evolution equation s with constraints in expansion-normalised variables, equation s~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega}.\nWe discuss the range of the individual variables and describe several subsets of the state space which remain invariant under the evolution. These correspond to specific Bianchi types, matter models, or families of models with additional symmetry. For a classification of Bianchi Lie group s in terms of their structure constants, we refer to Subsection~\\ref{subsect_appendixbianchiclassification}.\n\nOne sees from the evolution equation\\ of~$\\Omega$, equation ~\\eqref{eqn_evolutionomega}, that the sets $\\Omega=0$ and $\\Omega>0$ are invariant, and in the latter case the behaviour of~$\\Omega$ depends on the value of the constant~$\\gamma$. One distinguishes between the following cases:\n\\begin{defi}\n\tA solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} is called\n\t\\begin{itemize}\n\t\t\\item a vacuum solution if $\\Omega\\equiv0$ along the orbit,\n\t\t\\item an inflationary matter solution if $\\Omega>0$ along the orbit and $\\gamma\\in\\left[0,2\/3\\right)$,\n\t\t\\item a stiff fluid solution if $\\Omega>0$ along the orbit and $\\gamma=2$.\n\t\\end{itemize}\n\\end{defi}\nIn this paper we also cover the case $\\gamma\\in[2\/3,2)$. Of particular interest are the values $\\gamma=1$ and $\\gamma=4\/3$, which correspond to dust and radiation respectively.\nDue to the definition of the variable~$\\Omega$ as the expansion-normalised version of the energy density~$\\mu$, see equation ~\\eqref{eqn_definitionomega} below, the previous definition coincides with the distinction between vacuum and matter via $\\mu=0$ and $\\mu>0$ resp.\\ \n\n\nIn vacuum, one sees that the expression for~$q$, equation ~\\eqref{eqn_definitionqgeneral}, simplifies to\n\\begin{equation}\n\\label{eqn_definitionqvacuum}\n\tq=2(\\Sigma_+^2+\\tilde\\Sigma)=2(1-\\tilde A-\\tilde N),\n\\end{equation}\nand in the general case the following expressions are of use\n\\begin{equation}\n\\label{eqn_definitionqmatter}\n\\begin{split}\n\tq={}&\\frac32(2-\\gamma)(\\Sigma_+^2+\\tilde\\Sigma)+\\frac12(3\\gamma-2)(1-\\tilde A-\\tilde N),\\\\\n\t\t={}&2(1-\\tilde A-\\tilde N)-\\frac32(2-\\gamma)\\Omega,\\\\\n\t\t={}&2(\\Sigma_+^2+\\tilde\\Sigma)+\\frac12(3\\gamma-2)\\Omega.\n\\end{split}\n\\end{equation}\nWe note that the differential equations\\ with constraints~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} as well as the additional expressions for~$q$, equation s~\\eqref{eqn_definitionqvacuum}--\\eqref{eqn_definitionqmatter} are invariant under the symmetry\n\\begin{equation}\n\t(\\Delta,N_+) \\mapsto {-}(\\Delta,N_+).\n\\end{equation}\n\\begin{rema}\n\\label{rema_expnormvarconstraintspreserved}\n\tThe constraint equations~\\eqref{eqn_constraintgeneralone}--\\eqref{eqn_constraintgeneraltwo} are invariant under the evolution equations~\\eqref{eqns_evolutionbianchib} with~$q$ and~$\\tilde N$ as in~\\eqref{eqn_definitionqgeneral} and~\\eqref{eqn_definitiontilden}. For the first constraint, equation~\\eqref{eqn_constraintgeneralone}, this follows from\n\t\\begin{equation}\n\t\t(\\tilde\\Sigma\\tilde N-\\Delta^2-\\Sigma_+^2\\tilde A)'=4(q+\\Sigma_+-1)(\\tilde\\Sigma\\tilde N-\\Delta^2-\\Sigma_+^2\\tilde A).\n\t\\end{equation}\n\tThe second and fourth relation in~\\eqref{eqn_constraintgeneraltwo} are an immediate consquence of the evolution equations of~$\\tilde A$ in~\\eqref{eqns_evolutionbianchib} and of~$\\Omega$ in~\\eqref{eqn_evolutionomega}, which imply that~$\\tilde A=0$ and~$\\Omega=0$ are invariant sets.\n\n\tFor the remaining two inequalities~$\\tilde\\Sigma\\ge0$ and~$\\tilde N\\ge0$, notice that equality in both cases at some time implies~$\\Delta=0=\\Sigma_+\\tilde A$ at that time by the constraint equation~\\eqref{eqn_constraintgeneralone}.\n\tThe evolution equations for~$\\tilde\\Sigma$ and~$\\Delta$ in~\\eqref{eqns_evolutionbianchib} together with those for~$\\tilde N$ and~$\\Sigma_+\\tilde A$,\n\t\\begin{align}\n\t\t\\tilde N'={}&2(q+2\\Sigma_+)\\tilde N+4\\Delta N_+,\\\\\n\t\t(\\Sigma_+\\tilde A)'={}&(3q+4\\Sigma_+-2)\\Sigma_+\\tilde A-2\\tilde A\\tilde N,\n\t\\end{align}\n\tthen reveal that~$\\tilde\\Sigma\\equiv0\\equiv\\tilde N$ (as well as~$\\Delta\\equiv0\\equiv\\Sigma_+\\tilde A$) at all times. Further, $\\tilde\\Sigma$ and~$\\tilde N$ cannot change sign at different times, as~$\\tilde\\Sigma\\tilde N<0$ is excluded by the constraint equation~\\eqref{eqn_constraintgeneralone}. Hence, the inequalities~$\\tilde\\Sigma\\ge0$ and~$\\tilde N\\ge 0$ are preserved.\n\\end{rema}\n\n\\begin{rema}\n\\label{rema_statespacecompact}\n\tThe choice of expansion-normalised variables $(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)$ has two advantages. Firstly, normalisation yields a compact state space~\\eqref{eqn_constraintgeneralone}--\\eqref{eqn_omegageneral}, which facilitates the discussion of dynamical properties. In fact, equations ~\\eqref{eqn_constraintgeneraltwo},~\\eqref{eqn_omegageneral} and~\\eqref{eqn_definitionqmatter} in combination with\n\t$\\gamma\\in\\left[0,2\\right]$ yield\n\t\\begin{equation}\n\t\\label{eqn_constraintgeneralthree}\n\t\t\\Sigma_+\\in[{-}1,1],\\quad\n\t\t\\tilde\\Sigma\\in[0,1],\\quad\n\t\t\\tilde A\\in[0,1],\\quad\n\t\t\\tilde N\\in[0,1],\\quad\n\t\t\\Omega\\in[0,1],\\quad\n\t\tq\\in[{-}1,2].\n\t\\end{equation}\n\tThe second advantage is that the different Bianchi~B types are represented by invariant subsets of the same state space and can thus be discussed simultaneously.\n\t\\begin{table}[htp]\n\t\t\\centering\n\t\t\\caption{Bianchi~B invariant sets.}\n\t\\label{table_bianchibsubsets}\n\t\t\\begin{tabular}{ll}\n\t\t\tNotation & Restrictions \\\\\n\t\t\t\\hline\n\t\t\t\\noalign{\\vskip 1mm}\n\t\t\tB(VI$_{\\eta}$) & ${\\kappa}=\\nicefrac1{\\eta}<0$, $\\tilde A>0$ \\\\\n\t\t\tB$^\\pm$(VII$_{\\eta}$) & ${\\kappa}=\\nicefrac1{\\eta}>0$, $\\tilde A>0$, $N_+>0$ or $N_+<0$ \\\\\n\t\t\tB$^\\pm$(IV) & ${\\kappa}=0$, $\\tilde A>0$, $N_+>0$ or $N_+<0$ \\\\\n\t\t\tB(V) & ${\\kappa}=0$, $\\tilde A>0$, $\\Sigma_+=\\Delta=N_+=0$\n\t\t\\end{tabular}\n\t\\end{table}\nTable~\\ref{table_bianchibsubsets} lists the subsets and their names according to the Bianchi classification of Lie groups. We recall this classification in Subsection~\\ref{subsect_appendixbianchiclassification}. This table was already given by~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}, as were Tables~\\ref{table_bianchiasubsets} and~\\ref{table_bianchihighersymmetry}\\footnote{Note that only one of the three LRS subsets in Table~\\ref{table_bianchihighersymmetry} was mentioned in~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}, namely LRS Bianchi~II. LRS Bianchi~VI$_{{-}1}$ first appeared in~\\cite{wainwrightellis_dynamsystemsincosm}, it is also called LRS Bianchi~III in the literature.\n\n\tIn the literature, the subset LRS Bianchi~II in Table~\\ref{table_bianchihighersymmetry} is defined with $\\Delta^2=\\Sigma_+^2N_+^2$.\n\tConsidering the evolution equations of~$3\\Sigma_+^2-\\tilde\\Sigma$ and~$\\Sigma_+N_+\\pm\\Delta$, we realise however that it is the case with negative sign~$\\Sigma_+N_+-\\Delta=0$ which is preserved, not the one with positive sign, see also equations~\\eqref{eqn_derivativefunctionforscc}.\n\t}.\n\n\n\tIn Table~\\ref{table_bianchiasubsets}, two types of Bianchi~A models are given which appear as boundary sets of Bianchi~B invariant sets.\n\t\\begin{table}[htp]\n\t\t\\centering\n\t\t\\caption{Bianchi~A invariant sets.}\n\t\\label{table_bianchiasubsets}\n\t\t\\begin{tabular}{ll}\n\t\t\tNotation & Restrictions \\\\\n\t\t\t\\hline\n\t\t\t\\noalign{\\vskip 1mm}\n\t\t\tB$^\\pm$(II) & $\\tilde A=0$, $N_+>0$ or $N_+<0$ \\\\\n\t\t\tB(I) & $\\tilde A=\\Delta=N_+=0$\n\t\t\\end{tabular}\n\t\\end{table}\n\\end{rema}\nTable~\\ref{table_bianchihighersymmetry} lists several invariant subsets which describe Bianchi models with higher symmetry.\n\\begin{table}[htp]\n\t\\centering\n\t\\caption{Bianchi invariant sets with higher symmetry. (Abbreviations: FLRW= Friedman-Lema\u00eetre-Robertson-Walker, LRS= locally rotationally symmetric)}\n\\label{table_bianchihighersymmetry}\n\t\\begin{tabular}{lll}\n\t\tNotation & Class of models & Restrictions \\\\\n\t\t\\hline\n\t\t\\noalign{\\vskip 1mm}\n\t\t$S_{\\operatorname{LRS}}$(I) & LRS Bianchi~I & $\\Delta=\\tilde A=N_+=0$, $3\\Sigma_+^2=\\tilde\\Sigma$ \\\\\n\t\t$S_{\\operatorname{LRS}}^\\pm$(II) & LRS Bianchi~II & $\\tilde A=0$, $3\\Sigma_+^2=\\tilde\\Sigma$, $\\Delta=\\Sigma_+N_+$ \\\\\n\t\t$S_{\\operatorname{LRS}}^\\pm$(VI$_{{-}1}$)$=S_{\\operatorname{LRS}}^\\pm$(III) & LRS Bianchi~VI$_{{-}1}$ & ${\\kappa}={-}1$, $\\tilde A>0$, $3\\Sigma_+^2=\\tilde\\Sigma$, $\\Delta=\\Sigma_+N_+$ \\\\\n\t\t$S_0$(VI$_{\\eta}$) & Bianchi~VI$_{\\eta}$, $\\operatorname{tr} n=0$ & $\\Delta=N_+=0$, $3\\Sigma_+^2+{\\kappa}\\tilde\\Sigma=0$, $\\tilde A>0$ \\\\\n\t\t$S_{\\operatorname{FLRW}}$(V) & Bianchi~V FLRW & ${\\kappa}=0$, $\\Sigma_+=\\tilde\\Sigma=\\Delta=N_+=0$\\\\\n\t\t$S_{\\operatorname{FLRW}}^\\pm$(VII$_{\\eta}$) & Bianchi~VII$_{\\eta}$ FLRW & $\\Sigma_+=\\tilde\\Sigma=\\Delta=0$, ${\\kappa}\\tilde A=N_+^2>0$\n\t\\end{tabular}\n\\end{table}\n\nWe want to give particular attention to the locally rotationally symmetric (LRS) models. The property of local rotational symmetry in expansion-normalised variables is defined as follows:\n\\begin{defi}[Local rotational symmetry (expansion-normalised)]\n\\label{defi_lrsexpansionnorm}\n\tA solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_constraintgeneraltwo} is called locally rotationally symmetric (LRS) if\n\t\\begin{equation}\n\t\\label{eqn_lrsexpansionnorm}\n\t\t3\\Sigma_+^2=\\tilde\\Sigma, \\qquad\n\t\t\\Sigma_+N_+=\\Delta, \\qquad\n\t\t({\\kappa}+1)\\tilde A=0.\n\t\\end{equation}\n\\end{defi}\nComparison with the three LRS subsets given in Table~\\ref{table_bianchihighersymmetry} shows that the union of these subsets equals the union of all locally rotationally symmetric solutions.\nWe have now defined the notion of local rotational symmetry twice, in terms of expansion-normalised variables in Def.~\\ref{defi_lrsexpansionnorm} and in terms of initial data to Einstein's or\\-tho\\-go\\-nal\\ perfect fluid equations in Def.~\\ref{defi_lrsinitialdata}. In Subsection~\\ref{constr_highersymmetrysolutions}, we clarify why these two definitions are equivalent under suitable transformation between the initial data setting and the expansion-normalised variables.\n\nConsidering the derivatives\n\\begin{align}\n\\label{eqn_derivativefunctionforscc}\n\t(3\\Sigma_+^2-\\tilde\\Sigma)'={}&2(q-2)(3\\Sigma_+^2-\\tilde\\Sigma)-4N_+(\\Sigma_+N_+-\\Delta)+4\\tilde A\\Sigma_+(1+{\\kappa}),\\\\\n\t(\\Sigma_+N_+-\\Delta)'={}&2(q-2-2\\Sigma_++1)(\\Sigma_+N_+-\\Delta)+2N_+(3\\Sigma_+^2-\\tilde\\Sigma),\n\\end{align}\none sees that the three LRS subsets are invariant under the evolution equation s~\\eqref{eqns_evolutionbianchib}. For the remaining sets given in Tables~\\ref{table_bianchibsubsets}, \\ref{table_bianchiasubsets} and~\\ref{table_bianchihighersymmetry}, invariance follows by direct computation, see also Remark~\\ref{rema_expnormvarconstraintspreserved} for Bianchi VII$_\\eta$ FLRW.\n\n\\begin{rema}\n\\label{rema_singularconstraintequ}\n\tThe evolution equation s~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_definitiontilden}\n\tdefine a dynamical system in~$\\maths{R}^5$, and we will on several occasions apply dynamical systems methods to this. Whenever we restrict our attention to only these evolution equation s, i.\\,e.\\  without assuming the constraint equation s~\\eqref{eqn_constraintgeneralone}--\\eqref{eqn_constraintgeneraltwo}, we call this the \\textit{evolution equation s in the extended state space}.\n\n\tThe evolution in the physical, i.\\,e.\\  constrained, state space, is obtained via restriction to the set defined by the constraint equation s~\\eqref{eqn_constraintgeneralone}--\\eqref{eqn_constraintgeneraltwo}.\n\tThe first equation is invariant under the evolution and describes a submani\\-fold\\ as long as the gradient of its left-hand side\n\t\\begin{equation}\n\t\\label{eqn_gradientconstraintequ}\n\t\t({-}2\\Sigma_+\\tilde A\\,,\\,\\frac13(N_+^2-{\\kappa}\\tilde A)\\,,\\,{-}2\\Delta\\,,\\,{-}(\\Sigma_+^2+\\frac13{\\kappa}\\tilde\\Sigma)\\,,\\,\\frac23\\tilde\\Sigma N_+)\n\t\\end{equation}\n\tdoes not vanish, implying that the hypersurface is non-singular. The only exceptions to a non-vanishing gradient are:\n\t\\begin{itemize}\n\t\t\\item If ${\\kappa}>0$: $\\Sigma_+=\\tilde\\Sigma=\\Delta=0$, $N_+^2={\\kappa}\\tilde A$. This defines an invariant set of dimension one.\n\t\t\\item If ${\\kappa}=0$: $\\Sigma_+=N_+=\\Delta=0$. This defines an invariant set of dimension two.\n\t\t\\item If ${\\kappa}<0$: $\\Delta=\\tilde A=N_+=0$, ${\\kappa}\\tilde\\Sigma+3\\Sigma_+^2=0$. This defines an invariant set of dimension one.\n\t\\end{itemize}\n\\end{rema}\n\\begin{rema}\n\tThere is a similar set of expansion-normalised coordinates which is used to describe Bianchi class~A models, but does not apply to class~B models. It was introduced by Wainwright and Hsu in~\\cite{wainwrighthsu_dynamicalsystemsapproachbianchiorthogonalA} and motivated the definition of the present coordinates. In the cases of Bianchi~I and II, models can be decribed in both sets of variables.\n\n\tCertain Bianchi perfect fluid spacetimes with a Lie group\\ of type VI$_{{-}1\/9}$ cannot be described by the evolution equations~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega}. This is the case for the so-called 'exceptional' Bianchi~B perfect fluids, a notion which we explain in Remark~\\ref{rema_exceptionalityexplained}. Initial data sets in these spacetimes admit an additional degree of freedom compared to the 'non-exceptional' ones given in Definition~\\ref{defi_initialdatabianchib}. In these 'exceptional' cases as well, it is possible to introduce a set of expansion-normalised variables, this has been done in~\\cite{hewitthorwoodwainwright_asymptdynamexceptbianchicosm}. Due to the additional freedom, these spacetimes are described by evolution equations in six dimensions instead of five.\n\\end{rema}\n\n\n\n\\section{The Kasner parabola and the plane wave equilibrium points}\n\\label{section_kasnerparabolaplanewave}\n\n\nThe dynamical system for the expansion-normalised variables described by equation s~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} possesses a number of equilibrium points and sets, i.\\,e.\\  points where the right-hand side\\ of the evolution equations~\\eqref{eqns_evolutionbianchib} becomes zero. These equilibrium sets have been studied in~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}, to whom we refer for more details. For our present discussion, the Kasner parabola~$\\K$ and the plane wave equilibrium points~$\\planewave k$ are of importance, see Definition~\\ref{defi_kasnerparabola} and Definition~\\ref{defi_planewaveexpansionnorm}\n\nOn the Kasner parabola~$\\K$, one finds $\\Omega=0$. Furthermore, in case $\\gamma<2$, the set~$\\K$ is characterised by $q=2$. A closer look at the evolution equations reveals that the Kasner parabola is a curve in the $\\Sigma_+\\tilde\\Sigma$-plane consisting of individual equilibrium points.\nInformation about the local stability can be drawn from the linearised evolution equations in the extended five-dimensional space.\nWe give the explicit form of this vector field for points on the Kasner parabola~$\\K$ in Appendix~\\ref{subsect_appendixlinearisedevolutionkasner}.\nThe eigenvalues of this vector field are given in~\\eqref{eqn_eigenvalueskasner}.\nThe number of positive, negative and zero eigenvalues corresponds to the quali\\-tative behaviour of orbits close to the Kasner parabola. This is a result from dynamical systems theory, which we state in Appendix~\\ref{section_appendixdynamsystheo}, but do not make use of in this section. The eigenvalues to the linearised evolution equations will appear further down as the exponential decay rates of certain linear combination of the variables.\n\nWe notice that for two special points on the Kasner parabola~$\\K$ the number of zero eigenvalues is greater than one, namely the points Taub~1 and Taub~2, see Definition~\\ref{defi_taubpoints}.\nEspecially the latter point will play a dominant role in our discussion, as it is the limit point of locally rotationally symmetric solutions, which constitute exceptions to the Strong Cosmic Censorship conjecture. To the right of the point Taub~2 ($1\/2<\\Sigma_+\\le1$) all non-vanishing eigenvalues are positive, while to the left of Taub~2 ($-1<\\Sigma_+<1\/2$) exactly one of the four non-vanishing eigenvalues is negative.\nThis difference reflects a difference in qualitative behaviour which we explore in more detail in this paper.\n\nThe plane wave equilibrium points~$\\planewave {\\kappa}$ form a curve consisting of individual equilibrium points with $\\Omega=0$, as was the case for the Kasner parabola.\nUsing the function\n\\begin{equation}\n\\label{eqn_definitionfunctionz}\n\tZ\\coloneqq(1+\\Sigma_+)^2-\\tilde A,\n\\end{equation}\nwhich is discussed in~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}, the set~$\\planewave {\\kappa}$ can be characterised as follows: Direct computation shows that the constraint equation~\\eqref{eqn_constraintgeneralone} is equivalent to\n\\begin{equation}\n\\label{eqn_functionzextended}\n\t(\\tilde\\Sigma-\\tilde N)^2+4\\Delta^2+2(\\tilde\\Sigma+\\tilde N)\\Omega+(\\Omega-Z)^2=4Z(\\Sigma_+^2+\\frac12(\\tilde\\Sigma+\\tilde N)),\n\\end{equation}\nwhich due to the non-negativity of~$\\tilde\\Sigma$, $\\tilde N$ and~$\\Omega$, see~\\eqref{eqn_constraintgeneraltwo}, means that the function~$Z$ is non-negative, and vanishes if and only if\n\\begin{equation}\n\\label{eqn_functionzzeros}\n\t\\Omega=0,\\qquad \\Delta=0,\\qquad -\\Sigma_+(1+\\Sigma_+)=\\tilde\\Sigma=\\tilde N.\n\\end{equation}\nNote that the identity $-\\Sigma_+(1+\\Sigma_+)=\\tilde\\Sigma$ is a reformulation of $\\Omega=Z$, using the identities~\\eqref{eqn_functionzzeros}.\nConsequently, $Z=0$ characterises the plane wave equilibrium points together with the Kasner point Taub~1.\nIn case $\\tilde A=0$, $Z=0$ characterises exactly the point Taub~1.\nThe derivative of the function~$Z$ is\n\\begin{equation}\n\tZ'={-}2(2-q)Z+3(2-\\gamma)(1+\\Sigma_+)\\Omega\n\\end{equation}\ndue to the evolution equations ~\\eqref{eqns_evolutionbianchib}.\n\\begin{rema}\n\tBoth the Kasner parabola~$\\K$ and the plane wave equilibrium points~$\\planewave {\\kappa}$ can be interpreted in terms of initial data to Einstein's or\\-tho\\-go\\-nal\\ perfect fluid equations. This is done in detail in Subsection~\\ref{constr_highersymmetrysolutions}, where we explain how to establish a relation between initial data and solutions to Einstein's equations on the one hand and initial data and solutions to the evolution equation s~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} on the other. We find that\n\t\\begin{itemize}\n\t\t\\item The Kasner parabola~$\\K$ corresponds to vacuum Bianchi type~I initial data.\n\t\t\\item The set of plane wave equilibrium points~$\\planewave {\\kappa}$ together with the point Taub~1 correspond to plane wave equilibrium initial data as in Def.~\\ref{defi_planewaveinitialdata}.\n\t\t\\item The point Taub~1 corresponds to initial data of Bianchi type~I which is of plane wave equilibrium type.\n\t\t\\item The point Taub~2 corresponds to initial data of Bianchi type~I which is locally rotationally symmetric and the symmetric two-tensor~$k$ additionally satisfies~${k}_{11}>{k}_{22}$.\n\t\\end{itemize}\n\\end{rema}\n\nThe Kasner parabola and the plane wave equilibrium points are of central importance when investigating the asymptotic behaviour of orbits as $\\tau\\rightarrow{-}\\infty$. In~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}, the $\\alpha$-limit set of (non-constant, generic) orbits in vacuum and in inflationary matter ($\\gamma\\in\\left[0,2\/3\\right)$) are determined. We state and prove a refined version of their Prop.~5.1 and Prop.~5.2.\n\\begin{prop}[Alpha-limit sets in vacuum and inflationary matter]\n\\label{prop_alphalimitsets_vacuum_inflat}\n\tAssume vacuum, i.\\,e.\\  $\\Omega=0$. If $\\Gamma(\\tau)=(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)(\\tau)$, $\\tau\\in\\maths{R}$, is a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega}, then the $\\alpha$-limit set of~$\\Gamma$ satisfies\n\t\\begin{equation}\n\t\t\\alpha(\\Gamma)\\subset\\K\\cup\\planewave {\\kappa},\n\t\\end{equation}\n\twhich means that for every sequence $\\tau_n\\rightarrow{-}\\infty$ all accumulation points of $\\Gamma(\\tau_n)$ are contained in $\\K\\cup\\planewave {\\kappa}$. The only solutions with\n\t\\begin{equation}\n\t\t\\alpha(\\Gamma)\\cap\\planewave {\\kappa}\\not=\\emptyset\n\t\\end{equation}\n\tare constant solutions.\n\n\tAssume inflationary matter, i.\\,e.\\  $\\Omega>0$, $\\gamma\\in\\left[0,2\/3\\right)$. If $\\Gamma(\\tau)=(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)(\\tau)$, $\\tau\\in\\maths{R}$, is a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega}, then the $\\alpha$-limit set of~$\\Gamma$ satisfies\n\t\\begin{equation}\n\t\t\\alpha(\\Gamma)\\subset\\K\\cup\\planewave {\\kappa}\\cup\\{\\Sigma_+=\\tilde\\Sigma=\\Delta=\\tilde A=N_+=0\\}.\n\t\\end{equation}\n\tThere are no solutions whose $\\alpha$-limit set intersects both~$\\K\\setminus\\operatorname{T1}$ and~$\\planewave {\\kappa}$.\n\tThe only solution with\n\t\\begin{equation}\n\t\t\\{\\Sigma_+=\\tilde\\Sigma=\\Delta=\\tilde A=N_+=0\\}\\subset\\alpha(\\Gamma)\n\t\\end{equation}\n\tis the constant solutions.\n\\end{prop}\nAlongside with the proof of this proposition, we find a statement about asymptotic behaviour in the future time direction in case of inflationary matter: for~$\\tau\\rightarrow{+}\\infty$, all non-constant solutions converge to one point. Note that this is the only time that we consider this time direction, all other statements treat the case~$\\tau\\rightarrow{-}\\infty$.\n\\begin{prop}\n\\label{prop_omegalimitsetinflationary}\n\tAssume inflationary matter, i.\\,e.\\  $\\Omega>0$, $\\gamma\\in\\left[0,2\/3\\right)$, and consider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega}. Then\n\t\\begin{equation}\n\t\t\\lim_{\\tau\\rightarrow{+}\\infty}(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)(\\tau)=(0,0,0,0,0).\n\t\\end{equation}\n\\end{prop}\n\\begin{proof}[Proof of Prop.~\\ref{prop_alphalimitsets_vacuum_inflat}]\n\tWe start with the case of a vacuum solution $\\Omega=0$. If~$q<2$ and $Z>0$, then the derivative of~$Z$ defined in~\\eqref{eqn_definitionfunctionz} is negative, which means that the function~$Z$ is strictly monotone decreasing.\n\tThe $\\alpha$-limit set is therefore contained in the union of $\\{q=2\\}$ and $\\{Z=0\\}$. The first set is the Kasner parabola~$\\K$, while the second describes the plane wave equilibrium points $\\planewave {\\kappa}$ together with the point Taub~1.\n\tSuppose there is a solution with an $\\alpha$-limit point in~$\\planewave {\\kappa}\\cup\\operatorname{T1}$. Then there is a sequence of times $\\tau_k\\rightarrow{-}\\infty$ such that \n\t\\begin{equation}\n\t\tZ(\\tau_k)\\le\\frac1k.\n\t\\end{equation}\n\tIn combination with $Z\\ge0$ and monotonicity, this implies that~$Z$ is vanishing identically along the whole orbit. The orbit is therefore contained in the zero set of~$Z$, which are the plane wave equilibrium points together with the point Taub~1. Hence the orbit is the constant orbit.\n\n\tFor inflationary matter solutions $\\Omega>0$, $\\gamma\\in[0,2\/3)$, one reformulates the evolution equation\\ for~$\\Omega$ into the form\n\t\\begin{equation}\n\t\t\\Omega'=(4(\\Sigma_+^2+\\tilde\\Sigma)-(3\\gamma-2)(1-\\Omega))\\Omega\n\t\\end{equation}\n\tusing~\\eqref{eqn_definitionqmatter}. Due to the restrictions on the individual variables~\\eqref{eqn_constraintgeneralthree}, this shows that~$\\Omega$ is monotone increasing and implies that the $\\alpha$-limit set is contained in the union of $\\{\\Omega=0\\}$ and $\\{\\Omega=1\\}$, as the latter is equivalent to the bracket vanishing. It follows immediately from monotonicity that if there is an~$\\alpha$-limit point in $\\{\\Omega=1\\}$, then the whole solution is contained in this set.\n\tThe condition $\\Omega=1$ characterises the point $\\Sigma_+=\\tilde\\Sigma=\\Delta=\\tilde A=N_+=0$, which proves the last statement.\n\n\tFor $0<\\Omega<1$, one computes\n\t\\begin{equation}\n\t\t(\\frac{Z}{\\Omega})'={-}(6-3\\gamma)\\frac{Z}{\\Omega}+(6-3\\gamma)(1+\\Sigma_+),\n\t\\end{equation}\n\tand concludes from\n\t\\begin{equation}\n\t\t(e^{(6-3\\gamma)\\tau}\\frac{Z}{\\Omega})'=(6-3\\gamma)(1+\\Sigma_+)e^{(6-3\\gamma)\\tau}\\ge0\n\t\\end{equation}\n\tand the fact that both~$\\Omega$ and~$Z$ are non-negative\n\tthat the quantity $e^{(6-3\\gamma)\\tau}{Z}\/{\\Omega}$ decreases monotonically to some non-negative constant~$B_1$ as $\\tau\\rightarrow{-}\\infty$. \n\tThis has to be understood in the sense that the expression decreases when going backwards in time, here and on similar occurences of this formulation further down.\n\tMonotone convergence to~$B_1$ implies\n\t\\begin{equation}\n\t\t\\frac{Z}{\\Omega}=B_1e^{-(6-3\\gamma)\\tau}+\\int_{{-}\\infty}^\\tau(6-3\\gamma)(1+\\Sigma_+)e^{(6-3\\gamma)(s-\\tau)}ds,\n\t\\end{equation}\n\twhere the integral is well-defined and bounded by~$2$ due to the bound on~$\\Sigma_+$ from~\\eqref{eqn_constraintgeneralthree}.\n\n\tThere are two cases to consider. If~$B_1>0$, one can reformulate the previous equality into\n\t\\begin{equation}\n\t\te^{-(6-3\\gamma)\\tau}\\Omega=\\frac1{B_1}(Z-\\Omega\\int_{{-}\\infty}^\\tau(6-3\\gamma)(1+\\Sigma_+)e^{(6-3\\gamma)(s-\\tau)}ds),\n\t\\end{equation}\n\tand sees that the right-hand side\\ is bounded. \n\tThe evolution equation\\ for the left-hand side\\ is\n\t\\begin{equation}\n\t\t(e^{-(6-3\\gamma)\\tau}\\Omega)'=2(q-2)e^{-(6-3\\gamma)\\tau}\\Omega,\n\t\\end{equation}\n\twhich is of the form~$f'=g\\cdot f$. For~$C^1$ functions~$f$, $g$ and~$f$ bounded as~$\\tau\\rightarrow{-}\\infty$, integration of~$f'\/f$ implies that~$g$ is integrable on~$({-}\\infty,0)$. \n\tIn our case, one concludes that~$q-2$ is integrable on the interval $({-}\\infty,0)$.\n\tOne further knows from the evolution equations that the derivative of~$q-2$ is a polynomial in the expansion-normalised variables and consequently bounded, and thus can conclude that~$q\\rightarrow2$ as $\\tau\\rightarrow{-}\\infty$. The $\\alpha$-limit set is therefore contained in $\\{q=2\\}$, which characterises the Kasner parabola.\n\n\tIf $B_1=0$, then we have already argued above that\n\t\\begin{equation}\n\t\t\\frac Z\\Omega\\le2.\n\t\\end{equation}\n\tAs $\\Omega\\rightarrow0$ for $\\tau\\rightarrow{-}\\infty$ due to monotonicity and the fact that the $\\alpha$-limit set is contained in $\\{\\Omega=0\\}$, this implies $Z\\rightarrow0$. One concludes that the $\\alpha$-limit set is contained in the set $\\{Z=0\\}$, which characterises the plane wave equilibrium points~$\\planewave {\\kappa}$ together with the point Taub~1. This proves the first statement for inflationary matter models.\n\n\tThe two cases for~$B_1$ are mutually exclusive, and the only point satisfying both $q=2$ and $Z=0$ is the point Taub~1, which is contained in the Kasner parabola~$\\K$.\n\tThis concludes the proof.\n\\end{proof}\n\\begin{proof}[Proof of Prop.~\\ref{prop_omegalimitsetinflationary}]\n\tWe have found in the previous proof that~$\\Omega$ is monotone increasing with\n\t\\begin{equation}\n\t\t\\Omega'=(4(\\Sigma_+^2+\\tilde\\Sigma)-(3\\gamma-2)(1-\\Omega))\\Omega.\n\t\\end{equation}\n\tDue to the restrictions on the individual variables~\\eqref{eqn_constraintgeneralthree}, this shows that all solutions with~$\\Omega>0$ have to satisfy $\\Omega\\rightarrow1$ as $\\tau\\rightarrow{+}\\infty$, because the condition~$\\Omega=1$ is equivalent to the bracket vanishing. Due to the definition of~$\\Omega$ in equation~\\eqref{eqn_omegageneral} and the restriction of the individual variables in~\\eqref{eqn_constraintgeneralthree}, this concludes the proof.\n\\end{proof}\n\n\nIn the next proposition, we strengthen the result about the $\\alpha$-limit set: In inflationary matter models, there cannot be more than one $\\alpha$-limit point in the plane wave equilibrium points~$\\planewave {\\kappa}$, i.\\,e.\\  we find convergence.\n\\begin{prop}[Convergence to plane wave equilibrium points]\n\\label{prop_convergencetoplanewave}\n\tAssume inflationary matter, i.\\,e.\\  $\\Omega>0$, $\\gamma\\in\\left[0,2\/3\\right)$, and consider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} whose $\\alpha$-limit set is contained in $\\planewave {\\kappa}\\cup\\operatorname{T1}$. Then there is an $\\operatorname s\\in\\left[{-}1,0\\right]$ such that\n\t\\begin{equation}\n\t\t\\lim_{\\tau\\rightarrow{-}\\infty}(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)(\\tau)=(\\operatorname s,-\\operatorname s(1+\\operatorname s),0,(1+\\operatorname s)^2,\\nu_+(\\operatorname s)),\n\t\\end{equation}\n\twhere $\\nu_+^2(\\operatorname s)=(1+\\operatorname s)({\\kappa}(1+\\operatorname s)-3\\operatorname s)$.\n\\end{prop}\n\\begin{proof}\n\tIf the point Taub~1 is the only $\\alpha$-limit point, there is nothing to show. We assume therefore without loss of generality\\ that there is at least one $\\alpha$-limit point contained in~$\\planewave {\\kappa}$.\n \tDue to the assumption on the $\\alpha$-limit set, the solution satisfies $Z\\rightarrow0$ as $\\tau\\rightarrow{-}\\infty$. The zero set of~$Z$ is characterised by expressions~\\eqref{eqn_functionzzeros}, \n \tand inserting these into the third expression for~$q$ in~\\eqref{eqn_definitionqmatter}, we conclude that the solution satisfies\n\t\\begin{equation}\n\t\\label{eqn_convergenceqinflationary}\n\t\tq+2\\Sigma_+\\rightarrow0\n\t\\end{equation}\n\tas $\\tau\\rightarrow{-}\\infty$.\n\tAll plane wave equilibrium points~$\\planewave {\\kappa}$ satisfy ${-}1<\\Sigma_+\\le0$, which means that~$q$ converges to a non-negative value. In combination with the fact that~$3\\gamma-2$ is a strictly negative constant, this implies that the factor $2q-(3\\gamma-2)$ in the evolution equation ~\\eqref{eqn_evolutionomega} is strictly positive for $\\tau\\le\\tau_0$ sufficiently negative. Consequently, $\\Omega$ decays to zero exponentially as $\\tau\\rightarrow{-}\\infty$.\n\n\tWe have seen in the proof of Prop.~\\ref{prop_alphalimitsets_vacuum_inflat}, that the solutions under consideration satisfy $B_1=0$, and consequently $0\\le Z\\le2\\Omega$.\n\tTherefore, $Z$ decays to zero exponentially as well.\n\tBoundedness of the state space, see Remark~\\ref{rema_statespacecompact}, therefore reveals that the right-hand side\\ of equation~\\eqref{eqn_functionzextended} decays to zero exponentially, and therefore the same has to hold for all terms appearing on the left-hand side. Using the identities~\\eqref{eqn_functionzzeros} to rewrite~$\\Omega-Z\\rightarrow0$, we find \n\texponential decay for\n\t\\begin{equation}\n\t\t\\Sigma_+(1+\\Sigma_+)+\\tilde\\Sigma\\rightarrow0,\\qquad\n\t\t\\Delta\\rightarrow0,\n\t\\end{equation}\n\tand as a consequence also for $q+2\\Sigma_+$, applying the third expression in equation~\\eqref{eqn_definitionqmatter}. Inserting this into the evolution equation ~\\eqref{eqns_evolutionbianchib} shows the same decay for $\\Sigma_+'$, therefore $\\Sigma_+$ converges to some $\\operatorname s\\in\\left[{-}1,0\\right]$. The limiting values for the remaining variables follows from the definition of the plane wave equilibrium points~$\\planewave {\\kappa}$, Def.~\\ref{defi_planewaveexpansionnorm}.\n\\end{proof}\nConsider now a non-constant solution to~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} in vacuum or inflationary matter whose $\\alpha$-limit set has a non-empty intersection with~$\\K\\setminus\\operatorname{T1}$ instead.\nDue to Prop.~\\ref{prop_alphalimitsets_vacuum_inflat}, this implies that the whole $\\alpha$-limit set is contained in the Kasner parabola~$\\K$.\nThen\n\\begin{equation}\n\\label{eqn_convergenceratestepone}\n\t\\Sigma_+^2+\\tilde\\Sigma\\rightarrow1,\\quad\\Delta\\rightarrow0,\\quad \\tilde A\\rightarrow0,\\quad N_+\\rightarrow0,\\quad\\tilde N\\rightarrow0,\\quad \\Omega\\rightarrow0,\\quad q\\rightarrow 2,\n\\end{equation}\nas $\\tau\\rightarrow{-}\\infty$, as otherwise compactness of the state space, see Remark~\\ref{rema_statespacecompact}, would yield an $\\alpha$-limit point which does not lie on the Kasner parabola.\nA convergence result similar to the previous statement is achieved further down in Prop.~\\ref{prop_convergencetokasner}. Its proof needs some additional work.\n\nFor the discussion of more detailed properties of solutions close to the $\\alpha$-limit points, we frequently make use of the following lemma which appears with slightly different notation as~\\cite[Lemma~8]{ringstrom_curvblowupbianchiviiiandixvacuumspacetimes}.\n\\begin{lemm}\n\\label{lemm_decaylemmageneral}\n\tConsider a positive function $M:\\maths{R}\\rightarrow(0,\\infty)$ satisfying $M'=\\zeta M$, where $\\zeta:\\maths{R}\\rightarrow\\maths{R}$ and $\\zeta(\\tau)\\rightarrow\\iota$ as $\\tau\\rightarrow{-}\\infty$. Then for all $\\varepsilon>0$ there is a $\\tau_\\varepsilon>{-}\\infty$ such that $\\tau\\le \\tau_\\varepsilon$ implies\n\t\\begin{equation}\n\t\te^{(\\iota+\\varepsilon)\\tau}\\le M(\\tau)\\le e^{(\\iota-\\varepsilon)\\tau}.\n\t\\end{equation}\n\\end{lemm}\nThis lemma can be applied to the density parameter~$\\Omega$ without any further assumptions: In vacuum $\\Omega=0$ holds, while inflationary matter models satisfy $\\Omega>0$ with $\\gamma\\in\\left[0,2\/3\\right)$. If the $\\alpha$-limit set is contained in the Kasner parabola~$\\K$, one uses~\\eqref{eqn_convergenceratestepone} to find convergence to zero at rate\n\\begin{equation}\n\\label{eqn_decayomega}\n\te^{(6-3\\gamma+\\varepsilon)\\tau}\\le \\Omega(\\tau) \\le e^{(6-3\\gamma-\\varepsilon)\\tau},\n\\end{equation}\nfor $\\tau\\le\\tau_\\varepsilon$ sufficiently negative.\n\\begin{nota}\n\tWhen we are only interested in the upper bound as $\\tau\\rightarrow{-}\\infty$, we also make use of the big~$\\O$ notation, i.\\,e.\\  we write\n\t\\begin{equation}\n\t\t\\Omega=\\O(e^{(6-3\\gamma)\\tau}),\n\t\\end{equation}\n\tas $\\tau\\rightarrow{-}\\infty$, if we want to say that there is a $\\tau_0>{-}\\infty$ and a~$C_\\Omega>0$ such that $\\tau\\le \\tau_0$ implies\n\t$\n\t\t\\Omega(\\tau) \\le C_\\Omega e^{(6-3\\gamma)\\tau}\n\t$.\n\tAs we are in the present paper not interested in any other limit than $\\tau\\rightarrow{-}\\infty$, we frequently \n\tomit the range of~$\\tau$.\n\\end{nota}\nIf the function~$\\zeta$ in the previous lemma converges exponentially, the statement can be improved as follows.\n\\begin{lemm}\n\\label{lemm_decaylemmageneralimproved}\n\tConsider a postitive function $M:\\maths{R}\\rightarrow(0,\\infty)$ satisfying $M'=\\zeta M$, where $\\zeta:\\maths{R}\\rightarrow\\maths{R}$ and $\\zeta(\\tau)=\\iota+\\O(e^{\\xi\\tau})$ as $\\tau\\rightarrow{-}\\infty$, for some constants $\\xi>0$, $\\iota$. Then there are constants~$c_M,C_M>0$ and a~$\\tau_0$ such that $\\tau\\le \\tau_0$ implies\n\t\\begin{equation}\n\t\tc_M e^{\\iota\\tau}\\le M(\\tau)\\le C_M e^{\\iota\\tau}.\n\t\\end{equation}\n\\end{lemm}\n\\begin{proof}\n\tIntegration of\n\t\\begin{equation}\n\t\t(\\ln M)'=\\frac{M'}{M}=\\zeta=\\iota+\\O(e^{\\xi\\tau})\n\t\\end{equation}\n\tyields\n\t\\begin{equation}\n\t\tM(\\tau)=e^{\\iota\\tau}M(\\tau_0)e^{-\\iota\\tau_0+\\O(e^{\\xi\\tau_0})}.\n\t\\end{equation}\n\tThe last two factors are contained in some interval $\\left[c_M,C_M\\right]\\subset(0,\\infty)$ for~$\\tau$ smaller than a fixed number~$\\tau_0$.\n\\end{proof}\n\n\n\n\\section{Convergence to Taub~1}\n\\label{section_taubone}\n\n\nIn this section, we show that in vacuum and inflationary matter the only orbit with Taub~1 as an $\\alpha$-limit point is the constant orbit. As a consequence, this special Kasner point can be neglected when we determine in more detail the asymptotic behaviour close to the Kasner parabola.\n\\begin{prop}\n\\label{prop_taubone}\n\tAssume either vacuum or inflationary matter, i.\\,e.\\  either $\\Omega=0$ or $\\Omega>0$, $\\gamma\\in\\left[0,2\/3\\right)$, and consider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} such that the point Taub~1\n\t\\begin{equation}\n\t\t(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)=({-}1,0,0,0,0)\n\t\\end{equation}\n\tis contained in the $\\alpha$-limit set. Then the solution is the constant orbit.\n\\end{prop}\n\\begin{prop}\n\\label{prop_tauboneothermatter}\n\tAssume $\\Omega>0$ and $\\gamma\\in\\left[2\/3,2\\right)$, and consider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} converging to the point Taub~1\n\t\\begin{equation}\n\t\t(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)=({-}1,0,0,0,0)\n\t\\end{equation}\n\tas $\\tau\\rightarrow{-}\\infty$. Then the solution is the constant orbit.\n\\end{prop}\n\n\\begin{rema}\n\\label{rema_tauboneexcludedinnonvacuum}\n\tAs the point Taub~1 is contained in the vacuum set~$\\Omega=0$, we conclude from these propositions that if~$\\Omega>0$, then the point Taub~1 is not allowed as an $\\alpha$-limit point.\n\\end{rema}\n\\begin{proof}[Proof of Prop.~\\ref{prop_taubone}]\n\tThe proof revolves around the function~$Z$ defined by equation~\\eqref{eqn_definitionfunctionz}, and we start with the case of a vacuum solution $\\Omega=0$. As the point Taub~1 is an $\\alpha$-limit point but does not lie in~$\\planewave {\\kappa}$, Prop.~\\ref{prop_alphalimitsets_vacuum_inflat} yields that the $\\alpha$-limit set is contained in the Kasner parabola~$\\K$, which is characterised by $q=2$.\n\tIn the point Taub~1, the function~$Z$ vanishes. This point being an~$\\alpha$-limit point therefore implies that $Z\\rightarrow0$ along a time sequence~$\\tau_k\\rightarrow{-}\\infty$.\n\tIn the proof of Prop.~\\ref{prop_alphalimitsets_vacuum_inflat}, we have used the existence of such a time sequence together with monotonicity of the function~$Z$ to conclude that~$Z$ is vanishing constantly along the whole orbit. This argument applies to the present case, and we conclude that the orbit is contained in the zero set of~$Z$. The set~$Z=0$ equals the union of~$\\planewave k$ and the point Taub~1 and consists solely of equilibrium points. As a consequence, the solution has to be the constant one in  the point Taub~1, as this is the only point which satisfies both~$Z=0$ and~$q=2$.\n\n\tIn the case of matter $\\Omega>0$, one first realises that due to its evolution equation ~\\eqref{eqns_evolutionbianchib}, if~$\\tilde A$ vanishes at one time, then it vanishes along the whole orbit. One then reformulates the evolution of~$\\Sigma_+$ using the definition of~$\\tilde N$ from~\\eqref{eqn_definitiontilden} together with equation ~\\eqref{eqn_definitionqmatter} for~$q$ to find\n\t\\begin{equation}\n\t\t\\Sigma_+'={-}2\\Sigma_+\\tilde A+\\frac23(\\Sigma_++1){\\kappa}\\tilde A-\\frac23(\\Sigma_++1)N_+^2-\\frac32(2-\\gamma)\\Omega.\n\t\\end{equation}\n\tIf $\\tilde A=0$, then $\\Sigma_+'<0$, as~$\\Sigma_+$ is contained in the interval $[{-}1,1]$ by Remark~\\ref{rema_statespacecompact}. Hence, $\\Sigma_+$ is monotone decreasing, and an argument similar to the one for~$Z$ in the vacuum case applies:\n\tThe assumption on the $\\alpha$-limit set gives a sequence of times $\\tau_k\\rightarrow{-}\\infty$ such that \n\t\\begin{equation}\n\t\t\\Sigma_+(\\tau_k)\\le{-}1+\\frac1k.\n\t\\end{equation}\n\tIn combination with $\\Sigma_+\\ge{-}1$ and monotonicity, this shows that $\\Sigma_+={-}1$ along the whole orbit.\n\tThis in turn implies that~$Z$ vanishes along the orbit and concludes the proof for $\\tilde A=0$, as then $\\Omega=0$ due to~\\eqref{eqn_functionzzeros}, a contradiction.\n\n\tAssume therefore $\\tilde A>0$ and $\\Omega>0$.\n\tWe first prove convergence, i.\\,e.\\  that Taub~1 is the unique $\\alpha$-limit point, then show that the only orbit converging to this point is the constant one. From Prop.~\\ref{prop_alphalimitsets_vacuum_inflat} we know that the $\\alpha$-limit set is either contained in~$\\K$ or in~$\\planewave {\\kappa}\\cup\\operatorname{T1}$, and together with Prop.~\\ref{prop_convergencetoplanewave} this implies that we only have to show convergence for solutions where the $\\alpha$-limit set is contained in the Kasner parabola~$\\K$.\n\n\tAs shown in~\\eqref{eqn_decayomega}, $\\Omega$ then decays as\n\t\\begin{equation}\n\t\te^{(6-3\\gamma+\\varepsilon)\\tau}\\le \\Omega(\\tau) \\le e^{(6-3\\gamma-\\varepsilon)\\tau},\n\t\\end{equation}\n\tfor $\\tau\\le\\tau_\\varepsilon$ sufficiently small. Setting\n\t$\\tilde\\R\\coloneqq\\int_\\tau^0(q-2)ds$, one finds that $\\tilde\\R$ is non-positive for~$\\tau\\le0$ due to Remark~\\ref{rema_statespacecompact}. The derivative of this function is $\\tilde\\R'=2-q$. Hence\n\t\\begin{equation}\n\t\t\\frac{d}{d\\tau}(e^{2\\tilde\\R}Z)=e^{2\\tilde\\R}3(2-\\gamma)(1+\\Sigma_+)\\Omega=\\O(e^{(6-3\\gamma-\\varepsilon)\\tau})\n\t\\end{equation}\n\tfor $\\tau\\le\\tau_\\varepsilon$,\n\tsince all factors apart from~$\\Omega$ are at least bounded.\n\tTherefore, there exists a non-negative constant~$B_2$ such that\n\t\\begin{equation}\n\t\te^{2\\tilde\\R}Z=B_2+\\O(e^{(6-3\\gamma-\\varepsilon)\\tau}).\n\t\\end{equation}\n\tIn case $B_2>0$, the function~$Z$ is bounded away from zero, which excludes Taub~1 as an $\\alpha$-limit point.\n\tConsequently~$B_2=0$, which means\n\t\\begin{equation}\n\t\te^{2\\tilde\\R}Z=\\O(e^{(6-3\\gamma-\\varepsilon)\\tau}).\n\t\\end{equation}\n\tThe convergence of~$q$ to~$2$, see~\\eqref{eqn_convergenceratestepone}, implies $\\absval{q-2}<\\varepsilon$ for sufficiently negative times, hence\n\t\\begin{equation}\n\t\tZ=\\O(e^{(6-3\\gamma-2\\varepsilon)\\tau}).\n\t\\end{equation}\n\tAs $Z\\rightarrow0$, and additionally~$\\tilde A\\rightarrow0$ from the assumption that the $\\alpha$-limit set is contained in the Kasner parabola~$\\K$, one concludes that~$\\Sigma_+\\rightarrow{-}1$. The only point in~$\\K$ with this property is the point Taub~1, which implies convergence to this point.\n\n\tIt remains to exclude non-constant solutions with $\\tilde A>0$ and $\\Omega>0$ which converge to the point Taub~1 as $\\tau\\rightarrow{-}\\infty$.\n\tKnowing that $\\Sigma_+\\rightarrow{-}1$ and $q\\rightarrow2$, one can apply Lemma~\\ref{lemm_decaylemmageneral} to the evolution equation\\ of~$\\tilde A$, equation~\\eqref{eqns_evolutionbianchib}, to obtain\n\t\\begin{equation}\n\t\\label{eqn_tauboneslowdecaytildea}\n\t\te^{\\varepsilon\\tau}\\le \\tilde A(\\tau) \\le e^{-\\varepsilon\\tau}\n\t\\end{equation}\n\tfor $\\tau\\le\\tau_\\varepsilon$.\n\tUsing the decay of the function~$Z$ together with boundedness of the state space, Remark~\\ref{rema_statespacecompact}, in equation~\\eqref{eqn_functionzextended} yields $\\Delta^2=\\O(e^{(6-3\\gamma-\\varepsilon)\\tau})$.\n\tWith this, one computes\n\t\\begin{equation}\n\t\t(\\frac{\\tilde N}{\\tilde A})'=\\frac{4\\Delta N_+}{\\tilde A}=\\O(e^{(3-\\frac32\\gamma-2\\varepsilon)\\tau}),\n\t\\end{equation}\n\tas~$N_+$ is at least bounded, and finds\n\t\\begin{equation}\n\t\t\\frac{\\tilde N}{\\tilde A}=B_3+\\O(e^{(3-\\frac32\\gamma-2\\varepsilon)\\tau})\n\t\\end{equation}\n\tfor some constant~$B_3$. The constraint equation ~\\eqref{eqn_constraintgeneralone} then reads\n\t\\begin{equation}\n\t\t\\O(e^{(6-3\\gamma-2\\varepsilon)\\tau})=\\Delta^2=\\tilde\\Sigma\\tilde N-\\Sigma_+^2\\tilde A=(B_3\\tilde\\Sigma-\\Sigma_+^2)\\tilde A+\\O(e^{(3-\\frac32\\gamma-3\\varepsilon)\\tau}),\n\t\\end{equation}\n\tand therefore\n\t\\begin{equation}\n\t\t(B_3\\tilde\\Sigma-\\Sigma_+^2)\\tilde A=\\O(e^{(3-\\frac32\\gamma-3\\varepsilon)\\tau}).\n\t\\end{equation}\n\tHowever, due to~\\eqref{eqn_tauboneslowdecaytildea}~$\\tilde A$ decays at most as $e^{\\varepsilon\\tau}$, and the bracket on the left-hand side\\ converges to~$-1$, a contradiction. Thus $\\tilde A>0$ is not possible, which concludes the proof.\n\\end{proof}\n\\begin{proof}[Proof of Prop.~\\ref{prop_tauboneothermatter}]\n\tThe proof is similar to the one of Prop.~\\ref{prop_taubone} for inflationary matter, i.\\,e.\\  $\\Omega>0$ with~$\\gamma\\in[0,2\/3)$. In that setting, we first had to show that the point Taub~1 is the unique~$\\alpha$-limit point, but for the current statement, this holds by assumption.\n\tShowing that there are no non-constant solution converging to the point Taub~1 then hinged on the fact that\n\t\\begin{equation}\n\t\t3-\\frac32\\gamma-3\\varepsilon>0.\n\t\\end{equation}\n\tIn the case~$\\gamma\\in[2\/3,2)$, it is still possible to choose~$\\varepsilon>0$ sufficiently small that this holds. The argument at the end of the proof of~Prop.\\ref{prop_taubone} excluding non-constant solutions then applies without any change.\n\\end{proof}\n\n\n\n\\section{Convergence properties and asymptotic decay towards the Kasner parabola}\n\\label{section_asymptoticdecaykasner}\n\nThis section is the longest and most technical in our discussion of the evolution equations~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega}, and it is here that we prove the main statements we build upon in the following.\n\nWe focus our attention on solutions whose $\\alpha$-limit set is contained in the Kasner parabola~$\\K$. For vacuum models, all non-constant solutions satisfy this property, while in the case of inflationary matter we have to additionally assume that the $\\alpha$-limit set does not intersect the plane wave equilibrium points~$\\planewave {\\kappa}$, see Prop.~\\ref{prop_alphalimitsets_vacuum_inflat}.\n\nIn a first step we show that these non-constant solutions with $\\alpha$-limit set in~$\\K$ converge, i.\\,e.\\  every such solution has a unique accumulation point. This is done in Prop.~\\ref{prop_convergencetokasner}. An equivalent convergence result for inflationary matter solutions with an~$\\alpha$-limit point in~$\\planewave {\\kappa}$ has been obtained in Prop.~\\ref{prop_convergencetoplanewave}, and we therefore find convergence for all inflationary matter and vacuum solutions.\n\nThe main aim of this section is to now obtain decay and convergence rates of the individual variables under the assumption of convergence to a limit point on the Kasner parabola~$\\K$.\nThe behaviour we discover is exponential decay or convergence, and the different exponents coincide with specific eigenvalues to the linearised evolution equations in the extended state space, see~\\eqref{eqn_eigenvalueskasner}.\nWe further find that the rates of convergence depend on where the limit point is situated relative to the point Taub~2. For solutions converging to the point Taub~2 or to a limit point to the left of this point, the lowest order exponential terms are determined in Prop.~\\ref{prop_main_generalasymptoticproperties}. In certain situations, we can refine this statement to even include the second-lowest term, see Prop.~\\ref{prop_leftoftaub2additionaldecay}. Solutions which converge to such a point on the Kasner parabola necessarily have to satisfy that~$\\Delta N_+>0$ for sufficiently negative times or $\\Delta\\equiv0\\equiv N_+$, as we conclude from Lemma~\\ref{lemm_deltanplussigns} and Lemma~\\ref{lemm_deltanplusoppositesign}. In case~$\\Delta N_+<0$ for sufficiently negative times, we determine the exponential convergence rates in Lemma~\\ref{lemm_deltanplusrightoftaub2oppositesign}.\n\nWe remark that even though we prove convergence only in the case of vacuum and inflationary matter, we then drop this restriction on the matter in all the following statements and only assume convergence to a limit point on the Kasner parabola.\nThe results we show hold\nfor all matter models apart from, for some statements, the stiff fluid case $\\gamma=2$.\n\\begin{prop}[Convergence to the Kasner parabola]\n\\label{prop_convergencetokasner}\n\tAssume either vacuum or inflationary matter, i.\\,e.\\  either $\\Omega=0$ or $\\Omega>0$, $\\gamma\\in\\left[0,2\/3\\right)$, and consider a non-constant solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega}. In the inflationary matter case, assume additionally that the $\\alpha$-limit does not intersect the plane wave equilibrium points~$\\planewave {\\kappa}$.\n\tThen there is an $\\operatorname s\\in\\left({-}1,1\\right]$ such that\n\t\\begin{equation}\n\t\t\\lim_{\\tau\\rightarrow{-}\\infty}(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)(\\tau)=(\\operatorname s,1-\\operatorname s^2,0,0,0).\n\t\\end{equation}\n\\end{prop}\nWe prove this statement below.\n\\begin{prop}\n\\label{prop_main_generalasymptoticproperties}\n\tLet $\\gamma\\in[0,2)$ and consider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} converging to $(\\operatorname s,1-\\operatorname s^2,0,0,0)$. If $\\operatorname s\\in\\left[{-}1,1\/2\\right]$, then\n\t\\begin{equation}\n\t\t\\tilde A(3\\operatorname s^2+{\\kappa}(1-\\operatorname s^2))=0\n\t\\end{equation}\n\talong the whole orbit, and\n\t\\begin{align}\n\t\t\\Sigma_+={}&\\operatorname s+\\O(e^{(\\Pi-\\varepsilon)\\tau}),\\\\\n\t\t\\tilde\\Sigma={}&1-\\operatorname s^2+\\O(e^{(\\Pi-\\varepsilon)\\tau}),\\\\\n\t\t\\tilde N={}&\\O(e^{(4+4\\operatorname s-\\varepsilon)\\tau}),\\\\\n\t\tq={}&2+\\O(e^{(\\Pi-\\varepsilon)\\tau}),\\\\\n\t\\end{align}\n\tas $\\tau\\rightarrow{-}\\infty$, for every $\\varepsilon>0$. Here $\\Pi\\coloneqq\\min(6-3\\gamma,4+4\\operatorname s)$ if $\\Omega>0$, and $\\Pi\\coloneqq 4+4\\operatorname s$ if $\\Omega=0$. Furthermore, the following properties hold:\n\t\\begin{itemize}\n\t\t\\item If $\\Delta$ and $N_+$ do not both vanish identically, then $\\Delta N_+>0$ along the whole orbit and for all $\\varepsilon>0$ there is a $\\tau_\\varepsilon>{-}\\infty$ such that $\\tau\\le \\tau_\\varepsilon$ implies\n\t\t\\begin{align}\n\t\t\te^{(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)}+\\varepsilon)\\tau}\\le {}&\\absval{\\Delta}\\le e^{(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)}-\\varepsilon)\\tau},\\\\\n\t\t\te^{(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)}+\\varepsilon)\\tau}\\le {}&\\absval{N_+}\\le e^{(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)}-\\varepsilon)\\tau}.\n\t\t\\end{align}\n\t\t\\item Either $\\tilde A=0$ along the whole orbit, or for all $\\varepsilon>0$ there is a $\\tau_\\varepsilon>{-}\\infty$ such that $\\tau\\le \\tau_\\varepsilon$ implies\n\t\t\\begin{equation}\n\t\t\te^{(4+4\\operatorname s+\\varepsilon)\\tau}\\le \\tilde A\\le e^{(4+4\\operatorname s-\\varepsilon)\\tau}.\n\t\t\\end{equation}\n\t\t\\item Either $\\Omega=0$ along the whole orbit (vacuum), or for all $\\varepsilon>0$ there is a $\\tau_\\varepsilon>{-}\\infty$ such that $\\tau\\le \\tau_\\varepsilon$ implies\n\t\t\\begin{equation}\n\t\t\te^{(6-3\\gamma+\\varepsilon)\\tau}\\le\\Omega\\le e^{(6-3\\gamma-\\varepsilon)\\tau}.\n\t\t\\end{equation}\n\t\\end{itemize}\n\\end{prop}\nThe proof is divided into several steps which will have additional individual use later. The arguments revolve around the constraint equation ~\\eqref{eqn_constraintgeneralone} written in the form\n\\begin{equation}\n\\label{eqn_constraintforproofs}\n\t\\tilde\\Sigma N_+^2-3\\Delta^2=(3\\Sigma_+^2+{\\kappa}\\tilde\\Sigma)\\tilde A\n\\end{equation}\nwhich is then used to determine the asymptotic decay properties of the individual variables.\n\n\n\\begin{lemm}\n\\label{lemm_monotonicitysigmaplus}\n\tConsider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega}. If $\\Sigma_+\\in[0,1]$, then $\\Sigma_+'\\le0,$ i.\\,e.\\  $\\Sigma_+$ is monotonically decreasing.\n\\end{lemm}\n\\begin{proof}\n\tThis follows from inspection of the evolution equation ~\\eqref{eqns_evolutionbianchib} for~$\\Sigma_+'$, using the range of the variables given by the constraints~\\eqref{eqn_constraintgeneraltwo} and Remark~\\ref{rema_statespacecompact}.\n\\end{proof}\n\\begin{lemm}\n\\label{lemm_deltanplussigns}\n\tConsider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} whose $\\alpha$-limit set is contained in~$\\K$ and such that $\\tilde\\Sigma(\\tau)>\\delta>0$ for $\\tau\\le\\tau_0$. Then one of the following statements holds:\n\t\\begin{enumerate}\n\t\t\\item $\\Delta=N_+=0$ along the whole orbit,\n\t\t\\item There is $\\tau_1\\in\\maths{R}$ such that  $\\Delta N_+(\\tau)>0$ for all $\\tau\\le\\tau_1$,\n\t\t\\item There is $\\tau_1\\in\\maths{R}$ such that  $\\Delta N_+(\\tau)<0$ for all $\\tau\\le\\tau_1$.\n\t\\end{enumerate}\n\\end{lemm}\nNote that this statement does not require any assumption on the matter model but holds for all values of~$\\gamma\\in[0,2]$.\n\\begin{proof}\n\tThe set $\\Delta=0=N_+$ is invariant under the evolution equations ~\\eqref{eqns_evolutionbianchib}, which means that every orbit with $\\Delta=0=N_+$ at one time will satisfy this property at all times.\n\tOne can therefore assume that $\\Delta^2+N_+^2>0$.\n\tThe evolution equations ~\\eqref{eqns_evolutionbianchib} for~$\\Delta$ and~$N_+$ yield\n\t\\begin{equation}\n\t\t(\\Delta N_+)'=6\\Delta^2+(3q+4\\Sigma_+-2)\\Delta N_++2(\\tilde\\Sigma-\\tilde N)N_+^2,\n\t\\end{equation}\n\tand due to the convergence relation~\\eqref{eqn_convergenceratestepone} and the assumption on~$\\tilde\\Sigma$, the coefficient to~$N_+^2$ is strictly positive for $\\tau\\le\\tau_2$ sufficiently negative. Therefore, for every time $\\tau_1<\\tau_2$ where the product $\\Delta N_+(\\tau_1)$ becomes zero, the derivative $(\\Delta N_+)'(\\tau_1)$ is strictly positive, which means that $\\Delta N_+$ changes sign from negative to positive. Consequently, the product $\\Delta N_+$ can become zero at most once, which concludes the proof.\n\\end{proof}\n\\begin{proof}[Proof of Prop.~\\ref{prop_convergencetokasner}]\n\tUsing Prop.~\\ref{prop_alphalimitsets_vacuum_inflat}, we can conclude that under the given assumptions, the $\\alpha$-limit points are contained in the Kasner parabola~$\\K$, both for the vacuum and the inflationary case.\n\tWe show in the following that no solution can have $\\alpha$-limit points with different $\\Sigma_+$-values. As points on the Kasner parabola are uniquely identified by their $\\Sigma_+$-value, this implies convergence.\n\n\tIn case $\\Sigma_+(\\tau_0)>0$ at some time $\\tau_0$, monotonicity of~$\\Sigma_+$ shown in Lemma~\\ref{lemm_monotonicitysigmaplus} contradicts the existence of two $\\alpha$-limit points with different~$\\Sigma_+$ values. Consequently, there is exactly one $\\alpha$-limit point, which is equivalent to convergence.\n\tRecall that in the proof of Prop.~\\ref{prop_taubone} we have computed the evolution of~$\\Sigma_+$ to be\n\t\\begin{equation}\n\t\\label{eqn_proofdeltanplussignsevolutionsigmaplus}\n\t\t\\Sigma_+'={-}2\\Sigma_+\\tilde A+\\frac23(\\Sigma_++1){\\kappa}\\tilde A-\\frac23(\\Sigma_++1)N_+^2-\\frac32(2-\\gamma)\\Omega.\n\t\\end{equation}\n\tHence, monotonicity of~$\\Sigma_+$ holds in general in case $\\tilde A=0$, as then~\\eqref{eqn_proofdeltanplussignsevolutionsigmaplus} gives $\\Sigma_+'\\le0$ due to $\\Sigma_+\\in\\left[{-}1,1\\right]$ and $\\Omega\\ge0$.\n\tAssume therefore that $\\Sigma_+\\le0$ and $\\tilde A>0$.\n\n\tAccording to Prop.~\\ref{prop_taubone}, either the orbit is the constant one in the point Taub~1, or this Kasner point is not contained in the $\\alpha$-limit set. The first case is excluded by the assumption. In the latter case, one can assume that for sufficiently negative times the orbit is bounded away from the point Taub~1, which implies that~$\\Sigma_+$ is bounded from below by some constant greater than~${-}1$ for sufficiently negative times~$\\tau$.\n\tConsequently, $\\tilde\\Sigma$ is bounded away from~$0$, as all possible $\\alpha$-limit points are contained in $\\K\\cap\\{\\Sigma_+\\le0\\}$ and the point Taub~1 is excluded.\n \tUsing additionally that $q\\rightarrow2$ due to~\\eqref{eqn_convergenceratestepone}, the term $2(q+2\\Sigma_+)$ is bounded from below by some suitable constant $D_1>0$ for sufficiently negative times, and the evolution equation ~\\eqref{eqns_evolutionbianchib} for~$\\tilde A$ reads\n\t\\begin{equation}\n\t\t\\tilde A'\\ge D_1\\tilde A.\n\t\\end{equation}\n\tConsequently, one finds\n\t\\begin{equation}\n\t\t\\tilde A=\\O(e^{D_1\\tau}).\n\t\\end{equation}\n\n\tOne computes from the evolution equations ~\\eqref{eqns_evolutionbianchib} that\n\t\\begin{equation}\n\t\t(\\frac{N_+^2}{\\tilde A})'=\\frac{12\\Delta N_+}{\\tilde A}\n\t\\end{equation}\n\tand sees that according to Lemma~\\ref{lemm_deltanplussigns} this derivative does not change sign for~$\\tau$ sufficiently negative. As a consequence, the term ${N_+^2}\/{\\tilde A}$ either converges to a non-negative real number or diverges to~$\\infty$ as $\\tau\\rightarrow{-}\\infty$. In the latter case, the evolution of~$\\Sigma_+$ is dominated by~$N_+^2$ alone, in the sense that~$\\Sigma_+'$ as in~\\eqref{eqn_proofdeltanplussignsevolutionsigmaplus} has negative sign for sufficiently negative times. One concludes as for $\\Sigma_+>0$ that there is a unique $\\alpha$-limit point. If the limit of ${N_+^2}\/{\\tilde A}$ is finite, this means~$N_+^2=\\O(e^{D_1\\tau})$.\n\tIn combination with the decay estimate~\\eqref{eqn_decayomega} on~$\\Omega$, this yields that~$\\Sigma_+'$ is integrable and implies convergence.\n\\end{proof}\nAs a direct consequence of convergence which we have shown in Prop.~\\ref{prop_convergencetokasner}, we can apply Lemma~\\ref{lemm_decaylemmageneral} to~$\\tilde A$.\n\\begin{lemm}\n\\label{lemm_decaytildea}\n\tConsider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} converging to $(\\operatorname s,1-\\operatorname s^2,0,0,0)$ with $\\operatorname s\\in\\left[{-}1,1\\right]$. Then either $\\tilde A=0$ along the whole orbit, or there exist for every $\\varepsilon>0$ a $\\tau_\\varepsilon>{-}\\infty$ such that $\\tau\\le\\tau_\\varepsilon$ implies\n\t\\begin{equation}\n\t\te^{(4+4\\operatorname s+\\varepsilon)\\tau}\\le \\tilde A(\\tau)\\le e^{(4+4\\operatorname s-\\varepsilon)\\tau}.\n\t\\end{equation}\n\\end{lemm}\nThe next lemma is of a technical nature and will be used in the following.\n\\begin{lemm}\n\\label{lemm_functionintegraltwominusq}\n\tLet $\\gamma\\in[0,2)$ and consider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} converging to $(\\operatorname s,1-\\operatorname s^2,0,0,0)$ with $\\operatorname s\\in\\left[{-}1,1\\right]$. Then the function\n\t\\begin{equation}\n\t\t\\R\\coloneqq\\int_{{-}\\infty}^\\tau(2-q)ds\n\t\\end{equation}\n\tis well-defined.\n\\end{lemm}\n\\begin{proof}\n\tIf $\\operatorname s={-}1$, Prop.~\\ref{prop_taubone} and Prop.~\\ref{prop_tauboneothermatter} imply that the orbit is the constant orbit, in which case $\\R=0$ along the whole orbit. Assume therefore that $\\operatorname s>{-}1$.\n\tOne computes using the evolution equations ~\\eqref{eqns_evolutionbianchib}\n\t\\begin{equation}\n\t\\label{eqn_evolutionsigmaplusplusone}\n\t\\begin{split}\n\t(\\Sigma_++1)'={}&(q-2)\\Sigma_+-2\\tilde N\\\\\n\t\t={}&(q-2)(\\Sigma_++1)-(q-2)-2\\tilde N\\\\\n\t\t={}&(q-2)(\\Sigma_++1)+2\\tilde A+\\frac32(2-\\gamma)\\Omega.\n\t\\end{split}\n\t\\end{equation}\n\tSuppose that the function $\\tilde\\R\\coloneqq\\int_\\tau^0(q-2)ds$ is unbounded as $\\tau\\rightarrow{-}\\infty$, meaning that the integral tends to~${-}\\infty$. As\n\t\\begin{equation}\n\t\t\\frac{d}{d\\tau}(e^{\\tilde\\R}(\\Sigma_++1))=(2\\tilde A+\\frac32(2-\\gamma)\\Omega)e^{\\tilde\\R}\n\t\\end{equation}\n\tand~$\\tilde A$ and~$\\Omega$ decay as in Lemma~\\ref{lemm_decaytildea} and equation ~\\eqref{eqn_decayomega}, integration yields\n\t\\begin{equation}\n\t\te^{\\tilde\\R}(\\Sigma_+(\\tau)+1)=\\int_{{-}\\infty}^\\tau(2\\tilde A+\\frac32(2-\\gamma)\\Omega)e^{\\tilde\\R}ds=\\O(e^{(\\Pi-\\varepsilon)\\tau}),\n\t\\end{equation}\n\twhere $\\Pi\\coloneqq\\min(6-3\\gamma,4+4\\operatorname s)$ if $\\Omega>0$, and $\\Pi\\coloneqq 4+4\\operatorname s$ if $\\Omega=0$. The assumption on~$\\gamma$ ensures that~$\\Pi>0$.\n\tDue to convergence to the Kasner parabola which implies $2-q\\le\\varepsilon$ for sufficiently negative times, one finds that for some suitably chosen constant~$D_2>0$ and sufficiently negative times~$\\tau$\n\t\\begin{equation}\n\t\t\\tilde\\R\\ge\\varepsilon\\tau-D_2.\n\t\\end{equation}\n\tTherefore, one concludes\n\t\\begin{equation}\n\t\t\\Sigma_++1=\\O(e^{(\\Pi-2\\varepsilon)\\tau}),\n\t\\end{equation}\n\twhich is a contradiction to $\\Sigma_+\\rightarrow\\operatorname s>{-}1$.\n\tAs a consequence, the function $\\tilde\\R$ is bounded on $({-}\\infty,0)$, and\n\t\\begin{equation}\n\t\t\\R(\\tau)=\\int_{{-}\\infty}^\\tau(2-q)ds=\\int_{{-}\\infty}^0(2-q)ds+\\tilde\\R\n\t\\end{equation}\n\tis well-defined.\n\\end{proof}\n\nHaving found detailed decay properties for~$\\tilde A$ and~$\\Omega$, the next step is to determine the asymptotic behaviour of~$\\Delta$ and~$N_+$. Their decay rates are intertwined: One searches for a linear combination of~$\\Delta$ and~$N_+$ such that the evolution equation\\ has a form suitable for Lemma~\\ref{lemm_decaylemmageneral}, i.\\,e.\\ \n\\begin{equation}\n\t(\\Delta+\\hat{\\operatorname r} N_+)'=\\zeta(\\Delta+\\hat{\\operatorname r} N_+),\n\\end{equation}\nand~$\\zeta$ converging as $\\tau\\rightarrow{-}\\infty$. It turns out that the limit of~$\\zeta$ not only depends on the value of~$\\operatorname s$ but also on the sign of~$\\Delta N_+$, and one recovers exactly the eigenvalues $2+2\\operatorname s\\pm2\\sqrt{3(1-\\operatorname s^2)}$ of the linearised evolution equations on the Kasner parabola, see~\\eqref{eqn_eigenvalueskasner} and Appendix~\\ref{subsect_appendixlinearisedevolutionkasner}.\n\\begin{lemm}\n\\label{lemm_decaydeltanplus}\n\tConsider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} converging to $(\\operatorname s,1-\\operatorname s^2,0,0,0)$ with $\\operatorname s\\in\\left({-}1,1\\right)$. If $\\Delta N_+(\\tau)>0$ for all $\\tau\\le\\tau_0$, then for every $\\varepsilon>0$ there exists $\\tau_\\varepsilon>{-}\\infty$ such that $\\tau\\le\\tau_\\varepsilon$ implies\n\t\\begin{equation}\n\t\te^{(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)}+\\varepsilon)\\tau}\\le \\absval{\\Delta+\\sqrt{\\frac{1-\\operatorname s^2}3}N_+}\\le e^{(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)}-\\varepsilon)\\tau}.\n\t\\end{equation}\n\tIf on the other hand $\\Delta N_+(\\tau)<0$ for all $\\tau\\le\\tau_0$, then for every $\\varepsilon>0$ there exists $\\tau_\\varepsilon>{-}\\infty$ such that $\\tau\\le\\tau_\\varepsilon$ implies\n\t\\begin{equation}\n\t\te^{(2+2\\operatorname s-2\\sqrt{3(1-\\operatorname s^2)}+\\varepsilon)\\tau}\\le \\absval{\\Delta-\\sqrt{\\frac{1-\\operatorname s^2}3}N_+}\\le e^{(2+2\\operatorname s-2\\sqrt{3(1-\\operatorname s^2)}-\\varepsilon)\\tau}.\n\t\\end{equation}\n\n\tConsider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} converging to $(1,0,0,0,0)$. If $\\Delta N_+(\\tau)>0$ for all $\\tau\\le\\tau_0$, then for every $\\varepsilon>0$ there exists~$\\hat\\varepsilon>0$ and~$\\tau_\\varepsilon>{-}\\infty$ such that $\\tau\\le\\tau_\\varepsilon$ implies\n\t\\begin{equation}\n\t\te^{(4+\\varepsilon)\\tau}\\le \\absval{\\Delta+\\hat\\varepsilon N_+}\\le e^{(4-\\varepsilon)\\tau}.\n\t\\end{equation}\n\tIf on the other hand $\\Delta N_+(\\tau)<0$ for all $\\tau\\le\\tau_0$, then for every $\\varepsilon>0$ there exists~$\\hat\\varepsilon>0$ and~$\\tau_\\varepsilon>{-}\\infty$ such that $\\tau\\le\\tau_\\varepsilon$ implies\n\t\\begin{equation}\n\t\te^{(4+\\varepsilon)\\tau}\\le \\absval{\\Delta-\\hat\\varepsilon N_+}\\le e^{(4-\\varepsilon)\\tau}.\n\t\\end{equation}\n\\end{lemm}\nWe remark at this point that for this statement, no restriction on~$\\gamma$ is imposed.\n\\begin{proof}\n\tFor notational convenience, set\n\t\\begin{equation}\n\t\\label{eqn_definitionrfactor}\n\t\t\\hat{\\operatorname r}\\coloneqq\\sqrt{\\frac{1-\\operatorname s^2}3}\n\t\\end{equation}\n\tand note that\n\t\\begin{equation}\n\t\\label{eqn_propertyrfactor}\n\t\t2(1-\\operatorname s^2)=6\\hat{\\operatorname r}^2.\n\t\\end{equation}\n\tIn case $\\Delta N_+>0$ and $\\operatorname s\\in({-}1,1)$, one computes\n\t\\begin{equation}\n\t\t(\\Delta+\\hat{\\operatorname r} N_+)'=(2q+2\\Sigma_+-2+6\\hat{\\operatorname r})\\Delta\n\t\t+(\\hat{\\operatorname r} q+2\\hat{\\operatorname r}\\Sigma_++2\\tilde\\Sigma-2\\tilde N)N_+\n\t\\end{equation}\n\tand notices\n\t\\begin{align}\n\t\t&\\lim_{\\tau\\rightarrow{-}\\infty}(2q+2\\Sigma_+-2+6\\hat{\\operatorname r}) =2+2\\operatorname s+6\\hat{\\operatorname r},\\\\\n\t\t&\\lim_{\\tau\\rightarrow{-}\\infty}(\\hat{\\operatorname r} q+2\\hat{\\operatorname r}\\Sigma_++2\\tilde\\Sigma-2\\tilde N) =2\\hat{\\operatorname r}+2\\hat{\\operatorname r}\\operatorname s+2(1-\\operatorname s^2).\n\t\\end{align}\n\tTherefore\n\t\\begin{equation}\n\t\\label{eqn_evolutionrdeltaplusnplus}\n\t\\begin{split}\n\t\t(\\Delta+\\hat{\\operatorname r} N_+)'\n\t\t\t={}&(2+2\\operatorname s+6\\hat{\\operatorname r}+f_1)\\Delta+(2\\hat{\\operatorname r}+2\\hat{\\operatorname r}\\operatorname s+6\\hat{\\operatorname r}^2+f_2)N_+\\\\\n\t\t\t={}&(2+2\\operatorname s+6\\hat{\\operatorname r}+\\frac{f_1\\Delta+f_2N_+}{\\Delta+\\hat{\\operatorname r} N_+})(\\Delta+\\hat{\\operatorname r} N_+)\\\\\n\t\t\t={}&(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)}+\\frac{f_1\\Delta+f_2N_+}{\\Delta+\\hat{\\operatorname r} N_+})(\\Delta+\\hat{\\operatorname r} N_+),\n\t\\end{split}\n\t\\end{equation}\n\twith two functions $f_1,f_2$ converging to~$0$ as $\\tau\\rightarrow{-}\\infty$. We do not need the explicit form of these two functions here, but use them in the proof of Prop.~\\ref{prop_leftoftaub2additionaldecay}. Note that this computation makes use of the fact that~$\\Delta$ and~$N_+$ have the same sign.\n\tAs the last term in the bracket vanishes asymptotically,\n\tLemma~\\ref{lemm_decaylemmageneral} yields the decay of $\\Delta+\\hat{\\operatorname r} N_+$ in case both~$\\Delta$ and~$N_+$ are positive. If both are negative, the statement follows due to the invariance of the evolution equations~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} under a change of sign in these two variables.\n\n\tIn case~$\\operatorname s=1$, we find~$\\hat{\\operatorname r}=0$ and can no longer conclude that the quotient in the last line in equation~\\eqref{eqn_evolutionrdeltaplusnplus} vanishes asymptotically. We assume~$\\Delta N_+>0$ and compute\n\t\\begin{equation}\n\t\t(\\Delta+\\hat{\\eps} N_+)'=(2q+2\\Sigma_+-2+6\\hat{\\eps})\\Delta\n\t\t+(\\hat{\\eps} q+2\\hat{\\eps}\\Sigma_++2\\tilde\\Sigma-2\\tilde N)N_+,\n\t\\end{equation}\n\tfor~$\\hat{\\eps}>0$.\n\tBy similar argument as above, we obtain the requested statement.\n\n\tIn order to treat the cases where $\\Delta N_+<0$ it is enough to replace every occurence of~$\\hat{\\operatorname r}$ and~$\\hat{\\eps}$ by~${-}\\hat{\\operatorname r}$ and~${-}\\hat{\\eps}$, respectively.\n\\end{proof}\nThe decay of~$\\Delta$ and~$N_+$, depending on whether they have the same or opposite sign, determines the decay of the remaining variables.\n\\begin{lemm}\n\\label{lemm_deltanplussamesign}\n\tLet $\\gamma\\in[0,2)$ and consider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} converging to $(\\operatorname s,1-\\operatorname s^2,0,0,0)$ with $\\operatorname s\\in\\left[{-}1,1\\right]$.\n\tAssume that $\\Delta N_+(\\tau)>0$ for all $\\tau\\le\\tau_0$. Then\n\t\\begin{align}\n\t\t\\Sigma_+={}&\\operatorname s+\\O(e^{(\\Pi-\\varepsilon)\\tau}),\\\\\n\t\t\\tilde\\Sigma={}&1-\\operatorname s^2+\\O(e^{(\\Pi-\\varepsilon)\\tau}),\\\\\n\t\t\\Delta={}&\\O(e^{(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)}-\\varepsilon)\\tau}),\\\\\n\t\tN_+={}&\\O(e^{(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)}-\\varepsilon)\\tau}),\\\\\n\t\t\\tilde N={}&\\O(e^{(4+4\\operatorname s-\\varepsilon)\\tau}),\\\\\n\t\tq={}&2+\\O(e^{(\\Pi-\\varepsilon)\\tau}),\n\t\\end{align}\n\tas $\\tau\\rightarrow{-}\\infty$, for every $\\varepsilon>0$. Here $\\Pi\\coloneqq\\min(6-3\\gamma,4+4\\operatorname s)$ if $\\Omega>0$, and $\\Pi\\coloneqq 4+4\\operatorname s$ if $\\Omega=0$.\n\tFurthermore,\n\t\\begin{equation}\n\t\t\\tilde A(3\\operatorname s^2+{\\kappa}(1-\\operatorname s^2))=0\n\t\\end{equation}\n\tholds along the whole orbit, and $\\tilde A>0$ implies that\n\t\\begin{equation}\n\t\t3\\Sigma_+^2+{\\kappa}\\tilde\\Sigma =\\O(e^{(4+4\\operatorname s+4\\sqrt{3(1-\\operatorname s^2)}-\\varepsilon)\\tau})\n\t\\end{equation}\n\tas $\\tau\\rightarrow{-}\\infty$, for every $\\varepsilon>0$.\n\n\tUnder the additional restriction that $\\operatorname s\\in({-}1,1)$, there is for each $\\varepsilon>0$ a $\\tau_\\varepsilon>{-}\\infty$ such that $\\tau\\le \\tau_\\varepsilon$ implies\n\t\\begin{align}\n\t\\label{eqn_lowerbounddelta}\n\t\te^{(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)}+\\varepsilon)\\tau}\\le {}&\\absval{\\Delta},\\\\\n\t\\label{eqn_lowerboundnplus}\n\t\te^{(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)}+\\varepsilon)\\tau}\\le {}&\\absval{N_+}.\n\t\\end{align}\n\\end{lemm}\n\\begin{rema}\n\\label{rema_auxiliarydecayconstraintexpr}\n\tAs a consequence of this result, one finds that in the setting of this lemma\n\t\\begin{equation}\n\t\t\\frac13\\tilde\\Sigma N_+^2-\\Delta^2=(\\Sigma_+^2+\\frac {\\kappa}3\\tilde\\Sigma)\\tilde A=\\O(e^{(8+8\\operatorname s+4\\sqrt{3(1-\\operatorname s^2)}-\\varepsilon)\\tau})\n\t\\end{equation}\n\tboth for $\\tilde A=0$ and $\\tilde A>0$.\n\\end{rema}\n\\begin{proof}[Proof of Lemma~\\ref{lemm_deltanplussamesign}]\n\tIf $\\operatorname s={-}1$, then due to Prop.~\\ref{prop_taubone} and Prop.~\\ref{prop_tauboneothermatter} the solution is the constant orbit for which all the required properties hold. Let us therefore assume $\\operatorname s\\in({-}1,1)$.\n\tIt follows from Lemma~\\ref{lemm_decaydeltanplus} that for sufficiently negative~$\\tau$\n\t\\begin{equation}\n\t\t\\absval{\\Delta+\\hat{\\operatorname r} N_+}\\le e^{(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)}-\\varepsilon)\\tau},\n\t\\end{equation}\n\twith~$\\hat{\\operatorname r}$ as in equation ~\\eqref{eqn_definitionrfactor}. In particular $\\hat{\\operatorname r}>0$, which implies that~$\\Delta$ and~$\\hat{\\operatorname r} N_+$ have the same sign and yields the upper bound of~$\\Delta$ and~$N_+$ in the statement. In case~$\\operatorname s=1$, we can use the same argument with~$\\hat{\\operatorname r}$ replaced by~$\\hat\\varepsilon$.\n\n\tComparing the decay of~$N_+$ with the decay~$\\tilde A=\\O(e^{(4+4\\operatorname s-\\varepsilon)\\tau})$ from Lemma~\\ref{lemm_decaytildea}, we find\n\t\\begin{equation}\n\t\t\\tilde N=\\frac13(N_+^2-{\\kappa}\\tilde A)=\\O(e^{(4+4\\operatorname s-\\varepsilon)\\tau}).\n\t\\end{equation}\n\tDue to its definition~\\eqref{eqn_definitionqmatter}, the quantity~$q$ inherits its convergence rate either from the decay of~$\\Omega$ as in equation ~\\eqref{eqn_decayomega} or from the one of~$\\tilde N+\\tilde A$, whichever is slower:\n\t\\begin{equation}\n\t\tq=2(1-\\tilde A-\\tilde N)-\\frac32(2-\\gamma)\\Omega=2+\\O(e^{(\\Pi-\\varepsilon)\\tau})\n\t\\end{equation}\n\tfor $\\Pi\\coloneqq\\min(6-3\\gamma,4+4\\operatorname s)$.\n\tVacuum is defined by $\\Omega=0$, and one sees that the statement holds if one sets $\\Pi\\coloneqq 4+4\\operatorname s$ in this case.\n\tUsing the above information in the evolution equations~\\eqref{eqns_evolutionbianchib} for~$\\Sigma_+$ and~$\\tilde\\Sigma$ yields\n\t\\begin{equation}\n\t\t\\Sigma_+'=\\O(e^{(\\Pi-\\varepsilon)\\tau}),\\qquad \\tilde\\Sigma'=\\O(e^{(\\Pi-\\varepsilon)\\tau}),\n\t\\end{equation}\n\tand thus gives the convergence rates for~$\\Sigma_+$ and~$\\tilde\\Sigma$.\n\n\t\\smallskip\n\n\tWriting the constraint equation\\ as in~\\eqref{eqn_constraintforproofs} and applying the convergence rates of~$\\Sigma_+$ and~$\\tilde\\Sigma$, one sees that\n\t\\begin{equation}\n\t\\label{eqn_constraintinproofsamesign}\n\t\t\\tilde\\Sigma N_+^2-3\\Delta^2=(3\\Sigma_+^2+{\\kappa}\\tilde\\Sigma)\\tilde A=(3\\operatorname s^2+{\\kappa}(1-\\operatorname s^2)+\\O(e^{(\\Pi-\\varepsilon)\\tau}))\\tilde A.\n\t\\end{equation}\n\tThe left-hand side\\ is of order $\\O(e^{(4+4\\operatorname s+4\\sqrt{3(1-\\operatorname s^2)}-\\varepsilon)\\tau})$ due to the above. The right-hand side---if non-vanishing---consists of the bracket with its explicitly given decay and the factor~$\\tilde A$ which---if non-vanishing---decays at most as $e^{(4+4\\operatorname s+\\varepsilon)\\tau}$, see Lemma~\\ref{lemm_decaytildea}. In order for equation ~\\eqref{eqn_constraintinproofsamesign} to be consistent, either $\\tilde A=0$ or $3\\operatorname s^2+{\\kappa}(1-\\operatorname s^2)=0$ has to hold.\n\n\tFor the term in the bracket, one further computes from the evolution equations ~\\eqref{eqns_evolutionbianchib} and $\\tilde N=(N_+^2-{\\kappa}\\tilde A)\/3$ that\n\t\\begin{align}\n\t(3\\Sigma_+^2+{\\kappa}\\tilde\\Sigma)'={}&6\\Sigma_+\\Sigma_+'+{\\kappa}\\tilde\\Sigma'\\\\\n\t\t={}&6(q-2)\\Sigma_+^2-12\\Sigma_+\\tilde N+2{\\kappa}(q-2)\\tilde\\Sigma-4{\\kappa}\\Sigma_+\\tilde A-4{\\kappa}\\Delta N_+\\\\\n\t\t={}&2(q-2)(3\\Sigma_+^2+{\\kappa}\\tilde\\Sigma)-4\\Sigma_+N_+^2-4{\\kappa}N_+\\Delta\\\\\n\t\t={}&2(q-2)(3\\Sigma_+^2+{\\kappa}\\tilde\\Sigma)-4N_+(\\Sigma_+N_++{\\kappa}\\Delta).\n\t\\end{align}\n\tIn Lemma~\\ref{lemm_functionintegraltwominusq}, the function\n\t\\begin{equation}\n\t\t\\R(\\tau)=\\int_{{-}\\infty}^\\tau(2-q)ds\n\t\\end{equation}\n\tis found to be well-defined, hence one obtains\n\t\\begin{equation}\n\t\t\\frac{d}{d\\tau}(e^{2\\R}(3\\Sigma_+^2+{\\kappa}\\tilde\\Sigma))={-}e^{2\\R}4N_+(\\Sigma_+N_++{\\kappa}\\Delta),\n\t\\end{equation}\n\tand consequently\n\t\\begin{align}\n\t\te^{2\\R}(3\\Sigma_+^2+{\\kappa}\\tilde\\Sigma)\n\t\t\t={}&3\\operatorname s^2+{\\kappa}(1-\\operatorname s^2)-\\int_{{-}\\infty}^\\tau e^{2\\R}4N_+(\\Sigma_+N_++{\\kappa}\\Delta)ds\\\\\n\t\t\t={}&3\\operatorname s^2+{\\kappa}(1-\\operatorname s^2)+\\O(e^{(4+4\\operatorname s+4\\sqrt{3(1-\\operatorname s^2)}-\\varepsilon)\\tau})\n\t\\end{align}\n\tdue to the above. If $\\tilde A>0$, then the constant term on the right-hand side\\ vanishes due to our previous arguments, and one can conclude that\n\t\\begin{equation}\n\t\t3\\Sigma_+^2+{\\kappa}\\tilde\\Sigma=\\O(e^{(4+4\\operatorname s+4\\sqrt{3(1-\\operatorname s^2)}-\\varepsilon)\\tau}).\n\t\\end{equation}\n\tThis yields that the right-hand side\\ of equation ~\\eqref{eqn_constraintforproofs}\n\t\\begin{equation}\n\t\t\\frac13\\tilde\\Sigma N_+^2-\\Delta^2=(\\Sigma_+^2+\\frac {\\kappa}3\\tilde\\Sigma)\\tilde A\n\t\\end{equation}\n\tdecays exponentially to order at least $8+8\\operatorname s+4\\sqrt{3(1-\\operatorname s^2)}-\\varepsilon$, both for $\\tilde A=0$ and $\\tilde A>0$: In the first case, the right-hand side\\ vanishes identically, while in the latter case we have determined the decay properties for both factors on the right-hand side\\ individually.\n\n\t\\smallskip\n\n\tIt remains to show the lower bound for~$\\Delta$ and~$N_+$ in case~$\\operatorname s\\in({-}1,1)$. From the last argument we conclude that\n\t\\begin{equation}\n\t\t\\frac13\\tilde\\Sigma(\\frac{N_+}{\\Delta+\\hat{\\operatorname r} N_+})^2-(\\frac{\\Delta}{\\Delta+\\hat{\\operatorname r} N_+})^2=\\frac{(\\Sigma_+^2+\\frac {\\kappa}3\\tilde\\Sigma)\\tilde A}{({\\Delta+\\hat{\\operatorname r} N_+})^2}\\rightarrow0\n\t\\end{equation}\n\tas $\\tau\\rightarrow{-}\\infty$, due to the lower bound on the denominator found in Lemma~\\ref{lemm_decaydeltanplus}. As $1-\\operatorname s^2>0$, the variable~$\\tilde\\Sigma$ is bounded away from zero for sufficiently negative times. If one of the two squared terms tends to zero along a time sequence, so does the other one (along the same time sequence), which would imply that\n\t\\begin{equation}\n\t\t\\frac{\\Delta}{\\Delta+\\hat{\\operatorname r} N_+}+\\hat{\\operatorname r}\\frac{N_+}{\\Delta+\\hat{\\operatorname r} N_+}\\rightarrow0,\n\t\\end{equation}\n\ta contradiction. Consequently, both\n\t\\begin{equation}\n\t\t\\absval{\\frac{\\Delta}{\\Delta+\\hat{\\operatorname r} N_+}} \\qquad \\text{and}\\qquad \\absval{\\frac{N_+}{\\Delta+\\hat{\\operatorname r} N_+}}\n\t\\end{equation}\n\tare bounded from below by a positive constant for $\\tau\\le\\tau_\\varepsilon$ sufficiently negative. This gives the lower bounds on~$\\Delta$ and~$N_+$ and concludes the proof.\n\\end{proof}\n\\begin{lemm}\n\\label{lemm_deltanpluszero}\n\tLet $\\gamma\\in[0,2)$ and consider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} converging to $(\\operatorname s,1-\\operatorname s^2,0,0,0)$ with $\\operatorname s\\in[{-}1,1]$.\n\tAssume that $\\Delta=N_+=0$ along the orbit.\n\tThen\n\t\\begin{equation}\n\t\t\\tilde A(3\\operatorname s^2+{\\kappa}(1-\\operatorname s^2))=0\n\t\\end{equation}\n\tand\n\t\\begin{align}\n\t\t\\Sigma_+={}&\\operatorname s+\\O(e^{(\\Pi-\\varepsilon)\\tau}),\\\\\n\t\t\\tilde\\Sigma={}&1-\\operatorname s^2+\\O(e^{(\\Pi-\\varepsilon)\\tau}),\\\\\n\t\t\\tilde N={}&\\O(e^{(4+4\\operatorname s-\\varepsilon)\\tau}),\\\\\n\t\tq={}&2+\\O(e^{(\\Pi-\\varepsilon)\\tau}),\\\\\n\t\\end{align}\n\tas $\\tau\\rightarrow{-}\\infty$, for every $\\varepsilon>0$. Here $\\Pi\\coloneqq\\min(6-3\\gamma,4+4\\operatorname s)$ if $\\Omega>0$, and $\\Pi\\coloneqq 4+4\\operatorname s$ if $\\Omega=0$.\n\\end{lemm}\n\\begin{proof}\n\tThe first equation follows immediately from the constraint equation ~\\eqref{eqn_constraintforproofs} for $\\tau\\rightarrow{-}\\infty$. For the decay properties, there is nothing to show for $\\operatorname s={-}1$, as this is the constant orbit due to Prop.~\\ref{prop_taubone} and Prop.~\\ref{prop_tauboneothermatter}.\n\tIn the other cases, one simplifies equation~\\eqref{eqn_definitiontilden} for~$\\tilde N$, equation~\\eqref{eqn_definitionqmatter} for~$q$ and the evolution equations ~\\eqref{eqns_evolutionbianchib} for~$\\Sigma_+$ and~$\\tilde\\Sigma$ using $\\Delta=N_+=0$ to find\n\t\\begin{align}\n\t\t\\tilde N={}&-\\frac13{\\kappa}\\tilde A,\\\\\n\t\tq={}&2(1-\\tilde A+\\frac13{\\kappa}\\tilde A)-\\frac32(2-\\gamma)\\Omega,\\\\\n\t\t\\Sigma_+'={}&(q-2)\\Sigma_++\\frac23{\\kappa}\\tilde A,\\\\\n\t\t\\tilde\\Sigma'={}&2(q-2)\\tilde\\Sigma-4\\Sigma_+\\tilde A,\n\t\\end{align}\n\tthen inserts the decay of~$\\tilde A$ and~$\\Omega$ from Lemma~\\ref{lemm_decaytildea} and equation ~\\eqref{eqn_decayomega} respectively to conclude the proof.\n\\end{proof}\nFor an orbit with $\\Delta N_+\\ge0$ for sufficiently negative times, there is no a priori restriction on where its limit point on the Kasner parabola is located with respect to Taub~2. In the case of opposite signs, the limit point has to be situated to the right of Taub~2, as the next lemma shows.\n\\begin{lemm}\n\\label{lemm_deltanplusoppositesign}\n\tLet $\\gamma\\in[0,2)$ and consider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} converging to $(\\operatorname s,1-\\operatorname s^2,0,0,0)$. If $\\Delta N_+(\\tau_k)<0$ for a sequence $\\tau_k\\rightarrow{-}\\infty$, then $\\operatorname s\\in(1\/2,1]$.\n\\end{lemm}\n\\begin{proof}\n\tAs $\\Sigma_+\\in[{-}1,1]$ due to Remark~\\ref{rema_statespacecompact}, the limit value~$\\operatorname s$ is contained in the same interval.\n\tThe case $\\operatorname s={-}1$ can be excluded by Prop.~\\ref{prop_taubone} and Prop.~\\ref{prop_tauboneothermatter}.\n\tDue to Lemma~\\ref{lemm_deltanplussigns} and under the assumption that $\\operatorname s\\in({-}1,1)$, we know that~$\\Delta N_+$ has a fixed sign for sufficiently negative~$\\tau$, hence it is enough to show that orbits with $\\Delta N_+(\\tau)<0$ for $\\tau\\le\\tau_0$ cannot converge to a Kasner point with $\\operatorname s\\in({-}1,1\/2]$.\n\n\tConsider first the case $-1<\\operatorname s<1\/2$ and recall from Lemma~\\ref{lemm_decaydeltanplus}\n\t\\begin{equation}\n\t\t\\absval{\\Delta-\\hat{\\operatorname r} N_+}\\ge e^{(2+2\\operatorname s-2\\sqrt{3(1-\\operatorname s^2)}+\\varepsilon)\\tau}\n\t\\end{equation}\n\tfor~$\\tau$ sufficiently negative, where~$\\hat{\\operatorname r}$ is as in equation ~\\eqref{eqn_definitionrfactor}.\n\tThe bracket in the exponent is strictly negative for sufficiently small~$\\varepsilon$, which means that $\\absval{\\Delta-\\hat{\\operatorname r} N_+}$ grows exponentially as $\\tau\\rightarrow{-}\\infty$. This contradicts the fact that~$\\Delta$ and~$N_+$ converge to zero.\n\n\tWe can therefore assume that $\\operatorname s=1\/2$. One finds the special value $\\hat{\\operatorname r}=1\/2$ and notices\n\t\\begin{equation}\n\t\t2+2\\operatorname s-2\\sqrt{3(1-\\operatorname s^2)}=0.\n\t\\end{equation}\n\tHence\n\t\\begin{equation}\n\t\t\\absval{2\\Delta-N_+}\\ge e^{\\varepsilon\\tau}\n\t\\end{equation}\n\tfor~$\\tau$ sufficiently negative. This estimate holds for~$\\Delta$ and~$N_+$ individually, as the following argument shows: The constraint equation ~\\eqref{eqn_constraintforproofs} can be reformulated into\n\t\\begin{align}\n\t\t(3\\Sigma_+^2+{\\kappa}\\tilde\\Sigma)\\tilde A={}&\\tilde\\Sigma N_+^2-3\\Delta^2\\\\\n\t\t\t={}&(\\tilde\\Sigma N_+-\\frac34N_++\\frac34N_+-\\frac32\\Delta)N_++\\frac32(N_+-2\\Delta)\\Delta.\n\t\\end{align}\n\tThe right-hand side\\ divided by $3(N_+-2\\Delta)\/4$ equals\n\t\\begin{equation}\n\t\t(\\frac{4(\\tilde\\Sigma-\\frac34)}{3(N_+-2\\Delta)}N_++1)N_++2\\Delta=(1+f_3)N_++2\\Delta,\n\t\\end{equation}\n\twith an asymptotically vanishing function~$f_3$, while the left-hand side\\ divided by the same expression decays exponentially to order $\\O(e^{(4+4\\operatorname s-2\\varepsilon)\\tau})$ due to Lemma~\\ref{lemm_decaytildea}. This yields\n\t\\begin{equation}\n\t\t\\absval{N_+}\\ge e^{\\varepsilon\\tau},\\qquad \\absval{\\Delta}\\ge e^{\\varepsilon\\tau}.\n\t\\end{equation}\n\tUsing the definition of~$\\tilde N$, the decay of~$\\tilde A$ from Lemma~\\eqref{lemm_decaytildea}, and equation ~\\eqref{eqn_definitionqmatter} for~$q$, one additionally finds\n\t\\begin{equation}\n\t\t\\tilde N\\ge e^{\\varepsilon\\tau},\\qquad 2-q\\ge e^{\\varepsilon\\tau}.\n\t\\end{equation}\n\n\t\\smallskip\n\n\tThe rest of the proof aims at constructing a contradiction to the slow decay behaviour of~$N_+$. This becomes possible by considering the evolution equation s in more detail and relating the decay of several quantities.\n\tOne starts by noting that Taylor expansion of the square root applied to the constraint equation ~\\eqref{eqn_constraintforproofs} yields\n\t\\begin{equation}\n\t\\label{eqn_proofoppositesignsdelta}\n\t\t\\Delta={-}\\frac{\\sqrt3}{3}\\tilde\\Sigma^{\\nicefrac12}N_++\\O(e^{(6-2\\varepsilon)\\tau}),\n\t\\end{equation}\n\twhere we used Lemma~\\ref{lemm_decaytildea}, and the sign stems from our assumption $\\Delta N_+<0$.\n\tWe then multiply the constraint equation ~\\eqref{eqn_constraintgeneralone} by~$N_+^2$ and reformulate the resulting expression using equation~\\eqref{eqn_definitiontilden} defining~$\\tilde N$ and equation~\\eqref{eqn_definitionqmatter} in the form\n\t\\begin{equation}\n\t\t\\tilde N=\\frac12(2-q)-\\tilde A-\\frac34(2-\\gamma)\\Omega\n\t\\end{equation}\n\tto find\n\t\\begin{align}\n\t\t\\Delta^2N_+^2={}&\\tilde\\Sigma\\tilde NN_+^2-\\Sigma_+^2\\tilde AN_+^2\\\\\n\t\t={}&3\\tilde\\Sigma\\tilde N^2+{\\kappa}\\tilde\\Sigma\\tilde A\\tilde N-\\Sigma_+^2\\tilde AN_+^2\\\\\n\t\t={}&\\frac34(2-q)^2\\tilde\\Sigma+\\O(e^{(6-3\\gamma-\\varepsilon)\\tau}).\n\t\\end{align}\n\tThe last line follows from convergence to the point Taub~2, i.\\,e.\\  $\\operatorname s=1\/2$, in combination with Lemma~\\ref{lemm_decaytildea}: $\\tilde A$ decays as $\\O(e^{(6-\\varepsilon)\\tau})$, which is faster than the decay of~$\\Omega$, given by $\\O(e^{(6-3\\gamma-\\varepsilon)\\tau})$.\n\tUsing once more Taylor expansion of the square root yields\n\t\\begin{equation}\n\t\t\\Delta N_+={-}\\frac{\\sqrt3}{2}(2-q)\\tilde\\Sigma^{\\nicefrac12} +\\O(e^{(6-3\\gamma-2\\varepsilon)\\tau}).\n\t\\end{equation}\n\n\tIn the next step one determines the behaviour of~$\\Sigma_+$ and~$\\tilde\\Sigma$. The evolution of~$\\Sigma_++1$ has been computed in~\\eqref{eqn_evolutionsigmaplusplusone}. With the function~$\\R$ defined as in Lemma~\\ref{lemm_functionintegraltwominusq}, integration of $(e^\\R(\\Sigma_++1))'$ yields\n\t\\begin{equation}\n\t\te^{\\R}(\\Sigma_+(\\tau)+1)=\\frac32+\\int_{{-}\\infty}^\\tau(2\\tilde A+\\frac32(2-\\gamma)\\Omega)e^{\\R}ds,\n\t\\end{equation}\n\tand thus\n\t\\begin{equation}\n\t\\label{eqn_proofoppositesignssigmaplus}\n\t\t\\Sigma_++1=\\frac32e^{-\\R(\\tau)}+\\O(e^{(6-3\\gamma-2\\varepsilon)\\tau}).\n\t\\end{equation}\n\tFor~$\\tilde\\Sigma$, one finds\n\t\\begin{align}\n\t\t\\tilde\\Sigma'={}&2(q-2)\\tilde\\Sigma-4\\Delta N_+-4\\Sigma_+\\tilde A\\\\\n\t\t\t={}&2(q-2)(\\tilde\\Sigma-\\sqrt3\\tilde\\Sigma^{\\nicefrac12})+\\O(e^{(6-3\\gamma-2\\varepsilon)\\tau}).\n\t\\end{align}\n\tHence\n\t\\begin{equation}\n\t\t(\\tilde\\Sigma^{\\nicefrac12}-\\sqrt3)'\n\t\t=\\frac12\\frac{\\tilde\\Sigma'}{\\tilde\\Sigma^{\\nicefrac12}}\n\t\t=(q-2)(\\tilde\\Sigma^{\\nicefrac12}-\\sqrt3)+\\O(e^{(6-3\\gamma-2\\varepsilon)\\tau}),\n\t\\end{equation}\n\tfrom which one concludes\n\t\\begin{equation}\n\t\t\\frac{d}{d\\tau}(e^\\R(\\tilde\\Sigma^{\\nicefrac12}-\\sqrt3))\n\t\t\t=\\O(e^{(6-3\\gamma-2\\varepsilon)\\tau}).\n\t\\end{equation}\n\tIntegration yields\n\t\\begin{equation}\n\t\\label{eqn_proofoppositesignstildesigma}\n\t\t\\tilde\\Sigma^{\\nicefrac12}-\\sqrt3\n\t\t={-}\\frac{\\sqrt3}2e^{-\\R(\\tau)}+\\O(e^{(6-3\\gamma-2\\varepsilon)\\tau}).\n\t\\end{equation}\n\tOne can eliminate~$\\R$ using a linear combination of equations ~\\eqref{eqn_proofoppositesignssigmaplus} and~\\eqref{eqn_proofoppositesignstildesigma} and finds\n\t\\begin{equation}\n\t\\label{eqn_proofoppositesignscombination}\n\t\t\\Sigma_++\\sqrt3\\tilde\\Sigma^{\\nicefrac12}-2=\\O(e^{(6-3\\gamma-2\\varepsilon)\\tau}).\n\t\\end{equation}\n\n\tNow, we consider the evolution of~$N_+$. Due to equations ~\\eqref{eqn_proofoppositesignsdelta} and~\\eqref{eqn_proofoppositesignscombination} and the slow decay of~$N_+$, one finds\n\t\\begin{align}\n\t\tN_+'={}&(q+2\\Sigma_+)N_++6\\Delta\\\\\n\t\t\t={}&(q+2\\Sigma_+-2\\sqrt3\\tilde\\Sigma^{\\nicefrac12})N_++\\O(e^{(6-2\\varepsilon)\\tau})\\\\\n\t\t\t={}&(q-2+4\\Sigma_+-2+\\O(e^{(6-3\\gamma-2\\varepsilon)\\tau}))N_++\\O(e^{(6-2\\varepsilon)\\tau})\\\\\n\t\t\t={}&(q-2+4\\Sigma_+-2+\\O(e^{(6-3\\gamma-3\\varepsilon)\\tau}))N_+,\n\t\\end{align}\n\twhich, recalling the discussion of~$\\R$ in the proof of Lemma~\\ref{lemm_functionintegraltwominusq}, implies that there is a function~$f_4$ which is integrable on $({-}\\infty,0)$ and satisfies\n\t\\begin{equation}\n\t\tN_+'=(4\\Sigma_+-2+f_4)N_+.\n\t\\end{equation}\n\tSet\n\t\\begin{equation}\n\t\tF_4(\\tau)\\coloneqq\\int_{{-}\\infty}^\\tau f_4(s)ds\n\t\\end{equation}\n\tand compute\n\t\\begin{equation}\n\t\t\\frac{d}{d\\tau}(e^{-F_4}N_+)=(4\\Sigma_+-2)e^{-F_4}N_+.\n\t\\end{equation}\n\tAs $\\Sigma_+'\\le0$ asymptotically due to Lemma~\\ref{lemm_monotonicitysigmaplus}, every solution converging to the point Taub~2 satisfies $4\\Sigma_+\\le2$, which means that the function $\\absval{e^{-F_4}N_+}$ increases as $\\tau\\rightarrow{-}\\infty$. It follows that~$N_+$ does not converge to~$0$, a contradiction.\n\\end{proof}\nWith the results found above, we are finally in a position to prove Prop.~\\ref{prop_main_generalasymptoticproperties}.\n\\begin{proof}[Proof of Prop.~\\ref{prop_main_generalasymptoticproperties}]\n\tIn case $\\operatorname s={-}1$, the solution is the constant solution (Prop.~\\ref{prop_taubone} and Prop.~\\ref{prop_tauboneothermatter}), for which the statement trivially holds. Otherwise, the decay of~$\\Omega$ and~$\\tilde A$ follows from equation ~\\eqref{eqn_decayomega} and Lemma~\\ref{lemm_decaytildea}. According to Lemma~\\ref{lemm_deltanplussigns}, there are three cases to be considered regarding the sign of~$\\Delta N_+$. As $\\operatorname s\\in({-}1,1\/2]$, the case $\\Delta N_+<0$ is excluded by Lemma~\\ref{lemm_deltanplusoppositesign}, and the remaining two cases are discussed in Lemma~\\ref{lemm_deltanplussamesign} and~\\ref{lemm_deltanpluszero}.\n\\end{proof}\n\n\nIn the proof of the previous lemma, we have discussed the asymptotic behaviour of orbits converging to a Kasner point to the left of Taub~2, as $\\tau\\rightarrow{-}\\infty$. In particular, we could exclude $\\Delta N_+<0$ asymptotically.\nFor orbits converging to a Kasner point to the right of the point Taub~2, i.\\,e.\\  with~$\\Sigma_+$ converging to $\\operatorname s\\in(1\/2,1]$, we cannot exclude the negative sign. This stems from the fact that the non-positive eigenvalue $2+2\\operatorname s-2\\sqrt{3(1-\\operatorname s^2)}$ which was used to construct a contradiction in the proof of Lemma~\\ref{lemm_deltanplusoppositesign} becomes positive when changing from the Kasner arc to the left of Taub~2 to the one on the right. Orbits which satisfy~$\\Delta N_+<0$ asymptotically and converge to a point on the Kasner parabola to the right of Taub~2 exist, and as in the case of~$\\Delta N_+>0$ asymptotically, one finds that the rates of convergence are related to eigenvalues of the linearised evolution equation: For~$\\Delta N_+>0$ asymptotically, we found exponential decay of order~$2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)}$ (see Prop.~\\ref{prop_main_generalasymptoticproperties}), while for~$\\Delta N_+<0$ asymptotically, we obtain exponential decay of order~$2+2\\operatorname s-2\\sqrt{3(1-\\operatorname s^2)}$.\n\\begin{lemm}\n\\label{lemm_deltanplusrightoftaub2oppositesign}\n\tLet $\\gamma\\in[0,2)$ and consider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} converging to $(\\operatorname s,1-\\operatorname s^2,0,0,0)$ with $\\operatorname s\\in\\left(1\/2,1\\right]$.\n\tAssume that $\\Delta N_+(\\tau)<0$ for all $\\tau\\le\\tau_0$.\n\tThen\n\t\\begin{align}\n\t\t\\Sigma_+={}&\\operatorname s+\\O(e^{(\\overline{\\Pi}-\\varepsilon)\\tau}),\\\\\n\t\t\\tilde\\Sigma={}&1-\\operatorname s^2+\\O(e^{(\\overline{\\Pi}-\\varepsilon)\\tau}),\\\\\n\t\t\\Delta={}&\\O(e^{(2+2\\operatorname s-2\\sqrt{3(1-\\operatorname s^2)}-\\varepsilon)\\tau}),\\\\\n\t\tN_+={}&\\O(e^{(2+2\\operatorname s-2\\sqrt{3(1-\\operatorname s^2)}-\\varepsilon)\\tau}),\\\\\n\t\t\\tilde N={}&\\O(e^{(4+4\\operatorname s-4\\sqrt{3(1-\\operatorname s^2)}-\\varepsilon)\\tau}),\\\\\n\t\tq={}&2+\\O(e^{(\\overline{\\Pi}-\\varepsilon)\\tau})\n\t\\end{align}\n\tas $\\tau\\rightarrow{-}\\infty$, for every $\\varepsilon>0$. Here $\\overline{\\Pi}\\coloneqq\\min(6-3\\gamma,4+4\\operatorname s-4\\sqrt{3(1-\\operatorname s^2)})$ if $\\Omega>0$, and $\\overline{\\Pi}\\coloneqq4+4\\operatorname s-4\\sqrt{3(1-\\operatorname s^2)}$ if $\\Omega=0$.\n\n\tUnder the additional restriction that $\\operatorname s\\in(1\/2,1)$, there is for each $\\varepsilon>0$ a $\\tau_\\varepsilon>{-}\\infty$ such that $\\tau\\le \\tau_\\varepsilon$ implies\n\t\\begin{align}\n\t\te^{(2+2\\operatorname s-2\\sqrt{3(1-\\operatorname s^2)}+\\varepsilon)\\tau}\\le {}&\\absval{\\Delta},\\\\\n\t\te^{(2+2\\operatorname s-2\\sqrt{3(1-\\operatorname s^2)}+\\varepsilon)\\tau}\\le {}&\\absval{N_+}.\n\t\\end{align}\n\\end{lemm}\n\\begin{proof}\n\tIf $\\operatorname s={-}1$, then due to Prop.~\\ref{prop_taubone} and Prop.~\\ref{prop_tauboneothermatter} the solution is the constant orbit for which all the required properties hold. Let us therefore assume $\\operatorname s\\in({-}1,1]$.\n\tAs in the beginning of the proof of Lemma~\\ref{lemm_deltanplussamesign}, we conclude the upper bounds for~$\\Delta$ and~$N_+$ from Lemma~\\ref{lemm_decaydeltanplus}.\n\n\tThe decay for~$\\Omega$ and~$\\tilde A$ has been determined in equation ~\\eqref{eqn_decayomega} and Lemma~\\ref{lemm_decaytildea} independently of the sign of~$\\Delta N_+$. In particular, $\\tilde A$ decays faster than~$\\Delta^2$ and~$N_+^2$, and consequently the evolution is no longer dominated by~$\\tilde A$, but rather by those two variables. More precisely\n\t\\begin{align}\n\t\t\\tilde N={}&\\frac13(N_+^2-{\\kappa}\\tilde A)=\\O(e^{(4+4\\operatorname s-4\\sqrt{3(1-\\operatorname s^2)}-\\varepsilon)\\tau}),\\\\\n\t\tq-2={}&{-}2\\tilde A-2\\tilde N-\\frac32(2-\\gamma)\\Omega=\\O(e^{(\\overline{\\Pi}-\\varepsilon)\\tau}),\n\t\\end{align}\n\twith $\\overline{\\Pi}$ defined as in the statement of the lemma.\n\tUsing these properties in the evolution equation s for~$\\Sigma_+$ and~$\\tilde\\Sigma$ yields\n\t\t\\begin{equation}\n\t\t\\Sigma_+'=\\O(e^{(\\overline{\\Pi}-\\varepsilon)\\tau}),\\qquad \\tilde\\Sigma'=\\O(e^{(\\overline{\\Pi}-\\varepsilon)\\tau}),\n\t\\end{equation}\n\tand hence gives the convergence rates for~$\\Sigma_+$ and~$\\tilde\\Sigma$.\n\n\t\\smallskip\n\n\tIn order to show the lower bounds for~$\\Delta$ and~$N_+$ in case $\\operatorname s\\in({-}1,1)$, one considers the expression\n\t\\begin{equation}\n\t\t\\frac13\\tilde\\Sigma(\\frac{N_+}{\\Delta-\\hat{\\operatorname r} N_+})^2-(\\frac{\\Delta}{\\Delta-\\hat{\\operatorname r} N_+})^2=\\frac{(\\Sigma_+^2+\\frac {\\kappa}3\\tilde\\Sigma)\\tilde A}{({\\Delta-\\hat{\\operatorname r} N_+})^2},\n\t\\end{equation}\n\twhich is a reformulation of the constraint equation~\\eqref{eqn_constraintforproofs}. Due to the previous argument, the numerator of the right-hand side\\ decays as $\\O(e^{(4+4\\operatorname s-\\varepsilon)\\tau})$, while the denominator is bounded from below by $e^{(2+2\\operatorname s-2\\sqrt{3(1-\\operatorname s^2)}+\\varepsilon)\\tau}$ due to Lemma~\\ref{lemm_decaydeltanplus}. The left-hand side\\ consequently converges to~$0$ as $\\tau\\rightarrow{-}\\infty$. We conclude the same way we did when proving the lower bounds of Lemma~\\ref{lemm_deltanplussamesign}.\n\\end{proof}\n\nSo far, we have obtained exponential convergence rates with exponents that always included an~$\\varepsilon>0$. Such convergence rates hold for all variables.\nAs an immediate consequence, we can apply Lemma~\\ref{lemm_decaylemmageneralimproved} in order to eliminate the~$\\varepsilon$ in the decay of~$\\tilde A$ and~$\\Omega$.\n\\begin{lemm}\n\\label{lemm_improveddecaytildeaomega}\n\tLet $\\gamma\\in[0,2)$ and consider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} converging to $(\\operatorname s,1-\\operatorname s^2,0,0,0)$ with $\\operatorname s\\in\\left[{-}1,1\\right)$. Then the following holds:\n\t\\begin{itemize}\n\t\t\\item Either $\\tilde A=0$ along the whole orbit, or there are constants $c_{\\tilde A},C_{\\tilde A}>0$ and~$\\tau_0$ such that $\\tau\\le\\tau_0$ implies\n\t\t\\begin{equation}\n\t\t\tc_{\\tilde A} e^{(4+4\\operatorname s)\\tau}\\le \\tilde A(\\tau)\\le C_{\\tilde A} e^{(4+4\\operatorname s)\\tau}.\n\t\t\\end{equation}\n\t\t\\item Either $\\Omega=0$ along the whole orbit, or there are constants $c_{\\Omega},C_{\\Omega}>0$ and~$\\tau_0$ such that $\\tau\\le\\tau_0$ implies\n\t\t\\begin{equation}\n\t\t\tc_{\\Omega} e^{(6-3\\gamma)\\tau}\\le \\Omega(\\tau)\\le C_{\\Omega} e^{(6-3\\gamma)\\tau}.\n\t\t\\end{equation}\n\t\\end{itemize}\n\\end{lemm}\n\\begin{proof}\n\tIn case $\\operatorname s={-}1$, nothing has to be shown, as this is the constant orbit (Prop.~\\ref{prop_taubone} and Prop.~\\ref{prop_tauboneothermatter}).\n\tFor the remaining values, all that has to be shown to apply Lemma~\\ref{lemm_decaylemmageneralimproved} is that the expressions\n\t\\begin{equation}\n\t\t2(q+2\\Sigma_+),\\qquad 2q-(3\\gamma-2)\n\t\\end{equation}\n\tappearing in the evolution equations of~$\\tilde A$ and~$\\Omega$ converge at exponential rates.\n\tFrom Lemma~\\ref{lemm_deltanplussigns}, one knows that~$\\Delta N_+$ has constant sign for sufficiently negative times. The case $\\Delta N_+>0$ is covered in Lemma~\\ref{lemm_deltanplussamesign}. If $\\Delta=N_+=0$ along the whole orbit, then Lemma~\\ref{lemm_deltanpluszero} yields the result. In case $\\Delta N_+<0$ exponential convergence is shown in Lemma~\\ref{lemm_deltanplusrightoftaub2oppositesign}.\n\tNote however that this case of opposite sign can be excluded for $\\operatorname s\\in({-1},1\/2]$ by Lemma~\\ref{lemm_deltanplusoppositesign}.\n\\end{proof}\nThese more detailed estimates can now be used to determine the convergence of the remaining variables in more detail. So far, we have found the constant term and the slowest order of exponential convergence. This is improved in the following way: We can relate the slowest order of exponential convergence of several variables, and determine which is the next non-vanishing order.\n\nWe carry this out for orbits converging to the left of the point Taub~2, as it is in this case that we make use of the more detailed convergence properties in Section~\\ref{section_leftoftaubtwo}.\nWe point out, however, that the same approach can also be used on the remaining Kasner limit points.\n\\begin{prop}\n\\label{prop_leftoftaub2additionaldecay}\n\tLet $\\gamma\\in[0,2)$ and consider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} converging to $(\\operatorname s,1-\\operatorname s^2,0,0,0)$ with $\\operatorname s\\in({-1},1\/2)$. If $\\tilde A>0$, and if $\\Delta$ and $N_+$ do not both vanish identically along the solution, then $\\operatorname s=\\pm\\sqrt{{\\kappa}\/({\\kappa}-3)}$, and there are constants $\\alpha>0$, $\\omega\\ge0$, $\\beta_\\Delta\\not=0$ and $\\beta_{N_+}\\not=0$ such that \n\t\\begin{align}\n\t\t\\Delta={}&\\beta_\\Delta e^{(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)})\\tau}+\\O(e^{(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)}+\\Pi-\\varepsilon)\\tau}),\\\\\n\t\t\\tilde A={}&\\alpha e^{(4+4\\operatorname s)\\tau}+\\O(e^{(4+4\\operatorname s+\\Pi-\\varepsilon)\\tau}),\\\\\n\t\tN_+={}&\\beta_{N_+} e^{(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)})\\tau}+\\O(e^{(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)}+\\Pi-\\varepsilon)\\tau}),\\\\\n\t\t\\tilde N={}&{-}\\frac {\\kappa}3\\alpha e^{(4+4\\operatorname s)\\tau}+\\O(e^{(4+4\\operatorname s+\\Pi-\\varepsilon)\\tau}),\\\\\n\t\tq={}&2+2\\alpha(\\frac {\\kappa}3-1)e^{(4+4\\operatorname s)\\tau}-\\frac32(2-\\gamma)\\omega e^{(6-3\\gamma)\\tau}+\\O(e^{(2\\Pi-\\varepsilon)\\tau}),\n\t\\end{align}\n\tas $\\tau\\rightarrow{-}\\infty$, for every $\\varepsilon>0$. Here $\\Pi\\coloneqq\\min(6-3\\gamma,4+4\\operatorname s)$ if $\\Omega>0$, and $\\Pi\\coloneqq 4+4\\operatorname s$ if $\\Omega=0$, and irrespectively of whether~$\\Omega$ is positive or not\n\t\\begin{equation}\n\t\t\\beta_\\Delta=\\sqrt{(1-\\operatorname s^2)\/3}\\beta_{N_+}.\n\t\\end{equation}\n\tFurthermore, if $\\operatorname s\\not=0$, then\n\t\\begin{align}\n\t\\label{eqn_leftoftaub2additionaldecay_otherssigmaplus}\n\t\t\\Sigma_+={}&\\operatorname s+\\frac{\\alpha {\\kappa}}{6\\operatorname s}e^{(4+4\\operatorname s)\\tau}-\\frac{\\operatorname s}2\\omega e^{(6-3\\gamma)\\tau}+\\O(e^{(2\\Pi-\\varepsilon)\\tau}),\\\\\n\t\\label{eqn_leftoftaub2additionaldecay_otherstildesigma}\n\t\t\\tilde\\Sigma={}&1-\\operatorname s^2-\\alpha e^{(4+4\\operatorname s)\\tau}-(1-\\operatorname s^2)\\omega e^{(6-3\\gamma)\\tau}+\\O(e^{(2\\Pi-\\varepsilon)\\tau}),\n\t\\end{align}\n\totherwise\n\t\\begin{align}\n\t\\label{eqn_leftoftaub2additionaldecay_specialssigmaplus}\n\t\t\\Sigma_+={}&\\O(e^{(4+4\\sqrt3)\\tau}),\\\\\n\t\\label{eqn_leftoftaub2additionaldecay_specialstildesigma}\n\t\t\\tilde\\Sigma={}&1-\\alpha e^{4\\tau}-\\omega e^{(6-3\\gamma)\\tau}+\\O(e^{(2\\Pi-\\varepsilon)\\tau}).\n\t\\end{align}\n\tAdditionally, either $\\Omega\\equiv0$ and then $\\omega=0$, or\n\t\\begin{equation}\n\t\t\\Omega=\\omega e^{(6-3\\gamma)\\tau}+\\O(e^{(6-3\\gamma+\\Pi-\\varepsilon)\\tau})\n\t\\end{equation}\n\twith $\\omega>0$.\n\\end{prop}\n\n\\begin{proof}\n\tIt follows from Lemma~\\ref{lemm_deltanplusoppositesign} that $\\Delta N_+>0$ for sufficiently negative times, and Lemma~\\ref{lemm_deltanplussamesign} collects the convergence properties which we obtained so far. From the relation\n\t\\begin{equation}\n\t\t\\tilde A(3\\operatorname s^2+{\\kappa}(1-\\operatorname s^2))=0,\n\t\\end{equation}\n\twhich was shown there, we find the special value for~$\\operatorname s$. In particular, if~$\\kappa>0$ then there are no solutions converging to a point on the Kasner parabola with~$\\operatorname s\\in({-}1,1\/2)$.\n\tWe now deduce more precise decay properties for the individual variables.\n\n\t\\smallskip\n\n\tFor~$\\tilde A$, the decay established in Lemma~\\ref{lemm_improveddecaytildeaomega} implies that $e^{-(4+4\\operatorname s)\\tau}\\tilde A$ is bounded for sufficiently negative~$\\tau$. As\n\t\\begin{equation}\n\t\t(e^{-(4+4\\operatorname s)\\tau}\\tilde A)'=(2(q-2)+4(\\Sigma_+-\\operatorname s))e^{-(4+4\\operatorname s)\\tau}\\tilde A\n\t\\end{equation}\n\tand $2(q-2)+4(\\Sigma_+-\\operatorname s)=\\O(e^{(\\Pi-\\varepsilon)\\tau})$, the left-hand side\\ decays as $\\O(e^{(\\Pi-\\varepsilon)\\tau})$, and one finds\n\t\\begin{equation}\n\t\\label{eqn_auxiliarybetterdecaytildea}\n\t\te^{-(4+4\\operatorname s)\\tau}\\tilde A(\\tau)=\\lim_{\\tau\\rightarrow{-}\\infty}e^{-(4+4\\operatorname s)\\tau}\\tilde A(\\tau)+\\int_{{-}\\infty}^\\tau\\O(e^{(\\Pi-\\varepsilon)s})ds.\n\t\\end{equation}\n\tSetting\n\t\\begin{equation}\n\t\t\\alpha\\coloneqq \\lim_{\\tau\\rightarrow{-}\\infty}e^{-(4+4\\operatorname s)\\tau}\\tilde A(\\tau),\n\t\\end{equation}\n\tequation~\\eqref{eqn_auxiliarybetterdecaytildea} is equivalent to\n\t\\begin{equation}\n\t\t\\tilde A=\\alpha e^{(4+4\\operatorname s)\\tau}+\\O(e^{(4+4\\operatorname s+\\Pi-\\varepsilon)\\tau}).\n\t\\end{equation}\n\tThe exact value of~$\\alpha$ is not known, only that it is positive due to $\\tilde A>0$. But this form shows that there are no terms of exponential order between $4+4\\operatorname s$ and $4+4\\operatorname s+\\Pi$. By the same method one finds that\n\t\\begin{equation}\n\t\t\\Omega=\\omega e^{(6-3\\gamma)\\tau}+\\O(e^{(6-3\\gamma+\\Pi-\\varepsilon)\\tau})\n\t\\end{equation}\n\twith a constant $\\omega>0$, if not $\\Omega\\equiv0$ along the whole orbit.\n\n\t\\smallskip\n\n\tThe improved decay properties of~$\\tilde A$ and~$\\Omega$ together with the ones from Lemma~\\ref{lemm_deltanplussamesign} imply\n\t\\begin{align}\n\t\t\\tilde N={}&\\frac13(N_+^2-{\\kappa}\\tilde A)={-}\\frac {\\kappa}3\\alpha e^{(4+4\\operatorname s)\\tau}+\\O(e^{(4+4\\operatorname s+\\Pi-\\varepsilon)\\tau}),\\\\\n\t\tq={}&2(1-\\tilde A-\\tilde N)-\\frac32(2-\\gamma)\\Omega\\\\\n\t\t\t={}&2+2\\alpha(\\frac {\\kappa}3-1)e^{(4+4\\operatorname s)\\tau}-\\frac32(2-\\gamma)\\omega e^{(6-3\\gamma)\\tau}+\\O(e^{(2\\Pi-\\varepsilon)\\tau}),\n\t\\end{align}\n\tas $4\\sqrt{3(1-\\operatorname s^2)}\\ge4+4\\operatorname s$ if and only if\\ $\\operatorname s\\in\\left[{-}1,1\/2\\right]$.\n\tThe evolution equations\\ for~$\\Sigma_+$ and~$\\tilde\\Sigma$ thus read\n\t\\begin{align}\n\t\\Sigma_+'={}&(q-2)\\Sigma_+-2\\tilde N\\\\\n\t\t={}&2\\alpha(\\frac13{\\kappa}\\operatorname s-\\operatorname s+\\frac13{\\kappa}) e^{(4+4\\operatorname s)\\tau}-s\\omega \\frac32(2-\\gamma)e^{(6-3\\gamma)\\tau}+\\O(e^{(2\\Pi-\\varepsilon)\\tau}),\\\\\n\t\\tilde\\Sigma'={}&2(q-2)\\tilde\\Sigma-4\\Delta N_+-4\\Sigma_+\\tilde A\\\\\n\t\t={}&4\\alpha(\\frac13{\\kappa}-\\frac13{\\kappa}\\operatorname s^2-1+\\operatorname s^2-\\operatorname s )e^{(4+4\\operatorname s)\\tau}-3\\omega(1-\\operatorname s^2)(2-\\gamma) e^{(6-3\\gamma)\\tau}+\\O(e^{(2\\Pi-\\varepsilon)\\tau}).\n\t\\end{align}\n\tIntegrating these expressions and making use of\n\tthe relation $\\operatorname s^2={\\kappa}\/({\\kappa}-3)$\n\tyields that for $\\operatorname s\\not=0$ (or equivalently ${\\kappa}\\not=0$) one finds~\\eqref{eqn_leftoftaub2additionaldecay_otherssigmaplus} and~\\eqref{eqn_leftoftaub2additionaldecay_otherstildesigma}. In case~$\\operatorname s=0$ and~${\\kappa}=0$, one obtains equation~\\eqref{eqn_leftoftaub2additionaldecay_specialstildesigma} for $\\tilde\\Sigma$, and\n\t\\begin{equation}\n\t\t\\Sigma_+=\\O(e^{(2\\Pi-\\varepsilon)\\tau}).\n\t\\end{equation}\n\tNote that~$2\\Pi\\le8<4+4\\sqrt3$. We are going to obtain a stronger estimate further down.\n\n\t\\smallskip\n\n\tTo find the improved decay for the remaining variables~$\\Delta$ and~$N_+$, we recall from the proof of Lemma~\\ref{lemm_decaydeltanplus} that\n\t\\begin{equation}\n\t\\label{eqn_deltaplusrnplusforimproveddecay}\n\t\t(\\Delta+\\hat{\\operatorname r} N_+)'=(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)}+\\frac{f_1\\Delta+f_2N_+}{\\Delta+\\hat{\\operatorname r} N_+})(\\Delta+\\hat{\\operatorname r} N_+),\n\t\\end{equation}\n\twhich there appeared as equation ~\\eqref{eqn_evolutionrdeltaplusnplus}. Here, one has $\\hat{\\operatorname r}=\\sqrt{(1-\\operatorname s^2)\/3}$, and a closer look at the computation carried out in that proof reveals that the two functions~$f_1$, $f_2$ which vanish asymptotically as $\\tau\\rightarrow{-}\\infty$ have the form\n\t\\begin{align}\n\t\tf_1={}&2q-4+2\\Sigma_+-2\\operatorname s,\\\\\n\t\tf_2={}&\\hat{\\operatorname r} q-2\\hat{\\operatorname r}+2\\hat{\\operatorname r}\\Sigma_+-2\\hat{\\operatorname r}\\operatorname s+2\\tilde\\Sigma-2(1-\\operatorname s^2)-2\\tilde N.\n\t\\end{align}\n\tDue to the previous results, both functions decay as $\\O(e^{(\\Pi-\\varepsilon)\\tau})$.\n\tAs~$\\Delta$ and~$\\hat{\\operatorname r} N+$ have the same sign, we see that the quotient with the functions~$f_1$, $f_2$ in equation ~\\eqref{eqn_deltaplusrnplusforimproveddecay} inherits the asymptotic behaviour of these two functions, i.\\,e.\\  decays exponentially to order $\\O(e^{(\\Pi-\\varepsilon)\\tau})$.\n\tHence, we have\n\t\\begin{equation}\n\t\t(e^{-(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)})\\tau}(\\Delta+\\hat{\\operatorname r} N_+))'=e^{-(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)})\\tau}(\\Delta+\\hat{\\operatorname r} N_+)\\O(e^{(\\Pi-\\varepsilon)\\tau}),\n\t\\end{equation}\n\tand find\n\t\\begin{equation}\n\t\te^{-(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)})\\tau}(\\Delta+\\hat{\\operatorname r} N_+)=\\beta+\\O(e^{(\\Pi-\\varepsilon)\\tau}),\n\t\\end{equation}\n\tfor some constant $\\beta\\not=0$ having the same sign as~$\\Delta$ and~$N_+$. Consequently, we obtain\n\t\\begin{equation}\n\t\\label{eqn_improveddecaydeltaplusrnplus}\n\t\t(\\Delta+\\hat{\\operatorname r} N_+)=\\beta e^{(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)})\\tau}+\\O(e^{(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)}+\\Pi-\\varepsilon)\\tau}).\n\t\\end{equation}\n\tWe further see from Remark~\\ref{rema_auxiliarydecayconstraintexpr} that\n\t\\begin{equation}\n\t\t(\\sqrt{\\frac{\\tilde\\Sigma}{3}}N_++\\Delta)(\\sqrt{\\frac{\\tilde\\Sigma}{3}}N_+-\\Delta)=\\frac13(3\\Sigma_+^2+{\\kappa}\\tilde\\Sigma)\\tilde A=\\O(e^{(8+8\\operatorname s+4\\sqrt{3(1-\\operatorname s^2)}-\\varepsilon)\\tau}).\n\t\\end{equation}\n\tThe very first bracket is of order $e^{(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)})\\tau}$ and not faster.\n\tThus,\n\t\\begin{equation}\n\t\\label{eqn_relationdeltanplusimproveddecay}\n\t\t\\Delta=\\hat{\\operatorname r} N_++\\O(e^{(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)}+\\Pi-\\varepsilon)\\tau}).\n\t\\end{equation}\n\tTogether with equation~\\eqref{eqn_improveddecaydeltaplusrnplus}, this implies the decay expressions for~$\\Delta$ and~$N_+$ in the statement, with $\\beta_\\Delta=\\hat{\\operatorname r}\\beta_{N_+}$.\n\n\t\\smallskip\n\n\tIt remains to show equation~\\eqref{eqn_leftoftaub2additionaldecay_specialssigmaplus}. We know from the above that~$\\tilde N=N_+^2\/3=\\O(e^{(4+4\\sqrt{3})\\tau})$.\n\tUsing the function~$\\R$ defined as in Lemma~\\ref{lemm_functionintegraltwominusq},\n\twe integrate\n\t\\begin{equation}\n\t\t(\\Sigma_+e^{\\R})'={-}2\\tilde N e^{\\R}=\\O(e^{(4+4\\sqrt{3})\\tau})\n\t\\end{equation}\n\tand obtain\n\t\\begin{equation}\n\t\t\\Sigma_+e^{\\R}=\\O(e^{(4+4\\sqrt{3})\\tau}).\n\t\\end{equation}\n\tAs the function~$\\R$ is non-negative, this gives the decay for~$\\Sigma_+$ and concludes the proof.\n\\end{proof}\n\n\n\\section{Asymptotics at Taub~2}\n\\label{section_taubtwo}\n\nIn this section, we determine precisely which orbits converge to the point Taub~2, i.\\,e.\\  which solutions to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} satisfy\n\\begin{equation}\n\t\\lim_{\\tau\\rightarrow{-}\\infty}(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)(\\tau)=\\left(\\frac12,\\frac34,0,0,0\\right).\n\\end{equation}\nWe have to restrict ourselves to~$\\gamma\\in[0,2)$ in order to apply our results from the previous section. From Prop.~\\ref{prop_main_generalasymptoticproperties}, we know that along every solution converging to the point Taub~2\n\\begin{equation}\n\t\\tilde A({\\kappa}+1)=0\n\\end{equation}\nhas to hold, and we found precise decay conditions for all variables.\n\nWe deduce in the present section that only locally rotationally symmetric models of Bianchi type~I, II and~VI$_{{-}1}$ are possible.\nIn order to prove this statement, we show that the invariant set consisting of these three Bianchi types, compare Table~\\ref{table_bianchihighersymmetry} and Definition~\\ref{defi_lrsexpansionnorm}, contains all solution converging to the point Taub~2.\n\\begin{prop}\n\\label{prop_taub2specialandnonspecialkvalue}\n\tLet $\\gamma\\in[0,2)$ and consider a solution to~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} which satisfies\n\t\\begin{equation}\n\t\t\\lim_{\\tau\\rightarrow{-}\\infty}(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)(\\tau)=\\left(\\frac12,\\frac34,0,0,0\\right).\n\t\\end{equation}\n\tThen the solution is contained in the invariant set\n\t\\begin{equation}\n\t\t3\\Sigma_+^2=\\tilde\\Sigma\\qquad \\Sigma_+N_+=\\Delta.\n\t\\end{equation}\n\\end{prop}\nThe proof revolves around the function\n\\begin{equation}\n\t\\mathsf f=(3\\Sigma_+^2-\\tilde\\Sigma)^2+2(\\Sigma_+N_+-\\Delta)^2.\n\\end{equation}\nIt is constructed in such a way that its zero set characterises the invariant set from Prop.~\\ref{prop_taub2specialandnonspecialkvalue}.\nThe proof consists of showing that the function vanishes along all orbits converging to the point Taub~2. This is an adaptation of an approach which has already been used successfully in~\\cite[Sect.~4]{ringstrom_curvblowupbianchiviiiandixvacuumspacetimes} in the case of Bianchi~A vacuum models. However, our method of showing that the function~$\\mathsf f$ indeed vanishes is slightly different.\n\\begin{proof}\n\tThe derivatives of the constituents of~$\\mathsf f$ have been computed in~\\eqref{eqn_derivativefunctionforscc}. Combining this with\n\t\\begin{equation}\n\t\t\\tilde A(\\kappa+1)=0\n\t\\end{equation}\n\twhich holds due to Prop.~\\ref{prop_main_generalasymptoticproperties},\n\tthis immediately yields that the set $\\mathsf f=0$ is invariant.\n\tFrom the definition of~$\\mathsf f$ as the sum of two squares\n\twe see that either~$\\mathsf f\\equiv0$ or~$\\mathsf f>0$. In the first case, the statement follows. Assume therefore that~$\\mathsf f>0$ holds.\n\n\tLet us have a closer look at the derivatives of the two terms which compose $\\mathsf f$, see equation~\\eqref{eqn_derivativefunctionforscc}. As\n\t\\begin{equation}\n\t\t2\\Sigma_+\\rightarrow1,\\qquad N_+\\rightarrow0,\\qquad q\\rightarrow2,\n\t\\end{equation}\n\tdue to the assumption on the limit point, we conclude that\n\t\\begin{equation}\n\t\t\\mathsf f'\\le \\zeta\\mathsf f,\n\t\\end{equation}\n\tfor some function~$\\zeta$ which is integrable on~$({-}\\infty,T]$ for some sufficiently negative time~$T$. This follows from the convergence rates obtained in Prop.~\\ref{prop_main_generalasymptoticproperties}. Consequently, integration yields\n\t\\begin{equation}\n\t\t\\mathsf f(T)\\le e^C\\mathsf f(\\tau)\n\t\\end{equation}\n\tfor all~$\\tau\\le T$, where~$C>0$ is a constant. Due to~$\\mathsf f(\\tau)\\rightarrow0$ as~$\\tau\\rightarrow{-}\\infty$, this implies~$\\mathsf f(T)=0$, a contradiction.\n\\end{proof}\nAs a direct consequence of Prop.~\\ref{prop_taub2specialandnonspecialkvalue}, the following theorem characterises the models whose orbits converge to the Kasner point Taub~2.\n\\begin{theo}\n\\label{theo_taubtwocharacterisationoforbits}\n\tLet $\\gamma\\in[0,2)$ and consider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} converging to $\\left(1\/2,3\/4,0,0,0\\right)$ as $\\tau\\rightarrow{-}\\infty$. Then this solution is one of the following:\n\t\\begin{itemize}\n\t\t\\item Bianchi~I~LRS, i.\\,e.\\  $\\Delta=\\tilde A=N_+=0$, $3\\Sigma_+^2=\\tilde\\Sigma$,\n\t\t\\item Bianchi~II~LRS, i.\\,e.\\  $\\tilde A=0$, $3\\Sigma_+^2=\\tilde\\Sigma$, $\\Sigma_+N_+=\\Delta$,\n\t\t\\item Bianchi~VI$_{{-}1}$~LRS, i.\\,e.\\  ${\\kappa}={-}1$, $\\tilde A>0$, $3\\Sigma_+^2=\\tilde\\Sigma$, $\\Sigma_+N_+=\\Delta$.\n\t\\end{itemize}\n\\end{theo}\n\n\n\\section{Asymptotics to the left of Taub~2}\n\\label{section_leftoftaubtwo}\n\nIn this section, we discuss the set of solutions converging to a Kasner point to the left of Taub~2 as $\\tau\\rightarrow{-}\\infty$, i.\\,e.\\  with a limit point $(\\operatorname s,1-\\operatorname s^2,0,0,0)$ satisfying~${-}1<\\operatorname s<1\/2$.\nWe have shown in Prop.~\\ref{prop_main_generalasymptoticproperties} that for such orbits either $\\tilde A=0$ has to hold, or the limit value~$\\operatorname s$ and the parameter~$\\kappa$ have to satisfy the relation~$3\\operatorname s^2+\\kappa(1-\\operatorname s^2)=0$. This is the same restriction we appealed to in the previous section, when discussing convergence to the point Taub~2.\nIn other words, for a given~${\\kappa}$, any solution to equation s~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} converging to a point $(\\operatorname s,1-\\operatorname s^2,0,0,0)$ on the Kasner parabola with $-1<\\operatorname s<1\/2$ is either a Bianchi~A solution or has to converge to the limit point with $\\operatorname s=\\pm\\sqrt{{\\kappa}\/({\\kappa}-3)}$. This last special case, which can only occur if ${\\kappa}\\le0$, is discussed in more detail in this section, using concepts, notation and results from dynamical systems theory which we recall in Appendix~\\ref{section_appendixdynamsystheo}.\n\nThe object we are interested in is the submani\\-fold\\ called centre-unstable mani\\-fold~${\\mathcal C}^u$, as it contains the maximal negatively invariant set~$A^-(U)$ of a suitable open neighborhood~$U$ of the point $(\\operatorname s,1-\\operatorname s^2,0,0,0)$. This latter set consists of all points which remain in~$U$ under the evolution in the negative time direction. In particular, all solutions converging to a point in~$U$ as $\\tau\\rightarrow{-}\\infty$ are contained in this set~$A^-(U)$ for sufficiently negative times.\n\nTo determine the properties of the centre-unstable mani\\-fold~${\\mathcal C}^u$, we consider\nthe linearised evolution equations in the extended five-dimensional state space, by which we mean the linear approximation of the evolution equation s~\\eqref{eqns_evolutionbianchib}, with~$q$ and~$\\tilde N$ defined as in equations~\\eqref{eqn_definitionqgeneral} and~\\eqref{eqn_definitiontilden}, but without assuming the constraint equations~\\eqref{eqn_constraintgeneralone} and~\\eqref{eqn_constraintgeneraltwo}.\nThe corresponding matrix is a linear transformation of the five-dimensional tangent space to~$\\maths{R}^5$ at equilibrium points of the evolution. We give the explicit form of the linear mapping and its eigenvectors and -values in Appendix~\\ref{subsect_appendixlinearisedevolutionkasner}. For every Kasner point to the left of Taub~2, there are exactly four eigenvectors such that the corresponding eigenvalues are non-negative.\n\nThe centre-unstable mani\\-fold~${\\mathcal C}^u$ which we want to understand is tangent to the set~$E_m{}^c\\oplus E_m{}^u$ which in its turn is spanned by the eigenvectors to eigenvalues with non-negative real part. The information on the number of such eigenvalues therefore translates into properties of the centre-unstable mani\\-fold, and thus into information on the set which solutions converging to a Kasner point to the left of Taub~2 have to eventually be contained in. We find the following statement: There is a four-dimensional mani\\-fold\\ in~$\\maths{R}^5$ such that every solution converging to a point $(\\operatorname s,1-\\operatorname s^2,0,0,0)$ on the Kasner parabola with $-1<\\operatorname s<1\/2$ is contained in this mani\\-fold\\ for sufficiently negative times.\n\\begin{prop}\n\\label{prop_centreunstablelocallyleftoftaub2}\n\tConsider the evolution equation s~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_definitiontilden}\n\tin the extended state space, i.\\,e.\\  without assuming the constraint equation s~\\eqref{eqn_constraintgeneralone}--\\eqref{eqn_constraintgeneraltwo}.\n\tIf ${\\kappa}\\le0$ and $\\operatorname s\\coloneqq\\pm\\sqrt{{\\kappa}\/({\\kappa}-3)}\\in({-}1,1\/2)$, then\n\tthere is a neighborhood~$U$ of the point $(\\operatorname s,1-\\operatorname s^2,0,0,0)$ and a four-dimensional\n\t$C^1$~submani\\-fold\n\t~$M_{\\kasnerparabola}$ of~$\\maths{R}^5$ in~$U$ with the following properties:\n\t\\begin{itemize}\n\t\t\\item $M_{\\kasnerparabola}$ contains the point $(\\operatorname s,1-\\operatorname s^2,0,0,0)$ and in this point is tangent to those eigenvectors of the linearised evolution equations in the extended five-dimensional space, given in Appendix~\\ref{subsect_appendixlinearisedevolutionkasner}, which correspond to eigenvalues with non-negative real part.\n\t\t\\item Points in~$U$ are either contained in~$M_{\\kasnerparabola}$ or\n\t\ttheir evolution under equation s~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_definitiontilden} leaves~$U$ as $\\tau\\rightarrow{{-}\\infty}$.\n\t\\end{itemize}\n\\end{prop}\n\\begin{rema}\n\tWe see in the proof that we could change the regularity of the manifold~$M_{\\kasnerparabola}$ to~$C^r$, for some~$r<\\infty$. Further, as all eigenvalues on~$\\K$ are real, the mani\\-fold~$M_{\\kasnerparabola}$ is tangent to those eigenvectors which correspond to non-negative eigenvalues.\n\\end{rema}\n\\begin{rema}\n\tThis statement in particular applies to every solution to the evolution equation s~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} which converges to the point $(\\operatorname s,1-\\operatorname s^2,0,0,0)$, as $\\tau\\rightarrow{-}\\infty$, with limit value $\\operatorname s\\coloneqq\\pm\\sqrt{{\\kappa}\/({\\kappa}-3)}\\in({-}1,1\/2)$: There is a time~$\\tau_0$ such that the solution is contained in the neighborhood~$U$ for all times~$\\tau\\le\\tau_0$. Therefore, the solution has to be contained in the submani\\-fold~$M_{\\kasnerparabola}$ for~$\\tau\\le\\tau_0$.\n\\end{rema}\n\\begin{proof}\n\tEvery point on the Kasner parabola is a zero-dimensional mani\\-fold\\ consisting of equilibrium points of the evolution equation s. Using the notation of Appendix~\\ref{section_appendixdynamsystheo}, we can therefore apply Thm~\\ref{theo_centremftheory} to this zero-dimensional submani\\-fold\\ of equilibrium points to find a centre-unstable mani\\-fold~${\\mathcal C}^u$ near this point, and\n\ta neighborhood~$U$ of $(\\operatorname s,1-\\operatorname s^2,0,0,0)$ such that the maximal negatively invariant set~$A^-(U)$ is contained in~${\\mathcal C}^u$.\n\tThe point~$(\\operatorname s,1-\\operatorname s^2,0,0,0)$ is contained in the manifold~${\\mathcal C}^u$ by Def.~\\ref{defi_centremf}. Without loss of generality, we therefore restrict the manifold to~$U$.\n\n\tBy definition, the mani\\-fold~${\\mathcal C}^u$ is tangential to $E_m{}^c\\oplus E_m{}^u$, which are the subspaces of the tangent space at $(\\operatorname s,1-\\operatorname s^2,0,0,0)$ associated with eigenvalues on the imaginary axis and in the right half-plane:\n\t\\begin{equation}\n\t\t0\\qquad\n\t\t2(1+\\operatorname s+\\sqrt{3(1-\\operatorname s^2)})\\qquad\n\t\t4(1+\\operatorname s)\\qquad\n\t\t3(2-\\gamma).\n\t\\end{equation}\n\tThe evolution equations~\\eqref{eqns_evolutionbianchib} are polynomial and consequently~$C^\\infty$.\n\tWe can therefore apply Thm~\\ref{theo_centremftheory} with some finite~$r$, for example~$r=1$.\n\tConsequently, ${\\mathcal C}^u$ is a four-dimensional submani\\-fold, and it has the requested properties.\n\\end{proof}\n\\begin{rema}\n\tThis statement could have been achieved using results from the theory of dynamical systems which are less powerful than the one we used here, Thm~\\ref{theo_centremftheory}, as we only apply it to zero-dimensional mani\\-fold s and the individual points they contain. However, in Section~\\ref{section_asymptoticsplanewave}, we make use of this theorem again, this time using it to a fuller extent.\n\\end{rema}\n\nThe previous statement gives information on the centre-unstable mani\\-fold\\ corresponding to the evolution in the extended state space.\nIn order to understand the evolution in the non-extended state space, i.\\,e.\\  restricted to the set of points\nsuch that  the constraint equation s~\\eqref{eqn_constraintgeneralone} and~\\eqref{eqn_constraintgeneraltwo} are satisfied, we need to understand the relation between the mani\\-fold\\ we found and the constraint surface defined by equation~\\eqref{eqn_constraintgeneralone}. More precisely, we are interested in the codimension of their intersection, which is what we determine in Thm~\\ref{theo_leftoftaubtworesult}. As the constraint equation~\\eqref{eqn_constraintgeneralone} defines a set which becomes singular in certain points, see Remark~\\ref{rema_singularconstraintequ}, we cannot deduce this codimension solely from the knowledge about normal and tangent directions derived above. In addition, we use the convergence behaviour of the individual variables from Prop.~\\ref{prop_leftoftaub2additionaldecay}, where we determined the slowest order exponential term and the next non-vanishing one, for convergence to Kasner points to the left of the point Taub~2.\n\n\n\\begin{theo}\n\\label{theo_leftoftaubtworesult}\n\tLet~$0\\not={\\kappa}\\in\\maths{R}$, $\\gamma\\in[0,2)$ and consider the set of solutions to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} converging to $(\\operatorname s,1-\\operatorname s^2,0,0,0)$ with $\\operatorname s\\in\\left({-}1,1\/2\\right)$, as $\\tau\\rightarrow{-}\\infty$. If~$\\tilde A>0$, then\n\t\\begin{equation}\n\t\t\\operatorname s=\\pm\\sqrt{\\frac{\\kappa}{{\\kappa}-3}}\n\t\\end{equation}\n\tand the solution is contained in one of the following subsets:\n\t\\begin{itemize}\n\t\t\\item The invariant set satisfying $\\tilde A>0$, $\\Delta=0=N_+$, $3\\Sigma_+^2+{\\kappa}\\tilde\\Sigma=0$.\n\t\t\\item A countable union of $C^1$ submani\\-fold s satisfying $\\tilde A>0$, and~$\\Delta$, $N_+$ not both vanishing identically. The submani\\-fold s are contained either in the set of non-vacuum solutions or the set of vacuum solutions, and in the respective sets have codimension at least~one.\n\t\\end{itemize}\n\\end{theo}\n\\begin{proof}\n\tThe relation between the parameter~${\\kappa}$ and the value of~$\\operatorname s$ is an immediate consequence of Prop.~\\ref{prop_main_generalasymptoticproperties}, as we assume~$\\tilde A>0$. If the solution satisfies~$\\Delta=0=N_+$ at some time, then it does so at all times, as this property is invariant under the evolution equations~\\eqref{eqns_evolutionbianchib}. The relation between~$\\Sigma_+$ and~$\\tilde\\Sigma$ follows from equation~\\eqref{eqn_constraintgeneralone},\n\twhich concludes the proof for the first case.\n\tIt thus remains to calculate the maximal dimension and codimension of the submanifolds in the second case.\n\n\tConsider a solution as in the statement which satisfies that~$\\Delta$ and~$N_+$ do not both vanish identically. Due to convergence, the solution is contained in the neighborhood~$U$ from the previous proposition for sufficiently negative times, and therefore the solution has to lie in the submani\\-fold ~$M_{\\kasnerparabola}$ for sufficiently negative times.\n\tAt the same time, the solution satisfies the constraint equation ~\\eqref{eqn_constraintgeneralone}.\n\tWe therefore have to show that the constraint surface and the submani\\-fold ~$M_{\\kasnerparabola}$ either have an empty intersection or intersect transversally in a set of the correct dimension, when additionally restricted to $\\tilde A>0$ and~$\\Delta$, $N_+$ not both vanishing identically.\n\tTo do this, we compare the normal direction of the constraint surface to the vectors spanning~$M_{\\kasnerparabola}$. By counting the number of spanning vectors which are or\\-tho\\-go\\-nal\\ to the gradient direction, we find the dimension and properties of the set in question.\n\n\tAs we consider solutions with~$\\tilde A>0$ and $\\Delta$, $N_+$ not both vanishing, and\n\t\\begin{equation}\n\t\t\\kappa=\\frac{3\\operatorname s^2}{\\operatorname s^2-1}<0\n\t\\end{equation}\n\tby Prop.~\\ref{prop_main_generalasymptoticproperties}, we conclude from Remark~\\ref{rema_singularconstraintequ}\n\tthat the gradient of the constraint equation does not vanish along such solutions.\n\tConsequently, the set of points defined by equation~\\eqref{eqn_constraintgeneralone} is smooth along the solutions under consideration.\n\tHowever, due to the special value of~${\\kappa}$ the constraint surface becomes singular in the limit point~$(\\operatorname s,1-\\operatorname s^2,0,0,0)$.\n\tWe therefore cannot simply compute the scalar product between the (vanishing) gradient of the constraint equation and the vectors spanning the submani\\-fold ~$M_{\\kasnerparabola}$ to show transversality, at least not in the limit point itself. Instead, we consider the gradient in its general form~\\eqref{eqn_gradientconstraintequ}, but replace the fourth component using the constraint equation ~\\eqref{eqn_constraintforproofs} to obtain\n\t\\begin{equation}\n\t\t({-}2\\Sigma_+\\tilde A\\,,\\,\\frac13(N_+^2-{\\kappa}\\tilde A)\\,,\\,{-}2\\Delta\\,,\\,{-}\\frac1{3\\tilde A}(\\tilde\\Sigma N_+^2-3\\Delta^2)\\,,\\,\\frac23\\tilde\\Sigma N_+).\n\t\\end{equation}\n\tApplying the improved convergence properties found in Prop.~\\ref{prop_leftoftaub2additionaldecay} as well as Remark~\\ref{rema_auxiliarydecayconstraintexpr} shows that the decay behaviour of this gradient is\n\t\\begin{equation}\n\t\t({-}2\\operatorname s,{-}\\frac {\\kappa}3,0,0,0)\\cdot\\alpha e^{(4+4\\operatorname s)\\tau}+\\O(e^{(\\min(2+2\\operatorname s+2\\sqrt{3(1-\\operatorname s^2)},4+4\\operatorname s+\\Pi-\\varepsilon))\\tau}),\n\t\\end{equation}\n\twhere we used the special value of~${\\kappa}$. The last term here denotes a vector in~$\\maths{R}^5$ whose every component has the denoted decay. This decay is faster than the one of the first vector, which decays to order $4+4\\operatorname s$. Consequently, we can normalise the gradient vector by multiplying with $e^{{-}(4+4\\operatorname s)\\tau}\/\\alpha$\n\tto eliminate the highest order of decay. Rescaling in this manner gives a well-defined non-vanishing gradient direction even up to the singular point $(\\operatorname s,1-\\operatorname s^2,0,0,0)$.\n\n\tLet us now turn to~$M_{\\kasnerparabola}$. In the limit point~$(\\operatorname s,1-\\operatorname s^2,0,0,0)$, this four-dimensional submani\\-fold\\ is spanned by the eigenvectors to non-negative eigenvalues, see Appendix~\\ref{subsect_appendixlinearisedevolutionkasner} for the explicit form of these eigenvectors.\n\tDirect computation of the scalar product shows that the rescaled gradient of the constraint equation, i.\\,e.\\  the vector $({-}2\\operatorname s,{-}{\\kappa}\/3,0,0,0)$, is or\\-tho\\-go\\-nal\\ to the eigenvectors to eigenvalues $2+2\\operatorname s\\pm2\\sqrt{3(1-\\operatorname s^2)}$, $4(1+\\operatorname s)$ and $3(2-\\gamma)$, but not the eigenvector to~$0$.\n\n\tSimilarly, we can compare the spanning directions of~$M_{\\kasnerparabola}$ to the normal direction of the set~$\\Omega=0$, i.\\,e.\\  the gradient of~$\\Omega$. Due to equation~\\eqref{eqn_omegageneral}, this gradient is\n\t\\begin{equation}\n\t\t({-}2\\operatorname s,{-}1,0,\\frac{\\kappa}3-1,0)\n\t\\end{equation}\n\tfor the point on the Kasner parabola~$\\K$ with~$\\Sigma_+=\\operatorname s$. Direct computation of the scalar product shows that this gradient is or\\-tho\\-go\\-nal\\ to the eigenvectors to eigenvalues $2+2\\operatorname s\\pm2\\sqrt{3(1-\\operatorname s^2)}$, $4(1+\\operatorname s)$ and $0$, but not the eigenvector to~$3(2-\\gamma)$.\n\n\tAs the mani\\-fold~$M_{\\kasnerparabola}$ is $C^1$, the spanning vectors depend continuously on the point. Consequently, in a sufficiently small neighborhood of the point~$(\\operatorname s,1-\\operatorname s^2,0,0,0)$, the mani\\-fold~$M_{\\kasnerparabola}$ and the constraint surface intersect transversally in a submani\\-fold\\ of dimension at most~three if they intersect at all. The solutions for non-vacuum, i.\\,e.\\  $\\Omega>0$, form a set of dimension four in the constraint surface, see also Table~\\ref{table_bianchibsubsets}, as~$\\kappa<0$ by assumption. Hence, in a sufficiently small neighborhood the intersection of the mani\\-fold~$M_{\\kasnerparabola}$ and the constraint surface is of codimension at least one in the set of all non-vacuum solutions.\n\n\tFor vacuum solutions, we realise that one of the four eigenvectors in question is non-or\\-tho\\-go\\-nal\\ to the gradient of~$\\Omega$, and another one is non-or\\-tho\\-go\\-nal\\ to the rescaled gradient of the constraint equation. Consequently, restricting the mani\\-fold~$M_{\\kasnerparabola}$ first to the set~$\\Omega=0$ and then additionally to the constraint surface by an argument similar to the one for~$\\Omega>0$ yields that\n\tin a sufficiently small neighborhood of the point~$(\\operatorname s,1-\\operatorname s^2,0,0,0)$, the intersection of the mani\\-fold~$M_{\\kasnerparabola}$, the constraint surface and the set~$\\Omega=0$ is a submani\\-fold\\ of dimension at most~two. As all vacuum solutions form a set of dimension three, in a sufficiently small neighborhood this intersection is of codimension at least one in the set of all vacuum solutions.\n\n\tWe now apply the flow corresponding to the evolution equations and integer times to this intersection mani\\-fold. As the flow is a diffeomorphism coming from a polynomial evolution equation, the resulting set is a countable union of $C^1$ submani\\-fold s of codimension at least~one in the respective set of solutions, and by construction contains all solutions satisfying the properties listed in the statement. Further, the set~$\\Omega=0$ is invariant under the flow.\n\\end{proof}\n\\begin{rema}\n\tIn case~${\\kappa}=0$, we cannot apply the same reasoning, as the improved convergence properties from Prop.~\\ref{prop_leftoftaub2additionaldecay} and the relation between~$\\beta_\\Delta$ and~$\\beta_{N_+}$ found in the same proposition yield that the gradient~\\eqref{eqn_gradientconstraintequ} of the constraint equation decays as\n\t\\begin{equation}\n\t\t(0,0,{-}1,0,\\frac{\\sqrt3}3)\\cdot2\\beta_\\Delta e^{(2+2\\sqrt{3})\\tau}\n\t\t+\\O(e^{(2+2\\sqrt{3}+\\Pi-\\varepsilon)\\tau}),\n\t\\end{equation}\n\twhere we used Remark~\\ref{rema_auxiliarydecayconstraintexpr} for the fourth component.\n\tOne can normalise this vector by multiplication with $e^{{-}(2+2\\sqrt{3})\\tau}\/(2\\beta_\\Delta)$ but the resulting direction in the limit point~$(0,1,0,0,0)$ is or\\-tho\\-go\\-nal\\ to all four eigenvectors to non-negative eigenvalues. This means that the submani\\-fold~$M_{\\kasnerparabola}$ and the constraint surface do not intersect transversally, but are tangent in the limit point. This does not give any additional information on the set containing possible solutions.\n\\end{rema}\n\n\n\n\n\n\\section{Asymptotics to the right of Taub~2}\n\n\\label{section_rightoftaubtwo}\n\nIn this section, we turn our attention to solutions with a limit point on the Kasner parabola to the right of Taub~2, i.\\,e.\\  a limit point in $\\K\\cap\\{\\Sigma_+>1\/2\\}$.\nFor such points, all eigenvalues but one are positive as soon as $\\gamma<2$. One can therefore expect that this arc of equilibrium points acts as a source, even in the extended state space, as mentioned by Hewitt--Wainwright in~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}.\nAs one considers an arc of equilibrium points, it is desirable to make a more thorough analysis. We carry out this analysis, using not the signs of the eigenvalues but the explicit evolution equations.\nIn our understanding, an arc of the Kasner parabola can be considered a source if for any point on this arc, every orbit which\nenters a sufficiently small neighborhood of that point\nalso converges to the Kasner parabola as~$\\tau\\rightarrow{-}\\infty$, and the limiting point is close to that particular point on the arc. Corollary~\\ref{coro_rightoftaubtwovacuuminflationary} states that this holds for the arc of the Kasner parabola which lies to the right of the point Taub~2, for vacuum and inflationary matter models. In general, we are able to show that the~$\\Sigma_+$ coordinate cannot differ too much.\n\\begin{prop}\n\\label{prop_rightoftaub2source}\n\tLet $1\/2<\\bar\\operatorname s\\le1$ and $0<\\varepsilon<\\bar\\operatorname s-1\/2$. Then there is a neighborhood~$U$ of $(\\bar\\operatorname s,1-\\bar\\operatorname s^2,0,0,0)$\n\twith the following properties: For every solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} which intersects~$U$ at time~$\\tau_0$, the~$\\Sigma_+$-value\n\tsatisfies\n\t\\begin{equation}\n\t\t\\absval{\\Sigma_+(\\tau)-\\bar\\operatorname s}<\\varepsilon\n\t\\end{equation}\n\tfor all~$\\tau\\le\\tau_0$.\n\\end{prop}\nThis estimate holds in particular for the~$\\Sigma_+$-value of every $\\alpha$-limit point of the solution.\nIn case one knows that all $\\alpha$-limit points satisfying $\\Sigma_+>1\/2$ are located on the Kasner parabola, this immediately gives an even stronger statement:\n\\begin{coro}\n\\label{coro_rightoftaubtwovacuuminflationary}\n\tAssume either vacuum or inflationary matter, i.\\,e.\\  either $\\Omega=0$ or $\\Omega>0$, $\\gamma\\in\\left[0,2\/3\\right)$, let $1\/2<\\bar\\operatorname s\\le1$ and $0<\\varepsilon<\\bar\\operatorname s-1\/2$.\n\tThen there is a neighborhood~$U$ of $(\\bar\\operatorname s,1-\\bar\\operatorname s^2,0,0,0)$\n\tsuch that every solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} which intersects~$U$ converges to a Kasner point $(\\operatorname s,1-\\operatorname s^2,0,0,0)$ with $\\absval{\\operatorname s-\\bar\\operatorname s}<\\varepsilon$.\n\\end{coro}\n\\begin{proof}\n\tIt follows from the previous proposition in combination with Prop.~\\ref{prop_alphalimitsets_vacuum_inflat} and the fact that\n\t\\begin{equation}\n\t\t\\planewave {\\kappa} \\cap \\left\\{\\Sigma_+\\ge\\frac12\\right\\}=\\emptyset\n\t\\end{equation}\n\tthat the~$\\alpha$-limit set of every such solution is contained in the Kasner parabola~$\\K$ and does not intersect the plane wave equilibrium points~$\\planewave {\\kappa}$. Due to Prop.~\\ref{prop_convergencetokasner}, the solution converges to a Kasner point $(\\operatorname s,1-\\operatorname s^2,0,0,0)$, and the estimate on~$\\operatorname s$ follows from applying Prop.~\\ref{prop_rightoftaub2source} again.\n\\end{proof}\nTo prove the previous proposition, the main idea is to use the fact that to the right of Taub~2, the Kasner parabola has a slope which is steeper than~${-}1$. Then, one shows that one can bound solutions from below by some straight line with this slope. This is the statement of Lemma~\\ref{lemm_rightoftaub2deltanplusbothsigns} below and we provide a visualisation in Figure~\\ref{figure_rightoftaubtwo}.\nAs the convergence point has to lie above this line but below the Kasner parabola, one gains control over where the convergence point has to be situated exactly.\n\\begin{lemm}\n\\label{lemm_rightoftaub2deltanplusbothsigns}\n\tConsider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} such that $\\Sigma_+(\\tau_0)>1\/2$ for some fixed~$\\tau_0$,\n\tand let $\\varepsilon_1>0$. Then there is a $\\delta=\\delta(\\varepsilon_1)>0$ such that $\\tilde A(\\tau_0)<\\delta$ implies\n\t\\begin{equation}\n\t\\label{eqn_rightoftaub2estimates}\n\t\t\\Sigma_+(\\tau)\\ge\\Sigma_+(\\tau_0),\\qquad\n\t\t\\tilde\\Sigma(\\tau)\\ge{-}\\Sigma_+(\\tau)+(\\Sigma_++\\tilde\\Sigma)(\\tau_0)-\\varepsilon_1,\\qquad\n\t\t\\tilde A<\\delta e^{\\tau-\\tau_0},\n\t\\end{equation}\n\tfor all $\\tau\\le\\tau_0$.\n\\end{lemm}\nWe visualise the two first inequalities of this statement in Figure~\\ref{figure_rightoftaubtwo}: Given that the~$\\Sigma_+$- and~$\\tilde\\Sigma$-values are known at some time~$\\tau_0$, the solution at earlier times~$\\tau\\le\\tau_0$ has to be contained in the shaded area which is bounded by the vertical~$\\Sigma_+=const$ line, the straight line with slope ${-}1$ through the point~$(\\Sigma_+(\\tau_0),\\tilde\\Sigma(\\tau_0)-\\varepsilon_1)$, and the Kasner parabola~$\\K$.\n\\begin{figure}[!ht]\n\t\\begin{tikzpicture}[xscale=5.45,yscale=5.45]\n\t\\draw [->] (0,0) -- (1.2,0);\n\t\\draw [->] (0,0) -- (0,0.85);\n\t\\draw [gray, thick, domain=0.35:1] plot (\\x, {(\\x+1)*(1-\\x)});\n\t\\node [below left] at (0,0) {$0$};\n\t\\draw (1,-0.02) -- (1,0.02);\n\t\\node [below] at (1,-0.01) {$1$};\n\t\\draw (-0.02,0.75) -- (0.02,0.75);\n\t\\node [left] at (0,0.75) {$3\/4$};\n\t\\node [right] at (1.2,0) {$\\Sigma_+$};\n\t\\node [above] at (0,0.85) {$\\tilde\\Sigma$};\n\t\\draw (0.5,-0.02) -- (0.5,0.02);\n\t\\node [below] at (0.5,-0.01) {$1\/2$};\n\t\\draw[fill] (0.5,0.75) circle [radius=0.01];\n\t\\node [below left] at (0.5,0.75) {$\\operatorname{T2}$};\n\t\\node [above left] at (0.7,0.6) {$\\K$};\n\t\\newcommand{0.7}{0.7}\n\t\\newcommand{0.4}{0.4}\n\t\\path [fill=gray,fill opacity=0.7] plot [smooth,samples=100,domain=0.7:0.958](\\x,{(\\x+1)*(1-\\x)}) -- plot [smooth,samples=100,domain=0.958:0.7] (\\x,{-\\x +0.7 +0.4-0.06});\n\t\\draw [dotted] (0,0.4) -- (0.7,0.4);\n\t\\draw [dotted] (0.7,0) -- (0.7,0.3);\n\t\\draw (-0.02,0.4) -- (0.02,0.4);\n\t\\node [left] at (0,0.4) {$\\tilde\\Sigma(\\tau_0)$};\n\t\\draw [dotted] (0,0.34) -- (0.7,0.34);\n\t\\draw (-0.02,0.34) -- (0.02,0.34);\n\t\\node [left] at (0,0.34) {$\\tilde\\Sigma(\\tau_0)-\\varepsilon_1$};\n\t\\draw (0.7,-0.02) -- (0.7,0.02);\n\t\\node [below] at (0.7,-0.01) {$\\Sigma_+(\\tau_0)$};\n\t\\draw[fill] (0.7,0.4) circle [radius=0.012];\n\t\\node [above left] at (0.7,0.4) {$(\\Sigma_+,\\tilde\\Sigma)(\\tau_0)$};\n\t\\draw [dashed] (0.7,0.3)--(0.7,0.55);\n\t\\draw [dashed, domain=0.67:1.01] plot (\\x, {-\\x +0.7 +0.4-0.06});\n\t\\end{tikzpicture}\n\t\\caption{Visualisation of Lemma~\\ref{lemm_rightoftaub2deltanplusbothsigns}: For times~$\\tau\\le\\tau_0$, the projection of the solution to the~$\\Sigma_+\\tilde\\Sigma$-plane has to be contained in the shaded area.}\n\t\\label{figure_rightoftaubtwo}\n\\end{figure}\n\\begin{proof}\n\tThe estimate for~$\\Sigma_+$ is an immediate consequence of the monotonicity shown in Lemma~\\ref{lemm_monotonicitysigmaplus}. It then follows that $\\Sigma_+(\\tau)>1\/2$ for all~$\\tau\\le\\tau_0$, and\n\tequation~\\eqref{eqn_definitionqmatter} implies\n\t\\begin{equation}\n\t\tq=2(\\Sigma_+^2+\\tilde\\Sigma)+\\frac12(3\\gamma-2)\\Omega\\ge2\\Sigma_+^2-\\Omega\\ge{-}\\frac12\n\t\\end{equation}\n\tdue to Remark~\\ref{rema_statespacecompact}.\n\tUsing this estimate in the evolution equation\\ for~$\\tilde A$, equation ~\\eqref{eqns_evolutionbianchib},\n\tone finds\n\t\\begin{equation}\n\t\t\\tilde A(\\tau)\\le\\tilde A(\\tau_0)e^{\\tau-\\tau_0}\n\t\\end{equation}\n\tfor all~$\\tau\\le\\tau_0$.\n\tConsequently, the estimate on~$\\tilde A$ holds for every choice of~$\\delta$.\n\n\tTo prove the remaining estimate, we fix a time~$\\tau_1\\le\\tau_0$ and\n\tdistinguish between the two cases that~$\\Delta N_+(\\tau_1)\\ge0$ or~$\\Delta N_+(\\tau_1)<0$. We prove that in both cases\n\t\\begin{equation}\n\t\\label{eqn_evolutionsigmaplusplustildesigma}\n\t\t(\\Sigma_++\\tilde\\Sigma)'(\\tau_1)\\le M\\sqrt{\\tilde A(\\tau_1)},\n\t\\end{equation}\n\tfor some constant~$M>0$ independent of~$\\tau_1$. This estimate is then used to conclude the proof.\n\n\tWe start with the case $\\Delta N_+(\\tau_1)\\ge 0$.\n\tAs $\\Sigma_+(\\tau)>1\/2$ holds for all~$\\tau\\le\\tau_0$ due to the estimate on~$\\Sigma_+$, we find\n\t\\begin{equation}\n\t\t(\\Sigma_++\\tilde\\Sigma)'(\\tau_1)\\le0,\n\t\\end{equation}\n\twhich gives the desired statement.\n\n\n\tNext, we consider the case $\\Delta N_+(\\tau_1)<0$.\n\tThe constraint equation ~\\eqref{eqn_constraintgeneralone} in the form\n\t\\begin{equation}\n\t\t\\Delta^2=\\frac13\\tilde\\Sigma N_+^2-(\\Sigma_+^2+\\frac{\\kappa}3\\tilde\\Sigma)\\tilde A\n\t\\end{equation}\n\timplies that\n\t\\begin{equation}\n\t\t\\absval{\\Delta}\\le \\sqrt{\\frac{\\tilde\\Sigma}{3}}\\absval{N_+}+\\sqrt{\\frac{3+\\absval {\\kappa}}{3}}\\sqrt{\\tilde A},\n\t\\end{equation}\n\tusing $\\Sigma_+\\in\\left[{-}1,1\\right]$ and $\\tilde\\Sigma\\in\\left[0,1\\right]$.\n\tDue to the assumption on the sign of~$\\Delta N_+$, this implies\n\t\\begin{equation}\n\t\t0<{-}\\Delta N_+\\le \\sqrt{\\frac{\\tilde\\Sigma}{3}}N_+^2+\\sqrt{\\frac{3+\\absval {\\kappa}}{3}}\\absval{N_+}\\sqrt{\\tilde A}\n\t\\end{equation}\n\tat time~$\\tau_1$.\n\tWith this and equation ~\\eqref{eqn_definitionqmatter} for~$q$ one computes, suppressing the time~$\\tau_1$ for readibility, that\n\t\\begin{align}\n\t\t(\\Sigma_++\\tilde\\Sigma)'(\\tau_1)={}&{-}\\frac23(\\Sigma_++1)N_+^2-\\frac43\\tilde\\Sigma N_+^2-4\\Delta N_+\n\t\t\t-\\frac32(2-\\gamma)(\\Sigma_++2\\tilde\\Sigma)\\Omega\\\\\n\t\t\t&{}-2\\tilde A((3-\\frac {\\kappa}3)\\Sigma_++2(1-\\frac {\\kappa}3)\\tilde\\Sigma-\\frac {\\kappa}3)\\\\\n\t\t\t\\le{}&\\frac43({-}\\frac12(\\Sigma_++1)-\\tilde\\Sigma+\\sqrt{3\\tilde\\Sigma})N_+^2 -\\frac32(2-\\gamma)(\\Sigma_++2\\tilde\\Sigma)\\Omega+f_1\\sqrt{\\tilde A},\n\t\\end{align}\n\tfor a function~$f_1$ which is\n\tbounded due to the compactness on the state space, say\n\t\\begin{equation}\n\t\\absval{f_1}\\le M\n\t\\end{equation}\n\tfor some constant~$M>0$.\n\tThe term containing~$\\Omega$ is non-positive. For the first bracket, one easily sees that $\\Sigma_+\\in[1\/2,1]$ and $\\tilde\\Sigma\\in[0,3\/4]$ imply\n\t\\begin{align}\n\t\t-\\frac12(\\Sigma_++1){}&\\in\\left[{-}1,{-}\\frac34\\right],\\\\\n\t\t-\\tilde\\Sigma+\\sqrt{3\\tilde\\Sigma}{}&\\in\\left[0,\\frac34\\right],\n\t\\end{align}\n\thence the term containing~$N_+^2$ is non-positive as well. In total, this implies that\n\t\\begin{equation}\n\t(\\Sigma_++\\tilde\\Sigma)(\\tau_1)'\\le M\\sqrt{\\tilde A}.\n\t\\end{equation}\n\n\tWe now integrate inequality~\\eqref{eqn_evolutionsigmaplusplustildesigma} from~$\\tau<\\tau_0$ to~$\\tau_0$.\n\tUsing the estimate on~$\\tilde A$ yields\n\t\\begin{align}\n\t\t(\\Sigma_++\\tilde\\Sigma)(\\tau)\\ge{}&(\\Sigma_++\\tilde\\Sigma)(\\tau_0)-\\int_{\\tau}^{\\tau_0}M\\sqrt{\\tilde A(s)}ds\\\\\n\t\t\t\\ge{}&(\\Sigma_++\\tilde\\Sigma)(\\tau_0)-\\int_{\\tau}^{\\tau_0}M\\sqrt\\delta e^{(s-\\tau_0)\/2}ds\\\\\n\t\t\t\\ge{}&(\\Sigma_++\\tilde\\Sigma)(\\tau_0)-2M\\sqrt\\delta,\n\t\\end{align}\n\tand setting $\\delta\\coloneqq\\varepsilon_1^2(2M)^{{-}2}$ concludes the proof.\n\\end{proof}\n\\begin{proof}[Proof of Prop.~\\ref{prop_rightoftaub2source}]\n\tSet~$\\tilde U$ to be the neighborhood of the limiting point $(\\bar\\operatorname s,1-\\bar\\operatorname s^2,0,0,0)$ that satisfies\n\t\\begin{equation}\n\t\t\\absval{\\Sigma_+-\\bar\\operatorname s}<\\varepsilon.\n\t\\end{equation}\n\tOne now constructs an even smaller neighborhood~$U$ whose closure is contained in~$\\tilde U$ and such that all orbits starting in this smaller neighborhood are contained in~$\\tilde U$.\n\n\tThe construction proceeds as follows: Note that due to the restriction on the state space,\n\tequation~\\eqref{eqn_constraintgeneraltwo}, one finds that\n\t\\begin{equation}\n\t\t0\\le\\Sigma_+^2+\\tilde\\Sigma\\le1.\n\t\\end{equation}\n\tThat is, the orbit projected to the $(\\Sigma_+,\\tilde\\Sigma)$-plane lies below the Kasner parabola, which is the graph of a function with slope~$-2\\Sigma_+$. For~$\\Sigma_+$ in the interval $(1\/2,1)$, this slope is strictly less than~$-1$ and decaying.\n\tOne can therefore choose a constant $d<1-\\bar\\operatorname s^2$ such that the straight line with slope~${-}1$ through $(\\bar\\operatorname s,d)$\n\t\\begin{equation}\n\t\\label{eqn_proofrightoftaub2straightline}\n\t\t\\tilde\\Sigma-d={-}(\\Sigma_+-\\bar\\operatorname s)\n\t\\end{equation}\n\tintersects the Kasner parabola at some $\\bar\\operatorname s<\\Sigma_+<\\bar\\operatorname s+\\varepsilon$. Let $0<\\varepsilon_1<\\min((1-\\bar\\operatorname s^2-d)\/2,\\varepsilon)$, choose $\\delta=\\delta(\\varepsilon_1)$ as in Lemma~\\ref{lemm_rightoftaub2deltanplusbothsigns} and let~$U$ be the set defined by\n\t\\begin{equation}\n\t\t\\absval{\\Sigma_+-\\bar\\operatorname s}<\\varepsilon_1,\\qquad\n\t\t\\Sigma_++\\tilde\\Sigma>\\bar\\operatorname s+d+\\varepsilon_1,\\qquad\n\t\t\\tilde A<\\delta.\n\t\\end{equation}\n\n\tBy Lemma~\\ref{lemm_rightoftaub2deltanplusbothsigns}, any orbit which is contained in~$U$ at time~$\\tau_0$ satisfies\n\t\\begin{align}\n\t\t\\Sigma_+(\\tau)\\ge{}&\\Sigma_+(\\tau_0)>\\bar\\operatorname s-\\varepsilon_1,\\\\\n\t\t\\tilde\\Sigma(\\tau)\\ge{}&{-}\\Sigma_+(\\tau)+(\\Sigma_++\\tilde\\Sigma)(\\tau_0)-\\varepsilon_1>{-}\\Sigma_+(\\tau)+\\bar\\operatorname s+d,\n\t\\end{align}\n\tfor all~$\\tau\\le\\tau_0$.\n\tFrom the first inequality we conclude that~$\\Sigma_+(\\tau)>\\bar\\operatorname s-\\varepsilon$ at all times~$\\tau\\le\\tau_0$.\n\tThe second inequality implies that the graph of the solution lies above the straight line from equation~\\eqref{eqn_proofrightoftaub2straightline}. Both inequalities are visualised in Figure~\\ref{figure_rightoftaubtwo}. Because the graph also has to lie below the Kasner parabola, but we have chosen the constant~$d$ such that the straight line and the Kasner parabola intersect at some $\\bar\\operatorname s<\\Sigma_+<\\bar\\operatorname s+\\varepsilon$, this implies that\n\t\\begin{equation}\n\t\t\\Sigma_+(\\tau)<\\bar\\operatorname s+\\varepsilon\n\t\\end{equation}\n\tfor all times~$\\tau\\le\\tau_0$ and every $\\alpha$-limit point. This concludes the proof.\n\\end{proof}\n\n\\section{Asymptotics towards the plane wave equilibrium solutions}\n\n\\label{section_asymptoticsplanewave}\n\nIn this section, we use the theory of dynamical systems to determine the qualitative behaviour of solutions converging to the plane wave equilibrium points~$\\planewave {\\kappa}$ as $\\tau\\rightarrow{-}\\infty$.\nThe statements are not qualitatively new, as certain parts of~$\\planewave {\\kappa}$ have already been identified as ``saddles'' or ``sinks'' in~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}. Here, we state and prove more detailed properties of solutions converging to~$\\planewave {\\kappa}$. The approach we use here is similar to the one in Section~\\ref{section_leftoftaubtwo}, where we discussed the behaviour of solutions converging to a Kasner point situated to the left of the point Taub~2.\n\\begin{prop}\n\\label{prop_centreunstablelocallyplanewave}\n\tConsider the evolution equation s~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_definitiontilden}\n\tin the extended state space, i.\\,e.\\  without assuming the constraint equation s~\\eqref{eqn_constraintgeneralone}--\\eqref{eqn_constraintgeneraltwo}.\n\tLet~$K_1$, $K_2$ be compact subsets of the arc $\\planewave {\\kappa}\\cap\\{{-}1<\\Sigma_+<{-}(3\\gamma-2)\/4\\}$ and the arc $\\planewave {\\kappa}\\cap\\{{-}(3\\gamma-2)\/4<\\Sigma_+<0\\}$, if these are non-empty. Let~$K_3$ and~$K_4$ denote the points on~$\\planewave {\\kappa}$ with $\\Sigma_+={-}(3\\gamma-2)\/4$ and $\\Sigma_+=0$, resp.\\  Then there are neighborhoods~$U_i$ of~$K_i$, $i=1,\\ldots,4$, and $C^1$~submani\\-fold s~$M_{\\planewave {\\kappa},\\operatorname{left}}$, $M_{\\planewave {\\kappa},\\operatorname{right}}$, $M_{\\planewave {\\kappa},\\operatorname{special}}$, $M_{\\planewave {\\kappa},0}$ of~$\\maths{R}^5$ in~$U_1$, $U_2$, $U_3$, $U_4$ resp.\\  with the following properties:\n\t\\begin{itemize}\n\t\t\\item $M_{\\planewave {\\kappa},\\operatorname{left}}$ contains the set~$K_1$ and in these points is tangent to those eigenvectors of the linearised evolution equations in the extended five-dimensional space, given in Appendix~\\ref{subsect_appendixlinearisedevolutionplanewave}, which correspond to eigenvalues with non-negative real parts.\n\t\t\\item $M_{\\planewave {\\kappa},\\operatorname{right}}$ contains the set~$K_2$ and in these points\n\t\tis tangent to those eigenvectors which correspond to eigenvalues with non-negative real parts.\n\t\t\\item $M_{\\planewave {\\kappa},\\operatorname{special}}$ contains the point~$K_3$ and in this point\n\t\tis tangent to those eigenvectors\n\t\twhich correspond to eigenvalues with non-negative real parts.\n\t\t\\item $M_{\\planewave {\\kappa},0}$ contains the point~$K_4$ and in this point\n\t\tis tangent to those eigenvectors\n\t\twhich correspond to eigenvalues with non-negative real parts.\n\t\\end{itemize}\n\tFurther:\n\t\\begin{itemize}\n\t\t\\item Points in~$U_1$ are either contained in~$M_{\\planewave {\\kappa},\\operatorname{left}}$ or\n\t\ttheir evolution under equation s~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_definitiontilden} leaves~$U_1$ as $\\tau\\rightarrow{{-}\\infty}$, and similarly for the remaining~$U_i$.\n\t\\end{itemize}\n\tThe dimensions of the submani\\-fold s are\n\t\\begin{align}\n\t\t\\dim M_{\\planewave {\\kappa},\\operatorname{left}} ={}&2,\\\\\n\t\t\\dim M_{\\planewave {\\kappa},\\operatorname{right}} ={}&1,\\\\\n\t\t\\dim M_{\\planewave {\\kappa},\\operatorname{special}} ={}&2,\\\\\n\t\t\\dim M_{\\planewave {\\kappa},0}\n\t\t\t={}& \\left\\{\\begin{array}{ll} 1 & \\text{if}\\quad3\\gamma-2>0 ,\\\\\n\t\t\t2 & \\text{if}\\quad3\\gamma-2\\le0 \\end{array}\\right..\n\t\\end{align}\n\\end{prop}\n\\begin{rema}\n\tWe see in the proof that we could change the regularity of the submani\\-fold s to~$C^r$, for some~$r<\\infty$.\n\tFurther, the eigenvalues on~$\\planewave k$ which are not real have real part~${-}2(1+\\Sigma_+)$. The only possibility where this is non-negative for a point in one of the compact sets is if~$\\gamma=2$ and~$\\Sigma_+={-}1$. This implies that the solution converges to the point Taub~1 and hence is constant, see Prop.~\\ref{prop_tauboneothermatter}. For the current statement, this situation is not of interest, and excluding the case~$K_4=\\planewave k\\cap\\{\\Sigma_+={-}1\\}$, the submani\\-fold s are tangent to those eigenvectors which correspond to non-negative eigenvalues.\n\\end{rema}\nFor the proof, we make use of the concepts and notation introduced in Appendix~\\ref{section_appendixdynamsystheo}, see also the explanation in the beginning of Section~\\ref{section_leftoftaubtwo}. The set of points which remain in~$U_i$ under the evolution in negative time direction is the maximal negatively invariant set~$A^-(U_i)$, and we prove the proposition using properties of the centre-unstable mani\\-fold~${\\mathcal C}^u$.\n\\begin{proof}\n\tThe arc $\\planewave {\\kappa}\\cap\\{{-}1<\\Sigma_+<{-}(3\\gamma-2)\/4\\}$ is a mani\\-fold\\ in~$\\maths{R}^5$ consisting of equilibrium points of the evolution equation, with three eigenvalues of the linearised evolution equations lying in the left half-plane, one vanishing, and one lying in the right half-plane, see Def.~\\ref{defi_planewaveexpansionnorm} and the adjacent text.\n\n\tAccording to Thm~\\ref{theo_centremftheory}, there is a centre-unstable mani\\-fold~${\\mathcal C}^u$ near~$K_1$ and a neighborhood~$U_1$ of~$K_1$ such that the maximal negatively invariant set~$A^-(U_1)$ is contained in~${\\mathcal C}^u$.\n\tWithout loss of generality, we can restrict the manifold~${\\mathcal C}^u$ to the open set~$U_1$.\n\n\tThe mani\\-fold~${\\mathcal C}^u$ is by definition tangential to $E_m{}^c\\oplus E_m{}^u$, which are the subspaces of the tangent spaces at points on~$K_1$ associated with eigenvalues on the imaginary axis and in the right half-plane.\n\tThese are the eigenvalues~$0$ and ${-}4\\Sigma_+-(3\\gamma-2)$ whose eigenvectors span a two-dimensional subspace, which implies that~${\\mathcal C}^u$ is a submani\\-fold\\ of dimension~two.\n\tThe evolution equations~\\eqref{eqns_evolutionbianchib} are polynomial and consequently~$C^\\infty$, and the plane wave equilibrium points form a smooth curve. We can therefore apply Thm~\\ref{theo_centremftheory} with some finite~$r$, for example~$r=1$.\n\n\tThe proof for the arc $\\planewave {\\kappa}\\cap\\{{-}(3\\gamma-2)\/4<\\Sigma_+<0\\}$ proceeds in the same way, with~$0$ being the only eigenvalue with non-negative real part. The two individual points on the arc~$\\planewave {\\kappa}$ constitute (zero-dimensional) mani\\-fold s of equilibrium points on their own, to which we can also apply Thm~\\ref{theo_centremftheory}. In case $\\Sigma_+=0$, the value of~$\\gamma$ determines where the eigenvalue ${-}4\\Sigma_+-(3\\gamma-2)$ is situated in the complex plane.\n\\end{proof}\nAs in~Section~\\ref{section_leftoftaubtwo}, we now have to restrict the submani\\-fold s to the constraint surface. There are two main differences between the situation in that section and the present one: The dimension of all centre-unstable mani\\-fold s is now at most~two, and the constraint surface is singular only in the point $\\Sigma_+=0$, see Remark~\\ref{rema_singularconstraintequ}.\n\nOn the other hand,\nintersecting the centre-unstable~mani\\-fold s for plane wave equilibrium points~$\\planewave {\\kappa}$ with the constraint surface does not necessarily result in a lower dimension:\nWe know from the proof of the previous proposition that the centre-unstable mani\\-fold s are tangent to the eigenvectors to eigenvalues with positive or zero real parts, which are~$0$ and possibly ${-}4\\Sigma_+-(3\\gamma-2)$, depending on the relation between~$\\Sigma_+$ and~$\\gamma$, see Appendix~\\ref{subsect_appendixlinearisedevolutionplanewave}. It follows by direct computation that these eigenvectors are orthogonal to the gradient of equation~\\eqref{eqn_constraintgeneralone}.\nConsequently, the centre-unstable mani\\-fold s and the constraint surface do not intersect transversally.\nWe can only conclude that restricting the submani\\-fold s to the constraint surface yields submani\\-fold s of at most the same dimension, that is dimension at most~one or~two.\n\n\\begin{theo}\n\\label{theo_centreunstablegloballyplanewave}\n\tFor a given~${\\kappa}\\in\\maths{R}$ and~$\\gamma\\in[0,2)$, consider the set of solutions to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} converging to a point on~$\\planewave {\\kappa}$ as $\\tau\\rightarrow{-}\\infty$. Assume that the limit point satisfies~$\\Sigma_+=\\operatorname s$. Then\n\t\\begin{equation}\n\t\\label{eqn_inequalityplanewave}\n\t\t{\\kappa}(1+\\operatorname s)\\ge3\\operatorname s\n\t\\end{equation}\n\thas to hold. Furthermore:\n\t\\begin{itemize}\n\t\t\\item If~${-}(3\\gamma-2)\/4<\\operatorname s<0$, then the solution is the constant solution. In particular, the solution is a vacuum solution.\n\t\t\\item There is a countable family of $C^1$ submani\\-fold s $\\{L'_m\\}_{m\\in\\maths{N}}$ of dimension at most~two\n\t\t\t\tsuch that if $\\operatorname s\\le{-}(3\\gamma-2)\/4$ and~$\\operatorname s<0$,\n\t\t\t\tthen the solution is contained in $\\bigcup_{m\\in\\maths{N}}L'_m$.\n\t\t\\item There is a countable family of $C^1$ submani\\-fold s $\\{\\submfsplanewave'_m\\}_{m\\in\\maths{N}}$ of dimension at most~two\n\t\t\t\tsuch that if $\\operatorname s=0$,\n\t\t\t\tthen the solution is contained in $\\bigcup_{m\\in\\maths{N}}\\submfsplanewave'_m$.\n\t\\end{itemize}\n\\end{theo}\n\\begin{proof}\n\tThe relation between the parameter~${\\kappa}$ and the limiting value~$\\operatorname s$ follows from the definition of the plane wave equilibrium points, Def.~\\ref{defi_planewaveexpansionnorm}, as~$N_+^2\\ge0$.\n\n\tWe start with the case ${-}(3\\gamma-2)\/4<\\operatorname s<0$.\n\tFix a compact subarc~$K_2$ of the arc $\\planewave {\\kappa}\\cap\\{{-}(3\\gamma-2)\/4<\\Sigma_+<0\\}$ containing the limiting point of the solution. As the solution converges to a point in~$K_2$, it is contained in the neighborhood~$U_2$ from the previous statement for sufficiently negative times, and therefore the solution has to lie in the one-dimensional submani\\-fold ~$M_{\\planewave {\\kappa},\\operatorname{right}}$ for sufficiently negative times. The set of plane wave equilibrium points~$\\planewave {\\kappa}$ itself forms a one-dimensional submani\\-fold\\ as well. It consists of constant solutions, and~$K_2$ consequently must be contained in the centre-unstable mani\\-fold ~$M_{\\planewave {\\kappa},\\operatorname{right}}$.\n\tDue to the dimension, both submani\\-fold s coincide in a sufficiently small neighborhood of the limiting point, and the first part of the statement follows. The solution satisfies~$\\Omega=0$ due to the definition of~$\\planewave {\\kappa}$, see Def.~\\ref{defi_planewaveexpansionnorm} and below.\n\n\tFor the second case,\n\tchoose a countable family of compact subarcs~$K_m$, $m\\in\\maths{N}$, which exhaust the arc $\\planewave {\\kappa}\\cap\\{{-}1<\\Sigma_+<{-}(3\\gamma-2)\/4\\}$. For every~$K_m$ as well as for the\n\tpoint on~$\\planewave {\\kappa}$ with $\\Sigma_+={-}(3\\gamma-2)\/4$,\n\tconsider the submani\\-fold s found in Prop.~\\ref{prop_centreunstablelocallyplanewave}. Convergence implies that for sufficiently negative times the solution cannot escape the corresponding open neighborhoods, and it consequently lies in one of these submani\\-fold s for sufficiently negative times.\n\n\tStill without restricting to the constraint equations\\, we apply the flow corresponding to the evolution equation s and integer times to these submani\\-fold s. The flow resulting from a polynomial evolution equation implies that the regularity of the submani\\-fold s is preserved. As the graph of any solution is invariant under this flow, solutions to the evolution equation s in the extended state space are fully contained in the resulting family of submani\\-fold s~$L'_m$. By construction, the family is countable. In the extended state space, the statement about the dimension follows from Prop.~\\ref{prop_centreunstablelocallyplanewave} and the fact that the dimension of a submani\\-fold\\ is invariant under diffeomorphism s, consequently invariant under the flow. Restricting to the constraint equation s cannot increase the dimension, which concludes the proof in this case. The remaining case where~$\\operatorname s=0$ is treated in the same way.\n\\end{proof}\n\\begin{rema}\n\\label{rema_planewaveconvergencebianchiseparated}\n\tConsider the element of the plane wave equilibrium points~$\\planewave {\\kappa}$ which satisfies~$\\Sigma_+=\\operatorname s\\in({-}1,0]$.\n\tWe have found in the previous theorem that the question whether there is a solution to equations~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} converging to this point, and whether it is constant or not, depends on the relation between~$\\operatorname s$ and the group parameter~${\\kappa}$, as well as the relation between~$\\operatorname s$ and the matter parameter~$\\gamma$.\n\n\tIn case of vacuum~$\\Omega=0$, the only solutions converging to~$\\planewave {\\kappa}$ are the constant solutions, see Prop.~\\ref{prop_alphalimitsets_vacuum_inflat}.\n\tAssume that~$\\Omega>0$. All solutions have to be non-constant and in particular cannot converge to limit points with~${-}(3\\gamma-2)\/4<\\operatorname s<0$. We analyse the possible combinations of~$\\operatorname s$,~${\\kappa}$ and~$\\gamma$, depending on the Bianchi type.\n\t\\begin{itemize}\n\t\t\\item Bianchi class~A solutions cannot converge to the plane wave equilibrium points~$\\planewave {\\kappa}$, due to~$\\tilde A=0$ for class~A but non-vanishing on~$\\planewave {\\kappa}$.\n\t\t\\item Bianchi type VI$_{\\eta}$: As~${\\kappa}<0$ in this Bianchi type, solutions can only converge to limit points satisfying\n\t\t\\begin{equation}\n\t\t\t\\operatorname s\\le\\frac{{\\kappa}}{3-{\\kappa}}<0, \\qquad \\operatorname s\\le{-}(3\\gamma-2)\/4,\n\t\t\\end{equation}\n\t\tdue to inequality~\\eqref{eqn_inequalityplanewave} and~$\\Omega>0$. The solutions are then contained in the union of submanifolds~$L'_m$.\n\t\t\\item Bianchi type VII$_{\\eta}$: In this Bianchi type, the parameter~$\\kappa$ is positive.\n\t\tInequality~\\eqref{eqn_inequalityplanewave} is satisfied for all possible values of~$\\operatorname s$.\n\t\tThen either~$\\operatorname s\\le{-}(3\\gamma-2)\/4$ and~$\\operatorname s<0$, or~$\\operatorname s=0$ has to hold, and the solutions are contained in the union of submanifolds~$L'_m$ or~$\\submfsplanewave'_m$.\n\t\t\\item Bianchi type IV: As~${\\kappa}=0$, inequality~\\eqref{eqn_inequalityplanewave} is satisfied for all possible values of~$\\operatorname s$.\n\t\tThen either~$\\operatorname s\\le{-}(3\\gamma-2)\/4$ and~$\\operatorname s<0$, or~$\\operatorname s=0$ has to hold, and the solutions are contained in the union of submanifolds~$L'_m$ or~$\\submfsplanewave'_m$.\n\t\t\\item Bianchi type V: These models are defined by~${\\kappa}=0$, $\\tilde A>0$, $\\Sigma_+=\\Delta=N_+=0$, which implies that solutions can only converge to the limit point with~$\\operatorname s=0$.\n\t\tThe evolution equation~\\eqref{eqns_evolutionbianchib} for~$\\tilde\\Sigma$ reads\n\t\t\\begin{equation}\n\t\t\t\\tilde\\Sigma'=2(q-2)\\tilde\\Sigma\n\t\t\\end{equation}\n\t\ton Bianchi type~V solutions. As~$q\\le2$ and~$\\tilde\\Sigma\\ge0$, due to~\\eqref{eqn_constraintgeneralthree}, this implies that~$\\tilde\\Sigma$ is monotone decreasing or constant. As the only element of the plane wave equilibrium points~$\\planewave {\\kappa}$ with~$\\operatorname s=0$ satisfies~$\\tilde\\Sigma=0$,\n\t\tthis shows that~$\\tilde\\Sigma$ vanishes at all times.\n\t\tFurther, the element in the plane wave equilibrium point with~$\\operatorname s=0$ satisfies~$\\tilde A=1$. With the information on the other variables, the evolution equation of~$\\tilde A$ reads\n\t\t\\begin{equation}\n\t\t\t\\tilde A'=(3\\gamma-2)(1-\\tilde A)\\tilde A.\n\t\t\\end{equation}\n\t\tIts range, see~\\eqref{eqn_constraintgeneralthree} then implies that for~$\\gamma\\ge2\/3$, only the constant orbit is possible.\n\t\tIn case~$0\\le\\gamma<2\/3$, the solution is contained in~$\\Sigma_+=\\tilde\\Sigma=\\Delta=N_+=0$ and~$\\tilde A$ decreases mononotically from~$1$ to~$0$.\n\t\\end{itemize}\n\\end{rema}\n\n\n\n\\section{Equivalence of the initial data perspective and the expansion-normalised variables}\n\n\\label{section_equivalenceinitialdataexpansionnormalised}\n\nThe goal of this section is to justify the use of expansion-normalised variables, and show that under the correct transformation the evolution of these variables is equivalent to solving Einstein's equation for or\\-tho\\-go\\-nal\\ Bianchi~B perfect fluid initial data.\nAt the same time, we show how to construct, for given initial data $(G,{h},{k},\\mu_0)$ as in Def.~\\ref{defi_initialdatabianchib}, the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ and prove properties of this spacetime. This is done via the expansion-normalised variables $(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)$ for Bianchi class~B models.\n\nThe expansion-normalised variables in Bianchi class~B models were developed in~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}, motivated by a similar set of coordinates for Bianchi~A models introduced in~\\cite{wainwrighthsu_dynamicalsystemsapproachbianchiorthogonalA}.\nTheir deduction starts out with given structure constants~$\\gamma_{\\alpha\\beta}^\\delta$ of a suitably chosen four-dimensional or\\-tho\\-nor\\-mal\\ frame of the spacetime in question. To connect this to the initial data perspective, we have to understand how such a frame can be constructed from the knowledge of the metric~$h$ and the two-tensor~${k}$ on the Lie group ~$G$. In particular, this means choosing a suitable three-dimensional basis of the Lie al\\-ge\\-bra\\ associated with the Lie group.\nFor this reason, we collect the necessary background on three-dimensional Lie group s in Subsection~\\ref{subsect_appendixbianchiclassification}.\n\n\\smallskip\n\nWe recall the deduction of the expansion-normalised variables from given structure constants~$\\gamma_{\\alpha\\beta}^\\delta$ in Subsection~\\ref{subsect_appendixdeducenormalisedvariablesbianchib}.\nNote that this deduction already starts with the full spacetime, in particular with structure constants $\\gamma_{\\alpha\\beta}^\\delta$ in four dimensions. For initial data, this information is not available, only three-dimensional spacelike structure constants~$\\gamma_{ij}^k$ make sense. The structure constants~$\\gamma_{0i}^j$ require the existence of a timelike vector field~$e_0$.\nHowever, to begin with there is no such vector field.\nNonetheless, we can define objects~$\\tilde\\gamma_{0i}^j$ using the metric~$h$ and the two-tensor~${k}$ and need to make sure that they have the form required for the construction by~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}, which means that\nthe only non-vanishing ones are\n\\begin{equation}\n\t\\tilde\\gamma_{01}^1,\\quad \\tilde\\gamma_{0A}^B,\n\\end{equation}\n$A,B\\in\\{2,3\\}$.\nIn order to see that in our setting we can indeed choose a suitable basis such that the commutators have the required form, we discuss in more detail initial data sets where the three-dimensional Rie\\-mann\\-ian\\ mani\\-fold\\ is a Lie group\\ with left-invariant metric. This is done in Subsection~\\ref{subsect_liegrinitialdata}, and it is here that we explain the terms 'exceptional' and 'or\\-tho\\-go\\-nal ' as well as the reason for excluding Lie group s of type~VI$_{{-}1\/9}$.\n\nThe objects~$\\tilde\\gamma_{01}^1,\\tilde\\gamma_{0A}^B$ have to be understood as merely numbers, devoid of any geometric meaning. It is only a posteriori that we can interpret these numbers as structure constants of a suitable four-dimensional frame.\nWith these objects at our disposal, we can follow through with the transformation explained in Subsection~\\ref{subsect_appendixdeducenormalisedvariablesbianchib}. We wish to point out however that we make use only of the algebraic relations, not their geometric interpretation.\nThis yields initial data $(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)(0)$ for the evolution equations\\ in expansion-normalised variables, equations~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega}.\n\nIn order to avoid confusion and shorten notation, we are going to denote initial data to Einstein's field equation, i.\\,e.\\  initial data as in Def.~\\ref{defi_initialdatabianchib}, by \\textit{geometric initial data}, and initial data to the evolution equations in expansion-normalised variables, equations~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega}, by \\textit{dynamical initial data}.\nThe details of how to translate geometric initial data into dynamical initial data are given in the first part of the construction of the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment, Subsection~\\ref{constr_initialdatatoexpnorm}.\n\n\\smallskip\n\nOnce geometric initial data is translated into the expansion-normalised variables setting, i.\\,e.\\  into dynamical initial data, and we have obtained a solution in these variables, the main work lies in the construction of a global four-dimensional frame with structure constants behaving correctly over time, and such that our initially defined objects~$\\tilde\\gamma_{0i}^j$ are consistent with the geometric objects~$\\gamma_{0i}^j$ at the starting time~$t=0$, i.\\,e.\\  on the initial Cauchy hypersurface.\nFinally, we use this four-dimensional frame to construct a spacetime metric. This is done in the second and third part of the construction, Subsections~\\ref{constr_expnormtosolutionbasic} and~\\ref{constr_solutionbasictomghd}.\nHaving obtained through this construction a spacetime into which the initial data is embedded in the correct way, we investigate its properties in Subsection~\\ref{constr_propertiesdevelopmentinitialdata}.\nWe can show that this spacetime is in fact the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ of the geometric initial data.\n\n\n\\smallskip\n\nIn Subsection~\\ref{constr_highersymmetrysolutions}, we then consider Bianchi spacetimes with additional symmetries, namely local rotational symmetry and plane wave equilibrium solutions. We compare their definitions from the point of view of geometric initial data with their definitions in expansion-normalised variables and show that these definitions coincide, respectively.\n\n\\smallskip\n\nThe construction of expansion-normalised variables by~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB} which we explain in Subsection~\\ref{subsect_appendixdeducenormalisedvariablesbianchib} is related to the study of or\\-tho\\-go\\-nal ly transitively~$G_2$ cosmologies. Even though~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB} appeal to results having been obtained in this context, our construction does not make use of these additional statements but is self-contained. We do nonetheless give several remarks explaining the relations, but these are logically independent from\nthe construction presented here.\n\n\\smallskip\n\nIn the Bianchi class~A setting, results equivalent to what we achieve in this section have been obtained by~\\cite{ringstrom_bianchiixattr}, however with a somewhat different approach.\n\n\n\n\\subsection{Bianchi classification of three-dimensional Lie group s}\n\\label{subsect_appendixbianchiclassification}\n\n\nWe give a brief introduction to the classification of three-dimensional Lie group s proposed by Bianchi in 1903. The article~\\cite{krasinskibehrschueckingestabrookwahlquistellisjantzenkundt_bianchiclass} gives a historical overview and provides insights into how our modern understanding of this classification came to be.\nIn~\\cite{ellismaccallum_classofhomogcosmmodels}, the details are laid out, and for the first part of this subsection we refer to~\\cite[App.~E]{ringstrom_topologyfuturestabilityuniverse} for the details.\n\nFor a three-dimen\\-sional Lie group~$G$, let $\\{e_i\\}$, $i=1,2,3$, be a basis of the associated Lie algebra~$\\ggg$, and let~$\\gamma_{ij}^k$ denote the structure constants, i.\\,e.\\ \n\\begin{equation}\n\t\\liebr{e_i}{e_j}=\\gamma_{ij}^ke_k.\n\\end{equation}\nThe equivalent information is encoded in the symmetric matrix~$n$ and the vector~$a$ given by\n\\begin{equation}\n\\label{eqn_data3dimliegr}\n\tn^{ij}=\\frac12\\gamma_{kl}^{\\(i}\\epsilon^{j\\)kl},\\quad a_k=\\frac12\\gamma_{ki}^i,\n\\end{equation}\nwith $\\epsilon_{ijk}=\\epsilon^{ijk}$ the permutation symbol which satisfies $\\epsilon_{123}=1$ and is antisymmetric in all indices. The brackets~$^{()}$ denote symmetrisation, in fact $\\gamma_{kl}^{(i}\\epsilon^{j)kl}=(\\gamma_{kl}^{i}\\epsilon^{jkl}+\\gamma_{kl}^{j}\\epsilon^{ikl})\/2$.\nEquivalence follows from\n\\begin{equation}\n\\label{eqn_data3dimliegrback}\n\t\\gamma_{jk}^i=\\epsilon_{jkl}n^{li}+a_j\\delta^i_k-a_k\\delta^i_j.\n\\end{equation}\nThe structure constants have to satisfy the Jacobi identity, which is equivalent to the condition\n\\begin{equation}\n\\label{eqn_jacobiidentequiv}\n\tn a=0.\n\\end{equation}\nApplying a suitable change of basis yields~$n$ and~$a$ in a specific simplified form and gives the following classification of three-dimensional Lie group s, of which we sketch a proof further down in Lemma~\\ref{lemm_classthreedimliealg}. As there is a one-to-one correspondence between simply connected Lie group s and their Lie al\\-ge\\-bra s, see~\\cite[Thm~3.28]{warner_founddiffmfsliegrs}, we formulate this classification in terms of (simply connected) Lie group s.\n\\begin{enumerate}\n\t\\item In the case of simply connected unimodular Lie groups (Bianchi class~A) which are defined by $a=0$, one can choose~$n$ diagonal, i.\\,e.\\  $n=\\operatorname{diag}(\\nu_1,\\nu_2,\\nu_3)$, such that it falls in exactly one of the categories given in Table~\\ref{table_bianchiaclassification}.\n\n\tGiven a left-invariant metric in~$G$, the basis~$e_1,e_2,e_3$ producing~$a$ and~$n$ of this form can be chosen or\\-tho\\-nor\\-mal.\n\t\\begin{table}[htp]\n\t\t\\centering\n\t\t\\caption{Bianchi class~A}\n\t\\label{table_bianchiaclassification}\n\t\t\\begin{tabular}{l|c|c|c}\n\t\t\tType & $\\nu_1$ & $\\nu_2$ & $\\nu_3$ \\\\\n\t\t\t\\hline\n\t\t\tI & 0 & 0 & 0 \\\\\n\t\t\tII & + & 0 & 0 \\\\\n\t\t\tVI$_0$ & 0 & + & -- \\\\\n\t\t\tVII$_0$ & 0 & + & + \\\\\n\t\t\tVIII & -- & + & + \\\\\n\t\t\tIX & + & + & +\n\t\t\\end{tabular}\n\t\\end{table}\n\t\\item In the case of simply connected non-unimodular Lie groups (Bianchi class~B) which are defined by $a\\not=0$, one can choose $a_1\\not=0$, $a_2=a_3=0$, and~$n$ diagonal, i.\\,e.\\  $n=\\operatorname{diag}(\\nu_1,\\nu_2,\\nu_3)$. The Jacobi identity then implies $\\nu_1=0$. All possible types and their resp.\\  names are listed in Table~\\ref{table_bianchibclassification}.\n\n\tAgain, given a left-invariant metric in~$G$, the basis~$e_1,e_2,e_3$ can be chosen or\\-tho\\-nor\\-mal.\n\n\tLie groups of Bianchi type~VI and~VII have an additional degree of freedom which is captured in the quantity~$\\eta$. In the chosen basis this parameter satisfies\n\t\\begin{equation}\n\t\\label{eqn_binvparamdefi}\n\t\t\\eta \\nu_2\\nu_3=a_1^2\n\t\\end{equation}\n\tand is invariant under scaling.\n\tWe give a more geometric definition further down, see Lemma~\\ref{lemm_invardefibparaminv}.\n\t\\begin{table}[htp]\n\t\t\\centering\n\t\t\\caption{Bianchi class~B}\n\t\\label{table_bianchibclassification}\n\t\t\\begin{tabular}{l|c|c|c}\n\t\t\tType & $\\nu_1$ & $\\nu_2$ & $\\nu_3$ \\\\\n\t\t\t\\hline\n\t\t\tV & 0 & 0 & 0 \\\\\n\t\t\tIV & 0 & 0 & + \\\\\n\t\t\tVI$_{\\eta}$ & 0 & + & -- \\\\\n\t\t\tVII$_{\\eta}$ & 0 & + & +\n\t\t\\end{tabular}\n\t\\end{table}\n\\end{enumerate}\n\n\nIn order to understand the parameter~$\\eta$, we have to gather more detailed information on the structure of the Lie al\\-ge\\-bra s of the different Bianchi types.\n\\begin{lemm}\n\\label{lemm_abeliansubalgstructconstants}\n\tLet~$G$ a three-dimensional Lie group, and~$\\ggg$ the associated Lie al\\-ge\\-bra. Then~$\\ggg$ admits a two-dimensional Abelian subalgebra if and only if~$G$ is not of Bianchi type~VIII or~IX.\n\\end{lemm}\n\\begin{proof}\n\tChoose a basis $e_1,e_2,e_3$ of the Lie al\\-ge\\-bra\\ such that~$n$ and~$a$ are of one of the forms given in i) and ii). If the Lie group\\ is not of Bianchi type~II, VIII or~IX, then~$\\nu_1=0$. One computes that\n\t\\begin{equation}\n\t\t\\liebr{e_2}{e_3}=\\gamma_{23}^ie_i=(\\epsilon_{23l}n^{li}+a_2\\delta^i_3-a_3\\delta^i_2)e_i\n\t\t\t=\\epsilon_{231}n^{1i}e_i=\\nu_1e_1=0,\n\t\\end{equation}\n\twhich implies that $e_2,e_3$ span an Abelian subalgebra. For a Lie group\\ of Bianchi type~II one finds~$\\nu_2=0$, and~$e_1,e_3$ commute.\n\n\tAssume now that the Lie group\\ is of type~VIII or~IX and suppose that there are two linearly independent vectors~$b,c\\in\\ggg$ which commute, i.\\,e.\\ \n\t\\begin{equation}\n\t\tb=\\sum_{i=1}^3b_ie_i,\\qquad c=\\sum_{i=1}^3c_ie_i,\n\t\\end{equation}\n\tfor $b_i,c_i\\in\\maths{R}$, and\n\t\\begin{equation}\n\t\t\\liebr bc=0.\n\t\\end{equation}\n\tThis implies\n\t\\begin{equation}\n\t\t0=\\liebr bc=\\liebr{\\sum_{i=1}^3b_ie_i}{\\sum_{j=1}^3c_je_j}=\\sum_{i,j}b_ic_j\\liebr{e_i}{e_j}=\\sum_{i,j}b_ic_j\\gamma_{ij}^ke_k,\n\t\\end{equation}\n\thence for all $k=1,2,3$\n\t\\begin{equation}\n\t\\label{eqn_abeliansubgroupcondition}\n\t\t0=\\sum_{i,j}b_ic_j\\gamma_{ij}^k=\\sum_{i<j}(b_ic_j-b_jc_i)\\gamma_{ij}^k.\n\t\\end{equation}\n\tAs the two Bianchi types in question are of class~A, one finds from equation~\\eqref{eqn_data3dimliegrback} that\n\t\\begin{equation}\n\t\t\\gamma_{12}^k=\\delta_{3k}\\nu_3,\\qquad\n\t\t\\gamma_{13}^k=-\\delta_{2k}\\nu_2,\\qquad\n\t\t\\gamma_{23}^k=\\delta_{1k}\\nu_1.\n\t\\end{equation}\n\tAll~$\\nu_i$, $i=1,2,3$, are non-vanishing, and equation~\\eqref{eqn_abeliansubgroupcondition} is therefore equivalent to the system of equations\n\t\\begin{equation}\n\t\tb_2c_3-b_3c_2=0,\\qquad\n\t\tb_1c_3-b_3c_1=0,\\qquad\n\t\tb_1c_2-b_2c_1=0,\n\t\\end{equation}\n\tor in other words $b\\times c=0$, where~$\\times$ denotes the cross product. Hence,~$b$ and~$c$ are parallel, a contradiction.\n\\end{proof}\nIn case the Lie group\\ is of class~B, there is also a geometric way of defining the Abelian subalgebra. For this, we consider the adjoint~$\\operatorname{ad}:\\ggg\\rightarrow\\operatorname{End}(\\ggg)$, $\\operatorname{ad}_xy=\\liebr xy$, see~\\cite[Prop.~3.47]{warner_founddiffmfsliegrs}.\n\\begin{lemm}\n\\label{lemm_abeliansubalginvardefi}\n\tLet~$G$ a three-dimensional Lie group\\ of class~B and~$H$ the one-form \n\t\\begin{equation}\n\t\tH:\\ggg\\rightarrow\\maths{R}, \\quad x\\mapsto \\frac12\\operatorname{tr}(\\operatorname{ad}_x).\n\t\\end{equation}\n\tDenote by~$\\ggg_2$ the kernel of~$H$.\n\tThen~$\\ggg_2$ is an Abelian subalgebra and coincides with the subalgebra identified in Lemma~\\ref{lemm_abeliansubalgstructconstants}.\n\\end{lemm}\n\\begin{proof}\n\tWe choose a basis~$e_1,e_2,e_3$ of~$\\ggg$ and denote by~$e^k$, $k=1,2,3$, the dual basis. Setting~$a_k$, $k=1,2,3$, as in~\\eqref{eqn_data3dimliegr}, we find\n\t\\begin{equation}\n\t\tH = a_ke^k.\n\t\\end{equation}\n\tAs the Lie group\\ is of class~B, we can choose the basis~$e_1,e_2,e_3$ as above in ii), meaning that~$a_1\\not=0$, $a_2=a_3=0$, and $n$ is diagonal, i.\\,e.\\  $n=\\operatorname{diag}(0,\\nu_2,\\nu_3)$. With this, we find that the kernel~$\\ggg_2$ is spanned by the two basis elements~$e_2$ and~$e_3$.\n\tComparison with the proof of Lemma~\\ref{lemm_abeliansubalgstructconstants} shows that it is the same subalgebra we identified there.\n\\end{proof}\n\\begin{rema}\n\tGiven a class~B Lie group~$G$ with a left-invariant metric, we obtain the following useful interpretation of the kernel~$\\ggg_2$: it provides a splitting of the Lie al\\-ge\\-bra~$\\ggg_2$ into a two-dimensional Abelian subalgebra and the direction or\\-tho\\-go\\-nal\\ to it. For any or\\-tho\\-nor\\-mal\\ basis~$e_1,e_2,e_3$ of~$\\ggg$ such that~$e_2,e_3$ span~$\\ggg_2$, the basis element~$e_1$ is the one with the non-vanishing~$a_i$.\n\\end{rema}\n\n\\begin{rema}\n\\label{rema_frameforggg2rotationtodiagonalise}\n\tGiven a Lie group\\ of class~B with Lie al\\-ge\\-bra~$\\ggg$ and a left-invariant metric on~$G$, we have at our disposal the uniquely defined Abelian subalgebra~$\\ggg_2$. If instead the Lie group\\ is of class~A but not of type VIII or IX, we can fix an Abelian subalgebra, which exists due to Lemma~\\ref{lemm_abeliansubalgstructconstants}. In both cases, we can then introduce an or\\-tho\\-nor\\-mal\\ basis $e_1,e_2,e_3$ of~$\\ggg$ such that~$e_2,e_3$ span the Abelian subalgebra and~$e_1$ is or\\-tho\\-go\\-nal\\ to this span. Given~$\\ggg_2$, this basis is uniquely defined up to a rotation in the $e_2e_3$-plane and a choice of sign in~$e_1$.\n\n\tOne easily sees that it is equivalent to say that one uses the choice of or\\-tho\\-nor\\-mal\\ basis from~i) and~ii), and then allows for a rotation in the~$e_2e_3$-plane. This holds in all cases apart from the case of Bianchi type~II, where it is necessary to first switch the basis elements~$e_1$ and~$e_2$. In the later subsections, in particular for the deduction of the different variables in Subsection~\\ref{subsect_appendixdeducenormalisedvariablesbianchib}, this rotation in the~$e_2e_3$-plane is left as a gauge freedom.\n\\end{rema}\n\nWith this knowledge, we are now in a position to give an invariant definition of the parameter~$\\eta$ appearing in the Lie group s of type~VI$_\\eta$ and VII$_\\eta$, which are of class~B.\n\\begin{lemm}\n\\label{lemm_invardefibparaminv}\n\tLet~$G$ a three-dimensional Lie group\\ of type~VI$_\\eta$ or VII$_\\eta$, $\\eta\\not=0$ in either case, with associated Lie al\\-ge\\-bra~$\\ggg$, and~$\\ggg_2$ the uniquely defined Abelian subalgebra which is the kernel of the one-form~$H$.\n\tLet~$v_1\\in\\ggg\\setminus\\ggg_2$. Then\n\t\\begin{equation}\n\t\tA_2\\coloneqq \\operatorname{ad}_{v_1} |_{\\ggg_2}:\\ggg_2\\rightarrow\\ggg_2,\n\t\\end{equation}\n\tand a different choice~$\\hat v_1\\in\\ggg\\setminus\\ggg_2$ results into a map~$\\hat A_2$ differing from~$A_2$ by a constant non-zero multiple.\n\tFurther, setting\n\t\\begin{equation}\n\t\t\\eta\\coloneqq \\frac{(\\operatorname{tr} A_2)^2}{4\\det A_2-(\\operatorname{tr} A_2)^2}\n\t\\end{equation}\n\tis well-defined independently of the choice of~$v_1$ and consistent with equation~\\eqref{eqn_binvparamdefi}, i.\\,e.\\  under a choice of basis as in ii) one finds\n\t\\begin{equation}\n\t\t\\eta\\nu_2\\nu_3=a_1^2.\n\t\\end{equation}\n\\end{lemm}\n\\begin{proof}\n\tConsider a basis~$e_1,e_2,e_3$ as in~ii), i.\\,e.\\  satisfying~$a_1\\not=0$, $a_2=a_3=0$, and $n=\\operatorname{diag}(0,\\nu_2,\\nu_3)$. By Remark~\\ref{rema_frameforggg2rotationtodiagonalise}, the Abelian subalgebra~$\\ggg_2$ is spanned by~$e_2,e_3$. Consequently, we can express the given unit vector~$v_1$ as~$v_1=b_1e_1+\\sum_Ab_Ae_A$ and find\n\t\\begin{equation}\n\t\tA_2(e_B)=\\liebr{b_1e_1+\\sum_Ab_Ae_A}{e_B}=b_1\\liebr{e_1}{e_B}=b_1\\gamma_{1B}^Ce_C\\quad\\in\\ggg_2,\n\t\\end{equation}\n\tas~$\\liebr{e_2}{e_3}=0$ and~$\\gamma_{1A}^1=0$. This proves the first statement.\n\n\tFor the second statement, we realise that~$v_1,e_2,e_3$ also forms a basis of~$\\ggg$, hence~$\\hat v_1=c_1v_1+\\sum_Ac_Ae_A$ with~$c_1\\not=0$, and\n\t\\begin{equation}\n\t\t(A_2-\\hat A_2)(e_B)=\\liebr{e_1-c_1e_1}{e_B}=(1-c_1)A_2(e_B).\n\t\\end{equation}\n\n\tFrom the last computation, it even follows immediately that~$\\eta$ is independent of the choice of~$v_1$, and we can for all purposes assume that~$v_1=e_1$.\n\tDoing so, the linear mapping~$A_2$ in the chosen basis~$e_2,e_3$ is described by the matrix\n\t\\begin{equation}\n\t\t\\begin{pmatrix}\n\t\t\ta_1 & \\nu_3\\\\\n\t\t\t-\\nu_2 & a_1\n\t\t\\end{pmatrix},\n\t\\end{equation}\n\twhich implies\n\t\\begin{equation}\n\t\t\\operatorname{tr} A_2=2a_1,\\qquad \\det A_2=a_1^2+\\nu_2\\nu_3,\n\t\\end{equation}\n\tand yields\n\t\\begin{equation}\n\t\t\\eta=\\frac{a_1^2}{\\nu_2\\nu_3}.\n\t\\end{equation}\n\tThis is equivalent to the requested relation and in addition shows that~$\\eta$ is well-defined, as both~$\\nu_2$ and~$\\nu_3$ are non-vanishing for the Bianchi types in question.\n\\end{proof}\nWe conclude this subsection by proving that the types we listed in Tables~\\ref{table_bianchiaclassification} and~\\ref{table_bianchibclassification}, together with the parameter~$\\eta$, indeed provide a classification of all three-dimensional Lie al\\-ge\\-bra s.\n\\begin{lemm}\n\\label{lemm_classthreedimliealg}\n\tTwo three-dimensional Lie al\\-ge\\-bra s are isomorpic if and only if\\ they have the same Bianchi type, and in case of Bianchi type~VI or~VII additionally the same quantity~$\\eta$.\n\\end{lemm}\n\\begin{proof}\n\tGiven a Lie al\\-ge\\-bra\\ of either class, the matrix~$n$ and the vector~$a$ defined in equations~\\eqref{eqn_data3dimliegr} admit the form~$n=\\operatorname{diag}(\\nu_1,\\nu_2,\\nu_3)$ and~$a=(a_1,0,0)$ after applying a suitable change of basis.\n\tAs a consequence, the Lie al\\-ge\\-bra\\ falls in one of the types defined in Tables~\\ref{table_bianchiaclassification} and~\\ref{table_bianchibclassification}.\n\tFrom the transformation behaviour of~$n$ and~$a$ under a change of basis, see~\\cite[eq.~(E.4)]{ringstrom_topologyfuturestabilityuniverse}, we see that the number of non-vanishing diagonal elements of~$\\nu$ as well as the number of diagonal elements of~$\\nu$ having the same sign is fixed. As a conclusion, the given Lie al\\-ge\\-bra\\ cannot fall in two different of the types from Tables~\\ref{table_bianchiaclassification} and~\\ref{table_bianchibclassification}.\n\tFor Lie al\\-ge\\-bra s of class~A, scaling the basis such that~$\\nu_i\\in\\{{-}1,0,1\\}$, $i=1,2,3$, shows that each of the types in Table~\\ref{table_bianchiaclassification} has one unique representative, which concludes the proof for this class. For a Lie al\\-ge\\-bra\\ of type V or IV, one can apply scaling to achieve~$a_1=1$, $\\nu_3\\in\\{0,1\\}$, and thereby uniqueness. In case of a Lie al\\-ge\\-bra\\ of type~VI or~VII, the quantities~$a_1$ and~$\\nu_2,\\nu_3$ cannot be scaled independently of one another but have to satisfy equation~\\eqref{eqn_binvparamdefi}. Requesting~$\\nu_2=1$ and~$\\nu_3\\in\\{\\pm1\\}$ fixes~$a_1$ up to sign, which by a change of direction in~$e_1$ can be set to be positive. This concludes the proof.\n\\end{proof}\n\n\\subsection{The expansion-normalised variables from a four-dimensional spacetime point of view}\n\\label{subsect_appendixdeducenormalisedvariablesbianchib}\n\n\nFor this subsection and this subsection alone, we assume the existence of a four-dimensional spacetime from the start. That is, we assume that we already have a four-dimensional solution to Einstein's equations at our disposal, not only geometric initial data consisting of information on a three-dimensional mani\\-fold. With the four-dimensional information given, we recall the deduction of the expansion-normalised variables $(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)$ from given structure constants~$\\gamma_{\\alpha\\beta}^\\delta$, as it was developed in~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}.\nIn the following subsections, we then connect this to the initial data perspective we started with in the beginning of this paper.\n\nWe restrict ourselves to spacetimes with a Bianchi symmetry, i.\\,e.\\  with a three-dimen\\-sional Lie group~$G$ acting on the spacetime.\nWe assume a stress-energy tensor which is either that of vacuum or that of a perfect fluid, and additionally assume that the fluid velocity~$u$ is or\\-tho\\-go\\-nal\\ to the group orbits of~$G$.\n\n\nWe further assume that we are given an or\\-tho\\-nor\\-mal\\ frame $(e_0,e_1,e_2,e_3)$ such that the only non-zero structure constants are\n\\begin{equation}\n\\label{eqn_structureconstantsnonzero}\n\t\\gamma_{01}^1,\\quad \\gamma_{10}^1,\\quad \\gamma_{0A}^B,\\quad\\gamma_{A0}^B,\\quad\\gamma_{1A}^B,\\quad\n\t\\gamma_{A1}^B,\n\\end{equation}\nwith $A,B=2,3$, which is the setting considered in~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}.\n\\begin{rema}\n\tIn~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}, the property that the only non-vanishing structure constants are those in~\\eqref{eqn_structureconstantsnonzero} is justified by building upon the study of orthogonally transitive~$G_2$ cosmologies, which are a generalisation of or\\-tho\\-go\\-nal\\ Bianchi cosmologies. In~\\cite{wainwright_classschemenonrotinhomogcosm}, a suitably adapted or\\-tho\\-nor\\-mal\\ frame $(e_0,e_1,e_2,e_3)$ of a~$G_2$ spacetime is introduced, such that the vector fields $e_2,e_3$ are tangential to the group orbits of the two-dimensional symmetry group~$G_2$, the vector field~$e_1$ is spacelike, and the vector field~$e_0$ is aligned with the fluid velocity.\n\tThe study of orthogonally transitive~$G_2$ cosmologies is continued in~\\cite{hewittwainwright_orthogtransg2cosm} where the authors, following a construction proposed by~\\cite{maccallum_cosmmodelsgeometricpointofview}, decompose the structure constants and deduce the corresponding evolution equations\\ equivalent to the Einstein field equations\\ of an or\\-tho\\-go\\-nal\\ perfect fluid.\n\tThis decomposition is explicitly adapted to the setting of Bianchi~B orthogonal perfect fluid models by~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}.\n\n\tWe do not make use of these arguments but instead treat the non-vanishing of all structure constants other than those in~\\eqref{eqn_structureconstantsnonzero} as an assumption. In the following subsection, we discuss in more detail initial data sets where the three-dimensional mani\\-fold\\ is a Lie group. We notice that by excluding one specific Bianchi type, we can ensure that we end up in a setting where the structure constants have the requested form, see Lemma~\\ref{lemm_exceptionalbianchiasinitialdata} and the beginning of Subsection~\\ref{constr_initialdatatoexpnorm}.\n\\end{rema}\n\nWith structure constants as in~\\eqref{eqn_structureconstantsnonzero} at hand, we set\n\\begin{equation}\n\\label{eqns_prebasic_definitionpartone}\n\\begin{subaligned}\n\t\\sigma_{22}={}&{-}\\gamma_{02}^2+\\frac13(\\gamma_{01}^1+\\gamma_{02}^2+\\gamma_{03}^3), &\n\tn_{22}={}&{-}\\gamma_{13}^2 ,\\\\\n\t\\sigma_{23}={}&\\frac12(-\\gamma_{02}^3-\\gamma_{03}^2)=\\sigma_{32}, &\n\tn_{23}={}&\\frac12\\gamma_{12}^2-\\frac12\\gamma_{13}^3=n_{32} ,\\\\\n\t\\sigma_{33}={}&{-}\\gamma_{03}^3+\\frac13(\\gamma_{01}^1+\\gamma_{02}^2+\\gamma_{03}^3), \\qquad&\n\tn_{33}={}&\\gamma_{12}^3,\\\\\n\t\\theta={}&{-}\\gamma_{01}^1-\\gamma_{02}^2-\\gamma_{03}^3, &\n\ta_1={}&\\frac12(\\gamma_{12}^2+\\gamma_{13}^3) .\\\\\n\t\\Omega_1={}&\\frac12(\\gamma_{02}^3-\\gamma_{03}^2), &&\n\\end{subaligned}\n\\end{equation}\nThis gives a decomposition of the structure constants as proposed for general cosmological models by~\\cite{maccallum_cosmmodelsgeometricpointofview} and deduced in the present setting by~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}:\n\\begin{equation}\n\\label{eqns_commutators}\n\\begin{subaligned}\n\t\\liebr{e_0}{e_1}={}&(\\sigma_A{}^A-\\frac13\\theta)e_1,\\\\\n\t\\liebr{e_0}{e_A}={}&({-}\\sigma_A{}^B-\\frac13\\theta\\delta_A^B+\\Omega_1\\epsilon_A{}^B)e_B,\\\\\n\t\\liebr{e_1}{e_A}={}&(\\epsilon_{AB}n^{BC}+a_1\\delta_A^C)e_C,\\\\\n\t\\liebr{e_A}{e_B}={}&0,\n\\end{subaligned}\n\\end{equation}\nwith $A,B\\in\\{2,3\\}$, $\\epsilon_{AB}$ the two-dimensional permutation symbol, and~$\\delta_{AB}$ the Kronecker delta which is also used for lifting and lowering indices.\nThe non-vanishing structure constants~$\\gamma_{\\alpha\\beta}^\\delta$, see~\\eqref{eqn_structureconstantsnonzero}, of the frame elements on the one hand and the set of variables $(\\theta,\\sigma_{AB},\\Omega_1,a_1,n_{AB})$ as given in~\\eqref{eqns_commutators} on the other hand encode the same information.\n\nThe choice of or\\-tho\\-nor\\-mal\\ frame $(e_0,e_1,e_2,e_3)$ is not unique but allows for a rotation in the $e_2e_3$-plane. In the classification of Lie group s in Subsection~\\ref{subsect_appendixbianchiclassification}\nwe have chosen a specific frame which additionally diagonalises~$n_{AB}$, but for the remainder of this section, we leave this rotation as a gauge freedom. Having this freedom of rotation in choosing the frame however becomes an issue when constructing the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ in Subsections~\\ref{constr_initialdatatoexpnorm}--\\ref{constr_solutionbasictomghd}.\n\n\\begin{rema}\n\\label{rema_geometricmeaningthetasigmaAB}\n\tThe quantities appearing in the left column of~\\eqref{eqns_prebasic_definitionpartone} can be interpreted geometrically. In order to do so, assume that the spacetime is~$I\\times G$, with~$I$ an open interval described by a time parameter~$t$, and the frame is such that~$e_0=\\partial_t$, and~$e_i$, $i=1,2,3$, are tangential to every~$\\{t\\}\\times G$ and invariant under the action of~$G$.\n\n\tThe decomposition of structure constants in a spacetime was originally proposed by~\\cite{maccallum_cosmmodelsgeometricpointofview}.\n\tUpon comparison, one notices that the quantity~$\\theta_{ij}$ given there (but in our index convention) equals the second fundamental form of $\\{t\\}\\times G$ in the spacetime.\n\tConsequently, the trace~$\\theta$ of~$\\theta_{ij}$ equals the mean curvature of~$\\{t\\}\\times G$, and\n\t\\begin{equation}\n\t\t\\sigma_{ij}=\\theta_{ij}-\\frac{\\theta}{3}h_{ij}\n\t\\end{equation}\n\tis the trace-free part of the second second fun\\-da\\-men\\-tal form, where~$h$ is the spacetime metric restricted to~$\\{t\\}\\times G$.\n\n\tWe compute\n\t\\begin{equation}\n\t\t\\gamma_{0i}^i=\\scalprod{\\liebr{e_0}{e_i}}{e_i}=\\scalprod{\\nabla_{e_0}e_i-\\nabla_{e_i}e_0}{e_i}={-}\\theta_{ii},\n\t\\end{equation}\n\twhere~$\\nabla$ denotes the four-dimensional Levi-Civita\\ connection corresponding to the metric on~$I\\times G$, which we here denote by~$\\scalprod{\\cdot}{\\cdot}$.\n\tAs~$\\sigma_{ij}$ is trace free, we find\n\t\\begin{equation}\n\t\t0=\\sigma_{11}+\\sigma_{22}+\\sigma_{33}=\\theta_{11}-\\frac\\theta3+\\sigma_{22}+\\sigma_{33},\n\t\\end{equation}\n\tor equivalently\n\t\\begin{equation}\n\t\t{-}\\theta_{11}=\\sigma_A{}^A-\\frac13\\theta,\n\t\\end{equation}\n\twhich should be compared to the first equation in~\\eqref{eqns_commutators}.\n\n\tThe quantity~$\\Omega_1$ can be expressed in terms of the four-dimensional frame as follows:\n\t\\begin{equation}\n\t\t\\Omega_1=\\scalprod{e_3}{\\nabla_{e_0}e_2},\n\t\\end{equation}\n\twhere~$\\nabla$ denotes the four-dimensional Levi-Civita\\ connection. The quantity~$\\Omega_1$ thus describes a certain timelike derivative, but not one which is encoded in the second fundamental form~$\\theta_{ij}$.\n\\end{rema}\n\n\nFollowing the deduction of the expansion-normalised variables $(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)$ as given in~\\cite{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}, one in a first step replaces~$n_{AB}$ and~$\\sigma_{AB}$, $A,B\\in\\{2,3\\}$, by their trace and trace-free part (with respect to\\ the two-dimensional trace in the $23$-components), i.\\,e.\\  one defines\n\\begin{equation}\n\\label{eqns_prebasic_definitionparttwo}\n\\begin{subaligned}\n\t\\sigma_+={}&\\frac32\\delta^{AB}\\sigma_{AB}, &\n\tn_+={}&\\frac32\\delta^{AB}n_{AB} ,\\\\\n\t\\tilde\\sigma_{AB}={}&\\sigma_{AB}-\\frac13\\sigma_+\\delta_{AB}, \\qquad &\n\t\\tilde n_{AB}={}&n_{AB}-\\frac13n_+\\delta_{AB}.\n\\end{subaligned}\n\\end{equation}\nThe information encoded in these new variables is equivalent to that encoded in~$n_{AB}$, $\\sigma_{AB}$. Additionally, one sets\n\\begin{equation}\n\\label{eqns_basic_definitionpartone}\n\t{}^*\\tilde\\sigma_{AB}={}\\tilde\\sigma_A{}^C\\epsilon_{BC}, \\qquad\n\t{}^*\\tilde n_{AB}={}\\tilde n_A{}^C\\epsilon_{BC}.\n\\end{equation}\n\nAssuming that the only non-vanishing structure constants are those given in~\\eqref{eqn_structureconstantsnonzero} which are decomposed as above, and further assuming a stress-energy tensor of an or\\-tho\\-go\\-nal\\ perfect fluid~\\eqref{eqn_stressenergyperfectfluid} with linear equation of state~\\eqref{eqn_lineareqnofstate}, the evolution equations\\ for the variables $(\\theta,\\sigma_+,\\tilde\\sigma_{AB},a_1,n_+,\\tilde n_{AB})$ have been given explicitly in~\\cite[App.~A]{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}:\n\\begin{equation}\n\\label{eqns_prebasic_evolutionbianchib}\n\\begin{subaligned}\t\\dot\\theta={}&{-}\\frac13\\theta^2-\\frac23\\sigma_+^2-\\tilde\\sigma^{AB}\\tilde\\sigma_{AB}-\\frac12(3\\gamma-2)\\mu,\\\\\n\t\\dot\\sigma_+={}&{-}\\theta\\sigma_+-\\tilde n^{AB}\\tilde n_{AB},\\\\\n\t\\dot{\\tilde\\sigma}_{AB}={}&{-}\\theta\\tilde\\sigma_{AB}+2\\Omega_1{}^*\\tilde\\sigma_{AB}-\\frac23n_+\\tilde n_{AB}+2a_1{}^*\\tilde n_{AB},\\\\\n\t\\dot a_1={}&\\frac13(2\\sigma_+-\\theta)a_1,\\\\\n\t\\dot n_+={}&\\frac13(2\\sigma_+-\\theta)n_++3\\tilde\\sigma^{AB}\\tilde n_{AB},\\\\\n\t\\dot{\\tilde n}_{AB}={}&\\frac13(2\\sigma_+-\\theta)\\tilde n_{AB}+2\\Omega_1{}^*\\tilde n_{AB}+\\frac23n_+\\tilde\\sigma_{AB},\n\\end{subaligned}\n\\end{equation}\nwith constraint equation\n\\begin{equation}\n\\label{eqn_prebasic_constraint}\n\t3a_1\\sigma_+-\\frac32{}^*\\tilde\\sigma^{AB}\\tilde n_{AB}=0,\n\\end{equation}\n\\noeqref{eqn_prebasic_constraint\ndefining equation for~$\\mu$\n\\begin{equation}\n\\label{eqn_prebasic_mu}\n\t\\mu=\\frac13\\theta^2-\\frac13\\sigma_+^2-\\frac12\\tilde\\sigma^{AB}\\tilde\\sigma_{AB}-\\frac12\\tilde n^{AB}\\tilde n_{AB}-3a_1^2,\n\\end{equation}\nand auxiliary equation\n\\begin{equation}\n\\label{eqn_prebasic_evolutionmu}\n\t\\dot\\mu={-}\\gamma\\mu\\theta.\n\\end{equation}\n\\begin{rema}\n\\label{rema_equivalenceeinsteinperfectfluidprebasic}\n\tConsider a four-dimensional spacetime with an or\\-tho\\-nor\\-mal\\ frame~$(e_0,e_1,e_2,e_3)$.\n\tUnder the assumption that the only non-vanishing structure constants~$\\gamma_{\\alpha\\beta}^\\delta$ are those given in~\\eqref{eqn_structureconstantsnonzero} and they additionally satisfy\n\t\\begin{equation}\n\t\te_i(\\gamma_{\\alpha\\beta}^\\delta)=0,\n\t\\end{equation}\n\tthe system of evolution equations with constraints~\\eqref{eqns_prebasic_evolutionbianchib}--\\eqref{eqn_prebasic_evolutionmu}\n\tholds if and only if\\\n\tthe Ricci curvature\\ of the spacetime is invariant under a three-dimensional Lie group\\ action in the~$e_1e_2e_3$-space such that in the chosen basis the only non-vanishing terms are\n\t\\begin{equation}\n\t\tRic(e_0,e_0)=\\frac12(3\\gamma-2)\\mu,\\qquad Ric(e_i,e_i)=\\frac12(2-\\gamma)\\mu,\n\t\\end{equation}\n\t$i=1,2,3$, and the Jacobi identities hold in the $e_1e_2e_3$-space. This follows from direct computation,\n\texpressing the Ricci curvature\\ in terms of the structure constants~$\\gamma_{\\alpha\\beta}^\\delta$ and then replacing them by the variables $(\\theta,\\sigma_+,\\tilde\\sigma_{AB},a_1,n_+,\\tilde n_{AB})$.\n\tA Ricci curvature\\ of this form is equivalent to Einstein's equations of a perfect fluid~\\eqref{eqn_stressenergyperfectfluid} with linear equation of state~\\eqref{eqn_lineareqnofstate}.\n\n\tIn detail, one finds that the evolution equations for the variables~$\\theta$, $\\sigma_+$ and~$\\tilde\\sigma_{AB}$ correspond to the Ricci curvature\\ terms~$Ric(e_i,e_j)$ with~$i\\not=0\\not=j$, while the evolution of~$a_1$, $n_+$ and~$\\tilde n_{AB}$ follows from the Jacobi identities on the Lie group. The constraint equation~\\eqref{eqn_prebasic_constraint} is equivalent to the momentum constraint, and equation~\\eqref{eqn_prebasic_mu} defining~$\\mu$ is equivalent to the Hamiltonian constraint. The evolution of~$\\mu$ is equivalent to the matter equation of a perfect fluid with linear equation of state.\n\\end{rema}\n\n\n\nIn the next step of the construction, one introduces the variables\n\\begin{equation}\n\\label{eqns_basic_definitionparttwo}\n\\begin{subaligned}\n\t\\tilde\\sigma=\\frac32\\tilde\\sigma^{AB}\\tilde\\sigma_{AB}, \\qquad&\\tilde n={}\\frac32\\tilde n^{AB}\\tilde n_{AB},\\\\\n\t\\delta=\\frac32\\tilde\\sigma^{AB}\\tilde n_{AB}, \\qquad& ^*\\delta={}\\frac32\\,^*\\tilde\\sigma^{AB}\\tilde n_{AB},\n\\end{subaligned}\n\\end{equation}\nwhich are invariant under the freedom of rotation and satisfy $\\tilde\\sigma\\tilde n=\\delta^2+\\,^*\\delta^2$.\nThe constraint equation~\\eqref{eqn_prebasic_constraint}\nimplies $^*\\delta=3a_1\\sigma_+$, which can be used to eliminate~$^*\\delta$ in the following.\nOne further finds that there is a constant~${\\kappa}$ such that\n\\begin{equation}\n\\label{eqn_basic_tildenrelation}\n\t\\tilde n=\\frac13(n_+^2-9{\\kappa}a_1^2).\n\\end{equation}\nIn the case of Bianchi type~VI and~VII, this relation immediately follows from the definition of the Bianchi group parameter~$\\eta$ in equation~\\eqref{eqn_binvparamdefi}, and the constant satisfies~${\\kappa}=1\/{\\eta}$. In all other Bianchi cases, one computes directly from the form of~$n_{AB}$ and~$a_1$ that equation~\\eqref{eqn_basic_tildenrelation} holds for~${\\kappa}=0$.\nOne further sets\n\\begin{equation}\n\\label{eqn_basic_definitionpartthree}\n\t\\tilde a=9a_1^2\n\\end{equation}\nfor simplification. The resulting variables\n$(\\theta ,\\sigma_+ ,\\tilde{\\sigma},\\delta ,\\tilde{a},n_+)$\nare called the \\textit{basic variables} and evolve according to\n\\begin{equation}\n\\label{eqns_basic_evolutionbianchib}\n\\begin{subaligned}\n\t\\dot\\theta={}&{-}\\frac13\\theta^2-\\frac23(\\sigma_+^2+\\tilde\\sigma)-\\frac12(3\\gamma-2)\\mu,\\\\\n\t\\dot\\sigma_+={}&{-}\\theta\\sigma_+-\\frac23\\tilde n,\\\\\n\t\\dot{\\tilde\\sigma}={}&{-}2\\theta\\tilde\\sigma-\\frac43n_+\\delta-\\frac43\\tilde a\\sigma_+,\\\\\n\t\\dot\\delta={}&\\frac23(\\sigma_+-2\\theta)\\delta+\\frac23n_+(\\tilde\\sigma-\\tilde n),\\\\\n\t\\dot{\\tilde a}={}&\\frac23(2\\sigma_+-\\theta)\\tilde a,\\\\\n\t\\dot n_+={}&\\frac13(2\\sigma_+-\\theta)n_++2\\delta.\n\\end{subaligned}\n\\end{equation}\nThey satisfy\n\\begin{equation}\n\\label{eqn_basic_constraintnonnegative}\n\t\\tilde\\sigma\\ge0, \\qquad \\tilde a\\ge0, \\qquad\\tilde n\\ge0, \\qquad \\sigma_+^2+\\tilde\\sigma+\\tilde a+\\tilde n\\le \\theta^2,\n\\end{equation}\nand the constraint\n\\begin{equation}\n\\label{eqn_basic_constraint}\n\t\\tilde\\sigma \\tilde n -\\delta^2-\\tilde a\\sigma_+^2=0,\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eqn_basic_definitiontilden}\n\t\\tilde n=\\frac13(n_+^2-{\\kappa}\\tilde a).\n\\end{equation}\nThe matter $\\mu$ is given by\n\\begin{equation}\n\\label{eqn_basic_mu}\n\t\\mu=\\frac13(\\theta^2-\\sigma_+^2-\\tilde\\sigma-\\tilde a-\\tilde n),\n\\end{equation}\nand the auxiliary equation is satisfied\n\\begin{equation}\n\\label{eqn_basic_evolutionmu}\n\t\\dot\\mu={-}\\gamma\\theta\\mu.\n\\end{equation}\n\\begin{rema}\n\tOne sees that the evolution equations~\\eqref{eqns_prebasic_evolutionbianchib}--\\eqref{eqn_prebasic_evolutionmu} imply the evolution equations~\\eqref{eqns_basic_evolutionbianchib}--\\eqref{eqn_basic_evolutionmu}, provided relation~\\eqref{eqn_basic_definitiontilden} holds and the matter~$\\mu$ in the first set of equations is assumed to be non-negative.\n\n\tNote also that the relations in~\\eqref{eqn_basic_constraintnonnegative} and the constraint equation~\\eqref{eqn_basic_constraint} are preserved under the evolution equations~\\eqref{eqns_basic_evolutionbianchib}. This follows from similar arguments as for the expansion-normalised variables, see Remark~\\ref{rema_expnormvarconstraintspreserved}.\n\\end{rema}\n\n\\begin{lemm}\n\t\\label{lemm_excludethetazerobasic}\n\tConsider a solution to equations~\\eqref{eqns_basic_evolutionbianchib}--\\eqref{eqn_basic_evolutionmu}. Then either the solution is the trivial solution\n\t\\begin{equation}\n\t\t\\theta=\\sigma_+=\\tilde{\\sigma}=\\delta=\\tilde{a}=n_+=\\mu=0,\n\t\\end{equation}\n\tor $\\theta\\not=0$ at all times.\n\\end{lemm}\n\\begin{proof}\n\tUsing non-negativity of~$\\mu$ in equation~\\eqref{eqn_basic_mu} shows that if~$\\theta=0$ at one time then\n\t\\begin{equation}\n\t\t\\sigma_+=0,\\quad \\tilde\\sigma=0,\\quad \\tilde a=0,\\quad \\tilde n=0,\\quad \\mu=0\n\t\\end{equation}\n\tat this time, and from equations~\\eqref{eqn_basic_definitiontilden} and~\\eqref{eqn_basic_constraint}, even~$n_+$ and~$\\delta$ vanish at this time.\n\tThe evolution equations~\\eqref{eqns_basic_evolutionbianchib} imply that all variables vanish at all times.\n\\end{proof}\n\n\n\n\n\nIn the final step, the basic variables are normalised with appropriate powers of the rate of expansion scalar~$\\theta$. By excluding the trivial solution in basic variables, we ensure that $\\theta(t)\\not=0$ at all times~$t$, see Lemma~\\ref{lemm_excludethetazerobasic}. Setting then\n\\begin{equation}\n\\label{eqn_expansionnormalisation}\n\t\\Sigma_+=\\frac{\\sigma_+}{\\theta},\\quad \\tilde\\Sigma=\\frac{\\tilde\\sigma}{\\theta^2},\\quad\n\t\\Delta=\\frac{\\delta}{\\theta^2},\\quad \\tilde A=\\frac{\\tilde a}{\\theta^2},\\quad\n\tN_+=\\frac{n_+}{\\theta},\\quad\n\t\\tilde N=\\frac{\\tilde n}{\\theta^2},\n\\end{equation}\nand replacing the energy density~$\\mu$ by the density parameter~$\\Omega$ defined by\n\\begin{equation}\n\\label{eqn_definitionomega}\n\t\\Omega=\\frac{3\\mu}{\\theta^2}\n\\end{equation}\nyields the \\textit{expansion-normalised and dimensionless variables}\n$(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)$. After changing to the dimensionless time~$\\tau$ which satisfies\n\\begin{equation}\n\\label{eqn_evolutiontau}\n\t\\frac{dt}{d\\tau}=\\frac{3}{\\theta}\n\\end{equation}\nand some arbitrarily chosen initial condition,\nthey evolve according to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega}. The deceleration parameter~$q$ appearing in these evolution equations is related to the basic variable~$\\theta$ via\n\\begin{equation}\n\\label{eqn_evolutiontheta}\n\t\\theta'={-}(1+q)\\theta.\n\\end{equation}\n\\begin{rema}\n\\label{rema_equivalencebasicvarexpnormvar}\n\tIt follows by direct computation that the evolution in basic variables, equations~\\eqref{eqns_basic_evolutionbianchib}--\\eqref{eqn_basic_evolutionmu}, on the one hand and the evolution in expansion-normalised variables, equations \\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega}, together with equation~\\eqref{eqn_evolutiontheta} on the other hand are equivalent to one another, provided~$\\theta\\not=0$, the variables are related as in~\\eqref{eqn_expansionnormalisation}--\\eqref{eqn_definitionomega},\n\tand the time~$t$ is related to the dimensionless time~$\\tau$ as in equation~\\eqref{eqn_evolutiontau}.\n\\end{rema}\n\n\n\\subsection{Lie group s as initial data subsets}\n\\label{subsect_liegrinitialdata}\n\n\tIn this subsection, we consider initial data sets where the three-dimensional mani\\-fold\\ is a Lie group, and the initial metric and second fun\\-da\\-men\\-tal form\\ are invariant under the group action.\n\tIn addition to the purely three-dimensional Lie group\\ properties which we have discussed in Subsection~\\ref{subsect_appendixbianchiclassification}, the Hamilton and momentum constraint equations pose restrictions on the two-tensor~$k$ which we now investigate in detail.\n\tIn the following subsections, these restrictions are used to connect geometric initial data to expansion-normalised variables and the construction of the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment.\n\n\tWe start with initial data $(G,{h},{k},\\mu_0)$ as in Def.~\\ref{defi_initialdatabianchib}:\n\tA Lie group~$G$ of class~B or\n\tof type~I or~II,\n\ta left-invariant Rie\\-mann\\-ian\\ metric~${h}$ on~$G$, a left-invariant symmetric covariant two-tensor~${k}$ on~$G$, and a constant $\\mu_0\\ge0$, satisfying the constraint equations\n\t\\begin{align}\n\t\t\\overline R-{k}_{ij}{k}^{ij}+(\\operatorname{tr}_{h}{k})^2={}&2\\mu_0,\\\\\n\t\t\\overline\\nabla_i\\operatorname{tr}_{h}{k}-\\overline\\nabla^j{k}_{ij}={}&0.\n\t\\end{align}\n\tIn a first step, we fix a two-dimensional Abelian subalgebra~$\\ggg_2$ of the Lie al\\-ge\\-bra~$\\ggg$ corresponding to the Lie group~$G$. In case of a group of Bianchi class~B, this is the uniquely defined subalgebra which is the kernel of the one-form~$H$, see Lemma~\\ref{lemm_abeliansubalginvardefi}. In case of a group of Bianchi type~I or~II, the existence of such a subalgebra is ensured by Lemma~\\ref{lemm_abeliansubalgstructconstants}. We then choose an orthonormal basis $\\tilde e_2,\\tilde e_3$ of this subalgebra.\n\tHaving fixed~$\\ggg_2$, these two vectors can be chosen uniquely up to rotation and reflection.\n\tUsing the given initial metric~${h}$ to fix a third unit vector~$\\tilde e_1$ in the Lie al\\-ge\\-bra\\ orthogonal to the span of those two yields a basis of~$\\ggg$.\n\tThis choice of basis gives\n\tstructure constants~$\\gamma_{ij}^k$, $i,j,k\\in\\{1,2,3\\}$, or equivalently $n^{ij}$, $a_k$, see~\\eqref{eqn_data3dimliegr} and~\\eqref{eqn_data3dimliegrback} in Subsection~\\ref{subsect_appendixbianchiclassification}.\n\n\tFor a fixed~$\\ggg_2$, the chosen frame of the Lie al\\-ge\\-bra~$\\ggg$ is unique up to rotation and reflection in~$\\ggg_2$ in case of~$\\tilde e_2$, $\\tilde e_3$, and a choice of sign in case of~$\\tilde e_1$.\n\tWe can therefore use the frame from the classification of Lie group s in Subsection~\\ref{subsect_appendixbianchiclassification}, see also Remark~\\ref{rema_frameforggg2rotationtodiagonalise}. Note that in the case of a Bianchi type~II Lie group, we rename the basis elements such that~$\\tilde e_2$ and~$\\tilde e_3$ commute. For this special frame, the structure constants are such that~$a_2=a_3=0$, the matrix~$n$ is diagonal with~$n_{11}=0$, and~$a_1\\not=0$ for~$G$ of Bianchi class~B. The reflection in~$\\tilde e_1$ and as well as the rotation and reflection in~$\\ggg_2$ do not affect~$a_i$ or~$n_{1i}=n_{i1}$, $i=2,3$.\n\n\tLet us now have a closer look at the constraint equations, which the metric~$h$ and two-tensor~$k$ have to satisfy by Def.~\\ref{defi_initialdatabianchib}. As both tensors are left-invariant, the momentum constraint reduces to\n\t\\begin{equation}\n\t\t\\overline\\nabla^jk_{ij}=0.\n\t\\end{equation}\n\tIn terms of the chosen or\\-tho\\-nor\\-mal\\ frame, we find\n\t\\begin{equation}\n\t\t\\overline\\nabla_lk_{ij}\n\t\t=\\tilde e_l(k(\\tilde e_i,\\tilde e_j)) -k(\\overline\\nabla_{\\tilde e_l}\\tilde e_i, \\tilde e_j) -k(\\tilde e_i,\\overline\\nabla_{\\tilde e_l}\\tilde e_j)\n\t\t={-}\\overline\\Gamma_{li}^mk_{mj}-\\overline\\Gamma_{lj}^mk_{im},\n\t\\end{equation}\n\twhere~$\\overline\\Gamma_{ij}^k$ denote the Christoffel symbols~$\\overline\\Gamma_{ij}^k\\tilde e_k=\\overline\\nabla_{\\tilde e_i}\\tilde e_j$\n\tcorresponding to the Levi-Civita\\ connection~$\\overline\\nabla$ on the Rie\\-mann\\-ian\\ mani\\-fold~$(G,h)$.\n\tWe conclude that the momentum constraint is equivalent to\n\t\\begin{equation}\n\t\\label{eqn_momentumconstraintchristoffel}\n\t\t0={-}\\sum_{j=1}^3(\\overline\\Gamma_{ji}^mk_{mj}+\\overline\\Gamma_{jj}^mk_{im}).\n\t\\end{equation}\n\tAs a consequence of Koszul's formula, the Christoffel symbols can be expressed in terms of the structure constants via\n\t\\begin{equation}\n\t\\label{eqn_christoffelinstructureconstants}\n\t\t\\overline\\Gamma_{ij}^k=\\frac12({-}\\gamma_{jk}^i+\\gamma_{ki}^j+\\gamma_{ij}^k).\n\t\\end{equation}\n\tUsing the definitions for~$n^{ij}$ and $a_k$, equations~\\eqref{eqn_data3dimliegr}, as well as the symmetry of~$k$\n\twe then find\n\t\\begin{equation}\n\t\\label{eqn_threericcizeroione}\n\t\t0={-}2a_1k_{11}+a_1(k_{22}+k_{33})+(n_{33}-n_{22})k_{23}\n\t\\end{equation}\n\tfor the~$i=1$-component, while\n\tthe~$i=2$ and~$i=3$-component of equation~\\eqref{eqn_momentumconstraintchristoffel} read\n\t\\begin{equation}\n\t\\label{eqns_threericcizeroitwothree}\n\t\\begin{subaligned}\n\t\t0={}&{-}3a_1k_{12}-n_{33}k_{13}\\\\\n\t\t0={}&{+}n_{22}k_{12}-3a_1k_{13}.\n\t\\end{subaligned}\n\t\\end{equation}\n\tWe conclude that there is a non-vanishing solution~$(k_{12},k_{13})$ of the system of equations~\\eqref{eqns_threericcizeroitwothree} if and only if\n\t\\begin{equation}\n\t\t9 a_1^2+n_{22}n_{33}=0.\n\t\\end{equation}\n\tIn Bianchi class~B, this is only possible for Bianchi type~VI due to~$a_1^2>0$ and the signs of~$n_{ii}$,  compare Table~\\ref{table_bianchibclassification}.  Furthermore, the invariance of the parameter~$\\eta$, equation~\\eqref{eqn_binvparamdefi}, implies that only the case~$\\eta={-}1\/9$ allows for a non-vanishing solution. In total, we have shown the following:\n\t\\begin{lemm}\n\t\\label{lemm_exceptionalbianchiasinitialdata}\n\t\tLet~$G$ be a Lie group\\ of class~B, ${h}$ a left-invariant Rie\\-mann\\-ian\\ metric on~$G$, and ${k}$ a left-invariant symmetric covariant two-tensor on~$G$\n\t\tsatisfying the momentum constraint\n\t\t\\begin{equation}\n\t\t\t\\overline\\nabla_i\\operatorname{tr}_{h}{k}-\\overline\\nabla^j{k}_{ij}=0.\n\t\t\\end{equation}\n\t\tLet~$\\ggg_2$ be the kernel of the one-form~$H$ from Lemma~\\ref{lemm_abeliansubalginvardefi} which is an Abelian subalgebra of the Lie al\\-ge\\-bra~$\\ggg$ corresponding to~$G$. Let~$(\\tilde e_1,\\tilde e_2,\\tilde e_3)$ an or\\-tho\\-nor\\-mal\\ basis of~$\\ggg$ such that~$\\tilde e_2$ and~$\\tilde e_3$ span~$\\ggg_2$.\n\t\tIf~$k_{12}$ or $k_{13}$ is non-vanishing, then the Lie group~$G$ is of Bianchi type~VI$_{{-}1\/9}$.\n\t\\end{lemm}\n\tNote that in the previous Lemma, the subalgebra~$\\ggg_2$ is defined geometrically, as the kernel of a well-defined one-form. Consequently, whether\n\t\\begin{equation}\n\t\\label{eqn_nonexceptionalsigmaij}\n\t\tk_{12}=k_{21}=0=k_{13}=k_{31}\n\t\\end{equation}\n\tor not is also a well-defined geometric property, and independent of the exact choice of or\\-tho\\-nor\\-mal\\ basis, as long as~$\\tilde e_2$ and~$\\tilde e_3$ span~$\\ggg_2$.\n\n\tFor a Lie group\\ of Bianchi class~A, a basis can be chosen which satisfies the properties of i) in the classification and diagonalises both the metric and the second fun\\-da\\-men\\-tal form, see~\\cite[Cor.~9.14]{ringstrom_cauchyproblem}. Therefore in this basis the relations~\\eqref{eqn_nonexceptionalsigmaij}\n\thold as well.\n\n\tLemma~\\ref{lemm_exceptionalbianchiasinitialdata} is the reason why we have explicitly excluded Bianchi type~VI$_{{-}1\/9}$ Lie group s in the definition of geometric initial data, Def.~\\ref{defi_initialdatabianchib}: By doing so, we ensure that equations~\\eqref{eqn_nonexceptionalsigmaij} hold.\n\tIn fact, we do not have to exclude the special Bianchi type~VI$_{{-}1\/9}$ altogether. It can be included in the discussion as long as we adopt equation~\\eqref{eqn_nonexceptionalsigmaij} as an additional assumption.\n\t\\begin{rema}\n\t\\label{rema_exceptionalityexplained}\n\t\tIt has been shown in~\\cite{ellismaccallum_classofhomogcosmmodels} that condition~\\eqref{eqn_nonexceptionalsigmaij} is equivalent to `non-exceptio\\-nality' of a Bianchi spacetime.\n\t\tMore precisely, this reference considers the trace-free part\n\t\t\\begin{equation}\n\t\t\t\\sigma_{ij}= {k}_{ij}-\\frac{\\theta}3h_{ij}\n\t\t\\end{equation}\n\t\tof~$k$, where~$\\theta=\\operatorname{tr}_hk$ is the mean curvature, see also Remark~\\ref{rema_geometricmeaningthetasigmaAB}, and makes use of the equivalent formulation\n\t\t\\begin{equation}\n\t\t\t\\sigma_{12}=\\sigma_{21}=0=\\sigma_{13}=\\sigma_{31}.\n\t\t\\end{equation}\n\t\tThis notion of 'non-exceptionality' is a property of the four-dimensional spacetime, namely not being one of the 'exceptional' spacetimes which~\\cite{ellismaccallum_classofhomogcosmmodels} denote by Bbii. These 'exceptional' spacetimes are those which admit an or\\-tho\\-nor\\-mal\\ frame~$e_i$, $i=0,1,2,3$, such that~$e_1,e_2,e_3$ are tangential to the spacelike hypersurfaces and\n\t\t\\begin{equation}\n\t\t\\label{eqn_definitionBbii}\n\t\t\t\\sigma_{12}\\sigma_{13}\\not=0,\\qquad {\\eta}={-}1\/9,\\qquad n_{22}=\\frac{3a_1\\sigma_{13}}{\\sigma_{12}},\\qquad n_{33}={-}\\frac{3a_1\\sigma_{12}}{\\sigma_{13}}\n\t\t\\end{equation}\n\t\tholds.\n\n\t\tThe property of being `non-exceptional' is further related to the notion of or\\-tho\\-go\\-nal ly transitive~$G_2$ cosmologies, see~\\cite[Thm~3.1(i)]{wainwright_classschemenonrotinhomogcosm} for a characterisation of or\\-tho\\-go\\-nal\\ transitivity. In combination with several results from~\\cite{ellismaccallum_classofhomogcosmmodels}, in particular Lemma~4.1 and Thm~5.1 in that reference, we conclude the following: A Bianchi class~B spacetime admits an Abelian subgroup acting or\\-tho\\-go\\-nal ly transitively if and only if\\ it is `non-exceptional'.\n\t\tFor this reason, we use the terms 'or\\-tho\\-go\\-nal ' and 'non-exceptional' initial data interchangeably.\n\t\t \n\t\tIn terms of the Bianchi classification, only certain initial data sets for one specific parameter in one Bianchi type of class~B satisfy the 'exceptional' properties. However, due to the additional degree of freedom, the phase space effectively has a higher dimension than that of all remaining Bianchi class~B types.\n\t\tFor this reason, the notion 'exceptional' should be interpreted as 'having exceptional behaviour', not 'being special enough to discard'. In fact, towards the initial singularity such spacetimes are expected to show chaotic oscillatory behaviour, see~\\cite{hewitthorwoodwainwright_asymptdynamexceptbianchicosm}.\n\t\\end{rema}\n\n\n\\subsection{Construction part I: From geometric initial data to the expansion-normalised evolution equa\\-tions}\n\\label{constr_initialdatatoexpnorm}\n\n\tIn this subsection and the two following ones, we carry out the construction of the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ for given geometric initial data, by which we mean initial data to Einstein's field equations.\n\n\tWe start with initial data $(G,{h},{k},\\mu_0)$ as in Def.~\\ref{defi_initialdatabianchib}.\n\tWe fix a two-dimensional Abelian subalgebra~$\\ggg_2$ of the Lie al\\-ge\\-bra~$\\ggg$ corresponding to the Lie group~$G$. As we detailed in the previous subsection, this is the kernel of the one-form~$H$ from Lemma~\\ref{lemm_abeliansubalginvardefi} in case of a Lie group\\ of class~B, otherwise such a subalgebra exists due to Lemma~\\ref{lemm_abeliansubalgstructconstants}. We then choose an orthonormal basis $\\tilde e_1,\\tilde e_2,\\tilde e_3$ of~$\\ggg$ such that~$\\tilde e_2,\\tilde e_3$ span~$\\ggg_2$.\n\tSuch a choice of basis gives initial structure constants~$\\gamma_{ij}^k(0)$, $i,j,k\\in\\{1,2,3\\}$, and this information is equivalent to\\begin{equation}\n\t\tn^{ij}(0),\\qquad a_k(0)\n\t\\end{equation}\n\tdue to~\\eqref{eqn_data3dimliegr} and~\\eqref{eqn_data3dimliegrback} in Subsection~\\ref{subsect_appendixbianchiclassification}.\n\tWe have explained in the previous subsection that in this basis one has~$a_2=a_3=0$, the matrix~$n$ is diagonal with~$n_{11}=0$, and~$a_1\\not=0$, in case of a Lie group~$G$ of Bianchi class~B. For groups of type~I or~II, we can assume that~$a_1=a_2=a_3=0$ and the matrix~$n$ is diagonal with~$n=\\operatorname{diag}(0,n_{22},n_{33})$. In particular, the initial structure constants are such that the third and fourth equation in~\\eqref{eqns_commutators} hold.\n\n\n\tWe further set initial values for~$\\theta$ and~$\\sigma_{ij}$\n\t\\begin{equation}\n\t\\label{eqn_initialvaluesthetasigmaAB}\n\t\t\\theta(0)\\coloneqq\\operatorname{tr}_{h}{k},\\qquad \\sigma_{ij}(0)\\coloneqq {k}_{ij}-\\frac{\\theta(0)}3h_{ij},\n\t\\end{equation}\n\tin accordance with Remark~\\ref{rema_geometricmeaningthetasigmaAB}.\n\tFor the initial value~$\\Omega_1(0)$, we choose an arbitrary value. This is not determined by the geometric initial data. We will see further down that one can without loss of generality\\ assume $\\Omega_1\\equiv0$, but at this stage this is a non-determined variable.\n\tFrom Lemma~\\ref{lemm_exceptionalbianchiasinitialdata}, we conclude that\n\tthe~$12$- and~$13$-component of~$\\sigma_{ij}$ (or equivalently of~$k_{ij}$) vanish. In particular, we are now in a position to connect geometric initial data to the construction carried out in Subsection~\\ref{subsect_appendixdeducenormalisedvariablesbianchib}, as the only non-vanishing initial quantities are those appearing on the left-hand side\\ of equations~\\eqref{eqns_prebasic_definitionpartone}.\n\n\tWith the definitions made so far, we can introduce quantities\n\t\\begin{equation}\n\t\t\\tilde\\gamma_{01}^1(0)\\coloneqq\\sigma_A{}^A(0)-\\frac13\\theta(0),\\qquad\n\t\t\\tilde\\gamma_{0A}^B(0)\\coloneqq{-}\\sigma_A{}^B(0)-\\frac13\\theta(0)\\delta_A^B+\\Omega_1(0)\\epsilon_A{}^B.\n\t\\end{equation}\n\tWe stress that these are merely numbers at this point. Only after we have constructed the spacetime can they be interpreted as the structure constants on the hypersurface~$\\{0\\}\\times G$ of a suitable four-dimensional frame.\n\tIn combination with the three-dimensional structure constants~$\\gamma_{ij}^k(0)$ we found above, we have now constructed, at time~$t=0$, a set of numbers as appear on the~right-hand side\\ of equation s~\\eqref{eqns_commutators}.\n\n\tIn the next step, we apply the algebraic operations~\\eqref{eqns_prebasic_definitionparttwo} and~\\eqref{eqns_basic_definitionpartone} to the set of numbers\n\t\\begin{equation}\n\t\t(\\theta,\\sigma_{AB},\\Omega_1,a_1,n_{AB})(0)\n\t\\end{equation}\n\tto obtain a set of numbers\n\t\\begin{equation}\n\t\t(\\theta,\\sigma_+,\\tilde\\sigma_{AB},a_1,n_+,\\tilde n_{AB})(0).\n\t\\end{equation}\n\tDirect computation shows that the Hamiltonian constraint\n\t\\begin{equation}\n\t\t\\overline R-{k}_{ij}{k}^{ij}+(\\operatorname{tr}_{h}{k})^2=2\\mu_0\n\t\\end{equation}\n\twhich the geometric initial data has to satisfy\n\tis equivalent to equation~\\eqref{eqn_prebasic_mu}, using for example~\\cite[eq.~{[}E.12{]}]{ringstrom_topologyfuturestabilityuniverse} to compute the three-dimensional scalar curvature~$\\overline{R}$.\n\tThe~$i=1$-component of the momentum constraint implies equation~\\eqref{eqn_prebasic_constraint}, as can be concluded from equation~\\eqref{eqn_threericcizeroione}.\n\tWe therefore interpret~$(\\theta,\\sigma_+,\\tilde\\sigma_{AB},a_1,n_+,\\tilde n_{AB})(0)$ as initial data for the evolution equations~\\eqref{eqns_prebasic_evolutionbianchib}--\\eqref{eqn_prebasic_evolutionmu}. Note that from this point on, the arbitrary value~$\\Omega_1(0)$ does no longer appear.\n\tApplying now also the transformations~\\eqref{eqns_basic_definitionparttwo} and~\\eqref{eqn_basic_definitionpartthree}, one obtains a set of numbers\n\t\\begin{equation}\n\t\t(\\theta,\\sigma_+,\\tilde\\sigma,\\delta,\\tilde a,n_+)(0)\n\t\\end{equation}\n\twhich can be interpreted as initial data in basic variables. Note that equation~\\eqref{eqn_basic_definitiontilden} holds due to the underlying Lie group\\ structure of the geometric initial data we started with.\n\n\tWe assume that we have started out with initial data~$(G,{h},{k},\\mu_0)$\n\tsuch that after the steps carried out so far, the initial data in basic variables is different from the trivial data, i.\\,e.\\ \n\t\\begin{equation}\n\t\t(\\theta,\\sigma_+,\\tilde\\sigma,\\delta,\\tilde a,n_+)(0)\\not=(0,0,0,0,0,0).\n\t\\end{equation}\n\tFurther down in Lemma~\\ref{lemm_excludeminkowski}, we show that this restriction is equivalent to geometric initial data whose ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ is not isometric to a quotient of Minkowski space.\n\tGoing through the construction so far, looking for a solution to~\\eqref{eqns_prebasic_evolutionbianchib}--\\eqref{eqn_prebasic_evolutionmu} for the initial values we constructed, we can assume that $\\theta\\not=0$ at all times due to Lemma~\\ref{lemm_excludethetazerobasic}. We therefore assume that\n\t\\begin{equation}\n\t\t\\theta>0.\n\t\\end{equation}\n\tOnce we have obtained a spacetime which admits the geometric initial data induced on a hypersurface, this choice of sign corresponds to the choice of normal vector~$e_0$, or in other words fixes the orientation of time~$t$.\n\n\tEssentially, the transformation of geometric initial data into initial values in basic variables is a one-to-one correspondence up to sign. We can recover the signs from the geometric initial data, see the third part of the construction.\n\n\tIn the next step, we normalise the initial values via equations~\\eqref{eqn_expansionnormalisation} to obtain initial values\n\t\\begin{equation}\n\t\t(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)(0).\n\t\\end{equation}\n\tThese are interpreted as initial data to the evolution equations in expansion-normalised variables~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega}, where we also change to an expansion-normalised time~$\\tau$ which we choose to satisfy~$\\tau(t=0)=0$.\n\n\tIn total, we have achieved the following: Geometric initial data $(G,{h},{k},\\mu_0)$ as in Def.~\\ref{defi_initialdatabianchib} (or even of Bianchi type~VI$_{{-}1\/9}$ satisfying equation~\\eqref{eqn_nonexceptionalsigmaij}) is transformed into dynamical initial data $(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)(0)$ for the evolution equations~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} of expansion-normalised variables. In the process, an arbitrary number~$\\Omega_1(0)$ was chosen, but the resulting dynamical initial data is independent of the choice of~$\\Omega_1(0)$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Construction part II: From the expansion-normalised evolution equations to a solution in basic variables}\n\\label{constr_expnormtosolutionbasic}\n\n\tAt the end of the first part of the construction, Subsection~\\ref{constr_initialdatatoexpnorm}, we have obtained dynamical initial data $(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)(0)$. One can apply the evolution equations~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} for the expansion-normalised variables, which form a system of polynomial differential equations on a compact subset of~$\\maths{R}^5$, see Remark~\\ref{rema_statespacecompact}. Consequently, for given dynamical initial data, there exists a unique solution which is defined at all times $\\tau\\in({-}\\infty,\\infty)$. In this part of the construction, this solution is now translated back to a solution in basic variables, and then in the following subsection even further, to a spacetime solving the correct Einstein's equations and with correct initial data.\n\n\tIn order to retrieve basic variables at all times~$t$, one considers the evolution equation for the basic variable~$\\theta$, equation~\\eqref{eqn_evolutiontheta}. The initial value~$\\theta(0)$ is given from the geometric initial data, see equation~\\eqref{eqn_initialvaluesthetasigmaAB}. Consequently this evolution equation has a unique maximal solution~$\\theta(\\tau)$. As~$q\\in[{-}1,2]$ due to the relations in~\\eqref{eqn_constraintgeneralthree}, there are two cases to consider:\n\t\\begin{enumerate}\n\t\t\\item $q={-1}$ at some time. This corresponds to $\\gamma=0$ and $\\Omega=1$, which implies\n\t\t\\begin{equation}\n\t\t\t\\Sigma_+=\\tilde\\Sigma=\\Delta=\\tilde A=N_+=0.\n\t\t\\end{equation}\n\t\tThis point is an equilibrium point of the evolution equations ~\\eqref{eqns_evolutionbianchib}, from which we conclude that $q={-1}$ at all times and $\\theta=\\theta(0)$ constant.\n\t\t\\item $q>{-}1$ at all times.\n\t\tIn case of inflationary matter~$\\Omega>0$, $\\gamma\\in[0,2\/3)$, we have shown in Prop.~\\ref{prop_alphalimitsets_vacuum_inflat} that the only solution whose $\\alpha$-limit set contains the point $\\Sigma_+=\\tilde\\Sigma=\\Delta=\\tilde A=N_+=0$ is the constant solution, which is excluded in this case.\n\t\tConsequently, $1+q$ is bounded from below by a positive number for~$\\tau$ sufficiently negative. In vacuum and in the remaining matter cases~$\\Omega>0$, $\\gamma\\in[2\/3,2]$, equation~\\eqref{eqn_definitionqvacuum} and equation~\\eqref{eqn_definitionqmatter} even yield~$1+q>1$. With this, equation~\\eqref{eqn_evolutiontheta} implies that~$\\theta$ is monotone and, for sufficiently negative times, $\\absval\\theta$ can be estimated from above and below by exponential functions of~$\\tau$ with a negative exponent.\n\t\tDue to choice of sign we have made in the first part of the construction, $\\theta>0$ at all times and therefore~$\\theta$ monotone decaying.\n\t\tThus, $\\theta$ converges\n\t\tto some non-negative number~$\\theta_\\infty$\n\t\tas $\\tau\\rightarrow\\infty$, and diverges to~$\\infty$ as $\\tau\\rightarrow{-}\\infty$.\n\t\\end{enumerate}\n\tIn both cases, $\\theta$ is positive and defined for all~$\\tau\\in({-}\\infty,\\infty)$. One can therefore multiply the individual components of the solution $(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)(\\tau)$ with the corresponding power of~$\\theta$ such that equation ~\\eqref{eqn_expansionnormalisation} holds, and retrieves the remaining basic variables $(\\sigma_+ ,\\tilde{\\sigma},\\delta ,\\tilde{a},n_+)(\\tau)$. As was the case for~$\\theta$, these variables are defined at all times~$\\tau\\in({-}\\infty,\\infty)$.\n\n\tTo replace the expansion-normalised time~$\\tau$, we define a different time scale~$t$ via equation~\\eqref{eqn_evolutiontau} and the initial condition~$\\tau(t=0)=0$.\n\tDue to $\\theta>0$, this definition immediately yields that~$t$ and~$\\tau$ have the same time orientation. In case $q={-}1$, the time~$t$ is simply a rescaling of~$\\tau$, and the basic variables are defined at all times $t\\in({-}\\infty,\\infty)$. In case~$q>{-}1$, our discussion in~ii) implies that time~$\\tau\\rightarrow{-}\\infty$ corresponds to diverging~$\\theta$ and a finite time~$t$, while~$\\tau\\rightarrow\\infty$ corresponds to~$\\theta\\rightarrow\\theta_\\infty\\ge0$\n\tand $t\\rightarrow\\infty$.\n\n\tIn total, we obtain a curve in basic variables $(\\theta ,\\sigma_+ ,\\tilde{\\sigma},\\delta ,\\tilde{a},n_+)(t)$. By our construction, they are related to the solution in expansion-normalised variables $(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)(\\tau)$ exactly as in Subsection~\\ref{subsect_appendixdeducenormalisedvariablesbianchib}. We have noted there,\n\tin Remark~\\ref{rema_equivalencebasicvarexpnormvar}, that the evolution equations\\ for the expansion-normalised variables and the evolution equation s for the basic variables are equivalent. Therefore, the construction carried out here yields the maximal solution to equations~\\eqref{eqns_basic_evolutionbianchib}--\\eqref{eqn_basic_evolutionmu}. The resulting maximal interval of existence is $(t_-,\\infty)$, with $t_->{-}\\infty$ apart from when $\\gamma=0$ and $\\Omega=1$, in which case $t_-={-}\\infty$.\n\n\n\n\n\\subsection{Construction part III: From a solution in basic variables to a spacetime}\n\\label{constr_solutionbasictomghd}\n\n\tHaving found a solution in basic variables $(\\theta ,\\sigma_+ ,\\tilde{\\sigma},\\delta ,\\tilde{a},n_+)(t)$ via the previous two parts of the construction, Subsections~\\ref{constr_initialdatatoexpnorm} and~\\ref{constr_expnormtosolutionbasic}, we now translate this into a spacetime with suitable Lorentzian metric which is consistent with the geometric initial data~$(G,{h},{k},\\mu_0)$ we started with.\n\n\tRecall how the construction started: In order to obtain initial data in basic variables from the geometric initial data, one introduced a frame~$\\tilde e_1,\\tilde e_2,\\tilde e_3$ and then had to choose an arbitrary initial value~$\\Omega_1(0)$, whose information was lost in the first algebraic manipulation. This corresponded to introducing variables which were explicitly invariant under the choice of frame, in the sense that they did no longer depend on the freedom of rotation in the $\\tilde e_2\\tilde e_3$-plane. In order to now recover from a maximal solution $(\\theta ,\\sigma_+ ,\\tilde{\\sigma},\\delta ,\\tilde{a},n_+)(t)$, $t\\in(t_-,\\infty)$, a spacetime frame with four-dimensional structure constants and then a spacetime metric, one needs to break this gauge invariance and reverse this process by choosing a suitable frame. This is done in several steps:\n\t\\begin{enumerate}\n\t\t\\item Given an arbitrary, sufficiently smooth~$\\Omega_1$ defined on the same~$t$-time interval~$(t_-,\\infty)$ as the solution in basic variables, one retrieves the variables $(\\sigma_{AB},\\theta,n_{AB},a_1)(t)$.\n\t\t\\item With this, one defines a mani\\-fold\\ and constructs a frame~$e_0,e_1,e_2,e_3$.\n\t\t\\item Then, one checks that this construction holds through time, independently of the choice of~$\\Omega_1$.\n\t\t\\item Once the frame is obtained, one defines this frame as or\\-tho\\-nor\\-mal, which uniquely defines the corresponding spacetime metric on the manifold~$(t_-,\\infty)\\times G$.\n\t\t\\item Finally, one checks that in the resulting spacetime Einstein's equations are satisfied and the correct initial data induced.\n\t\\end{enumerate}\n\n\n\tWe begin with the first step, which is achieved via constructing a solution in the variables~$(\\theta,\\sigma_+,\\tilde\\sigma_{AB},a_1,n_+,\\tilde n_{AB})$ satisfying equations~\\eqref{eqns_prebasic_evolutionbianchib}--\\eqref{eqn_prebasic_evolutionmu}. With this solution at our disposal, we can immediately deduce the variables $(\\sigma_{AB},\\theta,n_{AB},a_1)(t)$. It is the first part which is more intricate, as we have to be careful about notation. We start with a solution in basic variables, which for the sake of precision, we denote by\n\t\\begin{equation}\n\t\t(\\theta_{\\operatorname{basic}} \\,,\\,\\sigma_{+,{\\operatorname{basic}}} \\,,\\,\\tilde{\\sigma}_{\\operatorname{basic}}\\,,\\,\\delta_{\\operatorname{basic}} \\,,\\,\\tilde{a}_{\\operatorname{basic}}\\,,\\,n_{+,{\\operatorname{basic}}})(t)\\,,\n\t\\end{equation}\n\tas the solution satisfies the evolution equations for basic variables, equations~\\eqref{eqns_basic_evolutionbianchib}--\\eqref{eqn_basic_evolutionmu}.\n\n\tWe want to find a solution $(\\theta,\\sigma_+,\\tilde\\sigma_{AB},a_1,n_+,\\tilde n_{AB})$ to equations~\\eqref{eqns_prebasic_evolutionbianchib}--\\eqref{eqn_prebasic_evolutionmu}. Even though we do not know that the functions~$\\theta_{\\operatorname{basic}}$, $\\sigma_{+,{\\operatorname{basic}}}$ and~$n_{+,{\\operatorname{basic}}}$ evolve correctly, we carry them over, but keep the subscript as a reminder.\n\tThe variable~$a_1$ is supposed to satisfy $\\tilde a=9a_1^2$, see equation~\\eqref{eqn_basic_definitionpartthree}. As $\\tilde a=0$ is an invariant set due to the evolution equations~\\eqref{eqns_basic_evolutionbianchib}, the sign of~$a_1$ cannot change and is determined by the initial data. From the knowledge of~$\\tilde a_{\\operatorname{basic}}$ and the initial data, we can uniquely define a function~$a_{1,{\\operatorname{basic}}}$. Upon comparison with the evolution equation for~$\\tilde a$ in~\\eqref{eqns_basic_evolutionbianchib}, we find that~$a_{1,{\\operatorname{basic}}}$ satisfies the evolution equation for~$a_1$ in~\\eqref{eqns_prebasic_evolutionbianchib}, if one replaces~$\\sigma_+$ and~$\\theta$ by~$\\sigma_{+,{\\operatorname{basic}}}$ and~$\\theta_{\\operatorname{basic}}$.\n\n\tNow consider the evolution equations\\ of~$\\tilde \\sigma_{AB}$ and~$\\tilde n_{AB}$ in~\\eqref{eqns_prebasic_evolutionbianchib}. Together, they form a system of linear differential equations whose coefficients are the function~$\\Omega_1$ and---after adding the subscript~${}_{\\operatorname{basic}}$ at all necessary places---quantities whose time development is known from the solution in basic variables. For every sufficiently smooth function~$\\Omega_1$, there is a maximal solution~$(\\tilde n_{AB,{\\operatorname{basic}}},\\tilde\\sigma_{AB,{\\operatorname{basic}}})$ to this system.\n\tDue to linearity of the differential equations, the maximal solution is defined on the whole time interval on which the coefficients are defined. By construction, this is the same time interval as for the solution in basic variables.\n\n\tNext, we consider the expressions\n\t\\begin{align}\n\t\tF_{\\tilde\\sigma}\\coloneqq{}&\\tilde\\sigma_{\\operatorname{basic}}-\\frac32\\tilde\\sigma^{AB}_{\\operatorname{basic}}\\tilde\\sigma_{AB,{\\operatorname{basic}}},\\\\\n\t\tF_{\\tilde n}\\coloneqq{}&\\tilde n_{\\operatorname{basic}}-\\frac32\\tilde n^{AB}_{\\operatorname{basic}}\\tilde n_{AB,{\\operatorname{basic}}},\\\\\n\t\tF_\\delta\\coloneqq{}&\\delta_{\\operatorname{basic}}-\\frac32\\tilde\\sigma^{AB}_{\\operatorname{basic}}\\tilde n_{AB,{\\operatorname{basic}}},\\\\\n\t\tF_{\\operatorname{constraint}}\\coloneqq{}&3a_{1,{\\operatorname{basic}}}\\sigma_{+,{\\operatorname{basic}}}-\\frac32\\,^*\\tilde\\sigma^{AB}_{\\operatorname{basic}}\\tilde n_{AB,{\\operatorname{basic}}},\n\t\\end{align}\n\twith~$\\tilde n_{{\\operatorname{basic}}}$ defined by equation~\\eqref{eqn_basic_definitiontilden}.\n\tThese expressions should be compared to the constraint equation~\\eqref{eqn_prebasic_constraint} and equations~\\eqref{eqns_basic_definitionparttwo}. The derivative of ~$(F_{\\tilde\\sigma},F_{\\tilde n},F_\\delta,F_{\\operatorname{constraint}})$ is a homogeneous system of equations, and as all four functions vanish initially due to their components having been constructed from the same geometric initial data, we see that all four functions vanish identically. In particular, the constructed functions~$\\tilde n_{{\\operatorname{basic}},AB}$ and~$\\tilde\\sigma_{{\\operatorname{basic}},AB}$ are related to the basic variables~$\\tilde\\sigma_{\\operatorname{basic}}$, $\\tilde n_{\\operatorname{basic}}$ and~$\\delta_{\\operatorname{basic}}$ as in equations~\\eqref{eqns_basic_definitionparttwo}, and we can replace these expressions at every occurence. This in particular implies that the function~$\\mu_{\\operatorname{basic}}$, defined via equation~\\eqref{eqn_basic_mu} in basic variables, also satisfies equation~\\eqref{eqn_prebasic_mu}. Further, we conclude from the argumentation above that the constraint equation~\\eqref{eqn_prebasic_constraint} holds for the functions indexed~${}_{\\operatorname{basic}}$.\n\n\tIt remains to check that the variables~$\\theta_{\\operatorname{basic}}$, $\\sigma_{+,{\\operatorname{basic}}}$ and~$n_{+,{\\operatorname{basic}}}$ satisfy the evolution equations~\\eqref{eqns_prebasic_evolutionbianchib}. To this end, consider the initial data to the evolution equations~\\eqref{eqns_prebasic_evolutionbianchib}--\\eqref{eqn_prebasic_evolutionmu} which we, in the construction in Subsection~\\ref{constr_initialdatatoexpnorm}, obtained from geometric initial data and then subsequently used to obtain initial data to the evolution in basic variables. There is a unique solution with this initial data, and in order to distinguish its individual variables from the ones constructed above, we denote this solution by\n\t\\begin{equation}\n\t\t(\\theta_{\\operatorname{i.d.}} \\,,\\,\\sigma_{+,{\\operatorname{i.d.}}} \\,,\\,\\tilde{\\sigma}_{\\operatorname{i.d.}}\\,,\\,\\delta_{\\operatorname{i.d.}} \\,,\\,\\tilde{a}_{\\operatorname{i.d.}}\\,,\\,n_{+,{\\operatorname{i.d.}}})(t)\\,.\n\t\\end{equation}\n\tThe initial values of these variables coincide with the initial values of the ones constructed, and uniqueness of systems of ordinary differential equations yields that~$\\theta_{\\operatorname{i.d.}}=\\theta_{\\operatorname{basic}}$ at all times, and equivalently for all other variables.\n\tWe conclude that via this construction, we have obtained the unique solution to equations~\\eqref{eqns_prebasic_evolutionbianchib}--\\eqref{eqn_prebasic_evolutionmu} to the initial data coming from the geometric initial data.\n\n\tIn this transformation, the maximal interval of the solution~$(\\theta,\\sigma_+,\\tilde\\sigma_{AB},a_1,n_+,\\tilde n_{AB})$ cannot exceed that of the solution in basic variables, as otherwise this would yield an extension of the solution in basic variables, a contradiction. Consequently, the maximal interval of existence of the solution after transformation coincides with the interval~$(t_-,\\infty)$.\n\tWith this solution at hand, we use equations~\\eqref{eqns_prebasic_definitionparttwo} and uniquely retrieve the variables~$\\sigma_{AB}$ and~$n_{AB}$, defined on the same interval.\n\n\t\\smallskip\n\n\tIn the next step, we construct a four-dimensional mani\\-fold\\ with a frame whose structure constants have the correct form.\n\tTo this end, we define scalar functions~$\\gamma_{\\alpha\\beta}^\\delta(t)$ on the time interval~$(t_-,\\infty)$\n\tvia setting\n\t\\begin{equation}\n\t\\label{eqns_structureconstantlikeobjectsalltimes}\n\t\\begin{subaligned}\n\t\t\\gamma_{01}^1(t)\\coloneqq{}&\\sigma_A{}^A(t)-\\frac13\\theta(t),\\\\\n\t\t\\gamma_{0A}^B(t)\\coloneqq{}&{-}\\sigma_A{}^B(t)-\\frac13\\theta(t)\\delta_A^B+\\Omega_1(t)\\epsilon_A{}^B,\\\\\n\t\t\\gamma_{1A}^C(t)\\coloneqq{}&\\epsilon_{AB}n^{BC}(t)+a_1(t)\\delta_A^C,\n\t\\end{subaligned}\n\t\\end{equation}\n\tand setting all other~$\\gamma_{\\alpha\\beta}^\\delta$ to vanish identically. The form of these scalar functions coincides with the form of the structure constants in equation~\\eqref{eqns_commutators}.\n\tBy construction, these objects are consistent with the initial data:\n\tAt time~$t=0$, the spacelike ones~$\\gamma_{ij}^k(0)$ coincide with the Lie group\\ structure constants we chose in Subsection~\\ref{constr_initialdatatoexpnorm}, and the remaining ones~$\\gamma_{0i}^j(0)$ coincide with the structure constant-like object~$\\tilde\\gamma_{0i}^j(0)$ defined there.\n\tWe construct now a four-dimensional frame whose structure constants coincide with these~$\\gamma_{\\alpha\\beta}^\\delta(t)$ at all times.\n\tNote that it is at this point that the structure constant-like object~$\\tilde\\gamma_{0i}^j(0)$ can finally be interpreted geometrically.\n\n\tFrom the initial data, we have obtained an initial frame $\\tilde e_1,\\tilde e_2,\\tilde e_3$ on the Lie group~$G$. We consider the mani\\-fold~$(t_-,\\infty)\\times G$ and in this mani\\-fold\\ extend this frame globally, i.\\,e.\\  time-independently to every $\\{t\\}\\times G$.\n\tFurther, we choose $e_0=\\partial_t$ globally. This gives a four-dimensional frame~$e_0,\\tilde e_1,\\tilde e_2,\\tilde e_3$, but the relation between its structure constants and the solution~$(\\theta ,\\sigma_{AB},\\Omega_1,a_1,n_{AB})(t)$ does not necessarily fulfill equations~\\eqref{eqns_commutators} at times other than~$t=0$. The final frame $e_0,e_1,e_2,e_3$ which does satisfy these relations can be constructed as follows.\n\n\tWe wish the frame vectors~$e_2$,~$e_3$ to be tangent to the subalgebra~$\\ggg_2$ which we have chosen in the beginning of Subsection~\\ref{constr_initialdatatoexpnorm}, which implies that we need to construct time-dependent functions~$f_A^B(t)$ such that\n\t\\begin{equation}\n\t\te_A=f_A^B\\tilde e_B, \\qquad f_A^B(0)=\\delta_A^B.\n\t\\end{equation}\n\tAssuming the existence of such functions gives the following commutator relation:\n\t\\begin{equation}\n\t\t\\liebr{e_0}{e_A}=\\liebr{e_0}{f_A^B\\tilde e_B}=e_0(f_A^B)\\tilde e_B+f_A^B\\liebr{e_0}{\\tilde e_B}=e_0(f_A^B)\\tilde e_B,\n\t\\end{equation}\n\twhere we seek to find\n\t\\begin{equation}\n\t\t\\liebr{e_0}{e_A}=(-\\sigma_A{}^B-\\frac13\\theta\\delta_A^B+\\Omega_1\\epsilon_A{}^B)e_B=(-\\sigma_A{}^B-\\frac13\\theta\\delta_A^B+\\Omega_1\\epsilon_A{}^B)f_B^C\\tilde e_C.\n\t\\end{equation}\n\tConsequently, solving the system\n\t\\begin{equation}\n\t\te_0(f_A^C)=(-\\sigma_A{}^B-\\frac13\\theta\\delta_A^B+\\Omega_1\\epsilon_A{}^B)f_B^C,\\qquad f_A^C(0)=\\delta_A^C,\n\t\\end{equation}\n\tyields the final frame vectors~$e_2$ and~$e_3$. Note that in order to do so, we have to fix a function~$\\Omega_1(t)$.\n\n\tFor the missing spatial frame component~$e_1$, we make the ansatz\n\t\\begin{equation}\n\t\te_1=f_1^1\\tilde e_1+f_1^A\\tilde e_A,\\qquad f_1^1(0)=1,\\quad f_1^A(0)=0.\n\t\\end{equation}\n\tAfter a computation similar to the above, the resulting commutator reads\n\t\\begin{equation}\n\t\t\\liebr{e_0}{e_1}=\\liebr{e_0}{f_1^1\\tilde e_1+f_1^A\\tilde e_A}=e_0(f_1^1)\\tilde e_1+e_0(f_1^A)\\tilde e_A.\n\t\\end{equation}\n\tComparison with the desired commutator\n\t\\begin{equation}\n\t\t\\liebr{e_0}{e_1}=(\\sigma_A{}^A-\\frac13\\theta)e_1=(\\sigma_A{}^A-\\frac13\\theta)(f_1^1\\tilde e_1+f_1^A\\tilde e_A)\n\t\\end{equation}\n\tgives the following systems:\n\t\\begin{align}\n\t\te_0(f_1^1)={}&(\\sigma_A{}^A-\\frac13\\theta)f_1^1,& \\qquad f_1^1(0)={}&1,\\\\\n\t\te_0(f_1^A)={}&(\\sigma_A{}^A-\\frac13\\theta)f_1^A,& \\qquad f_1^A(0)={}&0.\n\t\\end{align}\n\tIn particular, $f_1^A(t)=0$ at all times.\n\tThe solution to this system of equations is independent of~$\\Omega_1$, and yields the final frame vector~$e_1$.\n\n\tHaving found functions~$f_A^B,f_1^A,f_1^1$, we have constructed a frame which satisfies all but the third commutator relation in~\\eqref{eqns_commutators}. These remaining equations can be considered as constraint equations. From the construction of the final frame vectors, we obtain\n\t\\begin{align}\n\t\t\\liebr{e_1}{e_A}={}&\\liebr{f_1^1\\tilde e_1+f_1^C\\tilde e_C}{f_A^B\\tilde e_B}=f_1^1f_A^B\\liebr{\\tilde e_1}{\\tilde e_B}+f_1^Cf_A^B\\liebr{\\tilde e_C}{\\tilde e_B}=f_1^1f_A^B\\liebr{\\tilde e_1}{\\tilde e_B}\\\\\n\t\t={}&f_1^1f_A^B\\gamma_{1B}^C(0)\\tilde e_C,\n\t\\end{align}\n\twhich has to coincide with\n\t\\begin{equation}\n\t\t\\gamma_{1A}^Be_B=\\gamma_{1A}^Bf_B^C\\tilde e_C,\n\t\\end{equation}\n\t$\\gamma_{1A}^B$ as defined in~\\eqref{eqns_structureconstantlikeobjectsalltimes},\n\tin order for our construction to by consistent.\n\tThus, we have to check whether\n\t\\begin{equation}\n\t\\label{eqn_constraintincommutatorform}\n\t\tLHS_A^C\\coloneqq f_1^1f_A^B\\gamma_{1B}^C(0)=\\gamma_{1A}^Bf_B^C \\eqqcolon RHS_A^C\n\t\\end{equation}\n\tholds. Initially, the left-hand side\\ and right-hand side\\ coincide, and in order to prove that the relation holds at all times, it is therefore enough to prove an evolution equation of the form\n\t\\begin{equation}\n\t\t\\partial_t((LHS-RHS)_A^C)=(LHS-RHS)_A^B \\cdot M_B^C,\n\t\\end{equation}\n\tfor some time-dependent functions~$M_B^C$.\n\nUsing the derivatives of the functions~$f_1^1$, $f_1^A$ and~$f_A^B$ which we determined above, we find that the derivative of the left-hand side\\ of~\\eqref{eqn_constraintincommutatorform} reads\n\\begin{equation}\n\\label{eqn_constructionmghdconstraint}\n\\begin{split}\n\t\\partial_t(LHS_A^C)={}&\n\t\t\\dot f_1^1f_A^B\\gamma_{1B}^C(0)+f_1^1\\dot f_A^B\\gamma_{1B}^C(0)\n\t\t=\\gamma_{01}^1f_1^1f_A^B\\gamma_{1B}^C(0)+f_1^1\\gamma_{0A}^Df_D^B\\gamma_{1B}^C(0)\\\\\n\t\t={}&\\gamma_{01}^1LHS_A^C+\\gamma_{0A}^DLHS_D^C.\n\\end{split}\n\\end{equation}\nFor the right-hand side, we find\n\\begin{equation}\n\t\\partial_t(RHS_A^C)={}\n\t\t\\dot\\gamma_{1A}^Bf_B^C+\\gamma_{1A}^B\\dot f_B^C\n\t\t=\\dot\\gamma_{1A}^Bf_B^C+\\gamma_{1A}^B\\gamma_{0B}^Df_D^C\n\t\t=(\\dot\\gamma_{1A}^B+\\gamma_{1A}^D\\gamma_{0D}^B)f_B^C.\n\\end{equation}\nThe time-derivative of the structure constants can be obtained by expressing~$\\gamma_{1A}^B$ in terms of the variables $(\\theta,\\sigma_+,\\tilde\\sigma_{AB},a_1,n_+,\\tilde n_{AB})$, applying their evolution equations ~\\eqref{eqns_prebasic_evolutionbianchib}--\\eqref{eqn_prebasic_evolutionmu}, and converting the results back to structure constants. This yields\n\\begin{equation}\n\t\\dot\\gamma_{1A}^B+\\gamma_{1A}^D\\gamma_{0D}^B=\\gamma_{01}^1\\gamma_{1A}^B+\\gamma_{0A}^D\\gamma_{1D}^B,\n\\end{equation}\nand therefore\n\\begin{equation}\n\t\\partial_t(RHS_A^C)\n\t=(\\gamma_{01}^1\\gamma_{1A}^B+\\gamma_{0A}^D\\gamma_{1D}^B)f_B^C\n\t=\\gamma_{01}^1RHS_A^C+\\gamma_{0A}^DRHS_D^C,\n\\end{equation}\nwhich is the same structure we also found for the evolution of the left-hand side, equation ~\\eqref{eqn_constructionmghdconstraint}. We conclude that the constraint is preserved independently of the choice of~$\\Omega_1$. For all our purposes, we can assume $\\Omega_1\\equiv0$ at all times.\nThe above construction results in a time-dependent left-invariant frame $e_1(t),e_2(t),e_3(t)$ on the Lie group, which in combination with $e_0=\\partial_t$ yields a spacetime frame admitting the correct structure constants~\\eqref{eqns_commutators}.\nIn checking this consistency, we have completed the third step of our construction of the spacetime.\n\n\\smallskip\n\nThere is for every~$t$ a unique metric ${}^t\\overline g$ on $\\{t\\}\\times G$ such that the frame~$e_1(t),e_2(t),e_3(t)$ is orthonormal. With this, we define the spacetime\n\\begin{equation}\n\\label{eqn_constructedspacetime}\n\tI\\times G,\\qquad g=-dt^2+{}^t\\overline g,\n\\end{equation}\nwhere~$I$ is the existence interval of the basic variables and contains~$0$, i.\\,e.\\  $I=(t_-,\\infty)$, with ${-}\\infty<t_-<0$ apart from the case $\\gamma=0$, $\\Omega=1$, where $t_-={-}\\infty$.\n\n\\smallskip\n\nBy our construction above, the only non-vanishing structure constants of the four-dimensional frame~$(e_0,e_1,e_2,e_3)$ are those in~\\eqref{eqn_structureconstantsnonzero}, and as the variables composing the structure constants satisfy the evolution equations~\\eqref{eqns_prebasic_evolutionbianchib}--\\eqref{eqn_prebasic_evolutionmu}, we can deduce from Remark~\\ref{rema_equivalenceeinsteinperfectfluidprebasic} that the spacetime constructed in this way solves Einstein's field equations\\ for an or\\-tho\\-go\\-nal\\ perfect fluid with linear equation of state.\n\nOn the~$t=0$ timeslice, the frame vectors~$e_i$ and~$\\tilde e_i$ coincide, $i=1,2,3$. The metric induced on this timeslice by the spacetime metric~\\eqref{eqn_constructedspacetime} therefore coincides with the initial metric~$h$.\nDue to our choice of metric, the vector field~$e_0$ is the timelike unit normal to every~$\\{t\\}\\times G$ timeslice.\nIt follows from the construction that the second fun\\-da\\-men\\-tal form\\ of~the timeslice~$\\{0\\}\\times G$ in~$(I\\times G,g)$ coincides with the initial two-tensor~$k$. We conclude that the constructed spacetime induces the correct initial data.\nThis completes the fifth step, and as a consequence the following definition is reasonable.\n\\begin{defi}\n\\label{defi_mghd}\n\tGiven initial data as in Def.~\\ref{defi_initialdatabianchib}, we call a spacetime~$(I\\times G,g,\\mu)$ constructed as in the three parts of the construction, Subsections~\\ref{constr_initialdatatoexpnorm}--\\ref{constr_solutionbasictomghd}, a Bianchi~B development of the data.\n\\end{defi}\nDue to the choice of frame~$\\tilde e_2,\\tilde e_3$ in the beginning of the construction, the spacetime constructed this way is not necessarily unique.\n\nIn order to transform to expansion-normalised variables in Subsection~\\ref{constr_initialdatatoexpnorm}, we had to exclude initial data with zero mean curvature~$\\theta$. We now prove that this corresponds to excluding Bianchi~B developments which are part of Minkowski spacetime.\n\\begin{lemm}\n\\label{lemm_excludeminkowski}\n\tConsider initial data as in Def.~\\ref{defi_initialdatabianchib}. Then either\n\tits universal covering space is initial data for the four-dimensional Minkowski spacetime, or the mean curvature~$\\theta=\\operatorname{tr}_{h}k$ is non-vanishing at all times.\n\\end{lemm}\nThe proof proceeds similar to that of~\\cite[Lemma~20.6]{ringstrom_cauchyproblem} for Bianchi class~A developments.\n\\begin{proof}\n\tAs in the first part of the construction, Subsection~\\ref{constr_initialdatatoexpnorm}, we choose a suitable frame and carry out the transformations into initial data in basic variables, i.\\,e.\\  initial data for the evolution equations~\\eqref{eqns_basic_evolutionbianchib}--\\eqref{eqn_basic_evolutionmu}. From the construction, we see that the mean curvature\\ coincides with the initial value for~$\\theta$. Using Lemma~\\ref{lemm_excludethetazerobasic}, we find that if the initial value~$\\theta(0)$ vanishes, then the solution in basic variables is the trivial solution.\n\n\tWith this trivial solution in basic variables at hand, we retrace the steps of the third part of the construction, Subsection~\\ref{constr_solutionbasictomghd}, and recover a spacetime with an or\\-tho\\-nor\\-mal\\ frame~$(e_0,e_1,e_2,e_3)$ such that the initial data is correctly induced. We have been able to choose~$\\Omega_1\\equiv0$ in this construction, and from this we conclude that all structure constants~$\\gamma_{\\alpha\\beta}^\\delta$ vanish identically.\n\n\tConsider now the Lie group~$G$ with structure constants~$\\gamma_{ij}^k=0$. The three-dimensional Christoffel symbols~$\\overline\\Gamma_{ij}^k$ vanish due to equation~\\eqref{eqn_christoffelinstructureconstants}, and we conclude that the three-dimensional Ricci curvature\n\t\\begin{equation}\n\t\t\\overline{R}_{ij}=\\sum_{l=1}^3\\scalprod{\\overline R(e_i,e_l)e_l}{e_j}\n\t\t=\\sum_{k,l=1}^3(\\Gamma_{ll}^k\\Gamma_{ik}^j-\\Gamma_{il}^k\\Gamma_{lk}^j-\\gamma_{il}^k\\Gamma_{kl}^j)\n\t\\end{equation}\n\tis zero identically.\n\tAs~$G$ is three-dimensional, this implies that the Rie\\-mann\\-ian\\ curvature\\ tensor vanishes as well. As the metric is left-invariant, it is complete, from which it follows that~$G$ with this metric is isometric to a quotient of Euclidean space, by virtue of having the same dimension, index and curvature, see~\\cite[Prop.~8.23]{oneill_semiriemgeomappl}.\n\n\tWe have explained in Remark~\\ref{rema_geometricmeaningthetasigmaAB} that the second fun\\-da\\-men\\-tal form\\ of the timeslice~$\\{t\\}\\times G$ can be computed from~$\\theta$ and the trace-free variable~$\\sigma_{ij}$, which satisfies~$\\sigma_{12}=\\sigma_{13}=0$. We can therefore conclude that the second fun\\-da\\-men\\-tal form\\ vanishes. This concludes the proof.\n\\end{proof}\n\n\n\\subsection{Properties of Bianchi~B developments of initial data}\n\n\\label{constr_propertiesdevelopmentinitialdata}\n\nWe now show that a development as in Def.~\\ref{defi_mghd} of given geometric initial data, i.\\,e.\\  one which we obtain through our construction in the previous three subsections, is in fact the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ of the geometric initial data. The proof of global hyberbolicity works identically as in the case of Bianchi class~A and has been carried out in~\\cite[p.~217]{ringstrom_cauchyproblem}.\nWe only state the proposition:\n\\begin{prop}\n\\label{prop_developmentglobhyperbolic}\n\tLet $(G,{h},{k},\\mu_0)$ be initial data as in Def.~\\ref{defi_initialdatabianchib}, and~$(I\\times G,g,\\mu)$ a Bianchi~B development of the data.\n\tThen $\\{t\\}\\times G$ is a Cauchy hypersurface for every $t\\in I$.\n\\end{prop}\nFor the proof of maximality, we wish to apply the following proposition, which is part of~\\cite[Prop.~18.16]{ringstrom_cauchyproblem}:\n\\begin{prop}\n\\label{prop_mghdnecessaryproperties}\n\tLet $(M,g)$ be a connected and time oriented Lorentzian mani\\-fold\\ and assume that the following holds:\n\t\\begin{itemize}\n\t\t\\item $(M,g)$ is future geodesically complete and there are real numbers~$\\kappa_j$ such that  $\\kappa_j\\rightarrow\\infty$ as $j\\rightarrow\\infty$ and smooth spacelike Cauchy hypersurfaces~$\\Sigma_j$ in $(M,g)$ with constant mean curvature~$\\kappa_j$.\n\t\\end{itemize}\n\tAssume $(\\tilde M,\\tilde g)$ to be a time oriented and connected Lorentzian mani\\-fold\\ satisfying the timelike convergence condition: $Ric(v,v)\\ge0$ for all timelike vectors $v\\in TM$. Assume further $i:M\\rightarrow\\tilde M$ to be a smooth embedding such that  $i(M)$ is an open set, and that there is a Cauchy hypersurface~$S$ in $(M,g)$ such that  $i(S)$ is a Cauchy hypersurface in $(\\tilde M,\\tilde g)$. Then $i(M)=\\tilde M$.\n\\end{prop}\nWe further adapt~\\cite[Prop.~20.10]{ringstrom_cauchyproblem} to our setting:\n\\begin{lemm}\n\tConsider a Bianchi~B development~$(M=I\\times G,g,\\mu)$ of initial data as in Def.~\\ref{defi_initialdatabianchib}. Let $c:(s_-,s_+)\\rightarrow M$ be a future directed inextendible causal geodesic, and\n\t\\begin{equation}\n\t\tf_0(s)=\\scalprod{c'(s)}{e_0|_{c(s)}}.\n\t\\end{equation}\n\tThen either $\\theta\\equiv0$ and~$f_0$ is a constant, or\n\t\\begin{equation}\n\t\t\\frac{d}{ds}(f_0\\theta)\\ge C\\theta^2f_0^2,\n\t\\end{equation}\n\twith $C\\coloneqq\\min(1\/2-\\sqrt2\/3,\\gamma\/2)$ if~$\\mu>0$, and~$C\\coloneqq(2-\\sqrt2)\/3 $ if~$\\mu=0$.\n\\end{lemm}\n\\begin{proof}\n\tBy construction of the Bianchi~B development in Subsections~\\ref{constr_initialdatatoexpnorm}--\\ref{constr_solutionbasictomghd}, we know that the trace-free part~$\\sigma_{ij}$ of the second fun\\-da\\-men\\-tal form~$\\theta_{ij}$ of the submani\\-fold~$\\{t\\}\\times M$ satisfies\n\t\\begin{equation}\n\t\t\\sigma_{12}=\\sigma_{21}=0=\\sigma_{13}=\\sigma_{31}.\n\t\\end{equation}\n\n\tFor a fixed time~$s$, we apply a rotation in the~$e_2e_3$-plane to diagonalise~$\\sigma_{ij}$ (and simultaneously~$\\theta_{ij}$), then define\n\t\\begin{equation}\n\t\tf_i(s)=\\scalprod{c'(s)}{e_i|_{c(s)}},\n\t\\end{equation}\n\tfor~$i=1,2,3$.\n\n\tAs $f_0(s)=\\scalprod{c'(s)}{e_0|_{c(s)}}$ and~$c$ is a geodesic, we find\n\t\\begin{equation}\n\t\t\\frac{df_0}{ds}=\\scalprod{c'(s)}{\\nabla_{c'(s)}e_0}\n\t\t\t=\\sum_{i,j}f_i(s)f_j(s)\\scalprod{e_i}{\\nabla_{e_j}e_0}\n\t\t\t=\\sum_{i}f_i^2\\theta_{ii}\n\t\t\t=\\frac13\\theta\\sum_{i}f_i^2+\\sum_{i}f_i^2\\sigma_{ii},\n\t\\end{equation}\n\twhere we applied\n\t$\\nabla_{e_0}e_0=0$.\n\tThe diagonalisation we applied does not affect the basic variables~$(\\theta,\\sigma_+,\\tilde\\sigma,\\delta,\\tilde a,n_+)$, as they are invariant under rotation in the~$e_2e_3$-plane. We can therefore apply Lemma~\\ref{lemm_excludethetazerobasic} to see that~$\\theta=0$ at some time implies that\n\t\\begin{align}\n\t\t0=\\tilde\\sigma=\\frac34(\\sigma_{22}-\\sigma_{33})^2, \\qquad\n\t\t0=\\sigma_+=\\frac32(\\sigma_{22}+\\sigma_{33}),\n\t\\end{align}\n\tat all times. In particular $\\sigma_{ij}\\equiv0$, and from this we see that~$f_0$ is constant.\n\n\tThe Raychaudhuri equation\\ for an or\\-tho\\-go\\-nal\\ perfect fluid with linear equation\\ of state~\\eqref{eqn_lineareqnofstate} is\n\t\\begin{equation}\n\t\\label{eqn_raychaudhuri}\n\t\t\\dot\\theta+\\theta^{ij}\\theta_{ij}={-}\\frac\\mu2(3\\gamma-2),\n\t\\end{equation}\n\tand coincides with the evolution equation for~$\\theta$ in~\\eqref{eqns_prebasic_evolutionbianchib}. Here, the dot~$\\dot{ }$ denotes differentiation with respect to\\ the time parameter~$t$ of the foliation~$M=I\\times G$, in contrast to~$'$ denoting differentiation with respect to\\ the parameter~$s$ of the geodesic~$c$.\n\tThe Raychaudhuri equation\\ follows from the~$00$-component of the Ricci curvature\\ for a perfect fluid~\\eqref{eqn_stressenergyperfectfluid} with linear equation of state~\\eqref{eqn_lineareqnofstate}\n\tand holds independently of the Bianchi class, see~\\cite[Eq.~(25.10)]{ringstrom_topologyfuturestabilityuniverse}. The general expression in terms of an or\\-tho\\-nor\\-mal\\ tetrad for a perfect fluid can be found in~\\cite[Eq.~(82)]{maccallum_cosmmodelsgeometricpointofview}.\n\n\tCombining the Raychaudhuri equation ~\\eqref{eqn_raychaudhuri} with the fact that the surfaces have constant mean curvature, it follows that\n\t\\begin{equation}\n\t\\label{eqn_mghdprooffunctionfzero}\n\t\t\\frac{d}{ds}(f_0\\theta)=\\frac13\\theta^2\\sum_{i}f_i^2+\\sum_{i}\\theta\n\t\tf_i^2\\sigma_{ii}+\\frac13f_0^2\\theta^2+f_0^2\\sum_i\\sigma_{ii}^2+\\frac12f_0^2(3\\gamma-2)\\mu.\n\t\\end{equation}\n\tWe observe that apart from the last term including the energy density~$\\mu$ this coincides with the expression for vacuum Bianchi~A developments used in the proof of~\\cite[Lemma~20.10]{ringstrom_cauchyproblem}. In particular, in case this last term is non-negative, the identical argument works, yielding that\n\t\\begin{equation}\n\t\t\\frac{d}{ds}(f_0\\theta)\\ge\\frac{2-\\sqrt2}{3}\\theta^2f_0^2.\n\t\\end{equation}\n\tNote that $(2-\\sqrt2)\/3$ is in particular larger than the constant~$C$ we aim to achieve in case~$\\mu>0$.\n\n\tTo conclude the proof, we have to deal with the case that the last term in equation~\\eqref{eqn_mghdprooffunctionfzero} is negative, i.\\,e.\\  the case $\\mu>0$ and $0\\le\\gamma<\\frac23$.\n\tWithout loss of generality one can assume that $\\absval{\\sigma_{11}}\\le\\absval{\\sigma_{22}}\\le\\absval{\\sigma_{33}}$, and as~$\\sigma_{ij}$ is trace-free, one can estimate\n\t\\begin{equation}\n\t\t3\\sigma_{33}^2=2\\sigma_{33}^2+(\\sigma_{11}+\\sigma_{22})^2\\le 2\\sum_i\\sigma_{ii}^2.\n\t\\end{equation}\n\tConsequently,\n\t\\begin{equation}\n\t\t\\absval{\\sum_{i}f_i^2\\sigma_{ii}}\\le(\\frac23)^{1\/2}(\\sum_i\\sigma_{ii}^2)^{1\/2}\\sum_if_i^2.\n\t\\end{equation}\n\n\tWe compute that\n\t\\begin{equation}\n\t\\label{eqns_auxiliaryexpressionsumsigmaij}\n\t\t\\begin{subaligned}\n\t\t\t\\sigma_+^2+\\tilde\\sigma\n\t\t\t={}&\\frac94(\\sigma_{11})^2+\\frac34(\\sigma_{22}-\\sigma_{33})^2+\\frac32(\\sigma_{23})^2+\\frac32(\\sigma_{32})^2\\\\\n\t\t\t={}&\\frac32(\\sigma_{11})^2+\\frac34(\\sigma_{22}+\\sigma_{33})^2+\\frac34(\\sigma_{22}-\\sigma_{33})^2+\\frac32(\\sigma_{23})^2+\\frac32(\\sigma_{32})^2\\\\\n\t\t\t={}&\\frac32\\sigma^{ij}\\sigma_{ij},\n\t\t\\end{subaligned}\n\t\\end{equation}\n\tand as~$\\sigma_{ij}$ is diagonal by assumption, we can use the definition of~$\\mu$ in terms of the basic variables, equation~\\eqref{eqn_basic_mu} to find\n\t\\begin{equation}\n\t\t\\frac13\\theta^2-\\mu\n\t\t\t=\\frac13(\\sigma_+^2+\\tilde\\sigma+\\tilde a+\\tilde n)\n\t\t\t\\ge\\frac13(\\sigma_+^2+\\tilde\\sigma)\n\t\t\t=\\frac12\\sum_i\\sigma_{ii}^2.\n\t\\end{equation}\n\tWe conclude that the derivative of $f_0\\theta$ can be estimated by\n\t\\begin{equation}\n\t\\label{eqn_initialestimatef0thetawithoutgamma}\n\t\t\\frac{d}{ds}(f_0\\theta)\\ge\\frac13\\theta^2\\sum_{i}f_i^2+\\sum_{i}\\theta\n\t\tf_i^2\\sigma_{ii}+\\frac32f_0^2\\sum_i\\sigma_{ii}^2+\\frac32f_0^2\\gamma\\mu.\n\t\\end{equation}\n\tOn the other hand, we can use the fact that $3\\gamma-2<0$ to find\n\t\t\\begin{align}\n\t\t\\frac13\\theta^2+\\frac12(3\\gamma-2)\\mu ={}&\\frac13\\theta^2+\\frac12(3\\gamma-2)\\frac13(\\theta^2-\\sigma_+^2-\\tilde\\sigma-\\tilde a-\\tilde n)\\\\\n\t\t\\ge{}&\\frac12\\gamma\\theta^2+(\\frac13-\\frac12\\gamma)(\\sigma_+^2+\\tilde\\sigma)\\\\\n\t\t={}&\\frac12\\gamma\\theta^2+(\\frac12-\\frac34\\gamma)\\sum_i\\sigma_{ii}^2\n\t\\end{align}\n\tand start out with the estimate\n\t\\begin{equation}\n\t\\label{eqn_initialestimatef0thetawithgamma}\n\t\t\\frac{d}{ds}(f_0\\theta)\\ge\\frac13\\theta^2\\sum_{i}f_i^2+\\sum_{i}\\theta\n\t\tf_i^2\\sigma_{ii}+\\frac12\\gamma\\theta^2f_0^2+(\\frac32-\\frac34\\gamma)f_0^2\\sum_i\\sigma_{ii}^2.\n\t\\end{equation}\n\tWe now divide into the three cases $\\frac13\\theta^2\\le\\sum_i\\sigma_{ii}^2$, $\\frac16\\theta^2\\le\\sum_i\\sigma_{ii}^2\\le\\frac13\\theta^2$ and $\\sum_i\\sigma_{ii}^2\\le\\frac16\\theta^2$. In the first two cases, inequality~\\eqref{eqn_initialestimatef0thetawithoutgamma} and causality of the curve yields the desired estimate, while in the third case we apply inequality~\\eqref{eqn_initialestimatef0thetawithgamma}.\n\\end{proof}\n\\begin{lemm}\n\\label{lemm_bianchibdevelopmentproperties}\n\tConsider a Bianchi~B development\n\tof initial data as in Def.~\\ref{defi_initialdatabianchib}.\n\tIf $\\mu=0$ or~$\\mu>0$, $\\gamma>0$, then all inextendible causal geodesics are future complete and past incomplete.\n\tIf instead $\\mu>0$, $\\gamma=0$, then all inextendible causal geodesics are future complete.\n\\end{lemm}\n\\begin{proof}\n\tIn case $\\mu=0$ or~$\\mu>0$, $\\gamma>0$, the previous lemma implies that\n\t\\begin{equation}\n\t\t\\frac{d}{ds}(f_0\\theta)\\ge C\\theta^2f_0^2\n\t\\end{equation}\n\tfor a positive constant~$C$. Additionally, from equation~\\eqref{eqn_basic_mu} one finds\n\t\\begin{equation}\n\t\t\\theta^2=3\\mu+\\sigma_+^2+\\tilde\\sigma+\\tilde a+\\tilde n\\ge\\sigma_+^2+\\tilde\\sigma,\n\t\\end{equation}\n\tand in combination with the computation carried out in~\\eqref{eqns_auxiliaryexpressionsumsigmaij} concludes\n\t\\begin{equation}\n\t\t\\sigma^{ij}\\sigma_{ij}\\le\\frac23\\theta^2,\n\t\\end{equation}\n\twhich is identical to~\\cite[Eq.~(20.19)]{ringstrom_cauchyproblem}. These two inequalities are the only ingredients necessary for the proof of~\\cite[Lemma~20.12]{ringstrom_cauchyproblem}, which is the Bianchi class~A variant of the present lemma.\n\n\tIn case $\\mu>0$, $\\gamma=0$, the constant in the previous lemma is zero. Consequently, $f_0\\theta$ is non-decreasing, which is enough to conclude future completeness of inextendible causal geodesics, retracing the arguments in the proof of~\\cite[Lemma~20.12]{ringstrom_cauchyproblem}.\n\\end{proof}\n\\begin{coro}\n\\label{coro_mghdforgammapositive}\n\tConsider a Bianchi~B development\n\tof initial data as in Def.~\\ref{defi_initialdatabianchib}.\n\tIf $\\mu=0$ or~$\\mu>0$, $\\gamma>0$, then this development is isometric to the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ of the initial data.\n\\end{coro}\n\\begin{proof}\n\tThe case of vacuum~$\\mu=0$ is equivalent to~$Ric=0$, thus the timelike convergence condition is satisfied. The convergence properties of the mean curvature ~$\\theta$ have been discussed in Subsection~\\ref{constr_expnormtosolutionbasic}. With this, the statement is a direct consequence of Lemma~\\ref{lemm_bianchibdevelopmentproperties}, Prop.~\\ref{prop_developmentglobhyperbolic}, Prop.~\\ref{prop_mghdnecessaryproperties} and the definition of the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment.\n\n\tIn the matter case $\\mu>0$, $\\gamma>0$, Lemma~\\ref{lemm_bianchibdevelopmentproperties} yields future completeness of inextendible cau\\-sal geodesics. If the development was extendible towards the past as a globally hyperbolic mani\\-fold, then it would in particular be extendible as a semi-Rie\\-mann\\-ian\\ mani\\-fold.\n\tIn Lemma~\\ref{lemm_curvblowupricci}, we show that the Ricci curvature\\ contraction $R_{\\alpha\\beta}R^{\\alpha\\beta}$ is unbounded as $\\tau\\rightarrow{-}\\infty$, using the expansion-normalised variables. The statement translates into the setting of the Bianchi~B development~$I\\times G$, $I=(t_-,\\infty)$, when we translate the condition on the time~$\\tau$ into the condition that $t\\rightarrow t_-$. By an argument similar to~\\cite[Lemma~18.18]{ringstrom_cauchyproblem}, this contradicts extendibility.\n\\end{proof}\nThe only case not covered by this last statement is when~$\\mu>0$, $\\gamma=0$, in which case one can interpret the stress-energy tensor~$T_{\\alpha\\beta}$ as that of vacuum with a positive cosmological constant $\\Lambda>0$.\n\nWe have to distinguish between the two situations which we encountered in Subsection~\\ref{constr_expnormtosolutionbasic}: $q\\equiv{-}1$ and $q>{-}1$. In the first case, $\\theta=\\theta_0$ is constant and all expansion-normalised variables vanish at all times. One concludes from the construction of Bianchi~B development that the commutators have the form\n\\begin{equation}\n\t\\liebr{e_0}{e_i}={-}\\frac13\\theta_0 e_i,\\qquad\n\t\\liebr{e_i}{e_j}=0.\n\\end{equation}\nFrom the second relation, we can conclude that a left-invariant metric on the universal covering group of the Lie group ~$G$ is isometric to Euclidean three-space $(\\maths{R}^3,\\delta)$, while the first relation implies\n\\begin{equation}\n\t{}^tk=\\frac{\\theta_0}{3}\\cdot{}^tg,\n\\end{equation}\nwith~${}^tg$ and~${}^tk$ the metric and second fundamental form of $\\{t\\}\\times G$ with respect to\\ the development.\nCombined, the second fundamental form of $\\{t\\}\\times G$ equals\n\\begin{equation}\n\t{}^tk=\\frac{\\theta_0}{3}\\delta.\n\\end{equation}\nOne compares this with the spacetime\n\\begin{equation}\n\t(\\maths{R}\\times\\maths{R}^3\\,,\\,g={-}dt^2+e^{2Ht}\\delta),\n\\end{equation}\nwith $H=\\sqrt{\\Lambda\/3}$, which can be isometrically embedded in the well-known de~Sitter\\ spacetime, see~\\cite[eq.~(52)]{schroedinger_expandinguniverses}. Upon comparison, one realises that the~$t=const.$ slices in this spacetime have the same induced metric and second fun\\-da\\-men\\-tal form\\ as we computed for our case~$\\gamma=0$, $q\\equiv{-}1$. We conclude that\na Bianchi~B development, which by Prop.~\\ref{prop_developmentglobhyperbolic} is a globally hyperbolic development of the given initial data, corresponds to a development of a hypersurface in a quotient of the de~Sitter\\ spacetime.\nThis concludes the case~$\\gamma=0$, $q={-}1$.\n\nIn case~$q>{-}1$, we can show that the development we constructed cannot be isometrically embedded into a globally hyperbolic spacetime which extends the development to the past.\n\\begin{prop}\n\\label{prop_cosmconstpastinextend}\n\tConsider a Bianchi~B development~$(M=I\\times G,g,\\mu)$\n\tof initial data as in Def.~\\ref{defi_initialdatabianchib}.\n\tIf $\\mu>0$, $\\gamma=0$, and the mean curvature\\ of~$\\{t\\}\\times G$ covers the interval~$(\\theta_\\infty,\\infty)$, then the development is past inextendible as a globally hyperbolic spacetime.\n\\end{prop}\n\\begin{proof}\n\tSuppose that the opposite holds, i.\\,e.\\  there exists a proper globally hyperbolic extension~$(\\tilde M,\\tilde g,\\tilde\\mu)$ of the Bianchi~B development~$(M=I\\times G,g,\\mu)$, $I=(t_-,\\infty)$, to the past. Then there exists a point~$p$ in the interior of~$\\tilde M\\setminus M$.\n\tAs the timeslice~$\\{0\\}\\times G$ is a Cauchy hypersurface by Prop.~\\ref{prop_developmentglobhyperbolic}, the set\n\t\\begin{equation}\n\t\tJ^+(p)\\cap J^-(\\{0\\}\\times G)\n\t\\end{equation}\n\tis compact by~\\cite[Lemma~40, p.~423]{oneill_semiriemgeomappl}, where~$J^\\pm$ denote the causal future and past. Consequently, there exists a unit speed causal curve\n\t\\begin{equation}\n\t\tc:[s_-,s_+]\\rightarrow\\tilde M\n\t\\end{equation}\n\tmaximising the length between the point~$p$ and the timeslice~$\\{0\\}\\times G$. Denote by~$q$ the endpoint of this curve on~$\\{0\\}\\times G$.\n\n\tBy construction, the curve also maximises the length between~$q$ and every earlier timeslice~$\\{t\\}\\times G$, $t\\in(t_-,0)$ and therefore the causal vector~$c'$ is or\\-tho\\-go\\-nal\\ to every one of these hypersurfaces. We conclude that in~$M$, the curve~$c$ has the form\n\t\\begin{equation}\n\t\tc:s\\mapsto (t(s),q).\n\t\\end{equation}\n\n\tAssume that~$s_0\\in(s_-,s_+)$ is the curve parameter such that~$c(s_0)\\in\n\t\\partial M\\subset\\tilde M$ and choose a sequence of times~$s_n\\in(s_0,s_+)$ such that~$s_n\\searrow s_0$. Set~$t_n$ to be the time satisfying~$c(s_n)\\in\\{t_n\\}\\times M$, for every~$n\\in\\maths{N}$.\n\n\tLet now~$E_i$, $i=1,2,3$, be vector fields which are parallelly propagated along~$c$ and such that they form an or\\-tho\\-nor\\-mal\\ basis of~$T_{c(0)}(\\{0\\}\\times G)$. Consequently, they are or\\-tho\\-nor\\-mal\\ along the whole curve and span the or\\-tho\\-go\\-nal\\ complement to~$c'(s)$, for every $s\\in[s_-,s_+]$. Construct a local spacelike hypersurface~$\\Sigma_{c(s_-)}$ through the point~$c(s_-)$ which is tangent to this frame, for example by using the exponential map. Consider then a smooth function~$\\chi:[s_-,s_+]\\rightarrow[0,1]$ with $\\chi(s_-)=0$ and $\\chi(s)=1$ for all~$s\\ge s_0$. According to~\\cite[Lemma~18.4]{ringstrom_cauchyproblem}, for every~$i=1,2,3$ and every~$n\\in\\maths{N}$ there is a piecewise smooth variation~$\\boldsymbol x$ of~$c|_{[s_-,s_n]}$, such that the variation vector field of~$\\boldsymbol x$ is~$\\chi E_i$, i.\\,e.\\  there exists a continuous and piecewise smooth curve~$\\boldsymbol x:[s_-,s_n]\\times({-}\\delta,\\delta)\\rightarrow\\tilde M$ such that\n\t\\begin{itemize}\n\t\t\\item $\\boldsymbol x(s,0)=c(s)$ for all $s\\in[s_-,s_n]$,\n\t\t\\item $\\boldsymbol x_v(s,0)=\\chi(s) E_i$ for all $s\\in[s_-,s_n]$,\n\t\t\\item $\\boldsymbol x(s_-,\\cdot)$ is contained in~$\\Sigma_{c(s_-)}$ and $\\boldsymbol x(s_n,\\cdot)$ is contained in~$\\{t_n\\}\\times G$.\n\t\\end{itemize}\n\tNote that the subscript~$v$ denotes derivation with respect to the second component.\n\n\tWe see that~$\\boldsymbol x$ is a variation of~$c$ with fixed endpoint~$\\boldsymbol x(s_-,\\cdot)=c(s_-)=q$. By assumption, the curve~$c$ maximises the length between~$q$ and the hypersurface~$\\{t_n\\}\\times G$. We can therefore apply~\\cite[Lemma~18.7]{ringstrom_cauchyproblem} to find\n\t\\begin{equation}\n\t\t0\\ge\\int_{s_-}^{s_n}\\scalprod{R_{c'\\,\\chi\\cdot E_i}\\chi\\cdot E_i}{c'}ds+k_{\\{t_n\\}\\times G}[E_i(c(s_n)),E_i(c(s_n))]-k_\\Sigma[0,0],\n\t\\end{equation}\n\twhere we have used that~$E_i'=0$ and the properties of the function~$\\chi$. Here~$k_{\\{t_n\\}\\times G}$ and~$k_\\Sigma$ denote the second fun\\-da\\-men\\-tal form\\ of~$\\{t_n\\}\\times G$ and~$\\Sigma$, respectively.\n\tSumming over~$i$, we obtain the inequality\n\t\\begin{equation}\n\t\\label{eqn_estimatepastinextendibility}\n\t\t\\int_{s_-}^{s_n}\\chi^2(s)Ric(c'(s),c'(s))ds+\\theta(t_n)\\le0,\n\t\\end{equation}\n\twhere, as in the construction of a Bianchi~B development, $\\theta(t)$ denotes the mean curvature\\ of the timeslice~$\\{t\\}\\times M$.\n\n\tWe are interested in estimate~\\eqref{eqn_estimatepastinextendibility} in the limit~$n\\rightarrow\\infty$.\n\tDue to the assumption on extendibility, the Ricci curvature\\ is bounded along~$c$, consequently\n\t\\begin{equation}\n\t\t\\int_{s_-}^{s_+}\\absval{Ric(c'(s),c'(s))ds}<\\infty.\n\t\\end{equation}\n\tHowever, the mean curvature~$\\theta(t=t_n)$ diverges to~$\\infty$ as~$n\\rightarrow\\infty$ by construction of the development, which is a contradiction.\n\\end{proof}\nWe can conclude from the previous proof that in a Bianchi~B development with~$\\mu>0$, $\\gamma=0$ and~$\\theta$ covering~$(\\theta_\\infty,\\infty)$, a causal curve connecting the hypersurfaces~$\\{t_1\\}\\times G$ and~$\\{t_2\\}\\times G$ has length at most~$\\absval{t_1-t_2}$. In particular, causal curves have finite length towards the past. This stands in contrast to the de~Sitter\\ spacetime, where causal curves have infinite length towards the past.\n\nWith Prop.~\\ref{prop_cosmconstpastinextend} at hand, we can now prove that\nin the remaining case~$\\mu>0$ with $\\gamma=0$ but~$q>{-}1$,\nthe Bianchi~B development is isometric to the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment.\n\\begin{coro}\n\\label{coro_mghdforgammazero}\n\tConsider a Bianchi~B development\n\tof initial data as in Def.~\\ref{defi_initialdatabianchib}.\n\tIf~$\\mu>0$, $\\gamma=0$, and the development does not correspond to a development of a hypersurface in a quotient of the de~Sitter\\ spacetime, it\n\tis isometric to the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ of the initial data.\n\\end{coro}\n\\begin{proof}\n\tRecall the construction we carried out in Subsections~\\ref{constr_initialdatatoexpnorm}--\\ref{constr_solutionbasictomghd}. We argued in the second part of the construction that there are two different cases which can occur: $q\\equiv{-}1$ and~$q>{-}1$. We showed above that the first case results into a development which is part of the de~Sitter\\ spacetime, and is hence excluded by assumption.\n\tAs $q>{-}1$, the mean curvature~$\\theta(t)$ covers the interval~$(\\theta_\\infty,\\infty)$, see the end of Subsection~\\ref{constr_expnormtosolutionbasic}. As a result, we can apply Prop.~\\ref{prop_cosmconstpastinextend} to see that the Bianchi~B development cannot be extended to the past as a globally hyperbolic development.\n\tWe further know from Lemma~\\ref{lemm_bianchibdevelopmentproperties} that every timelike geodesic is future complete.\n\tThis concludes the proof.\n\\end{proof}\nWe collect the statements we made about Bianchi~B developments in the different cases in the following proposition.\n\\begin{prop}\n\\label{prop_mghdincompletedirections}\n\tGiven or\\-tho\\-go\\-nal\\ perfect fluid Bianchi class~B initial data as in Def.~\\ref{defi_initialdatabianchib}, consider the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment~$(\\tilde M,\\tilde g)$ to this data. In case the universal covering of this spacetime is not a part of the Minkowski or the de~Sitter\\ spacetime, it is isometric to every Bianchi~B development~$(M=I\\times G,g,\\mu)$, $I=(t_-,t_+)$, as in Def.~\\ref{defi_mghd}. The mean curvature~$\\theta$ of $\\{t\\}\\times G$ in~$M$ is positive, monotone, and tends to~$\\infty$ at~$t_-$,\n\twhere~${-}\\infty<t_-<0$ and~$t_+=\\infty$.\n\n\tConsider a causal geodesic in~$\\tilde M$. The behaviour of any geometric quantity in the incomplete direction of this geodesic is the same as the behaviour of this geometric quantity on~$\\{t\\}\\times G$ as~$t\\rightarrow t_-$, while the complete direction corresponds to~$t\\rightarrow t_+$.\n\\end{prop}\n\\begin{proof}\n\tIf we exclude spacetimes isometric to (quotients of)\n\tthe Minkowski spacetime, we can carry out the construction from Subsections~\\ref{constr_initialdatatoexpnorm}--\\ref{constr_solutionbasictomghd}, see also Lemma~\\ref{lemm_excludeminkowski}. We have seen in Corollaries~\\ref{coro_mghdforgammapositive} and~\\ref{coro_mghdforgammazero} that the resulting Bianchi~B development is isometric to the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ of the data, if in case~$\\gamma=0$ we exclude de~Sitter\\ spacetime.\n\n\tThe properties of $t_-,t_+$ and the convergence behaviour of~$\\theta$ are a consequence of the construction, see also at the end of Subsection~\\ref{constr_expnormtosolutionbasic}. The one case with~$t_-={-}\\infty$ mentioned there corresponds to~$\\gamma=0$ and~$\\Omega=1$, or in other words~$q\\equiv{-}1$, which implies de~Sitter\\ spacetime and is therefore excluded by assumption.\n\n\tConsider an inextendible causal geodesic in~$\\tilde M$. Using the isometry between~$\\tilde M$ and the Bianchi~B development~$M$, we obtain an inextendible causal geodesic in~$M$. All geometric quantities are invariant under isometries, and in addition constant on every timeslice~$\\{t\\}\\times G$ due to the metric being invariant under the action of the Lie group~$G$.\n\n\tAs in the proof of Corollary~\\ref{coro_mghdforgammazero}, the case~$q={-}1$ is excluded by assumption, and~$\\theta$ covers the interval~$(\\theta_\\infty,\\infty)$. Consequently, Prop.~\\ref{prop_cosmconstpastinextend} applies and in combination with\n\tLemma~\\ref{lemm_bianchibdevelopmentproperties} implies that the geodesic in~$M$ is future complete and past incomplete. The incomplete direction of an inextendible geodesic in both~$\\tilde M$ and~$M$ therefore corresponds to~$t\\rightarrow t_-$, and the complete direction to~$t\\rightarrow t_+$. This concludes the proof.\n\\end{proof}\n\n\n\\subsection{Solutions with additional symmetry}\n\n\\label{constr_highersymmetrysolutions}\n\n\n\nIn the results of this paper, several types of geometric initial data sets as well as invariant subsets are of interest which have additional properties. In particular, those with local rotational symmetry and those describing plane wave equilibrium solutions appear frequently.\nWe have defined both notions twice, first in the setting of geometric initial data and then in the setting of expansion-normalised variables. In this subsection, we show that these definitions are consistent provided we transform between geometric initial data sets and expansion-normalised variables the way we described in Subsections~\\ref{constr_initialdatatoexpnorm}--\\ref{constr_solutionbasictomghd}.\n\n\\smallskip\n\nThe notion of locally rotationally symmetric geometric initial data was given in Def.~\\ref{defi_lrsinitialdata}, where such initial data is defined via the properties of a specific basis of the associated Lie al\\-ge\\-bra.\nGiven more generally some or\\-tho\\-nor\\-mal\\ basis~$e_1,e_2,e_3$ of the Lie al\\-ge\\-bra~$\\ggg$ such that\n\\begin{equation}\n\t\\liebr{e_2}{e_1}=0=\\liebr{e_2}{e_3},\n\t\\qquad\n\t\\liebr{e_1}{e_3}\\parallel e_2,\n\t\\qquad\n\tk=\\operatorname{diag}(k_{11},k_{22},k_{33}=k_{11}),\n\\end{equation}\nbut not necessarily with~$e_2,e_3$ spanning a particular subalgebra, we find that these properties are preserved under a rotation in the~$e_1e_3$-plane. The term local rotational symmetry is thereful meaningful and should be compared to the definition of local rotational symmetry in Bianchi class~A spacetimes in~\\cite[Def.~19.16]{ringstrom_cauchyproblem} and in more general spacetimes in~\\cite{stewartellis_solseinsteinequsfluidlrs}.\n\nLet us now discuss Definition~\\ref{defi_lrsinitialdata} in connection with the different Bianchi types.\nFor a Lie group\\ of type~V or~IV, the form of the structure constants from Table~\\ref{table_bianchibclassification} reveals\n\\begin{equation}\n\t\\liebr{e_1}{e_2}=a_1e_2+\\nu_3e_3,\\qquad\n\t\\liebr{e_1}{e_3}=a_1e_3,\\qquad\n\t\\liebr{e_2}{e_3}=0,\n\\end{equation}\nfor a basis chosen as in the classification of Lie group s in Subsection~\\ref{subsect_appendixbianchiclassification}. Note that the basis elements~$e_2,e_3$ span~$\\ggg_2$, see Remark~\\ref{rema_frameforggg2rotationtodiagonalise}. There is no rotation in the~$e_2e_3$-plane such that the commutators of the rotated basis have the form requested for local rotational symmetry. Consequently, no initial data set of Bianchi type~V or~IV can have local rotational symmetry.\n\nLet us therefore assume that the Lie group\\ is of type~VI or~VII. When comparing the properties of local rotational symmetry from Def.~\\ref{defi_lrsinitialdata} with the decomposition of the structure constants in~\\eqref{eqns_commutators}, we see that the vanishing of~$\\liebr{e_2}{e_1}$ and~$\\liebr{e_2}{e_3}$ is equivalent to~$a_1={-}n_{32}$ and~$n_{33}=0$, while~$\\liebr{e_1}{e_3}\\parallel e_2$ holds if and only if~$a_1=n_{23}$.\nIn particular, we can consider the linear map~$A_2=\\operatorname{ad}_{v_1}|_{\\ggg_2}$, $v_1\\in\\ggg\\setminus\\ggg_2$, from Lemma~\\ref{lemm_invardefibparaminv} and conclude that local rotational symmetry implies~$\\det A_2=0$. The image of either~$\\operatorname{ad}_{v_1}$ or~$A_2$, which coincides with~$\\liebr{\\ggg}{\\ggg}$, is one-dimensional and spanned by~$e_2$. After a rotation in the~$e_2e_3$ plane, we can assume that~$n$ is diagonal, but lose the properties~$a_1=n_{23}={-}n_{23}$ and~$n_{33}=0$, see Remark~\\ref{rema_frameforggg2rotationtodiagonalise}. A vanishing determinant of~$A_2$ implies that~$\\eta={-}1$, hence the Lie group\\ can only be of type~VI$_{{-}1}$.\n\nAlternatively, let~$e_1,\\tilde e_2,\\tilde e_3$ a basis of the Lie al\\-ge\\-bra~$\\ggg$ with~$\\tilde e_2,\\tilde e_3$ spanning~$\\ggg_2$ and such that~$a=(a_1,0,0)$, $n=\\operatorname{diag}(0,\\nu_2,\\nu_3)$, i.\\,e.\\  a basis as in the classification of Lie group s in Subsection~\\ref{subsect_appendixbianchiclassification}. Let us further assume that~$\\eta={-}1$, i.\\,e.\\  the Lie group\\ is of type~VI$_{{-}1}$. In this case, we find from the matrix representation of~$A_2$ that\n\\begin{equation}\n\t\\liebr{\\ggg}{\\ggg}=\\operatorname{span}(\\tilde e_2\\pm\\sqrt{\\absval{\\frac{\\nu_2}{\\nu_3}}}\\tilde e_3),\n\\end{equation}\nwhere the sign depends on the sign of~$a_1$. We further find that if we set~$e_2$ and~$e_3$ to be unit vectors in~$\\ggg_2$, one spanning~$\\liebr{\\ggg}{\\ggg}$ and one or\\-tho\\-go\\-nal\\ to it, and by a change of sign in~$e_1$ ensure that~$a_1$ is negative, then we recover that the commutator of this new basis satisfy the properties\n\\begin{equation}\n\t\\liebr{e_2}{e_1}=0=\\liebr{e_2}{e_3},\n\t\\qquad\n\t\\liebr{e_1}{e_3}\\parallel e_2\n\\end{equation}\nfrom the definition of local rotational symmetry, Def.~\\ref{defi_lrsinitialdata}. Whether geometric initial with an associated Lie al\\-ge\\-bra\\ of type~VI$_{{-}1}$ is locally rotationally symmetric then depends solely on the two-tensor~$k$.\n\nGiven geometric initial data as in Def.~\\ref{defi_lrsinitialdata}, we can deduce from the commutators the corresponding basic variables via expressions~\\eqref{eqns_prebasic_definitionpartone},~\\eqref{eqns_prebasic_definitionparttwo},~\\eqref{eqns_basic_definitionpartone} and~\\eqref{eqns_basic_definitionparttwo} and find that local rotational symmetry implies that~$3\\sigma_+^2=\\tilde\\sigma$ and~$\\sigma_+n_+=\\delta$. Further, the Lie group\\ has to be of class~A or of type~VI$_{{-}1}$ due to the above discussion. The expansion-normalised variables result from applying~\\eqref{eqn_expansionnormalisation} and~\\eqref{eqn_definitionomega}, and one recovers the definition of locally rotationally solutions in expansion-normalised variables, Def.~\\ref{defi_lrsexpansionnorm}, as Bianchi class~A corresponds to~$\\tilde A=0$.\n\n\\smallskip\n\nFor the notion of plane wave equilibrium initial data, we turn to Def.~\\ref{defi_planewaveinitialdata}.\nIn terms of structure constants for the four-dimensional spacetime, the property~$\\gamma_{1A}^B+\\gamma_{1B}^A=-2k_{AB}$ is equivalent to\n\\begin{equation}\n\\label{eqn_defiplanewavestructconst}\n  \\gamma_{12}^2=\\gamma_{02}^2,\\qquad \\gamma_{13}^3=\\gamma_{03}^3,\\qquad\\gamma_{12}^3+\\gamma_{13}^2=\\gamma_{02}^3+\\gamma_{03}^2.\n\\end{equation}\nThis should be compared with the definition given in~\\cite{hervikkundurilucietti_homogplanewavestability}, where the special spacelike direction is set to be~$e_1$ instead of~$e_n$. Applying an appropriate permutation of the basis elements to fit with our choice of frame, the definition given there reads\n\\begin{equation}\n  \\gamma_{1A}^B=\\gamma_{0A}^B.\n\\end{equation}\nIn terms of geometric initial data, the right-hand side\\ is not defined, but its symmetrisation can be replaced by the expression~${-}k_{AB}$, see equations~\\eqref{eqns_commutators} and Remark~\\ref{rema_geometricmeaningthetasigmaAB}. This tensor is symmetric by definition, and also replacing the left-hand side\\ by its symmetrisation one obtains Def.~\\ref{defi_planewaveinitialdata}.\n\nTranslating structure constants into basic variables via expressions~\\eqref{eqns_prebasic_definitionpartone},~\\eqref{eqns_prebasic_definitionparttwo},~\\eqref{eqns_basic_definitionpartone}, and~\\eqref{eqns_basic_definitionparttwo}, and using the constraint equation~\\eqref{eqn_prebasic_constraint}, one finds that the relations~\\eqref{eqn_defiplanewavestructconst} carry information equivalent to\n\\begin{equation}\n  {-}\\sigma_+(\\theta+\\sigma_+)=\\tilde\\sigma=\\tilde n,\\qquad \\delta=0,\\qquad \\mu=0.\n\\end{equation}\nVia~\\eqref{eqn_expansionnormalisation} and~\\eqref{eqn_definitionomega} one obtains that this is equivalent to expansion-normalised variables satisfying~$Z=0$, see~\\eqref{eqn_functionzzeros}. This characterises the set~$\\planewave {\\kappa}$, Def.~\\ref{defi_planewaveexpansionnorm}, together with the point Taub~1, and\nis equivalent to dropping the condition $\\Sigma_+>{-}1$ in equation ~\\eqref{eqn_defiplanewave}.\n\n\n\\begin{rema}\n\\label{rema_taubpointsasinitialdata}\n\tIn terms of initial data, the Kasner parabola~$\\K$ corresponds to vacuum Bianchi type~I data. This can easily be deduced from Table~\\ref{table_bianchiasubsets} and using the condition for vacuum, $\\Omega=0$, in equation~\\eqref{eqn_omegageneral}.\n\n\tThe two special Taub points have the following characterisation:\n\t\\begin{itemize}\n\t\t\\item The point Taub~1 corresponds to initial data of Bianchi type~I which is of plane wave equilibrium type.\n\t\t\\item The point Taub~2 corresponds to initial data of Bianchi type~I which is locally rotationally symmetric  and additionally\n\t\tthe unique eigenvalue of~$k$ in the rotation plane is smaller than the unique eigenvalue along the rotation axis. In a basis satisfying the conditions for local rotation symmetry from Definition~\\ref{defi_lrsinitialdata}, this means that\n\t\t${k}_{11}<{k}_{22}$ holds.\n\t\\end{itemize}\n\tThis follows from the consistency check we carried out before. Geometric initial data being locally rotationally symmetric is equivalent to the corresponding expansion-normalised variables satisfying Def.~\\ref{defi_lrsexpansionnorm}, and geometric initial data of plane wave equilibrium type corresponds to Def.~\\ref{defi_planewaveexpansionnorm}, including the possibility that $\\Sigma_+={-}1$.\n\tIntersection with the Kasner parabola~$\\K$, which consist of all Bianchi type~I vacuum spacetimes, yields the characterisation: Taub~1 is the unique point on the Kasner parabola which satisfies the plane wave equilibrium point relations. Taub~2 is one of two intersection points between the Kasner parabola~$\\K$ and the set of LRS solutions, the one with~$\\Sigma_+>0$. This last property translates into $2{k}_{11}-{k}_{22}-{k}_{33}<0$, and as ${k}_{11}={k}_{33}$ for LRS geometric initial data, the statement follows.\n\\end{rema}\n\n\n\n\\section{Proof of the main theorems}\n\\label{section_proofmainthms}\n\n\nIn this final section, we give the proofs of the main theorems stated in the introduction. Four of them, Thm~\\ref{theo_fullmeasurelimitset}, Thm~\\ref{theo_curvblowupexpansionnormalisedmatter}, Thm~\\ref{theo_curvblowupexpansionnormalisedvacuum}, and Thm~\\ref{theo_curvblowupexpansionnormalisedcosmconst}, are stated in the setting of expansion-normalised variables, and these are proven first. The remaining statements, Thm~\\ref{theo_curvblowupinitialdatamatter} and Thm~\\ref{theo_curvblowupinitialdatavacuum}, are translated versions of Thm~\\ref{theo_curvblowupexpansionnormalisedmatter} and Thm~\\ref{theo_curvblowupexpansionnormalisedvacuum}, giving the results in terms of geometric initial data and the corresponding ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment.\n\nIn the previous sections, we have determined the asymptotic behaviour of solutions to the evolution equation s~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega}. We have found possible~$\\alpha$-limit sets and determined in detail the solutions which converge to the Kasner parabola~$\\K$ or the plane wave equilibrium points~$\\planewave {\\kappa}$ as~$\\tau\\rightarrow{-}\\infty$.\nThm~\\ref{theo_curvblowupexpansionnormalisedmatter}, Thm~\\ref{theo_curvblowupexpansionnormalisedvacuum}, and Thm~\\ref{theo_curvblowupexpansionnormalisedcosmconst} state that apart from a short list of exceptional solutions, either the Kretschmann scalar $R_{\\alpha\\beta\\gamma\\delta}R^{\\alpha\\beta\\gamma\\delta}$ or the contraction of the Ricci tensor with itself $R_{\\alpha\\beta}R^{\\alpha\\beta}$ or both become unbounded along solutions as~$\\tau\\rightarrow{-}\\infty$. This implies Strong Cosmic Censorship in the~$C^2$-sense.\nWe start with a discussion of these two geometric invariants and their form and asymptotic behaviour in terms of expansion-normalised variables, before we are then in a position to give the proof of these three theorems.\n\n\nWe then continue this section with a proof of Thm~\\ref{theo_fullmeasurelimitset}, where we show that apart from a `small` subset, all solutions to the evolution equation s~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} converge to a Kasner point to the right of Taub~2, as $\\tau\\rightarrow{-}\\infty$. For this proof as well, we heavily rely on the results on asymptotic behaviour of solutions which we obtained in the previous sections.\n\n\nAt the end of this section, we conclude with the proofs of the main statements in the initial data perspective, Thm~\\ref{theo_curvblowupinitialdatamatter} and Thm~\\ref{theo_curvblowupinitialdatavacuum}. The equivalent statements in expansion-normalised variables are given in Thm~\\ref{theo_curvblowupexpansionnormalisedmatter} and Thm~\\ref{theo_curvblowupexpansionnormalisedvacuum}. Using the transformation between this set of variables and the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ to given geometric initial data which we constructed in Subsections~\\ref{constr_initialdatatoexpnorm}--\\ref{constr_solutionbasictomghd}, we re-translate the statements back to the setting of geometric initial data.\nApplying the proof to~$\\mu>0$, $\\gamma=0$, i.\\,e.\\  the stress-energy tensor of a positive cosmological constant in vacuum, even justifies the statement given in Remark~\\ref{rema_curvblowupcosmconst} as a re-translated version of Thm~\\ref{theo_curvblowupexpansionnormalisedcosmconst}.\n\n\n\\smallskip\n\nWe start with computing several curvature expressions in terms of expansion-normalised variables.\nThe Weyl tensor of a four-dimensional mani\\-fold\\ with metric~$g$ is given by\n\\begin{equation}\n\tC_{\\alpha\\beta\\gamma\\delta}=R_{\\alpha\\beta\\gamma\\delta}-(g_{\\alpha[\\gamma}R_{\\delta]\\beta}-g_{\\beta[\\gamma}R_{\\delta]\\alpha})+\\frac13Sg_{\\alpha[\\gamma}g_{\\delta]\\beta},\n\\end{equation}\nsee~\\cite[Eq.~(3.2.28)]{wald_generalrelativity}.\nFor a spacetime satisfying Einstein's equation ~\\eqref{eqn_einsteineqn} with stress-energy tensor of a perfect fluid~\\eqref{eqn_stressenergyperfectfluid} with linear equation\\ of state~\\eqref{eqn_lineareqnofstate}, one computes that the contraction of the Ricci tensor with itself is\n\\begin{equation}\n\\label{eqn_contractionricciricci}\n\tR_{\\alpha\\beta}R^{\\alpha\\beta}=\\mu^2+3p^2=(1+3(\\gamma-1)^2)\\mu^2,\n\\end{equation}\nand finds the Kretschmann scalar\n\\begin{align}\nR_{\\alpha\\beta\\gamma\\delta}R^{\\alpha\\beta\\gamma\\delta}\n\t={}&C_{\\alpha\\beta\\gamma\\delta}C^{\\alpha\\beta\\gamma\\delta}+2R_{\\alpha\\beta}R^{\\alpha\\beta}-\\frac13S^2\\\\\n\t={}&C_{\\alpha\\beta\\gamma\\delta}C^{\\alpha\\beta\\gamma\\delta}+\\frac13(4+(3\\gamma-2)^2)\\mu^2.\n\\end{align}\nThe relation between the Weyl tensor and the expansion normalised variables $(\\Sigma_+,\\tilde\\Sigma,\\Delta,\\tilde A,N_+)$ has been computed in~\\cite[App.~B]{hewittwainwright_dynamicalsystemsapproachbianchiorthogonalB}, referring to~\\cite{ellismaccallum_classofhomogcosmmodels}:\n\\begin{equation}\n\\label{eqn_weyltensorexpansionnormalised}\n\tC_{\\alpha\\beta\\gamma\\delta}C^{\\alpha\\beta\\gamma\\delta}=\\theta^4(\\tilde E_{AB}\\tilde E^{AB}+\\frac23E_+^2)-\\theta^4(\\tilde H_{AB}\\tilde H^{AB}+\\frac23H_+^2),\n\\end{equation}\nwhere the electrical components are\n\\begin{align}\n\t\\tilde E_{AB}\\tilde E^{AB}={}&\\frac2{27}[\\tilde\\Sigma(2\\Sigma_+-1)^2+4(\\tilde A\\Sigma_++\\tilde N(\\tilde A+3\\tilde N-2\\tilde\\Sigma))\\\\\n\t\t& +4\\Delta^2-4N_+\\Delta(2\\Sigma_+-1)+4\\kappa\\tilde A\\tilde N],\\\\\n\tE_+={}&\\frac13(\\tilde \\Sigma -\\Sigma_+(1+\\Sigma_+)-2\\tilde N),\n\\end{align}\nand the magnetic ones are\n\\begin{align}\n\t\\tilde H_{AB}\\tilde H^{AB}={}&\\frac2{27}[9\\Sigma_+^2\\tilde N+6\\tilde\\Sigma\\tilde N+\\tilde\\Sigma\\tilde A+6\\Delta^2+12\\Sigma_+N_+\\Delta+4\\kappa\\tilde A\\tilde\\Sigma],\\\\\n\tH_+={}&{-}\\Delta.\n\\end{align}\n\\begin{lemm}\n\\label{lemm_curvblowupricci}\n\tConsider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega}. If $\\Omega>0$ and $\\gamma>0$, then\n\t\\begin{equation}\n\t\t\\lim_{\\tau\\rightarrow{-}\\infty}R_{\\alpha\\beta}R^{\\alpha\\beta}=\\infty.\n\t\\end{equation}\n\tIf $\\Omega>0$, $\\gamma=0$, or if $\\Omega=0$, then $R_{\\alpha\\beta}R^{\\alpha\\beta}$ remains bounded as $\\tau\\rightarrow{-}\\infty$.\n\\end{lemm}\nThe central idea of proof has already been given in~\\cite{ringstrom_asymptbianchiaspacetimes}, where the Bianchi~A case was discussed. It only relies upon equation ~\\eqref{eqn_contractionricciricci}, which follows immediately from the assumption on the stress-energy tensor and is independent of the Bianchi class of the Lie group.\nNote that only the evolution equations for~$\\theta$ and~$\\Omega$ come into play. In particular, this statement can be obtained without knowledge on the detailed asymptotic properties of the individual variables. \n\\begin{proof}\n\tThe contraction of the Ricci tensor with itself is given by~\\eqref{eqn_contractionricciricci}.\n\tThe density~$\\mu$ satisfies equation ~\\eqref{eqn_definitionomega}, and the evolution equations~\\eqref{eqn_evolutiontheta} and~\\eqref{eqn_evolutionomega} for~$\\theta$ and~$\\Omega$ yield\n\t\\begin{align}\n\t\t9\\mu^2(\\tau)={}&\\Omega^2\\theta^4(\\tau)\\\\\n\t\t\t={}&\\Omega(\\tau_0)^2\\theta(\\tau_0)^4\\exp(\\int_{\\tau_0}^\\tau -4-4q +4q -2(3\\gamma-2)ds)\\\\\n\t\t\t={}&\\Omega(\\tau_0)^2\\theta(\\tau_0)^4\\exp({-}6\\gamma(\\tau-\\tau_0)).\n\t\\end{align}\n\tAs $\\Omega>0$ is an invariant set by equation ~\\eqref{eqn_evolutionomega}, and $\\theta\\not=0$ by construction of the expansion-normalised coordinates in Subsections~\\ref{constr_initialdatatoexpnorm}--\\ref{constr_solutionbasictomghd}, the statement follows.\n\\end{proof}\n\\begin{lemm}\n\\label{lemm_curvblowupvacuum}\n\tAssume either vacuum or inflationary matter with $\\gamma=0$, i.\\,e.\\  $\\Omega=0$ or $\\Omega>0$, $\\gamma=0$, and consider a solution to equations ~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega}. Assume that the solution has an $\\alpha$-limit point $(\\Sigma_+,1-\\Sigma_+^2,0,0,0)$ with $\\Sigma_+\\notin\\{{-}1,1\/2\\}$. Then\n\t\\begin{equation}\n\t\t\\limsup_{\\tau\\rightarrow{-}\\infty}\\absval{R_{\\alpha\\beta\\gamma\\delta}R^{\\alpha\\beta\\gamma\\delta}}=\\infty.\n\t\\end{equation}\n\\end{lemm}\n\\begin{proof}\n\tVacuum is equivalent to vanishing Ricci curvature, consequently\n\t\\begin{equation}\n\t\tR_{\\alpha\\beta\\gamma\\delta}R^{\\alpha\\beta\\gamma\\delta}=C_{\\alpha\\beta\\gamma\\delta}C^{\\alpha\\beta\\gamma\\delta}\n\t\\end{equation}\n\tin this case,\n\twhile for $\\Omega>0$, $\\gamma=0$, the argument from the previous proof implies that\n\t\\begin{equation}\n\t\tR_{\\alpha\\beta\\gamma\\delta}R^{\\alpha\\beta\\gamma\\delta}-C_{\\alpha\\beta\\gamma\\delta}C^{\\alpha\\beta\\gamma\\delta}\n\t\t=2R_{\\alpha\\beta}R^{\\alpha\\beta}-\\frac13S^2=\\frac13(4+(3\\gamma-2)^2)\\mu^2\n\t\\end{equation}\n\tis a constant.\n\n\tDirect computation shows that on the Kasner parabola~$\\K$ one finds\n\t\\begin{equation}\n\t\t\\tilde E_{AB}\\tilde E^{AB}+\\frac23E_+^2-\\tilde H_{AB}\\tilde H^{AB}-\\frac23H_+^2=\\frac4{27}(2\\Sigma_+-1)^2(\\Sigma_++1),\n\t\\end{equation}\n\twhich is non-zero if and only if\\ $\\Sigma_+\\notin\\{{-}1,1\/2\\}$. The statement then follows from equation ~\\eqref{eqn_weyltensorexpansionnormalised} and the fact that $\\theta\\rightarrow\\infty$ as $\\tau\\rightarrow{-}\\infty$, due to equation ~\\eqref{eqn_evolutiontheta}.\n\\end{proof}\n\\begin{lemm}\n\\label{lemm_expressionforweyltensorvanishes}\n\tThe expression\n\t\\begin{equation}\n\t\t\\tilde E_{AB}\\tilde E^{AB}+\\frac23E_+^2-\\tilde H_{AB}\\tilde H^{AB}-\\frac23H_+^2\n\t\\end{equation}\n\tvanishes for the Kasner points Taub~1 and~2, for the plane wave equilibrium points~$\\planewave {\\kappa}$ and for the point $\\Sigma_+=\\tilde\\Sigma=\\Delta=\\tilde A=N_+=0$.\n\\end{lemm}\n\\begin{proof}\n\tThis follows from direct computation, which we in the case of the Taub points already carried out in the previous proof, and where the set~$\\planewave {\\kappa}$ is defined in Def.~\\ref{defi_planewaveexpansionnorm}.\n\\end{proof}\n\\begin{rema}\n\tOn the Taub points~1 and~2, on the plane wave equilibrium points~$\\planewave {\\kappa}$, and on the point $\\Sigma_+=\\tilde\\Sigma=\\Delta=\\tilde A=N_+=0$, the contraction of the Weyl tensor with itself, expression~\\eqref{eqn_weyltensorexpansionnormalised}, vanishes due to Lemma~\\ref{lemm_expressionforweyltensorvanishes}. Together with the result from Lemma~\\ref{lemm_curvblowupricci} and equation~\\eqref{eqn_contractionricciricci}, this implies that both the Kretschmann scalar $R_{\\alpha\\beta\\gamma\\delta}R^{\\alpha\\beta\\gamma\\delta}$ and the contraction of the Ricci tensor with itself $R_{\\alpha\\beta}R^{\\alpha\\beta}$ remain bounded for the constant solutions\n\tin the points Taub~1 and Taub~2, the plane wave equilibrium points and the point $\\Sigma_+=\\tilde\\Sigma=\\Delta=\\tilde A=N_+=0$.\n\\end{rema}\nWe now give the proofs of the theorems stated in expansion-normalised variables.\n\\begin{proof}[Proof of Thm~\\ref{theo_curvblowupexpansionnormalisedmatter}]\n\tThis statement is an immediate consequence of Lemma~\\ref{lemm_curvblowupricci}.\n\\end{proof}\n\\begin{proof}[Proof of Thm~\\ref{theo_curvblowupexpansionnormalisedvacuum}]\n\tProp.~\\ref{prop_alphalimitsets_vacuum_inflat} restricts the possible $\\alpha$-limit sets of solutions in vacuum~$\\Omega=0$ to the Kasner parabola~$\\K$ and the plane wave equilibrium points~$\\planewave {\\kappa}$.\n\tNon-constant solutions with $\\alpha$-limit set in~$\\planewave {\\kappa}$ are immediately excluded by the same statement.\n\tOne knows further from Prop.~\\ref{prop_convergencetokasner} that solutions with $\\alpha$-limit set in~$\\K$ have a unique $\\alpha$-limit point, i.\\,e.\\  they converge as $\\tau\\rightarrow{-}\\infty$.\n\tWe now discuss the different possible locations of limit points for solutions to equations~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} and check in which cases the Kretschmann scalar possibly remains bounded.\n\n\tLemma~\\ref{lemm_curvblowupvacuum} states that the Kretschmann scalar becomes unbounded upon convergence to every Kasner points but the points Taub~1 and~2.\n\tThe only solution converging to the point Taub~1 is the constant one, see Prop.~\\ref{prop_taubone}.\n\tAll solutions converging to Taub~2 are characterised in Thm~\\ref{theo_taubtwocharacterisationoforbits}.\n\tThis concludes the proof.\n\\end{proof}\n\\begin{proof}[Proof of Thm~\\ref{theo_curvblowupexpansionnormalisedcosmconst}]\n\tThe proof proceeds similarly to the previous one.\n\tProp.~\\ref{prop_alphalimitsets_vacuum_inflat} restricts the possible $\\alpha$-limit sets of solutions with~$\\Omega>0$, $\\gamma=0$, to the Kasner parabola~$\\K$, the plane wave equilibrium points~$\\planewave {\\kappa}$, and the point $\\Sigma_+=\\tilde\\Sigma=\\Delta=\\tilde A=N_+=0$.\n\tNon-constant solutions converging to the point $\\Sigma_+=\\tilde\\Sigma=\\Delta=\\tilde A=N_+=0$ are immediately excluded by the same proposition, which also states that solutions whose $\\alpha$-limit set intersects both~$\\K\\setminus\\operatorname{T1}$ and~$\\planewave {\\kappa}$ do not exist.\n\tOne knows further from Prop.~\\ref{prop_convergencetokasner} and Prop.~\\ref{prop_convergencetoplanewave} that solutions with $\\alpha$-limit set in either~$\\K$ or~$\\planewave {\\kappa}$ have a unique $\\alpha$-limit point, i.\\,e.\\  they converge as $\\tau\\rightarrow{-}\\infty$.\n\tWe now discuss the different possible locations of limit points and check in which cases the Kretschmann scalar possibly remains bounded.\n\n\tThe constant solution in the point $\\Sigma_+=\\tilde\\Sigma=\\Delta=\\tilde A=N_+=0$ satisfies~$\\Omega=1$. Due to the previous remark, the Kretschmann scalar\n\tremains bounded.\n\n\tLet us now turn to solutions converging to a plane wave equilibrium point, i.\\,e.\\  solutions contained in~$\\C$.\n\tWe apply Thm~\\ref{theo_centreunstablegloballyplanewave} and find that the case~${-}(3\\gamma-2)\/4<\\operatorname s<0$ is excluded by~$\\gamma=0$. Every solution in~$\\C$ is contained in the union of submani\\-fold s~$L'_m$ or~$\\submfsplanewave'_m$, and the limiting value~$\\operatorname s$ and the parameter~${\\kappa}$ have to satisfy relation~\\eqref{eqn_inequalityplanewave}. In Remark~\\ref{rema_planewaveconvergencebianchiseparated}, we list the possible values depending on the Bianchi type. Setting the union of submani\\-fold s~$\\{L_m\\}$ to include all~$L'_m$ and~$\\submfsplanewave'_m$ gives the statement.\n\n\tThe arguments regarding solutions converging to a point on the Kasner parabola~$\\K$ are identical to the ones given in the previous proof. Note that only the non-constant solutions, i.\\,e.\\  locally rotationally symmetric solutions of Bianchi type~I, II or~VI$_{{-}1}$ can satisfy~$\\Omega>0$. This concludes the proof.\n\\end{proof}\nFor the case of vacuum, our statement Thm~\\ref{theo_curvblowupexpansionnormalisedvacuum} makes precise and proves a claim by~\\cite[p.~165]{wainwrightellis_dynamsystemsincosm} saying that all solutions except those contained in the unstable manifold of the point Taub~2 have an initial curvature singularity.\n\n\\begin{proof}[Proof of Thm~\\ref{theo_fullmeasurelimitset}]\n\tThe three sets describing solutions of Bianchi type~VI$_{\\eta}$, VII$_{\\eta}$ and~IV are open sets of~$\\maths{R}^5$. Restricted to the constraint equation s~\\eqref{eqn_constraintgeneralone} and~\\eqref{eqn_constraintgeneraltwo}, they form open subsets of the set defined by these equations. Consequently, these three Bianchi sets are of dimension four. Restricting to vacuum~$\\Omega=0$ yields sets of dimension three. The set describing solutions of Bianchi type~V is a two-dimensional closed subset of~$\\maths{R}^5$ and contained in the set defined by the constraint equation s~\\eqref{eqn_constraintgeneralone} and~\\eqref{eqn_constraintgeneraltwo}. Restricted to vacuum~$\\Omega=0$, this Bianchi type yields a set of dimension~one. We prove the theorem by showing that all solutions with convergence behaviour different from the one in the statement are contained in countable unions of smooth submani\\-fold s of positive codimension.\n\n\tIndependently of the Bianchi type, the $\\alpha$-limit set of a solution to equation s~\\eqref{eqns_evolutionbianchib}--\\eqref{eqn_evolutionomega} with either $\\Omega=0$ or $\\Omega>0$, $\\gamma\\in\\left[0,2\/3\\right)$ is given in Prop.~\\ref{prop_alphalimitsets_vacuum_inflat}: We exclude constant solutions as they are contained in two smooth curve arcs together with one additional point, namely the Kasner parabola~$\\K$, the plane wave equilibrium points~$\\planewave \\kappa$ and the point Taub~1.\n\tIn Bianchi type~V, we additionally notice that the intersection between these two arcs and the subset defining this Bianchi type consists of two points.\n\n\tProp.~\\ref{prop_taubone} states that only the constant solution has Taub~1 as an $\\alpha$-limit point, so in particular we exclude solutions with Taub~1 as an $\\alpha$-limit point. We conclude from Prop.~\\ref{prop_alphalimitsets_vacuum_inflat} that the $\\alpha$-limit set is contained in either the Kasner parabola~$\\K$ without the point Taub~1 or in the plane wave equilibrium points~$\\planewave {\\kappa}$, and also see that latter set can only occur for inflationary matter, i.\\,e.\\  for $\\Omega>0$, $\\gamma\\in\\left[0,2\/3\\right)$.\n\n\tIn Prop.~\\ref{prop_convergencetokasner} and Prop.~\\ref{prop_convergencetoplanewave}, we further showed that the $\\alpha$-limit sets consist of exactly one point meaning that solutions converge as $\\tau\\rightarrow{-}\\infty$. All that remains to show now is that the set of solutions converging to a plane wave equilibrium point~$\\planewave {\\kappa}$ or a point on the Kasner parabola~$\\K$ with ${-}1<\\Sigma_+\\le1\/2$, resp.\\  the complement of this set of solutions, has the necessary properties.\n\n\t\\smallskip\n\n\tWe start with solutions converging to a point on the Kasner parabola~$\\K$.\n\tAccording to Thm~\\ref{theo_taubtwocharacterisationoforbits}, the only Bianchi class~B solutions converging to the point Taub~2 are those of Bianchi type~VI$_{{-}1}$ which are locally rotationally symmetric. They are characterised by\n\t\\begin{equation}\n\t\t{\\kappa}={-}1,\\qquad \\tilde A>0,\\qquad 3\\Sigma_+^2=\\tilde\\Sigma,\\qquad \\Sigma_+N_+=\\Delta,\n\t\\end{equation}\n\tand are solutions of Bianchi type~VI$_{{-}1}$. In particular, no Bianchi type~VII$_{\\eta}$, type~IV  or type~V solution converges to the point Taub~2. The set is a~$C^1$ submani\\-fold\\ of dimension three, and its restriction to vacuum of dimension~two. In particular, it forms a subset of positive codimension in the set of all Bianchi type~VI$_{{-}1}$ solutions, both in case of vacuum and inflationary matter.\n\n\tThe solutions converging to a Kasner point to the left of Taub~2 are given in Thm~\\ref{theo_leftoftaubtworesult}.\n\tIn case of Bianchi type~VI$_{\\eta}$,\n\tsolutions either satisfy $\\tilde A>0$, $\\Delta=0=N_+$, $3\\Sigma_+^2+{\\kappa}\\tilde\\Sigma=0$ (Bianchi type VI$_{\\eta}$ with $n^\\alpha{}_\\alpha=0$, see Table~\\ref{table_bianchihighersymmetry}), or have to be contained in a countable union of $C^1$ submani\\-fold s satisfying $\\tilde A>0$ and~$\\Delta$, $N_+$ not both vanishing identically.\n\tThese submani\\-fold s are contained either in the set of all non-vacuum solutions or the set of all vacuum solutions, and in the respective set have codimension at least~one.\n\tSolutions of Bianchi type~VII$_{\\eta}$ cannot converge to a Kasner point to the left of Taub~2 due to~$\\kappa>0$. In the remaining two cases, Bianchi type~IV and~V, the restriction~${\\kappa}=0$ implies that solutions can only converge to the Kasner point with~$\\operatorname s=0$.\n\n\tFor non-constant solutions converging to a plane wave equilibrium point, Thm~\\ref{theo_centreunstablegloballyplanewave} states that they converge to the arc $\\planewave {\\kappa}\\cap\\{\\Sigma_+\\le{-}(3\\gamma-2)\/4\\}$, which includes the point~$\\planewave {\\kappa}\\cap\\{\\Sigma_+=0\\}$ due to the assumption on~$\\gamma$. All solutions are contained in the countable union of $C^1$ submani\\-fold s~$L'_m$ or~$\\submfsplanewave'$ whose dimension is at most two. This is enough to identify them as subsets of positive codimension in the Bianchi type~VI$_{\\eta}$,~VII$_{\\eta}$ and~IV. For Bianchi type~V solutions, we have argued in Remark~\\ref{rema_planewaveconvergencebianchiseparated} that the solution is contained in the subset~$\\Sigma_+=\\tilde\\Sigma=\\Delta=N_+=0$, which is a subset of positive codimension in non-vacuum.\n\n\t\\smallskip\n\n\tCollecting the different partial results, we conclude the following: In the case of Bianchi type~VI$_{\\eta}$ or VII$_{\\eta}$, if a solution does not converge to a Kasner point to the right of Taub~2, then it is contained in a countable union of $C^1$ submani\\-fold s of positive codimension, both in case of vacuum and inflationary matter.\n\tTherefore, this set of exceptions has the right properties: its complement is of full measure and a countable intersection of open and dense sets.\n\tWe argue similarly in case of Bianchi type~IV, with the additional possibility of convergence to the Kasner point with~$\\operatorname s=0$. For the case of Bianchi type~V, convergence to Kasner point to the right of Taub~2 is not possible, as all such Bianchi solutions satisfy~$\\Sigma_+=0$ by definition. We have shown that all non-constant solutions converge to the point on the Kasner parabola~$\\K$ satisfying~$\\operatorname s=0$.\n\\end{proof}\n\nWe conclude this section with the proof of the two remaining theorems, which are stated in the setting of geometric initial data. As large portions of the proofs are identical, we combine them into one.\n\\begin{proof}\n[Proofs of Thm~\\ref{theo_curvblowupinitialdatamatter} and Thm~\\ref{theo_curvblowupinitialdatavacuum}]\n\tWe consider initial data which is of Bianchi class~B. This implies in particular that\n\tthe Minkowski spacetime or quotients of this space cannot occur as development of the data.\n\tThis excludes the case~$\\theta=\\operatorname{tr}_hk=0$, see Lemma~\\ref{lemm_excludeminkowski}, and we can therefore consider a development of the data~$(I\\times G,g,\\mu)$ as in Def.~\\ref{defi_mghd}, i.\\,e.\\  one constructed as in Subsections~\\ref{constr_initialdatatoexpnorm}--\\ref{constr_solutionbasictomghd}. According to Corollary~\\ref{coro_mghdforgammapositive}, this development is isometric to the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ of the given initial data. We further know from Prop.~\\ref{prop_mghdincompletedirections} that the behaviour of any geometric quantity in the incomplete direction of causal geodesics in~$M$ corresponds to the behaviour of this geometric quantity on~$\\{t\\}\\times G$ as~$t\\rightarrow t_-$.\n\n\tWe now switch to the point of view of expansion-normalised variables, as explained in Subsection~\\ref{constr_initialdatatoexpnorm}. The behaviour of any geometric quantity in the incomplete direction of causal geodesics in~$M$ consequently corresponds to the behaviour of this geometric quantity as~$\\tau\\rightarrow{-}\\infty$, see also the construction of the development, specifically the end of Subsection~\\ref{constr_expnormtosolutionbasic}.\n\n\tThe result in the matter case~$\\mu_0>0$, $\\gamma>0$, which is the setting of Thm~\\ref{theo_curvblowupinitialdatamatter} is an immediate consequence of Thm~\\ref{theo_curvblowupexpansionnormalisedmatter}, as the definition of the density parameter in equation~\\eqref{eqn_definitionomega} yields~$\\Omega>0$ at~$\\tau=0$, and this property is conserved by the evolution equation~\\eqref{eqn_evolutionomega}.\n\n\tConsider now the vacuum case~$\\mu_0=0$, i.\\,e.\\  the setting of Thm~\\ref{theo_curvblowupinitialdatavacuum}.\n\tIn the setting of expansion-normalised variables this corresponds to~$\\Omega=0$, and we have listed all exceptions to curvature\\ blow-up as $\\tau\\rightarrow{-}\\infty$ in Thm~\\ref{theo_curvblowupexpansionnormalisedvacuum}. To prove the theorem for geometric initial data, we have to carry over the individual exceptions to the geometric initial data perspective. As we only consider initial data with a Lie group\\ of Bianchi class~B, a number of these exceptions cannot occur by assumption, namely those assuming Bianchi type~I and~II. Note also that the point Taub~1 is a constant Bianchi type~I solution.\n\n\tWe have argued in Subsection~\\ref{constr_highersymmetrysolutions} that the definitions of local rotational symmetry in terms of geometric initial data, Def.~\\ref{defi_lrsinitialdata}, and in terms of expansion-normalised variables, Def.~\\ref{defi_lrsexpansionnorm}, carry equivalent information. This proves the statement.\n\\end{proof}\n\\begin{rema}\n\tWe can adapt the proof to justify the statement made in Remark~\\ref{rema_curvblowupcosmconst}. Assuming~$\\mu>0$ and~$\\gamma=0$, we switch to expansion-normalised variables. The de~Sitter\\\n\tspacetime or quotients of this space cannot occur as development of the data due to the assumption on the Bianchi class. It therefore follows from Prop.~\\ref{prop_mghdincompletedirections} that the spacetime constructed as in Subsections~\\ref{constr_initialdatatoexpnorm}--\\ref{constr_solutionbasictomghd} is in fact the ma\\-xi\\-mal glo\\-bally hy\\-per\\-bo\\-lic de\\-ve\\-lop\\-ment\\ of the given geometric initial data, otherwise the arguments are identical to those given in the previous proof. The exceptions to curvature\\ blow-up are LRS Bianchi~VI$_{{-}1}$ solutions and solutions converging to plane wave equilibrium points, see Thm~\\ref{theo_curvblowupexpansionnormalisedcosmconst}.\n\\end{rema}\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nImagine that you have edited a 15-minute movie to the picture-lock stage, and now you need to source a continuous music soundtrack to be added in the final stages of postproduction: what are your options? You could attempt to find appropriate music on a stock-audio site such as MusicBed \\cite{MusicBed}, but such sites tend to offer a large number of tracks that are five-minutes or less and very few that are over 10 minutes. Even if you do find a pre-recorded 15-minute music track to your liking, the chances are that you'd need to go back into edit to adjust the cut, so that key events in the movie fit well to the timeline imposed by the choice of music: but what if you don't want to do that (remember, your movie is already at picture-lock)? If you're a gifted {\\em auteur}, you could compose your own soundtrack, fitting the music to the video, but that will likely take some time; and maybe you just don't have the music skills. If instead you have money, you could pay a composer to write a specifically commissioned soundtrack, synchronised to the cut of the video. But what if none of these options are practicable? In this paper we sketch a solution for your woes: we have created a proof-of-concept automated system that analyses the visual content of a video and uses the results to compose a musical soundtrack that fits well to the sequence of events in the video, which we refer to as a {\\em picture-synched}\\\/ soundtrack. We have conducted a series of user studies, and the results demonstrate that our approach has merit and is worthy of further exploration. Because it's easier to refer to such a system via a proper noun, we have named our video-driven automated music composition system {\\em Barrington}.\\footnote{In honour of Barrington Pheloung (1954-2019), composer of clever themes, incidental music, and soundtracks for film and TV.}\n\nAnother way of framing the task performed by Barrington is as follows: in the production of promo videos for music singles, especially in genres such as disco, house, techno, and other forms of electronic dance music with a strongly repetitive beat structure, it is quite common to see videos edited in such a way that they're {\\em cut to the beat}, i.e.\\ where abrupt (hard-cut or jump-cut) transitions between scenes in the video are synchronised with the beat-pattern of the music. In those kind of promo videos, the pattern of switching between scenes within the movie is determined by the structure of the music: the music comes first, and the video is then cut to fit to the music. Barrington is a provisional system for automatically generating answers to the question posed by inverting this process: starting with a video, what music can be created that fits the style and structure of that video? Obviously, for any one video, there is an essentially infinite number of music tracks that could potentially be used as the soundtrack for that video, and deciding which ones are good, and which is best, is a subjective process requiring artistic judgement. Barrington incorporates a number of heuristics which are intended to improve the quality of its output. In its current proof-of-concept form, Barrington is certainly not offered as something that is likely to put professional music composers out of work anytime soon. More realistically, we think it likely that a later evolution of Barrington could conceivably be used to generate bespoke\/unique soundtracks as a service for consumers, and an even more mature version could plausibly be used in creating targeted advertising. For example, the same video-advert could be shown to Customer A with a rock-music picture-synched soundtrack, and to Customer B with a classical-music picture-synched soundtrack, with both soundtracks produced by Barrington on the basis of prior knowledge of the customers' respective tastes in music. \n\nIn Section \\ref{sec:bckgnd} we review relevant background material. The technical notes on the design and implementation of Barrington are given in Section  \\ref{sec:implement}. Then in Section \\ref{sec:trials} we give details of our user studies and discuss the results. We close the paper with a discussion of further work in Section \\ref{sec:concl}.\n\n\n\\section{Background}\n\\label{sec:bckgnd}\n\\input{DBC_background_JB.tex}\n\n\n\n\\section{Implementation}\n\\label{sec:implement}\n\\input{DBC_implementation_JB.tex}\n\n\n\\section{User Evaluation}\n\\label{sec:trials}\n\\input{DBC_trials_JB.tex}\n\n\n\\section{Future Work}\n\\label{sec:concl}\n\\input{DBC_conclusion.tex}\n\n\n\n\n\\bibliographystyle{ACM-Reference-Format}\n\n\n\n\\subsection{Design}\n\\label{sec:design}\n\n In traditional film-making, directors and editors work closely with composers to develop a soundtrack that fits with their narrative intent. Similarly, our intention was to create a tool that was not totally autonomous but rather could be guided by visual content creators.\n \n\n \n\n \n\n \n Barrington is primarily written in the Python 3 programming language, due to the availability of a wide range of open-source libraries; as is made clear later in this paper, some aspects of Barrington are controlled by shell-scripts consisting of operating-system commands.\n\n\\subsection{Scene Transition Detection}\nScene transitions are the most easily identifiable markers of progression in a video, and served as the starting point for the analysis. Scene transition detection, often also referred to as shot boundary detection, is a difficult task for computers, due to the variety of transition methods video editors have at their disposal. Abrupt transitions follow a predictable pattern, as the change in content from one frame to another is sudden and usually extensive. Gradual transitions often utilise visual effects to blend together two scenes into a smooth manner, such as cross-dissolves or fades, over any length of time, and with very little change in content from frame to frame. This variability makes it difficult to create a single method of detecting every type of transition. Transition detection algorithms typically look at pairs of frames, computing a score for each using some defined similarity measure, then evaluating these scores against some threshold to determine if a transition has occurred.\n\n\nFollowing \\cite{zhang1993automatic}, Barrington implements two basic scene detectors, one to detect abrupt transitions ({\\em cut detection}) and the other to detect fades to\/from black ({\\em fade detection}). \n\nCuts are detected by computing  the average change in hue, saturation and intensity values from one frame to the next. This aggregate value will be low between frames with little change in content, and high between frames with large changes. A threshold is specified such that frames above the threshold are marked as abrupt transitions. \n\nFades are detected using RGB intensity. For each frame, the R, G, and B pixel values are summed to produce an intensity score. A threshold is specified such that the RGB intensity values in the majority of frames within a scene lie above said threshold; if the RGB intensity for a frame falls below the threshold, it is marked as the start of the fade, and the next frame to return above the threshold is marked as the end. This remarkably simple technique works effectively, as any fade to\/from black will inevitably contain at least one black frame, with an RGB intensity value of 0.\n\n\nPySceneDetect \\cite{PySceneDetect} was the open-source library chosen to implement the scene detection algorithms described above. It provides methods for performing both fade detection and cut detection, each outputting a list of scenes identified with their start and end times. Barrington uses PySceneDetect's default threshold values of 12 and 30 for the fade and cut detection respectively, outputting a final list of scenes detected through both detection methods. The library is also able to export clips of individual scenes, which are used for scene-level analysis discussed later in Section \\ref{sec:scene-energy}.\n\n\n\n\n\n\\subsection{Loop-based Soundtracks}\n\nThe list of scenes output from scene detection provide a starting point on which to base the temporal structure of the soundtrack. Taking a simplistic view, we can split the overall soundtrack into sections commonly found in popular music, such as the intro, verse, chorus and coda (outro), and map each scene to a section of music: the first scene being the intro and the last the coda. Mapping the rest of the scenes is more complicated, as different videos will have different numbers of scenes.\n\nIn order to gain an understanding of how best to approach this, we first created a simple rule-based system that loops some pre-existing samples of music, with the middle scene (or middle pair of scenes for cases where the total number of scenes is even) arbitrarily chosen as the point at which the soundtrack peaks, and every other scene transition either introduces or removes a particular sample. This is analogous to the way music is composed professionally, as repetition is often a key part of musical structure. These loop-based sequenced soundtracks were an important early step in developing Barrington because they allowed us to explore and demonstrate video-driven sequence-assembly, but they offer only a coarse-grained approach to music composition. \n\nTo give Barrington more creative scope, we switched to using fully-fledged music generation systems. We initially used Magenta PolyphonyRNN, a recurrent neural network, based on the BachBot architecture described in \\cite{liang2016bachbot}. It is able to compose and complete polyphonic musical sequences using a long short-term memory (LSTM) generative model, without having to explicitly encode musical theory concepts. BachBot's success is based on modelling melodies and harmonisation independently. It is able to learn a representation of harmonisation based on samples from a corpus, and apply it to any melody. In user tests it was capable of producing music virtually indistinguishable from Bach's compositions. Unfortunately, the computational costs of PolyphonyRNN meant it was not suitable for our system: run-times lasting a week or more could be consumed in training the system, and the results were often disappointing. We therefore switched to IBM's Watson Beat. \n\n\n\\subsection{Music Generation: Watson Beat}\n\nIBM's Watson Beat \\cite{IBMWatsonBeat} is designed to be a compositional aid for musicians, helping them generate new ideas they can then adapt to compose a piece of music, although it is capable of producing a complete piece on its own. Watson Beat provides more granular control over the composition compared to PolyphonyRNN; it composes parts for multiple instruments at once, and allows users to define the composition's mood, instrumentation, tempo, time signature and energy. This made it a much more versatile tool for developing Barrington.\n\n\nA brief overview of the process of developing Watson Beat is available on IBM's website \\cite{IBMWatsonBeat}, in which the developers describe using supervised learning on a vast amount of data relating to each composition's pitch, rhythm, chord progression and instrumentation, as well as labeled information on emotions and musical genres. A key feature of Watson Beat is its ability to arrange the musical structure in a manner that provides the user control over individual sections of music, though it is unclear if this was manually encoded or learned from training data.\n\nWatson Beat requires two inputs to compose a piece of music: a MIDI file with a melody to base the composition on; and a \\textit{.ini} file that specifies the structure of each section of the composition. The \\textit{.ini} file for each composition must start with the following parameters:\n\n\\renewcommand{\\baselinestretch}{1}\\normalsize\n\\begin{quote}\n\\begin{itemize}\n\\item \\textbf{Composition duration}: in seconds.\n\\item \\textbf{Mood}: which is selected from one of the 8 pre-configured options.\n\\item \\textbf{Complexity}: three options (simple, semi-complex, or complex) which determines the complexity of the chord progressions in the composition.\n\\end{itemize}\n\\end{quote}\n\nThis is followed by a list of parameters for each section of the composition, which are explained below:\n\n\\renewcommand{\\baselinestretch}{1}\\normalsize\n\\begin{quote}\n\\begin{itemize}\n\\item \\textbf{Section ID}: a series of consecutive integers starting from 0 identifying the current section.\n\\item \\textbf{Time signature}.\n\\item \\textbf{Tempo}: in BPM.\n\\item \\textbf{Energy}: three options (low, medium or high) which represents the number of active instruments for the current section. The number of instruments (also referred to as layers) each energy option uses can vary, and depends on the selected mood. For example, low could signify 1-3 instruments for one mood option, but 2-5 for another.\n\\item \\textbf{Section duration}: in seconds (allows for a range to be specified, such as 10 to 20 seconds, for situations where a precise duration is not required).\n\\item \\textbf{Direction}: set to either up or down, and determines whether instrument layers will be added or removed during the section.\n\\item \\textbf{Slope}: one of three options (stay, gradual or steep) which control the rate of change of the Direction, that is, the rate at which layers are added or removed.\n\\end{itemize}\n\\end{quote}\n\n\n\nWatson Beat only requires a duration range to generate each section of music, and calculates a best-fit phrase length, tempo and time signature, from a pre-specified range of options for the given mood, for durations within this range. This calculation is non-deterministic as it utilises a randomly generated number to pick from the computed values. Given that section durations for the autogenerated soundtracks must match exactly with the video, the best-fit calculation is not ideal as it can often lead to an incorrect section duration. \n\nTo counter this, Barrington incorporates a Python function to compute the correct tempo and time signature required to produce the specified section duration. The function loops through every possible combination of parameters and selects those where the number of phrases is a whole number and the calculated duration equals the desired duration. These lists of valid combinations are generated for each scene, and then trimmed to only contain options that ensure a consistent tempo throughout the soundtrack. For each section, one of these possible combinations is randomly selected for use, ensuring variation in the resulting soundtracks. To generate the soundtrack, Barrington invokes a Bash shell script that runs Watson Beat, specifying the \\textit{.ini} file, an existing MIDI file with an input melody, and the output folder for the generated MIDI files.\n\nAs described thus far, Barrington is able to automatically produce synchronised soundtracks with discernibly different sections, only requiring the \\textit{energy}, \\textit{direction} and \\textit{slope} parameters to be manually input. These parameters, along with the composition's mood, play a role in the subjective interpretation of how the soundtrack supports the visual content, therefore it is important for users to have control over them. Nevertheless, for the system to operate at higher levels of autonomy, it must also be capable of determining these values itself, directly from the input video.\n\n\\subsection{Determining the Energy of a Scene}\n\\label{sec:scene-energy}\nThere are no definitive criteria to determine the energy of a scene of video or a section of music; the parameter is intrinsically dependant on the music generation tool being used. Watson Beat uses energy, direction and slope to define the number of instrument layers active in a given section of music, and how this number should vary through the composition, so energy values such as low, medium and high are only relevant in the context of a Watson Beat composition. Traditionally, tempo is used to convey energy - a high tempo is considered more energetic than a low tempo - so it stands to reason that any representation of energy derived through video analysis should control tempo, as well as the instrumentation parameters.\n\nA naive approach to this problem could be to use activity or `busyness' within a scene to determine energy. For example, a scene with lots of objects can be considered more energetic than one with very few. The downside of naive approximations is that once again there is no consistent way of comparing the results across different videos. Without a standardised metric, the only method of determining if a scene has  low or high energy is to compare it with others within the same video.\n\nTo test the effectiveness of a naive object-counting approach, we wrote a Python function that uses ImageAI \\cite{ImageAI}, an open-source computer vision library, and version 3 of the YOLO object detection model \n \\cite{DBLP:journals\/corr\/RedmonDGF15}. YOLO utilises a single convolutional neural network (CNN) for training and detection. which are often used for image classification tasks. In mathematics, a convolution expresses how the shape of some function is affected as it is shifted across a \\textit{static}, or non-moving, function, i.e. `blending' the two together. CNNs treat the input image as this static function and pass lots of different functions over it, each of these being a \\textit{filter} that looks for a specific feature in the image. Each filter is passed over small sections of the image at a time, to build up a \\textit{feature map} that tracks the locations at which various features occur across the image. \n \nYOLO is able to learn a general representation for different object classes, while optimising detection performance during training, resulting in faster, more precise detection. YOLO subdivides an image into a grid, using features from across the whole image to predict the bounding box for every object class, and a confidence score that encodes the probability of that box containing an object. The clips of individual scenes are processed one by one to produce a listing of the number of objects detected in each scene. Barrington then calculates the mean and standard deviation of the results to label the scenes low, medium or high energy: scenes with a number of objects fewer than one standard deviation below the mean are classified as low energy; scenes within the mean plus or minus one standard deviation as medium-energy; and scenes one standard deviation or greater above the mean as high-energy. A similar approach is used to match an associated tempo; the tempo range for a particular mood is split into three and matched the same way as the energy value, then a tempo within this range is selected from the list of valid tempo options. This approach does not work for all scenes, as it relies on a tempo within the sub-range to be valid for the particular scene duration, which may not be the case, but it is a useful first approximation. \n\nThe Watson Beat \\textit{direction} and \\textit{slope} parameters are selected by comparing each scene's energy value to the next, using a heuristic decision tree, details of which are given in \\emph{Anonymous Citation}.\n\n\nFor the final scene in the video, the function assumes the next scene has the same energy value as the current. Note that if `stay' is selected for the Slope parameter, it doesn't matter what direction is selected as the number of instrument layers stays constant through the section.\n\n\\subsection{Post-Processing}\n\nThe final step involves post-processing the MIDI files and exporting a video with the generated soundtrack. Watson Beat outputs a separate MIDI file for each instrument within each section of music, allowing composers to rearrange the piece and select the instruments to use when converting to an audio file. While this option is still available for more technical users, the Barrington default post-processing step converts these MIDI files into a single audio file containing the final soundtrack, as well as producing a copy of the input video with the new soundtrack attached.\n\nThe MIDI specification allows for instruments to be declared in a message at the start of a file, however the instrument labels used by Watson Beat do not map directly to instruments listed in the General MIDI 1 (GM1)\\footnote{\\url{https:\/\/www.midi.org\/specifications-old\/item\/gm-level-1-sound-set}} sound set. Barrington takes all the generated MIDI files as an input, assigning the correct tempo to each section using the reference structure, and the correct instrument using a simple mapping based on recommendations found in Watson Beat's documentation (\\url{https:\/\/github.com\/cognitive-catalyst\/watson-beat\/blob\/master\/customize_ini.md}). It then concatenates them into a single file and uses FluidSynth \\cite{FluidSynth}, an open-source software synthesizer, to play and record the soundtrack to audio. A Bash script exports the final video, using ffmpeg  \\cite{ffmpeg}, a cross-platform video conversion tool, to attach the soundtrack to the input video. Samples of output from Barrington are available online at \\emph{Anonymous Website}.\n\n\n\\subsection{Music Terminology}\n\nFor brevity, we assume that the reader has a basic understanding of Western music structure, such as the way in which music is composed of a number of  \\textit{bars} (or \\textit{measures}), where each bar represents a segment of time containing a specified number of beats. The \\textit{phrase length} is the number of bars that make up a phrase; phrases in most contemporary Western popular styles of music tend to contain four bars.\n\nThe number of beats in a bar is denoted by a piece's \\textit{time signature}, for example 3\/4, where the denominator indicates the duration of a beat, and the numerator indicates the number of beats in the bar. The time signature 3\/4 tells us that there are 3 beats, each with the length of a quarter (1\/4) note, in each bar. Similarly \\textit{tempo} dictates the speed of a piece of music and is usually indicated in beats per minute (BPM). \\textit{Monophonic} music is style of music that consists solely of a melody, without any harmonisation or other accompaniment. \\textit{Polyphonic} music consists of two or more parts, each with its own melody, with harmonisation across parts.\n\n\\subsection{Algorithmic Composition}\n\n\n\nThe following subsections cover the two music generation tools used in Barrington, both of which use neural-network machine learning techniques. For further details of neural networks, see e.g. \\cite{Bishop95}.\n\n\\subsubsection{Magenta PolyphonyRNN}\n\\label{sec:poly}\n\nPolyphonyRNN is a recurrent neural network, based on the BachBot architecture described in \\cite{liang2016bachbot}. It is able to compose and complete polyphonic musical sequences using a long short-term memory (LSTM) generative model, without having to explicitly encode musical theory concepts. BachBot's success is based on modelling melodies and harmonisation independently. It is able to learn a representation of harmonisation based on samples from a corpus, and apply it to any melody. In user tests it was capable of producing music virtually indistinguishable from Bach's compositions. LSTMs differ from conventional feed-forward neural networks in that they contain feedback connections, which connect outputs from hidden layers back to the input layer of the network, allowing the network to form a representation in an iterative manner. For a more thorough exploration of the benefits of feedback neural networks see Zamir et al.\\cite{zamir2017feedback}. LSTMs are discussed in more detail in \\cite{HochreiterSchmidhuber1997}.\n\n\n\n\\subsubsection{IBM Watson Beat}\n\\label{sec:watson}\n\nWatson Beat models polyphonic music in a similar manner to PolyphonyRNN. There is no published literature to confirm this, however, the code was released as an open-source project (\\url{https:\/\/github.com\/cognitive-catalyst\/watson-beat\/}). A brief overview of the process of developing Watson Beat is available on IBM's website \\cite{IBMWatsonBeat}, in which the developers describe using supervised learning on a vast amount of data relating to each composition's pitch, rhythm, chord progression and instrumentation, as well as labeled information on emotions and musical genres. A key feature of Watson Beat is its ability to arrange the musical structure in a manner that provides the user control over individual sections of music, though it is unclear if this was manually encoded or learned from training data.\n\n\\subsection{Video Analysis}\n\n\\subsubsection{Scene Transition Detection}\n\\label{scene-transition-detection}\n\nScene transition detection, often also referred to as shot boundary detection, is a difficult task for computers, due to the variety of transition methods video editors have at their disposal. Abrupt transitions follow a predictable pattern, as the change in content from one frame to another is sudden and usually extensive. Gradual transitions often utilise visual effects to blend together two scenes into a smooth manner, such as cross-dissolves or fades, over any length of time, with very little change in content from frame to frame. This variability makes it difficult to create a single method of detecting every type of transition. Transition detection algorithms typically look at pairs of frames, computing a score for each using some defined similarity measure, then evaluating these scores against some threshold to determine if a transition has occurred.\n\nAs scene transition detection is not the focus of this project, the ability to accurately detect complex transitions is not fundamental to the outcome, therefore currently Barrington uses only two basic transition detectors, one to detect abrupt transitions ({\\em cut detection}) and the other to detect fades to\/from black ({\\em fade detection}).\n\nThe similarity measure used for fade detection is RGB intensity. For each frame, the R, G, and B pixel values are summed to produce an intensity score. A threshold is specified such that the RGB intensity values in the majority of frames within a scene lie above said threshold; if the RGB intensity for a frame falls below the threshold, it is marked as the start of the fade, and the next frame to return above the threshold is marked as the end. This remarkably simple technique works effectively, as any fade to\/from black will inevitably contain at least one black frame, which will have an RGB intensity value of 0.\n\nCheng et al. \\cite{CHENG20012259} recommend the use of the HSI (aka HSV or HSL) \ncolour space for computer vision tasks on colour images, as it enables segmentation in the one-dimensional hue space, which is less computationally expensive than in 3D RGB space. Cut detection requires comparing two frames for changes in content, therefore the HSI space is more appropriate than RGB. The similarity measure for cut detection is computed by taking the average of the change in hue, saturation and intensity values from one frame to the next. This aggregate value will be low for frames with little change in content, and high for frames with large changes. A threshold is then specified such that frames above the threshold are marked as abrupt transitions. \n\nThe twin-comparison approach described in \\cite{zhang1993automatic} has been adapted to combine fade and cut detection for Barrington. RGB intensity and delta HSI values are computed for each frame, and both thresholds are monitored for scene transitions.\n\n\\subsubsection{Object Detection}\n\\label{sec:obj}\n\nObject detection has been the subject of intense research for many years in computer vision. Traditional machine learning approaches to object detection, such as the popular Viola-Jones method \\cite{Viola01robustreal-time}, involve first defining the features of the object, then training a classifier on a labelled dataset of images containing the object. Finally, the classifier's object detection performance is evaluated at different locations and scaling factors across an image.\n\nConvolutional neural networks (CNNs) are now considered by very many to be the preferred tool for image classification tasks. In mathematics, a convolution expresses how the shape of some function is affected as it is shifted across a \\textit{static}, or non-moving, function, i.e. \"blending\" the two together. CNNs treat the input image as this static function and pass lots of different functions over it, each of these being a \\textit{filter} that looks for a specific feature in the image. Each filter is passed over small sections of the image at a time, to build up a \\textit{feature map} that tracks the locations at which various features occur across the image. \n\nYOLO \\cite{DBLP:journals\/corr\/RedmonDGF15}, the method used in Barrington, utilises a single CNN for training and detection. Utilising dropouts, YOLO is able to learn a general representation for different object classes, while optimising detection performance during training, resulting in faster, more precise detection. YOLO subdivides an image into a grid, using features from across the whole image to predict the bounding box for every object class, and a confidence score that encodes the probability of that box containing an object.\n\n\n\n\n\\subsection{Methodology}\n\nUltimately, a system like Barrington is only useful if the end user enjoys its outputs, so this is a primary measure by which to judge success. Our goal was to determine how the Barrington-generated soundtracks affected a user's experience of a video. There are few, if any, objective methods for measuring this, therefore a user study was carried out to measure subjective responses to a selection of Barrington's outputs, adapting the procedure used by Ghinea and Ademoye \\cite{ghinea2012sweet}. They used a questionnaire to measure a user's quality of experience (QoE) of visual media when they were presented alongside stimuli in other sensory modalilties, such as olfaction. The QoE study they performed was particularly well suited for evaluating Barrington, as it also investigated the role of synchronisation of the different sensory stimuli on users' QoE, that is, how temporal offsets between the sensory stimuli affected users' reported QoE. A key feature of Barrington is the synchronisation of changes in auditory and visual content, so it is important to assess  whether users were able to detect this synchronisation and, if so, how it affected their QoE. \n\nAnother element of Barrington that we wanted to evaluate is the relationship between the \\emph{energy} parameter of a section of the soundtrack, and the corresponding scene's visual content. In particular, we wanted to test whether the object detection method discussed in Section \\ref{sec:scene-energy} is a suitable proxy for the mood or intensity of a scene. \n\nFinally, it is important to assess how Barrington's output compares with existing solutions in the form of royalty-free music that is not picture-synched. We wanted to test whether the soundtracks generated by Barrington produced greater QoE than royalty-free music whose structure is not synched with the video content.\n\nHow these key feautures affected users' QoE was tested using an approach inspired by a Dual-Stimulus Impairment Scale (DSIS), a standard video testing methodology recommended by the International Telecommunications Union (ITU) \\cite{series2012methodology}. In a DSIS test the viewer sees a reference video and a video with an \\textit{impairment} or alteration, and is asked to indicate on a Likert scale how irritating they find this impairment. Instead of only measuring a user's experience of the impaired video, we asked participants to rate both the reference and the impaired videos, in order to determine whether their QoE was affected. Participants were not informed which video was which, to prevent any bias when responding to the questions.\n\n\\subsubsection{Hypotheses}\n\nThe user evaluation was carried out to test two hypotheses. \n\nFirst, we wanted to investigate the effect of synching a soundtrack with video scene changes, hypothesising that:\\\\\n\\emph{H1} -- users will report higher QoE for videos combined with synchronised soundtracks generated by Barrington than for videos combined  with unsynchronised soundtracks. \n\nSecond, we sought to investigate whether the method of selecting the soundtrack's energy parameters was appropriate, hypothesising that:\\\\\n\\emph{H2} -- users will perceive a correlation between the auditory and visual content of the synchronised soundtracks generated by Barrington.\n\n\\subsubsection{Participants}\n\nThe 26 participants who volunteered to take part in this study consisted of 17 males and 9 females between the ages of 20 and 24. All the participants were either undergraduate or graduate students at \\emph {Anonymous University}, the majority of whom study in the Computer Science Department. Each participant was provided with information and task instruction sheets explaining  the goal of the project (which can be viewed in \\emph{Anonymous Citation}) before taking part, and was left alone in a private meeting room to proceed through the experiment at their own pace. \n\n\\subsubsection{Procedure}\nEach participant completed four tasks, the first three focusing on, respectively: picture-synchronisation, the relationship between auditory and visual energy, and a comparison with royalty-free soundtracks. The final task consisted of a short six item questionnaire relating to videos and soundtracks. \n\n\nFor the first three tasks, each participant was presented with two videos, the reference and the impaired, which had identical visual content but different soundtracks. They were asked to watch each video in turn and complete the seven-item questionnaire shown below. Each item on the questionnaire required participants to select one option from a five-point Likert scale. For statements 1-6, participants were asked to indicate their level of agreement (LoA) by choosing one of the following options: \\textit{Strongly agree}, \\textit{Agree}, \\textit{Neither agree nor disagree}, \\textit{Disagree}, or \\textit{Strongly disagree}. Statement 7 required participants to rate the synchronisation of the soundtrack and the scene transitions by selecting one of the following options: \\textit{Too early}, \\textit{Early}, \\textit{At an appropriate time}, \\textit{Late}, or \\textit{Too late}.\n\n\\begin{quote}\n\\begin{enumerate}\n\\item \\textit{The soundtrack added to the video experience}\\label{stm:1}\n\\item \\textit{The soundtrack was relevant to the video}\\label{stm:2}\n\\item \\textit{The soundtrack was enjoyable}\\label{stm:3}\n\\item \\textit{There were noticeable points in the soundtrack where the melody\/rhythm changed}\\label{stm:4}\n\\item \\textit{The mood\/intensity of the soundtrack reflected the mood\/intensity of the video content}\\label{stm:5}\n\\item \\textit{The synchronisation between the audio and video improved the overall experience}\\label{stm:6}\n\\item \\textit{In relation to the scene transitions in the video, the melody\/rhythm of the music changed}\\label{stm:7}\n\\end{enumerate}\n\\end{quote}\n\nIn the final task, participants were presented with the following six statements and asked to indicate their LoA on a five-point Likert scale: \\textit{Strongly agree}, \\textit{Agree}, \\textit{Neither agree nor disagree}, \\textit{Disagree}, or \\textit{Strongly disagree}.\n\n\\begin{quote}\n\\begin{enumerate}\n\\item \\textit{Soundtracks are an important aspect of video content}\\label{stm:4-1}\n\\item \\textit{Soundtracks improve the experience of watching video content}\\label{stm:4-2}\n\\item \\textit{The mood of a soundtrack should match the mood of the video content}\\label{stm:4-3}\n\\item \\textit{A bad soundtrack negatively affects the viewing experience}\\label{stm:4-4}\n\\item \\textit{There was an obvious relationship between the soundtracks and the video content in the previous examples}\\label{stm:4-5}\n\\item \\textit{The synchronisation between the soundtrack and video improved the overall experience}\\label{stm:4-6}\n\\end{enumerate}\n\\end{quote}\n\n\\subsubsection{Test Videos}\nWe created three 60-second duration, soundtrack-free videos for this study - one for each of the first three tasks. The videos consisted predominantly of open source wildlife and nature clips, chosen because they best complemented the \\textit{Inspire} mood that had been selected in Watston Beat and which generates calming, cinematic music. The only exceptions to this were a clip of a busy pedestrian crossing and a clip of a herd of wildebeest, which were used because they were a sharp contrast to the relatively motion-free nature scenes, and specifically chosen to output a high energy parameter when object detection was used. Each soundtrack-free video was processed by Barrington to create the soundtracks for the reference videos, while the process for creating each of the impaired videos is described below.\n\n{\\bf Task 1} investigated the effect of synchronisation of the video scene changes and the soundtrack on users' QoE and tested H1. The video used contained three scene transitions, which provided ample opportunity for the participants to determine if the audio and video content were synchronised. The reference video the soundtrack was synchronised with the video scene changes. In the impaired video, we altered the scene-transition timings so the soundtrack transitions were three seconds earlier than their corresponding scene transitions, then used the system to generate a new soundtrack based on the alteration.\n\n{\\bf Task 2}  investigated the effect of the value of the energy parameter and investigated H2 The video contains just two scenes, one with  low energy and the other high. The high energy scene contains lots of moving objects, which intuitively corresponds to an increased scene intensity. To create the impaired video the energy parameters were swapped from high to low and  {\\em vice versa}. (N.B. abrupt scene transitions were ignored in this video, due to the fact that one of the clips used consisted of multiple hard cuts within seconds of each other, which led to very short sections of music that clashed quite heavily and sounded bad.)\n\n{\\bf Task 3}  compares a video with a Barrington generated soundtrack to a video with a royalty-free soundtrack and also investigated H1. For the impaired video, a song was selected from Bensound \\cite{Bensound}, an online royalty-free music provider, and trimmed to the duration of the video.\n\n\n\n\n\\subsection{Results}\n\n\\subsubsection{Task 1: Synchronisation}\n\nTwo aspects of synchronisation were tested: a) the detection of auditory and visual transitions; and b) the effect of their synchronisation on the participants' enjoyment of the video. For the reference video in Task 1, 26\/26 (100\\%) participants responded that the music changed, or progressed, at an appropriate time in relation to scene transitions, while only 3\/26 (12\\%) believed the same for the impaired video, where the soundtrack transitioned three seconds prior to the visual scence transitions.\n\nThis initially suggests that participants were able to recognise the points at which musical transitions occurred, and determine whether they were in-sync with visual scene transitions. However, it is clear from the results in Figure \\ref{res:q1-7} that while participants were able to detect that the audio and visuals were not synchronised, they were poor at judging whether the music transition occurred before or after the scene transition. 46\\% of participants responded with \\emph{late} or \\emph{too late} to question \\ref{stm:7} even though the soundtrack was configured to transition three seconds earlier than the video.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{DBC_Fig4_2.png}\n\\caption{Participant responses to the statement \\textit{In relation to the scene transitions in the video, the melody\/rhythm of the music changed}, in Task 1 of the user study.}\n\n\\label{res:q1-7}\n\\end{figure}\n\nWhen looking at the response to this question across all videos synchronised by Barrington,\nFigure \\ref{res:sync_all} shows that the majority of participants thought the synchronisation was appropriately timed.\n\nIt's interesting to note the remaining participants all selected \\emph{late}, suggesting that participants may differ from one another in what they consider as being appropriately synchronised. Additionally, the videos for Task~2 elicited different timing responses, which is unexpected because the timing parameters for both were identical. \n\nThis is possibly a result of the musical transitions being too gradual, leaving participants unclear as to when a transition occurred. Or it is perhaps an indication that the end of a scene is not the best marker for a musical transition, although it is not immediately clear what an alternative may be.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{DBC_Fig4_3.png}\n\\caption{Participant responses to the statement \\textit{In relation to the scene transitions in the video, the melody\/rhythm of the music changed}, across all videos produced via Barrington.}\n\\label{res:sync_all}\n\\end{figure}\n\nFigure \\ref{res:q1-6} presents participant responses to statement \\ref{stm:6}, which ascertains whether the soundtrack has enhanced their enjoyment of the video. The impaired video, 1B, elicited a predominantly neutral response, which could suggest either that the delay between musical and visual transitions is not significant enough to adversely affect enjoyment, or that a lack of synchronisation does not adversely affect enjoyment. This makes intuitive sense, as one would assume a large proportion of videos on media platforms contain unsynchronised soundtracks. In contrast, the reference video 1A received predominantly positive responses, suggesting that synchronisation can enhance the quality of experience.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{DBC_Fig4_4.png}\n\\caption{Participant responses to the statement \\textit{The synchronisation between the audio and video improved the overall experience}, in Task 1 of the user study.}\n\\label{res:q1-6}\n\\end{figure}\n\nLooking at participant responses to statement \\ref{stm:4}, plotted in Figure \\ref{res:q1-4}, can give us further insight into why the above results neither supports nor disproves the hypothesis that synchronisation can enhance the experience.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{DBC_Fig4_5.png}\n\\caption{Participant responses to the statement \\textit{There were noticeable points in the soundtrack where the melody\/rhythm changed}, in Task 1 of the user study.}\n\\label{res:q1-4}\n\\end{figure}\n\nOnly 50\\% of participants were able to detect musical transitions in the impaired video 1B, compared with 73\\% for the reference video 1A. This discrepancy in responses is unexpected, as intuition suggests participants would respond the same way for 1A and 1B, given that both soundtracks contain three transitions. \n\nIt is possible this is due to the different sections within those specific soundtracks sounding very similar to one another, however it is not possible to determine this from the results of this study. Ambiguity regarding participants' interpretations of the word \\textit{changed}\\\/ could also have played a role, however none of the participants brought this up in the informal post-study feedback conversations. It seems that participants found it difficult to detect melody and rhythm, despite overall responding with a more positive perceived experience for picture-synched videos.\n\n\\subsubsection{Task 2: Energy and Mood\/Intensity}\n\nThis test was to determine whether the number of objects in a scene is a good proxy for how the soundtrack's energy levels should reflect the visual intensity of a scene. Figure \\ref{res:q2-5} shows the results for the reference and impaired videos in Task 2 of the study. Only a small proportion (27\\%) of participants provided the expected response disagreeing with statement \\ref{stm:5} for the impaired video, whereas 81\\% agreed for 1A and 54\\% agreed for 1B. The results imply that the energy parameter has a minimal effect on the perceived mood\/intensity of visual content, although it is promising to note that some participants did perceive a difference when the change was made, suggesting a more refined method could produce better results.\n\nThis raises the question of whether a soundtrack should reflect the mood\/intensity of visual content: Section \\ref{sec:task4} provides some insight on this.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{DBC_Fig4_6.png}\n\\caption{Participant responses to the statement \\textit{The mood\/intensity of the soundtrack reflected the mood\/intensity of the video content}, in Task 2 of the user study.}\n\\label{res:q2-5}\n\\end{figure}\n\n\\subsubsection{Task 3: Comparison with Royalty-free Music}\n\nIn Task 3, participants responded more positively towards the royalty-free soundtrack on statements relating to the mood and relevance of the soundtrack with respect to the visual content. Figure \\ref{res:q3-5} shows that 100\\% of participants agreed that the mood\/intensity of the royalty-free soundtrack reflected the mood of the visual content, while only 46\\% thought the same for the Barrington-generated soundtrack.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{DBC_Fig4_7.png}\n\\caption{Participant responses to the statement \\textit{The mood\/intensity of the soundtrack reflected the mood\/intensity of the video content}, in Task 3 of the user study.}\n\\label{res:q3-5}\n\\end{figure}\n\nThis result reinforces the prior suspicion that the object-detection proxy for the soundtrack's energy level captures a limited representation of the mood of a scene in a video. However, it could also be a reflection of the audio quality of the music produced by the system. The royalty-free soundtrack was professionally composed and instrumented, whereas the generated soundtracks use a basic software synthesizer to apply instrument sounds as a post-process, a limitation imposed by Watson Beat. When it comes to synchronisation, participants were in agreement that the timing of the generated soundtrack was more accurate than the royalty-free one (see Figure \\ref{res:q3-7}), although this did not have the expected result on user enjoyment, which once again could have been due to the quality of the soundtrack. Figure \\ref{res:q3-3} shows a negligible difference (4\\%, or 1 participant) in the enjoyment of the generated vs royalty-free soundtrack. This study was not intended to compare the quality of generated soundtracks against professionally produced royalty-free samples, but it is encouraging to note that the Watson Beat soundtracks were no less enjoyable than their counterparts.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{DBC_Fig4_8.png}\n\\caption{Participant responses to the statement \\textit{In relation to the scene transitions in the video, the melody\/rhythm of the music changed}, in Task 3 of the user study.}\n\\label{res:q3-7}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{DBC_Fig4_9.png}\n\\caption{Participant responses to the statement \\textit{The soundtrack was enjoyable}, in Task 3 of the user study.}\n\\label{res:q3-3}\n\\end{figure}\n\n\\subsubsection{Task 4: Participants' Reflection on Video Soundtracks}\n\\label{sec:task4}\n\nFigure \\ref{res:q4} presents a summary of the responses to the final task of the study that involved indicating a level of agreement with six statements. Generally speaking, participants agreed with all the statements, although some disagreed with the notion that soundtracks should match the mood of the visual content (statement \\ref{stm:4-3}) and likewise some did not detect a relationship between the auditory and visual content (statement \\ref{stm:4-5}) in the sample videos.\n\nThe responses to statement \\ref{stm:4-5} were understandable given some participants struggled to notice the temporal synchronisation, while the results from Task 2 suggested the number of objects in the scene was not a strong indicator of mood\/intensity.\n\nThe key takeaway from this task is that 100\\% of participants agree that soundtracks play an important role in experiencing video content. 62\\% of participants strongly agreed that synchronisation improves the experience. While these results suggest that Barrington is not yet performing as strongly as hoped for, there is a clear indication that the problems being tackled are relevant to potential users.\n\nA complete breakdown of responses to individual questions in this study can be found in \\emph{Anonymous Citation}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{DBC_Fig4_10.png}\n\\caption{Participant responses to all statements in Task 4 of the user study.}\n\\label{res:q4}\n\\end{figure}\n\n\\subsection{Reflections}\n\\label{sec:reflection}\n\nThis user study, while by no means conclusively proving that synchronisation of audio and video provides a better experience than unsynchronised content, was an important step in validating some of the work that has gone into this project. We used a fixed set of questions to disguise which video was the reference and which the impaired. Even when using subjective measures of quality, careful consideration is required to avoid ambiguous questions that could result in different interpretations by different participants. \n\n\nAdditionally, some interactive tasks could have been used instead of passive questionnaires. One example would be to ask participants to manually mark the points in a video at which they detect a scene transition, in order to objectively evaluate the performance of scene detection algorithms against a human. These interactive tasks could also form the basis of a dataset that could be used to train a machine to tackle some of the challenges with this topic.\n\nThis work has shown that temporal synchronisation of audio and video can lead to an enhanced user experience, and that Barrington's synchronisation method was accurate, as perceived by the  participants. Our user evaluation provides some support for hypothesis \\emph{H1}. Less successful was the object-counting proxy used to determine the energy of a scene; this method was not able to accurately convey the intended relationship between auditory and visual content to a viewer and we can reject hypothesis \\emph{H2}.\n\nThe limiting factor of this system is the music generation system used. Watson Beat is the most feature-packed of the open-source options currently available, but a lack of published literature or  documentation makes it a difficult tool to work with and adapt for more general use. As it stands, in terms of musical structure, there is little one can do beyond defining the number and duration of sections within a piece of music. More fine grained temporal control over musical components, such as chord progressions and melodies, would allow for the development of new methods to create a temporal mapping. There is the potential for scene transitions, or even content within a scene, to define how different chords within a piece of music flow. For example, an action shot of a child raising its arms could be used to trigger a musical buildup or the climax of a particular phrase of music, which would more effectively demonstrate audio-visual synchronisation.\n\nEmotionality is an extremely important aspect of music; significant work would be required to enable this system to associate moods and genres of music with visual content. Creating a dataset of images labelled with the same mood and emotion tags as the corpus of music used to train the music generation model would be an excellent first step in being able to associate visual and auditory moods using machine learning. For example, an image classifier trained on such a dataset could potentially identify the overarching mood of a video as happy or sad, using the kind of emotion detection described in \\cite{DBLP:journals\/corr\/abs-1712-02896}. The music generation model would then generate music in a style that has been labelled with the same mood, as opposed to the user-driven approach currently taken in this system.\n\nBarrington is able to automatically generate soundtracks for videos, with the only requirements being a mood and melody on which to base the composition. Scene transition detection and some scene-level object detection form a base structure for the composition to follow. Once generated, a simple post-processing step converts the MIDI representation of the composition into audio, using modifiable instrumentation, after which the soundtrack is ready for use. Processing components can be added or removed as required, and are easy to integrate; for example, we were able to add the object detection functionality in under 40 lines of code. A 30-second soundtrack takes under a minute to generate from an input video, making this a fast, versatile prototype system for testing ideas and a firm foundation for  further work. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMean first passage time (MFPT) of a diffusing Brownian particle is a heavily researched problem. Recently MFPT has been studied in context of Active Brownian Particles (ABPs) which under go self-propulsion in addition to the Brownian motion~\\cite{sharma2017escape}. An ABP performs persistent motion along the direction of an embedded unit vector which performs rotational diffusion. Often, the motion of ABPs is modelled on a coarse grained level by averaging out the orientational degree of freedom~\\cite{farage2015effective,wittmann2017effective}, resulting in a non Markovian equation of motion. Following this approach, the authors in Ref.~\\cite{sharma2017escape} studied the MFPT of an active Brownian particle diffusing in an external potential. In this paper, we preserve the orientational degree of freedom of the active particle and obtain its MFPT in 1-dimension. This allows us to verify the validity and accuracy of the coarse graining approach. We obtain the MFPT of an ABP in two scenarios, (a) freely diffusing ABP with stochastic resetting and (b) escape of an ABP over a potential barrier. For both these scenarios, we benchmark our findings with existing literature. \n\nDiffusion with stochastic resetting appears naturally in several search processes. A typical scenario would be to find a lost object with a random search strategy which is intermittently reset to its starting point. In Ref.~\\cite{evans2011diffusion}, the concept of stochastic resetting was applied to a freely diffusing Brownian particle. It was shown that with a fixed target located at an arbitrary location, the resetting of the diffusion particle at a given rate, leads to a finite mean first passage time (MFPT). This is in contrast to a freely diffusion particle for which the MFPT is infinite. The most interesting finding of their work was that there exists an optimal resetting rate that yields the minimum MFPT. The concept of stochastic resetting has been extended to diverse fields such as search strategies~\\cite{montanari2002optimizing}, microbiology~\\cite{cates2012diffusive} and even the search and hopping behavior of capuchin monkeys~\\cite{bartumeus2009optimal}. \n\nWe focus our study on active Brownian particles (ABPs), a more general situation where Brownian particles are a special case. This generalisation allows a more flavoured and realistic description in various fields, such as those described above. For example, in a biological context, the constituting components of a system are active agents such as molecular motors inside animal cells. These motors are active particles which, in addition to the directed active motion, undergo Brownian motion\\cite{molecular_motors}. Another example could be, on a macroscopic scale, the persistence of path in the context of the search of an object. While searching for a lost object, one persists on a certain time scale before resetting to the origin. Motivated by these considerations, we introduce activity as a key component of the motion of particle and study its impact on the MFPT.\n\nWe first investigate the MFPT of an ABP in 1-dimension with stochastic resetting using numerical simulations. The self propulsion in 1-dimension can be modelled by a two state model in which the particle moves with a constant velocity $v_0$ but switches the direction at a rate $\\tau^{-1}$ in a random uncorrelated fashion. Introducing stochastic resetting to an ABP, as for a passive particle, leads to a finite MFPT. Moreover, one expects to find a corresponding optimal resetting rate for which the MFPT is minimum. The main question we focus on in this study is whether, one can reduce this problem of ABP to a passive Brownian particle with effective transport coefficients. For a freely diffusing ABP, this would correspond to an effective diffusion constant which is determined by $v_0$ and $\\tau$. Can one express the optimal resetting rate in terms of this effective diffusion constant? \n\n\nThe mapping of ABPs to an effective equilibrium has been successful in describing the escape of an ABP over a potential barrier~\\cite{sharma2017escape}. In the coarse grained approach, the stochastic equation of motion of an ABP includes a correlated colored noise and a white Gaussian noise. The strength and autocorrelation of the colored noise is determined by the activity parameters, the propulsion strength $v_0$ and $\\tau$, the time scale of persistent motion of an ABP. We explicitly include the propulsion of ABP as a two state velocity and show that the MFPT of an ABP diffusing in an external potential can again be obtained as for a passive particle with effective diffusion constant and an effective potential; both of which are determined by the activity parameters. \n\n\n\n\\section{Model and theory}\\label{model}\nWe consider a one dimensional system of a single Brownian particle of size $d$\nwith coordinate $x$ and orientation specified by an embedded unit vector $\\boldsymbol{p}$. The orientation vector can point either along the positive $x$-axis or the negative one. The switching between these two states occurs at an average rate of $\\tau^{-1}$ in an independent, uncorrelated fashion. A constant self-propulsion of speed $v_0$ \nacts in the direction of orientation. \nThe motion can be modelled by the Langevin equation\n\\begin{align}\\label{full_langevin}\n\\dot{x} = v_0\\,\\boldsymbol{p}  + \\gamma^{-1}\\boldsymbol{F} + \\boldsymbol{\\xi}\n\\end{align}\nwhere $\\gamma$ is the friction coefficient and the force on particle is generated from the external potential $\\Phi(x)$ according to $\\boldsymbol{F}\\!=\\!-\\nabla \\Phi$. \nThe stochastic term $\\boldsymbol{\\xi}(t)$ is Gaussian distributed with zero mean and \nhas time correlation\t\n$\\langle\\boldsymbol{\\xi}(t)\\boldsymbol{\\xi}(t')\\rangle=2D_t\\delta(t-t')$ where $D_t$ is the translational diffusion coefficient. \n\nThe stochastic resetting is implemented as a Poissonian process with a mean rate $r$ such that at every resetting event, the particle is put back to its starting point ($x = 0$).\nWhen there is no external potential and no resetting, Eq.~\\eqref{full_langevin} describes the well known dichotomous diffusion~\\cite{balakrishnan2005connection}. The probability distribution of the particle can be obtained analytically in terms of modified Bessel functions. However, since we are interested in the MFPT, we consider the long time limit $t \\gg \\tau$, where it can be shown that the probability distribution reduces to a Gaussian with the diffusion constant \n\\begin{equation}\\label{effective_diffusion_coefficient}\nD =  D_t+D_a=D_t+\\frac{v_0^2 \\tau}{2}.\n\\end{equation}\n\nIn order to perform analytics for a finite resetting rate $r$, we make the assumption that the motion of an active particle can be described at all time by the ordinary diffusion equation with the diffusion constant given in Eq.~\\eqref{effective_diffusion_coefficient}. This is a reasonable assumption only under the self consistency condition that the MFPT is much larger than $\\tau$. As shown below, this assumption holds when the target is placed sufficiently far from the initial position of the ABP, such that the MFPT is much larger than $\\tau$. Under this assumption, the problem of finding the MFPT as well as the optimal resetting rate $r^{\\ast}$ reduces to that presented by Evans and Majumdar in Ref.~\\cite{evans2011diffusion}. Using the expression derived in Ref.~\\cite{evans2011diffusion}, it follows that the MFPT $T(x_0)$ is given as \n\\begin{align}\nT(x_0) = \\frac{1}{r}\\left(\\exp\\left(x_0\\sqrt \\frac{r}{D}\\right) - 1\\right),\n\\label{mfpt}\n\\end{align}\n where $x_0$ is the location of the target object for an ABP which starts at the origin. \n\nThe optimal resetting rate $r^{\\ast}$ for an ABP freely diffusing in one dimension is given as\n\\begin{align}\\label{optimal_rate}\nr^{\\ast} &=\\frac{z^{\\ast2}D}{x_0^2},\n\\end{align}\nwhere $z^{\\ast}$ is the solution of the transcendental equation $z^{\\ast}\/2=1-e^{-z^{\\ast}}$.\n\nThis optimal rate minimizes the MFPT of a particle to a target object in a one dimensional system. The first aim of this work is to test Eq.~\\eqref{optimal_rate} using Brownian dynamics simulations. In the second part of this work, we consider an ABP diffusing in an external potential. We particularly focus on the case of an ABP escaping a potential barrier. The escape problem is studied for the resetting rate set to 0. As in Ref.~\\cite{sharma2017escape}, we choose the potential to have the form\n\\begin{equation}\\label{potential}\n\\beta\\Phi(x)=\\frac{1}{2}\\omega_0 x^2 - \\alpha \\mid x \\mid^3,\n\\end{equation}\nwhere $\\beta = 1\/k_{\\rm B}T$, and $\\omega_0$ and $\\alpha$ are parameters. We are interested in the MFPT of a particle starting at the origin escaping over the barrier to be captured by a sink located sufficiently far from the barrier. In the effective equilibrium approach~\\cite{farage2015effective,sharma2017escape} to ABP in an external potential, one obtains an approximate Fokker-Planck equation with an effective external potential $\\Phi^{\\rm eff}(x)$ and an effective position-dependent diffusion constant $D(x)$:\n\\begin{align}\nD(x) &= D_t+\\frac{D_a}{1+\\frac{\\tau D_t}{d^2} \\beta \\Phi^{\\prime\\prime}(x)}\\, \\label{Dx},\\\\\n\\beta \\Phi^\\text{eff}(x)&=\\int_0^x dy\\,\\frac{\\beta \\Phi'(y)+D'(y)\/D_t}{D(y)\/D_t} \\label{Vx}.\n\\end{align}\n \nThe effective diffusion constant is determined jointly by the activity and the potential and it reduces to the expression in Eq.~\\eqref{effective_diffusion_coefficient} when there is no external potential. \n\nThe expressions in Eqs.~\\eqref{Dx} and~\\eqref{Vx} were obtained under the assumption~\\cite{farage2015effective,sharma2017escape,wittmann2017effective} that the stochastic process corresponding to time evolution of the orientation vector can be considered as a Gaussian noise process with a finite correlation time. In other words, one can disregard the explicit time evolution of the orientation vector of an ABP by including a colored noise term (exponentially correlated Gaussian noise) in the equation for the time evolution of the position of the ABP. There are two important conditions underlying this assumption: (1) The variance of the stochastic process corresponding to the orientation of an ABP decays exponentially in time independent of the spatial dimensions allowing the mapping to a colored Gaussian noise. However, the mapping is only approximate because the process is not Gaussian. This is most evident in one dimension where the process is a random telegraphic process and (2) the correlation time $\\tau$ is sufficiently small such that the particle's displacement during this time period is negligible. This is required to ensure that there is no significant change in the potential that the particle experiences over a time period $\\tau$.\n\n\n In Ref.~\\cite{sharma2017escape}, the authors validated the theoretical predictions of the effective equilibrium approach against the numerical simulations of a stochastic process driven by colored Gaussian noise. The strength and autocorrelation of the colored noise were determined by the activity parameters $v_0$ and $\\tau$. In our numerical simulations, we explicitly consider the orientation of the particle. Since we consider one dimensional system, the orientation can only take two values which is why we model it as a two state random telegraphic process. We will calculate MFPT and compare with the predictions of the effective equilibrium approach. For the sake of completeness, the expression for the inverse MFPT obtained in the effective equilibrium approach is given as~\\cite{sharma2017escape}\n \\begin{align}\\label{MFPT_EFF_EQ}\n T^{-1} =r_\\text{pass}\\exp{\\left({\\frac{D_a\\left(\\beta E_0 - \\omega_0\\tau \\right)}{D_t+D_a}}\\right)},\n \\end{align}\n where $r_{\\rm pass} = \\beta D_t\\omega_0\\exp(-\\beta E_0)\/(2\\pi)$ is the escape rate of a passive particle over the potential barrier (Eq.~\\eqref{potential}) and $\\beta E_0 = \\omega_0^3\/(54\\alpha^2)$ is the height of the barrier.\n\n\n\n\n\nBefore presenting the results, we give a brief description of the simulations performed in this study. We set the particle size to $d=1$. Time is always measured in units of $d^2\/D_t$ and we have chosen $D_t = 1$ which together with $k_{\\rm B}T = 1$ gives the friction coefficient $\\gamma = 1$. The Eq.~\\eqref{full_langevin} is integrated in time to generate the particle trajectory by advancing time in steps of $dt = 10^{-4}$ for the free diffusing particles and $dt=10^{-3}$ in the case with an external potential. The integral of the stochastic process $\\boldsymbol{\\xi}(t)$ over a time interval $dt$ is taken as a Gaussian distribution with zero mean and variance $2dt$. At every time step, the orientation of the particle is flipped with a probability $dt\/\\tau$. Similarly, stochastic resetting is implemented by resetting the particle to the origin with a probability of $rdt$ during every time step. The first passage time is calculated by averaging over  $10^5$ trajectories for a freely diffusing ABP. In the case of an external potential, the MFPT is obtained by averaging over $10^3-10^4$ trajectories. \n\n\n\\section{Results and discussion}\n\nWe first discuss the MFPT for a freely diffusing ABP with resetting. We investigate the effect of activity on the MFPT by considering two cases: (a) fixed value of $v_0 = 3$ and two different values for $\\tau=0.02$ and $\\tau = 0.05$ in Fig.~\\ref{free}(a) and (b) fixed $\\tau = 0.05$ and two different values of $v_0 = 1$ and $v_0=2$ in Fig.~\\ref{free}(b). In both cases, the target object is located at $x_0 = 2$. In Fig.~\\ref{free}, we plot the MFPT as a function of the resetting rate. As expected, the MFPT exhibits a minimum at a particular $r^{\\ast}$. As can be seen in Fig.~\\ref{free}, the theoretical predictions of Eq.~\\eqref{mfpt} are in good agreement with the numerically obtained values.\n\\begin{figure}[t]\n\\begin{tabular}{@{}c@{}}\n\\includegraphics[width=0.53\\textwidth]{fig2.jpeg} \\\\\n\\includegraphics[width=0.53\\textwidth]{fig1.jpeg}\n\\end{tabular}\n\\caption{Mean first passage time in (a) for different persistence times $\\tau$ and in (b) for different velocities $v_0$ (see legends). In both figures we also show the results for the Brownian case (red). The lines represent the theoretical predictions of Eq.(\\ref{mfpt}) and the filled circles are the simulations results. The black diamonds show the optimal resetting rate (Eq.(\\ref{optimal_rate})) and the corresponding MFPT using Eq.(\\ref{mfpt}). The arrow indicates the decreasing optimal rate with increasing activity.}\n\\label{free}\n\\end{figure}\nIn order to obtain the optimal resetting rate from simulations, one needs to determine the location of the minimum from the numerical data. However, as can be seen in Fig. \\ref{free}, an unambiguous determination of the minimum is difficult in simulations. Nevertheless, on plotting the analytical prediction for the optimal resetting rate (Eq. \\eqref{optimal_rate}) together with the numerically obtained data for MFPT, one can clearly see that the Eq. \\eqref{optimal_rate} provides an accurate measure for the optimal resetting rate. \n\n\n\nThe good agreement between the theoretical predictions of Eqs.~\\eqref{mfpt} and~\\eqref{optimal_rate} with the numerical results can be qualitatively understood in the following way. The MFPT shown in Fig.~\\ref{free} is much larger than $\\tau$, the time scale of the persistent motion of an ABP. With $x_0$ chosen to be sufficiently large and resetting rate to be sufficiently small, an active particle diffusing away from the origin will undergo several transitions in its orientation such that the motion of the particle would resemble that of a passively diffusing particle with an effective diffusion constant (Eq.~\\eqref{effective_diffusion_coefficient}). However, if $x_0 \\sim v_0\\tau$, one cannot consider the particle as performing pure diffusion as the discrete nature (in time) of the transitions from one orientation to the other cannot be ignored. In Fig. \\ref{bad_params} we show a case with the following choice of parameters: $v_0=8$, $\\tau=0.1$ and $x_0 = 1$. These parameters are chosen such that the object can be reached from the origin by the ABP on time scale of the order $\\tau$. \n\\begin{figure}[h!]\n\\includegraphics[width=0.53\\textwidth]{fig3.jpeg}\n\\caption{MFPT in a case where $x_0\\sim v_0\\tau$ (see legend). The line represents the theoretical prediction of Eq.(\\ref{mfpt}), the black diamond shows the optimal resetting rate from Eq.(\\ref{optimal_rate}) with the corresponding MFPT and the symbols are obtained from the numerical simulations. Clearly, the theoretical predictions strongly deviate from the numerical measurements. }\\label{bad_params}\n\\end{figure}\nAs shown in Fig. \\ref{bad_params} the particle reaches the object in a relative short time $t\\sim\\tau$. On such short time scale the ABP does not undergo enough orientational transitions and therefore does not admit a description in terms of an effective diffusion (passive).\n\n\nWe now discuss the MFPT for an ABP diffusing in an external potential (Eq.(\\ref{potential})). We compare the theoretical predictions of Eq.(\\ref{MFPT_EFF_EQ}) with the numerical simulations for a specific case where the parameters of the potential are set to be: $\\alpha=1$ and $\\omega_0=10$. These parameters are chosen to be the same as in Ref. \\cite{sharma2017escape}. These numerical results together with the theoretical predictions of Eq. \\eqref{MFPT_EFF_EQ} are shown in Fig. \\ref{log_plot}. As can be seen in the log-linear plot in Fig. \\ref{log_plot}, the MFPT decreases by orders of magnitude with increasing $D_a$, an indication of an exponential dependence on the activity. The exponential dependence of the MFPT on activity is captured in Eq. \\eqref{MFPT_EFF_EQ} which is derived in the effective equilibrium approach.  In the effective equilibrium approach, an ABP diffusing in an external potential is mapped to a passive Brownian particle with a modified diffusion constant as well a modified potential. Both the potential and the diffusion constant are determined by the activity parameters. The excellent agreement between the simulations and the theory suggest that a coarse grained description of an ABP diffusing in an external potential is well suited to investigate the MPFT of an ABP. The slight deviation of the numerics from the theoretical predictions for high $D_a$ is due to the fact that for high activities, the modified potential does not satisfy the assumptions (large potential barrier and locally quadratic potential) underlying Eq. \\ref{MFPT_EFF_EQ}. This can be seen in the inset of Fig \\ref{log_plot} where the effective potential is plotted for different values of $D_a$.\n\\begin{figure}[h!]\n\\includegraphics[width=0.53\\textwidth]{fig4.jpeg}\n\\caption{Mean first passage time for active particles overtaking an external potential of the form given by equation(\\ref{potential}), with the parameters $\\alpha=1$ and $\\omega_0=10$. The dashed line represents the theoretical prediction using Eq.~(\\ref{MFPT_EFF_EQ}) and the symbols are obtained from numerical simulations. There is an excellent agreement between the theoretical predictions and the numerical measurements. The simulations start to deviate from the theoretical results for large $D_a$. In the inset, the bare external potential, Eq.~(\\ref{potential}) and analytic effective potentials $\\Phi^{eff}(x)$, Eq.~(\\ref{Vx}), for the parameter $\\tau=0.01$ is shown for different values of $D_a$. The sink is positioned at $x_0=6$, indicated with a black arrow.}\\label{log_plot}\n\\end{figure}\n\n\n\n\n\\section{Conclusion}\nWe investigated the mean first passage time of an Active Brownian Particle in one dimension using numerical simulations. The particle moves with a constant propulsion speed and switches direction at an average rate $\\tau$. We studied the influence of a finite resetting rate $r$ on the MFPT to a fixed target of a single free ABP and mapped this result using an effective diffusion process approach. In order to perform analytics, we made the assumption that the motion of an ABP can be described at all time by the ordinary diffusion equation with the diffusion constant given in Eq.~\\eqref{effective_diffusion_coefficient}. This is a reasonable assumption only under the self consistency condition that the MFPT is much larger than $\\tau$, which is a limitation of this approach. The theoretical predictions are in good agreement with the simulations if the target is located at a distance much larger than the distance travelled by the particle over a period $\\tau$, i.e. $x_0 \\gg v_0\\tau$. As in the case of a passive Brownian particle, we found an optimal resetting rate $r^*$ for an active Brownian particle for which the target is found with the minimum average time. \n\nIn the case of an ABP diffusing in an external potential, we found excellent agreement between the theoretical prediction of the effective equilibrium approach and the computer simulations for ABP. We focussed on a single potential, but similar calculations can be made involving any other trapping potentials. As it was already mentioned in \\citep{sharma2017escape}, this excellent agreement suggests that in one dimension, the effective potential can yield accurate quantitative description of the escape process. The validity of the theoretical prediction is limited to low\/medium activities.\n\nIn this work we focussed on a simple 1-dimensional problem. It will be of interesting to extend this approach to higher dimensions  with the goal of treating more realistic problems.\n\n\n\n\\section{Acknowledgements}\nAlberto Scacchi thanks the Swiss National Science Foundation for financial\nsupport under the grant number 200021-153657\/2.\\bibliographystyle{apsrev}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\n\\noindent Andreianov, Donadello and Rosini developed in~\\cite{BorisCarlottaMax-M3AS} a macroscopic model, called here ADR, aiming at describing the behaviour of pedestrians at bottlenecks. The model is given by the Cauchy problem for a scalar hyperbolic conservation law in one space dimension with non-local point constraint of the form\n\\begin{subequations}\\label{eq:constrianed}\n\\begin{align}\\label{eq:constrianed1}\n    &\\partial_t\\rho + \\partial_xf(\\rho) = 0 && (t,x) \\in {\\mathbb{R}}_+\\times{\\mathbb{R}},\\\\\n    \\label{eq:constrianed3}\n    &\\rho(0,x) =\\bar\\rho(x) && x \\in {\\mathbb{R}},\\\\\n    \\label{eq:constrianed2}\n    &f\\left(\\rho(t,0\\pm)\\right) \\le p\\left( \\int_{{\\mathbb{R}}_-} w(x)~ \\rho(t,x)~{\\d} x\\right) && t \\in {\\mathbb{R}}_+,\n\\end{align}\n\\end{subequations}\nwhere $\\rho(t,x) \\in \\left[0,R\\right]$ is the (mean) density of pedestrians in $x \\in {\\mathbb{R}}$ at time $t \\in {\\mathbb{R}}_+$ and $\\bar\\rho  \\colon {\\mathbb{R}} \\to [0,R]$ is the initial (mean) density, with $R >0$ being the maximal density. \nThen, $f \\colon [0,R] \\to {\\mathbb{R}}_+$ is the flow considered to be bell-shaped, which is an assumption commonly used in crowd dynamics. A typical example of such flow is the so-called Lighthill-Whitham-Richards (LWR) flux~\\cite{LWR1, LWR2, Greenshields_1934} defined by \n$$f(\\rho)=\\rho\\,v_{\\max}\\left(1-\\dfrac{\\rho}{\\rho_{\\max}}\\right),$$where $v_{\\max}$ and $\\rho_{\\max}$ are the maximal velocity and the maximal density of pedestrians respectively. Throughout this paper the LWR flux will be used.\nNext $p \\colon {\\mathbb{R}}_+ \\to {\\mathbb{R}}_+$ prescribes the maximal flow allowed through a bottleneck located at $x=0$ as a function of the weighted average density in a left neighbourhood of the bottleneck and $w  \\colon {\\mathbb{R}}_- \\to {\\mathbb{R}}_+$ is the weight function used to average the density. \n\nFinally in \\eqref{eq:constrianed2}, $\\rho(t,0-)$ denotes the left measure theoretic trace along the constraint, implicitly defined by\n\\begin{align*}\n&\\lim_{\\varepsilon\\downarrow0}\\frac{1}{\\varepsilon} \\int_0^{+\\infty}  \\int_{-\\varepsilon}^0 \\modulo{\\rho(t,x) - \\rho(t,0-)}~\\phi(t,x)~{\\d} x~{\\d} t=0&\n&\\text{for all }\\phi \\in \\Cc\\infty({\\mathbb{R}}^2;{\\mathbb{R}}).\n\\end{align*}\nThe right measure theoretic trace, $\\rho(t,0+)$, is defined analogously.\\\\\n\n\\noindent In the last few decades, the study of the pedestrian behaviour through bottlenecks, namely at locations with reduced capacity, such as doors, stairs or narrowings, drawn a considerable attention. The papers \\cite{Schadschneider2008Evacuation, Cepolina2009532, Hoogendoorn01052005, Kopylow, Schreckenberg2006, Seyfried:59568, Zhang20132781} present results of empirical experiments. \nHowever, for safety reasons, experiments reproducing extremal conditions such as evacuation and stampede are not available.\nIn fact, the unique experimental study of a crowd disaster is proposed in~\\cite{Helbing_disaster}.\nThe available data show that the capacity of the bottleneck (i.e.~the maximum number of pedestrians that can flow through the bottleneck in a given time interval) can drop when high-density conditions occur upstream of the bottleneck. \nThis phenomenon is called \\emph{capacity drop} and can lead to extremely serious consequences in escape situations.\nIn fact, the crowd pressure before an exit can reach very high values, the efficiency of the exit dramatically reduces and accidents become more probable due to the overcrowding and the increase of the evacuation time (i.e.~the temporal gap between the times in which the first and the last pedestrian pass through the bottleneck). \nA linked phenomenon is the so-called \\emph{Faster Is Slower} (FIS) effect, first described in~\\cite{Helbing2000Simulating}. \nFIS effect refers to the jamming and clogging at the bottlenecks, that result in an increase of the evacuation time when the degree of hurry of a crowd is high. \nWe recall that the capacity drop and the FIS effect are both experimentally reproduced in~\\cite{Cepolina2009532,Soria20121584}.\nA further related (partly counter-intuitive) phenomenon is the so-called \\emph{Braess' paradox} for pedestrian flows~\\cite{hughes2003flow}.\nIt is well known that placing a small obstacle before an exit door can mitigate the inter-pedestrian pressure and, under particular circumstances, it reduces the evacuation time by improving the outflow of people.\n\nNote that as it happens for any first order model, see for instance~\\cite[Part~III]{Rosinibook} and the references therein, ADR can not explain the capacity drop and collective behaviours at the bottlenecks. Therefore one of the difficulties we have to face is that the constraint $p$ has to be deduced together with the fundamental diagram from the empirical observations. \n\nThe aim of this paper is to validate ADR by performing simulations in order to show the ability of the model to reproduce the main effects described above and related to capacity drop that are FIS and Braess' paradox. To this end we propose a numerical scheme for the model and prove its convergence. The scheme is obtained by adapting the local constrained finite volume method introduced in~\\cite{scontrainte} to the non-local case considered in ADR, using a splitting strategy. \n\nThe paper is organized as follows. In Section~\\ref{sec:model} we briefly recall the main theoretical results for ADR. In Section~\\ref{sec:NumericalMethod} we introduce the numerical scheme, prove its convergence and validate it with an explicit solution obtained in \\cite{BorisCarlottaMax-M3AS}. In Section~\\ref{sec:simulations} we perform simulations to show that ADR is able to reproduce the Braess' paradox and the FIS effect. In Subsection~\\ref{sec:Braess+FIS} we combine local and non-local constraints to model a slow zone placed before the exit. Conclusions and perspectives are outlined in Section~\\ref{sec:conclusions}.\n\n\n\\section{Well-posedness for the ADR model}\\label{sec:model}\n\nExistence, uniqueness and stability for the general Cauchy problem~\\eqref{eq:constrianed} are established in~\\cite{BorisCarlottaMax-M3AS} under the following assumptions\n\\begin{enumerate}[label=\\bfseries{(W)}]\n  \\item[\\textbf{(F)}]  $f$ belongs to $\\mathbf{Lip}\\left( [0,R]; \\left[0, +\\infty\\right[ \\right)$ and is supposed to be bell-shaped, that is $f(0) = 0 = f(R)$ and there exists $\\sigma \\in \\left]0,R\\right[$ such that $f'(\\rho)~(\\sigma-\\rho)>0$ for a.e.~$\\rho \\in [0,R]$.\n  \\item[\\textbf{(W)}] $w$ belongs to $\\L\\infty({\\mathbb{R}}_-;{\\mathbb{R}}_+)$, is an increasing map, $\\norma{w}_{\\L1({\\mathbb{R}}_-)} = 1$ and there exists $\\textrm{i}_{w} >0$ such that $w(x) = 0$ for any $x \\le -\\textrm{i}_{w}$.\n  \\item[\\textbf{(P)}] $p$ belongs to $\\mathbf{Lip} \\left( \\left[0,R\\right]; \\left]0,f(\\sigma)\\right] \\right)$ and is a non-increasing map.\n\\end{enumerate}\n\nThe regularity $w\\in \\L\\infty({\\mathbb{R}}_-;{\\mathbb{R}}_+)$ is the minimal requirement needed in order to prove existence and uniqueness of~\\eqref{eq:constrianed}. In this paper, we shall consider continuous $w$.  \n\nThe existence of solutions for the Riemann problem for~\\eqref{eq:constrianed} is proved in~\\cite{AndreianovDonadelloRosiniRazafisonProc} for piecewise constant $p$. \nHowever, such hypothesis on $p$ is not sufficient to ensure uniqueness of solutions, unless the flux $f$ and the efficiency $p$ satisfy a simple geometric condition, see~\\cite{AndreianovDonadelloRosiniRazafisonProc} for details.\nIn the present paper, we consider either continuous nonlinear $p$ or a piecewise constant $p$ that satisfies such geometric condition.\n\nThe definition of entropy solution for a Cauchy problem~\\eqref{eq:constrianed1}, \\eqref{eq:constrianed3} with a fixed \\emph{a priori} time dependent constraint condition\n\\begin{align}\\label{eq:constrianed2bis}\n    &f\\left(\\rho(t,0\\pm)\\right) \\le q(t) && t \\in {\\mathbb{R}}_+\n\\end{align}\nwas introduced in~\\cite[Definition~3.2]{ColomboGoatinConstraint} and then reformulated in~\\cite[Definition~2.1]{scontrainte}, see also~\\cite[Proposition~2.6]{scontrainte} and \\cite[Definition 2.2]{chalonsgoatinseguin}.\nSuch definitions are obtained by adding a term that accounts for the constraint in the classical definition of entropy solution given by Kruzkov in~\\cite[Definition~1]{Kruzkov}.\nThe definition of entropy solution given in~\\cite[Definition~2.1]{BorisCarlottaMax-M3AS} is obtained by extending these definitions to the framework of non-local constraints.\n\nThe following theorem on existence, uniqueness and stability of entropy solutions of the constrained Cauchy problem~\\eqref{eq:constrianed} is achieved under the hypotheses~\\textbf{(F)}, \\textbf{(W)} and~\\textbf{(P)}.\n\\begin{thm}[Theorem~3.1 in~\\cite{BorisCarlottaMax-M3AS}]\\label{thm:1}\n    Let~\\textbf{(F)}, \\textbf{(W)}, \\textbf{(P)} hold. Then, for any initial datum $\\bar\\rho \\in \\L\\infty({\\mathbb{R}};[0,R])$, the Cauchy problem~\\eqref{eq:constrianed} admits a unique entropy solution $\\rho$. Moreover, if $\\rho' = \\rho'(t,x)$ is the entropy solution corresponding to the initial datum $\\bar\\rho' \\in \\L\\infty({\\mathbb{R}};[0,R])$, then for all~$T>0$ and $L>{\\rm{i}_w}$, the following inequality holds\n\\begin{equation}\\label{eq:lipdepen-localized}\n\\norma{\\rho(T) - \\rho'(T)}_{\\L1([-L,L])} \\le e^{CT}\\norma{\\bar\\rho - \\bar\\rho'}_{\\L1(\\{\\modulo{x}\\le L+MT\\})},\n\\end{equation}\nwhere $M=\\mathrm{Lip}(f)$ and $C=2 \\mathrm{Lip}(p) \\norma{w}_{\\L\\infty({\\mathbb{R}}_-)}$.\n\\end{thm}\n\nThe total variation of the solution may in general increase due to the presence of the constraint.\nIn~\\cite{BorisCarlottaMax-M3AS} the authors provide an invariant domain $\\mathcal{D} \\subset \\L1 \\left({\\mathbb{R}} ; [0,R]\\right)$ such that if $\\bar\\rho$ belongs to $\\mathcal{D}$, then one obtains a Lipschitz estimate with respect to time of the $\\L1$ norm and an a priori estimate of the total variation of\n\\[\n\\Psi(\\rho) = \\mathrm{sign}(\\rho-\\sigma) [f(\\sigma) - f(\\rho)] = \\int_\\sigma^\\rho \\modulo{\\dot{f}(r)} \\, \\d r.\n\\]\n\n\\section{Numerical method for approximation of ADR}\\label{sec:NumericalMethod}\nIn this section we describe the numerical scheme  based on finite volume method that we use to solve~\\eqref{eq:constrianed}. \nThen we prove the convergence of our scheme and validate it by comparison with an explicit solution of~\\eqref{eq:constrianed}.\nIn what follows, we assume that \\textbf{(F)}, \\textbf{(W)} and~\\textbf{(P)} hold.\n\n\\subsection{Non-local constrained finite volume method}\n\nLet $\\Delta x$ and $\\Delta t$ be the constant space and time steps respectively.\nWe define the points $x_{j+1\/2}=j\\Delta x$, the cells $K_j=[x_{j-1\/2},x_{j+1\/2}[$ and the cell centers $x_j=(j-1\/2)\\Delta x$ for $j\\in{\\mathbb{Z}}$. We define the time discretization $t^n=n\\Delta t$.\nWe introduce the index $j_c$ such that $x_{j_c+1\/2}$ is the location of the constraint (a door or an obstacle). For $n\\in{\\mathbb{N}}$ and $j\\in{\\mathbb{Z}}$, we denote by $\\rho_j^n$ the approximation of the average\nof $\\rho(t^n,\\cdot~)$ on the cell $K_j$, namely\n\\begin{align*}\n&\\rho_j^0=\\frac{1}{\\Delta x}\\displaystyle\\int_{x_{j-1\/2}}^{x_{j+1\/2}}\\overline{\\rho}(x)\\,\\d x&\n&\\text{ and }&\n&\\rho_j^n\\simeq\\frac{1}{\\Delta x}\\displaystyle\\int_{x_{j-1\/2}}^{x_{j+1\/2}}\\rho(t^n,x)\\,\\d x\n&\\text{ if }n>0.\n\\end{align*}\n\nWe recall that for the classical conservation law~\\eqref{eq:constrianed1}-\\eqref{eq:constrianed3},\na standard finite volume method can be written into the form \n\\begin{equation}\n\\label{schema}\n\\rho_j^{n+1}=\\rho_j^n-\\frac{\\Delta t}{\\Delta x}\\left(\\mathcal{F}_{j+1\/2}^n-\\mathcal{F}_{j-1\/2}^n\\right),\n\\end{equation} \nwhere $\\mathcal{F}_{j+1\/2}^n=F\\left(\\rho_j^n,\\rho_{j+1}^n\\right)$ is a monotone, consistent numerical flux, that is, $F$ satisfies the following assumptions:\n\\begin{itemize}\n\\item $F$ is Lipschitz continuous from $[0,R]^2$ to ${\\mathbb{R}}$ with Lipschitz constant $\\mathrm{Lip}(F)$,\n\\item $F(a,a)=f(a)$ for any $a\\in[0,R]$,\n\\item $(a,b) \\in [0,R]^2 \\mapsto F(a,b) \\in {\\mathbb{R}}$ is non-decreasing with respect to $a$ and non-increasing with respect to $b$.\n\\end{itemize} \nWe also recall that in~\\cite{scontrainte} the numerical flux for the time dependent constraint~\\eqref{eq:constrianed2bis} is modified as follow in order to take into account the constraint condition\n\\begin{align}\n\\label{def.flux.num.contrainte}\n    &\\mathcal{F}_{j+1\/2}^n=\\left\\{\\begin{array}{l@{\\quad\\text{if }}l}\n    F\\left (\\rho_j^n,\\rho_{j+1}^n\\right )&j\\ne j_c,\\\\[6pt]\n    \\min\\left \\{F\\left(\\rho_j^n,\\rho_{j+1}^n\\right),q^n\\right \\}&j=j_c,\n       \\end{array}\\right.\n\\end{align}\nwhere $q^n$ is an approximation of $q(t^n)$.\nIn the present paper, when dealing with a Cauchy problem subject to a non-local constraint of the form~\\eqref{eq:constrianed2} we will use the approximation\n\\begin{equation}\n\\label{contrainte.approx}\nq^n=p\\left(\\Delta x\\sum_{j\\le j_c}w(x_j)\\,\\rho_j^n\\right).\n\\end{equation}\nRoughly speaking\n\\begin{itemize}\n\\item we apply the numerical scheme~\\eqref{schema} for the problem~\\eqref{eq:constrianed1}-\\eqref{eq:constrianed3},\n\\item we apply the numerical scheme~\\eqref{schema}-\\eqref{def.flux.num.contrainte} for the problem~\\eqref{eq:constrianed1}-\\eqref{eq:constrianed3}-\\eqref{eq:constrianed2bis},\n\\item we apply the numerical scheme~\\eqref{schema}-\\eqref{def.flux.num.contrainte}-\\eqref{contrainte.approx} for the problem~\\eqref{eq:constrianed}.\n\\end{itemize}\n\n\\subsection{Convergence of the scheme}\nLet us introduce the finite volume approximate solution $\\rho_{\\Delta}$ defined by\n\\begin{align}\\label{def.rho.delta}\n&\\rho_\\Delta(t,x)=\\rho_j^n&\n&\\mbox{for }x\\in K_j\\mbox{ and }t\\in [t^n,t^{n+1} [,\n\\end{align}\nwhere the sequence $(\\rho_{j}^n)_{j\\in{\\mathbb{Z}},\\,n\\in{\\mathbb{N}}}$ is obtained by the numerical scheme~\\eqref{schema}-\\eqref{def.flux.num.contrainte}. \nAnalogously, we also define the approximate constraint function\n\\begin{align}\n\\label{def.q.delta}\n&q_\\Delta(t)=q^n&\n&\\mbox{for }t\\in [t^n,t^{n+1} [.\n\\end{align}\n\nFirst, we prove a discrete stability estimate valid for any domain $Q = [0,T] \\times {\\mathbb{R}}$ with $T>0$, \nfor the scheme~\\eqref{schema}-\\eqref{def.flux.num.contrainte} applied to problem~\\eqref{eq:constrianed1}-\\eqref{eq:constrianed3}-\\eqref{eq:constrianed2bis}.\nThis estimate can be seen as the equivalent, in this framework, of the stability result established in~\\cite[Proposition~2.10]{scontrainte}.\n\n\\begin{pro}\n\\label{pro.diff.solutions}\nLet $\\overline{\\rho}$ be in $\\L\\infty({\\mathbb{R}};[0,R])$ and $q_\\Delta$, $\\hat{q}_\\Delta$ be piecewise constant functions of the form~\\eqref{def.q.delta}.\nIf $\\rho_\\Delta$ and $\\hat{\\rho}_\\Delta$ are the approximate solutions of~\\eqref{eq:constrianed1}-\\eqref{eq:constrianed3}-\\eqref{eq:constrianed2bis} corresponding, respectively, to $q_\\Delta$ and $\\hat{q}_\\Delta$ and constructed by applying the scheme~\\eqref{schema}-\\eqref{def.flux.num.contrainte}, then we have \n$$\\norma{\\rho_\\Delta-\\hat{\\rho}_\\Delta}_{\\L1(Q)}\\le2 T \\norma{q_\\Delta-\\hat{q}_\\Delta}_{\\L1([0,T])}.$$ \n\\end{pro}\n\n\\begin{proof}\nFor notational simplicity, let $N = \\lfloor T\/\\Delta t\\rfloor$. Let us also introduce $(\\tilde{\\rho}_j^n)_{j\\in{\\mathbb{Z}},\\,n\\in{\\mathbb{N}}}$ defined by,  \n$$\\tilde{\\rho}_j^{n+1}=\\rho_j^n-\\frac{\\Delta t}{\\Delta x}\\left(\\mathcal{\\tilde{F}}_{j+1\/2}^n-\\mathcal{\\tilde{F}}_{j-1\/2}^n\\right),\\quad\\mbox{for any }\nj\\in{\\mathbb{Z}},\\,n\\in{\\mathbb{N}},$$\nwhere $\\mathcal{\\tilde{F}}_{j+1\/2}^n$ is defined by \n\\begin{align*}\n    &\\mathcal{\\tilde{F}}_{j+1\/2}^n=\\left\\{\\begin{array}{l@{\\quad\\text{if }}l}\n    F\\left (\\rho_j^n,\\rho_{j+1}^n\\right )&j\\ne j_c,\\\\[6pt]\n    \\min\\left \\{F\\left(\\rho_j^n,\\rho_{j+1}^n\\right),\\hat{q}^n\\right \\}&j=j_c.\n       \\end{array}\\right.\n\\end{align*}\n Then using the definitions of $(\\rho_j^n)_{j\\in{\\mathbb{Z}},\\,n\\in{\\mathbb{N}}}$ and $(\\tilde{\\rho}_j^n)_{j\\in{\\mathbb{Z}},\\,n\\in{\\mathbb{N}}}$, we have for any $n=1,\\dots,N$, $$\\rho_j^n=\\tilde{\\rho}_j^n\\quad\\mbox{if}\\quad j\\notin\\{j_c,j_c+1\\}$$\nand  \\begin{align*}\n&\\rho_{j_c}^n-\\tilde{\\rho}_{j_c}^n=\n-\\frac{\\Delta t}{\\Delta x}\\left(\\min\\left \\{F\\left (\\rho_{j_c}^{n-1},\\rho_{j_c+1}^{n-1}\\right ),q^{n-1}\\right\\} +\\min\\left \\{F\\left (\\rho_{j_c}^{n-1},\\rho_{j_c+1}^{n-1}\\right ),\\hat{q}^{n-1}\\right \\}\\right),&\\\\\n&\\rho_{j_c+1}^n-\\tilde{\\rho}_{j_c+1}^n=\n\\frac{\\Delta t}{\\Delta x}\\left(\\min\\left \\{F\\left(\\rho_{j_c}^{n-1},\\rho_{j_c+1}^{n-1}\\right),q^{n-1}\\right \\} - \\min\\left \\{F\\left(\\rho_{j_c}^{n-1},\\rho_{j_c+1}^{n-1}\\right),\\hat{q}^{n-1}\\right \\} \\right),\n\\end{align*}\nwhich implies that \n\\begin{align*}\n&\\modulo{\\rho_{j_c}^n-\\tilde{\\rho}_{j_c}^n}\\le\\frac{\\Delta t}{\\Delta x} ~ \\modulo{q^{n-1}-\\hat{q}^{n-1}},&\n&\\modulo{\\rho_{j_c+1}^n-\\tilde{\\rho}_{j_c+1}^n}\\le\\frac{\\Delta t}{\\Delta x} ~ \\modulo{q^{n-1}-\\hat{q}^{n-1}}.\n\\end{align*}\nTherefore we deduce that, for any $n=1,\\dots,N$, \n\\begin{equation}\n\\label{pro.diff.solutions.eq1}\n\\sum_{j\\in{\\mathbb{Z}}}\\modulo{\\rho_{j}^n-\\tilde{\\rho}_{j}^n}\\le2\\frac{\\Delta t}{\\Delta x} ~ \\modulo{q^{n-1}-\\hat{q}^{n-1}}.\n\\end{equation}\nBesides, observe that the modification of the numerical flux at the interface $x_{j_c+1\/2}$ introduced in~\\eqref{def.flux.num.contrainte} does not affect the monotonicity of the scheme~\\eqref{schema}-\\eqref{def.flux.num.contrainte} (see~\\cite[Proposition~4.2]{scontrainte}). \nTherefore, for any $n=1,\\dots,N$, we have \n\\begin{equation}\n\\label{pro.diff.solutions.eq2}\n\\sum_{j\\in{\\mathbb{Z}}}\\modulo{\\tilde{\\rho}_j^n-\\hat{\\rho}_j^n}\\le\n\\sum_{j\\in{\\mathbb{Z}}}\\modulo{\\rho_j^{n-1}-\\hat{\\rho}_j^{n-1}}.\n\\end{equation} \nHence thanks to~\\eqref{pro.diff.solutions.eq1} and~\\eqref{pro.diff.solutions.eq2}, we can write\n\\begin{align*}\n\\sum_{j\\in{\\mathbb{Z}}}|\\rho_j^1-\\hat{\\rho}_j^1|&\\le\\sum_{j\\in{\\mathbb{Z}}}|\\rho_j^1-\\tilde{\\rho}_j^1|+\\sum_{j\\in{\\mathbb{Z}}}|\\tilde{\\rho}_j^1-\\hat{\\rho}_j^1|\\le2\\frac{\\Delta t}{\\Delta x} ~ \\modulo{q^0-\\hat{q}^0}+\\sum_{j\\in{\\mathbb{Z}}}|\\rho_j^0-\\hat{\\rho}_j^0|\n=2\\frac{\\Delta t}{\\Delta x} ~ \\modulo{q^0-\\hat{q}^0}.\n\\end{align*}\nThen an induction argument shows that for any $n=1,\\dots,N$, \n$$\\sum_{j\\in{\\mathbb{Z}}}\\modulo{\\rho_j^n-\\hat{\\rho}_j^n}\\le2\\frac{\\Delta t}{\\Delta x}\\sum_{k=0}^{n-1}|q^k-\\hat{q}^k|\\le\\frac{2}{\\Delta x}~\\|q_\\Delta-\\hat{q}_\\Delta\\|_{L^1([0,t^n])}.$$\nIn conclusion, we find that \n\\begin{align*}\n\\|\\rho_\\Delta-\\hat{\\rho}_\\Delta\\|_{L^1(Q)}&=\\Delta t\\,\\Delta x\\sum_{n=1}^N\\sum_{j\\in{\\mathbb{Z}}}|\\rho_j^n-\\hat{\\rho}_j^n|\n\\le 2 \\|q_\\Delta-\\hat{q}_\\Delta\\|_{L^1([0,T])}~\\sum_{n=1}^N \\Delta t\n\\le 2 T \\|q_\\Delta-\\hat{q}_\\Delta\\|_{L^1([0,T])}\n\\end{align*}\nand this ends the proof.\n\\end{proof}\n\n\n\\begin{comment}\n\\begin{proof}\nAssume first $n=1$. Then we easily see from the scheme~\\eqref{schema}-\\eqref{def.flux.num.contrainte} that we have $\\rho_j^1=\\hat{\\rho}_j^1$ for any $j\\not\\in\\left \\{ j_c, j_c+1\\right \\}$. \nMoreover we can write\n\\begin{align*}\n&\\rho_{j_c}^1-\\hat{\\rho}_{j_c}^1=\n-\\frac{\\Delta t}{\\Delta x}\\left(\\min\\left \\{F\\left (\\rho_{j_c}^0,\\rho_{j_c+1}^0\\right ),q^0\\right\\} -\\min\\left \\{F\\left (\\rho_{j_c}^0,\\rho_{j_c+1}^0\\right ),\\hat{q}^0\\right \\}\\right),&\\\\\n&\\rho_{j_c+1}^1-\\hat{\\rho}_{j_c+1}^1=\n\\frac{\\Delta t}{\\Delta x}\\left(\\min\\left \\{F\\left(\\rho_{j_c}^0,\\rho_{j_c+1}^0\\right),q^0\\right \\} - \\min\\left \\{F\\left(\\rho_{j_c}^0,\\rho_{j_c+1}^0\\right),\\hat{q}^0\\right \\} \\right),\n\\end{align*}\nand therefore\n\\begin{align*}\n&\\modulo{\\rho_{j_c}^1-\\hat{\\rho}_{j_c}^1}\\le\\frac{\\Delta t}{\\Delta x} ~ \\modulo{q^0-\\hat{q}^0},&\n&\\modulo{\\rho_{j_c+1}^1-\\hat{\\rho}_{j_c+1}^1}\\le\\frac{\\Delta t}{\\Delta x} ~ \\modulo{q^0-\\hat{q}^0}.\n\\end{align*}\nThen an induction argument allows to prove that for any $n\\in{\\mathbb{N}}$, $n\\ge1$, we have\n\\begin{align*}\n&\\rho_j^n=\\hat{\\rho}_j^n\\quad\\mbox{if }j\\not\\in \\left \\{j_c,\\,j_c+1\\right \\},&\n&\\modulo{\\rho_{j_c}^n-\\hat{\\rho}_{j_c}^n} \\le \\frac{\\Delta t}{\\Delta x} ~ \\modulo{q^{n-1}-\\hat{q}^{n-1}},&\n&\\modulo{\\rho_{j_c+1}^n-\\hat{\\rho}_{j_c+1}^{n}} \\le \\frac{\\Delta t}{\\Delta x} ~ \\modulo{q^{n-1}-\\hat{q}^{n-1}}.\n\\end{align*}\nTherefore we have \n\\begin{align*}\n\\norma{\\rho_{\\Delta}-\\hat{\\rho}_{\\Delta}}_{\\L1(Q)}&=\\Delta t ~ \\Delta x\\sum_{n=0}^{\\lfloor T\/\\Delta t\\rfloor}\\sum_{j\\in{\\mathbb{Z}}}\\modulo{\\rho_j^n-\\hat{\\rho}_j^n}\n=\\Delta t ~ \\Delta x\\sum_{n=0}^{\\lfloor T\/\\Delta t\\rfloor}\\left(\\modulo{\\rho_{j_c}^n-\\hat{\\rho}_{j_c}^n} + \\modulo{\\rho_{j_c+1}^n-\\hat{\\rho}_{j_c+1}^n}\\right)\\\\\n&\\le2 ~ \\Delta t^2\\sum_{n=1}^{\\lfloor T\/\\Delta t\\rfloor}\\modulo{q^{n-1}-\\hat{q}^{n-1}}\n\\le2 ~ \\Delta t ~ \\norma{q_\\Delta-\\hat{q}_\\Delta}_{\\L1([0,T])},\n\\end{align*}\nwhich ends the proof.\n\\end{proof}\n\\end{comment}\n\n\\begin{comment}\n\\begin{proof}\nFor notational simplicity, let $N = \\lfloor T\/\\Delta t\\rfloor$.\nBy definition we have that\n\\begin{align*}\n&\\norma{\\rho_{\\Delta}-\\hat{\\rho}_{\\Delta}}_{\\L1(Q)}=\n\\Delta t  ~ \\Delta x \\left [\\sum_{n=1}^N \\sum_{i=1-n}^{n} \\modulo{\\rho^n_{j_c+i} - \\hat{\\rho}^n_{j_c+i}}\\right ]\n\\end{align*}\nwhere by~\\eqref{CFL}\n\\begin{align*}\n&\\sum_{n=1}^N \\sum_{i=1-n}^{n} \\modulo{\\rho^n_{j_c+i} - \\hat{\\rho}^n_{j_c+i}}\n\\leq\n\\sum_{n=2}^N \\sum_{i=1-n}^{n} \n\\modulo{\\rho^{n-1}_{j_c+i} - \\hat{\\rho}^{n-1}_{j_c+i}}\n+2 ~ \\frac{\\Delta t}{\\Delta x} \\sum_{n=1}^N \\modulo{q^{n-1}-\\hat{q}^{n-1}}\n\\\\\n&+\\frac{\\Delta t}{\\Delta x} ~ \\mathrm{Lip}(F)\n\\sum_{n=2}^N \\sum_{i=1-n}^{n} \n\\left [\n\\modulo{\\rho^{n-1}_{j_c+i-1} - \\hat{\\rho}^{n-1}_{j_c+i-1}}\n+2\\modulo{\\rho^{n-1}_{j_c+i} - \\hat{\\rho}^{n-1}_{j_c+i}}\n+\\modulo{\\rho^{n-1}_{j_c+i+1} - \\hat{\\rho}^{n-1}_{j_c+i+1}}\n\\right ]\n\\\\\n&\\leq\n2 ~ \\frac{\\Delta t}{\\Delta x} \\sum_{n=1}^N \\modulo{q^{n-1}-\\hat{q}^{n-1}}\n+\\left [1+4\\frac{\\Delta t}{\\Delta x} ~ \\mathrm{Lip}(F)\\right ]\n\\left [\\sum_{n=2}^N \\sum_{i=1-n}^{n} \n\\modulo{\\rho^{n-1}_{j_c+i} - \\hat{\\rho}^{n-1}_{j_c+i}}\\right ]\n\\\\\n&\\leq\n2 ~ \\frac{\\Delta t}{\\Delta x} \\sum_{n=1}^N \\modulo{q^{n-1}-\\hat{q}^{n-1}}\n+3 \\sum_{n=1}^{N-1} \\sum_{i=1-n}^{n} \n\\modulo{\\rho^{n}_{j_c+i} - \\hat{\\rho}^{n}_{j_c+i}}\n\\end{align*}\n\\end{proof}\n\\end{comment}\n\n\\noindent Let us now notice that as in~\\cite[Proposition~4.2]{scontrainte}, under the CFL condition \n\\begin{equation}\n\\label{CFL}\n\\mathrm{Lip}(F) ~ \\frac{\\Delta t}{\\Delta x}\\le\\frac{1}{2},\n\\end{equation}\nwe have the $\\L\\infty$ stability of the scheme~\\eqref{schema}-\\eqref{def.flux.num.contrainte}-\\eqref{contrainte.approx} that is \n\\begin{align}\n\\label{stability.Linfty}\n&0\\le\\rho_\\Delta(t,x)\\le R&&\\mbox{for a.e.~}(t,x)\\in Q. \n\\end{align}\n\\noindent This stability result allows to prove the statement below. \n\n\\begin{pro}\n\\label{prop.estimation.bv.q}\nLet $q_\\Delta$ be defined by~\\eqref{contrainte.approx}-\\eqref{def.q.delta}. Then under the CFL condition~\\eqref{CFL}, for any $T>0$, there exists $C>0$ only depending on $T$, $f$, $F$, $p$, $w$ and $R$ such that:\n\\begin{equation}\n\\label{estimation.bv.q}\n\\modulo{q_\\Delta}_{BV([0,T])}\\le C.\n\\end{equation}\n\\end{pro}\n\n\\begin{proof}\nLet $N=\\lfloor{T\/\\Delta t}\\rfloor$ and $j_w$ be an integer such that supp$(w)\\subset\\underset{j_w\\le j\\le j_c}\\cup K_j$.\nThen for any $n=0,\\dots,N-1$, we have \n\\begin{align*}\n\\modulo{q^{n+1}-q^n}& = \\modulo{p\\left(\\Delta x\\sum_{j_w\\le j\\le j_c}w(x_j)\\rho_j^{n+1}\\right)-p\\left(\\Delta x\\sum_{j_w\\le j\\le j_c}w(x_j)\\rho_j^n\\right)}\\\\\n&\\le\\Delta x\\,\\mathrm{Lip}(p)\\left|\\sum_{j_{w}\\le j\\le j_c}w(x_j)(\\rho_j^{n+1}-\\rho_j^n)\\right|\n=\\Delta t\\,\\mathrm{Lip}(p)\\left|\\sum_{j_{w}\\le j\\le j_c}w(x_j)\\left(\\mathcal{F}_{j+1\/2}^n-\\mathcal{F}_{j-1\/2}^n\\right)\\right|.\n\\end{align*}\nNow, using a summation by part, we have\n\\begin{align*}\n\\sum_{j_{w}\\le j\\le j_c}w(x_j)\\left(\\mathcal{F}_{j+1\/2}^n-\\mathcal{F}_{j-1\/2}^n\\right)\n=w(x_{j_c})\\mathcal{F}_{j_c+1\/2}^n-w(x_{j_w})\\mathcal{F}_{j_w-1\/2} - \\!\\!\\!\\sum_{j_w\\le j\\le j_c-1}\\left(w(x_{j+1})-w(x_j)\\right)\\mathcal{F}_{j+1\/2}^n.\n\\end{align*}\nThen, it follows that \n$$|q^{n+1}-q^n|\\le\\Delta t\\,\\mathrm{Lip}(p)\\,\\|w\\|_{\\L\\infty({\\mathbb{R}}_-;{\\mathbb{R}})}\\sum_{j_w-1\\le j\\le j_c}|\\mathcal{F}_{j+1\/2}^n|.$$\nNow, from~\\eqref{def.flux.num.contrainte}, for any $j\\in{\\mathbb{Z}}$ we have the estimate\n\\begin{align*}\n\\modulo{\\mathcal{F}_{j+1\/2}^n} & \\le \\modulo{F(\\rho_{j}^n,\\rho_{j+1}^n)}\n\\le \\modulo{F(\\rho_{j}^n,\\rho_{j+1}^n)-F(\\rho_{j}^n,\\rho_{j}^n)} + \\modulo{f(\\rho_{j}^n)}\n\\le \\mathrm{Lip}(F) \\, \\modulo{\\rho_{j+1}^n-\\rho_{j}^n}+\\mathrm{Lip}(f)\\, \\modulo{\\rho_{j}^n}\n\\le R\\left (\\mathrm{Lip}(F)+\\mathrm{Lip}(f)\\right ).\n\\end{align*}\nHence we deduce that \n$$\\modulo{q_\\Delta}_{BV([0,T])}=\\sum_{n=0}^{N-1}\\modulo{q^{n+1}-q^n}\\le C,$$\nwhere $C=(j_c-j_w+2)\\,T\\,R\\,\\mathrm{Lip}(p)\\,\\norma{w}_{\\L\\infty({\\mathbb{R}}_-;{\\mathbb{R}})}\\,\\left (\\mathrm{Lip}(F)+\\mathrm{Lip}(f)\\right )$.\n\\end{proof}\n\nWe are now in a position to prove a convergence result for the scheme ~\\eqref{schema}-\\eqref{def.flux.num.contrainte}-\\eqref{contrainte.approx}.\n\n\\begin{thm}\n\\label{th.convergence}\nUnder the CFL condition~\\eqref{CFL}, the constrainted finite volume scheme~\\eqref{schema}-\\eqref{def.flux.num.contrainte}-\\eqref{contrainte.approx} converges in $\\L1(Q)$ to the unique entropy solution to~\\eqref{eq:constrianed}.\n\\end{thm}\n\n\\begin{proof}\nLet $(\\rho_\\Delta,q_\\Delta)$ be constructed by the scheme~\\eqref{schema}-\\eqref{def.flux.num.contrainte}-\\eqref{contrainte.approx}. \nProposition~\\ref{prop.estimation.bv.q} and Helly's lemma give the existence of a subsequence, still denoted $q_\\Delta$ and a constraint function $q\\in \\L\\infty([0,T])$ such that $q_\\Delta$ converges to $q$ strongly in $\\L1([0,T])$ as $\\Delta t\\to0$. Let $\\rho\\in \\L\\infty({\\mathbb{R}}_+\\times{\\mathbb{R}};[0,R])$ be the unique entropy solution to~\\eqref{eq:constrianed1}-\\eqref{eq:constrianed3}-\\eqref{eq:constrianed2bis} associated to $q$. \nIt remains to prove that the subsequence $\\rho_\\Delta$ converges to $\\rho$ strongly in $\\L1(Q)$ as $\\Delta t,\\,\\Delta x\\to0$. The uniqueness of the entropy solution to~\\eqref{eq:constrianed1}-\\eqref{eq:constrianed3}-\\eqref{eq:constrianed2bis} will then imply that the full sequence $\\rho_\\Delta$ converges to $\\rho$ and, as a consequence, the full sequence $q_\\Delta$ converges to $q=p\\left (\\int_{{\\mathbb{R}}_-}w(x)\\,\\rho(t,x)\\,\\d x\\right )$.\\\\ \nLet $\\hat{q}_\\Delta$ be a piecewise constant approximation of $q$ such that $\\hat{q}_\\Delta$ converges to $q$ strongly in $\\L1([0,T])$. Furthermore, we also introduce $\\hat{\\rho}_\\Delta$ constructed by the scheme~\\eqref{schema}-\\eqref{def.flux.num.contrainte} and associated to $\\hat{q}_\\Delta$. Now we have\n$$\\norma{\\rho-\\rho_\\Delta}_{\\L1(Q)}\\le\\norma{\\rho-\\hat{\\rho}_\\Delta}_{\\L1(Q)}+\\norma{\\rho_\\Delta-\\hat{\\rho}_\\Delta}_{\\L1(Q)}. $$ \nBut, thanks to~\\cite[Theorem~4.9]{scontrainte}, under the CFL condition~\\eqref{CFL}, $\\norma{\\rho-\\hat{\\rho}_\\Delta}_{\\L1(Q)}$ tends to $0$ as $\\Delta t$, $\\Delta x\\to0$. Furthermore, thanks to Proposition~\\ref{pro.diff.solutions}, we have \n$$\\norma{\\rho_\\Delta-\\hat{\\rho}_\\Delta}_{\\L1(Q)}\\le2 ~ T~ \\norma{q_\\Delta-\\hat{q}_\\Delta}_{\\L1([0,T])}$$\nwhich also shows that $\\norma{\\rho_\\Delta-\\hat{\\rho}_\\Delta}_{\\L1(Q)}$ tends to $0$ as $\\Delta t$, $\\Delta x\\to0$. \n\\end{proof}\n\n\n\n\\subsection{Validation of the numerical scheme}\\label{sec:validation}\n\n\\begin{figure}[htpb]\\tiny\n\\centering\n        {\\includegraphics[width=0.25\\textwidth]{1DevacuationFP}}\n\\caption{The functions ${[\\rho \\mapsto f(\\rho)]}$ and ${[\\xi \\mapsto p(\\xi)]}$ as in Section~\\ref{sec:validation}.}\n\\label{fig:validation1}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\begin{subfigure}[The solution in the $(t,x,\\rho)$-coordinates.]\n    {\\includegraphics[width=0.49\\textwidth]{giraffa3Dfull}\n    \\label{fig:gull1}}\n\\end{subfigure}\\hfill\n\\begin{subfigure}[The solution in the $(x,t)$-coordinates.]\n    {\\includegraphics[width=0.49\\textwidth]{giraffa3Daboveview}\n    \\label{fig:gull2}}\n\\end{subfigure}\\\\\n\\begin{subfigure}[The solution in the $(t,x,\\rho)$-coordinates for $0 ~\\le ~t \\le~15$.]\n    {\\includegraphics[width=0.49\\textwidth]{giraffa3Dbody}\n    \\label{fig:gull3}}\n\\end{subfigure}\\hfill\n\\begin{subfigure}[The solution in the $(t,x,\\rho)$-coordinates for $85 ~\\le ~t ~\\le ~87.5$.]\n    {\\includegraphics[width=0.49\\textwidth]{giraffa3Dhead}\n    \\label{fig:gull4}}\n\\end{subfigure}\n\\caption{Representation of the solution constructed in~\\cite[Section~6]{BorisCarlottaMax-M3AS} and described in Subsection \\ref{sec:validation}.}\\label{fig:validation2}\n\\end{figure}\n\nWe propose here to validate the numerical scheme~\\eqref{schema}-\\eqref{def.flux.num.contrainte}-\\eqref{contrainte.approx} using the Godounov numerical flux (see e.g.~\\cite{GodlewskiRaviartBook, LevequeBook}) which will be used in the remaining of this paper:\n\\begin{eqnarray*}\n    F(a,b) &= \\left\\{\n    \\begin{array}{l@{\\quad\\text{ if }}l}\n       \\underset{[a,b]}\\min f & a\\le b,\\\\\n        \\underset{[b,a]}\\max f & a>b.\n    \\end{array}\n    \\right.\n\\end{eqnarray*}\n\n\\noindent We consider the explicit solution to~\\eqref{eq:constrianed} constructed in~\\cite[Section~6]{BorisCarlottaMax-M3AS} by applying the wave front tracking algorithm. The set up for the simulation is as follows. Consider the domain of computation $[-6,1]$, take  a normalized flux $f(\\rho) = \\rho(1-\\rho)$ (namely the maximal velocity and the maximal density are assumed to be equal to one) and a linear weight function $w(x) = 2 (1 + x)\\,\\chi_{[-1, 0]}(x)$. Assume a uniform distribution of maximal density in $[x_A, x_B]$ at time $t=0$, namely $\\bar\\rho = \\chi_{[x_A, x_B]}$. The efficiency of the exit, $p$, see Figure~\\ref{fig:validation1}, is of the form\n\\begin{eqnarray*}\n    p(\\xi) &= \\left\\{\n    \\begin{array}{l@{\\quad\\text{ if }}l}\n       p_0 & 0\\le\\xi<\\xi_1,\\\\\n       p_1 & \\xi_1\\le\\xi<\\xi_2,\\\\\n       p_2 & \\xi_2\\le\\xi\\le 1.\n    \\end{array}\n    \\right.\n\\end{eqnarray*}\nThe explicit solution $\\rho$ corresponding to the values\n\\begin{align*}\n    &p_0 =0.21, && p_1 =0.168, && p_2 =0.021, && \\xi_1 \\sim 0.566,\n    &x_A = -5.75, && x_B =-2, && \\xi_2 \\sim 0.731,\n\\end{align*}\nis represented in Figure~\\ref{fig:validation2}. The above choices for the flux $f$ and the efficiency $p$ ensure that the solution to each Riemann problem is unique, see~\\cite{AndreianovDonadelloRosiniRazafisonProc}. We defer to~\\cite[Section~6]{BorisCarlottaMax-M3AS} for the details of the construction of the solution $\\rho$ and its physical interpretation.\n\\begin{figure}\n\\begin{subfigure}[$\\rho_\\Delta(0,x)$]\n    {\\begin{psfrags}\n    \\psfrag{rho}[c,b]{}\n    \\psfrag{x}[c,t]{\\tiny$x$}\n        \\includegraphics[width=0.22\\textwidth]{girafe_t0}\n    \\end{psfrags}\\label{fig:tiger1}}\n\\end{subfigure}~\n\\begin{subfigure}[$\\rho_\\Delta(1,x)$]\n    {\\begin{psfrags}\n    \\psfrag{rho}[c,b]{}\n    \\psfrag{x}[c,t]{\\tiny$x$}\n        \\includegraphics[width=0.22\\textwidth]{girafe_t1}\n    \\end{psfrags}\\label{fig:tiger2}}\n\\end{subfigure}~\n\\begin{subfigure}[$\\rho_\\Delta(7.325,x)$]\n    {\\begin{psfrags}\n    \\psfrag{rho}[c,b]{}\n    \\psfrag{x}[c,t]{\\tiny$x$}\n        \\includegraphics[width=0.22\\textwidth]{girafe_t2}\n    \\end{psfrags}\\label{fig:tiger3}}\n\\end{subfigure}~\n\\begin{subfigure}[$\\rho_\\Delta(10,x)$]\n    {\\begin{psfrags}\n    \\psfrag{rho}[c,b]{}\n    \\psfrag{x}[c,t]{\\tiny$x$}\n        \\includegraphics[width=0.22\\textwidth]{girafe_t3}\n    \\end{psfrags}\\label{fig:tiger4}}\n\\end{subfigure}\\\\\n\\begin{subfigure}[$\\rho(0,x)$]\n    {\\begin{psfrags}\n    \\psfrag{r}[c,c]{}\n    \\psfrag{x}[c,t]{\\tiny$\\vphantom{\\int}x$}\n    \\psfrag{z}[c,t]{\\tiny$\\vphantom{\\int}1$}\n    \\psfrag{0}[c,t]{\\tiny$\\vphantom{\\int}0$}\n    \\psfrag{1}[c,t]{\\tiny$\\vphantom{\\int}-1$}\n    \\psfrag{2}[c,t]{\\tiny$\\vphantom{\\int}-2$}\n    \\psfrag{3}[c,t]{\\tiny$\\vphantom{\\int}-3$}\n    \\psfrag{4}[c,t]{\\tiny$\\vphantom{\\int}-4$}\n    \\psfrag{5}[c,t]{\\tiny$\\vphantom{\\int}-5$}\n    \\psfrag{6}[c,t]{\\tiny$\\vphantom{\\int}-6$}\n    \\psfrag{a}[r,b]{\\tiny$0.0$}\n    \\psfrag{b}[r,c]{\\tiny$0.2$}\n    \\psfrag{c}[r,c]{\\tiny$0.4$}\n    \\psfrag{d}[r,c]{\\tiny$0.6$}\n    \\psfrag{e}[r,c]{\\tiny$0.8$}\n    \\psfrag{f}[r,c]{\\tiny$1.0$}\n        \\includegraphics[width=0.22\\textwidth]{1Devacuationxr0}\n    \\end{psfrags}\\label{fig:zebra1}}\n\\end{subfigure}~\n\\begin{subfigure}[$\\rho(1,x)$]\n    {\\begin{psfrags}\n    \\psfrag{r}[c,c]{}\n    \\psfrag{x}[c,t]{\\tiny$\\vphantom{\\int}x$}\n    \\psfrag{z}[c,t]{\\tiny$\\vphantom{\\int}1$}\n    \\psfrag{0}[c,t]{\\tiny$\\vphantom{\\int}0$}\n    \\psfrag{1}[c,t]{\\tiny$\\vphantom{\\int}-1$}\n    \\psfrag{2}[c,t]{\\tiny$\\vphantom{\\int}-2$}\n    \\psfrag{3}[c,t]{\\tiny$\\vphantom{\\int}-3$}\n    \\psfrag{4}[c,t]{\\tiny$\\vphantom{\\int}-4$}\n    \\psfrag{5}[c,t]{\\tiny$\\vphantom{\\int}-5$}\n    \\psfrag{6}[c,t]{\\tiny$\\vphantom{\\int}-6$}\n    \\psfrag{a}[r,b]{\\tiny$0.0$}\n    \\psfrag{b}[r,c]{\\tiny$0.2$}\n    \\psfrag{c}[r,c]{\\tiny$0.4$}\n    \\psfrag{d}[r,c]{\\tiny$0.6$}\n    \\psfrag{e}[r,c]{\\tiny$0.8$}\n    \\psfrag{f}[r,c]{\\tiny$1.0$}\n        \\includegraphics[width=0.22\\textwidth]{1Devacuationxr1}\n    \\end{psfrags}\\label{fig:zebra2}}\n\\end{subfigure}~\n\\begin{subfigure}[$\\rho(7.325,x)$]\n    {\\begin{psfrags}\n    \\psfrag{r}[c,c]{}\n    \\psfrag{x}[c,t]{\\tiny$\\vphantom{\\int}x$}\n    \\psfrag{z}[c,t]{\\tiny$\\vphantom{\\int}1$}\n    \\psfrag{0}[c,t]{\\tiny$\\vphantom{\\int}0$}\n    \\psfrag{1}[c,t]{\\tiny$\\vphantom{\\int}-1$}\n    \\psfrag{2}[c,t]{\\tiny$\\vphantom{\\int}-2$}\n    \\psfrag{3}[c,t]{\\tiny$\\vphantom{\\int}-3$}\n    \\psfrag{4}[c,t]{\\tiny$\\vphantom{\\int}-4$}\n    \\psfrag{5}[c,t]{\\tiny$\\vphantom{\\int}-5$}\n    \\psfrag{6}[c,t]{\\tiny$\\vphantom{\\int}-6$}\n    \\psfrag{a}[r,b]{\\tiny$0.0$}\n    \\psfrag{b}[r,c]{\\tiny$0.2$}\n    \\psfrag{c}[r,c]{\\tiny$0.4$}\n    \\psfrag{d}[r,c]{\\tiny$0.6$}\n    \\psfrag{e}[r,c]{\\tiny$0.8$}\n    \\psfrag{f}[r,c]{\\tiny$1.0$}\n        \\includegraphics[width=0.22\\textwidth]{1Devacuationxr2}\n    \\end{psfrags}\\label{fig:zebra3}}\n\\end{subfigure}~\n\\begin{subfigure}[$\\rho(10,x)$]\n    {\\begin{psfrags}\n    \\psfrag{r}[c,c]{}\n    \\psfrag{x}[c,t]{\\tiny$\\vphantom{\\int}x$}\n    \\psfrag{z}[c,t]{\\tiny$\\vphantom{\\int}1$}\n    \\psfrag{0}[c,t]{\\tiny$\\vphantom{\\int}0$}\n    \\psfrag{1}[c,t]{\\tiny$\\vphantom{\\int}-1$}\n    \\psfrag{2}[c,t]{\\tiny$\\vphantom{\\int}-2$}\n    \\psfrag{3}[c,t]{\\tiny$\\vphantom{\\int}-3$}\n    \\psfrag{4}[c,t]{\\tiny$\\vphantom{\\int}-4$}\n    \\psfrag{5}[c,t]{\\tiny$\\vphantom{\\int}-5$}\n    \\psfrag{6}[c,t]{\\tiny$\\vphantom{\\int}-6$}\n    \\psfrag{a}[r,b]{\\tiny$0.0$}\n    \\psfrag{b}[r,c]{\\tiny$0.2$}\n    \\psfrag{c}[r,c]{\\tiny$0.4$}\n    \\psfrag{d}[r,c]{\\tiny$0.6$}\n    \\psfrag{e}[r,c]{\\tiny$0.8$}\n    \\psfrag{f}[r,c]{\\tiny$1.0$}\n        \\includegraphics[width=0.22\\textwidth]{1Devacuationxr3}\n    \\end{psfrags}\\label{fig:zebra4}}\n\\end{subfigure}\\\\\n\\begin{subfigure}[$\\rho_\\Delta(11.939,x)$]\n    {\\begin{psfrags}\n    \\psfrag{rho}[c,b]{}\n    \\psfrag{x}[c,t]{\\tiny$x$}\n        \\includegraphics[width=0.22\\textwidth]{girafe_t4}\n    \\end{psfrags}\\label{fig:tiger5}}\n\\end{subfigure}~\n\\begin{subfigure}[$\\rho_\\Delta(85.2,x)$]\n    {\\begin{psfrags}\n    \\psfrag{rho}[c,b]{}\n    \\psfrag{x}[c,t]{\\tiny$x$}\n        \\includegraphics[width=0.22\\textwidth]{girafe_t5}\n    \\end{psfrags}\\label{fig:tiger6}}\n\\end{subfigure}~\n\\begin{subfigure}[$\\rho_\\Delta(85.5526,x)$]\n    {\\begin{psfrags}\n    \\psfrag{rho}[c,b]{}\n    \\psfrag{x}[c,t]{\\tiny$x$}\n        \\includegraphics[width=0.22\\textwidth]{girafe_t6}\n    \\end{psfrags}\\label{fig:tiger7}}\n\\end{subfigure}~\n\\begin{subfigure}[$\\rho_\\Delta(87.4981,x)$]\n    {\\begin{psfrags}\n    \\psfrag{rho}[c,b]{}\n    \\psfrag{x}[c,t]{\\tiny$x$}\n        \\includegraphics[width=0.22\\textwidth]{girafe_t7}\n    \\end{psfrags}\\label{fig:tiger8}}\n\\end{subfigure}\\\\\n\\begin{subfigure}[$\\rho(11.939,x)$]\n    {\\begin{psfrags}\n    \\psfrag{r}[c,c]{}\n    \\psfrag{x}[c,t]{\\tiny$\\vphantom{\\int}x$}\n    \\psfrag{z}[c,t]{\\tiny$\\vphantom{\\int}1$}\n    \\psfrag{0}[c,t]{\\tiny$\\vphantom{\\int}0$}\n    \\psfrag{1}[c,t]{\\tiny$\\vphantom{\\int}-1$}\n    \\psfrag{2}[c,t]{\\tiny$\\vphantom{\\int}-2$}\n    \\psfrag{3}[c,t]{\\tiny$\\vphantom{\\int}-3$}\n    \\psfrag{4}[c,t]{\\tiny$\\vphantom{\\int}-4$}\n    \\psfrag{5}[c,t]{\\tiny$\\vphantom{\\int}-5$}\n    \\psfrag{6}[c,t]{\\tiny$\\vphantom{\\int}-6$}\n    \\psfrag{a}[r,b]{\\tiny$0.0$}\n    \\psfrag{b}[r,c]{\\tiny$0.2$}\n    \\psfrag{c}[r,c]{\\tiny$0.4$}\n    \\psfrag{d}[r,c]{\\tiny$0.6$}\n    \\psfrag{e}[r,c]{\\tiny$0.8$}\n    \\psfrag{f}[r,c]{\\tiny$1.0$}\n        \\includegraphics[width=0.22\\textwidth]{1Devacuationxr4}\n    \\end{psfrags}\\label{fig:zebra5}}\n\\end{subfigure}~\n\\begin{subfigure}[$\\rho(85.2,x)$]\n    {\\begin{psfrags}\n    \\psfrag{r}[c,c]{}\n    \\psfrag{x}[c,t]{\\tiny$\\vphantom{\\int}x$}\n    \\psfrag{z}[c,t]{\\tiny$\\vphantom{\\int}1$}\n    \\psfrag{0}[c,t]{\\tiny$\\vphantom{\\int}0$}\n    \\psfrag{1}[c,t]{\\tiny$\\vphantom{\\int}-1$}\n    \\psfrag{2}[c,t]{\\tiny$\\vphantom{\\int}-2$}\n    \\psfrag{3}[c,t]{\\tiny$\\vphantom{\\int}-3$}\n    \\psfrag{4}[c,t]{\\tiny$\\vphantom{\\int}-4$}\n    \\psfrag{5}[c,t]{\\tiny$\\vphantom{\\int}-5$}\n    \\psfrag{6}[c,t]{\\tiny$\\vphantom{\\int}-6$}\n    \\psfrag{a}[r,b]{\\tiny$0.0$}\n    \\psfrag{b}[r,c]{\\tiny$0.2$}\n    \\psfrag{c}[r,c]{\\tiny$0.4$}\n    \\psfrag{d}[r,c]{\\tiny$0.6$}\n    \\psfrag{e}[r,c]{\\tiny$0.8$}\n    \\psfrag{f}[r,c]{\\tiny$1.0$}\n        \\includegraphics[width=0.22\\textwidth]{1Devacuationxr4,5}\n    \\end{psfrags}\\label{fig:zebra6}}\n\\end{subfigure}~\n\\begin{subfigure}[$\\rho(85.5526,x)$]\n    {\\begin{psfrags}\n    \\psfrag{r}[c,c]{}\n    \\psfrag{x}[c,t]{\\tiny$\\vphantom{\\int}x$}\n    \\psfrag{z}[c,t]{\\tiny$\\vphantom{\\int}1$}\n    \\psfrag{0}[c,t]{\\tiny$\\vphantom{\\int}0$}\n    \\psfrag{1}[c,t]{\\tiny$\\vphantom{\\int}-1$}\n    \\psfrag{2}[c,t]{\\tiny$\\vphantom{\\int}-2$}\n    \\psfrag{3}[c,t]{\\tiny$\\vphantom{\\int}-3$}\n    \\psfrag{4}[c,t]{\\tiny$\\vphantom{\\int}-4$}\n    \\psfrag{5}[c,t]{\\tiny$\\vphantom{\\int}-5$}\n    \\psfrag{6}[c,t]{\\tiny$\\vphantom{\\int}-6$}\n    \\psfrag{a}[r,b]{\\tiny$0.0$}\n    \\psfrag{b}[r,c]{\\tiny$0.2$}\n    \\psfrag{c}[r,c]{\\tiny$0.4$}\n    \\psfrag{d}[r,c]{\\tiny$0.6$}\n    \\psfrag{e}[r,c]{\\tiny$0.8$}\n    \\psfrag{f}[r,c]{\\tiny$1.0$}\n        \\includegraphics[width=0.22\\textwidth]{1Devacuationxr5}\n    \\end{psfrags}\\label{fig:zebra7}}\n\\end{subfigure}~\n\\begin{subfigure}[$\\rho(87.4981,x)$]\n    {\\begin{psfrags}\n    \\psfrag{r}[c,c]{}\n    \\psfrag{x}[c,t]{\\tiny$\\vphantom{\\int}x$}\n    \\psfrag{z}[c,t]{\\tiny$\\vphantom{\\int}1$}\n    \\psfrag{0}[c,t]{\\tiny$\\vphantom{\\int}0$}\n    \\psfrag{1}[c,t]{\\tiny$\\vphantom{\\int}-1$}\n    \\psfrag{2}[c,t]{\\tiny$\\vphantom{\\int}-2$}\n    \\psfrag{3}[c,t]{\\tiny$\\vphantom{\\int}-3$}\n    \\psfrag{4}[c,t]{\\tiny$\\vphantom{\\int}-4$}\n    \\psfrag{5}[c,t]{\\tiny$\\vphantom{\\int}-5$}\n    \\psfrag{6}[c,t]{\\tiny$\\vphantom{\\int}-6$}\n    \\psfrag{a}[r,b]{\\tiny$0.0$}\n    \\psfrag{b}[r,c]{\\tiny$0.2$}\n    \\psfrag{c}[r,c]{\\tiny$0.4$}\n    \\psfrag{d}[r,c]{\\tiny$0.6$}\n    \\psfrag{e}[r,c]{\\tiny$0.8$}\n    \\psfrag{f}[r,c]{\\tiny$1.0$}\n        \\includegraphics[width=0.22\\textwidth]{1Devacuationxr6}\n    \\end{psfrags}\\label{fig:zebra8}}\n\\end{subfigure}\\\\\n\\caption{With reference to Subsection \\ref{sec:validation}: The numerically computed solution $x \\mapsto \\rho_\\Delta(t,x)$ and the explicitly computed solution $x \\mapsto \\rho(t,x)$ at different fixed times $t$.}\\label{fig:validation3}\n\\end{figure}\n\n\\noindent A qualitative comparison between the numerically computed solution $x \\mapsto \\rho_\\Delta(t,x)$ and the explicitly computed solution $x \\mapsto \\rho(t,x)$ at different fixed times $t$ is in Figure~\\ref{fig:validation3}. We observe good agreements between $x \\mapsto \\rho(t,x)$ and $x \\mapsto\\rho_\\Delta(t,x)$. The parameters for the numerically computed solution are $\\Delta x=3.5\\times10^{-4}$ and $\\Delta t=7\\times10^{-5}$.\\\\\nA convergence analysis is also performed for this test. We introduce the relative $\\L1$-error for the density $\\rho$, at a given time $t^n$, defined by\n$$E_{\\L1}^n=\\left [\\sum_{j}\\left|\\rho(t^n,x_j)-\\rho_j^n\\right|\\right ]\\,\\Big\/\\left [\\sum_{j}\\left|\\rho(t^n,x_j)\\right|\\right ].$$\nIn Table~\\ref{erreurs}, we computed the relative $\\L1$-errors for different numbers of space cells at the fixed time $t=10$. We deduce that the order of convergence is approximatively $0.906$. As in~\\cite{scontrainte}, we observe that the modification~\\eqref{def.flux.num.contrainte} of the numerical flux does not affect the accuracy of the scheme. \n\n\\begin{table}[ht]\n\\begin{center}\n\\begin{tabular}{|c|c|}\n\\hline\nNumber of cells & $\\L1$-error\\\\\n\\hline\n$625$ & $9.6843\\times10^{-3}$\\\\\n\\hline\n$1250$ & $6.2514\\times10^{-3}$\\\\\n\\hline\n$2500$ & $3.4143\\times10^{-3}$\\\\\n\\hline\n$5000$ & $1.3172\\times10^{-3}$\\\\\n\\hline\n$10000$ & $1.03\\times10^{-3}$\\\\\n\\hline\n$20000$ & $4.2544\\times10^{-4}$\\\\\n\\hline\nOrder & $0.906$\\\\\n\\hline\n\\end{tabular}\n\\tiny\\caption{Relative $\\L1$-error at time $t=10$.}\n\\label{erreurs}\n\\end{center}\n\\end{table}\n\n\n\\section{Numerical simulations}\\label{sec:simulations}\n\n\nThis section is devoted to the phenomenological description of some collective effects in crowd dynamics related to capacity drop, namely the Braess' paradox and the Faster Is Slower (FIS) effect.\n\n\n\n\n\n\n\\subsection{Faster is Slower effect}\\label{sec:FIS}\n\nThe FIS effect was first described in~\\cite{Helbing2000Simulating, Parisi2005606} in the context of the room evacuation problem. The authors studied the evolution of the evacuation time as a function of the maximal velocity reached by the pedestrians, and they shown that there exists an optimal velocity for which the evacuation time attains a minimum. Therefore, any acceleration beyond the optimal velocity worses the evacuation time. \nFollowing the studies above, the curve representing the evacuation time as a function of the average velocity takes a characteristic shape \\cite[Figure 1c]{Parisi2005606}. \n\nThe first numerical tests we performed aim to verify if such shape is obtained starting from the ADR model. \nTo this end, we consider the corridor modeled by the segment [-6,1], with an exit at $x=0$. We consider the flux $f(\\rho)=\\rho \\, v_{\\max} \\, (1-\\rho)$ where $v_{\\max}$ is the maximal velocity of the pedestrians and the maximal density is equal to one. We use the same weight function as for the validation of the scheme, $w(x)=2(1+x)\\chi_{[-1,0]}(x)$ and, the same initial density, $\\bar{\\rho}=\\chi_{[-5.75,-2]}$. The efficiency of the exit $p$ is now given by the following continuous function\n\\begin{eqnarray}\n\\label{P_continuous}\n    p(\\xi) &= \\left\\{\n    \\begin{array}{l@{\\quad\\text{ if }}l}\n       p_0 & 0\\le\\xi<\\xi_1,\\\\[6pt]\n       \\displaystyle\\frac{(p_0-p_1)\\xi+p_1\\xi_1-p_0\\xi_2}{\\xi_1-\\xi_2} & \\xi_1\\le\\xi<\\xi_2,\\\\[10pt]\n       p_1 & \\xi_2\\le\\xi\\le1,\n    \\end{array}\n    \\right.\n\\end{eqnarray}  \nwhere\n\\begin{align*}\n    &p_0 =0.24, && p_1 =0.05, && \\xi_1=0.5, && \\xi_2=0.9.\n\\end{align*}\n\nThe space and time steps are fixed to $\\Delta x=5\\times10^{-3}$ and $\\Delta t=5\\times10^{-4}$. In Figure~\\ref{fis_f_p} are plotted the flux $f$ corresponding to the maximal velocity $v_{\\max}=1$ and the above efficiency of the exit.\n\n\\begin{figure}[ht]\n\\psfrag{rho, xi}[][][0.8]{$\\rho,\\,\\xi$}\n\\psfrag{f}[][][0.3]{$f$}\n\\psfrag{p}[][][0.3]{$p$}\n\\psfrag{f, p}[][][0.8]{$f,\\,p$}\n\\centering\\includegraphics[width=0.48\\hsize]{flux_constraint.eps}\n\\tiny \\caption {The normalized flux $\\rho\\to f(\\rho)$ and the constraint $\\xi\\to p(\\xi)$ defined in~\\eqref{P_continuous}.}\n\\label{fis_f_p}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\psfrag{velocity}[][][0.8]{$v_{\\max}$}\n\\psfrag{exit time}[][][0.8]{evacuation time}\n\\centering\\includegraphics[width=0.48\\hsize]{t_exit}\n\\tiny \\caption {With reference to Subsection \\ref{sec:FIS}: Evacuation time as a function of the velocity $v_{\\max}$.}\n\\label{fis_exit_times_velocities}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{subfigure}[$\\rho\\mapsto\\rho_\\Delta(0,t)$ for velocities $v_{\\max}\\le1$.]\n{\\begin{psfrags}\n\\psfrag{v1}[][][0.3]{$v_{\\max}=1$\\qquad}\n\\psfrag{v095}[][][0.3]{$v_{\\max}=0.95$\\qquad}\n\\psfrag{v099}[][][0.3]{$v_{\\max}=0.99$\\qquad}\n\\psfrag{rho}[][][0.6]{}\n\\psfrag{t}[][][0.8]{t}\n\\includegraphics[width=0.48\\hsize]{densities_doors_inf.eps}\n\\end{psfrags}}\n\\end{subfigure}~\n\\begin{subfigure}[$\\rho\\mapsto\\rho_\\Delta(0,t)$ for velocities $v_{\\max}\\ge1$.]\n{\\begin{psfrags}\n\\psfrag{v1}[][][0.3]{$v_{\\max}=1$\\qquad}\n\\psfrag{v101}[][][0.3]{$v_{\\max}=1.01$\\qquad}\n\\psfrag{v105}[][][0.3]{$v_{\\max}=1.05$\\qquad}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{t}[][][0.8]{t}\n\\includegraphics[width=0.48\\hsize]{densities_doors_sup.eps}\n\\end{psfrags}}\n\\end{subfigure}\n\\tiny \\caption{With reference to Subsection \\ref{sec:FIS}: Densities at the exit as a function of time for different velocities.}\n\\label{fis_densities_doors}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\psfrag{p1}[r][][0.3]{$\\beta=1$}\n\\psfrag{p2}[r][][0.3]{$\\beta=0.9$}\n\\psfrag{p3}[r][][0.3]{$\\beta=0.8$}\n\\psfrag{xi}[][][0.8]{$\\xi$}\n\\psfrag{p}[][][0.8]{$p_\\beta$}\n\\centering\\includegraphics[width=0.48\\hsize]{p_beta.eps}\n\\tiny \\caption {With reference to Subsection \\ref{sec:FIS}: The efficiencies $\\xi\\to p_\\beta(\\xi)$ for $\\beta=0.8, 0.9, 1$.}\n\\label{p_beta}\n\\end{figure}\n\n\\begin{figure}[ht]\\centering\n\\begin{subfigure}[Evacuation time as a function of $v_{\\max}$ for different amounts of initial densities.]\n{\\begin{psfrags}\n\\psfrag{exit time}[][][0.8]{evacuation time}\n\\psfrag{init1}[l][][0.3]{$\\bar{\\rho}$}\n\\psfrag{init08}[l][][0.3]{$\\bar{\\rho}_1$}\n\\psfrag{init06}[l][][0.3]{$\\bar{\\rho}_2$}\n\\psfrag{velocity}[][][0.8]{$v_{\\max}$}\n\\includegraphics[width=0.48\\hsize]{t_exit_init.eps}\n\\end{psfrags}}\n\\end{subfigure}\n\\begin{subfigure}[Evacuation time as a function of $v_{\\max}$ for different efficiencies of the exit.]\n{\\begin{psfrags}\n\\psfrag{exit time}[][][0.8]{evacuation time}\n\\psfrag{beta1}[c][][0.3]{$\\beta=1$}\n\\psfrag{beta09}[c][][0.3]{$\\beta=0.9$}\n\\psfrag{beta08}[c][][0.3]{$\\beta=0.8$}\n\\psfrag{velocity}[][][0.8]{$v_{\\max}$}\n\\includegraphics[width=0.48\\hsize]{t_exit_beta.eps}\n\\end{psfrags}}\n\\end{subfigure}\n\\begin{subfigure}[Evacuation time as a function of $v_{\\max}$ for different locations of the initial density.]\n{\\begin{psfrags}\n\\psfrag{exit time}[][][0.8]{evacuation time}\n\\psfrag{init}[l][][0.3]{$\\bar{\\rho}$}\n\\psfrag{t12}[l][][0.3]{$\\bar{\\rho}_3$}\n\\psfrag{t20}[l][][0.3]{$\\bar{\\rho}_4$}\n\\psfrag{velocity}[][][0.8]{$v_{\\max}$}\n\\includegraphics[width=0.48\\hsize]{t_exit_loc2.eps}\n\\end{psfrags}}\n\\end{subfigure}\n\\tiny \\caption {With reference to Subsection \\ref{sec:FIS}: Evacuation time as a function of $v_{\\max}$ for different parameters of the model.}\n\\label{fis_stabilities}\n\\end{figure}\n\nFigure~\\ref{fis_exit_times_velocities} represents the evacuation time as a function of the maximal velocity $v_{\\max}$, as $v_{\\max}$ varies in the interval $[0.1 , 5]$. As we can observe, the general shape described above is recovered. The numerical minimal evacuation time is $19.007$ and is obtained for $v_{\\max}=1$.\n\nIn addition, we reported in Figure~\\ref{fis_densities_doors} the density at the exit as a function of time for different values of the maximal velocity $v_{\\max}$ around the optimal one. We notice that the maximal density at the exit and the time length where the density is maximal increase with the velocity. This expresses the jamming at the exit that leads to the FIS effect. \n\n\\begin{figure}\n\\centering\n\\psfrag{t}[][][0.8]{evacuation time}\n\\psfrag{d}[][][0.8]{position of the obstacle}\n\\psfrag{A}[][][0.3]{with obstacle\\hspace{1.5cm}}\n\\psfrag{B}[][][0.3]{without obstacle\\hspace{2cm}}\n\\psfrag{D}[][][0.8]{$d$}\n\\includegraphics[width=0.48\\hsize]{braess.eps}\n\\tiny \\caption{With reference to Subsection \\ref{sec:Braess}: Evacuation time as a function of the position of the obstacle.}\n\\label{braess_exit_times}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\renewcommand{\\arraystretch}{1.5\n\\begin{tabular}{>{\\centering\\bfseries}m{0.05\\hsize} @{}>{\\centering}m{0.3\\hsize} @{}>{\\centering}m{0.3\\hsize} @{}>{\\centering\\arraybackslash}m{0.3\\hsize}}\n& Without obstacle & Obstacle at $d=-1.85$ & Obstacle at $d=-1.72$\\\\\n\\rotatebox{90}{$t=1$}&\n{\\begin{psfrags}\n\\psfrag{t=1}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(1,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{SO_1.eps}\n\\end{psfrags}}&\n{\\begin{psfrags}\n\\psfrag{t=1}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(1,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{d185_1.eps}\n\\end{psfrags}}&\n{\\begin{psfrags}\n\\psfrag{t=1}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(1,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{d172_1.eps}\n\\end{psfrags}}\n\\\\\n\\rotatebox{90}{$t=7$}&\n{\\begin{psfrags}\n\\psfrag{t=7}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(7,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{SO_7.eps}\n\\end{psfrags}}&\n{\\begin{psfrags}\n\\psfrag{t=7}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(7,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{d185_7.eps}\n\\end{psfrags}}&\n{\\begin{psfrags}\n\\psfrag{t=7}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(7,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{d172_7.eps}\n\\end{psfrags}}\n\\\\\n\\rotatebox{90}{$t=15$}&\n{\\begin{psfrags}\n\\psfrag{t=15}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(15,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{SO_15.eps}\n\\end{psfrags}}&\n{\\begin{psfrags}\n\\psfrag{t=15}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(15,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{d185_15.eps}\n\\end{psfrags}}&\n{\\begin{psfrags}\n\\psfrag{t=15}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(15,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{d172_15.eps}\n\\end{psfrags}}\n\\\\\n\\rotatebox{90}{$t=19$}&\n{\\begin{psfrags}\n\\psfrag{t=19}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(19,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{SO_19.eps}\n\\end{psfrags}}&\n{\\begin{psfrags}\n\\psfrag{t=19}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(19,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{d185_19.eps}\n\\end{psfrags}}&\n{\\begin{psfrags}\n\\psfrag{t=19}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(19,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{d172_19.eps}\n\\end{psfrags}}\n\\\\\n\\rotatebox{90}{$t=24.246$}&\n{\\begin{psfrags}\n\\psfrag{final}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(24.246,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{SO_24_246.eps}\n\\end{psfrags}}&\n{\\begin{psfrags}\n\\psfrag{final}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(24.246,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{d185_24_246.eps}\n\\end{psfrags}}&\n{\\begin{psfrags}\n\\psfrag{final}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(24.246,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{d172_24_246.eps}\n\\end{psfrags}}\n  \\end{tabular}\n\\caption{With reference to Subsection \\ref{sec:Braess}: Braess paradox simulations: density profiles at times $t=1$ (first line), $t=7$ (second line), $t=15$ (third line), $t=19$  (fourth line) and $t=24.246$ (last line).}\n\\label{Braess_snapshots}\n\\end{figure}\n\nThen we performed some series of tests to see how the general shape obtained in Figure~\\ref{fis_exit_times_velocities} changes with respect to variations of the parameters of the model. \n In Figure~\\ref{fis_stabilities}~(a), we show this variation when we consider different initial densities, namely, $\\bar\\rho$, $\\bar{\\rho}_1$ and $\\bar{\\rho}_2$ with $\\bar\\rho_1(x)=0.8\\chi_{[-5.75,-2]}$ and $\\bar\\rho_2(x)=0.6\\chi_{[-5.75,-2]}$. The general shape of the curves is conserved. We observe that the evacuation time increases with the initial amount of pedestrians while the optimal velocity decreases as the initial amount of pedestrians increases. The minimal evacuation time and the corresponding optimal maximal velocity are $12.259$ and $1.07$ for $\\bar{\\rho}_2$ and $15.691$ and $1.03$ for $\\bar{\\rho}_1$.\n \nNext we explore the case where the efficiency of the exit varies. We consider the function $p$ defined in~\\eqref{P_continuous} and the modification $p_\\beta$ such that $p_\\beta(\\xi)=p(\\beta\\xi)$. In Figure~\\ref{p_beta}, we plotted the functions $p$, $p_\\beta$ for $\\beta=0.8$ and $\\beta=0.9$. \nThen, in Figure~\\ref{fis_stabilities}~(b) are plotted the evacuation time curves corresponding to these three efficiencies of the exit. \nAs minimum evacuation times, we obtain $18.586$ and $18.827$ for $\\beta=0.8$, $0.9$ respectively. As expected, the minimal evacuation time increases with lower efficiency of the exit. The corresponding velocities are approximatively $1.06$ and $1.02$ respectively.\n\nFinally, we change the location of the initial density. In addition to the corridor $[-6,1]$, we consider two other corridors modeled by the segments $[-12,1]$ and $[-20,1]$. In these two corridors we take as initial densities $\\bar{\\rho}_3(x)=\\chi_{[-11.75,-8]}$ and $\\bar{\\rho}_4(x)=\\chi_{[-19.75,-16]}$ respectively. We have reported the obtained evacuation time curves in Figure~\\ref{fis_stabilities}~(c). As expected, the minimal evacuation time increases with the distance between the exit and the initial density location.     \n\n\\subsection{Braess' paradox}\\label{sec:Braess}\n\nThe presence of obstacles, such as columns upstream from the exit, may prevent the crowd density from reaching dangerous values and may actually help to minimize the evacuation time, since in a moderate density regime the full capacity of the exit can be exploited. \nFrom a microscopic point of view, the decrease of the evacuation time may seem unexpected, as some of the pedestrians are forced to chose a longer path to reach the exit. \n\nThe ADR model is able to reproduce the Braess' paradox for pedestrians, as we show in the following simulations. \nWe consider, as in the previous subsection, the corridor modeled by the segment $[-6,1]$ with an exit at $x=0$.\nWe compute the solution corresponding to the flux $f(\\rho)=\\rho(1-\\rho)$, the initial density $\\bar\\rho(x)=\\chi_{[-5.75,-2]}(x)$, the efficiency of the exit $p$ of the form~\\eqref{P_continuous} with the parameters \n\\begin{align*}\n    &p_0 =0.21, && p_1 =0.1, && \\xi_1=0.566, && \\xi_2=0.731\n\\end{align*}\nand the same weight function $w(x)=2(1+x)\\chi_{[-1,0]}(x)$. The space and time steps are fixed to $\\Delta x=5\\times10^{-3}$ and $\\Delta t=5\\times10^{-4}$. Without any obstacle, the numerical evacuation time is $29.496$. In these following simulations we place an obstacle at $x=d$, with $-2<d<0$.\nThe obstacle reduces the capacity of the corridor and can be seen as a door, which we assume larger than the one at $x=0$. Following these ideas we define an efficiency function $p_d(\\xi)=1.15 p(\\xi)$ and a weight function $w_d(x)=2(x-d+1)\\chi_{[d-1,d]}(x)$ associated to the obstacle. \n\nIn Figure~\\ref{braess_exit_times} we have reported the evolution of the evacuation time when the position of the obstacle varies in the interval $[-1.9,-0.01]$ with a step of 0.01. We observe that for $-1.8\\le d\\le-1.72$, the evacuation time is lower than in the absence of the obstacle. The optimal position of the obstacle is obtained for $d=-1.72$ and the corresponding evacuation time is $24.246$. We compare in Figure~\\ref{Braess_snapshots} five snapshots of the solution without obstacle and the solutions with an obstacle placed at $d=-1.72$ and $d=-1.85$. This latter location corresponds to a case where the evacuation time is greater than the one without an obstacle. In these snapshots, we see that the obstacle placed at $d=-1.85$ becomes congested very soon. This is due to the fact that the obstacle is too close to the location of the initial density. When the obstacle is placed at $d=-1.72$, it delays the congestion at the exit.\n  \n\\subsection{Zone of low velocity}\\label{sec:Braess+FIS}\n  \n\\begin{figure}\n\\centering\n\\begin{subfigure}[Evacuation time as a function of $\\lambda$.]\n{\\begin{psfrags}\n\\psfrag{t}[][][0.8]{evacuation time}\n\\psfrag{lambda}[][][0.8]{$\\lambda$}\n\\includegraphics[width=0.48\\hsize]{t_exit_coeff.eps}\n\\end{psfrags}\n\\end{subfigure}~\n\\begin{subfigure}[Evacuation time as a function of $d$.]\n{\\begin{psfrags}\n\\psfrag{t}[][][0.8]{evacuation time}\n\\psfrag{d}[][][0.8]{$d$}\n\\includegraphics[width=0.48\\hsize]{t_exit_d.eps}\n\\end{psfrags}\n\\end{subfigure}\n\n\\begin{subfigure}[Evacuation time as a function of $v_{\\max}$.]\n{\\begin{psfrags}\n\\psfrag{t}[][][0.8]{evacuation time}\n\\psfrag{v}[][][0.8]{$v_{\\max}$}\n\\includegraphics[width=0.48\\hsize]{t_exit_v.eps}\n\\end{psfrags}\n\\end{subfigure}\n\\tiny \\caption {With reference to Subection \\ref{sec:Braess+FIS}: Evacuation time as a function of different parameters of the model.}\n\\label{slowzone}\n\\end{figure}\n\n\\noindent In this section, we perform a series of simulations where the obstacle introduced in Subsection~\\ref{sec:Braess} is now replaced by a zone where the velocity of pedestrians is lower than elsewhere in the domain.\nThe effect we want to observe here is similar to the one we see in Braess' Paradox. Namely we prevent an high concentration of pedestrians in front of the exit by constraining their flow in an upstream portion of the corridor. In this case however the constraint is local, as the maximal value allowed for the flow only depends on the position in the corridor. \n\n\\begin{figure}\n\\centering\n\\renewcommand{\\arraystretch}{1.5\n\\begin{tabular}{>{\\centering\\bfseries}m{0.05\\hsize} @{}>{\\centering}m{0.3\\hsize} @{}>{\\centering}m{0.3\\hsize} @{}>{\\centering\\arraybackslash}m{0.3\\hsize}}\n& Without obstacle & Obstacle at $d=-1.72$ & Zone of low velocity centered at $d=-1.72$\\\\\n\\rotatebox{90}{$t=1$}&\n{\\begin{psfrags}\n\\psfrag{t=1}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(1,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{SO_1.eps}\n\\end{psfrags}}&\n{\\begin{psfrags}\n\\psfrag{t=1}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(1,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{d172_1.eps}\n\\end{psfrags}}&\n{\\begin{psfrags}\n\\psfrag{t=1}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(1,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{d172_1bis.eps}\n\\end{psfrags}}\n\\\\\n\\rotatebox{90}{$t=7$}&\n{\\begin{psfrags}\n\\psfrag{t=7}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(7,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{SO_7.eps}\n\\end{psfrags}}&\n{\\begin{psfrags}\n\\psfrag{t=7}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(7,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{d172_7.eps}\n\\end{psfrags}}&\n{\\begin{psfrags}\n\\psfrag{t=7}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(7,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{d172_7bis.eps}\n\\end{psfrags}}\n\\\\\n\\rotatebox{90}{$t=15$}&\n{\\begin{psfrags}\n\\psfrag{t=15}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(15,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{SO_15.eps}\n\\end{psfrags}}&\n{\\begin{psfrags}\n\\psfrag{t=15}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(15,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{d172_15.eps}\n\\end{psfrags}}&\n{\\begin{psfrags}\n\\psfrag{t=15}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(15,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{d172_15bis.eps}\n\\end{psfrags}}\n\\\\\n\\rotatebox{90}{$t=19$}&\n{\\begin{psfrags}\n\\psfrag{t=19}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(19,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{SO_19.eps}\n\\end{psfrags}}&\n{\\begin{psfrags}\n\\psfrag{t=19}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(19,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{d172_19.eps}\n\\end{psfrags}}&\n{\\begin{psfrags}\n\\psfrag{t=19}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(19,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{d172_19bis.eps}\n\\end{psfrags}}\n\\\\\n\\rotatebox{90}{$t=20.945$}&\n{\\begin{psfrags}\n\\psfrag{t=20.945}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(20.945,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{SO_20_945.eps}\n\\end{psfrags}}&\n{\\begin{psfrags}\n\\psfrag{t=20.945}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(20.945,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{d172_20_945.eps}\n\\end{psfrags}}&\n{\\begin{psfrags}\n\\psfrag{t=20.945}[r][][0.4]{$\\displaystyle\\vphantom{\\int^{\\int^{\\int}}}\\rho(20.945,x)$}\n\\psfrag{rho}[][][0.8]{}\n\\psfrag{x}[][][0.8]{$x$}\n\\includegraphics[width=\\hsize]{d172_finalbis.eps}\n\\end{psfrags}}\n  \\end{tabular}\n\\caption{With reference to Subsection \\ref{sec:Braess+FIS}: Braess' paradox and zone of low velocity simulations: density profiles at times $t=1$ (first line), $t=7$ (second line), $t=15$ (third line), $t=19$  (fourth line) and $t=20.945$ (last line).}\n\\label{column_snapshots}\n\\end{figure}\nWe consider again the corridor modeled by the segment $[-6,1]$ with an exit at $x=0$. The efficiency of the exit and the initial density are the same as in the previous subsection. Assume that the slow zone is of size one and is centred at $x=d$, where $-1.9\\le d\\le0$. Define the following function\n\n\\begin{eqnarray}\n\\label{def_k}\n    k(x) &= \\left\\{\n    \\begin{array}{l@{\\quad\\text{ if }}l}\n       1 & x\\le d-0.5,\\\\[6pt]\n       -2(x-d) & d-0.5\\le x\\le d,\\\\[10pt]\n       2(x-d) & d\\le x\\le d+0.5,\\\\[10pt]\n       1 & x\\ge d+0.5,\n    \\end{array}\n    \\right.\n\\end{eqnarray}  \nand the following velocity $v(x,\\rho)= \\left[\\lambda+(1-\\lambda) \\, k(x)\\right] v_{\\max} \\, (1-\\rho)$, where $\\lambda\\in[0,1]$ and $v_{\\max}\\ge1$ is the maximal velocity. With such velocity, the maximal velocity of pedestrians decreases in the interval $[d-0.5,d]$, reaching its minimal value $\\lambda \\, v_{\\max}$ at $x=d$. Then the velocity increases in the interval $[d,d+0.5]$ reaching the maximum value $v_{\\max}$, that corresponds to the maximal velocity away from the slow zone. Finally we consider the flux $f(x,\\rho)=\\rho\\,v(x,\\rho)$ and the space and time steps are fixed to $\\Delta x=5\\times10^{-3}$ and $\\Delta t=5\\times10^{-4}$.\n\n\\noindent Figure~\\ref{slowzone}~(a) shows the evolution of the evacuation time as a function of the parameter $\\lambda$ varying in the interval $[0.1,1]$ when the center of the slow zone is fixed at $d=-1.5$. We observe that the optimal minimal velocity in the slow zone is for $\\lambda=0.88$ and the corresponding evacuation time is $20.945$. Recalling that without the slow zone the evacuation time is $29.496$, we see that the introduction of the slow zone allows to reduce the evacuation time. In Figure~\\ref{slowzone}~(b), we show the evolution of the evacuation time when varying the center of the slow zone $d$ in the interval $[-1.9,0]$ and when the minimal and the maximal velocities are fixed and correspond to $\\lambda=0.88$ and $v_{\\max}=1$. We observe here that, unlike in the Braess paradox tests case, the evacuation time does not depend on the location of the slow zone, except when this latter is close enough to the exit. \nIndeed, when the slow zone gets too close to the exit, the evacuation time grows. This is due to the fact that pedestrians do not have time to speed up before reaching the exit.\n\nFix now $d=-1.5$ and $\\lambda=0.88$ and assume that $v_{\\max}$ varies in the interval $[0.1,5]$. The evolution of the evacuation time as a function of $v_{\\max}$ is reported in Figure~\\ref{slowzone}~(c). We observe that we get the characteristic shape already obtained in the FIS effect.\n\nFinally we present in Figure~\\ref{column_snapshots} five snapshots for three different solutions. The first two solutions are the ones computed in Subsection~\\ref{sec:Braess}, without obstacle and with an obstacle located at $d=~-1.72$ respectively. The third solution is computed with a zone of low velocity centered at $d=-1.72$, $\\lambda=0.88$ and $v_{\\max}=1$. In order to have a good resolution of this third solution, the space and time steps where fixed to $\\Delta x=3.5\\times10^{-4}$ and $\\Delta t=7\\times10^{-5}$.We note that in the case where a zone of low velocity is placed in the domain, we do not see the capacity drop, as the density of pedestrians never attains very high values in the region next to the exit.\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nQualitative features that are characteristic of pedestrians' macroscopic behaviour at bottlenecks (Faster is Slower, Braess' paradox) are reproduced in the setting of the simple scalar model with non-local point constraint introduced in~\\cite{BorisCarlottaMax-M3AS}. These effects are shown to be persistent for large intervals of values of parameters. The validation is done by means of a simple and robust time-explicit splitting finite volume scheme which is proved to be convergent, with experimental rate close to one. \n\nThe results presented in this paper allow to consider more complex models.  Indeed, as ADR is a first order model, it is not able to capture more complicated effects related to crowd dynamics. Typically, ADR fails to reproduce the amplification of small perturbations. This leads to consider second order model such as the model proposed by Aw, Rascle and Zhang~\\cite{Aw_SIAP_2000, Zhang_Method_2002} in the framework of vehicular traffic.\n\nAnother extension of this work is to consider the ADR model with constraints that are non-local in time. Such constraints allow to tackle optimal management problems in the spirit of \\cite{ColomboFacchiMaterniniRosini, CGRESAIM}.  \n\nFinally, this work can also be extended to two-dimensional models where experimental validations may be possible. \n \n\\section*{Acknowledgment}\n\nAll the authors are supported by French ANR JCJC grant CoToCoLa and Polonium 2014 (French-Polish cooperation program) No.331460NC. The first author is grateful to IRMAR, Universit\\'e de Rennes, for the hospitality during the preparation of this paper. The second author is also supported by the Universit\\'e de Franche-Comt\\'e, soutien aux EC 2014.\n\nProjekt zosta{\\l} sfinansowany ze \\'srodk\\'ow Narodowego Centrum Nauki przyznanych na podstawie decyzji nr: DEC-2011\/01\/B\/ST1\/03965.\n\n\n\n\\section*{References}\n\n\\bibliographystyle{elsarticle-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\IEEEPARstart{T}{here} is a constant drive to make sensors more power-efficient.\nUsers require embedded sensor systems such as wearable fitness trackers and battery-powered smart thermostats to be small with long battery life.\nIn resource-constrained sensor applications, engineers optimise code for a small\nmemory footprint and low computational overhead~\\cite{david2020tensorflow}. \n\n\nThis article describes new methods to optimise the computational overhead of the\nconversion routines for the BME680, a digital temperature, pressure, humidity, and\nindoor air quality sensor~\\cite{bme680manual}.\nDigital sensors require conversion routines to have meaningful outputs. Embedded sensor systems require low-power\nmicrocontrollers with sufficient random-access memory (RAM), flash storage, and\ncomputational performance to store and run the conversion routines to use sensors: \nOptimising the conversion routines will help meet these requirements under tighter memory size and power consumption constraints.\n\n\\begin{figure}[!t]\n   \\includegraphics[width=0.5\\textwidth]{Figures\/raw-sensor-data-conversion-approaches.pdf}%\n   \\caption{Block diagrams for the original and new learned\n        conversion routines.}\n\t\t\\label{fig:original_block_diagram}\n\\end{figure}\n\n\n\\subsection{Contributions}\n\\noindent\nThis article presents the following three contributions: \n\\begin{enumerate}\n    \\item A new approach and its implementation of alternative, machine-learned conversion routines for temperature,\n          pressure, and humidity. \n\t\t  We demonstrate the methods using a state-of-the-art sensor, the Bosch BME680 but the insights and methods are applicable to other sensors.\n    \\item Quantitative evaluation of the computational overhead, accuracy, RAM, and flash storage usage of these conversion routines.\n    \\item A Pareto analysis of accuracy against computational overhead for the conversion routines.\n\\end{enumerate}\n\n\\subsection{Why do Sensors Need Conversion Routines?}\n\\noindent\nTypical sensors with digital outputs have a transducer which is sensitive to the\nsignal they are designed to measure (the measurand), e.g., temperature, and produce a voltage which is\nconverted to a digital signal by an analogue-to-digital converter (ADC). The ADC\noutputs a single unsigned integer with a fixed number of bits (the measurement).\nObtaining a reading with meaningful units requires\na transformation to map these unsigned integers into the correct real number range.\nIf the transducer is non-linear then the conversion\nmust also invert the sensor transducer transfer function after the ADC has digitised its analogue output signal~\\cite{stretchSensor}.\n\n\n\\subsection{Manufacturer-Provided Conversion Routines}\n\\noindent\nThe top half of Figure~\\ref{fig:original_block_diagram} shows the BME680\nconversions provided by Bosch Sensortec~\\cite{bme680manual}. \nAn external microcontroller computes the conversion routines in real time each time the microcontroller samples the sensor.\nFor the BME680, the manufacturer provides calibration constants\nspecific to each sensor. These are parameters of the conversion\nroutine, which tune the conversion on a per-sensor basis to reduce the\nimpact of any manufacturing variability in sensor properties.\n\n\\subsection{Learning New Conversions from Data}\n\\noindent\nWe can use machine learning to learn alternative conversion\nroutines from data. The bottom half of Figure~\\ref{fig:original_block_diagram} shows a block diagram of the process.\n Using the original manufacturer-provided conversion routines and a set of\ncalibration constants, our method generates training datasets, to train the models.\nOur method saves the trained weights so that at inference time the models are a direct replacement for the manufacturer-provided conversion routines in Figure~\\ref{fig:original_block_diagram}. The calibration\nconstants used to generate the training data are specific to one physical\nsensor, therefore the trained weights will only be accurate for that sensor.\n\\section{Conclusion}\n\\noindent\nLookup tables with linear interpolation, quadratic and linear regression\nare all superior to the original conversion routines in terms of computational overhead. The Pareto frontiers for the conversions we tested were\nentirely occupied by function approximation methods: Sequence-based methods ranging from ARMA to RNNs, LSTMs, and GRUs, were not on the Pareto frontier. The function approximation methods avoid the overheads of sequence\nmodels including initialising internal state, the possibility that error will\nbe worse on some unusual sequences, and the possibility that models are\nunstable.\n\nFor temperature, linear regression provides a reduction in\ncomputational overhead of \\SI{62}{\\%} with an error of \\SI{0.0114}{C}. For pressure,\nquadratic regression provides a \\SI{71}{\\%} reduction in\ncomputational overhead and an error of \\SI{0.0280}{KPa}. For humidity, a linear\ninterpolation lookup table with 400 data points performs well with a \\SI{18}{\\%}\nreduction in computational overhead and an error of \\SI{0.0337}{\\%}.\n\nComputational overhead reductions translate almost linearly to energy savings and are therefore\nuseful in low-power systems using the BME680 sensor. There is no obvious reason why these methods will not \nwork with similar sensors and their conversion routines.\n\n\\section*{Acknowledgements}\n\\noindent\nThis research is supported by an Alan Turing Institute award TU\/B\/000096 under EPSRC grant EP\/N510129\/1, by EPSRC grant EP\/V047507\/1, and by the UKRI Materials Made Smarter Research Centre (EPSRC grant EP\/V061798\/1).\nWe thank Vasileios Tsoutsouras and Orestis Kaparounakis for assistance with the figures.\n\\section{Experimental Evaluation}\n\\noindent\nFor each of the models increasing the number of input dimensions increases the computational overhead, the humidity routines are slightly more complex than the those for pressure because they clamp the output between 0 and 100\\,\\%.\nThe sequence-based methods introduce additional computational overhead without providing a corresponding decrease in error. \nMore complicated sensor conversion routines using sequence-based methods could produce a reduction in the error that justifies the increased overhead.\nMachine learning models are most uncertain (or over confident) where they have the least training data~\\cite{foong2019inbetween}.\nWe trained the function approximation methods using a mesh of points that uniformly span the domain.\nThe model never had to generalise outside of the training data and therefore overfitting is not a problem. \nFor sequence-based methods the performance on real data and synthetic training and test data was similar indicating that overfitting was not a problem.\n\n\\noindent\n\\textbf{Temperature conversion routines:} Figure~\\ref{fig:pareto_plots}(a) shows the Pareto plot of error and computational\noverhead for the temperature conversion routine. Quadratic regression (\\textcolor{2}{\\ding{68}})\nPareto dominates the original conversion (its normalised RMS error\nwas of the order $10^{-12}$, rounded to $0$) and its computational overhead is\n\\SI{47}{\\%} lower than the original. Linear regression (\\textcolor{1}{\\ding{67}})\nachieves a \\SI{62}{\\%} reduction in overhead compared to the original.\nThere is a small increase in normalised RMS error, from approximately $10^{-12}$ to\n$0.00909$, which corresponds to an RMS error of \\SI{0.0114}{^\\circ C}.\nFigure~\\ref{fig:resource_usage}(a) shows that the linear and quadratic\nregression also have lower RAM and flash storage usage than any other\nmethod including the original.\n\n\\noindent\n\\textbf{Pressure conversion routines:} Figure~\\ref{fig:pareto_plots}(b) shows that the pressure\nPareto frontier includes the original (\\textcolor{0}{\\ding{66}}), the linear interpolation lookup\ntable with 400 entries (\\textcolor{2}{\\ding{78}}), the quadratic regression (\\textcolor{2}{\\ding{68}}), and the linear regression (\\textcolor{1}{\\ding{67}}). \nThe original conversion has zero error by definition since we used it as ground\ntruth. The linear interpolation lookup table with 400 entries (\\textcolor{2}{\\ding{78}}) has the next smallest\nerror of $0.0223$ with an overhead reduction of \\SI{49}{\\%}.\nAlthough it results in an increase in normalised error of $0.0350$ (i.e., \\SI{28.0}{Pa}), quadratic regression (\\textcolor{2}{\\ding{68}})\noffers a larger reduction in computational overhead of \\SI{71}{\\%} compared to\nthe original. Linear regression (\\textcolor{1}{\\ding{67}}) offers\na greater reduction in overhead of \\SI{85}{\\%} but with\nincreased error of up to $0.998$ which is \\SI{798}{Pa}.\nFigure~\\ref{fig:resource_usage}(b) shows that the linear and quadratic regression\nhave lower resource usage than the original but the linear interpolation lookup table with 400 entries (\\textcolor{2}{\\ding{78}}) has a high RAM usage of\n\\SI{1652}{bytes} compared to \\SI{60}{bytes} for the original conversion routine. This is because\nit stores a lookup table consisting of $400$ floating-point numbers in RAM.\nStoring the lookup table in flash storage would be possible but it would\nincrease the overhead due to the need to move the data between flash storage\nand memory. \n\n\n\n\\def \\pareto_chart_height{1.45in}\n\\begin{figure}[!t]\n    \\subfloat[Temperature.]{\\adjincludegraphics[trim=0 0 {0.615\\width} 0, clip, height=\\pareto_chart_height]{Figures\/pareto_chart.pdf}%\n        \\label{fig:temperature_pareto}}\n    \\subfloat[Pressure.]{\\adjincludegraphics[trim={0.385\\width} 0 {0.31\\width} 0, clip, height=\\pareto_chart_height]{Figures\/pareto_chart.pdf}%\n        \\label{fig:pressure_pareto}}\n    \\subfloat[Humidity.]{\\adjincludegraphics[trim={0.69\\width} 0 0 0, clip, height=\\pareto_chart_height]{Figures\/pareto_chart.pdf}%\n        \\label{fig:humidity_pareto}}\n\\vspace{-0.05in}\n\t\\begin{tcolorbox}[colframe=white,colback=white]\n\t\t\\resizebox{\\textwidth}{!}{%\n\t\t\\begin{tabular}{ll}\n\t\t\\textcolor{0}{\\ding{66}:~Original} & \\textcolor{1}{\\ding{67}:~Linear Regression} \\\\\n\t\t\\textcolor{2}{\\ding{68}:~Quadratic Regression} & \\textcolor{3}{\\ding{69}:~Linear Interpolation LUT 20} \\\\\n\t\t\\textcolor{4}{\\ding{70}:~Linear Interpolation LUT 10} & \\textcolor{5}{\\ding{71}:~Linear Interpolation LUT 3}\\\\\n\t\t\\textcolor{6}{\\ding{72}:~Gaussian Process 3} & \\textcolor{7}{\\ding{73}:~Gaussian Process 2} \\\\\n\t\t\\textcolor{8}{\\ding{74}:~Neural Network 3-3-1} & \\textcolor{9}{\\ding{75}:~Neural Network 3-1} \\\\\n\t\t\\textcolor{0}{\\ding{76}:~Neural Network 1-1} &  \\textcolor{1}{\\ding{77}:~Neural Network 1} \\\\\n\t\t\\textcolor{2}{\\ding{78}:~Linear Interpolation LUT 400} & \\textcolor{3}{\\ding{79}:~Linear Interpolation LUT 100} \\\\\n\t\t\\textcolor{4}{\\ding{80}:~Linear Interpolation LUT 9} & \\textcolor{5}{\\ding{81}:~Gaussian Process 9} \\\\\n\t\t\\textcolor{6}{\\ding{82}:~Gaussian Process 4} & \\textcolor{7}{\\ding{83}:~AR 1 MA 1} \\\\\n\t\t\\textcolor{8}{\\ding{84}:~AR 2 MA 1} & \\textcolor{9}{\\ding{85}:~AR 3 MA 1} \\\\\n\t\t\\textcolor{0}{\\ding{86}:~AR 3 MA 2} & \\textcolor{1}{\\ding{87}:~GRU 1 tanh sigmoid} \\\\\n\t\t\\textcolor{2}{\\ding{88}:~GRU 1 RELU sigmoid} & \\textcolor{3}{\\ding{89}:~GRU 1 RELU softsign} \\\\\n\t\t\\textcolor{4}{\\ding{90}:~Half GRU 1 RELU softsign} & \\textcolor{5}{\\ding{91}:~Simple RNN 1 tanh} \\\\\n\t\t\\textcolor{6}{\\ding{92}:~Simple RNN 1 RELU} &  \\\\\n\t\t\\end{tabular}}\n    \\end{tcolorbox}\n\\vspace{-0.25in}\n    \\caption{\n\t\\label{fig:pareto}\n\tPareto plots for each conversion routine where smaller overhead and normalised RMS error is desirable. Pareto\n    frontiers marked in black. See the caption of Figure~\\ref{fig:resource_usage} for an explanation of the omitted results.}\n\t\\vspace{-0.1in}\n    \\label{fig:pareto_plots}\n\\end{figure}\n\n\\noindent\n\\textbf{Humidity conversion routines:} Figure~\\ref{fig:pareto_plots}(c) shows the Pareto plot for the humidity conversion.\nThe Pareto frontier includes the same four methods as for the pressure: original (\\textcolor{0}{\\ding{66}}),\nthe linear interpolation lookup table with 400 entries (\\textcolor{2}{\\ding{78}}), quadratic (\\textcolor{2}{\\ding{68}}) and linear (\\textcolor{1}{\\ding{67}})\nregression. As before the original conversion (\\textcolor{0}{\\ding{66}}) has an error of $0$. The linear interpolation lookup table with 400 entries (\\textcolor{2}{\\ding{78}}) has normalised error of $0.0344$ and an overhead reduction of \\SI{18}{\\%}. \nQuadratic regression (\\textcolor{2}{\\ding{68}}) provides a larger\nreduction in overhead of \\SI{51}{\\%} with a normalised error of $0.626$. This error is\nmuch larger than that for the quadratic regression for the pressure conversion.\nLinear regression (\\textcolor{1}{\\ding{67}}) provides the best computational overhead with a\nreduction of \\SI{71}{\\%} compared to the original but the normalised error is high\nat $4.77$. Figure~\\ref{fig:resource_usage}(c) shows that the RAM usage of the linear interpolation lookup table with 400 entries (\\textcolor{2}{\\ding{78}}) is high, but the other methods on the Pareto frontier have lower resource usage than the original.\nIf we exclude the linear interpolation lookup table with 400 entries (\\textcolor{2}{\\ding{78}}), a smaller lookup table\nreplaces it on the Pareto frontier. Most methods had larger error on the humidity conversion than on the pressure\nconversion. The 400 entry linear interpolation lookup table (\\textcolor{2}{\\ding{78}}) was an exception.\n\n\\def\\resource_chart_height{2.03in}\n\\begin{figure}[!t]\n    \\subfloat[Temperature.]{\\adjincludegraphics[trim=0 0 {0.65\\width} 0, clip, height=\\resource_chart_height]{Figures\/resource_usage.pdf}%\n        \\label{fig:temperature_resource}}\n    \\subfloat[Pressure.]{\\adjincludegraphics[trim={0.35\\width} 0 {0.33\\width} 0, clip, height=\\resource_chart_height]{Figures\/resource_usage.pdf}%\n        \\label{fig:pressure_resource}}\n    \\subfloat[Humidity.]{\\adjincludegraphics[trim={0.67\\width} 0 0 0, clip, height=\\resource_chart_height]{Figures\/resource_usage.pdf}%\n        \\label{fig:humidity_resource}}\n    \\caption{RAM and flash storage usage of each model. The caption of Figure~\\ref{fig:pareto_plots} shows the key for the y-axis.\n\t\t\t The hatched regions correspond to models which are not appropriate for the conversion routine being modelled or where we had prior knowledge that the quadratic regression would Pareto dominate all the sequence based methods. \n\t\\vspace{-0.1in}}\n    \\label{fig:resource_usage}\n\\end{figure}\n\n\\section{Related Work}\n\\noindent\nAs far as we are aware there is no prior work on using machine learning to find\nconversion routines with reduced computational overhead and memory\nrequirements for an existing, commercially-available sensor. Two recent review\npapers discuss the use of machine learning for signal processing on the same\nphysical hardware as the sensor but this requires bespoke hardware with\nintegrated sensing and computing capability~\\cite{zhou2020near, ballard2021machine}. Our method\nof learning new sensor conversion routines from data can run on existing\nhardware without modification. Oldfrey et al. present a machine learning approach\nthat linearises the output of cheap, highly-non-linear stretch\nsensors~\\cite{stretchSensor} and Zhang et al. reconstruct a full\nspectrum from reduced data~\\cite{spectrometer}. Neither of these methods are\ndirectly applicable to the learning of sensor conversion routines for existing\ncommercially-available sensors from data.\n\\section{Evaluation Method}\n\\noindent\nWe measured the accuracy of the models using a sequence of synthetic data which was $1000$ points\nlong. We averaged each result over ten similar datasets.\nFor models that included an element of randomness such as\nneural networks, we averaged the results over five different random seeds. \n\\subsection{Accuracy Measure}\n\\noindent\nLet $y_\\mathrm{Predict}$ be the value predicted by the machine learning model, $y_\\mathrm{True}$ be the ground truth value from the original conversion routine, and $y_\\mathrm{True}^\\mathrm{(Max)}$ and $y_\\mathrm{True}^\\mathrm{(Min)}$ be the maximum and minimum values of the operating range of the sensor for the physical quantity of interest. \nWe used a normalised RMS error to quantify accuracy, where:\n\\small\n\\begin{equation}\\label{eq:normalised_error}\n    E_\\mathrm{RMS} = \\sqrt{\\sum_i \\left( (y_\\mathrm{Predict}^{(i)}-y_\\mathrm{True}^{(i)}) \\times\\frac{100}{y_\\mathrm{True}^\\mathrm{(Max)}-y_\\mathrm{True}^\\mathrm{(Min)} }\\right)^2} .\n\\end{equation}\n\\normalsize\nThis allows comparison between the measurements from different sensors\n(e.g., temperature -40 to 85\\,$^\\circ$C, pressure \\SI{30}{kPa} to \\SI{110}{kPa}, and humidity 0 to \\SI{100}{\\%}~\\cite{bme680manual}).\n\n\\subsection{Computational Overhead and Memory Usage Measures}\n\\noindent\nWe first implemented each model in Python making use of the TensorFlow library~\\cite{tensorflow}. \nWe then exported the trained weights and created \\lstinline{C}-language implementations of the inference functions based on these weights.\nWe measured overhead as the number of RISC-V instructions needed to\nconvert a single ADC sample into a physically-meaningful real-valued number with units. \nWe used the Sunflower embedded system emulator~\\cite{sunflower} to measure the number\nof dynamic instructions executed at run time by a binary compiled from a given \\lstinline{C}-language implementation:\nWe used a feature in Sunflower to precisely and deterministically measure the dynamic instructions\nexecuted for a given region of source code. Computational overhead translates almost linearly to energy consumption so we use computational overhead as the cost metric~\\cite{sunflower}. We measured the flash and RAM usage of the implementations of the models generated by each machine learning method by\ntaking the size of the \\code{.text} section as the flash storage requirement. We summed the\n\\code{.bss} and \\code{.data} section sizes to determine the RAM requirement for global\nvariables and manually counted the memory required for local variables.\n\n\n\\section{Tested Models}\n\\noindent\nWe evaluated both function approximation methods and time series methods.\n\\subsection{Function Approximation Methods}\\label{sec:function_approximation_methods}\n\\noindent\nWe trained most models with datasets generated by the mesh method described in\nSection~\\ref{sec:training_data} using the inverse refinement and 20 \ndiscretisation levels each for temperature, pressure, and humidity. \nWe used other datasets where explicitly mentioned.\nThe methods that we trained fall into five broad categories: \n\n\\subsubsection{Linear Regression}\\label{sec:linear_regression}\n\\noindent\nSimple linear-in-the-parameters regression using linear feature vectors.\nWe used a simple non-Bayesian linear regression.\n\n\\subsubsection{Quadratic Regression}\\label{sec:quadratic_regression}\n\\noindent\nA linear-in-the-parameters regression using quadratic feature vectors.\n\n\\subsubsection{Linear Interpolation Lookup Table}\\label{sec:lookup_table}\n\\noindent\nA lookup table that linearly interpolates between three nearby points.\nWe used bespoke datasets using the mesh method described in\nSection~\\ref{sec:training_data} without the inverse refinement. We did this to \nensure a regular, square grid in the unconverted (input) domain to simplify \nthe interpolation. We tested discretisation levels of 3, 10, and 20.\n\n\\subsubsection{Gaussian Process Regression}\\label{sec:gaussian_processes}\n\\noindent\nGaussian Process regression as described by Rasmussen~\\cite{gaussianProcessses}.\nWe used a Gaussian likelihood function and Gaussian kernel to solve the\nregression analytically. The training datasets used discretisation levels of two \nand three. We optimised the hyper-parameters in the kernel function to maximise the\nlog-likelihood of generating the data.\n\n\\subsubsection{Neural Networks}\\label{sec:neural_networks}\n\\noindent\nSimple feed-forward fully-connected neural networks. Every neuron uses the same activation function. \nA preliminary evaluation showed that the exponential, Gaussian kernel, sigmoid, RELU, hard sigmoid, tanh, and soft sign activation functions had dynamic instruction overheads of 109, 131, 122, 19, 33, 124, and 11 RISC-V instructions respectively. We therefore chose to use the RELU function because it is\ncommonly used and has low computational overhead (19 dynamic instructions). We trained models to\nminimise the RMS error (Equation~\\ref{eq:normalised_error}). To aid training, we rescaled the inputs and outputs to the range $[0, 1]$.\n\n\\subsection{Sequence-Based Models}\n\\noindent\nSequence-based models may be able to provide lower computational overhead\nby taking advantage of the fact that the sensors measure physical systems and\nwe have prior knowledge of what a sequence of sensor data might look like.\n\nWe trained all the sequence-based models to minimise the RMS error.\nWe generated $5000$ point training datasets using the\nsequence method from Section~\\ref{sec:training_data}. Training required\napproximately $20,000$ epochs. By comparison, the feed-forward\nneural networks required fewer than $10,000$ epochs.\nWe trained four categories of sequence-based models:\n\n\\subsubsection{Auto-Regressive Moving Average (ARMA)}\\label{sec:arma}\n\\noindent\nARMA models are usually used in systems with many wide sense stationarity\nassumptions and where the inputs are white noise. With these\nassumptions the optimal parameters can be\ncomputed analytically but in this case, with no simplifying assumptions, we\ntrained them iteratively using TensorFlow.\n\n\\subsubsection{Gated Recurrent Unit (GRU)}\\label{sec:gru}\n\\noindent\nA single GRU cell with a history vector of length one,\nproviding a scalar output as required. There are two different\nversions of the GRU: We used the version proposed by Chung et al.~\\cite{gru}, but both\nusually perform similarly~\\cite{gru}.\nLong short term memory (LSTM) neurons, used by Oldfrey et al.~\\cite{stretchSensor} are\nmore complicated than the GRU and have higher computational overhead.\n\n\\subsubsection{Half GRU}\\label{sec:half_gru}\n\\noindent\nA GRU cell without the output gate. The output is returned directly after applying the activation function.\nThis reduces computational overhead.\n\n\\subsubsection{Simple RNN}\\label{sec:simple_rnn}\n\\noindent\nThe simple RNN is the least-complex of all, with no gating. The non-linear activation function\nallows it to model more complex data than the ARMA models.\n\\section{Training Data}\\label{sec:training_data}\n\\noindent\nWe produced synthetic mesh datasets and sequence datasets for training and testing the models.\nThe synthetic datasets take the original conversion routines, provided by the\nmanufacturer~\\cite{bme680manual}, as ground truth.\nWe generated mesh datasets by producing a fine mesh of equally-spaced points\nacross the unconverted domain. We applied the original conversion routines\nto produce labels for all of these data points.\n\nTo produce datasets which are continuous sequences, we used Gaussian processes,\nadjusting the characteristics using a kernel function.\nWe used a five-halves Matern kernel function and\nset the length scale to $20$ times the virtual sampling period. The virtual\nsampling period is the constant interval between evaluating the\nGaussian process and has no link to any real sampling period. To ensure that\nthe generated sequence fills the required range, we apply a linear\ntransformation after generating the function. This makes the\namplitude parameter of the kernel function irrelevant. For multi-dimensional\ninputs, we evaluated a single dimensional sample function for each input.\nWe drew sample functions independently from the same single dimensional Gaussian process.\nWe applied the original conversions to provide labels for the dataset.\n\nAn optional refinement, the inverse refinement, ensures datasets\nprecisely match the sensor operating range. Instead of generating data\nin the unconverted domain, we generate data in the converted domain, where it\nis easy to enforce the limits of the operating range.\nWe compute the unconverted values by inverting the original conversion\nroutines.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{INTRODUCTION}\nRadiative loss via X-ray emission in galaxy clusters has long been supposed to be the origin of the substantial gas inflow, namely the ``cooling flow'' (see Fabian 1994 for a review). However, X-ray spectra from {\\em ASCA}\\\/\\ and {\\em XMM-Newton}\\\/\\ observations fail to show line emission from ions with intermediate or low temperatures, which implies that the cooling rate is only at most 20\\% of the previously assumed value (e.g., Kaastra et al. 2001; Peterson et al. 2001; Xu et al. 2002; Peterson et al. 2003; Tamura et al. 2003). {\\em Chandra}\\\/\\ observations have also confirmed such small cooling rates (McNamara et al. 2000). There must be a heating source in the cluster core region. The most popular candidate of the heating source is the AGN at cluster centers. The AGN dissipates energy via  bubbles, jets, sound waves and weak shocks evolved from sound waves (Fabian \\& Sanders 2009). In Perseus and Virgo Clusters, previous works have observed the sound waves and weak shocks generated by activities of central AGNs (Fabian et al. 2003; Ruszkowski et al. 2004; Forman et al. 2005). Besides the AGN activity, some other heating mechanisms have also been concerned in the last decade. Especially since the launch of {\\em Chandra}, shock fronts have been revealed clearly in the ICM of galaxy clusters through X-ray imaging-spectroscopic studies, which can efficiently convert kinetic energy of the gas into thermal energy. Such heating process may be associated with the feedback mechanism that balances the radiative cooling in galaxy clusters. \n\nThis work would study the weak shock front in the galaxy cluster A2556 by using the {\\em Chandra}\\\/\\ observation. A2556 (RA=23h13m03.3s, Dec=$-$21d37m40s, J2000.0; richness 1) is located at $z \\simeq 0.0871$ in the Aquarius supercluster (Batuski et al. 1999), an overdense region of the universe. It is moderately luminous in the X-ray band ($L_{\\rm X}\\simeq2\\times 10^{44}$ erg ${\\rm s^{-1}}$; Ebeling et al. 1996), possessing a velocity dispersion $\\simeq872$ km ${\\rm s^{-1}}$ (White et al. 1997) and a virial mass $\\simeq2.5\\pm0.1\\times10^{15}$ M$_{\\odot}$ (Reimers et al. 1996). \n\n\nWe describe the data reduction, imaging and spectroscopy analysis in \\S 2; we discuss and summarize our results in \\S 3 and  \\S 4, respectively. Throughout this paper, we adopt the cosmological parameters $H_{0}=71$ km s$^{-1}$ Mpc$^{-1}$, $\\Omega_{m} = 0.27$ and $\\Omega_{\\Lambda} = 0.73$, so that $1^{\\prime}$ corresponds to $96.7h_{71}^{-1}$ kpc at such distance. We utilize the solar abundance standards of Grevesse and Sauval (1998), where the iron abundance relative to hydrogen is $3.16\\times10^{-5}$ in number. Unless stated otherwise, all quoted errors are derived at the 68\\% confidence level.\n\n\n\n\\section{DATA ANALYSIS}\\label{ana}\n\n\n\\subsection{Observation and Data Reduction}\nThe {\\em Chandra}\\\/\\ observation of A2556 was carried out on October 5, 2001 (ObsID 2226) for a total exposure of 21.5 ks with CCDs 3, 5, 6, 7 and 8 of the Advanced  CCD Imaging Spectrometer (ACIS) in operation. Events were collected  with a frame time of 3.2 s and telemetered in the VeryFaint mode. The focal plane temperature was set to $-120$ $^{\\circ}$C. The center of A2556 was positioned nearly on the ACIS-S3 chip with about $0.578^{\\prime}$ offset. Most of the X-ray emissions of A2556 are covered by the S3 chip. Therefore, this work focuses on the data drawn from the S3 chip. The {\\em Chandra}\\\/\\\ndata analysis package CIAO software (version 3.4) is used to process the data from the S3 chip. We keep events with {\\em ASCA}\\\/\\ grades 0, 2, 3, 4 and 6, and remove all the bad pixels, bad columns, columns adjacent to bad columns and node boundaries. No background flare is found during the live operating time.\n\n\n\n\n\n\n\\subsection{X-ray Surface Brightness}\\label{xsb}\n\n\n\n\nWe show the raw {\\em Chandra}\\\/\\  ACIS-S3 image of A2556 in 0.7--7.0 keV band in Figure \\ref{ima}a, and have all the point sources excluded, which could be detected at the confidence level of $3\\sigma$ by the CIAO tool ${celldetect}$. The black cross in the center of Figure \\ref{ima}a indicates the X-ray peak of A2556. The core region of the A2556 shows no complex irregularity and should be composed of a single nucleus; the outer region shows a slight elongation toward the northwest. The X-ray morphology roughly shows a generally relax appearance, except that there exhibits an arm-like structure with an edge at approximately 160$h_{71}^{-1}$ kpc northeast of the cluster center, and the edge spans about $85^{\\circ}$ azimuthally from southeast to northeast. We highlight the edge span by two white lines in Figure \\ref{ima}a. Figure \\ref{ima}b shows the SDSS B-band image of A2556 with the Chandra 0.7--7.0 keV X-ray intensity contours overlaid. The X-ray peak consists with the centroid of the cD galaxy 2MASX J23130142-2138039 within $1^{\\prime\\prime}$. The cD galaxy shows an elongation in the same direction as the X-ray morphology, which may due to the nearby A2554. \n\n\nTo get more insight into the edge of this arm-like structure, we extract the exposure-corrected  surface brightness profile (SBP) from a series of narrow elliptical annuli, which are parallel to the east (E) region show in the Figure \\ref{ima}b. In order to get a comparison, we also extract SBPs of other three directions in Figure \\ref{ima}b, namely south (S), west (W) and north (N) regions. The energy band for these profiles is restricted to 0.7--7.0 keV. We show such extracted SBPs in Figure \\ref{sbp}. In region E, a substantial discontinuity of the SBP could be observed at approximately 160$h_{71}^{-1}$ kpc, and other three regions do not present such characteristic.\n\nTo quantitatively describe the discontinuity of the SBP in region E, we fit observed SBP with a modeled gas emissivity distribution, which corresponds to a function as\n\\begin{equation}\\label{er}\n\\varepsilon(r) = \\left\\{ \\begin{array}{ll}\n \\varepsilon_{1}\\left(r\/R_{\\rm cut}\\right)^{\\alpha_{1}} & \\textrm{$r\\le R_{\\rm cut}$}\\\\\n \\varepsilon_{2}\\left(r\/R_{\\rm cut}\\right)^{\\alpha_{2}} & \\textrm{$r>R_{\\rm cut}$},\\\\\n  \\end{array} \\right.\n\\end{equation}\nwhere $r$ represents the radius of the projected SBP; the gas emissivity distribution, $\\varepsilon(r)$,  is broken at the truncation radius, $R_{\\rm cut}$, and described with two power-law components; $\\varepsilon_{1}$, $\\varepsilon_{2}$, $\\alpha_{1}$ and $\\alpha_{2}$ are 4 fitting parameters. The solid line in Figure \\ref{sbp} shows the best-fit SBP obtained with Eq. \\ref{er}. The best-fit $R_{\\rm cut}=158.26^{+0.54}_{-0.42}$$h_{71}^{-1}$ kpc corresponding to the edge of the arm-like structure. $\\varepsilon_{1}$ and $\\varepsilon_{2}$ shall represent the plasma emissivities inside and outside of the edge, respectively. The best-fit emissivity jump $\\varepsilon_{1}\/\\varepsilon_{2}= 1.90^{+0.02}_{-0.04}$. Since the gas density is related to the emissivity by $\\varepsilon \\sim \\rho^{2}$, the density jump across the edge is calculated to be $\\rho_{1}\/\\rho_{2}=(\\varepsilon_{1}\/\\varepsilon_{2})^{1\/2}=1.37^{+0.02}_{-0.03}$.\n\n\n\\subsection{Spectral Analysis}\n\n\nIn the {\\em Chandra}\\\/\\ spectral analysis, there are two usual ways to obtain a background spectrum, including cosmic, instrumental and non X-ray particle components. Firstly, the observated dataset can be directly used to extract the background spectrum from a local and uncontaminated region on {\\em Chandra}\\\/\\ chips, which must be located far away enough from the source. Secondly, one can draw the background spectrum from the {\\em Chandra}\\\/\\ blank field datasets in the same detector region as where the source spectrum is extracted. Since the core radius of A2556 is 350$h_{71}^{-1}$ kpc (White et al. 1997) and it should cover most of ACIS-S3 CCD, we have to utilize the {\\em Chandra}\\\/\\ blank-sky template as the background for A2556. The template is tailored to match the actual pointing and the background spectrum is extracted and processed identically to the source spectrum. Then, the background spectrum is rescaled by normalizing its high energy end to the corresponding observed spectrum. Corresponding spectral redistribution matrix files (RMFs) and auxiliary response files (ARFs) are created with CIAO tools, $mkwarf$ and $mkacisrmf$, respectively. All spectra are rebinned to give a minimum of 20 raw counts per spectral bin, which allows $\\chi^2$ statistics to be applied. Since the contribution of the hard spectral component is expected to be rather weak, and also to minimize effects of the instrumental background at higher energies and calibration uncertainties at lower energies, the spectrum fitting is restricted to 0.7--7.0 keV .\n\n\n\n\\subsubsection{Temperature Across the Edge}\\label{Tpro}\n\nIn order to derive the temperature profile across the SBP edge, we extract the spectra from 4 elliptical annuli in region E of Figure \\ref{ima}b, so that the SBP edge lies exactly at the boundaries of certain elliptical annulus, and the elliptical annulus covers the azimuthal angle, where the edge is most prominent. We use the XSPEC 12.4.0 package to perform the deprojected spectral analyses. In spectral fitting process, the gas emission is modeled with an absorbed APEC component and the PROJCT model, which performs a 3-dimensional to 2-dimensional projection of prolate ellipsoidal shells onto elliptical annuli and evaluates the influence of outer ellipsoidal shells onto inner ones. The absorption column density $N_{\\rm H}$ is fixed at the Galactic value $N_{\\rm H}=2.07\\times 10^{20}$ cm$^{-2}$ (Dickey \\& Lockman 1990). Best-fit temperature values are shown in Figure \\ref{dis_t}a. Temperature behind the edge of arm-like structure is much higher than those of ambient regions. The calculated temperature jump across the edge is $T_{1}\/T_{2}=1.93^{+0.51}_{-0.72}$. Both gas density and temperature jumps across the edge of the arm-like structure indicate that the edge is a weak shock front.\n\nThe Mach number, $\\mathcal{M}$, of the weak shock front can be determined by employing the Rankine-Hugoniot shock relation (Landau \\& Lifshitz 1959)\n\\begin{equation} \\label{eq:RH1}\n  \\frac{\\rho_{1}}{\\rho_{2}} = \\frac{(1+\\gamma)M^2}{2+(\\gamma-1)M^2}\n\\end{equation}\nand\n\\begin{equation} \\label{eq:RH2}\n\\frac{T_{1}}{T_{2}}=\n\\frac{\\left( 2\\gamma M^2 - (\\gamma-1) \\right) \\left( 2+(\\gamma-1)M^2\\right)}{(\\gamma+1)^2 M^2},\n\\end{equation}\nwhere $T_{1}$ and $T_{2}$ denote the temperature behind (the ``post-shock''; region 1 in Figure \\ref{ima}b) and ahead (the ``pre-shock''; region 2 in Figure \\ref{ima}b) of the shock, respectively; $\\rho_{1}$ and $\\rho_{2}$ denote the density behind and ahead of the shock, respectively. For the intracluster gas, $\\gamma=5\/3$, which is the adiabatic index for monatomic gas. For weak shock fronts, the accuracy of $\\mathcal{M}$ derived from the density discontinuity is better than that from temperature discontinuity (Markevitch \\& Vikhlinin 2007). Therefore, with the density jump $\\rho_{1}\/\\rho_{2}=1.37_{-0.03}^{+0.02}$ and Eq. \\ref{eq:RH1}, we derive the Mach number $\\mathcal{M}=1.25_{-0.03}^{+0.02}$. According to Eq. \\ref{eq:RH2}, a temperature jump of $T_{1}\/T_{2}=1.24_{-0.01}^{+0.02}$ is indicated, which confines the spectroscopic value $1.93^{+0.51}_{-0.72}$ within errors.\n\nWe crosscheck the detected temperature jump by examining the 2-dimensional temperature distribution of the gas in A2556. In Figure \\ref{dis_t}b, the projected temperature map of A2556 (Gu et al. 2009) exhibits a substantial high temperature region exactly behind the edge of the arm-like structure, which is coincident with the temperature profile shown in Figure \\ref{dis_t}a.\n\n\n\n\\subsubsection{Classification of Weak Cool Core Cluster}\\label{cla}\n\nTo test whether a cooling flow model fits the spectrum of A2556, we apply four types of models to fit the spectrum in a circle region with radius of $237h_{71}^{-1}$ kpc, which center on the X-ray peak. The circle region covers nearly the whole {\\em Chandra}\\\/\\ ACIS-S3 CCD. Firstly, we use the WABS$\\times$APEC model (hereafter 1TA) and WABS$\\times$MEKAL model (hereafter 1TM) to describe the X-ray emission only by a single isothermal plasma. In these two models, temperatures and elemental abundances are allowed to vary. Secondly, in order to feature a cooling flow, we employ a single temperature MEKAL model plus a multi-phase cooling flow component MCKFLOW, where their abundances are tied together and allowed to vary; the temperature of MEKAL and the higher gas temperature of MCKFLOW are also tied together and allowed to vary. We consider two cases for this model: the lower bound on the gas temperature in the MCKFLOW is fixed at 0.1 keV, which means the cooling material is fully cooled below the X-ray emitting temperature (hereafter denote FC for fully cooling), and this component is set as a free parameter (hereafter denote VC for variable cooling).\n\n\n\nBest-fit spectral parameters of four models with 90\\% confidence level are presented in Table 1. As a general comment, all models do a reasonably good job at fitting the spectrum with reduced $\\chi^2$ values around 1.1. The 1TA and the 1TM give nearly the same spectral parameters.  The cooling rates for the FC and the VC are $19.99_{-10.14}^{+10.14}$ M$_{\\odot}$ yr$^{-1}$ and $20.13_{-9.85}^{+10.56}$ M$_{\\odot}$ yr$^{-1}$, respectively. Therefore, A2556 could be classified as a weak cool core cluster. This conclusion consists with the cooling time within $1.5^{\\prime}$ of A2556, 6.5 Gyr (Gu et al. 2009), according to which A2556 could also be classified as a weak cool core cluster (Mittal et al. 2009).\n\n\n\n\n\\section{DISCUSSION}\\label{dis}\n\nIn \\S 2, we present robust evidences for the existence of the weak shock front located at approximately 160$h_{71}^{-1}$ kpc northeast of the cluster center. As in the Perseus, the Hydra A, and several other clusters, the AGN activity might be the cause of the weak shock in A2556. However, partly due to the poor photons statistics, we fail to find robust evidence of AGN bubbles and disturbances on cluster scales. In this section, we shall discuss the other two possible origins of this weak shock front more particularly.\n\n\n\n\n\n\\subsection{ Lack of Supersonic Substructure}\nA2556 has a cD galaxy which is consistent with the X-ray peak within $1^{\\prime\\prime}$ as shown in Figure \\ref{ima}b. This indicates A2556 has no violent disturbance caused by major merger in recent $\\sim 10^9$ yrs. The difference in photometric magnitudes between two most luminous galaxies, namely the magnitude (luminosity) gap $\\Delta mag_{12}$, is the simplest measurable statistic to quantify the dynamical age of a system (Milosavljevi$\\acute{c}$ et al. 2006): larger $\\Delta mag_{12}$ indicates an older dynamical age.  According to the absolute total magnitudes of the brightest (2MASX J23130142-2138039) and second-brightest (2MASX J 23132498-2137390) galaxies from Smith et al (2004), $\\Delta mag_{12}=0.851$ in R band. Comparing it with the mean magnitude gap of 730 clusters (Milosavljevi$\\acute{c}$ et al. 2006), $\\langle \\Delta mag_{12}\\rangle \\simeq 0.75$, we conclude that A2556 is a dynamical old cluster.\n\nKrick et al. (2007) measured the flux, surface brightness profile, color and substructure in the diffuse intracluster light (ICL) of A2556. They concluded that this cluster is dynamical relaxed and shows no subcluster. We further examine the line-of-sight velocity distribution of 46 member galaxies in A2556 identified by Smith et al. (2004) and plot it in Figure \\ref{dis_v}, which shows a single velocity plateau from eye's impression. We can fit the velocity distribution with a single Gaussian profile, the best fit ($\\chi^2\/dof =27.87\/17$) gives the average velocity of $\\langle v\\rangle = 26165.29\\pm88.34$ km s$^{-1}$ and the corresponding variance of $\\sigma = 658.31\\pm88.37$ km s$^{-1}$. The Kolmogorov-Smirnov statistic tells observed distribution has a probability of 95.0\\% to be a single Gaussian profile, which indicates that A2556 is very unlikely to have subcluster.\n\n\nPreviously observed shock fronts in ICM of galaxy clusters, for instance bullet cluster (Markevitch et al. 2002) and A520 (Markevitch et al. 2005), could be attributed to supersonic infalling subclusters. However, there are indications that A2556 possibly has no subcluster. Therefore, the weak shock front in A2556 may not be formed by the supersonic infalling of the subcluster.\n\n\n\n\\subsection{Nonlinear Steepening Weak Shock in A2556}\n\nBesides the supersonic flow, the nonlinear steepening from normal acoustic wave could also generate shock front (Landau \\& Lifshitz 1959). Such nonlinear steepening is a possible reason for the weak shock front in A2556, since the supersonic infalling of the subcluster may not be the origin of the shock front in it.\n\nFirstly,  we would like to discuss the origin of normal acoustic waves in clusters. According to their numerical simulations Nagai et al. (2003) showed that even for the relatively relaxed cluster gas turbulent velocity in the ICM can reach up to about 20\\% - 30\\% of the local sound speed, and this result is not rule out by current observational upper limits (e.g., Sandsers 2011). Such turbulence generates acoustic wave in the ICM (Fujita et al. 2003), which can be described by Euler's equation. Since the turbulent velocity $v$ is comparable with the sound velocity $c_s$, the nonlinear term $({\\bf v}\\cdot {\\bf grad}){\\bf v}$ in Euler's equation cannot be neglected (Landau \\& Lifshitz 1959). This leads the acoustic wave to inevitably steepen into the shock wave. According to this mechanism, the ICM turbulence of A2556 could be the origin of the shock front we detected. \n\n\nSecondly, these shock waves would dissipate the energy and heat the surrounding gas of the clusters, and we perform a schematic calculation for the nonlinear steepening weak shock. Following the method of Stein and Schwartz (1972). When acoustic wave steepens, the crest overtakes the trough in a distance given by\n\\begin{equation}\n\\int\\frac{\\gamma+1}{2} vdt=\\int\\frac{\\gamma+1}{2}\\frac{v}{c_s}dL=\\frac{\\lambda}{2},\n\\end{equation}\nwhere $\\gamma=5\/3$ is the adiabatic index for monatomic gas, $\\lambda$ is the pulse width, $v$ is the pulse velocity amplitude, $c_s$ is the sound speed (for A2556,  $c_s=900$ km s$^{-1}$ according to its average temperature 3.2 keV), and $L$ is the distance, which the excited sound wave travels before it steepens into the shock front. Since energy flux of acoustic wave $F \\propto \\rho v^{2}$ and assuming acoustic wave prior to shock formation does not dissipate any energy, we get the relation $v \\propto \\rho^{-1\/2}$. Additionally, with a single $\\beta$ distribution $\\rho=\\rho_{0}[1+(L\/r_{c})^{2}]^{-3\\beta\/2}$ for the gas density in the ICM, where $r_{c}$ is the core radius, we obtain\n\\begin{equation}\\label{lam}\n \\lambda = 2\\int_{0}^{L}\\frac{\\gamma+1}{2}\\frac{v_{0}}{c_{s}}[1+(L\/r_{c})^2]^{3\\beta\/4}dL,\n\\end{equation}\nwhere $v_{0}$ is the initial velocity amplitude.\nSuppose the acoustic wave originates from the cluster center. The distance it travels before steepens to the shock front is $L \\simeq 160h_{71}^{-1}$ kpc in A2556. According to Eq. \\ref{lam}, $\\lambda \\simeq 400h_{71}^{-1}$ kpc, which means the period of acoustic wave generated by the turbulence of ICM in cluster center is $\\simeq$ 0.43 Gyr.\n\n\n\n\\section{SUMMARY}\\label{sum}\n\nThe {\\em Chandra}\\\/\\ observation of A2556 reveals a weak shock front located at approximately 160$h_{71}^{-1}$ kpc northeast of the cluster center with $\\mathcal{M}=1.25_{-0.03}^{+0.02}$. Since no subcluster associated with the weak shock front is observed in the optical or X-ray band, we cannot ascribe the formation of the shock front to the supersonic infalling of the subcluster. We suggest the steepening of the acoustic wave could be a possible origin of such shock front, and such acoustic wave could induce by the turbulence of the ICM in the cluster core. \n\nThe nonlinear steepening shock front should be an ubiquity phenomenon in the ICM. In contrast, the observation of nonlinear steepening weak shock front is rare. This may ascribe to that the weak shock wave is difficult to observe. However, to solve this question requires further investigation, including both deeper X-ray observation and extensive theoretic studies.\n\n\\section*{Acknowledgments}\nWe thank Hakon Dahle for kindly providing us with the magnitudes of member galaxies in A2556 that are used in this paper. We thank the anonymous referee for useful suggestions on the manuscript. Also, we thank Chunsheng Pei for helpful suggestions. This work was supported by the Ministry of Science and Technology of the People's Republic of China (973 Program; Grant Nos. 2009CB824900 and 2009CB824904), the National Science Foundation of China (Grant Nos. 10878001, 10973010, and 11125313), and the Shanghai Science and Technology Commission (Program of Shanghai Subject Chief Scientist; Grant Nos. 12XD1406200 and 11DZ2260700).\n\n\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{introduction}\nThe growth of interest towards complex networks during the last decade was mainly determined by the unexpected structural and dynamical consequences of the lack of spatial constraints in many real-world networks.\nFor instance, the emergence of a scale-free topology \\cite{guidosbook} and unconventional percolation properties \\cite{dynamicalprocessesoncomplexnetworks} is impossible in networks (such as regular lattices) that are entirely shaped by geometric constraints.\nHowever, many real networks are in general the result of a combination of spatial and non-spatial effects. So, even if not entirely, spatial constraints still partially and substantially affect network topology \\cite{spatialreview}. A remarkable example is that of transportation networks, that offer an insightful approach to some prominent and apparently unrelated scientific questions concerning the size and form of recurrent structures in Nature \\cite{west:4dimension,west:ontogeneticgrowth,west:scalingcities_1,Bejan2010,WoolleyMeza2011Complexity,Kaluza2010Complex}. \nThe ubiquitous allometric scaling laws characterizing the shape of living (e.g. vascular networks \\cite{west:allometricscaling}) and non-living (e.g. river networks \\cite{banavar:sizeandform}) systems have been demonstrated to be the result of efficient and optimized transportation processes, crucially dependent on the dimension of the embedding space \\cite{west:allometricscaling,banavar:sizeandform,QueirosConde2007719,Rocha20051281,BebHyn2007,Tero2010Rules}.\nEven more generally, when the embedding space is abstract rather than physical (as in ecological networks \\cite{citeulike:722073,fath:ecologicalfunction} and food webs \\cite{garla:universalscaling,zhang}), effective pseudo-spatial constraints (such as interspecific competition) allow to treat the network as an efficient transportation system \\cite{garla:universalscaling}.\nIn all these examples, the optimized character of many real transportation networks highlights the interplay between spatial (geometric constraints) and non-spatial (evolution towards efficiency) effects. \n\nThanks to the important results in graph theory and network science \\cite{guidosbook,dynamicalprocessesoncomplexnetworks}, it is now relatively easy to generate artificial networks with any desired combination of spatial and non-spatial effects \\cite{spatialreview,barthelemy,ginestra}. \nHowever, disentangling these two factors in real networks is still difficult, for at least two reasons.\nFirstly, one needs to specify a model incorporating spatial information and fit the model to real data. This immediately poses the problem of the (arbitrary) choice of the model's functional dependence on empirical spatial quantities. \nSecondly, one also needs to appropriately filter out a spurious component of spatial dependence which might be simply due to other non-spatial factors shaping the network.\nFor instance, in scale-free networks where many vertices with small degree (number of neighbours) coexist with a few vertices (hubs) with very large degree, the spatial position of the hubs will be the destination of many links, irrespective or the positions of the source vertices. This will induce an apparent spatial independence (hubs appear to have no preference to connect over short or long distances) even if the topology was instead generated by a space-dependent process. Conversely, since any two hubs are very likely to be connected (and also very intensely connected if the network is weighted), the distance between them would set an apparent spatial scale for connectivity, even if the network formation process was instead space-independent.\nIn general, the entity of the spurious mixture of spatial and non-spatial effects will depend on the detailed spatial distribution of vertices, and on the correlation between positions and topological properties.\n\nIn this article, we propose a solution to the above problem by jointly introducing meaures of spatial effects and adequate null models. Null models are meant to control for important topological properties, treated as non-spatial constraints, while being neutral with respect to spatial factors.\nThe most important null models are those that control for local topological properties which have an immediate structural effect, i.e. the degree sequence (the set of the degrees of all vertices in binary networks) or the strength sequence (the set of the sum of edge weights of all vertices in weighted networks).\nWe show that, besides filtering out spurious spatial effects, null models also suggest a natural and simple choice for the definition of (both global and local) spatial measures.\nThis enables us to analyse both binary and weighted networks in a similar manner.\nTaken together, measures and null models allow us to effectively disentangle spatial and non-spatial effects in real networks.  \n\nIn order to illustrate our approach, we apply it to the World Trade Web (WTW), the network of international import\/export trade relationships among world countries \\cite{mywtw,squartiniPRE2011a,squartiniPRE2011b}.\nThe choice of this particular network arises from the fact that the WTW is an excellent example of a network where spatial effects are expected but weak, and in some sense dominated by stronger non-spatial ones.\nOn one hand, well-known results in the economic literature show that international trade depends significantly on the geographic distances among countries. \nIn particular, the so-called Gravity Models \\cite{Feenstra_2001,anderson_2012} are able to reproduce quite well the intensity of trade between two countries as a function of their Gross Domestic Product (GDP) and their geographic distances. Including distances improves the fit significantly.\nOn the other hand, as has been recently found \\cite{fagiolo_gravity}, when such models are adjusted in order to predict the \\emph{existence} of a link along with its weight (thus, when the binary topology of the network is also concerned), they perform very badly.\nAt the binary level, the topology of the WTW can instead be excellently reproduced by the specification of local non-spatial constraints (the degree sequence in its more and more detailed forms: undirected \\cite{mywtw,squartiniPRE2011a}, directed \\cite{squartiniPRE2011a}, or reciprocated \\cite{mytriadic}). \nThis means that, once the degree sequence is specified, the topology of the entire network can be reproduced almost exactly (in other words, the degree sequence is highly informative about the topology of the whole network).\nIf the GDP of world countries is used as an alternative constraint replacing the degree, the same results are obtained \\cite{mywtw,mylikelihood}. Again, the GDP is a non-spatial constraint.\n\nTaken together, the above results appear to indicate that the influence of distances on the structure of the WTW becomes smaller as the focus is shifted from non-zero weights to the network as a whole. \nAs a result, we expect distances to play a significant but weaker role than the degree sequence.\nThis makes the WTW a challenging benchmark for our method.\nWe therefore use our approach in order to look for the residual spatial dependence of the WTW, once the degree sequence is controlled for.\nWe also perform an analogous analysis in the weighted case, by controlling for the strength sequence.\nWe find that our method is indeed capable to filter out the strong non-spatial effects and uncover significant spatial dependencies, despite the weakness of the latter.\nThis makes us confident that our approach can be successfully applied to any network, irrespective of the strength of spatial effects.\n\n\\section{Binary analysis\\label{sec:binary}}\nWe start by considering binary graphs. The extension to weighted networks will be presented in section \\ref{sec:weighted}.\nFor generality, we assume that the graph is directed, even if our discussion can be  applied to undirected graphs as well, by treating each undirected link as a pair of directed ones pointing in opposite direction.\nA binary directed graph with $N$ vertices is specified by a $N \\times  N$ adjacency matrix $\\mathbf{A}$. The entries $a_{ij}$ of $\\mathbf{A}$ are equal to $1$ if a directed link from vertex $i$ to vertex $j$ exists, and $0$ otherwise. The number of links in the network is given by $L= \\sum_i\\sum_{j \\ne i} a_{ij}$.\nIf the nodes of the network are embedded in a metric space, we can also define a symmetric distance matrix $\\mathbf{D}$, whose entries $d_{ij}$ are the distances between nodes $i$ and $j$. This matrix is symmetric, i.e. $d_{ij}=d_{ji}$  $\\forall i, j$, and additionally $d_{ii}=0$ $\\forall i$.\nFor future convenience, we also consider the list of all (off-diagonal) distances ordered from the smallest to the largest, and denote it by $V^{\\uparrow}=(d_1^\\uparrow, \\dots,d_{n}^\\uparrow, \\dots , d^\\uparrow_{N(N-1)})$, where $d^\\uparrow_n \\le d^\\uparrow_{n+1}$. Similarly, we consider the inverse ordering from the largest to the smallest distances, and denote it by $V^{\\downarrow}=(d_1^\\downarrow, \\dots,d_{n}^\\downarrow, \\dots , d^\\downarrow_{N(N-1)})$, where $d_n^\\downarrow \\ge d_{n+1}^\\downarrow$.\nNote that, since $d_{ij}=d_{ji}$, the distance between each pair of vertices (dyad) appears twice in both lists.\n\nJust to visualize a concrete example, imagine the subset of the World Trade Web (graph of the whole WTW would be confusing), whose vertices are the 27 countries belonging to the European Union (EU27).\nThe embedding space for this network is shown in Fig. \\ref{plotEU27}, where vertices are located at capital cities. Figure \\ref{plotcoord} shows the cities' coordinates $\\vec{r}_i=(x_i,y_i)$ (longitudes and latitudes) generating the metric distances $d_{ij}\\equiv d(\\vec{r}_i,\\vec{r}_j)$. \nOur goal is to introduce appropriate measures in order to check whether these distances have an effect on the topology of the embedded network.\n\\begin{figure}[t]\n                \\includegraphics[width=0.45\\textwidth]{europoints.eps}\n        \\caption{Color online. Map of the EU27 countries, with capital cities highlighted in red.\n        }\n        \\label{plotEU27}\n\\end{figure}\n\n\n\\subsection{Measuring global spatial effects\\label{sec:global}}\nIn order to define a global measure of spatial effects in real networks, we are in principle left with the arbitrary choice of any function $F(\\mathbf{A},\\mathbf{D})$ quantifying the dependence of the realized topology (i.e. the entries of the matrix $\\mathbf{A}$) on the spatial distances (i.e. the entries of $\\mathbf{D}$). \nHowever, we now show that a natural choice for the functional form exists, so that much of this arbitrariness can be removed. \n\nWe recall that our aim is to filter out the component of $F(\\mathbf{A},\\mathbf{D})$ that can be attributed to non-spatial effects. \nPractically, this means comparing $F(\\mathbf{A},\\mathbf{D})$ with its expected value $\\langle F(\\mathbf{A},\\mathbf{D})\\rangle$ under a null model where a given set of non-spatial constraints are specified.\nAs we mentioned, the most important null model in the binary case is one where the degree sequence is specified (and kept equal to the observed one), and the rest of the topology is completely random.\nThis null model is known as the \\emph{Configuration Model} (CM) \\cite{maslov}.\nIn directed networks, the corresponding \\emph{Directed Configuration Model} (DCM) is one where the in-degree $k^{in}_i=\\sum_{j\\ne i}a_{ji}$ (number of incoming links) and out-degree $k^{out}_i=\\sum_{j\\ne i}a_{ij}$ (number of outgoing links) of each vertex $i$ are simultaneously specified, and the network is otherwise random.\nBesides the DCM, we will also consider a more relaxed null model where the only non-spatial constraint is the total number of links $L$ in the network, i.e. the \\emph{Directed Random Graph} (DRG) \\cite{mymethod}, and a more stringent null model where there are additional constraints on the reciprocity, i.e. the \\emph{Reciprocated Configuration Model} (RCM) \\cite{mytriadic,mymultispecies}.\n\\begin{figure}[t]\n                \\includegraphics[width=0.45\\textwidth]{coordinates.eps}\n        \\caption{Color online. Longitudes and latitudes (deg) of the vertices in the EU27 trading network.\n        }\n        \\label{plotcoord}\n\\end{figure}\n\nA first way of implementing a null model is computational \\cite{maslov,bias}. For instance, in the CM and DCM, one starts with the original network and randomizes it through the iteration of some fundamental \\emph{rewiring move} that alters the topology but keeps the (in- and out-) degrees of all vertices fixed \\cite{maslov}. \nAs found recently \\cite{bias}, this approach produces biased results unless one uses a careful (but difficult to implement) acceptance probability of the attempted rewiring moves.\nEven with the correct acceptance probability, this approach is extremely time consuming since many iterations are necessary in order to produce a single randomized network, and many such randomized network variants are needed in order to sample the \\emph{microcanonical} ensemble of random graphs with degree sequence exactly equal to the observed one.\nThe quantities of interest -- $F(\\mathbf{A},\\mathbf{D})$ in our case -- should then be calculated on each randomized variant and averaged, the result being beyond mathematical control and entirely dependent on numerical simulations.\n\nA second way of implementing a null model is instead analytical \\cite{mymethod}. For the CM, one solves a system of $N$ ($2N$ for the DCM) nonlinear equations, and the solution is used to produce the exact probability matrix $\\mathbf{P}$, whose entry $p_{ij}$ represents the correct probability that vertex $i$ is connected to vertex $j$ in the \\emph{canonical} ensemble of graphs with \\emph{average} degree sequence equal to the observed one \\cite{mymethod}.\nUsing this approach, it is not necessary to generate any randomized network, since the matrix $\\mathbf{P}$ is already the \\emph{exact} expectation value of the adjancency matrix $\\mathbf{A}$ over the entire ensemble, i.e. $\\langle \\mathbf{A}\\rangle=\\mathbf{P}$ \\cite{mymethod}.\nThe same considerations apply to the other two binary null models we will consider, i.e. the DRG and the RCM, and also to weighted models (more details on all the models we adopted will be given later).\nThis property makes the analytical method extremely fast, and completely under mathematical control.\nGiven the above considerations, we choose the second approach to the problem. \n\nOur choice of the null model also automatically implies a natural choice for the function $F(\\mathbf{A},\\mathbf{D})$. Indeed, if we choose a linear function, then the expected value $\\langle F(\\mathbf{A},\\mathbf{D})\\rangle$ can be calculated \\emph{exactly} as\n\\begin{equation}\n\\langle F(\\mathbf{A},\\mathbf{D})\\rangle=F(\\langle \\mathbf{A}\\rangle,\\mathbf{D})=F(\\mathbf{P},\\mathbf{D})\n\\end{equation}\nwhich only requires the probability matrix $\\mathbf{P}$ (note that the distance matrix $\\mathbf{D}$ is constant and unaffected by the null model).\nThe simplest linear choice for a global measure that exploits the entire topological and spatial information (i.e. relevant to all pairs of vertices) is \n\\begin{equation}\nF\\equiv\\sum_{i=1}^N\\sum_{j\\ne i} a_{ij}d_{ij}\n\\label{eq:F}\n\\end{equation}\n(where we have dropped the dependence on $\\mathbf{A}$ and $\\mathbf{D}$ to simplify the notation), whose expected value reads\n\\begin{equation}\n\\langle F\\rangle =\\sum_{i=1}^N\\sum_{j\\ne i} p_{ij}d_{ij}\n\\end{equation}\nDue to the exact knowledge of $\\mathbf{P}$, $\\langle F\\rangle$ can be computed in a time as short as that required to compute $F$ on the real network, with no need to generate any realization of the null model.\n\\begin{figure}[t]\n                \\includegraphics[width=0.45\\textwidth]{g4.eps}\n        \\caption{Color online. Maximally shrunk configuration for the EU27 trading network with a constraint on the number of links: $f=0$ ($N=27$, $L=27$).\n        }\n        \\label{plotf=0}\n\\end{figure}\n\nAlthough not fundamentally different, we consider a slightly more complicated definition, for the unique purpose of having a normalized quantity between $0$ and $1$:\n\\begin{equation}\nf\\equiv \\frac{\\sum_{i}\\sum_{j\\ne i} a_{ij}d_{ij} - F_{min} }{F_{max} - F_{min}}\n\\label{eq:filling}\n\\end{equation}\nwhere $F_{min}$ and $F_{max}$ are the minimum and maximum values that $F$ can take, given the distance matrix $\\mathbf{D}$ and the total number of links $L$.\nExplicitly, in terms of the two lists $V^{\\uparrow}$ and $V^{\\downarrow}$ introduced above, the maximum and minimum values for $F$ read $F_{min}= \\sum_{n=1}^L d_n^\\uparrow$ and $F_{max}= \\sum_{n=1}^L d_n^\\downarrow$.\nThe former extreme ($f=0$) represents the case where the $L$ links are placed among the closest couples of nodes (maximally \\emph{shrunk} network). \nThe latter extreme ($f=1$) instead represents the case where the $L$ links are placed among the most distant couples of nodes (maximally \\emph{stretched} network). \nFor our previous example of the EU27 trading network, these two extremes are shown in Figs. \\ref{plotf=0} and \\ref{plotf=1} respectively, where for visualization purposes we have actually chosen a value of $L$ equal to $N=27$, much less than the real value (which would fill the plot with links). This would have been the case, for instance, if in the original network each vertex had exactly one outgoing link.\nNetworks in between the two extrema would have a value $0<f<1$.\nAs intuitively rendered by the figures, a larger value of $f$ implies a more pronounced filling of the available space. Therefore we denote $f$ as the (spatial) \\emph{filling} of the network represented by the matrix $\\mathbf{A}$.\n\\begin{figure}[t]\n                \\includegraphics[width=0.45\\textwidth]{g3.eps}\n        \\caption{Color online.Maximally stretched configuration for the EU27 trading network with a constraint on the number of links: $f=1$ ($N=27$, $L=27$).\n        }\n        \\label{plotf=1}\n\\end{figure}\n\nThe above choice of the normalization for $f$ is practical but somewhat arbitrary. For instance, while networks generated under the DRG model will achieve the two extremes, networks generated under the DCM (where the constraints are stricter than just the number of links) will in general do not.\nFor instance, Fig. \\ref{plotclosed} shows the maximally shrunk configuration available by imposing that each vertex has one outgoing link (for simplicity, without constraints on the number of incoming links). This produces a filling $f=0.06>0$.\nSimilarly, in Fig. \\ref{plotfar} we show the maximally stretched configuration under the same constraints ($f=0.87<1$).\nIn Fig. \\ref{plotclosed} the outgoing link of each vertex is directed towards the closest node, while in Fig. \\ref{plotfar} it is directed towards the most distant one.\n\\begin{figure}[t]\n                \\includegraphics[width=0.45\\textwidth]{g1.eps}\n        \\caption{Color online. Maximally shrunk configuration for the EU27 trading network with a constraint on the out-degrees:  $f=0.06$ ($N=27$, $k^{out}_i=1$ $\\forall i$).\n        }\n        \\label{plotclosed}\n\\end{figure}\n\nIn principle, depending on the null model or on the type of analysis, one could choose alternative and more convenient values for the constants $F_{min}$ and $F_{max}$, such that $f$ can indeed achieve the extreme values 0 and 1. \nHowever, we are interested not in the value of $f$ itself, but rather on its comparison with the expected value \n\\begin{equation}\n\\langle f\\rangle= \\frac{\\sum_{i}\\sum_{j\\ne i} p_{ij}d_{ij} - F_{min} }{F_{max} - F_{min}}\n\\end{equation}\nunder the null model adopted.\nThis makes the choice of the constants $F_{min}$ and $F_{max}$ irrelevant, so we can stick to the option which is easiest to calculate (we will keep the one based on the total number of links, i.e. the one naturally associated with the DRG).\nTherefore, we will focus on the following rescaled quantity, which we denote as the \\emph{filtered filling}:\n\\begin{equation}\n\\phi\\equiv \\frac{f  - \\langle f \\rangle }{ 1 - \\langle f \\rangle  }\n\\label{eq_phi}\n\\end{equation}\nThe above quantity is equal to zero whenever the observed filling ($f$ or $F$) coincides with the expected filling ($\\langle f\\rangle$ or $\\langle F\\rangle$) under the specified model.\nPositive (negative) values of $\\phi$ indicate a network which is more stretched (shrunk) than expected.\nThe above definition filters out the non-spatial effects encapsulated into the null model considered, i.e. the random value $\\langle f \\rangle$ that would be produced, given the spatial configuration of vertices, even in a network generated irrespective of the distances, and only determined by the topological constraints enforced. \nBeing a rescaled quantity, it also allows us to compare the degree of filling in networks with different numbers of vertices and links, or with different degree sequences.\nThis also means that we can consistently compare different snapshots of the same network, even if the topological properties of the latter change over time.\n\n\\subsection{Local spatial effects\\label{sec:local}}\nThe above considerations also suggest a natural (linear) choice for the definition of local measures of spatial effects.\nThe two possible building blocks for any linear quantity locally defined around vertex $i$, in analogy with eq.(\\ref{eq:F}), are the  local sums \n\\begin{equation}\nF^{out}_i\\equiv\\sum_{j\\ne i} a_{ij} d_{ij}\\qquad F^{in}_i\\equiv\\sum_{j\\ne i} a_{ji} d_{ij}\n\\end{equation}\n(note that $d_{ij}=d_{ji}$), whose expected values under the null model are\n\\begin{equation}\n\\langle F^{out}_i\\rangle=\\sum_{j\\ne i} p_{ij} d_{ij}\\qquad \\langle F^{in}_i\\rangle=\\sum_{j\\ne i} p_{ji} d_{ij}\n\\end{equation}\n\\begin{figure}[t]\n                \\includegraphics[width=0.45\\textwidth]{g2.eps}\n        \\caption{Color online. Maximally stretched configuration for the EU27 trading network with a constraint on the out-degrees:  $f=0.87$ ($N=27$, $k^{out}_i=1$ $\\forall i$).\n        }\n        \\label{plotfar}\n\\end{figure}\n\nAgain, although this is not strictly necessary, we can rescale $F^{out}_i$ and define the \\emph{local outward filling} as\n\\begin{equation}\nl^{out}_i\\equiv\\frac{\\sum_{j\\ne i}a_{ij}d_{ij}-(F^{out}_i)_{min}}{(F^{out}_i)_{max}-(F^{out}_i)_{min}}\n\\label{localfilling}\n\\end{equation}\nwhere $(F^{out}_i)_{max}$ and $(F^{out}_i)_{min}$ are the two extreme values of $F^{out}_i$ achieved in a maximally stretched and maximally shrunk network respectively, for some convenient reference situation.\nFor consistency with the global case, we choose the reference associated with the DRG, i.e. one where all vertices have on average the same out- and in-degree $\\overline{k^{out}}=\\overline{k^{in}}=L\/N$. \nIn this case we can write $(F^{out}_i)_{min}=\\sum_{n=1}^{L\/N}d_{i,n}^{\\uparrow}$ and $(F^{out}_i)_{max}=\\sum_{n=1}^{L\/N}d_{i,n}^{\\downarrow}$, where $d_{i,n}^{\\uparrow}$ and $d_{i,n}^{\\downarrow}$ are elements of the two ordered lists ${V_i^{\\uparrow}=(d_{i,1}^\\uparrow, \\dots,d_{i,n}^\\uparrow, \\dots , d^\\uparrow_{i,N-1})}$ and ${V_i^{\\downarrow}=(d_{i,1}^\\downarrow, \\dots,d_{i,n}^\\downarrow, \\dots , d^\\downarrow_{i,N-1})}$ of local distances from vertex $i$ to the other $N-1$ vertices (where $d^\\uparrow_{i,n} \\le d^\\uparrow_{i,n+1}$ and $d_{i,n}^\\downarrow \\ge d_{i,n+1}^\\downarrow$).\nNote that the above extremes are those achieved by the \\emph{typical} realizations of the DRG where all degrees are close to their expected value. \nThe actual extremes would be either the empty graph and the complete graph (in the canonical DRG), or the maximally shrunk and maximally stretched configuration as in the examples of Figs. \\ref{plotf=0} and \\ref{plotf=1} (in the microcanonical DRG). \nHowever, in both cases these extreme configurations, along with all the other non-typical ones where the degrees are far from their expected values, are produced with very small probability.\nAs for the global case, the above choice of $(F^{out}_i)_{max}$ and $(F^{out}_i)_{min}$ is arbitrary (and additionally, in this case $l_i^{out}$ can be even larger than $1$ for vertices with large out-degree in the real network).\nHowever, this limitation is not essential, since our aim is the comparison of eq.(\\ref{localfilling}) with its expected value\n\\begin{equation}\n\\langle l^{out}_i\\rangle=\\frac{\\sum_{j\\ne i}p_{ij}d_{ij}-(F^{out}_i)_{min}}{(F^{out}_i)_{max}-(F^{out}_i)_{min}}\n\\end{equation}\nwhere $(F^{out}_i)_{max}$ and $(F^{out}_i)_{min}$ will be treated as constants.\n\nIn complete analogy with eq.(\\ref{localfilling}), we can also define the \\emph{local inward filling} as\n\\begin{equation}\nl^{in}_i\\equiv\\frac{\\sum_{j\\ne i}a_{ji}d_{ij}-(F^{in}_i)_{min}}{(F^{in}_i)_{max}-(F^{in}_i)_{min}}\n\\end{equation}\nwhose expected value is \n\\begin{equation}\n\\langle l^{in}_i\\rangle=\\frac{\\sum_{j\\ne i}p_{ji}d_{ij}-(F^{in}_i)_{min}}{(F^{in}_i)_{max}-(F^{in}_i)_{min}}\n\\end{equation}\nNote that, due to the symmetry of the distances, $(F^{in}_i)_{min}=(F^{out}_i)_{min}$ and $(F^{in}_i)_{max}=(F^{out}_i)_{max}$ under the same reference as above.\nIn principle, as for the global filling $f$, we can combine the measured and expected values into a single filtered quantity analogous to eq.(\\ref{eq_phi}). \nHowever, in this case it is also instructive to compare $l^{out}_i$ and $l^{in}_i$ (which combine local spatial and topological information) with the out-degree $k^{out}_i$ and in-degree $k_i^{in}$ (which only embody topological information).\nTherefore we will avoid the introduction of additional quantities, and simply discuss the impact of spatial and non-spatical effects by directly comparing measured and expected values as a function of the degree.\n\nWe finally introduce a higher-order property which probes the effects of correlations between vertices.\n(Dis)assortativity is the (reduced) tendency of vertices with determined common topological properties to connect with each other \\cite{guidosbook}.\nA way to measure assortativity locally is comparing the degree of a vertex with the average degree of its neighbours.\nWe adapt this definition in order to take into account spatial effects as well, and introduce the following measures of \\emph{outward and inward assortativity}:\n\\begin{equation}\nA^{out}_i\\equiv\\frac{\\sum_{j\\ne i}a_{ij}l^{out}_{j}}{k_i^{out}}\\qquad A^{in}_i\\equiv\\frac{\\sum_{j\\ne i}a_{ji}l^{in}_{j}}{k_i^{in}}\n\\label{assort}\n\\end{equation}\nWe can approximate the expected values of the above quantities with\n\\begin{equation}\n\\langle A^{out}_i\\rangle=\\frac{\\sum_{j\\ne i}p_{ij}\\langle l^{out}_{j}\\rangle}{\\langle k_i^{out}\\rangle }\\qquad \\langle A^{in}_i\\rangle=\\frac{\\sum_{j\\ne i}p_{ji}\\langle l^{in}_{j}\\rangle}{\\langle k_i^{in}\\rangle}\n\\label{eassort}\n\\end{equation}\nUnder the DCM and RCM, $\\langle k_i^{in}\\rangle =k_i^{in}$ and $\\langle k_i^{out}\\rangle =k_i^{out}$ so the above expectations become particularly simple. Moreover, expected and observed values can be compared by plotting both quantities as a function of the degree of vertices.\n\n\\subsection{Binary null models}\nBefore presenting our results, we describe a bit more in detail the set of null models we adopt in our analysis, in order to highlight our motivations and also to clarify the interpretations we can get from the analysis we present immediately after.\n\nThe method we use is the analytical one proposed by Squartini and Garlaschelli \\cite{mymethod}. The method is based on the class of Exponential Random Graphs (ERGs) \\cite{newman:statmec} that mathematically represent maximum-entropy ensembles of networks with specified constraints, and uses the Maximum-Likelihood Principle (MLP) \\cite{mylikelihood} to fit ERGs to real networks.\nA constraint is a given topological property (e.g. the number of links), which can be evaluated on any graph $\\mathbf{G}$. \nDifferent choices of the constraints specify different models in the family of ERGs.\nFor a particular choice of constraints, the model is specified by a \\emph{graph Hamiltonian} $H(\\mathbf{G})$ which is a linear combination of the constraints, and hence a function of the particular graph $\\mathbf{G}$ on which it is evaluated.\nFor binary graphs, one can think of $\\mathbf{G}$ as the adjacency matrix $(\\mathbf{G}=\\mathbf{A})$, while for weighted graphs one can think of $\\mathbf{G}$ as the weight matrix $(\\mathbf{G}=\\mathbf{W})$, where $w_{ij}$ is the weight of the link from vertex $i$ to vertex $j$. \nIn the Hamiltonian, each of the constraints is coupled to a free parameter acting as a Lagrange multiplier.\n$H(\\mathbf{G})$ uniquely determines the occurrence probability $P(\\mathbf{G})=e^{-H(\\mathbf{G})}\/\\sum_{\\mathbf{G'}}e^{-H(\\mathbf{G'})}$ of each graph $\\mathbf{G}$ in the ensemble.\nThe free parameters are then chosen in such a way that the likelihood $P(\\mathbf{G^*})$ to generate the observed network $\\mathbf{G^*}$ is maximum. \nThis parameter choice automatically ensures that the expected value of each of the constraints is exactly equal to the observed value \\cite{mylikelihood}.\nThe fitted parameters are finally used to produce, among other quantities, the expected values of $\\mathbf{G}$ under the null model, i.e. the probability matrix $\\mathbf{P}=\\langle \\mathbf{A}\\rangle=\\langle \\mathbf{G}\\rangle$ or the matrix of expected weights $\\langle \\mathbf{W}\\rangle=\\langle \\mathbf{G}\\rangle$ \\cite{mymethod}.\nIn the present article, we use three null models for binary networks and two null models for weighted networks. \n\nThe first binary null model is the DRG where, as we mentioned, the only constraint is the total number of links $L$. \nIn terms of ERGs, the DRG is therefore defined by the following Hamiltonian \\cite{newman:statmec}: \n\\begin{equation}\nH_{DRG}(\\mathbf{G})=\\theta L(\\mathbf{G})\n\\label{eq_Hrandom}\n\\end{equation}\nFollowing the general method \\cite{mymethod}, we fit this model to a real network according to the MLP and obtain the resulting parameter value $\\theta^*$. The latter is used to produce the probability matrix $\\mathbf{P}$, which in this simple case has all entries equal to $p_{ij}=e^{-\\theta^*}\/(1+e^{-\\theta^*})=L\/N(N-1)$.\nWhen used to obtain the expected values of the space-dependent quantities we have defined in sections \\ref{sec:global} and \\ref{sec:local}, the DRG filters out the non-spatial effects due to the overall density of the network, i.e. the spurious spatial dependencies only due by chance when on average $L$ links are placed among vertices with a given spatial configuration. \nAlthough such overall density effects are important, the number of links does not represent a sufficiently informative property of real networks. Therefore the non-spatial effects embodied by the DRG are not highly informative as well. For this reason, we also consider more stringent models.\n\nThe second binary null model is the DCM, where the constraints are the in- and out-degree sequence \\cite{newman:statmec,garlruzz:networksymmetry1,ruzzgarl:networksymmetry2}:\n\\begin{eqnarray}\nH_{DCM}(\\mathbf{G})=\\sum_{i=1}^N \\left[ \\theta_i^{in} k^{in}_i(\\mathbf{G})+\\theta_i^{out}k^{out}_i(\\mathbf{G})\\right]\n\\label{eq_Hdirconf}\n\\end{eqnarray}\nThe MLP yields in this case a more complicated, but still exact, probability matrix $\\mathbf{P}$ whose entries are functions of the fitted values of $\\theta_i^{in}$ and $\\theta_i^{out}$.\nThe DCM filters out not only the overall density effects, but also the spurious spatial dependencies due to the different intrinsic connectivities of the vertices of the network. \nSince the striking heterogeneity of the degrees of vertices is one of the key properties of real networks, the DCM is a very important null model.\nThe non-spatial effects filtered out by it are very informative and robust.\nFor the particular case of the WTW, it was shown that the degree sequence is extremely informative, as the DCM very closely reproduces many higher-order topological properties such as the degree correlations and the clustering coefficients \\cite{squartiniPRE2011a}.\nIn our analysis, by filtering out the effects induced by the degree sequence, the DCM will filter out a significant amount of non-spatial effects present in the WTW.\n\nThe third and most stringent binary model is the RCM, where the constraint are the three \\emph{reciprocated degrees} of all vertices \\cite{mytriadic,mymethod,mymultispecies}. \nThe out-degree $k_i^{out}$ can be split in two contributions $k_i^{\\rightarrow}$ and $k_i^{\\leftrightarrow}$, representing the number of non-reciprocated outgoing links and the number of reciprocated (thus both outgoing and incoming) links of vertex $i$.\nSimilarly, the in-degree $k_i^{in}$ can be split in $k_i^{\\leftarrow}$ and $k_i^{\\leftrightarrow}$, where $k_i^{\\leftarrow}$ represents the number of non-reciprocated incoming links of vertex $i$.\nIf we specify these three degrees separately for each vertex, we obtain the Hamiltonian for the RCM:\n\\begin{equation}\nH_{RCM}(\\mathbf{G})=\\sum_{i=1}^N \\left[ \\theta_i^{\\leftarrow} k^{\\leftarrow}_i(\\mathbf{G})+\\theta_i^{\\rightarrow}k^{\\rightarrow}_i(\\mathbf{G})+\n\\theta_i^{\\leftrightarrow}k^{\\leftrightarrow}_i(\\mathbf{G})\\right]\n\\end{equation}\nAgain, the MLP yields a precise expression \\cite{mytriadic,mymethod} for the exact probability matrix $\\mathbf{P}$, which now involves the fitted values of the three sets of parameters $\\theta_i^{\\leftarrow}$, $\\theta_i^{\\rightarrow}$, and $\\theta_i^{\\leftrightarrow}$.\nThe RCM controls for the non-spatial effects due to the different connectivities of vertices, including their different intrinsic tendency to reciprocate links.\nThe reciprocity is another robust property of real directed networks \\cite{mytriadic,myreciprocity,mywreciprocity}.\nAs a consequence, the RCM is a further improvement with respect to the DCM in cotrolling for systematic topological effects.\nIn the particular case of the WTW, is was shown that, unlike the DCM, the RCM almost perfectly reproduces the occurrence of triadic motifs \\cite{mytriadic}.\nMore in general, we note that, in spatially embedded networks, both distance-dependence and reciprocity will tend to produce symmetric networks. \nFor instance, as shown in Figs. \\ref{plotf=0} and \\ref{plotf=1}, the two extremes of a maximally shrunk and maximally stretched network are strongly symmetric (and perfectly symmetric if the number of links is even and the matrix of distances is non-degenerate). In those examples, this effect is entirely due to the strong dependence on the (necessarily symmetric) distances, and not to any reciprocity effect.\nHowever, the same topology would be induced in the opposite situation of maximal reciprocity and no distance dependence.\nTherefore it is very important, for the specific problem of assessing spatial effects, to filter out the impact of reciprocity on real networks.\nThe RCM will accurately filter out such strong and systematic topological effects.\n\n\\subsection{Spatial effects in the binary WTW}\nHaving clarified our approach in the binary case, we can finally apply it to the topology of the WTW, which represents a challenging and instructive benchmark wherein spatial effects are expected but weak, as we have already discussed.\nWe use the database prepared by Gleditsch \\cite{data} from year 1948 to 2000.\nDuring this time span, the network exhibits an increase in its size $N_t$ (number of world countries) and connectance $c_t$ (link density), from $N_{1948}=86$ and $c_{1948}=0.39$ to $N_{2000}=187$ and $c_{2000}=0.57$. \nDespite the increase of $N_t$ (mainly due to the creation of new countries from former European colonies and the Soviet Union), the mean inter-country distance $\\mu_t$ and variance $\\sigma^2_t$ have been remarkably stable, going from $\\mu_{1948}=7516$ Km and $\\sigma^2_{1948}=2.3*10^7$ Km$^2$ to $\\mu_{2000}=7550$ Km and $\\sigma^2_{2000}=1.9*10^7$ Km$^2$. \nThis is not surprising, since the embedding space of the WTW (the Earth's surface) is bounded.\nHowever, as we now show, spatial effects have not remained constant.\n\\begin{figure}[t]\n\\begin{minipage}{0.45\\textwidth}\n        \\centering\n                \\includegraphics[scale=0.75]{phi.eps}\n        \\caption{Color online. Filtered binary filling in the WTW from 1948 to 2000. Non-spatial effects have been removed using the Random Graph model (red\/rhombus), the Directed Configuration Model (blue\/cirlces), and the Reciprocated Configuration Model (yellow\/squares).\n        }\n        \\label{plotfilling}\n\\end{minipage}\n\\end{figure}\n\nIn Fig. \\ref{plotfilling} we plot the temporal evolution of the filtered filling corresponding to all null models, i.e. the three quantities $\\phi_{DRG}(t)$, $\\phi_{DCM}(t)$ and $\\phi_{RCM}(t)$.\nAs a general observation, it should be noted that the negative values of $\\phi$ always indicate that the WTW is more spatially shrunk than expected on the basis of topological constraints.\nThis means that our method indeed succeeds in uncovering spatial effects that were expected in the WTW, but that could not detected by previous analyses where non-spatial properties of the real network were compared with the expectations of the above null models \\cite{mywtw,squartiniPRE2011a,squartiniPRE2011b,mytriadic}.\nMoreover, the entity of the spatial effects is moderate (i.e. the values of $\\phi$ range between $-0.10$ and $-0.25$, far from the extreme value $-1$). This confirms quantitatively the qualitative expectation that distances should play an appreciable but weak role.\n\nIt is also instructive to inspect the nontrivial temporal behaviour of the filtered spatial effects.\nWe start by comparing the results obtained under the DRG and the DCM.\nInterestingly, apart from the first decade, the trends of $\\phi_{DRG}$ and $\\phi_{DCM}$ are almost perfectly inverted.\nRoughly from 1960 to the mid 1970s, $\\phi_{DRG}$ fluctuated around an increasing trend, while $\\phi_{DCM}$ decreased steadily. \nConversely, from the late 1970s to 2000, $\\phi_{DRG}$ decreased markedly while $\\phi_{DCM}$ increased.\nThis result highlights the importance of the choice of topological constraints in null models.\nAs we discussed, while the number of links used by the DRG is not informative about the entire topology of the WTW, the degree sequence is, to a great extent \\cite{squartiniPRE2011a}. \nThus, by successfully reproducing the structure of the WTW, the DCM effectively filters out strong topological effects.\nThe net result is a surprising inversion of the trend generated under the DRG.\n\n\\begin{figure}[t]\n                \\includegraphics[width=0.45\\textwidth]{localout2000.eps}\n        \\caption{Color online. Local outward filling $l^{out}_i$ versus out-degree $k^{out}_i$ in the binary WTW (year 2000): observed (blue) and predicted under the DCM (yellow).\n        }\n        \\label{plotlocalout}\n\\end{figure}\n\nIt should be noted that the rise of $\\phi_{DCM}$ partly overlaps with the period during which the reciprocity also strongly increased in the WTW, i.e. from the late 1980s onward \\cite{ruzzgarl:networksymmetry2}. \nIt therefore becomes important to further control (using the RCM) for the strong reciprocity which, as we discussed, might have the same apparent impact of distance-dependence by enhancing the symmetry of the network.\nUsing the RCM is important in general, as it controls for more stringent topological constraints and achieves a better fit to any real directed network (including the WTW \\cite{mytriadic}), thus filtering out  non-spatial effects even more accurately than the DCM.\nAs shown in the figure, we find that the RCM yields almost exactly the same results as the DCM. \n\nWe can therefore conlude that, even after disentangling distance and reciprocity effects, global spatial dependencies are present throughout the evolution of the WTW, with varying intensity.\nIn particular, the WTW experienced a phase of spatial \\emph{shrinking} from 1960 to the mid 1970s, followed by a phase of spatial \\emph{stretching} until 2000.\nThe latter trend is probably the result of the economic globalization.\nIn any case, note that for a network starting from a shrunk configuration (i.e. with $\\phi<0$), the evolution towards a more stretched configuration means that distances play a smaller and smaller role (i.e. $\\phi$ getting closer to zero).\nThis means that, if the above trend will persist in time, spatial effects will become less and less important in the WTW.\n\n\\begin{figure}[t]\n                \\includegraphics[width=0.45\\textwidth]{localin2000.eps}\n        \\caption{Color online. Local inward filling $l^{in}_i$ versus in-degree $k^{in}_i$ in the binary WTW (year 2000): observed (blue) and predicted under the DCM (yellow).\n        }\n        \\label{plotlocalin}\n\\end{figure}\n\nWe deepen our analysis of the binary WTW by studying the local spatial quantities defined in eqs. (\\ref{localfilling})-(\\ref{eassort}).\nThis is important in order to understand whether individual vertices contribute uniformly to the global spatial effects we characterized above, or whether systematic heterogeneities are instead present.\nIn Figs. \\ref{plotlocalout} and \\ref{plotlocalin} we show the local outward filling $l^{out}_i$ versus the out-degree $k^{out}_i$ and the local inward filling $l^{in}_i$ versus the in-degree $k^{in}_i$, both for the WTW in year 2000 (we observed similar results for all snapshots).\nAlong with the observed quantities, we also plot the expectations under the DCM. \nNote that, due to the approximate symmetry of the WTW topology and the exact symmetry of geographic distances, the inward and outward quantities are very similar.\nFor brevity, given the similarity of the results generated by the DCM and RCM as shown above, we discard the RCM in what follows (this approach is justified for the specific case of the WTW, but is not a recipe to be followed in general).\nWe find that $l^{out}_i$ and $l^{in}_i$ are strongly dependent, but nonlinearly, on  $k^{in}_i$ and $k^{out}_i$ respectively. \nMoreover, for vertices with small and intermediate degree, the observed local (inward and outward) filling is significantly smaller than expected under the DCM. \nThis difference gradually reduces as the degree increases, and for vertices with large degree the observed and expected values coincide.\nThis means that poorly connected countries tend to trade more locally (i.e. to geographically closer countries) than expected only on the basis of the different numbers of trade partners of all countries.\nBy contrast, highly connected countries are less limited by distances, as clear in the extreme case of a country trading with all other countries, an `unavoidable' space-neutral configuration which coincides with the null model's prediction, irrespective of spatial dependencies.\nIt is interesting to notice that, since the number of trade partners (degree) is strongly correlated with the GDP \\cite{mywtw}, the above results can be rephrased in terms of the different resistance to trade that geographic distances impose on poor (strong resistance) and rich (weak or no resistance) countries.\nWe therefore find that the global spatial effects captured by $\\phi$ arise from the combination of heterogeneous contributions from different vertices, and are not  representative of the more diverse local patterns.\n\n\\begin{figure}[t]\n                \\includegraphics[width=0.45\\textwidth]{assortout2000.eps}\n        \\caption{Color online. Outward assortativity $A^{out}_i$ versus out-degree $k^{out}_i$ in the binary WTW (year 2000): observed (blue) and predicted under the DCM (yellow).\n        }\n        \\label{plotassortout}\n\\end{figure}\n\nWe finally study the distance-induced correlation profiles by plotting, in Figs. \\ref{plotassortout} and \\ref{plotassortin}, the outward and inward assortativity ($A^{out}_i$ and $A^{in}_i$) versus the out- and in-degree ($k^{out}_i$ and $k^{in}_i$), again for the WTW in year 2000.\nWe find clearly decreasing trends, indicating that countries with large degree trade with countries that are embedded in a more spatially shrunk trade neighborhood, while countries with few connections  trade with countries that have a more spatially stretched neighborhood (where by `neighborhood' of a vertex we mean the set of vertices topologically connected to that vertex via a link, and \\emph{not} the set of geographically close vertices).\nHowever, the expected values are also decreasing, even if systematically larger than the observed ones.\nThis means that the negative correlation between the degree of a vertex and the average filling of its neighbours is simply interpretable as a negative correlation between the degrees of neighboring vertices imposed by the specific degree sequence (well documented in the WTW \\cite{mywtw,squartiniPRE2011a}), through the relation between degree and filling already shown in Figs. \\ref{plotlocalout} and \\ref{plotlocalin}.\nTherefore we find that the mere decrease of the trend is not informative \\emph{per se} about spatial effects.\nRather, it is the comparison with the null model that conveys information: the systematic difference between the observed and expected value of the assortativity shows that, irrespective of their degree, all countries tend to trade with countries that are embedded in a more spatially shrunk trade neighborhood than expected on the basis of topological constraints.\nThis confirms that our method successfully disentangles spatial and non-spatial effects, even when both produce similar patterns. \n\n\\begin{figure}[t]\n                \\includegraphics[width=0.45\\textwidth]{assortin2000.eps}\n        \\caption{Color online. Inward assortativity $A^{in}_i$ versus in-degree $k^{in}_i$  in the binary WTW (year 2000): observed (blue) and predicted under the DCM (yellow).\n        }\n        \\label{plotassortin}\n\\end{figure}\n\n\\section{Weighted analysis\\label{sec:weighted}}\nIn this section we extend our formalism to weighted networks.\nAs we mentioned, a weighted graph $\\mathbf{G}$ is described by a weight matrix $\\mathbf{W}$ whose entries $w_{ij}$ represent the intensity of the link from vertex $i$ to vertex $j$ (including $w_{ij}=0$ if no link is there).\nAs for binary networks, we assume no self-loops, i.e. $w_{ii}=0$ $\\forall i$.\nIn weighted networks, the analogue of the degree is the \\emph{strength}. For a directed network, the \\emph{out-strength} $s^{out}_{i}=\\sum_{j\\ne i} w_{ij}$ is the total weight of the outgoing links of vertex $i$, and similarly the \\emph{in-strength} $s^{in}_{i}=\\sum_{j\\ne i} w_{ji}$ is the total weight of the incoming links of vertex $i$.\nThe total weight of a network is the sum of the weights of all directed links, i.e. $W=\\sum_{i=1}^N\\sum_{j\\ne i}w_{ij}=\\sum_{i=1}^N s^{out}_{i}=\\sum_{i=1}^N s^{in}_{i}$.\n \n \\subsection{Generalizing the binary approach}\nFor weighted networks, in order to filter out non-spatial effects encoded in structural constraints, we will use two null models generated according to the same analytical method \\cite{mymethod} we used so far. \n\nThe first null model is the Weighted Random Graph Model (WRG) \\cite{mymethod,garlWRG}, which is the analogue of the DRG. In this model, the only constraint is the total weight $W$, so its Hamiltonian reads\n\\begin{equation}\nH_{WRG}(\\mathbf{G})=\\theta W(\\mathbf{G})\n\\label{eq_Hwrg}\n\\end{equation}\nThe WRG filters out the non-spatial effects due to the overall intensity of connections in a real network. In the WRG, every vertex has on average the same out- and in-strength $\\overline{s^{out}}=\\overline{s^{in}}=W\/N$.\nThe out- and in-strengths of vertices follow a negative binomial distribution with the above mean value, and link weights follow a geometric distribution with mean $W\/N(N-1)$ \\cite{garlWRG}.\n\nThe second null model we consider is the Weighted Configuration Model (WCM) \\cite{mymethod}, which is the analogue of the DCM. The constraints are the out-strength and in-strength of all vertices:\n\\begin{eqnarray}\nH_{WCM}(\\mathbf{G})=\\sum_{i=1}^N \\left[ \\theta_i^{in}s^{in}_i(\\textbf{G})+\\theta_i^{out}s^{out}_i(\\textbf{G})\\right]\n\\label{eq_Hwcm}\n\\end{eqnarray}\nThe WCM controls for non-spatial effects due to the different intrinsic sizes or capacities of vertices.\nAs the DCM, the WCM is very important in order to filter out apparent spatial dependencies that are only due to the heterogeneity of the weights induced by the strength of vertices. As we discuss later, in the particular case of the WTW this model is particularly appropriate as it indirectly controls for the heterogeneity of the GDP of world countries. \n \nFor simplicity, in the weighted case we do not consider the equivalent of the RCM, i.e. a weighted model that controls for reciprocated and non-reciprocated interactions separately. In fact, we expect that, as in the binary case, such a model would be indistinguishable (as far as spatial effects are considered) from the WCM. For the interested reader, a detailed description of the reciprocated weighted configuration model is provided in ref. \\cite{mywreciprocity}. What follows can be directly extended in order to include that model as well.\n\nFollowing the general method \\cite{mymethod}, both the WRG and the WCM can be fitted to the real network to generate the exact matrix $\\langle \\mathbf{W}\\rangle$ of expected weights.\nAgain, this suggests that a natural choice for any function $F(\\mathbf{W},\\mathbf{D})$ aimed at measuring the effects of distances on the weighted structure of a network is the linear one, so that the expectation value can be obtained exactly as $\\langle F(\\mathbf{W},\\mathbf{D})\\rangle=F(\\langle \\mathbf{W}\\rangle,\\mathbf{D})$.\nTherefore it is easy to generalize the linear binary quantities we introduced in section \\ref{sec:binary} to the weighted case.\n Starting from the quantity\n\\begin{equation}\nF\\equiv\\sum_{i=1}^N\\sum_{j\\ne i} w_{ij}d_{ij}\n\\end{equation}\nwe introduce the rescaled global filling, in analogy with eq. (\\ref{eq:filling}), as\n\\begin{equation}\nf\\equiv\\frac{\\sum_{i}\\sum_{j\\ne i}w_{ij}d_{ij}-F_{min} }{F_{max}-F_{min}}\n\\label{w_filling}\n\\end{equation}\nwhere now $F_{min}$ and $F_{max}$ are chosen as the two extreme values that $F$ can take in a network with the same total weight $W$ as the original network, i.e. $F_{min}=Wd_1^\\uparrow$ and $F_{max}=Wd_1^\\downarrow$ ($d_1^\\uparrow$ and $d_1^\\downarrow$ are the smallest and largest distance between vertices respectively). This reference corresponds to the WRG, and is analogous to the choice of the DRG we made in the binary case.\n\nThe filtered filling that properly controls for non-spatial effects embodied in a null model is again\n\\begin{equation}\n\\phi\\equiv \\frac{f  - \\langle f \\rangle }{ 1 - \\langle f \\rangle  }\n\\label{eq_wphi}\n\\end{equation}\nwhere $\\langle f \\rangle$ is now obtained by replacing $w_{ij}$ with $\\langle w_{ij}\\rangle$ in eq. (\\ref{w_filling}). \nPositive (negative) values of $\\phi$ indicate a network which is more stretched (shrunk) than expected. For a space-independent network, $\\phi=0$.\n\nFor the local weighted quantities, from the sums \n\\begin{equation}\nF^{out}_i\\equiv \\sum_{j\\ne i}w_{ij}d_{ij}\\qquad F^{in}_i\\equiv \\sum_{j\\ne i}w_{ji}d_{ij}\n\\end{equation}\nwe define the \\emph{local outward filling} as\n\\begin{equation}\nl^{out}_i\\equiv\\frac{\\sum_{j\\ne i}w_{ij}d_{ij}-(F^{out}_i)_{min}}{(F^{out}_i)_{max}-(F^{out}_i)_{min}}\n\\label{localwfilling}\n\\end{equation}\nand the \\emph{local inward filling} as\n\\begin{equation}\nl^{in}_i\\equiv\\frac{\\sum_{j\\ne i}w_{ji}d_{ij}-(F^{in}_i)_{min}}{(F^{in}_i)_{max}-(F^{in}_i)_{min}}\n\\label{localwfillingin}\n\\end{equation}\nConsistently with the global case and in analogy with the binary one, we choose the extreme values to correspond to the maximally stretched and maximally shrunk configuration achieved by a \\emph{typical} realization of the WRG (where all strengths are close to their expected value $W\/N$), i.e. $(F^{out}_i)_{max}=(F^{in}_i)_{max}=Wd_{i,1}^{\\downarrow}\/N$ and $(F^{out}_i)_{min}=(F^{in}_i)_{min}=Wd_{i,1}^{\\uparrow}\/N$.\nThe expected values of the above quantities are \n\\begin{equation}\n\\langle l^{out}_i\\rangle=\\frac{\\sum_{j\\ne i}\\langle w_{ij}\\rangle d_{ij}-(F^{out}_i)_{min}}{(F^{out}_i)_{max}-(F^{out}_i)_{min}}\n\\end{equation}\n\\begin{equation}\n\\langle l^{in}_i\\rangle=\\frac{\\sum_{j\\ne i}\\langle w_{ji}\\rangle d_{ij}-(F^{in}_i)_{min}}{(F^{in}_i)_{max}-(F^{in}_i)_{min}}\n\\end{equation}\n\nFinally, we define the \\emph{local outward assortativity} and the \\emph{local inward assortativity} as \n\\begin{equation}\nA^{out}_i\\equiv\\frac{\\sum_{j\\ne i}w_{ij}l^{out}_{j}}{s_i^{out}}\\qquad A^{in}_i\\equiv\\frac{\\sum_{j\\ne i}w_{ji}l^{in}_{j}}{s_i^{in}}\n\\label{wassortatout}\n\\end{equation}\nThe expected values can be approximated as \n\\begin{equation}\n\\langle A^{out}_i\\rangle=\\frac{\\sum_{j\\ne i}\\langle w_{ij}\\rangle\\langle l^{out}_{j}\\rangle}{\\langle s_i^{out}\\rangle}\\qquad \\langle A^{in}_i\\rangle=\\frac{\\sum_{j\\ne i}\\langle w_{ji}\\rangle\\langle l^{in}_{j}\\rangle}{\\langle s_i^{in}\\rangle}\n\\label{ewassort}\n\\end{equation}\nFor the WCM, since $\\langle s_i^{in}\\rangle =s_i^{in}$ and $\\langle s_i^{out}\\rangle =s_i^{out}$, the observed and expected assortativities can be compared by plotting both quantities as functions of the strength of vertices.\n\n\\begin{figure}[t]\n                \\includegraphics[width=0.45\\textwidth]{wphi.eps}\n        \\caption{Color online. Filtered weighted filling in the WTW from 1948 to 2000. Non-spatial effects have been removed using the Weighted Random Graph model (yellow) and the Weighted Configuration Model (blue).\n        }\n        \\label{plotwfilling}\n\\end{figure}\n\n\\subsection{Spatial effects in the weighted WTW}\nWe now employ the quantities defined above in order to perform a weighted analysis of the WTW.\nThe evolution of $\\phi_{WRG}$ and $\\phi_{WCM}$ is shown in Fig. \\ref{plotwfilling}. The trends are the weighted counterparts of those shown previously in Fig. \\ref{plotfilling}.\nBoth quantities are always negative, indicating that, even after taking into account the intensity of trade flows, the WTW is still found to be more spatially shrunk than expected under non-spatial models.\nThis confirms the general lesson learnt from gravity models in the economic literature \\cite{Feenstra_2001,anderson_2012}, but with a big difference: while gravity models only take into account the observed non-zero flows, and completely disregard missing links \\cite{fagiolo_gravity}, here we are combining the weighted information of nonzero flows with the topological information of missing links, as our quantities exploit the full matrix $\\mathbf{W}$.\nThe end result is that spatial effects are significant but weak, as the values of $\\phi$ (especially under the WCM) are small.\n\n\\begin{figure}[b]\n                \\includegraphics[width=0.45\\textwidth]{wlocalout2000.eps}\n        \\caption{Color online. Weighted outward filling $l^{out}_i$ versus out-strength $s^{out}_i$ in the weighted WTW (year 2000): observed (blue) and predicted under the WCM (yellow).\n        }\n        \\label{plotwlocalout}\n\\end{figure}\n\nAt this point it is worth observing that the WCM, by controlling for the strengths of vertices (total exports and total imports of all countries), automatically controls for the GDP, since the linear correlation between the latter and the total exports (or imports) of a country is extremely high.\nThis important aspect of the WCM is similar to the GDP-dependence in gravity models, and makes the WCM a more accurate and economically meaningful null model than the WRG. \nIn any case, unlike gravity models, our approach is not intended to provide a spatial model of trade, but only to single out statistically robust space-dependencies that can be used in the future to develop an improved spatially explicit model.\n\nAs for the binary case, we find that different null models provide a very different filtering of non-spatial effects, even over time.\nApart from the first snapshot, $\\phi_{WCM}$ first increases, then remains almost constant (during the 1970s) and finally decreases, approximately following an inverted U shape with small fluctuations. By contrast, $\\phi_{WRG}$ first decreases and then displays large fluctuations around an approximately constant trend.\nQuite surprisingly, these trends are almost opposite to the corresponding binary trends, and in particular $\\phi_{WCM}$ appears to indicate an unexpected spatial shrinking of the network from 1980 onwards, depite the economic globalization.\nHowever, it should be noted that while the DCM reproduces the higher-order properties of the binary WTW with good accuracy \\cite{squartiniPRE2011a}, the WCM does not satisfactorily reproduce the observed weighted WTW \\cite{squartiniPRE2011b}. Therefore, even if the WCM accurately controls for the total import and exports (hence the GDP) of world countries, it does not entirely filter out other non-spatial effects, at least not as stringently as its binary counterpart.\n\n\\begin{figure}[b]\n                \\includegraphics[width=0.45\\textwidth]{wlocalin2000.eps}\n        \\caption{Color online. Weighted inward filling $l^{in}_i$ versus in-strength $s^{in}_i$ in the weighted WTW (year 2000): observed (blue) and predicted under the WCM (yellow).\n        }\n        \\label{plotwlocalin}\n\\end{figure}\n\nWe finally consider the local spatial effects in the weighted WTW. Figures \\ref{plotwlocalout}-\\ref{plotwassortin} show the quantities defined in Eqs. (\\ref{localwfilling})-(\\ref{ewassort}), where the null model used is the WCM.\nThe increasing trends mean that rich contries have a more spatially stretched trade neighborhood than poor countries.\nHowever, as opposed to the binary results, here we find that the observed and expected values of the local filling ($l_i^{out}$ and $l_i^{in}$) are almost always in close agreement, except for minor differences in the scattered values of vertices with small strength (Figs. \\ref{plotwlocalout} and \\ref{plotwlocalin}).\nThis means that the global spatial effects detected by $\\phi$ are due to the combination of local effects involving only vertices with very small values of the strength, i.e. the poorest countries with small GDP and small import\/export values.\nInterestingly, the richest countries appear not to be affected by distances at all, in both imports and exports.\n\n\\begin{figure}[t]\n                \\includegraphics[width=0.45\\textwidth]{wassortout2000.eps}\n        \\caption{Color online. Weighted outward assortativity $A^{out}_i$ versus out-strength $s^{out}_i$ in the weighted WTW (year 2000): observed (blue) and predicted under the WCM (yellow).\n        }\n        \\label{plotwassortout}\n\\end{figure}\n\nBy contrast, the observed assortativities are systematically larger than the expectations of the WCM (Figs. \\ref{plotwassortout} and \\ref{plotwassortin}). \nHowever, looking at the definition in Eq. (\\ref{ewassort}), we find that the cause of this divergence must be the difference between $w_{ij}$ and $\\langle w_{ij}\\rangle$, and \\emph{not} the tiny disagreement between the observed and expected values of $l_i^{out}$ and $l_i^{in}$ shown above (of course, the expected and observed strengths are identical by construction). \nThe strong difference between $w_{ij}$ and $\\langle w_{ij}\\rangle$ under the WCM has already been well documented in ref. \\cite{squartiniPRE2011b}.\nThis confirms the different interpretations we can get from the binary and weighted analysis of the WTW. \nWhile the remarkable agreement between the DCM and the observed topology of the WTW \\cite{squartiniPRE2011a} ensures that the DCM accurately controls for non-spatial effects, the disagreement between the WCM and the real WTW \\cite{squartiniPRE2011b} implies that the final differences are partly due to the disagreement itself (i.e. differences between observed and expected quantities, irrespective of spatial effects).\nStill, as far as the problem of controlling for the intrinsic heterogeneity of vertices (in terms of import, exports, and indirectly GDP), the WCM yields the correct answer.\n\n\\begin{figure}[t]\n                \\includegraphics[width=0.45\\textwidth]{wassortin2000.eps}\n        \\caption{Color online. Weighted inward assortativity $A^{in}_i$ versus in-strength $s^{in}_i$ in the weighted WTW (year 2000): observed (blue) and predicted under the WCM (yellow).\n        }\n        \\label{plotwassortin}\n\\end{figure}\n\n\\section{Conclusions}\nIn this paper we have introduced a simple framework to detect spatial effects in geometrically embedded networks. \nWe have proposed global and local measures of space dependence for both binary and weighted networks.\nOur approach makes intense use of null models intended to preserve non-spatial constraints that strongly and unavoidably affect the structure of any network, irrespective of any spatial arrangement.\nThis allowed us to filter out, from the measured spatial effects, a spurious component due either to the overall density and intensity of connections (as in the DRG and WRG), or to the intrinsic heterogeneity of vertices (as in the DCM, RCM and WCM).\n\nWe tested our approach on the WTW, which is affected by both non-spatial constraints \\cite{mywtw,squartiniPRE2011a,squartiniPRE2011b} and geographic distances \\cite{Feenstra_2001,anderson_2012}.\nThe dependence of the weighted WTW on distances is well documented by gravity models, however the latter are generally fitted to the observed non-zero weights of the network, and hence disregard the effects of distances on the creation of links, and hence on the binary topology \\cite{fagiolo_gravity}.\nBy constrast, our weighted analysis of the WTW takes into account both zero and non-zero weights, as it exploits the full matrix $\\mathbf{W}$.\nA recent study employed a more sophisticated class of Gravity Models (the so-called `Zero-Inflated' models) \\cite{fagiolo_gravity} which can also incorporate zero flows, thus aiming at reproducing topology and weights simultaneously.\nThe important result of this study is that even these generalized models fail in reproducing the properties of the WTW. \nIn other words, unless the topology is pre-specified, Gravity Models perform not satisfactorily.\nThese results show that, if the binary topology is taken into account, the WTW is less clearly affected by geographic distances than ordinarily thought.\nAt the same time, it has been shown that the binary structure of the WTW is strongly dependent on the degree-sequence \\cite{squartiniPRE2011a} and indirectly on the GDP \\cite{mywtw}.\n\nTaken together, the above results imply that spatial effects in the binary WTW, if present, are expected to be weak.\nThis makes the WTW an ideal candidate for testing our method, due to the challenge of disentangling spatial dependencies and strong topological constraints.\nIndeed, we showed that our method is able to detect weak spatial effects even in a network which is dominated by non-spatial constraints.\nOur results confirm the importance of the use of null models in the analysis of real-world networks, including spatially embedded ones.\n\n\\begin{acknowledgments}\nFR acknowledges the University of Siena for the financial support and the Instituut-Lorentz for Theoretical Physics of Leiden for the logistic, scientific and moral support received during the visit.\nDG acknowledges support from the Dutch Econophysics Foundation (Stichting Econophysics, Leiden, Netherlands) with funds from beneficiaries of Duyfken Trading Knowledge BV, Amsterdam, The Netherlands. \nRB acknowledges support from the FESSUD project on ``Financialisation, economy, society and sustainable development'', Seventh Framework Programme, EU. Authors warmly thank Tiziano Squartini for his kind and insightful comments that helped us improve a previous version of this paper.\n\\end{acknowledgments}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nInitial margin (IM) has become a topic of high relevance for the financial industry in recent years. The relevance stems from its wide implications to the way business will be conducted in the financial industry and in addition, for the need of approximation methods to calculate it.\n\n\\medskip\n\\noindent The exchange of initial margin is not new in the financial industry, however, after regulations passed in 2016 by Financial Regulatory Authorities \\cite{BCBS}, a wider set of financial institutions are required to post significant amounts of initial margin to each other.\n\n\\medskip\n\\noindent The initial margin amount is calculated at the netting set level following a set of rules depending on whether the trade has been done bilaterally \\cite{ISDA-SIMM} or facing a Clearing House \\cite{Henrard}. The calculation is performed on a daily basis and carried on throughout the life of the trade. \n\n\\medskip\n\\noindent Since each netting set consumes initial margin throughout its entire life, financial institutions need resources to fund this amount in the future, given that the margin received from its counterparty cannot be rehypothecated. To estimate the total need for funds, each counterparty needs to perform a forecast of the initial margin amount up to the time to maturity of the netting set.\n\n\\medskip\n\\noindent Forecasting the total forward initial margin amount is a computationally challenging problem. An exact initial margin calculation requires a full implementation of a nested Monte Carlo simulation, which is computationally onerous given the real-time demand  for the IM numbers as part of day-to-day business activities. \n\n\\subsection{Literature Review}\n\\medskip\n\\noindent As a response to this computational challenge, practitioners and academics have proposed several approximation methods to reduce the dimensionality of the problem, at the cost of losing a tolerable degree of accuracy. The most popular of these approaches have been the regression methods, since they offer simplicity and re usability of the Bank's legacy Monte Carlo engines, paired with good results. Inspired by early works of Longstaff-Schwartz \\cite{LS}, several banks have implemented some version of the polynomial regression as proposed by \\cite{Anfuso}, \\cite{Caspers} and \\cite{Justin}, or the Kernel regressions proposed by \\cite{Andersen-Exposure}, \\cite{Justin} and \\cite {Dahlgren}. Most recently, estimation by Neural Networks has been proposed by \\cite{Hernandez} and \\cite{DeepLearning}. \n\n\\medskip\n\\noindent These methods have not been developed exclusively to tackle the initial margin problem. These are well-known regression methodologies and have been used extensively in finance and engineering. As such, the current literature on initial margin have focused on the implementation and numerical validation of the regression methods from a practical point of view. The authors' main focus have been to show, the applicability and performance of the different methodologies when compared to a brute-force benchmark, without going deep into mathematical formalities.\n\n\\medskip\n\\noindent This paper aims to fill the gap between theory and practice. The underlying mathematical framework required to analyze the initial margin computation problem from a theoretical standpoint is presented here. \n\n\\subsection{Organization}\nThe rest of this paper is organized as follows. Section  2 translates the practitioner's definition of initial margin into a formal probability setting. Section 3-4 develop the assumptions needed to fit the initial margin problem into a classical problem of orthogonal projections into Hilbert spaces. The last section concludes by linking back the formal theory presented with the practitioner's suggested approaches. \n\n\\section{Description of the Initial Margin Problem}\n\n\\subsection{Definition of Forward Initial Margin}\n\n\\noindent Following the definition in \\cite {Green}, the aim of a Bank is to calculate the total cost associated with the posting of initial margin to a counterparty over the life of a netting set. Since this is a quantity that evolves in time, it is necessary to consider a forward diffusion model, together with a probability space and the relevant discounting and funding rates into the equation. \n\n\\medskip\n\\noindent Given a filtered probability space $(\\Omega, \\mathcal{F},\\left\\{{\\mathcal {F}}_{{t}}\\right\\}_{{t\\geq 0}}, \\mathbb{P})$ where as usual, $\\Omega$ is the set of all possible sample paths $\\omega$, $\\mathcal{F}$ is the sigma-algebra,$\\left\\{{\\mathcal {F}}_{{t}}\\right\\}$ is the filtration of events at time $t$ (i.e. the natural filtration to which risk factors are adapted to) and $\\mathbb{P}$ the probability measure defined on the elements of $\\mathcal{F}$. The total cost of posting initial margin over the entire life of a netting set is referred as the \\textit{Margin Valuation Adjustment} as defined below.\n\n\\begin{equation*}\n\\label{eq:MVA1}\nMVA_C(t) = \\mathbb{E}_{t}\\left[ \\int_{t}^{T} ((1-R_C)\\lambda (u)-S_I(u)) e^{\\int_{t}^{u}(r(s)+\\lambda_B(s)+\\lambda_C(s))ds}\\mathbb{E}_{u}\\left[IM_C(u)\\right] du \\right]  \n\\end{equation*}\n \n \n\\noindent where:\n\\begin{itemize}\n    \\item $\\mathbb{E}_{t}$ denotes an expectation conditional to information at time $t$, i.e. $\\mathbb{E}_{t}\\left[\\dots\\right] = \\mathbb{E}\\left[\\dots \\mid \\mathcal{F}_t\\right]$\n    \\item $R_C(t)$ is the recovery rate at time $t$ of the counterparty related to netting set $C$;\n    \\item $T$ is the final maturity of netting set $C$;\n    \\item $\\lambda (t)$ is the Bank's effective financing rate (i.e. borrowing spread) at time $t$;\n    \\item $S_I$ is the spread received on initial margin;\n    \\item $r(t)$ is the risk free rate (e.g. an OIS rate) at time $t$;\n    \\item $\\lambda_B(t),\\lambda_C(t)$ are the survival probabilities at time $t$ of the Bank and the counterparty related to netting set $C$, respectively;\n    \\item $IM_C(t)$ is the initial margin posted by the bank at time $t$ against netting set $C$.\n\\end{itemize}\n\n\\medskip\n\\noindent Note that the expectation $\\mathbb{E}_{t}$ is taken on the whole integral. This is because certain simulation setups, like the one in \\cite{AlbGimPie} and \\cite{AlbaneseKVA}, feature explicitly modeled credit factors that lead to stochastic credit spreads and survival probabilities. If the simulation setup instead considers deterministic credit factors, the formula can be simplified as follows: \n\n\n\n\\begin{eqnarray*}\nMVA_C(t) &=&  \\int_{t}^{T} ((1-R_C)\\lambda (u)\\\\\n&-&S_I(u))e^{\\int_{t}^{u}(\\lambda_B(s)+\\lambda_C(s))ds}\\mathbb{E}_{t}\\left[\\mathbb{E}_{u}\\left[IM_C(u)\\right]e^{\\int_{t}^{u}r(s)ds}\\right] du  \n\\end{eqnarray*}\n \n\\medskip\n\\noindent Here our quantity of interest is $\\mathbb{E}_{t}\\left[IM(t)\\right]$. This is the expectation of the initial margin at time $t$, taken across all random paths $\\omega$, conditional on the realization of the risk factors at time $t$. The initial margin is a path-wise random variable defined as \\footnote{This is the simplest definition of initial margin, for alternative definitions see \\cite{RiskBook}}\n\n\\begin{equation}\n\\label{eq:IM}\nP(\\Delta_t \\leq IM(\\omega,t) \\vert \\mathcal{F}_t) = p\n\\end{equation}\n\n\\medskip\n\\noindent Where:\n\\begin{itemize}\n    \\item $p$ denotes the desired quantile (usually the $99\\%$ quantile);\n    \\item $\\Delta(\\omega,t,\\delta_{IM})$ is defined (in its simplest form\\footnote{This is the simplest definition of netting set PnL, for further definitions see \\cite{RiskBook}.}) as the change, conditional to counterparty default, in the netting set's value (i.e. profit and loss (PnL)) over a time interval $(t,t+\\delta_{IM}]$:\n    \\begin{equation*}\n    \\label{eq:PnL}\n    \\Delta(\\omega,t,\\delta_{IM}) = V (\\omega,t+\\delta_{IM}) - V (\\omega,t)\n    \\end{equation*}\n\\end{itemize}\n\n\\noindent Note that the Initial Margin as defined above is specifically the \\emph{posted} IM, as the quantile is calculated on the upper-tail of the PnL distribution. In order to calculate the \\emph{received} IM, one can instead compute the $1\\%$ quantile on the same distribution.\n\n\\medskip\n\\noindent The following sections describe the computational methods developed in the current literature to estimate this quantity.\n\n\\subsection{The Brute Force Approach to Calculate Initial Margin}\n\nThe most exact method to calculate the initial margin requires a \\textit{brute force} implementation of a nested Monte Carlo simulation. Since this approach is computationally untractable, in practice it is only used as a ground truth benchmark to assess the performance of the approximation methodologies. \n\n\n\\medskip\n\\noindent Figure 1 exemplifies the situation. For a given point in time, $t_1$, an \\textit{outer} Monte Carlo simulation for the values of $V(\\omega,t_1)$ and an \\textit{inner} Monte Carlo simulation of the forward Probability Density Function (PDF) of $V(\\omega,t_1+\\delta_{IM})\\mid \\mathcal{F}_{t_1})$ have been performed. The difference in the value of the netting set between $t_1$ and $t_1+\\delta_{IM}$ time steps is $\\Delta(\\omega,t_1,\\delta_{IM})$. Since the PDF distribution of $(V(\\omega,t_1+\\delta_{IM})\\mid \\mathcal{F}_{t_1})$ is known, it is possible to obtain the PDF distribution of $\\Delta(\\omega,t_1,\\delta_{IM})$.  Then, one should take the $\\mathcal{Q}_{99}$ to obtain the initial margin.\n\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{fig1.PNG}\n\t\\caption{A simulation tree with two branches and two time steps}\n\t\\label{fig:nested-mc}\n\\end{figure}\n\n\n\\medskip\n\\noindent The initial margin at every time step $t$ is the $99^{th}$ percentile of the inner PDF scenarios. Both processes (inner and outer) are usually simulated using the same stochastic differential equation with a sufficiently large number of scenarios to ensure convergence. The number of scenarios become particularly important for the inner scenarios as the $Q_{99}$ is a tail statistic, therefore the number of scenarios on the tail should be sufficiently large.\n\n\\medskip\n\\noindent As every nested Monte Carlo problem, the \\textit{brute force} calculation of initial margin is computationally onerous given that the number of operations increase exponentially. This type of complexity is not possible to afford in real time, e.g. in a live trading scenario, thus the reason behind the development of approximation methods.\n\n\\section{Regression Methods for the Approximation of Initial Margin}\n\nThe regression proposed by Longstaff-Schwartz in \\cite{LS} is the traditional framework used in finance to reduce dimensions in nested Monte Carlo problems. In short, the authors propose to reduce the number of machine operations by simulating only the outer scenarios and fitting a regression to forecast the inner ones. Their regression uses a polynomial basis as the functional choice for the regression.\n\n\\medskip\n\\noindent  The use of regressions is justified since the authors' mathematical problem is the estimation of a conditional expectation on the inner scenarios, which is usually approximated with a linear regression. Initial margin computation problem posed in this paper requires approximation of a quantile. It will be shown that  the general Longstaff-Schwartz framework can be applied to this problem as well.\n\n\\medskip\n\\noindent Figure 2 provides a graphical description of the approach followed to approximate a nested Monte Carlo problem. The main idea is to simulate only the outer scenarios and assume a parametrized PDF for the inner ones. Under this setting, one can use a regression to estimate the required quantile.  \n\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=1.01\\linewidth]{Fig2.PNG}\n\t\\caption{Simplification of a nested Monte Carlo by applying distributional assumptions}\n\t\\label{fig:nested-mc1}\n\\end{figure}\n\n \\noindent From equation \\ref{eq:IM}, the inner scenarios are used to calculate the PDF of the change in the portfolio value, conditional on the information provided on each filtration set $\\mathcal{F}_t$. From this PDF, one can obtain the $Q_{99}$ that makes the initial margin amount. However, when only the outer scenarios are simulated, there is only a single point of the empirical distribution per branch left, which is not enough to build an entire PDF. To overcome this limitation, the relevant literature propose to assume a parametric distribution for the rest of the unknown inner scenarios. The majority of authors propose modelling the quantity  $[V(\\omega,t+\\delta_{IM})-V(\\omega,t)]$ with a local Normal distribution,  since it can be conveniently fitted by only two parameters, the mean and the variance. This choice is justified under the assumption that the simulated $V(\\omega,t)$ come from a Normal SDE and the time lapse between observations $\\delta_{IM}$ is small. Other authors like \\cite{Dickinson} propose a fat-tailed $\\mathcal{T}$ distribution.\n\n\\medskip\n\\noindent If it can be assumed that the inner scenarios follow a local Normal distribution, the only task left is the calculation of the distribution's expectation and variance to then obtain the initial margin as:\n\n\\begin{equation*}\n\\label{eq:IMnormal}\n(IM(\\omega,t) \\mid \\mathcal{F}_t) = \\sigma(\\omega,t) \\cdot \\Phi^{-1}(99\\%)\\mid \\mathcal{F}_t\n\\end{equation*}\n\n\\medskip\n\\noindent Where for simplicity and without loss of generalization, we have assumed that the data is centered and thus the expectation is zero. Under this assumption, the term $\\sigma(\\omega,t)$ can be written as:\n\n\\begin{equation}\n\\label{eq:devCondit}\n(\\sigma(\\omega,t)\\mid \\mathcal{F}_t) = \\sqrt{\\mathbb{E} \\left[ \\Delta^2(\\omega,t,\\delta_{IM}) \\mid \\mathcal{F}_t \\right]}\n\\end{equation}\n\n\\medskip\n\\noindent Equation \\ref{eq:devCondit} shows that, when restated in this way, a quantile estimation problem has been transformed into the conditional expectation problem, which ends up in the same setting as the Longstaff-Schwartz model. The next sections study different approximations of $\\mathbb{E} \\left[ \\Delta^2(\\omega,t,\\delta_{IM}) \\mid \\mathcal{F}_t \\right]$, which is the conditional expectation of the netting set's PnL under the information given at time $t$.\n\n\n\\section{Conditional Expectations as Orthogonal Projections  on Hilbert Spaces}\n\n\\subsection{Conditional Expectations as Orhtogonal Projections}\n\nThe task is to find the best (i.e. unbiased) estimator of the conditional expectation in equation \\ref{eq:devCondit} at every time step $t$. The proposal is to approximate the conditional expectation by regressing $ \\Delta^2(\\omega,t,\\delta_{IM})$ against a set of basis functions. However, this use of regressions to approximate conditional expectations needs to be formally justified. \n\n\\medskip\n\\noindent  The intuition behind the connection between conditional expectations and regressions is to frame conditional expectations as functions mapping a set of vectors in $\\mathcal{L}^2$ onto a given set of vectors in the same space. The the particular form of these functions is an arbitrary choice, and the coefficients are calibrated following the geometric properties of $\\mathcal{L}^2$ spaces. \n\n\n\\subsection{Properties of Conditional Expectations in Hilbert Spaces}\nThe connection between conditional expectations and regressions is proved in two steps. Firstly, the theoretical framework is described in order to characterize conditional expectations as orthogonal projections on Hilbert spaces, and, secondly, orthogonal projections are connected to regressions in an empirical step.\n\n\\medskip\n\\noindent  The conditional expectation is expressed as a function of a set of vectors $x_t$ obtained from the filtration, \n\n\\begin{equation}\n\\label{eq:RegDefn}\n \\mathbb{E} \\left[\\Delta^2(\\omega,t,\\delta_{IM}) \\mid \\mathcal{F}_t \\right]  = m(x_t) \n\\end{equation} \n\n\\noindent where $x_t$  a short for $span(x_1,...,x_T)$, an $\\mathcal{F}_t$-adapted process and $m(x_t)$ is a stochastic function with certain properties described in the following sections.\n\n\n\\medskip\n\\noindent  In equation \\ref{eq:RegDefn},  the set of $(x_1,...,x_T)$ regressors have been fixed by conditioning on information obtained from the filtration at time $t$, and therefore, $\\mathbb{E} \\left[\\Delta^2(\\omega,t,\\delta_{IM}) \\right]$ is projected on a fixed space. In this setting, a conditional expectation is a \\textit{function} of the vectors $(x_1,...,x_T)$ and not just a random variable, as it is typically defined in option pricing. By interpreting conditional expectations as functions, the properties inherited from the functional space they belong to can be used. In particular, the properties regarding orthogonal projections and decomposition into orthogonal bases from $\\mathcal{L}^2$ .\n\n\\medskip\n\\noindent In order to set up an appropriate functional space, the space of random bounded functions is considered on the filtered probability space $\\mathcal{L}^2$$(\\Omega,\\mathcal {F},\\left\\{{\\mathcal {F}}_{{t}}\\right\\}_{{t\\geq 0}}, \\mathbb{P})$. Thus, only square-integrable functions, satisfying $\\mathbb{E}[y^2] <\\infty$ for every $y \\in \\mathcal{L}^2$ (i.e. finite second moment), are considered. As such, the inner product of two elements $y_1$ and $y_2$ is $\\mathbb{E}[y_1 .y_2]$, with the induced norm $\\parallel y \\parallel = \\sqrt{\\mathbb{E}[y^2]}$. \n\n\\medskip\n\\noindent Following the definition \\ref{defn:orthoPrj} below, under $\\mathcal{L}^2$ spaces conditional expectations are equivalent to linear maps that perform orthogonal projections between a space and a subspace  \\footnote{While conditional expectations are well defined in $\\mathcal{L}^1$, this is not an inner product space, only the $\\mathcal{L}^2$ space defines an inner product that can be used to meet orthogonality requirement.}.\n\n\n\\medskip\n\\begin{definition}\n\\label{defn:orthoPrj}\nGiven a $\\sigma$-algebra $\\mathcal{F}$, there is a closed subspace $\\mathcal{L}^2$\n $(\\Omega,X, \\mathbb{P})$ $\\subset \\mathcal{L}^2$  $(\\Omega,\\mathcal {F}, \\mathbb{P})$, each equipped with its filtration. Suppose that $y \\in \\mathcal{L}^2$  $(\\Omega,\\mathcal {F}, \\mathbb{P})$, the conditional expectation $\\mathbb{E}[y \\mid X]$ is defined to be the orthogonal projection of $y$ onto the closed subspace $\\mathcal{L}^2(\\Omega,X, \\mathbb{P})$.\n\\end{definition}\n\n\\medskip\n\\noindent The next step is to find a functional form for the projection function and this is done by choosing an orthogonal basis for the $span (x_1,...,x_T)$. While the basis vectors can be chosen arbitrarily, the vector weights (i.e. coefficients) must be chosen so that the resulting projection vector is orthogonal to the original space\\footnote{The linear combination is guaranteed to exist since every Hilbert space admits an orthonormal basis (Zorn's Lemma) and it is unique (by the theorem of characterization of orthogonal projections (see \\cite{Luenberg}). Moreover, if the space projected onto is separable, the combination of basis functions needed to obtain this projected vector is finite. }. \n\n\n\n \\medskip\n\\noindent The challenge under a Monte Carlo simulation setting is to find a finite set of suitable basis functions to represent the projected vectors. The following sections expand on the most common empirical methodologies used for this mean. \n\n\\section{Approximation Methodologies for the Estimation of Conditional Expectations}\n\nThis section deals with the practical implementation of the theory above, in particular, the choice of regressors and basis functions. The described theory of Hilbert spaces can be considered as the \\textit {population} version, where conditional expectations are equivalent to orthogonal projections, while the simulated vectors are the \\textit{sample} version. The spaces are now Euclidean (i.e. finite) and conditional expectations are estimated with regressions. \n\n\\subsection{The Choice of Regressors}\n\nUnder a Monte Carlo simulation, the independent $X_t$ and dependent variables $Y_t$ are empirically generated following an arbitrary choice of SDE. In particular, the discrete time filtration for both is obtained.\n\n\\medskip\n\\noindent The key art of estimating regressions is the choice of the regressors $X_t$. The output of the Monte Carlo simulation contains several variables that can be used as regressors. As explained in the previous sections, in theory it is desirable to find an orthogonal basis to decompose the projected vectors, in practice, the choice of regressors is an empirical call. However, some minimal conditions must be met.\n\n\\subsection{Markovian Regressors}\nIn choosing the regressors, there should be a sound relationship between the projected vector $Y_t$ and the projection vectors $(X_t)_{0\\le t\\le T}$ obtained from the filtration. The second aspect to determine is the relevance of past data as predictors of the dependent variable, $Y_t$. This is, how much of the past history coming from the filtration is relevant. For Markov chains, only the most recent values of the state variables are necessary, while for non Markov, past realizations of the filtrations should be included as regressors.\n\n\\medskip\n\\noindent If the variables forming the filtration are Markovian, then the conditional expectation can be estimated as:\n\n\\begin{equation}\n\\label{eq:markovianPred}\n\\mathbb{E}[Y_{t+1} \\mid \\mathcal{F}_t] = \\mathbb{E}[Y_{t+1} \\mid X_t] \n\\end{equation}\n\n\\medskip\n\\noindent \nIf the $(X_t)_{0\\le t\\le T}$ process has stochastic independent increments, then it is a Markov process (\\cite{Luenberg}). \n\n\n\\subsection{The Choice of the Basis Functions}\n\nThe choice of basis functions depends on the projection subspace $span(x_1,...,x_T)$ and mostly, on the empirical relationship between the independent and dependent variables $X_t$ and $Y_t$.\n\n\\medskip\n\\noindent The underlying idea behind the different approximation methods is to find an approximation for $\\mathbb{E}[Y_{t+\\delta} \\arrowvert X = X_t] = m(X_t)$ that solves the below minimization problem:\n\n\\medskip\n\\noindent Let $Y_{t+\\delta}$ be a random vector in a Hilbert space and $(X_t)_{0\\le t\\le T}$ a vector in a closed Hilbert subspace. Additionally, assume $m(X_t)$ is a given a function of $span (X_t)$. Solve the minimization problem below to find $\\hat{\\beta} \\subset \\mathcal{R}^n$ such that:\n\n\\begin{equation}\n\\label{eq:orthoProj}\n\\underset{\\hat{\\beta}}{inf} \\; \\mathbb{E}[(Y_{t+\\delta} - m(X_t,\\hat{\\beta}))^2]\n\\end{equation}\n\n\\medskip\n\\noindent where the function $m(\\cdot )$ can be decomposed into basis functions as explained in Section 4.2. \n\n\\subsection{Empirical Relationship Between the Dependent and Independent Variables}\n\nThe choice of functional form to represent the relationship between the vectors $Y_t$ and $X_t$ is an empirical call, mostly based on data analysis. The question is, which would be the best choice of basis functions such that $m(X) \\colon X \\to Y$ agrees with our simulation data?\n\n\\medskip\n\\noindent In practice, under a nested Monte Carlo simulation, basis functions are selected arbitrarily so that the resulting functional form of the conditional expectation function represents appropriately the relationship between the vector $(Y_t)_n$ and the filtration vector $(X_t)_n$, where $n$ is the dimension of the simulated vectors at time $t$. \n\n\\medskip\n\\noindent  From the Monte Carlo simulation engine, one can easily obtain the simulated variables  $V(\\omega,t)$ and  $V(\\omega,t+\\delta_{IM})$ from which $\\Delta^2(\\omega,t,\\delta_{IM})$ can be calculated at every $t$. The natural choice used in the literature is to choose $V(\\omega,t)$ as the regressor for $\\Delta^2(\\omega,t,\\delta_{IM}) $. This assumes that $V(\\omega,t)$ is a good predictor for $\\Delta^2(\\omega,t,\\delta_{IM})$ as in equation \\ref{eq:markovianPred}. In what follows, the following terms will be used:\n\n\\bigskip\n$X_t = {V(\\omega,t)}_n = [V(\\omega_1,t), V(\\omega_2,t), ..., V(\\omega_n,t)]^T$\n\n$Y_{t +\\delta} = {\\Delta^2(\\omega,t,\\delta_{IM})}_n = \\left({V(\\omega,t+ \\delta_{IM})}_n- {V(\\omega,t)}_n\\right)^2$\n\n\\medskip\n\\noindent where $n$ is the number of simulated scenarios, $\\omega$. \n\n\\medskip\n\\noindent In order to choose the functional relationship between $V(\\omega,t)$ and $\\Delta^2(\\omega,t,\\delta_{IM})$ one must remember from the previous section that the only available data are the simulations from the outer scenarios, and from this information it is necessary to infer the relationship between the $V(\\omega,t)$ and $\\Delta^2(\\omega,t,\\delta_{IM})$ on the inner scenarios. The assumption is that the functional relationship between the $V(\\cdot)$ and $\\Delta^2(\\cdot)$ in the inner and outer scenarios is the same. In other words, it is assumed that the model we adjust with data coming from the outer scenarios simulation can be used to infer the inner scenarios. This assumption is based on the fact that the same SDE is used to simulate both the inner and outer scenarios (provided observer is at a given $t$).\n\n\\medskip\n\\noindent From the graphs in \\cite{Justin} we can study the relationship between $V(\\cdot)$ and $\\Delta^2(\\cdot)$ for different type of regression models. From the cloud of points, the functional form is rather un-conclusive.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=1.01\\linewidth]{GraphsJustin.PNG}\n\t\\caption{ A comparison of scatter plots of portfolio value vs simulated initial margin and PnL for initial margin models. Taken from \\cite{Justin}.}\n\t\\label{fig:nested-mc}\n\\end{figure}\n\n\n\\medskip\n\\noindent Lastly, a usual simplification is to assume that the set of basis functions does not change over time, but weights do. Hence, after a functional form of the conditional expectation has been chosen, the problem is to estimate the weights at every simulation time step $t$, conditional on the information given by the filtration up to that time.\n\n\\subsection{Necessary conditions on MtM processes}\n\n\\noindent In earlier sections it has been stated that the analysis should be restricted to square integrable functions with finite second moment (variance) - in the available literature this point is assumed to hold without proof, especially with respect to the squared PnL random variable $Y_t$. Now formal derivation of two conditions that a generic Mark-to-Market (MtM) random variable has to satisfy in order for $Y_t$ to be square integrable will be provided. It will be assumed that for the MtM to be a real random variable, as must be the case for the financial setting we are analysing.\n\n\\subsubsection{Theorems}\nThis section begins with stating of the theorems on square-integrable variables that will be used in the rest of the derivations.\n\n\\begin{theorem}\\label{theo_l2_var}\nIf a real random variable $X$ has finite mean and variance, it is square integrable:\n\\begin{equation*}\n    \\mathbb{E}[X] < \\infty, Var[X] < \\infty \\Longrightarrow \\mathbb{E}[\\lvert X \\rvert^2] < \\infty\n\\end{equation*}\n\\end{theorem}\n\n\\begin{proof}\nThe variance of $X$ is by definition:\n\\begin{equation*}\n    Var[X] = \\mathbb{E}[X^2] - \\mathbb{E}^2[X]\n\\end{equation*}\nIf $Var[X]<\\infty$, then all terms in the RHS must be finite. It is known that $\\mathbb{E}[X]<\\infty$, therefore $\\mathbb{E}^2[X]<\\infty$. It must follow that $\\mathbb{E}[X^2]<\\infty$ and therefore $\\mathbb{E}[\\lvert X\\rvert^2]<\\infty$.\n\n\\end{proof}\n\n\\begin{remark}\nNote that $\\mathcal{L}^2$ is formally defined as the space of square integrable functions. Since a space of random variables is considered, from now on these two concepts will be used interchangeably.\n\\end{remark}\n\n\\begin{theorem}\\label{theo_l2_sums}\nSums or scalar multiplications of square integrable variables are also square integrable.\n\\end{theorem}\n\n\\begin{proof}\nThe theorem for a generic sum $Z = X + Y$ of two square integrable random variables $X$ and $Y$ will be proved. Let's express the variance of $Z$:\n\\begin{multline*}\n    Var(Z) = \\mathbb{E}[(X+Y)^2] - \\mathbb{E}^2[X + Y]\\\\\n    = \\mathbb{E}[X^2] + \\mathbb{E}[Y^2] - \\mathbb{E}^2[X] - \\mathbb{E}^2[Y] - 2\\mathbb{E}[X]\\mathbb{E}[Y] + 2\\mathbb{E}[XY] \n\\end{multline*}\nIn the above, the first two terms are finite as $X$ and $Y$ are square-integrable. The third and fourth terms are also finite as $X$ and $Y$ must have finite mean. Finally, the Cauchy-Schwarz inequality can be used to infer that the last term must also be finite, as its absolute value is bounded by the square root of the product of two finite terms:\n\\begin{equation*}\n    \\lvert\\mathbb{E}[XY]\\rvert \\leq \\sqrt{\\mathbb{E}[X^2]\\mathbb{E}[Y^2]}\n\\end{equation*}\n Since the variance is finite, we can refer to Theorem \\ref{theo_l2_var} and conclude that $Z$ must also be square integrable. The result is trivially extendable, via induction arguments, to multiple sums of square integrable variables as well as scalar multiplications thereof.\n\\end{proof}\n\n\\subsubsection{Derivation of conditions}\nRecalling that the generic form of the squared PnL random variable $Y_{t +\\delta_{IM}}$ is:\n\\begin{multline*}\n    Y_{t +\\delta_{IM}} = {\\Delta^2(\\omega,t,\\delta_{IM})}_n = \\left({V(\\omega,t+ \\delta_{IM})}_n - {V(\\omega,t)}_n\\right)^2\\\\\n    = {V^2(\\omega,t+ \\delta_{IM})}_n -2{V(\\omega,t+ \\delta_{IM})}_n{V(\\omega,t)}_n + {V^2(\\omega,t)}_n\n\\end{multline*}\n    \n\\noindent It can be observed that, without loss of generality, $Y_t \\sim \\mathcal{O}({V^2(\\omega,t)}_n)$. By virtue of Theorem \\ref{theo_l2_sums} it can be concluded that if the generic squared MtM random variable ${V^2(\\omega,t)}_n$ is square-integrable, so will be $Y_t$. In turn, by virtue of Theorem \\ref{theo_l2_var}, this only leaves the need to state the conditions on ${V(\\omega,t)}_n$ for which the variance of ${V^2(\\omega,t)}_n$ is finite. The variance of ${V^2(\\omega,t)}_n$ is:\n\\begin{equation*}\n    Var({V^2(\\omega,t)}_n) = \\mathbb{E}[{V^4(\\omega,t)}_n] - \\mathbb{E}^2[{V^2(\\omega,t)}_n]\n\\end{equation*}\n\n\\noindent This variance is finite i.f.f. the two terms on the RHS are finite, yielding the following two conditions on ${V(\\omega,t)}_n$:\n\n\\begin{proposition}\n${V(\\omega,t)}_n$ must have finite mean and variance. This ensures that ${V^2(\\omega,t)}_n$ is square integrable and therefore $\\mathbb{E}^2[{V^2(\\omega,t)}_n] < \\infty$. \n\\end{proposition}\n\n\\begin{proposition}\n${V(\\omega,t)}_n$ must have finite fourth moment. This ensures that $\\mathbb{E}[{V^4(\\omega,t)}_n] < \\infty$. \n\\end{proposition}\n\n\\medskip\n\\subsection{Analytical Approximations for the Conditional Expectation Function}\n\nSeveral methods can be used to estimate the conditional expectation function $\\mathbb{E}[Y_{t+\\delta} \\arrowvert X = X_t]  = m(X_t)$, depending on the properties of the underlying data. A good summary of the available methodologies for nested Monte Carlo problems can be found in \\cite{Carrier}. The next sections discuss  the most popular approaches used in the context of initial margin. \n\n\\subsubsection{Algorithms Based on Linear Maps}\nIn the context of initial margin estimation, linear regression methods are proposed by \\cite{Anfuso}, \\cite{Caspers} and \\cite{Justin}. The problem setting is as described in equation \\ref{eq:orthoProj}. In the case where the Hilbert subspace $\\mathcal{F}_t$ is a linear vector space, the estimate can be computed solving a linear equation system. Linear maps include the ordinary least square regression and polynomial functions. \n\n\\medskip\n\\noindent The projection is done onto a linear independent set of random variables $(X_t)$. \n\n\\noindent where $m(X,\\hat{\\beta}))$ is a linear map of the form:\n\n\\begin{equation}\n\\label{eq:beta_linFunct}\nm(X_t,\\hat{\\beta}) = \\sum_{i=0}^{P} \\hat{\\beta}_i . \\phi_i(X_t) = \\mathbb{B}^T . \\phi_i(X_t)\n\\end{equation}\n\n\\noindent where $\\phi_i(X_t) $ are the $P$ set of linear basis functions arbitrarily chosen.\n\n\\medskip\n\\noindent A linear problem like this can be solved by inverting matrices. If $Y$ and $X$ are jointly Gaussian distributed, then the setting is of the well known Normal Equations. For example, in \\cite{Anfuso} $P=2$ and $\\phi_i(X_t) =  X_t^i$.\n\n\n\\medskip\n\\noindent In the non-linear methods below, the vectors spaces (and thus the function $m(X,\\hat{\\beta})$) are non-linear and thus, the Least Ordinary Square approach does not work. \n\n\n\\subsubsection{Algorithms Based on Non Parametric Regressions}\nAmong the vast amount of non-parametric algorithms available, kernel regressions have been the most popular ones. In particular the Naradaya-Watson kernel \\cite{Naradaya} has been used by \\cite{Andersen-Exposure}, \\cite{Justin} and \\cite{Dahlgren} in the context of initial margin estimation.\n\n\\medskip\n\\noindent In principle, Naradaya-Watson proposed an algorithm to approximate the empirical joint density distribution of two random variables. This is, instead of estimating a conditional expectation directly, the method estimates the entire joint pdf distribution,from which the conditional expectation is derived. The method calculates a locally weighted average  of the regressors given a choice of  a  kernel  function. The weights provide a smoothing feature to the fit as they decrease smoothly to zero with distance from the given point. The derivation of the Naradaya-Watson pdf estimator can be found in \\cite{KernelReg}. Under this method, the conditional expectation function can be approximated as  \\footnote{\"Kernel\" functions can be seen as data \"filters\" or \"envelopes\", in the sense that they define proximity between points. See \\cite{KernelReg} for a derivation.} \n\n\\begin{equation}\n\\label{eq:m_x_kernel}\n\\mathbb{E}[Y \\arrowvert X=x] = \\frac{\\sum_{i=1}^{N}K_h(x,x_i)y_i}{\\sum_{j=1}^{N}K_h(x,x_j)}\n\\end{equation}\n\n\\medskip\n\\noindent where $h$ is the bandwidth parameter that controls the degree of smoothing, $x$ is the centroid, $x_i, y_i$ are elements of the vectors $Y_t, X_t$ and $K(.)$ is a Kernel function satisfying:  :\n\n\n\\begin{enumerate}\n\t\\item symmetry: $k(-u)=k(u)$\n\t\\item positive semi-definiteness: $k(u) \\geq 0 \\, \\forall u$\n\t\\item $\\int_{-\\infty}^{+\\infty}k(u)du=1$\n\\end{enumerate}\n\n\\medskip\n\\noindent While equation \\ref{eq:m_x_kernel} is non-linear for the regressors, it can be shown that  the NW estimator is a particular case of the local estimation problem. The function above can be re-expressed as being \\textit{locally linear} (i.e. piece-wise linear) on a certain set of $\\beta$ parameters. Moreover, it will be shown that the conditional expectation estimator proposed by kernel regressions can be framed as a particular case of the orthogonal projection problem in equation \\ref{eq:orthoProj} with a modified objective function and a weighted norm. \n\n\\medskip\n\\noindent The local minimization problem works by setting a small neighborhood of a given centroid $x$ and assuming a parametric function within the selected region. The process is equivalent to partitioning the support of the regressors space producing smooth step functions. For every time step $t$ of the Monte Carlo simulation, consider a neighborhood of $x$ and assume a local functional relationship $m(.)$ between the $Y, X$ variables. If $m(.)$ is continuous and smooth, it should be somewhat constant over a small enough neighborhood (by Taylor expansion). While linear regressions assume a linear relationship between variables and minimize a global sum of square errors, kernel regressions assume a linear relationship only on a small neighborhood, focusing on the modeling of pice-wise areas of the observed variables, and repeating the procedure for all centroids. The trick is to determinze the size of the neighborhood to trade off error bias with model variance. The local minimization problem is now as follows:\n\n\\medskip\n\\noindent Given a random vector $Y_t \\in \\mathcal{H}$ and a deterministic vector $X_t  \\subset \\mathcal{H}$, and given the functions $K_h(\\cdot)$ and $m(\\cdot)$, find $\\hat{\\beta}$ such that:\n\n\\begin{equation*}\n\\label{eq:minKernel}\n\\underset{\\hat{\\beta}}{inf} \\; \\mathbb{E}[K_h(X_t, Y_{t+\\delta}).(Y_{t+\\delta} - m(X_t,\\hat{\\beta}))^2]\n\\end{equation*}\n\n\n\\medskip\n\\noindent In the local minimization problem, the model for $m(x)$ is a linear expansion of basis functions as in equation \\ref{eq:beta_linFunct}.\n\n\\begin{equation*}\nm(X_t,\\hat{\\beta}) = \\sum_{i=0}^{P} \\hat{\\beta}_i.\\phi_i(X_t)\n\\end{equation*}\n\n\\medskip\n\\noindent Let $p$ be the order of the $\\phi(X_t)$ polynomial, some examples of $m(.)$ are:\n\n\\begin{itemize}\n\t\\item if p = 0, then $m(X_t,\\hat{\\beta}) = \\beta_0$, i.e. the constant function (i.e. intercept only). This results in the NW estimate in equation \\ref{eq:m_x_kernel}.\n\t\\item if $p =1$ then $m(X_t,\\hat{\\beta}) = \\beta_0 + \\beta_1 (x_i -x)$ with both $x_i$ and the centroid $x$ $\\in X_t$   gives a local linear regression.\n\\end{itemize}\n\n\\medskip\n\\noindent The $\\hat{\\beta}$ are solved by weighted least squares. As with the linear case, the vector $W(Y_{t+\\delta} - m(X_t,\\hat{\\beta}))$ should be orthogonal to all other vectors of X, but now the relevant projection is orthogonal with respect to this new inner product: \n\n\\begin{equation}\n\\langle W(Y_{t+\\delta} - m(X_t,\\hat{\\beta})), m(X_t,\\hat{\\beta})\\rangle\n\\end{equation}\n\n\\medskip\n\\noindent where $W$ is a sequence of weights $ W(X)=\\operatorname {diag}\\left(K_{{h_{{\\lambda }}}}(X,X_{i})\\right)_{{N\\times N}}$.\n\n\n\\medskip\n\\noindent The solution is given by:\n\n\\begin{equation}\n\\hat{\\beta}= \\left( X^T W X \\right)^{-1}\\left( X^T W Y \\right)\n\\end{equation}\n\n\\medskip\n\\noindent We can obtain the same expression for $m(x)$ as in equation \\ref{eq:m_x_kernel}: \n\\begin{equation*} \nm(X_t) = \\beta_0 = \\left [ \\left( X^T W X \\right)^{-1} X^T W \\right ]Y  = \\sum_{i=1}^{N}w_iy_i\n\\end{equation*}\n\n\\medskip\n\\noindent The task is now to choose the $K_h(.)$ function. One common choice in the literature are the radial basis functions, in which each basis function depends only on the magnitude of the radial distance (often Euclidean) from a centroid $x_0 \\in X$. The most cited in the literature is the Gaussian kernel: \n\n\\begin{equation*}\nk(x,x_0) = \\exp\\left (\\frac{-\\parallel x-x_0 \\parallel}{2 \\sigma^2}\\right)\n\\end{equation*}\n\n\n\\subsubsection{Deep Neural Networks}\n\n\\noindent Deep Neural Networks (DNN) have been proposed to estimate initial margin by \\cite{Hernandez} and \\cite{DeepLearning}. Deep NN is an umbrella term for different related methodologies, the focus of this paper is on methods proposed on the initial margin literature, in particular the \\textit{feed-forward neural network} as implemented in \\cite{DeepLearning}.\n\n\\medskip \n\\noindent A DNN can be thought of as a nonlinear generalization of a linear model, where the parameters (called weights) have to be \"learned\" from the data. The idea is to create a nonlinear function from a stack of linear functions. In a feed-forward DNN, each layer projects onto the next layer of \\textit{neurons} (computational nodes). Each linear regression is called a \\textit{neuron}. Each neuron takes outputs from the previous layer as the input by employing linear transformation with weights and bias, and then applies an activation function to the result. In order to find the weights for each layer, the total sum of errors between the final output and the real observation  for each example as in equation \\ref{eq:orthoProj} is minimized by performing an orthogonal projection of the dependent variables onto a non-linear subspace.\n\n\\medskip \n\\noindent The theoretical basis for the approximation of a function by neural networks is given by the Universal Approximation Theorem (UAT) \\cite{Sodhi-DL}. This theorem states that any continuous function in a compact subset of $\\mathcal{R}^n$ can be approximated by a feed-forward network with a single hidden layer containing a finite number of neurons. However, in practice we might need more than one layer to improve the approximation. Recently, in \\cite{UAT+},  it has been shown that any continuous function on the $d_{in}$-dimensional unit cube can be approximated to arbitrary precision by DNN with ReLU activations on hidden layers with width of exactly  $d_{in}+1$. ReLU actiovation function is ReLU(x) $= \\max(0,x)$. The activation function for the output layer is linear.\n\n\\medskip \n\\noindent Following \\cite{DeepLearning} and \\cite{UAT+}, let's assume that in the feed-forward neural network there are an input layer, L hidden layers and an output layer. An example architecture diagram for this general neural network is provided in figure \\ref{fig:nn}.\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{NN1.png}\n\t\\caption{Feed-forward neural network architecture}\n\t\\label{fig:nn}\n\\end{figure}\n\n\\noindent Then the functional form for the approximation of the conditional expectation function is given by\n\n\\begin{equation*}\nm(X_t,W,b) = \\hat{Y_t} =  \\sum_{i=1}^{N_L} W_L^iA_L^i=\\sum_{i=1}^{N_L} W_L^i \\cdot\\phi_L\\left(\\sum_{j=1}^{N_{L-1}}W_{L-1}^jA_{L-1}^j+b_{L}^i\\right)\n\\end{equation*}\n\n\\medskip \n\\noindent where for each hidden layer l, $l=1..L$,  $\\phi_l$ is an activation function, $A^i_l$ is an activation output for the $i^{th}$ neuron, $W_l$ are weights and $b_l$ is the bias. Here, without loss of generality and for the ease of comparison, bias for the output layer, $b_{L+1}$ is assumed to be incorporated into the $W_L$. Note, that weights $W_0$ are applied to the input layer, $X_t$, and $A_0 = X_t$. The authors in \\cite{DeepLearning} use the classic ReLU function as the activation function for all hidden layers, $\\phi_l$. As in \\cite{UAT+}, activation function for the output layer, $\\phi_{L+1}$, is linear. \n\n\\medskip \n\\noindent  Except for the input layer, each node $i$ in the layer $l$ is a neuron (function) that performs a simple computation: $A_l^i=\\phi_l(\\sum_j W_{l-1}^jA_{l-1}^j +b_l^i)$, where $A_l^i$ is the output and $A_{l-1}^j$ is the $j^{th}$ input.\n\n\n\\subsubsection{Summary}\nThis section summarises theoretical formulations of the three approaches considered in Sections 5.6.1-5.6.3 and provides advantages and disadvantages from the practitioners' point of view in the table below.\n\\begin{center}\n\\begin{tabular}{|p{1.5cm}|p{3cm}|p{6cm}| }\n\\hline\n \\textbf{Method} & \\textbf{$m(X_t,\\hat{\\beta})$}&\\textbf{Pros and Cons}  \\\\ \n \\hline\n Linear  Maps&\\begin{tabular}{@{}p{3cm}@{}} \\\\ $\\sum_{i=0}^N \\hat{\\beta_i}\\cdot \\phi_i(X_t)$\\\\  \\\\$\\hat{\\beta}\\in R^N$ \\\\ \\\\$\\phi_i$ -  linear basis functions\\\\ \\\\ N - degree of polynomial \\end{tabular}& \\begin{tabular}{@{}p{6cm}@{}} {\\color{green}+} Same few calibrated parameters are used across all simulation scenarios and time steps.  \\\\\n {\\color{green}+} Computationally cheap compared to the two other methods.\\\\\n {\\color{red}-} Large oscillations in the calibrated parameters can be observed due to daily change in IM value at $t_0$.\\\\\n {\\color{red}-} In a distributed grid setup small simulation run is required to pre-compute $\\hat{\\beta_i}$ prior to the main MC simulation.  \\end{tabular}\\\\\n\\hline  \nKernel   Regressions (NW) &\\begin{tabular}{@{}p{3.1cm}@{}} \\\\ $\\sum_{i=0}^N \\hat{\\beta_i}\\cdot \\phi_i(X_t)$\\\\ \\\\$\\hat{\\beta_i}=\\frac{K_h(x_0,x_i)}{\\sum_{j=1}^{N}K_h(x_0,x_j)}$ \\\\  \\\\$\\phi_i= y_i$ \\\\  \\\\N - number of neighbourhood pairs ($x_i, y_i$) \\\\ \\\\ $K_h$ - Kernel function \\end{tabular} & \\begin{tabular}{@{}p{6cm}@{}}\n{\\color{green}+} Computationally cheap \\\\\n {\\color{green}+} Better empirical fit to data than linear regression. \\\\\n {\\color{green}+} Ensures functional smoothness. \\\\\n {\\color{red}-} Assumes parametric functional form.\\\\\n {\\color{red}-} It could be challenging to obtain sensitivities to model parameters and inputs.   \n \\end{tabular}\\\\\n\\hline  \nFeed-Forward Neural   Networks &\\begin{tabular}{@{}p{3cm}@{}} \\\\ $\\sum_{i=0}^N \\hat{\\beta_i}\\cdot \\phi_i(X_t)$\\\\  \\\\$\\hat{\\beta_i} = W^i_L$, weight applied to the output of the neuron i in the final hidden layer\\\\ \\\\ $\\phi_i(X_t) = A_L^i$ - activation of the neuron i in the final hidden layer   \\\\ \\\\N - number of neurons in the final hidden layer (plus 1 if incorporating bias) \\end{tabular}&  \\begin{tabular}{@{}p{6cm}@{}} {\\color{green}+} Once trained, predicted IM can be calculated extremely fast because it only involves small scale matrix multiplication and is done at the portfolio level.\\\\\n {\\color{green}+} $\\hat{\\beta_i}$ only need to be updated every quarter compared to daily in linear maps method. \\\\\n {\\color{red}-} Training requires portfolio sensitivities for each MC path and time-step per SIMM model requirements. \\\\\n {\\color{red}-} Data generation for training is computationally expensive and needs to be done offline.  \\end{tabular}\\\\\n\\hline  \n\\end{tabular}\n\\end{center}\n\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{s:int}\n\nLet $\\Gamma$ be a profinite group isomorphic to $\\mathbb{Z}_p$ and let $\\Lambda$ denote the completed group algebra $\\mathbb{Z}_p[[\\Gamma]]$. By a general structure theorem, if $M$ is a finitely generated torsion $\\Lambda$-module then one has a pseudo-isomorphism (that is, a homomorphism with finite kernel and cokernel)\n\\begin{equation} \\label{e:ps0} M\\,\\sim\\,\\bigoplus_i\\Lambda\/(p^{m_i})\\oplus\\bigoplus_j\\Lambda\/(f_j) \\end{equation}\n(where the $f_j$'s can be identified with certain irreducible polynomials after choosing a topological generator of $\\Gamma$ and hence fixing an isomorphism between $\\Lambda$ and the power series ring $\\mathbb{Z}_p[[t]]$). In particular \\eqref{e:ps0} makes it possible to define the number\n$$\\mu(M):=\\sum_im_i\\;,$$\nwhich is called the $\\mu$-invariant of $M$. The algebraic side of Iwasawa theory deals with modules over rings exemplified by $\\Lambda$ and understanding their $\\mu$-invariant has always been an interesting question. For example, in the setting above $\\mu(M)=0$ means that $M$ is finitely generated as a $\\mathbb{Z}_p$-module. More generally, let $M_{\\Gamma^{p^n}}$ denote the set of coinvariants for the subgroup of index $p^n$. Then \\eqref{e:ps0} implies the formula\n$$\\frac{\\log|M_{\\Gamma^{p^n}}|}{p^n\\log p}=\\mu(M)+o(1)\\,\\;\\;\\text{ as }n\\rightarrow\\infty\\,,$$\nprovided that $|M_{\\Gamma^{p^n}}|<\\infty$ holds for every $n$.\n\nThe $\\mu$-invariant was first studied in the following situation. Let $F$ be a number field and let $F^{cyc}$ denote its $\\mathbb{Z}_p$-cyclotomic extension: then one takes $\\Gal(F^{cyc}\/F)$ as $\\Gamma$ and $\\mathbb{Z}_p[[\\Gal(F^{cyc}\/F)]]$ as $\\Lambda$. The $\\Lambda$-module to be considered is $M(F^{cyc}\/F)$, the inverse limit of the $p$-parts of class groups in the intermediate extensions of $F^{cyc}\/F$. Iwasawa showed that $M(F^{cyc}\/F)$ is finitely generated and torsion and he conjectured the equality $\\mu\\big(M(F^{cyc}\/F)\\big)=0$. His conjecture was proved by Ferrero-Washington in the case when $F\/\\mathbb{Q}$ is an abelian extension, but it is still open in the general setting. On the other hand, Iwasawa \\cite{Iwa73} also gave examples of $\\mathbb{Z}_p$-extensions $L\/F$ of number fields such that $\\mu\\big(M(L\/F)\\big)>0$.\n\nThe $\\Lambda$-modules of interest in this paper are duals of Selmer groups of abelian varieties over function fields. The Iwasawa theory of abelian varieties was started by Mazur in \\cite{Maz72}, where he already gave some examples of elliptic curves with non-vanishing $\\mu$ over $\\mathbb{Q}^{cyc}$ (\\cite[\\S10]{Maz72}). More recently Coates and Sujatha have conjectured that the $\\mu$-invariant of an elliptic curve always vanishes in the cyclotomic extension of a number field if one replaces the Selmer group with the fine Selmer group (\\cite[Conjecture A]{CS05}; see also \\cite{Suj10} and \\cite{Suj11} for an introduction to these ideas).\n\n\\subsection{Our setting: $\\mathbb{Z}_p$-extensions of function fields} In the following, $K$ is a function field in one variable over a finite field of characteristic $p$. The $\\mathbb{Z}_p$-extension of $K$ we are going to consider is the arithmetic one, that is, the unique $\\mathbb{Z}_p$-extension obtained by enlarging the field of constants $\\mathbb{F}_{q}$, which is an obvious analogue of $F^{cyc}\/F$.\n\n\\subsubsection{$\\mu$-invariants over function fields: class groups} In  this paper, we study the $\\mu$-invariant attached to abelian varieties. Before discussing it, we review the situation for class groups, to place our results better in context. Let $h_n$ denote the cardinality of the $p$-part of the class group at the $n$-th layer of the extension. In the case of the arithmetic $\\mathbb{Z}_p$-extension, the growth of $h_n$ can be computed from the zeta function and Weil's work on the Riemann hypothesis gives some estimation. Interpreting class groups as sets of rational points in a Jacobian, geometric considerations and some simple Galois theory yield a more precise result: see \\cite{Ros73}. The outcome is that $\\log(h_n)$ is proportional to $n$, for $n$ large enough: that is, the relevant $\\mu$-invariant vanishes (\\cite[Proposition 5.4]{LZ97}). (It might be worth to recall that similar observations and the attempt to develop an analogue theory for number fields were Iwasawa's starting point: see \\cite{Tha94} for more on this function field-number field dialogue.)\n\nThings change rather dramatically if instead one takes a geometric $\\mathbb{Z}_p$-extension of $K$ - that is, an extension where the constant field is the same at any layer. Such extensions can ramify at infinitely many different places and in such a situation the limit of class groups is not a noetherian $\\Lambda$-module (\\cite[Theorem 2]{GK88}). Moreover, it turns out that in a geometric $\\mathbb{Z}_p$-extension $\\log(h_n)$ grows at least as $p^{2n}$ (\\cite[Theorem 1]{GK88}).\n\n\\begin{remark} In the function field setting there is also a different kind of ``cyclotomic'' extension, obtained by adding to $K$ the $\\mathfrak p^\\infty$-torsion of a rank 1 Drinfeld module, for $\\mathfrak p$ a place of $K$. It turns out that when $K$ is the rational function field (that is, $K=\\mathbb{F}_q(\\theta)$ ) and the cyclotomic extension is generated through the Carlitz module, an analogue of the Ferrero-Washington theorem holds (\\cite[Theorem 1.3]{ABBL20}). An important difference with the situation we discussed before is that the Galois group of the extension in \\cite{ABBL20} is isomorphic to an infinite product of copies of $\\mathbb{Z}_p$: hence the corresponding Iwasawa algebra is not even noetherian. A general theory for modules over such algebras is still lacking, so it is not completely clear what is the meaning (or the general definition) of a $\\mu$-invariant in this setting. The result in \\cite{ABBL20} consists in proving that a certain $p$-adic $L$-functions generates the ($\\chi$-part of the) Fitting ideal of a limit of class groups, and computing that the reduction of this $L$-function modulo $p$ is not zero. \\end{remark}\n\n\\subsubsection{$\\mu$-invariants over function fields: abelian varieties} For an abelian variety $A\/K$, Iwasawa theory over the arithmetic extension of $K$ was first developed in \\cite{OT09}. In particular, \\cite[Theorem 1.7]{OT09} proves that the dual of the Selmer group is a finitely generated torsion $\\Lambda$-module: thus one can apply \\eqref{e:ps0} and define a $\\mu$-invariant $\\mu_{A\/K}$. Moreover, \\cite[Theorem 1.8]{OT09} gives some conditions for its vanishing when $A$ is isotrivial: more precisely, \\begin{itemize}\n\\item $\\mu_{A\/K}=0$ always holds if, after base change by a finite extension, $A$ becomes isomorphic to a constant ordinary abelian variety;\n\\item in the supersingular case, $\\mu_{A\/K}=0$ is true if $A$ becomes constant over a finite extension of $K$ with invertible Hasse-Witt matrix. \\footnote{In \\cite[Theorem 1.8]{OT09}, this is given as an ``if and only if''. However, that is wrong: we discuss it in \\S\\ref{su:sic}.}  \\end{itemize}\nFinally, \\cite[Theorem 1.9]{OT09} shows an application of $\\mu_{A\/K}=0$ to non-commutative Iwasawa theory.\n\nWe are not aware of other works discussing the $\\mu$-invariants of abelian varieties over global function fields,\nexcept the paper of Ulmer that we will discuss below.\n\n\\subsubsection{Ulmer's notion of the dimension of $\\Sha$}\n\\label{ss:Ulmer}\nOutside the context of Iwasawa theory,\nUlmer \\cite[Section 4]{Ulm19} defines a certain number\ncalled the \\emph{dimension of $\\Sha(A)$} and denoted as $\\dim \\Sha(A)$,\nby looking at the asymptotic behavior of the order of the $p$-primary part\nof the Tate-Shafarevich group $\\Sha(A)$ of an abelian variety $A \/ K$\nunder extensions of the constant field $\\mathbb{F}_{q}$,\nassuming the finiteness of Tate-Shafarevich groups.\nHis aim is to study the so-called Brauer-Siegel problem for abelian varieties over function fields\nusing $\\dim \\Sha(A)$.\nThe relevance of his work to our context of Iwasawa theory\nlies in his suggestion \\cite[Remark 4.3 (5)]{Ulm19} of the potential relation\nbetween $\\dim \\Sha(A)$ and the $\\mu$-invariant $\\mu_{A \/ K}$.\n\nBy the result of Kato-Trihan \\cite{KT03},\nthe assumption of the finiteness of Tate-Shafarevich groups implies\nthe Birch and Swinnerton-Dyer formula for the leading coefficient\nof the Hasse-Weil $L$-function of $A \/ K$ at $s = 1$.\nUsing this formula, Ulmer \\cite[Proposition 4.2]{Ulm19} gives a formula for $\\dim \\Sha(A)$\nin terms of various other quantities including the degree of the Hodge bundle of the N\\'eron model of $A$\n(the function field analogue of the Faltings height)\nand slopes ($p$-adic valuations of the roots) of the numerator of the $L$-function of $A \/ K$.\nFrom this formula, he deduces that the limit defining $\\dim \\Sha(A)$ exists\n(as a real number a priori!)\\ and is an integer.\nHowever, there is a gap in the proof of this final integrality result,\nsince the corresponding integrality property of the quantity involving slopes is not properly justified there.\nIt only shows that $e \\cdot \\dim \\Sha(A)$ is an integer,\nwhere $q = p^{e}$.\nWe will see (and solve) this problem in the appendix of this paper.\n\nIn \\cite[Remark 4.3 (2)]{Ulm19}, Ulmer justifies the terminology ``dimension of $\\Sha$''\nin the special case where $A \/ K$ is the Jacobian of a proper smooth curve over $K$.\nIn this case, its proper flat regular model gives a surface $\\mathcal{S}$ over $\\mathbb{F}_{q}$.\nHe relates the definition of $\\dim \\Sha(A)$\nto the genuine dimension of the actual group scheme $\\underline{\\coh}^{2}(p^{\\infty})$ over $\\mathbb{F}_{q}$\ndefined by Artin \\cite{Art74b} and Milne \\cite{Mil76},\nwhich is the canonical group scheme structure on the second flat cohomology of $\\mathcal{S}$.\nUlmer \\cite[Remark 4.3 (3), (4)]{Ulm19} then suggests a relation\nbetween his formula for $\\dim \\Sha(A)$ for the Jacobian $A \/ K$\nand Milne's formula \\cite[the last equation of Section 6]{Mil75}\non Euler characteristics for the surface $\\mathcal{S}$,\nstill assuming the finiteness of Tate-Shafarevich groups.\n\nThis relation between Ulmer's work and Artin-Milne's work\ndoes not easily generalize to arbitrary abelian varieties over $K$,\nsince an arbitrary abelian variety is not necessarily a product factor\nbut only an isogeny factor of a Jacobian,\nand the degree of the needed isogeny might be a power of $p$ that kills\nthe (unipotent) identity component of $\\underline{\\coh}^{2}(p^{\\infty})$.\n\nBy \\cite{Gro68}, the finiteness of $\\Sha(A)$ for the Jacobian $A$ is equivalent\nto the finiteness of the Brauer group of $\\mathcal{S}$,\nthe latter of which is the Tate conjecture for divisors on $\\mathcal{S}$.\nThis conjecture is verified for several surfaces including the ones treated in \\cite{Ulm19}.\nIn general, it is a hard conjecture of motivic origin.\n\nNotice that the group scheme $\\underline{\\coh}^{2}(p^{\\infty})$ for a surface $\\mathcal{S}$ and\nthe $\\mu$-invariant $\\mu_{A \/ K}$ for an arbitrary abelian variety $A$ are unconditionally defined.\nThe general philosophy here is that\nin Iwasawa theory and other purely $p$-adic theories, we can do much more\n``without hard motivic conjectures'' (at least in the function field case).\n\n\n\\subsection{Our results: contents of this paper}\nWe assume that the field of constants of our $K$ contains $q$ elements. The same $q$ appears in our definition of the $\\mu$-invariant $\\mu_{A\/K}$, which is slightly different from the usual one (see Definition \\ref{d:mu}). In particular, it is not a priori obvious that $\\mu_{A\/K}$ is always an integer: this will be a consequence of Theorem \\ref{muissA}.\n\nFirst, we study the behavior with respect to base change (\\S\\ref{sub:bc}). We check that, if $L\/K$ is a finite Galois extension, we have $\\mu_{A\/K}\\leqslant\\mu_{A\/L}$ (Lemma \\ref{l:kl}) and we give conditions ensuring that $\\mu_{A\/K}=0$ implies $\\mu_{A\/L}=0$: in particular, we prove that this implication holds in a $p$-extension unramified outside of ordinary places (Theorem \\ref{t:ocht}).\n\nIn \\S\\ref{su:ts}, we observe that $\\mu_{A\/K}$ is determined by the Tate-Shafarevich group of $A$ over the unramified $\\mathbb{Z}_{p}$-extension. As a consequence, one can compute it from the cardinality of the $p^m$-torsion of $\\Sha$ in intermediate extensions.\nThis is a non-trivial step: the crucial fact here is the control theorem for Selmer groups \\cite[Theorem 4]{Tan10},\nwhich shows how to obtain information about the Selmer groups for $A$ over the intermediate extensions\nfrom the Selmer group for $A$ over the $\\mathbb{Z}_{p}$-extension.\nAssuming that the $p$-part of $\\Sha$ is always finite in our $\\mathbb{Z}_p$-tower, we deduce the limit formula  \\eqref{e:shafni} for $\\mu_{A\/K}$ by a somewhat technical argument (coming from the difference between $\\Sha$ and $\\Sel$).\nThis shows that Ulmer's $\\dim \\Sha(A)$ is equal to $\\mu_{A\/K}$, as suggested in \\cite[Remark 4.3 (5)]{Ulm19}.\n\nIn \\S\\ref{su:su}, following \\cite{Suz19}, we define an $\\mathbb{F}_q$-group scheme $\\mathcal{G}_A^{(1)}$ which represents the Tate-Shafarevich group $\\Sha(A \/ K \\bar{\\mathbb{F}}_{q})$.%\n\\footnote{This representation is a bit subtle.\nSee the first paragraph of \\S\\ref{su:su} for the precise meaning.\nThe subtleties exist but are ``controllable'',\nwhich is the content of the control theorem for $\\Sha$,\nnamely Proposition \\ref{p:ShaGone}.}\nWe prove the formula\n$$\\mu_{A\/K}=\\dim\\mathcal{G}_A^{(1)}$$ \n(Theorem \\ref{muissA}).\nWith $\\dim \\Sha(A) = \\mu_{A \/ K}$ (assuming the finiteness of Tate-Shafarevich groups),\nthis formula justifies the terminology ``dimension of $\\Sha$'' for an arbitrary abelian variety $A \/ K$.\nIt also shows that $\\mu_{A \/ K}$ (or $\\dim \\Sha(A)$) is an integer.\nUsing the group scheme $\\mathcal{G}_{A}^{(1)}$\n(and the related group schemes $\\mathcal{G}_{A}^{(i)}$),\nwe prove a control theorem for $\\Sha$ (Proposition \\ref{p:ShaGone}),\nwhich is not a consequence of the control theorem for $\\Sel$ mentioned earlier.\nThis gives another more direct proof of $\\mu_{A \/ K} = \\dim \\Sha(A)$.\n\nIn \\eqref{e:muform}, we reinterpret the formula in \\cite[Proposition 4.2]{Ulm19} on $\\dim \\Sha(A)$\n(which assumes the finiteness of Tate-Shafarevich groups and uses the BSD formula) as a formula on $\\mu_{A \/ K}$\nand hence give an upper bound \\eqref{e:upperbound} for $\\mu_{A\/K}$.\nSince both of the sides of the resulting $\\mu$-invariant formula \\eqref{e:muform} are unconditionally defined,\nit is then natural to ask if the formula \\eqref{e:muform} can be proved unconditionally.\nWe will give partial results in subsequent chapters.\n\nIn Chapter \\ref{s:JacSur}, we take $A$ to be the Jacobian of a curve $\\mathcal{S}_{K}$ over $K$. In this case, we define an $\\mathbb{F}_q$-group scheme $\\underline{\\Br}$ representing the Brauer group of the associated surface $\\mathcal{S}$ over $\\mathbb{F}_{q}$ using Artin-Milne's group scheme $\\underline{\\coh}^{2}(p^{\\infty})$. We show that the group schemes $\\mathcal{G}_{A}^{(1)}$ and $\\underline{\\Br}$ are isomorphic up to finite \\'etale groups (Proposition \\ref{p:ShaBr}). This implies that $\\mu_{A\/K} = \\dim \\underline{\\Br}$. As a consequence, we can show that in this case the formula \\eqref{e:muform} is equivalent to Milne's formula \\cite[the last equation of Section 6]{Mil75} (Corollary \\ref{c:MilUlm}), as suggested by Ulmer \\cite[Remark 4.3 (3), (4)]{Ulm19}. This proves \\eqref{e:muform} independently of the finiteness of $\\Sha$. As an application, in Propositions \\ref{p:trivmu} and \\ref{p:aborK3}, we give necessary and sufficient conditions for $\\mu_{A\/K}=0$ and (under some additional assumptions) $\\mu_{A\/K}=1$. Proposition \\ref{p:Shio} provides an explicit example of an elliptic curve with $\\mu$-invariant 1.\n\nChapter \\ref{s:ssav} deals with the case of semistable abelian varieties. Theorem \\ref{t:muss} shows that formula \\eqref{e:muform} holds with no condition on $\\Sha$ also in this setting. The proof is based on the fact that in this case the Iwasawa Main Conjecture holds (\\cite{LLTT16}) and so $\\mu_{A\/K}$ can be computed explicitly from the ($p$-adic) $L$-function.\n\nSummarizing the previous three chapters,\nthe $\\mu$-invariant formula \\eqref{e:muform} is true for the following three cases:\n\\begin{itemize}\n\t\\item if the Tate-Shafarevich groups are finite (by Ulmer),\n\t\\item if $A$ is a Jacobian, or\n\t\\item if $A$ is semistable everywhere.\n\\end{itemize}\n\nIn Chapter \\ref{s:muelliptic}, we specialize to $A$ being an elliptic curve. As explained in Theorem \\ref{t:tan}, this gives a simplified version of \\eqref{e:muform} and \\eqref{e:upperbound} and hence more instances of $\\mu_{A\/K}>0$ (Proposition \\ref{p:KodOne}). In \\S\\ref{su:isogen}, we investigate the variation of the $\\mu$-invariant in isogeny classes. If $p>2$, the Legendre form of the Weierstrass equation provides sufficient conditions for $\\mu_{A\/K}=0$ (Theorem \\ref{t:Legendretype}).\n\nIn Chapter \\ref{su:generic}, we prove that, at least for $p>3$, the vanishing of $\\mu_{A\/K}$ is the normal behavior in the following sense: for $n>\\frac{1}{2}(g-1)$ (where $g$ is the genus of the curve corresponding to $K$), we build an irreducible variety $X(n,\\bar{\\mathcal C})$ parameterizing elliptic curves over $K$ with discriminant of degree $12n$ and we show that the locus determined by $\\mu=0$ is open and dense (Theorem \\ref{t:GenVan}).\n\nFinally, Chapter \\ref{s:verify} contains some explicit examples where we can compute the $\\mu$-invariant.\nWith Theorem \\ref{t:tan} and Magma \\cite{BCP97},\nwe can routinely and quickly calculate the $\\mu$-invariant of\n(a quadratic base change of) an elliptic curve over a rational function field\nof small characteristic with mild complexity.\nWe demonstrate this in \\S \\ref{su:high} and \\S \\ref{su:nonss} among other examples.\n\nThe paper ends with an appendix explaining the gap in the literature mentioned in Section \\ref{ss:Ulmer}\nand proving a certain integrality property\nof the exact $p$-power factor of the $L$-function of $A$.\nThis shows that the slope term in the formula \\eqref{e:muform} is an integer.\nThis is not a consequence of the result $\\mu_{A \/ K} \\in \\mathbb{Z}$ in Theorem \\ref{muissA}\nsince \\eqref{e:muform} has not been verified unconditionally for all abelian varieties.\nThese two types of integrality should be considered as separate results.\n\n\\subsection*{Acknowledgments}\nThe authors are grateful to\nLuc Illusie, Bruno Kahn, Masato Kurihara, Kentaro Mitsui, Yukiyoshi Nakkajima,\nTadashi Ochiai, Atsushi Shiho, Douglas Ulmer and Christian Wuthrich\nfor helpful discussions.\nMany thanks also to the referee for the careful reading and the comments, which helped to significantly improve the paper.\n\n\\subsection{Notation}\\label{su:notation}\nLet $p>0$ be a prime number. Let $K$ be a function field in one variable with field of constants $\\mathbb{F}_q$, $q=p^e$. Write $\\mathbb{F}_{q,n}$, $\\mathbb{F}_{q,\\infty}$, $K_n^{(p)}$ and $K_\\infty^{(p)}$, for\n$\\mathbb{F}_{q^{p^n}}$, $\\bigcup_n \\mathbb{F}_{q,n}$, $K\\mathbb{F}_{q,n}$ and $K\\mathbb{F}_{q,\\infty}$ so that $K_n^{(p)}$ is the $n$th layer of $K_\\infty^{(p)}\/K$.\nDenote $\\Gamma:=\\Gal(K_\\infty^{(p)}\/K)$ and $\\Lambda:=\\mathbb{Z}_p[[\\Gamma]]$.\n\nThe cohomology groups will be usually Galois or \\'etale cohomology,\nexcept for cohomology of finite group schemes where we use the flat cohomology, denoted $\\coh^*_{fl}$ (see \\cite[II, \\S 1 and III, Definition 1.5]{Mil80} for its definition and \\cite[III]{Mil06} for the facts needed in this paper).\n\nLet $\\mathcal C$ be the smooth projective curve over $\\mathbb{F}_q$ having $K$ as its function field. Write $g_{\\mathcal C}$ for\nthe genus of $\\mathcal C$.  Let $A\/K$ be an abelian variety.\nFor an abelian group $H$, we denote its $p^n$-torsion subgroup by $H[p^n]$,\nincluding $n = \\infty$ (so that $H[p^{\\infty}]$ means the $p$-primary torsion part).\nHowever we let $A_{p^n}$ denote the kernel of the multiplication by $p^n$ on $A$\nand put $A_{p^\\infty}:=\\bigcup_n A_{p^n}$.\nFor a field extension $L\/K$, denote the $p^{n}$-Selmer group and the $p^{n}$-Tate-Shafarevich groups ($n=\\infty$ included) by\n$$\\Sel_{p^n}(A\/L):=\\ker (\\coh^1_{fl}(L,A_{p^n})\\longrightarrow \\bigoplus_w \\coh^1(L_w, A)),$$\n$$\\Sha_{p^n}(A\/L):=\\ker (\\coh^1(L,A)[p^{n}]\\longrightarrow \\bigoplus_w \\coh^1(L_w, A)),$$\nwhere $w$ runs through all places of $L$.\nBy dropping ``$[p^{n}]$'' in the latter equation,\nwe obtain the usual definition of the (full) Tate-Shafarevich group $\\Sha(A \/ L)$.\n\nLet ${}^\\vee$ denote the Pontryagin dual.\n\nWe will denote by $\\mathcal{A}$ the N$\\acute{\\text{e}}$ron model of $A$.\nThe $K \/ \\mathbb{F}_{q}$-trace of $A$ (\\cite{Con06}) is denoted by $\\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)$.\n\n\\section{The $\\mu$-invariant for abelian varieties}\\label{s:gen}\nLet $A\/K$ be an abelian variety and let $X_{A\/K}$ be the Pontryagin dual of the Selmer group $\\Sel_{p^\\infty}(A\/K_\\infty^{(p)})$.\nThen $X_{A\/K}$ is finitely generated and torsion over $\\Lambda$ (\\cite[Theorem 1.7]{OT09}).\nFrom a general structure theorem for such modules, there is an exact sequence\n\\begin{equation}\\label{e:x}\n0\\longrightarrow \\bigoplus_{i=1}^k \\Lambda\/(p^{\\mu_i}) \\oplus \\bigoplus_{j=1}^l \\Lambda\/(g_j)\\longrightarrow X_{A\/K}\n\\longrightarrow F\\rightarrow 0,\n\\end{equation}\nwhere $F$ is finite and each $g_j$ corresponds to a power of an irreducible distinguished polynomial in the isomorphism $\\Lambda\\simeq\\mathbb{Z}_p[[t]]$.\nThe following definition is different from the\nconvention, but it suits us well (see Theorem \\ref{muissA}).\n\\begin{definition} \\label{d:mu}\nWe define the $\\mu$-invariant of $X_{A\/K}$ as the non-negative rational number $\\mu_{A\/K}$ such that \n\\begin{equation} \\label{e:dfmu} q^{\\mu_{A\/K}}=\\prod_{i=1}^k p^{\\mu_i}\\,. \\end{equation}\n\\end{definition}\n\nReaders should note that on the left-hand side of \\eqref{e:dfmu} there is a power of $q$ but on the right-hand side there is a product of powers of $p$.\nWe will see in Theorem \\ref{muissA} that $\\mu_{A \/ K}$ is actually an integer.\n\nThe above is not specific to $X_{A \/ K}$.\nFor any finitely generated torsion $\\Lambda$-module $X$ appearing in this paper,\nwe define its $\\mu$-invariant by \\eqref{e:x} and \\eqref{e:dfmu}.\nIt is an integer divided by $e$ in general, where $q = p^{e}$.\n\n\\subsection{Base changes}\\label{sub:bc}\nThe snake lemma applied to the multiplication by $p$ on \\eqref{e:x} yields the exact sequence\n$$F[p]\\longrightarrow \\bigoplus_{i=1}^k\\Lambda\/(p) \\oplus \\bigoplus_{j=1}^l\\Lambda\/(g_j,p)   \\longrightarrow X_{A\/K}\/pX_{A\/K}\\longrightarrow F\/pF.$$\n\n\\begin{lemma}\\label{l:mu=0}\n$\\mu_{A\/K}=0$ if and only if $\\Sel_p(A\/K_\\infty^{(p)})$ is finite.\n\\end{lemma}\n\\begin{proof}\nBecause each $\\Lambda\/(g_j,p)$ is finite, the lemma follows via duality from the above exact sequence.\n\\end{proof}\n\n\\begin{lemma}\\label{l:kl}\nIf $L\/K$ is a finite Galois extension with $\\mu_{A\/L}=0$, then $\\mu_{A\/K}=0$.\n\\end{lemma}\n\\begin{proof}\nThe kernel of the restriction map\n$\\Sel_p(A\/K_\\infty^{(p)})\\longrightarrow \\Sel_p(A\/L_\\infty^{(p)})$ is finite\nsince it is a subgroup of the finite group\n$\\coh^1(L_\\infty^{(p)}\/K_\\infty^{(p)}, A_p(L_\\infty^{(p)}))$.\nBy Lemma \\ref{l:mu=0}, we obtain the result.\n\\end{proof}\n\nFor a Galois extension $L\/K$ with $G:=\\Gal(L_\\infty^{(p)}\/K_\\infty^{(p)})$, denote\n$$\\mathcal{N}_{L\/K}:=\\ker\\,[\\coh^1_{fl}(K_\\infty^{(p)},A_p)\\longrightarrow \\bigoplus_w \\coh^1(K_{\\infty,w}^{(p)},A)\/\\coh^1(G_w,A(L_{\\infty,w}^{(p)}))],$$\nwhere $w$ runs through all places of $K_\\infty^{(p)}$.\n\n\\begin{lemma}\\label{l:pext} Let $L\/K$ be a finite Galois $p$-extension.\nThen $\\mu_{A\/L}=0$, if and only $\\mathcal{N}_{L\/K}$ is finite.\n\\end{lemma}\n\\begin{proof} For simplicity write $M$ for $\\Sel_p(A\/L)$\nand let $M^\\vee$ be the Pontryagin dual.\nLet $I$ be the augmentation ideal of $\\mathbb{F}_p[G]$. If $M$ is infinite, then\\footnote{This should be well known for more general context. Here is a simple proof for our need. As in the commutative case, we  need to show that $M\/IM=0$ implies $M=0$. But this is obvious, since $G$ is a $p$-group, $I^m=0$ for some $m$.} so is $M^\\vee\/IM^\\vee$. Hence\nby duality $M^G$ is also infinite. The exact sequence\n$$\\xymatrix{0\\ar[r] & \\mathcal{N}_{L\/K}\\cap\\coh^1(G,A_p(L)) \\ar[r] & \\mathcal{N}_{L\/K} \\ar[r] & M^G \\ar[r] & \\coh^2(G,A_p(L))}$$\nimplies $\\mathcal{N}_{L\/K}$ is infinite. Conversely, if $\\mathcal{N}_{L\/K}$ is infinite, then so is $M^G$.\n\n\\end{proof}\n\n\n\\begin{lemma}\\label{l:local}\nLet $L\/K$ be a finite Galois $p$-extension with $G:=\\Gal(L_\\infty^{(p)}\/K_\\infty^{(p)})$. Then $\\mathcal{N}_{L\/K}$ is finite if and only if $\\mu_K=0$ and\n$\\bigoplus_w\\coh^1(G_w,A(L_{\\infty,w}^{(p)}))$ is finite.\n\\end{lemma}\n\\begin{proof} Write $W$ for $\\bigoplus_w\\coh^1(G_w,A(L_{\\infty,w}^{(p)}))$, $W$ is finite if and only if $W[p]$\nis finite. Consider the commutative diagram\n$$\\xymatrix{ & \\coh^1(K_\\infty^{(p)},A) \\ar[r]^-{\\mathscr{L}} & \\bigoplus_w \\coh^1(K_{\\infty,w}^{(p)},A)\\\\\n\\Sel_p(A\/K_\\infty^{(p)}) \\ar@{^{(}->}[r] & \\mathcal{N}_{L\/K} \\ar[r]^-{\\mathscr{L}_p} \\ar[u] & W[p]\\ar[u],}$$\nwhere $\\mathscr{L}$ and $\\mathscr{L}_p$ are localization maps.\nBy using the (generalized) Cassels-Tate exact sequence \\cite{GAT07}, one can deduce (see \\cite[Proposition 4.2]{Tan14})\nthat $\\mathscr L$ has co-kernel of finite co-rank. Hence the co-kernel of\n$\\mathscr{L}_p$ is finite.\n\n\\end{proof}\n\n\\begin{theorem}\\label{t:ocht}\nLet $L\/K$ be a finite Galois $p$-extension unramified outside ordinary places of $A\/K$. If $\\mu_{A\/K}=0$, then $\\mu_{A\/L}=0$.\n\\end{theorem}\n\\begin{proof}One can follow the proof of \\cite[Theorem 1.9]{OT09}, or apply Lemmas \\ref{l:kl}, \\ref{l:pext}, \\ref{l:local} and show\nthat  $W:=\\bigoplus_w\\coh^1(G_w,A(L_{\\infty,w}^{(p)}))$ has finite $p$-rank. If $L\/K$ is unramified at $w$, then $\\coh^1(G_w,A(L_{\\infty,w}^{(p)}))$ is finite, and is trivial for good places (see \\cite[Proposition I.3.8]{Mil06}).\nWe may assume that $L\/K$ is cyclic of degree $p$,\nso that by local class field theory, $L_v\/K_v$ is an intermediate extension of some $\\mathbb{Z}_p$-extension of $K_v$.\nBy \\cite[Theorem 2(c)]{Tan10} and the proof of Theorem 4, {\\em{op.\\ cit.}}, if $A$ has good ordinary or split multiplicative reduction\nat $w$, then $\\coh^1(G_w,A(L_{\\infty,w}^{(p)}))$ is finite. If $w$ is a non-split multiplicative place, then there is a constant field extension $K'\/K$\nsuch that $A\/K'_w$ is split multiplicative. Let $L'=K'L$. Then the kernel of the restriction map\n$$\\coh^1(G_w, A(L_{\\infty,w}^{(p)})) \\longrightarrow \\coh^1((L')_{\\infty,w}^{(p)}\/(K')_{\\infty,w}^{(p)}, A((L')_{\\infty,w}^{(p)})),$$\nwhere the target is finite, is contained in $\\coh^1((K')_{\\infty,w}^{(p)}\/K_{\\infty,w}^{(p)}, A((K')_{\\infty,w}^{(p)}))$ which is finite,\nsince $K'\/K$ is unramified.\n\n\\end{proof}\n\n\\subsection{The $L$-function} \\label{s:Lfun}\nWe collect here mostly well-known facts about the Hasse-Weil $L$-function of $A$\nand define several invariants related to it\nas preparations for subsequent sections.\n\nLet $L_{A}(s)$ be the (completed) Hasse-Weil $L$-function of $A$.\nSee for example \\cite[\\S 1]{Sch82} and \\cite[V, \\S 6]{Kah18a}\nfor lists of properties of $L_{A}(s)$.\nWe can write\n\t\\[\n\t\t\tL_{A}(s)\n\t\t=\n\t\t\t\\frac{P_{1}(t)}{P_{0}(t) P_{2}(t)},\n\t\\]\nwhere $P_{i}(t) \\in \\mathbb{Z}[t]$ is a polynomial in $t = q^{-s}$ with constant term $1$\nwhose reciprocal roots (with multiplicities) are Weil $q$-numbers $\\{\\alpha_{ij}\\}$ of weight $i + 1$\n(\\cite[V, 6.8.2]{Kah18a}).\n\n\\begin{proposition}\n\tThe function $P_{0}(t)^{-1}$ (with $t = q^{-s}$) is\n\tthe zeta function of $\\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)$,\n\tand $P_{2}(t) = P_{0}(q t)$\n\\end{proposition}\n\n\\begin{proof}\n\tBy \\cite[Satz 1]{Sch82},\n\t$P_{0}(t \/ q)$ is the characteristic polynomial of $q$-th power Frobenius on\n\t$\\mathrm{V}_{l}(\\mathcal{A}(\\mathcal{C} \\times_{\\mathbb{F}_{q}} \\bar{\\mathbb{F}}_{q}))$,\n\twhere $l$ is a prime different from $p$\n\tand $\\mathrm{V}_{l}$ denotes the $l$-adic Tate module tensored with $\\mathbb{Q}_{l}$.\n\tWe have $\\mathcal{A}(\\mathcal{C} \\times_{\\mathbb{F}_{q}} \\bar{\\mathbb{F}}_{q}) = A(K \\bar{\\mathbb{F}}_{q})$,\n\twhose quotient by the subgroup $\\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)(\\bar{\\mathbb{F}}_{q})$\n\tis a finitely generated abelian group by the Lang-N\\'eron theorem\n\t(\\cite[Theorem 7.1]{Con06}).\n\tHence $\\mathrm{V}_{l} A(K \\bar{\\mathbb{F}}_{q}) \\cong \\mathrm{V}_{l} \\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)(\\bar{\\mathbb{F}}_{q})$.\n\tThis implies the statement for $P_{0}(t)$.\n\tThe functional equation (\\cite[V, 6.8.1, 6.8.2]{Kah18a}) shows that\n\t$L_{A}(2 - s)$ is some exponential function times $L_{A}(s)$.\n\tHence the weight argument shows that $P_{2}(t) = P_0(q t)$.\n\\end{proof}\n\nLet $\\lambda_{j} \\in \\mathbb{Q}$ be the $q$-valuation of $\\alpha_{1 j}$\n(where the $q$-valuation means the $p$-adic valuation\nsuch that the valuation of $q$ is $1$).\n\n\\begin{definition} \\label{d:theta}\n\tWe define $\\theta = \\theta_{A}$ to be the non-negative rational number\n\tsuch that $q^{\\theta} P_{1}(t \/ q)$ is $p$-primitive\n\t(i.e.\\ all the coefficients are $p$-adically integral\n\tand some coefficients are $p$-adic units).\n\t\n\tWe also define $a = a_{A} := \\deg(P_{1})$.\n\\end{definition}\n\n\\begin{proposition} \\label{p:slope}\n\tWe have\n\t\t\\[\n\t\t\t\t\\theta\n\t\t\t=\n\t\t\t\t\\sum_{\\lambda_{j} < 1} (1 - \\lambda_{j})\n\t\t\t\\le\n\t\t\t\t\\frac{a}{2}.\n\t\t\\]\n\\end{proposition}\n\n\\begin{proof}\n\tThe equality can be obtained by a direct calculation or by noting that both sides are the absolute value of the height of the lowest point\n\tof the Newton polygon of the polynomial $P_{1}(t \/ q)$ with respect to the $q$-valuation.\n\t\n\tThe functional equation $L_{A}(s) \\leftrightarrow L_{A}(2 - s)$ (\\cite[V, 6.8.1, 6.8.2]{Kah18a})\n\timplies a functional equation $P_{1}(t) \\leftrightarrow P_{1}(q^{2} \/ t)$\n\tsince $\\{\\alpha_{i j}\\}$ are Weil numbers.\n\tHence the set $\\{\\alpha_{1 j}\\}$ (with multiplicities) is equal to\n\t$\\{q^{2} \/ \\alpha_{1 j}\\}$,\n\tso $\\{\\lambda_{j}\\} = \\{2 - \\lambda_{j}\\}$.\n\tWe have $\\lambda_{j} \\ge 0$ since $\\alpha_{i j}$ are algebraic integers.\n\tTherefore\n\t\t\\[\n\t\t\t\t\\sum_{\\lambda_{j} < 1} (1 - \\lambda_{j})\n\t\t\t\\le\n\t\t\t\t\\sum_{\\lambda_{j} < 1} 1\n\t\t\t\\le\n\t\t\t\t\\frac{1}{2} \\sum_{\\text{all } \\lambda_{j}} 1\n\t\t\t=\n\t\t\t\t\\frac{\\deg(P_{1})}{2}.\n\t\t\\]\n\\end{proof}\n\nLet $N = N_{A \/ K} = \\sum_{v} \\varepsilon_{v}^{l} \\cdot v$ be the global conductor of $A \/ K$,\nwhich is an effective divisor on $\\mathcal{C}$\n(\\cite[II, \\S 3]{Ray95}).\nBy the Ogg-Shafarevich formula \\cite[II, Th\\'eor\\`eme 3 (ii)]{Ray95}, we have\n\t\\begin{equation} \\label{e:OS}\n\t\t\ta\n\t\t=\n\t\t\t\\deg(N) + 4 \\dim(A) (g_{\\mathcal{C}} - 1) + 4 \\dim \\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A).\n\t\\end{equation}\n\n\\begin{remark}\n\tThe number $\\theta$ is actually an integer\n\tas we will see in the appendix.\n\tThis fact will not be used below.\n\\end{remark}\n\n\n\\subsection{The Tate-Shafarevich groups}\\label{su:ts}\nFor a finitely generated torsion Iwasawa module,\nif its Pontryagin dual is $p$-divisible, then its $\\mu$-invariant is zero.\nTherefore, the proposition below follows from the exact sequence\n$$0\\longrightarrow A(K_\\infty^{(p)})\\otimes \\mathbb{Q}_p\/\\mathbb{Z}_p\\longrightarrow \\Sel_{p^\\infty}(A\/K_\\infty^{(p)})\\longrightarrow \\Sha_{p^\\infty}(A\/K_\\infty^{(p)})\\longrightarrow 0.$$\n\n\\begin{proposition}\\label{p:ts}\nThe $\\mu$-invariant of $\\Sha_{p^\\infty}(A\/K_\\infty^{(p)})^\\vee$ equals $\\mu_{A\/K}$.\n\\end{proposition}\n\nNext, using the control theorem \\cite[Theorem 4]{Tan10} for Selmer groups,\nwe will give a formula for $\\mu_{A \/ K}$ in terms of the asymptotic behavior\nof Tate-Shafarevich groups for finite levels $A \/ K_{n}^{(p)}$,\nas suggested by Ulmer \\cite[Remark 4.3 (5)]{Ulm19},\nwith a variant that does not assume the finiteness of Tate-Shafarevich groups.\n\n\\begin{proposition}\\label{p:limint} We have\n\\begin{equation}\\label{e:sha}\n\\mu_{A\/K}=\\lim_{n\\rightarrow \\infty}\\frac{\\log |\\Sha_{p^m}(A\/K_n^{(p)})|}{{p^n}\\log q}\n\\end{equation}\nfor all large $m$.\n\\end{proposition}\n\\begin{proof}\nLet $d_j$ denote the degree of the polynomial $g_j$ in \\eqref{e:x}.\n Let $\\sigma\\in \\Gal(K_\\infty^{(p)}\/K)$ be a topological generator.\nSince $|\\Lambda\/(p^m, g_j)|=p^{md_j}$ and $|\\Lambda\/(p^m, \\sigma^{p^n}-1)|=p^{mp^n}$, for every fixed $m$ one has\n$$\\frac{\\log |\\Sel_{p^m}(A\/K_\\infty^{(p)})^{\\Gamma^{p^n}}|}{{p^n}\\log p}=\\sum_i \\lambda_i+\\mathrm{o}(1),\\; \\text{as}\\; n\\rightarrow\\infty,$$\nwith $\\lambda_i:=\\mathrm{min}\\{m,\\mu_i\\}$. By \\cite[Theorem 4]{Tan10}, we have the exact sequence \n$$\\xymatrix{0\\ar[r] & \\mathrm{Ker}_n\\ar[r] & \\Sel_{p^\\infty}(A\/K_n^{(p)}) \\ar[r] & \\Sel_{p^\\infty}(A\/K_\\infty^{(p)})^{\\Gamma^{p^n}}\n\\ar[r] & \\mathrm{Coker}_n\\ar[r] & 0}$$\nwith $|\\mathrm{Ker}_n|$ and $|\\mathrm{Coker}_n|$ bounded as $n$ varies. For the time being, we call this the control sequence\nof Selmer groups. It implies\n$$\\log |\\Sel_{p^m}(A\/K_\\infty^{(p)})^{\\Gamma^{p^n}}|=\\log |\\Sel_{p^m}(A\/K_n^{(p)})| +\\mathrm{O}(1),\\;\\text{as}\\;n,m\\;\\text{vary}.$$\nThe rank of of $A(K_n^{(p)})$ is bounded. Since $\\frac{\\log p^m}{{p^n}\\log p}=\\mathrm{o}(1)$, the exact sequence\n$$0\\longrightarrow A(K_n^{(p)})\\otimes \\mathbb{Q}_p\/\\mathbb{Z}_p\\longrightarrow \\Sel_{p^\\infty}(A\/K_n^{(p)})\\longrightarrow \\Sha_{p^\\infty}(A\/K_n^{(p)})\\longrightarrow 0$$\nleads to\n$$\\frac{\\log |\\Sha_{p^m}(A\/K_n^{(p)})|}{{p^n}\\log p}=\\sum_i \\lambda_i+\\mathrm{o}(1),\\; \\text{as}\\; n\\rightarrow\\infty.$$\nIf $m\\geq \\mu_i$ for all $i$, then the above tends to $\\sum_i \\mu_i$, as $n\\rightarrow\\infty$. \n\n\\end{proof}\nDenote\n$\\sigma_n:=\\sigma^{p^n}$, $\\omega_n=\\sigma_n-1$ and write $\\omega_n=\\nu_0\\cdot \\nu_1\\cdot\\cdots\\nu_n$, with\n$\\nu_0=\\omega_0$ and $\\nu_i=1+\\sigma_{i-1}+\\sigma_{i-1}^2+\\cdots+\\sigma_{i-1}^{p-1}$, for $i>0$.\nThen $\\nu_0,\\nu_1,\\dots,\\nu_n$ are relatively prime irreducible elements in $\\Lambda$.\n\\begin{lemma}\\label{l:serre}\nLet $g_j$ be as in \\eqref{e:x} and set $V=\\Lambda\/(g_j)$. If $g_j$ is relatively prime to all $\\omega_n$, then $V\/\\omega_nV$ is finite of order\n$p^{n d_j+\\mathrm{O}(1)}$. If $g_j$ is a power of some $\\nu_i$ and $\\omega'_n=\\omega_n\/\\nu_i$, for $n\\geq i$,\nthen $V\/\\omega'_nV$ is finite of order\n$p^{n d_j+\\mathrm{O}(1)}$.\n\\end{lemma}\n\\begin{proof} Since $V$ is a finite free $\\mathbb{Z}_p$-module, if $\\varphi$ is a $\\mathbb{Z}_p$-endomorphism on $V$, then the order\nof $V\/\\varphi(V)$ equals $|\\det \\varphi|_p^{-1}$, where $|\\;\\;|_p$ denote the normalized $p$-adic absolute value.\nThe proof of \\cite[Theorem 8]{Ser95}, Case iii, shows $|\\det \\nu_l|_p^{-1}=p^{d_j}$, for $l$ large enough\n(where Serre denoted our $\\nu_{l}$ by $\\nu_{l - 1}'$).\n\\end{proof}\n\n\\begin{proposition}\\label{p:liminf}\nIf $\\Sha_{p^\\infty}(A\/K_n^{(p)})$ is finite for all $n\\in\\mathbb N$, then \n\\begin{equation}\\label{e:shafni}\n\\mu_{A\/K}=\\lim_{n\\rightarrow \\infty}\\frac{\\log |\\Sha_{p^\\infty}(A\/K_n^{(p)})|}{{p^n}\\log q}.\n\\end{equation}\nThe right-hand side is equal to the ``\\emph{dimension of $\\Sha(A)$}'' denoted as $\\dim \\Sha(A)$\nin the sense of Ulmer \\cite[Proposition\/Definition 4.1]{Ulm19}.\n\\end{proposition}\n\\begin{proof}\nSince $\\Sha_{p^\\infty}(A\/K_n^{(p)})$ is the $p$ co-torsion part of $\\Sel_{p^\\infty}(A\/K_n^{(p)})$, via the aforementioned control sequence,\nwe can estimate its order by that of the $p$ co-torsion part of $\\Sel_{p^\\infty}(A\/K_\\infty^{(p)})^{\\Gamma^{p^n}}$, or by duality,\nthe order of the $p$-primary torsion part of $X_{A\/K}\/\\omega_n X_{A\/K}$. In view of \\eqref{e:x}, we need to deal with the elementary $\\Lambda$-module appearing in \\eqref{e:x}. If $V=\\Lambda\/(p^{\\mu_i})$, then $V\/\\omega_n V$ has oder $p^{\\mu_i p^n}$.\nIn the case where $V=\\Lambda\/(g_j)$, $g_j=\\nu_i^{l}$, the exact sequence\n\t\\[\n\t\t\t0\n\t\t\\to\n\t\t\t\\nu_i\\Lambda\/(\\nu_i^{l},\\omega_n)\n\t\t\\to\n\t\t\tV\/\\omega_n V\n\t\t\\to\n\t\t\t\\Lambda\/\\nu_i\\Lambda\n\t\t\\to\n\t\t\t0\n\t\\]\nshows $\\Lambda\/\\nu_i\\Lambda$ is the $\\mathbb{Z}_p$-free quotient of $V\/\\omega_n V$, while since $\\nu_i^{l-1}$ and $\\omega'_n$ \nare relatively prime, $\\nu_i\\Lambda\/(\\nu_i^{l},\\omega_n)\\simeq \\Lambda\/ (\\nu_i^{l-1},\\omega'_n)$ is the $p$-primary torsion part.\nThus, Lemma \\ref{l:serre} says, for every $g_j$ in \\eqref{e:x}, with $V=\\Lambda\/(g_j)$, the order of the $p$-primary torsion part of\n$V\/\\omega_n V$ is bounded by $p^{n d_j+\\mathrm{O}(1)}$. These lead to the desired\n$$\\frac{\\log |\\Sha_{p^\\infty}(A\/K_n^{(p)})|}{{p^n}\\log p}=\\sum_i\\mu_i+\\mathrm{o}(1), \\; \\text{as}\\; n\\rightarrow\\infty.$$\n\\end{proof}\n\n\\begin{remark} \\mbox{}\n\t\\begin{enumerate}\n\t\t\\item\n\t\t\tWithout the assumption of the finiteness of Tate-Shafarevich group,\n\t\t\tthe above proof shows that, if we replace  $\\Sha_{p^\\infty}(A\/K_n^{(p)})$\n\t\t\tby its $p$ co-torsion part, then \\eqref{e:shafni} also holds.\n\t\t\\item\n\t\t\tThe proof of Proposition \\ref{p:liminf} is complicated\n\t\t\tdue to the lack of a control theorem for $\\Sha$ at this point.\n\t\t\tIt is not clear how to deduce such a theorem from the (known) control theorem for $\\Sel$.\n\t\t\tThe difficulties mainly come from the fact that\n\t\t\t$H^{1}(\\Gamma^{p^{n}}, A(K_{\\infty}^{(p)}) \\otimes \\mathbb{Q}_{p} \/ \\mathbb{Z}_{p})$ is not finite in general.\n\t\t\tIn the next subsection, we will prove a control theorem for $\\Sha$\n\t\t\twithout using a control theorem for $\\Sel$,\n\t\t\tfrom which Proposition \\ref{p:liminf} will follow.\n\t\\end{enumerate}\n\\end{remark}\n\nIn the next subsection, we will use $\\Sha_{p^{\\infty}}(A \/ K \\bar{\\mathbb{F}}_{q})$\nin addition to $\\Sha_{p^\\infty}(A \/ K_{\\infty}^{(p)})$.\nTheir relation is given as follows.\n\n\\begin{proposition} \\label{p:inv}\n\tThe $\\Gal(\\bar{\\mathbb{F}}_{q} \/ \\mathbb{F}_{q, \\infty})$-invariant part of\n\t$\\Sha_{p^{\\infty}}(A \/ K \\bar{\\mathbb{F}}_{q})$ is\n\t$\\Sha_{p^\\infty}(A \/ K_{\\infty}^{(p)})$.\n\\end{proposition}\n\n\\begin{proof}\n\t The group $\\Gal(\\bar{\\mathbb{F}}_{q} \/ \\mathbb{F}_{q, \\infty})$ is pro-prime-to-$p$.\n\t Hence the $\\Gal(\\bar{\\mathbb{F}}_{q} \/ \\mathbb{F}_{q, \\infty})$-invariant part functor\n\t on $p$-primary torsion discrete $\\Gal(\\bar{\\mathbb{F}}_{q} \/ \\mathbb{F}_{q, \\infty})$-modules is an exact functor.\n\t This implies that the $\\Gal(\\bar{\\mathbb{F}}_{q} \/ \\mathbb{F}_{q, \\infty})$-invariant part of\n\t $\\coh^{1}(K \\bar{\\mathbb{F}}_{q}, A)[p^{\\infty}]$ is\n\t $\\coh^{1}(K_{\\infty}^{(p)}, A)[p^{\\infty}]$.\n\t On the other hand, we have\n\t \t\\[\n\t \t\t\t\\bigoplus_{\\bar{v}}\n\t \t\t\t\t\\coh^{1}((K \\bar{\\mathbb{F}}_{q})_{\\bar{v}}, A)\n\t \t\t\\cong\n\t \t\t\t\\bigoplus_{w}\n\t \t\t\t\t\\coh^{1}(K_{\\infty, w}^{(p)} \\otimes_{\\mathbb{F}_{q, \\infty}} \\bar{\\mathbb{F}}_{q}, A),\n\t \t\\]\n\t where $\\bar{v}$ and $w$ run through all places of\n\t $K \\bar{\\mathbb{F}}_{q}$ and $K_{\\infty}^{(p)}$, respectively.\n\t (Here note that $K_{\\infty, w}^{(p)} \\otimes_{\\mathbb{F}_{q, \\infty}} \\bar{\\mathbb{F}}_{q}$ is\n\t not a field but a finite product of fields).\n\t The group $\\Gal(\\bar{\\mathbb{F}}_{q} \/ \\mathbb{F}_{q, \\infty})$ acts on the right-hand side term by term.\n\t The $\\Gal(\\bar{\\mathbb{F}}_{q} \/ \\mathbb{F}_{q, \\infty})$-invariant part of\n\t $\\coh^{1}(K_{\\infty, w}^{(p)} \\otimes_{\\mathbb{F}_{q, \\infty}} \\bar{\\mathbb{F}}_{q}, A)[p^{\\infty}]$ is\n\t $\\coh^{1}(K_{\\infty, w}^{(p)}, A)[p^{\\infty}]$ by the same reasoning as before.\n\t Now the proposition follows from the definition of Tate-Shafarevich groups.\n\\end{proof}\n\n\\subsection{The Tate-Shafarevich group scheme}\\label{su:su}\n\nIn \\cite{Suz19}, a commutative perfect group scheme\n$$\\mathcal{G}^{(i)} = \\mathcal{G}_{A}^{(i)}:={\\bf H}^i(\\mathcal{C},\\mathcal{A})$$\nover $\\mathbb{F}_{q}$ for each integer $i$ is defined\n(where perfect means that the Frobenius is an isomorphism).\nIt is characterized by the property \\cite[Proposition 2.7.8]{Suz19}\nthat for any perfect field extension $k\/\\mathbb{F}_q$,\nwe have a canonical isomorphism\n\t\\[\n\t\t\t\\mathcal{G}_{A}^{(i)}(k)\n\t\t\\cong\n\t\t\t\\coh^i(\\mathcal C_{\\bar{k}}, \\mathcal{A})^{\\Gal({\\bar k}\/k)}\n\t\\]\nfunctorial in the variable $k$,\nwhere $\\mathcal{C}_{\\bar{k}} = \\mathcal{C} \\times_{\\mathbb{F}_{q}} \\bar{k}$ denotes the base change\nand the cohomology is taken in the \\'etale topology\n(or, equivalently, in the flat topology, as the N\\'eron model $\\mathcal{A}$ is smooth).\nWe call $\\mathcal{G}_{A}^{(1)}$ the Tate-Shafarevich scheme, because\n\t\\[\n\t\t\t\\mathcal{G}_{A}^{(1)}(\\bar{\\mathbb{F}}_{q})\n\t\t=\n\t\t\t\\coh^{1}(\\mathcal{C}_{\\bar{\\mathbb{F}}_{q}}, \\mathcal{A})\n\t\t\\cong\n\t\t\t\\Sha(A \/ K \\bar{\\mathbb{F}}_{q}),\n\t\\]\nwhere the last canonical isomorphism is \\cite[III, Lemma 11.5]{Mil06}.\nTaking the $\\Gal(\\bar{\\mathbb{F}}_{q} \/ \\mathbb{F}_{q, \\infty})$-invariant parts of the $p$-primary torsion parts of both of the sides\nand using Proposition \\ref{p:inv}, we have\n\\begin{equation} \\label{e:Gsha}\n\t\t\t\\mathcal{G}_{A}^{(1)}(\\mathbb{F}_{q,\\infty})[p^\\infty]\n\t\t\\cong\n\t\t\t\\Sha_{p^\\infty}(A \/ K_{\\infty}^{(p)}).\n\t\\end{equation}\nTherefore,\n\\begin{equation}\\label{e:gsh}\n(\\mathcal{G}_{A}^{(1)}(\\mathbb{F}_{q,\\infty})[p^\\infty])^\\vee\n\t\t\\cong\n\t\t\t\\Sha_{p^\\infty}(A \/ K_{\\infty}^{(p)})^\\vee.\n\\end{equation}\n\nThe group $\\mathcal{G}_{A}^{(1)}$ is the perfection\n(inverse limit along Frobenius morphisms)\nof a commutative smooth group scheme over $\\mathbb{F}_{q}$\nwith unipotent identity component and torsion cofinite component group $\\pi_{0}(\\mathcal{G}_{A}^{(1)})$\n(\\cite[Theorem 3.4.1 (2)]{Suz19}).\nHere cofinite means that\n$\\pi_{0}(\\mathcal{G}_{A}^{(1)})[n]$ is finite \\'etale for any $n \\ge 1$.\n\n\\begin{definition}\nDefine $s_{A \/ K}: = \\dim  \\mathcal{G}_{A}^{(1)}$.\n\\end{definition}\n\n\n\\begin{proposition} \\label{p:Gchar}\n\tLet $\\mathcal{G}$ be a commutative smooth group scheme over $\\mathbb{F}_{q}$\n\tsuch that its identity component $\\mathcal{G}^{0}$ is unipotent\n\tand its component group $\\pi_{0}(\\mathcal{G})$ is $p$-primary torsion and cofinite.\n\tThen the Iwasawa module $\\mathcal{G}(\\mathbb{F}_{q, \\infty})^{\\vee}$ is\n\tfinitely generated torsion over $\\Lambda$,\n\twith characteristic ideal generated by\n\t$q^{\\dim(\\mathcal{G})}$ times the characteristic polynomial of the Frobenius action\n\ton the finite-dimensional $\\mathbb{Q}_{p}$-vector space\n\t$\\pi_{0}(\\mathcal{G})(\\mathbb{F}_{q, \\infty})^{\\vee} \\otimes_{\\mathbb{Z}_{p}} \\mathbb{Q}_{p}$.\n\tIn particular, the $\\mu$-invariant of $\\mathcal{G}(\\mathbb{F}_{q,\\infty})^{\\vee}$ is $\\dim(\\mathcal{G})$.\n\\end{proposition}\n\n\\begin{proof}\n\tConsider the connected-\\'etale sequence\n\t$0 \\to \\mathcal{G}^{0} \\to \\mathcal{G} \\to \\pi_{0}(\\mathcal{G}) \\to 0$.\n\tThis induces an exact sequence\n\t\t\\[\n\t\t\t\t0\n\t\t\t\\to\n\t\t\t\t\\mathcal{G}^{0}(\\mathbb{F}_{q,\\infty})\n\t\t\t\\to\n\t\t\t\t\\mathcal{G}(\\mathbb{F}_{q,\\infty})\n\t\t\t\\to\n\t\t\t\t\\pi_{0}(\\mathcal{G})(\\mathbb{F}_{q, \\infty})\n\t\t\t\\to\n\t\t\t\t0\n\t\t\\]\n\tsince $\\coh^{1}(\\mathbb{F}_{q, \\infty}, \\mathcal{G}^{0})$\n\t(which is the direct limit of $\\coh^{1}(\\mathbb{F}_{q, n}, \\mathcal{G}^{0})$ in $n$)\n\tvanishes by Lang's theorem.%\n\t\\footnote{Recall the statement of Lang's theorem\n\t\\cite[III, \\S 5, 7.5 and 7.11]{DG70}:\n\tfor any commutative connected (not necessarily affine) algebraic group $G$ over a finite field $\\mathbb{F}_{q}$,\n\tthe cohomology $\\coh^{n}(\\mathbb{F}_{q}, G)$ vanishes for all $n \\ge 1$.}\n\tHence we have an exact sequence\n\t\t\\[\n\t\t\t\t0\n\t\t\t\\to\n\t\t\t\t\\pi_{0}(\\mathcal{G})(\\mathbb{F}_{q,\\infty})^{\\vee}\n\t\t\t\\to\n\t\t\t\t\\mathcal{G}(\\mathbb{F}_{q,\\infty})^{\\vee}\n\t\t\t\\to\n\t\t\t\t\\mathcal{G}^{0}(\\mathbb{F}_{q,\\infty})^{\\vee}\n\t\t\t\\to\n\t\t\t\t0\n\t\t\\]\n\tof $\\Lambda$-modules.\n\tSince characteristic ideals are multiplicative in short exact sequences\n\t(\\cite[Appendix A, Proposition 1]{CS06}),\n\twe may assume that $\\mathcal{G}$ is either connected or \\'etale.\n\tA commutative smooth connected unipotent group over $\\mathbb{F}_{q}$ is\n\ta finite successive extension of copies of $\\mathbf{G}_{a}$\n\t(\\cite[IV, \\S 2, 3.9]{DG70}).\n\tTherefore the connected case is further reduced to the case of $\\mathbf{G}_{a}$\n\tby a similar argument.\n\t\n\tAssume that $\\mathcal{G} = \\mathbf{G}_{a}$.\n\tThen $(\\mathbb{F}_{q,\\infty})^{\\vee}$ is a rank one free $\\mathbb{F}_{q}[[\\Gamma]]$-module.\n\tIn particular, it is a finitely generated torsion $\\Lambda$-module\n\twith characteristic ideal $(q)$.\n\tAssume that $\\mathcal{G}$ is \\'etale.\n\tThen $\\mathcal{G}(\\mathbb{F}_{q,\\infty})^{\\vee}$ is finitely generated over $\\mathbb{Z}_{p}$.\n\tHence it is a finitely generated torsion $\\Lambda$-module\n\twith characteristic ideal generated by\n\tthe characteristic polynomial of the Frobenius action\n\ton the finite-dimensional $\\mathbb{Q}_{p}$-vector space\n\t$\\mathcal{G}(\\mathbb{F}_{q,\\infty})^{\\vee} \\otimes_{\\mathbb{Z}_{p}} \\mathbb{Q}_{p}$.\n\\end{proof}\n\n\\begin{theorem}\\label{muissA} The $\\mu$-invariant of $(\\mathcal{G}^{(1)}(\\mathbb{F}_{q,\\infty})[p^\\infty])^{\\vee}$ is $s_{A \/ K}$.\nTherefore,\n$$\\mu_{A\/K}=s_{A \/ K}.$$\nIn particular, $\\mu_{A \/ K}$ (and hence $\\dim \\Sha(A)$ when Tate-Shafarevich groups are finite) is an integer.\n\\end{theorem}\n\n\\begin{proof}\n\tThis follows from \\eqref{e:gsh}, Proposition \\ref{p:ts}, and Proposition \\ref{p:Gchar}.\n\\end{proof}\n\nIn the proof of Propositions \\ref{p:limint} and \\ref{p:liminf},\nwe have used the control theorem for Selmer groups \\cite[Theorem 4]{Tan10}.\nThe group schemes $\\mathcal{G}^{(i)}$, their derived descent property and arithmetic duality give\na control theorem for Tate-Shafarevich groups,\nwhich we are now going to see below (Proposition \\ref{p:ShaGone}).\nFor $n\\ge 0$, consider the natural homomorphisms:\n\\[\n\t\t\\Sha(A \/ K_{n}^{(p)})\n\t\\overset{i_{n}}{\\longrightarrow}\n\t\t\\coh^{1}(\\mathcal{C}_{\\mathbb{F}_{q, n}}, \\mathcal{A})\n\t\\overset{j_{n}}{\\longrightarrow}\n\t\t\\mathcal{G}^{(1)}(\\mathbb{F}_{q,n}).\n\\]\nNote that the $p$-primary part of the last term is isomorphic to\n$\\Sha_{p^\\infty}(A \/ K_{\\infty}^{(p)})^{\\Gamma^{p^{n}}}$\nby \\eqref{e:Gsha}.\n\n\\begin{proposition} \\label{p:ShaGone}\nFor all $n\\ge 0$, the map $i_n$ is injective with finite cokernel of bounded order in $n$,\nand the map $j_n$ has finite kernel and cokernel of bounded order in $n$.\n\\end{proposition}\n\n\\begin{proof}\n\tThe first statement follows from the exact sequence\n\t\t\\[\n\t\t\t\t0\n\t\t\t\\to\n\t\t\t\t\\Sha(A \/ K_{n}^{(p)})\n\t\t\t\\to\n\t\t\t\t\\coh^{1}(\\mathcal{C}_{\\mathbb{F}_{q, n}}, \\mathcal{A})\n\t\t\t\\to\n\t\t\t\t\\bigoplus_{v \\in \\mathcal{C}_{\\mathbb{F}_{q, n}}}\n\t\t\t\t\t\\coh^{1} \\bigl(\n\t\t\t\t\t\t\\mathbb{F}_{q, n}(v), \\pi_{0}(\\mathcal{A}_{v})\n\t\t\t\t\t\\bigr)\n\t\t\\]\n\twritten in \\cite[III, Proposition 9.2]{Mil06}.\n\tFor the second statement, we have a canonical spectral sequence\n\t\t\\[\n\t\t\t\tE_{2}^{i j}\n\t\t\t=\n\t\t\t\t\\coh^{i} \\bigl(\n\t\t\t\t\t\\mathbb{F}_{q,n},\n\t\t\t\t\t\\mathcal{G}^{(j)}\n\t\t\t\t\\bigr)\n\t\t\t\\Longrightarrow\n\t\t\t\t\\coh^{i + j}(\\mathcal{C}_{\\mathbb{F}_{q,n}}, \\mathcal{A})\n\t\t\\]\n\tby \\cite[Proposition 2.7.8]{Suz19}.\n\tHence we have an exact sequence\n\t\t\\begin{align*}\n\t\t\t\t\t0\n\t\t\t&\t\\to\n\t\t\t\t\t\\coh^{1}(\\mathbb{F}_{q,n}, \\mathcal{G}^{(0)})\n\t\t\t\t\\to\n\t\t\t\t\t\\coh^{1}(\\mathcal{C}_{\\mathbb{F}_{q,n}}, \\mathcal{A})\n\t\t\t\t\\to\n\t\t\t\t\t\\mathcal{G}^{(1)}(\\mathbb{F}_{q,n})\n\t\t\t\\\\\n\t\t\t&\t\\to\n\t\t\t\t\t\\coh^{2}(\\mathbb{F}_{q, n}, \\mathcal{G}^{(0)})\n\t\t\t\t\\to\n\t\t\t\t\t\\coh^{2}(\\mathcal{C}_{\\mathbb{F}_{q, n}}, \\mathcal{A}).\n\t\t\\end{align*}\n\tThe natural morphism from $\\coh^{m}(\\mathbb{F}_{q,n}, \\mathcal{G}^{(0)})$\n\tto $\\coh^{m}(\\mathbb{F}_{q,n}, \\pi_{0}(\\mathcal{G}^{(0)}))$ is an isomorphism\n\tfor any $m \\ge 1$ since $\\coh^{m}(\\mathbb{F}_{q,n}, (\\mathcal{G}^{(0)})^{0}) = 0$ by Lang's theorem.\n\tThe group scheme $\\pi_{0}(\\mathcal{G}^{(0)})$ over $\\mathbb{F}_{q}$ is \\'etale\n\twith group of geometric points finitely generated\n\tby \\cite[Theorem 3.4.1 (2)]{Suz19}.\n\tHence $\\coh^{1}(\\mathbb{F}_{q,n}, \\pi_{0}(\\mathcal{G}^{(0)}))$ is finite of order bounded in $n$.\n\tTherefore the kernel of $\\coh^{1}(\\mathcal{C}_{\\mathbb{F}_{q, n}}, \\mathcal{A}) \\to \\mathcal{G}^{(1)}(\\mathbb{F}_{q, n})$\n\tis finite of order bounded in $n$.\n\t\n\tTo show that the cokernel of this map is also finite of order bounded in $n$,\n\tlet $B \/ K$ be the abelian variety dual to $A$ with N\\'eron model $\\mathcal{B}$.\n\tLet $\\mathcal{B}^{0}$ be the part of $\\mathcal{B}$ with connected fibers.\n\tLet\n\t\t\\[\n\t\t\t\t\\mathcal{F}^{(i)}\n\t\t\t=\n\t\t\t\t\\mathbf{H}^{i}(\\mathcal{C}, \\mathcal{B}^{0})\n\t\t\\]\n\tbe the group scheme defined in \\cite{Suz19}.\n\tThe group $\\coh^{2}(\\mathbb{F}_{q,n}, \\mathcal{G}^{(0)})$ is isomorphic\n\tto $\\coh^{1}(\\mathbb{F}_{q,n}, \\pi_{0}(\\mathcal{G}^{(0)}) \\otimes \\mathbb{Q} \/ \\mathbb{Z})$\n\tsince Galois cohomology is torsion in positive degrees.\n\tThe latter group $\\coh^{1}(\\mathbb{F}_{q,n}, \\pi_{0}(\\mathcal{G}^{(0)}) \\otimes \\mathbb{Q} \/ \\mathbb{Z})$\n\tis Pontryagin dual to the profinite completion of\n\t$\\Hom(\\pi_{0}(\\mathcal{G}^{(0)}) \/ \\mathrm{tor}, \\mathbb{Z})(\\mathbb{F}_{q, n})$\n\t($=$ the group of $\\mathbb{F}_{q, n}$-valued points of the dual $\\mathbb{Z}$-lattice of $\\pi_{0}(\\mathcal{G}^{(0)}) \/ \\mathrm{tor}$).\n\tThe height pairing gives a non-degenerate pairing\n\t\t\\[\n\t\t\t\t\\pi_{0}(\\mathcal{G}^{(0)}) \/ \\mathrm{tor}\n\t\t\t\\times\n\t\t\t\t\\pi_{0}(\\mathcal{F}^{(0)}) \/ \\mathrm{tor}\n\t\t\t\\to\n\t\t\t\t\\mathbb{Z}\n\t\t\\]\n\tof $\\mathbb{Z}$-lattices over $\\mathbb{F}_{q}$ (\\cite[Theorem 3.4.1 (6e), (7)]{Suz19}).\n\tHence we have injections\n\t\t\\[\n\t\t\t\t\\pi_{0}(\\mathcal{F}^{(0)})(\\mathbb{F}_{q, n}) \/ \\mathrm{tor}\n\t\t\t\\hookrightarrow\n\t\t\t\t\\bigl(\n\t\t\t\t\t\\pi_{0}(\\mathcal{F}^{(0)}) \/ \\mathrm{tor}\n\t\t\t\t\\bigr)(\\mathbb{F}_{q, n})\n\t\t\t\\hookrightarrow\n\t\t\t\t\\Hom(\\pi_{0}(\\mathcal{G}^{(0)}) \/ \\mathrm{tor}, \\mathbb{Z})(\\mathbb{F}_{q, n})\n\t\t\\]\n\twith finite cokernels of order bounded in $n$.\n\tTherefore we have a natural map\n\t\t\\[\n\t\t\t\t\\pi_{0}(\\mathcal{F}^{(0)})(\\mathbb{F}_{q, n})^{\\wedge}\n\t\t\t\\to\n\t\t\t\t\\coh^{2}(\\mathbb{F}_{q,n}, \\mathcal{G}^{(0)})^{\\vee}\n\t\t\\]\n\twith finite cokernel of order bounded in $n$,\n\twhere $\\wedge$ denotes the profinite completion.\n\t\n\tOn the other hand, we have a canonical exact sequence\n\t\t\\[\n\t\t\t\t0\n\t\t\t\\to\n\t\t\t\t\\mathcal{B}^{0}(\\mathcal{C}_{\\mathbb{F}_{q,n}})^{\\wedge}\n\t\t\t\\to\n\t\t\t\t\\coh^{2}(\\mathcal{C}_{\\mathbb{F}_{q,n}}, \\mathcal{A})^{\\vee}\n\t\t\t\\to\n\t\t\t\t\\mathrm T \\coh^{1}(\\mathcal{C}_{\\mathbb{F}_{q,n}}, \\mathcal{B}^{0})\n\t\t\t\\to\n\t\t\t\t0\n\t\t\\]\n\tby the duality result \\cite[Proposition 4.2.3]{Suz19},\n\twhere $\\mathrm T$ denotes the profinite Tate module.\n\tBy the connected-\\'etale sequence\n\t\t\\[\n\t\t\t\t0\n\t\t\t\\to\n\t\t\t\t(\\mathcal{F}^{(0)})^{0}\n\t\t\t\\to\n\t\t\t\t\\mathcal{F}^{(0)}\n\t\t\t\\to\n\t\t\t\t\\pi_{0}(\\mathcal{F}^{(0)})\n\t\t\t\\to\n\t\t\t\t0\n\t\t\\]\n\tand Lang's theorem (namely, $\\coh^{1}(\\mathbb{F}_{q, n}, (\\mathcal{F}^{(0)})^{0}) = 0$), the map\n\t\t\\[\n\t\t\t\t\\mathcal{F}^{(0)}(\\mathbb{F}_{q,n})\n\t\t\t=\n\t\t\t\t\\mathcal{B}^{0}(\\mathcal{C}_{\\mathbb{F}_{q,n}})\n\t\t\t\\to\n\t\t\t\t\\pi_{0}(\\mathcal{F}^{(0)})(\\mathbb{F}_{q,n})\n\t\t\\]\n\tis surjective.\n\t\n\tTherefore the cokernel of the map from $\\coh^{2}(\\mathcal{C}_{\\mathbb{F}_{q,n}}, \\mathcal{A})^{\\vee}$\n\tto $\\coh^{2}(\\mathbb{F}_{q,n}, \\mathcal{G}^{(0)})^{\\vee}$ is finite of order bounded in $n$.\n\tThus the kernel of the map from $\\coh^{2}(\\mathbb{F}_{q,n}, \\mathcal{G}^{(0)})$\n\tto $\\coh^{2}(\\mathcal{C}_{\\mathbb{F}_{q,n}}, \\mathcal{A})$ is finite of order bounded in $n$.\n\tThis implies that the cokernel of\n\t$\\coh^{1}(\\mathcal{C}_{\\mathbb{F}_{q, n}}, \\mathcal{A}) \\to \\mathcal{G}^{(1)}(\\mathbb{F}_{q, n})$\n\tis finite of order bounded in $n$.\n\\end{proof}\n\nThe above proof actually gives an explicit bound on the orders of (the $p$-primary part of) the kernel and cokernel of\n$i_{n}$ and $j_{n}$ for large $n$ in terms of the finite \\'etale groups $\\pi_{0}(\\mathcal{A}_{v})$,\n$\\pi_{0}(\\mathcal{G}^{(0)})_{\\mathrm{tor}}$ and $\\pi_{0}(\\mathcal{F}^{(0)})_{\\mathrm{tor}}$\nand the discriminant of the height pairing\non $\\pi_{0}(\\mathcal{G}^{(0)}) \/ \\mathrm{tor} \\times \\pi_{0}(\\mathcal{F}^{(0)}) \/ \\mathrm{tor}$.\n\nUsing this, we can give another proof of\nPropositions \\ref{p:limint} and \\ref{p:liminf}:\n\n\\begin{proof}[Another proof of Propositions \\ref{p:limint} and \\ref{p:liminf}]\n\tThe $\\Gamma^{p^{n}}$-invariant part of\n\t$\\mathcal{G}^{(1)}(\\mathbb{F}_{q, \\infty})$ is $\\mathcal{G}^{(1)}(\\mathbb{F}_{q,n}$).\n\tTherefore\n\t\t\\[\n\t\t\t\t\\lim_{n \\to \\infty}\n\t\t\t\t\t\\frac{\n\t\t\t\t\t\t\\log |\\mathcal{G}^{(1)}(\\mathbb{F}_{q,n})[p^m]|\n\t\t\t\t\t}{\n\t\t\t\t\t\tp^n\\log q\n\t\t\t\t\t}\n\t\t\t=\n\t\t\t\ts_{A \/ K}\n\t\t\\]\n\tfor all large $m$ by the same argument as the first part of the proof of Proposition \\ref{p:limint},\n\tand if $\\mathcal{G}^{(1)}(\\mathbb{F}_{q, n})[p^{\\infty}]$ is finite for all $n$, then\n\t\t\\[\n\t\t\t\t\\lim_{n \\to \\infty}\n\t\t\t\t\t\\frac{\n\t\t\t\t\t\t\\log |\\mathcal{G}^{(1)}(\\mathbb{F}_{q,n})[p^{\\infty}]|\n\t\t\t\t\t}{\n\t\t\t\t\t\tp^n\\log q\n\t\t\t\t\t}\n\t\t\t=\n\t\t\t\ts_{A \/ K}\n\t\t\\]\n\tby Iwasawa's formula \\cite[Theorem 8]{Ser95}.\n\tBy Proposition \\ref{p:ShaGone}, we can compare the asymptotic behavior of\n\t$|\\mathcal{G}^{(1)}(\\mathbb{F}_{q,n})[p^m]|$ and $|\\Sha_{p^m}(A\/K_n^{(p)})|$.\n\tThis, with Theorem \\ref{muissA}, gives\n\t\t\\[\n\t\t\t\t\\lim_{n\\rightarrow \\infty}\\frac{\\log |\\Sha_{p^m}(A\/K_n^{(p)})|}{{p^n}\\log q}\n\t\t\t=\n\t\t\t\t\\lim_{n \\to \\infty}\n\t\t\t\t\t\\frac{\n\t\t\t\t\t\t\\log |\\mathcal{G}^{(1)}(\\mathbb{F}_{q,n})[p^m]|\n\t\t\t\t\t}{\n\t\t\t\t\t\tp^n\\log q\n\t\t\t\t\t}\n\t\t\t=\n\t\t\t\ts_{A \/ K}\n\t\t\t=\n\t\t\t\t\\mu_{A \/ K}.\n\t\t\\]\n\tfor all large $m$, and if $\\Sha_{p^{\\infty}}(A \/ K_{n}^{(p)})$ is finite for all $n$, then\n\t\t\\begin{equation}\\label{e:Gone}\n\t\t\t\t\\lim_{n\\rightarrow \\infty}\\frac{\\log |\\Sha_{p^{\\infty}}(A\/K_n^{(p)})|}{{p^n}\\log q}\n\t\t\t=\n\t\t\t\t\\lim_{n \\to \\infty}\n\t\t\t\t\t\\frac{\n\t\t\t\t\t\t\\log |\\mathcal{G}^{(1)}(\\mathbb{F}_{q,n})[p^{\\infty}]|\n\t\t\t\t\t}{\n\t\t\t\t\t\tp^n\\log q\n\t\t\t\t\t}\n\t\t\t=\n\t\t\t\ts_{A \/ K}\n\t\t\t=\n\t\t\t\t\\mu_{A \/ K}.\n\t\t\\end{equation}\n\\end{proof}\n\n\\begin{remark} \\mbox{}\n\t\\begin{enumerate}\n\t\\item\n\t\tPropositions \\ref{p:Gchar} and \\ref{p:ShaGone},\n\t\ttogether with the properties of $\\mathcal{G}^{(1)}$ cited after \\eqref{e:gsh},\n\t\tgive another proof that $X_{A\/K}$ is finitely generated torsion over $\\Lambda$,\n\t\ti.e.\\ \\cite[Theorem 1.7]{OT09}.\n\t\\item\n\t\tTo seek for a number field analogue of the Tate-Shafarevich scheme,\n\t\tlet $E \/ \\mathbb{Q}$ be an elliptic curve and $p$ a prime number.\n\t\tAssume that $\\Sel_{p^{\\infty}}(E \/ \\mathbb{Q}^{cyc})^{\\vee}$ is torsion over\n\t\t$\\mathbb{Z}_{p}[[\\Gal(\\mathbb{Q}^{cyc} \/ \\mathbb{Q})]]$.\n\t\tDoes there exist the perfection of some smooth group scheme over $\\mathbb{F}_{p}$\n\t\twhose group of $\\mathbb{F}_{p, \\infty}$-valued points is isomorphic to\n\t\t$\\Sha_{p^{\\infty}}(E \/ \\mathbb{Q}^{cyc})$ as\n\t\t$\\mathbb{Z}_{p}[[\\Gal(\\mathbb{Q}^{cyc} \/ \\mathbb{Q})]] \\cong \\mathbb{Z}_{p}[[\\Gal(\\mathbb{F}_{p, \\infty} \/ \\mathbb{F}_{p})]]$-modules?\n\t\\end{enumerate}\n\\end{remark}\n\n\n\\subsection{The $\\mu$-invariant formula}\n\nWith the formula \\eqref{e:shafni} or \\eqref{e:Gone}\nand Proposition \\ref{p:slope},\nwe can now reinterpret Proposition 4.2 of \\cite{Ulm19} as follows:\n\n\\begin{corollary}\\label{muA}\n\tAssume that $\\Sha(A \/ K_{n}^{(p)})$ is finite for all $n$.\n\tWe have\n\t\t\\begin{equation} \\label{e:muform}\n\t\t\t\t\\mu_{A \/ K}\n\t\t\t=\n\t\t\t\t\t\\deg(\\mathcal{L})\n\t\t\t\t+\n\t\t\t\t\t\\dim(A)(g_{\\mathcal{C}} - 1)\n\t\t\t\t+\n\t\t\t\t\t\\dim(\\Tr_{K \/ \\mathbb{F}_{q}}(A))\n\t\t\t\t-\n\t\t\t\t\t\\theta,\n\t\t\\end{equation}\n\twhere $\\mathcal{L}$ is the invertible sheaf on $\\mathcal{C}$ defined by\n\t\t\\[\n\t\t\t\t\\mathcal{L}\n\t\t\t:=\n\t\t\t\to^{*} \\Omega_{\\mathcal{A} \/ \\mathcal{C}}^{\\dim(A)}\n\t\t\\]\n\t(here $o: \\mathcal{C} \\to \\mathcal{A}$ denotes the zero section of $\\mathcal{A} \\to \\mathcal{C}$).\n\\end{corollary}\n\nWe will prove below that the formula \\eqref{e:muform} also holds\nfor the following two cases without any assumption on $\\Sha$:\n$A$ is the Jacobian of a projective smooth curve over $K$;\nand $A$ is a semistable abelian variety over $K$.\n\n\\begin{remark}\n\tAs in Proposition \\ref{p:liminf},\n\tUlmer \\cite[4.1]{Ulm19} called the right-hand side of \\eqref{e:shafni}\n\tthe \\emph{dimension of $\\Sha(A)$}.\n\tHe justified this terminology in \\cite[4.3 (2)]{Ulm19}\n\tin the special case that $A$ is a Jacobian.\n\tFor a general abelian variety $A$,\n\tour definition $s_{A \/ K} = \\dim \\mathcal{G}_{A}^{(1)}$\n\tand the formula \\eqref{e:Gone} together justify the terminology.\n\\end{remark}\n\nThe term $\\deg(\\mathcal{L})$ is non-negative (\\cite[Theorem 2.6]{Yua18}). Also, recall that the term $\\theta$ in the formula of the above corollary is non-negative so that we get\nthe following upper bound on the $\\mu$-invariant (see \\cite[Proposition 4.4]{Ulm19} for another upper bound for $\\mu$):\n\n\\begin{equation} \\label{e:upperbound}\n\t\t\t\t\\mu_{A \/ K}\n\t\t\t\\le\n\t\t\t\t\t\\deg(\\mathcal{L})\n\t\t\t\t+\n\t\t\t\t\t\\dim(A)(g_{\\mathcal{C}} - 1)\n\t\t\t\t+\n\t\t\t\t\t\\dim(\\Tr_{K \/ \\mathbb{F}_{q}}(A)).\n\t\t\\end{equation}\\\n\nAbout the following definition, see also Theorem \\ref{c:szpiro} below.\n\n\\begin{definition} \\label{d:Sz}\n\tDefine the Szpiro difference as\n\t\t\\[\n\t\t\t\td\n\t\t\t=\n\t\t\t\td_{A}\n\t\t\t:=\n\t\t\t\t\t\\frac{\\deg(N)}{2}\n\t\t\t\t+\n\t\t\t\t\t\\dim(A) (g_{\\mathcal{C}} - 1)\n\t\t\t\t+\n\t\t\t\t\t\\dim \\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)\n\t\t\t\t-\n\t\t\t\t\t\\deg(\\mathcal{L}).\n\t\t\\]\n\tAlso define $b = b_{A} := a \/ 2 - \\theta$.\n\\end{definition}\n\nNote that $b$ is non-negative by Proposition \\ref{p:slope}\nand defined purely from the $L$-function of $A$.\nBy \\eqref{e:OS}, we can rewrite the formula \\eqref{e:muform} in terms of $b$ and $d$:\n\n\\begin{corollary} \\label{c:bminusd}\n\tThe formula \\eqref{e:muform} is equivalent to the formula\n\t$\\mu_{A \/ K} = b - d$.\n\\end{corollary}\n\n\\begin{remark}\n\tThe formula \\eqref{e:muform} can be understood as a certain kind of Euler characteristic formulas\n\tfor $L_{A}(s)$ at $s = 1$.\n\tTo see this, let $\\mathrm{Lie}(\\mathcal{A})$ be the vector bundle on $\\mathcal{C}$\n\tdual to $o^{*} \\Omega_{\\mathcal{A} \/ \\mathcal{C}}$.\n\tThen by the Riemann-Roch theorem, we have\n\t\t\\[\n\t\t\t\t\\chi(\\mathcal{C}, \\mathrm{Lie}(\\mathcal{A}))\n\t\t\t:=\n\t\t\t\t\\sum_{n}\n\t\t\t\t\t(-1)^{n} \\dim_{\\mathbb{F}_{q}} \\coh^{n}(\\mathcal{C}, \\mathrm{Lie}(\\mathcal{A}))\n\t\t\t=\n\t\t\t\t\\dim(A) (1 - g_{\\mathcal{C}}) - \\deg(\\mathcal{L}).\n\t\t\\]\n\tAlso let\n\t\t\\[\n\t\t\t\t\\chi^{0}(\\mathcal{C}, \\mathcal{A})\n\t\t\t:=\n\t\t\t\t\\sum_{n}\n\t\t\t\t\t(-1)^{n} \\dim \\mathcal{G}_{A}^{(n)}.\n\t\t\\]\n\tBy \\cite[Propositions 3.2.2, 3.2.3, Theorem 3.4.1 (2)]{Suz19},\n\twe have $\\dim \\mathcal{G}_{A}^{(0)} = \\dim \\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)$\n\tand $\\dim \\mathcal{G}_{A}^{(n)} = 0$ for $n \\ne 0, 1$.\n\tTherefore the formula \\eqref{e:muform} can be written as\n\t\t\\[\n\t\t\t\t\\theta\n\t\t\t=\n\t\t\t\t\t\\chi^{0}(\\mathcal{C}, \\mathcal{A})\n\t\t\t\t-\n\t\t\t\t\t\\chi(\\mathcal{C}, \\mathrm{Lie}(\\mathcal{A})).\n\t\t\\]\n\tCompare this with the Weil-\\'etale BSD formula \\cite[Proposition 8.4]{GS20}.\n\tThis presentation might be useful when one wants to generalize \\eqref{e:muform},\n\tfor example to motives over $K$ other than abelian varieties.\n\\end{remark}\n\n\n\\section{The $\\mu$-invariant for Jacobians} \\label{s:JacSur}\nIn this section, we will show that the formula (\\ref{e:muform}) holds for Jacobians\nwithout any hypothesis (Corollary \\ref{c:MilUlm})\nand will deduce some necessary and sufficient conditions\nabout the (non-)vanishing of the $\\mu$-invariant\n(Propositions \\ref{p:trivmu} and \\ref{p:aborK3}).\nA good reference on fibered surfaces is \\cite{Liu02}.\nLet $\\mathcal{S}$ be a projective smooth surface over $\\mathbb{F}_{q}$\nand $\\pi \\colon \\mathcal{S} \\to \\mathcal{C}$ a flat morphism over $\\mathbb{F}_{q}$.\nAssume that $\\pi_{\\ast} \\mathcal{O}_{\\mathcal{S}} = \\mathcal{O}_{\\mathcal{C}}$ and that\nthe generic fiber $\\pi_{K} \\colon \\mathcal{S}_{K} \\to \\spec K$ of $\\pi$ is smooth.\nLet $A = \\Pic_{\\mathcal{S}_{K} \/ K}^{0}$ be the Jacobian variety of $\\mathcal{S}_{K}$ over $K$.\nAny elliptic curve is an example of such $A$\nby the theory of regular models.\nWe do not assume that $\\pi$ admits a section.\n\n\n\\subsection{The Brauer group}\n\nRecall from \\cite[\\S 3]{Art74b}, \\cite[\\S 5]{Mil76} that there is a perfect group scheme\n$\\underline{\\coh}^{i}(p^{\\infty})$ over $\\mathbb{F}_{q}$ for each integer $i$\nsuch that for any perfect field extension $k \/ \\mathbb{F}_{q}$, we have\n\t\\[\n\t\t\t\\underline{\\coh}^{i}(p^{\\infty})(k)\n\t\t=\n\t\t\t\\coh_{fl}^{i}(\\mathcal{S}_{\\bar{k}}, \\mu_{p^{\\infty}})^{\\Gal(\\bar{k} \/ k)}\n\t\\]\nas a functor in the variable $k$,\nwhere $\\mathcal{S}_{\\bar{k}} = \\mathcal{S} \\times_{\\mathbb{F}_{q}} \\bar{k}$.\nIts identity component and component group are denoted by\n$\\underline{\\mathrm{U}}^{i}(p^{\\infty})$ and $\\underline{\\mathrm{D}}^{i}(p^{\\infty})$, respectively.\nThe group $\\underline{\\mathrm{U}}^{i}(p^{\\infty})$ is\nthe perfection of a commutative unipotent algebraic group over $\\mathbb{F}_{q}$.\nThe group $\\underline{\\mathrm{D}}^{i}(p^{\\infty})$ is\na $p$-primary torsion cofinite \\'etale group scheme over $\\mathbb{F}_{q}$.\n\nLet $\\underline{\\mathrm{NS}} = \\pi_{0}(\\Pic_{\\mathcal{S} \/ \\mathbb{F}_{q}})$.\nIt is an \\'etale group scheme over $\\mathbb{F}_{q}$\nsuch that its group of $\\bar{\\mathbb{F}}_{q}$-points is\nthe N\\'eron-Severi group $\\mathrm{NS}(\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}})$ of $\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}}$,\nwhich is a finitely generated abelian group.\nThe Kummer sequence defines a canonical injection\n$\\underline{\\mathrm{NS}} \\otimes \\mathbb{Q}_{p} \/ \\mathbb{Z}_{p} \\hookrightarrow \\underline{\\coh}^{2}(p^{\\infty})$\nof perfect group schemes.\nDefine $\\underline{\\mathrm{Br}}_{p^{\\infty}}$ to be its cokernel\n(a priori as a sheaf on the fppf site or the big \\'etale site of $\\mathbb{F}_{q}$).\n\n\\begin{proposition}\n\tThe sheaf $\\underline{\\mathrm{Br}}_{p^{\\infty}}$ is represented by\n\ta perfect group scheme over $\\mathbb{F}_{q}$.\n\tIts identity component is a quotient of $\\underline{\\mathrm{U}}^{2}(p^{\\infty})$\n\tby a finite \\'etale group.\n\\end{proposition}\n\n\\begin{proof}\n\tWe have an exact sequence\n\t\t\\[\n\t\t\t\t0\n\t\t\t\\to\n\t\t\t\t\\frac{\n\t\t\t\t\t\\underline{\\mathrm{U}}^{2}(p^{\\infty})\n\t\t\t\t}{\n\t\t\t\t\t\t(\\underline{\\mathrm{NS}} \\otimes \\mathbb{Q}_{p} \/ \\mathbb{Z}_{p})\n\t\t\t\t\t\\cap\n\t\t\t\t\t\t\\underline{\\mathrm{U}}^{2}(p^{\\infty})\n\t\t\t\t}\n\t\t\t\\to\n\t\t\t\t\\underline{\\mathrm{Br}}_{p^{\\infty}}\n\t\t\t\\to\n\t\t\t\t\\frac{\n\t\t\t\t\t\\underline{\\mathrm{D}}^{2}(p^{\\infty})\n\t\t\t\t}{\n\t\t\t\t\t\\underline{\\mathrm{NS}} \\otimes \\mathbb{Q}_{p} \/ \\mathbb{Z}_{p}\n\t\t\t\t}\n\t\t\t\\to\n\t\t\t\t0.\n\t\t\\]\n\tThe intersection\n\t$(\\underline{\\mathrm{NS}} \\otimes \\mathbb{Q}_{p} \/ \\mathbb{Z}_{p})\n\t\\cap \\underline{\\mathrm{U}}^{2}(p^{\\infty})$\n\tis a finite \\'etale group\n\tsince the unipotent group $\\underline{\\mathrm{U}}^{2}(p^{\\infty})$ over $\\mathbb{F}_{q}$ is\n\tkilled by a power of $p$.\n\tHence the first term in the above exact sequence is\n\tthe perfection of a connected unipotent algebraic group.\n\tThe third term is an \\'etale group scheme.\n\tTherefore $\\underline{\\mathrm{Br}}_{p^{\\infty}}$ is represented by\n\ta perfect group scheme,\n\tand the above exact sequence is its connected-\\'etale sequence.\n\\end{proof}\n\nFor a prime $l \\ne p$, define $\\underline{\\Br}_{l^{\\infty}}$ to be the \\'etale group scheme over $\\mathbb{F}_{q}$\nwhose group of $\\bar{\\mathbb{F}}_{q}$-points is $\\Br(\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}})[l^{\\infty}]$,\nwhere $\\Br$ denotes the usual Brauer group for schemes.\nDefine $\\underline{\\Br}$ to be the direct sum of\n$\\underline{\\Br}_{l^{\\infty}}$ over all primes $l$,\nwhich is again a perfect group scheme.\nFor any perfect field extension $k \/ \\mathbb{F}_{q}$, we have a canonical isomorphism\n\t\\[\n\t\t\t\\underline{\\mathrm{Br}}(k)\n\t\t\\cong\n\t\t\t\\mathrm{Br}(\\mathcal{S}_{\\bar{k}})^{\\Gal(\\bar{k} \/ k)}\n\t\\]\nfunctorial in $k$.\n\n\\begin{remark}\n\tThere is a natural map $\\mathrm{Br}(\\mathcal{S}) \\to \\underline{\\mathrm{Br}}(\\mathbb{F}_{q})$.\n\tIt can be shown that the kernel and cokernel of this map are finite,\n\tand the kernel and cokernel of the map\n\t$\\mathrm{Br}(\\mathcal{S}_{\\mathbb{F}_{q, n}}) \\to \\underline{\\mathrm{Br}}(\\mathbb{F}_{q, n})$\n\tare finite of order bounded in $n$,\n\tby an argument similar to the proof of Proposition \\ref{p:ShaGone},\n\tusing \\cite{Mil76} instead of \\cite{Suz19}.\n\tAn explicit bound for large $n$ involves the order of the torsion part of $\\underline{\\mathrm{NS}}$\n\tand the discriminant of the intersection pairing on $\\underline{\\mathrm{NS}} \/ \\mathrm{tor}$.\n\tWe do not use this fact, so we do not prove it.\n\tSee \\cite[Section 6]{Art74b} for a discussion about this map\n\tin the special case of an elliptic supersingular K3 surface over a large enough finite field.%\n\t\\footnote{Note however that \\cite[(6.1)]{Art74b} claims that\n\tthe map $\\mathrm{Br}(\\mathcal{S}) \\to \\underline{\\mathrm{Br}}(\\mathbb{F}_{q})$ is bijective for such $\\mathcal{S}$.\n\tThis is not correct: the map is injective, but\n\tthe cokernel is isomorphic to the discriminant group\n\t$\\mathrm{NS}(\\mathcal{S})^{\\ast} \/ \\mathrm{NS}(\\mathcal{S}) \\ne 0$\n\tof the intersection pairing on $\\mathcal{S}$,\n\tas everywhere else in \\cite[Section 6]{Art74b} correctly suggests.}\n\\end{remark}\n\n\\begin{definition}\n\tDefine $s_{\\mathcal{S}}$ to be the common integer\n\t\t\\[\n\t\t\t\t\\dim \\underline{\\coh}^{i}(p^{\\infty})\n\t\t\t=\n\t\t\t\t\\dim \\underline{\\mathrm{U}}^{i}(p^{\\infty})\n\t\t\t=\n\t\t\t\t\\dim \\underline{\\mathrm{Br}}_{p^{\\infty}}\n\t\t\t=\n\t\t\t\t\\dim \\underline{\\mathrm{Br}}.\n\t\t\\]\n\\end{definition}\n\n\\begin{remark}\\label{sdef}\nNote that the integer $s_{\\mathcal{S}}$  is also equal\nto the length of\n\t\\[\n\t\t\\coh^2(\\mathcal{S}, W \\mathcal{O}_\\mathcal{S})[p^{\\infty}] \\otimes_{W[[V]]} W((V))\n\t\\]\nas a $W((V))$-module,\nwhere $W = W(\\mathbb{F}_{q})$ and $V$ denotes the Verschiebung;\nsee \\cite[(1.3)]{Mil75} (see also \\cite[Lemma 2.1, Proposition 4.4]{MR15}).\nIf the formal Brauer group of $\\mathcal{S}$ is\npro-representable by a formal Lie group\n(such as when $\\Pic_{\\mathcal{S} \/ \\mathbb{F}_{q}}$ is smooth \\cite[(4.1)]{AM77}),\nthen its Dieudonn\\'e module is\ngiven by $\\coh^2(\\mathcal{S}, W \\mathcal{O}_\\mathcal{S})$ (\\cite[II, (4.3)]{AM77}).\nTherefore in this case, $s_{\\mathcal{S}}$ is also equal to the\ndimension of the unipotent part of the formal Brauer group.\n\\end{remark}\n\n\\begin{proposition} \\label{p:ShaBr}\n\tThere exists a canonical morphism\n\t$\\mathcal{G}_{A}^{(1)} \\to \\underline{\\mathrm{Br}}$\n\tof perfect group schemes over $\\mathbb{F}_{q}$\n\twith finite \\'etale kernel and cokernel.\n\tIn particular,\n\t\t\\[\n\t\t\t\ts_{A \/ K} = s_{\\mathcal{S}}\n\t\t\\]\n\t(which is equal to $\\mu_{A \/ K}$ by Theorem \\ref{muissA}).\n\\end{proposition}\n\n\\begin{proof}\n\tLet $j \\colon \\spec K \\hookrightarrow \\mathcal{C}$ be the inclusion.\n\tLet $\\mathrm{Pic}_{\\mathcal{S} \/ \\mathcal{C}}$ and $\\mathrm{Pic}_{\\mathcal{S}_{K} \/ K}$ be the Picard functors sheafified in the \\'etale topology\n\t(\\cite[Definition 2.2]{Kle05}, \\cite[8.1]{BLR90}).\n\tFor any perfect field extension $k \/ \\mathbb{F}_{q}$, we have canonical homomorphisms\n\t\t\\[\n\t\t\t\t\\Br(\\mathcal{S}_{k})\n\t\t\t\\to\n\t\t\t\t\\coh^{1}(\\mathcal{C}_{k}, \\Pic_{\\mathcal{S} \/ \\mathcal{C}})\n\t\t\t\\to\n\t\t\t\t\\coh^{1}(\\mathcal{C}_{k}, j_{\\ast} \\Pic_{\\mathcal{S}_{K} \/ K})\n\t\t\\]\n\tfunctorial in $k$, which are isomorphisms if $k$ is algebraically closed,\n\tby \\cite[Proposition (4.3), Equation (4.14 bis)]{Gro68}.\n\t\n\tOn the other hand, consider the exact sequence\n\t$0 \\to A \\to \\Pic_{\\mathcal{S}_{K} \/ K} \\to \\mathbb{Z} \\to 0$ over $K_{\\mathrm{et}}$.\n\tApplying $j_{\\ast}$, we have an exact sequence\n\t$0 \\to \\mathcal{A} \\to j_{\\ast} \\Pic_{\\mathcal{S}_{K} \/ K} \\to \\mathbb{Z}$ over $\\mathcal{C}_{\\mathrm{et}}$.\n\tLet $Q$ be the image of $j_{\\ast} \\Pic_{\\mathcal{S}_{K} \/ K} \\to \\mathbb{Z}$,\n\twhich is a $\\mathbb{Z}$-constructible \\'etale subsheaf of $\\mathbb{Z}$ on $\\mathcal{C}$.\n\tThe quotient $\\mathbb{Z} \/ Q$ is a skyscraper sheaf with finite stalks.\n\tConsider the exact sequence $0 \\to \\mathcal{A} \\to j_{\\ast} \\Pic_{\\mathcal{S}_{K} \/ K} \\to Q \\to 0$.\n\tFor any perfect field extension $k \/ \\mathbb{F}_{q}$, this induces an exact sequence\n\t\t\\[\n\t\t\t\tQ(\\mathcal{C}_{k})\n\t\t\t\\to\n\t\t\t\t\\coh^{1}(\\mathcal{C}_{k}, \\mathcal{A})\n\t\t\t\\to\n\t\t\t\t\\coh^{1}(\\mathcal{C}_{k}, j_{\\ast} \\Pic_{\\mathcal{S}_{K} \/ K})\n\t\t\t\\to\n\t\t\t\t\\coh^{1}(\\mathcal{C}_{k}, Q)\n\t\t\\]\n\tfunctorial in $k$.\n\tThe image of $Q(\\mathcal{C}_{k}) \\to \\coh^{1}(\\mathcal{C}_{k}, \\mathcal{A})$ is torsion\n\tsince $\\coh^{1}(\\mathcal{C}_{k}, \\mathcal{A})$ is torsion\n\tby the proof of \\cite[III, Lemma 11.5]{Mil06}.\n\tIf $k$ is algebraically closed, then $Q(\\mathcal{C}_{k})$ is finitely generated\n\tand $\\coh^{1}(\\mathcal{C}_{k}, Q)$ is finite\n\tby the properties of $Q$ seen above.\n\tTherefore as functors in $k$, the kernel and cokernel of\n\t\t\\[\n\t\t\t\t\\coh^{1}(\\mathcal{C}_{\\bar{k}}, \\mathcal{A})^{\\Gal(\\bar{k} \/ k)}\n\t\t\t\\to\n\t\t\t\t\\coh^{1}(\\mathcal{C}_{\\bar{k}}, j_{\\ast} \\Pic_{\\mathcal{S}_{K} \/ K})^{\\Gal(\\bar{k} \/ k)}\n\t\t\\]\n\tare represented by finite \\'etale group schemes over $\\mathbb{F}_{q}$.\n\t\n\tNow we have a homomorphism and an isomorphism\n\t\t\\[\n\t\t\t\t\\coh^{1}(\\mathcal{C}_{\\bar{k}}, \\mathcal{A})^{\\Gal(\\bar{k} \/ k)}\n\t\t\t\\to\n\t\t\t\t\\coh^{1}(\\mathcal{C}_{\\bar{k}}, j_{\\ast} \\Pic_{\\mathcal{S}_{K} \/ K})^{\\Gal(\\bar{k} \/ k)}\n\t\t\t\\overset{\\sim}{\\gets}\n\t\t\t\t\\Br(\\mathcal{S}_{\\bar{k}})^{\\Gal(\\bar{k} \/ k)}\n\t\t\\]\n\tfunctorial in $k$.\n\tThis gives the required morphism\n\t$\\mathcal{G}_{A}^{(1)} \\to \\underline{\\mathrm{Br}}$.\n\\end{proof}\n\n\n\\subsection{The trace and the Picard variety}\n\n\\begin{proposition} \\label{p:TrPic}\n\tLet $\\Pic_{\\mathcal{S} \/ \\mathbb{F}_{q}, \\mathrm{red}}^{0}$ be the reduced part of $\\Pic_{\\mathcal{S} \/ \\mathbb{F}_{q}}^{0}$.\n\tThen there exist canonical morphisms\n\t$\\Pic_{\\mathcal{C} \/ \\mathbb{F}_{q}}^{0} \\to \\Pic_{\\mathcal{S} \/ \\mathbb{F}_{q}, \\mathrm{red}}^{0} \\to \\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)$\n\tover $\\mathbb{F}_{q}$, which induce an exact sequence\n\t\t\\[\n\t\t\t\t0\n\t\t\t\\to\n\t\t\t\t\\Pic_{\\mathcal{C} \/ \\mathbb{F}_{q}}^{0}\n\t\t\t\\to\n\t\t\t\t\\Pic_{\\mathcal{S} \/ \\mathbb{F}_{q}, \\mathrm{red}}^{0}\n\t\t\t\\to\n\t\t\t\t\\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)\n\t\t\t\\to\n\t\t\t\t0\n\t\t\\]\n\tin the category of abelian varieties up to isogeny over $\\mathbb{F}_{q}$.\n\tIn particular, we have\n\t\t\\[\n\t\t\t\t\\dim \\Pic_{\\mathcal{S} \/ \\mathbb{F}_{q}, \\mathrm{red}}^{0}\n\t\t\t=\n\t\t\t\t\\dim \\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A) + g_{\\mathcal{C}}.\n\t\t\\]\n\\end{proposition}\n\n\\begin{proof}\n\tThe morphism\n\t$\\Pic_{\\mathcal{C} \/ \\mathbb{F}_{q}}^{0} \\to \\Pic_{\\mathcal{S} \/ \\mathbb{F}_{q}, \\mathrm{red}}^{0}$\n\tis the obvious one.\n\tTo define the second morphism,\n\tconsider the inclusion\n\t\t$\n\t\t\t\t\\mathcal{S}_{K}\n\t\t\t=\n\t\t\t\t\\mathcal{S} \\times_{\\mathcal{C}} K\n\t\t\t\\hookrightarrow\n\t\t\t\t\\mathcal{S} \\times_{\\mathbb{F}_{q}} K\n\t\t$\n\tover $K$.\n\tThis induces a morphism\n\t\t\\[\n\t\t\t\t(\\Pic_{\\mathcal{S} \/ \\mathbb{F}_{q}}) \\times_{\\mathbb{F}_{q}} K\n\t\t\t=\n\t\t\t\t\\Pic_{(\\mathcal{S} \\times_{\\mathbb{F}_{q}} K) \/ K}\n\t\t\t\\to\n\t\t\t\t\\Pic_{\\mathcal{S}_{K} \/ K}\n\t\t\\]\n\tand hence a morphism\n\t\t\\[\n\t\t\t\t(\\Pic_{\\mathcal{S} \/ \\mathbb{F}_{q}, \\mathrm{red}}^{0}) \\times_{\\mathbb{F}_{q}} K\n\t\t\t\\to\n\t\t\t\t\\Pic_{\\mathcal{S}_{K} \/ K}^{0}\n\t\t\t=\n\t\t\t\tA\n\t\t\\]\n\tover $K$.\n\tThe universal property of the trace then induces a morphism\n\t$\\Pic_{\\mathcal{S} \/ \\mathbb{F}_{q}, \\mathrm{red}}^{0} \\to \\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)$\n\tover $\\mathbb{F}_{q}$.\n\t\n\tTo prove the second claim, it is enough to show that\n\tthese morphisms induce an exact sequence\n\t\t\\[\n\t\t\t\t0\n\t\t\t\\to\n\t\t\t\t\\Pic_{\\mathcal{C} \/ \\mathbb{F}_{q}}^{0}(\\bar{\\mathbb{F}}_{q})\n\t\t\t\\to\n\t\t\t\t\\Pic_{\\mathcal{S} \/ \\mathbb{F}_{q}}^{0}(\\bar{\\mathbb{F}}_{q})\n\t\t\t\\to\n\t\t\t\t\\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)(\\bar{\\mathbb{F}}_{q})\n\t\t\t\\to\n\t\t\t\t0\n\t\t\\]\n\tup to finite groups.\n\tBy \\cite[Equation (4.14)]{Gro68}, there exists a canonical exact sequence\n\t\t\\[\n\t\t\t\t0\n\t\t\t\\to\n\t\t\t\tE\n\t\t\t\\to\n\t\t\t\t\\Pic_{\\mathcal{S} \/ \\mathcal{C}}(\\mathcal{C}_{\\bar{\\mathbb{F}}_{q}})\n\t\t\t\\to\n\t\t\t\t\\Pic_{\\mathcal{S}_{K \/ K}}(K \\bar{\\mathbb{F}}_{q})\n\t\t\t\\to\n\t\t\t\t0\n\t\t\\]\n\tfor some finitely generated group $E$.\n\tSince $\\Br(\\mathcal{C}_{\\bar{\\mathbb{F}}_{q}}) = 0$,\n\twe have a canonical exact sequence\n\t\t\\[\n\t\t\t\t0\n\t\t\t\\to\n\t\t\t\t\\Pic(\\mathcal{C}_{\\bar{\\mathbb{F}}_{q}})\n\t\t\t\\to\n\t\t\t\t\\Pic(\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}})\n\t\t\t\\to\n\t\t\t\t\\Pic_{\\mathcal{S} \/ \\mathcal{C}}(\\mathcal{C}_{\\bar{\\mathbb{F}}_{q}})\n\t\t\t\\to\n\t\t\t\t0\n\t\t\\]\n\tby \\cite[Equation (4.5)]{Gro68}.\n\tHence we have an exact sequence\n\t\t\\[\n\t\t\t\t0\n\t\t\t\\to\n\t\t\t\t\\Pic(\\mathcal{C}_{\\bar{\\mathbb{F}}_{q}}) \\oplus E\n\t\t\t\\to\n\t\t\t\t\\Pic(\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}})\n\t\t\t\\to\n\t\t\t\t\\Pic_{\\mathcal{S}_{K \/ K}}(K \\bar{\\mathbb{F}}_{q})\n\t\t\t\\to\n\t\t\t\t0.\n\t\t\\]\n\tThe group $\\Pic_{\\mathcal{S} \/ \\mathbb{F}_{q}}^{0}(\\bar{\\mathbb{F}}_{q})$ is the divisible subgroup of\n\t$\\Pic(\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}})$ and the quotient of $\\Pic(\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}})$\n\tby $\\Pic_{\\mathcal{S} \/ \\mathbb{F}_{q}}^{0}(\\bar{\\mathbb{F}}_{q})$ is $\\mathrm{NS}(\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}})$,\n\twhich is finitely generated.\n\tOn the other hand,\n\tthe group $\\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)(\\bar{\\mathbb{F}}_{q})$\n\tis the divisible subgroup of $A(K \\bar{\\mathbb{F}}_{q})$\n\tand the quotient of $A(K \\bar{\\mathbb{F}}_{q})$ by\n\t$\\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)(\\bar{\\mathbb{F}}_{q})$ is finitely generated\n\tby the Lang-N\\'eron theorem (\\cite[Theorem 7.1]{Con06}).\n\tSince $\\Pic_{\\mathcal{S}_{K} \/ K}(K \\bar{\\mathbb{F}}_{q}) \/ A(K \\bar{\\mathbb{F}}_{q})$ injects into $\\mathbb{Z}$,\n\tthis implies that the group $\\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)(\\bar{\\mathbb{F}}_{q})$\n\tis the divisible subgroup of $\\Pic_{\\mathcal{S}_{K} \/ K}(K \\bar{\\mathbb{F}}_{q})$\n\tand the quotient of $\\Pic_{\\mathcal{S}_{K} \/ K}(K \\bar{\\mathbb{F}}_{q})$ by\n\t$\\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)(\\bar{\\mathbb{F}}_{q})$ is finitely generated.\n\tTherefore the above sequence induces an exact sequence\n\t\t\\[\n\t\t\t\t0\n\t\t\t\\to\n\t\t\t\t\\Pic_{\\mathcal{C} \/ \\mathbb{F}_{q}}^{0}(\\bar{\\mathbb{F}}_{q})\n\t\t\t\\to\n\t\t\t\t\\Ker \\bigl(\n\t\t\t\t\t\t\\Pic_{\\mathcal{C} \/ \\mathbb{F}_{q}}^{0}(\\bar{\\mathbb{F}}_{q})\n\t\t\t\t\t\\twoheadrightarrow\n\t\t\t\t\t\t\\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)(\\bar{\\mathbb{F}}_{q})\n\t\t\t\t\\bigr)\n\t\t\t\\to\n\t\t\t\t\\mathbb{Z} \\oplus E,\n\t\t\\]\n\tand the morphism from the middle kernel term to $\\mathbb{Z} \\oplus E$ has finite image.\n\tThis proves the second claim.\n\\end{proof}\n\n\n\\subsection{The $L$-function and the zeta function}\n\nWrite the zeta function of $\\mathcal{S}$ as\n\t\\[\n\t\t\tZ(\\mathcal{S}, s)\n\t\t=\n\t\t\t\\frac{\n\t\t\t\tP_{\\mathcal{S}, 1}(t) P_{\\mathcal{S}, 3}(t)\n\t\t\t}{\n\t\t\t\tP_{\\mathcal{S}, 0}(t) P_{\\mathcal{S}, 2}(t) P_{\\mathcal{S}, 4}(t)\n\t\t\t},\n\t\\]\nwhere $P_{\\mathcal{S}, i}(t) \\in \\mathbb{Z}[t]$ is a polynomial in $t = q^{-s}$ with constant term $1$\nwhose reciprocal roots are Weil $q$-numbers $\\{\\alpha_{\\mathcal{S}, ij}\\}$ of weight $i$.\nDefine $\\theta_{\\mathcal{S}}$ to be the non-negative number such that\n$q^{\\theta_{\\mathcal{S}}} P_{\\mathcal{S}, 2}(t \/ q)$ is $p$-primitive.\nLet $\\lambda_{\\mathcal{S}, j}$ be the $q$-valuation of $\\alpha_{\\mathcal{S}, 2 j}$.\nThen again we have\n\t\\[\n\t\t\t\\theta_{\\mathcal{S}}\n\t\t=\n\t\t\t\\sum_{\\lambda_{\\mathcal{S}, j} < 1} (1 - \\lambda_{\\mathcal{S}, j}).\n\t\\]\n\n\\begin{proposition} \\label{p:Lcomp}\n\tThe zeros and poles of the rational function\n\t$P_{2, \\mathcal{S}}(t \/ q) \/ P_{1}(t \/ q)$\n\tare roots of unity.\n\tIn particular, we have $\\theta_{\\mathcal{S}} = \\theta_{A}$.\n\\end{proposition}\n\n\\begin{proof}\n\tLet $l \\ne p$ be a prime number.\n\tWe have an exact sequence\n\t\t\\begin{equation} \\label{e:NSBr}\n\t\t\t\t0\n\t\t\t\\to\n\t\t\t\t\\mathrm{NS}(\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}}) \\otimes \\mathbb{Q}_{l}\n\t\t\t\\to\n\t\t\t\t\\coh^{2}(\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}}, \\mathbb{Q}_{l}(1))\n\t\t\t\\to\n\t\t\t\t\\mathrm{V}_{l}(\\Br(\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}}))\n\t\t\t\\to\n\t\t\t\t0\n\t\t\\end{equation}\n\tof $l$-adic representations over $\\mathbb{F}_{q}$,\n\twhere $\\mathrm{V}_{l}$ denotes the $l$-adic Tate module tensored with $\\mathbb{Q}_{l}$\n\tand the cohomology in the middle term is the continuous cohomology.\n\tThe characteristic polynomial of geometric Frobenius\n\ton the middle term $\\coh^{2}(\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}}, \\mathbb{Q}_{l}(1))$ is $P_{\\mathcal{S}, 2}(t \/ q)$.\n\tSince the Galois action on the left term\n\t$\\mathrm{NS}(\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}}) \\otimes \\mathbb{Q}_{l}$\n\tfactors through a finite quotient,\n\tits Frobenius eigenvalues are roots of unity.\n\t\n\tOn the other hand,\n\tlet $\\mathcal{A}^{0}$ be the part of $\\mathcal{A}$ with connected fibers.\n\tFor any $n \\ge 0$, the sequence\n\t$0 \\to \\mathcal{A}^{0}[l^{n}] \\to \\mathcal{A}^{0} \\overset{l^{n}}{\\to} \\mathcal{A}^{0} \\to 0$\n\tis exact by \\cite[7.3\/1, 2]{BLR90}.\n\tHence it induces an exact sequence\n\t\t\\[\n\t\t\t\t0\n\t\t\t\\to\n\t\t\t\t\\mathcal{A}^{0}(\\mathcal{C}_{\\bar{\\mathbb{F}}_{q}}) \\otimes \\mathbb{Z} \/ l^{n} \\mathbb{Z}\n\t\t\t\\to\n\t\t\t\t\\coh^{1}(\\mathcal{C}_{\\bar{\\mathbb{F}}_{q}}, \\mathcal{A}^{0}[l^{n}])\n\t\t\t\\to\n\t\t\t\t\\coh^{1}(\\mathcal{C}_{\\bar{\\mathbb{F}}_{q}}, \\mathcal{A}^{0})[l^{n}]\n\t\t\t\\to\n\t\t\t\t0\n\t\t\\]\n\tof $\\mathbb{Z} \/ l^{n} \\mathbb{Z}$-representations over $\\mathbb{F}_{q}$.\n\tTaking the inverse limit in $n$ and the tensor product with $\\mathbb{Q}_{l}$,\n\twe have an exact sequence\n\t\t\\begin{equation} \\label{e:MWSha}\n\t\t\t\t0\n\t\t\t\\to\n\t\t\t\t\\frac{A(K \\bar{\\mathbb{F}}_{q})}{\\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)(\\bar{\\mathbb{F}}_{q})} \\otimes \\mathbb{Q}_{l}\n\t\t\t\\to\n\t\t\t\t\\coh^{1}(\\mathcal{C}_{\\bar{\\mathbb{F}}_{q}}, \\mathrm{V}_{l}(\\mathcal{A}))\n\t\t\t\\to\n\t\t\t\t\\mathrm{V}_{l}(\\coh^{1}(\\mathcal{C}_{\\bar{\\mathbb{F}}_{q}}, \\mathcal{A}))\n\t\t\t\\to\n\t\t\t\t0\n\t\t\\end{equation}\n\tof $l$-adic representations over $\\mathbb{F}_{q}$.\n\tBy \\cite[Satz 1]{Sch82},\n\tthe characteristic polynomial of geometric Frobenius\n\ton the middle term $\\coh^{1}(\\mathcal{C}_{\\bar{\\mathbb{F}}_{q}}, \\mathrm{V}_{l}(\\mathcal{A}))$ is $P_{1}(t \/ q)$.\n\tSince the Galois action on the left term\n\t$A(K \\bar{\\mathbb{F}}_{q}) \/ \\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)(\\bar{\\mathbb{F}}_{q}) \\otimes \\mathbb{Q}_{l}$\n\tfactors through a finite quotient,\n\tits Frobenius eigenvalues are roots of unity.\n\t\n\tNow the morphism in Proposition \\ref{p:ShaBr} induces an isomorphism\n\t\t\\[\n\t\t\t\t\\mathrm{V}_{l}(\\coh^{1}(\\mathcal{C}_{\\bar{\\mathbb{F}}_{q}}, \\mathcal{A}))\n\t\t\t\\overset{\\sim}{\\to}\n\t\t\t\t\\mathrm{V}_{l}(\\Br(\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}})).\n\t\t\\]\n\tThe result follows from this.\n\\end{proof}\n\n\n\\subsection{Comparison and criteria}\n\nDefine\n\t\\[\n\t\t\t\\chi(\\mathcal{S}, \\mathcal{O}_{\\mathcal{S}})\n\t\t=\n\t\t\t\\sum_{n} (-1)^{n} \\dim_{\\mathbb{F}_{q}} \\coh^{n}(\\mathcal{S}, \\mathcal{O}_{\\mathcal{S}}).\n\t\\]\nBy \\cite[Equation (14)]{LLR04}, we have\n\t\\[\n\t\t\t\\chi(\\mathcal{S}, \\mathcal{O}_{\\mathcal{S}})\n\t\t=\n\t\t\t\\deg(\\mathcal{L}) + (1 - \\dim(A)) (1 - g_{\\mathcal{C}}).\n\t\\]\nBy Milne's formula \\cite[the last equation of Section 6]{Mil75} on Euler characteristics\nor Crew's formula \\cite[Proposition 6.3.9]{Ill83} on Hodge-Witt numbers, we have\n\t\\begin{equation} \\label{e:Milne}\n\t\t\t\\sum_{\\lambda_{\\mathcal{S}, j} < 1} (1 - \\lambda_{\\mathcal{S}, j})\n\t\t=\n\t\t\t\t\\chi(\\mathcal{S}, \\mathcal{O}_{\\mathcal{S}})\n\t\t\t-\n\t\t\t\t(1 - \\dim \\Pic_{\\mathcal{S} \/ \\mathbb{F}_{q}, \\mathrm{red}}^{0} + s_{\\mathcal{S}}).\n\t\\end{equation}\nCombining these with Propositions \\ref{p:ShaBr} and \\ref{p:TrPic}, we have:\n\n\\begin{corollary} \\label{c:MilUlm}\n\tThe formula \\eqref{e:Milne} is equivalent to the formula\n\t\\eqref{e:muform} for the Jacobian $A$.\n\tIn particular, the formula \\eqref{e:muform} holds for $A$\n\twithout any assumption on $\\Sha$ (or the reduction type of $A$).\n\tIn particular, it holds for any elliptic curve $A \/ K$.\n\\end{corollary}\n\nThe equivalence between Milne's formula for $\\mathcal{S}$\nand the formula \\eqref{e:muform} for a Jacobian $A$ is\nsuggested by Ulmer \\cite[Remark 4.3 (3), (4)]{Ulm19}.\nThe above corollary verifies his suggestion.\n\nWe will give some criteria and cases of $\\mu = 0$ and $\\mu > 0$.\n\n\\begin{proposition}\\label{p:trivmu}\n\tThe following are equivalent:\n\t\\begin{itemize}\n\t\t\\item[(i)] $\\mu_{A \/ K} = 0$.\n\t\t\\item[(ii)] $\\coh_{fl}^{2}(\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}}, \\mu_{p})$ is finite.\n\t\t\\item[(iii)] $\\Br(\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}})[p]$ is finite.\n\t\t\\item[(iv)] $\\mathcal{S}$ is of Hodge-Witt type, i.e.\\\n\t\t$\\coh^i(\\mathcal{S},W \\Omega^j_{\\mathcal{S}})$ is a finitely generated $W(\\mathbb{F}_q)$-module for any $i,j$.\n\t\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}\n\tLet $W = W(\\mathbb{F}_{q})$.\n\tFrom the definition of $s_{\\mathcal{S}}$,\n\twe know that $\\mu_{A \/ K} = s_{\\mathcal{S}} = 0$ if and only if $\\underline{\\coh}^{2}(\\mu_{p^{\\infty}})$ is \\'etale.\n\tAs $\\underline{\\mathrm{U}}^{2}(\\mu_{p^{\\infty}})$ is always unipotent,\n\tthis is equivalent to $\\underline{\\coh}^{2}(\\mu_{p^{\\infty}})[p]$ being \\'etale.\n\tThe group scheme $\\underline{\\coh}^{1}(\\mu_{p^{\\infty}})$ is always \\'etale\n\t(\\cite[the paragraph after Theorem (3.1)]{Art74b}, \\cite[Theorem 5.2]{Mil76}).\n\tHence the exact sequence\n\t\t\\[\n\t\t\t\t0\n\t\t\t\\to\n\t\t\t\t\\underline{\\coh}^{1}(\\mu_{p^{\\infty}}) \/ p \\underline{\\coh}^{1}(\\mu_{p^{\\infty}})\n\t\t\t\\to\n\t\t\t\t\\underline{\\coh}^{2}(\\mu_{p})\n\t\t\t\\to\n\t\t\t\t\\underline{\\coh}^{2}(\\mu_{p^{\\infty}})[p]\n\t\t\t\\to\n\t\t\t\t0\n\t\t\\]\n\tshows that $\\underline{\\coh}^{2}(\\mu_{p^{\\infty}})[p]$ is \\'etale\n\tif and only if $\\underline{\\coh}^{2}(\\mu_{p})$ is \\'etale.\n\tSince $\\underline{\\coh}^{2}(\\mu_{p})$ is the perfection of a smooth algebraic group,\n\tthis is equivalent to its group of geometric points $\\coh_{fl}^{2}(\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}}, \\mu_{p})$ being finite.\n\tThe group $\\Pic(\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}}) \/ p \\Pic(\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}})$ is always finite.\n\tHence $\\coh_{fl}^{2}(\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}}, \\mu_{p})$ is finite if and only if\n\t$\\Br(\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}})[p]$ is finite. This proves the first three equivalences.\n\t\n\tFor the equivalence of (i) and (iv), first recall that\n\t$\\coh^2(\\mathcal{S}, W \\mathcal{O}_{\\mathcal{S}})$ is finitely generated over $W[[V]]$,\n\tits $p$-primary torsion submodule $\\coh^2(\\mathcal{S}, W \\mathcal{O}_{\\mathcal{S}})[p^{\\infty}]$ is killed by a power of $p$,\n\tand the quotient of $\\coh^2(\\mathcal{S}, W \\mathcal{O}_{\\mathcal{S}})$\n\tby $\\coh^2(\\mathcal{S}, W \\mathcal{O}_{\\mathcal{S}})[p^{\\infty}]$ is finite free over $W$\n\tby \\cite[II, Corollaire 2.11 and Th\\'eor\\`eme 2.13]{Ill79}.\n\tIn particular, $\\mu_{A \/ K} = s_{\\mathcal{S}} = 0$ if and only if\n\t$\\coh^2(\\mathcal{S}, W \\mathcal{O}_{\\mathcal{S}})[p^{\\infty}] \\otimes_{W[[V]]} W((V))$ has length $0$ as $W((V))$-module,\n\ti.e.\\ is trivial.\n\t\n\tTo prove the equivalence, first assume that $\\coh^2(\\mathcal{S}, W \\mathcal{O}_{\\mathcal{S}})[p^{\\infty}] \\otimes_{W[[V]]}\n\tW((V))$ is zero. This implies that $\\coh^2(\\mathcal{S}, W \\mathcal{O}_{\\mathcal{S}})[p] \\otimes_{W[[V]]} W((V))$,\n\tor $\\coh^2(\\mathcal{S}, W \\mathcal{O}_{\\mathcal{S}})[p] \\otimes_{k[[V]]} k((V))$ is zero.\n\tSince $\\coh^2(\\mathcal{S}, W \\mathcal{O}_{\\mathcal{S}})[p]$ is finitely generated over $k\n\t[[V]]$, this implies that $\\coh^2(\\mathcal{S}, W \\mathcal{O}_{\\mathcal{S}})[p]$ is finite-dimensional over $k$.\n\tThen $\\coh^2(\\mathcal{S}, W \\mathcal{O}_{\\mathcal{S}})[p^{\\infty}]$ has finite length over $W$.\n\tHence $\\coh^2(\\mathcal{S}, W \\mathcal{O}_{\\mathcal{S}})$ is\n\tfinitely generated over $W$. Since $\\mathcal{S}$ is a surface,\n\tby Nygaard's theorem (\\cite[II, Corollaire 3.14]{Ill79}),\n\twe know that the slope spectral sequence for $\\mathcal{S}$ degenerates at $E_1$. Since\n\tcrystalline cohomology is finitely generated over $W$, this implies that\n\tall the $E_1$-terms are finitely generated over $W$.\n\tHence $\\mathcal{S}$ is of Hodge-Witt type.\n\t\n\tConversely, assume that $\\mathcal{S}$ is of Hodge-Witt type.\n\tThen $\\coh^2(\\mathcal{S}, W \\mathcal{O}_{\\mathcal{S}})[p^{\\infty}]$ has\n\tfinite length over $W$. Since the $V$-action is topologically nilpotent,\n\tthis implies that $\\coh^2(\\mathcal{S}, W \\mathcal{O}_{\\mathcal{S}})[p^{\\infty}]$ is killed by a power of $V$. This proves the claim.\n\\end{proof}\n\n\\begin{remark}\nIt is a non-trivial result \\cite[III, Th\\'eor\\`eme (4.13)]{IR83} that\nif $\\mathcal{S}$ is ordinary in the sense of Illusie-Raynaud \\cite[IV, D\\'efinition (4.12)]{IR83},\nthen it is of Hodge-Witt type.\nThe reciprocal statement does not hold however.\n\\end{remark}\n\n\\begin{proposition}\\label{p:aborK3}\n\tAssume that $\\mathcal{S}$ is relatively minimal over $\\mathcal{C}$\n\tand has Kodaira dimension $\\le 0$.\n\tThen $\\mu_{A \/ K} \\ne 0$ if and only if $\\mu_{A \/ K} = 1$\n\tif and only if $\\mathcal{S}$ is a supersingular abelian surface\n\tor a supersingular K3 surface.\n\\end{proposition}\n\n\\begin{proof}\n\tWe will use the Kodaira-Enriques classification (\\cite[\\S 6, \\S 7]{Lie13}).\n\tIf the Kodaira dimension of $\\mathcal{S}$ is $- \\infty$,\n\tthen $\\mathcal{S}$ is of Hodge-Witt type by\n\t\\cite[\\S 5.8, Proposition 5.8.1]{Jos14},\n\tso $\\mu_{A \/ K} = 0$.\n\tIf $\\mathcal{S}$ is an abelian surface or a K3 surface,\n\tthen it is not of Hodge-Witt type if and only if it is supersingular\n\tby \\cite[Proposition 5.9.1]{Jos14},\n\tin which case the formal Brauer group of $\\mathcal{S}$ is the formal completion of the additive group\n\tby \\cite[II, \\S 7.1, 7.2]{Ill79}\n\tand hence $\\mu_{A \/ K} = 1$\n\tby Remark \\ref{sdef}.\n\tIf $\\mathcal{S}$ is (quasi-)hyperelliptic, then\n\t$\\dim \\Pic_{\\mathcal{S} \/ \\mathbb{F}_{q}, \\mathrm{red}}^{0} = 1$\n\tand $\\chi(\\mathcal{S}, \\mathcal{O}_{\\mathcal{S}}) = 0$\n\tby the table in Introduction of \\cite{BM77}\n\t(with $\\Pic_{\\mathcal{S} \/ \\mathbb{F}_{q}}^{0}$ reduced or not).\n\tHence by \\cite[Corollary 5]{Suw83}, we know that $\\mathcal{S}$ is of Hodge-Witt type.\n\tIf $\\mathcal{S}$ is Enriques, then it is of Hodge-Witt type\n\tby \\cite[II, Corollaire 7.3.3 (a)]{Ill79}.\n\\end{proof}\n\nAn explicit example is given as follows.\n\n\\begin{proposition} \\label{p:Shio}\n\tSuppose $p \\equiv 3 \\mod 4$.\n\tLet $\\mathcal{S} \\to \\mathcal{C} = \\mathbb{P}_{\\mathbb{F}_{q}}^{1}$ be Shioda's modular elliptic surface of level $4$\n\t(\\cite[Appendices A-C]{Shi72}, \\cite[\\S 3]{Shi75}; see also \\cite{Shi20}),\n\twhose generic fiber $\\mathcal{S}_{K} = A$ is the elliptic curve defined by the equation\n\t\t\\[\n\t\t\t\ty^{2}\n\t\t\t=\n\t\t\t\tx (x - 1)\n\t\t\t\t\\left(\n\t\t\t\t\tx -\n\t\t\t\t\t\\frac{1}{4} \\left(\n\t\t\t\t\t\tt + \\frac{1}{t}\n\t\t\t\t\t\\right)^{2}\n\t\t\t\t\\right),\n\t\t\\]\n\twhere $t$ is the coordinate of $\\mathbb{P}_{\\mathbb{F}_{q}}^{1}$.\n\tThen $\\mu_{A \/ K} = 1$, and $A$ has semistable reduction everywhere.\n\\end{proposition}\n\n\\begin{proof}\n\tBy \\cite[Appendix A, viii)]{Shi72}, \\cite[\\S 3, Theorem 1, Corollary 1]{Shi75}\n\t(see also \\cite[Theorem 1.1 (1), (2)]{Shi20}),\n\twe know that $\\mathcal{S}$ is a supersingular K3 surface\n\tsemistable everywhere over $\\mathcal{C}$.\n\tHence this proposition is a special case of the previous proposition.\n\\end{proof}\n\nNot much can be said systematically about the case of positive Kodaira dimension,\nexcept Proposition \\ref{p:KodOne}, Section \\ref{su:generic} and\nan example in Section \\ref{su:high} below.\n\n\\begin{remark} \\mbox{}\n\t\\begin{enumerate}\n\t\\item\n\t\tIn the proof of Corollary \\ref{c:MilUlm}, we saw that\n\t\tCrew's formula on Hodge-Witt numbers of $\\mathcal{S}$ (with $i = 0$)\n\t\tis equivalent to the $\\mu$-invariant formula \\eqref{e:muform} for the Jacobian $A \/ K$.\n\t\tBy \\cite[(6.3.5-8)]{Ill83},\n\t\tCrew's formula is applicable to any object of $D^{b}_{c}(R)$,\n\t\twhere $R$ is the Raynaud ring.\n\t\tWe used this formula for the de Rham-Witt cohomology\n\t\t$R \\Gamma(\\mathcal{S}, W \\Omega_{\\mathcal{S}}) \\in D^{b}_{c}(R)$.\n\t\tOn the other hand, Illusie, at the end of \\cite{Ill83},\n\t\texpressed his hope to define a theory of de Rham-Witt complexes\n\t\twith coefficients in $F$-crystals.\n\t\t\n\t\tWith these observations in mind, one may speculate that\n\t\tthere should be a ``de Rham-Witt complex with coefficients in the N\\'eron model $\\mathcal{A}$'',\n\t\t``$W \\Omega_{\\mathcal{C}} \\otimes_{W \\mathcal{O}_{\\mathcal{C}}} D(\\mathcal{A})$'',\n\t\tcorresponding to an arbitrary abelian variety $A \/ K$,\n\t\tso that Crew's formula for\n\t\t\t$\n\t\t\t\t\tR \\Gamma \\bigl(\n\t\t\t\t\t\t\\mathcal{C},\n\t\t\t\t\t\tW \\Omega_{\\mathcal{C}} \\otimes_{W \\mathcal{O}_{\\mathcal{C}}} D(\\mathcal{A})\n\t\t\t\t\t\\bigr)\n\t\t\t\t\\in\n\t\t\t\t\tD^{b}_{c}(R)\n\t\t\t$\n\t\tcontains the $\\mu$-invariant formula \\eqref{e:muform} for $A \/ K$ as a special case.\n\t\\item\n\t\tLet $\\mathcal{S}$ and $A = \\Pic^{0}_{\\mathcal{S}_{K} \/ K}$ as above.\n\t\tConsider the canonical domino associated with the graded $R$-module\n\t\t\t$\n\t\t\t\t\t\\coh^{2}(\\mathcal{S}, W \\mathcal{O}_{\\mathcal{S}})\n\t\t\t\t\\to\n\t\t\t\t\t\\coh^{2}(\\mathcal{S}, W \\Omega_{\\mathcal{S}}^{1})\n\t\t\t$\n\t\tas in \\cite[Proposition 2.5.2 (ii)]{Ill83}.\n\t\tIt admits a finite filtration whose successive subquotients are isomorphic to\n\t\telementary dominoes\n\t\t\t$\n\t\t\t\t\\underline{\\underline{\\mathrm{U}}}_{i_{1}},\n\t\t\t\t\\dots,\n\t\t\t\t\\underline{\\underline{\\mathrm{U}}}_{i_{m}}\n\t\t\t$\n\t\t\\cite[\\S 2.2.2]{Ill83}.\n\t\tThe number $m$ of the elementary dominoes is\n\t\tthe dimension of this domino in the sense of \\cite[paragraph after Proposition 2.5.2]{Ill83},\n\t\twhich is equal to $\\mu_{A \/ K}$ by Remark \\ref{sdef}, Proposition \\ref{p:ShaBr}\n\t\tand \\cite[\\S 2.5, b2)]{Ill83}.\n\t\t\n\t\tAs a more precise relation,\n\t\tcan we read off the set of integers $i_{1}, \\dots, i_{m}$\n\t\tfrom the Iwasawa theory of $A \/ K$?\n\t\tWhen $\\mathcal{S}$ is a supersingular K3 surface (so $m = 1$),\n\t\tthe number $i = i_{1} = i_{m}$ is called the Artin invariant and usually denoted by $\\sigma_{0}$\n\t\t(\\cite[\\S 2.2.2 b)]{Ill83}, \\cite[Equation (4.6)]{Art74b}),\n\t\twhich satisfies the property that the discriminant of the intersection pairing\n\t\ton the geometric N\\'eron-Severi group of $\\mathcal{S}$ is $- p^{2 \\sigma_{0}}$ (where $p \\ne 2$).\n\t\tFor a more general $\\mathcal{S}$,\n\t\tthe integers $i_{1}, \\dots, i_{m}$ might be related to\n\t\tthe geometric Cassels-Tate discriminant group $\\delta_{\\mathrm{CT}}$\n\t\t\\cite[Theorem 3.4.1 (6g)]{Suz19} and\n\t\tthe analogue of Artin's period map\n\t\t\\cite[paragraphs after Theorem 3.4.1]{Suz19}.\n\t\t\n\t\tIt is an interesting problem to see\n\t\tif the numbers $i_{1}, \\dots, i_{m}$ define a nice stratification\n\t\ton the moduli of elliptic curves with given $\\deg(\\Delta)$ in Section \\ref{su:generic},\n\t\tgeneralizing Artin's filtration on the moduli of supersingular K3 surfaces \\cite[\\S 7]{Art74b}.\n\t\\end{enumerate}\n\\end{remark}\n\n\n\\section{The $\\mu$-invariant for semistable abelian varieties} \\label{s:ssav}\n\nThe aim of this section is to prove that\nthe $\\mu$-invariant formula \\eqref{e:muform} holds for semistable abelian varieties $A$\nwithout any assumption on $\\Sha$.\n\n\\subsection{Statement and the $p$-adic $L$-function}\n\n\\begin{theorem} \\label{t:muss}\n\tLet $A$ be a semistable abelian variety over $K$.\n\tThen the formula \\eqref{e:muform} holds for $A$.\n\\end{theorem}\n\nWe will prove this below.\nLet $Q(\\Lambda)$ be the fraction field of $\\Lambda$.\nLet $\\mathcal{Z}$ be a finite set of places of $K$\nsuch that $A$ has good reduction over $\\mathcal{C} \\setminus \\mathcal{Z}$.\nView the Frobenius element $\\mathrm{Frob}_{q} \\in \\Gal(\\mathbb{F}_{q,\\infty}\/\\mathbb{F}_q)$\nsending $x$ to $x^q$ as an element in $\\Gamma$.\nFor each place $v$, set $\\mathrm{Frob}_{v} = \\mathrm{Frob}_{q}^{\\deg(v)}$\nand $q_{v} = q^{\\deg(v)}$.\nLet $P_{v}(t)$ be the Euler factor at $v$ of the $L$-function of $A$,\nwhich is a polynomial in $t = q^{-s}$ with $\\mathbb{Z}$-coefficients and constant term $1$.\nLet $L_{A, \\mathcal{Z}}(s)$ be the $L$-function of $A$ without the Euler factors at points of $\\mathcal{Z}$,\nso that\n\t\\[\n\t\t\tL_{A, \\mathcal{Z}}(s)\n\t\t=\n\t\t\tL_{A}(s) \\cdot\n\t\t\t\\prod_{v \\in \\mathcal{Z}}\n\t\t\t\tP_{v}(q_{v}^{-s})\n\t\t=\n\t\t\t\\frac{\n\t\t\t\tP_{1}(q^{-s})\n\t\t\t}{\n\t\t\t\tP_{0}(q^{-s})\n\t\t\t\tP_{2}(q^{-s})\n\t\t\t}\n\t\t\t\\prod_{v \\in \\mathcal{Z}}\n\t\t\t\tP_{v}(q_{v}^{-s}).\n\t\\]\nPut\n\t\\begin{equation} \\label{e:padicL}\n\t\t\t\\mathcal{L}_{A\/K^{(p)}_\\infty}\n\t\t:=\n\t\t\t\\frac{\n\t\t\t\tP_{1}(q^{-1} \\mathrm{Frob}_{q}^{-1})\n\t\t\t}{\n\t\t\t\tP_{0}(q^{-1} \\mathrm{Frob}_{q}^{-1})\n\t\t\t\tP_{2}(q^{-1} \\mathrm{Frob}_{q}^{-1})\n\t\t\t}\n\t\t\t\\prod_{v \\in \\mathcal{Z}}\n\t\t\t\tP_{v}(q_{v}^{-1} \\mathrm{Frob}_{v}^{-1})\n\t\\end{equation}\nand regard it as an element in $Q(\\Lambda)$.\nIt is actually a rational function of the variable $\\mathrm{Frob}_{q}$ with $\\mathbb{Q}$-coefficients\n(i.e.\\ an element of the rational function field $\\mathbb{Q}(\\mathrm{Frob}_{q})$)\nand the substitution $\\mathrm{Frob}_{q}^{-1} \\mapsto q^{-s}$ turns it into\n$L_{A, \\mathcal{Z}}(s + 1)$.\n\n\\begin{proposition}\n\tThe element $\\mathcal{L}_{A\/K^{(p)}_\\infty}$ coincides with\n\tthe $p$-adic $L$-function of \\cite{LLTT16}.\n\\end{proposition}\n\n\\begin{proof}\n\tBy \\cite[\\S 3.1.2]{LLTT16},\n\tthe $p$-adic $L$-function is defined as\n\t\t\\begin{equation}\n\t\t\t\\prod_{i = 0}^{2}\n\t\t\t\t\\det\\nolimits_{\\Lambda[1\/p]} \\bigl(\n\t\t\t\t\t1 - \\Phi^{i}_{0} \\otimes [\\mathrm{Frob}_q^{-1}],\n\t\t\t\t\tP^{i}_{0} \\otimes_{\\mathbb{Z}_p} \\Lambda[1\/p]\n\t\t\t\t\\bigr)^{(-1)^{i + 1}},\n\t\t\\end{equation}\n\twhere $P^{i}_{0} := \\coh^{i}_{\\mathrm{log-crys},c}(\\mathcal C^\\sharp\/\\mathbb{Z}_p,D(A))$\n\tendowed with its Frobenius operator $\\Phi^{i}_0$\n\t(Here $D(A)$ is the log Dieudonn\\'e crystal\n\tassociated with the semistable abelian variety $A$ following \\cite{KT03}\n\tand $\\mathcal{C}^\\sharp$ is the log-scheme with underlying scheme $\\mathcal{C}$\n\tand log-structure induced by $\\mathcal{Z}$).\n\tOn the other hand, by \\cite[\\S 3.2.1]{KT03}, the right-hand side of \\eqref{e:padicL} is\n\t\t\\[\n\t\t\t\\prod_{i = 0}^{2}\n\t\t\t\t\\det\\nolimits_{\\mathbb{Q}_p}(\n\t\t\t\t\t1-\\mathrm{Frob}_q^{-1} \\Phi^{i}_0,\n\t\t\t\t\tP^{i}_0 \\otimes \\mathbb{Q}_p\n\t\t\t\t)^{(-1)^{i + 1}}.\n\t\t\\]\n\\end{proof}\n\nThen by \\cite[Theorem 1.1 (2)]{LLTT16},\nfor a characteristic element $c_{A\/K^{(p)}_\\infty}$ of $X_{A \/ K}$,\n\t\\begin{equation} \\label{e:IMC}\n\t\t\t\\mathcal{L}_{A\/K^{(p)}_\\infty}\n\t\t=\n\t\t\tu \\cdot \\star \\cdot c_{A\/K^{(p)}_\\infty},\n\t\\end{equation}\nwhere $u\\in\\Lambda^{\\times}$, and $\\star$ is given as follow.\nFor each $v \\in \\mathcal{Z}$, let $T_{v}$ be the maximal torus of $\\mathcal{A}_{v}^{0}$\nand set $B_{v} = \\mathcal{A}_{v}^{0} \/ T_{v}$\n(which is an abelian variety over $k(v)$ since $A$ is semistable).\nLet $g_{v} = \\dim(B_{v})$.\nLet $\\{\\beta_{i}^{(v)}\\}$ be the eigenvalues of $\\mathrm{Frob}_{v}$\non the $l$-adic Tate module of $B_{v}$ (for any $l \\ne p$)\nviewed as elements of $\\bar{\\mathbb{Q}}$ and hence $\\bar{\\mathbb{Q}}_{p}$.\n(Here, as usual, the Tate module $\\mathrm{T}_{l}(B_{v})$ means\n$\\mathrm{T}_{l}(B_{v}(\\bar{\\mathbb{F}}_{q}))$\nwith the natural action of the $q_{v}$-th power Frobenius $\\mathrm{Frob}_{v}$.)\n\nLet $\\zeta_{1}, \\dots, \\zeta_{m}$ be the eigenvalues of $\\mathrm{Frob}_{q}$\non the $p$-adic Tate module of $A(K_{\\infty}^{(p)})$.\nThen\n\t\\[\n\t\t\t\\star\n\t\t=\n\t\t\t\\frac{\n\t\t\t\tq^{- \\dim(A)(\\deg(\\mathcal{Z}) + g_{\\mathcal{C}} - 1) - \\deg(\\mathcal{L})} \\cdot\n\t\t\t\t\\prod_{v \\in \\mathcal{Z}} \\prod_{i = 1}^{2 g(v)}\n\t\t\t\t\t(\\beta_{i}^{(v)} - \\mathrm{Frob}_{v}^{-1})\n\t\t\t}{\n\t\t\t\t\\prod_{i = 1}^{m}\n\t\t\t\t\t(1 - \\zeta_{i}^{-1} \\mathrm{Frob}_{q})\n\t\t\t\t\t(1 - \\zeta_{i}^{-1} \\mathrm{Frob}_{q}^{-1})\n\t\t\t}.\n\t\\]\n(The term $\\delta$ in the original formula is equal to $\\deg(\\mathcal{L})$ here\nby \\cite[Proposition 3.3]{GS20} for example.)\n\n\n\\subsection{Local factors and the trace}\n\nTo prove the theorem, we will simplify both sides of \\eqref{e:IMC}.\nFor each $v \\in \\mathcal{Z}$, let $t(v) = \\dim(T_{v})$, so that $t(v) + g(v) = \\dim(A)$.\nLet $P_{v, \\mathrm{tor}}(t)$ (resp.\\ $P_{v, \\mathrm{ab}}(t)$) be\nthe characteristic polynomial of $q_{v} \\mathrm{Frob}_{v}^{-1}$\non $\\mathrm{V}_{l}(T_{v})$ (resp.\\ $\\mathrm{V}_{l}(B_{v})$),\nso that $P_{v}(t) = P_{v, \\mathrm{tor}}(t) P_{v, \\mathrm{ab}}(t)$.\nWe denote by $\\equiv$ the equality in\n\t$\n\t\t\tQ(\\Lambda \\otimes_{\\mathbb{Z}_{p}} \\mathcal{O})^{\\times}\n\t\t\/\n\t\t\t(\\Lambda \\otimes_{\\mathbb{Z}_{p}} \\mathcal{O})^{\\times}\n\t$,\nwhere $\\mathcal{O}$ is the ring of integers of some large enough finite extension of $\\mathbb{Q}_{p}$.\nNote that $Q(\\Lambda)^{\\times} \/ \\Lambda^{\\times}$ injects into\n\t$\n\t\t\tQ(\\Lambda \\otimes_{\\mathbb{Z}_{p}} \\mathcal{O})^{\\times}\n\t\t\/\n\t\t\t(\\Lambda \\otimes_{\\mathbb{Z}_{p}} \\mathcal{O})^{\\times}\n\t$.\n\n\\begin{lemma}\n\tFor any $v \\in \\mathcal{Z}$, we have\n\t\t\\begin{gather*}\n\t\t\t\t\tP_{v, \\mathrm{tor}}(q_{v}^{-1} \\mathrm{Frob}_{v}^{-1})\n\t\t\t\t\\equiv\n\t\t\t\t\tq_{v}^{- t(v)},\n\t\t\t\t\\quad\n\t\t\t\t\tP_{v, \\mathrm{ab}}(q_{v}^{-1} \\mathrm{Frob}_{v}^{-1})\n\t\t\t\t\\equiv\n\t\t\t\t\tq_{v}^{- g(v)}\n\t\t\t\t\t\\prod_{i = 1}^{2 g(v)}\n\t\t\t\t\t\t(\\beta_{i}^{(v)} - \\mathrm{Frob}_{v}^{-1}),\n\t\t\t\\\\\n\t\t\t\t\tP_{0}(q^{-1} \\mathrm{Frob}_{q}^{-1})\n\t\t\t\t\\equiv\n\t\t\t\t\t\tq^{- \\dim \\Tr_{K \/ \\mathbb{F}_{q}}(A)}\n\t\t\t\t\t\\cdot\n\t\t\t\t\t\t\\prod_{i = 1}^{m}\n\t\t\t\t\t\t\t(1 - \\zeta_{i}^{-1} \\mathrm{Frob}_{q}^{-1}),\n\t\t\t\\\\\n\t\t\t\t\tP_{2}(q^{-1} \\mathrm{Frob}_{q}^{-1})\n\t\t\t\t\\equiv\n\t\t\t\t\t\\prod_{i = 1}^{m}\n\t\t\t\t\t\t(1 - \\zeta_{i}^{-1} \\mathrm{Frob}_{q}),\n\t\t\\end{gather*}\n\\end{lemma}\n\n\\begin{proof}\n\tFor $P_{v, \\mathrm{tor}}$,\n\tlet $\\{\\gamma_{j}^{(v)}\\}$ be the eigenvalues of $\\mathrm{Frob}_{v}$ on $\\mathrm{V}_{l}(T_{v})$.\n\tThen $q_{v}^{-1} \\gamma_{j}^{(v)}$ are roots of unity.\n\tWe have\n\t\t\\begin{align*}\n\t\t\t\t\tP_{v, \\mathrm{tor}}(q_{v}^{-1} \\mathrm{Frob}_{v}^{-1})\n\t\t\t&\t=\n\t\t\t\t\t\\prod_{j} (1 - (\\gamma_{j}^{(v)})^{-1} \\mathrm{Frob}_{v}^{-1})\n\t\t\t\\\\\n\t\t\t&\t=\n\t\t\t\t\tq_{v}^{- t(v)}\n\t\t\t\t\t\\prod_{j} (q_{v} -  q_{v} (\\gamma_{j}^{(v)})^{-1} \\mathrm{Frob}_{v}^{-1}).\n\t\t\\end{align*}\n\tEach factor $q_{v} -  q_{v} (\\gamma_{j}^{(v)})^{-1} \\mathrm{Frob}_{v}^{-1}$ is a unit\n\tin $\\Lambda \\otimes_{\\mathbb{Z}_{q}} \\mathcal{O}$ for some $\\mathcal{O}$.\n\t\n\tFor $P_{v, \\mathrm{ab}}$, we have\n\t\t\\[\n\t\t\t\tP_{v, \\mathrm{ab}}(q_{v}^{-1} \\mathrm{Frob}_{v}^{-1})\n\t\t\t=\n\t\t\t\t\\prod_{i} (1 - (\\beta_{i}^{(v)})^{-1} \\mathrm{Frob}_{v}^{-1})\n\t\t\t=\n\t\t\t\t\\prod_{i} (\\beta_{i}^{(v)})^{-1}\n\t\t\t\t\\prod_{i} (\\beta_{i}^{(v)} - \\mathrm{Frob}_{v}^{-1}).\n\t\t\\]\n\tThe product $\\prod_{i} (\\beta_{i}^{(v)})$ is a Weil $q_{v}$-number of weight $2 g(v)$\n\tthat is in $\\mathbb{Z}$.\n\tHence it is $\\pm q_{v}^{g(v)}$.\n\t\n\tFor $P_{0}$, first notice that by the Lang-N\\'eron theorem (\\cite[Theorem 7.1]{Con06}),\n\tthe group $A(K_{\\infty}^{(p)}) \/ \\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)(\\mathbb{F}_{q, \\infty})$\n\tis a finitely generated abelian group.\n\tTherefore the $p$-adic Tate module of $\\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)(\\mathbb{F}_{q, \\infty})$\n\tinjects into that of $A(K_{\\infty}^{(p)})$ with finite cokernel.\n\tHence $\\zeta_{i}$ are the eigenvalues of $\\mathrm{Frob}_{q}$\n\ton the $p$-adic Tate module of $\\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)(\\mathbb{F}_{q, \\infty})$.\n\tIn particular, they are $p$-adic units.\n\tLet $g_{0} = \\dim \\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)$.\n\t\n\tBy \\cite[Proposition 2.9]{Tan14} and the proof of \\cite[Lemma 2.2.1]{LLTT16},\n\twe know that $\\zeta_{1}, \\dots, \\zeta_{m}$ are part of the eigenvalues\n\t$\\zeta_{1}, \\dots, \\zeta_{m}, \\dots, \\zeta_{2 g_{0}}$\n\tof $\\mathrm{Frob}_{q}$ on the $l$-adic Tate module of $\\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)$ (for any $l \\ne p$)\n\tand\n\t\t\\begin{equation} \\label{e:pandl}\n\t\t\t\t\\prod_{i = 1}^{m}(\\zeta_{i} - \\mathrm{Frob}_{q}^{-1})\n\t\t\t\\equiv\n\t\t\t\t\\prod_{i = 1}^{2 g_{0}}(\\zeta_{i} - \\mathrm{Frob}_{q}^{-1}).\n\t\t\\end{equation}\n\tTogether with the relation between $P_{0}$ and $\\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)$\n\twe saw in Section \\ref{s:Lfun}, we have\n\t\t\\begin{align*}\n\t\t\t&\t\tP_{0}(q^{-1} \\mathrm{Frob}_{q}^{-1})\n\t\t\t\t=\n\t\t\t\t\t\\prod_{i = 1}^{2 g_{0}}\n\t\t\t\t\t\t(1 - \\zeta_{i}^{-1} \\mathrm{Frob}_{q}^{-1})\n\t\t\t\t\\equiv\n\t\t\t\t\tq^{- g_{0}}\n\t\t\t\t\t\\prod_{i = 1}^{2 g_{0}}\n\t\t\t\t\t\t(\\zeta_{i} - \\mathrm{Frob}_{q}^{-1})\n\t\t\t\\\\\n\t\t\t&\t\\equiv\n\t\t\t\t\tq^{- g_{0}}\n\t\t\t\t\t\\prod_{i = 1}^{m}\n\t\t\t\t\t\t(\\zeta_{i} - \\mathrm{Frob}_{q}^{-1})\n\t\t\t\t\\equiv\n\t\t\t\t\tq^{- g_{0}}\n\t\t\t\t\t\\prod_{i = 1}^{m}\n\t\t\t\t\t\t(1 - \\zeta_{i}^{-1} \\mathrm{Frob}_{q}^{-1}).\n\t\t\\end{align*}\n\t\n\tFor $P_{2}$, first notice that \\eqref{e:pandl} also holds\n\twith $\\mathrm{Frob}_{q}$ replaced by $\\mathrm{Frob}_{q}^{-1}$\n\tsince $\\mathrm{Frob}_{q} \\leftrightarrow \\mathrm{Frob}_{q}^{-1}$ is\n\tan automorphism of $\\Lambda$.\n\tAlso, $\\{\\zeta_{i}\\}_{i = 1}^{2 g_{0}} = \\{q \/ \\zeta_{i}\\}_{i = 1}^{2 g_{0}}$\n\t(counting the multiplicities)\n\tby the functional equation for the zeta function of $\\mathrm{Tr}_{K \/ \\mathbb{F}_{q}}(A)$.\n\tUsing these, we have\n\t\t\\begin{align*}\n\t\t\t&\t\tP_{2}(q^{-1} \\mathrm{Frob}_{q}^{-1})\n\t\t\t\t=\n\t\t\t\t\t\\prod_{i = 1}^{2 g_{0}}\n\t\t\t\t\t\t(1 - q \\zeta_{i}^{-1} \\mathrm{Frob}_{q}^{-1})\n\t\t\t\t=\n\t\t\t\t\t\\prod_{i = 1}^{2 g_{0}}\n\t\t\t\t\t\t(1 - \\zeta_{i} \\mathrm{Frob}_{q}^{-1})\n\t\t\t\\\\\n\t\t\t&\t\\equiv\n\t\t\t\t\t\\prod_{i = 1}^{2 g_{0}}\n\t\t\t\t\t\t(\\zeta_{i} -  \\mathrm{Frob}_{q})\n\t\t\t\t\\equiv\n\t\t\t\t\t\\prod_{i = 1}^{m}\n\t\t\t\t\t\t(\\zeta_{i} -  \\mathrm{Frob}_{q})\n\t\t\t\t\\equiv\n\t\t\t\t\t\\prod_{i = 1}^{m}\n\t\t\t\t\t\t(1 -  \\zeta_{i}^{-1} \\mathrm{Frob}_{q}).\n\t\t\\end{align*}\n\\end{proof}\n\nUsing this lemma, we see that \\eqref{e:IMC} is equivalent to\n\t\\[\n\t\t\tc_{A \/ K_{\\infty}^{(p)}}\n\t\t\\equiv\n\t\t\tq^{\\deg(\\mathcal{L}) + \\dim(A) (g_{\\mathcal{C}} - 1) + \\dim \\Tr_{K \/ \\mathbb{F}_{q}}(A)}\n\t\t\t\\cdot P_{1}(q^{-1} \\mathrm{Frob}_{q}^{-1}).\n\t\\]\nThe exact $p$-power factor of $c_{A \/ K_{\\infty}^{(p)}}$ is $q^{\\mu_{A \/ K}}$ by definition.\nThe exact $p$-power factor of $P_{1}(q^{-1} \\mathrm{Frob}_{q}^{-1})$ is $q^{- \\theta_{A}}$ by definition.\nComparing the exact $p$-power factors of both sides, we obtain Theorem \\ref{t:muss}.\n\n\n\\section{The $\\mu$-invariant for elliptic curves}\\label{s:muelliptic}\nFor the rest of the paper, we deal with the case where $\\dim A=1$.\nLet $\\Delta:=\\sum_{v\\in \\mathcal{C}} \\delta_v\\cdot v$ the global discriminant of $A\/K$, where $\\delta_v$ is the valuation of the minimal discriminant of $A$ at $v$ (\\cite{Sil86}, VII.1).\n\nWhen it becomes necessary to emphasize the roles played by $A$ and $K$,\nwe shall use $\\Delta_{A\/K}$ to denote it.\nBy \\cite[Example 2.2]{Con06},\nwe know $\\Tr_{K \/ \\mathbb{F}_{q}}(A) = 0$ unless $A$ is constant,\nin which case $\\Tr_{K \/ \\mathbb{F}_{q}}(A)=A$ (\\cite[Theorem 6.4 (1)]{Con06}).\nIn the case $A$ is non-constant, we have $P_{0}(t) = P_{2}(t) = 1$,\nso we simply write $P(t) = P_{1}(t)$.\n\n\\subsection{The $\\mu$-invariant formula simplified}\nWe have $\\deg(\\mathcal{L}) = \\deg(\\Delta) \/ 12$\nby \\cite[\\S 2.2.1, Footnote 1]{LLTT16}.\nTherefore the formula \\eqref{e:muform}\n(true for $A$ by Corollary \\ref{c:MilUlm}) and the inequality \\eqref{e:upperbound}\nbecome the following statement.\n\n\\begin{theorem} \\label{t:tan}\n\t\t\\[\n\t\t\t\t\\mu_{A \/ K}\n\t\t\t=\n\t\t\t\t\t\\frac{\\deg(\\Delta)}{12} + g_{\\mathcal{C}} - 1 - \\theta\n\t\t\t\t\\le\n\t\t\t\t\t\\frac{\\deg(\\Delta)}{12} + g_{\\mathcal{C}} - 1\n\t\t\\]\n\tif $A$ is non-constant and\n\t\t\\[\n\t\t\t\t\\mu_{A \/ K}\n\t\t\t=\n\t\t\t\t\tg_{\\mathcal{C}}  - \\theta\n\t\t\t\t\\le\n\t\t\t\t\tg_{\\mathcal{C}}\n\t\t\\]\n\tif $A$ is constant.\n\\end{theorem}\n\nLet $\\pi \\colon \\mathcal{S} \\to \\mathcal{C}$ be the minimal elliptic surface\nassociated with $A \/ K$.\nBy \\cite[Chapter 7, Lemma 14]{Fri98} (adapted to positive characteristic),\n\t\\[\n\t\t\t\t\\dim H^{1}(\\mathcal{S}, \\mathcal{O}_{\\mathcal{S}})\n\t\t\t=\n\t\t\t\tg_{\\mathcal{C}},\n\t\t\\quad\n\t\t\t\t\\dim H^{2}(\\mathcal{S}, \\mathcal{O}_{\\mathcal{S}})\n\t\t\t=\n\t\t\t\t\\deg(\\mathcal{L}) + g_{\\mathcal{C}} - 1\n\t\\]\nif $\\mathcal{L}$ is non-trivial, and\n\t\\[\n\t\t\t\t\\dim H^{1}(\\mathcal{S}, \\mathcal{O}_{\\mathcal{S}})\n\t\t\t=\n\t\t\t\tg_{\\mathcal{C}} + 1,\n\t\t\\quad\n\t\t\t\t\\dim H^{2}(\\mathcal{S}, \\mathcal{O}_{\\mathcal{S}})\n\t\t\t=\n\t\t\t\tg_{\\mathcal{C}}\n\t\\]\nif $\\mathcal{L}$ is trivial.\nThe canonical bundle formula\n$\\Omega_{\\mathcal{S}}^{2} \\cong \\pi^{\\ast}(\\Omega_{\\mathcal{C}}^{1} \\otimes \\mathcal{L})$\nimplies that the Kodaira dimension of $\\mathcal{S}$ is $- \\infty$, $0$, $1$\nif $\\deg(\\mathcal{L}) + 2 g_{\\mathcal{C}} - 2 < 0$,\n$\\deg(\\mathcal{L}) + 2 g_{\\mathcal{C}} - 2 = 0$,\n$\\deg(\\mathcal{L}) + 2 g_{\\mathcal{C}} - 2 > 0$, respectively\n(see for instance \\cite[Lecture 3, Proposition 4.3]{Ulm11} and the paragraphs after%\n\\footnote{We take the opportunity to mention a mistake in \\cite[Lecture 3 (6.3)]{Ulm11}.\nThe correct formula can be found in\n\\cite[Equation (2.9) and Lemma 6]{Shi92}.}).\n\nLet $\\bar{r}$ be the geometric Mordell-Weil rank of $A$,\ni.e.\\ the rank of\n$A(K \\bar{\\mathbb{F}}_{q}) \/ \\Tr_{K \/ \\mathbb{F}_{q}}(A)(\\bar{\\mathbb{F}}_{q})$.\nRecall from Definition \\ref{d:theta} that $a = a_{A}$ denotes the degree of the polynomial $P_{1}(t)$.\nThen \\eqref{e:MWSha} shows that $\\bar{r} \\le a$.\n\n\\begin{proposition} \\label{p:KodOne}\n\tAssume that the minimal elliptic surface $\\mathcal{S} \\to \\mathcal{C}$ over $\\mathbb{F}_{q}$ associated with $E \/ K$\n\thas Kodaira dimension $1$\n\tand that $\\bar{r} = a$.\n\tThen $\\mu_{A \/ K} > 0$.\n\\end{proposition}\n\n\\begin{proof}\n\tBy the proof of Proposition \\ref{p:Lcomp}\n\tand by \\eqref{e:NSBr}, \\eqref{e:MWSha},\n\twe know that $a - \\bar{r} = \\deg P_{\\mathcal{S}, 2} - \\rho_{\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}}}$,\n\twhere $\\rho_{\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}}}$ is the rank of $\\mathrm{NS}(\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}})$.\n\tHence the assumption implies that $\\rho_{\\mathcal{S}_{\\bar{\\mathbb{F}}_{q}}} = \\deg P_{\\mathcal{S}, 2}$,\n\ti.e.\\ $\\mathcal{S}$ is Shioda supersingular.\n\tIn particular (cf.\\ the paragraph after \\cite[Definition 9.11]{Lie13}),\n\tit is Artin supersingular, i.e.\\ $\\lambda_{\\mathcal{S}, i} = 1$ for all $i$,\n\tor $\\theta_{\\mathcal{S}} = 0$.\n\tBy Proposition \\ref{p:Lcomp},\n\twe know that $\\theta_{A} = 0$.\n\tWe have $\\deg(\\mathcal{L}) + 2 g_{\\mathcal{C}} - 2 > 0$ as above.\n\tHence if $g_{\\mathcal{C}} = 0$, then $\\deg(\\mathcal{L}) \\ge 3$ and $A$ is non-constant;\n\tif $g_{\\mathcal{C}} = 1$, then $\\deg(\\mathcal{L}) \\ge 1$ and $A$ is non-constant;\n\tor else $g_{\\mathcal{C}} \\ge 2$.\n\tApplying Theorem \\ref{t:tan}, we know $\\mu_{A \/ K} > 0$.\n\\end{proof}\n\n\n\\subsection{The $j$-invariant}\n\nLet $j_A$ denote the $j$-invariant of $A$.\nRecall the Szpiro difference $d = d_{A}$\nfrom Definition \\ref{d:Sz}.\n\n\\begin{theorem}[Pesenti-Szpiro {\\cite[Th\\'eor\\`eme 0.1]{PS00}}] \\label{c:szpiro}\n\tIf $j_A$ is not a $p$th power or $A$ is isotrivial, then $d\\geq 0$.\n\\end{theorem}\n\nSince $\\mu_{A \/ K} = b - d$ by Corollary \\ref{c:bminusd},\nthis theorem allows us to bound $\\mu_{A \/ K}$ purely in terms of the $L$-function:\n\n\\begin{corollary}\n\tIf $j_A$ is not a $p$th power or $A$ is isotrivial, then $\\mu_{A \/ K} \\le b$.\n\\end{corollary}\n\nFor the rest of this section, we further assume that\nthe elliptic curve $A$ is \\emph{non-isotrivial}.\nWe will study the behavior of the $j$-invariant and the $\\mu$-invariant under isogeny.\n\n\\begin{lemma}\\label{l:j} The following conditions are equivalent\n\\begin{enumerate}\n\\item[(a)] The group scheme $A_p$ has a non-trivial $\\acute{\\text{e}}$tale subgroup.\n\\item[(b)] The $j$-invariant $j_A$ is a $p$th power of an element in $K$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nIf $E$ is a non-trivial $\\acute{\\text{e}}$tale subgroup scheme of $A_p$, then it is of order $p$ and $B:=A\/E$\nis isogenous to $A$ by the natural homomorphism $\\varphi:A\\longrightarrow B$.\nBecause the group scheme $C= \\varphi(A_p)$ is connected and hence $B\/C=A$ is isomorphic to $B^{(p)}$,\n$\\varphi$ is isomorphic to the Verschiebung. This shows\n$j_A=j_{B^{(p)}}$ is a $p$th power.\nConversely, if $j_A=j_0^p$ and let $B\/K$ be an elliptic curve with $j_B=j_0$,\nthen $j_A=j_{B^{(p)}}$ and hence $A$ and $B^{(p)}$ are isomorphic over the separable closure of $K$ \\cite[III.1.4(b),A.1.2(b)]{Sil86}.\nSince $B^{(p)}_p$ contains a non-trivial $\\acute{\\text{e}}$tale subgroup scheme, so does $A$.\n\\end{proof}\n\n\\begin{lemma}\\label{l:isogeny}\nEvery non-isotrivial elliptic curve is isogenous to an elliptic curve with $j$-invariant not a $p$-th power.\n\\end{lemma}\n\\begin{proof} Let $\\mathcal E$ be the maximal  $\\acute{\\text{e}}$tale subgroup of $A_{p^\\infty}$. Then $\\mathcal E$ is finite and\n$(A\/\\mathcal E)_p$ has no non-trivial $\\acute{\\text{e}}$tale subgroup.\n\\end{proof}\n\n\n\\subsection{The isogeny class}\\label{su:isogen}\n\nConsider the connected-\\'etale sequence\n$$0\\longrightarrow A_{p^n}^0\\longrightarrow A_{p^n}\\longrightarrow \\pi_0(A_{p^n})\\longrightarrow 0,$$\nwhere $A_{p^n}^0$ and $\\pi_0(A_{p^n})$ are Cartier dual to each other. Since $\\pi_0(A_{p^n})$ is of $p$-rank $1$, there exists\na unique filtration\n$$0\\subset G_1\\subset \\cdots \\subset G_{n-1}\\subset G_{n}=\\pi_0(A_{p^n})$$\nwith $G_i$ of order $p^i$, and by duality\nthe filtration\n$$0\\subset H_1\\subset \\cdots \\subset H_{n-1}\\subset H_{n}=A_{p^n}^0$$\nwith $H_i$ of order $p^i$. If $\\varphi:A\\longrightarrow B$\nis an isogeny of degree $p^n$, then $E:=\\ker \\varphi\\subset A_{p^n}$ with $E^0\\subset A_{p^n}^0$ and $\\pi_0(E)\\subset \\pi_0(A_{p^n})$.\nBy the above filtrations, $\\varphi$ is decomposed into a sequence\n$$\n\\xymatrix{A \\ar[r]^-{\\phi_1} & A_1 \\ar[r]^-{\\phi_2} &\\cdots \\ar[r]^-{\\phi_c} & A_c \\ar[r]^-{\\psi_1} & B_1\\ar[r]^-{\\psi_2} & \\cdots\n\\ar[r]^{\\psi_e} & B_e \\ar[r]^-{\\psi_{e+1}}\n & B}\n$$\nwhere each $\\phi_i$ is a Frobenius, $\\psi_j$ a Verschiebung, all of degree $p$. Suppose both $c$ and $e$ are positive.\nThen up to isomorphisms, the segment $\\xymatrix{A_{c-1}\\ar[r]^-{\\phi_c} & A_c \\ar[r]^-{\\psi_1} & B_1}$ equals to the multiplication by $p$.\nHence $\\varphi$ can be written as $\\varphi'\\circ [p]$ with $\\varphi'$ of less degree than $\\varphi$. By repeating this procedure, we prove the\nfollowing lemma.\n\n\\begin{lemma}\\label{l:piso}\nIf $\\varphi:A\\longrightarrow B$ is an isogeny of degree $p^n$ for some $n$, then either $B=A^{(p^m)}$ or $A=B^{(p^m)}$\nfor some $m$.\n\\end{lemma}\n\nSuppose $A\/K$ has semi-stable reduction everywhere. Then\n\\begin{equation}\\label{e:delta}\n\\deg \\Delta_{A^{(p^m)}\/K}=p^m \\deg\\Delta_{A\/K},\n\\end{equation}\nwhile $N_{A^{(p^m)}\/K}=N_{A\/K}$ and the Hasse-Weil $L$-function for both curves are the same. Hence Theorem \\ref{t:tan} says\n\\begin{equation}\\label{e:apm}\n\\mu_{A^{(p^m)}\/K}=\\mu_{A\/K}+(p^m-1)\\cdot \\frac{\\deg \\Delta_{A\/K}}{12}.\n\\end{equation}\n\n\n\n\n\\begin{lemma}\\label{l:l}\nIf $\\varphi:B\\longrightarrow A$ is an isogeny of degree $l$ with $(l,p)=1$, then\n$$\\deg \\Delta_{B\/K}=\\deg \\Delta_{A\/K}.$$\nHence $\\mu_{B\/K}=\\mu_{A\/K}$.\n\\end{lemma}\n\\begin{proof}\nChoose an invariant differential $\\omega'$ of $A\/K$ and denote $\\omega:=\\varphi^*\\omega'$.\nFor each place $v$, let $\\omega_v$ and $\\omega_v'$ respectively be local N$\\acute{\\text{e}}$ron differentials\nof $B$ and $A$.  Then we have (see \\cite[\\S 1.2]{Tan95})\n$$\\deg \\Delta_{B\/K}=12\\sum_v \\ord_v{\\frac{\\omega}{\\omega_v}}\\cdot \\deg v,$$\nand\n$$\\deg \\Delta_{A\/K}=12\\sum_v \\ord_v{\\frac{\\omega'}{\\omega'_v}}\\cdot \\deg v.$$\nLet $\\psi:A\\longrightarrow B$ denote the dual isogeny. Write $\\varphi^*\\omega'_v=\\alpha_v\\cdot \\omega_v$, $\\psi^*\\omega_v=\\beta_v\\cdot\\omega'_v$, $\\alpha_v,\\beta_v\\in\\O_v$. But, since $\\psi\\circ\\varphi=\\left[l\\right]$, we have $\\alpha_v\\cdot\\beta_v=l$. Hence $\\alpha_v\\in\\O_v^*$.\nConsequently,\n$$\\ord_v\\frac{\\omega'}{\\omega'_v}=\\ord_v\\frac{\\varphi^*\\omega'}{\\varphi^*\\omega'_v}=\\ord_v\\frac{\\omega}{\\alpha_v\\cdot\\omega_v}=\\ord_v\\frac{\\omega}{\\omega_v},$$\nso $\\deg \\Delta_{B\/K}=\\deg \\Delta_{A\/K}$.\n\\end{proof}\nWe summarize the above discussion below.\n\\begin{proposition}\\label{p:isoclass}\nAmong an isogeny class of elliptic curves with semi-stable reduction everywhere, those curves with $j$-invariant not a $p$-th power have minimal $\\mu$-invariant.\n\\end{proposition}\n\nThe elliptic curve $A$ in Proposition \\ref{p:Shio} has $j$-invariant not a $p$-th power.\nHence by Proposition \\ref{p:isoclass},\nwe cannot replace $A$ by an isogenous elliptic curve\nfor which the $\\mu$-invariant is zero.\nThis should be compared with Greenberg's conjecture \\cite[Conjecture 1.11]{Gre99}\non $\\mu$-invariants of elliptic curves over $\\mathbb{Q}$.\nTo recall this conjecture, let $E$ be an elliptic curve over $\\mathbb{Q}$.\nAssume that $\\Sel_{p^{\\infty}}(E \/ \\mathbb{Q}^{cyc})^{\\vee}$ is torsion over $\\mathbb{Z}_{p}[[\\Gal(\\mathbb{Q}^{cyc} \/ \\mathbb{Q})]]$.\nThen Greenberg's conjecture claims the existence of a $\\mathbb{Q}$-isogenous elliptic curve $E'$\nsuch that $\\Sel_{p^{\\infty}}(E' \/ \\mathbb{Q}^{cyc})^{\\vee}$ has $\\mu$-invariant zero.\n\n\n\\subsection{Elliptic curves with trivial $\\mu$-invariant}\\label{su:trivmu}\n\nIn this subsection, we produce infinitely many elliptic curves with trivial $\\mu$-invariant.\n\nAssume that $\\mathrm{char}(K) \\ne 2$ and $A\/K$ is defined by the Legendre form\n(see \\cite[\\S III.1.7]{Sil86})\n\\begin{equation}\\label{e:legend}\nY^2=X(X-1)(X-\\lambda),\n\\end{equation}\nwith $\\lambda\\in K$, so that $F_0:= \\mathbb{F}_q(\\lambda)\\subset K$. The equation \\eqref{e:legend} defines an elliptic curve $E_0\/F_0$ such that $A\/K$ is just its base change to $K$. Put $t^2=\\lambda$, $F=\\mathbb{F}_q(t)$ and let $E\/F$ be the elliptic curve defined by\n\\begin{equation}\\label{e:legsemi}\n Y^2=X(X-1)(X-t^2)\n\\end{equation}\nThen $E\/F$ is the base change of $E_0$ to $F$. Let $K'=KF$, then $A\/K'$ is the base change of $E$ to $K'$.\n\n\\begin{definition}\\label{d:Legendreptype}\nAn elliptic curve is of Legendre $p$-type, if\n\\begin{enumerate}\n\\item $char(K)\\neq 2$ and $A$ has the Legendre invariant $\\lambda\\in K$.\n\\item $K\/F_0$, where $F_0=\\mathbb{F}_q(\\lambda)$, is a Galois $p$-extension unramified at supersingular places of $E_{0} \/ F_{0}$.\n\\end{enumerate}\n\\end{definition}\n\n\n\\begin{theorem}\\label{t:Legendretype} If $A\/K$ is of Legendre $p$-type, then $\\mu_{A\/K}=0$.\n\\end{theorem}\n\n\n\n\\begin{proof} By Lemma \\ref{l:kl}, it is enough to show $\\mu_{A\/K'}=0$.\nHere $K'=KF$ and $F=\\mathbb{F}_q(t)$ with $t^2=\\lambda$.\nLet $E_{0} \/ F_{0}$ and $E \/ F$ be the elliptic curves\ndefined by \\eqref{e:legend} and \\eqref{e:legsemi}, respectively.\nBy the condition (2), if a place of $F$ ramifies in $K'$,\nthen the corresponding place of $F_{0}$ is\neither an ordinary or additive place for $E_{0} \/ F_{0}$.\nThe only place where $E_{0} \/ F_{0}$ has additive reduction is where $\\lambda = \\infty$\n(see the proof of Lemma \\ref{l:ss} below).\nThe corresponding place $t = \\infty$ of $F$ is an ordinary place for $E\/F$ (see Lemma \\ref{l:ss}).\nTherefore $K' \/ F$ is a Galois $p$-extension unramified outside ordinary places of $E \/ F$.\nNow Theorem \\ref{t:ocht} says we only need to prove $\\mu_{E\/F}=0$.\nLemma \\ref{l:ss} below says $E\/F$ has semi-stable reduction everywhere, and by simple computation (see \\S\\ref{sub:example}),\nwe find $\\deg(\\Delta_{E \/ F})=12$, and hence $\\mu_{E \/ F} = 0$\nby Theorem \\ref{t:tan}.\nThus $\\mu_{A\/K} = 0$.\n\\end{proof}\n\n\\begin{remark}\n\tLet $A \/ K$ be an elliptic curve satisfying the condition (1)\n\tsuch that $\\mu_{A \/ K} = 0$.\n\tThe previous theorem is a type of results\n\tsaying $\\mu_{A \/ L} = 0$ for\n\tsome extension $L \/ K$ unramified at supersingular places of $A \/ K$.\n\tBut $L \/ K$ being ramified at a supersingular place does not always imply that $\\mu_{A \/ L} > 0$.\n\tFor example, let $K=\\mathbb{F}_3(t)$ and $A$ defined by $Y^2=X(X-1)(X-t^2)$.\n\tThere is one supersingular place which is the zero of $(t^2+1)$.\n\tThe extension $L=K(s)$ of $K$ with $s^2=1+t^2$ is a quadratic extension ramified at $(t^2+1)$.\n\tHowever, Magma gives a trivial $\\mu$-invariant for $A\/L$ (see \\S\\ref{su:high} below).\n\\end{remark}\n\n\n\\section{$\\mu = 0$ for generic elliptic curves} \\label{su:generic}\nIn this section, we show that ``generic'' elliptic curves over $K$ with $p > 3$\nin a certain precise sense have trivial $\\mu$-invariant\n(Theorem \\ref{t:GenVan}).\nThis partially extends the result \\cite[Lemma (5.10)]{Art74b}\nthat the generic fibers of general Weierstrass fibrations which are K3 have $\\mu = 0$.\nWe do not assume that the elliptic curves are non-isotrivial or semistable.\n\n\\subsection{An elliptic curve having $\\mu = 0$ with given $\\deg(\\Delta)$}\n\nWe first produce some (isotrivial) elliptic curves having $\\mu = 0$ with given $\\deg(\\Delta)$.\n\n\\begin{proposition} \\label{p:NonEmp}\n\tAssume $p > 2$.\n\tLet $n$ be a non-negative integer.\n\tThen there exist a finite constant extension $K'$ of $K$\n\tand an elliptic curve $E \/ K'$ such that $\\deg(\\Delta_{E \/ K'}) = 12 n$ and $\\mu_{E \/ K'} = 0$.\n\\end{proposition}\n\n\\begin{proof}\n\tLet $K'$ be a finite constant extension of $K$\n\thaving $2 n$ places $v_{1}, \\dots, v_{2 n}$ of degree $1$.\n\tWe first show that by replacing $K'$\n\tby a larger finite constant extension of $K$,\n\twe can choose an element $f \\in K'$\n\tsuch that $v_{i}(f)$ is odd for any $i$\n\tand $v(f)$ is even for any other place $v$ of $K'$.\n\tConsider the exact sequence\n\t\t\\[\n\t\t\t\t (K \\bar{\\mathbb{F}}_{q})^{\\times}\n\t\t\t\\to\n\t\t\t\t\\bigoplus_{v} \\mathbb{Z}\n\t\t\t\\to\n\t\t\t\t\\Pic(\\mathcal{C}_{\\bar{\\mathbb{F}}_{q}})\n\t\t\t\\to\n\t\t\t\t 0,\n\t\t\\]\n\twhere the middle direct sum is over all places of $K \\bar{\\mathbb{F}}_{q}$,\n\tthe first map is the sum of the valuations and\n\tthe second map is $1 \\mapsto v$ for each summand corresponding to a place $v$.\n\tSince $\\Pic^{0}(\\mathcal{C}_{\\bar{\\mathbb{F}}_{q}})$ is\n\tthe group of geometric points of the Jacobian of $\\mathcal{C}$ and hence divisible,\n\tthis gives an exact sequence\n\t\t\\[\n\t\t\t (K \\bar{\\mathbb{F}}_{q})^{\\times} \\to \\bigoplus_{v} \\mathbb{Z} \/ 2 \\mathbb{Z} \\to \\mathbb{Z} \/ 2 \\mathbb{Z} \\to 0,\n\t\t\\]\n\twhere the second map is the summation map.\n\tConsider the element of $\\bigoplus_{v} \\mathbb{Z} \/ 2 \\mathbb{Z}$\n\tsuch that the component corresponding to each $v_{i}$ is $1$\n\tand the other components are zero.\n\tThis element belongs to the kernel of the summation map.\n\tHence there exists an element $f \\in (K \\bar{\\mathbb{F}}_{q})^{\\times}$\n\tsuch that $v_{i}(f) \\equiv 1 \\mod 2$ for any $i$ and $v(f) \\equiv 0 \\mod 2$ for any other $v$.\n\tReplace $K'$ by a larger finite constant extension of $K$ so that $f$ is in $K'$.\n\tThis choice of $K'$ and $f$ does the job.\n\n\tLet $E_{0} \\colon y^{2} = x^{3} + a x^{2} + b x + c$ be\n\tany constant ordinary elliptic curve over $K'$.\n\tLet $E \\colon f y^{2} = x^{3} + a x^{2} + b x + c$ be\n\tits quadratic twist by $f$.\n\tThis $E$ has $\\mu = 0$ by \\cite[Theorem 1.8 (1)]{OT09}.\n\tWe show $\\deg(\\Delta_{E \/ K'}) = 12 n$.\n\tLet $v$ be any place of $K'$.\n\tIf $v(f)$ is even and hence $f = u \\pi^{m}$ for some unit $u$ at $v$,\n\tuniformizer $\\pi$ at $v$ and integer $m$,\n\tthen $E$ is isomorphic to $u y^{2} = x^{3} + a x^{2} + b x + c$ over $K'_{v}$,\n\twhich has good reduction at $v$.\n\tIf $v(f)$ is odd, then similarly, $E$ is isomorphic to\n\t$y^{2} = x^{3} + \\pi a x^{2} + \\pi^{2} b x + \\pi^{3} c$ over $K_{v}$\n\tfor some uniformizer $\\pi$ at $v$.\n\tIts discriminant is the discriminant of $E_{0}$ times $\\pi^{6}$.\n\tHence this equation is minimal at $v$ and $v(\\Delta_{E \/ K'}) = 6$.\n\tTherefore, by the choice of $f$,\n\twe know that $\\deg(\\Delta_{E \/ K'}) = 6 \\cdot 2 n = 12 n$.\n\\end{proof}\n\n\\subsection{Moduli of elliptic curves with given $\\deg(\\Delta)$}\n\nTo precisely define ``generic'' elliptic curves,\nwe need a parameter space for elliptic curves over $K$ with given $\\deg(\\Delta)$.\nWe follow \\cite{Kas77}.\nSee also \\cite{Mir81} and \\cite{Sei87},\nwhich work over positive characteristic as well.\n\nFor a perfect field extension $k \/ \\mathbb{F}_{q}$ and\nan abelian variety $A$ over the fraction field of $K \\otimes_{\\mathbb{F}_{q}} k$\nwith N\\'eron model $\\mathcal{A}$ over $\\mathcal{C}_{k}$,\nwe define $\\mu_{A}$ to be the dimension of the group scheme\n$\\mathbf{H}^{1}(\\mathcal{C}_{k}, \\mathcal{A})$ over $k$.\nIf $A$ is an elliptic curve and\nthe associated minimal elliptic surface $\\mathcal{S}$ has smooth Picard scheme,\nthen this $\\mu_{A}$ is equal to the dimension of the unipotent part\nof the formal Brauer group of $\\mathcal{S}$\nby Remark \\ref{sdef} and Proposition \\ref{p:ShaBr}\n(which are still true with $\\mathbb{F}_{q}$ replaced by $k$ by the same proof).\nWhen $k = \\bar{\\mathbb{F}}_{q}$, we denote $\\bar{\\mathcal{C}} = \\mathcal{C}_{\\bar{\\mathbb{F}}_{q}}$.\nIn what follows, any fiber product is taken over $\\bar{\\mathbb{F}}_{q}$ when ``$\\times$'' has no specified base.\nFor any integer $n$,\nlet $\\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}$ be the degree $n$ component of\nthe Picard scheme $\\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}$,\nwhich parametrizes the line bundles of degree $n$.\nLet $\\mathcal{P}$ be a Poincar\\'e sheaf on\n$\\bar{\\mathcal{C}} \\times \\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}$\n(\\cite[Exercise 4.3]{Kle05}, \\cite[8.2\/4]{BLR90})\n(rigidified at some closed point of $\\bar{\\mathcal{C}}$).\nIts restriction to $\\bar{\\mathcal{C}} \\times \\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}$\nfor any $n$ is denoted by $\\mathcal{P}_{n}$.\nDenote the second projection\n\t$\n\t\t\t\\bar{\\mathcal{C}} \\times \\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}\n\t\t\\twoheadrightarrow\n\t\t\t\\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}\n\t$\nby $\\mathrm{pr}_{2}$.\n\nIf $n > 2 g_{\\mathcal{C}} - 2$,\nthen for any line bundle $\\mathscr{L}$ of degree $n$ on $\\bar{\\mathcal{C}}$\n(that is, any element of $\\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}(\\bar{\\mathbb{F}}_{q})$),\nthe dimension of $\\coh^{0}(\\bar{\\mathcal{C}}, \\mathscr{L})$ is $n + 1 - g_{\\mathcal{C}}$\nby the Riemann-Roch theorem,\nwhich does not depend on $\\mathscr{L}$.\nHence the direct image sheaf $\\mathrm{pr}_{2 \\ast} \\mathcal{P}_{n}$ on\n$\\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}$\nhas a constant rank over all points of $\\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}$.\nTherefore Grauert's theorem (\\cite[III, Corollary 12.9]{Har77}) implies that\n$\\mathrm{pr}_{2 \\ast} \\mathcal{P}_{n}$ is a locally free sheaf\nwhose fiber at any point $\\mathscr{L}$ of $\\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}$ is given by\n$\\coh^{0}(\\bar{\\mathcal{C}}, \\mathscr{L})$.\nLet\n\t$\n\t\t\t\\mathbf{V}((\\mathrm{pr}_{2 \\ast} \\mathcal{P}_{n})^{\\vee})\n\t\t\\twoheadrightarrow\n\t\t\t\\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}\n\t$\nbe the total space of $\\mathrm{pr}_{2 \\ast} \\mathcal{P}_{n}$\n(where $\\vee$ denotes the dual bundle),\nwhich is the relative $\\spec$ of the symmetric algebra of\n$(\\mathrm{pr}_{2 \\ast} \\mathcal{P}_{n})^{\\vee}$.\nThe closed points of $\\mathbf{V}((\\mathrm{pr}_{2 \\ast} \\mathcal{P}_{n})^{\\vee})$ are\ngiven by pairs $(\\mathscr{L}, s)$,\nwhere $\\mathscr{L}$ is a line bundle of degree $n$ on $\\bar{\\mathcal{C}}$\nand $s \\in \\coh^{0}(\\bar{\\mathcal{C}}, \\mathscr{L})$.\n\nAssume $p > 3$.\nLet $n > (g_{\\mathcal{C}} - 1) \/ 2$ be an integer.\nLet $\\mathcal{P}_{n}^{\\otimes 4}$ and $\\mathcal{P}_{n}^{\\otimes 6}$\nbe the tensor powers of $\\mathcal{P}_{n}$,\nwhich are line bundles on $\\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}$.\nSince $6 n \\ge 4 n > 2 g_{\\mathcal{C}} - 2$,\nwe have a vector bundle\n\t$\n\t\t\\mathbf{V}((\n\t\t\t\\mathrm{pr}_{2 \\ast}(\n\t\t\t\t\\mathcal{P}_{n}^{\\otimes 4} \\oplus \\mathcal{P}_{n}^{\\otimes 6}\n\t\t\t)\n\t\t)^{\\vee})\n\t$\nover $\\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}$ as above.\nThe closed points of\n\t$\n\t\t\\mathbf{V}((\n\t\t\t\\mathrm{pr}_{2 \\ast}(\n\t\t\t\t\\mathcal{P}_{n}^{\\otimes 4} \\oplus \\mathcal{P}_{n}^{\\otimes 6}\n\t\t\t)\n\t\t)^{\\vee})\n\t$\nare given by triples $(\\mathscr{L}, g_{2}, g_{3})$,\nwhere $\\mathscr{L}$ is a line bundle of degree $n$ on $\\bar{\\mathcal{C}}$\nand $g_{2} \\in \\coh^{0}(\\bar{\\mathcal{C}}, \\mathscr{L}^{\\otimes 4})$,\n$g_{3} \\in \\coh^{0}(\\bar{\\mathcal{C}}, \\mathscr{L}^{\\otimes 6})$.\nFollowing the paragraphs after \\cite[Theorem 1]{Kas77},\nwe make the following definition:\n\n\\begin{definition}\n\tDefine $Y(n, \\bar{\\mathcal{C}})$ to be the set of triples $(\\mathscr{L}, g_{2}, g_{3})$ as above\n\tsatisfying the following two conditions:\n\t\\begin{itemize}\n\t\t\\item\n\t\t\t$4 g_{2}^{3} - 27 g_{3}^{2} \\ne 0$ as an element of\n\t\t\t$\\coh^{0}(\\bar{\\mathcal{C}}, \\mathscr{L}^{\\otimes 12})$.\n\t\t\\item\n\t\t\tFor every closed point $v$ of $\\bar{\\mathcal{C}}$,\n\t\t\t$\\min(3 \\ord_{v}(g_{2}), 2 \\ord_{v}(g_{3})) < 12$.\n\t\\end{itemize}\n\\end{definition}\n\nThen $Y(n, \\bar{\\mathcal{C}})$ is an open subset of the set of closed points of\n\t$\n\t\t\\mathbf{V}((\n\t\t\t\\mathrm{pr}_{2 \\ast}(\n\t\t\t\t\\mathcal{P}_{n}^{\\otimes 4} \\oplus \\mathcal{P}_{n}^{\\otimes 6}\n\t\t\t)\n\t\t)^{\\vee})\n\t$.\nAdding generic points, we obtain an open subscheme of\n\t$\n\t\t\\mathbf{V}((\n\t\t\t\\mathrm{pr}_{2 \\ast}(\n\t\t\t\t\\mathcal{P}_{n}^{\\otimes 4} \\oplus \\mathcal{P}_{n}^{\\otimes 6}\n\t\t\t)\n\t\t)^{\\vee})\n\t$,\nwhich we denote by the same symbol $Y(n, \\bar{\\mathcal{C}})$ by abuse of notation.\nBelow we only consider $Y(n, \\bar{\\mathcal{C}})$ as a scheme.\n\nConsider the action of $\\mathbb{G}_{m, \\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}}$\non\n\t$\n\t\t\\mathbf{V}((\n\t\t\t\\mathrm{pr}_{2 \\ast}(\n\t\t\t\t\\mathcal{P}_{n}^{\\otimes 4} \\oplus \\mathcal{P}_{n}^{\\otimes 6}\n\t\t\t)\n\t\t)^{\\vee})\n\t$\nminus the zero section given by\n$\\lambda \\cdot (\\mathscr{L}, g_{2}, g_{3}) = (\\mathscr{L}, \\lambda^{4} g_{2}, \\lambda^{6} g_{3})$.\nThe quotient of\n\t$\n\t\t\\mathbf{V}((\n\t\t\t\\mathrm{pr}_{2 \\ast}(\n\t\t\t\t\\mathcal{P}_{n}^{\\otimes 4} \\oplus \\mathcal{P}_{n}^{\\otimes 6}\n\t\t\t)\n\t\t)^{\\vee})\n\t$\nminus the zero section by this action of $\\mathbb{G}_{m, \\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}}$\nis the (total space of) weighted projective bundle associated with the vector bundle\n$\\mathrm{pr}_{2 \\ast}(\\mathcal{P}_{n}^{\\otimes 4} \\oplus \\mathcal{P}_{n}^{\\otimes 6})$\nover $\\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}$\nwith weight $(4, 6)$ (\\cite[\\S 1]{Dol82}),\nwhich is a universal geometric quotient (\\cite[\\S 1.2.1]{Dol82}).\nThe subscheme $Y(n, \\bar{\\mathcal{C}})$ is stable under the action of\n$\\mathbb{G}_{m, \\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}}$.\n\n\\begin{definition}\n\tDefine $X(n, \\bar{\\mathcal{C}})$ to be the quotient of $Y(n, \\bar{\\mathcal{C}})$\n\tby this action of $\\mathbb{G}_{m, \\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}}$.\n\\end{definition}\n\nLet $k$ be an algebraically closed extension field of $\\bar{\\mathbb{F}}_{q}$.\nBy \\cite[Theorem 2]{Kas77}\n(which also works in any characteristic $p > 3$, since\nminimal Weierstrass equations still have only rational double points by\n\\cite[Chapter 9, Theorem 4.35 (a)]{Liu02} and duality%\n\\footnote{The exact statement can also be found in Corollary 8.4 of Conrad's notes\n``Minimal models for elliptic curves'' available at\n\\texttt{http:\/\/math.stanford.edu\/\\~{}conrad\/papers\/minimalmodel.pdf}}),\nthe set of $k$-points of $X(n, \\bar{\\mathcal{C}})$\ncorresponds bijectively to the isomorphism classes of\nminimal elliptic surfaces $\\mathcal{S} \\to \\mathcal{C}_{k}$\nwith section and $\\chi(\\mathcal{S}, \\mathcal{O}_{\\mathcal{S}}) = n$.\nIt also corresponds bijectively to the isomorphism classes of\nelliptic curves over the fraction field of $K \\otimes_{\\mathbb{F}_{q}} k$ with $\\deg(\\Delta) = 12 n$\nby the paragraph after Theorem \\ref{t:tan}.\nFor a (not necessarily closed) point $a \\in X(n, \\bar{\\mathcal{C}})$,\nwe define its $\\mu$-invariant $\\mu_{a}$ to be\nthe $\\mu$-invariant of the elliptic curve\ncorresponding to a geometric point (the spectrum of an algebraically closed field) lying over $a$.\nThis defines a function on the underlying set of the scheme $X(n, \\bar{\\mathcal{C}})$\nvalued in non-negative integers.\n\n\\subsection{Statement and proof}\n\nNow the following theorem states that general points of $X(n, \\bar{\\mathcal{C}})$\ncorrespond to elliptic curves with trivial $\\mu$-invariant.\n\n\\begin{theorem} \\label{t:GenVan}\n\tLet $p > 3$ and $n > (g_{\\mathcal{C}} - 1) \/ 2$ be as above.\n\tThen the $\\mu$-invariant function $a \\mapsto \\mu_{a}$ on $X(n, \\bar{\\mathcal{C}})$ as defined above\n\tis upper semicontinuous.\n\tThe (open) locus where $\\mu_{a} = 0$ is dense.\n\\end{theorem}\n\nNote that if $g_{\\mathcal{C}} = 0$, then there is no restriction on $n$.\nWe will prove this theorem below.\nWe need some preparations.\nWe first define a universal family of Weierstrass models over $Y(n, \\bar{\\mathcal{C}})$.\n\nLet $h \\colon Y(n, \\bar{\\mathcal{C}}) \\to \\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}$\nbe the composite of the inclusion\n\t$\n\t\t\tY\n\t\t\\hookrightarrow\n\t\t\t\\mathbf{V}((\n\t\t\t\t\\mathrm{pr}_{2 \\ast}(\n\t\t\t\t\t\\mathcal{P}_{n}^{\\otimes 4} \\oplus \\mathcal{P}_{n}^{\\otimes 6}\n\t\t\t\t)\n\t\t\t)^{\\vee})\n\t$\nand the natural projection\n\t$\n\t\t\t\\mathbf{V}((\n\t\t\t\t\\mathrm{pr}_{2 \\ast}(\n\t\t\t\t\t\\mathcal{P}_{n}^{\\otimes 4} \\oplus \\mathcal{P}_{n}^{\\otimes 6}\n\t\t\t\t)\n\t\t\t)^{\\vee})\n\t\t\\twoheadrightarrow\n\t\t\t\\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}\n\t$,\nwhich is smooth.\nLet\n\t\\[\n\t\t\th_{\\bar{\\mathcal{C}}}\n\t\t\\colon\n\t\t\t\\bar{\\mathcal{C}} \\times Y(n, \\bar{\\mathcal{C}})\n\t\t\\to\n\t\t\t\\bar{\\mathcal{C}} \\times \\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}\n\t\\]\nbe the product of the identity on $\\bar{\\mathcal{C}}$ with $h$.\nLet\n\t$\n\t\t\\mathbb{P} \\bigl(\n\t\t\th_{\\bar{\\mathcal{C}}}^{\\ast}(\n\t\t\t\t\\mathcal{P}_{n}^{\\otimes 2} \\oplus \\mathcal{P}_{n}^{\\otimes 3} \\oplus \\mathcal{O}\n\t\t\t)^{\\vee}\n\t\t\\bigr)\n\t$\nbe the (total space of) projective bundle associated with the vector bundle\n\t$\n\t\th_{\\bar{\\mathcal{C}}}^{\\ast}(\n\t\t\t\\mathcal{P}_{n}^{\\otimes 2} \\oplus \\mathcal{P}_{n}^{\\otimes 3} \\oplus \\mathcal{O}\n\t\t)^{\\vee}\n\t$\non $\\bar{\\mathcal{C}} \\times Y(n, \\bar{\\mathcal{C}})$,\nwhere $\\mathcal{O}$ is the structure sheaf of\n$\\bar{\\mathcal{C}} \\times \\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}$.\n\n\\begin{definition}\n\tDefine $\\mathcal{S}^{\\ast}$ to be the closed subscheme of\n\t\t$\n\t\t\t\\mathbb{P} \\bigl(\n\t\t\t\th_{\\bar{\\mathcal{C}}}^{\\ast}(\n\t\t\t\t\t\\mathcal{P}_{n}^{\\otimes 2} \\oplus \\mathcal{P}_{n}^{\\otimes 3} \\oplus \\mathcal{O}\n\t\t\t\t)^{\\vee}\n\t\t\t\\bigr)\n\t\t$\n\tdefined by the homogeneous Weierstrass equation\n\t$y^{2} z = x^{3} - g_{2} x z^{2} - g_{3} z^{3}$,\n\twhere $(\\mathscr{L}, g_{2}, g_{3})$ is any point of $Y(n, \\bar{\\mathcal{C}})$\n\t(so $g_{2}$ is a global section of $\\mathscr{L}^{\\otimes 4}$\n\tand $g_{3}$ is a global section of $\\mathscr{L}^{\\otimes 6}$)\n\tand $x, y, z$ are coordinates of\n\t$\\mathcal{P}_{n}^{\\otimes 2} \\oplus \\mathcal{P}_{n}^{\\otimes 3} \\oplus \\mathcal{O}$\n\t(or $\\mathscr{L}^{\\otimes 2} \\oplus \\mathscr{L}^{\\otimes 3} \\oplus \\mathcal{O}$).\n\\end{definition}\n\nThe equations for the relative affine patches\n(affine over $\\bar{\\mathcal{C}} \\times Y(n, \\bar{\\mathcal{C}})$) are given by\n$y^{2} = x^{3} - g_{2} x - g_{3}$ in\n\t$\n\t\t\\mathbb{V} \\bigl(\n\t\t\th_{\\bar{\\mathcal{C}}}^{\\ast}(\n\t\t\t\t\\mathcal{P}_{n}^{\\otimes 2} \\oplus \\mathcal{P}_{n}^{\\otimes 3}\n\t\t\t)^{\\vee}\n\t\t\\bigr)\n\t$\n(for the locus $z \\ne 0$) and\n$z = x^{3} - g_{2} x z^{2} - g_{3} z^{3}$ in\n\t$\n\t\t\\mathbb{V} \\bigl(\n\t\t\th_{\\bar{\\mathcal{C}}}^{\\ast}(\n\t\t\t\t\\mathcal{P}_{n}^{\\otimes -1} \\oplus \\mathcal{P}_{n}^{\\otimes -3}\n\t\t\t)^{\\vee}\n\t\t\\bigr)\n\t$\n(for the locus $y \\ne 0$).\nThe composite of the inclusion\n\t$\n\t\t\t\\mathcal{S}^{\\ast}\n\t\t\\hookrightarrow\n\t\t\t\\mathbb{P} \\bigl(\n\t\t\t\th_{\\bar{\\mathcal{C}}}^{\\ast}(\n\t\t\t\t\t\\mathcal{P}_{n}^{\\otimes 2} \\oplus \\mathcal{P}_{n}^{\\otimes 3} \\oplus \\mathcal{O}\n\t\t\t\t)^{\\vee}\n\t\t\t\\bigr)\n\t$\nand the natural projection\n\t$\n\t\t\t\\mathbb{P} \\bigl(\n\t\t\t\th_{\\bar{\\mathcal{C}}}^{\\ast}(\n\t\t\t\t\t\\mathcal{P}_{n}^{\\otimes 2} \\oplus \\mathcal{P}_{n}^{\\otimes 3} \\oplus \\mathcal{O}\n\t\t\t\t)^{\\vee}\n\t\t\t\\bigr)\n\t\t\\twoheadrightarrow\n\t\t\t\\bar{\\mathcal{C}} \\times Y(n, \\bar{\\mathcal{C}})\n\t$\ndefines a morphism\n$\\mathcal{S}^{\\ast} \\to \\bar{\\mathcal{C}} \\times Y(n, \\bar{\\mathcal{C}})$.\n\n\\begin{lemma} \\label{l:UniWei} \\mbox{}\n\t\\begin{itemize}\n\t\t\\item\n\t\t\tThe morphism $\\mathcal{S}^{\\ast} \\to \\bar{\\mathcal{C}} \\times Y(n, \\bar{\\mathcal{C}})$ is\n\t\t\tprojective and flat.\n\t\t\\item\n\t\t\tThe fibers of the morphism\n\t\t\t$\\mathcal{S}^{\\ast} \\to Y(n, \\bar{\\mathcal{C}})$\n\t\t\tare normal projective surfaces with only rational double points.\n\t\t\\item\n\t\t\tFor any geometric point $b$ of $Y(n, \\bar{\\mathcal{C}})$,\n\t\t\tlet $\\mathcal{S}_{b}^{\\ast}$ be the fiber of\n\t\t\t$\\mathcal{S}^{\\ast} \\to Y(n, \\bar{\\mathcal{C}})$ over $b$\n\t\t\tand $\\mathcal{S}_{b}$ the minimal resolution of singularities of $\\mathcal{S}_{b}^{\\ast}$.\n\t\t\tThen for any algebraically closed field $k$ over $\\mathbb{F}_{q}$,\n\t\t\tthe map $b \\in Y(n, \\bar{\\mathcal{C}})(k) \\mapsto [\\mathcal{S}_{b} \\to \\mathcal{C}_{k}]$\n\t\t\tgives a bijection between $X(n, \\bar{\\mathcal{C}})(k) = Y(n, \\bar{\\mathcal{C}})(k) \/ k^{\\times}$\n\t\t\tand the set of isomorphism classes of minimal elliptic surfaces over $k$\n\t\t\tfibered over $\\mathcal{C}_{k}$ with section and\n\t\t\t$\\chi(\\mathcal{S}_{b}, \\mathcal{O}_{\\mathcal{S}_{b}}) = n$.\n\t\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\n\tThe flatness follows from the above description of the relative affine patches.\n\tThe rest is \\cite[Theorems 1 and 2]{Kas77}\n\t(valid in any characteristic $p > 3$ as noted before).\n\\end{proof}\n\nNote that a blowup of $\\mathcal{S}^{\\ast}$ does not give\na family over $Y(n, \\bar{\\mathcal{C}})$ with fibers $\\mathcal{S}_{b}$\ndue to exceptional divisors.\nA simultaneous resolution of singularities for a family of surfaces\ndoes not always exist as a scheme (\\cite{Art74a}).\nThat is why we work with the non-smooth surfaces $\\mathcal{S}_{b}^{\\ast}$.\n\n\\begin{lemma} \\label{l:PicFib}\n\tFor any geometric point $b$ of $Y(n, \\bar{\\mathcal{C}})$,\n\tthe object $\\Pic_{\\mathcal{S}_{b}^{\\ast} \/ b}^{0}$ is\n\tan abelian variety of dimension independent of $b$.\n\\end{lemma}\n\n\\begin{proof}\n\tIf $n = 0$, then $\\deg(\\Delta) = 0$ and $g_{\\mathcal{C}} = 0$.\n\tHence $\\mathcal{S}_{b} = \\mathcal{S}_{b}^{\\ast} \\to \\mathcal{C}_{b}$ is\n\ta smooth family of elliptic curves over a projective line,\n\twhich is necessarily a constant family.\n\tTherefore $\\Pic_{\\mathcal{S}_{b}^{\\ast} \/ b}^{0}$ is an elliptic curve.\n\tLet $n > 0$.\n\tThe morphism $\\mathcal{S}_{b}^{\\ast} \\to \\mathcal{C}_{b}$ has a section\n\tcorresponding to $(x : y : z) = (0 : 1 : 0)$.\n\tHence the morphism\n\t$\\Pic_{\\mathcal{C}_{b} \/ b} \\to \\Pic_{\\mathcal{S}_{b}^{\\ast} \/ b}$\n\tis injective.\n\tThe group scheme $\\Pic_{\\mathcal{S}_{b}^{\\ast} \/ b}^{0}$ is projective\n\tby the (geometric) normality of $\\mathcal{S}_{b}^{\\ast}$ and \\cite[Theorem 5.4]{Kle05}.\n\tIts tangent space at zero is $\\coh^{1}(\\mathcal{S}_{b}^{\\ast}, \\mathcal{O}_{\\mathcal{S}_{b}^{\\ast}})$,\n\twhich is isomorphic to $\\coh^{1}(\\mathcal{S}_{b}, \\mathcal{O}_{\\mathcal{S}_{b}})$\n\tsince $\\mathcal{S}_{b}^{\\ast}$ has only rational singularities\n\tby Lemma \\ref{l:UniWei}.\n\tThe dimension of this space is $g_{\\mathcal{C}}$\n\tby $\\deg(\\mathcal{L}) = n > 0$ and the paragraph after Theorem \\ref{t:tan}.\n\tTherefore $\\Pic_{\\mathcal{C}_{b} \/ b}^{0}$ and $\\Pic_{\\mathcal{S}_{b}^{\\ast} \/ b}^{0}$\n\thave the same tangent space.\n\tHence $\\Pic_{\\mathcal{C}_{b} \/ b}^{0} \\overset{\\sim}{\\to} \\Pic_{\\mathcal{S}_{b}^{\\ast} \/ b}^{0}$,\n\twhich is an abelian variety of dimension $g_{\\mathcal{C}}$.\n\\end{proof}\n\n\\begin{lemma} \\label{l:BrRes}\n\tFor any geometric point $b$ of $Y(n, \\bar{\\mathcal{C}})$, we have\n\t$\\hat{\\Br}_{\\mathcal{S}_{b}^{\\ast} \/ b} \\overset{\\sim}{\\to} \\hat{\\Br}_{\\mathcal{S}_{b} \/ b}$,\n\twhich is a formal Lie group.\n\\end{lemma}\n\n\\begin{proof}\n\tThe same argument as the previous lemma shows that\n\t$\\Pic_{\\mathcal{S}_{b} \/ b}^{0}$\n\tis an abelian variety.\n\tHence the formal Brauer group of $\\mathcal{S}_{b}$ is a formal Lie group\n\tby \\cite[II, Corollary (4.1)]{AM77} and the paragraphs after.\n\tBy the same argument as \\cite[(2.1)]{Art74b}, the morphism\n\t$\\hat{\\Br}_{\\mathcal{S}_{b}^{\\ast} \/ b} \\to \\hat{\\Br}_{\\mathcal{S}_{b} \/ b}$\n\tis an isomorphism.\n\\end{proof}\n\n\\begin{lemma} \\label{l:RelPic}\n\t$\\Pic_{\\mathcal{S}^{\\ast} \/ Y(n, \\bar{\\mathcal{C}})}^{0}$ is represented\n\tby a smooth scheme over $Y(n, \\bar{\\mathcal{C}})$.\n\tIn particular, the formal completion of $\\Pic_{\\mathcal{S}^{\\ast} \/ Y(n, \\bar{\\mathcal{C}})}$\n\talong the zero section is formally smooth.\n\\end{lemma}\n\n\\begin{proof}\n\tThe morphism $\\mathcal{S}^{\\ast} \\to Y(n, \\bar{\\mathcal{C}})$ is projective and flat\n\tand its fibers are normal surfaces\n\tby Lemma \\ref{l:UniWei}.\n\tHence the sheaf $\\Pic_{\\mathcal{S}^{\\ast} \/ Y(n, \\bar{\\mathcal{C}})}$ is represented by\n\ta separated scheme locally of finite type by \\cite[Theorem 4.8]{Kle05}.\n\tThe fibers of $\\Pic_{\\mathcal{S}^{\\ast} \/ Y(n, \\bar{\\mathcal{C}})}^{0}$\n\tare abelian varieties of the same dimension by Lemma \\ref{l:PicFib}.\n\tTherefore \\cite[Proposition 5.20]{Kle05} implies that\n\t$\\Pic_{\\mathcal{S}^{\\ast} \/ Y(n, \\bar{\\mathcal{C}})}^{0}$ is smooth over $Y(n, \\bar{\\mathcal{C}})$.\n\\end{proof}\n\n\\begin{lemma} \\label{l:RelBr}\n\tThe formal Brauer group $\\hat{\\Br}_{\\mathcal{S}^{\\ast} \/ Y(n, \\bar{\\mathcal{C}})}$\n\tof $\\mathcal{S}^{\\ast} \\to Y(n, \\bar{\\mathcal{C}})$ exists\n\tas a formally smooth formal group scheme over $Y(n, \\bar{\\mathcal{C}})$.\n\tAny point of $Y(n, \\mathcal{C})$ has an affine open neighborhood $\\mathcal{U} = \\spec R$\n\twhere $\\hat{\\Br}_{\\mathcal{S}^{\\ast} \/ Y(n, \\bar{\\mathcal{C}})}$\n\tis given by a formal group law over $R$.\n\\end{lemma}\n\n\\begin{proof}\n\tThe morphism $\\mathcal{S}^{\\ast} \\to Y(n, \\bar{\\mathcal{C}})$ is\n\tcohomologically flat in dimension zero by \\cite[(7.8.6)]{Gro63}\n\tsince it is flat with geometrically reduced fibers\n\tby Lemma \\ref{l:UniWei}.\n\tThe same lemma says that these fibers are normal projective surfaces.\n\tTherefore, with Lemma \\ref{l:RelPic},\n\twe may apply \\cite[Corollary (4.1)]{AM77} and the paragraphs after\n\tto see that $\\hat{\\Br}_{\\mathcal{S}^{\\ast} \/ Y(n, \\bar{\\mathcal{C}})}$\n\texists as a formally smooth formal group scheme over $Y(n, \\bar{\\mathcal{C}})$.\n\tBy \\cite[II, (2.4) and (3.2)]{AM77},\n\tthe Lie algebra of $\\hat{\\Br}_{\\mathcal{S}^{\\ast} \/ Y(n, \\bar{\\mathcal{C}})}$\n\tis the second direct image of the structure sheaf of $\\mathcal{S}^{\\ast}$\n\tby the morphism $\\mathcal{S}^{\\ast} \\to Y(n, \\bar{\\mathcal{C}})$.\n\tFor any point $b \\in Y(n, \\bar{\\mathcal{C}})$,\n\tthe dimension of\n\t\t$\n\t\t\t\t\\coh^{2}(\\mathcal{S}_{b}^{\\ast}, \\mathcal{O}_{\\mathcal{S}_{b}^{\\ast}})\n\t\t\t\\cong\n\t\t\t\t\\coh^{2}(\\mathcal{S}_{b}, \\mathcal{O}_{\\mathcal{S}_{b}})\n\t\t$\n\tdepends only on $n$ and $g_{\\mathcal{C}}$ but not on $b$\n\tby the paragraph after Theorem \\ref{t:tan}.\n\tTherefore by Grauert's theorem (\\cite[III, Corollary 12.9]{Har77}),\n\t$\\mathrm{Lie}(\\hat{\\Br}_{\\mathcal{S}^{\\ast} \/ Y(n, \\bar{\\mathcal{C}})})$\n\tis a vector bundle on $Y(n, \\mathcal{C})$.\n\tOn each open affine $\\mathcal{U} = \\spec R$ of $Y(n, \\mathcal{C})$\n\twhere $\\mathrm{Lie}(\\hat{\\Br}_{\\mathcal{S}^{\\ast} \/ Y(n, \\bar{\\mathcal{C}})})$ is free,\n\tthe formal group $\\hat{\\Br}_{\\mathcal{S}^{\\ast} \/ Y(n, \\bar{\\mathcal{C}})}$\n\tis given by a formal group law over $R$\n\tby \\cite[II, Corollary 2.32]{Zin84}.\n\\end{proof}\n\nBy pullback, the function $a \\mapsto \\mu_{a}$ on $X(n, \\bar{\\mathcal{C}})$ defines\na function $b \\mapsto \\mu_{b}$ on $Y(n, \\bar{\\mathcal{C}})$.\nThe value $\\mu_{b}$ is equal to the dimension of the unipotent part of\nthe geometric fiber of $\\hat{\\Br}_{\\mathcal{S}^{\\ast} \/ Y(n, \\bar{\\mathcal{C}})}$ at $b$\nby Remark \\ref{sdef}, Proposition \\ref{p:ShaBr} and Lemma \\ref{l:BrRes}.\n\n\\begin{lemma} \\label{l:usc}\n\tThe function $b \\mapsto \\mu_{b}$ on $Y(n, \\bar{\\mathcal{C}})$ is upper semicontinuous.\n\\end{lemma}\n\n\\begin{proof}\n\tSet $\\mathcal{Y} = Y(n, \\bar{\\mathcal{C}})$,\n\t$\\mathcal{G} = \\hat{\\Br}_{\\mathcal{S}^{\\ast} \/ Y(n, \\bar{\\mathcal{C}})}$\n\tand $d = \\dim(\\mathcal{G})$.\n\tFor any non-negative integer $m$,\n\tlet $\\mathcal{G}[F^{m}]$ be the kernel of the $p^{m}$-th power relative Frobenius morphism\n\t$\\mathcal{G} \\to \\mathcal{G}^{(p^{m})}$ over $\\mathcal{Y}$.\n\tThen $\\mathcal{G}[F^{m}]$ is finite flat over $\\mathcal{Y}$ by Lemma \\ref{l:RelBr}.\n\tHence the intersection $\\mathcal{G}[F^{m}] \\cap \\mathcal{G}[p^{d}]$ of\n\t$\\mathcal{G}[F^{m}]$ and the $p^{d}$-torsion part $\\mathcal{G}[p^{d}]$\n\tis finite over $\\mathcal{Y}$.\n\t\n\tFor any $b \\in \\mathcal{Y}$,\n\tlet $\\mathcal{G}_{b, \\mathrm{uni}}$ be the unipotent part of the fiber $\\mathcal{G}_{b}$\n\tand $\\mathcal{G}_{b, \\mathrm{div}}$ the $p$-divisible quotient of $\\mathcal{G}_{b}$\n\t(\\cite[V, \\S 8, Theorem 5.36]{Zin84}).\n\tLet $h_{b}$ be the height of $\\mathcal{G}_{b, \\mathrm{div}}$.\n\tLet $h_{b, m}$ be the log base $p$ of the degree of the finite group scheme\n\t$\\mathcal{G}_{b}[F^{m}] \\cap \\mathcal{G}_{b}[p^{d}]$ over $b$.\n\tWe first show that\n\t\t\\[\n\t\t\t\tm \\mu_{b}\n\t\t\t\\le\n\t\t\t\th_{b, m}\n\t\t\t\\le\n\t\t\t\tm \\mu_{b} + d h_{b}.\n\t\t\\]\n\tThe exact sequence\n\t\t$\n\t\t\t\t0\n\t\t\t\\to\n\t\t\t\t\\mathcal{G}_{b, \\mathrm{uni}}\n\t\t\t\\to\n\t\t\t\t\\mathcal{G}_{b}\n\t\t\t\\to\n\t\t\t\t\\mathcal{G}_{b, \\mathrm{div}}\n\t\t\t\\to\n\t\t\t\t0\n\t\t$\n\tinduces an exact sequence\n\t\t\\[\n\t\t\t\t0\n\t\t\t\\to\n\t\t\t\t\\mathcal{G}_{b, \\mathrm{uni}}[F^{m}] \\cap \\mathcal{G}_{b, \\mathrm{uni}}[p^{d}]\n\t\t\t\\to\n\t\t\t\t\\mathcal{G}_{b}[F^{m}] \\cap \\mathcal{G}_{b}[p^{d}]\n\t\t\t\\to\n\t\t\t\t\\mathcal{G}_{b, \\mathrm{div}}[F^{m}] \\cap \\mathcal{G}_{b, \\mathrm{div}}[p^{d}].\n\t\t\\]\n\tSince $\\mathcal{G}_{b, \\mathrm{uni}}$ is killed by $p^{d}$,\n\tthe first term\n\t$\\mathcal{G}_{b, \\mathrm{uni}}[F^{m}] \\cap \\mathcal{G}_{b, \\mathrm{uni}}[p^{d}]$\n\tis equal to $\\mathcal{G}_{b, \\mathrm{uni}}[F^{m}]$,\n\twhose log base $p$ of the degree is $m \\mu_{b}$.\n\tThe third term\n\t$\\mathcal{G}_{b, \\mathrm{div}}[F^{m}] \\cap \\mathcal{G}_{b, \\mathrm{div}}[p^{d}]$\n\tis contained in $\\mathcal{G}_{b, \\mathrm{div}}[p^{d}]$,\n\twhose log base $p$ of the degree is $d h_{b}$.\n\tThese show the desired inequalities.\n\t\n\tSet $m = m_{b} = d h_{b} + 1$.\n\tLet $\\mathcal{U}_{b}$ be the set of points $c \\in \\mathcal{Y}$\n\tsuch that $h_{c, m} \\le h_{b, m}$.\n\tBy the finiteness of $\\mathcal{G}[F^{m}] \\cap \\mathcal{G}[p^{d}]$ over $\\mathcal{Y}$\n\tand Nakayama's lemma,\n\twe know that $\\mathcal{U}_{b}$ is an open subset of $\\mathcal{Y}$\n\tand hence an open neighborhood of $b$.\n\tFor any $c \\in \\mathcal{U}_{b}$,\n\twe have\n\t\t\\[\n\t\t\t\tm \\mu_{c}\n\t\t\t\\le\n\t\t\t\th_{c, m}\n\t\t\t\\le\n\t\t\t\th_{b, m}\n\t\t\t\\le\n\t\t\t\tm \\mu_{b} + d h_{b}\n\t\t\t=\n\t\t\t\tm (\\mu_{b} + 1) - 1,\n\t\t\\]\n\tso $\\mu_{c} \\le \\mu_{b}$.\n\tThis shows that $b \\mapsto \\mu_{b}$ is upper semicontinuous.\n\\end{proof}\n\nNow we prove Theorem \\ref{t:GenVan}.\n\n\\begin{proof}[Proof of Theorem \\ref{t:GenVan}]\n\tBy Lemma \\ref{l:usc},\n\tthe locus of $Y(n, \\bar{\\mathcal{C}})$\n\twhere $\\mu_{b} = 0$ (or equivalently, $\\mu_{b} < 1$) is open.\n\tWe show that this locus is dense.\n\tThe scheme $\\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}$ is irreducible\n\tsince for any choice of a closed point $P$ of $\\bar{\\mathcal{C}}$,\n\tthe map $D \\mapsto D + P$ of divisors defines an isomorphism\n\t\t$\n\t\t\t\t\\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{0}\n\t\t\t\\overset{\\sim}{\\to}\n\t\t\t\t\\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{n}\n\t\t$\n\tfrom the Jacobian variety $\\Pic_{\\bar{\\mathcal{C}} \/ \\bar{\\mathbb{F}}_{q}}^{0}$ of $\\bar{\\mathcal{C}}$.\n\tHence the vector bundle\n\t\t$\n\t\t\t\\mathbf{V}((\n\t\t\t\t\\mathrm{pr}_{2 \\ast}(\n\t\t\t\t\t\\mathcal{P}_{n}^{\\otimes 4} \\oplus \\mathcal{P}_{n}^{\\otimes 6}\n\t\t\t\t)\n\t\t\t)^{\\vee})\n\t\t$\n\tover it is also irreducible.\n\tHence so is its non-empty open subscheme $Y(n, \\bar{\\mathcal{C}})$.\n\tThe locus where $\\mu = 0$ on $X(n, \\bar{\\mathcal{C}})$ or $Y(n, \\bar{\\mathcal{C}})$ is non-empty\n\tby Proposition \\ref{p:NonEmp}.\n\tA non-empty open subset of an irreducible topological space is dense.\n\tTherefore the locus of $Y(n, \\bar{\\mathcal{C}})$ with $\\mu_{b} = 0$ is dense.\n\t\n\tAs we saw before, $X(n, \\bar{\\mathcal{C}})$ is\n\ta universal geometric quotient of $Y(n, \\bar{\\mathcal{C}})$.\n\tIn particular, $X(n, \\bar{\\mathcal{C}})$ has a quotient topology of $Y(n, \\bar{\\mathcal{C}})$.\n\tHence the function $a \\mapsto \\mu_{a}$ on $X(n, \\bar{\\mathcal{C}})$ is\n\talso upper semicontinuous,\n\tand the locus of $X(n, \\bar{\\mathcal{C}})$ with $\\mu_{a} = 0$ is dense.\n\\end{proof}\n\nIf the condition $n > (g_{\\mathcal{C}} - 1) \/ 2$ is not satisfied,\nthe scheme structure on the set $X(n, \\bar{\\mathcal{C}})$ is much more complicated;\nsee \\cite{Sei87}.\nIn particular, it is not necessarily irreducible.\nIt is not clear if the theorem extends to this case.\n\n\\begin{remark}\n\tThe variety $X(n, \\bar{\\mathcal{C}})$ over $\\bar{\\mathbb{F}}_{q}$ can be naturally defined\n\tover a finite extension of $\\mathbb{F}_{q}$ where $\\mathcal{C}$ has a rational point\n\t(so that a Poincar\\'e sheaf is well-defined).\n\tAssume that $\\mathcal{C}$ has a rational point\n\tand denote the resulting variety over $\\mathbb{F}_{q}$ by $X(n, \\mathcal{C})$.\n\tThe residue fields at closed points of $X(n, \\mathcal{C})$ are finite.\n\tTherefore Theorem \\ref{t:GenVan} says that\n\tthere are ``many'' elliptic curves over finite constant extensions of $K$\n\twith given $\\deg(\\Delta)$ and $\\mu = 0$,\n\twhere this $\\mu$ means the ``usual'' $\\mu$-invariant\n\tas defined in Definition \\ref{d:mu}.\n\tMore specifically, the $\\mu=0$ locus\n\t$X(n, \\mathcal{C})_{\\mu = 0}$ is a dense open subvariety of $X(n, \\mathcal{C})$.\n\tHence the set of closed point of $X(n, \\mathcal{C})_{\\mu = 0}$\n\twith the induced Zariski topology is dense open in\n\tthe set of closed points of $X(n, \\mathcal{C})$.\n\\end{remark}\n\n\n\\section{Computations}\\label{s:verify}\n\nIn this section, we give a certain number of examples where we are able to calculate the $\\mu$-invariant.\n\n\\subsection{Legendre curves}\nAssume that $\\mathrm{char}(K)\\not=2$ and $A$ is defined by the Legendre form,\n\\begin{equation}\\label{e:legendre}\ny^2=x(x-1)(x-f).\n\\end{equation}\nFor each non-zero $g\\in K$, let $(g)_0$ and $(g)_\\infty$ denote the divisors of zeros and poles of $g$.\nWrite\n$$N=\\sum_v n_v\\cdot v,$$\nand\n$$\\Delta=\\sum_v\\delta_v\\cdot v.$$\n\nFor a place $v$, let $\\mathbb{F}_v$ denote the residue field and choose a local parameter $\\pi_v$.\nSuppose $v$ does not divide $(f)_\\infty$ and let $\\bar f\\in\\mathbb{F}_v$ denote the residue class of $f$ at $v$.\nThen the equation\n\\begin{equation}\\label{e:bar}\ny^2=x(x-1)(x-\\bar f)\n\\end{equation}\ndefines a curve over $\\mathbb{F}_v$ having at worst a node. Hence it actually defines $\\bar A$, the reduction of $A$ at $v$,\nso \\eqref{e:legendre} is a minimal equation.\nFurthermore, if $v$ does not divide $(f)_0+(f-1)_0$, then $\\bar A$ is an elliptic curve, in this case, we have\n\\begin{equation}\\label{e:nd0}\nn_v=0,\\;\\;\\text{and}\\;\\;\\delta_v=0.\n\\end{equation}\nIn the case where $v$ divides $(f)_0$, the equation \\eqref{e:bar} reads $y^2=x^2(x-1)$. Put $y=\\xi\\cdot x$, then\nfor $(x,y)\\not=(0,0)$, the singularity of $\\bar A$, we have $x=\\xi^2+1$, $y=\\xi(\\xi^2+1)$. Thus, $A$ has split multiplicative reduction if\n$-1$ is a square in $\\mathbb{F}_v$; non-split multiplicative reduction, if $-1$ is not a square. Let $m_v$ be the coefficient of $v$ in $(f)_0$\nso that $f=\\pi^{m_v}u$, for some $u\\in\\O_v$. Then the discriminant of \\eqref{e:legendre} equals\n$$(\\pi_v^{m_v}u(\\pi_v^{m_v}u-1))^2=\\pi^{2m_v}u',\\;\\;u'\\in\\O_v^*.$$\nThus, in this case, we have\n\\begin{equation}\\label{e:nd1}\nn_v=1,\\;\\;\\text{and}\\;\\;\\delta_v=2m_v.\n\\end{equation}\n\nIf $v$ divides $(f-1)_0$, the equation \\eqref{e:bar} becomes $y^2=x(x-1)^2$. By putting $y=\\xi\\cdot (x-1)$\nand by an argument similar to the above, we deduce that $A$ has split multiplicative reduction at $v$ and if $m_v$ is the coefficient of $v$ in\n$(f-1)_0$, then also\n\\begin{equation}\\label{e:nd2}\nn_v=1,\\;\\;\\text{and}\\;\\;\\delta_v=2m_v.\n\\end{equation}\n\n\nCall a divisor $C=\\sum c_v\\cdot v$ even, if all $c_v$ are even integers.\n\n\\begin{lemma}\\label{l:ss}\nThe elliptic curve $A$ has semi-stable reduction at all places, if and only if $(f)_\\infty$ is even.\n\\end{lemma}\n\\begin{proof}\nSuppose $m_v>0$ is the coefficient of a place $v$ in $(f)_\\infty$ so that $f=\\pi^{-m_v}u$, for some $u\\in\\O_v^*$.\nIf $m_v=2e$ is even, then by replacing $(x,y)$ with $(x\\pi_v^{-2e},y\\pi_v^{-3e})$, the equation \\eqref{e:legendre} becomes\n$$y^2=x(x-\\pi_v^{2e})(x-u).$$\nThis turns out to be a minimal equation with multiplicative reduction.\nIn this case,\n\\begin{equation}\\label{e:nd3}\nn_v=1,\\;\\;\\text{and}\\;\\;\\delta_v=2m_v.\n\\end{equation}\nOn the other hand, if $m_v=2e-1$ is odd, then by replacing $(x,y)$\nwith $(x\\pi_v^{-2e},y\\pi_v^{-3e})$, the equation \\eqref{e:legendre} becomes\n$$y^2=x(x-\\pi_v^{2e})(x-u\\pi_v),$$\nwhich is also a minimal equation (see \\cite[\\S VI.1]{Sil86}) with additive reduction.\n\\end{proof}\n\nIf $(f)_\\infty$ is even, then the equalities \\eqref{e:nd0}, \\eqref{e:nd1}, \\eqref{e:nd2}, \\eqref{e:nd3} together implies\n\\begin{equation}\\label{e:ddelta}\n\\deg \\Delta=6\\deg f.\n\\end{equation}\n\n\\subsection{A simple example}\\label{sub:example}\nLet $K=\\mathbb{F}_q(t)$. Here we demonstrate the case of lower degree $f$\\footnote{Here the degree means that of $(f)_0$, or $(f)_\\infty$.}.\nTake $f(t)=\\frac{g(t)}{t^2}$, $g(t)\\in \\mathbb{F}_q(t)$, $ t\\nmid g(t)$, $\\deg g(t)\\leq 2$, so $\\deg f=2$ and hence\n$$\\deg \\Delta=12.$$\nTherefore $\\mu_{A \/ K} = 0$\nby Theorem \\ref{t:tan}.\nThe associated surface is rational\n(hence has trivial Brauer group over $\\bar{\\mathbb{F}}_{q}$)\nby the paragraph after Theorem \\ref{t:tan}.\n\n\n\\subsection{Computation using Magma}\\label{su:high}\n\nIn this subsection, we examine the $\\mu$-invariant of an elliptic curve\nover $K$ other than a rational function field using Magma \\cite{BCP97}.\nWe limit ourselves to an elliptic curve over a quadratic extension of a rational\nfunction field since Magma cannot treat curves over cubic extensions or\nhigher at the time we wrote the paper.\n\nLet $\\mathcal{C}\/\\mathbb{F}_7$ be the supersingular elliptic curve with Weierstrass equation $Y^2=X^3-5X$. We denote by $K$ its function field. It is a quadratic extension of the rational function field $\\mathbb{F}_7(x)$. Let $E\/\\mathbb{F}_7(x)$ be the elliptic curve given by the Legendre form $Y^2=X(X-1)(X-1\/x^2)$. Then, the $L$-function of $E\/K$ can be computed as the product $L_{E}(s).L_{tw(E)}(s)$, where $tw(E)\/\\mathbb{F}_7(x)$ is the quadratic twist of $E$ by the element $x^3+5x$ of $\\mathbb{F}_7(x)$. The integers $\\deg(\\Delta)$ and $\\theta$ of $E\/K$ are computed as follows: First to compute $\\theta$ we just need to know $P(T)$ such that $P(7^{-s})=L_{E}(s)$. This information is obtained by the following Magma command:\n\\begin{enumerate}\n\\item[] \\verb#F<x> := FunctionField(GF(7));#\n\\item[] \\verb#f := (1\/x^2);#\n\\item[] \\verb#E := EllipticCurve([0,-1-f,0,f,0]);#\n\\item[] \\verb#Etw := QuadraticTwist(E,x^3-5*x);#\n\\item[] \\verb#LFunction(Etw);#\n\\item[] \\verb#LFunction(E);#\n\\end{enumerate}\nWe obtain\n\\begin{enumerate}\n\\item[] \\verb#-117649*T^6 + 1715*T^4 - 35*T^2 + 1#\n\\item[] \\verb#1#\n\\end{enumerate}\nBy hand computation, we know that $7^{2} P(7^{-1} T)$ is $7$-primitive,\nso $\\theta = 2$.\n\nNext, to implement the curve $E\/K$, we use the Magma command:\n\\begin{enumerate}\n\\item[] \\verb#S<z> := PolynomialRing(F);#\n\\item[] \\verb#K<t> := ext<F |z^2-x^3-5*x>;#\n\\item[] \\verb#EK:=BaseChange(E,K);#\n\\end{enumerate}\nThe integer $\\deg(\\Delta_{E\/K})$ is obtained by the following command:\n\\begin{enumerate}\n\\item[] \\verb#SK:=BadPlaces(EK);#\n\\item[] \\verb#&+[LocalInformation(EK,v)[2]*Degree(v):v in SK];#\n\\end{enumerate}\nWe obtain $\\deg(\\Delta_{E\/K})=24$ and conclude that the $\\mu$-invariant of $E\/K$ is zero\nby Theorem \\ref{t:tan}.\nThe associated surface has Kodaira dimension $1$\nby the paragraph after Theorem \\ref{t:tan}.\n\n\\subsection{An example of non-semistable reduction}\\label{su:nonss}\nWe consider the curve $y^2=x(x-1)(x-f(t))$\nwith $f:= (1+t+t^2)\/(1+t)\\in\\mathbb{F}_3[t]$ on Magma. The output is the following\n\\vskip5pt\n\\noindent\n\\begin{enumerate}\n\\item[(a)] The bad places are $( t + 1)_0$, $(1\/t)_0$, those are additive places, and $(t + 2)_0$, $(t)_0$, multiplicative places.\n\\item[(b)] The $j$-invariant is\n$$j=\\frac{t^{12} + t^9 + 2t^6 + 2t^3 + 1}{t^{10} + t^9 + 2t^8 + t^7 + 2t^6 + t^5 + t^4}=\\frac{t^{12} + t^9 + 2t^6 + 2t^3 + 1}{(t+1)^2(t-1)^4t^4}.$$\nOne can compute by\nhand that\n$$\\Delta=8(t+1)_0+8(1\/t)_0+4(t)_0+4(t+2)_0,$$\nso $\\deg \\Delta=24$. The equation \\eqref{e:ddelta} is not satisfied here. Note that $j=c_4^3\/\\Delta$, but in this case,\n$\\Delta$ is not the exact ``denominator''\nof $j$.\n\\item[(c)] $P(T)=-9T^2+1$, so $\\theta=0$.\n\\end{enumerate}\n\nTherefore $\\mu = 1$.\nThe associated surface is a K3 surface,\nand the result $\\mu = 1$ shows that it is supersingular.\n\n\\subsection{A supersingular isotrivial curve} \\label{su:sic}\n\nWe give an example of an isotrivial non-constant elliptic curve\nwith everywhere good supersingular reduction and $\\mu = 0$\nover a function field with non-invertible Hasse-Witt matrix.\nAs any finite extension of such a function field has non-invertible Hasse-Witt matrix,\nthis gives a counterexample to the ``only if'' direction of \\cite[Theorem 1.8 (2)]{OT09}.\n\nAssume $p \\ne 2$.\nLet $\\mathcal{C}' \\to \\mathcal{C}$ be a degree $2$ isogeny between supersingular elliptic curves over $\\mathbb{F}_{q}$.\nLet $K' \/ K$ be the corresponding extension of function fields.\nLet $A_{0} \/ K$ be a constant supersingular elliptic curve.\nLet $A \/ K$ be the quadratic twist of $A_{0}$ by $K' \/ K$,\nwhich has good reduction everywhere since $K' \/ K$ is unramified everywhere.\nThe elliptic surfaces over $\\mathbb{F}_{q}$ corresponding to $A_{0} \/ K$ and $A_{0} \/ K'$\nare both supersingular abelian surfaces.\nTherefore $\\mu_{A_{0} \/ K} = \\mu_{A_{0} \/ K'} = 1$\nby Proposition \\ref{p:aborK3}.\nAs $p \\nmid [K' : K] = 2$,\nthe $\\Lambda$-module $\\Sel_{p^{\\infty}}(A_{0} \/ K_{\\infty}'^{(p)})$ is isomorphic to\n$\\Sel_{p^{\\infty}}(A \/ K_{\\infty}^{(p)}) \\oplus \\Sel_{p^{\\infty}}(A_{0} \/ K_{\\infty}^{(p)})$.\nTherefore $\\mu_{A \/ K} = \\mu_{A_{0} \/ K'} - \\mu_{A_{0} \/ K} = 0$.\nThis $A \/ K$ thus gives a desired example.\n\nOne checks that the elliptic surface corresponding to $A \/ K$ is a hyperelliptic surface.\nAs such, we have $\\mu_{A \/ K} = 0$ also by Proposition \\ref{p:aborK3}.\nMore generally, for any hyperelliptic surface $\\mathcal{S}$\nobtained as a quotient of the product of two supersingular elliptic curves (\\cite[Theorem 4]{BM77}),\nthe generic fiber $A \/ K$ of its Albanese morphism $\\mathcal{S} \\to \\mathcal{C}$ gives a desired example.\nThis includes the case of $p = 2$.\n\nWhat is true about the ``only if'' direction of \\cite[Theorem 1.8 (2)]{OT09}\nand what the proof of this statement in \\cite{OT09} actually says is that\nfor a \\emph{constant} non-zero supersingular abelian variety $A \/ K$,\nif $\\mu_{A \/ K} = 0$, then the Hasse-Witt matrix of $K$ is invertible.\n\n\\subsection{Large rank curves of Ulmer and Shioda}\nWe compute the $\\mu$-invariants of the elliptic curves with large ranks\nconsidered by Ulmer \\cite[Theorem 1.5]{Ulm02} and Shioda \\cite[Remark 10]{Shi86}.\nThis is an application of the method of proof of Proposition \\ref{p:KodOne}.\n\nUlmer's elliptic curve is defined by the equation\n$y^{2} + x y = x^{3} - t^{d}$ over $K = \\mathbb{F}_{p}(t)$,\nwhere $p$ is an arbitrary prime, $d = p^{n} + 1$ and $n$ a positive integer.\nAs explained in \\cite[\\S 10.2]{Ulm02},\nits geometric Mordell-Weil rank $\\bar{r}$ attains the geometric bound,\ni.e., $\\bar{r} = a$ ($= \\deg(P_{1}(t))$; see \\eqref{e:OS}).\nThis implies $\\theta = 0$ by the proof of Proposition \\ref{p:KodOne}.\nTherefore $\\mu = \\deg(\\Delta) \/ 12 - 1$ by Theorem \\ref{t:tan}.\nThe calculations in \\cite[\\S 2]{Ulm02} show that\n$\\deg(\\Delta) \/ 12 = \\lceil d \/ 6 \\rceil$ (the least integer $\\ge d \/ 6$).\nHence $\\mu = \\lceil d \/ 6 \\rceil - 1$.\n\nOn the other hand, Shioda's elliptic curve is defined by the equation\n$y^{2} = x^{3} + x + t^{d}$ over $K = \\mathbb{F}_{p}(t)$,\nwhere $p \\equiv -1 \\mod 4$, $d = (p^{\\nu} + 1) \/ 2$\nand $\\nu$ is a positive odd integer.\nLet $\\mathcal{S} \/ \\mathbb{F}_{p}$ be the elliptic surface corresponding to this curve.\nThen its geometric Lefschetz number\n\t$\n\t\t\t\\lambda(\\mathcal{S}_{\\bar{\\mathbb{F}}_{p}})\n\t\t=\n\t\t\t\\deg P_{\\mathcal{S}, 2} - \\rho_{\\mathcal{S}_{\\bar{\\mathbb{F}}_{p}}}\n\t$\nis zero as shown in \\cite[Remark 10]{Shi86}.\nTherefore $\\theta = 0$ again by the proof of Proposition \\ref{p:KodOne}.\nTherefore $\\mu = \\deg(\\Delta) \/ 12 - 1$ by Theorem \\ref{t:tan}.\nA calculation similar to \\cite[\\S 2]{Ulm02} shows that\n$\\deg(\\Delta) \/ 12 = \\lceil d \/ 6 \\rceil$.\nHence $\\mu = \\lceil d \/ 6 \\rceil - 1$.\n\n\n\\subsection{Other examples in the literature} Numerous other computations can be found in \\cite{Ulm19} (for example \\cite{Ulm19}, Remark 4.5, Theorem 5.1, Corollary 6.5 and its applications in Sections 10--12).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA decomposition of the complete graph on $n$ points into \ntriangles is called a \\emph{Steiner triple system} \nof order $n$ and is denoted by $\\STS{n}$. The vertex sets of \nthe triples used are called the \\emph{blocks} of the Steiner \ntriple system. Thus, equivalently, an $\\STS{n}$ is a pair\n$(X,B)$, where $X$ is an $n$-element set of \\emph{points}\nand $B$ is a collection of $3$-element subsets of $X$\ncalled \\emph{blocks}, such that every pair of points is contained \nin exactly one block. It is well known that an $\\STS{n}$ \nexists if and only if $n \\equiv 1,3 \\pmod{6}$. A decomposition of a proper subgraph of the complete graph on $n$ points into\ntriangles is called a \\emph{partial Steiner triple system}\nof order $n$ and is denoted by $\\PSTS{n}$. The \\emph{size} of a $\\PSTS{n}$ is the number of blocks it contains.\n\nAn $\\STS{n}$ is \\emph{sequenceable} if the points can be sequenced so \nthat no proper segment can be partitioned into blocks.\nSuch a sequence is called an \\emph{admissible sequence}.\nIf  an $\\STS{n}$ has no admissible sequence, then we say it is\n\\emph{nonsequenceable}.  \nFor example, $\\{\\mathit{abd}, \\mathit{bce}, \\mathit{cdf}, \\mathit{deg}, \\mathit{efa}, \\mathit{fgb}, \\mathit{gac}\\}$,\nthe unique (up to isomorphism) $\\STS{7}$ has the admissible\nsequence $\\mathit{abcdefg}$. \nA fascinating study of sequenceable partial \nSteiner triple systems can be found in~\\cite{man41}.\n\nIn \\cite{KS} a related problem is examined. In this article the authors ask if the vertices of a Steiner triple system can be sequenced such that no length $\\ell$ segment of the sequence contains a block. They obtain results for $\\ell=3$ and $4$.\n\n\n\n\nA set of disjoint blocks in an $\\STS{n}$ is called a \n\\emph{partial parallel class}.\nClearly any partial parallel class contains at most \n$\\lfloor v\/3 \\rfloor$ blocks. A partial parallel class \ncontaining all the points of a design is called a \\emph{parallel \nclass} and a partial parallel class containing all but one \nof the points of a design is called an \\emph{almost parallel class}. \n\n\\begin{theorem}\\label{one}\nSuppose a  $\\PSTS{n}$ has the property, for $n-1$ distinct points $x$, that \nthere is an almost parallel class\nthat does not contain $x$. Then the $\\PSTS{n}$\n is nonsequenceable.\n\\end{theorem}\n\n\\begin{proof}\nConsider any sequence $x_1x_2\\cdots x_n$ of the vertices \nof such a $\\PSTS{n}$.   There is either an almost parallel  class that \ndoes not contain $x_1$ or one that does not contain $x_n$.  \nThus the blocks of the first almost  parallel class, if it exists, would\npartition the segment $x_2\\cdots x_{n-1}x_n$,\nand the latter would partition\nthe segment $x_1x_2\\cdots x_{n-1}$.\n\\end{proof}\n\n\\section{Example applications of Theorem~\\ref{one}}\\label{examples}\n\\begin{description}[leftmargin=1em]\n\\item[\\boldmath\\STS{13}:]\nDeveloping the base blocks\n\n\\hfill\\begin{minipage}{.875\\textwidth}\\begin{flushleft}\n$\\{0,2,7\\}$,\n$\\{0,1,4\\}$,\n\\end{flushleft}\\end{minipage}\n\nmodulo 13 generates an $\\STS{13}$ that contains the almost parallel class \n\n\\hfill\\begin{minipage}{.875\\textwidth}\\begin{flushleft}\n\\noinden\n\\{7,8,11\\}$,\n\\{3,5,10\\}$,\n\\{2,4,9\\}$,\n\\{1,6,12\\}$.\n\\end{flushleft}\\end{minipage}\n\n\nBecause any translate of this almost parallel class is again an almost parallel class, there is an almost parallel class missing any desired point.\nThus, by Theorem~\\ref{one} this $\\STS{13}$ is nonsequenceable.\n\n\\item[\\boldmath\\STS{19}:]\nDeveloping the base blocks \n\n\\hfill\\begin{minipage}{.875\\textwidth}\\begin{flushleft}\n$\\{0,1,6\\}$,\n$\\{0,2,10\\}$,\n$\\{0,3,7\\}$,\n\\end{flushleft}\\end{minipage}\n\nmodulo 19 generates an $\\STS{19}$ that contains the almost parallel class \n\n\\hfill\\begin{minipage}{.875\\textwidth}\\begin{flushleft}\n\\noinden\n$\\{1,3,11\\}$,\n$\\{2,15,16\\}$,\n$\\{4,17,18\\}$,\n$\\{5,8,12\\}$,\n$\\{6,9,13\\}$,\n$\\{7,10,14\\}$.\n\\end{flushleft}\\end{minipage}\n\n\nThus by Theorem~\\ref{one} this $\\STS{19}$ is nonsequenceable.\n\n\n\\item[\\boldmath\\STS{25}:]\nDeveloping the base blocks\n\n\\hfill\\begin{minipage}{.875\\textwidth}\\begin{flushleft}\n\\noinden\n$\\big\\{(0,0),(0,1),(2,3)\\big\\}$,\n$\\big\\{(0,0),(1,2),(2,0)\\big\\}$,\n$\\big\\{(0,0),(1,0),(3,1)\\big\\}$,\n\\end{flushleft}\\end{minipage}\n\nmodulo 5 independently in both coordinates generates a\n$\\STS{25}$ with vertices $\\Integers{5}\\times\\Integers{5}$\nthat contains the almost parallel class\n\n\\hfill\\begin{minipage}{.875\\textwidth}\\begin{flushleft}\n\\noinden\n$\\big\\{(0,1),(0,2),(2,4)\\big\\}$,\n$\\big\\{(1,0),(3,2),(1,4)\\big\\}$,\n$\\big\\{(1,1),(4,1),(0,3)\\big\\}$,\n$\\big\\{(2,0),(2,3),(3,4)\\big\\}$,\n$\\big\\{(2,1),(2,2),(4,4)\\big\\}$,\n$\\big\\{(3,0),(1,2),(1,3)\\big\\}$,\n$\\big\\{(3,1),(4,2),(3,3)\\big\\}$,\n$\\big\\{(4,0),(4,3),(0,4)\\big\\}$.\n\\end{flushleft}\\end{minipage}\n\n\nIt follows from Theorem~\\ref{one} that\nthis $\\STS{25}$ is nonsequenceable.\n\\item[\\boldmath\\STS{31}:]\nDeveloping the base blocks \n\n\\hfill\\begin{minipage}{.875\\textwidth}\\begin{flushleft}\n\\noinden\n$\\{0,5,11\\}$,\n$\\{0,4,12\\}$,\n$\\{0,3,13\\}$,\n$\\{0,2,9\\}$,\n$\\{0,1,15\\}$,\n\\end{flushleft}\\end{minipage}\n\nmodulo 31 generates an $\\STS{31}$ that contains the almost parallel class \n\n\\hfill\\begin{minipage}{.875\\textwidth}\\begin{flushleft}\n\\noinden\n$\\{11,15,23\\}$,\n$\\{10,26,27\\}$,\n$\\{9,14,20\\}$,\n$\\{8,13,19\\}$,\n$\\{7,25,28\\}$,\n$\\{5,21,22\\}$,\n$\\{4,24,29\\}$,\n$\\{3,6,16\\}$,\n$\\{2,12,30\\}$,\n$\\{1,17,18\\}$.\n\\end{flushleft}\\end{minipage}\n\n\nThus by Theorem~\\ref{one} this $\\STS{31}$ is nonsequenceable.\n\\item[\\boldmath\\STS{43}:]\nDeveloping the base blocks\n\n\\hfill\\begin{minipage}{.875\\textwidth}\\begin{flushleft}\n\\noinden\n$\\{0,1,16\\}$,\n$\\{0,2,14\\}$,\n$\\{0,3,11\\}$,\n$\\{0,4,37\\}$,\n$\\{0,5,25\\}$,\n$\\{0,7,24\\}$,\n$\\{0,9,22\\}$,\n\\end{flushleft}\\end{minipage}\n\nmodulo 43 generates an $\\STS{43}$ that contains the almost \nparallel class \n\n\\hfill\\begin{minipage}{.875\\textwidth}\\begin{flushleft}\n\\noindent$\\{14,15,30\\}$,\n$\\{1,28,29\\}$,\n$\\{2,20,25\\}$,\n$\\{3,23,41\\}$,\n$\\{4,33,35\\}$,\n$\\{5,34,36\\}$,\n$\\{6,19,40\\}$,\n$\\{7,39,42\\}$,\n$\\{8,26,31\\}$,\n$\\{9,27,32\\}$,\n$\\{10,37,38\\}$,\n$\\{11,17,21\\}$,\n$\\{12,18,22\\}$,\n$\\{13,16,24\\}$.\n\\end{flushleft}\\end{minipage}\n\n\nThus by Theorem~\\ref{one} this $\\STS{43}$ is nonsequenceable.\n\n\\end{description}\n\n\\section{Constructions}\n\nA group divisible design with blocks of size 3, having \n$u$ groups of size $g$ and $v$ groups of size $m$ (denoted by $\\GDD{3}{g^um^v}$),\nis a decomposition of the complete multipartite graph\n\\[\n K_{\\underbrace{g,g,\\ldots,g}_u,\\underbrace{m,m,\\ldots,m}_v}\n\\]\ninto triangles. The triangles (or triples ) used in the decomposition are the blocks of the $\\textsc{gdd}$.\nIf $6\\mid m$ and $6 \\mid g$, then a $\\GDD{3}{m^u}$ and\na $\\GDD{3}{g^um^1}$ exist for all $u\\geq 3$  \n\\cite{HB555,HB2211}.\n\n\\begin{theorem}\\label{A}\nThere exists a nonsequenceable $\\STS{n}$ for all $n  \\equiv  1 \\pmod{6}$ except for $n=7$.\n\\end{theorem}\n\\begin{proof}\nFirst suppose $n = 1+6k$ with $k=2u$.\nIf $u=1$, then $n=13$ and if $u=2$, then $n=25$.  \nFor both orders $n=13$ or $25$, there is an example of a nonsequenceable $\\STS{n}$ given in Section~\\ref{examples}. \n\nIf $u \\geq 3$, there exist a $\\GDD{3}{12^u}$ with groups \n(partite sets) $G_1,G_2,\\ldots , G_u$.\nLet $X$ be a new point.\nIn each group $G_i$, fill in \n$G_i \\cup \\{X\\}$\nwith the  nonsequenceable\n$\\STS{13}$  found in Section~\\ref{examples}.\n\nWe now show this $\\textsc{STS}$ satisfies the the conditions of Theorem~\\ref{one}.\nClearly it is satisfied for the point X. \nFor any other point, say $y \\in G_i$, take the almost\nparallel class in the $i$-th $\\STS{13}$ that misses $y$, \nand for all $j \\neq i$,\ntake the almost parallel class in the $j$-th $\\STS{13}$ \nthat misses $X$. The union\nof these partial parallel classes is an almost parallel class that misses $y$.\n\nNow suppose $n = 1+6k$ with $k=2u+3$.\nIf $u = -1$, then $n=7$  and, as noted in the Theorem, a\nnonsequenceable $\\STS{7}$ does not exist. \nIf $n=0,1$ or $2$, then $n=19,31$ or $43$ respectively.\nThere is an example of a nonsequenceable $\\STS{n}$ given in of Section~\\ref{examples} for each $n \\in \\{19,31,43\\}$.\nIf $u \\geq 3$, there exists\na $\\GDD{3}{12^u18^1}$ with groups (partite sets)\n$G_1,G_2,\\ldots ,G_u,M$, where $|G_i|=12$ for all $i$ and $|M|=18$.\nLet $X$ be a new point.\nIn each group $G_i$, fill in \n$G_i \\cup \\{X\\}$ \nwith the  nonsequenceable  $\\STS{13}$ \ngiven in Section~\\ref{examples}. On $M \\cup \\{X\\}$, place the nonsequenceable  $\\STS{19}$ found in Section~\\ref{examples}. The rest of the proof is the same as \nfor $k$ even, and we leave it for the reader to check the details.\n\\end{proof}\n\n\\begin{corollary}\nIf $0 \\leq a \\leq (n-1)\/3$, then\nthere exists a nonsequenceable $\\PSTS{n}$ \nof size $\\frac{1}{3}\\binom{n}{2}-a$,\nfor all $n  \\equiv  1 \\pmod{6}$ except for $n=7$.\n\\end{corollary}\n\\begin{proof}\nThe $\\STS{n}$ constructed in Theorem~\\ref{A} has, for each point $x$, an almost parallel class that does not contain $x$. Fixing any point $x_0$\nand removing $a$ of the blocks in the almost parallel class that does\nnot contain $x_0$  constructs a $\\PSTS{n}$\nof size $\\frac{1}{3}\\binom{n}{2}-a$ that still satisfies \nTheorem~\\ref{one}.\n\\end{proof}\n\n\\section*{Closing remarks.}\nBrian Alspach gave a talk on August 9, 2018 entitled ``Strongly Sequenceable \nGroups'' at the 4th Kliakhandler Conference held at MTU. In this \ntalk, among other things, the notion of sequencing diffuse posets was introduced and the following research problem was posed:\n\\begin{quotation}\n``Given a triple system of order $n$ with $\\lambda=1$, define a poset $P$ by letting its elements be the triples and any union of disjoint triples. This poset is not diffuse in general, but it is certainly possible that $P$ is sequenceable.''\n\\end{quotation}\nOur article shows surprisingly that there are in fact triple systems where \nthe poset $P$ is nonsequenceable. However, it still remains an interesting open question to construct a $\\STS{n}$ that is nonsequenceable when $n \\equiv 3 \\pmod{6}$.\nA more ambitious research problem would be to determine necessary and sufficient conditions for when an $\\STS{n}$ is sequenceable.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\\label{sec:introduction}\n\nSince the possibility of the restoration of broken symmetries in the\nvery early universe was suggested \\cite{KL}, the cosmological\nconsequences of the accompanying phase transitions have been\ninvestigated by a number of authors \\cite{phase}. As Kibble first\npointed out, topological defects of various types may be formed,\ndepending on the pattern of the symmetry breaking \\cite{Kib} (see also\nRef.~\\cite{defect} for review). If the symmetry group $G$ breaks into\na subgroup $H$ and the coset space $G\/H$ is not simply-connected,\nstrings are formed. Strings as local defects have been extensively\ninvestigated in the context of structure formation theory \\cite{Bra}.\nOn the other hand, global strings are predicted in the axion theory of\nthe strong {\\it CP} problem \\cite{PQ}, and they may have radiated\nsignificant amount of axions as the universe cools down from the\nPeccei-Quinn breaking scale to the energy scale of QCD \\cite{davis}.\nAlso, in condensed-matter systems, analogous vortices are\nobserved \\cite{Zur}.\n\nMost studies of the formation of strings have employed Monte-Carlo\nsimulations. Vachaspati and Vilenkin \\cite{VV} first examined the\ninitial configuration of a string network. They found that it has the\nstatistical properties of a Brownian random walk (see also\nRef.~\\cite{SF}), and found that a string network is made of closed\nloops with a scale-invariant size distribution and infinite strings\noccupying $80\\%$ of the total string length. Since that time, many\nauthors have reexamined the problem in more realistic situations than\nthe original, focusing especially on the existence and the fraction\n(if any) of infinite strings \\cite{form}. However, as seen from the\nfact that the above mentioned studies were carried out exclusively by\nthe Monte-Carlo simulations, little work on the dynamical aspects of\nstring formation has been done.\n\nOne exception is Borrill's paper \\cite{Bor}, where the simulation of\nthe bubble nucleation process in a first order phase transition is\nperformed using a lattice-free method. However, in his approach the\nsubsequent evolution of the string network could not be followed. In\nRef.~\\cite{Hod} the Langevin-equation approach is considered. But in\nthat study the effect of the expansion of the universe was considered\nonly partially. This is quiet different from the real dynamics. Ye and\nBrandenberger \\cite{YE}, on the other hand, performed a numerical\nsimulation properly taking cosmic expansion into account. But there, a\nMexican-hat potential was adopted from the outset, and the simulation\ndomain was so small that the whole system fell within one horizon\nbefore they could conclude that the network had reached the scaling\nregime. The evolution of local (gauged) strings has been investigated\nby using the Nambu-Goto action. There are many analytical \\cite{ana}\nand numerical \\cite{num} approaches, and it has been confirmed that\nthe string networks relax to the scaling solution by intercommutation\nwhich generates closed loops that eventually disappear by radiating\ngravitational waves or particles.  On the other hand, as for global\nstrings, the Kalb-Ramond action \\cite{KR} must be used instead of the\nNambu-goto action in order to examine the evolution. But in that\napproach it is difficult to include the long-range force between\ninfinite strings.  Here, we use an approach based on a complex scalar\nfield.\n\nIn this paper, we simulate both the formation and evolution of the\nglobal string network by solving the dynamics of the complex scalar\nfield directly. Starting from a symmetry-restored stage, we follow the\ndynamics of the complex scalar field, fully considering the expansion\nof the universe and the associated change of the effective potential,\nwhich exhibits spontaneous symmetry breaking below the critical\ntemperature.\n\nThe simulation is made in 2+1 dimensions as a first step to analyze\nthe full 3+1-dimensional analysis, because the spatial resolution and\ntotal cosmic time are limited due to our limited computing power. Note\nthat in 2+1 dimensions strings are point-like, and each such string\ncorresponds to an infinite string in 3+1 dimensions so that the energy\nloss mechanism is pair-annihilation rather than loop-formation through\nintercommutation.  The direct emission of goldstone modes from strings\nis suppressed because of the symmetry of the system. In a forth-coming\npaper we will report the results of our simulations in 3+1 dimensions\nand their cosmological implications \\cite{YKY}.\n\nThe remainder of this paper is organized as follows. In the next\nsection, we establish the formulation for the numerical simulation. In\n\\S \\ref{sec:results}, we give the result of the numerical simulation\nand judge whether during the evolution the scaling behavior is\nobserved as expected from the analogy of local strings. In \\S\n\\ref{sec:model}, the results obtained from the simulation are\nexplained analytically using a phenomenological model. Finally, \\S\n\\ref{sec:summary} is devoted to summary and discussion.\n\n\n\n\\section{Formulation}\n\n\\label{sec:formulation}\n\nWe consider the following Lagrangian density for a complex scalar\nfield $\\Phi(x)$: \n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  {\\cal L}[\\Phi] = \\partial}  \\newcommand{\\abs}[1]{\\vert{#1}\\vert_{\\mu}\\Phi \\partial}  \\newcommand{\\abs}[1]{\\vert{#1}\\vert^{\\mu}\\Phi^{\\dagger} - V[\\Phi] \\:.\n\\eeq\n\n\\noindent\nHere, the potential $V[\\Phi]$ is given by\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  V[\\Phi] = \\frac{\\lambda}{2}\\,(\\Phi\\Phi^{\\dagger} - \\eta^2)^2 \\:.\n\\eeq\n\n\\noindent\nThen the one-loop finite temperature effective potential $V_{\\rm\neff}[\\Phi]$ in the high temperature limit is written as\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  V_{\\rm eff}[\\Phi] = \\frac{\\lambda}{2}(\\Phi\\Phi^{\\dagger} - \\eta^2)^2 \n                 + \\frac{\\lambda}{3}T^2\\,\\Phi\\Phi^{\\dagger} \\:.\n\\eeq\n\n\\noindent\nFor $T > T_{c} = \\sqrt{3}\\eta$, the potential $V_{\\rm eff}$ has a\nminimum at $\\Phi = 0$, and the $U(1)$ symmetry is restored. On the\nother hand, new minima $|\\Phi|_{min} = \\eta\\sqrt{1-(T\/T_{c})^2}$\nappear and the symmetry is broken for $T < T_{c}$ (Fig.\\ \n\\ref{fig:potential}). In this case the phase transition is of second\norder.\n\nIn the expanding universe the effective Lagrangian density ${\\cal L}_{\\rm\n  eff}[\\Phi]$ is given by\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  {\\cal L}_{\\rm eff}[\\Phi] = g_{\\mu\\nu}(\\partial}  \\newcommand{\\abs}[1]{\\vert{#1}\\vert^{\\mu}\\Phi)(\\partial}  \\newcommand{\\abs}[1]{\\vert{#1}\\vert^{\\nu}\\Phi)^{\\dagger}\n                 - V_{\\rm eff}[\\Phi] \\:,\n\\eeq \n\n\\noindent\nwhere $g_{\\mu\\nu}$ is identified with the Robertson-Walker metric.\nThen the equation of motion is\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  \\ddot{\\Phi}(x) + 3H\\dot{\\Phi}(x) - \\frac{1}{a(t)^2}\\nabla^2\\Phi(x)\n   = - V'_{\\rm eff}[\\Phi] \\:,\n\\eeq\n\n\\noindent\nwhere the prime represents the derivative $\\partial}  \\newcommand{\\abs}[1]{\\vert{#1}\\vert\/\\partial}  \\newcommand{\\abs}[1]{\\vert{#1}\\vert\\Phi^{\\dagger}$,\nand $a(t)$ is a scale factor.  The Hubble parameter $H = \\dot\na(t)\/a(t)$ and the cosmic time $t$ are given by\n\\begin{eqnarray}}  \\newcommand{\\eea}{\\end{eqnarray}\n  H^2 = \\frac{8\\pi}{3 M_{\\rm pl}^2} \\frac{\\pi^2}{30} g_{*} T^4 \\:,\n   ~~~~~\n  t = \\frac{1}{2H} = \\frac{\\xi}{T^2} \\:,\n\\eea\n\n\\noindent\nwhere $M_{\\rm pl}$ is the Plank mass, $g_{*}$ is the total number of\ndegrees of freedom for the relativistic particles, and radiation\ndomination is assumed. We define the dimensionless parameter $\\zeta$\nas\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  \\zeta \\equiv \\frac{\\xi}{\\eta}  = \\left(}  \\newcommand{\\rmk}{\\right) \\frac{45M_{\\rm pl}^2}\n      {16\\pi^3g_{*} \\eta^2}\n  \\rmk^{1\/2} \\:.\n\\eeq\n\n\\noindent\nIn our simulation, we take $\\zeta = 10$, which corresponds to $\\eta\n\\sim (10^{15}-10^{16})$ GeV, but the essential result is independent\nof this choice. The energy density of each point is written as\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  \\rho(x) = \\dot\\Phi(x) \\dot\\Phi^{\\dagger}(x) \n           + \\frac{1}{a(t)^2}\\nabla\\Phi(x) \\cdot \\nabla\\Phi^{\\dagger}(x) \n            + V_{\\rm eff}[\\Phi]  \\:.\n\\eeq\n\nWe simulate the system in 2+1 dimensions from the initial time\n$t_{i} = t_{c}\/4$, that is, $T_{i} = 2\\,T_{c}$, to the final\ntime $t_{f} = 150\\,t_{i} = 37.5\\,t_{c}$, when $T_{f} \\sim T_{c} \/\n6.1$. Since the $U(1)$ symmetry is restored at the initial time $t =\nt_{i}$, we adopt as the initial condition the thermal equilibrium\nstate with the mass squared,\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  m^2 = \\left. \\frac{d^2 V_{\\rm eff}[|\\Phi|]}{d|\\Phi|^2} \n        \\right|_{|\\Phi|=0} \\:,\n\\eeq\n\n\\noindent\nwhich is the inverse curvature of the potential at the origin at $t =\nt_{i}$. In the thermal equilibrium state, $\\Phi$ and $\\dot\\Phi$ are\nGaussian distributed with the correlation functions,\n\\begin{eqnarray}}  \\newcommand{\\eea}{\\end{eqnarray}\n  \\left\\langle} \\newcommand{\\ra}{\\right\\rangle \\beta|\\Phi(\\vect x)\\Phi^{\\dagger}(\\vect y)|\\beta\n                             \\ra_{\\rm equal\\hbox{-}time} &=&\n   \\int \\frac{d\\vect k}{(2\\pi)^3} \\frac1{2\\sqrt{\\vect k^2 + m^2}}\n           \\coth{\\frac{\\beta\\sqrt{\\vect k^2 + m^2}}{2}}\n             e^{i\\vect k \\cdot (\\vect x-\\vect y)} \\:, \\nonumber}  \\newcommand{\\eqn}[1]{\\beq {#1}\\eeq  \\\\\n             && \\\\\n  \\left\\langle} \\newcommand{\\ra}{\\right\\rangle \\beta|\\dot\\Phi(\\vect x)\\dot\\Phi^{\\dagger}(\\vect y)|\\beta\n                             \\ra_{\\rm equal\\hbox{-}time} &=&\n   \\int \\frac{d\\vect k}{(2\\pi)^3} \\frac{\\sqrt{\\vect k^2 + m^2}}{2}\n           \\coth{\\frac{\\beta\\sqrt{\\vect k^2 + m^2}}{2}}\n             e^{i\\vect k \\cdot (\\vect x-\\vect y)} \\:. \\nonumber}  \\newcommand{\\eqn}[1]{\\beq {#1}\\eeq \\\\\n             &&\n\\eea\n\n\\noindent \nThe functions $\\Phi(\\vect x)$ and $\\dot\\Phi(\\vect y)$ are uncorrelated\nfor $\\vect x \\ne \\vect y$. We generate these fields for the initial\ncondition in the momentum space, because the corresponding fields\n$\\tilde{\\Phi}(\\vect k)$ and $\\tilde{\\dot\\Phi}(\\vect k)$ are\nuncorrelated there.\n\nHereafter we measure the scalar field in units of $t_{i}^{-1}$, $t$\nand $x$ in units of $t_{i}$, and the energy density in units of\n$t_{i}^{-4}$. The equation of motion and the total energy density are\ngiven by\n\\begin{eqnarray}}  \\newcommand{\\eea}{\\end{eqnarray}\n  && \\hspace{-1.8cm} \n   \\ddot{\\Phi}(x) + \\frac{3}{2t}\\dot{\\Phi}(x) - \\frac{1}{t}\\nabla^2\\Phi(x)\n   = - \\left(}  \\newcommand{\\rmk}{\\right) |\\Phi|^2 + \\frac{25}{9t} \n                                - \\frac{25}{36} \\rmk \n    \\Phi^{\\dagger} \\:, \\\\\n  && \\hspace{-1.8cm} \n   \\rho(x) = \\dot\\Phi(x) \\dot\\Phi^{\\dagger}(x) \n           + \\frac{1}{t}\\nabla\\Phi(x) \\cdot \\nabla\\Phi^{\\dagger}(x) \n            + \\frac12 \\left(}  \\newcommand{\\rmk}{\\right) |\\Phi|^2 - \\frac{25}{36} \\rmk^2\n             + \\frac{25}{9t} |\\Phi|^2 \\:,\n\\eea\n\n\\noindent\nwhere $\\lambda$ is set to unity.\\footnote{Strictly speaking, the high\n  temperature approximation is valid only when $\\lambda$ is much\n  smaller than unity. However, a change of $\\lambda$ only scales the\n  physical quantity and is not relevant to the final result.} The\nscale factor $a(t)$ is normalized as $a(1) = 1$. We arrange $512\n\\times 512$ lattice points for our simulations. The lattice spacing\n$\\delta x$ is set to 0.1, and its physical length $\\delta\nx_{\\rm{phys}}$ grows as $a(t)\\delta x = 0.1\\,t^{1\/2}$. The time step\n$\\delta t$ is chosen to be $0.01$. At sufficiently low temperatures,\nthe typical width of the string $d$ is nearly $1\/\\eta$.  Then,\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n\\hspace{-0.4 cm}\n  \\frac{H^{-1}}{\\delta x_{\\rm{phys}}} =\n         20\\,t^{1\/2}, \\qquad\n         (\\sim 245 \\:\\: {\\rm at} \\:\\: t = t_{f})\n\\eeq\n\n\\vspace{0.2 cm}\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  \\frac{d}{\\delta x_{\\rm{phys}}} = \n         12\\,t^{-1\/2}.  \\qquad\n         (\\sim 0.98 \\:\\: {\\rm at} \\:\\: t = t_{f})\n\\eeq\n\n\\noindent\nEven at the final time $t_{f}$, the resolution of the simulation is\nsufficiently good, and the simulation box is larger than the horizon\nscale, so that we can safely impose a periodic boundary condition.\n\nUnder these circumstances, we simulated the system using the second\norder leap-frog method and the Crank-Nicholson scheme over 100 runs\nwith a different random realization of the initial conditions.\n\n\n\n\\section{Results}\n\n\\label{sec:results}\n\nFirst we depict in Figs.\\ \n\\ref{fig:evolution1}$\\sim$\\ref{fig:evolution4} how strings are\ngenerated as the symmetry is spontaneously broken and the gradient and\nkinetic energy density decreases.  In these figures, the total energy\ndensity at each lattice point is depicted at $t = 1, 50, 100, 150$.\nOne can see that initially the field fluctuates very strongly and the\ngradient and kinetic energy dominates over the potential energy. But\nas the universe expands and the temperature decreases, the false\nvacuum energy becomes dominant, and the string configurations become\nstable.\n\nNext, in order to judge whether the string network really enters a\nscaling regime in this case, we count the number of strings per\nhorizon volume at various times. Since spacetime is discretized in our\nsimulations, a point at which $\\Phi = 0$ corresponding to a string\ncore is not necessarily situated at a lattice point. In the worst\ncase, a point at which $\\Phi = 0$ lies at the center of a plaquette.\nTherefore in order to identify a string, we stipulate that a lattice\nis identified with a part of a string if the potential energy density\nthere is larger than that corresponding to the field value of a static\nstring solution at $r = \\delta x_{\\rm{phys}}\/\\sqrt{2}$,\\footnote{In\n  fact, the potential energy is peaked so highly at the string that\n  the identification of the string depends very little on this\n  criterion.} which is obtained by solving the equation\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  \\frac{\\partial}  \\newcommand{\\abs}[1]{\\vert{#1}\\vert^2\\rho}{\\partial}  \\newcommand{\\abs}[1]{\\vert{#1}\\vert r^2} + \\frac{1}{r}\\frac{\\partial}  \\newcommand{\\abs}[1]{\\vert{#1}\\vert\\rho}{\\partial}  \\newcommand{\\abs}[1]{\\vert{#1}\\vert r}\n   - \\frac{\\rho}{r^2} - V'_{\\rm eff}[\\rho] = 0 \\:,\n\\eeq\n\n\\noindent\nwith $\\Phi(r, \\theta) \\equiv \\rho(r)e^{i\\theta}$. Then we regard a\nconnected region of lattices satisfying the above condition as one\nstring. Around $t = 20$, the kinetic and the gradient energy is\nsufficiently suppressed in comparison with the vacuum energy of the\nstring so that we can identify the string unambiguously. We carried\nout this identification procedure every 10 steps after $t = 20$.\n\nFigure\\ \\ref{fig:simulation} displays that time evolution of the\nnumber of strings per horizon volume averaged over 100 realizations.\nIt is clear that the number is not constant but gradually increases\nwith the cosmic time. Since the rate of increase is a decreasing\nfunction of time, one may think that we have observed only a\nrelaxation stage and that it will eventually approach a constant.  In\norder to judge if this is the case, we run a different set of\nsimulations with different initial conditions, in which strings and\nanti-strings are arranged by turns every two lattices, like a\nchecker-board, and $\\dot\\Phi$ is set to 0 for all lattices. The\ninitial string number per horizon volume in this case is as large as\n$100$.  The result is depicted in Fig.\\ \\ref{fig:fitting} together\nwith the previous result. The number per horizon volume rapidly\ndecreases once and then rebounds and continues to increase, as in the\nprevious case. This result implies that the upward trend in the\nprevious simulation is genuine, because, if it were spurious, it would\ngradually decrease and approach the asymptotic constant value from\nabove. In the next section we present a phenomenological analytic\nmodel to explain why a departure from the scaling solution is\nobtained.\n\n\n\n\\section{Analytic explanation}\n\n\\label{sec:model}\n\nIn the present simulations in 2+1 dimensions, strings are represented\nby point-like quasi-particles and anti-particles depending on the\ndirection of the winding. Here, strings cannot intercommute with each\nother to form loops, but rather they can only pair-annihilate upon\ncollision. Between a string and an anti-string works an attractive\nforce proportional to the inverse separation, which is a consequence\nof the extended field gradient as a global defect. Seen from the\nsimulation result, as the system relaxes sufficiently and enters a\nquasi-equilibrium state, a string usually remains at rest because it\nloses kinetic energy due to cosmic expansion. Once within the horizon,\na pair of strings feel the above long-range force, and they start\nmoving toward each other.  Therefore, we have only to elucidate the\ndynamics of a pair of particles obeying such a force in an expanding\nuniverse.\n\nLet $n(t)$ be the average number of strings per unit physical volume\nat the cosmic time $t$. When the system is in a quasi-equilibrium\nstate, the probability per unit time $P(t)$ that a string annihilates\nwith an anti-string is written by \\begin{equation}}  \\newcommand{\\eeq}{\\end{equation} P(t) \\sim \\frac{1}{T(t)} \\:,\n\\eeq\n\n\\noindent\nwhere $T(t)$ is the period required for a pair of strings at rest with\nmean separation, $\\sim 1\/n(t)$, to pair-annihilate. Since $n(t)$ is\ndiluted by cosmic expansion and the above annihilation process, it\nsatisfies the Boltzmann equation,\n\\begin{eqnarray}}  \\newcommand{\\eea}{\\end{eqnarray}\n  \\frac{dn(t)}{dt} &=& - P(t)\\,n(t) - 2 H n(t) \\:, \\nonumber}  \\newcommand{\\eqn}[1]{\\beq {#1}\\eeq \\\\\n                   &=& - \\frac{n(t)}{T(t)} -\n                         \\frac{n(t)}{t} \\:.\n\\label{eqn:evolution}\n\\eea           \n\nIn order to determine the time dependence of $T(t)$, we investigate\nthe relative motion of particles subject to an attractive force\nproportional to the inverse separation in an expanding universe, which\nis described by the equation\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  \\frac{d^2r}{dt^2} + \\frac{1}{t}\\frac{dr}{dt} = -\\frac{2\\pi}{a(t)^2r}\n  \\:,\n\\eeq\n\n\\noindent\nwhere $r$ is the comoving separation. In terms of the physical\nseparation $x \\equiv a(t) r = (t\/t_{i})^{1\/2} r$, this equation reads\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  \\frac{d^2x}{dt^2} + \\frac{1}{4}\\frac{x}{t^2} = -\\frac{2\\pi}{x} \\:.\n\\label{eqn:diff}\n\\eeq\n\n\\noindent\nFor the moment, we neglect the dissipation term due to cosmic\nexpansion. This is justified below. Then integration of Eq.\\ \n(\\ref{eqn:diff}) over $t$ gives\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  \\frac12 \\left(}  \\newcommand{\\rmk}{\\right) \\frac{dx}{dt} \\rmk^2 \n        + 2\\pi \\ln \\left(}  \\newcommand{\\rmk}{\\right) \\frac{x}{x_{0}} \\rmk = 0 \\:,\n\\eeq\n\n\\noindent\nwhere $\\frac{dx}{dt} = 0$ and $ x = x_{0}$ at $t = 0$. Then the period\n$T$ is estimated as\n\\begin{eqnarray}}  \\newcommand{\\eea}{\\end{eqnarray}\n  T &=& \\int_{0}^{T} dt = \\int_{x_{0}}^{d} \n      \\frac{-dx}{\\sqrt{-4\\pi \\ln \\left(}  \\newcommand{\\rmk}{\\right) \\frac{x}{x_{0}} \\rmk}} \\nonumber}  \\newcommand{\\eqn}[1]{\\beq {#1}\\eeq \\\\\n    &=& \\frac{1}{2\\sqrt{\\pi}}\n        \\left \\{ }  \\newcommand{\\rhk}{\\right \\} \\, \\left[}  \\newcommand{\\rkk}{\\right]\\,-2x \\sqrt{\\ln \\left(}  \\newcommand{\\rmk}{\\right) \\frac{x_{0}}{x} \\rmk}\n                \\,\\rkk_{d}^{x_{0}}\n        + 2 \\int_{d}^{x_{0}} \\sqrt{\\ln \\left(}  \\newcommand{\\rmk}{\\right) \\frac{x_{0}}{x} \\rmk} dx\n          \\,\\rhk \\nonumber}  \\newcommand{\\eqn}[1]{\\beq {#1}\\eeq \\\\  \n    &\\sim& \\sqrt\\frac{1}{\\pi}\\,x_{0}\\,\\sqrt{\\ln \\left(}  \\newcommand{\\rmk}{\\right) \\frac{x_{0}}{d} \\rmk}\n\\:,\n\\label{eqn:period}\n\\eea\n\n\\noindent\nwhere $d$ is the core radius. Before inserting this into Eq.\\ \n(\\ref{eqn:evolution}) to solve for $n(t)$, we justify the omission of\nthe dissipation term. A pair of strings at rest initially with\nseparation $x_{0}$ approach a separation $x$ in a time interval $t$\ngiven by\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  t \\sim \\sqrt\\frac{1}{\\pi}\\,x_{0}\\,\\sqrt{\\ln \\left(}  \\newcommand{\\rmk}{\\right) \\frac{x_{0}}{x} \\rmk} \\:.\n\\eeq\n\n\\noindent\nThen the dissipation term, $x\/(4t^2)$, reads $\\pi x \/(4 x_{0}^2\n\\ln(x_{0}\/{x}))$. Since $x$ is smaller than $x_{0}$, the term from the\npotential force in Eq.\\ (\\ref{eqn:diff}), $2\\pi\/x$, always dominates\nover the dissipation term.\n\nTaking $x_{0}$ as the mean separation, $1\/\\sqrt{n(t)}$, Eq.\\ \n(\\ref{eqn:evolution}) becomes\n\\begin{eqnarray}}  \\newcommand{\\eea}{\\end{eqnarray}\n  \\frac{dn(t)}{dt} &=& - \\frac{\\sqrt{\\pi}\\,n(t)^{\\frac32}}\n                 {\\sqrt{\\ln \\left(}  \\newcommand{\\rmk}{\\right) \\frac{1}{d\\sqrt{n(t)}} \\rmk}}  -\n                         \\frac{n(t)}{t} \\nonumber}  \\newcommand{\\eqn}[1]{\\beq {#1}\\eeq \\\\\n                   &\\sim& - \\frac{\\sqrt{\\pi}\\,n(t)^{\\frac32}} \n                 {\\sqrt{\\ln \\left(}  \\newcommand{\\rmk}{\\right) \\frac{A t}{d} \\rmk}}  -\n                         \\frac{n(t)}{t} \\:,\n\\eea \n\n\\noindent\nwhere $1\/\\sqrt{n(t)}$ is nearly proportional to the cosmic time from\nthe simulation result, and the proportional coefficient is $A$. \n~Substituting $f(t)$ for $n(t) t$, we have\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  \\frac{1}{t} \\frac{df}{dt} = - \\frac{\\sqrt{\\pi} f^{3\/2}}\n                             {\\sqrt{\\ln \\left(}  \\newcommand{\\rmk}{\\right) \\frac{A t}{d} \\rmk}}\n                              \\frac{1}{t^{3\/2}}  \\:.\n\\eeq\n\n\\noindent\nIntegrating over both sides gives\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  \\frac{1}{f^{1\/2}} \\sim \\sqrt{\\pi}\\, B  \\frac{t^{1\/2}}\n                             {\\sqrt{\\ln \\left(}  \\newcommand{\\rmk}{\\right) \\frac{A t}{d} \\rmk}}\n                               + C  \\:,\n\\eeq\n\n\\noindent\nwhere $B$ represents a numerical factor $\\sim {\\cal O}(1)$ and $C$ is an\nintegration constant determined from the initial conditions.\n\nThen $n(t)$ is given by\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  n(t) \\sim \\frac{D}{t^2}\\, \n         \\frac{ \\ln{\\left(}  \\newcommand{\\rmk}{\\right) \\frac{t}{E} \\rmk} }\n              { \\left[}  \\newcommand{\\rkk}{\\right]\\,1 + \\frac{C D^{1\/2}}{t^{1\/2}} \n                    \\sqrt{\\ln{\\left(}  \\newcommand{\\rmk}{\\right) \\frac{t}{E} \\rmk}} \\,\\rkk^{2}  } \n\\:,\n\\eeq\n\n\\noindent\nwhere $D = (\\pi B^{2})^{-1}, E \\sim d\/A \\sim d\/\\sqrt{D}$.\n~Therefore the number of strings per horizon volume is given by\n\\begin{eqnarray}}  \\newcommand{\\eea}{\\end{eqnarray}\n  n(t) H^{-2} &\\sim& 4D\\,\n         \\frac{ \\ln{\\left(}  \\newcommand{\\rmk}{\\right) \\frac{t}{E} \\rmk} }\n              { \\left[}  \\newcommand{\\rkk}{\\right]\\,1 + \\frac{C D^{1\/2}}{t^{1\/2}} \n                    \\sqrt{\\ln{\\left(}  \\newcommand{\\rmk}{\\right) \\frac{t}{E} \\rmk}} \\,\\rkk^{2}  }\n  \\nonumber}  \\newcommand{\\eqn}[1]{\\beq {#1}\\eeq \\\\\n  &\\Longrightarrow& 4D\\,\n         \\ln{\\left(}  \\newcommand{\\rmk}{\\right) \\frac{t}{E} \\rmk} \\:.\n   \\qquad \\qquad  (\\mbox{asymptotically})\n\\eea\n\n\\noindent\nTaking $D \\sim 0.1$, $E \\sim 0.4$~($d \\sim 1.0$) and $C \\sim -0.09$, we\ncan fit the results of the simulation by the above formula. This is\ndepicted in Fig.\\ \\ref{fig:fitting}. The number of strings per horizon\nvolume is not a constant, but proportional to $(\\ln t)$ asymptotically.\nNote that only the constant $C$ depends on the initial conditions.\nTaking $C \\sim 0.8$ with almost the same $D$ and $E$, that from the\nchecker-board initial conditions can also be fitted. This implies that\nour fitting formula is consistent.\n\n\n\n\\section{Summary and Discussion}\n\n\\label{sec:summary}\n\nIn this paper, we have reported the dynamics of a complex scalar field\nfrom the symmetric phase to the broken phase in 2+1 dimensions.\nStarting from the thermal equilibrium state, we have observed how\nstrings are formed as the temperature decreases and the symmetry is\nspontaneously broken. We also find that, as the string system grows,\nit does not go into a scaling regime, where the number of strings per\nhorizon volume stays constant, but the number increases in proportion\nto $(\\ln t)$ asymptotically.  This can be interpreted as follows.  Our\nsimulation was performed in 2+1 dimensions and strings are in fact\npoint-like. Thus, the dominant energy loss process is not the formation\nof loops through the intercommutation of infinite strings, but rather\npair-annihilation of straight strings. They move under logarithmic\npotentials, which causes deviation from scale-invariance.\n\n\n\n\\section*{Acknowledgments}\nMY is grateful to Professor K. Sato for his continuous encouragement\nand Dr.\\ T.\\ Shiromizu and Professor M.\\ Morikawa for their useful\ncomments. JY would like to thank Professor Andrei Linde for his\nhospitality at Stanford University, where part of this work was done.\nThis work was partially supported by the Japanese Grant-in-Aid for\nScientific Research from the Monbusho, Nos.\\ 10-04558~(MY), 09740334~(JY)\nand 10640250~(MK). JY acknowledges support from the Monbusho.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nA \\textit{lattice polytope} is a convex polytope all of whose vertices have integer coordinates.\nIn the present paper, we discuss two algebraic properties of lattice polytopes, which are called the integer decomposition property and the level property.\nFirst, we begin with the definitions of these properties.\n\\subsection{IDP polytopes and level polytopes}\nLet ${\\mathcal P} \\subset {\\NZQ R}^N$ be a lattice polytope and let $\\dim({\\mathcal P})$ denote the dimension of ${\\mathcal P}$. \nWe say that a lattice polytope ${\\mathcal P}$ has \\textit{the integer decomposition property} if for each integer $n \\geq 1$, \n\\[\nn{\\mathcal P} \\cap {\\NZQ Z}^N=(n-1){\\mathcal P} \\cap {\\NZQ Z}^N+{\\mathcal P} \\cap {\\NZQ Z}^N, \n\\]\nwhere $n{\\mathcal P}$ is the $n$th dilated polytope of ${\\mathcal P}$, i.e., $n{\\mathcal P}=\\{n{\\bold x} : {\\bold x} \\in {\\mathcal P} \\}$.\nA lattice polytope which has the integer decomposition property is called \\textit{IDP}.\nIDP polytopes turn up in many fields of mathematics such algebraic geometry, where they correspond to projectively normal embeddings of toric varieties, and commutative algebra, where they correspond to standard graded Cohen-Macaulay domains (see \\cite{BG}).\nMoreover, the integer decomposition property is particularly important in the theory and application of integer programing \\cite[\\S 22.10]{integer}.\n\nNow, we see a connection between IDP polytopes and commutative algebras.\nLet $K$ be a field.\nGiven a graded noetherian commutative ring with $A=\\oplus_{i =0}^{\\infty}A_i$ with $A_0=K$, we say $A$ is {\\it standard graded} if $A=K[A_1]$, i.e., $A$ is generated by $A_1$ as a $K$-algebra and {\\it semi-standard graded} if $A$ is finitely generated as a $K[A_1]$-module.\nWe associate a lattice polytope ${\\mathcal P}$ with a semi-standard graded $K$-algebra.\nLet $K[{\\bf X}^{\\pm 1}, T]=K[X_1^{\\pm1}, \\ldots, X_N^{\\pm 1}, T]$ be the Laurent polynomial ring in $N+1$ variables over $K$.\nWe define the $K$-algebra $K[{\\mathcal P}]$ as follows:\n\\[\nK[{\\mathcal P}]=K[{\\bf X}^{{\\bold a}}T^n : {\\bold a} \\in n{\\mathcal P} \\cap {\\NZQ Z}^N, n \\in {\\NZQ Z}_{\\geq 0}] \\subset K[{\\bf X}^{\\pm 1}, T],\n\\]\nwhere for a lattice point ${\\bold a}=(a_1,\\ldots,a_N) \\in {\\NZQ Z}^N$, ${\\bf X}^{{\\bold a}}T^n=X_1^{a_1}\\cdots X_N^{a_N}T^n$ denotes a Laurent monomial in $K[{\\bf X}^{\\pm 1}, T]$.\nIt is known that $K[{\\mathcal P}]$ is a semi-standard graded normal Cohen-Macaulay domain of dimension $\\dim({\\mathcal P})+1$ by setting $\\deg({\\bf X}^{{\\bold a}}T^n)=n$.\nMoreover, $K[{\\mathcal P}]$ is standard graded if and only if ${\\mathcal P}$ is IDP.\nWe call this graded $K$-algebra $K[{\\mathcal P}]$ the {\\it Ehrhart ring} of ${\\mathcal P}$.\nRefer the reader to \\cite{BG} for the detailed information about Ehrhart rings.\n\nNext, we recall what level polytopes are.\nFor a subset  $A \\subset {\\NZQ R}^N$, let $\\inter(A)$ denote the relative interior of $A$ with respect to the affine subspace of ${\\NZQ R}^N$ spanned by $A$.\nWe say that a lattice polytope ${\\mathcal P}$ is \\textit{level} of index $r$,\nif $r = \\min \\{t \\in {\\NZQ Z}_{>0} : \\inter(t{\\mathcal P}) \\cap {\\NZQ Z}^N \\neq \\emptyset  \\}$ and for each integer $n \\geq r$, \n\\[\n\\inter(n{\\mathcal P}) \\cap {\\NZQ Z}^N=\\inter(r{\\mathcal P}) \\cap {\\NZQ Z}^N+(n-r){\\mathcal P} \\cap {\\NZQ Z}^N.\n\\]\nIn particular, if ${\\mathcal P}$ is level of index $r$ and $|\\inter(r{\\mathcal P}) \\cap {\\NZQ Z}^N|=1$,\nthen ${\\mathcal P}$ is called \\textit{Gorenstein} of index $r$.\nThe Gorenstein polytopes give important examples in combinatorial commutative algebra, mirror symmetry and tropical geometry (for details we refer to \\cite{Batyrev,JK}). \nOn the other hand, the level property is a generalization of the Gorenstein property and it has only fairly recently been examined for certain classes polytopes (e.g., \\cite{Hig,HY,KO}).\n\nNow, we see a connection between level polytopes and commutative algebras.\nLet $R$ be a Cohen-Macaulay graded ring with canonical module $\\omega_R$.\nThen the number \n\\[\na(R)=-\\min\\{\ni : (\\omega_{R})_i \\neq 0\n\\}\n\\]\nis called the {\\it $a$-invariant} of $R$.\nIt then follows that $a(k[{\\mathcal P}])=-\\min \\{t \\in {\\NZQ Z}_{>0} : \\inter(t{\\mathcal P}) \\cap {\\NZQ Z}^N \\neq \\emptyset  \\}$.\nWe say that $R$ is level if the canonical module $\\omega_R$ of $R$ is generated by elements of the same degree.\nThe notion of level rings was introduced by Stanley \\cite{Sta77}. By virtue of Danilov \\cite{Dan} and Stanley \\cite{Sta},\nwe know that the Ehrhart ring $K[{\\mathcal P}]$ of ${\\mathcal P}$ is level (of $a$-invariant $-r$) if and only if ${\\mathcal P}$ is level (of index $r$).\n\n\\subsection{Motivations and main results}\nIn the present paper, we discuss when a lattice polytope is IDP or level.\nFor instance, every lattice polygon is IDP and level.\nOne of the most famous results of this problem is the following:\n\\begin{Theorem}[{\\cite[Theorem 1.3.3]{BGT}}]\n\\label{orginal}\n\tLet ${\\mathcal P} \\subset {\\NZQ R}^N$ be a lattice polytope.\nThen we obtain the following:\n\t\\begin{enumerate}\n\t\\item[{\\rm (1)}]\t For any positive integer $n \\geq \\dim({\\mathcal P})-1$, $n{\\mathcal P}$ is IDP;\n \\item[{\\rm (2)}] For any positive integer $n \\geq \\dim({\\mathcal P})+1$, $n{\\mathcal P}$ is level of index $1$.\n\\end{enumerate}\n\\end{Theorem}\nThis theorem says that ``large polytopes\" are IDP and level.\nFurthermore, the following result also says that ``large polytopes\" are IDP:\n\\begin{Theorem}[{\\cite[Corollary 2.6]{HH}}]\n\t\\label{thm:hh}\nLet ${\\mathcal P} \\subset {\\NZQ R}^N$ be a lattice polytope of dimension $d$.\nIf every edge of ${\\mathcal P}$ has lattice length $\\geq 2d(d+1)$, then $P$ is IDP.\n\\end{Theorem}\nThis theorem follows from the fact that  $2$-convex-normal lattice polytopes are IDP ({\\cite[Corollarly 5]{HH}}).\nA lattice polytope ${\\mathcal P} \\subset  {\\NZQ R}^N$  is called \\textit{$2$-convex-normal} if \n$2{\\mathcal P}={\\mathcal P} \\cap {\\NZQ Z}^N+{\\mathcal P}$.\nIt is known that lattice polytopes of dimension $d$ each of whose edges has lattice length $\\geq 2d(d+1)$ are $2$-convex-normal (\\cite[Theorem 1.2]{Gub}).\nHence Theorem \\ref{thm:hh} follows from this fact.\n\nNow, we recall two well-known constructions of lattice polytopes.\n\\begin{Definition}{\\em\n\t\tLet ${\\mathcal P}_1,\\ldots,{\\mathcal P}_m \\subset {\\NZQ R}^N$ be lattice polytopes.\n\t\tThe \\textit{Minkowski sum} ${\\mathcal P}_1+\\cdots+{\\mathcal P}_m$ of ${\\mathcal P}_1$, $\\ldots,{\\mathcal P}_m$ is defined by\n\t\t\\[\n\t\t{\\mathcal P}_1+\\cdots+{\\mathcal P}_m=\\{{\\bold a}_1+\\cdots+{\\bold a}_m \\in {\\NZQ R}^N \\mid {\\bold a}_1 \\in {\\mathcal P}_1,\\ldots, {\\bold a}_m \\in {\\mathcal P}_m \\} \\subset {\\NZQ R}^N.\n\t\t\\]\n\t\tThe \\textit{Cayley sum} ${\\mathcal P}_1*\\cdots*{\\mathcal P}_m$ of ${\\mathcal P}_1$, $\\ldots,{\\mathcal P}_m$ is defined by\n\t\t\\[{\\mathcal P}_1*\\cdots*{\\mathcal P}_m=\\textnormal{conv}(\\{{\\bold e}_1\\} \\times {\\mathcal P}_1 \\union \\cdots \\union\\{{\\bold e}_m\\} \\times {\\mathcal P}_m ) \\subset {\\NZQ R}^{m}\\times {\\NZQ R}^{N} = {\\NZQ R}^{m+N},\n\t\t\\]\n\t\twhere ${\\bold e}_1,\\ldots,{\\bold e}_m$ are the canonical unit coordinate vectors of ${\\NZQ R}^m$.\n\t}\n\\end{Definition}\n\nIn the present paper, we  consider the integer decomposition property and the level property of Minkowski sums and Cayley sums.\nFirst, we discuss relations between Minkowski sums and Cayley sums for the IDP and the level property.\nLet ${\\mathcal P}_1,\\ldots,{\\mathcal P}_m \\subset {\\NZQ R}^N$ be lattice polytopes.\nWe say that the tuple $({\\mathcal P}_1,\\ldots,{\\mathcal P}_m)$ is \\textit{IDP} if for any subset $\\emptyset \\neq I \\subset [m]:=\\{1,\\ldots,m\\}$,\nthe equation\n\\[\n\\left(\\sum_{i \\in I}{\\mathcal P}_{i} \\right) \\cap {\\NZQ Z}^N=\\sum_{i \\in I}({\\mathcal P}_i \\cap {\\NZQ Z}^N)\n\\]\n is satisfied.\nWhen $m=2$, this notion is introduced and discussed in \\cite{HH}.\nIn particular, in \\cite{Oda}, Oda asked for which pair of lattice polytopes ${\\mathcal P}, {\\mathcal Q} \\subset  {\\NZQ R}^N$, $({\\mathcal P},{\\mathcal Q})$ is IDP.\nThe following theorem is the first main theorem of the present paper.\n\\begin{Theorem}\n\t\\label{thm;relation}\n\tLet ${\\mathcal P}_1,\\ldots,{\\mathcal P}_m \\subset {\\NZQ R}^N$ be lattice polytopes.\n\tThen we obtain the following:\n\t\\begin{enumerate}\n\t\n\t\n\t\t\\item [{\\rm (1)}] The Cayley sum ${\\mathcal P}_1*\\cdots*{\\mathcal P}_m$ is IDP if and only if each ${\\mathcal P}_i$ is IDP and for any nonnegative integers $a_1,\\ldots, a_m$, the tuple $(a_1 {\\mathcal P}_1,\\ldots, a_m {\\mathcal P}_m)$ is IDP.\n\t\t\tIn the case, the Minkowski sum $\\sum_{i=1}^ma_i{\\mathcal P}_i$ is IDP.\n\t\t\\item[{\\rm (2)}] If ${\\mathcal P}_1*\\cdots*{\\mathcal P}_m$ is level of index $m$, then ${\\mathcal P}_1+\\cdots+{\\mathcal P}_m$ is level of index $1$.\n\t\\end{enumerate}\n\\end{Theorem}\nIt is known that ${\\mathcal P}_1*\\cdots*{\\mathcal P}_m$ is Gorenstein of index $m$ if and only if ${\\mathcal P}_1+\\cdots+{\\mathcal P}_m$ is Gorenstein of index $1$.\nHowever, the reverse of Theorem \\ref{thm;relation} (2) does not hold in general (Example \\ref{counter}). \n\nNext, we discuss when Minkowski sums and Cayley sums are IDP or level.\nIn \\cite{Hig}, Higashitani considered when the Minkowski sums of dilated polytopes are IDP or level.\nIn fact, he showed the following:\n\\begin{Theorem}[{\\cite[Theorem 2.1]{Hig}}]\n\t\\label{Higa}\n\tLet ${\\mathcal P}_1,\\ldots,{\\mathcal P}_m \\subset {\\NZQ R}^N$ be lattice polytopes. For each $i$, let $n_i$ be a positive integer.\n\tThen we obtain the following:\n\t\\begin{enumerate}\n\t\t\t\\item[{\\rm (1)}] If for each $i$, $n_i \\geq \\dim ({\\mathcal P}_i)$, then $n_1{\\mathcal P}_1+\\cdots+n_m{\\mathcal P}_m$ is IDP;\n\t\t\\item[{\\rm (2)}] If for each $i$, $n_i \\geq \\dim ({\\mathcal P}_i)+1$, then $n_1{\\mathcal P}_1+\\cdots+n_m{\\mathcal P}_m$ is level of index $1$.\n\t\\end{enumerate}\n\\end{Theorem}\n\nTheorems \\ref{thm;relation} and \\ref{Higa} naturally lead us to consider the following question:\n\\begin{Question}\n\t\\label{ques}\n\tLet ${\\mathcal P}_1,\\ldots,{\\mathcal P}_m \\subset {\\NZQ R}^N$ be lattice polytopes. For each $i$, let $n_i$ be a positive integer.\n\t\\begin{enumerate}\n\t\t\\item[{\\rm (1)}] If for each $i$, $n_i \\geq \\dim ({\\mathcal P}_i)$, then is $n_1{\\mathcal P}_1*\\cdots*n_m{\\mathcal P}_m$ IDP?\n\t\t\\item[{\\rm (2)}] If for each $i$, $n_i \\geq \\dim ({\\mathcal P}_i)+1$, then is $n_1{\\mathcal P}_1*\\cdots*n_m{\\mathcal P}_m$ level of index $m$?\n\t\\end{enumerate}\n\\end{Question}\nThe goal of the present paper is to give a complete answer to this question.\nIn order to solve Question \\ref{ques} (1), we focus on $2$-convex-normal lattice polytopes.\nIn fact, we will show when the Minkowski sums and the Cayley sums of $2$-convex-normal lattice polytopes are IDP (Theorem \\ref{thm:2cn}).\nBy showing for any lattice polytope ${\\mathcal P}$ and for any positive integer $n \\geq \\dim ({\\mathcal P})$, $n{\\mathcal P}$ is $2$-convex-normal (Lemma \\ref{lem:2cn}),\nwe obtain an answer to Question \\ref{ques} (1) (Corollary \\ref{cor:IDPdi} and Example \\ref{ex:IDPcounter}).\n\nNext, in order to solve Question \\ref{ques} (2), we introduce a class of level polytopes.\nFor a lattice polytope ${\\mathcal P} \\subset {\\NZQ R}^N$, we consider the following condition:\n\\begin{equation}\n\\label{2cl}\n\\inter(2{\\mathcal P})=\\inter({\\mathcal P})+({\\mathcal P} \\cap {\\NZQ Z}^N) \\tag{0.1}.\n\\end{equation}\nThen such polytopes are always level (Proposition \\ref{prop:2cl}).\nIn particular, we will show that the Minkowski sum and the Cayley sum of lattice polytopes with interior lattice points satisfying the condition (\\ref{2cl}) are always level (Theorem \\ref{thn:2cl}).\nBy showing for any lattice polytope ${\\mathcal P}$ and for any positive integer $n \\geq \\dim ({\\mathcal P})+1$, $n{\\mathcal P}$ has interior lattice points and satisfies the condition (\\ref{2cl}) (Lemma \\ref{lem:2cl}),\nwe obtain an answer to Question \\ref{ques} (2) (Corollary \\ref{cor:leveldi}).\n\n\n\n\n\\section{Relations between Minkowski sums and Cayley sums}\n\\label{sec:relation}\nIn this section, we discuss relations between Minkowski sums and Cayley sums.\nIn particular, we prove Theorem \\ref{thm;relation}.\nFirst, we consider a relation between  points in Minkowski sums and that in Cayley sums.\n\\begin{Lemma}\n\t\\label{decomp1}\n\tLet ${\\mathcal P}_1,\\ldots,{\\mathcal P}_m \\subset {\\NZQ R}^N$ be convex polytopes.\n\tThen for any nonnegative real numbers $a_1,\\ldots,a_m$, we have\n\t$$(a_1+\\cdots+a_m)({\\mathcal P}_1*\\cdots*{\\mathcal P}_m) \\cap \t\n\t\\left( \\left\\{\\sum_{i=1}^{m}a_i {\\bold e}_i\\right\\} \\times {\\NZQ R}^{N} \\right)\n\t=\\left\\{\\sum_{i=1}^{m}a_i {\\bold e}_i\\right\\} \\times \\sum_{i=1}^{m} a_i {\\mathcal P}_i.$$\n\\end{Lemma}\n\n\\begin{proof}\n\tFor each $1 \\leq i \\leq m$, let ${\\bold v}_{i1},\\ldots,{\\bold v}_{ir_i}$ be the vertices of ${\\mathcal P}_i$, where $r_i$ is the number of vertices of ${\\mathcal P}_i$.\n\tFix positive real numbers $a_1,\\ldots,a_m$.\n\tGiven a point  ${\\bold x} \\in \\sum_{i=1}^{m} a_i {\\mathcal P}_i$,\n\twe can write \n\t${\\bold x}=\\sum_{i=1}^{m}\\sum_{j=1}^{r_i}\\lambda_{ij}{\\bold v}_{ij}$,\n\twhere for any $1 \\leq i \\leq m$ and $1 \\leq j \\leq r_i$, $0 \\leq \\lambda_{ij} \\leq 1$ and $\\sum_{j=1}^{r_i}\\lambda_{ij}=a_i$.\n\tThen one has \n\t$$(a_1 {\\bold e}_1+\\cdots+ a_m {\\bold e}_m,{\\bold x}) =\\sum_{i =1}^m\\sum_{j=1}^{r_i}\\lambda_{ij}({\\bold e}_{i}, {\\bold v}_{ij}) \\in {\\NZQ R}^{m+N}.$$\n\tSince each $({\\bold e}_{i}, {\\bold v}_{ij})$ is a vertex of ${\\mathcal P}_1*\\cdots*{\\mathcal P}_m$, we obtain $(a_1 {\\bold e}_1+\\cdots+ a_m {\\bold e}_m,{\\bold x}) \\in (a_1+\\cdots+a_m)({\\mathcal P}_1*\\cdots*{\\mathcal P}_m)$. \n\tHence it follows \n\t$$(a_1+\\cdots+a_m)({\\mathcal P}_1*\\cdots*{\\mathcal P}_m) \\cap \t\n\t\\left( \\left\\{\\sum_{i=1}^{m}a_i {\\bold e}_i\\right\\} \\times {\\NZQ R}^{N} \\right)\n\t\\supset \\left\\{\\sum_{i=1}^{m}a_i {\\bold e}_i\\right\\} \\times \\sum_{i=1}^{m} a_i {\\mathcal P}_i.$$\n\n\t\n\tConversely, take a point ${\\bold x} \\in (a_1+\\cdots+a_m)({\\mathcal P}_1*\\cdots*{\\mathcal P}_m)$. \n\tThen we can write ${\\bold x}=\\sum_{i=1}^{m}\\sum_{j=1}^{r_i}\\nu_{ij}({\\bold e}_i, {\\bold v}_{ij})$,\n\twhere for any $1 \\leq i \\leq m$ and $1 \\leq j \\leq r_i$, $0 \\leq \\nu_{ij}$ and $\\sum_{i=1}^{m}\\sum_{j=1}^{r_i}\\nu_{ij}=a_1+\\cdots+a_m$.\n\tAssume that $ {\\bold x} \\in \\left\\{\\sum_{i=1}^{m}a_i {\\bold e}_i\\right\\} \\times {\\NZQ R}^{N}$.\n\tThen for any $1 \\leq i \\leq m$, one has $\\sum_{j=1}^{r_i}\\nu_{ij}=a_i$.\n\tHence we obtain ${\\bold x} \\in \\{\\sum_{i=1}^{m}a_i {\\bold e}_i\\} \\times \\sum_{i=1}^{m} a_i {\\mathcal P}_i$.\n\tThis implies that\n\t$$(a_1+\\cdots+a_m)({\\mathcal P}_1*\\cdots*{\\mathcal P}_m) \\cap \t\n\t\\left( \\left\\{\\sum_{i=1}^{m}a_i {\\bold e}_i\\right\\} \\times {\\NZQ R}^{N} \\right)\n\t\\subset \\left\\{\\sum_{i=1}^{m}a_i {\\bold e}_i\\right\\} \\times \\sum_{i=1}^{m} a_i {\\mathcal P}_i,$$\tas desired.\n\\end{proof}\n\nSimilarly, we can obtain the following. \n\\begin{Lemma}\n\t\\label{decomp2}\n\tLet ${\\mathcal P}_1,\\ldots,{\\mathcal P}_m \\subset {\\NZQ R}^N$ be convex polytopes.\n\tThen for any positive real numbers $a_1,\\ldots,a_m$, we have\n\t$${\\rm int}((a_1+\\cdots+a_m)({\\mathcal P}_1*\\cdots*{\\mathcal P}_m)) \\cap \t\n\t\\left( \\left\\{\\sum_{i=1}^{m}a_i {\\bold e}_i\\right\\} \\times {\\NZQ R}^{N} \\right)\n\t=\\left\\{\\sum_{i=1}^{m}a_i {\\bold e}_i\\right\\} \\times {\\rm int} \\left(\\sum_{i=1}^{m} a_i {\\mathcal P}_i \\right).$$\n\\end{Lemma}\n\t\n\nNow, we prove Theorem \\ref{thm;relation}.\n\\begin{proof}[Proof of Theorem \\ref{thm;relation}]\n\t(1) Suppose that ${\\mathcal P}_1*\\cdots*{\\mathcal P}_m$ is IDP. Since ${\\mathcal P}_1,\\ldots,{\\mathcal P}_m$ is faces of ${\\mathcal P}_1*\\cdots*{\\mathcal P}_m$, each ${\\mathcal P}_i$ is IDP.\n\tFix nonnegative integers $a_1,\\ldots,a_m$ and take ${\\bold x} \\in (n\\sum_{i=1}^m a_i{\\mathcal P}_i) \\cap {\\NZQ Z}^N$ with a positive integer $n$.\n\tThen by Lemma \\ref{decomp1}, we have \n\t$$\\left(n\\sum_{i=1}^m a_i {\\bold e}_i,{\\bold x}\\right) \\in n(a_1+\\cdots+a_m)({\\mathcal P}_1*\\cdots*{\\mathcal P}_m) \\cap  \\left(\\left\\{n\\sum_{i=1}^m a_i {\\bold e}_i \\right\\} \\times {\\NZQ R}^N \\right).$$\n\tSince ${\\mathcal P}_1*\\cdots*{\\mathcal P}_m$ is IDP, we can write\n\t$(n\\sum_{i=1}^m a_i {\\bold e}_i,{\\bold x}) ={\\bold x}_1+\\cdots+{\\bold x}_{n(a_1+\\cdots+a_m)}$, where each ${\\bold x}_i \\in {\\mathcal P}_1*\\cdots*{\\mathcal P}_m \\cap {\\NZQ Z}^{m+N}$.\n\tIn particular, we may assume that for any $1 \\leq k \\leq n$, $1 \\leq i \\leq m$ and $1 \\leq j \\leq a_i$, ${\\bold x}_{k(a_1+\\cdots+a_{i-1}+j)} \\in \\{{\\bold e}_i\\} \\times ({\\mathcal P}_i \\cap {\\NZQ Z}^N)$.\n\tHence it follows that the tuple $(a_1{\\mathcal P}_1,\\ldots,a_m {\\mathcal P}_m)$ is IDP.\n\tMoreover, we can write \n\t$(n\\sum_{i=1}^m a_i {\\bold e}_i,{\\bold x}) = {\\bold y}_1+\\cdots+{\\bold y}_{n}$, where ${\\bold y}_1,\\ldots,{\\bold y}_n \\in  \\{n\\sum_{i=1}^m a_i {\\bold e}_i\\} \\times (\\sum_{i=1}^{m} a_i {\\mathcal P}_i \\cap {\\NZQ Z}^{N})$.\n\tThis implies that the Minkowski sum $\\sum_{i=1}^{m} a_i {\\mathcal P}_i$ is IDP.\n\t\n\tConversely, suppose that each ${\\mathcal P}_i$ is IDP and for any nonnegative integers $a_1,\\ldots, a_m$, the tuple $(a_1 {\\mathcal P}_1,\\ldots, a_m {\\mathcal P}_m)$ is IDP.\n\tTake ${\\bold x} \\in n({\\mathcal P}_1*\\cdots*{\\mathcal P}_m) \\cap {\\NZQ Z}^{m+N}$ with some positive integer $n$.\n\tIt then follows from Lemma \\ref{decomp1} that there are nonnegative integers $a_1,\\ldots,a_m$ with $a_1+\\cdots+a_m=n$ and ${\\bold y} \\in (\\sum_{i=1}^{m} a_i {\\mathcal P}_i)\\cap {\\NZQ Z}^N$ such that $(\\sum_{i=1}^m a_i {\\bold e}_i, {\\bold y})={\\bold x}$. \n\tSince the tuple $(a_1 {\\mathcal P}_1,\\ldots, a_m {\\mathcal P}_m)$ is IDP, there are ${\\bold y}_1,\\ldots,{\\bold y}_m \\in {\\NZQ Z}^N$ such that each ${\\bold y}_i \\in a_i{\\mathcal P}_i \\cap {\\NZQ Z}^N$ and ${\\bold y}={\\bold y}_1+\\cdots+{\\bold y}_m$. Moreover since each ${\\mathcal P}_i$ is IDP, for any $1 \\leq i \\leq m$, there are ${\\bold y}^{(i)}_1,\\ldots,{\\bold y}^{(i)}_{a_i} \\in {\\mathcal P}_i \\cap {\\NZQ Z}^N$ such that ${\\bold y}_i={\\bold y}^{(i)}_1+\\cdots+{\\bold y}^{(i)}_{a_i}$.\n\tTherefore, ${\\mathcal P}_1*\\cdots*{\\mathcal P}_m$ is IDP since each $({\\bold e}_i, {\\bold y}^{(i)}_j) \\in {\\mathcal P}_1*\\cdots*{\\mathcal P}_m \\cap {\\NZQ Z}^{m+N}$ and ${\\bold x}=\\sum_{i=1}^{m}\\sum_{j=1}^{a_i}({\\bold e}_i, {\\bold y}^{(i)}_j)$.\n\t\n\t\n\t(2)\n\tEvery interior lattice point of $m({\\mathcal P}_1*\\cdots*{\\mathcal P}_m)$ belongs to $\\{{\\bold e}_1+\\cdots+{\\bold e}_m\\} \\times {\\NZQ Z}^{N}$.\n\tHence by Lemma  \\ref{decomp2}, we know that ${\\mathcal P}_1+\\cdots+{\\mathcal P}_m$ has interior lattice points.\n\tLet ${\\bold a} \\in \\text{int}(n(P_1+\\cdots+P_m)) \\cap {\\NZQ Z}^N$ with a positive integer $n$.\n\tThen by Lemma \\ref{decomp2}, one has $$(n({\\bold e}_{1}+\\cdots+{\\bold e}_m),{\\bold a}) \\in \\text{int}(nm({\\mathcal P}_1*\\cdots*{\\mathcal P}_m)) \\cap  (\\{n({\\bold e}_{1}+\\cdots+{\\bold e}_m)\\} \\times {\\NZQ R}^{N}).$$\n\tSince ${\\mathcal P}_1*\\ldots*{\\mathcal P}_m$ is level of index $m$,\n\twe can write $(n({\\bold e}_{1}+\\cdots+{\\bold e}_m),{\\bold a})={\\bold b}_1+{\\bold b}_2$, where ${\\bold b}_1 \\in \\text{int}(m({\\mathcal P}_1*\\cdots*{\\mathcal P}_m)) \\cap {\\NZQ Z}^{m+N}$ and ${\\bold b}_2 \\in  (n-1)m({\\mathcal P}_1*\\cdots*{\\mathcal P}_m)\\cap {\\NZQ Z}^{m+N}$.\n\tThen  by Lemmas \\ref{decomp1} and  \\ref{decomp2}, ${\\bold b}_1 \\in \\{{\\bold e}_{1}+\\cdots+{\\bold e}_m\\} \\times \\textnormal{int}({\\mathcal P}_1+\\cdots+{\\mathcal P}_m)$\n\tand ${\\bold b}_2 \\in \\{(n-1)({\\bold e}_{1}+\\cdots+{\\bold e}_m)\\} \\times (n-1)({\\mathcal P}_1+\\cdots+{\\mathcal P}_m)$.\n\tHence ${\\mathcal P}_1+\\cdots+{\\mathcal P}_m$ is level of index $1$, as desired.\n\\end{proof}\nNow, we focus on the case where $m=2$ and the integer decomposition property.\nThe following conjecture by Oda is well-known.\n\n\\begin{Conjecture}[Oda Conjecture]\n\t\\label{conj:oda}\n\tLet ${\\mathcal P}, {\\mathcal Q} \\subset {\\NZQ R}^N$ be lattice polytopes. If ${\\mathcal P}$ is smooth and the normal fan of ${\\mathcal Q}$ coarsens that of ${\\mathcal P}$, then $({\\mathcal P}, {\\mathcal Q})$ is IDP.\n\\end{Conjecture}\n\nHence from Theorem \\ref{thm;relation} (1) this conjecture is equivalent to the following.\n\\begin{Conjecture}\n\t\tLet ${\\mathcal P}, {\\mathcal Q} \\subset {\\NZQ R}^N$ be lattice polytopes. If ${\\mathcal P}$ is smooth and the normal fan of ${\\mathcal Q}$ coarsens that of ${\\mathcal P}$, then ${\\mathcal P}*{\\mathcal Q}$ is IDP.\n\\end{Conjecture}\n\nConjecture \\ref{conj:oda} holds when $N=2$. Moreover, in the case, we do not need the smoothness assumption on ${\\mathcal P}$.\nIn fact,\n\\begin{Theorem}[{\\cite[Theorem 1.1]{HNP}}]\n\tLet ${\\mathcal P},{\\mathcal Q} \\subset {\\NZQ R}^2$ be lattice polygons such that the normal fan of ${\\mathcal Q}$ coarsens that of ${\\mathcal P}$.\n\tThen $({\\mathcal P},{\\mathcal Q})$ is IDP.\n\\end{Theorem}\n\nHence we obtain the following.\n\\begin{Corollary}\n\tLet ${\\mathcal P},{\\mathcal Q} \\subset {\\NZQ R}^2$ be lattice polygons such that the normal fan of ${\\mathcal Q}$ coarsens that of ${\\mathcal P}$.\n\tThen ${\\mathcal P}*{\\mathcal Q}$ is IDP.\n\\end{Corollary}\nNext, we recall the following result on the Gorenstein property of Minkowski sums and Cayley sums.\n\\begin{Theorem}[{\\cite[Th. 2.6]{BN}}]\n\tLet ${\\mathcal P}_1,\\ldots,{\\mathcal P}_m \\subset {\\NZQ R}^N$ be lattice polytopes with $\\dim({\\mathcal P}_1+\\cdots+{\\mathcal P}_m)=N$.\n\tThen ${\\mathcal P}_1*\\cdots*{\\mathcal P}_m$ is Gorenstein of index $m$ if and only if  ${\\mathcal P}_1+\\cdots+{\\mathcal P}_m$ is Gorenstein of index $1$.\n\\end{Theorem}\nLet us note that Gorenstein polytopes are level.\nHowever, the reverse of Theorem \\ref{thm;relation} (2) does not hold in general.\n\\begin{Example}\n\t\\label{counter}\n\t{\\em\n\t\tLet ${\\mathcal P}_1$ be the line segment from $(1,0)$ to $(0,1)$ and \n\t\t${\\mathcal P}_2$ the line segment from $(1,1)$ to $(-h, -nh)$ with positive integers $h$ and $n$.\n\t\tThen since the dimension of ${\\mathcal P}_1+{\\mathcal P}_2$ is $2$, ${\\mathcal P}_1+{\\mathcal P}_2$ is level of index $1$.\n\t\tHowever, from \\cite[Theorem 4.5]{HY}, we know that ${\\mathcal P}_1*{\\mathcal P}_2$ is not level.\n\t\tHence the reverse of Theorem \\ref{thm;relation} (2) does not hold in general.\n\t}\n\\end{Example}\n\n\\section{Minkowski sums and Cayley sums of $2$-convex-normal lattice polytopes}\nIn this section, we will give an answer to Question \\ref{ques} (1).\nFirst, we show when the Minkowski sums and the Cayley sums of $2$-convex-normal lattice polytopes are IDP.\n\\begin{Theorem}\n\t\\label{thm:2cn}\n\tLet ${\\mathcal P}_1,\\ldots,{\\mathcal P}_m \\subset {\\NZQ R}^{N}$ be $2$-convex-normal lattice polytopes. Then we obtain the following:\n\t\\begin{enumerate}\n\t\t\\item[{\\rm (1)}] The Minkowski sum ${\\mathcal P}_1+\\cdots+{\\mathcal P}_m$ is IDP;\n\t\t\\item[{\\rm (2)}] The Cayley sum ${\\mathcal P}_1*\\cdots*{\\mathcal P}_m$ is IDP if (and only if) the tuple $({\\mathcal P}_1,\\ldots,{\\mathcal P}_m)$ is IDP.\n\t\\end{enumerate}\n\\end{Theorem}\n\\begin{proof}\n\t(1) Given a lattice point  ${\\bold x} \\in n({\\mathcal P}_1+\\cdots+{\\mathcal P}_m)$ with a positive integer $n$,\n\tthere exist $m$ points ${\\bold x}_1,\\ldots,{\\bold x}_m$ such that for each $i$, ${\\bold x}_i \\in n{\\mathcal P}_i$ and ${\\bold x}={\\bold x}_1+\\cdots+{\\bold x}_m$.\n\tMoreover, since each ${\\mathcal P}_i$ is $2$-convex-normal, for each $i$, we can write\n\t${\\bold x}_i={\\bold y}_{i1}+\\cdots+{\\bold y}_{in}$,\n\twhere ${\\bold y}_{i1} \\in {\\mathcal P}_i$ and for any $2 \\leq j \\leq n$, ${\\bold y}_{ij} \\in {\\mathcal P}_i \\cap {\\NZQ Z}^N$.\n\tFor $1 \\leq j \\leq n$, set ${\\bold z}_j={\\bold y}_{1j}+\\cdots+{\\bold y}_{mj}$.\n\tThen ${\\bold z}_2,\\ldots,{\\bold z}_n$ are lattice points in ${\\mathcal P}_1+\\cdots+{\\mathcal P}_m$.\n\tMoreover since ${\\bold x}$ is a lattice point,  ${\\bold z}_1$ must be a lattice point in ${\\mathcal P}_1+\\cdots+{\\mathcal P}_m$.\n\tHence ${\\mathcal P}_1+\\cdots+{\\mathcal P}_m$ is IDP.\n\t\n\t(2) Thanks to Theorem \\ref{thm;relation} (1), we should show that if $({\\mathcal P}_1,\\ldots,{\\mathcal P}_m)$ is IDP,\n\tthen ${\\mathcal P}_1*\\cdots*{\\mathcal P}_m$ is IDP.\n\tIn order to prove this, we use the induction on $m$.\n\tWhen $m=1$, this is clear.\n\tAssume that $m \\geq 2$.\n\tLet ${\\bold a} \\in n({\\mathcal P}_1*\\cdots*{\\mathcal P}_m) \\cap {\\NZQ Z}^{m+N}$ with a positive integer $n$.\n\tThen we can write \n\t$${\\bold a}=(t_1{\\bold e}_1,{\\bold a}_1)+\\cdots+(t_m{\\bold e}_m, {\\bold a}_m),$$\n\twhere ${\\bold a}_i \\in t_i {\\mathcal P}_i$, $t_i \\in {\\NZQ Z}_{\\geq 0}$ and $\\sum_{i=1}^{m}t_i=n$.\n\tSet $I=\\{ i : t_i \\geq 1\\}$.\n\tSince each ${\\mathcal P}_i$ is $2$-convex-normal, by using Lemma \\ref{decomp1}, ${\\bold a}$ can be written like\n\t$${\\bold a}=(\\sum_{i \\in I} {\\bold e}_i,{\\bold c})+{\\bold c}_1+\\cdots+{\\bold c}_{n-|I|},$$\n\twhere ${\\bold c} \\in (\\sum_{i \\in I} {\\mathcal P}_i) \\cap {\\NZQ Z}^N$ and ${\\bold c}_i \\in ({\\mathcal P}_1*\\cdots*{\\mathcal P}_m) \\cap {\\NZQ Z}^{m+N}$. \n\tSince $({\\mathcal P}_1,\\ldots,{\\mathcal P}_m)$ is IDP, \n\t$(\\sum_{i \\in I} {\\bold e}_i,{\\bold c})$ can be decomposed into $|I|$ lattice points belonging to \n\t${\\mathcal P}_1*\\cdots* {\\mathcal P}_m$. \n\tThus, ${\\mathcal P}_1*\\cdots*{\\mathcal P}_m$ is IDP, as desired.\n\\end{proof}\n\nNext, we see when a dilated polytope is $2$-convex-normal.\n\\begin{Lemma}\n\t\\label{lem:2cn}\n\tLet ${\\mathcal P} \\subset {\\NZQ R}^N$ be a lattice polytope.\n\tThen for any integer $n \\geq \\dim({\\mathcal P})$, \n\t$n {\\mathcal P}$ is $2$-convex-normal.\n\\end{Lemma}\n\n\n\\begin{proof}\n\tSet $d=\\dim({\\mathcal P})$.\n\tWhen $d=0$, this is clear.\n\tWe assume that $d \\geq 1$.\n\tLet ${\\bold a} \\in 2n{\\mathcal P}$.\n\tBy Carath\\'{e}odory's Theorem (cf. {\\cite[Corollary 7.1j]{integer}}), there exist $d+1$ affinely independent vertices ${\\bold v}_0,\\ldots,{\\bold v}_d$ of ${\\mathcal P}$ such that ${\\bold a}=\\sum_{i=0}^{d}\\lambda_i{\\bold v}_i$, where $\\lambda_i \\geq 0$ and $\\sum_{i=0}^{d}\\lambda_i=2n$.\n\n\n\tFrom the facts that $n+1 \\geq d+1$ and $\\sum_{i=1}^d (\\lambda_i- \\lfloor \\lambda_i \\rfloor) \\leq n$, ${\\bold a}$ can be written like\n\t${\\bold a}={\\bold b}+{\\bold v}_{j_1}+\\cdots+{\\bold v}_{j_n}$, where ${\\bold b} \\in n{\\mathcal P}$.\n\tSince ${\\bold v}_{j_1}+\\cdots+{\\bold v}_{j_n}  \\in (n{\\mathcal P}\\cap {\\NZQ Z}^N)$, one has ${\\bold a} \\in n{\\mathcal P} +(n{\\mathcal P} \\cap {\\NZQ Z}^N)$.\n\tTherefore, $n{\\mathcal P}$ is $2$-convex-normal.\n\\end{proof}\n\nFrom Theorem \\ref{thm:2cn} and Lemma \\ref{lem:2cn} we obtain the following corollary:\n\\begin{Corollary}\n\t\\label{cor:IDPdi}\n\tLet ${\\mathcal P}_1,\\ldots,{\\mathcal P}_m \\subset {\\NZQ R}^N$ be lattice polytopes. For each $i$, let $n_i$ be a positive integer with $n_i \\geq \\dim ({\\mathcal P}_i)$.\n\tThen we obtain the following:\n\t\\begin{enumerate}\n\t\t\\item[{\\rm (1)}] {\\rm (\\cite{Hig})} $n_1{\\mathcal P}_1+\\cdots+n_m{\\mathcal P}_m$ is IDP;\n\t\t\\item[{\\rm (2)}] \n\t\t$n_1{\\mathcal P}_1*\\cdots*n_m{\\mathcal P}_m$ is IDP if (and only if) $(n_1{\\mathcal P}_1,\\ldots,n_m{\\mathcal P}_m)$ is IDP. \n\t\\end{enumerate}\n\\end{Corollary}\n\nNow, we see that Question \\ref{ques} (1) is not true in general.\n\\begin{Example}\n\t\\label{ex:IDPcounter}\n\t{\\em \n\tLet ${\\mathcal P}_1 \\subset {\\NZQ R}^2$ be the line segment from $(0,0)$ to $(1,2)$ and ${\\mathcal P}_2 \\subset {\\NZQ R}^2$ the line segment from $(0,0)$ to $(1,0)$.\n\tThen for any positive integer $n_1,n_2$, $(n_1{\\mathcal P}_1,n_2{\\mathcal P}_2)$ is not IDP.\n\tIndeed, one has $(1,1) \\in (n_1{\\mathcal P}_1+n_2{\\mathcal P}_2) \\cap {\\NZQ Z}^2$.\n\tOn the other hand, since \n\t\\[\n\t(n_1 {\\mathcal P}_1 \\cap {\\NZQ Z}^2)+(n_2 {\\mathcal P}_2 \\cap {\\NZQ Z}^2)=\n\t\\{\n\t(a+b,2a) : a,b \\in {\\NZQ Z}, 0 \\leq a \\leq n_1, 0 \\leq b \\leq n_2\n\t\\},\n\t\\]\n\twe obtain $(1,1) \\notin (n_1 {\\mathcal P}_1 \\cap {\\NZQ Z}^2)+(n_2 {\\mathcal P}_2 \\cap {\\NZQ Z}^2)$.\n\tHence $(n_1{\\mathcal P}_1,n_2{\\mathcal P}_2)$ is not IDP.\n\tThus, it follows from Theorem \\ref{thm;relation} (1)  that $n_1{\\mathcal P}_1*n_2{\\mathcal P}_2$ is not IDP.\n\tTherefore, Question \\ref{ques} (1) in not true in general.\n}\n\\end{Example}\n\nFinally, we give another corollary of Theorem \\ref{thm:2cn}.\n\\begin{Corollary}\n\tLet ${\\mathcal P}_1,\\ldots,{\\mathcal P}_m \\subset {\\NZQ R}^N$ be lattice polytopes. Suppose that for each $i$, every edge of ${\\mathcal P}_i$ has lattice length $\\geq 2\\dim({\\mathcal P}_i)(\\dim({\\mathcal P}_i)+1)$.\n\tThen we obtain the following:\n\t\\begin{enumerate}\n\t\t\\item[{\\rm (1)}]  ${\\mathcal P}_1+\\cdots+{\\mathcal P}_m$ is IDP;\n\t\t\\item[{\\rm (2)}] \n\t\t${\\mathcal P}_1*\\cdots*{\\mathcal P}_m$ is IDP if (and only if) $({\\mathcal P}_1,\\ldots,{\\mathcal P}_m)$ is IDP. \n\t\\end{enumerate}\n\\end{Corollary}\n\\begin{proof}\n\tThis follows from Theorem \\ref{thm:2cn} and {\\cite[Corollary 6]{HH}}.\n\\end{proof}\n\n\n\n\n\\section{Minkowski sums and Cayley sums of level polytopes}\nIn this section, we give an answer to Question \\ref{ques} (2).\nFirst, we show that a lattice polytope satisfying the condition (\\ref{2cl}) is level.\n\\begin{Proposition}\n\t\\label{prop:2cl}\n\tLet ${\\mathcal P} \\subset {\\NZQ R}^N$ be a lattice polytope which satisfies the condition {\\rm (\\ref{2cl})}.\n\tThen ${\\mathcal P}$ is level.\n\\end{Proposition}\n\n\\begin{proof}\nLet $r = \\min \\{t \\in {\\NZQ Z}_{>0} : \\inter(t{\\mathcal P}) \\cap {\\NZQ Z}^N \\neq \\emptyset  \\}$.\n\tFix a positive integer $n \\geq r+1$ and take an interior lattice point ${\\bold x}$ in $n {\\mathcal P}$.\n\tSince for lattice polytopes ${\\mathcal P}_1,\\ldots,{\\mathcal P}_m \\subset {\\NZQ R}^N$, ${\\rm int}({\\mathcal P}_1+\\cdots+{\\mathcal P}_m)={\\rm int}({\\mathcal P}_1)+\\cdots+{\\rm int}({\\mathcal P}_m)$, from the condition (\\ref{2cl}), one has\n\t\\[\n\t{\\rm int}(n{\\mathcal P})={\\rm int}(r{\\mathcal P})+\\underbrace{({\\mathcal P} \\cap {\\NZQ Z}^N)+\\cdots+({\\mathcal P} \\cap {\\NZQ Z}^N)}_{n-r}.\n\t\\]\n\tHence there exist ${\\bold x}_1 \\in {\\rm int}(r{\\mathcal P})$ and ${\\bold x}_2,\\ldots,{\\bold x}_{n-r+1} \\in {\\mathcal P} \\cap {\\NZQ Z}^N$ such that ${\\bold x}={\\bold x}_1+{\\bold x}_2+\\cdots+{\\bold x}_{n-r+1}$.\n\tIn particular, from ${\\bold x} \\in {\\NZQ Z}^N$, one has ${\\bold x}_1 \\in {\\rm int}(r{\\mathcal P}) \\cap {\\NZQ Z}^N$.\n\tTherefore, since ${\\bold x}_2+\\cdots+{\\bold x}_{n-r+1} \\in (n-r){\\mathcal P} \\cap {\\NZQ Z}^N$, ${\\mathcal P}$ is level of index $r$.\n\\end{proof}\nNow, we show when the Minkowski sums and the Cayley sums of lattice polytopes with interior lattice points which satisfy the condition (\\ref{2cl}) are level.\n\\begin{Theorem}\n\t\\label{thn:2cl}\n\tLet ${\\mathcal P}_1,\\ldots,{\\mathcal P}_m \\subset {\\NZQ R}^N$ be  lattice polytope with  interior lattice points satisfying the condition {\\rm (\\ref{2cl})}.\n\tThen we obtain the following:\n\t\\begin{enumerate}\n\t\t\\item[{\\rm (1)}] The Minkowski sum ${\\mathcal P}_1+\\cdots+{\\mathcal P}_m$ is level of index $1$;\n\t\t\\item[{\\rm (2)}] The Cayley sum ${\\mathcal P}_1*\\cdots*{\\mathcal P}_m$ is level of index $m$.\n\t\\end{enumerate}\n\\end{Theorem}\n\\begin{proof}[Proof of Theorem \\ref{thn:2cl}]\nFrom Theorem \\ref{thm;relation} (2), we should show only the claim (2).\nSince each ${\\mathcal P}_i$ has interior lattice points, from Lemma \\ref{decomp2}, $m({\\mathcal P}_1*\\cdots*{\\mathcal P}_m)$ has interior lattice points.\nIn particular, $\\min \\{t \\in {\\NZQ Z}_{\\geq 1} : {\\rm int}(t({\\mathcal P}_1*\\cdots*{\\mathcal P}_m)) \\cap {\\NZQ Z}^{m+N} \\neq \\emptyset \\}=m$.\n\tLet $n \\geq m+1$ and ${\\bold a} \\in \\text{int}(n({\\mathcal P}_1*\\cdots* {\\mathcal P}_m)) \\cap {\\NZQ Z}^{m+N}$,\n\tand for $1 \\leq i \\leq m$, ${\\bold v}_{i1},\\ldots,{\\bold v}_{ir_i}$ the vertices of ${\\mathcal P}_i$, where $r_i$ is the number of vertices of $P_i$.\n\tThen since ${\\bold a}$ is an interior lattice point in $n({\\mathcal P}_1*\\cdots*{\\mathcal P}_m)$, we can write \n\t$${\\bold a}=\\sum\\limits_{i=1}^{m} \\sum_{j=1}^{r_i}\\lambda_{ij}({\\bold e}_i,{\\bold v}_{ij}),$$\n\twhere for any $1 \\leq i \\leq m$ and $1 \\leq j \\leq r_i$, $0<\\lambda_{ij}$, and $\\sum_{i=1}^{m} \\sum_{j=1}^{r_i}\\lambda_{ij}=n$.\n\tFor each $1 \\leq i \\leq m$, set\n\t$t_i=\\sum_{j=1}^{r_i}\\lambda_{ij} \\in {\\NZQ Z}_{\\geq 1}$ and ${\\bold a}'_i=\\sum_{j=1}^{r_i}\\lambda_{ij}({\\bold e}_i,{\\bold v}_{ij}) \\in \\text{int}(t_i(\\{{\\bold e}_i \\} \\times {\\mathcal P}_i))$.\n\tSince each ${\\mathcal P}_i$ satisfies the condition (\\ref{2cl}), for each $i$, there exist ${\\bold b}_{i1} \\in {\\rm int}(\\{{\\bold e}_i\\} \\times {\\mathcal P}_i)$ and ${\\bold b}_2,\\ldots,{\\bold b}_{t_i} \\in (\\{{\\bold e}_i \\} \\times {\\mathcal P}_i) \\cap {\\NZQ Z}^{m+N}$ with ${\\bold a}'_i={\\bold b}_{i1}+{\\bold b}_{i2}+\\cdots+{\\bold b}_{it_i}$.\n\tHence one has\n\t\\[\n\t{\\bold a}=({\\bold b}_{11}+\\cdots+{\\bold b}_{m1})+\\sum_{i=1}^m \\sum_{j=2}^{t_i}{\\bold b}_{ij}.\n\t\\]\n Then it follows that ${\\bold b}_{11}+\\cdots+{\\bold b}_{m1} \\in  {\\rm int}(m ({\\mathcal P}_1*\\cdots*\\cdots{\\mathcal P}_m))$ and $\\sum_{i=1}^m \\sum_{j=2}^{t_i}{\\bold b}_{ij}\\in (n-m)({\\mathcal P}_1*\\cdots*{\\mathcal P}_m) \\cap {\\NZQ Z}^{N+m}$.\n Since ${\\bold a}$ is a lattice point, ${\\bold b}_{11}+\\cdots+{\\bold b}_{m1}$ must be a lattice point.\n Hence one has ${\\bold b}_{11}+\\cdots+{\\bold b}_{m1}  \\in {\\rm int}(m ({\\mathcal P}_1*\\cdots*{\\mathcal P}_m)) \\cap {\\NZQ Z}^{m+N}$.\n\tTherefore, ${\\mathcal P}_1*\\cdots*{\\mathcal P}_m$ is level of index $m$, as desired.\n\\end{proof}\n\nNext, we see when a dilated polytope satisfies the condition (\\ref{2cl}).\n\\begin{Lemma}\n\t\\label{lem:2cl}\n\tLet ${\\mathcal P} \\subset {\\NZQ R}^N$ be a lattice polytope.\n\tThen for any integer $n \\geq \\dim({\\mathcal P})+1$,\n$n{\\mathcal P}$ has interior lattice points and  satisfies the condition {\\rm (\\ref{2cl})}.\n\\end{Lemma}\n\n\\begin{proof}\n\tSet $d=\\dim({\\mathcal P})$\n\tand\n\t${\\bold a} \\in \\textnormal{int}(2n{\\mathcal P})$.\n\tThen there exist $d'+1$ affinely independent vertices ${\\bold v}_0,\\ldots,{\\bold v}_{d'}$ of ${\\mathcal P}$ such that ${\\bold a}=\\sum_{i=0}^{d'}\\lambda_i{\\bold v}_i$, where\n\t$d' \\leq d$, for each $1 \\leq i \\leq d'$, $\\lambda_i > 0$, and $\\sum_{i=0}^{d'}\\lambda_i=2n$.\n\tSince $2n \\geq d+2 \\geq d'+2$, there exists an index $j$ such that $\\lambda_j > 1$.\n\tHence ${\\bold a}=(\\sum_{i=0}^{d}\\lambda_i{\\bold v}_i-{\\bold v}_j)+{\\bold v}_j$ and $(\\sum_{i=0}^{d}\\lambda_i{\\bold v}_i-{\\bold v}_j) \\in \\textnormal{int}((2n-1){\\mathcal P})$.\n\tFrom the fact that $n+1 \\geq d+2$, ${\\bold a}$ can be written like\n\t${\\bold a}={\\bold a}'+{\\bold v}_{j_1}+\\cdots+{\\bold v}_{j_n}$, where ${\\bold a}' \\in \\textnormal{int}(n{\\mathcal P})$.\n\tHence ${\\bold a} \\in \\textnormal{int}(n{\\mathcal P})+(n{\\mathcal P} \\cap {\\NZQ Z}^N)$, as desired.\n\\end{proof}\n\nFrom Theorem \\ref{thn:2cl} and Lemma \\ref{lem:2cl} we obtain the following corollary.\n\\begin{Corollary}\n\t\\label{cor:leveldi}\n\tLet ${\\mathcal P}_1,\\ldots,{\\mathcal P}_m \\subset {\\NZQ R}^N$ be lattice polytopes. For each $i$, let $n_i$ be a positive integer with $n_i \\geq \\dim ({\\mathcal P}_i)+1$.\n\tThen we obtain the following:\n\t\\begin{enumerate}\n\t\t\\item[{\\rm (1)}] {\\rm (\\cite{Hig})} $n_1{\\mathcal P}_1+\\cdots+n_m{\\mathcal P}_m$ is level of index $1$;\n\t\t\\item[{\\rm (2)}] \n\t\t$n_1{\\mathcal P}_1*\\cdots*n_m{\\mathcal P}_m$ is level of index $m$. \n\t\\end{enumerate}\n\\end{Corollary}\nTherefore, Question \\ref{ques} (2) is true.\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nNoninvasive electroencephalography (EEG) based brain computer interfaces (BCIs) have shown promising results for establishing a safe alternative channel between the people with severe speech\/muscle impairment and their environment~\\cite{mur14}. BCIs can be used for different applications such as communication, environment control and wheelchair navigation~\\cite{mur14,mog12}. For communication, letter by letter typing BCIs have been the subject of extensive research and development in the field~\\cite{mur14,mog12,far88,sel03,orh12}. For instance, P300-matrix speller, which was first introduced by Donchin and Farewell, typically uses a letter by letter typing BCI~\\cite{far88}. Even though matrix-based presentation scheme is very commonly used and various paradigms for this scheme were developed to improve typing speed and accuracy~\\cite{sel06b,all03,jin11,tow10,tow12,jin15,li16,yeo14}, it has been shown the matrix-based presentation paradigms are highly gaze dependent and they cannot operate well for the population with covert attention\\cite{tre10}. As a minimally gaze dependent alternative, rapid serial visual presentation (RSVP) paradigm has been offered in which all the symbols are presented in a pseudo-random order at a predefined location of the screen in a rapid serial manner~\\cite{acq10,acq13,orh12,orh13,orh12b}. Generally, the performance comparison of different users on typing with row and column presentation (RCP), single character presentation (SCP), and RSVP paradigms, have demonstrated that at least for healthy population, the best presentation paradigm should be defined separately for each individual~\\cite{mog15}.\n\nWe have developed a noninvasive EEG-based typing BCI which enables the user to choose among different matrix-based presentation paradigms and RSVP paradigm~\\cite{mog15}. This system detects the user intent in a recursive Bayesian state estimation (RBSE) in which the state represents the user target. To improve the detection performance, we have incorporated a 6-gram language model that provides context priors to be probabilistically fused with the EEG likelihoods. Our earlier studies demonstrated a great benefit in using the language model both in terms of typing speed and accuracy~\\cite{mog15,orh12}. In our system, a user is presented with a sequence of symbols and an event related potential in response to the desired character, in the recorded EEG, can indicate the user intent. To achieve a confident decision, our system might query the user with one or multiple sequences until a predefined confidence threshold is attained or an upper-bound on the number of sequences is reached ~\\cite{orh12b,mog15,mog15b}.  In earlier versions of the system, each sequence contained the entire set of the symbols. However, experimentally, we observed that this method is very inefficient and adaptive subset selection methods are necessary to further improve the typing speed and accuracy.   \n\nHere, we describe a sequence design strategy using the active learning concept. Active learning will be built upon RBSE for dynamic and time effective sequence optimization. The proposed query design and inference mechanism is denoted as active-RBSE framework. In this framework, we develop query strategies to actively choose a subset of stimuli for every presentation sequence. Specifically, in this manuscript, we propose to use a query function that employs the observed EEG evidence along with context information. According to this function, quires are adaptively specialized for the user, at every iteration, during each target detection step. We show that this query function is a modular and monotonic set function and accordingly the optimal solution for this function can be achieved with guaranteed convergence. We utilized real EEG data, to run Monte-Carlo simulation of our system when active-RBSE framework was utilized. The results show that, the proposed query function along with the time efficient optimization framework can improve the online typing performance in terms of speed and accuracy for both matrix-based presentation and RSVP paradigms. \n\\section{General BCI system specifications}\nIn a previous study, we have shown that popular matrix-based presentation paradigms (namely 1. row and column presentation (RCP) paradigm, 2. single character presentation (SCP) paradigm), and rapid serial visual presentation (RSVP) paradigm in general provide comparable performances and the optimal presentation paradigm is different for each user~\\cite{mog15}. Based on our previous results, we have decided to include all three presentation paradigms in the current design of our letter by letter BCI typing interface.  \n\nWe assume that the user intent is to select a character $x$ from a finite vocabulary set $\\mathcal{V}$. In a typical typing scenario we can define $\\mathcal{V}=\\{A,~B,~,\\cdots,~Z\\}\\cup \\{<,~\\_\\}$ where $<$ represents the backspace for error correction and $\\_$ is the space symbol. In a matrix presentation paradigm, one can propose to flash every arbitrary subset of vocabulary ${A}\\subseteq\\mathcal{V}$ in a ``\\textit{trial}''. For example, in SCP the paradigm is such that, trials are singletons of the vocabulary set $A\\in \\mathcal{V},\\,\\,\\, |{A}|=1$. Accordingly, we can record the EEG response to that trial $\\mathbf{e}({A})$ for detection. A set of trials $\\Phi=\\{{A}_1,~{A}_2,~\\cdots,{A}_{|\\Phi|}\\}$ which is presented to the user in a rapid serial manner is called a ``\\textit{sequence}''. After every sequence the system attempts to infer the  target character intended by the user, but, due to low signal to noise ratio (SNR) of EEG usually the system need to query the user with more sequences to reach a decision with a predefined confidence level. On the other hand, in order to limit the time spent on typing each character the system will make a decision, regardless of confidence threshold, after a predefined number of sequences. In our system a set of sequences which lead to a decision is called an ``\\textit{epoch}''. \n \\subsection{Probabilistic Graphical Model}\nThe proposed probabilistic graphical model (PGM) illustrated in Fig~\\ref{fig:PGM}. In this figure, $x_k$ represents the system state at epoch $k$, $\\Phi_k^s$ is the set of trials selected (i.e. query set) to be flashed in the $s^{\\rm th}$ sequence, $|\\Phi_k^s|$ is the number of trials (i.e. the size of query set), $C_k$ is the context information, which in a typing scenario can be provided by a language model, and $e(A_j^s)$ is the EEG observation for the $j^{\\rm th}$ trial in sequence $s$. In our model the label $y_{x_k}(A_j^s)$ for trial $A_j^s$ is a deterministic function of $x_k$, i.e.  $y_{x_k}(A_j^s):=\\mathbbm{1}(x_k\\in A_j^s)$, then $y_{x_k}(A_j^s)=1$ if $x_k\\in A_j^s$ or $y_{x_k}(A_j^s)=0$ otherwise. Finally, the maximum number of sequences allowed in an epoch is denoted as $m_s$.\n    \\begin{figure}[!ht]\n        \\begin{center}\n            \\scalebox{0.24}{\n            \\input{PGM}}\n        \\end{center}\n        \\vspace{-5mm}\n        \\caption{Proposed probabilistic graphical model representing the \n        $k^{\\rm th}$ epoch. Here, the dashed lines show a deterministic \n        relation, the solid lines define a probabilistic correspondence, and red rectangles represent conditional independence.}\n        \\label{fig:PGM}\n    \\end{figure}\n    \n Active-RBSE is built upon this graphical model and Within this framework, a generic AL and maximum a posteriori (MAP) inference loop will iterate by alternating between the following two steps:\n    \\begin{eqnarray}\n        \\text{Query:} \n            \\widehat{\\Phi}_k^{s+1}= \n            &\\arg \\max_{\\Phi_k^{s+1}} g(\\Phi_k^{s+1}) \\hspace{5pt} \\text{s.t.} \\hspace{5pt} \n            \\Phi_k^{s+1} \\in \\mathcal{F}_k \\subseteq 2^{\\mathcal{V}}\n            \\label{subeqn:querystrategy}\\\\\n        \\text{Inference:} \\hat{x}_k = &\n            \\arg\\max_{\\text{x}}~P(x_k=\\text{x}|\\mathcal{E}^s, C, \\{\\Phi_k^i\\}_{i=1}^s)\n            \\label{subeqn:mapinference}\n    \\end{eqnarray}\nHere, $\\Phi_k^{s+1}$ is a potential query set restricted to the set of feasible queries at time $k$, $\\mathcal{F}_k$, which is a subset of all possible queries, $2^{\\mathcal{V}}$, the power set of $\\mathcal{V}$. The quality of a query from the perspective of AL is measured by the set function $g$. Note that we focus on MAP inference in this manuscript. Next, we describe how to obtain the MAP the inference and then in Section~\\ref{sec:queryOptimization}, we will explain the query strategy. \n\n \\subsubsection{Recursive Bayesian State Estimation and Intent Inference}\nOur systems employs the maximum a-posteriori (MAP) inference mechanism to detect the user intent. We use the well-known Bayes rule of \"\\textit{posterior }$\\propto$ \\textit{prior } $\\times$ \\textit{likelihood}\" to estimate the posterior probability mass function (PMF) over the state space. For letter by letter typing, we use an n-gram language model (LM) to estimate the prior PMF over the character set, and the likelihood is obtained from EEG observations. Given the system state we assume, EEG measurements for sequences are independent from each others (see Fig~\\ref{fig:PGM}). Based on this assumption we can update the posterior PMF recursively, after every sequence.   \n\nThe prior PMF from the LM along with the EEG observation likelihood is used in our system to calculate the posterior PMF over the vocabulary set after each sequence. Assume $1\\leq s\\leq m_s$ sequences have been shown to the user and define $\\mathcal{E}^s=\\{E^i\\}_{i=1}^s$, where $E^i=\\{e(A^i_j)|~j=1,~\\cdots,~|{\\Phi}_k^i|\\}$. Similarly, take $Y_{x_k}^i=\\{y_{x_k}(A_j^i)|~j=1,~\\cdots,~|\\Phi_k^i|\\}$ then define $\\mathcal{Y}_{x_k}^s=\\{Y_{x_k}^i\\}_{i=1}^s$. The MAP framework estimates the user intent by solving the following optimization problem:\n    \\begin{equation}\n        \\hat{x}_k=\\arg\\max_{\\text{x}}~~P(x_k=\\text{x}|\\mathcal{E}^s,~ C, \\{\\Phi_k^i\\}_{i=1}^s)\n        \\label{eq:MAPInference}\n    \\end{equation}\nThe posterior probability defined in (\\ref{eq:MAPInference}) can be factorized in terms of likelihood and context prior using the assumptions imposed in Figure~\\ref{fig:PGM}.\n    \\begin{equation}\n    \\small\n        \\begin{split}\n           & P(x_k=\\text{x}|\\mathcal{E}^s,~ C, \\{\\Phi_k^i\\}_{i=1}^s)=\\frac{p(x_k=\\text{x},~\\mathcal{E}^s|~C,\\{\\Phi_k^i\\}_{i=1}^s)}{p(\\mathcal{E}^s|~C, \\{\\Phi_k^i\\}_{i=1}^s)}\\\\\n            &\\propto p(\\mathcal{E}^s|~x_k=\\text{x}, \\{\\Phi_k^i\\}_{i=1}^s)\\cdot P(x_k=\\text{x}|~C)\n        \\end{split}\n        \\label{eq:PosteriorEstimatior1}\n    \\end{equation}\nBut, for a given $x_k$ the $\\mathcal{Y}_{x_k}^s$ for $\\{\\Phi_k^i\\}_{i=1}^s$ is deterministically defined. Hence, according to the conditional independence of  $\\mathcal{E}^s$ and context information defined in PGM we obtain;\n    \\begin{equation}\n    \\small\n    \\begin{split}\n    &p(\\mathcal{E}^s|~x_k=\\text{x}; \\{\\Phi_k^i\\}_{i=1}^s) = p(\\mathcal{E}^s|~\\mathcal{Y}_{x_k}^s; ~\\{\\Phi_k^i\\}_{i=1}^s)=\\\\&\\prod_{\\substack{\n            i=1,~\\ldots,~s\\\\\n            j=1,~\\ldots,~|\\Phi_k^i|}}^s {p(e(A_j^i)|y_{x_k}(A_j^i); ~\\{\\Phi_k^i\\}_{i=1}^s)}\n    \\end{split}\n            \\label{eq:likelihood}\n    \\end{equation}\nThen we can rewrite (\\ref{eq:MAPInference}) a\n    \\begin{equation}\n        \\begin{split}\n            &P(x_k=\\text{x}|\\mathcal{E}^s,~ C, \\{\\Phi_k^i\\}_{i=1}^s)\\propto \\\\& \\prod_{\\substack{\n            i=1,~\\ldots,~s\\\\\n            j=1,~\\ldots,~|\\Phi_k^i|}}^s {p(e(A_j^i)|y_{x_k}(A_j^i);~\\{\\Phi_k^i\\}_{i=1}^s)} \\cdot P(x_k=\\text{x}|~C)\\propto \\\\\n            &\\prod_{\\substack{\n            i=1,~\\ldots,~s\\\\\n            \\{j|~y_{x_k}(A_j^i)=1\\}}} {\\frac{p(e(A_j^i)|y_{x_k}(A_j^i)=1; ~\\{\\Phi_k^i\\}_{i=1}^s)}{p(e(A_j^i)|y_{x_k}(A_j^i)=0; ~\\{\\Phi_k^i\\}_{i=1}^s)}} \\cdot P(x_k=\\text{x}|~C)\n        \\end{split}\n        \\label{eq:posteriorWithLikelihoods}\n    \\end{equation}\nIn the next two subsections, we will describe how to obtain the context probability $P(x_k=\\text{x}|~C)$ and the class conditional probabilities $p(e(A_j^i)|y_{x_k}(A_j^i)=1; ~\\{\\Phi_k^i\\}_{i=1}^s)$ and $p(e(A_j^i)|y_{x_k}(A_j^i)=0; ~\\{\\Phi_k^i\\}_{i=1}^s)$.\n\\subsubsection{n-gram Language Model for Context-based Prior Estimation}\nIn our system we utilize an n-gram language model (LM) to estimate the prior probabilities over the vocabulary set. The n-gram LM used in our system is essentially a Markov model of order $n-1$ which estimates a PMF over the state space for upcoming character, give the $n-1$ previously typed letters. \n Let us define $C=\\{x_l^*\\}_{l=n-1,~\\cdots,~1}$ as the order set of $n-1$ preceding typed characters where, $x^*_{l}$ represents the character at lag $l$. Then the probability of current character can be defined as follows:\n    \\begin{equation}\n    \\small\n        P(x|C)=P(x|\\{x_l^*\\}_{l=n-1,~\\cdots,~1})=\\frac{P(x,~x_{n-1}^*,~\\cdots,~x_1^*)}{P(x|~x_{n-1}^*,~\\cdots,~x_1^*)}\n        \\label{eq:LM}\n    \\end{equation}\nThe LM used in our system, is trained on the NY Times portion of English Gigaword corpus~\\cite{roa10}. \nThis LM has shown to enhance the BCI typing performance in terms of typing accuracy and speed~\\cite{orh12b,orh13}.\n\n \\subsubsection{Class Conditional Distributions for EEG observations}\nTo estimate the class conditional distributions we collect labeled data in a \"calibration session\". Typically, during a calibration session the user is presented with $100$ sequences. Prior to presentation the user is asked to focus on a predefined target character during that sequence\n\nIn our system, we record the signal from 16 EEG channels. To improve the SNR we apply a bandpass filter with the pass band of $[1.5,42]$ Hz and a notch filter at $60$ Hz to further attenuate the line noise. According to the pass band of the filter and data acquisition sampling rate ($256$ Hz), we down-sample the signal by a factor of $2$ to eliminate non-informative time samples while avoiding aliasing. The preprocessed EEG in a time window of $[0,500)$ms from the onset of each stimulus is assigned to that trial as it's EEG measurement at each channel. Subsequently, we eliminate directions with zero or negligible variances by applying principle component analysis (PCA) on these EEG measurements at each channel separately. Finally, we concatenate the measurements from every channel to form the EEG feature vector $\\mathbf{f}_j^s$ for $j^{\\rm th}$ trial at sequence $s$ in the $k^{\\rm th}$ epoch.\n\nEEG is assumed to be a Gaussian process~\\cite{fau07,zho08,mog15}, hence, we could utilize quadratic discriminant analysis (QDA) to project the EEG feature vector onto a one dimensional space with minimum expected classification risk. But QDA requires a full rank class conditional covariance matrix estimation while for a typical setup of our system this is not feasible because the feature vector dimensionality is relatively higher than number of observations at each class. Instead, we utilize regularized discriminant analysis (RDA) which applies regularization and shrinkage on estimated class conditional covariance matrices and makes them invertible~\\cite{RDA}. \n\nAssume $\\mathbf{f}_i$ is an $m$ dimensional feature vector and $y_i\\in \\{0,1\\}$ is the binary label for $\\mathbf{f}_i$ then the maximum likelihood estimator for class conditional covariance matrix for class $k\\in \\{0,1\\}$ is;\n   \\begin{equation}\n        \\begin{split}\n            &\\boldsymbol{\\upmu}_k =\\frac{1}{N_k}\\sum_{i=1}^N\n                {\\mathbf{f}_i\\delta(y_i,k)} \\\\\n            &\\mathbf{\\Sigma}_k =\\frac{1}{N_k}\\sum_{i=1}^N\n                {(\\mathbf{f}_i-\\boldsymbol{\\upmu}_k)\n                (\\mathbf{f}_i-\\boldsymbol{\\upmu}_k)^T\\delta(y_i,k)} \n        \\end{split}\n    \\end{equation}\nfor which $\\delta(.,.)$ represents the Kronecker-$\\delta$ function and $N_k$ is the number of observations in class $k$. Then the regularization and shrinkage steps are applied as follows:\n    \\begin{equation}\n        \\begin{split}\n            &\\widehat{\\mathbf{\\Sigma}}_k(\\lambda)=\\frac{(1-\\lambda)\n                N_k\\mathbf{\\Sigma}_k+(\\lambda)\\sum_{k=0}^1{N_k\\mathbf{\\Sigma}_k}}\n                {(1-\\lambda) N_k+(\\lambda)\\sum_{k=0}^1{N_k}}  \\\\\n            &\\widehat{\\mathbf{\\Sigma}}_k(\\lambda,\\gamma)=\n                (1-\\gamma)\\widehat{\\mathbf{\\Sigma}}_k(\\lambda)+(\\gamma) \n                \\frac{1}{m}tr[\\widehat{\\mathbf{\\Sigma}}_k(\\lambda)]\\mathbf{I}_m\n        \\end{split}\n    \\end{equation}\nwhere $\\lambda,\\gamma\\in [0,1]$ are the shrinkage and regularization parameters, $tr[\\cdot]$ is the trace operator and $\\mathbf{I}_m$ is an identity matrix of size $m\\times m$. To optimize the values of $\\lambda$ and $\\gamma$, in our system we apply $10$-fold cross validation to maximize the area under the receiver operating characteristics (ROC) curve (AUC). In our system the \"EEG evidence\" for trial $A_j^i$ is computed as follows.\n    \\begin{equation}\n        e(A_j^i)=\\log\\left(\\frac{f_\\mathcal{N}\n        (\\mathbf{f}_j^i;\\boldsymbol{\\upmu}_1,\n        \\widehat{\\mathbf{\\Sigma}}_1(\\lambda,\\gamma))}\n        {f_\\mathcal{N}(\\mathbf{f}_j^i;\\boldsymbol{\\upmu}_0,\n        \\widehat{\\mathbf{\\Sigma}}_0(\\lambda,\\gamma))} \\right)\n    \\end{equation}\nHere, $\\mathbf{f}_j^i$ is the EEG feature vector for trial $A_j^i$, $f_\\mathcal{N}(\\mathbf{f};\\boldsymbol{\\upmu},{\\mathbf{\\Sigma}})$ is the Gaussian probability density function with mean $\\boldsymbol{\\upmu}$ and covariance $\\mathbf{\\Sigma}$, and $\\lambda,\\gamma\\in [0,1]$ are the shrinkage and regularization parameters. To optimize the values of $\\lambda$ and $\\gamma$, in our system we apply $10$-fold cross validation to maximize the area under the receiver operating characteristics (ROC) curve (AUC).\nTo obtain the class conditional probability distributions $p(e(A_j^i)|y_{x_k}(A_j^i)=1; ~\\{\\Phi_k^i\\}_{i=1}^s)$ and $p(e(A_j^i)|y_{x_k}(A_j^i)=0; ~\\{\\Phi_k^i\\}_{i=1}^s)$ over the EEG evidence we further apply kernel density estimation (KDE) with a Gaussian kernel of a bandwidth which is computed using Silverman rule of thumb~\\cite{sil86}.\n\\section{Query set Optimization Using Active Learning}\\label{sec:queryOptimization}\nIn this section, we propose that optimizing the query set using the prior information from the language model and the EEG signal in response to earlier sequences in a specific epoch can improve the typing performance for that epoch. As a claim to this proposition in our earlier studies we have shown that for RSVP paradigm, it is inefficient to present the whole or randomly selected subsets of vocabulary at every sequence~\\cite{mog15b}. Here, we use active query selection inspired by the active learning concept to define a combinatorial optimization problem, which exploits previously acquired information to appoint the query set elements in a timely manner. \n\\subsection{Objective Function}\nIn this section, we will consider a specific selection of $g(.)$ to use in active-RBSE framework, specifically in (\\ref{subeqn:querystrategy}). Let us hypothesize that we know the actual user intent $x_k^*$ for current epoch (i.e. epoch $k$). Our objective is to define an optimal query set for sequence $s+1$ while $s$ sequences have been already queried for inference of $x_k^*$ but the confidence threshold is not attained yet. Then, before presenting the $(s+1)^{\\rm th}$ sequence, we can obtain a prediction of posterior PMF for a given $\\Phi_k^i$ as follows. \n   \n   \n   \n   \n   \n   \n   \n   \n   \nWe define a function $g:\\mathcal{V},2^{2^{\\mathcal{V}}}\\to \\mathbb{R}$ as:\n\\begin{equation}\n    \\begin{split}\n        &g(\\mathbf{x}, \\Phi_k^{s+1})= P(x_k=\\text{x}|\\mathcal{E}^s,~ C, \\{\\Phi_k^i\\}_{i=1}^{s+1}, ~x_k^*=\\text{x})\\\\\n        &=\\int_{\\tilde{E}^{s+1}}P(x_k=\\text{x},\\tilde{E}^{s+1}|\\mathcal{E}^s,~ C,\\{\\Phi_k^i\\}_{i=1}^{s+1}, ~x_k^*=\\text{x}) d(\\tilde{E}^{s+1})\\\\\n        &=\\mathbb{E}_{\\tilde{E}^{s+1}|\\Phi_k^{s+1},~x_k^*} [P(x_k=\\text{x}|\\tilde{E}^{s+1},~\\mathcal{E}^s,~ C,\\{\\Phi_k^i\\}_{i=1}^s, ~x_k^*)]\\\\\n        &=\\mathbb{E}_{\\tilde{E}^{s+1}|\\Phi_k^{s+1},~x_k^*} \\left[\\frac{\\Pi^{s+1}(\\text{x})\\cdot p(\\tilde{E}^{s+1}|x_k=\\text{x},~\\Phi_k^{s+1})}{\\sum_{\\text{v}\\in \\mathcal{V}} \\Pi^{s+1}(\\text{v})\\cdot p(\\tilde{E}^{s+1}|x_k=\\text{v},~\\Phi_k^{s+1})}\\right]\n   \\end{split}\n   \\label{eq:SubObj}\n  \\end{equation}\n where $\\Pi^{s+1}(\\text{x})=P(x_k=\\text{x}|\\mathcal{E}^s,~ C, \\{\\Phi_k^i\\}_{i=1}^s)$ represents the prior probability of $\\text{x}\\in \\mathcal{V}$ before observing sequence $s+1$. Moving from the third line to the fourth of (\\ref{eq:SubObj}), we use the following\n\\begin{equation}\n\\begin{split}\n    &P(x_k=\\text{x}|\\tilde{E}^{s+1},~\\mathcal{E}^s,~ C, \\{\\Phi_k^i\\}_{i=1}^{s+1})= \\\\\n    &\\frac{\\Pi^{s+1}(\\text{x})\\cdot p(\\tilde{E}^{s+1}|x_k=\\text{x},~\\Phi_k^{s+1})}{\\sum_{\\text{v}\\in \\mathcal{V}} \\Pi^{s+1}(\\text{v})\\cdot p(\\tilde{E}^{s+1}|x_k=\\text{v},~\\Phi_k^{s+1})}\n\\end{split}\n\\end{equation}\nwhere the denominator is the normalization constant.  \n        \nNote that, $g(\\mathbf{x}, \\Phi_k^{s+1})$ computes the posterior probability of hypothesized target for a particular $\\Phi_k^{s+1}$ given previously observed EEG and context. But during the current epoch, $\\mathbf{x}$ is yet to be estimated; and hence, we marginalize out the dependency on this unobserved random variable by computing the expected value of $g(\\mathbf{x}, \\Phi_k^{s+1})$ with respect to the most recent estimate of state space posterior PMF, $\\Pi^{s+1}(\\text{x})$.\n\n\nAccordingly, The objective function for query set selection is then defined as follows.\n    \\begin{equation}\n        \\widehat{\\Phi}_k^{s+1}=\\arg\\max_{\\Phi_k^{s+1}} ~~\\mathbb{E}_{\\Pi^{s+1}(\\text{x})} \\left[ g(\\mathbf{x}, \\Phi_k^{s+1}) \\right]\n    \\end{equation}\n\\subsection{Solution of the Optimization Problem}\nIn equation (\\ref{eq:SubObj}), for fixed $\\text{x}$ and $\\Phi_k^{s+1}$, the argument inside the expectation is only a function of $\\tilde{E}^{s+1}$. Accordingly, we define $\\tilde{\\boldsymbol{\\sigma}}=\\left[\\tilde{\\sigma}(A_1^{s+1}),~\\cdots,~\\tilde{\\sigma}(A_{|\\Phi_k^{s+1}|}^{s+1})\\right]$, where\\[ \\tilde{\\sigma}(A_j^i)=\\frac{p(\\tilde{e}(A_j^i)|1)}{p(\\tilde{e}(A_j^i)|0)}\\] and use \\eqref{eq:posteriorWithLikelihoods}, to specify $\\mathcal{F}:\\mathbb{R}^{|\\Phi_k^{s+1}|}\\to \\mathbb{R}$ such that,\n    \\begin{equation}\n        \\begin{split}\n            \\mathcal{F}(\\tilde{\\boldsymbol{\\sigma}})=\\frac{\\Pi^{s+1}(\\text{x})\\cdot \\prod_{\n            \\{j|~y_{\\text{x}}(A_j^{s+1})=1\\}} \\tilde{\\sigma}(A_j^{s+1}) }{\\sum_{\\text{v}\\in \\mathcal{V}} \\Pi^{s+1}(\\text{v})\\cdot \\prod_{\n            \\{j|~y_{\\text{v}}(A_j^{s+1})=1\\}} \\tilde{\\sigma}(A_j^{s+1})}\n        \\end{split}\n        \\label{eq:F}\n    \\end{equation}\nTo simplify the optimization --the reason for the simplification is described later in this section--, we approximate the $g(\\mathbf{x}, \\Phi_k^{s+1})$ using the  Taylor series expansion and the function defined in (\\ref{eq:F}).\n    \\begin{equation}\n        \\begin{split}\n           &g(\\mathbf{x}, \\Phi_k^{s+1})=\\mathbb{E}_{\\tilde{E}^{s+1}|\\Phi_k^{s+1},~x_k^*} \\left[\\mathcal{F}(\\tilde{\\boldsymbol{\\sigma}})\\right]=\\\\\n           &\\mathbb{E}_{\\tilde{E}^{s+1}|\\Phi_k^{s+1},~x_k^*} \\left[ \\mathcal{F}(\\boldsymbol{\\upmu}_{\\boldsymbol{\\sigma}})+\\left(\\tilde{\\boldsymbol{\\sigma}}-\\boldsymbol{\\upmu}_{\\boldsymbol{\\sigma}}\\right)\\cdot \\nabla\\mathcal{F}(\\boldsymbol{\\upmu}_{\\boldsymbol{\\sigma}})+\\cdots \\right]\n        \\end{split}\n    \\end{equation}\nwhere $\\boldsymbol{\\upmu}_{\\boldsymbol{\\sigma}}=\\mathbb{E}_{\\tilde{E}^{s+1}|\\Phi_k^{s+1},~x_k^*} [\\tilde{\\boldsymbol{\\sigma}}]$. We propose to substitute the original function $g(\\mathbf{x}, \\Phi_k^{s+1})$ with its locally suboptimal linear approximation around the $\\boldsymbol{\\upmu}_{\\boldsymbol{\\sigma}}$. The proposed linear approximation is widely used in the field of signal processing~\\cite{kay08}, especially when higher order central moments of the distribution are negligible. Typically, in our system, the estimated class conditional distributions are unimodal with small variance hence we assume this approximation is justifiable. Accordingly, we have:\n    \\begin{equation}\n        \\begin{split}\n            &g(\\mathbf{x}, \\Phi_k^{s+1})\\approx\\hat{g}(\\mathbf{x}, \\Phi_k^{s+1})=\\\\\n            &\\mathcal{F}(\\boldsymbol{\\upmu}_{\\boldsymbol{\\sigma}})+\\mathbb{E}_{\\tilde{E}^{s+1}|\\Phi_k^{s+1},~x_k^*} \\left[\\left(\\tilde{\\boldsymbol{\\sigma}}-\\boldsymbol{\\upmu}_{\\boldsymbol{\\sigma}}\\right)\\right]\\cdot \\nabla\\mathcal{F}(\\boldsymbol{\\upmu}_{\\boldsymbol{\\sigma}})=~\\mathcal{F}(\\boldsymbol{\\upmu}_{\\boldsymbol{\\sigma}})\n        \\end{split}\n        \\label{eq:linearApprox}\n    \\end{equation}\nSince we define  $\\boldsymbol{\\upmu}_{\\boldsymbol{\\sigma}}=\\mathbb{E}_{\\tilde{E}^{s+1}|\\Phi_k^{s+1},~x_k^*} [\\tilde{\\boldsymbol{\\sigma}}]$, the second term in equation~\\eqref{eq:linearApprox} is equal to zero.\nThe next step, in our approach is to compute $\\boldsymbol{\\upmu}_{\\boldsymbol{\\sigma}}=\\mathbb{E}_{\\tilde{E}^{s+1}|\\Phi_k^{s+1},~x_k^*} [\\tilde{\\boldsymbol{\\sigma}}]$. Recall that according to the proposed graphical model in Figure~\\ref{fig:PGM}, for a given $\\mathcal{Y}_{x_k^*}^{s+1}$ the EEG evidence for different symbols, $A_j^{s+1}\\in \\Phi_k^{s+1} $, are independent from each other. Hence, $\\tilde{\\sigma}(A_i^{s+1})$ is independent from $\\tilde{\\sigma}(A_j^{s+1})$ $\\forall i, j=1,\\ldots, \\left|\\Phi_k^{s+1}\\right|$ such that $i\\neq j$. $\\tilde{\\sigma}(A_j^{s+1})$ is evaluated at samples from the following distributions:\n\\[ \\left\\{\\begin{array}{ll} \\tilde{e}(A^{s+1}_j)\\sim p(e(.)|1),&\\text{if}~~x_k^*\\in A^{s+1}_j\\\\\n\\tilde{e}(A^{s+1}_j)\\sim p(e(.)|0),&\\text{if}~~x_k^*\\not\\in A^{s+1}_j\n\\end{array}\\right.\\]\nHence, by defining $\\boldsymbol{\\upmu}_{\\boldsymbol{\\sigma}}=[\\hat{\\sigma}(A_1^{s+1}),~\\cdots,~\\hat{\\sigma}(A_{|\\Phi_k^{s+1}|}^{s+1})]$, such that\n\\begin{equation}\n\\small\n        \\hat{\\sigma}(A_j^{s+1})=\\left\\{\\begin{array}{ll}\n        \\widehat{\\sigma^+}=\\mathbb{E}_{e(.)|1}\n        \\left[\\frac{p\\left(e(A_j^{s+1})|1\\right)}\n        {p\\left(e(A_j^{s+1})|0\\right)}\\right],\n        &\\text{if}~ x_k^*\\in A^{s+1}_j  \\\\\\\\\n        \\widehat{\\sigma^-}=\\mathbb{E}_{e(.)|0}\\left[\n        \\frac{p\\left(e(A_j^{s+1})|1\\right)}\n        {p\\left(e(A_j^{s+1})|0\\right)}\\right],\n         &\\text{if}~  x_k^*\\not\\in A^{s+1}_j \n        \\end{array}\\right.\n        \\label{eq:pointEstims}\n    \\end{equation}\nwe can estimate  $ \\hat{g}(\\mathbf{x}, \\Phi_k^{s+1})$ \\footnote{Note that the approximation in \\eqref{eq:linearApprox}, also corresponds to defining a point estimate of the EEG scores by calculating their mean value as computed in (\\ref{eq:pointEstims}).} as follows:\n    \\begin{equation}\n    \\small\n        \\begin{split}\n            \\hat{g}(\\mathbf{x}, \\Phi_k^{s+1})=\\frac{\\Pi^{s+1}(\\text{x})\\cdot \\widehat{\\sigma^+}^{c^+_{\\text{x,x}}(\\Phi_k^{s+1})}}{\\sum_{\\text{v}\\in \\mathcal{V}}\\Pi^{s+1}(\\text{v})\\cdot \\widehat{\\sigma^+}^{c^+_{\\text{x,v}}(\\Phi_k^{s+1})}\\cdot \\widehat{\\sigma^-}^{c^-_{\\text{x,v}}(\\Phi_k^{s+1})}}\n        \\end{split}\n        \\label{eq:postpredict2}\n    \\end{equation}\nin which, \\[\\begin{split}\n            &c^+_{\\text{x,v}}(\\Phi_k^{s+1}) = \\sum_{j=1}^{|\\Phi_k^{s+1}|}{y_{\\text{x}}(A_j^{s+1})\\cdot y_{\\text{v}}(A_j^{s+1})} ~~\\text{and}\\\\\n            &c^-_{\\text{x,v}}(\\Phi_k^{s+1}) = \\sum_{j=1}^{|\\Phi_k^{s+1}|}{(1-{y}_{\\text{x}}(A_j^{s+1}))\\cdot y_{\\text{v}}(A_j^{s+1})}\n        \\end{split}\\] where $y_{x}(A_j^{s+1}) \\in \\{0,1\\}$.\nWe approximate $g(\\mathbf{x},\\Phi_k^{s+1})$ with $\\hat{g}(\\mathbf{x},\\Phi_k^{s+1})$ and rewrite the optimization problem as follows:\n    \\begin{equation}\n        \\begin{split}\n            \\widehat{\\Phi}_k^{s+1}&=\\arg\\max_{\\Phi_k^{s+1}} ~~\\mathbb{E}_{\\Pi^{s+1}(\\text{x})} \\left[ \\hat{g}(\\mathbf{x},\\Phi_k^{s+1}) \\right]\\\\\n            &=\\arg\\max_{\\Phi_k^{s+1}} ~~\\log\\left(\\mathbb{E}_{\\Pi^{s+1}(\\text{x})} \\left[ \\hat{g}(\\mathbf{x},\\Phi_k^{s+1}) \\right]\\right)\n        \\end{split}\n        \\label{eq:linearApprox2}\n    \\end{equation}\nHere since logarithm is a monotonically increasing function, taking the logarithm of the cost function does not change the solution. Then, to solve the optimization problem defined in (\\ref{eq:linearApprox2}), we define a lower-bound for the objective function using Jensen's inequality as follows:\n    \\begin{equation}\n        \\begin{split}\n            &\\log\\left(\\mathbb{E}_{\\Pi^{s+1}(\\text{x})} \\left[ \\hat{g}(\\mathbf{x},\\Phi_k^{s+1}) \\right]\\right)\\geq \\mathbb{E}_{\\Pi^{s+1}(\\text{x})} \\left[ \\log\\left(\\hat{g}(\\mathbf{x},\\Phi_k^{s+1})\\right) \\right]=\\\\\n            & \\mathbb{E}_{\\Pi^{s+1}(\\text{x})} \\left[ \\log\\left(\\frac{\\Pi^{s+1}(\\text{x})\\cdot \\widehat{\\sigma^+}^{c^+_{\\text{x,x}}(\\Phi_k^{s+1})}}{\\sum_{\\text{v}\\in \\mathcal{V}}\\Pi^{s+1}(\\text{v})\\cdot \\widehat{\\sigma^+}^{c^+_{\\text{x,v}}(\\Phi_k^{s+1})}\\cdot \\widehat{\\sigma^-}^{c^-_{\\text{x,v}}(\\Phi_k^{s+1})}}\\right)\\right]\\\\\n            & =\\mathbb{E}_{\\Pi^{s+1}(\\text{x})} \\left[\\log(\\Pi^{s+1}(\\text{x}))+c^+_{\\text{x,x}}(\\Phi_k^{s+1})\\log(\\widehat{\\sigma^+}) \\right]-\\\\\n            &\\mathbb{E}_{\\Pi^{s+1}(\\text{x})}\\left[  \\log \\left(\\sum_{\\text{v}\\in \\mathcal{V}}\\Pi^{s+1}(\\text{v})\\cdot \\widehat{\\sigma^+}^{c^+_{\\text{x,v}}(\\Phi_k^{s+1})}\\cdot \\widehat{\\sigma^-}^{c^-_{\\text{x,v}}(\\Phi_k^{s+1})}\\right)\\right]\n        \\end{split}\n    \\end{equation}\nIn our system, the class-conditional distributions are typically unimodal with small variances and different mean values, accordingly, we assume, $\\widehat{\\sigma^+}>1$ and $\\widehat{\\sigma^-}<1$. Furthermore, in our system we introduce an upper bound $|\\Phi_k^{s+1}|\\leq m_t$ to limit the number of trials in a sequence and provide a practical presentation paradigm. Then we get, \n    \\[\n        \\begin{split}\n            &\\widehat{\\sigma^+}^{c^+_{\\text{x,v}}(\\Phi_k^{s+1})}\\leq (\\widehat{\\sigma^+})^{m_t}~~\\text{and}~~\\widehat{\\sigma^-}^{c^-_{\\text{x,v}}(\\Phi_k^{s+1})}\\leq \\widehat{\\sigma^-}^{0}=1.\n        \\end{split}\n    \\]\nAs a result $\\widehat{\\sigma^+}^{c^+_{\\text{x,v}}(\\Phi_k^{s+1})}\\cdot \\widehat{\\sigma^-}^{c^-_{\\text{x,v}}(\\Phi_k^{s+1})}\\leq (\\widehat{\\sigma^+})^{m_t}$. Hence,\n    \\begin{equation}\n        \\begin{split}\n            & \\mathbb{E}_{\\Pi^{s+1}(\\text{x})} \\left[\\log(\\Pi^{s+1}(\\text{x}))+c^+_{\\text{x,x}}(\\Phi_k^{s+1})\\log(\\widehat{\\sigma^+}) \\right]-\\\\\n            &\\mathbb{E}_{\\Pi^{s+1}(\\text{x})}\\left[  \\log \\left(\\sum_{\\text{v}\\in \\mathcal{V}}\\Pi^{s+1}(\\text{v})\\cdot \\widehat{\\sigma^+}^{c^+_{\\text{x,v}}(\\Phi_k^{s+1})}\\cdot \\widehat{\\sigma^-}^{c^-_{\\text{x,v}}(\\Phi_k^{s+1})}\\right)\\right] \\geq \\\\\n            & \\mathbb{E}_{\\Pi^{s+1}(\\text{x})} \\left[\\log(\\Pi^{s+1}(\\text{x}))+c^+_{\\text{x,x}}(\\Phi_k^{s+1})\\log(\\widehat{\\sigma^+}) \\right]-\\\\\n            &\\mathbb{E}_{\\Pi^{s+1}(\\text{x})}\\left[  \\log \\left((\\widehat{\\sigma^+})^{m_t}\\right)\\right].\n        \\end{split}\n        \\label{eq:defLowerBound}\n    \\end{equation}\nNote that in~\\eqref{eq:defLowerBound}, the only term that is a function of $\\Phi_k^{s+1}$ is $c^+_{\\text{x,x}}(\\Phi_k^{s+1})$; therefore, from~\\eqref{eq:linearApprox2},    \n    \\begin{equation}\n        \\begin{split}\n            \\widehat{\\Phi}_k^{s+1}&\\approx\\arg\\max_{\\Phi_k^{s+1}} ~~Q=\\mathbb{E}_{\\Pi^{s+1}(\\text{x})} \\left[c^+_{\\text{x,x}}(\\Phi_k^{s+1})\\log(\\widehat{\\sigma^+}) \\right]\n            \\label{eq:lowerB}\n        \\end{split}\n    \\end{equation}\n\\subsection{Combinatorial Optimization}\nThe approximated objective function defined in (\\ref{eq:lowerB}) is a modular and monotonic set function $Q:2^{2^{\\mathcal{V}}}\\to \\mathbb{R}$; therefore the optimization defined in \\eqref{eq:lowerB} has guaranteed convergence properties~\\cite{nem78}. Here, we prove that $Q$ is a monotone modular set function.\\\\\n\n\\textbf{Definition 1 .} (Discrete derivative~\\cite{kra12})  \n        \n        \\textit{ Assume a set function $f:2^{\\mathcal{W}}\\to \\mathbb{R},~B\\subseteq\\mathcal{W},$ and $w\\in \\mathcal{W}$, then $\\Delta_f(w|B):=f(B\\cup\\{w\\})-f(B)$ is ``discrete derivative'' of $f$ at $B$ with respect to $w$.\n        }\\\\\n        \n\\textbf{Definition 2 .} (Modularity~\\cite{kra12})  \n        \n        \\textit{A function $f:2^{\\mathcal{W}}\\to \\mathbb{R}$ is ``modular'' if for every $B_1\\subseteq B_2\\subseteq\\mathcal{W}$ and $w\\in \\mathcal{W}\\setminus B_2$, \\[\\Delta(w|B_1)=\\Delta(w|B_2)\\] or equivalently \\[f(B_1\\cap B_2)+f(B_1\\cup B_2)=f(B_1)+f(B_2).\\]\n        }\n\n\\textbf{Lemma 1 .}  \n        \n        \\textit{Take $\\mathcal{W}=2^{\\mathcal{V}}$, then the function $Q:2^{\\mathcal{W}}\\to \\mathbb{R}$ as define in \\eqref{eq:lowerB} is a modular set function.\n        }\\\\\n\n\\small\n\\begin{proof}\nAssume $\\Phi_1\\subseteq\\Phi_2\\subseteq 2^{\\mathcal{V}}$ and $A\\in 2^{\\mathcal{V}}\\setminus \\Phi_2$, then \n    \\[\\begin{split}\n        &\\Delta_Q(A|\\Phi_1)=\\\\\n        &\\mathbb{E}_{\\Pi^{s+1}(\\text{x})} \\left[c^+_{\\text{x,x}}(\\Phi_1\\cup \\{A\\})\\log(\\widehat{\\sigma^+}) \\right]-\\mathbb{E}_{\\Pi^{s+1}(\\text{x})} \\left[c^+_{\\text{x,x}}(\\Phi_1)\\log(\\widehat{\\sigma^+}) \\right]=\\\\\n        & \\mathbb{E}_{\\Pi^{s+1}(\\text{x})}\\left[c^+_{\\text{x,x}}(\\Phi_1\\cup \\{A\\})\\log(\\widehat{\\sigma^+})-c^+_{\\text{x,x}}(\\Phi_1)\\log(\\widehat{\\sigma^+})\\right]\n    \\end{split}\\]\nSince $A\\notin\\Phi_1$, we use the definition of $c^+_{\\text{x,x}}(.)$ to write\n    \\[\\begin{split}\n        &c^+_{\\text{x,x}}(\\Phi_1\\cup \\{A\\})=c^+_{\\text{x,x}}(\\Phi_1)+c^+_{\\text{x,x}}(\\{A\\})\\Rightarrow \\Delta_Q(A|\\Phi_1)=\\\\\n        &\\mathbb{E}_{\\Pi^{s+1}(\\text{x})}\\left[(c^+_{\\text{x,x}}(\\Phi_1)+c^+_{\\text{x,x}}(\\{A\\}))\\log(\\widehat{\\sigma^+})-c^+_{\\text{x,x}}(\\Phi_1)\\log(\\widehat{\\sigma^+})\\right]=\\\\\n        & \\mathbb{E}_{\\Pi^{s+1}(\\text{x})}\\left[c^+_{\\text{x,x}}(\\{A\\})\\log(\\widehat{\\sigma^+})\\right]\n    \\end{split}\\]\nSimilarly as $A\\notin\\Phi_2$, we have \n    \\[\\begin{split}\n        &\\Delta_Q(A|\\Phi_2)=\\mathbb{E}_{\\Pi^{s+1}(\\text{x})}\\left[c^+_{\\text{x,x}}(\\{A\\})\\log(\\widehat{\\sigma^+})\\right]\\Rightarrow\\\\\n        &\\Delta_Q(A|\\Phi_1)=\\Delta_Q(A|\\Phi_2)\n    \\end{split}\\]\n\\end{proof}\n\n\\normalsize\n\\textbf{Definition 3 .} (Monotonicity~\\cite{kra12})\n\n\\textit{A set function $f:2^{\\mathcal{W}}\\to \\mathbb{R}$ is ``monotone'' if for every $B_1\\subseteq B_2\\subseteq\\mathcal{W}$, we get $f(B_1)\\leq f(B_2)$.}\\\\\n\n\\textbf{Lemma 2 .}  \n        \n        \\textit{Take $\\mathcal{W}=2^{\\mathcal{V}}$, then the function $Q:2^{\\mathcal{W}}\\to \\mathbb{R}$ as define in \\eqref{eq:lowerB} is a monotone set function.\n        }\\\\\n\n\\small\n\\begin{proof}\nAssume $\\Phi_1\\subseteq\\Phi_2\\subseteq 2^{\\mathcal{V}}$, and define $\\Phi_3=\\Phi_2\\setminus\\Phi_1$, then  $\\Phi_3\\cup\\Phi_1=\\Phi_2$ and we can write:\n    \\[\\begin{split}\n        Q(\\Phi_2)=\\mathbb{E}_{\\Pi^{s+1}(\\text{x})} \\left[c^+_{\\text{x,x}}(\\Phi_1\\cup \\Phi_3)\\log(\\widehat{\\sigma^+}) \\right]\n    \\end{split}\\]\nMoreover, $\\Phi_3\\cap\\Phi_1=\\varnothing$ then according to the definition of $c^+_{\\text{x,x}}(.)$ we have\n    \\[\\begin{split}\n         &Q(\\Phi_2)=\\mathbb{E}_{\\Pi^{s+1}(\\text{x})} \\left[(c^+_{\\text{x,x}}(\\Phi_1)+ c^+_{\\text{x,x}}(\\Phi_3))\\log(\\widehat{\\sigma^+}) \\right]=\\\\\n         &\\mathbb{E}_{\\Pi^{s+1}(\\text{x})} \\left[c^+_{\\text{x,x}}(\\Phi_1)\\log(\\widehat{\\sigma^+}) \\right]+\\mathbb{E}_{\\Pi^{s+1}(\\text{x})} \\left[ c^+_{\\text{x,x}}(\\Phi_3)\\log(\\widehat{\\sigma^+}) \\right]=\\\\\n         &Q(\\Phi_1)+\\mathbb{E}_{\\Pi^{s+1}(\\text{x})} \\left[ c^+_{\\text{x,x}}(\\Phi_3)\\log(\\widehat{\\sigma^+}) \\right]\n    \\end{split}\\]\nBased on our assumption, $\\widehat{\\sigma^+}\\geq 1\\Rightarrow \\log(\\widehat{\\sigma^+})\\geq 0$. Also due to definition, $c^+_{\\text{x,x}}(.)\\geq 0$. Hence \n    \\[\\begin{split}\n        &\\mathbb{E}_{\\Pi^{s+1}(\\text{x})} \\left[ c^+_{\\text{x,x}}(\\Phi_3)\\log(\\widehat{\\sigma^+}) \\right]\\geq 0\\Rightarrow\\\\\n        &\\mathbb{E}_{\\Pi^{s+1}(\\text{x})} \\left[ c^+_{\\text{x,x}}(\\Phi_3)\\log(\\widehat{\\sigma^+}) \\right]+Q(\\Phi_1)\\geq 0+Q(\\Phi_1)\\Rightarrow\\\\\n        &Q(\\Phi_2)\\geq Q(\\Phi_1)\n    \\end{split}\\]\n\\end{proof}\n\\normalsize\nIt has previously been shown that a deterministic greedy algorithm can provide a good approximation of the optimal solution for an $NP$-hard optimization problem with submodular and monotone objective functions within a guaranty bound from the optimal solution. Moreover, the greedy algorithm attains the optimal solution when the objective function is a modular monotone set functions~\\cite{nem78}. \n\nFor a fixed number of trials, $N_t$, in each sequence, \\emph{deterministic greedy algorithm} is described in Algorithm~\\ref{alg:Greedy}. This algorithm provides the global solution to the optimization problem defined in (\\ref{eq:lowerB})\n    \\begin{algorithm}[ht!]\n           \\caption{Greedy algorithm for maximization of $Q$}\n        \\label{alg:Greedy}\n            \\textbf{Input:} The size of sequence set $N_t$.\\\\\n            \\textbf{Output:} Estimated sequence set $\\widehat{\\Phi}_k^{s+1}$.\\\\\n            \\noindent\\rule{.5\\columnwidth}{.5pt}\\\\\n            \\SetNlSkip{.5em}\n            \\DontPrintSemicolon\n            \\tcc{\\footnotesize Initializations}\n            \\nl$\\widehat{\\Phi}_k^{s+1}\\leftarrow \\varnothing$\\;\n            \\tcc{\\footnotesize  Starting the Iterations}\n            \\nl\\For{$i=1\\to N_t$}{\n            \\tcc{\\footnotesize Adding a the next optimal $A\\in \n            2^\\mathcal{V}\\setminus \\widehat{\\Phi}_k^{s+1}$}\n            \\nl$\\widehat{\\Phi}_k^{s+1}\\leftarrow \\widehat{\\Phi}_k^{s+1}\\cup\\{{\\arg\\max_{A}~\\Delta_Q(A|\\widehat{\\Phi}_k^{s+1})}\\}$ where $A\\in\n            2^\\mathcal{V}\\setminus \\widehat{\\Phi}_k^{s+1}$.\\;\n            }\n            \\nl\\textbf{return} $\\widehat{\\Phi}_k^{s+1}$\\;\n    \\end{algorithm}\n\\section{Experimental Resrults Results}\\label{sec:expResults}\nIn this study, we used EEG data collected from 12 healthy individuals according to an IRB approved protocol for an earlier study~\\cite{mog15}. The data was acquired from 16 EEG locations: Fp1, Fp2, F3, F4, Fz, Fc1, Fc2, Cz, P1, P2, C1, C2, Cp3, Cp4, P5 and P6 according to the International 10\/20 configuration. To record the data we utilized g.USBamp bio-signal amplifier at the rate of 256Hz with active g.Butterfly electrodes. Each user performed three calibration sessions, one for each presentation paradigm i.e 1. RCP, 2.SCP and, 3. RSVP, at the presentation rate of $150$ms inter-trial-interval (ITI)\\footnote{Here the ITI is referred to the time between the onsets of two consecutive trials.}. A calibration session of our system, consists of 100 sequences and prior to each sequence we ask the user to intent a  predefined character during that sequence. These data sets were used to obtain a the system parameters to be used in Monte-Carlo simulations of our BCI.\n\nFor each calibration data set we first estimate class conditional distributions over the target and non-target EEG evidences. We use the samples drawn from these distributions to perform $20$ Monte-Carlo simulations of the system. In each simulation, the system types missing words in $10$ different phrases with different difficulty levels \\footnote{Lower levels consist of copying phrases that have letters which are assigned high probabilities by the language model. As the level increases, the language model probabilities become increasingly adversarial. Level $3$ is neutral on average.} from 1 (the easiest) to 5 (the most difficult). These phrases are selected uniformly across five difficulty levels. Here we report the results for simulated online performance of our system under proposed and baseline methods. One should note that the proposed method does not affect the calibration session as its goal is to enhance the system online performance by including the prior knowledge available in sequence design to avoid non-informative and irrelevant queries while obtaining most information about more probable choices. \n\nWe report the results in terms of: (I) total typing duration (TTD) for typing $10$ phrases --which is inversely proportional to typing speed--, and (II) probability of phrase completion (PPC) --which is a measure of typing accuracy.\n\n\\subsection{RSVP Paradigm}\\label{subsec:RSVPResults}\nWe used two sets of Monte-Carlo simulations (1) with random trial selection, and (2) with optimal query selection, to assess the effect active-RBSE on online system performance. The maximum number of sequences $m_t=8$ and number of trials within a sequence $k=14$, were selected based on our earlier experimental study~\\cite{mog15b,mog15}. \n    \\begin{figure}[ht!]\n        \\centering\n        \\includegraphics[width=.49\\textwidth]{RSVP_totalTypingDuration}\n        \\vspace{-.2in}\n        \\caption{Scatter plot of average TTD in minutes from $20$ Monte-Carlo simulations. The horizontal axis shows the TTD when the sequences are selected randomly and the vertical axis represents the TTD for the optimal sequence selection. The height of the box around every point shows the standard deviation of TTD for random sequences and the width is the variance when sequences are optimized.}\n        \\label{fig:RSVPTotalTypingDuration}\n    \\end{figure} \nThe TTD for active RSVP (ARSVP) vs. the random RSVP paradigm is presented in a scatter plot in Figure~\\ref{fig:RSVPTotalTypingDuration}. In this figure, the horizontal axis represents TTD for random RSVP and the vertical axis show the TTD for typing with ARSVP. The width and the height of the box around each data point in the scatter plot shows the standard deviation of TTD from $20$ Monte-Carlo simulations for random and optimal sequences scenarios, respectively. \nAccording to this figure $9$ out of $12$ users could achieve a higher typing speed with optimal sequence selection. Wilcoxon signed-rank test result confirms a statistically significant improvement $(P < 0.03)$ in TTD among users.\n\nIn Monte-Carlo simulations, a phrase is assumed to be successfully completed, if the system types the correct phrase in a predefined duration with no more than five consecutive mistakes; otherwise, that phrase is assumed to be incomplete. Using this setup, the PPC obtained from the simulation sets are presented in Figure~\\ref{fig:RSVPPofPhraseComp}. \n\\begin{figure}[ht!]\n        \\centering\n        \\includegraphics[width=.49\\textwidth]{PofPhraseComp_RSVP}\n         \\vspace{-.1in}\n        \\caption{Average probability of phrase completion with 90\\% confidence intervals for RSVP paradigm. The confidence interval is calculated by fitting a Beta distribution on PPC obtained from $20$ Monte-Carlo simulations.}\n        \\label{fig:RSVPPofPhraseComp}\n    \\end{figure}\nIn this figure, the horizontal axis represents the AUC values for different participants. The green ``$*$'' points represent the averaged PPC from $20$ Monte-Carlo simulations and the error-bars represent the $90\\%$ area under a beta distribution fitted to different PPCs obtained from ARSVP paradigm. Similarly, using the PPC values obtained from simulations with random RSVP paradigm, the mean PPC value, the red ``o'', and $90\\%$ standard deviations, the red bars, are computed. \n\nThe results in Figure~\\ref{fig:RSVPPofPhraseComp} show that the optimal query strategy improves the typing accuracy. This effect is more clear for the AUCs$\\in [0.7,0.9]$, and usually this range includes most of the user in healthy population. We measured the consistency of this improvement for all participants by performing Wilcoxon signed-rank test on average PPCs. The result demonstrates statistically significant improvement in PPC with $P < 0.003$.  \n\\subsection{Matrix-based Presentation Paradigm with Overlapping Trials}\nLet us define a function $\\mathbf{c}:2^{\\mathcal{V}}\\to {\\{1,0\\}}^{\\mathcal{V}}$ such that $\\mathbf{c}(A_i)=[\\mathbbm{1}\\{v_1\\in A_i\\},~\\cdots,~\\mathbbm{1}\\{v_{|\\mathcal{V}|}\\in A_i\\}]^T$. Accordingly, we define a $|\\mathcal{V}|\\times k$ code matrix $\\mathbf{C}=[\\mathbf{c}(A_1),~\\cdots,~\\mathbf{c}(A_k)]$. Then each row of the $\\mathbf{C}$ matrix assigns a code word to each member of the vocabulary set which demonstrates its presence in trials of a sequence. \n\nRCP is the most widely used matrix-based presentation paradigm in which the trials have overlaps with $A_i\\cap A_j\\leq 1,~~\\forall~i,j\\in \\{1,\\cdots,|\\Phi_k^{s+1}|\\},~i\\neq j$. If we define the number of rows and columns as $N_r$ and $N_c$ respectively, then the RCP paradigm offers unique codewords of length  $N_r$ + $N_c$ with two nonzero elements. In our experiments, we utilized a $4 \\times 7$ background matrix of characters to efficiently distribute the $28$ symbols of our vocabulary set over the space available in wide-screen layout. In this layout $N_r=4$ and $N_c=7$ and the codewords length is $11$ for the RCP paradigm. \n\nIn this study, we propose to define the search space such that each letter is uniquely identifiable from each sequence. When considering matrix presentation paradigms for ERP-based BCIs, one needs to consider some conditions for sequence set design, to satisfy the requirements for eliciting ERP. Here note that, in oppose to RSVP-based paradigms, matrix-based presentation paradigms can benefit from visual evoked potentials (VEPs) so we allowed for more frequent flashes of the same character in matrix based presentation paradigm~\\cite{che13}. This can improve the typing speed by reducing the length of sequences. Accordingly, we propose to define the feasible set such that a unique code word is assigned to each symbol while, each symbol is presented with a probability of less than $0.5$ in each sequence. \n\nWe set the codeword length $k=6$, to get enough codewords with at most $3$ non-zero elements. This setup offers ${6 \\choose 3}+{6 \\choose 2}+{6 \\choose 1}=41$ unique code words to be assigned to each member of the vocabulary set. \n    \\begin{figure}[ht!]\n        \\centering\n        \\includegraphics[width=.49\\textwidth]{RCP_totalTypingDuration}\n        \\vspace{-.2in}\n        \\caption{Scatter plot of total typing duration of $10$ phrases in terms of minutes. The horizontal axis shows the TTD when the sequences are selected based on RCP paradigm and the vertical axis represents the TTD for optimal sequence selection. The height of the box around every point shows the standard deviation of TTD for RCP paradigm and the width is the variance when sequences are optimized.}\n        \\label{fig:SCPTotalTypingDuration}\n    \\end{figure}  \nThe TTD comparison of RCP and actively learnt presentation (ALP) paradigm are presented in Figure~\\ref{fig:SCPTotalTypingDuration}. The scatter plot suggests that the ALP can offer shorter TTD. The benefit of ALP is more visible for the lower AUCs which are demonstrated by the points concentrated in the center of the figure. Although the TTD improvement due to our proposed method is clear from this figure but, we performed Wilcoxon signed-rank test between average TTDs of ALP and RCP among all participants to demonstrate statistical evidence. Result confirms the statistical significant with $P<0.0005$. \n    \\begin{figure}[ht!]\n        \\centering\n        \\includegraphics[width=.49\\textwidth]{PofPhraseComp_RCP}\n         \\vspace{-.1in}\n        \\caption{Average probability of phrase completion with 90\\% confidence intervals.}\n        \\label{fig:SCPPofPhraseComp}\n    \\end{figure}\nFigure~\\ref{fig:SCPPofPhraseComp}, shows the effect of ALP on PPC in contrast to RCP. In RCP paradigm the participants AUCs are generally higher and consequently the PPC are above $\\%95$ event without any sequence optimization. Thus our method did not offer any significant improvement for this case ($P>0.75$). As a conclusion, from Figures~\\ref{fig:SCPTotalTypingDuration} and ~\\ref{fig:SCPPofPhraseComp}, ALP can significantly reduce the TTD while preserving the PPC.\n    \nIn a real typing scenario based on the assumptions, one can further propose to present a smaller subset of vocabulary while applying more restriction on the feasible space to prevent from repetition blindness or reduce the probability of each character in a sequence. \n\\subsection{Matrix-based Presentation Paradigm with Single Character Trials}\nIn our earlier studies on optimal sequence length for RSVP paradigm, we have shown that the best typing performance can be achieved when not all letters but a subset of vocabulary is presented in each sequence~\\cite{mog15b}. The matrix SCP paradigm is closely related to RSVP paradigm in the sense that each trial consists of a single letter and each letter will presented at most once in each sequence. Hence, we assume here that the best typing performance for SCP can be achieved with sequences of length $14$, similar to RSVP paradigm. Consequently, we assess the typing performance for optimizing the sequences of length $k=14$ where $|A_i|=1$, and compare it to typing performance obtained from the standard SCP paradigm in which all the vocabulary set would be flashed in every sequence. \n\nThe results are summarized in Figures~\\ref{fig:SingleTotalTypingDuration} \\&~\\ref{fig:SinglePofPhraseComp}. Figure~\\ref{fig:SingleTotalTypingDuration} represents the scatter plot of TTD. The horizontal axis shows the TTD value for standard SCP paradigm and the vertical axis shows the TTD when the adaptive single character presentation (ASCP) paradigm is used for typing. This figure suggests that, the typing speed of a SCP paradigm can be significantly improved by optimally selecting smaller subset of characters $(P< 0.01)$.\n \\begin{figure}[ht!]\n        \\centering\n        \\includegraphics[width=.49\\textwidth]{SCP_totalTypingDuration}\n        \\vspace{-.2in}\n        \\caption{Scatter plot of TTD of $10$ phrases in terms of minutes. The horizontal axis shows the TTD when the sequences are selected based on SCP paradigm and the vertical axis represents the TTD for optimal sequence selection. The height of the box around every point shows the standard deviation of TTD for SCP paradigm and the width is the variance when sequences are optimized.}\n        \\label{fig:SingleTotalTypingDuration}\n    \\end{figure}  \nMore interestingly, the PPC comparison of ASCP and SCP as demonstrated in Figure~\\ref{fig:SinglePofPhraseComp}, presents statistically significant improvements, with $P<0.008$, when an optimized subset of characters are used instead of full vocabulary set.\n    \\begin{figure}[ht!]\n        \\centering\n        \\includegraphics[width=.49\\textwidth]{PofPhraseComp_Single}\n         \\vspace{-.1in}\n        \\caption{Average probability of phrase completion with 90\\% confidence intervals.}\n        \\label{fig:SinglePofPhraseComp}\n    \\end{figure}\n\\section{Discussion and Future Work}\nIn this manuscript, we presented the active-RBSE framework which utilizes active-learning concept to optimize query sets in a noninvasive ERP-based BCIs. This framework is a mathematical establishment of our experimental observations in RSVP paradigm. Our goal was to demonstrate the usefulness of active-RBSE and providing a rough generalization to matrix-based presentation paradigms. \n\nFor that, we used $36$ supervised data sets collected from $12$ healthy participants who utilized a language-model-assisted letter-by-letter typing interface with three different presentation paradigms of: RSVP, SCP, and RCP. Initial assessment of active-RBSE framework through Monte-Carlo simulations demonstrated that this framework offers a significant improvement in terms of typing speed and accuracy over well known existing presentation paradigms. We aim to experimentally validate these results in our future work. Also, we believe that this framework needs further improvements and analysis at least in three different categories. \n\n(I). The alternative bound on the original objective function used for solving optimizations problem does not factor-in overlaps between trials. This effect is mainly due to replacing the normalization factor with a fix upper-bound. In addition we need to provide the error introduced in the objective function value when the optimum point is estimated with this alternative bound.  \n\n(II). The objective function presented in this manuscript was a particular choice based on intuition. This also corresponds to minimizing the expected value of R\\'enyi entropy of order infinity obtained from predicted posterior PMFs. In future we can optimize the queries based on well established information theoretic concepts such as Shannon and R\\'enyi entropy for exploration and\/or exploitation. Moreover, we can relax the fixed sequence length assumption by allowing the algorithm to decide on query set size in every decision cycle. \n\n(III). This mathematical formulation allows us to consider human-in-the-loop effect both for decision making and query optimization by including physiological factors such as repetition blindness and effect of neighboring flashes which will be included in our future works.  \n\nIn our vision active-RBSE framework is not limited to EEG-based typing BCIs, we expect to utilize this framework not only under different sensor modalities and BCI applications but also for many other systems which utilize recursive querying in a sticky state estimation scenario.\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA black hole responds to test field perturbations by producing quasinormal modes (QNMs) that dissipate over time as the black hole returns to its initial state of equilibrium \\cite{DeyChakrabarti2019}. Similar to the well-known phenomenon of standing waves, these QNMs oscillate with a set of discrete resonant frequencies, known as quasinormal frequencies (QNFs), which are represented by complex-valued scalars. The real part is the physical oscillation frequency of the QNM and the imaginary part is proportional to its damping rate. Since these frequencies depend only on the physical parameters of their source, they are ideal for probing the characteristics of black holes \\cite{KonoplyaZhidenko2011, IyerWill1987}.\n\nMathematically, black hole QNMs are solutions to second-order differential equations describing the space-time evolution of perturbing fields in the vicinity of a black hole. Given that many of these equations are analytically intractable, several approximation techniques have arisen to solve them over the past decades. These methods have been outlined concisely in recent review articles such as Ref.~\\cite{KonoplyaZhidenko2011} and include a modified form of the asymptotic iteration method \\cite{Choetal2012} initiated by one of the authors of this proceeding.\n\nThis proceeding, however, is focused on investigating physics-informed neural networks (PINNs) to determine whether they could compute black hole QNMs. PINNs were recently introduced in the literature and are derived from traditional artificial neural networks. These machine learning algorithms are known for their capabilities as universal function approximators \\cite{Raissietal2019}. We have considered them in our study of black hole perturbation equations because of their capacity to solve differential equations. The key to their operation is the inclusion of physical constraints in the loss function, which is used to steer them towards accurate solutions. Note that PINNs have been shown to be more effective at solving inverse problems than forward-modelling problems where they are currently surpassed by existing numerical mesh-based techniques \\cite{Karniadarkisetal2021}. Inverse problems are differential equations with unknown parameters that can be determined from observational data of the solution. On the other hand, forward problems are well-posed differential equations that require only the boundary and\/or initial conditions of a given physics problem for the PINNs to solve them.\n\n\\section{Perturbations of spherically symmetric black holes} \n\nTo arrive at the differential equations of black hole perturbations, we begin by considering the space-times of spherically symmetric black holes given generally as \\cite{CardosoLemos2003}:\n  \n\\begin{equation}\nds^2 = -f(r) dt^2 + f(r)^{-1} dr^2 + r^2(d\\theta^2 + \\mathrm{sin}^2\\theta d\\phi^2).\n\\end{equation}\n\nHere $r$ is the radial distance from the centre of the black hole and $f(r)$ is a metric function which, for Schwarzschild-de Sitter (SdS) and Reissner-Nordstr\\\"{o}m-de Sitter (RNdS) black holes is \\cite{CardosoLemos2003, Molina2003}:\n\n\\begin{equation}\nf(r) = 1 - \\frac{2M}{r} + \\frac{Q^2}{r^2} - \\frac{\\Lambda r^3}{3},\n\\end{equation}\n\nwhere $M$ and $Q$ are the mass and electric charge of the black hole in geometrical units, and $\\Lambda$ is the cosmological constant. The SdS black hole has no electric charge; thus, the term $Q$ in the metric function vanishes. For de-Sitter black holes, the cosmological constant is positive as these space-times have positive curvature as $r \\rightarrow +\\infty$. \n\nBuilding on these black hole space-times, the perturbation equations for given perturbing fields are derived from the equations of motion of these fields. For example, the perturbation equation for an electromagnetic test field in the vicinity of a black hole is obtained from the source-free Gauss-Amp\\`{e}re law of Maxwell's equations \\cite{dInverno1992, Carroll1997}:\n\n\\begin{equation}\nF^{\\mu\\nu}_{\\ \\ ;\\nu} = \\frac{1}{\\sqrt{|g|}} \\partial_{\\nu}\\left(\\sqrt{|g|} F^{\\mu\\nu} \\right) = 0.\n\\end{equation} \n\nHere $F^{\\mu\\nu}$ is the electromagnetic field tensor and $|g|$ is the determinant of the metric tensor of the curved space-time around the black hole. When we consider metric tensors of spherically symmetric black holes, the equations of motion for perturbing fields simplify to \"Schr\\\"{o}dinger-like\" black hole perturbation equations given generally as \\cite{CardosoLemos2003}:\n\n\\begin{equation}\n\\frac{d^2 \\psi(r)}{d x^2} +\\ (\\omega^2 - V(r))\\psi(r) = 0,\n\\end{equation}\n\nwhere $x$ is the \"tortoise\" coordinate that is related to $r$ by the relation: $dr\/dx = f(r)$ \\cite{KonoplyaZhidenko2011}. The black hole effective potential, $V(r)$, for massless scalar perturbations of SdS and RNdS black holes is given as \\cite{Choetal2012, Molina2003}:\n\n\\begin{equation}\nV(r) = f\\left[ \\frac{l(l + 1)}{r^2} + \\left(\\frac{2M}{r^3} - \\frac{2Q^2}{r^4}- \\frac{2\\Lambda}{3} \\right)\\right],\n\\end{equation}\n\nwhere $l$ is the multipole number and $Q = 0$ for a SdS black hole. When black hole QNMs have negative temporal dependence they satisfy the boundary conditions given by $\\psi(x) \\sim \\mathrm{exp}(\\pm i\\omega x)$ as $x \\rightarrow \\pm \\infty$. Physically, this implies that the QNMs can only radiate inward at the horizon and outward at spatial infinity \\cite{Leaver1985, Stein2019} at a time $t > 0$.\n\nDue to the nature of the effective potential $V(r)$, equation (4) is generally difficult to solve analytically except for some special cases such as those considered here for simplicity. These space-times are the SdS and RNdS black holes in the near extremal limit where the cosmological horizons of the space-times are very close (in the $r$ coordinate) to their event horizons.\n\nRef.~\\cite{Molina2003} demonstrated that for any de-Sitter black hole space-time in the near extremal limit, the metric function $f(r)$ can be written as:\n\n\\begin{equation}\nf(r(x)) =  \\frac{(r_2 - r_1)\\kappa_1}{2 \\mathrm{cosh}^2{\\kappa_1 x}} + \\mathcal{O}(\\delta^3),  \n\\end{equation}\n\nwhere $\\delta = (r_2 - r_1)\/r_1$, $\\kappa_1$ is the surface gravity at the horizon, $r_1$ and $r_{2}$ are two consecutive positive roots of $f(r)$,  and $x$ is the tortoise coordinate whose domain lies within ($r_1$, $r_2$). For both the SdS and RNdS, $r_1$ and $r_2$ are the event and cosmological horizons, respectively, with $r_2 > r_1$.\nConsequently, in the near extremal limit the two black holes share the same mathematical expression for the metric function, which in turn results in a single expression for the effective potential valid for both SdS and RNdS black holes \\cite{CardosoLemos2003, Molina2003}:\n\n\\begin{equation}\nV(x) = \\frac{V_0}{\\cosh^2(\\kappa_{_b}x)}.\n\\end{equation}\n\n Here $V_0 = \\kappa_b^2 l(l+1)$ for massless scalar perturbations and $\\kappa_b$ is the surface gravity associated with the black hole horizon. With the effective potential in this form, the black hole perturbation equations yield analytic solutions.\n\nIn the case of massless scalar perturbations of SdS and RNdS black holes, the exact analytic expressions for the QNMs (denoted by $\\psi(x)$) and QNFs (denoted by $\\omega$) are, respectively \\cite{CardosoLemos2003, Molina2003, FerrariMashhoon1984}:\n\n\\begin{equation}\n\\psi(x) =[\\xi(\\xi -1)]^{i\\omega\/2\\kappa_{_b}} \\cdot  {}_2F_1\\left(1 + \\beta + i\\frac{\\omega}{\\kappa_{_b}}, - \\beta + i\\frac{\\omega}{\\kappa_{_b}}; 1 + i\\frac{\\omega}{\\kappa_{_b}}; \\xi\\right),\n\\end{equation}\n\n\\begin{equation}\n\\frac{\\omega}{\\kappa_{b}} = \\sqrt{\\left(l(l + 1) - \\frac{1}{4}\\right)} - i\\left(n + \\frac{1}{2}\\right),\\quad n = 0, 1, 2, ... \n\\end{equation}\n\nwhere $\\xi^{-1} = 1 + \\exp{(-2\\kappa_{_b}x)}$ and $\\beta = -1\/2 + (1\/4 - V_0\/\\kappa_{_b}^2)^{1\/2}$. \n\nWith these exact solutions, a dataset was generated to provide \"observational biases\" \\cite{Karniadarkisetal2021} for training the PINNs to solve the perturbation equations given as inverse problems.\n\nAnother problem similar to the perturbations of SdS and RNdS black holes, which was discussed in Ref.~\\cite{Choetal2012}, involves a simpler form of the inverted P\\\"{o}schl-Teller potential given as:\n\n\\begin{equation}\nV(x) = \\frac{1}{2\\mathrm{cosh}^2(x)}.\n\\end{equation}\n\nThe aforementioned boundary conditions of black hole QNMs are also satisfied by the QNMs of this problem. In this case, the exact form of the QNMs, obtained in Ref.~\\cite{ChoHo2007} using quasi-exactly solvable theory, are:\n\n\\begin{equation}\n\\psi(x) = (\\mathrm{cosh}(x))^{(i + 1)\/2}\\chi_n(\\mathrm{sinh}(x)), \n\\end{equation}\n\\begin{equation}\n\\omega_n = \\pm \\frac{1}{2} - i(n + \\frac{1}{2}),\n\\end{equation}\n\nwhere $\\chi_n$ is a polynomial of degree $n$ in $\\mathrm{sinh}(x)$. This problem and the previously mentioned one pertaining to near extremal SdS and RNdS black holes were used to test the accuracy of PINNs in computing QNFs.\n\n\\section{Computing QNMs with PINNs} \n\nThe PINN algorithm, and the steps taken to implement it on black hole perturbation equations, have two components \\cite{Karniadarkisetal2021}:\n\n\\begin{enumerate}\n    \\item A physics-uninformed artificial neural network of a particular architecture, such as a feed-forward neural network (FNN). It represents the neural network solution to the differential equation (figure 1 (left)).\n   \\item A \\textit{loss-function} that measures the deviation of the neural network solution from the physical constraints of the problem (figure 1 (right)). During the training phase, these constraints force the PINN solution to satisfy the boundary and initial conditions of the problem in addition to a labelled training dataset. At the same time, the neural network is forced to minimise a residual associated with the differential equation.\n\\end{enumerate}\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=0.9\\linewidth]{PINNmodel.pdf}\n\\small\\caption{\\  A schematic of the PINN set-up that was built to solve the perturbation equation with the effective potential given by an inverted P\\\"{o}schl-Teller potential.}    \n\\end{center}\n\\end{figure}\n\nThe algorithm for building PINN models is outlined in figure 2. Initially, the physics equations governing the problem are all defined and included as arguments for \\texttt{deepxde.data.PDE(...)}. The neural network part is set up by defining the specific neural network architecture, the number of hidden layers, nodes per layer and the non-linear activation function used in the nodes. The PINN model is the sum of the neural network (denoted by \\texttt{net} in figure 2) and the physical constraints (denoted by \\texttt{data} in figure 2) set as arguments for \\texttt{deepxde.Model}. The parameters for training, such as the specific choice for the optimiser, are set up using the function \\texttt{Model.Compile(...)}. Thereafter, the PINN model can be run for a specified number of training epochs using the command \\texttt{Model.Train(...)}.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=0.7\\linewidth]{PINNalgorithm.pdf}\n\\caption{\\ The PINN algorithm in DeepXDE \\cite{Luetal2019}.}\n\\end{center}\n\\end{figure}\n\nThe same steps were followed in customising PINNs to solve our inverse problem involving massless scalar perturbations of SdS and RNdS black holes with an inverted P\\\"{o}schl-Teller potential. Since the domain of the problem in the original form is infinite, a new co-ordinate, $y=\\mathrm{tanh}(x)$, was used to give a finite domain (-1,1) which is easier to implement in code. Consequently, the QNM boundary conditions become $\\psi(y) = \\left(1 \\mp y\\right)^{-i\\omega\/2}$, as $y \\rightarrow \\pm 1$. Thus, in terms of  $y$, the perturbation equations for the near extremal SdS and RNdS black holes are:\n\n\\begin{equation}\n \\kappa_{_b}^2 (1 - y^2)^2 \\cdot \\frac{d^2\\psi(y)}{dy^2} - 2 \\kappa_{_b}^2 y (1 - y^2) \\cdot \\frac{d\\psi(y)}{dy} + \\left[ \\omega^2 - V_0 (1 - y^2)\\right]\\psi(y) = 0.\n\\end{equation}\n\nThis equation takes the same form in the case of the problem with an effective potential given by equation (10), except that $\\kappa_{_b} = 1$ and $V_0 = 1\/2$. For easier implementation within the DeepXDE package the equations were split into real and imaginary parts. Figure 1 illustrates the PINN that was used to solve the perturbation equations. \n\nThe training hyperparameters used in all our computations were the following: FNN with 3 hidden layers, 20 nodes per layer; \\textit{tanh} as a non-linear activation function; Adam optimiser, which is a standard optimisation method in neural networks \\cite{KingmaBa2015}; 20 000 training epochs; training data consisting of 100 domain points, and a dataset with 50 uniformly distributed true values of the QNMs ($\\psi(y)$) in the \"spatial\" domain [-0.9,0.9].\n \n\\section{Results}\n\nThe results for the first inverse problem with the SdS and RNdS black holes, and the second inverse problem with an inverted P\\\"{o}schl-Teller potential given by equation (10), and are listed in tables 1 and 2, respectively. To measure the accuracy of the PINN approximations, we used the L2 relative as a metric \\cite{Luetal2019}. For the SdS and RNdS cases, we obtained at most 4 digit decimal accuracy with errors increasing with higher values of $l$. As seen in table 2, the accuracy of the PINN model for the other problem decreased with higher values of $n$.\n\n\\begin{table}[ht!]\n\\caption{\\label{Table 1} PINN approximations of the QNFs (in geometrical units) for massless scalar perturbations of SdS and RNdS black holes.}\n\\begin{center}\n\\begin{tabular}{cccccccccccr}\n\\br\n  & && \\multicolumn{2}{c}{PINN approximation}  && \\multicolumn{2}{c}{Exact \\cite{CardosoLemos2003, Molina2003}} && $L_2$ \\\\ \n$n$ & $l$ &&   $\\omega_{_{Re}}\/\\kappa_{_b}$   &   $\\omega_{_{Im}}\/\\kappa_{_b}$  &&   $\\omega_{_{Re}}\/\\kappa_{_b}$    &  $\\omega_{_{Im}}\/\\kappa_{_b}$ &&  relative error \\\\\n\\mr\n0 & 1 && 1.32287 & -0.49997 && 1.32288 & -0.50000 && 0.00214\\%\\\\ \n & 2 && 2.39790 & -0.50002 && 2.39792 & -0.50000 && 0.00106\\%\\\\ \n & 3 && 3.42761 & -0.50032 && 3.42783 & -0.50000 && 0.01105\\%\\\\ \n & 4 && 4.44096 & -4.98165 && 4.44410 & -0.50000 && 0.08128\\%\\\\\n & 5 && 5.43513 & -0.50716 && 5.45436 & -0.50000 && 0.37463\\%\\\\ \n\\br\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}[ht!]\n\\caption{\\label{Table 2} PINN approximations of the QNFs (in geometrical units) for the \"Schr\\\"{o}dinger-like\" equation with the simple effective potential given by equation (10).}\n\\begin{center}\n\\begin{tabular}{cccccccccccr}\n\\br\n && \\multicolumn{2}{c}{PINN approximation}  && \\multicolumn{2}{c}{ Exact \\cite{Choetal2012, ChoHo2007}} && $L_2$ \\\\ \n $n$ &&   $\\omega_{_{Re}}$   &   $\\omega_{_{Im}}$  && $\\omega_{_{Re}}$ & $\\omega_{_{Im}}$ &&  relative error \\\\ \n\\mr\n 0 && 0.49984 & -0.50002 && 0.50000 & -0.50000 && 0.02270\\%\\\\ \n 1 && 0.49961 & -1.49877 && 0.50000 & -1.50000 && 0.08174\\%\\\\ \n 2 && 0.49807 & -2.49714 && 0.50000 & -2.50000 && 0.13540\\%\\\\ \n 3 && 5.01544 & -3.53140 && 0.50000 & -3.50000 && 0.88922\\%\\\\\n\\br\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\section{Conclusion}\n\nThis proceeding has investigated the accuracy of PINNs in solving perturbation equations with effective potentials expressed in the form of inverted P\\\"{o}schl-Teller potentials. This work was preliminary to the larger goal of our project, which is to implement PINNs for differential equations of more general black hole perturbation scenarios. So far, our results indicate that PINNs can accurately compute QNFs, for a readily implementable set of hyperparameters. Further work on more general space-times will entail fine-tuning our choice of hyperparameters using a grid-search algorithm similar to the technique employed in Ref.~\\cite{Kadeethumetal2020}. Increasing the performance of our PINN models may ultimately require us overcoming the spectral bias that is inherent in fully connected neural networks, an attribute that renders them less accurate for higher frequency solution functions \\cite{Wangetal2020}.\n\n\\ack\nAMN is supported by the Faculty of Science at the University of Johannesburg. GEH was supported by the GES, and ASC is partially supported by the National Research Foundation South Africa.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nTransverse momentum-dependent parton distributions (TMDs) \\cite{Boer:2011fh}\nconstitute one of the pillars on which the three-dimensional tomography of\nhadrons rests. Together with the three-dimensional spatial information\nderived from generalized parton distributions (GPDs), they permit a\ncomprehensive reconstruction of hadron substructure and thus\nhave a bearing on seminal topics in hadron physics, such as\norbital angular momentum contributions to nucleon spin, or\nspin-orbit correlations in hadrons. Through the selection of\nparticular parton spin and transverse momentum components, a\nvariety of effects can be probed, including naively time-reversal odd\n(T-odd) quantities such as the Sivers and Boer-Mulders functions;\nthese only exist by virtue of initial or final state interactions\nin corresponding physical processes, introducing a preferred chronology\nin the description of the process. For example, in semi-inclusive\ndeep inelastic scattering (SIDIS), the operative element is final-state\ninteractions between the struck quark and the hadron remnant; on\nthe other hand, in the Drell-Yan (DY) process, initial state interactions\nbefore the lepton pair production enable T-odd effects. TMDs thus\nin general have to be considered in the context of specific physical\nprocesses, within a factorization framework appropriate for the\nprocess in question, separating the hard reaction from the TMD and\nother elements such as fragmentation functions. In the case of T-odd\neffects, the process-dependence manifests itself in the prediction\nof a sign change of the Sivers and Boer-Mulders functions between\nthe SIDIS and DY processes \\cite{signchange}.\n\nIn view of the fundamental importance of TMDs and the rich spectrum of\neffects that can be probed, TMDs have been, and continue to be the\ntarget of a variety of experimental efforts. Deep-inelastic scattering\nexperiments performed by COMPASS \\cite{COMPASS}, HERMES \\cite{HERMES} and\nJefferson Lab \\cite{jlab} have yielded TMD data including evidence\nfor the T-odd Sivers effect. Complementary Drell-Yan experiments at\nCOMPASS \\cite{compassdy} and Fermilab \\cite{fnaldy} are envisaged,\nwhich could, in particular, test the aforementioned sign change between\nthe SIDIS and DY processes. Related transverse single-spin\nasymmetries have been measured at RHIC in polarized proton-proton\ncollisions \\cite{BNL}. Further experimental efforts at RHIC are projected\nto provide insight into strong QCD evolution effects expected for the\nSivers TMD \\cite{rhic}. TMDs furthermore constitute a central focus\nof the proposed Electron-Ion Collider facility \\cite{eicwhite}.\n\nTo complement these efforts, providing nonperturbative QCD input from\nfirst principles to the analysis of TMD effects, a project to calculate\nTMD observables within lattice QCD was initiated and developed in\n\\cite{straightlett,straightlinks,tmdlat}. The present work constitutes\na continuation of this project. As described in detail below, the formal\ndefinition of TMDs is based on nonlocal operators, specifically quark\nbilocal operators with a gauge connection that takes the shape of a staple.\nThe path followed by the gauge connection is in principle infinite in\nlength, and thus it cannot be straightforwardly treated in terms of an\noperator product expansion, as is commonly done, e.g., for ordinary parton\ndistributions or generalized parton distributions. In view of this genuinely\nnonlocal character of the operators, lattice QCD explorations of\ncorresponding hadronic matrix elements directly at the nonlocal operator\nlevel were undertaken in \\cite{straightlett,straightlinks}, concentrating\ninitially on the simpler case of straight gauge links connecting the quark\noperators. The nonlocal nature of the operators in particular raises novel\nquestions regarding regularization and renormalization, which were\naddressed in considerable detail in \\cite{straightlinks}. Whereas\nthese questions deserve further study, the aforementioned explorations\nsuggest that it is a viable working assumption to treat nonlocal lattice\noperators in analogy to the fashion in which they are treated in continuum\nQCD \\cite{collbook}, namely, by absorbing divergences into multiplicative\nsoft factors. These soft factors can then be canceled in appropriate\nratios; this scheme was used to construct TMD observables in the\nsubsequent investigation \\cite{tmdlat}, and will be used in the\npresent work. Formally related studies of nonlocal lattice operators,\nin which a gauge link in the (Euclidean) time direction originates from\nthe propagation of a heavy auxiliary quark, have been carried out in\n\\cite{detmold}; also, a direct approach to light-cone distribution\namplitudes based on nonlocal lattice operators was laid out in\n\\cite{vbraun}. Moreover, the comprehensive framework for investigating\nparton physics within lattice QCD put forward in \\cite{Ji_1} and\ndeveloped and explored in \\cite{Ji_2,Ji_3,steffens} relies\non a direct treatment of such nonlocal lattice operators.\n\nThe present work focuses on a TMD observable related to the Boer-Mulders\neffect in a pion. Lattice QCD studies of pion structure, predominantly\nfocusing on form factors, have been previously reported in\n\\cite{bonnet,ff1,ff2,pion_GPD,ff3,ff4,ff5,ff6,ff7}. Choosing the pion as\nthe hadron state is motivated by the principal goal of the investigation\npresented here, namely, understanding the behavior of TMDs as a function\nof an evolution parameter quantifying the rapidity difference between\nthe hadron momentum and a vector describing the trajectory of the struck\nquark. Details are furnished further below. In the previous nucleon\nstudy \\cite{tmdlat}, no definite conclusions regarding the limit of\nlarge rapidity difference proved possible. By virtue of its lower mass,\nthe pion provides a larger rapidity difference at given momentum, and\nthis choice of hadron state thus aids in approaching the limit of\nphysical interest. In addition, the spinless nature of the pion permits\nadditional spatial averaging to suppress statistical uncertainties.\nIndeed, the chief advance of the present work lies in providing\nquantitative insight into the limit of large evolution parameter.\nPreliminary accounts of this work were given in\n\\cite{latt13,qcdevol14,lc2014}.\n\n\\section{Definition of TMD observables}\n\\subsection{Correlation functions}\nQuark transverse momentum-dependent parton distributions (TMDs) can be\ndefined in terms of the fundamental correlator\n\\begin{equation}\n\\widetilde{\\Phi }^{[\\Gamma ]}_{\\mbox{\\scriptsize unsubtr.} } (b,P,\\ldots )\n\\equiv \\frac{1}{2} \\langle P | \\ \\bar{q} (0) \\\n\\Gamma \\ {\\cal U} [0,\\eta v, \\eta v+b,b] \\ q(b) \\ |P\\rangle \\ \\ ,\n\\label{spacecorr}\n\\end{equation}\nwhere $P$ denotes the momentum of the hadron state; the present work\nfocuses on pions, and thus no spin is attached to the state. $\\Gamma $\nrepresents an arbitrary Dirac $\\gamma $ matrix structure. The quark\noperators at positions $0$ and $b$ are connected by the gauge link\n${\\cal U} [0,\\eta v, \\eta v+b,b]$, which connects the points listed in\nits argument by straight-line segments; thus, the gauge link has the\nshape of a staple, cf.~Fig.~\\ref{fig_2a}, with the unit vector $v$\nspecifying the staple direction and $\\eta $ its length. One is\nultimately interested in the limit $\\eta \\rightarrow \\infty $, which\nin a concrete lattice calculation is of course reached by extrapolation.\nThis gauge link form incorporates final state interactions between the\nstruck quark and the hadron remnant in semi-inclusive deep inelastic\nscattering (SIDIS) \\cite{pijlman}, and analogously initial state\ninteractions in the Drell-Yan process (DY).\n\\begin{figure}[b]\n\\centerline{\\psfig{file=StapleLink2.pdf,width=7cm }}\n\\caption{Gauge link structure ${\\cal U} [0,\\eta v, \\eta v+b,b]$ in the\ncorrelator (\\ref{spacecorr}). In a concrete lattice calculation, the\nlimit $\\eta \\rightarrow \\infty $ has to be taken numerically.}\n\\label{fig_2a}\n\\end{figure}\nThe ellipsis in the argument of\n$\\widetilde{\\Phi }^{[\\Gamma ]}_{\\mbox{\\scriptsize unsubtr.} } $ indicates\nthat the correlator will depend on various further parameters, related,\ne.g., to regularization, specified below as needed.\n\nFourier transformation of (\\ref{spacecorr}),\n\\begin{eqnarray}\n\\Phi^{[\\Gamma ]} (x,k_T ,P,\\ldots ) &=&\n\\int \\frac{d^2 b_T }{(2\\pi )^2 } \\int \\frac{d(b\\cdot P)}{(2\\pi )P^{+} }\n\\exp \\left( ix(b\\cdot P) -ib_T \\cdot k_T \\right)\n\\left. \\frac{\\widetilde{\\Phi }^{[\\Gamma ]}_{\\mbox{\\scriptsize unsubtr.} }\n(b,P,\\ldots )}{\\widetilde{\\cal S} (b_T^2 ,\\ldots )}\n\\ \\right|_{b^{+} =0} \\ \\ \\ \\ \\ \\ \\ \\ \\,\n\\label{fundcorr}\n\\end{eqnarray}\nleads to the momentum space correlator $\\Phi^{[\\Gamma ]} $, which\nultimately will be parametrized in terms of TMDs, cf.~below. The\nposition space correlator (\\ref{spacecorr}), as written, requires\nregularization not only of the quark operator self-energies, but also\nof the self-energy of the Wilson line (this is indicated by the\nsubscript ``unsubtr.''). The regularization of the latter is effected\nby dividing by the soft factor $\\widetilde{\\cal S} $. The detailed\nstructure of the soft factor depends on the concrete factorization\napproach employed. For example, in the scheme developed in\n\\cite{collbook,spl1}, which, for reasons discussed further below,\nprovides the phenomenological framework for the present study, the\nsoft factor takes the form\n\\begin{equation}\n\\widetilde{\\cal S} (b_T^2 ,\\ldots ) =\n\\sqrt{\\frac{\\tilde{S}_{(0)} (b_T , +\\infty , -\\infty )\n\\tilde{S}_{(0)} (b_T , y_s , -\\infty )}{\\tilde{S}_{(0)}\n(b_T , +\\infty , y_s )} }\n\\end{equation}\nwhere $\\tilde{S}_{(0)} (b_T ,\\ldots )$ is a vacuum expectation\nvalue of Wilson line structures extending, initially, into space-like\ndirections; some of them remain at finite rapidity $y_s $, whereas\nothers are taken to the light-cone limit, i.e., infinite rapidity.\nThis particular form of the soft factor, containing more than one\nrapidity, cannot be cast in a Lorentz frame in which it exists at\na single time, and for this reason, it is not suited for evaluation\nwithin lattice QCD. However, the observables that will be defined\nfurther below are ratios in which the soft factors cancel \\cite{tmdlat}.\nThere is, therefore, no obstacle to evaluating those observables\nwithin lattice QCD, and the detailed form of the soft factor is\nimmaterial. An alternative construction of TMD soft factors which,\nin principle, is amenable to lattice QCD evaluation has been put\nforward in \\cite{Ji_3} within the framework laid out in\n\\cite{Ji_1}.\n\nAs written in (\\ref{fundcorr}), the transverse components $b_T $\nof the quark separation $b$ are Fourier conjugate to the quark transverse\nmomentum $k_T $, while the longitudinal component $b\\cdot P$ is\nFourier conjugate to the longitudinal momentum fraction $x$. The present\nwork will be confined to the case $b\\cdot P=0$, corresponding to evaluating\nthe integral with respect to $x$ over the correlator $\\Phi^{[\\Gamma ]} $\nand TMDs derived from it. It should be stressed, however, that there is\nno obstacle to extending calculations of the type presented here to a\nscan of the $b\\cdot P$-dependence\\footnote{In a practical calculation,\nthe range of accessible $b\\cdot P$ is limited by the available $b$ and\n$P$, $|b\\cdot P| \\leq |{\\bf P} | \\sqrt{-b^2 } $ (where ${\\bf P} $ denotes\nthe spatial momentum), leading to an\nincreasing systematic uncertainty at small $x$.}, yielding upon\nFourier transformation the $x$-dependence of $\\Phi^{[\\Gamma ]} $ and\nthe TMDs under consideration. Studies of the $b\\cdot P$-dependence in\nthe straight gauge link ($\\eta v=0$) case have already been carried\nout in \\cite{straightlinks}, and further investigations in this direction\nare planned for future work. A related proposal to obtain the $x$-dependence\nof parton distributions has been put forward and explored in\n\\cite{Ji_1,Ji_2,Ji_3,steffens}.\n\nFinally, (\\ref{fundcorr}) is evaluated at $b^{+} =0$, in accordance\nwith the standard phenomenological framework, which employs a Lorentz frame\nin which the hadron of mass $m_h $ propagates with a large momentum in\n3-direction, $P^{+} \\equiv (P^0 +P^3 )\/\\sqrt{2} \\gg m_h $; then, the\nquark momentum components scale such that the correlator (\\ref{fundcorr})\nand TMDs derived from it are principally functions of the quark\nlongitudinal momentum fraction $x=k^{+} \/P^{+} $ and the quark\ntransverse momentum vector $k_T $, whereas the dependence on the\ncomponent $k^{-} \\equiv (k^0 -k^3 )\/\\sqrt{2} \\ll m_h $ becomes irrelevant\nin this limit. Correspondingly, (\\ref{fundcorr}) is regarded as having been\nintegrated over $k^{-} $, implying that the conjugate variable $b^{+} $\nis to be set to zero, as written.\n\nBefore continuing with the parametrization of $\\Phi^{[\\Gamma ]} $ in\nterms of TMDs, it is important to note a further specification with\nrespect to the staple direction $v$. The Wilson lines along the legs of\nthe staple represent an effective, resummed description of gluon exchanges\nbetween, in the case of SIDIS, the struck quark and the hadron remnant\n\\cite{pijlman}. Accordingly, their direction $v$ should be taken to follow\nthe path of the ejected quark, close to the light cone from the point of\nview of the hadron. Whereas, at tree level, there is no obstacle to the\nmost straightforward choice, namely, a light-like $v$, beyond tree level,\nthis choice is associated with rapidity divergences \\cite{rapidrev}.\nVarious schemes have been advanced to treat these divergences\n\\cite{collbook,spl1,Ji:2004wu,idilbi}, and the equivalence between\nsome of them discussed in \\cite{tmdequiv}. In particular, the\nscheme advanced in \\cite{collbook,spl1} effects regularization by\ntilting the staple direction $v$ slightly off the light cone into\nthe spacelike region. This feature is crucial for a concrete\nimplementation of a lattice QCD evaluation of matrix elements of\nthe type (\\ref{spacecorr}), as will be detailed further below.\nThus, the phenomenological framework providing the backdrop for the\npresent treatment is, specifically, the one advanced in \\cite{collbook,spl1}.\nChoosing $v$ to be spacelike for the purposes of regularization implies\na dependence of the calculation on an additional parameter, characterizing\nproximity of the staple to the light cone. Here, this parameter will be\nchosen as\n\\begin{equation}\n\\hat{\\zeta} = \\frac{v\\cdot P}{|v| |P| } \\ ,\n\\label{zetahat}\n\\end{equation}\nwhere the absolute length of a four-vector $w$ is denoted by\n$|w|=\\sqrt{| w^2 |} $. In terms of $\\hat{\\zeta} $, the light-cone limit\ncorresponds to $\\hat{\\zeta} \\rightarrow \\infty $. The generated lattice\ndata thus have to be extrapolated to the double limit\n$\\eta \\rightarrow \\infty $, $\\hat{\\zeta} \\rightarrow \\infty $,\ni.e., the limit of infinite staple length, and staple direction\nconverging toward the light cone.\n\nAlternative to this purely kinematic characterization, also the parameter\n$\\zeta = 2m_h \\hat{\\zeta} $ is frequently employed \\cite{cs},\nand viewed as a dynamical scale, to be compared to $\\Lambda_{QCD} $;\nperturbative evolution equations in $\\zeta $ can then be derived at\nsufficiently large $\\zeta $, cf., e.g., \\cite{spl2}. The question\nwhether one has reached the asymptotic regime appropriate for the\ndefinition of TMDs presumably has both kinematic and dynamical aspects.\nThe applicability of perturbation theory, e.g., for determining\nevolution equations is a dynamical issue most adequately characterized\nby considering the dimensionful parameter $\\zeta $ in relation to the\ncharacteristic QCD scale, i.e., requiring $\\zeta \\gg \\Lambda_{QCD} $.\nOn the other hand, this by itself does not necessarily guarantee\nkinematics close to the light cone, $\\hat{\\zeta} \\gg 1$. Presumably,\nboth conditions need to be taken into account in general. In the\npresent work, no quantitative connection to perturbative evolution is\nattempted; the numerical data do not reach values of $\\zeta $\nwhich lie clearly within the perturbative regime. The maximum spatial\nmomentum employed is $1.17\\, \\mbox{GeV} $, and the maximum value of\n$\\zeta $ is $\\zeta =2.1\\, \\mbox{GeV} $, corresponding to\n$\\hat{\\zeta} =2.03$. Instead, the dependence of the lattice data on the\nkinematical variable $\\hat{\\zeta} $ will be studied empirically, including\nexploration of ad hoc ans\\\"atze for the large-$\\hat{\\zeta} $\nbehavior, allowing for corresponding extrapolations. Characterizing\nthe large-$\\hat{\\zeta} $ limit is in fact the primary goal of the present\ninvestigation. This limit was seen to present a considerable challenge\nin the previous study \\cite{tmdlat}, which considered nucleon TMDs;\nno definite statements concerning the large-$\\hat{\\zeta} $ behavior\nproved possible. The present work, focusing on pions, permits accessing\nhigher $\\hat{\\zeta} $ both by virtue of the lighter hadron mass\n(note that the hadron mass enters the denominator of (\\ref{zetahat})),\nand by employing additional spatial averaging facilitated by the spinless\nnature of the pion, enhancing statistics. As will be seen further below,\nthe data extracted in the present investigation are of sufficient\nquality to yield a signal for the $\\hat{\\zeta} \\rightarrow \\infty $\nlimit of the generalized Boer-Mulders shift defined in eq.~(\\ref{gbmshift})\nbelow. This constitutes the main advance of this work.\n\n\\subsection{Parametrizations}\nReturning to the momentum space correlator $\\Phi^{[\\Gamma ]} $, its\nparametrization in terms of the relevant Lorentz structures yields,\nat leading twist,\n\\begin{eqnarray}\n\\Phi^{[\\gamma^{+} ]} (x,k_T ,P,\\ldots ) &=&\nf_1 (x,k_T ,P,\\ldots ) \\label{tmdunpol} \\\\\n\\Phi^{[i\\sigma^{i+} \\gamma^{5} ]} (x,k_T ,P,\\ldots ) &=&\n-\\frac{\\epsilon^{-+ij} k_j }{m_{\\pi } }\nh_1^{\\perp } (x,k_T ,P,\\ldots ) \\ . \\label{tmdbm}\n\\end{eqnarray}\nFor spinless particles such as the pion, there are only two leading twist\nTMDs, in contrast to the eight which arise for spin-$\\frac{1}{2} $\nparticles \\cite{tmd1,tmd2,tmd3}. The TMD $f_1 $ is simply the unpolarized\nquark distribution, whereas the Boer-Mulders function \\cite{bm}\n$h_1^{\\perp } $ encodes the distribution of transversely polarized\nquarks in the pion. The Boer-Mulders function is odd under time\nreversal (T-odd). Physically, it only exists by virtue of the final\nand initial state interactions, in the SIDIS and DY processes,\nrespectively, which break the symmetry of the processes under time\nreversal. Formally, it is the introduction of the additional vector\n$v$ describing the staple direction in the staple-shaped gauge link\nwhich breaks the symmetry; the Boer-Mulders function vanishes for\na straight gauge link, $\\eta v=0$. Correspondingly, the generalized\nBoer-Mulders shift defined in eq.~(\\ref{gbmshift}) below will be\nan odd function of $v$.\n\nOn the other hand, one can also decompose the position space correlator\n$\\widetilde{\\Phi }^{[\\Gamma ]}_{\\mbox{\\scriptsize unsubtr.} } $ into\ninvariant amplitudes \\cite{tmdlat}. The general decomposition\nis (the combinations corresponding specifically to the leading twist\nTMDs (\\ref{tmdunpol}) and (\\ref{tmdbm}) will be considered further\nbelow):\n\\begin{eqnarray}\n\\frac{1}{2} \\widetilde{\\Phi }^{[\\mathds{1}]}_{\\mbox{\\scriptsize unsubtr.} }\n&=& m_{\\pi } \\widetilde{A}_{1} \\\\\n\\frac{1}{2} \\widetilde{\\Phi }^{[\\gamma^{\\mu }]}_{\\mbox{\\scriptsize unsubtr.} }\n&=& P^{\\mu } \\widetilde{A}_{2} -im_{\\pi }^{2} b^{\\mu } \\widetilde{A}_{3}\n+\\frac{m_{\\pi }^{2} }{v\\cdot P} v^{\\mu } \\widetilde{B}_{1} \\\\\n\\frac{1}{2}\n\\widetilde{\\Phi }^{[\\gamma^{\\mu } \\gamma^{5} ]}_{\\mbox{\\scriptsize unsubtr.} }\n&=& \\frac{im_{\\pi }^{2} }{v\\cdot P} \\epsilon^{\\mu \\nu \\rho \\sigma } P_{\\nu }\nb_{\\rho } v_{\\sigma } \\widetilde{B}_{4} \\\\\n\\frac{1}{2}\n\\widetilde{\\Phi }^{[i\\sigma^{\\mu \\nu }\n\\gamma^{5} ]}_{\\mbox{\\scriptsize unsubtr.} }\n&=& im_{\\pi } \\epsilon^{\\mu \\nu \\rho \\sigma } P_{\\rho } b_{\\sigma }\n\\widetilde{A}_{4} - \\frac{m_{\\pi } }{v\\cdot P} \\epsilon^{\\mu \\nu \\rho \\sigma }\nP_{\\rho } v_{\\sigma } \\widetilde{B}_{2} + \\frac{im_{\\pi }^{3} }{v\\cdot P}\n\\epsilon^{\\mu \\nu \\rho \\sigma } b_{\\rho } v_{\\sigma } \\widetilde{B}_{3}\n\\end{eqnarray}\nThe present treatment focuses on the special case $b\\cdot P=0$, which\nin the context of TMDs, defined in a frame in which $b^{+} =0$ and\n$v_T = P_T =0$, also implies $b\\cdot v=0$ \\cite{tmdlat}.\nUnder these constraints, the above relations are readily inverted. The\namplitudes needed below are, explicitly,\n\\begin{eqnarray}\n\\widetilde{A}_{2} &=& \\frac{1}{1+\\hat{\\zeta}^{2} } \\frac{1}{2m_{\\pi }^{2} }\n\\left( P_{\\mu } - \\frac{v\\cdot P}{v^2 } v_{\\mu } \\right)\n\\widetilde{\\Phi }^{[\\gamma^{\\mu }]}_{\\mbox{\\scriptsize unsubtr.} }\n\\label{ampa2} \\\\\n\\widetilde{B}_{1} &=& \\frac{\\hat{\\zeta}^{2} }{1+\\hat{\\zeta}^{2} }\n\\frac{1}{2m_{\\pi }^{2} }\n\\left( P_{\\mu } - \\frac{m_{\\pi }^{2} }{v\\cdot P} v_{\\mu } \\right)\n\\widetilde{\\Phi }^{[\\gamma^{\\mu }]}_{\\mbox{\\scriptsize unsubtr.} }\n\\label{ampb1} \\\\\n\\widetilde{A}_{4} &=& i\\frac{1}{1+\\hat{\\zeta}^{2} }\n\\frac{1}{4b^2 m_{\\pi }^{3} }\n\\left( P^{\\kappa } - \\frac{v\\cdot P}{v^2 } v^{\\kappa } \\right)\nb^{\\lambda } \\epsilon_{\\kappa \\lambda \\mu \\nu }\n\\widetilde{\\Phi }^{[i\\sigma^{\\mu \\nu }\n\\gamma^{5} ]}_{\\mbox{\\scriptsize unsubtr.} } \\\\\n\\widetilde{B}_{3} &=& -i\\frac{\\hat{\\zeta}^{2} }{1+\\hat{\\zeta}^{2} }\n\\frac{1}{4b^2 m_{\\pi }^{3} }\n\\left( P^{\\kappa } - \\frac{m_{\\pi }^{2} }{v\\cdot P} v^{\\kappa } \\right)\nb^{\\lambda } \\epsilon_{\\kappa \\lambda \\mu \\nu }\n\\widetilde{\\Phi }^{[i\\sigma^{\\mu \\nu }\n\\gamma^{5} ]}_{\\mbox{\\scriptsize unsubtr.} }\n\\label{ampb3}\n\\end{eqnarray}\nNote that $\\widetilde{B}_{1} $ and $\\widetilde{B}_{3} $ are regular for\n$v\\cdot P \\rightarrow 0$ owing to the $\\hat{\\zeta}^{2} $ prefactor.\nOf particular interest are the leading twist objects\n\\begin{eqnarray}\n\\frac{1}{2P^{+} }\n\\widetilde{\\Phi }^{[\\gamma^{+} ]}_{\\mbox{\\scriptsize unsubtr.} } &=&\n\\widetilde{A}_{2B} \\label{amp2b} \\\\\n\\frac{1}{2P^{+} }\n\\widetilde{\\Phi }^{[i\\sigma^{i+} \\gamma^{5} ]}_{\\mbox{\\scriptsize unsubtr.} }\n&=& -im_{\\pi } \\epsilon^{-+ij} b_j \\widetilde{A}_{4B}\n\\label{amp4b}\n\\end{eqnarray}\n(where $i,j$ denote transverse spatial indices),\ngiven in terms of the amplitude combinations\n\\begin{eqnarray}\n\\widetilde{A}_{2B} &=& \\widetilde{A}_{2} +\n\\left( 1-\\sqrt{1+\\hat{\\zeta }^{-2} } \\right) \\widetilde{B}_{1} \\\\\n\\widetilde{A}_{4B} &=& \\widetilde{A}_{4} -\n\\left( 1-\\sqrt{1+\\hat{\\zeta }^{-2} } \\right) \\widetilde{B}_{3}\n\\label{a4bdef}\n\\end{eqnarray}\nAlso these combinations are regular for $\\hat{\\zeta} \\rightarrow 0$ by\nvirtue of the $\\hat{\\zeta}^{2} $ prefactors in (\\ref{ampb1}),(\\ref{ampb3}).\nNote that, in the case of vanishing spatial momentum, one cannot identify\n``forward'' and ``backward'' directions for $v$; there is then only a\nsingle branch in $|\\eta v|$, the sign of which is a matter of definition.\nAlthough only the combinations $\\widetilde{A}_{2B} $ and\n$\\widetilde{A}_{4B} $ appear in (\\ref{amp2b}),(\\ref{amp4b}),\nfor the numerical analysis to follow, it will be valuable\nto be able to consider $\\widetilde{A}_{2} $, $\\widetilde{B}_{1} $,\n$\\widetilde{A}_{4} $ and $\\widetilde{B}_{3} $ individually, not just\nthose combinations.\n\nGiven that $\\widetilde{\\Phi }^{[\\Gamma ]}_{\\mbox{\\scriptsize unsubtr.} } $\nand $\\Phi^{[\\Gamma ]} $ are related via a Fourier transformation,\nthe quantities arising in the respective decompositions\n(\\ref{tmdunpol})-(\\ref{tmdbm}) and (\\ref{amp2b})-(\\ref{amp4b}) must\nbe similarly related, i.e., the amplitudes $\\widetilde{A}_{iB} $\nmust be related to Fourier-transformed TMDs. Indeed, denoting\n$x$-moments of generic Fourier-transformed TMDs by\n\\begin{eqnarray}\n\\tilde{f}^{[m](n)} (b_T^2 ,\\ldots ) &=& n! \\left( -\\frac{2}{m_h^2 }\n\\partial_{b_T^2 } \\right)^{n} \\int_{-1}^{1} dx\\, x^{m-1}\n\\int d^2 k_T \\,\ne^{ib_T \\cdot k_T } \\ f(x,k_T^2 ,\\ldots )\n\\\\\n&=& \n\\frac{2\\pi n!}{(m_h^2)^n}\\, \\int_{-1}^{1} dx\\, x^{m-1}\n\\int d |k_T | \\, |k_T | \\,\n\\left( \\frac{|k_T |}{|b_T |} \\right)^n \nJ_n(|b_T | |k_T |) f(x, k_T^2, \\ldots )\n\\label{eq10}\n\\end{eqnarray}\nwhere $J_n $ denotes the Bessel functions of the first kind, one finds\n\\cite{tmdlat}\n\\begin{eqnarray}\n\\tilde{f}_{1}^{[1](0)} (b_T^2 , \\hat{\\zeta } ,\\ldots , \\eta v\\cdot P)\n&=& 2\\widetilde{A}_{2B} (-b_T^2 ,b\\cdot P=0, b\\cdot v=0,\n\\hat{\\zeta } ,\\eta v\\cdot P) \/ \\widetilde{\\cal S} (b_T^2 ,\\ldots )\n\\label{ampdec1} \\\\\n\\tilde{h}_{1}^{\\perp [1](1)} (b_T^2 ,\\hat{\\zeta } ,\\ldots ,\\eta v\\cdot P)\n&=& 2\\widetilde{A}_{4B} (-b_T^2 ,b\\cdot P=0, b\\cdot v=0,\n\\hat{\\zeta } ,\\eta v\\cdot P) \/ \\widetilde{\\cal S} (b_T^2 ,\\ldots )\n\\label{ampdec2}\n\\end{eqnarray}\nNote the appearance of the soft factors on the right hand sides.\n\n\\subsection{Boer-Mulders shift}\nAs already indicated further above, one obtains an observable in which\nthe soft factors cancel by forming a suitable ratio, namely, the\n``generalized Boer-Mulders shift''\n\\begin{equation}\n\\langle k_y \\rangle_{UT} (b_T^2 , \\ldots ) \\equiv m_{\\pi }\n\\frac{\\tilde{h}_{1}^{\\perp [1](1)}\n(b_T^2 ,\\ldots )}{\\tilde{f}_{1}^{[1](0)} (b_T^2 , \\ldots )}\n= m_{\\pi } \\frac{\\widetilde{A}_{4B} (-b_T^2 ,0,0,\\hat{\\zeta } ,\n\\eta v\\cdot P)}{\\widetilde{A}_{2B} (-b_T^2 ,0,0,\\hat{\\zeta } ,\n\\eta v\\cdot P)}\n\\label{gbmshift}\n\\end{equation}\nNote that ratios of this type also cancel $\\Gamma $-independent\nmultiplicative field renormalization constants attached to the quark\noperators in (\\ref{spacecorr}) at finite physical separation $b$.\nIt should be emphasized that the construction thus far is a\ncontinuum perturbative QCD construction. That this construction\ncarries across into lattice QCD, i.e., that the lattice operators\nare similarly regularized and renormalized by multiplicative soft\nfactors which cancel in ratios, is a working assumption which was\nalready explored in considerable detail in \\cite{straightlinks},\nand which will be investigated further in future work. Physically,\nthis assumption appears plausible at least at separations substantially\nlarger than the lattice spacing, where the lattice operators are\nexpected to approximate the corresponding continuum operators.\n\nTo interpret the generalized Boer-Mulders shift, note that\nthe $b_T \\rightarrow 0$ limit of the quantities defined in\n(\\ref{eq10}) formally corresponds to $k_T^{2} $-moments of TMDs,\n\\begin{eqnarray}\n\\tilde{f}^{[m](n)} (0,\\ldots ) &=& \\int_{-1}^{1} dx\\, x^{m-1}\n\\int d^2 k_T \\ \\left( \\frac{k_T^2 }{2m_h^2 }\\right )^{n} \\\nf(x,k_T^2 ,\\ldots )\n\\label{ktmom}\n\\end{eqnarray}\nThus, in the formal $b_T \\rightarrow 0$ limit, the generalized\nBoer-Mulders shift reduces to the ``Boer-Mulders shift''\n\\begin{equation}\n\\langle k_y \\rangle_{UT} (0, \\ldots ) = \nm_{\\pi } \\frac{\\tilde{h}_{1}^{\\perp [1](1)}\n(0, \\ldots )}{\\tilde{f}_{1}^{[1](0)} (0, \\ldots )} =\n\\left. \\frac{\\int dx \\int d^2 k_T \\, k_y\n\\Phi^{[\\gamma^{+} + s^j i\\sigma^{j+} \\gamma^{5} ]}\n(x,k_T ,P,\\ldots )}{\\int dx \\int d^2 k_T \\,\n\\Phi^{[\\gamma^{+} + s^j i\\sigma^{j+} \\gamma^{5} ]}\n(x,k_T ,P,\\ldots )} \\right|_{s_T =(1,0)}\n\\label{bmshift}\n\\end{equation}\nwhich, in view of the structure of the right-hand side, formally takes the\nform of the average transverse momentum in $y$-direction of quarks\npolarized in the transverse (``$T$'') $x$-direction, in an unpolarized\n(``$U$'') pion, normalized to the corresponding number of valence quarks.\nThis is in accord with the $({\\bf s}_{T} \\times {\\bf b}_{T})\\cdot {\\bf P} $\nstructure of the correlator (\\ref{amp4b}), where ${\\bf s}_{T} $ represents\nthe quark spin. The numerator in (\\ref{bmshift}) sums over the contributions\nfrom quarks and antiquarks, whereas the denominator contains the difference\nbetween quark and antiquark contributions, thus giving the number of valence\nquarks \\cite{straightlinks,muldtan}. It should be noted, however, that the\n$k_T^{2} $-moments of TMDs (\\ref{ktmom}) appearing in (\\ref{bmshift})\nare in general divergent \\cite{bacchetta} at large $k_T $ and thus\nnot well-defined absent further regularization. The generalized quantity\n(\\ref{gbmshift}) is a natural regularization, with finite $b_T $\neffectively acting as a regulator through the associated Bessel weighting,\ncf.~(\\ref{eq10}). This Bessel weighting also is advantageous in the\nanalysis of experimental asymmetries \\cite{bweight1,bweight2}. In the\npresent work, lattice QCD data for the generalized Boer-Mulders shift\n(\\ref{gbmshift}) will be obtained and presented at finite $b_T $.\nThe path by which these data can be obtained proceeds via lattice QCD\nevaluation of the fundamental correlator (\\ref{spacecorr}) for a range\nof Dirac and staple link structures, extraction of the relevant invariant\namplitudes (\\ref{ampa2})-(\\ref{ampb3}), and construction of the ratio\n(\\ref{gbmshift}). As already mentioned further above, for this lattice\nQCD calculational scheme to be viable, it is necessary to employ a\nphenomenological framework such as the one advanced in \\cite{collbook,spl1},\nin which all separations in the correlator (\\ref{spacecorr}) are spacelike,\nincluding the staple direction $v$. Such a scheme implies dependence on\nthe Collins-Soper-type evolution parameter $\\hat{\\zeta }$,\ncf.~(\\ref{zetahat}), quantifying proximity of the staple to the\nlight cone. The principal focus of the present investigation is,\nindeed, the dependence of the generalized Boer-Mulders shift on\n$\\hat{\\zeta }$, including its asymptotic $\\hat{\\zeta } \\rightarrow \\infty $\nbehavior.\n\n\\section{Lattice QCD calculations}\nLattice QCD employs a Euclidean time coordinate, serving to project\nout hadronic ground states via the associated exponentially decaying\ntime evolution. As a consequence, when evaluating matrix elements of\noperators in hadronic states, no Minkowski time separations in those\noperators can be accomodated; one is restricted to operators which\nare defined at one single time. This is the reason why it is imperative\nto employ a framework in which all separations in the fundamental\ncorrelator (\\ref{spacecorr}) are spacelike. Only in this case is\nthere no obstacle to boosting the problem to a Lorentz frame in which\nthe operator in (\\ref{spacecorr}) exists at a single time, and\nperforming the lattice calculation in that particular frame.\n\nThe decomposition of the resulting correlators into invariant\namplitudes, cf.~(\\ref{ampa2})-(\\ref{ampb3}), is a further crucial\nelement of the present treatment. Expressed in this fashion, the\nresults of the lattice calculation are immediately applicable also in\nthe original Lorentz frame in which (\\ref{spacecorr}) was initially\ndefined. Finally, as already discussed above, the construction of\nratios of amplitudes in which soft factors cancel serves to connect\nthe results to phenomenological observables such as the generalized\nBoer-Mulders shift (\\ref{gbmshift}).\n\nThe lattice QCD data for the present exploration were obtained within a\nmixed action scheme employing domain wall valence quarks on an $N_f =2+1$\ndynamical asqtad quark gauge ensemble provided by the MILC collaboration\n\\cite{milc}. Since the principal focus lies on understanding the systematics\nof the large $\\hat{\\zeta } $ limit, which proved inaccessible in previous\ninvestigations, a fairly high pion mass, $m_{\\pi}=518\\, \\mbox{MeV} $, was\nchosen for this study to alleviate statistical fluctuations. Further\ndetails of the ensemble are given in Table~\\ref{Tab_1}.\n\\begin{table}\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n$L^3\\times T$ & $a$(fm) & $am_{u,d}$ & $am_s$ &\n$m_{\\pi}^{\\mbox{\\tiny DWF} }$ (MeV) & $m_N^{\\mbox{\\tiny DWF} } $ (GeV) &\n\\#conf. & \\#meas. \\\\\n\\hline\n$20^3\\times 64$ & 0.11849(14)(99) & 0.02  & 0.05 &\n518.4(07)(49) & 1.348(09)(13) & 486 & 3888 \\\\\n\\hline\n\\end{tabular}\n\\caption{Parameters of the lattice ensemble. Note that the lattice spacing\n$a$ was determined in a different scheme than in \\cite{LHPC_1,LHPC_2}, and\nconsequently it, as well as the pion and nucleon masses quoted, differ\nslightly from the ones given in the aforementioned references \\cite{tmdlat}.\nThe first error quoted for the pion and nucleon masses is statistical, the\nsecond stems from the conversion to physical units using $a$. The bare\nasqtad quark masses are denoted $m_{u,d,s} $. Eight measurements were\nmade on each gauge configuration.}\n\\label{Tab_1}\n\\end{table}\nThis mixed action scheme, including the specific ensemble employed\nhere, has been used extensively by the LHP Collaboration for studies\nof hadron structure, cf., e.g., \\cite{LHPC_1,LHPC_2}. It also provided\nthe basis for the previous nucleon TMD investigation reported in\n\\cite{tmdlat}.\n\nTo extract the correlator (\\ref{spacecorr}), one evaluates both\nthree-point functions $C_{\\mbox{\\tiny 3pt} } $ and two-point\nfunctions $C_{\\mbox{\\tiny 2pt} } $ with pion sources and sinks\nof definite spatial momentum\\footnote{In practice, momentum\nconservation eliminates the need for projection at the source,\nprovided one projects onto zero momentum transfer at the\noperator insertion in $C_{\\mbox{\\tiny 3pt} } $ instead.} ${\\bf P} $,\n\\begin{eqnarray}\nC_{\\mbox{\\tiny 3pt} } [\\hat{O} ] (t_i ,t,t_f ,P) &=&\n\\sum_{{\\bf x}_{i} ,{\\bf x}_{f} }\ne^{-i({\\bf x}_{f} -{\\bf x}_{i} ) \\cdot {\\bf P} }\n\\langle \\phi (t_f ,{\\bf x}_{f} ) \\hat{O} (t)\n\\phi^{\\dagger } (t_i ,{\\bf x}_{i} ) \\rangle \\\\\nC_{\\mbox{\\tiny 2pt} } (t_i ,t_f ,P) &=&\n\\sum_{{\\bf x}_{i} ,{\\bf x}_{f} }\ne^{-i({\\bf x}_{f} -{\\bf x}_{i} ) \\cdot {\\bf P} }\n\\langle \\phi (t_f ,{\\bf x}_{f} )\n\\phi^{\\dagger } (t_i ,{\\bf x}_{i} ) \\rangle\n\\end{eqnarray}\nwhere $t_i $, $t$ and $t_f $ are source time, operator insertion time\nand sink time, respectively, and $\\phi $ denotes an interpolating field\nwith the quantum numbers of the pion. Wuppertal-smeared quark fields\nwere employed to construct these pion sources and sinks, which were\nseparated by $t_f -t_i = 9a$. Only connected contractions contributing to\n$C_{\\mbox{\\tiny 3pt} } $ were evaluated; disconnected contributions, which\nare expected to be small, are omitted in all results presented below.\nIn the case of the $\\pi^{+} $ meson, $u$-quark and $\\bar{d} $-quark\ndistributions coincide; contrary to the nucleon case, there is therefore\nno nontrivial $u-d$ quark combination in which disconnected contributions\nexactly cancel.\n\nThe fundamental correlator (\\ref{spacecorr}) is then obtained from\nplateaus in $t$ for $t_i \\ll t \\ll t_f $ in the three-point to two-point\nfunction ratio \\cite{bonnet},\n\\begin{equation}\n\\widetilde{\\Phi }^{[\\Gamma ]}_{\\mbox{\\scriptsize unsubtr.} } =\nE(P) \\frac{C_{\\mbox{\\tiny 3pt} } [\\hat{O} ]\n(t_i ,t,t_f ,P)}{C_{\\mbox{\\tiny 2pt} } (t_i ,t_f ,P)}\n\\label{3to2rat}\n\\end{equation}\nwhere $E(P)$ is the energy of the pion state and $\\hat{O} $ is taken to\nbe the operator in (\\ref{spacecorr}). It should be noted that, at the\nemployed source-sink separation of $t_f -t_i = 9a = 1.07\\, \\mbox{fm} $,\nsignificant excited state contaminations in the plateaus extracted from\n(\\ref{3to2rat}) cannot be excluded a priori. This issue was not\ninvestigated in the present exploratory study at the fairly high pion mass \n$m_{\\pi } =518\\, \\mbox{MeV} $. However, in future work at lower pion\nmasses, where excited state contaminations are exacerbated, it will\npresent an additional challenge.\n\nThe set of combinations of pion momenta and staple-shaped gauge link\npaths used is listed in Table~\\ref{stapletable}. It should be noted\nthat, in the case of either ${\\bf b} $ or ${\\bf v} $ extending into a\ndirection which does not coincide with a lattice axis, there is more\nthan one optimal approximation of the corresponding continuum path by\na lattice link path; e.g., if ${\\bf b} = 2({\\bf e}_{1} + {\\bf e}_{2} )$,\nwhere ${\\bf e}_{i} $ denotes the lattice link vector in $i$-direction,\nboth the sequence of links\n$({\\bf e}_{1} ,{\\bf e}_{2} ,{\\bf e}_{1} ,{\\bf e}_{2} )$ and the sequence\nof links $({\\bf e}_{2} ,{\\bf e}_{1} ,{\\bf e}_{2} ,{\\bf e}_{1} )$ equally\nwell approximate the continuum path. In such a case,\n$\\widetilde{\\Phi }^{[\\Gamma ]}_{\\mbox{\\scriptsize unsubtr.} } $ was\nalways averaged over all equivalent lattice link paths, for both\n${\\bf b} $ and ${\\bf v} $ vectors. This symmetry improvement of the\nlattice operators is important to preserve the manifest time-reversal\ntransformation properties present for the continuum staple-shaped gauge\nlink path operators.\n\nNote also that, in the mixed action scheme used for these calculations,\nbefore evaluating domain wall propagators for valence quarks, the asqtad\ngauge configurations are HYP-smeared to reduce dislocations (or rough\nfields) that would otherwise allow right-handed states on one domain\nwall to mix with left-handed states on the other domain wall. The\nlattice gauge link paths in (\\ref{spacecorr}) were constructed using\nthose same HYP-smeared gauge configurations. This has the advantageous\nconsequence that renormalization constants are closer to their tree-level\nvalues, while it would have no effect in the continuum limit. On the other\nhand, as shown in ref.~\\cite{steffens}, there are significant differences\nbetween 0 and 2 steps of HYP-smearing in the direct calculation of parton\ndistributions using straight gauge link paths instead of staples\n(see Figs.~3-5 therein), so the optimal use of HYP smearing requires\nfurther study.\n\n\\begin{table}\n\\begin{tabular}{|c|c|c|}\n\\hline\n${\\bf b} \/a $ & $\\eta {\\bf v} \/a $\n& ${\\bf P} \\cdot aL \/(2\\pi )$ \\\\\n\\hline\\hline\n$n\\cdot (0,0,1), n=-9,\\ldots,9$ &\n$\\pm n^{\\prime } \\cdot (1,0,0)$\n& $(0,0,0)$, $(-1,0,0)$, $(-2,0,0)$, \\\\\n& & $(0,-2,0)$, $(-1,-1,0)$, $(-2,-1,0)$ \\\\\n\\cline{2-3}\n& $\\pm n^{\\prime } \\cdot (0,1,0)$\n& $(0,0,0)$, $(-1,0,0)$, $(-2,0,0)$, \\\\\n& & $(0,-2,0)$, $(-1,-1,0)$, $(-2,-1,0)$ \\\\\n\\cline{2-3}\n& $\\pm n^{\\prime } \\cdot (1,1,0)$\n& $(0,0,0)$, $(-1,0,0)$, $(-2,0,0)$, \\\\\n& & $(0,-2,0)$, $(-1,-1,0)$, $(-2,-1,0)$ \\\\\n\\cline{2-3}\n& $\\pm n^{\\prime } \\cdot (1,-1,0)$\n& $(0,0,0)$, $(-1,0,0)$, $(-2,0,0)$, $(0,-2,0)$, $(-2,-1,0)$ \\\\\n\\cline{2-3}\n& $\\pm n^{\\prime } \\cdot (2,1,0)$ & $(-2,-1,0)$ \\\\\n\\cline{2-3}\n& $\\pm n^{\\prime } \\cdot (2,-1,0)$ & $(-2,-1,0)$ \\\\\n\\cline{1-3}\n$n\\cdot (0,1,0), n=-9,\\ldots,9$ &\n$\\pm n^{\\prime } \\cdot (1,0,0)$\n& $(0,0,0)$, $(-1,0,0)$, $(-2,0,0)$, $(0,0,-2)$ \\\\\n\\cline{2-3}\n& $\\pm n^{\\prime } \\cdot (0,0,1)$\n& $(0,0,0)$, $(-1,0,0)$, $(-2,0,0)$, $(0,0,-2)$ \\\\\n\\cline{2-3}\n& $\\pm n^{\\prime } \\cdot (1,0,1)$\n& $(0,0,0)$, $(-1,0,0)$, $(-2,0,0)$, $(0,0,-2)$ \\\\\n\\cline{2-3}\n& $\\pm n^{\\prime } \\cdot (1,0,-1)$\n& $(0,0,0)$, $(-1,0,0)$, $(-2,0,0)$, $(0,0,-2)$ \\\\\n\\cline{1-3}\n$n\\cdot (1,0,0), n=-9,\\ldots,9$ & \n$\\pm n^{\\prime } \\cdot (0,1,0)$\n& $(0,-2,0)$, $(0,0,-2)$ \\\\\n\\cline{2-3}\n& $\\pm n^{\\prime } \\cdot (0,0,1)$\n& $(0,-2,0)$, $(0,0,-2)$ \\\\\n\\cline{2-3}\n& $\\pm n^{\\prime } \\cdot (0,1,1)$\n& $(0,-2,0)$, $(0,0,-2)$ \\\\\n\\cline{2-3}\n& $\\pm n^{\\prime } \\cdot (0,1,-1)$\n& $(0,-2,0)$, $(0,0,-2)$ \\\\\n\\cline{1-3}\n$n\\cdot (0,1,1), n=-4,\\ldots,4$ & $\\pm n^{\\prime } \\cdot (1,0,0)$ &\n$(0,0,0)$, $(-1,0,0)$, $(-2,0,0)$ \\\\\n\\cline{1-3}\n$n\\cdot (0,1,-1), n=-4,\\ldots,4$ & $\\pm n^{\\prime } \\cdot (1,0,0)$ &\n$(0,0,0)$, $(-1,0,0)$, $(-2,0,0)$, $(-1,-1,-1)$ \\\\\n\\cline{2-3}\n& $\\pm n^{\\prime } \\cdot (0,1,1)$ &\n$(-1,-1,-1)$ \\\\\n\\cline{1-3}\n$n\\cdot (1,0,1), n=-4,\\ldots,4$ & $\\pm n^{\\prime } \\cdot (0,1,0)$ &\n$(0,-2,0)$ \\\\\n\\cline{1-3}\n$n\\cdot (1,0,-1), n=-4,\\ldots,4$ & $\\pm n^{\\prime } \\cdot (0,1,0)$ &\n$(0,-2,0)$, $(-1,-1,-1)$ \\\\\n\\cline{2-3}\n& $\\pm n^{\\prime } \\cdot (1,0,1)$ &\n$(-1,-1,-1)$ \\\\\n\\cline{1-3}\n$n\\cdot (1,1,0), n=-4,\\ldots,4$ & $\\pm n^{\\prime } \\cdot (0,0,1)$ &\n$(0,0,-2)$ \\\\\n\\cline{1-3}\n$n\\cdot (1,-1,0), n=-4,\\ldots,4$ & $\\pm n^{\\prime } \\cdot (0,0,1)$ &\n$(0,0,-2)$, $(-1,-1,0)$, $(-1,-1,-1)$ \\\\\n\\cline{2-3}\n& $\\pm n^{\\prime } \\cdot (1,1,0)$ &\n$(-1,-1,0)$, $(-1,-1,-1)$ \\\\\n\\cline{1-3}\n$n\\cdot (0,2,1), n=-3,\\ldots,3$ & $\\pm n^{\\prime } \\cdot (1,0,0)$ &\n$(0,0,0)$, $(-1,0,0)$, $(-2,0,0)$ \\\\\n\\cline{1-3}\n$n\\cdot (0,2,-1), n=-3,\\ldots,3$ & $\\pm n^{\\prime } \\cdot (1,0,0)$ &\n$(0,0,0)$, $(-1,0,0)$, $(-2,0,0)$ \\\\\n\\cline{1-3}\n$n\\cdot (0,1,2), n=-3,\\ldots,3$ & $\\pm n^{\\prime } \\cdot (1,0,0)$ &\n$(0,0,0)$, $(-1,0,0)$, $(-2,0,0)$ \\\\\n\\cline{1-3}\n$n\\cdot (0,1,-2), n=-3,\\ldots,3$ & $\\pm n^{\\prime } \\cdot (1,0,0)$ &\n$(0,0,0)$, $(-1,0,0)$, $(-2,0,0)$ \\\\\n\\cline{1-3}\n$n\\cdot (2,0,1), n=-3,\\ldots,3$ & $\\pm n^{\\prime } \\cdot (0,1,0)$ &\n$(0,-2,0)$ \\\\\n\\cline{1-3}\n$n\\cdot (2,0,-1), n=-3,\\ldots,3$ & $\\pm n^{\\prime } \\cdot (0,1,0)$ &\n$(0,-2,0)$ \\\\\n\\cline{1-3}\n$n\\cdot (1,0,2), n=-3,\\ldots,3$ & $\\pm n^{\\prime } \\cdot (0,1,0)$ &\n$(0,-2,0)$ \\\\\n\\cline{1-3}\n$n\\cdot (1,0,-2), n=-3,\\ldots,3$ & $\\pm n^{\\prime } \\cdot (0,1,0)$ &\n$(0,-2,0)$ \\\\\n\\cline{1-3}\n$n\\cdot (2,1,0), n=-3,\\ldots,3$ & $\\pm n^{\\prime } \\cdot (0,0,1)$ &\n$(0,0,-2)$ \\\\\n\\cline{1-3}\n$n\\cdot (2,-1,0), n=-3,\\ldots,3$ & $\\pm n^{\\prime } \\cdot (0,0,1)$ &\n$(0,0,-2)$ \\\\\n\\cline{1-3}\n$n\\cdot (1,2,0), n=-3,\\ldots,3$ & $\\pm n^{\\prime } \\cdot (0,0,1)$ &\n$(0,0,-2)$ \\\\\n\\cline{1-3}\n$n\\cdot (1,-2,0), n=-3,\\ldots,3$ & $\\pm n^{\\prime } \\cdot (0,0,1)$ &\n$(0,0,-2)$, $(-2,-1,0)$ \\\\\n\\cline{2-3}\n& $\\pm n^{\\prime } \\cdot (2,1,0)$ &\n$(-2,-1,0)$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Sets of staple-shaped gauge link paths and spatial pion momenta\n${\\bf P} $ used on the lattice. Gauge link paths are characterized by\nthe quark separation vector ${\\bf b} $ and the staple vector\n$\\eta {\\bf v} $, cf.~Fig.~\\ref{fig_2a}. The surveyed range of $\\eta $,\nparameterized in the table by the integer $n^{\\prime } $, was always\nchosen to extend from zero to well beyond the point where a numerical\nsignal ceases to be discernible. The maximal magnitude of the\nCollins-Soper parameter $\\hat{\\zeta} $ attained in these sets is\n$|\\hat{\\zeta} |=2.03$, for ${\\bf P} \\cdot aL\/(2\\pi)=(-2,0,0)$ paired\nwith $\\eta {\\bf v} \/a =\\pm n^{\\prime } \\cdot (1,0,0)$ and rotations\nthereof.}\n\\label{stapletable}\n\\end{table}\n\n\\section{Numerical results}\n\\subsection{SIDIS and DY limits}\nThe first step in the analysis of the obtained data concerns the behavior\nas a function of staple length $\\eta $. For ease of notation, both positive\nand negative $\\eta $ are considered for a fixed $v\\cdot P > 0$ to\ndistinguish staples oriented in the forward and backward directions with\nrespect to the pion momentum. Of particular physical interest is the\nasymptotic behavior for $\\eta \\rightarrow \\pm \\infty $, corresponding to\nthe SIDIS and DY limits. Fig.~\\ref{vseta1} displays results for the\n$u$-quark generalized Boer-Mulders shift as a function of $\\eta |v|$ at a\nfixed $\\hat{\\zeta } =1.01$, with each of the four panels corresponding\nto a successively larger transverse quark separation $|b_T |$. The T-odd\nbehavior of the observable is evident. As the SIDIS and DY limits are\napproached, a clear plateau behavior in $\\eta |v|$ is observed up to\nmoderate values of $|b_T |$; as $|b_T |$ rises, statistical uncertainties\nincrease (as indicated by the jackknife error estimates in the plots),\nand the identification of the plateaus becomes more tenuous, cf., e.g.,\nthe lower right panel, for $|b_T |=0.48\\, \\mbox{fm} $. Plateau values\nare extracted by averaging over the regions $7a \\leq \\eta |v| \\leq 9a$\nand $-7a \\geq \\eta |v| \\geq -9a$, respectively, as indicated by the fit\nlines in the plots; finally, the SIDIS and DY limits are obtained imposing\nT-oddness, i.e., the two plateau values in each plot are averaged with a\nrelative minus sign to yield the asymptotic SIDIS and DY estimates also\ndisplayed in the panels (open symbols). The asymptotic values slightly\ndecrease in magnitude as $|b_T |$ rises.\n\n\\begin{figure}\n\\psfig{file=bm_u_zetahat1p01_b0p12_vseta.pdf,width=8.5cm}\n\\hfill\n\\psfig{file=bm_u_zetahat1p01_b0p24_vseta.pdf,width=8.5cm}\n\\\\\n\\psfig{file=bm_u_zetahat1p01_b0p36_vseta.pdf,width=8.5cm}\n\\hfill\n\\psfig{file=bm_u_zetahat1p01_b0p48_vseta.pdf,width=8.5cm}\n\\caption{Generalized Boer-Mulders shift for $u$-quarks as a function of\nstaple extent $\\eta |v|$, for fixed $\\hat{\\zeta } =1.01$; the panels\nillustrate data obtained at a succession of quark separations $|b_T |$.\nPlateau fits and extraction of asymptotic values (open symbols) are\ndescribed in the main text.}\n\\label{vseta1}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\psfig{file=bm_u_sidis_vsb.pdf,width=8.5cm} }\n\\caption{Generalized Boer-Mulders shift for $u$-quarks in the SIDIS limit\nas a function of the quark separation $|b_T |$, for three different\nvalues of the Collins-Soper parameter $\\hat{\\zeta } $. The shown\nuncertainties are statistical jackknife errors.}\n\\label{vsb}\n\\end{figure}\n\nFig.~\\ref{vsb} summarizes the results obtained in the SIDIS limit\nas a function of the quark separation $|b_T |$, for three different\nvalues of the Collins-Soper parameter $\\hat{\\zeta } $. Note that\nthe data at small $|b_T |$, up to $|b_T | \\approx 2a = 0.24\\, \\mbox{fm} $,\nmay be affected by discretization artefacts, but at larger $|b_T |$,\nthe data are expected to well approximate the continuum limit. For\nlarger $\\hat{\\zeta } $, cf.~also further examples below, the\nstatistical fluctuations rapidly increase, and no useful signal\nwas obtained beyond $|b_T | =0.5\\, \\mbox{fm} $ in the case\n$\\hat{\\zeta } =2.03$. The data appear to approach well-defined\nlimits as either $|b_T |$ or $\\hat{\\zeta } $ becomes large. The\nbehavior as a function of $\\hat{\\zeta } $ will be discussed in\ngreater detail below; the behavior as $|b_T |$ becomes large seems\nplausible: Physically, once $|b_T |$ exceeds the size of the pion,\nthe correlator (\\ref{spacecorr}) cannot anymore probe correlations\ninside the pion; it only contains vacuum-vacuum and vacuum-pion\ncorrelations. The $|b_T |$-dependence of these correlations is then\nexpected to be dominated by the typical exponential fall-off with\n$|b_T |$ observed in the vacuum\\footnote{Heuristically, the expectation\nvalue of the gauge link staple, which, after integrating out the quark\nfields, may be thought of as being completed into a closed loop by the\n(fluctuating) world lines of dynamically propagating quarks, is expected\nto be determined by the chromodynamic flux piercing the loop. This is,\ne.g., the origin of the Wilson loop area law demonstrating confinement\nin Yang-Mills theory \\cite{greensite}. Once $|b_T |$ exceeds the size\nof the pion, at most one of the legs of the staple can traverse the pion;\nthe other runs entirely within the vacuum. Consider now varying $|b_T |$\nby shifting the latter leg; the area being added or removed from the\nloop lies purely within the vacuum. Only vacuum chromodynamic flux is\nbeing added or subtracted, while the chromodynamic flux influenced by\nthe pion remains fixed. Thus, the variation of the expectation value\nwith $|b_T |$ is determined purely by vacuum properties (which is not\nto say that the expectation value becomes entirely independent of the\nproperties of the pion; only its $|b_T |$-dependence does). This\nargument is unchanged if one subsequently averages over different\npositions of the leg of the staple traversing the pion.}; neglecting\nall other dependences in comparison, and canceling the dominant\nbehavior in the ratio (\\ref{gbmshift}) leads to the expectation\nof a constant asymptotic behavior in $|b_T |$.\n\nAs $\\hat{\\zeta } $ increases, one furthermore would expect the\naforementioned constant to converge to an asymptotic limit. Remarkably,\nhowever, even the $\\hat{\\zeta } =0$ data appear to already approach\nthe same large-$|b_T |$ constant as the data at higher $\\hat{\\zeta } $.\nPossibly, this may be understood as a consequence of the Lorentz\ninvariance of the vacuum; only vacuum-vacuum and vacuum-pion\ncorrelations are probed at large $|b_T |$, and it seems plausible\nthat these would be independent of the pion momentum. It would be\ndesirable to gain a more definite understanding of this property.\nThe most remarkable feature of the data, however, is the apparent\ntendency of the generalized Boer-Mulders shift to become constant\nin $|b_T |$ as $\\hat{\\zeta } $ is increased, not only for asymptotic\nvalues of $|b_T |$, but for all $|b_T |$. No obvious reason for this\nbehavior at low to intermediate $|b_T |$ is apparent, and it would be\nvery interesting to develop an understanding of it. The constant\nbehavior implies that, in the relevant transverse momentum $|k_T |$ range\ncorresponding to the probed range of $|b_T |$, the transverse momentum\nspectrum of polarized quarks is the same as for unpolarized ones.\n\n\\subsection{Evolution in $\\hat{\\zeta } $}\n\\label{zetasec}\nA special focus of the present investigation is the behavior as a function\nof $\\hat{\\zeta } $ and the large $\\hat{\\zeta } $ limit, i.e., studying\nin detail the behavior of the sequence of data seen in Fig.~\\ref{vsb}\nat a fixed value of $|b_T |$. Fig.~\\ref{vseta2} shows two further panels\nanalogous to the ones in Fig.~\\ref{vseta1}, but with $\\hat{\\zeta } $\nvarying between the panels and $|b_T | =0.36\\, \\mbox{fm} $ fixed instead.\nThus, in terms of a $\\hat{\\zeta } $ sequence, the lower left panel of\nFig.~\\ref{vseta1} lies in between the two panels displayed in\nFig.~\\ref{vseta2}. As already mentioned further above, for pion spatial\nmomentum ${\\bf P} =0$, corresponding to the left panel in Fig.~\\ref{vseta2},\nthere is only one branch as a function of $\\eta |v|$, as shown. The right\npanel in Fig.~\\ref{vseta2}, corresponding to $\\hat{\\zeta } =2.03$,\nillustrates the rapid deterioration of signal as the pion momentum is\nincreased. Nevertheless, at the moderate $|b_T |$ used here, a plateau\ncan still be extracted.\n\n\\begin{figure}\n\\psfig{file=bm_u_zetahat0_b0p36_vseta.pdf,width=8.5cm}\n\\hfill\n\\psfig{file=bm_u_zetahat2p03_b0p36_vseta.pdf,width=8.5cm}\n\\caption{Generalized Boer-Mulders shift for $u$-quarks as a function of\nstaple extent $\\eta |v|$, analogous to Fig.~\\ref{vseta1}, but at fixed\n$|b_T | =0.36\\, \\mbox{fm} $ and $\\hat{\\zeta } $ varying between the panels\ninstead. In terms of a $\\hat{\\zeta } $ sequence, the lower left panel of\nFig.~\\ref{vseta1} lies in between the two panels shown here. For pion\nspatial momentum ${\\bf P} =0$ (left panel), there is only one branch as\na function of $\\eta |v|$.}\n\\label{vseta2}\n\\end{figure}\n\n\\begin{figure}\n\\psfig{file=bm_u_sidis_b0p36_vszetahat.pdf,width=8.5cm}\n\\hfill\n\\psfig{file=bm_u_sidis_b0p34_vszetahat.pdf,width=8.5cm}\n\\caption{Generalized Boer-Mulders shift for $u$-quarks in the SIDIS limit\nas a function of the Collins-Soper parameter $\\hat{\\zeta } $, for fixed\nquark separation $|b_T | =0.36\\, \\mbox{fm} $ (left) and\n$|b_T | =0.34\\, \\mbox{fm} $ (right). Filled symbols correspond to\nthe full generalized Boer-Mulders shift, whereas open symbols correspond\nto the partial contribution obtained by replacing $\\widetilde{A}_{4B} $\nin (\\ref{gbmshift}) by $\\widetilde{A}_{4} $, for further discussion,\ncf.~main text. Spatial pion momenta ${\\bf P} $ along various directions,\nas labeled, were employed; labels in general subsume more than one\nmagnitude along the specified axis, cf.~Table~\\ref{stapletable} for the\ncomplete set of momenta for which data were generated. Data labeled\n${\\bf P} \\sim (1,0,0)$ include averaging over 90 degree rotations\nthereof, i.e., momenta along any of the three spatial lattice axes\nare included. A given value of $\\hat{\\zeta } $ can be accessed using\ndifferent pairs of ${\\bf P} $ and staple direction $v$; in such cases,\ndata corresponding to different ${\\bf P} $ directions are slightly\noffset from one another horizontally for better visibility.\nThe shown uncertainties are statistical jackknife errors.}\n\\label{vszeta3x}\n\\end{figure}\n\nFig.~\\ref{vszeta3x} summarizes all results obtained in the SIDIS limit\nat $|b_T | =0.36\\, \\mbox{fm} $ as a function of the Collins-Soper\nparameter $\\hat{\\zeta } $ (left panel), and analogous data for the\nnearby value $|b_T | =0.34\\, \\mbox{fm} $ (right panel).\nFrom the figure, the good rotational properties of the calculation\nare evident; a given value of $\\hat{\\zeta } $ can\nbe accessed using different directions of the pion spatial momentum\n${\\bf P} $ and the staple direction $v$, including both on- and off-axis\ndirections, as shown. The corresponding results coincide, indicating\nthat potential lattice artefacts are under control, and thus buttressing\nthe physical significance of the data obtained. To assess the asymptotic\nbehavior at large $\\hat{\\zeta } $, it is advantageous to consider not\nonly the full generalized Boer-Mulders shift, but also the partial\ncontribution obtained by replacing $\\widetilde{A}_{4B} $ in\n(\\ref{gbmshift}) by $\\widetilde{A}_{4} $, omitting the contribution\nfrom $\\widetilde{B}_{3} $, cf.~eq.~(\\ref{a4bdef}). Both quantities\nare displayed Fig.~\\ref{vszeta3x}; as is evident from the figure,\nthe partial $\\widetilde{A}_{4} $ contribution vanishes at\n$\\hat{\\zeta } =0$, but monotonically increases in magnitude as\n$\\hat{\\zeta } $ is increased. By contrast, the remaining contribution\nfrom $\\widetilde{B}_{3} $ to the full generalized Boer-Mulders shift,\ndominant at $\\hat{\\zeta } =0$, decreases in magnitude as $\\hat{\\zeta } $ \nrises. This matches the behavior expected from eq.~(\\ref{a4bdef}),\naccording to which the contribution from $\\widetilde{B}_{3} $ becomes\ninsignificant for $\\hat{\\zeta } \\rightarrow \\infty $ under the assumption\nthat the amplitude $\\widetilde{B}_{3} $ stays finite. Altogether, the\nfull generalized Boer-Mulders shift also decreases in magnitude as\n$\\hat{\\zeta } $ increases. Thus, the behavior of the data suggests that,\nby considering both quantities, one has access to both lower and upper\nbounds for the generalized Boer-Mulders shift, considerably increasing\nthe confidence in the extrapolations to large $\\hat{\\zeta } $ discussed\nbelow. The comparison between the partial and full quantities also permits\nan assessment of the extent to which evolution in $\\hat{\\zeta } $ has\nprogressed towards the asymptotic limit. Evidently, according to\nFig.~\\ref{vszeta3x}, already about half of the magnitude of the full\ngeneralized Boer-Mulders shift is subsumed in the partial\n$\\widetilde{A}_{4} $ contribution at $\\hat{\\zeta } \\approx 2$.\nA significant part of the $\\hat{\\zeta } $ evolution has thus already\nbeen achieved at that value of $\\hat{\\zeta } $. In this respect, the\npresent study yields a much clearer picture than was obtained in the\nprevious nucleon investigation \\cite{tmdlat}.\n\n\\begin{figure}\n\\psfig{file=bm_u_sidis_b0p36_vszetahat_extrap_pow1.pdf,width=8.5cm}\n\\hfill\n\\psfig{file=bm_u_sidis_b0p36_vszetahat_extrap_pow2.pdf,width=8.5cm}\n\\caption{Least-squares fits to the generalized Boer-Mulders shift data\nshown in Fig.~\\ref{vszeta3x} (left). Separate fits to filled and open\ndata points are shown. Left panel displays fits obtained\nwith ansatz $c+d\/\\hat{\\zeta } $; right panel displays fits obtained with\nansatz $c+d\/\\hat{\\zeta }^{2} $. Error bands show statistical jackknife\nuncertainties of the fits. Data points that lie outside the horizontal\nrange of the displayed fit curves were not included in fits; in the\ncase of the open symbols, one less data point was taken into account\nbecause it already lies in the region of an apparent inflection point,\nat which the simple asymptotic fit ansatz certainly is not applicable\nanymore. The asymptotic values displayed include the separate fits to\nthe filled and open data points (filled and open squares), as well as\nthe results of combined fits (filled circles); for details, cf.~main text.\nThe shown uncertainties are statistical jackknife errors.}\n\\label{fit1}\n\\end{figure}\n\n\\begin{figure}\n\\psfig{file=bm_u_sidis_b0p34_vszetahat_extrap_pow1.pdf,width=8.5cm}\n\\hfill\n\\psfig{file=bm_u_sidis_b0p34_vszetahat_extrap_pow2.pdf,width=8.5cm}\n\\caption{Least-squares fits to the generalized Boer-Mulders shift data\nshown in Fig.~\\ref{vszeta3x} (right). Separate fits to filled and open\ndata points are shown. Left panel displays fits obtained\nwith ansatz $c+d\/\\hat{\\zeta } $; right panel displays fits obtained with\nansatz $c+d\/\\hat{\\zeta }^{2} $. Error bands show statistical jackknife\nuncertainties of the fits. Data points that lie outside the horizontal\nrange of the displayed fit curves were not included in fits. The\nasymptotic values displayed include the separate fits to the filled\nand open data points (filled and open squares), as well as the results\nof combined fits (filled circles); for details, cf.~main text. The shown\nuncertainties are statistical jackknife errors.}\n\\label{fit2}\n\\end{figure}\n\n\\begin{figure}\n\\psfig{file=bm_u_sidis_b0p27_vszetahat_extrap_pow1.pdf,width=8.5cm}\n\\hfill\n\\psfig{file=bm_u_sidis_b0p27_vszetahat_extrap_pow2.pdf,width=8.5cm}\n\\caption{Least-squares fits to generalized Boer-Mulders shift data at\n$|b_T | =0.27\\, \\mbox{fm} $, analogous to Figs.~\\ref{fit1} and \\ref{fit2}.\nThe data were all obtained using spatial pion momenta ${\\bf P} $ along\nthe lattice axes (${\\bf P} \\sim (1,0,0)$ in the nomenclature of\nFig.~\\ref{vszeta3x}, including multiples and 90 degree rotations thereof,\ncf.~Table~\\ref{stapletable} for the complete set of momenta for which\ndata were generated). Separate fits to filled and open\ndata points are shown. Left panel displays fits obtained with ansatz\n$c+d\/\\hat{\\zeta } $; right panel displays fits obtained with ansatz\n$c+d\/\\hat{\\zeta }^{2} $. Error bands show statistical jackknife\nuncertainties of the fits. Data points that lie outside the horizontal\nrange of the displayed fit curves were not included in fits. The\nasymptotic values displayed include the separate fits to the filled\nand open data points (filled and open squares), as well as the results\nof combined fits (filled circles); for details, cf.~main text. The\nshown uncertainties are statistical jackknife errors.}\n\\label{fit3}\n\\end{figure}\n\nTo obtain quantitative statements about the large $\\hat{\\zeta } $ limit,\nleast-squares fits\\footnote{Whereas the data at higher $\\hat{\\zeta } $\ngenerally display the larger statistical uncertainties, the data at\nlower $\\hat{\\zeta } $ are likely to deviate systematically to a larger\ndegree from any putative simple asymptotic behavior. It therefore\ndoes not seem appropriate to bias the fits towards the lower\n$\\hat{\\zeta } $ data by performing a $\\chi^{2} $ fit taking into\naccount the statistical uncertainties.} to the $\\hat{\\zeta } $-dependences\nof the data were performed, using power-correction ans\\\"atze of the form\n$c+d\/\\hat{\\zeta } $ and $c+d\/\\hat{\\zeta }^{2} $. It should be noted that\nthe values of $\\hat{\\zeta } $ for which data were obtained in the present\nstudy do not reach clearly into the perturbative regime within which a\nreliable connection to perturbative evolution can be guaranteed. The\naforementioned fit functions should therefore be regarded as ad hoc\nans\\\"atze. On the one hand, separate fits to the full generalized\nBoer-Mulders shift data and to the partial $\\widetilde{A}_{4} $\ncontribution were performed; on the other hand, a combined fit to\nboth quantities with a common constant $c$ (but, of course, separate\ncoefficients $d$) was carried out. The results are shown in\nFigs.~\\ref{fit1}-\\ref{fit3}, and the asymptotic values summarized\nin Table~\\ref{fittab}. Fig.~\\ref{fit1} displays the fits to the data\nof Fig.~\\ref{vszeta3x} (left), at $|b_T | =0.36\\, \\mbox{fm} $, whereas\nFig.~\\ref{fit2} displays the fits to the data of Fig.~\\ref{vszeta3x}\n(right), at $|b_T | =0.34\\, \\mbox{fm} $. Fig.~\\ref{fit3} additionally\ndisplays fits to data obtained at a lower $|b_T |$, namely,\n$|b_T | =0.27\\, \\mbox{fm} $.\n\n\\begin{table}\n\\begin{tabular}{|c|c||c|c|c|c|}\n\\hline\n& Fit function\n& Full B.-M. shift & Contribution $\\widetilde{A}_{4} $ &\nCombined fit & RMS deviation of \\\\\n& & (GeV) & only (GeV) & (GeV) & combined fit (GeV) \\\\\n\\hline\\hline\n$|b_T |=0.36\\, \\mbox{fm} $ & $c+d\/\\hat{\\zeta } $ &\n-0.146(26) & -0.141(36) & -0.145(25) & 0.00755 \\\\\n\\hline\n$|b_T |=0.36\\, \\mbox{fm} $ & $c+d\/\\hat{\\zeta }^{2} $ &\n-0.166(16) & -0.110(22) & -0.148(15) & 0.01695 \\\\\n\\hline\\hline\n$|b_T |=0.34\\, \\mbox{fm} $ & $c+d\/\\hat{\\zeta } $ &\n-0.145(33) & -0.112(33) & -0.128(29) & 0.01466 \\\\\n\\hline\n$|b_T |=0.34\\, \\mbox{fm} $ & $c+d\/\\hat{\\zeta }^{2} $ &\n-0.157(19) & -0.084(19) & -0.121(16) & 0.02315 \\\\\n\\hline\\hline\n$|b_T |=0.27\\, \\mbox{fm} $ & $c+d\/\\hat{\\zeta } $ &\n-0.153(37) & -0.122(29) & -0.138(28) & 0.00975 \\\\\n\\hline\n$|b_T |=0.27\\, \\mbox{fm} $ & $c+d\/\\hat{\\zeta }^{2} $ &\n-0.170(25) & -0.095(20) & -0.133(19) & 0.03855 \\\\\n\\hline\n\\end{tabular}\n\\caption{Asymptotic values of least-squares fits to the\n$\\hat{\\zeta } $-dependences of the full generalized Boer-Mulders\nshift and the partial $\\widetilde{A}_{4} $ contribution separately,\nas well as a combined fit. These are the values displayed at\n$\\hat{\\zeta } =\\infty $ in Figs.~\\ref{fit1}-\\ref{fit3}. Uncertainties\nquoted are statistical jackknife errors. The right-hand column gives\nthe root mean square deviation, per degree of freedom, of the combined\nfit from the data means.}\n\\label{fittab}\n\\end{table}\n\nThe fits using the form $c+d\/\\hat{\\zeta } $ are superior to those\nusing the form $c+d\/\\hat{\\zeta }^{2} $. In the former case,\nthe asymptotic values obtained by analyzing the full generalized\nBoer-Mulders shift data and the partial $\\widetilde{A}_{4} $ contribution\nseparately agree within uncertainties, and also with the result of the\ncombined fit. In the latter case, the asymptotic values obtained with\nthe separate fits differ significantly. Not surprisingly, the results\nof the combined fits using the two fit ans\\\"atze are quite similar;\nby construction, they settle roughly in the middle between the two\nseparate quantities. Nevertheless, also in the combined fits, the\n$c+d\/\\hat{\\zeta } $ ansatz is more favorable, as evidenced by the\nlast column in Table~\\ref{fittab}, which gives the root mean square\ndeviation, per degree of freedom, of the combined fit from the data\nmeans. The deviation is considerably larger for the $c+d\/\\hat{\\zeta }^{2} $\nfits; it is of the order of, or larger than, the jackknife statistical\nuncertainty of the asymptotic value. By contrast, for the\n$c+d\/\\hat{\\zeta } $ ansatz, the deviation is considerably smaller\nthan the jackknife statistical uncertainty of the asymptotic value.\nThese observations favor the $c+d\/\\hat{\\zeta } $ fit ansatz.\n\nNote that the asymptotic values for the generalized Boer-Mulders shift\nobtained using the $c+d\/\\hat{\\zeta } $ ansatz both at\n$|b_T | =0.36\\, \\mbox{fm} $ and at $|b_T | =0.34\\, \\mbox{fm} $,\nas well as at the lower separation $|b_T | =0.27\\, \\mbox{fm} $,\nall coincide within uncertainties. These extrapolations thus do not\nmodify the observation made further above in connection with\nFig.~\\ref{vsb}, namely, that the generalized Boer-Mulders shift\nappears to become constant in $|b_T |$ as $\\hat{\\zeta } $ becomes\nlarge.\n\nAltogether, it is encouraging to observe that the data generated\nin the present study are of sufficient quality to ensure that the signal\nsurvives extrapolation to the $\\hat{\\zeta } \\rightarrow \\infty $ limit,\nwith an uncertainty as low as, roughly, 20\\% for the selected parameter\nvalues analyzed above. It is possible to tentatively discriminate\nbetween different asymptotic models. Further improvements of the\nanalysis are to be expected by confronting lattice data with\nperturbative evolution equations for the observables considered\nhere.\n\n\\subsection{Pion-proton comparison}\nIn \\cite{mb_1}, the question was posed how similar Boer-Mulders functions\nare in different hadrons, and also for different flavors. The present\nstudy permits addressing this question in passing, since Boer-Mulders\ndata are now available on the same lattice ensemble both for nucleons\n\\cite{tmdlat} and for pions, obtained here, with corresponding parameters,\nfacilitating a comparison. Fig.~\\ref{pionproton} provides a sample\njuxtaposition of the $\\pi^{+} $-meson and proton $u$-quark generalized\nBoer-Mulders shifts generated on the same ensemble, at identical transverse\nquark separation $|b_T | =0.36\\, \\mbox{fm} $, and identical spatial hadron\nmomentum ${\\bf P}$ and staple direction $v$, as a function of staple\nextent. This corresponds to identical $\\zeta = 2m_h \\hat{\\zeta } $,\nbut of course $\\hat{\\zeta } $ then differs because of the appearance\nof the hadron mass in the denominator of (\\ref{zetahat}). In this\nparticular juxtaposition, the behavior of the two shifts is seen to\nbe very similar. In the proton, the approach to the large $|\\eta |$\nplateaus appears to be slightly faster; the plateaus lie within 10\\%\nof one another. One could argue that the more appropriate comparison is one\nat identical $\\hat{\\zeta } $, and not at identical $\\zeta $, as shown here.\nHowever, using identical $\\hat{\\zeta } $ makes the correspondence hardly\nless favorable. If one interpolates the SIDIS limit pion data at\n$|b_T | =0.36\\, \\mbox{fm} $ shown in Fig.~\\ref{vszeta3x} (left) from\n$\\hat{\\zeta } =1.01$ to $\\hat{\\zeta } =0.39$, the $\\hat{\\zeta } $ value\ncorresponding to the proton data in Fig.~\\ref{pionproton} (right), then\nthe pion data are enhanced in magnitude by around 20\\%, rendering the\npion data about 10\\% higher in magnitude than the proton data, as\nopposed to 10\\% lower in the comparison at equal $\\zeta $.\n\nOf course, it should be noted that a close quantitative correspondence\nto the degree observed here for the $u$-quark generalized Boer-Mulders\nshifts is not generic. In the $\\pi^{+} $-meson, the $\\bar{d} $-quark\nBoer-Mulders shift is identical to the $u$-quark one, whereas in the\nproton, the $d$-quark Boer-Mulders shift is appreciably higher in\nmagnitude compared to the $u$-quark one \\cite{tmdlat}. The corresponding\ncomparison would therefore not reveal a similarly close quantitative\nagreement. Nevertheless, qualitative features such as the signs of the\ndifferent flavor contributions remain in correspondence.\n\n\\begin{figure}\n\\psfig{file=bm_u_zetahat1p01_b0p36_vseta_squeezed.pdf,width=8.5cm}\n\\hspace{0.63cm}\n\\psfig{file=m020_run36_U_BoerMulders.pdf,width=8.6cm}\n\\caption{Generalized Boer-Mulders shift for $u$-quarks in a\n$\\pi^{+} $-meson (left) and a proton \\cite{tmdlat} (right), as a\nfunction of staple extent $\\eta |v|$, for quark separation\n$|b_T | =0.36\\, \\mbox{fm} $. The data in the two panels were\nobtained using the same spatial hadron momentum ${\\bf P}$ and\nstaple direction $v$. This corresponds to identical\n$\\zeta = 2m_h \\hat{\\zeta } $, but, consequently, differing $\\hat{\\zeta } $,\nas labeled in the panels, cf.~main text for further discussion.\nPlateau fits and extraction of asymptotic values are analogous to\nFig.~\\ref{vseta1}. In the proton case, taken from the analysis\ndescribed in \\cite{tmdlat}, a wider fit range was used.}\n\\label{pionproton}\n\\end{figure}\n\n\\section{Summary and outlook}\nBuilding on the previous study \\cite{tmdlat} of transverse\nmomentum-dependent parton distributions in the nucleon, the present\nwork focused in particular on the evolution of TMD ratios such as\nthe generalized Boer-Mulders shift (\\ref{gbmshift}) as a function\nof the Collins-Soper parameter $\\hat{\\zeta } $, cf.~(\\ref{zetahat}).\nTo this end, the behavior of this TMD observable in a pion was\nexplored; the pion, by virtue of its lower mass, allows one to\naccess higher $\\hat{\\zeta } $, and the spinless nature of the pion\nfacilitates spatial averaging to increase statistics. Indeed, in\ncontrast to the nucleon study \\cite{tmdlat}, which did not\nyield definite conclusions concerning the large-$\\hat{\\zeta } $ limit,\nthe present investigation provides data of sufficient quality to perform\ntentative extrapolations to $\\hat{\\zeta } =\\infty $. While the generated\ndata do not reach into the $\\hat{\\zeta } $ region in which a clear\nconnection to perturbative evolution equations is given, the ad hoc\nextrapolations explored here lead to estimates of the\n$\\hat{\\zeta } =\\infty $ limit of the generalized Boer-Mulders shift\n(\\ref{gbmshift}) to within an uncertainty of as low as roughly 20\\%, for\nfavorable values of the transverse quark separation $b_T $,\ncf.~Table~\\ref{fittab}. The confidence in these extrapolations is\nbuttressed in particular by the ability to partition the generalized\nBoer-Mulders shift into two separate contributions, as discussed in\ndetail in section \\ref{zetasec}, such that one is provided with both\nan upper and a lower bound for the asymptotic behavior.\n\nHaving obtained data for the generalized Boer-Mulders shift in the pion,\nthe present study, together with \\cite{tmdlat}, in passing also permits\na juxtaposition of pion and nucleon TMD observables. A close correspondence\nbetween the $u$-quark generalized Boer-Mulders shifts in a $\\pi^{+} $-meson\nand a proton is observed, corroborating the behavior conjectured in\n\\cite{pion_GPD,mb_1}.\n\nGoing forward, the success in determining the characteristics of the\n$\\hat{\\zeta } $ evolution of the generalized Boer-Mulders shift in a pion\nsuggests that analogous results will also be accessible for nucleons in\nfuture lattice studies with higher statistics, focused on obtaining\nsignals at higher nucleon momenta. To this end, it will be useful to\nexplore schemes of generating nucleon interpolating fields with a\nfavorable overlap with higher momentum states. Further efforts currently\nin progress are concerned with the universality of TMD ratios of the\ntype considered here under variation of the lattice discretization scheme;\nthis is expected to provide empirical support for the working assumption\nemployed here, that the lattice TMD operators are regularized and\nrenormalized in analogy to the corresponding continuum operators,\nby multiplicative soft factors which cancel in TMD ratios. Moreover,\nlattice TMD calculations are in progress at lighter pion masses,\nwith a view to transferring the exploratory results obtained here to the\nphysical light quark mass regime.\n\n\\section*{Acknowledgments}\nThis work benefited from fruitful discussions with T.~Bhattacharya,\nD.~Boer, V.~Braun, M.~Diehl, R.~Edwards, R.~Gupta, X.~Ji, V.~Papavassiliou,\nS.~Pate, A.~Prokudin, J.~Qiu, T.~Rogers, X.~Wang, B.~Yoon and J.~Zhang.\nThe lattice calculations performed in this work relied on the Chroma\nsoftware suite \\cite{chroma} and employed computing resources provided by\nthe U.S.~DOE through USQCD at Jefferson Lab. Support by the\nHeisenberg-Fellowship program of the DFG (P.H.), SFB\/TRR-55 and the\nAlexander von Humboldt-Stiftung (A.S.), as well as by the U.S.~DOE through\ngrants DE-FG02-96ER40965 (M.E.) and DE-FG02-94ER40818 (J.N.), and through\ncontract DE-AC05-06OR23177, under which Jefferson Science Associates, LLC,\noperates Jefferson Laboratory (B.M.), is acknowledged.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nAbout one hundred  years have passed since R.~Fuchs showed that \nthe sixth Painlev\\'e  equation is represented as a monodromy preserving deformation \\cite{Fuchs1}. \nGarnier showed that every type of the Painlev\\'e  equations is also \nrepresented as a monodromy preserving deformation \\cite{Garnier}. \nAlthough the monodromy data is preserved by the deformations,  \nit is difficult to calculate the monodromy data in general. \n\nRiemann calculated the monodromy group for the Euler-Gauss hypergeometric equation.\nMoreover,   monodromy groups of the Euler-Gauss hypergeometric equation \nare the polyhedron  groups when it has algebraic solutions \\cite{Schw}. \nSchwarz' fifteen algebraic solutions are obtained from  simple hypergeometric \nequations by rational transformations of the independent variable. \nWe will study an analogue of Schwarz' solutions in the Painlev\\'e  case.\n\nIn this paper we call a linear equation  whose isomonodromic \ndeformation gives a Painlev\\'e  equation as {\\it the Painlev\\'e  type}. \nFor  the linear equation of the Painlev\\'e  type, we can calculate the monodromy data \nexplicitly if we substitute particular solutions of the Painlev\\'e  equations.\nHistorically, the first example of such solutions  is \nPicard's solutions of the sixth Painlev\\'e   equation \\cite{Pi}.\nR.~Fuchs calculated the monodromy group of the linear equation \ncorresponding to Picard's solutions   \\cite{Fuchs:11}. \nIt seems that R.~Fuchs' paper \\cite{Fuchs:11}, whose title is  the same  as \nthe famous paper \\cite{Fuchs1}, has been  forgotten for long years.\nRecently Mazzocco found again his result independently but \nhis paper was not referred in her paper  \\cite{Ma1}. \nIn \\cite{Fuchs:11}, R.~Fuchs proposed the following problem: \n\\begin{quote}\n{\\bf R.~Fuchs' Problem}\\quad \nLet $y(t)$ be an algebraic solution $y(t)$ of a Painlev\\'e  equation. \nFind a suitable transformation $x=x(z,t)$ such that the corresponding \nlinear differential equation \n$$\\frac{d^2 v}{dz^2}=Q(t,y(t),z)v$$\nis changed  to the form without the deformation parameter $t$:\n$$\\frac{d^2 u}{dx^2}=\\tilde{Q}(x)u.$$\nHere $v= \\sqrt{ {dz}\/{dx}}\\ u.$ (See the lemma \\ref{keylemma}.)\n\\end{quote}\nPicard's solution is algebraic if it corresponds to rational points of \nelliptic curves. For three, four and six divided points, \nthe genus of algebraic Picard's solutions is zero. \n R.~Fuchs showed that for algebraic Picard's solutions whose genus are zero  \nthe corresponding linear equations are reduced the Euler-Gauss hypergeometric \nequations by suitable rational transformations.\n \nIf R.~Fuchs' problem is true, algebraic solutions of the Painlev\\'e  equations can be considered as\na kind of a generalization of Schwarz' solutions.  Schwarz' solutions are \nconstructed by rational transformations which change hypergeometric equations \nto hypergeometric equations. Algebraic solutions of the Painlev\\'e  equations are \nconstructed by rational transformations which change hypergeometric equations \nto linear equations of the Painlev\\'e  type and we can calculate the monodromy data of \nthe linear equation explicitly.  \n\n\nRecently, Kitaev, jointing with Andreev,   constructed many algebraic solutions of the \nsixth Painlev\\'e  equation, which include  known ones and new ones, by rational transformations \n of the hypergeometric equations \\cite{AK}, \\cite{Kitaev:05}.  \nAt least now, we do not know whether R.~Fuchs' problem is true or not for the \nsixth Painlev\\'e  equation. We do not have negative example for R.~Fuchs' problem. \nKitaev's transformation is a generalization of classical work by Goursat \n\\cite{Goursat}. \nGoursat found  many rational transformations which keep the hypergeometric equations. \nHis list is incomplete and Vid\\=unas made a complete list of rational transformations \\cite{Vidunas}.\n\nIn this paper we will study a confluent version of \\cite{AK} and \nwe show that  R,~Fuchs' problem is true for the Painlev\\'e  equation from the first \nto the fifth.  \nWe will classify all  rational  transformations which change \nthe confluent hypergeometric equations to linear equations  of  \nthe Painlev\\'e  type from the first to the fifth.  Compared with the sixth Painlev\\'e  equation,\nwe obtain less transformations since the Poincar\\'e rank of irregular singularities\nof the linear equations \nof the Painlev\\'e  type is up to three. We have up to sixth order rational \nfunctions which change the confluent hypergeometric equations to linear equations of \nthe Painlev\\'e  type. \n\nWe show such rational transformations correspond  to almost all  of algebraic solutions \nof the Painlev\\'e  equations from the first to fifth up to   the B\\\"acklund transformations.\nThe cases of the degenerate fifth Painlev\\'e  equation \nand the Laguerre type solution of the  fifth Painlev\\'e  equation are  exceptional. \nWe need exponential type transformations since the monodromy data is decomposable. \n\nThe parabolic cylinder equation (the Weber equation),\nthe Bessel equation and the Airy equation are reduced to the confluent hypergeometric \nequations by rational transformations of the independent variable (see \\eqref{bessel},\n\\eqref{weber} and \\eqref{airy}). These classical formula can be considered \nas confluent version of the  Goursat transformations. Our result is \nan analogue of such classical formula  in the Painlev\\'e  analysis.\n\n\nMoreover, we obtain four non-algebraic solutions \nby rational transformations from the confluent hypergeometric equations.\nThey are called as symmetric solutions of the Painlev\\'e  equations \nwhich are fixed points of simple transformations of the Painlev\\'e  equations. \nThe symmetric solutions are not classical solutions in Umemura's sense, \nbut it is a kind of generalization of classical solutions.\n\n\nIn the section \\ref{PA}, we will review the Painlev\\'e  equations and their special solutions. \nIn the section \\ref{chg}, we will review confluent hypergeometric equations since \nwe treat degenerate confluent hypergeometric equations. \nIn the section \\ref{iso}, we will give a list of linear equations which correspond \nto the Painlev\\'e  equations. In the section  \\ref{rational}, we will show that \nR.~Fuchs' problem is true for the Painlev\\'e  equations from the first to the fifth.\n\n\nThe authors give   thanks to Professor Kazuo Okamoto, Professor Alexander Kitaev and Mr. Kazuo Kaneko for \nfruitful discussions. \n\n\\section{Remarks on the Painlev\\'e  equations}\\label{PA}\nIn order to fix the notation, we will review the Painlev\\'e  equations.\n\n\\subsection{Remarks on P2 and P34}\nThe thirty-fourth Painlev\\'e  equation P34($a$)\n$$y^{\\prime\\prime}=\\frac{(y^{\\prime })^2}{2y}+2y^2 -ty -\\frac{a}{2y} $$\nappeared in Gambier's list of second order nonlinear equations without \nmovable singularities \\cite{Gambier}. It is known that P34 is equivalent to\nthe second Painlev\\'e  equation P2($\\alpha$)\n$$y^{\\prime\\prime}= 2y^3 + t y +\\alpha.$$\nP2($\\alpha$) can be written in the Hamiltonian form \n\\begin{equation}\\label{2:ha2}\n{\\cal H}_{II}: \\\n\\left\\{\\begin{array}{l}\n   q^{\\prime } = -q^2 +p -\\frac t2, \\\\\n   p^{\\prime } = 2pq+ \\left(\\alpha+\\frac12\\right), \\\\\n\\end{array}\n\\right.\n\\end{equation}\nwith the Hamiltonian\n $$ H_{II}= {1 \\over 2}p^2- \\left(q^2+{t \\over 2} \\right) p-\\left(\\alpha+\\frac12\\right) q.$$\nIf we eliminate $p$ from \\eqref{2:ha2}, we obtain P2($\\alpha$). \nIf we eliminate $q$ from \\eqref{2:ha2}, we obtain the thirty-fourth \nPainlev\\'e  equation P34($(\\alpha+1\/2)^2$).\nTherefore P2($\\alpha$) and P34($(\\alpha+1\/2)^2$) are equivalent as nonlinear \ndifferential equations, but we will distinguish these two equations from the isomonodromic viewpoint. \n\n\\subsection{Remarks on P3 and P5}\n\nIn the following we distinguish the three types of \nthe third Painlev\\'e  equation P3$(\\alpha, \\beta, \\gamma, \\delta)$:\n\\begin{eqnarray*}\n y^{\\prime\\prime}&=& \\frac{1}{ y}{y^{\\prime}}^2-\\frac{y^{\\prime}}{ t}+\n\\frac{\\alpha y^2 + \\beta} {t}+\\gamma y^3 + \\frac{\\delta }{ y},\n\\end{eqnarray*}\nbecause they have different forms of isomonodromic deformations:\n\\begin{itemize}\n\t\\item $D_8^{(1)}$ if $\\alpha\\not=0, \\beta\\not=0$, $\\gamma=0, \\delta=0$,\n\t\\item $D_7^{(1)}$ if $\\delta=0, \\beta\\not=0$ or $\\gamma=0,  \\alpha\\not=0$,\n\t\\item $D_6^{(1)}$ if $\\gamma\\delta\\not=0$.\n\\end{itemize}\nIn the case $\\beta=0, \\delta=0$ (or $\\alpha=0, \\gamma=0$), the third Painlev\\'e  equation \nis a quadrature, and we exclude this case from the Painlev\\'e  family.  $D_j^{(1)} (j=6,7,8)$ mean \nthe affine Dynkin diagrams corresponding to Okamoto's initial value spaces.  \nIn this paper we omit the upper index $(1)$ for simplicity and denote $D_6, D_7, D_8$.\nFor details, see \\cite{KOOS}.\n By   suitable scale transformations\n$t \\to ct, y \\to dt$, we can fix $\\gamma=4, \\delta=-4$ for $D_6$. \nand $\\gamma=2$ for $D_7$. \nWe will use another form of the third Painlev\\'e  equation P3${}^\\prime(\\alpha, \\beta, \\gamma, \\delta)$ \n\\begin{eqnarray*}\nq^{\\prime\\prime}&=& \n\\frac{1}{q}{q^{\\prime}}^2-\\frac{q^{\\prime}}{x}+\n\\frac{q^2(\\alpha+\\beta)} {4x^2}+\\frac{\\gamma}{4x} + \\frac{\\delta }{4q},\n\\end{eqnarray*}\nsince P3${}^\\prime$ is more sympathetic to isomonodromic deformations than P3. \nWe can change P3 to P3${}^\\prime$ by $x=t^2, ty = q.$\n\\par\\bigskip\n\nFor the fifth equations P5$(\\alpha, \\beta, \\gamma, \\delta)$ \n\\begin{eqnarray*}\ny^{\\prime\\prime}&=& \\left(\\frac{1}{2y}+\\frac{1}{y-1} \\right){y^{\\prime}}^2-\\frac{1}{t}\n{y^{\\prime}}+\\frac{(y-1)^2}{t^2}\\left(\\alpha y +\\frac{\\beta}{y}\\right)\\\\\n&{}& \\hskip 2cm +\\gamma {y\\over t}+\\delta {y(y+1)\\over y-1},\n\\end{eqnarray*}\nwe assume that $\\delta\\not=0$. \nWhen $\\delta=0, \\gamma\\not=0$, the fifth equation is equivalent to the third equation of \nthe $D_6$ type \\cite{Gromak}. We denote deg-P5$(\\alpha, \\beta, \\gamma, 0)$ if $\\delta=0$. \nBy a suitable scale transformation \n$t \\to ct$, we can fix $\\delta=-1\/2$ for P5 and  $\\gamma=-2$ for deg-P5. \nLet $q$ is a solution of  P3${}^\\prime (4(\\alpha_1-\\beta_1),-4(\\alpha_1+\\beta_1-1), 4, \\ -4)$.\nThen\n$$ y =\\frac{tq'-q^2-(\\alpha_1+\\beta_1)q-t}{tq'+q^2-(\\alpha_1+\\beta_1)q-t}$$\nis a solution of  deg-P5$( {\\alpha_1^2}\/2, -{\\beta_1^2}\/2, -2,0)$. \n\n\nP5$(\\alpha, \\beta, 0, 0)$ is quadrature and we exclude this case from the Painlev\\'e  family.\n\n\\subsection{Special solutions from P1 to P5}\nWe will review special solutions of the  Painlev\\'e  equations. \nAlthough generic solutions of the  Painlev\\'e  equations are transcendental, \nthere exist some special solutions which reduce to known functions.\nUmemura defined a class of `known functions', which are\ncalled {\\it classical solutions} \\cite{U:stras}. He also gave a method \nhow to classify classical solutions of the Painlev\\'e  equations. \n\nThere exist two types of classical solutions of Painlev\\'e  equations, \none is algebraic and the second is the Riccati type. \nUp to now, all of classical solutions  are classified \nexcept algebraic  solutions of the sixth Painlev\\'e  equations. \nWe will list up all of classical solutions of P1 to P5. \n\n\\begin{theorem}\\label{classical}\n1) All solutions of P1 are transcendental.\n\n\\par\\medskip\\noindent\n2) P2($0$) has a rational solution $y=0$. \nP2($-1\/2$) has a Riccati type solution $y=-u'\/u$. \nHere $u$ is any solution of the Airy equation $ u^{\\prime\\prime}+tu\/2=0$.\n\n\\par\\medskip\\noindent\n2) P34($(\\alpha+1\/2)^2$)  is equivalent to P2($\\alpha$).  \nP34($1\/4$) has a rational solution $y=t\/2$. P34($1$) has the Riccati type solutions. \n\n\\par\\medskip\\noindent\n4) P4($0,-2\/9$) has a rational solution $y=-2t\/3$.  \nP4($1-s, -2s^2$) has a Riccati type solution $y=-u'\/u$. \nHere $u$ is any solution of the Hermite-Weber equation \n$ u^{\\prime\\prime}+2t u^{\\prime}+2s u  =0$. If $s=1$, \nP4($0,-2$) has a rational solution $y=-2t$, which reduces to the \nHermite polynomial.\n\n\\par\\medskip\\noindent\n5) P3${}^{\\prime}(D_6)$($a,-a, 4,-4$)  has an algebraic solution $y=-\\sqrt{t}$.\nP3$^{\\prime}(D_6)$($4h, 4(h+1),4,-4$) has a Riccati type solution $y=u'\/u$.\nHere $u$ is any solution  of\n$ tu^{\\prime\\prime}+(2h+1) u^{\\prime}-4t u =0$. \n\n\\par\\medskip\\noindent\n6) P3${}^\\prime$($D_7$)($\\alpha, \\beta, \\gamma, 0$) does not have the Riccati type solution. \nP3${}^\\prime$($D_7$)($0, -2, 2,0$) has an algebraic solution $y=  t^{1\/3}$. \n\n\\par\\medskip\\noindent\n7) P3${}^\\prime$($D_8$)($\\alpha, \\beta, 0, 0$) does not have the Riccati type solution. \nP3${}^\\prime$($D_8$) ($8h,$ $-8h,0,0$) has an algebraic solution $y= -\\sqrt{t} $. \n\n\\par\\medskip\\noindent\n8) P5($a, -a, 0, \\delta$) has a rational solution $y=-1$.  \nP5($ (\\kappa_0+s)^2\/2, -\\kappa_0^2\/2, -(s+1), -1\/2$) has the Riccati type solutions \n$y=-tu'\/ (\\kappa_0+s)u$. \nHere $u$ is any solution of \n$ t^2 u^{\\prime\\prime}+ t( t-s-2\\kappa_0+1) u^{\\prime}+\\kappa_0(\\kappa_0+s) u 2=0$. \nIf $\\kappa_0=1 $, \nP5($ (s+1)^2\/2, -1\/2, -(s+1), -1\/2$) has a rational solution $y= t\/(s+1)+1$, \nwhich reduces to the Laguerre polynomial.\n\n\\par\\medskip\\noindent\n9) deg-P5$( {\\alpha_1^2}\/2, -{\\beta_1^2}\/2, -2,0)$ is equivalent to \nP3($D_6$)$(4(\\alpha_1-\\beta_1),-4(\\alpha_1+\\beta_1-1), 4, \\ -4)$. \ndeg-P5($h^2\/2,-8,-2,0$) has an algebraic solution $y=1+2\\sqrt{t}\/h$.\ndeg-P5($\\alpha, 0, \\gamma, 0$) has the Riccati type solutions. \n\n\\par\\medskip\\noindent\n10) All of classical solutions of  P1 to P5 are equivalent to the \nabove solutions up to the B\\\"acklund transformations.\n\\end{theorem} \n\n\n\\subsection{Symmetric solutions for P1, P2, P34 and P5}\nWe will review symmetric solutions \\cite{AVK}, \\cite{KK}. \nThe first, second, thirty-fourth and fourth Painlev\\'e  equations\n\\begin{eqnarray*}\n&\\textrm{P1}\\qquad  &y^{\\prime\\prime}=6y^2 + t,\\\\\n&\\textrm{P2}(\\alpha)\\qquad  &y^{\\prime\\prime}= 2y^3+  t y + \\alpha,\\\\\n&\\textrm{P34}(a)\\qquad  & y^{\\prime\\prime}=\\frac{(y^{\\prime })^2}{2y}+2y^2 -ty -\\frac{a}{2y},\\\\\n&\\textrm{P4}(\\alpha,\\beta)\\qquad  &y^{\\prime\\prime}= \\frac{1}{2y}{y^{\\prime}}^2+\\frac{3}{ 2}y^3+ 4 t y^2 \n+2(t^2-\\alpha)y +\\frac{\\beta}{y},\\\\\n\\end{eqnarray*}\nhave a simple symmetry:\n\\begin{eqnarray*}\n&\\textrm{P1}\\qquad &y \\to \\zeta^3 y, \\quad t \\to \\zeta t,  \\quad (\\zeta^5=1)\\\\\n&\\textrm{P2}\\qquad &y \\to \\omega y, \\quad t \\to \\omega^2 t, \\quad  (\\omega^3=1)\\\\\n&\\textrm{P34}\\qquad &y \\to \\omega y, \\quad t \\to \\omega t, \\quad  (\\omega^3=1)\\\\\n&\\textrm{P4}\\qquad &y \\to - y, \\quad t \\to - t, \n\\end{eqnarray*}\nThere exist symmetric solutions which are invariant under the action of the cyclic group.\nThe symmetric solutions are studied by Kitaev \\cite{AVK} for P1 and P2 and  by Kaneko \n\\cite{KK} for P4.  Since these symmetric solutions exist for any parameter of the Painlev\\'e  equations,\nthey are not algebraic for generic parameters. \n\n\n\\begin{theorem}\\label{symmetric} 1) For P1, we have two symmetric solutions\n\\begin{eqnarray*}\ny &=& \\frac16 t^3 +\\frac1{336}t^8+\\frac1{26208}t^{13}+\\frac{95}{224550144}t^{18}+\\cdots,\\\\\ny &=& t^{-2} -\\frac1{6}t^3+\\frac1{264}t^{8}-\\frac1{19008}t^{13}+\\cdots.\n\\end{eqnarray*}\n\n2) For P2($\\alpha$), we have three symmetric solutions\n\\begin{eqnarray*}\ny &=& \\frac{\\alpha}2 t^2 +\\frac{\\alpha}{40} t^5  +\\frac{10\\alpha^3+\\alpha}{40}t^{8}+\\cdots,\\\\\ny &=&  t^{-1}-\\frac{\\alpha+1}{4} t^3   +\\frac{(\\alpha+1)(3\\alpha+1)}{112}t^{5}+\\cdots,\\\\\ny &=& - t^{-1} -\\frac{\\alpha-1}{4} t^3  -\\frac{(\\alpha-1)(3\\alpha-1)}{112}t^{5}+\\cdots.\n\\end{eqnarray*}\nThey are equivalent to each other by the B\\\"acklund transformations.\n\n2) For P34($a^2$), we have three symmetric solutions\n\\begin{eqnarray*}\ny &=& a t  +\\frac{a(2a-1)}{8} t^4  +\\frac{a(2a-1)(10a-3)}{560}t^{7}+\\cdots,\\\\\ny &=& -a t  +\\frac{a(2a+1)}{8} t^4 -+\\frac{a(2a+1)(10a+3)}{560}t^{7}+\\cdots,\\\\\ny &=& \\frac2{t^2}+\\frac t2 -\\frac{4a^2-9}{224} t^4 -\\frac{4a^2-9}{5600}t^7+\\cdots.\n\\end{eqnarray*}\nEach solution corresponds to the symmetric solutions of P2, respectively, by \n$$y_{34}=y_2^\\prime+y_2^2+\\frac t2.$$\n\n\n4) For P4($\\alpha, -8\\theta_0^2$), we have four symmetric solutions\n\\begin{eqnarray*}\ny &=& \\pm 4\\theta_0\\left(t -\\frac{2 \\alpha}{3} t^3 \n     +\\frac{2}{15} (\\alpha^2+12\\theta_0^2\\pm \\theta_0+1 )t^{5}+\\cdots\\right),\\\\\ny &=& \\pm t^{-1}+\\frac{2}{3}(\\pm\\alpha-2) t^3 \n       \\mp \\frac{2}{45}(-7\\alpha^2\\pm 16\\alpha+36\\theta_0^2-4 )t^{5}+\\cdots.\n\\end{eqnarray*}\nThey are equivalent to each other by the B\\\"acklund transformations.\n\\end{theorem}\n\n\\par\\noindent\nWe remark that there are no B\\\"acklund transformation for P1.\nWe may think symmetric solutions as a generalization of Umemura's classical \nsolutions. We notice that symmetric solutions are not classical functions \nexcept for special parameters. For example, the first solution of P2($\\alpha$) \nis transcendental for a generic parameter, but it is a rational solution $y=0$ \nfor $\\alpha=0$. It is rational if and only if $\\alpha\/3 $ is an integer. \nThe symmetric solutions exist for \nany parameters and classical solutions exist only for special parameters.\n\n\\section{Confluent hypergeometric equation}\\label{chg}\nIn this section we review Kummer's confluent hypergeometric equation. \nIt is known that there exist two standard forms, Kummer's form and Whittaker's form  \nfor the confluent hypergeometric equation.  At the first we will use  Kummer's standard form.\n\nWe review   irregular singularities to fix our notations. \nFor a rational function $a(x)$, we set\n$${\\rm ord}_{x=c}a(x)=\\ ({\\rm pole\\ order\\ of}\\ a(x)\\ {\\rm at}\\ x=c).$$\nA linear differential equation \n$$\\frac{d^2u}{dx^2}+p(x)\\frac{d u}{dx }+q(x)=0$$\nhas an irregular singularity at $x=c$ if and only if\n${\\rm ord}_{x=c}p(x)>1$ or ${\\rm ord}_{x=c}q(x)>2$. \nThe Poincar\\'e rank $r$ of the irregular singularity   $x=c$ is \n$$r=\\max \\{{\\rm ord}_{x=c}p(x), 1\/2 \\cdot{\\rm ord}_{x=c}q(x)  \\}-1.$$\nWe think that a regular singularity has the Poincar\\'e rank $r=0$. \n\n\nKummer's confluent hypergeometric equation is\n\\begin{equation}\\label{2:chg}\nx\\frac{d^2u}{dx^2}+(c-x)\\frac{du}{dx}-au=0, \n\\end{equation}\nwhich has a regular singularity at $x=0$ and an irregular singularity \nwith the Poincar\\'e rank $1$ at $x=\\infty$. \nWe set $x\\to \\varepsilon x, a \\to 1\/\\varepsilon$ in \\eqref{2:chg}  and \ntake the limit $\\varepsilon \\to 0$. Then we get a degenerate confluent \nhypergeometric equation (the confluent hypergeometric limit equation)\n\\begin{equation}\\label{2:dchg}\nx\\frac{d^2u}{dx^2}+c\\frac{du}{dx}-u=0, \n\\end{equation}\nwhich has a regular singularity at $x=0$ and an irregular singularity \nwith the Poincar\\'e rank $1\/2$ at $x=\\infty$. \nThe solution of \\eqref{2:dchg} is\n$$y=C\\ {}_0F_{1}(c;x)+D\\ x^{1-c}{}_0F_{1}(2-c;x).$$\n\\eqref{2:dchg} is reduced to \\eqref{2:chg} by Kummer's second formula\n$${}_0F_{1}\\left(c; {x^2}\/{16}\\right)= e^{-x\/2}\\ {}_{1}F_1 (c-1\/2,2c-1;x).$$\nIt is also related to the Bessel function \n\\begin{equation}\\label{bessel}\n{}_0F_{1}\\left(c; {x^2}\/{16}\\right)= \\Gamma(c)(-ix\/4)^{1-c}J_{c-1}(-ix\/2).\n\\end{equation}\n\nLater we will use  $SL$-type equations in the section\\ref{pa_type}. \n$SL$-type of the confluent hypergeometric equation is called the Whittaker equation:\n\\begin{eqnarray}\nW_{k,m}: & \\dfrac{d^2u}{dx^2}= \\left(\\dfrac{ 1}{4}-\\dfrac{k}{x}+\\dfrac{m^2-\\frac{1}{4}}{x^2} \\right)u=0,\\label{sl:chg}\\\\\nDW_{m}:&  \\dfrac{d^2u}{dx^2}= \\left(\\dfrac{ 1}{x}+\\dfrac{m^2-\\frac{1}{4}}{x^2} \\right)u=0.\\label{sl:dchg}\n\\end{eqnarray}\nIn $W_{k,m}$, the parameters $k, m$ correspond to \n $k=c\/2-a, \\quad m= (c-1)\/2$ \nin \\eqref{2:chg}. In $DW_{m}$, the parameter  $m$ corresponds to \n $ m=(c-1)\/2$ in \\eqref{2:dchg}. \n\n\n\\section{Linear equations of the Painlev\\'e  type}\\label{iso}\nWe call a linear equation of the second order whose isomonodromic \ndeformation gives a Painlev\\'e  equation \nas {\\it the Painlev\\'e  type}. \nIn this section we will list up all of linear equations of the Painlev\\'e  type.\nSince a linear equation of the second order is equivalent to a $2\\times 2$ system of \nlinear equations of the first order, we use a single equation. It is easy to \nrewrite as a $2\\times 2$ system. \n\n\\subsection{Singularity type}\n\nLinear differential equations of  the Painlev\\'e  type have \nthe following types of singular points. \n\\begin{center}\n\\begin{tabular}{ r|l }\nP6\\ &  $(0)^4$  \\\\\n\\hline\nP5\\ & $(0)^2(1)$ \\\\\n\\hline\nP3$(D_6)$\\ & $(1)^2$ or $(0)^2(1\/2)$ \\\\\n\\hline\nP3$(D_7)$\\ & $(1)(1\/2)$  \\\\\n\\hline\nP3$(D_8)$\\ & $(0)(2)$ \\\\\n\\hline\nP4\\ & $(0)(2)$ \\\\\n\\hline\nP34\\ &   $(0)(3\/2)$ \\\\\n\\hline\nP2\\ & $(3)$  \\\\\n\\hline\nP1\\ & $(5\/2)$ \n \\end{tabular} \n\\end{center}\nHere $(m)^k$ means $k$ singular points with the Poincar\\'e rank $m$, and \n$(0)$ means a regular singularity.  The Painlev\\'e  third equation\nhas two type of singularities. $(1)^2$ is a standard one, and $(0)^2(1\/2)$ is \ndeg-P5. \n\nTwo different types of linear equations are used for the isomonodromic deformation of \nthe Painlev\\'e  second equation. One is Garnier's form \\cite{Garnier}, which is the same one used by \nOkamoto \\cite{Okamoto}\nand Miwa-Jimbo \\cite{JM2}, The second one is Flaschka-Newell's form \\cite{FN}, which is \nis equivalent to the type $(0)(3\/2)$   by a rational transformation \\cite{Kapaev}.\nIt is natural that Flaschka-Newell's form as an isomonodromic deformation for P34. \nWe will study Garnier's form and Flaschka-Newell's form in a succeeding paper.\n\n\n\n\\subsection{List of equations of the Painlev\\'e  type}\\label{pa_type}\n\nIn the following, we will list up all of linear equations of the Painlev\\'e  type.\nWe use linear equations of $SL$-type:\n\\begin{equation}\\label{SL}  \n\\frac{d^2 u}{d z^2}=p(z,t)u.\n\\end{equation}\nIsomonodromic deformation for a linear equation of $SL$-type is given\nby the compatibility condition of the following system \\cite{Okamoto}:\n\\begin{equation}\\label{SL:mpd}\\begin{aligned} \n\\frac{\\partial^2 u}{\\partial x^2}&=V(x,t)u,\\\\\n\\frac{\\partial u}{\\partial t}=& A(x,t)\\frac{\\partial  u}{\\partial x}\n                             -\\frac12\\frac{\\partial A(x,t)}{\\partial x }u.\n\\end{aligned}\\end{equation}\nWe will list up $V(x,t)$ and $A(z)$ of linear equations of Painlev\\'e  type. We corrected \nmisprints in \\cite{Okamoto}.  The compatibility condition gives the Painlev\\'e  equation, \nwhich turn out a Hamiltonian system  with the Hamiltonian $K_J$ in the following list.\n$(q,p)$ are canonical coordinates and $q$ satisfies the Painlev\\'e  equation.\n\\par\\bigskip\\noindent\n{\\it Type $(5\/2)$}\\ : the first Painlev\\'e  equation P1 \\\\\n$$V(z,t)=4z^3 +2t z +2 K_{\\rm I} +\\frac{3}{4(z-q)^2}  -\\frac{p}{z-q}$$\n$$A(z)= \\frac 12\\cdot \\frac{1}{z-q}$$\n$$K_{\\rm I} =\\frac12 p^2 -2q^3 -t q.$$\n\n\\par\\bigskip\\noindent\n{\\it Type $(3)$}\\ : the second Painlev\\'e  equation P2($\\alpha$)\\\\\n$$V(z,t)=z^4 +t z^2 +2\\alpha z +2 K_{\\rm II} +\\frac{3}{4(z-q)^2}  -\\frac{p}{z-q}$$\n$$A(z)= \\frac 12\\cdot \\frac{1}{z-q}$$\n$$K_{\\rm II}  =\\frac12 p^2 -\\frac12 q^4 -\\frac12 t q^2 -\\alpha q.$$\n\n\\par\\bigskip\\noindent\n{\\it Type $(1)(3\/2)$}\\ : the thirty-fourth Painlev\\'e  equation P34($\\alpha$)\\\\\n$$V(z,t)=\\frac z2 -\\frac t2 +\\frac{\\alpha-1}{4z^2} -\\frac{ K_{\\rm XXXIV}}z\n +\\frac{3}{4(z-q)^2} -\\frac{pq}{z(z-q)}$$\n$$A(z)= - \\frac{z}{z-q}$$\n$$K_{\\rm XXXIV}  =-qp^2 +p +\\frac{q^2}2 -\\frac{tq}{2} +\\frac{\\alpha-1}{4q}.$$\n\n\n\\par\\bigskip\\noindent\n{\\it Type $(1)(2)$}\\ : the fourth Painlev\\'e  equation P4($\\alpha, \\beta$)\\\\\n$$V(z,t)=\\frac{{a_0}}{z^2}+\\frac{K_{\\rm VI}}{2z} +a_1 +\\left(\\frac{z+2t}4\\right)^2 \n+\\frac{3}{4(z-q)^2}  -\\frac{pq}{z(z-q)},$$\n$$A(z)=   \\frac{2z}{z-q},$$\n$$K_{\\rm VI}  =2qp^2-2p -\\frac{{a_0}}{q} -2a_1 q-2q\\left(\\frac{q+2t}4\\right)^2,$$\n$$a_0= -\\frac\\beta 8-\\frac14,\\ a_1=-\\frac\\alpha4.$$\n\n\\par\\bigskip\\noindent\n{\\it Type $(1)^2$}\\ : the third Painlev\\'e  equation P3${}^\\prime$($\\alpha, \\beta, \\gamma, \\delta$)\\\\\n$$V(z,t)=\\frac{a_0t^2}{z^4}+\\frac{a_0^\\prime t }{z^3}-\\frac{t K_{\\rm III}^\\prime}{ z^2}+\n\\frac{a_\\infty^\\prime}{ z } + a_\\infty\n+\\frac{3}{4(z-q)^2}-\\frac{pq}{z(z-q)},$$\n$$A(z)=   \\frac{ qz}{t(z-q)},$$ \n$$t K_{\\rm III}^\\prime = q^2p^2 -qp -\\frac{a_0 t^2}{ q^2}\n-\\frac{a_0^\\prime t}{ q}- {a_\\infty^\\prime q} -a_\\infty  {q^2},$$\n$$a_0= -\\frac\\delta{16},\\ a_0^\\prime=-\\frac\\beta8,\\ \na_\\infty= \\frac\\gamma{16},\\ a_\\infty^\\prime=\\frac\\alpha8.$$\n\n\\par\\bigskip\\noindent\n{\\it Type $(0)^2(1)$}\\ : the fifth Painlev\\'e  equation P5($\\alpha, \\beta, \\gamma, \\delta$)\\\\\n$$V(z,t)=\\frac{a_1 t^2}{(z-1)^4}+\\frac{K_{\\rm V} t}{(z-1)^2 z}+\\frac{a_2  t}{(z-1)^3}-\\frac{p\n   (q -1) q }{ z (z-1)(z-q )}+\\frac{a_\\infty}{(z-1)^2}+\\frac{{a_0}}{z^2}+\\frac{3}{4\n   (z-q )^2}$$\n$$A(z)=\\frac{q-1}t \\cdot \\frac{z(z-1) }{ z-q }$$\n$$ tK_{\\rm V}= q(q -1)^2   \\left[-\\frac{a_1 t^2}{(q-1)^4}-\\frac{a_2 t}{(q -1)^3}+p^2-\n   \\left(\\frac{1}{q }+\\frac{1}{q -1}\\right)p -\\frac{a_\\infty}{(q -1)^2}-\\frac{a_0}{q \n   ^2}\\right] $$\n$$a_0= -\\frac\\beta 2-\\frac14,\\ a_1=-\\frac\\delta2  ,\\ a_2=-\\frac\\gamma2,\\ a_\\infty=\\frac12(\\alpha+\\beta)-\\frac34$$\n\n\\par\\bigskip\nIf $\\gamma=0$ or  $\\delta=0$ for P3, the type of the linear equation is $(1)(1\/2)$. \nWe will take  $\\delta=0$ as a standard form, which reduces to P3($D_7$)($\\alpha, \\beta, \\gamma,0$).\nIf $\\gamma=0$ and  $\\delta=0$ for P3, the type of the linear equation is $ (1\/2)^2$, which reduces to\nP3($D_8$)($\\alpha, \\beta,0,0$). \nIf $\\delta=0$ for P5, the type of the linear equation is $(0)^2(1\/2)$, which reduces to\ndeg-P5($\\alpha, \\beta, \\gamma,0$). \nWe omit P6 since we treat the cases of  irregular singularities   in this paper.\n\n\n\n\\section{Rational transform of $W_{k,m}$ and $DW_m$}\\label{rational}\n\nIn this section, we will classify all of rational transformations \nof independent variables, which change $W_{k,m}$ or $DW_{m}$ into linear \ndifferential equations of the Painlev\\'e  type.\n\nThe following simple lemma is a key of classification.\n\\begin{lemma}\nFor a linear differential equation \n\\begin{equation}\\label{2ndnormal}\n\\frac{d^2u}{dx^2}=p(x)u(x),\n\\end{equation}\nwe take  a new independent variable $z$ defined by a rational transform $x=x(z)$.\nIf $x=c$ is a regular singularity, $z_c=x^{-1}(c)$ is also a regular singularity \nof the transformed equation. When $z_c$ is a $n$-th branched point and the distance\nof the two exponent at $x=c$ is $1\/n$, $z=z_c$ is an apparent singularity and \n$z=z_c$ can be reduced to a regular point by a suitable change $y=v(z)u$.\n\nIf $x=c$ is an irregular singularity of the Poincar\\'e rank $m$ and \n$z_c$ is a $n$-th branched point, $z=z_c$ is an irregular singularity of the Poincar\\'e \nrank $mn$.\n\\end{lemma}\n\nThe proof is obvious. Since we will start from the confluent hypergeometric equations, \nwe take a care of $x=0, \\infty$ as branched points. For $x=x(z)$, \nif  $x^{-1}(0)$ are   branch points with $\\mu_1,\\mu_2,\\cdots$ order and \n$x^{-1}(\\infty)$ are   branch points with $\\nu_1,\\nu_2,\\cdots$ order, \nwe  call  $x(z)$ as a type $(\\mu_1+\\mu_2+\\cdots |\\nu_1+\\nu_2+\\cdots)$.\nWe allow $\\mu_j$ or $\\nu_j$ equals to one. $\\sum \\mu_j= \\sum\\nu_j$ is \nthe order of the rational function $x(z)$. \n\n\\begin{theorem}\\label{th:cover} By a rational transform $x=x(z)$, $W_{k,m}$ or $DW_m$\nchanges to a linear equation of the Painlev\\'e  type or a confluent hypergeometric equation \nif and only if one of the following cases occur. \n\\par\\medskip\n\n1) Double cover \n\\par\\medskip\n\\begin{tabular}{ c|c|c|c }\n $W_{k,m}$\\   & $(2|2)$     & $(0)(2)$   & P4-sym \\\\   \n $W_{k, 1\/4}$\\ & $(2|2)$     &   $(2)$    & Weber\\\\\t\n $W_{k, 1\/4}$ \\ & $(2|1+1)$   & $(1)^2 $   & D6-alg \\\\\t\n $W_{0, 1\/2}$\\   & $(1+1|2)$   & $(0)(2) $   & P4-Her \\\\\t\n $DW_{m}$\\    & $(2|2)$ & $(0)(1) $ & Kummer \\\\\t\n $DW_{m}$\\    & $(1+1|2)$ & $(1)^2(2) $ & P5-rat \\\\\t\n $DW_{1\/4}$\\  & $(2|1+1)$ & $(1\/2)^2$ & D8-alg \\\\\t\n\\end{tabular}\n\n\\par\\medskip\n2) Cubic cover\n\\par\\medskip\n\\begin{tabular}{ r|c|c|c }\n $W_{k,1\/3}$\\   & $(3|3)$     & $(3)$   & P2-sym \\\\\t\t\n $DW_{m}$\\ & $(3|3)$      &   $(1)(3\/2)$    & P34-sym \\\\\t\n $DW_{1\/6}$\\ & $(3|3)$      &   $(3\/2)$    & Airy \\\\\t\n $DW_{1\/4}$\\ & $(2+1|3)$   &   $(0)(3\/2)$    & P34-rat \\\\\t\n $DW_{1\/6}$\\ & $(3|2+1)$   &   $(1)(1\/2)$    & D7-alg  \t\n\\end{tabular}\n\\par\\medskip\n\n\n3) Quartic cover\n\\par\\medskip\n\\begin{tabular}{ r|c|c|c }\n $DW_{1\/6}$\\   & $(3+1|4)$     & \\hskip 5mm  $(3) \\hskip 5mm $   & P4-rat \n\\end{tabular}\n\\par\\medskip\n\n4) Quintic cover \n\\par\\medskip\n\\begin{tabular}{ r|c|c|c }\n $DW_{1\/5}$\\    & \\hskip 3mm $(5|5)$ \\hskip 3mm {} &\\hskip 3mm  $(5\/2) $ \\hskip 3mm &  P1-sym \\\\\n  $DW_{1\/10}$\\    & \\hskip 3mm $(5|5)$ \\hskip 3mm {} &\\hskip 3mm  $(5\/2) $ \\hskip 3mm &  P1-sym \t\t\t\n\\end{tabular}\n\n5) Sextic cover \n\\par\\medskip\n\\begin{tabular}{ r|c|c|c }\n $DW_{1\/6}$\\    & $(3+3|6)$     &\\hskip 5mm $(3)\\hskip 5mm  $ & P2-rat \n\\end{tabular}\n\\par\\medskip\\noindent\nHere the first column is the starting linear equation. The second column is \nthe type of a rational transform. The third column is the singularity type of \nof the transformed linear equation. The fourth column is the solution of \nthe Painlev\\'e  equation.\n\\end{theorem}\n\\par\\noindent\n{\\it Proof.}  The Poincare rank of irregular singularities of\n equations of the Painlev\\'e  type is at most three. Therefore we can take \n up to a cubic cover of  $W_{k,m}$ and we can take  up to a sextic cover of  $DW_{m}$.\n In each cases, we can classify the covering map directly.\n \\hfill \\boxed{} \n \\par\\bigskip\\noindent \nThe other types of confluent equations, such as the Weber (parabolic cylinder) equation\nand the Airy equation, appear in the table. Therefore covering of such equations are\nalso included in the covering of $W_{k,m}$ or $DW_{m}$.  We consider the Bessel equation as a \nspecial case of $W_{0,m}$. \nWe will explain each case in the next subsection.  The theorem \\ref{th:cover} shows that \nit is necessary to distinguish  P2 and P34 from the isomonodromic viewpoint. \n \n\n\n\\subsection{Exact form}\\label{exact}\nIn this section we will write down algebraic solutions obtained by rational transformations\nfrom $W_{k,m}$ or $DW_{m}$ explicitly.\nThe following lemma is a key to calculate exact transformations.\n\n\\begin{lemma}\\label{keylemma} For the equation\n\\begin{equation*} \n \\frac{d^2u}{dx^2}=Q(x)u,\n\\end{equation*}\nwe set \n$$x=x(z), \\ u(x)= \\sqrt{\\frac{dx}{dz}} v(z).$$\nThen $v$ satisfies\n\\begin{equation}\\label{transformed}\n\\frac{d^2v}{dz^2} =\\left( Q(x(z))(x'(z))^2- \\frac12\\{z,x\\} \\right)v.\n\\end{equation}\nHere $\\{z,x\\}$ is the Schwarzian derivative\n$$\\{z,x\\}=\\frac{z^{\\prime\\prime\\prime}}{z'}\n     -\\frac32 \\left(\\frac{z^{\\prime\\prime }}{z'} \\right).$$\n\\end{lemma}\n\\noindent \nWe will take  $Q(x)$ as \\eqref{sl:chg} or \\eqref{sl:dchg}. Then we calculate  $V(z,t)$ \nin the left hand side of \\eqref{transformed} in each case of the theorem \\ref{th:cover}. \nWe should take a suitable coefficients of the rational function $x=x(z)$ \nto coincide  with the equation of the Painlev\\'e  type.  In the following $(q,p)$ is the \ncanonical coordinates in the section \\ref{pa_type}.\n\n\\subsubsection{Double cover}\n1) P4-sym $(2|2)$ \\\\ \\noindent\nFor $W_{k,m}$, we set $x=z^2\/4$.  Then  \n$$V(z,t)= -k+ \\frac{16m^2-1}{4 z^2}+\\frac {z^2}{16},$$\nwhich is the case $t=0$ for the symmetric solution $ q(t)=2(4m+1)t+O(t^3)$ \nof P4$(4k,-2(4m+1)^2)$.\n\n\\par\\bigskip\\noindent\n2) Weber $(2|2)$ \\\\ \\noindent\nThis is a special case of P4-sym.\nFor $W_{k,1\/4}$, we set $x=z^2\/2$.  Then  \n$$V(z,t)= \\frac{z^2}{4}  -2k,$$\nwhich is the parabolic cylinder equation for $D_{2k-1\/2}(z)$. \nThis shows the  well-known formula\n\\begin{equation}\\label{weber}\nD_{2k-1\/2}(z)= 2^k z^{-1\/2}W_{k,-1\/4}\\left(\\frac{z^2}2 \\right).\n\\end{equation}\n\n\\par\\bigskip\\noindent\n3) $D_6$-alg $(2|1+1)$ \\\\ \\noindent\nFor $W_{k,1\/4}$, we set $x=(z-\\sqrt{t})^2\/z$.  Then  \n\\begin{eqnarray*}\nV(z,t)= \\frac{1}{4} + \\frac{t^2}{4z^2}-\\frac{kt}{z^3}\n-\\frac{8t+32k\\sqrt{t}+3}{16 z^2}\n-\\frac{k}{z }+ \\frac{3}{ 4(z+\\sqrt{t})^2}-\\frac{3}{ 4z(z+\\sqrt{t})},\n\\end{eqnarray*}\nwhich gives the algebraic solution $ q(t)=-\\sqrt{t}$ of P3$(-8k,8k,4,-4)$.\n\n\\par\\bigskip\\noindent\n4) P4-Her $(1+1|2)$ \\\\ \\noindent\nFor  $W_{0, 1\/2}$, we set $x =z (z+4t)\/4$. Then  \n$$V(z,t)= \\left(\\frac{z+2t}{4} \\right)^2+\\frac{4}{ 3(z+2t)},$$\nwhich gives the rational solution $ q(t)=-2t$ of P4($0,-2$). \n\n\\par\\bigskip\\noindent\n5) Kummer's second formula $(2|2)$ \\\\ \\noindent\nFor $DW_{m}$, we set $x=z^2\/16$.  Then  \n$$V(z,t)= \\frac{1}{4} + \\frac{16m^2-1}{4 z^2},$$\nwhich is $W_{0, 2m}$. This is Kummer's second formula.\n\n\\par\\bigskip\\noindent\n6) P5-rat $(1+1|2)$  \\\\ \\noindent\nFor $DW_{m}$, we set $x= h t^2 z  \/4(z-1)^2$.  Then  \n$$V(z,t)= \\frac{h t^2}{(z-1)^4} + \\frac{16m^2-1+h t^2}{4 z(z-1)^2}\n-\\frac3{4z} + \\frac{4m^2-1}{4z^2} + \\frac{3(z+2)}{4(z+1)^2},$$\nwhich gives the rational solution $ q(t)=-1$ of  P5($2m^2, -2m^2, 0, -2h$).\n\n\\par\\bigskip\\noindent\n7) $D_8$-alg $(2|1+1)$ \\\\ \\noindent\nFor $DW_{1\/4}$, we set $x=h (z-\\sqrt{t})^2\/z$.  Then  \n$$V(z,t)= \\frac {h t}{z^3} +\\frac {32 h \\sqrt{t}-3}{16z^2}\n+\\frac{h}z+ \\frac{3}{4(z+\\sqrt{t})^2}-\\frac{3}{4z(z+\\sqrt{t})},$$\nwhich gives the algebraic solution $ q(t)=-\\sqrt{t}$ of P3$(8h,-8h,0,0)$.\n\n\n\\subsubsection{Cubic cover}\n1) P2-sym $(3|3)$\\\\ \\noindent\nFor $W_{k,1\/3}$, we set $x=2z^3\/3$.  Then  \n$$V(z,t)= z^4 -6kz +\\frac{3}{4z^2},$$\nwhich is the symmetric solution $ q(0)=0, p(0)=0$ of P2$(-3k)$.\n\n\\par\\bigskip\\noindent\n2) P34-sym $(3|3)$\\\\ \\noindent\nFor $DW_{m}$, we set $x= z^3\/18 $.  Then  \n$$V(z,t)=\\frac z 2 +\\frac{36m^2-1}{4z^2},$$\nwhich is the symmetric solution $ q(0)=0$ of P34$(3 (12m^2 -1))$.  \nWe have two symmetric solutions with $ q(0)=0$, and both give the same \nequation when $t=0$.\n\n\\par\\bigskip\\noindent\n3) Airy $(3|3)$\\\\ \\noindent \nThis is a special case of P34-sym.\nFor $DW_{1\/6}$, we set $x=  z^3\/9 $.  Then  \n$$V(z,t)= z,$$\nwhich is  the Airy equation. It is know that \n\\begin{equation}\\label{airy}\n{\\rm Ai}(x)=\\frac1{ 3^{2\/3} \\Gamma(\\frac23) } {}_0F_1\\left(\\frac23; \\frac{z^3}9  \\right)\n-\\frac{x}{3^{1\/3}\\Gamma(\\frac13)}{}_0F_1 \\left(\\frac43; \\frac{z^3}9  \\right).\n\\end{equation}\n\n\n\\par\\bigskip\\noindent\n4) P34-rat $(2+1|3)$\\\\ \\noindent \nFor $DW_{1\/4}$, we set $x=  z(z-3t\/2)^3\/18 $.  Then \n$$V(z,t)=  \\frac z 2-\\frac  t2 +\\frac{t^3\/4+1}{2tz}-\\frac{3}{16z^2} \n          +\\frac{3}{4(z-t\/2)^2}-\\frac{1}{2t(z-t\/2)},$$\nwhich gives the rational solution $ q(t)=t\/2$ of P34$( 1\/4)$.\n\n\n\\par\\bigskip\\noindent\n5) $D_7$-alg $(3|2+1)$\\\\ \\noindent\nFor $DW_{1\/6}$, we set $x =\\left(z+2 t^{1\/3} \\right)^3\/32z$. Then  \n$$V(z,t)= \\frac{ t}{4z^3}-\\frac{16+27 t^{2\/3}}{72z^2}\n+\\frac{2}{ 3 t^{1\/3} z }+ \\frac18   +\\frac{3}{4 (  z-  t^{1\/3})^2}\n- \\frac{2}{3t^{1\/3}(  z-  t^{1\/3}) },$$\nwhich is the algebraic solution $ q(t)=  t^{1\/3}  $ of P3($0, -2, 2,0$).\n\n\\subsubsection{Quartic cover}\nIf the covering degree is more than three, we start only from $DW_{m}$. \nIf  a quartic covering map $x=x(z)$ splits to $x= (\\bar{x}(z))^2$, \nthe covering through  $W_{0,2m}$, which is Kummer's case 5.1.4. \nSuch case occurs in case that all of $\\mu_j, \\nu_j$ are even.\n$(4|4), (4|2+2), (2+2|4)$ corresponds to $(2|2), (2|1+1), (1+1|2)$, respectively.\n\n\\par\\bigskip\\noindent\n1) P4-rat $(3+1|4)$ \\\\ \\noindent\nFor $DW_{1\/6}$, we set  $   x=\\frac{1}{256} z \\left(z+  8 t\/3 \\right)^3$.\nThen \n$$V(z,t)= -\\frac{2}{9 z^2}+\\frac{2t^3}{27 z} +\\left(\\frac{z+2t}4\\right)^2\n+\\frac{27}{4(3z+2t)^2}-\\frac1{z(3z+2t)}$$ \nwhich gives the rational solution $q(t)=-2t\/3$ of P4$(0, -2\/9)$.\n\n\\subsubsection{Quintic cover}\n1) P1-sym $(3+1|4)$ \\\\ \\noindent\nFor $DW_{1\/5}$, we set $x =4z^5\/25$. Then  \n$$V(z,t)= \\frac{3}{4 z^2}+4 z^3,$$\nwhich is the symmetric solution $ y(0)=0, y'(0)=0, t=0$ of P1.\n\\par\\bigskip \\noindent\nFor $DW_{1\/10}$, we set $x =4z^5\/25$. Then  \n$$V(z,t)=  4 z^3,$$\nwhich is the symmetric solution $ y(t)=1\/t^2+\\cdots$ and $t=0$ of P1. \nIf we substitute $ y(t)=1\/t^2+\\cdots$ in $Q(z,t)$, $Q(z,t)$ may have a pole \n$t=0$ but this pole is apparent. \n\n\\subsubsection{Sextic cover}\nAs the same as the quartic cover, we omit the case $(6|6), (4+2|6)$. \n\\par\\bigskip\\noindent\n1) P2-rat $(3+3|6)$ \\\\ \\noindent\nFor $DW_{1\/6}$, we set $x =\\frac1{36} (z^2+t)^3$. Then  \n$$V(z,t)= \\frac{3}{4 z^2}+tz^2+ z^4,$$\nwhich is the rational solution $q=0$ of P2$(0)$.\n\n\n\\subsection{R.~Fuchs' Problem}\n\nCompared with the theorem \\ref{classical} and the theorem \\ref{th:cover}, \nwe obtain all of algebraic solutions and symmetric solutions except \nalgebraic solutions of deg-P5 and the Laguerre solutions of  P5. \nThese two solutions are not obtained from $W_{k,m}$ or $DW_{m}$ by rational transformations, \nbut it is  obtained by exponential type transformations. \n\n\\subsubsection{Split case}\n1) For deg-P5-alg, we start from\n$$\\frac{d^2 u}{dx^2} =\\frac {h^2-1}{4 x^2}u.$$\nWe set $x=e^{4\\sqrt{tz\/(z-1)}\/h }(\\sqrt{z}+\\sqrt{z-1} )\/ (\\sqrt{z}-\\sqrt{z-1} )$. Then \n\\begin{eqnarray*}\nV(z,t)=\\frac{t}{ (z-1)^3}-\\frac{4h^2-13 }{16(z-1)^2}-\\frac3{16z^2}\n        -\\frac{2(h+2\\sqrt{t})^2-5}{8z(z-1)^2}\\\\\n   +\\frac{3}{4(z-2\\sqrt{t}\/h-1)^2}-\\frac{3+8\\sqrt{t}\/h}{4z(z-1)(z-1-2\\sqrt{t}\/h)},\n\\end{eqnarray*} \nwhich gives an algebraic solution $y=1+2\\sqrt{t}\/h$ for P5($h^2\/2,-8,-2,0$).\n\n\\par\\bigskip\\noindent\n2)  For P5-Lag, we start from\n$$\\frac{d^2 u}{dx^2} =\\frac {h^2-1}{4 x^2}u.$$\nWe set $x=e^{t\/(h(z-1))}(z-1) $. Then \n$$V(z,t)=\\frac{t^2}{4(z-1)^4}-\\frac{ht}{2(z-1)^3}+\\frac{h^2\/4-1}{(z-1)^2}\n         -\\frac{3}{4(z-t\/h-1)^2}-\\frac{ht}{(z-1)^2(z-t\/h-1)},$$\nwhich gives a rational solution $y= t\/h+1$ for P5($h^2\/2,-1\/2,-h,-1\/2$).\n\nIn these two cases, the monodromy group of  \n\\begin{equation}\\label{split}\n\\frac{d^2u}{dz^2}= V(z,t)u(z)\n\\end{equation} \nis diagonal.   Therefore we cannot reduce them to $W_{k,m}$ or $DW_{m}$.\n\n\\subsubsection{Summary}\nWe summarize our result:\n\\begin{theorem}\\label{final} By  any rational transformation from confluent hypergeometric \nequations $W_{k,m}$ or $DW_{m}$ to equations of the Painlev\\'e  type,  we get algebraic solutions or symmetric \nsolutions of P1, P2, P34, P3, P4, P5 except deg-P5-alg and P5-Lag. \nConversely, any algebraic solution except deg-P5-alg and P5-Lag or symmetric solution  \nfrom P1 to P5 is obtained by  a rational transformation of $W_{k,m}$ or $DW_{m}$. \nFor deg-P5-alg and P5-Lag, we can reduce them to  differential equations with \n constant coefficients by exponential type transformations.\n\\end{theorem} \n\n\n\\par\\noindent\nIn this sense, R.~Fuchs' problem is true for P1 to P5.  We obtain rational \nsolutions and symmetric solutions of P2 and  P34 in different way. \nIn \\cite{AVK}, Kitaev studied a symmetric solution of P2, but his result is \non   a symmetric solution of P34 from our viewpoint since he used  Flaschka-Newell's form. \nSymmetric solutions of P2  is studied in \\cite{KK2} by using   Miwa-Jimbo's form.\n\nThe authors do not know R.~Fuchs' problem is true or not in the case the sixth Painlev\\'e  equations. \nRecent work by A.~V.~Kitaev \\cite{Kitaev:05}  give partial, but affirmative \nanswers to R.~Fuchs' problem.\nWe do not know any negative examples of R.~Fuchs' problem.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nArtificial neural networks are often ill-conditioned systems in that a small change in the inputs can cause significant changes in the outputs \\citep{szegedy}.\nThis results in poor robustness and vulnerability under adversarial attacks which has been reported on a variety of networks including image classification \\citep{carlini,goodfellow}, speech recognition \\citep{kreuk2018fooling,alzantot2018did,carlini2018audio}, image captioning \\citep{chen2017show} and natural language processing \\citep{gao2018black,ebrahimi2017hotflip}.\nThese issues bring up both theoretical questions of how neural networks generalize \\citep{kawaguchi2017generalization,xu2012robustness} and practical concerns of security in applications \\citep{akhtar2018threat}.\n\nA number of remedies have been proposed for these issues and will be discussed in Section~\\ref{sec:liter}.\nWhite-box defense is particularly difficult and many proposals have failed.\nFor example, \\citet{athalye2018obfuscated} reported that out of eight recent defense works, only \\citet{mit} survived strong attacks.\nSo far the mainstream and most successful remedy is that of adversarial training \\citep{mit}.\nHowever, as will be shown in Tables~\\ref{tbl:mnist} and \\ref{tbl:cifar}, the robustness by adversarial training diminishes when a white-box attacker \\citep{carlini} is allowed to use more iterations.\n\nThis paper explores a different approach and demonstrates that a combination of the following three conditions results in enhanced robustness:\n1) the Lipschitz constant of a network from inputs to logits is no greater than 1 with respect to the $L_2$-norm;\n2) the loss function explicitly maximizes \\emph{confidence gap}, which is the difference between the largest and second largest logits of a classifier;\n3) the network architecture restricts confidence gaps as little as possible. We will elaborate.\n\nThere are previous works that achieve the first condition \\citep{cisse,hein} or bound responses to input perturbations by other means \\citep{kolter,raghu,haber}.\nFor example, Parseval networks \\citep{cisse} bound the Lipschitz constant by requiring each linear or convolution layer be composed of orthonormal filters.\nHowever, the reported robustness and guarantees are often under weak attacks or with low noise magnitude, and none of these works has demonstrated results that are comparable to adversarial training.\n\nIn contrast, we are able to build MNIST and CIFAR-10 classifiers, without needing any adversarial training, that exceed the state of the art \\citep{mit} in robustness against white-box $L_2$-bounded adversarial attacks.\nThe defense is even stronger if adversarial training is added.\nWe will refer to these networks as \\emph{$L_2$-nonexpansive neural networks} (L2NNNs).\nOur advantage comes from a set of new techniques:\nour weight regularization, which is key in enforcing the first condition, allows greater degrees of freedom in parameter training than the scheme in \\citet{cisse}; \na new loss function is specially designed for the second condition;\nwe adapt various layers in new ways for the third condition, for example norm-pooling and two-sided ReLU, which will be presented later.\n\nLet us begin with intuitions behind the second and third conditions.\nConsider a multi-class classifier.\nLet $g\\left({\\bf x}\\right)$ denote its confidence gap for an input data point ${\\bf x}$.\nIf the classifier is a single L2NNN,\\footnote{This is only an example and we recommend building a classifier as multiple L2NNNs, see Section~\\ref{sec:loss}.} we have a guarantee\\footnote{See Lemma~\\ref{th:sqrt2} in Appendix~\\ref{sec:proofs} for the proof of the guarantee.} that the classifier will not change its answer as long as the input ${\\bf x}$ is modified by no more than an $L_2$-norm of $g\\left({\\bf x}\\right)\/\\sqrt{2}$.\nTherefore maximizing the average confidence gap directly boosts robustness and this motivates the second condition.\nTo explain the third condition, let us introduce the notion of preserving distance: the distance between any pair of input vectors with two different labels ought to be preserved as much as possible at the outputs, while we do not care about the distance between a pair with the same label.\nLet $d\\left({\\bf x_1},{\\bf x_2}\\right)$ denote the $L_2$-distance between the output logit-vectors for two input points ${\\bf x_1}$ and ${\\bf x_2}$ that have different labels and that are classified correctly.\nIt is straightforward to verify the condition\\footnote{See Lemma~\\ref{th:gapsqrt2} in Appendix~\\ref{sec:proofs} for the proof of the condition.} of $g\\left({\\bf x_1}\\right)+g\\left({\\bf x_2}\\right) \\leq \\sqrt{2} \\cdot d\\left({\\bf x_1},{\\bf x_2}\\right)$.\nTherefore a network that maximizes confidence gaps well must be one that preserves distance well.\nUltimately some distances are preserved while others are lost, and ideally the decision of which distance to lose is made by parameter training rather than by artifacts of network architecture.\nHence the third condition involves distance-preserving architecture choices that leave the decision to parameter training as much as possible, and this motivates many of our design decisions such as Sections~\\ref{sec:relu} and \\ref{sec:pool}.\n\nIn practice we employ the strategy of divide and conquer and build each layer as a nonexpansive map with respect to the $L_2$-norm.\nIt is straightforward to see that a feedforward network composed of nonexpansive layers must implement a nonexpansive map overall.\nHow to adapt subtleties like recursion and splitting-reconvergence is included in the appendix.\n\nBesides being robust against adversarial noises, L2NNNs have other desirable properties.\nThey generalize better from noisy training labels than ordinary networks: for example, when 75\\% of MNIST training labels are randomized, an L2NNN still achieves 93.1\\% accuracy on the test set, in contrast to 75.2\\% from the best ordinary network.\nThe problem of exploding gradients, which is common in training ordinary networks, is avoided because the gradient of any output with respect to any internal signal is bounded between -1 and 1.\nUnlike ordinary networks, the confidence gap of an L2NNN classifier is a quantitatively meaningful indication of confidence on individual data points, and the average gap is an indication of generalization.\n\n\n\n\\section{Conclusions and future work}\n\nIn this work we have presented $L_2$-nonexpansive neural networks which are well-conditioned systems by construction.\nPractical techniques are developed for building these networks.\nTheir properties are studied through experiments and benefits demonstrated, including that our MNIST and CIFAR-10 classifiers exceed the state of the art in robustness against white-box adversarial attacks, that they are robust against partially random training labels, and that they output confidence gaps which are strongly correlated with robustness and generalization.\nThere are a number of future directions, for example, other applications of L2NNN, L2NNN-friendly neural network architectures, and the relation between L2NNNs and interpretability.\n\n\n\\section{Related work} \\label{sec:liter}\n\nAdversarial defense is a well-known difficult problem \\citep{szegedy,goodfellow,carlini,athalye2018obfuscated,gilmer2018adversarial}.\nThere are many avenues to defense \\citep{carlini2017adversarial,meng2017magnet}, and here we will focus on defense works that fortify a neural network itself instead of introducing additional components.\n\nThe mainstream approach has been adversarial training, where examples of successful attacks on a classifier itself are used in training \\citep{tramer2017ensemble,zantedeschi2017efficient}.\nThe work of \\citet{mit} has the best results to date and effectively flattens gradients around training data points, and, prior to our work, it is the only work that achieves sizable white-box defense.\nIt has been reported in \\citet{mindistort} that, for a small network, adversarial training indeed increases the average minimum $L_1$-norm and $L_{\\infty}$-norm of noise needed to change its classification.\nHowever, in view of results of Tables~\\ref{tbl:mnist} and \\ref{tbl:cifar}, adversarial-training results may be susceptible to strong attacks.\nThe works of \\citet{drucker1992improving,ross2017improving} are similar to adversarial training in aiming to flatten gradients around training data set but use different mechanisms.\n\nWhile the above approaches fortify a network around training data points, others aim to bound a network's responses to input perturbations over the entire input space.\nFor example, \\citet{haber} models ResNet as an ordinary differential equation and derive stability conditions.\nOther examples include \\citet{kolter,raghu,wong2018scaling} which achieved provable guarantees against $L_{\\infty}$-bounded attacks.\nHowever there exist scalability issues with respect to network depth, and the reported results so far are against relatively weak attacks or low noise magnitude.\nAs shown in Table~\\ref{tbl:li}, we can match their measured $L_{\\infty}$-bounded defense.\n\nControlling Lipschitz constants also regularizes a network over the entire input space.\n\\citet{szegedy} is the seminal work that brings attention to this topic.\n\\citet{bartlett} proposes the notion of spectrally-normalized margins as an indicator of generalization, which are strongly related to our confidence gap.\n\\citet{pascanu2013difficulty} studies the role of the spectral radius of weight matrices in the vanishing and the exploding gradient problems.\n\\citet{yoshida2017spectral} proposes a method to regularize the spectral radius of weight matrices and shows its effect in reducing generalization gap.\nThe work on Parseval networks \\citep{cisse} shows that it is possible to control Lipschitz constants of neural networks through regularization.\nThe core of their work is to constrain linear and convolution layer weights to be composed of Parseval tight frames, i.e., orthonormal filters, and thereby force the Lipschitz constant of these layers to be 1; they also propose to restrict aggregation operations.\nThe reported robustness results of \\citet{cisse}, however, are much weaker than those by adversarial training in \\citet{mit}.\nWe differ from Parseval networks in a number of ways.\nOur linear and convolution layers do not require filters to be orthogonal to each other and subsume Parseval layers as a special case, and therefore provide more freedom to parameter training.\nWe use non-standard techniques, e.g. two-sided ReLU, to modify various network components to maximize confidence gaps while keeping the network nonexpansive, and we propose a new loss function for the same purpose.\nWe are unable to obtain Parseval networks for a direct comparison, however it is possible to get a rough idea of what the comparison might be by looking at Table~\\ref{tbl:les} which shows the impacts of those new techniques.\nThe work of \\citet{hein} makes an important point regarding guarantees provided by local Lipschitz constants, which helps explain many observations in our results, including why adversarial training on L2NNNs leads to lasting robustness gains.\nThe regularization proposed by \\citet{hein} however is less practical and again introduces reliance on the coverage of training data points.\n\n\n\\section{$L_2$-nonexpansive neural networks} \\label{sec:tech}\n\nThis section describes how to adapt some individual operators in neural networks for L2NNNs.\nDiscussions on splitting-reconvergence, recursion and normalization are in the appendix.\n\n\\subsection{Weights} \\label{sec:w}\n\nThis section covers both the matrix-vector multiplication in a fully connected layer and the convolution calculation between input tensor and weight tensor in a convolution layer. The convolution calculation can be viewed as a set of vector-matrix multiplications: we make shifted copies of the input tensor and shuffle the copies into a set of small vectors such that each vector contains input entries in one tile; we reshape the weight tensor into a matrix by flattening all but the dimension of the output filters; then convolution is equivalent to multiplying each of the said small vectors with the flattened weight matrix. Therefore, in both cases, a basic operator is ${\\bf y} = W {\\bf x}$.\nTo be a nonexpansive map with respect to the $L_2$-norm, a necessary and sufficient condition is\n\\begin{equation} \\label{eq:radius}\n\\begin{aligned}\n{\\bf y}^{\\mathrm T}{\\bf y} \\leq {\\bf x}^{\\mathrm T}{\\bf x} \\quad \\Longrightarrow \\quad {\\bf x}^{\\mathrm T}W^{\\mathrm T}W{\\bf x} & \\leq {\\bf x}^{\\mathrm T}{\\bf x},\\quad \\forall {\\bf x}\\in \\mathbb{R}^N \\\\\n\\rho\\left( W^{\\mathrm T}W \\right) & \\leq 1\n\\end{aligned}\n\\end{equation}\nwhere $\\rho$ denotes the spectral radius of a matrix.\n\nThe exact condition of (\\ref{eq:radius}) is difficult to incorporate into training. Instead we use an upper bound:\\footnote{The spectral radius of a matrix is no greater than its natural $L_{\\infty}$-norm. $W^{\\mathrm T}W$ and $WW^{\\mathrm T}$ have the same non-zero eigenvalues and hence the same spectral radius.}\n\\begin{equation} \\label{eq:bound}\n\\rho\\left( W^{\\mathrm T}W \\right) \\leq b\\left(W\\right) \\triangleq \\min \\left( r(W^{\\mathrm T}W), r(WW^{\\mathrm T}) \\right)\n, \\quad\n\\textrm{where } r \\left( M \\right) = \\max_i\\sum_j \\left| M_{i,j} \\right|\n\\end{equation}\nThe above is where our linear and convolution layers differ from those in \\citet{cisse}: they require $WW^{\\mathrm T}$ to be an identity matrix, and it is straightforward to see that their scheme is only one special case that makes $b\\left(W\\right)$ equal to 1.\nInstead of forcing filters to be orthogonal to each other, our bound of $b\\left(W\\right)$ provides parameter training with greater degrees of freedom.\n\nOne simple way to use (\\ref{eq:bound}) is replacing $W$ with $W^\\prime = W \/ \\sqrt{b\\left(W\\right)}$ in weight multiplications, and this would enforce that the layer is strictly nonexpansive.\nAnother method is described in the appendix.\n\nAs mentioned, convolution can be viewed as a first layer of making copies and a second layer of vector-matrix multiplications.\nWith the above regularization, the multiplication layer is nonexpansive.\nHence we only need to ensure that the copying layer is nonexpansive.\nFor filter size of $K_1$ by $K_2$ and strides of $S_1$ by $S_2$, we simply divide the input tensor by a factor of $\\sqrt{ \\lceil K_1 \/ S_1 \\rceil \\cdot \\lceil K_2 \/ S_2 \\rceil }$.\n\n\\subsection{ReLU and others} \\label{sec:relu}\n\nLet us turn our attention to the third condition from Section~\\ref{sec:intro}.\nReLU, tanh and sigmoid are nonexpansive but do not preserve distance well.\nThis section presents a method that improves ReLU and is generalizable to other nonlinearities.\nA different approach to improve sigmoid is in the appendix.\n\nTo understand the weakness of ReLU, let us consider two input data points A and B, and suppose that a ReLU in the network receives two different negative values for A and B and outputs zero for both.\nComparing the A-B distance before and after this ReLU layer, there is a distance loss and this particular ReLU contributes to it.\nWe use \\emph{two-sided ReLU} which is a function from $\\mathbb{R}$ to $\\mathbb{R}^2$ and simply computes ReLU($x$) and ReLU($-x$).\nTwo-sided ReLU has been studied in \\citet{shang2016understanding} in convolution layers for accuracy improvement.\nIt is straightforward to verify that two-sided ReLU is nonexpansive with respect to any $L_p$-norm and that it preserves distance in the above scenario.\nWe will empirically verify its effectiveness in increasing confidence gaps in Section~\\ref{sec:results}.\n\nTwo-sided ReLU is a special case of the following general technique.\nLet $f(x)$ be a nonexpansive and monotonically increasing scalar function, and note that ReLU, tanh and sigmoid all fit these conditions.\nWe can define a function from $\\mathbb{R}$ to $\\mathbb{R}^2$ that computes $f(x)$ and $f(x)-x$.\nSuch a new function is nonexpansive with respect to any $L_p$-norm\\footnote{See Lemma~\\ref{th:twoside} in Appendix~\\ref{sec:proofs} for the proof of nonexpansiveness.} and preserves distance better than $f(x)$ alone.\n\n\\subsection{Pooling} \\label{sec:pool}\n\nThe popular max-pooling is nonexpansive, but does not preserve distance as much as possible.\nConsider a scenario where the inputs to pooling are activations that represent edge detection, and consider two images A and B such that A contains an edge that passes a particular pooling window while B does not.\nInside this window, A has positive values while B has all zeroes.\nFor this window, the A-B distance before pooling is the $L_2$-norm of A's values, yet if max-pooling is used, the A-B distance after pooling becomes the largest of A's values, which can be substantially smaller than the former.\nThus we suffer a loss of distance between A and B while passing this pooling layer.\n\nWe replace max-pooling with norm-pooling, which was reported in \\citet{boureau2010theoretical} to occasionally increase accuracy.\nInstead of taking the max of values inside a pooling window, we take the $L_2$-norm of them.\nIt is straightforward to verify that norm-pooling is nonexpansive\\footnote{See Lemma~\\ref{th:pool} in Appendix~\\ref{sec:proofs} for the proof of nonexpansiveness.} and would entirely preserve the $L_2$-distance between A and B in the hypothetical scenario above.\nOther $L_p$-norms can also be used.\nWe will verify its effectiveness in increasing confidence gaps in Section~\\ref{sec:results}.\n\nIf pooling windows overlap, we divide the input tensor by $\\sqrt{K}$ where $K$ is the maximum number of pooling windows in which an entry can appear, similar to convolution layers discussed earlier.\n\n\\subsection{Loss function} \\label{sec:loss}\n\nFor a classifier with $K$ labels, we recommend building it as $K$ overlapping L2NNNs, each of which outputs a single logit for one label.\nIn an architecture with no split layers, this simply implies that these $K$ L2NNNs share all but the last linear layer and that the last linear layer is decomposed into $K$ single-output linear filters, one in each L2NNN.\nFor a multi-L2NNN classifier, we have a guarantee\\footnote{See Lemma~\\ref{th:multi2} in Appendix~\\ref{sec:proofs} for the proof of the guarantee. The guarantee in either Lemma~\\ref{th:sqrt2} or Lemma~\\ref{th:multi2} is only a loose guarantee and it has been shown in \\citet{hein} that a larger guarantee exists by analyzing local Lipschitz constants, though it is expensive to compute.} that the classifier will not change its answer as long as the input ${\\bf x}$ is modified by no more than an $L_2$-norm of $g\\left({\\bf x}\\right)\/2$, where again $g\\left({\\bf x}\\right)$ denotes the confidence gap.\nAs mentioned in Section~\\ref{sec:intro}, a single-L2NNN classifier has a guarantee of $g\\left({\\bf x}\\right)\/\\sqrt{2}$.\nAlthough this seems better on the surface, it is more difficult to achieve large confidence gaps.\nWe will assume the multi-L2NNN approach.\n\nWe use a loss function with three terms, with trade-off hyperparameters $\\gamma$ and $\\omega$:\n\\begin{equation} \\label{eq:lossall}\n\\mathcal{L} = \\mathcal{L}_a + \\gamma\\cdot\\mathcal{L}_b + \\omega\\cdot\\mathcal{L}_c\n\\end{equation}\n\nLet $y_1,y_2,\\cdots,y_K$ be outputs from the L2NNNs. The first loss term is\n\\begin{equation} \\label{eq:loss0}\n\\mathcal{L}_a = \\textrm{softmax-cross-entropy}\\left( u_1y_1, u_2y_2, \\cdots, u_Ky_K, \\textrm{label} \\right)\n\\end{equation}\nwhere $u_1,u_2,\\cdots,u_K$ are trainable parameters. The second loss term is\n\\begin{equation} \\label{eq:loss1}\n\\mathcal{L}_b = \\textrm{softmax-cross-entropy}\\left( vy_1, vy_2, \\cdots, vy_K, \\textrm{label} \\right)\n\\end{equation}\nwhere $v$ can be either a trainable parameter or a hyperparameter.\nNote that $u_1,u_2,\\cdots,u_K$ and $v$ are not part of the classifier and are not used during inference. The third loss term is\n\\begin{equation} \\label{eq:loss2}\n\\mathcal{L}_c = \\frac{\\textrm{average}\\left( \\log\\left( 1 - \\textrm{softmax}\\left(zy_1, zy_2, \\cdots, zy_K\\right)_\\textrm{label}\\right)  \\right)}{z}\n\\end{equation}\nwhere $z$ is a hyperparameter.\n\nThe rationale for the first loss term (\\ref{eq:loss0}) is that it mimics cross-entropy loss of an ordinary network.\nIf an ordinary network has been converted to L2NNNs by multiplying each layer with a small constant, its original outputs can be recovered by scaling up L2NNN outputs with certain constants, which is enabled by the formula (\\ref{eq:loss0}).\nHence this loss term is meant to guide the training process to discover any feature that an ordinary network can discover.\nThe rationale for the second loss term (\\ref{eq:loss1}) is that it is directly related to the classification accuracy.\nMultiplying L2NNN outputs uniformly with $v$ does not change the output label and only adapts to the value range of L2NNN outputs and drive towards better nominal accuracy.\nThe third loss term (\\ref{eq:loss2}) approximates average confidence gap: the log term is a soft measure of a confidence gap (for a correct prediction), and is asymptotically linear for larger gap values.\nThe hyperparameter $z$ controls the degree of softness, and has relatively low impact on the magnitude of loss due to the division by $z$; if we increase $z$ then (\\ref{eq:loss2}) asymptotically becomes the average of minus confidence gaps for correct predictions and zeroes for incorrect predictions.\nTherefore loss (\\ref{eq:loss2}) encourages large confidence gaps and yet is smooth and differentiable.\n\nA notable variation of (\\ref{eq:lossall}) is one that combines with adversarial training.\nOur implementation applies the technique of \\citet{mit} on the first loss term (\\ref{eq:loss0}): we use distorted inputs in calculating $\\mathcal{L}_a$.\nThe results are reported in Tables~\\ref{tbl:mnist} and \\ref{tbl:cifar} as Model 4.\nAnother possibility is to use distorted inputs in calculating $\\mathcal{L}_a$ and $\\mathcal{L}_b$, while $\\mathcal{L}_c$ should be based on original inputs.\n\n\n\\section{Experiments} \\label{sec:results}\n\nExperiments are divided into three groups to study different properties of L2NNNs.\nOur MNIST and CIFAR-10 classifiers are available at\\\\\n\\url{http:\/\/researcher.watson.ibm.com\/group\/9298}\n\n\\begin{figure}[th]\n\\centering\n\\includegraphics[width=0.1\\columnwidth]{1k}\n\\includegraphics[width=0.1\\columnwidth]{10k}\n\\caption{Attacks on Model 2 found after 1K and 10K iterations: the same 0 recognized as 5.}\n\\label{fig:zero}\n\\end{figure}\n\n\\subsection{Robustness} \\label{sec:defense}\n\nThis section evaluates robustness of L2NNN classifiers for MNIST and CIFAR-10 and compares against the state of the art \\citet{mit}.\nThe robustness metric is accuracy under white-box non-targeted $L_2$-bounded attacks. The attack code of \\citet{carlini} is used.\nWe downloaded the classifiers\\footnote{At {\\scriptsize\\rurl{github.com\/MadryLab\/mnist_challenge}} and {\\scriptsize\\rurl{github.com\/MadryLab\/cifar10_challenge}}.\nThese models (Model 2's in Tables~\\ref{tbl:mnist} and \\ref{tbl:cifar}) were built by adversarial training with $L_{\\infty}$-bounded adversaries \\citep{mit}.\nTo the best of our knowledge, \\citet{mittradeoff} from the same lab is the only paper in the literature that reports on models trained with $L_2$-bounded adversaries, and it reports that training with $L_2$-bounded adversaries resulted in weaker $L_2$ robustness than the $L_2$ robustness results from training with $L_{\\infty}$-bounded adversaries in \\citet{mit}.\nTherefore we choose to compare against the best available models, even though they were trained with $L_{\\infty}$-bounded adversaries.\nNote also that our own Model 4's in Tables~\\ref{tbl:mnist} and \\ref{tbl:cifar} are trained with the same $L_{\\infty}$-bounded adversaries.} of \\citet{mit} and report their robustness against $L_2$-bounded attacks in Tables~\\ref{tbl:mnist} and \\ref{tbl:cifar}.\\footnote{In reading Tables~\\ref{tbl:mnist} and \\ref{tbl:cifar}, it is worth remembering that the norm of after-attack accuracy is zero, and for example the 7.6\\% on MNIST is currently the state of the art.}\nNote that their defense diminishes as the attacks are allowed more iterations.\nFigure~\\ref{fig:zero} illustrates one example of this effect: the first image is an attack on MNIST Model 2 (0 recognized as 5) found after 1K iterations, with noise $L_2$-norm of 4.4, while the second picture is one found after 10K iterations, the same 0 recognized as 5, with noise $L_2$-norm of 2.1.\nWe hypothesize that adversarial training alone provides little absolute defense at the noise levels used in the two tables: adversarial examples still exist and are only more difficult to find.\nThe fact that in Table~\\ref{tbl:cifar} Model 2 accuracy is lower in the 1000x10 row than the 10K row further supports our hypothesis.\n\n\\begin{table}[th]\n\\caption{Accuracies of MNIST classifiers under white-box non-targeted attacks with noise $L_2$-norm limit of 3.\nMaxIter is the max number of iterations the attacker uses.\nModel 1 is an ordinarily trained model.\nModel 2 is the model from \\citet{mit}.\nModel 3 is L2NNN without adversarial training.\nModel 4 is L2NNN with adversarial training.}\n\\label{tbl:mnist}\n\\centering\n\\begin{tabular}{lcccc}\n\\toprule\nMaxIter & Model1 & Model2 & Model3 & Model4 \\\\\n\\midrule\nNatural & 99.1\\% & 98.5\\% & 98.7\\% & 98.2\\% \\\\\n100     & 70.2\\% & 91.7\\% & 77.6\\% & 75.6\\% \\\\\n1000    & 0.05\\% & 51.5\\% & 20.3\\% & 24.4\\% \\\\\n10K     &    0\\% & 16.0\\% & 20.1\\% & 24.4\\% \\\\\n100K    &    0\\% &  9.8\\% & 20.1\\% & 24.4\\% \\\\\n1M      &    0\\% &  7.6\\% & 20.1\\% & 24.4\\% \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[th]\n\\caption{Accuracies of CIFAR-10 classifiers under white-box non-targeted attacks with noise $L_2$-norm limit of 1.5.\nMaxIter is the max number of iterations the attacker uses, and 1000x10 indicates 10 runs each with 1000 iterations.\nModel 1 is an ordinarily network.\nModel 2 is the model from \\citet{mit}.\nModel 3 is L2NNN without adversarial training.\nModel 4 is L2NNN with adversarial training.}\n\\label{tbl:cifar}\n\\centering\n\\begin{tabular}{lcccc}\n\\toprule\nMaxIter & Model1 & Model2 & Model3 & Model4 \\\\\n\\midrule\nNatural & 95.0\\% & 87.1\\% & 79.2\\% & 77.2\\% \\\\\n100     &    0\\% & 13.9\\% & 10.2\\% & 20.8\\% \\\\\n1000    &    0\\% &  9.4\\% & 10.1\\% & 20.4\\% \\\\\n10K     &    0\\% &  9.0\\% & 10.1\\% & 20.4\\% \\\\\n1000x10 &    0\\% &  8.7\\% & 10.1\\% & 20.4\\% \\\\\n100K    &    0\\% &  NA    & 10.1\\% & 20.4\\% \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\nIn contrast, the defense of the L2NNN models remain constant when the attacks are allowed more iterations, specifically MNIST Models beyond 10K iterations and CIFAR-10 Models beyond 1000 iterations.\nThe reason is that L2NNN classifiers achieve their defense by creating a confidence gap between the largest logit and the rest, and that half of this gap is a lower bound of $L_2$-norm of distortion to the input data in order to change the classification.\nHence L2NNN's defense comes from a minimum-distortion guarantee.\nAlthough adversarial training alone may also increase the minimum distortion limit for misclassification, as suggested in \\citet{mindistort} for a small network, that limit likely does not reach the levels used in Tables~\\ref{tbl:mnist} and \\ref{tbl:cifar} and hence the defense depends on how likely the attacker can reach a lower-distortion misclassification.\nConsequently when the attacks are allowed to make more attempts the defense with guarantee stands while the other diminishes.\n\nFor both MNIST and CIFAR-10, adding adversarial training boosts the robustness of Model 4.\nWe hypothesize that adversarial training lowers local Lipschitz constants in certain parts of the input space, specifically around the training images, and therefore makes local robustness guarantees larger \\citep{hein}.\nTo test this hypothesis on MNIST Models 3 and 4, we measure the average $L_2$-norm of their Jacobian matrices, averaged over the first 1000 images in the test set, and the results are 1.05 for Model 3 and 0.83 for Model 4.\nNote that the $L_2$-norm of Jacobian can be greater than 1 for multi-L2NNN classifiers.\nThese measurements are consistent with, albeit does not prove, the hypothesis.\n\n\\begin{table}[th]\n\\caption{Ablation studies: MNIST model without weight regularization;\none without $\\mathcal{L}_c$ loss;\none with max-pooling instead of norm-pooling;\none without two-sided ReLU;\nGap is average confidence gap.\nR-Accu is under attacks with 1000 iterations and with noise $L_2$-norm limit of 3.}\n\\label{tbl:les}\n\\centering\n\\begin{tabular}{lccc}\n\\toprule\n        & Accu. & Gap & R-Accu. \\\\\n\\midrule\nno weight reg.          & 99.4\\% & 68.3 & 0\\%    \\\\\nno $\\mathcal{L}_c$ loss & 99.2\\% & 2.2  & 8.9\\%  \\\\\nno norm-pooling         & 98.8\\% & 1.3  & 9.9\\%  \\\\\nno two-sided ReLU       & 98.0\\% & 2.5  & 15.1\\% \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo test the effects of various components of our method, we build models for each of which we disable a different technique during training.\nThe results are reported in Table~\\ref{tbl:les}.\nTo put the confidence gap values in context, our MNIST Model 3 has an average gap of 2.8.\nThe first one is without weight regularization of Section~\\ref{sec:w} and it becomes an ordinary network which has little defense against adversarial attacks; its large average confidence gap is meaningless.\nFor the second one we remove the third loss term (\\ref{eq:loss2}) and for the third one we replace norm-pooling with regular max-pooling, both resulting in smaller average confidence gap and less defense against attacks.\nFor the fourth one, we replace two-sided ReLU with regular ReLU, and this leads to degradation in nominal accuracy, average confidence gap and robustness.\nParseval networks \\citep{cisse} can be viewed as models without $\\mathcal{L}_c$ term, norm-pooling or two-sided ReLU, and with a more restrictive scheme for weight matrix regularization.\n\nModel 3 in Table~\\ref{tbl:mnist} and the second row of Table~\\ref{tbl:les} are two points along a trade-off curve that are controllable by varying hyperparameter $\\omega$ in loss function (\\ref{eq:lossall}).\nOther trade-off points have nominal accuracy and under-attack accuracy of (98.8\\%,19.1\\%), (98.4\\%,22.6\\%) and (97.9\\%,24.7\\%) respectively.\nSimilar trade-offs have been reported by other robustness works including adversarial training \\citep{mittradeoff} and adversarial polytope \\citep{wong2018scaling}.\nIt remains an open question whether such trade-off is a necessary part of life, and please see Section~\\ref{sec:random} for further discussion on the L2NNN trade-off.\n\n\\begin{table}[th]\n\\caption{Accuracy of L2NNN classifiers under white-box non-targeted attacks with 1000 iterations and with noise $L_{\\infty}$-norm limit of $\\epsilon$.}\n\\label{tbl:li}\n\\centering\n\\begin{tabular}{lccc}\n\\toprule\n     & $\\epsilon$ & Model3 & Model4 \\\\\n\\midrule\nMNIST    & 0.1    & 90.9\\% & 92.4\\% \\\\\nMNIST    & 0.3    & 7.0\\%  & 44.0\\% \\\\\nCIFAR-10 & $\\nicefrac{8}{256}$  & 32.3\\% & 42.5\\% \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\nAlthough we primarily focus on defending against $L_2$-bounded adversarial attacks in this work,\nwe achieve some level of robustness against $L_{\\infty}$-bounded attacks as a by-product.\nTable~\\ref{tbl:li} shows our results, again measured with the attack code of \\citet{carlini}.\nThe $\\epsilon$ values match those used in \\citet{raghu,kolter,mit}.\nOur MNIST $L_{\\infty}$ results are on par with \\citet{raghu,kolter} but not as good as \\citet{mit}.\nOur CIFAR-10 Model 4 is on par with \\citet{mit} for $L_{\\infty}$ defense.\n\n\\subsection{Meaningful outputs} \\label{sec:interp}\n\nThis section discusses how to understand and utilize L2NNNs' output values. We observe strong correlation between the confidence gap of L2NNN and the magnitude of distortion needed to force it to misclassify, and images are included in appendix.\n\nIn the next experiment, we sort test data by the confidence gap of a classifier on each image.\nThen we divide the sorted data into 10 bins and report accuracy separately on each bin in Figure~\\ref{fig:bars}.\nWe repeat this experiment for Model 2 \\citep{mit} and our Model 3 of Tables~\\ref{tbl:mnist} and \\ref{tbl:cifar}.\nNote that the L2NNN model shows better correlation between confidence and robustness:\nfor MNIST our first bin is 95\\% robust and second bin is 67\\% robust.\nThis indicates that the L2NNN outputs are much more quantitatively meaningful than those of ordinary neural networks.\n\n\\begin{figure}[th]\n\\centering\n\\includegraphics[width=\\columnwidth]{bars}\n\\caption{Accuracy percentages of classifiers on test data bin-sorted by the confidence gap.}\n\\label{fig:bars}\n\\end{figure}\n\nIt is an important property that an L2NNN has an easily accessible measurement on how robust its decisions are.\nSince robustness is easily measurable, it can be optimized directly, and we believe that this is the primary reason that we can demonstrate the robustness results of Tables~\\ref{tbl:mnist} and \\ref{tbl:cifar}.\nThis can also be valuable in real-life applications where we need to quantify how reliable a decision is.\n\nOne of the other practical implications of this property is that we can form hybrid models which use L2NNN outputs when the confidence is high and a different model when the confidence of the L2NNN is low.\nThis creates another dimension of trade-off between nominal accuracy and robustness that one can take advantage of in an application.\nWe built such a hybrid model for MNIST with the switch threshold of 1.0 and achieved nominal accuracy of 99.3\\%, where only 6.9\\% of images were delegated to the alternative classifier.\nWe built such a hybrid model for CIFAR-10 with the switch threshold of 0.1 and achieved nominal accuracy of 89.4\\%, where 25\\% of images were delegated.\nTo put these threshold values in context, MNIST Model 3 has an average gap of 2.8 and CIFAR-10 Model 3 has an average gap of 0.34.\nIn other words, if for a data point the L2NNN confidence gap is substantially below average, the classification is delegated to the alternative classifier, and this way we can recover nominal accuracy at a moderate cost of robustness.\n\n\\subsection{Generalization versus memorization} \\label{sec:random}\n\nThis section studies L2NNN's generalization through a noisy-data experiment where we randomize some or all MNIST training labels.\nThe setup is similar to \\citet{zhang_iclr17}, except that we added three scenarios where 25\\%, 50\\% and 75\\% of training labels are scrambled.\n\n\\begin{table}[th]\n\\caption{Accuracy comparison of MNIST classifiers that are trained on noisy data.\nRand is the percentage of training labels that are randomized.\nWD is weight decay.\nDR is dropout.\nES is early stopping.\nGap1 is L2NNN's average confidence gap on training set and Gap2 is that on test set.}\n\\label{tbl:rand}\n\\centering\n\\begin{tabular}{lcccccccc}\n\\toprule\nRand & \\multicolumn{5}{c}{Ordinary network}      & \\multicolumn{3}{c}{L2NNN} \\\\\n     & Vanilla & WD     & DR     & ES     & WD+DR+ES &        & Gap1 & Gap2 \\\\\n\\midrule\n 0    & 99.4\\% & 99.0\\% & 99.2\\% & 99.0\\% & 99.3\\%   & 98.7\\% & 2.84 & 2.82   \\\\\n 25\\% & 90.4\\% & 91.1\\% & 91.8\\% & 96.2\\% & 98.0\\%   & 98.5\\% & 0.64 & 0.63   \\\\\n 50\\% & 65.5\\% & 67.7\\% & 72.6\\% & 81.0\\% & 88.3\\%   & 96.0\\% & 0.58 & 0.60   \\\\\n 75\\% & 41.5\\% & 44.9\\% & 41.8\\% & 75.2\\% & 66.4\\%   & 93.1\\% & 0.86 & 0.89   \\\\\n 100\\%&  9.7\\% &  9.1\\% &  9.4\\% & NA     & NA       & 11.9\\% & 0.09 & 0.01   \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[th]\n\\caption{Training-accuracy-versus-confidence-gap trade-off points of L2NNNs on 50\\%-scrambled MNIST training labels.}\n\\label{tbl:fifty}\n\\centering\n\\begin{tabular}{cccc}\n\\toprule\n\\multicolumn{2}{c}{on training set} & \\multicolumn{2}{c}{on test set} \\\\\n Accu. & Gap & Accu. & Gap \\\\\n\\midrule\n 98.7\\% & 0.17   & 79.0\\% & 0.12   \\\\\n 96.5\\% & 0.21   & 79.3\\% & 0.18   \\\\\n 89.4\\% & 0.22   & 86.3\\% & 0.20   \\\\\n 70.1\\% & 0.36   & 93.4\\% & 0.37   \\\\\n 66.1\\% & 0.45   & 93.7\\% & 0.47   \\\\\n 59.8\\% & 0.58   & 96.0\\% & 0.60   \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\nTable~\\ref{tbl:rand} shows the comparison between L2NNNs and ordinary networks.\nDropout rate and weight-decay weight are tuned for each WD\/DR run, and each WD+DR+ES run uses the combined hyperparameters from its row.\nIn early-stopping runs, 5000 training images are withheld as validation set and training stops when loss on validation set stops decreasing.\nThe L2NNNs do not use weight decay, dropout or early stopping.\nL2NNNs achieve the best accuracy in all three partially-scrambled scenarios, and it is remarkable that an L2NNN can deliver 93.1\\% accuracy on test set when three quarters of training labels are random.\nMore detailed data and discussions are in the appendix.\n\nTo illustrate why L2NNNs generalize better than ordinary networks from noisy data, we show in Table~\\ref{tbl:fifty} trade-off points between accuracy and confidence gap on the 50\\%-scrambled training set.\nThese trade-off points are achieved by changing hyperparameters $\\omega$ in (\\ref{eq:lossall}) and $v$ in (\\ref{eq:loss1}).\nIn a noisy training set, there exist data points that are close to each other yet have different labels.\nFor a pair of such points, if an L2NNN is to classify both points correctly, the two confidence gaps must be small.\nTherefore, in order to achieve large average confidence gap, an L2NNN must misclassify some of the training data.\nIn Table~\\ref{tbl:fifty}, as we adjust the loss function to favor larger average gap, the L2NNNs are forced to make more and more mistakes on the training set.\nThe results suggest that loss is minimized when an L2NNN misclassifies some of the scrambled labels while fitting the 50\\% original labels with large gaps, and parameter training discovers this trade-off automatically.\nHence we see in Table~\\ref{tbl:fifty} increasing accuracies and gaps on the test set.\nThe above is a trade-off between memorization (training-set accuracy) and generalization (training-set average gap), and we hypothesize that L2NNN's trade-off between nominal accuracy and robustness, reported in Section~\\ref{sec:defense}, is due to the same mechanism.\nTo be fair, dropout and early stopping are also able to sacrifice accuracy on a noisy training set, however they do so through different mechanisms that tend to be brittle, and Table~\\ref{tbl:rand} suggests that L2NNN's mechanism is superior.\nMore discussions and the trade-off tables for 25\\% and 75\\% scenarios are in the appendix.\n\nAnother interesting observation is that the average confidence gap dramatically shrinks in the last row of Table~\\ref{tbl:rand} where the training is pure memorization.\nThis is not surprising again due to training data points that are close to each other yet have different labels.\nThe practical implication is that after an L2NNN model is trained, one can simply measure its average confidence gap to know whether and how much it has learned to generalize rather than to memorize the training data.\n\n\n\n\\section{$L_2$-nonexpansive network components} \\label{sec:parts}\n\n\\subsection{Additional methods for weight regularization}\n\nThere are numerous ways to utilize the bound of (\\ref{eq:bound}).\nThe main text describes a simple method of using $W^\\prime = W \/ \\sqrt{b\\left(W\\right)}$ to enforce strict nonexpansiveness.\nThe following is an alternative.\n\nApproximate nonexpansiveness can be achieved by adding a penalty to the loss function whenever $b\\left(W\\right)$ exceeds 1, for example:\n\\begin{equation} \\label{eq:loss}\n\\mathcal{L}_W = \\min \\left( l(W^{\\mathrm T}W), l(WW^{\\mathrm T}) \\right)\n, \\textrm{ where } l \\left( M \\right) = \\sum_i \\max\\left(\\sum_j \\left| M_{i,j} \\right|-1,0\\right)\n\\end{equation}\nThe sum of (\\ref{eq:loss}) losses over all layers becomes a fourth term in the loss function (\\ref{eq:lossall}), multiplied with one additional hyperparameter. This would lead to an approximate L2NNN with trade-offs between how much its layers violate (\\ref{eq:radius}) with surrogate (\\ref{eq:bound}) versus other objectives in the loss function.\n\nIn practice, we have found that it is beneficial to begin neural network training with the regularization scheme of (\\ref{eq:loss}), which allows larger learning rates, and switch to the first scheme of using $W^\\prime$, which avoids artifacts of an extra hyperparameter, when close to convergence.\nOf course if the goal is building approximate L2NNNs one can use (\\ref{eq:loss}) all the way.\n\n\\subsection{Sigmoid and others}\n\nSigmoid is nonexpansive as is, but does not preserve distance as much as possible. A better way is to replace sigmoid with the following operator\n\\begin{equation} \\label{eq:sigmoid}\ns\\left(x\\right) = t \\cdot \\mathrm{sigmoid} \\left( \\frac {4x}{t}  \\right)\n\\end{equation}\nwhere $t>0$ is a trainable parameter and each neuron has its own $t$.\nIn general, the requirement for any scalar nonlinearity is that its derivative is bounded between -1 and 1. If a nonlinearity violates this condition, a shrinking multiplier can be applied. If the actual range of derivative is narrower, as in the case of sigmoid, an enlarging multiplier can be applied to preserve distance.\n\nFor further improvement, (\\ref{eq:sigmoid}) can be combined with the general form of the two-sided ReLU of Section~2.2.\nThen the new nonlinearity is a function from $\\mathbb{R}$ to $\\mathbb{R}^2$ that computes $s(x)$ and $s(x)-x$.\n\n\\subsection{Splitting and reconvergence}\n\nThere are different kinds of splitting in neural networks. Some splitting is not followed by reconvergence. For example, a classifier may have common layers followed by split layers for each label, and such an architecture can be viewed as multiple L2NNNs that overlap at the common layers and each contain one stack of split layers. In such cases, no modification is needed because there is no splitting within each individual L2NNN.\n\nSome splitting, however, is followed by reconvergence. In fact, convolution and pooling layers discussed earlier can be viewed as splitting, and reconvergence happens at the next layer. Another common example is skip-level connections such as in ResNet. Such splitting should be viewed as making two copies of a certain vector. Let the before-split vector be ${\\bf x}_0$, and we make two copies as\n\\begin{equation} \\label{eq:split}\n\\begin{aligned}\n{\\bf x}_1 & = t \\cdot {\\bf x}_0 \\\\\n{\\bf x}_2 & = \\sqrt{1-t^2} \\cdot {\\bf x}_0\n\\end{aligned}\n\\end{equation}\nwhere $t \\in \\left[0,1\\right]$ is a trainable parameter.\n\nIn the case of ResNet, the reconvergence is an add operator, which should be treated as vector-matrix multiplication as in Section~2.1, but with much simplified forms. Let ${\\bf x}_1$ be the skip-level connections and $f\\left({\\bf x}_2\\right)$ be the channels of convolution outputs to be added with ${\\bf x}_1$, we perform the addition as\n\\begin{equation} \\label{eq:reconv}\n{\\bf y} =  t \\cdot {\\bf x}_1 + \\sqrt{1-t^2} \\cdot f\\left({\\bf x}_2\\right)\n\\end{equation}\nwhere $t \\in \\left[0,1\\right]$ is a trainable parameter and could be a common parameter with (\\ref{eq:split}).\n\nResNet-like reconvergence is referred to as aggregation layers in \\citet{cisse} and a different formula was used:\n\\begin{equation}\n{\\bf y} =  \\alpha \\cdot {\\bf x}_1 + \\left(1-\\alpha\\right) \\cdot f\\left({\\bf x}_2\\right)\n\\end{equation}\nwhere $\\alpha \\in \\left[0,1\\right]$ is a trainable parameter.\nBecause splitting is not modified in \\citet{cisse}, their scheme may seem approximately equivalent to ours if a common $t$ parameter is used for (\\ref{eq:split}) and (\\ref{eq:reconv}).\nHowever, there is a substantial difference: in many ResNet blocks, $f\\left({\\bf x}_2\\right)$ is a subset of rather than all of the output channels of convolution layers, and our scheme does not apply the shrinking factor of $\\sqrt{1-t^2}$ on channels that are not part of $f\\left({\\bf x}_2\\right)$ and therefore better preserve distances.\nIn contrast, because splitting is not modified, at reconvergence the scheme of \\citet{cisse} must apply the shrinking factor of $1-\\alpha$ on all outputs of convolution layers, regardless of whether a channel is part of the aggregation or not.\nTo state the difference in more general terms, our scheme enables splitting and reconvergence at arbitrary levels of granularity and multiplies shrinking factors to only the necessary components.\nWe can also have a different $t$ per channel or even per entry.\n\nTo be fair, the scheme of \\citet{cisse} has an advantage of being nonexpansive with respect to any $L_p$-norm.\nHowever, for $L_2$-norm, it is inferior to ours in preserving distances and maximizing confidence gaps.\n\n\\subsection{Recursion} \\label{sec:rnn}\n\nThere are multiple ways to interpret recurrent neural networks (RNN) as L2NNNs. One way is to view an unrolled RNN as multiple overlapping L2NNNs where each L2NNN generates the output at one time step. Under this interpretation, nothing special is needed and recurrent inputs to a neuron are simply treated as ordinary inputs.\n\nAnother way to interpret an RNN is to view unrolled RNN as a single L2NNN that generates outputs at all time steps. Under this interpretation, recurrent connections are treated as splitting at their sources and should be handled as in (\\ref{eq:split}).\n\n\\subsection{Normalization}\n\nNormalization operations are limited in an L2NNN.\nSubtracting mean is nonexpansive and allowed, and subtract-mean operation can be performed on arbitrary subsets of any layer.\nSubtracting batch mean is also allowed because it can be viewed as subtracting a bias parameter.\nHowever, scaling, e.g., division by standard deviation or batch standard deviation is only allowed if the multiplying factors are between -1 and 1.\nTo satisfy this in practice, one simple method is to divide all multiplying factors in a normalization layer by the largest of their absolute values.\n\n\\section{MNIST images}\n\n\\begin{figure}[th]\n\\centering\n\\includegraphics[width=0.87\\columnwidth]{biggap}\\\\\nGap $\\quad$\\,\\, 5.1 $\\quad$\\,\\, 4.4 $\\quad$\\,\\, 5.1 $\\quad$\\,\\, 4.6 $\\quad$\\,\\, 5.0 $\\quad$\\,\\, 4.4 $\\quad$\\,\\, 4.6 $\\quad$\\,\\, 4.5 $\\quad$\\,\\, 4.0 $\\quad$\\,\\, 3.1 \\\\\n\\includegraphics[width=0.87\\columnwidth]{biggap2}\\\\\nMstk $\\quad$\\, 5 $\\quad\\quad$\\, 8 $\\quad\\quad$\\, 3 $\\quad\\quad$\\, 5 $\\quad\\quad$\\, 9 $\\quad\\quad$\\, 8 $\\quad\\quad$ 2 $\\quad\\quad$\\, 5 $\\quad\\quad$ 0 $\\quad\\quad$ 7 \\, \\\\\nDist $\\quad$\\, 4.8 $\\quad$\\,\\, 3.6 $\\quad$\\,\\, 4.8 $\\quad$\\,\\,\\, 3.4 $\\quad$\\,\\, 4.1 $\\quad$\\,\\,  3.7 $\\quad$\\,\\, 4.0 $\\quad$\\,\\, 4.5$\\quad$\\,\\,\\, 3.8 $\\quad$\\,\\, 2.3\n\\caption{Original and distorted images of MNIST digits in test set with the largest confidence gaps.\nMstk denotes the misclassified labels. Dist denotes the $L_2$-norm of the distortion noise.}\n\\label{fig:digits1}\n\\end{figure}\n\n\\begin{figure}[th]\n\\centering\n\\includegraphics[width=0.9\\columnwidth]{smallgap}\\\\\nGap $\\quad$\\,\\,\\,\\, 0.03 $\\quad$\\,\\, 0.2 $\\quad$\\,\\, 0.1 $\\quad$\\,\\, 0.03 $\\quad$\\, 0.03 $\\quad$       0.001 $\\quad$\\, 0.3 $\\quad$\\,   0.06 \\,\\,\\,\\,\\, 0.01 \\,\\,\\,\\, 0.005\\\\\n\\includegraphics[width=0.9\\columnwidth]{smallgap2}\\\\\nMstk $\\quad$\\,\\,\\,\\, 8 $\\quad\\quad$\\,\\, 6 $\\quad\\quad$\\, 8 $\\quad\\quad$\\, 5 $\\quad\\quad$\\,\\, 9 $\\quad\\quad$\\,\\, 3 $\\quad\\quad$\\,\\,\\, 5 $\\quad\\quad$\\, 2 $\\quad\\quad$\\, 3 $\\quad\\quad$ 7 \\,\\,\\, \\\\\nDist $\\quad$\\,\\,\\, 0.04 $\\quad$\\,\\, 0.3 $\\quad$\\,\\, 0.1 $\\quad$\\,\\, 0.02 $\\quad$\\, 0.03 $\\quad$        0.001 $\\quad$\\,   0.3 $\\quad$\\,   0.05 \\,\\,\\,\\,\\,   0.02 \\,\\,\\,\\, 0.01\n\\caption{Original and distorted images of MNIST digits in test set with the smallest confidence gaps.\nMstk denotes the misclassified output label. Dist denotes the $L_2$-norm of the distortion noise.}\n\\label{fig:digits2}\n\\end{figure}\n\nLet us begin by showing MNIST images with the largest confidence gaps in Figure~\\ref{fig:digits1} and those with the smallest confidence gaps in Figure~\\ref{fig:digits2}.\nThey include images before and after attacks as well as Model 3's confidence gap, the misclassified label and $L_2$-norm of the added noise.\nThe images with large confidence gaps seem to be ones that are most different from other digits, while some of the images with small confidence gaps are genuinely ambiguous.\nIt's worth noting the strong correlation between the confidence gap of L2NNN and the magnitude of distortion needed to force it to misclassify.\nAlso note that our guarantee states that the minimum $L_2$-norm of noise is half of the confidence gap, but in reality the needed noise is much stronger than the guarantee.\nThe reason is that the true local guarantee is in fact larger due to local Lipschitz constants, as pointed out by \\citet{hein}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.1\\columnwidth]{orig}\n\\includegraphics[width=0.1\\columnwidth]{1k}\n\\includegraphics[width=0.1\\columnwidth]{10k}\n\\includegraphics[width=0.1\\columnwidth]{mine_1m}\n\\caption{Original image of 0; attack on Model 2 \\citep{mit} found after 1K iterations; attack on Model 2 found after 10K iterations; attack on Model 3 (L2NNN) found after 1M iterations. The latter three all lead to misclassification as 5.}\n\\label{fig:zeroes}\n\\end{figure}\n\nFigure~\\ref{fig:zeroes} shows additional details regarding the example in Figure~\\ref{fig:zero}.\nThe first image is the original image of a zero.\nThe second image is an attack on Model 2 \\citep{mit} found after 1K iterations, with noise $L_2$-norm of 4.4.\nThe third is one found after 10K iterations for Model 2, with noise $L_2$-norm of 2.1.\nThe last image is the best attack on our Model 3 found after one million iterations, with noise $L_2$-norm of 3.5.\nThese illustrates the trend shown in Table~\\ref{tbl:mnist} that the defense by adversarial training diminishes as the attacks are allowed more iterations, while L2NNNs withstand strong attacks and it requires more noise to fool an L2NNN.\nIt's worth noting that the slow degradation of Model 2's accuracy is an artifact of the attacker \\citep{carlini}: when gradients are near zero in some parts of the input space, which is true for MNIST Model 2 due to adversarial training, it takes more iterations to make progress.\nIt is conceivable that, with a more advanced attacker, Model 2 could drop quickly to 7.6\\%.\nWhat truly matter are the robust accuracies where we advance the state of the art from 7.6\\% to 24.4\\%.\n\n\\section{Details of scrambled-label experiments}\n\nFor ordinary networks in Table~\\ref{tbl:rand}, we use two network architectures.\nThe first has 4 layers and is the architecture used in \\citet{mit}.\nThe second has 22 layers and is the architecture of Models 3 and 4 in Table~\\ref{tbl:mnist}, which includes norm-pooling and two-sided ReLU.\nResults of ordinary networks using these two architectures are in Tables~\\ref{tbl:shallow} and \\ref{tbl:deep} respectively.\nThe ordinary-network section of Table~\\ref{tbl:rand} is entry-wise max of Tables~\\ref{tbl:shallow} and \\ref{tbl:deep}.\n\n\\begin{table}[th]\n\\caption{Accuracies of non-L2NNN MNIST classifiers that use a 4-layer architecture and that are trained on training data with various amounts of scrambled labels.\nRand is the percentage of training labels that are randomized.\nWD is weight decay.\nDR is dropout.\nES is early stopping.}\n\\label{tbl:shallow}\n\\centering\n\\begin{tabular}{lccccc}\n\\toprule\nRand & \\multicolumn{5}{c}{Ordinary network}    \\\\\n     & Vanilla & WD     & DR     & ES     & WD+DR+ES  \\\\\n\\midrule\n 0    & 98.9\\% & 99.0\\% & 99.2\\% & 99.0\\% & 99.3\\%  \\\\\n 25\\% & 82.5\\% & 91.1\\% & 91.8\\% & 79.1\\% & 98.0\\%   \\\\\n 50\\% & 57.7\\% & 67.7\\% & 72.6\\% & 66.4\\% & 88.3\\%  \\\\\n 75\\% & 32.1\\% & 44.9\\% & 41.8\\% & 52.7\\% & 66.4\\%   \\\\\n 100\\%&  9.5\\% &  8.9\\% &  9.4\\% & NA     & NA    \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[th]\n\\caption{Accuracies of non-L2NNN MNIST classifiers that use a 22-layer architecture and that are trained on training data with various amounts of scrambled labels.\nRand is the percentage of training labels that are randomized.\nWD is weight decay.\nDR is dropout.\nES is early stopping.}\n\\label{tbl:deep}\n\\centering\n\\begin{tabular}{lccccc}\n\\toprule\nRand & \\multicolumn{5}{c}{Ordinary network}    \\\\\n     & Vanilla & WD     & DR     & ES     & WD+DR+ES  \\\\\n\\midrule\n 0    & 99.4\\% & 99.0\\% & 99.0\\% & 99.0\\% & 99.0\\%  \\\\\n 25\\% & 90.4\\% & 86.5\\% & 89.8\\% & 96.2\\% & 90.3\\%   \\\\\n 50\\% & 65.5\\% & 62.5\\% & 63.7\\% & 81.0\\% & 83.1\\%  \\\\\n 75\\% & 41.5\\% & 38.2\\% & 40.2\\% & 75.2\\% & 61.9\\%   \\\\\n 100\\%&  9.7\\% &  9.1\\% &  8.8\\% & NA     & NA      \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\nIn Tables~\\ref{tbl:shallow} and \\ref{tbl:deep}, dropout rate and weight-decay weight are tuned for each WD\/DR run, and each WD+DR+ES run uses the combined hyperparameters from its row.\nIn early-stopping runs, 5000 training images are withheld as validation set and training stops when loss on validation set stops decreasing.\nEach ES or WD+DR+ES entry is an average over ten runs to account for randomness of the validation set.\nThe L2NNNs do not use weight decay, dropout or early stopping.\n\n\\begin{table}[th]\n\\caption{Training-accuracy-versus-confidence-gap trade-off points of L2NNNs on 25\\%-scrambled MNIST training labels.}\n\\label{tbl:twentyfive}\n\\centering\n\\begin{tabular}{cccc}\n\\toprule\n\\multicolumn{2}{c}{on training set} & \\multicolumn{2}{c}{on test set} \\\\\n Accu. & Gap & Accu. & Gap \\\\\n\\midrule\n 99.6\\% & 0.12   & 92.6\\% & 0.10   \\\\\n 97.6\\% & 0.20   & 95.7\\% & 0.17   \\\\\n 78.6\\% & 0.31   & 98.2\\% & 0.30   \\\\\n 77.2\\% & 0.64   & 98.5\\% & 0.63   \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[th]\n\\caption{Training-accuracy-versus-confidence-gap trade-off points of L2NNNs on 75\\%-scrambled MNIST training labels.}\n\\label{tbl:seventyfive}\n\\centering\n\\begin{tabular}{cccc}\n\\toprule\n\\multicolumn{2}{c}{on training set} & \\multicolumn{2}{c}{on test set} \\\\\n Accu. & Gap & Accu. & Gap \\\\\n\\midrule\n 97.9\\% & 0.07   & 49.8\\% & 0.03   \\\\\n 93.0\\% & 0.09   & 59.2\\% & 0.05   \\\\\n 75.9\\% & 0.10   & 70.0\\% & 0.08   \\\\\n 58.0\\% & 0.18   & 80.4\\% & 0.17   \\\\\n 46.2\\% & 0.29   & 86.8\\% & 0.30   \\\\\n 40.1\\% & 0.44   & 89.8\\% & 0.46   \\\\\n 34.7\\% & 0.86   & 93.1\\% & 0.89   \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\nTable~\\ref{tbl:twentyfive} shows L2NNN trade-off points between accuracy and confidence gap on the 25\\%-scrambled training set.\nTable~\\ref{tbl:seventyfive} shows L2NNN trade-off points between accuracy and confidence gap on the 75\\%-scrambled training set.\nLike Table~\\ref{tbl:fifty}, they demonstrate the trade-off mechanism between memorization (training-set accuracy) and generalization (training-set average gap).\n\nTo be fair, dropout and early stopping are also able to sacrifice accuracy on a noisy training set.\nFor example, the DR run in the 50\\%-scrambled row in Table~\\ref{tbl:shallow} has 67.5\\% accuracy on the training set and 72.6\\% on the test set.\nHowever, the underlying mechanisms are very different from that of L2NNN.\nDropout \\citep{dropout} has an effect of data augmentation, and, with a noisy training set, dropout can create a situation where the effective data complexity exceeds the network capacity.\nTherefore, the parameter training is stalled at a lowered accuracy on the training set, and we get better performance if the model tends to fit more of original labels and less of the scrambled labels.\nThe mechanism of early stopping is straightforward and simply stops the training when it is mostly memorizing scrambled labels.\nWe get better performance from early stopping if the parameter training tends to fit the original labels early.\nThese mechanisms from dropout and early stopping are both brittle and may not allow parameter training enough opportunity to learn from the useful data points with original labels.\nThe comparison in Table~\\ref{tbl:rand} suggests that they are inferior to L2NNN's trade-off mechanism as discussed in Section~\\ref{sec:random} and illustrated in Tables~\\ref{tbl:fifty}, \\ref{tbl:twentyfive} and \\ref{tbl:seventyfive}.\nThe L2NNNs in this paper do not use weight decay, dropout or early stopping, however it is conceivable that dropout may be complementary to L2NNNs.\n\n\\section{Proofs}\\label{sec:proofs}\n\n\\newtheorem{lemma}{Lemma}\n\\begin{lemma}\\label{th:sqrt2}\nLet $g\\left({\\bf x}\\right)$ denote a single-L2NNN classifier's confidence gap for an input data point ${\\bf x}$.\nThe classifier will not change its answer as long as the input ${\\bf x}$ is modified by no more than an $L_2$-norm of $g\\left({\\bf x}\\right)\/\\sqrt{2}$.\n\\end{lemma}\n\n\\begin{proof}\nLet ${\\bf y}\\left({\\bf x}\\right) = \\left[ y_1\\left({\\bf x}\\right),y_2\\left({\\bf x}\\right),\\cdots,y_K\\left({\\bf x}\\right) \\right]$ denote logit vector of a single-L2NNN classifier for an input data point ${\\bf x}$.\nLet ${\\bf x_1}$ and ${\\bf x_2}$ be two input vectors such that the classifier outputs different labels $i$ and $j$.\nBy definitions, we have the following inequalities:\n\\begin{equation}\n\\begin{aligned}\ny_i\\left({\\bf x_1}\\right) - y_j\\left({\\bf x_1}\\right) & \\geq g\\left({\\bf x_1}\\right) \\\\\ny_i\\left({\\bf x_2}\\right) - y_j\\left({\\bf x_2}\\right) & \\leq 0\n\\end{aligned}\n\\end{equation}\nBecause the classifier is a single L2NNN, it must be true that:\n\\begin{equation}\n\\begin{aligned}\n\\| {\\bf x_2} - {\\bf x_1} \\|_2 & \\geq \\| {\\bf y}\\left({\\bf x_2}\\right) - {\\bf y}\\left({\\bf x_1}\\right) \\|_2 \\\\\n & \\geq \\sqrt{ \\left( y_i\\left({\\bf x_2}\\right) - y_i\\left({\\bf x_1}\\right) \\right)^2 + \\left( y_j\\left({\\bf x_2}\\right) - y_j\\left({\\bf x_1}\\right) \\right)^2 } \\\\\n & =    \\sqrt{ \\left( y_i\\left({\\bf x_1}\\right) - y_i\\left({\\bf x_2}\\right) \\right)^2 + \\left( y_j\\left({\\bf x_2}\\right) - y_j\\left({\\bf x_1}\\right) \\right)^2 } \\\\\n & \\geq \\sqrt{\\frac{ \\left( y_i\\left({\\bf x_1}\\right) - y_i\\left({\\bf x_2}\\right) + y_j\\left({\\bf x_2}\\right) - y_j\\left({\\bf x_1}\\right) \\right)^2 }{2}} \\\\\n & =   \\sqrt{\\frac{ \\left( \\left( y_i\\left({\\bf x_1}\\right) - y_j\\left({\\bf x_1}\\right) \\right) + \\left( y_j\\left({\\bf x_2}\\right) - y_i\\left({\\bf x_2}\\right) \\right) \\right)^2 }{2}} \\\\\n & \\geq \\sqrt{\\frac{ \\left( g\\left({\\bf x_1}\\right) + 0 \\right)^2 }{2}} \\\\\n & = g\\left({\\bf x_1}\\right)\/\\sqrt{2}\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\n\n\\begin{lemma} \\label{th:gapsqrt2}\nLet $g\\left({\\bf x}\\right)$ denote a classifier's confidence gap for an input data point ${\\bf x}$.\nLet $d\\left({\\bf x_1},{\\bf x_2}\\right)$ denote the $L_2$-distance between the output logit-vectors for two input points ${\\bf x_1}$ and ${\\bf x_2}$ that have different labels and that are classified correctly.\nThen this condition holds: $g\\left({\\bf x_1}\\right)+g\\left({\\bf x_2}\\right) \\leq \\sqrt{2} \\cdot d\\left({\\bf x_1},{\\bf x_2}\\right)$.\n\\end{lemma}\n\n\\begin{proof}\nLet ${\\bf y}\\left({\\bf x}\\right) = \\left[ y_1\\left({\\bf x}\\right),y_2\\left({\\bf x}\\right),\\cdots,y_K\\left({\\bf x}\\right) \\right]$ denote logit vector of a classifier for an input data point ${\\bf x}$.\nLet $i$ and $j$ be the labels for ${\\bf x_1}$ and ${\\bf x_2}$.\nBy definitions, we have the following inequalities:\n\\begin{equation}\n\\begin{aligned}\ny_i\\left({\\bf x_1}\\right) - y_j\\left({\\bf x_1}\\right) & \\geq g\\left({\\bf x_1}\\right) \\\\\ny_j\\left({\\bf x_2}\\right) - y_i\\left({\\bf x_2}\\right) & \\geq g\\left({\\bf x_2}\\right)\n\\end{aligned}\n\\end{equation}\nTherefore,\n\\begin{equation}\n\\begin{aligned}\nd\\left({\\bf x_1},{\\bf x_2}\\right) & \\triangleq \\| {\\bf y}\\left({\\bf x_2}\\right) - {\\bf y}\\left({\\bf x_1}\\right) \\|_2 \\\\\n & \\geq \\sqrt{ \\left( y_i\\left({\\bf x_2}\\right) - y_i\\left({\\bf x_1}\\right) \\right)^2 + \\left( y_j\\left({\\bf x_2}\\right) - y_j\\left({\\bf x_1}\\right) \\right)^2 } \\\\\n & =    \\sqrt{ \\left( y_i\\left({\\bf x_1}\\right) - y_i\\left({\\bf x_2}\\right) \\right)^2 + \\left( y_j\\left({\\bf x_2}\\right) - y_j\\left({\\bf x_1}\\right) \\right)^2 } \\\\\n & \\geq \\sqrt{\\frac{ \\left( y_i\\left({\\bf x_1}\\right) - y_i\\left({\\bf x_2}\\right) + y_j\\left({\\bf x_2}\\right) - y_j\\left({\\bf x_1}\\right) \\right)^2 }2} \\\\\n & =   \\sqrt{\\frac{ \\left( \\left( y_i\\left({\\bf x_1}\\right) - y_j\\left({\\bf x_1}\\right) \\right) + \\left( y_j\\left({\\bf x_2}\\right) - y_i\\left({\\bf x_2}\\right) \\right) \\right)^2 }2} \\\\\n & \\geq \\sqrt{\\frac{ \\left( g\\left({\\bf x_1}\\right) + g\\left({\\bf x_2}\\right) \\right)^2 }2} \\\\\n & = \\frac{ g\\left({\\bf x_1}\\right) + g\\left({\\bf x_2}\\right) }{\\sqrt{2}}\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\n\\begin{lemma} \\label{th:ineq}\nFor any $a \\geq 0$, $b \\geq 0$, $p \\geq 1$, the following inequality holds: $a^p + b^p \\leq \\left( a+b \\right)^p$.\n\\end{lemma}\n\n\\begin{proof}\nIf $a$ and $b$ are both zero, the inequality holds.\nIf at least one of $a$ and $b$ is nonzero:\n\\begin{equation}\n\\begin{aligned}\na^p + b^p & = \\left( a+b \\right)^p\\cdot\\left( \\frac{a}{a+b} \\right)^p + \\left( a+b \\right)^p\\cdot\\left( \\frac{b}{a+b} \\right)^p \\\\\n & \\leq \\left( a+b \\right)^p\\cdot\\frac{a}{a+b} + \\left( a+b \\right)^p\\cdot\\frac{b}{a+b} \\\\\n & = \\left( a+b \\right)^p\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\n\n\\begin{lemma} \\label{th:twoside}\nLet $f(x)$ be a nonexpansive and monotonically increasing scalar function.\nDefine a function from $\\mathbb{R}$ to $\\mathbb{R}^2$: ${\\bf h}(x)=[f(x),f(x)-x]$.\nThen ${\\bf h}(x)$ is nonexpansive with respect to any $L_p$-norm.\n\\end{lemma}\n\n\\begin{proof}\nFor any $x_1>x_2$, by definition we have the following inequalities:\n\\begin{equation}\n\\begin{aligned}\nf(x_1)-f(x_2) & \\geq 0 \\\\\nf(x_1)-f(x_2) & \\leq x_1-x_2\n\\end{aligned}\n\\end{equation}\nFor any $p \\geq 1$, invoking Lemma~\\ref{th:ineq} with $a=f(x_1)-f(x_2)$ and $b=x_1-x_2-f(x_1)+f(x_2)$, we have:\n\\begin{equation}\n\\begin{aligned}\n\\left((f(x_1)-f(x_2)\\right)^p + \\left(x_1-x_2-f(x_1)+f(x_2)\\right)^p & \\leq \\left( x_1-x_2 \\right)^p \\\\\n\\left(\\left((f(x_1)-f(x_2)\\right)^p + \\left(x_1-x_2-f(x_1)+f(x_2)\\right)^p\\right)^{1\/p} & \\leq  x_1-x_2 \\\\\n\\left(\\lvert f(x_1)-f(x_2) \\rvert^p + \\lvert (f(x_1)-x_1)-(f(x_2)-x_2) \\rvert^p\\right)^{1\/p} & \\leq x_1-x_2 \\\\\n\\| {\\bf h}(x_1) - {\\bf h}(x_2) \\|_p & \\leq x_1-x_2\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\n\\begin{lemma} \\label{th:pool}\nNorm-pooling within each pooling window is a nonexpansive map with respect to $L_2$-norm.\n\\end{lemma}\n\n\\begin{proof}\nLet ${\\bf x_1}$ and ${\\bf x_2}$ be two vectors with the size of a pooling window.\nBy triangle inequality, we have\n\\begin{equation}\n\\begin{aligned}\n\\| {\\bf x_1} - {\\bf x_2} \\|_2 + \\| {\\bf x_1} \\|_2 & \\geq \\| {\\bf x_2} \\|_2 \\\\\n\\| {\\bf x_1} - {\\bf x_2} \\|_2 + \\| {\\bf x_2} \\|_2 & \\geq \\| {\\bf x_1} \\|_2\n\\end{aligned}\n\\end{equation}\nTherefore,\n\\begin{equation}\n\\begin{aligned}\n\\| {\\bf x_1} - {\\bf x_2} \\|_2 & \\geq \\| {\\bf x_2} \\|_2 - \\| {\\bf x_1} \\|_2 \\\\\n\\| {\\bf x_1} - {\\bf x_2} \\|_2 & \\geq \\| {\\bf x_1} \\|_2 - \\| {\\bf x_2} \\|_2 \n\\end{aligned}\n\\end{equation}\nTherefore,\n\\begin{equation}\n\\| {\\bf x_1} - {\\bf x_2} \\|_2 \\geq \\lvert \\| {\\bf x_1} \\|_2 - \\| {\\bf x_2} \\|_2 \\rvert\n\\end{equation}\n\\end{proof}\n\n\n\\begin{lemma} \\label{th:multi2}\nLet $g\\left({\\bf x}\\right)$ denote a multi-L2NNN classifier's confidence gap for an input data point ${\\bf x}$.\nThe classifier will not change its answer as long as the input ${\\bf x}$ is modified by no more than an $L_2$-norm of $g\\left({\\bf x}\\right)\/2$.\n\\end{lemma}\n\n\\begin{proof}\nLet ${\\bf y}\\left({\\bf x}\\right) = \\left[ y_1\\left({\\bf x}\\right),y_2\\left({\\bf x}\\right),\\cdots,y_K\\left({\\bf x}\\right) \\right]$ denote logit vector of a multi-L2NNN classifier for an input data point ${\\bf x}$.\nLet ${\\bf x_1}$ and ${\\bf x_2}$ be two input vectors such that the classifier outputs different labels $i$ and $j$.\nBy definitions, we have the following inequalities:\n\\begin{equation}\n\\begin{aligned}\ny_i\\left({\\bf x_1}\\right) - y_j\\left({\\bf x_1}\\right) & \\geq g\\left({\\bf x_1}\\right) \\\\\ny_i\\left({\\bf x_2}\\right) - y_j\\left({\\bf x_2}\\right) & \\leq 0\n\\end{aligned}\n\\end{equation}\nFor a multi-L2NNN classifier, each logit is a nonexpansive function of the input, and it must be true that:\n\\begin{equation}\n\\begin{aligned}\n\\| {\\bf x_2} - {\\bf x_1} \\|_2 & \\geq \\lvert y_i\\left({\\bf x_1}\\right) - y_i\\left({\\bf x_2}\\right) \\rvert \\\\\n\\| {\\bf x_2} - {\\bf x_1} \\|_2 & \\geq \\lvert y_j\\left({\\bf x_2}\\right) - y_j\\left({\\bf x_1}\\right) \\rvert\n\\end{aligned}\n\\end{equation}\nTherefore,\n\\begin{equation}\n\\begin{aligned}\n\\| {\\bf x_2} - {\\bf x_1} \\|_2 & \\geq \\frac{ \\lvert y_i\\left({\\bf x_1}\\right) - y_i\\left({\\bf x_2}\\right) \\rvert + \\lvert y_j\\left({\\bf x_2}\\right) - y_j\\left({\\bf x_1}\\right) \\rvert }{2} \\\\\n & \\geq \\frac{ \\lvert y_i\\left({\\bf x_1}\\right) - y_i\\left({\\bf x_2}\\right) + y_j\\left({\\bf x_2}\\right) - y_j\\left({\\bf x_1}\\right) \\rvert }{2} \\\\\n & = \\frac{ \\lvert \\left( y_i\\left({\\bf x_1}\\right) - y_j\\left({\\bf x_1}\\right) \\right) + \\left( y_j\\left({\\bf x_2}\\right) - y_i\\left({\\bf x_2}\\right) \\right) \\rvert }{2} \\\\\n & \\geq \\frac{ \\lvert g\\left({\\bf x_1}\\right) + 0 \\rvert }{2} \\\\\n & = g\\left({\\bf x_1}\\right)\/2\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:level1}Introduction}\n$Introduction-$ In recent years, several acheivements have been reported~\\cite{Displacement, Goetz1, Goetz2, Pogorzalek, SanzIEEE, Federov}, which are now the potential building blocks of microwave quantum communication protocols. This novel research area is not only useful for free space communications, but also a candidate for chip-to-chip communication, required by distributed quantum computing. The latter represents an alternative paradigm to increasing the number of qubits in a single quantum processor, and aims at solving larger quantum algorithms in a distributed form between different processors with lower number of qubits. Nowadays, one of the best quantum platforms suited for quantum computing is superconducting circuits, which interact via microwave photons. The main challenge concerns the efficient distribution of such microwave states between circuits, the two main options being direct state transfer and teleportation. Concerning the former, several experiments have been done in a single cryogenic environment~\\cite{Axline, Campagne, Kurpiers, Leung, Roch, Narla, Dickel}, as well as with microwave to optical conversion~\\cite{Forsch, Rueda}. Recently, a successful transfer of transmon qubits has been reported, via a cryogenic waveguide coherently linking two dilution refrigerators separated by five meters, with average transfer and target state fidelities of 85.8 \\% and 79.5 \\%, respectively, in terms of discrete variables~\\cite{SuperQLAN}. \n\nSo far, microwave teleportation between two fridges has not been investigated experimentally, neither in terms of discrete variables nor continuous variables (CVs). However, a teleportation scheme in the microwave regime in terms of CVs has been proposed before~\\cite{Roberto}. Taking into account the limitations of the previous protocol, we investigate the feasibility of an experimental implementation of microwave quantum teleportation of Gaussian states in real conditions, e.g. fridge and free space, in terms of CVs based on a clear physical formalism. Indeed, in quantum information processing with CVs, Gaussian states play important roles~\\cite{Adesso1, Ferraro}, which are well-known and commonly-used experimentally. Here, we focus on teleportation of single-mode Gaussian states via entangled two-mode Gaussian states.\n\n$Quantum\\;\\;Teleportation\\;\\;with\\;\\;CVs-$A preliminary CV model for teleportation was proposed by Vaidman~\\cite{Vaidman}, and then developed by Braunstein and Kimble~\\cite{Braunstein}. The latter represented a conditional teleportation protocol, whereas an unconditional one was proposed by Furusawa et al.~\\cite{Furusawa1}. \n\nQuantum teleportation uses quantum entanglement and classical communication to transfer quantum information between two distant parties, Alice and Bob. The ideal scenario involves Alice and Bob sharing a maximally-entangled state, which in CVs translates to a two-mode squeezed vacuum (TMSV) state with infinite squeezing $r$. In a realistic scenario, the squeezing $r$ has technological limitations. In such protocol, Alice attempts to send a Gaussian state, with covariance matrix $ V'_{\\text{in}}$ and first moments $ \\bar{x}^{\\prime}_{\\text{in}}=( x_{\\text{in}}, p_{\\text{in}})$ (see Appendix) to Bob. The input state can also be described by a harmonic oscillator mode $\\alpha_{\\text{in}}=\\frac{x_{\\text{in}}+ip_{\\text{in}}}{\\sqrt{2}}$. If this process is successful, then Alice and Bob share a Gaussian entangled state with covariance matrix ${ V_{\\text{TMSS}}}$ and null first moments~\\cite{Serafini}. Then, after entangling the input state with TMSS, Alice makes a double homodyne measurement of both quadratures (i.e. a heterodyne detection) and modulates the classical results, $X_{\\text{u}}$ and $P_{\\text{v}}$, in the form of a single mode $\\delta=\\frac{X_{\\text{u}}+iP_{\\text{v}}}{\\sqrt{2}}$ and then sends it to Bob.  Finally, Bob applies a unitary displacement with a function of $\\delta$ to his share of the original entangled state to reconstruct the input state (see Fig.~\\ref{fig1}). Different teleportation protocols in terms of CVs are discussed in Ref.~\\cite{Mancini}. In the following, we focus basically on the Braunstein-Kimble protocol~\\cite{Braunstein} for teleportation of Gaussian states.\n\n\\begin{figure}\n\\includegraphics[scale=0.3]{Teleportation2.pdf}\n\\caption{The Braunstein-Kimble Protocol for the Teleportation of Gaussian States. In this protocol, two parties, Alice and Bob, share an entangled quantum state with two modes $\\alpha_1$ and $\\alpha_2$. In fact, Alice wants to send a single-mode Gaussian state (i.e. mode $\\alpha_{\\text{in}}$) to Bob via the teleportation mechanism where she cannot send the original mode $\\alpha_{\\text{in}}$ to Bob directly but a classical mode $\\delta$ is sent instead. Finally, Bob can reconstruct the input state by applying the displacement $D(\\delta)$ operator on his mode $\\alpha_2$ to construct the input state as $\\alpha_{\\text{out}}\\approx \\alpha_{\\text{in}}$.}\n\\label{fig1}\n\\end{figure}\n\n\n\n$Measurement\\;\\;by\\;\\;Alice-$When Alice receives the mixture of input mode and one mode of the entangled state, she performs a double homodyne detection (see Fig.~\\ref{fig2}), which is the optimal measurement for the teleportation protocol. In this case, the results of the measurement (which are two classical values, $X_u$ and $P_v$) will be modulated as a single mode $\\delta$ to be sent to Bob. As seen in Fig.~\\ref{fig2}, two modes $\\alpha_{\\text{in}}$ and $\\alpha_1$ are passing through a 50:50 beamsplitter, and then again each output beam again passes through two other 50:50 beamsplitters while a classical local oscillator mode $\\alpha_{\\text{LO}_x}$ enters in the top beamsplitter and another classical local oscillator mode $\\alpha_{\\text{LO}_p}$ passes through the other beamsplitter. According to Fig.~\\ref{fig2} in each line for quantum modes we use the annihiliation operators, and $\\alpha$ as classical coherent state for the local oscillator. Finally, there would be four outputs at the end, where there is a detector at each output which can measure the current produced due to the collision of photons to the detector.\n\\begin{figure}\n\\includegraphics[scale=0.35]{Heterodyne.pdf}\n\\caption{Alice's measurement is performed via double homodyne detection. First, two modes $\\alpha_{\\text{in}}$ and $\\alpha_1$ passing through a 50:50 beamsplitter, and then again each output beam again passes through two other 50:50 beamsplitters while a classical local oscillator mode $\\alpha_{\\text{LO}_x}$ enters in the top beamsplitter and another classical local oscillator mode $\\alpha_{\\text{LO}_p}$ passes through the other beamsplitter. Then, Alice performs a double homodyne detection where the classical result of the measurement will be modulated as a single mode $\\delta$ to be sent to Bob.}\n\\label{fig2}\n\\end{figure}\n\n\nIn the symplectic representation (see Appendix), we consider two symplectic operators in the double-homodyne detection circuit depicted in Fig.~\\ref{fig2}, $ S_{\\text{h1}}= \\mathbb{I}_2 \\oplus B_{\\text{S}}(1\/2)_4 \\oplus \\mathbb{I}_2$ and $ S_{\\text{h2}}= B_{\\text{S}}(1\/2)_4 \\oplus B_{\\text{S}}(1\/2)_4$ where $\\mathbb{I}_2$ is the identity 2$\\times$2matrix, and $B_{\\text{S}}(1\/2)_4$ is the 50:50 beamsplitter operator, i.e. a 4$\\times$4 matrix. The input first moment in the double-homodyne detection is ${ {\\bf \\bar{x'}}_{\\text{in}}}=( x_{\\text{LO}_x}, p_{\\text{LO}_x}, x_{\\text{in}}, p_{\\text{in}}, e^{r}x_1, e^{-r}p_1, x_{\\text{LO}_p}, p_{\\text{LO}_p})^{ T}$, and therefore the output's first moments at the photodetectors right before the measurement are $ { \\bar{x'}}_{\\text{out}}= S_{\\text{h2}}S_{\\text{h1}}{ \\bar{x'}}_{\\text{in}}$, which gives ${ { \\bar{x'}}_{\\text{out}}}=\\sqrt{2}( {(x_{u'}), (p_{u'}), (x_{u''}), (p_{u''}), (x_{v'}), (p_{v'}), (x_{v''}), (p_{v''}))^T}$.\n\nHere, we assume that the detectors are ideal, therefore we define the produced current as $i= \\langle \\hat{n}\\rangle=\\langle \\hat{a}^\\dagger\\hat{a}\\rangle=\\langle 1\/2(\\hat{x}^2+\\hat{p}^2)-1\/2\\rangle=1\/2(x^2+p^2)-1\/2$. Then, the difference between the currents from each detector after the beam splitters is measured~\\cite{Braunstein2}. The current from the top beamsplitter is the difference between the currents in $u'$ and $u''$, i.e.    \n$i_1=\\langle\\hat{a}^{\\dagger}_{u'}\\hat{a}_u'\\rangle-\\langle\\hat{a}^{\\dagger}_{u''}\\hat{a}_{u''}\\rangle$,\nand in the other two arms $i_2=\\langle\\hat{a}^{\\dagger}_{v'}\\hat{a}_{v'}\\rangle-\\langle\\hat{a}^{\\dagger}_{v''}\\hat{a}_{v''}\\rangle$\nAssuming $\\alpha_{\\text{LO}_x}=|\\alpha_{\\text{LO}_x}|e^{i\\theta_x}$, one obtains $i_1=|\\alpha_{\\text LO_x}|(x_{\\text{in}}+e^rx_1)$ by letting $\\theta_x=0$ that consequently can obtain $X_u=\\frac{i_1}{|\\alpha_{\\text LO_x}|}=x_{\\text{in}}+e^rx_1=\\text{Re}(\\delta)$. Similarly $\\alpha_{\\text{LO}_p}=|\\alpha_{\\text{LO}_p}|e^{i\\theta_p}$, if we let $\\theta_p=\\pi\/2$ then it turns to $i_2=|\\alpha_{\\text LO_p}|(p_{\\text{in}}-e^{-r}p_1)$, and therefore $P_v=\\frac{i_2}{|\\alpha_{\\text LO_p}|}=p_{\\text{in}}-e^{-r}p_1=\\text{Im}(\\delta)$. By using classical coherent light with similar amplitudes for both local oscillators, one can write $|\\alpha_{\\text LO_x}|=|\\alpha_{\\text LO_p}|=|\\alpha_{\\text LO}|$, so the currents $i_1$ and $i_2$ can be modulated into a single mode as $\\delta=X_u+iP_v$, to be sent and received by Bob. On the other side, at the same time that Alice measures her state, the state at Bob collapses to $\\alpha_2=\\frac{e^rx_2+ie^{-r}p_2}{\\sqrt{2}}$ with $x_2=-x_1$ and $p_2=p_1$. Comparing $\\delta$ and $\\alpha_2$ we realize that $\\delta+\\alpha_2=\\alpha_{\\text{in}}$. Once Bob receives this information, he can reconstruct Alice's input state by applying the displacement $D(\\delta)$ on his mode, i.e. $D(\\delta) |\\alpha_2\\rangle=|\\delta+\\alpha_2\\rangle=|\\alpha_{\\text{out}}\\rangle\\approx |\\alpha_{\\text{in}}\\rangle$. In the symplectic representation, this means that the first moments of Bob, ${\\bar{x}}_2=(x_2, p_2)^T$ should be displaced with ${ \\Delta}=(X_u, P_v)^T$, which means\n\\begin{equation}\n{ \\bar{x}}_2 \\rightarrow { \\bar{x}}_2+{ \\Delta}.\n\\end{equation}\nThe performance of the teleportation protocol can be measured by the teleportation fidelity $F$. It can be computed via a symplectic approach~\\cite{Weedbrook, Mancini}, i.e. $F=2\/\\sqrt{\\text{det}({ \\Gamma})}$, in which $ \\Gamma=2{ V'_{\\text{in}}+\\mathbb{Z}A\\mathbb{Z}+B-C\\mathbb{Z}-\\mathbb{Z}^TC^T}$ where $A$, $B$, and $C$ are the block matrices of two-mode squeezed states (TMSS) in symplectic representation, i.e. shown by a block matrix as\n$ V_{\\text{TMSS}}=\\left[A, C; C^T, B\\right]$, where $ A=A^T$, $ B=B^T$ and $ C$ is a $2\\times 2$ real matrix, and $T$ denotes transpose~\\cite{Serafini}. If the input is a squeezed coherent state, i.e. ${V'_{\\text{in}}}=\\big(\\begin{smallmatrix} e^{2y} & 0\\\\ 0 & e^{-2y} \\end{smallmatrix}\\big)$ where $y$ is the squeezing level of the input, the fidelity in general form is obtained as \n\\begin{equation}\t\nF=\\frac{1}{\\sqrt{(e^{-2y}+(2n+1)\\sigma)(e^{+2y}+(2n+1)\\sigma)}}.\n\\end{equation}\nwhere $\\sigma=\\exp{(-2r)}$ is the variance of the resource, and $n$ is the number of thermal photons in the two-mode squeezed thermal states (TMSTS) resource as a general form for TMSS. The particular case $n=0$ represents the fidelity for the general case when the resource is TMSV (see Appendix). One can compute the average fidelity for the teleportation of an arbitrary squeezed displaced vacuum state by integration over fidelity in the range 0 and 1.\n\n\n$\\;Microwave\\;\\;Quantum\\;\\;Teleportation-$ Microwave quantum communication, as an exciting line of research with potentially broad applications in science and industry, is an accessible technology due to the recent achievements of circuit quantum electrodynamics (cQED). Since thermal noise in the microwave domain are much larger than in the optical one, losses have to be taken into account, which can significantly affect the quality of the teleportation protocol. In order to suppress thermal fluctuations, macroscopic superconducting circuit operate at low temperatures, i.e. $T<10-100$mK~\\cite{Roberto}. In cQED, superconducting Josephson junctions are nonlinear elements which have essential applications in quantum computation and quantum information. Recently, path-entanglement between propagating quantum microwaves as TMSS was generated via Josephson parametric amplifiers (JPAs) and hybrid ring \\cite{Pogorzalek} to be used to perform microwave protocol equivalent to the traditional ones in optical quantum teleportation. In this protocol, a TMSS is generated from two single-mode squeezed states, with squeezing in orthogonal quadratures, generated by two JPAs, J$_1$ and J$_2$ at the same squeezing level $r$ with possible endogenous thermal photons, and then sending them through a hybrid ring, which is a microwave beam splitter~\\cite{Roberto, Pogorzalek}. In general, the quality of the entanglement between the two modes is affected by thermal fluctuations on the JPA during the generation of the single-mode squeezed states. Then, after the interaction of one mode with the input state via 50:50 beamsplitter, the outputs are connected to other two JPAs, J$_3$ and J$_4$, which operate as amplifiers with equal gain $g_{\\text J}=\\exp(2r_{\\text J})$ with squeezing parameter $r_{\\text J}$ (see Fig.~\\ref{fig3}). The final step is the measurement by Alice via heterodyne detection. The method is the same as what we explained before (and in Appendix) but with taking amplifications and losses into account. Transfer efficiencies are modelled by beamsplitters with reflectivities $\\epsilon$, $\\eta$, $\\kappa$, and $\\nu$, where the losses are indeed $1- \\epsilon$, $1-\\eta$, $1-\\kappa$, and $1-\\nu$, respectively. The reflectivity $\\eta$ represents the interaction of TMSS in free space. Other losses are related to JPAs and amplifiers. For simplicity, we choose $\\epsilon_1=\\epsilon$, $\\epsilon_2=1$, $\\eta_1=\\eta$, and $\\eta_2=1$. If the single-mode operator J$_{\\text {in}}$ squeezes the input coherent state with squeezing parameter $y$, e.g. ${ J}_{\\text {in}}=[e^{-y}, 0; e^{y}, 0]$, it produces a squeezed coherent state as $\\alpha_{in}=(e^{-y}x_{\\text{in}}+ie^{y}p_{\\text{in}})\/\\sqrt{2}$, which is the state to be teleported. Following the procedure in heterodyne detection, the output components can be obtained as $ X_u =  |\\alpha_{\\text{LO}}| \\sqrt{\\nu\\kappa g_{\\text {J}}} [ e^{-y} x_{\\text{in}}+(e^rx_1 \\sqrt{\\eta\\epsilon}+\\zeta_{x_1})]\\cos (\\theta_x)$ and $P_u =  |\\alpha_{\\text{LO}}| \\sqrt{\\frac{\\nu\\kappa}{g_{\\text {J}}}} [e^{ y} p_{\\text{in}}+(e^{-r}p_1 \\sqrt{\\eta\\epsilon}+\\zeta_{p_1})] \\sin (\\theta _x)$\nas the real and imaginary parts, respectively. By letting $\\theta_x=0$ the current $i_1$ turns to \n\n\\begin{equation}\nX_u= [|\\alpha_{\\text{LO}}|(\\sqrt{\\nu\\kappa g_{\\text {J}}}  (e^rx_1\\sqrt{\\eta\\epsilon}+e^{- y} x_{\\text{in}})+\\zeta_{x})]  \n\\end{equation}\n\nand $P_u=0$ where $\\zeta_{x}$ is the noise term, i.e. $\\zeta_{x}=x_{\\text{th-1}}\\sqrt{\\nu\\kappa\\eta(1-\\epsilon)}+x_{\\text{th-2}}\\sqrt{\\nu\\kappa(1-\\eta)}+x_{\\text{th-3}}\\sqrt{\\nu(1-\\kappa)}+x_{\\text{th-4}}\\sqrt{1-\\nu}$ where $x_{\\text{th-i}}$ ($i$=1, 2, 3, 4) are thermal quadratures at temperatures $T_1$, $T_2$, $T_3$, and $T_4$ respectively. If there are no losses ($\\epsilon=\\eta=\\kappa=\\nu=1$), then the current is $i_1= |\\alpha_{\\text{LO}}|\\sqrt{g_{\\text {f}}}  (e^rx_1+e^{- y} x_{\\text{in}})$. Similarly for current $i_2$, one can obtain  $X_v= |\\alpha_{\\text{LO}}|\\sqrt{\\frac{\\nu\\kappa}{g_{\\text {J}}}}  [e^{- y} x_{\\text{in}}-(e^{r}x_1 \\sqrt{\\eta\\epsilon}-\\zeta_{x_2}) ] \\cos (\\theta_p)$ and $P_v= |\\alpha_{\\text{LO}}| \\sqrt{\\nu\\kappa g_{\\text {J}}} [e^{ y} p_{\\text{in}}-(e^{-r}p_1 \\sqrt{\\eta\\epsilon}-\\zeta_{p_2})] \\sin (\\theta _p)$. Again, under similar conditions, but letting $\\theta_p=\\pi\/2$ the current $i_2$ turns to \n\\begin{equation}\nP_v= |\\alpha_{\\text{LO}}|[\\sqrt{\\nu\\kappa g_{\\text {J}}}  ( e^{ y} p_{\\text{in}}-e^{-r}p_1\\sqrt{\\eta\\epsilon})+\\zeta_{p}]\n\\end{equation}\nand $X_v=0$, where the noise term $\\zeta_{p}$ is $\\zeta_{p}=p_{\\text{th-1}}\\sqrt{\\nu\\kappa\\eta(1-\\epsilon)}+p_{\\text{th-2}}\\sqrt{\\nu\\kappa(1-\\eta)}+p_{\\text{th-3}}\\sqrt{\\nu(1-\\kappa)}+p_{\\text{th-4}}\\sqrt{1-\\nu}$. If there is no any loss, then the noise term turns to zero, and $i_2=|\\alpha_{\\text{LO}}|\\sqrt{g_{\\text {J}}}( e^{ y} p_{\\text{in}}-e^{-r}p_1)$. \n \n The above results are in agreement with the heterodyne outputs (see the Appendix) by letting $g_{\\text {J}}=1$ and $y=0$ for coherent states in the BK protocol. In reality, there are always losses for microwave signals. So, it is desirable to reduce the losses as far as possible to have a successful quantum protocol, otherwise the noisy signal received by Bob makes it harder to reconstruct the input state. In fact, the realization of a microwave single-photon detector is a difficult task due to the low energy of microwave photons, therefore measuring a quadrature of a weak microwave signal is hard. Thus, amplification of the signal is required. Cryogenic high electronic mobility transistor (HEMT) amplifiers are routinely used in quantum microwave experiments because of their large gains in a relatively broad frequency band. Basically, HEMT amplifiers are phase insensitive and add a significant amount of noise photons that may disrupt the protocol (see section E. on Amplification). These signals are digitized with analog-to-digital (ADC) converters and sent to a computer for digital data processing~\\cite{Pogorzalek}. In terms of ADC in heterodyne detection, quadrature moments are $I_1$, $ I_2$, $Q_1$, and $Q_2$ (see Fig.~\\ref{fig3}) which are calculated and averaged in the computer. The $I$ and $Q$ components can be described in terms of continuous variables $x$ and $p$, so they can be written as $I_i=\\sqrt{\\hbar \\omega_iBRg_{\\text H}}x_i$ and $Q_i=\\sqrt{\\hbar \\omega_iBRg_{\\text H}}p_i$, where $R=50\\Omega$, $B$ is the measurement bandwidth set by a digital filter, and $g_{\\text H}$ is the HEMT gain.\n  \n\\begin{figure}\n\\includegraphics[scale=0.32]{Teleportationex3.pdf}\n\\caption{Microwave quantum teleportation circuit for Gaussian states}\n\\label{fig3}\n\\end{figure}\nThe modulated classical signal $\\delta=I_1+iQ_2=\\sqrt{\\hbar \\omega_iBRg_{\\text H}}(X_u+iP_v)$ is communicated classically with Bob, where\n\\begin{eqnarray}\n\\nonumber I_1 &=&  \\sqrt{\\hbar \\omega_iBRg_{\\text H}}   [|\\alpha_{\\text{LO}}|(\\sqrt{\\nu\\kappa g_{\\text {J}}}  (e^{- y} x_{\\text{in}}+\\sqrt{\\eta\\epsilon}e^rx_1+\\zeta'_{x})], \\\\\nQ_2 &=&   \\sqrt{\\hbar \\omega_iBRg_{\\text H}} [|\\alpha_{\\text{LO}}|(\\sqrt{\\nu\\kappa g_{\\text {J}}}  ( e^{ y} p_{\\text{in}}-\\sqrt{\\eta\\epsilon}e^{-r}p_1+\\zeta'_{p})].\n\\end{eqnarray}\n\nwhere $\\zeta'_{x}=\\frac{\\zeta_{x}}{\\sqrt{\\nu\\kappa g_{\\text J}}}$, and $\\zeta'_{p}=\\frac{\\zeta_{p}}{\\sqrt{\\nu\\kappa g_{\\text J}}}$. The above states $I_1$ and $Q_2$ can be sent (or prepared at distant) to Bob via remote state preparation (RSP)\\cite{Pogorzalek}. Then Bob displaces his mode, i.e. $\\alpha_2=(e^rx_2+ie^{-r}p_2)\/\\sqrt{2}$, according to the received signal. In fact, after the measurement the values of entangled states already turn to $x_2=-x_1$ and $p_2=p_1$. The displacement on Bob's side is implemented with a directional coupler and is described as an asymmetric beam splitter with transmissivity $\\tau$, which is $B_S(\\tau)_4=[\\sqrt{1-\\tau}\\mathbb{I}_2, \\sqrt{\\tau}\\mathbb{I}_2; -\\sqrt{\\tau}\\mathbb{I}_2, \\sqrt{1-\\tau}\\mathbb{I}_2]$ with $\\tau=1-10^{\\beta\/10}$, where $\\beta$ is the coupling strength expressed in decibels (dB)\\cite{Pogorzalek, Roberto}.  Letting the first moments before the beam splitter as ${ x_{\\tau}}=(I_1, Q_2, e^rx_2, e^{-r}p_2 )^T$ by adjusting the parameter $\\tau$ as\n\n\\begin{equation}\n\\tau=\\frac{\\epsilon\\eta}{2}=1-\\frac{1}{|\\alpha_{\\text{LO}}|^2(\\hbar \\omega_iBR\\nu\\kappa g_{\\text {J}}g_{\\text H} )}\n\\end{equation}\nand letting $\\Lambda=|\\alpha_{\\text{LO}}|^2\\hbar \\omega_iBR\\nu\\kappa g_{\\text {J}}g_{\\text H}$ therefore $\\tau=1-\\frac{1}{\\Lambda}$. Since the maximum value of $\\tau=\\frac{\\epsilon\\eta}{2}=1\/2$ then it makes a restriction on $\\Lambda$ as $1<\\Lambda \\leq 2$ to have a feasible protocol of teleportation.\n\nThe coupling strength turns to                                                                               \n\n\\begin{equation}\n\\beta=10\\log\\frac{1}{\\Lambda}\n\\end{equation}\n\nFinally, after the operation $B_S(\\tau)_4{ x_{\\tau}}$, the state will be reconstructed in the upper arm after the beam splitter as $\\sqrt{2}\\alpha_{in}=e^{-                                                                                                                                                                                                                                                                                                                                                y}x_{\\text{in}}+ie^{y}p_{\\text{in}}$ with noise term $\\zeta=\\zeta'_{x}+i\\zeta'_{p}$ where the components are                             \n \\begin{eqnarray}                                                                        \n\\nonumber e^{-y}x_{\\text{in}} +\\zeta'_{x}= \\frac{1 }{\\sqrt{\\Lambda}}I_1+ \\sqrt{\\tau} e^rx_2  \\\\\ne^{y}p_{\\text{in}}+\\zeta'_{p} = \\frac{1 }{\\sqrt{\\Lambda}}Q_2+\\sqrt{\\tau} e^{-r}p_2 \n\\end{eqnarray}\n\n In a lossless protocol, $\\eta=\\epsilon=\\kappa=\\nu=1$, the noise term disappears and the state can be reconstructed perfectly, but in reality the amount of loss is significant in the microwave regime. The magnitudes of noise terms $\\zeta'_{x}$ and $\\zeta'_{p}$ can be obtained experimentally via calibration of the setup by letting the zero input values, i.e. $e^{-y}x_{\\text{in}}=0$ and $e^{y}p_{\\text{in}}=0$. \n\n\n\\begin{widetext}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{l c c}\n\\hline\nParameter &Symbol &Value\\\\\n\\hline\nFrequency  & $\\omega_i$ & 5 GHz\\\\\nMeasurement bandwidth & $B$ & 420 kHz\\\\\nResistance & $R$ & 50 $\\Omega$\\\\\nHEMT gain & $g_{\\text H}$  & $10^4$\\\\\nAmplification gain of JPA & $g_{\\text J}$ & $10^2$ \\\\\nTransfer efficiency (at T$_1$=40mK) & $\\epsilon$ & 0.95 \\\\\nTransfer efficiency (at T$_2$=300K, Free space) & $\\eta$ & 0.10\\\\\nTransfer efficiency (at T$_2$=4K, Fridge) & $\\eta$ & 0.90\\\\\nTransfer efficiency (at T$_3$=4K) & $\\kappa$ & 0.65\\\\\nTransfer efficiency (at T$_4$=100mK) & $\\nu$ & 0.75\\\\\n\nLocal oscillator mode amplitude & $|\\alpha_{\\text{LO}}|$ &  $10^6$V\/m\\\\\nAmplification squeezing& $r_{\\text{J}}$ & 2.30\\\\ \nSqueezing parameter of the TMSS & $r$ &   1.32 \\\\\nTransmissivity (for T$_2$=300K, Free space) & $\\tau$ & 0.095\\\\\nTransmissivity (for T$_2$=4K, Fridge) & $\\tau$ & 0.427\\\\\nCoupling strength (for T$_2$=300K, Free space)& $\\beta$ & -0.41\\\\\nCoupling strength (for T$_2$=4K, Fridge)& $\\beta$ & -2.40\\\\\nCoefficient (for T$_2$=300K, Free space) & $\\Lambda$ & 1.10 \\\\\nCoefficient (for T$_2$=4K, Fridge) & $\\Lambda$ &  1.74\\\\\nNoise (zero input, for T$_2$=300K, Free space)& $\\zeta'_x$ & (0.954)$<I_1>$+(1.152)$<x_2>$\\\\\nNoise (zero input, for T$_2$=300K, Free space)& $\\zeta'_p$  & (0.954)$<Q_2>$+(0.082)$<p_2>$ \\\\\nNoise (zero input, for $_2$=4K, Fridge)& $\\zeta'_x$ & (0.758)$<I_1>$+(2.444)$<x_2>$\\\\\nNoise (zero input, for T$_2$=4K, Fridge)& $\\zeta'_p$  &  (0.758)$<Q_2>$+(0.174)$<p_2>$\\\\\n\n\\hline\n\\end{tabular}\n\\caption{Some approximations for the values in the microwave teleportation circuit}\n\\end{table}\n\n\\end{widetext}\n\n\n \n\n\n\n\n\n\n$Amplification-$ The amplification of signals is an essential feature in microwave communication in open air. In the quantum regime, HEMT amplifiers are suited for experiments in the microwave regime. A commercial HEMT usually has $g_{H}=10^{4}$, and working at 5 GHz frequencies it introduces between $n\\sim 10-100$ thermal photons~\\cite{Roberto} which can have destructive effects on the protocol. One alternative for amplification is the replacement of each HMET with two additional JPAs in the same arm (see Fig.~\\ref{fig4}) to reach the same gain of HEMT but with significant reduction of noise. In this case, the parameter $\\Lambda$ turns to $\\Lambda'$ where $\\Lambda'_{\\text{J}}=|\\alpha_{\\text{LO}}|^2\\hbar \\omega_iBR\\nu\\kappa (g_{\\text {J}})^3$ and therefore the state can be reconstructed based on the circuit in Fig.~\\ref{fig4}, as follows\n\n\\begin{eqnarray}                                                                        \n\\nonumber e^{-y}x_{\\text{in}} +\\zeta'_{x}= \\frac{1 }{\\sqrt{\\Lambda'_{\\text{J}}}}I_1+ \\sqrt{\\tau} e^rx_2  \\\\\ne^{y}p_{\\text{in}}+\\zeta'_{p} = \\frac{1 }{\\sqrt{\\Lambda'_{\\text{J}}}}Q_2+\\sqrt{\\tau} e^{-r}p_2 \n\\end{eqnarray}\n\n\n\\begin{figure}\n\\includegraphics[scale=0.3]{Teleportationex4.pdf}\n\\caption{Replacement of each HEMT with two JPAs in each arm to reach the same gain but with significant lower thermal noise.}\n\\label{fig4}\n\\end{figure}\n\nBased on the above theoretical proposed protocols and physical formalism, an unconditional microwave quantum teleportation is applicable in real conditions. However, quantum microwave signals are very fragile and there are experimental limitations in free space as well as some fundamental bounds \\cite{SanzIEEE, Pirandolanew} that should be studied as the next research prospect. \n\n\n\n\n\n\n\n\n\n\\begin{acknowledgments}\nThe author greatly acknowledges financial support from the projects QMiCS (820505) of the EU Flagship on Quantum Technologies as well as EU project EPIQUS (899368). Also, the author is very grateful for very helpful and constructive discussions with Mikel Sanz, Yasser Omar, Frank Deppe, and Shabir Barzanjeh.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nThe recent upgrade to the Compact Array Broadband Backend (CABB) correlator on the Australia Telescope Compact Array (ATCA) has increased the bandwidth from 128\\,MHz to 2\\,GHz (for more details see paper by Emil Lenc in these proceedings). The increased sensitivity this provides allows us not only to go deeper, but to also observe large samples of sources in smaller amounts of time. To this effect, we have observed a sample of 1138 X-ray selected QSOs at 20\\,GHz as a comprehensive study of the high-frequency radio luminosity distribution of QSOs.\n\nQSOs are often classified into two broad categories; radio-loud and radio-quiet, but the underlying distribution of radio luminosity has long been debated in the literature. There are two opposing views - the first is that the distribution of radio-loudness is bimodal, i.e.\\ there are distinct radio-loud and radio-quiet populations, with 10--20\\% of QSOs being radio-loud\\cite{kellermann}. The second view is that there is a broad, continuous distribution with no clear dividing line between radio-loud and radio-quiet QSOs\\cite{cirasuolo}. Understanding this distribution can provide new insights into the high frequency radio properties of these sources while also allowing us to investigate the difference between X-ray selected and radio-selected QSO populations.\n\nWe are currently studying two samples of QSOs; one selected in X-rays using the {\\it ROSAT} All Sky Survey (RASS), and the other selected in the radio from the Australia Telescope 20\\,GHz (AT20G) survey\\cite{murphy}. The two samples overlap in redshift, but only a small number of objects appear in both samples. We aim to compare the physical properties of radio and X-ray-selected QSOs in the redshift range $z<1$. As a first step, we have used the ATCA to make deep 20\\,GHz observations of the $1138$ QSOs from the RASS X-ray sample which were not detected at the 40--50\\,mJy flux limit of the AT20G survey. This will allow us to determine whether their radio luminosity distribution is continuous with that of radio-selected QSOs in the same redshift range, or is clumped at much lower radio luminosities (implying a bimodal distribution). \n\nBy observing at 20\\,GHz we pick up the central core component of the AGN and hence see the most recent activity. At lower frequencies we observe a higher fraction of emission from radio lobes; relics of past activity integrated over large timescales which therefore confuse our sample.\n\n\\section*{Sample Selection}\n\nTargets were seleced from the RASS--6dFGS catalogue\\cite{mahony}; a catalogue of 3405 AGN selected from the ROSAT ALL Sky Survey (RASS) Bright Source Catalogue\\cite{voges} that were observed as part of the 6dF Galaxy Survey (6dFGS)\\cite{jones}. Sources were selected if the 6dFGS spectrum exhibited broad emission features indicative of a QSO or Type 1 AGN. We also set a redshift cutoff of z<1 to minimize any evolutionary effects. We then searched the Australia Telescope 20\\,GHz (AT20G) survey for any known 20 GHz radio sources which were excluded from our target list. This leaves a final sample of 1138 X-ray selected QSOs at z<1. Example 6dFGS spectra and corresponding 20 GHz observations are shown in Figure \\ref{example}. Since this is a large, low-redshift QSO sample which spans a wide range in optical luminosity, it can provide a definitive test of whether the $z<1$ QSO population is bimodal in its radio properties. Selecting sources that have 6dFGS spectroscopic information not only provides a uniform sample, but will also give us a wealth of extra information. Hence we can also study the optical spectral line properties, black hole masses, multiwavelength properties (X-ray -- optical --radio) and how these vary with redshift. \n\n\\begin{figure}[h]\n\\begin{minipage}{0.6\\linewidth}\n\\centerline{\\epsfig{file=g0150-5322.eps, width=\\linewidth}}\n\\end{minipage}\n\\begin{minipage}{0.4\\linewidth}\n\\centerline{\\epsfig{file=overlay0150-5322.eps, width=0.8\\linewidth}}\n\\end{minipage}\n\\begin{minipage}{0.6\\linewidth}\n\\centerline{\\epsfig{file=g0311-2046.eps, width=\\linewidth}}\n\\end{minipage}\n\\begin{minipage}{0.4\\linewidth}\n\\centerline{\\epsfig{file=overlay0311-2046.eps, width=0.8\\linewidth}}\n\\end{minipage}\n\\caption{Example RASS selected QSOs. The left image shows the 6dFGS spectrum. The arrows denote where spectral features would occur at that redshift, but not all of these are necessarily observed. On the right are the corresponding optical (B-band) images with 20\\,GHz contours overlaid. The top source was observed in October 2008 (47.3\\,mJy) with the old correlator and the bottom source was observed in October 2009 (1.64\\,mJy) using CABB. In both images the contours represent 10\\% intervals in flux, with the central contour corresponding to 90\\%. }\\label{example}\n\\end{figure}\n\n\\section*{Observations}\n\nWe observed 1138 X-ray selected QSOs at 20 GHz with the ATCA in a compact, hybrid configuration from 2008 -- 2010. The observations were carried out in a two-step process; all objects were observed for 2$\\times$40s cuts and the objects not detected in that time were then reobserved for 2$\\times$5min. These observations are summarised in Table 1. \n\\begin{table}\n\\begin{center}\n\\caption{List of observations for this program. All the observations from 2008--2010 used the Hybrid 168m array configuration to obtain better u,v coverage. The October 2008 run was using the old correlator (128\\,MHz bandwidth) and all other runs used CABB. \\label{tab1}}\n\\begin{tabular}{lcccc}\n\\hline\n{\\bf Date of Observations} & \\multicolumn{2}{c}{\\bf No. Sources} & {\\bf Time on Source} & {\\bf Detection limit} \\\\\n& {\\bf Observed} & {\\bf Detected} & & \\\\\n\\hline\nOctober 2008 & 135 & 16 (12\\%) & 2$\\times$40s & 3\\,mJy \\\\\nApril 2009 & 417 &  103 (25\\%) & 2$\\times$40s & 0.9\\,mJy \\\\\nOctober 2009 & 586 & 100 (17\\%) & 2$\\times$40s & 0.9\\,mJy \\\\\n& 377 & 58 (15\\%) & 2$\\times$5m & 0.5\\,mJy \\\\\nMarch 2010 & 122 & 21(17\\%) & 2$\\times$5m & 0.5\\,mJy \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[h]\n \\centerline{\\epsfig{file=zrlum_split2.ps, width=0.85\\linewidth}}\n\\caption{20 GHz radio luminosity against redshift for our sample of X-ray selected QSOs. The green diamonds are sources that were observed using the old correlator on the Compact Array, including sources observed as part of the AT20G survey, while the blue circles show the sources that were detected at the 5 sigma threshold using CABB. The red triangles denote the upper limits for sources that weren't detected with CABB. }\n\\label{zrlum}\n\\end{figure}\n\nFigure \\ref{zrlum} shows preliminary results from these observations. The increased sensitivity provided by CABB is immediately obvious when comparing the 20\\,GHz luminosities of sources detected using the old correlator (green diamonds) and those detected with CABB (blue circles). Figure \\ref{hist} shows preliminary distributions of both the radio luminosity, and the R-parameter\\cite{kellermann} which is often used as a measure of the `radio-loudness' of a QSO. This is defined as the ratio of the radio to optical flux; in this case the 20\\,GHz radio flux divided by the optical B--band flux. In these figures, the red solid line indicates the distribution that was obtained using the old correlator on the ATCA whilst the solid black line shows the distributions obtained using CABB. The dashed line indicates the upper limits of the sources that were not detected. \n\n\\begin{figure}[h]\n\\begin{minipage}{0.5\\linewidth}\n \\centerline{\\epsfig{file=split_rlumdist.ps, width=\\linewidth}}\n\\end{minipage}\n\\begin{minipage}{0.5\\linewidth}\n \\centerline{\\epsfig{file=split_Rhist.ps, width=\\linewidth}}\n\\end{minipage}\n\\caption{{\\it(Left):} The radio luminosity distribution of our X-ray selected sample of QSOs at z<1. {\\it(Right):} The `radio-loudness' distribution of this sample. The `radio-loudness' was determined using the Kellermann R-parameter which is defined as the ratio of the radio to optical fluxes. Radio-loud QSOs have R>10, while radio-quiet QSOs generally have R<1. Since all of our objects have optical spectroscopy and reliable redshifts, this allows us to compare both ways of determining whether a QSO is radio-loud and investigate any differences between them.}\n\\label{hist}\n\\end{figure}\n\nThese figures, along with the detection limits noted in Table \\ref{tab1} highlight the significant improvement that CABB has achieved.\n\n\\section*{Future Work}\nThe results presented here are very preliminary and the data reduction and analysis is ongoing\\cite{mahony2}. However, the strict selection criteria and multiwavelength information provides a wealth of data allowing for a comprehensive study of these sources. In particular, work in progress includes:\n\\begin{itemize}\n\\item A statistical analysis of the data and resulting radio luminosity distributions.\n\\item Stacking experiments of the non-detected sources to study the average properties of radio-quiet X-ray QSOs. \n\\item We also have data at 5 and 9 GHz for a subset of this sample, allowing us to investigate whether the radio luminosity distributions change with frequency.\n\\end{itemize}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nTime series exhibiting spatial dependencies are present in many domains including ecology, meteorology, biology, medicine, economics, traffic, and vision. The observations can come from multiple sources e.g. GPS, satellite imagery, video cameras, etc. Two main difficulties when modeling spatio-temporal data come from their size - sensors can cover very large space and temporal lags - and from the complexity of the data generation process. Reducing the dimensionality and uncovering the underlying data generation process naturally leads to consider latent dynamic models. This has been exploited both in statistics \\cite{cressie2011} and in machine learning (ML) \\cite{bahadori2014fast,koppula2013learning}.\n\nDeep learning has developed a whole range of models for capturing relevant information representations for different tasks and modalities. For dynamic data, recurrent neural networks (RNN) handles complex sequences for tasks like classification, sequence to sequence prediction, sequence generation and many others \\cite{Bengio2008, chung2015recurrent, li2015gated}.\n\nThese models are able to capture meaningful features of sequential data generation processes, but the spatial structure, essential in many applications, has been seldom considered in Deep Learning. Very recently, convolutional RNNs \\cite{NIPS2015_5955,srivastava2015} and video pixel networks \\cite{kalchbrenner2017} have been used to handle both spatiality and temporality, but for video applications only.\n\nWe explore a general class of deep spatio-temporal models by focusing on the problem of time series forecasting of spatial processes for different types of data. Although the target in this paper is forecasting, the model can easily be extended to cover related tasks like spatial forecasting (kriging \\cite{stein2012interpolation}) or data imputation.\n\n\nThe model, denoted Spatio-Temporal Neural Network (STNN), has been designed to capture the dynamics and correlations in multiple series at the spatial and temporal levels. This is a dynamical system model with two components: one for capturing the spatio-temporal dynamics of the process into latent states, and one for decoding these latent sates into actual series observations.\n\nThe model is tested and compared to state of the art alternatives, including recent RNN approaches, on spatio-temporal series forecasting problems: disease prediction, traffic forecasting, meteorology and oceanography. Besides a comparative evaluation on forecasting tasks, the ability of the model to discover relevant spatial relations between series is also analyzed.\n\nThe paper is organized as follow: in section \\ref{rw} we introduce the related work in machine learning and spatio-temporal statistics. The model is presented in section \\ref{model} with its different variants. The experiments are described in section \\ref{xp} for both forecasting \\ref{xp-forecast} and relations discovery \\ref{xp-rel}.\n \n\\section{Related Work}\n\\label{rw}\nThe classical topic of time series modeling and forecasting has given rise to an extensive literature, both in statistics and machine learning. In statistics, classical linear models are based on auto-regressive and moving average components. Most assume linear and stationary time dependencies with a noise component \\cite{de200625}. In machine learning, non linear extensions of these models based on neural networks were proposed as early as the nineties, opening the way to many other non linear models developed both in statistics and ML, like kernel methods \\cite{muller1999using} for instance.\n\nDynamical state space models, such as recurrent neural networks, have been used for time series forecasting in different contexts since the early nineties \\cite{connor1994recurrent}. Recently, these models have witnessed important successes in different areas of sequence modeling problems, leading to breakthrough in domains like speech \\cite{graves2013speech}, language generation \\cite{Sutskever2011}, translation \\cite{cho2014learning} and many others. A model closely related to our work is the dynamic factor graph model \\cite{mirowski2009dynamic} designed for multiple series modeling. Like ours, it is a generative model with a latent component that captures the temporal dynamics and a decoder for predicting the series. However, spatial dependencies are not considered in this model, and the learning and inference algorithms are different.\n\nRecently, the development of non parametric generative models has become a very popular research direction in Deep Learning, leading to different families of innovative and promising models. For example, the Stochastic Gradient Variational Bayes algorithm (SGVB) \\cite{kingma2013auto} provides a framework for learning stochastic latent variables with deep neural networks, and has recently been used by some authors to model time series \\cite{bayer2014learning, Chung2015, krishnan2015deep}. In our context, which requires to model explicitly both spatial and temporal dependencies between multiple time series, variational inference as proposed by such models is still intractable, especially when the number of series grows, as in our experiments.\n\nSpatio-temporal statistics have already a long history \\cite{cressie2011,wikle2010general}. The traditional methods rely on a descriptive approach using the first and second-order moments of the process for modeling the spatio-temporal dependencies. More recently, dynamical state space models, where the current state is conditioned on the past have been explored \\cite{Wikle2015}. For these models, time and space can be either continuous or discrete. The usual way is to consider discrete time, leading to the modeling of time series of spatial processes. When space is continuous, models are generally expressed by linear integro-difference equations, which is out of the scope of our work. With discrete time and space, models come down to general vectorial autoregressive formulations. These models face a curse of dimensionality in the case of a large number of sources. Different strategies have been adopted to solve this problem, such as embedding or parameter reduction. This leads to model families that are close to the ones used in machine learning for modeling dynamical phenomena, and incorporate a spatial components. An interesting feature of these approaches is the incorporation of prior knowledge inspired from physical models of space-time processes. This  consists in taking inspiration from prior background of physical phenomena, e.g. diffusion laws in physics, and using this knowledge as guidelines for designing dependencies in statistical models. In climatology, models taking into account both temporal and geographical components have also been used such as Gaussian Markov Random Fields\\cite{rue2005gaussian} \n\nIn the machine learning domain, spatio-temporal modeling has been seldom considered, even though some spatio-temporal models have been proposed \\cite{ceci2017predictive}. \\cite{bahadori2014fast} introduce a tensor model for kriging and forecasting. \\cite{koppula2013learning} use conditional random fields for detecting activity in video, where time is discretized at the frame level and one of the tasks is the prediction of future activity. Brain Computer Interface (BCI) is another domain for spatio-temporal data analysis with some work focused on learning spatio-temporal filters \\cite{Dornhege2005,ren2014convolutional}, but this is a very specific and different topic.\n\n\\section{The STNN Model}\n\\label{model}\n\\subsection{Notations and Task}\nLet us consider a set of $n$ temporal series, $m$ is the dimensionality of each series and $T$ their length \\footnote{We assume that all the series have the same dimensionality and length. This is often the case for spatio-temporal problems otherwise this restriction can be easily removed.}. $m=1$ means that we consider $n$ univariate series, while $m>1$ correspond to $n$ multivariate series each with $m$ components. We will denote $X$ the values of all the series between time $1$ and time $T$. $X$ is then a $\\mathbb{R}^{ T \\times n \\times m}$ 3-dimensional tensor, such that $X_{t,i,j}$ is the value of the j-th component of series $i$ at time $t$. $X_t$ will denote a slice of $X$ at time $t$ such that $X_t \\in \\mathbb{R}^{n \\times m}$ denotes the values of all the series at time $t$.\n\nFor simplicity, we first present our model in a mono-relational setting. An extension to multi-relational series where different relations between series are observed is described in section \\ref{multi-modal-rel}. We consider that the spatial organization of the sources is captured through a matrix $W \\in \\mathbb{R}^{n \\times n}$. Ideally, $W$ would indicate the mutual influence between sources, given as a prior information. In practice, it might be a proximity or similarity matrix between the sources: for geo-spatial problems, this might correspond to the inverse of a physical distance - e.g. geodesic - between sources. For other applications, this might be provided through local connections between sources using a graph structure (e.g. adjacency matrix for connected roads in a traffic prediction application or graph kernel on the web). In a first step, we make the hypothesis that $W$ is provided as a prior on the spatial relations between the series. An extension where weights on these relations are learned is presented in section \\ref{learn-spt-rel}. \n\nWe consider in the following the problem of spatial time series forecasting i.e predicting the future of the series, knowing their past. We want to learn a model $f : \\mathbb{R}^{ T \\times n \\times m} \\times \\mathbb{R}^{n \\times n} \\rightarrow \\mathbb{R}^{\\tau \\times n \\times m}$ able to predict the future at $\\tau$ time-steps of the series based on $X$ and on their spatial dependency.\n\n\\subsection{Modeling Time Series with Continuous Latent Factors}\n\\label{mts}\nLet us first introduce the model in the simpler case of multiple time series prediction, without considering spatial relations. The model has two components.\n\nThe first one captures the dynamic of the process and is expressed in a latent space. Let $Z_t$ be the latent representation, or latent factors, of the series at time $t$. The dynamical component writes $Z_{t+1}=g(Z_t)$. The second component is a decoder which maps latent factors $Z_t$ onto a prediction of the actual series values at $t$: $\\tilde{X_t} = d(Z_t)$, $\\tilde{X_t}$ being the prediction computed at time $t$. In this model, both the representations $Z_t$ and the parameters of the dynamical and decoder components are learned. Note that this model is different from the classical RNN formulations \\cite{hochreiter1997long, cho2014learning}. The state space component of a RNN with self loops on the hidden cells writes $Z_{t+1}=g(Z_t,X'_t)$, where $X'_t$ is the ground truth $X_t$ during training, and the predicted value $\\tilde{X_t}$ during inference. In our approach, latent factors $Z_t$ are learned during training and are not an explicit function of past inputs as in RNNs: the dynamics of the series are then captured entirely in the latent space.\n\nThis formal definition makes the model more flexible than RNNs since not only the dynamic transition function $g(.)$, but also the state representations $Z_t$ are learned from data. A similar argument is developed in \\cite{mirowski2009dynamic}. It is similar in spirit to Hidden Markov models or Kalman filters.\n\n\n\n\n\n\n\n\n\\paragraph{Learning problem}\nOur objective is to learn the two mapping functions $d$ and $g$ together with the latent factors $Z_t$, directly from the observed series. We formalize this learning problem with a bi-objective loss function that captures the dynamics of the series in the latent space and the mapping from this latent space to the observations. Let $\\mathcal{L}(g,d,Z)$ be this objective function:\n\\begin{equation}\n\\begin{gathered}\n \\mathcal{L}(d,g,Z) = \\frac{1}{T}\\sum\\limits_t \\Delta(d(Z_t),X_t) \\hfill \\text{ (i)}\\\\+ \\lambda \\frac{1}{T} \\sum\\limits_{t=1}^{T-1} || Z_{t+1} - g(Z_t) ||^2 \\hfill \\text{ (ii)}\n\\end{gathered}\n\\label{eqfold}\n\\end{equation}\n\nThe first term (i) measures the ability of the model to reconstruct the observed values $X_t$ from the latent factor $Z_t$. It is based on loss function $\\Delta$ which measures the discrepancy between predictions $d(Z_t)$ and ground truth $X_t$. The second term (ii) aims at capturing the dynamicity of the series in the latent space. This term forces the system to learn latent factors $Z_{t+1}$ that are as close as possible to $g(Z_t)$. Note that in the ideal case, the model converges to a solution where $Z_{t+1}=g(Z_t)$, which is the classical assumption made when using RNNs. The hyper-parameter $\\lambda$ is used here to balance this constraint and is fixed by cross-validation. The solution $d^*,g^*,Z^*$ to this problem is computed by minimizing $\\mathcal{L}(d,g,Z)$:\n\\begin{equation}\nd^*,g^*,Z^* = \\arg \\min\\limits_{d,g,Z} \\mathcal{L}(d,g,Z)\n\\label{eqf}\n\\end{equation}\n\n\\paragraph{Learning algorithm}\nIn our setting, functions $d$ and $g$, described in the next section, are differentiable parametric functions. Hence, the learning problem can be solved end-to-end with Stochastic Gradient Descent (SGD) techniques\\footnote{In the experiments, we used the Nesterov's Accelerated Gradient (NAG) method \\cite{sutskever2013importance}.} directly from \\eqref{eqf}. At each iteration, a pair $(Z_t, Z_{t+1})$ is sampled, and $Z_t$, $Z_{t+1}$, $g$ and $d$ are updated according to the gradient of \\eqref{eqfold}. Training can also be performed via mini-batch, meaning that for each iteration several pairs $(Z_t, Z_{t+1})$ are sampled, instead of a single pair. This results in a high learning speed-up when using GPUs which are the classical configuration for running such methods.\n\\paragraph{Inference} Once the model is learned, it can be used to predict future values of the series. The inference method is the following: the latent factors of any future state of the series is computed using the $g$ function, and the corresponding observations is predicted by using $d$ on these factors. Formally, let us denote $\\tilde{Z}_\\tau$ the predicted latent factors at time $T+\\tau$. The forecasting process computes $\\tilde{Z}_{\\tau}$ by successively applying the $g$ function $\\tau$ times on the learned vector $Z_T$:\n\\begin{equation}\n\\tilde{Z}_{\\tau} = g \\circ g \\circ ... \\circ g(Z_T)\n\\end{equation}\nand then computes the predicted outputs $\\tilde{X}_{\\tau}$:\n\\begin{equation}\n\\tilde{X}_{\\tau}=d(\\tilde{Z_{\\tau}})\n\\end{equation}\n\n\\subsection{Modeling Spatio-Temporal Series}\n\\label{msts}\nLet us now introduce a spatial component in the model. We consider that each series has its own latent representation at each time step. $Z_t$ is thus a $n \\times N$ matrix such that $Z_{t,i} \\in \\mathbb{R}^N$ is the latent factor of series $i$ at time $t$, $N$ being the dimension of the latent space. This is different from approaches like \\cite{mirowski2009dynamic} or RNNs for multiple series prediction, where $Z_t$ would be a single vector common to all the series. The decoding and dynamic functions $d$ and $g$ are respectively mapping $\\mathbb{R}^{n \\times N}$ to $\\mathbb{R}^{n \\times m}$ and $\\mathbb{R}^{n \\times N}$ to $\\mathbb{R}^{n \\times N}$.\n\nThe spatial information is integrated in the dynamic component of our model through a matrix $W \\in \\mathbb{R}_+^{n \\times n}$. In a first step, we consider that $W$ is provided as prior information on the series' mutual influences. In \\ref{ref}, we remove this restriction, and show how it is possible to learn the weights of the relations, and even the spatial relations themselves, directly from the observed data. The latent representation of any series at time $t+1$ depends on its own latent representation at time $t$ (intra-dependency) and on the representations of the other series at $t$ (inter-dependency). Intra-dependency will be captured through a linear mapping denoted $\\Theta^{(0)} \\in \\mathbb{R}^{N \\times N}$ and inter-dependency will be captured by averaging the latent vector representations of the neighboring series using matrix $W$, and computing a linear combination denoted $\\Theta^{(1)} \\in \\mathbb{R}^{N \\times N}$ of this average. Formally, the dynamic model $g(Z_t)$ is designed as follow:\n\\begin{equation}\n\tZ_{t+1} = h(Z_t \\Theta^{(0)}+ W Z_t \\Theta^{(1)})\n    \\label{dynamic}\n\\end{equation}\nHere, $h$ is a non-linear function. In the experiments we set $h=tanh$ but $h$ could also be a more complex parametrized function like a multi-layer perceptron (MLPs)  for example -- see section \\ref{xp}. The resulting optimization problem over $d$, $Z$, $\\Theta^{(0)}$ and $\\Theta^{(1)}$ writes:\n\n\\begin{equation}\n\\begin{gathered}\nd^*, Z^*, \\Theta^{(0)*}, \\Theta^{(1)*} =\\\\ \\argmin_{d,Z,\\Theta^{(0)}, \\Theta^{(1)}} \\frac{1}{T}\\sum\\limits_t \\Delta(d(Z_t),X_t) +\\\\\n\\lambda \\frac{1}{T} \\sum\\limits_{t=1}^{T-1} || Z_{t+1} - h(Z_t \\Theta^{(0)}+ W Z_t \\Theta^{(1)}) ||^2\\\\\n\\text{  with } Z_t \\in \\mathbb{R}^{n \\times N}\n\\end{gathered}\n\\label{eqm}\n\\end{equation}\n\n\\begin{table*}[ht]\n\\centering\n\\begin{tabular}{|c||c||c|c|c|c|c|c|c|} \\hline\nDataset & Task & $n$ & $m$ & nb relations & time-step & total length & training length & \\#folds  \\\\ \\hline \\hline\nGoogle Flu &  Flu trends & 29 & 1 & 1 to 3 & weeks& $\\approx$  10 years & 2 years & 50 \\\\ \nGHO (25 datasets) & Number of deaths & 91 & 1 & 1 to 3 & years& 45 years & 35 years & 5 \\\\ \\hline \nWind & Wind speed and orientation & 500 & 2 & 1 to 3 & hours& 30 days & 10 days & 20 \\\\\nPST & Temperature & 2520 & 1 & 8  &  months & $\\approx$ 33 years& 10 years & 15 \\\\ \n\\hline\nBejing & Traffic Prediction & 5000 & 1 &  1 to 3 & 15 min& 1 week & 2 days & 20 \\\\ \\hline\n\\end{tabular}\n\n\\caption{Datasets statistics. $n$ is the number of series, $m$ is the dimension of each series, $timestep$ corresponds to the duration of one time-step and \\textit{\\#folds} corresponds to the number of temporal folds used for validation. For each fold, evaluation has been made on the next 5 values at $T+1,T+2,...,T+5$. The relation columns specifies the number of different relation types used in the experiments i.e the number of $W^{(r)}$ matrices used in each dataset. 1 to 3 means that the best among 1 to 3 relations was selected using cross validation}\n\\label{datasets}\n\\end{table*}\n\n\n\\subsection{Modeling different types of relations}\n\\label{multi-modal-rel}\nThe model in section \\ref{msts} considers that all the spatial relations are of the same type (e.g. source proximity). For many problems, we will have to consider different types of relations. For instance, when sensors correspond to physical locations and the target is some meteorological variable, the relative orientation or position of two sources may imply a different type of dependency between the sources. In the experimental section, we consider problems with relations based on the relative position of sources, $north, south, west, east, ...$. The multi-relational framework generalizes the previous formulation of the model, and allows us to incorporate more abstract relations, like different measures of proximity or similarity between sources. For instance, when sources are spatiality organized in a graph, it is possible to define different graph kernels, each one of them modeling a specific similarity. The following multi-relational formulation is based on adjacency matrices, and can directly incorporate different graph kernels. \n\n\n\n\nEach possible relation type is denoted \\textit{r} and is associated to a matrix $W^{(r)} \\in \\mathbb{R}_+^{n \\times n}$. For now, and as before, we consider that the $W^{(r)}$ are provided as prior knowledge. Each type of relation \\textit{r} is associated to a transition matrix $\\Theta^{(r)}$. This learned matrix captures the spatio-temporal relationship between the series for this particular type of relation. The model dynamics writes:\n\\begin{equation}\nZ_{t+1} = h( Z_t \\Theta^{(0)}+\\sum\\limits_{r \\in \\mathcal{R}} W^{(r)} Z_t \\Theta^{(r)})\n\\label{dynamic-multi-rel}\n\\end{equation}\nwhere $\\mathcal{R}$ is the set of all possible types of relations. The learning problem is similar to equation \\eqref{eqm} with $Z_{t+1}$ replaced by the expression in \\eqref{dynamic-multi-rel}. The corresponding model is illustrated in figure \\ref{networks}. This dynamic model aggregates the latent representations of the series for each type of relation, and then applies $\\Theta^{(r)}$ on this aggregate. Each $\\Theta^{(r)}$ is able to capture the dynamics specific to relation $(r)$.\n\n\n\n\n\\begin{figure}[ht]\n  \\begin{center}\n   \t\\includegraphics[width=1\\linewidth]{STNN_2.png} \\\\\n  \\end{center}\n  \\caption{Architecture of the STNN model as described in Section \\ref{multi-modal-rel}}.\n  \\label{networks}\n\\end{figure}\n\n\\section{Learning the relation weights and capturing spatio-temporal correlations}\n\\label{learn-spt-rel}\n\\label{ref}\nIn the previous sections, we made the hypothesis that the spatial relational structure and the strength of influence between series were provided to the model through the $W^{(r)}$ matrices. We introduce below an extension of the model where weights on these relations are learned. This model is denoted STNN-R(efining). We further show that, with a slight modification, this model can be extended to learn both the relations and their weights directly from the data, without any prior. This extension is denoted  STNN-D(iscovering).\n\n\n\n\\begin{table*}[t]\n\\centering\n\\begin{tabular}{|c||c|c|c|c|c|c|}\n\\hline\n\\multirow{2}{*}{Models} & \\multicolumn{2}{c|}{Disease}& Car Traffic &\\multicolumn{2}{|c}{Geographical}\\\\\n\\hhline{~------}\n& Google Flu & GHO\\footnote{Results shown here correspond to the average for all diseases in the dataset. The detail is in the supplementary material for space convenience.} & Beijing &Speed&Direction&PST \\\\\n\\hhline{=:=:=:=:=:=:=}\nMEAN & .175& .335&.201&0.191&0.225&.258\\\\\\hline\nAR &  .101$\\pm .004$&$.299 \\pm .008$&$.075 \\pm .003$&$.082 \\pm .005$&0.098$\\pm .016$&$.15\\pm .002$\\\\\\hline\nVAR-MLP &  $.095 \\pm .004$& $.291 \\pm .004$ & $.07 \\pm .002$&$.071 \\pm .005$&0.111$\\pm 0.14$&$.132\\pm .003$\\\\\\hline\nDFG &  $.095 \\pm .008$& $.288 \\pm .002$ & $.068 \\pm .005$&$.07 \\pm .004$&$.092 \\pm .006$&$.99\\pm .019$\\\\\\hline\nRNN-tanh &  $.082 \\pm .008$& $.287 \\pm .011$ &   $.075 \\pm .006$&$.064 \\pm .003$&$.09 \\pm .005$&$.141 \\pm .01$\\\\\\hline\nRNN-GRU &  $.074 \\pm .007$ &$.268 \\pm .07$&$.074\\pm .002$&$.059 \\pm .009$&$.083 \\pm .005$&$.104\\pm .008$\\\\\\hline\nSTNN &  $.066 \\pm .006$ &$\\textbf{.261}\\pm .009$&$.056\\pm .003$&$\\textbf{.047}\\pm .008$&$\\textbf{.061}\\pm .008$&$.095\\pm .008$\\\\\\hline\nSTNN-R &  $\\textbf{.061} \\pm .008$ &$\\textbf{.261}\\pm .01$&$\\textbf{.055}\\pm .004$&$\\textbf{.047}\\pm .008$&$\\textbf{.061}\\pm .008$&$\\textbf{.08}\\pm .014$\\\\\\hline\nSTNN-D &  $.073 \\pm .007$ & $.288 \\pm .09$&$.069\\pm .01$&$.059\\pm .008$&$.073\\pm .008$&$.109\\pm .015$  \\\\\\hline\n\n\\end{tabular}\n\\caption{\\label{tab:widgets}Average RMSE for the different datasets computed for T+1, T+2,...,T+5. Standard deviation was computed by re-training the models on different seeds.}\n\\label{table1}\n\\end{table*}\n\n\\begin{table*}[h]\n\\begin{center}\n\\begin{tabular}{|p{4.5cm}|c|c|c|c|c|c|}\n\n\\hline\nDisease \/ Model  & AR & VAR-MLP& RNN-GRU &Mean&DFG&STNN-R\\\\\n\\hline\n\nAll causes&0.237&0.228&0.199&0.35&0.291&\\textbf{0.197}\\\\\n\\hline\nTuberculosis&0.407&0.418&\\textbf{0.37}&0.395&0.421&0.377\\\\\n\\hline\nCongenital syphilis&0.432&0.443&0.417&0.459&0.422&\\textbf{0.409}\\\\\n\\hline\nDiphtheria&0.406&0.396&0.387&0.404&0.419&\\textbf{0.385}\\\\\n\\hline\nMalignant neoplasm of esophagus&0.355&\\textbf{0.341}&\\textbf{0.341}&0.363&0.372&0.345\\\\\n\\hline\nMalignant neoplasm of stomach&0.44&0.434&0.431&0.455&0.452&\\textbf{0.43}\\\\\n\\hline\n&0.267&0.254&0.282&0.303&0.301&\\textbf{0.253}\\\\\n\\hline\nMalignant neoplasm of intestine&0.281&0.29&0.278&0.314&0.305&\\textbf{0.275}\\\\\n\\hline\nMalignant neoplasm of rectum&0.501&0.499&\\textbf{0.481}&0.504&0.509&0.498\\\\\n\\hline\nMalignant neoplasm of larynx&0.321&0.313&0.32&0.314&0.329&\\textbf{0.310}\\\\\n\\hline\nMalignant neoplasm of breast&0.375&0.375&0.382&0.394&0.38&\\textbf{0.36}\\\\\n\\hline\nMalignant neoplasm of prostate&0.111&0.113&\\textbf{0.109}&0.184&0.138&\\textbf{0.109}\\\\\n\\hline\nMalignant neoplasm of skin&0.253&0.243&0.227&0.264&0.256&\\textbf{0.221}\\\\\n\\hline\nMalignant neoplasm of bones&0.103&0.099&0.097&0.204&0.173&\\textbf{0.08}\\\\\n\\hline\nMalignant neoplasm of all other and unspecified sites &0.145&0.157&\\textbf{0.147}&0.164&0.169&0.156\\\\\n\\hline\nLymphosarcoma&0.15&0.132&0.13&0.231&0.135&\\textbf{0.122}\\\\\n\\hline\nBenign neoplasms &0.366&0.362&0.332&0.398&0.331&\\textbf{0.331}\\\\\n\\hline\nAvitaminonsis&0.492&0.474&0.449&0.571&0.58&\\textbf{0.414}\\\\\n\\hline\nAllergic disorders&0.208&0.217&0.221&0.342&0.24&\\textbf{0.202}\\\\\n\\hline\nMultiple sclerosis&0.061&0.057&0.061&0.242&0.152&\\textbf{0.056}\\\\\n\\hline\nRheumatic fever&0.325&0.31&0.287&0.345&0.313&\\textbf{0.256}\\\\\n\\hline\nDiseases of arteries&0.302&0.301&0.269&0.345&0.328&\\textbf{0.238}\\\\\n\\hline\nInfluenza&0.141&0.141&0.155&0.23&0.217&\\textbf{0.125}\\\\\n\\hline\nPneumonia&0.119&0.128&\\textbf{0.1}&0.187&0.187&\\textbf{0.1}\\\\\n\\hline\nPleurisy&0.246&0.246&0.247&0.29&0.272&\\textbf{0.245}\\\\\n\\hline\nGastro-enteritis&0.386&0.369&\\textbf{0.291}&0.394&0.398&0.295\\\\\n\\hline\nDisease of teeth&0.344&0.312&0.305&0.413&0.361&\\textbf{0.302}\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{RMSE of STNN-R over the 25 datasets in GHO for T+1, T+2,...,T+5}\n\\label{fl4}\n\\end{table*}\n\nWe will first introduce the STNN-R extension. Let $\\Gamma^{(r)} \\in \\mathbb{R}^{n \\times n}$ be a matrix of weights such that $\\Gamma^{(r)}_{i,j}$ is the strength of the relation between series $i$ and $j$ in the relation $r$. Let us extend the formulation in Equation \\eqref{dynamic-multi-rel} as follows:\n\\begin{equation}\nZ_{t+1} = h( Z_t \\Theta^{(0)}+\\sum\\limits_{r \\in \\mathcal{R}} (W^{(r)}\\odot \\Gamma^{(r)}) Z_t \\Theta^{(r)})\n\\label{gamma-multi-rel}\n\\end{equation}\nwhere $\\Gamma^{(r)}$ is a matrix to be learned, $W^{(r)}$ is a prior i.e a set of observed relations, and $\\odot$ is the element-wise multiplication between two matrices. \n The learning problem can be now be written as:\n\\begin{equation}\n\\begin{gathered}\nd^*, Z^*, \\Theta^*, \\Gamma^* =\\\\ \\argmin_{d,Z,\\Gamma} \\frac{1}{T}\\sum\\limits_t \\Delta(d(Z_t),X_t)+ \\gamma |\\Gamma| \\\\+ \\lambda \\frac{1}{T} \\sum\\limits_{t=1}^{T-1} || Z_{t+1} - h(\\sum\\limits_{r \\in \\mathcal(R)} (W^{(r)} \\odot \\Gamma^{(r)}) Z_t \\Theta^{(r)}) ||^2\n\\end{gathered}\n\\label{eqf}\n\\end{equation}\n\nwhere $|\\Gamma^{(r)}|$ is a $l_1$ regularizing term that aims at sparsifying $\\Gamma^{(r)}$. We thus add an hyper-parameter $\\gamma$ to tune this regularization factor.\n\nIf no prior is available, then simply removing the $W^{(r)}s$ from equation \\eqref{gamma-multi-rel} leads to the following model:\n\\begin{equation}\nZ_{t+1} = h( Z_t \\Theta^{(0)}+\\sum\\limits_{r \\in \\mathcal{R}} \\Gamma^{(r)} Z_t \\Theta^{(r)})\n\\label{gamma-multi-rel-no-W}\n\\end{equation}\n\nwhere $\\Gamma^{(r)}$ is no more constrained by the prior $W^{(r)}$ so that it will represent both the relational structure and the relation weights. Both models are learned with SGD, in the same way as described in \\ref{mts}. The only difference is that a gradient step on the $\\Gamma^{(r)}s$ is added.\n\n\\section{Experiments}\n\\label{xp}\n\\begin{figure*}[t]\n\\begin{center}\n\\begin{tabular}{ccc}\nGround Truth & RNN-GRU & STNN-R \\\\\n\\includegraphics[width=0.3\\linewidth]{truth2} &\n\\includegraphics[width=0.3\\linewidth]{predicrecur.png} & \n\\includegraphics[width=0.3\\linewidth]{predicted_selec.png} \n\\end{tabular}\n\\end{center}\n\\caption{Prediction of wind speed over around 500 stations on the US territory. prediction is shown at time-step $T+1$ for RNN-GRU (centre) and STNN-R (right).}\n\\label{meteo}\n\\end{figure*}\n\n\\begin{figure*}\n  \\centering\n  \\begin{tabular}{ccc}\n     Ground Truth & RNN-GRU & STNN-R \\\\\n     \\includegraphics[width=0.30\\linewidth]{t1.png} \n      & \\includegraphics[width=0.30\\linewidth]{r1.png}\n      & \\includegraphics[width=0.30\\linewidth]{o1.png}\\\\ \n   \\includegraphics[width=0.30\\linewidth]{t2.png} \n      & \\includegraphics[width=0.30\\linewidth]{r2.png}\n      & \\includegraphics[width=0.30\\linewidth]{o2.png}\\\\ \n         \\includegraphics[width=0.30\\linewidth]{t3.png} \n      & \\includegraphics[width=0.30\\linewidth]{r3.png}\n      & \\includegraphics[width=0.30\\linewidth]{o3.png}\n  \\end{tabular}\n  \\caption{Example of a 3 months prediction of Pacific temperature. Left column is the ground truth, central and right columns correspond respectively to RNN-GRU and STNN-R predictions at horizon $T+1$, $T+2$ and $T+3$ (top to bottom).}\n\\label{oceano}\n\\end{figure*}\n\nExperiments are performed on a series of spatio-temporal forecasting problems representative of different domains. We consider predictions within a $+5$ horizon i.e. given a training series of size $T$, the evaluation of the quality of the model will be made over $T+1$ to $T+5$ time steps. The different model hyper-parameters are selected using a time-series cross-validation procedure called rolling origin as in \\cite{ben2014boosting,ganeshapillai2013learning}. This protocol makes use of a sliding window of size $T'$: on a series of length $T$, a window of size $T'$  shifted several times in order to create a set of train\/test folds. The beginning of the $T'$ window is used for training and the remaining for test. The value of $T'$ is fixed so that it is large enough to capture the main dynamics of the different series. Each series was re-scaled between $0$ and $1$.\n\nWe performed experiments with the following models:\\\\\n(i) \\textbf{Mean}: a simple heuristic which predicts future values of a series with the mean of its observed past values computed on the $T'$ training steps of each training fold.\\\\\n(ii) \\textbf{AR}: a classical univariate Auto-Regressive model. For each series and each variable of the series, the prediction is a linear function of $R$ past lags of the variable, $R$ being a hyper-parameter tuned on a validation set.\\\\\n(iii) \\textbf{VAR-MLP}: a vectorial auto-regressive  model where the predicted values of the series at time $t+1$ depend on the past values of all the series for a lag of size $R$. The predictive model is a multi-layer perceptron with one hidden layer. Its performance were uniformly better than a linear VAR. Here again the hidden layer size and the lag $R$ were set by validation\\\\\n(iv) \\textbf{RNN-tanh}: a vanilla recurrent neural network with one hidden layer of recurrent units and tanh non-linearities. As for the \\textbf{VAR-MLP}, one considers all the series simultaneously, i.e. at time $t$ the RNN receives as input $X_{t-1}$ the values of all the series at $t-1$ and predicts $X_{t}$. A RNN is a dynamical state-space model but its latent state $Z_t$ explicitly depends through a functional dependency both on the preceding values of the series $X_{t-1}$ and on the preceding state $Z_{t-1}$. Note that this model has the potential to capture the spatial dependencies since all the series are considered simultaneously, but does not model them explicitly.\\\\\n(v) \\textbf{RNN-GRU}: same as the \\textbf{RNN-tanh}, but recurrent units is replaced with gated recurrent units (GRU) units, which are considered state of the art for many sequence prediction problems today \\footnote{We also performed tests with LSTM and obtained similar results as GRU.}. We have experimented with several architectures, but using more than one layer of GRU units did not improve the performance, so we used 1 layer in all the experiments.\\\\\n(vi) \\textbf{Dynamic Factor Graph (DFG)}: the model proposed in \\cite{mirowski2009dynamic} is the closest to ours but uses a joint  vectorial latent representation for all the series as in the RNNs, and does not explicitly model the spatial relations between series.\\\\\n(vii) \\textbf{STNN}: our model where $g$ is the function described in equation \\eqref{dynamic-multi-rel}, $h$ is the $tanh$ function, and $d$ is a linear function. Note that other architectures for $d$ and $g$ have been tested (e.g. multi-layer perceptrons) without improving the quality of the prediction. The $\\lambda$ value has been set by cross validation.\\\\\n(viii and ix) \\textbf{STNN-R} and \\textbf{STNN-D}: For the forecasting experiments the $\\gamma$ value of the $L_1$ penalty  (see equation \\eqref{eqf}) wer set to $0$ since higher value decreased the performance, a phenomena often observed in other models such as $L_1$-regularized SVMs. The influence of $\\gamma$ on the discovered spatial structure is further discussed and illustrated in figure \\ref{sparsity}. \n\nThe complete set of hyper-parameters values for the different models is given in appendix. \n\n\\subsection{Datasets}\n\\label{datasets}\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}\n  \\begin{axis}[\n     \n      ybar=2pt,\n      width=7cm,,\n      enlargelimits=0.25,\n      legend style={at={(0.5,-0.15)},\n        anchor=north,legend columns=-1},\n      ylabel={RMSE },\n      symbolic x coords={0.0001,0.001,0.01,0.1,1,10,100},\n      xtick=data,\n      ticklabel style = {font=\\tiny},\n     \n      nodes near coords align={vertical},\n      ]\n  \\addplot coordinates {(0.0001,0.034) (0.001,0.026) (0.01,0.025) (0.1,0.021) (1,0.027) (10,0.031) (100,0.040)};\n  \\legend{$\\lambda$}\n  \\end{axis}\n\\end{tikzpicture}\n\\caption{RMSE on Google Flu w.r.t $\\lambda$}\n\\label{fl1}\n\\end{figure}\n\nThe different forecasting problems and the corresponding datasets are described below. The dataset characteristics are provided in table \\ref{datasets}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.9\\linewidth]{flucourbe}\n\\end{center}\n\\caption{RMSE on the Google Flu dataset at horizon $T+1$ to $T+13$}\n\\label{curve}\n\\end{figure}\n\n\\begin{itemize}\n\\item \\textbf{Disease spread forecasting:} The \\textbf{Google Flu} dataset contains for 29 countries, about ten years of weekly estimates of influenza activity computed by aggregating Google search queries (see \\url{http:\/\/www.google.org\/flutrends}). We extract binary relations between the countries, depending on whether or not they share a border, as a prior $W$.\n\\item \\textbf{Global Health Observatory (GHO): } This dataset made available by the Global Health Observatory, (\\url{http:\/\/www.who.int\/en\/}) provides the number of deaths for several diseases. We picked 25 diseases corresponding to \\textbf{25 different datasets}, each one composed of 91 time series corresponding to 91 countries (see table \\ref{datasets}). Results are averages over all the datasets. As for Google Flu, we extract binary relations $W$ based on borders between the countries.\n\\item \\textbf{Geo-Spatial datasets:} The goal is to predict the evolution of geophysical phenomena measured on the surface of the Earth. \\\\The \\textbf{Wind} dataset (\\url{www.ncdc.noaa.gov\/}) consists of hourly summaries of meteorological data. We predict wind speed and orientation for approximately 500 land stations on U.S. locations. In this dataset, the relations correspond to a thresholded spatial proximity between the series.  Given a selected threshold value $d$, two sources are connected ($w_{i,j}=1$) if their distance is below $d$ and not connected ($w_{i,j}=0$) otherwise. \\\\The \\textbf{Pacific Sea Temperature (PST)} dataset represents gridded (at a 2 by 2 degrees resolution, corresponding to 2520 spatial locations) monthly Sea Surface Temperature (SST) on the Pacific for 399 consecutive months from January 1970 through March 2003. The goal is to predict future temperatures at the different spatial locations. Data were obtained from the Climate Data Library at Columbia University (\\url{http:\/\/iridl.ldeo.columbia.edu\/}). Since the series are organized on a 2D grid, we extract 8 different relations : one for each cardinal direction (north, north-west, west, etc...). For instance, the relation $north$, is associated to a binary adjacency matrix $W^{(north)}$ such that $W^{(north)}_{i,j}$ is set to $1$ if and only if source $j$ is located 2 degree at the north of source $i$ (the pixel just above on the satellite image).\n\\item \\textbf{Car Traffic Forecasting:} The goal is to predict car traffic on a network of streets or roads. We use the \\textbf{Beijing dataset } provided in \\cite{yuan2011driving,yuan2010t} which consists of GPS trajectories for $\\sim 10500$ taxis during a week, for a total of 17 millions of points corresponding to road segments in Beijing. From this dataset, we extracted the traffic-volume aggregated on a 15 min window for 5,000 road segments. The objective is to predict the traffic at each segment. We connect two sources if they correspond to road segments with a shared crossroad.\n\\end{itemize}\n\nFor all the datasets but PST (i.e. Google Flu, GHO, Wind and Bejing), we defined the relational structure using a simple adjacency matrix $W$. Based on this matrix, we defined $K$ different relations by introducing the powers of this matrix: $W^{(1)}=W$, $W^{(2)}=W \\times W$, etc. In our setting $K$ took values from 1 to 3 and the optimal value for each dataset has been selected during the validation process. \n\n\n\\subsection{Forecasting Results}\n\\label{xp-forecast}\n\n\\begin{figure}[t]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=0.75\\linewidth]{correl_all} \\\\\n\\includegraphics[width=0.75\\linewidth]{correl_l1strong} \\\\\n\\includegraphics[width=0.75\\linewidth]{correl_l1_verystrong.png} \n\\end{tabular}\n\\end{center}\n\\caption{Illustrations of correlations $\\Gamma$ discovered by the STNN-D model, with $\\gamma$ in $\\{0.01,0.1,1\\}$ ( from top to bottom).}\n\\label{sparsity}\n\\end{figure}\n\n\\begin{figure}[t]\n\\includegraphics[width=\\linewidth]{pst_static_refined_1.png}\n\\caption{Spatial correlations extracted by the STNN-R model on the PST dataset. The color of each pixel correspond to the principal relation extracted by the model.}\n\\label{alpah_refined_pst}\n\\end{figure}\n\n\n\nA quantitative evaluation of the different models and the baselines, on the different datasets is provided in table \\ref{table1}. All the results are average prediction error for $T+1$ to $T+5$ predictions. The score function used is the Root Mean Squared Error (RMSE). A first observation is that STNN and STNN-R models which make use of prior spatial information significantly outperform all the other models on all the datasets. For example, on the challenging PST dataset, our models increase by $23\\%$ the performance of the GRU-RNN baseline. The increase is more important when the number of series is high (geo-spatial and traffic datasets) than when it is small (disease datasets). In these experiments, STNN-D is on par with RNN-GRU. The two models do not use prior information on spatial proximity. STNN makes use of a more compact formulation than RNN-GRU for expressing the series mutual dependency but the results are comparable. Vectorial AR logically improves on mono-variable AR (not shown here) and non linear MLP-VAR improves on linear VAR.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{pst_dyn_angle_icdm.png}\n\\caption{Dynamic spatio-temporal relations extract from the PST dataset on the training set. The color represents the actual sea surface temperature. The arrows represent the extracted spatial relations that evolve through time.}\n\\label{arrows}\n\\end{figure}\n\nWe also provide in table \\ref{fl4} the score for each of the 25 diseases of the GHO dataset. STNN-R obtained the best performance compared to STNN and STNN-D. It outperforms state-of-the-art methods in 20 over 25 datasets, and is very close to the RNN-GRU model on the 5 remaining diseases where RNN-GRU performs best. It thus shows that our model is able to benefit from the neighbour information in the proximity graph. \n\nFigures \\ref{meteo} and \\ref{oceano} illustrate the prediction of \\textit{STNN-R} and \\textit{RNN-GRU} on the meteorology and on the oceanography datasets along with the ground truth. Clearly on these datasets, STNN qualitatively performs much better than RNNs by using explicit spatial information. STNN is able to predict fine details corresponding to local interactions when RNNs produce a much more noisy prediction. These illustrations are representative of the general behavior of the two models.\n\nWe also provide the performance of the models at different prediction horizons $T+1, T+2,.... T+13$ on figure \\ref{curve} for the Google Flu dataset. Results show that STNN performs better than the other approaches for all prediction horizons and is thus able to better capture longer-term dependencies.\n\nFigure \\ref{fl1} illustrates the RMSE of the STNN-R model when predicting at $T+1$ on the Google Flu dataset for different values of $\\lambda$. One can see that the best performance is obtained for an average value of $\\lambda$: low values corresponding to weak temporal constraints do not allow the model to learn the dynamicity of the series while high values degrade the performance of STNN. \n\n\\subsection{Discovering the Spatial Correlations}\n\\label{xp-rel}\n\nIn this subsection, we illustrate the ability of STNN to discover relevant spatial correlations on different datasets. Figure \\ref{sparsity} illustrates the values of $\\Gamma$ obtained by STNN-D where no structure (e.g. adjacency matrix $W$) is provided to the model on the PST dataset. Each pixel corresponds to a particular time series and the figure shows the correlation $\\Gamma_{i,j}$ discovered between each series $j$ with a series $i$ roughly located at the center of the picture. The darker a pixel is, the higher the absolute value of $\\Gamma_{i,j}$ is (note that black pixels correspond to countries and not sea). Different levels of sparsity are illustrated from low (up) to high (down). Even if the model does not have any knowledge about the spatial organization of the series (no $W$ matrix provided), it is able to re-discover this spatial organization by detecting strong correlations between close series, and low ones for distant series. \n\nFigure \\ref{alpah_refined_pst} illustrates the correlations discovered on the PST dataset. We used as priors 8 types of relations corresponding to the 8 cardinal directions (South, South-West, etc...). In this case, STNN-R learns weights (i.e $\\Gamma^{(r)}$) for each relation based on the prior structure. For each series, we plot the direction with the highest learned weight. The strongest direction for each series is illustrated by a specific color in the figure. For instance, a dark blue pixel indicates that the stronger spatial correlation learned for the corresponding series is the North-West direction. The model extracts automatically relations corresponding to temperature propagation directions in the pacific, providing relevant information about the spatio-temporal dynamics of the system. \\\\ \\linebreak\n\n\n\nThe model can be adapted to different situations. Figure \\ref{arrows} represents the temporal evolution of the spatial relations on the PST dataset. For this experiment, we have  slightly changed the STNN-R model by making the $\\Gamma^{(r)}$ time dependent according to:\n\\begin{equation}\n\\Gamma^{(r)}_{t,j,i} = f_r(Z_t^i)\n\\end{equation}\nThis means that with this modified model, the spatial relation weights depend on the current latent state of the corresponding series and may evolve with time.\nIn the experiment, $f_r$ is a logistic function. On figure \\ref{arrows}, the different plots correspond to successive time steps. The color represent the actual sea surface temperatures, and the arrows represent the direction of the stronger relation weights $\\Gamma^{(r)}_{t}$ among the eight possible directions (N, NE, etc). One can see that the model captures coherent dynamic spatial correlations such as global currents directions or rotating motions that gradually evolve with time.\n\n\n\n\n\n\n\\section{Conclusion}\nWe proposed a new latent model for addressing multivariate spatio-temporal time series forecasting problems. For this model the dynamics are captured in a latent space and the prediction makes use of a decoder mechanism. Extensive experiments on datasets representative of different domains show that this model is able to capture spatial and temporal dynamics, and performs better than state of the art competing models. This model is amenable to different variants concerning  the formulation of spatial and temporal dependencies between the sources.\n\nFor the applications, we have concentrated on forecasting (time based prediction). The same model could be used for interpolating (space based prediction or kriging) or for data imputation when dealing with time-series with missing values.\n\n\\section{Acknowledgments}\nLocust project ANR-15-CE23-0027-01, funded by Agence Nationale de la Recherche. \n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n The archetypical ultracompact \\hbox{H{\\sc ii}}\\ region (UC \\hbox{H{\\sc ii}}) W3(OH) has been\n the topic of numerous studies targeted at understanding the phenomena\n involved in the formation of massive stars and the interaction of\n these objects with their environment.  In particular, interferometric\n observations at radio and millimeter wavelengths have revealed an\n ever more detailed picture of the dense gas associated with W3(OH)\n (e.g., Baudry et al.\\ 1993; Bloemhof et al.\\ 1992; Reid et al.\\ 1995;\n Turner \\& Welch 1984; Wilner et al.\\ 1995; Wink et al.\\ 1994;\n Wyrowski et al.\\ 1997).\n \n The observations suggest that there are several sites of very recent\n and on-going star formation: the most prominent of these are the UC\n \\hbox{H{\\sc ii}}\\ itself, which is ionized by a young O star, and a region\n showing hot molecular line and dust emission that is associated with\n strong water maser emission. The latter region is located at a\n projected distance of $\\approx 0.06$~pc to the east of the UC \\hbox{H{\\sc ii}},\n assuming a distance of 2.2 kpc for W3(OH).  Most of the dense neutral\n gas in the W3(OH) complex appears to be associated with the \\WAT\\ \n maser\/``hot core'' source, whereas the most luminous young star in\n the region seems to be the exciting star of the UC \\hbox{H{\\sc ii}}.\n \n There are several lines of evidence suggesting that there are one or\n more young stars embedded in the dense molecular gas near the \\WAT\\ \n maser source: Turner \\&\\ Welch (1984) found a compact source of\n emission in the HCN $J = 1-0$ transition toward the \\WAT\\ maser\n position (in the following referred to as W3(OH)-TW).  The HCN line\n showed broad wings suggestive of mass loss from a young embedded\n star.  Alcolea et al.\\ (1992) measured the \\WAT\\ maser proper motions\n and found them to be consistent with a bipolar outflow along an E-W\n axis in the plane of the sky.  Mauersberger et al.\\ (1986a,b) used\n their ammonia observations to demonstrate the presence of dense, hot\n ($> 160$~K) gas close to the water masers and this was confirmed by\n the studies of Wink et al.\\ (1994) and Helmich et al.\\ (1994).\n However, the most surprising result was perhaps that of Reid et al.\\ \n (1995), who studied weak elongated centimeter continuum emission\n toward the TW source. They concluded that there was strong evidence\n for this being synchrotron emission from a ``jet-like'' structure\n aligned with the outflow indicated by the proper motions of the \\WAT\\ \n masers associated with W3OH-TW.  Moreover, the synchrotron emission\n centroid was found to be coincident with the center of expansion of\n the \\WAT\\ outflow as determined by Alcolea et al.\\ (1992).  All of\n these facts suggest the presence of a relatively luminous embedded\n star which drives a collimated outflow.\n \n In this paper, we present sensitive, sub-arcsecond resolution\n interferometric observations of the W3(OH) complex at a frequency of\n 220 GHz; results from simultaneous observations at 107 GHz will be\n presented elsewhere. We find an excellent coincidence between the\n structures seen at millimeter wavelengths in thermal dust emission\n and those seen in non-thermal emission (see Wilner et al.\\ 1998) at\n centimeter wavelengths.\n\n \\section{Observations}\n \n Our observations were made with the 5 element Plateau de Bure\n Interferometer in three configurations: B1-N13 on 1998 January 6\/7,\n A1 on February 7, and B2 on February 17. The phase center was\n $\\alpha(J2000)$ = $02^{\\rm h}27^{\\rm m}04{\\rlap.}{^{\\rm s}}284$,\n $\\delta(J2000)$ = +$61^{\\circ}52'24{\\rlap.}{''}55$, which is between\n the positions of W3(OH) and W3(OH)-TW. The total observing time on\n source was 17.1 hours covering a baseline range from 30 to 410~m. We\n used the dual-frequency receiver systems to simultaneously observe\n the \\METH\\ $J_k = 3_1-4_0A^+$ and \\CEIO\\ ($J = 2-1$) lines.  Due to\n good winter weather conditions, the 220 GHz system temperature was in\n the range 200 to 400~K and the radio seeing on all days better than\n 0\\farcs5. On-source integrations of 20~min were interspersed with\n phase calibrator observations on the quasar 0224+671.  For bandpass\n and flux density calibration, the sources 3C454.3, 3C273 and 3C84\n were used and the absolute flux density scale was established by\n observing MWC~349, for which a flux density of 1.66~Jy was assumed at\n 220 GHz.  From the day-to-day variance in W3(OH)'s continuum flux\n density, we estimate that our 220 GHz flux density scale is accurate\n to within 20\\%.\n \n The data were processed using the GILDAS software package.  To remove\n the effects of short-term atmospheric fluctuations on the\n interferometer phases, phase corrections derived from 220 GHz total\n power measurements were applied to the 107 and 220 GHz data (see\n Bremer et al.\\ 1996).  Then, correcting for phase drifts on long\n timescales was done in the standard manner by observing a calibrator\n source. The phase solutions for the 107~GHz data were subtracted,\n appropriately scaled, from the 220~GHz data.  The RMS noise of the\n fits to the residual 220 GHz phase was found to be only 10\\arcdeg\\ on\n average.  After gridding and Fast Fourier Transform of the $uv-$data,\n a 220 GHz beam size of 0\\farcs83$\\times$0\\farcs55 (FWHM) was\n determined for natural weighting with a position angle of\n 100$\\arcdeg$. From all correlator units, spectral line data cubes\n were built and checked for frequency ranges free of line emission in\n order to produce a continuum map. All data were deconvolved using the\n CLEAN algorithm and the resulting dynamical range limited RMS noise\n in the 220 GHz continuum maps is 20~mJy.\n\n \\section{Results}\n\n  \\subsection{220 GHz continuum measurements}\n  \n  In Fig.~\\ref{cmap}, we present our 220 GHz continuum map of W3(OH)\n  superimposed upon the 8~GHz VLA map of Wilner et al.\\ (1998).  For a\n  better comparison with the latter, our 220 GHz map was restored with\n  a circular beam of 0\\farcs5 diameter (FWHM) and is thus slightly\n  superresolved. On both maps, the main features are the UC \\hbox{H{\\sc ii}}\\ to\n  the west and the water maser source 6\\arcsec \\ to the east.  We\n  checked the alignment of the maps by computing their correlation and\n  found them to coincide within 0\\farcs05. Toward the \\hbox{H{\\sc ii}}\\ region,\n  the main emission mechanism even at 220 GHz is free-free emission\n  from the ionized gas. In fact, in agreement with the measurements of\n  Wyrowski et al.\\ (1997), we observe an integrated flux density of\n  $3.1 \\pm 0.5$~Jy at 220~GHz toward the \\hbox{H{\\sc ii}}\\ region as compared to a\n  value of $3.0 \\pm 0.15$~Jy at 107~GHz.  Combining the 3~mm\n  measurements with results from the literature allows us to\n  extrapolate the flux density to 220~GHz assuming optically thin\n  free--free emission and we find $3.0 \\pm 0.3$~Jy.  This value is\n  consistent with the actually measured flux density at 220~GHz and we\n  place a conservative upper limit of 0.5 Jy on any contribution from\n  emission from dust associated with the UC \\hbox{H{\\sc ii}}\\ region.\n  \n  Most strikingly, Fig.~\\ref{cmap} reveals the small-scale spatial\n  coincidence of dust emission features (shown in grey-scale) with\n  radio continuum sources detected toward the TW-object in the 3.6~cm\n  map of Wilner et al.\\ (1998, shown as contours). As seen on the\n  inset to Fig.~\\ref{cmap}, the easternmost component A is coincident\n  with the centroid of the elongated structure seen with the VLA.\n  There is also a rough correspondence between components B and C of\n  the 220 GHz map with features seen at 3.6~cm.  In particular, it\n  seems clear that dust emission source A is associated with the\n  non-thermal radio source detected at centimeter wavelengths.\n  \n  We note here that there is also evidence in Fig.~\\ref{cmap} for\n  extended emission surrounding components A, B, and C.  Our\n  observations are not sensitive to structures extended over size\n  scales larger than about 5\\arcsec\\ and hence it is likely that we\n  are missing flux.  We consider therefore that the total integrated\n  flux density measured toward the water maser position (1.6~Jy from\n  Fig.~\\ref{cmap}) is a lower limit.  Estimates of the integrated\n  fluxes in components A, B, and C are rendered difficult due to,\n  both, the presence of the extended emission and blending obvious on\n  Fig.~\\ref{cmap}. Consequently, we only quote peak flux densities of\n  components A, B, and C measured in our 0\\farcs5 beam in\n  Table~\\ref{peaks}.\n\n  \\subsection{Line measurements near 220 GHz}\n  \n  Compared to previous, similar, observations, our data have much\n  improved sensitivity and angular resolution and thus allow a new\n  discussion of the molecular line emission distribution in the W3(OH)\n  region.  The observations discussed in Wyrowski (1997) clearly\n  showed that there is a dichotomy between oxygen-containing molecules\n  related to methanol (essentially methyl formate and dimethyl ether)\n  and nitrogen-containing species such as ethyl and vinyl cyanide.\n  The former are seen both toward the UC \\hbox{H{\\sc ii}}\\ and the water masers\n  whereas the latter are {\\it only} seen toward the water masers.\n  This is shown with improved angular resolution in\n  Fig.~\\ref{line-overlays} where we compare naturally weighted\n  integrated intensity maps of lines detected near 220 GHz from a\n  variety of species.  We note in particular the fact that the\n  $24_{2,22}-23_{2,21}$ transition of \\ETHCN \\ and the $v_7=2$,\n  $J=24-23$, transition of \\CYAC\\ (respectively 135~K and 772~K above\n  ground) peak precisely in the direction of continuum component C and\n  are not detected toward the \\hbox{H{\\sc ii}}\\ region. In contrast, the\n  transitions which we detect from \\METH, \\MEFORM, and \\FORM \\ are\n  seen both toward the ionized gas and toward the water maser\n  concentration. The similarity of the intensity distributions in all\n  three cases suggests a chemical link between these molecules as\n  indeed discussed by Blake et al.\\ (1987) in the context of Orion.\n  Finally, we note the curious behavior of the $22_{2,20}-22_{1,21}$\n  transition of \\SOTW \\ which, in contrast to e.g. \\METH, is seen\n  toward the eastern rather than the western border of the \\hbox{H{\\sc ii}}\\ \n  region.\n  \n  We also detected the $K_{a}=0,2,3,4$ components of the $J=10-9$ HNCO\n  transitions at excitations of 50--750~K above ground.  The relative\n  populations of these levels is thought to be determined by a\n  combination of collisions and radiative transitions induced by the\n  FIR radiation field within the hot cores (Churchwell et al.\\ 1986).\n  We have attempted to use the relative intensities of these lines as\n  a ``thermometer'' measuring the temperature of the hot core by\n  assuming, first, that all transitions are optically thin (which is\n  justified by the fact that the $K_{a}=0$ line has a brightness\n  temperature of less than 20~K) and, second, that the level\n  populations are maintained by the radiation temperature of the dust.\n  Since the dust is optically thick at the HNCO FIR pump wavelengths\n  of 50--330~$\\mu\\hbox{m}$, the radiation temperature equals the dust\n  temperature.  We thus estimate rotational temperatures as a function\n  of position over the water maser source.  In practise, the derived\n  level column densities are consistent with a single rotational\n  temperature at each point and allow us (assuming LTE) to infer the\n  dust (and gas) temperature.  The resulting temperature map is shown\n  in Fig.~\\ref{hnco-results}, where one sees that there are two 200~K\n  peaks roughly coincident with our continuum peaks A and C.  Between\n  these there is a ``plateau'' where the temperature is of order\n  150~K. We interpret this as evidence that there are two energy\n  sources (young proto-B stars) embedded in the molecular gas whose\n  positions are roughly defined by the temperature peaks and which are\n  responsible for the cm-emission.\n  \n  Moreover, we can use the temperature distribution thus derived to\n  deduce the dust and, thus, the hydrogen column density distribution\n  from our continuum map.  The resulting hydrogen column density map\n  is also shown on Fig.~\\ref{hnco-results} where the resulting gas\n  distribution appears to be {\\it single-peaked} and extended with\n  dimensions of $0.02\\times 0.01$~pc. Using the formula given by\n  Mezger et al.\\ (1990) we calculate a total mass of 15~$\\hbox{M}_\\odot$\\ for\n  this structure. Thus, the triple-peaked appearance of our continuum\n  map (Fig.~\\ref{cmap}) is caused by an interplay of the temperature\n  and column density distributions.  This interpretation is supported\n  by our \\CEIO\\ $(2-1)$ integrated intensity distribution\n  (Fig.~\\ref{hnco-results}) which has the same general form as our\n  hydrogen column density map but is more sensitive to cooler material\n  further from the embedded stars.\n\n \\section{Discussion and Conclusions}\n \n Our new 220 GHz interferometer data suggest the presence of two\n embedded young stars lying slightly offset from the center of the gas\n clump associated with the water maser complex close to W3(OH). The\n mass of high temperature gas in the region is of order 15~$\\hbox{M}_\\odot$,\n although our \\CEIO \\ map makes it clear that there is more cooler gas\n at larger distances from the protostars.  The luminosity of the\n hypothesised embedded stars coincident with positions A and C can be\n crudely inferred from the observed temperature distribution\n (Fig.~\\ref{hnco-results}) by applying the Stefan-Boltzmann law. This\n way we find $3\\times 10^4$~\\solum\\ for the gas clump with a factor of\n 3 uncertainty due to temperature errors.  This (Panagia 1973)\n suggests embedded B0 stars.  It is of course likely that there are\n other lower mass objects associated with the complex and it is\n possible that there is more mass in stars than in gas.\n \n Finally, we note that perhaps the most remarkable result of this\n study has been the coincidence of continuum source A with the\n ``synchrotron jet'' of Wilner et al.\\ (1998). What is the\n significance of this?  It seems reasonable that the dense hot core\n gas plays a role in confining the ``jet''. The magnetic pressure of\n the medium in which the relativistic electrons radiate ( B$^{2}\/(8\\pi) \n \\sim \\, 4\\, 10^{-6}$ erg \\percc, Reid et al.\\ 1995) is of the same\n order as the thermal pressure in the molecular gas for a density of\n $10^8$ \\percc \\ and thus such confinement seems feasible.  The origin\n of the relativistic electrons is unclear but there are analogous\n cases known. An example is the the jet in Cepheus A (Garay et al.\\ \n 1996), whose radio emission is thought to be produced in shocks\n resulting from the interaction of the jet with the confining medium.\n The explanation in the case of W3(OH) may be similar.\n\n \\acknowledgments\n \n We thank D. Wilner and M. Reid for providing the VLA data in advance\n of publication and an anonymous referee for helpful comments.  CMW\n acknowledges travel support from CNR grant 97.00018.CT02 and ASI\n grant ARS-96-66. He would also like to thank the Max Planck Institut\n f{\\\"u}r Radioastronomie for hospitality during the course of this\n work.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nString theory is a promising candidate for a consistent quantum theory of gravity and needless to say that the characteristics of black holes (BH) in string theory would be utmost interest. The predictions of string theory differ from those of general relativity and one of the reasons for this difference is the presence of a scalar field called dilaton that can change the properties of the BH geometries.\n\nThe spherically symmetric static charged BH solution in low energy heterotic string theory in four dimension was found by Gibbons and Maeda~[1] and independently by Garfinkle, Horowitz, Strominger~[2], which from now on will be referred to as the Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) solution. These works generated enormous interest in the dilatonic charged BHs (see, e.g., [3-16] and references therein). In particular, the GMGHS BH spacetime can be described either in the Einstein frame (EF) or in the conformally related string frame (SF). (Unless the frame is specifically mentioned, the solution will be understood to be in EF). In EF, the action is in the form of the Einstein-Hilbert action, while in the SF strings directly couple to the metric as $e^{2\\phi }g_{\\mu \\nu }$, where $\\phi $ is the dilaton and $g_{\\mu \\nu }$ is the EF metric. Even though the solutions in the two frames are related by a conformal transformation so that they are mathematically isomorphic to each other~[3], there are differences in some of the physical properties of the BH solutions in these two frames~[4]. For instance, Sagnac delay for rotating Kerr-Sen metric of heterotic string theory was studied and an estimate of assumed terrestrial dilatonic charge was given in [5]. Strong gravitational lensing by charged stringy BHs do not produce any significant string effect on the Schwarzshild BH as was shown by Bhadra [6]. Timelike geodesics of particles around GMGHS BH were investigated in [7]. The magnetically charged GMGHS interior spacetime was studied in [8]. Particle collision near the horizon of GMGHS BH was studied in [9]. Quasinormal mode frequencies in the string BH were evaluated by using WKB approximation with P\\\"{o}schl-Teller potential in [10]. Some notable works on the implications of electric charge on various relativistic observables in the context of general relativistic BHs can be found in [17-21].\n\nIn addition to the above studies, which are by no means exhaustive, the accretion disk properties could be yet another diagnostic for distinguishing various types of objects. The first comprehensive study of accretion disks using a Newtonian approach was made in [22]. Later a general relativistic model of thin accretion disk was developed in three seminal papers, by Novikov and Thorne [23], Page and Thorne [24] and Thorne [25] under the assumption that the disk is in a steady-state, that is, the mass accretion rate $\\dot{M}_{0}$ is constant in time and does \\textit{not} depend of the radius of the disk. The disk is further supposed to be in hydrodynamic and thermodynamic equilibrium, which ensure a black body electromagnetic spectrum and properties of emitted radiation. The thin accretion disk model further assumes that individual particles are moving on Keplerian orbits, but for this to be true the central object should have weak magnetic field, otherwise the orbits in the inner edge of the disk will be deformed. More recently, the properties of the accretion disk around exotic central objects, such as wormholes (WH) [26,27], and non-rotating or rotating quark, boson or fermion stars, brane-world BHs, gravastars or naked singularities (NS) [28-44], $f(R)$ modified gravity models [44-46] have been studied. One of the most promising method to distinguish the types of astrophysical objects and their spin through their accretion disk properties is the profile analysis of iron line for different spacetimes [47-52]. While various accreting objects have been considered, the study of emissivity properties of the GMGHS objects, for which the celebrated Page-Thorne model is the most ideal one, have somehow been left out in the literature, to our knowledge.\n\nOne recent work by Bahamonde and Jamil [11] pertains to fluid motion (as opposed to particle motion) in different spherically symmetric spacetimes, where the authors focused on the radial variation of the fluid velocity, the density and the accretion rate\\ $\\dot{M}_{0}$ of the fluid to the GMGHS BH. The work in [11], though useful in its own right, is distinct from the Page-Thorne emission model since it studied only the non-emissivity aspects of the fluid flow with the predicted mass accretion rate $\\dot{M}_{0}$ depending on the radius of the disk. Also the critical radius $r_{\\textmd{\\scriptsize{c}}}$ used in [11] is not the marginally stable radius $r_{\\textmd{\\scriptsize{ms}}}$ required in the Page-Thorne model. On the other hand, we are motivated by the understanding that genuine observable signatures of the accretion disk should be provided by an analysis of the properties of radiation emerging from the surface of the disk for which the Page-Thorne model is most suitable.\n\nThe present paper is thus devoted to studying the kinematic and emissivity properties of a central object represented by stringy GMGHS solutions (not necessarily BHs) in the EF and SF using the Page-Thorne model. We shall analyze the luminosity spectra, flux of radiation, temperature profile, efficiency etc. In particular, we wish to see how the kinematic and accretion features compare between EF and SF including their extreme counterparts and with similar features from Reissner-Nordstr\\\"{o}m and Schwarzschild BH of general relativity. We shall assume for numerical illustration a toy model of a central object with mass $15M_{\\odot }$ and accretion rate $\\dot{M}_{0}=10^{18}$ gm.sec$^{-1}$, which could be a BH, a WH or a NS.\n\nThe paper is organized as follows: In Sec.2, we give a brief preview of the GMGHS solutions and in Sec.3, summarize the main formulas relating to the accretion phenomenon to be used in the paper. In Sec.4, we present the kinematic and accretion properties in two frames and compare how they differ from those for the Reissner-Nordstr\\\"{o}m and Schwarzschild BH. In Sec.5, the obtained results are summarized. We take units such that $8\\pi G=c=1$, metric signature ($-,+,+,+$) and greek indices run from $0$ to $3$.\n\n\n\n\n\\section{GMGHS solutions}\n\nIn this section, a brief preview of the action and the static spherically symmetric dilaton BHs are given. In the EF, the GMGHS action is [1,2]\n\\begin{equation}\nS_{\\textmd{\\scriptsize{EF}}}=\\int d^{4}x\\sqrt{-g}\\left[ R_{(g)}-2(\\bigtriangledown \\phi\n)^{2}-e^{-2\\phi }F_{\\mu \\nu }F^{\\mu \\nu }\\right],\n\\end{equation}%\nwhere $\\phi $ is a dilaton, $R_{(g)}$ is the scalar curvature related to $g_{\\mu \\nu }$, and $F_{\\mu \\nu }$ is the Maxwell field. The line element\nrepresenting a 4-dimensional charged dilatonic GMGHS BH in the EF is given by\n\\begin{eqnarray}\nd\\tau _{\\textmd{\\scriptsize{Mag,EF}}}^{2}&=&-\\left( 1-\\frac{2M}{r}\\right) dt^{2}+\\left( 1-\\frac{2M}{r}\\right)^{-1}dr^{2} \\nonumber\\\\\n&&+r\\left( r-\\frac{Q^{2}}{M}\\right) (d\\theta^{2}+ \\sin^{2}{\\theta} d\\varphi^{2}),\n\\end{eqnarray}%\nwhere $M$ is the mass and $Q$ is the magnetic charge. The Maxwell field is given by\n\\begin{equation}\nF=Q\\sin {\\theta }d\\theta \\wedge d\\varphi\n\\end{equation}%\nand the dilaton field $\\phi $ is defined as\n\\begin{equation}\ne^{-2\\phi }=e^{-2\\phi _{0}}\\left( 1-\\frac{Q^{2}}{Mr}\\right) ,\n\\end{equation}%\nwhere $\\phi _{0}$ is the asymptotic value of the dilaton. As we consider only asymptotically flat cases, we will assume $\\phi _{0}\\equiv 0$. Solution (2) describes BHs of mass $M$ and charge $Q$ when $Q\/M$ is sufficiently small. For $Q=\\sqrt{2}M$, the event horizon $r=2M$ becomes singular. (Figs.1,4,5 show the properties of this singularity at $r=2M$: potential $V_{\\textmd{\\scriptsize{eff}}}$, Flux of radiation and Temperature diverge).\n\nSince the metric, for fixed $\\theta $ and $\\varphi $, is the same as that of Schwarzschild, $r=2M$ is a regular event horizon only when $Q<\\sqrt{2}M$. Note that the area goes to zero at $r=Q^{2}\/M$ $<2M$ causing this surface to be singular since the Kretschmann scalar and the dilaton $\\phi $ diverge there. However, the surface is hidden under $r=2M$ and no information can emerge from it to outside observers (All observable quantities also diverge on that surface). The dilatonic charge for the charged BH (2) is\n\\begin{equation}\nD=-\\frac{Q^{2}}{2M},\n\\end{equation}%\nwhere $D$ is not a new free parameter in (2) since once the asymptotic value of $\\phi $ is fixed, it is determined by $M$ and $Q$, and is always negative. The dilaton charge is also responsible for a long-range, attractive force between BHs.\n\nElectrically charged solutions may be obtained by a duality rotation defined by\n\\begin{equation}\n\\tilde{F}_{\\mu \\nu }=\\frac{1}{2}e^{-2\\phi }\\epsilon _{\\mu \\nu }^{\\lambda\n\\rho }F_{\\lambda \\rho }.\n\\end{equation}%\nThe equations of motion (2) are invariant under $F\\rightarrow \\tilde{F}$ and $\\phi \\rightarrow -\\phi $. Such solutions can therefore be obtained by simply changing the sign of $\\phi $ while keeping the metric fixed. This implies that the dilaton charge is $D$ is positive for electrically charged BHs in the EF.\n\nThe effective action and equations of motion can be further modified by applying the conformal transformation $\\widetilde{g}_{\\mu \\nu }= e^{2\\phi}g_{\\mu\\nu }$ thus obtaining the action in the SF\n\\begin{equation}\nS_{\\textmd{\\scriptsize{SF}}}=\\int d^{4}x\\sqrt{-\\widetilde{g}}e^{-2\\phi }\\left[ R_{(\\widetilde{g})}-4(\\bigtriangledown \\phi )^{2}- F_{\\mu\\nu}F^{\\mu\\nu}\\right],\n\\end{equation}%\nin which the space-time coordinates have been left unchanged and $R_{(\\widetilde{g})}$ is the Ricci curvature from $\\widetilde{g}_{\\mu \\nu }$. Upon transforming to SF one obtains magnetically charged GMGHS BH metric is given by [15,16]:\n\\begin{eqnarray}\nd\\tau _{\\textmd{\\scriptsize{Mag,SF}}}^{2}&=&-\\frac{\\left( 1-\\frac{2M}{r}\\right) }{\\left( 1-\\frac{Q^{2}}{Mr}\\right) }dt^{2}+\\frac{dr^{2}}{\\left(1- \\frac{2M}{r}\\right)\\left(1-\\frac{Q^{2}}{Mr}\\right)} \\nonumber\\\\\n&&+r^{2}(d\\theta ^{2}+\\sin^{2}{\\theta} d\\varphi ^{2}).\n\\end{eqnarray}%\n\nThis is the metric that appears in the string $\\sigma $ model. The Kretschmann scalar diverges at $r=Q^{2}\/M$, hence it is a singular surface. For SF, the statement that the horizon is singular when $Q^{2}=2M^{2}$ is actually irrelevant. This is because strings do not couple to the metric $g_{\\mu \\nu }$ but rather to $e^{2\\phi }g_{\\mu \\nu }$. For $Q^{2}<2M^{2}$, this again describes a BH with an event horizon at $r_{\\textmd{\\scriptsize{eh}}}=2M$. We have simply rescaled the metric by a conformal factor, which is finite everywhere outside (and on) the horizon. However, at the extremal value $Q^{2}=2M^{2}$, the metric becomes\n\\begin{eqnarray}\nd\\tau _{\\textmd{\\scriptsize{WH,SF}}}^{2}&=&-dt^{2}+\\left( 1-2M\/r\\right)^{-2}dr^{2} \\nonumber\\\\\n&&+r^{2}(d\\theta ^{2}+\\sin^{2}{\\theta} d\\varphi ^{2}).\n\\end{eqnarray}%\nThe geometry of a $t=$ const. surface in this spacetime is identical to that\nof a static slice in the extreme Reissner-Nordstr\\\"{o}m metric. But the\nhorizon, along with the singularity inside it, have completely disappeared.\nIn its place have appeared a WH. This metric, with $r>2M$, is globally\nstatic and geodesically complete and has all the properties of a traversable\nMorris-Thorne WH with a regular throat at $r_{\\textmd{\\scriptsize{th}}}=2M$ with redshift\nfunction $\\Phi =0$ and a shape function $b(r)=4M\\left( 1-\\frac{M}{r}\\right) $\nhaving an imbedding surface%\n\\begin{eqnarray}\nz(r)&=&4\\sqrt{M}\\left[ \\sqrt{r-M}\\right. \\nonumber\\\\\n&&\\left.-\\sqrt{M}\\textmd{arctanh}\\left\\{ \\sqrt{\\frac{r}{%\nM}-1}\\right\\} \\right].\n\\end{eqnarray}%\n\nThe solution of the electrically charged GMGHS solution in the SF is given\nby [15,16]\n\\begin{eqnarray}\nd\\tau _{\\textmd{\\scriptsize{Elec,SF}}}^{2}&=&-\\frac{\\left( 1+\\frac{Q^{2}-2M^{2}}{Mr}\\right) }{%\n\\left( 1+\\frac{Q^{2}}{Mr}\\right) ^{2}}dt^{2}+\\frac{dr^{2}}{\\left( 1+\\frac{Q^{2}-2M^{2}}{Mr}\\right) } \\nonumber\\\\\n&&+r^{2}(d\\theta ^{2}+\\sin^{2}{\\theta} d\\varphi ^{2}).\n\\end{eqnarray}%\nThis solution describes a BH, when $Q^{2}<2M^{2}$, but in the extremal case (%\n$Q^{2}=2M^{2}$) the resulting solution is\n\\begin{eqnarray}\nd\\tau _{\\textmd{\\scriptsize{NS,SF}}}^{2}&=&-\\left( 1+2M\/r\\right)\n^{-2}dt^{2}+dr^{2} \\nonumber\\\\\n&&+r^{2}(d\\theta ^{2}+\\sin^{2}{\\theta} d\\varphi ^{2}).\n\\end{eqnarray}%\nwhich describes a singularity at $r=0$ since the Kretschmann scalar diverges\nthere but this divergence is not covered by an event horizon, and so the\npoint $r=0$ represents a NS.\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Veff1}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Veff2} \\\\\n\\vspace{0.5cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Veff3}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{VeffRN}\n\\end{center}\n\\caption{The effective potential $V_{\\textmd{\\scriptsize{eff}}}(r)$ for a GMGHS BH in the EF (top left hand), magnetically charged one in SF (top right hand), electrically charged one in the SF (bottom left hand) and Reissner-Nordstr\\\"{o}m BH (bottom right hand). The specific angular momentum of the orbiting particle is chosen to be $\\widetilde{L}=4M$. The potentials for $Q<\\protect\\sqrt{2}M$ do not show appreciable difference with those of the Reissner-Nordstr\\\"{o}m and Schwarzschild BH (Fig.1d). At the extreme limit, $Q=\\protect\\sqrt{2}M$, the potential diverges at the NS radii (Figs.1a,c) and at the WH throat (Fig.1b). The potential coincides with that of the Schwarzschild asymptotically.}\n\\label{Veff}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{SpAngulMom1}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{SpAngulMom2} \\\\\n\\vspace{0.5cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{SpAngulMom3}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{SpAngulMomRN}\n\\end{center}\n\\caption{The specific angular momentum $\\tilde{L}(r)$ of the orbiting particle as a function of the radial coordinate $r$ (in cm) for a GMGHS BH in EF (top left hand), magnetically charged in the SF (top right hand), electrically charged in the SF (bottom left hand) and Reissner-Nordstr\\\"{o}m BH (bottom right hand) plotted for different values of $Q$. We see that $\\tilde{L}\\rightarrow 0$ at the NS radii $r\\rightarrow 2M$ (Fig.2a), $r\\rightarrow 0$ (Fig.2c), but $\\tilde{L}\\rightarrow 0$ at the WH throat (Fig.2b). Fig.2d displays the behavior only for BHs. All plots show no appreciable difference with those for Schwarzshild BH either in the far field of the accreting object.}\n\\label{SpAngulMom}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{SpEnergy1}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{SpEnergy2} \\\\\n\\vspace{0.5cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{SpEnergy3}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{SpEnergyRN}\n\\end{center}\n\\caption{The specific energy $\\tilde{E}(r)$ of the orbiting particles as a function of the radial coordinate $r$ (in cm) for a GMGHS BH in EF (top left hand), magnetically charged in the SF (top right hand), electrically charged in the SF (bottom left hand) and Reissner-Nordstr\\\"{o}m BH (bottom right hand) plotted for different values of $Q$. We see that $\\tilde{E}\\rightarrow 0$ at the NS radii $r\\rightarrow 2M$ (Fig.3a), $r\\rightarrow 0$ (Fig.3c), but $\\tilde{E}$ assumes a constant value at the WH throat (Fig.3b). Fig.3d displays the behavior only for BHs, and there are no appreciable differences in the far field of observation.}\n\\label{SpEnergy}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Flux1}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Flux2} \\\\\n\\vspace{0.5cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Flux3}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{FluxRN}\n\\end{center}\n\\caption{The time averaged radiation flux $F(r)$ as a function of the radial coordinate $r$ (in cm) radiated by the disk around a GMGHS BH in EF (top left hand), magnetically charged in the SF (top right hand), electrically charged in the SF (bottom left hand) and Reissner-Nordstr\\\"{o}m BH (bottom right hand) plotted for different values of $Q$. We see that $F(r)\\rightarrow \\infty $ at NS radii $2M$ coinciding with $r_{\\textmd{\\scriptsize{ms}}}$ (Fig.4a), $r=r_{\\textmd{\\scriptsize{ms}}}\\rightarrow 0$ (Fig.4c), but $F(r)$ assumes a finite value at the WH throat (Fig.4b). Fig.4d displays the behavior only for BHs. All plots except that of WH show no appreciable differences at the asymptotic limit.}\n\\label{Flux}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Temp1}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Temp2} \\\\\n\\vspace{0.5cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Temp3}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{TempRN}\n\\end{center}\n\\caption{Temperature distribution $T(r)$ of of the accretion disk for a GMGHS BH in EF (top left hand), magnetically charged in the SF (top right hand), electrically charged in the SF (bottom left hand) and Reissner-Nordstr\\\"{o}m BH (bottom right hand) plotted for different values of $Q$. We see that $T(r)\\rightarrow \\infty $ at NS radii $2M$ coinciding with $r_{\\textmd{\\scriptsize{ms}}}$ (Fig.5a), $r=r_{\\textmd{\\scriptsize{ms}}}\\rightarrow 0$ (Fig.5c), but $T(r)$ assumes a finite value at the WH throat (Fig.5b). Fig.5d displays the behavior only for BHs. All plots except that of WH show no appreciable differences at asymptotic distances of observation.}\n\\label{Temp}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Lum1}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Lum2} \\\\\n\\vspace{0.5cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Lum3}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{LumRN}\n\\end{center}\n\\caption{The emission spectra $\\protect\\nu L(\\protect\\nu )$ vs frequency ($\\protect\\nu $ in Hz) of the accretion disk with inclination $i=0^{0}$ for a GMGHS BH in EF (top left hand), magnetically charged in the SF (top right hand), electrically charged in the SF (bottom left hand) and Reissner-Nordstr\\\"{o}m BH (bottom right hand) plotted for different values of $Q$. \\ Fig. 6a shows that, at $Q\\leq \\protect\\sqrt{2}M$, $\\protect\\nu L(\\protect\\nu )$ behaves like a BH of general relativity at the low frequency range but assumes a steady value until $\\protect\\nu \\sim 10^{19}$ Hz, when it begins to decline to the Schwarzschild profile (not shown). Similar behavior is seen in Fig.6c. Fig.6b shows the emission spectra of the disk around WH throat. At low $\\protect\\nu $, the features are almost indistinguishable from those of BHs but at higher frequency, the spectra diminishes considerably. Fig.6d displays the behavior only for BHs. These features are consistent with efficiency of conversion (see summary).}\n\\label{Lum}\n\\end{figure*}\n\n\n\n\n\n\\section{Thin accretion disk}\n\nThe accretion disk is formed by particles moving in circular orbits around a\ncompact object, with the geodesics determined by the space-time geometry\naround the object, be it a WH, BH or NS. For a static and spherically\nsymmetric geometry the metric is generically given by\n\n\\begin{equation}\nd\\tau ^{2}=-g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\\theta \\theta }d\\theta\n^{2}+g_{\\varphi \\varphi }d\\varphi ^{2}.\n\\end{equation}%\nAt and around the equator, i.e., when $\\left\\vert \\theta -\\pi \/2\\right\\vert\n\\ll 1,$we assume, with Harko \\textit{et al.} [27], that the metric functions\n$g_{tt},g_{rr},g_{\\theta \\theta }$ and $g_{\\varphi \\varphi }$ depend only on\nthe radial coordinate $r$. The radial dependence of the angular velocity $%\n\\Omega $, of the specific energy $\\widetilde{E}$, and of the specific\nangular momentum $\\widetilde{L}$ of particles moving in circular orbits in\nthe above geometry are given by\n\\begin{eqnarray}\n&&\\frac{dt}{d\\tau }=\\frac{\\widetilde{E}}{g_{tt}}, \\\\\n&&\\frac{d\\varphi }{d\\tau }=\\frac{\\widetilde{L}}{g_{\\varphi \\varphi }}, \\\\\n&&g_{tt}g_{rr}\\left( \\frac{dr}{d\\tau }\\right) ^{2}+V_{\\textmd{\\scriptsize{eff}}%\n}\\left( r\\right) =\\widetilde{E}^{2}.\n\\end{eqnarray}%\nFrom the last equation, the effective potential $V_{\\textmd{\\scriptsize{eff}}}(r)$\ncan be obtained in the form\n\n\\begin{equation}\nV_{\\textmd{\\scriptsize{eff}}}\\left( r\\right) =g_{tt}\\left( 1+\\frac{\\widetilde{L}^{2}}{g_{\\varphi \\varphi }}\\right).\n\\end{equation}%\nExistence of circular orbits at any arbitrary radius $r$ in the equatorial\nplane demands that $V_{\\textmd{\\scriptsize{eff}}}\\left( r\\right) =0$ and $dV_{%\n\\textmd{\\scriptsize{eff}}}\/dr=0$. These conditions allow us to write\n\\begin{eqnarray}\n&&\\widetilde{E}=\\frac{g_{tt}}{\\sqrt{g_{tt}-g_{\\varphi \\varphi }\\Omega ^{2}}},\n\\\\\n&&\\widetilde{L}=\\frac{g_{\\varphi \\varphi }\\Omega }{\\sqrt{g_{tt}-g_{\\varphi\n\\varphi }\\Omega ^{2}}}, \\\\\n&&\\Omega =\\frac{d\\varphi }{dt}=\\sqrt{\\frac{g_{tt,r}}{g_{\\varphi \\varphi ,r}}}.\n\\end{eqnarray}\n\nStability of orbits depend on the signs of $d^{2}V_{\\textmd{\\scriptsize{eff}}}\/dr^{2}$, while the condition $d^{2}V_{\\textmd{\\scriptsize{eff}}}\/dr^{2}=0$ gives the inflection point or \\textit{marginally stable} (ms) orbit or innermost stable circular orbit (ISCO) at $r=r_{\\textmd{\\scriptsize{ms}}}$. For the solutions under consideration, we explicitly find the $r_{\\text{ms}}$ as under:\n\n\\begin{equation}\nV_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Mag,EF}}}= - \\frac{(r-2M)^{2}(2Mr-Q^{2})}{%\nrJ},\n\\end{equation}%\n\\begin{eqnarray}\nV_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Mag,EF}}\\;\\prime\\prime} &=&\n-\\frac{4M^{3}(r-6M)}{r(Mr-Q^{2})J} \\nonumber\\\\\n&&-\\frac{8M^{2}Q^{2}(3Mr-Q^{2})}{r^{3}(Mr-Q^{2})J},\n\\end{eqnarray}%\nwhere\n\\begin{eqnarray}\nJ=2Mr(r-3M)-Q^{2}(r-4M). \\nonumber\n\\end{eqnarray}\nSolving $V_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Mag,EF}}\\;\\prime\\prime}=0$, we find\n\\begin{eqnarray}\nr_{\\textmd{\\scriptsize{ms}}}^{\\textmd{\\scriptsize{Mag,EF}}}&=&2M+\\left( \\frac{2}{M}\\right) ^{\\frac{1}{3}%\n}(2M^{2}-Q^{2})^{\\frac{2}{3}} \\nonumber \\\\\n&&+2^{\\frac{2}{3}}\\{M(2M^{2}-Q^{2})\\}^{\\frac{1}{3}%\n}.\n\\end{eqnarray}\n\nSimilarly, the effective potential $V_{\\textmd{\\scriptsize{eff}}}(r)$ and $V_{\\textmd{\\scriptsize{eff}}}^{\\prime \\prime}(r)$ for magnetic GMGHS spacetime in SF is\n\\begin{equation}\nV_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Mag,SF}}}= -\\frac{2M(r-2M)^{2}}{J},\n\\end{equation}%\n\\begin{equation}\nV_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Mag,SF}}\\;\\prime\\prime}=-\\frac{%\n2M(r-6M)(2M^{2}-Q^{2})}{r(Mr-Q^{2})J}.\n\\end{equation}%\nSolving $V_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Mag,SF}}\\;\\prime\\prime}=0$, we find\n\\begin{equation}\nr_{\\textmd{\\scriptsize{ms}}}^{\\textmd{\\scriptsize{Mag,SF}}}=6M.\n\\end{equation}\n\nThe effective potential $V_{\\textmd{\\scriptsize{eff}}}(r)$ and $V_{\\textmd{\\scriptsize{eff}}}^{\\prime\n\\prime }(r)$ for electric GMGHS spacetime in SF\n\\begin{equation}\nV_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Elec,SF}}}=-\\frac{2Mr\\{M(r-2M)+Q^{2}\\}^{2}}{(Q^{2}+Mr)P},\n\\end{equation}%\n\\begin{equation}\nV_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Elec,SF}}\\;\\prime\\prime}=-\\frac{2MN}{%\nr(Mr+Q^{2})^{3}P}.\n\\end{equation}%\nSolving $V_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Elec,SF}}\\;\\prime\\prime}=0$, we find the\nmarginally stable orbit\n\\begin{eqnarray}\nr_{\\textmd{\\scriptsize{ms}}}^{\\textmd{\\scriptsize{Elec,SF}}}&=&\\frac{2M^{2}-Q^{2}}{M}+2^{\\frac{4}{3}%\n}M^{2}(2M^{2}-Q^{2})K^{-\\frac{1}{3}} \\nonumber \\\\\n&&+\\frac{2^{\\frac{2}{3}}K^{\\frac{1}{3}}}{%\n2M^{2}+Q^{2}},\n\\end{eqnarray}%\nwhere\n\\begin{eqnarray}\nP &=&2M^{2}r(r-3M)-Q^{2}(2M^{2}-3Mr-Q^{2}), \\\\\nN &=&2M^{5}r^{2}(r-6M)+M^{3}Q^{2}r^{3}+M^{2}Q^{4}\\left(4M^{2} \\right.  \\nonumber \\\\\n&& \\left.-3Mr+3r^{2}\\right)-Q^{6}(4M^{2}-3Mr-Q^{2}), \\\\\nK &=&16M^{9}-8M^{5}Q^{4}+MQ^{8}+MQ^{2}(Q^{4}-4M^{4})^{\\frac{3}{2}}.\n\\end{eqnarray}\n\nWe assume geometrically thin accretion disk, which means that the disk height $H$\nabove the equator is much smaller than the characteristic radius $R$ of the\ndisk, $H\\ll R$. The disk is assumed to be in hydrodynamical equilibrium\nstabilizing its vertical size, with the pressure and vertical entropy\ngradient being negligible. An efficient cooling mechanism via heat loss by\nradiation over the disk surface is assumed to be functioning in the disk,\nwhich prevents the disk from collecting the heat generated by stresses and\ndynamical friction. The thin disk has an inner edge defined by the \\textit{%\nmarginally stable} circular radius $r_{\\textmd{\\scriptsize{ms}}}$, while the\norbits at higher radii are Keplerian.\n\nIn the above approximation, Page and Thorne [24], using the rest mass\nconservation law, showed that the time averaged rate of rest mass accretion $%\ndM_{0}\/dt$ is independent of the radius: $\\dot{M}_{0}\\equiv dM_{0}\/dt=-2\\pi\nru^{r}\\Sigma =$ \\textmd{const}. (Here $t$ and $r$ are the coordinate time\nand radial coordinates respectively, $u^{r}$ is the radial component of the\nfour velocity $u^{\\mu }$ of the accreting particles and $\\Sigma $ is the\naveraged surface density of the disk). In the steady-state thin disk model,\nthe orbiting particles have $\\Omega $ , $\\widetilde{E}$ and $\\widetilde{L}$\nthat depend only on the radii of the orbits. We omit other technical details\nhere (see [27]), but quote only the relevant formulas below.\n\nThe flux $F$ of the radiant energy over the disk can be expressed in terms\nof $\\Omega $ , $\\widetilde{E}$ and $\\widetilde{L}$ as [22-24]\n\n\\begin{equation}\nF(r)= -\\frac{\\dot{M}_{0}}{4\\pi \\sqrt{-g}}\\frac{\\Omega _{,r}}{%\n\\left( \\widetilde{E}-\\Omega \\widetilde{L}\\right) ^{2}}\\int_{r_{\\textmd{\\scriptsize{ms}}%\n}}^{r}\\left( \\widetilde{E}-\\Omega \\widetilde{L}\\right) \\widetilde{L}_{,r}dr.\n\\end{equation}\nThe disc is supposed to be in thermodynamical equilibrium, so the radiation\nflux emitted by the disk surface will follow Stefan-Boltzmann law:%\n\\begin{equation}\nF\\left( r\\right) =\\sigma T^{4}\\left( r\\right) ,\n\\end{equation}%\nwhere $\\sigma $ is the Stefan-Boltzmann constant. The observed luminosity $%\nL\\left( \\nu \\right) $ has a redshifted black body spectrum [36]\n\n$$L_{\\nu }=4\\pi \\textmd{d}^{2}I(\\nu )=\\frac{8\\pi h\\cos {i}}{c^{2}}\\int_{r_{\\textmd{\\scriptsize{in}}}}^{r_{\\textmd{\\scriptsize{f}}}}\\int_{0}^{2\\pi }\\frac{\\nu _{e}^{3}rdrd\\varphi }{\\textmd{Exp}\\left[ \\frac{h\\nu _{e}}{k_{B}T}\\right] -1},$$\nwhere $i$ is the disk inclination angle, d is the distance between the\nobserver and the center of the disk, $r_{\\textmd{\\scriptsize{in}}}$ and $r_{\\textmd{\\scriptsize{f}}}$ are\nthe inner and outer radii of the disc, $h$ is the Planck constant, $\\nu _{e}$\nis the emission frequency, $I(\\nu )$ is the Planck distribution, and $k_{B}$\nis the Boltzmann constant. The observed photons are redshifted and their\nfrequency $\\nu $ is related to the emitted ones in the following way $\\nu\n_{e}=(1+z)\\nu $. The redshift factor $(1+z)$ has the form [27]:\n\n\\begin{equation}\n(1+z)=\\frac{1+\\Omega r\\sin {\\varphi }\\sin {i}}{\\sqrt{g_{tt}-\\Omega\n^{2}g_{\\varphi \\varphi }}},\n\\end{equation}%\nwhere the light bending effect is neglected.\n\nAnother important characteristic of the accretion disk is its efficiency $%\n\\epsilon$, which quantifies the ability with which the central body\nconverts the accreting mass into radiation. The efficiency is measured at\ninfinity and it is defined as the ratio of two rates: the rate of energy of\nthe photons emitted from the disk surface and the rate with which the\nmass-energy is transported to the central body. If all photons reach\ninfinity, an estimate of the efficiency is given by the specific energy of\nthe accreting particles measured at the marginally stable orbit [25]:\n\n\\begin{equation}\n\\epsilon =1-\\widetilde{E}\\left( r_{\\textmd{\\scriptsize{ms}}}\\right) .\n\\end{equation}%\nThe Eqs.(18-20,33-36) are valid for any static spherically symmetric spacetime and\nhence valid both in the EF and SF since spherical symmetry is preserved\nunder conformal transfomation.\n\n\n\n\n\n\\section{GMGHS solutions: kinematic and accretion features}\n\nWe shall consider for illustration a central compact object of mass $%\nM=15M_{\\odot }$ with an accretion rate $\\dot{M_{0}}=10^{18}$gm.sec$^{-1}$\nand assume that it possesses magnetic charge $Q$ with the spacetime\ndescribed by stringy BH solutions (2), (8) and (11).\n\nWe examine how kinematic features of the concerned solutions differ from\nthose of Schwarzschild BH. First, the plot for $V_{\\textmd{\\scriptsize{eff}}}$ are shown in\nFigs.1a-1c, for $r\\in \\lbrack r_{\\textmd{\\scriptsize{eh}}}$,$\\infty )$, while the parameter\n$Q$ is taken to assume values $0.5M$, $M$, $1.2M$ and $\\sqrt{2}M$. The plots\nreveal that all potentials show finite maxima for $Q^{2}<2M^{2}$ but diverge\nat $r_{\\textmd{\\scriptsize{eh}}}^{\\textmd{\\scriptsize{Mag,EF}}}=r_{\\textmd{\\scriptsize{eh}}}^{%\n\\textmd{\\scriptsize{Mag,SF}}}=2M$ in the extreme case $Q^{2}=2M^{2}$ and at $r_{%\n\\textmd{\\scriptsize{eh}}}^{\\textmd{\\scriptsize{Elec,SF}}}=0$ corresponding to a NS, see\nafter Eq.(12). The effective potentials of the Schwarzschild and\nReissner-Nordstr\\\"{o}m solution are plotted in Fig.1d for comparison. An\ninteresting feature is that all the potentials in the EF and SF show smooth\nasymptotic fall-off approaching the Schwarzschild curve from above. In\ncontrast, the potential $V_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Elec,SF}}}$, after reaching a\nlocal maximum at $r_{\\textmd{\\scriptsize{ms}}}$, dips below the Schwarzschild curve\nintersecting it at a radius given by (Fig.1c):\n\\begin{equation}\nr|_{V_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Elec,SF}}}=V_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Sch}}}}=\\frac{%\n4M^{2}-Q^{2}+\\sqrt{16M^{4}+Q^{4}}}{2M}.\n\\end{equation}\n\nSecond, we can observe similar features in Figs.2a-2d demonstrating the\neffect of $Q$ on specific angular momenta $\\widetilde{L}$. Note that, in all\nthese plots, the distance scale is $r\\in \\lbrack r_{\\textmd{\\scriptsize{eh}}}$,$\\infty )$\nexcept for $Q^{2}=2M^{2}$, when $r_{\\textmd{\\scriptsize{eh}}}=0$ (singularity). However, in\ncase of $\\widetilde{L}^{\\textmd{\\scriptsize{Elec,SF}}}$, its intersection point\nwith that of the Schwarzschild BH differs from the intersection point of\neffective potential and is given by\n\\begin{equation}\nr|_{\\widetilde{L}^{\\textmd{\\scriptsize{Elec,SF}}}=\\widetilde{L}^{\\textmd{\\scriptsize{Sch}%\n}}}=\\frac{8M^{2}-Q^{2}+\\sqrt{32M^{4}+Q^{4}}}{2M}.\n\\end{equation}\n\nThird, Figs.3a-3d show influence of $Q$ on the specific orbital energies.\nFig.3a shows that energies of a particle orbiting a Schwarzschild BH is\nhigher than that orbiting a magnetic\\ GMGHS BH at the same radius in the EF.\nA similar behavior is seen for orbits in the electric\\ GMGHS BH in the SF\n(Fig.3c). This behavior is similar to that in the Reissner-Nordstr\\\"{o}m\ncase (Fig.3d). However, exactly the opposite behavior is shown by the orbits\nin the magnetic\\ GMGHS BH in the SF, where orbital energies for particles in\nthe Schwarzschild BH spacetime are at the lowest (Fig.3b).\n\nFinally, we focus on the observable emissivity features such as the\nradiation flux, luminosity, temperature and conversion efficiency. Figs.4a-d\ndisplay the flux of radiation $F\\left( r\\right) $ emitted by the disk\nbetween $r_{\\textmd{\\scriptsize{ms}}}$ and received at an arbitrary radius $r$ [Eq.(33)],\nFigs 5a-d show variation of temperature over the disk from $r_{\\textmd{\\scriptsize{ms}}}$\nto an arbitrary radius and Figs.6a-d show observed luminosity variations on\na logarithmic scale over different frequency ranges. The conversion\nefficiency $\\epsilon $ of the accreting mass into radiation, measured at\ninfinity is given by Eq.(36). In Tab.1, we show the radiation properties of\nthe accretion disks using marginally stable orbits $r_{\\textmd{\\scriptsize{ms}}}$ and $%\n\\epsilon $ with the parameter $Q$ in the range used in the previous plots.\n\n\\begin{table*}[!ht]\n\\caption{The $r_{\\textmd{\\scriptsize{ms}}}$ and the efficiency $\\epsilon $ for GMGHS and Reissner-Nordstr\\\"{o}m BHs, all having a mass $M=15M_{\\odot }$\n         with the accretion rate $\\dot{M}_{0}=10^{18}$ gm.sec$^{-1}$. The general relativistic Schwarzschild BH corresponds to $Q=0$.}\n  \\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n  \\hline\n  $Q$ & \\multicolumn{2}{c|}{Magnetic, EF} & \\multicolumn{2}{c|}{Magnetic, SF}\n  & \\multicolumn{2}{c|}{Electric, SF} & \\multicolumn{2}{c|}{Reissner-Nordstr\\\"{o}m BH} \\\\\n  \\cline{2-9}\n     & $r_{\\textmd{\\scriptsize{ms}}}$ [$M$] & $\\epsilon $ & $r_{\\textmd{\\scriptsize{ms}}}$ [$M$] & $\\epsilon $\n  & $r_{\\textmd{\\scriptsize{ms}}}$ [$M$] & $\\epsilon $ & $r_{\\textmd{\\scriptsize{ms}}}$ [$M$] & $\\epsilon $ \\\\\n  \\hline\n  & \\multicolumn{8}{c|}{BH} \\\\\n  $0$ (Sch) & 6.0000 & 0.0572 & \\multicolumn{6}{c|}{} \\\\\n  $0.5M$ & 5.7426 & 0.0607 & 6.0000 & 0.0506 & 5.2746 & 0.0715 & 5.6066 &\n  0.0608 (BH) \\\\\n  $M$ & 4.8473 & 0.0768 & 6.0000 & 0.0298 & 3.2743 & 0.1361 & 4.0000 & 0.0814\n  (extr. BH) \\\\\n  $1.2M$ & 4.1644 & 0.0950 & 6.0000 & 0.0170 & 2.1163 & 0.2080 & 2.4548 &\n  0.1149 (NS) \\\\\n  - & \\multicolumn{2}{c|}{NS} & \\multicolumn{2}{c|}{WH}\n  & \\multicolumn{2}{c|}{NS} &  &  \\\\\n  $\\sqrt{2}M$ & 2.0000 & 1.0000 & 6.0000 & 0 & 0 & 1.0000 & 2.3129 & 0.0773\n  (NS) \\\\ \\hline\n  \\end{tabular}\n\\end{table*}\n\nTable 1 demonstrates the variation in the location of the inner disk edge\nwith the changing charge $Q$. For GMGHS BH in EF, we notice that the higher\nvalues of charge $Q$ are, the closer are the marginally stable orbits to the\ncenter. However, for magnetically charged GMGHS BH in SF, we see that its\nconversion efficiency for $Q\\sim 0.3M$ mimics that of the Schwarzschild BH,\nboth being $0.0572.$ For electrically charged GMGHS BH in SF, the efficiency\nincreases to over $20\\%$, whereas, interestingly, that for magnetically\ncharged GMGHS BH in SF the efficiency decreases to lower than $2.45\\%$.\nThese features are characteristic of the frames chosen for describing BHs.\n\n\\begin{table*}[!ht]\n \\caption{Comparison with BH of luminosity spectra from accretion disk around\n  different extreme GMGHS central objects.}\n \\begin{tabular}{|c|c|c|c|c|}\n  \\hline\n  $\\nu$[1\/s] & \\multicolumn{4}{c|}{$\\nu L(\\nu)$[erg s$^{-1}$]} \\\\\n  \\cline{2-5}\n            & \\multicolumn{2}{c|}{Naked singularity} & Wormhole & Schwarzschild \\\\\n  \\cline{2-4}\n             & Magnetic, EF    & Electric, SF          & Magnetic, SF & black hole\\\\\n  \\hline\n  $10^{12}$  & $1.15\\times 10^{24}$   & $1.63\\times 10^{24}$  & $8.75\\times 10^{22}$   & $1.15\\times 10^{24}$   \\\\\n  $10^{13}$  & $1.13\\times 10^{26}$   & $1.60\\times 10^{26}$  & $8.20\\times 10^{24}$   & $1.11\\times 10^{26}$   \\\\\n  $10^{14}$  & $1.07\\times 10^{28}$   & $1.51\\times 10^{28}$  & $6.21\\times 10^{26}$   & $9.84\\times 10^{27}$   \\\\\n  $10^{15}$  & $9.08\\times 10^{29}$   & $1.31\\times 10^{30}$  & $8.95\\times 10^{27}$   & $5.57\\times 10^{29}$   \\\\\n  $10^{16}$  & $4.57\\times 10^{31}$   & $1.02\\times 10^{32}$  & $1.13\\times 10^{19}$   & $1.90\\times 10^{29}$   \\\\\n  $10^{17}$  & $2.13\\times 10^{31}$   & $8.43\\times 10^{33}$  & $\\sim 0$               & $1.01\\times 10^{0}$    \\\\\n  $10^{18}$  & $3.34\\times 10^{15}$   & $6.77\\times 10^{35}$  & $\\sim 0$               & $\\sim 0$               \\\\\n  $10^{19}$  & $\\sim 0$               & $8.31\\times 10^{36}$  & $\\sim 0$               & $\\sim 0$               \\\\\n  $10^{20}$  & $\\sim 0$               & $1.10\\times 10^{27}$  & $\\sim 0$               & $\\sim 0$               \\\\\n  $10^{21}$  & $\\sim 0$               & $\\sim 0$              & $\\sim 0$               & $\\sim 0$               \\\\\n  \\hline\n\\end{tabular}\n\\end{table*}\n\nThe Figs.1-6 show different kinematic and emissivity parameters for\ndifferent values of $Q$ related to dilatonic charge $D$ [Eq.(5)]. We have\nassumed the central mass to be $M=15M_{\\odot },$ and mass accretion rate $%\n\\dot{M}_{0}=10^{18}$ gm$.$sec$^{-1}$.\n\n\n\n\n\n\\section{Summary}\n\nThe spacetime structure of GMGHS BHs differ from that of the Schwarzschild BH in many important ways, which are expected to show up in their kinematic and accretion disk properties. In the present paper, we analyzed thin accretion disk properties around magnetically and electrically charged GMGHS BHs in EF and in SF. The physical parameters describing the disk such as the effective potential, radiation flux, temperature and emissivity profiles have been explicitly obtained for several values of the parameter $Q$, that in turn correspond to several values for dilationic charge $D$ for a given mass $M$. All the astrophysical quantities related to the observable properties of the accretion disk in the two frames have been compared with those for the Schwarzschild BH of the same mass. We considered as a toy model a stellar sized compact object of mass $M=15M_{\\odot}$ with an accretion rate $\\dot{M}_{0}=10^{18}$ gm.sec$^{-1}$ and assumed that its accretion properties can be described by the those of the GMGHS spacetimes including their extreme limits of NS and WHs. Our aim has been to examine whether the kinematic and emissivity properties significantly change when the central object changes.\n\nThe main conclusions of our analyses are as follows: Kinematic properties were already analyzed in Sec.4 with corresponding figures and need not be repeated here. Suffice it to say that at the NS radius, all kinematic quantities diverge, as expected. The emissivity properties of GMGHS BHs do not appreciably differ from those of the Schwarzschild or Reissner-Nordstr\\\"{o}m BH for $Q<\\sqrt{2}M$. This conclusion is in accord with the strong lensing properties of GMGHS BH studied by Bhadra [6]. However, in the extreme limit $Q=\\sqrt{2}M$, the GMGHS BH in the EF yields NS. In the SF, there are two GMGHS solutions, one is electrically charged and the other is magnetically charged. In the extreme limit the former yields NS and the latter yields WHs. These latter types of geometries are interesting in their own right since the existence of NS is associated with the no-hair theorem and a recent work includes accretion properties in the JNW NS [42,43]. Also, the horizonless WHs are seriously considered as candidates for mimicking initial post-merger ring down signals characteristic of BH horizon [55-60]. While the emissivity properties of ordinary GMGHS BHs in EF and in SF do not differ appreciably from those of the Schwarzschild BH, their extreme counterparts studied here show that they differ quite significantly from those of the Schwarzschild BH. These differences provide yet another avenue to distinguish between ordinary and extremal objects.\n\nA very interesting result, qualitatively similar to the one obtained by Torres [36] for boson stars, is presented in Table 2 and correspondingly in Figs.6a-d for our toy model. The table shows the difference with Schwarzschild BH, when NS and WH are concerned. From these, we see that for BHs ($Q<\\sqrt{2}M$), and WHs in the SF ($Q=\\sqrt{2}M$) the spectra decay rapidly for $\\nu >10^{16}$ Hz but for NS ($Q=\\sqrt{2}M$), the luminosity (Figs. 6a,6c) does not decay until a frequency $\\nu >10^{18}$ Hz (for magnetic NS in EF) beyond which it becomes nearly invisible. The same thing happens for electric NS in SF beyond $\\nu>10^{20}$ Hz. In either case, there is almost an \\textit{infinite} increase in observed luminosity compared to nearly invisible WH and BH at the same frequency. Thus NS should be the brightest objects in the sky (like QSOs). This conclusion is supported by the efficiency $\\epsilon =1$ for NS in Table 1. It thus seems that, at the singular radius $r=r_{\\textmd{\\scriptsize{ms}}}=2M$ (first column of the Table, see Figs. 6a,c), all infalling matter is minced into radiation all of which then escape to us.\n\nIt would be of interest to examine the physical reason as to why such rapid decays in the luminosity spectra occur at higher frequencies but it is evident that observable distinctions exist among different objects described by GMGHS. Understandably, a more adequate and realistic model for the central object should include spin. A choice for this purpose could be the spinning Kerr-Sen solution [61] but it is expected that spin might not drastically alter the main conclusions derived here. We keep it as an open problem for future work.\n\n\n\n\n\n\n\n\n\\begin{acknowledgments}\nWe thank an anonymous referee for useful suggestions. The reported study was funded by RFBR according to the research project No. 18-32-00377.\n\\end{acknowledgments}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThere has been much hope that one might be able to use  AdS\/CFT\n\\cite{AdS\/CFT} to describe the real systems after certain amount of\ndeformations. In fact it has been suggested that the fireball in\nRelativistic Heavy Ion Collision (RHIC)\ncan be explained from dual gravity point of view\n\\cite{SZ,Nastase,SSZ,Aharony}, since the quark-gluon plasma (QGP)\ncreated there are in the strong coupling region \\cite{RHIC-1,RHIC-2}.\nAlthough the YM theory described by the standard AdS\/CFT\nis large-$N_{c}$ ${\\cal N}=4$ SYM theory,\nthere are many attempts to construct models closer to QCD\n\\cite{AdS\/QCD}.\nSUSY is not very relevant in the finite temperature context\nbecause it is broken completely.\n\n\nSince the RHIC fireball is expanding, we need to understand AdS\/CFT\nin time dependent situations. Recently, Janik and Peschanski\n\\cite{Janik, Janik-2} discussed this problem in non-viscous cases.\nThey use the conservation law and conformal invariance together with\nthe holographic renormalization \\cite{HSS,Skend} to express the bulk\ngeometry from the given boundary data. As a result, the bulk\ngeometry reproduces the basic features of Bjorken theory\n\\cite{Bjorken}. Theses results were generalized  to the cases where\nshear viscosity is included \\cite{sin-shin}. Indeed, it had been\npointed out that inclusion of shear viscosity is very important in\nthe analysis of real RHIC physics since it plays an essential role\nin the elliptic flow (see for example,\n\\cite{KH,bulk-zero,Teaney,Kolb,Hirano-Heinz,Zhang}). The shear\nviscosity at the strong coupling limit was calculated for the ${\\cal\nN}=4$ SYM systems in \\cite{PSS-1} using AdS\/CFT, which fits the\nperfect fluidity of RHIC QGP (see \\cite{RHIC-2,bulk-zero,Hirano} and\nthe references therein).\n\n\nIn this paper, we consider a holographic dual of strongly interacting\n${\\cal N}=4$ large-$N_{c}$ SYM fluid with non-isotropic\nthree-dimensional expansion which is relevant to\n``Little Bang'' of RHIC.\nWe will first make a simplest generalization of the Bjorken's\none-dimensional expansion to three-dimensional cases.\nThe resulting local rest frame of the fluid is described by\nthe Kasner metric\nwhose extreme limit reproduces the Bjorken hydrodynamics.\nInterestingly, the hydrodynamic equation and the\nequation of state will be shown to be independent of\nthe non-isotropy parameters.\n\nFurther more, using holographic\nrenormalization, we will establish the holographic dual\ngeometry of the anisotropically expanding fluid in the Kasner\nspacetime and discuss its physical consequences by extending\nthe work of \\cite{Janik,sin-shin}.\nInterestingly, the gravity dual carries more information than\nthe hydrodynamics which is our input.\nThe integration constants coming from\nthe hydrodynamic equation cannot be determined by the hydrodynamics\nitself. However, we will show that such quantities can be determined\nby considering the dual geometry. Therefore it provides a method to\nextract useful information on the macroscopic properties of the\nexpanding fluid in terms of microscopic data.\n\nKasner spacetime is a curved spacetime in general.\nHowever, we find that the Kasner spacetime\ncan be a well-controled approximation of a local rest frame\nof an anisotropically expanding fluid on Minkowski spacetime.\nTherefore, the Kasner spacetime provides a useful framework\nfor analysing the three-dimensional expansion because of\nthe simplification of the hydrodynamic equations.\n\n\nThe organization of the paper is the following. In Section\n\\ref{hydro},\nwe introduce Kasner spacetime as a local rest frame of a fluid\nunder three-dimensional anisotropic expansion.\nWe also establish the hydrodynamics based on that local rest\nframe. Section \\ref{gravity} gives analyses in the gravity\ndual in the late time regime.\nWe extend the results of \\cite{Janik,sin-shin} to the\ncase of our interest: the three-dimensional anisotropic expansion\nwith shear viscosity.\nWe show that the dual geometry determines some of the hydrodynamic\nquantities in terms of the initial condition and the fundamental\nconstants. In Section \\ref{flow} we show that the hydrodynamics\non the Kasner spacetime\n(and hence its gravity dual) describes\nan elliptic flow on the flat spacetime\nwithin a well-controlled approximation.\nWe analyze the properties of the flow.\nWe conclude in the final section.\nThe definition of the approximation and its justification are\ngiven in appendix.\n\n\n\\section{Hydrodynamics in Kasner space and anisotropic expansion \\label{hydro}}\n\nThe remarkable success of Bjorken's hydrodynamics needs to be\nextended for more realistic cases of\nthree-dimensional expansions. The Bjorken's basic assumption is\nexistence of the so-called ``central rapidity region'' (CPR). The\nlocal rest frame of the fluid can be given by\npropertime($\\tau$)-rapidity($y$), whose relationship with the\ncartesian coordinate is\n$(X^{0},X^{1},X^{2},X^{3})=(\\tau \\cosh y,\\tau \\sinh y,X^{2},X^{3})$.\nWe have chosen the collision axis to\nbe in the $x^{1}$ direction. The Minkowski metric in this coordinate\nhas the form\n\\begin{eqnarray}\nds^{2}=-d\\tau^{2}+\\tau^{2}dy^{2}+(dX^{2})^{2}+(dX^{3})^{2}.\n\\label{g00}\n\\end{eqnarray}\nOur starting point is an observation that the above metric describes\na one-dimensional Hubble expansion, i.e., an expansion of the\nuniverse instead of that of the fluid, in which the rapidity plays\nthe role of the co-moving coordinate. The hydrodynamic equations can\nbe derived from the covariant conservation of the energy-momentum\ntensor with the above metric.\n\nThe real expansion in RHIC is not one-dimensional but a\nthree-dimensional one, although the dominant flow is along the\ncollision axis. The idea is that since the one-dimensional\napproximation of RHIC fireball expansion is seen as a Hubble flow, a\nthree-dimensional Hubble flow may describe the RHIC plasma better.\nLet us begin with an ansatz for a local rest frame given by\n\\begin{eqnarray}\nds^{2}=-d\\tau^{2}+\\tau^{2a}(dx^{1})^{2}\n+\\tau^{2b}(dx^{2})^{2}+\\tau^{2c}(dx^{3})^{2}.\n\\label{kasner}\n\\end{eqnarray}\nHere $x^{i},i=1, 2, 3$ denote the co-moving coordinates.\nThe $a, b$ and $c$ are arbitrary constants for a moment. However, as\nwe will show in Section \\ref{gravity}, they satisfy \\begin{equation}\na+b+c=1,\\quad a^2+b^2+c^2=1, \\label{bdeq} \\end{equation} if we impose conformal\ninvariance of the fluid. Under the above conditions, the metric is\ncalled Kasner metric describing a homogeneous but anisotropic\nexpansion of the Universe. In short, we identify the ``Little Bang''\nin ``SYM version of RHIC'' with a Big Bang with the homogeneous but\nanisotropic expansion described by the Kasner metric.\nWe use this metric in the late time regime\nbecause we describe only the late time\nevolution of the fluid where hydrodynamics is valid.\nTherefore the initial\nsingularity of the metric is not a relevant feature for us. Since we\ntake $x^1$ as the longitudinal direction, a realistic set up is to\nchoose\n$a\\sim 1$ and $b,c \\sim 0$ (see section \\ref{flow}).\n\nWe first establish the hydrodynamics in the Kasner metric.%\n\\footnote{We use the late time approximation, which is employed in\n\\cite{sin-shin}, where the macroscopic quantities are assumed to\nevolve sufficiently slowly. } We assume that the expansion is\nanisotropic but homogeneous, and the physical quantities depend only\non $\\tau$. Using the fact that the energy-momentum tensor is\ndiagonal on the local rest frame, and using the symmetry in transverse\ncoordinates, we can write \\begin{equation} T^{\\mu}_{~~\\nu}={\\rm\ndiag}(-\\rho,f_1,f_2,f_3), \\label{general} \\end{equation} where $\\rho$ is the\nenergy density of the fluid. By use of the conservation law $\n\\nabla_{\\mu}T^{\\mu\\nu}=0$ we get \\begin{equation} {\\dot \\rho}+\\rho\/\\tau+\\sum_i\na_if_i\/\\tau=0,\\label{consv} \\end{equation} and from the conformal invariance\n$T^{\\mu}_{\\mu}=0$, we get \\begin{equation} -\\rho+\\sum f_i=0. \\label{cfinv} \\end{equation} To\nget the above results, we have used the following non-zero\n$\\Gamma^{\\mu}_{\\alpha\\beta}$'s: \\begin{equation} \\Gamma^{\\tau}_{11}=a\\tau^{2a-1},\n\\quad \\Gamma^{\\tau}_{22}=b\\tau^{2b-1},\\quad\n\\Gamma^{\\tau}_{33}=c\\tau^{2c-1} , \\quad \\Gamma^{1}_{1\\tau}=a\/\\tau,\n\\quad \\Gamma^{2}_{2\\tau}=b\/\\tau, \\quad \\Gamma^{1}_{3\\tau}=c\/\\tau.\n\\end{equation}\n\nOn the other hand, the energy-momentum tensor in the framework of relativistic hydrodynamics is known to be\n\\begin{eqnarray}\nT^{\\mu\\nu}=(\\rho+P)u^{\\mu}u^{\\nu}+Pg^{\\mu\\nu}+\\tau^{\\mu\\nu},\n\\label{T-ideal}\n\\end{eqnarray}\nwhere $P$ is the pressure of the fluid, $u^{\\mu}=(\\gamma, \\gamma\n\\vec{v})$ is the four-velocity field in terms of the local fluid\nvelocity $\\vec{v}$, and  $\\tau^{\\mu\\nu}$ is the dissipative term. In\na frame where the energy three-flux vanishes,\n$\\tau^{\\mu\\nu}$ is given in terms of the bulk viscosity\n$\\xi$ and the shear viscosity $\\eta$ by\n\\begin{eqnarray}\n\\tau^{\\mu\\nu}\n=-\\eta\n(\\bigtriangleup^{\\mu\\lambda}\\nabla_{\\lambda}u^{\\nu}\n+\\bigtriangleup^{\\nu\\lambda}\\nabla_{\\lambda}u^{\\mu}\n-\\frac{2}{3}\\bigtriangleup^{\\mu\\nu}\\nabla_{\\lambda}u^{\\lambda})\n-\\xi \\bigtriangleup^{\\mu\\nu}\\nabla_{\\lambda}u^{\\lambda},\n\\label{T-dissp}\n\\end{eqnarray}\nunder the assumption that $\\tau^{\\mu\\nu}$ is of first order in\ngradients. We have defined the three-frame projection tensor as\n$\\bigtriangleup^{\\mu\\nu}=g^{\\mu\\nu}+u^{\\mu}u^{\\nu}$.\nFor the conformal invariance, we set $\\xi=0$.\nNotice that the bulk viscosity in the realistic RHIC setup is\nalso negligible. (See for example, Ref. \\cite{bulk-zero}.)\n\nThe four-velocity of the fluid at any point in  the local\nrest frame is\n$u^{\\mu}=(1,0,0,0)$, and this makes the energy-momentum tensor\ndiagonal. Using $\\Delta^{\\mu\\nu}={\\rm\ndiag}(0,\\tau^{-2a},\\tau^{-2b},\\tau^{-2c})$, $\\nabla_\\lambda\nu^\\nu=\\Gamma^\\nu_{\\lambda 0}={\\rm diag}(0,a\/\\tau,b\/\\tau,c\/\\tau)$ and\n$\\nabla_\\nu u^\\nu =(a+b+c)\/\\tau$, we get the mixed energy-momentum\ntensor:\n\\begin{eqnarray}\nT^{\\mu}_{~~\\nu}\n=\n\\left(\n  \\begin{array}{cccc}\n  - \\rho&   0&   0&   0\\\\\n      0&  P-\\frac{2}{3}(3a-1)\\frac{\\eta}{\\tau}\n      &   0&   0\\\\\n      0&   0&\n  P-\\frac{2}{3}(3b-1)\\frac{\\eta}{\\tau} &   0\\\\\n      0&   0&   0&\n       P-\\frac{2}{3}(3c-1)\\frac{\\eta}{\\tau} \\\\\n  \\end{array}\n\\right). \\label{T-diag}\n\\end{eqnarray}\nBy identifying (\\ref{T-diag}) with (\\ref{general}), we obtain\n\\begin{equation}\nf_i= p-(a_i-{1\\over3})\\frac{2\\eta}{\\tau},\\end{equation}\nwhere $a_i=a,b,c$ for $i=1,2,3$ respectively.\nInserting these into (\\ref{consv}) and (\\ref{cfinv}),\nwe get\n\\begin{equation}\n{\\dot \\rho}+{1\\over\\tau}\\rho+\n\\sum_i{a_i\\over\\tau}\\left(p-(a_i-{1\\over 3}){2\\eta\\over\\tau}\\right)=0,\n\\label{consv-2}\n\\end{equation}\nand\n\\begin{equation}\n-\\rho+3p-2\\eta\\left({{\\sum_ia_i-1}\\over\\tau}\\right)=0.\n\\label{cfinv-2}\n\\end{equation}\nUsing the second equation, we may write the first equation in terms\nof the energy density as\n\\begin{equation} {\\dot\n\\rho}+\\left(1+\\frac{1}{3}\\sum_i{a_i} \\right){\\rho\\over\\tau} =\n\\left(\\sum_i a_i^2-\\frac{1}{3}(\\sum_ia_i)^2\\right)\n{2\\eta\\over\\tau^2} . \\end{equation}\nAs we will discuss in Section \\ref{gravity},\n{\\it if we impose the conformal invariance}, we get the conditions\n\\begin{equation}\n\\sum_i a_i=1, \\;\\;\\; \\sum_ia_i^2=1,\n\\label{abc-cond}\n\\end{equation}\nunder which\nthe equation of state and the conservation law become: \\begin{eqnarray}\np&=&{\\rho\\over 3},\\nonumber \\\\\n\\frac{d\\rho}{d\\tau}+\\frac{4}{3}\\frac{\\rho}{\\tau}&=&\\frac{4}{3}\\frac{\\eta\n}{\\tau^2}. \\label{consv2}\\end{eqnarray} Remarkably, the equations for fluid\ndynamics are completely {\\it independent of the parameters $a,b,c$}\nin this case. In fact the dynamical law should not depend on the\ninitial conditions ($a,b$ and $c$) that reflects the initial\ncollision geometry. In this respect, (\\ref{abc-cond})\nleads us a satisfactory consequence.\n\nNotice that both of $\\rho$ and $\\eta$ depend on the proper time\n$\\tau$ in general.\nLet's assume that the shear viscosity evolves by\n$\\eta= {\\eta_{0}}\/{\\tau^{\\beta}},$\nwhere $\\eta_{0}$ is a positive constant.\nThe solution of (\\ref{consv2}) is then given by\n\\begin{eqnarray}\n\\rho(\\tau) &=&\\frac{\\rho_{0}}{\\tau^{4\/3}}\n+\\frac{4\/3}{1\/3-\\beta} \\frac{\\eta_{0}}{\\tau^{1+\\beta}}\n \\:\\:\\:\\:({\\rm for}\\:\\: \\beta\\neq 1\/3),\n \\label{solution}\n \\\\ \\nonumber\n &=&\\frac{\\rho_{0}}{\\tau^{4\/3}}\n+\\frac{4\\eta_{0}}{3} \\frac{\\ln\\tau}{\\tau^{4\/3}}\n\\:\\:\\:\\:({\\rm for}\\:\\: \\beta=1\/3),\n\\end{eqnarray}\nwhere $\\rho_{0}$ is a positive constant.\nFor $\\beta \\leq 1\/3$ case,\nthe viscous corrections in the hydrodynamic quantities become\ndominant in the late time, which invalidates the hydrodynamic\ndescription. If $\\beta > 1\/3$, the shear viscosity term\nis subleading in the late time as we expect.\nTherefore we will consider only the latter case from now on.\n\nThe proper-time dependence of the temperature $T$\ncan be read off by assuming\n the Stefan-Boltzmann's law $\\rho \\propto T^{4}$:\n\\begin{eqnarray}\nT= T_{0}\\left( \\frac{1}{\\tau^{1\/3}} +\n\\frac{1\/3}{1\/3-\\beta}\\frac{\\eta_{0}}{\\rho_{0}}\n\\frac{1}{\\tau^{\\beta}} +\\cdots \\right).\n\\label{T-general}\n\\end{eqnarray}\nIn the {\\em static} finite temperature system of strongly coupled\n${\\cal N}=4$ SYM theory, it is known that $\\eta \\propto T^{3}$\n\\cite{PSS-1}. Let us assume that the result is valid\n in the slowly varying non-static cases.\nThen we can set $\\beta=1$ hence\n\\begin{eqnarray}\n \\eta=\\frac{\\eta_{0}}{\\tau}.\n\\label{beta1}\n\\end{eqnarray}\nWe know $\\rho\\sim T^4$ and\n$\\eta\\sim T^{3}$ can   be consistent only if there is an additional\nterm in (\\ref{beta1}), but the correction term is negligible in our\ncase.\nThe temperature behavior is then given by\n\\begin{eqnarray}\nT=T_{0}\\left( \\frac{1}{\\tau^{1\/3}} -\n\\frac{1}{2}\\frac{\\eta_{0}}{\\rho_{0}} \\frac{1}{\\tau}\n+\\cdots \\right).\n\\end{eqnarray}\n\nWe can evaluate the entropy change in the presence of shear\nviscosity by using hydrodynamics.\nThe conservation of energy-momentum tensor can be rewritten as\n\\begin{eqnarray}\n \\frac{d(\\sqrt{g}\\rho)}{d\\tau}+ \\frac{d\\sqrt{g}}{d\\tau}P =\n \\frac{4}{3}\\frac{\\sqrt{g}\\eta}{\\tau^2}\n, \\label{3deom}\n\\end{eqnarray}\nwhere $\\sqrt{g}=\\tau$ is the volume element in the co-moving coordinate.\nBy integrating over the unit volume in the co-moving coordinate,\nand using the thermodynamic relation between the entropy and energy-work,\n\\begin{equation}\ndE+PdV=TdS,\n\\end{equation}\n(\\ref{3deom}) can be written as\n\\begin{eqnarray}\nT\\frac{d(\\sqrt{g} s)}{d\\tau}=\\frac{4}{3}\\frac{\n\\eta\\sqrt{g}}{\\tau^2},\n\\end{eqnarray}\nwhere $s$ denotes the entropy density and $\\sqrt{g} s \\equiv S$\nis the entropy per unit  co-moving volume.\nNotice that in the absence of viscosity, the entropy per unit\nco-moving volume is constant.\nNow, the entropy per unit co-moving volume has the time dependence:\n\\begin{eqnarray}\nS(\\tau) &=& S_0+ \\frac{4}{3} \\int_0^\\tau d\\tau \\frac{ \\eta\\sqrt{g}}{\n\\tau^{2} T}\n\\nonumber \\\\\n&=& S_\\infty - 2\\frac{\\eta_{0}}{T_{0}}\\tau^{-2\/3} + \\cdots,\n\\label{entropy}\n\\end{eqnarray}\nwhere $S_\\infty = S(\\infty)$. The dissipation creates entropy but\nits rate slows down with time. Notice that all these arguments are\ncompletely in parallel with the case of the one-dimensional\nexpansion.\n\nThe hydrodynamics does not calculate the constants $S_\\infty$ and\n$T_0$ in terms of the initial condition. It is  important to  point\nout that by embedding the hydrodynamics into AdS\/CFT, we can\ndetermine these parameters in terms of the initial condition\n$\\rho_{0}$ and the fundamental constants of the theory.\n\n\\section{Holographic dual of anisotropic expansion }\n\\label{gravity}\n\n\nIn this section, we will find a five-dimensional metric that is dual\nto the hydrodynamic description of the YM fluid in the previous\nsection. Some of the parameters of hydrodynamics will be determined\nas a consequence. The basic strategy is to use the Einstein's\nequation together with the boundary condition given by the\nfour-dimensional energy-momentum tensor \\cite{HSS,Skend,Janik}.\nWe consider general asymptotically AdS metrics in the\nFefferman-Graham coordinate:\n\\begin{eqnarray}\nds^{2}=\nr_{0}^{2}\n\\frac{g_{\\mu\\nu}dx^{\\mu}dx^{\\nu}+dz^{2}}{z^{2}},\n\\label{FG-metric}\n\\end{eqnarray}\nwhere $x^{\\mu}=(\\tau,x^1,x^{2},x^{3})$ in our case.\n$r_{0}\\equiv (4\\pi g_{s} N_{c} \\alpha'^{2})^{1\/4}$\nis the length scale given by the string coupling $g_{s}$ and\nthe number of the colors $N_{c}$.\nThe four-dimensional metric $g_{\\mu\\nu}$ is expanded\nwith respect to $z$ in the following form \\cite{HSS,Skend}:\n\\begin{eqnarray}\ng_{\\mu\\nu}(\\tau, z)=\ng^{(0)}_{\\mu\\nu}(\\tau)+z^{2}g^{(2)}_{\\mu\\nu}(\\tau)\n+z^{4}g^{(4)}_{\\mu\\nu}(\\tau)\n+z^{6}g^{(6)}_{\\mu\\nu}(\\tau)+\\cdots .\n\\label{expansion}\n\\end{eqnarray}\nHere $g^{(0)}_{\\mu\\nu}$ is the physical four-dimensional metric for\nthe gauge theory on the boundary, which is given by (\\ref{kasner}) in\nthe present case. The $g_{\\mu\\nu}^{(n)}$'s depend only on $\\tau$\nbecause our physical quantities are assumed to depend only on\n$\\tau$.\n\n$g^{(2)}_{\\mu\\nu}$ is related to the conformal anomaly of the\nYM theory in the following way \\cite{HSS}:\n\\begin{eqnarray}\n\\langle T^{\\mu}_{\\mu} \\rangle\n=-\\frac{1}{16\\pi G_{5}}\n\\left[\n({\\rm Tr}g^{(2)})^{2}-{\\rm Tr}(g^{(2)})^{2}\n\\right],\n\\label{anomaly}\n\\end{eqnarray}\nwhere $G_{5}$ is the 5d Newton's constant given by\n$G_{5}=8\\pi^{3}\\alpha'^{4}g_{s}^{2}\/r_{0}^{5}$ so that\n\\begin{eqnarray}\n\\frac{4\\pi G_{5}}{r_{0}^{3}}=\\frac{2\\pi^2}{N_c^2}\n\\end{eqnarray}\nin our notation. Since we are dealing with ${\\cal N}=4$ SYM theory\non $R^{1,3}$, we should require the conformal invariance: $\\langle\nT^{\\mu}_{\\mu} \\rangle=0$. The most natural choice is given by\n\\begin{eqnarray}\ng^{(2)}_{\\mu\\nu}=0.\n\\label{g2}\n\\end{eqnarray}\nThis is equivalent to the Ricci flat condition for the\nfour-dimensional metric:\n\\begin{eqnarray}\nR_{\\mu\\nu}=0,\n\\label{ricciflat}\n\\end{eqnarray}\nthrough the relationship \\cite{HSS}\n\\begin{eqnarray}\ng^{(2)}_{\\mu\\nu}\n=\\frac{1}{2}\n\\left(\nR_{\\mu\\nu}-\\frac{1}{6}R g^{(0)}_{\\mu\\nu}\n\\right).\n\\label{g2-g0}\n\\end{eqnarray}\nSince\n\\begin{equation}\nR_{00}=(\\sum_i a_i-\\sum_i a_i^2)\/\\tau^2, \\quad \\quad\nR_{ii}=a_i(\\sum_j a_j-1)\\tau^{2a_i-2},\n\\end{equation}\nthe Ricci flat condition (\\ref{ricciflat}) gives\nthe Kasner condition (\\ref{bdeq}).\n\nOne should notice that we are {\\em not} solving the four-dimensional\nEinstein's equation in the presence of $T_{\\mu\\nu}$. The Kasner\nmetric is not a consequence of the gravitational effect of\n$T_{\\mu\\nu}$, but an effective description of the spacetime\nexpansion satisfying the conformal invariance of strong gauge theory\ninteraction.\n\n\nWe can identify the first non-trivial data in (\\ref{expansion}),\n$g^{(4)}_{\\mu\\nu}$, with the energy-momentum tensor at the boundary\n\\cite{HSS}:\n\\begin{eqnarray}\ng^{(4)}_{\\mu\\nu}\n=\\frac{4\\pi G_{5}}{r_{0}^{3}} \\langle T_{\\mu\\nu} \\rangle,\n\\end{eqnarray}\nin our notation.\nFor the time being, we set\n$4\\pi G_{5}=1$ and $r_{0}=1$.\nThe higher-order terms in (\\ref{expansion}) are determined by\nsolving the Einstein's equation with the negative cosmological\nconstant $\\Lambda=-6$ \\cite{HSS} (see also \\cite{Janik}):\n\\begin{eqnarray}\nR_{MN}-\\frac{1}{2}G_{MN}R-6G_{MN}=0,\n\\label{Eeq}\n\\end{eqnarray}\nwhere the metric and the curvature tensor are the five-dimensional\nones of (\\ref{FG-metric}). $g_{\\mu\\nu}^{(2n)}$ is described by\n$g_{\\mu\\nu}^{(2n-2)}, g_{\\mu\\nu}^{(2n-4)}, \\cdots, g_{\\mu\\nu}^{(0)}$\nthrough the Einstein's equation in five dimensions. In other words,\nwe can obtain the higher-order terms in (\\ref{expansion})\nrecursively by starting with the initial data $g_{\\mu\\nu}^{(0)}$\n($\\sim$Kasner) and $g_{\\mu\\nu}^{(4)}$ ($\\sim T_{\\mu\\nu}$).\n\nLet us come back to our main interest  to obtain\nthe bulk geometry in the presence of shear viscosity.\nThe energy-momentum tensor for $\\beta=1$ is\nwritten by using  (\\ref{solution}) as\n\\begin{eqnarray}\nT^{\\mu}_{\\nu}=\\left(\n  \\begin{array}{cccc}\n    -\\frac{\\rho_{0}}{\\tau^{4\/3}}\n    +\\frac{2\\eta_{0}}{\\tau^{2}}\n       & 0  & 0  & 0  \\\\\n     0 &   \\frac{\\rho_{0}}{3\\tau^{4\/3}}\n    -\\frac{2a\\eta_{0}}{\\tau^{2}}   & 0  & 0  \\\\\n     0 & 0  &  \\frac{\\rho_{0}}{3\\tau^{4\/3}}\n    -\\frac{2b\\eta_{0}}{\\tau^{2}}  & 0  \\\\\n     0 & 0  & 0  &  \\frac{\\rho_{0}}{3\\tau^{4\/3}}\n    -\\frac{2c\\eta_{0}}{\\tau^{2}} \\\\\n  \\end{array}\n\\right).\n\\label{Tij-viscous}\n\\end{eqnarray}\nWe find that the metric components,\n$g_{\\tau\\tau}$, $g_{yy}\/\\tau^{2}$, $g_{xx}$\nhave the following structure by solving the Einstein's equation\nrecursively:\n\\begin{eqnarray}\nf^{(1)}(v)+\\eta_{0}h^{(1)}(v)\/\\tau^{2\/3}+\n{\\tilde f}^{(2)}(v)\/\\tau^{4\/3}+\\cdots ,\n\\label{expansion2}\n\\end{eqnarray}\nNote that {\\it the viscosity dependent terms exist at the order of\n$\\tau^{-2\/3}$ and these are more important than the higher-order\nterms neglected in} \\cite{Janik}. We are considering the late time\nregion $\\tau \\gg 1$. But to see the effects of the viscosity, we\nneed to keep the terms at least to the order of $\\tau^{-2\/3}$. In\nthis paper we consider the viscosity effects to the minimal order.\n\nNow we solve the Einstein's equation recursively.\nThe power series that appear in the solution\ncan be re-summed to give a compact form of the metric.\nAfter some hard work,\nwe  can get the late time 5d bulk geometry given by%\n\\footnote{ One should keep in mind that we are\nlooking for the late time geometry; the metric (\\ref{our-geometry})\nis correct only to the order of $\\gamma$ and the $O(\\gamma^2)$\ncontributions are not unambiguously determined. The representation\nof (\\ref{our-geometry}) is chosen since it makes the volume of the\nhorizon finite.}\n\\begin{eqnarray}\nds^{2}\n&=&\n\\frac{1}{z^{2}}\n\\left\\{\n-\\frac{(1-\\frac{\\rho z^{4}}{3})^{2}}{1+\\frac{\\rho z^{4}}{3}}\nd\\tau^{2}\n+\n\\left(1+\\frac{\\rho z^{4}}{3}\\right)\n\\sum_{i=1}^3\n\\left(\\frac{1+\\frac{\\rho z^{4}}{3}}{1-\\frac{\\rho z^{4}}{3}}\\right)^{(1-3a_i)\\gamma}\n\\tau^{2a_i}(dx^i)^{2}\n\\right\\}+\\frac{dz^{2}}{z^{2}},\n\\label{our-geometry}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\gamma \\equiv \\frac{\\eta_{0}}{\\rho_{0}\\tau^{2\/3}}\n ~~{\\rm and}~~\n\\rho=\n\\frac{\\rho_{0}}{\\tau^{4\/3}}\n-\n\\frac{2\\eta_{0}}{\\tau^{2}}.\n\\label{def-gamma-1}\n\\end{eqnarray}\nNotice that the energy-momentum tensor (\\ref{Tij-viscous}) can NOT\nbe written in terms of the whole $\\rho(\\tau)$. It is an amusing\nsurprise to see that the final metric nevertheless can be written in\nterms of $\\rho(\\tau)$  (apart from the powers) in a compact form.\nThis implies that the position of the horizon can be determined\nsolely by the energy density.\n\n\\subsection{Macroscopic Parameters from Gravity Dual }\nThe Hawking temperature in the adiabatic approximation\nis given by\n\\begin{equation}\nT(\\tau)=\\sqrt{2}\/(\\pi z_{0}(\\tau)),\n\\end{equation}\nwhere\n\\begin{equation}\nz_{0}(\\tau)=[3\/\\rho(\\tau)]^{1\/4}\n\\end{equation}\n is the time-dependent position of the horizon.\n Using  these, we obtain\n\\begin{eqnarray}\n\\rho=\n\\frac{3}{8}\\pi^{2}N_{c}^{2}T^{4}(\\tau),\n\\end{eqnarray}\nby restoring the normalization of $\\rho$: $\\rho\\to 4\\pi G_{5}\/r_0^3 \\cdot \\rho$.\nThe entropy per unit co-moving volume\nis given by\n\\begin{eqnarray}\nS\n&=& \\lim_{z\\to z_0(\\tau)} \\frac{1}{4G_{5}} \\sqrt{\\prod_{i=1}^3\\left(1+\\frac{\\rho z^{4}}{3}\\right)\n\\left(\\frac{1+\\frac{\\rho z^{4}}{3}}{1-\\frac{\\rho z^{4}}{3}}\\right)^{(1-3a_i)\\gamma}\n\\tau^{2a_i} }\\nonumber \\\\\n&=&\\frac{1}{4G_{5}} \\frac{2\\sqrt{2}\\tau r_{0}^{3}}{z_{0}^{3}(\\tau)}\n\\nonumber \\\\\n&=&\nN_c^2 \\frac{\\sqrt{2}}{\\pi} \\left(\\frac{2\\pi^2}{N_{c}^{2}}\\frac{\\rho_0}{3}\\right)^{3\/4}\n\\left(\n1-\\frac{3}{2} \\frac{\\eta_{0}}{\\rho_{0}\\tau^{2\/3}}\n+O(\\tau^{-4\/3})\n\\right).\n\\label{s-evolve-ads}\n\\end{eqnarray}\nNotice that {\\it the entropy is completely independent of geometric\n parameters (i.e., $a_i$).}  Without conformal invariance this result is not guaranteed.\nAs we emphasized in \\cite{sin-shin}, the value of $S$ at\n$\\tau=\\infty$, which cannot be determined by hydrodynamics alone, is\nprecisely determined to be\n\\begin{eqnarray}\nS_{\\infty}\n=N_c^2 \\frac{\\sqrt{2}}{\\pi} \\left(\\frac{2\\pi^2}{N_{c}^{2}}\\frac{\\rho_0}{3}\\right)^{3\/4}\n\\label{Sinfty}\n\\end{eqnarray}\nin terms of the initial condition $\\rho_{0}$ and the microscopic\ngauge theory parameter $N_c$. Similarly, \\begin{equation}\nT(\\tau)=\\frac{T_0}{\\tau^{1\/3}}(1-2\\gamma(\\tau))^{1\/4}, \\quad {\\rm\nwith} ~~\nT_0=\\frac{\\sqrt{2}}{\\pi}\\left(\\frac{2\\pi^2}{N_{c}^{2}}\\frac{\\rho_0}{3}\\right)^{1\/4}.\n\\label{T0} \\end{equation} These parameters $S_\\infty, T_0$  are precisely the\nquantities used in macroscopic theory (hydrodynamics) which should\nbe provided by a microscopic theory like QCD. What we are showing\nhere is that by considering the AdS\/CFT dual of hydrodynamics, one\ncan determine such quantities.\n\nLet us check consistency of (\\ref{s-evolve-ads}) and (\\ref{entropy}).\nApparent time dependence agrees in the leading order.\nIn fact one can do more.\nThe normalized entropy-creation rate from the gravity dual result (\\ref{s-evolve-ads})\n is given by\n\\begin{eqnarray}\n\\frac{1}{S}\\frac{dS}{d\\tau}\n=\\frac{\\eta_{0}}{\\rho_{0}\\tau^{5\/3}}+O(\\tau^{-7\/3})\n\\label{rate-ads},\n\\end{eqnarray}\n and that from hydrodynamics result (\\ref{entropy}) is\n\\begin{eqnarray}\n\\frac{1}{S}\\frac{dS}{d\\tau}\n=\\frac{4}{3}\\frac{\\eta_{0}}{T_{0}S_{\\infty}\\tau^{5\/3}}\n+O(\\tau^{-7\/3}).\n\\label{rate-hydro}\n\\end{eqnarray}\nComparing\n(\\ref{rate-ads}) and (\\ref{rate-hydro}), we obtain\n\\begin{eqnarray}\nS_{\\infty}=\\frac{4}{3}\\frac{\\rho_{0}}{T_{0}}\n=\n\\left. \\frac{4}{3}\\frac{\\rho \\tau}{T}\\right|_{\\tau=\\infty}. \\label{consistency}\n\\end{eqnarray}\nThis is  the  consistency condition that is required to be checked.%\n\\footnote{In fact this is the relationship among the entropy, the\nenergy (per unit co-moving volume) and the temperature obtained by\nthermodynamics at $\\tau=\\infty$ where the system reaches thermal\nequilibrium.} With  use of $S_\\infty$ and $T_0$ given in eqn's\n(\\ref{Sinfty}) and (\\ref{T0}) we can check that the consistency\ncondition (\\ref{consistency}) is indeed satisfied.\n\n\n\\section{Flow of RHIC fireball and Kasner spacetime \\label{flow}}\n\n\nSo far, we have established the gravity dual of\nthe Yang-Mills system in Kasner spacetime.\nNow we would like to suggest a relevance of\nour model to description of the elliptic flow in RHIC experiments.\nWe have seen in section \\ref{hydro} that the hydrodynamic\ndescription of the three-dimensional expansion in the\nKasner spacetime is as simple as that of the Bjorken expansion\nin the flat spacetime.\nTherefore, it is great if we can apply such a simple formalism\nto anisotropically expanding RHIC fireball.\nHowever, from  the realistic point of view, we sacrificed the flatness of\n4d spacetime for the simplicity of the fluid dynamics of 3 dimensional\nexpansion.\nOne immediate question is when and under which condition we can justify it.\nThe hydrodynamics\non the Kasner spacetime provides a well-approximated\n description of a three-dimensional expansion in the flat\nspacetime  if curvature\nwhich is small enough. Notice also that\nthe spacetime symmetry crucial to our problem, which\nis uniformity of the spacetime, is maintained in Kasner spacetime.\n\n\\begin{figure}\n\\centerline{\\epsfig{file=flow1.eps,width=6.0cm}}\n\\caption{\\small The available region of $(b,c)$ is on the ellipse\nbetween $(0,0)$ and $(1,0)$. $(b,c)=(0,0)$ is   Bjorken point}\n\\label{sus}\n\\end{figure}\n\nIn figure 1, we show the allowed region of the anisotropic\nparameters.\n$(b,c)=(0,0)$ corresponds to the Bjorken expansion.\nThe non-zero components of the Riemann tensor of the Kasner\nspacetime are\n\\begin{equation}\nR_{0i0i}=(1-a_i)a_i \\tau^{2a_i-2}, \\quad\nR_{ijij}=a_ia_j\\tau^{2a_i+2a_j-2},\n\\end{equation}\nand the non-flatness is directly related to the distance\nfrom the Bjorken point $(a,b,c)=(1,0,0)$ on the parameter space.\nIn order to keep the deviation from the flat spacetime small,\nwe restrict ourselves within the vicinity of the Bjorken point,\n\\begin{equation}\na\\simeq 1, \\quad b\\simeq 0, \\quad c\\simeq 0,\n\\end{equation}\nthat corresponds to almost central collisions in RHIC.\n\nIn appendix, we show\n what approximation is necessary to reach kasner space\n starting from a flat spacetime.\nThe conditions are:\n1) the fluid is produced by almost central collision (small $b$),\n2) we consider only  the central part of the fluid (small $x^\\perp$), and\n3) we consider only  the late time regime.\n For more detail, see appendix \\ref{appendix}.\n\nWith these limitations in mind,\nlet us consider how the elliptic flow can be described\nwithin our framework.\nWe can choose $b \\ge c$ without any loss of generality.\nBy considering the intersection of the plane and the unit\nsphere in the $a, b, c$ space\nof (\\ref{bdeq}), we can see that\n\\begin{equation}\na>b>-c>0.\n\\end{equation}\nThis means that one of the transverse directions must\n{\\it contract} and the others\nexpand so that the expansion is elliptical whose eccentricity\ngrows and eventually saturate to 1.\n\\footnote{\nThere is no contraction in the realistic RHIC QGP.\nThe contraction in the present model is due to the conformal\ninvariance which is unavoidable for ${\\cal N}=4$ SYM theory.\nNotice that this volume-preserving nature indicates our\n${\\cal N}=4$ SYM plasma is more ``liquid-like'' than the\nRHIC QGP.\n}\n\nThe transverse expansion has a natural interpretation as\nthe elliptic flow, one of the most concrete evidence for the\nstrong nature of the interaction.\nLet $\\varepsilon$ be the eccentricity defined by%\n\\footnote{ The $\\varepsilon$ defined here contains an extra minus\nsign comparing to other literature such as \\cite{KH,bulk-zero}. We\ndefine its positivity by the direction of the $v_2$ evolution. }\n\\begin{equation}\n\\varepsilon=\\frac{\\langle X_2^2-X_3^2 \\rangle_{X}} {\\langle\nX_2^2+X_3^2 \\rangle_{X}},\n\\label{epsion-def}\n\\end{equation} where $X^{i}$ is\nthe cartesian coordinate and\n$\\langle\\cdots\\rangle_{X} \\equiv \\int \\cdots \\rho \\:dX_{2}dX_{3}$.\nOn the other hand, $v_2$, the quantity experimentally characterizing\nthe elliptic flow, is\n defined by (see for example, \\cite{KH,bulk-zero,Zhang})\n\\begin{eqnarray}\n\\frac{1}{N}\\frac{dN}{d\\phi}\n=\n\\frac{1}{2\\pi}\n\\left(\nv_{0}+2v_{2}\\cos(2\\phi)+2v_{4}\\cos(4\\phi)+\\cdots\n\\right),\n\\end{eqnarray}\nwhere $N$ is the number of the partons and $\\phi$ is the angular\ncoordinate on the transverse momentum plane. It can be calculated\nfrom the following identification,\n\\begin{eqnarray}\nv_2=\n\\frac{\\int d^{2}P_{T}\n\\left(\\frac{P_2^2-P_3^2}{P_2^2+P_3^2 }\\right)\n\\frac{dN}{d^{2}P_{T}}}{\\int d^{2}P_{T}\\frac{d N}{d^{2}P_{T}}},\\label{v2def}\n\\end{eqnarray}\nwhere $P^{i}$ is the momentum of the fluid in the $X^{i}$ coordinate\nand $d^{2}P_{T}=dP^{2}dP^{3}$.\nTo consider $v_2$ in the present model, we introduce a\ncoarse-grained (i.e., averaged over a small volume)\nmomentum flow,\n\\begin{equation} P^i=K^i(x,\\tau). \\end{equation} At each fixed time $\\tau$, this can be\nconsidered as a mapping from the $P^i$-space to the\n$x^i$-space.\\footnote{ In the co-moving coordinate, where the fluid is\nat rest, we do not have any flow. So $P^i(x)$ is the coarse-grained\nmomentum field in Minkowski space written as a function of co-moving\ncoordinate. } Now $v_2$ can be expressed as an integral over the\nco-moving coordinates:\n\\begin{equation} {  v}_2= \\frac{\\int dx^{2}dx^{3}\n\\left(\\frac{P_2^2-P_3^2}{P_2^2+P_3^2 }\\right)_x \\rho_N(x)}{\\int\ndx^{2}dx^{3}\\rho_N(x)},\n\\label{v2-comp}\n\\end{equation}\nwhere $\\rho_N(x)=\\frac{dN}{dx^2 dx^3}$ is the particle density\nin the transverse space.\nNotice that the Jacobians cancel out.\n\nTo proceed, we need an explicit expression for $K(x)$. We first\nrelate the coordinates of the Kasner spacetime and the usual\nMinkowski spacetime.\nWe can identify the flat space variable $X^i$ for $i=2, 3$ by\n\\begin{eqnarray}\nX^i \\simeq \\tau^{a_i}x^i ~~{\\rm near}\n~~x^i=0,\n\\label{x-trans}\n\\end{eqnarray}\nwhich is nothing but (\\ref{transform}) under (\\ref{approximation}).\nThen the fluid momentum in the flat space is given by \\begin{equation}\nP^{i} =\\rho  {dX^i\\over d\\tau}=\\rho  \\:a_i\\tau^{a_i-1} x^{i}. \\end{equation}\nHere $\\rho =m\\rho_N$ is a mass density with some proper mass\nparameter $m$. Since we treat it as a constant from now on, it is\nirrelevant in the calculation below.\n\nA few technical remarks are in order:\n\\begin{enumerate}\n \\item One should notice that the fluid momentum\n$P^i(x)$ is different from the individual particle momentum. By\nreplacing the momentum by fluid momentum, we expect a small\ndeviation from the original $v_2$. So one may want to call the final\nexpression by $\\bar  v_2$. However, this is precisely what we should\nhave when we describe the system by hydrodynamics where everything\nis to be defined in the coarse-grained level.\n  \\item Since our approximation is valid in the small $x^{i}$ region,\nthe integrals in (\\ref{v2-comp}) are now defined within\nthe small $x^{i}$ region by introducing a cut off radius.\n\\end{enumerate}\n\nThe resulting $v_2$ can be calculated in our model to yield\n\\begin{equation} {\nv}_2=\\frac{b\\tau^{b}+ c\\tau^{c} }{b\\tau^{b}- c\\tau^{c}},\n\\label{v2}\n\\end{equation}\nfor $b>0, c<0$. Comparing this with the result for the\neccentricity in the small $x^{i}$ region\n\\begin{equation} \\varepsilon=\\frac{\n\\tau^{2b} -\\tau^{2c} }{ \\tau^{2b}+ \\tau^{2c}},\n\\label{ecc}\n\\end{equation}\nwe plot the time evolution of $v_2$ in figure 2.\n\nNotice that the hydrodynamics describes relatively late time\nregime and we do not consider the time region where $v_2$ is\nnegative.\nAt the time of  zero $v_2$, $\\varepsilon $ is negative\nand its absolute value decreases\nto zero. That is, the fireball core becomes more round in the\ntransverse space.\nAfter that initial period, $\\varepsilon$ becomes positive\nand grows in the same direction of growing $v_2$.\nThis is qualitatively the same behaviour as those given in\n\\cite{KH,bulk-zero} and \\cite{Zhang}.\nNotice that there is only one independent parameter $b$\nthat can be used to parameterize the initial eccentricity.\n\n\\begin{figure}\n\\centerline{\\epsfig{file=flow2.eps,width=6.0cm}}\n\\caption{\\small Time evolution of the elliptic flow parameters.\nDotted line is the eccentricity and the solid line is the $v_2$ for\n$b=0.1$.}\n\\label{sus}\n\\end{figure}\n\n\\section{Discussion}\nIn this paper, we extended the Bjorken hydrodynamics to the case of\nanisotropic three-dimensional expansion. The Ricci flat condition\nsuggested by the conformal invariance imposes the condition on the\nanisotropy parameters. As a consequence, the four-dimensional\nboundary metric becomes precisely that of Kasner universe.\nOur hydrodynamic equation of motion is independent\nof the anisotropic parameters, so that it is the\nsame as that obtained by Bjorken.\n\nAlthough the Kasner spacetime is a curved spacetime, we found\nthat it gives a well-approximated local rest frame\nof an anisotropically expanding fluid on the flat spacetime.\nWe obtained the eccentricity and $v_{2}$ of the elliptic flow\nof the fluid.\nThe expansion in this setting\nhas a deviation from that of realistic RHIC fireballs,\nsince one of the\ntransverse direction contracts as a consequence of the\nKasner condition. This deviation is essentially due to\nthe conformal invariance. Whether one can lessen the condition\nof the conformal invariance is an important issue,\nwhich we will treat in other publication.\n\n\nWe also extended the ``falling horizon solution'' obtained in\n\\cite{Janik,sin-shin} to the case of three-dimensionally expanding\nfluid. It is very important to figure out how to use the bulk metric\napart from reading off the horizon location, which gives the cooling\nrate and the entropy creation rate. It is also interesting to\nworkout the details of the Hawking evaporation and Wilson line\ncalculations for the external quark-antiquark in our time-dependent\nmetric. We will come back to these issues in later publication.\n\n\n\\noindent\n{\\large\\bf Acknowledgments}\\\\\nWe thank T. Hirano and E. Shuryak for valuable comments\n on the manuscript and interesting suggestions.\nS.N. thanks the Yukawa Institute for Theoretical Physics at Kyoto University.\nDiscussions during the YITP workshop YITP-W-99-99 on\n``thermal quantum field theories and their applications''\nwere useful to improve the section 4.\nThis work was supported by\nthe SRC Program of the KOSEF through the Center for Quantum\nSpace-time(CQUeST) of Sogang University with grant number R11 - 2005\n-021 and also by KOSEF Grant R01-2004-000-10520-0.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe high-sensitivity data provided by visible-light space-based transit surveys, designed to detect the minute decrease in flux as a planet transits across its host star, can be used to look for variability along the entire orbital phase of a star-planet system. Beyond the transit, the phase curve includes the secondary eclipse, when the planet's day-side hemisphere is occulted by the host star, and sinusoidal brightness modulations across the orbital phase. \n\nWhile the transit depth is sensitive primarily to the planet-star radius ratio, the secondary eclipse depth is determined by the planet's thermal emission and the geometric albedo in the observed bandpass. Modulations along the orbital phase, in visible light, are shaped by the gravitational interaction between the star and planet, as well as the longitudinal variations of the planet's brightness. \n\nMore specifically, the shape of the measured phase curve is a superposition of the effects of four main processes, described briefly and somewhat simplistically below, where we assume the transit to be at zero orbital phase, the orbit to be circular, and the planet's rotation to be synchronized with the orbit (i.e., tidally locked), as expected for short period systems \\citep{mazeh2008}:\n(1) Beaming, or Doppler boosting, where the periodic red- and blue-shifting of the host star's spectrum observed in the bandpass follows its orbital motion around the system's center of mass \\citep[e.g.,][]{shakura1987, loeb2003, zucker2007, shporer2010}. The shape of the photometric variability reflects the orbital radial velocity (RV) curve, albeit with the opposite sign, and therefore can be described as a sine at the orbital period.\n(2) Tidal distortion of the host star by the planet \\citep[e.g.,][]{morris1985, morris1993, pfahl2008, jackson2012}, which leads to a cosine modulation at the first harmonic of the orbital period, commonly referred to as ellipsoidal modulation.\n(3) Thermal emission from the planet's atmosphere, where a tidally-locked planet's day-side hemisphere (facing the star) is hotter than the planet's night-side hemisphere (facing away from the star), resulting in a cosine modulation at the orbital period. This process dominates the phase curve modulations in the near-infrared \\citep[e.g.,][]{knutson2012, wong2015, wong2016}, but for highly irradiated planets it can also be detected at visible wavelengths \\citep{snellen2009}.\n(4) Stellar light reflected by the planet's atmosphere, which due to the geometric configuration of the system reaches maximum at the phase of secondary eclipse and minimum at mid-transit, producing a cosine variation at the orbital period \\citep[e.g.,][]{jenkins2003, shporer2015}. Both the thermal emission and reflected light modulations (processes 3 and 4 above) are expected to have the same schematic shape, but a different amplitude. Hence, when combined they are commonly referred to as the atmospheric phase component \\citep{parmentier2017}. \n\nThe summary above shows that the orbital phase curve is sensitive to the star-planet mass ratio and characteristics of the planet's atmosphere, including the geometric albedo and thermal emission, along with their longitudinal distribution. For a review of visible-light orbital phase curves, see \\citealt{shporer2017} and references therein. \n\nWe present here our analysis of the {\\it TESS}\\ orbital phase curve of WASP-18b\\ (TIC  100100827, TOI 185, \\citealt{hellier2009, southworth2009}). This massive 10.4~\\ensuremath{M_{\\rm Jup}}\\ gas giant planet orbits its host star at a very short orbital period of 0.94 days, which establishes favorable conditions for a strong phase curve signal at visible wavelengths and allows us to study the atmosphere of a gas giant planet with high surface gravity subjected to high stellar irradiation. Our analysis of the visible-light orbital phase curve adds to previous measurements in the near-infrared of the phase curve \\citep{maxted2013} and the secondary eclipse \\citep{nymeyer2011, maxted2013, sheppard2017, arcangeli2018}.\n\nThe {\\it TESS}\\ observations are described in \\secr{obs}, and our data analysis is described in \\secr{dataanal}. We present our results in \\secr{res} and discuss their implications in \\secr{dis}. We conclude with a brief summary in \\secr{sum}.\n\n\n\\section{Observations}\n\\label{sec:obs}\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{stamps.eps}\n\\caption{Stacked {\\it TESS}\\ images of the 11$\\times$11~pixel stamps centered around WASP-18 (red cross) for Sectors 2 and 3. The photometric extraction aperture used by the SPOC pipeline is outlined in red. The positions of two nearby $T\\sim 12.5$ mag stars, which have been deblended by the SPOC pipeline, are denoted with blue crosses: TIC 100100823 (A) and TIC 100100829 (B).\n}\n\\label{fig:stamp}\n\\end{figure}\n\nThe WASP-18 system was observed by Camera 2 of the {\\it TESS}\\ spacecraft during Sector 2 (from 2018 August 22 to 2018 September 20) and Sector 3 (from 2018 September 20 to 2018 October 18). WASP-18 is listed in the {\\it TESS}\\ input catalog (TIC; \\citealt{stassun2018}) as ID 100100827 and included in the list of pre-selected target stars, which are observed with a 2-minute cadence using an 11$\\times$11 pixel subarray centered on the target. The photometric data were processed through the Science Processing Operations Center (SPOC) pipeline \\citep{jenkins2016}, hosted at the NASA Ames Research Center, which is largely based on the predecessor \\textit{Kepler} mission pipeline \\citep{jenkins2010, jenkins2017}. The stacked {\\it TESS}\\ subarrays produced by the SPOC pipeline are shown in Figure~\\ref{fig:stamp}. Outlined in red are the optimal apertures used to extract the WASP-18 light curve in each sector.\n\nFor the results presented in this paper, we use the Presearch Data Conditioning (PDC, \\citealt{smith2012, stumpe2014}) light curves from the SPOC pipeline. The data files include quality flags that indicate when photometric measurements may have been affected by non-nominal operating conditions on the spacecraft or may otherwise yield unreliable flux values. Most of the flagged points occur in the vicinity of momentum dumps, when the spacecraft thrusters are engaged to reset the onboard reaction wheels. Momentum dumps occurred every 2--2.5 days and lasted up to about half an hour. Data taken during or near these momentum dumps typically display anomalous fluxes in the raw photometric time series. We also note that a large portion of data taken during Sector 3 observations suffered from poor pointing and other non-nominal instrumental behavior and were assigned NaN flux values by the SPOC pipeline; these points were removed, resulting in shorter segments of usable data in the two physical orbits of Sector 3.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\linewidth]{plot1.pdf}\n\\caption{Plot of the median-normalized and outlier-removed Presearch Data Conditioning (PDC) Sector 2 and 3 light curves of the WASP-18 system. The gaps in the middle of each sector's time series separate the two physical orbits of the {\\it TESS}\\ spacecraft contained within the sector. We have removed 30 minutes of data from the start of each orbit's light curve. The points highlighted in red show severe residual uncorrected systematics in the fits and are removed in the joint analysis presented in this paper. The phase curve variation of the system is clearly visible, as are residual long-term trends in the data. Those trends can include long-term photometric variability of the target or systematics introduced by the instrument and\/or analysis.}\n\\label{fig:lightcurves}\n\\end{figure*}\n\nWe remove all flagged data points from the light curves. Then, we apply a moving median filter to the photometric time series with a width of 16 data points and remove \\sig{3} outliers, while masking out the transits and secondary eclipses. We also remove the first 30 minutes of data from each orbit's light curve as these segments display residual ramp-like systematic artifacts. The resulting median-normalized light curves are shown in Figure~\\ref{fig:lightcurves}. Even without detrending, the orbital phase variation is clearly discernible, as are the transits and, upon closer inspection, the secondary eclipses. The gaps in the middle of each sector's time series is due to the data downlink, separating the two physical orbits within each {\\it TESS}\\ Sector.\n\nWhen fitting each orbit's light curve separately, several sections of the Sector 3 data display obvious features in the residuals that are not well-corrected by the systematics model we use (Section~\\ref{subsec:systematics}). These sections occur before momentum dumps and at the end of an orbit's light curve. In the final light curve fits presented in this paper, we have carefully inspected the individual orbit light curve fits and removed regions with strong discernible uncorrected systematics. In the first orbit of Sector 3, we remove all points after BJD = 2,458,392, while in the second orbit, we remove 0.75~days worth of data before the last three momentum dumps and 0.5~days worth of data from the end of the time series. These regions are highlighted in red in Figure~\\ref{fig:lightcurves}.\n\nFor Sector 2 data, trimming and outlier filtering remove 1.3\\% and 1.6\\% of the data from the two orbits, respectively. The additional removal of regions in Sector 3 data with uncorrected residuals entails a significantly higher percentage of removed points in the final Sector 3 light curves used in the joint analysis: 37\\% and 29\\% for the two orbits, respectively.\n\nIn addition to the phase curve modulation, there are clear residual long-term trends in the light curves at the level of several hundreds of parts per million (ppm). Previous analyses of {\\it TESS}\\ transit light curves have corrected for these and other flux variations by fitting a basis spline across the out-of-occultation (transit and secondary eclipse) light curve \\citep{huang2018,vanderspek2019} or using a Gaussian Process (GP) model \\citep{wang2019}, thereby removing all non-transit variability. Since such methods would remove the astrophysical phase variations of interest here, we do not utilize them and instead define a detrending model in our fits (Section~\\ref{subsec:systematics}).\n\nWe have also carried out a parallel analysis using the Simple Aperture Photometry (SAP) light curves, which are not corrected for systematics by the SPOC pipeline. One particular characteristic of these data that is not manifested in the PDC light curves is significant flux ramps and periods of increased photometric scatter lasting up to a day at the start of each orbit's photometric time series and preceding each momentum dump. These features are not adequately detrended by the polynomial model we use in this work (Section~\\ref{subsec:systematics}), so we choose to trim one day worth of data prior to each momentum dump, as well as the first day of data for each orbit. All in all, this removes almost 40\\% of the SAP time series from the phase curve analysis. \n\nThe best-fit astrophysical parameters from our analysis of SAP light curves are consistent with the results from the PDC light curves at much better than the \\sig{1} level, with most parameters lying within 0.1-0.2$\\sigma$. Meanwhile, the data trimming and larger red noise in the SAP light curves lead to parameter estimate uncertainties that are as much as 100\\% larger than those derived from fitting the PDC light curves. Given the demonstrated consistency between the PDC and SAP light curve analyses and the poorer quality of the latter, we have decided to present the results from our PDC light curve analysis in this paper.\n\n\n\\section{Data Analysis}\n\\label{sec:dataanal}\n\nIn this work, we utilize the ExoTEP pipeline to analyze the {\\it TESS}\\ PDC light curves for the WASP-18 system. ExoTEP is a highly-modular Python-based tool in development for extracting and analyzing all types of time series photometric datasets of relevance in exoplanet science --- primary transit and secondary eclipse light curves and full-orbit phase curves. The pipeline allows the user to execute joint fits of datasets from multiple instruments (e.g., \\textit{Kepler}, \\textit{Hubble}, \\textit{Spitzer}, and {\\it TESS}) and customize the handling of limb-darkening and systematics models in a self-consistent way. So far, the main application of the ExoTEP pipeline has been in transmission spectroscopy \\citep[e.g.,][see first reference for a detailed technical description]{wong2018,benneke2018,chachan2018}. This work is the first application of the ExoTEP pipeline to time series photometry that includes orbital phase curves.\n\n\\subsection{Transit and eclipse model}\n\\label{subsec:transit}\n\nExoTEP models both transits and secondary eclipses using the BATMAN package \\citep{kreidberg2015}. In our analysis, we fit for the planet-star radius ratio $R_{p}\/R_{s}$ and the relative brightness of the planet's day-side hemisphere $f_{p}$, which determine the planetary transit and secondary eclipse depths, respectively. In order to obtain updated values, we allow the transit geometry parameters --- impact parameter $b$ and scaled orbital semi-major axis $a\/R_{s}$ --- to vary and fit for a new transit ephemeris --- specific mid-transit time $T_{0}$ and orbital period $P$. The zeroth epoch, to which we assign $T_{0}$, is designated to be the transit event closest to the center of the combined time series.\n\nFor the transit model, we use a standard quadratic limb-darkening law and fix the coefficients to values calculated for the {\\it TESS}\\ bandpass by \\citet{claret2017}. Assuming the stellar properties for WASP-18 listed in \\citet{stassun2017} and \\citet{torres2012} ($T_{\\mathrm{eff}}=6431\\pm48$~K, $\\log g=4.47\\pm0.13$, $\\mathrm{[Fe\/H]}=0.11\\pm0.08$; see also Table~\\ref{tab:knownparams}), we take the coefficient values tabulated for the nearest-neighbor set of stellar properties ($T_{\\mathrm{eff}}=6500$~K, $\\log g=4.50$, $\\mathrm{[Fe\/H]}=0.1$): $u_{1}=0.2192$ and $u_{2}=0.3127$. \nUtilizing coefficients for other similar combinations of stellar properties does not significantly affect the results of our fit, yielding changes to the fitted parameter values of at most \\sig{0.2}.\nWhen experimenting with fitting for the quadratic limb-darkening coefficients, we obtain modestly constrained estimates that differ from the \\citet{claret2017} coefficient values at the 3.1--3.4$\\sigma$ level, while likewise maintaining the other fitted astrophysical parameter estimates at statistically consistent values. On the other hand, fitting for a single coefficient assuming a linear limb-darkening law yields estimates that differ significantly from the modeled values in \\citet{claret2017} and introduces strong correlations between the limb-darkening coefficient and the transit depth and transit geometry parameters.\n\nIn the fits presented here, we fix the orbital eccentricity $e$ and argument of periastron $\\omega$ to the values obtained by \\citet{nymeyer2011}: $e=0.0091$ and $\\omega=269^{\\circ}$. For further discussion of orbital eccentricity, see Section~\\ref{sec:res}.\n\n\n\n\\begin{deluxetable}{lllc}\n\\tablewidth{0pc}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{\n    Known Parameters\n    \\label{tab:knownparams}\n}\n\\tablehead{\n    \\multicolumn{1}{c}{~~~~~~~~Parameter~~~~~~~~} &\n    \\multicolumn{1}{c}{Value}                     &\n    \\multicolumn{1}{c}{Error}    &\n    \\multicolumn{1}{c}{Source}    \n}\n\\startdata\n$\\ensuremath{T_{\\rm eff}}$ (K)  \\dotfill  & 6431 & 48 & \\cite{stassun2017} \\\\\n$\\ensuremath{R_s}$ (\\ensuremath{R_\\sun}) \\dotfill  & 1.26 & 0.04 & Stassun et~al., in prep.\\tablenotemark{a}\\\\\n$\\ensuremath{M_s}$ (\\ensuremath{M_\\sun}) \\dotfill  & 1.46 & 0.29 & \\cite{stassun2017} \\\\\n$\\log g$ \\dotfill & 4.47 & 0.13 & \\cite{stassun2017} \\\\\n${\\rm [Fe\/H]}$ \\dotfill & 0.11 & 0.08 & \\cite{torres2012} \\\\\n$K_{RV}$ (\\ensuremath{\\rm m\\,s^{-1}})   \\dotfill  & 1816.6 & $_{-6.3}^{+6.1}$ & \\cite{knutson2014}  \n\\enddata\n\\tablenotetext{a}{Stellar radius is derived using Gaia DR2 data.}\n\\end{deluxetable}\n\n\\subsection{Phase curve model}\n\\label{subsec:phase}\n\nWe model the out-of-occultation variation of the system brightness as a third-order harmonic series in phase \\citep[e.g.,][]{carter2011}:\n\\begin{equation}\\label{eq:prefull}\n\\psi'(t) = 1+\\bar{f_{p}}+\\sum\\limits_{k=1}^{3}A_{k}\\sin(k\\phi(t))+\\sum\\limits_{k=1}^{3}B_{k}\\cos(k\\phi(t)).\\end{equation}\nHere, we have normalized the flux such that the average brightness of the star alone is unity. The baseline relative planetary brightness $\\bar{f_{p}}$ is the average of the planet's apparent flux across its orbit. The phase function $\\phi(t)$ is derived from the time series via the relation $\\phi(t) = 2\\pi(t-T_{0})\/P$, where $t$ is time.\n\nSeveral of the harmonic terms in the phase curve model are attributed to various physical processes on the star or planet. The star's brightness is modulated by the beaming effect and ellipsoidal variation. These two processes produce phase curve signals at the fundamental of the sine ($A_{1}$) and the first harmonic of the cosine ($B_{2}$), respectively. A tidally-locked hot Jupiter has a fixed day-side hemisphere facing the star, which produces a variation in its apparent brightness due to the changing viewing geometry. This atmospheric brightness component produces a signal at the fundamental of the cosine ($B_{1}$). See the discussion in Section~\\ref{sec:dis} for more details concerning the astrophysical implications of the phase curve terms.\n\nWe can separate the total system phase curve model into terms describing the star's brightness variation $\\psi_{*}(t)$ and terms describing the planet's brightness variation $\\psi_{p}(t)$:\n\\begin{align}\n\\psi_{p}(t) & = f_{p}-|B_{1}|+B_{1}\\cos(\\phi(t)), \\\\\n\\psi_{*}(t) & = 1 + \\sum\\limits_{k=1}^{3}A_{k}\\sin(k\\phi(t))+\\sum\\limits_{k=2}^{3}B_{k}\\cos(k\\phi(t)).\n\\end{align}\nWe have assigned all terms without a direct corresponding physical process to the star's phase modulation. These other terms can become significant if there are discernible phase shifts in the aforementioned modulation signals. We have also used the fact that the average brightness of the planet is the maximum measured brightness, which occurs at secondary eclipse, minus the semi-amplitude of the atmospheric variation: $\\bar{f_{p}}\\equiv f_{p}-|B_{1}|$. \n\nThe separation of stellar and planetary phase curve terms is important when including the transit and eclipse light curves $\\lambda_{t}(t)$ and $\\lambda_{e}(t)$, since only the brightness modulation of the occulted region on the star or planet is removed from the total system flux. This correction is particularly consequential during secondary eclipse, when the atmospheric brightness component is completely blocked by the star ($\\lambda_{e}=0$), while the ellipsoidal and beaming modulations on the star are unaffected ($\\lambda_{t}=1$). From here, we can write down the full phase curve model, including eclipses (transit and secondary eclipse) and re-normalized such that the average out-of-eclipses flux is unity:\n\\begin{equation}\\label{full}\n\\psi(t) = \\frac{\\lambda_{t}(t)\\psi_{*}(t)+\\lambda_{e}(t)\\psi_{p}(t)}{1+f_{p}-|B_{1}|}.\n\\end{equation}\n\nWhen compared to the standard approach in the literature, which simply multiplies the system phase curve model in \\eqr{prefull} and the occultation light curves together \\citep[e.g.,][]{carter2011}, this more detailed and physical model deviates most significantly during the ingress and egress of secondary eclipse, where the discrepancy for the WASP-18 system can be as large as several tens of ppm. While this level of model discrepancy is well within the noise of the light curves analyzed in this work, studies with higher signal-to-noise photometry would benefit from our more careful treatment of the phase curve model.\n\n\\subsection{Modeling long-term trends}\n\\label{subsec:systematics}\n\nAs can be seen in Figure~\\ref{fig:lightcurves}, the {\\it TESS}\\ PDC light curves from the SPOC pipeline show long-term trends at the level of several hundreds of ppm. Those can be caused by low-frequency residual systematics and\/or long-term stellar variability. \n \nFor the detrending model, we use a polynomial in time of the form\n\\begin{equation} \nS^{\\lbrace i\\rbrace}_{n}(t) = \\sum\\limits_{k=0}^{n}c^{\\lbrace i\\rbrace}_{k}(t-t_{0})^{k},\n\\end{equation}\nwhere $t_{0}$ is the first time stamp in the light curve from orbit $i$, and $n$ is the order of the polynomial model. In the following, we assign $i=1,2$ to the two orbits of the {\\it TESS}\\ spacecraft that comprise the full Sector 2 data (those correspond to physical orbits 11 and 12, where the numbering started during commissioning) and $i=3,4$ to the two orbits contained in Sector 3 data (corresponding to physical orbits 13 and 14). The complete phase curve model is therefore\n\\begin{equation}\nf(t) = S^{\\lbrace i\\rbrace}_{n}(t)|_{i=1,2,3,4} \\times \\psi(t).\n\\end{equation}\n\nTo determine the optimal polynomial order for each orbit, we carry out phase curve fits of individual orbit light curves and choose the order that minimizes the Bayesian Information Criterion (BIC), which is defined as $\\mathrm{BIC} \\equiv k\\log n-2\\log L$, where $k$ is the number of free parameters in the fit, $n$ is the number of data points, and $L$ is the maximum log-likelihood.\n\nFor the first orbit, we find that a 7th-order polynomial model minimizes the BIC, while for the second orbit, we use a 9th-order polynomial. For the two shorter Sector 3 orbits light curves, we use a 5th-order and 3rd-order polynomial, respectively. \n\nThe use of a time-dependent detrending model when fitting for a time-varying astrophysical signal can sometimes introduce artificial biases into the parameter estimates. When selecting polynomials of similar order, the best-fit astrophysical parameters do not vary by more than 0.2$\\sigma$. To further assess the effects of our use of polynomial detrending, we carry out a special joint fit of all four orbital light curves without using any detrending model, instead using a simple multiplicative normalization factor for each orbit's light curve. The resultant estimates of the transit\/eclipse depths and phase curve amplitudes are consistent with the results of the full joint fit with detrending models (\\secr{res} and \\tabr{modelparams}) at better than the \\sig{1} level. This test demonstrates that the results presented in this paper are highly robust to the particular choice of detrending model.\n\n\\subsection{Contaminating sources}\n\\label{subsec:contamination}\n\nSince the primary objective of the {\\it TESS}\\ mission is to carry out a survey of nearby, bright stars in search of transiting planets, the camera focus is set so as to spread a point source's flux over several pixels to adequately sample the point spread function and achieve high photometric precision. As a benchmark, approximately 50\\% and 90\\% of a star's flux is contained within a 1$\\times$1 and 4$\\times$4~pixel region around the centroid, respectively \\citep{ricker2015}. Given the pixel scale of $21''$, this indicates that a star's pixel response function (PRF) occupies a significant on-sky area, raising the possibility of contaminating sources overlapping with the target PRF and blending with the extracted photometry.\n\nThere are two moderately bright stars within the {\\it TESS}\\ input catalog that are in the vicinity of WASP-18 --- TIC 100100823 and TIC 100100829. These two nearby sources lie $73''$ and $83''$ away from the target and have {\\it TESS}\\ magnitudes of $T=12.65$ mag and $T=12.50$ mag, respectively. When compared to WASP-18 ($T=8.83$ mag) the nearby sources are 34 and 29 times fainter, respectively.\n\nThe optimal apertures selected by the SPOC pipeline for photometric extraction are shown in Figure~\\ref{fig:stamp}. The locations of the two nearby stars are also indicated. The SPOC pipeline uses a model PRF derived from commissioning data to remove the flux from neighboring sources located on each target's subarray. The relative uncertainty of the deblending process due to imperfections in the PRF model and intrinsic variations in the PRFs of different sources across the detector is estimated to be at the level of a few percent \\citep{jenkins2010}. \n\nIn the Sector 2 pixel stamp, TIC 100100823 is 0.66~pixels away from the edge of the aperture at its closest point. Using the aforementioned benchmark estimates of a point source's flux distribution on the detector, we determine that the central 50\\% of its undeblended flux (from the 1$\\times$1~pixel region around its centroid) was not included in the aperture. Another 40\\% of its flux was distributed in the remaining 15 pixels in the surrounding 4$\\times$4~pixel region. Tracing a 4$\\times$4~pixel region centered on the location of TIC 100100823, with edges aligned with the pixel boundaries, we can estimate that the fractional area lying within the extraction aperture is $\\sim$0.25. Assuming that the flux was evenly distributed across the 15-pixel region to obtain a generous upper limit, we predict that at most $0.25\\times 40\\% = 10\\%$ of the star's flux lay within the science aperture prior to deblending. Multiplying this upper limit with the estimated uncertainty in the deblending process and the flux of the star relative to WASP-18 gives a relative deblending contamination contribution of roughly 0.01\\%. The other similarly-bright nearby star, TIC 100100829, is further away from the edge of the aperture, so its level of residual contamination in the extracted photometry is smaller. For Sector 3, the nearby stars are situated farther away from the edge of the optimal aperture, and thus the expected deblending contamination contribution is even more negligible.\n\nAs detailed in the following section, the relative uncertainties we obtain for the fitted astrophysical parameters that would be directly affected by such contamination  --- phase curve amplitudes and transit\/eclipse depth --- lie above the 0.13\\% level (in fact, for all parameters except for transit depth, the relative uncertainties are greater than 3\\%). This means that any effect stemming from uncorrected contamination is overshadowed by the much larger intrinsic uncertainties in the parameter estimates, given the sensitivity of the data. Therefore, we do not consider contamination in our phase curve analysis.\n\n\n\n\\begin{deluxetable}{lll}\n\\tablewidth{0pc}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{\n    Model Parameters\n    \\label{tab:modelparams}\n}\n\\tablehead{\n    \\multicolumn{1}{c}{~~~~~~~~~~Parameter~~~~~~~~~~} &\n    \\multicolumn{1}{l}{Value}                     &\n    \\multicolumn{1}{l}{Error}    \n}\n\\startdata\n\\sidehead{\\textit{Fixed Parameters}}\n$e$\\tablenotemark{a}   \\dotfill  & 0.0091 & --- \\\\\n$\\omega$\\tablenotemark{a} ($^{\\circ}$) \\dotfill  & 269 & --- \\\\\n$u_{1}$\\tablenotemark{b} \\dotfill & 0.2192 & --- \\\\\n$u_{2}$\\tablenotemark{b} \\dotfill & 0.3127 & --- \\\\ \n\n\\sidehead{\\textit{Fitted Parameters}}\n$R_p\/R_s$   \\dotfill  & 0.09716 & $_{-0.00013}^{+0.00014}$ \\\\\n$f_p$ (ppm) \\dotfill  & 341 & $_{-18}^{+17}$ \\\\\n$T_0$ (BJD$_{\\mathrm{TDB}}$)      \\dotfill  & 2458375.169883 & $_{-0.000025}^{+0.000026}$ \\\\\n$P$ (days)  \\dotfill  & 0.9414526 & $_{-0.0000015}^{+0.0000016}$\\\\\n$b$         \\dotfill  & 0.318 & $_{-0.019}^{+0.018}$\\\\\n$a\/R_s$     \\dotfill  & 3.562 & $_{-0.023}^{+0.022}$\\\\\n$A_1$ (ppm) \\dotfill  & 21.0  & 4.5 \\\\\n$A_2$ (ppm) \\dotfill  & -4.0   & 4.6 \\\\\n$A_3$ (ppm) \\dotfill  & -14.0  & 4.6 \\\\\n$B_1$ (ppm) \\dotfill  & -174.4 & $_{-6.2}^{+6.4}$ \\\\\n$B_2$ (ppm) \\dotfill  & -190.5 & $_{-5.9}^{+5.8}$ \\\\\n$B_3$ (ppm) \\dotfill  & -3.9  & 6.1\\\\\n$\\sigma_{1}$ (ppm) \\dotfill  & 527.6 & $_{-4.1}^{+3.9}$ \\\\\n$\\sigma_{2}$ (ppm) \\dotfill  & 528.6 &  3.8\\\\\n$\\sigma_{3}$ (ppm) \\dotfill  & 526.5 &  5.6\\\\\n$\\sigma_{4}$ (ppm) \\dotfill  & 521.4 &  $_{-5.2}^{+5.3}$\\\\\n\\sidehead{\\textit{Derived Parameters}} \nTransit depth\\tablenotemark{c} (ppm)  \\dotfill  & 9439 & $_{-26}^{+27}$\\\\\n$i$ ($^{\\circ}$)    \\dotfill  & 84.88 & 0.33 \\\\\n$R_p$ ($R_{\\mathrm{Jup}}$)\\dotfill & 1.191 & 0.038\\\\\n$a$ (AU)\\dotfill & 0.02087 & 0.00068\n\\enddata\n\\tablenotetext{a}{Best-fit values from \\citet{nymeyer2011}.}\n\\tablenotetext{b}{Tabulated in \\citet{claret2017}.}\n\\tablenotetext{c}{Calculated as $(R_p\/R_s)^2$.}\n\\end{deluxetable}\n\n\\begin{figure*}\n\\includegraphics[width=\\linewidth]{plot2.pdf}\n\\caption{Top panel: The phase-folded light curve, after correcting for long-term trends, binned in 5-minute intervals (black points), along with the best-fit full phase curve model from our joint analysis (red line). Middle panel: same as top panel, but with an expanded vertical axis to detail the fitted phase curve modulation and secondary eclipse signal. The contributions to the phase curve from ellipsoidal ($B_{2}$), atmospheric brightness ($B_{1}$), and beaming ($A_{1}$) modulations are plotted individually with black solid, dot-dash, and dotted lines. Bottom panel: plot of the corresponding residuals from the best-fit model.}\n\\label{fig:bestfit}\n\\end{figure*}\n\n\\subsection{Model fit}\n\\label{sec:fit}\n\nWe carry out a joint fit of the WASP-18 light curves from Sectors 2 and 3. We fit for $R_p\/R_s$, $f_{p}$, $T_{0}$, $P$, $b$, $a\/R_{*}$, $A_{1-3}$, $B_{1-3}$, $c^{\\lbrace 1\\rbrace}_{0-7}$, $c^{\\lbrace 2\\rbrace}_{0-9}$, $c^{\\lbrace 3\\rbrace}_{0-5}$, and $c^{\\lbrace 4\\rbrace}_{0-3}$. In addition, we fit for a uniform per-point uncertainty for each orbit --- $\\sigma_{1-4}$ --- which ensures that the resultant reduced $\\chi^2$ value is near unity and self-consistently produces realistic uncertainties on the other parameters given the intrinsic scatter of the light curves. The total number of free astrophysical, long-term trend, and noise parameters is 44. \n\nThe ExoTEP pipeline simultaneously computes the best-fit values and posterior distributions for all parameters using the affine-invariant Markov Chain Monte Carlo (MCMC) ensemble sampler \\texttt{emcee} \\citep{emcee}. We set the number of walkers to four times the number of free parameters: 176. To facilitate the convergence of the chains we initialize the walkers with values close to those of the best fit to light curves of individual physical orbits. The length of each walker's chain is 100,000 steps, and we discard the first 80\\% of each chain before calculating posterior distributions. The chains are plotted and visually inspected to ensure convergence. We also re-run the entire process, to confirm that the resultant parameter estimates are consistent to well within $0.1\\sigma$.\n\n\\section{Results}\n\\label{sec:res}\n\nThe results of our joint analysis are listed in Table~\\ref{tab:modelparams}, where we include astrophysical and noise parameters. The median and \\sig{1} uncertainties derived from the posterior distributions are given. The combined phased light curve with long-term trends removed is plotted in Figure~\\ref{fig:bestfit} along with the best-fit full phase curve model.\n\nComparing with other published values in the literature \\citep{hellier2009, southworth2009, nymeyer2011}, we find that our results generally lie in good agreement with previous estimates. The calculated values of the orbital parameters $i$ and $a\/R_{*}$ are the most precise to date and are well within \\sig{1} of the values presented in the discovery paper \\citep{hellier2009} and follow-up photometric studies \\citep{southworth2009}. We obtain an updated and refined mid-transit time with a precision of 2.2~seconds and a period estimate that is consistent at better than the \\sig{0.2} level with the results of \\citet{hellier2009} as well as recent studies that fitted ephemerides across all previously measured transit times \\citep{wilkins2017,mcdonald2018}.\n\nThe measured transit depth $(R_{p}\/R_{*})^{2}$ of $9439_{-26}^{+27}$~ppm is significantly (\\sig{3.3}) larger than the estimate of \\citet{hellier2009} of $8750\\pm 210$~ppm. A separate analysis of the photometry and radial velocity (RV) data obtained by \\citet{hellier2009} incorporating additional RV measurements yielded a somewhat larger transit depth estimate of $9160^{+200}_{-120}$~ppm \\citep{triaud2010}, which is more consistent (\\sig{1.4}) with the value derived here. \\citet{southworth2009} carried out a thorough analysis of WASP-18b transit light curves using a variety of limb-darkening laws and fitting methodologies and obtained a wide range of transit depths (9300--9800~ppm) consistent with the estimate from our analysis.\n\nThe most notable results from our phase curve analysis are the $19\\sigma$ detection of a secondary eclipse in the {\\it TESS}\\ bandpass and the robust detection of phase curve variations corresponding to beaming ($A_{1}$), atmospheric brightness ($B_{1}$), and ellipsoidal ($B_{2}$) modulations (See table~\\ref{tab:modelparams}). These three leading phase curve harmonics are plotted individually in the middle panel of Figure~\\ref{fig:bestfit}.\n\nIn our analysis, we have fixed the orbital eccentricity $e$ and argument of periastron $\\omega$ to the most recent literature values \\citep{nymeyer2011}. Since the orbit is very nearly circular, we have experimented with carrying out fits where we fix $e=0$, as well as fits where we allow $e$ and $\\omega$ to vary freely. In both cases, we obtain parameter estimates that are consistent with the best-fit values listed in Table~\\ref{tab:modelparams} at better than the $0.9\\sigma$ level. \n\nIn the free-eccentricity fits, we obtain relatively weak eccentricity constraints: $e=0.0015_{-0.0011}^{+0.0038}$ and $\\omega=156_{-69}^{+115}$~deg. These estimates are consistent with those published in \\citet{nymeyer2011} at better than the \\sig{2} level. In their analysis of high signal-to-noise thermal infrared secondary eclipse light curves from \\textit{Spitzer}, they obtain orbital eccentricity estimates that are much more precise than what we can constrain from our analysis. Therefore, we have decided to fix $e$ and $\\omega$ to their best-fit values in the joint fits described in Section~\\ref{sec:res}.\n\nThe fitted per-point uncertainties for the light curves $\\sigma_{1-4}$ are nearly identical at 520--530~ppm, indicating that the noise level in the data is consistent across the four orbits. The standard deviations of the residuals binned at 1-hour intervals in time are 130, 143, 142, and 158~ppm for the four orbits, respectively, which are generally consistent with the benchmark prediction of 123~ppm calculated for a $T=8.83$~mag target based on simulated photometry for the {\\it TESS}\\ mission \\citep{sullivan2015,stassun2017}. The larger binned residuals in the Sector 3 light curves (particularly in the last orbit) indicate suboptimal photometric performance, as discussed in \\secr{obs}.\n\n\n\\section{Discussion}\n\\label{sec:dis}\n\nAmong the sinusoidal phase curve coefficients fitted in our model, only three have a high statistical significance (with a signal to noise ratio of well over 3, see \\tabr{modelparams}) --- those corresponding to the fundamental, $A_1$ and $B_1$, and the first harmonic of the cosine, $B_2$. As discussed in Section~\\ref{subsec:phase}, these terms are attributed to the beaming, atmospheric, and ellipsoidal components, respectively. Each component is discussed in Sections~\\ref{sec:beam}--\\ref{sec:atm} below. \n \nFor the beaming and ellipsoidal components we show in Sections~\\ref{sec:beam} and \\ref{sec:tide} that their amplitudes and signs are consistent with theoretical expectations. While that consistency can be expected, there are systems where that is not the case for one of those phase components and cases where the mass ratios derived from the two amplitudes do not agree \\citep[e.g.,][see also \\citealt{shporer2017} Section 3.4]{vankerkwijk2010, carter2011, shporer2011, barclay2012, bloemen2012, esteves2013, faigler2015, rappaport2015}. One possible explanation for that disagreement is a phase shift in the atmospheric component \\citep[e.g.,][]{faigler2015, shporer2015}. Another possible explanation is poor modeling of the tidal distortions of hot stars, given their fast rotation and lack of convective zone in their atmospheres \\citep{pfahl2008}.\n\nOur phase curve model includes three other components --- the first harmonic of the sine, $A_2$, and the second harmonic terms, $A_3$ and $B_3$ (see \\tabr{modelparams}).\nThe first harmonic of the sine, $A_2$, is not statistically significant which is consistent with theoretical expectations as we are not aware of any astrophysical process associated with that coefficient \\citep[e.g.,][]{faigler2011, shporer2017}.  \nThe second harmonic of the cosine ($B_3$) is expected as a higher-order term of the ellipsoidal distortion modulation, and the fitted amplitude, although not statistically significant, is in agreement with expectations (see more details in \\secr{tide}).\nFinally, the sine amplitude of the second harmonic, $A_3$, is $-14.0 \\pm 4.6$ ppm, at the \\sig{3} level. While it is smaller in amplitude and lower in statistical significance than the coefficients associated with the beaming, ellipsoidal, and atmospheric components ($A_1$, $B_2$, and $B_1$, respectively, see \\tabr{modelparams}), the astrophysical origin of that coefficient is not immediately clear. It is interesting to note that a similar phase component was measured at a statistically significant level for Kepler-13Ab \\citep{esteves2013, shporer2014}, although that host star is a hot A-type star \\citep{shporer2014} while WASP-18 is a mid F-type star.\n\nWe have experimented with fitting a simplified phase curve model without the second harmonic terms, i.e., including only the fundamental ($A_{1}$, $B_{1}$) and first harmonic ($A_{2}$, $B_{2}$) terms. We do not find that any of the fitted astrophysical parameter estimates change by more than $0.3\\sigma$. This indicates that our phase curve analysis is not significantly affected by the inclusion of the second harmonic terms.\n\n\\subsection{Beaming}\n\\label{sec:beam}\n\nThe sine component of the fundamental, $A_1$, is expected to result from the beaming modulation. We derive the expected amplitude using:\n\\begin{equation}\n    A_{\\rm beam} = \\alpha_{\\rm beam} 4 \\frac{K}{c} \\ ,\n    \\label{eq:beam}\n\\end{equation}\nwhere $K$ is the orbital RV semi-amplitude, and $c$ the speed of light. The $\\alpha_{\\rm beam}$ coefficient is of order unity and depends on the target's spectrum in the observed bandpass (for a more detailed description of the nature of this coefficient, see \\citealt{shporer2017}). \n\nAssuming the target is a blackbody and integrating across the {\\it TESS}\\ bandpass, we derive an expected beaming modulation amplitude of $A_{\\rm beam}$ = $18 \\pm 2$~ppm, using the known parameters of the system (see \\tabr{knownparams}). This is consistent with the measured value of $21.0 \\pm 4.5$~ppm. Examining the shape of the phase curve in Figure~\\ref{fig:bestfit}, the presence of a significant beaming modulation signal is seen as the difference between the two brightness maxima within the orbit.\n\n\n\\subsection{Tidal interaction}\n\\label{sec:tide}\n\n\nThe ellipsoidal modulation measured in the phase curve is the result of tidal interaction between the planet and the star. To estimate the expected value of that photometric modulation amplitude, we use the following approximate equation \\citep{morris1985, morris1993}: \n\\begin{equation}\n    A_{\\rm ellip} = \\alpha_{\\rm ellip} \\frac{M_p \\sin^2 i}{M_s}\\left(\\frac{R_s}{a}\\right)^3  \\ \\ ,\n    \\label{eq:aellip}\n\\end{equation}\n    where $M_s$ and $R_s$ are the host star mass and radius, respectively, $M_p$ the planet mass, $a$ the orbital semi-major axis, and $i$ the orbital inclination angle. The $\\alpha_{\\rm ellip}$ coefficient is derived from the linear limb darkening coefficient $u$ and gravity darkening coefficient $g$:\n\\begin{equation}\n    \\alpha_{\\rm ellip} = 0.15 \\frac{(15+u)(1+g)}{3-u} \\  \\ .\n\\end{equation}\nWe estimate $u$ and $g$ using the known parameters of the host star (see \\tabr{knownparams}) and the tables of \\cite{claret2017} to arrive at $\\alpha_{\\rm ellip} = 1.10 \\pm 0.05$. This gives an expected amplitude of $A_{\\rm ellip} = 172.5 \\pm 14.6$~ppm (uncertainty derived from uncertainty of known parameters), in good agreement with the measured amplitude of $|B_2|=190.5_{-5.8}^{+5.9}$ ppm.\n\nAs noted earlier, \\eqr{aellip} is an approximation, as it is the leading term in a Fourier series \\citep{morris1985, morris1993}. We have calculated the next term in the series, the coefficient of the cosine of the 2nd harmonic of the orbital period \\citep{morris1985}, to be $-11.8 \\pm 1.8$ ppm. This is in agreement with the measured amplitude of $B_3 = -3.9 \\pm 6.1$ ppm (see \\tabr{modelparams}), although the latter is not statistically significant.\n\n\\subsection{Atmospheric characterization}\n\\label{sec:atm}\n\nThe drop in flux during secondary eclipse, as the planet is occulted by its host star, is due to the blocking of (1) starlight reflected by the planet's atmosphere and (2) thermal emission from the planet's atmosphere, since, given the strong stellar irradiation, the planet's thermal emission is expected to be significant at visible wavelengths. To estimate the planet's thermal emission in the {\\it TESS}\\ band, we use the atmospheric model of \\cite{arcangeli2018}, derived by fitting to the measured secondary eclipse depths in the four Spitzer\/IRAC bands (centered at 3.6, 4.5, 5.8, and 8.0~\\ensuremath{\\mu \\rm m}) and HST\/WFC3 (1.1--1.7~\\ensuremath{\\mu \\rm m}). Integrating that atmospheric model across the {\\it TESS}\\ bandpass gives an expected thermal emission of 327~ppm. It is difficult to accurately quantify the uncertainty on the expected thermal emission and we estimate it to be at the few percents level (the uncertainty on the star-planet radii ratio derived here, which contributes to the thermal emission uncertainty, is only 0.14\\%).\n\nWe measure a secondary eclipse depth of $341_{-18}^{+17}$~ppm. While this depth is significantly shallower than the $z'$ band eclipse depth of $682 \\pm 99$ ppm measured by \\cite{kedziora2019}, it is consistent with the expected thermal emission predicted by the atmospheric models of \\cite{arcangeli2018}.\n\nSince the difference between the measured secondary eclipse depth and the expected thermal emission in the {\\it TESS}\\ bandpass is only at a \\sig{0.8} significance, we cannot claim a detection of reflected light, and therefore place a \\sig{2} upper limit of 35~ppm on the relative contribution of reflected light to the atmospheric brightness. This upper limit is consistent with the upper limit on reflected polarized light from the planet measured by \\cite{bott2018}.\n\nThe upper limit on the reflected light translates to an upper limit on the geometric albedo in the {\\it TESS}\\ bandpass of $A_g <$ 0.048 (\\sig{2}), using the model parameters measured here (specifically $R_p\/R_s$ and $a\/R_s$). This upper limit is consistent with low visible-light geometric albedo measured for other short period gas giant planets, like WASP-12b ($A_g < $ 0.064 at 97.5\\% confidence, \\citealt{bell2017}), HD~209458b ($A_g = 0.038 \\pm 0.045$, \\citealt{rowe2008}), TrES-2b ($A_g = 0.0253 \\pm 0.0072$, \\citealt{kipping2011}), Qatar-2b ($A_g < 0.06$ at \\sig{2}, \\citealt{dai2017}), and others \\citep[e.g.,][]{heng2013, esteves2015, angerhausen2015}.\n\nThe small geometric albedo suggests that the bond albedo is also small since the {\\it TESS}\\ bandpass is close to the wavelength region where the host star is brightest. The low albedo is consistent with a correlation between decreasing albedo and increasing planet mass, suggested by \\citet{zhang2018}, although there is currently no theoretical mechanism to explain such a correlation.\n\n\nThe measurements of the secondary eclipse depth and the atmospheric phase component amplitude allow us to estimate the flux from the planet's night side --- the hemisphere facing away from the star, visible to the observer during transit. The night-side flux is the difference between the secondary eclipse depth $f_p$ and the full amplitude of the atmospheric brightness modulation $2 \\times |B_{1}|$, which is $-8\\pm 22$ ppm (see \\tabr{modelparams}). Therefore, we do not measure statistically significant flux from the planet's night side, and place a \\sig{2} upper limit of 43~ppm on the night side's brightness in the {\\it TESS}\\ bandpass, which is 13\\% of the day-side brightness. Measurements of the orbital phase curve in the near-infrared at 3.6 and 4.5~\\ensuremath{\\mu \\rm m}\\ were also unable to detect flux from the night side \\citep{maxted2013}. This points to very low efficiency of longitudinal heat distribution from the day side to the night side, as suggested by other authors \\citep{nymeyer2011,maxted2013} and consistent with similar findings for other highly irradiated hot Jupiters \\citep[e.g.,][]{wong2015,wong2016} and theoretical models \\citep{perez2013, komacek2017}.\n\n\nWe have assumed here that the atmospheric phase variability is a sinusoidal modulation, where the maximum is coincident with the phase of secondary eclipse, and the minimum occurs at mid-transit. Deviations from this simplistic model have been observed, where the planet's surface brightness distribution is such that the brightest region is shifted away from the sub-stellar point, leading to a shift between the phase of maximum light and the center of secondary eclipse \\citep[e.g.,][]{demory2013, shporer2015, hu2015, faigler2015, parmentier2016}. In our phase curve analysis a phase shift of the atmospheric phase component will manifest itself as a deviation of the beaming phase component amplitude from the theoretically predicted value, since the beaming and atmospheric components are the sine and cosine of the fundamental (the orbital period). The beaming amplitude is consistent with the predicted amplitude (see \\secr{beam}), and we therefore place an upper limit on a phase shift of 2.9~deg (\\sig{2}), derived from the fitted values and uncertainties of $A_1$ and $B_1$ (see \\tabr{modelparams}). \n\nIn principle, since the observed modulation is a superposition of modulations due to reflected light and thermal emission, each of the two processes may have a phase shift that is canceled out in the combined light. However, phase curves in the near-infrared do not show a phase shift of the thermal emission, down to 5--10~deg \\citep{maxted2013}. Hence, we can rule out a phase shift of the reflected light modulation in the optical. The lack of a phase shift is consistent with the inefficient heat redistribution from the day-side to the night-side hemisphere.\n\n\n\n\n\n\\subsection{Detecting non-transiting WASP-18b-like objects}\n\\label{sec:nontransit}\n\nThe clear sinusoidal modulations seen along the orbital phase (see \\figr{lightcurves}) suggests that they could be identified even if the system were not in a transiting configuration, i.e., with a smaller orbital inclination angle. If so, this in turn raises the possibility of detecting non-transiting but otherwise similar systems, as explored by \\citet[][see also \\citealt{faigler2012,faigler2013,talor2015, millholland2017}]{faigler2011} using a phase curve analysis.\n\nTo test that possibility, we have removed all data points within the transit and secondary eclipse and have carried out a period analysis of the remaining light curve, similar to the analysis done in \\citet[][see also \\citealt{shporer2014}]{shporer2011} for the Kepler-13 system. \\figr{ls} shows both the Lomb-Scargle periodogram \\citep{lomb1976, scargle1982} and the double-harmonic periodogram (following \\citealt{faigler2011}). Both periodograms are dominated by variability at the orbital period. This demonstrates that variability at the orbital period can be clearly detected for non-transiting but otherwise similar systems. This also further confirms the prediction of \\citet[][see their Figure 10]{shporer2017} that the orbital phase curve variability of systems such as WASP-18, containing massive planets on short periods, can be detected in {\\it TESS}\\ data.\n\nIt is important to note that the detection of periodic variability does not uniquely reveal the nature of the system as a star-planet system, since similar variability can be induced by a stellar-mass companion, stellar pulsations, and\/or the combination of stellar activity (in the form of starspots) and stellar rotation. However, a star rotating at a period as short as the WASP-18b orbital period would show large rotational broadening of the spectral lines, with a rotation velocity over 50~\\ensuremath{\\rm km\\,s^{-1}}, which can be measured with a single stellar spectrum. Furthermore, as shown by \\citet{faigler2011}, the measured amplitudes can be compared with the expected amplitudes based on the stellar parameters.\n\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.4\\textwidth]{ls.eps}\n\\includegraphics[width=0.4\\textwidth]{beer.eps}\n\\caption{Lomb-Scargle (top) and double-harmonic (bottom) periodograms of the lightcurves where data within the transits and secondary eclipse were removed, and normalized by their standard deviation (estimated through the median absolute deviation, or MAD). Both periodograms are dominated by the orbital frequency and its harmonics and sub-harmonic, marked by short vertical green lines on the x-axis. \n\\label{fig:ls}}\n\\end{centering}\n\\end{figure}\n\n\\subsection{TTV analysis}\n\\label{sec:ttv}\n\nTo complement our analysis of the phase curve we have also analyzed the individual transit times in order to look for transit timing variations (TTV). That analysis was divided into long term TTV, using all available transit times in the literature spanning about a decade, and short term TTV, within the time scale of the {\\it TESS}\\ Sector 2 and 3 data. Both analyses are described below.\n\n\\subsubsection{Long term TTV}\n\\label{sec:ttvlong}\n\nMost hot Jupiters are vulnerable to tidal orbital decay, and their\norbits should be shrinking \\citep[e.g.,][]{levrard2009,matsumura2010}. However, directly observing tidal inspiral has proven an obstinate challenge; there have not yet been any unambiguous detections of orbital decay due to tides (see \\citealt{maciejewski2016,patra2017,bailey2019}\nfor discussion of the most promising case yet).\nWASP-18b is a promising target in the search for tidal orbital decay.\nCompared to other hot Jupiters, it has a particularly high planet to\nstar mass ratio, and an exceptionally small separation from its host\nstar. It was realized quite early that if the stellar tidal dissipation rates inferred from binary star systems were applicable in WASP-18, the system would undergo orbital decay on a timescale of only megayears (\\citealt{hellier2009} and references therein). \\citet{wilkins2017} recently searched for orbital decay of WASP-18b, concatenating previous observations with new data. Within the context of the ``constant phase lag'' model for tidal interaction \\citep{zahn1977}, they reported a limit on the modified stellar tidal quality factor $Q_\\star' \\geq 1\\times10^6$, at 95\\% confidence.\n\nThe {\\it TESS}\\ observations provide new transit times that let us extend WASP-18b\norbital decay search. \\tabr{tt} lists the transit and occultation times we used in our long-term timing analysis, where most of the archival times were already compiled by \\citet{wilkins2017}. We require that each archival time (i) originate from the peer-reviewed literature, and (ii) be based on observations of a single transit or occultation observed in its entirety. We adopt the methods and equations described by \\cite{bouma2019}, who performed a similar study in the context of WASP-4b. To measure the {\\it TESS}\\ transit times, each individual transit is isolated to a window of $\\pm 3$ transit durations. Each transit window is then fitted simultaneously for a local linear trend in relative flux, the mid-transit time, and the depth. The remaining transit parameters are fixed to those found in \\tabr{modelparams}. Given the abundance of transits, we omitted three {\\it TESS}\\ transits that had significant gaps due to momentum wheel dumps, close to BJDs of 2,458,359.2, 2,458,376.1, and 2,458,390.2.\nAfter deriving the new transit times and uncertainties, we fitted the full timing dataset with two competing models: a linear ephemeris model and a quadratic ephemeris model. \\figr{O_minus_C} left panel shows the times and best-fitting models where the linear ephemeris was subtracted. The difference in the Bayesian Information Criterion between the best-fitting linear and quadratic models is $-3.8$, so there is no evidence to prefer the quadratic model over the linear model. The relevant limits from the quadratic model, at 95\\% confidence, can be expressed as:\n\\begin{align}\n  -8.78 < \\dot{P}  &< 4.98 \\ \\mathrm{ms}\\,\\mathrm{year}^{-1}, \\label{eq:pdot} \\\\\n  Q_\\star' &> 1.73 \\times 10^6. \\label{eq:q}\n\\end{align}\nAs shown in \\eqr{pdot} our analysis does not find a long term period derivative. And, as seen in \\eqr{q} we do not find the stellar quality factor of $Q\\star' \\sim 5\\times10^5$ suggested by \\citet{mcdonald2018}, while our result is a modest improvement relative to the results of \\citet{wilkins2017}.\n\nThe linear ephemeris we derived by fitting the decade-long transit timings data set is:\n\\begin{align}\n& P = 0.941452419 \\pm 0.000000021 \\, {\\rm day}, \\\\\n& T_0 \\, ({\\rm BJD_{TDB}}) = 2458002.354726 \\pm 0.000023.\n\\end{align}\n\n\n\n\\begin{figure*}[t]\n\t\\begin{center}\n\t\t\\leavevmode\n\t\t\\includegraphics[width=0.47\\textwidth]{data_bestfit_OminusC_occ-tra.pdf}\n\t\t\\includegraphics[width=0.47\\textwidth]{modl_sinu.pdf}\n\t\\end{center}\n\t\\vspace{-0.7cm}\n\t\\caption{Long-term (left panel) and short-term (right panel) TTV search. Both panels show the residual timing after subtracting the linear ephemeris fitted to the decade-long transit timing data set listed in \\tabr{tt}. In the left panel, the best-fit constant-period model (blue) and constant period derivative model (orange) provide comparable fits to the data, but the latter model has one extra free parameter. In the right panel, the blue curves are fair samples drawn from the posterior of a quasi-periodic model used to fit the measured timing residuals. \n    }\n\t\\label{fig:O_minus_C}\n\\end{figure*}\n\n\\input{WASP-18b_transit_time_table.tex}\n\n\\subsubsection{Short term TTV}\n\\label{sec:ttvshort}\n\nThe existence of a third body in a star-planet system such as WASP-18 may introduce deviations of the observed transit timings from the expected timings assuming a Keplerian orbit \\citep[e.g.,][]{agol2005, holman2005}. Generally known as TTVs, these data features can allow one to probe the system for planetary companions. While these variations can have a wide range of dependencies on the orbital and planetary parameters, the expected generic shape of the TTVs is quasi-sinusoidal.\n\nWe analyzed the 44 transit timing inferred from {\\it TESS}\\ Sectors 2 and 3 data, listed in \\tabr{tt}, to test whether the observed timings contain any evidence for sinusoidal variations. Towards this purpose, we subtract from observed timings the expected timings based on the linear ephemeris derived in \\secr{ttvlong}. Those timing residuals are plotted in \\figr{O_minus_C} right panel and we attempt to model those data by sampling from the posterior probability distribution of two TTV models. The first model (the null model) has only a single parameter which is an offset fitted to the timing residuals. The second model has four parameters where in addition to the timing offset it includes the period, phase, and amplitude of the sinusoidal TTV. \n\nIn order to perform the sampling and calculate the associated Bayesian evidences, we use \\texttt{dynesty}\\footnote{\\url{https:\/\/github.com\/joshspeagle\/dynesty}}. The resulting Bayesian evidence of the sinusoidal TTV model relative to that of the null model is -3, indicating that the null hypothesis should be readily preferred. Furthermore, the posterior-mean reduced $\\chi^2$ (i.e., root mean square of the residuals per degree of freedom) of the null and alternative models are 1.18 and 1.22, respectively. Hence, we conclude that there is no evidence for TTVs in the {\\it TESS}\\ Sectors 2 and 3 data of WASP-18b.\n\n\n\\section{Summary}\n\\label{sec:sum}\n\nWe have presented here an analysis of the full orbital phase curve of the WASP-18 system measured by {\\it TESS}\\ in Sectors 2 and 3. The per-point residual scatter of the 2-minute data is 520--530~ppm, yielding a binned scatter of 130--160~ppm per 1-hour exposure. This noise level is consistent with the expected noise level of {\\it TESS}\\ data for this $T$=8.83~mag target.\n\nWe detect at high significance beaming and ellipsoidal modulations that are consistent with theoretical predictions. We robustly measure a secondary eclipse depth of $341_{-18}^{+17}$ ppm, which when combined with the expected thermal emission in the {\\it TESS}\\ bandpass leads to a null detection of reflected light and a \\sig{2} upper limit of 0.048 on the geometric albedo in the {\\it TESS}\\ bandpass. The low optical geometric albedo is consistent with that of other hot Jupiters in similar wavelength range, especially highly-irradiated hot Jupiters \\citep{schwartz2015}.\n\nWe do not detect a phase shift in the atmospheric phase curve component, with an upper limit of 2.9~deg (\\sig{2}), indicating that the phase of maximum light is well-aligned with the phase of secondary eclipse. In addition, we do not detect light from the planet's night-side hemisphere, with an upper limit of 43~ppm (\\sig{2}), or 13\\% of the day-side brightness. These findings indicate very inefficient distribution of incident energy from the day-side hemisphere to the night-side hemisphere and are consistent with results based on previously published phase curve measurements (both in the near-infrared and in the optical) of similarly highly-irradiated hot Jupiters \\citep[e.g.,][]{wong2015,wong2016} and with theoretical expectations \\citep{perez2013,komacek2017}.\n\nThe clear detection of the WASP-18 phase curve modulations demonstrate that {\\it TESS}\\ data are sensitive to the photometric variations of systems with massive short period planets, a sensitivity that increases when data from several {\\it TESS}\\ sectors is combined. For such objects, the {\\it TESS}\\ phase curve can be used for atmospheric characterization, as shown here, as well as independent mass estimates from the constraints derived from RV analyses.\n\nTo complement our study of the phase curve we have also searched for TTVs, both long term using all available measured transit times spanning about a decade,  and also short term within the time span of the {\\it TESS}\\ Sectors 2 and 3 data. In both cases we do not find a statistically significant deviation of the transit timings from a linear ephemeris.\n\n\\acknowledgments\n\nWe acknowledge the use of {\\it TESS}\\ Alert data, which is currently in a beta test phase. These data are derived from pipelines at the {\\it TESS}\\ Science Office and at the {\\it TESS}\\ Science Processing Operations Center.\nFunding for the {\\it TESS}\\ mission is provided by NASA's Science Mission directorate.\nThis paper includes data collected by the {\\it TESS}\\ mission, which are publicly available from the Mikulski Archive for Space Telescopes (MAST).\nResources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center.\nThis research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with NASA under the Exoplanet Exploration Program.\nI.W.~is supported by a Heising-Simons \\textit{51 Pegasi b} postdoctoral fellowship.\nC.X.H. and M.N.G.~acknowledge support from MIT's Kavli Institute as a Torres postdoctoral fellow.\nT.D.~acknowledges support from MIT's Kavli Institute as a Kavli postdoctoral fellow.\n\n{\\it Facilities:} \n\\facility{\\textit{TESS}}\n\n{\\it Software:} \n\\texttt{astrobase} \\citep{bhatti_astrobase_2018},\n\\texttt{astropy} \\citep{astropy2018},\n\\texttt{BATMAN} \\citep{kreidberg_batman_2015},\n{\\scshape dynesty} (\\url{https:\/\/github.com\/joshspeagle\/dynesty}),\n\\texttt{emcee} \\citep{emcee},\n\\texttt{numpy} \\citep{vanderwalt2011}, \n\\texttt{matplotlib} \\citep{hunter2007}, \n\\texttt{pandas} \\citep{mckinney-proc-scipy-2010},\n\\texttt{scipy} \\citep{jones_scipy_2001}.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection*{Acknowledgements}\n\nWe gratefully acknowledge financial support from Trinity College, Cambridge to A.L.G., from the EPSRC (UK) to A.L.G.\\ and M.J.C., and from the US NSF to D.A.D.\\ under grant DMR 09-03225. We thank D.\\ A.\\ Keen (Rutherford Appleton Laboratory) for useful discussions, and acknowledge the University of Cambridge's CamGrid infrastructure for computational resources.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nWhen a liquid is cooled below its freezing point it is supposed to freeze.\nUsually, impurities or the solid boundaries of the liquid provide preferential\nsites for the formation of the solid phase.\nHowever, even in the absence of impurities, small nuclei of the new phase may\nbe formed within the bulk metastable liquid.\nThis mechanism of formation of the solid phase is called homogeneous\nnucleation.\\cite{debenedettibook,kashchievbook}\nHomogeneous nucleation is an activated process since the formation of a\ncritical nucleus requires the surmounting of a free energy barrier.\nAfter that, the crystalline nucleus can grow (nucleation-and-growth mechanism).\nIn general, at moderate supercooling, the limiting step is the formation of\nthe critical cluster rather than the crystal growth. The most relevant\nquantity to characterize nucleation is the nucleation rate, {\\em i.e.} the\nnumber of nucleating clusters per unit time and volume.\n\nWater freezing is arguably the most important liquid-to-solid transition. \nFor example, ice formation in atmospheric clouds is a key factor to the global radiation \nbudget and to  \nclimate change.\\cite{review_ice_formation_clouds_2005,baker97,demott10}\nWater freezing is also a big issue, for instance, in the cryopreservation of cells and tissues.\\cite{cryopres}\nMoreover, ice formation is relevant to microbiology\\cite{hirano00},\nfood industry\\cite{maki74,bacterial_ice_nucleation}, materials \nscience\\cite{michaelides07}, geology\\cite{weathering_book} \nor physics.\\cite{pruppacher1995,debenedettibook,taborek,knopf11,nature_valeria,galli_mw,riechers13}\n\nDespite its great importance, our understanding of water freezing is far from \ncomplete. Not even homogeneous nucleation, the simplest \nconceivable mechanism by which ice can be formed, is fully understood.\nOne of the reasons for this is the need to perform experiments \nwith small droplets (10-100$~\\mu$m) \nto avoid heterogeneous nucleation.\\cite{kramer1999,stockel2005,murray2010,mishima10}\nThis, and the time that the droplets can be stabilized, sets the order of magnitude\nthat can be probed for the nucleation rate, $J$. Thus, experimental measurements for \n$\\log_{10} (J \/$(m$^{-3}$s$^{-1}$)) typically \nrange between $4$ and and $14$ . This corresponds to a temperature window spanning from 239~K to 233~K, \nthe latter often referred to as ``homogeneous nucleation temperature''.\\cite{debenedetti03} \nOur knowledge of the nucleation rate outside this temperature window is limited\nto extrapolations based on CNT.  Such extrapolations\nmust be taken with care since the uncertainties in the nucleation rate and the\nnarrow range of temperatures for which J can be measured lead to important\ndifferences in the estimated value of the interfacial free energy and\/or the\nkinetic prefactor.\\cite{murray2010}\nMoreover, it has not been possible so far to observe a \ncritical ice nucleus in experiments because critical nuclei \nare relatively small and short-lived. Therefore, we only have \nestimates of the critical cluster size based on experimental measurements\nof $J$.\\cite{pruppacher1995,furukawa1996,jeffery1997,kramer1999,kiselev01,micelas}\nThe purpose of this paper is to fill these gaps by obtaining\nthe first estimate of the size of the critical\ncluster and of the nucleation rate at high temperatures which \nis not entirely based on theoretical extrapolations from measurements\nat low temperatures. We will make use of computer simulations\nto achieve these goals.\n\nComputer simulations are a valuable tool to investigate nucleation\\cite{S_1997_277_01975,Nature_2001_409_1020} since they\nprovide a microscopic description of the process.\nIt is therefore somehow surprising that the number of simulation studies dealing\nwith ice nucleation is rather small.\\cite{sear2012} On the one hand, it has\nbeen shown that ice nucleation can occur spontaneously (without the aid of \nspecial simulation techniques) when \nan electric field is applied\\cite{kusalik_electric_field}, when  crystallization is assisted by a \nsubstrate\\cite{koga01,moore10} or by an interface\\cite{subsurface}, when coarse-grained models with accelerated dynamics are \nsimulated at high supercoolings,\\cite{nature_valeria,valeria_jcp_2010,valeria_pccp_2011} or when small systems \nare simulated\\cite{matsumoto02,PhysRevLett.88.195701,st2_recent}. \nOn the other hand, if nucleation does not\nhappen spontaneously, rare event techniques must be used. \nThe number of such works is limited and the agreement\nbetween different groups is not entirely satisfactory.\nRadhakrishnan and\nTrout\\cite{trout03,trout_prl_2003}, Quigley and Rodger\\cite{quigley08} and Brukhno et\nal.\\cite{anwar_tip4p_nucleation}  determined\nthe free energy barrier for the formation of ice critical clusters with the\nTIP4P water model at 180~K (50 degrees below\nthe model's melting temperature), but mutually consistent results were not found.\nReinhardt and Doye\\cite{doye12} and Li {\\em et al.}\\cite{galli_mw} evaluated\nthe nucleation rate of the mW model at 55 K below freezing finding a \ndiscrepancy of \nsix orders of magnitude. \nVery recently, \nReinhardt {\\em et al.} investigated ice nucleation at moderate supercoolings,\\cite{doyejcp2013} \nto estimate the free energy of formation of small pre-critical clusters.  \nIt is almost certain that more ice nucleation studies are on the way and,\nhopefully, the discrepancies will become smaller.\n\nNone of the studies mentioned in the previous paragraph deal with large systems at moderate \nsupercoolings like the present investigation does. \nBy supercooling, $\\Delta T$, we mean \nthe difference between the melting temperature and the temperature of interest. \nNote that the melting temperature of a model does not necessarily coincide with \nthe experimental melting temperature or with the melting temperature of other models.\nIn this work we determine, by means of\ncomputer simulations, the size of critical ice clusters and the nucleation rate \nfor $\\Delta T$ ranging from 15 to 35~K. \n\\edu{In this way we provide, for the first time, nucleation rates for $\\Delta T$ lower\nthan 35 K, where experimental measurements  are not currently feasible (CNT based\nestimates of $J$ can in principle be made for any supercooling but, to the best of our knowledge, \nthere are no such estimates available  for $\\Delta T < 30 K$).\\cite{pruppacher1995,kashchievbook,murray2010}}\nOur simulations predict that for $\\Delta T < 20$~K\nit is impossible that homogeneous ice nucleation takes place.\nTherefore, ice must necessarily nucleate heterogeneously \nfor supercoolings lower than 20~K.\nMoreover, we can directly compare our results for the \nlargest studied supercoolings to the experimental measurements. \nWe find, within uncertainty, a good agreement \nwith experimental nucleation rates. \nWe \npredict that the radius of the critical cluster goes from $\\sim$40~\\AA (8000 molecules)\nat $\\Delta T$ {\\em ca.} 15~K to $\\sim$17~\\AA (600 molecules)\nat $\\Delta T$ {\\em ca.} 35~K.   \nWe also estimate the surface free energy via CNT. \nWe obtain, in agreement with \npredictions based on experimental measurements,\\cite{pruppacher1995,zobrist07,alpert11} that the surface free energy decreases \nwith temperature. An extrapolation of the interfacial free energy to the melting\ntemperature gives a value of $\\sim$29~mN\/m, in reasonable \nagreement with experimental results\\cite{gamma_exp}, and with\ncalculations by simulation.\\cite{gammadavid}\n\nWe use two simple, yet realistic, water models; namely \nTIP4P\/2005\\cite{TIP4P2005} and TIP4P\/Ice\\cite{TIP4PICE}. The \nmelting temperature\\cite{TIP4P2005,TIP4PICE} and the ability of these models to \npredict properties of real water has already been well established.\\cite{vega11}\nThe results obtained for both water models are quite similar provided that they are compared at the\nsame $\\Delta T$.\n\n\n\n\\section{Methodology}\n\nTo evaluate the size of critical ice clusters we follow a similar approach\nto that proposed by Bai and Li\\cite{bai06} to calculate the solid-liquid\ninterfacial energy for a Lennard-Jones system. They employ spherical crystal\nnuclei embedded in the supercooled liquid and determine the temperature at\nwhich the solid neither grows nor melts.\nThe key issue of this methodology is that determining the melting temperature\nof a solid cluster embedded in its corresponding supercooled liquid water is\nequivalent to the determination of the critical size of the cluster for a\ncertain given temperature.\nThus, in a sense, this methodology can be regarded as the extension to nucleation\nphenomena of the well known direct coexistence technique.\\cite{woodcook2}\nA similar method was applied to water by Pereyra et al.\\cite{carignano} They\ninserted an infinitely long (through periodical boundary conditions) ice\ncylinder in water and determined the melting temperature of the\ncylinder.\nRecently, the approach of Bai and Li has been used to investigate  \nthe nucleation of clathrate \nhydrates.\\cite{jacobson11,knott12}\n\nHere we shall implement this methodology to study a three-dimensional spherical\nice cluster embedded in supercooled water. This follows closely the\nexperimental situation where the incipient ice embryo is fully immersed into\nliquid water.\nSuch {\\it brute force} approach requires very large systems\n(containing up to 2$\\times 10^5$ water molecules). \nHowever, molecular dynamics simulations can be efficiently parallelised so that it is \nnowadays possible to deal with such system size.\nThe methodology can then be implemented in a rather straightforward way, and\nis particularly useful at moderate supercooling,\nwhere other techniques (such as umbrella sampling\\cite{ARPC_2004_55_333,auer_frenkel}, Forward Flux Sampling\\cite{PRL_2005_94_018104} or \nTransition Path Sampling\\cite{ARPC_2002_53_0291}) may become numerically too expensive.\n\nOnce we calculate the critical cluster size we make use of \nCNT \\cite{ZPC_1926_119_277_nolotengo,becker-doring,kelton} in its version\nfor spherical clusters to \nestimate the surface free energy, $\\gamma$:\n\\begin{equation}\n\\label{ncrit}\n\\gamma=\\left(\\frac{3 N_{c} \\rho^2_s |\\Delta \\mu|^3}{32 \\pi}\\right)^{1\/3}\n\\end{equation}\nwhere  $\\rho_s$ is the number density of the solid and $\\Delta \\mu$ is \nthe chemical potential difference between the metastable liquid and the solid at the temperature under consideration.\nThis expression allows us to obtain a value for $\\gamma$ associated to each cluster.\nCNT can also be used to estimate the height of the nucleation free-energy barrier,  $\\Delta G_c$: \n\\begin{equation}\n\\label{eq_G_cnt}\n\\Delta G_c  = \\frac{16 \\pi \\gamma^3}{3 \\rho^2_s |\\Delta \\mu|^2}.\n\\end{equation}\nOnce $\\Delta G_c$ is known, we can use the following CNT-based expression to evaluate the \nnucleation rate\\cite{auerjcp}:\n\\begin{equation}\nJ=Z f^{+} \\rho_f \\exp(-\\Delta G_{c}\/(k_B T))\n\\label{eqrate}\n\\end{equation}\nwhere $Z$ is the Zeldovich factor, $Z=\\sqrt{(|\\Delta G^{''}|_{N_c}\/(2\\pi k_B T))}$,\nand $f^{+}$ is the attachment rate of particles to the critical cluster.\nThe CNT form of the Zeldovich factor is \n\\begin{equation}\nZ=\\sqrt{|\\Delta \\mu|\/(6 \\pi k_B T N_c)},\n\\end{equation}\nwhich can be obtained from our calculations of $N_c$.\nWe follow Ref.~\\onlinecite{auerjcp} to calculate $f^{+}$ as a diffusion coefficient of the \ncluster at the top of the barrier: \n\\begin{equation}\nf^{+}=\\frac{<(N(t)-N_c)^2>}{2t}. \n\\label{eqattach}\n\\end{equation}\nTherefore, in order to obtain nucleation rates we combine CNT predictions with simulations\nof the critical clusters. \n\nBy using the methodology here described, the nucleation rate of clathrate hydrates \nhas been recently calculated.\\cite{knott12} The validity of this approach \nrelies on the ability of CNT to make good estimates of the free energy barrier\nfrom measured values of the critical cluster size. CNT is expected to work well for big \ncritical clusters. We are confident that the cluster sizes we deal with in this work \nare big enough for CNT to produce meaningful predictions. \nWe discuss why in Sec.~\\ref{validity}. \n\n\n\\section{Technical details}\n\n\\subsection{Simulation details}\nWe carry out $NpT$ GROMACS\\cite{hess08}  molecular dynamics simulations (MD)  \nof a system that consists of one spherical ice-Ih cluster surrounded by  supercooled water  molecules.  \nWe use two different rigid non-polarizable models of water:   \nTIP4P\/2005\\cite{TIP4P2005} and TIP4P\/Ice.\\cite{TIP4PICE}\nTIP4P\/2005 is a  model that provides a quantitative account of many water\nproperties\\cite{vega09,vega11} including not only the well known thermodynamic\nanomalies but also the dynamical ones.\\cite{pi09,gonzalez10}\nTIP4P\/Ice was designed to reproduce the melting temperature, the densities and\nthe coexistence curves of several ice phases. \nOne of the main differences between the two models is \ntheir ice Ih melting temperature at 1~bar: $T_m = 252$~K for TIP4P\/2005 and\n$T_m = 272$~K for TIP4P\/Ice.\nWe evaluate long range  electrostatic interactions using the smooth\nParticle Mesh Ewald method\\cite{essmann95} and  truncate both the LJ and real\npart of the Coulombic interactions at 9~\\AA. \nWe preserve the rigid geometry of the water model  by\nusing constraints. All simulations are run at the constant pressure of\n$p = 1$~bar, using an isotropic Parrinello-Rahman barostat\\cite{parrinello81}\nand  at constant temperature, using the velocity-rescaling\nthermostat.\\cite{bussi07}\nWe set the MD time-step to  3~fs.\n\n\\subsection{Order parameter}\nTo determine the time evolution of the cluster size, we use the \nrotationally invariant order parameters proposed by Lechner and Dellago, $\\bar{q_{i}}$.\\cite{dellago}\nIn Fig.~\\ref{q6q4} we show the $\\bar{q}_{4},\\bar{q}_{6}$ values for 5000~molecules of either liquid water, \nice Ih or ice Ic at 1~bar and 237~K for TIP4P\/2005. The cut-off distance  to identify neighbors  for the calculation\nof $\\bar{q_{i}}$ is $3.5$~\\AA\\ between the oxygen atoms. \nThis approximately  corresponds to the position of the first minimum of the \noxygen-oxygen pair correlation function in the liquid phase. \n\\begin{figure}[h!]\n\\includegraphics[width=0.4\\textwidth,clip=]{q6q4_1-108_T237K.eps}\\\\\n\\caption{Values of $\\bar{q}_6$ and $\\bar{q}_4$\\cite{dellago} \nfor 5000~molecules of the liquid phase (blue), of ice-Ih (red), and  of ice-Ic (green) \nat 237~K for the TIP4P\/2005 model.} \n\\label{q6q4}\n\\end{figure}\n\nFrom Fig.~\\ref{q6q4} it is clear that $\\bar{q}_{6}$ alone is enough to\ndiscriminate between solid-like and fluid-like molecules, as already suggested\nin Ref.~\\onlinecite{doye_tip4p_2005}. As a threshold to separate the \nliquid from the solid clouds in Fig.~\\ref{q6q4} we \nchoose $\\bar{q}_{6,t}=0.358$, represented as a horizontal\ndashed line in the figure.  \nThis threshold separates the liquid from both ice Ih and Ic. Therefore, \neven though we prepare the clusters with ice-Ih structure, ice-Ic molecules would \nbe detected as solid-like should they appear as the clusters grow.\nUnlike Refs.~\\onlinecite{ghiringhelli08} and \\onlinecite{li13} we do not consider\nas solid-like particles on the surface which are neighbor to solid-like particles. \nOnce molecules are labelled either as\nsolid or liquid-like, the solid cluster is found by means of a clustering\nalgorithm that uses a cut-off of 3.5~\\AA\\ to find neighbors of the same\ncluster. \n\n\\subsection{Initial configuration}\nWe prepare the initial configuration by inserting a spherical ice-Ih cluster\n(see Fig.~\\ref{cluster} for a cluster of 4648~molecules) into a configuration\nof supercooled water with $\\sim 20$~times as many molecules as the cluster. \n\\begin{figure}[h!]\n\\includegraphics[width=0.4\\textwidth,clip=]{cluster4648.eps}\n\\caption{Snapshot of a spherical ice-Ih cluster of 4648~molecules.} \n\\label{cluster}\n\\end{figure}\nTo obtain the cluster, we  simply cut a spherical portion of a large\nequilibrated  ice Ih crystal. \nNext, we insert the ice cluster in the supercooled liquid \nremoving the liquid molecules that overlap with the cluster.\nFinally, we equilibrate the system for about 0.2~ns at 200~K.\nThis time is long enough to equilibrate the cluster-liquid interface \n(see Supporting Information). \nWe then perform simulations for three different system\/cluster sizes labeled as H (Huge), L (Large) and B (Big) \n(see Table~\\ref{systemsize}). As far as we are aware, the studied system size are \nbeyond any previous numerical study of ice nucleation. Calculations were performed in the Spanish super-computer \nTirant. \nFor system H we use 150~nodes yielding 0.72~ns\/day; \nfor system L, 50~nodes at 1.5~ns\/day and, \nfor system B, 32~nodes at 4.7~ns\/day. \n\nOur order parameter allows us to correctly identify as solid-like the great\nmajority of the molecules belonging to the cluster shown in Fig.~\\ref{cluster}\n(4498 out of 4648). \nFig.~\\ref{cluster-po}(a) shows that indeed most molecules of the inserted ice cluster are detected as solid-like (red) \nas opposed to liquid-like (blue). Notice that most blue particles in Fig.~\\ref{cluster-po}(a) are located at the interface. \nThis is not surprising giving that our order parameter was tuned to distinguish between liquid-like and solid-like particles in the bulk.  \nFig. ~\\ref{cluster-po}(a) corresponds to the cluster {\\em just} inserted in the liquid. \nAfter 0.2~ns of equilibration our order parameter detects that the number of molecules in the cluster drops down to 3170. \nTo explain the origin of this drop we show in Fig.~\\ref{cluster-po}(b) \na snapshot of the 4648 inserted molecules after the 0.2~ns equilibration period. \nClearly, the drop comes from the fact that the outermost layer of molecules of the inserted cluster becomes liquid-like during equilibration.\nBy removing the liquid-like molecules from Fig.~\\ref{cluster-po}(b) one can easily identify again the hexagonal \nchannels typical of ice (Fig.~\\ref{cluster-po}(c)).    \nTherefore, the drop from 4648 to 3170~molecules in the ice cluster is due to the equilibration of the ice-water interface. \nThe size of the equilibrated clusters, $N_c$, is given in Table~\\ref{systemsize}. \n\n\\begin{figure}[h!]\n\\includegraphics[width=0.2\\textwidth,clip=]{cluster0.eps}(a)\n\\includegraphics[width=0.2\\textwidth,clip=]{cluster200ps.eps}(b)\\\\\n\\includegraphics[width=0.2\\textwidth,clip=]{cluster200psnaked.eps}(c)\n\\caption{Snapshot of the 4648~molecules inserted as an ice cluster just after insertion (a),\nand after 0.2~ns equilibration (b).  \nMolecules are colored in red if detected as solid-like and \nin blue if detected as liquid-like. In (c) only solid-like molecules of snapshot (b) are shown.} \n\\label{cluster-po}\n\\end{figure}\n\n\\begin{table}[h!]\n\\caption{Total\nnumber of molecules in the system, $N_t$ (ice cluster + surrounding liquid water molecules) \nand number of molecules of the inserted spherical ice cluster, $N_i$ for the \nthree configurations prepared. $N_c$ is the number of molecules in the ice cluster after equilibration of the interface.  \nThe radius of the equilibrated clusters $r_c$ in \\AA ~is also presented.}\n\\label{systemsize} \n\\centerline{\n\\begin{tabular}{ccccccc}\n\\hline \n System   &   $N_{t}$   &     $N_{i}$   & $N_{c}^{2005}$ &  $N_{c}^{Ice}$ & $r_{c}^{2005}$ & $r_{c}^{Ice}$  \\\\\n\\hline\nB   & 22712  & 1089  & \t\t600  & 600 & 16.7 & 16.8 \\\\\nL   & 76781  & 4648  & \t\t3170  & 3167 & 29.1 & 29.2  \\\\\nH   & 182585 & 9998  & \t\t7931  & 7926 &  39.5 & 39.7 \\\\\n\\hline \n\\end{tabular}}\n\\end{table}\n\nOnce the interface is equilibrated for $0.2$ ns, the number of molecules in the cluster grows\nor shrinks (depending on the temperature) at a much slower rate (typically\nrequiring several nanoseconds as it is shown in Fig.~\\ref{figlargest}). \nThe initial time in our simulations corresponds to the configuration equilibrated after 0.2~ns. \nWe run MD simulations of the system with the equilibrated interface at \nseveral temperatures below the bulk melting temperature of the model.\nThe objective is to find a temperature range within which the cluster\ncan be considered to be critical.\nThe temperature range is comprised between the lowest temperature at which the solid cluster\nmelts and the highest at which it grows.  \nWe monitor the number of molecules in the cluster \nand the global potential energy to find whether the cluster melts or grows.\n\n\\section{Results}\n\n\\subsection{Size of the critical clusters}\n\nIn Fig.~\\ref{figlargest} we represent the number of molecules in the ice cluster\nversus time for system H, TIP4P\/2005. \n\\begin{figure}[h!]\n\\includegraphics[width=0.45\\textwidth,clip=]{nbigvstime_7805.eps}\n\\caption{Number of molecules in the ice cluster versus time for system H and the TIP4P\/2005 potential. Results are shown for different temperatures as\nindicated in the legend.} \\label{figlargest}\n\\end{figure}\nDepending on the temperature the cluster either grows (230~K and 235~K) or\nshrinks (240~K).  The highest temperature at which the cluster grows is 235~K\nand the lowest temperature at which it melts is 240~K. Hence, a cluster of\n$\\sim$7900~molecules (as detected by our order parameter) is critical at\n$237.5 \\pm 2.5$~K. \nAn analogous result can be obtained by monitoring the potential energy of the\nsystem as a function of time.\n(see Supporting Information).\nA decrease in the energy corresponds to the cluster's growth \nwhereas an increase in the energy corresponds to its melting. \nBy doing this analysis for both models (TIP4P\/2005 and TIP4P\/Ice ) and for the\nthree cluster sizes (H, L, and B ), \nwe obtain the results summarized in Table~\\ref{tabcluster}. \n\n\\begin{table}[h!]\n\\caption{\nWe report the\ntemperature ($T$ in K) for which the prepared ice clusters are found to be  critical,  \nthe supercooling ($\\Delta T$ in K) for the corresponding water model, \nthe chemical potential difference between the fluid and the solid ($\\Delta \\mu$ in kcal\/mol), \nthe liquid-solid surface free energy ($\\gamma$ in mN\/m) estimated from Eq.~\\ref{ncrit},\nand the nucleation free energy barrier height ($\\Delta G_{c}$ in $k_B T$) estimated from Eq.~\\ref{eq_G_cnt}.  \n\\label{tabcluster}} \n\\centerline{\n\\begin{tabular}{ccccccccc}\n\\hline \n  Model & System         & $N_c$ & $T$  & $\\Delta T$ & $\\Delta \\mu$ & $\\gamma$ & $\\Delta G_{c}$\\\\\n\\hline\nTIP4P\/2005 & B &  600  & 222.5  & 29.5   &  0.114      &   20.4    &   77 \\\\\nTIP4P\/2005 & L &  3170 & 232.5  & 19.5   &  0.080      &   24.9    &  275 \\\\\nTIP4P\/2005 & H &  7931 & 237.5  & 14.5   &  0.061      &   25.9    &  515 \\\\\n\\hline\nTIP4P\/Ice & B  & 600  & 237.5   & 34.5   &  0.133      &   23.6    &   85 \\\\\nTIP4P\/Ice & L  & 3167 & 252.5   & 19.5   &  0.083      &   25.4    &  261 \\\\\nTIP4P\/Ice & H  & 7926 & 257.5   & 14.5   &  0.063      &   26.3    &  487 \\\\\n\\hline \n\\end{tabular}}\n\\end{table}\n\n\nFor the temperatures explored in this work (from about 15~K to 35~K below the\nmelting temperature of both TIP4P\/2005 and TIP4P\/Ice) the size of the ice\ncritical cluster ranges from nearly 8000 (radius of 4 nm) to about 600 molecules (radius of 1.7 nm). \nThis compares reasonably well with a \ncritical cluster radius of $\\sim$ 1.3 nm obtained by \napplying CNT to \nexperimental measurements at a supercooling of about 40 K.\\cite{micelas,kiselev01} \n\\edu{Our results are also consistent with CNT based estimates of the critical size at lower\nsupersaturations.\\cite{bogdan,kashchievbook} For instance, in Fig. \n15.7 of Ref. \\cite{kashchievbook}, a critical\ncluster size ranging from 1000 to 300 molecules is predicted for 25 K $<\\Delta T <$ 30 K.} \nAn interesting remark is\nthat the temperatures of the TIP4P\/Ice are basically shifted 20~K above the\nthe corresponding ones for TIP4P\/2005 with the same nucleus size. \nThis is precisely the difference between the melting temperatures of both\nmodels and, thus, the supercoolings are very similar for a given ice cluster size in both models.\nThis is more clearly shown in Fig.~\\ref{masterfigure} where the size of\nthe critical cluster is plotted as a function of the the difference between the\nmelting temperature of the model and the temperature of interest $\\Delta T=T_m-T$.\nWe observe that, within our error bar, the critical cluster size of both models \nscales in the same way with respect to their melting temperatures. This is not so surprising since \nTIP4P\/2005 and TIP4P\/Ice present a similar charge  distribution and mainly  differ in \nthe choice of the potential parameters.\n\\begin{figure}[h!]\n\\includegraphics[width=0.45\\textwidth,clip=,angle=0]{new_nc_vs_deltat.eps}\n\\caption{Critical cluster size versus $\\Delta T$ for the studied water models.\nNotice that the points corresponding to both models at low supercooling are essentially on top \nof each other.}\n\\label{masterfigure}\n\\end{figure}\n\nIn previous works\\cite{vega07,vega09,vega11} we observed that, for a number of properties, \nthe values of TIP4P\/2005 lie in the middle of the values obtained for TIP4P and TIP4P\/Ice.\nTherefore, it is expected that TIP4P gives similar results to TIP4P\/2005 and TIP4P\/Ice regarding\nthe dependence of $N_c$ with $\\Delta T$. \nMatsumoto et al.\\cite{matsumoto02} studied ice nucleation at 230~K and a density of 0.96~g\/cm$^3$ using the\nTIP4P model. This thermodynamic state point corresponds to a pressure of about -1000~bar and $\\Delta T$ 5 K.\\cite{sanz_prl} \nBy extrapolating the data of Fig.~\\ref{masterfigure} to $\\Delta T=5 K$\none gets a critical cluster of the order of hundreds of thousand molecules. \nTherefore, \nit is likely that the results obtained by Matsumoto et al.\\cite{matsumoto02}, although pioneering \nand useful to learn about the \nice nucleation pathway, may suffer from system size effects and may not be valid\nto estimate either the size of the critical cluster or the nucleation rate. \n\n\nIn an important paper, Koop et al.\\cite{koop_nature} showed that the\nhomogeneous nucleation rate (and therefore the temperature of homogeneous\nnucleation) of pure water and of water solutions can be described quite well by\na function that depends only on the water activity.\nThis conclusion has been confirmed in more recent experiments.\\cite{knopf11}\nAlthough the nucleation rate for an aqueous solution is the same as for pure\nwater, the freezing points are different. One is then tempted to suggest that\nthe size of the critical cluster at the homogeneous nucleation temperature\ncould be the same for pure water and for aqueous solutions. Moreover, the fact\nthat thermodynamics is sufficient to predict the rate seems to indicate that\nthe water mobility is also determined by the free energy of water. A\nmicroscopic study of the relationship between crystallization rates, structure\nand thermodynamics of water which may explain the empirical findings of Koop\nand coworkers has recently been presented in Ref.~\\onlinecite{nature_valeria}.\n\n\\subsection{Interfacial free energy and free energy barrier}\n\nOnce the size of the critical cluster is known, \none can use Eq.~\\ref{ncrit} to estimate the solid-liquid interfacial free energy. \nSince ice density changes little with temperature\\cite{mendu_eos}, the density at coexistence is used in our calculations\n($\\rho_{m,TIP4P\/Ice}$=0.906 g\/cm$^3$ and $\\rho_{m,TIP4P\/2005}$=0.921 g\/cm$^3$). \nFor most substances it is possible to approximate $\\Delta \\mu$ by $\\Delta h_m\n(T_m-T)\/T_m$, where $\\Delta h_m$ is the melting enthalpy and $T_m$ is the\nmelting temperature. For water, however, this may not be a good approximation because $\\Delta\nh$ significantly changes with temperature as a manifestation of the anomalous\nsharp increase of the  heat capacity of water as temperature\ndecreases.\\cite{jeffery1997,Kumar05062007} Hence, one needs to do a proper\nevaluation of the chemical potential difference between both phases to get the\nsurface free energy from Eq.~\\ref{ncrit}. \nWe have calculated $\\Delta \\mu$ at every temperature by means of standard\nthermodynamic integration \\cite{frenkel96} from the coexistence \ntemperature, at which $\\Delta \\mu = 0$. \nIn Table~\\ref{tabcluster} we report the values we obtain for $\\Delta \\mu$\nand $\\gamma$. \n\nFirst of all we note that $\\gamma$ decreases with temperature for both models.\nThis is in qualitative agreement with experimental estimates of the behavior\nof $\\gamma$ with $T$.\\cite{pruppacher1995,zobrist07,alpert11} A more quantitative comparison is\nnot possible in view of the large discrepancies between different estimates\n(see Fig.~10 in Ref.~\\onlinecite{pruppacher1995}).  \nMotivated by the fact that the interfacial free energy can only be measured at\ncoexistence, we extrapolate our results to the melting temperature. \nTo do that, we take the two largest clusters and evaluate the slope of\n$\\gamma(T)$.\nWe get a value for the slope of $\\sim$0.18~mN\/(m~K) for both models, in\nvery good agreement with a recent calculation for the TIP4P\/2005\nmodel.\\cite{doyejcp2013} \nWith a linear extrapolation we get a value for $\\gamma$ at $T_m$ of\n$\\sim$28.7~mN\/m for both models, which can be compared to experimental\nmeasurements.\nIn contrast with the vapor-liquid surface tension, the value of\n$\\gamma$ for the solid-fluid interface is not well established.\nExperimental values range from 25 to 35~mN\/m.\\cite{pusztai2002} \nOur calculated data for $\\gamma$ at coexistence lies in the middle of that\nrange, so our models predict a surface free energy which is consistent with\ncurrent experimental data.\nWe now compare our estimated $\\gamma$ to direct calculations from simulations\nusing a planar interface.  The value of $\\gamma$ depends on the plane in\ncontact with the liquid. Since the cluster used here is spherical \nwe shall compare with the average of the values obtained for the basal and prismatic planes. \nDavidchak et al. computed $\\gamma$ for a planar fluid-solid interface using two models similar to those\nused in this work: TIP4P and TIP4P-Ew.\nFor TIP4P, in an initial publication the authors reported a value of  $\\gamma\n= 23.9$~mN\/m\\cite{gammadavid_old} that was later on modified (after improving\ntheir methodology) to $\\gamma = 26.5$~mN\/m.\\cite{gammadavid} For the TIP4P-Ew\\cite{horn04}\nDavidchak et al. reported (using the improved methodology) a value of\n$27.6$~mN\/m.\\cite{gammadavid} TIP4P-Ew is known to predict water properties in\nrelatively close agreement to those of TIP4P\/2005. Therefore, our results are also\nconsistent with the calculations reported in the literature for similar models.\nTo conclude, our values of $\\gamma$ seem to be\nreasonable estimates of the interfacial free-energy of the planar ice-water\ninterface.\n\nTo estimate the height of the nucleation free-energy barrier \nwe make use of Eq.~\\ref{eq_G_cnt}.\nOur results are summarized in Table~\\ref{tabcluster}. \nIn view of the height of the nucleation barrier for the clusters of systems\nL and H, around 250 and 500 $k_B T$ respectively, it seems virtually impossible\nto observe homogeneous nucleation of ice for supercoolings lower than \n20~K.  \nThe height of the nucleation barrier provides an estimate of the \nconcentration of critical clusters in the metastable fluid as \n$\\rho_f \\exp(-\\Delta G_{c}\/(k_B T))$, where $\\rho_f$ is the number density \nof the fluid. \nFor $\\Delta G_{c} = 250$~$k_B T$, one critical cluster would appear on average\nin a volume $\\sim$10$^{60}$ times larger than the volume of the whole\nhydrosphere.  From the values of $\\Delta G_{c}$ of  Table~\\ref{tabcluster}  we may infer why \nspontaneous ice nucleation has never been observed in previous studies of\nsupercooled water with the TIP4P\/2005 model\\cite{abascal10,abascal11,pettersson11}. Our \nresults show that the free energy\nbarrier for nucleation even for temperatures as low as 35~K below melting is\nstill of about 80 $k_B T$. This is much larger than  the typical barrier found in studies where spontaneous \ncrystallization occurs in brute force simulations \\cite{hs_filion,lundrigan:104503} (about $18 k_B T$). \nIt is worth mentioning that neither Shevchuk and  Rao\\cite{rao} nor Overduin and Patey\\cite{patey_sc} \nfind any evidence of ice nucleation in TIP4P models after runs of several microseconds which is consistent with the\nresults of this work. \nOur results may be of great interest to studies in which the competition between the \ncrystallization time and the equilibration time of water is crucial\\cite{limmer13,poole13,liu12}.  \n\n\n\\subsection{Nucleation rate}\n\nAlthough the free energy barriers alone provide a strong indication that ice\ncan not appear on our planet  via homogeneous nucleation at moderate\nsupercoolings ($\\Delta T < 20$~K), it is worth calculating the nucleation rate,\n$J$, to confirm such statement. \nThe nucleation rate takes into account not only the concentration of the \nclusters but also the speed at which these are formed. \nMoreover, the supercoolings for the smallest clusters we investigate are\ncomparable to those where most experimental measurements of $J$ have been made\n($\\Delta T \\sim$~35~K).\\cite{taborek,pruppacher1995,knopf11,riechers13,pruppacher_book}\n\nTo calculate the nucleation rate we use Eq.~\\ref{eqrate}. \nFirst, we compute $f^{+}$ from Eq.~\\ref{eqattach} by running 30 simulations of\nthe cluster at the temperature at which it was determined to be critical. We\nmonitor $(N(t)-N_c)^2$ and average it over all the runs.  In Fig.~\\ref{attach}\nwe plot $<(N(t)-N_c)^2>$ versus time for the system L, TIP4P\/2005.\nFrom the slope at long times we can infer $f^{+}$.\\cite{auerjcp} We get \n$f^{+} = 70\\cdot10^9$~$s^{-1}$. The Zeldovich factor for this particular case is $1.77\\cdot10^{-3}$, \nand the density of the liquid is 0.977~g\/cm$^3$. With this, we have all the ingredients needed\nto calculate the nucleation rate via Eq.~\\ref{eqrate}. The final result for this \ncase is $\\log_{10} (J \/$(m$^{-3}$s$^{-1}$))=-83. \n\nThe same procedure is used to calculate the nucleation rate for the rest of the\nsystems described in Table~\\ref{systemsize}. The results for the nucleation\nrate as a function of the supercooling are presented in Fig.~\\ref{rate}\nand compared to the experimental measurements of Pruppacher\\cite{pruppacher1995}\nand Taborek\\cite{taborek}.\nThe horizontal dashed line corresponds to the nucleation rate required for the\nappearance of one critical cluster in the volume of Earth's hydrosphere in\nthe age of the universe, which we call ``impossible nucleation rate''.\nThe vertical line shows at which temperature the impossible nucleation rate\nline intercepts the upper limit of our error bars (grey and orange shadows for\nTIP4P\/2005 and TIP4P\/Ice respectively). \nIn view of this figure we can confidently claim what the free energy barriers\npreviously hinted: it is impossible that ice nucleates homogeneously in our\nplanet for $\\Delta T < 20$~K. \nIn other words, heterogeneous nucleation must take place in order for water to\nfreeze for supercoolings lower than 20~K. This is consistent with the fact\nthat, when heterogeneous nucleation is suppressed, moderately supercooled water\ncan remain  metastable long enough for its thermodynamic properties to be\nmeasured.\\cite{speedy76,hare_sorensen_eos_supercooled_water,mishima10,debenedetti03,holten12} \nFrom our results it is also clear that ice formation should not be expected in\nbrute force molecular dynamics simulations at moderate supercoolings (provided\nthat the system is large enough not to be affected by finite size\neffects).\\cite{matsumoto02} To  observe  ice formation in brute force\nsimulations the nucleation rate should be higher than $\\log_{10} (J\n\/$(m$^{-3}$s$^{-1}$)) = 32 (this number is obtained assuming the formation of\nice after running about 100~ns in a system of about 50nm$^3$, which are typical\nvalues in computer simulations of supercooled water).  Notice also that the\nmaximum in the isothermal compressibility at room\npressure\\cite{abascal10,abascal11} found at about $\\Delta T=20$~K for the\nTIP4P\/2005 model can not be the ascribed to the transient formation of ice\nas the nucleation rate of ice at this temperature is negligible. \n\nAnother interesting aspect of Fig.~\\ref{rate} is the comparison with\nexperiment.  Both models give nucleation rates that reproduce the experimental\nmeasurements within the uncertainty of our method.  This excellent result\nbrings confidence in  the ability of the selected models to predict relevant\nquantities for the nucleation of ice such as the nucleation rate, the critical\ncluster size, and the surface free energy. \n\n\\edu{We also include in Fig.~\\ref{rate}\na green dashed line that corresponds to\nthe CNT based estimates of $J$ shown in Fig. 13.6 of Ref. \\cite{kashchievbook} The agreement \nbetween CNT, simulations and experiments is quite satisfactory.   \nTo the best of our knowledge, there are no CNT estimates of $J$ available for\nsupersaturations lower than 30 K to compare our results with.\\cite{pruppacher1995,kashchievbook,murray2010}} \n\n\\begin{figure}[h!]\n\\includegraphics[width=0.45\\textwidth,clip=,angle=0]{figura30_trajs.eps}\n\\caption{$<(N(t)-N_c)^2>$ versus time for configuration L, TIP4P\/2005. The  attachment rate $f^{+}$ is obtained as half the\nvalue of the slope. The curve above is obtained as an average over 30 trajectories. \nIn approximately half of these trajectories the critical \ncluster ended up growing, whereas it eventually melted in the other half. \n}\n\\label{attach}\n\\end{figure}\n\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.45\\textwidth,clip=,angle=0]{ultimissimamitica.eps}\n\\caption{Nucleation rate as a function of the supercooling.\n\\edu{Symbols correspond to our simulation results and to \nexperimental measurements as indicated in the legend. The green dashed line corresponds\nto CNT estimates of $J$.\\cite{kashchievbook}} The grey and orange shadows \nrepresent the estimated error bars for TIP4P\/2005 and TIP4P\/Ice respectively interpolated\nby splines. The horizontal \\edu{dotted} line indicates the rate given by the growth of one cluster in\nthe age of the universe in all the water of the Earth's hydrosphere. The vertical \n\\edu{dotted} line indicates the supercooling below which homogeneous nucleation is impossible.}\n\\label{rate}\n\\end{figure}\n\nBy using Forward Flux Sampling\\cite{PRL_2005_94_018104}, Li {\\em et al.}\ndetermined $J$ for the mW model of water for temperatures between 35 and 55~K\nbelow the model's melting temperature.\\cite{galli_mw} Since we are\ninterested in ice nucleation at moderate supersaturation, our study deals with\nlower supercoolings \n($14.5$~K$ < \\Delta T < 34.5$~K). \nNonetheless, our highest supercooling (34.5~K) is very close to the lowest one of  \nLi {\\em et al.} (35~K) so we can compare both results. \nThe values of Li {\\em et al.} for $J$ are 5-8 orders of\nmagnitude below the experimental ones when compared at the same absolute\ntemperature (the deviation increases when the comparison is made at the same degree of\nsupercooling). The nucleation rates calculated in this work for TIP4P\/2005 and\nTIP4P\/ice are similar (although slightly larger) to those for the mW model. Initially this may\nappear surprising as the mW model is a coarse grained model of water with no\nhydrogens, which makes its dynamics faster than that of both real water and TIP4P-like models.\\cite{mw}\nHowever the free energy barrier of mW may be larger, compensating this kinetic effect. \nIn fact the interfacial free energy of mW has been found\\cite{galli_mw} to be $\\gamma=31$~mN\/m \n(larger than the values found in this work for TIP4P\/2005 and TIP4P\/Ice).\nThis high value of $\\gamma$ may be partially compensated by a significant overestimate\nof the ice density of this model (0.978 g\/cm$^3$ to be compared to the experimental result 0.91 g\/cm$^3$).\nThe net balance is that the values of $J$ of the mW model are similar,\nalthough somewhat lower, than those for TIP4P\/2005 and TIP4P\/ice.\n\nAs for the size of the critical cluster, we find that it is of about 600\nwater molecules for TIP4P\/Ice at 237.5~K ($\\Delta T = 34.5$~K). \nLi {\\em et al.} have reported a critical cluster size of about 850\nmolecules for the mW model at 240~K ($\\Delta T = 35$~K).\nBoth results are compatible since Li {\\em et al.} include in the ice cluster molecules which \nare neighbor to the solid cluster, and we do not. \nIn summary, our results for TIP4P\/2005 and TIP4P\/Ice \nare consistent with Li's {\\em et al.} for mW. \n\n\n\n\\section{Discussion}\n\\subsection{Validity and possible sources of error}\n\\label{validity}\n\nThe methodology we have used is subject to two main error sources: the determination\nof the cluster size and the location of the temperature at which the clusters\nare found to be critical. Moreover, our approach relies on the validity of \nCNT. In the following paragraphs we discuss the extent to which our \nresults may be affected by these issues.  \n\nIn nucleation studies, the size of the largest solid cluster is usually\nconsidered a good reaction coordinate.\nTo identify the cluster, we first need to distinguish between liquid-like and solid-like molecules.\nThe chosen criterion should be able to identify the majority of molecules of the bulk solid as solid-like, \nand the majority of molecules of the  bulk fluid  as liquid-like. \nOne could in principle find several criteria  that successfully perform this task. \nHowever,  when interfaces are present in the system \n(as in the case of a solid-liquid\\cite{hs_filion} \nor a solid-vapor\\cite{maria06b,conde08} interface) depending on the chosen criterion \none might assign differently the interfacial molecules (see for instance \nReferences~\\onlinecite{doye12} and \\onlinecite{galli_mw} \nfor an illustration of this problem for the mW water model).  \n\nHow does the choice of a criterion to distinguish liquid from solid-like molecules affect our results? \nWhether the cluster grows or shrinks for a given temperature does\nnot depend on the particular choice of the order parameter (see Supporting Information).\nThe same trend can be obtained by monitoring \nglobal  thermodynamic properties of the system, such as the total potential energy (see Supporting Ingormation). \nTherefore,  the fact that the cluster shown in Fig.~\\ref{cluster-po} \nis critical at 232.5~K is independent on the particular choice of the  criterion to distinguish liquid from solid-like molecules.  \n\nA different problem arises if one asks the question: how many ice molecules  are \npresent in Fig.3b? Different criteria provide different answers  even though the configuration \npresented in Fig.3b is unique. \nSince the origin of this arbitrarity is due to the interfacial region, it is expected that\nthe arbitrarity will become smaller as the ice cluster becomes larger. \nHowever, for the system sizes considered in this work the interface region still matters. \nTo take this effect into account we have estimated the error bars in Fig.~\\ref{rate}\nconsidering an arbitrarity of 60\\% in the labeling of {\\em interfacial} molecules. \nThis would affect the value of $\\gamma$ by  7\\%, and the free-energy barriers height \nby up to 20\\%. \nAlthough this estimated error seems large, it is worth pointing out that differences between the free-energy barrier estimated by\ndifferent groups may be, in the case of water, much larger than that.\\cite{trout_prl_2003,trout03,quigley08,anwar_tip4p_nucleation} \nIn summary we  conclude that the liquid\/solid criterion chosen in this work provides reasonable estimates of \n$\\gamma$, and when used within the CNT framework allows to interpret our simulations results \nin a rather straightforward way. \n\nAnother important error source in the calculation of $J$ is the location of the temperature \nat which a cluster is critical. As we show in Fig.~\\ref{figlargest}, by performing runs at different temperatures we identify, within a certain\nrange, the temperature that makes critical a given ice cluster. \nWe assign the temperature in the middle of the range to the corresponding cluster, but the temperature\nthat really makes the cluster critical \ncould in principle be any other within the range. \nThis uncertainty \nhas a strong contribution to the error bars in Fig~\\ref{rate}, particularly at low supercoolings,\nwhere the variation of $J$ with $T$ is very steep.  \nThis error could, in principle, be easier to reduce than that coming from the arbitrarity in the determination\nof the number of particles in the cluster. One simply has to do more runs to narrow \nthe temperature range. However, these simulations are very expensive given the large system \nsizes we are dealing with. \nIt is interesting to point out that temperature control is also seen as a major\nerror source in experiments.\\cite{riechers13} \n\nOur results for $\\gamma$, $\\Delta G_c$, and $J$ rely on the validity of CNT. \nClassical Nucleation Theory is expected to break down for small clusters, when \nthe view of nucleation as a competition between bulk and surface free energies starts to be\nquestionable (in clusters of a few hundred particles most molecules are placed at\nthe surface). However, for the large cluster sizes investigated in this work \nit seems reasonable to assume that CNT works well. The satisfactory comparison \nof our estimate of $\\gamma$ with that obtained in simulations of a flat interface\\cite{gammadavid}\nis certainly encouraging in this respect. \nMoreover, we have applied the methodology described in this paper \nto calculate the nucleation rate of the mW water model\nand we get, within error, the same nucleation rate as in Ref.\\cite{galli_mw}  This is a very\nstringent test to our approach, given that in Ref\\cite{galli_mw} \na method that relies neither on CNT nor on the definition of the\ncluster size was used (Forward Flux Sampling). This comparison is made for a supercooling of 35 K, the deepest\ninvestigated in this work. \nFor lower supercoolings, where the critical cluster is larger,\nthe methodology is expected to be even more robust.\nThe advantage of the approach used here is that it allows to estimate (at a \nreasonable computational cost) critical cluster sizes and \nnucleation rates at low and moderate supercooling. \n\n\n\\subsection{Novelty}\n\nIn this paper\nwe provide values for the homogeneous nucleation rate of ice at moderate \nsupercoolings ($\\Delta T < 33$~K). For the first time, this is done without \nextrapolating from measurements at high supercoolings.\nThe experimental determination of $J$ is limited to a narrow temperature window \nat high supercoolings\n(between 233~K and 239~K). In that window, $J$ can be directly measured \nwithout introducing any type of approximation. \nIt only requires the knowledge of the droplet volume, the cooling\nrate and the fraction of freezing events.  \nDifferences in the value of $J$ \nbetween different experimental groups are\nrelatively small (between one and two orders of magnitude). \nTherefore, the experimental\nvalue of $J$ is well established for the narrow range of temperatures in which the current experimental\ntechniques can probe the nucleation rate.\\cite{taborek,pruppacher1995,knopf11,riechers13,pruppacher_book}\nTo obtain values of $J$ outside that temperature window \none can either extrapolate the data or make an estimate via CNT. \nAn extrapolation from such a narrow temperature window would not be very reliable because  $J$ changes sharply with T. \nIn turn, an estimate of $J$ based on CNT relies in the knowledge of the interfacial free energy. \nUnfortunately, our current knowledge of $\\gamma$ for the water-ice\ninterface is far from satisfactory in at least three respects. Firstly, the\ncalculated values of different groups using CNT differ\nsignificantly (see for instance Tables I and II in Ref~\\onlinecite{murray2010}). \nSecondly, the values obtained for $\\gamma$ from CNT seem to\nbe different from those determined for a planar ice-water interface at the\nmelting point (see for instance Fig.~8 in Ref~\\onlinecite{bartell94}).\nFinally, there is even no consensus about the\nvalue of $\\gamma$ for a planar interface at the melting point of water, a\nmagnitude that in principle could be obtained from direct experiments without\ninvoking CNT (values between 25 and 35~mN\/m have been reported). A look to\nFig.~10 of the classic paper of Pruppacher\\cite{pruppacher1995}\nis particularly useful. It shows the enormous\nuncertainty that exists at any temperature about the value of $\\gamma$ for the\nice-water interface. Since $\\gamma$ enters in the estimation of $J$ as\na power of three in an exponential term, the enormous scatter implies that,\nat this moment, there is no reliable estimate of the value of $J$ for moderately supercooled water arising from\nCNT.\nIn other words, you can get many different estimates of $J$ from the different\nestimates of $\\gamma$ shown in the paper by Pruppacher. \\edu{In addition, to the best of our knowledge, \nno one has estimated $J$ using CNT for supersaturations lower than 30 K. \\cite{pruppacher1995,kashchievbook,murray2010}}\n\n\\edu{Regarding the critical nucleus size}, it is not possible at the moment to \nmeasure it experimentally\nby direct observation. \nTherefore, the prediction of the critical cluster at moderate and experimentally accessible supercoolings\nis a novel result. \nSince the TIP4P\/2005 has been quite successful in describing a number of\nproperties of water (notably including the surface tension for the vapor-liquid\nequilibrium) we believe that the values reported here for $\\gamma$ and $J$ from our\nanalysis of the critical cluster are a reasonable estimate for the\ncorresponding values for real water.\n\n\n\\subsection{Summary and outlook}\n\nWe have studied homogeneous ice nucleation by means of computer simulations \nusing the TIP4P\/2005 and TIP4P\/Ice  water models. \nThis is the first calculation of the size of the critical cluster and \nthe nucleation rate at moderate supercoolings (14.5-35~K). \nBoth models give similar results when compared at the same supercooling. \n\nTo determine the size of the critical cluster, we use a numerical approach in\nthe spirit of direct coexistence methods.\nWe prepare an initial configuration by inserting a large ice cluster (about\n10000, 4600 and 1000~molecules) in an equilibrated sample of liquid water.\nThen, we let the interface equilibrate for 0.2 ns at 200~K.  Finally, we\nperform molecular dynamic runs at several temperatures to detect either the\nmelting or the growth of the inserted cluster by monitoring its size.  We find\nthat the size of the critical cluster varies from $\\sim$8000~molecules\n(radius$ = 4$~nm) at 15~K below melting to $\\sim$600 molecules (radius$ =\n1.7$~nm) at 35~K below melting.  \n\nWe use CNT to estimate\nthe interfacial free energy and the nucleation free energy barrier.\nOur predictions show that \nthe interfacial free energy decreases as the supercooling increases, in agreement\nwith experimental predictions. An extrapolation of the interfacial free energy to the melting\ntemperature gives a value of 29(3)~mN\/m, which is in reasonable agreement \nwith experimental measurements\nand with estimates obtained from computer simulations for TIP4P-like models.\nWe get free energy barriers higher than 250~kT\nfor supercoolings lower than 20~K. This strongly suggests that homogeneous ice nucleation\nfor supercoolings lower than 20~K is virtually impossible. \nWe confirm this by calculating the nucleation rate. To do that we compute, by means of molecular \ndynamics\nsimulations,  \nthe rate at which particles attach to the critical clusters. \nThese calculations show that, indeed, for supercoolings lower than 20~K it is impossible\nthat ice nucleates homogeneously. According to this prediction, ice nucleation must necessarily \nbe heterogeneous for supercoolings lower than 20~K. The nucleation rate we obtain at higher \nsupercoolings (30-35~K) agrees, within the statistical uncertainty of our methodology, \nwith experimental measurements. \n\nIt would be interesting to extend this work in several directions. Modifying the shape of the inserted \ncluster (inserting for instance a small crystal with planar faces) or even inserting a block of \ncubic ice Ic to analyse whether this cluster may be more stable as suggested by some studies\\cite{huang_bartell,murray2010} \nare interesting issues that deserve further studies. \nSecondly, it would be of interest to consider other water models, to analyse the possible similarities\/differences\nwith respect to nucleation of different potential models varying significantly either in the charge distribution \nas TIP5P\\cite{mahoney00} or in the way the tetrahedral order is induced as in the mW model.\\cite{mw}\nAnalyzing the behaviour at higher degrees of supercooling than those presented here is another interesting problem as \nwell as the determination of the growth rate of ice.\\cite{kusalik_growth_ice}\nWe foresee that all these issues  will be the centre of significant activity in the near future. \\\\ \n\n\n\n\n\\section{Supporting Information to ``Homogeneous ice nucleation at moderate supercooling from molecular simulation\"}\n\\subsection{Equilibration of the initial configuration}\n\nThe preparation protocol used  for all cluster sizes is the following.\nAfter having inserted the ice cluster in the supercooled liquid, we remove the liquid molecules \noverlapping with the solid ones. Next, we equilibrate the system for about 0.2~ns at 200 K.\nTo make sure that the chosen 0.2~ns is a proper equilibration time, long enough to allow \nfor annealing mismatches at the interface, we run a simulation \nstarting from the initial configuration at time zero. \nIn Fig.~\\ref{fig:purity}, we represent the cluster size for the Large system of\nthe TIP4P\/2005 model as a function of time.  \n\\begin{figure}[h!]\n\\centerline{\\includegraphics[clip,width=0.5\\columnwidth]{200_equilibrado.eps}}\n\\caption{Equilibration for the L system of the TIP4P\/2005 model. After an initial drop of the cluster size due to equilibration of the interphase, the cluster size changes very slowly with time.}\n\\label{fig:purity}\n\\end{figure}\nThe figure shows that, even though the dynamics is very slow at such low\ntemperatures,  the chosen equilibration time is long enough. This result is\nindependent on the chosen cluster size or water model potential.\n\n\\subsection{Choice of the order parameter to distinguish between liquid\/solid particles}\n\nThe use of an alternative order parameter to identify solid-like particles\n($\\bar{q}_{3}$) does not affect the observed response of the cluster to\ntemperature. This is shown in Fig.~\\ref{q3op}, where the number of particles in\nthe cluster is monitored with two different order parameters for three\ndifferent temperatures.  Both order parameters allow to conclude that the\ninserted cluster is critical between 235 and 240 K.\nObviously, the number of particles that belong to the cluster does depend on the order parameter. \nThe order parameter we use in this work should at least work well for the inner particles \ngiven that, according to Fig.~1 of the main text, it is able to \ndiscriminate between bulk liquid and bulk solid particles. The main ambiguity in the number of particles\nbelonging to the cluster comes from those particles that lie in the interface. \nIn view of Fig.~3 of the main text, it seems that our order parameter is doing reasonably well \nin identifying such particles either. Nonetheless, we have considered an error as large\nas 60\\% in the identification of the {\\em interfacial particles} to estimate the error of the nucleation \nrate. In this way the unavoidable ambiguity in the determination of the cluster size is reflected\nin the error bar of $J$.\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.5\\columnwidth]{comparetwoorderparams.eps}\n\\caption{Number of particles in the cluster versus time for system size H, TIP4P\/2005 model. Black curves correspond to 230 K, red ones to 235 K and green to 240 K. \nSolid lines correspond to the analysis made with the order parameter described in the main text. Dashed lines correspond to the use of an alternative order parameter.  \nWith such order parameter particles are considered as neighbors if their oxygen atoms are closer than 3 ~\\AA\\ and are labelled as solid-like whenever \ntheir $\\bar{q}_{3}$ is larger than 0.28.}\n\\label{q3op}\n\\centering\n\\end{figure}\n\n\\subsection{Cluster size and potential energy versus time for all system sizes and model potentials studied}\n\nIn order to determine the temperature at which the cluster was critical, we\nevaluated the highest temperature at which the cluster grows and the lowest\ntemperature at which it melts. In Fig.~4 of the main text we represented the\nnumber of molecules in the ice cluster versus time for system H simulated with\nTIP4P\/2005 potential. In what follows, we present the results of the cluster\nsize versus time for all sizes (B,L and H) for both the TIP4P\/2005\n(Fig.\\ref{fig:cluster-energy-2005}) and TIP4P\/Ice water models\n(Fig.\\ref{fig:cluster-energy-ice}).  An analogous result could have been\nobtained by monitoring the potential energy of the system as a function of time\n(see the right panels at Figs.~\\ref{fig:cluster-energy-2005} and\n\\ref{fig:cluster-energy-ice}).\n\\begin{figure}[h!]\n\\includegraphics[clip,width=0.5\\columnwidth]{22712-clusters-TIP4P2005.eps}%\n\\includegraphics[clip,width=0.5\\columnwidth]{deltaE-661-tip4p2005.eps}\n\\includegraphics[clip,width=0.5\\columnwidth]{76781-clusters-TIP4P2005.eps}%\n\\includegraphics[clip,width=0.5\\columnwidth]{deltaE-4648-tip4p2005.eps}\n\\includegraphics[clip,width=0.5\\columnwidth]{182585-clusters-TIP4P2005.eps}%\n\\includegraphics[clip,width=0.5\\columnwidth]{deltaE-7805-tip4p2005.eps}\n\\caption{Left-hand panels: Number of molecules in the ice cluster versus time  for system B (top), L (middle) and H (bottom) simulated with the TIP4P\/2005 potential. \nRight-hand panels: energy difference (between the energy at time $t$ and the one at time zero) versus time.  Results are shown for different temperatures as indicated in the legend.}\n\\label{fig:cluster-energy-2005}\n\\end{figure}\n\\begin{figure}[h!]\n\\includegraphics[clip,width=0.5\\columnwidth]{22712-clusters-tip4pice.eps}%\n\\includegraphics[clip,width=0.5\\columnwidth]{deltaE-661-tip4pice.eps}\n\\includegraphics[clip,width=0.5\\columnwidth]{76781_clusters_tip4pice.eps}%\n\\includegraphics[clip,width=0.5\\columnwidth]{deltaE-4648-tip4pice.eps}\n\\includegraphics[clip,width=0.5\\columnwidth]{182585-clusters-tip4pice.eps}%\n\\includegraphics[clip,width=0.5\\columnwidth]{deltaE-7805-tip4pice.eps}\n\\caption{Same as Fig. \\ref{fig:cluster-energy-2005} but for TIP4P\/Ice instead of TIP4P\/2005.}\n\\label{fig:cluster-energy-ice}\n\\end{figure}\n\nThe energy is much less sensitive to changes in the cluster size than the order parameter. \nThis is due to the fact that the number of molecules in the cluster is a small fraction of the total number of molecules.  \nNonetheless, by making a linear fit to the time evolution of the energy\nwe obtain in all cases consistent results with the analysis based in the order parameter: \nthe cluster grows when the slope is negative and shrinks when it is positive. \n\n\\section{Acknowledgments}\nE. Sanz acknowledges financial support from the EU grant 322326-COSAAC-FP7-PEOPLE-2012-CIG \nand from the Spanish grant Ramon y Cajal. C. Valeriani acknowledges financial support\nfrom the EU grant 303941-ANISOKINEQ-FP7-PEOPLE-2011-CIG and from the Spanish grant Juan de la Cierva. \nFundings also come from MECD Project FIS2010-16159 and from CAM  MODELICO P2009\/ESP\/1691. \nAll authors acknowledge the use of the super-computational facility Tirant at Valencia  \nfrom the Spanish Supercomputing Network (RES), along with the technical support and the\ngenerous allocation of CPU time to carry out this project (through projects QCM-2012-2-0017,\nQCM-2012-3-0038 and QCM-2013-1-0047).\nOne of us (C. Vega) would like to dedicate this paper to the memory of Prof. Tomas Boublik.\nWe thank the three referees for their useful comments and to A. Reinhardt and J.P.K. Doye for sending us a preprint\nof their work prior to publication. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe reconstruction problem for ordered sets asks if it is possible to reconstruct the isomorphism type of\na given ordered set from its collection\n(the ``deck\")\nof one-point-deleted subsets.\nIn \\cite{Sanun}, Sands asked\nif ordered sets might even be reconstructible from\nthe the collection of subsets obtained by erasing single maximal elements (the ``maximal deck\").\nThe negative answer to Sands' question in\n\\cite{KRex}\ntogether with the paper \\cite{KRtow} were the starting point for serious investigation of\nthe reconstruction problem for ordered sets.\n(Reconstruction of graphs and other relations has a\nlonger history, see \\cite{BoHem,Manv,Pouinterdit,Stockwindmill}.)\nSince then, results that show reconstructibility given\ncertain types of information (for example, see \\cite{IlRaCG}) as well as\nresults on reconstructibility of certain classes of ordered sets (for example, see \\cite{RaSch})\nhave been proved. For a more comprehensive survey of results\nand references available to date, consider \\cite{JXsurvey,Schbook}.\n\n\n\nRecently in \\cite{Schexam} it has been shown that even the maximal and minimal decks together\nare not sufficient to reconstruct ordered sets.\nMoreover, it was shown in \\cite{Schexam} that there are families of $\\displaystyle{ 2^{O(\\sqrt[3]{n} )} } $\npairwise nonisomorphic ordered sets of size $n$ that\nall have equal maximal and minimal decks.\n\nThe desire to\nrely on only a limited, focused amount of information in a reconstruction proof,\nand the fact that almost all ordered sets are reconstructible from two identifiable maximal cards\n(see \\cite{Schrigid}, Corollary 3.10)\nmotivates extensions of Sands' question.\nWhat subsets of the deck have a reasonable chance to\neffect reconstruction? In \\cite{Schmoreexam} it was shown that the maximal deck plus the minimal deck plus one deck\nobtained by removing points of a rank $k$ that contains no extremal elements are not sufficient for\nreconstruction.\nThe examples in \\cite{Schmoreexam}\nare somewhat limited. They\ncould not be extended to an ambiguity with more than two sets.\nIt also appeared as if they could\nnot be extended to more than two maximal elements, two minimal elements and two elements of rank $k$\nor to more than one middle rank producing equal decks.\nImmediately two questions arise.\n\\begin{itemize}\n\\item\nAre ordered sets reconstructible from the maximal deck, the minimal deck and a deck obtained\nby removing points of rank $k$ if one of these decks has at least three cards?\n\n\\item\nAre ordered sets reconstructible from the maximal deck, the minimal deck and\ntwo decks that were obtained by removing points of ranks $k$ and $l$ with $k\\not= l $?\n\n\\end{itemize}\n\nIn this paper these two questions are answered negatively, even if the neighborhood\ndecks are equal and for any two isomorphic cards the neighborhoods of the\nremoved elements are also isomorphic.\nThe present examples provide new guidance as to what kind of\npartial information is at least needed to\nreconstruct ordered sets.\nIn particular, they show that information derived from\n``small\" ranks, and even from many small ranks, is not sufficient\nto effect reconstruction.\nAnalysis of the examples also leads to results that underscore the role of rigidity in\norder reconstruction (cf. Section \\ref{rigsect}).\nIdeas on what types of information to consider next are given in the conclusion.\n\n\n\\section{Basic Definitions and Preliminaries}\n\nAn {\\bf ordered set} is a set $P$ equipped with a reflexive, antisymmetric and\ntransitive relation $\\leq $, the order relation.\nThroughout this paper we will assume that all ordered sets involved are finite.\nElements $x,y\\in P$ are called {\\bf comparable} iff $x\\leq y$ or $y\\leq x$.\nAn {\\bf antichain} is an ordered set in which each element is only comparable to itself.\nA {\\bf chain} is an ordered set\nin which any two elements are comparable.\nThe {\\bf length} of a chain is its number of elements minus $1$.\nAn element $m\\in P$ is called {\\bf maximal} iff for all $x$ comparable to $m$ we\nhave $x\\leq m$. {\\bf Minimal} elements are defined dually.\nThe {\\bf rank} of an element $x\\in P$ is the length of the longest chain that has a minimal element\nas its smallest element and $x$ as its largest element.\n\n\nThe {\\bf dual} $P^d $ of an ordered set $P$ is the ordered set obtained by reversing all\ncomparabilities.\nThe {\\bf dual rank} of an element $x\\in P$ is the length of the longest chain that has a maximal element\nas its largest element and $x$ as its smallest element.\n\n\nA function $f:P\\to Q$ from the ordered set $P$ to the ordered set $Q$ is called {\\bf order-preserving}\niff for all $x,y\\in P$ we have that $x\\leq y$ implies $f(x)\\leq f(y)$.\nThe function $\\varphi :P\\to Q$ is called an {\\bf (order) isomorphism} iff $\\varphi $ is bijective, order-preserving\nand $\\varphi ^{-1} $ is order-preserving, too.\nAn order isomorphism with equal domain and range is called an {\\bf (order) automorphism}.\nAn ordered set with exactly one order automorphism (the identity)\nis called {\\bf rigid}.\n\n\n\n\nFor precise overall reconstruction terminology, cf. \\cite{Schbook,Schrigid,Schmoreexam}.\nFor the purposes of this paper, a {\\bf card} of an ordered set is a subset\nwith one point deleted.\nIf the deleted element is of rank $k$ we shall also call the card a {\\bf rank $k$ card}.\nThe set of all cards obtained by erasing elements of rank $k$ is called the\n{\\bf rank $k$ deck}.\nThe set of all rank $k$ decks is called the {\\bf ranked deck}.\nA rank $k$ card is {\\bf marked} iff there is a function that indicates the\nrank of each element in the original set.\nThe set of all marked rank $k$ cards is the {\\bf marked rank $k$ deck} and the\nset of all marked rank $k$ decks is the {\\bf marked ranked deck}.\nA {\\bf maximal card} is a subset in which a maximal element is erased\nand a {\\bf minimal card} is a subset in which a minimal element is erased.\nThe sets of maximal and minimal cards respectively are called the {\\bf maximal deck} and the\n{\\bf minimal deck}.\n{\\bf Marked maximal cards} are maximal cards for which there is a function that indicates which\nelements are maximal in the original set. The set of all marked maximal cards is called the\n{\\bf marked maximal deck}.\n{\\bf Marked minimal cards} and the {\\bf marked minimal deck} are defined dually.\nIsomorphic cards will also be called {\\bf equal cards}, because their isomorphism\nclasses are equal. Decks will be called {\\bf equal} iff there is a\nbijection such that each card is isomorphic to its image.\nMarked cards will be called equal iff there is an isomorphism that preserves the marked property\n(rank in the original set or maximality\/minimality in the original set). Marked decks will be called\nequal iff there is a bijection such that any set is isomorphic to its image and each\nisomorphism also preserves the marked property.\nThe {\\bf up-set} of an element $x$ is the set $\\uparrow x=\\{ p\\in P:p\\geq x\\} $\nand the {\\bf down-set} is $\\downarrow x=\\{ p\\in P:p\\leq x\\} $.\nThe {\\bf neighborhood} of an element $x$ is the set\n$\\updownarrow x=\\uparrow x\\cup \\downarrow x$.\nThe set of all neighborhoods of points of rank $k$ is called the\n{\\bf rank $k$ neighborhood deck}.\n\n\nWe are concerned with results that show what type of information is {\\em not}\nsufficient to effect reconstruction. Therefore, throughout we\nconstruct nonisomorphic ordered sets such that between some of their cards\nthere are isomorphisms with certain\nproperties.\n\n\n\n\\section{Pairs of Nonisomorphic Ordered Sets with Equal Maximal and Minimal Decks and\n$n+1$ Non-Extremal Ranks for Which the Rank $k$ Decks are Equal}\n\n\nIn this section we describe the fundamental construction used to build the examples.\nLemma \\ref{severalmiddle} gives the overall idea, which is an extension of the work in \\cite{Schmoreexam}.\nLemma \\ref{severalmiddle} is also quite similar to\nthe examples in \\cite{Pouinterdit}.\nIn a way, Lemma \\ref{severalmiddle} is reminiscent of a ``vertical M\\\"obius strip\".\n\n\n\nLemma \\ref{getsmallerR} then shows that sets as needed in the construction in\nLemma \\ref{severalmiddle} actually exist.\nIn the following, we explore some features of the construction as well as variations\nthat lead to examples with\nother properties.\n\n\n\\begin{lem}\n\\label{severalmiddle}\n\nLet $Q$ be an ordered set such that\n\\begin{enumerate}\n\\item\n$Q$ has exactly two maximal elements $d$ and $p$,\n\n\\item\n$d$ and $p$ have the same rank,\n\n\\item\n\\label{Qpsi}\nThere is an isomorphism $\\psi :Q\\setminus \\{ d\\} \\to Q\\setminus \\{ p\\} $\nwith $\\psi (p)=d$,\n\n\\item\n$Q$ has two minimal elements $a$ and $b$,\n\n\\item\n\\label{Qpsia}\n$Q\\setminus \\{ a\\} $ has an automorphism $\\psi ^a $ with $\\psi ^a\n(p) = d$, $\\psi ^a (d) = p$, and $\\psi ^a (b) = b$,\n\n\\item\n\\label{Qpsib}\n$Q\\setminus \\{ b\\} $ has an automorphism $\\psi ^b $ with $\\psi ^b\n(p) = d$ and $\\psi ^b (d) = p$, and $\\psi ^b (a) = a$,\n\n\\item\n$Q$ is rigid.\n\n\\setcounter{itemtransfer}{\\value{enumi}}\n\n\\end{enumerate}\n\nLet $R$ be an ordered set such that\n\\begin{enumerate}\n\\setcounter{enumi}{\\value{itemtransfer}}\n\n\\item\n\\label{Rtwomax}\n$R$ has exactly two maximal elements $d$ and $p$,\n\n\\item\n\\label{Rtwomaxrnk}\n$d$ and $p$ have the same rank,\n\n\\item\n\\label{Rtwomin}\n$R$ has exactly two minimal elements $\\overline{d}$ and\n$\\overline{p}$,\n\n\n\\item\n\\label{Rpaut}\n$R\\setminus \\{ \\overline{p} \\} $ has an automorphism\n$\\psi ^{\\overline{p} } $ such that $\\psi ^{\\overline{p} }  (d)=p $,\n$\\psi ^{\\overline{p} }  (p)=d $, and $\\psi ^{\\overline{p} }  (\\overline{d} )=\\overline{d}  $,\n\n\\item\n\\label{Rdaut}\n$R\\setminus \\{ \\overline{d} \\} $ has an automorphism\n$\\psi ^{\\overline{d} } $ such that $\\psi ^{\\overline{d} }  (d)=p $,\n$\\psi ^{\\overline{d} }  (p)=d $,  and $\\psi ^{\\overline{d} }  (\\overline{p} )=\\overline{p}  $,\n\n\n\\item\n\\label{Rswit}\n$R$ has an automorphism $\\varphi $ with $\\varphi (d)=p$, $\\varphi (p)=d$,\n$\\varphi (\\overline{d})=\\overline{p}$, $\\varphi (\\overline{p})=\\overline{d}$,\n\n\\item\n\\label{notwistR}\n$R$ has no automorphism that is the identity on the minimal elements and not the\nidentity on the maximal elements.\n\n\n\\end{enumerate}\n\nLet $\\tilde{Q} $ be the dual of $Q$ and let $R_1 , \\ldots , R_n $ be isomorphic copies of\n$R$ such that $Q, \\tilde{Q} , R_1 , \\ldots , R_n $ are all mutually disjoint.\nLet the elements of $R_i $ be distinguished by subscripts $i$, that is,\nthe maximal and minimal elements of $R_i $ are $d_i $, $p_i $ and $\\overline{d} _i $, $\\overline{p} _i $.\nLet the elements of $\\tilde{Q}$ similarly be distinguished by tildes.\n\nDefine $P_1 $ to be the ordered set obtained from\n$Q, R_1 , \\ldots , R_n , \\tilde{Q} $ as follows (also cf. Figure \\ref{switstck}).\n\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\n\\begin{enumerate}\n\\item\nAll non-maximal elements of $Q$ are below all non-minimal elements of $R_1 $.\n\n\\item\nThe element $d$ is identified with the element $\\overline{d} _1 $ and\nthe element $p$ is identified with the element $\\overline{p} _1 $.\nCall the thus obtained elements $d_0 $ and $p_0 $, respectively.\n\n\\item\nFor $i=1, \\ldots , n-1$,\nall non-maximal elements of $R_i $ are below all non-minimal elements of $R_{i+1} $.\n\n\\item\nFor $i=1, \\ldots , n-1$,\nthe element $d_i $ is identified with the element $\\overline{d} _{i+1} $ and\nthe element $p_i $ is identified with the element $\\overline{p} _{i+1} $.\nCall the thus obtained elements $d_i $ and $p_i $, respectively.\n\n\\item\nAll non-maximal elements of $R_n $ are below all non-minimal elements of $\\tilde{Q} $.\n\n\\item\n\\label{lastmerge}\nThe element $d_n $ is identified with the element $\\tilde{d} $ and\nthe element $p_n $ is identified with the element $\\tilde{p} $.\nCall the thus obtained elements $d_n $ and $p_n $, respectively.\n\n\\item\nPlus all comparabilities forced by transitivity.\n\n\\end{enumerate}\n\n\nDefine $P_2 $ to be the ordered set obtained from\n$Q, R_1 , \\ldots , R_n , \\tilde{Q} $ in the same way as $P_1 $ except that\n\\ref{lastmerge} is replaced with the following (also cf. Figure \\ref{switstck}).\n\n\n\\begin{enumerate}\n\\item[\\ref{lastmerge}']\nThe element $d_n $ is identified with the element $\\tilde{p} $ and\nthe element $p_n $ is identified with the element $\\tilde{d} $.\nCall the thus obtained elements $d_n $ and $p_n $, respectively.\n\n\n\\end{enumerate}\n\n\n\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\n\nThen\n\\begin{enumerate}\n\\item\n\\label{P1P2notiso}\n$P_1 $ and $P_2 $ are not isomorphic,\n\\item\n\\label{P1P2eqmarkedextr}\n$P_1 $ and $P_2 $ have\nequal marked maximal and minimal decks,\n\n\\item\n\\label{dipieqmark}\nFor $i=0, \\ldots , n$\nthe\ncard $P_1 \\setminus \\{ d_i \\} $ is isomorphic to the card $P_2\\setminus \\{ d_i \\} $ and\nthe\ncard $P_1 \\setminus \\{ p_i \\} $ is isomorphic to the card $P_2\\setminus \\{ p_i \\} $,\nand for all isomorphisms $\\Phi $ of cards and all\nelements $x\\in P_1 $ we have that ${\\rm rank} _{P_2 } (\\Phi (x) )={\\rm rank} _{P_1 } (x)$.\n\n\\item\n\\label{P1P2eqnremhood}\nFor all the above mentioned isomorphic cards $P_1 \\setminus \\{ x\\} $\nand $P_2 \\setminus \\{ x\\} $ the\nneighborhoods $\\updownarrow _{P_1 } x $ and $\\updownarrow _{P_2 } x $\nare isomorphic.\n\n\\end{enumerate}\n\n\\renewcommand{\\theenumi}{\\arabic{enumi}}\n\n\\end{lem}\n\n\n\\begin{figure}\n\n\\centerline{\n\\unitlength 1.1mm\n\\linethickness{0.4pt}\n\\begin{picture}(110.00,146.00)\n\\put(20.00,10.00){\\line(1,-1){5.00}}\n\\put(25.00,5.00){\\line(1,1){5.00}}\n\\put(30.00,10.00){\\line(1,-1){5.00}}\n\\put(35.00,5.00){\\line(1,1){5.00}}\n\\put(40.00,10.00){\\line(0,1){10.00}}\n\\put(40.00,20.00){\\line(-1,1){5.00}}\n\\put(35.00,25.00){\\line(-1,-1){5.00}}\n\\put(30.00,20.00){\\line(-1,1){5.00}}\n\\put(25.00,25.00){\\line(-1,-1){5.00}}\n\\put(20.00,20.00){\\line(0,-1){10.00}}\n\\put(30.00,17.00){\\makebox(0,0)[cc]{\\footnotesize $Q$}}\n\\put(30.00,13.00){\\makebox(0,0)[cc]{\\footnotesize {\\em not} symmetric}}\n\\put(25.00,15.00){\\vector(1,0){10.00}}\n\\put(35.00,5.00){\\circle*{2.00}}\n\\put(25.00,5.00){\\circle*{2.00}}\n\\put(25.00,2.00){\\makebox(0,0)[cc]{\\footnotesize $a$}}\n\\put(35.00,2.00){\\makebox(0,0)[cc]{\\footnotesize $b$}}\n\\put(37.00,25.00){\\makebox(0,0)[lc]{\\footnotesize $p_0 = p=\\overline{p}_1 $}}\n\\put(23.00,25.00){\\makebox(0,0)[rc]{\\footnotesize $d_0 = d=\\overline{d}_1 $}}\n\\put(20.00,30.00){\\line(1,-1){5.00}}\n\\put(25.00,25.00){\\line(1,1){5.00}}\n\\put(30.00,30.00){\\line(1,-1){5.00}}\n\\put(35.00,25.00){\\line(1,1){5.00}}\n\\put(40.00,30.00){\\line(0,1){10.00}}\n\\put(40.00,40.00){\\line(-1,1){5.00}}\n\\put(35.00,45.00){\\line(-1,-1){5.00}}\n\\put(30.00,40.00){\\line(-1,1){5.00}}\n\\put(25.00,45.00){\\line(-1,-1){5.00}}\n\\put(20.00,40.00){\\line(0,-1){10.00}}\n\\put(30.00,37.00){\\makebox(0,0)[cc]{\\footnotesize $R_1 $}}\n\\put(30.00,33.00){\\makebox(0,0)[cc]{\\footnotesize symmetric}}\n\\put(35.00,25.00){\\circle*{2.00}}\n\\put(25.00,25.00){\\circle*{2.00}}\n\\put(37.00,45.00){\\makebox(0,0)[lc]{\\footnotesize $p_1 =\\overline{p}_2 $}}\n\\put(23.00,45.00){\\makebox(0,0)[rc]{\\footnotesize $d_1=\\overline{d}_2 $}}\n\\put(20.00,50.00){\\line(1,-1){5.00}}\n\\put(25.00,45.00){\\line(1,1){5.00}}\n\\put(30.00,50.00){\\line(1,-1){5.00}}\n\\put(35.00,45.00){\\line(1,1){5.00}}\n\\put(40.00,50.00){\\line(0,1){10.00}}\n\\put(40.00,60.00){\\line(-1,1){5.00}}\n\\put(35.00,65.00){\\line(-1,-1){5.00}}\n\\put(30.00,60.00){\\line(-1,1){5.00}}\n\\put(25.00,65.00){\\line(-1,-1){5.00}}\n\\put(20.00,60.00){\\line(0,-1){10.00}}\n\\put(30.00,57.00){\\makebox(0,0)[cc]{\\footnotesize $R_2 $}}\n\\put(30.00,53.00){\\makebox(0,0)[cc]{\\footnotesize symmetric}}\n\\put(35.00,45.00){\\circle*{2.00}}\n\\put(25.00,45.00){\\circle*{2.00}}\n\\put(37.00,65.00){\\makebox(0,0)[lc]{\\footnotesize $p_2 =\\overline{p}_3 $}}\n\\put(23.00,65.00){\\makebox(0,0)[rc]{\\footnotesize $d_2=\\overline{d}_3 $}}\n\\put(35.00,65.00){\\circle*{2.00}}\n\\put(25.00,65.00){\\circle*{2.00}}\n\\put(30.00,75.00){\\makebox(0,0)[cc]{$\\vdots $}}\n\\put(37.00,85.00){\\makebox(0,0)[lc]{\\footnotesize $p_{n-2} =\\overline{p}_{n-1} $}}\n\\put(23.00,85.00){\\makebox(0,0)[rc]{\\footnotesize $d_{n-2} =\\overline{d}_{n-1} $}}\n\\put(20.00,90.00){\\line(1,-1){5.00}}\n\\put(25.00,85.00){\\line(1,1){5.00}}\n\\put(30.00,90.00){\\line(1,-1){5.00}}\n\\put(35.00,85.00){\\line(1,1){5.00}}\n\\put(40.00,90.00){\\line(0,1){10.00}}\n\\put(40.00,100.00){\\line(-1,1){5.00}}\n\\put(35.00,105.00){\\line(-1,-1){5.00}}\n\\put(30.00,100.00){\\line(-1,1){5.00}}\n\\put(25.00,105.00){\\line(-1,-1){5.00}}\n\\put(20.00,100.00){\\line(0,-1){10.00}}\n\\put(30.00,97.00){\\makebox(0,0)[cc]{\\footnotesize $R_{n-1} $}}\n\\put(30.00,93.00){\\makebox(0,0)[cc]{\\footnotesize symmetric}}\n\\put(35.00,85.00){\\circle*{2.00}}\n\\put(25.00,85.00){\\circle*{2.00}}\n\\put(37.00,105.00){\\makebox(0,0)[lc]{\\footnotesize $p_{n-1} =\\overline{p}_n $}}\n\\put(23.00,105.00){\\makebox(0,0)[rc]{\\footnotesize $d_{n-1} =\\overline{d}_n $}}\n\\put(20.00,110.00){\\line(1,-1){5.00}}\n\\put(25.00,105.00){\\line(1,1){5.00}}\n\\put(30.00,110.00){\\line(1,-1){5.00}}\n\\put(35.00,105.00){\\line(1,1){5.00}}\n\\put(40.00,110.00){\\line(0,1){10.00}}\n\\put(40.00,120.00){\\line(-1,1){5.00}}\n\\put(35.00,125.00){\\line(-1,-1){5.00}}\n\\put(30.00,120.00){\\line(-1,1){5.00}}\n\\put(25.00,125.00){\\line(-1,-1){5.00}}\n\\put(20.00,120.00){\\line(0,-1){10.00}}\n\\put(30.00,117.00){\\makebox(0,0)[cc]{\\footnotesize $R_n $}}\n\\put(30.00,113.00){\\makebox(0,0)[cc]{\\footnotesize symmetric}}\n\\put(35.00,105.00){\\circle*{2.00}}\n\\put(25.00,105.00){\\circle*{2.00}}\n\\put(37.00,125.00){\\makebox(0,0)[lc]{\\footnotesize $p_n =\\tilde{p} $}}\n\\put(23.00,125.00){\\makebox(0,0)[rc]{\\footnotesize $d_n=\\tilde{d} $}}\n\\put(35.00,125.00){\\circle*{2.00}}\n\\put(25.00,125.00){\\circle*{2.00}}\n\\put(20.00,130.00){\\line(1,-1){5.00}}\n\\put(25.00,125.00){\\line(1,1){5.00}}\n\\put(30.00,130.00){\\line(1,-1){5.00}}\n\\put(35.00,125.00){\\line(1,1){5.00}}\n\\put(40.00,130.00){\\line(0,1){10.00}}\n\\put(40.00,140.00){\\line(-1,1){5.00}}\n\\put(35.00,145.00){\\line(-1,-1){5.00}}\n\\put(30.00,140.00){\\line(-1,1){5.00}}\n\\put(25.00,145.00){\\line(-1,-1){5.00}}\n\\put(20.00,140.00){\\line(0,-1){10.00}}\n\\put(30.00,137.67){\\makebox(0,0)[cc]{\\footnotesize $\\tilde{Q} $}}\n\\put(30.00,133.00){\\makebox(0,0)[cc]{\\footnotesize {\\em not} symmetric}}\n\\put(25.00,135.00){\\vector(1,0){10.00}}\n\\put(37.00,145.00){\\makebox(0,0)[lc]{\\footnotesize $\\tilde{b} $}}\n\\put(23.00,145.00){\\makebox(0,0)[rc]{\\footnotesize $\\tilde{a} $}}\n\\put(35.00,145.00){\\circle*{2.00}}\n\\put(25.00,145.00){\\circle*{2.00}}\n\\put(5.00,145.00){\\makebox(0,0)[cc]{$P_1 $}}\n\\put(90.00,10.00){\\line(1,-1){5.00}}\n\\put(95.00,5.00){\\line(1,1){5.00}}\n\\put(100.00,10.00){\\line(1,-1){5.00}}\n\\put(105.00,5.00){\\line(1,1){5.00}}\n\\put(110.00,10.00){\\line(0,1){10.00}}\n\\put(110.00,20.00){\\line(-1,1){5.00}}\n\\put(105.00,25.00){\\line(-1,-1){5.00}}\n\\put(100.00,20.00){\\line(-1,1){5.00}}\n\\put(95.00,25.00){\\line(-1,-1){5.00}}\n\\put(90.00,20.00){\\line(0,-1){10.00}}\n\\put(100.00,17.00){\\makebox(0,0)[cc]{\\footnotesize $Q$}}\n\\put(100.00,13.00){\\makebox(0,0)[cc]{\\footnotesize {\\em not} symmetric}}\n\\put(95.00,15.00){\\vector(1,0){10.00}}\n\\put(105.00,5.00){\\circle*{2.00}}\n\\put(95.00,5.00){\\circle*{2.00}}\n\\put(95.00,2.00){\\makebox(0,0)[cc]{\\footnotesize $a$}}\n\\put(105.00,2.00){\\makebox(0,0)[cc]{\\footnotesize $b$}}\n\\put(107.00,25.00){\\makebox(0,0)[lc]{\\footnotesize $p_0 = p=\\overline{p}_1 $}}\n\\put(93.00,25.00){\\makebox(0,0)[rc]{\\footnotesize $d_0 = d=\\overline{d}_1 $}}\n\\put(90.00,30.00){\\line(1,-1){5.00}}\n\\put(95.00,25.00){\\line(1,1){5.00}}\n\\put(100.00,30.00){\\line(1,-1){5.00}}\n\\put(105.00,25.00){\\line(1,1){5.00}}\n\\put(110.00,30.00){\\line(0,1){10.00}}\n\\put(110.00,40.00){\\line(-1,1){5.00}}\n\\put(105.00,45.00){\\line(-1,-1){5.00}}\n\\put(100.00,40.00){\\line(-1,1){5.00}}\n\\put(95.00,45.00){\\line(-1,-1){5.00}}\n\\put(90.00,40.00){\\line(0,-1){10.00}}\n\\put(100.00,37.00){\\makebox(0,0)[cc]{\\footnotesize $R_1 $}}\n\\put(100.00,33.00){\\makebox(0,0)[cc]{\\footnotesize symmetric}}\n\\put(105.00,25.00){\\circle*{2.00}}\n\\put(95.00,25.00){\\circle*{2.00}}\n\\put(107.00,45.00){\\makebox(0,0)[lc]{\\footnotesize $p_1 =\\overline{p}_2 $}}\n\\put(93.00,45.00){\\makebox(0,0)[rc]{\\footnotesize $d_1=\\overline{d}_2 $}}\n\\put(90.00,50.00){\\line(1,-1){5.00}}\n\\put(95.00,45.00){\\line(1,1){5.00}}\n\\put(100.00,50.00){\\line(1,-1){5.00}}\n\\put(105.00,45.00){\\line(1,1){5.00}}\n\\put(110.00,50.00){\\line(0,1){10.00}}\n\\put(110.00,60.00){\\line(-1,1){5.00}}\n\\put(105.00,65.00){\\line(-1,-1){5.00}}\n\\put(100.00,60.00){\\line(-1,1){5.00}}\n\\put(95.00,65.00){\\line(-1,-1){5.00}}\n\\put(90.00,60.00){\\line(0,-1){10.00}}\n\\put(100.00,57.00){\\makebox(0,0)[cc]{\\footnotesize $R_2 $}}\n\\put(100.00,53.00){\\makebox(0,0)[cc]{\\footnotesize symmetric}}\n\\put(105.00,45.00){\\circle*{2.00}}\n\\put(95.00,45.00){\\circle*{2.00}}\n\\put(107.00,65.00){\\makebox(0,0)[lc]{\\footnotesize $p_2 =\\overline{p}_3 $}}\n\\put(93.00,65.00){\\makebox(0,0)[rc]{\\footnotesize $d_2=\\overline{d}_3 $}}\n\\put(105.00,65.00){\\circle*{2.00}}\n\\put(95.00,65.00){\\circle*{2.00}}\n\\put(100.00,75.00){\\makebox(0,0)[cc]{$\\vdots $}}\n\\put(107.00,85.00){\\makebox(0,0)[lc]{\\footnotesize $p_{n-2} =\\overline{p}_{n-1} $}}\n\\put(93.00,85.00){\\makebox(0,0)[rc]{\\footnotesize $d_{n-2} =\\overline{d}_{n-1} $}}\n\\put(90.00,90.00){\\line(1,-1){5.00}}\n\\put(95.00,85.00){\\line(1,1){5.00}}\n\\put(100.00,90.00){\\line(1,-1){5.00}}\n\\put(105.00,85.00){\\line(1,1){5.00}}\n\\put(110.00,90.00){\\line(0,1){10.00}}\n\\put(110.00,100.00){\\line(-1,1){5.00}}\n\\put(105.00,105.00){\\line(-1,-1){5.00}}\n\\put(100.00,100.00){\\line(-1,1){5.00}}\n\\put(95.00,105.00){\\line(-1,-1){5.00}}\n\\put(90.00,100.00){\\line(0,-1){10.00}}\n\\put(100.00,97.00){\\makebox(0,0)[cc]{\\footnotesize $R_{n-1} $}}\n\\put(100.00,93.00){\\makebox(0,0)[cc]{\\footnotesize symmetric}}\n\\put(105.00,85.00){\\circle*{2.00}}\n\\put(95.00,85.00){\\circle*{2.00}}\n\\put(107.00,105.00){\\makebox(0,0)[lc]{\\footnotesize $p_{n-1} =\\overline{p}_n $}}\n\\put(93.00,105.00){\\makebox(0,0)[rc]{\\footnotesize $d_{n-1} =\\overline{d}_n $}}\n\\put(90.00,110.00){\\line(1,-1){5.00}}\n\\put(95.00,105.00){\\line(1,1){5.00}}\n\\put(100.00,110.00){\\line(1,-1){5.00}}\n\\put(105.00,105.00){\\line(1,1){5.00}}\n\\put(110.00,110.00){\\line(0,1){10.00}}\n\\put(110.00,120.00){\\line(-1,1){5.00}}\n\\put(105.00,125.00){\\line(-1,-1){5.00}}\n\\put(100.00,120.00){\\line(-1,1){5.00}}\n\\put(95.00,125.00){\\line(-1,-1){5.00}}\n\\put(90.00,120.00){\\line(0,-1){10.00}}\n\\put(100.00,117.00){\\makebox(0,0)[cc]{\\footnotesize $R_n $}}\n\\put(100.00,113.00){\\makebox(0,0)[cc]{\\footnotesize symmetric}}\n\\put(105.00,105.00){\\circle*{2.00}}\n\\put(95.00,105.00){\\circle*{2.00}}\n\\put(107.00,125.00){\\makebox(0,0)[lc]{\\footnotesize $p_n =\\tilde{d} $}}\n\\put(93.00,125.00){\\makebox(0,0)[rc]{\\footnotesize $d_n=\\tilde{p} $}}\n\\put(105.00,125.00){\\circle*{2.00}}\n\\put(95.00,125.00){\\circle*{2.00}}\n\\put(90.00,130.00){\\line(1,-1){5.00}}\n\\put(95.00,125.00){\\line(1,1){5.00}}\n\\put(100.00,130.00){\\line(1,-1){5.00}}\n\\put(105.00,125.00){\\line(1,1){5.00}}\n\\put(110.00,130.00){\\line(0,1){10.00}}\n\\put(110.00,140.00){\\line(-1,1){5.00}}\n\\put(105.00,145.00){\\line(-1,-1){5.00}}\n\\put(100.00,140.00){\\line(-1,1){5.00}}\n\\put(95.00,145.00){\\line(-1,-1){5.00}}\n\\put(90.00,140.00){\\line(0,-1){10.00}}\n\\put(100.00,137.67){\\makebox(0,0)[cc]{\\footnotesize $\\tilde{Q} $}}\n\\put(100.00,133.00){\\makebox(0,0)[cc]{\\footnotesize {\\em not} symmetric}}\n\\put(107.00,145.00){\\makebox(0,0)[lc]{\\footnotesize $\\tilde{a} $}}\n\\put(93.00,145.00){\\makebox(0,0)[rc]{\\footnotesize $\\tilde{b} $}}\n\\put(105.00,145.00){\\circle*{2.00}}\n\\put(95.00,145.00){\\circle*{2.00}}\n\\put(75.00,145.00){\\makebox(0,0)[cc]{$P_2 $}}\n\\put(105.00,135.00){\\vector(-1,0){10.00}}\n\\end{picture}\n}\n\n\\caption{Ordered sets $P_1 $ and $P_2 $ as constructed in Lemma \\protect\\ref{severalmiddle}.\nThe mentioned symmetry is symmetry along the vertical axis.}\n\\label{switstck}\n\n\\end{figure}\n\n\n{\\bf Proof.}\nTo prove \\ref{P1P2notiso}, we assume that $P_1 $ and $P_2 $ are isomorphic.\nSo suppose that $\\Phi :P_1 \\to P_2 $ is an isomorphism.\nThen, because $Q$ is rigid, we have that $\\Phi (a)=a$, $\\Phi (b)=b$,\n$\\Phi (d_0 ) = d_0 $ and\n$\\Phi (p_0 )= p_0 $.\nBy property \\ref{notwistR},\nthis implies that\n$\\Phi (d_1 ) = d_1 $,\n$\\Phi (p_1 )= p_1 $, $\\ldots $, $\\Phi (d_n ) = d_n $,\n$\\Phi (p_n )= p_n $.\nBut then, because in $P_1 $ we have $d_n = \\tilde{d} $ and $p_n = \\tilde{p}$, while\nin $P_2 $ we have $d_n = \\tilde{p} $ and $p_n = \\tilde{d}$, $\\Phi |_{\\tilde{Q} } $ would be an\nautomorphism of $\\tilde{Q}$ with\n$\n{ \\Phi |_{\\tilde{Q} } ( \\tilde{d} ) = \\tilde{p} } $ and\n${ \\Phi |_{\\tilde{Q} } ( \\tilde{p} ) = \\tilde{d} } $.\nThis is a contradiction to the rigidity of $\\tilde{Q} $.\nTherefore, $P_1 $ and $P_2 $ cannot be isomorphic.\n\n\n\nTo show\n\\ref{P1P2eqmarkedextr},\nfirst note that\n$P_1 \\setminus \\{ a\\} $ is isomorphic to $P_2 \\setminus \\{ a\\} $.\nTo see this\nlet $\\varphi _i $ denote the automorphism for $R_i $ guaranteed by\nproperty \\ref{Rswit}.\nWe define\n\n$$\\Phi (x):=\\cases{ \\psi ^a (x); & if $x\\in Q\\setminus \\{ a\\} $, \\cr\n\\varphi _i (x); & if $x\\in R_i $, \\cr\nx; & if $x\\in \\tilde{Q}\\setminus \\{ \\tilde{d}, \\tilde{p} \\} $. \\cr\n} $$\n\nThe function $\\Phi $ is well-defined and bijective between\n$P_1 \\setminus \\{ a\\} $ and $P_2 \\setminus \\{ a\\} $ and it maps minimal elements of $P_1 $\nto minimal elements of $P_2 $.\nTo see that $\\Phi $ is order-preserving both ways, let $x<y$ in $P_1$.\nIt is trivial to infer that $\\Phi (x)<\\Phi (y)$ is equivalent to $x<y$ unless\n$x\\in \\{ \\tilde{d} , \\tilde{p} \\} $.\nAssume without loss of generality that $x=\\tilde{d} = d_n $.\nThen $\\Phi (x) =p_n = \\tilde{d} =x$.\nSince $y>x$ we have $y\\in \\tilde{Q} $ and thus $\\Phi (y)=y$. The other direction, as well as the proof for\n$x=\\tilde{p} = p_n $ is similar.\n\n\nWe have shown that\n$P_1 \\setminus \\{ a\\} $ is isomorphic to $P_2 \\setminus \\{ a\\} $ and that the isomorphism\npreserves the marked property ``minimality\".\nSimilarly, $P_1 \\setminus \\{ b\\} $ is isomorphic to $P_2 \\setminus \\{ b\\} $\n(and minimal elements of $P_1 $ are mapped to minimal elements of $P_2 $) via\n\n$$\\Phi (x):=\\cases{ \\psi ^b (x); & if $x\\in Q\\setminus \\{ b\\} $, \\cr\n\\varphi _i (x); & if $x\\in R_i $, \\cr\nx; & if $x\\in \\tilde{Q}\\setminus \\{ \\tilde{d}, \\tilde{p} \\} $. \\cr\n} $$\n\nThe proof that $P_1 $ and $P_2 $ have equal marked maximal decks is similar.\nThe set\n$P_1 \\setminus \\{ \\tilde{a} \\} $ is isomorphic to $P_2 \\setminus \\{ \\tilde{a} \\} $ via\n\n$$\\Phi (x):=\\cases{ \\psi ^{\\tilde{a} } (x); & if $x\\in \\tilde{Q}\\setminus \\{ \\tilde{a}\\} $, \\cr\nx; & if $x\\not\\in \\tilde{Q} $,  \\cr\n} $$\n\nwhere $\\psi ^{\\tilde{a} } $ denotes the automorphism of $\\tilde{Q} \\setminus \\{ \\tilde{a} \\} $\nthat is guaranteed by the dual of property \\ref{Qpsia}.\nClearly $\\Phi $ maps maximal elements of $P_1 $ to maximal elements of $P_2 $.\nThe set\n$P_1 \\setminus \\{ \\tilde{b} \\} $ is isomorphic to $P_2 \\setminus \\{ \\tilde{b} \\} $\n(and maximal elements of $P_1 $ are mapped to maximal elements of $P_2 $)\nvia\n\n$$\\Phi (x):=\\cases{ \\psi ^{\\tilde{b} } (x); & if $x\\in \\tilde{Q}\\setminus \\{ \\tilde{b}\\} $, \\cr\nx; & if $x\\not\\in \\tilde{Q} $,  \\cr\n} $$\n\nwhere $\\psi ^{\\tilde{b} } $ denotes the automorphism of $\\tilde{Q} \\setminus \\{ \\tilde{b} \\} $\nthat is guaranteed by the dual of property \\ref{Qpsib}.\n\n\nIn regards to \\ref{P1P2eqnremhood} note that the above isomorphisms show that\nfor $x\\in \\{ a,b,\\tilde{a},\\tilde{b} \\} $\nthe\nneighborhoods $\\updownarrow _{P_1 } x $ and $\\updownarrow _{P_2 } x $\nare isomorphic.\n(For example, the isomorphism between $P_1 \\setminus \\{ b\\} $ and\n$P_2 \\setminus \\{ b\\} $ provides an isomorphism\nbetween\n$\\updownarrow _{P_1 } a= \\uparrow _{P_1 } a$ and\n$\\updownarrow _{P_2 } a =\\uparrow _{P_2 } a$.)\n\n\nFor \\ref{dipieqmark}, let it be stated here that it is easy to see that\nall isomorphisms $\\Phi $ constructed in the following satisfy\n${\\rm rank} _{P_2 } (\\Phi (x) )={\\rm rank} _{P_1 } (x)$ for all\nelements $x\\in P_1 $.\n\n\n\nNow first notice that\nthe set\n$P_1 \\setminus \\{ d_n \\} $ is isomorphic to $P_2 \\setminus \\{ d_n \\} $ via\n$$\\Phi (x):=\\cases{ \\tilde{\\psi }  (x); & if $x\\in \\tilde{Q}\\setminus \\{ \\tilde{d}\\} $, \\cr\nx; & if $x\\not\\in \\tilde{Q} $,   \\cr\n} $$\n\nwhere $\\tilde{\\psi } $ is the isomorphism guaranteed by the dual of\nproperty \\ref{Qpsi}.\nThe set\n$P_1 \\setminus \\{ p_n \\} $ is isomorphic to $P_2 \\setminus \\{ p_n \\} $ via\n$$\\Phi (x):=\\cases{ \\left( \\tilde{\\psi } \\right) ^{-1} (x); & if $x\\in \\tilde{Q}\\setminus \\{ \\tilde{p}\\} $, \\cr\nx; & if $x\\not\\in \\tilde{Q} $.   \\cr\n} $$\n\n\n\n\n\nFinally let $i\\in \\{ 0,\\ldots , n-1\\} $.\nFor $R_i $ denote the automorphisms $\\psi ^{\\overline{p} }  $ and $\\psi ^{\\overline{d} }  $\nof the respective cards of $R$\nguaranteed by properties\n\\ref{Rpaut} and \\ref{Rdaut} by $\\psi ^{\\overline{p} }  _i $ and $\\psi ^{\\overline{d} }  _i $.\nThen\nthe set\n$P_1 \\setminus \\{ p_i \\} $ is isomorphic to $P_2 \\setminus \\{ p_i \\} $ via\n\n$$\\Phi (x):=\\cases{\nx; & if $x\\in \\tilde{Q}\\setminus \\{ \\tilde{d}, \\tilde{p} \\} $ or $x\\in R_j $ for $j\\leq i$ or $x\\in Q$, \\cr\n\\varphi _j (x); & if $x\\in R_j $ for $j>i+1$, \\cr\n\\psi ^{\\overline{p} }  _i (x) ; & if $x\\in R_{i+1} \\setminus \\{ p_i \\} $.\\cr\n} $$\n\nAll parts of the definition of isomorphism are readily verified.\nSimilarly,\nthe set\n$P_1 \\setminus \\{ d_i \\} $ is isomorphic to $P_2 \\setminus \\{ d_i \\} $ via\n\n$$\\Phi (x):=\\cases{\nx; & if $x\\in \\tilde{Q}\\setminus \\{ \\tilde{d}, \\tilde{p} \\} $ or $x\\in R_j $ for $j\\leq i$ or $x\\in Q$, \\cr\n\\varphi _j (x); & if $x\\in R_j $ for $j>i+1$, \\cr\n\\psi ^{\\overline{d} }  _i (x) ; & if $x\\in R_{i+1} \\setminus \\{ d_i \\} $.\\cr\n} $$\n\nTo finish the proof of \\ref{P1P2eqnremhood}, similar to\nwhat was said after the proof of \\ref{P1P2eqmarkedextr},\nthe above isomorphisms show that\nfor $x\\in \\{ d_0 , p_0 , \\ldots , d_{n-1} , p_{n-1} \\} $\nthe\nneighborhoods $\\updownarrow _{P_1 } x $ and $\\updownarrow _{P_2 } x $\nare isomorphic.\nJust use the isomorphism between the cards on which the respective\n``other\" element of that rank has been erased.\n\\hfill\\ \\rule{2mm}{2mm} \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}\n\n\\centerline{\n\\unitlength 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$}}\n\\put(83,73){\\makebox(0,0)[cc]{$c_t $}}\n\\put(17.5,37){\\makebox(0,0)[cc]{$A_1 $}}\n\\put(52.5,53){\\makebox(0,0)[cc]{$B_1 $}}\n\\put(97.5,53){\\makebox(0,0)[cc]{$C_1 $}}\n\\put(132.5,37){\\makebox(0,0)[cc]{$D_1 $}}\n\\put(27.5,37){\\makebox(0,0)[cc]{$A_2 $}}\n\\put(62.5,53){\\makebox(0,0)[cc]{$B_2 $}}\n\\put(107.5,53){\\makebox(0,0)[cc]{$C_2 $}}\n\\put(142.5,37){\\makebox(0,0)[cc]{$D_2 $}}\n\\put(37.5,37){\\makebox(0,0)[cc]{$A_{1,2} $}}\n\\put(72.5,53){\\makebox(0,0)[cc]{$B_{1,2} $}}\n\\put(87.5,53){\\makebox(0,0)[cc]{$C_{1,2} $}}\n\\put(122.5,37){\\makebox(0,0)[cc]{$D_{1,2} $}}\n\\put(75,10){\\circle*{2}}\n\\put(75,90){\\circle*{2}}\n\\put(85,10){\\circle*{2}}\n\\put(85,90){\\circle*{2}}\n\\put(75,7){\\makebox(0,0)[cc]{$\\overline{d}$}}\n\\put(75,93){\\makebox(0,0)[cc]{${d}$}}\n\\put(85,7){\\makebox(0,0)[cc]{$\\overline{p}$}}\n\\put(85,93){\\makebox(0,0)[cc]{${p}$}}\n\\put(15,85){\\makebox(0,0)[lc]{$A=A_1 \\cup A_2 \\cup A_{1,2} $}}\n\\put(15,75){\\makebox(0,0)[lc]{$B=B_1 \\cup B_2 \\cup B_{1,2} $}}\n\\put(145,85){\\makebox(0,0)[rc]{$C=C_1 \\cup C_2 \\cup C_{1,2} $}}\n\\put(145,75){\\makebox(0,0)[rc]{$D=D_1 \\cup D_2 \\cup D_{1,2} $}}\n\\end{picture}\n}\n\n\\caption{An ordered set $R$ as needed in Lemma \\protect\\ref{severalmiddle} and constructed in\nLemma \\ref{getsmallerR}.\nThe middle levels form an ordered set of height $1$.\nThe maximal element $d$ is above all maximal elements of the\nmiddle levels {\\em except} the circled maximal elements\nof the middle levels.\nThe maximal element $p$ is above all maximal elements of the\nmiddle levels {\\em except} the boxed maximal elements\nof the middle levels.\nSimilarly,\nthe minimal element $\\overline{d}$ is below all minimal elements of the\nmiddle levels {\\em except} the circled minimal elements and\nthe minimal element $\\overline{p}$ is below all minimal elements of the\nmiddle levels {\\em except} the boxed minimal elements.\nWithin the middle levels, the connected dotted arches indicate that the\nelements immediately above the upper arch and the elements immediately below the lower arch form a complete\nbipartite.\n}\n\\label{flatswit}\n\n\\end{figure}\n\n\n\n\n\\begin{lem}\n\\label{getsmallerR}\n\nThere is\nan ordered set $R$ as described in Lemma \\ref{severalmiddle}.\n\n\\end{lem}\n\n\n{\\bf Proof.}\nLet $R$ be the ordered set indicated in Figure \\ref{flatswit}.\nWe claim that $R$ is a set as desired in the description of the set $R$ in\nLemma \\ref{severalmiddle}.\n\n\nClaims \\ref{Rtwomax}, \\ref{Rtwomaxrnk}, \\ref{Rtwomin} are trivial.\nThe rest of the proof relies on the following, easy to verify, properties\nof the sets $A$, $B$, $C$ and $D$.\nFirst, $A\\cup \\{ c_t , \\overline{d} ,\\overline{p} \\} $ is dually isomorphic to\n$B\\cup \\{ c_b , d,p\\} $ and\n$D\\cup \\{ c_t , \\overline{d} ,\\overline{p} \\} $ is dually isomorphic to\n$C\\cup \\{ c_b , d,p\\} $.\nTherefore, properties proved for $B$ and $C$ will hold dually for $A$ and $D$.\nThe set $B\\cup \\{ c_b , d,p\\} $ is rigid.\nThis is because any automorphism of $B\\cup \\{ c_b , d,p\\} $ must map $c_b $ to itself\nand the sets $B_1 $, $B_2 $ and $B_{1,2} $ to themselves, respectively.\nThis implies that the automorphism must be the identity on $B$ and then it must be the identity on\n$\\{ d,p\\} $ also.\n\nMoreover,\nthere is exactly one isomorphism\nfrom\n$B\\cup \\{ c_b , d,p\\} $ to $C\\cup \\{ c_b , d,p\\} $. This isomorphism\nmaps $d$ to $p$, $p$ to $d$, $c_b $ to $c_b $ and $B_{1,2} $ to $C_{1,2} $, $B_{1} $ to $C_{1} $ and\n$B_{2} $ to $C_{2} $.\nFurthermore, there is exactly one isomorphism from\n$B\\cup \\{ c_b , d\\} $ to $C\\cup \\{ c_b , d\\} $.\nThis isomorphism maps $d$ to $d$, $c_b $ to $c_b $,\n$B_{1,2} $ to $C_{1} $, $B_{2} $ to $C_{1,2} $ and $B_{1} $ to $C_{2} $.\nSimilarly, there is exactly one isomorphism\nfrom\n$B\\cup \\{ c_b , p\\} $ to $C\\cup \\{ c_b , p\\} $.\nThis isomorphism maps $p$ to $p$, $c_b $ to $c_b $,\n$B_{1,2} $ to $C_{2} $, $B_{2} $ to $C_{1} $ and $B_{1} $ to $C_{1,2} $.\nThese facts and their duals will be used freely in the following.\n\n\n\n\nFor Claim \\ref{Rpaut} define $\\psi ^{\\overline{p} } $ to be\n\\begin{enumerate}\n\\item\n$\\psi ^{\\overline{p} } (\\overline{d} ):=\\overline{d} $,\n$\\psi ^{\\overline{p} } (d ):=p $,\n$\\psi ^{\\overline{p} } (p ):=d $,\n\n\\item\nOn $A\\cup \\{ c_t \\} $, $\\psi ^{\\overline{p} } $ is\nthe restriction of\nthe unique isomorphism from\n$A\\cup \\{ c_t , \\overline{d} \\} $ to\n$D\\cup \\{ c_t , \\overline{d} \\} $,\n\n\\item\nOn $D\\cup \\{ c_t \\} $, $\\psi ^{\\overline{p} } $ is\nthe restriction of\nthe unique isomorphism from\n$D\\cup \\{ c_t , \\overline{d} \\} $ to\n$A\\cup \\{ c_t \\overline{d} \\} $,\n\n\\item\nOn $B\\cup \\{ c_b \\} $, $\\psi ^{\\overline{p} } $ is\nthe restriction of the unique\nisomorphism from\n$B\\cup \\{ c_b, d,p \\} $ to\n$C\\cup \\{ c_b, d,p \\} $,\n\n\n\n\\item\nOn $C\\cup \\{ c_b \\} $, $\\psi ^{\\overline{p} } $ is\nthe restriction of the unique\nisomorphism from\n$C\\cup \\{ c_b, d,p \\} $ to\n$B\\cup \\{ c_b, d,p \\} $.\n\n\\end{enumerate}\n\nThen $\\psi ^{\\overline{p} } $ is as desired.\nClaim \\ref{Rdaut} is proved similarly.\n\nFor Claim \\ref{Rswit} define $\\varphi $ to be\n\\begin{enumerate}\n\\item\n$\\varphi (\\overline{d} ):=\\overline{p} $,\n$\\varphi (\\overline{p} ):=\\overline{d} $,\n$\\varphi (d ):=p $,\n$\\varphi (p ):=d $,\n\n\n\\item\nOn $A\\cup \\{ c_t \\} $, $\\varphi $ is\nthe restriction of the unique\nisomorphism from\n$A\\cup \\{ c_t , \\overline{d}, \\overline{p} \\} $ to\n$D\\cup \\{ c_t, \\overline{d}, \\overline{p} \\} $,\n\n\n\\item\nOn $D\\cup \\{ c_t \\} $, $\\varphi $ is\nthe restriction of the unique\nisomorphism from\n$D\\cup \\{ c_t, \\overline{d}, \\overline{p} \\} $ to\n$A\\cup \\{ c_t \\overline{d}, \\overline{p} \\} $,\n\n\\item\nOn $B\\cup \\{ c_b\\} $, $\\varphi $ is\nthe restriction of the unique\nisomorphism from\n$B\\cup \\{ c_b, d,p \\} $ to\n$C\\cup \\{ c_b , d,p \\} $,\n\n\n\\item\nOn $C\\cup \\{ c_b\\} $, $\\varphi $ is\nthe restriction of the unique\nisomorphism from\n$C\\cup \\{ c_b , d,p \\} $ to\n$B\\cup \\{ c_b, d,p \\} $.\n\n\\end{enumerate}\n\n\nFinally, for Claim \\ref{notwistR} suppose the automorphism\n$\\Psi :R\\to R$\nis the identity on the minimal elements.\nThen, because the unique isomorphism between\n$A\\cup \\{ c_t, \\overline{d},\\overline{p} \\} $ and $D\\cup \\{ c_t ,\\overline{d},\\overline{p} \\} $\nswitches $\\overline{d} $ and $\\overline{p} $,\n$\\Psi $ must map $A \\cup \\{c_t \\} $ to $A \\cup \\{ c_t\\} $ and\n$D  \\cup \\{ c_t\\} $ to $D \\cup \\{ c_t\\}  $.\nTherefore, $\\Psi $ must map\n$B \\cup \\{ c_b\\} $ to $B \\cup \\{ c_b\\}  $ and\n$C \\cup \\{ c_b\\}  $ to $C \\cup \\{ c_b\\}  $.\nSince $B\\cup \\{ c_b , d , p \\} $ is\nrigid, this means that\n$\\Psi $ must fix $d $ and $p $.\n\\hfill\\ \\rule{2mm}{2mm} \n\n\n\\vspace{.1in}\n\n\nThe set $R$ in Figure \\ref{flatswit} has an additional property that will allow us to\nprove further properties of our examples.\n\n\\begin{lem}\n\\label{takeoutc}\n\nThe sets $R\\setminus \\{ c_b\\} $ and $R\\setminus \\{ c_t \\} $\nwith $R$ as in Figure \\ref{flatswit} each have an automorphism\n$\\Psi $ with $\\Psi (\\overline{d} )=\\overline{d} $, $\\Psi (\\overline{p} )=\\overline{p} $,\n$\\Psi (d)=p$ and $\\Psi (p)=d$.\n\n\\end{lem}\n\n{\\bf Proof.}\nFor $R\\setminus \\{ c_t \\} $ we define $\\Psi (\\overline{d} ):=\\overline{d} $, $\\Psi (\\overline{p} ):=\\overline{p} $,\n$\\Psi (d):=p$ and $\\Psi (p):=d$. We let\n$\\Psi $ map $A_{1,2} $ to $D_{1,2} $, $A_{2} $ to $D_{1} $, $A_{1} $ to $D_{2} $ and vice versa.\n(Visually, in each case the map is obtained by sliding one wedge horizontally onto the other.)\n\nOn $B\\cup \\{ c_b, d,p\\} $ we define $\\Psi $ to be the unique isomorphism from\n$B\\cup \\{ c_b, d,p\\} $ to $C\\cup \\{ c_b, d,p\\} $.\nFinally, on $C\\cup \\{ c_b, d,p\\} $ we define $\\Psi $ to be the unique isomorphism from\n$C\\cup \\{ c_b, d,p\\} $ to $B\\cup \\{ c_b, d,p\\} $.\n\nFor $R\\setminus \\{ c_b \\} $ we let $\\Psi $ be the identity on $\\{ \\overline{d} , \\overline{p} , c_t \\} \\cup A\\cup D$.\nWe let $\\Psi (d)=p$ and $\\Psi (p)=d$. Finally $\\Psi $ maps $B_{1,2} $ to itself, $B_1 $ to $B_2 $, $B_2 $ to $B_1 $,\n$C_{1,2} $ to itself, $C_1 $ to $C_2 $, $C_2 $ to $C_1 $, each in such a way that the comparabilities\nwith the appropriate maximal elements are preserved.\n\\hfill\\ \\rule{2mm}{2mm} \n\n\n\n\n\n\n\\vspace{.1in}\n\n\n\nWith Lemmas \\ref{severalmiddle}\nand \\ref{getsmallerR} proved, we can state our first main result.\nAside from insights on ranks and maximal and minimal cards, we also see that\nour construction yields pairs of nonisomorphic sets\nfor which a significant\nnumber of cards is isomorphic.\n\n\n\n\\begin{define}\n\nFor two ordered sets $P_1 $ and $P_2 $ with $n$ elements each, define the\n{\\bf equal card ratio} $ECR(P_1 , P_2 )$ to be the number of\nisomorphic cards divided by the size of the set.\n\n\\end{define}\n\n\n\n\n\n\\begin{theorem}\n\\label{tallones}\n\nThere is a sequence of pairs of ordered sets $(P_1 ^n , P_2 ^n )$\nsuch that\n\\begin{enumerate}\n\\item\n$P_1 ^n $ is not isomorphic to $P_2 ^n $,\n\\item\n\\label{eqmaxmindeck}\n$P_1 ^n $ and $P_2 ^n $ have equal marked maximal and minimal decks,\n\\item\n\\label{eqrankkdecks}\nThere are ranks $k_0, \\ldots , k_n $ such that $P_1 ^n $ and $P_2 ^n $ have\nequal marked rank $k_i $ decks,\n\\item\nFor any\npair of isomorphic cards as in parts \\ref{eqmaxmindeck} and \\ref{eqrankkdecks},\nthe neighborhoods of the respective removed elements are isomorphic,\n\\item\n\\label{onlyfourleft}\nThere are only four ranks that do not produce any isomorphic cards,\n\\item\n\\label{ecr>0}\n$\\displaystyle{ \\liminf _{n\\to \\infty } ECR(P_1 ^n , P_2 ^n )\\geq {1\\over 10} } $,\n\\item\n\\label{equalnhooddecks}\nFor every rank $k$, the rank $k$ neighborhood decks of $P_1 ^n $ and $P_2 ^n $\nare equal.\n\\end{enumerate}\n\n\\end{theorem}\n\n\n{\\bf Proof.}\nTo construct sets as indicated,\nwith notation as in Lemma \\ref{severalmiddle}, we make the following choices.\n\\begin{enumerate}\n\\item\nFor all $n$, as the set $Q$, use a fixed set $Q$ as in the proof of Theorem 5.3 of \\cite{Schmoreexam}.\nThese sets have height $3$.\n\\item\nFor all $n$, as the set $R$, use a fixed set $R$ as guaranteed by\nLemma \\ref{getsmallerR}.\n\\item\nThe ranks $k_0 , \\ldots , k_n $ are\n$k_i = {\\rm height} (Q)+i\\cdot {\\rm height} (R)$.\n\\end{enumerate}\n\nThis construction, independent of the choices for $Q$ and $R$,\nyields a sequence of sets that satisfy all parts of this theorem except\npossibly parts \\ref{onlyfourleft}, \\ref{ecr>0} and \\ref{equalnhooddecks}.\nFor these parts the construction must be done with the set $R$ indicated in the proof of\nLemma \\ref{getsmallerR}.\n\nFor part \\ref{onlyfourleft} note that\nby Lemma \\ref{takeoutc} the erasure of an element $c_t $ or $c_b $\nat corresponding ranks produces isomorphic marked cards for $P_1 ^n $ and $P_2 ^n $.\nThis means that only at the ranks $1$ and $2$ and at the two ranks immediately below the\nmaximal elements will there be no elements whose removal produces isomorphic cards.\n\n\n\nFor part \\ref{ecr>0},\nfirst note that because each pair of sets has at least $4n+6$ equal cards we have\n$$\\displaystyle{ ECR(P_1 ^n , P_2 ^n )\\geq {4n+6\\over n(|R|-2)+2|Q|-2}\n\\ontop{n\\to \\infty }{\\longrightarrow } {4\\over |R|-2}. } $$\n\nThat is,\nour lower bound on\n$\\displaystyle{\n\\liminf _{n\\to \\infty } ECR(P_1 ^n , P_2 ^n )} $ will solely depend on the number of elements that $R$ has.\n\nWith the set $R$ as given in the proof of Lemma \\ref{getsmallerR} we have $|R|=42$ and so\n$\\displaystyle{ \\liminf _{n\\to \\infty } ECR(P_1 ^n , P_2 ^n )\\geq {4\\over 42-2}={1\\over 10} .} $\n\nFor part \\ref{equalnhooddecks} we know that\nthe neighborhoods $\\updownarrow _{P_1 ^n } x$ and $\\updownarrow _{P_2 ^n } x$\nare isomorphic for $x\\in \\{ a,b, \\tilde{a}, \\tilde{b},d_0 , p_0 , \\ldots , d_n , p_n \\} $.\nThis leaves the non-extremal elements of the $R_i $ and the non-extremal\nelements of $Q$ and $\\tilde{Q} $.\n\nWe first consider the non-extremal elements of the $R_i $.\nFor $x\\in A\\cup D$\nwe have that $\\uparrow _R x\\setminus \\{ x\\} $ is a four-crown if $x$ is minimal and\na two-antichain if $x$ is maximal.\nSimilarly,\n$\\uparrow _R x\\setminus \\{ x\\} $ is\na two-antichain for the maximal elements of $C$ and $D$ that are below both\n$d$ and $p$ as well as for $c_t $.\nThis means that $\\updownarrow _R x$ has an automorphism that fixes\n$\\overline{d} $ and $\\overline{p} $ and that switches $d$ and $p$,\nwhich means that for these elements\n(in any of the $R_i $) we have that\n$\\updownarrow _{P_1 ^n } x$ and $\\updownarrow _{P_2 ^n } x$\nare isomorphic.\nFor the minimal elements of $B_{1,2} $ and $C_{1,2} $, call them\n$b_{1,2} $ and $c_{1,2} $, the set\n$\\uparrow _R x\\setminus \\{ x\\} $ is the disjoint union of two 2-chains\nand for these elements (in any $R_i $) we have\n$\\updownarrow _{P_1 ^n } b_{1,2} $ is isomorphic to\n$\\updownarrow _{P_2 ^n } c_{1,2} $\nand\n$\\updownarrow _{P_1 ^n } c_{1,2} $ is isomorphic to\n$\\updownarrow _{P_2 ^n } b_{1,2} $.\nFor the minimal elements of $B_1 $ and $B_2 $, call them\n$b_1 $ and $b_2 $,\nthe set\n$\\uparrow _R x\\setminus \\{ x\\} $ is an ``N\"\nand for these elements (in any $R_i $) we have\n$\\updownarrow _{P_1 ^n } b_{1} $ is isomorphic to\n$\\updownarrow _{P_2 ^n } b_{2} $.\nThe corresponding elements of $C$ are handled similarly.\nThe remaining maximal elements of $B$ and $C$ are below exactly one of $d$ and $p$,\ncall this element $m$.\nFor these elements (in any $R_i $) we prove that\n$\\updownarrow _{P_1 ^n } m $ is isomorphic to\n$\\updownarrow _{P_2 ^n } m $ as follows.\nIn an $R_i $ with $i<n$ this is because\n$R_{i+1} \\setminus \\{ m\\} $ has an automorphism that\nswitches the maximal elements of $R_{i+1} $.\nIn $R_n $ this is because $\\tilde{Q} \\setminus \\{ \\tilde{d} \\} $ is\nisomorphic to $\\tilde{Q} \\setminus \\{ \\tilde{p} \\} $.\nFinally,\n$\\uparrow _R c_b \\setminus \\{ c_b \\} $ is\nmade up of two sets ``M\" with their maximal elements identified.\nThis means that\n$\\updownarrow _{P_1 ^n } c_b $ is isomorphic to\n$\\updownarrow _{P_2 ^n } c_b $.\n\nTo consider the non-extremal elements of $Q$, we\nneed to use sets $Q$ as in\n\\cite{Schmoreexam}.\nFor all non-extremal elements $x$ of $Q$ the same ideas as above show that\n$\\updownarrow _{P_1 ^n } x$ and $\\updownarrow _{P_2 ^n } x$\nare isomorphic.\nIf the strict upper bounds of $x$ in $Q$ are a set\n(such as a two antichain, a four crown or the disjoint union of\ntwo $2$-chains)\nthat allows\ninterchanging the maximal elements, we get an isomorphism.\nIf not, the strict upper bounds in $Q$ are a singleton, in which\ncase the switch can be done in $R_1 $, or the strict upper bounds\nform an ``N\". By choosing the appropriate\npermutations in the construction of the set $Q$, this last situation\ncan be avoided. This takes care of the non-extremal elements of $Q$.\nThe argument for the non-extremal elements of $\\tilde{Q} $ is the dual\nof the above.\n\\hfill\\ \\rule{2mm}{2mm} \n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}\n\n\n\\centerline{\n\\unitlength 1.00mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi\n\\begin{picture}(146,93)(0,0)\n\\put(50,50){\\line(1,-4){2.5}}\n\\put(15,50){\\line(1,-4){2.5}}\n\\put(60,50){\\line(1,-4){2.5}}\n\\put(25,50){\\line(1,-4){2.5}}\n\\put(70,50){\\line(1,-4){2.5}}\n\\put(35,50){\\line(1,-4){2.5}}\n\\put(85,50){\\line(1,-4){2.5}}\n\\put(120,50){\\line(1,-4){2.5}}\n\\put(95,50){\\line(1,-4){2.5}}\n\\put(130,50){\\line(1,-4){2.5}}\n\\put(105,50){\\line(1,-4){2.5}}\n\\put(140,50){\\line(1,-4){2.5}}\n\\put(52.5,40){\\line(1,4){2.5}}\n\\put(17.5,40){\\line(1,4){2.5}}\n\\put(62.5,40){\\line(1,4){2.5}}\n\\put(27.5,40){\\line(1,4){2.5}}\n\\put(72.5,40){\\line(1,4){2.5}}\n\\put(37.5,40){\\line(1,4){2.5}}\n\\put(87.5,40){\\line(1,4){2.5}}\n\\put(122.5,40){\\line(1,4){2.5}}\n\\put(97.5,40){\\line(1,4){2.5}}\n\\put(132.5,40){\\line(1,4){2.5}}\n\\put(107.5,40){\\line(1,4){2.5}}\n\\put(142.5,40){\\line(1,4){2.5}}\n\\put(50,50){\\circle*{1}}\n\\put(15,50){\\circle*{1}}\n\\put(60,50){\\circle*{1}}\n\\put(25,50){\\circle*{1}}\n\\put(70,50){\\circle*{1}}\n\\put(35,50){\\circle*{1}}\n\\put(85,50){\\circle*{1}}\n\\put(120,50){\\circle*{1}}\n\\put(95,50){\\circle*{1}}\n\\put(130,50){\\circle*{1}}\n\\put(105,50){\\circle*{1}}\n\\put(140,50){\\circle*{1}}\n\\put(52.5,40){\\circle*{1}}\n\\put(17.5,40){\\circle*{1}}\n\\put(62.5,40){\\circle*{1}}\n\\put(27.5,40){\\circle*{1}}\n\\put(72.5,40){\\circle*{1}}\n\\put(37.5,40){\\circle*{1}}\n\\put(87.5,40){\\circle*{1}}\n\\put(122.5,40){\\circle*{1}}\n\\put(97.5,40){\\circle*{1}}\n\\put(132.5,40){\\circle*{1}}\n\\put(107.5,40){\\circle*{1}}\n\\put(142.5,40){\\circle*{1}}\n\\put(55,50){\\circle*{1}}\n\\put(20,50){\\circle*{1}}\n\\put(65,50){\\circle*{1}}\n\\put(30,50){\\circle*{1}}\n\\put(75,50){\\circle*{1}}\n\\put(40,50){\\circle*{1}}\n\\put(90,50){\\circle*{1}}\n\\put(125,50){\\circle*{1}}\n\\put(100,50){\\circle*{1}}\n\\put(135,50){\\circle*{1}}\n\\put(110,50){\\circle*{1}}\n\\put(145,50){\\circle*{1}}\n\\put(75,50){\\line(1,-4){5}}\n\\put(80,30){\\line(1,4){5}}\n\\put(80,30){\\circle*{2}}\n\\qbezier(55,50)(57.5,31.5)(80,30)\n\\qbezier(105,50)(102.5,31.5)(80,30)\n\\qbezier(80,30)(67.5,35)(65,50)\n\\qbezier(80,30)(92.5,35)(95,50)\n\\put(75,50){\\circle{2}}\n\\put(40,50){\\circle{2}}\n\\put(60,50){\\circle{2}}\n\\put(25,50){\\circle{2}}\n\\put(90,50){\\circle{2}}\n\\put(125,50){\\circle{2}}\n\\put(95,50){\\circle{2}}\n\\put(130,50){\\circle{2}}\n\\put(69,49){\\framebox(2,2)[cc]{}}\n\\put(34,49){\\framebox(2,2)[]{}}\n\\put(54,49){\\framebox(2,2)[cc]{}}\n\\put(19,49){\\framebox(2,2)[]{}}\n\\put(84,49){\\framebox(2,2)[cc]{}}\n\\put(119,49){\\framebox(2,2)[]{}}\n\\put(109,49){\\framebox(2,2)[cc]{}}\n\\put(144,49){\\framebox(2,2)[]{}}\n\\put(80,20){\\circle*{2}}\n\\qbezier(20,50)(23,24)(80,20)\n\\qbezier(140,50)(137,24)(80,20)\n\\qbezier(80,20)(33,26)(30,50)\n\\qbezier(80,20)(127,26)(130,50)\n\\qbezier(80,20)(44.5,28)(40,50)\n\\qbezier(80,20)(115.5,28)(120,50)\n\\bezier{40}(15,49)(27.5,47)(40,49)\n\\bezier{40}(145,49)(132.5,47)(120,49)\n\\bezier{40}(75,41)(62.5,43)(50,41)\n\\bezier{40}(85,41)(97.5,43)(110,41)\n\\put(55,41.75){\\line(-3,1){20}}\n\\put(105,41.75){\\line(3,1){20}}\n\\put(83,27){\\makebox(0,0)[cc]{$c_b $}}\n\\put(83,17){\\makebox(0,0)[]{$c_t $}}\n\\put(17.5,53){\\makebox(0,0)[]{$A_1 $}}\n\\put(52.5,53){\\makebox(0,0)[cc]{$B_1 $}}\n\\put(97.5,53){\\makebox(0,0)[cc]{$C_1 $}}\n\\put(132.5,53){\\makebox(0,0)[]{$D_1 $}}\n\\put(27.5,53){\\makebox(0,0)[]{$A_2 $}}\n\\put(62.5,53){\\makebox(0,0)[cc]{$B_2 $}}\n\\put(107.5,53){\\makebox(0,0)[cc]{$C_2 $}}\n\\put(142.5,53){\\makebox(0,0)[]{$D_2 $}}\n\\put(37.5,53){\\makebox(0,0)[]{$A_{1,2} $}}\n\\put(72.5,53){\\makebox(0,0)[cc]{$B_{1,2} $}}\n\\put(87.5,53){\\makebox(0,0)[cc]{$C_{1,2} $}}\n\\put(122.5,53){\\makebox(0,0)[]{$D_{1,2} $}}\n\\put(75,80){\\circle*{2}}\n\\put(75,90){\\circle*{2}}\n\\put(85,80){\\circle*{2}}\n\\put(85,90){\\circle*{2}}\n\\put(75,77){\\makebox(0,0)[cc]{$\\overline{d}$}}\n\\put(75,93){\\makebox(0,0)[cc]{${d}$}}\n\\put(85,77){\\makebox(0,0)[cc]{$\\overline{p}$}}\n\\put(85,93){\\makebox(0,0)[cc]{${p}$}}\n\\put(15,85){\\makebox(0,0)[lc]{$A=A_1 \\cup A_2 \\cup A_{1,2} $}}\n\\put(15,75){\\makebox(0,0)[lc]{$B=B_1 \\cup B_2 \\cup B_{1,2} $}}\n\\put(145,85){\\makebox(0,0)[rc]{$C=C_1 \\cup C_2 \\cup C_{1,2} $}}\n\\put(145,75){\\makebox(0,0)[rc]{$D=D_1 \\cup D_2 \\cup D_{1,2} $}}\n\\put(75,80){\\line(0,1){10}}\n\\put(75,90){\\line(1,-1){10}}\n\\put(85,80){\\line(0,1){10}}\n\\put(85,90){\\line(-1,-1){10}}\n\\end{picture}\n}\n\n\n\\caption{Another construction for middle level sets ``$R$\", which allows adjacent\nranks $k$, $k+1$ with equal rank $k$ and rank $k+1$ decks. Here, $\\overline{d} $ is above all maximal elements of\n$B\\cup C$, {\\em except} those that are circled.\n$\\overline{p} $ is above all maximal elements of\n$B\\cup C $, {\\em except} those that are boxed.\nBy transitivity, $d$ and $p$ are above all elements of $B\\cup C $. Moreover,\n$d$\nis above all maximal elements of\n$A\\cup D$, {\\em except} those that are circled.\n$p $ is above all maximal elements of\n$A\\cup D$, {\\em except} those that are boxed.\n}\n\\label{flatswit2}\n\n\\end{figure}\n\n\n\n\n\\begin{remark}\n\\label{consecstack}\n\n{\\rm\nIt may be considered unsatisfying that the ordered sets in\nLemma \\ref{severalmiddle} are such that the ranks $k$ that yield equal rank $k$ decks are\nseparated from each other.\nA small modification in the construction can also produce adjacent ranks\nsuch that the rank $k$ decks of two nonisomorphic sets are equal.\nThe ordered set in Figure \\ref{flatswit2}\nhas the same symmetry properties as the sets $R$\nin Lemma \\ref{severalmiddle}, except\nthat $\\overline{d} $ and $\\overline{p} $ are not minimal.\n\n\nIf these sets are now ``stacked\" as indicated in Figure \\ref{tower3}, with\n$d$'s merged with $\\overline{d} $'s and $p$'s merged with $\\overline{p} $'s we obtain a\ntower structure as in Lemma \\ref{severalmiddle} without $Q$ and $\\tilde{Q} $ attached.\nThe only difference is that the points that\neventually yield equal cards are concentrated in the four crown tower.\nReplacing stretches of sets $R$ in a set as in Lemma \\ref{severalmiddle} with\nsets as in Figure \\ref{tower3} introduces stretches in which\narbitrarily many consecutive ranks produce equal rank $k$ decks.\nAlso note that\nthe elements of the four crown tower in Figure \\ref{tower3}\nare the only elements with rank exceeding 2 and that we can again show that the\nresulting sets will have equal rank $k$ neighborhood decks for all $k$.\n}\n\n\\end{remark}\n\n\n\n\n\n\n\n\n\\begin{figure}\n\n\n\\unitlength 1.00mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi\n\\begin{picture}(137,81)(0,0)\n\\put(65,80){\\line(0,-1){10}}\n\\put(65,70){\\line(1,1){10}}\n\\put(75,80){\\line(0,-1){10}}\n\\put(75,70){\\line(-1,1){10}}\n\\put(65,80){\\circle*{2}}\n\\put(75,80){\\circle*{2}}\n\\put(65,70){\\line(0,-1){10}}\n\\put(65,60){\\line(1,1){10}}\n\\put(75,70){\\line(0,-1){10}}\n\\put(75,60){\\line(-1,1){10}}\n\\put(65,70){\\circle*{2}}\n\\put(75,70){\\circle*{2}}\n\\put(65,60){\\circle*{2}}\n\\put(75,60){\\circle*{2}}\n\\put(65,40){\\line(0,-1){10}}\n\\put(65,30){\\line(1,1){10}}\n\\put(75,40){\\line(0,-1){10}}\n\\put(75,30){\\line(-1,1){10}}\n\\put(65,40){\\circle*{2}}\n\\put(75,40){\\circle*{2}}\n\\put(65,30){\\circle*{2}}\n\\put(75,30){\\circle*{2}}\n\\put(70,50){\\makebox(0,0)[cc]{$\\vdots $}}\n\\put(47,15){\\framebox(8,5)[cc]{}}\n\\put(55,20){\\line(1,-1){5}}\n\\put(60,15){\\framebox(8,5)[cc]{}}\n\\put(85,15){\\framebox(8,5)[cc]{}}\n\\put(85,20){\\line(-1,-1){5}}\n\\put(72,15){\\framebox(8,5)[cc]{}}\n\\put(65,5){\\circle*{2}}\n\\put(75,5){\\circle*{2}}\n\\put(30,25){\\makebox(0,0)[cc]{$\\vdots $}}\n\\put(110,25){\\makebox(0,0)[cc]{$\\vdots $}}\n\\put(65,80){\\vector(-3,-2){52}}\n\\put(65,70){\\vector(-3,-2){38}}\n\\put(65,60){\\vector(-3,-2){35}}\n\\put(65,40){\\vector(-4,-3){25}}\n\\bezier{236}(65,70)(31.33,38)(18.33,37)\n\\put(19,37.1){\\vector(-4,-1){1}}\n\\put(75,80){\\vector(3,-2){52}}\n\\put(75,70){\\vector(3,-2){38}}\n\\put(75,60){\\vector(3,-2){35}}\n\\put(75,40){\\vector(4,-3){25}}\n\\bezier{236}(75,70)(108.67,38)(121.67,37)\n\\put(121,37.1){\\vector(4,-1){1}}\n\\bezier{352}(65,30)(23,17.67)(65,5)\n\\bezier{352}(75,30)(117,17.67)(75,5)\n\\bezier{364}(65,5)(15.33,1)(3,40.33)\n\\bezier{364}(75,5)(124.67,1)(137,40.33)\n\\put(21,15){\\framebox(8,5)[cc]{}}\n\\put(29,20){\\line(1,-1){5}}\n\\put(34,15){\\framebox(8,5)[cc]{}}\n\\put(12,31){\\framebox(8,5)[cc]{}}\n\\put(20,36){\\line(1,-1){5}}\n\\put(25,31){\\framebox(8,5)[cc]{}}\n\\put(5,40){\\framebox(8,5)[cc]{}}\n\\put(13,45){\\line(1,-1){5}}\n\\put(18,40){\\framebox(8,5)[cc]{}}\n\\put(111,15){\\framebox(8,5)[cc]{}}\n\\put(111,20){\\line(-1,-1){5}}\n\\put(98,15){\\framebox(8,5)[cc]{}}\n\\put(120,31){\\framebox(8,5)[cc]{}}\n\\put(120,36){\\line(-1,-1){5}}\n\\put(107,31){\\framebox(8,5)[cc]{}}\n\\put(127,40){\\framebox(8,5)[cc]{}}\n\\put(127,45){\\line(-1,-1){5}}\n\\put(114,40){\\framebox(8,5)[cc]{}}\n\\end{picture}\n\n\n\\caption{A construction to combine sets as in Figure \\protect\\ref{flatswit2}\nto obtain sets in which many consecutive ranks have equal rank $k$ decks.\nFine structure of the involved sets is only indicated roughly\nby showing with arrows which boxes (boxes are stand ins for the sets $A$, $B$, $C$, $D$)\ngo with which points. Crossovers ``point on left to box on right\"\nand symmetrically, as well as points $c_b $ and $c_t $ are omitted.\nThe bottom set (oval with some structure inside) can be a set $R$ or a set $Q$.\nThe two bottom points are lower bounds of all elements in the boxes.\nNone of the boxes have any comparabilities between them unless they are connected,\nand the connection signifies comparability between the minimal and the maximal elements in\nthe indicated direction.\nThis implies that the elements of the four crown tower are the only elements\nin this structure whose rank exceeds $2$.\n}\n\\label{tower3}\n\n\\end{figure}\n\n\n\n\n\n\n\\section{A Folding Operation}\n\n\nThe construction that leads to Theorem \\ref{tallones} produces\nordered sets that are ``tall\"\nin the sense that their height can exceed their width by an arbitrary factor.\nMoreover, the ranks that produce equal rank $k$ decks are, with $2$ elements,\nas small as can be.\nIt is possible to produce ``wider\" examples by folding the examples in Theorem \\ref{tallones}\nappropriately.\nThis idea is explored in this section.\n\n\n\n\n\\begin{define}\n\\label{foldoper}\n\nLet $P$ be a connected finite ordered set.\nAn antichain $A\\subseteq P$ will be called a {\\bf seam}\niff the removal of\n\\begin{enumerate}\n\\item\nAll comparabilities $x<y$\nsuch that there are $a,b\\in A$ with $x<a$ and $b<y$, and of\n\\item\nThe antichain $A$ and all comparabilities involving points of $A$,\n\\end{enumerate}\ndisconnects the ordered set $P$.\nCall the resulting ordered set the {\\bf $A$-separation of $P$}.\nA seam will be called {\\bf foldable}\niff\n\\begin{enumerate}\n\\item[F)]\nFor all $x,y\\in P$ we have that if $x<y$ and there are $a,b\\in A$ such that\n$x<a$ and $b<y$, then there is a $c\\in A$ such that $x<c<y$.\n\\end{enumerate}\n\n\nA seam will be called {\\bf breakable}\niff\n\\begin{enumerate}\n\\item[B)]\nFor all $x,y\\in P$ we have that if there are $a,b\\in A$ such that\n$x<a$ and $b<y$, then $x<y$.\n\\end{enumerate}\n\n\n\nLet $A$ be a foldable or breakable seam in\n$P$, and let $F$\n(the part to be folded)\nand $S$ (the part that will stay as is) be nonempty unions of components\nof the $A$-separation of $P$ such that\nin $P$\nno element of $S$ is above any element of $F$.\nWe define $P_{F,S} $\n(also cf. Figure \\ref{foldant})\nto be the ordered set obtained from $P$ by\n\\begin{enumerate}\n\\item\nErasing all comparabilities $x<y$ in $P$\nsuch that there are $a,b\\in A$ with $x<a$ and $b<y$,\n\n\\item\nKeeping the remaining comparabilities in $S\\cup A$ as is,\n\n\\item\nReversing all comparabilities in $F\\cup A$.\n\n\\end{enumerate}\n\n(It is easy to see that this ``folding operation\" produces an ordered set.)\n\n\\end{define}\n\n\n\\begin{figure}\n\n\\centerline{\n\\unitlength 1.10mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi\n\\begin{picture}(130,41)(0,0)\n\\put(15,10){\\line(1,-1){5}}\n\\put(20,5){\\line(1,1){5}}\n\\put(25,10){\\line(1,-1){5}}\n\\put(30,5){\\line(1,1){5}}\n\\put(35,10){\\line(0,1){10}}\n\\put(35,20){\\line(-1,1){5}}\n\\put(30,25){\\line(-1,-1){5}}\n\\put(25,20){\\line(-1,1){5}}\n\\put(15,20){\\line(0,-1){10}}\n\\put(30,5){\\circle*{2}}\n\\put(20,5){\\circle*{2}}\n\\put(32,25){\\makebox(0,0)[lc]{\\footnotesize $a_2 $}}\n\\put(18,25){\\makebox(0,0)[rc]{\\footnotesize $a_1 $}}\n\\put(15,30){\\line(1,-1){5}}\n\\put(20,25){\\line(1,1){5}}\n\\put(25,30){\\line(1,-1){5}}\n\\put(30,25){\\line(1,1){5}}\n\\put(35,35){\\line(-1,1){5}}\n\\put(30,40){\\line(-1,-1){5}}\n\\put(25,35){\\line(-1,1){5}}\n\\put(20,40){\\line(-1,-1){5}}\n\\put(30,25){\\circle*{2}}\n\\put(20,25){\\circle*{2}}\n\\put(30,40){\\circle*{2}}\n\\put(20,40){\\circle*{2}}\n\\put(20,25){\\line(-1,-1){5}}\n\\put(25,32.67){\\makebox(0,0)[cc]{$F$}}\n\\put(25,15){\\makebox(0,0)[cc]{$S$}}\n\\put(10,40){\\makebox(0,0)[cc]{$P$}}\n\\put(20,5){\\line(-3,1){15}}\n\\put(30,5){\\line(-4,1){20}}\n\\put(13,12){\\makebox(0,0)[rc]{\\tiny (further parts of $S$)}}\n\\put(30,40){\\line(3,-1){15}}\n\\put(20,40){\\line(4,-1){20}}\n\\put(37,33){\\makebox(0,0)[lc]{\\tiny (further parts of $F$)}}\n\\put(70,10){\\line(1,-1){5}}\n\\put(75,5){\\line(1,1){5}}\n\\put(80,10){\\line(1,-1){5}}\n\\put(85,5){\\line(1,1){5}}\n\\put(90,10){\\line(0,1){10}}\n\\put(70,20){\\line(0,-1){10}}\n\\put(85,5){\\circle*{2}}\n\\put(75,5){\\circle*{2}}\n\\put(80,15){\\makebox(0,0)[cc]{$S$}}\n\\put(75,5){\\line(-3,1){15}}\n\\put(85,5){\\line(-4,1){20}}\n\\put(68,12){\\makebox(0,0)[rc]{\\tiny (further parts of $S$)}}\n\\put(100,27){\\makebox(0,0)[cc]{\\footnotesize $a_2 $}}\n\\put(90,27){\\makebox(0,0)[cc]{\\footnotesize $a_1 $}}\n\\put(100,25){\\circle*{2}}\n\\put(90,25){\\circle*{2}}\n\\put(100,25){\\line(-2,-1){10}}\n\\put(70,20){\\line(4,1){20}}\n\\put(90,25){\\line(-2,-1){10}}\n\\put(80,20){\\line(4,1){20}}\n\\put(120,10){\\line(-1,-1){5}}\n\\put(115,5){\\line(-1,1){5}}\n\\put(110,10){\\line(-1,-1){5}}\n\\put(105,5){\\line(-1,1){5}}\n\\put(105,5){\\circle*{2}}\n\\put(115,5){\\circle*{2}}\n\\put(110,13.33){\\makebox(0,0)[cc]{$F$}}\n\\put(115,5){\\line(3,1){15}}\n\\put(105,5){\\line(4,1){20}}\n\\put(122,12){\\makebox(0,0)[lc]{\\tiny (further parts of $F$)}}\n\\put(110,11){\\makebox(0,0)[cc]{\\tiny (folded)}}\n\\put(80,12.33){\\makebox(0,0)[cc]{\\tiny (stayed)}}\n\\put(65,25){\\makebox(0,0)[cc]{$P_{F,S} $}}\n\\put(20,30){\\vector(0,1){5}}\n\\put(30,30){\\vector(0,1){5}}\n\\put(20,10){\\vector(0,1){10}}\n\\put(30,10){\\vector(0,1){10}}\n\\put(75,10){\\vector(0,1){10}}\n\\put(85,10){\\vector(0,1){10}}\n\\put(115,15){\\vector(0,-1){5}}\n\\put(105,15){\\vector(0,-1){5}}\n\\put(15,35){\\line(0,-1){5}}\n\\put(35,30){\\line(0,1){5}}\n\\put(100,10){\\line(0,1){5}}\n\\put(100,15){\\line(-1,1){10}}\n\\put(90,25){\\line(2,-1){20}}\n\\put(110,15){\\line(-1,1){10}}\n\\put(100,25){\\line(2,-1){20}}\n\\put(120,15){\\line(0,-1){5}}\n\\end{picture}\n}\n\n\\caption{The folding operation in Definition \\protect\\ref{foldoper}.}\n\\label{foldant}\n\n\\end{figure}\n\n\nThe folding operation of Definition \\ref{foldoper} is well behaved with respect to\nisomorphism, isomorphism of cards and isomorphism of neighborhoods\nas the next lemma shows.\n\n\n\n\\begin{lem}\n\\label{foldthem}\n\nLet $P,P'$ be connected ordered sets and let $k\\in {\\bf  N}$\nbe such that $A:=\\{ a\\in P: {\\rm rank } _P (a)=k\\} \\subseteq P$ and\n$A':=\\{ a'\\in P': {\\rm rank } _{P'} (a')=k\\} \\subseteq P'$\nare both foldable seams\n(or both breakable seams)\nin $P$ and $P'$ respectively.\nMoreover assume that\nwe can choose\n$F$, $S$, $F'$ and $S'$ as follows.\n\n\n\\begin{enumerate}\n\\item\n$F$ and $S$ are nonempty unions of components\nof the $A$-separation of $P$ such that\n$F$ contains no points that are below any element of $A$ and\n$S$ contains no points that are above any element of $A$.\n\\item\n$F'$ and $S'$ are nonempty unions of components\nof the $A'$-separation of $P'$ such that\n$F'$ contains no points that are below any element of $A'$ and\n$S'$ contains no points that are above any element of $A'$.\n\\item\n\\label{dualrank1}\nAll elements of $P$ that have dual rank $\\geq k$ are in $S\\cup A$\nand all elements of $S$ have rank less than $k$.\n\\item\n\\label{dualrank2}\nAll elements of $P'$ that have dual rank $\\geq k$ are in $S'\\cup A'$\nand all elements of $S$ have rank less than $k$.\n\n\\item\n\\label{nocrossover1}\nNo component of $F\\cup A$ is isomorphic to the dual of a component of $S'\\cup A'$.\n\\item\n\\label{nocrossover2}\nNo component of $F'\\cup A'$ is isomorphic to the dual of a component of $S\\cup A$.\n\n\\end{enumerate}\n\nThen the following hold.\n\n\\begin{enumerate}\n\\item\n\\label{isoequiv}\n$P$ is isomorphic to $P'$ iff $P_{F,S} $ is isomorphic to\n$P' _{F',S'} $.\n\\item\n\\label{isofoldedcards}\nFor all points $p\\in P$ and $p'\\in P'$\nsuch that\n$P\\setminus \\{ p\\} $ is isomorphic to\n$P'\\setminus \\{ p'\\} $\nvia an isomorphism that maps $A\\setminus \\{ p\\} $ to $A'\\setminus \\{ p'\\} $,\n$S\\setminus \\{ p\\} $ to $S'\\setminus \\{ p'\\} $\nand $F\\setminus \\{ p\\} $ to $F'\\setminus \\{ p'\\} $,\nwe have that\n$P_{F,S} \\setminus \\{ p\\} $ is isomorphic to\n$P'_{F',S'} \\setminus \\{ p'\\} $.\n\\item\n\\label{isofoldednhoods}\nFor all points $p\\in P$ and $p'\\in P'$ such that\n${\\rm rank}_P (p)={\\rm rank}_{P'} (p')$\nand\nthe neighborhood $\\updownarrow _P p $\nis isomorphic to the neighborhood\n$\\updownarrow _{P' } p'$, we have that\n${\\rm rank}_{P_{F,S}} (p)={\\rm rank}_{P'_{F',S'}} (p')$\nand the neighborhood\n$\\updownarrow _{P_{F,S} } p $\nis isomorphic to the neighborhood\n$\\updownarrow _{P'_{F',S'} } p'$.\n\\end{enumerate}\n\n\\end{lem}\n\n\n{\\bf Proof.}\nFor part \\ref{isoequiv} first let $P_{F,S} $ be isomorphic to $P'_{F',S'} $\nvia the isomorphism $\\Phi :P_{F,S} \\to P'_{F',S'} $.\nBy hypotheses \\ref{dualrank1} and \\ref{dualrank2}\nwe have that $A=\\{ p\\in P_{F,S} : {\\rm rank } _{P_{F,S} } (p)=k\\} $ and\n$A'=\\{ p'\\in P'_{F,S} : {\\rm rank } _{P'_{F,S} } (p')=k\\} $.\nTherefore $\\Phi [A]=A'$.\nThen by hypotheses\n\\ref{nocrossover1}\nand\n\\ref{nocrossover2}\nwe have that $\\Phi |_{S\\cup A} $ is an isomorphism between $S\\cup A$ and $S'\\cup A'$\nand $\\Phi |_{F\\cup A} $ is an isomorphism between $F\\cup A$ and $F'\\cup A'$, where the\nrespective sets carry the orders that are induced by\n$P_{F,S} $ and by $P' _{F',S'} $, respectively.\nSince the folding construction only reverses the orders on $F\\cup A$ and $F'\\cup A'$,\n$\\Phi |_{S\\cup A} $ is an isomorphism between $S\\cup A$ and $S'\\cup A'$\nand $\\Phi |_{F\\cup A} $ is an isomorphism between $F\\cup A$ and $F'\\cup A'$, where the\nrespective sets carry the orders that are induced by $P$ and by\n$P'$, respectively.\nNow if both $A$ and $A'$ are foldable seams, then all comparabilities\nbetween elements of $F$ and $S$ (and of $F'$ and $S'$) are induced by transitivity through an\nelement of $A$ ($A'$ respectively).\nThis implies that $\\Phi $ is an isomorphism between $P$ and $P'$.\nIf both $A$ and $A'$ are breakable seams, then all comparabilities\nbetween elements of $F$ and $S$ (and of $F'$ and $S'$) that are related to any element of\n$A$ ($A'$ respectively) are present in $P$ ($P'$ respectively) and these are all comparabilities\nbetween elements of $F$ and $S$ (and of $F'$ and $S'$).\nAgain $\\Phi $ is an isomorphism between $P$ and $P'$.\n\nFor the converse, let $P $ be isomorphic to $P' $\nvia the isomorphism $\\Psi :P \\to P' $.\nThen\n$\\Psi |_{S\\cup A} $ is an isomorphism between $S\\cup A$ and $S'\\cup A'$\nand $\\Psi |_{F\\cup A} $ is an isomorphism between $F\\cup A$ and $F'\\cup A'$, where the\nrespective sets carry the orders that are induced by $P$ and by\n$P'$, respectively.\nSince there are no comparabilities between elements of $F$ and $S$ ($F'$ and $S'$ respectively)\nin $P_{F,S} $ ($P'_{F',S'} $), this means that\n$\\Psi $ is also an isomorphism between $P_{F,S} $ and $P'_{F',S'} $.\n\n\n\nFor part \\ref{isofoldedcards},\nlet $\\Psi :P\\setminus \\{ p\\} \\to P\\setminus \\{ p'\\} $ be an isomorphism\nthat maps $A\\setminus \\{ p\\} $ to $A'\\setminus \\{ p'\\} $, $S\\setminus \\{ p\\} $ to $S'\\setminus \\{ p'\\} $\nand $F\\setminus \\{ p\\} $ to $F'\\setminus \\{ p'\\} $.\nThen $\\Psi $ is a bijection between\n$P_{F,S} \\setminus \\{ p\\} $ and $P'_{F',S'} \\setminus \\{ p'\\} $\nthat is an isomorphism between\n$(F\\cup A)\\setminus \\{ p\\} $ and\n$(F'\\cup A')\\setminus \\{ p'\\} $ and between\n$(S\\cup A)\\setminus \\{ p\\} $ and\n$(S'\\cup A')\\setminus \\{ p'\\} $, respectively.\nThus\n$\\Psi $ is an isomorphism between\n$P_{F,S} \\setminus \\{ p\\} $ and\n$P'_{F',S'} \\setminus \\{ p'\\} $.\n\nFor part \\ref{isofoldednhoods} first note that\nbecause the neighborhoods are isomorphic,\nthe dual rank of\n$p$ in $P$ is equal to the dual rank of $p'$ in $P'$.\nSince $p$ and $p'$ have the same rank in their respective unfolded sets,\ntheir ranks in the folded sets is either their\noriginal rank or their original dual rank.\nEither way, their ranks in the folded sets are equal.\nIf the original rank of $p$ and $p'$ is less than $k$, then\ntheir neighborhoods in the folded sets are obtained by\ndiscarding all elements of rank greater than $k$ from the\noriginal neighborhoods.\nIf the original rank of $p$ and $p'$ is greater than $k$, then\ntheir neighborhoods in the folded sets are obtained by\ndiscarding all elements of rank less than $k$ from the\noriginal neighborhoods and dualizing the order.\nIf $p\\in A$ and $p'\\in A'$, then their neighborhoods\nin the folded sets are obtained\nfrom the original neighborhoods by folding at\n$p$ or $p'$, respectively.\nIn all cases the neighborhoods in the folded sets are isomorphic.\n\\hfill\\ \\rule{2mm}{2mm} \n\n\n\n\n\n\n\n\n\\begin{theorem}\n\\label{populatedlevels}\n\nFor every sequence of natural numbers $\\{ s_n \\} _{n\\geq 1} $,\nthere is a sequence of pairs of ordered sets $(Q_1 ^n , Q_2 ^n )$\nsuch that\n\\begin{enumerate}\n\\item\n$Q_1 ^n $ is not isomorphic to $Q_2 ^n $,\n\\item\n\\label{eqmaxmindeck2}\n$Q_1 ^n $ and $Q_2 ^n $ have equal maximal and minimal decks,\n\\item\n\\label{eqrankkdecks2}\nThere are ranks $r_0, \\ldots , r_{n} $ such that $Q_1 ^n $ and $Q_2 ^n $ have\nequal marked rank $r_i $ decks,\nand the ranks $r_1 , \\ldots ,r_n $ do not contain\nextremal elements,\n\\item\n\\label{eqcardeachrnk}\nFor each rank $k$ there is at least one element of rank $k$ in $Q_1 ^n $ and $Q_2 ^n $\nsuch that\nremoval of these elements produces isomorphic marked rank $k$ cards,\n\n\\item\n\\label{eqcardeqremnhood}\nFor all the above mentioned isomorphic cards $Q_1 ^n \\setminus \\{ x\\} $\nand $Q_2 ^n \\setminus \\{ x\\} $ the\nneighborhoods $\\updownarrow _{Q_1 ^n } x $ and $\\updownarrow _{Q_2 ^n } x $\nare isomorphic.\n\n\n\n\n\\item\n$\\displaystyle{ \\liminf _{n\\to \\infty } ECR(Q_1 ^n , Q_2 ^n )\\geq {1\\over 10}} $,\n\n\n\\item\n\\label{equalnhooddecksforQ}\nFor every rank $k$, the rank $k$ neighborhood decks of $Q_1 ^n $ and $Q_2 ^n $\nare equal.\n\n\n\\item\n$Q_1 ^n $ and $Q_2 ^n $ have at least $s_n $ maximal elements, at least\n$s_n $ minimal elements and for each $k_i $ at least $s_n $ elements of rank $k_i $.\n\\end{enumerate}\n\n\\end{theorem}\n\n\n{\\bf Proof.} This result is a direct consequence of Theorem \\ref{tallones} and\nLemma \\ref{foldthem}.\nWithout loss of generality, assume that $s_n $ is even\n(otherwise replace it with $s_n +1$).\nNow let $\\displaystyle{ t_n :={s_n \\over 2}(n+2) } $,\nconsider the pair of sets\n$\\displaystyle{\\left( H^{s_n \\over 2} , K^{s_n \\over 2} \\right) :=\\left( P_1 ^{t_n } , P_2 ^{t_n } \\right) } $\nfrom Theorem \\ref{tallones},\nand let $k_0 , \\ldots , k_{t_n } $ be the ranks mentioned in Theorem \\ref{tallones}.\nNow\nfor $\\displaystyle{ i \\in \\left\\{ {s_n \\over 2}, \\ldots , 1\\right\\} } $,\nobtain $\\left( H^{i-1} , K^{i-1} \\right) $ from $\\left( H^{i} , K^{i} \\right) $ by folding the sets\n$H^{i} , K^{i} $ as indicated in Lemma \\ref{foldthem} at rank\n$[{\\rm height} (Q)+i(n+2) {\\rm height}(R)]$.\nAt each fold, the\nheight of the part $S$ that stays as is is greater than the height of the part $F$ that is folded and\nboth parts are connected. Thus there cannot be any isomorphisms between duals and\nLemma \\ref{foldthem} applies without a problem.\n\nIn a last step, apply the dual of Lemma \\ref{foldthem} at rank $[{\\rm height} Q]$\n(which is possible because the height of $Q$ can be chosen to be equal to the height of $R$).\n\n\nLet $(Q_1 ^n , Q_2 ^n )$ be the pair of ordered sets thus obtained.\nBy Lemma \\ref{foldthem}, the pair $(Q_1 ^n , Q_2 ^n )$ is as desired.\nThe last step of folding up the bottom set $Q$ guarantees\npart \\ref{eqcardeachrnk}.\n\\hfill\\ \\rule{2mm}{2mm} \n\n\n\n\n\\begin{remark}\n\n{\\rm\nWhile the examples in Theorem \\ref{populatedlevels} are not counterexamples to the reconstruction conjecture,\nthey show that the subtlety of order reconstruction reaches beyond tools available today.\nAll parameters that we know to be reconstructible are equal for these sets.\nMoreover, several parameters, such as maximal and minimal decks as well as rank $k$ decks, plus appropriate\nmarkings, are equal also.\nThese parameters\nhave not yet been proven to be reconstructible.\n}\n\n\\end{remark}\n\n\\begin{remark}\n\\label{allbut2}\n\n{\\rm\nUsing sets as in Remark \\ref{consecstack} and their duals and\nfolding appropriately, using Lemma \\ref{foldthem} and its dual,\nit is now possible to produce pairs of nonisomorphic ordered sets\nof arbitrary height\nfor\nwhich all ranks except ranks $1$ and $2$ produce equal rank $k$ decks, for which all\nranks have arbitrarily many elements and for which even in ranks $1$ and $2$ there\nare arbitrarily many elements whose removal produces isomorphic cards.\nThe equal card ratio will approach at least $\\displaystyle{ 1\\over 10} $\nwith the constructions\navailable in this paper.\n}\n\n\\end{remark}\n\n\n\\begin{figure}\n\n\\centerline{\n\\unitlength 1.00mm\n\\linethickness{0.4pt}\n\\begin{picture}(123.00,64.00)\n\\put(10.00,30.00){\\circle*{2.00}}\n\\put(20.00,30.00){\\circle*{2.00}}\n\\put(30.00,30.00){\\circle*{2.00}}\n\\put(40.00,30.00){\\circle*{2.00}}\n\\put(50.00,30.00){\\circle*{2.00}}\n\\put(60.00,30.00){\\circle*{2.00}}\n\\put(70.00,30.00){\\circle*{2.00}}\n\\put(80.00,30.00){\\circle*{2.00}}\n\\put(90.00,30.00){\\circle*{2.00}}\n\\put(100.00,30.00){\\circle*{2.00}}\n\\put(110.00,30.00){\\circle*{2.00}}\n\\put(120.00,30.00){\\circle*{2.00}}\n\\put(10.00,40.00){\\circle*{2.00}}\n\\put(20.00,40.00){\\circle*{2.00}}\n\\put(30.00,40.00){\\circle*{2.00}}\n\\put(40.00,40.00){\\circle*{2.00}}\n\\put(50.00,40.00){\\circle*{2.00}}\n\\put(60.00,40.00){\\circle*{2.00}}\n\\put(70.00,40.00){\\circle*{2.00}}\n\\put(80.00,40.00){\\circle*{2.00}}\n\\put(90.00,40.00){\\circle*{2.00}}\n\\put(100.00,40.00){\\circle*{2.00}}\n\\put(110.00,40.00){\\circle*{2.00}}\n\\put(120.00,40.00){\\circle*{2.00}}\n\\put(10.00,40.00){\\line(1,-1){10.00}}\n\\put(20.00,30.00){\\line(0,1){10.00}}\n\\put(20.00,40.00){\\line(1,-1){10.00}}\n\\put(30.00,30.00){\\line(0,1){10.00}}\n\\put(10.00,30.00){\\line(0,1){10.00}}\n\\put(120.00,40.00){\\line(-1,-1){10.00}}\n\\put(110.00,30.00){\\line(0,1){10.00}}\n\\put(110.00,40.00){\\line(-1,-1){10.00}}\n\\put(100.00,30.00){\\line(0,1){10.00}}\n\\put(120.00,30.00){\\line(0,1){10.00}}\n\\put(50.00,60.00){\\circle*{2.00}}\n\\put(80.00,60.00){\\circle*{2.00}}\n\\put(10.00,40.00){\\line(2,1){40.00}}\n\\put(50.00,60.00){\\line(-3,-2){30.00}}\n\\put(20.00,40.00){\\line(3,1){60.00}}\n\\put(80.00,60.00){\\line(-5,-2){50.00}}\n\\put(120.00,40.00){\\line(-2,1){40.00}}\n\\put(80.00,60.00){\\line(3,-2){30.00}}\n\\put(110.00,40.00){\\line(-3,1){60.00}}\n\\put(50.00,60.00){\\line(5,-2){50.00}}\n\\put(40.00,40.00){\\line(1,-1){10.00}}\n\\put(50.00,30.00){\\line(0,1){10.00}}\n\\put(50.00,40.00){\\line(1,-1){10.00}}\n\\put(60.00,30.00){\\line(0,1){10.00}}\n\\put(40.00,30.00){\\line(0,1){10.00}}\n\\put(90.00,40.00){\\line(-1,-1){10.00}}\n\\put(80.00,30.00){\\line(0,1){10.00}}\n\\put(80.00,40.00){\\line(-1,-1){10.00}}\n\\put(70.00,30.00){\\line(0,1){10.00}}\n\\put(90.00,30.00){\\line(0,1){10.00}}\n\\put(50.00,10.00){\\circle*{2.00}}\n\\put(80.00,10.00){\\circle*{2.00}}\n\\put(40.00,30.00){\\line(1,-2){10.00}}\n\\put(50.00,10.00){\\line(0,1){20.00}}\n\\put(50.00,30.00){\\line(3,-2){30.00}}\n\\put(80.00,10.00){\\line(-1,1){20.00}}\n\\put(90.00,30.00){\\line(-1,-2){10.00}}\n\\put(80.00,10.00){\\line(0,1){20.00}}\n\\put(80.00,30.00){\\line(-3,-2){30.00}}\n\\put(50.00,10.00){\\line(1,1){20.00}}\n\\bezier{70}(39.00,41.30)(65.00,45.30)(91.00,41.30)\n\\put(50.00,60.00){\\line(0,-1){17.37}}\n\\put(80.00,60.00){\\line(0,-1){17.37}}\n\\bezier{30}(39.00,38.70)(50.00,37.30)(61.00,38.70)\n\\bezier{30}(9.00,31.30)(20.00,32.70)(31.00,31.30)\n\\put(50.00,10.00){\\line(-2,1){36.33}}\n\\put(80.00,10.00){\\line(-3,1){54.50}}\n\\put(47.00,38.00){\\line(-4,-1){24.00}}\n\\bezier{30}(9.00,28.70)(20.00,27.30)(31.00,28.70)\n\\bezier{30}(91.00,38.70)(80.00,37.30)(69.00,38.70)\n\\bezier{30}(121.00,31.30)(110.00,32.70)(99.00,31.30)\n\\put(80.00,10.00){\\line(2,1){36.33}}\n\\put(50.00,10.00){\\line(3,1){54.50}}\n\\put(83.00,38.00){\\line(4,-1){24.00}}\n\\bezier{30}(121.00,28.70)(110.00,27.30)(99.00,28.70)\n\\put(10.00,35.00){\\circle*{2.00}}\n\\put(20.00,35.00){\\circle*{2.00}}\n\\put(30.00,35.00){\\circle*{2.00}}\n\\put(40.00,35.00){\\circle*{2.00}}\n\\put(50.00,35.00){\\circle*{2.00}}\n\\put(60.00,35.00){\\circle*{2.00}}\n\\put(70.00,35.00){\\circle*{2.00}}\n\\put(80.00,35.00){\\circle*{2.00}}\n\\put(90.00,35.00){\\circle*{2.00}}\n\\put(100.00,35.00){\\circle*{2.00}}\n\\put(110.00,35.00){\\circle*{2.00}}\n\\put(120.00,35.00){\\circle*{2.00}}\n\\bezier{192}(10.00,30.00)(18.00,37.00)(30.00,40.00)\n\\bezier{192}(40.00,30.00)(48.00,37.00)(60.00,40.00)\n\\bezier{192}(120.00,30.00)(112.00,37.00)(100.00,40.00)\n\\bezier{192}(90.00,30.00)(82.00,37.00)(70.00,40.00)\n\\put(50.00,6.00){\\makebox(0,0)[cc]{$\\overline{d} $}}\n\\put(80.00,6.00){\\makebox(0,0)[cc]{$\\overline{p} $}}\n\\put(50.00,64.00){\\makebox(0,0)[cc]{${d} $}}\n\\put(80.00,64.00){\\makebox(0,0)[cc]{${p} $}}\n\\put(7.00,32.00){\\makebox(0,0)[cc]{$A$}}\n\\put(37.00,32.00){\\makebox(0,0)[cc]{$B$}}\n\\put(93.00,32.00){\\makebox(0,0)[cc]{$C$}}\n\\put(123.00,32.00){\\makebox(0,0)[cc]{$D$}}\n\\end{picture}\n}\n\n\\caption{An ordered set $R$ as needed in Lemma \\protect\\ref{severalmiddle}.\nConnected dotted arches again indicate a complete bipartite structure.\nThe sets in this figure can be used to construct sets with equal maximal and minimal\ndecks for which the ratio of the number of extremal elements to the size of the set is as large as\npossible to date.\n}\n\\label{switch4}\n\n\\end{figure}\n\n\n\\begin{remark}\n\n{\\rm\nAside from results in this paper, the only construction to obtain sets with\nequal maximal decks with $k>3$ cards is due to\nIlle and Rampon (cf. \\cite{JXsurvey}, Section 8.2.2).\nThe size of their sets is exponential in the number of maximal elements.\nThe size of the sets in\nTheorem \\ref{populatedlevels} is linear in the number of maximal\nelements.\nIn this construction, to date the largest ratio of extremal elements to the size of the set is\nachieved with sets $R$ as in Figure \\ref{switch4}. The ratio approaches $\\displaystyle{1\\over 19} $ and it is\nachieved by folding at every merge of sets $R$ and at the merge of the top set $R$ with $\\tilde{Q} $.\nWith sets as presented earlier, the ratio approaches $\\displaystyle{ 1\\over 20} $.\n}\n\n\\end{remark}\n\n\n\n\\section{Further Consequences of the Folding Construction}\n\n\n\nAside from giving access to examples on reconstruction, the folding construction of Definition \\ref{foldoper}\nalso allows access to some results that can simplify the start of a reconstruction proof.\n\nRecall that two elements $x,y$ of an ordered set are called {\\bf adjacent} iff\n$x<y$ and for all $z\\in P$ with $x\\leq z\\leq y$ we have $z\\in \\{ x,y\\} $\n(or the dual of this statement).\nAn ordered set $P$ is called {\\bf graded} iff there is a function\n$g:P\\to {\\bf  N} $ such that if $x,y\\in P$ are adjacent then $|g(x)-g(y)|=1$.\nIf the function $g$ can be chosen to be the function that assigns each element its rank, we say $P$\nis {\\bf graded by the rank function}.\n\n\n\n\\begin{prop}\n\\label{shrinkheight}\n\nIf ordered sets of height $2$ are reconstructible, then\nall ordered sets that are graded by the rank function are reconstructible from their\nmarked ranked decks.\n\n\\end{prop}\n\n\n{\\bf Proof.}\nSuppose $P $ and $P' $ were two nonisomorphic ordered sets that are graded by their\nrank functions and which have equal marked ranked decks.\nNote that in an ordered set that is graded by the rank function, every rank $P_k $ is\na foldable seam.\nRepeatedly folding $P $ and $P' $ as indicated in\nLemma \\ref{foldthem}, at the rank that is one less than the height of the set\nand stopping when the\nresulting sets have height $2$, would create\ntwo nonisomorphic ordered sets of height $2$ with the same deck.\n(Equality of the marked ranked decks is needed here to make sure that all\nhypotheses in part \\ref{isofoldedcards} of the conclusion of Lemma \\ref{foldthem}\nare satisfied.)\nWe have a contradiction to the assumption that\nordered sets of height $2$ are reconstructible.\n\\hfill\\ \\rule{2mm}{2mm} \n\n\n\\begin{remark}\n\\label{ht1remark}\n\n{\\rm\nNote that except for extremely symmetric sets, we could actually fold\nuntil we have an ordered set of height $1$ in the proof of Proposition \\ref{shrinkheight}.\nThis observation underscores once more the importance of understanding more about the\nreconstruction of ordered sets of height $1$.\n}\n\n\\end{remark}\n\n\n\n\n\\begin{prop}\n\\label{onlyoneseam}\n\nIf all ordered sets with at most one foldable or breakable\nseam are reconstructible from their\nmarked ranked decks, then\nall ordered sets are reconstructible from their\nmarked ranked decks\n\n\\end{prop}\n\n{\\bf Proof.}\nSuppose $P_1 $ and $P_2 $ are two nonisomorphic ordered sets that have equal marked ranked decks.\nFind the two consecutive seams for which the difference in the ranks is maximal.\nThen fold all seams except these two.\nIn a final step, one of these two can be folded also.\nThe result are two nonisomorphic ordered sets with equal marked ranked decks and\nat most one foldable or breakable seam.\n\\hfill\\ \\rule{2mm}{2mm} \n\n\n\n\\begin{remark}\n\n{\\rm Proposition \\ref{onlyoneseam} is reminiscent of the result in \\cite{Yongzhi}.\nRemoval of a foldable seam disconnects the\ncovering graph, while removal of a breakable seam is a step towards\ndisconnecting it (cross-connections still need to be erased).\nThis means for reconstruction work, some type of ``vertical connectivity\"\ncould be assumed in any attempt to reconstruct ordered sets.\nUnlike the result in \\cite{Yongzhi}, there is no restriction on the size of the separating set.\n}\n\n\\end{remark}\n\n\n\\begin{remark}\n\n{\\rm\nSimilar to Remark \\ref{ht1remark}, in the proof of Proposition \\ref{onlyoneseam},\nunless we encounter a highly symmetrical situation, both seams\ncan be folded at the end of the proof, leaving ordered sets without seams.\n}\n\n\\end{remark}\n\n\n\n\n\\section{The Role of Rigidity}\n\\label{rigsect}\n\nWe have seen that the rigid bottom and top sets $Q$ and $\\tilde{Q} $ play a crucial role in the\ndevelopment of the examples presented here.\nWithout these rigid ``anchors\" the sets $P_1 $ and $P_2 $ would ``untwist\" and be isomorphic to each other.\nThus it is reasonable to shed some light on the role of rigidity in reconstruction.\nThe following results show that\nisomorphism between various types of rigid substructures immediately leads to an isomorphism between the\nsets.\nSuch rigid substructures are often recognizable from a maximal card or would be\nrecognizable if the marked ranked deck is available.\nThus the presence of such structures leads to reconstructibility.\nConsequently, if there is a counterexample to the order reconstruction conjecture,\nthen it must be made up largely of non-rigid structures.\nFor other properties that a counterexample must have, cf. \\cite{Schrigid,Schmoreexam}.\n\n\nIn particular, the results in this section seem to indicate that any\ndevelopment of counterexamples based on the present examples will\nneed to\n\n\\begin{itemize}\n\\item\nRemove the rigid ``anchors\" without letting the two sets become isomorphic,\n\\item\nAvoid the introduction of rigid structures in middle levels.\n\\end{itemize}\n\nTo state the results in this section, for an ordered set $P$ we denote\n\n\\begin{eqnarray*}\nP_k & := & \\{ x\\in P: {\\rm rank} (x)= k\\} \\\\\nP_{k\\downarrow } & := & \\{ x\\in P: {\\rm rank} (x)\\leq k\\} \\\\\nP_{k\\uparrow } & := & \\{ x\\in P: {\\rm rank} (x)\\geq k\\} .\n\\end{eqnarray*}\n\n\n\n\n\n\n\n\n\\begin{prop}\n\\label{rigidtop}\n\nLet $k\\geq 0$ and let $P,Q$ be ordered sets with equal marked rank $k$ decks.\nIf\n\\begin{enumerate}\n\\item\n$|P_k |>1$ and $|Q_k |>1 $, and\n\\item\nNo two elements of rank $k$\nhave the same strict upper and lower bounds\nin either $P$ or $Q$, and\n\\item\n$P\\setminus P_k $ and $Q\\setminus Q_k $ are both rigid,\n\\end{enumerate}\nthen $P$ is isomorphic to $Q$.\n\n\\end{prop}\n\n{\\bf Proof.}\nLet $P\\setminus \\{ p\\} $ and $Q\\setminus \\{ q\\} $ be isomorphic rank $k$ cards and\nlet $\\psi :P\\setminus \\{ p\\} \\to Q\\setminus \\{ q\\} $ be an isomorphism that\npreserves the original rank of each element.\nLet $P\\setminus \\{ p'\\} $ and $Q\\setminus \\{ q'\\} $ be isomorphic rank $k$ cards\nwith $p\\not= p'$, $q\\not= q'$ and\nlet $\\phi :P\\setminus \\{ p'\\} \\to Q\\setminus \\{ q'\\} $ be an isomorphism that\npreserves the original rank of each element.\nThen $\\phi |_{P\\setminus P_k } =\\psi |_{P\\setminus P_k } $.\n\n\nIf $\\psi (x)=q'$ and $x\\not= p'$, then $\\psi (p')\\not= q'$, which means $y:=\\phi ^{-1} (\\psi (p'))\\not= p'$.\nThis means that $y$ and $p'$ have the same sets of strict upper and lower bounds, a contradiction.\nThus $\\psi (p')=q'$ and symmetrically $\\phi (p)=q$.\n\nNow for any element $x\\in P_k \\setminus \\{ p,p'\\} $, the inequality $\\psi (x)\\not= \\phi (x)$ would imply\n$\\phi ^{-1} (\\psi (x))\\not= x$ has the same strict upper and lower bounds as $x$, a contradiction.\nThus\nfor all $x\\in P_k \\setminus \\{ p,p'\\} $, we have $\\psi (x)= \\phi (x)$.\n\nVia $\\phi |_{P\\setminus P_k } =\\psi |_{P\\setminus P_k } $ we immediately conclude that\n$$\\Phi (x):=\\cases{\n\\psi (x); & for $x\\not= p$, \\cr\n\\phi (p); & for $x=p$, \\cr\n} $$\nis an isomorphism between $P$ and $Q$.\n\\hfill\\ \\rule{2mm}{2mm} \n\n\n\n\n\n\n\\begin{define}\n\nLet $P$ be an ordered set and let $0<k<l$.\nThe\nsubset $P_{k\\uparrow } \\cap P_{l\\downarrow } $ of $P$\nis called a {\\bf rigid separator} iff\n$P_{k\\uparrow } \\cap P_{l\\downarrow } $\nis rigid, there are elements in $P$ of rank $>l$ and\nno element of rank $<k$ is a lower cover of an element of rank $>l$.\n\n\\end{define}\n\n\n\n\n\\begin{prop}\n\\label{rigsep}\n\n\nLet $P$ and $Q$ be ordered sets and let $0<k<l$. If\n\\begin{enumerate}\n\\item\n$P_{k\\uparrow } \\cap P_{l\\downarrow } $\nand $Q_{k\\uparrow } \\cap Q_{l\\downarrow } $ are rigid separators,\n\\item\nThere is an isomorphism $\\phi :P\\setminus \\{ M_P \\} \\to Q\\setminus \\{ M_Q \\}  $,\nwhere\n$M_P $ and $M_Q $ denote maximal elements of rank $r>l$ in $P$ and $Q$, respectively, and where\n${\\rm rank} _Q (\\phi (x)) ={\\rm rank} _P (x)$ for all $x$,\n\\item\nThere is an isomorphism\n$\\psi :P\\setminus \\{ m_P \\} \\to Q\\setminus \\{ m_Q \\} $, where\n$m_P $ and $m_Q $ denote\nminimal elements in $P$ and $Q$, respectively,\nand where\n${\\rm rank} _Q (\\psi (x)) ={\\rm rank} _P (x)$ for all $x$,\n\\end{enumerate}\nthen\n$P$ is isomorphic to $Q$.\n\n\\end{prop}\n\n\n\n{\\bf Proof.}\nBecause we must have\n$\\phi |_{P_{k\\uparrow } \\cap P_{l\\downarrow } } =\\psi |_{P_{k\\uparrow } \\cap P_{l\\downarrow } } $\nand because there are no adjacencies that ``cross\" $P_{k\\uparrow } \\cap P_{l\\downarrow } $\nthe function\n$$\\Phi (x):=\\cases{\n\\psi (x); & if ${\\rm rank} (x)>l$, \\cr\n\\phi (x); & if ${\\rm rank} (x)\\leq l$, \\cr\n} $$\nis an isomorphism between $P$ and $Q$.\n\\hfill\\ \\rule{2mm}{2mm} \n\n\n\n\n\\section{Conclusion}\n\n\n\n\n\nThe examples presented in this paper show that even substantial partial information\non the deck of an ordered set is not sufficient to\neffect reconstruction. In particular (see Theorem \\ref{populatedlevels}),\nthe maximal deck plus the minimal deck plus\n$n+1$ rank $k$ decks, plus the rank $k$ neighborhood decks\nare not sufficient for reconstruction\neven if all these decks have substantially more than $2$ cards.\nMoreover, by Remark \\ref{allbut2} even all rank $k$ decks except for the ones for\nrank $1$ and rank $2$ are not sufficient to effect reconstruction. Again the\nabsolute number of cards within these ranks is immaterial.\n\nFuture reconstruction research has to take these facts into account.\nProof attempts that do not consider enough information from the deck will not succeed.\nOn the other hand, the examples presented here show what kind of\ninformation might effect reconstruction or lead to a counterexample.\n\n\n\n\\begin{enumerate}\n\\item\nAll examples of ordered sets with equal maximal, minimal and\nsome equal rank $k$ decks so far have ``small waists\".\nThat is, the ranks that have equal decks all have fewer elements\n(by at least a factor $6$, even if we use sets $R$ as in Figure \\ref{switch4})\nthan other ranks nearby.\nIt thus should be instructive to investigate what can be\nconcluded from equality of rank $k$ decks, where $k$ is such that\nno other rank has more elements than the $k^{\\rm th} $ rank.\n\n\\item\nAlternatively, the construction of examples could be expanded to the\npoint where the overlap between the decks of two nonisomorphic sets\nbecomes so large that the decks would indeed have to be equal.\nThe present examples have been shown to be quite malleable. They might point the\nway towards examples with similarly strong properties and\nlarger equal card ratios or maybe even a counterexample\noverall. The largest equal card ratios in ordered sets observed so far slightly exceed 50\\%\n(see \\cite{JHthes}), but the examples do not have the equal subdecks that the examples presented here have.\n\n\\item\nAlong these lines, would it be true that\nif there is a sequence of pairs of nonisomorphic ordered sets $(P_1 ^n , P_2 ^n )$\nsuch that $ECR (P_1 ^n , P_2 ^n ) \\to 1$ as $n\\to \\infty $, then\nthere is a counterexample to the reconstruction conjecture?\n\n\\item\nOn p. 185 of \\cite{Manv}, P. Stockmeyer is quoted to have said\nonly {\\em half} in jest that ``The reconstruction conjecture [for graphs]\nis not true, but the smallest counterexample has 87 vertices and will\nnever be found.\" The present examples seem to show that if there is a\ncounterexample to the order reconstruction conjecture, it would have to be\nof substantial size and complexity.\nAt the same time, the approach of analyzing macrostructure\nas in Lemma \\ref{severalmiddle}\nand microstructure as in Lemma \\ref{getsmallerR}\nseparately may allow to\nbreak down the complexity to manageable stages.\n\n\n\n\\item\nFinally, Proposition \\ref{onlyoneseam} shows that we can concentrate on\nordered sets that have a certain type of ``vertical connectivity\".\n\n\n\n\n\\end{enumerate}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Background: computational methods for epidemiology}\n\\label{sec:models}\nEpidemiological models fall in two broad classes: statistical models that are largely data driven and mechanistic models that are based on underlying theoretical principles developed by scientists on how the disease spreads.\n\nData-driven models use statistical and machine learning methods to forecast outcomes, such as case counts, mortality and hospital demands. This is a very active area of research, and a broad class of techniques have been developed, including auto-regressive time series methods, Bayesian techniques and deep learning  \\cite{adhikari:kdd19, perone:ssrn20, desai:hs19, Reich2019AccuracyOR, funk:epidemics18, murray:ihme}. \nMechanistic models of disease spread within a population~\\cite{Brauer-2008,Ne03, marathe:cacm13,ekmsw-2006} use mechanistic (also referred to as procedural or algorithmic) \nmethods to describe the evolution of an epidemic through a population. The most common of these is the SIR type models. Hybrid models that combine mechanistic models with data driven machine learning approaches are also starting to become popular, e.g.,~\\cite{Wang2019DEFSIDL}.\n\n\\subsection{Mass action compartmental models}\nThere are a number of models, which are referred to as SIR class of models. These partition a population of $N$ agents into three sets, each corresponding to a disease state, which is one of: susceptible ($S$), infective ($I$) and removed or recovered ($R$). \nThe specific model then specifies how susceptible individuals become infectious, and then recover. In its simplest form (referred to as the basic compartmental model)~\\cite{Brauer-2008,Ne03, marathe:cacm13}, the population is assumed to be completely mixed. \nLet $S(t)$, $I(t)$ and $R(t)$ denote the number of people who are susceptible, infected and recovered states at time $t$, respectively.\nLet $s(t)=S(t)\/N$, $i(t)=I(t)\/N$ and $r(t)=R(t)\/N$; then,\n$s(t)+i(t)+r(t)=1$. Then, the SIR model can be described by the following system of ordinary differential equations\n\\[\n\\frac{ds}{dt} = \\beta si, \\qquad \\frac{di}{dt} = \\beta si - \\gamma i, \\qquad \\frac{dr}{dt} = \\gamma i,\n\\]\nwhere $\\beta$ is referred to as the transmission rate, and $\\gamma$ is the recovery rate.\nA key parameter in such a model is the ``reproductive number'', denoted by $R_0=\\beta\/\\gamma$.\nAt the start of an epidemic, much of\nthe public health  effort is focused on estimating $R_0$ from observed  infections~\\cite{lipsitch:science03}.\n\n Mass action compartmental models have been the workhorse for epidemiologists\n and have been widely used for over 100 years. Their strength comes from their simplicity, both analytically and from the standpoint of understanding the\n outcomes. Software systems have been developed to solve such models and a number of associated tools have been built to support analysis using such models.\n \n\\subsection{Structured metapopulation models}\nAlthough simple and powerful, mass action compartmental models\ndo not capture the inherent heterogeneity of the underlying populations.\nSignificant amount of research has been conducted to extend the model, usually in two broad ways.  \nThe first involves structured metapopulation models---these construct an abstraction of the mixing patterns in the population into $m$ different sub-populations, e.g., age groups and small geographical regions, and attempt to capture the heterogeneity in mixing patterns across subpopulations. In other words, the model has states $S_j(t), I_j(t), R_j(t)$ for each subpopulation $j$. The evolution of a compartment $X_j(t)$ is determined by mixing within and across compartments.\nFor instance, survey data on mixing across age groups~\\cite{Mossong2008SocialCA} has been used to construct age structured metapopulation models~\\cite{medlock+optvacc09}. More relevant for our paper are spatial metapopulation models, in which the subpopulations are connected through airline and commuter flow networks~\\cite{balcan:pnas2009, srini:ploscb19, gomes:plos-curr14, zhang:pnas17, chinazzi:science20}. \n\n\\medskip\n\\noindent \n\\textbf{Main steps in constructing structured metapopulation models:} This depends on the disease, population and the type of question being studied. The key steps in the development of such models for the spread of diseases over large populations include\n\\begin{itemize}\n\\item \nConstructing subpopulations and compartments: the entire population $V$ is partitioned into subpopulations $V_j$, within which the mixing is assumed to be complete. Depending on the disease model, there are $S_j, E_j, I_j, R_j$ compartments corresponding to the subpopulation $V_j$ (and more, depending on the disease)---these represent the number of individuals in $V_j$ in the corresponding state\n\\item \nMixing patterns among compartments: state transitions between compartments might depend on the states of individuals within the subpopulations associated with those compartments, as well as those who they come in contact with. For instance, the $S_j\\rightarrow E_j$ transition rate might depend on $I_k$ for all the subpopulations who come in contact with individuals in $V_j$. Mobility and behavioral datasets are needed to model such interactions.\n\\end{itemize}\n\n\nSuch models are very useful at  the early days of the outbreak, when the disease dynamics are\ndriven to a large extent by mobility---these can be captured more easily within such models,\nand there is significant uncertainty in the disease model parameters. They can also model coarser interventions such as reduced mobility between spatial units and reduced mixing rates. However, these models become less useful to model the effect of detailed interventions\n(e.g., voluntary home isolation, school closures) on disease spread in and across communities. \n\n \n\\subsection{Agent based network models}\nAgent-based networked models (sometimes just called as agent-based models)  \nextend metapopulation models further by explicitly capturing the \ninteraction structure of the underlying populations. Often such models are also resolved at the level of single individual entities (animals, humans etc.).\nIn this class of models,  the epidemic dynamics can be modeled as a diffusion process on a specific undirected contact network $G(V,E)$ \non a population $V$ -- each edge $e=(u,v)\\in E$ implies that\nindividuals (also referred to as nodes) $u, v\\in V$ come into contact\\footnote{%\nNote that though edge $e$ is\nrepresented as a tuple $(u, v)$, it actually denotes the set $\\{u, v\\}$, as\nis common in graph theory.}\nLet $N(v)$ denote the set of neighbors of $v$. \nFor instance, in the graph in Figure \\ref{fig:network-ex}, we have $V=\\{a, b, c, d\\}$\nand $E=\\{(a, b), (a, c), (b, d), (c d)\\}$. Node $a$ has $b$ and $c$ as neighbors, so $N(a)=\\{b, c\\}$.\nThe SIR model on the graph $G$ is a dynamical process in which\neach node is in one of $S$, $I$ or $R$ states.  Infection can\npotentially spread from $u$ to $v$ along edge $e=(u,v)$ with a probability of\n$\\beta(e,t)$ at time instant $t$\nafter $u$ becomes infected, conditional on node $v$ remaining uninfected\nuntil time $t$--- this is a discrete version of the rate of infection for the\nODE model discussed earlier. \nWe let $I(t)$ denote the set of nodes that become\ninfected at time $t$. \nThe (random) subset of edges on which the infections spread represents\na disease outcome, and is referred to as a \\emph{dendogram}.\nThis dynamical system starts with a configuration in which there are one or more nodes\nin state {\\sf I} and reaches a fixed point in which all nodes are in states {\\sf S} or {\\sf R}.\nFigure \\ref{fig:network-ex} shows an example of the SIR model on a network.\n\\medskip \n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.5\\linewidth]{figs\/sir-nw.pdf}\n\\caption{The SIR process on a graph. The contact graph $G=(V, E)$ is defined on\na population $V=\\{a, b, c, d\\}$. The node colors white, black and grey represent\nthe Susceptible, Infected and Recovered states, respectively. Initially, only node $a$\nis infected, and all other nodes are susceptible. A possible outcome at $t=1$\nis shown, in which node $c$ becomes infected, while node $a$ recovers. Node $a$ tries\nto independently infect both its neighbors $b$ and $c$, but only node $c$ gets\ninfected--- this is indicated by the solid edge $(a, c)$.\nThe probability of getting\nthis outcome is $(1-p(a,b))p(a,c)$.}\n\\label{fig:network-ex}\n\\end{center}\n\\end{figure*}\n\n\\medskip\n\\noindent \n\\textbf{Main steps in setting up an agent based model.} While the specific steps depend on the disease, the population, and the type of question being studied, the general process involves the following steps:\n\\begin{itemize}\n\\item \nConstruct a network representation $G$: the set $V$ is the population in a region, and is available from different sources, such as Census and Landscan. However, the contact patterns are more difficult to model, as no real data is available on contacts between people at a large scale. Instead, researchers have tried to model activities and mobility, from which contacts can be inferred, based on co-location. Multiple approaches have been developed for this, including random mobility based on statistical models, and very detailed models based on activities in urban regions, which have been estimated through surveys, transportation data, and other sources, e.g.,~\\cite{ekmsw-2006,barrett:wsc09,eubank2004modelling, longini05:science,Ferg20}.\n\\item \nDevelop models of within-host disease progression: such models can be represented as finite state probabilistic timed transition models, which are designed in close coordination with biologists, epidemiologists, and parameterized using detailed incidence data (see~\\cite{marathe:cacm13} for discussion and additional pointers).\n\\item \nDevelop high performance computer (HPC) simulations to study epidemic dynamics in such models, e.g.,~\\cite{barrett2008episimdemics, bisset2009epifast, deodhar2012enhancing, grefenstette2013fred}. Typical public health analyses involve large experimental designs, and the models are stochastic; this necessitates use of such HPC simulations on large computing clusters.\n\\item \nIncorporate interventions and behavioral changes: interventions include closure of schools and workplaces~\\cite{Ferg20,halloran:pnas08} and vaccinations~\\cite{eubank2004modelling}, whereas behavioral changes include individual level social distancing, changes in mobility, and use of protective measures.\n\\end{itemize}\n\n\nSuch a network model captures the interplay between the three components of \ncomputational epidemiology: ($i$) individual behaviors of agents, \n($ii$) unstructured, heterogeneous multi-scale networks, \nand ($iii$) the dynamical processes on these networks.\nIt is based on the hypothesis that a better understanding of the\ncharacteristics of the underlying network and individual behavioral\nadaptation can give better insights into contagion dynamics and\nresponse strategies.\nAlthough computationally expensive and data intensive, network-based epidemiology\nalters the types of questions that can be posed, providing\nqualitatively different insights into disease dynamics and public health policies.\nIt also allows policy makers to formulate\nand investigate potentially novel and context specific  interventions. \n\n\n\\subsection{Models for epidemic forecasting}\nLike projection approaches, models for epidemic forecasting can be broadly classified into two broad groups: ($i$) statistical and machine learning based data driven models, ($ii$) causal or mechanistic models -- see~\\cite{desai:hs19,Reich2019AccuracyOR,nsoesie2014systematic,chretien2014influenza,kandula2019near,tabataba2017framework,brooks2020pancasting}  and the references therein for the current state of the art in this rapidly evolving field.\n\nStatistical methods employ statistical and time-series based methodologies to\nlearn patterns in historical epidemic data and leverage those patterns for forecasting. Of course the simplest yet useful class is called\n{\\em method of analogues}. One simply compares the current epidemic with one of the earlier outbreaks and then uses the best match to forecast the\ncurrent epidemic. Popular statistical methods for forecasting influenza\nlike illnesses (that includes COVID-19) \ninclude e.g. generalized linear models (GLM), autoregressive integrated moving average (ARIMA), and generalized autoregressive moving average (GARMA)~\\cite{kandula2019near,LANLforecasts,IHMEcovid2020forecasting}.\nStatistical  methods are fast, but they crucially depend on the availability of training data.  Furthermore, since they are purely data driven, they do\nnot capture the underlying causal mechanisms. As a result epidemic dynamics affected by behavioral adaptations are usually hard to capture.\nArtificial neural networks (ANN) have gained increased\nprominence in epidemic forecasting due to their self-learning ability without prior knowledge~(See \\cite{Wang2019DEFSIDL,wang2020tdefsi,adhikari:kdd19} and the\nreferences therein). \nSuch models have used a wide variety of data as surrogates\nfor producing forecasts. This includes: ($i$) social media data, ($ii$) weather data, ($iii$) incidence curves and ($iv$) demographic data.\n\nCausal models can be used for epidemic forecasting in a natural \nmanner~\\cite{funk:epidemics18,nsoesie2014systematic,\nfadikar2018calibrating,tabataba2017epidemic,chretien2014influenza,yamana2017individual}.\nThese models, calibrate the internal model parameters using the disease\nincidence data seen until a given day and then execute the model forward in time to produce the future time series. Compartmental as well as agent-based models can be used to produce such forecasts. The choice of the models depends on the specific question at hand and the computational and data resource constraints.\nOne of the key ideas in forecasting is to develop ensemble models -- models that combine forecasts from multiple models~\\cite{chakraborty2014forecasting,Reich2019AccuracyOR,yamana2017individual,tabataba2017epidemic}. The idea which originated in the domain of weather forecasting has found methodological advances in the machine learning literature. Ensemble models\ntypically show better performance than the individual models.\n\n\n\\section{Biographies}\n\n\\begin{wrapfigure}{r}{0.16\\textwidth}\n\\vspace{-13pt}\n\\centering\n\\includegraphics[width=0.14\\textwidth]{figs\/aniruddha_adiga_webedited.jpg}\n\\end{wrapfigure}\n\nAniruddha Adiga is a Postdoctoral Research Associate at the NSSAC Division of the Biocomplexity Institute and initiative. He completed his PhD from the Department of Electrical Engineering, Indian Institute of Science (IISc), Bangalore, India and has held the position of Postdoctoral fellow at IISc and North Carolina State University, Raleigh, USA. \nHis research areas include signal processing, machine learning, data mining, forecasting, big data analysis etc. At NSSAC, his primary focus has been the analysis and development of forecasting systems for epidemiological signals such as influenza-like illness and COVID-19 using auxiliary data sources.\\\\   \n\n\\begin{wrapfigure}{l}{0.18\\textwidth}\n\\vspace{-10pt}\n\\centering\n\\includegraphics[width=0.16\\textwidth]{figs\/Devdatt_Dubhashi-3.jpg}\n\\end{wrapfigure}\n\\noindent\nDevdatt Dubhashi is a Professor in the Data Science and AI Division in the Department of computer Science and Engineering, Chalmers University Sweden. His research interests are in machine learning, algorithms and cognitive science. \nHe received his Ph.D. in Computer Science at Cornell University USA and has held positions at the Max Planck Institute for Computer Science, Saarbr\\\"ucken Germany, BRICS (Basic Research in computer Science) a center of the Danish National Science Foundation at the University of Aarhus Denmark and at the Indian Institute of Technology, Delhi India.\\\\\n\n\\begin{wrapfigure}{r}{0.15\\textwidth}\n\\vspace{-14pt}\n\\centering\n\\includegraphics[width=0.13\\textwidth]{figs\/Bryan-Lews.png}\n\\end{wrapfigure}\nBryan Lewis is a research associate professor in the Network Systems Science and Advanced Computing division. His research has focused on understanding the transmission dynamics of infectious diseases within specific populations through both analysis and simulation. Lewis is a computational epidemiologist with more than 15 years of experience in crafting, analyzing, and interpreting the results of models in the context of real public health problems. As a computational epidemiologist, for more than a decade, Lewis has been heavily involved in a series of projects forecasting the spread of infectious disease as well as evaluating the response to them in support of the federal government. These projects have tackled diseases from Ebola to pandemic influenza and melioidosis to cholera.\\\\\n\n\n\n\\begin{wrapfigure}{l}{0.17\\textwidth}\n\\vspace{-13pt}\n\\centering\n\\includegraphics[width=0.14\\textwidth]{figs\/Madhav-Picture-Feb2019.pdf}\n\\end{wrapfigure}\nMadhav Marathe is a Distinguished Professor in \nBiocomplexity, the division director of the Networks, Simulation Science and Advanced Computing (NSSAC) Division at the Biocomplexity Institute and  \nInitiative, and a Professor in the Department of Computer Science at the University of Virginia (UVA).  \nHis research interests are in network\nscience, computational epidemiology, AI, foundations of computing,\nsocially coupled system science and high performance computing. \nBefore joining UVA, he held positions at Virginia Tech and  the Los Alamos National Laboratory. He is a Fellow of the IEEE, ACM, SIAM and AAAS.\\\\\n\n\\noindent\n\\begin{wrapfigure}{r}{0.15\\textwidth}\n\\vspace{-13pt}\n\\centering\n\\includegraphics[width=0.13\\textwidth]{figs\/Srini_Headshot_2019.jpg}\n\\end{wrapfigure}\nSrinivasan (Srini) Venkatramanan is a Research Scientist at the Biocomplexity Institute \\& Initiative, University of Virginia and his research focuses on developing, analyzing and optimizing computational models in the field of network epidemiology. He received his PhD from the Department of Electrical and Communication Engineering, Indian Institute of Science (IISc), and did his postdoctoral research at Virginia Tech. His areas of interest include network science, stochastic modeling and big data analytics. He has used in-silico models of society to study the spread of infectious diseases and invasive species. Recent research includes modeling and forecasting emerging infectious disease outbreaks (e.g., Ebola, COVID-19), impact of human mobility on disease spread and resource allocation problems in the context of epidemic control (e.g., seasonal influenza vaccines). \\\\\n\n\n\\begin{wrapfigure}{l}{0.15\\textwidth}\n\\vspace{-14pt}\n\\centering\n\\includegraphics[width=0.13\\textwidth]{figs\/anil.jpg}\n\\end{wrapfigure}\n\\medskip\n\\noindent\nAnil Vullikanti is a Professor in the Biocomplexity Institute and the Department of Computer Science\nat the University of Virginia. He got his PhD at the Indian Institute of Science, and was a postdoctoral associate at the Max Planck Institute for Computer Science, Saarbr\\\"ucken, Germany, and the Los Alamos National Lab. His research interests are in the broad areas of network science, dynamical systems, combinatorial optimization, and distributed computing, and their applications to computational epidemiology and social networks.\n\\section{Models and Policy making}\n\\label{sec:discussion}\n\\noindent\n\\textbf{Were some of the models wrong?} \\ \nIn a recent opinion piece\\footnote{\\emph{Indian Express}, July 30, 2020}, Professor Vikram Patel of the Harvard School of Public Health makes a stinging criticism of modelling:\n\\begin{quote}\n    Crowning these scientific disciplines is the field of modelling, for it was its estimates of mountains of dead bodies which fuelled the panic and led to the unprecedented restrictions on public life around the world. None of these early models, however, explicitly acknowledged the huge assumptions that were made,\n\\end{quote}\nA similar article in NY Times recounted the \\emph{mistakes} in COVID-19 response in Europe\\footnote{NY Times July 20, 2020: \\url{https:\/\/www.nytimes.com\/2020\/07\/20\/world\/europe\/coronavirus-mistakes-france-uk-italy.html}}; also see~\\cite{avery2020policy}.\n\n\\medskip\n\\noindent \n\\textbf{Our point of view.} \\\nIt is indeed important to ensure that assumptions underlying mathematical models be made transparent and explicit. But we respectfully \ndisagree with Professor Patel's statement: most of the \\emph{good} models tried to be very explicit about their assumptions. The mountains of deaths that are being referred to are explicitly calculated when {\\bf no} interventions are put in place and are often used as a worst case scenario. Now, one might argue that the authors be explicit and state that this worst case scenario will never occur in practice.\nForecasting dynamics in social systems is inherently challenging: individual\nbehavior, predictions and epidemic dynamics {\\em co-evolve}; this coevolution immediately implies that a dire prediction can lead to extreme change in\nindividual and collective behavior leading to reduction in the incidence numbers. Would one say forecasts were wrong in such a case or they were\ninfluential in ensuring the worst case never happens? None of this implies that one should not explicitly state the assumption underlying their model.\nOf course our experience is that policy makers,\nnews reporters and common public are looking exactly for such a forecast --\nwe have been constantly asked \"when will peak occur\" or \"how many people are likely to die\". A few possible ways to overcome this tension between the unsatiable appetite for forecasts and the inherent challenges that lie\nin doing this accurately, include: \n\\begin{itemize}\n    \\item \nWe believe that in general it might not be prudent to provide long term\nforecasts for such systems. \n    \\item\nState the assumptions underlying the models as clearly as possible. Modelers need to be much more disciplined about this. They also need to ensure that the models are transparent and can be reviewed broadly (and expeditiously).\n    \\item\nAccept that the forecasts are provisional and that they will be revised\nas new data comes in, society adapts, the virus adapts and we understand the biological impact of the pandemic.\n   \\item\nImprove surveillance systems that would produce data that the models can use more effectively. Even with data, it is very hard to estimate the prevalence of  COVID-19 in society.\n\\end{itemize}\nCommunicating scientific findings and risks is an important topical\narea in this context, see~\\cite{fischhoff2019evaluating,metcalf2020mathematical,adam2020modelling,vaezi2020infodemic}. \n\n\\medskip\n\\noindent \n\\textbf{Use of models for evidence-based policy making.} \\\nIn a new book, \\cite{KK20}, \\emph{Radical Uncertainty}, economists John Kay and Mervyn King (formerly Governor of the Bank of England) urge caution when using complex models. They argue that models should be valued for the insights they provide but not relied upon to provide accurate forecasts. The so-called \"evidence--based policy\" comes in for criticism where it relies on models but also supplies a false sense of certainty where none exists, or seeks out the evidence that is desired \\emph{ex ante} -- or \"cover\" -- to justify a policy decision. \\emph{\"Evidence based policy has become policy based evidence\".} \n\n\\medskip\n\\noindent \n\\textbf{Our point of view.} \\\nThe authors make a good point here. But again, everyone, from public\nto citizens and reporters clamour for a forecast. We argue that this can\nbe addressed in two ways:($i$) viewing the problem from the lens \nof control theory so that we forecast only to control the deviation\nfrom the path we want to follow and ($ii$) not insisting on exact numbers but general trends. As Kay and King opine, the value of models, especially in the face of \\emph{radical uncertainty}, is more in exploring alternative scenarios resulting from different policies:\n\\begin{quote}\n  a model is useful only if the person using it understands that it does not represent the \"the world as it really is\" but is a tool for exploring ways in which a decision might or might not go wrong. \n\\end{quote}\n\n\n\\medskip\n\\noindent \n\\textbf{Supporting science beyond the pandemic.} \\\nIn his new book \\emph{The Rules of Contagion}, Adam Kucharski \\cite{K20} draws on lessons from the past. In 2015 and 2016, during the Zika outbreak, researchers planned large-scale  clinical studies and vaccine trials. But these were discontinued as soon as the infection ebbed.  \n\\begin{quote}\n    This is a common frustration in outbreak research; by the time the infections end, fundamental questions about the contagion can remain unanswered. That's why building long term research capacity is essential.\n\\end{quote}\n\n\\medskip\n\\noindent \n\\textbf{Our point of view.} \\\nThe author makes an important point. We hope that today, after witnessing the devastating impacts of the pandemic on the economy and society, the correct lessons will be learnt: sustained investments need to be made in the field to be ready for the impact of the next pandemic.\n\n\\textbf{Concluding remarks}\nThe paper discusses a few important computational models \ndeveloped by researchers in the US, UK and Sweden for COVID-19 pandemic planning and response. The models have been used by \npolicy makers and public health officials in their respective \ncountries to assess the evolution of the pandemic, design and analyze\ncontrol measures and study various what-if scenarios.\nAs noted, all models faced challenges due to availability of data,\nrapidly evolving pandemic and unprecedented control measures put in place.\nDespite these challenges, we believe that mathematical models can provide\nuseful and timely information to the policy makers. On on hand the\nmodelers need to be transparent in the description of their models, \nclearly state the limitations and carry out detailed sensitivity \nand uncertainty quantification. Having these models reviewed independently\nis certainly very helpful. On the other hand, policy makers should be aware\nof the fact that using mathematical models \nfor pandemic planning, forecast\nresponse rely on a number of assumptions and lack data to over these\nassumptions. \n\n\n\\section*{Acknowledgments}\nThe authors would like to thank members of the Biocomplexity COVID-19 Response Team and Network Systems Science and Advanced Computing (NSSAC) Division for their thoughtful comments and suggestions related to epidemic modeling and response support. We thank members of the Biocomplexity Institute and Initiative, University of Virginia for useful discussion and suggestions. \nThis work was partially supported by National Institutes of Health (NIH) Grant 1R01GM109718, NSF BIG DATA Grant IIS-1633028, NSF DIBBS Grant ACI-1443054, \nNSF Grant No.: OAC-1916805, NSF Expeditions in Computing Grant CCF-1918656, CCF-1917819,  NSF RAPID CNS-2028004, NSF RAPID OAC-2027541, US Centers for Disease Control and Prevention 75D30119C05935, DTRA subcontract\/ARA S-D00189-15-TO-01-UVA. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the funding agencies.\n\n\n\\section{Introduction}\n\n\n\\iffalse\n\\textcolor{blue}{Maybe the following para is not needed. Also might not need some of the previous para, as it is fairly common knowledge now. I have\nleft his as is but moved things around. Over time the article becomes archival and so good to have it.}\nThe word epidemic is derived from two Greek works {\\em epi} (upon) and {\\em demos} (people) \nmeaning ``upon people''.\nAn epidemic is an occurrence in a community or region, of cases of an\nillness, specified health behavior or other health related events\nclearly in excess of normal expectancy. A pandemic refers to incidence of\nthe disease of global proportions, e.g., the H1N1 outbreak in 2009.\nIn contrast, an endemic disease is one\nwhose cases are constantly occurring in the population. \n{\\em Epidemiology} is the study of space-time patterns of illness in a population and\nthe factors that contribute to these patterns. It plays an essential\nrole in public health through the elucidation of the processes that\nlead to ill health as well as the evaluation of strategies designed to\npromote good health. Epidemiologists are primarily concerned with\npublic health data, which includes the design of studies, evaluation\nand interpretation of public health data, and the maintenance of data\ncollection systems. \n\\fi\n\n\nThe ongoing COVID-19 pandemic is the most significant pandemic since the 1918 Influenza pandemic. It has already caused over 21 Million confirmed cases and 758,000 deaths\\footnote{The numbers reported are as of August 14, 2020.\nSee \\url{https:\/\/coronavirus.jhu.edu\/map.html} and\n\\url{https:\/\/nssac.bii.virginia.edu\/covid-19\/dashboard\/} for most up to date\nsurveillance information}. \nThe economic impact is already in trillions of dollars.\nAs in other pandemics, researchers and public health policy makers are\ninterested in questions such as\\footnote{see \\url{https:\/\/www.nytimes.com\/news-event\/coronavirus}},\n($i$) How did it start?\n($ii$) How is it likely to progress and how can we control it?\n($iii$) How can we intervene while balancing public health and economic impact ?\n($iv$) Why did some countries do better than other countries thus far into the pandemic?\nIn particular, models and their projections\/forecasts have received unprecedented attention. With a multitude of modeling frameworks, underlying assumptions, available datasets and the region\/timeframe being modeled, these projections have varied widely, causing confusion among end-users and consumers. We believe an overview (non-exhaustive) of the current modeling landscape will benefit the readers and also serve as a historical record for future efforts. \n\n\\subsection{Role of models} \\ Models have been used by mathematical epidemiologists\nto support a broad range of policy questions. Their use during COVID-19\nhas been widespread. In general, the type and form of models used in epidemiology depends on the phase of the epidemic.  \nBefore an epidemic, models are used for planning and identifying critical gaps and prepare plans to detect and respond in the event of a pandemic. At the start of a pandemic, policy makers are interested in asking questions such as: ($i$) where and how did the pandemic start, ($ii$) risk of its spread in the region, ($iii$) risk of importation in other regions of the world, ($iv$) basic understanding of the pathogen and its epidemiological characteristics. As the pandemic takes hold researchers begin investigating: ($i$) various intervention and\ncontrol strategies; usually pharmaceutical interventions do not work in the event of a pandemic and thus non-pharmaceutical interventions are most appropriate, ($ii$) forecasting the epidemic incidence rate, hospitalization rate and mortality rate, ($iii$) efficiently allocating scarce medical resources to treat the patients and ($iv$) understanding the change in individual and collective behavior and adherence to public policies.\nAfter the pandemic starts to slow down, modelers are interested in developing models related to recovery and long term impacts caused by the pandemic.\n\nAs a result comparing models needs to be done with care. When comparing models: one needs to specify: ($a$) the purpose of the model, ($b$) the end user to whom the model is targeted, ($c$) the spatial and temporal resolution of the model, ($d$) and the underlying assumptions and limitations. We illustrate these issues by summarizing a few key methods for \\emph{projection and forecasting} of disease outcomes in the US and Sweden.\n\n\n\n\n\\medskip\n\\noindent\n\\textbf{Organization.} \\ The paper is organized as follows. In \nSection~\\ref{sec:models} we give preliminary definitions. \nSection~\\ref{sec:imperial} discusses US and UK centric models developed by researchers at the Imperial College.\nSection~\\ref{sec:usa-models} discusses metapopulation models focused on the US that were developed by our group at UVA and the models developed by researchers at Northeastern University. \nSection~\\ref{sec:sweden-models} describes models developed Swedish researchers for studying the outbreak in Sweden. In Section~\\ref{sec:forecasting} \nwe discuss methods developed for forecasting.\nSection~\\ref{sec:discussion} contains discussion, model limitations and \nconcluding remarks. In a companion paper that appears in this special issue, we address certain complementary issues related to pandemic planning and response, including role of data and analytics.\n\n\\medskip\n\\noindent\n\\textbf{Important note.} \\ The primary purpose of the paper\nis to highlight some of the salient computational \nmodels that are currently being used to support COVID-19 pandemic response.\nThese models, like all models, have their strengths and weaknesses---they have all faced challenges arising from the lack of timely data.\nOur goal is \\textbf{not} to pick winners and losers among these model; \neach model has been used by policy makers and continues to be used to advice various agencies. Rather, our goal is to introduce to the reader\na range of models that can be used in such situations. A simple model\nis no better or worse than a complicated model. The suitability of\na specific model for a given question needs to be evaluated by the \ndecision maker and the modeler.\n\n\n\n\n\n\n\n\n\\section{Models from the Imperial College Modeling Group}\n\\label{sec:imperial}\n\\noindent\n\\textbf{Background.} \\ The modeling group led by Neil Ferguson\nwas to our knowledge the first model to study the  impact of \nCOVID-19 across two large countries: US and UK. \nThe basic model was first developed in 2005 -- it was used to inform\npolicy pertaining to H5N1 pandemic and was one of three models used\nto informal the federal pandemic influenza plan and led to the now well\naccepted targeted layered containment (TLC) strategy. It was adopted to COVID-19 as discussed below.\n\n\\noindent\n\\textbf{Model Structure.} \\ The basic model structure consists of \ndeveloping a set of households based on census information for a given country. Individual members of the household interact with other members\nof the household. The data to produce these households is obtained using Census information for these countries. The paper is not clear if a\nnational census is used or more detailed state wise census information is used. Schools, workplaces and random meeting points are then added.\nThe number of schools, workplaces and random meeting points are used to represent the potential places of interaction between individuals. \nCensus data is used to assign age and household sizes.  \nData on average class sizes and staff-student ratios were used to generate a synthetic population of schools distributed proportional to local population density. Data on the distribution of workplace size was used to generate workplaces with commuting distance data used to locate workplaces appropriately across the population. Individuals are assigned to each of these locations at the start of the simulation. The gravity style kernel is used to decide how far a person can go in terms of attending\nwork, school or community interaction place. The number of contacts between individuals at school, work and community meeting points are calibrated to produce a given attack rate.\n\nEach individual has an associated disease transmission model. The disease\ntransmission model parameters are based on data collected when the pandemic\nwas evolving in Wuhan; see page 4 of \\cite{ferguson-report9}.\n\nFinally, the model also has rich set of interventions. These include:\n($i$) case isolation, ($ii$) voluntary home quarantine, ($iii$) Social distancing of those over 70 years, ($iv$) social distancing of the entire population, ($v$) closure of schools and universities; see page 6 \\cite{ferguson-report9}. The code hsa now been available but the details of how such interventions are implemented remains a bit unclear. This is important as the interpretation of these interventions can have substantial\nimpact on the outcome.\n\n\\noindent\n\\textbf{What did the model suggest.} \\ \nThe Imperial college (IC Model) model was one of the first models to evaluate the COVID-19 pandemic using detailed agent-based model. The predictions made by the model were quite dire. The results show that to be able to reduce $R$ to close to 1 or below, a combination of case isolation, social distancing of the entire population and either household quarantine or school and university closure are required.  The model had tremendous impact --\nUK and US both decide to start considering complete lock downs -- a\npolicy that was practically impossible to even talk about earlier in Western world. The paper came out around the same time that Wuhan epidemic was raging and the epidemic in Italy had taken the turn for the worse. This made the\nmodel results even more critical. So far the numbers have not played out per model forecasts -- this is a general issue that we will discuss across all models.\n\n\\noindent\n\\textbf{Strengths and Limitations.} \\\nIC model was one of the first model by a very reputed group to report the\npotential impact of COVID-19 with and without interventions. The model\nwas far more detailed than other models that were published by then.\nThe authors also took great care parameterizing the model with the best disease transmission data that was available until then. The model also considered a very rich set of interventions and was one of the first to analyze pulsing intervention. The work also had a few weaknesses. The\nrepresentation of the underlying social contact network was relatively \nsimple. Second, the details of how interventions were represented were\nnot made clear. Since then the modelers have made their code open and\nthe research community has witnessed an intense debate on the pros and cons of various modeling assumptions and the resulting software system. \nWe believe that despite some criticisms that were valid, overall the results\nrepresented a breakthrough in terms of when the results were put out and\nthe level of details the model had. \n\n\\section{Spatial metapopulation models}\n\\label{sec:metapop}\n\n\\noindent\n\\textbf{Background.}\nThis approach is an alternative to detailed agent based models, and has been used in\nmodeling the spread of multiple diseases, including Influenza~\\cite{balcan:pnas2009, srini:ploscb19},\nEbola~\\cite{gomes:plos-curr14} and Zika~\\cite{zhang:pnas17}. It has been adapted for\nstudying the importation risk of SARS-CoV-2 across the world~\\cite{chinazzi:science20}.\nStructured metapopulation models construct a simple abstraction of the mixing patterns in the population,\nin which the entire plane is decomposed into fully connected geographical regions,\nrepresenting subpopulation, which are connected through airline and commuter flow networks.\nThus, they lack the rich detail of agent based models, but have fewer parameters, and\nare therefore, easy to set up and scale to large regions.\n\n\\noindent\n\\textbf{Model structure.}\nHere, we summarize GLEaM~\\cite{balcan:pnas2009} and PatchSim~\\cite{srini:ploscb19}.\nGLEaM uses two classes of datasets-- population estimates and mobility.\nPopulation data is used from the ``Gridded Population of the World''~\\cite{ciesin}, which\ngives an estimated population value at a $15\\times 15$ minutes of arc (referred\nto as a ``cell'') over the entire planet.\nTwo different kinds of mobility processes are considered-- airline travel and commuter flow.\nThe former captures long-distance travel, whereas the latter captures localized mobility.\nAirline data is obtained from the International Air Transport Association (IATA)~\\cite{iata}, and\nthe Official Airline Guide (OAG)~\\cite{oag}. There are about 3300 airports world-wide; these\nare aggregated at the level of urban regions served by multiple airport (e.g., as in London).\nA voronoi tesselation is constructed with the resulting airport locations as centers, and\nthe population cells are assigned to these cells, with a 200 mile cutoff from the center.\nThe commuter flows connect cells at a much smaller spatial scale.\nWe represent this mobility pattern as a directed graph on the cells, and refer to it\nas the mobility network.\n\nIn the basic SEIR model, the subpopulation in each cell $j$ is partitioned into \ncompartments $S_j, E_j, I_j$ and $R_j$, corresponding to the disease states.\nFor each cell $j$, we define\nthe force of infection $\\lambda_j$ as the rate at which a susceptible individual in the\nsubpopulation in cell $j$ becomes infected---this is determined by the interactions the\nperson has with infectious individuals in cell $j$ or any cell $j'$ connected in the mobility network.\nAn individual in the susceptible compartment $S_j$ becomes infected with probability\n$\\lambda_j\\Delta t$ and enters the compartment $E_j$, in a time interval $\\Delta t$.\nFrom this compartment, the individual moves to the $I_j$ and then the $R_j$ compartments,\nwith appropriate probabilities, corresponding to the diease model parameters.\n\nThe PatchSim~\\cite{srini:ploscb19} model has a similar structure, except that it uses\nadministrative boundaries (e.g., counties), instead of\na voronoi tesselation, which are connected using the mobility network. Further, the disease model\nis deterministic, i.e., the transitions across the compartments occur at fixed rates.\n\n\\noindent\n\\textbf{What did the models suggest?}\n\n\\noindent\n\\textbf{Strength and limitations.}\nStructured metapopulation models provide a good tradeoff between the power and realism of\ndetailed agent-based models, and simplicity and need for fewer inputs for modeling,\nand scalability. This is especially true in the early days of the outbreak, when the disease dynamics are\ndriven to a large extent by mobility, which can be captured more easily within such models,\nand there is significant uncertainty in the disease model parameters.\nHowever, once the outbreak has spread, it is harder to model detailed interventions\n(e.g., social distancing), which are much more localized, and hard to model by a single parameter.\n\n\n\\section{Models in Sweden}\n\\label{sec:sweden-models}\nSweden was an outlier amongst countries in that it decided to implement public health interventions without a lockdown. Schools and universities were not closed and restaurants and bars remained open. Instead Swedish citizens\nimplemented \"work from home\" policies where possible.Moderate social distancing based on individual responsibility and without police enforcement was employed but emphasis was attempted to be placed on shielding the over 65 age group.\n\n\\subsection{Simple model}\n\n\\textbf{Background.} \\ \nStatistician Tom Britton developed a very simple model with a  focus  on predicting the number of infected over time in Stockholm.\n\n\\noindent\n\\textbf{Model structure.} \\ \nBritton \\cite{Britton2020} used a very simple SIR General epidemic model. It is used to make a coarse grain prediction of the behaviour of the outbreak based on knowing the basic reproduction number $R_0$  and the doubling time $d$ in the initial phase of the epidemic. Calibration to calendar time was done using the observed number of case fatalities, together with estimates of the time between infection to death, and the infection fatality risk. Predictions were made assuming no change of behaviour, as well as for the situation where preventive measures are put in place at one specific time--point. \n\n\\noindent\n\\textbf{What did the model suggest.} \\ One of the controversial predictions from this model was that the number of infections in the Stockholm area would quickly rise towards attaining \\emph{herd immunity} within a short period. However, mass testing carried out in Stockholm during June indicated a far smaller percentage of infections.\n\n\\noindent\n\\textbf{Strength and Limitations.} \\ \nBritton's model was intended as a quick and simple method to estimate and predict an on-going epidemic outbreak both with and without preventive measures put in place. It was intended as a complement to more realistic and detailed modelling. The estimation-prediction methodology is much simpler and straight- forward to implement for this simple model. It is more transparent to see how the few model assumptions affect the results, and it is easy to vary the few parameters to see their effect on predictions so one could see which parameter- uncertainties have biggest impact on predictions, and which parameter-uncertainties are less influential.\n\n\\subsection{Compartmentalized Models}\n\n\\noindent\n\\textbf{Background.} \\ \nA group around statistician Joacim Rockl\\\"ov developed a model to estimate the impact of COVID-19 on the Swedish population at the municipality level, considering demography and\nhuman mobility under various scenarios of mitigation and suppression. They attempted to estimate the\ntime course of infections, health care needs, and the mortality in relation to the Swedish ICU capacity, as well as the costs of care, and compared alternative policies and counterfactual\nscenarios.\n\n\\noindent\n\\textbf{Model structure.} \\ \\cite{Rocklov2020} used a SEIR compartmentalized model with age structured compartments (0-59, 60-79, 80+) susceptibles, infected, in-patient carem, ICU and recovered populations based on Swedish population data at the municipal level. It also incorporated inter-municipality travel using a radiation model. Parameters were calibrated based on a combination of values available from international literature and fitting to available outbreak data. The effect of a number of different intervention strategies were considered ranging from no intervention to modest social distancing and finally to imposed isolation of various groups. \n\n\\noindent\n\\textbf{What did the model suggest.} \\ \nThe model predicted an estimated death toll of around 40,000 for the strategies based only on social distancing and between 5000 and 8000 for policies imposing stricter isolation. It predicted ICU cases of upto 10,000 without much intervention and upto 6000 with modest social distancing, way above the available capacity of about 500 ICU beds. \n\\noindent\n\\textbf{Strength and limitations.} \\ \nThe model showed a good fit against the reported COVID-19 related deaths in Sweden up to 20th of April, 2020,\nHowever, the predictions of the total deaths and ICU demand turned out to be way off the mark.\n\n\\subsection{Agent Based microsimulations}\n\n\\noindent\n\\textbf{Background.} \\ \nFinally, \\cite{Gardner2020}used an individual-based model parameterized on Swedish demographics to assess the anticipated spread of COVID-19.\n\\noindent\n\\textbf{model structure.} \\ \n\\cite{Gardner2020} employed the individual agent-based model based on work by Ferguson et al \\cite{Ferg20}. Individuals are randomly assigned an age based on Swedish\ndemographic data and they are also assigned a household. Household size is normally distributed around the average household size in Sweden in 2018, 2.2 people per household. Households placed on a lattice using high-resolution population data from Landscan and census dara from the Statstics Sweden and each household is additionally allocated to a city based on\nthe closest city centre by distance and to a county based on city designation. Each individual is placed in a school or workplace at a rate similar to the current participation\nin Sweden.\n\nTransmission between individuals occurs through contact at each\nindividual's workplace or school, within their household, and in their communities. Infectiousness is thus a property dependent on contacts from household members,\nschool\/workplace members and community members with a probability based on household distances. Transmissibility was calibrated against data for the period 21\nMarch \u2013 6 April to reproduce either the doubling time reported using pan-European data35 or the growth in reported Swedish deaths for that period\n\nVarious types of interventions were studied including the policy implemented in Sweden by the public health authorities as well as more aggressive interventions approaching full lockdown.\n\n\\noindent\n\\textbf{What did the model suggest.} \\ \nTheir prediction was that \"under conservative epidemiological parameter estimates, the current Swedish public-health strategy will result in a peak intensive-care load in May that exceeds pre-pandemic capacity by over 40-fold, with a median mortality of 96,000 (95\\% CI 52,000 to 183,000)\". \n\n\\noindent\n\\textbf{Strength and limitations.} \\ \nThis model was based on adapting the well known Imperial model discussed in section~\\ref{sec:imperial} to Sweden and considered a wide range of intervention strategies. Unfortunately the predictions of the model were woefully off the mark on both counts: the deaths by June 18 are under 5000 and at he peak the ICU infrastructure had at least 20\\% unutilized capacity.\n\\section{Forecasting Models}\nIn the US, the Centers for Disease Control and Prevention (CDC) provides 1-,2-,3-, and 4-week ahead forecasts on cumulative deaths at the national level and across all states. The forecasts are provided by 21 (as of June 24, 2020) modelling groups and the CDC reports the cumulative deaths based on an ensemble of these forecasts.\n\\subsection{COVID-19 ForecastHub}\n\\noindent\n\\textbf{Model structure} It has been observed previously in the case of other infectious disease that an ensemble of forecasts from multiple models tends to be better than any individual contributing model \\cite{yamana2017individual}. Most of the models considered in the COVID-19 ForecastHub are variants of the SIER model with two models relying on agent-based models and one deep-learning-based model. The simulation outcomes are heavily dependent on the intervention scenarios considered. Some models assume that the effects of the interventions are reflected in the observed data and do not explicitly encode it in the model. Models such as [] incorporate the effects of interventions by suitably varying the contact rate while models [IHME,] incorporate the change in mobility patterns to reflect in the simulations. The current ensemble model equally weight the submitted forecasts.   \n\\section{Models from the Imperial College Modeling Group (UK Model)}\n\\label{sec:imperial}\n\n\\noindent\n\\textbf{Background.} \\ The modeling group led by Neil Ferguson\nwas to our knowledge the first model to study the  impact of \nCOVID-19 across two large countries: US and UK, see ~\\cite{Ferg20}. \nThe basic model was first developed in 2005 -- it was used to inform\npolicy pertaining to H5N1 pandemic and was one of the three models used\nto inform the federal pandemic influenza plan and led to the now well\naccepted targeted layered containment (TLC) strategy. It was adapted to COVID-19 as discussed below. The model was widely discussed and covered in the scientific as well as popular press~\\cite{adam2020modelling}.\nWe will refer to this as the IC-model.\n\n\\medskip\n\\noindent\n\\textbf{Model Structure.} \\ The basic model structure consists of \ndeveloping a set of households based on census information for a given country. The structure of the model is largely borrowed from their earlier work, see~\\cite{halloran:pnas08,ferguson2006strategies}.\nLandscan data was used to spatially distribute the population. Individual members of the household interact with other members\nof the household. The data to produce these households is obtained using Census information for these countries. Census data is used to assign age and household sizes. \nDetails on the resolution of census data and the dates was not clear. \nSchools, workplaces and random meeting points are then added.\nThe school data for US was obtained from the National Centre of Educational\nStatistics, while for UK schools were assigned randomly based on population density.\n Data on average class sizes and staff-student ratios were used to generate a synthetic population of schools distributed proportional to local population density. Data on the distribution of workplace size was used to generate workplaces with commuting distance data used to locate workplaces appropriately across the population. Individuals are assigned to each of these locations at the start of the simulation. The gravity style kernel is used to decide how far a person can go in terms of attending\nwork, school or community interaction place. The number of contacts between individuals at school, work and community meeting points are calibrated to produce a given attack rate.\n\nEach individual has an associated disease transmission model. The disease\ntransmission model parameters are based on data collected when the pandemic\nwas evolving in Wuhan; see page 4 of \\cite{Ferg20}.\n\nFinally, the model also has rich set of interventions. These include:\n($i$) case isolation, ($ii$) voluntary home quarantine, ($iii$) Social distancing of those over 70 years, ($iv$) social distancing of the entire population, ($v$) closure of schools and universities; see page 6 \\cite{Ferg20}. The code was recently released and is being analyzed. This is important as the interpretation of these interventions can have substantial impact on the outcome.\n\n\\medskip\n\\noindent\n\\textbf{Model predictions.} \\ \nThe Imperial college (IC Model) model was one of the first models to evaluate the COVID-19 pandemic using detailed agent-based model. The predictions made by the model were quite dire. The results show that to be able to reduce $R$ to close to 1 or below, a combination of case isolation, social distancing of the entire population and either household quarantine or school and university closure are required.  The model had tremendous impact --\nUK and US both decide to start considering complete lock downs -- a\npolicy that was practically impossible to even talk about earlier in the Western world. The paper came out around the same time that Wuhan epidemic was raging and the epidemic in Italy had taken a turn for the worse. This made the\nmodel results even more critical. \n\n\\medskip\n\\noindent\n\\textbf{Strengths and Limitations.} \\\nIC model was one of the first model by a reputed group to report the\npotential impact of COVID-19 with and without interventions. The model\nwas far more detailed than other models that were published until then.\nThe authors also took great care parameterizing the model with the best disease transmission data that was available until then. The model also considered a very rich set of interventions and was one of the first to analyze pulsing intervention. On the flip side, the\nrepresentation of the underlying social contact network was relatively \nsimple. Second, often the details of how interventions were represented were\nnot clear. Since the publication of their article, the modelers have made their code open and\nthe research community has witnessed an intense debate on the pros and cons of various modeling assumptions and the resulting \nsoftware system, see~\\cite{chawla2020critiqued}. \nWe believe that despite certain valid criticisms, overall, the results\nrepresented a significant advance in terms of the \nwhen the results were put out and the level of details incorporated in the models.\n\n\\section{Spatial metapopulation models: Northeastern and UVA models (US Models)}\n\\label{sec:usa-models}\n\n\\noindent\n\\textbf{Background.}\nThis approach is an alternative to detailed agent based models, and \nhas been used in modeling the spread of multiple diseases, including Influenza~\\cite{balcan:pnas2009, srini:ploscb19}, Ebola~\\cite{gomes:plos-curr14} and Zika~\\cite{zhang:pnas17}. \nIt has been adapted for studying the importation risk of COVID-19 \nacross the world~\\cite{chinazzi:science20}.\nStructured metapopulation models construct a simple abstraction \nof the mixing patterns in the population, in which the entire region under study is decomposed into fully connected geographical regions,\nrepresenting subpopulations, which are connected through airline and commuter flow networks. Thus, they lack the rich detail of agent based models, \nbut have fewer parameters, and are therefore, easy to set up and scale to large regions.\n\n\\medskip\n\\noindent\n\\textbf{Model structure.}\nHere, we summarize GLEaM~\\cite{balcan:pnas2009} (Northeastern model) and PatchSim~\\cite{srini:ploscb19} (UVA model).\nGLEaM uses two classes of datasets-- population estimates and mobility.\nPopulation data is used from the ``Gridded Population of the World''~\\cite{lloyd2017high}, which gives an estimated population value at a $15\\times 15$ minutes of arc (referred to as a ``cell'') over the entire planet. Two different kinds of mobility processes are considered-- airline travel  and commuter flow. The former captures long-distance travel, whereas the latter  captures localized mobility. Airline data is obtained from the International Air Transport Association (IATA)~\\cite{IATA}, and\nthe Official Airline Guide (OAG)~\\cite{oag}. There are about 3300 airports world-wide; these are aggregated at the level of urban regions served by multiple airport (e.g., as in London). A Voronoi tessellation is constructed with the resulting airport locations as centers, and the population cells are assigned to these cells, with a 200 mile cutoff from the center.\nThe commuter flows connect cells at a much smaller spatial scale.\nWe represent this mobility pattern as a directed graph on the cells, \nand refer to it as the mobility network.\n\n\n\nIn the basic SEIR model, the subpopulation in each cell $j$ is partitioned into  compartments $S_j, E_j, I_j$ and $R_j$, corresponding to the disease states. For each cell $j$, we define the force of infection \n$\\lambda_j$ as the rate at which a susceptible individual in the\nsubpopulation in cell $j$ becomes infected---this is determined by the interactions the person has with infectious individuals in cell $j$ or any cell $j'$ connected in the mobility network. An individual in the susceptible compartment $S_j$ becomes infected with probability $\\lambda_j\\Delta t$ and enters the compartment $E_j$, in a time interval $\\Delta t$. From this compartment, the individual moves to the $I_j$ and then the $R_j$ compartments, with appropriate probabilities, corresponding to the \ndisease model parameters.\n\nThe PatchSim~\\cite{srini:ploscb19} model has a similar structure, except that it uses administrative boundaries (e.g., counties), instead of\na Voronoi tesselation, which are connected using a mobility network. The mobility network is derived by combining commuter and airline networks, to model time spent per day by individuals of region (patch) $i$ in region (patch) $j$. Since it explicitly captures the level of connectivity through a commuter-like mixing, it is capable of incorporating week-to-week and month-to-month variations in mobility and connectivity. In addition to its capability to run in deterministic or stochastic mode, the open source implementation \\cite{NSSACPat38:online} allows fine grained control of disease parameters across space and time. Although the model has a more generic force of infection mode of operation (where patches can be more general than spatial regions), we will mainly summarize the results from the mobility model, which was used for COVID-19 response. \n\n\n\\medskip\n\\noindent\n\\textbf{What did the models suggest?}\nGLEaM model is being used in a number of COVID-19 related studies and analysis. In ~\\cite{kraemer2020effect} the Northeastern University team \nused the model to understand the spread of COVID-19 within China and relative risk of importation of the disease internationally. \nTheir analysis suggested that the spread of COVID-19 out of Wuhan into other parts of mainland China was not contained well due to the delays induced by detection and official reporting.\nIt is hard to interpret the results. The paper suggested that international importation could be contained substantially by strong travel ban. While it might have delayed the onset of cases, the subsequent spread across the world suggest that we were not able to arrest the spread effectively.\nThe model is also used to provide\nweekly projections (see \\url{https:\/\/covid19.gleamproject.org\/}); this site does not appear to be maintained for the most current forecasts\n(likely because the team is participating in the CDC forecasting group).\n\nThe  PatchSim model is being used to support federal agencies as well\nas the state of Virginia. Due to our past experience, we have refrained from providing longer term forecasts, instead focusing on short term projections. The model is used within a \\emph{Forecasting via Projection Selection} approach, where a set of counterfactual scenarios are generated based on on-the-ground response efforts and surveillance data, and the best fits are selected based on historical performance. While allowing for future scenarios to be described, they also help provide a reasonable narrative of past trajectories, and retrospective comparisons are used for metrics such as 'cases averted by doing X'. These projections are revised weekly based on stakeholder feedback and surveillance update. Further discussion of how the model is used by the Virginia Department of Health each week can be found at \\url{https:\/\/www.vdh.virginia.gov\/coronavirus\/covid-19-data-insights\/#model}.\n\n\n\n\\medskip\n\\noindent\n\\textbf{Strength and limitations.}\nStructured metapopulation models provide a good tradeoff between the realism\/compute of detailed agent-based models and simplicity\/speed of mass action compartmental models and need far fewer inputs for modeling, and scalability. This is especially true in the early days of the outbreak, when the disease dynamics are\ndriven to a large extent by mobility, which can be captured more easily within such models, and there is significant uncertainty in the disease model parameters. However, once the outbreak has spread, it is harder to model detailed interventions (e.g., social distancing), which are much more localized. Further, these are hard to model using a single parameter. Both GLeaM and PatchSim models also faced their share of challenges in projecting case counts due to rapidly evolving pandemic, inadequate testing, a lack of understanding of the number of asymptomatic cases and assessing the compliance levels of the population at large.\n\n\n\\section{Models by  KTH, Ume\u00e5 and Uppsala researchers (Swedish Models)}\n\\label{sec:sweden-models}\nSweden was an outlier amongst countries in that it decided to implement public health interventions without a lockdown. Schools and universities were not closed, and restaurants and bars remained open. Swedish citizens\nimplemented ``work from home'' policies where possible. Moderate social distancing based on individual responsibility and without police enforcement was employed but emphasis was attempted to be placed on shielding the 65+ age group.\n\n\\subsection{Simple model}\n\n\\medskip\n\\noindent\n\\textbf{Background.} \\ \nStatistician Tom Britton developed a very simple model with a  focus  on predicting the number of infected over time in Stockholm.\n\n\\medskip\n\\noindent\n\\textbf{Model structure.} \\ \nBritton \\cite{Britton2020} used a very simple SIR general epidemic model. It is used to make a coarse grain prediction of the behaviour of the outbreak based on knowing the basic reproduction number $R_0$  and the doubling time $d$ in the initial phase of the epidemic. Calibration to calendar time was done using the observed number of case fatalities, together with estimates of the time between infection to death, and the infection fatality risk. Predictions were made assuming no change of behaviour, as well as for the situation where preventive measures are put in place at one specific time--point. \n\n\\medskip\n\\noindent\n\\textbf{Model predictions.} \\ One of the controversial predictions from this model was that the number of infections in the Stockholm area would quickly rise towards attaining \\emph{herd immunity} within a short period. However, mass testing carried out in Stockholm during June indicated a far smaller percentage of infections.\n\n\\medskip\n\\noindent\n\\textbf{Strength and Limitations.} \\ \nBritton's model was intended as a quick and simple method to estimate and predict an on-going epidemic outbreak both with and without preventive measures put in place. It was intended as a complement to more realistic and detailed modelling. The estimation-prediction methodology is much simpler and straight-forward to implement for this simple model. It is more transparent to see how the few model assumptions affect the results, and it is easy to vary the few parameters to see their effect on predictions so that one could see which parameter-uncertainties have biggest impact on predictions, and which parameter-uncertainties are less influential.\n\n\\subsection{Compartmentalized Models I: FHM Model}\n\n\\textbf{Background.} \\\nThe Public Health Authority (FHM) of Sweden produced a model\nto study the spread of COVID-19 in four regions in Sweden: Dalarna, Sk\\aa ne, Stockholm, and V\\\"astra G\\\"otaland.\\cite{FHM20}.\n\n\\medskip\n\\noindent\n\\textbf{Model structure.} \\ \nIt is a standard compartmentalized SEIR model and within each compartment it is homogeneous, so individuals are assumed to have the same characteristics and act in the same way. Data used in the fitting of the model include point prevalences found by PCR-testing in Stockholm at two different time points.\n\n\\medskip\n\\noindent\n\\textbf{Model predictions.} \\ \nThe model estimated the number of infected individuals at different time points and the date with the largest number of infectious individuals. It predicted that by July 1, 8.5\\% (5.9 \u2013 12.9\\%) of the population in Dalarna will have been infected, 4\\% (2.4 \u2013 9.9\\%) of the population in Sk\u00e5ne will have been infected, 19\\% (17.7 \u2013 20.2\\%) of the population in Stockholm will have been infected, and 9\\% (6.3 \u2013 12.2\\%) of the population in V\u00e4stra G\u00f6taland will have been infected.  It was hard to test these predictions because of the great uncertainty in immune response to SARS-CoV-2 -- prevalence of antibodies was surprisingly low but recent studies show that mild cases never seem to develop antibodies against SARS-CoV-2, but only T-cell-mediated immunity \\cite{karolinska}.\n\nThe model also investigated the effect of increased contacts during the summer that stabilises in autumn. It found that if the contacts in Stockholm and Dalarna increase by less than 60\\% in comparison to the contact rate in the beginning of June, the second wave will not exceed the observed first wave.\n\n\\medskip\n\\noindent\n\\textbf{Strength and limitations.} \\ \nThe simplicity of the model is a strength in ease of calibration and understanding but it is also a major limitation in view of the well known characteristics of COVID-19: since it is primarily transmitted through droplet infection, the social contact structure in the population is of primary importance for the dynamics of infection. The compartmental model used in this analysis does not account for variation in contacts, where few individuals may have many contacts while the majority have fewer. The model is also not age--stratified, but COVID-19 strikingly affects different age groups differently; e.g., young people seem to get milder infections. In this model, each infected individual has the same infectivity and the same risk of becoming a reported case, regardless of age.  Different age groups normally have varied degrees of contacts and have changed their behaviour differently during the COVID-19 pandemic. This is not captured in the model.\n\n\\subsection{Compartmentalized Models II}\n\n\\medskip\n\\noindent\n\\textbf{Background.} \\ \nA group around statistician Joacim Rockl\\\"ov developed a model to estimate the impact of COVID-19 on the Swedish population at the municipality level, considering demography and\nhuman mobility under various scenarios of mitigation and suppression. They attempted to estimate the\ntime course of infections, health care needs, and the mortality in relation to the Swedish ICU capacity, as well as the costs of care, and compared alternative policies and counterfactual\nscenarios.\n\n\\medskip\n\\noindent\n\\textbf{Model structure.} \\ \\cite{Rocklov2020} used a SEIR compartmentalized model with age structured compartments (0-59, 60-79, 80+) susceptibles, infected, in-patient care, ICU and recovered populations based on Swedish population data at the municipal level. It also incorporated inter-municipality travel using a radiation model. Parameters were calibrated based on a combination of values available from international literature and fitting to available outbreak data. The effect of a number of different intervention strategies were considered ranging from no intervention to modest social distancing and finally to imposed isolation of various groups. \n\n\\medskip\n\\noindent\n\\textbf{Model predictions.} \\ \nThe model predicted an estimated death toll of around 40,000 for the strategies based only on social distancing and between 5000 and 8000 for policies imposing stricter isolation. It predicted ICU cases of upto 10,000 without much intervention and upto 6000 with modest social distancing, way above the available capacity of about 500 ICU beds. \n\n\\medskip\n\\noindent\n\\textbf{Strength and limitations.} \\ \nThe model showed a good fit against the reported COVID-19 related deaths in Sweden up to 20th of April, 2020,\nHowever, the predictions of the total deaths and ICU demand turned out to be way off the mark.\n\n\\subsection{Agent Based microsimulations}\n\n\\noindent\n\\textbf{Background.} \\ \nFinally, \\cite{Gardner2020,kamerlin2020managing}used an individual-based model parameterized on Swedish demographics to assess the anticipated spread of COVID-19.\n\n\\medskip\n\\noindent\n\\textbf{Model structure.} \\ \n\\cite{Gardner2020} employed the individual agent-based model based on work by Ferguson et al \\cite{Ferg20}. Individuals are randomly assigned an age based on Swedish\ndemographic data and they are also assigned a household. Household size is normally distributed around the average household size in Sweden in 2018, 2.2 people per household. Households were placed on a lattice using high-resolution population data from Landscan and census dara from the Statstics Sweden and each household is additionally allocated to a city based on\nthe closest city centre by distance and to a county based on city designation. Each individual is placed in a school or workplace at a rate similar to the current participation\nin Sweden.\n\nTransmission between individuals occurs through contact at each\nindividual's workplace or school, within their household, and in their communities. Infectiousness is thus a property dependent on contacts from household members,\nschool\/workplace members and community members with a probability based on household distances. Transmissibility was calibrated against data for the period 21\nMarch \u2013 6 April to reproduce either the doubling time reported using pan-European data or the growth in reported Swedish deaths for that period. Various types of interventions were studied including the policy implemented in Sweden by the public health authorities as well as more aggressive interventions approaching full lockdown.\n\n\\medskip\n\\noindent\n\\textbf{Model predictions.} \\ \nTheir prediction was that \"under conservative epidemiological parameter estimates, the current Swedish public-health strategy will result in a peak intensive-care load in May that exceeds pre-pandemic capacity by over 40-fold, with a median mortality of 96,000 (95\\% CI 52,000 to 183,000)\". \n\n\\medskip\n\\noindent\n\\textbf{Strength and limitations.} \\ \nThis model was based on adapting the well known Imperial model discussed in section~\\ref{sec:imperial} to Sweden and considered a wide range of intervention strategies. Unfortunately the predictions of the model were woefully off the mark on both counts: the deaths by June 18 are under 5000 and at the peak the ICU infrastructure had at least 20\\% unutilized capacity.\n\n\n\n\\section{Forecasting Models}\n\\label{sec:forecasting}\nForecasting is of particular interest to policy makers as they attempt to provide actual counts. Since the surveillance systems have relatively stabilized in recent weeks, the development of forecasting models has gained traction and several models are available in the literature. In the US, the Centers for Disease Control and Prevention (CDC) has provided a platform for modelers to share their forecasts which are analyzed and combined in a suitable manner to produce ensemble multi-week forecasts for cumulative\/incident deaths, hospitalizations and more recently cases at the national, state, and county level. Probabilistic forecasts are provided by 36 teams as of July 28, 2020 (there were 21 models as of June 24, 2020) and the CDC with the help of \\cite{Reichlab} has developed uniform ensemble model for multi-step forecasts \\cite{CDChub}.\n\n\\subsection{COVID-19 Forecast Hub ensemble model}\nIt has been observed previously for other infectious diseases that an ensemble of forecasts from multiple models perform better than any individual contributing model \\cite{yamana2017individual}. In the context of COVID-19 case count modeling and forecasting, a multitude of models have been developed based on different assumptions that capture specific aspects of the disease dynamics (reproduction number evolution, contact network construction, etc.). The models employed in the CDC Forecast Hub can be broadly classified into three categories, data-driven, hybrid models, and mechanistic models with some of the models being open source. \n\n\\medskip\n\\noindent\n\\textbf{Data-driven models:} They do not model the disease dynamics but attempt to find patterns in the available data and combine them appropriately to make short-term forecasts. In such data-driven models it is hard to incorporate interventions directly, hence, the machine is presented with a variety of exogenous data sources such as mobility data, hospital records, etc. with the hope that its effects are captured implicitly. Early iterations of Institute of Health Metrics and Evaluation (IHME) model \\cite{IHMEcovid2020forecasting} for death forecasting at state level employed a statistical model that fits a time-varying Gaussian error function to the cumulative death counts and is parameterized to control for maximum death rate, maximum death rate epoch, and growth parameter (with many parameters learnt using data from outbreak in China). The IHME models are undergoing revisions (moving towards the hybrid models) and updated implementable versions are available at \\cite{IHMEgithub}. The University of Texas at Austin\nCOVID-19 Modeling Consortium model \\cite{UTwoody2020projections} uses a very similar statistical model as \\cite{IHMEcovid2020forecasting} but employs real-time mobility data as additional predictors and also differ in the fitting process. The Carnegie Mellon Delphi Group employs the well known auto-regressive (AR) model that employs lagged version of the case counts and deaths as predictors and determines a sparse set that best describes the observations from it by using LASSO regression \\cite{delphi}. \\cite{deepGTcovid} is a deep learning model which has been developed along the lines of \\cite{adhikari:kdd19} and attempts to learn the dependence between death rate and other available syndromic, demographic, mobility and clinical data.  \n\n\\medskip\n\\noindent\n\\textbf{Hybrid models}: These methods typically employ statistical techniques to model disease parameters which are then used in epidemiological models to forecast cases. Most statistical models \\cite{IHMEcovid2020forecasting, UTwoody2020projections} are evolving to become hybrid models. A model that gained significant interest is the Youyang Gu (YYG) model and uses a machine learning layer over an SEIR model to learn the set of parameters (mortality rate, initial R$_0$,\n  post-lockdown R) specific to a region that best fits the region's observed data. The authors (YYG) share the optimal parameters, the SEIR model  and the evaluation scripts with general public for experimentation~\\cite{yygeval}. Los Alamos National Lab (LANL) model \\cite{LANLforecasts} uses a statistical model to determine how the number of COVID-19 infections changes over time. The second process maps the number of infections to the reported data. The number of deaths are a fraction of the number of new cases obtained and is computed using the observed mortality data.\n\n\\medskip\n\\noindent  \n\\textbf{Mechanistic models:} GLEaM and JHU models are county-level stochastic SEIR model dynamics. The JHU model incorporates the effectiveness of state-wide intervention policies on social distancing through the R$_0$ parameter. More recently model outputs from UVA's PatchSim model were included as part of a multi-model ensemble (including autoregressive and LSTM components) to forecast weekly confirmed cases. \n\n\\section{Comparative analysis across modeling types}\nWe end the discussion of the models above by qualitatively comparing\nmodel types. As discussed in the preliminaries, \nat one end of the spectrum are models that are largely data driven: these models range from simple statistical models (various forms\nof regression models) to the more complicated deep learning models.\nThe difference in such model lies in the amount of training data needed,\nthe computational resources needed and how complicated the mathematical\nfunction one is trying to fit to the observed data. These models are strictly data driven and hence unable to capture the constant behavioral\nadaptation at an individual and collective level. On the other end of the\nspectrum SEIR, meta-population and agent-based networked models\nare based on the underlying procedural representation of the dynamics -- in theory they are able to represent behavioral adaptation endogenously.\nBut both class of models face immense challenges due to the availability of data as discussed below.\n\n\\begin{enumerate}\n    \\item \n    Agent-based and SEIR models were used in all the three countries in the early part of the outbreak and continue to be used for counter-factual analysis. The primary reason is the lack of \n    surveillance and disease specific data and hence purely data driven models were not easy to use. SEIR models lacked heterogeneity but were simple to program and analyze. Agent-based models were more computationally intensive, required a fair bit of data to instantiate the model but captured the heterogeneity of the underlying countries. \n    By now it has become clear that use of such models for long term forecasting is challenging and likely to lead to mis-leading results. The fundamental reason is adaptive human behavior and lack of data about it.\n    \\item\n    Forecasting on the other hand has seen use of data driven methods as well as causal methods. Short term forecasts have been generally reasonable. Given the intense interest in the pandemic, a lot of data is also becoming available for researchers to use. This helps in validating some of the models further.  Even so, real-time data on\n    behavioral adaptation and compliance remains very hard to get and is one of the central modeling challenges.\n\\end{enumerate}\n\n\n \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{#1}\\setcounter{equation}{0}}\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n\\newcommand{\\nsect}{\\setcounter{equation}{0}\n\\def\\theequation{\\thesection.\\arabic{equation}}\\section}\n\\newcommand{\\nsubsect}{\\setcounter{equation}{0}\n\\def\\theequation{\\thesubsection.\\arabic{equation}}\\subsection}\n\n\\relax\n\n\\def\\d{\\partial}\n\\def\\a{\\alpha'}\n\\def\\R{\\mathcal{R}}\n\\def\\O{\\mathcal{O}}\n\n\\title{Tensorial perturbations and stability of spherically symmetric $d$--dimensional black holes in string theory}\n\n\\author{Filipe Moura\n\\\\\nCentro de Matem\\'atica da Universidade do Minho, \\\\Escola de Ci\\^encias, Campus de Gualtar, \\\\4710-057 Braga, Portugal\\\\\n\\\\\n\\email{fmoura@math.uminho.pt}\n}\n\n\n\\abstract{We compute the tensorial perturbations to a general spherically symmetric metric in $d$ dimensions with string--theoretical corrections quadratic in the Riemann tensor, from which we derive their respective potential. We use this result to study the stability of corresponding black hole solutions under such perturbations.\n}\n\n\n\n\\begin{document}\n\n\n\n\n\\vfill\n\n\\eject\n\n\n\\section{Introduction and Summary}\n\\indent\n\nIn string theory there is a large variety of black hole solutions in four and more spacetime dimensions. Studying the stability of such exact solutions is a very important subject: if a stationary black hole solution is stable under perturbations, it implies that such solution describes a possible final state of dynamical evolution of a gravitating system. If, on the other hand, it is shown to be unstable, that indicates the existence of a different branch of solutions which the original solution may decay into, which means that one can anticipate a wider variety of black hole solutions. The analysis of perturbations also gives information about physical properties of the black hole solutions, an example being the spectra of quasinormal modes. It is interesting to study how such properties, including the stability, are affected in the context of string theory.\n\nThe analysis of linear perturbations of stationary black holes in arbitrary $d$ spacetime dimensions was initiated in \\cite{iks00, ik03a, ik03c} for static, nonrotating black holes (without and with charge), for the three kinds of metric perturbations: tensor, vector and scalar. The analysis of stability was performed, and stability was obtained for asymptotically flat \\cite{ik03b, ik03d} and anti--de Sitter black holes \\cite{Konoplya:2008rq}, while an instability was found in de Sitter, for large charge and $\\Lambda$ \\cite{Konoplya:2008au}.\n\nMore recently the stability analysis was performed for rotating Myers--Perry black holes, first for tensor perturbations, where an instability was found in anti--de Sitter \\cite{Kunduri:2006qa, Kodama:2009rq}, but not for asymptotically flat black holes. Considering time--dependent perturbations, for the first time an instability was found for asymptotically flat black holes, for sufficiently rapid rotations (the ultraspinning instability) \\cite{Dias:2009iu, Dias:2010eu, Dias:2010maa}. Later the result was generalized for the case of anti--de Sitter \\cite{Dias:2010gk}.\n\nThe analysis of perturbations and stability was also extended to Lovelock theories in $d$ dimensions, considering corrections quadratic \\cite{dg04,dg05a} and cubic \\cite{Takahashi:2009dz, Takahashi:2010ye, Takahashi:2010gz} in the Riemann tensor. Detailed studies of black hole perturbations in Lovelock theories were\nperformed, in the context of the AdS\/CFT correspondence, in \\cite{Camanho:2009vw,Camanho:2009hu,Camanho:2010ru}. Depending on the gravitational potential resulting from the couplings of  the theory, different kinds of instabilities were found, which could imply either causality violations or plasma instabilities in the holographic dual thermal field theory.\n\nAlthough superstring theories also require higher order corrections to the Einstein--Hilbert action (actually to supergravity, which is the low energy effective theory coming from superstrings), there are important differences between string and Lovelock theory effective actions. In Lovelock theories, the actions are made just with powers of the Riemann curvature tensor (i.e. one can have an action just with the metric field and its derivatives); in string theory, besides the graviton we always have to consider at least the dilaton field which, as we will see later in this article (after eq. (\\ref{bdfe})), cannot be set to zero in the presence of higher order terms. Besides, Lovelock actions are not seen as ``effective'' in the sense of string perturbation theory, but as ``exact''. To be precise, in perturbative string theory one only works up to the order of the perturbation constant (in the case of higher order string corrections, this is the inverse string tension $\\a$) which appears in the effective action, neglecting all the higher order terms which are meaningless in such effective action, while in Lovelock theories one considers exact solutions to the equations of motion, no matter the order of the constant which appears in such solutions, even if the constant only appears to first order in the lagrangian (multiplying the higher order term) and in the field equations. Because of these differences, results and conclusions which are obtained in Lovelock theories may be very different from those in string theory.\n\nString theory is the most promising candidate for a consistent description of quantum gravity; therefore, it is natural to study its effects on the stability of its black hole solutions. The study of tensorial perturbations to the metric and the respective stability of black holes with higher order corrections in the context of string theory was initiated in \\cite{Moura:2006pz}, concerning a particular spherically symmetric solution. In this article we extend such study to incorporate the most general spherically symmetric black hole solutions with gravitational corrections to first order in $\\a$ in string theory in arbitrary $d$ dimensions.\n\nThe article is organized as follows: in section 2, we review the general formalism for field perturbations, and then we compute the tensorial perturbation equations for the most general spherically symmetric background metric in $d$ dimensions. In section 3, we apply these perturbation equations to the field equations obtained from a string theory effective action with string corrections to first order in $\\a$, i.e. quadratic in the Riemann tensor. We obtain the master equation for the perturbation variable and the potential for tensorial perturbations including such corrections. Then in section 4 we review the ``S--deformation'' approach and we obtain a criterion for the gravitational stability of black holes under tensor perturbations from the master equation. Finally on section 5 we apply this criterion to study the stability of two different solutions with leading $\\a$ corrections in $d$ dimensions: the dilatonic compactified black hole and the double--charged black hole.\n\n\n\\section{General setup of the perturbation theory}\n\n\n\\subsection{Perturbations on a $(d-2)$--sphere}\n\\indent\n\nWe will study the behavior, under gravitational perturbations, of string--corrected black hole solutions in a generic spacetime dimension $d$. For such analysis we use the framework developed by Ishibashi and Kodama \\cite{iks00,ik03a,ik03c} for black holes. This framework applies to generic spacetimes of the form $\\mathcal{M}^{d} = \\mathcal{N}^{d-n} \\times \\mathcal{K}^n$, with coordinates $\\left\\{ x^\\mu \\right\\} = \\left\\{ y^a, \\theta^i \\right\\}$. In here $\\mathcal{K}^n$ is a manifold with constant sectional curvature $K.$ The metric in the total space $\\mathcal{M}^{d}$ is then written as\n\n\\be \\label{ikmetric}\ng = g_{ab}(y)\\ dy^a\\ dy^b + r^2(y)\\ \\gamma_{ij}(\\theta)\\ d\\theta^i\\ d \\theta^j.\n\\ee\nFor our purposes, we take $n = d-2$ and the manifold $\\mathcal{K}^n$, describing the geometry of the black hole event horizon, will be a $(d-2)$--sphere (thus, with $K=1$). Also, $\\mathcal{N}^{d-n}$ coordinates will be $\\left\\{ y^a \\right\\} = \\left\\{ t, r \\right\\},$ with $\\left\\{ r, \\theta^i \\right\\}$ being the usual spherical coordinates so that $r(y) = r$ and $\\gamma_{ij}(\\theta)\\ d\\theta^i d\\theta^j = d\\Omega_{d-2}^2$.\n\nDefining generic perturbations to the metric as $h_{\\mu\\nu} = \\delta g_{\\mu\\nu}, \\, h^{\\mu\\nu} = -\\delta g^{\\mu\\nu},$\nwe get for the variation of the Levi-Civita connection\n\n\\be \\label{deltagamma}\n\\delta \\Gamma_{\\mu\\nu}^\\rho = \\frac{1}{2} \\left( \\nabla_\\mu {h_{\\nu}}^{\\rho} + \\nabla_\\nu {h_{\\mu}}^{\\rho} - \\nabla^{\\rho} h_{\\mu\\nu} \\right)\n\\ee\nFrom this variation and the Palatini equation\n\n\\be\n\\delta {\\R^{\\rho}}_{\\sigma\\mu\\nu} = \\nabla_\\mu\\ \\delta \\Gamma_{\\nu\\sigma}^\\rho - \\nabla_\\nu\\ \\delta \\Gamma_{\\mu\\sigma}^\\rho,  \\label{palatini}\n\\ee\none can easily derive the variation of the Riemann tensor:\n\n\\be \\label{palatiniexp}\n\\delta \\R_{\\rho\\sigma\\mu\\nu} = \\frac{1}{2} \\left( {\\R_{\\mu\\nu\\rho}}^{\\lambda} h_{\\lambda\\sigma} - {\\R_{\\mu\\nu\\sigma}}^{\\lambda} h_{\\lambda \\rho} - \\nabla_\\mu \\nabla_\\rho h_{\\nu\\sigma} + \\nabla_\\mu \\nabla_\\sigma h_{\\nu\\rho} - \\nabla_\\nu \\nabla_\\sigma h_{\\mu\\rho} + \\nabla_\\nu \\nabla_\\rho h_{\\mu\\sigma} \\right).\n\\ee\n\nGeneral tensors, of rank at most equal to two, can be uniquely decomposed in tensor, vector and scalar components, according to their tensorial behavior on the $(d-2)$--sphere, the geometry of the black hole event horizon \\cite{ik03a}. In particular, this is also true for the perturbations to the metric, but one should note that metric perturbations of tensor type only exist for dimensions $d>4$, unlike perturbations of vector and scalar type, which also exist for $d=4.$ This is because the 2--sphere does not admit any tensor harmonics \\cite{Higuchi:1986wu}. Tensor perturbations are therefore intrinsically higher--dimensional.\n\n\\subsection{Tensorial perturbations of a spherically symmetric static metric}\n\\indent\n\nIn this work we will only consider tensor type gravitational perturbations to the metric field, for $\\a$--corrected $\\R^{\\mu\\nu\\rho\\sigma} \\R_{\\mu\\nu\\rho\\sigma}$ black holes in string theory. One should consider perturbations to all the fields present in the low--energy effective action (in our case, the metric and the dilaton), but, as we will show later, one can consistently set tensor type perturbations to the dilaton field to zero. These metric perturbations were studied in \\cite{ik03a}, where it is shown that they can be written as\n\n\\be \\label{htensor}\nh_{ij} = 2 r^2 (y^a)\\ H_T (y^a)\\ \\mathcal{T}_{ij} (\\theta^i), \\quad h_{ia} = 0, \\quad h_{ab} = 0,\n\\ee\n\n\\noindent\nwith $\\mathcal{T}_{ij}$ satisfying\n\n\\be \\label{propt}\n\\left( \\gamma^{kl} D_k D_l + k_T \\right) \\mathcal{T}_{ij} = 0, \\quad D^i \\mathcal{T}_{ij} = 0, \\quad g^{ij} \\mathcal{T}_{ij} = 0.\n\\ee\n\n\\noindent\nHere, $D_i$ is the covariant derivative on the $(d-2)$--sphere, associated to the metric $\\gamma_{ij}$. Thus, the tensor harmonics $\\mathcal{T}_{ij}$ are the eigentensors of the $(d-2)$--laplacian $D^2$, whose eigenvalues are given by $k_T + 2 = \\ell \\left( \\ell + d - 3 \\right)$, with $\\ell = 2,3,4,\\ldots$. It should be further noticed that the expansion coefficient $H_T$ is gauge--invariant by itself. This is rather important: when dealing with linear perturbations to a system with gauge invariance one might always worry that final results could be an artifact of the particular gauge one chooses to work with. Of course the simplest way out of this is to work with gauge--invariant variables, and this is precisely implemented in the Ishibashi--Kodama framework \\cite{iks00, ik03a, ik03c}. As it was noticed in \\cite{Moura:2006pz}, the Ishibashi--Kodama gauge--invariant variables are also valid for higher derivative theories as long as diffeomorphisms keep implementing gauge transformations. This is because up to now we have only chosen the background metric we wish to perturb: so far, no choice of equations of motion has been done.\n\nNow we consider a static, spherically symmetric background metric. Such a metric is clearly of the type (\\ref{ikmetric}), and is given by\n\n\\be \\label{schwarz}\nds^2 = -f(r)\\ dt^2  + g^{-1}(r)\\ dr^2 + r^2 d\\Omega^2_{d-2}.\n\\ee\n\n\\noindent\nThe nonzero components of the Riemann tensor for this metric are\n\n\\bea \\label{ikriemann}\n\\R_{trtr} &=& \\frac{1}{2} f^{\\prime \\prime} + \\frac{1}{4} \\frac{f^\\prime g^\\prime}{g} - \\frac{1}{4} \\frac{f'^2}{f}\\,, \\nonumber \\\\\n\\R_{itjt} &=& \\frac{1}{2} \\frac{gf^\\prime}{r} g_{ij}\\,, \\nonumber \\\\\n\\R_{irjr} &=& - \\frac{1}{2} \\frac{g^\\prime}{rg} g_{ij}\\,, \\nonumber \\\\\\\n\\R_{ijkl} &=& \\frac{1}{r^2} \\big( 1 - g \\big) \\Big( g_{ik} g_{jl} - g_{il} g_{jk} \\Big).\n\\eea\n\n\\noindent\n\nOne first needs to obtain the variation of the Riemann tensor under generic perturbations of the metric. If one collects the expressions for $h_{\\mu\\nu}$ given in (\\ref{htensor}), their covariant derivatives, and further the components of the Riemann tensor given in (\\ref{ikriemann}), and replaces them on the Palatini equation (\\ref{palatiniexp}), one obtains\n\n\\bea\n\\delta\\R_{ijkl} &=& \\Big( \\big( 3 g - 1 \\big) H_T + r g \\partial_r H_T \\Big) \\Big( g_{il} \\mathcal{T}_{jk} - g_{ik} \\mathcal{T}_{jl} - g_{jl} \\mathcal{T}_{ik} + g_{jk} \\mathcal{T}_{il} \\Big) \\nonumber \\\\\n&+& r^2 H_T \\Big( D_i D_l \\mathcal{T}_{jk} - D_i D_k \\mathcal{T}_{jl} - D_j D_l \\mathcal{T}_{ik} + D_j D_k \\mathcal{T}_{il} \\Big), \\label{drtensori} \\\\\n\\delta\\R_{itjt} &=& \\left( - r^2 \\partial_t^2 H_T + \\frac{1}{2} r^2 f f' \\partial_r H_T + r f f' H_T \\right) \\mathcal{T}_{ij}\\,, \\\\\n\\delta\\R_{itjr} &=& \\left(- r^2 \\partial_t \\partial_r H_T -r \\partial_t H_T + \\frac{1}{2} r^2 \\frac{f'}{f} \\partial_t H_T \\right) \\mathcal{T}_{ij}\\,, \\\\\n\\delta\\R_{irjr} &=& \\left( - r \\frac{g'}{g} H_T - \\frac{1}{2} r^2 \\frac{g'}{g} \\partial_r H_T - 2 r \\partial_r H_T - r^2 \\partial^2_r H_T \\right) \\mathcal{T}_{ij}\\,, \\\\\n\\delta\\R_{abcd} &=& 0,\n\\eea\nand further\n\\bea\n\\delta \\mathcal{R}_{ij} &=& \\frac{r^2}{f} \\left( \\partial^2_t H_T \\right) \\mathcal{T}_{ij} - r^2 g \\left( \\partial^2_r H_T \\right) \\mathcal{T}_{ij} - \\frac{1}{2} r^2 \\left(f^\\prime + g^\\prime\\right) \\left( \\partial_r H_T \\right) \\mathcal{T}_{ij} - r \\left(f^\\prime + g^\\prime\\right) H_T \\mathcal{T}_{ij} \\nonumber \\\\ &-&\\left(d-2\\right) r g \\left( \\partial_r H_T \\right) \\mathcal{T}_{ij}\n+ 2\\left(d-3\\right) (1-g) H_T \\mathcal{T}_{ij} + \\left(k_T +2 \\right) H_T \\mathcal{T}_{ij} \\,, \\\\\n\\delta \\mathcal{R}_{ia} &=& 0, \\quad\n\\delta \\mathcal{R}_{ab} = 0, \\quad\n\\delta \\mathcal{R} = 0. \\label{drtensora}\n\\eea\n\\noindent\n\nThese are the equations we will need in order to perturb the $\\a$--corrected field equations.\n\n\\section{Gravitational perturbations to the $\\a$--corrected field equations}\n\n\\subsection{Analysis on the Einstein frame}\n\\indent\n\nThe $d$--dimensional effective action with $\\a$ corrections we will be dealing with is given, in the Einstein frame, by\n\\be \\label{eef} \\frac{1}{16 \\pi G} \\int \\sqrt{-g} \\left( \\R -\n\\frac{4}{d-2} \\left( \\d^\\mu \\phi \\right) \\d_\\mu \\phi +\n\\mbox{e}^{\\frac{4}{d-2} \\phi} \\frac{\\lambda}{2}\\\n\\R^{\\mu\\nu\\rho\\sigma} \\R_{\\mu\\nu\\rho\\sigma} \\right) \\mbox{d}^dx .\n\\ee\n\nHere $\\lambda = \\frac{\\a}{2}, \\frac{\\a}{4}$ and $0$, for\nbosonic, heterotic and type II strings, respectively. We are only\nconsidering gravitational terms: we can consistently settle all\nfermions and gauge fields to zero for the moment. That is not the\ncase of the dilaton, as it can be seen from the resulting field equations:\n\n\\bea\n\\nabla^2 \\phi - \\frac{\\lambda}{4}\\ \\mbox{e}^{\\frac{4}{2-d} \\phi} \\left(\n\\R_{\\rho\\sigma\\lambda\\tau} \\R^{\\rho\\sigma\\lambda\\tau} \\right) &=&\n0, \\label{bdfe} \\\\ \\R_{\\mu\\nu} + \\lambda\\ \\mbox{e}^{\\frac{4}{2-d}\n\\phi} \\left( \\R_{\\mu\\rho\\sigma\\tau} {\\R_{\\nu}}^{\\rho\\sigma\\tau} -\n\\frac{1}{2(d-2)} g_{\\mu\\nu} \\R_{\\rho\\sigma\\lambda\\tau}\n\\R^{\\rho\\sigma\\lambda\\tau} \\right) &=& 0. \\label{bgfe} \\eea\n\nThe correction term we are considering in (\\ref{eef}) is $\\R_{\\rho\\sigma\\lambda\\tau} \\R^{\\rho\\sigma\\lambda\\tau}$, the square of the Riemann tensor, which we generically designate as $\\R^2$: since we are not considering the Ricci tensor in the corrections (it would only contribute at a higher order in $\\lambda$), there is no possible confusion. From (\\ref{bdfe}) one sees that this correction term acts as a source for the dilaton and, therefore, one cannot set the dilaton to zero without setting this term to zero too. Still, as it was shown in \\cite{Moura:2009it} and we will review later (see eq. (\\ref{fr2})), for a spherically symmetric metric like (\\ref{schwarz}), at order $\\lambda=0$ the dilaton is a constant (which can be always set to 0). The dilaton only gets nonconstant terms at order $\\lambda;$ this is why we could neglect terms which are quadratic in $\\phi$ while deriving these field equations, since we are only working perturbatively to first order in $\\lambda.$\n\nIn the present context, any black hole solution is built perturbatively in $\\lambda,$ and a solution will only be valid in regions where $r^2 \\gg \\lambda$, \\textit{i.e.}, any perturbative solution is only valid for black holes whose event horizon is much bigger than the string length.\n\nWe want to study scattering processes associated to solutions to the field equations above and, therefore, they are the ones which we will perturb.\n\nThe perturbation of the $\\a$--corrected field equations (\\ref{bdfe}) and (\\ref{bgfe}) has already been taken in \\cite{Moura:2006pz}, for a spherically symmetric metric like (\\ref{schwarz}) but with $f(r)=g(r);$ here we consider the general case (\\ref{schwarz}) and apply the results to concrete metrics.\n\nBy perturbing (\\ref{bdfe}) and (\\ref{bgfe}) one gets\n\n\\bea\n\\delta \\nabla^2 \\phi &-& \\frac{\\lambda}{4}\\ \\mbox{e}^{\\frac{4}{2-d} \\phi}\\ \\delta \\left( \\R_{\\rho\\sigma\\lambda\\tau} \\R^{\\rho\\sigma\\lambda\\tau} \\right) + \\frac{\\lambda}{d-2}\\ \\mbox{e}^{\\frac{4}{2-d} \\phi}\\ \\R_{\\rho\\sigma\\lambda\\tau} \\R^{\\rho\\sigma\\lambda\\tau}\\ \\delta \\phi = 0, \\label{pbdfe} \\\\\n\\delta \\R_{ij} &+& \\lambda\\ \\mbox{e}^{\\frac{4}{2-d} \\phi} \\left[ \\delta \\left( \\R_{i\\rho\\sigma\\tau} {\\R_{j}}^{\\rho\\sigma\\tau} \\right) - \\frac{1}{2(d-2)} \\R_{\\rho\\sigma\\lambda\\tau} \\R^{\\rho\\sigma\\lambda\\tau}\\ h_{ij} \\right. \\nonumber \\\\\n&-& \\left. \\frac{1}{2(d-2)}\\ g_{ij}\\ \\delta \\left( \\R_{\\rho\\sigma\\lambda\\tau} \\R^{\\rho\\sigma\\lambda\\tau} \\right) \\right] + \\frac{4}{d-2}\\ \\R_{ij}\\ \\delta \\phi = 0. \\label{pbgfe}\n\\eea\n\n\\noindent\nUsing the explicit form of the Riemann tensor (\\ref{ikriemann}) together with the variations (\\ref{htensor}) and (\\ref{drtensori}--\\ref{drtensora}), one can compute the terms in (\\ref{pbdfe}) and (\\ref{pbgfe}).\n\nUsing these variations, it is a simple computation to verify that $\\delta \\left( \\R_{\\rho\\sigma\\lambda\\tau} \\R^{\\rho\\sigma\\lambda\\tau} \\right) \\equiv 0.$ From this fact and (\\ref{pbdfe}), we see that one can consistently set $\\delta \\phi=0,$ as expected for a tensorial perturbation of a scalar field. The derivation is explicitly given in article \\cite{Moura:2006pz}, to which we refer the reader. Eq. (\\ref{pbdfe}) does not give us any other relevant information.\n\nCollecting the several expressions, the result for (\\ref{pbgfe}) finally becomes\n\n\\bea\n&&\n\\left( 1 - 2 \\lambda \\frac{f'}{r} \\right) \\frac{r^2}{f} \\partial^2_t H_T - \\left( 1 - 2 \\lambda \\frac{g'}{r} \\right) r^2 g \\,\\partial^2_r H_T \\nonumber \\\\\n&&\n- \\left[(d-2) r g + \\frac{1}{2} r^2 \\left(f'+g'\\right) + 4 \\lambda (d-4) \\frac{g \\left( 1 - g \\right)}{r} - 4 \\lambda g g' - \\lambda r \\left( f'^2 + g'^2 \\right) \\right] \\partial_r H_T \\nonumber \\\\\n&&\n+ \\left[ \\left( \\ell \\left( \\ell + d - 3 \\right) -2 \\right) \\left(1+\\frac{4\\lambda}{r^2}\\left(1-g\\right)\\right) + 2(d-2)- 2(d-3)g- r\\left(f' + g'\\right) \\right. \\nonumber \\\\\n&&\n+ \\left. \\lambda \\left( 8 \\frac{1 - g}{r^2} + 2 \\left( d - 3 \\right)\\frac{\\left( 1 - g \\right)^2}{r^2} - \\frac{r^2}{d-2} \\left[f'' + \\frac{1}{2} \\left( \\frac{f' g'}{g} - \\frac{f'^2}{f}\\right)\\right]^2 \\right) \\right] H_T = 0. \\label{master0}\n\\eea\n\n\\noindent\nThis is a second order partial differential equation for the perturbation function $H_T.$ If we now divide (\\ref{master0}) by $\\left( 1 - 2 \\lambda \\frac{f'}{r} \\right) \\frac{r^2}{f},$ we obtain an equation of the form\n\\be \\label{dottigen}\n\\partial^2_t H_T - F^2(r)\\ \\partial^2_r H_T + P(r)\\ \\partial_r H_T + Q(r)\\ H_T = 0\n\\ee\nwith\n\\bea\nF &=& \\sqrt{\\frac{1 - 2 \\lambda \\frac{g'}{r}}{1 - 2 \\lambda \\frac{f'}{r}} fg}, \\nonumber \\\\\nP &=& - \\frac{f}{1 - 2 \\lambda \\frac{f'}{r}} \\left[ (d-2) \\frac{g}{r} + \\frac{1}{2} \\left(f'+g'\\right) + \\frac{2 \\lambda}{r} \\left( 2 ( d - 4 ) \\frac{g \\left( 1 - g \\right)}{r^2}\n- 2 \\frac{g g'}{r} - \\frac{1}{2} \\left(f'^2+g'^2\\right) \\right) \\right], \\nonumber \\\\\nQ &=& \\frac{f}{1 - 2 \\lambda \\frac{f'}{r}} \\left[ \\frac{\\ell \\left( \\ell + d - 3 \\right)}{r^2} - \\frac{f'+g'}{r} + 2 (d-3) \\frac{1-g}{r^2} \\right. \\nonumber \\\\ &+& \\left. \\frac{\\lambda}{r^2} \\left( 4 \\ell \\left( \\ell + d - 3 \\right) \\frac{1-g}{r^2} + 2 (d-3) \\frac{\\left( 1-g \\right)^2}{r^2} - \\frac{r^2}{d-2} \\left[f'' + \\frac{1}{2} \\left( \\frac{f' g'}{g} - \\frac{f'^2}{f}\\right)\\right]^2\n\\right) \\right]. \\label{fpqnp}\n\\eea\n\\noindent\nFor our purposes, we would like to re--write the above equation (\\ref{dottigen}) in a more tractable form, as a master equation. In order to achieve so, we follow a procedure similar to the one in \\cite{dg05a}, defining a gauge--invariant ``master variable'' for the gravitational perturbation as\n\n\\be\n\\Phi = k(r) H_T, \\quad k(r) = \\frac{1}{\\sqrt{F}} \\exp \\left( - \\int \\, \\frac{P}{2F^2} \\, dr \\right), \\label{k}\n\\ee\nand replacing $\\partial\/\\partial r$ by $\\partial\/\\partial r_*,$ $r_*$ being the tortoise coordinate defined in this case by $dr_* = \\frac{dr}{F(r)}.$ It is then easy to see that an equation like (\\ref{dottigen}) may be written as a master equation:\n\n\\be\n\\frac{\\partial^2 \\Phi}{\\partial r_*^2} - \\frac{\\partial^2 \\Phi }{\\partial t^2} = \\left( Q + \\frac{F'^2}{4} - \\frac{F F''}{2} - \\frac{P'}{2} + \\frac{P^2}{4 F^2} + \\frac{P F'}{F} \\right) \\Phi \\equiv V_{\\textsf{T}} \\left[ f(r), g(r) \\right] \\Phi. \\label{potential0}\n\\ee\nThis is the equation we shall be working with for the rest of the article. We see that the field equations for the tensorial perturbations of a background spherically symmetric metric like (\\ref{schwarz}) in $d$ dimensions can be reduced to a single second order partial differential equation, in $(r_*,t)$ coordinates, for a master variable $\\Phi$ which is a simple combination of gauge--invariant variables. This result had been obtained in \\cite{ik03a} for Einstein gravity; here, we see that it is also valid in the presence of string corrections quadratic in the Riemann tensor.\n\n\nIn order to explicitly compute the potential $V_{\\textsf{T}} \\left[ f(r), g(r) \\right],$ one should first simplify the expressions from (\\ref{fpqnp}). To begin with, one can judiciously use the field equation (\\ref{bgfe}) to derive the relation\n$$\\lambda \\R_{abcd} \\R^{abcd} = 2 g^{ij} \\R_{ij} + \\lambda \\R_{ijkl} \\R^{ijkl}.$$\nHere we used the fact that, in the presence of a metric like (\\ref{schwarz}), the dilaton field is of order $\\lambda,$ as it was explicitly shown in \\cite{Moura:2009it}. This way we could neglect all the dilaton terms, which would only contribute at least to order ${\\mathcal{O}} (\\lambda^2)$). Using the above relation and the explicit form of the Riemann tensor (\\ref{ikriemann}), we can obtain the (on--shell) relation\n\\be \\label{f''}\n\\lambda \\left[f'' + \\frac{1}{2} \\left( \\frac{f' g'}{g} - \\frac{f'^2}{f}\\right)\\right]^2 -\\frac{2 (d-3) (d-2)}{r^2} \\left(\\frac{\\lambda (1-g)}{r^2}+1\\right) (1-g)+\\frac{(d-2)}{r} \\left(\\frac{g f'}{f}+g'\\right)=0.\n\\ee\nWe may use the relation above in order to remove the $\\left[f'' + \\frac{1}{2} \\left( \\frac{f' g'}{g} - \\frac{f'^2}{f}\\right)\\right]^2$ term from $Q$ in (\\ref{fpqnp}). Also, although the expressions in (\\ref{fpqnp}) are non--polynomial in $\\lambda$, we can expand them and take only the first order terms, since that is the order in $\\lambda$ to which we are working. A simple power series expansion yields\n\n\\bea\nF &=& \\sqrt{fg}\\left(1 +\\lambda \\frac{f'-g'}{r}\\right), \\nonumber \\\\\nP &=& - f \\left[ (d-2) \\frac{g}{r} + \\frac{1}{2} \\left(f'+g'\\right) + \\frac{\\lambda}{r^2}  \\left(4 (d-4) \\frac{g(1-g)}{r}+r g'\\left(f'-g'\\right)-4 g g' +2 (d-2) g f' \\right) \\right], \\nonumber \\\\\nQ &=& \\frac{\\ell \\left( \\ell + d - 3 \\right)}{r^2} f + \\frac{(g-f)f'}{r} +\n\\frac{2\\lambda}{r^2} \\left[ \\frac{\\ell \\left( \\ell + d - 3 \\right)}{r} f \\left( 2 \\frac{1-g}{r} + f' \\right) + (g-f) f'^2 \\right].\n\\label{fpq}\n\\eea\nReplacing the expressions from (\\ref{fpq}) on $V_{\\textsf{T}} \\left[ f(r), g(r) \\right]$ given in (\\ref{potential0}), one finally obtains\n\n\\bea V_{\\textsf{T}} [f(r),g(r)] &=&\n\\frac{1}{16 r^2 f g}\n\\left[(16 \\ell (\\ell +d-3) f^2 g+ r^2 f^2 f'^2 +3 r^2 g^2 f'^2-2 r^2 f (f+g) f' g' -4 r^2 fg (g-f) f'' \\right. \\nonumber \\\\\n&+& \\left. 16r f g^2 f' +4 r (d-6)f^2 g f' +4 (d-2)r f^2 g g' +4 (d-4) (d-2) f^2 g^2 \\right] \\nonumber \\\\\n&+&\\frac{\\lambda}{8 r^4 f g} \\left[32 \\ell (\\ell +d-3) f^2 (1-g) g +16 \\ell (d+\\ell -3) f^2 g f' r -r^3 f^2 f'^2 \\left(f'-g'\\right) \\right. \\nonumber \\\\\n&+& 3 r^3 g^2 f'^2 \\left(f'-g'\\right)-2 r^3 f g f' \\left(f'-g'\\right) g' -4 r^3 f^2 g f' \\left(f''-g''\\right)-2 r^3 f^2 g g' \\left(f''-g''\\right) \\nonumber \\\\\n&+& 2 r^3 f g^2 \\left(-3 f' f''+2 g' f''+f' g''\\right) -4 r^3 f^2 g^2 \\left(f'''-g'''\\right) +18 r^2 f g^2 f'^2 -12 r^2 f^2 g f'^2 \\nonumber \\\\\n&-& 10 r^2 f^2 g g'^2 -2 r^2 f g^2 f' g' +2 r^2 (4 d-13) f^2 g f' g' +8 r^2 f^2 g^2 f'' +8 (d-5) r^2 f^2 g^2 g''\n \\nonumber \\\\\n&+& 4 r(d-4)^2 f^2 g^2 (f' + g') +8r f^2 g^2(g'-f') +8 (d-4) r f^2 g (f' + g'-4g g') \\nonumber \\\\\n&+& \\left. 16 (d-5) (d-4) f^2 g^2 (1-g) \\right].\n\\label{potential}\n\\eea\nTaking $f=g$ in (\\ref{schwarz}), (\\ref{potential}) matches the result of \\cite{Moura:2006pz}, as it should. Also, at order $\\lambda=0,$ (\\ref{potential}) matches the Einstein-Hilbert potential obtained in \\cite{ik03a}.\n\nEquation (\\ref{potential}) gives the generic expression for the potential for tensor--type gravitational perturbations of any kind of static, spherically symmetric $\\R^2$ string--corrected black hole in $d$--dimensions of the form (\\ref{schwarz}). This is also one of the main results of this article. Knowing this potential, we are now ready to study the stability of such black holes under those perturbations.\n\n\\subsection{Analysis in different frames}\n\\label{diffr}\n\\indent\n\nUnder a conformal transformation, the metric and Riemann tensor transform as:\n\\bea \\label{confgen}\ng_{\\mu\\nu} &\\rightarrow& \\exp \\left( \\Omega \\right) g_{\\mu\\nu}, \\\\\n{\\R_{\\mu\\nu}}^{\\rho\\sigma} &\\rightarrow& \\exp \\left( -\\Omega \\right) \\left( {\\R_{\\mu\\nu}}^{\\rho\\sigma} - 2 {\\delta_{\\left[\\mu\\right.}}^{\\left[\\rho\\right.} \\nabla_{\\left.\\nu \\right]} \\nabla^{\\left.\\sigma \\right]} \\Omega + {\\delta_{\\left[\\mu\\right.}}^{\\left[\\rho\\right.} \\left(\\nabla_{\\left.\\nu \\right]} \\Omega \\right) \\nabla^{\\left.\\sigma \\right]} \\Omega - {\\delta_{\\left[\\mu\\right.}}^{\\left[\\rho\\right.} \\delta_{\\left.\\nu\\right]}^{\\left. \\, \\, \\, \\, \\sigma\\right]} \\left(\\nabla^{\\lambda}\\Omega\\right) \\nabla_{\\lambda}\\Omega \\right).\n\\eea\n\nIn our perturbation analysis we assumed an action\/solution taken in the Einstein frame. But the action obtained directly from string perturbation theory comes in the string frame as\n\\be \\label{esf}\n\\frac{1}{16 \\pi G} \\int \\sqrt{-g}\\ \\mbox{e}^{-2 \\phi} \\Big( \\R + 4 \\left( \\d^\\mu \\phi \\right) \\d_\\mu \\phi + \\frac{\\lambda}{2}\\ \\R^{\\mu\\nu\\rho\\sigma} \\R_{\\mu\\nu\\rho\\sigma} \\Big) \\mbox{d}^dx.\n\\ee\nTaking a dilaton--dependent conformal transformation (\\ref{confgen}) with $\\Omega=\\frac{4}{d-2} \\phi,$ we obtain from (\\ref{esf}) the action (\\ref{eef}) in the Einstein frame, plus some terms with derivatives of the dilaton, which would only contribute at higher orders in $\\lambda,$ since as we saw the dilaton itself is already of order $\\lambda.$\n\nIn general, if $Y(\\R)$ is a scalar polynomial in the Riemann tensor representing the higher derivative corrections, with conformal weight $w$ and the convention that $w \\left( g_{\\mu\\nu} \\right) = +1,$ an arbitrary action with $\\a$ corrections is (just the gravitational sector)\n\\be\n\\frac{1}{16 \\pi G} \\int \\sqrt{-g} \\Big( \\R - \\frac{4}{d-2} \\left( \\d^\\mu \\phi \\right) \\d_\\mu \\phi + z Y(\\R) \\Big) \\mbox{d}^dx,\n\\ee\nwith $z$ being, up to a numerical factor, the suitable power of the inverse string tension $\\a$ for $Y(\\R)$.\n\nIn a different frame, after the conformal transformation (\\ref{confgen}), this action becomes\n\\be\n\\frac{1}{16 \\pi G} \\int \\sqrt{-g}\\ \\mbox{e}^{\\frac{d-2}{2} \\Omega} \\left( \\widetilde{\\R} - \\frac{4}{d-2} \\left( \\d^\\mu \\phi \\right) \\d_\\mu \\phi + z\\ \\mbox{e}^{\\left( 1 + w \\right) \\Omega} Y(\\widetilde{\\R}) \\right) \\mbox{d}^dx.\n\\ee\nAgain, if the conformal transformation (\\ref{confgen}) is dilaton--dependent, it will generate from $Y(\\widetilde{\\R})$ some dilaton terms which are of higher order in $\\lambda.$ Besides that, other lower order parcels containing the dilaton may be generated, affecting the numerical factor (and sign) of its kinetic term, as one can see comparing (\\ref{esf}) to (\\ref{eef}). These dilaton terms are the only change in the graviton field equation generated by a dilaton--dependent conformal transformation (\\ref{confgen}).\n\nThe field equation (\\ref{bgfe}) we considered was obtained after removing the trace term $-\\frac{1}{2} g_{\\mu\\nu} \\R,$ obtained after contracting (\\ref{bgfe}) itself. This procedure is totally independent of the chosen frame. Other dilaton terms were removed by considering the dilaton field equation, which obviously depends on the choice of frame (equation (\\ref{bdfe}) is valid in the Einstein frame). Still, as we have seen under tensorial perturbations the dilaton $\\phi$ is inert, and so is its field equation, in any chosen frame. Indeed the arguments in the discussion after (\\ref{bgfe}), namely the fact that $\\delta \\left( \\R_{\\rho\\sigma\\lambda\\tau} \\R^{\\rho\\sigma\\lambda\\tau} \\right) \\equiv 0$, do not depend on the choice of frame. This means that, in order to study tensorial perturbations, only (\\ref{bgfe}) is relevant, and the results obtained from it are valid in any chosen frame, namely the equation governing tensorial perturbations of the metric (\\ref{master0}) (or, equivalently, (\\ref{potential0})).\n\nFinally we should mention that, although the dilaton field equation looks different in the string and in the Einstein frames (see \\cite{cmp89}), their spherically symmetric $d$--dimensional solution is the same in both frames. Indeed the string frame dilaton solution obtained in \\cite{cmp89} is equivalent to the Einstein frame dilaton solution from \\cite{Moura:2009it}, apart from the normalization at infinity (which in \\cite{Moura:2009it} was taken to be zero - this is the solution we use). This is not obvious if one looks at the two solutions, but it can be verified case by case in $d$ using symbolic computation software. The reason becomes clear after comparing the two field equations and neglecting terms of higher order in $\\lambda.$ This fact makes the change from the string to the Einstein frame (and vice versa) unambiguous.\n\n\\section{General analysis of perturbative stability}\n\\indent\n\nIn order to study the stability of a solution, we use the ``S--deformation approach'', first introduced in \\cite{ik03b} and later further developed in \\cite{dg04, dg05a}. Let us briefly review this technique in the following (for more details we refer the reader to the original discussion in \\cite{ik03b}).\n\nAfter having obtained the potential $V_{\\textsf{T}}$ for the master equation (\\ref{potential0}), one assumes that its solutions are of the form $\\Phi(x,t) = e^{i\\omega t} \\phi(x)$, such that $\\frac{\\partial\\Phi}{\\partial t} = i\\omega \\Phi$. In this way the master equation (\\ref{potential0}) may be written in Schr\\\"odinger form, for a generic potential $V(x),$ as\n\n\\be\n\\left[ - \\frac{d^2}{dx^2} + V(x) \\right] \\phi(x) \\equiv A \\phi(x) = \\omega^2\n\\phi(x).\n\\ee\n\n\\noindent\nA given solution of the gravitational field equations will then be perturbatively stable if and only if the operator $A$ defined above has no negative eigenvalues for $x \\in {\\mathbb{R}}$ \\cite{ik03b}. The above condition is equivalent to the positivity, for any given $\\phi,$ of the inner product \\cite{ik03b}\n\n\\be\n\\left \\langle \\phi \\left| A \\phi \\right \\rangle \\right. = \\int_{-\\infty}^{+\\infty} \\phi^\\dagger (x) \\left[ - \\frac{d^2}{dx^2} + V(x) \\right] \\phi(x)\\ dx,\n\\ee\n\n\\noindent\nwhich, after some integrations by parts and further algebra, may be rewritten as\n\n\\be\n\\left \\langle \\phi \\left| A \\phi \\right \\rangle \\right.  = \\int_{-\\infty}^{+\\infty} \\left[ \\left| \\frac{d\\phi}{dx} \\right|^2 + V(x) \\left|\\phi \\right|^2 \\right] dx = \\int_{-\\infty}^{+\\infty} \\left[ \\left| D \\phi \\right|^2 + \\widetilde{V}(x) \\left| \\phi \\right|^2 \\right] dx.\n\\ee\n\n\\noindent\nHere we have defined $D = \\frac{d}{dx} + S$ and $\\widetilde{V}(x) = V(x) + f \\frac{dS}{dr} - S^2$, with $S$ a completely arbitrary function. Taking $S = - \\frac{F}{k} \\frac{dk}{dr},$ with $k(r)$ given by (\\ref{k}), we simply obtain $\\widetilde{V}(x) = Q$ and\n\n\\be\n\\left \\langle \\phi \\left| A \\phi \\right \\rangle \\right. = \\int_{-\\infty}^{+\\infty} \\left|D \\phi \\right|^2 dx + \\int_{-\\infty}^{+\\infty} Q(x) \\left|\\phi \\right|^2 dx.\n\\ee\n\n\\noindent\nThe second term of the expression above may be written as ($R_H$ being the radius of the event horizon)\n\n\\be\n\\int_{R_H}^{+\\infty} \\frac{Q(r)}{F(r)} \\left|\\phi \\right|^2 dr,\n\\ee\n\n\\noindent\nSince $\\left|\\phi \\right|^2$ and $\\left|D \\phi \\right|^2$ are positive, perturbative stability of a given black hole solution then follows if one can prove that $\\frac{Q(r)}{F(r)}$ is a positive function for $r \\ge R_H.$ Here we note that, by definition, the horizon is the largest root of $f(r),$ i.e $f>0$ for $r>R_H.$ Here we assume the same is valid for $g,$ i.e. the classical part of $F=F_0 + \\lambda F_1,$ given by $F_0=\\sqrt{fg},$ is well defined and positive for $r>R_H.$ This does not mean that $F$ is necessarily positive: close to the horizon, $F_0=\\sqrt{fg}$ is expected to have very small values, which we cannot assume to be larger than the $\\lambda$--corrections, as one usually does. These corrections, from (\\ref{fpq}) given by $F_1= \\sqrt{fg} \\frac{f'-g'}{r},$ may well be negative if $f'<g',$ i.e. if $f$ grows slower than $g.$ This means that, very close to the horizon, for a small range, it may happen that $F$ is negative (that would not affect the metric, as $F$ is not part of it). This way, it is more rigorous to verify the sign of $\\frac{Q(r)}{F(r)}.$ From (\\ref{fpq}) we have, to first order in $\\lambda,$\n\\bea\n\\frac{Q(r)}{F(r)} &=& \\frac{1}{r^2 \\sqrt{fg}} \\Big[ \\ell \\left( \\ell + d - 3 \\right)f + r (g-f)f' \\nonumber \\\\\n&+& \\frac{\\lambda}{r^2} \\left[ \\ell \\left( \\ell + d - 3 \\right) f \\left( 4 (1-g) +r (f'+g') \\right) + r^2 (g-f) f' (f'+g') \\right] \\Big].\n\\label{qf}\n\\eea\n\n\\noindent\nFor a given black hole solution of (\\ref{bgfe}), the positiveness of this expression will assure its stability under tensor perturbations of the metric.\n\n\\section{Application to concrete string-corrected black hole solutions}\n\\indent\n\nWe now apply our result to the study of the stability of a few specific black hole solutions in string theory. Articles \\cite{dg04,dg05a} were the first to apply this formalism to black holes with $\\R^2$ corrections in $d$ dimensions, but to solutions which are not in the context of string theory.\n\nThe most general static, spherically symmetric metric in $d$ dimensions for pure Einstein-Hilbert\ngravity in vacuum is the Tangherlini solution \\cite{Tangherlini:1963bw}, of the form (\\ref{schwarz}) with $f(r)=g(r) \\equiv f_0^T(r),$ where\n\\be\nf_0^T(r) =1 - \\left(\\frac{R_H}{r}\\right)^{d-3}, \\label{tangher}\n\\ee\n$R_H$ being the horizon radius.\n\nThe first solution of such kind with string theoretical $\\R^2$ corrections was obtained by Callan, Myers and Perry \\cite{cmp89}. It is of the form (\\ref{schwarz}), with $f(r)=g(r) \\equiv g_{CMP}(r),$ where\n\\be\ng_{CMP}(r) = \\left(1-\\left(\\frac{R_H}{r}\\right)^{d-3} \\right) \\left[1- \\lambda\n\\frac{(d-3)(d-4)}{2} \\frac{R_H^{d-5}}{r^{d-1}}\n\\frac{r^{d-1}-R_H^{d-1}}{r^{d-3} - R_H^{d-3}} \\right]. \\label{fr2x}\n\\ee\nIt is clearly a deformation of the Tangherlini metric (\\ref{tangher}). The stability of such solution under tensor perturbations was proven in \\cite{Moura:2006pz}.\n\nNext we will apply the method to two previously unstudied cases.\n\n\\subsection{The string-corrected dilatonic $d$--dimensional black hole}\n\\subsubsection{Description of the solution}\n\\indent\n\nThe Callan--Myers--Perry solution expresses the effect of the string $\\R^2$ corrections, but it does not consider any other string effects, namely the fact that string theories live in $d_S$ spacetime dimensons ($d_S=10$ or 26 on heterotic or bosonic strings, respectively), and have to be compactified to $d$ dimensions on a $d_S - d$--dimensional manifold. When passing from the string to the Einstein frame, the volume of the compactification manifold becomes spatially varying. In the simple case when such manifold is a flat torus, that volume depends only on the $d$--dimensional part of the dilaton $\\phi$ and, after solving the $\\a$--corrected field equation (\\ref{pbgfe}) the metrics of the compactification manifold and of the $d$--dimensional spacetime decouple.\n\nThe explicit solution was worked out in \\cite{Moura:2009it}. The $d$--dimensional part of the metric is of the form (\\ref{schwarz}), with $g(r)$ being equal to $g_{CMP}(r)$ given by (\\ref{fr2x}) and\n\\be\nf(r) = g(r) + 4 \\left( 1 -\n\\left(\\frac{R_H}{r}\\right)^{d-3} \\right)\n\\frac{d_s-d}{\\left(d_s-2\\right)^2} \\left(\\phi - r \\phi'\\right).\n\\label{fnew}\n\\ee\n\nThe general solution for the dilaton, in the classical background of the spherically symmetric Tangherlini black hole (\\ref{tangher}), was also obtained in \\cite{Moura:2009it}. As we previously mentioned, this solution is necessarily of order $\\lambda:$ it is given by $\\phi(r) = \\frac{\\lambda}{R_H^2} \\varphi(r),$ with\n\\bea \\label{fr2}\n\\varphi(r) &:=& \\frac{(d-2)^2}{4}\n\\left[\\ln\\left( 1 - \\left(\\frac{R_H}{r}\\right)^{d-3} \\right) -\\frac{d-3}{2}\\left(\\frac{R_H}{r}\\right)^2 - \\frac{d-3}{d-1}\n\\left(\\frac{R_H}{r}\\right)^{d-1} \\right. \\nonumber \\\\\n&+& \\left. B \\left(\\left(\\frac{R_H}{r}\\right)^{d-3};\\, \\frac{2}{d-3}, 0 \\right)\\right]\n\\eea\nwith $B(x;\\,a,b)=\\int_0^x t^{a-1}\\,(1-t)^{b-1}\\,dt \\!$ being the incomplete Euler beta function. Its derivative is given by\n\\be\n\\varphi'\\left(r\\right)=\\frac{(d-3)\n(d-2)^2}{4} \\frac{R_H^{d-3}}{r^{d-2}}\n\\frac{1-\\left(\\frac{R_H}{r}\\right)^{d-1}}{1-\\left(\\frac{R_H}{r}\\right)^{d-3}}. \\label{fr3}\n\\ee\n\nBoth $\\varphi(r)$ and $\\varphi'(r)$ are well--defined functions everywhere, namely at the horizon: \\footnote{The digamma function is given by $\\psi(z)=\\Gamma'(z)\/\\Gamma(z),$ $\\Gamma(z)$ being the usual $\\Gamma$ function. For positive $n,$ one defines $\\psi^{(n)}(z)=d^n\\,\\psi(z)\/d\\,z^n.$ This definition can be extended for other values of $n$ by fractional calculus analytic continuation. These are meromorphic functions of $z$ with no branch cut discontinuities.\n\n$\\gamma$ is Euler's constant, defined by $\\gamma=\\lim_{n \\to \\infty} \\left(\\sum_{k=1}^n \\frac{1}{k} - \\ln n \\right),$ with numerical value $\\gamma \\approx 0.577216.$}\n\\bea\n\\varphi\\left(r_H\\right)&=&- \\frac{(d-2)^2}{8 (d-1)} \\left(d^2-2d+2 (d-1) \\left(\\psi^{(0)}\\left(\\frac{2}{d-3}\\right) + \\gamma \\right) -3\\right), \\label{firh} \\\\\n\\varphi'\\left(r_H\\right)&=& \\frac{(d-2)^2 (d-1)}{4 R_H} \\label{firhd}.\n\\eea\n\nThe derivative (\\ref{fr3}) is a positive function, which means the dilaton solution (\\ref{fr2}) is negative and strictly monotonic. It grows up to 0 at infinity, as it can be seen from the asymptotic expansion\n\\be\n\\varphi(r) \\approx -\\frac{(d-2)^2}{4} \\left(\\frac{R_H}{r}\\right)^{d-3}\n-\\frac{(d-2)^2}{8} \\left(\\frac{R_H}{r}\\right)^{2d-6}\n+ \\frac{(d-2)(d-3)}{8} \\left(\\frac{R_H}{r}\\right)^{2d-4}+ {\\mathcal{O}} \\left( \\left( \\frac{R_H}{r}\\right)^{2d-3} \\right),\n\\label{dildeser}\n\\ee\nobtained from\n\\bea\n&& B \\left(\\left(\\frac{R_H}{r}\\right)^{d-3};\\, \\frac{2}{d-3}, 0 \\right)+ \\ln\\left( 1 - \\left(\\frac{R_H}{r}\\right)^{d-3} \\right) \\nonumber \\\\\n&=& \\frac{d-3}{2} \\left(\\frac{R_H}{r}\\right)^2 - \\left(\\frac{R_H}{r}\\right)^{d-3} + \\frac{d-3}{d-1} \\left(\\frac{R_H}{r}\\right)^{d-1}-\\frac{1}{2} \\left(\\frac{R_H}{r}\\right)^{2(d-3)} + {\\mathcal{O}} \\left( \\left( \\frac{R_H}{r}\\right)^{2d-5} \\right).\n\\eea\n\n\\subsubsection{Study of the stability}\n\\indent\n\nIn order to check for the stability of this metric under tensor perturbations, we split $\\frac{Q}{F},$ given in (\\ref{qf}), in their \"classical\" and $\\lambda$--corrected parts: $\\frac{Q}{F} = \\left.\\frac{Q}{F}\\right|_0 + \\lambda \\left.\\frac{Q}{F}\\right|_1.$ Since $\\left.\\frac{Q}{F}\\right|_1$ is already multiplied by $\\lambda,$ the order at which we are working, it must be evaluated using the functions corresponding to the $\\lambda=0$ Tangherlini solution (\\ref{tangher}). Using $f(r)=g(r) = f_0^T(r),$ and since for $r>R_H$ clearly $f_0^T >0,$ from (\\ref{qf}) we get\n$$r^4 \\sqrt{fg} \\left.\\frac{Q}{F}\\right|_1 = \\frac{2}{r^2} \\frac{\\ell \\left( \\ell + d - 3 \\right)}{r} f_0^T \\left( 2 \\frac{1-f_0^T}{r} + f_0^{T'} \\right).$$\nWe only need to compute\n$$2 \\frac{1-f_0^T}{r} + f_0^{T'}= (d-3) \\frac{R_H^{d-3}}{r^{d-2}},$$\nwhich is always a positive quantity. This way, $\\left.\\frac{Q}{F}\\right|_1$ is always positive.\n\n$r^2 \\sqrt{fg} \\left.\\frac{Q}{F}\\right|_0=\\frac{\\ell \\left( \\ell + d - 3 \\right)}{r^2} f + \\frac{(g-f)f'}{r} $ must be computed with the full, $\\lambda$--corrected metric (therefore, it will also include $\\lambda$ corrections).\n\nBy definition, the horizon is the largest root of $f(r).$ Since asymptotically $f(r) \\rightarrow 1,$ $f(r)$ must be positive for $r>R_H.$\n\nAs we have seen, the dilaton solution (\\ref{fr2}) is negative and its derivative (\\ref{fr3}) is a positive function.\nTherefore $\\phi - r \\phi'$ is a negative quantity. Since $1 - \\left(\\frac{R_H}{r}\\right)^{d-3} >0,$ from (\\ref{fnew}) we get $g-f>0$ (always for $r>R_H$). Also from (\\ref{fnew}) $g-f$ is of order $\\lambda,$ i.e. its classical part is zero. This way, in the term $(g-f)f'$ of $\\left.\\frac{Q}{F}\\right|_0,$ one can take just the classical part $f_0^{T'}$ to compute $f'$: its $\\lambda$ correction would only contribute to order $\\lambda^2.$ But, as we know, $f_0^{T'}>0.$ This means $\\left.\\frac{Q}{F}\\right|_0$ (and therefore $\\frac{Q}{F}$) is indeed positive for $r>R_H.$ This way, the dilatonic $d$--dimensional black hole solution (\\ref{fnew}) is indeed stable under tensorial perturbations.\n\n\\subsection{The double--charged black hole}\n\\label{giveon}\n\\subsubsection{Description of the solution}\n\\indent\n\nIn article \\cite{Giveon:2009da} one can find black holes in any dimension formed by a fundamental string compactified on an internal circle with any momentum $n$ and winding $w,$ both at leading order and with leading $\\a$\ncorrections. One starts with the Callan--Myers--Perry solution in the string frame \\cite{cmp89}, which is of the form (\\ref{schwarz}), with $f, g$ replaced by $f^{CMP}_S, g^{CMP}_S,$ given by\n\\bea\nf^{CMP}_S(r)&=&f_0^T \\left(1+2 \\frac{\\lambda}{R_H^2} \\mu(r)\\right), \\nonumber \\\\\ng^{CMP}_S(r)&=&f_0^T \\left( 1-2 \\frac{\\lambda}{R_H^2} \\epsilon(r)\\right), \\nonumber \\\\\n\\epsilon(r)&=&\\frac{d-3}{4} \\frac{\\left(\\frac{R_H}{r}\\right)^{d-3}}{1-\\left(\\frac{R_H}{r}\\right)^{d-3}} \\left[\\frac{(d-2)(d-3)}{2}\n-\\frac{2(2d-3)}{d-1} + (d-2)\\left(\\psi^{(0)}\\left(\\frac{2}{d-3}\\right) + \\gamma \\right)\\right. \\nonumber \\\\ &+& \\left. d \\left(\\frac{R_H}{r}\\right)^{d-1} +\\frac{4}{d-2} \\varphi(r) \\right], \\label{ep} \\\\\n\\mu(r)&=&-\\epsilon(r)+\\frac{2}{d-2} (\\varphi(r)-r \\varphi'(r)). \\label{mu}\n\\eea\n$f_0^T$ is given by (\\ref{tangher}) and $\\varphi(r)$ is given by (\\ref{fr2}). This dilaton solution $\\varphi(r)$ is such that $\\epsilon(r), \\mu(r)$ are finite at the horizon $r=R_H$ \\cite{Moura:2009it}: indeed, from (\\ref{firh}), (\\ref{firhd}) and the definitions (\\ref{ep}), (\\ref{mu}) one gets\n\\be\n\\epsilon(R_H)= -\\frac{d-1}{2}, \\, \\mu(R_H)=-\\frac{3d\\left(d-3\\right)\\left(d-\\frac{5}{3}\\right)}{4\\left(d-1\\right)}\n-\\frac{d-2}{2}\\left(\\psi^{(0)}\\left(\\frac{2}{d-3}\\right) + \\gamma \\right).\n\\ee\n\nIn the same way as $\\varphi(r)$, also $\\epsilon(r)$ in (\\ref{ep}) is negative (always for $r>R_H$): it has a negative value at the horizon and grows up to 0 at asymptotic infinity. Indeed one can easily show (for instance, constructing a table with Mathematica) that, for relevant values of $d,$ the first line in (\\ref{ep}) obeys $\\frac{(d-2)(d-3)}{2}-\\frac{2(2d-3)}{d-1} + (d-2)\\left(\\psi^{(0)}\\left(\\frac{2}{d-3}\\right) + \\gamma \\right)<0.$ There is a positive contribution from $d \\left(\\frac{R_H}{r}\\right)^{d-1}$, but it decreases with increasing $r,$ differently from other negative contributions. Another remarkable fact from $\\epsilon(r)$ is that, for most of its range, it has very small numerical values (when compared to $\\varphi(r)$). This is due to the overall factor $\\frac{\\left(\\frac{R_H}{r}\\right)^{d-3}}{1-\\left(\\frac{R_H}{r}\\right)^{d-3}},$ because of which the absolute value of $\\epsilon(r)$ decreases much faster than that of $\\varphi(r).$ Because of that, $\\mu(r)$ which, from (\\ref{mu}), is a difference of negative quantities, is also negative in all its range. This negativeness of $\\epsilon(r), \\mu(r)$ can be checked simply by plotting these two functions of $r\/R_H$ for all relevant values of $d.$\n\nThis Callan--Myers--Perry metric in the string frame is lifted to an additional dimension by adding an extra coordinate, taken to be compact (this means to produce a uniform black string). One then performs a boost along this extra direction, with parameter $\\alpha_w$, and $T$--dualizes around it (to change string momentum into winding), obtaining a $(d+1)$--dimensional black string winding around a circle. Finally one boosts one other time along this extra direction, with parameter $\\alpha_p$, in order to add back momentum charge. One finally obtains a spherically symmetric black hole in $d$ dimensions with two electrical charges.\n\nThe whole process is worked out in detail in \\cite{Giveon:2009da}; the final metric, in the string frame, is of the form (\\ref{schwarz}), with $f, g$ given by\n\\bea\nf_S(r)&=&\\frac{f_0^T}{\\Delta(\\alpha_n)\\Delta(\\alpha_w)}\n\\left[1+\\frac{2 \\lambda}{R_H^2} \\frac{\\mu(r)}{\\Delta(\\alpha_n)\\Delta(\\alpha_w)}\n-\\frac{2 \\lambda}{R_H^2} \\mu(r) \\frac{\\sinh^2(\\alpha_n)\\sinh^2(\\alpha_w)}{\\Delta(\\alpha_n)\\Delta(\\alpha_w)} \\left(\\frac{R_H}{r}\\right)^{2(d-3)}\\right. \\\\\n&+&\\frac{2 \\lambda}{R_H^2} \\mu(r)\\left(\\frac{\\sinh^{2}\\alpha_n}{\\Delta(\\alpha_n)}\n+\\frac{\\sinh^{2}\\alpha_w}{\\Delta(\\alpha_w)}\\right)+\\left. \\frac{\\lambda}{R_H^2} (d-3)^2 f_0^T \\left(\\frac{R_H}{r}\\right)^{2(d-2)}\n\\frac{\\sinh^2(\\alpha_n)\\sinh^2(\\alpha_w)}{\\Delta(\\alpha_n)\\Delta(\\alpha_w)}\\right], \\nonumber\\\\\n\\Delta\\left(x\\right)&:=&1+\\left(\\frac{R_H}{r}\\right)^{d-3}\\sinh^2x, \\label{delta} \\\\\ng_S(r)&=&f_0^T\\left( 1-2\\frac{\\lambda}{R_H^2} \\epsilon(r)\\right).\n\\eea\nThe dilaton in this case is given by\n\n\\bea\ne^{-2\\phi}&=&\\sqrt{\\Delta(\\alpha_n)\\Delta(\\alpha_w)}\n\\left[1-2 \\frac{\\lambda}{R_H^2} \\varphi(r)-\\frac{\\lambda}{R_H^2} \\mu(r) f_0^T \\left(\\frac{\\sinh^2\\alpha_n}{\\Delta(\\alpha_n)}\n+\\frac{\\sinh^2\\alpha_w}{\\Delta(\\alpha_w)}\\right) \\right. \\nonumber\\\\\n&-&\\left.\\frac{\\lambda}{R_H^2} \\frac{(d-3)^2}{2} f_0^T \\left(\\frac{R_H}{r}\\right)^{2(d-2)}\n\\frac{\\sinh^2(\\alpha_n)\\sinh^2(\\alpha_w)}{\\Delta(\\alpha_n)\\Delta(\\alpha_w)}\\right], \\label{conf}\n\\eea\nwith $\\varphi(r)$ still given by (\\ref{fr2}).\n\nFor later purposes it will useful to have $f=g,$ at least to order $\\lambda=0.$ This can be achieved with a conformal transformation which changes the frame, like we have seen in section \\ref{diffr}, defining a new metric\n\\be\ng_{\\mu\\nu}^I = e^{-2\\phi} g_{\\mu\\nu}^S. \\label{frs}\n\\ee\nThis way we get a solution of the form\n\\bea\nf(r) &=& f_0^I(r) \\left(1+ \\frac{\\lambda}{R_H^2} f_c(r) \\right), \\, \\, g(r)= f_0^I(r) \\left(1+ \\frac{\\lambda}{R_H^2} g_c(r) \\right), \\label{fcgc} \\\\\nf_0^I&=&\\frac{f_0^T}{\\sqrt{\\Delta(\\alpha_n)\\Delta(\\alpha_w)}}, \\label{fcgi}\n\\eea\n$f_0^T$ being given by (\\ref{tangher}). $f_c, g_c$ are given by\n\\bea\nf_c^I(r) &=& \\frac{1}{2 \\Delta(\\alpha_n)\\Delta(\\alpha_w)} \\Big(-4 \\Delta(\\alpha_n)\\Delta(\\alpha_w) \\varphi(r) + 2\\left(2-f_0^T\\right) \\left(\\Delta(\\alpha_n) \\sinh^2(\\alpha_w) + \\Delta(\\alpha_w) \\sinh^2(\\alpha_n)\\right)\\mu(r) \\nonumber\\\\\n&+& 4 \\left. \\left(1- \\left(\\frac{R_H}{r}\\right)^{2(d-3)} \\sinh^2(\\alpha_w) \\sinh^2(\\alpha_n) \\right) \\mu(r) + (d-3)^2 f_0^T \\left(\\frac{R_H}{r}\\right)^{2(d-2)} \\sinh^2(\\alpha_w) \\sinh^2(\\alpha_n) \\right), \\nonumber\\\\\ng_c^I(r)&=&\\frac{1}{2 \\Delta(\\alpha_n)\\Delta(\\alpha_w)} \\Big( 2 \\left(\\Delta(\\alpha_n) \\sinh^2(\\alpha_w) + \\Delta(\\alpha_w) \\sinh^2(\\alpha_n)\\right)\\mu(r) f_0^T \\nonumber\\\\\n&+& \\left. (d-3)^2 f_0^T \\left(\\frac{R_H}{r}\\right)^{2(d-2)} \\sinh^2(\\alpha_w) \\sinh^2(\\alpha_n)+ 4 \\Delta(\\alpha_n)\\Delta(\\alpha_w) \\left(\\varphi(r)-\\epsilon(r)\\right) \\right). \\label{fcgorb}\n\\eea\nSince, as discussed also in section \\ref{diffr}, the analysis of the stability under tensorial perturbations is independent of the chosen frame, this is the form of the metric we will take.\n\n\\subsubsection{Study of the stability}\n\\indent\n\nIn order to prove the stability of this solution, we need to show the positivity of (\\ref{qf}), as we have seen. Again it is simpler to split this expression in its \"classical\" and $\\lambda$--corrected parts: $\\frac{Q}{F} = \\left.\\frac{Q}{F}\\right|_0 + \\lambda \\left.\\frac{Q}{F}\\right|_1,$ with $\\left.\\frac{Q}{F}\\right|_1$ being evaluated using the $\\lambda=0$ parts of the corresponding functions. For this concrete solution,\n\\bea\nr^2 \\sqrt{fg} \\left.\\frac{Q}{F}\\right|_0 &=& \\frac{\\ell \\left( \\ell + d - 3 \\right)}{r^2} f + \\frac{(g-f)f'}{r} \\label{qf0} \\\\\nr^4 \\sqrt{fg} \\left.\\frac{Q}{F}\\right|_1 &=& \\frac{2}{r^2} \\ell \\left( \\ell + d - 3 \\right) f_0^I \\left[ 2 (1-f_0^I) + r (f_0^I)' \\right].\n\\eea\n$f_0^I(r)$ vanishes at $r=R_H,$ and from there it grows monotonically to the asymptotic value 1. Therefore one has both $(f_0^I)'>0$ and $1-f_0^I>0$ and therefore, in the range we are interested, $\\left.\\frac{Q}{F}\\right|_1>0.$\n\n$\\left.\\frac{Q}{F}\\right|_0$ given by (\\ref{qf0}) must be computed with the full, $\\lambda$ corrected metric. In this case, from (\\ref{fcgc}) and (\\ref{fcgorb}) we get\n\\bea\ng-f &=& \\frac{2 \\lambda f_0^T(r)}{\\left(\\Delta(\\alpha_w) \\Delta(\\alpha_n) \\right)^{\\frac{3}{2}}}\n\\Big[ \\Delta(\\alpha_w) \\Delta(\\alpha_n) \\left(2\\varphi(r)-\\epsilon(r)\\right) + \\left(\\left(\\frac{R_H}{r}\\right)^{2(d-3)} \\sinh^2(\\alpha_w) \\sinh^2(\\alpha_n) -1 \\right) \\mu(r) \\nonumber\\\\\n&-& \\left(\\frac{R_H}{r}\\right)^{d-3} \\left(\\sinh^2(\\alpha_n) \\Delta(\\alpha_w) + \\sinh^2(\\alpha_w) \\Delta(\\alpha_n) \\right) \\mu(r) \\Big]\n\\eea\nwhich, using the definitions (\\ref{mu}), (\\ref{delta}) and (\\ref{fcgi}), may be simplified to\n\\be\ng-f = 2 \\lambda \\frac{f_0^T(r)}{\\sqrt{\\Delta(\\alpha_w) \\Delta(\\alpha_n)}}\n\\Big[2\\varphi(r)-\\epsilon(r)- \\mu(r)\\Big] = \\frac{4}{d-2} \\lambda f_0^I(r)\n\\left[\\left(d-3\\right)\\varphi(r)+r \\varphi'(r)\\right].\\label{gmf}\n\\ee\n\nIn any other frame but the one we chose by (\\ref{frs}), this expression would be much more complicated.\n\n$\\Delta(\\alpha_w),\\, \\Delta(\\alpha_n)$ are strictly positive functions, as one can see from the definition (\\ref{delta}), and so is therefore $f_0^I(r)$. Therefore in order to analyze the positivity of (\\ref{gmf}) one needs to concentrate only on the factor inside brackets, $\\left(d-3\\right)\\varphi(r)+r \\varphi'(r).$ One can already anticipate that the final result for the stability will not depend on the magnitude of the charges\/boost parameters $\\alpha_w, \\alpha_n,$ a quite remarkable fact.\n\nAs we previously mentioned, the dilaton solution $\\varphi(r)$ is negative, but its derivative $\\varphi'(r)$ is positive: $\\varphi(r)$ have a negative value at the horizon and grow up to 0 at asymptotic infinity. The difference $g-f$ is therefore a sum of a positive and a negative term; now, does this mean that $g-f$ does not have a fixed sign, i.e. its sign may change for different values of $r?$ Or is there any dominant term in all the range $r>R_H$, such that the sign of $g-f$ does not change in such range? If this is the case, then which is that sign?\n\nClose to the horizon we have\n\\bea\ng-f &=& -\\frac{(d-3)(d-2)}{2 \\cosh(\\alpha_n) \\cosh(\\alpha_w)} \\frac{\\lambda}{R_H^2} \\nonumber \\\\\n&\\times& \\left(\\frac{8}{d-1} +d^2 -6d +1 +2(d-3) \\left(\\psi^{(0)}\\left(\\frac{2}{d-3}\\right) + \\gamma \\right)\\right) \\frac{r-R_H}{R_H} + {\\mathcal{O}} \\left( \\left( r-R_H \\right)^2 \\right)\n\\eea\nOne can easily show (again, for instance, constructing a table with Mathematica) that, for relevant values of $d,$ one has $\\frac{8}{d-1} +d^2 -6d +1 +2(d-3) \\left(\\psi^{(0)}\\left(\\frac{2}{d-3}\\right) + \\gamma \\right) <0$ and, therefore, in some neighborhood of $R_H$ (but of course with $r>R_H$) one has $g-f>0$ (for $r=R_H$ evidently $g=f$).\n\nFor large $r$ we obtain the following asymptotic expansion:\n\\be\ng-f = \\frac{d-3}{2} \\frac{\\lambda}{R_H^2} \\left(\\frac{R_H}{r}\\right)^{2(d-3)} \\left[d-2 - (d-1) \\left(\\frac{R_H}{r}\\right)^2 \\right] + {\\mathcal{O}} \\left( \\left(\\frac{R_H}{r}\\right)^{2d-3} \\right) .\n\\ee\nIn the limit $r\\rightarrow\\infty$ one has $g=f=1,$ i.e. $g-f=0.$ For large $r$ we see that the leading correction is positive: one also has $g-f>0,$ as one had close to the horizon. The same conclusions are reached when one restricts themselves to $\\left(d-3\\right)\\varphi(r)+r \\varphi'(r):$ positive at the horizon and vanishing at infinity. It remains to analyze the behavior of $\\left(d-3\\right)\\varphi(r)+r \\varphi'(r)$ in the intermediate range (neither close to $R_H$ nor asymptotically). Rather we analyze its derivative:\n\\be\n\\left(\\left(d-3\\right)\\varphi(r)+r \\varphi'(r)\\right)'=-\\frac{(d-3)(d-2)^2}{4r} \\left(\\frac{R_H}{r}\\right)^{2d-6} \\frac{d-3-(d-1) \\left(\\frac{R_H}{r}\\right)^{2} +2\\left(\\frac{R_H}{r}\\right)^{d-1}}{\\left(1-\\left(\\frac{R_H}{r}\\right)^{d-3}\\right)^2}. \\label{gmflinha}\n\\ee\nOne can easily check that, for relevant values of $d$, and for $r>R_H,$ $d-3-(d-1) \\left(\\frac{R_H}{r}\\right)^{2} +2\\left(\\frac{R_H}{r}\\right)^{d-1}>0.$ The other factors in (\\ref{gmflinha}) are clearly positive, except for the overall minus sign. Therefore one has $\\left(\\left(d-3\\right)\\varphi(r)+r \\varphi'(r)\\right)'<0$ for $r>R_H$ (in particular this derivative has no zeroes). This means $\\left(d-3\\right)\\varphi(r)+r \\varphi'(r)$ is a positive function which decreases to zero asymptotically. The behavior of $g-f$ is the same.\n\nSince for this solution $g-f$ is of order $\\lambda$ but it vanishes at the classical level, $f'$ in $\\left.\\frac{Q}{F}\\right|_0$ must be computed with the classical metric, exactly from the same argument we gave for the dilatonic solution (\\ref{fnew}) (the $\\lambda$--correction of the metric to $f'$ would only contribute at order $\\lambda^2$.) As we have mentioned, at order $\\lambda=0$ we have $(f_0^I)'>0;$ from our previous result, this way, to order $\\lambda,$ $(g-f) f' >0.$\n\nSince, by definition, $f(R_H)=0$ and $f>0$ for $r>R_H,$ we conclude that $\\left.\\frac{Q}{F}\\right|_0$  given by (\\ref{qf0}) is indeed positive for $r>R_H.$ The same is valid for $\\frac{Q}{F}$ given by (\\ref{qf}). This way, the double--charged $d$--dimensional black hole solution (\\ref{fcgc}) is indeed stable under tensorial perturbations, for any value of the magnitude of the charges\/boost parameters $\\alpha_w, \\alpha_n$.\n\n\\section{Discussion and Future Directions}\n\\indent\n\nWe have computed the tensorial perturbations for the most general spherically symmetric metric in $d$ dimensions. We applied it to the field equations resulting from a string effective action with $\\R^2$ corrections. We have shown that the master equation for the perturbation variable is of second order, regardless of the presence of the higher order terms in the lagrangian and in the field equations. We have also obtained the corresponding $\\a$ corrections to the potential for these perturbations. From these results we have studied the stability of two different $d$--dimensional black hole solutions with $\\R^2$ corrections in string theory. In both cases we concluded that they were stable under such perturbations.\n\nThese results are to be compared with the corresponding ones in Lovelock theory. As we have mentioned in the introduction, for tensorial perturbations of solutions to these theories several instabilities have been found \\cite{dg04,dg05a,Takahashi:2009dz,Takahashi:2010ye,Takahashi:2010gz}, depending on the dimensionality of spacetime (the necessary power of curvature needed in order to get a Lovelock theory also depends on $d$). Even more interesting are two other aspects of Lovelock theories which were found on these articles: the instabilities manifest themselves mainly on shorter scales (for ``small'' black holes), and there are domains of the parameters (the coupling constants of the higher order terms) in which the linear perturbation theory breaks down and is not applicable. The justification for such facts lies on the properties of Lovelock theories we mentioned: they are seen as exact and not effective theories; the dependence of the solutions on the coupling constants goes beyond perturbation theory, i.e. the order at which they appear in the lagrangian does not matter for such dependence, which is often nonlinear. This is the reason for the nonapplicability of the linear perturbation theory in certain domains.\n\nThe string--theoretical solutions we have considered are perturbative in $\\a$: their dependence on $\\a$ is of the same order in which $\\a$ appears on the lagrangian (first order). This is why linear perturbation theory is fully applicable to these solutions we have studied, but one must keep in mind that their stability which we have shown is just perturbative. Nothing in our work guarantees that, if one considers higher order corrections in $\\a$ to these solutions, an instability does not appear.\n\nThere is a lot of work still to be done in studying perturbations and stability of black holes in higher dimensions, already in classical Einstein gravity. With respect to static black holes for which the formalism of Ishibashi and Kodama is applicable, there is a lot to be understood concerning scalar perturbations (for generic horizon topologies) and tensor perturbations (for non--maximally symmetric black holes). For a recent review see \\cite{Ishibashi:2011ws}. There is even more to be done concerning the perturbations and stability of non--static black holes.\n\nWith respect to perturbative string--theoretical black holes, the main question one can ask is the following: do string $\\a$ corrections preserve the stability properties of the corresponding classical black hole solutions, or do they introduce a new behavior? The examples we have analyzed in this article provide some evidence in favor of the first answer, but they represent by far an extremely limited range of solutions for more definite conclusions to be drawn. Still a lot of work remains to be done, with other string--corrected solutions but even also with those solutions we have taken here, considering other kinds of metric perturbations (vector and scalar).\n\nKnowing the master equation and potential for tensor--type gravitational perturbations for these solutions, as we do, another study that can be made is the determination of the corresponding spectrum of quasinormal modes, including the leading $\\a$ corrections. We leave this for a future work.\n\n\\section*{Acknowledgments}\nThe author wishes to acknowledge useful discussions with Akihiro Ishibashi and Jos\\'e Sande Lemos. This work has been supported by CMAT - U.Minho through the FCT Pluriannual Funding Program, and by FCT and FEDER through CERN\/FP\/123609\/2011 project.\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=1.1]{small-diagram.pdf}\n    \\caption{A simplified diagram of the experiments followed by this work towards the comparison of classical vs transfer learning from synthetic data for speaker recognition. A more detailed diagram can be found in Figure \\ref{fig:method}}\n    \\label{fig:smalldiagram}\n\\end{figure}\nData scarcity is an issue that arises often outside of the lab, due to the large amount of data required for classification activities. This includes speaker classification in order to enable personalised Human-Machine (HMI) and Human-Robot Interaction (HRI), a technology growing in consumer usefulness within smart device biometric security on devices such as smartphones and tablets, as well as for multiple-user smarthome assistants (operating on a per-person basis) which are not yet available. Speaker recognition, i.e., autonomously recognising a person from their voice, is a well-explored topic in the state-of-the-art within the bounds of data availability, which causes difficulty in real-world use. It is unrealistic to expect a user to willingly provide many minutes or hours of speech data to a device unless it is allowed to constantly record daily life, something which is a modern cause for concern with the virtual home assistant. In this work, we show that data scarcity in speaker recognition can be overcome by collecting only several short spoken sentences of audio from a user and then using extracted Mel-Frequency Cepstral Coefficients (MFCC) data in both supervised and unsupervised learning paradigms to generate synthetic speech, which is then used in a process of transfer learning to better recognise the speaker in question.\n\nAutonomous speaker classification can suffer issues of data scarcity since the user is compared to a large database of many speakers. The most obvious solution to this is to collect more data, but with Smart Home Assistants existing within private environments and potentially listening to private data, this produces an obvious problem of privacy and security~\\cite{logsdon2018alexa,dunin2020alexa}. Not collecting more data on the other hand, presents an issue of a large class imbalance between the speaker to classify against the examples of other speakers, producing lower accuracies and less trustworthy results~\\cite{babbar2019data}, which must be solved for purposes such as biometrics since results must be trusted when used for security. In this study, weighting of errors is performed to introduce balance, but it is noted that the results still have room for improvement regardless. \n\n\n\nData augmentation is the idea that useful new data can be generated by algorithms or models that would improve the classification of the original, scarce dataset. A simple but prominent example of this is the warping, flipping, mirroring and noising of images to better prepare image classification algorithms~\\cite{wang2017effectiveness}. A more complex example through generative models can be seen in recent work that utilise methods such as the Generative Adversarial Network (GAN) to create synthetic data which itself also holds useful information for learning from and classification of data~\\cite{zhu2018emotion,frid2018synthetic}. Although image classification is the most common and most obvious application of generative models for data augmentation, recent works have also enjoyed success in augmenting audio data for sound classification~\\cite{yang2018se,madhu2019data}. This work extends upon a previous conference paper that explored the hypothesis \\textit{``can the synthetic data produced by a generative model aid in speaker recognition?\"}~\\cite{bird2020overcoming} within which a Character-Level Recurrent Neural Network (RNN), as a generative model, produced synthetic data useful for learning from in order to recognise the speaker. This work extends upon these preliminary experiments by the following:\n\\begin{enumerate}\n    \\item The extension of the dataset to more subjects from multiple international backgrounds and the extraction of the MFCCs of each subject;\n    \\item Benchmarking of a Long Short Term Memory (LSTM) architecture for 64, 128, 256 and 512 LSTM units in one to three hidden layers towards reduction of loss in generating synthetic data. The best model is selected as the candidate for the LSTM data generator;\n    \\item The inclusion of OpenAI's GPT-2 model as a data generator in order to compare the approaches of supervised (LSTM) and attention-based (GPT-2) methods for synthetic data augmentation for speaker classification.\n\\end{enumerate}\nThe scientific contributions of this work, thus, are related to the application of synthetic MFCCs for improvement of speaker recognition. A diagram of the experiments can be observed in Figure \\ref{fig:smalldiagram}. To the authors' knowledge, the paper that this work extends is the first instance of this research being explored\\footnote{to the best of our knowledge and based on literature review.}. The best LSTM and the GPT-2 model are tasked with generating 2,500, 5,000, 7,500, and 10,000 synthetic data objects for each subject after learning from the scarce datasets extracted from their speech. A network then learns from these data and transfers their weights to a network aiming to learn and classify the real data, and many show an improvement. For all subjects, the results show that several of the networks perform best after experiencing exposure to synthetic data. \n\nThe remainder of this article is as follows. Section \\ref{sec1:background} initially explores important scientific concepts of the processes followed by this work and also the current State-of-the-Art in the synthetic data augmentation field. Following this, Section \\ref{sec2:method} then outlines the method followed by this work including data collection, synthetic data augmentation, MFCC extraction to transform audio into a numerical dataset, and finally the learning processes followed to achieve results. The final results of the experiments are then discussed in Section \\ref{sec:results}, and then future work is outlined and conclusions presented in Section \\ref{sec4:futureworkconclusion}.\n\n\n\n\\section{Background and Related Work}\n\\label{sec1:background}\n\n\nVerification of a speaker is the process of identifying a single individual against many others by spoken audio data~\\cite{poddar2017speaker}. That is, the recognition of a set of the person's speech data $X$ specifically from a speech set $Y$ where $X \\in Y$. In the simplest sense, this can be given as a binary classification problem; for each data object $o$, is $o \\in X$? Is the speaker to be recognised speaking, or is it another individual? Speaker recognition is important for social HRI~\\cite{mumolo2003distant} (the robot's perception of the individual based on their acoustic utterances), Biometrics~\\cite{ratha2001automated}, and Forensics~\\cite{rose2002forensic} among many others. In~\\cite{hasan2004speaker}, researchers found relative ease of classifying 21 speakers from a limited set, but the problem becomes more difficult as it becomes more realistic, where classifying a speaker based on their utterances is increasingly difficult as the dataset grows~\\cite{nagrani2017voxceleb,yadav2018learning,zeinali2019but}. In this work, the speaker is recognised from many thousands of other examples of human speech from the Flickr8k speakers dataset. \n\n\\subsection{LSTM and GPT-2}\nLong Short Term Memory (LSTM) is a form of Artificial Neural Network in which multiple RNNs will learn from previous states as well as the current state. Initially, the LSTM selects data to delete: \n\\begin{equation}\n    \\mathop f\\nolimits_{\\text{t}} = \\sigma \\left( {\\mathop W\\nolimits_{f} \\cdot \\left[ {\\mathop h\\nolimits_{t - 1} ,\\mathop x\\nolimits_{t} } \\right] + \\mathop b\\nolimits_{f} } \\right),\n\\end{equation}\n\\noindent where $W_f$ are the weights of the units, $h_{t=1}$ is the output at $t=1$, $x_t$ are inputs and $b_f$ is an applied bias. Data to be stored is then selected based on input $i$, generating $C_t$ values:\n\\begin{equation}\n\\mathop o\\nolimits_{\\text{t}} = \\sigma \\left( {\\mathop W\\nolimits_{i} \\cdot \\left[ {\\mathop h\\nolimits_{t - 1} ,\\mathop x\\nolimits_{t} } \\right] + \\mathop b\\nolimits_{i} } \\right),\n\\end{equation}\n\n\\begin{equation}\n\\mathop {\\hat{C}}\\nolimits_{\\text{t}} = \\tanh \\left( {\\mathop W\\nolimits_{c} \\cdot \\left[ {\\mathop h\\nolimits_{t - 1} ,\\mathop x\\nolimits_{t} } \\right] + \\mathop b\\nolimits_{c} } \\right).\n\\end{equation}\n\nA convolutional operation updates values:\n\\begin{equation}\nC_{t}  =  f_{t} * C_{t-1} + i_{t} *  \\tilde{C}_{t} .\n\\end{equation}\n\nOutput $o_t$ is presented, and the hidden state is updated:\n\\begin{equation}\n\\mathop o\\nolimits_{\\text{t}} = \\sigma \\left( {\\mathop W\\nolimits_{o} \\cdot \\left[ {\\mathop h\\nolimits_{t - 1} ,\\mathop x\\nolimits_{t} } \\right] + \\mathop b\\nolimits_{o} } \\right),\n\\end{equation}\n\\begin{equation}\nh_{t} = o_{t} * tanh(C_{t}).\n\\end{equation}\nDue to the observed consideration of previous data, it is often found that time dependent data are very effectively classified due to the memory-like nature of the LSTM. LSTM is thus a particularly powerful technique in terms of speech recognition~\\cite{graves2013hybrid} due to the temporal nature of the data~\\cite{belin2000voice}. \\\\\n\nIn addition to LSTM, this study also considers OpenAI's Generative Pretrained Transformer 2 (GPT-2) model~\\cite{radford2018improving,radford2019language} as a candidate for producing synthetic MFCC data to improve speaker recognition. The model in question, \\textit{335M}, is much deeper than the LSTMs explored at 12 layers. In the OpenAI paper, the GPT modelling is given for a set of samples $x_1, x_2, ..., x_n$ composed of variable symbol-sequence lengths $s_1, s_2, ..., s_n$ factorised by joint probabilities over symbols as the product of conditional probabilities~\\cite{jelinek1980interpolated,bengio2003neural}:\n\\begin{equation}\n    p(x) =  \\prod_{i=1}^n p(s_{i}  | s_{1}, ..., s_{n-1}).\n\\end{equation}\nAttention is given to various parts of the input vector:\n\\begin{equation}\n    Attention(Q,K,V) = softmax \\left(  \\frac{QK^T}{  \\sqrt{d_{k}}  }  \\right) V,\n\\end{equation}\nwhere Q is the \\textit{query} i.e., the single object in the sequence, in this case, a word. K are the keys, which are vector representations of the input sequence, and V are values as vector representations of all words in the sequence. In the initial encoder, decoder, and attention blocks $Q=V$ whereas later on the attention block that takes these outputs as input, $Q \\neq V$ since both are derived from the block's 'memory'. In order to combine multiple queries, that is, to consider previously learnt rules from text, multi-headed attention is presented:\n\\begin{equation}\n\\begin{aligned}\n    MultiHead(Q,K,V) = Concatenate(head_{1}, ..., head_{h})W^{O} \\\\\n    head_{i} = Attention(QW^{Q}_{i}, KW^{K}_{i}, VW^{V}_{i}).\n\\end{aligned}\n\\end{equation}\nAs in the previous equation, it can be seen that previously learnt $h$ projections $d_Q, d_K$ and $d_V$ are also considered given that the block has multiple heads. The above equations and further detail on both attention and multi-headed attention can be found in \\cite{vaswani2017attention}. The unsupervised nature of the GPT models is apparent since $Q, K$ and $V$ are from the same source. \n\nImportantly, GPT produces output with consideration not only to input, but also to the task. The GPT-2 model has been shown to be a powerful state-of-the-art tool for language learning, understanding, and generation; researchers noted that the model could be used with ease to generate realistic propaganda for extremist terrorist groups~\\cite{solaiman2019release}, as well as noting that generated text by the model was difficult to detect~\\cite{gehrmann2019gltr,wolff2020attacking,adelani2020generating}. The latter aforementioned papers are promising, since a difficulty of detection suggests statistical similarities, which are likely to aid in the problem of improving classification accuracy of a model by exposing it to synthetic data objects output by such a model. \n\n\\subsection{Dataset Augmentation through Synthesis}\n\\label{subsec:dataaugmentation}\n\nSimilarities between real-life experiences and imagined perception have shown in psychological studies that the human imagination, though mentally augmenting and changing situations~\\cite{beres1960perception}, aids in improving the human learning process~\\cite{egan1989memory,heath2008exploring,macintyre2012emotions,egan2014imagination}. The importance of this ability in the learning process shows the usefulness of data augmentation in human learning, and as such, is being explored as a potential solution to data scarcity and quality in the machine learning field. Even though the synthetic data may not be realistic alone, minute similarities between it and reality allow for better pattern recognition. \n\nThe idea of data augmentation as the first stage in fine-tune learning is inspired by the aforementioned findings, and follows a similar approach. Synthetic data is generated by learning from the real data, and algorithms are exposed to them in a learning process prior to the learning process of real data; this is then compared to the classical approach of learning from the data solely, where the performance of the former model compared to the latter shows the effect of the data augmentation learning process. Much of the work is recent, many from the last decade, and a pattern of success is noticeable for many prominent works when comparing the approach to the classical method of learning from real data alone. \n\nAs described, the field of exploring augmented data to improve classification algorithms is relatively young, but there exist several prominent works that show success in applying this approach. When augmented data from the SemEval dataset is learned from by a Recurrent Neural Network (RNN), researchers found that the overall best F-1 score was achieved for relation classification in comparison to the model only learning from the dataset itself~\\cite{xu2016improved}. Due to data scarcity in the medical field, classification of liver lesions~\\cite{frid2018synthetic} and Alzheimer's Disease~\\cite{shin2018medical} have also shown improvement when the learning models (CNNs) considered data augmented by Convolutional GANs. In Natural Language Processing, it was found that word augmentation aids to improve sentence classification by both CNN and RNN models~\\cite{kobayashi2018contextual}. The \\textit{DADA} model has been suggested as a strong method to produce synthetic data for improvement of data-scarce problems through the Deep Adversarial method via a dedicated discriminator network aiming to augment data specifically~\\cite{zhang2019dada} which has noted success in machine learning problems~\\cite{barz2020deep}. \n\nData augmentation has shown promise in improving multi-modal emotion recognition when considering audio and images~\\cite{huang2018multimodal}, digital signal classification~\\cite{tang2018digital}, as well as for a variety of audio classification problems such as segments of the Hub500 problem~\\cite{park2019specaugment}. Additionally, synthetic data augmentation of mel-spectrograms have shown to improve acoustic scene recognition~\\cite{yang2018se}. Realistic text-to-speech is achieved by producing realistic sound such as the Tacotron model~\\cite{wang2017tacotron} when considering the textual representation of the audio being considered, and reverse engineering the model to produce audio based on text input. A recent preliminary study showed GANs may be able to aid in producing synthetic data for speaker recognition~\\cite{chien2018adversarial}. \n\nThe temporal models considered in this work to generate synthetic speech data have recently shown success in generating acoustic sounds~\\cite{eck2002finding} and accurate timeseries data~\\cite{senjyu2006application}, written text~\\cite{pawade2018story,sha2018order}, artistic images~\\cite{gregor2015draw}. Specifically, temporal models are also observed to be successful in generating MFCC data~\\cite{wang2017rnn}, which is the data type considered in this work. Many prominent works in speech recognition consider temporal learning to be highly important~\\cite{fernandez2007application,he2019streaming,sak2014long} and for generation of likewise temporal data~\\cite{valentini2016investigating,wang2017rnn,tachibana2018efficiently} (that is, which this study aims to perform). If it is possible to generate data that bares similarity to the real data, then it could improve the models while also reducing the need for large amounts of real data to be collected. \n\n\n\\section{Proposed Approach}\n\\label{sec2:method}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.9]{overalldiagram_new.pdf}\n    \\caption{A diagram of the experimental method in this work. Note that the two networks being directly compared are classifying the same data, with the difference being the initial weight distribution either from standard random distribution or transfer learning from GPT-2 and LSTM produced synthetic data.}\n    \\label{fig:method}\n\\end{figure*}\n\nThis section describes the development of the proposed approach, which can be observed overall in Figure \\ref{fig:method}. For each test, five networks are trained in order to gain results. Firstly, a network simply to perform the speaker classification experimen without transfer learning (from a standard random weight distribution). Firstly, a network simply to perform the speaker classification experiment. Produced by LSTM and GPT-2, synthetic data is used to train another network, of which the weights are used to train a final network as an initial distribution to perform the same experiment as described in the first network (classifying the speaker's real data from Flickr8k speakers). Thus, the two networks leading to the final classification score in the diagram are directly comparable since they are learning from the same data, they differ only in initial weight distribution (where the latter network has weights learnt from synthetic data).\n\n\n\n\\subsection{Real and Synthetic Data Collection}\nThe data collected, as previously described, presents a binary classification problem. That is, whether or not the individual in question is the one producing the acoustic utterance. \n\nThe large corpus of data for the \\textit{``not the speaker\"} class is gathered via the Flickr8k dataset~\\cite{harwath2015deep} which contains 40,000 individual utterances describing 8,000 images by a large number of speakers which is unspecified by the authors. MFCCs are extracted (described in Section \\ref{subsec:mfcc}) to generate temporal numerical vectors which represent a short amount of time from each audio clip. 100,000 data objects are selected through 50 blocks of 1,000 objects and then 50,000 other data objects selected randomly from the remainder of the dataset. This is performed so the dataset contains individual's speech at length as well as short samples of many other thousands of speakers also. \n\n\\begin{table*}[]\n\\footnotesize\n\\centering\n\\caption{Information regarding the data collection from the seven subjects}\n\\label{tab:subjects}\n\\begin{tabular}{@{}lllllll@{}}\n\\toprule\n\\textbf{Subject}           & \\textbf{Sex} & \\textbf{Age} & \\textbf{Nationality} & \\textbf{Dialect} & \\textbf{Time Taken (s)} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Real\\\\Data\\end{tabular}} \\\\ \\midrule\n\\textit{\\textbf{1}}        & M            & 23           & British              & Birmingham              & 24                      & 4978                           \\\\\n\\textit{\\textbf{2}}        & M            & 24           & American             & Florida          & 13                      & 2421                           \\\\\n\\textit{\\textbf{3}}        & F            & 28           & Irish                & Dublin                  & 12                      & 2542                           \\\\ \n\\textit{\\textbf{4}}        & F            & 30           & British                & London                  & 12                      & 2590                           \\\\ \n\\textit{\\textbf{5}}        & F            & 40           & British                & London                  & 10                      & 2189                           \\\\ \n\\textit{\\textbf{6}}        & M            & 21           & French                & Paris                  & 8                      & 1706                           \\\\ \n\\textit{\\textbf{7}}        & F            & 23           & French                & Paris                  & 9                      & 1952                           \\\\ \n\\midrule\n\\textit{\\textbf{Flickr8K}} &              &              &                      &                         &                         & 100000                       \\\\ \\bottomrule\n\\end{tabular}\n\\end{table*}\n\n\n\nTo gather data for recognising speakers, seven subjects are considered. Information on the subjects can be seen in Table \\ref{tab:subjects}. Subjects speak five random Harvard Sentences sentences from the \\textit{IEEE recommended practice for speech quality measurements}~\\cite{rothauser1969ieee}, and so contain most of the spoken phonetic sounds in the English language~\\cite{bird2019phoneme2}. Importantly, this is a user-friendly process, because it requires only a few short seconds of audio data. The longest time taken was by subject 1 in 24 seconds producing 4978 data objects and the shortest were the two French individuals who required 8 and 9 seconds respectively to speak the five sentences. All of the audio data were recorded using consumer-available recording devices such as smartphones and computer headsets. Synthetic datasets are generated following the learning processes of the best LSTM and the GPT-2 model, where the probability of the next character is decided upon depending on the learning algorithm and are generated in blocks of 1,000 within a loop and the final line is removed (since it was often within the cutoff point of the 1,000-character block). Illogical lines of data (those that did not have 26 comma separated values and a class) were removed, but were observed to be somewhat rare as both the LSTM and GPT-2 models had learnt the data format relatively well since it was uniform throughout. The format, throughout the datasets, was a uniform 27 comma separated values where the values were all numerical and the final value was `1' followed by a line break character.\n\n\\subsection{Feature Extraction}\n\\label{subsec:mfcc}\n\nThe nature of acoustic data is that the previous and following points of data from a single point in particular are also related to the class. Audio data is temporal in this regard, and thus classification of a single point in time is an extremely difficult problem~\\cite{xiong2003comparing,bird2019phoneme}. In order to overcome this issue features are extracted from the wave through a sliding window approach. The statistical features extracted in this work are the first 26 Mel-Frequency Cepstral Coefficients due to findings in the scientific state of the art arguing for their prominence over other methods~\\cite{muda2010voice,sahidullah2012design}. Sliding windows are set to a length of 0.025 seconds with a step of 0.01 seconds and extraction is performed as follows:\n\\begin{enumerate}\n\\item The Fourier Transform of time window data $\\omega$ is calculated:\n\\begin{equation}\nX(j\\omega)=\\int_{-\\infty}^\\infty x(t) e^{-j\\omega t} dt.\n\\end{equation}\n\\item The powers from the FT are mapped to the Mel-scale, that is, the human psychological scale of audible pitch~\\cite{stevens1937scale} via a triangular temporal window. \n\\item The power spectrum is considered and $log$s of each of their powers are taken.\n\\item The derived Mel-log powers are then treated as a signal, and a Discrete Cosine Transform (DCT) is calculated:\n\\begin{equation}\n\\begin{aligned}\nX_k =  \\sum_{n=0}^{N-1} x_n  cos  \\begin{bmatrix}  \\frac{\\pi}{N} (n+ \\frac{1}{2}) k \\end{bmatrix} \\\\     \nk=0,...,N-1,\n\\end{aligned}\n\\end{equation}\n\\noindent where $x$ is the array of length $N$, $k$ is the index of the output coefficient being calculated, where $N$ real numbers $x_{0} ... x_{n-1}$ are transformed into the $N$ real numbers $X_{0} ... X_{n-1}$. \n\\end{enumerate}\nThe amplitudes of the spectrum produced are taken as the \\textit{MFCCs}. The resultant data then provides a mathematical description of wave behaviour in terms of sounds, each data object made of 26 attributes produced from the sliding window are then treated as the input attributes for the neural networks for both speaker recognition and also synthetic data generation (with a class label also). \n\nThis process is performed for all of the selected Flickr8K data as well as the real data recorded from the subjects. The MFCC data from each of the 7 subjects' audio recordings is used as input to the LSTM and GPT-2 generative models for training and subsequent data augmentation.\n\n\n\\subsection{Speaker Classification Learning Process}\nFor each subject, the Flickr data and recorded audio forms the basis dataset and the speaker recognition problem. Eight datasets for transfer learning are then formed on a per-subject basis, which are the aforementioned data plus $2500, 5000, 7500$ and $10000$ synthetic data objects generated by either the LSTM or the GPT-2 models. LSTM has a standard dropout of 0.2 between each layer.\n\nThe baseline accuracy for comparison is given as ``Synth. Data: 0\" later in Table \\ref{tab:bigresults} which denotes a model that has not been exposed to any of the synthetic data. This baseline gives scores that are directly comparable to identical networks with their initial weight distributions being those trained to classify synthetic data generated for the subject, which is then used to learn from the real data. As previously described, the two sets of synthetic data to expose the models to during pre-training of the real speaker classification problem are generated by either an LSTM or a GPT-2 language model. Please note that due to this, the results presented have no baring on whether or not the network could classify the synthetic data well or otherwise, the weights are simply used as the initial distribution for the same problem. If the pattern holds that the transfer learning networks achieve better results than the networks which have not been trained on such data, it argues the hypothesis that speaker classification can be improved when considering either of these methods of data augmentation. This process can be observed in Figure \\ref{fig:method}. \n\nFor the Deep Neural Network that classifies the speaker, a topology of three hidden layers consisting of 30, 7, and 29 neurons respectively with ReLu activation functions and an ADAM optimiser~\\cite{kingma2014adam} is initialised. These hyperparameters are chosen due to a previous study that performed a genetic search of neural network topologies for the classification of phonetic sounds in the form of MFCCs~\\cite{bird2020optimisation}. The networks are given an unlimited number of epochs to train, only ceasing through a set early stopping callback of 25 epochs with no improvement of ability. The best weights are restored before final scores are calculated. This is allowed in order to make sure that all models stabilise to an asymptote and reduce the risk of stopping models prior to them achieving their potential best abilities. \n\nClassification errors are weighted equally by class prominence since there exists a large imbalance between the speaker and the rest of the data. All of the LSTM experiments performed in this work were executed on an Nvidia GTX980Ti GPU, while the GPT-2 experiment was performed on an Nvidia Tesla K80 GPU provided by Google Colab. \n\n\\section{Results}\n\\label{sec:results}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.5]{128-lstms.pdf}\n    \\caption{The training processes of the best performing models in terms of loss, separated for readability purposes. Results are given for a benchmarking experiment on all of the dataset rather than an individual.}\n    \\label{fig:128-lstms-benchmark}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.5]{bad-lstms.pdf}\n    \\caption{The training processes of LSTMs with 64, 256, and 512 units in 1-3 hidden layers, separated for readability purposes. Results are given for a benchmarking experiment on all of the dataset rather than an individual.}\n    \\label{fig:bad-lstms-benchmark}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.5]{gpt-graph.pdf}\n    \\caption{The training process of the GPT-2 model. Results are given for a benchmarking experiment on all of the dataset rather than an individual.}\n    \\label{fig:gpt-benchmark}\n\\end{figure}\n\n\\begin{table}[]\n\\centering\n\\caption{Best epochs and their losses for the 12 LSTM Benchmarks and GPT-2 training process. All models are benchmarked on the whole set of subjects for 100 epochs each in order to search for promising hyperparameters.}\n\\label{tab:benchmarking-models}\n\\begin{tabular}{@{}lll@{}}\n\\toprule\n\\textbf{Model}    & \\textbf{Best Loss} & \\textbf{Epoch} \\\\ \\midrule\nLSTM(64)          & 0.88           & 99             \\\\\nLSTM(64,64)       & 0.86           & 99             \\\\\nLSTM(64,64,64)    & 0.85           & 99             \\\\\nLSTM(128)         & 0.53  & 71             \\\\\nLSTM(128,128)     & 0.53            & 80             \\\\\nLSTM(128,128,128) & \\textbf{0.52}            & 93             \\\\\nLSTM(256)         & 0.83           & 83             \\\\\nLSTM(256,256)     & 0.82            & 46             \\\\\nLSTM(256,256,256) & 0.82            & 39             \\\\\nLSTM(512)         & 0.81           & 33             \\\\\nLSTM(512,512)     & 0.81           & 31             \\\\\nLSTM(512,512,512) & 0.82           & 25             \\\\\nGPT-2             & 0.92               & 94             \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTable \\ref{tab:benchmarking-models} shows the best results discovered for each LSTM hyperparameter set and the GPT-2 model. Figures \\ref{fig:128-lstms-benchmark} and \\ref{fig:bad-lstms-benchmark} show the epoch-loss training processes for the LSTMs separated for readability purposes and Figure \\ref{fig:gpt-benchmark} shows the same training process for the GPT-2 model. These generalised experiments for all data provide a tuning point for synthetic data to be generated for each of the individuals (given respective personally trained models). LSTMs with 128 hidden units far outperformed the other models, which were also sometimes erratic in terms of their attempt at loss reduction over time. The GPT-2 model is observed to be especially erratic, which is possibly due to its unsupervised attention-based approach. \n\nAlthough some training processes were not as smooth as others, manual exploration showed that acceptable sets of data were able to be produced.\n\n\\subsection{Transfer Learning for Data-scarce Speaker Recognition}\n\n\\begin{table*}[]\n\\centering\n\\caption{Results of the experiments for all subjects. Best models for each Transfer Learning experiment are bold, and the best overall result per-subject is also underlined. Red font denotes a synthetic data-exposed model that scored lower than the classical learning approach. }\n\\label{tab:bigresults}\n\\footnotesize\n\\begin{tabular}{@{}llllllllll@{}}\n\\toprule\n\\multicolumn{1}{c}{}                                   &                                                                                  & \\multicolumn{4}{l}{\\textbf{LSTM}}                                                                                                           & \\multicolumn{4}{l}{\\textbf{GPT-2}}                                                                                                                         \\\\ \\cmidrule(l){3-10} \n\\multicolumn{1}{c}{\\multirow{-2}{*}{\\textbf{Subject}}} & \\multirow{-2}{*}{\\textbf{\\begin{tabular}[c]{@{}l@{}}Synth.\\\\ Data\\end{tabular}}} & \\textit{\\textbf{Acc.}}       & \\textit{\\textbf{F1}}        & \\textit{\\textbf{Prec.}}     & \\multicolumn{1}{l|}{\\textit{\\textbf{Rec.}}}      & \\textit{\\textbf{Acc.}}                & \\textit{\\textbf{F1}}                 & \\textit{\\textbf{Prec.}}              & \\textit{\\textbf{Rec.}}               \\\\ \\midrule\n                                                       & \\textit{\\textbf{0}}                                                              & 93.57                        & 0.94                        & 0.93                        & \\multicolumn{1}{l|}{0.93}                        & 93.57                                 & 0.94                                 & 0.93                                 & 0.93                                 \\\\\n                                                       & \\textit{\\textbf{2500}}                                                           & {\\ul \\textbf{99.5}}          & {\\ul \\textbf{$\\sim$1}}      & {\\ul \\textbf{$\\sim$1}}      & \\multicolumn{1}{l|}{{\\ul \\textbf{$\\sim$1}}}      & 97.32                                 & 0.97                                 & 0.97                                 & 0.97                                 \\\\\n                                                       & \\textit{\\textbf{5000}}                                                           & 97.37                        & 0.97                        & 0.97                        & \\multicolumn{1}{l|}{0.97}                        & \\textbf{97.77}                        & \\textbf{0.98}                        & \\textbf{0.98}                        & \\textbf{0.98}                        \\\\\n                                                       & \\textit{\\textbf{7500}}                                                           & \\textbf{99.33}               & \\textbf{0.99}               & \\textbf{0.99}               & \\multicolumn{1}{l|}{\\textbf{0.99}}               & 99.2                                  & 0.99                                 & 0.99                                 & 0.99                                 \\\\\n\\multirow{-5}{*}{\\textbf{1}}                           & \\textit{\\textbf{10000}}                                                          & 99.1                         & 0.99                        & 0.99                        & \\multicolumn{1}{l|}{0.99}                        & \\textbf{99.3}                         & \\textbf{0.99}                        & \\textbf{0.99}                        & \\textbf{0.99}                        \\\\ \\cmidrule(r){1-1}\n                                                       & \\textit{\\textbf{0}}                                                              & 95.13                        & 0.95                        & 0.95                        & \\multicolumn{1}{l|}{0.95}                        & 95.13                                 & 0.95                                 & 0.95                                 & 0.95                                 \\\\\n                                                       & \\textit{\\textbf{2500}}                                                           & \\textbf{99.6}                & \\textbf{$\\sim$1}            & \\textbf{$\\sim$1}            & \\multicolumn{1}{l|}{\\textbf{$\\sim$1}}            & 99.5                                  & $\\sim$1                              & $\\sim$1                              & $\\sim$1                              \\\\\n                                                       & \\textit{\\textbf{5000}}                                                           & \\textbf{99.5}                & \\textbf{$\\sim$1}            & \\textbf{$\\sim$1}            & \\multicolumn{1}{l|}{\\textbf{$\\sim$1}}            & 99.41                                 & 0.99                                 & 0.99                                 & 0.99                                 \\\\\n                                                       & \\textit{\\textbf{7500}}                                                           & {\\ul \\textbf{99.7}}          & {\\ul \\textbf{$\\sim$1}}      & {\\ul \\textbf{$\\sim$1}}      & \\multicolumn{1}{l|}{{\\ul \\textbf{$\\sim$1}}}      & {\\ul \\textbf{99.7}}                   & {\\ul \\textbf{$\\sim$1}}               & {\\ul \\textbf{$\\sim$1}}               & {\\ul \\textbf{$\\sim$1}}               \\\\\n\\multirow{-5}{*}{\\textbf{2}}                           & \\textit{\\textbf{10000}}                                                          & \\textbf{99.42}               & \\textbf{0.99}               & \\textbf{0.99}               & \\multicolumn{1}{l|}{\\textbf{0.99}}               & 99.38                                 & 0.99                                 & 0.99                                 & 0.99                                 \\\\ \\cmidrule(r){1-1}\n                                                       & \\textit{\\textbf{0}}                                                              & 96.58                        & 0.97                        & 0.97                        & \\multicolumn{1}{l|}{0.97}                        & 96.58                                 & 0.97                                 & 0.97                                 & 0.97                                 \\\\\n                                                       & \\textit{\\textbf{2500}}                                                           & \\textbf{99.2}                & \\textbf{0.99}               & \\textbf{0.99}               & \\multicolumn{1}{l|}{\\textbf{0.99}}               & 98.41                                 & 0.98                                 & 0.98                                 & 0.98                                 \\\\\n                                                       & \\textit{\\textbf{5000}}                                                           & 98.4                         & 0.98                        & 0.98                        & \\multicolumn{1}{l|}{0.98}                        & \\textbf{99}                           & \\textbf{0.99}                        & \\textbf{0.99}                        & \\textbf{0.99}                        \\\\\n                                                       & \\textit{\\textbf{7500}}                                                           & \\textbf{99.07}               & \\textbf{0.99}               & \\textbf{0.99}               & \\multicolumn{1}{l|}{\\textbf{0.99}}               & 98.84                                 & 0.99                                 & 0.99                                 & 0.99                                 \\\\\n\\multirow{-5}{*}{\\textbf{3}}                           & \\textit{\\textbf{10000}}                                                          & 98.44                        & 0.98                        & 0.98                        & \\multicolumn{1}{l|}{0.98}                        & {\\ul \\textbf{99.47}}                  & {\\ul \\textbf{0.99}}                  & {\\ul \\textbf{0.99}}                  & {\\ul \\textbf{0.99}}                  \\\\ \\cmidrule(r){1-1}\n                                                       & \\textit{\\textbf{0}}                                                              & 98.5                         & 0.99                        & 0.99                        & \\multicolumn{1}{l|}{0.99}                        & 98.5                                  & 0.99                                 & 0.99                                 & 0.99                                 \\\\\n                                                       & \\textit{\\textbf{2500}}                                                           & {\\color[HTML]{FE0000} 97.86} & {\\color[HTML]{FE0000} 0.98} & {\\color[HTML]{FE0000} 0.98} & \\multicolumn{1}{l|}{{\\color[HTML]{FE0000} 0.98}} & \\textbf{99.42}                        & \\textbf{0.99}                        & \\textbf{0.99}                        & \\textbf{0.99}                        \\\\\n                                                       & \\textit{\\textbf{5000}}                                                           & \\textbf{99.22}               & \\textbf{0.99}               & \\textbf{0.99}               & \\multicolumn{1}{l|}{\\textbf{0.99}}               & {\\color[HTML]{FE0000} 97.75}          & {\\color[HTML]{FE0000} 0.98}          & {\\color[HTML]{FE0000} 0.98}          & {\\color[HTML]{FE0000} 0.98}          \\\\\n                                                       & \\textit{\\textbf{7500}}                                                           & {\\color[HTML]{FE0000} 97.6}  & {\\color[HTML]{FE0000} 0.98} & {\\color[HTML]{FE0000} 0.98} & \\multicolumn{1}{l|}{{\\color[HTML]{FE0000} 0.98}} & {\\color[HTML]{FE0000} \\textbf{98.15}} & {\\color[HTML]{FE0000} \\textbf{0.98}} & {\\color[HTML]{FE0000} \\textbf{0.98}} & {\\color[HTML]{FE0000} \\textbf{0.98}} \\\\\n\\multirow{-5}{*}{\\textbf{4}}                           & \\textit{\\textbf{10000}}                                                          & 99.22                        & 0.99                        & 0.99                        & \\multicolumn{1}{l|}{0.99}                        & {\\ul \\textbf{99.56}}                  & {\\ul \\textbf{$\\sim$1}}               & {\\ul \\textbf{$\\sim$1}}               & {\\ul \\textbf{$\\sim$1}}               \\\\ \\cmidrule(r){1-1}\n                                                       & \\textit{\\textbf{0}}                                                              & 96.6                         & 0.97                        & 0.97                        & \\multicolumn{1}{l|}{0.97}                        & 96.6                                  & 0.97                                 & 0.97                                 & 0.97                                 \\\\\n                                                       & \\textit{\\textbf{2500}}                                                           & \\textbf{99.47}               & \\textbf{0.99}               & \\textbf{0.99}               & \\multicolumn{1}{l|}{\\textbf{0.99}}               & 99.23                                 & 0.99                                 & 0.99                                 & 0.99                                 \\\\\n                                                       & \\textit{\\textbf{5000}}                                                           & 99.4                         & 0.99                        & 0.99                        & \\multicolumn{1}{l|}{0.99}                        & \\textbf{99.83}                        & \\textbf{$\\sim$1}                     & \\textbf{$\\sim$1}                     & \\textbf{$\\sim$1}                     \\\\\n                                                       & \\textit{\\textbf{7500}}                                                           & 99.2                         & 0.99                        & 0.99                        & \\multicolumn{1}{l|}{0.99}                        & {\\ul \\textbf{99.85}}                  & {\\ul \\textbf{$\\sim$1}}               & {\\ul \\textbf{$\\sim$1}}               & {\\ul \\textbf{$\\sim$1}}               \\\\\n\\multirow{-5}{*}{\\textbf{5}}                           & \\textit{\\textbf{10000}}                                                          & 99.67                        & $\\sim$1                     & $\\sim$1                     & \\multicolumn{1}{l|}{$\\sim$1}                     & \\textbf{99.78}                        & \\textbf{$\\sim$1}                     & \\textbf{$\\sim$1}                     & \\textbf{$\\sim$1}                     \\\\ \\cmidrule(r){1-1}\n                                                       & \\textit{\\textbf{0}}                                                              & 97.3                         & 0.97                        & 0.97                        & \\multicolumn{1}{l|}{0.97}                        & 97.3                                  & 0.97                                 & 0.97                                 & 0.97                                 \\\\\n                                                       & \\textit{\\textbf{2500}}                                                           & \\textbf{99.8}                & \\textbf{$\\sim$1}            & \\textbf{$\\sim$1}            & \\multicolumn{1}{l|}{\\textbf{$\\sim$1}}            & 99.75                                 & $\\sim$1                              & $\\sim$1                              & $\\sim$1                              \\\\\n                                                       & \\textit{\\textbf{5000}}                                                           & 99.75                        & $\\sim$1                     & $\\sim$1                     & \\multicolumn{1}{l|}{$\\sim$1}                     & {\\color[HTML]{FE0000} 96.1}           & {\\color[HTML]{FE0000} 0.96}          & {\\color[HTML]{FE0000} 0.96}          & {\\color[HTML]{FE0000} 0.96}          \\\\\n                                                       & \\textit{\\textbf{7500}}                                                           & {\\color[HTML]{000000} 97.63} & {\\color[HTML]{000000} 0.98} & {\\color[HTML]{000000} 0.98} & \\multicolumn{1}{l|}{{\\color[HTML]{000000} 0.98}} & {\\ul \\textbf{99.82}}                  & {\\ul \\textbf{$\\sim$1}}               & {\\ul \\textbf{$\\sim$1}}               & {\\ul \\textbf{$\\sim$1}}               \\\\\n\\multirow{-5}{*}{\\textit{\\textbf{6}}}                  & \\textit{\\textbf{10000}}                                                          & 99.67                        & $\\sim$1                     & $\\sim$1                     & \\multicolumn{1}{l|}{$\\sim$1}                     & \\textbf{99.73}                        & \\textbf{$\\sim$1}                     & \\textbf{$\\sim$1}                     & \\textbf{$\\sim$1}                     \\\\ \\cmidrule(r){1-1}\n                                                       & \\textit{\\textbf{0}}                                                              & 90.7                         & 0.91                        & 0.91                        & \\multicolumn{1}{l|}{0.91}                        & 90.7                                  & 0.91                                 & 0.91                                 & 0.91                                 \\\\\n                                                       & \\textit{\\textbf{2500}}                                                           & \\textbf{99.86}               & \\textbf{$\\sim$1}            & \\textbf{$\\sim$1}            & \\multicolumn{1}{l|}{\\textbf{$\\sim$1}}            & 99.78                                 & $\\sim$1                              & $\\sim$1                              & $\\sim$1                              \\\\\n                                                       & \\textit{\\textbf{5000}}                                                           & \\textbf{99.89}               & \\textbf{$\\sim$1}            & \\textbf{$\\sim$1}            & \\multicolumn{1}{l|}{\\textbf{$\\sim$1}}            & 99.86                                 & $\\sim$1                              & $\\sim$1                              & $\\sim$1                              \\\\\n                                                       & \\textit{\\textbf{7500}}                                                           & \\textbf{99.91}               & \\textbf{$\\sim$1}            & \\textbf{$\\sim$1}            & \\multicolumn{1}{l|}{\\textbf{$\\sim$1}}            & 99.84                                 & $\\sim$1                              & $\\sim$1                              & $\\sim$1                              \\\\\n\\multirow{-5}{*}{\\textit{\\textbf{7}}}                  & \\textit{\\textbf{10000}}                                                          & {\\ul \\textbf{99.94}}         & {\\ul \\textbf{$\\sim$1}}      & {\\ul \\textbf{$\\sim$1}}      & \\multicolumn{1}{l|}{{\\ul \\textbf{$\\sim$1}}}      & 99.73                                 & $\\sim$1                              & $\\sim$1                              & $\\sim$1                              \\\\ \\midrule\n\\textit{\\textbf{Avg.}}                                 &                                                                                  & 98.43                        & 0.98                        & 0.98                        & 0.98                                             & 98.40                                 & 0.98                                 & 0.98                                 & 0.98                                 \\\\ \\bottomrule\n\\end{tabular}\n\\end{table*}\n\nTable \\ref{tab:bigresults} shows all of the results for each subject, both with and without exposure to synthetic data. Per-run, the LSTM achieved better results over the GPT-2 in 14 instances whereas the GPT-2 achieved better results over the LSTM in 13 instances. Of the five runs that scored lower than no synthetic data exposure, two were LSTM and three were GPT-2. Otherwise, 51 of the 56 experiments all outperformed the original model without synthetic data exposure and every single subject experienced their best classification result in all cases when the model had been exposed to synthetic data. The best score on a per-subject basis was achieved by exposing the network to data produced by the LSTM three times and the GPT-2 five times (both including Subject 2 where both were best at 99.7\\%). The maximum diversion of training accuracy to validation accuracy was $\\sim1\\%$ showing that although high results were attained, overfitting was relatively low; with more computational resources, k-fold and LOO cross validation are suggested as future works to achieve more accurate measures of variance within classification. \n\nThese results attained show that speaker classification can be improved by exposing the network to synthetic data produced by both supervised and attention-based models and then transferring the weights to the initial problem, which most often scores lower without synthetic data exposure in all cases but five although those subjects still experienced their absolute best result through synthetic data exposure regardless. \n\n\n\\subsection{Comparison to other methods of speaker recognition}\n\\begin{table}[]\n\\caption{Comparison of the best models found in this work and other classical methods of speaker recognition (sorted by accuracy)}\n\\label{tab:bigcomparison}\n\\footnotesize\n\\begin{tabular}{@{}clllll@{}}\n\\toprule\n\\multicolumn{1}{l}{\\textbf{Subject}} & \\textbf{Model}                         & \\textbf{Acc.}  & \\textbf{F-1} & \\textbf{Prec.} & \\textbf{Rec.} \\\\ \\midrule\n\\multirow{7}{*}{\\textbf{1}}          & \\textit{\\textbf{DNN (LSTM TL 2500)}}   & \\textbf{99.5}  & \\textbf{$\\sim$1}   & \\textbf{$\\sim$1}     & \\textbf{$\\sim$1}    \\\\\n                                     & \\textit{\\textbf{DNN (GPT-2 TL 5000)}}  & 97.77          & 0.98         & 0.98           & 0.98          \\\\\n                                     & \\textit{\\textbf{SMO}}                  & 97.71          & 0.98         & 0.95           & 0.95          \\\\\n                                     & \\textit{\\textbf{Random Forest}}        & 97.48          & 0.97         & 0.97           & 0.97          \\\\\n                                     & \\textit{\\textbf{Logistic Regression}}  & 97.47          & 0.97         & 0.97           & 0.97          \\\\\n                                     & \\textit{\\textbf{Bayesian Network}}     & 82.3           & 0.87         & 0.96           & 0.82          \\\\\n                                     & \\textit{\\textbf{Naive Bayes}}          & 78.96          & 0.84         & 0.953          & 0.77          \\\\ \\cmidrule(r){1-1}\n\\multirow{7}{*}{\\textbf{2}}          & \\textit{\\textbf{DNN (LSTM TL 7500)}}   & \\textbf{99.7}  & \\textbf{$\\sim$1}   & \\textbf{$\\sim$1}     & \\textbf{$\\sim$1}    \\\\\n                                     & \\textit{\\textbf{DNN (GPT-2 TL 7500)}}  & 99.7           & $\\sim$1            & $\\sim$1              & $\\sim$1             \\\\\n                                     & \\textit{\\textbf{SMO}}                  & 98.94          & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{Logistic Regression}}  & 98.33          & 0.98         & 0.98           & 0.98          \\\\\n                                     & \\textit{\\textbf{Random Forest}}        & 98.28          & 0.98         & 0.98           & 0.98          \\\\\n                                     & \\textit{\\textbf{Bayesian Network}}     & 84.9           & 0.9          & 0.97           & 0.85          \\\\\n                                     & \\textit{\\textbf{Naive Bayes}}          & 76.58          & 0.85         & 0.97           & 0.77          \\\\ \\cmidrule(r){1-1}\n\\multirow{7}{*}{\\textbf{3}}          & \\textit{\\textbf{DNN (GPT-2 TL 10000)}} & \\textbf{99.47} & \\textbf{0.99}   & \\textbf{0.99}     & \\textbf{0.99}    \\\\\n                                     & \\textit{\\textbf{DNN (LSTM TL 2500)}}   & 99.2           & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{SMO}}                  & 99.15          & 0.99         & 0.99           & 0.98          \\\\\n                                     & \\textit{\\textbf{Logistic Regression}}  & 98.85          & 0.99         & 0.99           & 0.98          \\\\\n                                     & \\textit{\\textbf{Random Forest}}        & 98.79          & 0.99         & 0.99           & 0.98          \\\\\n                                     & \\textit{\\textbf{Bayesian Network}}     & 91.49          & 0.94         & 0.98           & 0.92          \\\\\n                                     & \\textit{\\textbf{Naive Bayes}}          & 74.37          & 0.83         & 0.96           & 0.74          \\\\ \\cmidrule(r){1-1}\n\\multirow{7}{*}{\\textbf{4}}          & \\textit{\\textbf{DNN (GPT-2 TL 10000)}} & \\textbf{99.56} & \\textbf{$\\sim$1}   & \\textbf{$\\sim$1}     & \\textbf{$\\sim$1}    \\\\\n                                     & \\textit{\\textbf{DNN (LSTM TL 5000)}}   & 99.22          & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{Logistic Regression}}  & 98.66          & 0.99         & 0.98           & 0.98          \\\\\n                                     & \\textit{\\textbf{SMO}}                  & 98.66          & 0.99         & 0.98           & 0.98          \\\\\n                                     & \\textit{\\textbf{Random Forest}}        & 98             & 0.98         & 0.98           & 0.98          \\\\\n                                     & \\textit{\\textbf{Bayesian Network}}     & 95.53          & 0.96         & 0.98           & 0.96          \\\\\n                                     & \\textit{\\textbf{Naive Bayes}}          & 88.74          & 0.92         & 0.97           & 0.89          \\\\ \\cmidrule(r){1-1}\n\\multirow{7}{*}{\\textbf{5}}          & \\textit{\\textbf{DNN (GPT-2 TL 10000)}} & \\textbf{99.85} & \\textbf{$\\sim$1}   & \\textbf{$\\sim$1}     & \\textbf{$\\sim$1}    \\\\\n                                     & \\textit{\\textbf{DNN (LSTM TL 10000)}}  & 99.67          & 1            & 1              & 1             \\\\\n                                     & \\textit{\\textbf{Logistic Regression}}  & 98.86          & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{Random Forest}}        & 98.7           & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{SMO}}                  & 98.6           & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{Naive Bayes}}          & 90.55          & 0.94         & 0.98           & 0.9           \\\\\n                                     & \\textit{\\textbf{Bayesian Network}}     & 88.95          & 0.93         & 0.98           & 0.89          \\\\ \\cmidrule(r){1-1}\n\\multirow{7}{*}{\\textbf{6}}          & \\textit{\\textbf{DNN (GPT-2 TL 7500)}}  & \\textbf{99.82} & \\textbf{$\\sim$1}   & \\textbf{$\\sim$1}     & \\textbf{$\\sim$1}    \\\\\n                                     & \\textit{\\textbf{DNN (LSTM TL 2500)}}   & 99.8           & $\\sim$1            & $\\sim$1              & $\\sim$1             \\\\\n                                     & \\textit{\\textbf{Logistic Regression}}  & 99.1           & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{Random Forest}}        & 98.9           & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{SMO}}                  & 98.86          & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{Naive Bayes}}          & 90.52          & 0.94         & 0.98           & 0.9           \\\\\n                                     & \\textit{\\textbf{Bayesian Network}}     & 89.27          & 0.93         & 0.98           & 0.89          \\\\ \\cmidrule(r){1-1}\n\\multirow{7}{*}{\\textbf{7}}          & \\textit{\\textbf{DNN (LSTM TL 10000)}}  & \\textbf{99.91} & \\textbf{$\\sim$1}   & \\textbf{$\\sim$1}     & \\textbf{$\\sim$1}    \\\\\n                                     & \\textit{\\textbf{DNN (GPT-2 TL 5000)}}  & 99.86          & $\\sim$1            & $\\sim$1              & $\\sim$1             \\\\\n                                     & \\textit{\\textbf{SMO}}                  & 99.4           & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{Logistic Regression}}  & 99.13          & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{Random Forest}}        & 99             & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{Bayesian Network}}     & 88.67          & 0.93         & 0.98           & 0.89          \\\\\n                                     & \\textit{\\textbf{Naive Bayes}}          & 86.9           & 0.91         & 0.98           & 0.87          \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[]\n\\caption{Average performance of the chosen models for each of the 7 subjects.}\n\\label{tab:average-comparison}\n\\centering\n\\begin{tabular}{@{}lllll@{}}\n\\toprule\n\\textbf{Model}                        & \\textbf{Avg acc} & \\textbf{F-1} & \\textbf{Prec.} & \\textbf{Rec.} \\\\ \\midrule\n\\textit{\\textbf{DNN (LSTM TL)}}       & 99.57            & $\\sim$1      & $\\sim$1        & $\\sim$1       \\\\\n\\textit{\\textbf{DNN (GPT-2 TL)}}      & 99.43            & $\\sim$1      & $\\sim$1        & $\\sim$1       \\\\\n\\textit{\\textbf{SMO}}                 & 98.76            & 0.99         & 0.98           & 0.98          \\\\\n\\textit{\\textbf{Logistic Regression}} & 98.63            & 0.99         & 0.98           & 0.98          \\\\\n\\textit{\\textbf{Random Forest}}       & 98.45            & 0.98         & 0.98           & 0.98          \\\\\n\\textit{\\textbf{Bayesian Network}}    & 88.73            & 0.92         & 0.98           & 0.89          \\\\\n\\textit{\\textbf{Naive Bayes}}         & 83.80            & 0.89         & 0.97           & 0.83          \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTable \\ref{tab:bigcomparison} shows a comparison of the models proposed in this paper to other state-of-the-art methods of speaker recognition. Namely, they are Sequential Minimal Optimisation (SMO), Logistic Regression, Bayesian Networks, and Naive Bayes. It can be observed that, although in some cases close, the DNN fine tuned from synthetic data generated by both the LSTM and GPT-2 achieve higher scores than other methods. Finally, Table \\ref{tab:average-comparison} shows the average scores for the chosen models for each of the seven subjects. \n\n\n\n\n\n\\section{Conclusion and Future Work}\n\\label{sec4:futureworkconclusion}\nTo finally conclude, this work found strong success for all 7 subjects when improving the classification problem of speaker recognition by generating augmented data by both LSTM and OpenAI GPT-2 models. Future work aims to solidify this hypothesis by running the experiments for a large range of subjects and comparing the patterns that emerge from the results. \n\nThe experiments in this work have provided a strong argument for the usage of deep neural network transfer learning from MFCCs synthesised by both LSTM and GPT-2 models for the problem of speaker recognition. One of the limitations of this study was hardware availability since it was focused on those available to consumers today. The Flickr8k dataset was thus limited to 8,000 data objects and new datasets created, which prevents a direct comparison to other speaker recognition works which often operate on larger data and with hardware beyond consumer availability. It is worth noting that the complex nature of training LSTM and GPT-2 models to generate MFCCs is beyond that of the task of speaker recognition itself, and as such, devices with access to TPU or CUDA-based hardware must perform the task in the background over time. The tasks in question took several minutes with the two GPUs used in this work for both LSTM and GPT-2 and as such are not instantaneous. As previously mentioned, although it was observed that overfitting did not occur too strongly, it would be useful in future to perform similar experiments with either K-fold or leave-one-out Cross Validation in order to achieve even more accurate representations of the classification metrics. In terms of future application, the then-optimised model would be the implemented within real robots and smarthome assistants through compatible software.\n\nIn this work, seven subjects were benchmarked with both a tuned LSTM and OpenAI's GPT-2 model. In future, as was seen with related works, a GAN could also be implemented in order to provide a third possible solution to the problem of data scarcity in speaker recognition as well as other related speech recognition classification problems - bias is an open issue that has been noted for data augmentation with GANs~\\cite{hu2019exploring}, and as such, this issue must be studied if a GAN is implemented for problems of this nature. This work further argued for the hypothesis presented, that is, data augmentation can aid in improving speaker recognition for scarce datasets. Following the 14 successful runs including LSTMs and GPT-2s, the overall process followed by this experiment could be scripted and thus completely automated, allowing for the benchmarking of many more subjects to give a much more generalised set of results, of which will be more representative of the general population. Additionally, samples spoken from many languages should also be considered in order to provide language generalisation, rather than just the English language spoken by multiple international dialects in this study. Should generalisation be possible, future models may require only a small amount of fine-tuning to produce synthetic speech for a given person rather than training from scratch as was performed in this work. In more general lines of thought, literature review shows that much of the prominent work is recent, leaving many fields of machine learning that have not yet been attempted to be improved via the methods described in this work. \n\n\\bibliographystyle{ieeetr}\n\n\\section{Introduction}\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=1.1]{small-diagram.pdf}\n    \\caption{A simplified diagram of the experiments followed by this work towards the comparison of classical vs transfer learning from synthetic data for speaker recognition. A more detailed diagram can be found in Figure \\ref{fig:method}}\n    \\label{fig:smalldiagram}\n\\end{figure}\nData scarcity is an issue that arises often outside of the lab, due to the large amount of data required for classification activities. This includes speaker classification in order to enable personalised Human-Machine (HMI) and Human-Robot Interaction (HRI), a technology growing in consumer usefulness within smart device biometric security on devices such as smartphones and tablets, as well as for multiple-user smarthome assistants (operating on a per-person basis) which are not yet available. Speaker recognition, i.e., autonomously recognising a person from their voice, is a well-explored topic in the state-of-the-art within the bounds of data availability, which causes difficulty in real-world use. It is unrealistic to expect a user to willingly provide many minutes or hours of speech data to a device unless it is allowed to constantly record daily life, something which is a modern cause for concern with the virtual home assistant. In this work, we show that data scarcity in speaker recognition can be overcome by collecting only several short spoken sentences of audio from a user and then using extracted Mel-Frequency Cepstral Coefficients (MFCC) data in both supervised and unsupervised learning paradigms to generate synthetic speech, which is then used in a process of transfer learning to better recognise the speaker in question.\n\nAutonomous speaker classification can suffer issues of data scarcity since the user is compared to a large database of many speakers. The most obvious solution to this is to collect more data, but with Smart Home Assistants existing within private environments and potentially listening to private data, this produces an obvious problem of privacy and security~\\cite{logsdon2018alexa,dunin2020alexa}. Not collecting more data on the other hand, presents an issue of a large class imbalance between the speaker to classify against the examples of other speakers, producing lower accuracies and less trustworthy results~\\cite{babbar2019data}, which must be solved for purposes such as biometrics since results must be trusted when used for security. In this study, weighting of errors is performed to introduce balance, but it is noted that the results still have room for improvement regardless. \n\n\n\nData augmentation is the idea that useful new data can be generated by algorithms or models that would improve the classification of the original, scarce dataset. A simple but prominent example of this is the warping, flipping, mirroring and noising of images to better prepare image classification algorithms~\\cite{wang2017effectiveness}. A more complex example through generative models can be seen in recent work that utilise methods such as the Generative Adversarial Network (GAN) to create synthetic data which itself also holds useful information for learning from and classification of data~\\cite{zhu2018emotion,frid2018synthetic}. Although image classification is the most common and most obvious application of generative models for data augmentation, recent works have also enjoyed success in augmenting audio data for sound classification~\\cite{yang2018se,madhu2019data}. This work extends upon a previous conference paper that explored the hypothesis \\textit{``can the synthetic data produced by a generative model aid in speaker recognition?\"}~\\cite{bird2020overcoming} within which a Character-Level Recurrent Neural Network (RNN), as a generative model, produced synthetic data useful for learning from in order to recognise the speaker. This work extends upon these preliminary experiments by the following:\n\\begin{enumerate}\n    \\item The extension of the dataset to more subjects from multiple international backgrounds and the extraction of the MFCCs of each subject;\n    \\item Benchmarking of a Long Short Term Memory (LSTM) architecture for 64, 128, 256 and 512 LSTM units in one to three hidden layers towards reduction of loss in generating synthetic data. The best model is selected as the candidate for the LSTM data generator;\n    \\item The inclusion of OpenAI's GPT-2 model as a data generator in order to compare the approaches of supervised (LSTM) and attention-based (GPT-2) methods for synthetic data augmentation for speaker classification.\n\\end{enumerate}\nThe scientific contributions of this work, thus, are related to the application of synthetic MFCCs for improvement of speaker recognition. A diagram of the experiments can be observed in Figure \\ref{fig:smalldiagram}. To the authors' knowledge, the paper that this work extends is the first instance of this research being explored\\footnote{to the best of our knowledge and based on literature review.}. The best LSTM and the GPT-2 model are tasked with generating 2,500, 5,000, 7,500, and 10,000 synthetic data objects for each subject after learning from the scarce datasets extracted from their speech. A network then learns from these data and transfers their weights to a network aiming to learn and classify the real data, and many show an improvement. For all subjects, the results show that several of the networks perform best after experiencing exposure to synthetic data. \n\nThe remainder of this article is as follows. Section \\ref{sec1:background} initially explores important scientific concepts of the processes followed by this work and also the current State-of-the-Art in the synthetic data augmentation field. Following this, Section \\ref{sec2:method} then outlines the method followed by this work including data collection, synthetic data augmentation, MFCC extraction to transform audio into a numerical dataset, and finally the learning processes followed to achieve results. The final results of the experiments are then discussed in Section \\ref{sec:results}, and then future work is outlined and conclusions presented in Section \\ref{sec4:futureworkconclusion}.\n\n\n\n\\section{Background and Related Work}\n\\label{sec1:background}\n\n\nVerification of a speaker is the process of identifying a single individual against many others by spoken audio data~\\cite{poddar2017speaker}. That is, the recognition of a set of the person's speech data $X$ specifically from a speech set $Y$ where $X \\in Y$. In the simplest sense, this can be given as a binary classification problem; for each data object $o$, is $o \\in X$? Is the speaker to be recognised speaking, or is it another individual? Speaker recognition is important for social HRI~\\cite{mumolo2003distant} (the robot's perception of the individual based on their acoustic utterances), Biometrics~\\cite{ratha2001automated}, and Forensics~\\cite{rose2002forensic} among many others. In~\\cite{hasan2004speaker}, researchers found relative ease of classifying 21 speakers from a limited set, but the problem becomes more difficult as it becomes more realistic, where classifying a speaker based on their utterances is increasingly difficult as the dataset grows~\\cite{nagrani2017voxceleb,yadav2018learning,zeinali2019but}. In this work, the speaker is recognised from many thousands of other examples of human speech from the Flickr8k speakers dataset. \n\n\\subsection{LSTM and GPT-2}\nLong Short Term Memory (LSTM) is a form of Artificial Neural Network in which multiple RNNs will learn from previous states as well as the current state. Initially, the LSTM selects data to delete: \n\\begin{equation}\n    \\mathop f\\nolimits_{\\text{t}} = \\sigma \\left( {\\mathop W\\nolimits_{f} \\cdot \\left[ {\\mathop h\\nolimits_{t - 1} ,\\mathop x\\nolimits_{t} } \\right] + \\mathop b\\nolimits_{f} } \\right),\n\\end{equation}\n\\noindent where $W_f$ are the weights of the units, $h_{t=1}$ is the output at $t=1$, $x_t$ are inputs and $b_f$ is an applied bias. Data to be stored is then selected based on input $i$, generating $C_t$ values:\n\\begin{equation}\n\\mathop o\\nolimits_{\\text{t}} = \\sigma \\left( {\\mathop W\\nolimits_{i} \\cdot \\left[ {\\mathop h\\nolimits_{t - 1} ,\\mathop x\\nolimits_{t} } \\right] + \\mathop b\\nolimits_{i} } \\right),\n\\end{equation}\n\n\\begin{equation}\n\\mathop {\\hat{C}}\\nolimits_{\\text{t}} = \\tanh \\left( {\\mathop W\\nolimits_{c} \\cdot \\left[ {\\mathop h\\nolimits_{t - 1} ,\\mathop x\\nolimits_{t} } \\right] + \\mathop b\\nolimits_{c} } \\right).\n\\end{equation}\n\nA convolutional operation updates values:\n\\begin{equation}\nC_{t}  =  f_{t} * C_{t-1} + i_{t} *  \\tilde{C}_{t} .\n\\end{equation}\n\nOutput $o_t$ is presented, and the hidden state is updated:\n\\begin{equation}\n\\mathop o\\nolimits_{\\text{t}} = \\sigma \\left( {\\mathop W\\nolimits_{o} \\cdot \\left[ {\\mathop h\\nolimits_{t - 1} ,\\mathop x\\nolimits_{t} } \\right] + \\mathop b\\nolimits_{o} } \\right),\n\\end{equation}\n\\begin{equation}\nh_{t} = o_{t} * tanh(C_{t}).\n\\end{equation}\nDue to the observed consideration of previous data, it is often found that time dependent data are very effectively classified due to the memory-like nature of the LSTM. LSTM is thus a particularly powerful technique in terms of speech recognition~\\cite{graves2013hybrid} due to the temporal nature of the data~\\cite{belin2000voice}. \\\\\n\nIn addition to LSTM, this study also considers OpenAI's Generative Pretrained Transformer 2 (GPT-2) model~\\cite{radford2018improving,radford2019language} as a candidate for producing synthetic MFCC data to improve speaker recognition. The model in question, \\textit{335M}, is much deeper than the LSTMs explored at 12 layers. In the OpenAI paper, the GPT modelling is given for a set of samples $x_1, x_2, ..., x_n$ composed of variable symbol-sequence lengths $s_1, s_2, ..., s_n$ factorised by joint probabilities over symbols as the product of conditional probabilities~\\cite{jelinek1980interpolated,bengio2003neural}:\n\\begin{equation}\n    p(x) =  \\prod_{i=1}^n p(s_{i}  | s_{1}, ..., s_{n-1}).\n\\end{equation}\nAttention is given to various parts of the input vector:\n\\begin{equation}\n    Attention(Q,K,V) = softmax \\left(  \\frac{QK^T}{  \\sqrt{d_{k}}  }  \\right) V,\n\\end{equation}\nwhere Q is the \\textit{query} i.e., the single object in the sequence, in this case, a word. K are the keys, which are vector representations of the input sequence, and V are values as vector representations of all words in the sequence. In the initial encoder, decoder, and attention blocks $Q=V$ whereas later on the attention block that takes these outputs as input, $Q \\neq V$ since both are derived from the block's 'memory'. In order to combine multiple queries, that is, to consider previously learnt rules from text, multi-headed attention is presented:\n\\begin{equation}\n\\begin{aligned}\n    MultiHead(Q,K,V) = Concatenate(head_{1}, ..., head_{h})W^{O} \\\\\n    head_{i} = Attention(QW^{Q}_{i}, KW^{K}_{i}, VW^{V}_{i}).\n\\end{aligned}\n\\end{equation}\nAs in the previous equation, it can be seen that previously learnt $h$ projections $d_Q, d_K$ and $d_V$ are also considered given that the block has multiple heads. The above equations and further detail on both attention and multi-headed attention can be found in \\cite{vaswani2017attention}. The unsupervised nature of the GPT models is apparent since $Q, K$ and $V$ are from the same source. \n\nImportantly, GPT produces output with consideration not only to input, but also to the task. The GPT-2 model has been shown to be a powerful state-of-the-art tool for language learning, understanding, and generation; researchers noted that the model could be used with ease to generate realistic propaganda for extremist terrorist groups~\\cite{solaiman2019release}, as well as noting that generated text by the model was difficult to detect~\\cite{gehrmann2019gltr,wolff2020attacking,adelani2020generating}. The latter aforementioned papers are promising, since a difficulty of detection suggests statistical similarities, which are likely to aid in the problem of improving classification accuracy of a model by exposing it to synthetic data objects output by such a model. \n\n\\subsection{Dataset Augmentation through Synthesis}\n\\label{subsec:dataaugmentation}\n\nSimilarities between real-life experiences and imagined perception have shown in psychological studies that the human imagination, though mentally augmenting and changing situations~\\cite{beres1960perception}, aids in improving the human learning process~\\cite{egan1989memory,heath2008exploring,macintyre2012emotions,egan2014imagination}. The importance of this ability in the learning process shows the usefulness of data augmentation in human learning, and as such, is being explored as a potential solution to data scarcity and quality in the machine learning field. Even though the synthetic data may not be realistic alone, minute similarities between it and reality allow for better pattern recognition. \n\nThe idea of data augmentation as the first stage in fine-tune learning is inspired by the aforementioned findings, and follows a similar approach. Synthetic data is generated by learning from the real data, and algorithms are exposed to them in a learning process prior to the learning process of real data; this is then compared to the classical approach of learning from the data solely, where the performance of the former model compared to the latter shows the effect of the data augmentation learning process. Much of the work is recent, many from the last decade, and a pattern of success is noticeable for many prominent works when comparing the approach to the classical method of learning from real data alone. \n\nAs described, the field of exploring augmented data to improve classification algorithms is relatively young, but there exist several prominent works that show success in applying this approach. When augmented data from the SemEval dataset is learned from by a Recurrent Neural Network (RNN), researchers found that the overall best F-1 score was achieved for relation classification in comparison to the model only learning from the dataset itself~\\cite{xu2016improved}. Due to data scarcity in the medical field, classification of liver lesions~\\cite{frid2018synthetic} and Alzheimer's Disease~\\cite{shin2018medical} have also shown improvement when the learning models (CNNs) considered data augmented by Convolutional GANs. In Natural Language Processing, it was found that word augmentation aids to improve sentence classification by both CNN and RNN models~\\cite{kobayashi2018contextual}. The \\textit{DADA} model has been suggested as a strong method to produce synthetic data for improvement of data-scarce problems through the Deep Adversarial method via a dedicated discriminator network aiming to augment data specifically~\\cite{zhang2019dada} which has noted success in machine learning problems~\\cite{barz2020deep}. \n\nData augmentation has shown promise in improving multi-modal emotion recognition when considering audio and images~\\cite{huang2018multimodal}, digital signal classification~\\cite{tang2018digital}, as well as for a variety of audio classification problems such as segments of the Hub500 problem~\\cite{park2019specaugment}. Additionally, synthetic data augmentation of mel-spectrograms have shown to improve acoustic scene recognition~\\cite{yang2018se}. Realistic text-to-speech is achieved by producing realistic sound such as the Tacotron model~\\cite{wang2017tacotron} when considering the textual representation of the audio being considered, and reverse engineering the model to produce audio based on text input. A recent preliminary study showed GANs may be able to aid in producing synthetic data for speaker recognition~\\cite{chien2018adversarial}. \n\nThe temporal models considered in this work to generate synthetic speech data have recently shown success in generating acoustic sounds~\\cite{eck2002finding} and accurate timeseries data~\\cite{senjyu2006application}, written text~\\cite{pawade2018story,sha2018order}, artistic images~\\cite{gregor2015draw}. Specifically, temporal models are also observed to be successful in generating MFCC data~\\cite{wang2017rnn}, which is the data type considered in this work. Many prominent works in speech recognition consider temporal learning to be highly important~\\cite{fernandez2007application,he2019streaming,sak2014long} and for generation of likewise temporal data~\\cite{valentini2016investigating,wang2017rnn,tachibana2018efficiently} (that is, which this study aims to perform). If it is possible to generate data that bares similarity to the real data, then it could improve the models while also reducing the need for large amounts of real data to be collected. \n\n\n\\section{Proposed Approach}\n\\label{sec2:method}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.9]{overalldiagram_new.pdf}\n    \\caption{A diagram of the experimental method in this work. Note that the two networks being directly compared are classifying the same data, with the difference being the initial weight distribution either from standard random distribution or transfer learning from GPT-2 and LSTM produced synthetic data.}\n    \\label{fig:method}\n\\end{figure*}\n\nThis section describes the development of the proposed approach, which can be observed overall in Figure \\ref{fig:method}. For each test, five networks are trained in order to gain results. Firstly, a network simply to perform the speaker classification experimen without transfer learning (from a standard random weight distribution). Firstly, a network simply to perform the speaker classification experiment. Produced by LSTM and GPT-2, synthetic data is used to train another network, of which the weights are used to train a final network as an initial distribution to perform the same experiment as described in the first network (classifying the speaker's real data from Flickr8k speakers). Thus, the two networks leading to the final classification score in the diagram are directly comparable since they are learning from the same data, they differ only in initial weight distribution (where the latter network has weights learnt from synthetic data).\n\n\n\n\\subsection{Real and Synthetic Data Collection}\nThe data collected, as previously described, presents a binary classification problem. That is, whether or not the individual in question is the one producing the acoustic utterance. \n\nThe large corpus of data for the \\textit{``not the speaker\"} class is gathered via the Flickr8k dataset~\\cite{harwath2015deep} which contains 40,000 individual utterances describing 8,000 images by a large number of speakers which is unspecified by the authors. MFCCs are extracted (described in Section \\ref{subsec:mfcc}) to generate temporal numerical vectors which represent a short amount of time from each audio clip. 100,000 data objects are selected through 50 blocks of 1,000 objects and then 50,000 other data objects selected randomly from the remainder of the dataset. This is performed so the dataset contains individual's speech at length as well as short samples of many other thousands of speakers also. \n\n\\begin{table*}[]\n\\footnotesize\n\\centering\n\\caption{Information regarding the data collection from the seven subjects}\n\\label{tab:subjects}\n\\begin{tabular}{@{}lllllll@{}}\n\\toprule\n\\textbf{Subject}           & \\textbf{Sex} & \\textbf{Age} & \\textbf{Nationality} & \\textbf{Dialect} & \\textbf{Time Taken (s)} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Real\\\\Data\\end{tabular}} \\\\ \\midrule\n\\textit{\\textbf{1}}        & M            & 23           & British              & Birmingham              & 24                      & 4978                           \\\\\n\\textit{\\textbf{2}}        & M            & 24           & American             & Florida          & 13                      & 2421                           \\\\\n\\textit{\\textbf{3}}        & F            & 28           & Irish                & Dublin                  & 12                      & 2542                           \\\\ \n\\textit{\\textbf{4}}        & F            & 30           & British                & London                  & 12                      & 2590                           \\\\ \n\\textit{\\textbf{5}}        & F            & 40           & British                & London                  & 10                      & 2189                           \\\\ \n\\textit{\\textbf{6}}        & M            & 21           & French                & Paris                  & 8                      & 1706                           \\\\ \n\\textit{\\textbf{7}}        & F            & 23           & French                & Paris                  & 9                      & 1952                           \\\\ \n\\midrule\n\\textit{\\textbf{Flickr8K}} &              &              &                      &                         &                         & 100000                       \\\\ \\bottomrule\n\\end{tabular}\n\\end{table*}\n\n\n\nTo gather data for recognising speakers, seven subjects are considered. Information on the subjects can be seen in Table \\ref{tab:subjects}. Subjects speak five random Harvard Sentences sentences from the \\textit{IEEE recommended practice for speech quality measurements}~\\cite{rothauser1969ieee}, and so contain most of the spoken phonetic sounds in the English language~\\cite{bird2019phoneme2}. Importantly, this is a user-friendly process, because it requires only a few short seconds of audio data. The longest time taken was by subject 1 in 24 seconds producing 4978 data objects and the shortest were the two French individuals who required 8 and 9 seconds respectively to speak the five sentences. All of the audio data were recorded using consumer-available recording devices such as smartphones and computer headsets. Synthetic datasets are generated following the learning processes of the best LSTM and the GPT-2 model, where the probability of the next character is decided upon depending on the learning algorithm and are generated in blocks of 1,000 within a loop and the final line is removed (since it was often within the cutoff point of the 1,000-character block). Illogical lines of data (those that did not have 26 comma separated values and a class) were removed, but were observed to be somewhat rare as both the LSTM and GPT-2 models had learnt the data format relatively well since it was uniform throughout. The format, throughout the datasets, was a uniform 27 comma separated values where the values were all numerical and the final value was `1' followed by a line break character.\n\n\\subsection{Feature Extraction}\n\\label{subsec:mfcc}\n\nThe nature of acoustic data is that the previous and following points of data from a single point in particular are also related to the class. Audio data is temporal in this regard, and thus classification of a single point in time is an extremely difficult problem~\\cite{xiong2003comparing,bird2019phoneme}. In order to overcome this issue features are extracted from the wave through a sliding window approach. The statistical features extracted in this work are the first 26 Mel-Frequency Cepstral Coefficients due to findings in the scientific state of the art arguing for their prominence over other methods~\\cite{muda2010voice,sahidullah2012design}. Sliding windows are set to a length of 0.025 seconds with a step of 0.01 seconds and extraction is performed as follows:\n\\begin{enumerate}\n\\item The Fourier Transform of time window data $\\omega$ is calculated:\n\\begin{equation}\nX(j\\omega)=\\int_{-\\infty}^\\infty x(t) e^{-j\\omega t} dt.\n\\end{equation}\n\\item The powers from the FT are mapped to the Mel-scale, that is, the human psychological scale of audible pitch~\\cite{stevens1937scale} via a triangular temporal window. \n\\item The power spectrum is considered and $log$s of each of their powers are taken.\n\\item The derived Mel-log powers are then treated as a signal, and a Discrete Cosine Transform (DCT) is calculated:\n\\begin{equation}\n\\begin{aligned}\nX_k =  \\sum_{n=0}^{N-1} x_n  cos  \\begin{bmatrix}  \\frac{\\pi}{N} (n+ \\frac{1}{2}) k \\end{bmatrix} \\\\     \nk=0,...,N-1,\n\\end{aligned}\n\\end{equation}\n\\noindent where $x$ is the array of length $N$, $k$ is the index of the output coefficient being calculated, where $N$ real numbers $x_{0} ... x_{n-1}$ are transformed into the $N$ real numbers $X_{0} ... X_{n-1}$. \n\\end{enumerate}\nThe amplitudes of the spectrum produced are taken as the \\textit{MFCCs}. The resultant data then provides a mathematical description of wave behaviour in terms of sounds, each data object made of 26 attributes produced from the sliding window are then treated as the input attributes for the neural networks for both speaker recognition and also synthetic data generation (with a class label also). \n\nThis process is performed for all of the selected Flickr8K data as well as the real data recorded from the subjects. The MFCC data from each of the 7 subjects' audio recordings is used as input to the LSTM and GPT-2 generative models for training and subsequent data augmentation.\n\n\n\\subsection{Speaker Classification Learning Process}\nFor each subject, the Flickr data and recorded audio forms the basis dataset and the speaker recognition problem. Eight datasets for transfer learning are then formed on a per-subject basis, which are the aforementioned data plus $2500, 5000, 7500$ and $10000$ synthetic data objects generated by either the LSTM or the GPT-2 models. LSTM has a standard dropout of 0.2 between each layer.\n\nThe baseline accuracy for comparison is given as ``Synth. Data: 0\" later in Table \\ref{tab:bigresults} which denotes a model that has not been exposed to any of the synthetic data. This baseline gives scores that are directly comparable to identical networks with their initial weight distributions being those trained to classify synthetic data generated for the subject, which is then used to learn from the real data. As previously described, the two sets of synthetic data to expose the models to during pre-training of the real speaker classification problem are generated by either an LSTM or a GPT-2 language model. Please note that due to this, the results presented have no baring on whether or not the network could classify the synthetic data well or otherwise, the weights are simply used as the initial distribution for the same problem. If the pattern holds that the transfer learning networks achieve better results than the networks which have not been trained on such data, it argues the hypothesis that speaker classification can be improved when considering either of these methods of data augmentation. This process can be observed in Figure \\ref{fig:method}. \n\nFor the Deep Neural Network that classifies the speaker, a topology of three hidden layers consisting of 30, 7, and 29 neurons respectively with ReLu activation functions and an ADAM optimiser~\\cite{kingma2014adam} is initialised. These hyperparameters are chosen due to a previous study that performed a genetic search of neural network topologies for the classification of phonetic sounds in the form of MFCCs~\\cite{bird2020optimisation}. The networks are given an unlimited number of epochs to train, only ceasing through a set early stopping callback of 25 epochs with no improvement of ability. The best weights are restored before final scores are calculated. This is allowed in order to make sure that all models stabilise to an asymptote and reduce the risk of stopping models prior to them achieving their potential best abilities. \n\nClassification errors are weighted equally by class prominence since there exists a large imbalance between the speaker and the rest of the data. All of the LSTM experiments performed in this work were executed on an Nvidia GTX980Ti GPU, while the GPT-2 experiment was performed on an Nvidia Tesla K80 GPU provided by Google Colab. \n\n\\section{Results}\n\\label{sec:results}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.5]{128-lstms.pdf}\n    \\caption{The training processes of the best performing models in terms of loss, separated for readability purposes. Results are given for a benchmarking experiment on all of the dataset rather than an individual.}\n    \\label{fig:128-lstms-benchmark}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.5]{bad-lstms.pdf}\n    \\caption{The training processes of LSTMs with 64, 256, and 512 units in 1-3 hidden layers, separated for readability purposes. Results are given for a benchmarking experiment on all of the dataset rather than an individual.}\n    \\label{fig:bad-lstms-benchmark}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.5]{gpt-graph.pdf}\n    \\caption{The training process of the GPT-2 model. Results are given for a benchmarking experiment on all of the dataset rather than an individual.}\n    \\label{fig:gpt-benchmark}\n\\end{figure}\n\n\\begin{table}[]\n\\centering\n\\caption{Best epochs and their losses for the 12 LSTM Benchmarks and GPT-2 training process. All models are benchmarked on the whole set of subjects for 100 epochs each in order to search for promising hyperparameters.}\n\\label{tab:benchmarking-models}\n\\begin{tabular}{@{}lll@{}}\n\\toprule\n\\textbf{Model}    & \\textbf{Best Loss} & \\textbf{Epoch} \\\\ \\midrule\nLSTM(64)          & 0.88           & 99             \\\\\nLSTM(64,64)       & 0.86           & 99             \\\\\nLSTM(64,64,64)    & 0.85           & 99             \\\\\nLSTM(128)         & 0.53  & 71             \\\\\nLSTM(128,128)     & 0.53            & 80             \\\\\nLSTM(128,128,128) & \\textbf{0.52}            & 93             \\\\\nLSTM(256)         & 0.83           & 83             \\\\\nLSTM(256,256)     & 0.82            & 46             \\\\\nLSTM(256,256,256) & 0.82            & 39             \\\\\nLSTM(512)         & 0.81           & 33             \\\\\nLSTM(512,512)     & 0.81           & 31             \\\\\nLSTM(512,512,512) & 0.82           & 25             \\\\\nGPT-2             & 0.92               & 94             \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTable \\ref{tab:benchmarking-models} shows the best results discovered for each LSTM hyperparameter set and the GPT-2 model. Figures \\ref{fig:128-lstms-benchmark} and \\ref{fig:bad-lstms-benchmark} show the epoch-loss training processes for the LSTMs separated for readability purposes and Figure \\ref{fig:gpt-benchmark} shows the same training process for the GPT-2 model. These generalised experiments for all data provide a tuning point for synthetic data to be generated for each of the individuals (given respective personally trained models). LSTMs with 128 hidden units far outperformed the other models, which were also sometimes erratic in terms of their attempt at loss reduction over time. The GPT-2 model is observed to be especially erratic, which is possibly due to its unsupervised attention-based approach. \n\nAlthough some training processes were not as smooth as others, manual exploration showed that acceptable sets of data were able to be produced.\n\n\\subsection{Transfer Learning for Data-scarce Speaker Recognition}\n\n\\begin{table*}[]\n\\centering\n\\caption{Results of the experiments for all subjects. Best models for each Transfer Learning experiment are bold, and the best overall result per-subject is also underlined. Red font denotes a synthetic data-exposed model that scored lower than the classical learning approach. }\n\\label{tab:bigresults}\n\\footnotesize\n\\begin{tabular}{@{}llllllllll@{}}\n\\toprule\n\\multicolumn{1}{c}{}                                   &                                                                                  & \\multicolumn{4}{l}{\\textbf{LSTM}}                                                                                                           & \\multicolumn{4}{l}{\\textbf{GPT-2}}                                                                                                                         \\\\ \\cmidrule(l){3-10} \n\\multicolumn{1}{c}{\\multirow{-2}{*}{\\textbf{Subject}}} & \\multirow{-2}{*}{\\textbf{\\begin{tabular}[c]{@{}l@{}}Synth.\\\\ Data\\end{tabular}}} & \\textit{\\textbf{Acc.}}       & \\textit{\\textbf{F1}}        & \\textit{\\textbf{Prec.}}     & \\multicolumn{1}{l|}{\\textit{\\textbf{Rec.}}}      & \\textit{\\textbf{Acc.}}                & \\textit{\\textbf{F1}}                 & \\textit{\\textbf{Prec.}}              & \\textit{\\textbf{Rec.}}               \\\\ \\midrule\n                                                       & \\textit{\\textbf{0}}                                                              & 93.57                        & 0.94                        & 0.93                        & \\multicolumn{1}{l|}{0.93}                        & 93.57                                 & 0.94                                 & 0.93                                 & 0.93                                 \\\\\n                                                       & \\textit{\\textbf{2500}}                                                           & {\\ul \\textbf{99.5}}          & {\\ul \\textbf{$\\sim$1}}      & {\\ul \\textbf{$\\sim$1}}      & \\multicolumn{1}{l|}{{\\ul \\textbf{$\\sim$1}}}      & 97.32                                 & 0.97                                 & 0.97                                 & 0.97                                 \\\\\n                                                       & \\textit{\\textbf{5000}}                                                           & 97.37                        & 0.97                        & 0.97                        & \\multicolumn{1}{l|}{0.97}                        & \\textbf{97.77}                        & \\textbf{0.98}                        & \\textbf{0.98}                        & \\textbf{0.98}                        \\\\\n                                                       & \\textit{\\textbf{7500}}                                                           & \\textbf{99.33}               & \\textbf{0.99}               & \\textbf{0.99}               & \\multicolumn{1}{l|}{\\textbf{0.99}}               & 99.2                                  & 0.99                                 & 0.99                                 & 0.99                                 \\\\\n\\multirow{-5}{*}{\\textbf{1}}                           & \\textit{\\textbf{10000}}                                                          & 99.1                         & 0.99                        & 0.99                        & \\multicolumn{1}{l|}{0.99}                        & \\textbf{99.3}                         & \\textbf{0.99}                        & \\textbf{0.99}                        & \\textbf{0.99}                        \\\\ \\cmidrule(r){1-1}\n                                                       & \\textit{\\textbf{0}}                                                              & 95.13                        & 0.95                        & 0.95                        & \\multicolumn{1}{l|}{0.95}                        & 95.13                                 & 0.95                                 & 0.95                                 & 0.95                                 \\\\\n                                                       & \\textit{\\textbf{2500}}                                                           & \\textbf{99.6}                & \\textbf{$\\sim$1}            & \\textbf{$\\sim$1}            & \\multicolumn{1}{l|}{\\textbf{$\\sim$1}}            & 99.5                                  & $\\sim$1                              & $\\sim$1                              & $\\sim$1                              \\\\\n                                                       & \\textit{\\textbf{5000}}                                                           & \\textbf{99.5}                & \\textbf{$\\sim$1}            & \\textbf{$\\sim$1}            & \\multicolumn{1}{l|}{\\textbf{$\\sim$1}}            & 99.41                                 & 0.99                                 & 0.99                                 & 0.99                                 \\\\\n                                                       & \\textit{\\textbf{7500}}                                                           & {\\ul \\textbf{99.7}}          & {\\ul \\textbf{$\\sim$1}}      & {\\ul \\textbf{$\\sim$1}}      & \\multicolumn{1}{l|}{{\\ul \\textbf{$\\sim$1}}}      & {\\ul \\textbf{99.7}}                   & {\\ul \\textbf{$\\sim$1}}               & {\\ul \\textbf{$\\sim$1}}               & {\\ul \\textbf{$\\sim$1}}               \\\\\n\\multirow{-5}{*}{\\textbf{2}}                           & \\textit{\\textbf{10000}}                                                          & \\textbf{99.42}               & \\textbf{0.99}               & \\textbf{0.99}               & \\multicolumn{1}{l|}{\\textbf{0.99}}               & 99.38                                 & 0.99                                 & 0.99                                 & 0.99                                 \\\\ \\cmidrule(r){1-1}\n                                                       & \\textit{\\textbf{0}}                                                              & 96.58                        & 0.97                        & 0.97                        & \\multicolumn{1}{l|}{0.97}                        & 96.58                                 & 0.97                                 & 0.97                                 & 0.97                                 \\\\\n                                                       & \\textit{\\textbf{2500}}                                                           & \\textbf{99.2}                & \\textbf{0.99}               & \\textbf{0.99}               & \\multicolumn{1}{l|}{\\textbf{0.99}}               & 98.41                                 & 0.98                                 & 0.98                                 & 0.98                                 \\\\\n                                                       & \\textit{\\textbf{5000}}                                                           & 98.4                         & 0.98                        & 0.98                        & \\multicolumn{1}{l|}{0.98}                        & \\textbf{99}                           & \\textbf{0.99}                        & \\textbf{0.99}                        & \\textbf{0.99}                        \\\\\n                                                       & \\textit{\\textbf{7500}}                                                           & \\textbf{99.07}               & \\textbf{0.99}               & \\textbf{0.99}               & \\multicolumn{1}{l|}{\\textbf{0.99}}               & 98.84                                 & 0.99                                 & 0.99                                 & 0.99                                 \\\\\n\\multirow{-5}{*}{\\textbf{3}}                           & \\textit{\\textbf{10000}}                                                          & 98.44                        & 0.98                        & 0.98                        & \\multicolumn{1}{l|}{0.98}                        & {\\ul \\textbf{99.47}}                  & {\\ul \\textbf{0.99}}                  & {\\ul \\textbf{0.99}}                  & {\\ul \\textbf{0.99}}                  \\\\ \\cmidrule(r){1-1}\n                                                       & \\textit{\\textbf{0}}                                                              & 98.5                         & 0.99                        & 0.99                        & \\multicolumn{1}{l|}{0.99}                        & 98.5                                  & 0.99                                 & 0.99                                 & 0.99                                 \\\\\n                                                       & \\textit{\\textbf{2500}}                                                           & {\\color[HTML]{FE0000} 97.86} & {\\color[HTML]{FE0000} 0.98} & {\\color[HTML]{FE0000} 0.98} & \\multicolumn{1}{l|}{{\\color[HTML]{FE0000} 0.98}} & \\textbf{99.42}                        & \\textbf{0.99}                        & \\textbf{0.99}                        & \\textbf{0.99}                        \\\\\n                                                       & \\textit{\\textbf{5000}}                                                           & \\textbf{99.22}               & \\textbf{0.99}               & \\textbf{0.99}               & \\multicolumn{1}{l|}{\\textbf{0.99}}               & {\\color[HTML]{FE0000} 97.75}          & {\\color[HTML]{FE0000} 0.98}          & {\\color[HTML]{FE0000} 0.98}          & {\\color[HTML]{FE0000} 0.98}          \\\\\n                                                       & \\textit{\\textbf{7500}}                                                           & {\\color[HTML]{FE0000} 97.6}  & {\\color[HTML]{FE0000} 0.98} & {\\color[HTML]{FE0000} 0.98} & \\multicolumn{1}{l|}{{\\color[HTML]{FE0000} 0.98}} & {\\color[HTML]{FE0000} \\textbf{98.15}} & {\\color[HTML]{FE0000} \\textbf{0.98}} & {\\color[HTML]{FE0000} \\textbf{0.98}} & {\\color[HTML]{FE0000} \\textbf{0.98}} \\\\\n\\multirow{-5}{*}{\\textbf{4}}                           & \\textit{\\textbf{10000}}                                                          & 99.22                        & 0.99                        & 0.99                        & \\multicolumn{1}{l|}{0.99}                        & {\\ul \\textbf{99.56}}                  & {\\ul \\textbf{$\\sim$1}}               & {\\ul \\textbf{$\\sim$1}}               & {\\ul \\textbf{$\\sim$1}}               \\\\ \\cmidrule(r){1-1}\n                                                       & \\textit{\\textbf{0}}                                                              & 96.6                         & 0.97                        & 0.97                        & \\multicolumn{1}{l|}{0.97}                        & 96.6                                  & 0.97                                 & 0.97                                 & 0.97                                 \\\\\n                                                       & \\textit{\\textbf{2500}}                                                           & \\textbf{99.47}               & \\textbf{0.99}               & \\textbf{0.99}               & \\multicolumn{1}{l|}{\\textbf{0.99}}               & 99.23                                 & 0.99                                 & 0.99                                 & 0.99                                 \\\\\n                                                       & \\textit{\\textbf{5000}}                                                           & 99.4                         & 0.99                        & 0.99                        & \\multicolumn{1}{l|}{0.99}                        & \\textbf{99.83}                        & \\textbf{$\\sim$1}                     & \\textbf{$\\sim$1}                     & \\textbf{$\\sim$1}                     \\\\\n                                                       & \\textit{\\textbf{7500}}                                                           & 99.2                         & 0.99                        & 0.99                        & \\multicolumn{1}{l|}{0.99}                        & {\\ul \\textbf{99.85}}                  & {\\ul \\textbf{$\\sim$1}}               & {\\ul \\textbf{$\\sim$1}}               & {\\ul \\textbf{$\\sim$1}}               \\\\\n\\multirow{-5}{*}{\\textbf{5}}                           & \\textit{\\textbf{10000}}                                                          & 99.67                        & $\\sim$1                     & $\\sim$1                     & \\multicolumn{1}{l|}{$\\sim$1}                     & \\textbf{99.78}                        & \\textbf{$\\sim$1}                     & \\textbf{$\\sim$1}                     & \\textbf{$\\sim$1}                     \\\\ \\cmidrule(r){1-1}\n                                                       & \\textit{\\textbf{0}}                                                              & 97.3                         & 0.97                        & 0.97                        & \\multicolumn{1}{l|}{0.97}                        & 97.3                                  & 0.97                                 & 0.97                                 & 0.97                                 \\\\\n                                                       & \\textit{\\textbf{2500}}                                                           & \\textbf{99.8}                & \\textbf{$\\sim$1}            & \\textbf{$\\sim$1}            & \\multicolumn{1}{l|}{\\textbf{$\\sim$1}}            & 99.75                                 & $\\sim$1                              & $\\sim$1                              & $\\sim$1                              \\\\\n                                                       & \\textit{\\textbf{5000}}                                                           & 99.75                        & $\\sim$1                     & $\\sim$1                     & \\multicolumn{1}{l|}{$\\sim$1}                     & {\\color[HTML]{FE0000} 96.1}           & {\\color[HTML]{FE0000} 0.96}          & {\\color[HTML]{FE0000} 0.96}          & {\\color[HTML]{FE0000} 0.96}          \\\\\n                                                       & \\textit{\\textbf{7500}}                                                           & {\\color[HTML]{000000} 97.63} & {\\color[HTML]{000000} 0.98} & {\\color[HTML]{000000} 0.98} & \\multicolumn{1}{l|}{{\\color[HTML]{000000} 0.98}} & {\\ul \\textbf{99.82}}                  & {\\ul \\textbf{$\\sim$1}}               & {\\ul \\textbf{$\\sim$1}}               & {\\ul \\textbf{$\\sim$1}}               \\\\\n\\multirow{-5}{*}{\\textit{\\textbf{6}}}                  & \\textit{\\textbf{10000}}                                                          & 99.67                        & $\\sim$1                     & $\\sim$1                     & \\multicolumn{1}{l|}{$\\sim$1}                     & \\textbf{99.73}                        & \\textbf{$\\sim$1}                     & \\textbf{$\\sim$1}                     & \\textbf{$\\sim$1}                     \\\\ \\cmidrule(r){1-1}\n                                                       & \\textit{\\textbf{0}}                                                              & 90.7                         & 0.91                        & 0.91                        & \\multicolumn{1}{l|}{0.91}                        & 90.7                                  & 0.91                                 & 0.91                                 & 0.91                                 \\\\\n                                                       & \\textit{\\textbf{2500}}                                                           & \\textbf{99.86}               & \\textbf{$\\sim$1}            & \\textbf{$\\sim$1}            & \\multicolumn{1}{l|}{\\textbf{$\\sim$1}}            & 99.78                                 & $\\sim$1                              & $\\sim$1                              & $\\sim$1                              \\\\\n                                                       & \\textit{\\textbf{5000}}                                                           & \\textbf{99.89}               & \\textbf{$\\sim$1}            & \\textbf{$\\sim$1}            & \\multicolumn{1}{l|}{\\textbf{$\\sim$1}}            & 99.86                                 & $\\sim$1                              & $\\sim$1                              & $\\sim$1                              \\\\\n                                                       & \\textit{\\textbf{7500}}                                                           & \\textbf{99.91}               & \\textbf{$\\sim$1}            & \\textbf{$\\sim$1}            & \\multicolumn{1}{l|}{\\textbf{$\\sim$1}}            & 99.84                                 & $\\sim$1                              & $\\sim$1                              & $\\sim$1                              \\\\\n\\multirow{-5}{*}{\\textit{\\textbf{7}}}                  & \\textit{\\textbf{10000}}                                                          & {\\ul \\textbf{99.94}}         & {\\ul \\textbf{$\\sim$1}}      & {\\ul \\textbf{$\\sim$1}}      & \\multicolumn{1}{l|}{{\\ul \\textbf{$\\sim$1}}}      & 99.73                                 & $\\sim$1                              & $\\sim$1                              & $\\sim$1                              \\\\ \\midrule\n\\textit{\\textbf{Avg.}}                                 &                                                                                  & 98.43                        & 0.98                        & 0.98                        & 0.98                                             & 98.40                                 & 0.98                                 & 0.98                                 & 0.98                                 \\\\ \\bottomrule\n\\end{tabular}\n\\end{table*}\n\nTable \\ref{tab:bigresults} shows all of the results for each subject, both with and without exposure to synthetic data. Per-run, the LSTM achieved better results over the GPT-2 in 14 instances whereas the GPT-2 achieved better results over the LSTM in 13 instances. Of the five runs that scored lower than no synthetic data exposure, two were LSTM and three were GPT-2. Otherwise, 51 of the 56 experiments all outperformed the original model without synthetic data exposure and every single subject experienced their best classification result in all cases when the model had been exposed to synthetic data. The best score on a per-subject basis was achieved by exposing the network to data produced by the LSTM three times and the GPT-2 five times (both including Subject 2 where both were best at 99.7\\%). The maximum diversion of training accuracy to validation accuracy was $\\sim1\\%$ showing that although high results were attained, overfitting was relatively low; with more computational resources, k-fold and LOO cross validation are suggested as future works to achieve more accurate measures of variance within classification. \n\nThese results attained show that speaker classification can be improved by exposing the network to synthetic data produced by both supervised and attention-based models and then transferring the weights to the initial problem, which most often scores lower without synthetic data exposure in all cases but five although those subjects still experienced their absolute best result through synthetic data exposure regardless. \n\n\n\\subsection{Comparison to other methods of speaker recognition}\n\\begin{table}[]\n\\caption{Comparison of the best models found in this work and other classical methods of speaker recognition (sorted by accuracy)}\n\\label{tab:bigcomparison}\n\\footnotesize\n\\begin{tabular}{@{}clllll@{}}\n\\toprule\n\\multicolumn{1}{l}{\\textbf{Subject}} & \\textbf{Model}                         & \\textbf{Acc.}  & \\textbf{F-1} & \\textbf{Prec.} & \\textbf{Rec.} \\\\ \\midrule\n\\multirow{7}{*}{\\textbf{1}}          & \\textit{\\textbf{DNN (LSTM TL 2500)}}   & \\textbf{99.5}  & \\textbf{$\\sim$1}   & \\textbf{$\\sim$1}     & \\textbf{$\\sim$1}    \\\\\n                                     & \\textit{\\textbf{DNN (GPT-2 TL 5000)}}  & 97.77          & 0.98         & 0.98           & 0.98          \\\\\n                                     & \\textit{\\textbf{SMO}}                  & 97.71          & 0.98         & 0.95           & 0.95          \\\\\n                                     & \\textit{\\textbf{Random Forest}}        & 97.48          & 0.97         & 0.97           & 0.97          \\\\\n                                     & \\textit{\\textbf{Logistic Regression}}  & 97.47          & 0.97         & 0.97           & 0.97          \\\\\n                                     & \\textit{\\textbf{Bayesian Network}}     & 82.3           & 0.87         & 0.96           & 0.82          \\\\\n                                     & \\textit{\\textbf{Naive Bayes}}          & 78.96          & 0.84         & 0.953          & 0.77          \\\\ \\cmidrule(r){1-1}\n\\multirow{7}{*}{\\textbf{2}}          & \\textit{\\textbf{DNN (LSTM TL 7500)}}   & \\textbf{99.7}  & \\textbf{$\\sim$1}   & \\textbf{$\\sim$1}     & \\textbf{$\\sim$1}    \\\\\n                                     & \\textit{\\textbf{DNN (GPT-2 TL 7500)}}  & 99.7           & $\\sim$1            & $\\sim$1              & $\\sim$1             \\\\\n                                     & \\textit{\\textbf{SMO}}                  & 98.94          & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{Logistic Regression}}  & 98.33          & 0.98         & 0.98           & 0.98          \\\\\n                                     & \\textit{\\textbf{Random Forest}}        & 98.28          & 0.98         & 0.98           & 0.98          \\\\\n                                     & \\textit{\\textbf{Bayesian Network}}     & 84.9           & 0.9          & 0.97           & 0.85          \\\\\n                                     & \\textit{\\textbf{Naive Bayes}}          & 76.58          & 0.85         & 0.97           & 0.77          \\\\ \\cmidrule(r){1-1}\n\\multirow{7}{*}{\\textbf{3}}          & \\textit{\\textbf{DNN (GPT-2 TL 10000)}} & \\textbf{99.47} & \\textbf{0.99}   & \\textbf{0.99}     & \\textbf{0.99}    \\\\\n                                     & \\textit{\\textbf{DNN (LSTM TL 2500)}}   & 99.2           & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{SMO}}                  & 99.15          & 0.99         & 0.99           & 0.98          \\\\\n                                     & \\textit{\\textbf{Logistic Regression}}  & 98.85          & 0.99         & 0.99           & 0.98          \\\\\n                                     & \\textit{\\textbf{Random Forest}}        & 98.79          & 0.99         & 0.99           & 0.98          \\\\\n                                     & \\textit{\\textbf{Bayesian Network}}     & 91.49          & 0.94         & 0.98           & 0.92          \\\\\n                                     & \\textit{\\textbf{Naive Bayes}}          & 74.37          & 0.83         & 0.96           & 0.74          \\\\ \\cmidrule(r){1-1}\n\\multirow{7}{*}{\\textbf{4}}          & \\textit{\\textbf{DNN (GPT-2 TL 10000)}} & \\textbf{99.56} & \\textbf{$\\sim$1}   & \\textbf{$\\sim$1}     & \\textbf{$\\sim$1}    \\\\\n                                     & \\textit{\\textbf{DNN (LSTM TL 5000)}}   & 99.22          & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{Logistic Regression}}  & 98.66          & 0.99         & 0.98           & 0.98          \\\\\n                                     & \\textit{\\textbf{SMO}}                  & 98.66          & 0.99         & 0.98           & 0.98          \\\\\n                                     & \\textit{\\textbf{Random Forest}}        & 98             & 0.98         & 0.98           & 0.98          \\\\\n                                     & \\textit{\\textbf{Bayesian Network}}     & 95.53          & 0.96         & 0.98           & 0.96          \\\\\n                                     & \\textit{\\textbf{Naive Bayes}}          & 88.74          & 0.92         & 0.97           & 0.89          \\\\ \\cmidrule(r){1-1}\n\\multirow{7}{*}{\\textbf{5}}          & \\textit{\\textbf{DNN (GPT-2 TL 10000)}} & \\textbf{99.85} & \\textbf{$\\sim$1}   & \\textbf{$\\sim$1}     & \\textbf{$\\sim$1}    \\\\\n                                     & \\textit{\\textbf{DNN (LSTM TL 10000)}}  & 99.67          & 1            & 1              & 1             \\\\\n                                     & \\textit{\\textbf{Logistic Regression}}  & 98.86          & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{Random Forest}}        & 98.7           & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{SMO}}                  & 98.6           & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{Naive Bayes}}          & 90.55          & 0.94         & 0.98           & 0.9           \\\\\n                                     & \\textit{\\textbf{Bayesian Network}}     & 88.95          & 0.93         & 0.98           & 0.89          \\\\ \\cmidrule(r){1-1}\n\\multirow{7}{*}{\\textbf{6}}          & \\textit{\\textbf{DNN (GPT-2 TL 7500)}}  & \\textbf{99.82} & \\textbf{$\\sim$1}   & \\textbf{$\\sim$1}     & \\textbf{$\\sim$1}    \\\\\n                                     & \\textit{\\textbf{DNN (LSTM TL 2500)}}   & 99.8           & $\\sim$1            & $\\sim$1              & $\\sim$1             \\\\\n                                     & \\textit{\\textbf{Logistic Regression}}  & 99.1           & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{Random Forest}}        & 98.9           & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{SMO}}                  & 98.86          & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{Naive Bayes}}          & 90.52          & 0.94         & 0.98           & 0.9           \\\\\n                                     & \\textit{\\textbf{Bayesian Network}}     & 89.27          & 0.93         & 0.98           & 0.89          \\\\ \\cmidrule(r){1-1}\n\\multirow{7}{*}{\\textbf{7}}          & \\textit{\\textbf{DNN (LSTM TL 10000)}}  & \\textbf{99.91} & \\textbf{$\\sim$1}   & \\textbf{$\\sim$1}     & \\textbf{$\\sim$1}    \\\\\n                                     & \\textit{\\textbf{DNN (GPT-2 TL 5000)}}  & 99.86          & $\\sim$1            & $\\sim$1              & $\\sim$1             \\\\\n                                     & \\textit{\\textbf{SMO}}                  & 99.4           & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{Logistic Regression}}  & 99.13          & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{Random Forest}}        & 99             & 0.99         & 0.99           & 0.99          \\\\\n                                     & \\textit{\\textbf{Bayesian Network}}     & 88.67          & 0.93         & 0.98           & 0.89          \\\\\n                                     & \\textit{\\textbf{Naive Bayes}}          & 86.9           & 0.91         & 0.98           & 0.87          \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[]\n\\caption{Average performance of the chosen models for each of the 7 subjects.}\n\\label{tab:average-comparison}\n\\centering\n\\begin{tabular}{@{}lllll@{}}\n\\toprule\n\\textbf{Model}                        & \\textbf{Avg acc} & \\textbf{F-1} & \\textbf{Prec.} & \\textbf{Rec.} \\\\ \\midrule\n\\textit{\\textbf{DNN (LSTM TL)}}       & 99.57            & $\\sim$1      & $\\sim$1        & $\\sim$1       \\\\\n\\textit{\\textbf{DNN (GPT-2 TL)}}      & 99.43            & $\\sim$1      & $\\sim$1        & $\\sim$1       \\\\\n\\textit{\\textbf{SMO}}                 & 98.76            & 0.99         & 0.98           & 0.98          \\\\\n\\textit{\\textbf{Logistic Regression}} & 98.63            & 0.99         & 0.98           & 0.98          \\\\\n\\textit{\\textbf{Random Forest}}       & 98.45            & 0.98         & 0.98           & 0.98          \\\\\n\\textit{\\textbf{Bayesian Network}}    & 88.73            & 0.92         & 0.98           & 0.89          \\\\\n\\textit{\\textbf{Naive Bayes}}         & 83.80            & 0.89         & 0.97           & 0.83          \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTable \\ref{tab:bigcomparison} shows a comparison of the models proposed in this paper to other state-of-the-art methods of speaker recognition. Namely, they are Sequential Minimal Optimisation (SMO), Logistic Regression, Bayesian Networks, and Naive Bayes. It can be observed that, although in some cases close, the DNN fine tuned from synthetic data generated by both the LSTM and GPT-2 achieve higher scores than other methods. Finally, Table \\ref{tab:average-comparison} shows the average scores for the chosen models for each of the seven subjects. \n\n\n\n\n\n\\section{Conclusion and Future Work}\n\\label{sec4:futureworkconclusion}\nTo finally conclude, this work found strong success for all 7 subjects when improving the classification problem of speaker recognition by generating augmented data by both LSTM and OpenAI GPT-2 models. Future work aims to solidify this hypothesis by running the experiments for a large range of subjects and comparing the patterns that emerge from the results. \n\nThe experiments in this work have provided a strong argument for the usage of deep neural network transfer learning from MFCCs synthesised by both LSTM and GPT-2 models for the problem of speaker recognition. One of the limitations of this study was hardware availability since it was focused on those available to consumers today. The Flickr8k dataset was thus limited to 8,000 data objects and new datasets created, which prevents a direct comparison to other speaker recognition works which often operate on larger data and with hardware beyond consumer availability. It is worth noting that the complex nature of training LSTM and GPT-2 models to generate MFCCs is beyond that of the task of speaker recognition itself, and as such, devices with access to TPU or CUDA-based hardware must perform the task in the background over time. The tasks in question took several minutes with the two GPUs used in this work for both LSTM and GPT-2 and as such are not instantaneous. As previously mentioned, although it was observed that overfitting did not occur too strongly, it would be useful in future to perform similar experiments with either K-fold or leave-one-out Cross Validation in order to achieve even more accurate representations of the classification metrics. In terms of future application, the then-optimised model would be the implemented within real robots and smarthome assistants through compatible software.\n\nIn this work, seven subjects were benchmarked with both a tuned LSTM and OpenAI's GPT-2 model. In future, as was seen with related works, a GAN could also be implemented in order to provide a third possible solution to the problem of data scarcity in speaker recognition as well as other related speech recognition classification problems - bias is an open issue that has been noted for data augmentation with GANs~\\cite{hu2019exploring}, and as such, this issue must be studied if a GAN is implemented for problems of this nature. This work further argued for the hypothesis presented, that is, data augmentation can aid in improving speaker recognition for scarce datasets. Following the 14 successful runs including LSTMs and GPT-2s, the overall process followed by this experiment could be scripted and thus completely automated, allowing for the benchmarking of many more subjects to give a much more generalised set of results, of which will be more representative of the general population. Additionally, samples spoken from many languages should also be considered in order to provide language generalisation, rather than just the English language spoken by multiple international dialects in this study. Should generalisation be possible, future models may require only a small amount of fine-tuning to produce synthetic speech for a given person rather than training from scratch as was performed in this work. In more general lines of thought, literature review shows that much of the prominent work is recent, leaving many fields of machine learning that have not yet been attempted to be improved via the methods described in this work. \n\n\\bibliographystyle{ieeetr}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe chromatic number $\\chi(G)$ of a graph $G=(V,E)$ is the minimal cardinal $\\varkappa$ for which there exists a vertex coloring with $\\varkappa$ colors. There is a long history of structure theorems deriving from large chromatic number assumptions, see e.g., \\cite{komjath}. The main topic of this paper will be the following conjecture proposed by Erd\\\"os-Hajnal-Shelah \\cite[Problem 2]{EHS} and Taylor \\cite[Problem 43, page 508]{Taylorprob43}.\n\\begin{conjecture}[Strong Taylor's Conjecture]\nFor any graph $G$ with $\\chi(G)>\\aleph_0$ there exists an $n\\in\\mathbb{N}$ such that $G$ contains all finite subgraphs of $\\text{Sh}_n(\\omega)$.\n\\end{conjecture}\nWhere, for a caridnal $\\kappa$, the shift graph $\\text{Sh}_n(\\kappa)$ is the graph whose vertices are increasing $n$-tuples of ordinals less than $\\kappa$, and we put an edge between $s$ and $t$ if for every $1\\leq i\\leq n-1$, $s(i)=t(i-1)$ or vice-versa. The shift graphs $\\text{Sh}_n(\\kappa)$ have large chromatic numbers depending on $\\kappa$, see Fact \\ref{F:Sh-high chrom} below. Consequently, if the strong Taylor's conjecture holds for a graph $G$, it has elementary extensions of unbounded chromatic number (having the same family of finite subgraphs). \n\nThe strong Taylor's conjecture was refuted in \\cite[Theorem 4]{HK}. See \\cite{komjath} and the introduction of \\cite{1196} for more historical information.\n\nIn \\cite{1196} we initiated the study of variants of the strong Taylor's conjecture for some classes of graphs with \\emph{stable} first order theory (stable graphs). Stablility theory, which is the study of stable theories and originated in the works of the third author in the 60s and 70s, is one of the most influential and important subjects in modern model theory. Examples of stable theories include abelian groups, modules, algebraically closed fields, graph theoretic trees, or more generally superflat graphs \\cite{PZ}. Stablility also had an impact in combinatorics, e.g. \\cite{MS} and \\cite{CPT} to name a few.\n\nMore precisely, in \\cite{1196} we proved the strong Taylor's conjecture for $\\omega$-stable graphs and variants of the conjecture for superstable graphs (replacing $\\aleph_0$ by $2^{\\aleph_0}$) and for stable graphs which are interpretable in a stationary stable theory (replacing $\\aleph_0$ by $\\beth_2(\\aleph_0)$). As there exist stable graphs that are not interpretable in a stationary stable structure, see \\cite[Proposition 5.22, Remark 5.23]{1196}, we asked what is the situation in general stable graphs and in this paper we answer it with the following theorem.\n\n\\begin{theorem*}[Corollary \\ref{C:main corollary}]\nLet $G=(V,E)$ be a stable graph. If $\\chi(G)>\\beth_2(\\aleph_0)$ then $G$ contains all finite subgraphs of $\\text{Sh}_n(\\omega)$ for some $n\\in \\mathbb{N}$.\n\\end{theorem*}\n\nThe key tool in proving the results for $\\omega$-stable graphs and superstable graphs is that every large enough saturated model is an Ehrenfeucht\u2013Mostowski model (EM-model) in some bounded expansion of the language. \n\nAn EM-model is a model which is the definable closure of an indiscernible sequence and was originally used by Ehrenfeucht\u2013Mostowski in order to find models with many automorphisms \\cite{EM}. It was shown by Lascar \\cite[Section 5.1]{lascar} that every saturated model of cardinality $\\aleph_1$ in an $\\omega$-stable theory is an EM-model in some countable expansion of the language, and was later generalized to any cardinality by Mariou \\cite[Theorem C]{mariou}. This was  generalized for superstable theories by Mariou \\cite[Theorem 3.B]{mariouthesis} and in an unpublished preprint by the third author \\cite{Sh:1151}.\n\nIt was shown by Mariou \\cite[Theorem 3.A]{mariouthesis} that in a certain sense the existence of such saturated EM-models for a stable theory necessarily implies that the theory is superstable. Consequently a different tool is needed in order to prove the theorem for general stable theories. \n\nIn the stationary stable case, we use a variant of representations of structures in the sense of \\cite{919}. However, this method did not seem to easily adjust to the general stable case.\n\nIn this paper we resolved this problem by generalizing the notion of EM-models to \\emph{infinitary EM-models} and show that such saturated models exist for any stable theory in Theorem \\ref{T:existence of gen em model in stable}. The definition is a bit technical, so here we will settle with an informal description: \n\nIn an EM-model every element is given by a term and a finite sequence of elements from the generating indiscernible sequence. Analogously, in an infinitary EM-model every element is given by some ``term'' with infinite (but bounded) arity and a suitable sequence of elements from an indiscernible sequence. \n\nWe prove that the existence of saturated infinitary EM-models characterizes stability.\n\n\\begin{theorem*}[Theorem \\ref{T:existence of gen em model in stable}]\nThe following are equivalent for a complete $\\mathcal{L}$-theory $T$:\n\\begin{enumerate}\n\\item $T$ is stable.\n\n\\item Let $\\kappa,\\mu$ and $\\lambda$ be cardinals satisfying $\\kappa=\\cf(\\kappa)\\geq \\kappa(T)+\\aleph_1$, $\\mu^{<\\kappa}=\\mu\\geq 2^{\\kappa+|T|}$ and $\\lambda=\\lambda^{<\\kappa}\\geq \\mu$ and let $T\\subseteq T^{sk}$ be an expansion with definable Skolem functions such that $|T|=|T^{sk}|$ in a language $\\mathcal{L}\\subseteq \\mathcal{L}^{sk}$. Then there exists an infinitary EM-model  $M^{sk}\\models T^{sk}$ based on $(\\alpha,\\lambda)$, where $\\alpha\\in \\kappa^U$ for some set $U$ of cardinality at most $\\mu$, such that $M=M^{sk}\\restriction \\mathcal{L}$ is saturated of cardinality $\\lambda$.\n\\end{enumerate}\n\\end{theorem*}\n\nSection \\ref{s:em-models} is the only purely model theoretic section and is the only place where stability is used. The results of this section (more specifically Theorem \\ref{T:existence of gen em model in stable}) are only used in Section \\ref{s:conclusion}. In Section \\ref{S:order type graphs} we study graphs on (perhaps infinite) increasing sequences whose edge relation is determined by the order type. Aiming to prove that if the chromatic number is large, then one can embed shift graphs, we analyze several different cases. The last case we deal with in Section \\ref{S:order type graphs} turns out to be rather complicated, so we devote all of Section \\ref{S:PCF} to it. There, we employ ideas inspired by PCF theory to get a coloring of small cardinality. Section \\ref{s:conclusion} concludes. \n\n\n\\section{Preliminaries}\nWe use small latin letters $a,b,c$ for tuples and capital letters $A,B,C$ for sets. We also employ the standard model theoretic abuse of notation and write $a\\in A$ even for tuples when the length of the tuple is immaterial or understood from context.\n\nFor any two sets $A$ and $J$, let $A^{\\underline{J}}$ be the set of injective functions from $J$ to $A$ (where the notation is taken from the falling factorial notation), and if $(A,<)$ and $(J,<)$  are both linearly ordered sets, let $(A^{\\underline J})_<$ be the subset of $A^{\\underline J}$ consisting of strictly increasing functions. If we want to emphasize the order on $J$ we will write $(A^{\\underline{(J,<)}})_<$.  \n\nThroughout this paper, we interchangeably use sequence notation and function notation for elements of $A^{\\underline J}$, e.g. for $f\\in A^{\\underline J}$, $f(i)=f_i$. For any sequence $\\eta$ we denote by $\\Rg(\\eta)$ the underlying set of the sequence (i.e. its image). If $(A,<^A)$ and $(B,<^B)$ are linearly ordered sets, then the most significant coordinate of the lexicographic order on $A\\times B$ is the left one.\n\n\\subsection{Stability}\nWe use fairly standard model theoretic terminology and notation, see for example \\cite{TZ,guidetonip}. We gather some of the needed notions. For stability, the reader can also consult with \\cite{classification}.\n\nWe denote by $\\tp(a\/A)$ the complete type of $a$ over $A$. Let $(I,<)$ be a linearly ordered set. A sequence $\\langle a_i: i\\in I\\rangle$ inside a first order structure is \\emph{indiscernible} if for any $i_1<\\dots<i_k$ and $j_1<\\dots<j_k$ in $I$,\n\\[\\tp(a_{i_1},\\dots,a_{i_k})=\\tp(a_{j_1},\\dots, a_{j_k}).\\]\nA structure $M$ is \\emph{$\\kappa$-saturated}, for a cardinal $\\kappa$, if any type $p$ over $A$ with $|A|< \\kappa$ is realized in $M$. The structure $M$ is \\emph{saturated} if it is $|M|$-saturated.\nA \\emph{monster model} for $T$, usually denoted by $\\mathbb{U}$, is a large saturated model containing all sets and models (as elementary substructures) we will encounter\\footnote{There are set theoretic issues in assuming that such a model exists, but these are overcome by standard techniques from set theory that ensure the generalized continuum hypothesis from some point on while fixing a fragment of the universe. The reader can just accept this or alternatively assume that $\\mathbb{U}$ is merely $\\kappa$-saturated and $\\kappa$-strongly homogeneous for large enough $\\kappa$.}.  All subsets and models will be \\emph{small}, i.e. of cardinality $<|\\mathbb{U}|$.\n\nA first theory $T$ is \\emph{stable} if there does not exist a model $M\\models T$, a formula $\\varphi(x,y)$ and elements $\\langle a_i\\in M: i<\\omega\\rangle$ such that $M\\models \\varphi(a_i,a_j)\\iff i<j$. An equivalent definition is that there exists some $\\kappa\\geq |T|$ such that for all $M\\models T$ with $|M|\\leq\\kappa$ the cardinality of complete types over $M$ is at most $\\kappa$. For any such $\\kappa$, $T$ has a saturated model of cardinality of $\\kappa$ \\cite[Theorem VIII.4.7]{classification}. \n\nEvery indiscernible sequence in a stable theory is \\emph{totally indiscernible}, i.e. in the notation above, for any $i_1,\\dots,i_k$ and $j_1,\\dots,j_k$ in $I$,\n\\[\\tp(a_{i_1},\\dots,a_{i_k})=\\tp(a_{j_1},\\dots, a_{j_k}).\\]\nOther than these notions we will also require basic understanding in forking. See the above references for more information.\n\n\\subsection{Graph theory}\nHere we gather some facts on graphs and the chromatic number of graphs (all can be found in \\cite{1196}).\n\nBy a \\emph{graph} we mean a pair $G=(V,E)$ where $E\\subseteq V^2$ is symmetric and irreflexive. A \\emph{graph homomorphism} between $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ is a map $f:V_1\\to V_2$ such that $f(e)\\in E_2$ for every $e\\in E_1$. If $f$ is injective we will say that $f$ embeds $G_1$ into $G_2$ a subgraph. If in addition we require that $f(e)\\in E_2$ if and only if $e\\in E_1$ we will say that $f$ embeds $G_1$ into $G_2$ as an induced subgraph.\n\n\n\\begin{definition}\nLet $G=(V,E)$ be a graph.\n\\begin{enumerate}\n\\item For a cardinal $\\varkappa$, a \\emph{vertex coloring} (or just coloring) of cardinality $\\varkappa$ is a function $c:V\\to \\varkappa$ such that $x\\mathrel{E} y$ implies $c(x)\\neq c(y)$ for all $x,y\\in V$.\n\\item The \\emph{chromatic number} $\\chi(G)$ is the minimal cardinality of a vertex coloring of $G$.\n\\end{enumerate}\n\\end{definition}\n\nThese are the basic properties of $\\chi(G)$ that we will require.\n\n\\begin{fact}\\cite[Lemma 2.3]{1196}\\label{F:basic prop}\nLet $G=(V,\\mathrel{E})$ be a graph.\n\\begin{enumerate}\n\n\\item If $V=\\bigcup_{i\\in I} V_i$ then $\\chi(G)\\leq \\sum_{i\\in I} \\chi(V_i, E\\restriction V_i)$.\n\\item If $E=\\bigcup_{i\\in I} \\mathrel{E}_i$ (with the $E_i$ being symmetric)  then $\\chi(G)\\leq \\prod_{i\\in I}\\chi(V,E_i)$.\n\\item If $\\varphi:H\\to G$ is a graph homomorphism then $\\chi(H)\\leq \\chi(G)$.\n\\item If $\\varphi:(H,E^H)\\to (G,E^G)$ is a surjective graph homomorphism with $e\\in E^H\\iff \\varphi(e)\\in E^G$ then $\\chi(H)=\\chi(G)$.\n\\end{enumerate}\n\\end{fact}\n\n\\begin{example} For any finite number $1\\leq r$ and any linearly ordered set $(A,<)$, let $\\text{Sh}_r(A)$, or $\\text{Sh}_r(A,<)$ if we want to emphasize the order, (\\emph{the shift graph on $A$}) be the following graph: its set of vertices is the set $(A^{\\underline r})_<$ of increasing $r$-tuples, $s_0,\\dots,s_{r-1}$, and we put an edge between $s$ and $t$ if for every $1\\leq i\\leq r-1$, $s(i)=t(i-1)$, or vice-versa. It is an easy exercise to show that $\\text{Sh}_r(A)$ is a connected graph. If $r=1$ this gives $K_{A}$, the complete graph on $A$.\n\\end{example}\n\n\n\\begin{fact}\\cite[Fact 2.6]{1196}\\cite[Proof of Theorem 2]{EH-shift}\\label{F:Sh-high chrom}\nLet $2\\leq r<\\omega$ be a natural number and $\\varkappa$ be a cardinal, \\[\\chi\\left(\\text{Sh}_r(\\beth_{r-1}\\left(\\varkappa\\right)^{+})\\right)\\geq\\varkappa^{+}.\\]\n\\end{fact}\n\nFinally, the following fact is a very useful tool.\n\\begin{fact}\\cite[Proposition 3.2]{1196}\\label{F:homomorphism is enough}\nLet $G=(V,E)$ be a graph and assume there exists an homomorphism of graphs $t:\\text{Sh}_k(\\omega)\\to G$. Then there exists $n\\leq k$, such that \n\\begin{itemize}\n\\item[($\\dagger$)] $G$ contains all finite subgraphs of $\\text{Sh}_n(\\omega)$.\n\\end{itemize}\n\nConsequently, if $H$ is a graph that contains all finite subgraphs of $\\text{Sh}_k(\\omega)$, for some $k$, and $t:H\\to G$ is a homomorphism of graphs, then there exists some $n\\leq k$ such that $G$ satisfies ($\\dagger$).\n\\end{fact}\n\n\n\n\\section{Infinitary EM-models and stability}\\label{s:em-models}\nLet $T$ be a first order theory and $\\mathbb{U}$ a monster model for $T$.\n\nAn \\emph{EM-Model} for $T$ is a model that is the definable closure of an indiscernible sequence (possibly in some expansion of the theory which admits Skolem functions).\n\nEvery element in an EM-model is of the form $t(a_{i_1},\\dots, a_{i_n})$, where $t$ is a term (in the expanded language) and $a_{i_1},\\dots, a_{i_n}$ are elements of the indiscernible sequence. In other words, to any element we may associate a pair $(i,\\eta)$, where $i<|T|$ (this codes the term $t_i(\\bar x_i)$) and $\\eta$ is an increasing sequence of cardinality $|\\bar x_i|$. \n\nMariou \\cite{mariouthesis,mariou} and Shelah \\cite{Sh:1151} proved that if $T$ is $\\omega$-stable or even superstable then it has an EM-model in some expansion of the language whose restriction to the original language is saturated. For general stable theories, as we will see in this section, one needs to allow ``terms'' with, possibly, infinite arity to get a parallel result.\n \nLet $\\kappa\\geq \\aleph_0$ be a regular cardinal (which we think of as a bound on the arity) and let $\\mu$ be a cardinal (which we think of as a bound on the number of terms). Let $\\alpha\\in \\kappa^\\mu$ be a function assigning to each function its arity.\n\n\n\\begin{definition}\\label{D:alpha-I}\nLet $\\kappa$ a cardinal, $(I,<)$ a linearly ordered set, $U$ a set and $\\alpha\\in \\kappa^U$.\nLet $a=\\langle a_{i,\\eta}:i\\in U ,\\,\\eta\\in (I^{\\underline{\\alpha_i}})_<\\rangle$ be a sequence of tuples from $\\mathbb{U}$.\n\nWe say that $a$ is \\emph{$(\\alpha,I)$-indiscernible} if for every  $\\langle i_j\\in U:j<k\\rangle$,  $\\langle \\eta_j\\in (I^{\\underline{\\alpha_{i_j}}})_<:j<k\\rangle$ and $\\langle \\rho_j\\in (I^{\\underline{\\alpha_{i_j}}})_<:j<k\\rangle$ if\nthere exists a partial isomorphism of $(I,<)$ mapping $\\langle \\eta_j:j<k\\rangle$ to $\\langle \\rho_j:j<k\\rangle$ then $\\langle a_{i_j,\\eta_j}:j<k\\rangle$ and $\\langle a_{i_j,\\rho_j}:j<k\\rangle$ have the same type.\n\\end{definition}\n\nRecall that given a subset $A\\subseteq \\mathbb{U}$ and an ultrafilter $\\mathcal{D}$ on $A$ we may define the global average type $p_\\mathcal{D}=\\mathrm{Av}(\\mathcal{D},\\mathbb{U})$ by\n\\[p_\\mathcal{D}\\vdash \\varphi(x,b)\\iff \\varphi(A,b)\\in \\mathcal{D}.\\] Obviously, $p_\\mathcal{D}$ is finitely satisfiable in $A$. \n\\begin{remark}\\label{R:global-ultrafilter}\nIf $\\mathcal{D}$ is an ultrafilter on $A$ and $A\\subseteq B$ then $\\{U\\in B: \\exists V\\in \\mathcal{D} (V\\subseteq U)\\}$ is the a unique ultrafilter $\\mathcal{D}^\\prime$ on $B$ containing $\\mathcal{D}$ and $p_{\\mathcal{D}}=p_{\\mathcal{D}^\\prime}$. \n\n\\end{remark}\n\nFor any linearly ordered set $(I,<)$ and $A\\subseteq B$, we say that $\\langle a_i:i\\in I\\rangle$ realizes $(p_{\\mathcal{D}})^{\\otimes I}|B$ if for any $k\\in I$, $a_k\\models p_{\\mathcal{D}}|B\\langle a_i :i<k\\in I\\rangle$.\n\n\\begin{proposition}\\label{P:existence-of-indisc}\nAssume that $\\mathbb{U}$ has definable Skolem functions. Let $\\kappa\\geq \\aleph_0$ a regular cardinal and $\\mu^{<\\kappa}=\\mu\\geq 2^{\\kappa+|\\mathcal{L}|}$ a cardinal.\n\nLet $\\alpha\\in \\kappa^\\mu$ be any function and let $(I,<)$ be any infinite linear order. \n\\begin{enumerate}\n\\item  There exist $U\\subseteq \\mu\\times \\mu$ and an $(\\alpha^\\prime,I)$-indiscernible sequence \\[a=\\langle a_{i,k,\\eta}: (i,k)\\in U,\\, \\eta\\in (I^{\\underline{\\alpha_i}})_<\\rangle,\\] where $\\alpha^\\prime\\in\\kappa^U$ is defined by $\\alpha^\\prime_{(i,k)}=\\alpha_i$, for $(i,k)\\in U$, such that $\\mathbb{U}\\restriction \\dcl(\\Rg (a))\\prec \\mathbb{U}$.\n\n\\item For $j<\\mu$ and $\\eta\\in (I^{\\underline{\\alpha_j}})_<$, if $A\\subseteq F_{j,\\eta}=\\dcl (\\{a_{i,k,\\nu}: (i,k)\\in U,\\, i<j,\\, \\nu\\in (\\Rg(\\eta)^{\\underline{\\alpha_i}})_<\\})$ with $|A|<\\kappa$ and non-algebraic $p\\in S(A)$ then there exists $k<\\mu$ with $(j,k)\\in U$ such that $a_{j,k,\\eta}\\models p$. Moreover, if $p$ is finitely satisfiable in $A$ then so is $\\tp(a_{j,k,\\eta}\/F_{j,\\eta})$. \n\n\\item  If in addition, $(I,<)$ is well-ordered and $\\alpha$ satisfies $\\alpha_i=(i\\mod \\kappa)$ then \n\\begin{enumerate}\n\\item for any $A\\subseteq \\dcl(\\Rg(a))$ with $|A|<\\kappa$ there exist $i<\\mu$ and $\\eta\\in (I^{\\underline{\\alpha_i}})_<$ satisfying $A\\subseteq F_{i,\\eta}$;\n\\item $\\dcl(\\Rg(a))$ is $\\kappa$-saturated.\n\\item Assume that $(I,<)$ is a cardinal with $\\cf(I)\\geq \\kappa$. For any infinite $A\\subseteq B \\subseteq \\dcl(\\Rg(a))$, with $|B|<\\kappa$, there is a non principal ultrafilter $\\mathcal{D}$ on $A$ such that $(p_{\\mathcal{D}})^{\\otimes I}|B$ is realized in $\\dcl(\\Rg(a))$. \n\n\\end{enumerate}\n\\end{enumerate}\n\n\n\n\\end{proposition}\n\\begin{proof}\nSince $\\mathbb{U}$ has definable Skolem functions, for any $A\\subseteq \\mathbb{U}$, $\\dcl(A)$ is an elementary substructure of $\\mathbb{U}$.\n\nLet $j<\\mu$ and assume we found $\\{U_{i}\\subseteq \\mu:i<j\\}$ and $a_{<j}=\\langle a_{i,k,\\eta}:i<j,\\, k\\in U_i,\\, \\eta\\in (I^{\\underline{\\alpha_i}})_<\\rangle $ such that $a_{<j}$ is $((\\alpha^\\prime)^{<j}, I)$-indiscernible, where $((\\alpha^\\prime)^{<j})_{i,k}=\\alpha_i$ for $i<j$ and $k\\in U_i$.\n\nIf $(I^{\\underline{\\alpha_j}})_<=\\emptyset$ then there is nothing to do. Otherwise, fix some $\\eta^*\\in (I^{\\underline{\\alpha_j}})_<$ and a $\\dcl$-closed subset \n\\[A_{j,\\eta^*}\\subseteq C_j=\\dcl(\\{a_{i,k,\\nu}:i<j,\\, k\\in U_i,\\, \\nu\\in (I^{\\underline{\\alpha_i}})_<\\})\\] of cardinality at most $\\mu$.\n\n\nFor any $A\\subseteq A_{j,\\eta^*}$ with $|A|<\\kappa$ and a non-algebraic type $p\\in S_1(A)$ we choose an extension $\\tilde p$ of $p$ to $A_{j,\\eta^*}$ and a non-principal ultrafilter $\\mathcal{D}(p)$ on $A_{j,\\eta^*}$, such that $\\tilde p = p_{\\mathcal{D}(p)}|A_{j,\\eta^*}$, in the following way\n\\begin{list}{\u2022}{}\n\\item if $p$ is finitely satisfiable in $A$ then let $\\mathcal{D}(p)$  be any (necessarily unique) ultrafilter (on $A_{j,\\eta^*}$) extending the filter $\\{B\\subseteq A_{j,\\eta^*}:\\varphi(A,a)\\subseteq B,\\, \\varphi(x,a)\\in p\\}$. We let $\\tilde p=p_{\\mathcal{D}(p)}|A_{j,\\eta^*}$;\n\\item otherwise, let $\\tilde p$ be any non-algebraic extension of $p$ to $A_{j,\\eta^*}$. Since $A_{j,\\eta^*}$ is a model, $\\tilde p$ is finitely satisfiable in $A_{j,\\eta^*}$. Let $\\mathcal{D}(p)$ be any non-principal ultrafilter extending $\\{\\varphi(A_{j,\\eta^*},b):\\varphi(x,b)\\in \\tilde p\\}$.\n\\end{list}\n\nWe note that there are at most $\\mu^{<\\kappa}\\leq\\mu$ subsets $A\\subseteq A_{j,\\eta^*}$ with $|A|<\\kappa$ and for all such $A$ there are at most $2^{\\kappa+|\\mathcal{L}|}\\leq \\mu$ types on $A$.\n\n\nLet $U_j\\subseteq \\mu$ be such that $\\langle (p_{j,k,\\eta^*},\\mathcal{D}_{j,k,\\eta^*}):k\\in U_j\\rangle$ enumerates the set of pairs $(\\tilde p,\\mathcal{D}(p))$ for non-algebraic $p\\in S_1(A)$.\n\nBy the induction hypothesis, any partial order-isomorphism $\\pi$ of $I$ induces a partial elementary map $\\widehat \\pi$ whose domain is \n\\[\\dcl\\left( \\{a_{i,k,\\nu}:i<j,\\, k\\in U_i,\\, \\Rg(\\nu)\\subseteq \\mathrm{Dom}(\\pi)\\}\\right),\\]\nmapping $a_{i,k,\\nu}\\mapsto a_{i,k,\\pi(\\nu)}$, where $\\pi(\\nu)=\\pi\\circ \\nu$. Note that for any $\\pi_1,\\pi_2$, if $\\pi_1\\circ \\pi_2$ makes sense then $\\widehat{\\pi_1\\circ \\pi_2}=\\widehat{\\pi_1}\\circ \\widehat{\\pi_2}$.\n\nAs a result, for any $\\rho\\in (I^{\\underline{\\alpha_j}})_<$ the unique order-isomorphism $\\pi_{\\eta^*,\\rho}:\\eta^*\\to \\rho$ induces a partial elementary map, $\\widehat{\\pi_{\\eta^*,\\rho}}: A_{j,\\eta^*}\\to A_{j,\\rho}$, where $A_{j,\\rho}=\\widehat\\pi(A_{j,\\eta^*})$, which is given by $a_{i,k,\\nu}\\mapsto a_{i,k,\\pi_{\\eta^*,\\rho}(\\nu)}$. For every $k\\in U_j$ let $p_{j,k,\\rho}=\\widehat{\\pi_{\\eta^*,\\rho}}(p_{j,k,\\eta^*})\\in S(A_{j,\\rho})$ and let $\\mathcal{D}_{j,k,\\rho}=\\widehat{\\pi_{\\eta^*,\\rho}}(\\mathcal{D}_{j,k,\\eta^*})$. \n\\begin{claim}\nFor any $\\rho\\in (I^{\\underline{\\alpha_j}})_<$ and $\\pi$ a partial isomorphism of $I$ whose domain contains  $\\Rg(\\rho)$, $\\widehat{\\pi}(A_{j,\\rho})=A_{j,\\pi(\\rho)}$, $\\widehat{\\pi}(p_{j,k,\\rho})=p_{j,k,\\pi(\\rho)}$ and $\\widehat{\\pi}(\\mathcal{D}_{j,k,\\rho})=\\mathcal{D}_{j,k,\\pi(\\rho)}$.\n\\end{claim}\n\\begin{claimproof}\nThere is no harm in restricting $\\pi$ to $\\Rg(\\rho)$.  Let $\\pi_{\\eta^*,\\rho}:\\eta^*\\to \\rho$ be the unique order-isomorphism, so $\\pi\\circ\\pi_{\\eta^*,\\rho}$ is the unique isomorphism from $\\eta^*$ to $\\pi(\\rho)$ and thus equal to $\\pi_{\\eta^*,\\pi(\\rho)}$. Hence $\\pi=\\pi_{\\eta^*,\\pi(\\rho)}\\circ \\pi_{\\eta^*,\\rho}^{-1}$. The result follows.\n\\end{claimproof}\n\n\nLet $((I^{\\underline{\\alpha_j}})_<,<^{\\text{lex}})$ be the lexicographic ordering and let $(U_j,<)$ be the order induced from $\\mu$. By induction on $k\\in U_j$, by compactness we may find a sequence $\\langle a_{j,k,\\eta}: \\eta\\in (I^{\\underline{\\alpha_j}})_<\\rangle$ satisfying that for any $\\eta\\in (I^{\\underline{\\alpha_j}})_<$\n\\[a_{j,k,\\eta}\\models p_{\\mathcal{D}_{j,k,\\eta}}|B_{j,k,\\eta},\\]\nwhere\n\\[B_{j,k,\\eta}=C_j\\cup \\{a_{j,l,\\rho}: l<k,\\, l\\in U_j,\\rho\\in (I^{\\underline{\\alpha_j}})_<\\}\\cup \\{a_{j,k,\\rho}:\\rho<^{\\text{lex}}\\eta\\}\\]\n\nWe show $((\\alpha^\\prime)^{\\leq j},I)$-indiscernibility by induction on $\\{(i,l):i\\leq j,\\, l\\in U_i\\}$ (with the lexicographic ordering). In other words, we assume that for any $\\langle (i_r,l_r): r<n\\rangle$, with $(i_r,l_r)<(j,k)$ and $l_r\\in U_r$, $\\langle \\eta_r\\in (I^{\\underline{\\alpha_{i_r}}})_<:r<n\\rangle$ and a partial isomorphism $\\pi$ of $I$, whose domain contains $\\bigcup \\{\\Rg(\\eta_r):r<n\\}$,\n\\[\\tp(a_{i_0,l_0,\\eta_0},\\dots,a_{i_{n-1},l_{n-1},\\eta_{n-1}})=\\tp(a_{i_0,l_0,\\pi(\\eta_0)},\\dots,a_{i_{n-1},l_{n-1},\\pi(\\eta_{n-1})}).\\]\nWe wish to show the same statement for $(i_r,l_r)\\leq (j,k)$.\n\nWe prove by induction on $n$ that for any $\\bar b\\subseteq \\{a_{i,l,\\eta}: (i,l)<(j,k),\\,\\eta\\in (I^{\\underline{\\alpha_i}})_<,\\, l\\in U_{i}\\}$, any $\\eta_{n-1}<^{\\text{lex}}\\dots<^{\\text{lex}}\\eta_0\\in (I^{\\underline{\\alpha_j}})_<$ and any partial isomorphism $\\pi$ of $(I,<)$ whose domain contains \n\\[\\Rg(\\eta_0)\\cup\\dots\\cup\\Rg(\\eta_{n-1})\\cup\\bigcup\\{\\Rg(\\eta):a_{i,l,\\eta}\\in \\bar b\\},\\]\n\\[\\tp(a_{j,k,\\eta_0},\\dots,a_{j,k,\\eta_{n-1}}, \\bar b)=\\tp(a_{j,k,\\pi(\\eta_0)},\\dots,a_{j,k,\\pi(\\eta_{n-1})},\\widehat{\\pi}(\\bar b)).\\]\n\nLet $\\varphi(x_0,\\dots,x_{n-1},\\bar b)$ be some formula, where $\\bar b$ is as above.\nWe show that \\[\\varphi(x_0,\\dots,x_{n-1},\\bar b)\\in p_{\\mathcal{D}_{j,k,\\eta_0}}\\otimes\\dots\\otimes p_{\\mathcal{D}_{j,k,\\eta_{n-1}}}\\iff\\]\\[\\varphi(x_0,\\dots,x_{n-1},\\pi(\\bar b))\\in p_{\\mathcal{D}_{j,k,\\pi(\\eta_0)}}\\otimes\\dots\\otimes p_{\\mathcal{D}_{j,k,\\pi(\\eta_{n-1})}}.\\]\nIndeed, if $\\varphi(x_0,\\dots,x_{n-1},\\bar b)\\in p_{\\mathcal{D}_{j,k,\\eta_0}}\\otimes\\dots\\otimes p_{\\mathcal{D}_{j,k,\\eta_{n-1}}}$ then by the choice of the $a_{j,k,\\eta}$'s,  $\\varphi(a_{j,k,\\eta_0},\\dots,a_{j,k,\\eta_{n-1}},\\bar b)$ holds and thus $X=\\varphi(A_{j,\\eta_0},a_{j,k,\\eta_1},\\dots,a_{j,k,\\eta_{n-1}},\\bar b)\\in \\mathcal{D}_{j,k,\\eta_0}$. By the Claim, $\\widehat\\pi(X)\\in \\mathcal{D}_{j,k,\\pi(\\eta_0)}$. By the induction hypothesis (on $n$), $\\widehat\\pi$ is elementary on $a_{j,k,\\eta_1}\\cup \\dots\\cup a_{j,k,\\eta_{n-1}}\\cup \\bar b$ and as a result, \\[\\widehat\\pi(X)=\\varphi(A_{j,\\pi(\\eta_0)},a_{j,k,\\pi(\\eta_1)},\\dots,a_{j,k,\\pi(\\eta_{n-1})},\\widehat \\pi(\\bar b))\\in \\mathcal{D}_{j,k,\\pi(\\eta_0)},\\]\n and as $\\pi$ preserves $<^{\\text{lex}}$,\n\\[\\varphi(x_0,a_{j,k,\\pi(\\eta_1)},\\dots,a_{j,k,\\pi(\\eta_{n-1})},\\widehat \\pi(\\bar b))\\in p_{\\mathcal{D}_{j,k,\\pi(\\eta_0)}}|B_{j,k,\\pi(\\eta_0)}.\\] \nAs a result, by the choice of the $a_{j,k,\\eta}$'s,\n\\[\\varphi(a_{j,k,\\pi(\\eta_0)},a_{j,k,\\pi(\\eta_1)},\\dots,a_{j,k,\\pi(\\eta_{n-1})},\\widehat \\pi(\\bar b)) \\text{ holds,}\\] and thus \n \\[\\varphi(x_0,\\dots,x_{n-1},\\widehat\\pi(\\bar b))\\in p_{\\mathcal{D}_{j,k,\\pi(\\eta_0)}}\\otimes\\dots\\otimes p_{\\mathcal{D}_{j,k,\\pi(\\eta_{n-1})}}.\\]\nThis proves $(1)$, i.e. $a=\\langle a_{i,k,\\eta}: i<\\mu,\\, k\\in U_i,\\, \\eta\\in (I^{\\underline{\\alpha_i}})_<\\rangle$ is $(\\alpha^\\prime,I)$-indiscernible.\n\nTo prove $(2)$ and $(3)$ recalling the beginning of the proof of $(1)$, let \\[A_{j,\\eta}=F_{j,\\eta}=\\dcl (\\{a_{i,k,\\nu}: i<j, k\\in U_{i}, \\nu\\in (\\Rg(\\eta)^{\\underline{\\alpha_i}})_<\\}).\\]\n$(2)$ follows immediately once we observe the following:\n\\begin{list}{\u2022}{}\n\\item $|A_{j,\\eta}|\\leq \\mu$. This follows from the following inequalities \\[\\mu\\cdot {|\\alpha_j|^{<\\kappa}}\\leq \\mu\\cdot \\kappa^{<\\kappa}\\leq \\mu.\\] \n\\item For any order-preserving partial isomorphism $\\pi$ of $I$, whose domain contains $\\Rg(\\eta)$, by the induction hypothesis on $j$, $\\widehat\\pi(A_{j,\\eta})=A_{j,\\pi(\\eta)}.$ \n\\end{list}\n\n\nNow assume that $(I,<)$ is well ordered, that $\\alpha$ satisfies $\\alpha_i=(i\\mod \\kappa)$ and let $A\\subseteq a$ with $|A|<\\kappa$. Since $(I,<)$ is well-ordered there exist an ordinal $\\beta$ and an order isomorphism, $\\eta:\\beta\\to \\bigcup_{a_{j,l,\\nu}\\in A} \\Rg(\\nu)$. Since $\\kappa$ is a regular cardinal and for every $a_{j,l,\\nu}\\in A$ we have $|\\Rg(\\nu)|=|\\alpha_j|<\\kappa$, it follows that $\\beta<\\kappa$.\n\nSince $\\kappa<2^\\kappa\\leq\\mu$ and $\\mu^{<\\kappa}=\\mu$ (so $\\cf(\\mu)\\geq \\kappa$), $\\widehat j=\\sup_{a_{j,l,\\nu}\\in A} j<\\mu$. Let $i=\\widehat j\\cdot \\kappa+\\beta<\\mu$. By the choice of $\\alpha$, $\\alpha_i=\\beta$ and $\\eta\\in (I^{\\underline{\\alpha_i}})_<$. This implies that $A\\subseteq F_{i,\\eta}$. This gives $(3.a)$. Now $(3.b)$ follows by $(2)$. \n\nItem $(3.c)$ follows from the construction, we elaborate. Let $A\\subseteq B$ with $|B|<\\kappa$. By $(3.a)$ there exists $j<\\mu$ and $\\eta\\in (I^{\\underline{\\alpha_j}})_<$ such that $B\\subseteq F_{j,\\eta}$. Let $p\\in S^{f.s.}(A)$ a non-algebraic type which is finitely satisfiable in $A$. As $\\cf(I)\\geq\\kappa$, there is some $\\Rg(\\eta)<\\gamma^*\\in I$ and by the Claim above for any $\\Rg(\\eta)<\\gamma\\in I$,\n\\[F_{j+1,\\eta^\\frown\\gamma}=\\pi_{\\eta^\\frown \\gamma^*,\\eta^\\frown \\gamma}(F_{j+1,\\eta^\\frown\\gamma^*}).\\]\nLet $\\mathcal{D}$ be the non-principle ultrafilter on $F_{j+1,\\eta^\\frown\\gamma^*}$ corresponding to $p$, as chosen above. Hence $p_{\\mathcal{D}}$ is finitely satisfiable in $A$.\nObserve that since $A\\subseteq F_{j,\\eta}$, $\\pi_{\\eta^\\frown\\gamma^*,\\eta^\\frown\\gamma}$ fixes $A$ point-wise. It follows by Remark \\ref{R:global-ultrafilter} that for every $\\Rg(\\eta)<\\gamma\\in I$, $p_{\\mathcal{D}}=p_{\\mathcal{D}_{j+1,k,\\eta^\\frown\\gamma}}$. \n\nLet $k\\in U_j$ be such that $\\tp(a_{j+1,k,\\eta^\\frown\\gamma^*}\/F_{j+1,\\eta^\\frown\\gamma^*})=p_{\\mathcal{D}}|F_{j+1,\\eta^\\frown\\gamma^*}$.  By the choice of elements, for any $\\Rg(\\eta)<\\gamma\\in I$,\n\\[a_{j+1,k,\\eta^\\frown\\gamma}\\models p_{\\mathcal{D}}|F_{j,\\eta}\\langle a_{j+1,k,\\eta^\\frown\\delta}: \\Rg(\\eta)<\\delta<\\gamma\\rangle.\\] We end by noting that since we are assuming that $(I,<)$ is a cardinal and $\\cf(I)\\geq \\kappa>\\alpha_j$, $|\\{\\gamma:\\Rg(\\eta)<\\gamma\\in I\\}|=|I|$.\n\\end{proof}\n\n\n\nIn stable theories, for any infinite indiscernible sequence $I$ over some set $A$ one may take the limit type defined by \n\\[\\lim(I)=\\{\\varphi(x,c): \\text{$\\varphi(a,c)$ holds for cofinitely many $a\\in I$}\\}.\\]\nIt is a consistent complete type by stability. It is obviously finitely satisfiable in $I$. Moreover, if $\\mathcal{D}$ is a non-principal ultrafilter on $I$, then $p_\\mathcal{D}=\\lim(I)$. We often write $\\lim(I\/A)=\\lim(I)|A$.\n\nThe following is \\cite[Lemma III.3.10]{classification}, we give a proof for completeness.\n\\begin{lemma}\\label{L:saturation}\nLet $T$ be a stable theory and $M\\models T$. If $M$ is $(\\kappa(T)+\\aleph_1)$-saturated and every countable indiscernible sequence over $A\\subseteq M$, with $|A|<\\kappa(T)$, in $M$ can be extended to one of cardinality $\\lambda$ then $M$ is $\\lambda$-saturated.\n\\end{lemma}\n\\begin{proof}\nWe may assume that $\\lambda>\\kappa(T)+\\aleph_1$.\nBy passing to $M^{eq}$ (and $T^{eq}$) there is no harm in assuming that $T$ eliminates imaginaries. \nLet $p\\in S(C)$ with $C\\subseteq  M$ and $|C|<\\lambda$. Let $B\\subseteq C$ with $|B|<\\kappa(T)$ such that $p$ does not fork over $B$. Let $q\\supseteq p$ be its non-forking global extension. Since $M$ is $\\kappa(T)$-saturated, we may find a sequence of elements $S=\\langle b_i :i<\\omega \\rangle\\subseteq M$ satisfying $b_i\\models q|B\\langle b_j:j<i\\rangle$. Note that $q|BS$ is stationary by \\cite[Corollary III.2.11]{classification}. \n\nSince $M$ is $(\\kappa(T)+\\aleph_1)$-saturated, we may find a Morley sequence $I=\\langle a_i:i<\\omega\\rangle$ of $q$ over $SB$, i.e. $a_i\\models q|SBa_{<i}$ and $a_i\\in M$. It follows that $I$ is also a Morley sequence of $q$ over $\\acl(B)$\\footnote{It is standard to see that $I$ is independent and indiscernible over $\\acl(B)$. On the other hand, since $q|BS$ is stationary, it isolates a complete type over $\\acl(BS)$.}. Let $I\\subseteq J\\subseteq M$ be an indiscernible sequence (over $B$) of cardinality $\\lambda$. As a result, $J$ is also a Morley sequence of $q$ over $\\acl(B)$.\n\nBy \\cite[Lemma III.1.10(2)]{classification}, $\\lim(J\/M)=q|M$ and in particular $\\lim(J\/C)=p$. By \\cite[Corollary III.3.5(1)]{classification}, there is $J_0\\subseteq J$ with $J\\setminus J_0$ indiscernible over $C$ and $|J_0|\\leq \\kappa(T)+|C|<\\lambda$. In particular, $|J\\setminus J_0|\\geq \\aleph_0$ and thus for every $c\\in J\\setminus J_0$, $p=\\tp(c\/C)$.\n\\end{proof}\n\n\\begin{definition}\nLet $T$ be a theory. We say that $M\\models T$ is an \\emph{infinitary EM-model based on $(\\alpha,I)$} if $M=\\dcl(a)$, where $a$ is an $(\\alpha,I)$-indiscernible sequence for $(\\alpha,I)$ as in Definition \\ref{D:alpha-I}.\n\\end{definition}\n\n\n\\begin{lemma}\\label{L:underlying set of Inf-EM is Inf-EM}\nLet $T$ be a any theory.\nLet $\\kappa\\geq \\aleph_0$ be a cardinal, $(I,<)$ a linearly ordered set, $\\alpha\\in \\kappa^U$, where $U$ is a set. If $a=\\langle a_{i,\\eta}:i\\in U,\\, \\eta\\in (I^{\\underline{\\alpha_i}})_<\\rangle$ is an $(\\alpha,I)$ indiscernible sequence, in some model $M\\models T$, then there exists some set $\\widehat U$, with $|\\widehat{U}|\\leq  |T|\\cdot |U|\\cdot \\kappa^{<\\kappa}$, and $\\widehat{\\alpha}\\in \\kappa^{\\widehat{U}}$ and an $(\\widehat{\\alpha},I)$-indiscernible sequence $b$ whose underlying set is $\\dcl(a)$.\n\\end{lemma}\n\\begin{proof}\nFor any $p\\subseteq \\kappa$ let $\\varphi_p:\\otp(p)\\to p$ be the unique order isomorphism. Let $\\mathcal{F}$ be the collection of all $\\emptyset$-definable functions. We consider the family $\\widehat U$ of tuples  $(f(\\bar v), s_0,p_0,\\dots,s_{|\\bar v|-1},p_{|\\bar v|-1})$ satisfying\n\\begin{list}{\u2022}{}\n\\item $f(\\bar v)\\in \\mathcal{F}$, \n\\item $s_0,\\dots, s_{|\\bar v|-1}\\in U$,\n\\item for any $i<|\\bar v|$, $p_i\\subseteq \\kappa$ with $\\otp(p_i)=\\alpha_{s_i}$,\n\\item $\\bigcup_{i<|\\bar v|}p_i\\in \\text{Ord}$.\n\\end{list}\nWe note that $|\\widehat U|\\leq |T|\\cdot |U|^{<\\aleph_0}\\cdot (\\kappa^{<\\kappa})^{<\\aleph_0}\\leq |T|\\cdot |U|\\cdot \\kappa^{<\\kappa}.$\nLet $\\widehat{\\alpha}\\in \\kappa^{\\widehat U}$ be the function mapping $x=(f(\\bar v), s_0,p_0,\\dots,s_{|\\bar v|-1},p_{|\\bar v|-1})$ to $\\bigcup_{i<|\\bar v|} p_i<\\kappa$.  For any $x=(f(\\bar v), s_0,p_0,\\dots,s_{|\\bar v|-1},p_{|\\bar v|-1})\\in\\widehat{U}$ and $\\eta\\in (I^{\\underline{\\widehat{\\alpha}_x}})_<$ set \\[b_{x,\\eta}=f(a_{s_0,\\eta\\restriction p_0\\circ \\varphi_{p_0}},\\dots,a_{s_{|\\bar v|-1},\\eta\\restriction p_{|\\bar v|-1}\\circ \\varphi_{p_{|\\bar v|-1}}}).\\] Note that for any $j<|\\bar v|$, $(\\eta\\restriction p_j)\\circ \\varphi_{p_j}\\in (I^{\\underline{\\alpha_{s_j}}})_<$.\n\nLet $b=\\langle b_{x,\\eta}: x\\in \\widehat{U},\\, \\eta\\in (I^{\\underline{\\widehat{\\alpha}_x}})_<\\rangle$. We  will show that $b$ is $(\\widehat \\alpha,I)$-indiscernible.\n\nLet $\\langle x_j\\in \\widehat{U}: j<k\\rangle$, $\\langle \\eta_j\\in (I^{\\underline{\\widehat{\\alpha}_{x_j}}})_<:j<k\\rangle$ and $\\pi$ be a partial isomorphism of $(I,<)$ whose domain contains $\\bigcup_{j<k}\\Rg(\\eta_j)$. For $j<k$, we write $b_{x_j,\\eta_j}=f_j(a_{s_{j,0},\\eta_j\\restriction p_{j,0}\\circ \\varphi_{p_{j,0}}},\\dots,a_{s_{j,|\\bar v_j|-1},\\eta_j\\restriction p_{j,|\\bar v_j|-1}\\circ \\varphi_{p_{j,|\\bar v_j|-1}}})$.\n\nSince $a$ is $(\\alpha,U)$-indiscernible, the type of \n\\[\\langle a_{s_{j,0},\\eta_j\\restriction p_{j,0}\\circ \\varphi_{p_{j,0}}},\\dots,a_{s_{j,|\\bar v|-1},\\eta_j\\restriction p_{j,|\\bar v_j|-1}\\circ \\varphi_{p_{j,|\\bar v_j|-1}}}:j<k\\rangle\\] is equal to the type of\n\\[\\langle a_{s_{j,0},\\pi(\\eta_j\\restriction p_{j,0}\\circ \\varphi_{p_{j,0}})},\\dots,a_{s_{j,\\bar v|-1},\\pi(\\eta_j\\restriction p_{j,|\\bar v_j|-1}\\circ \\varphi_{p_{j,|\\bar v_j|-1}})}:j<k\\rangle,\\]\nand consequently the type of $\\langle b_{x_j,\\eta_j}:j<k\\rangle$ is the equal to the type of $\\langle b_{x_j,\\pi(\\eta_j)}:j<k\\rangle$.\n\nFinally, let $c\\in \\dcl(a)$. I.e. there is a definable function $f(\\bar v)\\in \\mathcal{F}$, and $a_{i_0,\\eta_0},\\dots a_{i_{|\\bar v|-1},\\eta_{|\\bar v|-1}}\\in a$ such that $c= f(a_{i_0,\\eta_0},\\dots a_{i_{|\\bar v|-1},\\eta_{|\\bar v|-1}})$. Let $r=\\bigcup_{i<|\\bar v|}\\Rg(\\eta_{i})$ and $\\psi:r\\to \\otp(r)$ be the unique order isomorphism. For any $j<|\\bar v|$ set $p_j=\\psi(\\Rg(\\eta_j))$. Now note that for $x=(f(\\bar v),i_0,p_0,\\dots,i_{|\\bar v|-1},p_{|\\bar v|-1})$. So for $\\eta=\\psi^{-1}$, $c=b_{x,\\eta}$ (because e.g. $\\eta\\restriction p_0\\circ \\varphi_{p_0}=\\psi^{-1}\\restriction \\psi(\\Rg(\\eta_0))\\circ \\varphi_{\\psi(\\Rg(\\eta_0))}=\\psi^{-1}\\restriction \\psi(\\Rg(\\eta_0))\\circ \\psi\\circ \\eta_0=\\eta_0$).\n\\end{proof}\n\n\n\\begin{theorem}\\label{T:existence of gen em model in stable}\nThe following are equivalent for a complete $\\mathcal{L}$-theory $T$:\n\\begin{enumerate}\n\\item $T$ is stable.\n\n\\item Let $\\kappa,\\mu$ and $\\lambda$ be cardinals satisfying $\\kappa=\\cf(\\kappa)\\geq \\kappa(T)+\\aleph_1$, $\\mu^{<\\kappa}=\\mu\\geq 2^{\\kappa+|T|}$ and $\\lambda=\\lambda^{<\\kappa}\\geq \\mu$ and let $T\\subseteq T^{sk}$ be an expansion with definable Skolem functions such that $|T|=|T^{sk}|$ in a language $\\mathcal{L}\\subseteq \\mathcal{L}^{sk}$. Then there exists an infinitary EM-model  $M^{sk}\\models T^{sk}$ based on $(\\alpha,\\lambda)$, where $\\alpha\\in \\kappa^U$ for some set $U$ of cardinality at most $\\mu$, such that $M=M^{sk}\\restriction \\mathcal{L}$ is saturated of cardinality $\\lambda$.\n\n\\item Let $\\kappa,\\mu$ and $\\lambda$ be cardinals satisfying $\\kappa=\\cf(\\kappa)\\geq \\kappa(T)+\\aleph_1$, $\\mu^{<\\kappa}=\\mu\\geq 2^{\\kappa+|T|}$ and $\\lambda=\\lambda^{<\\kappa}\\geq \\mu$. Then there exists a saturated model of cardinality $\\lambda$.\n\n\\end{enumerate}\n\n\\end{theorem}\n\\begin{remark}\nFor example, the assumptions in (2) hold for $\\lambda=\\mu=2^{\\kappa+|T|}$ for any $\\kappa=\\cf(\\kappa)\\geq \\kappa(T)+\\aleph_1$. \n\\end{remark}\n\\begin{proof}\n$(1)\\implies (2)$. \nIn the following, the superscript $^{sk}$ means that we work in $T^{sk}$.\n \nWe apply Proposition \\ref{P:existence-of-indisc}(1,3) with $\\mathbb{U}$ there being a monster model for $T^{sk}$ and $(I,<)=(\\lambda,<)$.  Consequently, there exists an $(\\alpha^\\prime,\\lambda)$-indiscernible sequence $a$, where $\\alpha^\\prime$ is as in the proposition.\nLet $M^{sk}=\\dcl^{sk}(a)$ and $M=M^{sk}\\restriction \\mathcal{L}$. Note that $|M^{sk}|=|M|=\\mu\\cdot \\lambda^{<\\kappa}=\\lambda$.\n\nTowards applying Lemma \\ref{L:saturation}, note that $M$ is indeed $(\\kappa(T)+\\aleph_1)$-saturated by Proposition \\ref{P:existence-of-indisc}(3.b) and the assumption on $\\kappa$.  Let $I\\subseteq M$ be an infinite countable indiscernible sequence over some $B\\subseteq M$ with $|B|<\\kappa(T)\\leq \\kappa$.\n\nSince $\\lambda<\\lambda^{\\cf(\\lambda)}$, necessarily $\\cf(\\lambda)\\geq \\kappa$ so by Proposition \\ref{P:existence-of-indisc}(3.c) there is a  non principal ultrafilter $\\mathcal{D}$ on $I$ and elements $\\langle a_i\\in \\dcl(\\Rg(a)):i<\\lambda\\rangle$ satisfying that \n\\[a_i\\models p^{sk}_{\\mathcal{D}}|BI\\langle a_k: k<i\\rangle\\] for any $i<\\lambda$.\nLet $p_{\\mathcal{D}}$ be the restriction of $p^{sk}_{\\mathcal{D}}$ to $\\mathcal{L}$. Thus $p_{\\mathcal{D}}=\\lim(I)$ and for every $i<\\lambda$\n\\[a_i\\models \\lim(I)|BI\\langle a_k: k<i\\rangle.\\]\nBy stability, $I+\\langle a_i:i<\\lambda\\rangle$ is indiscernible over $B$ (see also \\cite[Exercise 2.25]{guidetonip} and \\cite[Lemma III.1.7(2)]{classification}). By Lemma \\ref{L:saturation}, $M$ is saturated.\n\n\n$(2)\\implies (3)$ is obvious.\n\n$(3)\\implies (1)$. Let $\\kappa$ be any cardinal satisfying $\\kappa=\\cf(\\kappa)\\geq \\kappa(T)+\\aleph_1$ and let $\\lambda = \\mu = \\beth_\\kappa(\\kappa)$. Then $\\lambda^{<\\kappa} = \\lambda$ because $\\kappa$ is regular. Indeed, any function from some $\\xi<\\kappa$ to $\\lambda$ is a function to $\\beth_\\alpha(\\kappa)$ for some $\\alpha<\\kappa$. So $\\lambda^\\xi = \\sup_{\\alpha<\\kappa} (\\beth_\\alpha(\\kappa)^\\xi)$. But $\\sup_{\\alpha<\\kappa}(\\beth_\\alpha(\\kappa)^\\xi)= \\sup_{\\alpha<\\kappa}(\\beth_{\\alpha+1}(\\kappa)^\\xi)$, and $(\\beth_{\\alpha+1}(\\kappa))^\\xi=(2^{\\beth_\\alpha(\\kappa)})^\\xi = 2^{\\beth_\\alpha(\\kappa)\\cdot \\xi} = 2^{\\beth_\\alpha(\\kappa)}$ because $\\kappa>\\xi$. Consequently, $\\lambda^\\xi = \\lambda$ and $\\lambda^{<\\kappa} = \\lambda$.\n\n\nHence, by (3), there is a saturated model of size $\\lambda$. On the other hand, since $\\lambda$ is singular (of cofinality $\\kappa<\\lambda$), $\\lambda^{<\\lambda}>\\lambda$ and as a result by \\cite[Theorem VIII.4.7]{classification},  $T$ is $\\lambda$-stable (and hence stable).\n \n\\end{proof}\n\n\\section{Order-Type graphs with large chromatic number} \\label{S:order type graphs}\nIn this section we discuss graphs whose vertices are (possibly infinite) increasing sequences, where the edge relation is determined by the order type. More specifically, our main interest in this section is the following type of graphs.\n\\begin{definition}\nLet $(I,<)$ and $(J,<)$ be linearly ordered sets and $\\bar a\\neq \\bar b\\in (I^{\\underline J})_<$ be increasing sequences. We define a graph $E^J_{\\bar a,\\bar b}$ and a directed graph $D_{\\bar a,\\bar b}^J$ on $(I^{\\underline J})_<$ by:\n\\begin{itemize}\n\\item $\\bar c \\mathrel{E}^J_{\\bar a,\\bar b} \\bar d \\iff \\otp(\\bar c,\\bar d)=\\otp(\\bar a,\\bar b)\\vee \\otp(\\bar d,\\bar c)=\\otp(\\bar a,\\bar b)$\n\\item $\\bar c \\mathrel{D}^J_{\\bar a,\\bar b} \\bar d \\iff \\otp(\\bar c,\\bar d)=\\otp(\\bar a,\\bar b).$\n\\end{itemize}\nWe omit $J$ from $E^J_{\\bar a,\\bar b}$ and $D^J_{\\bar a,\\bar b}$ when it is clear from the context.\n\nWe call these graphs the \\emph{(directed) order-type graphs}.\n\\end{definition}\n\n\\begin{remark}\nAlthough it will not define a graph, we sometimes use the notation $D_{\\bar a,\\bar b}$ and $E_{\\bar a,\\bar b}$ even if $\\bar a=\\bar b$.\n\\end{remark}\n\nIn Section \\ref{ss:embed-shift} we isolate a family of order-type graphs whose members contain all finite graphs of $\\text{Sh}_m(\\omega)$ for a certain integer $m$ (Corollary \\ref{C:k-ord-cov-implies shift}). In Section \\ref{ss:orde-type-large-chrom} we show that order-type graphs with large chromatic number fall into this family (Theorem \\ref{T:shift graphs in order-type-graphs}).\n\n\\subsection{Embedding shift graphs into order-type graphs}\\label{ss:embed-shift}\n\\begin{definition}\nLet $(I,<)$ and $(J,<)$ be linearly ordered sets, $\\bar a,\\bar b\\in (I^{\\underline J})_<$ be increasing sequences and $0<k<\\omega$.\n\nWe say that $\\langle \\bar a,\\bar b\\rangle$ is \\emph{k-orderly} if \nthere exists a finite partition $Conv(\\Img(\\bar a)\\cup\\Img(\\bar b))=C_0\\cup\\dots \\cup C_k$ by convex increasing subsets satisfying \nthat for every $n<k$ and $i\\in J$, $a_i\\in C_n\\iff b_i\\in C_{n+1}$;\n\\end{definition}\n\nRecall the following from \\cite{1196}.\n\\begin{definition}\nFor any linearly ordered set $(A,<)$ and $k\\geq 1$, let $\\text{LSh}_k(A)$ be the directed graph $((A^{\\underline k})_<,D)$, were $(\\eta,\\rho)\\in D$ if and only if $\\eta(i)=\\rho(i-1)$ for $0<i<k$ (if $k>1$) and $\\eta(0)<\\rho(0)$ (if $k=1$).\n\\end{definition}\n\n\n\n\\begin{lemma}\\label{L:k-orderly}\nLet $0<k<\\omega$ be an integer, $\\alpha,\\delta$ be ordinals, $(I,<)$ any infinite linearly ordered set satisfying $(\\delta\\times (2\\cdot \\alpha +1)^k,<_{lex})\\subseteq (I,<)$. Let $\\bar a,\\bar b\\in (I^{\\underline{\\alpha}})_<$.\nIf $\\langle \\bar a,\\bar b\\rangle $ is $k$-orderly then there exists a function $\\varphi:\\text{LSh}_k(\\delta)\\to (I^{\\underline{\\alpha}})_<$, satisfying that for any $\\eta,\\rho\\in \\text{LSh}_k(\\delta)$, if $(\\eta,\\rho)\\in D$ then $\\varphi(\\eta)\\mathrel{D}_{\\bar a,\\bar b} \\varphi(\\rho)$.\n\\end{lemma}\n\\begin{proof}\nAssume that $Conv(\\Img(\\bar a),\\Img(\\bar b))=C_0\\cup\\dots\\cup C_k$, as in the definition.\n\nLet $\\alpha^*=\\alpha\\cup\\{\\beta^-:\\beta<\\alpha\\}\\cup\\{\\infty\\}$, where the $\\beta^-$'s are immediate predecessors and $\\infty$ is a maximal element, i.e. for any $\\beta< \\gamma< \\alpha$\n\\begin{list}{\u2022}{}\n\\item $\\gamma<\\beta^-<\\beta$,\n\\item  $\\gamma^-<\\beta^-$ if and only $\\gamma<\\beta$ and \n\\item $\\gamma<\\infty$.\n\\end{list}\nFor any $S\\subseteq \\alpha$, let $S^*$ be $S\\cup\\{s^-:s\\in S\\}\\cup\\{0^-,\\infty\\}$.\n\nFor any $x\\in \\left(\\alpha^*\\right)^n$, we denote $x^-$ the immediate predecessor of $x$ in the lexicographic order if it exists, and otherwise let $x^-=x$. Note that for any $x=(x_0,\\dots,x_{n-1})\\neq(0^-,\\dots,0^-)\\in \\left(\\alpha^*\\right)^n$, if the maximal $l<n$ with $x_l\\neq 0^-$ satisfies $x_l< \\alpha$ then $x$ has an immediate predecessor.\n\nNote that the order type of $\\alpha^*$ is $2\\cdot \\alpha +1$, so by the assumption on $I$ we may replace $I$ by an isomorphic copy to get that $(\\delta\\times (\\alpha^*)^k,<_{lex})\\subseteq (I,<)$.\n\nFor any $0\\leq i\\leq k-1$ let $S_i=\\{\\beta< \\alpha: a_\\beta\\in C_i\\}$ and \nlet $\\mathcal{G}=\\{\\bar g=\\langle g_i: S_i\\cup\\{\\infty\\} \\to (S_{k-1}^*\\times\\dots\\times S_0^*,<_{lex}): i<k\\rangle: \\text{ $g_i$ increasing}\\}$.\n\nFor any $\\bar g \\in\\mathcal{G}$ and $\\eta\\in (\\delta^{\\underline k})_<$ let $f_{\\eta,\\bar g}\\in (I^{\\underline \\alpha})_<$ be defined by\n\\[f_{\\eta,\\bar g}(\\beta)=(\\eta(n_\\beta),g_{n_\\beta}(\\beta))\\in \\delta\\times (S^*_{k-1}\\times\\dots\\times S_0^*)\\subseteq I,\\] where $\\beta\\in S_{n_\\beta}$. We note that $f_{\\eta,\\bar g}$ is increasing: if $n_{\\beta_1}<n_{\\beta_2}$ then $\\eta(n_{\\beta_1})<\\eta(n_{\\beta_2})$. If $n_{\\beta_1}=n_{\\beta_2}$ then the results follows since $g_{n_{\\beta_1}}=g_{n_{\\beta_2}}$ is increasing.\n\n\\begin{claim}\nThere exists $\\bar g\\in \\mathcal{G}$ such that for any $\\eta,\\rho\\in \\text{Sh}_k(\\delta)$ satisfying $\\eta(i)=\\rho(i-1)$ for $0<i<k$ (if $k>1$) or $\\eta(0)<\\rho(0)$ (if $k=1$), $f_{\\eta,\\bar g}\\mathrel{D}_{\\bar a,\\bar b} f_{\\rho,\\bar g}$.\n\\end{claim}\n\\begin{claimproof}\nFor the purpose of this proof, for $1\\leq i\\leq k$ let $\\pi_i:S^*_{k-1}\\times\\dots\\times S_0^*\\to S^*_{k-1}\\times\\dots\\times S^*_{k-i}$ be the projection on the first $i$ coordinates. \nWe choose increasing functions $g_i:S_i\\cup\\{\\infty\\} \\to S_{k-1}^*\\times\\dots S^*_{k-i}\\times \\{0^-\\}\\times\\dots\\times \\{0^-\\}$ by downwards induction on $i<k$. Define $g_{k-1}$ by setting $g_{k-1}(\\beta)=(\\beta,0^-,\\dots,0^-)$ for $\\beta\\in S_{k-1}\\cup\\{\\infty\\}$.\n\n\n Assume that $g_i$ was defined and we want to define $g_{i-1}$. \n\nFor any $\\beta\\in S_{i-1}$ if there is $\\gamma\\in S_i$ minimal such that $b_\\beta\\leq a_\\gamma$ then define\n\\[g_{i-1}(\\beta)=\n\\begin{cases}\ng_i(\\gamma)=(\\pi_{k-i}(g_i(\\gamma)),0^-,0^-,\\dots, 0^-) & \\text{ if $a_\\gamma=b_\\beta$} \\\\\n(\\pi_{i}(g_i(\\gamma))^-,\\beta,0^-,\\dots, 0^-) & \\text{ otherwise.}\n\\end{cases}\n\\]\nIf such a minimal $\\gamma\\in S_i$ does not exists then we define\n\\[g_{i-1}(\\beta)=(\\pi_{i}(g_i(\\infty)),\\beta,0^-,\\dots,0^-).\\]\nLastly, \\[g_{i-1}(\\infty)=(\\pi_{i}(g_i(\\infty)),\\infty,0^-,\\dots,0^-).\\]\n\n\\begin{subclaim}\nFor any $0\\leq i\\leq k-1$, and for every $\\beta\\in S_i$, $\\pi_{i}(g_i(\\beta))$ has an immediate predecessor. I.e.,  for every $\\gamma<\\beta\\in S_i$, $\\pi_{i}(g_i(\\beta))>\\pi_{i}(g_i(\\beta))^-$.\n\nFor any $0\\leq i\\leq k-1$, $g_i$ is increasing.\n\\end{subclaim}\n\\begin{subclaimproof}\nThis is straightforward and follows, by downwards induction, that for any $\\beta\\in S_i$, if $g_i(\\beta)=(x_0,\\dots,x_{k-1})$ then the maximal $l<k$ such that $x_l\\neq 0^-$ satisfies that $x_l<\\alpha$. \n\nThe fact that the $g_i$s are increasing now follows by downwards induction.\n\\end{subclaimproof}\n\nThe main observation is that for any $1\\leq i<k$\n\\[(\\dagger)\\, \\otp(\\langle a_\\beta: \\beta\\in S_i\\rangle, \\langle b_\\beta: \\beta\\in S_{i-1}\\rangle)=\\otp(\\langle g_i(\\beta): \\beta\\in S_i\\rangle, \\langle g_{i-1}(\\beta): \\beta\\in S_{i-1}\\rangle ).\\]\nTo that end, let $1\\leq i<k$. Since $g_i$ and $g_{i-1}$ are increasing it is enough to compare $a_{\\beta_1}$ and $b_{\\beta_2}$, where $\\beta_1\\in S_i$ and $\\beta_2\\in S_{i-1}$.\n\n\\begin{list}{\u2022}{}\n\\item If $a_{\\beta_1}=b_{\\beta_2}$ then $\\beta_1\\in S_i$ is the minimal such that $b_{\\beta_2}\\leq a_{\\beta_1}$ and thus by definition $g_{i-1}(\\beta_2)=g_i(\\beta_1)$.\n\\item Assume $a_{\\beta_1}<b_{\\beta_2}$. If there does not exist a minimal $\\gamma\\in S_i$ with $b_{\\beta_2}\\leq a_\\gamma$  then\n\\[g_{i-1}(\\beta_2)=(\\pi_{i}(g_i(\\infty)),\\beta_2,0^-,\\dots,0^-)>\\]\\[(\\pi_{i}(g_i(\\beta_1)),0^-,\\dots,0^-)=g_{i}(\\beta_1).\\]\nOtherwise, let $\\beta_1< \\gamma\\in S_i$ be minimal such that $b_{\\beta_2}\\leq a_\\gamma$. If $b_{\\beta_2}=a_\\gamma$ then $g_{i-1}(\\beta_2)=g_i(\\gamma)>g_i(\\beta_1).$\n\nIf $b_{\\beta_2}<a_\\gamma$ then \n\\[g_{i-1}(\\beta_2)=(\\pi_{i}(g_i(\\gamma))^-,\\beta_2,0^-,\\dots, 0^-)>\\]\\[ (\\pi_{i}(g_i(\\beta_1)),0^-,0^-,\\dots, 0^-)=g_i(\\beta_1).\\]\n\\item Assume $a_{\\beta_1}>b_{\\beta_2}$ and let $\\gamma\\in S_i$ be minimal such that $a_\\gamma\\geq b_{\\beta_2}$, so $\\gamma\\leq \\beta_1$. If $a_\\gamma=b_{\\beta_2}$ then $\\gamma<\\beta_1$ and\n$g_{i-1}(\\beta_2)=g_i(\\gamma)<g_i(\\beta_1).$\n\nIf $a_\\gamma>b_{\\beta_2}$ then\n\\[g_{i-1}(\\beta_2)=(\\pi_{i}(g_i(\\gamma))^-,\\beta_2,0^-,\\dots, 0^-)<\\]\\[(\\pi_{i}(g_i(\\beta_1)),0^-,0^-,\\dots, 0^-)=g_i(\\beta_1).\\]\n\n\\end{list}\nThis proves $(\\dagger)$. Let $\\eta,\\rho\\in \\text{LSh}_k(\\delta)$ be as in the statement of the lemma. We proceed to prove that $f_{\\eta,\\bar g} \\mathrel{D}_{\\bar a,\\bar b} f_{\\rho,\\bar g}$.\n\nLet $\\beta_1,\\beta_2< \\alpha$ and assume that $\\beta_1\\in S_{n_1}$ and $\\beta_2\\in S_{n_2}$, for some $0\\leq n_1,n_2\\leq k-1$. Note that if $b_{\\beta_2}\\in C_n$, for $0<n\\leq k$, then $n_2=n-1$. Assume that $k>1$.\n\\begin{list}{\u2022}{}\n\\item Assume that $0<n_1<k$, $b_{\\beta_2}\\in C_{n_1}$. So $n_2=n_1-1$. Assume that $a_{\\beta_1}\\mathrel{\\square} b_{\\beta_2}$, where $\\square\\in\\left\\{ <,>,=\\right\\}$. By $(\\dagger)$, $g_{n_1}(\\beta_1)\\mathrel{\\square} g_{n_2}(\\beta_2)$ and as a result\n\\[f_{\\eta,\\bar g}(\\beta_1)=(\\eta(n_1),g_{n_1}(\\beta_1))=(\\rho(n_1-1),g_{n_1}(\\beta_1))\\mathrel{\\square} (\\rho(n_1-1),g_{n_2}(\\beta_2))=\\]\\[(\\rho(n_2),g_{n_2}(\\beta_2))=f_{\\rho,\\bar g}(\\beta_2).\\]\n\\item If $b_{\\beta_2}\\in C_n$ for some $n_1<n<k$ then necessarily, $a_{\\beta_1}<b_{\\beta_2}$ and $n_2=n-1\\geq n_1$. Consequently,\n\\[f_{\\eta,\\bar g}(\\beta_1)=(\\eta(n_1),g_{n_1}(\\beta_1))< (\\eta(n_1+1),g_{n_2}(\\beta_2))=(\\rho(n_1),g_{n_2}(\\beta_2))\\leq \\]\\[(\\rho(n_2),g_{n_2}(\\beta_2))=f_{\\rho,\\bar g}(\\beta_2).\\]\n\n\\item If $b_{\\beta_2}\\in C_n$ for some $n<n_1$ then necessarily $0<n<n_1$, $a_{\\beta_1}>b_{\\beta_2}$ and $n_2=n-1<n_1-1$. Hence\n\\[f_{\\eta,\\bar g}(\\beta_1)=(\\eta(n_1),g_{n_1}(\\beta_1))= (\\rho(n_1-1),g_{n_1}(\\beta_1))>(\\rho(n_2),g_{n_2}(\\beta_2))=\\]\\[f_{\\rho,\\bar g}(\\beta_2).\\]\n\n\\item If $b_{\\beta_2}\\in C_k$ then necessarily $n_2=k-1$ and $a_{\\beta_1}<b_{\\beta_2}$.\nAs a result\n\\[f_{\\eta,\\bar g}(\\beta_1)=(\\eta(n_1),g_{n_1}(\\beta_1))\\leq (\\eta(k-1),g_{n_1}(\\beta_1))=(\\rho(k-2),g_{n_1}(\\beta_1))<\\]\n\\[(\\rho(k-1),g_{n_2}(\\beta_2))=(\\rho(n_2),g_{n_2}(\\beta_2))=f_{\\rho,\\bar g}(\\beta_2).\\]\n\n\\end{list}\nIf $k=1$ then $n_1=n_2=0$ and\n\\[f_{\\eta,\\bar g}(\\beta_1)=(\\eta(0),g_{n_1}(\\beta_1))<(\\rho(0),g_{n_2}(\\beta_2))=f_{\\rho,\\bar g}(\\beta_2).\\]\n\\end{claimproof}\n\nWe may now define a map $\\varphi: \\text{Sh}_k(\\delta)\\to (I^{\\underline{\\alpha}})_<$ by letting for $\\eta\\in \\text{Sh}_k(\\delta)$, $\\varphi(\\eta)=f_{\\eta,\\bar g}\\in (I^{\\underline \\alpha})_<$. This maps satisfies the requirements by the previous claim.\n\\end{proof}\n\n\n\\begin{definition}\\label{D:k-ord-covered}\nLet $(I,<)$ and $(J,<)$ be linearly ordered sets and $\\bar a, \\bar b\\in (I^{\\underline J})_<$ be increasing sequences. \n\nWe say that $\\{\\bar a,\\bar b\\}$ is \\emph{k-orderly covered} if there exists an increasing partition of $J$ into convex sets $\\langle J_\\varepsilon: \\varepsilon\\in S\\rangle$ for some $S\\subseteq J$, such that for every $\\varepsilon\\in S$, exactly one of the following holds\n\\begin{enumerate}\n\\item $\\langle \\bar{a}\\restriction J_\\varepsilon,\\bar{b}\\restriction J_\\varepsilon\\rangle$ is $k_\\varepsilon$-orderly for some $0<k_\\varepsilon\\leq k$;\n\\item $\\langle \\bar{b}\\restriction J_\\varepsilon,\\bar{a}\\restriction J_\\varepsilon\\rangle$ is $k_\\varepsilon$-orderly for some $0<k_\\varepsilon\\leq k$; \n\\item $|J_\\varepsilon|=1$ and $\\bar{a}\\restriction J_\\varepsilon=\\bar{b}\\restriction J_\\varepsilon$.\n\\end{enumerate}\nMoreover, for every $\\varepsilon<\\varepsilon^\\prime\\in S$, $\\Img (\\bar a \\restriction J_\\varepsilon )<\\Img(\\bar b \\restriction J_{\\varepsilon^\\prime})$ and $\\Img(\\bar b\\restriction J_\\varepsilon) <\\Img(\\bar a \\restriction J_{\\varepsilon^\\prime})$.\n\\end{definition}\n\n\\begin{corollary}\\label{C:k-ord-cov-implies shift}\nLet $\\alpha$ be an ordinal, $(I,<)$ any infinite linearly ordered set with $(|\\alpha|^+ +\\aleph_0,<)\\subseteq (I,<)$. Let $\\bar a\\neq \\bar b\\in (I^{\\underline{\\alpha}})_<$ be some fixed sequences.\nIf $\\{\\bar a,\\bar b\\}$ is $k$-orderly covered then $((I^{\\underline{\\alpha}})_<,E_{\\bar a,\\bar b})$ contains all finite subgraphs of $\\text{Sh}_m(\\omega)$ for some $m\\leq k$.\n\\end{corollary}\n\\begin{proof}\nLet $\\langle J_\\varepsilon: \\varepsilon\\in S\\rangle$ be an increasing partition of $\\alpha$ as in Definition \\ref{D:k-ord-covered}, where $S\\subseteq \\alpha$. Since $\\bar a\\neq \\bar b$, there exists $\\varepsilon\\in S$ such that $|J_\\varepsilon|>1$.\n\nFor any $\\varepsilon\\in S$, with $|J_\\varepsilon|>1$, we say that $J_\\varepsilon$ is\n\\begin{list}{\u2022}{}\n\\item of type $A$ if $\\langle \\bar a\\restriction J_\\varepsilon,\\bar b\\restriction J_\\varepsilon\\rangle$ is $k_\\varepsilon$-orderly, and \n\\item of type $B$ if $\\langle \\bar b\\restriction J_\\varepsilon,\\bar a\\restriction J_\\varepsilon\\rangle$ is $k_\\varepsilon$-orderly.\n\\end{list}\n\nLet $N<\\omega$ be some natural number. By replacing $I$ with an isomorphic copy, we may assume that $(\\alpha \\times (N\\times (2\\alpha +1)^k),<_{lex})\\subseteq (I,<)$.  Let $\\varepsilon\\in S$ and let $I_\\varepsilon=\\{\\varepsilon\\}\\times (N\\times (2\\alpha+1)^{k})$. \n\nIf $|J_\\varepsilon|=1$ then we let $\\varphi_\\varepsilon:\\text{LSh}_1 (N)\\to ((I_\\varepsilon)^{\\underline{J_\\varepsilon}})_<$ be such that $\\varphi_\\varepsilon(\\eta)$ is the constant function giving $(\\varepsilon,0,\\dots,0)$.\n\nFor any $\\varepsilon\\in S$ let $E^\\varepsilon_{\\bar a,\\bar b}= E_{\\bar a\\restriction J_\\varepsilon, \\bar b\\restriction J_\\varepsilon}$ and $D^\\varepsilon_{\\bar a,\\bar b}= D_{\\bar a\\restriction J_\\varepsilon, \\bar b\\restriction J_\\varepsilon}$ and similarly $E^\\varepsilon_{\\bar b,\\bar a}$ and $D^\\varepsilon_{\\bar b,\\bar a}$.\n\nIf $|J_\\varepsilon|>1$ and $J_\\varepsilon$ is of type $A$ then let $\\varphi_\\varepsilon:\\text{LSh}_{k_\\varepsilon}(N)\\to(((I_\\varepsilon)^{\\underline{J_\\varepsilon}})_<,D^\\varepsilon_{\\bar a,\\bar b})$ be as supplied by Lemma \\ref{L:k-orderly}. I.e., for any $\\eta,\\rho\\in (N^{\\underline{k_\\varepsilon}})_<$, if $\\eta(i)=\\rho(i-1)$ for $0<i<k_\\varepsilon$ (if $k_\\varepsilon>1$) and $\\eta(0)<\\rho(0)$ (if $k_\\varepsilon=1$) then $\\otp(\\varphi_\\varepsilon(\\eta),\\varphi_\\varepsilon(\\rho))=\\otp(\\bar a\\restriction J_\\varepsilon,\\bar b\\restriction J_\\varepsilon)$. \n\nIf $|J_\\varepsilon|>1$ and $J_\\varepsilon$ is of type $B$ then let $\\widehat{\\varphi_\\varepsilon}:\\text{LSh}_{k_\\varepsilon}(N)\\to (((I_\\varepsilon)^{\\underline{J_\\varepsilon}})_<,D^\\varepsilon_{\\bar b,\\bar a})$ be as supplied by Lemma \\ref{L:k-orderly}. I.e, for any $\\eta,\\rho\\in (N^{\\underline{k_\\varepsilon}})_<$, if $\\eta(i)=\\rho(i-1)$ for $0<i<k_\\varepsilon$ (if $k_\\varepsilon>1$) and $\\eta(0)<\\rho(0)$ (if $k_\\varepsilon=1$) then $\\otp(\\widehat{\\varphi_\\varepsilon}(\\eta),\\widehat{\\varphi_\\varepsilon}(\\rho))=\\otp(\\bar b\\restriction J_\\varepsilon,\\bar a\\restriction J_\\varepsilon)$.  \n\nBy composing with the isomorphism $\\text{RSh}_{k_\\varepsilon}(N)\\to \\text{LSh}_{k_\\varepsilon}(N)$ mapping $(x_0,\\dots, x_{k_\\varepsilon-1})$ to $(N-1-x_{k_\\varepsilon-1},\\dots,N-1-x_0)$, we arrive to a directed graph homomorphism $\\varphi_\\varepsilon:\\text{RSh}_{k_\\varepsilon}(N)\\to (((I_\\varepsilon)^{\\underline{J_\\varepsilon}})_<,D^\\varepsilon_{\\bar b,\\bar a})$. By definition this map can be seen as a directed graph homomorphism $\\varphi_\\varepsilon:\\text{LSh}_{k_\\varepsilon}(N)\\to (((I_\\varepsilon)^{\\underline{J_\\varepsilon}})_<,D^\\varepsilon_{\\bar a,\\bar b})$.\n\nFor $1\\leq m\\leq k$ let $\\pi_m:(N^{\\underline k})_<\\to (N^{\\underline m})_<$ be the projection on the first $m$ coordinates. Note that it is a directed graph homomorphism $\\text{LSh}_k(N)\\to \\text{LSh}_m(N)$. We now define $\\varphi:\\text{LSh}_k(N)\\to ((I^{\\underline{\\alpha}})_<,D_{\\bar a,\\bar b})$. For any $\\eta\\in (N^{\\underline k})_<$, $\\varepsilon\\in S$ and $\\beta< \\alpha$, let $\\varepsilon(\\beta)\\in S$ be such that $\\beta\\in J_\\varepsilon$. We define\n\\[\\varphi(\\eta)(\\beta)= (\\varepsilon(\\beta),\\varphi_{\\varepsilon(\\beta)}(\\pi_{k_\\varepsilon}(\\eta))(\\beta)).\\]\nSince, for any $\\varepsilon\\in S$ and $\\eta\\in (N^{\\underline k})_<$, $\\varphi_{\\varepsilon}(\\pi_{k_\\varepsilon}(\\eta))$ is increasing, it is clear that $\\varphi(\\eta)$ is increasing as well.\n\nAssume that $\\eta,\\rho\\in \\text{LSh}_k(N)$ are connected, i.e., $\\eta(i)=\\rho(i-1)$ for $0<i<k$ (if $k>1$) and $\\eta(0)<\\rho(0)$ (if $k=1$). It is routine to check that $\\otp(\\varphi(\\eta),\\varphi(\\rho))=\\otp(\\bar a,\\bar b)$. As a result, $\\varphi$ is also a graph homomorphism between $\\text{Sh}_k(N)$ to $((I^{\\underline \\alpha})_<,E_{\\bar a,\\bar b})$.\n\nWe have proved that for every $N<\\omega$ there exists a graph homomorphism $\\varphi_N:\\text{Sh}_k(N)\\to ((I^{\\underline \\alpha})_<,E_{\\bar a,\\bar b})$. By compactness, we may find a graph homomorphism $\\text{Sh}_k(\\omega)\\to \\mathcal{H}$ for some elementary extension $((I^{\\underline \\alpha})_<,E_{\\bar a,\\bar b})\\prec (\\mathcal{H},E)$. By Fact \\ref{F:homomorphism is enough}, there exists $m\\leq k$ such that $((I^{\\underline \\alpha})_<,E_{\\bar a,\\bar b})$ contains all finite subgraphs of $\\text{Sh}_m(\\omega)$.\n\\end{proof}\n\n\n\\subsection{Analyzing order-type graphs with large chromatic number}\\label{ss:orde-type-large-chrom}\nThe main goal of this section is to prove that every order-type graph of large enough chromatic number is $k$-orderly covered for some $k$, i.e. we will prove the following.\n\\begin{theorem}\\label{T:shift graphs in order-type-graphs}\nLet $\\alpha$ be an ordinal, $(\\theta,<)$ an infinite ordinal with $|\\alpha|^+ +\\aleph_0<\\theta$. Let $\\bar a\\neq \\bar b\\in (\\theta^{\\underline{\\alpha}})_<$ be some fixed sequences.\n\nLet $G=((\\theta^{\\underline{\\alpha}})_<,E_{\\bar a,\\bar b})$. If $\\chi(G)>\\beth_2(\\aleph_0)$ then $G$  contains all finite subgraphs of $\\text{Sh}_m(\\omega)$ for some $m\\in \\mathbb{N}$.\n\\end{theorem}\n\nIn order to achieve this we will need to analyze the order-type of two infinite sequences. The tools developed here, we believe, may be useful in their own right.\n\nWe fix some ordinals $\\alpha$ and $\\theta$ with $\\theta$ infinite and $\\bar a\\neq \\bar b\\in (\\theta^{\\underline{\\alpha}})_<$ increasing sequences.\n\nWe partition $\\alpha=J_0\\cup J_+\\cup J_-$, where\n\\[J_0=\\{\\beta< \\alpha: a_\\beta=b_\\beta\\},\\, J_+=\\{\\beta < \\alpha: a_\\beta<b_\\beta\\},\\, J_-=\\{\\beta< \\alpha: b_\\beta<a_\\beta\\}.\\]\nLet $R$ be the minimal convex equivalence relation on $\\alpha$ containing\n\\[\\{(\\beta,\\gamma): a_\\beta=b_\\gamma\\},\\, \\{(\\beta,\\gamma): a_\\beta<a_\\gamma\\leq b_\\beta\\} \\text{ and } \\{(\\beta,\\gamma): b_\\beta< b_\\gamma\\leq a_\\beta\\}.\\]\n\n\\begin{lemma}\\label{L:equiv classes are disjoint}\nLet $A,B\\in \\alpha\/R$ and assume that $A<B$. Then $\\Img(\\bar a \\restriction A)< \\Img(\\bar b\\restriction B)$ and $\\Img(\\bar b\\restriction A)<\\Img(\\bar a \\restriction B)$.\n\\end{lemma}\n\\begin{proof}\nWe will show that $\\Img(\\bar a \\restriction A)< \\Img(\\bar b\\restriction B)$, the other assertion follows similarly. Let $\\beta\\in A$ and $\\gamma\\in B$, so $\\beta<\\gamma$.  If $a_\\beta\\geq b_\\gamma$ then $b_\\beta< b_\\gamma\\leq a_\\beta$ and hence $\\beta \\mathrel{R} \\gamma$, contradiction. \n\\end{proof} \n\n\\begin{lemma}\\label{L:each class has equal direction}\n\\begin{enumerate}\n\\item For any $\\beta\\in J_0$, $[\\beta]_R\\subseteq J_0$;\n\\item For any $\\beta\\in J_+$, $[\\beta]_R\\subseteq J_+$;\n\\item For any $\\beta\\in J_-$, $[\\beta]_R\\subseteq J_-$.\n\\end{enumerate}\n\nMoreover, $[\\beta]_R=\\{\\beta\\}$ for $\\beta\\in J_0$.\n\\end{lemma}\n\\begin{proof}\nTo prove (1), (2) and (3) it is sufficient to prove a weaker version where we assume that $\\beta=\\min [\\beta]_R$.\n\nWe show $(2)$, items $(1)$ and $(3)$ are proved similarly. Assume that $[\\beta]_R\\not\\subsetneq J_+$. Let $X= \\{ \\delta<\\alpha: \\delta\\in [\\beta]_R\\wedge (\\forall \\beta\\leq x\\leq \\delta)(a_x<b_x)\\}$ (in (1) we replace $a_x<b_x$ by $a_x=b_x$ and in (3) by $a_x>b_x$). By the assumptions, $X$ is a non empty initial segment of $[\\beta]_R$ and $Y=[\\beta]_R\\setminus X$ is non-empty convex.\n\nWe will show that both $X$ and $Y$ are closed under the relations defining $R$ and thus derive a contradiction to the minimality of $R$.\n\nAssume that $a_y=b_z$ with $y\\in X$ and $z\\in Y$. Since $X$ is an initial segment, $y<z$. Consequently, $a_y<b_y<b_z$, contradiction. Now assume that $z\\in X$ and $y\\in Y$, so there exists $z<x\\leq y$ with $a_x\\geq b_x$ and as a result $a_z<b_z<b_x\\leq a_x\\leq a_y=b_z$, contradiction.\n\nAssume that $a_y<a_z\\leq b_y$ with $y\\in X$ and $z\\in Y$, so $y<z$. Hence there is some $y<x\\leq z$ with $a_x\\geq b_x$ hence $a_y<a_z\\leq b_y<b_x\\leq a_x\\leq a_z$, contradiction. Now assume that $z\\in X$ and $y\\in Y$, so $z<y$. This implies that $a_z<a_y<a_z$, contradiction.\n\nAssume that $b_y<b_z\\leq a_y$ with $y\\in X$ and $z\\in Y$. Consequently, $a_y<b_y<b_z\\leq a_y$, contradiction.\n Now assume that $z\\in X$ and $y\\in Y$, so $z<y$.  As a result, $b_z<b_y<b_z$ \n \nFinally, we show the moreover. Assume note, so by $(1)$ it is easy to see that both $\\{\\beta\\}$ and $[\\beta]_R\\setminus \\{\\beta\\}$ are closed under the relations generating $R$. This contradicts the minimality of $R$.\n\\end{proof}\n\nBy Lemma \\ref{L:each class has equal direction}, $R\\restriction J_+$ is an equivalence relation on $J_+$. For any $A\\in J_+\/R$ we construct a set $Z_A\\subseteq A$. We construct a sequence $\\delta^A_n$ for $n<\\omega$ as follows. Let $\\delta^A_0=\\min A$  and assume that $\\delta^A_n$ has been chosen. Let $\\delta^A_{n+1}\\in A$ be the minimal index satisfying $b_{\\delta^A_n}\\leq a_{\\delta^A_{n+1}}$ if such exists, otherwise stop. Let $Z_A=\\langle \\delta^A_n: n<n_A\\rangle$, where $n_A\\leq \\omega$. Note that $Z_A$ is a strictly increasing sequence because $A\\in J_+\/R$. Furthermore, set  $C^A=Conv(\\Img(\\bar a\\restriction A)\\cup \\Img(\\bar b\\restriction A))$ and\n\\begin{itemize}\n\\item $C^A_0=[a_{\\delta^A_0},b_{\\delta^A_0})$;\n\\item If $n_A=\\omega$ then for any $0<n<\\omega$ we set $C^A_n=[b_{\\delta^A_{n-1}},b_{\\delta^A_{n}})$;\n\\item If $n_A<\\omega$ then for any $0<n<n_A$ set $C^A_n=[b_{\\delta^A_{n-1}},b_{\\delta^A_{n}})$ and $C^A_{n_A}=\\{x\\in C^A: b_{\\delta^A_{n_A-1}}\\leq x\\}$.\n\\end{itemize}\n\n\\begin{lemma}\\label{L:we are k-orderely covered}\nLet $A\\in J_+\/R$.\n\\begin{enumerate}\n\\item If $n_A=\\omega$ then $A=\\bigcup_{n<\\omega} [\\delta^A_0,\\delta^A_n]$. \n\\item If $n_A=\\omega$ then $C^A=\\bigcup_{n<\\omega}C_n^A$.\n\\item For every $\\beta\\in A$ and $0\\leq n<n_A$, $a_\\beta\\in C_n^A \\iff b_\\beta\\in C_{n+1}^A$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n\\begin{enumerate}\n\\item Let $X=\\bigcup_{n<\\omega}[\\delta^A_0,\\delta^A_n]$. Since the $\\delta^A_n$'s are chosen from $A$ and $A$ is convex, $X\\subseteq A$. As in the proof of Lemma \\ref{L:each class has equal direction}, it is enough to show that that both $X$ and $Y=A\\setminus X$ are closed under the relations defining $R$.\n\nIf $x,y\\in A$ satisfy that $a_x=b_y$ then since $b_y=a_x<b_x$ we conclude that $y<x$ and thus if $x\\in X$ then $y\\in X$. Now if we assume that $y\\in X$, e.g. $y<\\delta^A_n$, then $a_x=b_y<b_{\\delta^A_n}\\leq a_{\\delta^A_{n+1}}$ so $x<\\delta^A_{n+1}$.\n\nAssume that $x,y\\in A$ satisfy $a_x<a_y\\leq b_x$. If $x\\in X$, e.g. $x<\\delta^A_n$, then $a_x<a_y\\leq b_x<b_{\\delta^A_n}\\leq a_{\\delta^A_{n+1}}$ so $y<\\delta^A_{n+1}$. If $y\\in X$ then since $x<y$ we conclude that $x\\in X$ as well.\n\nAssume that $x,y\\in A$ satisfy $b_x<b_y\\leq a_x$. If $y\\in X$ then since $x<y$ it follows that $x\\in X$ as well. Assume that $y\\in Y$, i.e. $y\\geq \\delta^A_n$ for all $n$. But then $b_{\\delta^A_n}\\leq b_y\\leq a_x<b_x$ for all $n$. This implies that $\\delta^A_n< x$ for all $n$ and hence $x\\notin X$ as well.\n\n\\item The right-to-left inclusion is straightforward. For the other inclusion, let $x\\in C^A$. Since $\\delta^A_0=\\min A$ and $A\\in J_+\/R$, $a_{\\delta^A_0}=\\min C^A$ and hence $a_{\\delta^A_0}\\leq x$. If there exists $n<\\omega$ with $x<b_{\\delta^A_n}$ then for the minimal such $n$, $x\\in C^A_{n}$. Otherwise, since $a_\\beta<b_\\beta$ for any $\\beta\\in A$, we may assume that $x\\leq b_\\beta$ for some $\\beta\\in A$. Hence $x\\leq b_{\\delta^A_n}$ for some $n<\\omega$ by $(1)$. \n\n\\item Let $\\beta\\in A$ and $n$ be as in the statement. Assume that $n_A=\\omega$ is infinite ($n_A<\\omega$ is similar).\n\nLet $a_\\beta\\in C_n^A$. First assume $n=0$, i.e. $a_{\\delta^A_0}\\leq a_\\beta<b_{\\delta^A_0}$. It is always true that $b_{\\delta^A_0}\\leq b_\\beta$. If $\\beta\\geq \\delta^A_1$ then $a_{\\delta^A_1}\\leq a_\\beta<b_{\\delta^A_0}$, contradicting the choice of $\\delta^A_1$.\n\nNow, if $n>0$ then $b_{\\delta^A_{n-1}}\\leq a_\\beta<b_{\\delta^A_{n}}$ and thus by definition of $\\delta^A_n$, $\\delta^A_{n}\\leq \\beta$ so $b_{\\delta^A_{n}}\\leq b_\\beta$. If, on the other hand, $\\beta\\geq \\delta^A_{n+1}$ then $a_{\\delta^A_{n+1}}\\leq a_\\beta<b_{\\delta^A_{n}}$, contradiction. Hence $b_{\\delta^A_{n}}\\leq b_\\beta<b_{\\delta^A_{n+1}}$. \n\nLet $b_\\beta\\in C^A_{n+1}$. By $(2)$, $a_\\beta\\in C^A_k$ for some $k<\\omega$. Using the above we conclude that $b_\\beta\\in C^A_{k+1}$ and thus $k+1=n+1$, i.e. $k=n$.\n\\end{enumerate}\n\\end{proof}\n\n\\begin{lemma}\\label{L:special-sequence}\nFor any $A\\in J_+\/R$ there exist an increasing sequence $\\langle \\zeta_n^A\\in A:n<n_A\\rangle$, satisfying that for every $n$ with $n+1<n_A$\n\\[a_{\\zeta^A_{n+1}}\\leq b_{\\zeta^A_{n}},\\]\nand for every $n$ with $n+2<n_A$\n\\[ b_{\\zeta^A_{n}}<a_{\\zeta^A_{n+2}}.\\]\n\\end{lemma}\n\\begin{proof}\nLet $n$ be such that $n+1<n_A$. Assume for now that $b_{\\delta_n}\\neq a_{\\delta_{n+1}}$ (and hence $b_{\\delta_n}<a_{\\delta_{n+1}}$) and assume towards a contradiction that \n\\begin{itemize}\n\\item[(*)] for any $\\epsilon\\in (\\delta_n,\\delta_{n+1})$, $b_{\\epsilon}<a_{\\delta_{n+1}}$.\n\\end{itemize}\n Note that this implies that for any such $\\varepsilon$, $a_{\\delta_n}<a_\\varepsilon<b_{\\delta_{n}}<b_{\\varepsilon}<a_{\\delta_{n+1}}$. Let $X=\\{\\beta \\in A:\\beta <\\delta_{n+1}\\}$ and $Y=A\\setminus X$. This gives a convex partition of $A$, we will show that both $X$ and $Y$ are closed under the relations defining $R$.\n\nLet $\\beta,\\gamma\\in A$ with $a_\\beta=b_\\gamma$. If $\\beta<\\delta_{n+1}$ and $\\gamma\\geq \\delta_{n+1}$ then $b_{\\delta_{n+1}}\\leq b_\\gamma=a_\\beta< a_{\\delta_{n+1}}$, contradiction. Now assume that $\\gamma< \\delta_{n+1}$ and $\\beta\\geq \\delta_{n+1}$. If $\\gamma\\leq \\delta_n$ then $a_{\\delta_{n+1}}\\leq a_\\beta=b_\\gamma\\leq b_{\\delta_n}$, contradiction. If $\\gamma>\\delta_n$ then $a_{\\delta_{n+1}}\\leq a_\\beta=b_\\gamma<a_{\\delta_{n+1}}$ since $\\gamma\\in (\\delta_n,\\delta_{n+1})$ and by (*), contradiction.\n\nLet $\\beta,\\gamma\\in A$ with $a_\\beta<a_\\gamma \\leq b_\\beta$. Assume that $\\beta< \\delta_{n+1}$ and $\\gamma\\geq \\delta_{n+1}$. If $\\beta\\leq \\delta_n$ then  $a_{\\delta_{n+1}}\\leq a_\\gamma\\leq b_\\beta\\leq b_{\\delta_{n}}$, contradiction. If $\\beta\\in (\\delta_n,\\delta_{n+1})$ then $b_\\beta<a_{\\delta_{n+1}}\\leq a_\\gamma\\leq b_\\beta$ by $(*)$, contradiction. Note that we cannot have $\\gamma<\\delta_{n+1}$ and $\\beta\\geq \\delta_{n+1}$ since $\\beta<\\gamma$ by assumption.\n\nLet $\\beta,\\gamma\\in A$ with $b_\\beta<b_\\gamma\\leq a_\\beta$. If $\\beta< \\delta_{n+1}$ and $\\gamma\\geq \\delta_{n+1}$ then $b_{\\delta_{n+1}}\\leq b_\\gamma\\leq a_\\beta<a_{\\delta_{n+1}}<b_{\\delta_{n+1}}$, contradiction. As before, $\\gamma <\\delta_{n+1}$ and $\\beta\\geq \\delta_{n+1}$ is not possible since $\\beta<\\gamma$.\n\nAs a result, we may conclude that for all $n$ such that $n+1<n_A$ we may find $\\gamma_n\\in (\\delta_n,\\delta_{n+1}]$ satisfying $a_{\\delta_n}<a_{\\gamma_n}\\leq b_{\\delta_n}\\leq a_{\\delta_{n+1}}\\leq b_{\\gamma_n}$ (if $b_{\\delta_n}=a_{\\delta_{n+1}}$ choose $\\gamma_n=\\delta_{n+1}$, otherwise use the above).\n\nLet $I=\\{\\delta_n,\\gamma_n: n+1<n_A\\}$. The crucial property is  that for every $\\gamma\\in I\\setminus \\{\\sup I\\}$ there is some $\\beta \\in I$ satisfying $a_\\gamma<a_\\beta\\leq b_\\gamma$. We note that for every $n$ such that $n+1<n_A$ if $\\gamma\\leq \\delta_n$ then $\\beta\\leq \\delta_{n+1}$.  Indeed, otherwise $a_{\\delta_{n+1}}<a_\\beta\\leq b_\\gamma\\leq b_{\\delta_n}$, contradiction.\n\nWe construct a sequence $\\langle \\zeta_n: n<k\\rangle$ for some $k\\leq \\omega$ as follows. Define $\\zeta_0=\\delta_0$ and for every $n$ let $\\zeta_{n+1}\\in I$ be maximal\\footnote{If $n_A$ is finite then such a maximal element clearly exists. Otherwise, for $\\zeta \\in I$ there is some $n<\\omega$ such that $b_{\\zeta}<a_{\\delta_n}$, and hence $\\zeta\\in \\{\\xi\\in I: a_\\xi\\leq b_\\zeta\\}$ is finite.} with $a_{\\zeta_{n}}<a_{\\zeta_{n+1}}\\leq b_{\\zeta_{n}}$, if exists. Obviously, this is an increasing sequence. We claim that $k\\geq n_A$. By induction on $n<k$ with $n<n_A$, $\\zeta_n\\leq \\delta_n$. In particular if $n+1<n_A$, $\\zeta_{n+1}$ exists. Finally we note that by maximality, for all $n+2<n_A$, $a_{\\zeta_{n+1}}\\leq b_{\\zeta_{n}}< a_{\\zeta_{n+2}}$.\n\\end{proof}\n\n\nFor $A\\in J_-\/R$ we make dual (i.e. exchanging the roles of $\\bar a$ and $\\bar b$) constructions and similar properties hold.\n\n\\begin{corollary}\\label{C:embed shift if k bounds n_A}\nLet $\\alpha$ be an ordinal, $(\\theta,<)$ an infinite ordinal with $|\\alpha|^+ +\\aleph_0<\\theta$. Let $\\bar a\\neq \\bar b\\in (\\theta^{\\underline{\\alpha}})_<$ be some fixed sequences.\n\nLet $G=((\\theta^{\\underline{\\alpha}})_<,E_{\\bar a,\\bar b})$. If there exists $0<k<\\omega$ with $n_A\\leq k$ for all $A\\in (J_+\\cup J_-)\/R$ then $G$ contains all finite subgraphs of $\\text{Sh}_m(\\omega)$ for some $m\\leq k$.\n\\end{corollary}\n\\begin{proof}\nBy Lemma \\ref{L:equiv classes are disjoint} and Lemma \\ref{L:we are k-orderely covered}(3), $\\{\\bar a,\\bar b\\}$ is $k$-orderly covered in the sense of Definition \\ref{D:k-ord-covered}. Now apply Corollary \\ref{C:k-ord-cov-implies shift}.\n\\end{proof}\n\nThe aim of the rest of this section is to prove that $\\{n_A: A\\in (J_+\\cup J_-)\/R\\}$ has a finite bound. From now on we will only need the sequences defined in Lemma \\ref{L:special-sequence}.\n\n\\begin{lemma}\\label{L:each n_A is finite}\nIf $\\chi(G)>2^{\\aleph_0}$ then for any $A\\in (J_+\\cup J_-)\/R$, $n_A<\\omega$.\n\\end{lemma}\n\\begin{proof}\nWe assume that $A\\in J_+\/R$, the proof for $A\\in J_-\/R$ is similar. Assume towards a contradiction that $n_A=\\omega$. We will show that $\\chi(G)\\leq 2^{\\aleph_0}$.\n\nLet $S=\\{\\beta\\leq \\theta:\\cf(\\beta)=\\aleph_0\\}$. \nLet $\\langle \\zeta_l=\\zeta^A_l\\in A:l<\\omega\\rangle$ be the sequence supplied by Lemma \\ref{L:special-sequence}. \n\nFor any $\\gamma\\in S$ choose an increasing sequence of ordinals $\\langle \\alpha_{\\gamma,n}:n<\\omega\\rangle\\subseteq \\gamma$ with limit $\\gamma$. We define a coloring map $c:(\\theta^{\\underline{\\alpha}})_<\\to 2^{\\aleph_0\\times\\aleph_0}$.\nFor any $\\bar f\\in (\\theta^{\\underline{\\alpha}})_<$ let $\\gamma(\\bar f)=\\sup\\{f_{\\zeta_l}:l<\\omega\\}\\in S$ and\n\\[c(\\bar f)=\\{(l,n):l,n<\\omega,\\, f_{\\zeta_l}<\\alpha_{\\gamma(\\bar f),n}\\}.\\]\nTo show that it is a legal coloring, let $\\bar f,\\bar g\\in (\\theta^{\\underline{\\alpha}})_<$ such that $\\otp(\\bar f,\\bar g)=\\otp(\\bar a,\\bar b)$.\nBy assumption $f_{\\zeta_l}<f_{\\zeta_{l+1}}\\leq g_{\\zeta_l}\\leq f_{\\zeta_{l+2}}$ for $l<\\omega$, and hence $\\gamma(\\bar f)=\\gamma(\\bar g)$. By definition, there is some $n<\\omega$ such that $f_{\\zeta_0}<\\alpha_{\\gamma(\\bar f),n}$ and let $k$ be the minimal such that $f_{\\zeta_{k+1}}\\geq \\alpha_{\\gamma(\\bar f),n}$. So by minimality of $k$,\n\\[f_{\\zeta_k}<\\alpha_{\\gamma(\\bar f),n}\\leq f_{\\zeta_{k+1}}\\leq g_{\\zeta_{k}}\\]\nand hence $(k,n)\\in c(\\bar f)$ but $(k,n)\\notin c(\\bar g)$ so $c(\\bar f)\\neq c(\\bar g)$.\n\\end{proof}\n\n\nThe next lemma requires a more complicated argument: Section \\ref{S:PCF} below. Let us introduce some notation. \n\nFix some sequence $\\langle A_\\varepsilon \\in (J_+\/R : \\varepsilon<\\omega \\rangle$. \nFor any $\\varepsilon<\\omega$, let $J_\\varepsilon=\\{ \\zeta_n^{A_\\varepsilon}\\in A_\\varepsilon: n<n_{A_\\varepsilon}\\}$ be the sequence supplied by Lemma \\ref{L:special-sequence} applied to $A_\\varepsilon$. \nLet $J=\\bigcup_{\\varepsilon<\\omega}J_\\varepsilon$,\n $\\Omega=(\\theta^{\\underline J})_<$, \n $R'=\\{(\\bar c,\\bar d)\\in \\Omega^2: \\otp(\\bar c,\\bar d)=\\otp(\\bar a\\restriction J,\\bar b\\restriction J)\\}$ and \n $\\chi=\\beth_2(\\aleph_0)$. \n Note that $R'$ is an irreflexive relation on $\\Omega$ satisfying that if $f_1 \\mathrel{R}' f_2$, $f_1,f_2\\in \\Omega$, then for every $\\varepsilon<\\omega$ and $i\\in J_\\varepsilon$, the following hold:\n\\[f_1(i)<f_2(i)\\tag{1}\\] \nand for any $i\\in J_\\varepsilon$ with $\\suc{i}\\in J_\\varepsilon$\n\\[f_1(\\suc{i})\\leq f_2(i)\\tag{2},\\] and for any $i\\in J_\\varepsilon$ with $\\suc{\\suc{i}}\\in J_\\varepsilon$,\n\\[f_2(i)<f_1(\\suc{\\suc{i}})\\tag{3},\\] where $\\suc{i}$ is the successor of $i$ in $J_\\varepsilon$.\n\nUnder these assumptions (or more generally under Assumption \\ref{A:R}), we will prove in Conclusion \\ref{Con:coloring-pcf} that \n\\begin{itemize}\n    \\item[(*)] If $n_{A_\\varepsilon} < \\omega$ for all $\\varepsilon<\\omega$, then there exists a function $c:\\Omega\\to \\chi$ satisfying that if $f_1,f_2\\in \\Omega$ and $f_1 \\mathrel{R}' f_2$ then $c(f_1)\\neq c(f_2)$. In other words, there exists a coloring of the directed graph $(\\Omega,R')$ of cardinality $\\chi$. \n\\end{itemize}\n\n\\begin{lemma}\\label{L:all n_A are bounded}\nIf $\\chi(G)>\\beth_2(\\aleph_0)$ then the set $\\{n_A: A\\in (J_+\\cup J_-)\/R\\}$ is bounded.\n\\end{lemma}\n\\begin{proof}\nBy Lemma \\ref{L:each n_A is finite}, for any $A\\in (J_+\\cup J_-)\/R$, $n_A<\\omega$. We will show that $\\{n_A: A\\in J_+\/R\\}$ and $\\{n_A: A\\in J_-\/R\\}$ are both bounded.\n\nAssume that $\\{n_A: A\\in J_+\/R\\}$ is unbounded. Let $\\{A_\\varepsilon\\in J_+\/R:\\varepsilon<\\omega\\}$ be a family of convex equivalence classes such that $\\varepsilon<n_{A_\\varepsilon}$. \n\n\n\nBy (*), there exists a function $c:\\Omega\\to \\beth_2(\\aleph_0)$ satisfying that if $f_1,f_2\\in \\Omega$ and $f_1\\mathrel{R}' f_2$ then $c(f_1)\\neq c(f_2)$. Let $H=(\\Omega,(R')^{sym})$ be the graph induced by $R'$ (i.e. $(\\bar c, \\bar d)\\in (R')^{sym} \\iff (\\bar c,\\bar d)\\in R'\\vee (\\bar d,\\bar c)\\in R'$). The map $c$ induces a coloring on $H$ and hence $\\chi(H)\\leq \\beth_2(\\aleph_0)$. Since the map $(\\theta^{\\underline \\alpha})_< \\to \\Omega$ given by $\\eta\\mapsto \\eta\\restriction J$ is a graph homomorphism, $\\chi(G)\\leq \\beth_2(\\aleph_0)$ and this contradicts the assumption.\n\nIf on the other hand $\\{n_A: A\\in J_-\/R\\}$ is unbounded then we proceed as above but using $R''=\\{(\\bar c,\\bar d)\\in \\Omega^2: \\otp(\\bar c,\\bar d)=\\otp(\\bar b\\restriction J,\\bar a\\restriction J)\\}$ and the dual construction (replacing $R'$ by $R''$ in (*)) mentioned above instead and arrive at a similar contradiction.\n\\end{proof}\n\nFinally, we may conclude:\n\\proof[proof of Theorem \\ref{T:shift graphs in order-type-graphs}]\nThis is a direct consequence of Corollary \\ref{C:embed shift if k bounds n_A} and Lemma \\ref{L:all n_A are bounded}.\n\\qed\n\n\n\n\n\\section{Coloring increasing functions}\\label{S:PCF}\nThis section's main result is Conclusion \\ref{Con:coloring-pcf}, used in the final stage of the previous section. We prove that under mild conditions on a directed graph, namely Assumption \\ref{A:R}, on a family of strictly increasing functions there exists a coloring of small cardinality.\n\nLet $\\kappa=\\cf(\\kappa)$ be a regular cardinal and $(J,<)$ a well order of cofinality $\\kappa$. Let $\\sigma=(2^{\\kappa})^+$, $\\theta$ be an ordinal and $\\chi=\\chi^{\\kappa}+\\chi^{<\\sigma}$ a cardinal. \n\nLet $\\langle J_\\varepsilon: \\varepsilon <\\kappa\\rangle$ be an increasing partition of $J$ into  finite convex sets. \\iffalse with $\\sup J_\\varepsilon\\in J_\\varepsilon$ for all $\\varepsilon<\\kappa$.\\fi Assume that $\\sup_{\\varepsilon<\\kappa} |J_\\varepsilon|=\\omega$. Let $\\mathcal{D}$ be a non-principal ultrafilter on $\\kappa$ containing the filter generated by $\\{\\{\\varepsilon<\\kappa: |J_\\varepsilon|\\geq n\\}: n<\\omega\\}$.\n\nLet $\\Omega$ be the set of functions from $J$ to $\\theta$ that are strictly increasing on each $J_\\varepsilon$ ($\\varepsilon <\\kappa$). Let $\\mathcal{H}=(\\theta+1)^\\kappa$.\n\n\\begin{assumption}\\label{A:R}\n$R$ is an irreflexive relation on $\\Omega$ satisfying that if $f_1 \\mathrel{R} f_2$, $f_1,f_2\\in \\Omega$, then for every $\\varepsilon<\\kappa$ and $i\\in J_\\varepsilon$\n\\[f_1(i)<f_2(i)\\tag{1}\\] and for any $i\\in J_\\varepsilon$ with $\\suc{i}\\in J_\\varepsilon$\n\\[f_1(\\suc{i})\\leq f_2(i)\\tag{2},\\]\nand for any $i\\in J_\\varepsilon$ with $\\suc{\\suc{i}}\\in J_\\varepsilon$,\n\\[f_2(i)<f_1(\\suc{\\suc{i}})\\tag{3},\\]\nwhere $\\suc{i}$ is the successor of $i$ in the finite set $J_\\varepsilon$.\n\n\n\\end{assumption}\n\nWe say a subset $X$ of $\\Omega$ is \\emph{trivial} if $f_1\\not\\mathrel{R} f_2$ for any $f_1,f_2\\in X$.\n\n\\begin{definition}\\label{D:def of approximation}\nAn approximation $\\textbf{a}$ is a partition $\\Omega=\\bigcup_{s\\in S_\\textbf{a}}\\Omega_{s}^\\textbf{a}$ (so all the $\\Omega^{\\textbf{a}}_s$'s are non-empty), $\\rho_{s}^\\textbf{a}\\in \\chi^{<\\sigma}$ and $h^\\textbf{a}_s\\in \\mathcal{H}$ ($s\\in S_\\textbf{a}$) satisfying\n\\begin{enumerate}\n\\item for every $s\\in S_\\textbf{a}$ and $f\\in\\Omega^\\textbf{a}_s$, $\\{\\varepsilon<\\kappa: \\Rg(f\\restriction J_\\varepsilon)\\subseteq h^\\textbf{a}_s(\\varepsilon)\\}\\in \\mathcal{D}$\n\\item if $s\\neq t\\in S_\\textbf{a}$ and $\\rho_s^\\textbf{a}=\\rho^\\textbf{a}_t$ then for every $f_1\\in \\Omega^\\textbf{a}_s$ and $f_2\\in \\Omega^\\textbf{a}_t$, $f_1\\not\\mathrel{R} f_2$.\n\\end{enumerate}\n\\end{definition}\n\nWe want to define when one approximation is better than the other.\n\\begin{definition}\\label{D:order on approx}\nFor two approximations $\\textbf{a}$ and $\\textbf{b}$ we will say that $\\textbf{a}\\trianglelefteq_g \\textbf{b}$ if there exists a surjective function $g: S_\\textbf{b} \\to S_\\textbf{a}$ satisfying\n\\begin{enumerate}\n\\item for any $s\\in S_\\textbf{a}$, $\\{\\Omega^\\textbf{b}_t: t\\in g^{-1}(s)\\}$ is a partition of $\\Omega^\\textbf{a}_s$.\n\\item if $s\\in S_\\textbf{a}$ and $\\Omega^\\textbf{a}_s$ is trivial then $g^{-1}(s)$ is a singleton $t\\in S_\\textbf{b}$ satisfying $h^\\textbf{a}_s=h^\\textbf{b}_t$ and $\\Omega^\\textbf{a}_s=\\Omega^\\textbf{b}_t$ (in particular $\\Omega^\\textbf{a}_t$ is also trivial).\n\\item for $t\\in S_\\textbf{b}$,  $\\{\\varepsilon<\\kappa: h^\\textbf{b}_t(\\varepsilon)\\leq h^\\textbf{a}_{g(t)}(\\varepsilon)\\}\\in \\mathcal{D}$.\n\\item for $t\\in S_\\textbf{b}$, $\\rho^\\textbf{a}_{g(t)}$ is an initial segment of $\\rho^\\textbf{b}_t$.\n\\end{enumerate}\n\nWe will say that $\\textbf{a}\\triangleleft_g \\textbf{b}$ if $\\textbf{a}\\trianglelefteq_g \\textbf{b}$ and in addition for every $t\\in S_\\textbf{b}$, either $\\Omega^\\textbf{b}_t$ is trivial or $\\{\\varepsilon<\\kappa :h^\\textbf{b}_t(\\varepsilon)<h^\\textbf{a}_{g(t)}(\\varepsilon)\\}\\in \\mathcal{D}$. \n\\end{definition}\n\nThe following is clear.\n\\begin{lemma}\\label{L:composition-approx}\nLet $\\textbf{a},\\textbf{b}$ and $\\textbf{c}$ be approximations. If $\\textbf{a}\\trianglelefteq_g \\textbf{b}$ and $\\textbf{b}\\trianglelefteq_h\\textbf{c}$ then $\\textbf{a}\\trianglelefteq_{g\\circ h} \\textbf{c}$. If, in addition, either $\\textbf{a} \\triangleleft_g \\textbf{b}$ or $\\textbf{b} \\triangleleft_h\\textbf{c}$ then $\\textbf{a} \\triangleleft_{g\\circ h}\\textbf{c}$.\n\\end{lemma}\n\n\\begin{proposition}\\label{P:approx-small-cof}\nLet $\\textbf{a}$ be an approximation. Then there exists an approximation $\\textbf{a}\\trianglelefteq_g \\textbf{b}$ satisfying the following.\n\\begin{enumerate}\n\\item If $t\\in S_\\textbf{b}$, $\\Omega^\\textbf{a}_{g(t)}$ is non-trivial and $\\{\\varepsilon<\\kappa: 0<\\cf(h^\\textbf{a}_{g(t)}(\\varepsilon))\\leq \\chi\\}\\in \\mathcal{D}$ then either $\\Omega^\\textbf{b}_t$ is trivial or $\\{\\varepsilon<\\kappa: h^\\textbf{b}_t(\\varepsilon)<h^\\textbf{a}_{g(t)}(\\varepsilon)\\}\\in\\mathcal{D}$.\n\\item If $t\\in S_\\textbf{b}$ satisfies that $\\{\\varepsilon<\\kappa:0<\\cf(h^\\textbf{a}_{g(t)}(\\varepsilon))\\leq \\chi\\}\\notin \\mathcal{D}$ then $h^\\textbf{b}_t=h^\\textbf{a}_{g(t)}$ and $\\Omega^\\textbf{b}_t=\\Omega^\\textbf{a}_{g(t)}$.\n\\end{enumerate}\nLastly, for $t\\in S_\\textbf{b}$, if $\\rho^\\textbf{a}_{g(t)}\\in \\chi^{\\xi}$, with $\\xi<\\sigma$, then $\\rho^\\textbf{b}_t\\in \\chi^{\\xi+1}$.\n\\end{proposition}\n\\begin{proof}\nWe partition $S_\\textbf{a}$ in the following way. Let $S_1=\\{s\\in S_\\textbf{a}:  (\\forall^{\\mathcal{D}}\\varepsilon<\\kappa)(0<\\cf(h^\\textbf{a}_s(\\varepsilon))\\leq \\chi) \\text{ and } \\Omega^\\textbf{a}_s \\text{ is non-trivial}\\}$ and $S_0=S_\\textbf{a}\\setminus S_1$ is the rest.\n\nFix any $s\\in S_1$. For any $\\varepsilon<\\kappa$, if $0<\\cf(h^\\textbf{a}_s(\\varepsilon))\\leq \\chi$ we choose a an unbounded subset $C_{s,\\varepsilon}\\subseteq h^\\textbf{a}_s(\\varepsilon)$ of order type $\\cf(h^\\textbf{a}_s(\\varepsilon))$, and we set $C_{s,\\varepsilon}=\\{h^\\textbf{a}_s(\\varepsilon))\\}$, otherwise.\n\nSet $A_s=\\{\\varepsilon<\\kappa: 0<\\cf(h^\\textbf{a}_s(\\varepsilon))\\leq \\chi\\}$. Note that $A_s\\in \\mathcal{D}$.\n\nLet $H_s=\\{h\\in \\mathcal{H}: \\text{if $\\varepsilon\\in A_s$ then $h(\\varepsilon)\\in C_{s,\\varepsilon}$ and $h(\\varepsilon)=h^\\textbf{a}_s(\\varepsilon)$, otherwise}\\}$. Since $\\chi^{\\kappa}=\\chi$, $|H_s|\\leq \\chi$ and hence there is some $\\xi_s\\leq \\chi$ and an enumeration $\\langle h_{s,\\xi}:\\xi<\\xi_s\\rangle$ of $H_s$.\n\nBy induction on $\\xi<\\xi_s$ we define\n\\[\\Omega_{s,\\xi}=\\{f\\in \\Omega^\\textbf{a}_s: (\\forall^\\mathcal{D}\\varepsilon<\\kappa)(\\Rg(f\\restriction J_\\varepsilon)\\subseteq h_{s,\\xi}(\\varepsilon))\\}\\setminus \\bigcup_{\\alpha<\\xi}\\Omega_{s,\\alpha}\\]\nand for $\\xi=\\xi_s$ \n\\[\\Omega_{s,\\xi}=\\{f\\in \\Omega^\\textbf{a}_s:(\\forall^\\mathcal{D} \\varepsilon<\\kappa)(h^\\textbf{a}_s(\\varepsilon) \\text{ is a sucessor and }h^\\textbf{a}_s(\\varepsilon)-1=\\max\\Rg(f\\restriction J_\\varepsilon)\\}.\\]\nWe claim that $\\Omega^\\textbf{a}_s=\\bigsqcup_{\\xi\\leq \\xi_s}\\Omega_{s,\\xi}$. Let $f\\in \\Omega^\\textbf{a}_s$. Note that for every $\\varepsilon\\in A_s$ either (a) there an ordinal $\\gamma\\in C_{s,\\varepsilon}$ such that $\\Rg(f\\restriction J_\\varepsilon)\\subseteq \\gamma$ or (b) there is no such $\\gamma$. We may find $A_s^\\prime\\subseteq A_s$ satisfying that $A_s^\\prime\\in\\mathcal{D}$ and that for all $\\varepsilon\\in A_s^\\prime$ (a) holds or that for all $\\varepsilon\\in A_s^\\prime$ (b) holds.\n\nAssume that (a) holds for all $\\varepsilon\\in A_s^\\prime$ and let $\\langle \\gamma_\\varepsilon : \\varepsilon<A_s^\\prime\\rangle$ witness this. Define a function $h\\in H_s$ by setting for all $\\varepsilon\\in A_s^\\prime$, $h(\\varepsilon)=\\gamma_\\varepsilon$. For $\\varepsilon\\notin A_s^\\prime$ choose arbitrary $h(\\varepsilon)$ as long as $h\\in H_s$. Let $\\xi<\\xi_s$ be minimal such that $(\\forall^\\mathcal{D}\\varepsilon<\\kappa)(\\Rg(f\\restriction J_\\varepsilon)\\subseteq h_{s,\\xi}(\\varepsilon))$, so $f\\in \\Omega_{s,\\xi}$.\n\nNow, assume that (b) holds for all $\\varepsilon\\in A_s^\\prime$. As $\\textbf{a}$ is an approximation (see Definition \\ref{D:def of approximation}(1)) we may assume that for all $\\varepsilon\\in A_s^\\prime$, $\\Rg(f\\restriction J_\\varepsilon)\\subseteq h^\\textbf{a}_s(\\varepsilon)$ but we cannot find any $\\gamma\\in C_{s,\\varepsilon}$ satisfying $\\Rg(f\\restriction J_\\varepsilon)\\subseteq \\gamma$. For any $\\varepsilon \\in A_s^\\prime$, because $J_\\varepsilon$ is finite this implies that $\\cf(h_s^{\\textbf{a}}(\\varepsilon))=1$, i.e. that $h^\\textbf{a}_s(\\varepsilon)$ is  a successor ordinal and that $h^\\textbf{a}_s(\\varepsilon)-1=\\max\\Rg(f\\restriction J_\\varepsilon)$. Hence $f\\in \\Omega_{s,\\xi_s}$.\n\nLet $S_\\textbf{b}=\\{(s,\\xi):s\\in S_1,\\, \\xi\\leq \\xi_s,\\, \\Omega_{s,\\xi}\\neq \\emptyset\\}\\cup S_0$ and let $g:S_\\textbf{b}\\to S_\\textbf{a}$ be the function defined by $g(s,\\xi)=s$ for $s\\in S_1$ and $g(s)=s$ otherwise. For any $s\\in S_0$ let $\\Omega^\\textbf{b}_s=\\Omega^\\textbf{a}_s$, $h^\\textbf{b}_s=h^\\textbf{a}_s$ and $\\rho^\\textbf{b}_s={\\rho^\\textbf{a}_s}^\\frown\\langle 0\\rangle $. For $s\\in S_1$, if $\\xi\\leq \\xi_s$ we set $\\Omega^\\textbf{b}_{(s,\\xi)}=\\Omega_{s,\\xi}$ and $\\rho^\\textbf{b}_{(s,\\xi)}={\\rho^\\textbf{a}_s}^{\\frown} \\langle \\xi\\rangle$. Finally, for $\\xi<\\xi_s$ we set $h^\\textbf{b}_{(s,\\xi)}=h_{s,\\xi}$ and for $\\xi=\\xi_s$ we set $h^\\textbf{b}_{(s,\\xi)}=h^\\textbf{a}_s$.\n\n\\begin{claim}\n$\\textbf{b}$ is an approximation and $\\textbf{a}\\trianglelefteq_g \\textbf{b}$.\n\\end{claim}\n\\begin{claimproof}\nWe first show that $\\textbf{b}$ is an approximation. Items $(1)$ and $(2)$ from the definition follow since $\\textbf{a}$ is an approximation and the construction above. For example, if $(s_1,\\xi_1)\\neq (s_2,\\xi_2)\\in S_\\textbf{b}$ and $\\rho^\\textbf{b}_{(s_1,\\xi_1)}=\\rho^\\textbf{b}_{(s_2,\\xi_2)}$ then since $\\xi_1=\\xi_2$ necessarily $s_1\\neq s_2$ and $\\rho^\\textbf{a}_{s_1}=\\rho^\\textbf{a}_{s_2}$ so we may use the fact that $\\textbf{a}$ is an approximation.\n\nFinally, $\\textbf{a}\\trianglelefteq_g \\textbf{b}$ by the construction.\n\\end{claimproof}\n\nShowing $(1)$ from the statement of the proposition boils down to showing that $\\Omega^\\textbf{b}_{(s,\\xi_s)}=\\Omega_{s,\\xi_s}$ is trivial. This follows from Assumption \\ref{A:R}(1).\n\\end{proof}\n\n\\begin{proposition}\\label{P:approx-large-cof}\nLet $\\textbf{a}$ be an approximation. Then there exists an approximation $\\textbf{a}\\trianglelefteq_g \\textbf{b}$ satisfying the following. \n\\begin{enumerate}\n\\item If $t\\in S_\\textbf{b}$ with $\\{\\varepsilon<\\kappa: \\cf(h^\\textbf{a}_{g(t)}(\\varepsilon))>\\chi\\}\\in \\mathcal{D}$ then either $\\Omega^\\textbf{b}_t$ is trivial or $\\{\\varepsilon<\\kappa: h^\\textbf{b}_t(\\varepsilon)<h^\\textbf{a}_{g(t)}(\\varepsilon)\\}\\in \\mathcal{D}$.\n\\item If $t\\in S_\\textbf{b}$ satisfies that $\\{\\varepsilon<\\kappa:\\cf(h^\\textbf{a}_{g(t)}(\\varepsilon))> \\chi\\}\\notin \\mathcal{D}$ then $h^\\textbf{b}_t=h^\\textbf{a}_{g(t)}$ and $\\Omega^\\textbf{b}_t=\\Omega^\\textbf{a}_{g(t)}$.\n\\end{enumerate}\n\nLastly, for $t\\in S_\\textbf{b}$, if $\\rho^\\textbf{a}_{g(t)}\\in \\chi^{\\xi}$, with $\\xi<\\sigma$, then $\\rho^\\textbf{b}_t\\in \\chi^{\\xi+1}$.\n\\end{proposition}\n\\begin{proof}\nLet $S_1=\\{s\\in S_\\textbf{a}: \\Omega^\\textbf{a}_s \\text{ is non-trivial and } (\\forall^{\\mathcal{D}} \\varepsilon<\\kappa)(\\cf(h^\\textbf{a}_s(\\varepsilon))>\\chi\\}$ and $S_0=S_\\textbf{a}\\setminus S_1$. Fix any $s\\in S_1$. Let $A_s=\\{\\varepsilon<\\kappa: \\cf(h^\\textbf{a}_s(\\varepsilon))>\\chi\\}$, so $A_s\\in \\mathcal{D}$.\n\nLet $\\mathcal{D}_s=\\{D\\in \\mathcal{D}: D\\subseteq A_s\\}$ be the induced ultrafilter on $A_s$. Consider the ultraproduct $\\prod_{\\varepsilon\\in A_s}h^\\textbf{a}_s(\\varepsilon)\/\\mathcal{D}_s$. We may consider it as a linearly ordered set, ordered by $<_{\\mathcal{D}_s}$.\n\n\\begin{claim}\\label{C: sequence in ultraproduct}\nThere exists a sequence $H_s=\\langle h_{s,\\beta}\\in \\mathcal{H}:\\beta<\\beta_s\\rangle$ satisfying\n\\begin{enumerate}\n\\item for all $\\varepsilon\\in A_s$ and $\\beta<\\beta_s$, $h_{s,\\beta}(\\varepsilon)<h^\\textbf{a}_s(\\varepsilon)$;\n\\item for all $\\varepsilon\\in \\kappa\\setminus A_s$ and $\\beta<\\beta_s$, $h_{s,\\beta}(\\varepsilon)=h^\\textbf{a}_s(\\varepsilon)$;\n\\item $\\langle (h_{s,\\beta}\\restriction A_s)\/\\mathcal{D}_s:\\beta<\\beta_s\\rangle$ is $<_{\\mathcal{D}_s}$ increasing and cofinal in  $\\prod_{\\varepsilon\\in A_s}h^\\textbf{a}_s(\\varepsilon)\/\\mathcal{D}_s$.\n\\item for any $f\\in \\Omega^\\textbf{a}_s$ there exists $\\beta<\\beta_s$ such that $\\{\\varepsilon<\\kappa: \\Rg(f\\restriction J_\\varepsilon)\\subseteq h_{s,\\beta}(\\varepsilon)\\}\\in\\mathcal{D}$.\n\\end{enumerate}\n\\end{claim}\n\\begin{claimproof}\nFirst we choose a well-ordered increasing cofinal sequence in $\\prod_{\\varepsilon\\in A_s}h^\\textbf{a}_s(\\varepsilon)\/\\mathcal{D}_s$ and then choose a sequence of representatives  $\\langle h_{s,\\beta}\\restriction A_s:\\beta<\\beta_s\\rangle$. To get (2), set $h_{s,\\beta}(\\varepsilon)=h_s^{\\textbf{a}}(\\varepsilon)$ for any $\\varepsilon\\in \\kappa\\setminus A_s$. This gives us (1)--(3).\n\nWe show (4). Let $f\\in \\Omega_s^\\textbf{a}$. Since $\\textbf{a}$ is an approximation, the set $X_{s,f}=\\{\\varepsilon\\in A_s: \\Rg(f\\restriction J_\\varepsilon)\\subseteq h^{\\textbf{a}}_s(\\varepsilon)\\}$ is in $\\mathcal{D}$. Let $h_f: A_s\\to \\text{Ord}$ be the function defined by mapping $\\varepsilon\\in X_{s,f}$ to $\\max \\Rg(f\\restriction J_\\varepsilon)+1$ and $\\varepsilon\\in A_s\\setminus X_{s,f}$ to $0$. Note that for any $\\varepsilon\\in A_s$, $h_f(\\varepsilon)<h_s^{\\textbf{a}}(\\varepsilon)$. Indeed, if for $\\varepsilon\\in X_{s,f}$, $h_f(\\varepsilon)=h_s^{\\textbf{a}}(\\varepsilon)$ then $h_s^{\\textbf{a}}(\\varepsilon)$ is a successor contradicting $\\varepsilon\\in A_s$. Similarly (and even easier), this holds if $\\varepsilon\\in A_s\\setminus X_{s,f}$. It follows that for some $\\beta<\\beta_s$, $h_f\/\\mathcal{D}_s\\leq_{\\mathcal{D}_s} (h_{s,\\beta}\\restriction A_s)\/\\mathcal{D}_s$ and it is easy to check that this $\\beta$ satisfies (4).\n\\end{claimproof}\n\nFor any $f\\in \\Omega^\\textbf{a}_s$, $n<\\omega$ and $\\beta<\\beta_s$ let $B_n(f,h_{s,\\beta})=\\{\\varepsilon<\\kappa: |\\{i\\in J_\\varepsilon: f(i)\\geq h_{s,\\beta}(\\varepsilon)\\}|\\leq n\\}$. By Claim \\ref{C: sequence in ultraproduct}(4), we may set $\\beta_{s,n}(f)=\\min \\{\\beta: B_n(f,h_{s,\\beta})\\in \\mathcal{D}\\}$.  Note that $\\beta_{s,n}(f)\\geq \\beta_{s,n+1}(f)$. Let $\\beta_s(f)=\\min \\{\\beta_{s,n}(f):n<\\omega\\}$ and let $n_s(f)=\\min \\{n<\\omega: (\\forall k\\geq n)(\\beta_{s,k}(f)=\\beta_{s,n}(f)\\}$.\n\n\\begin{claim}\\label{C:=n_s(f)}\n$\\{\\varepsilon<\\kappa: |\\{i\\in J_\\varepsilon: f(i)\\geq h_{s,\\beta_s(f)}(\\varepsilon)\\}|=n_s(f)\\}\\in \\mathcal{D}$.\n\\end{claim}\n\\begin{claimproof}\nCall this set $Y_{s,f}$. Note that $Y_{s,f}\\subseteq B_{n_s(f)}(f,h_{s,\\beta_s(f)})$.\n\nIf $n_s(f)=0$ then $Y_{s,f}=B_0(f,h_{s,\\beta_{s,0}(f)})\\in \\mathcal{D}$. Assume $n_s(f)>0$. If $Y_{s,f}\\notin \\mathcal{D}$ then $\\{\\varepsilon<\\kappa: |\\{i\\in J_\\varepsilon: f(i)\\geq h_{s,\\beta_s(f)}(\\varepsilon)\\}|\\leq n_s(f)-1\\}\\in \\mathcal{D}$. So $\\beta_{s,n_s(f)-1}\\leq \\beta_s(f)=\\beta_{s,n_s(f)}(f)$, contradiction.\n\\end{claimproof}\n\nFor $s\\in S_1$, $\\beta<\\beta_s$ and $n<\\omega$, let $\\Omega_{(s,\\beta,n)}=\\{f\\in \\Omega^\\textbf{a}_s: \\beta_s(f)=\\beta,\\, n_s(f)=n\\}$.\nLet $S_\\textbf{b}=\\{(s,\\beta,n):s\\in S_1,\\, \\beta< \\beta_s,\\, n<\\omega,\\, \\Omega_{(s,\\beta,n)}\\neq\\emptyset \\}\\cup S_0$ and let $g:S_\\textbf{b}\\to S_\\textbf{a}$ be the function defined by $g(s,\\beta,n)=s$ for $s\\in S_1$ and $g(s)=s$ otherwise. For any $s\\in S_0$ let $\\Omega^\\textbf{b}_s=\\Omega^\\textbf{a}_s$, $h^\\textbf{b}_s=h^\\textbf{a}_s$ and $\\rho^\\textbf{b}_s={\\rho^\\textbf{a}_s}^\\frown\\langle 0\\rangle $. For $s\\in S_1$, $\\beta<\\beta_s$ and $n<\\omega$, we set $\\Omega^\\textbf{b}_{(s,\\beta,n)}=\\Omega_{(s,\\beta,n)}$, $\\rho^\\textbf{b}_{(s,\\beta,n)}={\\rho^\\textbf{a}_s}^\\frown \\langle n\\rangle$ and \n\\[\nh^\\textbf{b}_{(s,\\beta,n)}=\n\\begin{cases}\nh_{s,\\beta} & n=0 \\\\\nh^\\textbf{a}_s & n>0\n\\end{cases}\n.\\]\n\n\\begin{claim}\n$\\textbf{b}$ is an approximation and $\\textbf{a}\\trianglelefteq_g \\textbf{b}$.\n\\end{claim}\n\\begin{claimproof}\nWe check that $\\textbf{b}$ satisfies $(1)$ and $(2)$ from the definition. $(1)$ follows by the choice of $h^\\textbf{b}_{(s,\\beta,n)}$.\n\nWe are left with $(2)$. Let $t\\in S_0$ and $(s,\\beta,n)$ with $s\\in S_1$. If $\\rho^\\textbf{b}_t=\\rho^\\textbf{b}_{(s,\\beta,n)}$ then $\\rho^\\textbf{a}_t=\\rho^\\textbf{a}_s$ so the result follows since $\\textbf{a}$ is an approximation. Let $(s_1,\\beta_1,n_1)\\neq (s_2,\\beta_2,n_2)\\in S_\\textbf{b}$. If $\\rho^\\textbf{b}_{(s_1,\\beta_1,n_1)}=\\rho^\\textbf{b}_{(s_2,\\beta_2,n_2)}$ then $\\rho^\\textbf{a}_{s_1}=\\rho^\\textbf{a}_{s_2}$. If $s_1\\neq s_2$ then the results follows since $\\textbf{a}$ is an approximation. So assume that $s=s_1=s_2$ and $n=n_1=n_2$. Assume that $\\beta_1<\\beta_2<\\beta_s$ and let $f_1\\in \\Omega^\\textbf{b}_{(s,\\beta_1,n)}$ and $f_2\\in \\Omega^\\textbf{b}_{(s_,\\beta_2,n)}$. We need to show that $f_1\\not\\mathrel{R} f_2$ and $f_2\\not\\mathrel{R} f_1$.\n\nBy choice of $\\beta_1=\\beta_s(f_1)$ and $n=n_s(f_1)$, $B_n(f_1,h_{s,\\beta_1})\\in \\mathcal{D}$. On the other hand, since $\\beta_1<\\beta_2=\\beta_s(f_2)\\leq \\beta_{s,n+2}(f_2)$, $B_{n+2}(f_2,h_{s,\\beta_1})\\notin \\mathcal{D}$. I.e. $\\kappa\\setminus B_{n+2}(f_2,h_{s,\\beta_1})\\in \\mathcal{D}$. Let $\\varepsilon\\in B_n(f_1,h_{s,\\beta_1})\\cap (\\kappa\\setminus B_{n+2}(f_2,h_{s,\\beta_1}))$ and let $i_l$ be the $(n+l)$-th element of $J_\\varepsilon$ from the end, for $l=1,2,3$.\nAs a result,\n\\[f_1(i_3)<f_1(i_1)<h_{s,\\beta_1}(\\varepsilon)\\leq f_2(i_3).\\] Consequently, $f_1\\not\\mathrel{R} f_2$ by Assumption \\ref{A:R}(3) (since $\\suc{\\suc{i_3}}=i_1$) and $f_2\\not\\mathrel{R} f_1$ by Assumption \\ref{A:R}(1).\n\n$\\textbf{a}\\trianglelefteq_g \\textbf{b}$ by construction.\n\\end{claimproof}\n\nTo complete the proof, we note that for $(s,\\beta,n)\\in S_\\textbf{b}$ with $n>0$, $\\Omega^\\textbf{b}_{(s,\\beta,n)}$ is trivial. Let $f_1,f_2\\in \\Omega^\\textbf{b}_{(s,\\beta,n)}$. By Claim \\ref{C:=n_s(f)}, and the assumptions on $\\mathcal{D}$, we may find $\\varepsilon<\\kappa$ such that for $l=1,2$ the following holds\n\\begin{enumerate}\n\\item $|\\{i\\in J_\\varepsilon: f_l(i)\\geq h_{s,\\beta}(\\varepsilon)\\}|=n$ and\n\\item $|J_\\varepsilon|>n$.\n\\end{enumerate}\nSince $n>0$, letting $i$ be the $n+1$-th element from the end of $J_\\varepsilon$ we have that $f_l(i)<h_{s,\\beta}(\\varepsilon)\\leq f_l(\\suc{i})$, for $l=1,2$. If $f_1\\mathrel{R} f_2$ then $f_1(\\suc{i})\\leq f_2(i)<h_{s,\\beta}(\\varepsilon)$ by Assumption \\ref{A:R}(2), contradiction.\n\n\\end{proof}\n\n\\begin{proposition}\\label{P:approximation,proper-successor}\nLet $\\textbf{a}$ be an approximation.  Then there exists an approximation $\\textbf{c}$ and a surjective function $r:S_\\textbf{c}\\to S_\\textbf{a}$ such that $\\textbf{a} \\triangleleft_r \\textbf{c}$.\nMoreover, for $t\\in S_\\textbf{c}$, if $\\rho^\\textbf{a}_{r(t)}\\in \\chi^{\\xi}$, with $\\xi<\\sigma$, then $\\rho^\\textbf{c}_t\\in \\chi^{\\xi+2}$.\n\\end{proposition}\n\\begin{proof}\nLet $\\textbf{a}\\trianglelefteq_g \\textbf{b}$ be the approximation supplied by Proposition \\ref{P:approx-small-cof} and let $\\textbf{b}\\trianglelefteq_f \\textbf{c}$ be the approximation supplied by Proposition \\ref{P:approx-large-cof}.\nNote that $\\textbf{a}\\trianglelefteq_{g\\circ f}\\textbf{c}$ by Lemma \\ref{L:composition-approx}, we claim that $\\textbf{a} \\triangleleft_{g\\circ f} \\textbf{c}$.\n\nLet $t\\in S_\\textbf{c}$. Since $\\textbf{a}$ is an approximation and $\\sup_{\\varepsilon <\\kappa} |J_\\varepsilon|=\\omega$, we cannot have that $\\{\\varepsilon<\\kappa: \\cf(h_{gf(t)}^\\textbf{a}(\\varepsilon))=0\\}\\in \\mathcal{D}$, see Definition \\ref{D:def of approximation}(1). \n\nAssume that $\\{\\varepsilon<\\kappa: 0<\\cf(h_{gf(t)}^\\textbf{a}(\\varepsilon))\\leq \\chi\\}\\in \\mathcal{D}$. If $\\Omega^\\textbf{a}_{gf(t)}$ is trivial then so is $\\Omega^\\textbf{c}_t$, so assume not. By Proposition \\ref{P:approx-small-cof}(1) applied to $f(t)\\in S_\\textbf{b}$, either $\\Omega_{f(t)}^\\textbf{b}$ is trivial (and thus so is $\\Omega^\\textbf{c}_t$) or $\\{\\varepsilon<\\kappa: h_{f(t)}^\\textbf{b}<h^\\textbf{a}_{gf(t)}(\\varepsilon)\\}\\in \\mathcal{D}$. If it is the latter then, since $\\{\\varepsilon<\\kappa: h^\\textbf{c}_t(\\varepsilon)\\leq h^\\textbf{b}_{f(t)}(\\varepsilon)\\}\\in \\mathcal{D}$, we conclude that $\\{\\varepsilon<\\kappa:h^\\textbf{c}_t(\\varepsilon)<h^\\textbf{a}_{gf(t)}(\\varepsilon)\\}\\in \\mathcal{D}$.\n\nAssume that $\\{\\varepsilon<\\kappa:\\cf(h^\\textbf{a}_{gf(t)}(\\varepsilon))>\\chi\\}\\in \\mathcal{D}$. In particular, since by Proposition \\ref{P:approx-small-cof}(2), $h^\\textbf{a}_{gf(t)}=h^\\textbf{b}_{f(t)}$, $\\{\\varepsilon<\\kappa:\\cf(h^\\textbf{b}_{f(t)}(\\varepsilon))>\\chi\\}\\in \\mathcal{D}$. Assuming that $\\Omega^\\textbf{c}_{t}$ is not trivial, the by Proposition \\ref{P:approx-large-cof}(1), $\\{\\varepsilon<\\kappa: h^\\textbf{c}_t(\\varepsilon)<h^\\textbf{b}_{f(t)}(\\varepsilon)\\}\\in \\mathcal{D}$. Since $\\{\\varepsilon<\\kappa: h^\\textbf{b}_{f(t)}(\\varepsilon)\\leq h^\\textbf{a}_{gf(t)}(\\varepsilon)\\}\\in \\mathcal{D}$ we conclude that $\\{\\varepsilon<\\kappa:h^\\textbf{c}_t(\\varepsilon)<h^\\textbf{a}_{gf(t)}(\\varepsilon)\\}\\in \\mathcal{D}$, as needed.\n\nThe moreover part follows immediately from the construction.\n\\end{proof}\n\n\n\\begin{lemma}\\label{L:approx-limit}\nLet $\\delta<\\sigma$ be a limit ordinal and $\\langle \\textbf{a}_\\alpha:\\alpha<\\delta\\rangle$ a sequence of approximations. Assume that for $\\alpha<\\beta<\\delta$, $\\textbf{a}_\\alpha\\trianglelefteq_{g_{\\alpha,\\beta}}\\textbf{a}_\\beta$ and that for $\\alpha<\\beta<\\gamma<\\delta$, $g_{\\alpha,\\beta}\\circ g_{\\beta,\\gamma}=g_{\\alpha,\\gamma}$. Then the inverse limit exists, i.e. there are $(\\textbf{a}_\\delta, \\langle g_{\\alpha,\\delta}: \\alpha<\\delta\\rangle)$ such that $\\textbf{a}_\\alpha\\trianglelefteq_{g_{\\alpha,\\delta}}\\textbf{a}_\\delta$ and for $\\alpha<\\beta<\\delta$, $g_{\\alpha,\\beta}\\circ g_{\\beta,\\delta}=g_{\\alpha,\\delta}$. \n\nIn particular, if $\\textbf{a}_\\alpha \\triangleleft_{g_{\\alpha,\\beta}}\\textbf{a}_\\beta$ for some $\\alpha<\\beta<\\delta$ then $\\textbf{a}_\\alpha \\triangleleft_{g_{\\alpha,\\delta}}\\textbf{a}_\\delta$.\n\nFurthermore, for any $t\\in S_{\\textbf{a}_\\delta}$, if $\\xi=\\sup\\{\\zeta: \\rho^{\\textbf{a}_\\alpha}_{g_{\\alpha,\\delta}(t)}\\in \\chi^\\zeta,\\, \\alpha<\\delta\\}$ then $\\rho^{\\textbf{a}_\\delta}_t\\in \\chi^{\\xi+1}$.\n\\end{lemma}\n\\begin{proof}\nFor every $f\\in \\Omega$ let $t_f\\in \\prod_{\\alpha<\\delta}S_{\\textbf{a}_\\alpha}$ be the function defined by $t_f(\\alpha)=s$ if and only if $f\\in \\Omega^{\\textbf{a}_\\alpha}_s$. Note that for $\\alpha<\\beta<\\delta$, since for any $s\\in S_{\\textbf{a}_\\alpha}$, $\\{\\Omega^{\\textbf{a}_\\beta}_t:t\\in g_{\\alpha,\\beta}^{-1}(s)\\}$ is a partition of $\\Omega^{\\textbf{a}_\\alpha}_s$, necessarily $g_{\\alpha,\\beta}(t_f(\\beta))=t_f(\\alpha)$.\n\nLet $S_*=\\{t_f: f\\in \\Omega\\}$ and for any $t\\in S_*$, let $\\Omega_t=\\{f\\in \\Omega: t_f=t\\}$. Clearly, it is a partition of $\\Omega$. Furthermore, note that if $t_1,t_2\\in S_*$ and $\\alpha<\\delta$ is such that $t_1(\\alpha)=t_2(\\alpha)$ then $t_1(\\alpha^\\prime)=t_2(\\alpha^\\prime)$ for any $\\alpha^\\prime \\leq\\alpha$. \n\nLet $S_0=\\{t\\in S_*: (\\exists \\alpha <\\delta) (\\Omega^{\\textbf{a}_\\alpha}_{t(\\alpha)} \\text{ is trivial})\\}$ and $S_1=S_*\\setminus S_0$. \n\nFor any $t\\in S_1$ and $\\varepsilon<\\kappa$, let $A_{t,\\varepsilon}=\\{h^{\\textbf{a}_\\alpha}_{t(\\alpha)}(\\varepsilon):\\alpha<\\delta\\}$. Obviously, $1\\leq |A_{t,\\varepsilon}|\\leq |\\delta|<\\sigma\\leq\\chi$.\n\nFor every $(t,h\/\\mathcal{D})\\in S_1\\times\\prod_{\\varepsilon<\\kappa} A_{t,\\varepsilon}\/\\mathcal{D} $ let $\\Omega_{(t,h\/\\mathcal{D})}=\\{f\\in \\Omega_t: \\forall^\\mathcal{D} \\varepsilon<\\kappa,\\, h(\\varepsilon)=\\min \\{x\\in A_{t,\\varepsilon}:\\Rg(f\\restriction J_\\varepsilon)\\subseteq x\\}\\}$. Let $S_{\\textbf{a}_{\\delta}}=\\{(t,h\/\\mathcal{D})\\in S_1\\times\\prod_{\\varepsilon<\\kappa} A_{t,\\varepsilon}\/\\mathcal{D}: \\Omega_{(t,h\/\\mathcal{D})}\\neq \\emptyset\\}\\cup S_0$. For $t\\in S_0$, set $\\Omega^{\\textbf{a}_\\delta}_t=\\Omega^{\\textbf{a}_\\alpha}_{t(\\alpha)}$ and $h^{\\textbf{a}_\\delta}_t=h^{\\textbf{a}_\\alpha}_{t(\\alpha)}$, where $\\alpha<\\delta$ is minimal such that $\\Omega^{\\textbf{a}_\\alpha}_{t(\\alpha)}$ is trivial. For $(t,h\/\\mathcal{D})\\in S_{\\textbf{a}_\\delta}\\setminus S_0$, set $\\Omega^{\\textbf{a}_\\delta}_{(t,h\/\\mathcal{D})}=\\Omega_{(t,h\/\\mathcal{D})}$. \n\nNote that if $t\\in S_1$ then $\\{\\varepsilon<\\kappa: \\exists x\\in A_{t,\\varepsilon}( \\Rg(f\\restriction J_\\varepsilon)\\subseteq x)\\}\\in \\mathcal{D}$ because this set contains $\\{\\varepsilon<\\kappa: \\Rg(f\\restriction J_\\varepsilon)\\subseteq h^{\\textbf{a}_0}_{t(0)}(\\varepsilon)\\}$, which is $\\mathcal{D}$ since $\\textbf{a}_0$ is an approximation and $f\\in \\Omega^{\\textbf{a}_0}_{t(0)}$. Thus for any $t\\in S_1$ and for every $f\\in \\Omega_t$ there is a unique $h\/\\mathcal{D}\\in \\prod_{\\varepsilon<\\kappa}A_{t,\\varepsilon}\/\\mathcal{D}$ such that $f\\in \\Omega_{(t,h\/\\mathcal{D})}^{\\textbf{a}_\\delta}$. We choose $h^{\\textbf{a}_\\delta}_{(t,h\/\\mathcal{D})}$ to be any representative of the class $h\/\\mathcal{D}$.\n\nFor any $t\\in S_0$ and $\\alpha<\\delta$, $g_{\\alpha,\\delta}(t)=t(\\alpha)$, and for every $(t,h\/\\mathcal{D})\\in S_{\\textbf{a}_\\delta}\\setminus S_0$  and $\\alpha<\\delta$, $g_{\\alpha,\\delta}((t,h\/\\mathcal{D}))=t(\\alpha)$. Note that it already follows that for $\\alpha<\\beta<\\delta$, $g_{\\alpha,\\beta}\\circ g_{\\beta,\\delta}=g_{\\alpha,\\delta}$.\n\nFor any $t\\in S_0$ let \\[\\rho^{\\textbf{a}_\\delta}_t=\\bigcup\\{\\rho_{t(\\alpha)}^{\\textbf{a}_\\alpha}:\\alpha<\\delta\\}^\\frown \\langle 0\\rangle.\\] \n\nNow let $(t,h\/\\mathcal{D})\\in S_{\\textbf{a}_\\delta}\\setminus S_0$. Since $\\chi^{\\kappa}=\\chi$, there exists some $\\gamma_t\\leq \\chi$ and an enumeration of $\\prod_{\\varepsilon<\\kappa} A_{t,\\varepsilon}\/\\mathcal{D}$ as  $\\langle h_{t,\\gamma}\/\\mathcal{D}: \\gamma<\\gamma_t\\rangle$. Now for $(t,h\/\\mathcal{D})\\in S_{\\textbf{a}_\\delta}$ set \\[{\\rho^{\\textbf{a}_\\delta}_{(t,h\/\\mathcal{D})}=\\bigcup\\{\\rho_{t(\\alpha)}^{\\textbf{a}_\\alpha}:\\alpha<\\delta\\}}^\\frown \\langle\\gamma\\rangle,\\] where $h\/\\mathcal{D}=h_{t,\\gamma}\/\\mathcal{D}$. Assume that $t=t_f$ for some $f\\in \\Omega$ and let $\\alpha<\\beta<\\delta$. Since $g_{\\alpha,\\beta}(t(\\beta))=t(\\alpha)$, $\\rho^{\\textbf{a}_\\alpha}_{t(\\alpha)}$ is an initial segment of $\\rho^{\\textbf{a}_\\beta}_{t(\\beta)}$. This implies that $\\rho^{\\textbf{a}_\\delta}_{(t,h\/\\mathcal{D})}\\in \\chi^{\\xi+1}$, where $\\xi=\\sup\\{\\zeta: \\rho^{\\textbf{a}_\\alpha}_{g_{\\alpha,\\delta}(t)}\\in \\chi^\\zeta,\\, \\alpha<\\delta\\}$.\n\nWe check that $\\textbf{a}_\\delta$ is an approximation. Item (1) of Definition \\ref{D:def of approximation} follows from the definition of $\\Omega^{\\textbf{a}_\\delta}_{t}$ and the choice of $h^{\\textbf{a}_\\delta}_t$, for $t\\in S_{\\textbf{a}_\\delta}$. \n\nWe show item (2). Let $(t_1,h_1\/\\mathcal{D})\\neq (t_2,h_2\/\\mathcal{D})\\in S_{\\textbf{a}_\\delta}\\setminus S_0$, assume that $\\rho^{\\textbf{a}_\\delta}_{(t_1,h_1\/\\mathcal{D})}=\\rho^{\\textbf{a}_\\delta}_{(t_2,h_2\/\\mathcal{D})}$ and let $f_1\\in \\Omega^{\\textbf{a}_\\delta}_{(t_1,h_1\/\\mathcal{D})},\\, f_2\\in \\Omega^{\\textbf{a}_\\delta}_{(t_2,h_2\/\\mathcal{D})}$. If $t_1\\neq t_2$ then there exists some $\\alpha<\\delta$ such that $t_1(\\alpha)\\neq t_2(\\alpha)$. But $\\rho^{\\textbf{a}_\\alpha}_{t_1(\\alpha)}=\\rho^{\\textbf{a}_\\alpha}_{t_2(\\alpha)}$ and hence $f_1\\not\\mathrel{R} f_2$. Assume that $t_1=t_2$. Since $\\rho^{\\textbf{a}_\\delta}_{(t_1,h_1\/\\mathcal{D})}=\\rho^{\\textbf{a}_\\delta}_{(t_2,h_2\/\\mathcal{D})}$, $h_1\/\\mathcal{D}=h_2\/\\mathcal{D}$ which gives a contradiction. Let $(t,h\/\\mathcal{D})\\in S_{\\textbf{a}_\\delta}\\setminus S_0$ and $s\\in S_0$. If $s\\neq t$ then the same argument as above applies. On the other it cannot be that $s=t$ be the definition of $S_0$. If $t_1\\neq t_2\\in S_0$ then the same arguments as above applies.\n\nFinally, we show that $\\textbf{a}_\\alpha\\trianglelefteq_{g_{\\alpha,\\delta}}\\textbf{a}_\\delta$, for $\\alpha<\\delta$. Items (1), (2) and (4) are straightforward. We show item (3). Let $t\\in S_0$ and let $\\alpha^\\prime<\\delta$ be minimal such that $\\Omega^{\\textbf{a}_{\\alpha^\\prime}}_{t(\\alpha^\\prime)}$ is trivial. If $\\alpha^\\prime\\leq \\alpha$ then $h_t^{\\textbf{a}_\\delta}=h^{\\textbf{a}_{\\alpha^\\prime}}_{t(\\alpha^\\prime)}=h^{\\textbf{a}_{\\alpha}}_{t(\\alpha)}$. If $\\alpha<\\alpha^\\prime$ then $\\{\\varepsilon<\\kappa: h^{\\textbf{a}_\\delta}_t(\\varepsilon)\\leq h^{\\textbf{a}_\\alpha}_{t(\\alpha)}(\\varepsilon)\\}\\in \\mathcal{D}$ because $h_t^{\\textbf{a}_\\delta}=h^{\\textbf{a}_{\\alpha^\\prime}}_{t(\\alpha^\\prime)}$. Now let $(t,h\/\\mathcal{D})\\in S_{\\textbf{a}_\\delta}\\setminus S_0$. Since $\\Omega^{\\textbf{a}_\\delta}_{(t,h\/\\mathcal{D})}$ is non-empty, we may choose some function $f\\in \\Omega^{\\textbf{a}_\\delta}_{(t,h\/\\mathcal{D})}$. On the one hand, since $f\\in \\Omega^{\\textbf{a}_\\alpha}_{t(\\alpha)}$ and since $\\textbf{a}_\\alpha$ is an approximation, $\\{\\varepsilon<\\kappa: \\Rg(f\\restriction J_\\varepsilon)\\subseteq h^{\\textbf{a}_\\alpha}_{t(\\alpha)}(\\varepsilon)\\}\\in \\mathcal{D}$. On the other hand, since $f\\in \\Omega^{\\textbf{a}_\\delta}_{(t,h\/\\mathcal{D})}$, $\\{\\varepsilon<\\kappa: h(\\varepsilon)=\\min \\{x\\in A_{t,\\varepsilon}:\\Rg(f\\restriction J_\\varepsilon)\\subseteq x\\}\\}\\in \\mathcal{D}$. Combining these observation with the fact that $h^{\\textbf{a}_\\alpha}_{t(\\alpha)}\\in A_{t,\\varepsilon}$, it follows that $\\{\\varepsilon<\\kappa: h(\\varepsilon)\\leq h^{\\textbf{a}_\\alpha}_{t(\\alpha)}(\\varepsilon)\\}\\in \\mathcal{D}$ since it contains the intersection of both sets.\n\\end{proof}\n\n\\begin{conclusion}\\label{Con:coloring-pcf}\nThere exists a function $c:\\Omega\\to \\chi$ satisfying that if $f_1,f_2\\in \\Omega$ and $f_1 \\mathrel{R} f_2$ then $c(f_1)\\neq c(f_2)$.\n\\end{conclusion}\n\\begin{proof}\nWe define $(\\textbf{a}_\\xi, \\langle g_{\\zeta,\\xi} : \\zeta<\\xi\\rangle)$ satisfying that $\\textbf{a}_\\zeta\\triangleleft_{g_{\\zeta,\\xi}}\\textbf{a}_\\xi$ (for $\\zeta<\\xi$), by induction on $\\xi<\\sigma=(2^\\kappa)^+$.\n\\begin{list}{\u2022}{}\n\\item If $\\xi=0$ then $\\Omega^{\\textbf{a}_0}=\\Omega$, $S_a=\\{0\\}$, $\\Omega_{0}^{\\textbf{a}_0}=\\Omega^{\\textbf{a}_0}$, $\\rho^{\\textbf{a}_0}_0=\\emptyset$ and $h_0^{\\textbf{a}_0}\\in \\mathcal{H}$ be the constant function $\\theta$.\n\\item If $\\xi=\\alpha +1$ for some $\\alpha<\\xi$ then let $\\textbf{a}_{\\alpha}\\triangleleft_{g_{\\alpha,\\xi}}\\textbf{a}_\\xi$ be the approximation supplied by Proposition \\ref{P:approximation,proper-successor}. For any $\\zeta\\leq \\alpha$ we define $g_{\\zeta,\\xi}=g_{\\zeta,\\alpha}\\circ g_{\\alpha,\\xi}$.\n\\item If $\\xi$ is a limit ordinal we apply Lemma \\ref{L:approx-limit}. \n\\end{list}\n\nIt follows by induction, and using Proposition \\ref{P:approximation,proper-successor}, Lemma \\ref{L:approx-limit}, that for $\\alpha<\\beta<\\sigma$ and $t\\in S_{\\textbf{a}_\\alpha}$ and $s\\in S_{\\textbf{a}_\\beta}$, if $\\rho^{\\textbf{a}_\\alpha}_t\\in \\chi^{\\lambda_1}$ and $\\rho^{\\textbf{a}_\\beta}_s\\in \\chi^{\\lambda_2}$ then $\\lambda_1<\\lambda_2$ and hence $\\rho^{\\textbf{a}_\\alpha}_t\\neq \\rho^{\\textbf{a}_\\beta}_s$.\n\n\\begin{claim}\n\\phantomsection\n\\begin{enumerate}\n\\item For any $\\rho\\in \\chi^{<\\sigma}$, \\[\\Omega_\\rho:=\\bigcup\\{\\Omega^{\\textbf{a}_\\xi}_s: \\xi<\\sigma,\\, s\\in S_{\\textbf{a}_\\xi},\\, \\Omega^{\\textbf{a}_\\xi}_s \\text{ is trivial and } \\rho^{\\textbf{a}_\\xi}_s=\\rho\\}\\] is trivial. \n\\item $\\Omega=\\bigcup_{\\rho\\in \\chi^{<\\sigma}}\\Omega_\\rho$.\n\\end{enumerate}\n\\end{claim}\n\\begin{claimproof}\n$(1)$ Let $\\rho\\in \\chi^{<\\sigma}$ and $f_1,f_2\\in \\Omega_\\rho$. By definition, there exist $s_1\\in S_{\\textbf{a}_{\\xi_1}}$ and $s_2\\in S_{\\textbf{a}_{\\xi_2}}$ such that $f_1\\in \\Omega^{\\textbf{a}_{\\xi_1}}_{s_1}$ and $f_2\\in \\Omega^{\\textbf{a}_{\\xi_2}}_{s_2}$. As noted above, since $\\rho^{\\textbf{a}_\\xi}_{s_1}=\\rho^{\\textbf{a}_{\\xi}}_{s_2}$, necessarily, $\\xi=\\xi_1=\\xi_2$.  If $s_1=s_2$ then $f_1\\not\\mathrel{R} f_2$ since $\\Omega^{\\textbf{a}_{\\xi}}_{s_1}=\\Omega^{\\textbf{a}_{\\xi}}_{s_2}$ is trivial. If $s_1\\neq s_2$ then by the definition of an approximation, since $\\rho^{\\textbf{a}_\\xi}_{s_1}=\\rho^{\\textbf{a}_{\\xi}}_{s_2}$, $f_1\\not\\mathrel{R} f_2$.\n\n$(2)$ Assume there exists some $f\\in \\Omega\\setminus \\bigcup_{\\rho\\in \\chi^{<\\sigma}}\\Omega_\\rho$.\nWe construct a a sequence of function $\\langle h_\\xi\\in \\mathcal{H}: \\xi<\\sigma \\rangle$ satisfying that for any $\\alpha<\\beta<\\sigma$, $h_\\beta<_\\mathcal{D} h_\\alpha$.\n\nFor any $\\xi<\\sigma$ let $h_\\xi=h^{\\textbf{a}_\\xi}_t$ for the unique $t\\in S_{\\textbf{a}_\\xi}$ such that $f\\in \\Omega^{\\textbf{a}_\\xi}_t$. By assumption, $\\Omega^{\\textbf{a}_\\xi}_t$ is non-trivial for any such $\\xi$ (otherwise $f\\in \\Omega_\\rho$ for $\\rho=\\rho_t^{\\textbf{a}_\\xi}$). For any $\\alpha<\\beta<\\sigma$, since $\\textbf{a}_\\alpha\\triangleleft_{g_{\\alpha,\\beta}} \\textbf{a}_\\beta$, $h_\\beta<_\\mathcal{D} h_\\alpha$.\n\nWe color pairs of function $\\{(h_\\alpha,h_\\beta): \\alpha<\\beta<\\sigma\\}$ by $\\kappa$ colors, by setting that $(h_\\alpha,h_\\beta)$ has color $\\varepsilon_{\\alpha,\\beta}<\\kappa$ if $\\varepsilon_{\\alpha,\\beta}$ is the minimal $\\varepsilon$ for which $h_\\beta(\\varepsilon)<h_\\alpha(\\varepsilon)$. We know that such an $\\varepsilon$ exists, since $h_\\beta<_\\mathcal{D} h_\\alpha$. By Erd\\\"os-Rado there exists a subset $A\\subseteq \\sigma$ of cardinality $\\kappa^+$ and $\\varepsilon<\\kappa$ such that for every $\\alpha<\\beta\\in A$, $h_\\beta(\\varepsilon)<h_\\alpha(\\varepsilon)$. This contradicts the fact that the ordinals are well-ordered.\n\\end{claimproof}\n\nRecalling that $\\chi^{<\\sigma}=\\chi$ (as cardinals), we may now define $c:\\Omega\\to \\chi$ by choosing for every $f\\in \\Omega$ some $\\rho\\in \\chi^{<\\sigma}$ such that $f\\in \\Omega_\\rho$ and setting $c(f)=\\rho$. \n\\end{proof}\n\n\\section{Conclusion: stable graphs}\\label{s:conclusion}\n\nWe combine the results of the previous sections to conclude.\n\n\\begin{theorem}\\label{T:Taylor for infinitary EM-models}\nLet $\\mathcal{L}$ be a first order language containing a binary relation $E$. Let $T$ be an $\\mathcal{L}$-theory specifying that $E$ is a symmetric and irreflexive relation. Let $G=(V;E,\\dots)\\models T$ be an infinitary EM-model based on $(\\alpha,\\theta)$, where $\\alpha\\in \\kappa^{U}$ for some set $U$, $\\kappa\\geq \\aleph_0$ a cardinal and $\\theta$ an ordinal with $\\kappa<\\theta$. Let $\\varkappa>2^{2^{<(\\kappa+\\aleph_1)}}+|T|\\cdot |U|$ be a regular cardinal.\n\nIf $\\chi(G)\\geq \\varkappa$ then $G$ contains all finite subgraphs of $\\text{Sh}_n(\\omega)$ for some $n\\in \\mathbb{N}$.\n\\end{theorem}\n\\begin{proof}\n\nBy Lemma \\ref{L:underlying set of Inf-EM is Inf-EM} there exists some $(\\widehat \\alpha,\\theta)$-indiscernible sequence $b=\\langle b_{i,\\eta}:i\\in \\widehat{U},\\, \\eta\\in (\\theta^{\\underline{\\widehat{\\alpha}_i}})_<\\rangle$ whose underlying set is $V$, where $\\widehat \\alpha\\in \\kappa^{\\widehat U}$ and  $\\widehat U$ is a set such that $|\\widehat{U}|\\leq |T|\\cdot |U|\\cdot \\kappa^{<\\kappa}.$ \n\nLet $B=\\{(i,\\eta):i\\in \\widehat U,\\, \\eta\\in (\\theta^{\\underline{\\widehat{\\alpha}_i}})_<\\}$ and $R=\\{((i_1,\\eta_1),(i_2,\\eta_2)): b_{i_1,\\eta_1} \\mathrel{E} b_{i_2,\\eta_2}\\}$. Since $(i,\\eta)\\mapsto b_{i,\\eta}$ is surjective and $((i_1,\\eta_1),(i_2,\\eta_2))\\in R \\iff (b_{i_1,\\eta_1} b_{i_2,\\eta_2})\\in E$, $\\chi(B,R)=\\chi(G)\\geq \\varkappa$ (by Fact \\ref{F:basic prop}(4)). Moreover, by Fact \\ref{F:homomorphism is enough} it is enough to prove the conclusion for the graph $(B,R)$.\n\nFor any $i\\in \\widehat{U}$ let $B_i=\\{(i,\\eta):\\eta\\in (\\theta^{\\underline{\\alpha_i}})_<\\}$. By Fact \\ref{F:basic prop}(1), since $B=\\bigcup_{i\\in \\widehat{U}} B_i$, $\\varkappa\\leq \\chi(B,R)\\leq \\sum_{i\\in \\widehat{U}}\\chi(B_i,R\\restriction B_i)$. By definition\\footnote{As $2^{<\\kappa}=\\sup\\{2^\\mu:\\mu<\\kappa\\}$, if $2^{<\\kappa}<\\kappa$ then $2^{2^{<\\kappa}}\\leq 2^{<\\kappa}$.} $\\kappa\\leq 2^{<\\kappa}$ which implies $\\kappa^{<\\kappa}\\leq \\kappa^{\\kappa}=2^{\\kappa}\\leq 2^{2^{<\\kappa}}$ and thus $\\varkappa>|\\widehat U|$. Since $\\varkappa$ is a regular cardinal there exists $i\\in \\widehat U$ with $\\chi(B_i,R\\restriction B_i)\\geq \\varkappa$. As a result, it is enough to prove the conclusion for the graph $((\\theta^{\\underline{\\widehat{\\alpha}_i}})_<,S)$, where $S=\\{(\\eta_1,\\eta_2):(i,\\eta_1)\\mathrel{R} (i,\\eta_2)\\}$.\n\nFor $P=\\{\\otp(\\bar a,\\bar b):(\\bar a,\\bar b)\\in S\\}$, by $(\\widehat{\\alpha},\\theta)$-indiscernibility $S=\\bigcup_{p\\in P} \\{(\\bar c,\\bar d)\\in ((\\theta^{\\underline{\\widehat{\\alpha}_i}})_<)^2 :\\otp(\\bar c,\\bar d)=p\\vee \\otp(\\bar d,\\bar c)=p\\}$. \n\nBy Fact \\ref{F:basic prop}(2), \n\\[\\varkappa\\leq \\chi((\\theta^{\\underline{\\widehat{\\alpha}_i}})_<,S))\\leq \\prod_{p\\in P}\\chi ((\\theta^{\\underline{\\widehat{\\alpha}_i}})_<,P_p),\\] where $P_p=\\{(\\bar c,\\bar d)\\in ((\\theta^{\\underline{\\widehat{\\alpha}_i}})_<)^2 :\\otp(\\bar c,\\bar d)=p\\vee \\otp(\\bar d,\\bar c)=p\\}$. \nAssume towards a contradiction that $\\chi ((\\theta^{\\underline{\\widehat{\\alpha}_i}})_<,P_p)\\leq \\beth_2(\\aleph_0)$ for all $p\\in P$. Hence $\\varkappa\\leq \\beth_2(\\aleph_0)^{2^{|\\alpha_i|+\\aleph_0}}\\leq \\beth_2(|\\alpha_i|+\\aleph_0)$. Since $|\\alpha_i|+\\aleph_0<\\kappa+\\aleph_1$ and $\\varkappa>2^{2^{<(\\kappa+\\aleph_1)}}$, we derive a contradiction.\n\nConsequently, there exists $p\\in P$ with $\\chi ((\\theta^{\\underline{\\widehat{\\alpha}_i}})_<,P_p)>\\beth_2(\\aleph_0)$ and we may conclude by Theorem \\ref{T:shift graphs in order-type-graphs}.\n\\end{proof}\n\n\\begin{corollary}\\label{C:main corollary}\nLet $G=(V,E)$ be a stable graph. If $\\chi(G)>\\beth_2(\\aleph_0)$ then $G$ contains all finite subgraphs of $\\text{Sh}_n(\\omega)$ for some $n\\in \\mathbb{N}$.\n\\end{corollary}\n\\begin{proof}\nLet $G=(V,E)$ be stable graph, $T=Th(G)$ and $T^{sk}$ be a complete expansion of $T$ with definable Skolem functions in the language $E\\in \\mathcal{L}^{sk}$.  \n\nWe apply Theorem \\ref{T:existence of gen em model in stable} with $\\kappa=\\aleph_1$, $\\mu=2^{\\aleph_1}$ and $\\lambda=2^{\\max\\{\\mu,|G|\\}}$. We get an infinitary EM-model $\\mathcal{G}^{sk}\\models T^{sk}$ based on $(\\alpha,\\lambda)$, where $\\alpha \\in \\kappa^U$ for some set $U$ of cardinality at most $\\mu$, such that $\\mathcal{G}=\\mathcal{G}^{sk}\\restriction \\{E\\}$ is saturated of cardinality $\\lambda$. Since $\\mathcal{G}$ is saturated of cardinality $>|G|$, we may embed $G$ as an elementary substructure of $\\mathcal{G}$. Since $\\chi(\\mathcal{G})\\geq \\chi(G)>\\beth_2(\\aleph_0)$ and the conclusion is an elementary property, it is enough to show it for $\\mathcal{G}$. \n\nSince $2^{2^{<(\\kappa+\\aleph_1)}}+|T|+|U|\\leq 2^{2^{\\aleph_0}}+\\aleph_0+\\mu\\leq \\beth_2(\\aleph_0)+2^{\\aleph_1}=  \\beth_2(\\aleph_0)$, Theorem \\ref{T:Taylor for infinitary EM-models} applies with $\\theta=\\lambda$ and $\\varkappa=(\\beth_2(\\aleph_0))^+$.\n\\end{proof}\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\tElectroencephalography (EEG) is a widely used tool for the assessment of neural activity in the human brain \\cite{Brette2012}. To estimate the area of the brain responsible for the measured data, one has to simulate the electric potential as induced by hypothetical current sources in the brain, i.e. the EEG forward problem has to be solved. While quasi-analytical solutions to the differential equation underlying the forward problem exist, these are only available in simplified geometries such as the multi-layer sphere model\\cite{DeMunck1993}. One thus requires numerical methods \n to incorporate\n accurate representations of the head's shape and volume conduction properties. \n Popular\n approaches are the boundary element method (BEM)\\cite{Mosher1999,Gramfort2011,Makarov2020} and the finite element method (FEM)\\cite{Medani2015,Vallaghe2010}. Here, we will focus on the FEM due to its flexibility in modeling complex geometries with inhomogeneous and anisotropic compartments \\cite{He2020,Vermaas2020,Beltrachini2018a,VanUitert2004,Schimpf2002,Wolters2007,Nuesing2016}.\n    Efficient solvers and the transfer matrix approach \\cite{Lew2009,Wolters2004}\n    allow significantly reduced computational costs\n\t\n\tWhen \n employing FEM,\n one usually choses between either a hexahedral or tetrahedral discretization of the head. Both choices come with their own strengths and limitations. The mesh creation requires a classification of the MRI into tissue types. This segmentation data often comes in the form of binary maps with voxels of around 1mm resolution, allowing for quick and simple hexahedral mesh generation. However, as head tissue surfaces are smooth, approximating them with regular hexahedra is bound to be inaccurate. While methods for geometry adaptation exist \\cite{Wolters2007}, the resulting meshes still have a (reduced) angular pattern. Furthermore, when applying a standard continuous Galerkin FE scheme, areas with very thin compartments may suffer from leakage effects where current can bypass the insulating effects of the skull \\cite{Sonntag2013}. To alleviate this, flux based methods, like the discontinuous Galerkin method, offer a robust alternative \\cite{Engwer2017a}. These however severely increase the number of DOF and thus necessary computational effort.\n\t\n\tSurface-based tetrahedral FEM approaches on the other hand are able to accurately model the curvature of smooth tissue surfaces. Creating high quality tetrahedra, e.g. ones fulfilling a delaunay criterion, require tissue surface representations in the form of triangulations first. These triangulations have to be free of self-intersections and are often nested, usually leading to modeling inaccuracies such neglecting skull holes or an artificial separation of gray matter and skull. Therefore, we will not discuss surface-based tetrahedral FEM approaches throughout this work. \n\n    In \\cite{Rice2013}, the impact of prone vs supine subject positioning on EEG amplitudes was investigated. In the small group study, average differences of up to eighty percent were found. These were accompanied by differences in MRI-based CSF-thickness estimation of up to thirty percent underlining the importance of correctly modeling CSF-thickness and areas of contact between skull and brain surfaces.\n\t\n\tRecently, an unfitted discontinuous Galerkin method (UDG) \\cite{Bastian2009} was introduced to solve the EEG forward problem\\cite{Nuesing2016}. Rather than working with mesh elements that are tailored to the geometry, it uses a background mesh which is cut by level set functions, each representing a tissue surface. It was shown to outperform the accuracy of a discontinuous Galerkin approach on a hexahedral mesh, while not being limited by the assumptions necessary to create tetrahedral meshes.\n\n\tExtending the ideas of the UDG method, this paper introduces a multi-compartment formulation of the CutFEM \\cite{Burman2015} for EEG source analysis. Compared to UDG, it operates on a simpler trial function space and adds a ghost penalty based on \\cite{Burman2010}, stabilizing small mesh elements. \n \n    The paper is structured as follows. After introducing the theory behind CutFEM, three successively more realistic scenarios are tested. These scenarios include a multi-layer sphere model, followed by realistic brain tissues embedded in spherical skull and scalp compartments. Finally, a fully realistic five compartment head model is used for source analysis of the P20\/N20 component of measured somatosensory evoked potentials (SEP). Comparison results from different FEM and meshing approaches will be considered throughout the scenarios.\n\t\n\t\\section{Theory}\n\t\\subsection{CutFEM}\n\tDeviating from classical, fitted FEM-approaches, where \n the mesh cells resolves tissue boundaries,\n CutFEM uses a level set based representation of domain surfaces. \n\tLet $\\Omega = \\bigcup_i \\Omega_i$ be the head domain divided into $m$ disjunct open subdomains, e.g. gray matter, white matter, CSF, skull, skin. The level set function for compartment $i$ is then defined as\n\t$$ \\Phi_i(x) \\left\\{ \\begin{array}{ l l }\n\t< 0, \\text{ if $x \\in \\Omega_i$} \\\\\n\t= 0, \\text{ if $x \\in \\partial \\Omega_i$} \\\\\n\t> 0, \\text{ else}\n\t\\end{array} \\right.\n\t$$\n\tand $\\mathcal{L}_i = \\{x \\in \\Omega : \\Phi_i(x) = 0\\}$ \n denotes its (zero) level set.\n\tWe proceed by defining a background domain $\\hat{\\Omega} \\subset \\mathbb{R}^3$ covering the head domain $\\Omega$. This background is then tesselated, yielding a regular hexahedral mesh $\\mathcal{T}(\\hat{\\Omega})$, the fundamental or background mesh.\n\tTaking on the level set representation, submeshes $\\mathcal{T}_h^i \\subset \\mathcal{T}_h(\\hat{\\Omega}) $ are created from the background mesh, containing all cells that have at least partial support within the respective subdomain $\\Omega_i$. This results in an overlap of submeshes at compartment interfaces. For each submesh we define a conforming $\\mathbb{Q}_1$ space $V_h^i$. Thus, up to this point, each submesh is treated the way a conforming Galerkin method would treat the entire mesh.\n\t\n\tThe difference then lies in restricting the trial and test functions to their respective compartment, effectively cutting them off at the boundary and giving rise to the name CutFEM.\n   \n    A fundamental mesh cell intersected by a level set $\\mathcal{L}_i$ is called a cut cell. Their respective fundamental cells are contained in multiple compartements and thus have more DOF.\n    On the other hand, compared to classical conforming discretizations, a coarser mesh resolution can be chosen, as the mesh does not have to follow small geometric features.\n\n\n\tAs the trial functions are only continuous on their respective compartment and cut off at the boundary, using them to approximate the electric potential requires internal coupling conditions at the tissue interfaces.\n\tWe\n define the internal skeleton as the union of all subdomain interfaces\n\t\\begin{equation}\n    \\begin{split}\t    \n\t\\Gamma = \\bigcup \\Big\\{\\bar\\Omega_i \\cap \\bar\\Omega_j :\n ~i \\neq j,~\\mu_{d-1}(\\bar\\Omega_i \\cap \\bar\\Omega_j)>0 \\Big\\}.\\label{eq:1}\n\t\\end{split}\n \\end{equation}\n\t$\\mu_{d-1}$ is the d-1 dimensional measure in d-dimensional space. For two sets $E,F$ sharing a common interface (an element of $\\Gamma$) and a, possibly discontinuous, function $u$ operating on them we can define a scalar- or vector-valued jump operator as $\\llbracket u \\rrbracket := u|_E \\cdot n_E + u|_F \\cdot n_F$ with $n_E, n_F$ the outer unit normal of the respective set. Additionally, a (skew-)weighted average can be stated as\n\t\\begin{align}\n\t\\{u  \\} &= \\omega_E u|_E + \\omega_F u|_F\\label{eq:2}\\\\\n\t\\{u  \\}^* &= \\omega_F u|_E + \\omega_E u|_F.\\label{eq:3}\n\t\\end{align}\n\twith \n\t$\\omega_E = \\frac{\\delta_E}{\\delta_E + \\delta_F}$,\n\t$\\delta_E = n_E^t\\sigma_E n_E.$\n\tHere, $\\sigma_E$ refers to the symmetric $3 \\times 3$, positive definite electric conductivity tensor on $E$. Note that $\t\\llbracket uv \\rrbracket  = \\llbracket u\\rrbracket \\{v \\}^* + \\{u \\} \\llbracket v \\rrbracket.$ The purpose of these definitions will become clear when deriving the weak formulation for our forward model.\n\t\n\tTypically, the EEG forward problem for the electric potential $u$ induced by a neural source term $f$ is derived from the quasi-static formulation of Maxwell's equations  \\cite{Brette2012}.\n\n\t\\begin{align}\n\t\\nabla \\cdot \\sigma \\nabla u &= f, \\; \\text{ in } \\bigcup\\limits_i \\Omega_i\\label{eq:4}\\\\\n    \\langle \\sigma \\nabla u,n\\rangle &= 0, \\; \\text{ on } \\partial\\bar{\\Omega}\\label{eq:5}\\\\\n    \\intertext{And in addition we require continuity of the electric potential and the electrix current}\n    \\llbracket u \\rrbracket &= 0, \\; \\text{ on } \\Gamma\\label{eq:6}\\\\\n\t\\llbracket \\sigma \\nabla u \\rrbracket &= 0, \\; \\text{ on } \\Gamma. \\label{eq:7}\n\t\\end{align} \n\n\tAs trial and test space we employ $V_h$ as direct sum of all $V_h^i$. \n\t\n\tThe weak formulation can be obtained by multiplying with a test function, integrating and applying subdomain wise integration by parts. This yields\n\t$$\n\t\\sum_{i}(\\int_{ \\Omega_i} \\sigma \\nabla u_h^i \\nabla v_h^i - f v_h^idx)  -  \\int_{\\Gamma} \\{\\sigma \\nabla u_h \\}\\llbracket v_h\\rrbracket dS = 0,\n\t$$\n\twhere the jump formula for a product of two functions as well as \\eqref{eq:7} were used.\n\t$u_h^i$ is the restriction of $u_h \\in V$ to $V_i$. A symmetry term $ \\pm  \\int_{\\Gamma} \\{\\sigma \\nabla v_h \\}\\llbracket u_h\\rrbracket dS$ is added to end up with either a symmetric or non-symmetric bilinearform.\n\t\n\tTo incorporate \\eqref{eq:6} \n   \n\ta Nitsche penalty term\\cite{Nitsche1971} is added, which weakly couples the domains and ensures coercivity\\cite{Burman2015}:\n\t\\begin{equation}\n\tP_{\\gamma}(u,v) = \\gamma \\nu_k\\int_{\\Gamma} \\frac{\\hat{\\sigma}}{\\hat{h}} \\llbracket u_h\\rrbracket \\llbracket v_h\\rrbracket dS.\\label{eq:8}\n\t\\end{equation}\n\n    Here\n    $\\nu_k, \\hat{h}, \\hat{\\sigma}$ are scaling parameters based on the ratio of cut cell area on each interfaces' side, dimension, degree of trial functions used and conductivity. See \\cite{DiPietro2012} for a further discussion of these. $\\gamma$ is a free parameter to be discussed later. \n\t\n\tA challenge is the shape of the cut-cells. Heavily distorted and sliver-like snippets significantly impact the stability\n   \n    and\n    lead to a deterioration of the stiffness matrix condition number. To alleviate this, a ghost penalty \\cite{Burman2010} term is used, which weakly couples the solution on the snippets to their neighbors. Said coupling takes place on the interfaces of all the fundamental mesh cells cut by a level set. Let\n    \\begin{equation}\n    \\begin{split}\n\t\\hat{\\Gamma} = \\cup \\Big\\{\\partial E_i: E_i \\in \\mathcal{T}_h, E_i \\cap \\Gamma \\neq \\emptyset \\}.\\label{eq:9}\n    \\end{split}\n\t\\end{equation}\n\tNote the difference between $\\Gamma$ and $\\hat{\\Gamma}$. $\\Gamma$  operates on compartment interfaces, $\\hat{\\Gamma}$ on faces of the fundamental mesh.\n\tThe ghost penalty is then defined as\n\t\\begin{equation}\n\ta^G (u_h,v_h) = \\gamma_G \\int_{\\hat{\\Gamma}} \\hat{h} \\llbracket \\sigma \\nabla u_h \\rrbracket \\llbracket \\nabla v_h \\rrbracket dS,\\label{eq:10}\n\t\\end{equation}\n\twhere\n\t$\\gamma_G$ is again a free parameter, usually a couple orders of magnitude smaller than $\\gamma$.\n\t\n\tThe weak CutFEM EEG-forward problem can now be stated as finding the electric potential $u_h \\in V_h$ such that\n\t\\begin{equation}\n\ta(u_h,v_h) + a_{n\/s}^N(u_h,v_h) + a^G(u_h,v_h) = l(v_h) \\; \\forall v_h \\in V_h,\\label{eq:11}\n\t\\end{equation}\n\twith\n\t\\begin{align*}\n        a(u_h,v_h) &= \\sum_{i} \\int_{ \\Omega_i} \\sigma \\nabla u_h^i \\nabla v_h^idx,\\\\\n        l(v_h) &=  \\sum_{i} \\int_{ \\Omega_i} f v_h^idx\n    \\end{align*}\n\tand\n\t\\begin{align*}\n\ta_{n\/s}^{N}(u_h,v_h) :=  &- \\int_{\\Gamma}\\{\\sigma \\nabla u_h \\}\\llbracket v_h\\rrbracket  \\pm \\int_{\\Gamma} \\{\\sigma \\nabla v_h \\} \\llbracket u_h \\rrbracket dS\\\\\n\t   &+ \\gamma \\nu_k\\int_{\\Gamma} \\frac{\\hat{\\sigma}}{\\hat{h}} \\llbracket u_h\\rrbracket \\llbracket v_h\\rrbracket dS.\n\t\\end{align*}\n\tIn the following we will refer to these two variants as NWIPG\/SWIPG,\n short for non-symmetric\/symmetric weighted interior penalty Galerkin method.\n\t\n\tIn \\cite{Oden1998,Guzman2009} it was shown that the non-symmetric DG-methods may result in a sub-optimal convergence rate in the L2-norm (full convergence in H1), a result that also extends to CutFEM\\cite{Burman2012}.\n\tHowever, while SWIPG is coercive only if $\\gamma$ is chosen sufficiently large\\cite{Burman2012}, NWIPG does not have such a limitation.\n\tTherefore, we will employ the NWIPG method throughout this paper due to its stability with regard to the selection of $\\gamma$.\n \n\t\\paragraph{Integration over the cut domains}\t\n\tFundamental cells that are cut by level sets, the cut cells, can be integrated over by employing a topology preserving marching cubes algorithm (TPMC) \\cite{Engwer2017}. The initial cell is divided into a set of snippets, each completely contained within one subdomain. These snippets are of a simple geometry and therefore easy to integrate over.  Thus, integrals over the fundamental cell or subdomain boundaries are replaced by integrals over the snippets or their boundaries. The trial functions are effectively cut off at the compartment boundaries.\n\t\n\tSee Fig. 1 for an overview of the reconstruction steps. Note that the trial functions are coupled to their respective submesh, not to the TPMC reconstruction of the domain. The latter only determines the area over which the functions are integrated.\n\t\n\t\\begin{figure}[t!]\n\t\t\\centering\n\t\n\t\n\t\n\t\t\\subfloat{\\includegraphics[width=1\\linewidth]{graphics\/PosterTPMC.png}}\n\t\t\\caption[Level sets over fundamental mesh and TPMC reconstruction]{Left: Fundamental mesh with two spherical level sets, topology preserving marching cubes reconstruction. Center: Overlapping submeshes for the two compartments enclosed by the level sets. Right: trial function space for the inner compartment with white dots representing degrees of Freedom, cut area that the DOF are restricted to.}\n\t\\end{figure}\n\t\n\tStarting on the fundamental mesh, the algorithm is applied once per level set. Each following iteration is applied on the cut cells of the previous iteration, i.e. first the fundamental mesh is cut, then the resulting snippets are cut.  This ensures the correct handling of mesh cells that are cut by multiple level sets.\n\n\t\\paragraph{Source model and transfer matrix}\n\tFollowing the principle of St. Venant, the source term $f$ will be approximated by a set of monopoles. Where fitted FEM use mesh vertices as monopole locations, this is not feasible for CutFEM as fundamental cells may have vertices not belonging to the source compartment. Rather, only gray matter cut cells are used and the locations are based on a Gauss-Legendre quadrature rule. For more information on the Venant source model, see \\cite{Medani2015}, \\cite{Buchner1997}.\n\t\n\tFor an accurate source analysis it is necessary to compute the EEG-forward solution for a large number, i.e. tens of thousands, of possible sources. However, the electric potential induced by a source is only of interest at a set of predetermined points, namely the electrodes at the scalp. So, rather than solving \\eqref{eq:11} for each source individually, a transfer matrix approach \\cite{Gencer2004,Wolters2004} is employed, significantly reducing the amount of computation time needed.\n\t\n\n\n\n\n\n\t\\section{Methods}\n\t\n\n\n\t\\subsection{Head models}\n\tFor numerical evaluations three progressively more realistic scenarios were created, two sphere models, one of which contains realistic brain tissues, and a five compartment model created from anatomical data. For each model, we will compare CutFEM and a geometry-adapted hexahedral CG-FEM approach (\\textit{Hex}) with a node shift for the geometry-adaptation of 0.33 \\cite{Wolters2007}. In the first model, the UDG approach of \\cite{Nuesing2016} will also be added to the comparisons. To balance computational load, \\textit{Hex} will use 1mm meshes whereas for CutFEM and UDG we use a 2mm background mesh.\n\t\\paragraph{Shifted spheres}\n\tThe first scenario contains the four spherical compartments brain, CSF, skull and scalp. The brain sphere will be shifted to one side, simulating a situation where the subject lies down and the brain sinks to the back of the skull. Conductivities were chosen according to \\cite{McCann2019} with the exception that CSF and brain use the same conductivity. In terms of volume conduction the model is thus indistinguishable from a 3-layer concentric sphere model and analytical solutions \\cite{DeMunck1993} are available as benchmark. Conductivity values and radii of the compartments can be found in Table 1.\n\t\\begin{table}[htp]\n\t\t\\begin{minipage}[c]{0.5\\textwidth}\n\t\t\t\\centering\n\t\t\t\\begin{tabular}{l|ccc}\n\t\t\t\t& Radius [mm] & Center [mm]       & $\\sigma$ [S\/m] \\\\\n                \\hline\n\t\t\t\tScalp & 92     & (127 127 127) & 0.43     \\\\\n\t\t\t\tSkull & 86     & (127 127 127) & 0.01     \\\\\n\t\t\t\tCSF   & 80     & (127 127 127) & 0.33     \\\\\n\t\t\t\tBrain & 78     & (129 127 127) & 0.33    \\\\[0.5ex]\n\t\t\t\\end{tabular}\n\t\t\\end{minipage}\\hfill\n\t\t\\begin{minipage}[c]{0.5\\textwidth}\n\t\t\t\\centering\n\t\t\n\t\t\t\\caption[Radii, Center and Conductivities for the Shifted Sphere Model]{Radii, center and conductivities for the shifted sphere model.}\n\t\t\\end{minipage}\n\t\\end{table}\n\t\n\t TPMC was applied twice, once on the fundamental mesh and once on the resulting cut cells. Note that this additional refinement step does not change the number of trial functions of the model. A total of 13.000 Evaluation points were distributed evenly throughout the inner sphere and lead fields for both radial and tangential source directions were computed at each point. For CutFEM, a combination of $\\gamma = 16$ and $\\gamma_G = 0.1$ has shown promising results. For UDG, no ghost penalty was implemented and $\\gamma = 4$ was chosen, following {Nuesing2016}.\n\t\n\t\\paragraph{Spheres containing realistic brain}\n\tIn the previous section, the level set functions could be computed analytically up to an arbitrary accuracy. In a realistic scenario where the segmentation quality is limited by the MRI resolution as well as partial volume effects and MRI artefacts, this is not the case. An easy way to pass level sets to CutFEM lies in using tissue probability maps (TPM), a typical intermediate result \\cite{Ashburner2014} from segmentation which provides for each voxel the probability that it is located in a certain compartment. \n\t\n\tTo examine the performance of CutFEM when used together with TPM's, another sphere model is employed, this time containing realistic gray and white matter compartments obtained from MRI scans of a human brain.\tThe subject was a healthy 24 year-old male from whom T1- and T2-weighted MRI scans were acquired using a 3 Tesla MR Scanner (MagnetomTrio, Siemens, Munich, Germany) with a 32-channel head coil. For the T1, a fast gradient-echo pulse sequence (TFE) using water selective excitation to avoid fat shift (TR\/TE\/FW = 2300\/3.51 ms\/8\u00b0, inversion pre-pulse with TI = 1.1 s, cubic voxels of 1 mm edge length) was used. For the\tT2, a turbo spin echo pulse sequence (TR\/TE\/FA = 3200\/408 ms\/90\u00b0, cubic voxels, 1 mm edge length) was used. TPM's were extracted from both T1- and T2-MRI using SPM12 \\cite{Ashburner2014} as integrated into fieldtrip \\cite{Oostenveld2011}. For each voxel, the average of both TPM's was computed and a threshold probability of 0.4 was set as zero-line.\n\t\n\tThe inner skull surface was defined as the minimal sphere containing the entire segmented brain with CSF filling the gaps. The spherical skull and scalp were chosen to have a thickness of 6mm. The same conductivities as before were used with CSF, gray and white matter being identical and again 200 sensors were placed on the scalp surface.\n\t\\paragraph{Realistic 5 compartment head model}\n\tAs an extension of the previous model, realistic 5 compartment head models were created using the same anatomical data, replacing the spherical skin, skull and CSF by realistic segmentations. Again, level sets were created from probability maps. To obtain smooth skull and scalp surfaces in the TPM case, binary maps of skull and skin were created following the procedure in \\cite{Antonakakis2020}. The level sets of skull\/skin were then calculated as an average of the binary map and the T1\/T2 TPM again with a threshold of 0.4. Following \\cite{Antonakakis2020}, the level sets were cut off below the neck to reduce computational load while maintaining a realistic current flow below the skull.\n\tAgain, lead fields from hexahedral meshes were created for comparison. DOF, number of cut cells\/mesh elements and the resulting number of snippets can be found in Table 2.\n\t\\begin{table}[t!]\n\t\t\\centering\n\t\t\\begin{tabular}{l|lll}\n\t\t\t& DOF & Cut cells\/elements & snippets         \\\\ \\hline\n\t\t\tCutFEM & 917,463     & 716,994 & 7,950,120     \\\\\n\t\t\t$\\textit{Hex}$ & 3,909,303     & 3,475,138 & -   \n\t\t\\end{tabular}\n\t\t\\centering\n\t\n\t\t\\caption[Number of DOF, Cut cells and snippets]{Number of degrees of freedom\/snippets\/cut cells for CutFEM, number of degrees of freedom\/elements for hexahedral mesh.}\n\t\\end{table}\n\t\\subsection{Forward and inverse comparisons}\n\tFor the two spherical scenarios, analytical forward solutions were calculated as reference. For the realistic cases, somatosensory evoked potentials were recorded and a dipole scan was performed as described in detail in (b). \n\t\n\tThe two latter scenarios including realistic gray\/white matter use a regular 2mm source grid created using Simbio \\footnote{www.mrt.uni-jena.de\/simbio}. It was ensured that the sources are located inside the gray matter compartment for both approaches (\\textit{Hex} + CutFEM). The resulting source space contains 58.542 different dipole locations with no orientation constraint being applied.\n\t\\paragraph{Error measures}\n\t\n\tTwo different metrics were employed to quantify the observed errors, the relative difference measure (RDM) and the magnitude error (MAG) \\cite{Wolters2007}. \n\t\n\tThe RDM measures the difference in potential distribution at the scalp electrodes.\n\t\\begin{equation}\n\tRDM (\\%)(u^{ana}, u^{num}) = 50*||\\frac{u^{ana}}{||u^{ana}||_2} -\\frac{u^{num}}{||u^{num}||_2}||_2.\\label{eq:12}\n\t\\end{equation}\n\tIt ranges from $0$ to $100$, the optimal value being $0$.\n\tMAG determines the differences in signal strength at the electrodes.\n\t\\begin{equation}\n\tMAG(u^{ana}, u^{num}) = 100*(\\frac{||u^{num}||_2}{||u^{ana}||_2}-1).\\label{eq:13}\n\t\\end{equation}\n\t\n\tMeasured in percent, its optimal value is $0$. It is unbounded from above and bound by $-100$ from below. $u^{ana}, u^{num} \\in \\mathbb{R}^s$ contain the analytical and numerical potential at the $s$ different sensor locations.\n\t\n\tCutFEM is implemented into the DUNEuro toolbox\\footnote{https:\/\/www.medizin.uni-muenster.de\/duneuro} \\cite{Schrader2021} where the FEM calculations were performed. Analytical EEG solutions were calculated using the fieldtrip toolbox \\cite{Oostenveld2011}.\n\t\n\tFor a comparison of runtime and memory usage, the forward calculation is split into 5 steps. The time necessary to create a driver, i.e. the time DUNEuro needs to setup the volume conductor, the times needed to assemble the stiffness matrix and AMG solver, the transfer matrix solving process using Dune-ISTL \\cite{Bastian2021} and the calculation of the final lead field matrix. All Computations are performed on a bluechip workstation with an AMD Ryzen Threadripper 3960X and 128 GB RAM. 16 threads are used to calculate the 200 transfer matrix\/lead field columns in parallel.\n\t\n\t\\paragraph{Somatosensory data and dipole scan}\n\tTo investigate CutFEM's influence on source reconstruction, an electric stimulation of the median nerve was performed on the same subject the anatomical data was acquired from. \n\tThe stimuli were monophasic square-wave pulses of 0.5ms width in random intervals between 350-450ms. The stimulus strength was adjusted such that the right thumb moved clearly.\n\tEEG data was measured using an 80 channel cap (EASYCAP GmbH, Herrsching, Germany, 74 channel EEG plus additional 6 channels EOG to detect eye artifacts). EEG positions were digitized using a Polhemus device (FASTRAK, Polhemus Incorporated, Colchester, Vermont, U.S.A.).\n\t2200 stimuli were digitally filtered between 20 to 250Hz (50Hz notch) and averaged to improve signal to noise ratio. A single dipole scan was conducted over the whole source space using the data at the peak and the CutFEM lead field.\n\t\n\tThe P20\/N20 component typically exerts a high signal to noise ratio and a strongly dipolar topography, making it an ideal candidate for a dipole scan approach as motivated for example by \\cite{Buchner1994}. \n\t\n    \\begin{figure*}[tp]\n\t\t\\centering\n\t\t\\subfloat{\\includegraphics[width=1\\linewidth]{graphics\/stretchedRadSphere.png}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[width=1\\linewidth]{graphics\/stretchedTanSphere.png}}\n\t\t\\caption[EEG forward modeling errors for \\textit{Hex} and unfitted FEM approaches]{EEG forward modeling errors for \\textit{Hex} and unfitted FEM approaches in a shifted sphere scenario Top: Errors for tangential source directions. Bottom: Errors for radial source directions. Errors are in percent and grouped by eccentricities. The green line marks optimal error values. The grey area indicates the physiologically most realistic eccentricities.}\n\t\\end{figure*}\n\t\n\\section{results}\n\n    The first investigated model is the shifted sphere scenario, where the brain sphere was moved within the CSF-sphere until there was exactly one contact point between skull and brain (see \\uproman{3} A). When comparing number of DOF and RAM usage, it is clear that CutFEM is by far the most memory efficient approach, using about a fifth of the number of trial functions and about a tenth of the amount of RAM as UDG (Table 3). \\textit{Hex} also uses significantly more resources than CutFEM.\n    \n    Regarding computation time, as UDG has to solve a significantly larger system, each iteration step in the solution phase takes longer than for CutFEM. As most time is spent on solving the system, CutFEM is overall around 16 minutes or 34 percent faster than UDG. The same cannot be said for comparisons to the standard \\textit{Hex} approach. While each iteration of the solver required less time than for \\textit{Hex}, it required an average of 92 iterations compared to 14 for \\textit{Hex}. The unfitted approaches spend less time calculating the final lead field as the time needed to locate each dipole within the 2mm background mesh is lower than for the 1mm hexahedral mesh. In total, the hexahedral CG was only faster than CutFEM by a negligible 3 percent or 52 seconds.\n    \n    \\begin{table}[ht]\n\t\t\\begin{minipage}[c]{0.5\\textwidth}\n\t\t\t\\centering\n\t\t\t\\begin{tabular}{l|lll}\n\t\t\t\t               & CutFEM & UDG      & \\textit{Hex}        \\\\ \\hline\n\t\t\t\tnumber DOF    & 552 985   & 3 601 824 &  3 341 280 \\\\\n\t\t\t\tmax. RAM used   & 6.91 GB   & 64.77 GB & 40.2GB \\\\ \\hline\n\t\t\t\tDriver setup    & 44s       & 45s      & 52s     \\\\\n\t\t\t\tMatrix assembly & 319s      & 161s     & 25s     \\\\\n\t\t\t\tSolver setup    & 353s      & 235s     & 45s     \\\\\n\t\t\t\tSolving         & 1111s     & 2367s    & 1550s   \\\\\n\t\t\t\tLead field      & 22s       & 20s      & 125s   \\\\ \\hline\n\t\t\t\tTotal time      & 1849s     & 2828s    & 1797s   \\\\\n\n\t\t\t\\end{tabular}\n\t\t\\end{minipage}\\hfill\n\t\t\\begin{minipage}[c]{0.4\\textwidth}\n\t\t\t\\centering\n\t\t\n\t\t\t\\caption[Computation times, RAM\/DoF usage]{Computation times, RAM\/degree of freedom usage in the shifted sphere model. }\n\t\t\\end{minipage}\n\t\\end{table}\n    \nError comparisons between CutFEM, UDG and \\textit{Hex} can be found in Fig. 2. CutFEM outperforms \\textit{Hex} in all eccentricity categories and for both radial and tangential source directions. As the pyramidal cells that give rise to the EEG potential are located in layer \\uproman{5} of the gray matter \\cite{Murakami2006}, eccentricities corresponding to 1-2mm distance to the skull are physiologically the most relevant. For eccentricities between 0.96 and 0.98 and both source directions CutFEM has average RDM\/MAG values of 0.18\\% and -0.06\\%, comparable to UDGs 0.17\\% and -0.2\\% and significantly lower than \\textit{Hex}'s 0.94\\% and 1.57\\%.\n    \nThe most pronounced differences are at low eccentricities or when looking at magnitudes. CutFEM performance is similar for both radial and tangential source directions, UDG shows similar or slightly better results at low eccentricities. However, except for radial RDM's, UDG deteriorates faster at high eccentricities above 0.98. As both operate on the same cut mesh, the larger variance in the UDG results can most likely be explained by CutFEM's use of the ghost penalty stabilization. The overall largest absolute error values for CutFEM are 3.08 $\\%$ RDM and 8.21 $\\%$ MAG, underlining its performance with regard to outliers. Due to the similar numerical accuracy of CutFEM and UDG, we will only compare CutFEM and \\textit{Hex} in the following scenarios.\n\n\\subsection{Sphere containing realistic brain}\nThe results in the previous section were achieved using analytically computed level sets. Deviating from this, we will now use use a semi-realistic case where realistic brain compartments are contained within spheres. \nAgain, several different penalty parameters were tried, showing that a combination of  $\\gamma = 40$ and a ghost penalty of $\\gamma_g = 0.5$ yields good results for CutFEM.\n\t\nThe results are presented in Fig 3. Note that eccentricity is stated with respect to the distance to the skull. As source points are only inside the gray matter the number of source points at high eccentricities is much lower. The eccentricity groups 0.98, 0.985, 0.99, 0.995 were thus combined into one group containing 136 points.\n\n\t    \\begin{figure*}[tp]\n\t\t\\centering\n\t\t\\subfloat{\\includegraphics[width=1\\linewidth]{graphics\/stretchedRad.png}}\n\t\t\\hfill\n\t\t\\subfloat{\\includegraphics[width=1\\linewidth]{graphics\/stretchedTan.png}}\n\t\t\\caption[EEG forward model errors for CG- and CutFEM - Realistic brain]{Overview of different EEG-errors for 5 layer continuous Galerkin- and CutFEM approaches using realistic brain compartments contained in spherical skull and scalp shells. Top: Errors for tangential source directions. Bottom: Errors for radial source directions. Errors are in percent and grouped by eccentricities. The green line marks optimal error values. The grey area indicates the physiologically most realistic eccentricities.}\n\t\\end{figure*}\n\t\nMuch like before, CutFEM remains well below 1.5 and 2 percent RDM and MAG respectively, whereas \\textit{Hex} has for nearly all eccentricities higher median values and more outliers going up to more than 1.5\\% RDM and 4\\% MAG. CutFEM is again more stable with regard to outliers and especially when looking at magnitudes, differences between the two methods are in the several percent range.\n\t\nOverall, it can be stated that CutFEM is about as fast as and more accurate than \\textit{Hex} and about as accurate as and faster than UDG.\n\t\n\\subsection{Realistic 5 compartment head model}\nFor the final part of this paper, two lead fields, one from CutFEM, one from hexahedral CG, were created using realistic 5 compartment head models including gray and white matter, CSF, skull and scalp tissues. Somatosensory evoked potentials were acquired from a medianus stimulation of the right hand.\n\t\n\\paragraph{Lead field differences} Before looking at inverse reconstructions, we will investigate the differences between the forward results. As the same source space and electrodes were used for both models, we can again compute MAG and RDM values. In the absence of an analytical solution, these measurements cannot capture errors but rather differences between the methods without making a clear statement which is the more accurate. \n\t\nFor visualization purposes, for each gray matter centerpoint of the \\textit{Hex} mesh the closest source point is identified, RDM and MAG are computed for each spatial direction and averages over the directions are calculated. The result can be seen in Fig. 4.  In both measures, the highest differences can be observed in inferior areas near the foramen magnum and optic channels or in superior areas. Overall the difference in potential distribution was 9.40 $\\pm$ 4.15$\\%$ and the difference in magnitude 18.94 $\\pm$ 12.03$\\%$. Interestingly, with a correlation coefficient of only 0.22,  high RDM values do not necessarily coincide with high MAG values.\n\n\t\\paragraph{Reconstruction of somatosensory stimulation}\n\tFinally, the CutFEM lead field was used to perform a source reconstruction of the P20 component of an electric wrist stimulation. A dipole scan was conducted over the entire source space, the result of which can be seen in Fig. 4. $93.03\\%$ of the data could be explained by a dipole with a strength of 5.8nAm. From the literature \\cite{Buchner1994}, one expects the P20 component to be located in Brodmann Area 3b, that is in the anterior wall of the postcentral gyrus (and oriented towards the motor cortex).\n\n\t\\section{Discussion}\n\tThe purpose of this paper is to introduce CutFEM, an unfitted FEM for applications in EEG forward modeling. After discussing the mathematical theory behind CutFEM and implementational aspects, three progressively more realistic scenarios are introduced, ranging from a multi-layer sphere model to the reconstruction of somatosensory evoked potentials.     \n\n\tAt similar computation times, CutFEM shows preferable results when compared to a geometry-adapted hexahedral CG-FEM \\cite{Wolters2007} in both a shifted sphere scenario and a sphere model with realistic brain tissues. While CutFEM requires significantly less DOF, both methods require similar computation times due to the different number of solver iterations. Thus, a thorough investigation of different iterative solver techniques such as multigrid methods and possibly a modification of the ghost penalty will be part of future work.\n \n    Compared to UDG \\cite{Bastian2009}, it is shown that CutFEM combined with a ghost penalty leads to a decrease in outlier values at high eccentricities as well as a significant reduction in memory consumption and computation time. \n\t\n\tUsing a realistic five compartment head model, CutFEM correctly localizes the somatosensory P20 in the expected Brodmann area 3b. Especially in applications such as presurgical epilepsy diagnosis such accurate reconstructions might contribute significantly to the correct localization of the irritative zone \\cite{Neugebauer2022}. The employed somatosensory experiment featured clear peaks and a high signal to noise ratio, making it an ideal candidate for an initial study. Further investigations and a larger study size are necessary to determine CutFEM's contribution to accurate source reconstructions when used with noisier data and\/or more advanced inverse methods.\n\n    In \\cite{Vallaghe2010}, a trilinear immersed FEM approach was introduced that, like CutFEM, employs level sets as tissue surfaces. Rather than using a Nitsche-based coupling, continuity of the electric potential is enforced by modifying the trial function space. Compared to CutFEM, no free parameters such as $\\gamma$ and $\\gamma_G$ are introduced but the absence of overlapping submeshes means that there is no increased resolution in areas with complex geometries.\n\n    In \\cite{Windhoff2013,Nielsen2018} the process of building a tetrahedral mesh from segmentation data is investigated. Surface triangulations that are free of topological defects, self-intersections or degenerate angles have to be created before volumetric meshing. The authors show that it is possible to create such high quality surfaces and subsequent tetrahedral meshes for realistic head models, however they may come at the cost of modeling inaccuracies such as the separation of gray matter and skull by a thin layer of CSF.\n    \n    A main advantage of CutFEM is its flexibility with regard to the anatomical input data. Level sets can be created from a variety of sources, such as tissue probability maps, binary images or surface triangulations. This simplifies the question of how to create a mesh from segmentation data.\n    \n        \\begin{figure*}[tp]\n\t\t\\centering\n\t\t\\subfloat[][]{\\includegraphics[width=0.27\\linewidth]{graphics\/realRdmHexTpm.png}}\n\t\t\\subfloat[][]{\\includegraphics[width=0.27\\linewidth]{graphics\/realMAGHexTpm.png}}\n\t\t\\subfloat[][]{\\includegraphics[width=0.27\\linewidth]{graphics\/P20CutPaper.png}}\n\t\t\\caption[RDM\/MAG differences interpolated on gray matter + P20 reconstruction]{CutFEM vs \\textit{Hex} lead field differences in distribution (a) and magnitude (b). Differences are interpolated onto gray matter. (c): The CutFEM based dipole reconstruction of the P20 medianus stimulation.}\n\t\\end{figure*}\n\t\n    \\section{Conclusion}\n\tCutFEM performed well both when the underlying head model was created using analytical level sets or realistic segmentation results. Application to an inverse reconstruction of a somatosensory evoked potential yielded findings that are in line with the literature. The level sets underlying CutFEM impose few restrictions on the compartments, thus allowing for more simplified segmentation routines when compared to other FEM approaches using surface triangulations.\n\n    \\section{Author Contributions}\n    \\textbf{Conceptualization}: C. Engwer, A. Westhoff, Carsten H. Wolters.\n    \\textbf{Methodology}: C. Engwer, T. Erdbruegger,  A. Westhoff, C.H. Wolters.\n    \\textbf{Software}: C. Engwer, T. Erdbruegger, A. Westhoff.\n    \\textbf{Investigation}: T. Erdbruegger.\n    \\textbf{Writing \u2013 original draft}: T. Erdbruegger.\n    \\textbf{Writing \u2013 review and editing}: Y. Buschermoehle, C. Engwer, T. Erdbruegger,  J. Gross, M. Hoeltershinken, R. Lencer, J.O. Radecke, C.H. Wolters.\n    \\textbf{Supervision}: C. Engwer, C.H. Wolters.\n    \\textbf{Funding acquisition}: A. Buyx, J. Gross, R. Lencer, S. Pursiainen, F. Wallois, C.H. Wolters.\n\t\\bibliographystyle{ieeetr}\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLarge-scale multi-label text classification (\\textsc{lmtc}\\xspace) is the task of assigning to each document all the relevant labels from a large set, typically containing thousands of labels (classes). Applications include building web directories \\citep{Partalas2015LSHTCAB}, labeling scientific publications with concepts from ontologies \\cite{Tsatsaronis2015}, assigning diagnostic and procedure labels to medical records \\cite{Mullenbach2018,Rios2018-2}. We focus on legal text processing, an emerging \\textsc{nlp} field with many applications (e.g., legal judgment \\cite{Nallapati2008,Aletras2016}, contract element extraction \\cite{Chalkidis2017}, obligation extraction \\cite{Chalkidis2018b}), but limited publicly available resources.\n\nOur first contribution is a new publicly available legal \\textsc{lmtc}\\xspace dataset, dubbed \\textsc{eurlex57k}\\xspace, containing 57k English \\textsc{eu}\\xspace legislative documents from the \\textsc{eur-lex}\\xspace portal, tagged with $\\mathtt{\\sim}$4.3k labels (concepts) from the European Vocabulary (\\textsc{eurovoc}\\xspace).\\footnote{See \\url{https:\/\/eur-lex.europa.eu\/} for \\textsc{eur-lex}\\xspace, and \\url{https:\/\/publications.europa.eu\/en\/web\/eu-vocabularies} for \\textsc{eurovoc}\\xspace.} \\textsc{eurovoc}\\xspace contains approx.\\ 7k labels, but most of them are rarely used, hence they are under-represented (or  absent) in \\textsc{eurlex57k}\\xspace, making the dataset also appropriate for few- and zero-shot learning. \\textsc{eurlex57k}\\xspace can be viewed as an improved version of the dataset released by \\citet{Mencia2007}, which has been widely used in \\textsc{lmtc}\\xspace research, but is less than half the size of \\textsc{eurlex57k}\\xspace (19.6k documents, 4k \\textsc{eurovoc}\\xspace labels) and more than ten years old. \n\nAs a second contribution, we experiment with several neural classifiers on \\textsc{eurlex57k}\\xspace, including the Label-Wise Attention Network of \\citet{Mullenbach2018}, called \\textsc{cnn-lwan}\\xspace here, which was reported to achieve state of the art performance in \\textsc{lmtc}\\xspace on medical records. We show that a simpler \\textsc{bigru} with self-attention \\cite{Xu2015} outperforms \\textsc{cnn-lwan}\\xspace by a wide margin on \\textsc{eurlex57k}\\xspace. However, by replacing the \\textsc{cnn}\\xspace encoder of \\textsc{cnn-lwan}\\xspace with a \\textsc{bigru}\\xspace, we obtain even better results on \\textsc{eurlex57k}\\xspace. Domain-specific \\textsc{word2vec}\\xspace \\cite{Mikolov2013} and context-sensitive \\textsc{elmo}\\xspace embeddings \\cite{Peters2018} yield further improvements. We thus establish strong baselines for \\textsc{eurlex57k}\\xspace.\n\nAs a third contribution, we investigate which zones of the documents are more informative on \\textsc{eurlex57k}\\xspace, showing that considering only the title and recitals of each document leads to almost the same performance as considering the full document. This allows us to bypass \\textsc{bert}\\xspace's \\cite{bert} maximum text length limit and fine-tune \\textsc{bert}\\xspace, obtaining the best results for all but zero-shot learning labels. To our knowledge, this is the first application of \\textsc{bert}\\xspace to an \\textsc{lmtc}\\xspace task, which provides further evidence of the superiority of pretrained language models with task-specific fine-tuning, and establishes an even stronger baseline for \\textsc{eurlex57k}\\xspace and \\textsc{lmtc}\\xspace in general.\n\n\\section{Related Work}\n\\label{sec:relatedwork}\n\nYou et al.\\ \\shortcite{You2018} explored \\textsc{rnn}\\xspace-based methods with self-attention on five \\textsc{lmtc}\\xspace datasets that had also been considered by \\citet{Liu2017}, namely \\textsc{rcv1}\\xspace \\cite{Lewis2004}, Amazon-13K, \\cite{McAuley2013}, Wiki-30K and Wiki-500K \\cite{Zubiaga2012}, as well as the previous \\textsc{eur-lex}\\xspace dataset \\cite{Mencia2007}, reporting that attention-based \\textsc{rnn}\\xspace{s} produced the best results overall (4 out of 5 datasets). \n\n\\citet{Mullenbach2018} investigated the use of label-wise attention in \\textsc{lmtc}\\xspace for medical code prediction on the \\textsc{mimic-ii}\\xspace and \\textsc{mimic-iii}\\xspace datasets \\cite{Johnson2017}. Their best method, Convolutional Attention for Multi-Label Classification, called \\textsc{cnn-lwan}\\xspace here, employs one attention head per label and was shown to outperform weak baselines, namely logistic regression, plain \\textsc{bigru}\\xspace{s}, \\textsc{cnn}\\xspace{s} with a single convolution layer. \n\n\\citet{Rios2018-2} consider few- and zero-shot learning on the \\textsc{mimic} datasets. They propose Zero-shot Attentive \\textsc{cnn}\\xspace, called \\textsc{zero-cnn-lwan}\\xspace here, a method similar to \\textsc{cnn-lwan}\\xspace, which also exploits label descriptors. Although \\textsc{zero-cnn-lwan}\\xspace did not outperform \\textsc{cnn-lwan}\\xspace overall on \\textsc{mimic-ii}\\xspace and \\textsc{mimic-iii}\\xspace, it had much improved results in few-shot and zero-shot learning, among other variations of \\textsc{zero-cnn-lwan}\\xspace that exploit the hierarchical relations of the labels with graph convolutions.\n\nWe note that the label-wise attention methods of \\citet{Mullenbach2018} and \\citet{Rios2018-2} were not compared to strong generic text classification baselines, such as attention-based \\textsc{rnn}\\xspace{s} \\cite{You2018} or Hierarchical Attention Networks (\\textsc{han}\\xspace{s}) \\cite{Yang2016}, which we investigate below.\n\n\\section{The New Dataset}\n\\label{sec:dataset}\n\nAs already noted, \\textsc{eurlex57k}\\xspace contains 57k legislative documents from \\textsc{eur-lex}\\xspace\\footnote{Our dataset is available at \\url{http:\/\/nlp.cs.aueb.gr\/software_and_datasets\/EURLEX57K}, with permission of reuse under European Union\\copyright, \\url{https:\/\/eur-lex.europa.eu}, 1998--2019.} with an average length of 727 words (Table~\\ref{tab:dataset}).\\footnote{See Appendix~\\ref{app:dataset} for more statistics.} Each document contains four major zones: the \\emph{header}, which includes the title and name of the legal body enforcing the legal act; the \\emph{recitals}, which are legal background references; the \\emph{main body}, usually organized in articles; and the \\emph{attachments} (e.g., appendices, annexes).\n\n\\begin{table}[t]\n\\centering\n\\footnotesize\n\\begin{tabular}{lcccc}\n  Subset & Documents ($D$) & Words\/$D$ & Labels\/$D$ \\\\\n\\hline\n  Train & 45,000 & 729  & 5\\\\\n  Dev. & 6,000 & 714  & 5 \\\\\n  Test & 6,000 & 725  & 5\\\\\\hline\n  Total & 57,000 & 727 & 5\\\\\n  \\hline\n\\end{tabular}\n\\caption{Statistics of the \\textsc{\\textsc{eur-lex}\\xspace} dataset.}\n\\label{tab:dataset}\n\\end{table}\n\nSome of the \\textsc{lmtc}\\xspace methods we consider need to be fed with documents split into smaller units. These are often sentences, but in our experiments they are \\emph{sections}, thus we preprocessed the raw text,  respectively. We treat the header, the recitals zone, each article of the main body, and the attachments as separate sections. \n\nAll the documents of the dataset have been annotated by the Publications Office of \\textsc{eu}\\xspace\\footnote{See \\url{https:\/\/publications.europa.eu\/en}.} with multiple concepts from \\textsc{eurovoc}\\xspace. \nWhile \\textsc{eurovoc}\\xspace includes approx.\\ 7k concepts (labels), only 4,271 (59.31\\%) are present in \\textsc{eurlex57k}\\xspace, from which only 2,049 (47.97\\%) have been assigned to more than 10 documents. Similar distributions were reported by \\citet{Rios2018-2} for the \\textsc{mimic} datasets.\nWe split \\textsc{eurlex57k}\\xspace into training (45k documents), development (6k), and test subsets (6k). We also divide the 4,271 labels into \\emph{frequent} (746 labels), \\emph{few-shot} (3,362), and \\emph{zero-shot} (163), depending on whether they were assigned to more than 50, fewer than 50 but at least one, or no training documents, respectively.\n\n\\begin{figure*}[ht]\n  \\centering\n    \\includegraphics[width=1\\textwidth]{methods.png}\n  \\caption{Illustration of (a) \\textsc{bigru-att}\\xspace, (b) \\textsc{han}\\xspace, (c) \\textsc{bigru-lwan}\\xspace, and (d) \\textsc{bert}\\xspace.}\n  \\vspace*{-4mm}\n  \\label{fig:methods}\n\\end{figure*}\n\n\\section{Methods}\n\\label{sec:methods}\n\n\\paragraph{Exact Match, Logistic Regression:} \nA first naive baseline, Exact Match, assigns only labels whose descriptors can be found verbatim in the document. A second one uses Logistic Regression with feature vectors containing \\textsc{tf-idf}\\xspace scores of $n$-grams ($n=1,2,\\dots, 5$).\n\n\\paragraph{\\textsc{bigru-att}\\xspace:}\n\nThe first neural method is a \\textsc{bigru} with self-attention \\cite{Xu2015}. Each document is represented as the sequence of its word embeddings, which go through a stack of \\textsc{bigru}s (Figure~\\ref{fig:methods}a). A document embedding ($h$) is computed as the sum of the resulting context-aware embeddings ($h = \\sum_i a_i h_i$), weighted by the self-attention scores ($a_i$), and goes through a dense layer of $L=4,271$ output units with sigmoids, producing $L$ probabilities, one per label.\n\n\\paragraph{\\textsc{han}\\xspace:}\n\nThe Hierarchical Attention Network \\cite{Yang2016} is a strong baseline for text classification. We use a slightly modified version, where a \\textsc{bigru}\\xspace with self-attention reads the words of each section, as in \\textsc{bigru-att}\\xspace but separately per section, producing section embeddings. A second-level \\textsc{bigru}\\xspace with self-attention reads the section embeddings, producing a single document embedding ($h$) that goes through a similar output layer as in \\textsc{bigru-att}\\xspace (Figure~\\ref{fig:methods}b).\n\n\\paragraph{\\textsc{cnn-lwan}\\xspace, \\textsc{bigru-lwan}\\xspace:}\n\nIn the original Label-Wise Attention Network (\\textsc{lwan}\\xspace) of \\citet{Mullenbach2018}, called \\textsc{cnn-lwan}\\xspace here, the word embeddings of each document are first converted to a sequence of vectors $h_i$ by a \\textsc{cnn}\\xspace encoder.  A modified version of \\textsc{cnn-lwan}\\xspace that we developed, called \\textsc{bigru-lwan}\\xspace, replaces the \\textsc{cnn}\\xspace encoder with a \\textsc{bigru}\\xspace (Figure~\\ref{fig:methods}c), which converts the word embeddings into context-sensitive embeddings $h_i$, much as in \\textsc{bigru-att}\\xspace. Unlike \\textsc{bigru-att}\\xspace, however, both \\textsc{cnn-lwan}\\xspace and \\textsc{bigru-lwan}\\xspace use $L$ independent attention heads, one per label, generating $L$ document embeddings ($h^{(l)} = \\sum_i a_{l,i} h_i$, $l=1, \\dots, L$) from the sequence of vectors $h_i$ produced by the \\textsc{cnn}\\xspace or \\textsc{bigru}\\xspace encoder, respectively. Each document embedding ($h^{(l)}$) is specialized to predict the corresponding label and goes through a separate dense layer ($L$ dense layers in total) with a sigmoid, to produce the probability of the corresponding label.\n\n\\paragraph{\\textsc{zero-cnn-lwan}\\xspace, \\textsc{zero-bigru-lwan}\\xspace:}\n\\citet{Rios2018-2} designed a model similar to \\textsc{cnn-lwan}\\xspace, called \\textsc{zacnn}\\xspace in their work and \\textsc{zero-cnn-lwan}\\xspace here, to deal with rare labels. In \\textsc{zero-cnn-lwan}\\xspace, the attention scores ($a_{l,i}$) and the label probabilities are produced by comparing the $h_i$ vectors that the \\textsc{cnn}\\xspace encoder produces and the label-specific document embeddings ($h^{(l)}$), respectively, to label embeddings. Each label embedding is the centroid of the pretrained word embeddings of the label's descriptor; consult \\citet{Rios2018-2} for further details. By contrast, \\textsc{cnn-lwan}\\xspace and \\textsc{bigru-lwan}\\xspace do not consider the descriptors of the labels. We also experiment with a variant of \\textsc{zero-cnn-lwan}\\xspace that we developed, dubbed \\textsc{zero-bigru-lwan}\\xspace, where the \\textsc{cnn}\\xspace encoder is replaced by a \\textsc{bigru}\\xspace.\n\n\\paragraph{\\textsc{bert}\\xspace:} \n\\textsc{bert}\\xspace \\citep{bert} is a language model based on Transformers \\citep{Vaswani2017} pretrained on large corpora. \nFor a new target task, a task-specific layer is added on top of \\textsc{bert}. The extra layer is trained jointly with \\textsc{bert} by fine-tuning on task-specific data. We add a dense layer on top of \\textsc{bert}, with sigmoids, that produces a probability per label. Unfortunately, \\textsc{bert}\\xspace can currently process texts up to 512 wordpieces, which is too small for the documents of \\textsc{eurlex57k}\\xspace. Hence, \\textsc{bert}\\xspace can only be applied to truncated versions of our documents (see below). \n\n\\begin{table*}[ht!]\n\\centering\n{\n\\footnotesize\\addtolength{\\tabcolsep}{-2pt}\n\\begin{tabular}{lccccccccc}\n  \\hline\n  & \\multicolumn{3}{c}{\\textsc{All Labels}} & \\multicolumn{2}{c}{\\textsc{Frequent}} & \\multicolumn{2}{c}{\\textsc{Few}} & \\multicolumn{2}{c}{\\textsc{Zero}} \\\\ \n  & $RP@5$ & $nDCG@5$ & Micro-$F1$ & $RP@5$ & $nDCG@5$ & $RP@5$ & $nDCG@5$ & $RP@5$ & $nDCG@5$ \\\\\n  \\cline{2-10}\n  Exact Match & 0.097 & 0.099 & 0.120 & 0.219 & 0.201 & 0.111 & 0.074 & 0.194 & 0.186 \\\\\n  Logistic Regression & 0.710 & 0.741 & 0.539 & 0.767 & 0.781 & 0.508 & 0.470 & 0.011 & 0.011 \\\\\n  \\hline\n  \\textsc{bigru-att}\\xspace & 0.758 & 0.789 & 0.689 & 0.799 & 0.813 & 0.631 & 0.580 & 0.040 & 0.027\\\\\n  \\textsc{han}\\xspace & 0.746 & 0.778 & 0.680 & 0.789 & 0.805 & 0.597 & 0.544 & 0.051 & 0.034\\\\\n  \\hline\n \\textsc{cnn-lwan}\\xspace & 0.716 & 0.746 & 0.642 & 0.761 & 0.772 & 0.613 & 0.557 & 0.036  & 0.023 \\\\\n  \\textsc{bigru-lwan}\\xspace & \\textbf{0.766} & \\textbf{0.796} & \\textbf{0.698} & \\textbf{0.805} & \\textbf{0.819} & \\textbf{0.662} & \\textbf{0.618} & 0.029 & 0.019\\\\\n   \\hline\n  \\textsc{zero-cnn-lwan}\\xspace & 0.684 & 0.717 & 0.618 & 0.730 & 0.745 & 0.495 & 0.454 & 0.321 & 0.264 \\\\\n  \\textsc{zero-bigru-lwan}\\xspace & 0.718 & 0.752 & 0.652  & 0.764 & 0.780 & 0.561 & 0.510 & \\textbf{0.438} & \\textbf{0.345} \\\\\n  \\hline\n  \\hline\n \\textsc{bigru-lwan-l2v} & 0.775 & 0.804 & 0.711 & 0.815 & 0.828 & 0.656 & 0.612  & 0.034 & 0.024 \\\\\n\\hline\n\\textsc{bigru-lwan-l2v}* & 0.770 & 0.796 & 0.709 & 0.811 & 0.825 & 0.641 & 0.600 & 0.047 & 0.030\\\\\n\\textsc{bigru-lwan-elmo}* & 0.781 & 0.811 & 0.719 & 0.821 & 0.835 & 0.668 & 0.619 & 0.044 & 0.028\\\\\n\\textsc{bert-base}\\xspace * & \\textbf{0.796} & \\textbf{0.823} & \\textbf{0.732} & \\textbf{0.835} & \\textbf{0.846} & \\textbf{0.686} & \\textbf{0.636} & 0.028 & 0.023\\\\\n\\hline\n\\end{tabular}\n}\n\\caption{Results on \\textsc{eurlex57k}\\xspace for all, frequent, few-shot, zero-shot labels. Starred methods use the first 512 document tokens; all other methods use full documents. Unless otherwise stated, \\textsc{glove}\\xspace embeddings are used.}\n\\vspace*{-4mm}\n\\label{tab:results}\n\\end{table*}\n\n\\section{Experiments}\n\\label{sec:experiments}\n\n\\paragraph{Evaluation measures:} Common \\textsc{lmtc}\\xspace evaluation measures are precision ($P@K$) and recall ($R@K$) at the top $K$ predicted labels, averaged over test documents, micro-averaged F1 over all labels, and $nDCG@K$ \\cite{Manning2009}. However, $P@K$ and $R@K$ unfairly penalize methods when the gold labels of a document are fewer or more than $K$, respectively. Similar concerns have led to the introduction of $\\mathrm{R}\\text{-}\\mathrm{Precision}$ and $nDCG@K$ in Information Retrieval \\cite{Manning2009}, which we believe are also more appropriate for \\textsc{lmtc}\\xspace. Note, however, that $\\mathrm{R}\\text{-}\\mathrm{Precision}$ requires the number of gold labels per document to be known beforehand, which is unrealistic in practical applications. Therefore we propose using $\\mathrm{R}\\text{-}\\mathrm{Precision}@K$ ($RP@K$), where $K$ is a parameter. This measure is the same as $P@K$ if there are at least $K$ gold labels, otherwise $K$ is reduced to the number of gold labels. \n\nFigure~\\ref{fig:kappa_comparison} shows $RP@K$ for the three best systems, macro-averaged over test documents. Unlike $P@K$, $RP@K$ does not decline sharply as $K$ increases, because it replaces $K$ by the number of gold labels, when the latter is lower than $K$. \nFor $K=1$, $RP@K$ is equivalent to $P@K$, as confirmed by Fig.~\\ref{fig:kappa_comparison}. For large values of $K$ that almost always exceed the number of gold labels, $RP@K$ asymptotically approaches $R@K$, as also confirmed by Fig.~\\ref{fig:kappa_comparison}.\\footnote{See Appendix~\\ref{app:evaluation} for a more detailed discussion on the evaluation measures.} In our dataset, there are 5.07 labels per document, hence  $K=5$ is reasonable.\\footnote{Evaluating at other values of $K$ lead to similar conclusions (see Fig.~\\ref{fig:kappa_comparison} and Appendix~\\ref{app:experiments}).}\n\n\\begin{figure}[ht]\n\\centering\n\\begin{tikzpicture}[scale=1.0]\n    \\begin{axis}[\n        xlabel={$K$ top predictions},\n        ylabel={},\n        xmin=1, xmax=10,\n        ymin=0.2, ymax=1,\n        xtick={1,2,3,4,5,6,7,8,9,10},\n        ytick={0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0},\n        legend style={at={(0.24, 0.03)}, anchor=south west},\n        ymajorgrids=true,\n        grid style=dashed,\n    ]\n    \n    \\addplot[\n            color=black,\n            ]\n            coordinates {\n            (1,0.921)(2,0.878)(3,0.830)(4,0.797)(5,0.781)(6,0.780)(7,0.799)(8,0.818)(9,0.832)(10,0.845)};\n            \\addlegendentry{\\textsc{bigru-lwan}\\xspace (\\textsc{elmo}\\xspace)}\n            \n    \\addplot[\n            color=black,\n            mark=o,\n            ]\n            coordinates {\n            (1,0.916)(2,0.867)(3,0.821)(4,0.787)(5,0.770)(6,0.771)(7,0.791)(8,0.808)(9,0.823)(10,0.836)};\n            \\addlegendentry{\\textsc{bigru-lwan}\\xspace (\\textsc{l2v}\\xspace)}\n            \n    \\addplot[\n            color=black,\n            mark=square,\n            ]\n            coordinates {\n            (1,0.922)(2,0.880)(3,0.839)(4,0.810)(5,0.795)(6,0.796)(7,0.814)(8,0.832)(9,0.845)(10,0.857)};\n            \\addlegendentry{\\textsc{bert-base}\\xspace}\n     \n    \\addplot[\n        color=red,\n                ]\n        coordinates {\n        (1,0.921)(2,0.875)(3,0.811)(4,0.740)(5,0.674)(6,0.611)(7,0.552)(8,0.501)(9,0.456)(10,0.419)};\n\n    \\addplot[\n        color=green,\n                ]\n        coordinates {\n        (1,0.209)(2,0.391)(3,0.531)(4,0.632)(5,0.705)(6,0.756)(7,0.789)(8,0.813)(9,0.830)(10,0.844)};\n    \n    \\addplot[\n        color=red,\n        mark=o,\n        ]\n        coordinates {\n        (1,0.913)(2,0.863)(3,0.803)(4,0.734)(5,0.669)(6,0.607)(7,0.549)(8,0.498)(9,0.454)(10,0.417)};\n\n    \\addplot[\n        color=green,\n        mark=o,\n        ]\n        coordinates {\n        (1,0.207)(2,0.386)(3,0.527)(4,0.627)(5,0.700)(6,0.752)(7,0.787)(8,0.811)(9,0.828)(10,0.842)};\n    \n    \\addplot[\n        color=red,\n        mark=square,\n        ]\n        coordinates {\n        (1,0.922)(2,0.877)(3,0.820)(4,0.753)(5,0.686)(6,0.623)(7,0.562)(8,0.510)(9,0.463)(10,0.424)};\n\n    \\addplot[\n        color=green,\n        mark=square,\n        ]\n        coordinates {\n        (1,0.210)(2,0.393)(3,0.539)(4,0.642)(5,0.718)(6,0.771)(7,0.804)(8,0.829)(9,0.844)(10,0.856)};\n     \n\\end{axis}\n\\end{tikzpicture}\n\\caption{$R@K$ (green lines), $P@K$ (red), $RP@K$ (black) of the best methods (\\textsc{bigru-lwan}\\xspace{s} (\\textsc{l2v}\\xspace), \\textsc{bigru-lwan}\\xspace{s} (\\textsc{elmo}\\xspace), \\textsc{bert-base}\\xspace), for $K=1$ to 10. All scores macro-averaged over test documents.}\n\\label{fig:kappa_comparison}\n\\end{figure}\n\n\n\\paragraph{Setup:} \nHyper-parameters are tuned using the \\textsc{hyperopt} library selecting the values with the best loss on development data.\\footnote{We implemented all methods in \\textsc{keras} (\\url{https:\/\/keras.io\/}). Code available at \\url{https:\/\/github.com\/iliaschalkidis\/lmtc-eurlex57k.git}. See Appendix~\\ref{app:hyperopt} for details on hyper-parameter tuning.} For the best hyper-parameter values, we perform five runs and report mean scores on test data. For statistical significance tests, we take the run of each method with the best performance on development data, and perform two-tailed approximate randomization tests \\cite{Dror2018} on test data.\\footnote{We perform 10k iterations, randomly swapping in each iteration the responses (sets of returned labels) of the two compared systems for 50\\% of the test documents.} Unless otherwise stated, we used 200-D pretrained \\textsc{glove}\\xspace embeddings \\cite{pennington2014glove}.\n\n\\paragraph{Full documents:}\n\\label{sec:overall}\nThe first five horizontal zones of Table~\\ref{tab:results} report results for full documents. The naive baselines are weak, as expected. Interestingly, for all, frequent, and even few-shot labels, the generic \\textsc{bigru-att}\\xspace performs better than \\textsc{cnn-lwan}\\xspace, which was designed for \\textsc{lmtc}\\xspace. \\textsc{han}\\xspace also performs better than \\textsc{cnn-lwan}\\xspace for all and frequent labels. However, replacing the \\textsc{cnn}\\xspace encoder of \\textsc{cnn-lwan}\\xspace with a \\textsc{bigru}\\xspace (\\textsc{bigru-lwan}\\xspace) leads to the best results, indicating that the main weakness of \\textsc{cnn-lwan}\\xspace is its vanilla \\textsc{cnn}\\xspace encoder.\n\nThe zero-shot versions of \\textsc{cnn-lwan}\\xspace and \\textsc{bigru-lwan}\\xspace outperform all other  methods on zero-shot labels (Table~\\ref{tab:results}), in line with the findings of \\citet{Rios2018-2}, because they exploit label descriptors,\nbut more importantly because they have a component that uses prior knowledge as is (i.e., label embeddings are frozen). Exact Match also performs better on zero-shot labels, for the same reason (i.e., the prior knowledge is intact). \\textsc{bigru-lwan}\\xspace, however, is still the best method in few-shot learning. All the differences between the best (bold) and other methods in Table~\\ref{tab:results} are statistically significant ($p < 0.01$).\n\nTable~\\ref{tab:embs} shows that using \\textsc{word2vec}\\xspace embeddings trained on legal texts (\\textsc{l2v}\\xspace) \\cite{Chalkidis2018} or \\textsc{elmo}\\xspace embeddings \\cite{Peters2018} trained on generic texts further improve the performance of \\textsc{bigru-lwan}\\xspace. \n\n\\paragraph{Document zones:} \nTable~\\ref{tab:zones} compares the performance of \\textsc{bigru-lwan}\\xspace on the development set for different combinations of document zones (Section~\\ref{sec:dataset}): \\emph{header} (\\emph{H}), \\emph{recitals}  (\\emph{R}), \\emph{main body} (\\emph{MB}), full text. Surprisingly \\emph{H+R} leads to almost the same results as full documents,\\footnote{The approximate randomization tests detected no statistically significant difference in this case ($p = 0.20$).} indicating that \\emph{H+R} provides most of the information needed to assign \\textsc{eurovoc}\\xspace labels.\n\n\\begin{table}[ht!]\n\\centering\n\\footnotesize\\addtolength{\\tabcolsep}{-2pt}\n\\begin{tabular}{lccc}\n  & $RP@5$ & $nDCG@5$ & Micro-$F1$ \\\\\n\\cline{2-4}\n\\textsc{glove}\\xspace & 0.766 & 0.796 & 0.698 \\\\\n\\textsc{l2v}\\xspace & 0.775 & 0.804 & 0.711 \\\\\n\\textsc{glove}\\xspace + \\textsc{elmo}\\xspace & 0.777 & 0.808 & 0.714\\\\\n\\textsc{l2v}\\xspace + \\textsc{elmo}\\xspace & \\textbf{0.781} & \\textbf{0.811} & \\textbf{0.719}\\\\\n\\end{tabular}\n\\caption{\\textsc{bigru-lwan}\\xspace with \\textsc{glove}\\xspace, \\textsc{l2v}\\xspace, \\textsc{elmo}\\xspace.}\n\\label{tab:embs}\n\\end{table}\n\\vspace{-0.5em}\n\\begin{table}[ht!]\n\\centering\n\\footnotesize\\addtolength{\\tabcolsep}{-2pt}\n\\begin{tabular}{lcccc}\n  & $\\mu_{words}$ & $RP@5$ & $nDCG@5$ & Micro-$F1$ \\\\\n\\cline{2-5}\n\\emph{H} & 43 & 0.747 & 0.782 & 0.688 \\\\\n\\emph{R}  & 317 & 0.734 & 0.765 & 0.669 \\\\\n\\emph{H+R} & 360 & \\underline{0.765} & \\underline{0.796} & \\underline{0.701} \\\\\n\\emph{MB} & 187 & 0.643 & 0.674 & 0.590 \\\\\n\\emph{Full} & 727 & \\textbf{0.766} & \\textbf{0.797} & \\textbf{0.702} \\\\\n\\end{tabular}\n\\caption{\\textsc{bigru-lwan}\\xspace with different document zones.}\n\\vspace*{-4mm}\n\\label{tab:zones}\n\\end{table}\n\n\n\\paragraph{First 512 tokens:} Given that \\emph{H+R} contains enough information and is shorter than 500 tokens in 83\\% of our dataset's documents, we also apply \\textsc{bert}\\xspace to the first 512 tokens of each document (truncated to \\textsc{bert}\\xspace's max.\\ length), comparing to \\textsc{bigru-lwan}\\xspace also operating on the first 512 tokens. Table~\\ref{tab:results} (bottom zone) shows that \\textsc{bert}\\xspace outperforms all other methods, even though it considers only the first 512 tokens. It fails, however, in zero-shot learning, since it does not have a component that exploits prior knowledge as is (i.e., all the components are fine-tuned on training data).\n\n\\section{Limitations and Future Work}\n\nOne major limitation of the investigated methods is that they are unsuitable for \\emph{Extreme} Multi-Label Text Classification where there are hundreds of thousands of labels \\cite{Liu2017,Zhang2018,Wydmuch2018}, as opposed to the \\textsc{lmtc}\\xspace setting of our work where the labels are in the order of thousands. We leave the investigation of methods for extremely large label sets for future work. Moreover, \\textsc{rnn} (and \\textsc{gru}\\xspace) based methods have high computational cost, especially for long documents. We plan to investigate more computationally efficient methods, e.g., dilated \\textsc{cnn}\\xspace{s} \\cite{KalchbrennerESO16} and Transformers \\cite{Vaswani2017, Dai2019}. We also plan to experiment with hierarchical flavors of \\textsc{bert}\\xspace to surpass its length limitations. Furthermore, experimenting with more datasets e.g., \\textsc{rcv1}\\xspace, Amazon-13K, Wiki-30K, \\textsc{mimic-iii}\\xspace will allow us to confirm our conclusions in different domains. Finally, we plan to investigate Generalized Zero-Shot Learning \\cite{Liu2018}. \n\n\\section*{Acknowledgements}\nThis work was partly supported by the Research Center of the Athens University of Economics and Business.\n\n\\vspace{-3mm}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nNuclear resonant scattering (NRS) of synchrotron radiation (SR) has become\nan established method for the study of nuclear hyperfine interaction during\nthe last two decades \\cite{Gerdau2000,Roehlsberger2004}. The spectrum is\nconventionally recorded as the time response of the nuclear ensemble\nfollowing a short resonant synchrotron pulse, which simultaneously excites\nall resonant transitions between hyperfine-split nuclear sublevels. The\nobserved beating frequencies are characteristic for the hyperfine fields in\nthe specimen. As an alternative to nuclear resonant forward scattering of SR\nin time domain, a heterodyne detection scheme was suggested\n\\cite{Coussement1996,Labbe2000}. The two scatterers, viz. the one\nunder investigation and as a reference sample a single-line M\\\"{o}ssbauer \nabsorber, are mounted on a M\\\"{o}ssbauer drive. The heterodyne spectrum is\nthe full time integral of the delayed counts, plotted as a function of the\nDoppler velocity of the reference sample. An advantage of this experimental\nsetup is the similarity of the spectra to those in the conventional\nenergy-domain M\\\"{o}ssbauer spectroscopy \\cite{Coussement1996,Labbe2000}.\nAlthough conventional M\\\"{o}ssbauer spectroscopy delivers similar\ninformation on hyperfine interactions, the special properties of SR like\nhigh collimation, high degree of polarization and high brilliance increase\nthe number of possible applications of NRS of SR.\nFurthermore, the heterodyne setup allows for dense bunch\nmodes of the synchrotron (with bunch separation time much shorter than the\nnuclear lifetime), which are not suitable for time differential NRS\nexperiments.\n\nUndistorted time integration of the nuclear response can only be performed\nif the large non-resonant intensity contribution is extinguished.\nExperimentally, this can be achieved by using radiation from a nuclear\nmonochromator \\cite{Smirnov1997} or by applying a polarizer\/analizer setup \n\\cite{Labbe2000}. An alternative approach, namely, ``stroboscopic detection'',\nis based on appropriate time gating \\cite{Callens2002,Callens2003},\ni.e., integration of the delayed time response in a periodic time window. The\nperiod $t_p$ of the observation time window after the SR pulse is chosen so\nthat $1\/t_p$ falls within the frequency range of the hyperfine interactions\nin the investigated specimen. This leads to new type of periodic resonances\nat certain Doppler velocities that are shifted from the M\\\"{o}ssbauer\nresonances by $mh\/t_p$, with $h$ being Planck's constant, and $m$ an integer\nnumber indicating the stroboscopic order \\cite{Callens2002,Callens2003}. The\nperiod $t_p$ should be selected according to the hyperfine spectral range,\nthe synchrotron bunch period and the detector dead time \n\\cite{Inge2004,Callens2003}.\n\nSo far the theory of stroboscopic detection scheme has only been developed \nand discussed in detail for forward scattering geometry. The several applications \nof NRS in surface and thin-film magnetism that make use of the grazing incidence geometry \n\\cite{Roehlsberger2004,Chumakov1999,Roehlsberger1999,Deak1999a,Roehlsberger2003,Sladecek2002}\ncall for computer programs that easily allow to fit data obtained by stroboscopic\ndetection as well. \nIn this geometry, the interferences of the SR plane waves, scattered from the surface and\ninterfaces of a stratified sample, provide information on the value,\ndirection and topology of the internal fields in the sample with\nnanometer depth resolution. \n\nRecently, interesting experiments have been performed using stroboscopic detection in\nthe grazing \nincidence case \\cite{Roehlsberger2010,Roehlsberger2012}, which demonstrates the \npotential of this method. \n\nGrazing-incident NRS of SR, often called Synchrotron \nM\\\"{o}ssbauer Reflectometry (SMR) \\cite{Gerdau2000,Deak01,Deak02}, has been\nestablished in both time and angular regime \\cite{Chumakov1999,Deak02,Nagy1999}, \nas time differential (TD) and time integral (TI) SMR, respectively. In the forward scattering \nchannel, the prompt electronic scattering homogeneously contributes to the stroboscopic \nspectrum and does not affect the spectral shape. For \nother scattering channels, including grazing incidence reflection,\nthe interference of the electronic and nuclear scattering provides further information. \nThe stroboscopic SMR line shape may considerably differ from the forward M\\\"{o}ssbauer\nspectrum, calling for a specialized computer code.\n\nThe dynamical theory of x-ray scattering gives a self-consistent description\nof the radiation field in all scattering channels of the system of\nscatterers, taking all orders of multiple scattering into account. Theories\nthat expand the coherent elastic scattering to the case of sharp nuclear\nresonances \\cite{Afanasev1963,Kagan1964,Hannon68,Hannon69,Hannon3} have been\napplied to\nvarious scattering geometries. The simplest cases are the\none-beam cases, such as forward and off-Bragg scattering, and the two-beam\ncases, the Bragg-Laue scattering \\cite{Hannon69,Sturhahn1994} and the grazing\nincidence scattering \\cite{Roehlsberger2003,Hannon69,Hannon85,Andreeva1,Deak96}. \nIn the grazing incidence limit, an optical model was derived from the dynamical theory \n\\cite{Hannon69,Hannon85}, which has been implemented in numerical calculations \n\\cite{Roehlsberger2003}. The reflectivity formulae given by \\citeasnoun{Deak96} and \n\\citeasnoun{Deak01} are suitable for fast numerical calculations in order to \nactually fit the experimental data \\cite{Spiering00} and, as has been shown \n\\cite{Deak1999}, this optical method is equivalent to that of the other approaches \nin the literature \\cite{Roehlsberger2003,Hannon85}.\n\nThe aim of the present paper is to develop the concept of the\nheterodyne\/stro\\-bo\\-scop\\-ic \ndetection and to\nestablish the formula that can be applied to any scattering channel, \nlike forward scattering, Bragg, off-Bragg and grazing incidence scattering.\n\nThis paper is\norganized as follows. In the second section, the\nheterodyne\/stroboscopic intensity formula for the propagation of $\\gamma $%\n--photons in a medium containing both electronic and resonant nuclear\nscatterers is derived. The equivalence to the previously discussed calculations for\nthe forward channel \\cite{Callens2002,Callens2003}\nare shown and \nthe important specific case of the stroboscopic grazing incidence reflection\nare outlined. In the third section, features of\nthe grazing incidence case\nare demonstrated by least-squares fitted\nexperimental stroboscopic SMR spectra on isotope-periodic $\\left[ ^{\\mathrm{%\nnat}}\\mathrm{Fe}\/^{57}\\mathrm{Fe}\\right] $ and antiferromagnetically ordered \n$\\left[ ^{57}\\mathrm{Fe}\/\\mathrm{Cr}\\right] $ multilayer films.\n\n\\section{Heterodyne\/Stroboscopic detection of Nuclear\\newline\nResonance Scattering}\n\n\\subsection{General considerations}\n\nThe setup of a heterodyne\/stroboscopic NRS of SR experiment includes two\nscatterers, the investigated specimen and an additional reference sample\n\\cite{Coussement1996,Labbe2000,Callens2002,Callens2003}, the latter being\nmounted on a M\\\"{o}ssbauer drive (in forward scattering geometry). The \nM\\\"{o}ssbauer drive provides a Doppler-shift $E_v=\\left( v\/c\\right) E_0$ of the\nnuclear energy levels, with $c,$ $v$ and $E_0$ being the velocity of light, \nthe velocity of the drive and the energy of the M\\\"{o}ssbauer\ntransition, respectively.\nThe polarization dependence of the  nuclear scatterers is described \nadopting the notation of \n\\citeasnoun{Sturhahn1994}, by $2\\times 2$ transmissivity and reflectivity matrices\ncommonly called \\textit{scattering matrices}.\nThe scattering of the synchrotron photons on\nthe specimen and the reference sample is described by the \n\\textit{total scattering matrix} $T_\\tau \\left( E,E_v\\right)$,\n\\begin{equation}\nT_\\tau \\left( E,E_v\\right) =T_\\tau ^{\\left( \\mathrm{s}\\right) }\\left(\nE\\right) T^{\\left( \\mathrm{r}\\right) }\\left( E-E_v\\right) ,  \\label{tottrans}\n\\end{equation}\na product of the scattering matrices of the reference sample $T^{\\left(\n\\mathrm{r}\\right)} $ and of the \ninvestigated specimen $T^{\\left(\\mathrm{s}\\right)}$ \\cite{Blume68} in the energy domain.\nThe index $\\tau$ specifies the open scattering channel \\cite{Hannon69,Sturhahn1994}.\nThe scattering matrix \n$T^{\\left( \\mathrm{r}\\right) }\\left( E-E_v\\right) \\,$ of the reference sample\ndepends on the Doppler-shifted energy $E-E_v$, where the channel index $\\tau $\nis omitted for forward scattering. Note that both the electrons and \nthe resonant M\\\"{o}ssbauer nuclei scatter the $\\gamma $--photons coherently so\nthe scattering matrices have a resonant nuclear\nand nearly energy-independent electronic contribution. At\nenergies being far from the M\\\"{o}ssbauer resonances $\\left( E\\rightarrow\n\\pm \\infty \\right) $ on a hyperfine scale, the individual scattering\nmatrices $T^{\\left( \\mathrm{s},\\mathrm{r}\\right) }\\left( E\\rightarrow \\pm\n\\infty \\right) $, and thus their product $T_\\tau \\left( E\\rightarrow \\infty\n,E_v\\right) \\equiv T_{\\tau ,\\infty }\\,$in Eq.~(\\ref{tottrans}), approach\nthe non-resonant electronic contribution: \n\\begin{equation}\nT_{\\tau ,\\infty }=T_{\\tau ,\\mathrm{el}}^{\\left( \\mathrm{s}\\right) }T_{%\n\\mathrm{el}}^{\\left( \\mathrm{r}\\right) }.  \\label{eltottrans}\n\\end{equation}\nSince the reference is mounted in forward geometry, its scattering matrix $%\nT^{\\left( \\mathrm{r}\\right) }\\left( E\\right) $ is the matrix exponential\n\\begin{equation}\nT^{\\left( \\mathrm{r}\\right) }\\left( E\\right) =\\exp \\left[ ikd^{\\left( \n\\mathrm{r}\\right) }n^{\\left( \\mathrm{r}\\right) }\\left( E\\right) \\right] ,\n\\label{Blumeexp}\n\\end{equation}\nwhere $n^{\\left( \\mathrm{r}\\right) }\\left(E\\right)$ is the index of refraction,\n$d^{\\left( \\mathrm{r}\\right) }$ the thickness and $k$ the vacuum\nwave number of the incident radiation \\cite{Blume68,Lax51}. The index of refraction \nis related to the susceptibility matrix $\\chi$ \\cite{Deak01,Deak96} through  \n\\begin{equation}\nn^{\\left( \\mathrm{r}\\right) }\\left( E\\right) \\equiv I+\\frac{\\chi ^{\\left( \n\\mathrm{r}\\right) }\\left( E\\right) }2,  \\label{khidef}\n\\end{equation}\nwhere $I$\\ is the unit matrix and $\\chi ^{\\left( \\mathrm{r}\\right) }=\\frac{4\\pi\nN^{\\left( \\mathrm{r}\\right) }}{k^2}f^{\\left( \\mathrm{r}\\right) }$, with $N^{\\left( \\mathrm{r}\\right) }$\nand $f^{\\left( \\mathrm{r}\\right) }$ being the density of the scattering centers\nand the $2\\times 2$\\ coherent forward scattering amplitude \\cite{Blume68},\nrespectively. The susceptibility is the sum of the electronic and the\nnuclear susceptibilities,\n\\begin{equation}\n\\chi ^{\\left( \\mathrm{r}\\right) }\\left( E\\right) =\\chi _{\\mathrm{el}%\n}^{\\left( \\mathrm{r}\\right) }+\\chi _{\\mathrm{nuc}}^{\\left( \\mathrm{r}\\right)\n}\\left( E\\right) .  \\label{khisum}\n\\end{equation}\nWith Eqs.~(\\ref{Blumeexp}), (\\ref{khidef}) and (\\ref{khisum}), the\ntransmissivity of the reference is expressed as a product of electronic\nand nuclear transmissivities,\n\\begin{equation}\nT^{\\left( \\mathrm{r}\\right) }\\left( E\\right) =T_{\\mathrm{el}}^{\\left( \n\\mathrm{r}\\right) }\\tilde{T}_{\\mathrm{nuc\\vphantom{l}}}^{\\left( \\mathrm{r}\\right)\n}\\left( E\\right) ,  \\label{sepel}\n\\end{equation}\nwhere \n\\begin{equation}\n\\tilde{T}_{\\mathrm{nuc}}^{\\left( \\mathrm{r}\\right) }\\left( E\\right) =\\exp\n\\left( ikd^{\\left( \\mathrm{r}\\right) }\\frac{\\chi _{\\mathrm{nuc}}^{\\left( \n\\mathrm{r}\\right) }\\left( E\\right) }2\\right) .  \\label{expnukl}\n\\end{equation}\n\n$T_\\tau^{\\left(\\mathrm{s}\\right)}\\left(E\\right)$ is determined from the \nrespective theory of wave propagation of channel $\\tau $ (forward, Bragg-Laue,\ngrazing incidence, etc. scattering), i.e., from the dynamical theory\n\\cite{Hannon68,Hannon69,Hannon3,Sturhahn1994,Hannon85}.\n\nIn forward scattering, due to the\nexponential expression in (\\ref{Blumeexp}), the total transmissivity $\nT\\left( E,E_v\\right) =T_\\infty \\tilde{T}_{\\mathrm{nuc}}^{\\left( \\mathrm{r}\n\\right) }\\left( E-E_v\\right) \\tilde{T}_{\\mathrm{nuc}}^{\\left( \\mathrm{s}\n\\right) }\\left( E\\right) $ is proportional to $T_\\infty$. Therefore, in\nthis special case, the electronic scattering is a simple multiplicative\nfactor, which does not affect the spectral shape. \n\nThe intensity $I_{\\tau }$ allowing for a general\npolarization state of the incident beam, the $2\\times 2$ polarization \ndensity matrix $\\rho$ (\\possessivecite{Blume68}), is given by \n\\begin{equation}\nI_{\\tau }\\left( E,E_{v}\\right) =\\mathrm{Tr}\\left[ T_{\\tau }^{\\dagger }\\left(\nE,E_{v}\\right) T_{\\tau }\\left( E,E_{v}\\right) \\rho \\right] .\n\\label{Blumeint}\n\\end{equation}\n\nThe beating time response to a single short polychromatic photon bunch of SR  \nis obtained by the Fourier transform of the energy domain scattering matrices\n\\cite{Gerdau2000,Hannon3},\n\\begin{equation}\nT_{\\tau }\\left( t,E_{v}\\right) =\\frac{1}{\\sqrt{2\\pi }\\hbar }\\int \\mathrm{d}E%\n\\text{\\thinspace }\\left[ T_{\\tau }\\left( E,E_{v}\\right) -T_{\\tau ,\\infty }%\n\\right] \\exp \\left( -i\\frac{E}{\\hbar }t\\right),   \\label{timetrans}\n\\end{equation}\nwhere, by subtracting the constant $T_{\\tau ,\\infty }$, the Dirac\ndelta--like prompt $(t=0)$ and the delayed $\\left( t>0\\right) $ time\nresponses are separated \\cite{Sturhahn1994}. We note that Eq.~(\\ref{timetrans})\nis valid only for delayed times $t>0$ after the SR bunch $(t=0)$, but\n$T_{\\tau }\\left( t,E_{v}\\right) = 0$ for $t<0$\\,!\nIn the same way as for Eq.~(\\ref{Blumeint}),\nthe delayed intensity in time domain becomes\n\\begin{equation}\nI_{\\tau }\\left( t,E_{v}\\right) =\\mathrm{Tr}\\left[ T_{\\tau }^{\\dagger }\\left(\nt,E_{v}\\right) T_{\\tau }\\left( t,E_{v}\\right) \\rho \\right] .\n\\label{Blumeinttim}\n\\end{equation}%\n\nFor a heterodyne\/stroboscopic NRS of SR experiment a time window function is introduced,\nwhich can be described by boxcar functions, namely, $S(t)=1$ \nfor $mt_{\\mathrm{B}}+t_{1}<t<mt_{\\mathrm{B}}+t_{2}$ and $S(t)=0$ otherwise, with a  time\ninterval $t_{\\mathrm{B}}$ between the synchrotron bunches and an integer number $m$.\nThe periodic time window function is expanded in Fourier series,\n\\begin{equation}\nS\\left( t\\right) =\\sum_{-\\infty }^{\\infty }s_{m}\\exp \\left( im\\Omega\nt\\right) ,  \\label{fourwin}\n\\end{equation}%\nwhere $\\Omega =\\tfrac{2\\pi }{t_{\\mathrm{B}}}$ is the angular frequency of\nthe SR bunches \\cite{Callens2002,Callens2003}. \n\nThe total delayed photon rate $D_{\\tau }\\left( E_{v}\\right)$ of one bunch is \n\\begin{equation}\nD_{\\tau }\\left( E_{v}\\right) =\\int\\limits_{-\\infty }^{\\infty }\\mathrm{d}t%\n\\text{\\thinspace }S\\left( t\\right) I_{\\tau }(t,E_{v}),  \\label{stroboint}\n\\end{equation}%\nthe integral of the intensity $I_{\\tau }\\left( t,E_{v}\\right)$ times $S(t)$.\nSince there is no coherence between photons generated by different electron \nbunches, the integral of the contribution of one bunch reveals the correct \ncontribution of multiple bunches with periodicity of $t_{\\mathrm{B}}$.\n\nCombining Eqs.~(\\ref{timetrans}), \n(\\ref{Blumeinttim}), (\\ref{stroboint}) and (\\ref{fourwin}), the delayed count rate can be written as \n\\begin{equation}\nD_{\\tau }\\left( E_{v}\\right) =\\sum_{-\\infty }^{\\infty }s_{m}\\delta _{\\tau\n,m}\\left( E_{v}\\right) ,  \\label{finstrobo}\n\\end{equation}%\nwhere \n\\begin{equation}\n\\delta _{\\tau ,m}\\left( E_{v}\\right) =\\frac{1}{\\hbar }\\int \\mathrm{d}E%\n\\mathrm{Tr}\\left\\{ \\left[ T_{\\tau }^{\\dagger }\\left( E-m\\varepsilon\n,E_{v}\\right) -T_{\\tau ,\\infty }^{\\dagger }\\right] \\left[ T_{\\tau }\\left(\nE,E_{v}\\right) -T_{\\tau ,\\infty }\\right] \\rho \\right\\}  \\label{defalfa}\n\\end{equation}%\nand \n\\begin{equation}\n\\varepsilon =\\hbar \\Omega .  \\label{epsdef}\n\\end{equation}%\nSince the time window $S\\left( t\\right) $ and the intensity $D_{\\tau }\\left(\nE_{v}\\right) $ are real functions, $S_{m}^{}=S_{-m}^{\\ast }$ and $\\delta\n_{m}^{}=\\delta _{-m}^{\\ast }$\nhold, and Eq.~(\\ref{finstrobo}) can\nbe rewritten as \n\\begin{equation}\nD_{\\tau }\\left( E_{v}\\right) =s_{0}\\delta _{\\tau\n,0}+\\sum\\limits_{m=1}^{\\infty }2\\mathrm{Re}\\left( s_{m}\\delta _{\\tau\n,m}\\right) .  \\label{riet2}\n\\end{equation}%\nThe result in Eqs.~(\\ref{finstrobo})--(\\ref{epsdef}) is a direct\ngeneralization of the intensity formula (\\ref{Blumeint}) to the\nheterodyne\/stroboscopic NRS of SR for any observed channel $\\tau$ in the\napplied experimental geometry. This expression has already been derived for\nthe case of forward scattering \\cite{Callens2002,Callens2003}. The $m=0$ term\nwas called the ``heterodyne spectrum''\n\\cite{Coussement1996,Callens2002}, while the $m>0$ terms were called\n``stroboscopic resonances'' \nof order $m$ \\cite {Callens2002}. Nevertheless, the stroboscopic resonances are not restricted\nto the forward scattering case. They also appear in other experimental\ngeometries, including, as we\n show below, in the grazing incidence\nscattering geometry.\n\n\\subsection{Grazing incidence geometry}\n\nIn what follows, stroboscopic SMR spectra will be discussed. In terms of\nthe dynamical theory, grazing incidence is a two-beam case. The $\\tau =0^{+}$\ntransmission and the $\\tau =0^{-}$ reflection channels are open\n\\cite{Roehlsberger2004,Hannon69,Sturhahn1994}, the latter being observed in SMR.\nClose to the electronic total reflection, the reflected intensity is high.\nTherefore, SMR is an experimentally fairly instructive special case. The\nreflection from the surface of the specimen is a multiple coherent\nscattering process of the (SR) photons on atomic electrons and resonant \nM\\\"{o}ssbauer nuclei \\cite{Deak01,Hannon85,Deak96}. Like in the forward\ncase, this scattering is independent of the atomic positions in the\nreflecting medium, such that the scattering is described\nby its index of refraction $n\\left( E\\right) $ \\cite{Deak96,Lax51}. Henceforth, \nin compliance with the literature \\cite{Roehlsberger2003,Deak01,Deak96}, \nin the general theory, the scattering matrix\n$T_\\tau ^{\\left( \\mathrm{s}\\right) }\\left( E\\right) $\nwill be replaced by the $2\\times 2$ reflectivity matrix $R^{\\left( \\mathrm{s}%\n\\right) }\\left( E,\\theta \\right) $, where $\\theta $ is the angle of \nincidence. This takes into account the interferences of the reflected\nradiation from the surfaces and interfaces between the layers with different\nrefraction index. The methods of calculating the reflectivity matrix can be\nfound in the literature \\cite{Roehlsberger2003,Deak01,Deak96}. Accordingly, \nthe total scattering matrix of the specimen and the reference from \nEq.~(\\ref{tottrans}) is \n\\begin{equation}\nT\\left( E,E_v,\\theta \\right) =R^{\\left( \\mathrm{s}\\right) }\\left( E,\\theta\n\\right) T^{\\left( \\mathrm{r}\\right) }\\left( E-E_v\\right) .\n\\label{reftottrans}\n\\end{equation}\nSimilarly, for energies being far from the M\\\"{o}ssbauer resonances, Eq.~(%\n\\ref{eltottrans}) reads \n\\begin{equation}\nT_\\infty \\left( \\theta \\right) =R_{\\mathrm{el}}^{\\left( \\mathrm{s}\\right)\n}\\left( \\theta \\right) T_{\\mathrm{el}}^{\\left( \\mathrm{r}\\right) }.\n\\label{refeltottrans}\n\\end{equation}\nInserting $T\\left( E,E_v\\right) $ and $T_\\infty $ into Eq.~(\\ref{finstrobo}%\n), the delayed count rate $D\\left( E_v,\\theta \\right)$ of the\nheterodyne\/stroboscopic spectrum for grazing incidence (stroboscopic SMR\nintensity) on the specimen is calculated.\n\nCombining Eqs.~(\\ref{sepel}), (\\ref{expnukl}) and (\\ref{defalfa}) \nreveal\n\\begin{eqnarray}\n\\delta _m\\left( E_v,\\theta \\right) & = & \\frac{A^{\\left( \\mathrm{r}\\right) }}%\n\\hbar \\int \\mathrm{d}E\\mathrm{Tr}\\left\\{ \\left[ \\tilde{T}^{\\dagger }\\left(\nE-E_v-m\\varepsilon \\right) R^{\\dagger }\\left( E-m\\varepsilon \\right) -R_{%\n\\mathrm{el}}^{\\dagger }\\right]\\right.  \\notag \\\\\n& & \\qquad \\times \\left.\\left[ R\\left( E\\right) \\tilde{T}\\left( E-E_v\\right) -R_{\\mathrm{el}}%\n\\right] \\rho \\right\\} ,  \\label{dlform}\n\\end{eqnarray}\nwhere $A^{\\left( \\mathrm{r}\\right) }=\\left| T_{\\mathrm{el}}^{\\left( \\mathrm{r%\n}\\right) }\\right| ^2$ is the electronic absorption of the reference sample.\nFor the sake of simplicity, the indices on the right hand side have been\nomitted, so that $\\tilde{T}_{\\mathrm{nuc}}^{\\left( \\mathrm{r}\\right)\n}\\rightarrow $$\\tilde{T}$, $R_{\\mathrm{el}}^{\\left( \\mathrm{s}\\right)\n}\\left( \\theta \\right) \\rightarrow R_{\\mathrm{el}}$ and $R^{\\left( \\mathrm{s}%\n\\right) }\\left( E,\\theta \\right) \\rightarrow R\\left( E\\right) $. Note that\nall reflectivities are those of the specimen, and all transmissivities are\nthose of the reference sample. With the relevant angular parameter $\\theta $\nfor grazing incidence, Eq.~(\\ref{finstrobo}) reads \n\\begin{equation}\nD\\left( E_v,\\theta \\right) =\\sum_{-\\infty }^\\infty s_m\\delta _m\\left(\nE_v,\\theta \\right) .  \\label{simpstrobo}\n\\end{equation}\nThe observed nuclear, as well as stroboscopic, resonances can be interpreted\nin a straightforward manner using Eq.~(\\ref{dlform}). Indeed, far from the\nresonances, $R\\left( E\\rightarrow \\infty \\right) =R_{\\mathrm{el}}$ and $%\n\\tilde{T}\\left( E\\rightarrow \\infty \\right) =1$, and the differences in the\nsquare brackets in (\\ref{dlform}) vanish. We expect a significant\ncontribution to the energy integral only if at least one energy argument of each\nbracket is close to resonance, i.e., either \n\\begin{subequations}\n\\label{co1}\n\\begin{align}\nE-E_v-m\\varepsilon &\\simeq 0 \\quad \\text{and}  \\label{co1a} \\\\\nE &\\simeq E_i  \\label{co1b}\n\\end{align}\nor \n\\end{subequations}\n\\begin{subequations}\n\\label{co2}\n\\begin{align}\nE-m\\varepsilon & \\simeq E_i \\quad \\text{and}  \\label{co2a} \\\\\nE-E_v & \\simeq 0  \\label{co2b}\n\\end{align}\nare fulfilled, where $E_i$ is the energy of the\n\\textit{i}$^{\\mathrm{th}}$ M\\\"{o}ssbauer \nresonance of the specimen. The $m^{\\mathrm{th}}$ term of the sum in\nEq.~(\\ref{simpstrobo}) contributes considerably if the Doppler velocity is\nnear to the corresponding shifted M\\\"{o}ssbauer resonance. In this case: \n\\end{subequations}\n\\begin{subequations}\n\\label{fincond}\n\\begin{align}\nE_v =E_i-m\\varepsilon +\\Delta ,  \\label{finconda} \\\\\nE_v =E_i+m\\varepsilon +\\Delta .  \\label{fincondb}\n\\end{align}\nHere, $\\Delta $ is a small deviation (of the order of the resonance line width)\\\nfrom the energy $E_i-m\\varepsilon $ or $E_i+m\\varepsilon $, ensuring the\nappearance of stroboscopic resonances also in grazing incidence geometry. In\nthe case of $m=0$, all four conditions of Eqs.~(\\ref{co1}) and (\\ref{co2})\nmay be true simultaneously. This means that, for $m=0$, nuclear scattering in\nboth samples, i.e., ``the radiative coupling of the samples'' \n\\cite{Callens2003}, also contributes. Hence, the dynamical line broadening\n(coherent speed-up) is the most effective in the heterodyne spectrum\n(= baseline and resonances of stroboscopic order 0).\n\nIn order to perform computer simulations of stroboscopic spectra, Eqs.~(\\ref%\n{defalfa}) and (\\ref{dlform}) were calculated for the forward scattering and\nSMR cases, respectively. Eqs.~(\\ref{defalfa}), (\\ref{dlform}) and (\\ref%\n{riet2}) were implemented in the evaluation program EFFI \\cite{Deak01,Spiering00}.\nThis program allows for least-square fitting\nof stroboscopic spectra. Moreover, they can be fitted \\textit{simultaneously}\nwith other types of spectra of the same specimen, such as forward\nscattering, grazing incidence, conventional M\\\"{o}ssbauer and other spectra\nof the implemented theory \\cite{Deak01,Spiering00}. This way, the fit\nconstraints on the common parameters become very general, as already\ndescribed \\cite{Deak01,Spiering00}.\n\n\\section{Experimental results and discussion}\n\nIn order to test the feasibility of this new reflectometric scheme, we investigated two film\nspecimens, a $^{\\mathrm{nat}}\\mathrm{Fe}\/^{57}\\mathrm{Fe}$ isotopic and a $%\n^{57}$Fe\/Cr antiferromagnetic multilayer, in grazing\nincidence reflection geometry, using the $14.4~\\mathrm{keV}$ M\\\"{o}ssbauer\ntransition of $^{57}$Fe nuclei. The experiments were performed at the BL09XU\nnuclear resonance beam line of SPring-8 \\cite{Yoda2001}. The\nexperimental setup is shown in Fig.~\\ref{expsetup}. The synchrotron was\noperated in the 203-bunch mode, corresponding to a bunch separation time of $%\n23.6\\ \\mathrm{ns}$. The SR was monochromated by a Si(422)\/Si(12~2~2) double\nchannel-cut high resolution monochromator with $6~\\mathrm{meV}$ resolution.\nIt was incident on the K$_4$[$^{57}$Fe(CN)$_6$] single line pelleted\nreference sample of effective thickness 11, and on the multilayer specimen\ndownstream mounted in grazing incidence geometry (Fig.~\\ref{expsetup}). The \nM\\\"{o}ssbauer drive was operated in constant acceleration mode, with a\nmaximum velocity of $v_{\\mathrm{max}}=20.24~\\mathrm{mm\/s}$. This maximum was\ncalibrated by fitting the velocity separation of the stroboscopic orders in\na forward scattering stroboscopic spectrum of a single line \n$^{57}\\mathrm{Fe}$-enriched stainless steel absorber \n\\cite{Callens2002,Callens2003}. The delayed radiation was detected using \nthree $2~\\mathrm{ns}$ dead time\nHamamatsu avalanche photo diodes (APD) in series. To record the delayed\nintensity, a two-dimensional data acquisition system was used. Each count\nwas indexed according to the time elapsed after the synchrotron pulse (1024\nchannels), as well as to the velocity of the reference (1024 channels).\nThese stroboscopic SMR data were time integrated using appropriate time\nwindows of $t_p=7.87~\\mathrm{ns}$ period and $3.93~\\mathrm{ns}$ length\n\\cite{Callens2002,Callens2003}. Since the energy is measured in mm\/s, the shift of\nthe first stroboscopic order, Eq.~(\\ref{epsdef}), can be rewritten as \n\\end{subequations}\n\\begin{equation}\n\\varepsilon ~\\left[ \\mathrm{mm}\/\\mathrm{s}\\right] =1000\\frac{\\lambda ~\\left[ \n\\mathrm{nm}\\right] }{t_p~\\left[ \\mathrm{ns}\\right] }.  \\label{stroboshift}\n\\end{equation}\nWith the wavelength $\\lambda \\approx 0.086~\\mathrm{nm}$ for the M\\\"{o}%\nssbauer transition of $^{57}$Fe, the separation between the neighbouring\nstroboscopic orders can be calculated to be $\\varepsilon \\approx 10.93~%\n\\mathrm{mm}\/\\mathrm{s}$. Note that this is the range of the hyperfine\nsplitting in case of $\\alpha -\\mathrm{Fe}$ (outer line separation is $10.62~%\n\\mathrm{mm}\/\\mathrm{s}$ at room temperature), and the stroboscopic orders\nwould only slightly overlap in case of a sample of low effective thickness\nin forward scattering. However, in case of grazing incidence near the\ncritical angle of total external reflection due to the enhanced nuclear and\nelectronic multiple scattering, the M\\\"{o}ssbauer lines become extremely\nbroad and\\ a strong overlap of the stroboscopic orders is expected. This\ninterference and partial overlap are manifested in rather complex resonance\nline shapes and an intriguing angular dependence of the delayed intensity in\nthe various stroboscopic orders.\n\nBoth multilayers were prepared under ultra-high vacuum conditions by\nmolecular beam epitaxy at the IMBL facility in IKS Leuven. The\\thinspace $[^{%\n\\mathrm{nat}}\\mathrm{Fe}\/^{57}\\mathrm{Fe]}_{10}$ was prepared at room\ntemperature onto a Zerodur glass substrate. The first layer and all other $%\n^{57}\\mathrm{Fe}$-layers were 95.5\\% isotopically enriched, and were\ngrown from a Knudsen cell. The natural Fe layers, which have a $^{57}%\n\\mathrm{Fe}$-concentration of 2.17\\%, were grown from an electron gun\nsource. The nominal layer thickness was $3.15~\\mathrm{nm}$ throughout the\nmultilayer stack for both $^{\\mathrm{nat}}\\mathrm{Fe}$ and $^{57}\\mathrm{Fe}$%\n. Conversion electron M\\\"{o}ssbauer spectra showed a pure $\\alpha -\\mathrm{Fe%\n}$ spectrum. This spectrum was compared to a transmission M\\\"{o}ssbauer\nspectroscopy spectrum of a natural iron calibration specimen, which was\nprovided by Amersham. Both hyperfine magnetic fields were fitted to be\nidentical within the experimental error of 0.04\\%, and no sign of any second\nphase contamination was found.\n\nPreparation and characterization of the MgO(001)\/[$^{57}$Fe\/Cr]$_{20}$\nmultilayer sample has been described earlier \n\\cite{BottyanBSF1,Nagy02a,Tancziko2004}. The layering was verified as epitaxial\nand periodic, with thicknesses of $2.6~\\mathrm{nm}$ for the $^{57}$Fe layer,\nand $1.3~\\mathrm{nm}$ for the Cr layer. SQUID magnetometry showed dominantly\nantiferromagnetic coupling between neighboring Fe layers. According to\nprevious studies on this multilayer \\cite{BottyanBSF1,Nagy02a,Tancziko2004},\nthe magnetizations in Fe align to the [100] and [010] perpendicular easy\ndirections in remanence, respectively corresponding to the [110] and \n[$\\overline{1}$10] directions of the MgO substrate. The layer magnetizations\nwere aligned antiparallel in the consecutive Fe layers by applying a\nmagnetic field (1.6~T) above the saturation value (0.96~T) in the Fe[010]\neasy direction of magnetization, and then releasing the field to remanence.\nThis alignment is global, the antiferromagnetic domains were only different\nin the layer sequence of the parallel\/antiparallel orientations \n\\cite{Nagy02a}.\n\n\\subsection{Stroboscopic SMR on a $^{\\mathrm{nat}}$Fe\/$^{57}$Fe multilayer}\n\nSince in a $^{\\mathrm{nat}}\\mathrm{Fe}\/^{57}\\mathrm{Fe}$ isotope-periodic\nmultilayer the hyperfine field of $^{57}\\mathrm{Fe}$ is that \nof $\\alpha -\\mathrm{Fe}$ throughout the sample, this multilayer is \nparticularly suitable for studying the modification of the resonance \nline shapes due to interference between nuclear and electronic scattering \n\\cite{Deak1999a,Deak1994,Chumakov1991,Chumakov1993}. Fig.~\\ref{fefe} shows \nresults for the multilayer saturated in a transversal magnetic field of 50 mT. \nPanel \\textit{a} and \\textit{b} give the prompt electronic and delayed TISMR curves,\nrespectively. The stroboscopic SMR spectra at the angles indicated by the\narrows are given in panel \\textit{c} to \\textit{e}. The peak in the delayed\nreflectivity at the total reflection angle in panel \\textit{b} is a special\nfeature of SMR described earlier \\cite{Chumakov1999,Deak1994,Baron1994}. In\npanels \\textit{c} to \\textit{e}, the four resonance lines of the $+1$ and $-1$\nstroboscopic orders (right and left sides, respectively) partially overlap\nwith the $0^{\\mathrm{th}}$ order in the central part of the spectrum.\n\nThe delicate interplay between electronic and nuclear scattering is\ndemonstrated by the considerable difference between the stroboscopic SMR\nspectra \\textit{c} to \\textit{e} in Fig.~\\ref{fefe}, which are taken at only\nslightly different grazing angles. In contrast to the symmetric forward\nscattering spectra \\cite{Callens2002,Callens2003}, the stroboscopic SMR\nspectra are asymmetric due to the interference between the electronic and\nnuclear scattering. They also display both ``absorption-like'' and\n``dispersion-like'' resonance line shape contributions. In case of decreased\nnuclear scattering strength and of the same electronic reflectivity (cf.\npanels \\textit{d} and \\textit{e} in Fig.~\\ref{fefe}), the signal to baseline\nratio of the central part (heterodyne spectrum) decreases as compared to the\nsignal to baseline ratio of stroboscopic orders $\\pm 1$\\ in the spectrum\nwings.\n\nThe full lines are simultaneous least squares fits, using the theory\noutlined above and the computer code EFFI \\cite{Spiering00}. The\ninterference between nuclear and electronic scattering makes it possible to\nfit the layer structure in this isotope-periodic multilayer. The fitted\nvalue of the total thickness of pure $\\alpha -\\mathrm{Fe}$ is $42.5\\ \\mathrm{%\nnm,}$ comprised of nine times $1.49\\ \\mathrm{nm}$ of $^{\\mathrm{nat}}\\mathrm{%\nFe}$ and $3.23\\ \\mathrm{nm}$ of $^{57}\\mathrm{Fe,}$ with $0.4\\ \\mathrm{nm}$\ncommon roughness at the interfaces. In order to achieve the simultaneous\nfit, displayed by the full line in Figure \\ref{fefe}, we had to assume that\nhalf a bilayer on top and bottom ($^{\\mathrm{nat}}\\mathrm{Fe}$ and $^{57}%\n\\mathrm{Fe}$, respectively) was modified. The transversal hyperfine magnetic\nfield was fixed to $33.08\\ \\mathrm{T}$ in the nine $^{57}\\mathrm{Fe}$\/$^{%\n\\mathrm{nat}}\\mathrm{Fe}$ bi-layers in the middle of the multilayer, which\nis the room temperature value for $\\alpha -\\mathrm{Fe}$.\n\n\\subsection{Stroboscopic SMR of an antiferromagnetic $^{57}$Fe\/Cr multilayer}\n\nFig.~\\ref{fecraf} and Fig.~\\ref{fecrfm} display similar sets of spectra of\nan $^{57}$Fe\/Cr antiferromagnetically coupled epitaxial multilayer on\nMgO(001). The dots are the experimental data points, while the continuous\nlines are simultaneous fits to a model structure of $\\big[ {}^{57}\\mathrm{Fe}%\n\\left( 2.6~\\mathrm{nm}\\right) \/$ $\\mathrm{Cr}\\left( 1.3~\\mathrm{nm}\\right) %\n\\big] {}^{}_{20}\\:\\!$, based on the respective theory.\n\nNon-resonant reflectivity, TISMR and stroboscopic SMR spectra were recorded\nfirst with the Fe layer magnetizations parallel\/antiparallel (Fig.~\\ref%\n{fecraf}) to the $k-$vector of the SR beam. The stroboscopic spectra were\ntaken at the angles of total reflection (c), at the antiferromagnetic (d)\nand at the structural Bragg peak (e) positions. After this, a magnetic field\nof 20 mT was applied to the multilayer in longitudinal direction. This is\nknown to flop the magnetizations to the perpendicular Fe(010) easy axis \nof the magnetization \\cite{BottyanBSF1,Tancziko2004}. Non-resonant\nreflectivity, TISMR and stroboscopic SMR spectra at the same angular\npositions were again collected (Fig.~\\ref{fecrfm}).\n\nThe major difference between Figs.~\\ref{fecraf} and \\ref{fecrfm} is the\npresence, respectively absence, of the AF Bragg peak in the delayed\nreflectivity curve b. This antiferromagnetic alignment, i.e., the\nlongitudinal hyperfine field of alternating sign in consecutive Fe layers,\nis justified by the simultaneous fit in Fig.~\\ref{fecraf}. In Fig.~\\ref%\n{fecrfm}, the fitted Fe magnetizations are perpendicular to the wave vector\nof the SR. Indeed, the scattering amplitudes depend on the angle of the wave\nvector and the direction of the hyperfine magnetic field. In the case of\nperpendicular orientation, this angle is 90 degrees for consecutive layer\nmagnetizations and no AF contrast can be observed. In case of\nparallel\/anti--parallel orientations, however, the angles with respect to\nthe wave vector of SR are 0 and 180 degrees, respectively. Therefore, the\nhyperfine contrast is present and the AF Bragg peak is visible in panel b of\nFig.~\\ref{fecraf}.\n\nThe count rate at the baseline of a stroboscopic SMR spectrum, measured at a\ncertain grazing angle $\\theta $, is closely related to the TISMR spectrum at\nthis angle. Therefore, the respective experimental count rates of the\nstroboscopic SMR spectrum at the AF Bragg peak positions (panel d) differ by\nalmost two orders of magnitude. Spectrum 3d is also the only spectrum for\nwhich no considerable enhanced dynamic broadening can be observed.\n\nNote that, in panels d, the zeroth order resonances are considerably enhanced\nwith respect to the $\\pm 1\\,$order stroboscopic resonances. This can be\nexplained by an enhanced radiative coupling of the samples. Since the\nradiative coupling does not contribute to the $\\pm 1\\,$order stroboscopic\nresonances, it only influences the baseline and the central resonances.\n\nAt the multilayer Bragg reflections (panel e), and at the total reflection\npeak (panel c), the suppression of the higher stroboscopic orders is much\nsmaller, which means that the radiative coupling term is not dominating here.\nThese spectra also show a left\/right asymmetry due to the variation of the\nphase of the total scattering amplitude with energy. This latter allows for\nphase determination of the scattering amplitude from a set of stroboscopic\nSMR spectra, which work will be published later.\n\n\\section{Summary}\n\nIn summary, the concept of heterodyne\/stroboscopic detection of nuclear\nresonance scattering was outlined for a general scattering channel, with\nspecial emphasis on the grazing incidence reflection case. In any\nnon-forward scattering channel, the electronic scattering influences the NRS\nspectral shape, while in forward scattering, this is a mere multiplicative\nfactor. The interplay between electronic and nuclear scattering, as a\nfunction of the scattering angle, facilitates the determination of the\nelectronic and nuclear scattering amplitudes. The code of the present theory\nhas been merged into the EFFI program \\cite{Spiering00}, and was used in\nsimultaneous data fitting of x-ray reflectivity, time integral reflectivity\nand stroboscopic SMR spectra. Similar to time differential SMR, stroboscopic\nSMR spectra have been shown to be sensitive to the direction of the\nhyperfine fields of the individual layers. Therefore, it is possible to\napply this method to the study of magnetic multilayers and thin films. The\nexperiments on $\\left[ ^{57}\\mathrm{Fe}\\left( 2.6~\\mathrm{nm}\\right) \/%\n\\mathrm{Cr}\\left( 1.3~\\mathrm{nm}\\right) \\right] _{20}$ and $\\left[ ^{%\n\\mathrm{nat}}\\mathrm{Fe}\/^{57}\\mathrm{Fe}\\right] _{10}$ multilayers\ndemonstrated that stroboscopic detection of synchrotron M\\\"{o}ssbauer\nreflectometry of $^{57}$Fe-containing thin films is feasible in dense bunch\nmodes,\nwhich are not necessarily suitable for time differential nuclear resonance\nscattering experiments on $^{57}$Fe.\n\n    \n    \n    \n    \n    \n    \n\n    \n\n\\ac\n{The authors gratefully acknowledge the beam time supplied free of charge by\nthe Japan Synchrotron Radiation Institute (JASRI) for experiment No:\n2002B239-ND3-np. Our gratitude goes to Dr. A.Q. Baron (SPring-8, JASRI) for\nkindly supplying the fast Hammamatsu APD detectors and Dr. Johan Dekoster\n(IKS\\ Leuven) for preparing the multilayer samples for the experiment.\nSupport by the Flemish-Hungarian inter-governmental project BIL14\/2002, the\nDYNASYNC Framework Six project of the European Commission (contract number:\nNMP4-CT-2003-001516) the Fund for Scientific research--Flanders (G.0224.02),\nthe Inter-University Attraction Pole (IUAP P5\/1) and the Concerted Action of\nthe KULeuven (GOA\/2004\/02) is gratefully acknowledged. L.~De\\'{a}k and R.~Callens \nthank the Deutscher Akademischer Austauschdienst (DAAD) and the\nFWO-Flanders, respectively, for financial support.}\n\n    \n    \n\n\\referencelist[oldstrobo-140807]\n\n    \n    \n    \n\n    \n    \n\n    \n\n\\begin{figure}\n\\includegraphics{Fig-1.eps}\n\\caption{Experimental setup for stroboscopic Synchrotron M\\\"{o}ssbauer Reflectometry.}\n\\label{expsetup}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\includegraphics{Fig-2.eps}\n\\caption{Prompt electronic (a) and delayed nuclear reflectivity (b) curves\nas well as stroboscopic SMR spectra (c) to e), of a $\\left[ ^{\\mathrm{nat}}%\n\\mathrm{Fe}\/^{57}\\mathrm{Fe}\\right] _{10}$ isotopic multilayer at grazing\nangles indicated by the arrows. Vertical dotted lines in panels c) to e)\nindicate the center of the zero and $\\pm 1$ order stroboscopic bands\nseparated by $\\protect\\varepsilon \\approx 10.93~\\mathrm{mm}\/\\mathrm{s}$ for\nthe applied observation window period.}\n\\label{fefe}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics{Fig-3.eps}\n\\caption{Prompt electronic (a) and delayed nuclear (b) reflectivity curves\nas well as stroboscopic SMR spectra (c to e) of a $\\mathrm{MgO(001)}\/\\left[\n^{57}\\mathrm{Fe}\/\\mathrm{Cr}\\right] _{20}$ antiferromagnetic multilayer at\nvarious angles indicated by arrows in b). The consecutive Fe layer\nmagnetizations are aligned parallel\/antiparallel with to the SR beam.\nVertical dotted lines in panels c) to e) indicate the center of the zero and \n$\\pm 1$ order stroboscopic bands separated by $\\protect\\varepsilon \\approx\n10.93~\\mathrm{mm}\/\\mathrm{s}$ for the applied observation window period.}\n\\label{fecraf}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics{Fig-4.eps}\n\\caption{Prompt electronic (a) and delayed nuclear (b) reflectivity curves\nas well as stroboscopic SMR spectra (c to e) of a $\\left[ ^{57}\\mathrm{Fe}%\n\\left( 2.6\\,\\mathrm{nm}\\right) \/\\mathrm{Cr}\\left( 1.3\\,\\mathrm{nm}\\right) %\n\\right] _{20}\/\\mathrm{MgO}$ antiferromagnetic multilayer at various angles\nindicated by arrows. The consecutive Fe layer magnetizations are aligned\nperpendicular to the SR beam. Vertical dotted lines in panels c) to e)\nindicate the center of the zero and $\\pm 1$ order stroboscopic bands\nseparated by $\\protect\\varepsilon \\approx 10.93~\\mathrm{mm}\/\\mathrm{s}$ for\nthe applied observation window period.}\n\\label{fecrfm}\n\\end{figure}\n\n\\end{document}                   \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\n\nConsider a probability distribution $p$ over a countable alphabet $\\mathcal{X}$. Let $X^n$ be a sample of $n$ independent observations from $p$. Let $N_u(X^n)$ be the number of appearance of the symbol $u$ in the sample.  Unobserved (or unseen) events refer to outcomes which do not appear in the sample, $\\mathcal{X}_0(X^n)=\\{u\\;|\\;N_u(X^n)=0\\}$. \nIn this work we study inference of the unobserved. Specifically, we are interested in simultaneous confidence intervals (SCIs) for the parameters       $\\mathcal{M}_p(X^n)=\\{p(u)\\;|\\;N_u(X^n)=0\\}$. This problem is of high interest in variety of domains. For example, in $2006$ the New Zealand Ministry of Health reported new cancer cases among local populations \\cite{NZ_data}. Specifically, they focused on minorities in different age groups. Their study distinguished between $75$ cancer types. Their report indicated several interesting findings. For example, for Maori females under the age of $30$, there has been a total of $58$ new cancer cases that year. Importantly, out of the $75$ studied cancer types, only $37$ were observed. Does that mean that the remaining $38$ unobserved types of cancer are unlikely to appear in this population? The answer is obviously no, and we would like to infer the likelihood of these unobserved cancer types. Naturally, this example is just a special case of a broader problem, where multiple outcomes are studied and the sample size is limited.\n\n\nThe classical approach for the studied problem is based on the \\textit{rule-of-three}. The rule-of-three (ROT) suggests that for a sample of $n$ observations from a Bernoulli distribution, an approximate CI of level $1-\\alpha$ is given by $[0,-\\log(\\alpha)\/n]$ (as shown in Section \\ref{Previous_Work}). Extending this result to a multinomial  setup requires a simple multiplicity correction. That is, \na simultaneous CI (of level $1-\\alpha$) for the unobserved satisfies $[0,-\\log(\\alpha\/k)\/n]$. As we can see, the obtained CI grows with the alphabet size $k$ and may be too conservative. More importantly, it requires the knowledge of $k$, which is not always available. For example, it is well-known that the number of cancer types is much greater than $75$, despite the report above \\cite{SEER_data}. \n\nIn this work we introduce a novel selective inference scheme for unobserved events. That is, we construct CIs only for the events that do not appear in the sample, while refraining from multiplicity correction over the entire alphabet size. To the best of our knowledge, this work is the first to directly address this basic problem. We distinguish between two setups. We first study the case where the alphabet size $k$ is unknown and even unbounded. We obtain a simple closed-form CI that is independent of $k$ and, most importantly, does not grow with it (as opposed to the ROT). Next, we focus on the setup where the alphabet size $k$ is known. Here, we introduce an efficient computational routine which utilizes the alphabet size and further improves our proposed CI. Then, we show that our results are tight. That is, we show that the length of our proposed CI cannot be further reduced (up to a negligible scale). We demonstrate the performance of our proposed scheme in synthetic and real-world experiments, and show it significantly improves upon the alternative.  Finally, we apply our results to large alphabet inference. We introduce a novel simultaneous CI scheme for large alphabet distributions which outperforms currently known methods while maintaining the prescribed coverage rate. \n\n\n\n\n\n\n\\section{Previous Work}\n\\label{Previous_Work}\nConsider a set of fixed and unknown parameters  $\\theta_1,...\\theta_k$. Given a confidence level $1-\\alpha$, a simultaneous confidence region for $\\{\\theta_j\\}_{j=1}^k$ is defined as a collection $\\{T_j(X^n)\\}_{j=1}^k$ such that \n\n\\begin{align}\\label{classical}\n    P(\\cup_j\\{\\theta_j\\notin T_j(X^n)\\})\\leq \\alpha.\n\\end{align}\nIn words, the probability that all $\\theta_j$ simultaneously reside within their corresponding CI  $T_j(X^n)$ is not smaller than $1-\\alpha$. Selective inference generalizes this framework and considers a subset of parameters of interest, selected during the experiment. For example, consider a linear regression problem with feature selection. Naturally, the parameters of interest are those selected by the model. In other words, we would like to infer on a (random) subset of parameters (the \\textit{selected parameters}) and not the entire collection.\n\n\nThe problem of inference over selected hypotheses, parameters, or models was  recognized about seven decades ago (see \\cite{ben2017concentration} for a detailed discussion). One of the first major contributions to the problem is due to Benjamini and Yekutieli, who considered the problem of constructing CIs for selected parameters \\cite{benjamini2005false}. In their work, they showed that conditional coverage, following any selection rule for any set of (unknown) values for the parameters, is impossible to achieve. This means we cannot simply infer on the chosen parameters, given that they were selected. Benjamini and Yekutieli suggested an alternative viewpoint to the problem; instead of controlling the conditional coverage, the obstacle to avoid is that of making a false coverage statement. Specifically, given a selection rule, three outcomes are possible at each experiment; either a covering CI is constructed, a non-covering CI is constructed, or the interval is not constructed at all. Therefore, even though a $1-\\alpha$ CI does not offer selective (conditional) coverage, the probability of constructing a non-covering CI is at most $\\alpha$, \n\\begin{align}\\label{SoS}\n    P(\\cup_j\\{\\theta_j\\notin T_j(X^n), \\theta_j\\text{ is selected}\\})\\leq \\alpha.\n\\end{align}\nThis formulation is also known as \\textit{Simultaneous over Selected} (SoS) (Equation (4) in \\cite{benjamini2019confidence}). Selective inference was extensively studied over the years. In \\cite{benjamini2005false}, Benjamini and Yekulieli relaxed (\\ref{SoS}) and defined the \\textit{false coverage rate} (FCR) as the expected proportion of parameters not covered by their CIs among the selected parameters. They introduced several controlling procedures for this formulation in different setups. The FCR framework was generalized and applied to a variety of applications. Lee et al. \\cite{lee2016exact} and Tibshirani et al. \\cite{tibshirani2016exact} constructed confidence interval for parameters selected by the Lasso and by forward step-wise selection, respectively. Berk et al. \\cite{berk2013valid} addressed the problem of inference when the model is selected because a pre-specified explanatory variable had the highest statistical significance, which restricts the family over which simultaneous coverage is required. Weinstein and Yekutieli \\cite{weinstein2020selective} designed FCR intervals that try to avoid covering zero. See \\cite{benjamini2019confidence} for additional references and examples. Benjamini et al. revisited the problem of constructing confidence intervals for seleceted parameters in \\cite{benjamini2019confidence}. They defined four controlling formulations, namely SoS (as appears in (\\ref{SoS})), FCR, \\textit{Simultaneous over all Possible selections} (SoP) and \\textit{Conditional over Selected} (CoS),  (See (1)-(4) in \\cite{benjamini2019confidence}). They focused their attention to SoS and studied the problem of SoS-controlling CIs for the $r$ largest parameters (of a given collection of parameters). \nA similar framework was also studied by Katsevich and Ramdas \\cite{katsevich2020simultaneous}, who addressed simultaneous selective inference in testing under SoP. Specifically, They considered making selective inference statements on many selection rules, guaranteeing these statements hold simultaneously with high probability. \n\nIn this work we study interval estimation of the unseen. This task may be viewed as a selective inference problem, as we construct CIs only for the parameters of the events that are missing from the sample. Since the selected parameters are data-dependent we focus our attention to the SoS framework (\\ref{SoS}). In this sense, our work is an application of SoS-controlling CIs for this important problem.\n\n\nEstimation of the unseen has been extensively studied over the years. Interestingly, most work focus on point estimation, in a variety of setups and applications. Perhaps the first major contribution to this problem dates back to Laplace in the $18^{th}$ century \\cite{laplace2012pierre}. In his work, Laplace studied the \\textit{sunrise problem};  given that the sun raised every morning until today, what is the probability that it will rise tomorrow? Laplace addressed the problem of unobserved events by adding a single count to all $k$ events in the alphabet (including the unobserved). Then, the desired estimate is simply the empirical frequency,   $1\/(n+k)$. This scheme is also known as the  \\textit{rule of succession}. The Laplace estimator was later generalized to a family of \\textit{add-constant} estimators. An add-$c$ estimator assigns to a symbol that appeared $t$ times a probability proportional to $t+c$, where $c$ is a pre-defined constant.  Add-constant estimators hold many desirable properties, mostly in terms of their simplicity and interpretability \\cite{orlitsky2003always}.\nHowever, when the alphabet size $k$ is large compared to the sample size $n$, add-constant estimators perform quite poorly \\cite{orlitsky2003always}. Furthermore, add-$c$ estimators require the knowledge of the alphabet size $k$, which is not always available. Additional caveats of add-$c$ estimators are discussed in  \\cite{gale1994s}.\n\n\nMany years after Laplace, a major milestone was established in the work of Good and Turing \\cite{good1953population},  while trying to break the Enigma Cipher during World War II \\cite{orlitsky2003always}. The Good-Turing (GT) framework suggests that unobserved events shall be assigned a probability proportional to the number of events with a single appearance in the sample. This approach introduced a significant improvement compared to known estimators at the time. Furthermore, its promising performance and practical appeal have led many researchers to study and generalize these ideas. \n\nUnseen estimation is highly related to the \\textit{missing mass} problem; the total probability of symbols which do not appear in the sample. Estimating the missing mass is a basic problem in statistics and related fields (see \\cite{battiston2020consistent} and references therein). Further, it  corresponds to an important prediction  task. Namely, the problem of estimating the likelihood of encountering a future event which does not appear in the sample. Here too, the most popular approach is the GT estimator   (which is, again, proportional to the number of events with a single appearance in the sample). A variety of results were introduced over the years, focusing on the  properties of the missing mass the GT estimator. This includes, for example, asymptotic normality and large deviations \\cite{gao2013moderate}, admissibility and concentration properties\n\\cite{ben2017concentration}, expectation, consistency and convergence rates \\cite{mcallester2000convergence,drukh2005concentration,mossel2019impossibility,painsky2021refined,painsky2022convergence,painsky2022data}, Bayesian estimation schemes \n\\cite{lijoi2007bayesian,favaro2016rediscovery,favaro2012new}, and the\nestimation of the missing mass in the context of feature models under minimax risk \\cite{ayed2019good}.\n\nAs mentioned above, there are many results on point and interval estimation of the missing mass. Yet, these results only apply to the total mass of the unseen. Notice that this problem is fundamentally different than ours. Specifically, we are interested in inferring the probability of each unobserved outcome (as required, for example, in the Maori cancer study), and not their sum. To the best of our knowledge, inference of the unobserved is currently considered only in the simple binomial case ($k=2$). The \\textit{rule-of-three} (ROT) suggests that for a sample of $n$ identical samples from a Bernoulli distribution with a parameter $\\theta$, the edge of the CI is given by $P(X^n=0) = \\alpha$ which leads to $(1-\\theta)^n = \\alpha$. This implies  $n\\log(1-\\theta)\\approx -n\\theta=\\log(\\alpha)$, where the approximation follows from $\\log(1-\\theta)\\approx-\\theta$, for $\\theta$ close to zero. Plugging $\\alpha=0.05$ results in a one-sided CI of approximately $3\/n$ (henceforth, rule-of-three). As mentioned in Section \\ref{intro}, the ROT may be generalized to the control all the events that do not appear in the sample by applying a simple Bonferroni correction, leading to a CI of $[0,-\\log(\\alpha\/k)\/n]$. The ROT was further extended in different setups. For example, the Vysochanskij-Petunin inequality \\cite{vysochanskij1980justification} shows that the ROT holds for unimodal distributions with finite variance, beyond just the binomial distribution. To the best of our knowledge, our contribution is the first to directly address SCIs for unobserved events in the multinomial setup. \n\n\\section{Problem Statement}\n\\label{defs}\nDenote the collection of \\textit{missing probabilities} as \n\\begin{align}\n\\mathcal{M}_p(X^n)=\\{p(u)\\;|\\; N_u(X^n)=0\\}.\n\\end{align}\nWe are interested in simultaneous one-sided CIs for the parameters in $\\mathcal{M}_p(X^n)$. This corresponds to constructing a CI only for the greatest element in the set. Hence, our statistic of interest follows \n\\begin{align}\\label{M_max}\n    M_{max}(X^n)=\\max_{u \\in \\mathcal{X}} \\mathcal{M}_p(X^n)=\\max_{u \\in \\mathcal{X}}\\{p(u)\\mathbbm{1}(N_u(X^n)=0)\\},\n\\end{align}\nwhere $\\mathbbm{1}(\\cdot)$ is the indicator function. Notice that $M_{max}(X^n)$ depends on the (unknown) probability $p$, which is omitted from the syntax for brevity. Our goal is to construct a one-sided CI for $M_{max}(X^n)$, in a confidence level of $1-\\alpha$. Specifically, we are interested in $T(X^n)$ such that\n$P(M_{max}(X^n)\\geq T(X^n))\\leq \\alpha.$\nUnfortunately, the maximum operator is an involved,  non-smooth functional. Therefore, we begin our analysis by representing $M_{max}(X^n)$ as the limit of an $r$-norm. Specifically, given a sample $X^n$ and a fixed parameter $r\\geq 1$, we define   \n\\begin{align} \\label{M_r}\n    M_r(X^n)\\triangleq\\sum_{u\\in\\mathcal{X}}p^r(u)\\mathbbm{1}(N_u(X^n)=0).\n\\end{align}\nThe $r$-norm of the missing probabilities follows\n\\begin{align}\n    ||\\{p^r(u)\\mathbbm{1}(N_u(X^n)=0)\\}_{u \n    \\in \\mathcal{X}}||_r\\triangleq (M_r(X^n))^{1\/r}.\n\\end{align}\nConsequently, we have that \n\\begin{align}\\label{r-norm lim}\n\\lim_{r\\rightarrow \\infty} (M_r(X^n))^{1\/r}=\\max_{u \\in \\mathcal{X}}p(u)\\mathbbm{1}(N_u(X^n)=0)\\triangleq M_{max}(X^n)\n\\end{align}\nand \n$(M_t(X^n))^{1\/t}\\leq (M_{r}(X^n))^{1\/r}$\nfor any $1\\leq r\\leq t$ \\cite{maddox1988elements}. This means that $M_{max}(X^n)\\leq (M_r(X^n))^{1\/r}$ for every $r\\geq 1$.\nTherefore, a $(1-\\alpha)$-level confidence interval for $(M_r(X^n))^{1\/r}$ is also an  $(1-\\alpha)$-level confidence interval for $M_{max}(X^n)$,\n\\begin{align}\\label{CI r-norm}\n    P\\left(M_{max}(X^n)\\geq T(X^n)\\right)\\leq P\\left((M_r(X^n))^{1\/r}\\geq T(X^n)\\right)\\leq \\alpha.\n\\end{align}\nDenote the expected value of $M_r(X^n)$ as\n$E_{r,n}(p)\\triangleq\\mathbb{E}_{X^n \\sim p}(M_r(X^n)).$\nDefine the worst-case (supremum) of $E_{r,n}(p)$ over $\\mathcal{P}$ as\n\n\\begin{equation}\\label{E_r(P)}\nE_{r,n}(\\mathcal{P}) \\triangleq \\sup_{p \\in \\mathcal{P}} E_{r,n}(p).\n\\end{equation}  \nIn this work we focus on two sets of probability distributions $\\mathcal{P}$. Let $\\Delta_k$ be the set of all distributions of an alphabet size $k$, while  $\\Delta$ be the set of all distributions over any countable alphabet $\\mathcal{X}$ (that is, $k\\rightarrow \\infty$). Markov's inequality suggests that for every $\\lambda>0$,\n\\begin{align}\\label{Markov}\n    P(M_r(X^n) \\geq \\lambda)\\leq \\frac{E_{r,n}(p)}{\\lambda}\\leq \\frac{E_{r,n}(\\mathcal{P})}{\\lambda},\n\\end{align}\nwhere the second inequality follows from (\\ref{E_r(P)}). Setting  $\\alpha=E_{r,n}(\\mathcal{P})\/\\lambda$, we have that  \n\\begin{align}\\label{CI}\n    &P\\left(M_r(X^n)\\geq \\frac{E_{r,n}(\\mathcal{P})}{\\alpha}\\right)=P\\left(\\left(M_r(X^n)\\right)^{1\/r}\\geq \\left(\\frac{E_{r,n}(\\mathcal{P})}{\\alpha}\\right)^{1\/r}\\right)\\leq \\alpha.\n\\end{align}\nPlugging $\\mathcal{P}=\\Delta$ (alternatively, $\\mathcal{P}=\\Delta_k$), we obtain a one-sided confidence interval for $\\left(M_r(X^n)\\right)^{1\/r}$ (and henceforth, $M_{max}(X^n)$) which holds for every $p \\in \\Delta$ (alternatively, $\\Delta_k$). Notice that the obtained confidence interval is independent of the sample $X^n$, similarly to the ROT. This makes it a  robust non-random scheme, that generalizes the ROT for the multinomial setup. Further, notice that for $r=1$, (\\ref{CI}) is a CI for the missing mass. In that sense, our proposed framework generalizes the missing mass problem, and introduces CIs for any $r$-norm of the missing probabilities, $M_r(X^n)$. Interestingly, point estimation of $M_r(X^n)$ was recently studied by Chandra and Thangaraj in quite a different context \\cite{chandra2021estimation}. In their work, they showed that for $r \\in (1,\\infty)\/\\mathbb{N}$,\n\\begin{align}\\label{chandra}\n    \\min_{\\hat{M}_r (X^n)}\\max_{p\\in \\Delta} \\mathbb{E}(M_r(X^n)-\\hat{M}_r(X^n))^2\\leq O\\left(\\frac{1}{n^{2(r-1)}}\\right)\n\\end{align}\nwhere $\\mathbb{N}$ is the set of natural numbers and $O(\\cdot)$ is the standard big O notation \\cite{bachmann1894analytische}. Similarly, for $n\\geq2r$ and $r\\in \\mathbb{N}$, they attained a bound of $O\\left({1}\/{n^{2r-1}}\\right)$. \nWe discuss these bounds and compare them to our results later in Section \\ref{unbounded_alphabet}. \n\nTaking a closer look at our derivation steps, one may wonder if the obtained data-independent CI is too conservative. Later in Section \\ref{tightness}, we show that the proposed CI is indeed tight, in the sense that there exists a distribution $p$ for which (\\ref{CI}) is attained with equality. This is not the typical notion of tightness in inference literature, where one would expect a CI to be tight for every $p$. Yet, this approach is quite common in the study of the unobserved. For example, notice that the popular ROT is also only tight in this sense, where only $\\theta=-\\log(\\alpha)\/n$ attains it equality. Additional examples in missing mass literature are discussed in \\cite{rajaraman2017minimax,ben2017concentration,acharya2018improved,painsky2022generalized}. Let us proceed with our analysis and study $E_{r,n}(\\mathcal{P})$ for both bounded ($\\mathcal{P}=\\Delta_k$) and unbounded ($\\mathcal{P}=\\Delta$) alphabets.  \n\n\\section{Unbounded Alphabet Size}\n\\label{unbounded_alphabet}\n\nWe begin our analysis with the unbounded alphabet setup. First, the expected value of $M_r(X^n)$ satisfies\n\\begin{align}\\label{E(p) explicit}\n    E_{r,n}(p)=\\sum_{u\\in \\mathcal{X}} p^r(u)\\mathbb{E}_{X^n \\sim p} \\left(\\mathbbm{1}(N_u(X^n)=0) \\right)=\\sum_{u\\in \\mathcal{X}} p^r(u)(1-p(u))^n.\n\\end{align}\nWe would now like to bound (\\ref{E(p) explicit}) from above, for every possible $p\\in \\Delta$. For this purpose, we introduce the following proposition.\n\\begin{proposition} \\label{prop1}\nLet $p$ be a distribution over a countable alphabet $\\mathcal{X}$. Let $\\phi:[0,1] \\rightarrow \\mathbb{R}$. Then, \n    $\\sum_{u} p(u)\\phi(p(u))\\leq \\max_{q \\in [0,1]}\\phi(q).$\nFurther, equality is achieved for a uniform $p$. \n\\end{proposition}\n\\begin{proof}\nLet $Y \\sim p$ and define a random variable $T(u)$, such that $T(u)=\\phi(p(u))$. Then,\n   \n        $$\\mathbb{E}(T(Y))=\\sum_{u,\\in \\mathcal{X}} p(u)\\phi(p(u))\\leq \\max_{q \\in [0,1]}\\phi(q),$$\n        where in the last inequality, the expectation of a random variable is bounded from above by its maximal value.\n        Notice that equality is achieved if all $p(u)$'s are equal.\n     \n   \n\\end{proof}\n\\noindent Applying Proposition \\ref{prop1} to (\\ref{E(p) explicit}) we obtain\n\\begin{align}\\label{basic_unbounded}\n    E_{r,n}(p)=\\sum_{u\\in \\mathcal{X}} p^r(u)(1-p(u))^n\\leq \\max_{q\\in[0,1]} q^{r-1}(1-q)^n=(q_{r,n}^*)^{r-1}(1-q_{r,n}^*)^n,\n\\end{align}\nwhere $q_{r,n}^*=(r-1)\/(r-1+n)$.\nFurther, equality is obtained for $p(u)=q_{r,n}^*$, which implies an alphabet size of $k=(r-1+n)\/(r-1)$.\nTo conclude, for a given $r\\geq1$ and an unbounded alphabet size, we have that\n\\begin{align}\\label{temp}\n    E_{r,n}(\\Delta)=(q_{r,n}^*)^{r-1}(1-q_{r,n}^*)^n,\n\\end{align}\nwhich further implies that $E_{r,n}(\\Delta)=O(1\/n^{r-1})$. \nIn addition, the distribution which attains the above is a uniform distribution over an alphabet size $k=(r-1+n)\/(r-1)$.  Finally, a one-sided confidence interval for $(M_r(X^n))^{1\/r}$ is necessarily a one-sided confidence interval for $M_{max}(X^n)$, for every $r\\geq 1$. This leads to the following theorem. \n\n\\begin{theorem}\\label{theorem_analytical}\nLet $p$ be a probability distribution over a countable alphabet $\\mathcal{X}$. Let $X^n$ be $n$ independent samples from $p$. Let $M_{max}(X^n)$ be the maximum over the set of missing probabilities, as defined in (\\ref{M_max}). Then, the following holds,   \n\\begin{align}\\label{CI_unbounded}\n    P\\left(M_{max}(X^n)\\geq \\min_{r\\geq 1}\\left((q_{r,n}^*)^{r-1}(1-q_{r,n}^*)^n\/\\alpha  \\right)^{1\/r}\\right)\\leq\\alpha \n\\end{align}\nwhere $q_{r,n}^*=(r-1)\/(r-1+n)$. \n\\end{theorem}\n\nNotice that the obtained one-sided CI (\\ref{CI_unbounded}) is independent of the alphabet size $k$. This means that our proposed CI is constant, and does not grow with $k$, as opposed to the Bonferroni-corrected ROT (see Section \\ref{Previous_Work}). As we further examine our results, we observe that for every fixed $r\\geq 1$, the $r$-norm of the missing probabilities $M_r(X^n)$ satisfies \n\\begin{align}\\label{ours}\n    P\\left(M_{r}(X^n)\\geq E_{r,n}(\\Delta)\/\\alpha  \\right)\\leq\\alpha \n\\end{align}\nwhere $E_{r,n}(\\Delta)=O(1\/n^{r-1})$. Let us compare this result to \\cite{chandra2021estimation}. Applying Markov's inequality to (\\ref{chandra})  we obtain \n\\begin{align}\n    P\\left(|M_{r}(X^n)-\\hat{M}_r(X^n)|\\geq \\lambda\\right)\\leq\\max_{p\\in\\Delta} \\mathbb{E}\\left(M_{r}(X^n)-\\hat{M}_r(X^n)\\right)^2\/\\lambda^2.\n\\end{align}\nInterestingly, for $r>1$ this leads to a one-sided CI of length $O(1\/n^{r-1})$, similarly to (\\ref{ours}). However, we emphasize that (\\ref{chandra}) is obtained by a data-dependent estimator of $M_r(X^n)$, which also depends on $r$. This means that the choice of $r$ which minimizes the CI for $M_{max}(X^n)$ (as in (\\ref{CI_unbounded})) also depends on the sample and is henceforth invalid. However, this analysis emphasizes the tightness of our bound (\\ref{basic_unbounded}) and its resulting CI for $M_{max}(X^n)$, even if we compare it to a data-dependent scheme. \n\n\\section{Bounded Alphabet Size}\n\nLet us now study the case where the alphabet size is bounded from above. This is a typical setup, for example, in experimental studies where the number of outcomes is known a priori to the experiment. As discussed in Section \\ref{defs}, our proposed CI depends $E_{r,n}(\\mathcal{P})$, where $\\mathcal{P}=\\Delta_k$ in this setup. Therefore, our goal is to  maximize $E_{r,n}(p)$ over $p \\in \\Delta_k$,\n\\begin{align} \\label{bounded_objective}\nE_{r,n}(\\Delta_k)=\\max_{p\\in \\Delta_k} \\sum_{u\\in\\mathcal{X}}p^r(u)(1-p(u))^n.\n\\end{align}\nUnfortunately, this optimization problem does not hold a closed form solution. However, we show that it may be efficiently evaluated from  simple optimization considerations. We begin our analysis with the following property.\n\\begin{property}\\label{property1}\nLet $E_{r,n}(p)=\\sum_{u \\in \\mathcal{X}} p^r(u)(1-p(u))^n$. Let $$t^*=\\frac{r}{r+n},\\;\\;t_{1,2}=t^*\\pm \\frac{1}{r+n}\\sqrt{\\frac{rn}{r+n-1}}.$$\nAssume $0\\leq t_1\\leq t^*\\leq t_2\\leq 1$. \nFor $r\\geq 1$, the summand, $h(t)=t^r(1-t)^n$ satisfies the following:\n\\begin{enumerate}\n    \\item $h(t)$ has a local maximum at $t^*$\n    \\item $h(t)$ is concave in $t$, for $t_1 \\leq t \\leq t_2$.\n    \\item $h(t)$ is convex in $t$, for $0\\leq t\\leq t_1$ and $t_2\\leq t \\leq 1$ .\n    \n\\end{enumerate}\n\\end{property}\nThe proof of the above directly follows from the derivatives of the summand, $h(t)=t^r(1-t)^n$, and is located in Appendix A. Property \\ref{property1} shows that the function, $p^r(u)(1-p(u))^n$, consists of three separate regions, characterized by their concavity and convexity. This allows us to characterize the maximum of our objective.\n\n\n\\begin{theorem}\\label{theorem2}\nLet $p^* \\in \\Delta_k$ be the maximizer of $E_{r,n}(p)=\\sum_u p^r(u)(1-p(u))^n$ over $\\Delta_k$. Then, for $r\\geq 1$ the following holds.\n\\begin{enumerate}\n\\item $p^*(u)=p^*(v)$ for every  $p^*(u),p^*(v)\\in \\left[t_1,t_2\\right]$.\n\\item There exists at most a single $p^*(u)$ such that $p^*(u)\\in \\left(0,t_1\\right)$\n\n\\item There exists at most a single $p^*(u)$ such that $p^*(u)\\in \\left(t_2,1\\right]$\n\n\\end{enumerate}\n\\end{theorem}\nIn words, all $p^*(u)$ that are located in the concave region are identical, and there exists at most a single $p^*(u)$ in the interior of the convex regions. These properties are a direct consequence of the convexity\/concavity regions of the summand. The detailed proof is located in Appendix B. Proposition \\ref{theorem2} shows that the maximizer of $E_{r,n}(p)$ over $\\Delta_k$ depends on not more than four free parameters. Surprisingly, this results holds for every $k$. In other words, we may numerically evaluate $E_{r,n}(\\Delta_k)$ by considering only four free parameters, for every given $k$. This allows us to numerically evaluate the CI in a relatively small computational cost, even when the dimension of the problem increases, for every examined $r\\geq 1$, and choose the value of $r$ which minimizes the CI (similarly to (\\Ref{CI_unbounded})).\n\n\\section{Tightness Analysis}\\label{tightness}\nThe derivation of the proposed CI utilizes several relaxations and inequalities, such as the $r$-norm (\\ref{CI r-norm}) and Markov inequality (\\ref{Markov}). Therefore, it is of a reasonable concern that the obtained CI is over pessimistic. Here, we show that this it not the case. Specifically, we show that there exists a  distribution $p\\in \\Delta$ for which the proposed CI is (almost) tight.  \n\nAs we revisit our analysis in the unbounded alphabet setup (\\ref{basic_unbounded}), we observe that  $E_{r,n}(p)\\leq (q^*_{r,n})^{r-1}(1-q^*_{r,n})^n$, where equality holds if $p(u)=q^*_{r,n}$. In words, $E_{r,n}(p)$ attains its maximum for a uniform distribution over an alphabet of size $k^*=1\/q^*_{r,n}$. This means that in practice, even if the alphabet size $k$ is known to be greater than $k^*$, the worst-case distribution which attains $E_{r,n}(p)$ with equality is a uniform distribution with $p(u)=q^*_{r,n}$ for $k^*$ symbols, and $p(u)=0$ for the remaining alphabet. Interestingly, this type of distributions also attain Markov inequality with equality. Specifically, let $\\mathcal{U}_k$ be the set of uniform distributions over an alphabet size $m\\leq k$.  Then, for every $p_m\\in\\mathcal{U}_k$, we have $M_{max}(X^n)\\in\\{0,1\/m\\}$ and $P(M_{max}(X^n)\\geq 1\/m)=m{\\mathbb{E}_{X^n\\sim p}M_{max}(X^n)}$. This motivates exploring $\\mathcal{U}_k$ as a set of distributions for which our proposed CI may be tight. \n\nAs mentioned in Section \\ref{defs}, deriving an exact CI for $M_{max}(X^n)$, even when the underlying distribution $p$ is known, is not an easy task. However, we now show it is possible in several special cases, such as $p_m\\in \\mathcal{U}_k$. We begin with the following proposition.   \n\n\\begin{proposition} \\label{prop2}\n    Let $X^n$ be a sample of $n$ independent observations from $p_m \\in \\mathcal{U}_k$. Then, \n    \\begin{align}\\label{worst-case}\n   P\\left(M_{max}(X^n) \\geq \\frac{1}{m}\\right)=1-\\frac{m!S(n,m)}{m^n}\n   \\end{align}\nwhere $S(n,m)=\\frac{1}{m!}\\sum_{j=0}^m(-1)^j\\binom{m}{j}(m-j)^n$ is \\textit{Stirling number of the second kind}.\n\\end{proposition}\n   \nThe proof of Proposition \\ref{prop2} utilizes simple combinatorial properties. Specifically, given that $p_m$ is a uniform distribution over an alphabet size $m$, we have that $M_{max}(X^n)=1\/m$ if and only if there exists at least one symbol that do not appear in the sample, where all symbols are equiprobable. A detailed proof is provided in  Appendix C. \n\n\nNow, define $m_\\alpha$ as the largest value of $m$ for which $1-\\frac{m!S(n,m)}{m^n}\\leq \\alpha$. Then, $1\/m_\\alpha$ is the $\\alpha$-quantile of $M_{max}(X^n)$. This means that for $X^n \\sim p_{m_\\alpha}$, we cannot set any constant $c<1\/m_\\alpha$ such that $P(M_{max}(X^n)\\geq c)\\leq \\alpha$. In other words, $p_{m_\\alpha}$ requires a CI larger than $1\/m_{\\alpha}$ in order to control $M_{max}(X^n)$ in a confidence level of at least $1-\\alpha$. The implications of this result are fairly simple. In order to control every $p\\in \\Delta$ in a confidence level of $1-\\alpha$, a (constant) confidence interval of size of at least $1\/m_\\alpha$ is inevitable. In other words, $p_{m_\\alpha}$ is the distribution which requires the tightest CI (among all the distributions in $\\mathcal{U}_k$), and it is therefore the most challenging to control. We denote it as the \\textit{worst-case} distribution.  In the following section we compare our proposed CIs with $1\/m_\\alpha$ and show that the difference is practically negligible.   \n\n\\iffalse\nThen, we denote $p_{m_\\alpha}\\in \\mathcal{U}_k$ as the \\textit{$\\alpha$-level distribution}, for which \n\nfor which the $\\alpha$-confidence interval (namely, $1\/m_\\alpha$) is the smallest. In other words, we found a distribution $p_{m_\\alpha}$ such that \n\n\n\n$p_{m_\\alpha}$ is the distribution which requires the tightest CI (among all the distributions in $\\mathcal{U}_k$), and it is therefore the most challenging to control. We denote it as the \\textit{worst-case} distribution.  \n\n\n       \\fi\n\n\\section{Experiments}\\label{experiments}\nWe now illustrate the performance of our proposed CIs in synthetic and real-world experiments. First, we study six example distributions, which are common benchmarks for probability estimation and related problems \\cite{orlitsky2015competitive}. The Zipf's law distribution is a typical benchmark in large alphabet probability estimation; it is a commonly used heavy-tailed distribution, mostly for modeling natural (real-world) quantities in physical and social sciences, linguistics, economics and others fields \\cite{saichev2009theory}. The Zipf's law distribution follows $p(u;s,k)={u^{-s}}\/{\\sum_{v=1}^k v^{-s}}$ where $k$ is the alphabet size and $s$ is a skewness parameter.  Additional examples of commonly used heavy-tailed distributions are the geometric distribution, $p(u;\\alpha)=(1-\\alpha)^{u-1}\\alpha$, the negative-binomial distribution (specifically, see \\cite{efron1976estimating}), $p(u;l,r)= \\binom{u+l-1}{u} r^u(1-r)^l$ and the beta-binomial distribution $p(u;k,\\alpha,\\beta)= \\binom{k}{u}{B(u+\\alpha,k-u+\\beta)}\/{B(\\alpha,\\beta)}$. Notice that the support of the geometric and the negative binomial distributions is infinite. Therefore, for the purpose of our experiments, we truncate them to an alphabet size $k$ and normalize accordingly. Additional example distributions are the uniform, $p(u)=1\/k$, and the worst-case distribution, which is simply a uniform distribution over an alphabet size $1\/m_\\alpha$, as discussed in Section \\ref{tightness}. \n\nIn each experiment we draw $n=1000$ samples, and compare the lengths of different CIs for an increasing alphabet size. Figure \\ref{simulated_exp} illustrates the results we achieve. The red curve on top \ncorresponds to the Bonferroni-corrected ROT, as discussed in Section \\ref{Previous_Work}. As expected, it grows logarithmically with the alphabet size $k$. The blue curve below it is our proposed CI, for a known alphabet size $k$, while the blue dashed curve corresponds to the unbounded alphabet size. As we can see, the bounded $k$ curve is of similar length the ROT CI, for smaller values of $k$. However, as the alphabet size increases, it converges to the unobounded $k$ performance, as expected.  It is also evident that while the ROT CI grows with $k$, our proposed schemes are fixed, and demonstrate significantly shorter confidence intervals, while maintaining the desired coverage rate. As we examine the value of $r$-norm which minimizes our CI, we observe that it increases with $k$ (for the bounded $k$ scheme) and converges to approximately $r=10$. Finally, the black curve at the bottom is an Oracle CI, who knows the underlying distribution $p$. Specifically, the Oracle CI is simply the $\\alpha$-quantile of $M_{max}(X^n)$ in the case where $p$ is known. This serves us as a lower bound, for the best we can achieve in each experiment. We focus our attention to the worst-case distribution, which we study in detail in Section \\ref{tightness}. As we can see, the Oracle CI is almost identical to our proposed scheme in this setup. This means that our CIs are universally tight, in the sense that there exists a  distribution for which they we cannot be shorten.\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width =0.8\\textwidth,bb=10 70 750 540,clip]{simulated_six_rotated.pdf}\n\\caption{confidence intervals lengths in six synthetic experiments. We use the following parameters: Zipf's Law: $s=1.01$, Geometric: $\n\\alpha=0.4$, Negative-Binomial: $l=1, r=0.003$, Beta-Binomial: $\\alpha=\\beta=2$. The red curve on top is the rule-of-three CI, the blue curve below it is our proposed CI for a known alphabet size $k$. The blue dashed curve is our proposed method for unbounded $k$ and the black curve at the bottom is an Oracle CI, who knows the underlying distribution $p$.}\n\\label{simulated_exp}\n\\end{figure}\n\n\n\nNext, we turn to real-world experiments. Here, we follow \\cite{orlitsky2016optimal} and study three application domains. Notice that in these real-world settings, the true underlying probability is unknown. Hence, the missing probabilities refers to the frequency of symbols, in the full data-set, that do not appear in the sample. We begin with a corpus linguistic experiment. For this purpose we study a collection of word frequencies in English. \nSpecifically, we consider a list of word frequencies, collected from open source subtitles \\cite{subtitles1,subtitles2}. This list describes the frequency each word appears in the text, based on hundreds of millions of samples. We randomly sample $n$ words (with replacement) from the list, and construct a CI for the missing probabilities. The left chart of Figure \\ref{real_world_exp} demonstrates the CI of our proposed scheme, compared to the Bonferroni-corrected ROT. Notice we focus on the unbounded $k$ scheme, as it is more robust and may better describe the alphabet size in this setup (all the words in the English language). Next, we focus on a biota analysis. Gao et al. \\cite{gao2007molecular} considered the forearm skin biota of six subjects. They identified a total of $1{,}221$ clones consisting of $182$ different species-level operational taxonomic units (SLOTUs). As above, we sample $n$ out of the $1{,}221$ clones with replacement, and construct CI for the missing probabilities of the distinct SLOTUs found. The middle chart of  Figure \\ref{real_world_exp} demonstrates the results we achieve. Finally, we study census data. The right chart of Figure  \\ref{real_world_exp} considers the $2000$ United States Census \\cite{us2014frequently}, which lists the frequency of the top $1000$ most common last names in the United States. Here too, we sample $n$ names and construct corresponding CIs. Similarly to the synthetic experiments, our proposed scheme demonstrates shorter CIs than the ROT in the three examined setups, where the difference is typically more evident in the small $n$ regimes.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width =0.8\\textwidth,bb=10 140 745 470,clip]{real_world_updated_rotated.pdf}\n\\caption{confidence intervals lengths in three real-world experiments.}\n\\label{real_world_exp}\n\\end{figure}\n\n\n\n\n\\section{Application to Large Alphabet Inference}\\label{multinomial_intro}\n\nInference and estimation of unseen events is a corner-stone of large alphabet probability modeling. Here, the goal is to address not only the unobserved events, but the entire (very) large collection of possible outcomes. Specifically, the large alphabet regime considers multinomial distributions in cases where $k$ is much larger than $n$ (or at least comparable to it). This problem too is highly important to data-driven science and engineering. Its applications span a variety of disciplines  including  information retrieval \\cite{song1999general}, spelling correction \\cite{church1991probability}, word-sense disambiguation \\cite{gale1992method}, language modeling  \\cite{chen1999empirical}, learning theory \\cite{makur2020estimation} and many others. In this section we demonstrate the favorable properties of our proposed CI, as we apply it to large alphabet inference.  \n\nThere exists a large body of work on point estimation of large alphabet distributions. The GT probability estimator (based on the GT scheme described above) is perhaps the most popular estimator for this important task. While the GT estimator performs well in general, it is known to be sub-optimal for outcomes that frequently appear. Consequently, several modifications have been proposed, including the Jelinek-Mercer, Katz, Witten-Bell and Kneser-Ney estimators \\cite{chen1999empirical}. In language modeling for example, GT is usually used to estimate the probability of infrequent words, whereas the probability of frequent words is estimated by their empirical frequency. Different properties of the GT probability estimator were extensively studied over the years \\cite{mcallester2000convergence,drukh2005concentration,orlitsky2015competitive}.\nDespite this broad body of work, large alphabet inference has not received much attention. Current methods focus on two basic setups. The first considers an asymptotic regime, where $k$ is fixed the sample size $n$ is very large \\cite{quesenberry1964large,goodman1964simultaneous}. The second line of work addresses a fixed $n$, where the alphabet size $k$ is relatively small. Here, the most popular SCI scheme is arguably of Sison and Glaz \\cite{SisonGlaz1995}. In their work, Sison and Glaz (SG) proposed a method which utilizes Edgeworth expansions to approximate the desired distribution. Through extensive simulations, they showed that their method leads to smaller SCIs while maintaining a coverage rate closer to the desired level, compared to known methods at the time. Unfortunately, the SG scheme does not perform well in cases where the expected symbol counts are disparate \\cite{MayJohnson1997Properties}. Recently, \\cite{marton2022good} introduced a bootstrap framework for the case where both $k$ and $n$ are large. Yet, this approach is based on bootstrap sampling and does not provide solid theoretical guarantees. To the best of our knowledge, no method directly addresses the large alphabet regime, with provable performance guarantees.\n\n\nAs above, let $\\mathcal{X}$ be a countable alphabet. Here, we assume that the alphabet size $k$ is finite and known. Denote $p=p_1,\\dots,p_k$ as the unknown probability distribution over $\\mathcal{X}$, while $X^n$ is a collection of $n$ samples from  $\\mathcal{X}$. Let $S(X^n)$ be an $(1-\\alpha)$-level confidence region (CR) for $p$. That is,  \n$P(p \\in S(X^n))\\geq 1-\\alpha.$\nThe most popular form of a CR is the case where $S(X^n)$ is rectangular. That is, $S(X^n) = T_1(X^n), T_2(X^n), \\dots, T_k(X^n)$, where $T_j(X^n)=[a_j,b_j]$ for $j=1,\\dots,k$ and $0 \\leq a_j\\leq b_j\\leq 1$. This implies SCIs for the collection of the $k$ parameters, similarly to (\\ref{classical}). In fact, all the multinomial inference schemes mentioned above are rectangular CRs.   \n\n\nThe most basic rectangular CR may be obtained from a binomial viewpoint. That is, one may construct a binomial CI for each symbol independently and correct for multiplicity using a Bonferroni correction. Hence, the obtained SCIs are just a collection of binomial CIs (of confidence level $\\alpha\/k$), for every symbol in the alphabet. Naturally, this approach controls the prescribed confidence level (for every $n$ and $k$), but may be over pessimistic and result in large volume SCIs. \nNotice that such an approach also applies for unobserved symbols. Specifically, a binomial CI for symbols with zero counts is obtained by a Bonferroni-corrected ROT, as described in Section \\ref{Previous_Work}. \n\nLet us now introduce our proposed large alphabet inference scheme. We distinguish between observed and unobserved symbols. Assume that $n\\leq k$ and let $\\alpha_1,\\alpha_2 \\in [0,1]$ be two fixed constants. First, we set a binomial CI of level $1-\\alpha_1\/n$ for all the symbols that appear in the sample, while unobserved symbols are set a naive CI, $T_j(X^n)=[0,1]$. We have that\n\\begin{align}\n    P(p \\notin S(X^n))&=P(\\cup_{j=1}^k\\{p_j \\notin T_j(X^n)\\})\\leq \\sum_{j=1}^kP(p_j \\notin T_j(X^n))=\\\\\\nonumber\n    &\\sum_{j|N_j(X^n)=0}P(p_j \\notin T_j(X^n))+\\sum_{j|N_j(X^n)>0}P(p_j \\notin T_j(X^n))=\\\\\\nonumber\n    &\\sum_{j|N_j(X^n)>0}P(p_j \\notin T_j(X^n))\\leq n \\cdot \\frac{\\alpha_1}{n}=\\alpha_1\n\\end{align}\nwhere the first inequality follows from the union bound and the second inequality is due to $|\\{j|\\;N_j(X^n)>0\\}|\\leq n$ (that is, the number of symbols that appear in the sample is not greater than the sample size). Notice that in the case where $n>k$, we may define binomial CI of level $1-\\alpha_1\/k$ for all the symbols that appear in the sample, and the above still holds. \n\nNow, let $A_{n}=\\min_{r\\geq 1}\\left((q_{r,n}^*)^{r-1}(1-q_{r,n}^*)^n\/\\alpha_2  \\right)^{1\/r}$ be our proposed ($1-\\alpha_2$)-level CI for unobserved events (Theorem \\ref{theorem_analytical}). We would like to simultaneously control the events $M_{max}(X^n)\\leq A_{n}$ and $p \\in S(X^n)$ at a confidence level of $1-\\alpha$. Therefore, we set $\\alpha_1=\\alpha(1-c)$ and $\\alpha_2=\\alpha c$ for some $c\\in [0,1]$. We have, \n\\begin{align}\\label{eq2}\n    &P\\left(\\left\\{p \\notin S(X^n)\\}\\right\\}\\cup\\left\\{M_{max}(X^n)\\geq  A_n\\right\\}\\right)\\leq\\\\\\nonumber\n    &P(\\cup_{j=1}^k\\{p \\notin S(X^n)\\})+P(M_{max}(X^n)\\geq A_n)\\leq \\alpha(1-c)+\\alpha c=\\alpha.\n\\end{align}\nNotice that by simultaneously controlling both of the terms above, we may replace the naive unit intervals of the unobserved events with $[0,A_n]$. This implies the following scheme (Algorithm \\ref{alg:Ours}) for constructing the desired SCIs.\n\n\\begin{algorithm}[H]\n\t\\begin{algorithmic}[1]\n\t\t\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\t\t\\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n\t\t\\REQUIRE  A sample $X^n$, alphabet size $k$ and a confidence level $1-\\alpha$.\n\t\t\\STATE Set $c\\in[0,1]$\n\t\t\\STATE Construct a Binomial CI of level $1-\\alpha(1-c)\/n$ for all the symbols that appear in $X^n$ \n\t\t\\STATE Construct a CI for unobserved events (following Theorem $1$ or $2$) of level $1-\\alpha c$, for all the symbols that do not appear in $X^n$\n\t\\end{algorithmic}\n\t\\caption{Our Proposed Large Alphabet SCI's for Multinomial Proportions}\n\t\\label{alg:Ours}\n\\end{algorithm} \n\n\n\nThe scheme above introduces a simple analytical framework for constructing SCIs over large alphabets. The parameter $c$ defines an inference trade-off between observed and unobserved events. Specifically, for larger values of $k$ we expect many unobserved events which corresponds to a larger value of $c$. On the other hand, if $k$ is comparable to $n$, we would probably prefer a lower value of $c$. Therefore, choosing a reasonable value for $c$ is of a natural concern. Unfortunately, the choice of $c$ also depends on the unknown underlying distributions $p$. For example, a uniform $p$ results in fewer unobserved events than a degenerate $p$. Therefore, we cannot set a $c$ value that minimizes the SCIs uniformly, for every possible $p$. However, we  show it is possible to set a value for $c$, so that our proposed scheme  provably improves upon alternative methods.\n\nTypically, the performance of a CR is measured by its expected volume. That is, given two CRs, we say the one outperforms the other if its expected volume is smaller, while maintaining the prescribed confidence level. However, notice that in the large alphabet regime, the volume of a CR rapidly decays with the alphabet size $k$. For example, the volume of a rectangular CR with a fixed length of $L<1$ for each of its parameters demonstrates an exponential decay, $L^k$. Therefore, we focus on the log of the volume in this regime. Specifically, in each of our following experiments we measure the average log volume, as we cannot directly assess the volume by numerical means. Further, for the same reasons, we focus on the expected log volume as our analytical figure of merit.  \n\n\\begin{theorem} \\label{T3}\n    Let $p$ be a probability distribution over an alphabet $\\mathcal{X}$ of size $k$. Let $X^n$ be $n$ independent samples from $p$. Denote $A^{BC}_n=-\\log(\\alpha\/k)\/n$ as the Bonferroni-corrected CI for unobserved events.\n    Let $A_{n,c}=\\min_{r\\geq 1}\\left((q_{r,n}^*)^{r-1}(1-q_{r,n}^*)^n\/\\alpha c  \\right)^{1\/r}$ be our proposed CI, for a confidence level of $1-\\alpha c$. Define $z_0=z_{1-{\\alpha}\/{2k}}$ and $z_c=z_{1-{\\alpha(1-c)}\/{2n}}$, where $z_a$ is the $a$ quantile of a standard normal distribution. Assume there exists $c\\in[0,1]$ such that \n    \\begin{enumerate}[label=(\\alph*)]\n        \\item $k\\left(1-\\left(1-\\frac{1}{k}\\right)^n\\right)(z_c-z_0)+k\\left(1-\\frac{1}{k}\\right)^n(A_{n,c}-A^{BC}_n)\\leq 0$\n        \\item $(z_c-z_0)+(k-1)(A_{n,c}-A^{BC}_n)\\leq 0$\n    \\end{enumerate}\n    Then, for every $p\\in \\Delta_k$ , the following (approximately) holds, $$\\mathbb{E}\\log V_c\\leq \\mathbb{E}\\log V_0,$$ \n    where $V_c$ is the volume of our proposed CR (with a choice of $c$ that satisfies the above), $V_0$ is the volume of the Bonferroni-corrected CR and the approximation follows from Wald intervals for the Binomial proportions \\cite{de1820theorie}.\n\\end{theorem}\nTheorem \\ref{T3} establishes an important property of our proposed CR. Given a sample size $n$ and an alphabet size $k$, we seek a constant $c\\in[0,1]$ that satisfies the (a) and (b). This  requires a simple grid search over the unit interval. Assuming we find such $c$, then we are guaranteed that Algorithm \\ref{alg:Ours} outperforms the Bonferroni-corrected CR, for every $p \\in \\Delta_k$. The proof of Theorem \\ref{T3} is located in Appendix D.\n\n\\subsection{Large Alphabet Experiments}\nLet us now demonstrate the performance of our suggested inference scheme. We focus on two benchmark distributions which represent two extreme cases. Specifically, we study the heavy-tailed Zipf's law distribution (with $s=1.01$) and the benchmark uniform distribution. In each experiment we draw $n=1000$ samples, and evaluate the \nlog-volume of different CRs (for $\\alpha=0.05$), as $k$ increases. We repeat this process $1000$ times to obtain an averaged log-volume. We focus on the Bonferroni-corrected CR (denoted BC in the figures that follow), the Sison-Glaz (SG) scheme \\cite{sison1995simultaneous} and our proposed method (Algorithm \\ref{alg:Ours}). To configure our method, we set $c$ as the largest value within the unit interval that satisfies conditions (a) and (b), for every $k$ and $n$. We justify this choice later in this section. Figure \\ref{log_volume} demonstrates the results we achieve. \n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width =0.47\\textwidth,bb=70 180 530 600,clip]{examine_ab_size_new.pdf}\n\\caption{log-volume of CR for Zipf's Law and Uniform distributions, for $n=1000$.}\n\\label{log_volume}\n\\end{figure}\n\nFirst, it is evident that our proposed scheme outperforms the Bonferroni-corrected CR as $k$ grows. The SG method is omitted from Figure \\ref{log_volume} as it fails to provide the prescribed confidence level (see Appendix E). It may appear from Figure \\ref{log_volume} that the difference between the Zipf's Law and the uniform distribution is negligible. The reason for this phenomenon is fairly simple. For $k>>n$, most symbols do not appear in the sample (regardless to the underlying distribution). In this case, the volume of the CR is dominated by the CI of the unobserved events. This CI is fixed and independent of the sample, for both inference schemes. \nOn the other hand, there is a difference in the log-volume for smaller alphabets. However, it is less visible from Figure \\ref{log_volume}, and demonstrated more clearly in Figure \\ref{simulated_exp_c} below. To complete the picture, we examine the coverage rate of the examined inference schemes. The results are reported in Appendix E for brevity.  As we can see, both the Bonferroni-corrected and our proposed method obtain the prescribed $0.95$ confidence level as desired, while SG fails to do so.   \n\n\\iffalse\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width =0.7\\textwidth,bb=50 150 530 620,clip]{examine_coverage.pdf}\n\\caption{CR coverage for a Zipf's Law distribution. The sample size is $n=1000$.}\n\\label{simulated_exp_coverage}\n\\end{figure}\n\\fi\nFinally, we examine the performance (and sensitivity) of our suggested scheme for the choice of $c$. The upper charts of Figure \\ref{simulated_exp_c} correspond to a Zipf's Law distribution ($s=1.01)$ with $k=1000$ (right) and $k=20000$ (left). The lower charts correspond to a uniform distribution with the same alphabet sizes. We use $n=1000$ samples as above. First, it is evident that for large $k$, the performance of our proposed scheme improves as $c$ grows. This is not quite surprising as there are more unobserved events in this setup. For a relatively smaller $k$, we still observe a significant improvement over the Bonferroni-corrected scheme for a large span of $c$ values, for both distributions. Despite the above, we emphasize that any choice of $c$ that satisfies conditions (a) and (b) is guaranteed to improve upon the Bonferroni-corrected scheme. Therefore, for simplicity, we choose the largest possible $c$ so that the improvement is more evident for larger alphabets.   \n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width =0.58\\textwidth,bb=30 110 550 680,clip]{examine_c.pdf}\n\\caption{The choice of $c$ in Zipf's Law and Uniform setups. The sample size is $n=1000$.}\n\\label{simulated_exp_c}\n\\end{figure}\n\n\n\\section{Discussion}\nIn this work we introduce an interval estimation framework for the probability of symbols that do not appear in the sample. Our suggested framework is an SoS inference scheme, designed to simultaneously control the selected parameters. We distinguish between two setups, depending on the alphabet size. First, we \nconsider the case where the alphabet size $k$ is unknown and possible unbounded. This setup is of special interest in many real-world applications, as described throughout the manuscript. \nWe introduce a closed-form expression for the CI, which is independent of the alphabet size. Second, we study the case where the alphabet size in known (or bounded). Here, we derive an efficient numerical routine which improves upon the unbounded $k$ solution in cases where $k$ is relatively small. It is important to emphasize that in both setups, the proposed CI is independent of the sample, similarly to the ROT. This makes it a robust framework, which is easy to apply. Next, we show that our proposed CIs are (almost) tight, in the sense that there exists a probability distribution $p$, for which we cover the missing probabilities at a confidence level (almost) equal to $1-\\alpha$. We compare our proposed scheme to currently known methods, showing  significant improvement in synthetic and real-world experiments. Finally, we apply our proposed CI to large alphabet inference. Specifically, we introduce a novel scheme that provably improves upon the alternatives while controlling the desired coverage rate. \n\nTo conclude, we revisit the motivating example in Section \\ref{intro}. The $2006$ New Zealand Ministry of Health report indicated $58$ new cancer cases among Maori female under the age of $30$. Specifically, out of the $75$ studied cancer types, only $37$ were observed. Using the Bonferroni-corrected ROT, we obtain a confidence interval of $[0,0.126]$ for the unobserved cancer types. Applying our proposed scheme (under the more robust unbounded $k$ assumption), we obtain a shorter CI of length $[0,0.089]$. Similarly, for Pacific Islands men of the same age group, an additional part of the report indicated only $11$ new cancer cases in $2006$, where each case is of a different type. Here, the Bonferroni-corrected ROT suggests a CI of $[0,0.244]$ while our proposed scheme obtains a CI of $[0,0.15]$. As we can see, our interval estimation scheme demonstrates a significant improvement. This makes it an favorable alternative for this important problem, which applies to many applications.\n\nFinally, our proposed framework may be generalized to consider the collection of probabilities with $i$ appearances in the sample. This would allow us to control more events of interest, and further improve out large alphabet inference scheme. We consider this direction for our future work. \n\n\n\n\\begin{appendices}\n\\section{A Proof for Property \\ref{property1}}\n\nLet $r\\geq 1$ and  $h(t)=t^r(1-t)^n$. Then, the optimum of $h(t)$ satisfies\n\\begin{align}\n    \\frac{dh(t)}{dt}=t^{r-1}(1-t)^{n-1}(r(1-t)-nt)=0\n\\end{align}\nThis implies that $t^*\\triangleq r\/(r+n)$ is a local optimum. Further, \n\\begin{align}\\nonumber\n    \\frac{d^2h(t)}{dt^2}=&t^{r-2}(1-t)^{n-2}\\bigg(r(r-1)(1-t)^2-2nrt(1-t)+n(n-1)t^2\\bigg)=\\\\\\nonumber\n    &t^{r-2}(1-t)^{n-2}\\bigg(t^2\\big(r(r-1)+2nr+n(n-1)\\big)+t\\big(-2r(r-1)-2nr\\big)+r(r-1)\\bigg).\n\\end{align}\nDenote the roots of the quadratic form \n$$z(t)=t^2\\big(r(r-1)+2nr+n(n-1)\\big)+t\\big(-2r(r-1)-2nr\\big)+r(r-1)$$\nas $t_1$ and $t_2$. Simple calculus shows that \n$$t_{1,2}=t^*\\pm \\frac{1}{r+n}\\sqrt{\\frac{rn}{r+n-1}}.$$\n\nAs we can, $z(t)$ is quadratic and convex in $t$. This means that $z(t)<0$ for $t_1<t<t_2$ and $z(t)>0$ elsewhere. This implies that $h(t)$ is concave for $t_1<t<t_2$, and convex for $0\\leq t\\leq t_1$ and $t_2\\leq t \\leq 1$. Further, $h(t^*)<0$ which implies that $t^*$ is a local maximum. \n\n\\section{A Proof for Theorem \\ref{theorem2}}\n\nWe prove Theorem \\ref{theorem2} by a series of properties. \n\n\\begin{property}\\label{property2}\nLet $p^* \\in \\Delta_k$ be the maximizer of $E_{r,n}(p)=\\sum_u h(p(u))$ where $h(p(u))\\triangleq p^r(u)(1-p(u))^n$. Then, $p^*(u)=p^*(v)$ for all $p^*(u),p^*(v)\\in \\left[t_1,t_2\\right]$.\n\\end{property}\n\n\\begin{proof}\nBy negation, assume there exists $p^*(u) \\neq p^*(v)$ such that $p^*(u),p^*(v)\\in \\left[t_1,t_2\\right]$. Define \n\\begin{equation}\n   \\tilde{p}(l)=\n    \\begin{cases}\n      p^*(l) & l\\neq u,v\\\\\n      \\frac{p^*(u)+p^*(v)}{2} & l=u,v    \\end{cases}\n\\end{equation}\nThen, \n\\begin{align}\n    \\sum_l h(\\tilde{p}(l))=&\\sum_{l \\neq u,v}  h(\\tilde{p}(l))+\\sum_{l = u,v}  h(\\tilde{p}(l))=\\sum_{l \\neq u,v}  h(p^*(l))+2h\\left(\\frac{p^*(u)+p^*(v)}{2}\\right)>\\\\\\nonumber\n    &\\sum_{l \\neq u,v}  h(p^*(l))+h(p^*(u))+h(p^*(v))=\\sum_u h(p^*(u)).\n\\end{align}\nwhere the inequality follows from the concavity of $h(p(l))$ for every $p(l) \\in \\left[t_1,t_2\\right]$. Therefore, we found $\\tilde{p}\\in \\Delta_k$ for which $\\sum_l h(\\tilde{p}(l))>\\sum_l h(p^*(l))$, which contradicts the optimality of $p^*$. \n\\end{proof}\n\n\n\\begin{property}\\label{property4}\nLet $p^* \\in \\Delta_k$ be the maximizer of $E_{r,n}(p)=\\sum_u h(p(u))$, where $h(p(u))\\triangleq p^r(u)(1-p(u))^n$. Then, there exists at most a single $p^*(u)$ such that $p^*(u)\\in \\left(0,t_1\\right)$.\n\\end{property}\n\\begin{proof}\nBy negation, assume there exist $p^*(u)$ and $p^*(v)$ such that $p^*(u),p^*(v)\\in \\left(0,t_1\\right)$. Assume, without loss of generality, that  $p^*(v)\\leq p^*(u)$. Define $\\delta= p^*(v)>0$. \n\nLet us first assume that $p^*(u)+\\delta<t_1$. The function  $h(p(u))$ is convex for $p(u) \\in \\left[0,t_1\\right]$ and strictly convex for $p(u) \\in \\left[0,t_1\\right)$. Therefore, we have\n\n\\begin{align}\n    &h\\left(p^*(u)+\\delta\\right)> h\\left(p^*(u)\\right)+\\delta h'\\left(p^*(u)\\right)\\\\\\nonumber\n    &h\\left(p^*(v)-\\delta\\right)\\geq h\\left(p^*(v)\\right)-\\delta h'\\left(p^*(v)\\right).\n\\end{align}\nwhere $h'(p(u))={dh(p(u))}\/{dp(u)}$. Putting together the above, we have \n\\begin{align}\n    h\\left(p^*(u)+\\delta\\right)+&h\\left(p^*(v)-\\delta\\right)>h\\left(p^*(u)\\right)+h\\left(p^*(v)\\right)+\\delta\\left(h'\\left(p^*(u)\\right)-h'\\left(p^*(v)\\right)\\right).\n\\end{align}\nWe observe that $h'(p(u))$ is an increasing function in $p(u)$, for $p(u)\\in \\left(0,t_1\\right)$, as its derivative, ${d^2 h(p(u))}\/{dp^2(u)}$ is positive in this range. Therefore, $h'\\left(p^*(u)\\right)\\geq h'\\left(p^*(v)\\right)$ and \n\\begin{align}\n    h\\left(p^*(u)+\\delta\\right)+h\\left(p^*(v)-\\delta\\right)> h\\left(p^*(u)\\right)+h\\left(p^*(v)\\right).\n\\end{align}\nTherefore, we found $\\tilde{p}\\in \\Delta_k$ such that \n\\begin{equation}\n   \\tilde{p}(l)=\n    \\begin{cases}\n      p^*(l) & l\\neq u,v\\\\\n      0 & l=v\\\\\n      p^*(u)+\\delta & l=u\\\\\n      \\end{cases}\n\\end{equation}\nand $\\sum_l h(\\tilde{p}(l))> \\sum_l h(p^*(l)) $, which contradicts the optimality of $p^*$.\n\n\nNow, assume that $p^*(u)+\\delta\\geq t_1$. Then, define $\\tilde{\\delta}=t_1-p^*(u)>0$. We have\n\\begin{align}\n    h\\left(p^*(u)+\\tilde{\\delta}\\right)\\geq h\\left(p^*(u)\\right)+\\tilde{\\delta} h'\\left(p^*(u)\\right)\n\\end{align}\n\\begin{align}\n    h\\left(p^*(v)-\\tilde{\\delta}\\right)> h\\left(p^*(v)\\right)-\\tilde{\\delta} h'\\left(p^*(v)\\right).\n\\end{align}\nPutting together the above, we have \n\\begin{align}\n    h\\left(p^*(u)+\\tilde{\\delta}\\right)+&h\\left(p^*(v)-\\tilde{\\delta}\\right)>h\\left(p^*(u)\\right)+h\\left(p^*(v)\\right)+\\tilde{\\delta}\\left(h'\\left(p^*(u)\\right)-h'\\left(p^*(v)\\right)\\right).\\nonumber\n\\end{align}\nAs above, we observe that $h'(p(u))$ is an increasing function in $p(u)$, for $p(u)\\in \\left(0,t_1\\right)$. Therefore, $h'\\left(p^*(u)\\right)\\geq h'\\left(p^*(v)\\right)$ and \n\\begin{align}\n    h\\left(p^*(v)-\\tilde{\\delta}\\right)+h\\left(p^*(u)+\\tilde{\\delta}\\right)> h\\left(p^*(v)\\right)+h\\left(p^*(u)\\right).\n\\end{align}\nTherefore, we found $\\tilde{p}\\in \\Delta_k$ such that \n\\begin{equation}\n   \\tilde{p}(l)=\n    \\begin{cases}\n      p^*(l) & l\\neq u,v\\\\\n      p^*(v)-\\tilde{\\delta} & l=v\\\\\n      t_1 & l=u\\\\\n      \\end{cases}\n\\end{equation}\nand $\\sum_l h(\\tilde{p}(l))> \\sum_l h(p^*(l)) $, which again contradicts the optimality of $p^*$.\n\\end{proof}\n\n\n\\begin{property}\\label{property3}\nLet $p^* \\in \\Delta_k$ be the maximizer of $E_{r,n}(p)=\\sum_u h(p(u))$, where $h(p(u))\\triangleq p^r(u)(1-p(u))^n$. Then, there exists at most a single $p^*(u)$ such that $p^*(u)\\in \\left(t_2,1\\right]$.\n\\end{property}\n\\begin{proof}\nBy negation, assume there exist $p^*(u)$ and $p^*(v)$ such that $p^*(u),p^*(v)\\in \\left(t_2,1\\right]$. Assume, without loss of generality, that  $p^*(v)\\leq p^*(u)$. Define $\\delta= p^*(v)-t_2>0$. The function  $h(p(u))$ is convex for $p(u) \\in \\left[t_2,1\\right]$ and strictly convex for $p(u) \\in \\left(t_2,1\\right]$. Therefore, we have\n\\begin{align}\n    h\\left(t_2\\right)\\geq h\\left(p^*(v)\\right)-\\delta h'\\left(p^*(v)\\right)\n\\end{align}\n\\begin{align}\n    h\\left(p^*(u)+\\delta\\right)> h\\left(p^*(u)\\right)+\\delta h'\\left(p^*(u)\\right)\n\\end{align}\n\nPutting together the above, we have \n\\begin{align}\n    h\\left(t_2\\right)+&h\\left(p^*(u)+\\delta\\right)> h\\left(p^*(v)\\right)+h\\left(p^*(u)\\right)+\\delta\\left(h'\\left(p^*(u)\\right)-h'\\left(p^*(v)\\right)\\right).\n\\end{align}\nWe observe that $h'(p(u))$ is an increasing function in $p(u)$, for $p(u)\\in \\left(t_2,1\\right]$, as its derivative, ${d^2 h(p(u))}\/{dp^2(u)}$ is positive in this range. Therefore, $h'\\left(p^*(u)\\right)\\geq h'\\left(p^*(v)\\right)$ and \n\\begin{align}\n    h\\left(t_2\\right)+h\\left(p^*(u)+\\delta\\right)> h\\left(p^*(v)\\right)+h\\left(p^*(u)\\right).\n\\end{align}\nTherefore, we define $\\tilde{p}\\in \\Delta_k$ such that \n\\begin{equation}\n   \\tilde{p}(l)=\n    \\begin{cases}\n      p^*(l) & l\\neq u,v\\\\\n      p^*(l)-\\delta & l=v\\\\\n      p^*(l)+\\delta & l=u\\\\\n      \\end{cases}\n\\end{equation}\nand $\\sum_l h(\\tilde{p}(l))> \\sum_l h(p^*(l)) $, which contradicts the optimality of $p^*$.\\end{proof}\n\n\\section{A Proof for Proposition \\ref{prop2}}\nLet $X^n$ be a sample of $n$ independent observations from $p_m \\in \\mathcal{U}_k$. This means that $M_{max}(X^n)={0,1\/m}$, and $M_max=1\/m$ if and only if there exists at least one symbol that do not appear in the sample, where all symbols are equiprobable. Therefore, the probability that $M_max=1\/m$ equals the probability of placing $n$ balls in $m$ identical bins, where at least a single bin remains empty. Equivalently, \n\n\\begin{align}\\label{worst-case_appendix}\n   P\\left(M_{max}(X^n) = \\frac{1}{m}\\right)=1-\\frac{m!S(n,m)}{m^n},\n   \\end{align}\nwhere $m!S(n,m)$ is the number of combinations of placing $n$ distinguishable balls in $m$ distinguishable bins, where no bin is empty, and $m^n$ is the total number of combinations of placing $n$ distinguishable balls into $m$ distinguishable bins \\cite{graham1989concrete}.  \n\\section{A proof for Theorem \\ref{T3}}\\label{T3_appendix}\n\\begin{proof}\n   Let $L_i$ be the length of the CI for a symbol that appears $i$ times in the sample. The Bonferroni-corrected CR satisfies\n\\begin{align}\n    L^{BC}_0=-\\log(\\alpha\/k)\/n\\quad,\\quad L^{BC}_i=2z_{1-\\frac{\\alpha}{2k}}\\sqrt{\\frac{i\/n(1-i\/n)}{n}}\\;\\; \\forall i>0.\n\\end{align}\nNotice that we use a normal approximation for the binomial CI to simplify our derivation. Next, given a fixed $c \\in [0,1]$, the CI of our proposed method satisfies \n\\begin{align}\\nonumber\n    L^{c}_0=\\min_{r\\geq 1}\\left((q_{r,n}^*)^{r-1}(1-q_{r,n}^*)^n\/(c\\alpha)  \\right)^{1\/r}\\quad,\\quad L^{c}_i=2z_{1-\\frac{\\alpha(1-c)}{2n}}\\sqrt{\\frac{i\/n(1-i\/n)}{n}}\\;\\; \\forall i>0,\n\\end{align}\nwhere $q_{r,n}^*$ is defined in Theorem $1$. Notice that for simplicity, we use the result in Theorem $1$ although the alphabet size $k$ is known. The expected log-volume of a rectangular CR satisfies\n\\begin{align}\n    \\mathbb{E}\\log V= \\mathbb{E}\\log \\prod_{i=0}^nL_i^{\\sum_u\\mathbbm{1}(N_u(X^n)=i)}=\\sum_{i=0}^n\\sum_u\\mathbb{E}\\left(\\mathbbm{1}(N_u(X^n)=i)\\right)\\log L_i.\n\\end{align}\nWe would like to find $c\\in [0,1]$ such that $\\mathbb{E}\\log V_c\\leq \\mathbb{E}\\log V_0$. We have,\n\\begin{align}\n    \\mathbb{E}\\log V_c-\\mathbb{E}\\log V_0=&\\sum_{i=0}^n\\sum_u\\mathbb{E}\\left(\\mathbbm{1}(N_u(X^n)=i)\\right)\\left(\\log L_i^c-\\log L_i^{BC}\\right)=\\\\\\nonumber\n    &\\sum_u\\mathbb{E}\\left(\\mathbbm{1}(N_u(X^n)=0)\\right)\\left(\\log L_0^c-\\log L_0^{BC}\\right)+\\\\\\nonumber\n    &\\sum_{i=1}^n\\sum_u\\mathbb{E}\\left(\\mathbbm{1}(N_u(X^n)=i)\\right)\\left(\\log L_i^c-\\log L_i^{BC}\\right)=\\\\\\nonumber\n    &\\sum_u\\mathbb{E}\\left(\\mathbbm{1}(N_u(X^n)=0)\\right)\\left(\\log L_0^c-\\log L_0^{BC}\\right)+\\\\\\nonumber\n    &\\left(\\log z_{1-\\frac{\\alpha(1-c)}{2n}}-\\log z_{1-\\frac{\\alpha}{2k}}\\right)\\sum_{i=1}^n\\sum_u\\mathbb{E}\\left(\\mathbbm{1}(N_u(X^n)=i)\\right)=\\\\\\nonumber\n    &\\sum_u\\mathbb{E}\\left(\\mathbbm{1}(N_u(X^n)=0)\\right)\\left(\\log L_0^c-\\log L_0^{BC}\\right)+\\\\\\nonumber\n    &\\left(\\log z_{1-\\frac{\\alpha(1-c)}{2n}}-\\log z_{1-\\frac{\\alpha}{2k}}\\right)\\left(k-\\sum_u\\mathbb{E}\\left(\\mathbbm{1}(N_u(X^n)=0)\\right)\\right),\n\\end{align}\nwhere the last equality follows from $\\sum_{i=0}^n \\mathbbm{1}(N_u(X^n)=i)=k$. Notice we have that $\\mathbb{E}\\left(\\mathbbm{1}(N_u(X^n)=0)\\right)=(1-p_u)^n$. Therefore, \n\\begin{align}\\label{eq1a}\n    \\mathbb{E}\\log V_c-\\mathbb{E}\\log V_0=&\\sum_u(1-p_u)^n\\left(\\log L_0^c-\\log L_0^{BC}\\right)+\\\\\\nonumber\n    &\\left(k-\\sum_u(1-p_u)^n\\right)\\left(\\log z_{1-\\frac{\\alpha(1-c)}{2n}}-\\log z_{1-\\frac{\\alpha}{2k}}\\right).\n\\end{align}\nNotice that (\\ref{eq1a}) is linear in $\\sum_u(1-p_u)^n$. Further, simple calculus shows that $\\sum_u(1-p_u)^n$ attains its maximum for a uniform distribution, while its minimum is attained for a degenerate distribution. Therefore, $k(1-1\/k)^n\\leq \\sum_u(1-p_u)^n\\leq k-1$. This means that \n\\begin{align}\n    &\\mathbb{E}\\log V_c-\\mathbb{E}\\log V_0\\leq\\\\\\nonumber\n    &\\max\\bigg\\{  k(1-1\/k)^n\\left(\\log L_0^c-\\log L_0^{BC}\\right)+k\\left(1-(1-1\/k)^n\\right)\\left(\\log z_{1-\\frac{\\alpha(1-c)}{2n}}-\\log z_{1-\\frac{\\alpha}{2k}}\\right),\\\\\\nonumber\n    &\\quad\\quad\\quad (k-1)\\left(\\log L_0^c-\\log L_0^{BC}\\right)+\\left(\\log z_{1-\\frac{\\alpha(1-c)}{2n}}-\\log z_{1-\\frac{\\alpha}{2k}}\\right)\\bigg\\}.\n\\end{align}\nWe require that $\\mathbb{E}\\log V_c-\\mathbb{E}\\log V_0 \\leq0$. This holds if both arguments of the max above are non positive, as stated in conditions (a) and (b). \n\\end{proof}\n\n\\newpage\n\n\\section{Coverage Rate for large Alphabet SCIs}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width =0.45\\textwidth,bb=50 150 530 620,clip]{examine_coverage.pdf}\n\\caption{CR coverage for a Zipf's Law distribution. The sample size is $n=1000$.}\n\\label{simulated_exp_coverage}\n\\end{figure}\n\n\\end{appendices}\n\n\\section*{Acknowledgements}\nThis research is supported by the Israel Science Foundation \ngrant number 963\/21. The author thanks Ruth Heller and Yoav Benjamini for helpful discussions. \n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\noindent\nThe multilinear Kakeya problem was introduced in \\cite{B}, and its\nstudy began in earnest in \\cite{BCT}, where the natural conjecture was established \nup to the endpoint.\nWorking in $\\mathbb{R}^n$, we suppose that we are given $n$ transverse families \n$\\mathcal{T}_1, \\dots, \\mathcal{T}_n$ of $1$-tubes, which means that each $T \\in \\mathcal{T}_j$ \nis a $1$-neighbourhood of a doubly-infinite line in $\\mathbb{R}^n$ with direction \n$e(T) \\in \\mathbb{S}^{n-1}$, and that the directions $e(T)$ for $T \\in \\mathcal{T}_j$ all lie within a \nsmall fixed neighbourhood (depending only on the dimension $n$) of the $j$'th standard basis vector $e_j$. \n\n\\medskip\n\\noindent\nThe question is whether for each $q \\geq 1\/(n-1)$ we have the inequality\n\\begin{equation*}\\label{MKC}\n\\int_{\\mathbb{R}^n} \\left(\\sum_{T_{1} \\in \\mathcal{T}_1} a_{T_1} \\chi_{T_1}(x) \\dots \n\\sum_{T_{n} \\in \\mathcal{T}_n} a_{T_n} \\chi_{T_n}(x) \\right)^q \\; dx\n\\leq C_{n,q} \\left(\\sum_{T_{1} \\in \\mathcal{T}_1} a_{T_1} \\dots \n\\sum_{T_{n} \\in \\mathcal{T}_n} a_{T_n}\\right)^q\n\\end{equation*}\nfor nonnegative coefficients $a_{T_j}$. In \\cite{BCT} this was proved for each $q > 1\/(n-1)$ using a \nheat-flow technique which, because of certain error terms arising, did not apply at the endpoint \n$q = 1\/(n-1)$. (For further background on this problem consult \\cite{BCT}.)\n\n\\medskip\n\\noindent\nMore recently, Guth in \\cite{G} established the endpoint case $q = 1\/(n-1)$ using completely different \ntechniques motivated in part by the polynomial method used by Dvir \\cite{Dv} to solve the finite field \nKakeya set problem, but which also relied upon a fairly heavy dose of algebraic topology, and which were \ntherefore perhaps a little intimidating to the analyst or combinatorialist. In particular, Guth used the \ntechnology of cohomology classes, cup products and the Lusternik--Schnirelmann vanishing lemma in \nestablishing his result. We believe\nthat the endpoint multilinear Kakeya theorem is of such significance and importance that a proof free of \nthese techniques should be available, and so the purpose of this paper is to provide an argument leading to\nGuth's result which does not rely upon such sophisticated algebraic topology, but whose input is instead the \nBorsuk--Ulam theorem. It is hoped therefore that this paper might lead\nto further exploitation of Guth's techniques in a more flexible\nsetting. (For some recent works applying the multilinear\nKakeya perspective in other contexts, see \\cite{B1}, \\cite{B2} and \\cite{B3}.) \n\n\\medskip\n\\noindent\nThe Borsuk--Ulam theorem, while topological in nature, nevertheless has many proofs accessible \nto the analyst -- see for example \\cite{Mat}, and also \\cite{C} for a recent such proof. (See Section \n\\ref{sec:BU} below for its statement.) The use of the Borsuk--Ulam theorem in the context of Kakeya theorems is \nby now natural, as it can be considered as a topological analogue of the elementary linear-algebraic statement that \nthere are no linear injections $T: V \\to W$ if $V$ and $W$ are finite-dimensional vector spaces \nwith dim $V > $ dim $W$; this was a key element of Dvir's solution \\cite{Dv} of the finite field \nKakeya problem. It also features explicitly in Guth's warm-up discussion to the full result of \\cite{G}.\n\n\\medskip\n\\noindent\nIn order to proceed, \nwe place matters in a more general context which does not impose conditions on the directions \nof the tubes, nor requires the level of multilinearity to equal the dimension of the underlying \neuclidean space. Thus we now suppose that \nwe are given $d$ arbitrary families of $1$-tubes $\\mathcal{T}_1, \\dots, \\mathcal{T}_d$ in $\\mathbb{R}^n$, \nwhere $d \\leq n$. For $v_1, \\dots v_d \\in \\mathbb{R}^n$ let $v_1 \\wedge \\cdots \\wedge \nv_d$ denote the unsigned (i.e. nonnegative) $d$-dimensional volume of the parallelepiped whose sides are \ngiven by the vectors $v_1, \\dots, v_d$. \n\n\n\\begin{theorem}[The Multilinear Kakeya Theorem]\\label{main}\nLet $2 \\leq d \\leq n$. Then there exists a constant $C_{d,n}$ such that if $\\mathcal{T}_1, \\dots, \n\\mathcal{T}_d$ are families of $1$-tubes in $\\mathbb{R}^n$, we have\n\\begin{equation}\n\\begin{aligned}\n\\int_{\\mathbb{R}^n} & \\left(\\sum_{T_{1} \\in \\mathcal{T}_1} a_{T_1} \\chi_{T_1}(x) \\dots \n\\sum_{T_{d} \\in \\mathcal{T}_d} a_{T_d} \\chi_{T_d}(x) \\; e(T_1) \\wedge \\dots \\wedge e(T_d) \\right)^{1\/(d-1)} \n\\; dx \\\\\n& \\leq C_{d,n} \\left(\\sum_{T_{1} \\in \\mathcal{T}_1} a_{T_1} \\dots \n\\sum_{T_{d} \\in \\mathcal{T}_d} a_{T_d}\\right)^{1\/(d-1)}.\n\\end{aligned}\n\\end{equation}\n\\end{theorem}\n\n\\medskip\n\\noindent\n(The case $d=2$ is of course trivial.) \n\n\\medskip\n\\noindent\nThe situation where the level of multilinearity is less than the ambient euclidean dimension was already \naddressed in \\cite{BCT}, where once again the result was established up to the endpoint. The incorporation \nof the factor $e(T_1) \\wedge \\dots \\wedge e(T_d)$ on the left-hand side is natural in view of the\naffine-invariant formulation of the Loomis--Whitney inequality, and was considered \nin Section 7 of \\cite{BG}, where Theorem \\ref{main} was first proved. Indeed, when $d=n$, the statement of \nTheorem \\ref{main} is affine-invariant.\\footnote{The multilinear Kakeya theorem can also be cast in the following equivalent form when $d=n$. For a unit vector \n$\\omega \\in \\mathbb{R}^n$ let $\\Pi_\\omega$ denote the hyperplane in $\\mathbb{R}^n$ which is \nperpendicular to $\\omega$ and which contains the origin. Let $\\pi_\\omega : \\mathbb{R}^n \\to \\Pi_\\omega$\nbe the orthogonal projection map. Then for nonnegative $g_j$ we have\n$$ \\int_{\\mathbb{R}^n} \\left(\\int_{\\mathbb{S}^{n-1}} \\dots \\int_{\\mathbb{S}^{n-1}}\n\\left(\\prod_{j=1}^n g_j(\\omega_j, \\pi_{\\omega_j} x)\\right)\\omega_1 \\wedge \\dots \\wedge \\omega_n \\; d\\sigma(\\omega_1) \\dots d \\sigma(\\omega_n)\\right)^{1\/(n-1)} dx $$\n$$\\leq C_{n} \\prod_{j=1}^n \\left(\\int_{\\mathbb{S}^{n-1}} \\int_{\\Pi{_{\\omega_j}}} g_j(\\omega_j, \\xi) \nd\\xi d \\sigma(\\omega_j)\\right)^{1\/(n-1)}.$$} \nA variant of Theorem \\ref{main} \nwhere lines are replaced by algebraic curves of bounded degree was also proved in \\cite{BG} (and can \nlikewise be established by replacing Guth's original argument for Theorem \\ref{algtop0} below by \nthat of the current paper). On the other hand, the results of \\cite{BCT} have a somewhat more general scope\nin so far as they apply to $1$-neighbourhoods of $k$-planes for arbitrary $k$, rather than just\n$1$-neighbourhoods of lines, i.e. tubes, as in the present discussion.\n\n\\medskip\n\\noindent\nThe principal notion that Guth employs in proving the endpoint theorem is that of the \n{\\em visibility}   vis$\\,(Z)$ of a hypersurface $Z \\subseteq \\mathbb{R}^n$ -- see Section \n\\ref{sect:vis} below for the definition, which differs from Guth's in\nso far as in our treatment it (roughly)\nscales as does $(n-1)$-dimensional Hausdorff measure $\\mathcal{H}_{n-1}$ -- \nand the centrepiece of Guth's argument is the following result:\n\n\\begin{theorem}\\label{algtop0}\nGiven a nonnegative function\n$M$ defined on the lattice $\\mathcal{Q}$ of unit cubes of $\\mathbb{R}^n$,\nthere exists a non-zero polynomial $p$ such that\n$$ {\\rm{deg }} \\; p \\leq C_n \\left(\\sum_{Q \\in \\mathcal{Q}} M(Q)^n\\right)^{1\/n}$$\nand such that if we set $Z = Z_p = \\{x \\in \\mathbb{R}^n \\; : \\; p(x) =\n0\\}$, then for all $Q \\in \\mathcal{Q}$ we have \n$$ {\\rm{vis }}\\; (Z \\cap Q) \\geq C_n M(Q).$$\n\\end{theorem}\n\n\\medskip\n\\noindent\nIt is in the proof of this result that Guth uses algebraic-topological techniques, and \nthe main contribution of the present paper is to provide a proof of Theorem \\ref{algtop0} which \ndoes not use such topological machinery, but is instead a consequence\nof the Borsuk--Ulam theorem. (In fact, in our proof of Theorem\n\\ref{algtop0}, we do not use the Borsuk--Ulam theorem {\\em per se} but\ninstead an equivalent Lusternik--Schnirelmann type covering statement. See Section \\ref{sec:BU}.)\nOn the other hand, we must acknowledge that many of the arguments and\nconstructions of the present paper are inspired by those of Guth's approach.\n\n\\medskip\n\\noindent\nIn view of the connection between visibility and $(n-1)$-dimensional Hausdorff measure, and as a warm-up \nto our proof of Theorem \\ref{algtop0}, we indicate how the Borsuk--Ulam theorem can be used to \nestablish the following morally weaker variant of Theorem \\ref{algtop0}.\n\n\\begin{proposition}\\label{warmup}\nGiven a finitely supported function $M$ defined on the lattice $\\mathcal{Q}$\nof unit cubes of $\\mathbb{R}^n$ and taking nonzero \nvalues\nin $[1, \\infty)$, there exists a\nnon-zero polynomial $p$ such that \n$$ {\\rm{deg }} \\; p \\leq C_n \\left(\\sum_{Q\\in\\mathcal{Q}} M(Q)^n\\right)^{1\/n}$$\nand such that for all $Q\\in\\mathcal{Q}$ \n$$ \\mathcal{H}_{n-1} \\; (Z \\cap Q) \\geq C_n M(Q).$$\n\\end{proposition}\n\n\\begin{proof} Break up each $Q$ into $\\sim M(Q)^n$ congruent subcubes $S$; note that \naltogether we have $\\sim \\sum_Q M(Q)^n$ small cubes $S$ of various sizes. Consider the map \n$$ F: p \\mapsto \\left\\{\\int_{\\{p > 0\\} \\cap S} 1  -\\int_{\\{p < 0\\} \\cap S} 1 \\right\\}_{S}$$\ndefined on the vector space $\\mathcal{P}_k$ of polynomials of degree at most $k$ in $n$ real\nvariables, which has dimension $\\sim k^n$. Clearly $F$ is continuous, homogeneous of degree \n$0$ and odd.\n\n\\medskip\n\\noindent\nSo we can think of $F$ as \n$$F : \\mathbb{S}^N\\rightarrow \\mathbb{R}^J $$\nwhere $N \\sim k^n$ and $J \\sim \\sum_Q M(Q)^n$.\n\n\n\\medskip\n\\noindent\nSo provided $N \\geq J$ -- which we can arrange if $k \\sim \\left(\\sum_Q\nM(Q)^n\\right)^{1\/n}$ --  the Borsuk--Ulam theorem tells us that \n$F$ vanishes at some $p$. This means that the zero set $Z$ of $p$ exactly bisects each $S$. \n\n\\medskip\n\\noindent\nNow if $S$ is a subcube of $Q$, $S$ will have volume $\\sim M(Q)^{-n}$ and\ndiameter $\\sim M(Q)^{-1}$ and hence any bisecting surface will meet\nit in a set of $(n-1)$-dimensional measure $\\gtrsim\nM(Q)^{-(n-1)}$. This will be true for each of the $M(Q)^n$ disjoint $S$'s whose union\nis $Q$, so $Z$ will meet $Q$ in a set of $(n-1)$-dimensional measure $\\gtrsim M(Q)^n \\times\nM(Q)^{-(n-1)} = M(Q)$, as was needed.\n\\end{proof}\n\n\\medskip\n\\noindent\nIn the proof we used the ``geometrically obvious'' fact that a\nhypersurface bisecting the unit cube must have large surface area\ninside the cube. For a discussion of this in the context of the unit\nball, see Lemma \\ref{bisect} in the Appendix. Note that in the statements of both Theorem \\ref{algtop0} \nand Proposition \\ref{warmup}, a polynomial has the desired properties if and only if\nany non-zero scalar multiple of it does; for this reason we may choose to search for a suitable \npolynomial within the unit sphere of the class of polynomials of a given degree. \n\n\\medskip\n\\noindent\nProposition \\ref{warmup} is morally weaker than Theorem \\ref{algtop0} because not only does it place stronger \nconditions on $M$, but more importantly, in many situations of interest, we have \n${\\rm{vis }}\\; (Z \\cap Q) \\leq C_n \\mathcal{H}_{n-1} \\; (Z \\cap Q)$ -- see \\eqref{geomxx} below.\n\n\\medskip\n\\noindent\nOn an informal level, the fundamental difference between the proof of Theorem \\ref{algtop0} and that of Proposition \n\\ref{warmup} is that, roughly speaking, we no longer chop each cube $Q$ into $\\sim M(Q)^n$ congruent \n{\\em subcubes}, but we instead select, for each $Q$, an {\\em ellipsoid} \n$E(Q)$ of volume $\\sim M(Q)^{-n}$, so that $\\sim M(Q)^n$ translates of $E(Q)$ essentially tessellate $Q$. \nHowever the {\\em shape} and {\\em orientation} of the ellipsoid $E(Q_0)$ will depend not only on the \nvalue of $M(Q_0)$ but on the whole ensemble $\\{M(Q)\\}_Q$, and is in effect an output of the \nBorsuk--Ulam theorem at the same time as it produces the desired \npolynomial. At the risk of over-simplifying matters, \nwe now give an informal example which illustrates why, if we want the broad thrust of the proof of Proposition \\ref{warmup}\nto work in the context of Theorem \\ref{algtop0}, the shape of the ellipsoid selected {\\em must} depend on the totality of \nthe function $M(Q)$. This example may be safely ignored on a first reading of the paper.\n\n\\medskip\n\\noindent\n{\\bf Informal example.} Let $n=2$ and consider the function $M(Q)$ which is supported on a row of \n$N$ unit cubes centred at $(k-1\/2, 1\/2)$ for $1 \\leq k \\leq N$, and takes\nthe value $N^{1\/2}$ on each of these cubes. \nThen $\\left(\\sum_Q M(Q)^2\\right)^{1\/2} = N$. Consider the polynomial \n$$p(x) = (x_1 - 1) \\dots (x_1 - N)\\cdot(x_2 - 1\/2N)(x_2 - 3\/2N) \\dots (x_2 - (2N-1)\/2N)$$\nwhich has degree $2N$, and let $Z$ denote its zero set. For a subset\n$Z' \\subseteq Z$ of $Z$, and for each $Q$\\footnote{In this example we assume that the right-hand edge of a rectangle\nbelongs to the rectangle while the left-hand edge does not.}\nin the support of $M$, consider the projections counted with \nmultiplicities of $Z' \\cap Q$, in the directions of the two standard basis vectors $e_1$ and $e_2$; let their\ntotal lengths be $a_1(Z' \\cap Q)$ and $a_2(Z' \\cap Q)$ respectively, and let \n$$W(Z' \\cap Q) := \\{a_1(Z' \\cap Q)a_2(Z' \\cap Q)\\}^{1\/2}$$ \nbe their geometric mean.  Now it transpires that the quantity $W(Z' \\cap Q)$ is closely related to \nvis$(Z' \\cap Q)$ -- see Section \\ref{sect:vis} below -- and we shall\npretend (for the rest of this example) that $W$ \nreally is the visibility. Note that if $Z_1$ and $Z_2$ are disjoint subsets of $Z \\cap Q$ we \nhave\\footnote{The corresponding property for visibility fails.}\n$$W(Z_1 \\cup Z_2) \\geq W(Z_1) + W(Z_2).$$ \nNow $a_1(Z \\cap Q) = 1$ and $a_1(Z \\cap Q) = N$ so that $W(Z \\cap Q) \n= N^{1\/2}$. We consider \nwhether it is possible to break up each $Q$ into rectangles $R_j$ of area $1\/N$\nso that if $p$ bisects each rectangle then we can deduce that\n$N^{1\/2} =  W(Z \\cap Q)$ by using\n$W(Z \\cap Q) \\geq \\sum_j W(Z \\cap R_j)$. \n\n\\smallskip\n\\noindent\nFirstly, we could break up $Q$ into subcubes $R_j$ of side \n$N^{-1\/2}$. Now for all $R_j$ except those which meet the right-hand edge of $Q$ we shall have \n$W(Z \\cap R_j) = 0$, while for those which do meet the right hand edge of\n$Q$ we have $W(Z \\cap R_j) = N^{-1\/4}$, which only gives\n$W(Z \\cap Q) \\geq \\sum_j W(Z \\cap R_j) = N^{1\/4}$; this is not adequate.\n\n\\smallskip\n\\noindent\nNext, we could try breaking up $Q$ into vertical rectangles $R_j$ of sides $1\/N \\times 1$. Only the \nrectangle $R_0$ meeting the right-hand edge of $Q$ will have a non-zero value of $W(Z \\cap R_j)$, \nand $W(Z \\cap R_0) = 1$, giving $W(Z \\cap Q) \\geq \\sum_j W(Z \\cap R_j) = 1$, which is even worse.\n\n\\smallskip\n\\noindent\nFinally, we could try breaking up $Q$ into horizontal rectangles $R_j$ of sides $1 \\times 1\/N$. In this case\neach $W(Z \\cap R_j) = N^{-1\/2}$, resulting in the desired $W(Z \\cap Q) \\geq \\sum_j W(Z \\cap R_j) \\geq N^{1\/2}$.\n\n\\smallskip\n\\noindent\nSo only the third decomposition into horizontal rectangles is compatible with our needs. Once we have accepted \nthat the polynomial $p$ above is more or less\n``canonical'' for this $M$, we are essentially forced to break \nup each $Q$ into horizontal rectangles of sides $1 \\times 1\/N$ in order for our strategy to be successful. \nCrucially, observe that this \ndecomposition reflects the global shape of the function $M$: if the support of $M$ had been along the \n$x_2$-axis we would have had to instead decompose each $Q$ into\nvertical rectangles. The decomposition must therefore be aligned with the ``global\nprofile'' of $M$.  \\hfill \\qedsymbol\n\n\\bigskip\n\\noindent\nTo simplify the constructions in the proof we actually stop just short of fully\ndeveloping the moral outline given above. In fact we do not spend any time\nconstructing ellipsoids at the scale $\\sim M(Q)^{-n}$ which would be needed\nto really nail down the zero set of the polynomial we get from\nthe Borsuk--Ulam theorem. Instead we let ourselves be satisfied with\nfinding a \\emph{good} polynomial $p$ with zero set $Z_p$ which satisfies\na given lower bound on the visibility\n$\\operatorname{vis}\\,(Z_p\\cap Q)$ for all cubes $Q$.\nThis we get by constructing ellipsoids for all \\emph{bad}\npolynomials,\nwhich are those polynomials whose zero sets\nhave visibility less than desired on some cube.\nUsing these ellipsoids we show that the bad polynomials cannot cover\nthe unit sphere in the space of polynomials and in this way we see that\nthere must be a good polynomial, which gives the lower bounds mentioned\nabove.\nHerein lies the reason for using the covering statement instead of the\nBorsuk--Ulam theorem itself.\nThis very informal outline is developed more fully in\nSection~\\ref{sec:outline}.\n\n\\medskip\n\\noindent\nFor completeness, we also indicate in the following sections how Theorem \\ref{main} \nfollows from Theorem \\ref{algtop0}, so that we give what is in essence a fully self-contained \nproof of Theorem \\ref{main} (subject to the appeal to the Borsuk--Ulam theorem).\nThroughout, $C$ and $c$ will denote generic constants which depend only \non the dimension $n$ and the degree of multilinearity $d \\leq n$; $P \\lesssim Q$ and $P \\gtrsim Q$\nmean $P \\leq C Q$ and  $P \\geq C Q$ respectively, and $P \\sim Q$ means both $P \\lesssim Q$ and $P \\gtrsim Q$.\n\n\\medskip\n\\noindent\n{\\em {\\bf {\\em Acknowledgements:}}} Both authors would like to thank Jon Bennett and Jim Wright for many illuminating \nconversations on the topics of this paper, and helpful remarks on early drafts of it. The first author would like to thank Larry Guth \nfor sharing his insights into the philosophy behind the endpoint multilinear Kakeya theorem in Hyderabad in August 2010.\n\n\\section{A Preliminary reduction}\n\\medskip\n\\noindent\nRecall that we have collections $ \\mathcal{T}_j$, $1 \\leq j \\leq d$,\nof $1$-tubes $T$ in $\\mathbb{R}^n$ with directions $e(T) \\in \\mathbb{S}^{n-1}$. \nLet $\\mathcal{Q}$ denote the lattice of unit cubes in $\\mathbb{R}^n$.\n\n\n\\begin{proposition}\\label{MK}\nIn order to prove Theorem \\ref{main}, it suffices to establish the following assertion:\nfor every finitely supported nonnegative function $M : \\mathcal{Q} \\to \\mathbb{R}$ satisfying\n$\\sum_Q M(Q)^n = 1$, there exist nonnegative functions \n$S_j : \\mathcal{Q} \\times \\mathcal{T}_j \\to \\mathbb{R}$ such that for all\n$T_j \\in \\mathcal{T}_j$ with $T_j \\cap Q \\neq \\emptyset$,\n\\begin{equation}\\label{need1}\ne(T_1)\\wedge \\dots \\wedge e(T_d) \\, M(Q)^n \\leq C S_1(Q,T_1) \\dots S_d(Q,T_d)\n\\end{equation}\nand, for all $j$ and all $T_j \\in \\mathcal{T}_j$ \n\\begin{equation}\\label{need2}\n\\sum_{Q \\,: \\, T_j \\cap Q \\neq \\emptyset} S_j(Q, T_j) \\leq C.\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nFirstly, if we can find $S_j$ as in the statement of the proposition, homogeneity dictates that\nfor every finitely supported nonnegative function $M : \\mathcal{Q} \\to \\mathbb{R}$\nthere exist nonnegative functions \n$S_j : \\mathcal{Q} \\times \\mathcal{T}_j \\to \\mathbb{R}$ such that for all\n$T_j \\in \\mathcal{T}_j$ with $T_j \\cap Q \\neq \\emptyset$,\n\\begin{equation}\\label{tyu}\ne(T_1)\\wedge \\dots \\wedge e(T_d) \\, M(Q)^n \\leq C S_1(Q,T_1) \\dots S_d(Q,T_d) \\left(\\sum_Q M(Q)^n\\right)^{(n-d)\/n}\n\\end{equation}\nand, for all $j$ and all $T_j \\in \\mathcal{T}_j$ \n\\begin{equation}\\label{uyt}\n\\sum_{Q \\,: \\, T_j \\cap Q \\neq \\emptyset} S_j(Q, T_j) \\leq C \\left( \\sum_Q M(Q)^n \\right)^{1\/n}.\n\\end{equation}\n\n\\medskip\n\\noindent\nSecondly, we note that by the $l^1$ nature of the right hand side in\nTheorem \\ref{main}, we may\nassume that the sets $\\mathcal{T}_j$ are finite and that all the coefficients $a_{T_j}$ are equal to $1$.\n\n\\medskip\n\\noindent\nFor a unit cube $Q$ let \n$$F(Q) = \\sum_{T_j \\in \\mathcal{T}_j \\mbox{ with }T_j \\cap Q \\neq \\emptyset} \ne(T_1)\\wedge \\dots \\wedge e(T_d).$$\nIt then suffices to prove\n$$ \\sum_Q F(Q)^{1\/(d-1)} \\leq C  \\left(\\# \\mathcal{T}_1 \\dots \\# \\mathcal{T}_d \\right)^{1\/(d-1)}.$$\nLet $M(Q)^n = F(Q)^{1\/(d-1)} = F(Q)^{(1\/(d-1) - 1\/d)d}$, so that \n\\begin{eqnarray*}\n\\begin{aligned}\n&\\sum_Q F(Q)^{1\/(d-1)}\n= \\sum_Q F(Q)^{1\/d} M(Q)^{n\/d} \\\\\n= & \\sum_Q \\left(\\sum_{T_j \\in \\mathcal{T}_j \\mbox{ with }T_j \\cap Q \\neq \\emptyset} \ne(T_1)\\wedge \\dots \\wedge e(T_d)\\right)^{1\/d}M(Q)^{n\/d} \\\\\n= &\\sum_Q \\left(\\sum_{T_j \\in \\mathcal{T}_j \\mbox{ with }T_j \\cap Q \\neq \\emptyset} \ne(T_1)\\wedge \\dots \\wedge e(T_d) \\; M(Q)^n \\right)^{1\/d} \\\\\n\\leq C & \\sum_Q \\left(\\sum_{T_j \\in \\mathcal{T}_j \\mbox{ with }T_j \\cap Q \\neq \\emptyset} \nS_1(Q,T_1) \\dots S_d(Q,T_d) \\left(\\sum_Q M(Q)^n\\right)^{(n-d)\/n} \\right)^{1\/d} \\\\\n= C & \\sum_Q \\left(\\prod_{j=1}^d \\sum_{T_j \\in \\mathcal{T}_j \\mbox{ with }T_j \\cap Q \\neq \\emptyset} \nS_j(Q,T_j)\\right)^{1\/d}\\left(\\sum_Q M(Q)^n\\right)^{(n-d)\/dn} \\\\\n\\leq C & \\prod_{j=1}^d \\left(\\sum_Q \\sum_{T_j \\in \\mathcal{T}_j \\mbox{ with }T_j \\cap Q \\neq \\emptyset} \nS_j(Q,T_j)\\right)^{1\/d}\\left(\\sum_Q M(Q)^n\\right)^{(n-d)\/dn} \\\\\n= C & \\prod_{j=1}^d \\left(\\sum_{T_j \\in \\mathcal{T}_j} \\sum_{Q \\mbox{ with }T_j \\cap Q \\neq \\emptyset}\nS_j(Q,T_j)\\right)^{1\/d}\\left(\\sum_Q M(Q)^n\\right)^{(n-d)\/dn} \\\\\n\\leq C & \\prod_{j=1}^d \\left( \\# \\mathcal{T}_j\\right)^{1\/d} \\left( \\sum_Q M(Q)^n \\right)^{1\/n}\n\\left(\\sum_Q M(Q)^n\\right)^{(n-d)\/dn}\\\\\n= C & \\prod_{j=1}^d \\left( \\# \\mathcal{T}_j\\right)^{1\/d} \\left( \\sum_Q F(Q)^{1\/(d-1)} \\right)^{1\/d}\n\\end{aligned}\n\\end{eqnarray*}\nwhere the inequalities follow from \\eqref{tyu}, H\\\"older's inequality and \\eqref{uyt} respectively.\nRearranging, we obtain\n$$ \\left(\\sum_Q F(Q)^{1\/(d-1)}\\right)^{(d-1)\/d} \n\\leq C  \\prod_{j=1}^d \\left( \\# \\mathcal{T}_j\\right)^{1\/d},$$\nfrom which the result follows.\n\\end{proof}\n\n\\medskip\n\\noindent\nInterestingly, the line of argument here can be reversed in certain\ncircumstances: assuming that the special case of the multilinear \nKakeya theorem for transverse families of tubes $\\mathcal{T}_j$\nholds, it follows that for all $M$ one can \nfind $S_j$ satisfying \\eqref{tyu} and \n\\eqref{uyt}. See \\cite{CV} for more details.\n\n\\section{Directional surface area and visibility}\\label{sect:vis}\n\n\\medskip\n\\noindent\nWe follow Guth \\cite{G} and Bourgain--Guth \\cite{BG} in defining the functions $S_j$ and \nestablishing their desired properties \\eqref{need1} and \\eqref{need2}. In order to do this \nsome geometric notions are required. We first recall the notion of directional surface area \n(termed ``directed volume'' by Guth) of a hypersurface $Z \\subseteq \\mathbb{R}^n$ in \nthe direction of a unit vector $e$. If the element of surface area of $Z$ is denoted by \n$dS = d \\mathcal{H}_{n-1}|_S$, and $e$ is a unit vector, the element of the component of \nsurface area of $Z$ perpendicular to $e$ is $ |e \\cdot n(x)| dS(x)$ where $n(x)$ is the \nunit normal at $x$ (which is assumed to make sense for $\\mathcal{H}_{n-1}$--almost every \n$x \\in Z$).\nThus the {\\bf directional surface area of $Z$ in the direction $e \\in \\mathbb{S}^{n-1}$} is defined\nas\n$$ \\mbox{ surf}_e(Z) = \\int_Z |e \\cdot n(x)|\\; dS(x).$$\nIf $Z$ is given by the graph of a function $\\Gamma : \\Omega \\subseteq \\mathbb{R}^{n-1} \\to \n\\mathbb{R}$ above the hyperplane $x_n = 0$, then its directional surface area in the direction \n$e_n$ is simply the $(n-1)$-dimensional area of $\\Omega$. If $Z$ is given by disjoint graphs of functions\nabove  the hyperplane $x_n = 0$ then its directional surface area in the direction $e_n$ is just \n$\\int_{\\mathbb{R}^{n-1}} J(y) \\; dy$ where $J(y)$ is the number of times the line through $y$ parallel \nto $e_n$  passes through $Z$. These considerations lead immediately to Guth's ``cylinder estimate'':\n\n\\begin{lemma}[Guth's cylinder estimate]\\label{cylinder}\nIf $T$ is a $1$-tube in $\\mathbb{R}^n$ and $Z = \\{x \\; : \\; p(x) = 0 \\}$ is the zero hypersurface \nof a non-zero polynomial $p$ of degree at most $k$, then \n$$ {\\rm{ surf}}_{e(T)}(Z \\cap T) \\leq C k.$$\n\\end{lemma}\n\n\n\\medskip\n\\noindent\nSecondly, we associate a fundamental centrally-symmetric convex body $K(Z)$ to a hypersurface $Z$. Indeed, with $\\mathbb{B}$ \ndenoting the unit ball of $\\mathbb{R}^n$, defin\n\\begin{equation}\\label{Kdefn}\nK(Z) := \\{ u \\in \\mathbb{B} \\; : \\; \\mbox{ surf}_{\\widehat{u}}(Z) \\leq 1\/|u|\\}.\n\\end{equation}\nHere $\\widehat{u}$ is the unit vector in the direction of $u$. (Notice that if $Z$ is such that \nsurf$_e(Z) \\geq 1$ for all unit vectors $e$, then the requirement that $u$ lie in $\\mathbb{B}$ is \nsuperfluous.) It is clear that $K(Z)$ is symmetric. To see that it is in fact convex, note that $u$ satisfies \n$\\mbox{ surf}_{\\widehat{u}}(Z) \\leq 1\/|u|$\nif and only if $\\int_Z | u \\cdot n | \\; dS \\leq 1$; this condition is clearly retained under \nconvex combinations of $u$'s. We then define \\footnote{Guth's definition of visibility is \nthe $n$'th power of the one given here, but we find the current definition more natural for \nthree reasons: firstly it allows us to emphasise the ``$L^n$''-aspect of the statement of \nTheorem \\ref{algtop0} which, at least when $n=d$, is no coincidence, and is a reflection of \nthe fact that the optimal $L^p$ estimate for the linear Kakeya problem in $\\mathbb{R}^n$ \nis conjectured to be at $p=n$; secondly it scales roughly as does $(n-1)$-dimensional Hausdorff \nmeasure which permits the comparison with Proposition \\ref{warmup}; and thirdly, in the theory of \nfinite-dimensional Banach spaces, if $K$ is a convex body in isotropic position,\nthe quantity $\\left(\\mbox{vol } K\\right)^{-1\/n}$ arises naturally as its isotropic constant.} the \n{\\bf visibility} of $Z$ as\n$$ \\mbox{ vis}(Z) := \\left(\\mbox{vol } K(Z)\\right)^{-1\/n}.$$\nNote that since $K(Z) \\subseteq \\mathbb{B}$ we always have $\\mbox{ vis}(Z) \\geq C$.\n\n\\medskip\n\\noindent\nThe next lemma allows us to relate visibilty to geometric means of directional surface areas.\n\n\\begin{lemma}\\label{linalg}\nSuppose that for all unit vectors $e \\in \\mathbb{R}^n$ we have $ 1 \\lesssim {\\rm{ surf}}_e (Z) \\lesssim D$.\nIf $v_1, \\dots , v_d,$ $( 1 \\leq d\\leq n )$ are unit vectors, then\n$$ \\left(v_1 \\wedge \\dots \\wedge v_d\\right)^{1\/n} {\\rm vis }(Z) \n\\leq C D^{(n-d)\/n} \\left({\\rm surf }_{v_1}(Z) \\dots \\, {\\rm surf }_{v_d}(Z)\\right)^{1\/n}.$$   \n\\end{lemma}\n\n\\begin{proof}\nWe may assume that $\\{v_1, \\dots , v_d\\}$ is linearly independent and we extend it to a \nbasis $\\{v_1, \\dots, v_n\\}$ where $v_{d+1}, \\dots , v_n$ are mutually orthogonal unit vectors \nwhich are also orthogonal to the span of $\\{v_1, \\dots, v_d\\}$. \n\n\\medskip\n\\noindent\nSince surf$_e(Z) \\gtrsim 1$ for all $e$, we have that \n$\\pm c v_j\/\\operatorname{surf}_{v_j}(Z) \\in K(Z)$ for all $j$, so that by convexity of $K(Z)$\n\\begin{eqnarray*}\n\\begin{aligned}\n\\mbox{vol }K(Z) &\\geq C \\; v_1 \\wedge \\dots \\wedge v_n \\; \\prod_{j=1}^n \\mbox{surf }_{v_j}(Z)^{-1} \\\\\n&=  C \\; v_1 \\wedge \\dots \\wedge v_d \\; \\prod_{j=1}^d \\mbox{surf }_{v_j}(Z)^{-1} \n\\prod_{j=d+1}^n \\mbox{surf }_{v_j}(Z)^{-1} \\\\\n&\\geq   C D^{-(n-d)} \\; v_1 \\wedge \\dots \\wedge v_d \\; \\prod_{j=1}^d \\mbox{surf }_{v_j}(Z)^{-1},\n\\end{aligned}\n\\end{eqnarray*}\nfrom which the result follows.\n\\end{proof}\n\n\n\\medskip\n\\noindent\nIt is not hard to show that under the assumption that surf$_e(Z) \\gtrsim 1$ for all $e$,\n$$ \\mbox{ vis}(Z) \\sim \\prod_{j=1}^n \\mbox{ surf}_{e_j}(Z)^{1\/n}$$\nwhere $e_1, \\dots, e_n$ are the principal directions of the John\nellipsoid associated to $K(Z)$ (i.e. the ellipsoid of maximal volume contained in $K(Z)$ --\nsee \\cite{John}) and hence  \n\\begin{equation}\\label{geomxx}\n\\mbox{vis}(Z) \\sim \\inf_{\\{f_j\\} \\mbox{ approx orthonormal }} \n\\prod_{j=1}^n \\mbox{ surf}_{f_j}(Z)^{1\/n}\n\\end{equation}\nwhere we say that the unit vectors $f_1, \\dots , \nf_n$ are  ``approximately orthonormal'' if their wedge product\nsatisfies $f_1 \\wedge \\dots \\wedge f_n \\geq c_n$ for a suitable\ndimensional constant $c_n$. By the arithmetic-geometric mean\ninequality the right-hand side of\n\\eqref{geomxx} is in turn dominated by $\\mathcal{H}_{n-1}(Z)$. This \nshows in particular that Theorem \\ref{algtop0} is morally stronger than\nProposition \\ref{warmup}. \n\n\\medskip\n\\noindent\nThe John ellipsoid $E$ of a symmetric convex body $K$ satisfies $E \\subseteq K \\subseteq\nn^{1\/2}E$, and combining the latter inclusion with Lemma \\ref{cylinder} we obtain:\n\n\\begin{lemma}\\label{three}\nLet $p$ be a non-zero polynomial such that for some unit vector $e$,\n$\\operatorname{surf}_{e}(Z_{p}\\cap Q) \\lesssim 1.$ Then \n$$\\operatorname{vis}(Z_{p} \\cap Q)^{n\/(n-1)} \\leq C \\, {\\rm { deg }}\\;  p.$$\n\n\n\\end{lemma}\n\\begin{proof}\n    Let $E$ be the John ellipsoid associated to $K(Z_{p}\\cap Q)$ and let\n    $l_1\\geq l_2\\geq \\dots \\geq l_n$ be the lengths of the principal\n    axes of $E$. Let $A = \\operatorname{vis}(Z_{p} \\cap Q)$. By\n    hypothesis and the fact that $K(Z_{p}\\cap Q)\\subseteq n^{1\/2} E$, we have $l_1\\gtrsim 1$.\n    Moreover we have $(l_1\\dots l_n)^{-1\/n}\\sim A$, so $l_2\\dots l_n \\lesssim A^{-n}$\n    and therefore $l_n\\lesssim A^{-n\/(n-1)}$. So if $e_n$ is the direction with which $l_n$ is associated,\n    we have that $\\operatorname{surf}_{e_n} (Z_{p}\\cap Q)\\gtrsim A^{n\/(n-1)}$.\n    The cylinder estimate now gives $\\operatorname{deg}p \\gtrsim A^{n\/(n-1)}$. \n\\end{proof}\n\n\n\n\\medskip\n\\noindent\nIn order to deal with a continuity issue later in the argument (in Lemma \\ref{convergence} of \nSection \\ref{Antipodes}), we need (as does Guth) to define variants of \nthe directional surface area and visibility which are continuous functionals of \n$Z = Z_p$ when the polynomial $p$ is allowed to vary. In view of the fact that the class\nof polynomials with the desired properties for Theorem \\ref{algtop0} is invariant under \nmultiplication by non-zero scalars, it is natural to consider the unit sphere of the class\n$\\mathcal{P}_k$ of polynomials of degree at most $k$ in $n$ real variables. \nIndeed, $\\mathcal{P}_k$ is a vector space of dimension $\\sim k^n$, and so its unit sphere \n$\\mathcal{P}_k^\\ast$ is homeomorphic to $\\mathbb{S}^N$ where $N = N(k) \\sim k^n$. So with \n$k$ fixed, we allow $p$ to vary within  $\\mathcal{P}_k^\\ast$.\\footnote{From now on we shall use \nthe notations $\\mathcal{P}_k^\\ast$ and $\\mathbb{S}^N$ where $N= N(k)$ interchangeably, the former \nwhen we are thinking of individual polynomials, the latter when continuity and topological \nconsiderations are foremost.} \nThe continuity property needed\nis most simply achieved by replacing $\\mbox{surf}_e(Z)$ for surfaces of the form\n$Z=Z_p\\cap U$ (where $p\\in \\mathbb{S}^N$ and $U$ is open in $\\mathbb{R}^n$) by\n$\\operatorname{surf}_{e,\\varepsilon}(Z)$ which we define as the average of\n$\\mbox{surf}_e(Z')$ with $Z'=Z_{p'}\\cap U$ over $p'$\nin a ball of radius $\\varepsilon$ centred at $p$ in $\\mathbb{S}^N$.\nFrom this we define $K_\\varepsilon(Z)$ and\n$\\operatorname{vis}_\\varepsilon(Z)$ in analogy to $K(Z)$ and $\\operatorname{vis}(Z)$.\nIn the argument we will have to choose $\\varepsilon$ sufficiently small so that these\nentities behave in certain ways similarly to the unmollified versions.\n\n\\medskip\n\\noindent\nIt is a routine matter to verify that $K_\\varepsilon(Z)$ is convex and that the\nthree lemmas of this section hold with these mollified variants. To be precise, \nfixing $k$ and the associated definitions of $\\operatorname{surf}_{e,\\varepsilon}(Z)$,\n$K_\\varepsilon(Z)$ and $\\operatorname{vis}_\\varepsilon(Z)$ as above\nfor $\\mathcal{P}_k^\\ast$, we have, (with implicit constants independent of $\\varepsilon > 0$):\n\n\\begin{lemma}\\label{cylinder-moll} \nIf $T$ is a $1$-tube in $\\mathbb{R}^n$ and $Z = \\{x \\; : \\; p(x) = 0 \\}$ is the zero hypersurface \nof a polynomial $ p \\in \\mathcal{P}_k^\\ast$, then for all $\\varepsilon>0$,\n$$ {\\rm{ surf}}_{e(T), \\varepsilon}(Z \\cap T) \\leq C k.$$\n\\end{lemma}\n\n\\begin{lemma}\\label{linalg-moll}\nSuppose that $p \\in \\mathcal{P}_k^\\ast$ and that $Z=Z_p\\cap U$ as above. \nAlso suppose that for some $\\varepsilon>0$ and\nall unit vectors $e \\in \\mathbb{R}^n$ we have \n$ 1 \\lesssim {\\rm{ surf}}_{e, \\varepsilon} (Z) \\lesssim D$.\nIf $v_1, \\dots , v_d,$ $( 1 \\leq d\\leq n )$ are unit vectors, then,\n$$ \\left(v_1 \\wedge \\dots \\wedge v_d\\right)^{1\/n} {\\rm vis_{\\varepsilon} }(Z) \n\\leq C D^{(n-d)\/n} \\left({\\rm surf }_{v_1, \\varepsilon}(Z) \\dots \\, {\\rm surf }_{v_d, \\varepsilon}(Z)\\right)^{1\/n}.$$\n\\end{lemma}\n\n\\begin{lemma}\\label{three-moll}\nSuppose $p \\in \\mathcal{P}_k^\\ast$ is such that for some $\\varepsilon >0$ and some unit vector $e$, \n$\\operatorname{surf}_{e, \\varepsilon}(Z_{p}\\cap Q) \\lesssim 1.$ Then \n$$\\operatorname{vis}_\\varepsilon(Z_{p} \\cap Q)^{n\/(n-1)}\\leq Ck.$$\n\\end{lemma}\n\n\\medskip\n\\noindent\nThe reader may wish to proceed with the unmollified variants in mind \non a first reading.\n\n\\section{Application of the main result to multilinear Kakeya}\n\\noindent\nThe version of Theorem \\ref{algtop0} that we will actually need is:\n\n\\begin{theorem}\\label{algtop}\nGiven a nonnegative function $M : \\mathcal{Q} \\to \\mathbb{R}$,\nthere exists a nonnegative integer $k$, a polynomial $p \\in \\mathcal{P}_k^\\ast$ and an $\\varepsilon>0$ \nsuch that \n$$ k \\leq C \\left(\\sum_Q M(Q)^n\\right)^{1\/n}$$\nand such that for all $Q \\in \\mathcal{Q}$ \n$$ \\operatorname{vis}_\\varepsilon(Z_p \\cap Q) \\geq C M(Q).$$\n\\end{theorem}\n\\noindent\n(Note that since it is always the case that $\\operatorname{vis}_\\varepsilon(Z_p \\cap Q) \\gtrsim 1$, the \nlatter condition has content only for those $Q$ with $M(Q) \\gtrsim 1$.)\n\n\\medskip\n\\noindent\nIn this section we show how this result implies the conditions of Proposition \\ref{MK}, that is, \ngiven a finitely supported nonnegative function $M : \\mathcal{Q} \\to \\mathbb{R}$ satisfying\n$\\sum_Q M(Q)^n = 1$, there exist nonnegative functions \n$S_j : \\mathcal{Q} \\times \\mathcal{T}_j \\to \\mathbb{R}$ such that \\eqref{need1} and \\eqref{need2} hold.\n\n\n\n\\medskip\n\\noindent\nGiven such a finitely supported nonnegative function $M(Q)$ with $\\sum_Q M(Q)^n = 1$, we define $M_0(Q) = \\lambda M(Q)$ for \nsome $\\lambda \\gg 1$ which is required to satisfy $\\lambda \\geq M(Q)^{-n}$ for all $Q$ in the support of $M$. \nApply Theorem \\ref{algtop} with data $M_0$ to obtain a $k$, a $p \\in \\mathcal{P}_k^\\ast$ and an \n$\\varepsilon > 0$ such that\n\\begin{equation}\n    \\label{deg} k \\leq C \\lambda\n\\end{equation}\nand\n\\begin{equation}\n    \\label{wer}\n    \\operatorname{vis}_\\varepsilon(Z_{p} \\cap Q) \\geq C \\lambda M(Q).\n\\end{equation}\n\n\n\\medskip\n\\noindent\nUsing \\eqref{deg}, \\eqref{wer} and our requirement on $\\lambda$, we have\n\\begin{eqnarray*}\n\\begin{aligned}\nk &\\leq C \\lambda = C \\lambda^{n\/(n-1)} \\lambda^{-1\/(n-1)} \\\\\n&\\leq C \\left(\\frac{\\operatorname{vis}_\\varepsilon(Z_{p} \\cap Q)}{M(Q)}\\right)^{n\/(n-1)}\n\\left(M(Q)^{-n}\\right)^{-1\/(n-1)}\\\\\n&= C \\operatorname{vis}_\\varepsilon(Z_{p} \\cap Q)^{n\/(n-1)}.\n\\end{aligned}\n\\end{eqnarray*}\n\n\\medskip\n\\noindent\nUsing Lemma~\\ref{three-moll} we deduce that for all cubes $Q$ in the support of $M$ and all unit vectors $e$ we have \n$\\operatorname{surf}_{e,\\varepsilon}(Z_{p}\\cap Q) \\gtrsim 1$.\\footnote{Another way of achieving this is to \nmultiply the polynomial which Theorem~\\ref{algtop}\nproduces with a polynomial whose zero set consists of\nhyperplanes parallel to the coordinate hyperplanes which pass through the cubes in the\nsupport of $M$. This has an insignificant effect on the degree of the polynomial provided $\\lambda$ is large enough.\nHowever, some care must be taken when considering how this augmentation interacts with the mollification.} This in turn will permit us to apply Lemma~\\ref{linalg-moll} (relating visibility to geometric means of directional surface areas) below.\n\n\n\\medskip\n\\noindent\nWe turn to the verification of \\eqref{need2}. By Lemma~\\ref{cylinder-moll} and \\eqref{deg} we have, for all $e \\in \\mathbb{S}^{n-1}$,\n\\begin{equation}\\label{awq}\n{\\rm surf }_{e,\\varepsilon}(Z_p \\cap Q) \\leq C \\lambda\n\\end{equation} \nand moreover\n\\begin{equation}\\label{qwa}\n\\sum_{Q \\, : \\, Q \\cap T_j \\neq \\emptyset} \\mbox{ surf}_{e(T_j),\\varepsilon}({Z_p} \\cap Q)\n\\leq C \\mbox{ surf}_{e(T_j),\\varepsilon}({Z_p} \\cap \\tilde{T}_j) \n\\leq C k \\leq C \\lambda , \n\\end{equation}\n(where $\\tilde{T}$ denotes the expansion of a tube $T$ about its axis by a dimensional factor).\n\n\\medskip\n\\noindent\nWe now define\n$$S_j(Q, T_j) : = \\lambda^{-1} \\mbox{ surf}_{e(T_j),\\varepsilon}({Z_p} \\cap Q)$$ \nand observe that \\eqref{qwa} immediately implies\n$$ \\sum_{Q \\, : \\, Q \\cap T_j \\neq \\emptyset} S_j(Q, T_j) \\leq C, $$\nwhich establishes \\eqref{need2}.\n\n\\medskip\n\\noindent\nOn the other hand, to see that \\eqref{need1} is satisfied, note that \n\\eqref{wer} and \\eqref{awq} together with Lemma \\ref{linalg-moll} give\n$$ C \\lambda M(Q) \\leq {\\rm vis_\\varepsilon }(Z_p \\cap Q) \\leq \n\\frac{C  \\lambda^{(n-d)\/n} \\left({\\rm surf }_{e(T_1),\\varepsilon}(Z_p \\cap Q) \n\\dots \\, {\\rm surf }_{e(T_d),\\varepsilon}(Z_p \\cap Q)\\right)^{1\/n}}\n{\\left(e(T_1) \\wedge \\dots \\wedge e(T_d)\\right)^{1\/n}},$$\nand so\n\n$$ S_1(Q,T_1) \\dots S_d(Q,T_d) = \\lambda^{-d}\\prod_{j=1}^d \\mbox{ surf}_{e(T_j),\\varepsilon}({Z_p} \\cap Q)$$\nis at least\n$$ C \\lambda^{-d}\\lambda^n M(Q)^n \\lambda^{d-n} e(T_1)\\wedge \\dots \\wedge e(T_d) \n= C M(Q)^n e(T_1)\\wedge \\dots \\wedge e(T_d),$$\nand thus \\eqref{need1} is established.\n\n\\medskip\n\\noindent\nConsequently, the multilinear Kakeya theorem is reduced to proving Theorem \\ref{algtop}.\n\n\\section{The Borsuk--Ulam theorem and a covering lemma}\\label{sec:BU}\n\n\\noindent\nThe Borsuk--Ulam theorem is as follows:\n\n\\begin{theorem}[Borsuk--Ulam]\nSuppose that $N \\geq J$ and that $F: \\mathbb{S}^N \\to \\mathbb{R}^J$ is continuous and satisfies \n$F(-x) = - F(x)$ for all $x \\in \\mathbb{S}^N$. Then there is an $x \\in \\mathbb{S}^N$ such that \n$F(x) = 0$.\n\\end{theorem}\n\\noindent\n\n\\medskip\n\\noindent\nFor a delightful discussion of this theorem and its applications, see \\cite{Mat}. See also \\cite{C}\nfor a recent proof of the Borsuk--Ulam theorem using only point-set topology and Stokes' theorem. \nIncluded in \\cite{Mat} there is a discussion of various equivalent forms of this theorem, some of which \n(known as Lusternik--Schnirelmann results) take the form of covering statements for \nthe sphere. In this section we formulate another such equivalent covering statement which we shall use \nin our proof of Theorem \\ref{algtop}.\n\n\n\\begin{lemma}\\label{covering}\nSuppose that $A_i \\subseteq \\mathbb{S}^N$ for $ 1 \\leq i \\leq J$, and suppose that for each $i$,\n$A_i \\cap (\\overline{-A_i}) = \\emptyset$. \nIf $J \\leq N$, then the $2J$ sets $A_i$ and $-A_i$ do not cover $\\mathbb{S}^N$. \n\\end{lemma}\n\n\\medskip\n\\noindent\nNote that no topological hypothesis on the sets $A_i$ is needed.\n \n\\begin{proof}\nLet $F_i(x) = d(x, -A_i) - d(x, A_i)$ for $1 \\leq i \\leq J$. Then, with $F: \\mathbb{S}^N \\to \\mathbb{R}^J$\ndefined by $F(x) = (F_1(x), \\dots , F_J(x))$, we have  that $F$ is continuous and $F(-x) = -F(x)$ for all $x$, \nso by the Borsuk--Ulam theorem there is an $x$ with $F(x) = 0$. We claim that this $x$ does not belong \nto any $A_i$ or to $-A_i$. For if $x \\in A_i$ we have\n$d(x, A_i) = 0$, hence $d(x, -A_i) = 0$, hence $x \\in\n\\overline{-A_i}$, a contradiction. Likewise, since by hypothesis $\n\\overline{A_i} \\cap (-A_i) = \\emptyset$, \n$x \\in -A_i$ gives  $x \\in \\overline{A_i}$, another contradiction.\n\\end{proof}\n\n\\medskip\n\\noindent\n{\\bf{Remark.}} The converse argument also holds: if we assume the assertion of the lemma, \n{\\em{but only for open sets}} $U_i$, we can recover the Borsuk--Ulam theorem. Indeed, suppose \n$F: \\mathbb{S}^N \\rightarrow \\mathbb{R}^J$ is continuous, $F(-x) = -F(x)$ and $N \\geq J$. Let\n$U_i$ be the open set $\\{x \\; : \\; F_i(x) > 0\\}$. Then $-U_i = \\{x \\; : \\; F_i(x) < 0\\}$, so that\n$\\overline{- U_i} \\subseteq \\{x \\, : \\, F_i(x) \\leq 0\\}$ and so\n$U_i \\cap (\\overline{-U_i}) = \\emptyset$.\nBy assumption there is an $x$ which is not in any of the $U_i$ or $-U_i$. So $F_i(x) \\leq 0$ for \nall $i$ and $F_i(x) \\geq 0$ for all $i$. Hence $F_i(x) = 0$ for all $i$, that is, $F(x) = 0$.\\footnote{That\nthe assertion of Lemma \\ref{covering} for open sets $U_i$ logically implies the same \nstatement for sets $A_i$ with no topological restrictions can easily be seen directly from the fact that the \nmetric space $\\mathbb{S}^N$ satisfies appropriate separation axioms. More precisely, if two subsets $A$ and $B$ \nof a metric space are separated in the sense that the closure of either does not meet the other, then there are \nopen sets $U$ and $V$ with  the same property such that $A \\subseteq U$ and $B \\subseteq V$.} \n\n\n\\section{Outline of the proof of Theorem \\ref{algtop}} \n\\label{sec:outline}\n\n\\noindent\nWe now describe the scheme of the proof of Theorem \\ref{algtop}. The function $M$ is given, \nand we will be working with the class $\\mathcal{P}_k^\\ast = \\mathbb{S}^N$\nof normalised polynomials \n$p : \\mathbb{R}^n \\to \\mathbb{R}$ of degree bounded by some $k \\in \\mathbb{N}$. Recall that $N \\sim k^n$.\nFor each such polynomial $p$, its zero set is the algebraic hypersurface \n$Z_p = \\{x \\, : \\, p(x) = 0 \\}$, \nand we let \n$$S(Q) = \\{ p \\in \\mathcal{P}_k^\\ast \\; : \\; \\operatorname{vis}_\\varepsilon(Z_p \\cap Q) \\leq M(Q)\\}.$$  \nFollowing Guth \\cite{G}, the aim is to show that if we take a suitable \n$k \\sim (\\sum_Q M(Q)^n)^{1\/n}$, and a suitable $\\varepsilon >0$, then we can find a \npolynomial in $\\mathcal{P}_k^\\ast$ which is not \nin any of the $S(Q)$. (Note that $S(Q) = \\emptyset$ \nfor those $Q$ such that $M(Q) \\lesssim 1$.) Our method to \nestablish this diverges somewhat from that of Guth, but there are of course many points of contact \nbetween the two lines of argument.\n\n\\medskip\n\\noindent\nLet, for $r \\geq 0$,\n$$S^{(r)}(Q) = \\{ p \\in \\mathcal{P}_k^\\ast \\; : \\; \\operatorname{vis}_\\varepsilon(Z_p \\cap Q) \\sim 2^{-r} M(Q)\\}.$$ \nThen \n$$ S(Q) = \\bigcup_{1 \\lesssim 2^r \\lesssim M(Q)}S^{(r)}(Q)$$ \nsince $S^{(r)}(Q) = \\emptyset$ for $r$ such that $2^r \\gtrsim M(Q)$.\n \n\\medskip\n\\noindent\nWe shall introduce a collection $\\mathcal{C}$ of ``colours'' $\\Theta$ whose cardinality is bounded by $C$. \nFor each colour $\\Theta$ we shall define subsets $S^{(r), \\Theta}(Q)$ of $S^{(r)}(Q)$\nwhich have the property that\n\\begin{equation}\\label{colours}\nS^{(r)}(Q) = \\bigcup_{\\Theta \\in \\mathcal{C}} S^{(r), \\Theta}(Q).\n\\end{equation}\n\n\\medskip\n\\noindent\nFor each fixed $Q$ and $r$ such that $1 \\lesssim 2^r \\lesssim M(Q)$ we will \ndefine an indexing set $\\mathcal{A}_{Q, r}$ of cardinality\n$C 2^{-rn} M(Q)^n$, and for each $\\alpha \\in \\mathcal{A}_{Q, r}$, \na subset $S^{(r), \\Theta}_\\alpha (Q)$ of $S^{(r), \\Theta}(Q)$ such that \n\\begin{equation}\\label{translates}\nS^{(r), \\Theta}(Q) = \\bigcup_{\\alpha \\in \\mathcal{A}_{Q, r}}\nS^{(r), \\Theta}_\\alpha (Q).\n\\end{equation}\nTo ensure that this decomposition is well-defined, it will transpire that $\\varepsilon$ must be taken to be small.\n\n\n\n\\medskip\n\\noindent\nFinally we shall decompose each $S^{(r), \\Theta}_\\alpha (Q)$ as\n$$ S^{(r), \\Theta}_\\alpha (Q) = S^{(r), \\Theta + }_\\alpha (Q) \\cup S^{(r), \\Theta - }_\\alpha (Q),$$\nwhere \n\\begin{equation}\\label{opposite}\nS^{(r), \\Theta -}_\\alpha(Q) = -  S^{(r), \\Theta +}_\\alpha(Q)\n\\end{equation} \nin such a way that for all $Q, r, \\Theta$ and $\\alpha$,\n\\begin{equation}\\label{qw}\nS^{(r), \\Theta + }_\\alpha (Q) \\cap \\overline{S^{(r), \\Theta - }_\\alpha (Q)} = \\emptyset.\n\\end{equation}\n\n\\medskip\n\\noindent\n(The closure here refers to the natural topology of $\\mathcal{P}_k^\\ast = \\mathbb{S}^N$.)\n\n\\medskip\n\\noindent\nThe reason for the introduction of colours is to ensure that\nthere is sufficient separation between the sets $S^{(r), \\Theta \\pm\n}_\\alpha (Q)$ and their antipodes for \\eqref{qw} to hold.\n\n\\medskip\n\\noindent\nIn summary then, \n\\begin{equation}\\label{union}\n\\bigcup_Q S(Q) = \\bigcup_Q \\bigcup_{1 \\lesssim 2^r \\lesssim M(Q)}\\bigcup_{\\Theta \\in \\mathcal{C}} \n\\bigcup_{\\alpha \\in \\mathcal{A}_{Q, r}} \\left(S^{(r), \\Theta + }_\\alpha (Q) \n\\cup S^{(r), \\Theta - }_\\alpha (Q)\\right),\n\\end{equation}\nwhere $S^{(r), \\Theta + }_\\alpha (Q)$ and $S^{(r), \\Theta - }_\\alpha (Q)$\nsatisfy \\eqref{opposite} and \\eqref{qw}. \n\n\\medskip\n\\noindent\nLemma \\ref{covering} then implies that if the cardinality of the set indexing the union on the \nright hand side of \\eqref{union} is less than or equal to $N$,\nthen the sets in the union cannot cover $\\mathbb{S}^N = \\mathcal{P}_k^\\ast$. \n\n\\medskip\n\\noindent\nNow the number of terms indexing the union is at most\n$$ C \\sum_Q \\sum_{r \\geq 0} \\sum_{\\Theta \\in \\mathcal{C}} \\, \\sum_{\\alpha \\in \\mathcal{A}_{Q, r}} \n1 \\leq C \\sum_Q \\sum_{r \\geq 0} 2^{-rn} M(Q)^n \\leq C \\sum_Q M(Q)^n.$$\nSo provided that $N \\gtrsim \\sum_Q M(Q)^n$, amongst the polynomials in $\\mathbb{S}^N =\n\\mathcal{P}_k^\\ast$, there will exist one which is not in any of the $S(Q)$. Since \n$N \\sim k^n$, we can therefore take $k$ with $k \\sim \\left(\\sum_Q M(Q)^n \\right)^{1\/n}$\nand a $p \\in \\mathcal{P}_k^\\ast$ which, for suitable $\\varepsilon > 0$, satisfies \n$\\operatorname{vis}_\\varepsilon(Z_p \\cap Q) > M(Q)$ for all $Q$, as was needed.\n\n\\medskip\n\\noindent\nIt remains now to define the various decompositions introduced above,\nand establish the assertions we have made concerning them.\n\n\n\n\\section{Colours}\n\\noindent\nIn this section we describe how to establish \\eqref{colours} in such a way that the indexing set \n$\\mathcal{C}$ has cardinality at most $C$.\n\n\\medskip\n\\noindent\nLet $\\mathcal{E}$ denote the class of centred ellipsoids in $\\mathbb{R}^n$,\nthat is images of the unit ball $\\mathbb{B}$\nby affine linear maps $A$. Each ellipsoid $A (\\mathbb{B})$ is determined by an orthonormal basis of \nprincipal axes or directions given by the normalised eigenvectors of $A^tA$, and corresponding semiaxes, \nthe squares of whose lengths are the eigenvalues of $A^tA$.\nThus $\\mathcal{E}$ is a manifold of dimension\n$n(n-1)\/2 + n = n(n+1)\/2$. \n\n\\medskip\n\\noindent\nLet $\\mathcal{K}$ denote the class of centrally symmetric convex bodies in $\\mathbb{R}^n$.\nBy the John ellipsoid theorem \\cite{John}, every member $K$ of $\\mathcal{K}$ is close to some ellipsoid $E$ \nin the sense that $n^{-1\/4} E \\subseteq K \\subseteq n^{1\/4} E$. \n\n\\medskip\n\\noindent\nThere is a natural metric (the Banach--Mazur metric) to put on the class $\\mathcal{K}$, given by\n$$d(K,L) = \\log \\inf \\{ \\alpha \\geq 1 \\, : \\, \\alpha^{-1}K \\subseteq L \\subseteq \\alpha K\\}.$$    \nThe John ellipsoid theorem asserts that every convex body is distant at most \n$(\\log n)\/4 \\lesssim 1$  from some ellipsoid. An ellipsoid with semiaxes of lengths $2^{k_1}, \\dots , 2^{k_n}$ where\n$k_1 + \\dots + k_n =0$ will be distant $ \\lesssim \\max |k_j| $ from the unit ball. Two congruent ellipses \nin $\\mathbb{R}^2$ with semiaxes of lengths $1$ and $N$ and principal directions differing by $\\theta$ will be \ndistant $\\lesssim \\theta N $ apart.\n\n\n\n\\medskip\n\\noindent\nTo set the scene for the covering property of ellipsoids that we need, note that in $\\mathbb{R}^N$, \ngiven a scale $\\rho > 0$ and a pre-assigned number $\\gamma > 1$, \nwe can find a set $\\mathcal{X}$ of $\\rho$-separated points $x_i \\in \\mathbb{R}^N$ such that every \npoint of $\\mathbb{R}^N$ is in some $B(x_i,\\rho)$, and such that $\\mathcal{X}$ can be partitioned \ninto $O_N(\\gamma^N)$ families (colours), so that points of $\\mathcal{X}$ of the same colour are \ndistant at least $ \\gamma \\rho$ from each other. This property expresses the idea that the \ndimensionality of $\\mathbb{R}^N$ as a metric space is $N$. We can then assign to each $x \\in \\mathbb{R}^N$ \none or more colours according to whether $d(x, x_i) < \\rho$ for some $x_i \\in \\mathcal{X}$ of that \nparticular colour. \n\n\\medskip\n\\noindent\nSimilarly it is not hard to verify that given $\\rho >0$ and $\\gamma >1$, there exists a $\\rho$-separated \nsubset $\\mathcal{E}_0$ of $\\mathcal{E}$ such that $\\{B(E, \\rho) \\, : \\, E \\in  \\mathcal{E}_0\\}$ \ncovers $\\mathcal{E}$ and such that we can partition $\\mathcal{E}_0$ into at most \n$O_n( \\gamma^{n(n+1)\/2})$ families (colours) such that any two ellipsoids in $\\mathcal{E}_0$ \nof the same colour are distant at least $\\gamma \\rho$ from each other.  \n\n\\medskip\n\\noindent\nChoosing $\\rho = 1$ and $\\gamma$ sufficiently large depending only on the dimension $n$, and using the \nJohn ellipsoid theorem, we obtain the following: \n\n\\begin{lemma}\\label{ellipsoid}\nSupose $\\alpha_n > 1 $ is sufficiently large.\nThen there exists a set $\\mathcal{E}_0 \\subseteq \\mathcal{E}$ with the property that for every \n$K \\in \\mathcal{K}$ there is an $E \\in \\mathcal{E}_0$ such that\n\\begin{equation}\\label{close}\n\\alpha_n^{-1} K \\subseteq E \\subseteq \\alpha_n  K\n\\end{equation}\nand such that the set $\\mathcal{E}_0$ can be partitioned into at most $C = C(n, \\alpha_n)$\ncolours in such a way that every $K \\in \\mathcal{K}$ satisfies \\eqref{close} for at most one \n$E \\in \\mathcal{E}_0$ {\\em of a given colour}.\n\\end{lemma}\n\n\n\\medskip\n\\noindent\nGiven $n$ we now fix $\\alpha_n$ sufficiently large, and fix our palette $\\mathcal{C}$ \nconsisting of at most $C(n, \\alpha_n)$ colours once and for all so that the conclusion of \nLemma \\ref{ellipsoid} holds. We say that two convex bodies $E$ and $K$ are {\\bf close} if \n\\eqref{close} holds. So every $K \\in \\mathcal{K}$ is close to some member of $\\mathcal{E}_0$, \nbut there is at most one $E \\in \\mathcal{E}_0$ of a given colour to which it is close.  \nFor a colour $\\Theta \\in \\mathcal{C}$ let \n$$\\mathcal{E}^{\\Theta}_0 = \\{ E \\in \\mathcal{E}_0 \\; : \\; E \\mbox{ is of colour  } \\Theta\\}.$$\n\n\\medskip\n\\noindent\nFinally, given $Q$ and $r \\geq 0$,\nlet\n$$S^{(r), \\Theta} (Q) = \\{ p \\in S^{(r)}(Q) \\, : \\, K_\\varepsilon(Z_p \\cap Q) {\\mbox{ is close to a member of }}\n\\mathcal{E}^{\\Theta}_0\\};$$\nthen we have\n$$S^{(r)}(Q) = \\bigcup_{\\Theta \\in \\mathcal{C}} S^{(r), \\Theta}(Q),$$\nand \\eqref{colours} is established.\n\n\n\\section{Translates}\\label{sec:translates}\n\\noindent\nWe now fix $Q$, $r \\geq 0$ and a colour $\\Theta \\in \\mathcal{C}$. In this section we establish \n\\eqref{translates} for suitable subsets $S^{(r), \\Theta}_\\alpha(Q) \\subseteq \nS^{(r), \\Theta}(Q)$ which are indexed by $\\alpha \\in \\mathcal{A}_{Q, r}$, where $\\mathcal{A}_{Q, r}$\nhas cardinality $\\sim 2^{-rn} M(Q)^n$. We can assume that  $S^{(r), \\Theta}(Q) \\neq \\emptyset$.\n\n\\medskip\n\\noindent\nIf $p \\in S^{(r), \\Theta}(Q)$, the convex body $K_\\varepsilon(Z_p \\cap Q) \\subseteq \\mathbb{B}$ \nhas volume $\\sim 2^{rn}\/M(Q)^{n} \\lesssim 1$, and it is close to a unique member $E(p)$ of $\\mathcal{E}^{\\Theta}_0$ \nof comparable volume. Hence we can fit $\\sim 2^{-rn} M(Q)^{n}$ disjoint parallel translates of $E(p)$\ninside $Q$, with the translations along the principal directions of $E(p)$. Likewise,\nif $\\eta < 1$ is a numerical scaling factor, we can fit $\\sim \\eta^{-n}2^{-rn} M(Q)^{n}$ disjoint parallel \ntranslates of $\\eta E(p)$ inside $Q$, with the translations again along the principal directions of $E(p)$.\nIndeed, if the lengths of the semiaxes of $E(p)$ are $l_1, \\dots, l_n\n\\leq c$, and the principal directions are \n$e_1, \\dots, e_n$, we can place the centres of the translated copies of $\\eta E(p)$ at the points  \n$x_Q + \\eta \\sum_j m_j l_j e_j$ for $m_j \\in 2 \\mathbb{Z}$ and $|m_j| \\leq c \\eta^{-1}l_j^{-1}$; \nhere $x_Q$ is the centre of $Q$.\nIn this construction\nthe number of translated copies equals the product\n\\begin{equation}\n    \\label{numtranslates}\n    c\\eta^{-n}(l_1\\dots l_n)^{-1}=c \\eta^{-n}2^{-rn} M(Q)^{n}.\n\\end{equation}\n\n\\begin{lemma}\n    \\label{manybisections}\nThere is a dimensional constant $C_n$ such that if $p \\in S^{(r), \\Theta}(Q)$\nand $\\eta < 1$, then $Z_p$ bisects \nat most $ C_n \\eta ^{-(n-1)} 2^{-rn} M(Q)^{n}$ disjoint translates of $\\eta E(p)$ in $Q$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $E(p)$ has principal directions $\\{e_j\\}$ and corresponding semiaxes with lengths $\\{l_j\\}$. \nIf $Z_p$ bisects a translate $\\eta E(p) + \\xi$ of $\\eta E(p)$, then for at least one $j$ we will \nhave\\footnote{Here and in the next two displayed equations we are using the {\\em unmollified} \nnotion of directional surface area $\\mbox{surf}_e$.}\n$$ \\mbox{ surf}_{e_j}(Z_p \\cap (\\eta E(p) + \\xi)) \\geq C_n \\operatorname{vol}(\\eta E(p) + \\xi) \/ \\eta l_j\n= C_n \\eta ^{n-1} \\operatorname{vol}E(p)\/l_j.$$\nThis is just the affine-invariant formulation of the fact that a hypersurface which bisects the \nunit ball must have large $(n-1)$-dimensional Hausdorff measure inside\nthe ball -- see Lemma \\ref{bisect} in the Appendix.\nSo \n$$\\sum_{j=1}^n l_j \\operatorname{surf}_{e_j}(Z_p \\cap (\\eta E(p) + \\xi))\n \\geq C_n \\eta ^{n-1} \\operatorname{vol}E(p).$$\nIf now $Z_p$ bisects as many as $ A 2^{-rn} M(Q)^{n}$ disjoint copies of $\\eta E(p)$ in $Q$, \nwe will have\n$$\\sum_{j=1}^n l_j \\operatorname{surf}_{e_j}(Z_p \\cap Q) \\geq \nC_n A \\eta^{n-1} \\operatorname{vol}E(p)\\,\\, 2^{-rn} M(Q)^{n}.$$\n\n\\medskip\n\\noindent\nIf $p'$ is another polynomial in $\\mathbb{S}^N$ sufficiently close to $p$,\nand if $p$ bisects an ellispoid $E$, we can conclude that $p'$ approximately bisects $E$ in the sense that\n$p'$ is positive on at least $40\\%$ of $E$ and negative on at least $40\\%$ of $E$.\nSince we are only interested in a finite number of ellipsoids here, namely the translates $\\eta E+\\xi$\nfor $E\\in\\mathcal{E}_0$ with $M(Q)^{-n}\\lesssim\\operatorname{vol}E\\lesssim 1$ for the relevant $Q$,\nthen by choosing $\\varepsilon$ small enough we will have this approximate bisection property for all\npolynomials which affect the value of $\\operatorname{surf}_{e,\\varepsilon}$ in the expressions above.\nTherefore we have the estimate\n$$\\sum_{j=1}^n l_j \\operatorname{surf}_{e_j,\\varepsilon}(Z_p \\cap Q) \\geq \nC_n A \\eta^{n-1} \\operatorname{vol}E(p)\\,\\, 2^{-rn} M(Q)^{n}.$$\n\n\\medskip\n\\noindent\nThe definition (cf. \\eqref{Kdefn}) of $K_\\varepsilon(Z_p \\cap Q)$ together with the fact that\n$K_\\varepsilon(Z_p \\cap Q)$ and $E(p)$ are close \nimplies that \n$$ l_j  \\operatorname{surf}_{e_j,\\varepsilon}(Z_p \\cap Q) \\leq C_n$$\nfor each $j$, and moreover, as $p \\in  S^{(r), \\Theta}(Q)$,\n$$\\operatorname{vol}E(p) \\sim 2^{rn}\/M(Q)^{n}.$$ \nTherefore $A$ must satisfy $A  \\leq C_n \\eta^{-(n-1)}$.\n\\end{proof}\n\n\\medskip\n\\noindent\nChoose $\\eta$ sufficiently small so that $C_n \\eta <c$.\nHere $C_n$ is the constant from Lemma~\\ref{manybisections}\nand $c$ is the constant in~\\eqref{numtranslates}.\nThen\nfor each $p \\in S^{(r), \\Theta}(Q)$, $Z_p$ can bisect only a \nproportion $C_n \\eta\/c < 1$ of the $c\\eta ^{-n} 2^{-rn} M(Q)^{n}$ disjoint\ncopies of $\\eta E(p)$ in $Q$. In particular, $Z_p$ will {\\em not} bisect\n{\\em all} of the available disjoint copies of $\\eta E(p)$ in $Q$.\n\n\\medskip\n\\noindent\nFor each $E(p) \\in \\mathcal{E}^{\\Theta}_0$ of volume $\\sim 2^{rn}\/M(Q)^{n}$,\nthe set of translated ellipsoids which are placed into $Q$ in the construction above\nis of cardinality $c2^{-rn} M(Q)^{n}$.\nWe label these ellipsoids with an index $\\alpha$ from $\\mathcal{A}_{Q,r}=\\{1,\\dots, c2^{-rn} M(Q)^{n}\\}$.\nNow take a polynomial $p\\in S^{(r), \\Theta}_\\alpha (Q)$.\nThen $p$ is close to a unique member $E(p)\\in \\mathcal{E}_0^\\Theta$ of volume comparable to\n$2^{rn}\/M(Q)^{n}$ and we can ask whether or not $p$ bisects the $\\alpha$'th translate\nof $\\eta E(p)$. Note that for this question to be meaningful the uniqueness in\nthe previous sentence is important and if we had not already restricted attention to a single colour then\nthe uniqueness would not hold.\n\n\n\n\n\\medskip\n\\noindent\nFor $\\alpha \\in \\mathcal{A}_{Q,r}$ we can thus define\n\\begin{eqnarray*}\nS^{(r), \\Theta}_\\alpha (Q) \n:= S^{(r), \\Theta}(Q) \\cap \\{Z_p \\mbox{ does not bisect the $\\alpha$'th translate of \n$\\eta E(p)$ in $Q$}\\} \n\\end{eqnarray*}\n\n\n\n\n\\medskip\n\\noindent\nThen, since $Z_p$ cannot bisect all of the translates of $\\eta  E(p)$ in $Q$, we have \n$$S^{(r), \\Theta}(Q) = \n\\bigcup_{\\alpha \\in \\mathcal{A}_{Q, r}} S^{(r), \\Theta}_\\alpha (Q)$$\nas required, where there are $c 2^{-rn} M(Q)^{n}$ terms in the indexing set \n$\\mathcal{A}_{Q, r}$. Thus \\eqref{translates} is established.\n\n\n\n\\section{Antipodes}\\label{Antipodes}\n\n\\medskip\n\\noindent\nIn this section we establish \\eqref{qw}.\n\n\\medskip\n\\noindent\nFix $Q$, $r \\geq 0$, $\\Theta \\in \\mathcal{C}$ and $\\alpha \\in \\mathcal{A}_{Q, r}$. \nFor $p \\in S^{(r), \\Theta}_\\alpha (Q)$ let $E(p)$ be as before the unique member of $\\mathcal{E}_0^\\Theta$ \nto which $K_\\varepsilon(Z_p \\cap Q)$ is close. Let\n$E_\\alpha = E_\\alpha(p)$ denote the $\\alpha$'th translate of $\\eta E(p)$ in $Q$.\nSince $Z_p$ does not bisect $E_\\alpha$,\nwe have either \n$$ \\mbox{ vol }(\\{p>0\\} \\; \\cap \\; E_\\alpha) > \\mbox{ vol}(\\{p<0 \\}\n\\; \\cap \\; E_\\alpha) $$\n(in which case we say that $p \\in S^{(r), \\Theta +}_\\alpha(Q)$), or \n$$ \\mbox{ vol }(\\{p>0\\} \\; \\cap \\; E_\\alpha) < \\mbox{ vol}(\\{p<0 \\}\n\\; \\cap \\; E_\\alpha), $$\n(in which case we say that $p \\in S^{(r), \\Theta -}_\\alpha(Q)$).\n\n\\medskip\n\\noindent\nThen\n$$ S^{(r), \\Theta}_\\alpha(Q) = S^{(r), \\Theta +}_\\alpha(Q) \\cup S^{(r), \\Theta -}_\\alpha(Q).$$ \nMoreover \n$$ S^{(r), \\Theta -}_\\alpha(Q) = -  S^{(r), \\Theta +}_\\alpha(Q),$$ \nand so to establish \\eqref{qw} we wish to show that for all $\\alpha$,  \n$$S^{(r), \\Theta +}_\n\\alpha(Q) \\cap \\overline{S^{(r), \\Theta -}_\\alpha(Q)} = \\emptyset.\n$$\n\n\\medskip\n\\noindent\nTo see this, suppose for a contradiction that for some\n$\\alpha \\in \\mathcal{A}_{Q, r}$ there is a $p \\in S^{(r),\n\\Theta +}_\\alpha(Q)$ and a sequence of $p_m \\in  S^{(r), \\Theta -}_\\alpha(Q)$\nwhich converges to $p$ in $\\mathbb{S}^N$. That is, we suppose that \n\\begin{equation}\\label{last}\n\\mbox{ vol }(\\{p>0\\} \\; \\cap \\; E_\\alpha (p)) > \\mbox{ vol}(\\{p<0 \\}\n\\; \\cap \\; E_\\alpha(p)) \n\\end{equation}\nand\n$$ \\mbox{ vol }(\\{p_m> 0\\} \\; \\cap \\; E_\\alpha (p_m)) < \\mbox{ vol}(\\{p_m <0 \\}\n\\; \\cap \\; E_\\alpha(p_m))$$\nwhere  $p_m$ converges to $p$ in $\\mathbb{S}^N$.\n\n\\begin{lemma}\\label{convergence}\nFix $Q$, $r$ and $\\Theta$. Suppose that $p \\in S^{(r), \\Theta}(Q)$, $p_m \\in S^{(r), \\Theta}(Q)$ \nfor $m \\in \\mathbb{N}$ and that $p_m$ converges to $p$ in $\\mathbb{S}^N$. Then for all sufficiently \nlarge $m$ we have $E(p_m) = E(p)$. If $\\alpha \\in \\mathcal{A}_{Q,r}$ and in addition \n$p, p_m \\in S^{(r), \\Theta}_\\alpha(Q)$, then for $m$ sufficiently large, \n$E_\\alpha(p_m) = E_\\alpha(p)$\n\\end{lemma}\n\n\\begin{proof}\nSince we are using the mollified version of the directional surface area and quantities defined in terms of it, \nthe convergence of $p_m$ to $p$ in $\\mathbb{S}^N$ implies that the convex bodies $K_\\varepsilon(Z_{p_m} \\cap Q)$\nconverge to $K_\\varepsilon(Z_{p} \\cap Q)$ as $m \\to \\infty$\\footnote{in the sense that there is a sequence $\\gamma_m \\geq 1$ \nwith $\\gamma_m \\to 1$ such that\n$\\gamma_m^{-1} K_\\varepsilon (Z_{p_m} \\cap Q) \\subseteq K_\\varepsilon(Z_{p} \\cap Q) \\subseteq \\gamma_m \nK_\\varepsilon(Z_{p_m} \\cap Q)$} and in particular $K_\\varepsilon(Z_{p_m} \\cap Q)$ and $K_\\varepsilon(Z_{p} \\cap Q)$\nare close for $m$ sufficiently large.\nSince $p$ and $p_m$ are members of $S^{(r), \\Theta}(Q)$ then\n$K_\\varepsilon(Z_{p_m} \\cap Q)$ and $K_\\varepsilon(Z_p \\cap Q)$\nmust be close to \\emph{some} member of $\\mathcal{E}_0^\\Theta$\nand\nthus, for $m$ sufficiently large, \nthey\nare close to the {\\em same} member of $\\mathcal{E}^{\\Theta}_0$, which \nmust be $E(p)$. In particular, for $m$ sufficiently large, we have $E(p_m) = E(p)$ and consquently \n$E_\\alpha(p_m) = E_\\alpha(p)$ for all $\\alpha$.\n\\end{proof}\n\\noindent\n(It is at the end of the proof of this lemma, and in the construction of the sets $S^{(r), \\Theta}_\\alpha(Q)$, \nwhere the relevance of the earlier decomposition into colours becomes clear.)\n\n\\medskip\n\\noindent\nSo, for $m$ sufficiently large we have\n$$ \\mbox{ vol }(\\{p_m> 0\\} \\; \\cap \\; E_\\alpha (p)) < \\mbox{ vol}(\\{p_m <0 \\}\n\\; \\cap \\; E_\\alpha(p))$$\nwhich, upon taking limits and using the fact that vol $(\\{ p = 0\\}) = 0$ as $p$ is non-zero,\nimplies\n$$ \\mbox{ vol }(\\{p > 0\\} \\; \\cap \\; E_\\alpha (p)) \\leq \\mbox{ vol}(\\{p < 0 \\}\n\\; \\cap \\; E_\\alpha(p)),$$\nwhich is in contradiction with \\eqref{last}. Hence\n$$S^{(r), \\Theta +}_\\alpha(Q) \\cap \\overline{S^{(r), \\Theta -}_\\alpha(Q)} = \\emptyset,$$\nand we are therefore finished.\n\n\\section{Appendix -- Bisecting balls}\\label{app}\n\n\\medskip\n\\noindent\nIn this appendix we indicate a simple proof of the (geometrically obvious) fact that a hypersurface\nwhich bisects the unit ball must have large surface area inside the\nball. Let $\\mathbb{B}$ be the closed unit ball in $\\mathbb{R}^n$ and suppose $p: \\mathbb{R}^n \\to\n\\mathbb{R}$ is a polynomial. Let\n$$E = \\{x \\in \\mathbb{B} \\, : \\, p(x) \\leq 0 \\}$$\nand \n$$F = \\{x \\in \\mathbb{B} \\, : \\, p(x) \\geq 0 \\}.$$\n\n\\begin{lemma}\\label{bisect}\nIf $ {\\rm vol} \\, (E) = a \\, {\\rm vol} \\, (\\mathbb{B})$ and\n$ {\\rm vol} \\, (F) = b \\, {\\rm vol} \\, (\\mathbb{B})$ \nwhere $a + b = 1$, then \n$$\\mathcal{H}_{n-1} (\\{ x \\in \\mathbb{B} \\, : \\, p(x) = 0 \\}) \n>  \\frac{1}{2}\\left(a^{(n-1)\/n} + b^{(n-1)\/n} -1 \\right)\n\\mathcal{H}_{n-1}(\\mathbb{S}^{n-1}).$$\n\\end{lemma}\n\n\\begin{proof}\nIt is easy to see that \n$ \\partial E \\cup \\partial F = \\mathbb{S}^{n-1} \\cup (E \\cap F)$\nand\n$ \\partial E \\cap \\partial F = E \\cap F.$\nSince\n$$ \\mathcal{H}_{n-1}( \\partial E \\cup \\partial F) = \n\\mathcal{H}_{n-1}( \\partial E) + \\mathcal{H}_{n-1}(\\partial F) \n- \\mathcal{H}_{n-1}( \\partial E \\cap \\partial F)$$\nand \n$$ \\mathcal{H}_{n-1} (\\mathbb{S}^{n-1} \\cup (E \\cap F)) =\n\\mathcal{H}_{n-1} (\\mathbb{S}^{n-1}) + \\mathcal{H}_{n-1} (E \\cap F)\n- \\mathcal{H}_{n-1} (\\mathbb{S}^{n-1} \\cap E \\cap F)$$\nwe have\n$$ 2 \\mathcal{H}_{n-1} (E \\cap F)\n= \\mathcal{H}_{n-1}( \\partial E) +\n\\mathcal{H}_{n-1}(\\partial F) -  \\mathcal{H}_{n-1} (\\mathbb{S}^{n-1})\n+ \\mathcal{H}_{n-1}\n(\\mathbb{S}^{n-1} \\cap E \\cap F),$$\nand so\n$$\\mathcal{H}_{n-1} (E \\cap F) \n\\geq \\frac{1}{2}\\left(\\mathcal{H}_{n-1}( \\partial E) +\n\\mathcal{H}_{n-1}(\\partial F) -  \\mathcal{H}_{n-1} (\\mathbb{S}^{n-1})\\right).$$\n\n\\medskip\n\\noindent\nBy the isoperimetric inequality we have\n$$\\mathcal{H}_{n-1}( \\partial E) \\geq a^{(n-1)\/n}\n\\mathcal{H}_{n-1}(\\mathbb{S}^{n-1})$$\nand \n$$\\mathcal{H}_{n-1}( \\partial F) \\geq b^{(n-1)\/n}\n\\mathcal{H}_{n-1}(\\mathbb{S}^{n-1}),$$\n(with strict inequality in at least one place) so that \n$$\\mathcal{H}_{n-1} (E \\cap F) \n> \\frac{1}{2}\\left(a^{(n-1)\/n} + b^{(n-1)\/n} -1 \\right)\n\\mathcal{H}_{n-1}(\\mathbb{S}^{n-1})$$\nas required. \n\\end{proof}\n\n\\medskip\n\\noindent\nSince for $n \\geq 2$ and $0 < a, b < 1$ with $a+b =1$ we have $ (a+b)^{(n-1)\/n} < a^{(n-1)\/n} + b^{(n-1)\/n}$,\nthis establishes the desired bound.\n\\medskip\n\\noindent\n \n\\medskip\n\\noindent\nThe following question may be of interest. \nLet $K \\subseteq \\mathbb{R}^n$ be a symmetric convex body, which we can normalise so that its John ellipsoid \nis the unit ball. Within the class of polynomial hypersurfaces $Z$ which cut $K$ in the proportions $a:b$, \nhow do we minimise the surface area of $Z \\cap K$?\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nLet $n,k\\in \\mathbb{N}$ and let $\\brho=(\\rho_1,\\dots,\\rho_k)$ with $1\\leq \\rho_\\ell\\leq n\\leq k$ for $\\ell\\in [k]:=\\{1,2,\\dots, k\\}$ such that $\\sum_{\\ell\\in [k]} \\rho_\\ell=n^2$. A {\\it $\\brho$-latin square} $L$ of order $n$ is an $n\\times n$ array filled with $k$ different {\\it symbols}, $[k]$, each occurring at most once in each row and at most once in each column, such that each symbol $\\ell$ occurs exactly $\\rho_\\ell$ times in $L$ for $\\ell\\in [k]$. An $r \\times s$ {\\it  $\\brho$-latin rectangle} on the set $[k]$ of symbols is an $r\\times s$ array in which each symbol in $[k]$ occurs at most once in each row and in each column, and in which   each symbol $\\ell$ occurs at most $\\rho_\\ell$ times  for $\\ell\\in [k]$. A {\\it latin square} (or {\\it rectangle}) is a $\\brho$-latin square (or {\\it rectangle}) with $\\brho=(n,\\dots,n)$. \n\nWe are interested in conditions that ensure that an $r \\times s$ $\\brho$-latin rectangle can be extended to a $\\brho$-latin square.  The cases   $(\\brho,s)=((n,\\dots,n),n)$, and $\\brho=(n,\\dots,n)$ were settled by Hall \\cite{MR13111}, and  Ryser \\cite{MR42361}, respectively. Most recently, the case $s=n$ was solved by Goldwasser,  Hilton,  Hoffman, and \\\"{O}zkan  \\cite{MR3280683} (See Theorem \\ref{rhohallthmgoldetal}).  For an old yet excellent survey on embedding latin squares, we refer the reader to  Lindner \\cite{MR1096296}. \n\nWe need a few pieces of notation before we can state our main result. The complement of a set $S$ is denoted by $\\bar S$,  $x \\dotdiv y:=\\max\\{0, x-y\\}$, and we write $x-y\\dotdiv z$ for $(x-y)\\dotdiv z$. If $L$ is an $r\\times s$  $\\brho$-latin rectangle, $e_\\ell$ will denote the number of occurrences of symbol $\\ell$ in $L$. We will always assume that $n>\\min \\{r,s\\}$, and that $[r], [s]$, and $[k]$ are the set of rows, the set of columns, and the set of symbols of $L$. \nLet $i$ be a row in $[r]$,  $j$ be a column in $[s]$,  $\\ell$ be a symbol in $[k]$, and let $I\\subseteq [r], J\\subseteq [s], K\\subseteq [k]$. Then $\\mu_K(i), \\mu_K(j), \\mu_I(\\ell)$, and $\\mu_J(\\ell)$ are the number of symbols in $K$ that are missing in row $i$, the number of symbols in $K$ that are missing in column $j$, the number of rows in $I$ where symbol $\\ell$ is missing, and the number of columns in $J$ where symbol $\\ell$ is missing, respectively, and \n\\begin{align*}\n    \\mu_K(I)&:=\\sum_{i\\in I}\\mu_K(i), & \\mu_I(K)&:=\\sum_{\\ell\\in K} \\mu_I(\\ell),\\\\\n    \\mu_K(J)&:=\\sum_{j\\in J}\\mu_K(j), & \\mu_J(K)&:=\\sum_{\\ell\\in K} \\mu_J(\\ell).\n\\end{align*}\nObserve that \n\\begin{align*}\n    &\\mu_K(I)= \\mu_I(K), &\\mu_{I}(\\ell)+\\mu_{\\bar I}(\\ell)=r-e_\\ell,\\\\\n    &\\mu_K(J)= \\mu_J(K), &\\mu_{J}(\\ell)+\\mu_{\\bar J}(\\ell)=s-e_\\ell.\n\\end{align*}\n\nUsing our notation, the main  result of  \\cite{MR3280683} can be stated  as follows. \n\\begin{theorem} \\cite{MR3280683} \\label{rhohallthmgoldetal}\nAn $r\\times n$  $\\brho$-latin rectangle $L$ can be completed to an $n\\times n$ $\\brho$-latin square  if and only if $\\rho_\\ell - e_\\ell\\leq n-r$ for $\\ell\\in [k]$, and either one of the following conditions hold.\n\\begin{align}\n    |J|(n-r) &\\leq \\sum_{\\ell\\in [k]} \\min \\Big\\{\\rho_\\ell-e_\\ell, \\mu_J(\\ell)\\Big\\}  &\\forall J\\subseteq  [n],\\label{orenec}\\\\\n    |J|(n-r) &\\geq \\sum_{\\ell\\in [k]} \\Big( \\rho_\\ell-e_\\ell\\dotdiv \\mu_{\\bar J}(\\ell)\\Big)  &\\forall J\\subseteq [n]. \\label{hoffmanc}\n\\end{align}\n\\end{theorem}\nRewriting \\eqref{orenec} for $\\bar J$ will result in \\eqref{hoffmanc}, so conditions \\eqref{orenec} and \\eqref{hoffmanc} are essentially the same (See Remark \\ref{ordenewcond}). In \\cite{MR3280683}, \\eqref{orenec} is being referred to as Hall's refined condition, which  is a special case of the more general Hall's constrained condition. If you think of a set $J$ of columns of $L$ as an $r\\times |J|$ subrectangle of $L$, Hall's constrained condition checks an  inequality similar to that of Hall's refined condition for all subsets of cells, rather than those subsets that form an $r\\times |J|$ rectangle. \n\n\n\n\n\nWe provide a short proof of   Theorem \\ref{rhohallthmgoldetal} (See Theorem \\ref{rhohallthm}). We also generalize Theorem \\ref{rhohallthmgoldetal} further by establishing an analogue of Ryser's Theorem for $\\brho$-latin rectangles. \n\nSuppose that an $r\\times s$  $\\brho$-latin rectangle $L$ is extended to an $n\\times n$ $\\brho$-latin square, and let us fix a symbol $\\ell\\in [k]$. On the one hand, there are $\\rho_\\ell-e_\\ell$ occurrences of symbol $\\ell$ outside the top left $r\\times s$ subsquare. On the other hand, there are at most $n-r$ occurrences of symbol $\\ell$ in the last $n-r$ rows and at most $n-s$ occurrences of symbol $\\ell$ in the last $n-s$ columns. Therefore, we must have\n\\begin{align} \\label{mainryserineq}\ne_\\ell \\geq r+s+\\rho_\\ell - 2n\\quad \\forall \\ell\\in [k].\n\\end{align}\nIt turns out that condition \\eqref{mainryserineq} is a special case of a more general necessary condition. Let \n\\begin{align*}\n    P_t=\\{\\ell \\in [k] \\ |\\ \\rho_\\ell-e_\\ell >  n-t  \\}\\quad \\mbox{for } t\\in \\{r,s\\}.\n\\end{align*}\nNote that $P_{\\min \\{r,s\\}}\\subseteq P_{\\max \\{r,s\\}}$. \n\n\n\\begin{definition}  \\label{fitdef} A sequence $\\{(a_\\ell,b_\\ell)\\}_{\\ell=1}^k$ is said to be {\\it fitting} if $a_\\ell,b_\\ell \\in\\mathbb{N}\\cup \\{0\\}$ for  $\\ell\\in [k]$, and the following conditions are satisfied.\n\\begin{align*}\n\\begin{cases}\n       \\displaystyle\\sum_{\\ell\\in [k]} a_\\ell=r(n-s)+|P_r|(n-r)-\\sum_{\\ell\\in P_r}(\\rho_\\ell-e_\\ell), \\\\[20pt]\n        \\displaystyle\\sum_{\\ell\\in [k]} b_\\ell=s(n-r)+|P_s|(n-s)-\\sum_{\\ell\\in P_s}(\\rho_\\ell-e_\\ell),\\\\[20pt]\n         a_\\ell+b_\\ell\\leq \n        \\begin{cases}\n            2n- r-s+ e_\\ell-\\rho_\\ell& \\mbox {if  } \\ell \\in P_{\\min \\{r,s\\}},\\\\ \n            n- \\max \\{r,s\\} & \\mbox {if  } \\ell \\in P_{\\max \\{r,s\\}}\\backslash P_{\\min \\{r,s\\}},\\\\\n            \\rho_\\ell -e_\\ell & \\mbox {if  } \\ell \\in \\bar P_{\\max \\{r,s\\}}.\n        \\end{cases}\n    \\end{cases}\n\\end{align*} \n\\end{definition}\nHere is our main result. \n\\begin{theorem} \\label{rhoryserthmfullversion}\nAn $r\\times s$  $\\brho$-latin rectangle $L$ can be completed to an $n\\times n$ $\\brho$-latin square  if and only if\n there exists a fitting sequence $\\{(a_\\ell,b_\\ell)\\}_{\\ell=1}^k$ \nsuch that any of the  following conditions\n\\begin{align*}\n    |I|(n-s) &\\leq \\sum_{\\ell\\in P_r} \\min \\Big\\{a_\\ell+\\rho_\\ell-e_\\ell-n+r, \\mu_I(\\ell)\\Big\\}+ \\sum_{\\ell\\in \\bar P_r} \\min \\Big\\{a_\\ell, \\mu_I(\\ell)\\Big\\}  &\\forall I\\subseteq  [r],\\\\\n    \\sum_{\\ell\\in (K\\cap P_r)} (\\rho_\\ell-e_\\ell) &\\leq \\sum_{i\\in [r]} \\min \\Big\\{n-s, \\mu_K(i)\\Big\\}- \\sum_{\\ell\\in K} a_\\ell +|K\\cap P_r|(n-r) &\\forall K\\subseteq  [k],\\\\\n  |I|(n-s) &\\geq \\sum_{\\ell\\in P_r} \\Big( a_\\ell+\\rho_\\ell+\\mu_I(\\ell)\\dotdiv n\\Big)+\\sum_{\\ell\\in \\bar P_r} \\Big( a_\\ell\\dotdiv \\mu_{\\bar I}(\\ell)\\Big)  &\\forall I\\subseteq [r],\\\\    \n     \\sum_{\\ell\\in (K\\cap P_r)}(\\rho_\\ell-e_\\ell)    &\\geq \\sum_{i\\in [r]} \\Big (n-s\\dotdiv \\mu_{\\bar K}(i)\\Big) + |K\\cap P_r|(n-r) - \\sum_{\\ell\\in K} a_\\ell &\\forall K\\subseteq [k],\\\\\n    |I|(n-s) &\\leq \\sum_{\\ell\\in (K\\cap P_r)} (\\rho_\\ell-e_\\ell) +\\sum_{\\ell\\in K} a_\\ell+ \\mu_I(\\bar K) -|K\\cap P_r|(n-r)  &\\forall I\\subseteq [r], K\\subseteq [k], \\\\\n    \\sum_{\\ell\\in (K\\cap P_r)} (\\rho_\\ell-e_\\ell) &\\leq |I|(n-s)+|K\\cap P_r|(n-r) + \\mu_K(\\bar I)- \\sum_{\\ell\\in K} a_\\ell   &\\forall I\\subseteq  [r], K\\subseteq [k],\n\\end{align*}\ntogether with any of the following conditions hold.\n\\begin{align*}\n    |J|(n-r) &\\leq \\sum_{\\ell\\in P_s} \\min \\Big\\{b_\\ell+\\rho_\\ell-e_\\ell-n+s, \\mu_J(\\ell)\\Big\\} + \\sum_{\\ell\\in \\bar P_s} \\min \\Big\\{b_\\ell, \\mu_J(\\ell)\\Big\\}  &\\forall J\\subseteq  [s],\\\\\n    \\sum_{\\ell\\in (K\\cap P_s)} (\\rho_\\ell-e_\\ell) &\\leq \\sum_{j\\in [s]} \\min \\Big\\{n-r, \\mu_K(j)\\Big\\}- \\sum_{\\ell\\in K} b_\\ell +|K\\cap P_s|(n-s)  &\\forall K\\subseteq  [k],\\\\\n  |J|(n-r) &\\geq \\sum_{\\ell\\in [k]} \\Big( b_\\ell+\\rho_\\ell+\\mu_J(\\ell)\\dotdiv n\\Big) +\\sum_{\\ell\\in \\bar P_s} \\Big( b_\\ell\\dotdiv \\mu_{\\bar J}(\\ell)\\Big)  &\\forall J\\subseteq [s],\\\\    \n     \\sum_{\\ell\\in (K\\cap P_s)}(\\rho_\\ell-e_\\ell)    &\\geq \\sum_{j\\in [s]} \\Big (n-r\\dotdiv \\mu_{\\bar K}(j)\\Big) + |K\\cap P_s|(n-s) - \\sum_{\\ell\\in K} b_\\ell &\\forall K\\subseteq [k],\\\\\n    |J|(n-r) &\\leq \\sum_{\\ell\\in (K\\cap P_s)} (\\rho_\\ell-e_\\ell) + \\sum_{\\ell\\in K} b_\\ell+ \\mu_J(\\bar K)-|K\\cap P_s|(n-s)  &\\forall J\\subseteq [s], K\\subseteq [k], \\\\\n    \\sum_{\\ell\\in (K\\cap P_s)} (\\rho_\\ell-e_\\ell) &\\leq |J|(n-r)+|K\\cap P_s|(n-s) + \\mu_K(\\bar J) - \\sum_{\\ell\\in K} b_\\ell &\\forall J\\subseteq  [s], K\\subseteq [k].\n\\end{align*}\n\\end{theorem}\n\n\nFor a much simpler generalization of Ryser's Theorem, see Corollary \\ref{cor2}.\n\\section{Tools}\nFor $x,y\\in \\mathbb{R}$, $x\\approx y$ means $\\lfloor y \\rfloor \\leq x\\leq \\lceil y \\rceil$. For a real-valued function $f$ on a domain $D$ and  $S\\subseteq D$,  $f(S) :=\\sum_{x\\in S}f(x)$. \n\n\nAll graphs under consideration are loopless, but they may have parallel edges.  For a graph $G=(V,E)$, $u,v\\in V$ and $S,T\\subseteq V$ with $S\\cap T$, $\\dg_G(u)$, $\\mult_G(uv)$,  $\\mult_G(uS)$, and $\\mult_G(ST)$ denote the number of edges incident with $u$,  the number of edges whose endpoints are $u$ and $v$,  the number of edges between $u$ and $S$, and the number of edges between $S$ and $T$, respectively. If the edges of $G$ are colored with $k$ colors (the set of colors is always $[k]$), then $G(\\ell)$ is the color class $\\ell$ of $G$ for $\\ell\\in [k]$. A bigraph $G$ with bipartition $\\{X, Y \\}$ will be denoted by $G[X, Y ]$, and for $S\\subseteq X$, $\\bar S:=X\\backslash S$. \n\n\nLet $G$ be a graph whose edges are colored, and let $\\alpha\\in V(G)$. By {\\it splitting} $\\alpha$ into $\\alpha_1,\\dots,\\alpha_p$, we obtain a new graph $F$ whose vertex set is $\\left(V(G)\\backslash \\{\\alpha\\}\\right) \\cup \\{\\alpha_1,\\dots, \\alpha_p\\}$ so that each edge $\\alpha u$ in $G$ becomes $\\alpha_i u$ for some $i\\in [p]$ in $F$. Intuitively speaking, when we {\\it split}  a vertex $\\alpha$ into $\\alpha_1,\\dots,\\alpha_p$, we share the edges incident with $\\alpha$ among $\\alpha_1\\dots,\\alpha_p$.   In this manner, $F$ is a {\\it detachment} of $G$, and $G$ is an {\\it amalgamation} of $F$ obtained by {\\it identifying} $\\alpha_1,\\dots, \\alpha_p$ by $\\alpha$. The following detachment lemma will be crucial in the proof of our main result. \n\\begin{lemma} \n\\label{amalgambahrod} \\cite{MR2946077}\n  Let $G$ be a graph whose edges are colored with $k$  colors, and let $\\alpha,\\beta$ be two vertices of $G$. There exists a graph $F$ obtained by splitting $\\alpha$ and $\\beta$ into $\\alpha_1,\\dots,\\alpha_p$, and  $\\beta_1,\\dots,\\beta_q$, respectively, such that the following conditions hold.\n\\begin{enumerate}\n     \\item [\\textup{(i)}]  $\\dg_{F(\\ell)}(\\alpha_i)\\approx \\dg_{G(\\ell)}(\\alpha)\/p$  for  $i\\in [p],\\ell\\in [k]$;\n     \\item [\\textup{(ii)}]  $\\dg_{F(\\ell)}(\\beta_j)\\approx \\dg_{G(\\ell)}(\\beta)\/q$ for  $j\\in [q],\\ell\\in [k]$;\n     \\item [\\textup{(iii)}]  $\\mult_F(\\alpha_i u)\\approx \\mult_G(\\alpha u)\/p$ for  $i\\in[p],u\\in V(G)\\backslash \\{\\alpha,\\beta\\}$;\n     \\item [\\textup{(iv)}]  $\\mult_F(\\beta_j u)\\approx \\mult_G(\\beta u)\/q$ for  $j\\in[q],u\\in V(G)\\backslash \\{\\alpha,\\beta\\}$;\n     \\item [\\textup{(v)}]  $\\mult_F(\\alpha_i \\beta_j)\\approx \\mult_G(\\alpha\\beta)\/(pq)$ for  $i\\in[p],j\\in[q]$.\n\\end{enumerate}  \n \n \n\\end{lemma}\nFor a hypergraph analogue, we refer the reader to \\cite{MR2942724}. To give the reader an idea about the usefulness of this detachment lemma, let us show how to construct $\\brho$-latin squares.   Theorem \\ref{rholatconstthm} can be viewed as an immediate consequence of Theorem \\ref{rhohallthmgoldetal} (See  \\cite[Theorem 5.1]{MR3280683}). \n\\begin{theorem} \\label{rholatconstthm} For every  $n,k, \\brho:=(\\rho_1,\\dots,\\rho_k)$ with $1\\leq \\rho_1,\\dots,\\rho_k\\leq n\\leq k$ and $\\sum_{\\ell\\in [k]} \\rho_\\ell=n^2$,  there exists a  $\\brho$-latin square of order $n$.\n\\end{theorem}\n\\begin{proof}\nLet  $G[\\{\\alpha\\}, \\{\\beta\\}]$ be a $k$-edge-colored bigraph with $\\mult_G(\\alpha \\beta)=n^2$, $\\mult_{G(\\ell)}(\\alpha \\beta)=\\rho_\\ell$ for $\\ell\\in [k]$. \nApplying the detachment lemma with $p=q=n$ yields the complete bigraph $F\\cong K_{n,n}$ whose colored edges corresponds to symbols in the desired $\\brho$-latin square of order $n$. \n\\end{proof}\n Let $f,g$ be  integer functions on the vertex set of a graph $G$ such that $0\\leq g(x)\\leq f(x)$ for all $x$. A {\\it $(g,f)$-factor} is a spanning subgraph $F$ of $G$ with the property that $g(x)\\leq \\dg_F(x)\\leq f(x)$ for each $x$, and an {\\it $f$-factor} is  an $(f,f)$-factor.  We need the following result which is known as  Ore's Theorem. For far reaching generalizations of Ore's Theorem, we refer the reader to  Lov\\'{a}sz's seminal paper \\cite{MR325464}. \n\\begin{theorem}\\cite{MR83725} \\label{OresThm}\nThe bipgraph $G[X,Y]$ has an $f$-factor  if and only if $f(X)=f(Y)$ and either one of the following conditions hold.\n\\begin{align*}\n    f(A)&\\leq \\sum_{u\\in Y} \\min\\Big\\{f(u), \\mult_G(uA)\\Big\\} &\\forall A\\subseteq X,\\\\\n    f(A) &\\leq f(B) + \\mult_G(A\\bar B) &\\forall A\\subseteq X, B\\subseteq Y.\n\\end{align*}\n\\end{theorem} \n\\begin{remark} \\label{ordenewcond} \\textup{\nLet us fix $A\\subseteq X$. For $u\\in Y$, we have\n\\begin{align*}\n     \\kappa (u)&:=\\min\\Big\\{f(u), \\mult_G(uA)\\Big\\}+ \\Big (f(u)\\dotdiv \\mult_G(u  A)\\Big)\\\\\n     &=\n     \\begin{cases}\n        \\mult_G(uA)+ \\Big (f(u)- \\mult_G(u  A)\\Big) & \\mbox{if  } f(u)\\geq  \\mult_G(uA) \\\\\n        f(u)+0 & \\mbox{if  } f(u)<  \\mult_G(uA)\n     \\end{cases}\\\\\n     &=f(u).\n\\end{align*}\nSo, $\\sum_{u\\in Y} \\kappa(u)=f(Y)$, and the first condition in Theorem \\ref{OresThm} is equivalent to\n\\begin{align*}\n        f(A)&\\leq f(Y)-\\sum_{u\\in Y} \\Big (f(u)\\dotdiv \\mult_G(u  A)\\Big),\n\\end{align*}\nwhich is equivalent to \n\\begin{align*}\n             f(\\bar A)\\geq     \\sum_{u\\in Y} \\Big (f(u)\\dotdiv \\mult_G(u  A)\\Big).\n\\end{align*}\n}\\end{remark}\n\n\n\\section{{\\normalfont Hall's Theorem for }\\protect\\boldmath$\\rho${\\normalfont-latin Rectangles} }\nHall \\cite{MR13111} showed that any $r\\times n$ latin rectangle can be extended to an $n\\times n$ latin square. Here we give a   short proof of the main result of \\cite{MR3280683} which generalizes Hall's theorem to $\\brho$-latin rectangles. \n\n\n\n\\begin{theorem} \\label{rhohallthm}\nAn $r\\times n$  $\\brho$-latin rectangle $L$ can be completed to an $n\\times n$ $\\brho$-latin square  if and only if $\\rho_\\ell - e_\\ell\\leq n-r$ for  $\\ell\\in [k]$, and any of the following conditions hold.\n\\begin{align*}\n    |J|(n-r) &\\leq \\sum_{\\ell\\in [k]} \\min \\Big\\{\\rho_\\ell-e_\\ell, \\mu_J(\\ell)\\Big\\}  &\\forall J\\subseteq  [n],\\\\\n    \\sum_{\\ell\\in K} (\\rho_\\ell-e_\\ell) &\\leq \\sum_{j\\in [n]} \\min \\Big\\{n-r, \\mu_K(j)\\Big\\}  &\\forall K\\subseteq  [k],\\\\%\\label{orenec1equiv}\\\\\n  |J|(n-r) &\\geq \\sum_{\\ell\\in [k]} \\Big( \\rho_\\ell-e_\\ell\\dotdiv \\mu_{\\bar J}(\\ell)\\Big)  &\\forall J\\subseteq [n],\\\\    \n     \\sum_{\\ell\\in K}(\\rho_\\ell-e_\\ell)    &\\geq \\sum_{j\\in [n]} \\Big (n-r\\dotdiv \\mu_{\\bar K}(j)\\Big)  &\\forall K\\subseteq [k],\\\\%\\label{anothercondequi}\\\\\n    |J|(n-r) &\\leq \\sum_{\\ell\\in K} (\\rho_\\ell-e_\\ell) + \\mu_J(\\bar K)  &\\forall J\\subseteq [n], K\\subseteq [k], \\\\%\\label{orecond2}\\\\\n    \\sum_{\\ell\\in K} (\\rho_\\ell-e_\\ell) &\\leq |J|(n-r) + \\mu_K(\\bar J)  &\\forall J\\subseteq  [n], K\\subseteq [k]\n\\end{align*}\n\\end{theorem}\n \\begin{proof} Recall that the set of symbols used is $[k]$. The  necessity of \n\\begin{align} \\label{ryscond1}\n    \\rho_\\ell - e_\\ell&\\leq n-r\\quad \\forall \\ell\\in [k]\n\\end{align}\nis immediate from ~\\eqref{mainryserineq}.  The necessity of the remaining conditions will be evident at the end of the proof. For now, let us assume that \\eqref{ryscond1} is satisfied.  Let $F[\\tilde X, Y]$ be the complete bigraph $K_{n,n}$ where $\\tilde X:=\\{x_1,\\dots,x_n\\}, Y:=\\{y_1,\\dots, y_n\\}$, and let $X=\\{x_1,\\dots, x_r\\}$. The edge $x_i y_{j}$ of $F$ is colored $\\ell$ if $L_{ij}=\\ell$ for $i\\in[r],j\\in[n]$. We have $\\dg_{F(\\ell)}(u)\\leq 1$ for $u\\in X\\cup Y,$    $e_\\ell=|E(F(\\ell))|\\leq \\rho_\\ell$ for $\\ell\\in [k]$, and $\\sum_{\\ell\\in [k]}e_\\ell=rn$.  \nLet  $G$ be the bigraph obtained from $F$ by amalgamating $x_{r+1},\\dots, x_n$ into a single vertex $\\alpha$, so $\\mult_G(\\alpha y_j)=n-r$ for $j\\in [n]$. Let  $\\Gamma [Y, [k]]$ be the simple bigraph whose edge set is $$\\{u \\ell\\ \\big|\\ u\\in Y, \\ell\\in [k],  \\dg_{F(\\ell)}(u)=0\\}.$$ \nFor    $j\\in [n]$,  $\\sum_{\\ell\\in [k]}\\dg_{F(\\ell)}(y_j)=r$, and for $\\ell\\in [k]$, there are $e_\\ell$ edges in $F(\\ell)$. Therefore, \n\\begin{align*\n\\begin{cases}\n\\dg_\\Gamma(y_j)=k-r & \\mbox{if  } j\\in [n], \\\\\n\\dg_{\\Gamma}(\\ell)=n-e_\\ell & \\mbox{if  } \\ell\\in [k].\n\\end{cases}\n\\end{align*}\nObserve that $L$ can be completed if and only if the uncolored edges of $F$ can be colored so that\n\\begin{align}\\label{colorcon1s}\n\\forall \\ell\\in [k], \\quad \\quad \\begin{cases}\n\\dg_{F(\\ell)}(u)\\leq 1 & \\mbox{if  } u\\in \\tilde X\\cup Y, \\\\\n|E(F(\\ell))|=\\rho_\\ell.\n\\end{cases}\n\\end{align}\nWe show that the coloring of  $F$ can be completed such that \\eqref{colorcon1s} holds if and only if the coloring of $G$ can be completed so that \n\\begin{align}\\label{colorcon2s}\n\\forall \\ell\\in [k],\\quad \\quad  \\begin{cases}\n\\dg_{G(\\ell)}(u) \\leq 1 & \\mbox{if  } u \\in X \\cup Y, \\\\\n\\dg_{G(\\ell)}(\\alpha) \\leq n-r, \\\\\n|E(G(\\ell))|= \\rho_\\ell.\n\\end{cases}\n\\end{align}\nTo see this, first assume that the coloring of  $F$ can be completed such that \\eqref{colorcon1s} holds. Identifying all the vertices in $\\tilde X\\backslash X$ of $F$ by $\\alpha$ we will get the graph $G$ satisfying \\eqref{colorcon2s}. Conversely, suppose that  we have a  coloring of $G$\nsuch that \\eqref{colorcon2s} holds. Applying   Lemma \\ref{amalgambahrod} to $G$, we get a graph $F'$ obtained by splitting $\\alpha$  into $\\alpha_1,\\dots,\\alpha_{n-r}$ such that the following hold.\n\\begin{enumerate}\n    \\item  [(i)] $\\dg_{F'(\\ell)}(\\alpha_i)\\approx\\dg_{G(\\ell)}(\\alpha)\/(n-r)\\leq 1$  for  $i\\in [n-r],\\ell\\in [k]$;\n    \\item  [(ii)] $\\mult_{F'}(\\alpha_i u)= \\mult_G(\\alpha u)\/(n-r)=1$ for  $i\\in[n-r],u\\in Y$.\n\\end{enumerate}\nSince $F'\\cong F$ and the coloring of $F'$ satisfies \\eqref{colorcon1s}, we are done. \n\nNow, we show that the coloring of  $G$ can be completed such that  \\eqref{colorcon2s} is satisfied if and only if there exists  a subgraph $\\Theta$ of $\\Gamma$  with $n(k-r)$ edges so that \n\\begin{align}\\label{colorcon3s}\n\\begin{cases}\n\\dg_\\Theta (y_j)=n-r  & \\mbox{if  }  j\\in [n], \\\\\n\\dg_\\Theta (\\ell)=\\rho_\\ell-e_\\ell  & \\mbox{if  } \\ell \\in [k].\n\\end{cases}\n\\end{align}\nTo prove this, suppose that the coloring of  $G$ can be completed such that \\eqref{colorcon2s} holds. Let  $\\Theta [ Y, [k]]$ be the bigraph whose edge set is \n$$\\{u\\ell\\ \\big|\\ u\\in  Y, \\ell\\in [k], \\mbox{ and  }\\alpha u\\in E(G(\\ell))   \\}.$$ \nObserve that $\\Theta \\subseteq \\Gamma$. Moreover, $\\dg_\\Theta(y_j)=\\mult_G(\\alpha y_j)=n-r$ for $j\\in [n]$, and $\\dg_{\\Theta}(\\ell)= |E(G(\\ell))|-e_\\ell=\\rho_\\ell-e_\\ell$ for $\\ell\\in [k]$, and so \\eqref{colorcon3s} holds.  Conversely, suppose that   $\\Theta\\subseteq \\Gamma$ satisfying \\eqref{colorcon3s} exists. For each $\\ell\\in [k]$,   if $\\ell y_j \\in E(\\Theta)$ for some $j\\in [n]$, we color an $\\alpha y_j$-edge in $G$ with $\\ell$. Since $\\dg_\\Theta(y_j)=n-r$ for $j\\in [n]$, all the edges between $\\alpha$ and $Y$ can be colored this way. Since $\\Theta$ is simple, $d_{G(\\ell)}(u)\\leq 1$ for $\\ell\\in [k]$ and $u\\in  Y$.  It is also clear that $|E(G(\\ell))|= \\rho_\\ell-e_\\ell+e_\\ell=\\rho_\\ell$, and by ~\\eqref{ryscond1} \n$\\dg_{G(\\ell)}(\\alpha)=\\dg_\\Theta(\\ell)=\\rho_\\ell-e_\\ell \\leq n-r$ for $\\ell\\in [k]$. \n\n\nLet \n\\begin{align*}\n    \\begin{cases}\n        f: V(\\Gamma)\\rightarrow \\mathbb{N}\\cup \\{0\\},\\\\\n        f(y_j)=n-r & \\mbox{for  } j\\in [n],\\\\\n        f(\\ell)=\\rho_\\ell-e_\\ell & \\mbox{for  }\\ell\\in [k].      \n    \\end{cases}\n\\end{align*}\n Since $f(Y)= f([k])$, by Ore's Theorem and Remark \\ref{ordenewcond}, $\\Gamma$ has an $f$-factor (and so $\\Theta\\subseteq \\Gamma$ satisfying \\eqref{colorcon3s} exists)  if and only if any of the following  conditions holds. \n\\begin{align*}\n    f(J)&\\leq \\sum_{\\ell\\in [k]} \\min\\Big\\{f(\\ell), \\mult_\\Gamma(\\ell J)\\Big\\} &\\forall J\\subseteq Y,\\\\\n    f(K)&\\leq \\sum_{u\\in Y} \\min\\Big\\{f(u), \\mult_\\Gamma(uK)\\Big\\} &\\forall K\\subseteq [k],\\\\\n    f(J)&\\geq \\sum_{\\ell\\in [k]} \\Big (f(\\ell)\\dotdiv \\mult_\\Gamma(\\ell \\bar J)\\Big)  &\\forall J\\subseteq Y,\\\\\n    f(K)&\\geq \\sum_{u\\in Y} \\Big (f(u)\\dotdiv \\mult_\\Gamma(u \\bar K)\\Big)  &\\forall K\\subseteq [k],\\\\\n     f(J) &\\leq f(K) + \\mult_\\Gamma(J\\bar K) &\\forall K\\subseteq [k], J\\subseteq Y,\\\\  f(K) &\\leq f(J) + \\mult_\\Gamma(K\\bar J) &\\forall K\\subseteq [k], J\\subseteq Y.\n\\end{align*}\nBut these conditions are equivalent to  those of Theorem \\ref{rhohallthm}. \n\\end{proof}\n\n\\section{{\\normalfont Ryser's Theorem for }\\protect\\boldmath$\\rho${\\normalfont-latin Rectangles} } \\label{rhorythmprfsec}\nSeventy years ago, Ryser \\cite{MR42361} showed that any $r\\times s$  latin rectangle can be extended to an $n\\times n$ latin square if and only if  $e_\\ell\\geq r+s-n$ for $\\ell\\in [n]$. We provide a generalization of this result to $\\brho$-latin rectangles by proving the following Theorem \\ref{rhoryserthmfullversion}. For a simpler generalization of Ryser's Theorem, see Corollary \\ref{cor2}.\n\n\n\\noindent \\textit{Proof of Theorem \\ref{rhoryserthmfullversion}.}\nRecall that the set of symbols used is $[k]$.  The existence of a fitting sequence $\\{(a_\\ell,b_\\ell)\\}_{\\ell=1}^k$  implies ~\\eqref{mainryserineq}. To see this, observe that by Definition \\ref{fitdef},\n$$\n0\\leq a_\\ell+b_\\ell\\leq 2n- r-s+ e_\\ell-\\rho_\\ell \\quad \\mbox {for } \\ell\\in P_{\\min\\{r,s\\}}. \n$$\nMoreover, using the definition of $P_{\\min\\{r,s\\}}$, we have\n$$\\rho_\\ell-e_\\ell \\leq n- \\min\\{r,s\\}\\leq 2n-r-s \\quad \\mbox{for } \\ell\\in \\bar P_{\\min\\{r,s\\}}.$$ \nThe necessity of all  conditions will be evident at the end of the proof. For now, let us assume that ~\\eqref{mainryserineq} --- whose necessity was established in the introduction --- holds. \nLet $F[\\tilde X,\\tilde Y]$ be the complete bigraph $K_{n,n}$ where $\\tilde X:=\\{x_1,\\dots,x_n\\}, \\tilde Y:=\\{y_1,\\dots, y_n\\}$, and let $X=\\{x_1,\\dots, x_r\\},Y=\\{y_1,\\dots,y_s\\}$.  The edge $x_i y_j$ of $F$ is colored $\\ell$ if $L_{ij}=\\ell$ for $i\\in[r],j\\in[s]$. We have $\\dg_{F(\\ell)}(u)\\leq 1$ for $u\\in X\\cup Y, \\ell\\in [k]$,  $e_\\ell=|E(F(\\ell))|\\leq \\rho_\\ell$ for $\\ell\\in [k]$, and $\\sum_{\\ell\\in [k]}e_\\ell=rs$ (See Figure \\ref{fig1rhoryser} (b)).  \nLet  $G$ be the bigraph obtained from $F$ by amalgamating $x_{r+1},\\dots, x_n$ into a single vertex $\\alpha$, and amalgamating $x_{s+1},\\dots, y_n$ into a single vertex $\\beta$, so $\\mult_G(\\alpha y_j)=n-r$ for $j\\in [s]$, $\\mult_G(\\beta x_i)=n-s$ for $ i\\in [r]$, and $\\mult_G(\\alpha \\beta)=(n-r)(n-s)$ (See Figure \\ref{fig1rhoryser} (c)).  Let  $\\Gamma [X\\cup Y, [k]]$ be the simple bigraph whose edge set is \n$$\\{u\\ell\\ \\big |\\ u\\in X\\cup Y, \\ell\\in [k],  \\dg_{F(\\ell)}(u)=0\\}.$$ \nFor  $i\\in [r]$, since $\\sum_{\\ell\\in [k]}\\dg_{F(\\ell)}(x_i)=s$, we have $\\dg_\\Gamma(x_i)=k-s$. Similarly, for  $j\\in [s]$,  $\\sum_{\\ell\\in [k]}\n\\dg_{F(\\ell)}(y_j)=r$, and so we have $\\dg_\\Gamma(y_j)=k-r$. For $\\ell\\in [k]$, there are $e_\\ell$ edges in $F(\\ell)$, and so  $\\mult_{\\Gamma}(\\ell X)=r-e_\\ell$ and $\\mult_{\\Gamma}(\\ell Y)=s-e_\\ell$ (Recall that by the latin property of $L$, $e_\\ell \\leq \\min \\{r,s\\}$). Therefore, $\\dg_\\Gamma(\\ell)=r+s-2e_\\ell$ for $\\ell\\in [k]$. In short, $\\Gamma$ meets the following properties (See Figure \\ref{fig1rhoryser} (d)).\n\\begin{align*\n\\begin{cases}\n\\dg_\\Gamma(x_i)=k-s & \\mbox{if  }  i\\in [r], \\\\\n\\dg_\\Gamma(y_j)=k-r & \\mbox{if  }  j\\in [s], \\\\\n\\mult_{\\Gamma}(\\ell X)=r-e_\\ell  & \\mbox{if  } \\ell \\in [k],\\\\\n\\mult_{\\Gamma}(\\ell Y)=s-e_\\ell  & \\mbox{if  } \\ell \\in [k],\\\\\n\\dg_\\Gamma(\\ell)=r+s-2e_\\ell  & \\mbox{if  } \\ell \\in [k].\n\\end{cases}\n\\end{align*}\n\\begin{figure}[p] \n  \\begin{adjustbox}{addcode={\\begin{minipage}{\\width}}{\\caption{\n      A $\\rho$-latin rectangle and three associated auxiliary bigraphs\n      }\\end{minipage}},rotate=90,center}  \n \\includegraphics[scale=1.2]{fig1} \\end{adjustbox}\n \\label{fig1rhoryser} \n\\end{figure}\n\n\n\n\n\n\n\nObserve that $L$ can be completed if and only if the uncolored edges of  $F$ can be colored so that\n\\begin{align}\\label{colorcon1}\n\\forall \\ell\\in [k],\\quad \\quad \\begin{cases}\n\\dg_{F(\\ell)}(u)\\leq 1 & \\mbox{if  }  u\\in \\tilde X \\cup \\tilde Y, \\\\\n|E(F(\\ell))|=\\rho_\\ell.\n\\end{cases}\n\\end{align}\n\nWe show that the coloring of  $F$ can be completed such that  \\eqref{colorcon1} is satisfied if and only if the coloring of $G$ can be completed such that,\n\\begin{align}\\label{colorcon2}\n\\forall \\ell\\in [k],\\quad \\quad  \\begin{cases}\n\\dg_{G(\\ell)}(u) \\leq 1 & \\mbox{if  } u \\in X \\cup Y, \\\\\n\\dg_{G(\\ell)}(\\alpha) \\leq n-r, \\\\\n\\dg_{G(\\ell)}(\\beta) \\leq n-s, \\\\\n|E(G(\\ell))|= \\rho_\\ell.\n\\end{cases}\n\\end{align}\nTo see this, first assume that the coloring of  $F$ can be completed so that  \\eqref{colorcon1} holds. Identifying all the  vertices in $\\tilde X\\backslash X$   by $\\alpha$ and all the vertices in $\\tilde Y\\backslash Y$   by $\\beta$, we will get the graph $G$ satisfying \\eqref{colorcon2}. Conversely, suppose that  we have a  coloring of $G$\nsuch that \\eqref{colorcon2} holds. Applying the detachment lemma to $G$, we get a graph $F'$ obtained by splitting $\\alpha$ and $\\beta$ into $\\alpha_1,\\dots,\\alpha_{n-r}$, and  $\\beta_1\\dots,\\beta_{n-s}$, respectively, such that the following hold.\n\\begin{enumerate}\n\\item [(i)] $\\dg_{F'(\\ell)}(\\alpha_i)\\approx\\dg_{G(\\ell)}(\\alpha)\/(n-r)\\leq 1$  for  $i\\in [n-r],\\ell\\in [k]$;\n\\item [(ii)] $\\dg_{F'(\\ell)}(\\beta_j)\\approx \\dg_{G(\\ell)}(\\beta)\/(n-s)\\leq 1$ for  $j\\in [n-s],\\ell\\in [k]$;\n\\item [(iii)] $\\mult_{F'}(\\alpha_i u)= \\mult_G(\\alpha u)\/(n-r)=1$ for  $i\\in[n-r],u\\in Y$;\n\\item [(iv)] $\\mult_{F'}(\\beta_j u)= \\mult_G(\\beta u)\/(n-s)=1$ for  $j\\in[n-s],u\\in X$;\n\\item [(v)] $\\mult_{F'}(\\alpha_i \\beta_j)= \\mult_G(\\alpha\\beta)\/\\big((n-r)(n-s)\\big)=1$ for  $i\\in[n-r],j\\in[n-s]$.\n\\end{enumerate}\n Since $F'\\cong F$ and the coloring of $F'$ satisfies \\eqref{colorcon1}, we are done. \n\nFor $\\ell\\in [k]$, since $\\rho_\\ell\\leq n$, we have that $\\rho_\\ell-e_\\ell-n+r\\leq r-e_\\ell$ and $\\rho_\\ell-e_\\ell-n+s\\leq s-e_\\ell$. We show that the coloring of  $G$ can be completed such that  \\eqref{colorcon2} is satisfied if and only if there exists  a subgraph $\\Theta$ of $\\Gamma$  with $n(r+s)-2rs$ edges so that \n\\begin{align}\\label{colorcon3}\n\\begin{cases}\n\\dg_\\Theta (x_i)=n-s & \\mbox{if  }  i\\in [r], \\\\\n\\dg_\\Theta (y_j)=n-r  & \\mbox{if  }  j\\in [s], \\\\\n\\dg_\\Theta (\\ell)\\leq \\rho_\\ell-e_\\ell  & \\mbox{if  } \\ell \\in [k],\\\\\n\\mult_\\Theta (\\ell X)\\geq \\rho_\\ell-e_\\ell-n+r  & \\mbox{if  } \\ell \\in [k],\\\\\n\\mult_\\Theta (\\ell Y)\\geq \\rho_\\ell-e_\\ell-n+s  & \\mbox{if  } \\ell \\in [k].\n\\end{cases}\n\\end{align}\n\n\nTo prove this, suppose that the coloring $G$ can completed such that  \\eqref{colorcon2} is satisfied. Let  $\\Theta [X\\cup Y, [k]]$ be the bigraph whose edge set is \n$$\\{u \\ell\\ \\big |\\ u\\in X\\cup Y, \\ell\\in [k], \\mbox{ and either }\\alpha u\\in E(G(\\ell)) \\mbox{ or } \\beta u \\in E(G(\\ell))  \\}.$$ \nIt is clear that  $\\Theta \\subseteq \\Gamma$.   \nMoreover, \n $\\dg_\\Theta(x_i)=\\mult_G(\\beta x_i)=n-s$ for $i\\in [r]$, $\\dg_\\Theta(y_j)=\\mult_G(\\alpha y_j)=n-r$ for $j\\in [s]$, and $\\dg_{\\Theta}(\\ell)\\leq |E(G(\\ell))|-e_\\ell=\\rho_\\ell-e_\\ell$ for $\\ell\\in [k]$. Notice that \n $$\\rho_\\ell=|E(G(\\ell))|=e_\\ell+\\mult_{G(\\ell)}(\\alpha \\beta) +\\mult_{G(\\ell)} (\\beta X) + \\mult_{G(\\ell)} (\\alpha Y)\\quad  \\mbox {for }\\ell\\in [k].$$\nThus,  for $\\ell\\in [k]$ we have \n \\begin{align*}\n\\mult_\\Theta (\\ell X) &=\\mult_{G(\\ell)} (\\beta X)   \\\\\n& = \\rho_\\ell-e_\\ell-\\mult_{G(\\ell)}(\\alpha \\beta) - \\mult_{G(\\ell)} (\\alpha Y)\\\\\n &=  \\rho_\\ell-e_\\ell-\\dg_{G(\\ell)}(\\alpha) \\\\\n& \\geq   \\rho_\\ell-e_\\ell-n+r,\\\\\n\\mult_\\Theta (\\ell Y)& =\\mult_{G(\\ell)} (\\alpha Y)   \\\\\n &=  \\rho_\\ell-e_\\ell-\\mult_{G(\\ell)}(\\alpha \\beta) - \\mult_{G(\\ell)} (\\beta X)\\\\\n& = \\rho_\\ell-e_\\ell-\\dg_{G(\\ell)}(\\beta) \\\\\n& \\geq   \\rho_\\ell-e_\\ell-n+s,\n\\end{align*}\nand so \\eqref{colorcon3} holds. Observe  that $\\rho_\\ell-e_\\ell  \\geq \\dg_\\Theta (\\ell)= \\mult_\\Theta (\\ell X)+ \\mult_\\Theta (\\ell Y)\\geq  \\rho_\\ell-e_\\ell-n+r +  \\rho_\\ell-e_\\ell-n+s$; this is  consistent with   ~\\eqref{mainryserineq}. \n\n\n\nConversely, suppose that   $\\Theta\\subseteq \\Gamma$ satisfying \\eqref{colorcon3} exists. For each $\\ell\\in [k]$, if $\\ell x_i \\in E(\\Theta)$ for some $i\\in [r]$, we color a $\\beta x_i$-edge in $G$ with $\\ell$, and  if $\\ell y_j \\in E(\\Theta)$ for some $j\\in [s]$, we color an $\\alpha y_j$-edge in $G$ with $\\ell$. Since $\\dg_\\Theta(x_i)=n-s$ for $i\\in [r]$, and $\\dg_\\Theta(y_j)=n-r$ for $j\\in [s]$, all the edges between $\\beta$ and $X$, and all the edges between $\\alpha$ and $Y$ can be colored this way. Since $\\Theta$ is simple, $d_{G(\\ell)}(u)\\leq 1$ for $\\ell\\in [k]$ and $u\\in X\\cup Y$.  Then, we color the $\\alpha \\beta$-edges so that\n$$\\mult_{G(\\ell)}(\\alpha \\beta) = \\rho_\\ell-e_\\ell-\\dg_\\Theta (\\ell)\\quad  \\mbox {for }\\ell\\in [k].$$\nThis can be done, because $ \\rho_\\ell-e_\\ell-\\dg_\\Theta (\\ell)\\geq 0$ for $\\ell\\in [k]$, and \n\\begin{align*}\n   \\sum_{\\ell\\in [k]}\\mult_{G(\\ell)}(\\alpha \\beta)&=n^2-rs-n(r+s)+2rs\\\\\n   &=(n-r)(n-s)\\\\&=\\mult_G(\\alpha \\beta).\n\\end{align*}\nFor $\\ell\\in [k]$, we have\n  \\begin{align*}\n\\dg_{G(\\ell)} (\\alpha) &=\\mult_{G(\\ell)} (\\alpha Y)+ \\mult_{G(\\ell)} (\\alpha \\beta)  \\\\\n &=  \\mult_\\Theta (\\ell Y)+\\rho_\\ell-e_\\ell-\\dg_\\Theta (\\ell)\\\\\n &=  \\rho_\\ell-e_\\ell-\\mult_\\Theta (\\ell X)\\\\\n& \\leq  n-r,\\\\\n\\dg_{G(\\ell)} (\\beta) &=\\mult_{G(\\ell)} (\\beta X)+ \\mult_{G(\\ell)} (\\alpha \\beta)  \\\\\n& =  \\mult_\\Theta (\\ell X)+\\rho_\\ell-e_\\ell-\\dg_\\Theta (\\ell)\\\\\n& =  \\rho_\\ell-e_\\ell-\\mult_\\Theta (\\ell Y)\\\\\n& \\leq  n-s,\n\\end{align*}\nand so \\eqref{colorcon2} is satisfied. \n\n\nLet $\\Gamma_1, \\Gamma_2$ be the subgraphs of $\\Gamma$ induced by $X\\cup [k]$, and  $Y\\cup [k]$, respectively. The existence of $\\Theta\\subseteq \\Gamma$ satisfying \\eqref{colorcon3} is equivalent to the existence of $\\Theta_1\\subseteq \\Gamma_1, \\Theta_2\\subseteq \\Gamma_2$ satisfying the following condition.\n\\begin{align*\n\\begin{cases}\n\\dg_{\\Theta_1} (x_i)=n-s & \\mbox{if  }  i\\in [r], \\\\\n\\dg_{\\Theta_1} (\\ell)\\geq \\rho_\\ell-e_\\ell-n+r  & \\mbox{if  } \\ell \\in [k],\\\\\n\\dg_{\\Theta_2} (y_j)=n-r  & \\mbox{if  }  j\\in [s], \\\\\n\\dg_{\\Theta_2} (\\ell)\\geq \\rho_\\ell-e_\\ell-n+s  & \\mbox{if  } \\ell \\in [k],\\\\\n   \\dg_{\\Theta_1} (\\ell)+\\dg_{\\Theta_2} (\\ell)\\leq \\rho_\\ell-e_\\ell   &  \\mbox{if  } \\ell \\in [k].\n\\end{cases}\n\\end{align*}\nThe three inequalities of  this  condition are equivalent to the existence of a sequence $\\{(a_\\ell,b_\\ell)\\}_{\\ell=1}^k$ with $a_\\ell,b_\\ell\\in \\mathbb{N} \\cup \\{0\\}$ for $\\ell\\in [k]$ such that the following three properties are satisfied (Recall that $P_{\\min \\{r,s\\}}\\subseteq P_{\\max \\{r,s\\}}$).\n\n\\begin{align*}\n\\dg_{\\Theta_1} (\\ell)&=\\begin{cases}\na_\\ell+\\rho_\\ell-e_\\ell-n+r  & \\mbox{if  } \\ell \\in P_r,\\\\\na_\\ell  & \\mbox{if  } \\ell \\in \\bar P_r,\n\\end{cases}\\\\\n\\dg_{\\Theta_2} (\\ell)&=\\begin{cases}\nb_\\ell+\\rho_\\ell-e_\\ell-n+s  & \\mbox{if  } \\ell \\in P_s,\\\\\nb_\\ell  & \\mbox{if  } \\ell \\in \\bar P_s,\n\\end{cases}\\\\\n\\rho_\\ell-e_\\ell&\\geq\n\\begin{cases}\na_\\ell+b_\\ell+2\\rho_\\ell-2e_\\ell-2n+r+s  & \\mbox{if  } \\ell \\in P_{\\min \\{r,s\\}},\\\\\na_\\ell+b_\\ell+\\rho_\\ell-e_\\ell-n+\\max \\{r,s\\}  & \\mbox{if  } \\ell \\in P_{\\max \\{r,s\\}}\\backslash P_{\\min \\{r,s\\}},\\\\\na_\\ell+b_\\ell  & \\mbox{if  } \\ell \\in \\bar P_{\\max \\{r,s\\}},\n\\end{cases}\n\\end{align*}\nwhere the last property simplifies to the following.\n\\begin{align} \\label{abseqineqpro}\n        a_\\ell+b_\\ell\\leq \\begin{cases}\n            2n- r-s+ e_\\ell-\\rho_\\ell& \\mbox {if  } \\ell \\in P_{\\min \\{r,s\\}},\\\\\n            n- \\max \\{r,s\\} & \\mbox {if  } \\ell\\in \\ell \\in P_{\\max \\{r,s\\}}\\backslash P_{\\min \\{r,s\\}},\\\\\n            \\rho_\\ell -e_\\ell & \\mbox {if  } \\ell \\in \\bar P_{\\max \\{r,s\\}}.\n                \\end{cases}\n\\end{align} \nWe define $f_1,f_2$ as follows. \n\\begin{align*}\n    \\begin{cases}\n        f_1: V(\\Gamma_1)\\rightarrow \\mathbb{N}\\cup \\{0\\},\\\\\n        f_1(x_i)=n-s & \\mbox{if  } i\\in [r],\\\\\n        f_1(\\ell)=a_\\ell+\\rho_\\ell-e_\\ell-n+r & \\mbox{if  }\\ell\\in P_r, \\\\      \n        f_1(\\ell)=a_\\ell & \\mbox{if  }\\ell\\in \\bar P_r,       \n    \\end{cases}\n\\end{align*}\n\\begin{align*}\n    \\begin{cases}\n        f_2:  V(\\Gamma_2)\\rightarrow \\mathbb{N}\\cup \\{0\\},\\\\\n        f_2(y_j)=n-r & \\mbox{for  } i\\in [s],\\\\\n        f_2(\\ell)=b_\\ell+\\rho_\\ell-e_\\ell-n+s & \\mbox{for  }\\ell\\in P_s,  \\\\ \n        f_2(\\ell)=b_\\ell & \\mbox{for  }\\ell\\in \\bar P_s.  \n    \\end{cases}\n\\end{align*}\n\n\n\nBy the preceding paragraph,  $\\Theta\\subseteq \\Gamma$ satisfying \\eqref{colorcon3} exists if and only if there exists a sequence $\\{(a_\\ell,b_\\ell)\\}_{\\ell=1}^k$  satisfying \\eqref{abseqineqpro} such that  $\\Gamma_1$ and $\\Gamma_2$ have an $f_1$-factor and $f_2$-factor, respectively. \nObserve that $f_1(X)=f_1([k])$ if and only if\n\\begin{align*}\n    r(n-s)=\\sum_{\\ell\\in P_r} (a_\\ell+\\rho_\\ell-e_\\ell-n+r) + \\sum_{\\ell\\in \\bar P_r} a_\\ell,\n\\end{align*}\nor equivalently,  \n\\begin{align} \\label{abseqsum1}\n     \\sum_{\\ell\\in [k]} a_\\ell=r(n-s)+|P_r|(n-r)-\\sum_{\\ell\\in P_r} (\\rho_\\ell-e_\\ell).\n\\end{align}\nSimilarly, $f_2(Y)=f_2([k])$ if and only if\n\\begin{align*}\n    s(n-r)=\\sum_{\\ell\\in P_s} (b_\\ell+\\rho_\\ell-e_\\ell-n+s) + \\sum_{\\ell\\in \\bar P_s} b_\\ell,\n\\end{align*}\nor equivalently,   \n\\begin{align}\\label{abseqsum2}\n     \\sum_{\\ell\\in [k]} b_\\ell=s(n-r)+|P_s|(n-s)-\\sum_{\\ell\\in P_s} (\\rho_\\ell-e_\\ell).\n\\end{align}\n\n\n\nBy Ore's Theorem, $\\Gamma_1$ has an $f_1$-factor  if and only if \\eqref{abseqsum1} together with any  of the following conditions hold.\n\\begin{align*}\n    f_1(I)&\\leq \\sum_{\\ell\\in [k]} \\min\\Big\\{f_1(\\ell), \\mult_{\\Gamma_1}(\\ell I)\\Big\\} &\\forall I\\subseteq X,\\\\\n    f_1(K)&\\leq \\sum_{u\\in X} \\min\\Big\\{f_1(u), \\mult_{\\Gamma_1}(uK)\\Big\\} &\\forall K\\subseteq [k],\\\\\n    f_1(I)&\\geq \\sum_{\\ell\\in [k]} \\Big (f_1(\\ell)\\dotdiv \\mult_{\\Gamma_1}(\\ell \\bar I)\\Big)  &\\forall I\\subseteq X,\\\\\n    f_1(K)&\\geq \\sum_{u\\in X} \\Big (f_1(u)\\dotdiv \\mult_{\\Gamma_1}(u \\bar K)\\Big)  &\\forall K\\subseteq [k],\\\\\n     f_1(I) &\\leq f_1(K) + \\mult_{\\Gamma_1}(I\\bar K) &\\forall K\\subseteq [k], I\\subseteq X,\\\\\n    f_1(K) &\\leq f_1(I) + \\mult_{\\Gamma_1}(K\\bar I) &\\forall K\\subseteq [k], I\\subseteq X,\n\\end{align*} \nSimilarly,  $\\Gamma_2$ has an $f_2$-factor  if and only if \\eqref{abseqsum2}  and any  of the following conditions hold.\n\\begin{align*}\n    f_2(J)&\\leq \\sum_{\\ell\\in [k]} \\min\\Big\\{f_2(\\ell), \\mult_{\\Gamma_2}(\\ell J)\\Big\\} &\\forall J\\subseteq Y,\\\\\n    f_2(K)&\\leq \\sum_{u\\in Y} \\min\\Big\\{f_2(u), \\mult_{\\Gamma_2}(uK)\\Big\\} &\\forall K\\subseteq [k],\\\\\n    f_2(J)&\\geq \\sum_{\\ell\\in [k]} \\Big (f_2(\\ell)\\dotdiv \\mult_{\\Gamma_2}(\\ell \\bar J)\\Big)  &\\forall J\\subseteq Y,\\\\\n    f_2(K)&\\geq \\sum_{u\\in Y} \\Big (f_2(u)\\dotdiv \\mult_{\\Gamma_2}(u \\bar K)\\Big)  &\\forall K\\subseteq [k],\\\\\n     f_2(J) &\\leq f_2(K) + \\mult_{\\Gamma_2}(J\\bar K) &\\forall K\\subseteq [k], J\\subseteq Y,\\\\\n    f_2(K) &\\leq f_2(J) + \\mult_{\\Gamma_2}(K\\bar J) &\\forall K\\subseteq [k], J\\subseteq Y.\n\\end{align*} \nThese conditions are equivalent to those of Theorem \\ref{rhoryserthmfullversion}.\n\\qed\n\\begin{remark} \\textup{\n\\begin{itemize}\n    \\item [(a)] Let us verify that Theorem \\ref{rhoryserthmfullversion} implies Theorem \\ref{rhohallthm}.  In Theorem \\ref{rhoryserthmfullversion}, let $s=n$. Recall that the existence of the fitting sequence $\\{(a_\\ell,b_\\ell)\\}_{\\ell=1}^k$ implies  that $\\rho_\\ell-e_\\ell \\leq n-r$ for $\\ell\\in [k]$. Therefore,  $P_r=\\emptyset, P_n=\\{\\ell\\in [k] \\ |\\ \\rho_\\ell>e_\\ell\\}$. We have\n    \\begin{align*}\n    \\begin{cases}\n       \\displaystyle\\sum_{\\ell\\in [k]} a_\\ell=0,\\\\ \\\\\n        \\displaystyle\\sum_{\\ell\\in [k]} b_\\ell=n(n-r)-\\sum_{\\ell\\in P_n}(\\rho_\\ell-e_\\ell)=n(n-r)-\\sum_{\\ell\\in [k]}(\\rho_\\ell-e_\\ell)=0,\\\\ \n         a_\\ell+b_\\ell\\leq \n        \\begin{cases}\n            n- r+ e_\\ell-\\rho_\\ell& \\mbox {if  } \\ell \\in \\emptyset,\\\\\n            0 & \\mbox {if  } \\ell \\in P_{n},\\\\\n            \\rho_\\ell -e_\\ell=0 & \\mbox {if  } \\ell \\in \\bar P_{n}.\n        \\end{cases}\n    \\end{cases}\n    \\end{align*}     \n    Hence, there is a unique fitting sequence, namely, $a_\\ell=b_\\ell=0$ for $\\ell\\in [k]$.\nConsequently, the first six conditions of Theorem \\ref{rhoryserthmfullversion} will be trivial, and the remaining six conditions will be identical to those of Theorem \\ref{rhohallthm}. \n\\item [(b)] Now, we show that Ryser's Theorem is  immediate from Theorem \\ref{rhoryserthmfullversion}. In Theorem \\ref{rhoryserthmfullversion},  if we let $k=n$ (so $L$ is a latin rectangle), then  the existence of $\\Theta\\subseteq \\Gamma$ satisfying \\eqref{colorcon3} is trivial. In fact, in this case, $\\Theta=\\Gamma$. \n\\end{itemize}\n}\\end{remark}\n\\begin{remark} \\textup{\nWe can restate Theorem \\ref{rhoryserthmfullversion} without introducing the sets $P_r$ and $P_s$. To do so, let us  rewrite $f_1(\\ell)=a_\\ell+(\\rho_\\ell-e_\\ell+r\\dotdiv n)$, and  $f_2(\\ell)=b_\\ell+(\\rho_\\ell-e_\\ell+s\\dotdiv n)$. The main condition in the definition of fitting sequence can be written in the following manner.\n\\begin{align*}\n\\begin{cases}\n       \\displaystyle\\sum_{\\ell\\in [k]} a_\\ell=r(n-s)-\\sum_{\\ell\\in [k]}(\\rho_\\ell-e_\\ell+r\\dotdiv n),\\\\[20pt] \n        \\displaystyle\\sum_{\\ell\\in [k]} b_\\ell=s(n-r)-\\sum_{\\ell\\in [k]}(\\rho_\\ell-e_\\ell+s\\dotdiv n),\\\\[20pt]  \n         a_\\ell+b_\\ell\\leq \\rho_\\ell-e_\\ell-(\\rho_\\ell-e_\\ell+r\\dotdiv n)-(\\rho_\\ell-e_\\ell+s\\dotdiv n).\n    \\end{cases}\n\\end{align*}\nThe first six conditions of Theorem \\ref{rhoryserthmfullversion} can be restated as follows.\n}\\end{remark}\n\\begin{align*}\n    |I|(n-s) &\\leq \\sum_{\\ell\\in [k]} \\min \\Big\\{a_\\ell+(\\rho_\\ell-e_\\ell+r\\dotdiv n), \\mu_I(\\ell)\\Big\\}  &\\forall I\\subseteq  [r],\\\\\n\\sum_{i\\in [r]} \\min \\Big\\{n-s, \\mu_K(i)\\Big\\}  &\\geq     \\sum_{\\ell\\in K} \\Big(a_\\ell+(\\rho_\\ell-e_\\ell+r\\dotdiv n)\\Big)&\\forall K\\subseteq  [k],\\\\\n  |I|(n-s) &\\geq \\sum_{\\ell\\in [k]} \\Big( a_\\ell+(\\rho_\\ell-e_\\ell+r \\dotdiv n)\\dotdiv  \\mu_{\\bar I}(\\ell)\\Big) &\\forall I\\subseteq [r],\\\\ \n\\sum_{i\\in [r]} \\Big (n-s\\dotdiv \\mu_{\\bar K}(i)\\Big)  &\\leq  \\sum_{\\ell\\in K}\\Big(a_\\ell+(\\rho_\\ell-e_\\ell+r \\dotdiv n)\\Big)     &\\forall K\\subseteq [k],\\\\\n    |I|(n-s) &\\leq \\sum_{\\ell\\in K}\\Big(a_\\ell+(\\rho_\\ell-e_\\ell+r \\dotdiv n)\\Big)  + \\mu_I(\\bar K)  &\\forall I\\subseteq [r], K\\subseteq [k], \\\\\n  |I|(n-s)  &\\geq   \\sum_{\\ell\\in K}\\Big(a_\\ell+(\\rho_\\ell-e_\\ell+r \\dotdiv n)\\Big) - \\mu_K(\\bar I)   &\\forall I\\subseteq  [r], K\\subseteq [k].\n\\end{align*}\nSubstituting $r$ by $s$, $s$ by $r$, $I$ by $J$, and $a_\\ell$ by $b_\\ell$ in these conditions, one can obtain the remaining six conditions. \n\\section{Corollaries} \nThroughout this section, we will use the same notation as in \nSection \\ref{rhorythmprfsec}.  Recall that in order to embed an $r\\times s$  $\\brho$-latin rectangle into an $n\\times n$ $\\brho$-latin square, it is necessary that\n$$\ne_\\ell\\geq r+s+\\rho_\\ell-2n \\quad \\forall \\ell\\in [k].\n$$\nWe show that imposing slightly stronger assumptions will lead to much simpler conditions than those of Theorem \\ref{rhoryserthmfullversion}. Most importantly, the sequence $\\{(a_\\ell,b_\\ell)\\}_{\\ell=1}^k$ will not be needed if we allow each $e_\\ell$ to be slightly bigger  than what  is necessary. \n\n\n\nWe will  make use of the following $(g,f)$-factor theorem. Here, $N_G(A)$ in the neighborhood of $A$ in $G$. \n\\begin{theorem} \\cite[Theorem 5]{MR3564794}, \\cite[Theorem 1]{MR1081839} \\label{gffacthmcomb}\nThe bipgraph $G[X,Y]$ has a  $(g,f)$-factor   if and only if either one of the following two conditions hold.\n\\begin{align*}\n    g(A)&\\leq \\sum_{u\\in N_G(A)} \\min \\Big\\{ f(u), \\mult_G(uA)\\Big\\}\n &\\forall A\\subseteq X, A\\subseteq Y,\\\\\nf(A)&\\geq  \\sum_{u\\notin A} \\Big( g(u) \\dotdiv \\dg_{G-A}(u) \\Big)  &\\forall A\\subseteq X\\cup Y.\n\\end{align*}\n\\end{theorem} \nSlight modification to the proof of \\cite[Theorem 1]{MR1081839} leads to the following  simpler criteria for the case when  $g(y)=0$ for $y\\in Y$: The  bigraph $G[X,Y]$ has a  $(g,f)$-factor  with $g(y)=0$ for $y\\in Y$  if and only if\n\\begin{equation} \\label{refinedgfthmmain}\n    f(B)\\geq  \\sum_{x\\in A} \\Big( g(x) \\dotdiv \\dg_{G-B}(x) \\Big) \\quad \\forall A\\subseteq X, B\\subseteq Y.\n\\end{equation}\n\n\nIn our first application of Theorem \\ref{rhoryserthmfullversion}, we assume that $e_\\ell\\geq \\rho_\\ell-n+\\max\\{r,s\\}$ for  $\\ell\\in [k]$. This, in particular implies that\n$$\nrs=\\sum_{\\ell\\in [k]}e_\\ell\\geq \\sum_{\\ell\\in [k]} \\big( \\rho_\\ell-n+\\max\\{r,s\\}\\big)=n^2-kn + k \\max\\{r,s\\},\n$$\nwhich leads to the following lower bound for the number of symbols.\n$$k\\geq \\frac{n^2-rs}{n-\\max\\{r,s\\}}\\geq n+\\max\\{r,s\\}.$$\n\n\n\\begin{corollary} \\label{corkbig}\nAn $r\\times s$  $\\brho$-latin rectangle with $e_\\ell\\geq \\rho_\\ell-n+\\max\\{r,s\\}$ for  $\\ell\\in [k]$ can  be completed to an $n\\times n$ $\\brho$-latin square if and only if any of the following conditions hold.\n\\begin{align*}\n |I|(n-s)+ |J|(n-r)&\\leq \\sum_{\\ell\\in [k]} \\min \\Big\\{ \\rho_\\ell-e_\\ell, \\mu_I(\\ell) + \\mu_J(\\ell)\\Big\\}\n &\\forall I\\subseteq [r], J\\subseteq [s],\\\\\n\\sum_{\\ell\\in K} (\\rho_\\ell-e_\\ell)&\\geq \\sum_{i\\in  I} \\Big( n-s \\dotdiv \\mu_{\\bar K}(i) \\Big)+\n  \\sum_{j\\in  J} \\big( n-r \\dotdiv \\mu_{\\bar K}(j) \\Big) &\\forall I\\subseteq [r], J\\subseteq [s], K\\subseteq [k].\n\\end{align*}\n\\end{corollary}\n\\begin{proof}\nSuppose that $e_\\ell\\geq \\rho_\\ell-n+\\max\\{r,s\\}$ for $\\ell\\in [k]$. Then \\eqref{mainryserineq} holds, and neither $\\rho_\\ell-e_\\ell-n+s$ nor $\\rho_\\ell-e_\\ell-n+r$ is positive for $\\ell\\in [k]$. Thus,    \\eqref{colorcon3} will be simplified to the following.\n\\begin{align*}\n\\begin{cases}\n\\dg_\\Theta (x_i)=n-s & \\mbox{if  }  i\\in [r], \\\\\n\\dg_\\Theta (y_j)=n-r  & \\mbox{if  }  j\\in [s], \\\\\n\\dg_\\Theta (\\ell)\\leq \\rho_\\ell-e_\\ell  & \\mbox{if  } \\ell \\in [k].\n\\end{cases}\n\\end{align*}\n Let \n\\begin{align*}\n    \\begin{cases}\n        g,f: V(\\Gamma)\\rightarrow \\mathbb{N}\\cup \\{0\\},\\\\\n        g(x_i)=f(x_i)=n-s & \\mbox {for } i\\in [r],\\\\\n        g(y_j)=f(y_j)=n-r & \\mbox {for } j\\in [s],\\\\\n        g(\\ell)=0  & \\mbox {for } \\ell\\in [k],\\\\\n        f(\\ell)=\\rho_\\ell-e_\\ell & \\mbox {for } \\ell\\in [k].\n    \\end{cases}\n\\end{align*} \nBy Theorem \\ref{gffacthmcomb}, $\\Gamma  [X\\cup Y, [k]]$ has a $(g,f)$-factor (and so $\\Theta\\subseteq \\Gamma$ satisfying \\eqref{colorcon3} exists, and consequently, $L$ can be completed) if and only if the following  conditions hold.\n\\begin{align*}\n g(I)+ g(J)&\\leq \\sum_{\\ell\\in [k]} \\min \\Big\\{ f(\\ell), \\mult_\\Gamma(\\ell I)+ \\mult_\\Gamma(\\ell J)\\Big\\}\n &\\forall I\\subseteq X, J\\subseteq Y,\\\\\n g(K)&\\leq \\sum_{u\\in X\\cup Y} \\min \\Big\\{ f(u), \\mult_\\Gamma(u K)\\Big\\}\n &\\forall K\\subseteq [k].\n\\end{align*}\nThe first condition is equivalent to the first condition of this corollary. The second condition is trivial for $g(K)=0$ for $K\\subseteq [k]$. By \\eqref{refinedgfthmmain}  $\\Gamma$ has a  $(g,f)$-factor if and only if\n\\begin{equation*}\n    f(K)\\geq  \\sum_{u\\in U} \\Big( g(u) \\dotdiv \\dg_{\\Gamma-K}(u) \\Big) \\quad \\forall U\\subseteq (X\\cup Y), K\\subseteq [k].\n\\end{equation*}\nThis  is equivalent to the second condition of this corollary.\n\\end{proof}\n\n\nIn our next application, we assume that $e_\\ell\\geq r+s-\\rho _\\ell$ for  $\\ell\\in [k]$, which implies the following.\n$$\nrs=\\sum_{\\ell\\in [k]}e_\\ell\\geq \\sum_{\\ell\\in [k]} \\big( r+s-\\rho _\\ell\\big)=k(r+s)-n^2,\n$$\nThis leads to the following upper bound for the number of symbols.\n$$k\\leq \\frac{n^2+rs}{r+s}.$$\n\nIn the next corollary, we use the fact that $(x\\dotdiv y)\\dotdiv z=x-y \\dotdiv z$.\n\\begin{corollary} \\label{cor2}\nAn $r\\times s$  $\\brho$-latin rectangle with $e_\\ell\\geq r+s-\\rho _\\ell$ for  $\\ell\\in [k]$ can  be completed to an $n\\times n$ $\\brho$-latin square if and only if any of  the following  two conditions,\n\\begin{align*}\n    \\sum_{\\ell\\in K} (\\rho_\\ell-e_\\ell+r\\dotdiv n) &\\leq \\sum_{i\\in [r]} \\min \\Big\\{ n-s, \\mu_K(i)\\Big\\}  & \\forall K\\subseteq [k],\\\\\n|I|(n-s)   &\\geq \\sum_{\\ell\\in [k]} \\big( \\rho_\\ell-e_\\ell+r- n \\dotdiv \\mu_{\\bar I}(\\ell) \\big)  & \\forall I\\subseteq [r],\n\\end{align*}\ntogether with any of the following two  hold.\n\\begin{align*}\n    \\sum_{\\ell\\in K} (\\rho_\\ell-e_\\ell+s\\dotdiv n) &\\leq \\sum_{j\\in [s]} \\min \\Big\\{ n-r, \\mu_K(j)\\Big\\}  & \\forall K\\subseteq [k],\\\\\n|J|(n-r)   &\\geq \\sum_{\\ell\\in  [k]} \\big( \\rho_\\ell-e_\\ell+s- n \\dotdiv \\mu_{\\bar J}(\\ell) \\big) &  \\forall J\\subseteq [s].\n\\end{align*}\n\\end{corollary}\n\\begin{proof} \nSuppose that $e_\\ell\\geq r+s-\\rho _\\ell$ for $\\ell\\in [k]$. Then \\eqref{mainryserineq} holds,  $\\rho_\\ell-e_\\ell\\geq r+s-2e_\\ell$ for $\\ell\\in [k]$, and so  \\eqref{colorcon3} will be simplified to the following.\n\\begin{align*}\n\\begin{cases}\n\\dg_\\Theta (x_i)=n-s & \\mbox{if  }  i\\in [r], \\\\\n\\dg_\\Theta (y_j)=n-r  & \\mbox{if  }  j\\in [s], \\\\\n\\mult_\\Theta (\\ell X)\\geq \\rho_\\ell-e_\\ell-n+r  & \\mbox{if  } \\ell \\in [k],\\\\\n\\mult_\\Theta (\\ell Y)\\geq \\rho_\\ell-e_\\ell-n+s  & \\mbox{if  } \\ell \\in [k].\n\\end{cases}\n\\end{align*}\nLet $\\Gamma_1, \\Gamma_2$ be the subgraphs of $\\Gamma$ induced by $X\\cup [k]$, and  $Y\\cup [k]$, respectively. Let  \n\\begin{align*}\n    \\begin{cases}\n        g_1, f_1,: V(\\Gamma_1)\\rightarrow \\mathbb{N}\\cup \\{0\\},\\\\\n        f_1(x_i)=g_1(x_i)=n-s &\\mbox {for } i\\in [r],\\\\\n        g_1(\\ell)=\\rho_\\ell-e_\\ell+r\\dotdiv n &\\mbox {for } \\ell\\in [k],\\\\\n        f_1(\\ell)=r-e_\\ell &\\mbox {for } \\ell\\in [k],\n    \\end{cases}\n\\end{align*}\n\\begin{align*}\n    \\begin{cases}\n        g_2, f_2,: V(\\Gamma_2)\\rightarrow \\mathbb{N}\\cup \\{0\\},\\\\\n        f_2(y_j)=g_2(y_j)=n-r &\\mbox {for } j\\in [s],\\\\\n        g_2(\\ell)=\\rho_\\ell-e_\\ell+s\\dotdiv n &\\mbox {for } \\ell\\in [k],\\\\\n        f_2(\\ell)=s-e_\\ell &\\mbox {for } \\ell\\in [k].\n    \\end{cases}\n\\end{align*}\nIt is clear that  $\\Theta\\subseteq \\Gamma$ satisfying \\eqref{colorcon3} exists if and only if $\\Gamma_1$ and $\\Gamma_2$ have an $f_1$-factor and $f_2$-factor, respectively. By Theorem  \\ref{gffacthmcomb}, $\\Gamma_1$ has a $(g_1,f_1)$-factor if and only if\n\\begin{align*}\n    g_1(K)&\\leq \\sum_{u\\in X} \\min \\Big\\{ f_1(u), \\mult_{\\Gamma_1}(u K)\\Big\\}\n &\\forall K\\subseteq [k],\\\\\n    g_1(I)&\\leq \\sum_{\\ell\\in [k]} \\min \\Big\\{ f_1(\\ell), \\mult_{\\Gamma_1}(\\ell I)\\Big\\}\n &\\forall I\\subseteq X,\n\\end{align*}\nand $\\Gamma_2$ has a $(g_2,f_2)$-factor if and only if\n\\begin{align*}\n    g_2(K)&\\leq \\sum_{u\\in Y} \\min \\Big\\{ f_2(u), \\mult_{\\Gamma_2}(u K)\\Big\\}\n &\\forall K\\subseteq [k],\\\\\n    g_2(J)&\\leq \\sum_{\\ell\\in [k]} \\min \\Big\\{ f_2(\\ell), \\mult_{\\Gamma_2}(\\ell J)\\Big\\}\n\\quad &\\forall J\\subseteq Y.\n\\end{align*}\nEquivalently, $\\Gamma_1$ has a $(g_1,f_1)$-factor if and only if\n\\begin{align*}\n    \\sum_{\\ell\\in K} (\\rho_\\ell-e_\\ell+r\\dotdiv n)&\\leq \\sum_{i\\in [r]} \\min \\Big\\{ n-s, \\mu_{K}(i)\\Big\\}\n &\\forall K\\subseteq [k],\\\\\n    |I|(n-s)&\\leq \\sum_{\\ell\\in [k]} \\min \\Big\\{ r-e_\\ell, \\mu_I(\\ell)\\Big\\}\n &\\forall I\\subseteq [r].\n\\end{align*}\nBut the second condition is trivial, for $\\mu_I(\\ell)\\leq r-e_\\ell$, and so   $\\sum_{\\ell\\in [k]}\\mu_I(\\ell)=\\mu_I([k])=|I|(n-s)$. Similarly, $\\Gamma_2$ has a $(g_2,f_2)$-factor if and only if\n\\begin{align*}\n    \\sum_{\\ell\\in K} (\\rho_\\ell-e_\\ell+s\\dotdiv n)&\\leq \\sum_{j\\in [s]} \\min \\Big\\{ n-r, \\mu_{K}(j)\\Big\\}\n &\\forall K\\subseteq [k],\\\\\n    |J|(n-r)&\\leq \\sum_{\\ell\\in [k]} \\min \\Big\\{ s-e_\\ell, \\mu_J(\\ell)\\Big\\}\n &\\forall J\\subseteq [s].\n\\end{align*}\n Again, since $\\mu_J(\\ell)\\leq s-e_\\ell$, we have  $\\sum_{\\ell\\in [k]}\\mu_J(\\ell)=\\mu_J([k])=|J|(n-r)$, and so the second condition is trivial.\n\nNow we modify $f_1$ and $f_2$ defined above so that for $\\ell\\in [k]$, $f_1(\\ell)=f_2(\\ell)=z$ where $z$ is a sufficiently large number. By Theorem \\ref{gffacthmcomb}, $\\Gamma_1$ has a $(g_1,f_1)$-factor if and only if \n\\begin{align} \\label{gf1faccond1}\nf_1(I)+f_1(K)&\\geq    \\sum_{\\ell\\in  \\bar K} \\Big( g_1(\\ell) \\dotdiv \\dg_{\\Gamma_1-(I\\cup K)}(\\ell) \\Big) \\nonumber\\\\\n&\\quad+ \\sum_{i\\in  \\bar I} \\Big( g_1(i) \\dotdiv \\dg_{\\Gamma_1-(I\\cup K)}(i) \\Big) \\quad \\forall I\\subseteq X, K\\subseteq [k].\n\\end{align}\nFor $K\\neq \\emptyset$, \\eqref{gf1faccond1} is trivial, and for $K=\\emptyset$, it simplifies to the following.\n\\begin{align*} \n|I|(n-s)&\\geq    \\sum_{\\ell\\in   [k]} \\Big( (\\rho_\\ell-e_\\ell+r\\dotdiv n) \\dotdiv \\mu_{\\bar I}(\\ell) \\Big)+ \\sum_{i\\in  \\bar I} \\Big( n-s  \\dotdiv \\mu_{[k]}(i) \\Big)\\\\\n&=\\sum_{\\ell\\in   [k]} \\Big( (\\rho_\\ell-e_\\ell+r\\dotdiv n) \\dotdiv \\mu_{\\bar I}(\\ell) \\Big) \\quad \\forall I\\subseteq [r].\n\\end{align*}\nSimilarly,  $\\Gamma_2$ has a $(g_2,f_2)$-factor if and only if\n\\begin{align*} \n|J|(n-r)\\geq  \\sum_{\\ell\\in   [k]} \\Big( (\\rho_\\ell-e_\\ell+s\\dotdiv n) \\dotdiv \\mu_{\\bar J}(\\ell) \\Big) \\quad \\forall J\\subseteq [s].\n\\end{align*}\n\\end{proof}\n\n\\begin{corollary}\nAn $r\\times s$  $\\brho$-latin rectangle satisfying the following condition can always be completed to an $n\\times n$ $\\brho$-latin square.  \n$$r+s-e_\\ell \\leq \\rho_\\ell \\leq e_\\ell+n-\\max\\{r,s\\}\\quad \\forall \\ell\\in [k].$$\n\\end{corollary}\n\\begin{proof}\nCondition \\eqref{mainryserineq} holds, and  we have $\\rho_\\ell-e_\\ell-n+r\\leq 0$, $\\rho_\\ell-e_\\ell-n+s\\leq 0$, and $\\rho_\\ell-e_\\ell\\geq r+s-2e_\\ell$ for $\\ell\\in [k]$. Hence,  ~\\eqref{colorcon3}  will be simplified to the following.\n\\begin{align*}\n\\begin{cases}\n\\dg_\\Theta (x_i)=n-s & \\mbox{if  }  i\\in [r], \\\\\n\\dg_\\Theta (y_j)=n-r  & \\mbox{if  }  j\\in [s].\n\\end{cases}\n\\end{align*}\nConsequently,  $\\Theta\\subseteq \\Gamma$ always  exists. \n\\end{proof}\n\n\n\n\\section*{Acknowledgement}\n We wish to thank J. L.  Goldwasser for a fruitful discussion that led to this publication, and the anonymous referees for their constructive criticism. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGeneralised complex oriented cohomology first appeared in the work of\nNovikov \\cite{Novikov} and Quillen \\cite{Quillen} who realised that formal\ngroups naturally enter in algebraic topology. Such a theory is known to be\ncompletely characterised by the isomorphism $h^{\\ast }(\\mathbb{C}P^{\\infty\n})\\cong h^{\\ast }(\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{pt})[x]$, where $x$ is the first Chern class of the\ncanonical line bundle over the infinite complex projective space $\\mathbb{C}%\nP^{\\infty }$, and the K\\\"{u}nneth formula, $h^{\\ast }(\\mathbb{C}P^{\\infty\n}\\times \\mathbb{C}P^{\\infty })\\cong h^{\\ast }(\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{pt})[x,y]$, which\nimplies that the first Chern class of the tensor product of two line bundles\nobeys a formal group law \\cite{Adams}. There are three known types of formal\ngroup laws which come from the one-dimensional connected algebraic groups,\nthe additive group, the multiplicative group, and elliptic curves,\ndescribing respectively (ordinary) cohomology, K-theory and elliptic\ncohomology.\n\nOn the other hand to each of the mentioned groups one can associate\nrational, trigonometric and elliptic solutions of the Yang-Baxter equation\nwhich are linked to the appropriate quantum groups. It was first suggested\nin \\cite{GKV} that there should be a connection between the latter and the\nmentioned generalised cohomology theories.\n\nThe study of solutions of the Yang-Baxter equation is at the heart of the\narea of quantum integrable systems. Based on earlier pioneering works of\nHans Bethe \\cite{Bethe} and Rodney Baxter \\cite{Baxter}, the Faddeev School \n\\cite{Faddeevetal} developed the \\emph{algebraic Bethe ansatz} or \\emph{%\nquantum inverse scattering method}, where starting from a solution of the\nYang-Baxter equation one constructs the quantum integrals of motion of the\nphysical system as a commutative subalgebra, now often called the \\emph{%\nBethe algebra}, within a larger non-commutative \\emph{Yang-Baxter algebra}.\nHistorically, Yang-Baxter algebras were the origin for the later definition\nof quantum groups by Drinfeld \\cite{Drinfeld} and Jimbo \\cite{Jimbo}. Using\nthe commutation relations of the Yang-Baxter algebra the Bethe ansatz\nculminates in the derivation of a coupled set of -- in our setting --\npolynomial equations, whose solutions describe the spectrum of the commuting\ntransfer matrices which generate the Bethe algebra. In general solving these\nequations analytically is regarded as an intractable problem within the\nintegrable systems community except for a few special cases.\n\nThe first instance were quantum integrability was used in the study of\nquantum cohomology of full flag varieties and quantum K-theory was in works\nof Givental and Kim and Givental and Lee; see \\cite{Givental,GK,Kim,Kim2}\nand \\cite{Givental2,Lee}. In recent work of Nekrasov and Shatashvili \\cite%\n{NekrasovShatashvili} which was further developed mathematically by\nBraverman, Maulik and Okounkov \\cite{BMO,MaulikOkounkov} it was established\nthat the Bethe ansatz equations of some well known integrable systems\nrelated to the quantum groups known as Yangians describe the quantum\ncohomology and quantum K-theory for a large class of algebraic varieties,\nthe Nakajima varieties. Particular examples are the cotangent spaces of\npartial flag varieties, see the work \\cite{Gorbetal}, the simplest case\nbeing the contangent space of the Grassmannian. This opens up an exciting\nnew perspective on the connection made in \\cite{GKV}.\n\nIn this article we shall instead investigate the above connection for the\nGrassmannians $\\limfunc{Gr}_{n,N}=\\limfunc{Gr}_{n}(\\mathbb{C}^{N})$\nthemselves rather than their cotangent spaces based on the earlier findings\nin \\cite{KorffStroppel}, \\cite{VicOsc} and \\cite{GoKo}; see also the work on\nnon-quantum $GL(N)$-equivariant cohomology in \\cite{Rimanyietal}. The\ndifficulty here is, that it is initially not clear which quantum group to\nexpect. So instead we start out with special solutions to the Yang-Baxter\nequation which are tied to certain exactly solvable or quantum integrable\nlattice models in statistical mechanics and consider their associated\nYang-Baxter algebras as our \\textquotedblleft quantum group\". Despite the\nmodels being physically motivated, they are special degenerations of the\nasymmetric six-vertex model which describes ferroelectrics such as ice,\ntheir resulting Bethe algebras -- for certain special cases -- describe\nrings which have been defined previously in the setting of algebraic\ntopology and geometry where they are of great mathematical interest.\nSpecialising the parameters of the quantum integrable model in different\nways, we are able to identify them as the \\emph{quantum equivariant\ncohomology} \\cite{Mihalcea} and the (non-equivariant) \\emph{quantum K-theory}\n\\cite{BM}\\ of the Grassmannians using the results in \\emph{loc. cit}. \n\nThese special cases prompt us to conjecture that our main result, the\ndescription of a complex oriented generalised quantum cohomology and its\nequivariant version for the Grassmannians, also covers the so far unknown\ncase of \\emph{equivariant quantum K-theory}. At the same time this\ndescription can be seen as solving the well-posed mathematical problem of\nfinding the solution to the Bethe ansatz equations: we state the coordinate\nring defined by the equations, identify a special basis in it and explicitly\ndescribe the multiplication of two basis elements in terms of a generalised\nSchubert calculus within the framework of Goresky-Kottwitz-MacPherson theory\nwhich we show to extend to the quantum case.\n\n\\subsection{Statement of results}\n\nDenote by $\\limfunc{Gr}_{n,N}=\\limfunc{Gr}_{n}(\\mathbb{C}^{N})$ the\nGrassmannian of $n$-dimensional hyperplanes in $\\mathbb{C}^{N}$ with $N\\geq\n3 $ and fix a maximal torus $\\mathbb{T}\\subset GL(N)$. We describe\ngeneralised $\\mathbb{T}$-equivariant quantum cohomology rings $qh_{n}^{\\ast\n}=qh^{\\ast }(\\limfunc{Gr}_{n,N};\\beta )$ for $n=0,1,\\ldots ,N$ using the\ntheory of exactly solvable lattice models in statistical mechanics \\cite%\n{Baxter}. While the latter appear in theoretical physics, we shall use them\nhere as abstract combinatorial objects -- they define a weighted counting of\nnon-intersecting lattice paths as described for $\\beta =0$ in \\cite{VicOsc}\n-- which can be rigorously defined in purely mathematical terms using\nYang-Baxter algebras. The weights or probabilities attached to the lattice\nmodels depend on\n\n\\begin{itemize}\n\\item a variable $\\beta $ (the anisotropy parameter of the six-vertex model)\nentering the multiplicative formal group law \\cite{Quillen}, \\cite%\n{Bukhshtaberetal} and its inverse, \n\\begin{equation}\nx\\oplus y=x+y+\\beta xy\\;\\;\\;\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{and}\\;\\;\\;x\\ominus y=\\frac{x-y}{1+\\beta y}%\n\\;,  \\label{group_law}\n\\end{equation}\n\n\\item a \\textquotedblleft quantum\" parameter $q$ (the twist parameter\nrelated to quasiperiodic boundary conditions on the lattice) as well as\n\n\\item the equivariant parameters $t=(t_{1},\\ldots ,t_{N})$ (so-called\ninhomogeneities in the lattice) which are connected to the natural $\\mathbb{T%\n}$-action on $\\limfunc{Gr}_{n,N}$.\n\\end{itemize}\n\nThe case $\\beta =0$, which corresponds to the additive group law and in\nphysics terminology to the so-called \\emph{free fermion} point of the\nlattice models, has been treated previously for the homogeneous case ($%\nt_{j}=0$) in \\cite{VicOsc} and recently been extended to the equivariant\nsetting in \\cite{GoKo}.\n\nOur approach does not require any background knowledge in statistical\nmechanics, the lattice models are constructed in terms of special solutions\nto the quantum (as opposed to classical) Yang-Baxter equation, hence they\nare called \\emph{quantum integrable}, and their description is purely\nalgebraic. However, we find it noteworthy that they are degenerations of the\nasymmetric six-vertex model -- as mentioned previously -- and their\ncombinatorial description analogous to the one in \\cite{VicOsc} provides a\npowerful computational tool. For the latter to work we require the\npreviously mentioned restriction $N\\geq 3$.\n\nFrom these special solutions of the Yang-Baxter equation we construct\nYang-Baxter algebras, which in our case are bi-algebras only and not full\nHopf algebras. The so-called \\emph{row-to-row transfer matrices} of the\nlattice model generate a commutative subalgebra within the larger\nnon-commutative Yang-Baxter algebra which decomposes into the direct sum $%\n\\tbigoplus\\nolimits_{n=0}^{N}qh_{n}^{\\ast }$ of rings, which have the\nfollowing presentation.\n\nSet $\\mathcal{R}(\\mathbb{T)}=\\mathcal{R}(t_{1},\\ldots ,t_{N})$ where $%\n\\mathcal{R}$ is the ring of rational functions in $\\beta $ which are regular\nat $\\beta =0$ and $\\beta =-1$. Define $qh_{n}^{\\ast }$ %\nto be the polynomial algebra generated by $\\{e_{r}\\}_{r=1}^{n}$, $%\n\\{h_{r}\\}_{r=1}^{N-n}$ over $\\mathcal{R}(\\mathbb{T},q\\mathbb{)}=\\mathbb{Z}%\n[\\![q]\\!]\\otimes \\mathcal{R}(\\mathbb{T)}$ subject to the relations obtained\nby expanding the following functional relation in the variable $x$,%\n\\begin{equation}\nh(x)e(\\ominus x)=\\left( \\tprod_{i=1}^{n}t_{i}\\ominus x\\right) \\left(\n\\tprod_{i=1}^{N-n}x\\ominus t_{i+n}\\right) (1+\\beta h_{1})+q,\n\\label{ideal def}\n\\end{equation}%\nwhere $1$ is the unit element and, setting $h_{0}=e_{0}=1,$ $%\nh_{N+1-n}=e_{n+1}=0$,%\n\\begin{eqnarray}\nh(x) &=&\\sum_{r=0}^{N-n}\\left( h_{r}+\\beta h_{r+1}\\right)\n\\tprod_{i=1}^{N-n-r}\\left( x\\ominus t_{N+1-i}\\right)  \\label{H def} \\\\\ne(x) &=&\\sum_{r=0}^{n}\\left( e_{r}+\\beta e_{r+1}\\right)\n\\tprod_{i=1}^{n-r}\\left( x\\oplus t_{i}\\right) \\;.  \\label{E def}\n\\end{eqnarray}%\nFor the non-experts we recall that the Grothendieck $K$-functor assigns to\neach smooth compact manifold $\\mathcal{X}$ a ring which is built out of\ncomplex vector bundles on $\\mathcal{X}$ \\cite{AA}. It is the value of this\nfunctor and its quantum analogue $QK$ for $\\mathcal{X}=\\limfunc{Gr}_{n,N}$\nwhich we shall simply refer to as (quantum) ``K-theory'' of the\nGrassmannians throughout this article.\n\nDenote by $\\{e^{\\varepsilon _{j}}\\}_{j=1}^{N}$ the (formal) exponentials\ngenerating the character ring of $\\mathfrak{gl}(N)$.\n\n\\begin{theorem}\nWe have the following special cases:\n\n\\begin{itemize}\n\\item[(i)] $qh_{n}^{\\ast }\/\\langle \\beta \\rangle $ is isomorphic to the\nequivariant quantum cohomology $QH_{\\mathbb{T}}^{\\ast }(\\limfunc{Gr}_{n,N})$\nin the presentation given by Mihalcea \\cite[Thm 1.1]{Mihalcea}.\n\n\\item[(ii)] $qh_{n}^{\\ast }\/\\langle \\beta +1,t_{1},\\ldots ,t_{N}\\rangle $ is\nisomorphic to $QK(\\limfunc{Gr}_{n,N}) $ as studied in \\cite{BM}.\n\n\\item[(iii)] $qh_{n}^{\\ast }\/\\langle \\beta +1,t_{j}+e^{\\varepsilon\n_{N+1-j}}-1,q\\rangle $ is isomorphic to $K_{\\mathbb{T}}(\\limfunc{Gr}_{n,N})$\nwhere $K_{\\mathbb{T}}$ denotes the equivariant K-functor.\n\\end{itemize}\n\\end{theorem}\n\nEach of the cases (i)--(iii) is interesting in its own right and we compare\nour findings against existing presentations of these rings in the\nliterature. In particular, in case (i) our results are linked to previous\n(unpublished) work by Peterson \\cite{Peterson} and the affine nil-Hecke ring\nof Kostant and Kumar \\cite{KostantKumar}: we explicitly construct a family\nof operators whose matrix elements give the structure constants of $%\nqh_{n}^{\\ast }$ and which for $\\beta =0$ can be identified with Peterson's\nbasis; see \\cite{GoKo} for details. The other cases can then be seen as a\ngeneralisation of this construction to $K$-theory.\n\nTo establish (ii) we compare our ring structure against the Pieri rules\nderived by Lenart \\cite{Lenart} for $q=0$ and the quantum Pieri and\nGiambelli formulae of Buch and Mihalcea \\cite{BM} for $q\\neq 0$. The new\nresult in our article is the coordinate ring presentation which follows from\n(\\ref{ideal def}).\n\nFinally, we show (iii) by defining a generalisation of\nGoresky-Kottwitz-MacPherson theory \\cite{GKM}: we identify McNamara's\nfactorial Grothendieck polynomials \\cite{McNamara} with localised Schubert\nclasses using the Bethe ansatz of quantum integrable models. We also derive\nexpressions for the localised opposite Schubert classes and identify the\npartition functions of our lattice models with classes for Richardson\nvarieties.\n\nBased on the above special cases we have the following:\n\n\\begin{conjecture}\n$qh_{n}^{\\ast }\/\\langle \\beta +1,t_{j}+e^{\\varepsilon _{N+1-j}}-1\\rangle $\ndescribes the value $QK_{\\mathbb{T}}(\\limfunc{Gr}_{n,N})$ of the equivariant\nquantum $K$-functor for the Grassmannians.\n\\end{conjecture}\n\n\\begin{remark}\nWe note that we can define $qh^*_n$ also over the ring of Laurent\npolynomials in $\\beta$ instead, which would introduce a natural $\\mathbb{Z}$%\n-grading. This suggests that our framework might also be used to describe\nthe actual $\\mathbb{Z}$-graded quantum equivariant K-theory which is\nobtained from the K-functor in conjunction with the Bott Periodicity\nTheorem. However, there is currently not sufficient evidence available to\nfurther substantiate this claim, hence we state this here as a mere\nobservation and not as a conjecture.\n\\end{remark}\n\nBesides providing a complete description of $QK_{\\mathbb{T}}(\\limfunc{Gr}%\n_{n,N})$, which has so far been missing in the literature, the new aspects\nin our approach are\n\n\\begin{enumerate}\n\\item that our ring is defined for general $\\beta $ which allows us to treat\nall these special cases at once in a unified setting of a quantum\ngeneralised cohomology theory as first defined in \\cite{CG} and\n\n\\item that we reveal an underlying quantum group structure in terms of\nYang-Baxter algebras which we show to commute with the natural symmetric\ngroup action on the idempotents of these rings.\n\\end{enumerate}\n\nAs a byproduct of our investigation we also derive new combinatorial results\nsuch as a generalised Jacobi-Trudy formula and Cauchy identity for factorial\nGrothendieck polynomials.\n\n\\subsection{Outline of the article}\n\n\\begin{description}\n\\item[Section 2] We introduce the necessary combinatorial objects and\nnotations we will use throughout the article. In particular, we review\nMcNamara's definition of factorial Grothendieck polynomials which play a\ncentral role in our approach and derive several new results which we need to\ndescribe our generalised cohomology ring for the Grassmannian.\n\n\\item[Section 3] Starting from special solutions to the Yang-Baxter\nequation, so-called $L$-operators, we define the Yang-Baxter algebra in\nterms of endomorphisms over some vector space $\\mathcal{V}$ which will be\nidentified with the direct sum of the generalised cohomology rings $%\n\\tbigoplus\\nolimits_{n=0}^{N}qh_{n}^{\\ast }$. We describe the commutation\nrelations of the Yang-Baxter algebra and define the transfer matrices which\ngenerate a commutative subalgebra, the so-called Bethe algebra. The action\nof the latter on $\\mathcal{V}$ is described combinatorially using toric skew\ndiagrams. We also show that the transfer matrices obey the functional\nrelation (\\ref{ideal def}).\n\n\\item[Section 4] We derive the spectrum of the Bethe algebra by constructing\ntheir eigenvectors and computing their eigenvalues using the algebraic Bethe\nansatz. Both, eigenvectors and eigenvalues, are described in terms of the\nsolutions of a set of coupled equations, called the Bethe ansatz equations,\nwhich we show can be solved in terms of formal power series in the quantum\ndeformation parameter $q$ of $qh_{n}^{\\ast }$. We then initially define the\ngeneralised cohomology ring by identifying the eigenbasis of the transfer\nmatrices as the primitive, central orthogonal idempotents of $qh_{n}^{\\ast }$%\n. We also define a bilinear form which turns $qh_{n}^{\\ast }$ into a\nFrobenius algebra. Having identified the eigenvectors of the transfer\nmatrices as idempotents, we then fix the analogue of the Schubert basis and\ndescribe the product in this geometrically motivated basis instead. This\nallows us to state a residue formula for the structure constants in the\nSchubert basis in terms of the solutions of the Bethe ansatz equations and\nshow that they obey a recurrence formula which is derived from an\nequivariant quantum Pieri-Chevalley formula for $qh_{n}^{\\ast }$.\n\n\\item[Section 5] Employing the description of $qh_{n}^{\\ast }$ in terms of\nits idempotents leads to a formulation of the ring in terms of column\nvectors whose components can be thought of as generalised localised Schubert\nclasses where the localisation points are identified with the solutions of\nthe Bethe ansatz equations. We show that these generalised Schubert classes\nobey generalised Goresky-Kottwitz-MacPherson conditions which derive from an\naction of the symmetric group. Interestingly, the latter emerges naturally\nfrom solutions of the Yang-Baxter equation discussed in Section 3, gives\nrise to a representation of a generalised Iwahori-Hecke algebra and commutes\nwith the action of the Yang-Baxter algebra. Using this framework of GKM\ntheory we prove the special cases mentioned in the introduction, that is we\nshow that our ring $qh_{n}^{\\ast }$ can be specialised to equivariant\nquantum cohomology and quantum K-theory. This section also gives the proof\nof the presentation of $qh_{n}^{\\ast }$ as polynomial algebra modulo the\nrelations (\\ref{ideal def}).\\bigskip\n\\end{description}\n\n\\noindent \\textbf{Acknowledgment}. The authors would like to thank the Max\nPlanck Institute for Mathematics Bonn, where part of this work was carried\nout, for hospitality. They are grateful to Leonardo Mihalcea and Alexander\nVarchenko for comments on a draft version of the article. C. K. also\ngratefully acknowledges discussions with Gwyn Bellamy, Christian Voigt, Paul\nZinn-Justin and would like to thank the organisers Anita Ponsaing and Paul\nZinn-Justin for their kind invitation to the workshop \\emph{Combinatorics\nand Integrability}, Presqu'\\^{\\i}le de Giens, 23-27 June 2014, where the\nresults of this article were presented.\n\n\\section{Preliminaries}\n\nThis section introduces the combinatorial notions needed in the description\nof Schubert calculus in the rest of this paper. We also collect known as\nwell as a number of new results on factorial Grothendieck polynomials.\n\n\\subsection{Minimal coset representatives}\n\nDenote by $\\mathbb{S}_{N}$ the symmetric group in $N$-letters and choose $%\nk,n\\in \\mathbb{N}_{0}$ such that $N=n+k$. A set of minimal length coset\nrepresentatives $w$ for classes $[w]$ in $\\mathbb{S}_{N}\/\\mathbb{S}%\n_{n}\\times \\mathbb{S}_{k}$ is given by the permutations for which $%\nw(1)<w(2)<\\cdots <w(n)$ and $w(n+1)<w(n+2)<\\cdots <w(N)$. For instance, the\ncoset representative with $w^{0}(n+1)=1,$ $w^{0}(n+2)=2,\\ldots ,$ $%\nw^{0}(N)=k $ and $w^{0}(1)=k+1,$ $w^{0}(2)=k+2,\\ldots ,$ $w^{0}(n)=N$ is\ngiven by \n\\begin{equation}\nw^{0}=s_{n}s_{n+1}\\cdots s_{N-1}~\\cdots ~s_{2}s_{3}\\cdots\ns_{k+1}~s_{1}s_{2}\\cdots s_{k}~.\n\\end{equation}\n\n\\subsection{Binary strings}\n\nThe $w$'s are in bijection with $01$-words or binary strings $%\nb(w)=b_{1}b_{2}\\cdots b_{N}\\in \\{0,1\\}^{N}$ of length $N$, where $%\nn=|b|:=\\sum_{j}b_{j}$ is the number of $1$-letters, $k=N-|b|$ the number of $%\n0$-letters and \n\\begin{equation}\nb_{j}(w)=\\left\\{ \n\\begin{array}{cc}\n1, & j\\in I_{w} \\\\ \n0, & j\\in \\lbrack N]\\backslash I_{w}%\n\\end{array}%\n\\right.\n\\end{equation}%\nwith $I_{w}:=\\{w(1),\\ldots ,w(n)\\}$ and $[N]=\\{1,2,\\ldots N\\}$. So, in the\ncase of $w^{0}$ we have $I_{w^{0}}=\\{k+1,\\ldots ,N\\}$ and $b(w^{0})=0\\cdots\n01\\cdots 1$ is the binary string with $k$ 0-letters in front, followed by $n$\n1-letters. The identity $w=1$ on the other hand corresponds to the binary\nstring $b(1)=1\\cdots 10\\cdots 0$ instead. Note that under the above\nbijection the natural $\\mathbb{S}_{N}$-action on $\\mathbb{S}_{N}\/\\mathbb{S}%\n_{n}\\times \\mathbb{S}_{k}$ via $s_{j}\\cdot \\lbrack w]=[s_{j}w]$ coincides\nwith the natural $\\mathbb{S}_{N}$-action on binary strings, where $s_{j}$\npermutes the $j$th and $(j+1)$th letter in $b(w)$.\n\n\\subsection{Boxed partitions\\label{sec:partitions}}\n\nEach binary string $b(w)$, and thus each minimal length representative $w$,\nis in one-to-one correspondence with a partition $\\lambda =(\\lambda\n_{1},\\lambda _{2},\\ldots ,\\lambda _{n})$ which has at most $n$ parts and for\nwhich $\\lambda _{1}\\leq k$. That is, the corresponding Young diagram lies in\na bounding box of height $n$ and width $k$ and we will denote this by\nwriting $\\lambda \\subset (k^{n})$. The bijection is given by the relation%\n\\begin{equation}\nw(i)=\\lambda _{n+1-i}+i,\\qquad i=1,2,\\ldots ,n\\;.  \\label{w2lambda}\n\\end{equation}%\nFor $w=w^{0}$ we have that $\\lambda =(k^{n})$, the partition with $n$ parts\nequal to $k$, and for $w=1$ we obtain the empty partition denoted by $%\n\\lambda =\\emptyset $. N.B. the bijection is defined for fixed $n,k$, so each\npartition comes with a bounding box of fixed dimensions. For different $n,k$\none may obtain the same partition $\\lambda $ but the dimensions of the\nbounding box will then be different. We therefore refer to $\\lambda $ as a \n\\emph{boxed partition} as the dimensions of the bounding box enter in the\nbijection (\\ref{w2lambda}).\n\nThroughout this article we will use these various labellings of the same\ncoset $[w]$ interchangeably writing $b(w)$, $\\lambda (w)$ for the images\nunder the above bijection and $b(\\lambda ),w(\\lambda )$ for the pre-images.\nBy abuse of notation we shall also write $s_{j}b$ and $s_{j}\\lambda $ for\nthe binary string and partition corresponding to the coset $[s_{j}w]$.\n\n\\subsection{Cylindric loops and toric skew diagrams\\label{sec:torictab}}\n\nWe briefly recall the definition of cylindric loops $\\lambda \\lbrack r]$\nassociated with a partition $\\lambda $ and \\emph{toric skew diagrams}; see \n\\cite{GesselKrattenthaler}, \\cite{Postnikov}.\n\nLet $\\lambda =(\\lambda _{1},\\ldots ,\\lambda _{n})\\in (n,k)$, then the\nassociated \\emph{cylindric loop} $\\lambda \\lbrack r]$ for any $r\\in \\mathbb{Z%\n}$ is defined as%\n\\begin{equation}\n\\lambda \\lbrack r]:=(\\ldots ,\\underset{r}{\\lambda _{n}+r+k},\\underset{r+1}{%\n\\lambda _{1}+r},\\ldots ,\\underset{r+n}{\\lambda _{n}+r},\\underset{r+n+1}{%\n\\lambda _{1}+r-k},\\ldots )\\;.  \\label{cyl_loop}\n\\end{equation}%\nFor $r=0$ the cylindric loop can be visualised as a path in $\\mathbb{Z}%\n\\times \\mathbb{Z}$ determined by the outline of the Young diagram of $%\n\\lambda $ which is periodically continued with respect to the vector $(n,-k)$%\n. For $r\\neq 0$ this line is shifted $r$ times in the direction of the\nlattice vector $(1,1)$.\n\nGiven two boxed partitions $\\lambda ,\\mu \\subset (k^{n})$ denote by $\\lambda\n\/d\/\\mu $ the set of squares between the two lines $\\lambda \\lbrack d]$ and $%\n\\mu \\lbrack 0]$ modulo integer shifts by $(n,-k)$, \n\\begin{equation}\n\\lambda \/d\/\\mu :=\\{\\langle i,j\\rangle \\in \\mathbb{Z}\\times \\mathbb{Z}\/(n,-k)%\n\\mathbb{Z}~|~\\lambda \\lbrack d]_{i}\\geq j>\\mu \\lbrack 0]_{i}\\}\\;.\n\\label{cyl_diagram}\n\\end{equation}%\nWe shall refer to $\\theta =\\lambda \/d\/\\mu $ as a \\emph{cylindric skew-diagram%\n} of degree $d=d(\\theta )$. Postnikov introduced \\cite{Postnikov} the\nterminology \\emph{toric skew-diagram }for those\\emph{\\ }$\\theta $ where the\nnumber of boxes in each row does not exceed $k$. Note that $\\lambda \/0\/\\mu\n=\\lambda \/\\mu $, that is cylindric or toric skew-diagrams contain ordinary\nskew diagrams as special cases.\n\nA cylindric skew diagram $\\theta $ which has at most one box in each column\nwill be called a \\emph{toric} \\emph{horizontal strip} and one which has at\nmost one box in each row a \\emph{toric} \\emph{vertical strip}. The \\emph{%\nlength} of such strips will be the number of boxes within the skew diagram,\nwhere we identify squares modulo integer shifts by $(n,-k)$ and choose as\nrepresentatives those squares $s=\\langle i,j\\rangle $ with $1\\leq j\\leq n$.\nIn what follows this identification is always understood implicitly if we\ntalk about a square in a toric strip.\n\n\n\\subsection{Bases in equivariant cohomology and K-theory\\label{sec:schubert}}\n\nWe are interested in describing equivariant quantum cohomology ($\\beta =0$)\nand K-theory ($\\beta =-1$) as special cases of our generalised cohomology\ntheory for $\\limfunc{Gr}_{n,N}$. The equivariant cohomology \\cite%\n{KostantKumar} and K-theory \\cite{KostantKumar2} of flag varieties -- of\nwhich Grassmannians are a special case -- was studied by Kostant and Kumar.\nThe equivariant quantum cohomology of flag varieties was computed in \\cite%\n{Givental}, \\cite{Kim}, \\cite{GK}, \\cite{Kim2} and quantum K-theory in \\cite%\n{Givental2}, \\cite{Lee}, \\cite{GL} and since then has been discussed by\nnumerous authors.\n\n\nSpecialising $\\beta =0$ we identify $\\mathcal{R}(\\mathbb{T)}$ with the\nequivariant cohomology $H_{\\mathbb{T}}^{\\ast }(\\limfunc{pt})=\\mathbb{Z}%\n(t_{1},\\ldots ,t_{N})$ of a point by mapping each $f_{\\beta }\\in \\mathcal{R}(%\n\\mathbb{T)}$ to its value at $\\beta =0$. Let $X_{\\lambda }$ and $X^{\\lambda\n} $ denote the \\emph{Schubert} and \\emph{opposite Schubert varieties} where $%\n\\lambda \\subset (k^{n})$. We also recall the definition of the \\emph{%\nRichardson variety} $X_{\\mu }^{\\lambda }=X_{\\mu }\\cap X^{\\lambda }$. All\nthree varieties are left invariant under the torus action. The corresponding\nSchubert classes $\\{[X_{\\lambda }]\\}_{\\lambda \\subset (k^{n})}$ and $%\n\\{[X^{\\lambda }]\\}_{\\lambda \\subset (k^{n})}$ form dual bases over $\\mathbb{Z%\n}[q]\\otimes \\mathbb{Z}(t_{1},\\ldots ,t_{N})$. Both bases are related by $%\nX^{\\lambda }=w_{0}\\cdot X_{\\lambda ^{\\vee }}$ where $\\lambda ^{\\vee }$ is\nobtained by reversing the binary string $b(\\lambda )$ and $w_{0}$ is the \n\\emph{long element} in $\\mathbb{S}_{N}$. One is interested in the\ncomputation of the \\emph{3-point genus 0 equivariant Gromov-Witten invariants%\n} $C_{\\lambda \\mu }^{\\nu }(t,q)$ which appear in the product%\n\\begin{equation}\n\\lbrack X_{\\lambda }][X_{\\mu }]=\\sum_{\\nu\\subset (k^n)}C_{\\lambda \\mu\n}^{\\nu}(t,q)[X_{\\nu }]  \\label{GWinv}\n\\end{equation}%\nand for the Grassmannian are monomials in $q$, i.e. $C_{\\lambda \\mu }^{\\nu\n}(t,q)=q^{d}C_{\\lambda \\mu }^{\\nu ,d}(t)$. The invariants for $d=0$ also\nappear in the expansion%\n\\begin{equation}\n\\lbrack X_{\\mu }^{\\lambda }]=\\sum_{\\nu }C_{\\mu \\nu }^{\\lambda ,0}(t)[X^{\\nu\n}]\\;,  \\label{Richardson}\n\\end{equation}%\nand, thus, $C_{\\mu \\nu }^{\\lambda ,0}(t)=c_{\\lambda \\mu }^{\\nu }(t)$ are the\nanalogue of Littlewood-Richardson coefficients for factorial Schur functions \n\\cite{MolevSagan}.\n\nIn the case of $K$-theory we specialise $\\beta =-1$ and set $%\nt_{j}=1-e^{\\varepsilon _{N+1-j}}$ where the (formal) exponentials $%\n\\{e^{\\varepsilon _{j}}\\}_{j=1}^{N}$ generate the character ring of $%\n\\mathfrak{gl}(N)$. Mapping each $f_{\\beta }\\in \\mathcal{R}(\\mathbb{T)}$ to\nits value at $\\beta =-1$ then gives us $K_{\\mathbb{T}}(\\limfunc{pt})=%\n\\limfunc{Rep}(\\mathbb{T})$, the representation ring of $\\mathbb{T}$ which is\ncanonically isomorphic to the group algebra of the free abelian group of\ncharacters $e^{\\varepsilon _{i}}$. The ring $K_{\\mathbb{T}}(\\limfunc{Gr}%\n_{n,N})$ is generated by the classes $[\\mathcal{O}_{\\lambda }]$ and $[%\n\\mathcal{O}^{\\lambda }]$ of the \\emph{structure sheaves} $\\mathcal{O}%\n_{\\lambda }$ and $\\mathcal{O}^{\\lambda }$ of the Schubert and opposite\nSchubert varieties within the Grothendieck group of coherent sheaves on the\nGrassmannian. Their product expansions%\n\\begin{equation}\n\\lbrack \\mathcal{O}_{\\lambda }][\\mathcal{O}_{\\mu }]=\\sum_{\\nu \\subset\n(k^{n})}c_{\\lambda \\mu }^{\\nu }(t)[\\mathcal{O}_{\\nu }],\\qquad \\lbrack \n\\mathcal{O}_{\\mu }][\\mathcal{O}^{\\lambda }]=\\sum_{\\nu \\subset (k^{n})}d_{\\mu\n\\nu }^{\\lambda }(t)[\\mathcal{O}^{\\nu }]  \\label{Kstructure}\n\\end{equation}%\ndefine the K-theoretic Littlewood-Richardson coefficients $c_{\\lambda \\mu\n}^{\\nu }(t)$ where in case of the Grassmannian $d_{\\lambda \\mu }^{\\nu\n}(t)=c_{\\lambda \\mu }^{\\nu }(t)$; see e.g. \\cite{Knutson} and references\ntherein. There are known positivity statements for these structure\nconstants, see \\cite{GR} and \\cite[Sec 5]{AGM} as well as references therein.\n\nWe shall refer to the K-classes $\\{[\\mathcal{O}^{\\lambda }]\\}$ as Schubert\nbasis or simply Schubert classes. In contrast to the case $\\beta =0$ the\nclasses $[\\mathcal{O}_{\\lambda }]$ and $[\\mathcal{O}^{\\lambda }]$ do not\nform dual bases in $K_{\\mathbb{T}}(\\limfunc{Gr}_{n,N})$ but instead one has\nto introduce additional classes $[\\xi ^{\\lambda }]$ which can also be\ndefined in terms of sheaves (see \\cite[Prop 2.1]{GrKu}). For the\nnon-equivariant case $K(\\limfunc{Gr}_{n,N})=K_{\\mathbb{T}}(\\limfunc{Gr}%\n_{n,N})\/\\langle t_{1},\\ldots ,t_{N}\\rangle $ one has the relation \\cite[Sec 8%\n]{BuchKtheory} \n\\begin{equation}\n\\lbrack \\xi ^{\\lambda }]=(1-[\\mathcal{O}_{1}])[\\mathcal{O}_{\\lambda ^{\\vee\n}}]  \\label{Kdualbasis}\n\\end{equation}%\nwhere $[\\mathcal{O}_{1}]$ is the K-class of the Schubert divisor. We shall\nstate the analogue of this relation for the equivariant case in a subsequent\nsection. %\n\n\n\\subsection{Discrete symmetries\\label{sec:symmetries}}\n\nThroughout this article we will make use of several involutions and a\nnatural $\\mathbb{Z}_{N}$-action defined on the set of cosets in $\\mathbb{S}%\n_{N}\/\\mathbb{S}_{n}\\times \\mathbb{S}_{k}$ where $k=N-n$ as before. These\nwill induce mappings between elements in the Schubert basis, in some cases\nfrom different rings, and since they in turn lead to non-trivial\ntransformation properties of the structure constants of $qh_{n}^{\\ast }$, we\nrefer to them as \\textquotedblleft symmetries\\textquotedblright .\n\n\\subsubsection{Poincar\\'{e} Duality}\n\nDefine an involution $\\vee :qh_{n}^{\\ast }\\rightarrow qh_{n}^{\\ast }$ by\nreversing a binary string, i.e. $b_{i}^{\\vee }=b_{N+1-i}$. We shall denote\nthe corresponding permutation and partition by $w^{\\vee }$ and $\\lambda\n^{\\vee }$, respectively. One easily verifies that the Young diagram of $%\n\\lambda ^{\\vee }$ is the complement of the Young diagram of $\\lambda $ in\nthe $n\\times k$ bounding box.\n\n\\subsubsection{Level-Rank Duality}\n\nDefine an involution $\\ast :qh_{n}^{\\ast }\\rightarrow qh_{N-n}^{\\ast }$ by\nswapping 0 and 1-letters in binary strings, i.e. $b_{i}^{\\ast }=1-b_{i}$.\nThe corresponding partition $\\lambda ^{\\ast }$ is obtained by taking first\nthe conjugate partition $\\lambda ^{\\prime }$ and then its complement in the\nbounding box or vice versa, i.e. $\\lambda ^{\\ast }=(\\lambda ^{\\prime\n})^{\\vee }=(\\lambda ^{\\vee })^{\\prime }$. So, in particular we can define\nthe composite involution $qh_{n}^{\\ast }\\rightarrow qh_{N-n}^{\\ast }$ by $%\n\\lambda \\mapsto \\lambda ^{\\prime }$ and shall denote the corresponding\nbinary string and permutation respectively by $b^{\\prime }$ and $w^{\\prime }$%\n.\n\n\n\\subsection{Set-Valued Tableaux and Grothendieck polynomials\\label%\n{sec:grothendieck}}\n\nWe recall some of the necessary combinatorial objects and the definition of\nfactorial Grothendieck polynomials. This is based on earlier work by Buch \n\\cite{BuchKtheory} and McNamara \\cite{McNamara}, but we shall also derive\nseveral new results which are not contained in the latter works.\n\nLet $n$ be some non-negative integer. We will use the notation $%\n[n]=\\{1,\\ldots ,n\\}$ and $\\mathbb{P}_{n}=\\mathbb{P}([n])$ for the power set\nof $[n]$, the set of all subsets of $[n]$. Denote by $\\theta $ a skew Young\ndiagram with at most $|\\theta |\\leq n$ boxes which we identify with a subset\nof $\\mathbb{Z}^{2}$.\n\n\\begin{definition}[Buch]\nA set-valued tableau is a map $T:\\theta \\rightarrow \\mathbb{P}_{n}$ such\nthat the following conditions hold%\n\\begin{equation}\n\\max T(i,j)\\leq \\min T(i,j+1)\\;\\;\\;\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{and}\\;\\;\\;\\max T(i,j)<\\min\nT(i+1,j)\\;.\n\\end{equation}\n\\end{definition}\n\nDenote by $|T|$ the sum over the cardinalities of all the subsets in the\nimage of $T$ and let $\\limfunc{SVT}(\\theta )$ be the set of all set-valued\ntableau of shape $\\theta $. Then we have the following definition of\nfactorial Grothendieck polynomials due to McNamara \\cite{McNamara} which is\nan extension of Buch's earlier realisation \\cite{BuchKtheory} of ordinary\n(skew) Grothendieck polynomials as sum over set-valued tableaux.\n\n\\begin{definition}\nThe factorial (skew) Grothendieck polynomial is the weighted sum%\n\\begin{equation}\nG_{\\theta }(x|t)=\\sum_{T}\\beta ^{|T|-|\\theta |}\\prod_{\\substack{ (i,j)\\in\n\\theta  \\\\ r\\in T(i,j)}}x_{r}\\oplus t_{r+j-i}  \\label{facG}\n\\end{equation}%\nover all set-valued tableaux $T\\in \\limfunc{SVT}(\\theta )$.\n\\end{definition}\n\nN.B. the factorial Grothendieck polynomials are in general defined for an \n\\emph{infinite} sequence $(t_j)_{j\\in\\mathbb{Z}}$ of parameters. For this\nsection only we shall assume these parameters to be nonzero for all $j$ but\nthen set $t_j=0$ unless $1\\leq j\\leq N$ and identify them with the\nequivariant parameters mentioned in the introduction.\n\nEmploying (\\ref{group_law}) define the $\\beta $-deformed factorial power \n\\begin{equation}\n(x_{j}|t)^{r}:=\\prod_{i=1}^{r}x_{j}\\oplus t_{i}\\;.  \\label{facpower}\n\\end{equation}%\nThe following determinant formula is stated in \\cite[Eqn (2.12)]{IN}. Its\nproof follows along similar lines as indicated in \\emph{loc. cit.} where the\nfocus is on the symplectic case.\n\n\\begin{proposition}[Ikeda-Naruse]\n\\begin{equation}\nG_{\\theta }(x|t)=\\frac{\\det \\left[ (x_{j}|t)^{\\theta _{i}+n-i}(1+\\beta\nx_{j})^{i-1}\\right] _{1\\leq i,j\\leq n}}{\\det [x_{j}^{n-i}]_{1\\leq i,j\\leq n}}\n\\label{facGdet}\n\\end{equation}%\nwhere the denominator is the Vandermonde determinant $\\Delta\n(x)=\\prod_{i<j}(x_{i}-x_{j})$.%\n\\end{proposition}\n\nWe recall the following known specialisations of factorial Grothendieck\npolynomials.\n\nSetting $t_{j}=0$ for all $j$ one recovers the (ordinary) \\emph{Grothendieck\npolynomial} which has the following determinant presentation,%\n\\begin{equation}\nG_{\\theta }(x)=\\frac{\\det \\left( x_{j}^{\\theta _{i}+n-i}(1+\\beta\nx_{j})^{i-1}\\right) _{1\\leq i,j\\leq n}}{\\det \\left( x_{j}^{n-i}\\right)\n_{1\\leq i,j\\leq n}}\\;.  \\label{G}\n\\end{equation}\n\nSetting $\\beta =0$ one obtains the \\emph{factorial Schur function} (see e.g. \n\\cite{Macdonald} and \\cite[Cap I.3, Ex. 20]{MacdonaldBook} as well as\nreferences in \\emph{loc. cit.}),%\n\\begin{equation}\ns_{\\theta }(x|t)=\\frac{\\det [(x_{j}|t)^{\\theta _{i}+n-i}]_{1\\leq i,j\\leq n}}{%\n\\det [(x_{j}|t)^{n-i}]_{1\\leq i,j\\leq n}}~,\\qquad (x_{j}|t)^{r}\\overset{%\n\\beta =0}{=}\\prod_{i=1}^{r}(x_{j}+t_{i})\\;.  \\label{facSchur}\n\\end{equation}\n\nWe collect further properties of factorial Grothendieck polynomials which we\nwill use throughout this article.\n\nWe use the determinant formula (\\ref{facGdet}) to derive the following\nequation which is a generalisation of the known straightening rule for Schur\nfunctions $s_{\\theta }$, $s_{\\ldots ,\\theta _{i},\\theta _{i+1},\\ldots\n}=-s_{\\ldots ,\\theta _{i+1}-1,\\theta _{i}+1,\\ldots }$ \\cite[Ch I.3]%\n{MacdonaldBook}. The latter -- through repeated application -- allows one to\nexpress a Schur function indexed by a composition in terms of Schur\nfunctions indexed by partitions. We will use the straightening rule for\nfactorial Grothendieck polynomials for the same purpose.\n\n\\begin{corollary}[straightening rule]\nWe have the following relations%\n\\begin{multline}\nG_{\\ldots ,\\theta _{i},\\theta _{i+1},\\ldots }=  \\label{Gstraight} \\\\\n-\\beta G_{\\ldots ,\\theta _{i}+1,\\theta _{i+1},\\ldots }-\\frac{1+\\beta\nt_{n+\\theta _{i}-i+1}}{1+\\beta t_{n+\\theta _{i+1}-i}}~(G_{\\ldots ,\\theta\n_{i+1}-1,\\theta _{i}+1,\\ldots }+\\beta G_{\\ldots ,\\theta _{i+1},\\theta\n_{i}+1,\\ldots }),\n\\end{multline}%\nwhere $G_{\\theta }=G_{\\theta }(x|t)$ with $\\theta =(\\theta _{1},\\ldots\n,\\theta _{n})$.\n\\end{corollary}\n\n\\begin{proof}\nWithout difficulty one verifies the identity%\n\\begin{equation*}\n(1+\\beta t_{m+1})(x|t)^{m}(1+\\beta x)^{r}=(x|t)^{m}(1+\\beta x)^{r-1}+\\beta\n(x|t)^{m+1}(1+\\beta x)^{r-1}\\;.\n\\end{equation*}%\nApplying the latter first to the $i$th and then to the $(i+1)$th row of the\ndeterminant in the numerator of (\\ref{facGdet}), the assertion follows.\n\\end{proof}\n\nGiven a boxed partition $\\lambda \\subset (k^{n})$ we introduce the shorthand\nnotations%\n\\begin{eqnarray}\nt_{\\lambda } &=&(t_{\\lambda _{n}+1},\\ldots ,t_{\\lambda _{i}+n+1-i},\\ldots\n,t_{\\lambda _{1}+n})  \\notag \\\\\n\\ominus t_{\\lambda } &=&(\\ominus t_{\\lambda _{n}+1},\\ldots ,\\ominus\nt_{\\lambda _{i}+n+1-i},\\ldots ,\\ominus t_{\\lambda _{1}+n})  \\label{tlambda}\n\\end{eqnarray}%\nwhere $\\ominus x:=0\\ominus x=-x\/(1+\\beta x)$ for any formal variable $x$;\ncompare with (\\ref{group_law}). The following is a generalisation of the\nVanishing Theorem for factorial Schur functions \\cite{MolevSagan} to\nfactorial Grothendieck polynomials; see \\cite[Thm 4.4]{McNamara}.\n\n\\begin{theorem}[McNamara]\nLet $\\lambda ,\\mu $ be partitions with at most $n$ parts then%\n\\begin{equation}\nG_{\\lambda }(\\ominus t_{\\mu }|t)=\\left\\{ \n\\begin{array}{cc}\n0, & \\lambda \\nsubseteq \\mu \\\\ \n\\tprod\\limits_{\\langle i,j\\rangle \\in \\lambda }t_{n+j-\\lambda _{j}^{\\prime\n}}\\ominus t_{\\lambda _{i}+n+1-i}, & \\lambda =\\mu%\n\\end{array}%\n\\right.  \\label{Gvanish}\n\\end{equation}%\nand in general $G_{\\lambda }(\\ominus t_{\\mu }|t)$ will be non-zero if $%\n\\lambda \\subset \\mu $.\n\\end{theorem}\n\nFollowing \\cite{McNamara} we introduce for simplicity the notation%\n\\begin{equation}\n\\Pi (x)=\\prod_{i=1}^{n}(1+\\beta x_{i})\\;.  \\label{Pi}\n\\end{equation}%\nWe recall the following results \\cite[Ex 4.2 and Prop 4.8]{McNamara}.\n\n\\begin{lemma}[McNamara]\nWe have the identity%\n\\begin{equation}\n1+\\beta G_{1}(x|t)=\\sum_{i=0}^{n}\\beta ^{i}e_{i}(x\\oplus t)=\\Pi (x)\\Pi\n(t_{\\emptyset })\\;,  \\label{G1}\n\\end{equation}%\nwhere the $e_{i}$'s denote the elementary symmetric polynomials.\n\\end{lemma}\n\n\\begin{proposition}[McNamara]\nWe have the expansion%\n\\begin{equation}\n\\Pi (x)G_{\\lambda }(x|t)=\\Pi (\\ominus t_{\\lambda })\\sum_{\\lambda\n\\rightrightarrows \\mu }\\beta ^{|\\mu \/\\lambda |}G_{\\mu }(x|t),\n\\label{G1Pieri}\n\\end{equation}%\nwhere the notation $\\lambda \\rightrightarrows \\mu $ indicates that the sum\nruns over all partitions $\\mu $ which contain $\\lambda $ and for which the\nskew diagram $\\mu \/\\lambda $ has at most one box in each column or row.\n\\end{proposition}\n\nDenote by $\\Lambda _{n}\\otimes \\mathbb{Z}(\\beta ,t_{1},\\ldots ,t_{N})$ the\nlinear space spanned by the monomial symmetric functions $\\{m_{\\lambda\n}\\}_{\\lambda \\subset (k^{n})}$, then the following result is \\cite[Thm 4.6]%\n{McNamara}.\n\n\\begin{theorem}[McNamara]\nThe set $\\{G_{\\lambda }(x|t)\\}$ with $\\lambda $ having at most $n$ parts is\na basis of $\\Lambda _{n}\\otimes \\mathbb{Z}(\\beta ,t_{1},\\ldots ,t_{N})$.\n\\end{theorem}\n\n\\subsubsection{New results for factorial Grothendieck polynomials}\n\nWe expect the Grothendieck polynomials indexed by partitions which either\nconsist of a single column, $\\lambda =1^{r}$, or row, $\\lambda =r$, to be\nthe elementary building blocks for general $\\lambda $. The following lemma\nstates a generating function for the $G_{1^{r}}(x|t)$'s.\n\n\\begin{proposition}\nWe have the equality%\n\\begin{equation}\n\\Pi (t_{\\emptyset })\\prod_{i=1}^{n}(u-\nx_{i})=(u|t)^{n}+\\sum_{r=1}^{n}(-1)^{r}G_{1^{r}}(x|t)(u|t)^{n-r}%\n\\prod_{i=1}^{r-1}(1+\\beta u\\oplus t_{n+1-i})  \\label{-e}\n\\end{equation}%\nand the identity%\n\\begin{equation}\nG_{1^{r}}(x|t)=\\sum_{j=1}^{n+1-r}\\frac{\\prod_{i=1}^{n}x_{i}\\oplus t_{j}}{%\n\\prod_{i=1,i\\neq j}^{n+1-r}t_{j}\\ominus t_{i}}~,  \\label{G1r}\n\\end{equation}%\nwhere $r=1,\\ldots ,n$.\n\\end{proposition}\n\n\\begin{proof}\nFirst one derives the following equality involving the Vandermonde\ndeterminant via induction,%\n\\begin{equation}\na_{n}(x|t)=\\det [(x_{j}|t)^{n-i}]_{1\\leq i,j\\leq n}=\\Delta\n(x)\\prod_{i=1}^{n}(1+\\beta t_{i})^{n-i}\\;.  \\label{vandermonde}\n\\end{equation}\nThen it follows that \n\\begin{equation*}\n\\frac{a_{n+1}(u,x_{1},\\ldots ,x_{n}|t)}{a_{n}(x_{1},\\ldots ,x_{n}|t)}=\\Pi\n(t_{\\emptyset })\\prod_{i=1}^{n}(u-x_{i})\\;.\n\\end{equation*}%\nExpanding the determinant $a_{n+1}(u,x_{1},\\ldots ,x_{n}|t)$ with respect to\nthe first column one obtains the first formula. Setting $u=\\ominus t_{i} $\nwith $i=1,2,\\ldots ,n$ results in a linear system with lower triangular\nmatrix which can be solved to obtain the second formula.\n\\end{proof}\n\nUsing the last result we now derive an alternative generating function for\nthe $G_{1^{r}}(x|\\ominus t)$'s which will play an important role in what\nfollows.\n\n\\begin{corollary}\nWe have \n\\begin{equation}\n\\prod_{i=1}^{n}(u\\oplus x_{i})=(u|t)^{n}+\\sum_{r=1}^{n}(u|t)^{n-r}(1+\\beta\n~u\\oplus t_{n+1-r})G_{1^{r}}(x|\\ominus t)\\,.  \\label{e}\n\\end{equation}%\nSetting $t_{j}=0$ for all $j$ this becomes \n\\begin{equation}\n\\prod_{i=1}^{n}(u\\oplus x_{i})=u^{n}+(1+\\beta\nu)\\sum_{r=1}^{n}u^{n-r}G_{1^{r}}(x_{1},\\ldots ,x_{n})  \\label{e0}\n\\end{equation}%\nwhich implies for $r=1,2,\\ldots ,n$ the identities%\n\\begin{equation}\ne_{r}(x_{1},\\ldots ,x_{n})=\\sum_{s=r}^{n}(-\\beta )^{s-r}\\binom{s-1}{s-r}%\nG_{1^{s}}(x_{1},\\ldots ,x_{n})\\;,  \\label{er2G1r}\n\\end{equation}%\nwhere $e_{r}(x_{1},\\ldots ,x_{n})$ are the elementary symmetric polynomials.\n\\end{corollary}\n\n\\begin{proof}\nLet $f(u)=\\prod_{i=1}^{n}(u\\oplus x_{i})$. This is a polynomial in $u$ of\ndegree $n$ with the coefficient of $u^{n}$ being $\\Pi (x)$. Setting\nsuccessively $u=\\ominus t_{1},\\ominus t_{2},\\ldots ,\\ominus t_{n}$ one finds%\n\\begin{eqnarray*}\nf(u) &=&(u|t)^{n}+(1+\\beta u)\\sum_{r=1}^{n}(u|t)^{n-r}(1+\\beta\nt_{n+1-r})f_{r} \\\\\n&=&(u|t)^{n}(1+\\beta f_{1})+(u|t)^{n-1}(f_{1}+\\beta f_{2})+\\cdots +f_{n}\n\\end{eqnarray*}%\nwith \n\\begin{equation*}\nf_{n+1-r}=\\sum_{i=1}^{r}\\frac{f(\\ominus t_{i})}{\\prod_{j=1,j\\neq\ni}^{r}t_{j}\\ominus t_{i}}~,\\quad r=1,2,\\ldots ,n\\;.\n\\end{equation*}%\nThe identity (\\ref{e}) then follows from (\\ref{G1}) and (\\ref{G1r}). Setting \n$t_{1}=\\cdots =t_{n}=0$ in (\\ref{e}) we arrive at (\\ref{e0}).\n\nFinally, we have \n\\begin{eqnarray*}\n\\prod_{i=1}^{n}(u-x_{i}) &=&(1+\\beta u)^{n}(-1)^{n}f(\\ominus u) \\\\\n&=&u^{n}+\\sum_{r=1}^{n}(-1)^{r}(1+\\beta\nu)^{r-1}u^{n-r}G_{1^{r}}(x_{1},\\ldots ,x_{n})\n\\end{eqnarray*}%\nand comparing powers of $u$ on both sides of the equality sign the last\nassertion now follows.\n\\end{proof}\n\nAs in the case of factorial Schur functions \\cite[Chap I.3, Ex. 20]%\n{MacdonaldBook} define a \\emph{shift operator} $\\tau $ by \n\\begin{equation}\n(x|\\tau ^{m}t)^{n}=\\prod_{j=1}^{n}(x\\oplus t_{j+m}),\\qquad m\\in \\mathbb{Z}\\,.\n\\label{shifted power}\n\\end{equation}%\nWe wish to derive an analogue of the Jacobi-Trudy identity for factorial\nSchur functions. To this end we require the following result first.\n\n\\begin{lemma}\nWe have the expression \n\\begin{equation}\nG_{r}(x|t)=\\sum_{i=1}^{n}(x_{i}|t)^{n+r-1}\\prod_{j\\neq i}\\frac{1}{%\nx_{i}\\ominus x_{j}}  \\label{Gr}\n\\end{equation}%\nand the following equality between determinants, \n\\begin{equation}\n\\det [G_{\\lambda _{i}-i+j}(x|\\tau ^{1-j}t)]_{1\\leq i,j\\leq n}=\\frac{\\det\n[(x_{j}|t)^{n+\\lambda _{i}-i}]_{1\\leq i,j\\leq n}}{\\det\n[(x_{j}|t)^{n-i}]_{1\\leq i,j\\leq n}}  \\label{JTG}\n\\end{equation}%\nwhere $(x|t)^{m}$ is defined in (\\ref{facpower}).\n\\end{lemma}\n\n\\begin{proof}\nThe proof follows along the same steps as the proof for the analogous\nidentities in the case of factorial Schur functions; see e.g. the section on\nthe \\textquotedblleft 6th variation\\textquotedblright\\ in \\cite{Macdonald}\nand \\cite[Chap I.3, Ex. 20]{Macdonald}. We therefore omit the details.\n\\end{proof}\n\nWhile it would be desirable to have a single determinant in the $G_{r}$'s\nexpressing the Grothendieck polynomial $G_{\\lambda }$, this seems in general\nnot possible. Instead we obtain an expression in terms of sums of\ndeterminants which involve the polynomials in (\\ref{JTG}) \n\\begin{equation}\nF_{\\lambda }(x|t)=\\frac{\\det [(x_{j}|t)^{n+\\lambda _{i}-i}]_{1\\leq i,j\\leq n}%\n}{\\det [(x_{j}|t)^{n-i}]_{1\\leq i,j\\leq n}}  \\label{Fdet}\n\\end{equation}%\nNote that $F_{\\lambda }(x|t)=s_{\\lambda }(x|t)$ for $\\beta =0$ and $%\nF_{\\lambda }(x|0)=s_{\\lambda }(x)$, that is the $F_{\\lambda }$'s do not\nspecialise to the ordinary (non-factorial) Grothendieck polynomial for $%\nt_{j}=0$. We shall therefore treat this case separately.\n\nBefore we can state the expansion formula of $G_{\\lambda }$ into $F_{\\lambda\n}$'s we require the following technical result.\n\n\\begin{lemma}\n\\begin{equation}\n(1+\\beta u)^{r}(u|\\ominus t)^{n-r}=\\sum_{i=0}^{r}(u|\\ominus t)^{n-i}\\Gamma\n_{i}(r,n)  \\label{facpower_id}\n\\end{equation}%\nwhere the coefficients are given by%\n\\begin{equation}\n\\Gamma _{i}(r,n)=\\beta ^{r-i}\\prod_{j=i}^{r-1}(1+\\beta t_{n-j})\\sum_{i-1\\leq\nj_{1}\\leq \\cdots \\leq j_{i}\\leq r-1}\\prod_{l=1}^{i}(1+\\beta t_{n-j_{l}})\n\\label{Gamma}\n\\end{equation}%\nExplicitly, \n\\begin{eqnarray*}\n\\Gamma _{0}(r,n) &=&\\beta ^{r}\\prod_{j=0}^{r-1}(1+\\beta t_{n-j}) \\\\\n\\Gamma _{1}(r,n) &=&\\beta ^{r-1}\\prod_{j=1}^{r-1}(1+\\beta\nt_{n-j})\\sum_{j=0}^{r-1}(1+\\beta t_{n-j}) \\\\\n\\Gamma _{2}(r,n) &=&\\beta ^{r-2}\\prod_{j=2}^{r-1}(1+\\beta\nt_{n-j})\\sum_{j=1}^{r-1}(1+\\beta t_{n-j})\\sum_{i=1}^{j}(1+\\beta t_{n-i}) \\\\\n&&\\vdots \\\\\n\\Gamma _{r}(r,n) &=&(1+\\beta t_{n+1-r})^{r}\n\\end{eqnarray*}\n\\end{lemma}\n\n\\begin{proof}\nUse the simple identity%\n\\begin{equation*}\n(1+\\beta u)(u|\\ominus t)^{n}=(1+\\beta t_{n+1})[(u|\\ominus t)^{n}+\\beta\n(u|\\ominus t)^{n+1}]\n\\end{equation*}%\nto find the recurrence relation%\n\\begin{equation*}\n\\Gamma _{i}(r,n)=(1+\\beta t_{n+1-r})(\\Gamma _{i-1}(r-1,n-1)+\\beta \\Gamma\n_{i}(r-1,n))\\;.\n\\end{equation*}%\nHere $\\Gamma _{i}=0$ for $i<0$. Defining%\n\\begin{equation*}\n\\Gamma _{i}(r,n)=\\gamma _{i}(r,n)\\beta ^{r-i}\\prod_{j=i}^{r-1}(1+\\beta\nt_{n-j})\n\\end{equation*}%\nThe recurrence relation simplifies to \n\\begin{equation*}\n\\gamma _{i}(r,n)=(1+\\beta t_{n+1-r})\\gamma _{i-1}(r-1,n-1)+\\gamma\n_{i}(r-1,n))\n\\end{equation*}%\nand can now be successively solved starting from $\\gamma _{0}(r,n)=1$.\n\\end{proof}\n\nWe now state a generalised Jacobi-Trudy identity for factorial Grothendieck\npolynomials which simplifies for $\\beta=0$ to the known Jacobi-Trudy\nidentity for factorial Schur functions. We state it for the parameters $%\n\\ominus t$ as it is in this form that we will use the identity later on in\nthis article, but making the replacement $t\\rightarrow \\ominus t$ in the\nformula and the coefficients (\\ref{Gamma}) is straightforward.\n\n\\begin{proposition}\n\\label{prop:Gexp}Let $x=(x_{1},\\ldots ,x_{n})$ and $\\lambda $ a partition\nwith at most $n$ parts. Then%\n\\begin{equation}\nG_{\\lambda }(x|\\ominus t)=\\sum_{\\alpha }\\beta ^{|\\alpha |}\\phi _{\\alpha\n}(\\lambda )F_{\\lambda +\\alpha }(x|\\ominus t),  \\label{G2F}\n\\end{equation}%\nwhere the sum runs over all \\emph{compositions} $\\alpha =(0,\\alpha\n_{2}\\ldots ,\\alpha _{n})$ with $0\\leq \\alpha _{i}\\leq i-1$ and%\n\\begin{equation}\n\\phi _{\\alpha }(\\lambda )=\\frac{\\prod_{i=2}^{n}\\varphi _{\\alpha\n_{i}}(\\lambda _{i})}{\\prod_{i=1}^{n}(1+\\beta t_{i})^{n-i}},\\qquad \\beta\n^{\\alpha _{i}}\\varphi _{\\alpha _{i}}(\\lambda _{i})=\\Gamma _{i-1-\\alpha\n_{i}}(i-1,n+\\lambda _{i}-1)\\;.  \\label{Gamma2}\n\\end{equation}\n\\end{proposition}\n\nN.B. the determinant formula (\\ref{Fdet}) for $F_{\\alpha }$ is well-defined\nfor \\emph{compositions} $\\alpha $ by which we mean finite sequences of\nnon-negative integers which are not necessarily weakly decreasing. Any such $%\nF_{\\alpha }$ can be expressed in terms of $F_{\\lambda }$'s indexed by\npartitions $\\lambda $ using the same straightening rules which hold for\nSchur functions,%\n\\begin{equation}\nF_{(\\ldots ,a,b,\\ldots )}=-F_{(\\ldots ,b-1,a+1,\\ldots )}\\qquad \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{%\nand\\qquad }F_{(\\ldots ,a,a+1,\\ldots )}=0\\;.  \\label{Fstraight}\n\\end{equation}%\nBoth rules should be obvious from (\\ref{Fdet}), the first rule follows from\nexchanging two rows in the determinant in the numerator of (\\ref{Fdet}),\nwhile the second is simply a result of two rows being linearly dependent.\n\n\\begin{proof}\nEmploy the previous lemma and the formula (\\ref{facGdet}) to find%\n\\begin{eqnarray*}\n(x_{j}|t)^{\\lambda _{i}+n-i}(1+\\beta x_{j})^{i-1} &=&\\sum_{\\alpha\n_{i}=0}^{i-1}(x_{j}|\\ominus t)^{n+\\lambda _{i}-1-\\alpha _{i}}\\Gamma _{\\alpha\n_{i}}(i-1,n+\\lambda _{i}-1) \\\\\n&=&\\sum_{\\alpha _{i}=0}^{i-1}\\beta ^{\\alpha _{i}}\\varphi _{\\alpha\n_{i}}(\\lambda _{i})(x_{j}|\\ominus t)^{n+\\lambda _{i}-i+\\alpha _{i}}\\;.\n\\end{eqnarray*}%\nThe assertion now follows from row-linearity of the determinant and (\\ref%\n{vandermonde}).\n\\end{proof}\n\n\\begin{example}\nLet $n=2$. Then the compositions $\\alpha $ in the sum in (\\ref{G2F}) are $%\n\\alpha =(0,0)$ and $\\alpha =(0,1)$. We find from (\\ref{Gamma}) and (\\ref%\n{Gamma2}) that $\\Gamma _{0}(0,\\lambda _{1}+1)=1$, $\\Gamma _{1}(1,\\lambda\n_{2}+2)=1+\\beta t_{\\lambda _{2}+1}$ and $\\Gamma _{0}(1,\\lambda _{2}+2)=\\beta\n(1+\\beta t_{\\lambda _{2}+1})$. Hence, we arrive at%\n\\begin{equation}\nG_{\\lambda _{1},\\lambda _{2}}(x|\\ominus t)=\\frac{1+\\beta t_{\\lambda _{2}+1}}{%\n1+\\beta t_{1}}~(F_{\\lambda _{1},\\lambda _{2}}+\\beta F_{\\lambda _{1},\\lambda\n_{2}+1})\\;.  \\label{G2Fex}\n\\end{equation}\n\\end{example}\n\nThe analogous expansion of $G_{\\lambda }$ for the non-factorial case\ncorresponds to an expansion into Schur functions instead.\n\n\\begin{proposition}\nSet $t_{j}=0$ then%\n\\begin{equation}\nG_{\\lambda }(x)=\\sum_{\\alpha }\\beta ^{|\\alpha |}\\prod_{i=1}^{n-1}\\binom{i}{%\n\\alpha _{i}}~s_{\\lambda +\\alpha }(x),\\;  \\label{G2Schur}\n\\end{equation}%\nwhere the sum runs over all \\emph{compositions} $\\alpha =(0,\\alpha\n_{1},\\ldots ,\\alpha _{n-1})$ with $0\\leq \\alpha _{i}\\leq i$ and $s_{\\alpha\n}(x)=\\det (x_{j}^{n+\\alpha _{i}-i})\/\\det (x_{j}^{n-i})$ is the (generalised)\nSchur function with $\\alpha $ being a composition.\n\\end{proposition}\n\n\\begin{proof}\nUse the binomial theorem and row-linearity of the determinant.\n\\end{proof}\n\nUsing the Yang-Baxter algebra we will prove below the following special case\nof a Cauchy identity.\n\n\\begin{proposition}\nLet $\\mu \\subset (k^{n})$ be a partition inside the $n\\times k$ bounding\nbox, then%\n\\begin{eqnarray}\n\\prod_{i=1}^{n}\\prod_{j\\in I_{\\mu ^{\\ast }}}x_{i}\\ominus t_{j}\n&=&\\sum_{\\lambda }G_{\\lambda }(x|\\ominus t)G_{\\lambda ^{\\vee }}(t_{\\mu\n}|\\ominus t^{\\prime })\\frac{\\Pi (t_{\\mu })}{\\Pi (t_{\\lambda })}\n\\label{Cauchy0} \\\\\n&=&\\sum_{\\lambda }G_{\\lambda }(x|\\ominus t)G_{\\lambda ^{\\ast }}(\\ominus\nt_{\\mu ^{\\ast }}|t)\\frac{\\Pi (t_{\\lambda ^{\\ast }})}{\\Pi (t_{\\mu ^{\\ast }})}\n\\label{Cauchy0a}\n\\end{eqnarray}%\nwhere the second equality follows from the stronger identity \n\\begin{equation}\nG_{\\lambda ^{\\vee }}(\\ominus t_{\\mu }|t^{\\prime })=G_{\\lambda ^{\\ast\n}}(t_{\\mu ^{\\ast }}|\\ominus t),\\qquad t^{\\prime }=w_{0}t\\;.\n\\label{Gduality0}\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nSee Corollary \\ref{cor:cauchy0} and Prop \\ref{prop:Gduality}.\n\\end{proof}\n\n\\section{Yang-Baxter Algebras}\n\nThis section contains the main algebraic setup for the definition of the\nhierarchy of generalised equivariant quantum cohomologies $qh_{n}^{\\ast }$.\nAs explained earlier these are realised as commutative subalgebras of a\nlarger non-commutative algebra, the Yang-Baxter algebra, which then\nnaturally acts on the direct sum $\\tbigoplus\\nolimits_{n=0}^{N}qh_{n}^{\\ast\n} $.\n\n\\subsection{Quantum space and spin bases}\n\nOne of the main ingredients of our approach is to realise the Schubert basis\nas vectors in an $N$-fold tensor product of fundamental $sl(2)$-modules by\ntaking the direct sum $\\tbigoplus\\nolimits_{n=0}^{N}qh_{n}^{\\ast }$. Let $V=%\n\\mathbb{Z}v_{0}\\oplus \\mathbb{Z}v_{1}$ and $\\sigma ^{-}=\\left( \n\\begin{smallmatrix}\n0 & 1 \\\\ \n0 & 0%\n\\end{smallmatrix}%\n\\right) ,\\;\\sigma ^{+}=\\left( \n\\begin{smallmatrix}\n0 & 0 \\\\ \n1 & 0%\n\\end{smallmatrix}%\n\\right) ,\\;\\sigma ^{z}=\\left( \n\\begin{smallmatrix}\n1 & 0 \\\\ \n0 & -1%\n\\end{smallmatrix}%\n\\right) $ the fundamental representation of $sl_{2}$, the Pauli matrices,\nacting on $V$ via $\\sigma ^{-}v_{1}=v_{0}$, $\\sigma ^{+}v_{0}=v_{1}$ and $%\n\\sigma ^{z}v_{\\alpha }=(-1)^{\\alpha }v_{\\alpha },~\\alpha =0,1$. We introduce\nthe abbreviations $V(t_{j}):=\\mathcal{R}(t_{j})\\otimes V$ etc. Here we have\ndropped the dependence on $\\beta $ in the notation to simplify formulae. We\nwill identify (as vector spaces) $\\tbigoplus\\nolimits_{0\\leq n\\leq\nN}qh_{n}^{\\ast }$ with the following tensor product \n\\begin{equation}\n\\mathcal{V}=\\tbigotimes_{j=1}^{N}V(t_{j})\\cong \\mathcal{R}\\mathbb{(}%\nt_{1},\\ldots ,t_{N})\\otimes V^{\\otimes N}~.  \\label{quantum_space}\n\\end{equation}%\nOn the latter space, which is called the \\emph{quantum space} in the area of\nquantum integrable systems, we will define the action of the Yang-Baxter\nalgebra. Define the following \\textquotedblleft spin\nbasis\\textquotedblright\\ $\\{v_{\\lambda (b)}\\}\\subset V^{\\otimes N}$ where $b$\nruns over all binary strings of length $N$ and%\n\\begin{equation}\nv_{\\lambda (b)}=v_{b_{1}}\\otimes v_{b_{2}}\\otimes \\cdots \\otimes v_{b_{N}}\\;.\n\\label{spin basis}\n\\end{equation}%\nWe will also need its dual basis which we shall denote by $\\{\\tilde{v}%\n_{\\lambda }\\}\\subset \\tilde{V}^{\\otimes N},$ with $\\tilde{V}$ being the dual\nspace of $V$, and use the familiar bracket notation $\\langle \\tilde{v}%\n_{\\lambda }|v_{\\mu }\\rangle =\\delta _{\\lambda \\mu }$.\n\nThere is a natural $U(sl_{2})$-action on $V^{\\otimes N}$. Fix a Cartan\nsubalgebra $\\mathfrak{h}$ then we have the decomposition $V^{\\otimes\nN}=\\tbigoplus\\limits_{0\\leq n\\leq N}V_{n}$ into $U(\\mathfrak{h})$-weight\nspaces where $V_{n}\\subset V^{\\otimes N}$ denotes the subspace which is\nspanned by $\\{v_{\\lambda (b)}\\}_{|b|=n}$, i.e. the basis vectors indexed by\nbinary strings with $n$ 1-letters. This induces an analogous decomposition\nof the quantum space $\\mathcal{V}$ into the subspaces $\\mathcal{V}_{n}=%\n\\mathcal{R}(\\mathbb{T})\\otimes V_{n}$. Below we shall identify for each\nsubspace $\\mathcal{V}_{n}$ the basis (\\ref{spin basis}) with the Schubert\nbasis in $qh_{n}^{\\ast }$. We now define so-called exactly solvable lattice\nmodels in $\\limfunc{End}\\mathcal{V}$ following the analogous steps as in \n\\cite[Sec 3]{VicOsc}.\n\n\\subsection{Solutions to the Yang-Baxter equation}\n\nLet $x=(x_{1},x_{2},\\ldots )$ be a set of commuting indeterminates which we\ncall \\emph{spectral parameters} and as in the case of the equivariant\nparameters $t_{j}$ set $V(x_{i}):=\\mathcal{R}(x_{i})\\otimes V$. Define the\nfollowing $L$-operators $V(x_{i})\\otimes V(t_{j})\\rightarrow V(x_{i})\\otimes\nV(t_{j})$ by setting%\n\\begin{equation}\nL(x_i|t_j)=\\left( \n\\begin{array}{cc}\n\\sigma^{+}\\sigma^{-}+x_{i}\\ominus t_{j}~\\sigma^{-}\\sigma^{+} & (1+\\beta\nx_{i}\\ominus t_{j})\\sigma^{+} \\\\ \n\\sigma^{-} & \\sigma^{-}\\sigma^{+}%\n\\end{array}%\n\\right)  \\label{L}\n\\end{equation}%\nand%\n\\begin{equation}\nL^{\\prime }(x_i|t_j)=\\left( \n\\begin{array}{cc}\n\\sigma^{-}\\sigma^{+}+x_{i}\\oplus t_{j}~\\sigma^{+}\\sigma^{-} & \\sigma^{+} \\\\ \n(1+\\beta x_{i}\\oplus t_{j})\\sigma^{-} & \\sigma^{+}\\sigma^{-}%\n\\end{array}%\n\\right)  \\label{L'}\n\\end{equation}%\nwhere the matrix notation is to be read as the decomposition $%\nL=\\sum_{a,b=0,1}e_{ab}\\otimes L_{ab}$ with respect to the first factor in $%\nV(x_{i})\\otimes V(t_{j})$ and $e_{ab}$ denote the $2\\times 2$ unit matrices.\n\nThe matrix elements of these $L$-operators can be identified with weights\nfor vertex configurations using the same conventions as in \\cite[Sec 3]%\n{VicOsc}. Namely, define $(L)_{\\varepsilon _{1}\\varepsilon\n_{2}}^{\\varepsilon _{1}^{\\prime }\\varepsilon _{2}^{\\prime }}$ and $%\n(L^{\\prime })_{\\varepsilon _{1}\\varepsilon _{2}}^{\\varepsilon _{1}^{\\prime\n}\\varepsilon _{2}^{\\prime }}$ via the expansion $Lv_{\\varepsilon\n_{1}}\\otimes v_{\\varepsilon _{2}}=\\sum_{\\varepsilon _{1}^{\\prime\n},\\varepsilon _{2}^{\\prime }=0,1}(L)_{\\varepsilon _{1}\\varepsilon\n_{2}}^{\\varepsilon _{1}^{\\prime }\\varepsilon _{2}^{\\prime }}v_{\\varepsilon\n_{1}^{\\prime }}\\otimes v_{\\varepsilon _{2}^{\\prime }}$ with $\\varepsilon\n_{i},\\varepsilon _{i}^{\\prime }=0,1$. Then the coefficients can be\nexplicitly computed from (\\ref{L}), (\\ref{L'}). They are the weights of the\nvertex configurations given in Figure \\ref{fig:5vmodels} where $\\varepsilon\n_{1},\\varepsilon _{2},\\varepsilon _{1}^{\\prime },\\varepsilon _{2}^{\\prime }$\nare the values of the W, N, E and S edge of the vertex.\n\n\\begin{figure}[tbp]\n\\begin{equation*}\n\\includegraphics[scale=0.32]{5vertexGrothendieck.eps}\n\\end{equation*}%\n\\caption{Graphical depiction of the matrix elements of the $L$-operators (%\n\\protect\\ref{L}) and (\\protect\\ref{L'}) as weighted vertex configurations.}\n\\label{fig:5vmodels}\n\\end{figure}\n\n\n\\begin{proposition}\n\\label{prop:ybe}The $L,L^{\\prime }$-operators satisfy Yang-Baxter equations\nof the type%\n\\begin{equation}\nR_{12}(x_i\\ominus x_{i^{\\prime}})L_{13}(x_i|t_j)L_{23}(x_{i^{\\prime }}|t_j)=\nL_{23}(x_{i^{\\prime }}|t_j)L_{13}(x_i|t_j)R_{12}(x_i\\ominus x_{i^{\\prime }})\n\\label{YBE}\n\\end{equation}%\nand \n\\begin{equation}\nr_{23}(t_j\\ominus t_{j^{\\prime}})L_{12}(x_i|t_j)L_{13}(x_i|t_{j^{\\prime}})=\nL_{13}(x_i|t_{j^{\\prime}})L_{12}(x_i|t_j)r_{23}(t_j\\ominus t_{j^{\\prime}})\n\\label{ybe}\n\\end{equation}%\nwhere $R,r$ can be identified with $4\\times 4$ matrices respectively in $%\nV(x_i)\\otimes V(x_{i^{\\prime}})$ and $V(t_j)\\otimes V(t_{j^{\\prime}})$ with\nrespect to the basis $\\{v_0\\otimes v_0, v_0\\otimes v_1,v_1\\otimes\nv_0,v_1\\otimes v_1\\}$ and are of the form%\n\\begin{equation}\n\\left( \n\\begin{array}{cccc}\na & 0 & 0 & 0 \\\\ \n0 & b & c & 0 \\\\ \n0 & c^{\\prime } & b^{\\prime } & 0 \\\\ \n0 & 0 & 0 & a^{\\prime }%\n\\end{array}%\n\\right) \\;.  \\label{R}\n\\end{equation}%\nThe matrix entries are given in the following table for each of the\nrespective cases with $r^{\\prime},R^{\\prime}$ denoting the respective\nmatrices for $L^{\\prime}$:%\n\\begin{equation}\n\\begin{tabular}{|l||c|c|c|c|c|c|}\n\\hline\n& $a$ & $b$ & $c$ & $c^{\\prime }$ & $b^{\\prime }$ & $a^{\\prime }$ \\\\ \n\\hline\\hline\n\\multicolumn{1}{|c||}{$R$} & $1$ & $0$ & $1$ & $1+\\beta x_{i^{\\prime\n}}\\ominus x_{i}$ & $x_{i^{\\prime }}\\ominus x_{i}$ & $1$ \\\\ \\hline\n\\multicolumn{1}{|c||}{$R^{\\prime}$} & $1$ & $x_{i}\\ominus x_{i^{\\prime }}$ & \n$1$ & $1+\\beta x_{i}\\ominus x_{i^{\\prime }}$ & $0$ & $1$ \\\\ \\hline\n\\multicolumn{1}{|c||}{$r=r^{\\prime }$} & $1$ & $0$ & $1+\\beta t_{j}\\ominus\nt_{j^{\\prime }}$ & $1$ & $t_{j}\\ominus t_{j^{\\prime }}$ & $1$ \\\\ \\hline\n\\end{tabular}\n\\label{ybesolns}\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nA straightforward but rather tedious and lengthy computation which we omit.\n\\end{proof}\n\n\\begin{remark}\nThe Lax operators (\\ref{L}) and (\\ref{L'}) are 5-vertex degenerations of the\nasymmetric 6-vertex model which is used to model ferroelectrics in external\nelectromagnetic fields \\cite{Baxter}. The solutions (\\ref{L}), (\\ref{L'})\nand (\\ref{ybesolns}) of the Yang-Baxter equation are special cases of this\nmore general model. It is known that solutions of the form (\\ref{R}) exist\nif the Boltzmann weights $(a,a^{\\prime },b,b^{\\prime },c,c^{\\prime })$ for\neach $R$-matrix in the Yang-Baxter equation yield constant values for the\nfollowing two ratios,%\n\\begin{equation}\n\\Delta =\\frac{aa^{\\prime }+bb^{\\prime }-cc^{\\prime }}{2ab},\\qquad \\Gamma =%\n\\frac{a^{\\prime }b^{\\prime }}{ab}\\;.  \\label{quadrics}\n\\end{equation}%\nThis statement is originally due to Baxter \\cite{Baxter} but can also be\nfound in e.g. \\cite{Bumpetal}. For (\\ref{L'}) we find $\\Delta =-\\beta \/2$\nand $\\Gamma =0$ and the same values apply also to (\\ref{L}) after\n\\textquotedblleft spin-reversal\\textquotedblright , i.e. exchanging 0 and\n1-letters. The special point $\\beta =0$ corresponds to the so-called free\nfermion point, while $\\beta =-1$ is the value where connections with the\nalternating sign matrix conjecture and counting of plane partitions have\nbeen made in the literature; see e.g. \\cite{Bressoud} and \\cite{PZJ} as well\nas references therein.\n\\end{remark}\n\nIn what follows we concentrate on the solutions of (\\ref{YBE}) but when\ndiscussing Goresky-Kottwitz-MacPherson theory towards the end of this\narticle, the solutions of (\\ref{ybe}) will become important.\n\n\\subsection{Monodromy matrices}\n\n\n\\begin{figure}[tbp]\n\\begin{equation*}\n\\includegraphics[scale=0.5]{EKT_moms.eps}\n\\end{equation*}%\n\\caption{Graphical depiction of the $L$-operators and the monodromy\nmatrices. Each operator $L_{ij}$ is represented by a vertex in the $i$th row\nand $j$th column. The square lattice on the right then represents the\noperator (\\protect\\ref{rowmom}) over the tensor product $W_{n}\\otimes \n\\mathcal{V}$ obtained by reading out the lattice rows right to left, $%\nM_{n}\\cdots M_{2}M_{1}$. Braiding two lattice rows or two lattice columns\nleads to the matrices $R_{i+1,i}$ and $r_{j+1,j}$, respectively.}\n\\label{fig:eqc_moms}\n\\end{figure}\n\nWe will now consider products of the $L$ and $L^{\\prime }$-operators where\nthe number of factors is linked to the dimension of the ambient space $N=n+k$\nof $\\limfunc{Gr}_{n,N},$ the dimension $n$ of the hyperplanes and their\nco-dimension $k$. Namely, we consider the so-called \\emph{auxiliary spaces }%\n\\begin{equation}\nW_{r}=\\tbigotimes_{i=1}^{r}V(x_{i})\\cong \\mathcal{R}(x_{1},\\ldots\n,x_{r})\\otimes V^{\\otimes r},\\quad \\quad r=n,k  \\label{aux_space}\n\\end{equation}%\nand associate the tensor product $W_{n}\\otimes \\mathcal{V}$ with an $n\\times\nN$ square lattice (the vicious walker model) and $W_{k}\\otimes \\mathcal{V}$\nwith an $k\\times N$ square lattice (the osculating walker model); compare\nwith \\cite{VicOsc}. Consider the following operator $\\mathcal{Z}%\n:W_{n}\\otimes \\mathcal{V}\\rightarrow W_{n}\\otimes \\mathcal{V}$ \n\\begin{equation}\n\\mathcal{Z}=M_{n}\\cdots M_{2}M_{1},\\qquad M_{i}=L_{iN}\\cdots L_{i2}L_{i1}\\;,\n\\label{rowmom}\n\\end{equation}%\nwhere $M_{i}=M(x_{i}|t):V(x_{i})\\otimes \\mathcal{V}\\rightarrow\nV(x_{i})\\otimes \\mathcal{V}$ is called \\emph{row-monodromy matrix} and the $%\nL_{ij}=L_{ij}(x_{i}|t_{j})$ operators in (\\ref{rowmom}) only act\nnon-trivially in the $i$th row and $j$th column of the lattice, i.e. the $i$%\nth factor in the tensor product $W_{n}$ and the $j$th factor in $\\mathcal{V}$%\n.\n\n\n\\begin{corollary}\nThe row monodromy matrices also obey the Yang-Baxter equation \n\\begin{equation}\nR_{12}(x_i\\ominus x_{i^{\\prime }})M_{13}(x_i|t)M_{23}(x_{i^{\\prime }}|t)=\nM_{23}(x_{i^{\\prime }}|t)M_{13}(x_i|t)R_{12}(x_i\\ominus x_{i^{\\prime }})\n\\label{mom_ybe}\n\\end{equation}%\nwhere $i,i^{\\prime }=1,\\ldots ,n$ and $j,j^{\\prime }=1,\\ldots ,N$. The\nanalogous identity holds for $M^{\\prime }$.\n\\end{corollary}\n\n\\begin{proof}\nThe Yang-Baxter equations for the monodromy matrices are obtained by\nrepeated applying (\\ref{YBE}) in the definition (\\ref{rowmom}).\n\\end{proof}\n\nThe equation in (\\ref{mom_ybe}) can be seen as definition of a subalgebra in \n$\\limfunc{End}\\mathcal{V}$. Namely, for any $i$ we can decompose the row\nmonodromy matrix $M=M(x_i|t)$ defined in (\\ref{rowmom}) over the auxiliary\nspace $V(x_{i})$ as follows,%\n\\begin{equation}\nM=\\sum_{a,b=0,1}e_{ab}\\otimes M_{ab},\\qquad (M_{ab})=\\left( \n\\begin{array}{cc}\nA(x_{i}|t) & B(x_{i}|t) \\\\ \nC(x_{i}|t) & D(x_{i}|t)%\n\\end{array}%\n\\right)  \\label{row_yba}\n\\end{equation}%\nwhere $e_{ab}$ are the $2\\times 2$ unit matrices and the matrix entries $%\nA,B,C,D$ are elements in $\\limfunc{End}\\mathcal{V}$. The latter generate the\nso-called \\emph{row Yang-Baxter algebra }$\\subset \\limfunc{End}\\mathcal{V}$\nwith the commutation relations of $A,B,C,D$ given in terms of the matrix\nelements (\\ref{ybesolns}) of $R$ via (\\ref{mom_ybe}). The row Yang-Baxter\nalgebra $A^{\\prime },B^{\\prime },C^{\\prime },D^{\\prime }$ for the monodromy\nmatrix $M^{\\prime }$ associated with $L^{\\prime }$ is defined analogously.\nSimilar to the $L,L^{\\prime}$-operators $A,B,C,D$ also have a graphical\nrepresentation: they can be depicted as a single lattice row with $N$\nvertices where the values of the outer horizontal edges are fixed; see\nFigure \\ref{fig:mom}. Matrix elements of the Yang-Baxter algebra generators\nthen correspond to weighted sums over all possible vertex configurations of\nFigure \\ref{fig:5vmodels} in such a lattice row subject to the respective\nboundary condition. \n\\begin{figure}[tbp]\n\\begin{equation*}\n\\includegraphics[scale=0.9]{mom.eps}\n\\end{equation*}%\n\\caption{Graphical depiction of the monodromy matrix. Its matrix elements\ncorrespond to weighted sums over the vertex configurations defined by the $L$%\n-operator. To obtain the $A,B,C,D$-operators set respectively $(a,b)=(0,0)$, \n$(1,0)$, $(0,1)$ and $(1,1)$. }\n\\label{fig:mom}\n\\end{figure}\n\n\\subsection{Commutation relations of the Yang-Baxter algebras}\n\nWe state a number of important commutation relations of the Yang-Baxter\nalgebra elements which we will employ in subsequent sections. The first will\nbe used when deriving the spectrum of the Bethe algebra and the Bethe ansatz\nequations.\n\n\\begin{lemma}\n\\label{lem:yba1}Let $y=(y_{1},\\ldots ,y_{n})$ be an $n$-tuple of pairwise\ndistinct variables and $x\\neq y_{i}$ for all $i=1,\\ldots ,n$. Then we have\nthe identities%\n\\begin{multline}\nA(x)B(y_{1})\\cdots B(y_{n})= \\\\\n\\frac{B(y_{1})\\cdots B(y_{n})A(x)}{(x\\ominus y_{1})\\cdots (x\\ominus y_{n})}%\n-\\sum_{i=1}^{n}\\frac{B(y_{1})\\cdots \\overset{i}{B(x)}\\cdots B(y_{n})A(y_{i})%\n}{(x\\ominus y_{i})\\prod_{j\\neq i}y_{i}\\ominus y_{j}}\n\\end{multline}%\nand%\n\\begin{multline}\nD(x)B(y_{1})\\cdots B(y_{n})= \\\\\n\\frac{B(y_{1})\\cdots B(y_{n})D(x)}{(y_{1}\\ominus x)\\cdots (y_{n}\\ominus x)}%\n+\\sum_{i=1}^{n}\\frac{B(y_{1})\\cdots \\overset{i}{B(x)}\\cdots B(y_{n})D(y_{i})%\n}{(x\\ominus y_{i})\\prod_{j\\neq i}y_{j}\\ominus y_{i}}\n\\end{multline}%\nAnalogous identities hold for the Yang-Baxter algebra elements $A^{\\prime\n},B^{\\prime },D^{\\prime }$.\n\\end{lemma}\n\n\\begin{proof}\nInduction in $n$. The case $n=1$ follows from (\\ref{YBE}), namely one\ndeduces via (\\ref{row_yba}) the commutation relations%\n\\begin{eqnarray*}\n(x\\ominus y)A(x)B(y) &=&B(y)A(x)-B(x)A(y) \\\\\n(y\\ominus x)D(x)B(y) &=&B(y)D(x)-(1+\\beta\\, y\\ominus x)~B(x)D(y)\\;.\n\\end{eqnarray*}%\nFor the induction step use is made of the fact that $B(x)B(y)=B(y)B(x)$,\nwhich again is a consequence of (\\ref{YBE}) and implies that the result is\nsymmetric in the $y_{i}$'s.\n\\end{proof}\n\nThe next lemma will be used to compute the bilinear form of our generalised\ncohomology ring.\n\n\\begin{lemma}\n\\label{lem:yba2}Let $(x_{1},\\ldots ,x_{n})$ and $(y_{1},\\ldots ,y_{n})$ be\nsome mutually pairwise distinct sets of variables. Then%\n\\begin{multline}\nC(x_{1})\\cdots C(x_{n})B(y_{n})\\cdots B(y_{1})= \\\\\n\\frac{1}{\\Pi (x)}\\sum_{w}w\\left( \\frac{\\Pi (x)D(y_{1})\\cdots\nD(y_{n})A(x_{1})\\cdots A(x_{n})}{\\prod_{1\\leq i,j\\leq n}x_{i}\\ominus y_{j}}%\n\\right)\n\\end{multline}%\nwhere the sum runs over the minimal length coset representatives $w$ of $%\n\\mathbb{S}_{2n}\/\\mathbb{S}_{n}\\times \\mathbb{S}_{n}$ which act in the\nobvious manner on the alphabet $\\{x_{1},\\ldots ,x_{n},y_{1},\\ldots ,y_{n}\\}$.\n\\end{lemma}\n\n\\begin{proof}\nBy induction in $n$. The case $n=1$ follows from the commutation relation%\n\\begin{equation*}\nC(x)B(y)=\\frac{D(y)A(x)-D(x)A(y)}{x\\ominus y}=\\frac{A(x)D(y)-A(y)D(x)}{%\nx\\ominus y}\n\\end{equation*}%\nwhich is a direct consequence of (\\ref{YBE}). For the induction step one\nuses the commutation relations%\n\\begin{equation*}\nO(x)O(y)=O(y)O(x),\\qquad O=A,B,C,D\n\\end{equation*}%\nand%\n\\begin{eqnarray*}\nC(x)D(y) &=&\\frac{D(y)C(x)-D(x)C(y)}{x\\ominus y} \\\\\nA(x)B(y) &=&\\frac{B(y)A(x)-B(x)A(y)}{x\\ominus y}\n\\end{eqnarray*}%\nall of which follow once more from (\\ref{YBE}). Note that these commutation\nrelations again imply that the result must be symmetric in the $x_{i}$'s and\nsymmetric in the $y_{i}$'s. This greatly simplifies the computation.\n\\end{proof}\n\n\nAnalogous commutation relations hold for the monodromy matrix $M^{\\prime}$\nand the generators $A^{\\prime},B^{\\prime}, C^{\\prime}, D^{\\prime}$. These\ncan be derived easily from the following result which relates both\nYang-Baxter algebras in a simple manner.\n\n\n\\begin{lemma}\n\\label{lem:yba3} Let $\\Theta :\\mathcal{V}\\rightarrow \\mathcal{V}$ be the\nlinear extension of the involution $v_{\\lambda }\\mapsto v_{\\lambda ^{\\prime\n}}$. Then we have the identity%\n\\begin{equation}\n\\Theta M_{ab}(x_i|t)=M_{ba}^{\\prime }(x_i|\\ominus t^{\\prime })\\Theta\n\\label{levelrankmom}\n\\end{equation}%\nwhere $\\ominus t^{\\prime }=(\\ominus t_{N},\\ldots ,\\ominus t_{2},\\ominus\nt_{1})$.\n\\end{lemma}\n\n\\begin{proof}\nRecall the definition of the matrix elements of the $L$-operator (\\ref{L})\nvia the expansion $Lv_{\\varepsilon _{1}}\\otimes v_{\\varepsilon\n_{2}}=\\sum_{\\varepsilon _{1}^{\\prime },\\varepsilon _{2}^{\\prime\n}=0,1}(L)_{\\varepsilon _{1}\\varepsilon _{2}}^{\\varepsilon _{1}^{\\prime\n}\\varepsilon _{2}^{\\prime }}v_{\\varepsilon _{1}^{\\prime }}\\otimes\nv_{\\varepsilon _{2}^{\\prime }}$ and similarly define $(L_{ij}^{\\prime\n})_{\\varepsilon _{1}\\varepsilon _{2}}^{\\varepsilon _{1}^{\\prime }\\varepsilon\n_{2}^{\\prime }}$. The matrix elements are the weights of the vertex\nconfigurations given in Figure \\ref{fig:5vmodels} where $\\varepsilon\n_{1},\\varepsilon _{2},\\varepsilon _{1}^{\\prime },\\varepsilon _{2}^{\\prime }$\nare the values of the W, N, E and S edge of the vertex as explained earlier.\nInterchanging 0 with 1-letters attached to the vertical lines going through\nthe vertex configurations displayed in Figure \\ref{fig:5vmodels} we find \n\\begin{equation}\nL(x_{i}|t_{j})_{\\varepsilon _{1}\\varepsilon _{2}}^{\\varepsilon _{1}^{\\prime\n}\\varepsilon _{2}^{\\prime }}=L^{\\prime }(x_{i}|\\ominus t_{j})_{\\varepsilon\n_{1}^{\\prime }(1-\\varepsilon _{2})}^{\\varepsilon _{1}(1-\\varepsilon\n_{2}^{\\prime })}  \\label{LRduality_L}\n\\end{equation}%\nfor all $\\varepsilon _{i},\\varepsilon _{i}^{\\prime }=0,1$ with $i=1,2$.\n\nThe assertion for the row monodromy matrix is now an immediate consequence\nof the definition (\\ref{rowmom}) and the identity (\\ref{LRduality_L}).\n\\end{proof}\n\n\\subsection{Transposed Yang-Baxter algebras}\n\nWe will also need to consider the action of the Yang-Baxter algebra in the\ndual quantum space. The transposed monodromy matrices can be explicitly\ncomputed.\n\nDefine another pair of $L$-operators \n\\begin{equation}\nL^{\\vee }(x_i|t_j)=\\left( \n\\begin{array}{cc}\n\\sigma^{+}\\sigma^{-}+x_{i}\\ominus t_{j}~\\sigma^{-}\\sigma^{+} & \\sigma^{+} \\\\ \n(1+\\beta x_{i}\\ominus t_{j})\\sigma^{-} & \\sigma^{-}\\sigma^{+}%\n\\end{array}%\n\\right)\n\\end{equation}%\nand%\n\\begin{equation}\nL^{\\ast }(x_i|t_j)=\\left( \n\\begin{array}{cc}\n\\sigma^{-}\\sigma^{+}+x_{i}\\oplus t_{j}~\\sigma^{+}\\sigma^{-} & (1+\\beta\nx_{i}\\oplus t_{j})\\sigma^{+} \\\\ \n\\sigma^{-} & \\sigma^{+}\\sigma^{-}%\n\\end{array}%\n\\right)\\;.\n\\end{equation}%\nEmploying the latter we define \\emph{dual row monodromy matrices} $%\nM^\\vee_i=M^\\vee(x_i)$as follows%\n\\begin{equation}\nM_{i}^{\\vee }=L_{i1}^{\\vee }L_{i2}^{\\vee }\\cdots L_{iN}^{\\vee }\n\\label{dual_mom}\n\\end{equation}%\nand, similarly, $M_i^{\\ast }$ where we use the $L^{\\ast }$-operators\ninstead. The dual monodromy matrices also obey the Yang-Baxter equation,\nwhere the $R$-matrix elements are given by similar expressions as in (\\ref%\n{ybesolns}). As we shall not need their explicit form we omit them here.\n\n\\begin{lemma}\nRecall the definitions of the row monodromy matrices $M_{i},M_{i}^{\\prime\n},M_{i}^{\\vee },M_{i}^{\\ast }$ as maps $V(x_{i})\\otimes \\mathcal{V}%\n\\rightarrow V(x_{i})\\otimes \\mathcal{V}$ for some $i$. Then%\n\\begin{equation}\nM_{i}^{1\\otimes t}=(M_{i}^{\\vee })^{t\\otimes 1}\\qquad \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{and}\\qquad\n(M_{i}^{\\prime })^{1\\otimes t}=(M_{i}^{\\ast })^{t\\otimes 1},\n\\label{trans_mom}\n\\end{equation}\nwhere the upper indices $1\\otimes t$ and $t\\otimes 1$ indicate the transpose\nin respectively the quantum space $\\mathcal{V}$ and the auxiliary space $%\nV(x_{i})$ with respect to the spin basis $\\{v_\\lambda\\}$.\n\\end{lemma}\n\n\\begin{proof}\nRecall the definition of $L_{ij}:V(x_{i})\\otimes V(t_{j})\\rightarrow\nV(x_{i})\\otimes V(t_{j})$ and take the transpose in the second factor to\nfind that%\n\\begin{equation*}\nL_{ij}^{1\\otimes t}=\\left( \n\\begin{array}{cc}\n\\sigma _{j}^{-}\\sigma _{j}^{+}+x_{i}\\oplus t_{j}~\\sigma _{j}^{+}\\sigma\n_{j}^{-} & (1+\\beta x_{i}\\oplus t_{j})\\sigma _{j}^{-} \\\\ \n\\sigma _{j}^{+} & \\sigma _{j}^{+}\\sigma _{j}^{-}%\n\\end{array}%\n\\right) _{i}=(L_{ij}^{\\vee })^{t\\otimes 1}\\;.\n\\end{equation*}%\nThus, we can deduce for the monodromy matrix $M_{i}:V(x_{i})\\otimes \\mathcal{%\nV}\\rightarrow V(x_{i})\\otimes \\mathcal{V}$%\n\\begin{eqnarray*}\nM_{i}^{1\\otimes t} &=&L_{iN}^{1\\otimes t}\\cdots L_{i2}^{1\\otimes\nt}L_{i1}^{1\\otimes t} \\\\\n&=&(L_{iN}^{\\vee })^{t\\otimes 1}\\cdots (L_{i2}^{\\vee })^{t\\otimes\n1}(L_{i1}^{\\vee })^{t\\otimes 1}=(M_{i}^{\\vee })^{t\\otimes 1}\n\\end{eqnarray*}%\nThe proof of the other identity is completely analogous.\n\\end{proof}\n\n\\subsection{Quantum deformation}\n\nWe discuss a slight generalisation of the previous results which will allow\nus to introduce additional (invertible) \\textquotedblleft quantum\nparameters\\textquotedblright\\ $q_{1},\\ldots ,q_{N}$ in the monodromy\nmatrices. Consider the extension $\\mathbb{Z}[\\![q_{1},\\ldots\n,q_{n}]\\!]\\otimes \\mathcal{V}$ as quantum space and $\\mathbb{Z}%\n[\\![q_{1}^{-1},\\ldots ,q_{n}^{-1}]\\!]\\otimes W_{r}$ as auxiliary space.\n\n\\begin{lemma}\nWe have the $q$-deformed Yang-Baxter equation%\n\\begin{equation}\nr_{23}(q)L_{12}(x_{i};t_{j})\\left( \n\\begin{smallmatrix}\n1 & 0 \\\\ \n0 & q%\n\\end{smallmatrix}%\n\\right) _{i}L_{13}(x_{i};t_{j^{\\prime }})=L_{13}(x_{i};t_{j^{\\prime\n}})\\left( \n\\begin{smallmatrix}\n1 & 0 \\\\ \n0 & q%\n\\end{smallmatrix}%\n\\right) _{i}L_{12}(x_{i};t_{j})r_{23}(q),  \\label{qybe}\n\\end{equation}%\nwhere%\n\\begin{equation}\nr(q)=\\left( \n\\begin{array}{cccc}\n1 & 0 & 0 & 0 \\\\ \n0 & 0 & \\frac{1+\\beta t_{j}}{1+\\beta t_{j^{\\prime }}} & 0 \\\\ \n0 & 1 & q^{-1}\\frac{t_{j}-t_{j^{\\prime }}}{1+\\beta t_{j^{\\prime }}} & 0 \\\\ \n0 & 0 & 0 & 1%\n\\end{array}%\n\\right) \\;  \\label{qr}\n\\end{equation}\n\\end{lemma}\n\nUsing this result we can generalise our previous formulae for the monodromy\nmatrices by setting%\n\\begin{equation}\nM_{i}(q_{1},\\ldots ,q_{N}):=L_{iN}\\left( \n\\begin{smallmatrix}\n1 & 0 \\\\ \n0 & q_{N}%\n\\end{smallmatrix}%\n\\right) _{i}\\cdots L_{i2}\\left( \n\\begin{smallmatrix}\n1 & 0 \\\\ \n0 & q_{2}%\n\\end{smallmatrix}%\n\\right) _{i}L_{i1}\\left( \n\\begin{smallmatrix}\n1 & 0 \\\\ \n0 & q_{1}%\n\\end{smallmatrix}%\n\\right) _{i}  \\label{qMom}\n\\end{equation}%\nand%\n\\begin{equation}\nm_{j}(q_{j}):=L_{nj}\\left( \n\\begin{smallmatrix}\n1 & 0 \\\\ \n0 & q_{j}%\n\\end{smallmatrix}%\n\\right) _{n}\\cdots L_{2j}\\left( \n\\begin{smallmatrix}\n1 & 0 \\\\ \n0 & q_{j}%\n\\end{smallmatrix}%\n\\right) _{2}L_{1j}\\left( \n\\begin{smallmatrix}\n1 & 0 \\\\ \n0 & q_{j}%\n\\end{smallmatrix}%\n\\right) _{1}\\;.  \\label{qmom}\n\\end{equation}%\nEmploying the same type of arguments as in our previous discussion, one\nshows that these deformed monodromy matrices satisfy the same type of\nYang-Baxter relations (\\ref{qybe}) as the non-deformed ones, the only\ndifference lies in the braid matrix $r$ which is now replaced by $r(q)$. For\ndiscussing the $q\\,$-deformation of the cohomology and $K$-theory of the\nGrassmannian we need to choose $q_{1}=q$ and $q_{2}=\\cdots =q_{N}=1$. We\nshall henceforth denote $\\mathbb{Z}[\\![q]\\!]\\otimes \\mathcal{V}$ by $%\n\\mathcal{V}^{q}$ and, similarly, $\\mathbb{Z}[\\![q]\\!]\\otimes \\mathcal{V}_{n}$\nby $\\mathcal{V}_{n}^{q}$.\n\n\\subsection{Row-to-row transfer matrices}\n\nWe now introduce periodic boundary conditions in the horizontal direction of\nthe lattice by taking the partial trace of the operator (\\ref{rowmom}) over\nthe auxiliary space $V^{\\otimes n}$. We obtain the following operator $Z_{n}:%\n\\mathcal{R}[x_{1},\\ldots ,x_{n}]\\otimes \\mathcal{V}^{q}\\rightarrow \\mathcal{R%\n}[x_{1},\\ldots ,x_{n}]\\otimes \\mathcal{V}^{q}$,%\n\\begin{equation}\nZ_{n}(x|t)=\\limfunc{Tr}_{V^{\\otimes n}}\\left( \n\\begin{smallmatrix}\n1 & 0 \\\\ \n0 & q%\n\\end{smallmatrix}%\n\\right) _{n}M_{n}\\cdots \\left( \n\\begin{smallmatrix}\n1 & 0 \\\\ \n0 & q%\n\\end{smallmatrix}%\n\\right) _{2}M_{2}\\left( \n\\begin{smallmatrix}\n1 & 0 \\\\ \n0 & q%\n\\end{smallmatrix}%\n\\right) _{1}M_{1}\\;.  \\label{Zdef}\n\\end{equation}%\nThe matrix elements of the latter are partition functions of our lattice\nmodels on a cylinder. We also define an operator $Z_{k}^{\\prime }$ using\ninstead the $L^{\\prime }$-operators and replacing $n\\rightarrow k$\neverywhere.\n\n\\begin{lemma}\nDenote by $H=Z_{1}=A+qD$ and $E=Z_{1}^{\\prime }=A^{\\prime }+qD^{\\prime }$.\nWe have the relations \n\\begin{equation}\nZ_{n}(x|t)=H(x_{n}|t)\\cdots H(x_{2}|t)H(x_{1}|t)  \\label{Z2H}\n\\end{equation}%\nand%\n\\begin{equation}\nZ_{k}^{\\prime }(x|t)=E(x_{k}|t)\\cdots E(x_{2}|t)E(x_{1}|t)\\;.  \\label{Z'2E}\n\\end{equation}%\nThe operators $H,E$ are called the \\emph{row-to-row transfer matrices}.\n\\end{lemma}\n\n\\begin{proof}\nThis is immediate from the definitions (\\ref{rowmom}), (\\ref{Zdef}) and the\nfact that the $L$-operators $L_{ij},L_{i^{\\prime }j^{\\prime }}$ commute if $%\ni\\neq i^{\\prime }$ and $j\\neq j^{\\prime }$.\n\\end{proof}\n\n\\begin{corollary}\n\\label{cor:levelrankEH}We have the following identity for the row-to-row\ntransfer matrices, $\\Theta H(x|t)\\Theta =E(x|\\ominus t^{\\prime })$.\n\\end{corollary}\n\n\\begin{proof}\nEmploy (\\ref{levelrankmom}) and the defining relations $H=A+qD$, $%\nE=A^{\\prime}+qD^{\\prime}$.\n\\end{proof}\n\nThe following statement shows that the transfer matrix generate a\ncommutative subalgebra - the so-called Bethe algebra - within the\nYang-Baxter algebra which we will identify with our generalised cohomology\nring. Because of the existence of this commutative subalgebra, which should\nbe thought of as the analogue of integrals of motion of a classical\nintegrable system described in terms of differential equations, the models\nare called (quantum) \\emph{integrable}.\n\n\\begin{proposition}[Integrability]\n\\label{prop:integrability} \\emph{All} the row-to-row transfer matrices\ncommute, that is we have that \n\\begin{equation}\nH(x_{i})H(x_{i^{\\prime }})=H(x_{i^{\\prime }})H(x_{i}),\\qquad\nE(x_{i})E(x_{i^{\\prime }})=E(x_{i^{\\prime }})E(x_{i})  \\label{integrability1}\n\\end{equation}%\nas well as%\n\\begin{equation}\nH(x_{i})E(x_{i^{\\prime }})=E(x_{i^{\\prime }})H(x_{i})\\;.\n\\label{integrability2}\n\\end{equation}%\nIn particular ,the operators $Z_{n}$, $Z_{k}^{\\prime }$ are symmetric in the \n$x$-variables.\n\\end{proposition}\n\n\\begin{proof}\nThe last assertion is a direct consequence of the Yang-Baxter equation (\\ref%\n{mom_ybe}):%\n\\begin{eqnarray*}\nZ_{n}(x_{1},\\ldots ,x_{n}|t) &=&\\limfunc{Tr}_{V^{\\otimes\nn}}(R_{i,i+1}M_{n}\\cdots M_{1}R_{i,i+1}^{-1}) \\\\\n&=&\\limfunc{Tr}_{V^{\\otimes n}}(M_{n}\\cdots R_{i,i+1}M_{i}M_{i+1}\\cdots\nM_{1}R_{i,i+1}^{-1}) \\\\\n&=&\\limfunc{Tr}_{V^{\\otimes n}}(M_{n}\\cdots M_{i+1}M_{i}R_{i,i+1}\\cdots\nM_{1}R_{i,i+1}^{-1}) \\\\\n&=&\\limfunc{Tr}_{V^{\\otimes n}}(M_{n}\\cdots M_{i+1}M_{i}\\cdots M_{1}) \\\\\n&=&Z_{n}(x_{1},\\ldots ,x_{i+1},x_{i},\\ldots ,x_{n}|t)\n\\end{eqnarray*}%\nThe proof for $Z_{k}^{\\prime }$ follows along the same lines. Setting $n=k=2$\nwe obtain (\\ref{integrability1}).\n\nTo prove (\\ref{integrability2}) one establishes the existence of additional\nsolutions of the Yang-Baxter equation, \n\\begin{equation}\nR_{12}^{\\prime \\prime }(x_{i}\\oplus x_{i^{\\prime\n}})M_{13}(x_{i}|t)M_{23}^{\\prime }(x_{i^{\\prime }}|t)=M_{23}^{\\prime\n}(x_{i^{\\prime }}|t)M_{13}(x_{i}|t)R_{12}^{\\prime \\prime }(x_{i}\\oplus\nx_{i^{\\prime }})  \\label{ybe2}\n\\end{equation}%\nwhere $R^{\\prime \\prime }$ is again of the form (\\ref{R}) with \n\\begin{equation}\n\\begin{tabular}{|l||c|c|c|c|c|c|}\n\\hline\n& $a$ & $b$ & $c$ & $c^{\\prime }$ & $b^{\\prime }$ & $a^{\\prime }$ \\\\ \n\\hline\\hline\n\\multicolumn{1}{|c||}{$R^{\\prime \\prime }$} & $x_{i}\\oplus x_{i^{\\prime }}$\n& $1$ & $1+\\beta ~x_{i}\\oplus x_{i^{\\prime }}$ & $1$ & $1$ & $0$ \\\\ \\hline\n\\end{tabular}%\n\\end{equation}%\nNote that $R^{\\prime \\prime }$ is singular. However, from the Yang-Baxter\nequations (\\ref{ybe2}) one derives the commutation relations%\n\\begin{eqnarray*}\nA(x|t)A^{\\prime }(y|t) &=&A^{\\prime }(y|t)A(x|t) \\\\\nA(x|t)D^{\\prime }(y|t)-A^{\\prime }(y|t)D(x|t) &=&D^{\\prime\n}(y|t)A(x|t)-D(x|t)A^{\\prime }(y|t)\n\\end{eqnarray*}%\nfor the row Yang-Baxter algebras. Employing the graphical calculus in terms\nof the vertex configurations in Figure \\ref{fig:5vmodels} one obtains the\nadditional relations \n\\begin{equation*}\nD(x|t)D^{\\prime }(y|t)=D^{\\prime }(y|t)D(x|t)=0\\;.\n\\end{equation*}%\nFrom these equalities we then easily deduce that $H(x|t)E(y|t)=E(y|t)H(x|t)$.\n\\end{proof}\n\n\\subsection{Combinatorial description of the transfer matrices}\n\nWe now describe the action of the transfer matrices, that is the action of\nthe commutative subalgebra, in the \\emph{spin basis} $\\{v_{\\lambda\n}\\}\\subset \\mathcal{V}_{n}$ for $n\\leq N\/2$ using toric horizontal and\nvertical strips; see the earlier section on preliminaries. For $n>N\/2$ the\naction can be deduced employing Cor \\ref{cor:levelrankEH}.\n\nWe interpret partitions and their associated cylindric loops as subsets of $%\n\\mathbb{Z}^{2}$. Given a toric horizontal strip $\\theta =\\lambda \/d\/\\mu $ of\ndegree $d$ denote by\n\n\\begin{itemize}\n\\item $\\mathcal{R}_{\\theta }$ the set which contains all squares $s=\\langle\ni,j\\rangle \\in \\mathbb{Z}^{2}$, $1\\leq i\\leq n$ such that the square\nimmediately left to it, $s^{\\prime }=\\langle i,j-1\\rangle $, is the\nrightmost square in a row of $\\lambda \\lbrack d]$ intersecting $\\theta $;\n\n\\item $\\mathcal{\\bar{C}}_{\\theta }$ the set which contains all the bottom\nsquares $s=\\langle i,j\\rangle ,~1\\leq j\\leq k$ from each column of $\\mu\n\\lbrack 0]$ which does not intersect $\\theta $ as well as the squares $%\ns=\\langle 1,j\\rangle $ in empty columns if $\\lambda _{1}+n<j\\leq N$ and $\\mu\n\\subset \\lambda $.\n\\end{itemize}\n\nLikewise, given a toric vertical strip $\\theta =\\lambda \/d\/\\mu $ denote by\n\n\\begin{itemize}\n\\item $\\mathcal{\\bar{R}}_{\\theta }$ the set which contains the square $%\ns=\\langle i,j\\rangle $ next to the rightmost square $s^{\\prime }=\\langle\ni,j-1\\rangle $ in each row of $\\mu $ not intersecting $\\theta $. This\nincludes squares $s=\\langle i,1\\rangle $ in empty rows for which $1\\leq i<n$;\n\n\\item $\\mathcal{C}_{\\theta }$ the set which contains the bottom squares from\neach column of $\\lambda \\lbrack d]$ which intersects $\\theta $.\n\\end{itemize}\n\n\\begin{proposition}\n\\label{prop:comb_transfer}We have the following combinatorial action of the\ntransfer matrices on $\\mathcal{V}_{n}^{q}$ in the basis $\\{v_{\\lambda }\\}$,%\n\\begin{eqnarray*}\nH(x|t)v_{\\mu } &=&\\sum_{\\substack{ \\theta =\\lambda \/d\/\\mu \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{ }  \\\\ \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{%\nhor strip}}}q^{d}\\left( \\prod_{s\\in \\mathcal{\\bar{C}}_{\\theta }}x\\ominus\nt_{n+c(s)}\\right) \\left( \\prod_{s\\in \\mathcal{R}_{\\theta }}(1+\\beta x\\ominus\nt_{(n+c(s))\\!\\!\\!\\!\\!\\!\\mod\\!\\! N}\\right) v_{\\lambda } \\\\\nE(x|t)v_{\\mu } &=&\\sum_{\\substack{ \\theta =\\lambda \/d\/\\mu \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{ }  \\\\ \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{%\nver strip}}}q^{d}\\left( \\prod_{s\\in \\mathcal{\\bar{R}}_{\\theta }}x\\oplus\nt_{n+c(s)}\\right) \\left( \\prod_{s\\in \\mathcal{C}_{\\theta }}(1+\\beta x\\oplus\nt_{n+c(s)}\\right) v_{\\lambda }\n\\end{eqnarray*}%\nwhere the degree $d$ of the toric strips is either zero or one and $c(s)=j-i$\nis the content of the square $s=\\langle i,j\\rangle $ in the Young diagram of \n$\\lambda $ or $\\mu $.\n\\end{proposition}\n\n\\begin{proof}\nThe proof of these formulae follows along similar lines as in \\cite{VicOsc}\nand we therefore only sketch the main argument. Consider a fixed matrix\nelement $\\langle \\lambda |H(x|t)|\\mu \\rangle :=\\langle \\tilde{v}_{\\lambda\n}|H(x|t)v_{\\mu }\\rangle $ which is simply the partition function for a\nsingle lattice row where the values of the upper and lower vertical edges\nhave been fixed in terms of the binary strings $b(\\mu )$ and $b(\\lambda )$,\nrespectively. Using the bijection between binary strings $b$ and boxed\npartitions $\\lambda (b)$ from Section \\ref{sec:partitions} one can translate\nthe various vertex configurations in Figure \\ref{fig:5vmodels}, which\nrepresent matrix elements of the $L$ and $L^{\\prime }$-operators, into the\noperation of adding boxes to the Young diagram of $\\mu $. For example, the\nfirst and second vertex configuration in the top row of Figure \\ref%\n{fig:5vmodels} leave the Young diagram of $\\mu $ unchanged, the fourth and\nfifth vertex configurations signal respectively the end and start of a\nhorizontal strip being added to $\\mu $, while the third vertex in the top\nrow corresponds to two boxes being added in the same row. Similarly, the\nfirst two vertex configurations in the bottom row of \\ref{fig:5vmodels} do\nnot add any boxes to $\\mu $, the fourth and fifth signal the start and end\nof a vertical strip, while the third vertex in the bottom row indicates that\ntwo boxes are added in the same column. Using these equivalences between\nhorizontal (vertical) strips the above formulae follow from the weights\nfixed via the definitions (\\ref{L}) and (\\ref{L'}).\n\\end{proof}\n\n\\begin{figure}[tbp]\n\\begin{equation*}\n\\includegraphics[scale=0.35]{gr13_a.eps}\n\\end{equation*}%\n\\caption{Lattice configurations for the projective space $\\mathbb{P}^2$ and\ntheir weights; see Figure \\protect\\ref{fig:5vmodels}. The values of the top\nedges of the vertices are fixed by the binary string 010. Below the weights\nare the corresponding toric skew diagrams; see Proposition \\protect\\ref%\n{prop:comb_transfer} and Example \\protect\\ref{ex:1}. The dotted boxes are\nthe cylindric continuation (\\protect\\ref{cyl_loop}) of the solid Young\ndiagrams; see Section \\protect\\ref{sec:torictab}}\n\\label{fig:gr13ex}\n\\end{figure}\n\n\\begin{example}\n\\label{ex:1} Consider the simplest non-trivial case $\\limfunc{Gr}_{1,3}=%\n\\mathbb{P}^{2}$, i.e. we set $N=3$ and $n=1$. In terms of binary strings $%\n\\mathcal{V}_{1}$ is spanned by $\\{v_{100},v_{010},v_{001}\\}$. We consider\nthe matrix elements of $H(x)$ in this basis, which can be visualised as a\nsum over all the possible vertex configurations shown in Figure \\ref%\n{fig:5vmodels} occurring in a single lattice row of length $N=3$. Drawing\nall these allowed lattice configurations with fixed binary strings $010$ and \n$001 $ on the top edges, we arrive at Figures \\ref{fig:gr13ex} and \\ref%\n{fig:gr13exb} with the product of the respective vertex weights shown below.\nWe now convert the binary strings into partitions with bounding box $1\\times\n2$ to obtain toric horizontal strips; see Section \\ref{sec:torictab}.\n\nStarting from the left in Figure \\ref{fig:gr13ex} the first lattice\nconfiguration is the matrix element $\\langle \\tilde{v}_{010}|H(x)v_{010}%\n\\rangle $. The binary string $010$ is the partition with one square at\nposition $\\langle 1,1\\rangle $ and we have $\\lambda =\\mu =(1)$, that is an\nempty horizontal strip where no box is added and $d=0$. Thus, $\\mathcal{R}%\n_{\\lambda \/\\mu }=\\emptyset $ and $\\mathcal{\\bar{C}}_{\\lambda \/\\mu\n}=\\{\\langle 1,1\\rangle ,\\langle 1,3\\rangle \\}$ where the last square in $%\n\\mathcal{\\bar{C}}_{\\lambda \/\\mu }$ belongs to an empty column with the\ncolumn number $j$ obeying the stated condition $1<j=3\\leq N$. According to\nProp \\ref{prop:comb_transfer} we arrive at the weight \n\\begin{equation*}\n\\langle \\tilde{v}_{010}|H(x)v_{010}\\rangle =(x\\ominus t_{1})(x\\ominus\nt_{3})\\,.\n\\end{equation*}\n\nThe next lattice configuration is the matrix element $\\langle \\tilde{v}%\n_{001}|H(x)v_{010}\\rangle $ with $\\lambda =(2),\\mu =(1)$. Thus, we have the\nhorizontal strip $\\theta =\\lambda \/\\mu $ with one square at $\\langle\n1,2\\rangle $ and $d=0$. The sets appearing in the formula of Prop \\ref%\n{prop:comb_transfer} are $\\mathcal{R}_{\\lambda \/\\mu }=\\{\\langle 1,3\\rangle\n\\} $ and $\\mathcal{\\bar{C}}_{\\lambda \/\\mu }=\\{\\langle 1,1\\rangle \\}$, since\nthe square $\\langle 1,3\\rangle $ is adjacent to the square $\\langle\n1,2\\rangle $ which appears in a row intersecting $\\lambda \/\\mu $ while the\nsquare $\\langle 1,1\\rangle $ is the bottom square in a column not\nintersecting $\\lambda \/\\mu $. Hence, \n\\begin{equation*}\n\\langle \\tilde{v}_{001}|H(x)v_{010}\\rangle =(x\\ominus t_{1})(1+\\beta\nx\\ominus t_{3})\\,.\n\\end{equation*}\n\nThe last lattice configuration in the top row is the matrix element $\\langle \n\\tilde{v}_{100}|H(x)v_{010}\\rangle $ with $\\lambda =(0),\\mu =(1)$. Now, we\nhave a toric strip with $d=1$, that is $\\lambda \/1\/\\mu =\\{\\langle 1,2\\rangle\n,\\langle 1,3\\rangle \\}$. The first column with the square at $\\langle\n1,1\\rangle $ now intersects $\\lambda \/1\/\\mu $, because the square at $%\n\\langle 2,1\\rangle $ is in the cylindric loop $\\lambda \\lbrack 1]$.\nTherefore, $\\mathcal{R}_{\\lambda \/1\/\\mu }=\\{\\langle 1,4\\rangle \\}$, $%\n\\mathcal{\\bar{C}}_{\\lambda \/1\/\\mu }=\\emptyset $ and \n\\begin{equation*}\n\\langle \\tilde{v}_{100}|H(x)v_{010}\\rangle =q(1+\\beta x\\ominus t_{1})\\,.\n\\end{equation*}%\nIn summary, we have the action (compare with Prop \\ref{prop:comb_transfer}), \n\\begin{equation*}\nH(x)v_{010}=(x\\ominus t_{1})(x\\ominus t_{3})v_{010}+(x\\ominus t_{1})(1+\\beta\nx\\ominus t_{3})v_{001}+q(1+\\beta x\\ominus t_{1})v_{100}\\;.\n\\end{equation*}%\nWe leave the verification of the weights in Figure \\ref{fig:gr13exb} to the\nreader.\n\\begin{figure}[tbp]\n\\begin{equation*}\n\\includegraphics[scale=0.35]{gr13_b.eps}\n\\end{equation*}%\n\\caption{Lattice configurations for $\\mathbb{P}^{2}$ and binary string $001$%\n, their weights and corresponding toric strips; see Figure \\protect\\ref%\n{fig:5vmodels}.}\n\\label{fig:gr13exb}\n\\end{figure}\n\\end{example}\n\nLet $t^{\\ast }=(\\ominus t_{N},\\ldots ,\\ominus t_{2},\\ominus t_{1})$ and on\neach $\\mathcal{V}_{n}^{q}$ define operators $\\{H_{r}\\}_{r=1}^{k}$ and $%\n\\{E_{r}\\}_{r=1}^{n}$ through the expansions%\n\\begin{eqnarray}\nH(x|t)|_{\\mathcal{V}_{n}^{q}} &=&(x|t^{\\ast })^{k}\\cdot \\mathbf{1}_{\\mathcal{%\nV}_{n}^{q}}+(1+\\beta x)\\sum_{r=1}^{k}H_{r}~\\frac{(x|t^{\\ast })^{k-r}}{%\n1+\\beta t_{n+r}},  \\label{Hr} \\\\\nE(x|t)|_{\\mathcal{V}_{n}^{q}} &=&(x|t)^{n}\\cdot \\mathbf{1}_{\\mathcal{V}%\n_{n}^{q}}+(1+\\beta x)\\sum_{r=1}^{n}E_{r}~(1+\\beta t_{n+1-r})(x|t)^{n-r}\\;,\n\\label{Er}\n\\end{eqnarray}%\nwhere $(x|t)^{r}=\\prod_{j=1}^{r}(x\\oplus t_{j})$ are the factorial powers (%\n\\ref{facpower}) with respect to the group law (\\ref{group_law}). Below we\nwill relate the operator coefficients in these expansions to the Pieri rules\nin $qh^\\ast_n$. Setting $\\beta=0$ they correspond to the generators in\nMihalcea's coordinate ring representation of equivariant quantum cohomology \n\\cite[Thm 1.1]{Mihalcea}.\n\n\\begin{corollary}\nThe operators $\\{E_{r}\\}_{r=1}^{n}\\cup \\{H_{r}\\}_{r=1}^{k}$ generate a\ncommutative subalgebra $\\subset \\limfunc{End}\\mathcal{V}_{n}^{q}$ and we\nhave the formulae ($t_{j}^{\\prime }=t_{N+1-j}$)%\n\\begin{eqnarray}\nH_{k+1-i} &=&\\sum_{j=1}^{i}\\frac{H(t_{j}^{\\prime })}{\\prod\\nolimits_{1\\leq\n\\ell \\neq j\\leq i}t_{j}^{\\prime }\\ominus t_{\\ell }^{\\prime }},\\qquad\ni=1,\\ldots ,k  \\label{Hr2} \\\\\nE_{n+1-i} &=&\\sum_{j=1}^{i}\\frac{E(\\ominus t_{j})}{\\prod_{1\\leq \\ell \\neq\nj\\leq i}t_{\\ell }\\ominus t_{j}},\\qquad i=1,\\ldots ,n\\;.  \\label{Er2}\n\\end{eqnarray}%\nIn particular, for $i=1$ we have $H_{k}=H(t_{N})$ and $E_{n}=E(t_{1})$.\n\\end{corollary}\n\n\\begin{proof}\nSetting $x=t_{i}$ in (\\ref{Hr}) and $x=\\ominus t_{i}$ in (\\ref{Er}) we\nobtain a linear system of equations expressing $H(t_{i})$ and $E(\\ominus\nt_{i})$ in terms of the (operator) coefficients $H_{r}$ and $E_{r}$\nrespectively. The corresponding matrices are lower triangular and therefore\ncan be easily inverted to produce the stated expressions.\n\nIt follows from Prop \\ref{prop:integrability} that all these operators\ncommute.\n\\end{proof}\n\nTogether with Prop \\ref{prop:comb_transfer} the last result allows one to\ncompute the action of $H_{r}$ and $E_{r}$ in the spin-basis $\\{v_{\\lambda\n}\\}\\subset \\mathcal{V}_{n}$.\n\n\\begin{example}\n\\label{ex:2} We continue Example \\ref{ex:1} with $\\limfunc{Gr}_{1,3}=\\mathbb{%\nP}^{2}$. It follows from (\\ref{Hr2}) that%\n\\begin{equation*}\nH_{1}=\\frac{H(t_{2})}{t_{2}\\ominus t_{3}}+\\frac{H(t_{3})}{t_{3}\\ominus t_{2}}%\n,\\qquad H_{2}=H(t_{3})\\;.\n\\end{equation*}%\nEmploying the weights shown in Figure \\ref{fig:gr13ex}, \n\\begin{eqnarray*}\n\\langle \\tilde{v}_{010}|H(x)v_{010}\\rangle &=&(x\\ominus t_{1})(x\\ominus\nt_{3}), \\\\\n\\langle \\tilde{v}_{001}|H(x)v_{010}\\rangle &=&(x\\ominus t_{1})(1+\\beta\nx\\ominus t_{3}) \\\\\n\\langle \\tilde{v}_{100}|H(x)v_{010}\\rangle &=&q(1+\\beta x\\ominus t_{1})\n\\end{eqnarray*}%\nwe arrive at the matrix elements \n\\begin{eqnarray*}\n\\langle \\tilde{v}_{010}|H_{1}v_{010}\\rangle &=&\\frac{(t_{2}\\ominus\nt_{1})(t_{2}\\ominus t_{3})}{t_{2}\\ominus t_{3}}+0=t_{2}\\ominus t_{1} \\\\\n\\langle \\tilde{v}_{001}|H_{1}v_{010}\\rangle &=&\\frac{(t_{2}\\ominus\nt_{1})(1+\\beta t_{2}\\ominus t_{3})}{t_{2}\\ominus t_{3}}+\\frac{t_{3}\\ominus\nt_{1}}{t_{3}\\ominus t_{2}}=1+\\beta ~t_{2}\\ominus t_{1} \\\\\n\\langle \\tilde{v}_{100}|H_{1}v_{010}\\rangle &=&q\\frac{(1+\\beta t_{2}\\ominus\nt_{1})}{t_{2}\\ominus t_{3}}+q\\frac{(1+\\beta t_{3}\\ominus t_{1})}{%\nt_{3}\\ominus t_{2}}=0\n\\end{eqnarray*}%\nFrom these we obtain,%\n\\begin{equation}\nH_{1}v_{1}=t_{2}\\ominus t_{1}v_{1}+(1+\\beta t_{2}\\ominus t_{1})v_{2}\\;.\n\\label{Gr13Pieri1}\n\\end{equation}%\nIn an analogous fashion one finds,%\n\\begin{equation*}\nH_{2}v_{1}=t_{3}\\ominus t_{1}v_{2}+q(1+\\beta t_{1}\\ominus t_{1})v_{\\emptyset\n}\n\\end{equation*}%\nand using the weights in Figure \\ref{fig:gr13exb} \n\\begin{gather}\nH_{1}v_{2}=t_{3}\\ominus t_{1}v_{2}+q(1+\\beta t_{3}\\ominus t_{1})v_{\\emptyset\n}  \\label{Gr13Pieri2} \\\\\nH_{2}v_{2}=(t_{3}\\ominus t_{2})(t_{3}\\ominus t_{1})v_{2}+q(t_{3}\\ominus\nt_{2})(1+\\beta t_{3}\\ominus t_{1})v_{\\emptyset }+q(1+\\beta t_{3}\\ominus\nt_{2})v_{1}  \\notag\n\\end{gather}%\nBelow we will define a product by $v_{r}\\circledast v_{s}=H_{r}v_{s}$. Upon\nsetting $\\beta =-1$ and $t_{4-i}=1-e^{\\varepsilon _{i}}$ with $i=1,2,3$ the\nabove formulae then match the product expansions for quantum equivariant\nK-theory of $\\mathbb{P}^{2}$ stated by Buch and Mihalcea in \\cite[Sec 5.5]%\n{BM}.\n\\end{example}\n\n\\subsubsection{Functional relation \\& quantum Pieri-Chevalley rule}\n\nThe coefficients (\\ref{Hr2}) and (\\ref{Er2}) of the transfer matrices are\nalgebraically dependent. We now derive the functional relation (\\ref{ideal\ndef}) which allows one to deduce this dependence and as a byproduct of our\ncomputation we give an explicit formula for the action of $H_{1}$ in the\nspin basis.\n\nLet $u_{j}=\\sigma _{j}^{-}\\sigma _{j+1}^{+}$ for $j=1,\\ldots ,N-1$ and $%\nu_{N}=q\\sigma _{1}^{+}\\sigma _{N}^{-}$. Define the following operator on $%\n\\mathcal{V}^{q},$%\n\\begin{equation}\n\\bar{H}_{1}=\\sum_{j=1}^{N}u_{j}+\\beta \\sum_{|j_{1}-j_{2}|\\func{mod}%\nN>1}u_{j_{1}}u_{j_{2}}+\\beta ^{2}\\sum_{|j_{a}-j_{b}|\\func{mod}%\nN>1}u_{j_{1}}u_{j_{2}}u_{j_{3}}+\\cdots  \\label{h1}\n\\end{equation}%\nas a formal power series in $\\beta $. Note that the sums only run over\nindices where $|j_{a}-j_{b}|\\func{mod}N>1$ which ensures that all the $u_{j}$%\n's in each monomial commute. Obviously, only finitely many terms act\nnon-trivially for finite $N$ and the series therefore terminates.\n\n\\begin{lemma}\nActing with $\\bar{H}_{1}$ on a spin basis vector $v_{\\mu }\\in \\mathcal{V}%\n_{n} $ one obtains%\n\\begin{equation}\n\\bar{H}_{1}v_{\\mu }=\\sum_{\\substack{ \\mu \\rightrightarrows ^{\\ast }\\lambda\n\\lbrack d]  \\\\ d=0,1}}q^{d}\\beta ^{|\\lambda \/d\/\\mu |-1}v_{\\lambda },\n\\label{h1action}\n\\end{equation}%\nwhere the sum runs over all boxed partitions $\\lambda \\subset (k^{n})$ such\nthat either $\\lambda \/0\/\\mu =\\lambda \/\\mu $ or $\\lambda \/1\/\\mu $ are toric\ndiagrams which contain at most one box in each column and row and $\\lambda\n\\neq \\mu $.\n\\end{lemma}\n\n\\begin{proof}\nUsing the bijection between binary strings and partitions detailed in\nSection \\ref{sec:partitions} and the definition of cylindric loops in\nSection \\ref{sec:torictab}, one proves that either $u_{j}v_{\\mu\n}=q^{d}v_{\\lambda }$ where one adds a box with coordinates $(x,y)$ and $%\nj=n+y-x$ to obtain $\\lambda $ (or $\\lambda \\lbrack 1]$ if $d=1$ and $j=N$)\nor $u_{j}v_{\\mu }=0$. The assertion then easily follows from the fact that\nall $u_{j}$'s in each monomial term commute.\n\\end{proof}\n\n\\begin{proposition}\nThe transfer matrices obey the following functional operator identity%\n\\begin{equation}\nH(x|t)E(\\ominus x|t)=(1+\\beta \\bar{H}_{1})\\prod_{j=1}^{N}(t_{j}\\ominus\nx)^{\\sigma _{j}^{+}\\sigma _{j}^{-}}(x\\ominus t_{j})^{\\sigma _{j}^{-}\\sigma\n_{j}^{+}}+q\\cdot 1\\;.  \\label{func_eqn1}\n\\end{equation}%\nIn particular, we have that $H(t_{j}|t)E(\\ominus t_{j}|t)=q\\cdot 1$ for all $%\nj=1,\\ldots ,N$ which amount to non-trivial identities between the\ncoefficients $\\{H_{r}\\}$ and $\\{E_{r}\\}$ defined in (\\ref{Hr}), (\\ref{Er}).\n\\end{proposition}\n\n\\begin{proof}\n\\begin{figure}[tbp]\n\\begin{equation*}\n\\includegraphics[scale=0.35]{func_eqn.eps}\n\\end{equation*}%\n\\caption{The vertex configurations corresponding to the operator $%\nL_{i+1j}^{\\prime }(\\ominus x_{i})L_{i,j}(x_{i})$.}\n\\label{fig:func_eqn}\n\\end{figure}\nA computation along similar lines as in \\cite{VicOsc}. The idea is to\nanalyse the action of $\\mathcal{L}_{j}=L_{1j}^{\\prime }(\\ominus\nx)L_{2j}(x):W(x)\\otimes V(t_{j})\\rightarrow W(x)\\otimes V(t_{j})$ where $%\nW(x)=V(\\ominus x)\\otimes V(x)=\\mathcal{R}(x)\\otimes V^{\\otimes 2}$ with\nrespect to the basis vectors%\n\\begin{eqnarray*}\nw_{0} &=&v_{0}\\otimes v_{0},\\quad w_{1}=v_{0}\\otimes v_{1}+v_{1}\\otimes\nv_{0}, \\\\\nw_{1^{\\prime }} &=&v_{0}\\otimes v_{1},\\quad w_{2}=v_{1}\\otimes v_{1}~.\n\\end{eqnarray*}%\nWe find that%\n\\begin{eqnarray*}\n\\mathcal{L}_{j}w_{0}\\otimes v_{0} &=&x\\ominus t_{j}~w_{0}\\otimes v_{0} \\\\\n\\mathcal{L}_{j}w_{0}\\otimes v_{1} &=&t_{j}\\ominus x~w_{0}\\otimes\nv_{1}+(1+\\beta t_{j}\\ominus x)w_{1}\\otimes v_{0}-\\beta t_{j}\\ominus\nx~w_{1^{\\prime }}\\otimes v_{0} \\\\\n\\mathcal{L}_{j}w_{1}\\otimes v_{0} &=&w_{1}\\otimes v_{0} \\\\\n\\mathcal{L}_{j}w_{1}\\otimes v_{1} &=&w_{1}\\otimes v_{1} \\\\\n\\mathcal{L}_{j}w_{1^{\\prime }}\\otimes v_{0} &=&w_1\\otimes v_0-x\\ominus\nt_{j}~w_{0}\\otimes v_{1} \\\\\n\\mathcal{L}_{j}w_{1^{\\prime }}\\otimes v_{1} &=&0 \\\\\n\\mathcal{L}_{j}w_{2}\\otimes v_{0} &=&(1+\\beta x\\ominus t_{j})w_{1}\\otimes\nv_{1}-\\beta x\\ominus t_{j}~w_{1^{\\prime }}\\otimes v_{1} \\\\\n\\mathcal{L}_{j}w_{2}\\otimes v_{1} &=&0\n\\end{eqnarray*}%\nThis action of the $\\mathcal{L}_{j}$ in the spin basis (\\ref{spin basis})\ncan be encoded in terms of the vertex configurations shown in Figure \\ref%\n{fig:func_eqn} with labels $0,1,1^{\\prime },2$ similarly as we did deduce\nthe action of $L$ and $L^{\\prime }$ from the vertex configurations in Figure %\n\\ref{fig:5vmodels}. Thus, the operator product $H(x)E(\\ominus x)$ can be\nwritten as the partial trace%\n\\begin{equation*}\nE(\\ominus x)H(x)=\\limfunc{Tr}_{V\\otimes V}\\left( \n\\begin{smallmatrix}\n1 & 0 & 0 & 0 \\\\ \n0 & q & 0 & 0 \\\\ \n0 & 0 & q & 0 \\\\ \n0 & 0 & 0 & q^{2}%\n\\end{smallmatrix}\n\\right) \\mathcal{L}_{N}\\cdots \\mathcal{L}_{2}\\mathcal{L}_{1}\n\\end{equation*}\nand its matrix elements in the quantum space $\\mathcal{V}_{n}^{q}$ are sums\nover the possible vertex configurations of Figure \\ref{fig:func_eqn} in a\nsingle lattice row of length $N$. This lattice row is closed and forms a\ncircle of circumference $N$, since the partial trace together with the\nmatrix containing the deformation parameter $q$ imposes quasi-periodic\nboundary conditions. Due to these periodicity conditions, one finds the\nfollowing constraints:\n\n\\begin{itemize}\n\\item the last vertex in the bottom row of Figure \\ref{fig:func_eqn} cannot\noccur;\n\n\\item the 2nd and 3rd vertex from the right in the top row always have to\ncome as a pair, but since one of them has weight zero their contribution can\nbe discarded;\n\n\\item configurations involving the second vertex from the left in the bottom\nrow do not contribute as they eventually lead to a vertex configuration\nshown at the 2nd position from the right in the top row which has weight\nzero;\n\n\\item the 2nd and 3rd vertex from the right in the bottom row always have to\ncome as an adjacent pair and it are these vertices which give rise to the\nterm involving $\\beta \\bar H_1$ as they correspond to shifting a 1-letter in\na binary string to the right.\n\\end{itemize}\n\nFrom these conditions, which can be checked graphically, one then deduces\nthe asserted identity (\\ref{func_eqn}) as only a very restricted number of\nvertices in Figure \\ref{fig:func_eqn} remain.\n\\end{proof}\n\n\\begin{corollary}[equivariant quantum Pieri-Chevalley rule]\nWe have the following explicit action of $H_{1}$ in terms of the basis $%\n\\{v_{\\lambda }\\}\\subset \\mathcal{V}_{n},$%\n\\begin{equation}\n(1+\\beta H_{1})v_{\\mu }=\\frac{\\Pi (t_{\\mu })}{\\Pi (t_{\\emptyset })}\\sum \n_{\\substack{ \\mu\\rightrightarrows\\lambda[d]  \\\\ d=0,1}}q^{d}\\beta ^{|\\lambda\n\/d\/\\mu |}v_{\\lambda },  \\label{H1}\n\\end{equation}%\nwhere the sum runs over all $\\lambda \\subset (k^{n})$ such that either $%\n\\lambda \/\\mu $ or $\\lambda \/1\/\\mu $ is a skew diagram which contains at most\none box in each column or row. Moreover, the identity (\\ref{func_eqn1}) can\nbe rewritten as \n\\begin{equation}\nH(x|t)E(\\ominus x|t)=\\prod_{j=1}^{n}(t_{j}\\ominus\nx)\\prod_{j=n+1}^{N}(x\\ominus t_{j})\\,(1+\\beta H_{1})+q\\cdot 1\\;.\n\\label{func_eqn}\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\nActing with the first term on the right hand side of (\\ref{func_eqn1}) on a\nbasis vector $v_{\\lambda }$ we obtain%\n\\begin{multline*}\n(1+\\beta \\bar{H}_{1})\\prod_{j=1}^{N}(t_{j}\\ominus x)^{\\sigma _{j}^{+}\\sigma\n_{j}^{-}}(x\\ominus t_{j})^{\\sigma _{j}^{-}\\sigma _{j}^{+}}v_{\\lambda }= \\\\\n\\left( \\tprod\\nolimits_{j\\in I_{\\lambda }}t_{j}\\ominus x\\right) \\left(\n\\tprod\\nolimits_{j\\in I_{\\lambda ^{\\ast }}}x\\ominus t_{j}\\right) ~(1+\\beta \n\\bar{H}_{1})v_{\\lambda }= \\\\\n\\left( \\tprod\\nolimits_{j=1}^{n}t_{j}\\ominus x\\right) \\left(\n\\tprod\\nolimits_{j=n+1}^{N}x\\ominus t_{j}\\right) ~\\frac{\\Pi (t_{\\lambda })}{%\n\\Pi (t_{\\emptyset })}(1+\\beta \\bar{H}_{1})v_{\\lambda }\n\\end{multline*}%\nOn the other hand using the expansions (\\ref{Hr}) and (\\ref{Er}) we see that\nthe coefficients of the leading factorial powers are%\n\\begin{eqnarray*}\nH(x|t) &=&(x|\\ominus t^{\\prime })^{k}(1+\\beta H_{1})+\\cdots \\\\\nE(x|t) &=&(x|t)^{n}(1+\\beta E_{1})+\\cdots\n\\end{eqnarray*}%\nfrom which we deduce the desired identities with help of the left hand side\nof (\\ref{func_eqn1}) and (\\ref{facpower_id}). Namely, we have%\n\\begin{eqnarray*}\n(-1)^{n}\\frac{(1+\\beta x)^{n}}{\\Pi (t_{\\emptyset })}~E(\\ominus x)\n&=&(x|\\ominus t)^{n}\\sum_{r=0}^{n}(-1)^{r}\\beta ^{r}(E_{r}+\\beta\nE_{r+1})+~\\ldots \\\\\n&=&(x|\\ominus t)^{n}\\cdot 1+~\\ldots\n\\end{eqnarray*}%\nwhere the omitted terms involve factorial powers $(x|\\ominus t)^{p}$ with $%\np<n$ and we have set $E_{0}=1$, $E_{n+1}=0$. Thus, \n\\begin{equation*}\n(-1)^{n}\\frac{(1+\\beta x)^{n}}{\\Pi (t_{\\emptyset })}~E(\\ominus\nx)H(x)=(x|\\ominus t)^{N}(1+\\beta H_{1})+~\\ldots\n\\end{equation*}%\nand the assertion follows.\n\\end{proof}\n\n\\begin{example}\n\\label{ex:3} We consider once more the example $\\limfunc{Gr}_{1,3}=\\mathbb{P}%\n^{2}$. It follows from (\\ref{Gr13Pieri1}) and (\\ref{Gr13Pieri2}) in Example %\n\\ref{ex:2} that%\n\\begin{eqnarray*}\n(1+\\beta H_{1})v_{010} &=&(1+\\beta t_{2}\\ominus t_{1})(v_{010}+v_{001}) \\\\\n(1+\\beta H_{1})v_{001} &=&(1+\\beta t_{3}\\ominus t_{1})(v_{001}+qv_{100})\\;.\n\\end{eqnarray*}\nWe compare this against the quantum Pieri-Chevalley rule (\\ref{H1}). The\nbinary string $010$ corresponds to the partition $\\mu =(1)$ with a single\nbox and $001$ to the partition $\\mu =(2)$. Thus, in the first case the only\npartitions $\\lambda $ for which $\\lambda \/\\mu $ contains at most a single\nbox in each row and column are $\\lambda =(1)$ and $\\lambda =(2)$. For $\\mu\n=(2)$ we obtain $\\lambda =(2)$ and $\\lambda =\\emptyset $, since in the\nlatter case the cylindric loop $\\lambda \\lbrack 1]$ contains 3 boxes and $%\n\\lambda \/1\/\\mu $ has one box in one row.\n\nLet us now consider the functional relation (\\ref{func_eqn}). For $N=3$ and $%\nn=1$ expand the transfer matrices into factorial powers as follows%\n\\begin{multline*}\nH(x)=(x\\ominus t_{2})(x\\ominus t_{3})(1+\\beta H_{1})+(x\\ominus\nt_{3})(H_{1}+\\beta H_{2})+H_{2} \\\\\n-\\frac{1+\\beta x}{1+\\beta t_{1}}E(\\ominus x)=(x\\ominus t_{1})(1+\\beta E_{1})-%\n\\frac{1+\\beta x}{1+\\beta t_{1}}E_{2}=(x\\ominus t_{1})-E_{1}\n\\end{multline*}%\nThe left hand side of (\\ref{func_eqn}) yields%\n\\begin{multline*}\n-\\frac{1+\\beta x}{1+\\beta t_{1}}E(\\ominus x)H(x)=(x|\\ominus t)^{3}(1+\\beta\nH_{1}) \\\\\n-(x\\ominus t_{2})(x\\ominus t_{3})[(1+\\beta H_{1})E_{1}-(1+\\beta t_{2}\\ominus\nt_{1})(H_{1}+\\beta H_{2})] \\\\\n+(x\\ominus t_{3})[(1+\\beta t_{3}\\ominus t_{1})H_{2}+(t_{2}\\ominus\nt_{1}-E_{1})(H_{1}+\\beta H_{2})] \\\\\n-E_{1}H_{2}(1-t_{3}\\ominus t_{1})\n\\end{multline*}%\nwhile the right hand side reads \n\\begin{multline*}\n-\\frac{1+\\beta x}{1+\\beta t_{1}}E(\\ominus x)H(x)=(x|\\ominus t)^{3}(1+\\beta\nH_{1})-\\frac{1+\\beta x}{1+\\beta t_{1}}~q \\\\\n=(x|\\ominus t)^{3}(1+\\beta H_{1})-(1+\\beta t_{3}\\ominus t_{1})q[1+\\beta\n(x\\ominus t_{3})]\n\\end{multline*}%\nComparing the coefficients of each factorial power we obtain%\n\\begin{gather*}\nE_{1}H_{2}(1-t_{3}\\ominus t_{1})=(1+\\beta t_{3}\\ominus t_{1})q \\\\\n(E_{1}-t_{2}\\ominus t_{1})(H_{1}+\\beta H_{2})-(1+\\beta t_{3}\\ominus\nt_{1})H_{2}=\\beta (1+\\beta t_{3}\\ominus t_{1})q \\\\\n(1+\\beta H_{1})E_{1}=(1+\\beta t_{2}\\ominus t_{1})(H_{1}+\\beta H_{2})\n\\end{gather*}%\nWe will see in a subsequent section that there is an easier way to describe\nthe ideal which avoids these rather complicated looking relations. However,\nin the non-equivariant limit where $t_{j}=0$ for $j=1,2,3$, they simplify to%\n\\begin{equation*}\nE_{1}=H_{1},\\qquad E_{1}^{2}=H_{2},\\qquad E_{1}^{3}=q\\;.\n\\end{equation*}\n\\end{example}\n\n\n\\section{Bethe vectors as idempotents}\n\nWe now consider the eigenvalue problem of the transfer matrices. Eigenvalues\nand eigenvectors can be explicitly contructed using the Yang-Baxter algebra,\nthis general approach is known as \\emph{quantum inverse scattering method }%\nor \\emph{algebraic Bethe ansatz}. Using the eigenvectors, called \\emph{Bethe\nvectors} in the quantum integrable systems literature, we then define for\neach subspace $\\mathcal{V}_{n}^{q}$ a generalised matrix ring $qh_{n}^{\\ast\n} $ by identifying appropriate renormalised versions of the Bethe vectors as\nits idempotents.\n\n\\subsection{Bethe vectors \\& factorial Grothendieck polynomials}\n\nLet $y=(y_{1},\\ldots ,y_{n})$ and $z=(z_{1},\\ldots ,z_{k})$ be some\nindeterminates. Recall McNamara's definition of factorial Grothendieck\npolynomials from Section \\ref{sec:grothendieck} and the definition of the\nYang-Baxter algebra (\\ref{row_yba}).\n\n\\begin{proposition}\nLet $\\lambda \\subset (k^{n})$. Then we have the following equalities for the \n$C$ and $B^{\\prime }$-operators,%\n\\begin{eqnarray}\nC(y_{1})\\cdots C(y_{n})v_{\\lambda } &=&G_{\\lambda }(y|\\ominus\nt)~v_{0}\\otimes \\cdots \\otimes v_{0}  \\label{Caction} \\\\\nB^{\\prime }(z_{1})\\cdots B^{\\prime }(z_{k})v_{\\lambda } &=&G_{\\lambda\n^{\\prime }}(z|t^{\\prime })~v_{1}\\otimes \\cdots \\otimes v_{1}\n\\label{B'action}\n\\end{eqnarray}%\nwhen acting on the basis vector $v_{\\lambda }$ in $\\mathcal{V}_{n}$.\n\\end{proposition}\n\n\\begin{proof}\nWe only sketch the proof leaving technical details to the reader to verify.\nSince $\\lambda \\subset (k^{n})$ the corresponding binary string $b(\\lambda )$\ncontains $n$ 1-letters. From the definition (\\ref{row_yba}) it follows that $%\nC(x):\\mathbb{Z}[x]\\otimes \\mathcal{V}_{n}\\rightarrow \\mathbb{Z}[x]\\otimes \n\\mathcal{V}_{n-1}$ and $B^{\\prime }:\\mathbb{Z}[x]\\otimes \\mathcal{V}%\n_{n}\\rightarrow \\mathbb{Z}[x]\\otimes \\mathcal{V}_{n+1}$. This implies that $%\nC(y_{1})\\cdots C(y_{n})v_{\\lambda }$ is a multiple of $v_{0}\\otimes \\cdots\n\\otimes v_{0}$ and $B^{\\prime }(z_{1})\\cdots B^{\\prime }(z_{k})v_{\\lambda }$\na multiple of $v_{1}\\otimes \\cdots \\otimes v_{1}$. Denote the\nproportionality factors, i.e. the respective matrix elements, by $\\langle\n0|C(y_{n})\\cdots C(y_{1})|\\lambda \\rangle $ and $\\langle N|B^{\\prime\n}(z_{k})\\cdots B^{\\prime }(z_{1})|\\lambda \\rangle $. Each can be identified\nwith a weighted sum $\\sum_{\\gamma }\\limfunc{wt}(\\gamma )$ over vertex\nconfigurations $\\gamma $ on a lattice with certain fixed boundary\nconditions; see Figure \\ref{fig:dualB} and \\ref{fig:dualC} for simple\nexamples with $N=4$, $n=k=2$. Here $\\limfunc{wt}(\\gamma )=\\prod_{\\mathrm{v}%\n\\in \\gamma }\\limfunc{wt}(\\mathrm{v})$ with $\\mathrm{v}$ being one of the\nvertex configurations in Figure \\ref{fig:5vmodels} and the respective weight \n$\\limfunc{wt}(\\mathrm{v})$ takes the values $a,a^{\\prime },b,b^{\\prime\n},c,c^{\\prime }$ as specified in the figure or zero if it is none of the\nallowed vertices. We now identify lattice configurations $\\gamma $ with\ncertain \\emph{sets} of set-valued tableaux.\n\nDefine a surjection $\\limfunc{SVT}(\\lambda )\\twoheadrightarrow \\limfunc{SST}%\n(\\lambda )$ which assigns to each set-valued tableau $\\mathcal{T}$ the\nsemi-standard tableau $T:\\lambda \\rightarrow \\lbrack n]$ with $T(i,j)=\\min \n\\mathcal{T}(i,j)$. Given a semistandard tableau $T$ with entries $\\leq n$,\nthere exists a unique \\textquotedblleft maximal\\textquotedblright\\ set\nvalued tableau $\\mathcal{T}^{\\max }$ in its pre-image that has the maximum\nnumber of entries $\\leq n$, i.e. $|\\mathcal{T}^{\\max }|\\geq |\\mathcal{T}|$\nfor all $\\mathcal{T}$ in the pre-image of $T$. The lattice path\nconfigurations are in one-to-one correspondence with these maximal\nset-valued tableaux (and therefore semi-standard tableaux) of shape $\\lambda \n$ and $\\lambda ^{\\prime }$. We state the bijections:\n\n\\begin{itemize}\n\\item \\emph{Vicious walkers}. Starting from the bottom, place for each\nvertex labelled with a \\emph{bullet} in lattice row $i$ a box labelled with $%\ni$ in the $j$th row of the Young tableau where $j$ is the total number of\npaths crossing the row to the right of the vertex. Vertices with a \\emph{%\nsquare} in row $i$ mean that an entry $i$ is placed within an existing box\nof the $j$th row of the Young tableau where $j$ is again the total number of\npaths crossing the row to the right of the vertex. The resulting set-valued\ntableau has shape $\\lambda $.\n\n\\begin{figure}[tbp]\n\\begin{equation*}\n\\includegraphics[scale=0.5]{dualb_grothendieck.eps}\n\\end{equation*}%\n\\caption{The lattice configurations corresponding to the $C$-operator.}\n\\label{fig:dualB}\n\\end{figure}\n\n\\item \\emph{Osculating walkers}. Consider the rightmost path and add in the\nfirst column (counting from left to right) of the Young diagram of $\\lambda\n^{\\prime }$ a box with the lattice row number where a vertex with a bullet\noccurs. If a vertex with a square occurs in row $i$ then place an $i$ into\nan existing box in this column. Then do the same for the next path writing\nthe lattice row numbers now in the second column of the Young tableau etc.\nIf there are no vertices with a bullet leave the column empty. The resulting\nset-valued tableau has shape $\\lambda ^{\\prime }$.%\n\\begin{figure}[tbp]\n\\begin{equation*}\n\\includegraphics[scale=0.5]{dualc_grothendieck.eps}\n\\end{equation*}%\n\\caption{The lattice configurations corresponding to the $B^{\\prime}$%\n-operator.}\n\\label{fig:dualC}\n\\end{figure}\n\\end{itemize}\n\nLet $\\gamma $ be a lattice configuration of the $C$-operator ($B^{\\prime }$%\n-operator) and denote by $T_{\\gamma }$ the corresponding semi-standard\ntableau under the surjection $\\limfunc{SVT}(\\lambda )\\twoheadrightarrow \n\\limfunc{SST}(\\lambda )$. Note that each vertex with a bullet contributes a\nfactor $y_{i}\\ominus t_{j}$ and each vertex with a square a factor $(1+\\beta\ny_{i}\\ominus t_{j})$; see Figure \\ref{fig:5vmodels}. Here$\\ i,j$ are the\nlattice row and column numbers where the vertex occurs and we number lattice\nrows decreasingly from top to bottom and lattice rows increasingly from left\nto right. This allows us to deduce the following result.\n\n\\begin{lemma}\nWe have the following identities%\n\\begin{eqnarray}\n\\limfunc{wt}(\\gamma _{C}) &=&\\prod_{\\substack{ \\langle i,j\\rangle \\in\n\\lambda  \\\\ r=T_{\\gamma }(i,j)}}\\left( y_{r}\\ominus t_{r+j-i}\\right) ~\\prod \n_{\\substack{ \\langle i,j\\rangle \\in \\lambda  \\\\ r\\in \\mathcal{T}_{\\gamma\n}^{\\max }(i,j)\\backslash T_{\\gamma }(i,j)}}(1+\\beta y_{r}\\ominus t_{r+j-i}) \n\\notag \\\\\n&=&\\sum_{\\mathcal{T}}\\beta ^{|\\mathcal{T}|-|\\lambda |}\\prod_{\\substack{ %\n\\langle i,j\\rangle \\in \\lambda ^{\\vee }  \\\\ r\\in \\mathcal{T}(i,j)}}%\ny_{r}\\ominus t_{r+j-i}\n\\end{eqnarray}%\nand%\n\\begin{eqnarray}\n\\limfunc{wt}(\\gamma _{B^{\\prime }}) &=&\\prod_{\\substack{ \\langle i,j\\rangle\n\\in \\lambda ^{\\prime }  \\\\ r=T_{\\gamma }(i,j)}}\\left( z_{r}\\oplus\nt_{r+j-i}\\right) ~\\prod_{\\substack{ \\langle i,j\\rangle \\in \\lambda ^{\\prime\n}  \\\\ r\\in \\mathcal{T}_{\\gamma }^{\\max }(i,j)\\backslash T_{\\gamma }(i,j)}}%\n(1+\\beta y_{r}\\oplus t_{r+j-i})  \\notag \\\\\n&=&\\sum_{\\mathcal{T}}\\beta ^{|\\mathcal{T}|-|\\lambda |}\\prod_{\\substack{ %\n\\langle i,j\\rangle \\in \\lambda ^{\\ast }  \\\\ r\\in \\mathcal{T}(i,j)}}%\nz_{r}\\oplus t_{r+j-i}^{\\prime },\n\\end{eqnarray}%\nwhere the sums run over all set-valued tableaux $\\mathcal{T}$ of shape $%\n\\lambda $ ($\\lambda ^{\\prime }$) which obey the condition that%\n\\begin{equation*}\n\\min \\mathcal{T}(i,j)=T_{\\gamma }(i,j)\\;.\n\\end{equation*}\n\\end{lemma}\n\nThus, it follows that \n\\begin{eqnarray*}\n\\langle 0|C(y_{n})\\cdots C(y_{1})|\\lambda \\rangle &=&\\sum_{\\gamma _{C}}%\n\\limfunc{wt}(\\gamma _{C})=\\sum_{\\mathcal{T}\\in \\limfunc{SVT}(\\lambda )}\\beta\n^{|\\mathcal{T}|-|\\lambda ^{\\vee }|}\\prod_{\\substack{ \\langle i,j\\rangle \\in\n\\lambda  \\\\ r\\in \\mathcal{T}(i,j)}}y_{r}\\ominus t_{r+j-i} \\\\\n\\langle N|B^{\\prime }(z_{k})\\cdots B^{\\prime }(z_{1})|\\lambda \\rangle\n&=&\\sum_{\\gamma _{B^{\\prime }}}\\limfunc{wt}(\\gamma _{B^{\\prime }})=\\sum_{%\n\\mathcal{T}\\in \\limfunc{SVT}(\\lambda ^{\\prime })}\\beta ^{|\\mathcal{T}%\n|-|\\lambda ^{\\vee }|}\\prod_{\\substack{ \\langle i,j\\rangle \\in \\lambda\n^{\\prime }  \\\\ r\\in \\mathcal{T}(i,j)}}z_{r}\\oplus t_{r+j-i}^{\\prime }\n\\end{eqnarray*}%\nwhich proves the assertion as the last two equations are McNamara's\ndefinition (\\ref{facG}) of factorial Grothendieck polynomials.\n\\end{proof}\n\nIntroduce the so-called off-shell Bethe vector in $\\mathbb{Z}[y_{1},\\ldots\n,y_{n}]\\otimes \\mathcal{V}_{n}$ and its dual%\n\\begin{eqnarray}\n|y_{1},\\ldots ,y_{n}\\rangle &=&B(y_{n}|t)\\cdots B(y_{1}|t)|0\\rangle\n\\label{Bethev} \\\\\n\\langle y_{1},\\ldots ,y_{n}| &=&\\langle 0|C^{\\vee }(y_{n}|t)\\cdots C^{\\vee\n}(y_{1}|t)  \\label{leftBethev}\n\\end{eqnarray}%\nwhere $|0\\rangle =v_{0}\\otimes \\cdots \\otimes v_{0}$ and $\\langle 0|=\\tilde{v%\n}_{0}\\otimes \\cdots \\otimes \\tilde{v}_{0}$. Similarly, we define for a $k$%\n-tuple $z=(z_{1},\\ldots ,z_{k})$ the complementary Bethe vector in $\\mathbb{Z%\n}[z_{1},\\ldots ,z_{k}]\\otimes \\mathcal{V}_{n}$ and its dual as%\n\\begin{eqnarray}\n|z_{1},\\ldots ,z_{k}\\rangle &=&C^{\\prime }(z_{k}|t)\\cdots C^{\\prime\n}(z_{1}|t)|N\\rangle  \\label{Bethev'} \\\\\n\\langle z_{1},\\ldots ,z_{k}| &=&\\langle N|B^{\\ast }(z_{k}|t)\\cdots B^{\\ast\n}(z_{1}|t)\\;.  \\label{leftBethev'}\n\\end{eqnarray}%\nwith $|N\\rangle =v_{1}\\otimes \\cdots \\otimes v_{1}$ and $\\langle N|=\\tilde{v}%\n_{1}\\otimes \\cdots \\otimes \\tilde{v}_{1}$. From (\\ref{YBE}) one deduces that\nthe $B,B^{\\ast },C^{\\prime },C^{\\vee }$-operators each commute for different\nvalues of the spectral parameters. Hence, we can conclude that the vectors (%\n\\ref{Bethev}), (\\ref{Bethev'}) as well as their dual versions are symmetric\nin the $y$'s and $z$'s.\n\nWe now identify the coefficients of the off-shell Bethe vectors with\nfactorial Grothendieck polynomials.\n\n\\begin{proposition}\nRecall the definitions of $\\lambda ^{\\vee }$, $\\lambda ^{\\ast }$ from Sec %\n\\ref{sec:partitions} and set once more $\\ominus t^{\\prime }=(\\ominus\nt_{N+1},\\ldots ,\\ominus t_{2},\\ominus t_{1})$. Then we have the identities%\n\\begin{eqnarray}\n|y_{1},\\ldots ,y_{n}\\rangle &=&\\Pi (y)\\sum_{\\lambda \\in (k^{n})}\\frac{%\nG_{\\lambda ^{\\vee }}(y_{1},\\ldots ,y_{n}|\\ominus t^{\\prime })}{\\Pi\n(t_{\\lambda })}~v_{\\lambda }  \\label{rightBethe} \\\\\n|z_{1},\\ldots ,z_{k}\\rangle &=&\\Pi (z)\\sum_{\\lambda \\in (k^{n})}G_{\\lambda\n^{\\ast }}(z_{1},\\ldots ,z_{k}|t)\\Pi (t_{\\lambda ^{\\ast }})v_{\\lambda },\n\\label{rightBethe'}\n\\end{eqnarray}%\nFor the dual vectors we obtain instead%\n\\begin{eqnarray}\n\\langle y_{1},\\ldots ,y_{n}| &=&\\sum_{\\lambda \\in (k^{n})}G_{\\lambda\n}(y_{1},\\ldots ,y_{n}|\\ominus t)~\\tilde{v}_{\\lambda }  \\label{leftBethe} \\\\\n\\langle z_{1},\\ldots ,z_{k}| &=&\\sum_{\\lambda \\in (k^{n})}G_{\\lambda\n^{\\prime }}(z_{1},\\ldots ,z_{k}|t^{\\prime })~\\tilde{v}_{\\lambda }\\;.\n\\label{leftBethe'}\n\\end{eqnarray}\n\\end{proposition}\n\n\\begin{proof}\nThe proof is very similar to the one of the previous identities with some\nminor changes in the bijections between lattice configurations of\nnon-intersecting paths and maximal set-valued tableaux.\n\nAs before the matrix elements $\\langle \\lambda |B(y_{n})\\cdots\nB(y_{1})|0\\rangle $ and $\\langle \\lambda |C^{\\prime }(z_{k})\\cdots C^{\\prime\n}(z_{1})|N\\rangle $ can each be identified with a weighted sum $\\sum_{\\gamma\n}\\limfunc{wt}(\\gamma )$ over lattice path configurations $\\gamma $ with\ncertain fixed boundary conditions; examples are provided in Figure \\ref%\n{fig:B} for $B$ with $N=9$, $n=5$ and Figure \\ref{fig:C} for $C^{\\prime }$\nwith $N=9,$ $k=5$. Also these configurations are in one-to-one\ncorrespondence with certain maximal set-valued tableaux (as defined\npreviously) and are respectively mapped onto semistandard tableaux of shape $%\n\\lambda ^{\\vee }$ and $(\\lambda ^{\\vee })^{\\prime }$ when taking the\nsmallest entry in each box. The bijections are now as follows:\n\n\\begin{itemize}\n\\item \\emph{Vicious walkers}. Starting now from the top, place for each\nvertex labelled with a bullet in lattice row $i$ a box labelled with $i$ in\nthe $j$th row of the Young tableau where $j$ is the total number of paths\ncrossing the row to the left of the vertex. For a vertex with a square in\nrow $i$ place an additional entry $i$ into an already existing box. The\nresulting tableau will now have shape $\\lambda ^{\\vee }$. \n\\begin{figure}[tbp]\n\\begin{equation*}\n\\includegraphics[scale=0.5]{b_grothendieck.eps}\n\\end{equation*}%\n\\caption{A lattice configuration corresponding to the $B$-operator. The\nvertex configurations above the dotted line are ``frozen'', i.e. there is no\nother choice possible which would yield a nonzero weight.}\n\\label{fig:B}\n\\end{figure}\n\n\\item \\emph{Osculating walkers}. Consider the leftmost path and write in the\nfirst column (counting from left to right) of the Young diagram of $(\\lambda\n^{\\vee })^{\\prime }$ the lattice row numbers where a vertex with a bullet\noccurs. If there is a vertex with a square in row $i$ place an $i$ into the\nlast added box in the same column. Then do the same for the next path\nwriting the lattice row numbers now in the second column etc. If there are\nno vertices with a bullet leave the column empty.%\n\\begin{figure}[tbp]\n\\begin{equation*}\n\\includegraphics[scale=0.5]{c_grothendieck.eps}\n\\end{equation*}%\n\\caption{A lattice configuration corresponding to the $C^{\\prime}$-operator.\nThe vertex configuration above the dotted line are ``frozen''.}\n\\label{fig:C}\n\\end{figure}\n\\end{itemize}\n\nLet $\\gamma $ be a lattice configuration of the $B$-operator ($C^{\\prime }$%\n-operator) and denote by $T_{\\gamma }$ the corresponding semistandard\ntableau. If we multiply the matrix element $\\langle \\lambda |B(y_{n})\\cdots\nB(y_{1})|0\\rangle $ with $\\Pi (t_{\\lambda })\/\\Pi (y)$ then according to\nFigure \\ref{fig:5vmodels} each vertex with a bullet contributes a factor $%\ny_{i}\\ominus t_{j}^{\\prime }$ and each vertex with a square a factor $%\n(1+\\beta y_{i}\\ominus t_{j}^{\\prime })$ in the vicious walker case. In the\nosculating walker case we divide $\\langle \\lambda |C^{\\prime\n}(z_{k}^{-1})\\cdots C^{\\prime }(z_{1}^{-1})|N\\rangle $ with $\\Pi (z)\\Pi\n(t_{\\lambda ^{\\ast }})$ to obtain respectively the factors $z_{i}\\oplus\nt_{j} $ and $(1+\\beta z_{i}\\oplus t_{j})$. Here$\\ i,j$ are the lattice row\nand column numbers where the vertex occurs. As before this implies the\nfollowing summation identities for the lattice weights \n\\begin{eqnarray*}\n&\\frac{\\Pi (t_{\\lambda })}{\\Pi (y)}&\\limfunc{wt}(\\gamma _{B})=\\sum_{\\mathcal{%\nT}}\\beta ^{|\\mathcal{T}|-|\\lambda ^{\\vee }|}\\prod_{\\substack{ \\langle\ni,j\\rangle \\in \\lambda ^{\\vee }  \\\\ r\\in \\mathcal{T}(i,j)}}y_{r}\\ominus\nt_{r+j-i}^{\\prime }, \\\\\n&&\\frac{\\limfunc{wt}(\\gamma _{C^{\\prime }})}{\\Pi (z)\\Pi (t_{\\lambda ^{\\ast\n}})}=\\sum_{\\mathcal{T}}\\beta ^{|\\mathcal{T}|-|\\lambda ^{\\vee }|}\\prod \n_{\\substack{ \\langle i,j\\rangle \\in \\lambda ^{\\ast }  \\\\ r\\in \\mathcal{T}%\n(i,j) }}z_{r}\\oplus t_{r+j-i},\n\\end{eqnarray*}%\nwhere the sums run over all set-valued tableaux $\\mathcal{T}$ of shape $%\n\\lambda ^{\\vee }$ ($\\lambda ^{\\ast }=(\\lambda ^{\\vee })^{\\prime }$) which\nobey the condition that $\\min \\mathcal{T}(i,j)=T_{\\gamma }(i,j)$. The final\nstep then uses again that the map $\\limfunc{SVT}(\\lambda ^{\\vee\n})\\twoheadrightarrow \\limfunc{SST}(\\lambda ^{\\vee })$ which assigns to each\nset-valued tableau $\\mathcal{T}$ the SST $T:\\lambda ^{\\vee }\\rightarrow\n\\lbrack n]$ with $T(i,j)=\\min \\mathcal{T}(i,j)$ is a surjection. Thus, it\nfollows that \n\\begin{eqnarray*}\n\\langle \\lambda |B(y_{n})\\cdots B(y_{1})|0\\rangle &=&\\frac{\\Pi (y)}{\\Pi\n(t_{\\lambda })}\\sum_{\\mathcal{T}\\in \\limfunc{SVT}(\\lambda ^{\\vee })}\\beta ^{|%\n\\mathcal{T}|-|\\lambda ^{\\vee }|}\\prod_{\\substack{ \\langle i,j\\rangle \\in\n\\lambda ^{\\vee }  \\\\ r\\in \\mathcal{T}(i,j)}}y_{r}\\ominus t_{r+j-i}^{\\prime }\n\\\\\n\\langle \\lambda |C^{\\prime }(z_{k}^{-1})\\cdots C^{\\prime\n}(z_{1}^{-1})|N\\rangle &=&\\Pi (z)\\Pi (t_{\\lambda ^{\\ast }})\\sum_{\\mathcal{T}%\n\\in \\limfunc{SVT}(\\lambda ^{\\ast })}\\beta ^{|\\mathcal{T}|-|\\lambda ^{\\vee\n}|}\\prod_{\\substack{ \\langle i,j\\rangle \\in \\lambda ^{\\ast }  \\\\ r\\in \n\\mathcal{T}(i,j)}}z_{r}\\oplus t_{r+j-i}\n\\end{eqnarray*}%\nThe identities for the dual Bethe vectors (\\ref{leftBethe}), (\\ref%\n{leftBethe'}) are obtained by a very similar argument noting from the\ndefinition (\\ref{dual_mom}) that the matrix elements of the transposed\noperators are obtained by reversing binary strings and swapping $%\nt\\leftrightarrow t^{\\prime }$.\n\\end{proof}\n\nWe are now in the position to proof a generalised Cauchy identity for\nfactorial Grothendieck polynomials; compare with \\cite[Thm 5.3]{MotegiSakai}\nand \\cite[Thm 9]{LN} for the non-factorial case which we obtain as a special\ncase.\n\n\\begin{corollary}\nSetting $e(x,y)=\\langle 0|C(x_{1})\\cdots C(x_{n})B(y_{n})\\cdots\nB(y_{1})|0\\rangle $ we have%\n\\begin{multline}\ne(x,y)=\\Pi (y)\\sum_{\\lambda \\subset (k^{n})}\\frac{G_{\\lambda }(x|\\ominus\nt)G_{\\lambda ^{\\vee }}(y|\\ominus t^{\\prime })}{\\Pi (t_{\\lambda })} \\\\\n=\\frac{1}{\\Pi (x)}\\sum_{w}w\\left( \\Pi (x)\\frac{\\prod_{i=1}^{n}%\n\\prod_{j=1}^{N}x_{i}\\ominus t_{j}}{\\prod_{1\\leq i,j\\leq n}x_{i}\\ominus y_{j}}%\n\\right)  \\label{norm}\n\\end{multline}%\nwhere - as in Lemma \\ref{lem:yba2} - the sum runs over the minimal length\ncoset representatives $w$ of $\\mathbb{S}_{2n}\/\\mathbb{S}_{n}\\times \\mathbb{S}%\n_{n}$ which act on $(x,y)$ in the obvious manner.\n\\end{corollary}\n\n\\begin{proof}\nNoting that \n\\begin{equation}\nA(x)|0\\rangle =\\left( \\prod\\limits_{j=1}^{N}x\\ominus t_{j}\\right) |0\\rangle\n\\qquad \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{and\\qquad }D(x)|0\\rangle =|0\\rangle  \\label{A0_and_D0}\n\\end{equation}%\nthe assertion is immediate from Lemma \\ref{lem:yba2} and the formulae (\\ref%\n{Caction}), (\\ref{rightBethe}).\n\\end{proof}\n\nNote that the limit $\\lim_{x_{i}\\rightarrow y_{i}}e(x,y)$ is well-defined as\ncan be seen from the definition of $e(x,y)$ as matrix element $\\langle\n0|C(x_{1})\\cdots C(x_{n})B(y_{n})\\cdots B(y_{1})|0\\rangle $ and (\\ref{norm}%\n). The poles in the last line of Equation (\\ref{norm}) cancel against zeroes\nin the numerator as $x_{i}\\rightarrow y_{i}$ after the sum over the $w$'s is\ntaken.\n\n\\begin{corollary}\n\\label{cor:cauchy0}Setting $y=t_{\\mu }$ we obtain%\n\\begin{equation}\n\\prod_{i=1}^{n}\\prod_{j\\in I_{\\mu ^{\\ast }}}(x_{i}\\ominus\nt_{j})=\\sum_{\\lambda \\subset (k^{n})}\\frac{\\Pi (t_{\\mu })}{\\Pi (t_{\\lambda })%\n}~G_{\\lambda ^{\\vee }}(t_{\\mu }|\\ominus t^{\\prime })G_{\\lambda }(x|\\ominus\nt)\\;.\n\\end{equation}%\nThis proves in particular (\\ref{Cauchy0}) and, thus, we obtain after setting\nalso $x=t_{\\mu }$,%\n\\begin{equation}\ne(t_{\\mu },t_{\\mu })=\\prod_{i\\in I_{\\mu }}\\prod_{j\\in I_{\\mu ^{\\ast\n}}}(t_{i}\\ominus t_{j})\\;.  \\label{norm0}\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\nSpecialising $y=t_{\\mu }$ in (\\ref{norm}) one easily sees that only the term\nwith $w$ being the identity survives in the last sum.\n\\end{proof}\n\n\\subsection{The Bethe ansatz equations}\n\nWe call the Bethe vectors (\\ref{Bethev}), (\\ref{Bethev'}) \\textquotedblleft\non-shell\\textquotedblright\\ if the indeterminates $y=(y_{1},\\ldots ,y_{n})$\nare \\emph{pairwise distinct} solutions to the following set of coupled Bethe\nansatz equations with $\\Pi (y)$ defined in (\\ref{Pi}),%\n\\begin{equation}\n(-1)^{n}\\frac{\\Pi (y)}{\\left( 1+\\beta y_{i}\\right) ^{n}}\\prod_{j=1}^{N}y_{i}%\n\\ominus t_{j}+q=0,\\qquad i=1,\\ldots ,n\\;.  \\label{BAE}\n\\end{equation}%\nWe define a second set of equations for the indeterminates $z=(z_{1},\\ldots\n,z_{k})$ in (\\ref{Bethev'}),%\n\\begin{equation}\n(-1)^{k}\\frac{\\Pi (z)}{\\left( 1+\\beta z_{i}\\right) ^{k}}\\prod_{j=1}^{N}z_{i}%\n\\oplus t_{j}+q=0,\\qquad i=1,\\ldots ,k\\;.  \\label{BAE'}\n\\end{equation}%\nThe origin of these equations is Lemma \\ref{lem:yba1} from which we deduce\nthat if (\\ref{BAE}) holds the Bethe vector (\\ref{Bethev}) is an eigenvector\nof $H=A+qD$; see below. By the same argument one shows that (\\ref{Bethev'})\nis an eigenvector of $E=A^{\\prime }+qD^{\\prime }$ if (\\ref{BAE'}) hold.\nObviously, one set of equations transforms into the other under the\nsubstitution $t=(t_{1},\\ldots ,t_{N})\\rightarrow \\ominus t^{\\prime\n}=(\\ominus t_{N},\\ldots ,\\ominus t_{1})$ and exchanging $n$ with $k$. This\nsubstitution is related to level-rank duality (\\ref{levelrankmom}) which we\nwill use below to relate the Bethe vectors (\\ref{Bethev}) with the vectors (%\n\\ref{Bethev'}). We shall therefore focus on the equations (\\ref{BAE}) only.\n\n\\begin{lemma}\n\\label{lem:BAE}The set of equations (\\ref{BAE}) has $\\binom{N}{n}$ \\emph{%\npairwise distinct }solutions \n\\begin{equation}\ny_{\\lambda }=(y_{\\lambda _{n}+1},\\ldots y_{\\lambda _{2}+n-1},y_{\\lambda\n_{1}+n})\\in \\mathbb{Z}[\\![q]\\!]\\otimes \\mathbb{Z}(t_{1},\\ldots ,t_{N}),\n\\label{ylambda}\n\\end{equation}%\nwhere $\\lambda \\subset (k^{n})$ and up to first order in $q$ we have%\n\\begin{equation}\ny_{i}=t_{i}+q~(-1)^{n-1}\\frac{(1+\\beta t_{i})^{n+1}}{\\Pi (t_{\\lambda\n})\\prod_{j\\neq i}t_{i}\\ominus t_{j}}+O(q^{2})\\;.  \\label{Brootexpansion}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nMake the ansatz $y_{i}=y_{i}^{(0)}+y_{i}^{(1)}q+y_{i}^{(2)}q^{2}+\\cdots $\nand set $q=0$ in (\\ref{BAE}). Because the equations are coupled setting $%\ny_{i^{\\prime }}^{(0)}=-\\beta ^{-1}$ with $i^{\\prime }\\neq i$ in the factor\nin front of the product in (\\ref{BAE}) is not a valid solution, since it\nwould imply a singular term in the equations with $i$ replaced by $i^{\\prime\n}$. Therefore, the only possible solution is $y_{i}^{(0)}=t_{i}$. Here the\nlabelling in terms of the index $i$ is a matter of choice but it will prove\nconvenient later on. Differentiating the equations with respect to $q$,%\n\\begin{equation*}\n\\frac{d}{dq}\\left( \\frac{(-1)^{n-1}\\Pi (y)}{\\left( 1+\\beta y_{i}\\right) ^{n}}%\n\\right) \\prod_{j=1}^{N}y_{i}\\ominus t_{j}+\\frac{(-1)^{n-1}\\Pi (y)}{\\left(\n1+\\beta y_{i}\\right) ^{n}}\\frac{d}{dq}\\prod_{j=1}^{N}y_{i}\\ominus t_{j}=1,\n\\end{equation*}%\nand setting $q=0$ afterwards we find%\n\\begin{equation*}\n\\frac{d}{dq}\\left. \\prod_{j=1}^{N}y_{i}\\ominus t_{j}\\right\\vert _{q=0}=\\frac{%\ny_{i}^{(1)}}{1-\\beta t_{i}}\\prod_{j\\neq i}t_{i}\\ominus t_{j}\n\\end{equation*}%\nand the formula (\\ref{Brootexpansion}) follows. Continuing in the same\nmanner by taking the $r$th derivative we see that the coefficient $\\Pi\n(t_{\\lambda })\/(1+\\beta t_{i})^{n}$ in front of the term \n\\begin{equation*}\n\\frac{d^{r}}{dq^{r}}\\left. \\prod_{j=1}^{N}y_{i}\\ominus t_{j}\\right\\vert\n_{q=0}\n\\end{equation*}%\nis alway nonzero and, hence, that the equations yield a rational solution in\nthe $t_{j}$'s for any $y_{i}^{(r)}$.\n\\end{proof}\n\n\\begin{lemma}\nGiven a partition $\\lambda =(\\lambda _{1},\\ldots ,\\lambda _{n})$ with $%\n\\lambda _{1}>k$ and a solution $y=y_{\\mu }$ of the Bethe ansatz equations (%\n\\ref{BAE}), one has the identity%\n\\begin{multline}\nG_{\\lambda }(y|\\ominus t)=  \\label{Greduction} \\\\\nq\\sum_{r=0}^{\\lambda _{1}-1-k}h_{\\lambda _{1}-1-k-r}(t_{1},\\ldots\n,t_{r+1})G_{(\\lambda _{2}-1,\\ldots ,\\lambda _{n}-1,r)}(y|\\ominus\nt)\\prod_{i=1}^{r}(1+\\beta t_{i}),\n\\end{multline}%\nwhere the $h_{r}$'s denote the complete symmetric functions and the\nfactorial Grothendieck polynomial on the right hand side is defined via (\\ref%\n{Gstraight}).\n\\end{lemma}\n\n\\begin{proof}\nRecall the determinant formula (\\ref{facGdet}) for factorial Grothendieck\npolynomials. Writing out the determinant in the numerator we find%\n\\begin{multline*}\na_{\\lambda }=\\left\\vert \n\\begin{array}{ccc}\n(y_{1}|\\ominus t)^{n+\\lambda _{1}-1} & \\cdots & (y_{n}|\\ominus t)^{n+\\lambda\n_{1}-1} \\\\ \n(y_{1}|\\ominus t)^{n+\\lambda _{2}-2}(1+\\beta y_{1}) &  & (y_{n}|\\ominus\nt)^{n+\\lambda _{2}-2}(1+\\beta y_{n}) \\\\ \n\\vdots &  & \\vdots \\\\ \n(y_{1}|\\ominus t)^{\\lambda _{n}}(1+\\beta y_{1})^{n-1} & \\cdots & \n(y_{n}|\\ominus t)^{\\lambda _{n}}(1+\\beta y_{n})^{n-1}%\n\\end{array}%\n\\right\\vert \\\\\n=\\frac{q}{\\Pi (y)}\\left\\vert \n\\begin{array}{ccc}\n(y_{1}|\\ominus t)^{n+\\lambda _{2}-2}(1+\\beta y_{1}) & \\cdots & \n(y_{n}|\\ominus t)^{n+\\lambda _{2}-2}(1+\\beta y_{n}) \\\\ \n\\vdots &  & \\vdots \\\\ \n(y_{1}|\\ominus t)^{\\lambda _{n}}(1+\\beta y_{1})^{n-1} &  & (y_{n}|\\ominus\nt)^{\\lambda _{n}}(1+\\beta y_{n})^{n-1} \\\\ \ny_{1}^{\\lambda _{1}-1-k}(1+\\beta y_{1})^{n} & \\cdots & y_{n}^{\\lambda\n_{1}-1-k}(1+\\beta y_{n})^{n}%\n\\end{array}%\n\\right\\vert\n\\end{multline*}%\nHere we have made use of (\\ref{BAE}), exchanged the first row with the last\nrow in the determinant and used row linearity of the determinant to pull out\nthe common factor in front. Note that $t_{j}=0$ for $j>N$, whence the powers\nin the bottom row are not factorial. To rewrite them as factorial powers we\nuse the equality%\n\\begin{equation*}\nx^{m}=\\sum_{r=0}^{m}(x|\\ominus t)^{m-r}h_{r}(t_{1},\\ldots\n,t_{m+1-r})\\prod_{i=1}^{m-r}(1+\\beta t_{i})\n\\end{equation*}%\nwhich is easily proved via induction using the known recursion relation%\n\\begin{equation*}\nh_{r+1}(t_{1},\\ldots ,t_{m+1-r})=h_{r}(t_{1},\\ldots\n,t_{m+1-r})t_{m+1-r}+h_{r+1}(t_{1},\\ldots ,t_{m-r})\n\\end{equation*}%\nof the complete symmetric functions. We leave this step to the reader.\n\nThus, after employing the above identity and column\/row linearity of the\ndeterminant we arrive at%\n\\begin{equation*}\na_{\\lambda }=q\\sum_{r=0}^{\\lambda _{1}-1-k}a_{(\\lambda _{2}-1,\\ldots\n,\\lambda _{n}-1,\\lambda _{1}-1-k-r)}h_{r}(t_{1},\\ldots ,t_{\\lambda\n_{1}-k-r})\\prod_{i=1}^{\\lambda _{1}-1-k-r}(1+\\beta t_{i})\n\\end{equation*}%\nwhich is the asserted identity (\\ref{Greduction}) after dividing by the\nVandermonde determinant.\n\\end{proof}\n\n\\begin{theorem}\nThe on-shell Bethe vectors (\\ref{Bethev}), (\\ref{Bethev'}) and (\\ref%\n{leftBethev}), (\\ref{leftBethev'}) form respectively right and left\neigenbases of the transfer matrices $H$ and $E$ in each subspace $\\mathcal{V}%\n_{n}^{q}$ with eigenvalue equations%\n\\begin{equation}\nH(x|t)|y_{\\mu }\\rangle =\\left( \\frac{\\prod\\limits_{j=1}^{N}x\\ominus\nt_{j}+(-1)^{n}q\\prod\\limits_{i\\in I_{\\mu }}\\left( 1+\\beta x\\ominus\ny_{i}\\right) }{\\prod\\limits_{i\\in I_{\\mu }}x\\ominus y_{i}}\\right) ~|y_{\\mu\n}\\rangle  \\label{specH}\n\\end{equation}%\nand%\n\\begin{equation}\nE(x|t)|z_{\\mu }\\rangle =\\left( \\frac{\\prod\\limits_{j=1}^{N}x\\oplus\nt_{j}+(-1)^{n}q\\prod\\limits_{i\\in I_{\\mu ^{\\ast }}}\\left( 1+\\beta x\\ominus\nz_{i}\\right) }{\\prod\\limits_{i\\in I_{\\mu ^{\\ast }}}x\\ominus z_{i}}\\right)\n~|z_{\\mu }\\rangle \\;.  \\label{specE}\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nHere we use the commutation relations of the Yang-Baxter algebra as per\nLemma \\ref{lem:yba1} and (\\ref{A0_and_D0}) from which we deduce that if (\\ref%\n{BAE}) holds the Bethe vector (\\ref{Bethev}) is an eigenvector of $H=A+qD$.\nThe computation follows along the same lines for (\\ref{Bethev'}) and the\nleft eigenvectors (\\ref{leftBethev}, \\ref{leftBethev'}).\n\nOne deduces that the eigenvalues must separate points and, hence, $\\langle\ny_{\\lambda }|y_{\\mu }\\rangle =\\langle z_{\\lambda }|z_{\\mu }\\rangle =0$ for $%\n\\lambda \\neq \\mu $. That these eigenvectors form a basis then follows from\nthe fact that there exist $\\dim \\mathcal{V}_{n}=\\binom{N}{n}$ solutions to\nthe equations (\\ref{BAE}); see Lemma \\ref{lem:BAE}.\n\\end{proof}\n\nNote that the above formulae simplify if $q=0$. Then the on-shell Bethe\nvectors with $y_{\\mu }=t_{\\mu }$ are given by%\n\\begin{equation}\n|t_{\\mu }\\rangle =\\sum_{\\lambda \\subset (k^{n})}G_{\\lambda ^{\\vee }}(t_{\\mu\n}|\\ominus t^{\\prime })\\frac{\\Pi (t_{\\mu })}{\\Pi (t_{\\lambda })}~v_{\\lambda }\n\\label{Bethev0}\n\\end{equation}%\nand form an eigenbasis of the transfer matrices with eigenvalues,%\n\\begin{eqnarray}\nH(x|t)|t_{\\mu }\\rangle &=&\\left( \\tprod\\limits_{j\\in I_{\\mu ^{\\ast\n}}}x\\ominus t_{j}\\right) |t_{\\mu }\\rangle  \\label{specH0} \\\\\nE(x|t)|t_{\\mu }\\rangle &=&\\left( \\tprod\\limits_{j\\in I_{\\mu }}x\\oplus\nt_{j}\\right) |t_{\\mu }\\rangle  \\label{specE0}\n\\end{eqnarray}%\nAs we will discuss below this special case describes generalised equivariant\ncohomology theory, $h_{n}^{\\ast }=qh_{n}^{\\ast }\/\\langle q\\rangle $ and we\nshow below that $h_{n}^{\\ast }\/\\langle \\beta +1\\rangle \\cong K_{\\mathbb{T}}(%\n\\limfunc{Gr}_{n,N})$.\n\n\\begin{proposition}\n\\label{prop:Gduality}The eigenvectors of $H$ and $E$ coincide under the\nsubstitution $z_{\\lambda ^{\\prime }}=\\ominus y_{\\lambda ^{\\vee }}$ and,\nthus, we have the equality%\n\\begin{equation}\nG_{\\lambda ^{\\prime }}(\\ominus y_{\\mu ^{\\ast }}|t)=G_{\\lambda }(y_{\\mu\n}|\\ominus t^{\\prime })  \\label{Gduality}\n\\end{equation}%\nfor each solution $y_{\\mu }$ of (\\ref{BAE}). In particular, for $q=0$ we\nhave $G_{\\lambda ^{\\prime }}(\\ominus t_{\\mu ^{\\ast }}|t)=G_{\\lambda }(t_{\\mu\n}|\\ominus t^{\\prime })$.\n\\end{proposition}\n\n\\begin{proof}\nUsing the identity (\\ref{levelrankmom}) when acting on the Bethe vectors we\nfind%\n\\begin{eqnarray*}\n\\Theta H(x|t)|y_{\\lambda }\\rangle &=&E(x|\\ominus t^{\\prime })\\Theta\n|y_{\\lambda }\\rangle \\\\\n\\Theta E(x|t)|z_{\\lambda }\\rangle &=&H(x|\\ominus t^{\\prime })\\Theta\n|z_{\\lambda }\\rangle\n\\end{eqnarray*}%\nand%\n\\begin{eqnarray*}\n\\Theta |y_{\\lambda }\\rangle &=&C^{\\prime }(y_{1}|\\ominus t^{\\prime })\\cdots\nC^{\\prime }(y_{n}|\\ominus t^{\\prime })|N\\rangle \\\\\n\\Theta |z_{\\lambda }\\rangle &=&B(z_{1}|\\ominus t^{\\prime })\\cdots\nB(z_{k}|\\ominus t^{\\prime })|0\\rangle\n\\end{eqnarray*}%\nThese identities together with the expansion (\\ref{Brootexpansion}) allows\nus to identify $z_{\\lambda ^{\\prime }}=\\ominus y_{\\lambda ^{\\vee }}$.\n\\end{proof}\n\n\\begin{corollary}\nThe eigenvalue equation (\\ref{specE}) of the $E$-transfer matrix simplifies\nin the Bethe roots (\\ref{ylambda}) to \n\\begin{equation}\nE(x|t)|y_{\\mu }\\rangle =\\prod\\limits_{i\\in I_{\\mu }}\\left( x\\oplus\ny_{i}\\right) ~|y_{\\mu }\\rangle \\;.  \\label{SpecE}\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\nSetting $x=t_{j}$ the functional equation (\\ref{func_eqn}) together with (%\n\\ref{specH}) implies%\n\\begin{equation*}\nq|y_{\\mu }\\rangle =E(\\ominus t_{j})H(t_{j})|y_{\\mu }\\rangle =\\frac{q}{%\n\\prod\\limits_{i=1}^{n}\\left( t_{j}\\ominus y_{i}\\right) }E(\\ominus\nt_{j})|y_{\\mu }\\rangle\n\\end{equation*}%\nfor all $j=1,2,\\ldots ,N$. Since the $t_{j}$'s are arbitrary variables and\nthe Bethe vectors form an eigenbasis the assertion follows.\n\\end{proof}\n\nSince the Bethe vectors (\\ref{Bethev}) and (\\ref{leftBethev}) form each an\neigenbasis they give rise to a resolution of the identity $\\boldsymbol{1}%\n=\\sum_{\\alpha \\in (n,k)}|y_{\\alpha }\\rangle \\langle y_{\\alpha }|$ where $%\n|y_{\\alpha }\\rangle \\langle y_{\\alpha }|$ denotes the orthogonal projector\nonto the eigenspace spanned by $|y_{\\alpha }\\rangle $. This elementary fact\nof linear algebra translates into the following non-trivial identities for\nfactorial Grothendieck polynomials evaluated at solutions of the Bethe\nansatz equations (\\ref{BAE}).\n\n\\begin{corollary}[orthogonality \\& completeness]\nFor all $\\lambda ,\\mu \\subset (k^{n})$ we have the identities%\n\\begin{equation}\n\\sum_{\\alpha \\subset (k^{n})}\\frac{\\Pi (y_{\\lambda })}{\\Pi (t_{\\alpha })}%\n\\frac{G_{\\alpha ^{\\vee }}(y_{\\lambda }|\\ominus t^{\\prime })G_{\\alpha\n}(y_{\\mu }|\\ominus t)}{e(y_{\\lambda },y_{\\lambda })}=\\delta _{\\lambda \\mu }\n\\label{Cauchy2}\n\\end{equation}%\nand%\n\\begin{equation}\n\\sum_{\\alpha \\subset (k^{n})}\\frac{\\Pi (y_{\\alpha })}{\\Pi (t_{\\lambda })}%\n\\frac{G_{\\lambda ^{\\vee }}(y_{\\alpha }|\\ominus t^{\\prime })G_{\\mu\n}(y_{\\alpha }|\\ominus t)}{e(y_{\\alpha },y_{\\alpha })}=\\delta _{\\lambda \\mu\n}\\;,  \\label{res of 1}\n\\end{equation}%\nwhere $\\delta _{\\lambda \\mu }$ denotes the Kronecker delta with $\\delta\n_{\\lambda \\mu }=1$ if $\\lambda =\\mu $ and $0$ otherwise.\n\\end{corollary}\n\n\\subsection{Generalised matrix algebras and Frobenius structures}\n\nFollowing the suggested construction in \\cite[Section 7]{Korffproc} we now\nintroduce a ring structure on each $\\mathcal{V}_{n}^{q}=\\mathbb{Z}%\n[\\![q]\\!]\\otimes \\mathcal{V}_{n}$ by interpreting the on-shell Bethe vectors\n(\\ref{Bethev}) as central orthogonal idempotents of a semisimple algebra:\nfor each $n=0,1,\\ldots ,N$ define $qh_{n}^{\\ast }=(\\mathcal{V}%\n_{n}^{q},\\circledast )$ by fixing the product $\\circledast $ as follows,%\n\\begin{equation}\nY_{\\lambda }\\circledast Y_{\\mu }=\\delta _{\\lambda \\mu }Y_{\\mu }~,\\qquad\nY_{\\lambda }=e(y_{\\lambda },y_{\\lambda })^{-1}|y_{\\lambda }\\rangle \\;,\n\\label{idempotent_def}\n\\end{equation}%\nwhere $e(y_{\\lambda },y_{\\lambda })$ is the matrix element defined in (\\ref%\n{norm}). Note that $e(y_{\\lambda },y_{\\lambda })$ is a power series in $q$\nwith nonzero constant term (\\ref{norm0}) according to (\\ref{Brootexpansion}%\n). The unit element is given by \n\\begin{equation}\nv_{\\emptyset }=\\sum_{\\lambda \\subset (k^{n})}Y_{\\lambda }\\;.  \\label{unit}\n\\end{equation}%\nThis determines $qh_{n}^{\\ast }$ via its Peirce decomposition \\cite{Peirce}.\nWe turn $qh_{n}^{\\ast }$ into a Frobenius algebra by introducing in addition\nthe following symmetric bilinear form $\\mathcal{V}_{n}^{q}\\times \\mathcal{V}%\n_{n}^{q}\\rightarrow \\mathcal{R}(\\mathbb{T},q)$,%\n\\begin{equation}\n(Y_{\\lambda },Y_{\\mu })=e(y_{\\lambda },y_{\\lambda })^{-1}\\delta _{\\lambda\n\\mu }\\;.  \\label{bilinear_form}\n\\end{equation}%\nBy definition this bilinear form is invariant with respect to the product (%\n\\ref{idempotent_def}) and non-degenerate, since the Bethe vectors form a\nbasis.\n\n\\subsection{A residue formula for the structure constants}\n\nWe now describe the resulting generalised matrix algebra $qh_{n}^{\\ast }$ in\nthe spin basis $\\{v_{\\lambda }\\}_{\\lambda \\subset (k^{n})}$. Introduce a\nfamily of operators $\\{\\boldsymbol{G}_{\\lambda }\\}_{\\lambda \\subset\n(k^{n})}\\subset \\limfunc{End}\\mathcal{V}_{n}^{q}$ via the following\neigenvalue equation \n\\begin{equation}\n\\boldsymbol{G}_{\\lambda }Y_{\\mu }=G_{\\lambda }(y_{\\mu }|\\ominus t)Y_{\\mu }\\;.\n\\label{BigG}\n\\end{equation}%\nThis \\emph{defines} the operators $\\boldsymbol{G}_{\\lambda }$, since the\nBethe vectors form an eigenbasis and the eigenvalues separate points. Recall\nfrom Section \\ref{sec:grothendieck} that the factorial Grothendieck\npolynomials form a basis \\cite[Thm 4.6]{McNamara}. Below we give an\nexplicit, basis independent construction of $\\boldsymbol{G}_{\\lambda }$ in\nterms of the transfer matrix $H(x)$.\n\n\\begin{corollary}\nIn the spin basis (\\ref{spin basis}) the product (\\ref{idempotent_def}) is\ngiven by \n\\begin{equation}\nv_{\\lambda }\\circledast v_{\\mu }=\\boldsymbol{G}_{\\lambda }v_{\\mu }=\\sum_{\\nu\n\\subset (k^{n})}C_{\\lambda \\mu }^{\\nu }(t,q)v_{\\nu },  \\label{combproduct}\n\\end{equation}%\nwhere the structure constants $C_{\\lambda \\mu }^{\\nu }(t,q)=\\langle \\nu |%\n\\boldsymbol{G}_{\\lambda }|\\lambda \\rangle $ are obtained in terms of the\nBethe roots (\\ref{ylambda}) via the residue formula%\n\\begin{equation}\nC_{\\lambda \\mu }^{\\nu }(t,q)=\\sum_{\\alpha \\subset (k^{n})}\\frac{\\Pi\n(y_{\\alpha })}{\\Pi (t_{\\nu })}\\frac{G_{\\lambda }(y_{\\alpha }|\\ominus\nt)G_{\\mu }(y_{\\alpha }|\\ominus t)G_{\\nu ^{\\ast }}(\\ominus y_{\\alpha ^{\\ast\n}}|t)}{e(y_{\\alpha },y_{\\alpha })}\\;.  \\label{Verlinde}\n\\end{equation}%\nSimilarly, the bilinear form (\\ref{bilinear_form}) can be expressed as%\n\\begin{equation}\n(v_{\\lambda },v_{\\mu })=\\sum_{\\alpha \\subset (k^{n})}\\frac{G_{\\lambda\n}(y_{\\alpha }|\\ominus t)G_{\\mu }(y_{\\alpha }|\\ominus t)}{e(y_{\\alpha\n},y_{\\alpha })}\\;.  \\label{bilinear_form_spin}\n\\end{equation}\n\\end{corollary}\n\n\\begin{remark}\nOur residue formula (\\ref{Verlinde}) is a generalisation of the\nBertram-Vafa-Intriligator formula for Gromov-Witten invariants. It holds\nalso true for $q=0$, where the Bethe roots are explicitly known, $%\ny_{i}=t_{i} $,%\n\\begin{equation}\nc_{\\lambda \\mu }^{\\nu }(t)=C_{\\lambda \\mu }^{\\nu }(t,0)=\\sum_{\\alpha \\subset\n(k^{n})}\\frac{\\Pi (t_{\\alpha })}{\\Pi (t_{\\nu })}\\frac{G_{\\lambda }(t_{\\alpha\n}|\\ominus t)G_{\\mu }(t_{\\alpha }|\\ominus t)G_{\\nu ^{\\ast }}(\\ominus\nt_{\\alpha ^{\\ast }}|t)}{\\prod_{i\\in I_{\\alpha },j\\in I_{\\alpha ^{\\ast\n}}}t_{i}\\ominus t_{j}}\\;.  \\label{Verlinde0}\n\\end{equation}%\nThe bilinear form (\\ref{bilinear_form_spin}) for $q=0$ reads%\n\\begin{equation}\n(v_{\\lambda },v_{\\mu })=\\sum_{\\alpha \\subset (k^{n})}\\frac{G_{\\lambda\n}(t_{\\alpha }|\\ominus t)G_{\\mu }(t_{\\alpha }|\\ominus t)}{\\prod_{i\\in\nI_{\\alpha },j\\in I_{\\alpha ^{\\ast }}}t_{i}\\ominus t_{j}}\\;.\n\\label{bilinear_form0}\n\\end{equation}\n\\end{remark}\n\n\\begin{proof}\nAccording to (\\ref{rightBethe}) and (\\ref{res of 1}) we have the inverse\nbasis transformation%\n\\begin{equation}\nv_{\\lambda }=\\sum_{\\mu \\subset (k^{n})}G_{\\lambda }(y_{\\mu }|\\ominus\nt)Y_{\\mu }\\;.  \\label{Bethe2spin}\n\\end{equation}%\nwhich allows us to compute%\n\\begin{eqnarray*}\nv_{\\lambda }\\circledast v_{\\mu } &=&\\sum_{\\rho ,\\sigma }G_{\\lambda }(y_{\\rho\n}|\\ominus t)G_{\\mu }(y_{\\sigma }|\\ominus t)Y_{\\rho }\\circledast Y_{\\sigma }\n\\\\\n&=&\\sum_{\\rho }G_{\\lambda }(y_{\\rho }|\\ominus t)G_{\\mu }(y_{\\rho }|\\ominus\nt)Y_{\\rho }=\\boldsymbol{G}_{\\lambda }v_{\\mu }=\\boldsymbol{G}_{\\mu\n}v_{\\lambda }\\;.\n\\end{eqnarray*}%\nThis proves the first assertion. Continuing the computation from the second\nline employing (\\ref{rightBethe}) we arrive at (\\ref{Verlinde}).\n\nThe expression (\\ref{bilinear_form_spin}) is also an immediate consequence\nof (\\ref{Bethe2spin}). Insert the latter and use the definition (\\ref%\n{bilinear_form}) to find the asserted identity (\\ref{bilinear_form_spin}).\n\\end{proof}\n\nAs is to be expected from our previous results (\\ref{LRduality_L}) and (\\ref%\n{levelrankmom}), the rings related by exchanging the dimension $n$ with the\ncodimension $k$ of the hyperplanes in the Grassmannian are closely related.\n\n\\begin{corollary}[level-rank duality]\nThe involution $qh_{n}^{\\ast }\\rightarrow qh_{k}^{\\ast }$ given by $%\nf(t,q)v_{\\lambda }\\mapsto f(\\ominus t^{\\prime },q)v_{\\lambda ^{\\prime }}$ is\na ring isomorphism over $\\mathcal{R}\\otimes \\mathbb{Z}[\\![q]\\!]$. That is,%\n\\begin{equation}\nC_{\\mu \\nu }^{\\lambda }(t,q)=C_{\\mu ^{\\prime }\\nu ^{\\prime }}^{\\lambda\n^{\\prime }}(\\ominus t^{\\prime },q)\\;.  \\label{levelrank}\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\nFirst we note that (\\ref{G1}) and (\\ref{Gduality}) imply the identity%\n\\begin{eqnarray*}\n\\frac{\\Pi (y_{\\lambda })}{\\Pi (t_{\\mu })} &=&\\frac{\\Pi (t_{\\emptyset })}{\\Pi\n(t_{\\mu })}~\\left( 1+\\beta G_{1}(y_{\\lambda }|\\ominus t)\\right) \\\\\n&=&\\frac{\\Pi (\\ominus t_{\\emptyset }^{\\prime })}{\\Pi (\\ominus t_{\\mu\n^{\\prime }}^{\\prime })}~\\left( 1+\\beta G_{1}(\\ominus y_{\\lambda ^{\\ast\n}}|t^{\\prime })\\right) =\\frac{\\Pi (\\ominus y_{\\lambda ^{\\ast }})}{\\Pi\n(\\ominus t_{\\mu ^{\\prime }}^{\\prime })}\\;.\n\\end{eqnarray*}%\nNote further that according to (\\ref{Brootexpansion}) the $k$-tuple $\\ominus\ny_{\\lambda ^{\\ast }}$ is obtained from solutions $y_{i}$ by replacing $%\nt=(t_{1},\\ldots ,t_{N})$ with $\\ominus t^{\\prime }=(\\ominus t_{N},\\ldots\n,\\ominus t_{1})$, i.e. the constant terms of the components of the solution $%\n\\ominus y_{\\lambda ^{\\ast }}$ are $\\ominus t_{\\lambda ^{\\prime }}^{\\prime }$\nwhich identifies the solution uniquely. Using the residue formula (\\ref%\n{Verlinde}) and (\\ref{Gduality}) we compute%\n\\begin{multline*}\nC_{\\mu \\nu }^{\\lambda }(t,q)=\\sum_{\\alpha \\subset (k^{n})}\\frac{\\Pi\n(y_{\\alpha })}{\\Pi (t_{\\nu })}\\frac{G_{\\lambda }(y_{\\alpha }|\\ominus\nt)G_{\\mu }(y_{\\alpha }|\\ominus t)G_{\\nu ^{\\vee }}(y_{\\alpha }|\\ominus\nt^{\\prime })}{e(y_{\\alpha },y_{\\alpha })}= \\\\\n\\sum_{\\alpha }\\frac{\\Pi (\\ominus y_{\\alpha ^{\\ast }})}{\\Pi (\\ominus t_{\\nu\n^{\\prime }}^{\\prime })}\\frac{G_{\\lambda ^{\\prime }}(\\ominus y_{\\alpha ^{\\ast\n}}|t^{\\prime })G_{\\mu ^{\\prime }}(\\ominus y_{\\alpha ^{\\ast }}|t^{\\prime\n})G_{\\nu ^{\\ast }}(\\ominus y_{\\alpha ^{\\ast }}|t)}{e(y_{\\alpha },y_{\\alpha })%\n}=C_{\\mu ^{\\prime }\\nu ^{\\prime }}^{\\lambda ^{\\prime }}(\\ominus t^{\\prime\n},q),\n\\end{multline*}%\nwhere in the last step we have used the definition (\\ref{norm}) to show that%\n\\begin{eqnarray*}\ne(y_{\\alpha },y_{\\alpha }) &=&\\sum_{\\lambda \\subset (k^{n})}\\frac{\\Pi\n(y_{\\alpha })}{\\Pi (t_{\\lambda })}G_{\\lambda }(y_{\\alpha }|\\ominus\nt)G_{\\lambda ^{\\vee }}(y_{\\alpha }|\\ominus t^{\\prime }) \\\\\n&=&\\sum_{\\lambda \\subset (k^{n})}\\frac{\\Pi (\\ominus y_{\\alpha ^{\\ast }})}{%\n\\Pi (\\ominus t_{\\lambda ^{\\prime }}^{\\prime })}G_{\\lambda ^{\\prime\n}}(\\ominus y_{\\alpha ^{\\ast }}|t^{\\prime })G_{\\lambda ^{\\ast }}(\\ominus\ny_{\\alpha ^{\\ast }}|t)=e(\\ominus y_{\\alpha ^{\\ast }},\\ominus y_{\\alpha\n^{\\ast }})\\;.\n\\end{eqnarray*}\n\\end{proof}\n\n\\subsection{A recurrence formula}\n\nWe now return to the result (\\ref{H1}) and show that the latter formula\ndescribes the multiplication with the class of the Schubert divisor, i.e.\nthat (\\ref{H1}) describes indeed the equivariant quantum Pieri-Chevalley\nrule for the generalised cohomology ring $qh_{n}^{\\ast }$.\n\n\\begin{corollary}\nLet $\\lambda =(1,0,\\ldots ,0)$ then%\n\\begin{equation}\n\\boldsymbol{G}_{1}=H_{1}  \\label{G1=H1}\n\\end{equation}%\nand the product $v_{1}\\circledast v_{\\lambda }=H_{1}v_{\\lambda }$ in the\nspin basis is given explicitly via (\\ref{H1}).\n\\end{corollary}\n\n\\begin{proof}\nEmploying the functional equation (\\ref{func_eqn}) and (\\ref{specH}), (\\ref%\n{SpecE}) we obtain%\n\\begin{multline*}\n\\prod_{j=1}^{n}(t_{j}\\ominus x)\\prod_{j=n+1}^{N}(x\\ominus t_{j})(1+\\beta\nH_{1})Y_{\\mu }=(H(x)E(\\ominus x)-q\\cdot 1)Y_{\\mu } \\\\\n=(-1)^{n}\\frac{\\Pi (y_{\\mu })}{(1+\\beta x)^{n}}\\prod_{j=1}^{N}(x\\ominus\nt_{j})~Y_{\\mu } \\\\\n=\\frac{\\Pi (y_{\\mu })}{\\Pi (t_{\\emptyset })}\\prod_{j=1}^{n}(t_{j}\\ominus\nx)\\prod_{j=n+1}^{N}(x\\ominus t_{j})~Y_{\\mu }\n\\end{multline*}%\nThus, according to (\\ref{G1Pieri}), (\\ref{BigG}) we have%\n\\begin{equation*}\n(1+\\beta H_{1})Y_{\\mu }=\\frac{\\Pi (y_{\\mu })}{\\Pi (t_{\\emptyset })}~Y_{\\mu\n}=(1+\\beta \\boldsymbol{G}_{1})Y_{\\mu }\n\\end{equation*}%\nand the assertion follows from the fact that the Bethe vectors form a basis.\n\\end{proof}\n\nAnalogous to the case of equivariant (quantum) cohomology one derives from\nthe quantum Pieri-Chevalley rule (\\ref{H1}) the following recurrence\nrelation for the structure constants.\n\n\\begin{corollary}[Recurrence relation]\nWe have the identity%\n\\begin{equation}\n\\left( \\Pi (t_{\\nu })-\\Pi (t_{\\lambda })\\right) C_{\\lambda \\mu }^{\\nu\n}=\\sum_{\\tilde{\\lambda}\/d^{\\prime }\/\\lambda }\\beta ^{|\\tilde{\\lambda}%\n\/d\/\\lambda |}C_{\\tilde{\\lambda}\\mu }^{\\nu }-\\sum_{\\nu \/d^{\\prime \\prime }\/%\n\\tilde{\\nu}}\\beta ^{|\\nu \/d^{\\prime \\prime }\/\\tilde{\\nu}|}\\Pi (t_{\\tilde{\\nu}%\n})C_{\\lambda \\mu }^{\\tilde{\\nu}},  \\label{recurr}\n\\end{equation}%\nwhere the sums run over all partitions $\\tilde{\\lambda}\\neq \\lambda ,\\tilde{%\n\\nu}\\neq \\nu $ such that respectively $\\tilde{\\lambda}\/d^{\\prime }\/\\lambda $\nand $\\nu \/d^{\\prime \\prime }\/\\nu $ are toric skew-diagrams with $d^{\\prime\n},d^{\\prime \\prime }$ either $0$ or $1$ and where each row and column\ncontains at most one box.\n\\end{corollary}\n\n\\begin{proof}\nThe derivation follows the same idea as in ordinary (quantum) cohomology;\nsee e.g. \\cite{KnutsonTao}. Since the product $\\circledast $ by definition\nis associative we have in light of (\\ref{G1=H1}) that%\n\\begin{equation*}\n\\lbrack (1+\\beta H_{1})v_{\\lambda }]\\circledast v_{\\mu }=(1+\\beta\nH_{1})(v_{\\lambda }\\circledast v_{\\mu })\\;.\n\\end{equation*}%\nApplying the Pieri-Chevalley rule (\\ref{H1}) on both sides of the equality\nsign and comparing coefficients the assertion follows.\n\\end{proof}\n\n\\begin{example}\n\\label{ex:4}Consider once more the simplest non-trivial case $\\limfunc{Gr}%\n_{1,3}=\\mathbb{P}^{2}$. Let $\\lambda =\\mu =(2)$ and $\\nu =\\emptyset $. Then $%\n\\Pi (t_{\\nu })=1+\\beta t_{1}$, $\\Pi (t_{\\lambda })=1+\\beta t_{3}$ and $%\n\\tilde{\\lambda}=\\emptyset ,$ $\\tilde{\\nu}=(2)$ with $d^{\\prime }=d^{\\prime\n\\prime }=1$ are the only boxed partitions which give rise to allowed\ncylindric skew diagrams. Therefore, we arrive at the relation%\n\\begin{equation*}\n\\beta (t_{1}-t_{3})C_{22}^{\\emptyset }=q\\beta C_{\\emptyset 2}^{\\emptyset\n}-q\\beta (1+\\beta t_{3})C_{22}^{2}=-q\\beta (1+\\beta t_{3})C_{22}^{2},\n\\end{equation*}%\nwhere we have used that $v_{\\emptyset }$ is the unit and we therefore must\nhave $C_{\\emptyset 2}^{\\emptyset }=0$. Similarly, setting $\\nu =1$ we obtain%\n\\begin{equation*}\n\\beta (t_{2}-t_{3})C_{22}^{1}=q\\beta C_{\\emptyset 2}^{1}-\\beta (1+\\beta\nt_{1})C_{22}^{\\emptyset }\\;.\n\\end{equation*}%\nThus, we end up with the recursion%\n\\begin{equation*}\nC_{22}^{\\emptyset }=q\\frac{1+\\beta t_{3}}{t_{3}-t_{1}}~C_{22}^{2},\\qquad\nC_{22}^{1}=\\frac{1+\\beta t_{1}}{t_{3}-t_{2}}~C_{22}^{\\emptyset }\n\\end{equation*}%\nwith $C_{22}^{2}=(t_{3}\\ominus t_{2})(t_{3}\\ominus t_{1})$. Thus,%\n\\begin{equation*}\nC_{22}^{\\emptyset }=q(t_{3}\\ominus t_{2})\\frac{1+\\beta t_{3}}{1+\\beta t_{1}}%\n,\\qquad C_{22}^{1}=q\\frac{1+\\beta t_{3}}{1+\\beta t_{2}}\n\\end{equation*}%\nwhich is in agreement with our earlier computation and the product expansion\nin \\cite[Sec 5.5]{BM} upon setting $t_{i}=1-e^{\\varepsilon _{4-i}}$ and $%\n\\beta =-1$.\n\\end{example}\n\n\\section{Localised Schubert classes and GKM theory}\n\nAn important result in (ordinary) equivariant quantum cohomology and\nequivariant K-theory is that the respective rings have a purely algebraic\nrealisation by restricting Schubert classes to the fixed points under the\ntorus action. This monomorphism becomes a ring isomorphism with respect to\npointwise multiplication if one imposes the Goresky-Kottwitz-MacPherson\n(GKM) conditions \\cite[Thm 1.2.2]{GKM}; see \\cite[Thm 3.13]{KostantKumar2}\nfor the analogous statement in K-theory. We now show that this algebraic\nrealisation naturally emerges from our lattice model approach for our\ngeneralised cohomology theories $qh_{n}^{\\ast }$.\n\n\\subsection{Generalised difference operators and Iwahori-Hecke algebras}\n\nWe recall that the ring $\\mathcal{R}(\\mathbb{T})=\\mathcal{R}(t_{1},\\ldots\n,t_{N})$ is naturally endowed with an $\\mathbb{S}_{N}$-action by permuting\nthe equivariant parameters. By abuse of notation we will identify\npermutations $w\\in \\mathbb{S}_{N}$ with their operators acting on $\\mathcal{R%\n}(\\mathbb{T})$. This $\\mathbb{S}_{N}$-action can be used to define a\nrepresentation of a generalised (affine) Hecke or Iwahori algebra $\\mathbb{H}%\n_{N}(\\beta )$.\n\n\\begin{definition}\nDenote by $\\mathbb{H}_{N}(\\beta )$ the associative unital algebra with the\nfollowing generators and relations%\n\\begin{equation}\n\\pi _{i}^{2}=\\beta \\pi _{i}\\qquad \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{and}\\qquad \\left\\{ \n\\begin{array}{cc}\n\\pi _{i}\\pi _{j}=\\pi _{j}\\pi _{i}, & (i-j)\\func{mod}N\\neq \\pm 1 \\\\ \n\\pi _{i}\\pi _{i+1}\\pi _{i}=\\pi _{i+1}\\pi _{i}\\pi _{i+1}, & \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{else}%\n\\end{array}%\n\\right.  \\label{Iwahori_def}\n\\end{equation}%\nwhere all indices are understood modulo $N$. Denote by $\\mathbb{H}_{N}^{%\n\\limfunc{fin}}(\\beta )$ the subalgebra generated by $\\{\\pi _{1},\\ldots ,\\pi\n_{N-1}\\}$.\n\\end{definition}\n\nThe subring $\\mathcal{R}[t_{1},\\ldots ,t_{N}]\\subset \\mathcal{R}(\\mathbb{T})$\nand $\\mathcal{R}(\\mathbb{T})$ itself are both $\\mathbb{H}_{N}^{\\limfunc{fin}%\n}(\\beta )$-modules with respect to the following action in terms of isobaric\ndivided difference operators%\n\\begin{equation}\n\\partial _{j}=(1+\\beta t_{j})~\\frac{1-s_{j}}{t_{j}-t_{j+1}}\\;,\n\\label{divided_difference}\n\\end{equation}%\nwhere $s_{j}$ is the simple transposition interchanging $t_{j}$ and $t_{j+1}$%\n. Note that setting $\\beta =0$ we obtain a representation of the nil-Coxeter\nalgebra $\\mathbb{A}_{N}=\\mathbb{H}_{N}(0)$ and when setting $\\beta =-1$ a\nrepresentation of the nil-Hecke algebra $\\mathbb{H}_{N}=\\mathbb{H}_{N}(-1)$.\n\n\\begin{proposition}[braid matrices]\nLet $p_{j}:V_{n}\\rightarrow V_{n}$ the operator which permutes vectors in\nthe $j$th and $(j+1)$th factor and acts everywhere else trivially, i.e. $%\np_{j}v_{b}=v_{s_{j}b}$. Then the matrices $\\{\\hat{r}%\n_{j}(t_{j},t_{j+1})=p_{j}r_{j+1,j}(t_{j+1}\\ominus t_{j})\\}_{j=1}^{N}$\\ act\non the standard basis $\\{v_{b}\\}_{|b|=n}$ via \n\\begin{equation}\n\\hat{r}_{j}(t_{j},t_{j+1})v_{b}=\\left\\{ \n\\begin{array}{cc}\n(1+\\beta t_{j+1}\\ominus t_{j})v_{b}+q^{-\\delta _{j,N}}t_{j+1}\\ominus\nt_{j}~v_{s_{j}b}, & b_{j}<b_{j+1} \\\\ \nv_{b}, & \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{else}%\n\\end{array}%\n\\right.  \\label{braid_action}\n\\end{equation}%\nMoreover, the $\\hat{r}_{j}$'s obey the relations%\n\\begin{multline}\n\\hat{r}_{j}(t_{j+1},t_{j+2})\\hat{r}_{j+1}(t_{j},t_{j+2})\\hat{r}%\n_{j}(t_{j},t_{j+1})=  \\label{braid_relation} \\\\\n\\hat{r}_{j+1}(t_{j},t_{j+1})\\hat{r}_{j}(t_{j},t_{j+2})\\hat{r}%\n_{j+1}(t_{j+1},t_{j+2})\n\\end{multline}%\nand%\n\\begin{gather}\n\\hat{r}_{j}^{2}-(2+\\beta t_{j+1}\\ominus t_{j})~\\hat{r}_{j}+(1+\\beta\nt_{j+1}\\ominus t_{j})~1=0  \\label{quad_relation} \\\\\n(s_{j}\\otimes 1)\\hat{r}_{j}=\\hat{r}_{j}^{-1}(s_{j}\\otimes 1)\\;.\n\\label{inv_relation}\n\\end{gather}%\nHere all indices are understood modulo $N$.\n\\end{proposition}\n\n\\begin{proof}\nIf we fix the basis $\\{v_{0}\\otimes v_{0},v_{0}\\otimes v_{1},v_{1}\\otimes\nv_{0},v_{1}\\otimes v_{1}\\}$ in $V_{j}\\otimes V_{j+1}$ then $\\hat{r}_{j}$\nreads as a matrix,%\n\\begin{equation}\n\\hat{r}_{j}(t_{j},t_{j+1})=\\left( \n\\begin{array}{cccc}\n1 & 0 & 0 & 0 \\\\ \n0 & 1+\\beta t_{j+1}\\ominus t_{j} & 0 & 0 \\\\ \n0 & q^{-\\delta _{j,N}}t_{j+1}\\ominus t_{j} & 1 & 0 \\\\ \n0 & 0 & 0 & 1%\n\\end{array}%\n\\right) _{j,j+1}\n\\end{equation}%\nUsing this matrix form one now verifies easily the various assertions.\n\\end{proof}\n\n\\begin{corollary}[symmetric group action]\nThe operators $\\boldsymbol{s}_{j}=(s_{j}\\otimes 1)\\hat{r}_{j}$ for $%\nj=1,\\ldots ,N-1$ define an action of the symmetric group $\\mathbb{S}_{N}$ on\nthe space $\\mathcal{V}_{n}$. For $q$ invertible, we have an action of the\naffine symmetric group with $\\boldsymbol{s}_{N}=(s_{N}\\otimes 1)\\hat{r}%\n_{N}(q^{-1})$ on $\\mathbb{Z}[q^{\\pm 1}]\\otimes \\mathcal{V}_{n}$, where $%\ns_{N} $ is the affine reflection in the level-zero representation on $%\n\\mathcal{R}(\\mathbb{T})$. Explicitly, one has in the spin-basis%\n\\begin{equation}\n\\boldsymbol{s}_{j}v_{b}=\\left\\{ \n\\begin{array}{cc}\n(1+\\beta t_{j}\\ominus t_{j+1})v_{b}+q^{-\\delta _{j,N}}t_{j}\\ominus\nt_{j+1}~v_{s_{j}b}, & b_{j}<b_{j+1} \\\\ \nv_{b}, & \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{else}%\n\\end{array}%\n\\right. \\;.  \\label{SNaction}\n\\end{equation}%\nNote that the $\\mathbb{S}_{N}$-action does not commute with the\nmultiplicative action of $\\mathcal{R}(\\mathbb{T})$ on $\\mathcal{V}_{n}$.\n\\end{corollary}\n\n\\begin{proof}\nThat the $\\boldsymbol{s}_{j}$ yield a representation of $\\mathbb{S}_{N}$\nfollows easily from our previous findings (\\ref{braid_relation}), (\\ref%\n{quad_relation}), and (\\ref{inv_relation}).\n\\end{proof}\n\nThe next result shows that the Yang-Baxter algebra (\\ref{row_yba}) commutes\nwith the action of the symmetric group. For the transfer matrices this\nextends to the action including the affine reflection depending on the\ndeformation parameter $q$; compare with \\ref{qybe}.\n\n\\begin{corollary}\nThe action on $\\mathbb{Z}[x]\\otimes \\mathcal{V}$ commutes with the action of\nthe row Yang-Baxter algebras, i.e.%\n\\begin{eqnarray}\n(1\\otimes \\boldsymbol{s}_{j})M(x|t) &=&M(x|t)(1\\otimes \\boldsymbol{s}_{j})\n\\label{SWD} \\\\\n(1\\otimes \\boldsymbol{s}_{j})M^{\\prime }(x|t) &=&M^{\\prime }(x|t)(1\\otimes \n\\boldsymbol{s}_{j}),\\qquad j=1,2,\\ldots ,N-1,  \\label{SWD'}\n\\end{eqnarray}%\nwhere $M,M^{\\prime }$ are the monodromy matrices in (\\ref{rowmom}) and (\\ref%\n{row_yba}) for $L$ and $L^{\\prime }$, respectively. In case of the transfer\nmatrices we have the additional relations%\n\\begin{equation}\n\\boldsymbol{s}_{N}H(x|t)=H(x|t)\\boldsymbol{s}_{N}\\quad \\;\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{and\\quad\\ }%\n\\boldsymbol{s}_{N}E(x|t)=E(x|t)\\boldsymbol{s}_{N}\n\\end{equation}%\nwith $\\boldsymbol{s}_{N}=(s_{N}\\otimes 1)\\hat{r}_{N}$ and $%\ns_{N}=s_{1}s_{2}\\cdots s_{N-2}s_{N-1}s_{N-2}\\cdots s_{2}s_{1}$.\n\\end{corollary}\n\n\\begin{remark}\nThe commutation of the symmetric group action (\\ref{SNaction}) with the\naction of the Yang-Baxter algebra (\\ref{row_yba}) is reminiscent of\nSchur-Weyl duality and we will explore this connection in a forthcoming\npublication.\n\\end{remark}\n\n\\begin{proof}\nThe commutation relations with the monodromy and transfer matrices follow\nfrom (\\ref{ybe}) and (\\ref{qybe}).\n\\end{proof}\n\n\\begin{proposition}[generalised divided difference operators]\nThe matrices%\n\\begin{equation}\n\\delta _{j}=\\frac{1-\\hat{r}_{j}}{t_{j}-t_{j+1}}~(1+\\beta t_{j})=\\left( \n\\begin{array}{cccc}\n0 & 0 & 0 & 0 \\\\ \n0 & \\beta & 0 & 0 \\\\ \n0 & q^{-\\delta _{jN}} & 0 & 0 \\\\ \n0 & 0 & 0 & 0%\n\\end{array}%\n\\right) _{j,j+1}\\;,  \\label{Iwahori_action}\n\\end{equation}%\ndefine an action of $\\mathbb{H}_{N}(\\beta )$ on the space $\\mathbb{Z}[q^{\\pm\n1}]\\otimes V^{\\otimes N}$.\n\\end{proposition}\n\n\\begin{proof}\nA straightforward computation using the explicit matrix representation given\nwhich follows from (\\ref{braid_action}).\n\\end{proof}\n\nNote that the action (\\ref{Iwahori_action}) commutes with the multiplicative\naction of $\\mathcal{R}(\\mathbb{T})$ on $\\mathcal{V}$.\n\n\n\\subsection{Localised Schubert classes}\n\nRecall that each boxed partition $\\mu \\subset (k^{n})$ can be identified\nwith the 01-word of length $N$ which has one-letters at positions $I_{\\mu }$%\n. Recall the natural $\\mathbb{S}_{N}$-action on 01-words, i.e. write $%\ns_{j}\\mu $ for the partition obtained by exchanging the $j$th and $(j+1)$th\nletter in the corresponding 01-word for $\\mu $.\n\n\\begin{theorem}[localised Schubert classes]\n\\label{thm:GKM}The sequence $[\\mathcal{O}_{\\lambda }]=([\\mathcal{O}_{\\lambda\n}]_{\\mu })_{\\mu \\subset (k^{n})}$ in $\\mathcal{R}(\\mathbb{T},q)^{\\binom{N}{n}%\n}$ with $[\\mathcal{O}_{\\lambda }]_{\\mu }:=G_{\\lambda }(y_{\\mu }|\\ominus t)$\nobeys the following generalised Goresky-Kottwitz-MacPherson condition%\n\\begin{equation}\n\\boldsymbol{s}_{j}[\\mathcal{O}_{\\lambda }]-[\\mathcal{O}_{\\lambda\n}]=(t_{j}\\ominus t_{j+1})\\delta _{j}^{\\ast }[\\mathcal{O}_{\\lambda }],\\;\n\\label{GKM}\n\\end{equation}%\nwhere $\\boldsymbol{s}_{j}$ denotes the $\\mathbb{S}_{N}$-action given by $(%\n\\boldsymbol{s}_{j}[\\mathcal{O}_{\\lambda }])_{\\mu }=s_{j}G_{\\lambda\n}(y_{s_{j}\\mu }|\\ominus t)$ and \n\\begin{equation}\n\\delta _{j}^{\\ast }[\\mathcal{O}_{\\lambda }]=\\left\\{ \n\\begin{array}{cc}\n\\beta \\lbrack \\mathcal{O}_{\\lambda }]+[\\mathcal{O}_{s_{j}\\lambda }], & \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{%\nif }j\\notin I_{\\lambda }\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{ and }(j+1)\\in I_{\\lambda } \\\\ \n0, & \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{else}%\n\\end{array}%\n\\right. \\;.  \\label{gendiffop}\n\\end{equation}\n\\end{theorem}\n\nTo prove the theorem we require the following result first.\n\n\\begin{lemma}\nLet $\\boldsymbol{s}_{j}=(s_{j}\\otimes 1)\\hat{r}_{j}$ be the $\\mathbb{S}_{N}$%\n-action (\\ref{SNaction}). Then \n\\begin{equation}\n\\boldsymbol{s}_{j}Y_{b}=Y_{s_{j}b},\\qquad j=1,2,\\ldots ,N-1\\;.\n\\label{diagSNaction}\n\\end{equation}%\nIn other words, the action (\\ref{SNaction}) is the natural diagonal $\\mathbb{%\nS}_{N}$-action on $\\mathcal{V}_{n}$ in the basis of Bethe vectors.\n\\end{lemma}\n\n\\begin{proof}\nConsider the action of $\\hat{r}_{j}$ on an off-shell Bethe vector. According\nto (\\ref{SWD}) we have%\n\\begin{equation*}\n\\hat{r}_{j}B(x_{1}|t)\\cdots B(x_{n}|t)|0\\rangle =B(x_{1}|s_{j}t)\\cdots\nB(x_{n}|s_{j}t)|0\\rangle \\;.\n\\end{equation*}%\nAccording to Lemma \\ref{lem:BAE} the Bethe roots are uniquely determined by\nthe constant term, $y_{\\lambda }=t_{\\lambda }+O(q)$, thus, we have%\n\\begin{equation*}\n\\boldsymbol{s}_{j}|y_{\\mu }\\rangle =s_{j}(B(y_{\\mu _{n}+1}|s_{j}t)\\cdots\nB(y_{\\mu _{1}+n}|s_{j}t))|0\\rangle =|s_{j}y_{\\mu }\\rangle =|y_{s_{j}\\mu\n}\\rangle \\;.\n\\end{equation*}%\nAn analogous argument shows that \n\\begin{equation*}\ns_{j}e(y_{\\mu },y_{\\mu })=\\langle\n0|\\tprod_{i=1}^{n}C(y_{i})\\tprod_{i=1}^{n}B(y_{i})|0\\rangle =e(y_{s_{j}\\mu\n},y_{s_{j}\\mu })~.\n\\end{equation*}\n\\end{proof}\n\nWe now prove the generalised GKM conditions (\\ref{GKM}).\n\n\\begin{proof}[Proof of Theorem \\protect\\ref{thm:GKM}]\nEmploy the expansion (\\ref{Bethe2spin}) and apply $\\boldsymbol{s}_{j}$ on\nboth sides of the equation. Then using (\\ref{SNaction}) on the left hand\nside and (\\ref{diagSNaction}) on the right hand side of the equality, we\nobtain%\n\\begin{equation*}\nv_{\\lambda }+(t_{j}\\ominus t_{j+1})\\delta _{j}v_{\\lambda }=\\sum_{\\mu \\subset\n(k^{n})}(s_{j}G_{\\lambda }(y_{s_{j}\\mu }|\\ominus t))Y_{\\mu }\\;.\n\\end{equation*}%\nComparing coefficients with respect to the basis of the Bethe vectors yields\n(\\ref{GKM}).\n\\end{proof}\n\nThe next result states a generating formula for localised Schubert classes\nusing the representation (\\ref{gendiffop}) of the Iwahori-Hecke algebra. For \n$q=0$ and $\\beta =-1$ this statement is originally due to Kostant and Kumar \n\\cite{KostantKumar2}.\n\nEmploying McNamara's Vanishing Theorem we easily find for $q=0$ that%\n\\begin{equation}\n\\lbrack \\mathcal{O}_{(k^{n})}]_{\\lambda }=G_{(k^{n})}(t_{\\lambda }|\\ominus\nt)=\\left\\{ \n\\begin{array}{cc}\n\\prod_{i=1}^{k}\\prod_{j=k+1}^{N}t_{j}\\ominus t_{i}, & \\lambda =(k^{n}) \\\\ \n0, & \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{else}%\n\\end{array}%\n\\right.  \\label{topclass}\n\\end{equation}%\nwhich gives us an explicit description for the top (localised) Schubert\nclass. For the quantum case with $q\\neq 0$ we have instead \n\\begin{equation}\n\\lbrack \\mathcal{O}_{(k^{n})}]_{\\lambda }=G_{(k^{n})}(y_{\\lambda }|\\ominus\nt)=\\prod_{j=1}^{k}\\prod\\limits_{i\\in I_{\\lambda }}y_{i}\\ominus t_{j}\n\\label{qtopclass}\n\\end{equation}%\nwhere $y_{\\lambda }$ is the solution (\\ref{ylambda}) of (\\ref{BAE}) and the\nvalues $[\\mathcal{O}_{(k^{n})}]_{\\lambda }$ at fixed points $y_{\\lambda }$\nwith $\\lambda \\neq (k^{n})$ are in general nonzero.\n\n\\begin{corollary}\nAny Schubert class $[\\mathcal{O}_{\\lambda }]$ can be obtained by successive\naction of the generalised difference operators $\\delta _{j_{1}}^{\\ast\n},\\delta _{j_{2}}^{\\ast },\\ldots ,\\delta _{j_{r}}^{\\ast }$ on the top class $%\n[\\mathcal{O}_{(k^{n})}]$ for some $j_{1},\\ldots ,j_{r}\\in \\lbrack N]$ such\nthat $w=s_{j_{1}}\\cdots s_{j_{r}}$ is a reduced word with $w(k^{n})=\\lambda $\nin terms of the natural $\\mathbb{S}_{N}$-action on 01-words.\n\\end{corollary}\n\n\\begin{proof}\nA direct consequence of (\\ref{gendiffop}) and the $\\mathbb{S}_{N}$-action on\nbinary strings.\n\\end{proof}\n\n\\begin{corollary}\nThe ring $qh_{n}^{\\ast }\/\\langle q,\\beta +1,t_{j}-1+e^{\\varepsilon\n_{N+1-i}}\\rangle $ is isomorphic to $K_{\\mathbb{T}}(\\limfunc{Gr}_{n,N})$,\nwhile the ring $qh_{n}^{\\ast }\/\\langle q,\\beta \\rangle $ is isomorphic to $%\nH_{\\mathbb{T}}^{\\ast }(\\limfunc{Gr}_{n,N})$. In both cases the isomorphism\nis given by $v_{\\lambda }\\mapsto \\lbrack \\mathcal{O}_{\\lambda }]$, that is\nthe spin basis (\\ref{spin basis}) is mapped onto Schubert classes.\n\\end{corollary}\n\n\\begin{proof}\nWorking in the basis of Bethe vectors we employ once more (\\ref{Bethe2spin})\nfor $q=0$ to find%\n\\begin{equation}\nv_{\\lambda }=\\sum_{\\mu }G_{\\lambda }(t_{\\mu }|\\ominus t)Y_{\\mu }\\;.\n\\label{Bethe2spin0}\n\\end{equation}%\nIn other words each Schubert class $[\\mathcal{O}_{\\lambda }]$ is identified\nwith the (finite) sequence $\\{G_{\\lambda }(t_{\\mu }|\\ominus t)\\}_{\\mu\n\\subset (k^{n})}$ where each boxed partition $\\mu $ labels a fixed point\nunder the torus action. The definition (\\ref{idempotent_def}) of $%\n\\circledast $ corresponds to pointwise multiplication of these sequences\nwhich satisfy the conditions (\\ref{GKM}) and can be successively generated\nfrom the top class (\\ref{topclass}). The assertion then follows from \\cite[%\nThm 1.2.2]{GKM} for $\\beta =0$ and from \\cite[Thm 3.13]{KostantKumar2} for $%\n\\beta =-1$.\n\\end{proof}\n\n\\begin{corollary}\nThe ring $qh_{n}^{\\ast }\/\\langle \\beta \\rangle $ is isomorphic to\nequivariant quantum cohomology $QH_{\\mathbb{T}}^{\\ast }(\\limfunc{Gr}_{n,N})$.\n\\end{corollary}\n\n\\begin{proof}\nConsider the equivariant quantum Pieri-Chevalley rule (\\ref{H1}). Rewriting\nit as \n\\begin{equation*}\nH_{1}v_{\\mu }=\\beta ^{-1}\\left( \\frac{\\Pi (t_{\\mu })}{\\Pi (t_{\\emptyset })}%\n-1\\right) v_{\\mu }+\\frac{\\Pi (t_{\\mu })}{\\Pi (t_{\\emptyset })}\\sum \n_{\\substack{ \\mu \\rightrightarrows ^{\\ast }\\lambda \\lbrack d]  \\\\ d=0,1}}%\nq^{d}v_{\\lambda }\n\\end{equation*}%\nwhere the sum runs over all $\\lambda \\subset (k^{n})$ such that $\\lambda\n\\neq \\mu $ and either $\\lambda \/\\mu $ or $\\lambda \/1\/\\mu $ is a skew diagram\nwhich contains at most\\emph{\\ }one box in each column or row. Setting $\\beta\n=0$ this simplifies to%\n\\begin{equation*}\nH_{1}v_{\\mu }=\\left( \\tsum_{i\\in I_{\\mu }}t_{i}-\\tsum_{i=1}^{n}t_{i}\\right)\nv_{\\mu }+\\sum_{\\substack{ \\lambda \/d\/\\mu =(1)  \\\\ d=0,1}}q^{d}v_{\\lambda\n}=v_{1}\\circledast _{\\beta =0}v_{\\mu }\n\\end{equation*}%\nwhere the sum now runs over all $\\lambda \\subset (k^{n})$ such that $\\lambda\n\\neq \\mu $ and either $\\lambda \/\\mu $ or $\\lambda \/1\/\\mu $ is a skew diagram\nwhich contains \\emph{exactly} one box. This is Mihalcea's equivariant\nquantum Pieri-Chevalley rule for $QH_{\\mathbb{T}}^{\\ast }(\\limfunc{Gr}%\n_{n,N}) $ which together with the usual grading, $v_\\lambda$ has degree $%\n|\\lambda|$ and $q$ has degree $N$, fixes the ring up to isomorphism; see \n\\cite[Cor 7.1]{MihalceaAIM}. An alternative proof which exploits the\npresentation of $QH_{\\mathbb{T}}^{\\ast }(\\limfunc{Gr}_{n,N})$ as Jacobi\nalgebra can be found in \\cite{GoKo}.\n\\end{proof}\n\n\\subsection{Equivariant quantum Pieri rules and Giambelli formula}\n\nAccording to its definition (\\ref{BigG}) the operator $\\boldsymbol{G}%\n_{\\lambda }$ is the multiplication operator which multiplies with a\nlocalised Schubert class. The following corollary states that for $\\lambda $\nbeing a single row or column this operator is given by the transfer matrices\n(\\ref{Hr}), (\\ref{Er}) in the spin-basis.\n\n\\begin{corollary}\nThe operators $\\{H_{r}\\}_{r=1}^{k}$ and $\\{E_{r}\\}_{r=1}^{n}$ defined\nrespectively in (\\ref{Hr}) and (\\ref{Er}) act on the Bethe vectors $|y_{\\mu\n}\\rangle $ by multiplication with $G_{r}(y_{\\mu }|\\ominus t)$ and $%\nG_{1^{r}}(y_{\\mu }|\\ominus t)$, respectively. That is,%\n\\begin{equation}\n\\boldsymbol{G}_{r}=H_{r}\\qquad \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{and\\qquad }\\boldsymbol{G}%\n_{1^{r}}=E_{r}\\;.  \\label{Pieri0}\n\\end{equation}\n\\end{corollary}\n\nNote in particular, that this implies for $q=0$ that the matrix elements $%\n\\langle \\lambda |H_{r}|\\mu \\rangle $, $\\langle \\lambda |E_{r}|\\mu \\rangle $\nin the basis $\\{v_{\\lambda }\\}$ give the coefficients in the equivariant\nPieri rules for $H_{\\mathbb{T}}^{\\ast }(\\limfunc{Gr}_{n,n+k})$ if $\\beta =0$\nand for $K_{\\mathbb{T}}(\\limfunc{Gr}_{n,n+k})$ if $\\beta =-1$.\n\n\\begin{proof}\nUsing (\\ref{SpecE}) and the expansions (\\ref{e}), (\\ref{Er}) we deduce that%\n\\begin{equation*}\nE_{r}|y_{\\mu }\\rangle =G_{1^{r}}(y|\\ominus t)|y_{\\mu }\\rangle \\;.\n\\end{equation*}%\nBut then (\\ref{levelrankmom}) together with (\\ref{Gduality}) gives%\n\\begin{eqnarray*}\nH_{r}(t)|y_{\\mu }\\rangle &=&H_{r}(t)\\Theta |y_{\\mu ^{\\ast }}\\rangle =\\Theta\nE_{r}(\\ominus t^{\\prime })|y_{\\mu ^{\\ast }}\\rangle \\\\\n&=&G_{1^{r}}(\\ominus y_{\\mu ^{\\ast }}|t^{\\prime })|y_{\\mu }\\rangle\n=G_{r}(y_{\\mu }|\\ominus t)|y_{\\mu }\\rangle \\;.\n\\end{eqnarray*}\n\\end{proof}\n\nIn light of the expansion (\\ref{G2F}) and (\\ref{JTG}), the last result\nallows us to express the operator (\\ref{BigG}) which corresponds to\nmultiplication with a Schubert class, in terms of the transfer matrix\ncoefficients $H_{r}$ from (\\ref{Hr}). The latter, as we have just seen,\ncorrespond to multiplication with a Chern class. Such a formula expressing a\ngeneral Schubert class in terms of Chern classes, is often called \\emph{%\nGiambelli formula} in the literature on cohomology.\n\n\\begin{corollary}[equivariant quantum Giambelli formula]\nFor $\\lambda \\subset (k^{n})$ a boxed partition define the operators $%\n\\boldsymbol{F}_{\\lambda }=\\det (\\tau ^{1-j}H_{\\lambda _{i}-i+j})$ where $%\n\\tau $ is the shift operator (\\ref{shifted power}) and $\\tau ^{p}H_{r}$ is\nthe coefficient in (\\ref{Hr}) with respect to the shifted factorial powers $%\n(x|\\tau ^{p}t^{\\ast })^{r}$. Then%\n\\begin{equation}\n\\boldsymbol{G}_{\\lambda }=\\sum_{\\alpha }\\beta ^{|\\alpha |}\\phi _{\\alpha\n}(\\lambda )\\boldsymbol{F}_{\\lambda +\\alpha }  \\label{BigG2F}\n\\end{equation}%\nwith the same conventions for $\\alpha $ and $\\phi _{\\alpha }(\\lambda )$ as\nin Prop \\ref{prop:Gexp} and $\\boldsymbol{F}_{\\lambda +\\alpha }$ is defined\nin terms of the straightening rule (\\ref{Fstraight}).\n\\end{corollary}\n\n\\begin{example}\nRecall the formula (\\ref{G2Fex}) for $n=2$. Then%\n\\begin{equation*}\n\\boldsymbol{G}_{\\lambda _{1},\\lambda _{2}}=\\frac{1+\\beta t_{\\lambda _{2}+1}}{%\n1+\\beta t_{1}}~\\left( \\boldsymbol{F}_{\\lambda _{1},\\lambda _{2}}+\\boldsymbol{%\nF}_{\\lambda _{1},\\lambda _{2}+1}\\right)\n\\end{equation*}%\nwhere%\n\\begin{equation*}\n\\boldsymbol{F}_{\\lambda _{1},\\lambda _{2}}=\\left\\vert \n\\begin{array}{cc}\nH_{\\lambda _{1}} & \\tau ^{-1}H_{\\lambda _{1}-1} \\\\ \nH_{\\lambda _{2}-1} & \\tau ^{-1}H_{\\lambda _{2}}%\n\\end{array}%\n\\right\\vert\n\\end{equation*}%\nand $H_{r}$ is given by (\\ref{Hr2}) while the negative shifted factorial\npower is defined as%\n\\begin{equation*}\n\\tau ^{-1}H_{k+1-i}=\\sum_{j=1}^{i}\\frac{H(t_{N-j})}{\\prod\\nolimits_{1\\leq\n\\ell \\neq j\\leq i}t_{N-j}\\ominus t_{N-\\ell }}\\;.\n\\end{equation*}\n\\end{example}\n\nFor certain choices the formula (\\ref{BigG2F}) considerably simplifies. We\nalready saw that for $\\lambda $ a single row or column we obtain $%\n\\boldsymbol{G}_{r}=H_{r}$ and $\\boldsymbol{G}_{1^{r}}=E_{r}$. Setting $%\n\\lambda =(k^{n})$ we find from (\\ref{SpecE}) and (\\ref{qtopclass}) that \n\\begin{equation*}\n\\prod_{i=1}^{k}E(\\ominus t_{i})~Y_{\\mu }=G_{(k^{n})}(y_{\\mu }|\\ominus\nt)Y_{\\mu }\n\\end{equation*}%\nand, hence, that $\\boldsymbol{G}_{(k^{n})}=\\prod_{i=1}^{k}E(\\ominus t_{i})$.\n\n\\begin{corollary}[Fusion matrices]\n\\label{cor:fusion_matrices}The matrices $\\{\\boldsymbol{G}_{\\lambda\n}\\}_{\\lambda \\subset (k^{n})}$ yield a faithful representation of $%\nqh_{n}^{\\ast }$, that is%\n\\begin{equation}\n\\boldsymbol{G}_{\\lambda }\\boldsymbol{G}_{\\mu }=\\sum_{\\nu \\subset\n(k^{n})}C_{\\lambda \\mu }^{\\nu }(t,q)\\boldsymbol{G}_{\\nu }\\;.\n\\label{fusion_matrices}\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\nThis is a direct consequence of $v_{\\lambda }=v_{\\lambda }\\circledast\nv_{\\emptyset }=\\boldsymbol{G}_{\\lambda }v_{\\emptyset }$ and the fact that\nthe $v_{\\lambda }$'s are linearly independent. Namely, assume $%\n0=\\sum_{\\lambda \\subset (k^{n})}c_{\\lambda }\\boldsymbol{G}_{\\lambda }$ for\nsome coefficients $c_{\\lambda }$. Then $0=\\sum_{\\lambda \\subset\n(k^{n})}c_{\\lambda }\\boldsymbol{G}_{\\lambda }v_{\\emptyset }=\\sum_{\\lambda\n\\subset (k^{n})}c_{\\lambda }v_{\\lambda }$ and, thus, we must have $%\nc_{\\lambda }=0$ for all $\\lambda \\subset (k^{n})$. The product expansion\nfollows from (\\ref{combproduct}), $v_{\\lambda }\\circledast v_{\\mu }=%\n\\boldsymbol{G}_{\\lambda }\\boldsymbol{G}_{\\mu }v_{\\emptyset }$.\n\\end{proof}\n\n\\subsection{Coordinate ring presentation}\n\nWe now prove the presentation of $qh_{n}^{\\ast }$ stated in the\nintroduction. Consider the polynomial algebra $\\mathcal{A}_{n}$ generated by \n$\\{e_{r}\\}_{r=1}^{n}\\cup \\{h_{r}\\}_{r=1}^{k}$ over $\\mathcal{R}(\\mathbb{T}%\n,q) $ subject to the relations given by (\\ref{ideal def}) with $e(x)$ and $%\nh(x)$ as in (\\ref{H def}) and (\\ref{E def}). Define $\\{g_{\\lambda\n}\\}_{\\lambda \\subset (k^{n})}\\subset \\mathcal{A}_{n}$ as just explained for\nthe $\\boldsymbol{G}_{\\lambda }$'s: set $f_{\\lambda }=\\det (\\tau\n^{1-j}h_{\\lambda _{i}-i+j})$, where the \\textquotedblleft shifted\ngenerators\\textquotedblright\\ $\\tau ^{p}h_{r}$ are obtained by expanding $%\nh(x)$ into shifted factorial powers $(x|\\tau ^{p}t^{\\ast })^{r}$, and then\nintroduce $g_{\\lambda }$ through the analogous expansion as in Prop \\ref%\n{prop:Gexp} and (\\ref{BigG2F}).\n\n\\begin{theorem}\n\\label{thm:coord_ring}The map $g_{\\lambda }\\mapsto v_{\\lambda }$ constitutes\nan algebra isomorphism $\\mathcal{A}_{n}\\cong qh_{n}^{\\ast }$.\n\\end{theorem}\n\n\\begin{proof}\nIntroduce auxiliary variables $\\xi =(\\xi _{1},\\ldots ,\\xi _{n})$ by setting%\n\\begin{equation*}\ne(x)=\\prod_{i=1}^{n}x\\oplus \\xi _{i}\\;.\n\\end{equation*}%\nDividing by $e(x)$ in (\\ref{ideal def}) one obtains $h(x)$ as a rational\nfunction in $x$, but as $h(x)$ -- by definition -- is polynomial in $x$ the\nresidues at the poles must vanish. This implies that the $\\xi _{i}$'s obey\nthe Bethe ansatz equations (\\ref{BAE}). Moreover, one deduces in a similar\nmanner as we did before that $e_{r}=G_{1^{r}}(\\xi |\\ominus t)$ and $%\nh_{r}=G_{r}(\\xi |\\ominus t)$. Thus, $g_{\\lambda }=G_{\\lambda }(\\xi |\\ominus\nt)$ according to Prop \\ref{prop:Gexp}. This then implies that the map $%\ng_{\\lambda }\\mapsto v_{\\lambda }$ is an algebra homomorphism and it is also\nsurjective. It remains to show that the dimension of $\\mathcal{A}_{n}$\nequals the dimension of $qh_{n}^{\\ast }$. Recall from Section \\ref%\n{sec:grothendieck} that each $G_{\\lambda }$ can be expressed via (\\ref{JTG}%\n), (\\ref{Fdet}) and (\\ref{G2F}) in terms of $G_{r}$'s and that the factorial\nGrothendieck polynomials $\\{G_{\\lambda }\\}$ with $\\lambda $ having at most $%\nn $ parts form a basis of $\\mathcal{R}(\\mathbb{T},q)[\\xi _{1},\\ldots ,\\xi\n_{n}]^{\\mathbb{S}_{n}}$, hence $\\mathcal{R}(\\mathbb{T},q)[\\xi _{1},\\ldots\n,\\xi _{n}]^{\\mathbb{S}_{n}}\\cong \\mathcal{R}(\\mathbb{T},q)[G_{1},G_{2},%\n\\ldots ]$. Therefore, we only have to show that each $G_{\\lambda }(\\xi\n|\\ominus t)$ with $\\lambda \\nsubseteq (k^{n})$ can be expressed as a linear\ncombination of the $\\{g_{\\mu }\\}_{\\mu \\subset (k^{n})}$. But since the $\\xi $%\n's obey (\\ref{BAE}), we can deduce that each $G_{\\lambda }(\\xi |\\ominus t)$\nwith $\\lambda \\nsubseteq (k^{n})$ can be \\textquotedblleft\nreduced\\textquotedblright\\ using multiple times (\\ref{Greduction}) until it\nis indexed by a composition where no part is greater than $k$. Then one\napplies repeatedly the straightening rule (\\ref{Gstraight}) to rewrite the\nresult as a linear combination of the $g_{\\mu }$'s with $\\mu \\subset (k^{n})$%\n.\n\\end{proof}\n\n\\subsubsection{A generalised rim-hook algorithm\\label{sec:rimhook}}\n\nOur proof of the last theorem contains an algorithm for the successive\ncomputation of the structure constants $C_{\\lambda \\mu }^{\\nu }(t,q)$\nwithout making use of the explicit solutions of the Bethe ansatz equations (%\n\\ref{BAE}) and the residue formula (\\ref{Verlinde}). Namely, starting from\nthe Pieri rule (\\ref{G1Pieri}) for $G_{1}$, one can use (\\ref{Greduction})\nand (\\ref{Gstraight}) to define a generalised version of the rim-hook\nalgorithm at $\\beta =0$ \\cite{BCF}; see \\cite{BBT} for a recent extension to\nthe equivariant case with $\\beta =0$. We shall demonstrate this only on a\nsimple example.\n\n\\begin{example}\nSet $G_{\\lambda }=G_{\\lambda }(\\xi |\\ominus t)$ and consider the following\nproduct expansions which follow from (\\ref{G1}) and (\\ref{G1Pieri}),%\n\\begin{eqnarray*}\nG_{1,0}\\cdot G_{1,0} &=&t_{3}\\ominus t_{2}~G_{1,0}+\\frac{1+\\beta t_{3}}{%\n1+\\beta t_{2}}\\left( G_{2,0}+G_{1,1}+\\beta G_{2,1}\\right) \\\\\nG_{1,0}\\cdot G_{1,1} &=&t_{3}\\ominus t_{1}~G_{1,1}+\\frac{1+\\beta t_{3}}{%\n1+\\beta t_{1}}~G_{2,1}\\;.\n\\end{eqnarray*}%\nFor $N=3$ and $n=2$ employ (\\ref{Greduction}) and (\\ref{Gstraight}) to find%\n\\begin{equation*}\nG_{2,0}=qG_{-1,0}=-q\\beta G_{0,0}=-q\\beta \\qquad \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{and}\\qquad\nG_{2,1}=qG_{0,0}=q\\;.\n\\end{equation*}%\nThis yields the following product expansion in $qh_{2}^{\\ast }$,%\n\\begin{eqnarray*}\ng_{1,0}\\cdot g_{1,0} &=&t_{3}\\ominus t_{2}~g_{1,0}+\\left( 1+\\beta\nt_{3}\\ominus t_{2}\\right) g_{1,1} \\\\\ng_{1,0}\\cdot g_{1,1} &=&t_{3}\\ominus t_{1}~g_{1,1}+\\left( 1+\\beta\nt_{3}\\ominus t_{1}\\right) q\n\\end{eqnarray*}%\nwhich because of $qh_{2}^{\\ast }\\cong qh_{1}^{\\ast }$ -- see (\\ref{levelrank}%\n) -- are equivalent to the products $g_{1}\\cdot g_{1}$ and $g_{1}\\cdot g_{2}$\nin $qh_{1}^{\\ast }$ which we computed in Example \\ref{ex:3}.\n\\end{example}\n\n\\subsection{Partition functions and Richardson varieties}\n\nWe provide another concrete example where a natural link between our lattice\nmodel approach and geometry occurs. Recall the definition of Richardson\nvarieties and the expansion (\\ref{Richardson}) for $\\beta =0$.\n\n\\begin{proposition}\nThe partition functions (\\ref{Z2H}), (\\ref{Z'2E}) have the expansions%\n\\begin{eqnarray}\n\\langle \\lambda |Z_{n}(x|t)|\\mu \\rangle &=&\\sum_{\\lambda \\subset\n(k^{n})}c_{\\mu \\nu }^{\\lambda }(t)G_{\\nu ^{\\vee }}(x|\\ominus t^{\\prime }) \\\\\n\\langle \\lambda |Z_{k}^{\\prime }(x|t)|\\mu \\rangle &=&\\sum_{\\lambda \\subset\n(k^{n})}c_{\\mu \\nu }^{\\lambda }(t)G_{\\nu ^{\\ast }}(x|t),\n\\end{eqnarray}%\nwhere the coefficients are explicity given by (\\ref{Verlinde0}).\n\\end{proposition}\n\n\\begin{proof}\nEmploying the result (\\ref{SpecE}) from the Bethe ansatz and the Cauchy\nidentity (\\ref{Cauchy0}) for factorial Grothendieck polynomials we find,%\n\\begin{multline*}\n\\langle \\lambda |Z_{\\lambda ,\\mu }^{\\prime }(x_{1},\\ldots ,x_{k}|t)|\\mu\n\\rangle =\\sum_{\\alpha \\subset (k^{n})}\\langle \\lambda |E(x_{1})\\cdots\nE(x_{k})|y_{\\alpha }\\rangle \\langle y_{\\alpha }|\\mu \\rangle \\\\\n=\\sum_{\\alpha \\subset (k^{n})}\\frac{\\Pi (t_{\\alpha })}{\\Pi (t_{\\lambda })}%\n\\frac{G_{\\mu }(t_{\\alpha }|\\ominus t)G_{\\lambda ^{\\vee }}(t_{\\alpha\n}|\\ominus t^{\\prime })}{e(t_{\\alpha },t_{\\alpha })}\\prod_{i=1}^{k}\\prod_{j%\n\\in I_{\\alpha }}(x_{i}\\oplus t_{j}) \\\\\n=\\sum_{\\nu \\subset (k^{n})}\\sum_{\\alpha \\subset (k^{n})}\\frac{\\Pi (t_{\\alpha\n})}{\\Pi (t_{\\lambda })}\\frac{G_{\\mu }(t_{\\alpha }|\\ominus t)G_{\\lambda\n^{\\vee }}(t_{\\alpha }|\\ominus t^{\\prime })}{e(t_{\\alpha },t_{\\alpha })}\\frac{%\n\\Pi (t_{\\nu })}{\\Pi (t_{\\alpha })}G_{\\nu }(t_{\\alpha }|\\ominus t)G_{\\nu\n^{\\ast }}(x|t) \\\\\n=\\sum_{\\nu \\subset (k^{n})}c_{\\mu \\nu }^{\\lambda }(t)G_{\\nu ^{\\ast }}(x|t)\n\\end{multline*}%\nThe identity for the vicious walker model now follows from level-rank\nduality (\\ref{levelrank}) for $q=0$ and (\\ref{levelrankmom}),%\n\\begin{eqnarray*}\n\\langle \\lambda |Z_{k}^{\\prime }(x|t)|\\mu \\rangle &=&\\langle \\lambda\n|E(x_{1}|t)\\cdots E(x_{k}|t)|\\mu \\rangle \\\\\n&=&\\langle \\lambda ^{\\prime }|H(x_{1}|\\ominus t^{\\prime })\\cdots\nH(x_{k}|\\ominus t^{\\prime })|\\mu ^{\\prime }\\rangle =\\langle \\lambda ^{\\prime\n}|Z_{k}(x|\\ominus t^{\\prime })|\\mu ^{\\prime }\\rangle \\;.\n\\end{eqnarray*}\n\\end{proof}\n\n\\begin{remark}\nWe expect that an analogous expansion of the partition function holds also\nfor the quantum case with $q\\neq 0$. However, we are currently lacking the\nnecessary quantum analogue of the identity (\\ref{Cauchy0}).\n\\end{remark}\n\n\\subsection{Opposite Schubert varieties and their classes}\n\nRecall the definition of the opposite Schubert class $[\\mathcal{O}^{\\lambda\n}]$ and the dual basis (\\ref{Kdualbasis}) from Section \\ref{sec:schubert}\nfor $q=0$. We state the analogous quantum relations for $qh_{n}^{\\ast }$.\n\nDefine a \\textquotedblleft opposite Schubert basis\\textquotedblright\\ $%\n\\{v^{\\lambda }\\}$ by setting%\n\\begin{equation}\nv^{\\lambda }=\\frac{\\Pi (t_{\\emptyset })}{\\Pi (t_{\\lambda })}\\sum_{\\mu\n\\subset (k^{n})}G_{\\lambda ^{\\vee }}(y_{\\mu }|\\ominus t^{\\prime })Y_{\\mu }\\;.\n\\label{op_spin_basis}\n\\end{equation}%\nEmploying the bilinear form (\\ref{bilinear_form}) we now identify the dual\nspin basis $\\{\\tilde{v}_{\\lambda }\\}$ in terms of the product (\\ref%\n{idempotent_def}).\n\n\\begin{proposition}\nWe have the relation%\n\\begin{equation}\n(v_{\\lambda },(1+\\beta H_{1})v^{\\mu })=\\delta _{\\lambda \\mu }\n\\label{dual_spin_basis}\n\\end{equation}%\nand the product expansion (c.f. (\\ref{Kstructure}))%\n\\begin{equation}\nv_{\\mu }\\circledast v^{\\lambda }=\\sum_{\\nu \\subset (k^{n})}C_{\\mu \\nu\n}^{\\lambda }(t,q)v^{\\nu }\\;.  \\label{dual_product}\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nFrom the definition (\\ref{op_spin_basis}), the identity (\\ref{G1}) and (\\ref%\n{G1=H1}) it follows that%\n\\begin{equation*}\n(1+\\beta H_{1})v^{\\mu }=\\sum_{\\alpha \\subset (k^{n})}\\frac{\\Pi (y_{\\alpha })%\n}{\\Pi (t_{\\lambda })}~G_{\\lambda ^{\\vee }}(y_{\\alpha }|\\ominus t^{\\prime\n})Y_{\\alpha }\\;.\n\\end{equation*}%\nThe assertion (\\ref{dual_spin_basis}) then follows from the definition (\\ref%\n{bilinear_form}) and (\\ref{res of 1}).\n\nTo find the product expansion we make once more use of (\\ref{Bethe2spin})\nand (\\ref{op_spin_basis}) to find%\n\\begin{equation*}\nv_{\\mu }\\circledast v^{\\lambda }=\\sum_{\\alpha \\subset (k^{n})}\\frac{\\Pi\n(t_{\\emptyset })}{\\Pi (t_{\\lambda })}~G_{\\mu }(y_{\\alpha }|\\ominus\nt)G_{\\lambda ^{\\vee }}(y_{\\alpha }|\\ominus t^{\\prime })Y_{\\alpha }\\;.\n\\end{equation*}%\nUsing (\\ref{Cauchy2}) we compute the expansion%\n\\begin{equation*}\nY_{\\alpha }=\\frac{\\Pi (y_{\\alpha })}{\\Pi (t_{\\emptyset })}\\sum_{\\lambda\n\\subset (k^{n})}\\frac{G_{\\lambda }(y_{\\alpha }|\\ominus t)}{e(y_{\\alpha\n},y_{\\alpha })}~v^{\\lambda }\\;.\n\\end{equation*}%\nInserting the latter into the previous equation we arrive at (\\ref%\n{dual_product}) by making use of (\\ref{Verlinde}).\n\\end{proof}\n\nIn light of the known relations (\\ref{Kstructure}) and (\\ref{Kdualbasis})\nfor the non-equivariant case, we conjecture based on (\\ref{dual_product})\nand (\\ref{dual_spin_basis}) the following:\n\n\\begin{conjecture}\nConsider the ring $K_{\\mathbb{T}}(\\limfunc{Gr}_{n,N})=qh_{n}^{\\ast }\/\\langle\nq,\\beta +1,t_{j}-1+e^{\\varepsilon _{N+1-i}}\\rangle $. The localised class of\nthe sheaf $\\mathcal{O}^{\\lambda }$ of the opposite Schubert variety is given\nby the expression%\n\\begin{equation}\n\\lbrack \\mathcal{O}^{\\lambda }]_{\\mu }=\\frac{\\Pi (t_{\\emptyset })}{\\Pi\n(t_{\\lambda })}~G_{\\lambda ^{\\vee }}(t_{\\mu }|\\ominus t^{\\prime })=\\frac{\\Pi\n(t_{\\lambda ^{\\ast }})}{\\Pi (t_{(k^{n})})}~G_{\\lambda ^{\\ast }}(\\ominus\nt_{\\mu ^{\\ast }}|t)\\;.\n\\end{equation}\n\\end{conjecture}\n\n\\subsection{The homogeneous limit $t_{j}=0$: quantum $K$-theory}\n\nThe inversion formulae (\\ref{Hr2}), (\\ref{Er2}) for the expansions (\\ref{Hr}%\n), (\\ref{Er}) do not hold true in the homogeneous limit when $t_{j}=0$ for\nall $j=1,\\ldots ,N$. We therefore need to discuss this case separately. We\nstart with the Pieri formulae, i.e. the action of the transfer matrices in\nthe spin basis.\n\nGiven a toric horizontal (vertical) strip $\\theta =\\nu \/d\/\\lambda $ denote\nby $c(\\theta )=|\\mathcal{C}_{\\theta }|$ the number of columns and by $%\nr(\\theta )=|\\mathcal{R}_{\\theta }|$ the number of rows which intersect the\nstrip.\n\n\\begin{corollary}[non-equivariant Pieri rules]\nSet $t_{j}=0$ for all $j$. Then%\n\\begin{eqnarray}\nH_{\\ell }v_{\\mu } &=&\\sum_{\\substack{ \\theta =\\lambda \/d\/\\mu \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{ }  \\\\ \n\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{toric hor strip}}}q^{d}\\beta ^{|\\theta |-\\ell }\\binom{r(\\theta )-1}{%\n|\\theta |-\\ell }v_{\\lambda }  \\label{Hr0} \\\\\nE_{\\ell ^{\\prime }}v_{\\mu } &=&\\sum_{\\substack{ \\theta =\\lambda \/d\/\\mu \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{\n}  \\\\ \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{toric ver strip}}}q^{d}\\beta ^{|\\theta |-\\ell ^{\\prime }}\\binom{%\nc(\\theta )-1}{|\\theta |-\\ell ^{\\prime }}v_{\\lambda }  \\label{Er0}\n\\end{eqnarray}%\nwhere $\\ell =1,\\ldots ,k$ and $\\ell ^{\\prime }=1,\\ldots ,n$.\n\\end{corollary}\n\n\\begin{proof}\nSetting $t_{j}=0$ for all $j$ the combinatorial action of the transfer\nmatrices on $\\mathcal{V}_{n}$ simplifies to%\n\\begin{eqnarray}\nH(x)v_{\\mu } &=&\\sum_{\\substack{ \\theta =\\lambda \/d\/\\mu \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{ } \\\\ \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{%\ntoric hor strip}}}q^{d}x^{k-|\\theta |}(1+\\beta x)^{r(\\theta )}v_{\\lambda }\n\\label{H0} \\\\\nE(x)v_{\\mu } &=&\\sum_{\\substack{ \\theta =\\lambda \/d\/\\mu \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{ } \\\\ \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{%\ntoric ver strip}}}q^{d}x^{n-|\\theta |}(1+\\beta x)^{c(\\theta )}v_{\\lambda }\n\\label{E0}\n\\end{eqnarray}%\nEmploying in addition that the expansions (\\ref{Hr}), (\\ref{Er}) of the\ntransfer matrices on $\\mathcal{V}_{n}$ in the variable $x$ now read%\n\\begin{eqnarray}\nH(x) &=&x^{k}\\cdot \\mathbf{1}_{\\mathcal{V}_{n}^{q}}+(1+\\beta x)\\sum_{\\ell\n=1}^{k}H_{\\ell }x^{k-\\ell }  \\label{Hexp0} \\\\\nE(x) &=&x^{n}\\cdot \\mathbf{1}_{\\mathcal{V}_{n}^{q}}+(1+\\beta x)\\sum_{\\ell\n=1}^{n}E_{\\ell }x^{n-\\ell }  \\label{Eexp0}\n\\end{eqnarray}%\nthe asserted formulae are easily deduced.\n\\end{proof}\n\nWe now turn to the Bethe ansatz computation. Since the matrix elements of\nthe $R$-matrix in (\\ref{mom_ybe}) do not depend on the $t_{j}$'s the\ncommutation relations in the row Yang-Baxter algebra, and in particular the\nrelations in Lemma \\ref{lem:yba1} and \\ref{lem:yba2}, are unchanged for $%\nt_{j}=0$. From this one deduces, along the same lines as before, that the\nBethe ansatz equations are obtained by formally setting $t_{j}=0$ in (\\ref%\n{BAE}),%\n\\begin{equation}\ny_{i}^{N}\\prod_{j\\neq i}\\frac{1+\\beta y_{j}}{1+\\beta y_{i}}%\n=(-1)^{n-1}q,\\qquad i=1,\\ldots ,n\\;.  \\label{BAE0}\n\\end{equation}%\nWe have the following result which replaces Lemma \\ref{lem:BAE} when $t_{j}=0\n$. Suppose $q^{1\/N}$ exists and set $\\zeta =\\exp (2\\pi \\imath \/N)$ where $%\n\\imath $ is the imaginary unit.\n\n\\begin{lemma}\nThe set of equations (\\ref{BAE0}) has $\\binom{N}{n}$ \\emph{pairwise distinct \n}solutions \n\\begin{equation}\ny_{\\lambda }=(y_{\\frac{n+1}{2}+\\lambda _{n}^{\\prime }-n},\\ldots ,y_{\\frac{n+1%\n}{2}+\\lambda _{1}^{\\prime }-1})\\in \\mathbb{C}[\\![\\beta, q^{\\frac{1}{N}}]\\!]%\n,  \\label{Bethe_root_0}\n\\end{equation}%\nwhere $\\lambda \\subset (k^{n})$ and up to first order in $\\beta $ we have%\n\\begin{equation}\ny_{j}=q^{\\frac{1}{N}}\\zeta ^{j}+\\beta ~(-1)^{n-1}q^{\\frac{2}{N}}\\zeta^j\\sum_{l\\neq j}(\\zeta^{l}-\\zeta^j)+O(\\beta ^{2})\\;.  \\label{Brootexpansion0}\n\\end{equation}\nMoreover, the $r$th term in this expansion is proportional to $q^{r\/N}$ and, thus, we can always force convergence for a given $\\beta$ provided we specialise q to a sufficiently small number.\n\\end{lemma}\n\n\\begin{proof}\nWe now make the ansatz $y_{j}=\\sum_{r\\geq 0}y_{j}^{(r)}\\beta $. Setting $%\n\\beta =0$ in (\\ref{BAE0}) we obtain the Bethe ansatz equations at the free\nfermion point which decouple. Clearly, each of the $n$ equations has then $N$\nsolutions and using the conventions from \\cite[Prop 10.4]{KorffStroppel} we\nset $y_{j}^{(0)}=q^{1\/N}\\zeta ^{j}$ with $j\\in \\{\\frac{n+1}{2}+\\lambda\n_{1}^{\\prime }-1,\\ldots ,\\frac{n+1}{2}+\\lambda _{n}^{\\prime }-n\\}$ for $%\n\\lambda \\subset (k^{n})$. By the analogous arguments as in the previous case\nwhen we expanded the Bethe roots with respect to $q$ we find by\ndifferentiating with respect to $\\beta $ and setting $\\beta =0$ afterwards\nthe desired expansion. In particular, when taking the $r$th derivative with\nrespect to $\\beta ,$ the coefficient $\\left( \\prod_{j\\neq i}\\frac{1+\\beta\ny_{j}}{1+\\beta y_{i}}\\right) _{\\beta =0}=1$ in front of the term $\\frac{d^{r}%\n}{d\\beta ^{r}}y_{j}^{N}|_{\\beta =0}$ is always nonzero. One then proves by induction \nthe stated dependence on $q^{1\/N}$.\n\\end{proof}\n\nThis lemma can be used to establish the completeness of the Bethe ansatz when $t_j=0$ and\nto derive the results analogous to (\\ref{Greduction}), (\\ref{specH}), (\\ref{specE}), (\\ref%\n{SpecE}) and (\\ref{Cauchy2}), (\\ref{res of 1}) by simply setting formally $%\nt_{j}=0$ in the respective formulae. Thus, extending the base field of our\nquantum space, $\\mathcal{V}^{q}=V^{\\otimes N}\\otimes\\mathbb{C}[\\![\\beta, q^{\\frac{1}{N}}]\\!]$, we can introduce an algebra structure via (\\ref{idempotent_def})\nas before by making use of the Bethe vectors. Note, however, that this\nextension to $\\mathbb{C}[\\![\\beta, q^{\\frac{1}{N}}]\\!]$ is only necessary if we require\nthe existence of idempotents. Alternatively, we can introduce the product\nstructure via (\\ref{combproduct}) by defining the analogue of the operator $%\n\\boldsymbol{G}_{\\lambda }$ for $t_{j}=0$ as follows.\n\nFor each $n=0,1,\\ldots ,N$ define operators $\\boldsymbol{s}_{\\lambda }\\in \n\\limfunc{End}(\\mathbb{Z}[\\![q]\\!]\\otimes V_{n})$ for $\\lambda \\subset\n(k^{n}) $ by%\n\\begin{equation}\n\\boldsymbol{s}_{\\lambda }=\\det (\\boldsymbol{e}_{\\lambda _{i}^{\\prime\n}-i+j}),\\qquad \\boldsymbol{e}_{r}=\\sum_{j=r}^{n}(-\\beta )^{j-r}\\binom{j-1}{%\nj-r}E_{j}~,  \\label{ncSchur}\n\\end{equation}%\nwhere $r=1,2,\\ldots ,n$; compare with (\\ref{er2G1r}). We set $\\boldsymbol{e}%\n_{0}$ to be the identity operator. Note that since the $E_{j}$'s mutually\ncommute so do the $\\boldsymbol{e}_{r}$'s, whence the determinant $%\n\\boldsymbol{s}_{\\lambda }$ is well-defined.\n\nConsider the commutative algebra $qh_{n}^{\\ast }\/\\langle t_{1},\\ldots\n,t_{N}\\rangle $ generated by $\\{H_{r}\\}_{r=1}^{k}\\cup \\{E_{r}\\}_{r=1}^{n}$\nwith $t_{j}=0$. For each $\\lambda \\subset (k^{n})$ define in analogy with (%\n\\ref{G2Schur}) the operators \n\\begin{equation}\n\\boldsymbol{G}_{\\lambda }=\\sum_{\\alpha }\\beta ^{|\\alpha |}\\prod_{i=1}^{n}%\n\\binom{i-1}{\\alpha _{i}}\\boldsymbol{s}_{\\lambda +\\alpha }~,\n\\label{Schubert0}\n\\end{equation}%\nwhere $\\boldsymbol{s}_{\\lambda +\\alpha }$ is defined in terms of the\nstraightening rules analogous to (\\ref{Fstraight}).\n\n\\begin{corollary}\nConsider $qh_{n}^{\\ast }\/\\langle t_{1},\\ldots ,t_{N}\\rangle $. The map $%\nv_{\\lambda }\\mapsto \\lbrack \\mathcal{O}_{\\lambda }]$ defines for $\\beta =0$\na ring isomorphism with $QH^{\\ast }(\\limfunc{Gr}_{n,N})$ and for $\\beta =-1$\nwith $QK(\\limfunc{Gr}_{n,N})$.\n\\end{corollary}\n\n\\begin{proof}\nRecall that the rings $QH^{\\ast }(\\limfunc{Gr}_{n,N})$ and $QK(\\limfunc{Gr}%\n_{n,N})$ are multiplicatively generated from the Chern classes (see \\cite%\n{SiebertTian} for the case of quantum cohomology and \\cite[Cor 5.7]{BM} for\nquantum K-theory) which under the above maps are identified with the\ncoefficients $H_r$ and $E_r$ defined in (\\ref{Hexp0}) and (\\ref{Eexp0}),\nrespectively. Thus, it suffices to show that the respective rings feature\nthe same Pieri rule, i.e. that the respective expansions of the product of\nsuch a Chern class with a general class coincide.\n\nSetting $t_{j}=0$ in the functional relation (\\ref{ideal def}), (\\ref%\n{func_eqn}) the resulting ring is well-defined and it follows from our\nprevious results (\\ref{combproduct}), Cor \\ref{cor:fusion_matrices} and Thm %\n\\ref{thm:coord_ring} that $qh_{n}^{\\ast }\/\\langle t_{1},\\ldots ,t_{N}\\rangle \n$ is isomorphic to the ring with product $v_{\\lambda }\\circledast v_{\\mu }=%\n\\boldsymbol{G}_{\\lambda }v_{\\mu }$ with $\\boldsymbol{G}_{\\lambda }$ given by\n(\\ref{Schubert0}). Here we implicitly used the fact that the transfer matrices $E$%\n, $H$ stay well-defined when setting formally $t_{j}=0$, which in turn can be deduced from the explicit expressions for the $L$-operators (\\ref{L}), (\\ref{L'}). Furthermore, from the\ndefinition (\\ref{Schubert0}) it follows that $\\boldsymbol{G}_{1^{r}}=E_{r}$\nand, thus, the ring structure is fixed by the Pieri rule (\\ref{Er0}) which\nfor $\\beta =0$ coincides with the Pieri rule of $QH^{\\ast }(\\limfunc{Gr}%\n_{n,N})$ \\cite[p. 293]{Bertram} and for $\\beta =-1$ with the Pieri rule of $%\nQK(\\limfunc{Gr}_{n,N})$ \\cite[Thm 5.4]{BM}.\n\\end{proof}\n\nThe functional relation (\\ref{func_eqn1}) when setting $t_{j}=0$ becomes,%\n\\begin{equation*}\n(-1)^{n}(1+\\beta x)^{n}H(x)E(\\ominus x)=x^{N}(1+\\beta\nH_{1})+q(-1)^{n}(1+\\beta x)^{n}\\;.\n\\end{equation*}%\nUsing the expansions (\\ref{Hexp0}), (\\ref{Eexp0}) and comparing powers on\nboth sides of the functional relation one arrives at the following explicit\nrelations between the generators \n\\begin{equation}\n\\sum_{a+b=N-r}(-1)^{a}\\boldsymbol{e}_{a}(H_{b}+\\beta H_{b+1})=\\left\\{ \n\\begin{array}{cc}\n0, & r=1,\\ldots ,k-1 \\\\ \nq(-1)^{n}\\binom{n}{N-r}\\beta ^{N-r}, & r=k,\\ldots ,N%\n\\end{array}%\n\\right. \\;.  \\label{ideal0}\n\\end{equation}%\nThe expression (\\ref{bilinear_form_spin}) of the bilinear form and the\ndefinition of the dual basis (\\ref{dual_spin_basis}) simplify to%\n\\begin{equation}\n(v_{\\lambda },v_{\\mu })=\\sum_{\\alpha \\subset (k^{n})}\\frac{G_{\\lambda\n}(y_{\\alpha })G_{\\mu }(y_{\\alpha })}{e(y_{\\alpha },y_{\\alpha })}\n\\label{bilinear_form2}\n\\end{equation}%\nand \n\\begin{equation}\n(v_{\\lambda },(1+\\beta H_{1})v_{\\mu ^{\\vee }})=\\delta _{\\lambda \\mu },\n\\label{K_dual_basis_2}\n\\end{equation}%\nbecause factorial Grothendieck polynomials are replaced with ordinary ones.\nIn particular, the opposite spin basis (\\ref{op_spin_basis}) simply becomes $%\nv^{\\lambda }=v_{\\lambda ^{\\vee }}$ when $t_{j}=0$. Note that for $\\beta =-1$\nthe definition (\\ref{bilinear_form2}) and the relation (\\ref{K_dual_basis_2}%\n) are different from \\cite[Thm 5.14]{BM}. This is not a contradiction, as\nthe invariance of the bilinear form only fixes it up to a multiplicative\nfactor, which with respect to the form defined in \\emph{loc. cit.}, is $%\n(1-q) $.\n\n\\begin{remark}\nIn the homogeneous limit the analogue of the Littlewood-Richardson rule for\nstable Grothendieck polynomials is known \\cite[Thm 5.4 and Cor 5.5]%\n{BuchKtheory}. 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+{"text":"\n\\section*{Acknowledgements}\n\\label{sec:ack}\nWe thank the anonymous referees as well as the editor for insightful reviews that greatly improved this work.\nQUBIC is funded by the following agencies. France: ANR (Agence Nationale de la Recherche) 2012 and 2014, DIM-ACAV (Domaine d'Interet Majeur-Astronomie et Conditions d'Apparition de la Vie), Labex UnivEarthS (Universit\u00e9 de Paris), CNRS\/IN2P3 (Centre National de la Recherche Scientifique\/Institut National de Physique Nucl\u00e9aire et de Physique des Particules), CNRS\/INSU (Centre National de la Recherche Scientifique\/Institut National des Sciences de l'Univers). Italy: CNR\/PNRA (Consiglio Nazionale delle Ricerche\/ Programma Nazionale Ricerche in Antartide) until 2016, INFN (Istituto Nazionale di Fisica Nucleare) since 2017. Argentina: MINCyT (Ministerio de Ciencia, Tecnolog\u00eda e Innovaci\u00f3n), CNEA (Comisi\u00f3n Nacional de Energ\u00eda At\u00f3mica), CONICET (Consejo Nacional de Investigaciones Cient\u00edficas y T\u00e9cnicas).\n\\section{Impact of atmospheric fluctuations on scales smaller than the synthesized beam cutoff angles.}\n\\label{app_impact_small_scale_atm}\n\n    In section~\\ref{sec_map_making_noise_structure} we have seen that the synthesized beam naturally filters-out large-scale fluctuations from atmospheric gradients. We have also explained that to understand the effect of atmosphere fluctuations on scales smaller than the synthesized beam cutoff angles (8.8\\,degrees at 150\\,GHz and 6\\,degrees at 220\\,GHz) it is necessary to build simulations based on atmosphere measurements taken at the site. Here we want to show that the available data taken at sites similar to that of QUBIC allows us to say that the instrument will be sensitive to fluctuations in a defined range of turbulence scales.\n\n    Fluctuations in the atmospheric radiation load are determined mainly by turbulent cells in the water vapor column that are distributed over scales within an interval $\\left[L_0^\\mathrm{min}, L_0^\\mathrm{max}\\right]$. The values of the minimum and maximum cutoff coherence scales, $L_0^\\mathrm{min}$ and $L_0^\\mathrm{max}$, depend on the physical properties of the atmosphere and are typical of each observations site.\n    \n    In a study conducted in the framework of the POLARBEAR experiment Errard et al.~\\cite{errard2015} studied the distribution of various atmosphere parameters at Atacama and showed that turbulence scales are distributed in the interval $L_0 \\in \\left[200\\,\\mathrm{m}, 800\\,\\mathrm{m}\\right]$ with a peak around 300\\,m (see the left panel of figure~11 in Errard et al.~\\cite{errard2015}). Assuming that the atmosphere at Alto Chorrillo is not very different from that at Atacama, we can use this distribution to assess quickly the impact of atmosphere fluctuations on scales smaller than the synthetic beam cutoff angles (8.8~degrees at 150\\,GHz and 6~degrees at 220\\,GHz).\n    \n    Let us consider a fluctuations correlation length, $L_0$. This length corresponds to angular scales $\\theta^\\mathrm{atm}_\\mathrm{corr}(z)\\sim L_0\/z$, where $z$ is the height above the telescope line-of-sight.\n\n    \\begin{figure}[!t]\n        \\begin{center}\n            \\includegraphics[width=14cm]{Figures\/bubble_size_dist}\n            \\caption{\\label{fig_small_scale_atmo}Relationship between the relative water vapor concentration and atmospheric turbulence correlations angular scales. Black lines are a family of curves $\\theta^\\mathrm{atm}_\\mathrm{corr}(z)$ corresponding to values of $L_0$ in the interval $\\left[200\\,\\mathrm{m}, 800\\,\\mathrm{m}\\right]$. The curves are marked with a transparency level that reflects the distribution in figure~11 of Errard et al.~\\cite{errard2015}. The yellow curve marks the case with $L_0^\\mathrm{max}=300\\,\\mathrm{m}$, corresponding to the most likely correlation scale. The red and blue horizontal lines mark the synthetic beam cutoff angles at 150 and 220\\,GHz. The green curve (right axis) reports the relative water vapor concentration versus height, assuming a height scale of 2000\\,m. The dashed red and blue lines are guides that show the water vapor concentration corresponding to the synthesized beam cutoff scales and the case $L_0^\\mathrm{max}$. See text for a more detailed description.}\n        \\end{center}\n    \\end{figure}\n    \n    In figure~\\ref{fig_small_scale_atmo} we show, in black, a family of curves $\\theta^\\mathrm{atm}_\\mathrm{corr}(z)$ corresponding to values of $L_0$ in the interval $\\left[200\\,\\mathrm{m}, 800\\,\\mathrm{m}\\right]$. The curves are marked with a transparency level that reflects the distribution in figure~11 of Errard et al.~\\cite{errard2015}. The yellow curve marks the case with $L_0^\\mathrm{max}=300\\,\\mathrm{m}$, corresponding to the most likely correlation scale. The red and blue horizontal lines mark the synthetic beam cutoff angles at 150 and 220\\,GHz, so that for small scale fluctuations we are interested in the regions below these lines. The green curve (right axis) reports the relative water vapor concentration versus height, assuming a height scale of 2000\\,m\\footnote{The water vapor scale height is the distance over which a the water vapor content decreases by a factor of $e$}, which can be considered representative for Alto Chorrillo (see, for example, the left panel of figure 1 in Mararieva et al.~\\cite{makarieva2013}). Finally, the dashed red and blue lines are guides that show the water vapor concentration corresponding to the synthesized beam cutoff scales and the case $L_0^\\mathrm{max}$.\n    \n    Let us take, for example, the most likely correlation scale, $L_0^\\mathrm{max}$. The largest angular scales connected to $L_0^\\mathrm{max}$ are the synthetic beam cutoff scales and, with these values, we see that we will be affected by fluctuations involving $\\sim$40\\% of the water vapor column at 150\\,GHz and slightly more than 20\\% at 220\\,GHz. It is worth noting that the bottom-left and top-right white areas of the plot correspond to angular scales where we do not expect significant fluctuations. For example, for $z = 2000\\,\\mathrm{m}$ we do not expect fluctuations on angular scales up to 6\\,degrees.\n    \n    From this simple analysis we can conclude that the effect from turbulent cells larger than 300\\,m will be increasingly negligible, while we may expect some impact from cells in the range $\\left[200\\,\\mathrm{m}, 300\\,\\mathrm{m}\\right]$. Of course this analysis does not answer yet the question of what effect we can expect on the final science from atmosphere fluctuations on these scales. A complete assessment will be performed on the basis of dedicated atmospheric measurements taken at the QUBIC site.\n\n\n\\section{Bolometric interferometry and QUBIC}\n\\label{sec:boloint}\n\\subsection{Bolometric interferometry}\n\\label{subsec:boloint}\n\nThe idea of bolometric interferometry dates back to the 19$^\\mathrm{th}$ century when Fizeau proposed an adding interferometer to measure the expected difference in the velocity of light traveling in moving media \\citep{1851fizeau}. The setup split the signal from the Sun into two pipes, each filled with water flowing in  opposite directions. The beams were then recombined and the interference fringes measured to determine the phase difference between them. In QUBIC, the pipes are replaced by an array of back-to-back feedhorns, and the beams are recombined by an arrangement of two mirrors \\citep[see O'Sullivan et al.][for details]{2020.QUBIC.PAPER8}.\n\nAn important advantage of the Fizeau arrangement is its inherent wide band performance. There is no wavelength selectivity which results in an ideal performance at any particular wavelength. This can be compared to the case for example of the Martin-Puplett interferometer \\citep{1970InfPh..10..105M} which introduces a quarter wavelength optical path difference by moving one of the corner mirrors, making it tuned to a particular wavelength.\n\nImaging radio interferometers generally work with electronic signals where multiplicative correlation is a natural operation with electronic components, both analog and digital.  The correlations between signals from different antennas build up a sampling of the ``\\uvplane''. The Fourier transform of the \\uvplane\\ is an image of the sky, usually called the ``dirty image'' because no compensation has been done for the under-sampling of the \\uvplane, and no calibration has been done \\citep[see for example Thompson, Moran and Swenson][]{2017isra.book.....T}. In a bolometric interferometer, this ``dirty image'' is formed directly by the optical combiner, and is recorded by the detector array.  QUBIC treats the entire band ``in one go'', whereas a traditional radio interferometer using narrow band electronic processors must split the band into many sub-bands, making large bandwidths expensive to realize in terms of processing and power requirements.\n\nThis method is equivalent to imaging the sky with an imager whose beam is the synthesized beam of the bolometric interferometer (the ``dirty beam'' of the interferometer), formed by the combination of all interference patterns of all possible pairs of horns in the aperture array~\\cite{2011APh....34..705Q}. In the case of the QUBIC Full Instrument, 400~horns form the synthesized beam shown in figure~\\ref{Fig:sbqubic} for one specific detector. This is the beam the detector located in this position in the focal plane scans the sky with. Note that the synthesized beam differs for different detectors in the focal plane (see~Mousset et al.~\\cite{2020.QUBIC.PAPER2} and O'Sullivan et al.~\\cite{ 2020.QUBIC.PAPER8} from this series of articles for details). The synthesized beam is very different from a classical imager beam as it exhibits multiple peaks (because the horn array has a finite extension), the angular distance between two peaks is driven by the ratio between the wavelength and the distance between two nearby horns while the angular resolution of each peak is driven by the ratio between the wavelength and the maximal distance between horns. This is detailed in O'Sullivan~et.~al~\\cite{2020.QUBIC.PAPER8} from this series of articles. In the case of QUBIC, with 400~horns, the angular resolution achieved is 23.5~arcminutes at 150~GHz.\nBeing the result of the summation of interference fringes, this synthesized beam significantly evolves with electromagnetic frequency, which is the basis of the spectral imaging capabilities of QUBIC (described briefly in section~\\ref{sec:specimg} and in detail in~Mousset et al.~\\cite{2020.QUBIC.PAPER2} in this series of articles).\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[ width = 1 \\hsize ]{Figures\/SB_QUBIC_150GHz.pdf}\n\\caption{QUBIC synthesized beam on the sky (left: laboratory measurement with the TD, right: simulations without optical aberrations) at 150~GHz. Note that the color scales are arbitrary units. Details on this measurement can be found in Torchinsky et al.~\\cite{2020.QUBIC.PAPER3} from this series of articles. The synthesized beam shrinks with increasing frequency as can be seen with the animated version of this image that can be found online at \\href{https:\/\/box.in2p3.fr\/index.php\/s\/bzPYfmtjQW4wCGj}{https:\/\/box.in2p3.fr\/index.php\/s\/bzPYfmtjQW4wCGj}.}\n\\label{Fig:sbqubic}\n\\end{figure}\n\n\n\n\n\\subsection{Self-Calibration}\n\\label{sec:selfcal}\n\nSelf calibration in aperture synthesis evolved from the idea of ``phase-closure''~\\cite{1974A&AS...15..417H,1976A&A....50...19F,1978ApJ...223...25R} in a phased-array radiotelescope. Signals from the individual antenna elements of a phased-array are combined together and phase differences between the signals are corrected such that the combination of all the signals is equivalent to the signal captured by a large single-dish antenna pointing to the desired direction. With the advent of digital correlators, the correlation coefficients could be saved and a more sophisticated processing resulted in improved calibration. This came to be known as ``self-calibration''~\\cite{1981MNRAS.196.1067C,1984ARA&A..22...97P}.\n\nBolometric interferometry can also take advantage of the self-calibration used in traditional aperture synthesis in radio astronomy. \nHowever, as a bolometric interferometer directly observes the dirty image, one needs to `go backwards'' from the dirty image and reconstruct the \\uvplane.\nThe detailed procedure and analysis technique are described in Bigot-Sazy~et~al.~\\cite{2013A&A...550A..59B}.\nThe general idea is to observe a polarized  artificial point source with the bolometric interferometer with only one pair of horns open at a time while the others are closed\\footnote{In practice, in order to keep a roughly constant loading on the detector array, we achieve this observation through measuring successively the point source with one horn closed, all the others being open, then only the second horn closed, then both closed and finally all horns open. We eventually combine these observations to achieve the same measurement as with a single pair of horns open.}, which can be achieved using the mechanical shutters (also called RF switches) developed for QUBIC that can open or close each horn (see Cavaliere~et~al.~\\cite{2020.QUBIC.PAPER7} in this series of articles for details on the QUBIC horns and RF switches). \nIn the absence of instrumental systematic effects, two ``redundant'' pairs of horns (same distance and orientation) would correspond to the exact same visibility (Fourier mode on the sky) and the bolometric interferometer would therefore measure the same quantity. A difference between the two can only come from instrumental effects\\footnote{Small-scale atmospheric fluctuations could also be responsible for a difference between measurements performed with ``redundant'' pairs of horns. However, we anticipate this to be a marginal effect because i) the artificial source has a strong emission (a fraction of Watt) and ii) self-calibration will be  performed scanning the source over long periods, averaging-out the atmospheric fluctuations.}.\nBy measuring in this manner all possible visibilities, one can fit parameters of a very general instrument model comprising hundreds of parameters (a few for each horn, bolometer) as well as for different electromagnetic frequencies if the source can be tuned in frequency. The number of constraints scales as the number of horns squared while the number of unknowns is proportional to the number of horns. As a result the problem is heavily over constrained for an instrument like QUBIC. The following instrumental systematics, expected to be the major ones for a bolometric interferometer, have been considered~\\cite{2013A&A...550A..59B}:\n\\begin{itemize}\n    \\item uncertainty on the exact location of the horns which would affect the shape of the synthesized beam;\n    \\item uncertainty on the exact location of the bolometers in the focal plane. Similarly, it would affect the shape of the synthesized beam because it gradually changes across the focal plane (see O'Sullivan et al.~\\cite{2020.QUBIC.PAPER8});\n    \\item relative gains of the bolometers;\n    \\item uncertainties on the primary beam (towards the sky) shape;\n    \\item uncertainties on the secondary beam (towards the detectors) shape;\n    \\item imperfections of the successive optical elements, each modeled by a Jones matrix containing differential gains for each polarization orientation as well as cross-polarization.\n\\end{itemize}\nWe have shown in Bigot-Sazy et al.~\\cite{2013A&A...550A..59B} that spending 1 second on self-calibration for each baseline would reduce the $E$ to $B$ power spectrum leakage by an order of magnitude for typical values for the above systematic effects. The leakage drops to two orders of magnitude with 100 seconds per baseline.\n\nSelf-calibration is a special feature of bolometric interferometry for measuring instrumental systematic effects and subsequently accounting for them in the data analysis process through a more accurate modeling of the synthesized beam than the purely theoretical one, hence improving the performance of the map-making (see section~\\ref{sims}).\n\n    \n\\subsection{Spectral Imaging}\n\\label{sec:specimg}\nSpectral imaging is described in detail in Mousset~et~al.~\\cite{2020.QUBIC.PAPER2} from this series of articles, but we provide here a short explanation of this specific feature offered by bolometric interferometry.\nAs mentioned in section~\\ref{subsec:boloint} and shown in figure~\\ref{Fig:sbqubic}, the multiply peaked shape of the synthesized beams evolves with frequency. A cut of the QUBIC synthesized beam is shown in figure~\\ref{Fig:sb_freq} (left) for two different frequencies. \n\\begin{figure}[t]\n\\centering\n\\includegraphics[ width =\\hsize ]{Figures\/sb_and_src.png}\n\\caption{{\\bf (left)} A cut of the QUBIC theoretical synthesized and primary beams at two different frequencies for a bolometer located at the center of the focal plane. The figure shows how the location of the multiple peaks significantly changes for a small frequency change. This behaviour is at the basis of the spectral imaging capabilities offered by bolometric interferometry. {\\bf (left)} signal detected by the same bolometer when scanning across an imaginary point source emitting 1 and 1.5 at 140 and 160 GHz respectively: we have both spatial and frequency information.}\n\\label{Fig:sb_freq}\n\\end{figure}\nAs the respective distance between peaks changes with frequency, the signal detected with a given bolometer will combine (through convolution with the synthesized beam) different directions on the sky for different incoming frequencies within the physical wide band. Let's imagine, as a simple toy model, that the sky consists in a single point source emitting only at two distinct frequencies 140 and 160 GHz with relative brightness 1 and 1.5. As the instrument scans over the point source, the response of the Time-Ordered-Data (TOD) will be the sum of the synthesized beam at 140 and 160 GHz scaled by 1 and 1.5 respectively. This is shown in figure~\\ref{Fig:sb_freq} (right) as a function of the angle from the point source. We can easily reconstruct the location of the source from the position of the central peak, as well as the spectrum of the source from the amplitudes of each of the peaks around the central one.\nThese signals from different frequencies will only be significantly different from each other if the corresponding side-peaks are more separated than their intrinsic width. \n\nThis shows how one can simultaneously recover spatial and frequency information within a wide physical band for a bolometric interferometer. Of course, the above toy-model corresponds to a source that is not extended in spatial nor frequency domain, but we have shown in Mousset et al.~\\cite{2020.QUBIC.PAPER2} that the exact same reasoning can be applied to diffuse signals with continuous Spectral Energy Distribution (SED).\nThis spectral imaging reconstruction is done at the map-making stage in the data-analysis pipeline with data collected within a wide bandwidth. It can be done by reconstructing maps in as many different sub-frequencies as allowed by the ratio between frequency shift and intrinsic peak width in the synthesized beam. We have shown in Mousset et al.~\\cite{2020.QUBIC.PAPER2} that the corresponding relative frequency resolution FWHM is $\\Delta_\\nu\/\\nu \\sim 1\/P$ where $P$ is the maximum number of apertures along an axis of the bolometric interferometer horn array. For a 20x20 horn-array, QUBIC therefore achieves 5 sub-bands with a frequency FWHM resolution of $\\Delta_\\nu\/\\nu\\sim 0.05$ within each of the 25\\% bandwidth frequency bands.\n\n\nIn the current context of search for primordial B-modes in the CMB originating from tensor perturbations from inflation, spectral imaging presents an opportunity to constrain foregrounds alternative to other current approaches. Previous analyses have indeed shown that B-modes measurements are largely dominated by foregrounds~\\cite{2014PhRvL.112x1101B,2015A&A...576A.104P,2015PhRvL.114j1301B,2018PhRvL.121v1301B} which can only be removed through their frequency behaviour which  is distinct from that of the CMB. Classical imagers usually approach this issue by multiplying the number of frequencies at which they observe the CMB. Galactic dust is currently the most worrying foreground at frequencies above 100\\,GHz. It can be constrained from the ground through wide atmospheric windows around 150 and 220~GHz. The noise severely increases at higher frequencies because of atmospheric emissivity. As a result, constraints on foregrounds can only be achieved through comparisons between these few, largely separated frequency bands. While their large separation in frequency may appear as an advantage as it increases lever arm, it is also a limitation as it prevents data analyses to consider realistic electromagnetic spectra for dust emission that could exhibit changes of slope or dust decorrelation between frequencies~\\cite{2017A&A...599A..51P, 2018PhRvL.121v1301B}. With spectral imaging, one can measure the spectrum of the foreground components {\\bf locally} (i.e., within the bandwidth) and avoid large extrapolations between distant frequencies. This local approach to foreground mitigation is complementary to the usual approach with largely separated frequency bands. This is studied in section~\\ref{sect:foreground_challenge} of this article. \n\n\n\n\\subsection{The QUBIC Instrument}\nIn order to achieve bolometric interferometry, QUBIC relies on an optical system consisting of back-to-back horns that select the relevant baselines and an optical combiner focusing on a bolometric focal plane. The optical combiner forms interference fringes while the bolometers average their powers over timescales much larger than the period of the electromagnetic light. This is therefore the optical equivalent of a (wide-band) correlator in classical interferometry. Being a bolometric device, the whole instrument operates at cryogenic temperatures thanks to a large cryostat described in Masi et al.~\\cite{2020.QUBIC.PAPER5} from this series of articles.\n\n\\begin{figure}[p!]\n\\centering\n$\\vcenter{\\hbox{\\includegraphics[ width = 0.9 \\hsize ]{Figures\/qubic_schematic_flipped.pdf}}}$\n$\\vcenter{\\hbox{\\includegraphics[ width = 0.7 \\hsize ]{Figures\/cryostat-qubic.pdf}}}$\n\\caption{Schematic of the QUBIC instrument \\textbf{(top)} and sectional cut of the cryostat \\textbf{(bottom)} showing the same sub-systems in their real configuration.}\n\\label{Fig:qubic_scheme}\n\\end{figure}\n\n\\begin{table}[t]\n    \\renewcommand{\\arraystretch}{1.}\n    \\begin{center}\n        \\caption{\\label{tab_qubic_params}QUBIC main parameters}\n        \\begin{tabular}{p{6.4cm} m{2.5cm} m{5.5cm}}\n            \\hline\n            Parameter& TD & FI \\\\\n            \\hline\n            \\hline\n            {\\bf Instrument}\\\\\n            \\hline\n            Frequency channels \\dotfill &150 GHz & 150 GHz \\& 220 GHz\\\\\n            Frequency range 150 GHz \\dotfill &[131-169] GHz &[131-169] GHz\\\\\n            Frequency range 220 GHz \\dotfill &- &[192.5-247.5] GHz\\\\\n            Window Aperture [m]\\dotfill & 0.56 & 0.56 \\\\\n            Number of horns\\dotfill &64 &400\\\\\n            Number of detectors\\dotfill &248 &992$\\times$2\\\\\n            Detector noise [$\\mathrm{W\/\\sqrt{Hz}}$]\\dotfill & 2.05$\\times 10^{-16}$ & 4.7$\\times 10^{-17}$ \\\\\n            Focal plane temp. [mK]\\dotfill &300 &300\\\\\n            Synthesized beam FWHM [deg]\\dotfill &0.68~\\cite{2020.QUBIC.PAPER3} &0.39 (150 GHz), 0.27 (220 GHz)\\\\\n            \\hline\n            \\hline\n            {\\bf Scanning Strategy (see sect.~\\ref{sims})}\\\\\n            \\hline\n            Elevation range [deg]\\dotfill &[30-70] &[30-70]\\\\\n            Azimuth scan width [deg]\\dotfill &$\\pm 17.5$ &$\\pm 17.5$\\\\\n            Azimuth scan speed [deg\/s]\\dotfill &0.4 &0.4\\\\\n            Sky Coverage\\dotfill &1.5\\% &1.5\\%\n        \\end{tabular}\n    \\end{center}\n\\end{table}\n\n\nA schematic of the design of QUBIC is shown in figure~\\ref{Fig:qubic_scheme} and the main instrument parameters are listed in table~\\ref{tab_qubic_params}. The sky signal first goes through a 56~cm diameter window made of Ultra-High-Molecular-Weight Polyethylene followed by a series of filters cutting off frequencies higher than the desired ones. The next optical component is a stepped rotating Half-Wave-Plate which modulates incoming polarization. This sub-system is described in D'Alessandro~et~al.~\\cite{2020.QUBIC.PAPER6} from this series of articles. A single polarization is then selected thanks to a polarizing grid. Although reflecting half of the incoming photons may appear as a regrettable loss, it is in fact one of the key features of QUBIC for handling instrumental systematics, especially polarization-related ones: a single polarization is selected just after polarization modulation by a wire-grid, while our bolometers are not sensitive to polarization due to the XY symmetry of the absorbing grid (see Piat et al~\\cite{2020.QUBIC.PAPER4}). As a result, any cross-polarization occurring after the polarizing grid (horns, uncontrolled reflections inside the optical combiner) are negligible. This has been confirmed during calibration by measuring the cross-polarization of our TES (see figure~\\ref{Fig:fringe}-right where we show a cross-polarization measurement below 0.4\\% at 95\\% C.L. for a TES). The next optical device is an array of 400 back-to-back corrugated horns made of an assembly of two 400-horns arrays, composed of 175 aluminium platelets (0.3 mm thick) chemically etched to reproduce the corrugations required for the horns to achieve the required performance. An array of mechanical shutters (RF switches) separates the two back-to-back horn arrays in order to be able to close or open horns for self-calibration (see section~\\ref{sec:selfcal}). Both front and back horns are identical, each with a field of view of 13~degrees FWHM with secondary lobes below $-$25~dB. The horns and switches are described in detail in Cavaliere~et. al~\\cite{2020.QUBIC.PAPER7} from this series of articles. The back-horns directly illuminate the two-mirrors off-axis Gregorian optical combiner (described in detail in O'Sullivan et al.~\\cite{2020.QUBIC.PAPER8}) that focuses the signal onto the two perpendicular focal planes, separated by a dichroic filter that splits the incoming waves into two wide bands centered at 150~GHz for the on-axis focal plane and 220~GHz for the  off-axis one. The focal planes are each equipped with 992~NbSi Transition-Edge-Sensors (the detection chain is described in detail in Piat et al.~\\cite{2020.QUBIC.PAPER4} from this series of articles) cooled down to 300~mK using a sorption fridge. A realistic view of the cryostat can be seen in the right panel of figure~\\ref{Fig:qubic_scheme}. The cryostat weights roughly 800~kg and is around 1.6~meter high for a 1.4~meter diameter.\n\nThe anticipated scanning strategy for QUBIC is back and forth azimuth scans at constant elevation, following the azimuth of the center of the field as a function of sidereal time and updating elevation regularly in order to have scans with as many angles as possible in sky coordinates. The latitude of the QUBIC site allows us to cover a wide range of angles. The detailed scanning strategy will be determined when observing the real sky based on optimal mitigation of atmospheric fluctuations. We show in figure~\\ref{Fig:coverage} a typical sky coverage obtained with the baseline scanning parameters given in table~\\ref{tab_qubic_params}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[ width = 0.64 \\hsize ]{Figures\/coverage_moll.png}\\includegraphics[ width = 0.36 \\hsize ]{Figures\/coverage.png}\n\\caption{Baseline sky coverage in Galactic coordinates (see table~\\ref{tab_qubic_params} for corresponding scanning strategy parameters). The coverage is cut at 10\\% from the maximum and achieves a sky fraction of 1.5\\%. The grid spacing in latitude and longitude are 10 and 20 degrees respectively.}\n\\label{Fig:coverage}\n\\end{figure}\n\n\n\\subsubsection{The QUBIC Technological Demonstrator}\n\\label{sec_qubic_td}\nThe QUBIC Technological Demonstrator (hereafter QUBIC TD) uses the same cryostat, cooling system, filters and general sub-system architecture as described above but with only 64~back-to-back horns and mirrors reduced according to the illumination of the 64~horns. It also uses a single 248~TES bolometer array operating at 150~GHz. The QUBIC TD has been used as an intermediate step before the Full Instrument (FI) in order to characterize and demonstrate bolometric interferometry in the laboratory. We have confirmed the expected behaviour of the instrument and all the anticipated specific features of bolometric interferometry during this calibration campaign which is detailed in Torchinsky~et~al.~\\cite{2020.QUBIC.PAPER3}. A selection of the most relevant results from the calibration includes:\n\\begin{itemize}\n    \\item the measurement of the bolometric interferometer synthesized beams multiple peaked shape in overall agreement with the theoretical prediction we made in Battistelli et al.~\\cite{2011APh....34..705Q}, shown in figure~\\ref{Fig:sb_freq}. The difference between theory and measurement in the locations and amplitudes of the peaks is expected from optical aberrations (see~\\cite{2020.QUBIC.PAPER8}) and can be calibrated from the data for the map reconstruction;\n    \\item the evolution as a function of frequency of the inter-peak separation in the synthesized beam (available online at \\href{https:\/\/box.in2p3.fr\/index.php\/s\/bzPYfmtjQW4wCGj}{https:\/\/box.in2p3.fr\/index.php\/s\/bzPYfmtjQW4wCGj}) which is at the basis of Spectral Imaging described briefly in section~\\ref{sec:specimg} and in detail in Mousset et al.~\\cite{2020.QUBIC.PAPER2}. This possibility is studied in detail in section~\\ref{sect:foreground_challenge} of this article;\n    \\item the measurement of individual fringe patterns using the mechanical shutters~\\cite{2020.QUBIC.PAPER7} shown in the left panel of figure~\\ref{Fig:fringe} in this article and a more detailed description in figure~13 of Torchinsky et al.~\\cite{2020.QUBIC.PAPER3}. As discussed in section~\\ref{sec:selfcal}, this measurement is at the basis of the self-calibration allowing QUBIC to achieve a novel control of instrumental systematic effects, as shown in Bigot-Sazy et al.~\\cite{2013A&A...550A..59B};\n    \\item Figure~\\ref{Fig:fringe} shows a measurement of the TD cross-polarization with an upper-limit 0.4\\% at 95\\% C.L. for a detector close to the center of the focal plane. As studied in details in D'Alessandro et al.~\\cite{2020.QUBIC.PAPER6} from this series of articles, we have found a median measured cross-polarization of 0.12\\% among our detectors and a 0.61\\% median 95\\% upper-limit.\n    77\\% of our detectors have a measured cross-polarization compatible with zero at the one-sigma level. Such low cross-polarization is an important feature for detecting a signal as small as the primordial B-modes. This is achieved in QUBIC thanks to its specific polarization design with a single polarization selected before the interferometer apertures and full-power detectors as discussed above in this section.\n    \\item The intrinsic detector noise has been measured to be 4.7$\\times 10^{-17}~\\mathrm{W\/\\sqrt{Hz}}$ but due to noise aliasing with the TD readout chain, the effective detector+readout noise is 2.05$\\times 10^{-16}~\\mathrm{W\/\\sqrt{Hz}}$. This noise aliasing will be resolved for the FI using Nyquist inductance in order to reduce the noise bandwidth of the TES (see Piat et al.~\\cite{2020.QUBIC.PAPER4}).\n\\end{itemize}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[ width = 0.36 \\hsize ]{Figures\/fringes.png}\\includegraphics[ width = 0.36 \\hsize ]{Figures\/fringes_simulation.png}\\includegraphics[ width = 0.28\\hsize ]{Figures\/hwp_plot_tes95.pdf}\n\\caption{{\\bf (left and center)} Fringe pattern measured (left) and simulated (center) with the QUBIC TD  (coordinates are in the on-axis focal plane and units arbitrary) using the mechanical shutters~\\cite{2020.QUBIC.PAPER7} in order to achieve a measurement equivalent to that of a single baseline (two horns open, all others closed). Units are arbitrary. {\\bf (right)} Cross-Polarization measured by rotating the Half-Wave-Plate and observing our polarized calibration source. This result is obtained with a TES bolometer close to the focal plane center and showing a high signal-to-noise ratio.}\n\\label{Fig:fringe}\n\\end{figure}\n\nThe QUBIC TD will be shipped to Argentina in mid-2021 and subsequently deployed in its observing site (see section~\\ref{Sec:site} for a description). The objective is to conduct a full characterization of bolometric interferometry on the sky throughout 2021. The performance expected from observation with the QUBIC TD is discussed in section~\\ref{Sec:TD_perf}.\n\n\\subsubsection{The QUBIC Full Instrument\\label{FI}}\nThe upgrade to the FI will happen after one year of operation of the TD and is therefore scheduled for early 2023. It will consist in:\n\\begin{itemize}\n    \\item replacing the 64~back-to-back horn array by one with 400 horns which is already manufactured and being characterized at the sub-system level (see Cavaliere~et~al.~\\cite{2020.QUBIC.PAPER7} from this series of articles);\n    \\item replacing the current mirrors by larger mirrors that are already manufactured and characterized (see O'Sullivan et al.~\\cite{2020.QUBIC.PAPER8});\n    \\item upgrading the single 248~TES array by eight of these (four at 150~GHz and four at 220~GHz) achieving two focal planes of 992~TES each. Note that detailed quasi-optical simulations have shown that the 150~GHz optimization of the back-short of the TES is also nearly optimal at 220~GHz allowing us to use identical detector designs for both frequencies~\\cite{2020JLTP..200..363P,PerbostPhD}. A new readout electronics will avoid the TD noise aliasing through the addition of Nyquist inductors~\\cite{2020.QUBIC.PAPER4}.\n    The additional focal plane also requires the installation of a dichroic filter at 45 degrees in between the two focal planes (light-blue in figure~\\ref{Fig:qubic_scheme}-left, and in red  in figure~\\ref{Fig:qubic_scheme}-right).\n\\end{itemize} \nThe back-to-back horns are common to both frequencies in the QUBIC design. They have been optimized to be single-moded at 150~GHz but are multi-moded in the 220~GHz band (see O'Sullivan et al.~\\cite{2020.QUBIC.PAPER8} section~2.3) resulting in a significantly higher throughput at 220~GHz. \n\nDespite stronger emission from the atmosphere, as can be found in Table~\\ref{tab_qubic_td_simulation_parameters}, simulations have shown (see figure~\\ref{Fig:bmodes} in  section~\\ref{sect_Bmodes}) that this will result in a similar sensitivity to B-modes at 220~GHz and at 150~GHz. \nSeveral effects contribute to this result. First, considering the intrinsic detector noise (see Piat et al.~\\cite{2020.QUBIC.PAPER4}), our noise budget is dominated by the detectors at 150~GHz and has almost equal contributions from photon-noise and detectors at 220~GHz (see table~\\ref{tab_qubic_td_simulation_parameters}). Because of this, the net sensitivity loss at 220~GHz due to higher emissivity from the atmosphere is not as large as expected from pure photon-noise. \nWe also collect more CMB photons at 220~GHz due to the horns higher throughput. \n\nAs a result, the signal to noise ratio in the TOD at 220~GHz is higher than for a single-moded channel. \nFinally, multimoded horns at 220~GHz also result in a flatter primary beam (see O'Sullivan et al.~\\cite{2020.QUBIC.PAPER8} figure~5) than at 150~GHz where the primary beam is Gaussian. This strongly impacts the spatial noise correlation in our maps (see section~\\ref{sims}) resulting in a reduction of our power spectra error-bars at the lowest multipoles. This effect occurs for both bands but is stronger at 220~GHz due to the higher amplitude of the synthesized beam secondary peaks.\n\n\\subsubsection{The QUBIC site}\n\\label{Sec:site}\nQUBIC will be installed in its final observing site in Argentina at the end of 2021\\footnote{It was anticipated to ship the instrument mid-2020 but the global shutdown caused by the COVID19 pandemic induced uncontrolled delays. As a consequence, the date mentioned here is subject to changes depending on the resolution of the COVID19 pandemic crisis.}. The site is located at the Alto Chorillos ($24\n^\\circ 11'11.7''$ S;\n$66^\\circ 28'40.8''$ W, altitude of 4869 m a.s.l.) about 45 minutes drive from the city of San Antonio de los Cobres in the Salta Province~\\cite{2018BAAA...60..107B}. This site has been studied for mm-wave astronomy for many years as it will also host the LLAMA 12~m antenna (\\href{https:\/\/www.llamaobservatory.org\/en\/}{https:\/\/www.llamaobservatory.org\/en\/}), 800~m away from QUBIC. The synergy between QUBIC and LLAMA at the site simplifies  the site preparatory works, logistics and deployment operations.\n\nThis site exhibits excellent quality sky for CMB studies: zenith optical depth at 210~GHz $\\tau_{210} <0.1$ for 50\\% of the time and $<0.2$ for 85\\% of the time as well as relatively quiet atmosphere (winds $<6$~ m\/s for 50\\% of the time). From the LLAMA site-testing data, we have determined an average atmospheric temperature of 270~K with an average emissivity 0.081 and 0.138 at 150 and 220~GHz respectively taken at our average elevation of 50 degrees. These values are assumed for the atmospheric background in the simulations presented in this article\\label{atm}. \n\n The whole instrument is oriented to any sky direction with elevation between 30 and 70~degrees (limitations due to Pulse-Tubes-Coolers) and any azimuth by an alt-azimuthal mount on the top of a well-adapted container. A fore-baffle will be placed at the window entrance with an absorptive inner surface in order to increase side lobes rejection for angles larger than 20~degrees from bore-sight direction. Finally, the instrument will be surrounded by a ground-shield in order to minimize brightness contrast between the sky and the ground. Figure~\\ref{Fig:dome} shows an artist view of the mounted instrument. The calibration source used for self-calibration (see Torchinsky et al.~\\cite{2020.QUBIC.PAPER3} for details) will be installed on a 50~m-high tower (telecommunication-like) placed 50~m North from the instrument. It will directly be seen at normal elevations (without the use of a mirror) in the far field of the instrument and in the direction opposite to the main CMB observations.  \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[ width = 0.9 \\hsize ]{Figures\/dome_GS_FB.JPG}\n\\caption{Sketch of the instrument, the cryostat on the alt-azimuthal mount as will be installed on the observing site.}\n\\label{Fig:dome}\n\\end{figure}\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nIn this article, first of a series of eight, we have given an overview of the QUBIC instrument. QUBIC is the first CMB polarimeter using a new technology called ``Bolometric Interferometry'' that combines the background limited sensitivity of bolometric detectors (transition edge sensors) with the clean measurement of interference fringes. QUBIC has been designed to observe the large-scale B-modes of the CMB to detect the elusive tensor perturbations (primordial gravitational waves) created during inflation. A first version of the instrument, with reduced number of horns (64) and detectors (248) operating at 150~GHz, the QUBIC Technological Demonstrator (TD), will be deployed in 2021 and is intended to demonstrate on-sky the capabilities of Bolometric Interferometry. The TD will subsequently be upgraded to the Full Instrument (FI) with the nominal 400~horn array and 992~detectors in each of the two focal planes operating at 150 and 220~GHz.\n\nWe have described the general design of QUBIC and have shown that it can be considered as a classical imager that would scan the sky with a beam composed of multiple peaks. These are separated by an angular distance given by the distance between two apertures in the interferometer horn array (in wavelength units), while the resolution of the peaks is given by the maximum size of the interferometer horn array (in wavelength units). We have emphasized two main features of QUBIC. First, the possibility of performing self-calibration~\\cite{2013A&A...550A..59B}, similarly as in a classical interferometer, allows us to control instrumental systematic effects. Second,  a bolometric interferometer has simultaneous spatial and spectral sensitivity, thanks to the spectral dependence of the synthesized beam within the physical band of the instrument. This is what allows us to carry out spectral imaging~\\cite{2020.QUBIC.PAPER2}.\n\nWe have used an extensive set of simulations to make forecasts of the QUBIC performance for both the primordial B-mode search and for testing foregrounds using spectral imaging. These preliminary forecasts assume the detector noise measured during the QUBIC calibration campaign~\\cite{2020.QUBIC.PAPER3} and a stable atmosphere corresponding to the QUBIC site in Argentina at 5000\\,m a.s.l. In this work we did not consider neither component separation nor the effect of fluctuations in the atmospheric load, both of which require a significant step forward in our pipeline that we are currently carrying out. However, we showed that regardless of component separation spectral imaging allows us to reconstruct the sky emission SED pixel-by-pixel and to recognize the presence of foregrounds residuals in the derived tensor-to-scalar ratio.\n\nThese simulations show that the QUBIC TD will be able to demonstrate spectral imaging with one year of sky integration on a field centered on the Galactic center. The QUBIC TD will measure the SED of bright regions in both intensity and polarization. After the upgrade to the FI, QUBIC will reach its nominal configuration allowing us to reach a sensitivity to B-modes corresponding to a 68\\% C.L. upper-limit on the effective tensor-to-scalar ratio (primordial tensors + dust) $\\sigma(r)=0.015$ with three years of sky integration. We have also shown how QUBIC FI will be able to put constraints on the dust contamination through direct measurement of the dust SED and properties in the reconstructed maps at multiple sub-frequencies within a physical band, or through a study of the evolution of the recovered tensor-to-scalar ratio in sub-bands as a function of frequency.\n\nFuture studies will explore in detail the impact of atmosphere fluctuations and how spectral imaging can lead to improved component separation using classical techniques as well as techniques specific to bolometric interferometry.\n\n\n\\subsubsection{Overview}\n\\label{sec_fg_challenge_overview}\n\n    \\input{overview_foreground_challenge}\n    \n\\subsubsection{QUBIC Technological Demonstrator expected performance}\n\\label{Sec:TD_perf}\n\n    \\input{qubic_td_performance}\n    \n\n\\subsubsection{QUBIC Full Instrument expected performance}\n    \\input{qubic_fi_performance}\n\n\n\n\\subsubsection{Ability to recognize the presence of dust residuals with spectro-imaging}\n\\label{sss:dust_residuals}\n\n    \\input{detect_dust_residuals}\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\nA phase of exponential expansion called ``Inflation'' in the early Universe was proposed as a solution to major problems with the standard Big-Bang model: the horizon, flatness and monopole problems~\\citep{1981PhRvD..23..347G,1982PhLB..108..389L}.  The horizon problem relates to the observed homogeneity of the Universe as evidenced by the smoothness of the Cosmic Microwave Background (CMB). The flatness problem is the fact that the observed curvature of\nspace-time is so close to flat in present times, that it must have been flat to enormous precision in the early universe.  The inflationary paradigm has been so successful in resolving the flatness and horizon problems, that it soon became a keystone component to the standard model of Cosmology. Observations of the CMB have provided several convincing arguments in favor of inflation: i) the series of acoustic peaks observed in the temperature power spectrum~\\cite{debernardis_2000, netterfield_2002} ruled out the alternative to inflation, topological defects, as the mechanism responsible for the dominant primordial density fluctuations ; ii) the observation of correlations between temperature and E-polarization of the CMB~\\cite{kogut_2003} as well as the observed phase opposition between peaks in the $TT$ and $EE$ power spectra~\\cite{2004Sci...306..836R, 2009ApJ...692.1247P} are evidence for adiabatic primordial perturbations, such as produced by inflation ; iii) the measurement of the CMB scalar spectral index slightly lower than one~\\cite{2020A&A...641A...6P} is a prediction from inflation. However, despite these strong arguments, all showing agreement between observations and inflation, there continues to be a lack of direct observational evidence of inflation despite the 40~years that have passed since it was first proposed.\n\nOne observable effect of Inflation is the generation of polarization B-modes in the CMB. This happens because during the exponential expansion of the Universe intense gravitational waves are generated.  The gravitational waves induce re-orientation of the primordial plasma such that B-modes are visible in the radiation released at the surface of last scattering, which is the CMB.  This weak signal has not been detected so far and  many projects are operating, or are proposed, to measure the polarization B-modes in the CMB. These projects include SPTPol~\\cite{sayre2019measurements}, POLARBEAR~\\cite{ade2014measurement}, ACTPol~\\cite{louis2017atacama}, BICEP2~\\cite{ade2018constraints}, CLASS~\\cite{dahal2020class}, POLARBEAR 2 + Simons Array~\\cite{Polarbear+simons2016}, advanced ACT~\\cite{AdvACT2016}, upgrade of the BICEP3\/Keck array~\\cite{grayson2016bicep3}. Planned experiments include Simons Observatory~\\cite{2019JCAP...02..056A},  PIPER~\\cite{piper2016}, LSPE~\\cite{addamo2020large}, CMB-S4~\\cite{cmbs4}, LiteBIRD \\cite{ hazumi2019} and PICO~\\cite{PICO}.\n\nThe use of bolometric interferometry to measure polarization B-modes in the CMB was proposed for a series of projects leading up to QUBIC. These are the Millimeter-wave Bolometric Interferometry~\\citep[MBI,][]{2002AIPC..616..126A,2003NewAR..47.1173T}, the Einstein Polarization Interferometer for Cosmology~\\citep[EPIC,][]{2006NewAR..50..999T}, the Background RAdiation INterferometer \\citep[BRAIN,][]{2007NewAR..51..256P}, and CMBPol\\citep{2009JPhCS.155a2003T}. Members of all these collaborations joined together to develop the \\mbox{Q \\& U Bolometric} Interferometer for Cosmology \\citep[QUBIC,][]{2009arXiv0910.0391K,2009AIPC.1185..506G,2011APh....34..705Q,2012MNRAS.423.1293B,2016arXiv160904372A}.\n\nThis paper is part of a special issue on QUBIC which includes details on all its design aspects, as well as on the performance of an advanced prototype, the Technological Demonstrator (TD).  The scientific overview and expected performance of the instrument are addressed here.  The other papers in the special issue address the following topics: the ability of bolometric interferometry to do spectral imaging~\\citep{2020.QUBIC.PAPER2}, the calibration and performance in the laboratory of the TD~\\citep{2020.QUBIC.PAPER3}, the performance of the detector array and readout electronics~\\citep{2020.QUBIC.PAPER4}, the cryogenic system performance~\\citep{2020.QUBIC.PAPER5}, the Half Wave Plate rotator system~\\citep{2020.QUBIC.PAPER6}, the back-to-back feedhorn-switch system~\\citep{2020.QUBIC.PAPER7}, and the optical design and performance~\\citep{2020.QUBIC.PAPER8}.\n\nThis paper is organized as follows.  A description of bolometric interferometry is provided in Section~\\ref{sec:boloint}. This is followed in Section~\\ref{sec:sciobj} with an overview of the scientific objectives, the current state-of-the-art of CMB polarization experiments, and the expected performance of QUBIC. Finally, conclusions are presented in Section~\\ref{sec:conclusions}.\n\n\n\n\n\n\\section{Science Objectives of QUBIC}\n\\label{sec:sciobj}\nQUBIC, being a bolometric interferometer, has distinctive features with respect to traditional imagers designed for observing the CMB and optimized for measuring primordial B-mode polarization. QUBIC is a clean spectral polarimeter well adapted to the purpose of detecting primordial B-modes because of the following main features:\n\\begin{itemize}\n    \\item it scans the sky with a synthesized beam formed by the feedhorn array, that exhibits multiple peaks (see Fig.~\\ref{Fig:sbqubic} and~\\ref{Fig:sb_freq}). Each of the synthesized beam peaks is well approximated by a Gaussian (above -20~dB) with a resolution of 23.5~arcminutes at 150~GHz; \n    \\item the angular separation on the sky between the synthesized beam peaks is given by the smallest distance between two peaks and is 8.8~degrees at 150~GHz. This distance scales as a function of frequency within the physical bandwidth of 25\\% in such a way that sub-frequency maps can be reconstructed using spectral imaging;\n    \\item a specific optical design with a polarizing grid before any optical component (except for the Half-Wave-Plate, filters and window) combined with full-power detectors on the focal plane make QUBIC largely immune to cross-polarization.\n\\end{itemize}\nAll of these specific features were studied in detail and are incorporated in the forecasts shown in this section. \n\n\\subsection{Data Analysis and Simulations for QUBIC performance forecasts\\label{sims}}\n\\subsubsection{Map-making and noise structure\\label{sec_map_making_noise_structure}}\nThe non-trivial shape of the synthesized beam (figures~\\ref{Fig:sbqubic} and~\\ref{Fig:sb_freq}) requires a specific map-making method which was developed for QUBIC. It is based on forward modeling to solve the inverse problem of map-making (see Mousset et al.~\\cite{2020.QUBIC.PAPER2} for details).  We start from a guess map of the sky and a detailed model of the instrument\\footnote{The software \nmakes heavy use of the massively parallel libraries developed by P.~Chanial pyoperators~\\cite{chanial2012pyoperators} (\\href{https:\/\/pchanial.github.io\/pyoperators\/}{https:\/\/pchanial.github.io\/pyoperators\/}) and pysimulators (\\href{https:\/\/pchanial.github.io\/pysimulators\/}{https:\/\/pchanial.github.io\/pysimulators\/}).}. The instrument model uses a detailed description of the synthesized beam. At first we can use an idealized instrument model inspired by an ideal synthesized beam as shown in figure~\\ref{Fig:sb_freq}, or a more realistic one as simulated including optical aberrations (see figures 11 and 12 in O'Sullivan et al.~\\cite{2020.QUBIC.PAPER8}). Then, such a description is expected to be gradually refined during the observation campaigns using information from self-calibration~\\cite{2013A&A...550A..59B} in order to incorporate instrumental systematic effects (from optics, electronics) as well observational effects such as ground-pickup or atmospheric contamination. Our software is designed to be able to incorporate a large variety of such systematic effects through the use of a number of time-domain, frequency-domain, or map-domain operators. We observe the guess map at iteration $i$ with the same scanning strategy used with the real data, obtaining simulated TOD that are compared to the real TOD with a $\\chi^2$. Using a preconditioned conjugate gradient we modify the guess map in an iterative manner until convergence. The final guess map is the solution of the map-making linear problem. There are two ways of accounting for systematics effects in the map-making: i) directly implementing their action in the operators that produce the simulated TOD in the case we have a model with known parameters from self-calibration and ii) fitting the unknown instrumental systematics parameters (for instance Jones matrices elements~\\cite{2007MNRAS.376.1767O}) as extra-parameters, along with the maps pixels, as part of the map-making.\n\nThe synthesized beam used for the instrument model during map-making is actually just a set of Dirac functions with the relevant amplitude at the location of the peaks of the synthesized beam (ideal, including optical aberrations or resulting from self-calibration). In such a way, and similarly as with an imager, the map-making does not attempt to deconvolve from the resolution of the peaks, but only from the multiple peaks.\n\nIf the synthesized beam model accounts for the realistic frequency dependence, one can use multiple maps at multiple frequencies within the physical bandwidth of the instrument and therefore reconstruct such sub-frequency maps. This is spectral imaging~\\cite{2020.QUBIC.PAPER2}. \n\nAnother specificity of this map-making is that the presence of multiple peaks separated by 8.8~degrees on the sky at 150~GHz (6~degrees at 220~GHz) makes QUBIC insensitive to modes on the sky larger than this separation. This occurs because the deconvolution from the multiple peaks relies on the measured signal difference between observations pointing to different directions where the peaks capture different amounts of power. For sky signals at angular scales larger than the angular distance between the peaks, such a difference vanishes. This naturally filters-out large-scale information be it from the sky itself, or from atmospheric gradients. This of course only applies to atmospheric fluctuations at scales larger than the angular distance between peaks in the synthesized beam 8.8 and 6 degrees at 150 and 220 GHz respectively.\n\nTo understand the impact of fluctuations on smaller scales we need to run simulations based on dedicated atmosphere measurements taken on site. In appendix~\\ref{app_impact_small_scale_atm}, however, we briefly discuss this issue and show how we expect a significant impact only from turbulence cells in a restricted range of scales.\n\nA careful study of the noise structure in end-to-end simulated maps shows two significant features that merit further explanation.\n    The first is sub-band correlations.  When performing spectral imaging, we show that nearby sub-bands exhibit a significant level of noise anti-correlation~\\cite{2020.QUBIC.PAPER2}, as displayed in figure~\\ref{Fig:nunu_corr} for the three Stokes parameters at 150~GHz and 5~sub-bands. The anti-correlation is strong with the nearest sub-bands but reduces significantly beyond. Similar correlation matrices are found at 220~GHz.\n    \\begin{figure}[!t]\n    \\centering\n    \\includegraphics[ width = \\hsize ]{Figures\/nunu_corr_5bands.png}\n    \\caption{QUBIC sub-bands correlation matrices for I, Q and U Stokes parameters maps obtained from end-to-end simulations at 150 GHz reconstructing the TOD onto 5 sub-bands (equally spaced in log within the physical 150 GHz bandwidth of QUBIC that ranges from 131 to 169 GHz) using spectral imaging and averaged over pixels in the maps. }\n    \\label{Fig:nunu_corr}\n    \\end{figure}\n\n   \n    The second feature worth noticing is spatial correlations.  Map-making with a multiply-peaked synthesized beam involves partial deconvolution because a given time sample in a detector's TOD receives power from distinct pixels in the sky with weights given by the shape of the synthesized beam.  As a result, we expect significant spatial noise correlations in our maps. This is confirmed by end-to-end simulations as shown in the left panel of figure~\\ref{Fig:space_corr}. Anti-correlation peaks, are expected, at an angle corresponding to the angular separation between the peaks in the synthesized beam ($\\theta_{\\mathrm{peaks}}$=8.8~degrees at 150~GHz and 6 degrees at 220~GHz). A similar 2pt-correlation function is found at 220~GHz, but with even higher correlation amplitude because the secondary peaks are higher due to the top-hat shape of the primary beam resulting from multimode optics at 220~GHz (see figure~4 in O'Sullivan et al.~\\cite{2020.QUBIC.PAPER8}).\n    \\begin{figure}[h]\n    \\centering\n    \\includegraphics[ width = 0.5\n    \\hsize ]{Figures\/spatial_correlation.png}\\includegraphics[ width = 0.5 \\hsize]{Figures\/clnoise.png}\n    \\caption{{\\bf (left)} QUBIC spatial noise 2pt-correlation function obtained from end-to-end simulations normalized by the variance in the maps $C(\\theta=0)$. The solid lines show an adjustment by a sine-wave modulated by an exponential with a Dirac function at $\\theta=0$ (the noise variance in the maps). The maximum anti-correlation is found as expected at the scale of the angular distance between two peaks of the synthesized beam (S.B.). The amplitude of the correlation is higher at 220~GHz than at 150~GHz because of the top-hat shape of the primary beam at 220~GHz. {\\bf (right)} Spatial noise correlation converted to multipole space. The straight-line at $C_\\ell=1$ shows the expected shape for white noise. The noise correlation results in a reduction of the noise for multipoles larger than $\\sim 40-50$, that is for angular scales $\\lessapprox\\theta_\\mathrm{peaks}$ (angular separation between the synthesized beam peaks). At lower multipoles (larger angular scales), we observe an increase of the noise. This is an advantage for measuring the recombination peak around $\\ell=100$ as discussed in section~\\ref{sect_Bmodes}}.\n    \\label{Fig:space_corr}\n    \\end{figure}\n\n    In the right panel of figure~\\ref{Fig:space_corr} we display  the spherical harmonics transform of the 2pt-correlation function:\n    \\begin{equation}\n        C_\\ell = 2\\pi\\int_{-1}^{1}C(x)P_\\ell(x)\\mathrm{d}x\n    \\end{equation}\n    where $x=\\cos\\theta$ and $P_\\ell$ are the Legendre polynomials.\n    This is our noise angular power spectrum which corresponds to the equivalent for QUBIC of typical white noise for a classical imager. The shape of this noise in Harmonic Space exhibits an excess with respect to white noise at very large scales (small multipoles, below $\\ell=40$  at 150 GHz and $\\ell=50$ at 220 GHz) and a significant reduction at smaller angular scales (larger multipoles). The scale of this transition is determined by the angular distance between peaks in the synthesized beam\\footnote{It is however not strictly equal to $\\pi\/\\theta_\\mathrm{peaks}$ because of the shape of the 2pt-correlation function and the non-trivial correspondence between angles and multipoles.}. Angular scales $\\gtrapprox\\theta_\\mathrm{peaks}$ are not well constrained due to the presence of the multiple peaks that are effectively deconvolved during the map-making. Conversely, angular scales smaller than this angular separation see their noise significantly reduced thanks to the positive correlation of the noise at these angles. \n    Because these angular scales correspond to those of the recombination peak in the B-mode spectrum, this specific noise feature for Bolometric Interferometry turns out to be a significant advantage for detecting primordial B-modes. This is discussed with more details from Monte-Carlo simulations in section~\\ref{sect_Bmodes} and visible in Figure~\\ref{Fig:bmodes}. Also, because our noise is not white, the RMS in the maps does not have direct significance. In our case, it is more meaningful to measure the noise level in the angular power-spectrum as we will detail in section~\\ref{sect_Bmodes}.\n    \n\\subsubsection{The QUBIC Fast Simulator}\nThe peculiar noise structure of the QUBIC maps has been studied in detail using a number\\footnote{40 end-to-end simulations were used for each of the considered configurations.} of end-to-end simulations run on supercomputers as they have large memory requirements\\footnote{With a 156.25~Hz sampling rate, QUBIC produces $156.25 \\times 3600 \\times 24 \\times 1024 \\times 2 \\times 8 \/1e9 \\simeq 220$~GB\/day.}. We extract from these simulations the main features of the noise discussed above:\n\\begin{itemize}\n    \\item noise scaling as a function of normalized coverage;\n    \\item correlations between reconstructed sub-bands (see figure~\\ref{Fig:nunu_corr});\n    \\item spatial (pixel-pixel) noise correlation measured on maps corrected for the noise scaling with respect to coverage;\n\\end{itemize}\nWe have built a \"Fast Simulator\" that directly produces maps with theses features: \n\\begin{enumerate}\n    \\item we start by creating (in harmonic space) noise maps with the observed spatial correlation (see figure~\\ref{Fig:space_corr}),\n    \\item we then make linear combinations of these noise maps in order to have I, Q and U maps for each sub-band with the appropriate correlation matrix (see figure~\\ref{Fig:nunu_corr}),\n    \\item finally, we scale the noise in the maps according to the scaling with respect to coverage. \n\\end{enumerate} \nThe overall noise normalization is adjusted to match that of the end-to-end simulations with the same integration time. We have checked in detail the accuracy of the Fast Simulator by performing the same noise structure analysis on the output maps and verifying that they lead to the same noise modeling as with the end-to-end simulations.\n\nThe Fast Simulator allows for fast production of maps with large-number statistics (thousands of realizations) and has been used extensively for the forecasts presented in this article.\n\nWe have used a simplified sky coverage obtained using random pointings on the sky from one time sample to another, reproducing the same sky fraction as the anticipated sky coverage shown in figure~\\ref{Fig:coverage}. While this allows obtaining a fast and efficient coverage of the QUBIC observed sky, it prevents one from simulating actual $1\/f$ noise from atmospheric or any time-domain instrumental fluctuations as successive time samples do not correspond to nearby pointings as in a more realistic scanning strategy. As a consequence,\nthe simulations presented in this article implicitly assume a stable atmosphere with no $1\/f$ noise, but account for the average loading from the atmosphere (see section~\\ref{overview} for a discussion of the seasonal and diurnal variations). \nWe will include realistic atmospheric fluctuations in our simulations when more detailed information than the average emissivity and temperature will be available from the TD data on the sky. In addition, as discussed in the second paragraph of this subsection, the shape of the QUBIC synthesized beam implies a low sensitivity on scales much larger than the angular distance between our multiple peaks, which will significantly reduce the impact of the (mostly large scales) atmospheric fluctuations.\n\nNo instrumental systematics are considered in this article, they will be included in future studies as well as how they can be mitigated through self-calibration. All results presented in this article should therefore be considered as idealized and intended to estimate the ultimate sensitivity achievable by a perfect instrument observing a stable atmosphere.\n\n\\subsection{The quest for CMB B-modes}\n\\label{sec:bmodeoverview}\n  \n\\subsubsection{Overview}\\label{overview}\nWe have performed simulations for a three-year observation of the sky using the QUBIC FI on a sky without any foregrounds, but with realistic instrumental noise~\\cite{2020.QUBIC.PAPER3,2020.QUBIC.PAPER4}) and with atmospheric background noise (assumed to be stable). The latter has been obtained from measurements performed over 3 years at the QUBIC site in Argentina from a tipper at 210~GHz used for the LLAMA radiotelescope\\footnote{ \\href{https:\/\/www.llamaobservatory.org\/}{https:\/\/www.llamaobservatory.org\/}} to be installed near QUBIC. Atmospheric background is averaged over the 9~best months of the year and corresponds to an atmospheric temperature of 270~K and emissivities 0.081 and 0.138 at 150 and 220~GHz respectively for an average observation elevation of 50 degrees. \n\nChanging the atmospheric parameters according to maximal diurnal and seasonal variations induces 15 and 25\\% change in the photon noise at 150 and 220 GHz respectively (with respect to the numbers given in Table~\\ref{tab_qubic_td_simulation_parameters}). This corresponds to a negligible change of sensitivity for the TD because our noise is dominated by that of the detectors. For the FI, the change in the total noise is 5\\% at 150 GHz and 20\\% at  220 GHz. As said before, more detailed simulations including the measured atmospheric fluctuations will be performed when these measurement are available with the instrument.\n\nWe have used the ``Fast Simulator'' described above to produce thousands of realizations of the noise in the maps incorporating the peculiar noise structure (spatial variations of the noise as a function of coverage as well as spatial and sub-band correlations). The parameters of our simulation are summarized in table~\\ref{tab_qubic_td_simulation_parameters}.\n\n\\begin{table}[t]\n    \\renewcommand{\\arraystretch}{1.}\n    \\begin{center}\n        \\caption{\\label{tab_qubic_td_simulation_parameters}Main instrumental and simulation parameters used in our computations. The noise value for TD is measured from the TES calibration data~\\cite{2020.QUBIC.PAPER3} while that for the FI is the intrinsic TES noise and assumes reduction of the noise aliasing in the readout chain found in the TD. We have used average atmospheric parameters at 50 degrees elevation accounting for maximal seasonal and diurnal variations corresponds to negligible change for the total noise for the TD, 5\\% and 20\\% for the FI at 150 and 220 GHz respectively (see text).}\n        \\begin{tabular}{p{6cm} p{3.7cm} m{3.8cm}}\n            \\hline\n            Parameter& Value TD & Value FI \\\\\n            \\hline\n            \\hline\n            Detector noise [$\\mathrm{W\/\\sqrt{Hz}}$]\\dotfill & 2.05$\\times 10^{-16}$ & 4.7$\\times 10^{-17}$ \\\\\n            \n            \\hline\n            Atmosphere$^1$ temperature [K]\\dotfill& 270 &270\\\\\n           \n            Atmosphere emissivity$^1$ at 150\\,GHz\\dotfill& 0.081 &0.081\\\\\n           \n            Photon noise [$\\mathrm{W\/\\sqrt{Hz}}$]\\dotfill & 2.6$\\times 10^{-17}$~(150 GHz) & 3.1$\\times 10^{-17}$~(150 GHz),\\hfill\\break 1.17$\\times 10^{-16}$(220 GHz)\\\\\n            \\hline\n            Total noise [$\\mathrm{W\/\\sqrt{Hz}}$]\\dotfill & 2.06$\\times 10^{-16}$(150 GHz) & 5.7$\\times 10^{-17}$~(150 GHz),\\hfill\\break 1.26$\\times 10^{-16}$~(220 GHz)\\\\\n            \\hline\n            Cumulated observation time [years]\\dotfill & 1 & 3  \\\\\n            \\hline\n            $r$ upper-limit (68\\% C.L., No FG) &- & 0.021~(150~GHz),\\hfill\\break 0.023~(220~GHz),\\hfill\\break 0.015~(Combined)\\\\\n            \\hline\n            \\multicolumn{3}{l}{$^1$The atmosphere is considered perfectly stable.}\\\\         \\end{tabular}\n    \\end{center}\n\\end{table}\n\n\nFor each realization, we have used NaMaster~\\footnote{\\href{https:\/\/github.com\/LSSTDESC\/NaMaster}{https:\/\/github.com\/LSSTDESC\/NaMaster}}~\\cite{2019MNRAS.484.4127A} to compute pure TT, EE, TE and BB power spectra on the residual maps (therefore noise-only maps) in order to compute the expected noise on the power spectra. Exploring various values for the minimum multipole $\\ell_{min}$, the size of the multipole bins $\\Delta_\\ell$ and minimum value of the relative coverage (normalized to 1 at maximum) of the sky that defines the region of the sky we keep for analysis, $\\mathrm{Cov}_c$, we have found the best configuration to be $\\ell_\\mathrm{min}=40$, $\\Delta_\\ell=30$ and $\\mathrm{Cov}_c=0.1$ (keeping all pixels with relative coverage above 0.1) at 150 GHz. We have kept the same configuration for the 220~GHz for the sake of simplicity. \n\n\\subsubsection{QUBIC Full Instrument expected performance}\n\\label{sect_Bmodes}\nAs remarked before, measuring the RMS of a map in the case of non-white noise is meaningless, and the actual measurement of the effective depth of our maps is done in $\\ell$-space. The resulting uncertainties from the Monte-Carlo on the BB polarization power spectrum ($\\Delta D_\\ell$) are shown in the left panel of figure~\\ref{Fig:bmodes}. \n\\begin{figure}[!t]\n\\centering\n\\includegraphics[ width = 0.49 \\hsize ]{Figures\/bmodes_zoom.png}\\includegraphics[ width = 0.49\\hsize ]{Figures\/likelihood_r.png}\n\\caption{{\\bf (left)} BB power spectrum error-bars $\\Delta D_\\ell$ on $D_\\ell = \\frac{\\ell(\\ell+1)}{2\\pi}C_\\ell$ on residual maps from the Fast-Simulator Monte-Carlo in the absence of foregrounds, atmospheric fluctuations and instrument systematic effects for an integration time on the sky of three years from the site in San Antonio de los Cobres, Argentina. As these are calculated on residual maps, they do not incorporate sample variance and only refer to the instrumental noise.\nThe reduction at low-$\\ell$ of our error-bars with respect to theoretical white-noise (dotted lines) is clearly visible and in agreement with the expected shape from the 2pt correlation function in figure~\\ref{Fig:space_corr} (dashed lines). The difference in the first bin is discussed in the text.\n{\\bf (right)} Posterior likelihood on the $r$ (assuming no foregrounds) using QUBIC FI (two bands) with three years integration on the sky (including both noise and sample variance). The latest Planck+Bicep\/Keck constraint from Tristram et al.~\\cite{Tristram_r} is shown with the blue arrow}\n\\label{Fig:bmodes}\n\\end{figure}\nWe have also displayed theoretical B-mode power spectra $D_\\ell$ (including lensing) for $r$=0, 0.01, and 0.044 (current best upper limit from Tristram et al.~\\cite{Tristram_r}). Besides the QUBIC error bars, we also plot the expected shape for white noise (dotted lines) and for the QUBIC noise (dashed lines) from figure~\\ref{Fig:space_corr}. \nIn both cases the theoretical error-bars are obtained through the well known formula:\n\\begin{equation}\n    \\Delta D_\\ell^\\mathrm{th} = \\frac{\\ell(\\ell+1)}{2\\pi}\\times \\sqrt{\\frac{2}{(2\\ell+1)f_\\mathrm{sky}\\Delta\\ell}}\\times \\frac{1}{B_\\ell^2}\\times \\frac{1}{W_\\ell^2} \\times C_\\ell^\\mathrm{noise}\n\\end{equation}\nwhere $f_\\mathrm{sky}$ is the fraction of the sky used for analysis ($f_\\mathrm{sky}=0.015$ in our case), $\\Delta\\ell$ is the width of the $\\ell$-space binning ($\\Delta\\ell=30$ in our case). $B_\\ell$ is the beam transfer function and $W_\\ell$ is that of the Healpix pixellisation ($N_\\mathrm{side}=256$). $C_\\ell^\\mathrm{noise}$ is the expected shape for the noise power spectrum that can be either a constant in the case of white noise or a constant multiplying the shape from figure~\\ref{Fig:space_corr} (right) in the case of QUBIC. We performed a fit of the above noise normalization for QUBIC to our Monte-Carlo error-bars leading to 2.7~and 3.7~$\\mu\\mathrm{K.arcmin}$ at 150~and 220~GHz respectively. This is shown as dashed lines in figure~\\ref{Fig:bmodes}. However these numbers are hardly comparable with the case of a standard imager for which the noise is white: for each frequency we overplot the expected shape for white noise with the same normalization. The significant noise reduction with respect to the white noise case is particularly visible at the scales of the recombination peak near $\\ell=100$, giving QUBIC an enhanced sensitivity at those scales. At scales larger than the separation between peaks in the synthesized beam however, the error-bars increase sharply.\nThe first bin at 220~GHz exhibits significantly larger error-bars for all spectra. This is not surprising as we have kept the same $\\ell_{min}=40$ for both channels. In reality, the multiple-peaked shape of our synthesized beam is such that we have little sensitivity to multipoles corresponding to angular scales larger than the distance between the peaks (8.8~and 6~degrees at 150~and 220~GHz respectively). As a result, the optimal $\\ell_{min}$ at 150~GHz is slightly too low for 220~GHz, resulting in larger error-bars for the first bin. This will be optimized when analyzing real data.\n\nThe right panel of figure~\\ref{Fig:bmodes} shows the posterior on the tensor-to-scalar ratio which, in the absence of foregrounds, was the only free-parameter for this power-spectrum-based likelihood (simple $\\chi^2$ accounting for sample variance~\\cite{hamimeche-lewis}) with all parameters but $r$ fixed to their fiducial values\\footnote{We have used a fiducial cosmology with parameters [$h=0.675$, $\\Omega_b h^2=0.022$, $\\Omega_c h^2=0.122$, $\\Omega_k=0$, $\\tau=0.06$, $A_s=2e-9$, $n_s=0.965$]}.\nWe calculate the likelihood at 150 and 220 GHz separately as well as jointly. These simulations show that QUBIC  has the statistical power (without foregrounds, atmospheric fluctuations and systematics) to constrain the B-modes down to a tensor-to-scalar ratio $r<0.015$ at 68\\% C.L. ($r<0.03$ at 95\\% C.L.) with three years integration on the sky from our site in Argentina. \n\nIn the presence of foregrounds,  the numbers above are to be understood as our statistical sensitivity to effective B-modes including the contribution from primordial tensors as well as dust polarization. \nComponent separation has not been included in the current forecasts and will be investigated in details in a future publication. In our study we have focused only on dust contamination, neglecting a possible contribution from synchrotron. This hypothesis is supported by two facts: (i) QUBIC will search for cosmological B-modes in a well defined sky patch that will correspond roughly to the one shown in figure~\\ref{Fig:coverage}, and (ii) in this sky patch the systematic error on $r$ from synchrotron contamination at 150\\,GHz is below 0.005 (see figure~6 in Krachmalnicoff et al.~\\cite{krachmalnicoff2016}), so well below our target $r<0.021$. Moreover, QUBIC will have the capability to check for any residual foregrounds contamination in the obtained tensor-to-scalar ratio by exploiting its spectral imaging features as described in section~\\ref{sss:dust_residuals}. Finally, in the QUBIC data analysis we will also exploit the wealth of data available at low frequency (WMAP, Planck, C-Bass, QUIJOTE and other data that will be publicly available) to further improve the robustness of our final results.\n\n\\subsection{The foregrounds challenge}\n\\label{sect:foreground_challenge}\n\n    \\input{foreground_challenge}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nHeterogeneous graphs (which subsume knowledge graphs) can model complex relationships among multiple types of real-world entities and have been widely used in studying biological and biomedical data. In a heterogeneous graph (\\emph{e.g.}, \\cite{fan2019metapath,zhang2019heterogeneous,nicholson2020constructing}), there can be multiple types of nodes (\\emph{e.g.}, some for compounds and others for targets, gene or diseases) and edges (\\emph{e.g.}, a compound treats a disease, a compound resembles another compound, or a gene interacts with another gene), making discovery of deep insights (\\emph{e.g.}, perform drug repurposing or identify drug-target interactions) become possible. Graph representation learning aims to construct low-dimensional embeddings (vectors) for all nodes in a graph, and has been a standard approach to further knowledge discovery and development on graphs. For instance, both predicting biomedical network links based on their structural and feature information and clustering for drug-target interaction and drug discovery are based on high-quality heterogeneous network representation. Recently, various heterogeneous graph representation learning methods have been proposed, \\emph{e.g.}, random-walk based methods Metapath2Vec \\cite{dong2017metapath2vec} and HIN2Vec \\cite{fu2017hin2vec}, message-passing based graph neural networks HAN \\cite{wang2019heterogeneous} and HGT \\cite{hu2020heterogeneous}, and relation learning based methods TransE \\cite{bordes2013translating}, ComplEx \\cite{trouillon2016complex}, and RotatE \\cite{sun2019rotate}. For a systematic review on heterogeneous graph embedding, readers are referred to a good survey \\cite{yang2020heterogeneous}.\n\nHowever, modern heterogeneous graphs can be of very large size in terms of the number of nodes or the number of edges. For instance, the benchmarking knowledge graph Freebase has over $12$ millions nodes across $8$ node types and about $63$ millions edges of $36$ edge types \\cite{yang2020heterogeneous}. Processing such large-scale heterogeneous graphs imposes high computational and storage demands. We identify two directions to meet the requirements: one approach is to develop efficient heterogeneous embedding algorithms. But this direction may not offset the effects of ever growing big heterogeneous graphs. The other is to sparsify\/compress big heterogeneous graphs while approximating some important network property and not significantly impacting the quality of graph embedding. In this work, we focus on the latter topic of sparsifying heterogeneous graphs.\n\nGraph sparsification deals with approximating an arbitrary graph by a sparse graph that usually has a linear number of edges in the number of vertices of the original graph while preserving some important property.\nSeveral notions of graph sparsification have been proposed, \\emph{e.g.}, graph spanners \\cite{PS89} that approximate the shortest path distance between every pair of vertices, cut sparsifiers \\cite{BK96} that approximately preserve all graph cut values, and spectral sparsifiers \\cite{ST11} that approximate the graph spectrum\/eigenvalues. Graph sparsification has been used to improve the computational cost of learning over graphs, \\emph{e.g.}, graph semi-supervised learning and spectral clustering \\cite{SWT16,ZLB21a,ZLB21b}. Although heterogeneous graphs can have much larger size since heterogeneity allows more nodes or edges to be included in one network, however, there has been no systematic study on sparsification techniques for heterogeneous graphs yet.\n\nIn this paper, we for the first time study the problem of heterogeneous graph sparsification and its algorithmic applications in graph representation learning. We propose an efficient algorithm for constructing provably smaller sparsifiers for heterogeneous graphs, which, when feeding into a heterogeneous graph embedding algorithm, can save significant computational and storage cost. However, interestingly, the quality of the learned representations using sparsifiers is comparable or even \\emph{better} than those based on the original graphs. We have provided theoretical analysis of the saving in time and space complexities of heterogeneous graph representation learning. Extensive experiments have also been performed to demonstrate that the developed method can improve the efficiency of heterogeneous graphs while not significantly affecting the accuracy of downstream machine learning tasks based on the learned embedding.\n\n\n{\\noindent \\bf Related Work.}\nSparsification over homogeneous graphs has been extensively studied, \\emph{e.g.}, \\cite{PS89,BK96,ST11} and used in machine learning \\cite{SWT16,CKL+18,ZSL+19a}. In heterogeneous graphs, however, how to sparsify various types of vertices and edges while preserving graph properties has received limited studies \\cite{jiang2021pre,yang2020heterogeneous}. Jiang \\emph{et al.} \\cite{jiang2021pre} developed self-supervised pre-training methods for heterogeneous graphs with relation-based sparsification for efficient pre-training. But their focus is on designing both the node- and schema-level pre-training tasks to preserve semantic and structural properties, instead of a comprehensive study on the sparsification method.\nTo the best of our knowledge, this work is the first systematical study on heterogeneous graph sparsification and its use in representation learning.\n\n\n\n\n\n\n\n\\section{Heterogeneous Graph Sparsifiers}\n\nIn this section, we develop a sampling-based sparsification technique for heterogeneous graphs, analyze its computational and space complexities, and then discuss other potential methods.\n\n{\\noindent \\bf Notations and Definitions.}\nA heterogeneous graph can be defined as $G(V,E,\\phi,\\pi,X,R)$ where $V$ is the vertex set, $E$ is the edge set, $\\phi$ specifies the type for each vertex, $\\pi$ assigns the type of each edge, and $X$ and $R$ include the node and edge features respectively. $\\phi$, $\\pi$, $X$, and $R$ can be omitted from the presentation if they are clear from the context. Let $E_{Out}(u)$ be the set of $u$'s out-edges and $E_{Out}^t(u)$ be the set of $u$'s out-edges of type $t$. Similarly, let $E_{In}(u)$ be the set of $u$'s in-edges and $E_{In}^t(u)$ be the set of $u$'s in-edges of type $t$. Let $|G(V,E)|=|E|$.\n\n{\\noindent \\bf Motivation.}\nGraph sparsification in general\/homogeneous graphs has received considerable attention and many sparsification algorithms have been developed. However, they can be separated into two types, \\emph{importance} based sampling methods and \\emph{heuristic} based sampling methods. In the former, graph edges are sampled according to some important measure. For example, spectral sparsifiers and cut sparsifiers can be obtained by non-uniformly sampling each edge based on its effective resistance and edge connectivities \\cite{SS11,FHH+11}, respectively. This type of approaches is often sound and has nice theoretical guarantees. However, such a design for heterogeneous graphs can be highly challenging considering the complexity in defining an importance measure across different types of edges and computing that measure in large-scale heterogeneous graphs. \n\nIn the latter, sampling is performed on all edges independently using only heuristics instead of complex importance measures. For example, $k$-neighbors sparsifiers are constructed by randomly sampling at most $k$ edges for each vertex where $k>0$ is a tuneable parameter. Though simple, heuristic based sparsifiers have performed practically good, \\emph{e.g.}, comparable to spectral sparsifiers in machine learning tasks over graphs \\cite{SWT16}. In this paper, we will work towards sparsifying heterogeneous graphs using simple yet effective heuristics.\n\n\n{\\noindent \\bf Details.}\nAs a warm-up, we consider the simple method of sampling a fixed number of edges and including in the sparsifier. Since node degrees can be quite different for different nodes, it is possible that a node of low degree has none of its edges sampled and become \\emph{isolated} in the sparsifier. Unfortunately, some heterogeneous embedding algorithms do not allow isolated nodes (through reporting errors such as \\cite{shi2018aspem,wang2019heterogeneous}) and thus this type of sparsifiers is not robust to embedding algorithms. \nFurthermore, the number of edges of different types for a node can be very different because of their semantic difference. Consider an extreme case: a vertex $u$ with $10K$ edges of type $1$ and only one edge of type $2$. The information carried by the edge of type $2$ is supposed to be greater than that in an individual edge of type $1$. When all edges are considered together, the single edge of type $2$ can be easily sampled away, resulting in information loss.\n\nWe propose to perform random sampling within each type of edges and then aggregate all sampled edges as the sparsfiier. In this way, the isolation problem is avoided and the distribution of edge types cannot affect the quality of the constructed sparsifier.\nSpecifically, we sparsify the neighborhood of each vertex according to ascending order of vertex degree. Since heterogeneous graphs are often directed, we sparsify out-edges and in-edges of a vertex separately. Consider out-edges $E_{Out}(u)$ of vertex $u$. For each edge type $t$ in $E_{Out}(u)$, if the number of edges of type $t$ is at most $k$, we include all of them to the sparsifier $H$. Here $k$ is an input parameter to control the size of the output sparsifier. Otherwise, we first calculate the number of edges in $E^t_{Out}(u)$ that are already included in the current $H$. Note that we process vertices one by one and the processing of some vertex before the current vertex may already add a few edges in $E_{Out}(u)$ to $H$. Let $x$ be the number of such edges, $x=|E_{Out}^t(u)\\cap H|$. Then we sample $k-x$ edges from $E_{Out}^t(u)-H$ uniformly at random and include the sampled edges in $H$. Since in heterogeneous graphs there can be many edges without edge weight, we keep the weight of a sampled edge unchanged, if any. The processing for in-edges is symmetric. The algorithms are summarized in Algorithms \\ref{alg:graph} and \\ref{alg:node1}.\n\n\n\\begin{algorithm}[t]\n\\small\n\\caption{Sparsify-Graph}\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n\\begin{algorithmic}[1]\n\\REQUIRE A graph $G$ and a parameter $k$\n\\ENSURE A sparsified graph\n\\STATE $H=\\emptyset$;\n\\STATE Sort vertices in ascending order of degree;\n\\FOR {each node $u$ in the sorted order}\n    \\STATE Sparsify-Node-Out($u$, $G$, $H$, $k$);\n    \\STATE Sparsify-Node-In($u$, $G$, $H$, $k$);\n\\ENDFOR\n\\RETURN $(V,H)$;\n\\end{algorithmic}\n\\label{alg:graph}\n\\end{algorithm}\n\n\\begin{algorithm}[t]\n\\small\n\\caption{Sparsify-Node-Out(-In)}\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n\\begin{algorithmic}[1]\n\\REQUIRE A node $u$ in a graph $G$, an edge set $H$, and a parameter $k$\n\\ENSURE Updated $H$\n\\FOR {each edge type $t$}\n    \\IF {$|E_{Out}^t(u)|<k$ \\COMMENT{respectively, $|E_{In}^t(u)|<k$}}\n        \\STATE Include $E_{Out}^t(u)$ into $H$;\n    \\ELSE\n        \\STATE Compute $x=|E_{Out}^t(u)\\cap H|$; \\COMMENT{$x=|E_{In}^t(u)\\cap H|$;}\n        \\STATE Sample $k-x$ edges from $E_{Out}^t(u)-H$ uniformly at random; let the sampled edges be $S$; \\{Sample $k-x$ edges from $E_{In}^t(u)-H$;\\} \n        \\STATE Include $S$ into $H$;\n    \\ENDIF\n\\ENDFOR\n\\RETURN $H$;\n\\end{algorithmic}\n\\label{alg:node1}\n\\end{algorithm}\n\n\nThe computational and space complexities of Algorithm \\ref{alg:graph} are $O(ktn)$ and the size (number of edges) of the resulting sparsifier is also $O(ktn)$, where $t$ is the total number of edge types. Let the computational complexity of a machine learning task in a heterogeneous graph $G$ be $T'(G)$. It is generally believed to be a function of the number of edges $m=|E|$, \\emph{i.e.}, $T'(G)=T(m)$. Although there are few studies on the time complexity of learning in heterogeneous graphs, in homogeneous graphs, the time complexities of graph-based learning, \\emph{e.g.}, supervised learning such as Laplacian smoothing, semi-supervised learning, and spectral clustering, have shown to be linear in the number of edges \\cite{CKL+18,ZSL+19a}. We believe that similar relations appear in heterogeneous graphs. Based on this assumption, our approach can reduce the computational cost from $T(m)$ to $O(ktn)+T(ktn)$. The improvement is significant because $m\\gg n$ usually holds for heterogeneous graphs, and $k$ and $t$ are only a small value. This analysis is applicable to the space complexity as well. We will confirm the improvement through empirical studies in Section \\ref{sec:experiment}.\n\n\n{\\noindent \\bf Discussion.}\nWe develop another method that also does not suffer from the isolation problem. The principle is to first randomly select, for every edge type $t$ of vertex $u$, one edge from $E^t_{Out}(u)$ (and $E^t_{In}(u)$, respectively) to include in the sparsifier. Let the number of edges included be $y$. We then randomly keep $k-y$ edges out of all $u$'s edges, regardless of edge types. This method guarantees that each edge type $t$ of $u$'s edges has at least one sampled edges and thus does not lose important information. This algorithm constructs sparsifiers of size $O(\\max(k,t)\\cdot n)$, which is smaller than Alg. \\ref{alg:graph} for the same value of $k$. Consider sparsifiers of similar size produced by both methods, their empirical performance in subsequent machine learning tasks are comparable. As will be shown in Section \\ref{sec:experiment}, sparsifiers produced by Alg. \\ref{alg:graph} generally performs better in the link prediction task while sparsifiers output by this method works better in node classification, although the margins are relatively small. \n\n\n\n\n\n\n\\section{Experiments}\n\\label{sec:experiment}\n\nIn this section, we perform extensive experiments to evaluate the proposed algorithms for sparsifying heterogeneous graphs and performance of the constructed sparsifiers in representation learning.\nThe algorithms were implemented in Python, and all experiments were performed using 3 cores in a compute node, which is equipped with 24 cores, 2.50 GHz Intel processors, and 256GB RAM.\n\n\n{\\noindent \\bf Experimental Setting.}\nWe use three different real-world datasets provided by the benchmark of \\cite{yang2020heterogeneous} and their statistics are summarized in Table \\ref{tab:datasets}. \\emph{PubMed} is a network of genes, diseases, chemicals, and species from NIH PubMed project (not the traditional publication network). \\emph{Yelp} is a review network of businesses, users, locations, and reviews. \\emph{Freebase} is a knowledge base of books, films, music, sports, people and so on. Since this is the first work on heterogeneous graph sparsification for representation learning, we also implement the algorithm in the discussion for comparison, which is denoted as \\emph{All-Types} (\\emph{ALT}). \n\nThe tested graphs are first processed by a sparsification algorithm to get sparsified graphs, which are then fed into a representation learning algorithm to get node embedding. To show the robustness of the proposed sparsification, five diverse algorithms including random-walk based \\emph{HIN2Vec} \\cite{fu2017hin2vec} and \\emph{AspEm} \\cite{shi2018aspem}, message-passing based \\emph{HAN} \\cite{wang2019heterogeneous} and \\emph{HGT} \\cite{hu2020heterogeneous}, and relation learning based \\emph{ComplEx} \\cite{trouillon2016complex}, are employed to generate the embedding. Finally, \\emph{link prediction} and \\emph{node classification} tasks are performed based on the learned embedding following experimental setup in the benchmark \\cite{yang2020heterogeneous}. In link prediction, $20\\%$ links are held out and the remaining $80\\%$ links are used as the training set. A linear Support Vector Machine (LinearSVC) \\cite{fan2008liblinear} is trained on the training links. For node classification, we train a LinearSVC based on the learned embedding on $80\\%$ of the labeled nodes and then predict on the remaining $20\\%$. \n\nThe efficiency for graph learning tasks are measured by the \\emph{training time} in seconds and \\emph{memory consumption} in term of the maximal residual size during model training. The effectiveness of link prediction is measured by the \\emph{AUC} (area under the ROC curve) and \\emph{MRR} (mean reciprocal rank). The effectiveness of node classification is measured by \\emph{macro-F1} (across all labels) and \\emph{micro-F1} (across all nodes). These measures are averaged over \\emph{five} runs and reported together with their standard deviation.\n\n\\begin{table}[t]\n\\center\n\\small\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\nDataset & $n$ & $m$ & $m\/n$ & {\\footnotesize Node Types} & $t$ \\\\\n\\hline\n\\emph{PubMed} & $63,109$ & $236,458$ & $3.7$ & $4$ & $10$ \\\\\n\\emph{Freebase} & $12,164,758$ & $62,058,902$ & $5.1$ & $8$ & $36$ \\\\\n\\emph{Yelp} & $82,465$ & $16,274,179$ & $197$ & $4$ & $4$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Statistics of the Tested Real-world Networks}\n\\label{tab:datasets}\n\\end{table}\n\n\\begin{table}[t]\n\\center\n\\small\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n$k$ & \\emph{PubMed} & \\emph{Freebase} & $k$ & \\emph{Yelp} \\\\\n\\hline\n$1$ & $0.45$ & $0.39$ & $20$ & $0.15$ \\\\ \\hline \n$2$ & $0.59$ & $0.55$ & $50$ & $0.28$ \\\\ \\hline\n$3$ & $0.67$ & $0.65$ & $100$ & $0.42$ \\\\ \\hline\n$5$ & $0.75$ & $0.73$ & $200$ & $0.59$ \\\\ \\hline\n$10$ & $0.86$ & $0.81$ & $500$ & $0.8$ \\\\ \\hline\n\\end{tabular}\n\\caption{The Sparsification Ratio ($|H|\/|G|)$}\n\\label{tab:sparsified}\n\\vspace{-0.2in}\n\\end{table}\n\n\n\n\\begin{figure*}[t]\n     \\center\n     \\begin{subfigure}[t]{0.32\\linewidth}\n        \\centering\\includegraphics[width=\\linewidth]{Figs\/PubMed-time.png}\n        \\vspace{-0.2in}\n        \\caption{\\small Training Time in \\emph{PubMed}}\n        \\label{fig:time-pubmed}\n     \\end{subfigure}\n     \\hspace{\\fill}\n     \\begin{subfigure}[t]{0.32\\linewidth}\n        \\centering\\includegraphics[width=\\linewidth]{Figs\/PubMed-ram.png}\n        \\vspace{-0.2in}\n        \\caption{\\small Memory Consumption in \\emph{PubMed}}\n        \\label{fig:ram-pubmed}\n     \\end{subfigure}\n     \\hspace{\\fill}\n     \\begin{subfigure}[t]{0.32\\linewidth}\n        \\centering\\includegraphics[width=\\linewidth]{Figs\/PubMed.png}\n        \\vspace{-0.2in}\n        \\caption{\\small \\emph{AUC} \\& \\emph{MRR} in \\emph{PubMed}}\n        \\label{fig:auc-pubmed}\n     \\end{subfigure}\n     \\hspace{\\fill}\n     \\begin{subfigure}[t]{0.32\\linewidth}\n        \\centering\\includegraphics[width=\\linewidth]{Figs\/PubMed-F1.png}\n        \\vspace{-0.2in}\n        \\caption{\\small Macro-F1 \\& Micro-F1 in \\emph{PubMed}}\n        \\label{fig:f1-pubmed}\n     \\end{subfigure}\n     \\hspace{\\fill}\n     \\begin{subfigure}[t]{0.32\\linewidth}\n        \\centering\\includegraphics[width=\\linewidth]{Figs\/Yelp-time.png}\n        \\vspace{-0.2in}\n        \\caption{\\small Training Time in \\emph{Yelp}}\n        \\label{fig:time-yelp}\n     \\end{subfigure}\n     \\hspace{\\fill}\n     \\begin{subfigure}[t]{0.32\\linewidth}\n        \\centering\\includegraphics[width=\\linewidth]{Figs\/Yelp-ram.png}\n        \\vspace{-0.2in}\n        \\caption{\\small Memory Consumption in \\emph{Yelp}}\n        \\label{fig:ram-yelp}\n     \\end{subfigure}\n     \\hspace{\\fill}\n     \\begin{subfigure}[t]{0.32\\linewidth}\n        \\centering\\includegraphics[width=\\linewidth]{Figs\/Yelp.png}\n        \\vspace{-0.2in}\n        \\caption{\\small \\emph{AUC} \\& \\emph{MRR} in \\emph{Yelp}}\n        \\label{fig:auc-yelp}\n     \\end{subfigure}\n     \\hspace{\\fill}\n     \\begin{subfigure}[t]{0.32\\linewidth}\n        \\centering\\includegraphics[width=\\linewidth]{Figs\/Yelp-F1.png}\n        \\vspace{-0.2in}\n        \\caption{\\small Macro-F1 \\& Micro-F1 in \\emph{Yelp}}\n        \\label{fig:f1-yelp}\n     \\end{subfigure}\n     \\hspace{\\fill}\n     \\begin{subfigure}[t]{0.32\\linewidth}\n        \\centering\\includegraphics[width=\\linewidth]{Figs\/Freebase-time.png}\n        \\vspace{-0.2in}\n        \\caption{\\small Training Time in \\emph{Freebase}}\n        \\label{fig:time-freebase}\n     \\end{subfigure}\n     \\hspace{\\fill}\n     \\begin{subfigure}[t]{0.32\\linewidth}\n        \\centering\\includegraphics[width=\\linewidth]{Figs\/Freebase-ram.png}\n        \\vspace{-0.2in}\n        \\caption{\\small Memory Consumption in \\emph{Freebase}}\n        \\label{fig:ram-freebase}\n     \\end{subfigure}\n     \\hspace{\\fill}\n     \\begin{subfigure}[t]{0.32\\linewidth}\n        \\centering\\includegraphics[width=\\linewidth]{Figs\/Freebase.png}\n        \\vspace{-0.2in}\n        \\caption{\\small \\emph{AUC} \\& \\emph{MRR} in \\emph{Freebase}}\n        \\label{fig:auc-freebase}\n     \\end{subfigure}\n     \\hspace{\\fill}\n     \\begin{subfigure}[t]{0.32\\linewidth}\n        \\centering\\includegraphics[width=\\linewidth]{Figs\/Freebase-F1.png}\n        \\vspace{-0.2in}\n        \\caption{\\small Macro-F1 \\& Micro-F1 in \\emph{Freebase}}\n        \\label{fig:f1-freebase}\n     \\end{subfigure}\n   \n     \\caption{\\small Performance in Different Datasets Under the Baseline Setting}\n     \\label{fig:results}\n     \\vskip -0.1in\n\\end{figure*}\n\n\n\n\n     \n\n\n\\begin{figure*}[t]\n     \\center\n     \\begin{subfigure}[t]{0.32\\linewidth}\n        \\centering\\includegraphics[width=\\linewidth]{Figs\/HIN2Vec_AUC_MRR_EMB.png}\n        \\vspace{-0.2in}\n        \\caption{\\small \\emph{HIN2Vec}}\n        \\label{fig:HIN2Vec}\n     \\end{subfigure}\n     \\hspace{\\fill}\n     \\begin{subfigure}[t]{0.32\\linewidth}\n        \\centering\\includegraphics[width=\\linewidth]{Figs\/HAN_AUC_MRR_EMB.png}\n        \\vspace{-0.2in}\n        \\caption{\\small \\emph{HAN}}\n        \\label{fig:HAN}\n     \\end{subfigure}\n     \\hspace{\\fill}\n     \\begin{subfigure}[t]{0.32\\linewidth}\n        \\centering\\includegraphics[width=\\linewidth]{Figs\/ComplEx_AUC_MRR_EMB.png}\n        \\vspace{-0.2in}\n        \\caption{\\small \\emph{ComplEx}}\n        \\label{fig:ComplEx}\n     \\end{subfigure}\n     \\hspace{\\fill}\n     \\begin{subfigure}[t]{0.32\\linewidth}\n        \\centering\\includegraphics[width=\\linewidth]{Figs\/ALT.png}\n        \\vspace{-0.2in}\n        \\caption{\\small Performance of \\emph{ComplEx} in different sparsifiers in \\emph{PubMed}}\n        \\label{fig:alt}\n     \\end{subfigure}\n     \\hspace{\\fill}\n     \\begin{subfigure}[t]{0.32\\linewidth}\n        \\centering\\includegraphics[width=\\linewidth]{Figs\/Att-HAN-tr.png}\n        \\vspace{-0.2in}\n        \\caption{\\small Time and Memory in Attributed \\emph{HAN}}\n        \\label{fig:att-han-tr}\n     \\end{subfigure}\n     \\hspace{\\fill}\n     \\begin{subfigure}[t]{0.32\\linewidth}\n        \\centering\\includegraphics[width=\\linewidth]{Figs\/Att-HAN.png}\n        \\vspace{-0.2in}\n        \\caption{\\small \\emph{AUC} \\& \\emph{MRR} in Attributed \\emph{HAN}}\n        \\label{fig:att-han}\n     \\end{subfigure}\n     \\hspace{\\fill}\n     \\begin{subfigure}[t]{0.32\\linewidth}\n        \\centering\\includegraphics[width=\\linewidth]{Figs\/Att-HGT-tr.png}\n        \\vspace{-0.2in}\n        \\caption{\\small Time and Memory in Attributed \\emph{HGT}}\n        \\label{fig:att-hgt-tr}\n     \\end{subfigure}\n     \\hspace{0.2in}\n     \\begin{subfigure}[t]{0.32\\linewidth}\n        \\centering\\includegraphics[width=\\linewidth]{Figs\/Att-HGT.png}\n        \\vspace{-0.2in}\n        \\caption{\\small \\emph{AUC} \\& \\emph{MRR} in Attributed \\emph{HGT}}\n        \\label{fig:att-hgt}\n     \\end{subfigure}\n     \\caption{\\small Results of Extra Sets of Experiments Under the \\emph{PubMed} Dataset}\n     \\label{fig:extra}\n     \\vskip -0.1in\n\\end{figure*}\n\n\n{\\noindent \\bf Baseline Setting.}\nIn the baseline setting, we focus on unattributed representation learning that does not exploit node attributes and use the embedding size $50$. We first show effectiveness of the developed algorithm in reducing the size of heterogeneous graphs, and then compare performance of learning tasks in the sparsified graphs and the full graph. Table \\ref{tab:sparsified} shows the \\emph{sparsification ratio}, \\emph{i.e.}, the ratio of the size of a sparsifier $H$ and the size of the full graph $G$, for varied values of $k$. It can be seen that our method consistently reduces the number of edges in the sparsifiers. The reduction becomes small with an increase in the value of $k$ since more edges are kept in the sparsifiers then. In general, the empirical performance matches our theory that the constructed sparsifiers have size $O(ktn)$.\n\nFigure \\ref{fig:results} plots performance of link prediction and node classification when applying different representation learning algorithms in the sparsified graphs and the full graph\\footnote{HAN reported out-of-memory errors in \\emph{Yelp} and \\emph{Freebase} and \\emph{AspEm} did not complete within 2 weeks in \\emph{Freebase}.}. The training times for the sparsifiers in the \\emph{PubMed} dataset are significantly improved to lower than $70\\%$ of the training time required in the full graph except for \\emph{AspEm}, as shown in Figure \\ref{fig:time-pubmed}. The training times for \\emph{HIN2Vec} even drop to lower than $44\\%$ for all values of $k$. Interestingly, the reductions in training times are \\emph{larger} than the sparsification ratio in Table \\ref{tab:sparsified}, \\emph{e.g.}, $85\\%$ when $k=10$. It means that the reductions in computational time are remarkably larger than those in the graph size. The reductions in larger graphs such as \\emph{Yelp} are even greater and drop to lower than $60\\%$ of the original time. The running time of \\emph{ComplEx} are consistently lower than $40\\%$ while maintaining a good performance as will be shown later. The improvement in memory consumption during model training is not that obvious as the training time in Figure \\ref{fig:ram-pubmed}. This may be due to that the additional overhead in the training excluding loading the graph data is comparable to the size of \\emph{PubMed}. But the improvement of memory consumption in larger graphs \\emph{Yelp} and \\emph{Freebase} become much clearer as in Figures \\ref{fig:ram-yelp} and \\ref{fig:ram-freebase} and can be lower than $70\\%$ of the original consumption.\n\nImportantly, as in Figures \\ref{fig:auc-pubmed}, \\ref{fig:auc-yelp}, and \\ref{fig:auc-freebase}, the performance \\emph{AUC} and \\emph{MRR} of link prediction are not significantly affected by the sparsification, especially for large values of $k$. Interestingly, the performance of \\emph{ComplEx} and \\emph{HIN2Vec} for quite a few settings can be even \\emph{better} than that based on the full graph. We analyze that it could be because noises in the original graph are also removed during the sparsifying process. Among the tested embedding algorithms, \\emph{ComplEx} consistently achieved the highest \\emph{AUC} and \\emph{MRR} values in different datasets. In Figures \\ref{fig:f1-pubmed}, \\ref{fig:f1-yelp}, and \\ref{fig:f1-freebase}, the \\emph{F1} values are generally comparable to the values in the full graph, and even better in some settings. Furthermore, graph (edge) sparsification appears to have more impacts on the performance of link prediction than on node classification. This seems to suggest that the task of node classification depends less on the edge connectivity compared to link prediction.\n\n\nDue to superior performance of \\emph{ComplEx}, we choose it as the default embedding model for comparing Alg. \\ref{alg:graph} and \\emph{ALT}. Figure \\ref{fig:alt} plots the performance of \\emph{ComplEx} based on the two types of sparsified graphs with respect to different sparsification ratios. It can be seen that \\emph{AUC} and \\emph{MRR} values of Alg. \\ref{alg:graph} are slightly better than those of \\emph{ALT}, while both \\emph{F1} values of \\emph{ALT} are larger by a small margin. In practice, users may select a sparsification method that is most suitable for their target application.\n\n\n\n\n\n{\\noindent \\bf Additional Experiments.}\nWe first vary the size of the learned embedding from $10$ to $200$ ($[10, 25, 50, 75, 100, 200]$) in \\emph{PubMed}. We observe that improvements in the training time and memory consumption are always significant, similar to the baseline setting. As expected, with an increase in the embedding size, the \\emph{AUC} and \\emph{MRR} are gradually improved due to a better learned embedding, as in Figures \\ref{fig:HAN}, \\ref{fig:HIN2Vec} and \\ref{fig:ComplEx}. For \\emph{HIN2Vec}, multiple sparsifies achieve higher \\emph{AUC}\/\\emph{MRR} than the full graph, those red lines in Figure \\ref{fig:HIN2Vec}. For \\emph{HAN} and \\emph{ComplEx}, the performance of the sparsifies with larger values of $k$, \\emph{e.g.}, $k=10$, are comparable to that of the full graph as shown in Figures \\ref{fig:HAN} and \\ref{fig:ComplEx}.\n\nFurthermore, since \\emph{HAN} and \\emph{HGT} can incorporate node attributes into the embedding learning, we perform attributed model training by taking node attributes into consideration. The training times and memory consumption in the attributed setting become slightly larger due to the processing of more information. But they are still much lower than training times and memory consumption in the full graph, as shown in Figures \\ref{fig:att-han-tr} and \\ref{fig:att-hgt-tr}. For \\emph{HAN}, the accuracy performance in attributed graphs (red and black lines) are better than in unattributed graphs, especially for \\emph{F1} values as shown in Figure \\ref{fig:att-han}. \\emph{HGT} appears to be strong in exploiting graph connectvitiy to generate accurate prediction even without the attribute information, as in Figure \\ref{fig:att-hgt}. Consistent with the baseline setting, sparsification always brings great benefits in reducing computational resources while producing good accuracy performance.\n\n\n\\section{Conclusion and Future Work}\nIn this work, we study the problem of sparsifying heterogeneous graphs while well approximating the original graphs for subsequent graph learning tasks. We develop an efficient sampling method which takes the distribution of different types of edges into consideration and perform computational and storage complexity analysis. Extensive experiments validate the efficiencies of our proposed algorithms while attaining promising performance in downstream graph learning tasks. As the future work, we will study sophisticated algorithms for sparsifying heterogeneous graphs, \\emph{e.g.}, by sampling edges according to edge (betweenness) centrality measures and keeping only important edges with high centrality. It is also an interesting direction to develop deep learning models to perform end-to-end heterogeneous graph sparsification and graph learning tasks.\n\n\n\\section*{Acknowledgment}\n\nWe thank graduate students Chandra Malgari and Divya Sanga for their help in the experiments and anonymous reviewers for their constructive comments.\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nElectron pumps are devices that aim to generate a reference DC electric current by moving electrons one at a time in response to a periodic control signal at frequency $f$. They potentially offer a simple and elegant traceability route for small currents, direct to the SI definition of the ampere\\cite{kaneko2016review,scherer2019single}. A class of pumps fabricated from semiconductor materials\\cite{kaestner2015non} has demonstrated accurate and robust current generation at roughly the part-per-million (ppm) accuracy level, for currents $I_{\\text{P}}$ up to $160$~pA \\cite{giblin2019evidence}. However, important questions must be answered before electron pumps can confidently be adopted as reference current standards at the uncertainty levels of primary electrical metrology. Most significantly, the robustness and device independence of the current needs to be demonstrated at at least the $0.1$~ppm level, over a range of device designs and operating parameters. To date, two studies have focused on the robustness of the current from GaAs pumps, at current levels of $\\sim 100$~pA, at uncertainty levels for each data point of $\\sim 2$~ppm\\cite{giblin2017robust} and $\\sim 0.5$~ppm\\cite{stein2016robustness}. \n\nHowever, blind measurement techniques which have been implemented in other metrology areas to remove bias \\cite{schlamminger2015summary} have not yet been applied to the study of electron pumps where the pump current is treated as an unknown and compared to a known reference current. Addressing unconscious experimenter bias is particularly important in experiments where the expectation of the result is strongly constrained; in this case, we expect $I_{\\text{P}} = ef$, and there is a possibility that in a non-blind measurement, the experimenter may unconsciously favour pump control parameters that yield this result. Particularly important in the electron pump context is the lack of reproducibility in attempts to realise a capacitance standard based on pumping a known number of electrons onto a cryogenic capacitor \\cite{keller1999capacitance,scherer2017electron}. The authors of Ref. \\onlinecite{scherer2017electron} were unable to reproduce the results of Ref. \\onlinecite{keller1999capacitance}, and identified components in the capacitance measurement uncertainty which had previously been under-estimated.\n\nEvaluating the robustness of the pump current presents a challenge due to the time-scales involved: the small currents require many hours of averaging to resolve $0.1$~ppm for a single data point, and the time-scale of the whole measurement campaign challenges the stability of the measurement system and the electron pump itself\\cite{giblin2019evidence}. To reduce the measurement time, or equivalently, to allow more data points to be measured within the timescale of a measurement campaign, the pump current should be increased as much as possible. Custom gate drive waveforms which slow down the electron capture process have been used to operate GaAs pumps accurately at much higher frequencies than were possible with sine wave drive \\cite{giblin2012towards,stein2015validation,stein2016robustness}. With these pumps the upper frequency limit for accurate pumping was $f \\sim 1$~GHz even with the custom waveforms. Silicon pumps, on the other hand, have demonstrated accurate pumping at $f=1$~GHz with sine wave drive \\cite{zhao2017thermal,giblin2020realisation}, and the possibility of increasing the frequency further while maintaining sub-ppm pumping accuracy using custom waveforms has not yet been explored. \n\n\\begin{figure}\n\\includegraphics[width=9cm]{AWGGraph}\n\\caption{\\label{AWGGraphFig}\\textsf{(a): Grey-scale derivative pump map using sine wave drive at 2.062 GHz, $P_{\\text{RF}}=13.2$~dBm. (b): Pump map using a waveform from an AWG at repetition rate $f=2$~GHz. One cycle of the AWG waveform is shown in the inset. Note that this waveform is subsequently amplified by an inverting amplifier to yield the correct polarity of gate voltage, whereby the negative voltage pulse on the entrance gate raises the entrance barrier to pump an electron. (c): Log-scale plots of the pump current along the horizontal dashed lines in plots (a) and (b).}}\n\\end{figure}\n\n\\section{Experimental method and blind protocol}\n\nWe investigate a single well-characterised sample of silicon pump which has previously been the subject of two precision measurement campaigns\\cite{yamahata2016gigahertz,giblin2020realisation}. The pump is a silicon nanowire-MOSFET, in which charge carriers are induced by a positive voltage applied to a global top gate \\cite{fujiwara2008nanoampere,yamahata2016gigahertz} which was set to $4$~V for all the measurements. Two finger gates, denoted the entrance gate and exit gate, define the region of the nanowire where a single electron can be trapped. Negative DC voltages $V_{\\text{ENT}}$ and $V_{\\text{EXIT}}$ applied to these gates define the pump operating point, and the periodic pump drive signal is added to $V_{\\text{ENT}}$ using a room-temperature bias-tee. A $50$~Giga samples\/s arbitrary waveform generator (AWG, Tektronix 70001A) was used to generate a custom waveform for the pump drive. Because the AWG output had a maximum peak-peak amplitude of $V_{\\text{AC}} = 0.5$~V, the output was amplified by a wide-band inverting RF amplifier with $+15$~dB gain before the bias-tee. The AWG is referenced to a $10$~MHz frequency reference derived from a hydrogen maser.\n\nFigure \\ref{AWGGraphFig} shows characterisation data using both sine wave drive and the custom AWG waveform at a repetition frequency of $2$~GHz. It is clear from the log-scale plots of figure 1 (c) that there is a substantial plateau along the exit gate axis when using the AWG drive waveform, but not when using sine wave drive. The inset to figure 1(b) shows the AWG waveform used for all the measurements reported in this paper. Characterisation data at other frequencies is inlcuded in supplementary sections A and B.\n\nThe experimental apparatus and methods used for this study are in many respects identical to that used in Ref. \\onlinecite{giblin2020realisation}. As in those experiments, the pump is cooled to a temperature close to 4 K by suspending it above a liquid helium surface. The pump current $I_{\\text{P}}$ is measured using a noise-optimised ultrastable low-noise current amplifier (ULCA) \\cite{krause2019noise}, with a precision digital voltmeter (DVM) recording the ULCA output. As in Ref. \\onlinecite{giblin2020realisation}, the DVM was calibrated roughly once every hour by switching its input to a Josephson voltage standard (JVS). A single precision measurement typically lasted between 8 and 10 hours and included between 7 and 11 voltmeter calibrations. To remove offset drifts in the measurement system during precision measurements, the pump drive signal was toggled on and off with a cycle time of $228$~s. Roughly the first $34$ seconds of each data segment (300 out of 1000 data points) following each on or off switch was rejected from the analysis to remove transient effects. More details of the measurement protocol are given in supplementary section C. \n\nThe pump current $I_{\\text{P}}$ is calculated from the on-off difference in the DVM voltages $\\Delta V$ using the equation $I_{\\text{P}} = \\Delta V \/ A_{\\text{TR}}$. Here, $A_{\\text{TR}}$ is the trans-resistance gain of the ULCA, nominally equal to $10^9$~V\/A. This gain is calibrated against the quantum Hall resistance (QHR) in 2 stages\\cite{drung2015ultrastable} and via some intermediate transfer standards, using a cryogenic current comparator (CCC) with relative uncertainty less than $0.1$~ppm \\cite{giblin2019interlaboratory}. Detailed calibration results are reported in supplementary section E. The measurement of the pump current was therefore traceable to the SI unit ampere via the JVS, the QHR, and the relationship $I=V\/R$. For characterisation measurements such as those reported in figures 1, 2a, and small filled points in figures 2b and 2c, no offset subtraction was performed: the pump drive signal was left on, and each data point is a single $20$ power line cycle DVM measurement. \n\nA blind protocol was implemented so that the lead experimenter could not see the true value of $I_{\\text{P}}$ while the measurements and data analysis were in progress. This is achieved by multiplying all the DVM readings by a hidden scaling factor $\\beta=1.00000387$, at a low level in the measurement software. An exception occurs when when the DVM is connected to the JVS for calibration, in which case $\\beta =1$. While tuning the pump and performing measurements, the experimenter does not know the scaling factor and can only access the scaled pump current $I_{\\text{P,B}} = \\beta \\Delta V \/ A_{\\text{TR}}$. Therefore, the tuning of the pump operating parameters and the choice of parameters for the precision measurements can only be made with reference to the flatness of the current plateau, not the deviation of the current from $ef$. The scaling factor was programmed and password-protected by a member of the team who was not otherwise involved in the experiments. The experimenter knew that it was constrained such that $|1 - \\beta| < 5 \\times 10^{-6}$ so that gross failures of the pump or apparatus would be apparent during characterization measurements. The scaling factor was revealed after the experiments were finished and data analysis, including analysis of the ULCA calibrations, completed.\n\n\\section{precision measurement campaign}\n\nThe aim of the measurements was to study the pump current as a function of control parameters $V_{\\text{ENT}}$, $V_{\\text{EXIT}}$ and $V_{\\text{AC}}$. To this end, a total of 67 precision measurements were made during a 7-week campaign, employing the apparatus and blind protocol described in section II. The measurements were divided into 17 `runs'. For most of the runs, several measurements were made while varying one control parameter. Runs 11-13 consisted of single measurements without varying a parameter. Further detail of the measurement chronology is given in supplementary section D. To monitor the stability of the pump, a `fingerprint' pump map was obtained before and after each run, apart from a few occasions when it was prevented by an experimental difficulty. For completeness, all of these pump maps are shown in supplementary section H. Additional line scans of current as a function of one or more control parameters were also measured to assess the optimal values of fixed control parameters for the next precision run. Typically, these scans were used to find the value of the control parameter that maximised the plateau width. They used a single $20$~PLC measurement for each data point, with a relative uncertainty of approximately $10$~ppm per data point. A pass \/ fail stationary mean statistical test, described in supplementary section G, was applied at the data analysis stage to each precision measurement to evaluate whether the current was stable during the measurement time. \n\n\\begin{figure}\n\\includegraphics[width=9cm]{PumpMapandLogPlots}\n\\caption{\\label{LogPlotFig}\\textsf{(a): Derivative pump map measured after precision run 4 and before precision run 5. (b) and (c): Line-scans of the pump current measured along the gate voltage axes indicated by solid colored lines in (a), and plotted on a logarithmic scale. The vertical dashed lines indicate the fixed value of entrance (exit) gate voltage used for the exit (entrance) gate scan. The diagonal dashed lines are guides to the eye extrapolating the exponential edges of the plateau. Larger filled points are the precision measurement data for runs 1-5, with the run number indicated in the plot legend. Data points indicated with a star ($*$) failed the stationary-mean test.}}\n\\end{figure}\n\n\\section{Results of precision measurements}\n\n\\subsection{Precision results}\n\nAfter run 5, the pump became less stable, (supplementary section H), making it difficult to establish the flatness of plateaus along $V_{\\text{ENT}}$ and $V_{\\text{EXIT}}$ axes. For this reason, we concentrate here on the data from runs 1-5, although the full precision data set is presented in supplementary figure S9. In figure \\ref{LogPlotFig}, we present the data from the first 5 precision runs. Panel (a) shows a pump map recorded between runs 4 and 5, and panels (b) and (c) show line-scans on a log scale which highlight the deviation of the current from the ideal value on the $1ef$ plateau. The fixed value of $V_{\\text{ENT}}$ ($V_{\\text{EXIT}}$) for the $V_{\\text{EXIT}}$ ($V_{\\text{ENT}}$) line-scan was adjusted in order to maximise the width of the plateau in the log-scale plot. The results of precision runs 1-5 are plotted as solid points in figures \\ref{LogPlotFig} (b) and (c). Runs 1-4 were $V_{\\text{EXIT}}$ scans, plotted in figure \\ref{LogPlotFig} (c), and run 5 was a $V_{\\text{ENT}}$ scan, plotted in figure \\ref{LogPlotFig} (b). The 18 data points along the $V_{\\text{EXIT}}$ axis (figure \\ref{LogPlotFig} (c)) define a plateau in agreement with an extrapolation of the standard-accuracy measurement. The precision data point marked with a star ($*$), failed the stationary-mean test, presumably because it was close to the edge of the plateau, and small fluctuations in offset charge, equivalent to shifts in $V_{\\text{EXIT}}$, caused fluctuations in the pumped current to be resolved on the time-scale of the precision measurement.\n\nThe precision $I_{\\text{P}}(V_{\\text{EXIT}})$ data for runs 1-4 (apart from the point that failed the stationary mean test) are re-plotted on a linear y-axis in figure 3a as $\\Delta I_{\\text{P}} = (I_{\\text{P}} - ef)\/ef$. The mean of these 17 points is $\\Delta I_{\\text{P}} = 0.22$~ppm, with a standard deviation $\\sigma$ of $0.14$~ppm. The individual data points have a mean combined uncertainty $\\langle U_{\\text~{T}}\\rangle$ of $0.102$~ppm, although the uncorrelated random uncertainty, $U_{\\text{A}}$, for each data point is smaller, in the range $0.08-0.09$~ppm. The scatter of the points is therefore slightly larger than what would be expected from the type A uncertainty of each point ($\\sigma > \\langle U_{\\text~{A}}\\rangle$), although not statistically incompatible with the assumption that the data is sampling a stationary mean along the plateau. We can therefore conclude that this data is consistent with a plateau along the $V_{\\text{EXIT}}$ axis, flat at the $0.1$~ppm level, but significantly offset from $ef$ by $0.22$~ppm. Figure 4 (b) shows the same data re-analysed with the first 700 data points rejected from the beginning of each 1000-point data segment instead of the standard 300. This was to test for the presence of a time constant in the current, as discussed in section V.\n\n\\begin{figure}\n\\includegraphics[width=9cm]{LinearHighAccGraph}\n\\caption{\\label{HighAccFig}\\textsf{Results of precision measurements for runs 1-4 as a function of $V_{\\text{EXIT}}$, expressed as $\\Delta I_{\\text{P}} = (I_{\\text{P}} - ef)\/ef$. The data have been analysed with (a): 300 and (b): 700 data points rejected from the start of each 1000-point data segment. Error bars show the combined standard uncertainty $U_{\\text{T}}$ The horizontal dashed lines show the weighted means of each data set. Arrows highlight run 1, measurement 4 and run 4, measurement 4. A breakdown of the uncertainty for these measurements is given in table I.}}\n\\end{figure}\n\nOnly one precision run was performed along the $V_{\\text{ENT}}$ axis before the interruption, illustrated by the heavy filled points in figure \\ref{LogPlotFig} (b). One data point, marked with a $*$, failed the stationary-mean test. It is not clear from this single run whether this data point indicates real structure to the plateau at level of $\\sim 5$~ppm, or if it is the result of a drift in the device state. The remaining 4 data points mark a plateau region which, combined with the stability of the pump map from runs 1-5, gives confidence that the fixed value of of $V_{\\text{ENT}}$ selected for runs 1-4 is in the middle of an experimentally-determined plateau. The mean of the 4 measurements from run 5 is $\\Delta I_{\\text{P}} = 0.15$~ppm, with a standard deviation of $0.09$~ppm. This is consistent with the deviation measured in runs 1-4 given the much smaller sample size.\n\n\\subsection{Uncertainty}\n\nIn table I, the breakdown of the uncertainty is given for two measurements indicated by arrows in figure 3a. The uncertainties due to the two stages of the ULCA calibration are presented as separate components, with the uncertainty due to the drift of the ULCA gains in between calibrations included in these two terms. This was significantly reduced by performing frequent ULCA calibrations, with more detail given in supplementary sections D and E. Run 1, measurement 4 is a typical representative measurement, and run 4, measurement 4 had the lowest combined uncertainty of the campaign. As in previous measurement campaigns, the type A uncertainty of the pump measurement is the largest single contribution, but the larger pump current achieved in this study has reduced $U_{\\text{A}}$ to below $0.1$~ppm and the uncertainty in the ULCA calibration is now a significant contribution. Specifically, the uncertainty in the output stage gain $R_{\\text{IV}}$ (nominal value $1$~M$\\Omega$) is limited by the $0.04$~ppm uncertainty in the $100$~k$\\Omega$ reference resistor traceable to the QHR via a chain of 3 intermediate measurements \\cite{giblin2018reevaluation,giblin2019interlaboratory}.\n\n\\begin{table}\n\\caption{\\label{UncertTable} Uncertainty breakdown for run 1, measurement 4, and run 4, measurement 4. All entries in the table are dimensionless relative uncertainties ($k=1$) in parts per million.}\n\\centering\n\\setlength{\\tabcolsep}{8pt}\n\\begin{tabular}{c c c}\nContribution & Meas. 1.4 & Meas. 4.4 \\\\\n\\hline\\hline\nULCA $G_{\\text{I}}$ Cal. & 0.024 & 0.024 \\\\ \nULCA $R_{\\text{IV}}$ Cal. & 0.062 & 0.043 \\\\ \nULCA Temp. corr. & 0.023 & 0.023 \\\\ \nDVM Cal. & 0.014 & 0.016 \\\\\nPump $U_{\\text{A}}$  & 0.088 & 0.061 \\\\\n\\hline\nTotal $U_{\\text{T}}$& 0.111 & 0.084 \\\\\n\\end{tabular}\n\\end{table}\n\n\\subsection{stability of the pump}\n\nThe measurement campaign was divided into two parts by an instrument issue which forced a period of 8 days' down-time between runs 5 and 6. During this time, the pump was thermally cycled to room temperature and back to $4$~K twice. From examination of the pump maps in supplementary section H, it is clear that the pump became less stable after this interruption, although even before the interruption, small changes in the `nose' (the onset of pumped current as $V_{\\text{EXIT}}$ is made less negative) of the pump map are visible. This contrasts with the data of Ref. \\onlinecite{giblin2020realisation} showing this sample of pump to be extremely stable over multiple cool-downs in different laboratories, when driven with a sine wave at $\\sim 1$~GHz. We conjecture that at least some of the changes visible in the pump maps during the present campaign may be due to changes in the transmission of the cryogenic microwave line at frequencies $\\gg 1$~GHz. This could plausibly arise due to changes in the temperature gradient along the line, and would affect the high frequency components of the AWG waveform, causing distortion of the waveform at the pump entrance gate. \n\n\n\\section{discussion}\n\nThe study was complicated by instability in the pump map, which made it difficult in the later parts of the measurement campaign to interpret the results as sampling a stable state of the device. However, enough results were obtained from runs 1-5 to establish that the pump current is invariant in the exit gate voltage at the level of $1$ part in $10^{7}$. Averages over these data points presented in the previous section give $\\Delta I_{\\text{P}} \\sim 2 \\times 10^{-7}$, a significant offset from the ideal current $I_{\\text{P}} = ef$. The flatness of the plateau suggests that the offset is due to an error in the measurement system which applies a constant offset to all the measurements, rather than an error due to the physics of the pump itself.\n\nOne possible cause of error is a time constant in the pump current. This was discussed in Ref. \\onlinecite{giblin2020realisation}, and could plausibly arise from heating due to the relatively large RF powers applied to the device gate. Repeating the data analysis of runs 1-4 with 700 data points rejected from the start of each segment instead of 300 did indeed yield an average pump current closer to $ef$, as shown in figure 3b. However, the larger type A uncertainties in this analysis make it difficult to draw a firm conclusion regarding possible time constants. Measurements with much longer on-off cycle times could potentially resolve this question, but require the $1\/f$ noise corner of the ULCA current measurement to be at frequencies well below $1$~mHz. ULCA units have demonstrated this performance in bench tests \\cite{krause2019noise}, but the cryogenic wiring involved in a pump measurement introduces additional sources of noise and drift. Another possible cause of error is a non-linearity in the gain of the ULCA. The ULCA input stage gain $G_{\\text{I}}$ is calibrated at a current of $6$~nA, and the pump current is $320$~pA. Comparisons of the gains of two ULCA units with different input stage designs, detailed in supplementary section F, set an upper limit to possible non-linearity of a few parts in $10^8$, so this is unlikely to cause errors of a part in $10^{7}$. \n\nPossibly the most important cause of error could arise from the CCC calibration of the ULCA $G_{\\text{I}}$. This could result from rectification of noise by the CCC's SQUID detector leading to different SQUID offsets for the two polarities of current used in the ULCA calibration \\cite{drung2014ultrastable}. One study on CCC performance in the low-flux regime\\cite{drung2015improving} concluded that noise pickup might lead to this type of error at SQUID flux levels below $1$~$\\mu \\Phi_{0}$, although this number was based on a limited number of measurements and is specific to a particular CCC design\\cite{goetz2014compact}, different in detail to the CCC used to calibrate the ULCA in our experiments. We calibrated the ULCA $G_{\\text{I}}$ using a CCC\\cite{giblin2019interlaboratory} with a $10000:10$ turns ratio, and a sensitivity of $6$~$\\mu$A~turns$\/\\Phi_{0}$. The current in the large winding was approximately $\\pm 5$~nA, giving a full-signal ampere-turns product of $100$~$\\mu$A~turns, corresponding to a flux of $16.7$~$\\Phi_{0}$. A flux of $1$~$\\mu \\Phi_{0}$ therefore corresponds to $0.06$~ppm of the full signal in the ULCA $G_{\\text{I}}$ calibration, three times smaller than the observed discrepancy in the electron pump current. However, no investigations have yet been carried out on the performance of our CCC in the low-flux regime, so the size of possible noise-rectification errors is not known. Low flux ratio accuracy tests such as those presented in Ref. \\onlinecite{drung2015improving} should provide useful information on the scale of possible errors. We note that if these errors are affecting the ULCA calibrations in our experiment, they are remarkably constant in time, as shown by the $\\sim 5 \\times 10^{-8}$ relative stability of the ULCA input gain over the duration of the measurement campaign illustrated in supplementary section E. This indicates that if noise is affecting the SQUID, its most likely source is the CCC bridge electronics, rather than external sources.\n\nThe upper frequency limit for accurate pumping with tunable-barrier pumps has previously been empirically established at around $1$~GHz \\cite{giblin2019evidence}. We have shown that this can be increased, albeit in a rather exceptional sample of pump. In this study, the practical upper frequency limit was determined by a combination of plateau rounding, and increased incidence of switching events which shifted the pump operating point in the $V_{\\text{ENT}}-V_{\\text{EXIT}}$ plane. This hints at device-physics factors which may limit the practical upper operation frequency, possibly charge traps which are activated by high frequency components present in the drive signal. Further investigation of more samples of pump could shed fruitful light on this question.\n\n\\section{conclusions}\n\nPrecision measurements have been made of the current from a silicon electron pump driven at a frequency of $2$~GHz using a custom drive waveform applied to the entrance gate. The pump current is invariant in exit gate voltage with a precision of $0.1$~ppm (32 aA), but offset by roughly $0.2$~ppm from the expected current corresponding to one electron for each pump cycle. The application of a blind measurement protocol provides added confidence that this result is not affected by experimenter bias. At this accuracy level, the measurement of the pump current challenges the state of the art in existing electrical metrology methods, with scaling of small currents using CCCs at low flux levels posing a particularly interesting problem. The recent demonstration of current plateaus due to the dual Josephson effect \\cite{shaikhaidarov2022quantized} raises the possibility of a metrological investigation of the dual Josephson effect in the near future, providing added motivation for a better understanding of low current scaling.\n\n\\begin{acknowledgments}\nThe authors would like to thank Colin Porter and Scott Wilkins for making the NPL primary Josephson voltage standard available, and for assistance with setting up the voltmeter calibration. This research was supported by the UK department for Business, Energy and Industrial Strategy. A.F. and G.Y. are supported by JSPS KAKENHI Grant Number JP18H05258.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Motivation}\n\n\\subsubsection{Centralized User\/Distributed (Hybrid) Model}\t\t\\label{motivation:hybrid:sec}\nIEEE 802.1 Time Sensitive Networking (TSN) provides a standardized\nframework of tools for providing deterministic ultra-low latency\n(ULL), e.g., for industrial control applications, automotive\nnetworking, and avionics communication\nsystems~\\cite{finn2018introduction,nas2019ult}. In particular, the\nIEEE 802.1Qbv Time Aware Shaper (TAS) has received extensive attention\nas a key tool for achieving deterministic ULL network service. The TAS\noperation requires careful planning of the synchronized time cycles\nand the gate times that are allocated to the scheduled traffic (ST)\nand the unscheduled best effort traffic (BE). The TAS parameter\nsettings specifying the timing characteristics (cycle time, gate slot\nallocations) are also commonly referred to as the Qbv schedule or the\nTAS schedule. For a given static networking scenario, the TAS\noperation with a properly configured Qbv schedule can ensure the\ndeterministic ULL required by demanding industrial and automotive\napplications~\\cite{nas2019per}.\n\nModern network scenarios often involve dynamic changes with varied use\ncases, such as changes in the network nodes and network topology, or\nthe traffic pattern. For instance, nodes or links may be dynamically\nadded or removed. Or, nodes may inject additional traffic flows or\ntraffic flows may terminate, or the latency requirements of flows may\nchange dynamically. Such dynamic changes have been included in the use\ncases defined by IEC\/IEEE 802.1 TSN TG~\\cite{usecas2018}.\n\nIn a typical industrial environment, sensors that periodically or\nsometimes sporadically send ambient measurements to a local gateway\nrequire certain Quality of Service (QoS) guarantees. In such a\nvolatile and dynamic environment, new machinery that requires\nprioritized execution (e.g., emergency cooling procedures or\nmaintenance tasks for network traffic tests) may be brought onto the\nfactory floor. To deal with such scenarios, the Time-Aware Shaper\n(TAS) Gate Control Lists (GCLs) in coordination with the Network\nManagement Entities (NMEs), e.g., Centralized Network Configuration\n(CNC), has to adapt to changing environment conditions by judiciously\napplying reconfiguration such that stream deadlines, QoS, and total\nstream utilization times (reported by a stream registration procedure)\nare satisfied.\n\nSimilarly, in a disaster management networking scenario, a network may\nhave physically lost large amounts of network resources (links and\nroutes) but needs to manage the flows that are currently present. A\ncentral orchestrator can prudently time share the reserved resources such\nthat a large number of streams can be completely serviced, i.e., the total\nexecution time for each stream is completed by the network.\n\nGenerally, in such dynamic networking scenarios, applying only\nadmission control will clearly guarantee (in accordance with a traffic\nshaper) the QoS metrics (of the admitted flows).  However, for a given\nstatic network configuration, the total number of admissible streams\nmay be well below the number of streams that seek network\nservice. Therefore, adding a dynamic reconfiguration strategy to\nmanage and configure the network appears to be a plausible and\nattractive solution that intuitively should lower capital and\noperational expenditures as it mitigates the over-provisioning of\nnetwork resources. The general idea for such an allocation scheme is\nto control network access in a timely and orderly fashion such that a\nmaximum number of streams can be effectively serviced.\n\nOur objective therefore is to maximize the number of admitted flows\n(i.e., tasks or streams) in such a dynamically changing and volatile\nenvironment whilst keeping the TSN QoS metric guarantees.  To the\nbest of our knowledge there are no prior detailed studies on a\ndynamic stream resource allocation and admission control policy in\nconjunction with a network reconfiguration policy being executed while\nflows are carried in a TAS time scheduled network. In this paper, we\nfocus on the IEEE 802.1Qbv~\\cite{IEEE8021Qbv} enhancements and design\na reconfiguration framework taking inspiration from the IEEE\n802.1Qcc~\\cite{IEEE8021Qcc} standards for managing, configuring, and\nreconfiguring a TSN network.\n\n\\subsubsection{Decentralized Model}\nThe IEEE 802.1Qcc standard specifies three models for configuring the\nTime-Aware Shaper (TAS) gating schedules (GCL\/GCE timing computation):\na fully-centralized model, a centralized network\/distributed user\nmodel (hybrid model, see Section~\\ref{motivation:hybrid:sec}), and a\nfully-distributed configuration model. The centralized model greatly\neases control and configuration messages sent across the network and\ncan precisely configure TAS schedules due to having complete knowledge\nof the network and full capabilities of each bridge. However the\ncentralized model suffers from common disadvantages, such as a\nsingle-point of failure, relatively large capital\/operational\n(CapEx\/OpEx) expenditures (as the centralized control may be\nsuperfluous in a small-scale network~\\cite{chen2017rap}), and adding\nunnecessary complexity to a small-scale network. Thus, a\nfully-distributed configuration model (e.g., SRP over MRP or RAP over\nLRP) may be attractive for some networks. The fully-distributed\nconfiguration model avoids the added complexity and single point of\nfailure of a centralized management entity.  Moreover, Chen et\nal.~\\cite{chen2017rap} have argued that the centralized configuration\nmodels can be an over-design for real-time applications with relaxed\nlatency requirement (order of magnitude of milliseconds). Chen et\nal. have also argued that the distributed model is more\nscalable. (However, studies of the fully distributed model with RAP\nover LRP targeted typically applications with relatively relaxed\nlatency requirements.)\n\nIn the absence of a Centralized Network Configuration (CNC) node, the\nTSN Task Group (TG) specifies the IEEE 802.1CS (Link-Local\nRegistration Protocol, LRP)~\\cite{IEEE8021CS} standard for\nregistration and distribution of application configuration parameters\nbetween point-to-point links targeting newly published TSN features. A\nlegacy protocol, such as the Stream Reservation Protocol\n(SRP)~\\cite{IEEE8021Qat} which is primarily used for AVB application,\nis intended to serve as the main resource reservation and admission\ncontrol protocol. However, extending and porting the SRP to be\nutilized for bridges that support TAS will not suffice since bandwidth\nreservation cannot directly apply TAS's time slot reservation\nnatively. Therefore, the Resource Allocation Protocol, IEEE 802.1Qdd\n(RAP)~\\cite{chen2017rap}, has been proposed to apply a distributed\nresource reservation that can exchange TSN features.\n\nIn our model, the switch computes the TAS time slot for all admitted\nstreams as follows. In the absence of admission control, we predefine\nthe TAS slot to be a minimum of 10\\% and a maximum of 90\\% of the\nCycle Time (CT), even when no streams are registered, so as to avoid\nstarvation of Best Effort (BE) traffic. With admission control, the\nstatic predefinition, which can potentially waste resources if unused,\ncan be eliminated. Essentially, as streams get registered, we keep\ntrack of the remaining load on each egress port until the load (which\ndepends on the slot size and CT) is negative (oversubscribed link).\nSuch a link over-subscription invokes a procedure call that increases\nthe slot time (by a step size of 1\\%, or more fine-grained increments)\nuntil the remaining load is positive. This procedure is iteratively\ncalled until all registered streams and the new stream are\nappropriately registered with sufficient Scheduled Traffic (ST) slot\ntime to transmit all frames during a single appropriately sized\nCT. Note that the TAS time slot is defined as the portion of the CT\nthat is allocated to high-priority ST traffic.\n\nOur proposed TAS configuration\/reconfiguration, is designed for the\nfully-distributed configuration model. In the distributed approach,\nthe GCE slot parameters are configured in a distributed manner by the\nswitches as per the distributed algorithm\/procedure explained in\nSec.~\\ref{tas:reconfig:prot}. In the centralized approach, the GCE\nslot parameters are configured centrally by the CNC with the\nCentralized User Configuration (CUC) node assisting in passing\nend-station (source\/sink, devices, etc.) capabilities and\nparameters. Similarly, the ``hybrid'' model also utilizes the CNC for\nconfiguration exchange and network side management (see\nSection~\\ref{motivation:hybrid:sec}).\n\nRegarding the differences between the hybrid model and the fully\ncentralized model, the main network-side difference is the way the\nUser\/Network configuration Information (UNI) is propagated. In the\nfully centralized model, the sources communicate Control Data Traffic\n(CDT) messages to\/from the CUC node, instead of having each source\ninteract directly with the CNC. According to the 802.1Qcc standard,\nthe general advantage is that the computation complexity (especially\nfor industrial\/automotive applications with computationally complex\nI\/O timing requiring detailed knowledge of the application's\nsoftware\/hardware within each end station) can be tolerated by having\nthe CUC handle end station discovery, retrieving end station\ncapabilities and user requirements, and configuring TSN features in\nend stations. The CNC is used to only manage and configure network\nside components. In terms of benefits, the configuration delay is\npotentially reduced, scheduling optimality increased, and hardware\nperformance overhead\/complexity reduced with a shorter network\nresponse time for networks involving command and control systems. The\ndevelopment and investigation of a reconfiguration approach for the\nfully centralized configuration model is left as future work.\n\nOur study focuses on the centralized network\/distributed user model\n(hybrid model) and the fully-distributed (decentralized) configuration\nmodel.  For brevity we refer to the centralized network\/distributed\nuser model (hybrid model) also as the centralized model or the\ncentralized topology.  We refer to the fully-distributed\n(decentralized) model also as the decentralized model or the\ndecentralized topology.\n\nOur proposed TAS reconfiguration architecture maps and propagates\nstream information and conducts dynamic TAS time slot reservations\n(including GCL\/GCE scheduling and provisioning) on local shared stream\ndatabase records to guarantee TSN QoS for all admitted streams and to\nmaximize stream admission using a reconfiguration strategy of TAS\ngating schedules within each bridge in a fully decentralized IEEE\n802.1Qcc model.\n\n\\subsection{Related Work} \\label{tsn:rel:sec}\nRaagaard et al.~\\cite{raagaard2017fog} presents a heuristic algorithm\nthat reconfigures TAS switches according to runtime network\nconditions. Feasible schedules are produced and forwarded using a\nconfiguration agent (composed of a Centralized User Configuration\n(CUC) and Centralized Network Configuration (CNC)). Raagaard et al's\nmodel places emphasis on appearing and disappearing synthetic flows in\na fog computing platform that takes into account the flow's properties\nand possible routes. Contrary to this approach our framework performs\nflow maximization with optimal reconfiguration based on firm bandwidth\ncomputation strategies at run-time. Further, we show the equilibrium\npoint for this algorithm.\n\nFurther related work that is complementary to\nour study has been conducted by\nPop et al.~\\cite{pop2018enabling}, Hackel et al.~\\cite{hackel2019software},\nHerlich et al.~\\cite{herlich2016proof},\nNayak et al.~\\cite{nayak2016time,nayak2017incremental,nayak2017routing},\nand Kobzan et al.~\\cite{kobzan2018secure}.\n\n\\subsection{Contributions}\nWe comprehensively evaluate the performance of TAS\nfor reconfigurations in the hybrid and fully distributed models\nwith respect to\nnetwork deployment parameters, such as, maximum window size for the\nGate Control List (GCL) repetition, gating ratio proportion, i.e.,\nGate Control Entry (GCE) proportion, to control delay perceived at the\nreceiving end, signaling impact on Scheduled Traffic (ST) and\nBest-Effort traffic (BE) classes, and packet loss rate experienced at\nthe receiving end. In particular, we make the following contributions:\n\\begin{itemize}\n\\item[i)] We design a CNC interface for a TSN network to globally\n  manage and configure TSN streams, including admission control and\n  resource reservation.\n\\item[ii)] We integrate the CNC in the control plane with TAS in the\n  data plane to centrally manage and shape traffic using the CNC as\n  the central processing entity for flow schedules as more flows are\n  added.\n\\item[iii)] We modify and test the model to operate in a distributed\n  fashion, i.e., the control and data planes are combined.\n\\item[iv)] We evaluate each design approach and for a range of numbers of\n  streams and sources with different TAS parameters. We show results\n  based on admission ratios, network signaling overhead, and QoS\n  metrics.\n\\end{itemize}\n\n\\subsection{Organization}\nThis article is organized as follows. Section~\\ref{tsn:back:sec}\nprovides background information and an overview of related work on the\n802.1 TSN standardization, focusing on the enhancements to scheduled\ntraffic and centralized management and\nconfiguration. Section~\\ref{tsn:design:sec} shows the complete\ntop-down design of the CNC (hybrid model) and main components that\nachieve ultra-low latencies and guaranteed QoS for a multitude of ST\nstreams. Similarly, Section~\\ref{tas:reconfig:prot} shows the approach\nused in implementing the decentralized (fully distributed) TAS\nreconfiguration model. The simulation setup as well as main\nparameters and assumptions are given in Section~\\ref{tsn:sim:setup} and\nresults are presented in Sections~\\ref{eval:cent:sec} and~\\ref{eval:dec:sec}. Finally conclusions and future work are outlined\nin Section~\\ref{concl:sec}.\n\n\\section{Background: IEEE 802.1 Time Sensitive Networking} \\label{tsn:back:sec}\n\n\\subsection{IEEE 802.1Qbv: Time Aware Shaper (TAS)}\nTAS's main operation is to schedule critical traffic streams in\nreserved time-triggered windows. In order to prevent lower priority\ntraffic, e.g., best effort (BE) traffic, from interfering with the\nscheduled traffic (ST) transmissions, ST windows are preceded by a\nso-called guard band. TAS is applicable for Ultra Low Latency (ULL)\nrequirements but needs to have all time-triggered windows\nsynchronized, i.e., all bridges from sender to receiver must be\nsynchronized in time. TAS utilizes a gate driver mechanism that\nopens\/closes according to a known and agreed upon time schedule for\neach port in a bridge. In particular, the Gate Control List (GCL)\nrepresents Gate Control Entries (GCEs), i.e., a sequence of on and off\ntime periods that represent whether a queue is eligible to transmit or\nnot.\n\nThe frames of\na given egress queue are eligible for transmission according to the\nGCL, which is synchronized in time through the 802.1AS time\nreference. Frames are transmitted according to the GCL\/GCE and\ntransmission selection decisions. Each individual software queue has\nits own transmission selection algorithm, e.g., strict priority\nqueuing. Overall, the IEEE 802.1Qbv transmission selection transmits a\nframe from a given queue with an open gate if: $(i)$ The queue\ncontains a frame ready for transmission, $(ii)$ higher priority\ntraffic class queues with an open gate do \\textit{not} have a frame to\ntransmit, and $(iii)$ the frame transmission can be completed before\nthe gate closes for the given queue. Note that these transmission\nselection conditions ensure that low-priority traffic is allowed to\n\\textit{start} transmission only if the transmission will \\textit{be\n  completed} by the start of the scheduled traffic window for\nhigh-priority traffic. Thus, this transmission selection effectively\nenforces a ``guard band'' to prevent low-priority traffic from\ninterfering with high-priority traffic~\\cite{finn2018introduction}.\n\n\\subsection{IEEE 802.1Qcc: Centralized Management and Configuration}\n\n\\begin{figure}[t!]  \\centering\n\t\\includegraphics[width=3.5in]{fig_cnc_topo.pdf} \\vspace{-0.5cm}\n\t\\caption{Illustration of Centralized Network Configuration (CNC): End stations interact with the network entities via the User-Network Interface (UNI). The CNC receives the requests, e.g., flow reservation requests, and provides corresponding management functions.  An optional CUC provides delay-optimized configuration, e.g., for closed-loop IACS applications.  The solid arrows represent the protocol, e.g., YANG or TLV, that is used as the UNI for exchanging configuration information between Talkers\/Listeners (users) and Bridges (network).  The dashed arrows represent the protocol, e.g., YANG or TLV, that transfers configuration information between edge bridges and the CNC.} \\label{fig_tsn_Qcc}\n\\end{figure}\n\nIEEE 802.1Qcc~\\cite{IEEE8021Qcc} provides a set of tools to globally\nmanage and control the network. In particular, IEEE~802.1Qcc enhances\nthe existing Stream Reservation Protocol (SRP) with a User Network\nInterface (UNI) which is supplemented by a Centralized Network\nConfiguration (CNC) node, as shown in Fig.~\\ref{fig_tsn_Qcc}. The UNI\nprovides a common method of requesting layer 2 services. Furthermore,\nthe CNC interacts with the UNI to provide a centralized means for\nperforming resource reservation, scheduling, and other types of\nconfiguration via a remote management protocol, such as\nNETCONF~\\cite{enns2006netconf} or RESTCONF~\\cite{bierman2017restconf};\nhence, 802.1Qcc is compatible with the IETF YANG\/NETCONF data modeling\nlanguage.\n\n\\section{Centralized Model Design and Framework Considerations} \\label{tsn:design:sec}\nThis section presents our design methodology and main signaling\nframework for the centralized network\/distributed user\nmodel (hybrid model). Our main goals behind designing the CNC is given by the\nfollowing constraints. Additionally, the CNC can be logically or\nphysically connected to the data-plane with in-band or out-of-band\nmanagement links.\n\\begin{enumerate}\n\\item Our focus is mainly on stream based network adaptation. By this\n  technique, fluctuating streams (already registered streams and new\n  incoming streams) and their requirements can be accommodated by the\n  network dynamically based on a single remote procedure call to the\n  CNC.\n\\item Identify and execute flow requirements by populating the\n  registration table. The control plane resource orchestration is\n  purely carried out by the monitoring of existing flows which have\n  been satisfied.\n\\item Optimizing resource allocation (maximize admitted streams\/flows)\n  based on a bounded latency and network utilization, i.e., prioritize\n  flows that request low network resources.\n\\end{enumerate}\nOur main assumption to accurately apply admission control and,\nconsequently, reconfiguration, is that each source must define a flow\nin terms of total resources needed (governed by the bandwidth\nrequirements) and the total time needed for the resource to be used\n(which in our traffic model is termed as the resource utilization\ntime). Essentially, the CNC uses this information (which is tagged in\nthe Ethernet frame header) to determine whether a frame is admitted or\nrejected.\n\n\\begin{figure}[t!] \\centering\n\\includegraphics[width=3.3in]{fig_cnc_framework.pdf} \\vspace{0.2cm}\n\\caption{Network Management Entity Framework for TSN Switches:\n  Centralized Network Configuration (CNC) is used to send and receive\n  Control Data Traffic (CDT) to configure routing segments and network\n  resources with the goal of maximizing network flows\/streams.}\n\t\\label{fig_cnc_reconfig}\n\\end{figure}\n\n\\subsection{Core Components}\nOur design is split into two layers, Control Plane and Data Plane,\nfollowing the decoupling SDN paradigm, thereby inheriting the benefits\nof the orthogonality of the two planes, as shown in\nFig.~\\ref{fig_cnc_reconfig}.\n\n\\subsubsection{Configuration Module}\nThe configuration module is the main component that interacts with the\nregistered flows and network components. It includes the global stream\nregistration table which records all approved stream transmitting in\nthe network, and the admission control element that encapsulates and\ndecapsulates CDT headers and forwards the information to the necessary\nmodule\/element.\n\n\\paragraph{Global Stream Registration Table}\nThe source streams (devices\/users) make a Remote Procedural Call (RPC)\nvia the stream registration interface for providing information that\ncan be mapped as a unique tuple structure identification $<Flow ID,\nBridge Gateway>$. Upon receiving the registration packet, i.e.,\nControl Data Traffic (CDT), the CNC determines whether the new stream\ncan be accepted in its stream table. To guarantee the QoS for all\nregistered streams, admission control principles are applied to all\nstreams according to the stream's path, required network resources,\nand available resources.\n\n\\paragraph{Admission Control}\nThe admission control element is the first element that the new\nstreams interacts with. The admission control element under the\nconfiguration module globally manages all streams transmitting in the\nTSN domain governed by the CNC. The admission control element extracts\nthe necessary information from the CDT packet and forwards the\ninformation according to the CDT type. The CNC apples several steps to\ndecide whether to accept or reject the stream transmission request.\n\\begin{enumerate}\n\\item The CNC checks the destination address(es) of the stream and\n  consults its resource manager module for network resources available\n  on the new stream's path, which is computed based on the path\n  computation element within the CNC.\n\\item According to the bandwidth required for the new stream\n  (calculated at the bridge gateway for the new stream), all links on\n  the path are checked to see if enough bandwidth is available for the\n  new stream.\n\\item In the event that not enough resources are available, the CNC\n  applies the TAS reconfiguration module to identify the bottleneck\n  link(s) and to check whether the gating ratio can be increased for that\n  specific traffic class whose current resource utilization and\n  deadline will not cause a late deadline by being added to the TAS\n  slot reservation.\n\\end{enumerate}\n\n\\subsubsection{Reconfiguration Module}\nThe reconfiguration module includes the flow scheduling element (for\nour network model, the Time-Aware Shaper (TAS) is used in the data\nplane), the reconfiguration calculus element which optimizes flow\nregistration according to each stream's total resource utilization and\nflow deadlines, and finally the path computation element which defines\nthe path for all stream according to the QoS constraint.\n\n\\paragraph{Flow Scheduling}\nThe flow scheduling element currently takes the Time-Aware Shaper into\nconsideration. Due to the TAS's requirements on time synchronization\nbetween network components (switches, hosts, etc.), the CNC follows\nthe same principle of scheduling flows according to a known timescale\n(initially set to be 50~$\\mu$s in our network model). The CNC then\npasses on this time synchronization information to the TSN enabled\nswitches within its domain. Any approved streams will transmit frames\naccording to the time scale specified by the flow scheduler in the\nCNC.\n\n\\paragraph{Reconfiguration Calculus}\nIn addition to centrally managing resources and providing admission\ncontrol policies to the network, the CNC can invoke the TAS\nreconfiguration strategy with the goal of borrowing BE time\nslots for pending ST traffic streams. This element consults the\nresource manager module on the bottleneck link and checks whether the\nadded stream will oversubscribe the link. The TAS reconfiguration\nincrementally (1\\% of total CT) increases the traffic class slot time\nand reserves it for the new stream.\n\n\\paragraph{Path Computation}\nFor large scale and complex LAN\/MAN topologies, it is often required\nto supplement streams with equal cost paths in the event of a path\ndisruption (e.g., link failure, stream saturation, and explicit\ncongestion). The CNC's path computation element is tasked with finding\nsuch paths as a fail-over approach to avoid any violations to any\nstream's QoS. Presently, our model has a rudimentary application of\npath computation, i.e., it is defined statically for all core network\ncomponents (shortest path), since the main emphasis was on\nreconfiguration based on stream characteristics as defined by the\nsource.\n\n\\subsubsection{Resource Manager Module}\nThe resource manager module centrally manages all network resources\nwithin the CNC's domain. It includes the network resource table that\nrecords all streams' usage of resources, and the resource allocation\nscheme element to which we delegate the task of calculating the required\nnetwork resources for a given stream according to an allocation\nscheme.\n\n\\paragraph{Network Resource Table}\nTo remove certain overheads on the configuration module, the network\nresource table operates in tandem with the global stream registration\ntable to accurately determine the required network resources (mainly\nbandwidth for our traffic model). It classifies streams based on\nperiodic stream properties. Any stream that has been approved by the\nCNC has a record attached to it in the network resource table.\n\n\\paragraph{Resource Allocation Scheme}\nSeveral allocation schemes can be implemented for all traffic classes\ndefined in the network. For periodic streams, the time slot given by\nthe flow scheduler (according to the TAS Cycle Time and number of\ntraffic classes) and the data rate defined by the source is used to\ncalculate the required bandwidth for each link on the path to the\ndestination (i.e., sink).\n\n\n\\subsubsection{Data Plane}\nThe data plane contains all core switches. Any TSN switch interfaced\nby the CNC is given a switch ID and has a local stream registration\ntable. The remaining switch elements compose the forwarding and\nqueuing operation with several traffic shapers (802.1Qbv TAS in our\nnetwork model).\n\n\\paragraph{Local Stream Registration Table}\nThis data plane registry contains the subset of source streams that\nare established for the corresponding bridge gateway and attached\nsources to each port. The CNC delegates some control to the bridge\ngateway to instruct and alert sources of any new network conditions\nand explicit changes.\n\n\\paragraph{Traffic Shaper --- Time-Aware Shaper}\nThe traffic shaper is the main shaping and scheduling mechanism that\ncontrols the gating schedules for all the traffic classes within the\nTSN domain. All bridges are synchronized to the same gating schedule\nGCL Cycle Time (CT) given by the CNC's flow schedule element (CT\nindicates the time period for the GCL to repeat).\n\n\n\\section{Decentralized Model Design and Framework Considerations} \\label{tas:reconfig:prot}\n\n\\begin{figure}[t!] \\centering\n\\includegraphics[width=3.5in]{fig_distributed_reconfig.pdf} \\vspace{-0.2cm}\n\\caption{A TSN fully distributed configuration model example\n  illustrating the general strategy and logic of each TSN switch with\n  TAS support. In the absence of a CNC to centrally manage network\n  parameters, each switch performs admission control and resource\n  reservation (according to the TAS time slot load) and propagates the\n  information to the next hop on the stream path. A single rejection on\n  one hop terminates the forwarding of the CDT, and sends another CDT\n  on the reverse path indicating the stream rejection outcome. If all\n  switches on the path accept the stream, then the source is notified\n  of the stream acceptance outcome and can begin forwarding in the\n  next TAS cycle. In our model, CDT traffic has higher priority than\n  non-CDT traffic (including ST). The formal definition of the CDT\n  traffic is left for future work. }\n\t\\label{fig_distributed_reconfig}\n\\end{figure}\n\nThis section presents our design methodology and framework for the TAS\nreconfiguration in the decentralized (fully distributed) model. Our\ncurrent proposed architecture generally follows the steps enumerated\nbelow and illustrated in Fig.~\\ref{fig_distributed_reconfig}. Our\ndescription focuses on the additions to the design of RAP over LRP,\ne.g., TAS slot computation\/reservations.\n\n\\begin{enumerate}\n\t\\item At each egress port (Port Identifier, PID), the TSN switch maintains a local stream registration table that includes information, such as flow ID, gateway (i.e., the first bridge that a talker is connected to), destination address(es), the traffic injection rate per GCL cycle time, and the calculated port bandwidth requirement. The traffic injection rate is not computed, rather the traffic injection rate is reported by the source (talker) to the network devices. It mainly indicates the bandwidth requirements of a stream. Bandwidth for a bridge egress port needed for a stream is computed using the ST injection rate (or ST rate), the average packet size, and the bridge TAS timing configuration (e.g., the CT and current traffic\tclass slot time). This information is carried and communicated between bridges using the CDT packet type identifier (or message type).\n\t\\item A source (talker) can send a stream transmission request, i.e., a CDT message of type Stream Transmission Request to\tregister its stream and use the TSN service for scheduled traffic.\n\t\\item Each switch maintains a resource manager module for each port. If the newly incoming stream is accepted (due to available resources and TAS slot space). The TAS slot size for a specific traffic class is governed by the CT and traffic class gating ratio (in time). The TAS ST slot can be configured\/reconfigured according to stream requests and terminations. The stream registration message is then propagated towards the next switch, and a map is maintained for the stream (and any other streams) pending approval.\n\t\\item If accepted by the last switch, then the stream registration record is added to the local stream registration table, and bandwidth resources are allocated for the stream and TAS slot space is modified (if necessary) on the reverse path. The main reason for allocating the resources in the reverse path is as follows. If we allocate the resources in the forward direction but a switch in the next hop rejects the stream (due to lack of resources),\tthen we have to release the resources reserved earlier for the stream. Therefore, we avoid the allocation until all hops provide\tassurance that the stream will be accommodated.\n\t\\item Each switch receiving the pending registration message adds the stream record to its local table, allocates the necessary resources and TAS slot reservation, and propagates the registration message towards the source gateway.\n\t\\item The source gateway receives the pending stream registration message and similarly allocates the resources and finally sends an approval granted message towards the source which prompts the source to start sending data in the next available TAS cycle.\n\\end{enumerate}\n\nWhile the previous enumerated points provide an overview of the procedural approach used, a key question remains, namely what happens if frames belonging to a given stream arrive after the gate for it's traffic class queue has been closed? Generally, the way the stream traffic end station (source\/talker) operates is by synchronizing to the switch's TAS ST slot time for the class the stream belongs to (in\tour case, it is ST) after the CDT stream registration is complete. Therefore, it will always send data traffic at the beginning of the ST slot. After the data traffic for the ST stream arrives at the first switch (where the gate for the ST queue will definitely be open due to source-network synchronization), the new ST traffic is queued at the back of the queue awaiting transmission. Since registration and resource allocation for all ST streams is enforced, the computed ST slot size will be large enough to transmit all ST packets (from all streams being serviced through that specific switch's egress port) queued that arrived from different sources during the cycle time. The delay bounds (a data traffic stream arriving before a specific time and not necessarily at the right time) is guaranteed since the cycle time for each switch along the path is configured to be large enough to transmit all ST frames registered on the multi hop path (up to 5 hops). Some potential ST traffic sent on cycle time $(t)$ can be transmitted at cycle time $(t+1)$, but would be blocked by some subsequent slots in the first cycle time belonging to different traffic classes (in our case, it is BE) leading to higher delays but still within tolerated delays since the cycle time is usually much smaller than the max delay bounds or threshold defined (e.g., 50 microseconds cycle time and 100 microseconds delay bounds).\n\nIn our simplified example in Fig.~\\ref{fig_distributed_reconfig}, each switch, upon initialization and with admission control, provisions all the CT to BE traffic. When the first stream transmission request occurs, the switches exchange the CDT message and start building the local stream registration table. A switch builds the GCE slot time for ST by mapping each stream request to its internal stream registration table upon granting the stream\tapproval. The switch builds the ST slot time iteratively by\tfollowing each stream admission\/termination. The switch passes each set of stream configuration parameters (UNI) through the 802.1Qcc registration and reservation protocol.\n\nST traffic stream load is viewed in two main ways in our implementation. $i)$ ST injection rate, and $ii)$ resource utilization time. The ST injection rate corresponds to the number of packets sent from an ST stream in one TAS cycle time, while the resource utilization time is the time that the stream reports to the network on how long the resources reserved will be used. The bandwidth requirement for a new stream request is calculated by taking the reported ST injection rate of the stream and multiplying that with the packet size for the stream. That quantity is then divided by the TAS slot time (at a specific switch). Essentially, an increase in the slot time corresponds to a decrease in the bandwidth requirement (since more time is given) and vice versa. Note that this only works for streams with periodic data transmissions and not for sporadic data streams since we cannot calculate how much time is needed for a sporadic data transmission. This procedure is only for ST traffic; BE traffic is transmitted with the leftover slot time after the ST traffic has been serviced.\n\n\\subsection{Core Components}\n\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=3.5in]{fig_stream_registration.pdf} \\vspace{-0.2cm}\n\t\\caption{The main logical steps performed by each switch along the stream's path are shown to apply stream registration and reservation. Each switch generally waits for an event (addition, removal, or pending) for each stream. For instance, a stream removal is usually based on the resource utilization time (stream life time) that was specified at stream establishment. The bridges that allocated resources for the stream can remove the stream after the stream life time has expired. For the cases of stream addition or pending, the event is the CDT message received (whether in the forward or reverse direction). Towards completing (finalizing, confirming) a stream reservation (registration), the pending event is the event for a CDT message in the reverse direction where each switch (not the last switch) waits for the approval (confirmation of reservation) of the next-hop switch.}\n\t\\label{fig_stream_registration}\n\\end{figure}\n\nThis section outlines the main components required to successfully implement stream admission control and resource reservation within switches that support the TAS traffic shaper in a distributed fashion. Fig.~\\ref{fig_stream_registration} illustrates the typical registration\/reservation procedure for all streams within the TSN domain.\n\n\\subsubsection{Admission Control}\nThe admission control element extracts the necessary information from the CDT packet and forwards the information according to the CDT type. The switch forwarding mechanism applies several steps to accurately decide whether to accept or reject the stream transmission request Note that the stream transmission request corresponds to a CDT message. In particular, the switch consults the\nresource manager module to check if enough resources (bandwidth) is available for the new stream that is calculated by the reported traffic injection rate, the maximum cycle time, and the traffic class's TAS slot time. A given stream's bandwidth requirement is calculated by multiplying the ST injection rate with the average packet size and dividing by the current ST slot size. Note that the traffic class TAS slot time is the time during which the TAS gate is open to transmit frames belonging to the considered class.  Also note that all GCEs are executed during each CT. If the CT is smaller than the aggregate of the GCEs, then we need to either increase the CT or reject streams that cause the exceedance of the CT.\n\n\\subsubsection{Flow Scheduling}\nThis element currently takes the Time-Aware Shaper into consideration. Due to the TAS's requirements on time synchronization between network components (switches, hosts, etc.), all switches\/hosts follow the TAS GCL timescale cycle time (e.g., 50~$\\mu$s). Depending on the number of traffic classes supported, the TAS cycle time can be divided into appropriate slots for each traffic class load. The TAS CT is divided among all the traffic classes (in our evaluation model, we consider two traffic classes, BE and ST). Currently, in our evaluations, the CT is initially predefined to 50 microseconds. Note that the CT could be changed\/configured dynamically. The dynamic adaptation of the CT with respect to new stream additions, application specifications, or other events is a topic for future work.\n\n\\subsubsection{Stream Registration Table}\nStream creation follows a Poisson process with a mean duration. Different scenarios with varying mean duration enables analysis of how reconfiguration works in multiple settings. The stream life-time is defined by the duration. For example, how the switches in the path are aware of the Poisson parameter from the edge switch. All GCE entries and Queues are based purely on this slots allocated based on the Poisson parameter. Although, our approach does not intend to minimize delay, we follow a delay bound strategy that leads to finding the limiting value. Implicitly, the limiting value can be minimized that can provide a minimized delay. Additionally, the stream registration table contains the characteristics of the source streams that are established for the corresponding bridge egress port. Each record gets populated (if accepted) on the reverse path taken by the stream's registration message (after reaching the destination switch).\n\n\\subsubsection{Traffic Shaper - Time-Aware Shaper}\nThe main shaping and scheduling mechanism that controls the gating schedules for all the traffic classes within the TSN domain. All bridges are synchronized to the same gating schedule GCL CT that is initially predefined by network administrator. Ideally, we want the CT to be large enough for all streams from all\ntraffic classes to be accommodated and small enough that the all streams QoS fits the delay requirements.\nIn our current evaluations, the CT is predefined at the widely used 50 microseconds.\n\n\\subsubsection{Reconfiguration Calculus}\nIn essence, the reconfiguration (dynamic configuration) of the TAS schedules (switch GCL\/GCE) for each egress port is dynamically invoked according to two principle events, $i)$ adding a new stream, and $ii)$\nremoving an existing stream. The switch's gating ratio (for a particular stream belonging to a defined traffic class) reports certain parameters (e.g., packet injection rate, maximum packet size, latency requirement, deadline, application response time, etc.) which are then used to check if enough slot time is available (which corresponds to attempting bandwidth reservation). In the event that no slots are  available, the GCE slot size is recomputed (according to the registered stream properties within the registration table) generally by allocating more resources from Best Effort Traffic.\nThe stream life time is reported by the source to the network as user\/network information (UNI). Each UNI is propagated by each switch along the path which allows the switch to register the stream and store the stream's resource utilization time, (stream life-time), among other critical information. Any information\tpertaining to the UNI of a stream is recorded in the stream\tregistration table. In terms of GCEs for TAS with support of ST and BE traffic classes, only two GCEs within a GCL (1\/0 (ST\/BE) for the\tfirst entry and 0\/1 (BE\/ST) for the second) are necessary with a total of three outbound queues for each egress channel port in a TSN switch. Two queues for each traffic class, and another queue for CDT traffic (signaling traffic). Upon initialization, each switch allocates $0$ resources to ST, and BE therefore gets all the leftover CT slot. As streams get registered, the ST slot time is recomputed\n(according to the stream packet size, ST injection rate, and current slot time, if the slot exists). If the stream is the first stream to the switch, i.e., ST slot is $0$, then the ST slot size is defined\nto be 1\\% of the CT at a minimum by borrowing necessary time slot from BE.\n\n\\subsubsection{Path Computation}\nWhile this module is fundamentally necessary in any switch (in a decentralized\/distributed network), we manually define static routing tables for destination addresses and associated ports on each switch.\nEssentially, we assume a manual procedure to compute paths, i.e., we assume that there is a path computation module that is used in both centralized and distributed configuration models. We make this assumption to simplify operations and place emphasis on the TAS reconfiguration technique.\n\n\\subsubsection{Network Resource Table}\nTo remove certain overheads on the configuration procedure, the network resource table operates in tandem with the stream registration table to accurately determine the required network resources (mainly bandwidth for our traffic model) per switch egress port. It classifies streams based on periodic and sporadic streams properties, though currently the focus is on periodic ST streams. Any approved stream by the switch has a record attached to it in the network resource table, located within each switch, which can be called to compute and store current and remaining link\/port loads for each switch. Each egress port has a network resource table. More details on the network resource table will be provided in the next iteration of this document.\n\n\\section{Performance Evaluation}  \\label{tsn:sim:setup}\n\n\\begin{figure}[t!]  \\centering\n\\includegraphics[width=3.3in]{fig_tsn_indctrl} \\vspace{-0.2cm}\n\\caption{Industrial control loop\n  topology~\\cite{guck2016function}. Each source generates stream data\n  with varying hop counts and packets rates unidirectionally or\n  bidirectionally across the six switches ultimately destined to a\n  sink} \\label{fig_tsn_indLoop}\n\\end{figure}\n\n\\subsection{System Overview and Simulation Setup}  \\label{tsn:eval:sec}\nThis section explains the simulation setup and model. Furthermore, the\ntopology and simulation scenarios will be presented. Throughout, we\nemploy the OMNet++~\\cite{varga2008overview} simulation environment.\n\n\\begin{table}\n\t\\caption{Simulation Parameters}\n\t\\begin{tabularx}{\\columnwidth}{|X|X|X|}\n\t\t\\hline\n\t\t\\multicolumn{1}{|c|}{\\textbf{Key}} & \\multicolumn{1}{|c|}{\\textbf{Symbol}} & \\multicolumn{1}{c|}{\\textbf{Value}} \\\\\n\t\t\\hline\n\t\tSimulation Duration & $Sim_{limit}$ &  $100$ seconds \\\\\n\t\t\\hline\n\t\tInitialized Cycle Time & $GCL_{CT}$ & $50~\\mu$s \\\\\n\t\t\\hline\n\t\tInitialized Gating Ratio & $ST^{R}_{init}$ & $20$\\% (i.e., 10~$\\mu$s) \\\\\n\t\t\\hline\n\t\tAverage Streams per Second & $\\pi$ & $1 - 20$ \\\\\n\t\t\\hline\n\t\tAverage stream duration & $\\tau$ & $2 - 5$ seconds \\\\\n\t\t\\hline\n\t\tNumber of Frames per Cycle & $\\gamma$ & $1$ \\\\\n\t\t\\hline\n\t\tBE Traffic Intensity & $\\rho_{L}$ & $0.1, 1.0, 2.0$ (low, mid, and high) \\\\\n\t\t\\hline\n\t\tST sources & $S$ & $6$ \\\\\n\t\t\\hline\n\t\tHurst Parameter & $H$ & $0.5$ \\\\\n\t\t\\hline\n\t\tQueue Size & $Q_{size}$ & $512$~Kb\n\t\n\t\n\t\n\t\t\\\\\n\t\t\\hline\n\t\\end{tabularx}\n\t\\label{table: simulation parameters}\n\\end{table}\n\n\\begin{table}[t!]\n\t\\centering\n\t\\footnotesize\n\t\\caption{Traffic proportions relative to number of hops for the industrial control loop topology}\n\t\\label{hops}\n\t\\begin{tabular}{cccccc}\n\t\t\\textbf{Hops \\#} & 1 & 2 & 3 & 4 & 5 \\\\\n\t\t\\textbf{Range Distribution} & 20 \\% & 20 \\% & 20 \\% & 20 \\% & 20 \\% \\\\\n\t\\end{tabular}\n\\end{table}\n\n\\subsubsection{Network Model}\t\\label{tas:sec:net}\nThe network topology is modeled around an industrial control loop\ntopology that consists of six core switches in a ring topology\nconnected to the CNC as shown in Fig.~\\ref{fig_tsn_indLoop}. Each\nswitch-to-switch link operates as a full-duplex Ethernet link with a\ncapacity (transmission bitrate) $R = 1$~Gbps. Each switch can act as a\ngateway for a number of traffic sources and one sink. The distance\nbetween two successive switches along the ring is fixed to 100~m and\nthe switch-to-switch propagation delay is set accordingly to\n0.5~$\\mu$s. All switches are configured to use 802.1Qbv TAS as the\ntraffic shaper for each switch-to-switch egress port whose flow\nschedule (ST gating ratio and cycle time) is configured by the CNC in\nthe centralized (hybrid) model and independently in the decentralized\n(fully distributed) model.\n\n\\subsubsection{Traffic Model}\t\t\\label{tas:sec:traffic}\nWe consider periodic (pre-planned) traffic and sporadic self-similar\nPoisson traffic for ST traffic and for BE traffic, respectively. To\nemulate dynamic conditions in the network, we employ several\ndistributed ST sources that generate $\\pi$ ST streams according to the\nnetwork and traffic parameters shown in Table~\\ref{table: simulation\n  parameters}. The stream generation follows a Poisson distribution\nwhere $\\pi$ represents the average number of generated streams per\nsecond. Each stream within a source injects packet traffic with packet\nsize $64$~bytes and has an ST traffic injection rate that is uniformly\ndistributed with a value ($\\gamma$) statically defined at stream\ncreation time and a destination address assigned by the number of\nswitch-to-switch hops as shown in Table.~\\ref{hops}. Furthermore, at\nthe stream creation time, each stream is given a start time (usually\nthe current runtime), and a finish time based on $\\tau$. We consider\nadmission as the completion of the flow from start to the finish time\nreported by the source. Each source is attached to a core TSN switch\ngateway (first hop switch). While the TSN switches operate with time\nsynchronization, the ST sources (outside the TSN domain) do not need\nto be synchronized. Therefore, the gateways can in the worst case\ndelay ST traffic by a maximum of 1 cycle times. However, note that the\nST traffic follows an isochronous traffic class as specified by\nIEC\/IEEE 60802 where the sources are synchronized with the network\nafter stream registration is initiated.\n\n\\subsection{Centralized Model Evaluation}\t\\label{eval:cent:sec}\nIn evaluating the proposed solution described in\nSection~\\ref{tsn:design:sec}, we consider both periodic and sporadic\nsources for ST and BE traffic, as described in\nSection~\\ref{tas:sec:traffic}. We evaluated the CNC with TAS shaper on\nthe industrial control loop for the unidirectional and bi-directional\ntopologies and results are collected for tests following the simulation\nparameters shown in Table~\\ref{table: simulation parameters}.\n\n\\subsubsection{Unidirectional Ring Topology}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_2_cenUni.pdf}\n\t\\caption{Centralized Unidirectional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 2$ under different BE loads~$\\rho_{L}$, ST stream rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_2}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_3_cenUni.pdf}\n\t\\caption{Centralized Unidirectional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 3$ under different BE loads~$\\rho_{L}$, ST stream rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_3}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_4_cenUni.pdf}\n\t\\caption{Centralized Unidirectional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 4$ under different BE loads~$\\rho_{L}$, ST stream rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_4}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_5_cenUni.pdf}\n\t\\caption{Centralized Unidirectional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 5$ under different BE loads~$\\rho_{L}$, ST stream rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_5}\n\\end{figure}\n\nFigs.~\\ref{fig_delay_2}--\\ref{fig_delay_5} show the average mean\ndelay for ST traffic and for BE traffic for the centralized\nunidirectional ring topologies. The average delays in general are low\nand stable for both BE and ST traffic. Since the CNC manages the ST\ntraffic streams and therefore guarantees the bandwidth rates needed to\ntransmit across a single switch hop in one CT, the ST delays are less\nthan $100~\\mu$s for all $\\tau$ values. The ST delays with\nreconfiguration active at the CNC experience higher delays than ``No\nReconfiguration'' since we essentially push more frames into the\nnetwork that increase the queuing delay.\nBE traffic\nexperiences much higher delays than ST. With the ``No Reconfiguration''\napproach, the BE traffic delay is near constant since the gating ratio is\nleft unchanged throughout the simulation. For the test with\nreconfiguration, the BE mean delay increases dramatically since we\ntend to accept and exhaust the CT with ST streams over the course of\nthe simulation run.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI01_cenUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI10_cenUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI20_cenUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Unidirectional Topology: Max delay as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_maxDelay}\n\\end{figure}\n\nAs mentioned in the introduction section, TSN needs to guarantee and\nbound the maximum delay in order to deterministically forward traffic\nacross a TSN domain. Fig.~\\ref{fig_maxDelay} shows the maximum delay\nevaluation for ST traffic. For the unidirectional ring topology with a\nmaximum of five hop streams, the reconfiguration approach maximum\ndelay is bounded at $0.105$~ms, while for the ``No reconfiguration''\napproach, the max delay is bounded at nearly $60\\mu$~s. For TAS and\nthe CNC's registration and reservation procedure, the guarantee is\napplied to bandwidth as a share of the egress port using time division\nmultiplexing.\nWith the parameters chosen empirically, the\nmaximum delays is capped to approximately $100\\mu$~s which is ideal\nfor the topology chosen and critical ST traffic.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI01_cenUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI10_cenUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI20_cenUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Unidirectional Topology: Stream Admission as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_admin}\n\\end{figure}\n\nWhile QoS metrics are important, another factor that determines the\nperformance gains is the admission ratio for the\nsystem. Fig.~\\ref{fig_admin} shows the stream admission ratio that\nresults for both reconfiguration and no reconfiguration. In general,\neach generated stream needs a data rate of about $10.24$~Mbps for a\n50~$\\mu$s CT for each egress port on the stream's path with one ST\npacket rate per CT and fixed packet size of $64$~B. With an egress\nport channel capacity of $R=10^{9}$, approximately $100$ streams can\nbe guaranteed. Compared to the ``no reconfiguration'' approach, the\nreconfiguration significantly improves the admission rates at the cost\nof higher BE traffic delays since the ST slot borrows BE time slots to\naccommodate the ST streams.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI01_cenUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI10_cenUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI20_cenUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Unidirectional Topology: Stream Signaling delay as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_signalDelay}\n\\end{figure}\n\nCDT traffic that requests transmission guarantees from the CNC\nexperiences some delay before being either admitted or\nrejected. Fig.~\\ref{fig_signalDelay} shows the average signaling\nlatency for ST stream registration. Since the control plane is out-of\nband from the data plane within the TSN domain, the delay is constant\nthroughout the simulation run.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI01_cenUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI10_cenUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI20_cenUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Unidirectional Topology: Stream average signaling Overhead as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_signalOverhead}\n\\end{figure}\n\nStream registration and reservation introduces some control plane\noverhead. Fig.~\\ref{fig_signalOverhead} shows the signaling\nperformance overhead. More specifically, the overhead is measured at\nthe CNC or both incoming and outgoing control (CDT)\ntraffic. Generally, the reconfiguration approach introduces more\nsignaling overhead; however, Ethernet generally has large bandwidths,\nthus the CDT traffic rates are minimal compared to the link\ncapacities. Furthermore, when $\\tau = 2$, we observe higher signaling\noverhead due to accepting larger numbers of streams (rejections are\ninexpensive compared to acceptance) both with and without\nreconfiguration.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI01_cenUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI10_cenUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI20_cenUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Unidirectional Topology: ST Total average throughput measured at the sink as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_avgTput_ST}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI01_cenUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI10_cenUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI20_cenUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Unidirectional Topology: BE Total average throughput measured at the sink as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_avgTput_BE}\n\\end{figure}\n\nObserving the network's throughput gain, Fig.~\\ref{fig_avgTput_ST} and\nFig.~\\ref{fig_avgTput_BE} show the average throughput measured at the\nsink for both ST and BE traffic. While the total throughput that can\nbe achieved is around $6$~Gbps, the maximum throughput allowed is\naround $2$~Gbps since some switches get bottlenecked faster than other\nswitches which restricts the addition of more flows. For the ``no\nreconfiguration'' approach, BE traffic is varied between $0.1$ traffic\nintensity to $2.0$. Fig.~\\ref{fig_avgTput_BE}[a]--[b] shows an average\nthroughput of $0.1$~Gbps and $1.0$~Gbps, respectively, which is expected\nand no frames are dropped. However, as the load reaches $2.0$~Gbps and\nmore ST streams are accepted, i.e., the BE traffic time slot is shortened,\nBE traffic starts to suffer and caps at around $1.2$~Gbps, as shown in\nFig.~\\ref{fig_avgTput_BE}[c]. With reconfiguration, BE tends to suffer\nmore since we shift the time slot of BE to ST (maximum of $90\\%$) and\nthe throughput drops to $0.1$~Gbps as shown in\nFig.~\\ref{fig_avgTput_BE}[b][c].\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI01_cenUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI10_cenUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI20_cenUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Unidirectional Topology: ST Frame loss ratio as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_lossProb_ST}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI01_cenUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI10_cenUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI20_cenUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Unidirectional Topology: BE Frame loss ratio as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_lossProb_BE}\n\\end{figure}\n\nTo show the performance of the CNC management of streams,\nFig.~\\ref{fig_lossProb_ST} and Fig.~\\ref{fig_lossProb_BE} show the\npacket loss ratio for ST and BE traffic in the network. Since the CNC\nmanages only ST streams, the TSN guarantees (which include zero packet\nloss since retransmissions are in general too expansive for ST\ntraffic) are only valid for ST streams. For BE under the\nreconfiguration approach, as the load for ST traffic increases, the\npacket loss increases as well. For the ``no reconfiguration''\napproach, the loss typically is constant even for high loads of BE\ntraffic.\n\nOverall, the reconfiguration approach certainly provides a means to\nmanage and to ensure that the number of ST streams is maximized\naccording to the link capacity. However, we observe from the results\nand the topology used that any bottleneck switch can generally reduce\nthe link utilization which can significantly drop the throughput, even\nif the delay and loss are guaranteed. Selecting different paths (if\none exists) and modeling the queue to ensure maximum ST stream delays\ncan help potential bottleneck links and increase throughput throughout\nthe network.\n\n\\subsubsection{Bi-Directional Ring Topology}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_2_cenBi.pdf}\n\t\\caption{Centralized Bi-directional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 2$ under different BE loads~$\\rho_{L}$, ST stream rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_2_bi}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_3_cenBi.pdf}\n\t\\caption{Centralized Bi-directional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 3$ under different BE loads~$\\rho_{L}$, ST stream rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_3_bi}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_4_cenBi.pdf}\n\t\\caption{Centralized Bi-directional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 4$ under different BE loads~$\\rho_{L}$, ST stream rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_4_bi}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_5_cenBi.pdf}\n\t\\caption{Centralized Bi-directional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 5$ under different BE loads~$\\rho_{L}$, ST stream rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_5_bi}\n\\end{figure}\n\nThe unidirectional ring topology certainly simplifies the complexities of calculating the ST slot window. However, to show the a more pronounced gain in stream utilization and admission, a bi-directional ring topology is used with static shortest path routes. The two port switch now has two paths to the destination according to the hop count specified in Table.~\\ref{hops}, where about $60\\%$ of streams take one port and the rest ($40\\%$) take the other port. Note that the edge links (switch to sink and source to switch) are given higher link capacities to avoid congestion at the edges where the CNC currently does not control (at least $2$~Gbps for the bi-directional ring). Fig.~\\ref{fig_delay_2_bi} - \\ref{fig_delay_5_bi} shows the average mean delay evaluation for both ST and BE traffic under different stream lifetime values, $\\tau$. Compared to the unidirectional topology, the bi-directional provides significantly better delay results since an extra port with full-duplex link support now provides extra capacity to service streams giving more slot reservations to BE even at high ST stream load.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI01_cenBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI10_cenBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI20_cenBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Bi-directional Topology: Max delay as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_maxDelay_bi}\n\\end{figure}\n\nIn terms of maximum delays, the bi-directional topology configuration produces higher than expected max delays due to the increasing ST stream acceptance without taking into account queue delays. For the bi-directional topology tests, the queue sizes were left the same (see Table.~\\ref{table: simulation parameters}) and the CNC typically guarantees an upper bound delay per cycle per hop, i.e., each switch hop constitutes a max delay of one CT (or $50\\mu$s in our tests). Fig.~\\ref{fig_maxDelay_bi} shows the maximum delay evaluation for ST traffic for the bi-directional ring topology with CNC present. With the ``no reconfiguration'' approach, and since the ST slot size is kept at the initialized value ($20\\%$ of CT or $10~\\mu$s), the maximum delay is constant at $50~\\mu$s. However, if reconfiguration is active at the CNC, then the maximum delay is bounded at $300~\\mu$s due to the high admission rate and larger queuing delay per cycle. Note that the maximum hop traversal for any ST stream is kept at $3$~hops due to the bi-directional topology.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI01_cenBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI10_cenBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI20_cenBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Bi-directional Topology: Stream Admission as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_admin_bi}\n\\end{figure}\n\nWhile the max delay suffers due to not taking into account queuing delay in general, the admission rate is much higher (by about $40\\%$) at high ST loads. Fig.~\\ref{fig_admin_bi} shows the stream admission ratio results. With $\\pi = 20$ and $\\tau = 5$, the admission rate is close to $90\\%$ for the bi-directional topology with reconfiguration active at the CNC. In contrast, the ``no reconfiguration'' approach improves slightly (by about $20\\%$) compared to the unidirectional ring since the initialized gating ratio is too restrictive and can largely underutilize the link. Note that while we show the admission rates for different BE loads, $\\rho_{L}$, the admission ratio does not change since TAS effectively segments the traffic at the egress switch\/port, i.e., BE traffic does not block at ST traffic.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI01_cenBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI10_cenBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI20_cenBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Bi-directional Topology: Stream Signaling delay as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_signalDelay_bi}\n\\end{figure}\n\nSimilar to the unidirectional ring, the bidirectional ring topology provides constant signaling delay due to the CNC out-of band signaling channels. Fig.~\\ref{fig_signalDelay_bi} shows the signaling delay for ST stream registration. Note that the average signaling delay is lower than in the unidirectional ring since the edge links, specifically the source to switch link, is larger than in the unidirectional ring hence the transmission delay is shorter.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI01_cenBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI10_cenBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI20_cenBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Bi-directional Topology: Stream average signaling Overhead as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_signalOverhead_bi}\n\\end{figure}\n\nSince the topology is effectively the same (albeit having another port to the switch), the signaling overhead in general is very similar to the unidirectional topology. Fig.~\\ref{fig_signalOverhead_bi} shows the signaling performance overhead. Note that while the hop traversal is reduced (since the stream can take one of two paths to the destination governed by hop traversal), the number of sent and received CDT frames are the same in general. Clearly, similar to the unidirectional topology, the reconfiguration approach generates more CDT traffic. Note that rejections in general are less costly in terms of sent and received frames in the network. Therefore, the higher the admission rate, the more overhead is observed in the control plane, though based on Fig.~\\ref{fig_signalOverhead_bi}, the overall overhead is not even close to $1$~Mbps and therefore is much lower compared to the channel capacity.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI01_cenBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI10_cenBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI20_cenBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Bi-directional Topology: ST Total average throughput measured at the sink as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_avgTput_ST_bi}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI01_cenBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI10_cenBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI20_cenBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Bi-directional Topology: BE Total average throughput measured at the sink as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_avgTput_BE_bi}\n\\end{figure}\n\nIn terms of overall throughput, Fig.~\\ref{fig_avgTput_ST_bi} and Fig.~\\ref{fig_avgTput_BE_bi} shows the average throughput measured at the sink for both ST and BE traffic for the bi-directional ring topology. Compared to the unidirectional ring, the throughput for the bi-directional ring is much higher (sink maximum capacity is around $12$~Gbps from all switch to sink channels).\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI01_cenBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI10_cenBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI20_cenBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Bi-directional Topology: ST Frame loss ratio as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_lossProb_ST_bi}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI01_cenBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI10_cenBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI20_cenBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Bi-directional Topology: BE Frame loss ratio as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_lossProb_BE_bi}\n\\end{figure}\n\nSimilar to the unidirectional ring topology, bi-directional topology achieves zero loss to ST streams while significantly improving the loss rate for BE traffic. Fig.~\\ref{fig_lossProb_ST_bi} and Fig.~\\ref{fig_lossProb_BE_bi} shows the packet loss ratio for ST and BE traffic in the network for the bi-directional ring topology. Maximum BE loss as high BE traffic intensity, $\\rho_{L} = 2.0$, is around $30\\%$ which is a significant reduction from the unidirectional topology (of around $90\\%$).\n\nIn contrast to the unidirectional topology, the bi-directional topology with reconfiguration active at the CNC achieves improved QoS metrics and admission rates. However, without modeling the queue characteristics and guaranteeing queuing delay, it is difficult to guarantee maximum delays. Modeling the ST queue at each egress port in conjunction with TAS configuration properties grants the possibility to produce deterministic forwarding plane at the data plane for ST streams. This part is left for the next iteration of this report.\n\n\\subsection{Decentralized Model Evaluation}\t\\label{eval:dec:sec}\nIn evaluating the proposed solution describes in more detail in section~\\ref{tas:reconfig:prot}, we consider both periodic and sporadic sources for ST and BE traffic as discussed in section~\\ref{tas:sec:traffic} respectively. We evaluated network with TAS shaper on the industrial control loop unidirectional and bi-directional topology and results are collected for tests following the simulation parameters shown in Table.\\ref{table: simulation parameters}.\n\n\\subsubsection{Unidirectional Ring Topology}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_2_distUni.pdf}\n\t\\caption{Decentralized Unidirectional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 2$ under different loads~$\\rho_{L}$, mean traffic rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_2_dec}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_3_distUni.pdf}\n\t\\caption{Decentralized Unidirectional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 3$ under different loads~$\\rho_{L}$, mean traffic rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_3_dec}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_4_distUni.pdf}\n\t\\caption{Decentralized Unidirectional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 4$ under different loads~$\\rho_{L}$, mean traffic rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_4_dec}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_5_distUni.pdf}\n\t\\caption{Decentralized Unidirectional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 5$ under different loads~$\\rho_{L}$, mean traffic rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_5_dec}\n\\end{figure}\n\nDecentralized model essentially transfers some of the CNC functions (e.g., TAS reconfiguration and resource reservation modules) from the centralized model down to the data plane switch's egress TAS enabled ports. The main difference between the centralized and decentralized models is the signaling performance which is now in-band and can affect data traffic. Fig.~\\ref{fig_delay_2_dec} - \\ref{fig_delay_5_dec} shows the average mean delay evaluation for both ST and BE traffic. While the CDT traffic is in-band, the average delay is about the same as the centralized topology average delay in Fig.~\\ref{fig_delay_2} - \\ref{fig_delay_5}. Typically, the ST stream's average delay is minimal to near constant for both the reconfiguration and ``no reconfiguration'' approaches. For BE, the ``no reconfiguration'' approach produces constant average delay for each BE $\\rho_{L}$ traffic intensity. When used with reconfiguration, the mean delay start to democratically increase since BE traffic time slots are begin reserved for ST streams.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI01_distUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI10_distUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI20_distUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Unidirectional Topology: Max delay as a results of TAS.}\n\t\\label{fig_maxDelay_dec}\n\\end{figure}\n\nIn terms of maximum delays under the unidirectional topology using the decentralized model, Fig.~\\ref{fig_maxDelay_dec} shows the maximum delay evaluation for ST traffic. In contrast to the average delay, the maximum delay does get affected by in-band CDT traffic. In the decentralized model, the CDT traffic is given the highest priority above both ST and BE traffic. Therefore, the maximum delays can reach about $50~\\mu$s more than the ST threshold of $100~\\mu$s.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI01_distUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI10_distUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI20_distUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Unidirectional Topology: Stream admission as a results of TAS.}\n\t\\label{fig_admin_dec}\n\\end{figure}\n\nThe admission rate is very similar to the centralized model since the network parameters used are identical. Fig.~\\ref{fig_admin_dec} shows the stream admission ratio results.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI01_distUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI10_distUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI20_distUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Unidirectional Topology: Stream average signaling delay as a results of TAS.}\n\t\\label{fig_signalDelay_dec}\n\\end{figure}\n\nIn contrast to the centralized model, the decentralized model's in-band CDT traffic implies varied stream signaling delays as shown in Fig.~\\ref{fig_signalDelay_dec} which shows the signaling delay for ST stream registration. As the number of streams generated ($\\pi$) increase, the overall average signaling delay decreases which is due to the increased rejections as more streams attempt to request network resources. In the decentralized model, a rejection by an intermediate bottlenecked switch implies a termination of the CDT traffic and a notification to any previous pending stream records to cancel the potential reservation and eventually notify the source of the rejection. If this rejection happens closer to the source in the CDT registration procedure, then the average delay will be much shorter when compared to an stream acceptance. In general, the average stream signaling delay is in the order of microseconds which is reasonable for most industrial control systems applications.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI01_distUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI10_distUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI20_distUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Unidirectional Topology: Stream Signaling Overhead as a results of TAS.}\n\t\\label{fig_signalOverhead_dec}\n\\end{figure}\n\nGenerally, the decentralized model produced greater signaling overhead than the centralized model since CDT traffic is measured at each data traffic port for incoming and outgoing as shown in Fig.~\\ref{fig_signalOverhead_dec}. Analogous to the signaling delay, the more ST streams accepted, the more overhead is observed. Therefore, as $\\tau$ increases and consequently, the more rejections occur, the lower the overhead. As shown in Fig.~\\ref{fig_signalOverhead_dec}, the results for the signaling overhead with reconfiguration shows a more pronounced difference with varied $\\tau$ values.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI01_distUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI10_distUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI20_distUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Unidirectional Topology: ST Total average throughput measured at the sink as a results of TAS.}\n\t\\label{fig_avgTput_ST_dec}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI01_distUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI10_distUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI20_distUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Unidirectional Topology: BE Total average throughput measured at the sink as a results of TAS.}\n\t\\label{fig_avgTput_BE_dec}\n\\end{figure}\n\nThroughput results are generally the same when compared to the unidirectional centralized model. Fig.~\\ref{fig_avgTput_ST_dec} and Fig.~\\ref{fig_avgTput_BE_dec} shows the average throughput measured at the sink for both ST and BE traffic.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI01_distUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI10_distUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI20_distUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Unidirectional Topology: ST Frame loss ratio as a results of TAS.}\n\t\\label{fig_lossProb_ST_dec}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI01_distUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI10_distUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI20_distUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Unidirectional Topology: BE Frame loss ratio as a results of TAS.}\n\t\\label{fig_lossProb_BE_dec}\n\\end{figure}\n\nSimilarly, the packet loss rate is nearly similar to the unidirectional centralized model as shown  in Fig.~\\ref{fig_lossProb_ST_dec} and Fig.~\\ref{fig_lossProb_BE_dec} for ST and BE traffic in the network respectively. The unidirectional topology with either the centralized or decentralized approach generally get bottlenecked much faster compared to the bi-directional results. Therefore, BE traffic quickly suffers as more ST streams request TAS slot reservation. In terms of added improvements, changing the cycle time (GCL time) to different values depending on both ST and BE traffic proportions can generally improve BE traffic whilst still guaranteeing ST streams. In typical industrial environments, ST streams have generally low data rates, are less frequent, and smaller in size than BE traffic, and therefore can be admitted quite easily without much effect on BE traffic.\n\n\\subsubsection{Bi-Directional Ring Topology}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_2_distBi.pdf}\n\t\\caption{Decentralized Bi-directional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 2$ under different loads~$\\rho_{L}$, mean traffic rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_2_dec_bi}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_3_distBi.pdf}\n\t\\caption{Decentralized Bi-directional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 3$ under different loads~$\\rho_{L}$, mean traffic rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_3_dec_bi}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_4_distBi.pdf}\n\t\\caption{Decentralized Bi-directional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 4$ under different loads~$\\rho_{L}$, mean traffic rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_4_dec_bi}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_5_distBi.pdf}\n\t\\caption{Decentralized Bi-directional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 5$ under different loads~$\\rho_{L}$, mean traffic rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_5_dec_bi}\n\\end{figure}\n\nIn the bi-directional topology using the decentralized model, in-band CDT traffic affects the data traffic similar to the unidirectional model. Fig.~\\ref{fig_delay_2_dec_bi} - \\ref{fig_delay_5_dec_bi} shows the average mean delay evaluation for both ST and BE traffic. As $\\tau$ is increased, i.e., the number of streams at any time increase, the BE slot reservations are reserved for ST streams which affects BE QoS mean delay.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI01_distBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI10_distBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI20_distBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Bi-directional Topology: Max delay as a results of TAS.}\n\t\\label{fig_maxDelay_dec_bi}\n\\end{figure}\n\nIn terms of maximum delay, Fig.~\\ref{fig_maxDelay_dec_bi} shows the maximum delay evaluation for ST traffic. While the reconfiguration approach looks very similar to the centralized model, the ``no reconfiguration'' approach gets affected by in-band CDT traffic that raises the max delay in some cases to $100~\\mu$s.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI01_distBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI10_distBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI20_distBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Bi-directional Topology: Stream admission as a results of TAS.}\n\t\\label{fig_admin_dec_bi}\n\\end{figure}\n\nAdmission rate is exactly the same as the centralized model as shown in Fig.~\\ref{fig_admin_dec_bi} given the same networking parameters.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI01_distBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI10_distBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI20_distBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Bi-directional Topology: Stream average signaling delay as a results of TAS.}\n\t\\label{fig_signalDelay_dec_bi}\n\\end{figure}\n\nFig.~\\ref{fig_signalDelay_dec_bi} shows the signaling delay for ST stream registration.It is observed that as the ST stream generation increases, so does the mean signaling delay up to a stream mean rate, $\\pi$. The mean delay starts to decrease as the load keeps increasing.\n\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI01_distBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI10_distBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI20_distBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Bi-directional Topology: Stream Signaling Overhead as a results of TAS.}\n\t\\label{fig_signalOverhead_dec_bi}\n\\end{figure}\n\nThe stream signaling overhead produces results that are the same as the decentralized model as shown Fig.~\\ref{fig_signalOverhead_dec_bi}. In general, the results between different $\\tau$ values are close since for any $\\tau$ value, almost all the streams are getting accepted generating the same overhead in total.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI01_distBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI10_distBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI20_distBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Bi-directional Topology: ST Total average throughput measured at the sink as a results of TAS.}\n\t\\label{fig_avgTput_ST_dec_bi}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI01_distBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI10_distBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI20_distBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Bi-directional Topology: BE Total average throughput measured at the sink as a results of TAS.}\n\t\\label{fig_avgTput_BE_dec_bi}\n\\end{figure}\n\nFig.~\\ref{fig_avgTput_ST_dec_bi} and Fig.~\\ref{fig_avgTput_BE_dec_bi} shows the average throughput measured at the sink for both ST and BE traffic. The average throughput results are nearly identical to the centralized model. While it is difficult to see, the throughput is slightly less than the centralized since we now incorporate in-band control traffic while reduces the link utilization for data traffic.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI01_distBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI10_distBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI20_distBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Bi-directional Topology: ST Frame loss ratio as a results of TAS.}\n\t\\label{fig_lossProb_ST_dec_bi}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI01_distBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI10_distBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI20_distBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Bi-directional Topology: BE Frame loss ratio as a results of TAS.}\n\t\\label{fig_lossProb_BE_dec_bi}\n\\end{figure}\n\nFig.~\\ref{fig_lossProb_ST_dec_bi} and Fig.~\\ref{fig_lossProb_BE_dec_bi} shows the packet loss ratio for ST and BE traffic in the network. Similar to all the different models and topologies, ST streams have zero frame drops as required by TSN. BE traffic results are nearly identical to the centralized model under bi-directional topology. Similarly, the overall performance is largely improved under the bi-directional topology due to the additional port.\n\nWhile the decentralized model certainly proves to operate nearly identical to the centralized one in terms of QoS metrics and overall admission rate, the main disadvantage is in-band CDT traffic which can lead to delayed ST streams particularly affecting guaranteed maximum delays violations. A work around is to service all the ST streams first, and then service CDT frames before servicing the BE traffic, though this might lead to additional signaling delays depending on the ST load. However, some applications have a more relaxed constraint in guaranteeing maximum delay but require complete segregation of traffic based on the class of service which can be handled using the decentralized model without the overhead complexities of a CNC device. Furthermore, the decentralized model can struggle to find alternate paths without full network visibility, as opposed to the centralized model which can easily reroute streams in the event of failures due to having full network visibility, though this is out of scope for this phase of the project. Moreover, while not tested in our model, adding new devices and removing devices in the decentralized model leads to information flooding across the network that adds complexity and can skew TAS schedules if not handled appropriately. However, in the centralized model, this ``plug and play'' feature can be easily extended to our TSN domain since the CNC has full control and management of the data plane and can adjust and coordinate any scheduling issues in a timely and controllable manner.\n\n\\section{Conclusions and Future Work}  \\label{concl:sec}\nThe IEEE 802.1Qcc framework and the 802.1Qbv traffic shaper enable the\nimplementation of a deterministic forwarding plane that provides\nstrict guarantees to any scheduled traffic service without any flow or\ncongestion control mechanism at the source.  Using an automated\nnetwork configuration is an imperative tool set to provide a unified\ncommunication platform based on commercial of the shelf (COTS) full-duplex\nEthernet with high bandwidth\/low complexity compared to Controller\nArea Networks (CANs), Local Interconnect Networks (LINs), and\nspecialized field-buses in industrial control system applications\n(e.g., industrial control, automotive, avionics).  Such network\ndesigns can form a contract with the source to forward mission\ncritical traffic and to automate the network configuration process using\n802.1Qcc for the full lifetime of the stream. Additionally, depending\non the forwarding plane port traffic shaper (e.g., TAS), the required\nschedules can be passed to the switch servers using general\nuser\/network information protocols (e.g., TLV, Yang, SNMP).\n\nIn this paper, we have investigated the impact of TAS reconfigurations\nin response on dynamic network conditions. We have demonstrated the\neffectiveness of TAS with and without the CNC, i.e., for centralized\n(hybrid) vs. decentralized (fully distributed) models.  We have\nexamined network QoS traffic characteristics when maximizing stream\nadmission to the network whilst reserving some BE time slots in the\nevent of high ST transmission requests.\n\nBased on the insights from the present study we outline the following\nfuture research directions. First, it would be interesting to\njudiciously change the GCL time for switches during reconfiguration\nwhilst satisfying QoS requirements.\nThe studied reconfiguration techniques should also be examined\nin alternate approaches for providing deterministic Qos,\ne.g.,~\\cite{nas2019tsn,seaman2019pat} as well as in the context of related\nQoS oriented routing approaches, e.g.~\\cite{chu2018pre,guc2017uni}.\n\nIn the wider context of QoS networking and related applications,\ndeterministic networking should be examined in the context of\nemerging multiple-access edge computing\n(MEC)~\\cite{doa2019pro,gao2019dyn,mar2019mod,sha2018lay},\nin particular MEC settings for low-latency\napplications~\\cite{elb2018tow,xia2019red,zha2018mob}.\nAs an alternative approach to coordinating the\nreconfigurations, emerging softwarized control paradigms, such as\nsoftware defined networking can be explored~\\cite{ami2018sdn,der2019cou,des2018min,kel2019ada,san2018inf}.\nRegarding the reliability aspects, a potential future research direction is to\nexplore low-latency network coding mechanisms, e.g.,~\\cite{ace2018har,coh2019ada,eng2018exp,gab2018cat,luc2018ful,ma2019high,wun2017cat}, to enhance networking protocols targeting reliable low-latency communication.\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection*{Preliminaries}\n\\subsubsection*{Introduction and preliminaries.}\nThis paper gives a characterization of the class of languages recognized by a model of quantum automata, by using tools from algebraic theory, in particular, varieties of languages.\nMany models of one-way quantum finite automata are present in the literature: the oldest is the Measure-Once model \\cite{BC01,BP02}, characterized by unitary evolution operators and a single measurement performed at the end of the computation. On the contrary, in other models, evolutions and measurements alternate along the computation \\cite{Aal06,KW97}. The model we study is the Measure-Only Quantum Automaton (\\textsc{MOn-1qfa}), introduced in \\cite{BMP10}, in which we allow only measurement operations, not evolution.\nAll these quantum models are generalized by Quantum Automata with Control Language \\cite{BMP03}.\n\nA \\textsc{MOn-1qfa}\\ over the alphabet $\\Sigma$ is a tuple of the form \n$\nA = \\langle \\Sigma \\cup \\{\\#\\},$ $(O_c)_{c\\in\n  \\Sigma\\cup\\{\\#\\}}, \\pi_0, F \\rangle\n$. \nThe complex $m$-dimensional vector $\\pi_0\\in\\mathbb{C}^{1\\times m}$, with unitary norm $||\\pi_0||=1$, \nis called the quantum initial state of $A$.\nFor every $c\\in\\Sigma$, $O_c\\in\\mathbb{C}^{m\\times m}$\nis (the representative matrix of) an idempotent Hermitian operator and denotes an observable.\nThe subset $F\\subseteq V(O_{\\#})$ of the eigenvalues of $O_{\\#}$ is called the spectrum of the quantum final accepting states of $A$.\\mbox{}\\\\\nThe computation dynamics of automaton $A$ is carried out in the following way: \nlet $x=x_1\\ldots x_n\\in\\Sigma^*$, suppose we start from $\\pi_0$, then $A$ measures the system with cascade observables \n$O_{x_1}, \\ldots, O_{x_n}$ (by applying the associated orthogonal projectors) \nand then performs the final measure with the end-word observable $O_{\\#}$, that is the observable of the final accepting states $F$ of $A$.\\\\\nThis last measure returns, as a result, an eigenvalue\n$r\\in V(O_{\\#})$, if $r\\in F$ then we say that the automaton $A$ accepts the word $x\\in\\Sigma^*$, otherwise that $A$ does not accept it. \nWhat is remarkable in this computation dynamics is the probability $p_A(x)$ that $A$ accepts $x=x_1\\cdots x_n$.\nIn the specific case of \\textsc{MOn-1qfa}s\\, it turns out to be of some interest to express $p_A(x)$ using the well-known formalism of quantum density matrices.\nWe say that a language $L$ is recognized by $A$ with isolated cut point $\\lambda$ iff for all $x\\in\\Sigma^*$\\, $p_A(x)>\\lambda \\Leftrightarrow x\\in L$ and there exists a constant value $\\delta>0$ such that $|p_A(x)-\\lambda|\\geq\\delta$.\n\nWe now recall some general definitions and results from the algebraic theory of automata and formal languages.\nFor more details, we refer the reader to, e.g. \\cite{Ei76,Pi86}.\nLet $L$ be a regular language and let $\\langle \\Sigma, Q, \\delta, q_0,\nF \\rangle$ be the minimal deterministic automaton recognizing $L$. For a word $w=\\sigma_1\\cdots \\sigma_n\\in\\Sigma^*$, we define its variation as \n$\\text{Var}_L(w) =  \\#\\{0\\leq k < n \\mid \\delta(\\sigma_1\\cdots \\sigma_k)\\neq \\delta(\\sigma_1\\cdots \\sigma_{k+1})\\}$.\nWe say that $L$ has finite variation iff $ \\text{sup}_{x\\in\\Sigma^*} \\text{Var}_L(x) < \\infty $.\nUsing results and similar techniques as in \\cite{BMP10}, it is not difficult to show  \nthat the class $\\textbf{LMO}(\\Sigma)$ of languages recognized by a \\textsc{MOn-1qfa}\\ with isolated cut point is a boolean algebra of regular languages in $\\Sigma^*$ with finite variation.\nWe say that a language $L\\in\\Sigma^*$ is literally idempotent iff for all $x,y\\in\\Sigma^*$ and $a\\in\\Sigma$, $xa^2y\\in L\\Leftrightarrow xay\\in L$; we say that $L$ is literally idempotent piecewise testable if and only if it lies\nin the boolean closure\nof the following class of languages: \n$\\Sigma^* a_1 \\Sigma^* a_2 \\Sigma^*\\cdots \\Sigma^* a_k \\Sigma^*$, for $a_1, a_2, \\ldots, a_k\\in\\Sigma$ and $a_1\\neq \\cdots \\neq a_k$. \nWe denote by $\\text{liId}$ the class of literally idempotent languages and by $\\text{liId}\\textbf{PT}$ the class of literally idempotent piecewise testable languages.\nFor any language $L$, we call $M(L)$ its syntactic monoid.\nWe say that a class of finite monoids $\\mathbf{A}$ is a (literal) pseudovariety if and only if it is closed under (literal) substructures, homomorphic images and finite direct products, \\cite{KP08}.\nLet $\\mathbf{A}$ be a class of monoids and let $\\Sigma$ be an alphabet. We denote by $V_{\\Sigma}(\\mathbf{A})$ \nthe class of regular languages on $\\Sigma$ having syntactic monoid in $\\mathbf{A}$. \nLet $L, R$ and $J$ be the Green's relations determined by left, right and two-sided ideals, respectively.\nIn this paper we denote by $\\mathbf{R}$ the pseudovariety of $R$-trivial finite monoids and by $\\mathbf{J}$ the pseudovariety of $J$-trivial finite monoids.\nWe also define $\\overline\\mathbf{J}$ as the literal pseudovariety of $J$-trivial syntactic monoids $M(L)$ such that the associated morphism $\\phi_L:\\Sigma^*\\rightarrow M(L)$ satisfies the literal idempotent condition $\\phi_L(\\sigma)\\phi_L(\\sigma)=\\phi_L(\\sigma)$, for every $\\sigma\\in\\Sigma$.\nWe say that a class of regular languages $V:\\Sigma\\rightarrow 2^{\\Sigma^*}$ is a $*$-variety of \\emph{Eilenberg} if $V(\\Sigma)$ is closed under boolean operations, right and left quotient, and inverse homomorphism. Replacing closure under inverse homomorphism by closure under inverse literal homomorphism, we get the notion of literal variety of languages.\nA fundamental result is due to Eilenberg, who showed that there exists a bijection $V_\\Sigma$ from the psuedovarieties of monoids and the $*$-varieties of Eilenberg of formal languages \\cite{Pi86}.\nIn \\cite{KP08}, Kl\\'ima and Pol\\'ak showed the following\n\\begin{theorem}\\label{thm:klimapolak} Let $L\\subseteq\\Sigma^*$. \nIt holds that  \n$L\\in\\text{liId}\\textbf{PT}$ if and only if \n$L\\in V_{\\Sigma}(\\mathbf{J})\\cap \\text{liId}(\\Sigma)$ if and only if  $L\\in V_{\\Sigma}\\left(\\overline{\\mathbf{J}}\\right)$.\n\\end{theorem}\n\\subsubsection*{Results.}\nWe give a direct proof that the class of finite variation regular languages is a $*$-variety of Eilenberg.\nMoreover, we observe that a regular language $L$ has finite variation if and only if its syntactic monoid is $R$-trivial.\nWe proceed further on with our analysis by showing that the class of \\textsc{MOn-1qfa}\\ over $\\Sigma$ is in fact a sub-class of Latvian Automata. This class of \nautomata has been fully characterized algebraically in \\cite{Aal06} as the class of automata recognizing exactly regular languages having syntactic monoids in the class $\\mathbf{BG}$ of block groups. \n\\begin{theorem}\nLet $A$ be a \\textsc{MOn-1qfa}\\ on $\\Sigma$ and let $L_A$ be a language recognized by $A$ with cut-point $\\lambda$ isolated by $\\delta$. \nThen there exists a Latvian automaton $A'$ recognizing \n$L_{A'}=L_{A}$ with cut-point $\\lambda' = \\frac{1}{2}$ isolated by $\\delta' = \\frac{\\delta}{2\\cdot \\text{max}(\\lambda, 1-\\lambda)}$.\n\\end{theorem}\nThis directly implies that $\\textbf{LMO}(\\Sigma)\\subseteq V_{\\Sigma}(\\mathbf{BG})$.\n\nCombining our analysis with the results of \\cite{Aal06} on block groups syntactic monoids and the results of \\cite{BMP10} \non finite variation languages, we prove the following\n\\begin{theorem}\nLet $L\\in\\textbf{LMO}(\\Sigma)$ be a language recognized by some \\textsc{MOn-1qfa}\\ with isolated cutpoint. Then its syntactic monoid $M(L)$ \nis an $R$-trivial block group, formally speaking $ M(L) \\in \\mathbf{BG}\\cap \\mathbf{R} $.\n\\end{theorem}\nSince an $R$-trivial block group is also $J$-trivial, and since $\\textbf{LMO}(\\Sigma)$ is a boolean algebra, we have $\\textbf{LMO}(\\Sigma)\\subseteq V_\\Sigma(\\mathbf{J})$. This, together with Theorem \\ref{thm:klimapolak} and the fact that languages in $\\textbf{LMO}(\\Sigma)$ are literally idempotent, leads to the following\n\\begin{theorem}\n$\\textbf{LMO}(\\Sigma)\\subseteq V_\\Sigma(\\overline\\mathbf{J})$.\n\\end{theorem}\n\nWe now show how languages in \\text{liId}\\textbf{PT}\\ can be recognized by \\textsc{MOn-1qfa}s.\nConsider the language \n$L[a_1, \\ldots, a_k] = \\Sigma^*a_1\\Sigma^*\\cdots\\Sigma^*a_k\\Sigma^*$,\nwhere $a_1,\\ldots, a_k \\in\\Sigma$, $a_i \\neq a_{i+1}$ for $1\\leq i< k$, and let $S = \\{a_1, \\ldots, a_k\\}$.\nFor every $\\alpha \\in S$, let $\\#\\alpha$ be the number of times that $\\alpha$ appears as a letter in the word $a_1a_2\\cdots a_k$.\nLet $j^{(\\alpha)}_1 < j^{(\\alpha)}_2 < \\cdots < j^{(\\alpha)}_{\\#\\alpha}$ be all the indexes such that $\\alpha =\na_{j^{(\\alpha)}_1} = \\ldots = a_{j^{(\\alpha)}_{\\#\\alpha}}$. We define, for every $\\alpha\\in S$, two orthogonal projectors of dimension $(k+1)\\times (k+1)$: the up operator $P^{(k)}_{\\nearrow}(\\alpha)$ and the down operator $P^{(k)}_{\\searrow}(\\alpha)$, such that\n\\[\n\\left(P^{(k)}_{\\nearrow}(\\alpha)\\right)_{rs} =\n\\left\\{\n\t\\begin{array}{cl}\n\t1 & \\mbox{ if } r=s \\mbox{ and } \\forall\\, 1\\leq i\\leq \\#\\alpha \\mbox{ it holds } r,s\\notin\\{j_{i}^{\\alpha}, j_{i}^{\\alpha}+1\\},\\\\\n\t\\frac{1}{2} & \\mbox{ if } \\exists\\, 1\\leq i\\leq \\#\\alpha \\mbox{ such that } r,s\\in\\{j_{i}^{\\alpha}, j_{i}^{\\alpha}+1\\},\\\\\n\t0 & \\mbox{ otherwise,}\n\t\\end{array}\n\\right.\n\\]\n\\[\n\\left(P^{(k)}_{\\searrow}(\\alpha)\\right)_{rs} =\n\\left\\{\n\t\\begin{array}{cl}\n\t\\frac{1}{2} & \\mbox{ if } r=s \\mbox{ and } \\exists\\, 1\\leq i\\leq \\#\\alpha \\mbox{ such that } r,s\\in\\{j_{i}^{\\alpha}, j_{i}^{\\alpha}+1\\},\\\\\n\t-\\frac{1}{2} & \\mbox{ if } r\\neq s \\mbox{ and } \\exists\\, 1\\leq i\\leq \\#\\alpha \\mbox{ such that } r,s\\in\\{j_{i}^{\\alpha}, j_{i}^{\\alpha}+1\\},\\\\\n\t0 & \\mbox{ otherwise.}\n\t\\end{array}\n\\right.\n\\]\nBy calling $e_j$ the boolean row vector such that $(e_j)_i=1\\Leftrightarrow i=j$, we define $A[a_1,\\ldots,a_k]=\\langle \\Sigma\\cup\\{\\#\\}, \\pi_0^{(k)}, \\{O_\\sigma^{(k)}\\}_{\\sigma\\in\\Sigma\\cup\\{\\#\\}}, F^{(k)} \\rangle$ as the \\textsc{MOn-1qfa}\\ where\n\\begin{itemize}\n\t\\item $\\pi_0^{(k)}=e_1\\in\\mathbb{C}^{1\\times(k+1)}$,\n\t\\item for $\\alpha\\in S$, the associated projectors of $O_\\alpha^{(k)}$ are $P^{(k)}_{\\nearrow}(\\alpha)$ and $P^{(k)}_{\\searrow}(\\alpha)$,\n\t\\item with each $O_\\sigma^{(k)}$ such that $\\sigma\\in\\Sigma\\setminus S$, we associate the projector $I_{(k+1)\\times(k+1)}$,\n\t\\item the projector of the accepting result of $O_\\#^{(k)}$ is $(e_{k+1})^Te_{k+1}$, i.e. the $(k+1)\\times(k+1)$ boolean matrix having a 1 only in the bottom right entry.\n\\end{itemize}\nA careful analysis of the behavior of $A[a_1, \\ldots, a_k]$ leads to the following\n\\begin{theorem}\\label{th:PTtoMON}\nThe automaton $A[a_1, \\ldots, a_k]$ recognizes $L[a_1, \\ldots,\na_k]$ with cutpoint $\\lambda = \\frac{1}{2^{2k+1}}$ isolated by \n$\\delta = \\frac{1}{2^{2(k+1)}}$.\n\\end{theorem}\nSince the class \\text{liId}\\textbf{PT}\\ is the boolean closure of languages of the form $L[a_1, \\ldots, a_k]$, and $\\textbf{LMO}(\\Sigma)$ is a boolean algebra, Theorem \\ref{th:PTtoMON} implies that all literally idempotent piecewise testable languages can be recognized by \\textsc{MOn-1qfa}s.\nThe observations made up to this point imply our main result:\n\\begin{theorem}\\label{CharacterizationResult}\n$\n\\textbf{LMO}(\\Sigma) = V_{\\Sigma}(\\overline{\\mathbf{J}}) = \\text{liId}\\textbf{PT}(\\Sigma).\n$\n\\end{theorem}\nTheorem \\ref{CharacterizationResult} allows us to prove the existence of a polynomial time algorithm for deciding $\\textbf{LMO}(\\Sigma)$ membership:\n\\begin{theorem}\\label{th:LMOalgo}\nGiven a regular language $L\\in\\Sigma^*$, the problem of determining whether $L\\in\\textbf{LMO}(\\Sigma)$ is decidable in time $O((|Q|+|\\Sigma|)^2)$, where $|Q|$ is the size of the minimal deterministic automaton for $L$.\n\\end{theorem}\nThis algorithm first constructs the minimal deterministic automaton $A_L$ for $L$ in time $O(|Q|\\log(|Q|))$ as shown in \\cite{H71}. Then, in time $O(|Q|)$, it checks whether $L$ is literally idempotent by visiting all the vertices in the graph of $A_L$. Finally, it verifies whether $L$ is piecewise testable in time $O((|Q|+|\\Sigma|)^2)$ with the technique shown in \\cite{T01}. The fact that $\\textbf{LMO}(\\Sigma)=\\text{liId}\\textbf{PT}(\\Sigma)$ completes the proof.\n\n\\subsubsection*{Acknowledgements:} The authors wish to thank Alberto Bertoni for the stimulating discussions which lead to the results of this paper.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{introduction}}\n\nThe theory of cosmic inflation\n\\cite{starobinsky-1980,guth-1981,linde-1982,albrecht-1982,linde-1983}\nstands now on solid observational foundations, as several of its key predictions\nhave been confirmed by precise measurements of the\ncosmic microwave background radiation \\cite{Planck-inflation-2015}.\n\nAccording to the inflationary paradigm, the universe undergoes a period of\naccelerated expansion in the early stages of its evolution,\nwhich is driven by a scalar field --- the inflaton --- slowly rolling down its potential.\nSuch an early inflationary period not only solves the flatness, horizon, homogeneity,\nisotropy, and primordial monopole problems,\nbut also provides the seeds for the formation of the observed large-scale structures\nof the universe.\n\nA period of accelerated expansion is not exclusive to the early stages of\nevolution of the universe. In fact, cosmological observations have shown, not\nwithout surprise, that accelerated expansion is also taking place at the\npresent time \\cite{riess-1998,perlmutter-1999}, implying the existence of an\nunknown form of energy\n--- dubbed as dark energy --- which accounts for a substantial part of the total\nenergy density of the universe \\cite{Planck-parameters-2015}.\nWithin the $\\Lambda$CDM concordance model, this dark energy is assumed to be\na cosmological constant.\nHowever, this simple explanation raises problems of its own \\cite{weinberg-1989},\na circumstance that led to the hypothesis that dark energy could be identified\nwith a scalar field \\cite{caldwell-1998}, as in the inflationary paradigm.\n\nIn addition to dark energy, the concordance cosmological model also includes\ncold dark matter, which accounts for about one quarter of the total energy density\nof the universe \\cite{Planck-parameters-2015}.\nAlthough the existence of dark matter has been inferred by its gravitational\neffects on a multiplicity of astrophysical and cosmological phenomena,\nit has so far eluded a direct detection and, after decades of intense experimental\nefforts, its physical nature remains a mystery \\cite{bertone-2018}.\nSuch circumstances led to the consideration of a wider range of dark-matter candidates,\nincluding the possibility that a scalar field, similar to those appearing in the models\nof inflation and dark energy, could play the role of dark matter.\n\nSince inflation, dark energy, and dark matter can all be identified with\nscalar fields, it is natural to try to unify these seemingly disparate phenomena\nunder the same theoretical roof using these fields.\n\nSuch a unified description was proposed in Refs.~\\cite{liddle-2006,liddle-2008}.\nThere, inflation was assumed to be of the usual (cold) type,\nfollowed by a post-inflationary reheating period,\nin which the decay of the inflaton field was required to be incomplete,\nleaving a remnant that behaved like cold dark matter.\nIn order to reduce the energy density of the remnant to the level required by\ncosmological observations, two possibilities were considered:\nmodification of the decay rate during the reheating process \\cite{liddle-2006} or\nintroduction of an additional period of thermal inflation,\ndriven by a separate field, at lower energy densities \\cite{liddle-2008}.\nIn what concerns dark energy, this scenario assumed a non-zero vacuum energy\nfor the inflaton\/dark-matter field, motivated by a combination of the string\nlandscape picture and the anthropic principle.\n\nA new scenario for a triple unification, in which dark energy is described not\nby a cosmological constant, but rather by a dynamical scalar field, was soon\nafterwards proposed \\cite{henriques-2009}. Within a two-scalar-field\ncosmological model inspired by supergravity, one of the fields played the role\nof dark energy, inducing the present accelerated expansion of the universe,\nwhile the second field played the roles of both inflaton and dark matter.\nBecause inflation was assumed to be of the warm type \\cite{berera-1995}, no\ndistinctive post-inflationary reheating phase was required; soon after the\nsmooth transition to the radiation-dominated era, the energy transfer from the\ninflaton field to the radiation bath ceased and the former began to oscillate\naround the minimum of the potential, thus mimicking the behavior of a\ncold-dark-matter fluid. However, despite its success in unifying inflation,\ndark energy, and dark matter within a single framework, this two-scalar-field\ncosmological model was not entirely satisfactory, since it accounted for just\na fraction of the dark matter content of the universe.\n\nThe purpose of the present article is to provide a unified description\nof inflation, dark energy, and dark matter in a more general setting, namely,\nwithin a two-scalar-field cosmological model\ngiven by the action\\footnote{Throughout this article we will adopt the natural\nsystem of units and use the notation\n$\\kappa\\equiv \\sqrt{8\\pi G}= \\sqrt{8\\pi}\/m_\\texttt{P}$,\nwhere $G$ is the gravitational constant and\n$m_\\texttt{P} = 1.22 \\times 10^{19}\\, {\\rm GeV}$ is the Planck mass.}\n\\begin{align}\n S = {} & \\int d^4x \\sqrt{-g} \\bigg[\n  \\frac{R}{2\\kappa^2} - \\frac12 (\\nabla \\phi)^2\n    \\nonumber\n  \\\\\n  & - \\frac12 e^{-\\alpha\\kappa\\phi} (\\nabla \\xi)^2\n  - e^{-\\beta\\kappa\\phi} V(\\xi) \\bigg],\n \\label{action 2SF}\n\\end{align}\nwhere $g$ is the determinant of the metric $g_{\\mu\\nu}$, $R$ is the Ricci\nscalar, $\\phi$ and $\\xi$ are scalar fields, and $\\alpha$ and $\\beta$ are\nindependent dimensionless parameters\\footnote{The triple unifications of\ninflation, dark energy, and dark matter proposed in\nRefs.~\\cite{liddle-2006,liddle-2008} and Ref.~\\cite{henriques-2009} correspond\nto models given by action~(\\ref{action 2SF}) with, respectively,\n$\\alpha=\\beta=0$, $\\phi=0$, and $V(\\xi)=V_0+\\frac12 M^2 \\xi^2$, where $V_0$\nand $M$ are arbitrary constants, and $\\alpha=0$, $\\beta=-\\sqrt2$, and\n$V(\\xi)=A_1[1-2A_2 \\exp(-\\sqrt2\\kappa \\xi)+A_3 \\exp(-2\\sqrt2\\kappa \\xi)]$,\nwhere the constants $A_i$ are related to fundamental quantities of the\nSalam-Sezgin six-dimensional supergravity theory.}. Such an action, with a\nnon-standard kinetic term and an exponential potential, arises in a great\nvariety of gravity theories, such as the Jordan-Brans-Dicke theory,\nKaluza-Klein theories, $f(R)$-gravity, and string theories (see\nRefs.~\\cite{berkin-1991,starobinsky-2001} for a derivation of the above action\nin the context of these theories). More recently, it has been shown that this\naction also arises in the context of hybrid metric-Palatini theories of\ngravity \\cite{harko-2012,tamanini-2013}.\n\nFor an appropriate choice of the potential $V(\\xi)$, the two-scalar-field\ncosmological model given by action~(\\ref{action 2SF}) allows for a unified\ndescription of inflation, dark energy, and dark matter, in which the scalar\nfield $\\xi$ plays the roles of both inflaton and dark matter, while the scalar\nfield $\\phi$ plays the role of dark energy.\n\nThe simplest potential providing such a triple unification has the form\n\\begin{align}\n V(\\xi) & = V_a + \\frac12 m^2 \\xi^2,\n  \\label{potential xi}\n\\end{align}\nwhere $V_a$ and $m$ are constants, related, respectively, to the energy density\nof dark energy and to the $\\phi$-dependent mass of the scalar field\n$\\xi$, defined as\n\\begin{align}\n M_\\xi^2 (\\phi) & = m^2 e^{-\\beta\\kappa\\phi}.\n   \\label{mass xi}\n\\end{align}\nWhile emphasizing that a triple unification such as the one proposed in this\narticle could be achieved by any potential whose expansion around its minimum\nhas the form $A+B\\xi^2+\\dots$, for definiteness we will use the potential\ngiven by Eq.~(\\ref{potential xi}).\n\nIn what follows, let us briefly outline the key aspects of the triple unification\nproposed in this article.\n\nInflation is assumed to be of the warm type.\nEnergy is continuously transferred from the inflaton field $\\xi$\n(and also from the dark-energy field $\\phi$) to a radiation bath,\nthereby ensuring that the energy density of the latter is substantial\n--- albeit sub-dominant --- throughout the inflationary expansion\nand that a smooth transition to a radiation-dominated era takes place\nwithout the need for a distinctive post-inflationary reheating phase.\n\nThe dissipation coefficients, mediating the energy trans\\-fer from the scalar\nfields to the radiation bath, have a generic dependence on the temperature,\nnamely, $\\Gamma \\propto T^p$ ($p$ constant), and, immediately after the end of\nthe inflationary period, are exponentially suppressed, becoming negligible\nshortly afterwards.\n\nShortly after the end of the inflationary period, the inflaton $\\xi$ decouples from\nradiation and begins to oscillate rapidly around the minimum of its potential,\nthus behaving on average like a pressureless nonrelativistic fluid, i.e.,\nlike cold dark matter.\n\nDue to the non-standard kinetic term and the exponential factor in the potential,\nthe energy density of cold dark matter depends explicitly on the scalar field $\\phi$,\nimplying that, in general, this quantity does not evolve exactly\nas ordinary baryonic matter.\n\nAfter a radiation-dominated era, encompassing the primordial nucleosynthesis period,\ncold dark matter, together with ordinary baryonic matter,\ndominates the dynamics of the universe, giving rise to a matter-dominated era,\nlong enough to allow for structure formation.\n\nAt recent times, the scalar field $\\phi$ finally emerges as the dominant component\nof the universe, giving rise to a second era of accelerated expansion,\nthus behaving like dark energy.\n\nThese key aspects of the proposed triple unification will be detailed\nin the body of the article.\n\nTo conclude this introductory section, let us point out that unified\ndescriptions of inflation, dark energy, and dark matter have been proposed in\nseveral other\ncontexts~\\cite{capozziello-2006,bose-2009,santiago-2011,odintsov-2019,lima-2019,ketov-2020,odintsov-2020}.\n\nThis article is organized as follows. The evolution equations for the\ntwo-scalar-field cosmological model are presented in the next section. For\nclarity, the cosmic evolution is divided into two stages, the first\ncorresponding to the inflationary period and the transition to the\nradiation-dominated era (Sect.~\\ref{1st stage}) and the second encompassing\nthe radiation-, matter-, and dark-energy-dominated eras (Sect.~\\ref{2nd\nstage}). The continuity of the different physical quantities at the transition\nbetween the first and second stages of evolution is analyzed in\nSect.~\\ref{transition}. Numerical solutions are presented in\nSect.~\\ref{numerical}, which is divided into three subsections. In the first,\nwe analyze the case $\\alpha=\\beta$, while the second is devoted to the case\n$\\alpha\\neq\\beta$. Dissipative effects during inflation are analyzed in the\nlast subsection. Finally, in Sect.~\\ref{conclusions}, we present our\nconclusions.\n\n\n\\section{Two-scalar-field cosmological model\\label{2SF model}}\n\nOur analysis of the cosmic evolution is divided into two stages: the first\ncorresponds to the inflationary period and the transition to the\nradiation-dominated era, while the second encompasses the radiation-, matter-,\nand dark-energy-dominated eras.\n\n\n\n\\subsection{First stage of evolution: the inflationary era\\label{1st stage}}\n\nWe assume inflation to be of the warm type (for reviews, see\nRefs.~\\cite{berera-2009,bastero-gil-2009}).\nIn this inflationary paradigm, a continuous transfer of energy from the\ninflaton field to radiation ensures that the energy density of the latter\nremains substantial --- albeit sub-dominant --- throughout the inflationary era.\nThis energy transfer also guarantees that the\ntransition to a radiation-dominated era takes place in a smooth manner.\nIt contrasts with the usual (cold) inflationary paradigm,\nin which radiation is severely diluted during inflation, a circumstance that\nmakes a distinctive post-inflationary reheating process necessary in order\nto recover the standard cosmic evolution.\n\nIn our model, radiation is described by a perfect fluid with an equation-of-state\nparameter $w_\\texttt{R}=p_\\texttt{R}\/\\rho_\\texttt{R}=1\/3$, where $p_\\texttt{R}$\nand $\\rho_\\texttt{R}$ are the pressure and the energy density of the fluid, respectively.\n\nThe energy density of radiation is sustained, during the inflationary period,\nby a continuous transfer of energy from the scalar fields $\\xi$ and $\\phi$,\nwhich is  accomplished by the introduction of dissipative terms with\ncoefficients $\\Gamma_\\xi$ and $\\Gamma_\\phi$ into the equations of motion. This\nenergy transfer prevents the radiation bath from being diluted, keeping the\nexpanding universe ``warm''.\n\nTo implement warm inflation, it would suffice to have a continuous and\nsignificative transfer of energy from the inflaton field $\\xi$ to the\nradiation bath, in which case one could simply set the dissipation coefficient\n$\\Gamma_\\phi$ to zero and ignore any energy exchange between the dark-energy\nfield $\\phi$ and radiation. However, as it follows from action (\\ref{action\n2SF}), a direct transfer of energy between the two scalar fields also takes\nplace (for nonvanishing $\\alpha$ and\/or $\\beta$). Therefore, it seems natural\nto also allow for a direct energy transfer from the scalar field $\\phi$ to the\nradiation bath, mediated by a non-zero $\\Gamma_\\phi$, although we shall\nemphasize that this is not essential for the implementation of the\nwarm-inflation scenario.\n\nInflation comes to an end when, due to an increase of dissipative effects,\nthe energy density of the radiation bath smoothly takes over and begins to\ndominate the evolution of the universe.\nAt this point, the dissipation coefficients $\\Gamma_\\xi$ and $\\Gamma_\\phi$ are\nexponentially suppressed and, consequently, the radiation bath decouples from\nthe scalar fields $\\xi$ and $\\phi$ and begins to evolve in the usual manner.\n\nLet us now present the equations governing the cosmic evolution during the\ninflationary period and the transition to a radiation-dominated era.\n\nWe assume a flat Friedman-Robertson-Walker universe\\footnote{Since the\ncurrent cosmological measurements constrain the present-time value of the\ncurvature density parameter $\\Omega_k$ to be very small\n\\cite{Planck-parameters-2015}, a spatially flat universe can be assumed without\nmuch loss of generality.}, given by the metric\n\\begin{align}\nds^2 = -dt^2 + a^2(t) d\\Sigma^2,\n  \\label{FRW metric}\n\\end{align}\nwhere $a(t)$ is the scale factor and $d\\Sigma^2$ is the metric of the\nthree-dimensional Euclidean space.\n\nThe equations of motion for the scalar fields $\\xi(t)$ and $\\phi(t)$ and for\nthe energy density of radiation $\\rho_\\texttt{R}(t)$ are then\n\\begin{align}\n & \\ddot{\\xi} + 3 \\frac{\\dot{a}}{a} \\dot{\\xi} - \\alpha\\kappa \\dot{\\phi}\\dot{\\xi}\n + \\frac{\\partial V}{\\partial \\xi} e^{(\\alpha-\\beta)\\kappa\\phi}\n = - \\Gamma_\\xi \\dot{\\xi} e^{\\alpha\\kappa\\phi}, \\label{ddot-xi-1}\n  \\\\\n & \\ddot{\\phi} + 3 \\frac{\\dot{a}}{a} \\dot{\\phi} + \\frac{\\alpha\\kappa}{2}\n \\dot{\\xi}^2 e^{-\\alpha\\kappa\\phi}\n -\\beta \\kappa V e^{-\\beta \\kappa \\phi}\n = - \\Gamma_\\phi \\dot{\\phi}, \\label{ddot-phi-1}\n  \\\\\n & \\dot{\\rho_\\texttt{R}} + 4 \\frac{\\dot{a}}{a} \\rho_\\texttt{R} = \\Gamma_\\xi\n \\dot{\\xi}^2 + \\Gamma_\\phi \\dot{\\phi}^2, \\label{dot-rho-1}\n\\end{align}\nwhile the Einstein equations for the scale factor $a(t)$ are given by\n\\begin{align}\n& \\bigg( \\frac{\\dot{a}}{a} \\bigg)^2 = \\frac{\\kappa^2}{3} \\bigg(\n \\frac{\\dot{\\phi}^2}{2} + \\frac{\\dot{\\xi}^2}{2} e^{-\\alpha\\kappa\\phi}\n + V e^{-\\beta\\kappa\\phi} + \\rho_\\texttt{R} \\bigg), \\label{dot-a-1}\n   \\\\\n& \\hspace{2.7mm}\\frac{\\ddot{a}}{a} =  - \\frac{\\kappa^2}{3} \\bigg( \\dot{\\phi}^2 +\n \\dot{\\xi}^2 e^{-\\alpha\\kappa\\phi} - V e^{-\\beta\\kappa\\phi} + \\rho_\\texttt{R} \\bigg),\n \\label{ddot-a-1}\n\\end{align}\nwhere an overdot denotes a derivative with respect to time $t$ and the\npotential $V$ is given by Eq.~(\\ref{potential xi}).\n\nNote that the above evolution equations differ from the usual ones in warm\ninflationary models in that they contain extra terms arising due to the presence,\nin action (\\ref{action 2SF}), of a non-standard kinetic term for the field $\\xi$.\n\nInstead of the comoving time $t$, let us use a new variable $u$, related to the\nredshift $z$,\n\\begin{align}\n  u = -\\ln \\left( \\frac{a_0}{a} \\right) = - \\ln (1+z),\n  \\label{variable u}\n\\end{align}\nwhere $a_0\\equiv a(u_0)$ denotes the value of the scale factor at the\npresent time $u_0=0$.\n\nWith this change of variables, the above equations for $\\xi$, $\\phi$, and\n$\\rho_\\texttt{R}$ become\n\\begin{align}\n  &  \\hspace{-0.5mm}  \\xi_{uu} = - \\bigg\\{\n     \\bigg[ \\frac{\\ddot{a}}{a} + 2 \\bigg( \\frac{\\dot{a}}{a} \\bigg)^2\n           +\\frac{\\dot{a}}{a} \\Gamma_\\xi e^{\\alpha\\kappa \\phi} \\bigg]\\xi_u\n    \\nonumber\n  \\\\\n  & \\hspace{8.5mm}\n    - \\alpha\\kappa  \\bigg( \\frac{\\dot{a}}{a} \\bigg)^2 \\phi_u \\xi_u\n    + m^2 \\xi e^{(\\alpha-\\beta)\\kappa \\phi}\n  \\bigg\\} \\bigg( \\frac{\\dot{a}}{a} \\bigg)^{-2},\n      \\label{Eq xi E1}\n  \\\\\n  & \\hspace{-1.0mm}\\phi_{uu} =  - \\bigg\\{\n     \\bigg[ \\frac{\\ddot{a}}{a} + 2 \\bigg( \\frac{\\dot{a}}{a} \\bigg)^2\n           +\\frac{\\dot{a}}{a} \\Gamma_\\phi\n     \\bigg]\\phi_u\n  + \\frac{\\alpha\\kappa}{2} \\bigg( \\frac{\\dot{a}}{a} \\bigg)^2 \\xi_u^2\n  e^{-\\alpha\\kappa\\phi} \\nonumber\n  \\\\\n  & \\hspace{8.5mm}\n   -\\beta\\kappa \\left( V_a +\\frac12 m^2 \\xi^2 \\right)\n   e^{-\\beta\\kappa\\phi}\n  \\bigg\\} \\bigg( \\frac{\\dot{a}}{a} \\bigg)^{-2},\n      \\label{Eq phi E1}\n  \\\\\n  & \\hspace{0mm} \\rho_{\\texttt{R}u} = - 4 \\rho_\\texttt{R}\n  + \\frac{\\dot{a}}{a} \\left( \\Gamma_\\xi \\xi_u^2 + \\Gamma_\\phi \\phi_u^2 \\right),\n      \\label{Eq rho E1}\n\\end{align}\nwhere the subscript $u$ denotes a derivative with respect to $u$; $\\dot{a}\/a$\nand $\\ddot{a}\/a$ are functions of $u$, $\\xi$, $\\xi_u$, $\\phi$, and $\\phi_u$,\ngiven by\n\\begin{align}\n \\left( \\frac{\\dot{a}}{a} \\right)^2 = 2\\kappa^2\n \\frac{ \\left( V_a + \\frac12 m^2 \\xi^2 \\right) e^{-\\beta\\kappa\\phi} +\n \\rho_\\texttt{R} }{6 - \\kappa^2 \\phi_u^2\n - \\kappa^2 \\xi_u^2 e^{-\\alpha\\kappa\\phi}}\n \\label{Eq friedman E1}\n\\end{align}\nand\n\\begin{align}\n& \\frac{\\ddot{a}}{a} = \\frac{\\kappa^2}{3}\n \\Bigg\\{ 2\\kappa^2 \\left[ \\left( V_a+\\frac12 m^2 \\xi^2 \\right)\n e^{-\\beta\\kappa\\phi} + \\rho_\\texttt{R} \\right] \\nonumber\n \\\\\n& \\hspace{6.4mm} \\times \\frac{\\phi_u^2 + \\xi_u^2 e^{-\\alpha\\kappa\\phi}}\n {\\kappa^2 \\phi_u^2 + \\kappa^2 \\xi_u^2 e^{-\\alpha\\kappa\\phi} - 6}\n \\nonumber\n \\\\\n& \\hspace{6.4mm} + \\left( V_a + \\frac12 m^2 \\xi^2 \\right)\n  e^{-\\beta\\kappa\\phi} - \\rho_\\texttt{R}\n    \\Bigg\\}.\n   \\label{Eq Dotdota E1}\n\\end{align}\n\nIn order to solve the above system of equations, one has to specify the\ndissipation coefficients $\\Gamma_\\xi$ and $\\Gamma_\\phi$.\n\nOver the years, a variety of forms has been adopted for these coefficients,\nfrom the simplest, based on general phenomenological considerations, to the\nmore elaborate ones, derived from microscopic quantum field theory. In\ngeneral, the dissipation coefficients $\\Gamma$ appearing in the literature are\nfunctions of the temperature $T$ and\/or the inflaton field $\\xi$, as, for\ninstance, $\\Gamma \\propto T^3\/\\xi^2$\n\\cite{bastero-gil-2013,lima-2019}, $\\Gamma \\propto T$\n\\cite{bastero-gil-2016,rosa-2019,rosa-2019b}, or $\\Gamma \\propto T^{-1}$\n\\cite{bastero-gil-2019}.\n\nIn this article, we will not be concerned with the specific microscopic models\nused to derive the dissipation coefficients. We will adopt instead a\nmodel-independent approach, assuming that, during inflation, these\ncoefficients have a generic dependence on the temperature of the radiation\nbath, namely, $\\Gamma \\propto T^p$. We also assume that, immediately after the\nend of the inflationary period, the dissipation coefficients are exponentially\nsuppressed, becoming negligible soon afterwards. In short, we assume\nthe dissipation coefficients $\\Gamma_\\xi$ and $\\Gamma_\\phi$ to be given by\n\\begin{align}\n \\Gamma_{\\xi,\\phi} = f_{\\xi,\\phi} \\times \\left\\{\n    \\begin{aligned}\n     & T^p,\n     & T\\geq T_\\texttt{E},\n    \\\\\n     & T^p  \\exp \\left[ 1- \\left( \\frac{T_\\texttt{E}}{T} \\right)^q \\right],\n     & T\\leq T_\\texttt{E},\n    \\end{aligned}\n \\right.\n \\label{gammas}\n\\end{align}\nwhere $T_\\texttt{E}$ is the temperature of the radiation bath at the end of\nthe inflationary period, $f_\\xi$ and $f_\\phi$ are positive constants with\ndimension $(\\textrm{mass})^{1-p}$ encoding the details of the microscopic\nmodels used to derive the dissipation coefficients, and $q>0$ and $p$ are\nparameters determining the temperature dependence of these coefficients.\n\nThe suppression of the dissipation coefficients immediately after the\ninflationary period is of paramount importance in our unification proposal in\norder to guarantee that the inflaton field $\\xi$ survives and that it has\nenough energy to mimic the behavior of cold dark matter in a way consistent\nwith cosmological observations (see Sect.~\\ref{2nd stage} below for details).\nA microscopic model in which this suppression is achieved naturally has been\nproposed recently \\cite{rosa-2019}, in the context of the warm little inflaton\nscenario \\cite{bastero-gil-2016}. More specifically, the dissipation\ncoefficient mediating the energy transfer from the inflaton field to\nradiation, which initially is proportional to $T$, becomes exponentially\nsuppressed when the temperature drops below a certain threshold value (roughly\ncoinciding with the end of the inflationary period), thus leading to a stable\ninflaton remnant. Such suppression of the dissipative effects below a\nthreshold temperature is also present in another warm-inflation model\n\\cite{bastero-gil-2019} in which the dissipation coefficient during inflation\nis proportional to $T^{-1}$.\n\nAs will be shown in Sect.~\\ref{num-dissipative},\nfor dissipation coefficients proportional to $T^p$, with $p>2$,\nsuppression of the dissipation coefficients below a threshold\ntemperature occurs naturally, as a result of the background dynamics,\nmaking it unnecessary, for such values of p, to introduce explicitly the\nexponential factor in Eq.~(\\ref{gammas}).\n\nFor our base scenario we will choose $p=1$,\ncorresponding to dissipation coefficients linearly dependent on the temperature,\nand $q=2$ (see Sect.~\\ref{num-base}); other values of the parameters $p$ and $q$\nwill be considered in Sect.~\\ref{num-dissipative}.\n\nFinally, let us recall that the temperature $T$ of the radiation bath is\nrelated to its energy density by\n\\begin{align}\n \\rho_\\texttt{R} = \\frac{\\pi^2}{30}g_* T^4,\n   \\label{temp-rho}\n\\end{align}\nwhere $g_*$ denotes the effective number of relativistic degrees of freedom at\ntemperature $T$. Assuming the standard model of particle physics and taking\ninto account that, at the relevant temperatures, all the degrees of freedom of\nthis model are relativistic and in thermal equilibrium, $g_*$ takes the value\n$106.75$.\n\nSolving Eqs.~(\\ref{Eq xi E1})--(\\ref{Eq Dotdota E1}) allows us to determine\nthe density parameters for radiation and for the scalar fields $\\xi$ and\n$\\phi$,\n\\begin{align}\n& \\Omega_\\texttt{R} = \\frac{\\rho_\\texttt{R}}{\\rho_c}=\n  \\frac{\\kappa^2}{3}\\rho_\\texttt{R} \\left( \\frac{\\dot{a}}{a} \\right)^{-2},\n\\\\\n& \\Omega_\\xi = \\frac{\\rho_\\xi}{\\rho_c}= \\frac{\\kappa^2}{6} \\bigg[\n  \\xi_u^2 \\, e^{-\\alpha\\kappa\\phi}\n  + m^2 e^{-\\beta\\kappa\\phi} \\xi^2\n  \\left( \\frac{\\dot{a}}{a} \\right)^{-2} \\bigg],\n\\\\\n& \\Omega_\\phi = \\frac{\\rho_\\phi}{\\rho_c}= \\frac{\\kappa^2}{3} \\bigg[\n  \\frac{\\phi_u^2}{2} + V_a e^{-\\beta\\kappa\\phi}\n  \\left( \\frac{\\dot{a}}{a} \\right)^{-2} \\bigg], \\label{Omega-phi-E1}\n\\end{align}\nas well as the effective equation-of-state parameter,\n\\begin{align}\n w_{\\rm eff} = \\frac13 \\bigg( \\Omega_\\texttt{R}\n + 3 \\Omega_\\phi \\frac{p_\\phi}{\\rho_\\phi}\n + 3 \\Omega_\\xi \\frac{p_\\xi}{\\rho_\\xi} \\bigg), \\label{eos parameter E1}\n\\end{align}\nwhere $\\rho_c=(3\/\\kappa^2)(\\dot{a}\/a)^2$ is the critical density and the\nenergy density and pressure of the scalar fields $\\xi$ and $\\phi$ are given by,\nrespectively,\n\\begin{align}\n \\rho_\\xi = \\frac12 \\left( \\frac{\\dot{a}}{a} \\right)^2 \\xi_u^2\n e^{-\\alpha\\kappa\\phi}\n + \\frac12 m^2 e^{-\\beta\\kappa\\phi} \\xi^2,\n  \\label{rho xi E1}\n\\\\\n p_\\xi = \\frac12 \\left( \\frac{\\dot{a}}{a} \\right)^2 \\xi_u^2\n e^{-\\alpha\\kappa\\phi}\n - \\frac12 m^2 e^{-\\beta\\kappa\\phi} \\xi^2,\n    \\label{p xi E1}\n\\end{align}\nand\n\\begin{align}\n \\rho_\\phi = \\frac12 \\left( \\frac{\\dot{a}}{a} \\right)^2 \\phi_u^2\n + V_a e^{-\\beta\\kappa\\phi},\n  \\label{rho phi E1}\n \\\\\n  p_\\phi = \\frac12 \\left( \\frac{\\dot{a}}{a} \\right)^2 \\phi_u^2\n - V_a e^{-\\beta\\kappa\\phi}.\n  \\label{p phi E1}\n\\end{align}\n\nIn this article we will be focused on the background dynamics of the\ntwo-scalar-field cosmological model and its capability to reproduce the\nmain stages of evolution of the universe, leaving the analysis of the\nprimordial spectrum of density perturbations and its agreement with\ncosmological observations to future work.\nNevertheless, we would like to emphasize here that warm-inflation models\nhave been shown to be consistent with cosmic microwave background (CMB) data\nfor a large variety of potentials and dissipation coefficients.\nIn particular, predictions of the tensor-to-scalar ratio and the spectral tilt\nof the primordial spectrum were shown to agree with cosmological data for models\nwith a quartic potential and dissipation coefficients proportional to $T$ and $T^3$ \\cite{benetti-2017,arya-2018,bastero-gil-2018,bastero-gil-2018a,motaharfar-2019},\nand also in the case of a quadratic potential and dissipation\ncoefficient proportional to $T^{-1}$ \\cite{bastero-gil-2019}.\nThe former models favor mostly the weak dissipative regime, while the latter is\nconsistent with strong dissipation.\n\nWe now turn to the description of the second stage of evolution, which\nencompasses the radiation-, matter-, and dark-energy-dominated eras.\n\n\n\n\n\\subsection{Second stage of evolution: the radiation-, matter-, and\ndark-energy-dominated eras\\label{2nd stage}}\n\nAs mentioned above, at the end of the inflationary period, the dissipation\ncoefficients $\\Gamma_\\xi$ and $\\Gamma_\\phi$ are exponentially suppressed and,\nsoon afterwards, become negligible, allowing us to set them exactly to zero.\nThis marks the end of the first stage of evolution.\n\nDuring the second stage of evolution, in the absence of dissipation, radiation\ndecouples from the scalar fields $\\xi$ and $\\phi$ and Eq.~(\\ref{Eq rho E1})\nyields the solution\n\\begin{align}\n \\rho_\\texttt{R}=\\rho_{\\texttt{R}0}\\,e^{-4u},\n   \\label{Eq rho E2}\n\\end{align}\nwhere $\\rho_{\\texttt{R}0}\\equiv \\rho_{\\texttt{R}}(u_0)$ denotes the energy\ndensity of radiation at the present time $u_0=0$.\n\nFor its part, the scalar field $\\xi$ begins to oscillate rapidly around its\nminimum, behaving like a nonrelativistic dark-matter fluid with equation of state\n$\\langle p_\\xi \\rangle=0$ \\cite{turner-1983},\nwhere the brackets $\\langle ... \\rangle$\ndenote the average over an oscillation\\footnote{These oscillations take place\nif the mass of the scalar field $\\xi$, given by Eq.~(\\ref{mass xi}),\nis much bigger than the Hubble parameter, $M_\\xi \\gg H \\equiv \\dot{a}\/a$,\na condition that can be easily satisfied by choosing a large\nenough value for the constant $m$.}.\n\nLet us derive an expression for the energy density of the dark-matter fluid\nin terms of $u$ and $\\phi(u)$.\nTo that end, we multiply Eq.~(\\ref{Eq xi E1}) by $\\xi_u$ and use the\ndefinition of $\\rho_\\xi$ given by Eq.~(\\ref{rho xi E1}) to obtain\n\\begin{align}\n& \\rho_{\\xi u}\n +3 \\left( \\frac{\\dot{a}}{a} \\right)^2 \\xi_u^2 e^{-\\alpha\\kappa\\phi}\n -\\frac{\\alpha\\kappa}{2} \\left( \\frac{\\dot{a}}{a} \\right)^2\n   \\xi_u^2 \\phi_u e^{-\\alpha\\kappa\\phi} \\nonumber\n \\\\\n& \\hspace{15mm} + \\frac{\\beta\\kappa}{2} m^2 \\xi^2 \\phi_u\n  e^{-\\beta\\kappa\\phi}\n = 0.\n\\end{align}\n\nAveraging over an oscillation period and taking into account that\n$\\big<p_\\xi\\big>=0$ implies $\\big< \\xi^2 \\big>\n=\\rho_\\xi m^{-2} e^{\\beta\\kappa\\phi}$ and\n$\\big< \\xi_u^2 \\big> = \\rho_\\xi (\\dot{a}\/a)^{-2} e^{\\alpha\\kappa\\phi}$,\nthe above equation can be written as\n\\begin{align}\n \\rho_{\\xi u} + 3 \\rho_\\xi\n - \\frac{(\\alpha-\\beta)\\kappa}{2} \\rho_\\xi \\phi_u = 0,\n\\end{align}\nyielding the solution\n\\begin{align}\n \\rho_\\xi = C e^{-3u} e^{\\frac{(\\alpha-\\beta)\\kappa}{2}\\phi},\n   \\label{rho dark matter}\n\\end{align}\nwhere $C$ is a constant whose value is fixed by current cosmological\nmeasurements [see Eq.~(\\ref{condition 2}) below].\n\nAs expected, the energy density of dark matter is proportional to $e^{-3u}$\n(or, in terms of the scale factor, proportional to $a^{-3}$), due to the fact that\nthe potential $V(\\xi)$ was chosen to be quadratic.\nBut it also depends directly on the scalar field $\\phi$,\nthrough an exponential factor, as a consequence of both the non-standard\nkinetic term of the scalar field $\\xi$ and the exponential potential\n[see action~(\\ref{action 2SF})].\nAs will be seen in Sect.~\\ref{numerical}, such dependence of $\\rho_\\xi$ on the\ndark-energy field $\\phi$ has implications on the cosmic evolution, leading\nto a non-simultaneous peaking of the energy densities\nof dark matter and ordinary baryonic matter.\n\nNow, taking into account the above expressions for the energy densities of\nradiation and dark matter,\nthe evolution Eqs.~(\\ref{Eq xi E1})--(\\ref{Eq Dotdota E1})\ncan be considerably simplified, yielding\n\\begin{align}\n \\phi_{uu} = {}& - \\bigg\\{\n  \\bigg[ \\frac{\\ddot{a}}{a} + 2 \\bigg( \\frac{\\dot{a}}{a} \\bigg)^2 \\bigg] \\phi_u\n  -\\beta\\kappa V_a e^{-\\beta\\kappa\\phi} \\nonumber\n\\\\\n  & +\\frac{(\\alpha-\\beta)\\kappa C}{2}\n  e^{\\frac{(\\alpha-\\beta)\\kappa}{2}\\phi} e^{-3u} \\bigg\\} \\left(\n\\frac{\\dot{a}}{a} \\right)^{-2}, \\label{Eq phi E2}\n\\end{align}\nwith\n\\begin{align}\n \\left( \\frac{\\dot{a}}{a} \\right)^2= {}&\n 2\\kappa^2 \\bigg[ V_a e^{-\\beta\\kappa\\phi}\n + \\left( \\rho_{\\texttt{BM}0}\n + C e^{\\frac{(\\alpha-\\beta)\\kappa}{2}\\phi} \\right) e^{-3u} \\nonumber\n\\\\\n & + \\rho_{\\texttt{R}0}e^{-4u} \\bigg]\n \\left( 6-\\kappa^2 \\phi_u^2 \\right)^{-1},\n\\label{Eq friedman E2}\n\\end{align}\nand\n\\begin{align}\n \\frac{\\ddot{a}}{a} = {}& \\frac{\\kappa^2}{6} \\bigg\\{\n 4\\kappa^2 \\bigg[ V_a e^{-\\beta\\kappa\\phi}\n     + \\left( \\rho_{\\texttt{BM}0}\n     + C e^{\\frac{(\\alpha-\\beta)\\kappa}{2}\\phi} \\right) e^{-3u} \\nonumber\n\\\\\n & + \\rho_{\\texttt{R}0} e^{-4u} \\bigg]\n \\phi_{u}^2 \\left( \\kappa^2 \\phi_u^2 - 6 \\right)^{-1}\n + 2 V_a e^{-\\beta\\kappa\\phi} \\nonumber\n\\\\\n & - \\left( \\rho_{\\texttt{BM}0}\n   + C e^{\\frac{(\\alpha-\\beta)\\kappa}{2}\\phi} \\right) e^{-3u}\n   - 2\\rho_{\\texttt{R}0} e^{-4u} \\bigg\\}.\n \\label{Eq Dotdota E2}\n\\end{align}\n\nIn the above equations, we have introduced ordinary baryonic matter, described\nas a perfect fluid with pressure $p_\\texttt{BM}=0$ and energy density\n\\begin{align}\n \\rho_\\texttt{BM} = \\rho_{\\texttt{BM}0}\\,e^{-3u},\n  \\label{rho baryonic matter}\n\\end{align}\nwhere, as usual, the subscript $0$ indicates present-time values.\n\nAgreement with current cosmological measurements \\cite{Planck-parameters-2015}\nrequires $\\rho_{\\texttt{R}0}=9.02\\times10^{-128}\\, m_\\texttt{P}^4$ and\n$\\rho_{\\texttt{BM}0}=8.19\\times10^{-125}\\, m_\\texttt{P}^4$, as well as\n\\begin{align}\n & \\frac12 \\bigg[ \\left( \\frac{\\dot{a}}{a} \\right)^2 \\phi_{u}^2 \\bigg]_{u=u_0}\n + V_a e^{-\\beta\\kappa\\phi_0} = \\rho_{\\texttt{DE}0},\n\\label{condition 1}\n\\\\\n& C e^{\\frac{(\\alpha-\\beta)\\kappa}{2}\\phi_0} = \\rho_{\\texttt{DM}0},\n\\label{condition 2}\n\\end{align}\nwhere the present-time energy densities of dark energy and dark matter are\n$\\rho_{\\texttt{DE}0}=1.13\\times10^{-123}\\, m_\\texttt{P}^4$ and\n$\\rho_{\\texttt{DM}0} = 4.25\\times10^{-124}\\, m_\\texttt{P}^4$,\nrespectively\\footnote{Note that these values for $\\rho_{\\texttt{R}0}$,\n$\\rho_{\\texttt{BM}0}$, $\\rho_{\\texttt{DM}0}$, and $\\rho_{\\texttt{DE}0}$\ncorrespond to a Hubble constant $H_0\\equiv(\\dot{a}\/a)_0= 1.17 \\times 10^{-61}\n\\, m_\\texttt{P}$ or, in more familiar units, $H_0=67\\,{\\rm km}\\,{\\rm\ns}^{-1}\\,{\\rm Mpc}^{-1}$.}.\n\nAs will be shown in Sect.~\\ref{numerical}, for a choice of $V_a$ and $C$\nsatisfying the above conditions, Eqs.~(\\ref{Eq phi E2})--(\\ref{Eq Dotdota E2})\ndescribe a radiation-dominated era, encompassing the primordial\nnucleosynthesis period, followed by an era dominated by the scalar field $\\xi$\n(dark matter) and ordinary baryonic matter, lasting long enough for structure\nformation to occur, and, finally, a dark-energy-dominated era, induced by the\nscalar field $\\phi$, during which the universe undergoes accelerated\nexpansion. The requirement that the transition from the radiation- to the\nmatter-dominated era does not occur too early in the cosmic history and,\nconsequently, does not conflict with primordial nucleosynthesis, as well as\nthe requirement that the expansion of the universe is accelerating at the\npresent time, imposes constraints on the parameters $\\alpha$ and $\\beta$,\nnamely, $|\\alpha-\\beta| \\lesssim 1$ and $|\\beta| \\lesssim 3\/2$.\n\nDuring the second stage of evolution, the density parameter for the scalar\nfield $\\phi$ is given by Eq.~(\\ref{Omega-phi-E1}), while the density\nparameters for radiation, baryonic matter, and the scalar field $\\xi$, are\ngiven by, respectively,\n\\begin{align}\n & \\Omega_\\texttt{R} = \\frac{\\rho_\\texttt{R}}{\\rho_c} =\n \\frac{\\kappa^2}{3}\\rho_{\\texttt{R}0} e^{-4u}\n \\left( \\frac{\\dot{a}}{a} \\right)^{-2},\n\\\\\n & \\Omega_\\texttt{BM} = \\frac{\\rho_\\texttt{BM}}{\\rho_c} =\n \\frac{\\kappa^2}{3}\\rho_{\\texttt{BM}0} e^{-3u}\n \\left( \\frac{\\dot{a}}{a} \\right)^{-2},\n\\\\\n & \\Omega_\\xi = \\frac{\\rho_\\xi}{\\rho_c} =\n \\frac{\\kappa^2}{3} C e^{\\frac{(\\alpha-\\beta)\\kappa}{2}\\phi} e^{-3u}\n \\left( \\frac{\\dot{a}}{a} \\right)^{-2}.\n\\end{align}\n\nThe effective equation-of-state parameter, during this stage of evolution, is\n\\begin{align}\nw_{\\rm eff} = \\frac13 \\bigg[ 1 - \\Omega_\\texttt{BM} - \\Omega_\\xi - \\Omega_\\phi\n\\bigg( 1- 3 \\frac{p_\\phi}{\\rho_\\phi} \\bigg) \\bigg], \\label{eos parameter E2}\n\\end{align}\nwhere $\\rho_\\phi$ and $p_\\phi$ are given by Eqs.~(\\ref{rho phi E1}) and\n(\\ref{p phi E1}), respectively.\n\nA detailed analysis of this stage of evolution for the case $\\alpha=2\/\\sqrt6$\nand arbitrary $\\beta$ can be found in Ref.~\\cite{sa-2020},\nwhere a unified description of dark matter and dark energy was proposed\nwithin the generalized hybrid metric-Palatini theory of gravity.\n\n\n\\subsection{Transition between the first and second stages of evolution\n\\label{transition}}\n\nAs we have just seen, the first stage of evolution, corresponding to the\ninflationary era, is described by Eqs.~(\\ref{Eq xi E1})--(\\ref{Eq Dotdota\nE1}), while the second stage of evolution, encompassing the radiation-, matter-,\nand dark-energy-dominated eras, is described by Eqs.~(\\ref{Eq phi E2})--(\\ref{Eq\nDotdota E2}).\n\nThe transition from the first to the second stage of evolution occurs shortly\nafter the end of inflation, at the beginning of the radiation-dominated era,\nwhen the dissipation coefficients $\\Gamma_\\xi$ and $\\Gamma_\\phi$, given by\nEq.~(\\ref{gammas}), are exponentially suppressed and become negligible. The\nmoment at which the dissipation coefficients are set exactly to zero marks the\nend of the first stage of evolution and the beginning of the second. In what\nfollows, this transition moment will be denoted by $u=u_*$.\n\nAt the transition between the first and second stages of evolution the\ndifferent physical quantities should be continuous.\n\nFor the scalar field $\\phi$ this is achieved by\nsimply requiring the initial value of the second stage to be equal to the\nfinal value of the first stage.\n\nFor the energy density of the scalar field $\\xi$, continuity at the\ntransition requires\n\\begin{align}\n \\rho_\\xi(u_*) =\n C e^{-3u_*}\n e^{\\frac{(\\alpha-\\beta)\\kappa}{2}\\phi_*},\n      \\label{continuity rho-xi}\n\\end{align}\nwhere $\\phi_*\\equiv\\phi(u_*)$ and $\\rho_\\xi(u_*)$ denotes the energy density\nof the $\\xi$ field during the first stage of evolution, given by Eq.~(\\ref{rho\nxi E1}), evaluated at $u=u_*$. Since the constant $C$ is fixed by\nEq.~(\\ref{condition 2}), satisfying the above continuity condition amounts to\nfix the value of $u_*$ or, equivalently, the duration of the first stage of\nevolution,\n\\begin{align}\n \\Delta^{[\\texttt{I}]}u \\equiv {} & u_*-u_i \\nonumber\n \\\\\n = {} & - \\frac13 \\ln \\frac{\\rho_\\xi(u_*)}{\\rho_{\\texttt{DM}0}}\n -\\frac{(\\alpha-\\beta)\\kappa}{6}[\\phi_0-\\phi_*] - u_i,\n  \\label{duration I}\n\\end{align}\nwhere $u_i$ denotes the value of $u$ at the beginning of the first stage of\nevolution.\n\nFinally, for the energy density of radiation, continuity at the transition\nrequires that\n\\begin{align}\n \\rho_\\texttt{R}(u_*) = \\rho_{\\texttt{R}0} e^{-4u_*},\n   \\label{continuity rho-R}\n\\end{align}\nwhere $\\rho_\\texttt{R} (u_*)$ denotes the energy density of radiation during\nthe first stage of evolution [i.e., the solution of Eq.~(\\ref{Eq rho E1})]\nevaluated at $u=u_*$. Because $\\rho_{\\texttt{R}0}$ is fixed by current\ncosmological measurements (see the discussion in Sect.~\\ref{2nd stage}), the\nabove continuity condition leads, in general, to a value of $u_*$ different\nfrom the one determined from Eq.~(\\ref{continuity rho-xi}). To avoid this and,\nat the same time, to maintain adherence to the convention $u_0=0$, the value\nof the variable $u$ at the beginning of the first stage of evolution, $u_i$,\nhas to be shifted by an appropriate amount, i.e., by an amount ensuring that\n$u_*$, determined from Eq.~(\\ref{continuity rho-xi}), also satisfies\nEq.~(\\ref{continuity rho-R}). This procedure fixes the duration of the second\nstage of evolution to be\n\\begin{equation}\n \\Delta^{[\\texttt{II}]}u \\equiv u_0-u_*\n = \\frac14 \\ln \\frac{\\rho_\\texttt{R}(u_*)}{\\rho_{\\texttt{R}0}}.\n  \\label{duration II}\n\\end{equation}\n\nNow, we can proceed to the numerical analysis of the equations of our\ntwo-scalar-field cosmological model.\n\n\n\\section{Numerical solutions\\label{numerical}}\n\nLet us solve numerically Eqs.~(\\ref{Eq xi E1})--(\\ref{Eq Dotdota E1}) and\nEqs.~(\\ref{Eq phi E2})--(\\ref{Eq Dotdota E2}), corresponding to the first and\nsecond stages of evolution, respectively, and present a unified description of\ninflation, dark energy, and dark matter within the two-scalar-field\ncosmological model given by action~(\\ref{action 2SF}).\n\nFor appropriate choices of the initial values for the variables $\\xi$,\n$\\xi_u$, $\\phi$, $\\phi_u$, and $\\rho_\\texttt{R}$ and the values of the\nconstants $\\alpha$, $\\beta$, $V_a$, $m$, $C$, $f_\\xi$, $f_\\phi$, $p$, and $q$ our\nmodel allows for a cosmic evolution consistent with current cosmological\nobservations.\n\nNote that, because of the symmetries of action~(\\ref{action 2SF}), we can assume\n$\\alpha\\geq0$ without loss of generality. The solutions corresponding to\nnegative values of $\\alpha$ can be obtained from the solutions with positive values\nusing the transformation $\\alpha\\rightarrow-\\alpha$, $\\beta\\rightarrow-\\beta$,\nand $\\phi\\rightarrow-\\phi$.\n\nFor clarity of presentation, this section is divided into three subsections,\nwhere we analyze the case $\\alpha=\\beta$, the case $\\alpha\\neq\\beta$, and the\ndependence of the dissipative effects on the parameters $p$ and $q$.\n\n\n\\subsection{Case $\\alpha=\\beta$\\label{num-base}}\n\nWe first analyze the case $\\alpha=\\beta=1$, choosing representative values for\nthe initial conditions, namely $\\xi(u_i)=0.52\\, m_\\texttt{P}$,\n$\\phi(u_i)=10^{-3}\\, m_\\texttt{P}$, $\\xi_u(u_i)=10^{-2}\\, m_\\texttt{P}$,\n$\\phi_u(u_i)=10^{-5}\\, m_\\texttt{P}$, $\\rho_\\texttt{R}(u_i)=0.25\\times\n10^{-12} \\,m_\\texttt{P}^4$, and for the parameters\\footnote{The parameters\n$f_\\xi$ and $f_\\phi$, which determine the amplitude of the energy transfer\nbetween the scalar fields and the radiation bath, are, in general, different.\nHere, for simplicity, they are assumed to be equal.}, namely,\n$V_a=7.64\\times10^{-123} \\, m_\\texttt{P}^4$, $m=10^{-5} \\, m_\\texttt{P}$,\n$f_\\xi=f_\\phi=2$, $p=1$, $q=2$. We call this case the base scenario.\n\n\nInitially, the scalar field $\\xi$ (the inflaton) slowly rolls down its\nquadratic potential, leading to an accelerated expansion of the universe ($60$\n$e$-folds). The energy scale of inflation, defined as\n\\begin{align}\nE_{\\rm inf}=\\left[ e^{-\\beta\\kappa\\phi(u_i)} V[\\xi(u_i)] \\right]^{1\/4},\n\\end{align}\nis $2.3\\times10^{16} \\, {\\rm GeV}$. Because of the dependence on the scalar\nfield $\\phi$, the inflaton mass $M_\\xi$ decreases from $10^{-5} \\,\nm_\\texttt{P}$ to $4.3\\times 10^{-6} \\, m_\\texttt{P}$ during this stage of\ncosmic evolution. Dissipative effects guarantee that energy is continuously\ntransferred from both fields $\\xi$ and $\\phi$ to the radiation bath,\npreventing it from being diluted away by accelerated expansion. Throughout the\ninflationary period, the dissipation ratios, defined as\\footnote{Since we are\nconsidering $\\Gamma_\\phi=\\Gamma_\\xi$, in what follows, for\nsimplicity, both dissipation ratios $Q_\\phi$ and $Q_\\xi$ will\nbe denoted by $Q$.}\n\\begin{align}\n Q_{\\xi,\\phi} = \\frac{\\Gamma_{\\xi,\\phi}}{3H},\n   \\label{q}\n\\end{align}\nremain larger than unity (strong dissipative regime) and the temperature of\nthe radiation bath decreases only slightly (from $3.5\\times10^{15}\\, {\\rm\nGeV}$ to $7.9\\times10^{14}\\, {\\rm GeV}$). The evolution of the relevant\nquantities during the inflationary period is shown in\nFig.~\\ref{Fig_inflation}.\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig1x.pdf}\n\\caption{Evolution of the inflaton field $\\xi$, its mass $M_\\xi$, the\ndissipation ratio $Q$, the temperature $T$ of the radiation bath, the Hubble\nparameter $H$, and the ratio $\\rho_\\texttt{R}\/\\rho_\\xi$ during the\ninflationary period, which extends from $u\\approx-124.7$ to $u\\approx-64.7$\n($60$ $e$-folds of expansion). The energy scale of inflation is $E_{\\rm\ninf}\\approx 2.3\\times10^{16} \\, {\\rm GeV}$.}\n \\label{Fig_inflation}\n\\end{figure}\n\nAt a certain point of the evolution, radiation emerges as the dominant\ncomponent of the universe and the inflationary period comes to an end (see\nFig.~\\ref{Fig_rhoxi-rhoR-E1}). At that moment, when the temperature of the\nradiation bath is $T_\\texttt{E} \\approx 7.9 \\times 10^{14} \\, {\\rm GeV}$, the\ndissipation coefficients $\\Gamma_\\xi$ and $\\Gamma_\\phi$ begin to decrease\nexponentially, implying that soon afterwards the dissipation ratio $Q$ becomes\nnegligible (see Fig.~\\ref{Fig-Gamma}). At $u=u_*\\approx-63.7$, the dissipation\ncoefficients $\\Gamma_\\xi$ and $\\Gamma_\\phi$ are set exactly to zero and the\ndynamics of the cosmic evolution becomes governed by Eqs.~(\\ref{Eq phi\nE2})--(\\ref{Eq Dotdota E2}). This marks the beginning of the second stage of\nevolution.\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig2x.pdf}\n\\caption{Evolution of the energy densities of the scalar field $\\xi$ and of\nradiation. During the inflationary period, due to dissipative effects, the\nlatter remains almost constant. At $u\\approx-64.7$, radiation emerges as the\ndominant component of the universe and the inflationary period comes to an end\n(inset plot). During the radiation-dominated era, the scalar field $\\xi$\nbehaves like a pressureless nonrelativistic fluid (dark matter), becoming\ndominant, together with ordinary baryonic matter, at $u\\approx-8.6$.}\n \\label{Fig_rhoxi-rhoR-E1}\n\\end{figure}\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig3x.pdf}\n\\caption{Evolution of the dissipation ratio $Q$. Strong dissipation, $Q\\gg1$,\nis maintained throughout inflation. At the end of the inflationary period\n($u\\approx-64.7$, vertical grey line), the dissipation coefficients\n$\\Gamma_\\xi$ and $\\Gamma_\\phi$ are exponentially suppressed, implying that\nsoon afterwards $Q$ becomes negligible. At $u=u_*\\approx-63.7$, the\ndissipation coefficients are set exactly to zero, marking the end of the first\nstage of evolution.}\n \\label{Fig-Gamma}\n\\end{figure}\n\nIn the scenario we have been considering, the initial conditions\nand the values of the parameters are such that a strong dissipative regime ($Q>1$)\nis maintained throughout inflation. However, as will be shown below in\nSect.~\\ref{num-dissipative},\na weak dissipative regime can also be obtained for other choices of initial\nconditions and values of the parameters.\n\nThe value of the energy density of the field $\\xi$ at the transition between\nthe first and second stages of evolution, $\\rho_\\xi(u_*)$, is of crucial\nimportance (see Fig.~\\ref{Fig_rhoxi-rhoR-E1}). It cannot be too large,\notherwise the radiation-dominated era will be too short, conflicting with\nprimordial nucleosynthesis, but it cannot be too small, otherwise the\nmatter-dominated era will not be long enough for structure formation to take\nplace or, worse, such an era may not even occur. As discussed above [see\nEq.~(\\ref{duration I})], in order to guarantee an adequate value of\n$\\rho_\\xi(u_*)$, the duration of the first stage of evolution should be chosen\ncarefully; in the example we have been considering (base scenario),\n$\\Delta^{[\\texttt{I}]}u \\approx 61.0$, implying $\\rho_\\xi(u_*) \\approx 4.2\n\\times 10^{-41} \\, m_\\texttt{P}^4$, which allows for radiation- and\nmatter-dominated eras with durations consistent with current cosmological\nobservations.\n\nLet us open here a parenthesis to briefly comment on a recent proposal of\nunification of inflation and dark matter \\cite{rosa-2019}, in which inflation\nis also assumed to be of the warm type. There, the potential of the inflaton\nfield, in addition to a quadratic term, also has a quartic one, which\ndominates for large field values, implying that this scalar field behaves,\nduring most of the radiation-dominated era, including the primordial nucleosynthesis\nperiod, as dark radiation. It begins to behave like dark matter\njust before the transition between the radiation- and the matter-dominated eras.\nThis contrasts with the situation in our model,\nin which the scalar field $\\xi$ behaves like dark matter from the very beginning\nof the radiation-dominated era [see Eq.~(\\ref{rho dark matter}) and\nFig.~\\ref{Fig_rhoxi-rhoR-E1}].\n\nReturning to our base scenario, we point out that, as discussed in\nSect.~\\ref{transition}, continuity of the energy density of radiation at the\ntransition between the first and second stages of evolution fixes the duration\nof the second stage to be $\\Delta^{[\\texttt{II}]}u \\approx 63.7$; at the\ntransition, $\\rho_\\texttt{R}(u_*) \\approx 4.2 \\times 10^{-17} \\,\nm_\\texttt{P}^4$.\n\nDuring the second stage of evolution, the energy densities $\\rho_\\texttt{R}$ and\n$\\rho_\\xi$ evolve according to Eqs.~(\\ref{Eq rho E2}) and (\\ref{rho dark matter}),\nleading successively to radiation- and dark-matter-dominated eras.\nMeanwhile, the scalar field $\\phi$ plays no\nsignificant role in the dynamics of the universe, only becoming dominant at\nrecent times ($u\\approx-0.3$) when it  induces a period of accelerated expansion\n(see Fig.~\\ref{Fig-Omega}).\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig4x.pdf}\n\\caption{Evolution of the density parameters for radiation, baryonic matter,\nand the scalar fields $\\xi$ (inflaton and dark matter) and $\\phi$ (dark\nenergy). The successive inflationary, radiation, matter, and dark energy eras\nare clearly delimited. At the present time, $u_0=0$, the density parameters\nare $\\Omega_{\\phi}(u_0)\\approx 0.69$, $\\Omega_{\\xi}(u_0)\\approx 0.26$,\n$\\Omega_{\\texttt{BM}}(u_0)\\approx0.05$, and\n$\\Omega_{\\texttt{R}}(u_0)\\approx5.5\\times10^{-5}$, in agreement with\ncosmological measurements.}\n \\label{Fig-Omega}\n\\end{figure}\n\nFor our choice of the initial conditions and parameters (and $C \\approx\n4.25\\times10^{-124} \\, m_\\texttt{P}^4$) the conditions given by\nEqs.~(\\ref{condition 1}) and (\\ref{condition 2}) are satisfied, implying that\nthe density parameters at the present time $u_0=0$ become\n$\\Omega_{\\phi}(u_0)\\approx 0.69$, $\\Omega_{\\xi}(u_0)\\approx 0.26$,\n$\\Omega_{\\texttt{BM}}(u_0)\\approx0.05$, and\n$\\Omega_{\\texttt{R}}(u_0)\\approx5.5\\times10^{-5}$, in agreement with\ncosmological measurements \\cite{Planck-parameters-2015}.\n\nIn the future (i.e., for $u>0$), the density parameters for radiation\n$\\Omega_{\\texttt{R}}$, baryonic matter, $\\Omega_{\\texttt{BM}}$,\nand dark matter $\\Omega_{\\xi}$ become negligible in comparison\nwith the density parameter for dark energy $\\Omega_{\\phi}$,\nimplying that the effective equation-of-state parameter $w_{\\rm eff}$\ntends to the value $-1+\\beta^2\/3$ (see Fig.~\\ref{Fig-weff}).\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig5x.pdf}\n\\caption{Evolution of the effective equation-of-state parameter $w_{\\rm eff}$,\nclearly showing the inflationary, radiation, matter, and dark-energy eras\n(the solid blue line corresponds to the base scenario, $\\alpha=\\beta=1$).\nThe value of $w_{\\rm eff}$ for $u\\rightarrow+\\infty$ depends on the parameter\n$\\beta$ (but not on $\\alpha$), such that, for $|\\beta|<\\sqrt2$, the universe\nenters a period of everlasting accelerated expansion. For $\\beta=3\/2$ this\naccelerated expansion is only temporary, since asymptotically $w_{\\rm eff}$\napproaches the value $-1\/4$.}\n \\label{Fig-weff}\n\\end{figure}\n\nIndeed, for negligible $\\Omega_{\\texttt{R}}$, $\\Omega_{\\texttt{BM}}$,\nand $\\Omega_{\\xi}$, Eq.~(\\ref{Eq phi E2}) simplifies considerably,\nbecoming\n\\begin{align}\n \\phi_{uu} =  {} &- \\frac{1}{2\\kappa}(\\kappa\\phi_u -\\beta) (6-\\kappa^2\\phi_u^2).\n\\end{align}\nThis differential equation admits an analytical solution, which, in\nthe limit $u\\rightarrow+\\infty$ and for $\\beta^2<6$, becomes\n$\\phi_u=\\beta\/\\kappa$.\nNow, inserting this asymptotic solution, in Eq.~(\\ref{Eq friedman E2}), one obtains\n$V_a e^{-\\beta\\kappa\\phi}=(6-\\beta^2)H^2\/(2\\kappa^2)$.\nFinally, substituting the above expressions for $\\phi$ and $\\phi_u$\ninto Eq.~(\\ref{eos parameter E2}), one obtains\n$w_{\\rm eff}=-1+\\beta^2\/3$ for $u\\rightarrow+\\infty$.\n\nThe asymptotic behavior of the effective equation-of-state parameter implies that,\nfor $|\\beta|<\\sqrt2$, the universe enters a period of everlasting\naccelerated expansion.\nFor values of $|\\beta|$ slightly above $\\sqrt2$, this accelerated expansion\nstill takes place, but does not last forever.\nFor instance, in the case $\\beta=3\/2$, shown in Fig.~\\ref{Fig-weff},\naccelerated expansion occurs at the present time ($w_{\\rm eff}<-1\/3$)\nand then ceases as $w_{\\rm eff}$ tends to $-1\/4$.\n\nIn summary, in the example we have been considering (base scenario),\ninflation, driven by the scalar field $\\xi$, begins at $u\\approx-124.7$ and\nextends for $60$ $e$-folds, till $u\\approx-64.7$. During this period, a\nradiation bath with temperature of about $10^{15} \\, {\\rm GeV}$ is sustained\nby a continuous and copious energy transfer from the scalar fields $\\xi$ and\n$\\phi$. At the end of inflation, the dissipation coefficients are\nexponentially suppressed; as a consequence, radiation decouples from the\nscalar fields $\\xi$ and $\\phi$ and begins to evolve in the usual manner,\ndominating the dynamics of the universe till $u\\approx-8.6$. In the absence of\ndissipative effects, the scalar field $\\xi$ oscillates around its minimum,\nbehaving like a pressureless nonrelativistic fluid (dark matter), and,\ntogether with ordinary baryonic matter, becomes dominant at $u\\approx-8.6$.\nThe scalar field $\\phi$ (dark energy), having played no significant role in\nthe dynamics of the universe during the preceding eras, finally becomes\ndominant at $u\\approx-0.3$, giving rise to an everlasting period of\naccelerated expansion of the universe.\n\nBecause we are only interested in models that support accelerated expansion at\nthe present time, we restrict our analysis to the cases $\\beta\\lesssim3\/2$.\nFurthermore, as pointed out above, we can assume $\\alpha\\geq0$ without loss of\ngenerality. Our numerical simulations show that, for values of $\\alpha=\\beta$\nlying in this interval, the cosmic evolution is quite similar to the base\nscenario, making it unnecessary to present here a detailed analysis. We just\nrefer the reader to the base scenario and also to Fig.~\\ref{Fig-weff}, where,\nthe cosmic evolution is outlined for three more cases, namely\n$\\alpha=\\beta=0,1\/2$ and $3\/2$.\n\n\n\n\\subsection{Case $\\alpha\\neq\\beta$\n\\label{num-deviations}}\n\nLet us start by considering a varying $\\beta$ for fixed $\\alpha$ (say $\\alpha=1$,\nas in the base scenario considered in the previous subsection).\nThe evolution of the density parameters for the scalar fields $\\xi$ and $\\phi$,\nas well as for radiation and baryonic matter, are shown in\nFigs.~\\ref{Fig_Omegas-beta05} and \\ref{Fig_Omegas-beta0} for the cases\n$\\beta=1\/2$ and $\\beta=0$, respectively,\nwhile the evolution of the effective equation-of-state parameter\nis shown in Fig.~\\ref{Fig-weff-beta} for $\\beta=0,1\/2,1$ and $3\/2$.\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig6x.pdf}\n\\caption{Evolution of the density parameters for the case $\\alpha=1$ and\n$\\beta=1\/2$. The transition from the radiation- to the matter-dominated era\ntakes place at $u\\approx-9.4$, a little earlier than in the base scenario.\nDuring the matter-dominated era, the density parameter of dark energy is a\nnon-negligible fraction (about $3\\%$)\nof the density parameter of matter (dark plus baryonic).\nThe density parameter of dark matter reaches its maximum value at $u\\approx -4.8$,\nwhile for baryonic matter this peaking occurs later, at $u\\approx -1.4$.}\n \\label{Fig_Omegas-beta05}\n\\end{figure}\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig7x.pdf}\n\\caption{Evolution of the density parameters for the case $\\alpha=1$ and\n$\\beta=0$. The transition from the radiation- to the matter-dominated era\ntakes place at $u\\approx-15.2$, much earlier than in the base\nscenario.\nDuring the matter-dominated era, the density parameter of dark energy is a\nsignificant fraction (about $20\\%$)\nof the density parameter of matter (dark plus baryonic).\nThe density parameter of dark matter reaches its maximum value at $u\\approx -7$,\nwhile for baryonic matter this peaking occurs much later, at $u\\approx -0.8$.}\n \\label{Fig_Omegas-beta0}\n\\end{figure}\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig8x.pdf}\n\\caption{Evolution of the effective equation-of-state parameter $w_{\\rm eff}$\nfor $\\alpha=1$ and different values of $\\beta$ (the solid blue line corresponds\nto the base scenario, $\\beta=1$).\nDuring the matter-dominated era, the value of $w_{\\rm eff}$ is approximately zero\nfor $\\beta=1\/2$ and $3\/2$, similarly to the base scenario (inset plot).\nHowever, for $\\beta=0$, the situation changes drastically, with\nthe value of $w_{\\rm eff}$ departing significantly from zero, due to a\ngreater influence of dark energy during this era.\nFor $\\beta=0,1\/2$, and~$1$, the universe initiates, at recent times, a\nperiod of everlasting accelerated expansion.\nFor $\\beta=3\/2$ accelerated expansion takes place at the present time $u_0=0$,\nbut does not last forever, as $w_{\\rm eff}$ asymptotically approaches the value~$-1\/4$.}\n \\label{Fig-weff-beta}\n\\end{figure}\n\nA first change in the cosmic evolution, as compared with the case $\\alpha=\\beta$,\nis related to the duration of~the matter-dominated era.\nThe more $\\beta$ differs from $\\alpha$,\nthe earlier the transition from a radiation to a matter-dominated universe\ntakes place and the longer the duration of the latter.\nThis effect is quite mild for $|\\alpha-\\beta|\\lesssim1$,\nhaving no implications on the viability of the cosmological solutions\n(see Fig.~\\ref{Fig_Omegas-beta05} and the inset of Fig.~\\ref{Fig-weff-beta}).\nHowever, for $|\\alpha-\\beta|\\gtrsim 1$, the effect becomes so strong that it\nbegins to conflict with primordial nucleosynthesis.\nFor instance, in the case $\\alpha=1$ and $\\beta=0$, the transition from the\nradiation- to the matter-dominated era takes place already at $u\\approx-15.2$\n(see Fig.~\\ref{Fig_Omegas-beta0}),\nquite near to the  primordial nucleosynthesis value $u\\approx -18$.\nIf one further increases the value of $|\\alpha-\\beta|$,\nconflict with primordial nucleosynthesis can only be avoided by dropping the\nrequirement that the present-time density parameter $\\Omega_{\\xi}(u_0)$ must be\nequal to the observational value $\\Omega_\\texttt{DM0}$.\nIn other words, one must accept that the\nscalar field $\\xi$ accounts only for part of the dark-matter content of the\nuniverse; the rest must be introduced by hand, together with ordinary baryonic\nmatter. In fact, this was the situation reported in Ref.~\\cite{henriques-2009},\nwhere, for $\\alpha=0$ and $\\beta=-\\sqrt{2}$, the scalar field contributed only\nabout 6\\% to the total matter content of the universe.\n\nAnother change in the cosmic evolution concerns the behavior of the scalar field\n$\\phi$ (dark energy).\nFor $\\alpha\\neq\\beta$, it starts to influence the dynamics of the universe much earlier,\nat the beginning of the matter-dominated era, and its energy density is\na non-negligible fraction of the total energy density throughout\nthe matter-dominated era.\nThe higher $|\\alpha-\\beta|$, the greater the fraction of dark energy during this era.\nFor instance, in the case $\\alpha=1$ and $\\beta=1\/2$,\nshown in Fig.~\\ref{Fig_Omegas-beta05}, the density parameter of dark energy is about\n$3\\%$ of the density parameter of matter (dark plus baryonic),\nwhile in the case $\\alpha=1$ and $\\beta=0$, shown in Fig.~\\ref{Fig_Omegas-beta0},\nthis percentage increases to $20\\%$.\nThis behavior of dark energy implies that, during the matter-dominated era,\nthe value of the effective equation-of-state parameter $w_{\\rm eff}$ differs from zero\n(see Fig.~\\ref{Fig-weff-beta}).\nNote, however, that for the cases $\\beta=1\/2$ and $3\/2$ the dark-energy fraction\nof the total energy density during the matter-dominated era is much smaller than\nin the case $\\beta=0$, meaning that the value of $w_{\\rm eff}$\nin these cases remains close to zero during the matter-dominated era.\n\nA third change occurring in the cosmic evolution is related to the peaking of\nthe energy densities of dark and baryonic matter. In the case $\\alpha=\\beta$,\nthe energy density of dark energy, given by Eq.~(\\ref{rho dark matter}),\nevolves exactly as the energy density of ordinary baryonic matter (i.e., as\n$e^{-3u}$ or, in terms of the scale factor, as $a^{-3}$), meaning that the\nratio between $\\rho_\\xi$ and $\\rho_\\texttt{BM}$ is constant throughout time\nand, consequently, the peaking of these two quantities occurs simultaneously.\nFor $\\alpha\\neq\\beta$ the situation is quite different. The ratio\n$\\rho_\\xi\/\\rho_\\texttt{BM}$ depends directly on the behavior of the scalar\nfield $\\phi$, namely, $\\rho_\\xi\/\\rho_\\texttt{BM} \\propto \\exp\n[(\\alpha-\\beta)\\kappa\\phi\/2]$. This implies that the peaking of the energy\ndensities of dark matter and ordinary baryonic matter does not occur at the\nsame time. For instance, in the case $\\alpha=1$ and $\\beta=0$, shown in\nFig.~\\ref{Fig_Omegas-beta0}, the density parameter of dark matter reaches its\nmaximum value at $u\\approx -7$, while for baryonic matter this peaking occurs\nmuch later, at $u\\approx -0.8$.\n\nAs seen above, agreement with current cosmological data requires\n$|\\alpha-\\beta| \\lesssim 1$, in order to guarantee that the radiation-dominated era\nlasts long enough to encompass primordial nucleosynthesis,\nand $|\\beta|\\lesssim 3\/2$, in order to guarantee accelerated expansion at the\npresent time ($|\\beta|<\\sqrt{2}$ to guarantee that this accelerated expansion\nlasts forever).\nFor such values of the parameters $\\alpha$ and $\\beta$,\nthe two-scalar-field cosmological model given by action~(\\ref{action 2SF})\nallows for a triple unification of inflation, dark energy, and dark matter\nwhich is, at least qualitatively, consistent with observations.\n\nThe above conclusions are confirmed by the numerical solutions obtained for\nother values of the parameters $\\alpha$ and $\\beta$.\nFor completeness, the evolution of the effective equation-of-state parameter for the\ncase $\\beta=1$ and varying $\\alpha$ is presented in Fig.~\\ref{Fig-weff-alpha}.\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig9x.pdf}\n\\caption{Evolution of the effective equation-of-state parameter $w_{\\rm eff}$\nfor $\\beta=1$ and different values of $\\alpha$ (the solid blue line corresponds\nto the base scenario, $\\alpha=1$).\nAt recent times, the universe experiences a second period of accelerated expansion;\nas clearly shown in the bottom right inset plot, independently of the value of\nparameter $\\alpha$, $w_{\\rm eff}$ asymptotically approaches $-2\/3$.\nDue to the earlier influence of dark energy, during the matter-dominated era,\n$w_{\\rm eff}$ differs significantly from zero in the\ncases $\\alpha=0$ and $\\alpha=2$ and mildly in the cases $\\alpha=1\/2$ and\n$\\alpha=3\/2$ (see upper left inset plot).}\n \\label{Fig-weff-alpha}\n\\end{figure}\n\n\n\\subsection{Dissipative effects \\label{num-dissipative}}\n\nWe conclude the present section with an analysis of the\ndissipative effects during both the inflationary period and the transition\nto the radiation-dominated era.\n\nSo far we have considered the dissipation coefficients $\\Gamma_\\xi$ and\n$\\Gamma_\\phi$ to depend linearly on the temperature, i.e., we have chosen\n$p=1$ in Eq.~(\\ref{gammas}). However, as already referred to in Sect.~\\ref{1st\nstage}, in the context of warm inflation several other possibilities have been\nconsidered, from the simplest, based on general phenomenological\nconsiderations, to the more elaborate ones, derived from microscopic quantum\nfield theory. Accordingly, we can extend our analysis to include other values\nof $p$, corresponding, in particular, to dissipation coefficients constant\nthroughout the inflationary period ($p=0$) and dissipation coefficients\ninversely proportional to the temperature ($p<0$).\n\nOur numerical simulations show that, for these values of $p$, the cosmic\nevolution proceeds in a similar way to the case $p=1$. In\nFig.~\\ref{Fig_Gammas_p} the evolution of the dissipation ratio $Q$ is shown\nfor $p=-1,0,1$. For comparison purposes, we choose the duration of the\ninflationary period to be the same in all three cases, which in turns requires\nthe choice $f_\\xi=f_\\phi=2$ for $p=1$, $f_\\xi=f_\\phi=3.1\\times10^{-4}\\,\nm_\\texttt{P}$ for $p=0$, and $f_\\xi=f_\\phi=3.8\\times10^{-8}\\, m_\\texttt{P}^2$\nfor $p=-1$ (the initial conditions and the values of the other parameters are\nthe same as those of the base scenario).\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig10x.pdf}\n\\caption{Evolution of the dissipation ratio $Q$ for $q=2$ and $p=-1,0,1$,\ncorresponding to dissipation coefficients inversely proportional to the\ntemperature, constant, and  proportional to the temperature, respectively.}\n \\label{Fig_Gammas_p}\n\\end{figure}\n\nIn warm inflation, the dissipative effects --- and the consequent energy\ntransfer from the inflaton to the radiation bath --- play an essential role,\nbut they should vanish soon after the end of the inflationary period, allowing\ncosmic evolution to further proceed in the usual way. In our cosmological\nmodel, this is achieved by assuming that, immediately after the end of\ninflation, the dissipation coefficients are suppressed by an exponential term\nparameterized by $q$ [see Eq.~(\\ref{gammas})]. In all the cases considered so\nfar, we have choose $q=2$. However, the suppression of dissipative effects\nafter the inflationary period can be chosen to proceed slower or faster. This\nis illustrated in Fig.~\\ref{Fig_Gammas_q}, where the dissipation ratio $Q$ is\nshown for different values of $q$, namely $q=1,2,3$, with $p=1$ and\n$f_\\xi=f_\\phi=2$.\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig11x.pdf}\n\\caption{Evolution of the dissipation ratio $Q$ for $p=1$ and $q=1,2,3$. The\nhigher the value of $q$, the faster the dissipation ratio $Q$ is suppressed\nafter the inflationary period.}\n \\label{Fig_Gammas_q}\n\\end{figure}\n\nLet us note that for $p>2$, the dissipation coefficients $\\Gamma_\\xi$ and\n$\\Gamma_\\phi$ are suppressed naturally after inflation, with no need for an\nexplicit exponential suppression term in Eq.~(\\ref{gammas}). This can be seen\nas follows.\n\nAfter the end of the inflationary period ($u>u_\\texttt{E}$), cosmic evolution is\ndominated by radiation.\nLet $\\tilde{u}>u_\\texttt{E}$ be the value of $u$ above which the dissipative terms in\nEq.~(\\ref{Eq rho E1}) are much smaller than the energy density of radiation,\ni.e., $(\\dot{a}\/a)(\\Gamma_\\xi \\xi_u^2 + \\Gamma_\\phi \\phi_u^2) \\ll\n\\rho_\\texttt{R}$. Then, for $u>\\tilde{u}$, the energy density of radiation\nevolves approximately as\n$\\rho_\\texttt{R}(u)\\simeq\\tilde{\\rho}_\\texttt{R}e^{-4(u-\\tilde{u})}$, where\n$\\tilde{\\rho}_\\texttt{R} \\equiv \\rho_\\texttt{R}(\\tilde{u})$. Inspection of the\nnumerical solutions of Eq.~(\\ref{Eq friedman E1}) reveals that, for\n$u>\\tilde{u}$, the Hubble parameter is given, in a good approximation, by $H\n\\simeq \\kappa (\\rho_\\texttt{R}\/3)^{1\/2}$. Then, from Eqs.~(\\ref{gammas})\nand (\\ref{q}) one obtains an approximate analytical\nexpression for the dissipation ratio as a function of $u$, valid for\n$u>\\tilde{u}$,\n\\begin{align}\n Q \\simeq\n A_1 \\exp \\left[ 1-(p-2)(u-\\tilde{u})- A_2 e^{q(u-\\tilde{u})} \\right],\n   \\label{Q-approx}\n\\end{align}\nwhere $A_1$ and $A_2$ are given by\n\\begin{align}\n A_1 = {}& \\frac{f_{\\xi,\\phi}}{\\sqrt3\\kappa}\n  \\left( \\frac{30}{\\pi^2 g_*} \\right)^{\\frac{p}{4}}\n  \\tilde{\\rho}_\\texttt{R}^{\\frac{p-2}{4}}, \\\\\n A_2 = {}& \\left( \\frac{\\rho_\\texttt{RE}}{\\tilde{\\rho}_\\texttt{R}}\n           \\right)^{\\frac{q}{4}},\n\\end{align}\nand $\\rho_\\texttt{RE} \\equiv \\rho_\\texttt{R}(u_\\texttt{E})$ denotes the energy\ndensity of radiation at the end of the inflationary period. For $p\\leq2$, the\ndissipation ratio $Q$ does not decrease if $q=0$, but for $p>2$ the parameter\n$q$ can be set to zero and suppression nevertheless takes place (see\nFig.~\\ref{Fig_Gammas_q0}). This result is in agreement with\nRef.~\\cite{lima-2019}, where, in the context of a quintessential inflationary\nmodel, the dissipation coefficient, assumed to depend both on the temperature\nand the inflaton field as $\\Gamma\\propto T^p \\xi^c$, is shown to be a\ndecreasing function of the number of $e$-folds for $p>2$.\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig12x.pdf}\n\\caption{Evolution of the dissipation ratio $Q$ for $q=0$ and $p=1,2,3$. For\n$p>2$, as a result of the background dynamics, suppression takes place even in\nthe absence of the exponential factor in Eq.~(\\ref{gammas}).}\n \\label{Fig_Gammas_q0}\n\\end{figure}\n\nSo far, the cases considered correspond to a strong dissipative regime, for which\n$Q>1$. However, it is of relevance to show that a weak dissipative regime can also\nbe obtained in our two-scalar-field cosmological model for other choices of the\ninitial conditions and the values of the parameters.\nIndeed, as already mentioned above, the confrontation of warm-inflation predictions\nwith the latest CMB data\n\\cite{bastero-gil-2019, benetti-2017,arya-2018,bastero-gil-2018,bastero-gil-2018a,motaharfar-2019}\nhas shown a preference for the weak dissipation regime for certain combinations\nof the inflaton's potential and the dissipation coefficients (for instance,\nfor a quartic potential and a dissipation coefficient proportional to $T$ or $T^3$).\n\nFrom Eq.~(\\ref{q}), it follows that a smaller value of the dissipation ratio\nat the onset of the first stage of evolution, $Q_\\xi(u=u_i)$, can be obtained\nby decreasing $f_\\xi$ and\/or increasing $\\xi(u_i)$ and $\\rho_\\texttt{R}(u_i)$.\nStarting from the base scenario, considered in detail in Sect.~\\ref{num-base},\nit is then straightforward to find a set of initial conditions and values of\nthe parameters corresponding to weak dissipation, as, for instance,\n$\\xi(u_i)=3.19\\, m_\\texttt{P}$, $\\phi(u_i)=10^{-3}\\, m_\\texttt{P}$,\n$\\xi_u(u_i)=10^{-2}\\, m_\\texttt{P}$, $\\phi_u(u_i)=10^{-5}\\, m_\\texttt{P}$,\n$\\rho_\\texttt{R}(u_i)=2.2 \\times 10^{-12}\\, m_\\texttt{P}^4$, $V_a=2.69 \\times\n10^{-123}\\, m_\\texttt{P}^4$, $m=10^{-5}\\, m_\\texttt{P}$, $f_\\xi=0.05$,\n$f_\\phi=25$, $p=1$, and $q=2$. The evolution of the dissipation ratio $Q_\\xi$\nfor this case is shown in Fig.~\\ref{Fig_Q_xi} (the effective equation-of-state\nparameter and the density parameters evolve similarly as in the base scenario,\nmaking it unnecessary to present them here).\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig13x.pdf}\n\\caption{Evolution of the dissipation ratio $Q_\\xi$. Weak dissipation is maintained\nthroughout most of the inflationary period. Towards its end, the dissipation ratio\nsharply increases, reaching values above unity, which facilitates the transition to\nthe radiation-dominated era.}\n \\label{Fig_Q_xi}\n\\end{figure}\n\nThe primordial spectrum of density perturbations of our unification model and\nits agreement with observational data, for the quadratic potential, given by\nEq.~(\\ref{potential xi}), and the dissipation coefficients, given by\nEq.~(\\ref{gammas}), both in the strong and weak dissipative regimes,\nwill be explored in future work.\n\n\n\\section{Conclusions\\label{conclusions}}\n\nIn this article we have presented a unified description of inflation, dark energy,\nand dark matter in a two-scalar-field cosmological model with a non-standard\nkinetic term and an exponential potential.\nSuch models arise in a great variety of gravity theories, such as the\nJordan-Brans-Dicke theory, Kaluza-Klein theories, $f(R)$-gravity, string theories,\nand hybrid metric-Palatini theories of gravity.\n\nIn the proposed triple unification, one of the scalar fields plays the role of\ninflaton and dark matter and the other plays the role of dark energy.\nMore specifically, inflation, assumed to be of the warm type, is driven by the\nscalar field $\\xi$, which, shortly after the end of the inflationary period,\ndecouples from radiation and begins to oscillate rapidly around the minimum\nof its potential, thus behaving like a cold-dark-matter fluid; the second scalar\nfield $\\phi$ emerges, at recent times, as the dominant component of the universe\nand gives rise to an era of accelerated expansion. Thus, seemingly disparate\nphenomena like inflation, dark energy, and dark matter are unified under the\nsame theoretical roof using scalar fields.\n\nThe two-scalar-field cosmological model given by action~(\\ref{action 2SF})\ncontains two dimensionless parameters, $\\alpha$ and $\\beta$ (for $\\alpha=0$,\nthe kinetic term for the scalar field $\\xi$ becomes canonical; for $\\beta=0$,\nthe direct coupling in the potential between the two scalar fields $\\xi$\nand $\\phi$ disappears). These parameters could, in principle, be chosen freely;\nhowever, as detailed in Sect.~\\ref{num-deviations}, the requirement that the\ntransition from the radiation- to the matter-dominated era does not occur too\nearly in the cosmic history and, consequently, does not conflict with primordial\nnucleosynthesis, as well as the requirement that the expansion of the universe is\naccelerating at the present time, imposes constraints on the parameters $\\alpha$\nand $\\beta$, namely, $|\\alpha-\\beta| \\lesssim 1$\nand $|\\beta| \\lesssim 3\/2$.\n\nFor such values of $\\alpha$ and $\\beta$, the picture emerging in our unified\ndescription of inflation, dark energy, and dark matter is consistent with the\nstandard cosmological model.\nIndeed, the inflationary period is followed by a radiation-dominated era that\nencompasses the primordial nucleosynthesis epoch; the matter-dominated era lasts\nlong enough for structure formation to occur; the transition to a\ndark-energy-dominated universe takes place in a recent past;\nand the density parameters for dark matter and dark energy, as well as for radiation\nand ordinary baryonic matter, evaluated at the present time, are in agreement with\ncurrent cosmological observations (see Fig.~\\ref{Fig-Omega} for the case $\\alpha=\\beta=1$,\nFig.~\\ref{Fig_Omegas-beta05} for the case $\\alpha=1$, $\\beta=1\/2$,\nand Fig.~\\ref{Fig_Omegas-beta0} for the case $\\alpha=1$, $\\beta=0$).\n\nOf crucial importance for consistency with the standard cosmological model is\nthe value of the energy density of the scalar field $\\xi$ at the moment when\nthis field begins to oscillate rapidly around its minimum, changing its\nbehavior from an inflaton field to a nonrelativistic dark-matter fluid (i.e.,\nat the transition from the first to second stage of cosmic evolution). If this\nenergy density is too large, the radiation-dominated era is too short,\nconflicting with primordial nucleosynthesis; if it is too small, the\nmatter-dominated era is not long enough for structure formation to take place\n(see Fig.~\\ref{Fig_rhoxi-rhoR-E1}). In our model, an appropriate value of the\nenergy density of the inflaton\/dark-matter field at the transition between the\nfirst and second stages of evolution is achieved by ensuring that, immediately\nafter the end of the inflationary period, the dissipation coefficients,\nresponsible for the energy transfer from the scalar fields $\\xi$ and $\\phi$ to\nthe radiation bath, are rapidly suppressed, becoming negligible soon\nafterwards. When this happens, the scalar field $\\xi$ decouples from radiation\nand, because its effective mass is larger than the Hubble parameter, begins to\noscillate rapidly around its minimum, behaving like dark matter.\n\nIn our cosmological model, the behavior of the dissipation coefficients is\ncontrolled by two parameters, $p$ and $q$. The first sets the temperature\ndependence of the coefficients and the second determines how fast these\ncoefficients are suppressed after the end of the inflationary period [see\nEq.~(\\ref{gammas})]. As we have shown, for $p>2$, the dissipation coefficients\nare suppressed naturally after inflation, with no need for an explicit\nsuppression term, allowing us to set $q=0$. This does not mean, however, that\nthe models with $p\\leq2$, requiring $q\\neq0$, are less admissible. Actually,\nsuch models, have been proposed in recent years\n\\cite{bastero-gil-2016,rosa-2019,rosa-2019b,bastero-gil-2019} and are quite\nsound, both theoretically and observationally.\n\nAs already mentioned above, shortly after the end of the inflationary period,\nthe scalar field $\\xi$ begins to oscillate rapidly around the minimum of its\npotential. We have derived an expression for its energy density during this\noscillating phase [see Eq.~(\\ref{rho dark matter})],\nshowing that it is proportional to\n$\\exp[(\\alpha-\\beta)\\kappa\\phi\/2]$.\nIn the case $\\alpha=\\beta$, this exponential equals unity and dark matter behaves\nexactly as ordinary baryonic matter, i.e., evolves as $a^{-3}$, where $a$ is\nthe scale factor. But for $\\alpha\\neq\\beta$ the situation is quite different;\nthe energy density of dark matter depends directly on the dark-energy field $\\phi$,\nleading to a non-simultaneous peaking of the energy densities of dark matter and\nordinary baryonic matter\n(see Figs.~\\ref{Fig_Omegas-beta05} and \\ref{Fig_Omegas-beta0}).\nFurthermore, in the case $\\alpha\\neq\\beta$, the energy density of the dark energy\nis a non-negligible fraction of the critical energy density throughout the\nmatter-dominated era.\n\nFinally, we have shown that the effective equation-of-state parameter $w_{\\rm\neff}$ tends, asymptotically, to $-1+\\beta^2\/3$, implying that, for\n$|\\beta|<\\sqrt2$, the universe enters a period of everlasting accelerated\nexpansion (for values of $|\\beta|$ slightly above $\\sqrt2$, this accelerated\nexpansion still takes place, but does not last forever).\n\nIn this article, we have proposed a triple unification of inflation, dark\nenergy, and dark matter in a two-scalar-field cosmological model. This is a\nfirst approach, intended to show that such a unification is, in principle,\npossible and that it reproduces, at least qualitatively, the main features of\nthe observed universe. We expect to explore and deepen this model in future\npublications.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\@startsection {section}{1}{\\z@}%\n\n\n\n\\newcommand{\\mc}[1]{\\textcolor{red}{#1}}\n\n\\begin{document}\n\n\\title{Rotating Black Holes in Higher Order Gravity}\n\n\\author{Christos Charmousis}\n\\affiliation{Laboratoire de Physique Th\\'eorique, CNRS, Univ.\\ Paris-Sud, \nUniversit\\'e Paris-Saclay, 91405 Orsay, France}\n\\author{Marco Crisostomi} \n\\affiliation{Laboratoire de Physique Th\\'eorique, CNRS, Univ.\\ Paris-Sud, \nUniversit\\'e Paris-Saclay, 91405 Orsay, France}\n\\affiliation{Institut de physique th\\'eorique, Univ.\\ Paris Saclay, CEA, \nCNRS, 91191 Gif-sur-Yvette, France}\n\\affiliation{AIM, CEA, CNRS, Univ.\\ Paris-Saclay, Univ.\\ Paris Diderot,\nSorbonne Paris Cit\\'e, F-91191 Gif-sur-Yvette, France}\n\\author{Ruth Gregory}\n\\affiliation{Centre for Particle Theory, Durham University, South Road,\nDurham, DH1 3LE, UK}\n\\affiliation{Perimeter Institute, 31 Caroline St., Waterloo, Ontario,\nN2L 2Y5, Canada}\n\\author{Nikolaos Stergioulas}\n\\affiliation{Department of Physics, Aristotle University of Thessaloniki, \n54124 Thessaloniki, Greece}\n\\date{\\small \\today}\n\n\\begin{abstract}\nWe develop a new technique for finding black hole solutions in modified \ngravity that have ``stealth'' hair, i.e., hair whose only gravitational effect\nis to tune the cosmological constant. We consider scalar-tensor theories \nin which gravitational waves propagate at the speed of light, and show that \nEinstein metrics can be painted with stealth hair provided there exists a\nfamily of geodesics always normal to spacelike surfaces. We also present\na novel scalar-dressed rotating black hole that has finite scalar field\nat both the black hole and cosmological event horizons.\n\\end{abstract}\n\n\n\\maketitle\n\n\n\nBlack Holes are a cornerstone in the study of General Relativity (GR), \nbe it theoretical or practical.\nFrom the theoretical perspective, we have several exact solutions \ncorresponding to black holes in GR, with charges such as mass, angular \nmomentum, and even acceleration or Taub NUT parameters. \nOf these, the most important is the Kerr solution\nthat describes the rotating black hole; most astrophysical \nblack holes are believed to be rotating, indeed, the first\ndetection of gravitational waves was from the merger of two \nspinning Kerr black holes \\cite{Abbott:2016blz}. \n\nAnother key objective in gravity is to explain our universe. One of\nthe main challenges in cosmology is to explain late time acceleration:\nIs it due to dark energy, modified gravity, or a very \nsmall cosmological constant~$\\Lambda$, finely tuned by some as \nyet undiscovered mechanism? One fruitful means of explaining \nthe small late time acceleration is to modify gravity in such a way as\nto induce (or tune) a cosmological constant, but if gravity is modified,\nthen it is vital to understand how these modifications affect black holes. \nIf it is not possible to construct astrophysically realistic black hole \nsolutions, then the theory cannot be considered viable. \n\nIn GR, the Kerr solution (see \\cite{Carter:1968ks,CARTER1968399}\nfor inclusion of a cosmological constant) describes the most general \naxisymmetric stationary rotating black hole, moreover, the event horizon \ntelescope \\cite{Doeleman:2009te} hopes soon to be able to directly \nimage the shadow of the black hole at the centre of our galaxy, possibly \nsetting constraints on deviations\nfrom Kerr \\cite{Johannsen:2016vqy}. It is important therefore to investigate \nwhether spinning black hole solutions exist in modified gravity, and if so, \ndo they carry non-trivial extra degrees of freedom?  \n\nIn this letter, we focus on a particular family of modified scalar-tensor \ntheories of gravity, developing a new technique to find, for the first time, \nastrophysically realistic rotating black hole solutions \nwith a nontrivial scalar field, or \\emph{stealth hair}.\nOur method is based on the Hamilton-Jacobi approach\nto finding geodesics. In brief, we prove that a GR solution can also\nbe a solution to modified gravity if the scalar field is the Hamilton-Jacobi\npotential for a geodesic congruence in the spacetime in question.\nThis allows for a wide range of physically interesting stealth solutions,\nin particular we present examples of rotating black holes with stealth hair.\n\nWe focus for definiteness on shift-symmetric scalar-tensor theories \nof gravity in the family of Degenerate Higher Order Scalar-Tensor (DHOST) \ntheories \\cite{Langlois:2015cwa,Crisostomi:2016czh,BenAchour:2016fzp}, \nalthough the technique can be generalised to \nother modifications of gravity. In particular we will focus on the subset of \ntheories where gravitational waves propagate at the speed of \nlight,  $c_T=1$, in accord with the recent multi-messenger neutron \nstar binary merger observation~\\cite{GBM:2017lvd}. \n\nThe most general shift-symmetric scalar-tensor theory of gravity in\nwhich gravitational waves propagate with the speed of light is\n\\begin{equation}\n\\begin{aligned}\n{\\cal L}\\, &= \\, K(X) + G(X) R \n+ A_3 \\phi^{\\mu} \\phi_{\\mu \\nu} \\phi^{\\nu} \\Box \\phi \\\\\n&+ A_4  \\phi^{\\mu} \\phi_{\\mu \\rho} \\phi^{\\rho \\nu} \\phi_{\\nu} \n+ A_5  (\\phi^{\\mu} \\phi_{\\mu \\nu} \\phi^{\\nu})^2 \\,, \n\\end{aligned}\n\\label{ESTlag}\n\\end{equation}\nwhere $K$, $G$ and $A_i$ are all functions of $X = (\\partial\\phi)^2$, \nand we abbreviate $\\partial_\\m \\phi$ as $\\phi_\\mu$, and  $\\nabla_\\n \\nabla_\\m \\phi$\nas $\\phi_{\\m\\n}$. In order to propagate a single scalar degree of freedom \nand avoid Ostrogradski instabilities, $A_{4,5}$ are constrained by\n\\begin{equation}\n\\begin{aligned}\nA_4&= -A_3 +\\frac{1}{8 G}(4 G_X + A_3 X)(12 G_X + A_3 X) \\,, \\\\\nA_5 &= \\frac{A_3}{2 G}(4 G_X + A_3 X)\\,,\n\\end{aligned}\n\\end{equation}\nwhere $G_{X} = \\partial G \/ \\partial X$. These constraints reduce the number of free \nfunctions to three ($K, G$ and $A_3$) and for simplicity we do not consider \nthe cubic Horndeski term in our Lagrangian~(\\ref{ESTlag}), though this can \neasily be reinstated if required.\n\nA family of exact static spherically symmetric solutions, with the scalar field \nplaying the role of dark energy, was initially found in a class of shift-symmetric \nHorndeski theories \\cite{Babichev:2013cya}. It has the nice feature of locally \ndescribing a Schwarzschild geometry while asymptotically approaching a \nself-tuned accelerating cosmology. Since these solutions acquire a metric \nsimilar to that of GR while having a non trivial scalar field, they have been \nwidely called {\\it stealth} solutions\\footnote{These solutions were extended \nand generalised in different modified gravity theories with similar properties, \nsee for example \\cite{Rinaldi:2012vy,Kobayashi:2014eva,Babichev:2017guv,\nChagoya:2016aar,Babichev:2017rti}.}. \nThey can be mapped via disformal transformations to stealth solutions \nof DHOST theories with unitary speed of gravitational waves \n\\cite{Babichev:2017lmw}, and are free of ghost and gradient instabilities \n\\cite{Babichev:2018uiw}. For spherically symmetric stealth solutions in \nDHOST theories see \n\\cite{Chagoya:2018lmv,BenAchour:2018dap,Motohashi:2019sen}.\n\nKnown stealth solutions are spherically symmetric and all feature \nthe same characteristic: a constant kinetic term, $X$, for the scalar field\nthat does not deform an underlying Einstein geometry.\nWe suspect that it is this feature of a constant magnitude of $\\partial \\phi$\nthat allows stealth hair, thus we look for an Einstein manifold,\n$R_{\\m\\n}= \\L g_{\\m\\n}$, admitting such a solution for $\\phi$.\nFirst, the equations of motion under these assumptions \nbecome \n\\begin{equation}\n\\begin{aligned}\n&\\left[ A_3(X_0) \\left({\\cal E}_3 - \\L X_0 \\right) \n- 2 \\left(K_X +4 \\L G_X\\right)|_{_{X_0}} \\right] \\phi_\\m \\phi_\\n\\\\\n& +\\left(K + 2\\L G \\right)|_{_{X_0}}g_{\\m\\n} = 0\n\\end{aligned}\n\\label{EOMg}\n\\end{equation}\nfor the metric and\n\\begin{equation}\n\\begin{aligned}\n&A_3(X_0) \\left( {\\cal E}_4 +2R_{\\m\\n\\rho\\s} \\phi^{\\n\\s}\\phi^\\m \\phi^\\rho \n-3 \\L X_0 \\Box \\phi \\right) \\\\\n& - 2 \\left( K_X +4 \\L G_X \\right)|_{_{X_0}} \\Box \\phi =0 \n\\end{aligned}\n\\label{EOMphi}\n\\end{equation}\nfor the scalar, where for compactness we have defined \n\\begin{equation}\n\\begin{aligned}\n{\\cal E}_3 &\\equiv \\left( \\Box \\phi \\right)^2 - \\left( \\phi_{\\m\\n} \\right)^2 \\,, \\\\\n{\\cal E}_4 &\\equiv \\left( \\Box \\phi \\right)^3 -3 \\Box \\phi \\left( \\phi_{\\m\\n} \\right)^2 \n+ 2 \\left( \\phi_{\\m\\n} \\right)^3 \\,.\n\\end{aligned}\n\\end{equation}\nSetting aside the problem of finding a solution for $\\phi$ momentarily,\nnote that in the above, $A_3$, $G$ etc.\\ are all constants,\nevaluated at some $X=X_0=(\\nabla\\phi)^2$. Since $\\phi_\\mu$ itself\nis not necessarily a constant vector, to satisfy the above in general \nwe must choose subspaces of the general parameter space for\n$\\{A_3,G,K\\}$. Starting with the top line of \\eqref{EOMphi}, we\ndeduce $A_3(X_0)=0$ (unless spacetime has very special\nsymmetries as we will see later), hence \n\\begin{equation}\n(K_X+4\\Lambda G_X)|_{X_0}=0 \\,, \\label{Gconstr}\n\\end{equation}\nwhere we emphasise that this is \\emph{at the specific value} of $X$,\n$X_0$. These two constraints now imply that \\eqref{EOMg} is\nsatisfied, provided we set $\\Lambda = -K\/(2G) |_{X_0}$. In other words,\nthe cosmological constant appearing in the Einstein manifold is no\nlonger the bare cosmological constant included in \nthe constant part of the $K$ function. This is the \\emph{self-tuning}\nproperty of these gravity theories. \n\nTo sum up: Given a general Lagrangian \\eqref{ESTlag}, we first look for \nzeros of $A_3$ that determine the value(s) of $X_0$, then ask that the \nderivatives of $G$ and $K$ are related at $X=X_0$ as required above. The \neffective cosmological constant is then fixed by the ratio of~$K$ to~$G$.\n\nHaving established the conditions under which Einstein-like metrics can be\nsolutions to modified gravity, we now make a key observation that allows \nus to construct a stealth solution on the Einstein manifold: \nGiven sufficient symmetry in a spacetime, the geodesic equation\n\\begin{equation}\n\\frac{d^2 x^\\m}{d \\l^2} + \\Gamma^\\m_{\\rho\\s} \\frac{d x^\\rho}{d \\l} \n\\frac{d x^\\s}{d \\l} = 0 \\label{geo}\n\\end{equation}\ncan be solved using a Hamilton-Jacobi potential $S$, such that the gradient\nof the potential gives the tangent vector of the geodesic\n\\begin{equation}\n\\frac{\\partial S}{\\partial x^\\m} = p_\\m =  g_{\\mu\\nu}\n\\frac{d x^\\n}{d \\l}  \\,.\n\\label{HJp}\n\\end{equation}\nTypically, this method is used to simplify the solution of a particular\ngeodesic (such as the orbit of a planet), however, the form of the potential\ncan be used over a wider range of co-ordinate values that in the case of\na hypersurface orthogonal geodesic congruence becomes effectively\nthe whole of the spacetime. Thus, given that $\\phi_\\mu$ has constant \nmagnitude, as does the tangent vector of an affinely parametrised geodesic, \nit is natural to make the identification\n\\begin{equation}\n\\phi \\leftrightarrow S \\,, \n\\label{parallel}\n\\end{equation}\nthe properties of the geodesic congruence then will ensure that $\\phi$\nhas the requisite properties to be a stealth solution to the extended\ngravity equations of motion. Moreover, this provides a nice physical \ninterpretation of the constants appearing in the solution. \n\nWe will now illustrate this technique and find a rotating black hole\nwith stealth hair. Consider the Kerr-(A)dS geometry \\cite{CARTER1968399}\n\\begin{equation}\n\\begin{aligned}\nds^2 &= - \\frac{\\Delta_r}{\\Xi^2 \\rho^2} \\left[ dt - a\\, \\sin^2\\theta d \\varphi \\right]^2 \n+ \\rho^2 \\left( \\frac{d r^2}{\\Delta_r} + \\frac{d \\theta^2}{\\Delta_\\theta} \\right) \\\\\n&+ \\frac{\\Delta_\\theta \\sin^2\\theta}{\\Xi^2 \\rho^2} \n\\left[ a\\, dt - \\left( r^2 + a^2 \\right) d \\varphi \\right]^2 \\,, \n\\end{aligned}\n\\label{Kerr}\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{aligned}\n\\Delta_r &= \\left( 1 - \\frac{r^2}{\\ell^2}\\right) \\left( r^2 + a^2 \\right) -2Mr \\,, \\;\\;\n\\Xi = 1 + \\frac{a^2}{\\ell^2} \\,, \\\\\n\\Delta_\\theta &= 1 + \\frac{a^2}{\\ell^2}\\cos^2\\theta \\,,\\qquad\n\\rho^2 = r^2 + a^2 \\cos^2\\theta \\,, \n\\end{aligned}\n\\end{equation}\n$M$ is the black hole mass, $a$ the angular momentum parameter \nand $\\ell=\\sqrt{3\/\\Lambda}$ is the de Sitter radius, related to the \neffective cosmological constant (for AdS reverse the sign of $\\ell^2$).\n\nApplying the Hamilton-Jacobi technique, we note that the components of \nthe metric \\eqref{Kerr} are independent of $t$ and $\\varphi$, thus $E=-p_t$ \nand $L_z=p_\\varphi$ are two constants of the motion, identified with \nthe energy and the azimuthal angular momentum respectively.\nA third constant of motion is the magnitude of the tangent vector \n$g^{\\m\\n}p_\\m p_\\n=X_0=-m^2$, associated with the rest mass of the\ntest particle\\footnote{Note, for illustration we take timelike geodesics,\nshould a spacelike congruence be required, substitute $m^2 \\to -m^2$\nin the derivation.}. Most importantly however, a fourth\nconstant was discovered by Carter \\cite{Carter:1968rr}\n(here generalised to include $\\Lambda$):\n\\begin{equation}\n\\begin{aligned}\n{\\cal Q} &= \\Delta_\\theta \\, p_\\theta^2 + m^2 a^2 \\cos^2\\theta \\\\\n&- \\Xi^2 \\left[ \\left( a \\, E - L_z\\right)^2 - \\frac{\\sin^2\\theta}{\\Delta_\\theta} \n\\left( a \\, E - \\frac{L_z}{\\sin^2\\theta}  \\right)^2 \\right] \\,,\n\\end{aligned}\n\\end{equation}\nwho demonstrated that the geodesic equation was separable. We can\ntherefore write\n\\begin{equation}\nS = - E\\, t + L_z \\varphi + S_r (r) + S_\\theta (\\theta) \\,, \\label{S}\n\\end{equation}\nwhere\n\\begin{equation}\nS_r = \\pm \\int \\frac{\\sqrt{R}}{\\Delta_r} dr \\,, \\qquad \nS_\\theta = \\pm \\int \\frac{\\sqrt{\\Theta}}{\\Delta_\\theta} d\\theta \\,,\n\\label{sdefns}\n\\end{equation}\nwith\n\\begin{figure}\n\\includegraphics[width=0.4\\textwidth]{fullpenrose.pdf} \n\\caption{\\label{fig:penrose} Contours of constant $\\phi$ for \n$a=GM=0.1\\ell$, $\\eta_c = 0.612$ in local Kruskal coordinates \nfor the future event horizons, $\\kappa_b U = -e^{-\\frac{\\kappa_b (t-r^\\star)}{2}} ,\n \\kappa_cV = - e^{-\\frac{\\kappa_c  (t+r^\\star)}{2}}$, ($\\kappa_i$ being\nthe absolute values of the surface gravities of each horizon).}\n\\end{figure}\n\\begin{eqnarray}\nR &=& \\Xi^2 \\left[ E \\left( r^2 + a^2 \\right) -a\\,L_z \\right]^2 \\nonumber \\\\ \n&-& \\Delta_r \\left[ {\\cal Q} +  \\Xi^2 \\left( a \\, E - L_z \\right)^2 + m^2 r^2  \\right] \\,, \\\\\n\\Theta &=& - \\Xi^2 \\sin^2\\theta \\left( a\\,E - \\frac{L_z}{\\sin^2\\theta} \\right)^2 \\nonumber \\\\\n&+& \\Delta_\\theta \\left[ {\\cal Q} +  \\Xi^2 \\left( a \\, E - L_z \\right)^2 \n- m^2 a^2 \\cos^2\\theta \\right] \\,.\n\\end{eqnarray}\n\nNow let us look for explicit solutions for the scalar field $\\phi=S$. This places\nfurther constraints on the potential, as we require $\\phi_\\mu$ to\nbe regular throughout the spacetime. Checking regularity on the axes\nrequires $\\partial S\/\\partial \\theta\\to0$ as $\\theta\\to 0,\\pi$, i.e.\\ \n$\\Theta \\propto \\sin^2\\theta$. This in turn requires $L_z=0$ and \n${\\cal Q} + \\Xi^2 a^2 E^2 = m^2 a^2$, and writing $\\Xi E=\\eta m$,\nwe get:\n\\begin{equation}\n\\begin{aligned}\n\\Theta &= a^2 m^2 \\sin^2\\theta \\left ( \\Delta_\\theta - \\eta^2 \\right) \\,, \\\\\nR &= m^2 (r^2+a^2)\\left (\\eta^2 (r^2+a^2) - \\Delta_r\\right ) \\,.\n\\end{aligned}\n\\label{RTheta}\n\\end{equation}\nThis has now reduced the parameter space to an overall scaling, $m$,\nand a ``relative energy'' $\\eta$, constrained to lie in $\\eta\\in[\\eta_c,1]$;\nthe upper limit coming from $\\Theta\\geq0$, and the lower limit from \n$R\\geq0$ in \\eqref{RTheta}.\n\\begin{figure}\n\\includegraphics[width=0.35\\textwidth]{nearhorizoncontours.pdf} \n\\caption{\\label{fig:phicont} \nContours of $\\phi$ at constant $v=t+r^\\star$ in the $\\{r,\\theta\\}$ plane \nnear the black hole horizon with the same parameter values\nas figure \\ref{fig:penrose}, taking $m=100$.\n}\n\\end{figure}\n\nAt first sight, it appears we have four distinct solutions coming from\nthe choice of signs in \\eqref{sdefns},\nhowever, an interesting restriction occurs when $\\eta=1$ or $\\eta_c$.\nIn this case $\\Theta$ (or $R$) vanishes for some value of $\\theta$\n(or $r$), and the branch choice changes. This is most easily seen for\n$\\eta=1$, here $\\Theta_1 = \\frac{m^2 a^4}{\\ell^2} \\sin^2\\theta \\cos^2\\theta$,\nand the natural root is $\\cos\\theta$ which changes sign across the \nhemisphere. The same phenomenon occurs for $R$, but this\nleads to an important consequence as we now discuss.\n\nInspection of \\eqref{sdefns} shows that \n$S_r \\sim \\pm m\\eta r^\\star$ near the event horizons, where\n$r^\\star = \\int dr(r^2+a^2)\/\\Delta_r$ is the tortoise coordinate,\ntherefore, if we interpret $\\sqrt{R}$ as being the positive root,\nour scalar field will be divergent at one or the other horizon\n(dependent on the branch choice). \nNote however, that for $\\eta_c$, $R$ has a quadratic zero at\nsome $r_0$: $R\\sim R''(r_0) (r-r_0)^2\/2$, thus the true root,\n$\\sqrt{R}\\sim(r-r_0)$, changes sign at $r_0$. This means\nthat for $\\phi$ to be differentiable, we must change the sign \nof $\\sqrt{R}$ across $r_0$ and set\n\\begin{equation}\nS_r = (H[r-r_0] - H[r_0-r]) \\int_{r_0}^r \\frac{|\\sqrt{R}|\\,dr}{\\Delta_r}\n\\end{equation}\nwhere $H$ is the Heaviside step function.\nThis now renders $\\phi$ finite at both future event horizons,\nand infinitely differentiable between the horizons as shown\nin figure \\ref{fig:penrose}.\n\nIt is worth emphasising this last point: All black hole solutions in\nthe literature for higher order scalar-tensor gravity\nare spherically symmetric, and\nhave scalar fields that diverge either on the black hole or\ncosmological event horizon. While this is not a physical\nproblem when $\\phi$ interacts with gravity only through its gradient, \nit is nonetheless a less appealing feature of these\nsolutions. Here, we have constructed a \\emph{rotating}\nblack hole with \\emph{finite} stealth scalar hair. This scalar\nwill be manifestly continuous across each horizon, and be\nstraightforward to analyse in perturbation theory.\nFinally, the integral for the $\\theta-$potential $S_\\theta$ gives\n\\begin{equation}\n\\begin{aligned}\n\\pm S_\\theta = &\\eta \\,\\log \\left [\n\\frac{ \\sqrt{1-\\eta^2+\\frac{a^2}{\\ell^2}\\cos^2\\theta}\n+ \\frac{a}{\\ell} \\cos\\theta}{\\sqrt{(1-\\eta^2)\\Delta_\\theta}} \\right]\\\\\n&\\;\\;\\;-\\log \\left [\n\\frac{ \\sqrt{1-\\eta^2+\\frac{a^2}{\\ell^2}\\cos^2\\theta}\n+ \\frac{a}{\\ell} \\cos\\theta}{\\sqrt{1-\\eta^2}} \\right]\\;,\n\\end{aligned}\n\\end{equation}\nleading to an ``off-centre'' behaviour in the scalar as shown in \nfigure \\ref{fig:phicont}.\n\nFor $\\eta>\\eta_c$, the radial function $R$ has no zero, and \nthe scalar field diverges on one horizon, in common with the \nknown solutions in the literature. The field also demonstrates a \nsimilar asymmetry in $\\theta$, except for $\\eta=1$, when \n\\begin{equation}\nS_\\theta =  \\pm \\frac{m\\ell}{2} \\log \\Delta_\\theta \\,,\n\\end{equation}\nrendering the angular variation symmetric about the equator\nas shown in figure \\ref{fig:nari}.\n\\begin{figure}\n\\includegraphics[width=0.35\\textwidth]{phicontoursnari.pdf} \n\\caption{\\label{fig:nari} \nContours of $\\phi$ for $\\eta=1$ at constant $v=t+r^\\star$ in the \n$\\{r,\\theta\\}$ plane. ($a=GM=0.22\\ell$, $m=100$).\n}\n\\end{figure}\n\nOur solution for $\\phi$ shows a clear dependence on both $\\theta$ and $r$,\nas well as the time dependence in common with known stealth solutions \nhaving spherical symmetry \\cite{Babichev:2013cya}. The key \ndifference is that we can construct a scalar that is finite on\nboth the black hole and cosmological horizon. To compare to\nsolutions in the literature, we take spherical symmetry ($a\\to0$)\nand find general solutions of the form $\\phi = -m \\eta t \\pm S_r$.\nOnce again, we have a finite-$\\phi$ solution for $\\eta_c^2 \n= 1-3(M\/\\ell)^{2\/3}$, \n\\begin{equation}\n\\phi = - m \\left [ \\eta_c t + \\int \\frac{r(r-r_0) \\sqrt{r(r+2r_0)}}{\\ell\\Delta_r} dr\\right]\\,,\n\\label{regso3phi}\n\\end{equation}\nwhere $r_0 = M^{1\/3} \\ell^{2\/3}$. It is interesting to compare this to\nthe time-dependent solution of a black hole in slow-roll inflation\n\\cite{Chadburn:2013mta,Gregory:2018ghc}. \nThere, the scalar profile $\\phi_{SR}\\propto T$ (for a suitable time\ncoordinate) is also finite at both horizons, but \\eqref{regso3phi}\nhas constant gradient, whereas $\\phi_{SR}$ \nsolves a wave equation, resulting in a slightly different\nradial profile between the horizons.\n\nAlso, note that in the case of spherical symmetry, we can relax our \nconstraint $A_3(X_0)=0$. In this case ${\\cal E}_3, \\,{\\cal E}_4$ and the \nRiemann tensor term appearing in (\\ref{EOMg},\\ref{EOMphi}) have a \nsimple form, and combine to require\n\\begin{equation}\n(K_X+4\\Lambda G_X+\\frac{3}{2}\\Lambda X A_3)|_{X_0}=0 \\,,\n\\end{equation}\nwith the same self-tuning condition for the cosmological \nconstant\\footnote{See \\cite{Babichev:2016kdt} for a precise analysis \nof self tuning conditions in spherical symmetry.}, however, note\nthat $A_3(X_0)\\neq0$ \\emph{requires} $\\eta\\equiv1$, thus we\nno longer have the $\\eta-$degree of freedom.\nWhat now happens is that spacetime\nbecomes foliated by surfaces of constant $\\phi$ that are\nflat (for $\\L=0$).\n\nAnother interesting possibility for spherical symmetry is a static \nsolution, found by setting $E=0$, hence $X_0>0$ and $\\phi$\ncorresponds to a congruence of spacelike geodesics:\n\\begin{equation}\n\\phi_s = S_r(r) = \\sqrt{X_0} \\int \\frac{r}{\\sqrt{\\Delta_r}}\\,dr \\,,\n\\end{equation}\nagreeing with a solution reported in \\cite{Motohashi:2019sen}.\n\nIt is also worth noting a side result of our analysis: A search for \nsolutions with $X=X_0$ and spherical symmetry in $c_T=1$ \ntheories (\\ref{ESTlag}) allows \\emph{only} for Einstein geometries. \nThis is unlike Horndeski theories (with $c_T\\neq 1$) where solutions \nof black holes and solitons were found that have $X=X_0$ but are \n\\emph{not} Einstein spaces (for a concise up to date review see \n\\cite{Lehebel:2018zga}). This is an interesting feature, hinting that \nsolutions of $c_T=1$ theories, where $X$ is not constant belong to \nbranches that eventually flow towards $X$ constant solutions with \nan Einstein space metric.\n\nTo sum up: we have presented the first exact solutions for a rotating black\nhole with scalar hair in shift-symmetric \nscalar-tensor theories with unitary speed for gravitational waves.\nOur method was based on an interesting correspondence between \nfamilies of black hole geodesics and stealth solutions in modified gravity. \nThe geodesic correspondence can be used to find other stealth solutions, \nfor instance for more general type D spacetimes,\nor even non-stealth solutions that do not exist in GR --\none only needs to compute the Hamilton-Jacobi potential for a \ngeodesic in the relevant spacetime. \nIt is also plausible that this technique can be extended to other modifications of \ngravity, such as vector-tensor theories where we know that certain stealth \nsolutions exist. Although we did not consider charged black holes, clearly one\ncan also use this method to find stealth Kerr-Newman solutions.\nPerhaps most importantly, we have\npresented a scalar solution that is finite at both the cosmological and \nevent horizons, thus manifestly extendible beyond the cosmological horizon.\nThe angular asymmetry of this solution could provide a distinctive\nsignature of this hair, although this would require a full perturbation \nanalysis beyond the scope of this investigation.\n\n\n\\SEC{Acknowledgments.} \\;\\;\nIt is a pleasure to thank Eugeny Babichev, Gilles Esposito-Far\\`ese, \nAntoine Leh\\'ebel, Karim Noui, Eric Gourgoulhon, \nKarim Van Aelst for many interesting discussions. CC and NS would like \nto acknowledge networking support by the GWverse COST Action CA16104, \n\\emph{Black holes, gravitational waves and fundamental physics}. \nCC thanks the Laboratory of Astronomy of AUTh in Thessaloniki for \nhospitality during the course of this work.\nMC is supported by the Labex P2IO and the Enhanced Eurotalents Fellowship.\nRG is supported in part by the STFC [consolidated grant ST\/P000371\/1], \nand in part by the Perimeter Institute. Research at Perimeter Institute is \nsupported by the Government of Canada through the Department of Innovation, \nScience and Economic Development and by the Province of Ontario \nthrough the Ministry of Research and Innovation.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe morphological  investigation of  galaxies in the early Universe can provide important information on their\nformation and evolutionary processes.\nFor instance, a disturbed and multi-clump morphology, especially in the cold phase, may suggest\nthe presence of  disc instabilities, and can give indication of feedback processes, as well as minor and\/or major merger events during the galaxy assembly \\citep[e.g.,][]{Tamburello:2015, Fiacconi:2016, Ceverino:2017, Pallottini:2017,Pallottini:2017a}.\nIn this context, the advent of facilities delivering high angular resolution observations has enabled us to probe the internal structure of galaxies  in the distant Universe, revealing that clumpy morphologies are more common at higher redshift than at $z=0$ \\citep[e.g.][]{Forster-Schreiber:2006,Genzel:2008}. \nFor instance, the fraction of  galaxies at $0.5<z<3$  exhibiting kpc-scale clumps with sustained star-formation activity is higher than 30\\% \\citep{Ravindranath:2006, Elmegreen:2009, Guo:2015}.\nAt higher redshifts the fraction of galaxies showing disturbed morphology or multi clumps is even higher. on On a sample of 51 Ly$\\alpha$ emitters (LAEs) and 16 Lyman break galaxies (LBGs) at $z>5.7$ \\cite{Jiang:2013} found that roughly half of the brightest galaxies ($M_{1500}$<-20.5 mag) are made of multiple components that may be merging.\nNear-infrared (NIR) high-resolution imaging have also revealed that irregular shapes with multi-clump morphology are prevalent in LAEs and LBGs at $z\\sim7$ within the epoch of reionization \\citep{Ouchi:2010, Sobral:2015, Matthee:2017, Bowler:2017}.\n\nAdditional identification of clumpy systems in the early epoch and a detailed characterisation of  high-$z$ clumps or satellites is fundamental to constrain galaxy assembly. \nThe extended Atacama Large Millimetre\/submillimetre Array (ALMA) configurations enable us to reach high-angular\nresolution\nand exploit far-infrared  (FIR) fine structure emission lines, such as [C{\\sc ii}]\\ at 158$\\mu$m, as\npowerful diagnostics to assess the morphology of primeval galaxies. [C{\\sc ii}]\\  is emitted primarily in the (mostly neutral) atomic and molecular gas associated with Photon\nDominated Regions (excited by the soft UV photons), but also in partly ionised regions and it is one of the\nprimary coolants of the ISM. Indeed, it is generally the strongest emission lines observed in the spectra of\ngalaxies. Since its first detection at high redshift \\citep{Maiolino:2005} this transition has then been\ndetected in large samples of distant galaxies. However, until recently, the [C{\\sc ii}]\\ emission was only detected\nin extreme environments, such as quasar host galaxies and SMGs, characterised by SFRs of several hundred\nsolar masses per year, not really representative of the bulk of the galaxy population at these epochs\n\\citep[e.g.][]{Maiolino:2009, Maiolino:2012, De-Breuck:2011, Wagg:2012, Gallerani:2012, Carilli:2013, Carniani:2013, Williams:2014, Riechers:2014, Yun:2015, Schreiber:2017, Trakhtenbrot:2017, Decarli:2017}.\nDetecting [C{\\sc ii}] \\ in ``normal'' galaxies has required the sensitivity delivered by ALMA.\nTo date, the [C{\\sc ii}]\\ line has been detected in several galaxies at $z>5$  and it is  spatially resolved in most of these targets \\citep{Capak:2015,Willott:2015, Maiolino:2015, Knudsen:2016, Pentericci:2016, Bradac:2016, Smit:2017, Carniani:2017a,Matthee:2017}. \n\\cite{Smit:2017} recently presented [C{\\sc ii}]\\ observations of two galaxies at $z\\sim7$ characterised by a gradient of velocity consistent with a undisturbed  rotating gas disk.\nHowever, most of the $z>5$ galaxies show  extended and clumpy [C{\\sc ii}]\\ emission  with velocities consistent with the systematic redshift of the galaxy ($|\\Delta$v$_{\\rm Ly\\alpha}|<500$~km\/s) but spatially offset relative to the rest-frame UV counterpart \\citep{Maiolino:2015, Willott:2015, Capak:2015,Carniani:2017a, Carniani:2017, Jones:2017}.\nIn many cases these offsets have been ignored or ascribed to astrometric uncertainties. However, based\non detailed astrometric analysis,\nit has been shown that most of these offsets are physical (a revised analysis will be given in this paper),\nhence they should be taken as an important signature of the evolutionary processes in the early phases of\ngalaxy formation.\nVarious scenarios have been proposed to explain the positional offsets between [C{\\sc ii}]\\ and star-forming regions such as\nstellar feedback clearing part of the  ISM, gas accretion, wet mergers, dust obscuration and variations of the\nionisation parameter \\citep[e.g.][]{Vallini:2015, Katz:2017}. \n\\cite{Barisic:2017} and \\cite{Faisst:2017a} have recently found (rest-frame) UV faint companions  whose locations is\nconsistent with the displaced [C{\\sc ii}]\\ emission, suggesting that the carbon line traces star-forming regions where the UV light is absorbed by dust.\n\nAs mentioned,\nmost previous studies have attempted to assess the nature of [C{\\sc ii}]\\ emission  in primeval galaxies neglecting the positional offsets between the FIR line and rest-frame UV emission.  \nThe goal of this paper is to assess the connection between [C{\\sc ii}]\\ and SFR in the early Universe by taking into\naccount the multi-clump morphology of galaxies at $z>5$ and by associating the components with their\nproper optical-UV counterparts (if detected). This is achieved by re-analysing ALMA [C{\\sc ii}]\\ observations of $z > 5$ star-forming galaxies, and by performing a detailed kinematical analysis of the [C{\\sc ii}]\\ line, in order to deblend the different components of the multi-clump systems. \nIn addition to previous ALMA observations, partly discussed in literature, we also make use of new ALMA data\ntargeting five $z\\sim6$ star-forming galaxies with SFR~$<20$~M$_{\\odot}$ yr$^{-1}$. In Section~\\ref{sec:sample} we detail the  sample\nand the analysis of ALMA observations. The morphological analysis is presented in Section~\\ref{sec:Multi-component\nsystems}, while the relation between the [C{\\sc ii}]\\ and SFR is discussed in Section~\\ref{sec:dispersion}. In\nSection~\\ref{sec:lyaEW} we investigate the connection between  [C{\\sc ii}]\\ luminosity and Ly$\\alpha$\\ strength in our sample.\nSection~\\ref{sec:spatially}  focuses on the spatial extension of the [C{\\sc ii}]\\ and UV emission\nand the correlation between [C{\\sc ii}]\\ surface brightness and SFR surface density. We discuss the findings in Section~\\ref{sec:discussion}, while the conclusions of this work are reported in  Section~\\ref{sec:conclusions}\n\nThroughout this paper we assume the following cosmological parameters: $H_0 = 67.8$ km s$^{-1}$ Mpc$^{-1}$ , $\\Omega_M = 0.308$, $\\Omega_\\Lambda$ = 0.685 \\citep{Planck:2016}\n\n\n\\section{Sample, observations and analysis}\\label{sec:sample}\n\n\\subsection{Archival data}\\label{sec:literature_sample}\n\n\\begin{table}\n \\centering\n  \\caption{Overview of the $z>5$ star-forming galaxies observed with ALMA used in this paper, ordered by name.}\n \n \\addtolength{\\tabcolsep}{-3pt}    \n  \\begin{tabular}{lccccc }\n  \\hline\n  \\hline\n   Target$^{(a)}$ & Ra$^{(b)}$ & Dec$^{(c)}$ &   Ref.$^{(d)}$  & [C{\\sc ii}]$^{(e)}$ & Clumpy$^{(f)}$  \\\\\n  \\hline\n \t\\multicolumn{6}{c}{\\emph{Literature Sample}} \\\\\n\t  \\hline\nBDF3299 & 337.0511  & -35.1665 & 1,2 &  \\cmark & \\cmark\\\\\nBDF512 & 336.9444 & -35.1188 & 1 & &  \\\\\nCLM1 & 37.0124 &  -4.2717 & 3 &  \\cmark &  \\\\\nCOSMOS13679 & 150.0990 & 2.3436 & 4 & \\cmark & \\\\\nCOSMOS24108 & 150.1972 & 2.4786 & 4 & \\cmark & \\cmark\\\\\nCOS-2987030247 & 150.1245 & 2.2173 & 5 & \\cmark &  \\\\\nCOS-3018555981 & 150.1245 & 2.2666 & 5 & \\cmark &  \\\\\nCR7   & 150.2417 & 1.8042 & 6 &  \\cmark & \\cmark\\\\\nHimiko & 34.4898 &  -5.1458 & 7,12 & \\cmark & \\cmark\\\\\nHZ8 & 150.0168 & 2.6266 & 8 & \\cmark  & \\cmark \\\\\nHZ7 & 149.8769 & 2.1341 &  8 & \\cmark  &   \\\\\nHZ6 & 150.0896 & 2.5864 & 8 & \\cmark & \\cmark \\\\\nHZ4 & 149.6188 & 2.0518 & 8 & \\cmark &  \\\\\nHZ3 & 150.0392 & 2.3371 & 8 & \\cmark &  \\\\\nHZ9 & 149.9654 & 2.3783 & 8 & \\cmark &  \\\\\nHZ10 & 150.2470 & 1.5554 & 8  & \\cmark & \\cmark \\\\\nHZ2 & 150.5170 & 1.9289 & 8 & \\cmark & \\cmark \\\\\nHZ1 & 149.9718 & 2.1181 & 8  & \\cmark &  \\\\\nIOK-1 & 200.9492 & 27.4155 & 9 & &  \\\\\nNTTDF6345 & 181.4039 & -7.7561 & 4 & \\cmark &  \\\\\nSDF46975 & 200.9292 & 27.3414  &  1 &  \\\\\nSXDF-NB1006-2 & 34.7357  & -5.3330 & 10 & & \\\\\nUDS16291 & 34.3561 & -5.1856 & 4 &  \\cmark & \\\\\nWMH5 &  36.6126 & -4.8773 & 3,11 &  \\cmark & \\cmark\\\\\n      \\hline\n \t\\multicolumn{6}{|c|}{\\emph{Additional new data}} \\\\\n\t  \\hline\nBDF2203 & 336.958 & -35.1472 &  & \\cmark &  \\\\\nGOODS3203 & 53.0928 & -27.8826 \\\\\nCOSMOS20521  & 150.1396 & 2.4269 &  & &  \\\\\nNTTDF2313 & 181.3804 & -7.6935 &  &  \\\\\nUDS4812 & 34.4768 & -5.2472 &  &  & \\\\\n\n \\hline\n\\end{tabular}\n\\\\\n\n\\begin{tablenotes}[flushleft]\n\\footnotesize\n\\item  {\\bf Notes}.\n{\\bf (a)} Name of the source.\n{\\bf(b, c)} J2000 coordinates.\n{\\bf(d)} References in which  ALMA observations are presented (\n[1] \\citealt {Maiolino:2015}; [2] \\citealt{Carniani:2017a};\n[3] \\citealt{Willott:2015}; [4] \\citealt{Pentericci:2016};\n[5] \\citealt{Smit:2017}; [6] \\citealt{Matthee:2017};\n[7] \\citealt{Ouchi:2013}; [8] \\citealt{Capak:2015};\n[9] \\citealt{Ota:2014}; [10] \\citealt{Ouchi:2013};\n[11] \\citealt{Jones:2017} [12] Carniani et al. 2017b)\n{\\bf(e)} Check mark, \\cmark, indicates that [C{\\sc ii}]\\ emission has been detected at the redshift of the galaxy.\n{\\bf(f)} Check mark, \\cmark, indicates that the galaxy has a clumpy morphology.\n\n\\end{tablenotes}\n\\label{tab:projects}\n\\end{table}\n\nThe sample is mainly drawn from the archive and literature by selecting only spectroscopically confirmed\nstar-forming galaxies at $z>5$ observed with ALMA in the [C{\\sc ii}]\\ line. \nWe limit our sample to those systems with  SFR  $\\lesssim\n100$ M$_{\\odot}$ yr$^{-1}$\\ since they are representative of the bulk of the galaxy population in the  early Universe \\citep[e.g.][]{Robertson:2015, Carniani:2015}.\nThe list of selected sources is given in Table~\\ref{tab:projects}.\nThe  sample does not include lensed systems \\citep{Knudsen:2016, Gonzalez-Lopez:2014, Bradac:2016, Schaerer:2015}\nsince  magnification factor uncertainties may lead to large errors on SFR and [C{\\sc ii}]\\ luminosity estimates, as well\nas on the morphology analysis. \n\n\nFor the purpose of our investigation, which focuses on the nature and implications of the positional offsets\nbetween [C{\\sc ii}]\\ and UV emission, we have retrieved and re-analysed  ALMA data revealing a [C{\\sc ii}]\\ detection at the systemic velocity of the galaxy (see Table~\\ref{tab:projects}). For these objects,  ALMA observations have been calibrated following the prescriptions presented in  previous  works. \n\nIn addition to the rest-frame FIR images, we have also used Hubble Space Telescope ({\\it HST}) and Visible and Infrared Survey Telescope for Astronomy (VISTA) NIR observations (rest-frame UV at $z>5$). \nALMA, {\\it HST} and VISTA data have been aligned based on the location of serendipitous sources detected in both\nALMA continuum and NIR  images by  assuming that the  millimetre emission of these sources is cospatial to\nthe near-infrared map. This is also supported by the fact that all foreground sources used for registering\nmillimetre and NIR images do not exhibit any   multi-clump or merger-like morphologies indicating that\nastrometric offsets between the ALMA and NIR images are likely associated to astrometric calibrations. \nFor those observations revealing the presence of two (or more) serendipitous sources  we have verified that the\nastrometric shift for each source is consistent with that estimated from the other source(s) in the same map.\nIn all cases we have checked that the estimated astrometric offset is consistent with those obtained by aligning NIR foreground sources and ALMA phase calibrators to their astrometric position from the GAIA Data Release 1 catalogue \\citep{Gaia-Collaboration:2016}. %\nFor those sources whose ALMA continuum map showing no serendipitous sources, we have matched the NIR foreground sources and ALMA phase calibrators to either GAIA Data Release~1 catalogue \\citep{Gaia-Collaboration:2016} or  AllWISE catalogue \\citep{Cutri:2013}. \nWe note that in all ALMA datasets the locations of the various phase calibrators are in agreement within the\nerror with the GAIA  and  AllWISE catalogues, implying that, when a systemic (not physical)\noffset is seen, this is generally due to some small astrometric uncertainties in the optical-NIR\ndata.\nThese astrometric issues have  been also discussed by \\cite{Dunlop:2016} who analysed ALMA images targeting the Hubble Ultra Deep Field.\nWe therefore applied the  astrometric shifts, which span a range between 0.1\\arcsec\\ and 0.25\\arcsec,  to the NIR images. \n\n\nWe note that additional systems with low SFRs have also been tentatively\ndetected in 14 [C{\\sc ii}]\\ line emitting candidates at $6 < z < 8$ \\citep{Aravena:2016a},\nwhich are not included in this analysis as they are not spectroscopically confirmed\nyet and $\\sim$60\\% of these objects are expected to be spurious .\n\n\n\n\n\\subsection{Additional new ALMA data}\n\\label{sec:additional}\n\n\nIn addition to the archival\/literature sample, we  have also included new  [C{\\sc ii}]\\ observations of five star-forming\ngalaxies  at $z\\sim6$  with a SFR$\\sim$10 M$_{\\odot}$ yr$^{-1}$\\  observed with ALMA in Cycles 3 and 4 (P.I.\nPentericci). The five new sources, listed in Table~\\ref{tab:projects}, have been selected from a sample of\n$>120$ LBGs at $z\\sim6-7$. These have been spectroscopically confirmed thanks to  recent ultra-deep spectroscopic\nobservations from the ESO Large Program CANDELSz7  \\citep[P.I. L. Pentericci;][]{DE-Barros:2017}. The proposed\nALMA programs aimed at observing [C{\\sc ii}]\\ emission in ten  star-forming galaxies at $z>6$ with UV luminosities\nlower than -21 mag (UV SFR $<20$ M$_{\\odot}$ yr$^{-1}$) and spectroscopic redshift uncertainties $<$0.03, but only five\ngalaxies have been observed. The observations and data calibrations are presented in\nthe Appendix~\\ref{sec:appA}. We have registered NIR images to ALMA observations by matching the location of\nthe foreground, serendipitous continuum\nsources  and ALMA calibrators to the position given by the GAIA Data Release 1 catalogue (Gaia Collaboration et al. 2016).\n\nWhile the continuum emission is not detected at the location of any of the five galaxies, we detect two  and one serendipitous sources in the COSMOS20521 and  BDF2313 continuum maps, respectively. \nThe positions of the serendipitous continuum sources is in agreement with the location of NIR foreground galaxies, thus confirming the astrometric shifts estimated from the catalogue.\n \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{Figure\/FigureSpectrumBDF.pdf}  %\n\\caption{ ALMA spectrum of BDF2203 showing a new [C{\\sc ii}]\\ detection. The velocity reference is set to the redshift defined by the Ly$\\alpha$.\nThe dotted grey lines shows the 1$\\sigma$ and -1$\\sigma$ per channel. The red line  indicates the best fit 1D Gaussian line profile.} \n \\label{fig:appFig1}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{Figure\/FigBDF.pdf}  %\n\\includegraphics[width=0.6\\columnwidth]{Figure\/FigBDFZoom.pdf}  %\n\\caption{Top: [C{\\sc ii}]\\ flux map  of BDF2203 obtained integrating the ALMA cube over the velocity range between -60 km\/s and 132 km\/s. Red solid contours are at levels of 2$\\sigma$,\n3$\\sigma$ and 4$\\sigma$, where $\\sigma=23$ mJy km\/s. Dashed black contours indicate negative values at -3 and -2 times the noise level in the same map.\nBottom: zoom of the central 2.5\\arcsec\\ region around BDF2203. [C{\\sc ii}]\\ flux map contours are over imposed on the NIR image. Red contours are at the same levels\nas in the top panel. The ALMA synthesised beam  in the lower left corner in both panels. }\n \\label{fig:mapBdf}\n\\end{figure}\n\nThe [C{\\sc ii}]\\ emission is undetected in all but one of\nthese sources, that is, BDF2203.\nFigure~\\ref{fig:appFig1} shows the spectrum of the detected [C{\\sc ii}]\\ emission,\nwith a spectral rebinning of 40 km s$^{-1}$, while the spectra of the non-detections are shown in the Appendix (Figure~\\ref{fig:appFig2}).\n\nFor the four non-detections, we assume a full-width-at-half-maximum of\nFWHM = 100 km s$^{-1}$, which is consistent with [C{\\sc ii}]\\ line widths observed in other $z>6$ galaxies\n\\citep[e.g.][]{Pentericci:2016,Carniani:2017a, Carniani:2017}, and we infer  3$\\sigma$ upper limits on the [C{\\sc ii}]\\ luminosity (Table~\\ref{tab:appTab1}). \n\nThe redshift of the  [C{\\sc ii}]\\ emission detected in BDF2203  is $z_{\\rm [CII]}=6.1224\\pm0.0005$, which is in agreement\nwith that inferred from Ly$\\alpha$ ($z_{\\rm Ly\\alpha}= 6.12\\pm0.03$). \nBy fitting a 1D Gaussian profile to the [C{\\sc ii}]\\ line we estimate a FWHM=$150\\pm50$ km\/s that is similar to those estimated in high-$z$ [C{\\sc ii}]-emitting galaxies \\citep[FWHM=50-250 km\/s;][]{Willott:2015, Pentericci:2016, Matthee:2017, Carniani:2017}.\nThe flux map of the [C{\\sc ii}]\\ line, extracted with a spectral width of 200 km\/s, is shown in  Figure~\\ref{fig:mapBdf}.\nThe emission is detected in the map with a S\/N=5 and has an integrated flux density of $140\\pm35$ mJy km\/s, which\ncorresponds to L$_{\\rm[CII]}$$=(12.5\\pm2.5)\\times10^{7}$~L$_{\\odot}$\\ at  z=6.12. By fitting a 2D gaussian profile to the flux map, we\nmeasure a  size  of ${\\rm(2.3\\pm0.5)\\arcsec\\times(1.2\\pm0.2)\\arcsec}$ with ${\\rm PA=88\\deg\\pm8\\deg}$ and a beam-deconvolved size of  ${\\rm(1.4\\pm0.8)\\arcsec\\times(0.5\\pm0.4)\\arcsec}$ with a ${\\rm PA=80\\deg\\pm20\\deg}$. The [C{\\sc ii}]\\ emission is thus marginally resolved and has a diameter size of $\\sim5$~kpc. The [C{\\sc ii}]\\ flux map overlaps with the rest-frame UV emission that\nhas an extension of about 1 kpc. The centroids  are separated by 0.25\\arcsec. However, given the ALMA beam size\n($1.90\\arcsec\\ \\times1.11\\arcsec$), such positional offset is consistent with the positional uncertainties $\\Delta\\theta\\approx\\frac{<beam \\ size>}{S\/N}\\sim0.2\\arcsec-0.4\\arcsec$. The low angular resolution and sensitivity of current observations are not sufficient to assess the morphology of the FIR line emission.\n\n\n\n\n\\subsection{SFR estimates and morphology analysis}\n\\label{sec:SFRestimates}\n\nThe continuum emission at rest-frame wavelengths around 158 $\\mu$m is associated to thermal emission from dust heated by\nthe UV emission of young stellar populations in galaxies.\nThe continuum emission is detected only in four sources within the selected sample. When compared with the typical UV-IR SED of galaxies, the\nweak rest-frame far-IR continuum indicates that these galaxies are on average characterised by low dust masses.\n\n\nThe spectral energy distribution (SED) of the thermal dust emission can be modelled with a greybody with dust\ntemperature T$_{\\rm d}$\\ and spectral index emissivity $\\beta$.  However, one single photometric measurement  is not sufficient\nto perform the SED fitting and constrain the two free parameters. We thus assume T$_{\\rm d}$\\ = 30K and  $\\beta$ = 1.5, which are\nconsistent with those observed in local dwarf galaxies \\citep{Ota:2014}, to estimate  the far-infrared (FIR) luminosity from the ALMA continuum observations. For those galaxies that are not detected in ALMA continuum images we infer a 3$\\sigma$ upper limit on ${\\rm L_{FIR} }$. \nWe note that the FIR emission strongly depends on the assumed dust temperature,  yielding to a luminosity uncertainty of $\\Delta \\log({\\rm L_{FIR}}) = 0.6 $ \\citep{Faisst:2017a}.\n\nFor each galaxy we can infer the star-formation rate from both the UV (SFR$_{\\rm UV}$) and  \n FIR (SFR$_{\\rm FIR}$) emission by adopting the calibrations presented in \\cite{Kennicutt:2012}.\nExcluding HZ9 and HZ10, all  sources in the  sample have ${\\rm SFR_{\\rm FIR}\/SFR_{\\rm UV} \\lessapprox 1}$ with an average value of ${\\rm SFR_{\\rm FIR}\/SFR_{\\rm UV}}\\approx0.6$ . Given that most of the FIR luminosity estimates are $3\\sigma$\nupper limits, the average SFR$_{\\rm FIR}$\/SFR$_{\\rm UV}$ is actually much lower than 0.6.\n\nGiven the small contribution of SFR$_{\\rm FIR}$ (either in the IR-detected sources or those with upper limits), we assume total ${\\rm SFR \\approx SFR_{\\rm UV}}$ with no dust correction for those galaxies without continuum detection. While we infer total ${\\rm SFR = SFR_{\\rm UV}+SFR_{\\rm FIR}}$ for those galaxies revealing continuum emission at 158 $\\mu$m.\n\n\n \nIn order to assess the multi-clump morphology of the [C{\\sc ii}]\\ emission in our sources, we perform a kinematical analysis on the retrieved ALMA cube, extracting channel\nmaps at different velocities relative to the redshift of the galaxies. This analysis enables us to disentangle the emission of complex systems having multiple\ncomponents at different velocities \\citep[e.g.][]{Carniani:2013, Riechers:2014}. We also estimate the size of the\nvarious UV and [C{\\sc ii}]\\ emission by fitting a 2D elliptical Gaussian profile to emission maps and by taking into account\nthe angular resolution of the observations.\n\nThe properties of the [C{\\sc ii}], FIR and UV emission, such as redshift, flux density, luminosity, SFR, and radius, are reported in Table~\\ref{tab:table1}\n\n\n\\section{Multi-components systems}\\label{sec:Multi-component systems}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=2\\columnwidth]{Figure\/FigureStamps}\n  %\n \\caption{Rest-frame UV images of the nine  $z>5$ star-forming galaxies showing multi-clump morphology in [C{\\sc ii}]\\ and\/or rest-frame UV emission. The galaxies are ordered by redshift\n and each stamp is 3\\arcsec\\ on the side, with North to the top and East to the left. The red, blue, and green contours show the  [C{\\sc ii}]\\ channel maps at different velocity intervals. \n The levels of the contours and channel maps velocities are indicated in the legends. The properties of each galaxy are reported in Table~\\ref{tab:table1}.}\n \\label{fig:fig1}\n\\end{figure*}\n\n\n\n\\cite{Carniani:2017a}  discuss the origin  of the positional and spectral offsets between optical emission and FIR lines   observed in  high-$z$ systems, suggesting that the observational properties can arise from  distinct regions (or components) of galaxies. In order to understand the nature of these offsets, we have re-analysed  ALMA [C{\\sc ii}]\\ observations for those galaxies showing [C{\\sc ii}]\\ detections and have performed a morphology analysis as discussed in Section~\\ref{sec:SFRestimates}.\nWe have found a clear multi-clump morphology in nine out of 21 $z>5$ galaxies having [C{\\sc ii}]\\ detections (Table~\\ref{tab:table1}). \nThe UV rest-frame images and [C{\\sc ii}]\\ maps are shown in Figure~\\ref{fig:fig1}, sorted by redshift.\nThe stamps are 3\\arcsec\\ across, or $\\sim$17.5~kpc at the average redshift of this sample ($<z>= 6$), and show\nthe contours of the [C{\\sc ii}]\\ maps, obtained from our analysis, superimposed on \nthe rest-frame UV emission in grayscale. \n\nNew {\\it HST} near-IR images of HZ6, HZ8, and HZ10 have already been presented in \\cite{Barisic:2017} and \\cite{Faisst:2017a} revealing multi-component structures.\nThe location of the individual rest-frame UV clumps  is consistent with the  peak positions  of the [C{\\sc ii}]\\ emission extracted at different velocities relative to the\nredshift of the brightest component (which is labelled ``a'' in all stamps).  \nThe [C{\\sc ii}]\\ emission detected in all individual clumps has a level of significance\\footnote{$\\sigma$ is the rms of the channel map in which we detect the [C{\\sc ii}]\\ emission} higher than 5$\\sigma$ and it is spatially  resolved (see Table~\\ref{tab:table1}). \nThe channel map analysis confirms that HZ8b and HZ10b (Hz8W and Hz10W in previous works) are at the same redshift of\nHZ8a and HZ10a, respectively.  We note that the kinematic properties of HZ10 are consistent with the analysis reported by \\cite{Jones:2017a}, who claim that the velocity gradient observed in this galaxy  matches a merger scenario rather than a rotating gas disk model.\n\n\nThe deeper {\\it HST} observations reveal also two faint companions close to the HZ2 galaxy within a projected distance of $\\sim$6~kpc.\nThese sources are not discussed in previous studies \\citep{Capak:2015,Barisic:2017}, since their redshifts were not\nfully spectroscopically confirmed by the [C{\\sc ii}]\\ line. \nHowever, a detailed kinematic investigation of the carbon line shows an extended emission with a morphology consistent with the rest-frame UV emission. \nAlthough the two faint companions are detected with a low level of significance ($\\sim3.5\\sigma$), the match between the three UV peaks and the [C{\\sc ii}]\\ emission supports the reliability of the ALMA detections. \nThe three sources, dubbed in this work as HZ2a, HZ2b and HZ2c, are  at the same redshift and form a multi-component system similar to that observed in HZ6.\nThe angular resolution and the low sensitivity of current ALMA observations are not sufficient to spatially resolve the [C{\\sc ii}]\\ emission in HZ2b and HZ2c.\nHZ2a is instead spatially resolved, with a diameter of 2.5 kpc.\n\n\nAn additional  star-forming galaxy with a complex morphology is WHM5 at $z=6.0695$, which appears to consist of multiple components in [C{\\sc ii}]: a compact  source,  seen also in dust emission, and an extended component at the location of the rest-frame UV emission \\citep{Willott:2015a, Jones:2017}. \nThe two components are separated by a projected distance of $\\sim3$ kpc and a velocity of $\\sim200$ km\/s. Both [C{\\sc ii}]\\\nemission components are spatially resolved ($\\sim$1.3 and $\\sim$2.8 kpc) in the ALMA observations with an angular resolution of 0.3\\arcsec\\ \\citep{Jones:2017}.\n\nAt higher redshift, the presence of multiple [C{\\sc ii}]\\ components has recently been reported in CR7 at $z$=6.6  by\n\\cite{Sobral:2015,Sobral:2017} and \\cite{Matthee:2015,Matthee:2017}. Three out of four detected [C{\\sc ii}]\\ clumps  coincide\nwith the location of UV clumps. In two cases the [C{\\sc ii}]\\ emission  is spatially resolved with a radius of $\\sim3-3.8$ kpc (see \\citealt{Matthee:2017} for details).\n\nAn additional multi-clump galaxy observed with ALMA is Himiko at $z\\sim6.695$ \\citep{Ouchi:2013}. The rest-frame UV image (see Figure~3 of \\citealt{Ouchi:2013})\nreveals that the galaxy comprises three sub-components with SFR spanning in range between 5 and 8 M$_{\\odot}$ yr$^{-1}$. The projected distance between the sources is of about 3-7 kpc.\nkpc. Out of the three sub-components have a Ly$\\alpha$\\ EW=68 \\AA\\, while the other two have EW less than 8 \\AA. Although early observations had reported non-detections\nof [C{\\sc ii}]\\ in this source, recently \\cite{Carniani:2017} have reported a clear detection with extended\/multi-clump morphology. The primary [C{\\sc ii}]\\ emission is\ncoincident with the Ly$\\alpha$ peak, while the UV clumps are much weaker in [C{\\sc ii}]\\ or even undetected. More generally the extended [C{\\sc ii}]\\ emission does not resembles the UV clumpy distribution.\n\nFor the remaining systems (COS24108 and BDF3299) showing a clear positional offsets between [C{\\sc ii}]\\ and UV emission in Figure~\\ref{fig:fig1}, we cannot speculate much more than what has been already done in  previous works due to the lack of deeper ALMA and\/or {\\it HST} observations.  \nAs discussed by \\citealt{Carniani:2017a},  these offsets are certainly associated with physically distinct sub-components, and the [C{\\sc ii}]\\ clumps with no\nUV counterpart may either be tracing star forming\nregions that are heavily obscured at UV wavelengths \\citep{Katz:2017}, or associated with accreting\/ejected gas.\n\n\n\n\n\\subsection{On the nature of the kpc-scale sub-components} \n\nThe deeper [C{\\sc ii}]\\ and  rest-frame UV observations have unveiled the real multi-component nature of nine star-forming galaxies at $z>5$ further highlighting and\n(partly) explaining\npositional offsets between UV and [C{\\sc ii}]\\ emission in previous studies \\citep{Capak:2015,Maiolino:2015,Willott:2015,Pentericci:2016,Carniani:2017a}. These results\nsuggest that future evidence of displaced FIR line and UV emission should be not ignored since it generally reveals the\npresence of sub-components with different physical properties.\n\nSuch sub-components can be ascribed to either satellites in the process of accreting \\citep{Pallottini:2017a} or clumps ejected by past galactic outflows \\citep{Gallerani:2016}.\nHowever, in many cases both the SFR$_{\\rm UV}$ and the size of the various sub-components are comparable to those estimated for\nthe central galaxies, hence suggesting a\nmajor merger scenario in many cases.\nFuture ALMA observations with higher resolution and sensitivity are necessary to detect further sub-components in these and other systems and to perform a detailed dynamical analysis, which allow us to to estimate the dynamic mass and assess the nature of sub-components.\n\n\n\n\n\n\n\n\\section{The L$_{\\rm[CII]}$\\ - SFR relation at z=5--7}\\label{sec:dispersion}\n\nA tight relation between the [C{\\sc ii}]\\ luminosity and the global SFR is seen in local galaxy observations, at least\nwhen excluding extreme (ULIRG-like) cases \\citep{De-Looze:2014, Kapala:2015, Herrera-Camus:2015}.\nThis finding makes the [C{\\sc ii}]\\ line a promising tool to investigate the properties of early galaxies and to\ntrace their star formation. \nHowever,  the behaviour of the [C{\\sc ii}]\\ line emission at $z>5$ seems to be more complex than observed in the local Universe.  \nPrevious studies have shown that only a fraction of [C{\\sc ii}]\\ detections of early galaxies agree with the local relation, while most\nhigh-$z$ galaxies are broadly scattered, with claims that most of them are [C{\\sc ii}]-deficient relative to the local relation. \nHowever, most of previous high-$z$ studies classified multi-component systems as single objects in the L$_{\\rm[CII]}$-SF diagram.\nIf we associate each clump and\/or galaxy with its proper UV counterparts (or lack thereof), then the resulting location on the L$_{\\rm[CII]}$-SFR diagram changes significantly for these objects \n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1.6\\columnwidth]{Figure\/FigureLcii_SFR.pdf}\\\\\\ \\\\\n  %\n \\caption{L$_{\\rm[CII]}$\\ as a function of SFR at $z=5-7$. Blue stars indicate the global SFR and [C{\\sc ii}]\\ luminosity for system having multiple sub-components,\n as discussed in Section~\\ref{sec:Multi-component systems}; in these cases the location of the individual sub-components are shown with yellow stars.\nRed circles show the location of the remaining high-$z$ galaxies (i.e. single systems) listed in Table~\\ref{tab:table1}, that do not exhibit  any positional offsets and\/or disturbed morphology.\nIn the bottom-right of the figure we show an error bar that is representative for the whole sample.\nThe green line is the local relation for local star-forming galaxies \\citep{De-Looze:2014}.}\n \\label{fig:fig2}\n\\end{figure*}\n\nFigure~\\ref{fig:fig2} shows  L$_{\\rm[CII]}$\\ as a function of SFR. The green line illustrates the local relation obtained\nby \\cite{De-Looze:2014} and its dispersion is given by the\nshaded area. Results for $z>5$ galaxies, as listed in Table~\\ref{tab:table1}, are shown with various symbols in\nFigure~\\ref{fig:fig2}. \nThe SFR estimation for the $z>5$ galaxies (and their sub-components) is discussed in Section~\\ref{sec:SFRestimates}.\nThe multiple-component objects (HZ2, HZ6, HZ8, HZ10, WHM5, Himiko, CR7, COS24108, and BDF3299) are split into several individual components with their own SFRs and L$_{\\rm[CII]}$. \nThe nine complex systems discussed in Section~\\ref{sec:Multi-component systems} are broken into 20 sub-components distributed on different\nregions of the L$_{\\rm[CII]}$-SFR plane. The location of the individual subcomponents is indicated with yellow stars, while the location of these\nsystems by integrating the whole [C{\\sc ii}]\\ and UV emission (i.e. ignoring that these are actually composed of different subsystems) is\nindicated with blue stars.\nThe four new [C{\\sc ii}]\\ non-detections presented  in Section~\\ref{sec:additional} fall below the local relation, while the L$_{\\rm[CII]}$\\ for BDF2203 places this galaxy along the \\cite{De-Looze:2014} relation.\n\nOnce the association between [C{\\sc ii}]\\ emission and optical-UV counterparts is properly done,\nwe find that the resulting distribution occupies a large area of the  L$_{\\rm[CII]}$-SFR plot with a large scatter  both above and below the local relation.\nAbout 19 objects of the total sample  are in agreement within 1 sigma with the local relation, but the remaining 24 systems have deviations, either above or below the relation, up to 3 sigma.\n\nIn order to quantify the L$_{\\rm[CII]}$-SFR offset of the high-$z$ sample from the relation found in the local population, we investigate the distribution of offsets relative to the local\nrelation. More specifically,\nfor each galaxy we calculate the offset from the relation as $\\Delta$Log([C{\\sc ii}])=Log([C{\\sc ii}])-Log([C{\\sc ii}]$_{\\rm expect-local}$), where Log([C{\\sc ii}]$_{\\rm expect-local}$) is the [C{\\sc ii}]\\ luminosity\nexpected from the local relation according to the SFR measured in the galaxy or sub-component.\nThe result of this distribution is shown in the left panel of Figure~\\ref{fig:fig2b}, while the right panel shows the distribution of the offsets.  \nIn contrast with some previous claims based on fewer targets, the $\\Delta$Log([C{\\sc ii}]) distribution, which includes both detections and upper limits, does not exhibit  any clear shift relative\nto the local relation (and whose distribution is shown with the dotted green histogram.\nThe number of objects below (23)  and above (20)  the L$_{\\rm[CII]}$-SFR relation are comparable. \n However, the dispersion of the\nis 0.48$\\pm$0.07, which is about two times larger than the uncertainty reported by \\cite{De-Looze:2014} for the local relation. Such larger dispersion may be associated to the presence of kpc-scale sub-components that are not common in the local Universe. However, we will discuss the possible origin of this dispersion in the next sections.\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\columnwidth]{Figure\/FigureCIIdispersion.pdf}\n  %\n\\caption{Deviations from the local L$_{\\rm[CII]}$-SFR relation as a function of SFR. The 1$\\sigma$  dispersion of the local relation is indicated with the shaded green region.\nThe distribution of the offsets is shown in the right panel. The black curve represents the Log([C{\\sc ii}])-Log([C{\\sc ii}]$_{\\rm expect-local}$) distribution from our sample, while the dotted\ngreen histogram is the distribution of the local relation. The dashed grey line shows a Gaussian fit to the high-$z$ distribution.}\n \\label{fig:fig2b}\n\\end{figure}\n\n\n\n\n\n\n\n\\section{The relation between Ly$\\alpha$\\ EW and [C{\\sc ii}]\\ emission}\\label{sec:lyaEW}\n\nIt is well known that Ly$\\alpha$\\ emission depends on the level of ionizing photons produced by star formation (or AGN activity) and radiative transfer effects in the ISM.\nModels and observations suggest that the Ly$\\alpha$\\ EW increases with decreasing  metallicity and dust content \\citep{Raiter:2010, Song:2014}. \nSince the [C{\\sc ii}]\\ emission is sensitive  on the ISM properties as well, and in particular the ISM heating through photoelectric ejection from dust grains,\nwe expect a relation between the [C{\\sc ii}]\\ luminosity and the Ly$\\alpha$\\ strength \\citep{Harikane:2017}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\columnwidth]{Figure\/FigureLcii_SFR_EW.pdf}\n  %\n\\caption{  L$_{\\rm[CII]}$\\ as a function of SFR at $z=5-7$. \nStars and circles represent sub-components and individual galaxies, respectively.\nSymbols are colour-coded according to  their  Ly$\\alpha$\\ EWs (not corrected for the inter-galactic medium  absorption), as indicated on the colour bar on the right.\nIn the bottom-right corner we show an error bar that is representative for the whole sample.\nThe green line are the local relation for local star-forming galaxies \\citep{De-Looze:2014} while its dispersion  is indicated by the shaded green region.}\n \\label{fig:EW}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\columnwidth]{Figure\/FigureDelta_EW.pdf}\n\\includegraphics[width=1\\columnwidth]{Figure\/FigureDelta_EWcorr.pdf}\n  %\n\\caption{ Top panel: Offset from the local L$_{\\rm[CII]}$-SFR relation  as a function of EW(Ly$\\alpha$).  The right hand axis shows the [C{\\sc ii}]\/SFR ratio corresponding to the\ndeviations from the local L$_{\\rm[CII]}$-SFR relation.\nThe 1$\\sigma$  dispersion of the local relation is indicated by the shaded green region. Star and circles indicate sub-components and individual galaxies, respectively.\n  The grey dashed relationship shows our linear fit (see text).\n  Bottom panel: Same as the top panel but the EW(Ly$\\alpha$)$^{\\rm int}$ has been corrected for the IGM absorption by following \\citealt{Harikane:2017}.}\n \\label{fig:EW2}\n\\end{figure}\n\nIn Figure~\\ref{fig:EW}  we show  L$_{\\rm[CII]}$\\ as function of SFR by colour-coding the different symbols according to their Ly$\\alpha$\\ EWs (not corrected for the inter-galactic medium  absorption). We include \nonly those sub-components and galaxies having a Ly$\\alpha$\\ EW measurement.\nThere is a weak tendency for galaxies with high EW(Ly$\\alpha$) to lie below the local L$_{\\rm[CII]}$-SFR relation, and vice-versa.\nThis is shown better in Figure~\\ref{fig:EW2} where the offset from the local L$_{\\rm[CII]}$-SFR relation\nis plotted as a function of EW(Ly$\\alpha$). Note that, since the slope of the local relation is one, plotting the deviation from the local relation\n($\\Delta\\log$([C{\\sc ii}])=log([C{\\sc ii}])-log([C{\\sc ii}]$_{\\rm expect-local}$)) is equivalent to plotting the ratio between [C{\\sc ii}]\\ luminosity and SFR, i.e. L$_{\\rm [CII]}$\/SFR, which is indeed given on the\nright hand axis of  Figure~\\ref{fig:EW}.\nAlthough there is a large dispersion, there is a tentative indication that the offset from the local relation (hence L$_{\\rm [CII]}$\/SFR) anti-correlates with Ly$\\alpha$\\ EW. A linear\nfit gives\n$$\n{\\rm \\Delta \\log([C\\textsc{ii}]) = (0.55\\pm0.20)-(0.44\\pm0.15)\\log(EW(Ly\\alpha)) } \n$$\nand the dispersion around this best-fit is 0.25~dex.\n\nThe result does not change significantly if we attempt to correct the EW(Ly$\\alpha$) for IGM absorption, by following the prescription given by \\cite{Harikane:2017}.\nIn this case the relation is shown in the bottom panel of Figure~\\ref{fig:EW} and the resulting best-fit linear relation is\n$$\n{\\rm \\Delta \\log([C\\textsc{ii}]) = (0.56\\pm0.20)-(0.41\\pm0.14)\\log(EW(Ly\\alpha)^{int})  \\ \\  (0.30 {\\it dex})}\n$$\nand the dispersion around this best-fit is even larger, 0.3~dex.\n\n\\cite{Harikane:2017} find a relation between L$_{\\rm [CII]}$\/SFR and EW(Ly$\\alpha$) steeper than ours, though consistent within errors. The steeper relation found by\n\\cite{Harikane:2017} is probably associated with the fact that in their work they combine the global properties of galaxies and do not extract the subcomponents.\n\nSome anti-correlation between deviation from the local L$_{\\rm[CII]}$-SFR relation and EW(Ly$\\alpha$) (or, equivalently, between L$_{\\rm[CII]}$-SFR and EW(Ly$\\alpha$)$^{\\rm int}$) is expected\nfrom the dependence of these quantities from the metallicity, either directly\nor through the associated dust content, as already predicted by some models \\citep[e.g.][]{Vallini:2015,Pallottini:2017, Matthee:2017}.\nIndeed, lower metallicity implies lower amount of carbon available for cooling, but also less dust content. Indeed, since the heating of PDRs occurs primarily\nthrough photoelectric effect on dust grains, the lower is the dust content the lower is the heating efficiency of the gas in the PDR, hence the lower is the\nemissions of the [C{\\sc ii}]\\ cooling line. On the other hand, the lower the dust content the lower is the absorption of the Ly$\\alpha$ resonant line, hence the higher\nis the EW(Ly$\\alpha$).\n\n\n\\section{Spatially resolved L$_{\\rm[CII]}$--SFR relation}\\label{sec:spatially}\n\n\nIn the previous sections, we show that the L$_{\\rm[CII]}$-SFR relation at $z>5$ has a intrinsic dispersion larger than observed in the local Universe.\nSuch a large scatter suggests that the [C{\\sc ii}]\\ luminosity may not be good tracer of the SFR at least at early epochs. \n\nRecent spatially resolved studies have claimed that the [C{\\sc ii}]-SFR relation is better behaved in terms  of SFR surface density and [C{\\sc ii}]\\ surface brightness  than in global  proprieties (L$_{\\rm[CII]}$\\ and SFR), since the surface brightness calibration is more closely related to the local UV field  \\citep{Herrera-Camus:2015, Smith:2017}.\nIt is thus worth to analyse the relation $\\Sigma_{\\rm[CII]}$-$\\Sigma_{\\rm SFR}$\\ at $z>5$. \n\nIn this section we compare the spatial extension of the [C{\\sc ii}]\\ and UV emission and, then, we investigate the correlation between [C{\\sc ii}]\\ surface brightness and SFR surface density.\n\n\n\n\\subsection{Spatial extension of the [C{\\sc ii}]\\ emission}\\label{sec:spatialextension}\n\nFigure~\\ref{fig:fig_ext} shows the extension of the [C{\\sc ii}]\\ emission compared with the extension of star formation traced by the UV counterpart.\n[C{\\sc ii}]\\ emission is generally much more extended than the UV emission tracing unobscured star formation. This discrepancy may be partially\nassociated with observational effects. Indeed, while the high angular resolution of {\\it HST} enables to resolve small clumps, it may have low\nsensitivity to diffuse, extended emission. However, even by smoothing the UV images in the deepest observations available to us, we still\ndo not recover the extension observed in [C{\\sc ii}]. On the other hand, the resolution of the [C{\\sc ii}]\\ observations may, in some case, smear out\nclumps and result in an overall extended distribution. However, in many cases the ALMA observations achieve a resolution comparable, or even\nhigher, than {\\it HST} at UV rest-frame wavelengths and, despite this, we measure clearly larger [C{\\sc ii}]\\ sizes. Moreover, when high angular resolution\nobservation are used, these do reveal that a significant fraction of the [C{\\sc ii}]\\ flux is resolved out on large scales \\citep[see e.g. discussion\nin ][]{Carniani:2017a}. In conclusion, we believe that the different sizes between [C{\\sc ii}]\\ and UV emission are tracing truly different distribution\nof the [C{\\sc ii}]\\ and UV emission on different scales.\n\nThere could be various explanations for these differences. If the star formation associated with the [C{\\sc ii}]\\ emission\non large scale is heavily obscured, the UV light does not trace this component \\citep{Katz:2017}. This is certainly a possibility, although \none might expect that the bulk of the obscuration should affect the central region more heavily than the outer parts, hence one would\nexpect the opposite trend. In alternative, the extended component of [C{\\sc ii}]\\ may not be directly associated with star forming regions, but with\ncircumgalactic gas, either\nin accretion or ejected by the galaxy, and which is illuminated by the strong radiation field produced by the galaxy. \n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\columnwidth]{Figure\/Ruv_vs_Rcii.pdf}\\\\\\ \\\\\n  %\n\\caption{Half-light radii of star formation regions, as measured from the (rest-frame) UV light, compared with the half-light radii of the associated [C{\\sc ii}]\\ emission. The dashed line indicates the 1:1 relation.}\n \\label{fig:fig_ext}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\columnwidth]{Figure\/FigureECII_ESFR}\\\\\\ \\\\\n  %\n\\caption{$\\Sigma_{\\rm[CII]}$ vs $\\Sigma_{\\rm SFR}$ \\ for $z>5$ star-forming galaxies detected both in [C{\\sc ii}]\\ and rest-frame UV. Symbols are as in Figure~\\ref{fig:fig2}. The grey line indicates the local\nrelation by \\citet{Herrera-Camus:2015}. The dashed line indicates the best linear fit on the $z>5$ sample.}\n \\label{fig:fig3}\n\\end{figure}\n\n\n\n\\subsection{Surface brightness}\\label{sec:surface brightness}\\label{sec:surface}\n\n\nOnce we have measured the extension of the [C{\\sc ii}]\\ emission and of the SFR regions, we can estimate the [C{\\sc ii}]\\ surface brightness and SFR surface density of each sub-component and of\nthe individual sources detected in [C{\\sc ii}]\\ and UV.\nFigure~\\ref{fig:fig3} shows the $\\Sigma_{\\rm[CII]}$-$\\Sigma_{\\rm SFR}$\\ relation, where we have included only those galaxies detected in both [C{\\sc ii}]\\ and UV emission. \nSystems that are not spatially resolved in [C{\\sc ii}]\\ (or UV) emission are indicated with lower limits. We also show the local relation by \\cite{Herrera-Camus:2015} and its dispersion. \nIn contrast to the L$_{\\rm[CII]}$-SFR diagram, there are no galaxies located significantly above the local relation, only a few galaxies\nare located on the local relation, and most galaxies spread largely below the local relation. This is primarily due to the\nlarge extension of the [C{\\sc ii}]\\ emission in these high redshift systems, as discussed in the previous section.\nBy fitting the $\\Sigma_{\\rm[CII]}$-$\\Sigma_{\\rm SFR}$\\ measurements for our sample we obtain the following relation:\n$$\n\\log(\\Sigma_{\\rm SFR}) = (0.63\\pm0.11)\\times(\\log(\\Sigma_{\\rm [CII]})-(25\\pm6)\n$$\nPart of the $\\Sigma_{\\rm[CII]}$\\ deficit may be ascribed to the metallicity of the gas.\nIndeed a similar deviations have been  observed in local low-metallicity galaxies, in which the $\\Sigma_{\\rm[CII]}$-$\\Sigma_{\\rm SFR}$\\ calibration over-predicts the $\\Sigma_{\\rm[CII]}$\\\nby up to a factor of six.\nWe note that \\cite{Faisst:2017a} estimated a  metallicity of 12+log(O\/H)$>$8.5 for HZ10 and HZ9, and  a metallicity of  12+log(O\/H)$<$8.5 for HZ1, HZ2, and HZ4. \nThe former agree with the local $\\Sigma_{\\rm[CII]}$-$\\Sigma_{\\rm SFR}$\\  relation while HZ1, HZ2, and HZ4 have low $\\Sigma_{\\rm[CII]}$-$\\Sigma_{\\rm SFR}$\\ ratio.\nThis interpretation is also supported by the fact that all CR7 clumps, which have  metallicities 12+log(O\/H)$<8.2$ (see discussion in \\citealt{Sobral:2017}) , are located well below the local relation.  \nThe $\\Sigma_{\\rm[CII]}$\\ deficit is also akin to the simulations by \\cite{Pallottini:2017a} who investigated  the physical properties of a simulated galaxy at $z=6$, with metallicity\n12+log(O\/H)=8.35.  Such galaxy has a ${\\rm \\log(\\Sigma_{\\rm[CII]}\/erg \\ s^{-1} \\ kpc^{-2})=40.797}$ and ${\\rm \\log(\\Sigma_{\\rm SFR}\/M_\\odot yr^{-1})= 1.027}$, which places the mock\ngalaxy below the  $\\Sigma_{\\rm[CII]}$-$\\Sigma_{\\rm SFR}$\\ relation found for local star-formation galaxies.\n\nHowever, the offset observed\nin high-$z$ galaxies is significantly larger (one order of magnitude or more) than the one observed in local low metallicity galaxies, so other effects\nare likely in place, which will be discussed in the next section.\nFinally, still within the context of the surface brightness and SFR surface density, we mention that, for local galaxies,\n\\cite{Smith:2017} found a dependence between the L$_{\\rm[CII]}$\/L$_{\\rm FIR}$ ratio and the SFR surface density, suggesting that the [C{\\sc ii}]\\ deficit increases strongly with increasing $\\Sigma_{\\rm SFR}$.  Therefore the $\\Sigma_{\\rm SFR}$\\  dependence has a strong impact also on the  \\mbox{L$_{\\rm[CII]}$-SFR} relation. \nIn our sample, we cannot verify this dependence as most galaxies have only an upper limit on the L$_{\\rm FIR}$. However, we can investigate  the relation  L$_{\\rm[CII]}$\/L$_{\\rm UV}$ and $\\Sigma_{\\rm SFR}$\\, since the contribution of SFR$_{\\rm FIR}$ to the total SFR is negligible (see discussion in Section~\\ref{sec:SFRestimates}).\nFigure~\\ref{fig:fig5} shows the  observed L$_{\\rm[CII]}$\/L$_{\\rm UV}$ ratio spanning a range between 0.002\\% to 0.4\\% (over two orders of magnitude),\nwhile the SFR surface density spans the range between 1 and 30 ${\\rm M_{\\odot} \\ yr^{-1} \\ kpc^{-2}}$.\nThere is only a very weak correlation between\nL$_{\\rm[CII]}$\/L$_{\\rm UV}$ and $\\Sigma_{\\rm SFR}$, with very large dispersion (much larger than what observed locally for the\nL$_{\\rm[CII]}$\/L$_{\\rm FIR}$ and $\\Sigma_{\\rm SFR}$ relation). This indicates that in these high-$z$ systems the L$_{\\rm[CII]}$\/L$_{\\rm UV}$ line ratio (and L$_{\\rm[CII]}$\/SFR) does not strongly depend on the\nareal density with which galaxies form stars at $z>5$, and that other effects (metallicity, and other phenomena discussed in the next section) may contribute\nto the very large dispersion.\n\n\n\n\n\n\\section{Discussion on the [C{\\sc ii}]-SFR scaling relations at z=5--7}\\label{sec:discussion}\n\nClearly galaxies at $z>5$ behave differently, relative to their local counterparts, for what concerns the  [C{\\sc ii}]\\ and SFR properties.\nSummarising the finding of the previous sections:\n1) both the \\mbox{L$_{\\rm[CII]}$-SFR} relation and the $\\Sigma_{\\rm[CII]}$--$\\Sigma_{\\rm SFR}$\\ relation have a scatter much larger than the local relations;\n2) contrary to some previous claims,  the \\mbox{L$_{\\rm[CII]}$--SFR} relation is not offset relative to the local relation, while the $\\Sigma_{\\rm[CII]}$--$\\Sigma_{\\rm SFR}$\\ relation is clearly\noffset by showing much lower $\\Sigma_{\\rm[CII]}$ \\ relative to local relation;\n3) the extension of the [C{\\sc ii}]\\ emission is larger than the extent of the star formation traced by the UV emission. \n\n\nThe larger scatter observed in the SFR-L$_{\\rm[CII]}$\\ relation is certainly\nindicative of a broader range of properties spanned by such primeval galaxies relative to the local population.\nIndeed, \\mbox{high-$z$} cosmological simulations show  that the [C{\\sc ii}]\\ emission strongly  depends on the gas metallicity, ionisation parameter, and  evolutionary stage of\nthe system and that all of these properties are expected to span a much broader range at high-$z$ with respect to local galaxies \\citep{Vallini:2015,Vallini:2016,Vallini:2017, Pallottini:2015, Pallottini:2017, Olsen:2017, Katz:2017}.\nRecently, \\cite{Lagache:2017} investigated the expected dispersion of L$_{\\rm[CII]}$-SFR relation in the distant Universe by  using a semi-analytical model of galaxy formation\nfor a large sample of simulated galaxies at $z>4$. They found a L$_{\\rm[CII]}$-SFR  correlation with a large scatter of 0.4-0.8 dex, which is in agreement with our result. They claim that such large dispersion is associated to the combined effects of  different gas contents,  metallicities,  and interstellar radiation fields in the simulated high-$z$ galaxies. \n\nAs we have shown, the mild anti-correlation with EW(Ly$\\alpha$), as well as the analysis of some individual galaxies for which the metallicity has been estimated,\ndoes suggest that the metallicity may play a role on the [C{\\sc ii}]\\ emission (either simply in terms of carbon abundance and\/or in terms of heating of the ISM through\nphotoelectric effect on dust grains, whose abundance scales with the metallicity). In particular, the lower metallicity of galaxies at $z>5$ \ncan explain some of the scatter towards low [C{\\sc ii}]\\ emission in both the L$_{\\rm[CII]}$-SFR relation and in the $\\Sigma_{\\rm[CII]}$-$\\Sigma_{\\rm SFR}$\\ relation. However, metallicity effects are\nunlikely to explain the scatter towards high [C{\\sc ii}]\\ emission in the  L$_{\\rm[CII]}$-SFR relation. Moreover, for what concerns the $\\Sigma_{\\rm[CII]}$-$\\Sigma_{\\rm SFR}$\\ relation, the offset\nand large spread toward low $\\Sigma_{\\rm[CII]}$\\ is probably too large to be entirely ascribed to metallicity.\n\nLarge variations in ionisation parameter can also contribute to the spread in [C{\\sc ii}]\\ emission \\citep{Gracia-Carpio:2011, Katz:2017}. In particular, if [C{\\sc ii}]\\ in\nprimeval galaxies  also traces circumgalactic gas in accretion and\/or expelled from the galaxy, and excited by the UV radiation of the central galaxy,\nthis would result into the observed larger [C{\\sc ii}]\\ sizes, hence lower $\\Sigma_{\\rm[CII]}$ \\ and lower ionisation parameter, and the latter would increase the total L$_{\\rm[CII]}$ \\ relative to the local\nrelation. On the other hand, young compact star forming, primeval galaxies would be characterised by higher ionisation parameter, which would reduce the [C{\\sc ii}]\\ emission,\nhence contributing to the spread towards low L$_{\\rm[CII]}$.\n\nAs mentioned in the previous sections, yet another possibility is that the UV emission associated with some of the [C{\\sc ii}]\\ clumps is heavily obscured. This would\nexplain the scatter above the L$_{\\rm[CII]}$-SFR relation, in the sense that for some of these systems the underlying SFR, as traced by the UV, is heavily underestimated. \nThis can be the case for a few galaxies. However, as we mentioned, most of these systems show no or weak continuum dust emission, indicative of low dust content.\n\nTo make further progress additional data at other wavelength will be, in the future, extremely valuable. In particular, JWST will enable to identify obscured stellar components\nas well as H$\\alpha$ emission associated with star formation. ALMA observations of other transitions, such has [OIII]88$\\mu$m (though observable only in some redshift ranges), has\nproved extremely useful to constrain these scenarios \\citep[e.g. ][]{Inoue:2016,Carniani:2017a}.\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\columnwidth]{Figure\/FigureLcii_ESFR}  %\n\\caption{L$_{\\rm[CII]}$\/L$_{\\rm UV}$ ratio as a function of the SFR surface density. Symbols are as in Figure~\\ref{fig:fig2}.}\n \\label{fig:fig5}\n\\end{figure}\n\n\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nIn this work we have investigated the nature of the [C{\\sc ii}]\\ emission in  star-forming galaxies at $z=5-7$. In particular, we have explored the positional offsets between  UV  and  FIR line emissions, the\npresence of multiple components,\nand the implications on the [C{\\sc ii}]-SFR scaling relations in the distant Universe, once the correct association between [C{\\sc ii}]\\ and UV emission is properly taken into account.\nWe have performed our investigation in a sample of 29 $z>5$ ``normal'' star-forming galaxies (SFR $<$ 100 M$_{\\odot}$~yr$^{-1}$) observed with ALMA in the [C{\\sc ii}]\\ line. In addition to the re-analysis of archival objects,\nwe have also included  new ALMA observations targeting five star-forming galaxies at $z\\sim6$ with SFR $\\sim$ 10 M$_{\\odot}$ yr$^{-1}$, resulting into a new detection. Our mains results are:\n\n\\begin{itemize}\n\n\n\n\\item The continuum emission around 158 $\\mu$m is not detected in most of the $z>5$ galaxies observed with ALMA, indicating a low dust content. By modelling the dust emission with a greybody spectrum\nwith dust temperature T$_{\\rm d}$ = 30 K and emissivity index $\\beta$ = 1.5, we have found that the SFR based on the FIR emission is, on average, lower than\nthe SFR measured from the  UV emission by at least a factor 0.6, but probably much more (due to several upper limits).\n\n\\item By accurately registering ALMA and NIR images, and by  kinematically discriminating multiple [C{\\sc ii}]\\ components, our analysis has revealed that the [C{\\sc ii}]\\ emission breaks into multiple\nsub-components in 9 out of the 21  galaxies having [C{\\sc ii}]\\ detections. In these nine targets we have observed the presence of 19 FIR-line emitting clumps. Only very few of these, if any, are associated\nwith the primary (brightest) UV counterpart, while the bulk of the [C{\\sc ii}]\\ is associated with fainter UV components.\nIn only three cases (COSMOS24108, Himiko, and BDF3299) the shallow NIR images have not enabled us to detect the UV counterparts associated with some of the [C{\\sc ii}]\\ clumps. \n\n\\item  We  have studied the relation between [C{\\sc ii}]\\  and SFR on the high-$z$ sample by  taking into account the presence of these sub-components and the proper associations\nbetween [C{\\sc ii}]\\ and UV components. The distribution of $z>5$ galaxies on the L$_{\\rm[CII]}$-SFR diagram follows the local relation, but the dispersion is 1.8 times larger than that observed in nearby galaxies.\n\n\\item The deviation from the local L$_{\\rm[CII]}$-SFR relation shows a weak anti-correlation with EW(Ly$\\alpha$) though shallower and with larger dispersion than what found in other\nstudies that did not account for the multi-component nature of these systems.\n\n\\item  Most of the objects in the high-$z$ sample are spatially resolved in [C{\\sc ii}]\\ and UV emission. \nThe extension of the [C{\\sc ii}]\\ emission is generally much larger than the extension of star forming regions traced by the UV emission.\n\n\\item In the $\\Sigma_{\\rm[CII]}$-$\\Sigma_{\\rm SFR}$\\ diagram $z>5$ galaxies are characterised by a large scatter with respect to local galaxies, and are mostly distributed below the local relation (i.e. fainter\n$\\Sigma_{\\rm[CII]}$\\ at a given $\\Sigma_{\\rm SFR}$). \n\n\\end{itemize}\n\n\nWe have suggested that a combination of different effects may be responsible for the different properties of high-$z$ galaxies in terms of [C{\\sc ii}]--SFR properties\nrelative to local galaxies. More specifically: 1) the low metallicity of high-$z$ galaxies may be responsible (also indirectly through the lower dust photoelectric heating) for part\nof the scatter towards lower [C{\\sc ii}]\\ emission relative to the local relations; 2) the presence of circumnuclear gas in accretion and\/or expelled from the galaxy may be responsible\nfor the larger size in [C{\\sc ii}]\\ relative to the SFR distribution and may also be responsible for the scatter of the L$_{\\rm[CII]}$-SFR distribution above the local relation\nas a consequence of lower ionisation parameter; 3) in compact young star forming regions the increased ionisation parameter and higher gas density may be responsible for\nthe suppression of [C{\\sc ii}]\\ for galaxies which are below the local relation; 4) dust obscuration may be responsible for both the different morphology between [C{\\sc ii}]\\ and UV emission\nand also for the scatter of sources above the local L$_{\\rm[CII]}$-SFR relation.\n\n\\section*{Acknowledgments}\n\nThis paper makes use of the following ALMA data: {\\small ADS\/JAO.ALMA\\#2012.1.00719.S, ADS\/JAO.ALMA\\#2012.A.00040.S, ADS\/JAO.ALMA\\#2013.A.00433.S ADS\/JAO.ALMA\\#2011.0.00115.S, ADS\/JAO.ALMA\\#2012.1.00033.S, ADS\/JAO.ALMA\\#2012.1.00523.S, ADS\/JAO.ALMA\\#2013.1.00815.S, ADS\/JAO.ALMA\\#2015.1.00834.S., ADS\/JAO.ALMA\\#2015.1.01105.S, and ADS\/JAO.ALMA\\#2016.1.01240.S}\n which can be retrieved from the ALMA data archive: https:\/\/almascience.eso.org\/ alma-data\/archive. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI\/NRAO and NAOJ.\nWe are grateful to G. Jones to for providing the [C{\\sc ii}]\\ flux maps of WHM5.\nR.M. and S.C. acknowledge support by the Science and Technology Facilities Council (STFC). R.M. acknowledges ERC Advanced Grant 695671 ``QUENCH''. AF acknowledges support from the ERC Advanced Grant INTERSTELLAR H2020\/740120. \n\n\n\\setlength{\\labelwidth}{0pt}\n\\bibliographystyle{mn2e}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction} \\label{sec1}\n\\mbox{} \\\\[-9mm]\n\nIn optical and acoustic radiation with boundary conditions given on a disk, the choice of basis functions to represent the boundary conditions is of great importance. Here one can consider the following requirements:\n\\bi{A.0}\n\\ITEM{A.} the basis functions should be appropriate given the physical background,\n\\ITEM{B.} the basis functions should be effective and accurate in their ability of representing functions vanishing outside the disk and arising in the particular physical context,\n\\ITEM{C.} the basis functions should have convenient analytic results for transformations that arise in a natural way in the given physical context.\n\\end{inspring}\nFor the case of optical diffraction with a circular pupil having both phase and amplitude non-uniformities, Zernike \\cite{ref1} introduced in 1934 his circle polynomials, denoted here as $Z_n^m(\\rho,\\vartheta)$ with radial variable $\\rho\\geq0$ and angular variable $\\vartheta$, vanishing for $\\rho>1$, which find nowadays wide-spread application in fields like optical engineering and lithography \\cite{ref2}--\\cite{ref7}, astronomy \\cite{ref8}--\\cite{ref10} and ophthalmology \\cite{ref11}--\\cite{ref13}. The Zernike circle polynomials are given for integer $n$ and $m$ such that $n-|m|$ is even and non-negative as\n\\begin{eqnarray} \\label{e1}\nZ_n^m(\\rho,\\vartheta) & = & R_n^{|m|}(\\rho)\\,e^{im\\vartheta}~,~~~~~~0\\leq\\rho<1\\,,~~0\\leq\\vartheta<2\\pi~, \\nonumber \\\\[2mm]\n& = & 0\\hspace*{2cm},~~~~~~\\rho>1~,\n\\end{eqnarray}\nwhere the radial polynomials $R_n^{|m|}$ are given by\n\\begin{equation} \\label{e2}\nR_n^{|m|}(\\rho)=\\rho^{|m|}\\,P_{\\frac{n-|m|}{2}}^{(0,|m|)}\\,(2\\rho^2-1)~,\n\\end{equation}\nwith\n$P_k^{(\\alpha,\\beta)}$ the general Jacobi polynomial as given in \\cite{ref14}, Ch.~22, \\cite{ref15}, Ch.~4 and \\cite{ref16}, Ch.~5, \\S4. The circle polynomials were investigated by Bhatia and Wolf \\cite{ref17} with respect to their appropriateness for use in optical diffraction theory, see issue A above, and were shown to arise more or less uniquely as orthogonal functions with polynomial radial dependence satisfying form invariance under rotations of the unit disk.\n\nThe orthogonality condition,\n\\begin{equation} \\label{e3}\n\\int\\limits_0^1\\int\\limits_0^{2\\pi}\\,Z_{n_1}^{m_1}(\\rho,\\vartheta)(Z_{n_2}^{m_2}(\\rho,\\vartheta))^{\\ast}\\,\\rho\\,d\\rho\\,d\\vartheta=\\frac{\\pi} {n+1}\\,\\delta_{m_1m_2}\\,\\delta_{n_1n_2}~,\n\\end{equation}\nwhere $n$ is either one of $n_1$ and $n_2$ at the right-hand side, together with the completeness, see \\cite{ref18}, App.~VII, end of Sec.~I, guarantees effective and accurate representation of square integrable functions on the unit disk in terms of their expansion coefficients with respect to the $Z_n^m$, see issue B above. The Zernike circle polynomials, notably those of azimuthal order $m=0$, were considered recently by Aarts and Janssen \\cite{ref19}--\\cite{ref22} for use in solving a variety of forward and inverse problems in acoustic radiation. This raised in the acoustic community \\cite{ref23} the question how the circle polynomials with $m=0$ compare to other sets of non-polynomial, radially symmetric, orthogonal functions on the disk. It was shown in \\cite{ref24} that the expansion coefficients, when using circle polynomials, properly reflect smoothness of the functions to be expanded in terms of decay and that they compare favourably in this respect with the expansion coefficients that occur when using orthogonal Bessel series expansions; the latter expansions are sometimes used both in the acoustic and the optical domain, see \\cite{ref25}--\\cite{refY}.\n\nThe present paper focuses on analytic properties, see issue C above, of basis functions, and in Sec.~\\ref{sec2} we present a number of such properties for the set of Zernike circle polynomials. In Sec.~\\ref{sec3} a generalization of the set of Zernike circle polynomials is introduced, viz.\\ the set of functions\n\\begin{eqnarray} \\label{e4}\nZ_n^{m,\\alpha}(\\rho,\\vartheta) & = & (1-\\rho^2)^{\\alpha}\\,\\rho^{|m|}\\,P_{\\frac{n-|m|}{2}}^{(\\alpha,|m|)} (2\\rho^2-1)\\,e^{im\\vartheta}~,~~~~~~0\\leq\\rho<1~, \\nonumber \\\\[2mm]\n& = & 0\\hspace*{6.1cm},~~~~~~\\rho>1~,\n\\end{eqnarray}\nwith $\\rho$, $\\vartheta$ and $n$, $m$ as before, see (\\ref{e1}), and parameter $\\alpha>{-}1$. These new Zernike-type functions arise sometimes more naturally in certain physical problems and can be more convenient when solving inverse problems in diffraction theory since the decay in the Fourier domain can be controlled by choosing the parameter $\\alpha$ appropriately. While issue A above should guide the choice of $\\alpha$, the matter of orthogonality and completeness is settled as in the classical case $\\alpha=0$. A more involved question is what becomes of the analytic properties, noted for the classical circle polynomials in Sec.~\\ref{sec2}, when $\\alpha\\neq0$. A major part of the present paper is concerned with answering this question, and this yields generalization of many of the results holding for the case $\\alpha=0$ and about which more specifics will be given at the end of Sec.~\\ref{sec3} where the basic properties of the generalized Zernike functions are presented.\n\n\\section{Analytic results for the classical circle polynomials} \\label{sec2}\n\\mbox{} \\\\[-9mm]\n\nThe Zernike circle polynomials were used by Nijboer in his 1942 thesis \\cite{ref27} for the computation of the point-spread functions in near best-focus planes pertaining to circular optical systems of low-to-medium numerical aperture. In that case, starting from a non-uniform pupil function\n\\begin{equation} \\label{e5}\nP(\\rho,\\vartheta)=A(\\rho,\\vartheta)\\,e^{i\\Phi(\\rho,\\vartheta)}~,~~~~~~0\\leq\\rho<1\\,,~~0\\leq\\vartheta<2\\pi~,\n\\end{equation}\nthe point-spread function $U(r,\\varphi\\,;\\,f)$ at defocus plane $f$ with polar coordinates $x+iy=r\\,e^{i\\varphi}$ is given in accordance with well established practices in Fourier optics as\n\\begin{equation} \\label{e6}\nU(r,\\varphi\\,;\\,f)=\\int\\limits_0^1\\int\\limits_0^{2\\pi}\\,e^{if\\rho^2}\\,e^{2\\pi i\\rho r\\cos(\\vartheta-\\varphi)}\\,P(\\rho,\\vartheta)\\,\\rho \\,d\\rho\\,d\\vartheta~.\n\\end{equation}\nIt was discovered by Zernike \\cite{ref1} that the Fourier transform of the circle polynomials,\n\\begin{equation} \\label{e7}\n({\\cal F}\\,Z_n^m)(r,\\varphi)=\\int\\limits_0^1\\int\\limits_0^{2\\pi}\\,e^{2\\pi i\\rho r\\cos(\\vartheta-\\varphi)}\\,Z_n^m(\\rho,\\vartheta)\\,\\rho\\,d\\rho\\,d\\vartheta\n\\end{equation}\nhas the closed-form result\n\\begin{equation} \\label{e8}\n({\\cal F}\\,Z_n^m)(r,\\varphi)=2\\pi\\,i^n\\,\\frac{J_{n+1}(2\\pi r)}{2\\pi r}\\,e^{im\\varphi}~,~~~~~~r\\geq0\\,,~~0\\leq\\varphi<2\\pi~,\n\\end{equation}\nwhere $J_{n+1}$ is the Bessel function of the first kind and order $n+1$. Thus, one expands the pupil function $P$ as\n\\begin{equation} \\label{e9}\nP(\\rho,\\vartheta)=\\sum_{n,m}\\,\\beta_n^m\\,Z_n^m(\\rho,\\vartheta)~,~~~~~~0\\leq\\rho<1\\,,~~0\\leq\\vartheta<2\\pi~,\n\\end{equation}\nand uses the result in (\\ref{e8}) to compute the point-spread function $U$ in (\\ref{e6}) in terms of the expansion coefficients $\\beta_n^m$. In the case of best focus, $f=0$, this gives a computation result for the point-spread function immediately. For small values of $f$, say $|f|\\leq1$, one expands the focal factor,\n\\begin{equation} \\label{e10}\ne^{if\\rho^2}=1+if\\rho^2-\\tfrac12\\,f^2\\rho^4-...\n\\end{equation}\nand spends some additional effort to write functions $\\rho^{2k}\\,Z_n^m(\\rho,\\vartheta)$ as a $k$-terms linear combination of circle polynomials with upper index $m$ to which (\\ref{e7}) applies. Also see \\cite{ref18}, Ch.~9, Secs.~2--4. These are the main features of the classical Nijboer-Zernike theory for computation of optical point-spread functions in the presence of aberrations.\n\nIn \\cite{ref28} Janssen has computed the point-spread function $U_n^m(r,\\varphi\\,;\\,f)$ pertaining to a single term $Z_n^m$ in the form\n\\begin{equation} \\label{e11}\nU_n^m(r,\\varphi\\,;\\,f)=2\\pi\\,i^{|m|}\\,V_n^{|m|}(r,f)\\,e^{im\\varphi}~,\n\\end{equation}\nwhere\n\\begin{eqnarray} \\label{e12}\nV_n^{|m|}(r,f) & = & \\int\\limits_0^1\\,e^{if\\rho^2}\\,R_n^{|m|}(\\rho)\\,J_{|m|}(2\\pi\\rho r)\\,\\rho\\,d\\rho~= \\nonumber \\\\[3.5mm]\n& = & e^{if}\\,\\sum_{l=0}^{\\infty}\\,\\Bigl(\\frac{{-}if}{\\pi r}\\Bigr)^l\\,\\sum_{j=0}^p\\, u_{lj}\\, \\frac{J_{|m|+l+2j+1}(2\\pi r)}{2\\pi r}\n\\end{eqnarray}\nwith explicitly given $u_{lj}$ ($p=\\frac12\\,(n-|m|)$). This result has led to what is called the extended Nijboer-Zernike (ENZ) theory of forward and inverse computation for optical aberrations. The theory in its present form allows computation of point-spread functions for general $f$ and for high-NA optical systems, including polarization and birefringence, as well as for multi-layer systems. See \\cite{ref29}--\\cite{ref33} and \\cite{ref34} for an overview. Furthermore, it provides a framework for estimating pupil functions $P$, in terms of expansion coefficients $\\beta$, from measured data $|U|^2$ of the intensity point-spread function in the focal region. The latter inverse problem has the basic assumption that $P$ deviates only mildly from being constant so that the term with $m=n=0$ in (\\ref{e9}) dominates the totality of all other terms. The theoretical intensity point-spread function can then, with modest error, be linearized around the leading term $|\\beta_0^0\\,U_0^0|^2$, and this leads, via a matching procedure with the measured data in the focal region, to a first estimate of the unknown coefficients. This procedure is made iterative by incorporating the totality of all deleted small cross-terms (involving the $\\beta_n^m$, $(m,n)\\neq(0,0)$, quadratically) in the matching procedure using the estimate of the $\\beta$'s from the previous steps. In practice one finds that pupil functions $P$ deviating from being constant by as much as 2.5 times the diffraction limit can be retrieved. See \\cite{ref31}, \\cite{ref32}, \\cite{ref34}, \\cite{ref35}, \\cite{ref36} for more details.\n\nThe result in (\\ref{e8}) is one evidence of computational appropriateness of the circle polynomials for forward and inverse problems, but there are others. Many of these are based on the basic NZ-result in (\\ref{e8}). In \\cite{ref37}--\\cite{ref38}, Cormack used this result to calculate the Radon transform ${\\cal R}_n^m$,\n\\begin{equation} \\label{e13}\n{\\cal R}_n^m(\\tau,\\psi)=\\int\\limits_{l(\\tau,\\psi)}\\,Z_n^m(\\nu,\\mu)\\,dl\n\\end{equation}\nof $Z_n^m$, with integration along the line $\\nu\\cos\\psi+\\mu\\sin\\psi=\\tau$ in the plane with $\\tau\\geq0$ and $\\psi\\in[0,2\\pi)$ and where $(\\nu,\\mu)=(\\rho\\cos\\vartheta,\\rho\\sin\\vartheta)$. The result is that\n\\begin{equation} \\label{e14}\n{\\cal R}_n^m(\\tau,\\psi)=\\frac{2}{n+1}\\,(1-\\tau^2)^{1\/2}\\,U_n(\\tau)\\,e^{im\\psi}~,~~~~~0\\leq\\tau\\leq1\\,,~~0\\leq\\psi<2\\pi\\,,\n\\end{equation}\nwhere $U_n$ is the Chebyshev polynomial of the second kind and degree $n$, \\cite{ref14}, Ch.~22. On this explicit form of the Radon transform of $Z_n^m$, Cormack based a method for estimating a function on the disk from its Radon transform by estimating its Zernike expansion coefficients through matching. (In 1979, Cormack was awarded the Nobel prize in medicine, together with Hounsfield, for their work in computerized tomography.) Cormack's result was used by Dirksen and Janssen \\cite{ref39} to find the integral representation\n\\begin{equation} \\label{e15}\nR_n^m(\\rho)=\\frac{1}{2\\pi}\\,\\int\\limits_0^{2\\pi}\\,U_n(\\rho\\cos\\vartheta)\\cos m\\vartheta\\,d\\vartheta\n\\end{equation}\n(integer $m,n\\geq0$) that displays, for any $\\rho\\geq0$, the value of the radial part at $\\rho$ in the form of the Fourier coefficient of a trigonometric polynomial. This formula (\\ref{e15}) can be discretized, error free when more that $m+n$ equidistant points in $[0,2\\pi]$ are used, and this yields a scheme of the DCT-type for computation of $R_n^m(\\rho)$.\n\nA further consequence of the basic NZ-result (\\ref{e8}) is the theory of shifted-and-scaled circle polynomials developed in \\cite{ref40}. For given $a\\geq0$, $b\\geq0$ with $a+b\\leq1$, there are developed explicit expressions for the coefficients $K_{nn'}^{mm'}(a,b)$ in the expansion\n\\begin{equation} \\label{e16}\nZ_n^m((a+b\\rho'\\cos\\vartheta',b\\rho'\\sin\\vartheta'))=\n\\sum_{n',m'}\\,K_{nn'}^{mm'}(a,b)\\,Z_{n'}^{m'}(\\rho'\\cos\\vartheta,\\rho'\\sin\\vartheta')~.\n\\end{equation}\nThis generalizes the result in \\cite{ref41},\n\\begin{equation} \\label{e17}\nR_n^m(\\varepsilon\\rho)=\\sum_{n'=m(2)n}\\,(R_n^{n'}(\\varepsilon)-R_n^{n'+2}(\\varepsilon))\\,R_{n'}^m(\\rho)~,\n\\end{equation}\non the Zernike expansion of scaled circle polynomials ($m\\geq 0$).\n\nThe basic NZ-result is also useful for the computation of various acoustic quantities that arise from a circular piston in a planar baffle. The complex amplitude $p({\\bf r},k)$ of the sound pressure at the field point ${\\bf r}=(x,y,z)$, $z\\geq0$, in front of the baffle plane $z=0$ due to a harmonic excitation $\\exp(i\\omega t)$ with wave number $k=\\omega\/c$, with $c$ the speed of sound, is given by Rayleigh's integral and King's integral as\n\\begin{eqnarray} \\label{e18}\np({\\bf r},k) & = & \\frac{i\\rho_0 ck}{2\\pi}\\,\\int\\limits_S\\,v(\\sigma)\\,\\frac{e^{-ikr'}}{r'}\\,dS~= \\nonumber \\\\[3.5mm]\n& = & i\\rho_0ck\\,\\int\\limits_0^{\\infty}\\,\\frac{e^{-z(u^2-k^2)^{1\/2}}}{(u^2-k^2)^{1\/2}}\\,J_0(wu)\\,V(u)\\,u\\,du~.\n\\end{eqnarray}\nHere $\\rho_0$ is the density of the medium, $v(\\sigma)$ is a non-uniform velocity profile assumed to depend on the radial variable $\\sigma=(\\nu^2+\\mu^2)^{1\/2}$ on the piston surface $S$ with center ${\\bf 0}$ and radius $a$, and $r'$ is the distance from the field point ${\\bf r}$ to the point $(\\nu,\\mu,0)$ on $S$. Furthermore, $w=(x^2+y^2)^{1\/2}$ is the distance from the field point ${\\bf r}$ to the $z$-axis, the root $(u^2-k^2)^{1\/2}$ has the value $i\\,\\sqrt{k^2-u^2}$ and $\\sqrt{u^2-k^2}$ for $0\\leq u\\leq k$ and $u\\geq k$, respectively with $\\sqrt{~~}$ non-negative in both cases, and $V(u)$ is the Hankel transform,\n\\begin{equation} \\label{e19}\nV(u)=\\int\\limits_0^a\\,J_0(u\\sigma)\\,v(\\sigma)\\,\\sigma\\,d\\sigma~,~~~~~~u\\geq0~,\n\\end{equation}\nof order 0 of $v(\\sigma)$.\n\nIn \\cite{ref19}, the on-axis pressure $p_{2l}({\\bf r}=(0,0,z),k)$ due to $v(\\sigma)=R_{2l}^0(\\sigma\/a)$, $0\\leq\\sigma\\leq a$, was shown from Rayleigh's integral and a special result on spherical Bessel functions to be given as\n\\begin{equation} \\label{e20}\np_{2l}({\\bf r}=(0,0,z),k)=\\tfrac12\\,\\rho_0\\,c(ka)^2({-}1)^l\\,j_l(kr_-)\\,h_l^{(2)}(kr_+)~,\n\\end{equation}\nwhere $j_l$ and $h_l^{(2)}$ are spherical Bessel functions, see \\cite{ref14}, Ch.~10 and in particular 10.1.45--46, and $r_{\\pm}=\\frac12\\,[(r^2+a^2)^{1\/2}\\pm r]$. This result was used in \\cite{ref19} for estimating a velocity profile from on-axis pressure data on the level of expansion coefficients with respect to radially symmetric circle polynomials.\n\nIn \\cite{ref20}, the King integral for the pressure is employed to express the pressure $p((1,0,0),k)$ at the edge, the reaction force $\\int\\limits_S\\,p\\,dS$ and the total radiated power $\\int\\limits_S\\,p(0)\\,v^{\\ast}(\\sigma)\\,dS$ in integral form. Expanding $v(\\sigma)$ into radially symmetric circle polynomials and using the basic NZ-result for an explicit expression  of the Hankel transform $V(u)$ in (\\ref{e19}), this gives rise to integrals\n\\begin{equation} \\label{e21}\n\\int\\limits_0^{\\infty}\\,\\frac{J_m(au)\\,J_{n+1}(au)}{(u^2-k^2)^{1\/2}}\\,du~,~~~~~~\\int\\limits_0^{\\infty}\\, \\frac{J_{m+1}(au)\\,J_{n+1}(au)}{(u^2-k^2)^{1\/2}\\,u}\\,du\n\\end{equation}\nwith integer $m,n\\geq0$ of same parity. These integrals have been evaluated as a power series in $ka$ in \\cite{ref20}.\n\nIn \\cite{ref21}, the problem of sound radiation from a flexible spherical cap on a rigid sphere is considered. The scaling theory of Zernike circle polynomials, appropriately warped so as to account for the spherical geometry of the problem, is used to bring the standard solution of the Helmholtz equation with axially symmetric boundary data in a semi-analytic form per warped Zernike term. This gives rise to a coefficient-based solution of the inverse problem of estimating an axially symmetric velocity profile on the cap from measured pressure data in the space around the sphere.\n\nFinally, returning to baffled-piston radiation, in \\cite{ref22} the impulse response $h({\\bf r},t)$, $t\\geq0$, at a field point ${\\bf r}=(x,y,z)$ in front of the baffle is considered. This $h({\\bf r},t)$ is obtained as a Fourier inversion integral with respect to wave number $k$ of the velocity potential $\\varphi({\\bf r},k)=(i\\rho_0ck)^{-1}\\,p({\\bf r},k)$, which can be evaluated, via Rayleigh's integral, as an integral of the velocity profile $v$ over all points in the baffle plane at a common distance $(c^2t^2-z^2)^{1\/2}$ from the field point. For the case that $v(\\sigma)=R_{2l}^0(\\sigma\/a)$, $0\\leq\\sigma\\leq a$, the latter integral can be evaluated explicitly using the addition theorem for Legendre polynomials. This explicit result can be used to compute impulse responses for general $v$ by expanding such a $v$ into radially symmetric circle functions. And, of course, an inverse problem, with a coefficient-based solution, can again be formulated.\n\n\\section{Generalized Zernike circle functions} \\label{sec3}\n\\mbox{} \\\\[-9mm]\n\nThe Zernike circle polynomials, see (\\ref{e1}), all have modulus 1 at the edge $\\rho=1$ of the unit disk while in both Optics and Acoustics it frequently occurs that the non-uniformity behaves differently towards the edge of the disk.\n\nIn optical design, it is often desirable to have a pupil function $P$ whose point-spread function $U$, see (\\ref{e6}), decays relatively fast outside the focal region where the specifications of the designer are to be met. The Zernike circle polynomials are discontinuous at the edge $\\rho=1$, and the corresponding point-spread functions $U_n^m(r,\\varphi\\,;\\,f)$ have poor decay, like $r^{-3\/2}$ as $r\\rightarrow\\infty$.\n\nIn the ENZ point-spread functions, see Sec.~\\ref{sec2}, the key assumption is that the pupil function's deviation from being constant is not large. When $|P|$ gets small at the edge, which happens, for instance, when the source is a pinhole with a positive diameter while the objective lens has a large NA, this basic assumption is not met. In such a case, aberration retrieval with the ENZ method may become cumbersome, even in its iterative version.\n\nIn the theory of acoustic radiation from a baffled, planar piston with radially symmetric boundary conditions, Streng \\cite{ref42} follows the approach of Bouwkamp \\cite{ref43} in solving the Helmholtz equation for this case and postulates normalized pressure functions on the disk of the form\n\\begin{equation} \\label{e22}\n\\sum_{n=0}^{\\infty}\\,a_n(1-\\rho^2)^{n+1\/2}~,~~~~~~0\\leq\\rho<1~.\n\\end{equation}\nIn the light of the later developments of this paper, it is interesting to note here that Bouwkamp himself prefers expansions that involve the functions\n\\begin{equation} \\label{e23}\n(1-\\rho^2)^{1\/2}\\,P_l^{(1\/2,0)}(2\\rho^2-1)=({-}1)^l\\,P_{2l+1}((1-\\rho^2)^{1\/2})~,~~~~~~0\\leq\\rho<1~.\n\\end{equation}\nSimilarly, Mellow \\cite{ref44} has velocity profiles on the disk of the form (normalized to the unit disk)\n\\begin{equation} \\label{e24}\nv(\\rho)=\\sum_{n=0}^{\\infty}\\,b_n(1-\\rho^2)^{n-1\/2}~,~~~~~~0\\leq\\rho<1~.\n\\end{equation}\nIn \\cite{ref44} and \\cite{ref45}--\\cite{ref46}, the postulates (\\ref{e22}), (\\ref{e24}) are used to solve design problems in acoustic radiation from a membrane in a circular disk in terms of the expansion coefficients $a_n$, $b_n$. In principle, these design problems could also be solved when radially symmetric circle polynomials instead of $(1-\\rho^2)^{n\\pm1\/2}$ were used as trial functions, but this would become cumbersome, the functions $(1-\\rho^2)^{\\pm1\/2}$ themselves already having a poorly convergent expansion with respect to the $R_{2l}^0(\\rho)$.\n\nIn the three instances just discussed, it would be quite helpful when the system of Zernike circle polynomials would be replaced by systems whose members exhibit an appropriate behaviour at the edge of the disk. These new systems should satisfy the requirements A, B and C mentioned in the beginning of Sec.~\\ref{sec1}. In this paper, the set of functions\n\\begin{eqnarray} \\label{e25}\nZ_n^{m,\\alpha}(\\rho,\\vartheta) & \\!\\!= & \\!\\!(1-\\rho^2)^{\\alpha}\\,\\rho^{|m|}\\,P_{\\frac{n-|m|}{2}}^{(\\alpha,|m|)}(2\\rho^2-1)\\, e^{im\\vartheta}~,~~~0\\leq\\rho<1\\,,~0\\leq\\vartheta<2\\pi \\nonumber \\\\[3mm]\n& \\!\\!= & \\!\\!0\\hspace*{6.1cm},~~~\\rho>1~,\n\\end{eqnarray}\nis considered for this purpose. In (\\ref{e25}), the parameter $\\alpha>{-}1$, and $n$ and $m$ are integers such that $n-|m|$ is even and non-negative, and the $P_k^{(\\alpha,\\beta)}$ are Jacobi polynomials as before. Observe that the radial parts\n\\begin{equation} \\label{e26}\nR_n^{|m|,\\alpha}(\\rho)=(1-\\rho^2)^{\\alpha}\\, \\rho^{|m|}\\, P_{\\frac{n-|m|}{2}}^{(\\alpha,|m|)}(2\\rho^2-1)~,~~~~~~0\\leq\\rho<1~,\n\\end{equation}\ndepend non-polynomially on $\\rho$, unless $\\alpha=0,1,...\\,$. The radial parts occur essentially in Tango \\cite{ref47}, Sec.~1, but there the factor $(1-\\rho^2)^{\\alpha}$ is replaced by $(1-\\rho^2)^{\\alpha\/2}$ with somewhat different restrictions on $\\alpha$ and $|m|$. There is also a relation with the disk polynomials as they occur, for instance, in the work of Koornwinder \\cite{ref48}:\n\\begin{equation} \\label{e27}\nZ_n^{m,\\alpha}(\\rho,\\vartheta)=(1-\\rho^2)^{\\alpha}\\,D_{\\frac{n+m}{2}\\,,\\,\\frac{n-m}{2}}^{\\alpha}(\\rho\\,e^{i\\vartheta})~,\n\\end{equation}\nwith\n\\begin{equation} \\label{28}\nD_{k,l}^{\\alpha}(\\rho\\,e^{i\\vartheta})=\\rho^{|k-l|}\\,P_{\\min(k,l)}^{(\\alpha,|k-l|)}(2\\rho^2-1)\\,e^{i(k-l)\\vartheta}\n\\end{equation}\nfor $k,l=0,1,...\\,$. Thus, the disk polynomials omit the factor $(1-\\rho^2)^{\\alpha}$ altogether.\n\nFurther definitions used in this paper are\n\\begin{equation} \\label{e29}\np=\\frac{n-|m|}{2}~,~~~~~~q=\\frac{n+|m|}{2}\n\\end{equation}\nfor integers $n$ and $m$ such that $n-|m|$ is even and non-negative, and the generalized Pochhammer symbol\n\\begin{equation} \\label{e30}\n(x)_y=\\frac{\\Gamma(x+y)}{\\Gamma(x)}~.\n\\end{equation}\n\nBy orthogonality of the $P_k^{(\\alpha,\\beta)}(x)$ with respect to the weight function $(1-x)^{\\alpha}(1+x)^{\\beta}$ on $[{-}1,1]$, there holds\n\\begin{eqnarray} \\label{e31}\n& \\mbox{} & \\frac{1}{\\pi}\\,\\int\\limits_0^1\\int\\limits_0^{2\\pi}\\,(1-\\rho^2)^{-\\alpha}\\,Z_{n_1}^{m_1,\\alpha}(\\rho,\\vartheta) (Z_{n_2}^{m_2,\\alpha}(\\rho,\\vartheta))^{\\ast}\\,\\rho\\,d\\rho\\,d\\vartheta~= \\nonumber \\\\[3.5mm]\n& & =~\\frac{(p+1)_{\\alpha}}{(p+|m|+1)_{\\alpha}}~\\frac{\\delta_{m_1m_2}\\,\\delta_{n_1n_2}}{n_1+\\alpha+1}~.\n\\end{eqnarray}\n\nWe note that for any integer $N=0,1,...$\n\\begin{equation} \\label{e32}\n\\rho^{|m|}\\,P_{\\frac{n-|m|}{2}}^{(\\alpha,|m|)}(2\\rho^2-1)\\,e^{im\\vartheta}\n\\end{equation}\nwith $n=0,1,...,N$ and $m={-}n,{-}n+2,...,n$ are $\\frac12\\,(N+1)(N+2)$ linearly independent functions of the form $\\sum_{i,j}\\,a_{ij}\\nu^i\\mu^j$ ($\\nu=\\rho\\cos\\vartheta$, $\\mu=\\rho\\sin\\vartheta$) where the summation is over integer $i,j\\geq0$ with $i+j\\leq N$. Therefore, by Weierstrass theorem the functions in (\\ref{e32}) are complete.\n\nThis paper focuses on establishing versions for the generalized Zernike functions of the analytic results that were presented for the classical circle polynomials in Sec.~\\ref{sec2}. Thus, in Sec.~\\ref{sec4}, the Fourier transform of $Z_n^{m,\\alpha}$ is computed in terms of Bessel functions and a Weber-Schafheitlin representation of the radial functions $R_n^{m,\\alpha}$ is given. In Sec.~\\ref{sec5}, the Radon transform of $Z_n^{m,\\alpha}$ is computed in terms of the Gegenbauer polynomials $C_n^{\\alpha+1}$, and a representation of the radial functions as Fourier coefficients of the periodic function $C_n^{\\alpha+1}(\\rho\\cos\\vartheta)$, $0\\leq\\vartheta<2\\pi$, for a fixed $\\rho\\in[0,1)$ is given. Thus a computation scheme of the DCT-type for the radial functions arises. Then, in Sec.~\\ref{sec6}, a scaling result is given. This result is somewhat more awkward than in the classical case since the radial functions $R_n^{m,\\alpha}(\\rho)$ have restrictions on their behaviour as $\\rho\\uparrow1$ while the scaled radial functions $R_n^{m,\\alpha}(\\varepsilon\\rho)$, to be considered with $0<\\varepsilon<1$, do not. Next, in Sec.~\\ref{sec7}, the new Zernike functions are expanded in terms of the classical circle polynomials, with an explicit expression for the expansion coefficients. This makes it possible to transfer all forward computation results from the ENZ theory and from the acoustic Nijboer-Zernike (ANZ) theory for the classical case to the new setting in a semi-analytic form. This is useful for those cases that a closed form or a simple semi-analytic form is not available or awkward to find in the new setting. In Secs.~\\ref{sec8}--\\ref{sec10}, the focus is on generalizing the results from the ANZ theory, as developed in \\cite{ref19}--\\cite{ref22} to the more general setting. This gives rise in Sec.~\\ref{sec8} to a power series representation of the basic integrals that occur when various acoustic quantities are computed from King's integral for the sound pressure (in the case of baffled-piston radiation). An inverse problem, in which the velocity profile is estimated in terms of its expansion coefficients from near-field measurements via Weyl's formula, is considered in Sec.~\\ref{sec9}. Finally, in Sec.~\\ref{sec10}, the trial functions as used by Streng and Mellow, following Bouwkamp's solution of the diffraction problem for a circular aperture, are compared with the $Z_{2l}^{0,\\alpha={\\pm}1\/2}$. In \\cite{ref46}, Mellow and K\\\"arkk\\\"ainen consider radiation from a disk with concentric rings, and for this an indefinite integral involving the radial functions $R_{2l}^{0,{\\pm}1\/2}(\\rho)$ is required. This integral is computed in closed form.\n\n\\section{Fourier transform of generalized Zernike circle functions} \\label{sec4}\n\\mbox{} \\\\[-9mm]\n\nIn this section, the 2D Fourier transform of $Z_n^{m,\\alpha}$ is computed. It is convenient to write here\n\\begin{equation} \\label{e33}\nZ_n^{m,\\alpha}(\\nu,\\mu)\\equiv Z_n^{m,\\alpha}(\\rho,\\vartheta)~,\n\\end{equation}\nwhere $\\nu+i\\mu=\\rho\\,e^{i\\vartheta}$ with $\\rho\\geq0$ and $0\\leq\\vartheta<2\\pi$. \\\\ \\\\\n{\\bf Theorem 4.1.}~~For $\\alpha>{-}1$ and integer $n$, $m$ such that $p=\\frac12\\,(n-|m|)$ is a non-negative integer, there holds\n\\begin{eqnarray} \\label{e34}\n& \\mbox{} & \\int\\limits\\!\\!\\int\\limits\\,e^{2\\pi i\\nu x+2\\pi i\\mu y}\\,Z_n^{m,\\alpha}(\\nu,\\mu)\\,d\\nu\\,d\\mu~= \\nonumber \\\\[3.5mm]\n& & =~2\\pi\\,i^n\\,2^{\\alpha}(p+1)_{\\alpha}\\,\\frac{J_{n+\\alpha+1}(2\\pi r)}{(2\\pi r)^{\\alpha+1}}\\,e^{im\\varphi}~,\n\\end{eqnarray}\nwhere $x+iy=r\\,e^{i\\varphi}$ with $r\\geq0$ and $0\\leq\\varphi<2\\pi$. Furthermore,\n\\begin{equation} \\label{e35}\n\\displaystyle \\int\\limits_0^1\\,R_n^{|m|,\\alpha}(\\rho)\\,J_{|m|}(2\\pi\\rho r)\\,\\rho\\,d\\rho=({-}1)^p\\,2^{\\alpha}(p+1)_{\\alpha}\\, \\frac{J_{n+\\alpha+1}(2\\pi r)}{(2\\pi r)^{\\alpha+1}}~,\n\\end{equation}\nand\n\\begin{equation} \\label{e36}\nR_n^{|m|,\\alpha}(\\rho)=({-}1)^p\\,2^{\\alpha}(p+1)_{\\alpha}\\,\\displaystyle \\int\\limits_0^{\\infty}\\, \\frac{J_{n+\\alpha+1}(t)\\,J_{|m|}(\\rho t)}{t^{\\alpha}}\\,dt~,~~~~~0\\leq\\rho<1~.\n\\end{equation}\n{\\bf Proof.}~~First consider the case that $m\\geq0$. It follows from \\cite{ref14}, 11.4.33 (Weber-Schafheitlin integral), with\n\\begin{equation} \\label{e37}\n\\mu=n+p+1=m+2p+\\alpha+1\\,,~~a=1\\,,~~\\nu=m\\,,~~b=\\rho\\in[0,1)\\,,~~\\lambda=\\alpha\n\\end{equation}\nthat\n\\begin{eqnarray} \\label{e38}\n& \\mbox{} & \\int\\limits_0^{\\infty}\\,\\frac{J_{n+\\alpha+1}(t)\\,J_m(\\rho t)}{t^{\\alpha}}\\,dt~= \\nonumber \\\\[3mm]\n& & =~\\frac{\\rho^m\\,\\Gamma(m+p+1)}{2^{\\alpha}\\, \\Gamma(m+1)\\,\\Gamma(p+\\alpha+1)}\\,F({-}p-\\alpha,m+p+1\\,;\\,m+1\\,;\\,\\rho^2)~. \\nonumber \\\\\n& & \\mbox{}\n\\end{eqnarray}\nNext from \\cite{ref14}, 15.3.3 with\n\\begin{equation} \\label{e39}\na=m+1+p+\\alpha\\,,~~~b={-}p\\,,~~~c=m+1\\,,~~~z=\\rho^2\n\\end{equation}\nit follows that\n\\begin{eqnarray} \\label{e40}\n& \\mbox{} & F({-}p-\\alpha,m+p+1\\,;\\,m+1\\,;\\,\\rho^2)~= \\nonumber \\\\[3mm]\n& & =~(1-\\rho^2)^{\\alpha}\\,F({-}p,m+1+\\alpha+p\\,;\\,m+1\\,;\\,\\rho^2)~= \\nonumber \\\\[3mm]\n& & =~(1-\\rho^2)^{\\alpha}\\,\\frac{p!}{(m+1)_p}\\,P_p^{(m,\\alpha)}(1-2\\rho^2)~,\n\\end{eqnarray}\nwhere in the last step \\cite{ref14}, 15.4.6 with $n=p$, $\\alpha=m$, $\\beta=\\alpha$ and $z=\\rho^2$ has been used. Therefore, for general integer $m$, $|m|\\leq n$,\n\\begin{eqnarray} \\label{e41}\n& \\mbox{} & \\int\\limits_0^{\\infty}\\,\\frac{J_{n+\\alpha+1}(t)\\,J_{|m|}(\\rho t)}{t^{\\alpha}}\\,dt~= \\nonumber \\\\[3.5mm]\n& & =~\\frac{({-}1)^p}{2^{\\alpha}(p+1)_{\\alpha}}\\,\\rho^{|m|}(1-\\rho^2)^{\\alpha}\\,P_p^{(\\alpha,|m|)}(2\\rho^2-1) = \\frac{({-}1)^p}{2^{\\alpha}(p+1)_{\\alpha}}\\,R_n^{|m|,\\alpha}(\\rho)~, \\nonumber \\\\\n& & \\mbox{}\n\\end{eqnarray}\nwhere $P_k^{(\\alpha,\\beta)}({-}x)=({-}1)^k\\,P_k^{(\\beta,\\alpha)}({-}x)$ and the definition (\\ref{e26}) of $R_n^{|m|,\\alpha}(\\rho)$ have been used. This establishes (\\ref{e36}).\n\nNext,\n\\begin{eqnarray} \\label{e42}\n& \\mbox{} & \\int\\limits_0^{\\infty}\\int\\limits_0^{2\\pi}\\,e^{-2\\pi i\\nu x-2\\pi i\\mu y}\\,\\frac{J_{n+\\alpha+1}(2\\pi r)} {(2\\pi r)^{\\alpha+1}}\\,e^{im\\varphi}\\,r\\,dr\\,d\\varphi~= \\nonumber \\\\[3.5mm]\n& & =~2\\pi\\,i^m\\,\\int\\limits_0^{\\infty}\\,J_m({-}2\\pi r\\rho)\\,\\frac{J_{n+\\alpha+1}(2\\pi r)}{(2\\pi r)^{\\alpha+1}}\\,r\\,dr\\,e^{im\\vartheta}~= \\nonumber \\\\[3.5mm]\n& & =~\\frac{({-}1)^{|m|}}{2\\pi}\\,\\int\\limits_0^{\\infty}\\,\\frac{J_{|m|}(t)\\,J_{n+\\alpha+1}(t)}{t^{\\alpha}}\\,dt\\, e^{im\\vartheta}~,\n\\end{eqnarray}\nwhere, subsequently, use is made of\n\\begin{equation} \\label{e43}\n\\nu x+\\mu y=\\rho\\,r\\cos(\\vartheta-\\varphi)~,\n\\end{equation}\n\\begin{equation} \\label{e44}\n\\frac{1}{2\\pi}\\,\\int\\limits_0^{2\\pi}\\,e^{-it\\cos\\vartheta}\\,e^{im\\vartheta}\\,d\\vartheta=i^m\\,J_m({-}t)=({-}i)^{|m|}\\,J_{|m|}(t)~,\n\\end{equation}\nsee \\cite{ref14}, Sec.~9.1, and where the substitution $t=2\\pi r$ has been used in the last step in (\\ref{e42}). Therefore, using (\\ref{e41}), the definitions (\\ref{e25})--(\\ref{e26}) and $n=|m|+2p$, it follows that\n\\begin{eqnarray} \\label{45}\n& \\mbox{} & \\int\\limits\\!\\!\\int\\limits\\,e^{-2\\pi i\\mu x-2\\pi i\\mu y}\\,\\frac{J_{n+\\alpha+1}(2\\pi r)}{(2\\pi r)^{\\alpha+1}}\\,r\\,dr\\,d\\varphi~= \\nonumber \\\\[3mm]\n& & =~\\frac{({-}i)^n}{2\\pi}~\\frac{1}{2^{\\alpha}(p+1)_{\\alpha}}\\,Z_n^{m,\\alpha}(\\rho,\\vartheta)~.\n\\end{eqnarray}\nThen (\\ref{e34}) follows by 2D Fourier inversion. Now also (\\ref{e35}) follows using the definitions (\\ref{e25})--(\\ref{e26}) in (\\ref{e34}) and proceeding as in (\\ref{e42})--(\\ref{e44}). \\\\ \\\\\n{\\bf Notes.} \\\\[-7mm]\n\\bi{1.0}\n\\ITEM{1.} The result in (\\ref{e34}) generalizes the case $\\alpha=0$ in (\\ref{e7}) to general $\\alpha>{-}1$.\n\\ITEM{2.} The result in (\\ref{e35}) gives the Hankel transform of order $|m|$ of the radial part $R_n^{|m|,\\alpha}$, and (\\ref{e36}) is what one gets by inverse Hankel transformation of order $|m|$. These two results are reminiscent of, but clearly different from, the results in (\\ref{e10}) and (\\ref{e11}) of \\cite{ref47}.\n\\ITEM{3.} By the asymptotics of the Bessel functions, see \\cite{ref14}, 9.2.1, it is seen that the Fourier transform of $Z_n^{m,\\alpha}$ decays as $r^{-\\alpha-3\/2}$ as $r\\rightarrow\\infty$.\n\\end{inspring}\n\n\\section{Radon transform of generalized Zernike circle functions; integral representation and DCT-formula for radial parts} \\label{sec5}\n\\mbox{} \\\\[-9mm]\n\nIn this section, the Radon transform of the generalized circle functions is expressed in terms of Gegenbauer polynomials. Furthermore, an integral representation involving these Gegenbauer polynomials for the radial parts is proved, and a method of the DCT-type for computation of the radial parts is shown to follow from this integral representation.\n\nThe Radon transform of $Z_n^{m,\\alpha}$ is given by\n\\begin{equation} \\label{e46}\n({\\cal R}\\,Z_n^{m,\\alpha})(\\tau,\\psi)=\\int\\limits_{l(\\tau,\\psi)}\\,Z_n^{m,\\alpha}(\\nu,\\mu)\\,dl\n=\\int\\limits\\,R_n^{|m|,\\alpha}(\\rho(t))\\,e^{im\\vartheta(t)}\\,dt~,\n\\end{equation}\nwith $l(\\tau,\\psi)$ and $\\rho$, $\\vartheta$ given in Fig.~1 for $\\tau\\geq0$ and $0\\leq\\psi<2\\pi$. Thus\n\\begin{equation} \\label{e47}\n\\rho(t)=(\\tau^2+t^2)^{1\/2}\\,,~~\\vartheta(t)=\\psi+\\arctan(t\/\\tau)=\\psi+{\\rm sgn}(t)\\arccos(\\tau\/\\rho(t))~.\n\\end{equation}\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=0.70\\linewidth]{Report2011Fig1.eps}\n\\caption{Line of integration $l(\\tau,\\psi)$ and integration point $P=Q+t({-}\\sin\\psi,\\cos\\psi)$ with $-\\infty<t<\\infty$, given in polar coordinates with respect to ${\\bf \\underline{0}}$ as $(\\rho\\cos\\vartheta,\\rho\\sin\\vartheta)$ with $\\rho=(\\tau^2+t^2)^{1\/2}$ and $\\vartheta=\\psi+\\arctan(t\/\\tau)$.}  \n\\end{center}\n\\end{figure}\nSubstituting $\\rho=\\rho(t)$, with $-\\infty<t<\\infty$ such that $\\tau\\leq\\rho(t)<1$ in the second integral in (\\ref{e46}) and using the second form of $\\vartheta(t)$ in (\\ref{e47}), it follows that\n\\begin{eqnarray} \\label{e48}\n({\\cal R}\\,Z_n^{m,\\alpha})(\\tau,\\psi) & = & 2e^{im\\psi}\\,\\int\\limits_{\\tau}^1\\,R_n^{|m|,\\alpha}(\\rho)\\cos [m\\arccos(\\tau\/\\rho)]\\,\\frac{\\rho\\,d\\rho}{\\sqrt{\\rho^2-\\tau^2}}~= \\nonumber \\\\[3.5mm]\n& = & 2e^{im\\psi}\\,\\int\\limits_{\\tau}^1\\,R_n^{|m|,\\alpha}(\\rho)\\, \\frac{T_m(\\tau\/\\rho)}{\\sqrt{1-(\\tau\/\\rho)^2}}\\,d\\rho~,\n\\end{eqnarray}\nwhere $T_m$ is the Chebyshev polynomial of the first kind and of degree $m$, see \\cite{ref14}, Ch.~22. \\\\ \\\\\n{\\bf Theorem 5.1.}~~There holds for $\\alpha>{-}1$ and integers $n$, $m$ such that $p=\\frac12\\,(n-|m|)$ is a non-negative integer that\n\\begin{equation} \\label{e49}\n({\\cal R}\\,Z_n^{m,\\alpha})(\\tau,\\psi)= \\frac{(p+1)_{\\alpha}}{(n+1)_{2\\alpha}}~\\frac{2^{2\\alpha+1}\\,\\Gamma(\\alpha+1)} {n+2\\alpha+1}\\,(1-\\tau^2)^{\\alpha+1\/2}\\,C_n^{\\alpha+1}(\\tau)\\,e^{im\\psi}\n\\end{equation}\nwhen $0\\leq\\tau\\leq1$ and $({\\cal R}\\,Z_n^{m,\\alpha})(\\tau,\\psi)=0$ for $\\tau>1$. Here $C_n^{\\alpha+1}$ is the Gegenbauer (or ultraspherical) polynomial corresponding to the weight function $(1-x^2)^{\\alpha+1\/2}$, $-1<x<1$, and of degree $n$, see \\cite{ref14}, Ch.~22. \\\\[2mm]\n{\\bf Proof.}~~Consider the case that $m\\geq0$ (the case that $m<0$ is obtained from this by complex conjugation). By\\cite{ref14}, 11.4.24,\n\\begin{eqnarray} \\label{e50}\n\\int\\limits_{-\\infty}^{\\infty}\\,e^{-i\\omega t}\\,J_m(t)\\,dt & = & \\frac{2({-}i)^m\\,T_m(\\omega)}{(1-\\omega^2)^{1\/2}}~,~~~~~~ \\omega^2<1~, \\nonumber \\\\[2mm]\n&= & 0\\hspace*{2.5cm},~~~~~~\\omega^2>1~,\n\\end{eqnarray}\nand so, from (\\ref{e48})\n\\begin{eqnarray} \\label{e51}\n({\\cal R}\\,Z_n^{m,\\alpha})(\\tau,\\psi) & = & e^{im\\psi}\\,\\int\\limits_0^1\\,R_n^{m,\\alpha}(\\rho)\\,\\left( i^m\\,\\int\\limits_{-\\infty}^{\\infty}\\,e^{-i\\tau t\/\\rho}\\,J_m(t)\\,dt\\right)\\,d\\rho~= \\nonumber \\\\[3.5mm]\n& = & e^{im\\psi}\\,i^m\\,\\int\\limits_{-\\infty}^{\\infty}\\,e^{-i\\tau s}\\,\\left(\\int\\limits_0^1\\,R_n^{m,\\alpha}(\\rho)\\,J_m(\\rho s) \\,\\rho\\,d\\rho\\right)\\,ds~= \\nonumber \\\\[3.5mm]\n& = & e^{im\\psi}\\,i^n\\,2^{\\alpha}(p+1)_{\\alpha}\\,\\int\\limits_{-\\infty}^{\\infty}\\,e^{-i\\tau s}\\, \\frac{J_{n+\\alpha+1}(s)} {s^{\\alpha+1}}\\,ds~,\n\\end{eqnarray}\nwhere (\\ref{e35}) has been used, together with $i^m({-}1)^p=i^n$.\n\nBy \\cite{ref49}, 1.12, item (10), there holds for even $n$ ($a=1$, $2n$ instead of $n$, $\\nu=\\alpha+1$)\n\\begin{eqnarray} \\label{e52}\n& \\mbox{} & 2\\,\\int\\limits_0^{\\infty}\\cos\\tau s\\,\\frac{J_{n+\\alpha+1}(s)}{s^{\\alpha+1}}\\,ds~= \\nonumber \\\\[3.5mm]\n& & =~({-}1)^{\\frac12 n}\\,\\frac{2^{\\alpha+1}\\,n!\\,\\Gamma(\\alpha+1)}{\\Gamma(n+2\\alpha+2)}\\, (1-\\tau^2)^{\\alpha+1\/2} \\,C_n^{\\alpha+1}(\\tau)~,~~~~~~0\\leq\\tau<1~, \\nonumber \\\\\n& \\mbox{} &\n\\end{eqnarray}\nwhile the integral in (\\ref{e52}) equals 0 when $\\tau>1$. By \\cite{ref49}, 2.12, item (10), there holds for odd $n$ ($a=1$, $2n+1$ instead of $n$, $\\nu=\\alpha+1$)\n\\begin{eqnarray} \\label{e53}\n& \\mbox{} & -2i\\,\\int\\limits_0^{\\infty}\\sin\\tau s\\,\\frac{J_{n+\\alpha+1}(s)}{s^{\\alpha+1}}\\,ds~= \\nonumber \\\\[3.5mm]\n& & =~{-}i({-}1)^{\\frac{n-1}{2}}\\,\\frac{2^{\\alpha+1}\\,n!\\,\\Gamma(\\alpha+1)}{\\Gamma(n+2\\alpha+2)}\\, (1-\\tau^2)^{\\alpha+1\/2}\\,C_n^{\\alpha+1}(\\tau)~,~~~~~~0\\leq\\tau<1~, \\nonumber \\\\\n& \\mbox{} &\n\\end{eqnarray}\nwhile the integral in (\\ref{e53}) equals 0 when $\\tau>1$. Using (\\ref{e52}) and (\\ref{e53}) in (\\ref{e51}), it follows upon some administration with $i^n$ and $({-}1)^{\\frac12 n}$, $({-}1)^{\\frac12 (n-1)}$ that for $0\\leq\\tau<1$\n\\begin{equation} \\label{e54}\n({\\cal R}\\,Z_n^{m,\\alpha})(\\tau,\\psi)=(p+1)_{\\alpha}\\, \\frac{2^{2\\alpha+1}\\,n!\\,\\Gamma(\\alpha+1)} {\\Gamma(n+2\\alpha+2)}\\,(1-\\tau^2)^{\\alpha+1\/2}\\,C_n^{\\alpha+1}(\\tau)\\,e^{im\\psi}~,\n\\end{equation}\nwhile $({\\cal R}\\,Z_n^{m,\\alpha})(\\tau,\\psi)=0$ when $\\tau>1$. The result follows now upon some further administration with Pochhammer symbols. \\\\ \\\\\n{\\bf Theorem 5.2.}~~There holds for $\\alpha>{-}1$ and integer $n$ and $m$ such that $n-|m|$ is even and non-negative\n\\begin{equation} \\label{e55}\n\\Bigl(\\!\\begin{array}{c} q+\\alpha \\\\ q\\end{array}\\!\\Bigr)(1-\\rho^2)^{-\\alpha}\\,R_n^{|m|,\\alpha}(\\rho)=\\frac{1}{2\\pi}\\,\\int\\limits_0^{2\\pi} \\,C_n^{\\alpha+1}(\\rho\\cos\\vartheta)\\,e^{-im\\vartheta}\\,d\\vartheta~,\n\\end{equation}\nwhere $p=\\frac12\\,(n-|m|)$, $q=\\frac12\\,(n+|m|)$. \\\\[2mm]\n{\\bf Proof.}~~There holds, taking $|\\nu|=\\tau$ and $\\psi=0$ or $\\pi$ according as $\\nu \\geq 0$ or $\\nu < 0$ in (\\ref{e49}), with separate consideration of even $m$ and odd $m$ in the case $\\nu < 0$,\n\\begin{equation} \\label{e56}\n\\int\\limits_{-\\sqrt{1-\\nu^2}}^{\\sqrt{1-\\nu^2}}\\,Z_n^{m,\\alpha}(\\nu,\\mu)\\,d\\mu=K_{nm}(1-\\nu^2)^{\\alpha+1\/2}\\,C_n^{\\alpha+1}(\\nu)~,\n\\end{equation}\nwhere\n\\begin{equation} \\label{e57}\nK_{nm}=\\frac{(p+1)_{\\alpha}}{(n+1)_{2\\alpha}}~\\frac{2^{2\\alpha+1}\\,\\Gamma(\\alpha+1)}{n+2\\alpha+1}~.\n\\end{equation}\nExpand\n\\begin{equation} \\label{e58}\nC_n^{\\alpha+1}(\\nu)=\\sum_{n',m'}\\,\\beta_{nn'}^{mm'}(1-\\rho^2)^{-\\alpha}\\,Z_{n'}^{m',\\alpha}(\\nu,\\mu)\n\\end{equation}\nas a function depending only on $\\nu$ with $\\nu^2+\\mu^2\\leq1$. By orthogonality, see (\\ref{e31}), there holds\n\\begin{equation} \\label{e59}\n\\beta_{nn'}^{mm'}=L_{n'm'}^{-1}\\,\\int\\limits\\hspace*{-6mm}\\int\\limits_{\\nu^2+\\mu^2\\leq1}\\,C_n^{\\alpha+1}(\\nu)\\, Z_{n'}^{m',\\alpha} (\\nu,\\mu)\\,d\\nu\\,d\\mu~,\n\\end{equation}\nwhere\n\\begin{equation} \\label{e60}\nL_{n'm'}=\\frac{\\pi}{n'+\\alpha+1}~\\frac{(p'+1)_{\\alpha}}{(p'+|m'|+1)_{\\alpha}}~.\n\\end{equation}\nUsing (\\ref{e56}), it is seen that\n\\begin{eqnarray} \\label{e61}\n\\beta_{nn'}^{mm'} & = & L_{n'm'}^{-1}\\,\\int\\limits_{-1}^1\\,C_n^{\\alpha+1}(\\nu)\\left(\\int\\limits_{-\\sqrt{1-\\nu^2}}^{\\sqrt{1-\\nu^2}}\\, Z_{n'}^{m',\\alpha}(\\nu,\\mu)\\,d\\mu\\right)\\,d\\nu~= \\nonumber \\\\[3.5mm]\n& = & L_{n'm'}^{-1}\\,K_{n'm'}\\,\\int\\limits_{-1}^1\\,(1-\\nu^2)^{\\alpha+1\/2}\\,C_n^{\\alpha+1}(\\nu)\\,C_{n'}^{\\alpha+1}(\\nu)\\,d\\nu~= \\nonumber \\\\[3.5mm]\n& = & L_{n'm'}^{-1}\\,K_{n'm'}\\,M_n\\,\\delta_{nn'}~,\n\\end{eqnarray}\nwhere $M_n$ follows from orthogonality of the $C^{\\alpha+1}$ as, see \\cite{ref16}, (7.8),\n\\begin{equation} \\label{e62}\nM_n=\\frac{\\pi\\cdot2^{-2\\alpha-1}\\cdot\\Gamma(n+2\\alpha+2)}{n!\\,(n+\\alpha+1)\\,\\Gamma^2(\\alpha+1)}~.\n\\end{equation}\nWith $\\nu=\\rho\\cos\\vartheta$ it then follows that\n\\begin{eqnarray} \\label{e63}\nC_n^{\\alpha+1}(\\rho\\cos\\vartheta) & = & \\sum_{m'}\\,\\beta_{nn}^{mm'}(1-\\rho^2)^{-\\alpha}\\,Z_n^{m',\\alpha}(\\rho,\\vartheta)~= \\nonumber \\\\[3.5mm]\n& = & \\sum_{m'}\\,\\beta_{nn}^{mm'}\\,\\rho^{|m'|}\\,P_{\\frac{n-|m'|}{2}}^{(\\alpha,|m'|)}(2\\rho^2-1)\\,e^{im\\vartheta}~.\n\\end{eqnarray}\nTherefore, see (\\ref{e26}),\n\\begin{equation} \\label{e64}\n\\beta_{nn'}^{mm'}\\,\\rho^{|m'|}\\,P_{\\frac{n-|m'|}{2}}^{(\\alpha,|m'|)}(2\\rho^2-1)=\\frac{1}{2\\pi}\\,\\int\\limits_0^{2\\pi}\\, C_n^{\\alpha+1}(\\rho\\cos\\vartheta)\\,e^{-im'\\vartheta}\\,d\\vartheta\n\\end{equation}\nfor integer $m'$ such that $p'':=\\frac12\\,(n-|m'|)$ is a non-negative integer. An explicit computation from (\\ref{e57}), (\\ref{e60}), (\\ref{e61}) yields now\n\\begin{eqnarray} \\label{e65}\n\\beta_{nn}^{mm'} & = & L_{nm'}^{-1}\\,K_{nm'}\\,M_n~= \\nonumber \\\\[3.5mm]\n& = & \\Bigl(\\frac{\\pi}{n+\\alpha+1}~\\frac{(p''+1)_{\\alpha}}{(p''+|m'|+1)_{\\alpha}}\\Bigr)^{-1}\\cdot\n\\frac{(p''+1)_{\\alpha}}{(n+1)_{2\\alpha}}~\\frac{2^{2\\alpha+1}\\,\\Gamma(\\alpha+1)}{n+2\\alpha+1}~\\cdot \\nonumber \\\\[3.5mm]\n& & \\cdot~\n\\frac{\\pi\\cdot2^{-2\\alpha-1}\\,\\Gamma(n+2\\alpha+2)}{n!\\,(n+\\alpha+1)\\,\\Gamma^2(\\alpha+1)}~= \\nonumber \\\\[3.5mm]\n& = & \\frac{(p''+|m'|+1)_{\\alpha}}{\\Gamma(\\alpha+1)}=\\frac{\\Gamma(q''+\\alpha+1)}{\\Gamma(q''+1)\\,\\Gamma(\\alpha+1)} =\\Bigl(\\!\\begin{array}{c} q''+\\alpha \\\\ q \\end{array}\\!\\Bigr)~,\n\\end{eqnarray}\nin which $q''=p''+|m'|=\\frac12\\,(n+|m'|)$. Now replace $m'$ by $m$ in (\\ref{e64}) and (\\ref{e65}), and (\\ref{e55}) results. \\\\ \\\\\n{\\bf Notes.} \\\\[-7mm]\n\\bi{1.0}\n\\ITEM{1.} Theorem 5.1 generalizes the case $\\alpha=0$ in (\\ref{e14}) to general $\\alpha>{-}1$. A further generalization, to orthogonal functions on spheres of general dimension $N$ instead of disks, is provided in \\cite{refX}, Theorem 3.1. The proof of Theorem~5.1 as given here follows rather closely the approach of Cormack in \\cite{ref37}--\\cite{ref38} which differs from the approach used in \\cite{refX}.\n\\ITEM{2.} Theorem 5.2 generalizes the case $\\alpha=0$ of the integral representation of the $R_n^{|m|}$ in \\cite{ref39}, (A.10) to the case of general $\\alpha>{-}1$. Furthermore, for a fixed $\\rho\\in(0,1)$, the formula (\\ref{e55}) can be discretized to\n\\begin{equation} \\label{e66}\n\\Bigl(\\!\\begin{array}{c} q+\\alpha \\\\ q \\end{array}\\!\\Bigr)(1-\\rho^2)^{-\\alpha}\\,R_n^{|m|,\\alpha}(\\rho)=\n\\frac1N\\,\\sum_{k=0}^N\\,C_n^{\\alpha+1}\\Bigl(\\rho\\cos\\frac{2\\pi k}{N}\\Bigr)\\,e^{-2\\pi i\\frac{k}{N}}\n\\end{equation}\nwhen $N$ is an integer $>\\:n+|m|$. This yields a method of the DFT-type to compute the radial parts fast and reliably.\n\\ITEM{3.} The integral representation in (\\ref{e55}) is an excellent starting point to derive the asymptotics of the radial parts, by stationary phase methods, etc., when $n,|m|\\rightarrow\\infty$ such that $n\/|m|\\rightarrow\\kappa\\in(0,1)$, with $\\alpha$ fixed and $\\rho$ bounded away from 0 and 1. See \\cite{ref50}, ENZ document, Sec.~7, item~4.\n\\end{inspring}\n\n\\section{Scaling theory for generalized Zernike circle functions} \\label{sec6}\n\\mbox{} \\\\[-9mm]\n\nScaling theory for generalized Zernike circle functions is compromised by the occurrence of the factor $(1-\\rho^2)^{\\alpha}$. In the first place, one has to restrict the scaling parameter $\\varepsilon$ in $Z_n^{m,\\alpha}(\\varepsilon\\rho,\\vartheta)$ to the range $0\\leq\\varepsilon<1$. While this restriction is quite natural, the scaling results, such as (\\ref{e17}), for the case that $\\alpha=0$ allows unrestricted values of $\\varepsilon$ due to polynomial form of the radial parts. Furthermore, the value of $Z_n^{m,\\alpha}(\\varepsilon\\rho,\\vartheta)$ at $\\rho=1$ (with $0\\leq\\varepsilon<1$) is in general finite and unequal to 0. This implies that the only natural candidate among the systems $(Z_{n'}^{m',\\alpha'} (\\rho,\\vartheta))_{n',m'}$ as expansion set for $Z_n^{m,\\alpha}(\\varepsilon\\rho,\\vartheta)$ is the case that $\\alpha'=0$. Finally, an extension to shift-and-scaling theory, as in \\cite{ref40}, seems also more cumbersome. Nevertheless, for the radial part there is the following result. \\\\ \\\\\n{\\bf Theorem 6.1.}~~Let $\\alpha>{-}1$, $0\\leq\\varepsilon<1$, and let $n$, $m$ be non-negative integers such that $p=\\frac12\\,(n-m)$ is a non-negative integer. Then\n\\begin{equation} \\label{e67}\nR_n^{m,\\alpha}(\\varepsilon\\rho)=\\sum_{n'=m,m+2,...}\\,C_{nn'}^{m,\\alpha}(\\varepsilon)\\,R_{n'}^m(\\rho)~,~~~~~~0\\leq\\rho<1~,\n\\end{equation}\nin which the $C$'s can be expressed as the sum of two hypergeometric functions ${}_2F_1$. Furthermore,\n\\begin{eqnarray} \\label{e68}\n& \\mbox{} & C_{nn'}^{m,\\alpha}(\\varepsilon)=\\frac{(p+1)_{\\alpha}}{(p'+1)_{\\alpha}}\\,\\Bigl(R_n^{n',\\alpha}(\\varepsilon)-\\frac{p'+\\alpha}{p'}\\, R_n^{n'+2,\\alpha}(\\varepsilon)\\Bigr)~, \\nonumber \\\\[3mm]\n& & \\hspace*{5.2cm}n'=m,m+2,...,n-2~,\n\\end{eqnarray}\nwhere $p'=\\frac12\\,(n-n')$. Finally, when $\\alpha$ is a non-negative integer, it holds that\n\\begin{equation} \\label{e69}\nC_{nn'}^{m,\\alpha}(\\varepsilon)=0~,~~~~~~n'=n+2\\alpha+2,n+2\\alpha+4,...~.\n\\end{equation}\n{\\bf Proof.}~~By the orthogonality condition (\\ref{e31}) for the case $\\alpha=0$, there holds for $n'=m,m+2,...$\n\\begin{equation} \\label{e70}\nC_{nn'}^{m,\\alpha}(\\varepsilon)=2(n'+1)\\,\\int\\limits_0^1\\,R_n^{m,\\alpha}(\\varepsilon\\rho)\\,R_{n'}^m(\\rho)\\,\\rho\\,d\\rho~.\n\\end{equation}\nNow, by (\\ref{e36}) and, subsequently (\\ref{e35}) with $\\alpha=0$, $2\\pi r=\\varepsilon t$,\n\\begin{eqnarray} \\label{e71}\n& \\mbox{} & \\int\\limits_0^1\\,R_n^{m,\\alpha}(\\varepsilon\\rho)\\,R_{n'}^m(\\rho)\\,\\rho\\,d\\rho~= \\nonumber \\\\[3.5mm]\n& & =~({-}1)^p\\,2^{\\alpha}(p+1)_{\\alpha}\\,\\int\\limits_0^1\\,\\left(\\int\\limits_0^{\\infty}\\, \\frac{J_{n+\\alpha+1}(t)\\,J_m(\\varepsilon\\rho t)} {t^{\\alpha}}\\,dt\\right)\\,R_{n'}^m(\\rho)\\,\\rho\\,d\\rho~= \\nonumber \\\\[3.5mm]\n& & =~({-}1)^p\\,2^{\\alpha}(p+1)_{\\alpha}\\,\\int\\limits_0^{\\infty}\\,\\frac{J_{n+\\alpha+1}(t)}{t^{\\alpha}}\\, \\left(\\int\\limits_0^1\\,R_{n'}^m(\\rho)\\,J_m(\\varepsilon\\rho t)\\,\\rho\\,d\\rho\\right)\\,dt~= \\nonumber \\\\[3.5mm]\n& & =~({-}1)^p\\,2^{\\alpha}(p+1)_{\\alpha}({-}1)^{(n'-m)\/2}\\,\\int\\limits_0^{\\infty}\\,\\frac{J_{n+\\alpha+1}(t)\\, J_{n'+1}(\\varepsilon t)} {t^{\\alpha}\\cdot\\varepsilon t}\\,dt~.\n\\end{eqnarray}\nUsing \\cite{ref14}, first item in 9.1.27,\n\\begin{equation} \\label{e72}\n\\frac{J_{n'+1}(z)}{z}=\\frac{1}{2(n'+1)}\\,(J_{n'}(z)+J_{n'+2}(z))~,\n\\end{equation}\nit is then found that\n\\begin{equation} \\label{e73}\nC_{nn'}^{m,\\alpha}(\\varepsilon)=({-}1)^{(n+n'-2m)\/2}\\,2^{\\alpha}(p+1)_{\\alpha}(I_{nn'}(\\alpha,\\varepsilon)+I_{n,n'+2}(\\alpha, \\varepsilon))~,\n\\end{equation}\nwhere\n\\begin{equation} \\label{e74}\nI_{nn''}(\\alpha,\\varepsilon)=\\int\\limits_0^{\\infty}\\,\\frac{J_{n+\\alpha+1}(t)\\,J_{n''}(\\varepsilon t)}{t^{\\alpha}}\\,dt~.\n\\end{equation}\nNow, for $n''=m,m+2,...,n\\,$, there holds by (\\ref{e35})\n\\begin{equation} \\label{e75}\nI_{nn''}(\\alpha,\\varepsilon)=\\frac{({-}1)^{p''}}{2^{\\alpha}(p''+1)_{\\alpha}}\\,R_n^{n'',\\alpha}(\\varepsilon)~,\n\\end{equation}\nwhere $p''=\\frac12\\,(n-n'')$. For $n''=n+2,n+4,...\\,$, there holds by (\\ref{e38})\n\\begin{equation} \\label{e76}\nI_{nn''}(\\alpha,\\varepsilon)=\\frac{\\varepsilon^m\\,\\Gamma(n''+p''+1)}{2^{\\alpha}\\,\\Gamma(n''+1)\\,\\Gamma(p''+\\alpha+1)} \\,F({-}p''-\\alpha,n''+p''+1\\,;\\,n''+1\\,;\\,\\varepsilon^2)~,\n\\end{equation}\nwhere again $p''=\\frac12\\,(n-n'')$. When $\\alpha$ is a non-negative integer, (\\ref{e76}) vanishes when $p''+\\alpha$ is a negative integer, i.e., when\n\\begin{equation} \\label{e77}\nn''=n+2\\alpha+2,n+2\\alpha+4,...~.\n\\end{equation}\nWhen $\\alpha$ is not an integer, infinitely many of the $I_{nn''}$ must be expected to be non-vanishing.\n\nFrom (\\ref{e73}) and (\\ref{e75})it is found for $n'=m,m+2,...,n-2$ that\n\\begin{eqnarray} \\label{e78}\nC_{nn'}^{m,\\alpha} & = & ({-}1)^{(n+n'-2m)\/2}\\,2^{\\alpha}(p+1)_{\\alpha}~\\cdot \\nonumber \\\\[3.5mm]\n& & \\cdot~\\left[\\frac{({-}1)^{(n-n')\/2}}{2^{\\alpha}\\Bigl(\\dfrac{n-n'}{2}+1\\Bigr)_{\\alpha}}\\, R_n^{n',\\alpha}(\\varepsilon) + \\frac{({-}1)^{(n-n'-2)\/2}}{2^{\\alpha}\\Bigl(\\dfrac{n-n'}{2}\\Bigr)_{\\alpha}}\\,R_n^{n'+2,\\alpha}(\\varepsilon)\\right]~= \\nonumber \\\\[3.5mm]\n& = & \\frac{(p+1)_{\\alpha}}{(p'+1)_{\\alpha}}\\,\\Bigl(R_n^{n',\\alpha}(\\varepsilon)-\\frac{p'+\\alpha}{p'}\\, R_n^{n'+2,\\alpha} (\\varepsilon)\\Bigr)~,\n\\end{eqnarray}\nwhere $p'=\\frac12\\,(n-n')$ and the proof is complete.\n\n\\section{Forward computation schemes for generalized Zernike circle polynomials from ordinary ENZ and ANZ} \\label{sec7}\n\\mbox{} \\\\[-9mm]\n\nIn this section, the generalized Zernike circle functions $Z_n^{m,\\alpha}$ are expanded with respect to the system $(Z_{n'}^{m'})_{m',n'}$ of classical circle polynomials. The azimuthal dependence is in all cases through the factor $\\exp(im\\vartheta)$, and this allows restriction of the attention to the radial parts only. \\\\ \\\\\n{\\bf Theorem 7.1.}~~Let $\\alpha>{-}1$ and let $n$, $m$ be non-negative integers such that $p=\\frac12\\,(n-m)$ is a non-negative integer. Then\n\\begin{equation} \\label{e79}\nR_n^{m,\\alpha}(\\rho)=\\sum_{k=0}^{\\infty}\\,C_k\\,R_{m+2k}^m(\\rho)~,~~~~~~0\\leq\\rho<1~,\n\\end{equation}\nwhere $C_k=0$ for $k=0,1,...,p-1$ and\n\\begin{eqnarray} \\label{e80}\nC_k & = & ({-}1)^{k-p}\\,\\frac{m+2k+1}{m+k+p+\\alpha+1}\\,\\Bigl(\\!\\begin{array}{c} m+p+k \\\\ p \\end{array}\\!\\Bigr) \\Bigl(\\!\\begin{array}{c} \\alpha \\\\ k-p \\end{array}\\!\\Bigr)\\,\/ \\nonumber \\\\[3.5mm]\n& & \/\\,\\Bigl(\\!\\begin{array}{c} m+k+p+\\alpha \\\\ m+k \\end{array}\\!\\Bigr)~= \\nonumber \\\\[3.5mm]\n& = & \\frac{m+2k+1}{m+k+p+\\alpha+1}~\\frac{({-}\\alpha)_{k-p}}{(k-p)!}~\\frac{(p+1)_{\\alpha}}{(m+k+p+1)_{\\alpha}}\n\\end{eqnarray}\nwhen $k=p,p+1,...~$. \\\\[2mm]\n{\\bf Proof.}~~By the orthogonality condition (\\ref{e31}) with $\\alpha=0$, there holds\n\\begin{equation} \\label{e81}\nC_k=2(m+2k+1)\\,\\int\\limits_0^1\\,R_n^{m,\\alpha}(\\rho)\\,R_{m+2k}^m(\\rho)\\,\\rho\\,d\\rho~.\n\\end{equation}\nUsing the definition (\\ref{e26}) of $R_{n=m+2p}^{m,\\alpha}$ and $R_{m+2k}^m$, and using in the integral in (\\ref{e81}) the substitution\n\\begin{equation} \\label{e82}\nx=2\\rho^2-1\\in[{-}1,1]\\,,~~\\rho^2=\\tfrac12\\,(1+x)\\,,~~1-\\rho^2=\\tfrac12\\,(1-x)\\,,~~\\rho\\,d\\rho=\\tfrac14\\,dx~,\n\\end{equation}\nthis becomes\n\\begin{eqnarray} \\label{e83}\n& \\mbox{} & \\hspace*{-5mm}C_k~= \\nonumber \\\\[3mm]\n& & \\hspace*{-5mm}=~2(m+2k+1)\\,\\int\\limits_0^1\\,\\rho^{2m}(1-\\rho^2)^{\\alpha}\\,P_p^{(\\alpha,m)}(2\\rho^2-1)\\,P_k^{(0,m)}(2\\rho^2-1)\\, \\rho\\,d\\rho~= \\nonumber \\\\[3.5mm]\n& & \\hspace*{-5mm}=~\\frac{m+2k+1}{2^{m+\\alpha+1}}\\,\\int\\limits_{-1}^1\\,(1-x)^{\\alpha}\\,(1+x)^m\\,P_p^{(\\alpha,m)}(x)\\,P_k^{(0,m)}(x)\\,dx~.\n\\end{eqnarray}\nBy Rodriguez' formula, see \\cite{ref15}, (4.3.1) or \\cite{ref16}, p.~161, there holds\n\\begin{equation} \\label{e84}\n(1-x)^{\\alpha}\\,(1+x)^m\\,P_p^{(\\alpha,m)}(x)=\\frac{({-}1)^p}{2^p\\,p!}\\,\\Bigl(\\frac{d}{dx}\\Bigr)^p\\, [(1-x)^{\\alpha+p}(1+x)^{m+p}]~.\n\\end{equation}\nThen, by $p$ partial integrations from (\\ref{e83}) and (\\ref{e84}),\n\\begin{equation} \\label{e85}\nC_k=\\frac{m+2k+1}{2^{m+p+\\alpha+1}\\,p!}\\,\\int\\limits_{-1}^1\\,(1-x)^{p+\\alpha}(1+x)^{p+m}\\Bigl(\\frac{d}{dx}\\Bigr)^p\\,P_k^{(0,m)} (x)\\,dx~.\n\\end{equation}\nNext, by using, see \\cite{ref15}, (4.2.17) or \\cite{ref16}, (4.14),\n\\begin{equation} \\label{e86}\n\\frac{d}{dx}\\,P_n^{(\\alpha,\\beta)}(x)=\\tfrac12\\,(n+\\alpha+\\beta+1)\\,P_{n-1}^{(\\alpha+1,\\beta+1)}(x)\n\\end{equation}\nrepeatedly, it is found that for $k\\geq p$\n\\begin{equation} \\label{e87}\n\\Bigl(\\frac{d}{dx}\\Bigr)^p\\,P_k^{(0,m)}(x)=\\frac{1}{2^p}~\\frac{(k+m+p)!}{(k+m)!}\\,P_{k-p}^{(p,m+p)}(x)~,\n\\end{equation}\nwhile this vanishes for $k<p$. Hence, $C_k=0$ for $k<p$ and for $k\\geq p$ it is found that\n\\begin{equation} \\label{e88}\nC_k=\\frac{m+2k+1}{2^{m+2p+\\alpha+1}}~\\frac{(k+m+p)!}{(k+m)!\\,p!}\\,\\int\\limits_{-1}^1\\,(1-x)^{p+\\alpha}(1+x)^{p+m}\\, P_{k-p}^{(p,m-p)}(x)\\,dx~.\n\\end{equation}\nNext, again by Rodriguez' formula,\n\\begin{equation} \\label{e89}\n(1-x)^p(1+x)^{m+p}\\,P_{k-p}^{(p,m+p)}(x)=\\frac{({-}1)^{k-p}}{2^{k-p}(k-p)!}\\,\\Bigl(\\frac{d}{dx}\\Bigr)^{k-p} \\,[(1-x)^k(1+x)^{k+m}]~.\n\\end{equation}\nThus, it is found from (\\ref{e88}) and (\\ref{e89}) by $k-p$ partial integrations that\n\\begin{eqnarray} \\label{e90}\nC_k & = & \\frac{m+2k+1}{2^{m+k+p+\\alpha+1}}~\\frac{(k+m+p)!\\,({-}1)^{k-p}}{(k+m)!\\,p!\\,(k-p)!}~\\cdot \\nonumber \\\\[3.5mm]\n& & \\cdot~\\int\\limits_{-1}^1\\,(1-x)^{\\alpha}\\,\\Bigl(\\frac{d}{dx}\\Bigr)^{k-p}\\,[(1-x)^k(1+x)^{k+m}]\\,dx~= \\nonumber \\\\[3.5mm]\n& = & \\frac{m+2k+1}{2^{m+k+p+\\alpha+1}}~\\frac{(k+m+p)!\\,({-}1)^{k-p}}{(k+m)!\\,p!\\,(k-p)!}\\cdot\\frac{\\Gamma(\\alpha+1)} {\\Gamma(\\alpha-k+p+1)}~\\cdot \\nonumber \\\\[3.5mm]\n& & \\cdot~\\int\\limits_{-1}^1\\,(1-x)^{\\alpha+p}(1+x)^{k+m}\\,dx~.\n\\end{eqnarray}\nThe remaining integral in (\\ref{e90}) can be evaluated in terms of $\\Gamma$-functions as\n\\begin{equation} \\label{e91}\n\\int\\limits_{-1}^1\\,(1-x)^{\\alpha+p}(1+x)^{k+m}\\,dx=2^{m+k+p+\\alpha+1}\\,\\frac{\\Gamma(k+m+1)\\,\\Gamma(p+\\alpha+1)} {\\Gamma(k+m+p+\\alpha+1)}~.\n\\end{equation}\nThe final result in (\\ref{e80}) then follows upon some further administration with binomials and Pochhammer symbols. \\\\ \\\\\n{\\bf Notes.} \\\\[-7mm]\n\\bi{1.0}\n\\ITEM{1.} When $\\alpha$ is a non-negative integer, $C_k$ vanishes for $k\\geq p+\\alpha$.\n\\ITEM{2.} For the case that $m=n=0$, so that $p=0$, the expansion of $(1-\\rho^2)^{\\alpha}$ is obtained, with expansion coefficients\n\\begin{equation} \\label{e92}\nC_k=({-}1)^k\\,\\frac{2k+1}{k+1}\\,\\Bigl(\\!\\begin{array}{c} \\alpha \\\\ k \\end{array}\\!\\Bigr)\\,\/\\,\\Bigl(\\!\\begin{array}{c} k+\\alpha+1 \\\\ \\alpha \\end{array}\\!\\Bigr)~,\n\\end{equation}\ncompare \\cite{ref19}, (10).\n\\ITEM{3.} A short-cut of the proof of Theorem~7.1 can be obtained by noting that the integral on the last line of (\\ref{e83}) is essentially equal to the expansion coefficient $D_p$ in the connection formula\n\\begin{equation} \\label{e93}\nP_k^{(0,m)}(x)=\\sum_{p=0}^k\\,D_p\\,P_p^{(\\alpha,m)}(x)~.\n\\end{equation}\nThese connection coefficients are given in \\cite{ref51}, Theorem~7.1.3 in terms of Pochhammer symbols. The proof of Theorem~7.1 as given here is ``self-contained'', in the sense that only basic properties of Jacobi polynomials are used, while the proof of \\cite{ref51}, Theorem~7.1.3 uses also some more advanced properties of the hypergeometric function ${}_3F_2$.\n\\end{inspring}\n\nThe result of Theorem~7.1 gives a means to transfer forward computation schemes from the ordinary ENZ or ANZ theory to the general setting. Below is an example of this. \\\\ \\\\\n{\\bf Theorem 7.2.}~~Let $\\alpha>{-}1$, and let $n$, $m$ be non-negative integers such that $n-m$ is even and non-negative. Then the through-focus point-spread function $U_n^{m,\\alpha}(r,\\varphi\\,;\\,f)$ corresponding to $Z_n^{m,\\alpha}(\\rho,\\vartheta)$, see (\\ref{e6}), is given by\n\\begin{equation} \\label{e94}\nU_n^{m,\\alpha}(r,\\varphi\\,;\\,f)=2\\pi\\,i^m\\,e^{im\\varphi}\\,\\sum_{k=0}^{\\infty}\\,C_k\\,V_{m+2k}^m(r,f)\n\\end{equation}\nwith $V_{m+2k}^m$ given in semi-analytic form in (\\ref{e12}) and $C_k$ given in (\\ref{e80}). \\\\[2mm]\n{\\bf Proof.}~~Just insert the expansion (\\ref{e79}) into the integral\n\\begin{equation} \\label{e95}\n\\int\\limits_0^1\\int\\limits_0^{2\\pi}\\,e^{if\\rho^2}\\,e^{2\\pi i\\rho r\\cos(\\vartheta-\\varphi)}\\,Z_n^{m,\\alpha}(\\rho,\\vartheta)\\,\\rho\\,d\\rho\\,d\\vartheta\n\\end{equation}\nfor $U_n^{m,\\alpha}$ and use (\\ref{e11}). \\\\ \\\\\n{\\bf Notes.} \\\\[-7mm]\n\\bi{1.0}\n\\ITEM{1.} In a similar fashion, the on-axis pressure $p_{2i}^{0,\\alpha}((0,0,z),k)$ due to the radially symmetric velocity profile $v(\\sigma)=Z_{2i}^{0,\\alpha}(\\sigma\/a,\\vartheta)$, see (\\ref{e18}), can be obtained from the on-axis pressures $p_{2j}((0,0,z),k)$ due to $Z_{2j}^0(\\sigma\/a,0)$, given in (\\ref{e20}), and the coefficients $C_j$, case $m=0$, in (\\ref{e80}).\n\\ITEM{2.} The availability of the through-focus point-spread functions $U_n^{m,\\alpha}$ per Theorem~7.2 makes it possible to do aberration retrieval, with pupil functions expanded in generalized circle functions, in the same spirit as this is done in the ordinary ENZ theory, see Sec.~\\ref{sec2}. Similarly, radially symmetric velocity profiles, expanded into radially symmetric generalized circle functions, can be retrieved with the same approach that is used in the ordinary ANZ theory, see \\cite{ref19}, Sec.~V.\n\\end{inspring}\n\n\\section{Acoustic quantities for baffled-piston radiation from King's integral with generalized circle functions as velocity profiles} \\label{sec8}\n\\mbox{} \\\\[-9mm]\n\nIn this section various acoustic quantities that arise from baffled-piston radiation with a velocity profile that is expanded into generalized Zernike circle functions are computed in semi-analytic form. The starting point is King's integral, second integral expression in (\\ref{e18}), in which $V(u)$ is the Hankel transform (\\ref{e19}) of order 0 of $v(\\sigma)$. Having expanded $v(\\sigma)$ in radially symmetric circle functions $Z_{2j}^{0,\\alpha}=R_{2j}^{0,\\alpha}$, the Hankel transforms\n\\begin{eqnarray} \\label{96}\nV_{2j}^{0,\\alpha}(u) & = & \\int\\limits_0^a\\,R_{2j}^{0,\\alpha}(\\sigma\/a)\\,J_0(u\\sigma)\\,\\sigma\\,d\\sigma~= \\nonumber \\\\[3.5mm]\n& = & ({-}1)^j\\,a^2\\,2^{\\alpha}(j+1)_{\\alpha}\\,\\frac{J_{2j+\\alpha+1}(au)}{(au)^{\\alpha+1}}~,~~~~~~j=0,1,...~,\n\\end{eqnarray}\nsee (\\ref{e35}) arise.\n\nThe acoustic quantities considered are \\\\\n-- pressure $p((a,0,0)\\,;\\,k)$ at edge of the radiator, \\\\[2mm]\n-- reaction force $F=\\int\\limits_S\\,p\\,dS$ on the radiator, \\\\\n-- the power output $P=\\int\\limits_S\\,p\\,v^{\\ast}\\,dS$ of the radiator.\n\nBy taking $z=0$, $w=a$ in King's integral, it is seen that\n\\begin{equation} \\label{e97}\np_{{\\rm edge}}=p((a,0,0)\\,;\\,k)=i\\rho_0ck\\,\\int\\limits_0^{\\infty}\\,\\frac{J_0(au)\\,V(u)}{(u^2-k^2)^{1\/2}}\\,u\\,du~.\n\\end{equation}\nIt was shown in \\cite{ref20}, Sec.~III from the integral result\n\\begin{equation} \\label{e98}\n\\int\\limits_0^a\\,J_0(\\sigma u)\\,\\sigma\\,d\\sigma=au^{-1}J_1(au)\n\\end{equation}\nby taking $z=0$ in King's integral and integrating over $w$, $0\\leq w<a$ that\n\\begin{equation} \\label{e99}\nF=\\int\\limits_S\\,p\\,dS=2\\pi i\\rho_0cka\\,\\int\\limits_0^{\\infty}\\,\\frac{J_1(au)\\,V(u)}{(u^2-k^2)^{1\/2}}\\,du~.\n\\end{equation}\nIt was shown in \\cite{ref20}, Sec.~IV, from the representation\n\\begin{equation} \\label{e100}\np((\\sigma,0,0),k)=i\\rho_0ck\\,\\int\\limits_0^{\\infty}\\, \\frac{V(u)}{(u^2-k^2)^{1\/2}}\\,J_0(\\sigma u)\\,u\\,du~,~~~~~~0\\leq\\sigma<\\infty~,\n\\end{equation}\nas a Hankel transform and by using Parseval's formula for Hankel transforms that\n\\begin{equation} \\label{e101}\nP=\\int\\limits_S\\,p\\,v^{\\ast}\\,dS=2\\pi i\\rho_0ck\\,\\int\\limits_0^{\\infty}\\,\\frac{V(u)\\,V^{\\ast}(u)}{(u^2-k^2)^{1\/2}}\\,u\\,du~.\n\\end{equation}\n\nBy inserting $V=V_{2j}^{0,\\alpha}$ into the integrals in (\\ref{e97}), (\\ref{e99}), (\\ref{e101}), it is seen that the integrals\n\\begin{equation} \\label{e102}\ni\\,\\int\\limits_0^{\\infty}\\,\\frac{J_{m+\\beta}(au)\\,J_{n+\\gamma+1}(au)}{(u^2-k^2)^{1\/2}\\,u^{\\beta+\\gamma}}\\,du\n\\end{equation}\narise. Explicitly, in (\\ref{e102}) take \\\\\n-- $m=0$, $\\beta=0$, $n=2j$, $\\gamma=\\alpha$ for (\\ref{e97}), \\\\\n-- $m=0$, $\\beta=1$, $n=2j$, $\\gamma=\\alpha$ for (\\ref{e99}), \\\\\n-- $m=2j_1$, $\\beta=\\alpha+1$, $n=2j_2$, $\\gamma=\\alpha$ with $j_1\\leq j_2$ for (\\ref{e101}). \\\\\nThese integrals will now be evaluated as a power series in $ka$ using the method of \\cite{ref20}, Appendix~A. \\\\ \\\\\n{\\bf Theorem 8.1.}~~Let $\\beta\\geq0$, $\\gamma>{-}1$ and let $n$, $m$ be non-negative integers such that $n-m$ is non-negative and even. Then\n\\begin{eqnarray} \\label{e103}\n& \\mbox{} & i\\,\\int\\limits_0^{\\infty}\\,\\frac{J_{m+\\beta}(au)\\,J_{n+\\gamma+1}(au)}{(u^2-k^2)^{1\/2}\\,u^{\\beta+\\gamma}}\\,du~= \\nonumber \\\\[3.5mm]\n& & =~\\frac{-({-}1)^p\\,a^{2\\varepsilon}}{2ka}\\,\\sum_{l=1}^{\\infty}\\,\\frac{(\\tfrac12\\,l+\\tfrac12)_{\\varepsilon}} {(\\tfrac12\\,l+\\varepsilon)_{\\delta}}~\\frac{({-}\\tfrac12\\,l+1-\\beta)_p({-}\\tfrac12\\,l+1)_q({-}ika)^l}{\\Gamma(\\tfrac12\\,l+p+\\gamma+1) \\,\\Gamma(\\tfrac12\\,l+q+2\\varepsilon+1)}~, \\nonumber \\\\[1mm]\n& \\mbox{} &\n\\end{eqnarray}\nwhere\n\\begin{equation} \\label{e104}\n\\varepsilon=\\tfrac12\\,(\\beta+\\gamma)\\,,~~\\delta=\\tfrac12\\,(\\beta-\\gamma)\\,,~~p=\\tfrac12\\,(n-m)\\,,~~q=\\tfrac12\\,(n+m)~.\n\\end{equation}\n{\\bf Proof.}~~The plan of the proof is entirely the same as that of the proofs of the results in \\cite{ref20}, Appendix~A, and so only the main steps with key intermediate results are given.\n\nWith $(u^2-k^2)^{1\/2}$ as defined below (\\ref{e18}) there holds\n\\begin{eqnarray} \\label{e105}\n& \\mbox{} & i\\,\\int\\limits_0^{\\infty}\\,\\frac{J_{m+\\beta}(au)\\,J_{n+\\gamma+1}(au)}{(u^2-k^2)^{1\/2}\\,u^{\\beta+\\gamma}}\\,du~= \\nonumber \\\\[3.5mm]\n& & =~\\int\\limits_0^k\\,\\frac{J_{m+\\beta}(au)\\,J_{n+\\gamma+1}(au)}{u^{\\beta+\\gamma}\\,\\sqrt{k^2-u^2}}\\,du+i\\,\\int\\limits_k^{\\infty}\\, \\frac{J_{m+\\beta}(au)\\,J_{n+\\gamma+1}(au)}{u^{\\beta+\\gamma}\\,\\sqrt{u^2-k^2}}\\,du~= \\nonumber \\\\[3.5mm]\n& & =~I_1+I_2~.\n\\end{eqnarray}\nAs to the integral $I_1$ in (\\ref{e105}), the product of the two Bessel functions is written as an integral, see \\cite{ref52}, beginning of \\S13.61, and \\cite{ref20}, (A8), from $-\\infty i$ to $\\infty i$, the order of integration is reversed and it is used that\n\\begin{equation} \\label{e106}\n\\int\\limits_0^k\\,\\frac{u^{2q+2s+1}}{\\sqrt{k^2-u^2}}\\,du=k^{2q+2s+1}\\,\\frac{\\Gamma(q+1+s)\\,\\Gamma(\\tfrac12)} {\\Gamma(q+\\tfrac32+s)}\n\\end{equation}\nto obtain\n\\begin{eqnarray} \\label{e107}\n& \\mbox{} & \\hspace*{-7mm}I_1=\\frac{\\tfrac12\\,\\Gamma(\\tfrac12)}{2\\pi i}\\,(\\tfrac12\\,a)^{2\\varepsilon}\\,(\\tfrac12\\,ka)^{2q}~\\cdot \\nonumber \\\\[3.5mm]\n& & \\hspace*{-7mm}\\cdot~\\int\\limits_{-\\infty i}^{\\infty i}\\,\\frac{\\Gamma({-}s)\\,\\Gamma(2q+2\\varepsilon+2s+2)\\,\\Gamma(q+1+s)(\\tfrac12\\,ka)^{2s+1}} {\\Gamma(m+\\beta+s+1)\\,\\Gamma(n+\\gamma+s+2)\\,\\Gamma(2q+2\\varepsilon+s+2)\\,\\Gamma(q+s+\\tfrac32)}\\,ds~. \\nonumber \\\\\n& \\mbox{} &\n\\end{eqnarray}\nThe choice of the integration contour is such that it has all poles of $\\Gamma({-}s)$ on its right and all poles of $\\Gamma(2q+2\\varepsilon+2s+2)$ on its left (this is possible since $q\\geq0$ and $\\beta+\\gamma>{-}2$). Closing the contour to the right, thereby enclosing all poles of $\\Gamma({-}s)$ at $s=j=0,1,...$ with residues $({-}1)^{j+1}\/j!$, and using the duplication formula of the $\\Gamma$-function to write\n\\begin{equation} \\label{e108}\n\\Gamma(2q+2\\varepsilon+2j+2)=\\frac{2^{2q+2\\varepsilon+2j+1}}{\\Gamma(\\tfrac12)}\\,\\Gamma(q+\\varepsilon+j+1)\\,\\Gamma(q+\\varepsilon+j+\\tfrac32)~,\n\\end{equation}\nit follows that\n\\begin{eqnarray} \\label{e109}\nI_1 & = & \\tfrac12\\,a^{2\\varepsilon}\\,\\sum_{j=0}^{\\infty}\\,({-}1)^j\\,\\frac{\\Gamma(q+j+1)}{j!}~\\frac{\\Gamma(q+\\varepsilon+j+\\tfrac32)} {\\Gamma(q+j+\\tfrac32)}~\\frac{\\Gamma(q+\\varepsilon+j)}{\\Gamma(m+\\beta+j+1)}~\\cdot \\nonumber \\\\[3.5mm]\n& & \\cdot~\\frac{(ka)^{2(q+j)+1}}{\\Gamma(n+\\gamma+j+2)\\,\\Gamma(2q+2\\varepsilon+j+2)}~.\n\\end{eqnarray}\nReplacing $j+q+1$ by $j=q+1,q+2,...\\,$, it is found after some administration with Pochhammer symbols (such as $(x-q)_q=({-}1)^q(1-x)_q$) that\n\\begin{equation} \\label{e110}\nI_1=\\frac{-({-}1)^p\\,a^{2\\varepsilon}}{2ka}\\,\\sum_{j=q+1}^{\\infty}\\,\\frac{(j+\\tfrac12)_{\\varepsilon}}{(j+\\varepsilon)_{\\delta}}~\\frac{({-}j+1)_q ({-}j+1-\\beta)_p({-}ika)^{2j}}{\\Gamma(j+p+\\gamma+1)\\,\\Gamma(j+q+2\\varepsilon+1)}~.\n\\end{equation}\nHere it may be observed that $({-}j+1)_q=0$ for $j=1,...,q\\,$, so that the summation in (\\ref{e110}) could start at $j=1$ as well.\n\nAs to the integral $I_2$ in (\\ref{e105}), again the integral representation for the product of two Bessel functions is used, the integration order is reversed, and it is used that\n\\begin{equation} \\label{e111}\n\\int\\limits_k^{\\infty}\\,\\frac{u^{2q+2s+1}}{\\sqrt{u^2-k^2}}\\,du=\\tfrac12\\,k^{2q+2s+1}\\,\\frac{\\Gamma(\\tfrac12)\\, \\Gamma({-}q-s-\\tfrac12)}{\\Gamma({-}q-s)}~.\n\\end{equation}\nThis yields\n\\begin{eqnarray} \\label{e112}\n& \\mbox{} & \\hspace*{-7mm}I_2=\\tfrac12\\,i\\,\\Gamma(\\tfrac12)(\\tfrac12\\,a)^{2\\varepsilon}~\\cdot \\nonumber \\\\[3.5mm]\n& & \\hspace*{-7mm}\\cdot\\,\\int\\limits_{-\\infty i}^{\\infty i}\\,\\frac{\\Gamma({-}s)}{\\Gamma({-}q-s)}~\\frac{\\Gamma(2q+2\\varepsilon+2s+2) \\,\\Gamma({-}q-s-\\tfrac12)(\\tfrac12\\,ka)^{2s+2q+1}} {\\Gamma(m+\\beta+s+1)\\,\\Gamma(n+\\gamma+s+2)\\,\\Gamma(2q+2\\varepsilon+s+2)}\\, ds~. \\nonumber \\\\\n& \\mbox{} &\n\\end{eqnarray}\nThe factor $\\Gamma({-}s)\/\\Gamma({-}q-s)$ is a polynomial since $q=0,1,...\\,$, and so the integrand has its poles at $s=j-q-\\tfrac12\\,$, $j=0,1,...\\,$, and at $s={-}r-q-\\varepsilon-1$, $r=0,1,...\\,$. Since $\\varepsilon=(\\beta+\\gamma)\/2>{-}1$, the integration contour can be chosen such that all poles $j-q-\\tfrac12\\,$, with residues $({-}1)^{j+1}\/j!$, lie to the right of it while all poles ${-}r-q-\\varepsilon-1$ lie to the left of it. Closing the contour to the right, and using the duplication formula again, to write\n\\begin{equation} \\label{e113}\n\\Gamma(2j+2\\varepsilon+1)=\\frac{2^{2j+2\\varepsilon}}{\\Gamma(\\tfrac12)}\\,\\Gamma(j+\\varepsilon+\\tfrac12)\\,\\Gamma(j+\\varepsilon+1)~,\n\\end{equation}\nit follows that\n\\begin{eqnarray} \\label{e114}\nI_2 & = & \\tfrac12\\,i\\,a^{2\\varepsilon}\\,\\sum_{j=0}^{\\infty}\\,({-}1)^j\\,\\frac{\\Gamma(j+1+\\varepsilon)}{\\Gamma(j+1)}~\\frac{\\Gamma({-}j+\\tfrac12+q)} {\\Gamma({-}j+\\tfrac12)}~\\frac{\\Gamma(j+\\varepsilon+\\tfrac12)}{\\Gamma(j+\\beta+\\tfrac12-p)}~\\cdot \\nonumber \\\\[3.5mm]\n& & \\cdot~\\frac{(ka)^{2j}}{\\Gamma(j+\\gamma+\\tfrac32+p)\\,\\Gamma(j+2\\varepsilon+\\tfrac32+q)}~.\n\\end{eqnarray}\nThen some administration with Pochhammer symbols (such as $\\Gamma(x)\/\\Gamma(x-p)=({-}1)^p(1-x)_p$) yields\n\\begin{equation} \\label{e115}\nI_2=\\frac{-({-}1)^p\\,a^{2\\varepsilon}}{2ka}\\,\\sum_{j=0}^{\\infty}\\,\\frac{(j+1)_{\\varepsilon}} {(j+\\varepsilon+\\tfrac12)_{\\delta}}~\\frac{({-}j+\\tfrac12)_q({-}j-\\beta+\\tfrac12)_p({-}ika)^{2j+1}} {\\Gamma(j+p+\\gamma+\\tfrac32)\\,\\Gamma(j+q+2\\varepsilon+\\tfrac32)}~.\n\\end{equation}\n\nThe result in (\\ref{e103}) is now obtained by adding $I_1$ in (\\ref{e110}), with summation starting at $j=1$, and $I_2$ in (\\ref{e115}), while observing that the terms $j$ in (\\ref{e110}) yield the terms in (\\ref{e103}) with even $l=2j$, $j=1,2,...\\,$, and that the terms $j$ in (\\ref{e115}) yield the terms in (\\ref{e103}) with odd $l=2j+1$, $j=0,1,...\\,$.\n\n\\section{Estimating generalized velocity profiles in baffled-piston radiation from near-field pressure data via Weyl's formula} \\label{sec9}\n\\mbox{} \\\\[-9mm]\n\nIn this section a brief sketch is given of how one can estimate, in the setting of baffled-piston radiation, a not necessarily radially symmetric velocity profile from near-field pressure data. The starting point is the Rayleigh integral for the pressure, first integral expression in (\\ref{e18}), that is written in normalized form as\n\\begin{equation} \\label{e116}\np(\\nu,\\mu\\,;\\,\\zeta)=\\int\\limits_{~S}\\hspace*{-2mm}\\int\\limits\\,v(\\nu',\\mu')\\,\\frac{e^{ikar'}}{r'}\\,d\\nu'\\,d\\mu'~,\n\\end{equation}\nwhere\n\\begin{equation} \\label{e117}\nr'=((\\nu-\\nu')^2+(\\mu-\\mu')^2+\\zeta^2)^{1\/2}\n\\end{equation}\nis the distance from the field point $(\\nu,\\mu,\\zeta)$, with $\\zeta\\geq0$, to the point $(\\nu',\\mu',0)$ on the radiating surface $S$, for which we take the unit disk. For a fixed value of $\\zeta>0$, the equation (\\ref{e116}) can be written as\n\\begin{equation} \\label{e118}\np(\\nu,\\mu\\,;\\,\\zeta)=(v\\,{\\ast\\ast}\\,W({\\cdot},{\\cdot}\\,;\\,\\zeta))(\\nu,\\mu)~,\n\\end{equation}\nwhere $\\ast\\ast$ denotes $2D$ convolution and\n\\begin{equation} \\label{e119}\nW(\\nu,\\mu\\,;\\,\\zeta)=\\frac{\\exp[ika(\\zeta^2+\\nu^2+\\mu^2)^{1\/2}]}{(\\zeta^2+\\nu^2+\\mu^2)^{1\/2}}~.\n\\end{equation}\nUsing the Fourier transform ${\\cal F}$, defined as\n\\begin{equation} \\label{e120}\n({\\cal F}q)(x,y)=\\int\\limits_{-\\infty}^{\\infty}\\int\\limits_{-\\infty}^{\\infty}\\,e^{2\\pi i\\nu x+2\\pi i\\mu y}\\,q(\\nu,\\mu)\\,d\\nu\\,d\\mu~,~~~~~~x,y\\in{\\Bbb R}~,\n\\end{equation}\nthe formula (\\ref{e118}) can be written as\n\\begin{equation} \\label{e121}\n{\\cal F}\\,[p({\\cdot},{\\cdot}\\,;\\,\\zeta)]={\\cal F}v\\cdot{\\cal F}\\,[W({\\cdot},{\\cdot}\\,;\\,\\zeta)]~.\n\\end{equation}\nBy Weyl's result on the representation of spherical waves, see \\cite{ref18}, Sec.~13.2.1, there holds\n\\begin{equation} \\label{e122}\n{\\cal F}\\,[W({\\cdot},{\\cdot}\\,;\\,\\zeta)](x,y)=2\\pi i\\, \\frac{\\exp\\,[i\\zeta((ka)^2-(2\\pi x)^2-(2\\pi y)^2)^{1\/2}]} {((ka)^2-(2\\pi x)^2-(2\\pi y)^2)^{1\/2}}~,\n\\end{equation}\nwith the same definition of the square root as the one that was used in connection with King's integral in (\\ref{e18}).\n\nNow assume that the unknown velocity profile vanishes outside the unit disk and that it has a $(1-\\nu^2-\\mu^2)^{\\alpha}$-behaviour at the edge of the unit disk, where $\\alpha>{-}1$. Then $v$ has an expansion\n\\begin{equation} \\label{e123}\nv(\\nu,\\mu)=\\sum_{n,m}\\,C_n^{m,\\alpha}\\,Z_n^{m,\\alpha}(\\nu,\\mu)\n\\end{equation}\nin generalized Zernike circle functions, with Fourier transform\n\\begin{equation} \\label{e124}\n{\\cal F}v=\\sum_{n,m}\\,C_n^{m,\\alpha}\\,{\\cal F}\\,Z_n^{m,\\alpha}~,\n\\end{equation}\nin which ${\\cal F}\\,Z_n^{m,\\alpha}$ is given explicitly in Sec.~\\ref{sec4}. Thus, when the pressure $p$ is measured in the near-field plane $(\\nu,\\mu\\,;\\,\\zeta)$ with $\\zeta$ fixed, one can estimate $v$ on the level of its expansion coefficients $C_n^{m,\\alpha}$ by adopting a matching approach in (\\ref{e121}), using Weyl's result in (\\ref{e122}) and the result of Sec.~\\ref{sec4} in (\\ref{e124}). \\\\ \\\\\n\n\\section{Comparison with trial functions as used in acoustic design by Mellow and K\\\"arkk\\\"ainen} \\label{sec10}\n\\mbox{} \\\\[-9mm]\n\nMellow and K\\\"arkk\\\"ainen are concerned with design problems in acoustic radiation from a resilient disk (radius $a$) in an infinite or finite baffle ($z=0$), see \\cite{ref45}--\\cite{ref46}. The front and rear pressure distributions $p_+$ and $p_-$, $p_-={-}p_+$, are assumed to be radially symmetric and to have the form\n\\begin{equation} \\label{e125}\n\\sum_{l=0}^{\\infty}\\,a_l(1-(\\sigma\/a)^2)^{l+1\/2}\n\\end{equation}\non the disk in accordance with the choice of trial functions used by Streng \\cite{ref42} which is based on the work of Bouwkamp \\cite{ref43}. In the case that the normal gradient of the pressure at $z=0$ is considered, as is done by Mellow in \\cite{ref44}, an expansion of the form\n\\begin{equation} \\label{e126}\n\\sum_{l=0}^{\\infty}\\,b_l(1-(\\sigma\/a)^2)^{l-1\/2}\n\\end{equation}\non the disk has to be considered.\n\nIn the design problem considered in \\cite{ref45}--\\cite{ref46}, the coefficients $a_l$ in (\\ref{e125}) are to be found such that the pressure gradient $\\frac{\\partial p}{\\partial z}\\,(w,z=0{+})$ equals a desired function $\\Phi(w)$ of the distance $w$ of a point in the baffle plane to the origin. In the design problem considered in \\cite{ref44}, the coefficients $b_l$ in (\\ref{e126}) are to be found such that $p(w,z=0{+})$ equals a desired function $\\Psi(w)$.\n\nThe pressure $p(w,z)$, $z\\geq0$ can be expressed in terms of the boundary data $p_+$, $p_-$ via the dipole version of King's integral. Similarly, via the common version of King's integral, the pressure $p(w,z)$, $z\\geq0$, can be expressed in terms of the normal gradient $\\frac{\\partial p}{\\partial z}\\,(w,0)$ of $p$ at $z=0$. Inserting the series expansion (\\ref{e125}) and (\\ref{e126}) into the appropriate version of King's integral, the integrals\n\\begin{equation} \\label{e127}\n\\int\\limits_0^{\\infty}\\,\\Bigl(\\frac{1}{\\mu}\\Bigr)^{l\\pm 1\/2}\\,J_0(w\\mu)\\,J_{l\\pm 1\/2+1}(a\\mu)\\,\\sigma^{\\pm 1}\\,d\\mu\n\\end{equation}\narise where $\\sigma={-}i(\\mu^2-k^2)^{1\/2}$ and where the $\\pm$ follows the sign choice in the exponent $l\\pm 1\/2$ of $(1-(\\sigma\/a)^2)$ in (\\ref{e125}) and (\\ref{e126}). \nTo obtain (\\ref{e127}), an explicit result, due to Sonine, for the Hankel transform of order 0 of the functions $(1-(\\sigma\/a)^2)^{l\\pm 1\/2}$ has been used. The integrals in (\\ref{e127}) are evaluated in the form of a double power series in $ka$ and $w\/a$ in \\cite{ref44}--\\cite{ref46}. Thus, having the pressure available in this semi-analytic form, comprising the coefficients $a_l$ or $b_l$, one can evaluate $\\frac{\\partial p}{\\partial z}\\,(w,z=0{+})$ and $p(w,z=0{+})$ and find the coefficients by requiring a best match\n with the desired function $\\Phi(w)$ or $\\Psi(w)$.\n\n In the approach of the present paper, the starting point would be an expansion of the form\n \\begin{equation} \\label{e128}\n \\sum_{l=0}^{\\infty}\\,c_l\\,R_{2l}^{0,{\\pm} 1\/2}(\\sigma\/a)\n \\end{equation}\n of the pressure ($+$-sign) or pressure gradient ($-$-sign) on the disk. Following the approach in \\cite{ref44}--\\cite{ref46}, using either form of King's integral, this gives rise to the integrals\n \\begin{equation} \\label{e129}\n \\int\\limits_0^{\\infty}\\,\\Bigl(\\frac{1}{\\mu}\\Bigr)^{\\pm 1\/2}\\,J_0(w\\mu)\\,J_{2l\\pm1\/2+1}(a\\mu)\\,\\sigma^{\\pm 1}\\,d\\mu~,\n \\end{equation}\n where now the result of Sec.~\\ref{sec4} on the Hankel transform of $R_{2l}^{0,{\\pm}1\/2}$ has been used. The integral in (\\ref{e129}) is of the same type as the one in (\\ref{e127}) and can be evaluated by the method given in \\cite{ref44}--\\cite{ref46}.\n\n\\subsection{Numerical considerations} \\label{subsec10.1}\n\\mbox{} \\\\[-9mm]\n\nIn either approach, it is required to find coefficients such that a best match occurs between the semi-analytically computed pressure gradient or pressure at $z=0{+}$, comprising the coefficients, and the desired functions $\\Phi$ or $\\Psi$. For any $L=1,2,...\\,$, the linear span of the function systems\n\\begin{equation} \\label{e130}\n\\{(1-(\\sigma\/a)^2)^{l\\pm 1\/2}\\,|\\,l=0,1,...,L-1\\}\n\\end{equation}\nand\n\\begin{equation} \\label{e131}\n\\{R_{2l}^{0,{\\pm}1\/2}(\\sigma\/a)\\,|\\,l=0,1,...,L-1\\}\n\\end{equation}\nis the same. So matching using the first $L$ functions in (\\ref{e125}), (\\ref{e126}) yields the same result for the best matching pressure gradient or pressure at $z=0{+}$ as matching using the first $L$ functions in (\\ref{e128}), in theory. For small values of $L$, one finds numerically practically the same result when either system in (\\ref{e130}), (\\ref{e131}) is used. In the case that large values $L$ of the number of coefficients to be matched are required, the approach based on (\\ref{e125}), (\\ref{e126}) is expected to experience numerical problems while the one based on (\\ref{e129}) is likely not to have such problems. This is due to the fact that the functions in (\\ref{e130}) are nearly linearly dependent while the ones in (\\ref{e131}), due to orthogonality, are not, and this is expected to remain so after the linear transformation associated with either version of King's integral. Furthermore, it is to be expected that the semi-analytic forms, used in the matching procedure, that arise from any of the terms $(1-(\\sigma\/a)^2)^{l\\pm1\/2}$ must be used with much higher truncation levels than those that arise from the terms $R_{2l}^{0,{\\pm}1\/2}(\\sigma\/a)$.\n\nAll this can be illustrated by comparing the expansion coefficients of a $(1-(\\sigma\/a)^2)^{k\\pm1\/2}$ with respect to the system in (\\ref{e131}) with those of an $R_{2p}^{0,{\\pm}1\/2}(\\sigma\/a)$ with respect to the system in (\\ref{e130}). There is the following general result. \\\\ \\\\\n{\\bf Theorem 10.1.}~~For $m=0,1,...\\,$, $\\alpha>{-}1$ and $k,p=0,1,...$ there holds\n\\begin{equation} \\label{e132}\n\\rho^m(1-\\rho^2)^{k+\\alpha}\\,e^{im\\vartheta}=\\sum_{l=0}^k\\,D_{m+2l,k}^{m,\\alpha}\\,Z_{m+2l}^{m,\\alpha}(\\rho,\\vartheta)~,\n\\end{equation}\n\\begin{equation} \\label{e133}\nZ_{m+2p}^{m,\\alpha}(\\rho,\\vartheta)=\\sum_{r=0}^p\\,E_{r,m+2p}^{m,\\alpha}\\,\\rho^m(1-\\rho^2)^{r+\\alpha}\\,e^{im\\alpha}~,\n\\end{equation}\nwhere\n\\begin{eqnarray} \\label{e134}\n& \\mbox{} & D_{m+2l,k}^{m,\\alpha}=\\frac{(\\alpha+1)_k}{(m+\\alpha+1)_k}~\\frac{m+2l+\\alpha+1}{m+k+l+\\alpha+1}~\\frac{({-}k)_l(m+\\alpha+1)_l} {(\\alpha+1)_l(m+k+\\alpha+1)_l}~, \\nonumber \\\\[2mm]\n& \\mbox{} &\n\\end{eqnarray}\n\\begin{equation} \\label{e135}\nE_{r,m+2p}^{m,\\alpha}=\\frac{(\\alpha+1)_p}{(1)_p}~\\frac{({-}p)_r(m+p+\\alpha+1)_r}{(\\alpha+1)_r(1)_r}~.\n\\end{equation}\n\\mbox{} \\\\\n{\\bf Proof.}~~The proof of (\\ref{e132}), (\\ref{e134}) is quite similar to the one of Theorem~7.1, and so only the main steps and key intermediate results are given. By orthogonality, see (\\ref{e31}), and the substitution in (\\ref{e82}), there holds\n\\begin{equation} \\label{e136}\nD_{m+2l,k}^{m,\\alpha}=\\frac{m+2l+\\alpha+1}{2^{m+k+\\alpha+1}}~\\frac{(l+m+1)_{\\alpha}}{(l+1)_{\\alpha}}\\,\\int\\limits_{-1}^1\\, (1-x)^{k+\\alpha}(1+x)^m\\,P_l^{(\\alpha,m)}(x)\\,dx~.\n\\end{equation}\nNext, Rodriguez' formula, (see (\\ref{e84})), is used, and $l$ partial integrations are performed. There results for $k\\geq l$\n\\begin{eqnarray} \\label{e137}\n& \\mbox{} & \\frac{1}{2^{m+k+\\alpha+1}}\\,\\int\\limits_{-1}^1\\,(1-x)^{k+\\alpha}(1+x)^m\\,P_l^{(\\alpha,m)}(x)\\,dx~= \\nonumber \\\\[3.5mm]\n& & =~\\frac{({-}1)^l}{2^{m+k+l+\\alpha+1}}~\\frac{\\Gamma(k+l)}{l!\\,\\Gamma(k+1-l)}\\,\\int\\limits_{-1}^1\\, (1-x)^{k+\\beta}(1+x)^{l+m}\\,dx~,\n\\end{eqnarray}\nwhile this vanishes for $k<l$. The remaining integral can be expressed in terms of $\\Gamma$-functions as in (\\ref{e91}), and then the result follows upon some administration with Pochhammer symbols.\n\nFrom the definition of $Z_{m+1p}^{m,\\alpha}$, one should find the $E$'s according to\n\\begin{equation} \\label{e138}\nP_p^{(\\alpha,m)}(2\\rho^2-1)=\\sum_r\\,E_{r,m+2p}^{m,\\beta}(1-\\rho^2)^r~.\n\\end{equation}\nBy \\cite{ref14}, 15.4.6, it holds that\n\\begin{eqnarray} \\label{e139}\nP_p^{(\\alpha,m)}(2\\rho^2-1) & = & P_p^{(\\alpha,m)}(1-2(1-\\rho^2))~= \\nonumber \\\\[3mm]\n& = & \\frac{(\\alpha+1)_p}{p!}\\,F({-}p,\\alpha+1+m+p\\,;\\,\\alpha+1\\,;\\,1-\\rho^2)~, \\nonumber \\\\\n& \\mbox{} &\n\\end{eqnarray}\nand then the $E$'s follow from the definition of $F$ in \\cite{ref14}, 15.1.1. \\\\ \\\\\n{\\bf Example.}~~For the case $m=0$, it is found that the ratio of the $r^{{\\rm th}}$ coefficient for $(1-\\rho^2)^{p+\\alpha}$ and the $r^{{\\rm th}}$ coefficient for $Z_{m+2p}^{0,\\alpha}(\\rho,\\vartheta)$ satisfy\n\\begin{equation} \\label{e140}\n\\frac{D_{2r,p}^{0,\\alpha}}{E_{r,2p}^{0,\\alpha}}=\\frac{(1)_p}{(\\alpha+1)_p}~\\frac{2r+\\alpha+1}{p+r+\\alpha+1}~ \\frac{(1)_r(\\alpha+1)_r}{((p+\\alpha+1)_r)^2}~,\n\\end{equation}\nshowing that the $E$'s are much larger than the $D$'s. Note also that $(1-\\rho^2)^{p+\\alpha}$ and $Z_{m+2p}^{0,\\alpha}(\\rho,\\vartheta)$ have the same $L^2$-norm, where the weight function $(1-\\rho^2)^{-\\alpha}$ on the unit disk is used.\n\n\\subsection{Indefinite integrals for multi-ring design} \\label{subsec10.2}\n\\mbox{} \\\\[-9mm]\n\nIn \\cite{ref46}, Mellow and K\\\"arkk\\\"ainen replace the disk by a ring, and in \\cite{ref46}, Subsec.~II.F, the total radiation force is expressed as an integral over this ring of $p_+-p_-$. This integral can be expressed explicitly in terms of the $a_l$ in (\\ref{e125}) since the functions $(1-(\\sigma\/a)^2)^{l+1\/2}$ have an analytic result for their integrals over a concentric ring. In the case that expansions involving $R_{2l}^{0,{\\pm}1\/2}$ are used, non-trivial integrals arise. \\\\ \\\\\n{\\bf Theorem 10.2.}~~There holds for $l=0,1,...$\n\\begin{eqnarray} \\label{e142e143}\n\\int\\limits\\,R_{2l}^{0,{-}1\/2}(\\rho)\\,\\rho\\,d\\rho & = & \\frac{-1}{4l+1}\\,(R_{2l}^{0,1\/2}(\\rho)+R_{2l-2}^{0,1\/2}(\\rho))~, \\\\[3.5mm]\n\\int\\limits\\,R_{2l}^{0,1\/2}(\\rho)\\,\\rho\\,d\\rho & = & \\frac{1}{4l+1}\\,\\Bigl[\\frac{2l+2}{4l+5}\\,R_{2l+2}^{0,1\/2}(\\rho)~+\\Bigr. \\nonumber \\\\[3.5mm]\n& & -~\\Bigl.\\frac{4l+3}{(4l+5)(4l+1)}\\,R_{2l}^{0,1\/2}(\\rho)-\\frac{2l+1}{4l+1}\\,R_{2l-2}^{0,1\/2}(\\rho)\\Bigr]~, \\nonumber \\\\\n& \\mbox{} &\n\\end{eqnarray}\nwhere $R_{-2}^{0,1\/2}(\\rho)\\equiv0$ has been set for the case $l=0$. \\\\[2mm]\n{\\bf Proof.}~~There holds by \\cite{ref15}, $\\alpha=0$ in the first item in (4.1.5),\n\\begin{equation} \\label{e144}\nP_{2l}(x)=({-}1)^l P_l^{({-}1\/2,0)}(1-2x^2)~,\n\\end{equation}\nwhere $P_{2l}$ denotes the Legendre polynomial of degree $2l$. Using this with $x=(1-\\rho^2)^{1\/2}$, it is seen that\n\\begin{eqnarray} \\label{e145}\nR_{2l}^{0,{-}1\/2}(\\rho) & = & (1-\\rho^2)^{-1\/2}\\,P_l^{({-}1\/2,0)}(2\\rho^2-1)~= \\nonumber \\\\[3mm]\n& = & ({-}1)^l(1-\\rho^2)^{-1\/2}\\,P_{2l}((1-\\rho^2)^{1\/2})~.\n\\end{eqnarray}\nFor $l=0$, it then follows that\n\\begin{equation} \\label{e146}\n\\int\\limits\\,R_0^{0,{-}1\/2}(\\rho)\\,\\rho\\,d\\rho=\\int\\limits\\,(1-\\rho^2)^{-1\/2}\\,\\rho\\,d\\rho={-}(1-\\rho^2)^{1\/2}={-}R_0^{0,1\/2}(\\rho)~.\n\\end{equation}\nFor $l=1,2,...$ it follows from (\\ref{e145}) and the substitution $\\tau=(1-\\rho^2)^{1\/2}$, $\\rho\\,d\\rho={-}\\tau\\,d\\tau$ that\n\\begin{equation} \\label{e147}\n\\int\\limits\\,R_{2l}^{0,{-}1\/2}(\\rho)\\,\\rho\\,d\\rho=({-}1)^{l+1}\\,\\int\\limits\\,P_{2l}(\\tau)\\,d\\tau~.\n\\end{equation}\nThen from \\cite{ref16}, (10.10),\n\\begin{equation} \\label{e148}\nP_k(\\tau)=\\frac{1}{2k+1}\\,(P_{k+1}'(\\tau)-P_{k-1}'(\\tau))~,\n\\end{equation}\nit follows that\n\\begin{equation} \\label{e149}\n\\int\\limits\\,R_{2l}^{0,{-}1\/2}(\\rho)\\,\\rho\\,d\\rho=\\frac{({-}1)^{l+1}}{4l+1}\\,(P_{2l+1}(\\tau)-P_{2l-1}(\\tau))~.\n\\end{equation}\nNext, by \\cite{ref15}, second item in (4.1.5),\n\\begin{equation} \\label{e150}\nP_{2l+1}(x)=({-}1)^l\\,x\\,P_l^{(1\/2,0)}(1-2x^2)~.\n\\end{equation}\nUsing this with $x=(1-\\rho^2)^{1\/2}$, it is seen that\n\\begin{eqnarray} \\label{e151}\nR_{2l}^{0,1\/2}(\\rho) & = & (1-\\rho^2)^{1\/2}\\,P_l^{(1\/2,0)}(2\\rho^2-1)~= \\nonumber \\\\[3mm]\n& = & ({-}1)^l\\,P_{2l+1}((1-\\rho^2)^{1\/2})~.\n\\end{eqnarray}\nNow (141) follows from (\\ref{e150}) by using $\\tau=(1-\\rho^2)^{1\/2}$.\n\nTo show (142), let $l=1,2,...\\,$. It follows from (\\ref{e151}) and the substitution $\\tau=(1-\\rho^2)^{1\/2}$, $\\rho\\,d\\rho={-}\\tau\\,d\\tau$ that\n\\begin{equation} \\label{e152}\n\\int\\limits\\,R_{2l}^{0,1\/2}(\\rho)\\,\\rho\\,d\\rho=({-}1)^{l+1}\\,\\int\\limits\\,\\tau\\,P_{2l+1}(\\tau)\\,d\\tau~.\n\\end{equation}\nThen by (\\ref{e148}) and partial integration\n\\begin{equation} \\label{e153}\n\\int\\limits\\,R_{2l}^{0,1\/2}(\\rho)\\,\\rho\\,d\\rho=\\frac{({-}1)^{l+1}}{4l+3}\\, \\Bigl[\\tau\\,P_{2l+2}(\\tau)-\\tau\\, P_{2l}(\\tau)-\\int\\limits\\,(P_{2l+2}(\\tau)-P_{2l}(\\tau))\\,d\\tau\\Bigr]~.\n\\end{equation}\nNext, \\cite{ref16}, (10.2),\n\\begin{equation} \\label{e154}\nx\\,P_k(x)=\\frac{k}{2k+1}\\,P_{k-1}(x)+\\frac{k+1}{2k+1}\\,P_{k+1}(x)~,\n\\end{equation}\nis used. It follows from (\\ref{e153}) and (\\ref{e148}), (\\ref{e154}) and some administration that\n\\begin{eqnarray} \\label{e155}\n& \\mbox{} & \\hspace*{-4mm}\\int\\limits\\,R_{2l}^{0,1\/2}(\\rho)\\,\\rho\\,d\\rho~= \\nonumber \\\\[3.5mm]\n& & \\hspace*{-4mm}=~\\frac{({-}1)^{l+1}}{4l+3}\\,\n\\Bigl[\\frac{2l+2}{4l+5}\\,P_{2l+3}(\\tau)+\\frac{4l+3}{(4l+5)(4l+1)}\\, P_{2l+1}(\\tau)-\\frac{2l+1}{4l+1}\\,P_{2l-1} (\\tau)\\Bigr]~. \\nonumber \\\\\n& \\mbox{} &\n\\end{eqnarray}\nThen (142) follows for $l=1,2,...$ by using (\\ref{e151}) and $\\tau=(1-\\rho^2)^{1\/2}$. Finally, for $l=0$ it follows from (\\ref{e151}) that\n\\begin{equation} \\label{e156}\n\\int\\limits\\,R_0^{0,1\/2}(\\rho)\\,\\rho\\,d\\rho=\\int\\limits\\,(1-\\rho^2)^{1\/2}\\,\\rho\\,d\\rho={-}\\tfrac13\\,(1-\\rho^2)^{3\/2}~,\n\\end{equation}\nwhile (142) with $l=0$ equals by (\\ref{e155})\n\\begin{equation} \\label{e157}\n-\\tfrac13\\,[\\tfrac35\\,P_1(\\tau)+\\tfrac25\\,P_3(\\tau)]={-}\\tfrac13\\,\\tau^3={-}\\tfrac13\\,(1-\\rho^2)^{3\/2}~.\n\\end{equation}\nThe proof is complete. \\\\ \\\\\n{\\bf Acknowledgement.}~~The author wishes to thank R.\\ Aarts, J.\\ Braat, S.\\ van Haver and T.\\ Mellow for stimulating discussions and comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe aim of this paper is to study the computational complexity of the\nfollowing optimization problem:\n\n\\defpb{\\textsc{Minimum} \\textsc{Soapy Union}{} (\\textsc{MinSU})}\n{a finite set $A$ and an indexed family $\\inda{X}$ of non-empty finite sets of rational integers.}\n{an indexed family $\\inda{t}$ of rational integers.}\n{the cardinality of $\\bigcup_{a \\in A} (X_a + t_a)$.}\n\nLet us name \\textsc{Soapy Union}{} (\\textup{SU}) the decision problem associated with \\textsc{MinSU}.\nThe names have been chosen by analogy with the \\textsc{Soapy Set Cover} problem \\cite{nicolas08hardness}.\nClearly, \\textup{SU}{} is a \\emph{number problem} \\cite{garey79computers}.\n\\textsc{MinSU}{} can be seen as a generalization of the \\textsc{Subset Matching} problem \\cite{clifford07approximate}:\noptimally solving \\textsc{Subset Matching} is equivalent to optimally solving the restriction of \\textsc{MinSU}{} to those instances $\\inda{X}$  such that the cardinality of $A$ equals~$2$.\n\n\n\\textsc{MinSU}{} naturally appears in the context of\nprotein shotgun sequencing~\\cite{bandeira04shotgun,bandeira08automated,bandeira08de-novo}.  (This problem must not be confused with the more\nwidely known peptide shotgun sequencing.)  Sequencing the protein means that\nwe want to determine its amino acid sequence.  We assume that no genomic\ninformation is available for the protein, so that its sequence cannot be\nderived from the genomic information.  This is the case for many proteins\neven in humans, monoclonal antibodies being an important\nexample~\\cite{bandeira08automated}.  Experimentally, the protein is cleaved\ninto short and overlapping peptides, which are then analyzed by tandem mass\nspectrometry.  To improve the quality of such spectra, one then asks for the\nmass of the unknown prefix (the~shift) of the spectrum, such that the\nresulting shifted spectra show a maximum agreement.  For real-world data the\nproblem is even more complicated than our definition of \\textsc{MinSU};\nbut our intractability results clearly indicate that it is unlikely to find a\npolynomial time algorithm for shotgun protein sequencing.\n\n\n\\paragraph{Contribution.} \n\nIn Section~\\ref{sec:msu-easy}, \nwe prove \nthat  \\textup{SU}{} belongs to $\\mathrm{NP}$ and \nthat \\textsc{MinSU}{} can be solved in polynomial time for fixed~$A$.\nIn Section~\\ref{sec:msu-hard}, we show that \\textup{SU}{} is strongly $\\mathrm{NP}$-hard; \nfurthermore, \nwe prove that there exists a real number $\\rho > 1$ such that if \\textsc{MinSU}{} is $\\rho$-approximable in pseudo-polynomial time then $\\PNP$. \n\n\\paragraph{Notation and definitions.} \n\nFor every finite set $S$, $\\card{S}$ denotes the cardinality of~$S$.\nFor all sets $A$ and $S$, $S^A$ denotes the set of all families  of elements of $S$ indexed by~$A$.\n\nThe ring of rational integers is denoted~$\\mathbb{Z}$.\nFor every integer $n \\ge 0$,\n$\\seg{1}{n}$ denotes the set of all $k \\in \\mathbb{Z}$ such that $1 \\le k \\le n$.\n\nA(n undirected) \\emph{graph} is a pair $G = (V, E)$, where $V$ is a finite set and $E$ is a set of $2$-element subsets of~$V$: \nthe elements of $V$ are the \\emph{vertices} of $G$,\nthe elements of $E$ are the \\emph{edges} of $G$,\nand for each edge $e \\in E$, the elements of $e$ are the \\emph{extremities} of~$e$.\n\nLet \\textsc{Min} be a minimization problem.\nThe \\emph{decision problem associated with} \\textsc{Min} is: \ngiven an instance $I$ of \\textsc{Min} and an integer $k \\ge 0$,  \ndecide whether there exists a solution of \\textsc{Min} on $I$ with measure at most~$k$.\n\n\n\n\\section{Membership} \\label{sec:msu-easy}\n\nFor each instance $\\inda{X}$ of \\textsc{MinSU}, \nthe set of all feasible solutions  of \\textsc{MinSU}{} on $\\inda{X}$ equals $\\mathbb{Z}^A$, which is infinite.\nTherefore,  \n\\textsc{MinSU}{} is not an \\emph{$\\mathrm{NP}$-optimization problem} \\cite{ausiello03complexity}, \nand thus the membership of \\textup{SU}{} in $\\mathrm{NP}$ is not completely trivial.\n\nLet $G = (V, E)$ be a graph.\nA \\emph{disconnection} of $G$ is \na pair $(B, C)$ such that \n$B \\ne \\emptyset$, \n$C \\ne \\emptyset$,  \n$B \\cap C = \\emptyset$,\n$V = B \\cup C$,\nand \nfor every $(b, c) \\in B \\times C$, $\\{ b, c \\} \\notin E$.\nA graph is called \\emph{disconnected} if it admits a disconnection.\nA graph that is not disconnected is called \\emph{connected}.\n\nLet $\\inda{Y}$ be an indexed family of sets.\nThe \\emph{intersection graph} of $\\inda{Y}$ is defined as follows: \nits vertex set equals $A$ and \nfor all $b$, $c \\in A$ with $b \\ne c$, \n$\\{ b, c \\}$  is one of its edges if, and only if, $Y_b \\cap Y_c \\ne \\emptyset$.\n\n\\begin{lemma} \\label{lem:discon-opt}\nLet $\\inda{Y}$ be an instance of \\textsc{MinSU}.\nIf the intersection graph of $\\inda{Y}$ is disconnected \nthen \nthere exists $\\inda{u} \\in \\mathbb{Z}^A$ such that\n\\begin{equation} \\label{eq:X+t=A}\n\\card{\\bigcup_{a \\in A}  (Y_a + u_a )}\n< \n\\card{\\bigcup_{a \\in A}  Y_a}   \\, . \n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nFor each subset $B \\subseteq A$, put $Y_{B} = \\bigcup_{b \\in B} Y_b$.\nLet $(B, C)$ be a disconnection of the intersection graph of $\\inda{Y}$.\nLet \n$r \\in Y_B$ \nand \n$s \\in Y_C$\nbe fixed.\nSet \n$u_b = - r$ for every~$b \\in B$ \nand \n$u_c = - s$ for every~$c \\in C$.\nOn the one hand, we have \n$Y_B \\cap Y_C = \\emptyset$\nand \n$Y_A = Y_B \\cup Y_C$,\nso \n\\begin{equation} \\label{eq:A=BC}\n\\card{Y_A }  = \\card{Y_B} + \\card{Y_C} \\, .\n\\end{equation}\nOn the other hand, we have \n$$\n\\bigcup_{a \\in A}  (Y_a + u_a ) = (Y_B - r) \\cup (Y_C - s) \n$$\nand \n$$\n(Y_B - r) \\cap (Y_C - s) \\ne \\emptyset \n$$\nbecause $0 \\in (Y_B - r) \\cap (Y_C - s)$; \nit follows \n\\begin{equation} \\label{eq:A+t=BC}\n \\card{\\bigcup_{a \\in A}  (Y_a + u_a ) }\n < \n  \\card{Y_B}\n  +\n  \\card{Y_C} \\, .\n\\end{equation}\nIt now suffices to combine Equations~\\eqref{eq:A=BC} and \\eqref{eq:A+t=BC} to obtain Equation~\\eqref{eq:X+t=A}.\n\\qed\n\\end{proof}\n\nLemma~\\ref{lem:discon-opt} can be restated as follows:\n\n\\begin{lemma} \\label{lem:opt-con}\nLet $\\inda{X}$ be an instance of \\textsc{MinSU}.\nFor any optimum solution $\\inda{t}$ of \\textsc{MinSU}{} on  $\\inda{X}$,\nthe intersection graph of $\\left( X_a + t_a \\right)_{a \\in A}$ is connected.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\inda{t} \\in \\mathbb{Z}^A$ be such that the intersection graph of $\\left( X_a + t_a \\right)_{a \\in A}$  is disconnected.\nSet $Y_a = X_a + t_a$ for each $a \\in A$.\nBy Lemma~\\ref{lem:discon-opt}, \nthere exists $\\left( u_a \\right)_{a \\in A} \\in \\mathbb{Z}^A$ such that  Equation~\\eqref{eq:X+t=A} holds.\nIt follows that $\\left( t_a + u_a \\right)_{a \\in A}$ is a better solution of \\textsc{MinSU}{} on $\\inda{X}$ than $\\inda{t}$.\n\\qed\n\\end{proof}\n\n\n\\begin{definition} \\label{def:SGpi}\nLet $G = (V, E)$ be a graph.\nPut $$\\tilde E = \\left\\{ (a, b) \\in V \\times V : \\{ a, b \\} \\in E \\right\\}\\, .$$\nAn \\emph{antisymmetric edge-weight function} on $G$ is \na function $\\varpi$ from $\\tilde E$ to $\\mathbb{Z}$ such that \n$\\varpi(b, c) = - \\varpi(c, b)$\nfor every $(b, c) \\in \\tilde E$.\nFor each antisymmetric edge-weight function $\\varpi$ on $G$, \ndefine $S(G, \\varpi)$ as the set of all $\\left( t_a \\right)_{a \\in V} \\in \\mathbb{Z}^V$ such that \n$t_b - t_c = \\varpi(b, c)$ \nfor all $(b, c) \\in \\tilde E$.\n\\end{definition}\n\n\nLet us comment Definition~\\ref{def:SGpi}.\nThe function $\\varpi$ assigns both a magnitude and an orientation to each edge of~$G$:\nfor all $a$, $b \\in V$ such that $\\{ a, b \\} \\in E$, \nthe magnitude of $\\{ a, b \\}$ is the absolute value of $\\varpi(a, b)$\nand \nthe orientation of $\\{ a, b \\}$ \nis determined by the sign of $\\varpi(a, b)$.\nIt is clear that for every $\\left( t_a \\right)_{a \\in V} \\in S(G, \\varpi)$ and every $u \\in \\mathbb{Z}$, \n$ \\left( t_a + u \\right)_{a \\in V} \\in S(G, \\varpi)$.\nIf $G$ is connected then \neither $S(G, \\varpi)$ is empty or there exists $\\left( t_a \\right)_{a \\in V} \\in \\mathbb{Z}^V$ such that \n$S(G, \\varpi) = \\left\\{ \\left( t_a + u \\right)_{a \\in V} : u \\in \\mathbb{Z} \\right\\}$.\nIf $G$ is connected and $S(G, \\varpi) \\ne \\emptyset$\nthen \nfor any $(b, u) \\in V \\times \\mathbb{Z}$,\nthe unique element $\\left( t_a \\right)_{a \\in V} \\in S(G, \\varpi)$ that satisfies $t_b = u$ is computable from $G$, $\\varpi$, $b$, and $u$ in polynomial time.\nA \\emph{closed walk} in $G$ is a finite sequence \n$(a_0, a_1, a_2, \\dotsc, a_k)$ \nsuch that \n$a_0 = a_k$\nand \n$\\{ a_{i - 1}, a_i \\} \\in E$ for every $i \\in \\seg{1}{k}$;\nthe \\emph{weight} of $(a_0, a_1, a_2, \\dotsc, a_k)$ under $\\varpi$ is defined as \n$\\varpi(a_0, a_1) +  \\varpi(a_1, a_2) + \\dotsb +   \\varpi(a_{k - 1}, a_k)$.\nA (simple) \\emph{cycle} in $G$ is a closed walk  $(a_0, a_1, a_2, \\dotsc, a_k)$  in $G$ such that \nfor all $i$, $j \\in \\seg{1}{k}$, \n$a_i = a_j$ implies $i = j$. \nThe following three conditions are equivalent:\n\\begin{enumerate}\n \\item The set $S(G, \\varpi)$ is non-empty. \n \\item The weight under $\\varpi$ of every closed walk in $G$ equals~$0$.\n\\item The weight under $\\varpi$ of every cycle in $G$ equals~$0$.\n\\end{enumerate}\nThe second and third conditions can be thought as abstract forms of Kirchhoff's voltage law.\n\nA  \\emph{tree} is a connected graph with one fewer edges than vertices, or equivalently, an acyclic connected graph.\nAn arbitrary graph $G = (V, E)$ is connected \nif, and only if, \nthere exists a subset $E' \\subseteq E$ such that $(V, E')$ is a tree \n($(V, E')$ is then called a \\emph{spanning tree} of~$G$). \n\n\\begin{lemma} \\label{lem:tree-Xa}\nLet $\\inda{X}$ be an instance of \\textsc{MinSU}. \nThere exist a tree $H$ with vertex set $A$ and an antisymmetric edge-weight function $\\varpi$ on $H$ that satisfy the following two conditions:\n\\begin{enumerate}\n \\item \\label{cond:range-varpi}\nEvery integer in the range of $\\varpi$ can be written as the difference of two elements of $\\bigcup_{a \\in A} X_a$.\n \\item \\label{cond:SUpi-opt}\nEvery element of $S(H, \\varpi)$ is an optimum solution of \\textsc{MinSU}{} on  $\\inda{X}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nLet $\\inda{t}$ be an optimum solution of \\textsc{MinSU}{} on $\\inda{X}$.\nLet $H$ be a spanning tree of the intersection graph of $\\left( X_a + t_a \\right)_{a \\in A}$: such a tree exists \nby Lemma~\\ref{lem:opt-con}.\nLet  $\\varpi$ be the antisymmetric edge-weight function on $H$ defined by:\nfor all $b$, $c \\in A$ such that $\\{ b, c \\}$ is an edge of $H$,  \n$\\varpi(b, c) = t_b - t_c$. \n\nFor all $b$, $c \\in A$, \n such that $\\{ b, c \\}$ is an edge of the intersection graph of $\\left( X_a + t_a \\right)_{a \\in A}$,\n$(X_b + t_b) \\cap (X_c + t_c)$ is non-empty, \nand thus \n$t_b - t_c$ belongs to $X_c - X_b$.\nTherefore, the first condition holds.\nNow, remark that  $S(H, \\varpi) = \\left\\{ \\left( t_a + u \\right)_{a \\in A} : u \\in \\mathbb{Z} \\right\\}$,\nso the second condition holds.\n\\qed\n\\end{proof}\n\n\n\\begin{theorem} \\label{th:SU-inNP}\n\\textup{SU}{} belongs to $\\mathrm{NP}$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\left(\\inda{X}, k \\right)$ be an arbitrary instance of \\textup{SU}.\nWe propose the following (non-deterministic) algorithm to decide whether $\\left(\\inda{X}, k \\right)$ is a yes-instance of \\textup{SU}:\n\\begin{itemize}\n \\item \nGuess a tree $H$ with vertex set $A$ and an antisymmetric edge-weight function $\\varpi$ on $H$ such that the first condition of Lemma~\\ref{lem:tree-Xa} holds.\n\\item \nCompute an element $\\inda{t} \\in S(H, \\varpi)$.\n\\item \nCheck whether the cardinality of $\\bigcup_{a \\in A} (X_a + t_a)$ is at most~$k$.\n\\end{itemize}\nBy Lemma~\\ref{lem:tree-Xa}, the algorithm is correct.\nMoreover, \nthe bit-length of the guess (\\emph{i.e}, the ordered pair $(H, \\varpi)$) \nis polynomial in \nthe bit-length of the input (\\emph{i.e}, the instance $\\inda{X}$), \nso the algorithm can be implemented in non-deterministic polynomial time.\n\\qed\n\\end{proof}\n\nLet $m$ be a positive integer and let $X$ be a subset of $\\mathbb{Z}$ such that $X = -X$.\nOn each given $m$-edge graph,\nthere are exactly $\\card{X}^m$ distinct antisymmetric edge-weight functions whose ranges are subsets of~$X$. \n\nLet $n$ be a positive integer and let $\\mathcal{T}_n$ denote the set of all trees with vertex set $\\seg{1}{n}$.\n\\emph{Cayley's formula} ensures $\\card{\\mathcal{T}_n} = n^{n - 2}$ \\cite{harary73graphical}. \nMoreover, every tree can be reconstructed in polynomial time from its \\emph{Pr\\\"ufer code} \\cite{harary73graphical},\nso $\\mathcal{T}_n$ is enumerable in $O \\left( n^{O(n)} \\right)$ time.\n\n\\begin{theorem}\nThere exists an algorithm  that, \nfor each instance $\\inda{X}$ of \\textsc{MinSU}{} given as input,\nreturns an optimum solution of \\textsc{MinSU}{} on $\\inda{X}$ in \n$O \\left( N^{O(\\card{A})} \\right)$ \ntime, where $N$ denotes the bit-length of $\\inda{X}$.\n\\end{theorem}\n\n\\begin{proof}\nPut $U = \\bigcup_{a \\in A} X_a$.\nLet $\\mathcal{H}$ denote the set of all ordered pairs of the form $(H, \\varpi)$, \nwhere \n$H$ is a tree with vertex set $A$ \nand \n$\\varpi$ is an antisymmetric edge-weight function on $H$ whose range is a subset of $U - U$.\nWe propose the following algorithm to solve \\textsc{MinSU}{} on $\\inda{X}$:\n\\begin{itemize}\n \\item For each $(H, \\varpi) \\in \\mathcal{H}$, compute an element of $S(H, \\varpi)$.\n \\item Return a best solution of \\textsc{MinSU}{} on $\\inda{X}$ among those computed at the previous step.\n\\end{itemize}\nBy Lemma~\\ref{lem:tree-Xa}, the algorithm returns an optimum solution of \\textsc{MinSU}{} on $\\inda{X}$.\nMoreover, remark that $\\card{\\mathcal{H}} = \\card{A}^{\\card{A} - 2} \\card{U - U}^{\\card{A} - 1}$\nand that $\\mathcal{H}$ is enumerable in $O \\left( N^{O(\\card{A})} \\right)$ time.\nTherefore, the algorithm can be implemented to run in $O \\left( N^{O(\\card{A})} \\right)$ time.\n\\qed\n\\end{proof}\n\n\n\n\\section{Hardness} \\label{sec:msu-hard}\n\nThe aim of this section is prove the hardness results for \\textsc{MinSU}.\n\nLet $G = (V, E)$ be a graph. \nA \\emph{vertex cover} of $G$ is a subset $C \\subseteq V$ such that $C \\cap e \\ne \\emptyset$ for every $e \\in E$: \na vertex cover is a subset of vertices that contains at least one extremity of each edge.\n\n\\defpb\n{\\textsc{Minimum} \\textsc{Vertex Cover}{} (\\textsc{MinVC})}\n{a graph~$G$.}\n{a vertex cover $C$ of~$G$.}\n{the cardinality of~$C$.}\nThe decision problem associated with \\textsc{MinVC}{} is named \\textsc{Vertex Cover}{} (\\textup{VC}).\nIt is well-known that \\textup{VC}{} is $\\mathrm{NP}$-complete \\cite{garey79computers}.\n\n\nTo prove that \\textup{SU}{} is (strongly) $\\mathrm{NP}$-complete, \nwe show that \\textup{VC}{} Karp-reduces to (a suitable restriction of) \\textup{SU}.\nThe following gadget plays a crucial role in our reduction as well as in other reductions \nthat can be found in the literature \\cite{nicolas08hardness,michael10complexity}:\n\n\\begin{definition}\nFor each integer $n \\ge 1$, \ndefine $R_n = \\left\\{ (i - 1) n^2 + i^2 : i \\in \\seg{1}{n} \\right\\}$.\n\\end{definition}\n              \nA \\emph{Golomb ruler} \\cite{gardner83wheels,sidon32satz,babcock53intermodulation} is a finite subset $R \\subseteq \\mathbb{Z}$ that satisfies the following three equivalent conditions:\n\\begin{itemize}\n \\item For every $t \\in \\mathbb{Z}$,  $t \\ne 0$ implies  $\\card{R \\cap (R + t)} \\le 1$.\n \\item For every integer $d > 0$, there exists at most one $(r, s) \\in R \\times R$ such that $r - s = d$.\n \\item For all $r_1$, $r_2$, $s_1$, $s_2 \\in R$, $r_1 + r_2 = s_1 + s_2$ implies $\\{ r_1, r_2 \\} = \\{ s_1, s_2 \\}$.\n\\end{itemize}\nActually, only the first condition is referred to in what follows.\nAmong other convenient properties our gadget sets are Golomb rulers:\n\n\\begin{lemma} \\label{lem:golomb}\nLet $n$ be a positive integer.\nThe following four properties hold.\n\\begin{enumerate}\n\\item \\label{ppty:subset} The least element of $R_n$ is $1$ and the greatest element of $R_n$ is~$n^3$.\n\\item  \\label{ppty:card} The cardinality of $R_n$ equals~$n$.\n\\item \\label{ppty:distn} \nThe distance between any two elements of $R_n$ is at least $n^2 + 3$.\n\\item \\label{ppty:Golomb} $R_n$ is a Golomb ruler.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nProperties~\\ref{ppty:subset} and~\\ref{ppty:card} are clear.\nProofs of Property~\\ref{ppty:Golomb} can be found in \\cite{nicolas08hardness,michael10complexity}.\nFinally, \nremark that for every $i \\ge 1$, \nwe have  \n$$\n\\left( i n^2 + {(i + 1)}^2 \\right)\n-\n \\left( (i - 1) n^2 + i^2 \\right) \n  =\n n^2 + 2i + 1 \n \\ge \n n^2 + 3 \\, .\n$$\nHence, the distance the distance between any two consecutive elements of $R_n$ is at least \n$n^2 + 3$, and thus Property~\\ref{ppty:distn} holds.\n\\qed\n\\end{proof}\n\n\n\\begin{theorem} \\label{th:MSU-NPC}\n\\textup{SU}{} is strongly $\\mathrm{NP}$-hard. \n\\end{theorem}\n\n\\begin{proof}\nPut  $f(x) = \\left( \\tfrac{1}{4} x + 2 \\right)^3 + \\tfrac{1}{2}x - 4$.\nLet \\textsc{Aux}{} denote the restriction of \\textup{SU}{} to those instances $\\left( \\inda{X}, k \\right)$ \nsuch that \nthe absolute value of every integer in \n$\\left\\{ k \\right\\} \\cup \\bigcup_{a \\in  A} X_a$ is at most $f\\left( \\max_{a \\in A} \\card{X_a} \\right)$.\nWe prove that  \\textsc{Aux}{} is $\\mathrm{NP}$-hard which implies the theorem.\nMore precisely, we show that \\textup{VC}{}  Karp-reduces to \\textsc{Aux}.\n\n\n\\paragraph{Presentation of the reduction.}\n\nLet $I$ be an arbitrary instance of \\textup{VC}.\nThe reduction maps $I$ to an instance $J$ of \\textup{SU}{} that is defined as follows.\nLet $G$, $V$, $E$, and $k$ be such that $I = (G, k)$ and $G = (V, E)$.\nLet $n$ denote the cardinality of $V$.\nWithout loss of generality, \nwe may assume \n$V = \\seg{1}{n}$ \nand \n$k < n$ because $I$ is a yes-instance of \\textup{VC}{} whenever $k \\ge n$. \nLet $\\left( y_e \\right)_{e \\in E}$, $\\left( z_e \\right)_{e \\in E} \\in V^E$ be such that $e = \\left\\{ y_e,  z_e \\right\\}$ for every $e \\in E$.\nSet \n \\begin{gather*}\n A = \\{ \\emptyset \\} \\cup E \\, , \\\\\n s =  {(n + 4)}^3\\,, \\\\\n R =  R_{n + 4}\\,, \\\\\n X_\\emptyset = (V - s - n) \\cup  (R -  s) \\cup (R + n) \\cup (V + s + n)\\,, \\\\\n X_e =  \\left\\{ z_e - n \\right\\} \\cup R  \\cup  \\left\\{ y_e + s  \\right\\} \n\\end{gather*}\nfor each $e \\in E$, and \n\\begin{gather*}\nJ = \\left( \\inda{X}, \\card{X_\\emptyset} + k \\right) \\, . \n\\end{gather*}\nClearly, $J$ is computable from  $I$ in polynomial time.\n\n\\paragraph{An instance of \\textsc{Aux}.}\n Let us prove that $J$ is in fact an instance of \\textsc{Aux}.\nWith the help of Lemmas~\\ref{lem:golomb}.\\ref{ppty:subset} and~\\ref{lem:golomb}.\\ref{ppty:card},\nit is easy to see \nthat $1 - s - n$ is the least element of $\\bigcup_{a \\in A} X_a$,\nthat $s + 2n$ is the greatest element of $\\bigcup_{a \\in A} X_a$, and \nthat the cardinality of $X_\\emptyset$ equals $4n + 8$. \nThe latter property implies $\\card{X_\\emptyset} + k < 5 n + 8$.\nHence, \nthe absolute value of every integer in \n$\\left\\{ \\card{X_\\emptyset} + k \\right\\} \\cup \\bigcup_{a \\in A} X_a$ \nis at most $s + 2n$.\nNow, remark that $s + 2n = f(\\card{X_\\emptyset}) \\le f\\left( \\max_{a \\in A} \\card{X_a} \\right)$. \n\n\\paragraph{Correctness of the reduction.}\nIt remains to prove that \n$I$ is a yes-instance of \\textup{VC}{} \nif, and only if,\n$J$ is a yes-instance of \\textup{SU}.\n\n\\begin{lemma} \\label{lem:Xens}\nFor every $e \\in E$,\nit holds true that \n\\begin{enumerate}\n \\item \\label{ppty:s} $(X_e -  s) \\setminus  X_\\emptyset = \\left\\{  y_e  \\right\\}$ and that\n \\item \\label{ppty:n} $(X_e +  n) \\setminus  X_\\emptyset =  \\left\\{  z_e  \\right\\}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n We only prove Property~\\ref{ppty:s} because Property~\\ref{ppty:n} can be proven in the same way. \nPut $Y  =  \\left\\{ z_e - s - n \\right\\} \\cup (R - s)$.\nIt is clear that \n$X_e - s  =  Y  \\cup  \\left\\{ y_e  \\right\\}$  \nand \n$Y  \\subseteq X_\\emptyset$. \nTherefore, we have \n\\begin{equation} \\label{eq:Xe-Y-y}\n X_e - s = \\left\\{ y_e  \\right\\} \\setminus X_\\emptyset \\, .\n\\end{equation}\nMoreover,  it follows from Lemma~\\ref{lem:golomb}.\\ref{ppty:subset} that \n\\begin{itemize}\n \\item the greatest element of ${(V - s - n)} \\cup {(R - s)}$ equals $0$ and that\n \\item the least element of ${(R + n)} \\cup {(V + s + n)}$ equals $n + 1$.\n\\end{itemize}\nTherefore, $X_\\emptyset$ does not contain any element of $\\seg{1}{n}$.\nIn particular, $y_e$ does not belong to $X_\\emptyset$.\nCombining the latter fact with Equation~\\eqref{eq:Xe-Y-y}, \nwe obtain  Property~\\ref{ppty:s}.\n\\qed\n\\end{proof}\n\n\n\\begin{lemma} \\label{lem:tpms}\nFor every $t \\in \\mathbb{Z}$, \n$\\card{(R + t) \\setminus X_\\emptyset} <  n$ implies $t \\in \\left\\{ - s, + n \\right\\}$. \n\\end{lemma}\n\n\\begin{proof}\nLet us first bound from above the cardinality of $(R + t) \\cap X_\\emptyset$.\nFor each $\\tau \\in \\mathbb{Z}$, put\n$\nP_\\tau = (R  + t) \\cap (V + \\tau)\n$\nand  \n$\nQ_\\tau = (R  + t) \\cap (R + \\tau).\n$\nFirst, \nit follows from Lemma~\\ref{lem:golomb}.\\ref{ppty:distn} that \n$\\card{P_\\tau} \\le 1$.\nSecond, \n$\\tau \\ne t$ implies $\\card{Q_\\tau} \\le 1$ by Lemma~\\ref{lem:golomb}.\\ref{ppty:Golomb}.\nAnd third, \nit holds that\n$$(R + t) \\cap X_\\emptyset =  P_{- s - n} \\cup Q_{-s} \\cup Q_n \\cup P_{s + n} \\, .$$\nNow, assume $t \\notin \\{ - s, + n \\}$.\nFrom the preceding three facts, \nwe deduce that\n$$\n\\card{(R + t) \\cap X_\\emptyset} \n\\le \n\\card{P_{-s - n}} + \n\\card{Q_{-s}} + \n\\card{Q_n} + \n\\card{P_{s + n}} \n\\le \n4 \\,. \n$$\n(In fact, \nit is not hard to see that $\\card{(R + t) \\cap X_\\emptyset} \\le 2$ holds: \n$t >  +n$ implies  $P_{- s - n} = Q_{-s} = \\emptyset$,  \n$- s <  t < +n$ implies $P_{- s - n}  = P_{s + n} = \\emptyset$, and \n$t < - s$ implies $Q_n = P_{s + n} = \\emptyset$.)\nSince $\\card{R + t} = n + 4$ by Lemma~\\ref{lem:golomb}.\\ref{ppty:card}, \nwe finally get \n$$\n\\card{(R + t) \\setminus X_\\emptyset} = n + 4 - \\card{(R + t) \\cap X_\\emptyset }  \\ge  n  \\, .\n$$ \n\\qed \n\\end{proof}\n\n\n\\paragraph{(If).} \n\nAssume that $I$ is a yes-instance of \\textup{VC}.\nThen, there exists a vertex cover $C$ of $G$ with $\\card{C} \\le k$.\nPut \n$F = \\left\\{ e \\in E : y_e \\in C \\right\\}$.\nSet\n$t_\\emptyset = 0$, \n$t_e = -s$ for each $e \\in F$, and \n$t_e = +n$ for each $e \\in E \\setminus F$.\nOn the one hand, \nit holds that \n\\begin{equation} \\label{eq:Xa-ta-X0}\n\\card{\\bigcup_{a \\in A} (X_a + t_a) } = \\card{X_\\emptyset} + \\card{\\bigcup_{e \\in E} (X_e + t_e ) \\setminus X_\\emptyset}  \n\\end{equation}\nbecause $t_\\emptyset = 0$.\nOn the other hand,\nit follows from Lemma~\\ref{lem:Xens} that \n\\begin{equation} \\label{eq:Xe-te-ye-ze}\n\\bigcup_{e \\in E} (X_e + t_e) \\setminus X_\\emptyset \n= \n\\left\\{ y_e : e \\in F \\right\\} \\cup \\left\\{ z_e : e \\in E \\setminus F \\right\\} \\, .\n\\end{equation}\nSince the right-hand side of Equation~\\eqref{eq:Xe-te-ye-ze} is a subset of~$C$, \nwe have\n\\begin{equation} \\label{eq:Xe-te-X0-k}\n\\card{\\bigcup_{e \\in E} (X_e + t_e) \\setminus X_\\emptyset  } \\le  k \\, .\n\\end{equation}\nWe then get   \n\\begin{equation} \\label{eq:J-yes}\n \\card{\\bigcup_{a \\in A} (X_a + t_a) } \\le \\card{X_\\emptyset} + k \n\\end{equation}\nby combining Equations~\\eqref{eq:Xa-ta-X0} and~\\eqref{eq:Xe-te-X0-k}.\nHence, $J$ is a yes-instance of \\textup{SU}.\n\n\\paragraph{(Only if).} \nAssume that $J$ is a yes-instance of \\textup{SU}.\nThen, there exists $\\inda{t} \\in \\mathbb{Z}^A$ such that Equation~\\eqref{eq:J-yes} holds.\nReplacing \n$\\inda{t}$ with $\\left( t_a - t_\\emptyset \\right)_{a \\in A}$ \nleaves the cardinality of $\\bigcup_{a \\in A} (X_a + t_a)$ unchanged; \ntherefore, \nwe may assume that $t_\\emptyset = 0$;\nin particular, Equation~\\eqref{eq:Xa-ta-X0} holds.\n\nPut \n$$\nC = \\bigcup_{e \\in E} (X_e + t_e) \\setminus X_\\emptyset  \\, .\n$$\nCombining Equations~\\eqref{eq:Xa-ta-X0} and~\\eqref{eq:J-yes}, \nwe obtain Equation~\\eqref{eq:Xe-te-X0-k}, or equivalently, $\\card{C} \\le k$.\nNow, let us prove that $C$ is a vertex cover of~$G$.\nConsider an arbitrary edge~$e \\in E$.\nSince we have\n$$\n(R + t_e) \\setminus X_\\emptyset \n \\subseteq \n(X_e + t_e) \\setminus X_\\emptyset \\subseteq C \\, ,\n$$ \nit follows from Lemma~\\ref{lem:tpms} that $t_e \\in \\{  -s,  +n \\}$.\nConsequently, Lemma~\\ref{lem:Xens} ensures that some extremity of $e$ belongs to $(X_e + t_e) \\setminus X_\\emptyset$,\nand this extremity is \\emph{a fortiori} in~$C$.\nHence, $I$ is a yes-instance of \\textup{VC}.\n\\qed\n\\end{proof}\n\n\nA graph $G = (V, E)$ is called \\emph{cubic} if for every vertex $v \\in V$, \nthe degree of $v$ in $G$ \n(\\emph{i.e.}, the cardinality of $\\left\\{  w \\in V : \\{ v, w \\} \\in E \\right\\}$) \nequals~$3$. \nLet \\textsc{MinVC}$3$ denote the restriction of \\textsc{MinVC}{} to cubic graphs.\n\\textsc{MinVC}$3$  is $\\mathrm{APX}$-complete under L-reduction \\cite{alimonti00APX};\nmoreover, if \\textsc{MinVC}$3$ is $\\frac{100}{99}$-approximable in polynomial time then $\\PNP$ \\cite{chlebik06complexity}.\n\nTo prove that \\textsc{MinSU}{} is ``strongly'' $\\mathrm{APX}$-hard,\nwhich is a better result than Theorem~\\ref{th:MSU-NPC},\nwe show that \\textsc{MinVC}$3$ L-reduces to a suitable restriction of \\textsc{MinSU}.\nIn fact, we simply adapt the proof of Theorem~\\ref{th:MSU-NPC}.\n \n\n\\begin{theorem} \\label{th:MSU-APX}\nThere exists a real constant $\\rho > 1$ such that \nif \\textsc{MinSU}{} is $\\rho$-approximable in pseudo-polynomial time then $\\PNP$.\n\\end{theorem}\n\n\\begin{proof}\nLet $f$ be as in the proof of Theorem~\\ref{th:MSU-NPC} \nand \nlet \\textsc{MinAux}{} denote the restriction of \\textsc{MinSU}{} to those instances $\\inda{X}$ such that \nthe absolute value of every integer in $\\bigcup_{a \\in  A} X_a$ is at most $f\\left( \\max_{a \\in A} \\card{X_a} \\right)$.\nWe prove that \\textsc{MinAux}{} is $\\mathrm{APX}$-hard, \nwhich implies the theorem \nbecause \nevery pseudo-polynomial-time approximation algorithm for \\textsc{MinSU}{} \nis \n a polynomial-time approximation algorithm for \\textsc{MinAux}.\nMore precisely, we show that \n\\textsc{MinVC}$3$ {L-reduces} \\cite{ausiello03complexity} to \\textsc{MinAux}.\nWe use the notation of the proof of Theorem~\\ref{th:MSU-NPC}. \n\n\n\\paragraph{From a graph to an instance of \\textsc{Aux}.}\n\nLet $\\tau$ denote the minimum cardinality of a vertex cover of~$G$.\nLet $\\upsilon$ denote the minimum cardinality of $\\bigcup_{a \\in A} (X_a + t_a)$ over all $\\inda{t} \\in \\mathbb{Z}^A$.\nClearly, \n$\\inda{X}$ is computable from $G$ in polynomial time ($\\inda{X}$ is independent of $k$),\n$\\inda{X}$ is an instance of \\textsc{MinAux}, and \n$\\upsilon = \\card{X_\\emptyset} + \\tau = 4n + 8 + \\tau$.\n\nNow, assume that $G$ is cubic and $n \\ge 24$.\nThe first assumption implies $3 \\tau \\ge \\card{E} \\ge n$.\nIt follows \n$$\n4n + 8 \n= \\left( 4 + \\frac{8}{n}  \\right) n \n\\le \\left( 12  + \\frac{24}{n}  \\right) \\tau\n\\le 13\\tau \\, ,\n$$\nand thus $\\upsilon \\le 14 \\tau$.\n\n\n\\paragraph{From a solution of \\textsc{Aux}{} to a vertex cover.}\n\nLet $\\inda{t} \\in \\mathbb{Z}^A$. \nPut \n$$k = \\card{\\bigcup_{a \\in A} (X_a + t_a)} - \\card{X_\\emptyset}\\,.$$\nThere exists a  vertex cover $C$ of $G$ \nthat satisfies \n$\\card{C} \\le  k$, \n or equivalently, \n$$\\card{C} - \\tau \\le  \\card{\\bigcup_{a \\in A} (X_a + t_a)} - \\upsilon \\, . $$\nMoreover, such a vertex cover is computable from  $G$ and $\\inda{t}$ in polynomial time:\n\\begin{itemize}\n \\item if $k  \\ge n$ then set $C = V$ and \n \\item if $k  < n$ then set $C = \\bigcup_{e \\in E} (X_e + t_e - t_\\emptyset) \\setminus X_\\emptyset$.\n\\end{itemize}\n\n\n\\paragraph{Conclusion.}\n\nLet $\\varepsilon$ be a positive real number.\nIf \\textsc{MinSU}{} is ${(1 + \\varepsilon)}$-approximable in pseudo-polynomial time \nthen \\textsc{MinVC}$3$ is ${(1 + 14\\varepsilon)}$-approximable in polynomial time.\nTherefore, if \\textsc{MinSU}{} is $\\frac{1387}{1386}$-approximable in pseudo-polynomial then $\\PNP$.\n\\qed\n\\end{proof}\n\n \nAn immediate corollary of Theorem~\\ref{th:MSU-APX} is that \\textsc{MinSU}{} does not\nadmit any (pseudo-)polynomial time approximation scheme.\n\n\n\n\\section{Open questions}\n\nThe following three questions remain open:\nDoes there exist a constant $\\rho > 1$ such that \n\\textsc{MinSU}{} is $\\rho$-approximable in (pseudo-)polynomial time?\nIs \\textup{SU}{} \\emph{fixed-parameter tractable} \\cite{flum06parameterized} with respect to parameter $\\card{A}$?\nIs \\textup{SU}{} solvable in polynomial time for bounded $\\max_{a \\in A} \\card{X_a}$?\n\n\n\n\n\n\n\\bibliographystyle{abbrv}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\IEEEPARstart{W}{ith} the rapid standardization process of 5G\nnetworks~\\cite{busari20185g}, millimeter waves (mm-waves) have garnered great attention worldwide from industry, academia, and government.\nA major issue is to better understand the propagation characteristics of mm-wave signals.\nMany mm-wave channel measurement campaigns have recently been carried out in urban and suburban environments~\\cite{rappaport2013millimeter, rappaport2017small, zhang201728ghzicc, zhang2018improving}. \n{\n    However, very limited effort has been put into validating and improving current channel models in overcoming vegetation blockage. This is a key element of sensor data collection in forestry and agriculture~\\cite{wu2017propagation} for preventing cost incurred by under\/over-deployment of the sensors and improving their communication performance.}\nIn~\\cite{papazian1992wideband}, a constant excess path loss of around 25~dB was observed at 28.8~GHz through a pecan orchard for paths with roughly 8 to 20 trees. \nMore recent works~\\cite{zhang2018improving, rappaport201573} reported low attenuation values per unit foliage depth of 0.07~dB\/m at 28~GHz and of 0.4~dB\/m with 3~dB deviation at 73~GHz, respectively.\n{\n    In~\\cite{shaik2016millimeter}, attenuation with a dual-slope structure was observed for out-of-leaf measurements at 15~GHz, 28~GHz and 38~GHz in forest environments.}\nMoreover, even though a variety of modeling approaches have been considered, most of them ignore site-specific geographic features~\\cite{zhang2018improving}.\nA comprehensive analysis for attenuation in vegetation is required to validate those observations\nand make improvements\nto mm-wave propagation modeling.\n\n{We explore this gap by investigating mm-wave propagation at 28~GHz through a coniferous forest in Boulder, Colorado, where we recorded a total of 1415 basic transmission loss measurements.}\nA comprehensive model comparison is provided to elucidate the pros and cons of different modeling approaches for predicting signal attenuation through vegetation. Novel site-specific models with consistently better performance than existing models are developed. \n\n\\section{Measurement Setup}\n\nThe measurement system in our previous work~\\cite{zhang201728ghzicc} was utilized. The receiver (RX), with a chip rate of 399.95 megachips per second, was installed in a backpack and powered by a lithium-ion polymer battery for portability purposes. As illustrated in \\Cref{fig_measurement_overviews}\\subref{fig_rx_tracks}, the transmitter (TX) was set up at the edge of the forest, while the RX was moved in the coniferous forest to continuously record the signal along with the GPS location information. Basic transmission losses were computed accordingly. The TX antenna was adjusted before each signal recording activity to point to the middle area of the track to be covered. Beam alignment was achieved at the RX side using a compass.\nWe also obtained satellite images from Google Maps and LiDAR data from the United States Geological Survey (USGS). Tree locations were manually labeled accordingly. Foliage regions were automatically extracted by comparing the LiDAR data with USGS terrain elevation data. \nThese site-specific geographic features of the forest are illustrated in \\Cref{fig_measurement_overviews}\\subref{fig_tree_locaitons_zoomed_in}. \n{\n    Boulder is a semi-arid environment with low humidity and minimal rainfall. Measurements were performed on a warm spring day under mostly sunny conditions.\n}\n\n\n\\begin{figure}[t]\n\t\\captionsetup[subfigure]{justification=centering}\n\t\\centering\n\t\\subfloat[RX tracks illustrated with basic transmission loss results\\vspace{-1.5mm}]{\n    {\\parbox{3in}{\\centering\\includegraphics[height=2in]{.\/eps\/zhang_WCL2018-1396_fig1.pdf}}}\n\t\t\\label{fig_rx_tracks}\n    }\n    \\hfil   \n\t\\subfloat[Site-specific information available for the measurement site]{\n\t\t{\\parbox{3in}{\\centering\\includegraphics[height=2in,trim={0.01in 0.19in 0 0.19in},clip]{.\/eps\/zhang_WCL2018-1396_fig2.pdf}}}\n\t\t\\label{fig_tree_locaitons_zoomed_in}\n\t}\n\t\\caption{\n\t    Overview of the measurement campaign. \n    \t(a) The TX was installed at the edge of a coniferous forest. The RX followed 10 different tracks.\n        One basic transmission loss result was computed for each second of the recorded signal to match the GPS data. (b) We have zoomed into the dotted-square area in (a) to better illustrate the site-specific features. \n        {\n            Overlaid on top of the satellite image are\n            LiDAR data,\n            foliage regions, and\n            trunk locations, respectively.\n           \n    }}\n\t\\label{fig_measurement_overviews}\n    \\vspace{-1mm}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Foliage Analysis for the Coniferous Forest}\n\nWe compared three empirical foliage analysis models: the partition-dependent attenuation factor (AF) model~\\cite{durgin1998measurements}, the ITU-R obstruction by woodland model~\\cite{itu2016vegetation}, and Weissberger's model~\\cite{weissberger1982initial}.\nTo tune these models,\nfour parameters were computed for each measurement location: the distance between the TX and the RX, the number of tree trunks within the first Fresnel zone, the foliage depth along the line-of-sight (LoS) path, and the foliage area within the first Fresnel zone. These computations were performed in a three-dimensional (3D) reference system using Universal Transverse Mercator coordinates $(x, y)$ and altitude. Based on these results, site-specific models were introduced to improve path loss predictions.\n\n\nAll channel models considered here generate excess attenuation values on top of a site-general channel model.\nWe use the free-space path loss (FSPL) model as the baseline generic model. The path loss $PL$ in dB at the RX location $s$ is then composed of two parts:\n\\begin{equation}\n\t\\nonumber PL(s) = FSPL\\left[ d(s) \\right] + EPL(s) \\,\\text{,} \n\\end{equation}\nwhere $FSPL\\left[ d(s) \\right]$ is the FSPL in dB at a RX-to-TX distance of $d$ at $s$, and $EPL(s)$ is the excess path loss in dB at $s$.\n\n\\subsection{The AF Propagation Model~\\cite{durgin1998measurements}}\n\nThe partition-dependent AF propagation model takes advantage of site-specific information by assuming that each instance of one type of obstacle along the LoS path will incur a constant excess path loss.\nIn our case, we counted the number of trees, $N(s)$, along the LoS path to $s$ and added a constant excess path loss in dB, $L_0$, for each of the trees, as follows:\n\\begin{equation}\n\t\\nonumber EPL(s) = N(s) \\cdot L_0 \\,\\text{.}\n\\end{equation}\n\nConsidering the forest size and the number of RX locations involved, it is extremely difficult and time-consuming to count $N$ at each $s$ on-site. In our work, we simplified the trees, making them vertical lines rather than estimating the cylinder of each tree. Then, the number of trees within the first Fresnel zone for each $s$ was estimated based on manually labeled trunk locations\nand used as the number of obstacle trees. \n\n\\newcommand{2.7in}{2.7in}\n\\newcommand{-2mm}{-2mm}\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=2.7in\/560*640]{.\/eps\/zhang_WCL2018-1396_fig3.pdf}\n    \\vspace*{-2mm}\n\t\\caption{The AF propagation model degenerates to a constant-loss-per-tree model in our case. Its predictions fit the shape of the measurement results but have a poor overall accuracy. \n\n    }\n\t\\label{model_comparison_partition_dependent}\n    \\vspace{-1mm}\n\\end{figure}\n\n\\Cref{model_comparison_partition_dependent} shows the predictions obtained from \nthe AF model. The unknown constant $L_0$ was fit according to the measurement data, resulting in a value of 6.47~dB per tree.\nAs can be seen, the AF model closely follows the shape formed by the measurement results. However, it suffers in predicting the correct amount of excess path loss in general. This is expected because \n{\n    we have only considered the trunk locations for counting trees, but their physical sizes\n   \n    also play a critical role in attenuating the signal.}\nThe root mean squared error (RMSE) for the AF model compared with the measurements is 27.96~dB, achieving a 11.47~dB improvement over the FSPL model but still significantly worse than those for the other two empirical models discussed below.\n{\n   \n   \n    We observe that it may be possible to improve the AF model by classifying trees into different size categories and assigning each category a loss value.  \n    However, in our case, trees grew in clusters, making it extremely challenging to distinguish individual canopies and to properly classify trees.}\n\n\n\n\\subsection{ITU-R Obstruction by Woodland Model~\\cite{itu2016vegetation}}\n\nThe ITU-R obstruction by woodland model assumes one terminal (the TX or the RX) is located within woodland or similar extensive vegetation, which fits well our measurement scenario. Instead of the number of trees, the ITU model uses the length of the path within the woodland in meters, $d_w(s)$, \n{\nwhich is the distance from the woodland edge to the terminal in the woodland,}\nto estimate the excess path loss:\n\\begin{equation}\n\\label{equ_itu}\n\tEPL(s) = A_m \\left[1-\\text{exp}(-d_w(s)\\cdot\\gamma \/ A_m)\\right] \\text{,}\n\\end{equation}\nwhere $\\gamma\\approx 6$~dB\/m is the typical specific attenuation for very short vegetative paths at 28~GHz, and $A_m$ is the maximum attenuation in dB.\nThe most distinguishing feature of this model is the upper limit $A_m$ imposed on the excess path loss.\n\nSince the TX was installed approximately 15~m away from the forest, this offset has been taken away from the 3D RX-to-TX distance to estimate $d_w(s)$, with the negative values clipped to zero.\nAlso, $A_m$ is yet to be determined in \\cite{itu2016vegetation} for 28~GHz signals, so we fitted it to our measurement results to obtain the best possible performance, which yielded $A_m \\approx 34.5$~dB. The resulting predictions are plotted in \\Cref{fig_itu_results}. The ITU model exhibits the best fit among the empirical models considered, with an overall RMSE of 20.08~dB. \nHowever, it clearly overestimates the path loss for locations with $d_w$ smaller than 30~m. At those locations, the LoS path may be clear or blocked by only a couple of trees, differing from a typical woodland blockage scenario. \nOn the other hand, the ITU model underestimates the path loss for large $d_w$. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=2.7in]{.\/eps\/zhang_WCL2018-1396_fig4.pdf}\n    \\vspace*{-2mm}\n\t\\caption{Predictions from the ITU obstruction by woodland model. As a comparison, the predictions from one site-specific model, which is covered in \\Cref{section_site_specific_models}, are also shown. The site-specific model follows the measurements better than the ITU model at the lower and higher ends of $d_w$.}\n\t\\label{fig_itu_results}\n   \n\\end{figure}\n\n\\subsection{Weissberger's Model~\\cite{weissberger1982initial}}\n\n\nWeissberger's model, or Weissberger's modified exponential decay (WMED) model, can be formulated as follows:\n\\newcommand{-0.8em}{-0.8em}\n\\begin{equation}\n\t\\nonumber EPL(s) = {\n\t\t\\begin{cases}\n\t\t\t0.45\\,f_c^{{0.284}}\\,d_f(s)\t\t\t &\\hspace{-0.8em}{\\mbox{, if }}0<d_f(s)\\leqslant 14\\\\\n            1.33\\,f_c^{{0.284}}\\,d_f^{{0.588}}(s) &\\hspace{-0.8em}{\\mbox{, if }}14<d_f(s)\\leqslant 400\n\t\t\\end{cases}\n        }\n\\end{equation}\nwhere $f_c$ is the carrier frequency,\nand $d_f(s)$ is the foliage depth in meters along the LoS path for the RX location $s$. The model treats locations with\n$d_f(s)$ $>$14 differently from those with less foliage blockage.\n\n\n\n\nWe have taken an image processing approach to \n{\n    automatically}\nobtain the site-specific foliage depth, $d_f(s)$, \n{\n    which is the sum total distance for the intersections of the direct path and the foliage regions.}\nBoth the LiDAR data and the terrain elevation data from the USGS were rasterized onto the same set of reference location points. The foliage regions were then extracted by thresholding their difference,\nresulting in the foliage regions illustrated in \\Cref{fig_measurement_overviews}\\subref{fig_tree_locaitons_zoomed_in}. Along the LoS path, the ratio of the number of foliage region pixels over the total number of pixels was calculated and multiplied with the corresponding 3D RX-to-TX distance to get the foliage depth for each RX location. \n\n\\Cref{fig_foliage_depth_based_models}\\subref{fig_weiss_results} compares the predictions from the WMED model with the measurement results. \nOverall, the WMED model gives a reasonably good RMSE value of 22.19~dB. \n\n\n\n\n\n\n\n\n\\newcommand{\\modelPerformFigWidth-1mm} %{\\modelPerformFigWidth}{2.7in-1mm}\n\\begin{figure}[t]\n\t\\centering\n\t\\subfloat[Results from Weissberger's model\\vspace{-1.5mm}]{\n\t\t\\includegraphics[width=\\modelPerformFigWidth-1mm} %{\\modelPerformFigWidth]{.\/eps\/zhang_WCL2018-1396_fig5.pdf}\n\t\t\\label{fig_weiss_results}\n\t}\n\t\\hfil\n\t\\subfloat[Results from site-specific models]{\n\t\t\\includegraphics[width=\\modelPerformFigWidth-1mm} %{\\modelPerformFigWidth]{.\/eps\/zhang_WCL2018-1396_fig6.pdf}\n\t\t\\label{fig_site_spec_results}\n\t}\n\t\\caption{{\n\tPredictions from foliage-depth-based models. Results from \\textit{model C}, which makes predictions based on the foliage area in the first Fresnel zon\n\t, are also plotted in (b) for comparison.\n   \n   \n   \n   \n   \n    }}\n\t\\label{fig_foliage_depth_based_models}\n    \\vspace{-1mm}\n\\end{figure}\n\n\\subsection{Site-specific Models}\n\\label{section_site_specific_models}\n\nUsing high-precision publicly available geographic information, existing channel models can be tuned with well-estimated site-specific parameters.\nAs a result, simple but powerful site-specific models can be constructed as alternatives. We refer to these as \"site-specific\" models because their performance depends heavily on \nthe accuracy of\nthe parameters evaluated for each site.\n\n\n\nBy combining the idea of \n{\n    evaluating the blockage condition individually for each $s$ from the AF model} \nand the two-slope modeling approach in the WMED model, we constructed \\textit{model A-I}:\n\\renewcommand{-0.8em}{-0.9em}\n\\begin{equation}\n\t\\nonumber EPL(s) = {\n\t\t\\begin{cases}\n\t\t\td_f(s)\\cdot L_1 \\,\\,\\text{,}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{\\mbox{if }}0\\leqslant\\,&\\hspace{-0.8em}d_f(s)\\leqslant D_f\\\\\n            D_f L_1 + \\left[ d_f(s)-D_f \\right] L_2\t\\,\\,{\\mbox{, if }}&\\hspace{-0.8em}d_f(s)>D_f\n\t\t\\end{cases}\n        }\n\\end{equation}\nwhere $d_f(s)$ is the foliage depth in meters at $s$, $L_1$ and $L_2$ are two constants for adjusting the extra loss in dB caused by each meter of foliage, and $D_f$ is the boundary determining when $L_2$ will take effect. The upper bound from the ITU model can be imposed by setting $L_2=0$ to form \\textit{model A-II}:\n\\renewcommand{-0.8em}{-0.8em}\n\\begin{equation}\n\t\\nonumber EPL(s) = {\n\t\t\\begin{cases}\n\t\t\td_f(s)\\cdot L_1 \t&\\hspace{-0.8em}{\\mbox{, if }}0\\leqslant d_f(s)\\leqslant D_f\\\\\n            D_f \\cdot L_1 \t&\\hspace{-0.8em}{\\mbox{, if }}d_f(s)>D_f\n\t\t\\end{cases}\n        }\n\\end{equation}\n\nWe also reused the ITU model in \\Cref{equ_itu} with site-specific foliage depth to form \\textit{model B}. That is, $d_f(s)$ is used instead of $d_w(s)$, and parameters $A_m$ and $\\gamma$ are set according to the measurements. \n\nFor a fair performance comparison for these three models, we used the WMED boundary $D_f = 14$ for \\textit{model A-I} to leave only two adjustable parameters. After fitting these models to our data, we found $L_1 \\approx 2.39$~dB\/m and $L_2 \\approx 0.12$~dB\/m for \\textit{model A-I}, $L_1 \\approx 2.09$~dB\/m and $D_f \\approx 17.87$~m for \\textit{model A-II}, along with $A_m \\approx 38.04$~dB and $\\gamma \\approx 4.47$~dB\/m for \\textit{model B}. \nThe resulting predictions are plotted in \\Cref{fig_foliage_depth_based_models}\\subref{fig_site_spec_results}. The corresponding RMSE\nvalues are summarized in \\Cref{tab_overall_perf}, together with those for the traditional models as references. Note that the site-specific models perform very similarly, and each unit of foliage depth tends to contribute less to the excess loss as foliage depth grows. \\textit{Model A-I} does not limit the excess loss as the other two site-specific models do, but it performs slightly better than \\textit{model A-II} in terms of RMSE. Overall, \\textit{model B} performs the best, but computationally, it is more demanding because of its exponential form. \n\nWe can further push the best RMSE performance to 19.18~dB with \\textit{Model C}\n\\begin{equation}\n\t\\centering\n\t\\nonumber EPL(s) = {\n\t\t\\begin{cases}\n            0 \\,\\,\\text{,}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,&{\\mbox{if }a_f(s)=0} \\\\\n\t\t\ta_f(s)\\cdot L_1 + L_0 \\,\\,\\text{,}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,&{\\mbox{if }}0<a_f(s)\\leqslant A_f\\\\\n            A_f L_1 + \\left[ a_f(s)-A_f \\right] L_2\t\\,\\,\\text{,}&{\\mbox{if }}a_f(s)>A_f\n\t\t\\end{cases}\n        }\n\\end{equation}\n{where foliage area $a_f(s)$ is the sum total area for the intersections between the first Fresnel zone at RX location $s$ and the foliage regions;} \n$L_0$ (dB), $L_1$ (dB\/m$^2$), and $L_2$ (dB\/m$^2$) are constants adjusting the excess loss contribution; and $A_f$ is the boundary determining when the foliage is deep enough for $L_2$ to take effect. According to our measurement results, we have $L_0 \\approx 19.14$~dB, $L_1 \\approx 2.09$~dB\/m$^2$, $L_2 \\approx 0.06$~dB\/m$^2$, and $A_f \\approx 18.02$~m$^2$. This model has a sudden jump at the origin. Its prediction results are also shown in \\Cref{fig_foliage_depth_based_models}\\subref{fig_site_spec_results} for reference.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\subfloat[Regional RMSE improvement over the ITU model\\vspace{-1.5mm}]{\n\t\t\\includegraphics[width=\\modelPerformFigWidth-1mm} %{\\modelPerformFigWidth]{.\/eps\/zhang_WCL2018-1396_fig7.pdf}\n\t\t\\label{fig_regional_perf_itu}        \n\t}\n\t\\hfil\n\t\\subfloat[Regional RMSE improvement over the WMED model]{\n\t\t\\includegraphics[width=\\modelPerformFigWidth-1mm} %{\\modelPerformFigWidth]{.\/eps\/zhang_WCL2018-1396_fig8.pdf}\n\t\t\\label{fig_regional_perf_wmed}\n\t}\n\t\\caption{Regional performance improvement for site-specific models using a window size of 10~m. (a) Compared with the ITU model, site-specific models work significantly better for locations close to the TX and reasonably better for those far away. However, models \\textit{A-I}, \\textit{A-II}, and \\textit{B} suffer a severe performance degradation for $d_w$$\\in[35,  60]$~m, which is less of an issue for \\textit{model C}. (b) Compared with the WMED model, site-specific models again work reasonably better for extreme cases. A performance deterioration is observed at a foliage depth of around 80~m, where models \\textit{A-I} and \\textit{C} are less influenced.}\n\t\\label{fig_regional_perfs}\n    \\vspace{-1mm}\n\\end{figure}\n\n{\n    The most important feature for these models is that \n    they are fully automatic and thus can be applied in large-scale wireless communication networks. \n    Site-specific information was fetched from Google and USGS servers. Foliage information was extracted, and channel modeling performed, by our automated algorithms.}\n   \nAnother advantage of our site-specific models is their consistently good performance throughout the whole dataset. \nTo demonstrate this, regional RMSE improvements over the ITU and WMED models are evaluated in terms of $d_w$ and $d_f$, respectively, as summarized in \\Cref{fig_regional_perfs}. For our dataset, the ITU model works very well, as shown in \\Cref{tab_overall_perf}. However, according to \\Cref{fig_regional_perfs}\\subref{fig_regional_perf_itu}, the ITU model suffers from an RMSE degradation of as much as 20~dB compared with the site-specific models in the low-vegetation-coverage region ($d_w$$<$30~m).  \nFor large $d_w$, this value is observed to be as much as 6~dB. A visual comparison for predictions from the ITU model and \\textit{model A-I} is provided in \\Cref{fig_itu_results}, where \\textit{model A-I} clearly works better for extreme cases at the low and high ends of $d_w$. Similar comparisons have been carried out for the WMED model in \\Cref{fig_foliage_depth_based_models}\\subref{fig_weiss_results}  and \\Cref{fig_regional_perfs}\\subref{fig_regional_perf_wmed}. The WMED model slightly underestimates the path loss at RX locations with a small $d_f$ and overestimates it at \nlarge $d_f$. \n\n\\newcommand{1.1}{1.1}\n\\begin{table}[t]\n\t\\renewcommand{\\tabcolsep}{2.2pt\n\t\\renewcommand{\\arraystretch}{1.1}\n    \\vspace{-1.5mm}\n\t\\caption{Overall Performance}\n\t\\label{tab_overall_perf}\n\t\\centering\n    \\begin{tabular}{|>{\\bfseries}c||c|c|c|c|c|c|c|c|}\n    \\hline\n                       & \\textbf{Baseline} & \\multicolumn{3}{c|}{\\textbf{Traditional}} & \\multicolumn{4}{c|}{\\textbf{Site-Specific}} \\\\ \\hline\n    \\textbf{Model}     & FSPL \t  & AF  & ITU   & WMED & \\textit{A-I}  & \\textit{A-II}  & \\textit{B}  & \\textit{C} \\\\ \\hline\n    \\textbf{RMSE (dB)} & 39.43 \t  & 27.96 & 20.08 & 22.19       & 19.96 & 20.02 & 19.93 & 19.18 \\\\ \\hline\n   \n    \\end{tabular}\n    \\vspace{-1.5mm}\n\\end{table}\n\n\n\n\n\n\n\\section{Conclusion}\n\n{\n    A comprehensive channel model comparison for attenuation through vegetation was conducted using measurements in a coniferous forest near Boulder, Colorado. Inspired by the results, we developed novel site-specific models for consistent improvement in prediction accuracy through shallow to deep vegetation blockages. They are fully automatic, easy to implement, and feasibly applicable to machine learning frameworks. \n}\n\n\n\\nocite{}\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nLocalisation of a source of biological or chemical agent dispersing\nin the atmosphere is an important problem for national security\nand environmental monitoring applications \\cite{kendal}. Wind, as\nthe dominant transport mechanism in the atmosphere,\ncan generate strong turbulent motion, causing the released agent to\ndisperse as a plume whose spread increases with the downwind\ndistance \\cite{arya_98}. Assuming a constant release of the\ncontaminant, the problem involves estimation of source parameters:\nits location and intensity (release-rate). Two types of measurements\nare generally at disposal for source localisation: (i) the\nconcentration measurements at spatially distributed sensor\nlocations; (ii) the average wind speed and wind direction (typically\navailable from a nearby meteorological station).\n\nMany references are available on the topic of biochemical source\nlocalisation, assuming un-quantised (analog) concentration\nmeasurements. Standard solutions are based on optimisation\ntechniques, such as the nonlinear least squares \\cite{matthes_05} or\nsimulated annealing \\cite{thomson_07}. These methods are unreliable\ndue to local minima or poor convergence; in addition, they provide\nonly point estimates, without uncertainty intervals. The preferred\nalternative is the use of Bayesian techniques; they result in the\nposterior probability density function (PDF) of the source parameter\nvector, thereby providing an uncertainty measure to any point\nestimate derived from it. Most Bayesian methods for source\nestimation are based on Markov chain Monte Carlo (MCMC) technique,\nassuming either Gaussian or log-Gausiian likelihood function of\nmeasurements \\cite{keats_07,humphries_12,ortner,senocak_08}.\nRecently, a likelihood-free Bayesian method for source localisation\nwas proposed in \\cite{ristic_15}. \n\n\nBinary sensor networks have become widespread in environmental monitoring applications\nbecause binary sensors generate as little as one bit of information.\nSuch binary sensors allow inexpensive sensing with minimal communication\nrequirements \\cite{aslam_03}. In the context of binary sensor\nnetworks, an excellent overview of non-Bayesian chemical source\nlocalisation techniques is presented in \\cite{chen2008greedy}. Best\nachievable accuracy of source localisation using binary sensors has\nbeen discussed in \\cite{Ristic2014}.\n\nPrior work in using binary sensor data for biochemical source localisation assumes that the detection threshold\nof the sensor is known. It is a reasonable assumption for a commercial sensor whose sensitivity is specified (for example, in  parts per\nmillion by volume (ppm$_v$) or grams per cubic meter) by the manufacturer and when the sensor is well calibrated. However, we consider at least two scenarios where the detection threshold of a binary sensor may not be accurately known. The first scenario is when a sensor's detection threshold goes off calibration due to environmental conditions such as temperature or humidity or ageing of the sensor. The second scenario is where the sensor is a human rather than a device. For example, imagine a person smelling a strong odour such as due to a gas leak or a decomposing animal carcass. When the person moves around, the smell will be detected in some locations but not in others, producing a binary measurement sequence without knowing the exact value of the threshold in ppm or g\/m$^3$. In this paper, we develop a Bayesian algorithm that carries out source parameter estimation based on such\n{\\em binary concentration measurements} where the sensor threshold is unknown. A Monte Carlo technique, importance sampling, is applied to calculate the posterior PDF approximately. The method is successfully\nvalidated using three experimental datasets obtained under different wind conditions.\n\n\n\n \n\n\\section{Models}\n\\label{s:2}\n\n\\subsection{Dispersion model}\nTo solve the source localisation problem described above, we propose a solution  formulated in the Bayesian framework which relies on\ntwo mathematical models: the atmospheric dispersion model and the\nconcentration measurement model. A dispersion model mathematically describes the physical processes that govern the\natmospheric dispersion of the released agent within the plume. The\nprimary purpose of a dispersion model is to calculate the mean\nconcentration of emitted material at a given sensor location. A\nplethora of dispersion models are in use today\n\\cite{holmes_morawska_06} to account for specific weather\nconditions, terrain, source height, etc. In this paper, we adopt a\ntwo-dimensional dispersion model of ``particle encounters'' in a\nturbulent flow, described in \\cite{vergassola_07}. During a certain sensing\nperiod, each sensor  experiences a Poisson distributed number of\n``encounters'' with released particles. The binary nature of\nmeasurements indicates that a sensor reading of binary ``1'' or a ``positive detection''  corresponds to\nthe number of such encounters exceeding a particular threshold.\n\n If a binary sensor with a particular threshold makes\npositive detections (binary ``1'') at some locations and zero detections (binary ``0'') at other locations due to a source of \na certain release rate, the measurements at these locations will be the same even if both the source release rate and the sensor detection\nthreshold were scaled up or down together by the same amount; it is  the ratio between the source release rate and the sensor threshold that determines \nwhich sensor locations will have positive or zero readings. Therefore, when we estimate the source parameters using binary data from a sensor whose\ndetection threshold is unknown, it is not possible to estimate the absolute value of the source release rate; only the release rate normalised by the assumed sensor threshold can be estimated. Nevertheless, the source location, which is actually the parameter of main interest, can be estimated.  Without loss of generality, in our \nexperiments, we assumed the sensor to output binary ``1'' if it encounters at least one particle during a sensing period and output a binary  ``0'' otherwise. \n\nLet us assume that the biochemical source is located at  $(x_0, y_0)$, with a normalised release rate of $Q_0$.  The particles released from the source propagate with the isotropic diffusivity $D$, but can  also be advected by wind. We assume the released particles to have an average lifetime of  $\\tau$. While the wind speed is typically available from meteorological data from a nearby measuring station, we use this speed as the prior guess for a Bayesian estimate of the true, effective wind speed affecting the advection of particles. Accordingly, let us  assume that the mean wind speed is $V$ and  the mean wind  direction coincides with the direction of the $x$ axis. We denote the PDF of the wind speed by $\\pi(V)$.  A spherical sensor of\nsmall size $a$ at a location with coordinates $(x,y)$, non-coincidental with the source location $(x_0,y_0)$, will experience a series\nof encounters with the released particles. \n\n    \nDenoting the parameter vector we wish to estimate, consisting of the source coordinates  ($x_0$, $y_0$), normalised source release rate $Q_0$, and the wind speed $V$, by  $\\mbox{\\boldmath$\\theta$} = [x_0\\;y_0\\;\nQ_0\\; V]^\\intercal$, the rate of particle encounters by the sensor at the  $i$th location (where $i=1,\\dots,M$) with coordinates $(x_i,y_i)$ can be modelled\nas \\cite{vergassola_07}:\n\\begin{equation}\nR(x_i,y_i|\\mbox{\\boldmath$\\theta$}) =\n\\frac{Q_0}{\\ln\\left(\\frac{\\lambda}{a}\\right)}\\,\\exp\\left[{\\frac{(x_0-x_i)V}{2D}}\\right]\\cdot\nK_0\\left(\\frac{d_i(\\mbox{\\boldmath$\\theta$})}{\\lambda}\\right)  \\label{e:disp}\n\\end{equation}\nwhere $D$, $\\tau$ and $a$ are known environmental and sensor parameters,\n\\begin{equation}\nd_i(\\mbox{\\boldmath$\\theta$}) = \\sqrt{(x_i-x_0)^2+(y_i-y_0)^2}\n\\end{equation}\nis the distance from the source to  $i$th sensor location, $K_0$ is the modified\nBessel function of order zero, and\n\\begin{equation}\n\\lambda = \\sqrt{\\frac{D\\tau}{1+\\frac{V^2\\tau}{4D}}}.\n\\end{equation}\n\n\n\\subsection{Measurement model}\nThe stochastic process of sensor encounters with released particles\nis modelled by a Poisson distribution. The probability that \nsensor at location $(x_i,y_i)$ encounters\n$z\\in\\mathbb{Z}^+\\cup\\{0\\}$ particles ($z$ is a non-negative\ninteger) during a time interval $t_0$ is then:\n\\begin{equation}\n\\mathcal{P}(z; \\mu_i) = \\frac{(\\mu_i)^{z}}{z!}e^{-\\mu_i}\n\\label{e:likf}\n\\end{equation}\nwhere $\\mu_i =  t_0\\cdot R(x_i,y_i|\\mbox{\\boldmath$\\theta$}) $ is the mean concentration\nat $(x_i,y_i)$. Equation (\\ref{e:likf}) represents the full\nspecification of the likelihood function of parameter vector $\\mbox{\\boldmath$\\theta$}$,\ngiven the sensor encounters $z$ counts at the  $i$th position.\n\nHowever, because the actual sensor is binary, the measurement model is\n\\begin{equation}\nb_i = \\begin{cases} 1, & \\text{if } z =1,2,3,\\dots\\\\\n0, & \\text{if } z = 0.\n\\end{cases}\n\\end{equation}\nNote that $b_i$ is a Bernoulli random variable with the parameter\n\\begin{eqnarray} q_i(\\mbox{\\boldmath$\\theta$}) & = & \\text{Pr}\\{b_i=1\\}\\\\\n   & = & \\sum_{z=1}^{\\infty} \\mathcal{P}(z;\\mu_i) \\\\\n   & = & 1 -  \\mathcal{P}(0;\\mu_i) \\\\\n   & = & 1 - e^{-\\mu_i}.\n   \\end{eqnarray}\nThe likelihood function for the sensor when it is at the $i$th location is then:\n\\begin{equation}\np(b_i|\\mbox{\\boldmath$\\theta$}) = [q_i(\\mbox{\\boldmath$\\theta$})]^{b_i} \\, [1-q_i(\\mbox{\\boldmath$\\theta$})]^{1-b_i}.\n\\end{equation}\nAssuming sensor measurements are conditionally independent, the likelihood\nfunction of the parameter vector $\\mbox{\\boldmath$\\theta$}$,  given the binary\nmeasurement vector $\\mathbf{b} = [b_1,\\dots,b_M]^\\intercal$, is a product:\n\\begin{equation}\np(\\mathbf{b}|\\mbox{\\boldmath$\\theta$}) = \\prod_{i=1}^M  [q_i(\\mbox{\\boldmath$\\theta$})]^{b_i} \\,\n[1-q_i(\\mbox{\\boldmath$\\theta$})]^{1-b_i}. \\label{e:lik}\n\\end{equation}\n\n\n\\section{Parameter estimation}\n\nThe estimation problem is formulated in the Bayesian framework. The goal\nis to compute the posterior PDF: the probability distribution of the\nparameter vector $\\mbox{\\boldmath$\\theta$}$  conditional on the measurement vector\n$\\mathbf{b}$. The posterior PDF provides a complete probabilistic\ndescription of the information contained in the measurements about\nthe parameter vector $\\mbox{\\boldmath$\\theta$}$. The basic elements required to compute the\nposterior distribution of are: (i) the prior distribution for the\nparameter vector $\\pi(\\mbox{\\boldmath$\\theta$})$ and (ii) the likelihood function\n$p(\\mathbf{b}|\\mbox{\\boldmath$\\theta$})$. Given these quantities, Bayes rule can be used to\nfind the posterior PDF  as\n\\begin{equation}\np(\\mbox{\\boldmath$\\theta$}|\\mathbf{b}) = \\frac{p(\\mathbf{b}|\\mbox{\\boldmath$\\theta$})\\pi(\\mbox{\\boldmath$\\theta$})}{\\int p(\\mathbf{b}|\\mbox{\\boldmath$\\theta$})\\pi(\\mbox{\\boldmath$\\theta$})\nd\\mbox{\\boldmath$\\theta$}}.\n\\end{equation}\nThe prior distribution $\\pi(\\mbox{\\boldmath$\\theta$})$ is typically non-Gaussian. For\nexample, the source position can be restricted to polygon regions,\nwhile $Q_0$ and $V$ are strictly positive random variables.\nQuantities of interest related to $\\mbox{\\boldmath$\\theta$}$ (e.g., the posterior mean,\nvariance) can be computed from the posterior PDF.\n\n\\begin{figure*}[t!]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[height=7.0cm]{data1_a.eps} &\n\\includegraphics[height=7.0cm]{data1_d.eps}\\\\\n\\includegraphics[height=5.3cm]{data1_bc.eps} &\n\\includegraphics[height=5.3cm]{data1_ef.eps}\n\\end{tabular}\n \\caption{\\footnotesize Estimation results for dataset 1: the left column is the prior PDF; the right column is the posterior\n PDF. Marginalised prior PDF $\\pi(x_0,y_0)$ in (a) and the marginalised posterior PDF $p(x_0,y_0|\\mathbf{b})$ in\n (d), approximated by random samples (indicated by scattered red dots).\nSensor locations (green squares are positive readings,\n blue circles are non-detections), as well as the building contours (black lines), also indicated in (a) and (d).\n Wind direction coincides with the $x$ axis. True source marked in (d) at $(-298.4,-342.6)$ (grey asterix). Figures (b) and (e) show the histograms corresponding to\n $\\pi(Q_0)$ and $p(Q_0|\\mathbf{b})$, respectively. Figures (c) and (f) show the histograms corresponding to $\\pi(V)$ and $p(V|\\mathbf{b})$, respectively.}\n \\label{f:1}\n\\end{figure*}\n\nOptimal Bayesian estimation is generally impossible because the\nposterior PDF cannot be found in closed-form; this is certainly the\ncase for the likelihood function specified in Sec.\\ref{s:2} and a\nnon-Gaussian prior $\\pi(\\mbox{\\boldmath$\\theta$})$. Hence we apply a Monte Carlo\napproximation technique of the optimal Bayesian estimation, known as\nimportance sampling \\cite{robert_casella}. This technique\napproximates the posterior PDF by a weighted random sample\n$\\{w_n,\\mbox{\\boldmath$\\theta$}_n\\}_{1\\leq n \\leq N}$, which is created as follows.\nFirst, a sample $\\{\\mbox{\\boldmath$\\theta$}_n\\}_{1\\leq n \\leq N}$ is drawn from an\nimportance distribution $\\varrho$, i.e., $\\mbox{\\boldmath$\\theta$}_n\\sim \\varrho(\\mbox{\\boldmath$\\theta$})$,\nfor $n=1,\\dots,N$. The unnormalised  weights are then computed as:\n\\begin{equation}\n\\tilde{w}_n = \\frac{\\pi(\\mbox{\\boldmath$\\theta$}_n)}{\\varrho(\\mbox{\\boldmath$\\theta$}_n)}\\,p(\\mathbf{b}|\\mbox{\\boldmath$\\theta$}_n)\n\\end{equation}\nfor $n=1,\\dots,N$. Finally, the weights are normalised, i.e., $w_n =\n\\tilde{w}_n \/\\sum_{n=1}^N \\tilde{w}_n$, for $n=1,\\dots,N$. The\nchoice of importance distribution $\\varrho$ plays a significant role\nin the convergence of point estimators based on the approximation\n$\\{\\mbox{\\boldmath$\\theta$}_n\\}_{1\\leq n \\leq N}$. Ideally $\\varrho$ should resemble the\nposterior. Since the posterior is unknown, good importance\ndistributions are often designed iteratively (population Monte Carlo\n\\cite{robert_casella},  progressive correction \\cite{musso_et_al_00,\nmorelande_ristic_rad}). As $N\\rightarrow \\infty$, however, the\nchoice of $\\varrho$ is less relevant and even the prior $\\pi$ may be\nused as an (admittedly inefficient) importance distribution; this is\ndone in our implementation for convenience. Furthermore, once the weighted random\nsample $\\{w_n,\\mbox{\\boldmath$\\theta$}_n\\}_{1\\leq n \\leq N}$ is computed, resampling\n(with replacement) is carried out \\cite{pfbook} resulting in a\nsample with uniform weights. This last step was carried out mainly\nto improve the effect of visualisation of the posterior PDF (see\nfigures in the next section). In our implementation, the prior PDF\nis adopted as:\n\\begin{equation}\n\\pi(\\mbox{\\boldmath$\\theta$}) = \\pi(x_0,y_0)\\, \\pi(Q_0)\\, \\pi(V)\n\\end{equation}\nwhere: $\\pi(x_0,y_0)$ is a uniform distribution over designated\npolygon areas (e.g., buildings); $\\pi(Q_0)$ is a Gamma distribution\nwith shape $k$ and scale parameter $\\eta$; $\\pi(V)$ is a normal\ndistribution with mean $\\bar{V}$ and standard deviation $\\sigma_V$.\n\n\n\n\n\n\n\\section{Experimental results}\n\n\n\nThree experimental datasets collected using a single moving binary sensor with unknown detection threshold were used in the paper to demonstrate the algorithm performance(further details of the experiments cannot be revealed on security grounds).  In all cases, algorithm parameters were\nadopted as follows: $a=0.2$ m, $D=1$ m$^2$\/s, $\\tau = 1000$ s,\n$t_0=1$ s,  sample size $N=5000$, shape parameter $k=3$, scale\nparameter $\\eta=7$, standard deviation of wind speed $\\sigma_V =\n0.2$ m\/s. The wind conditions were different for the three datasets.\nThe mean wind direction was specified by angle $\\alpha$, measured\nanticlockwise from the $x$ axis. The values of wind parameters\n$(\\bar{V},\\alpha)$ were $(0.28 \\text{ m\/s}, 195^\\circ)$, $(0.12\n\\text{m\/s},-10^\\circ)$ and $(0.14 \\text{m\/s},150^\\circ)$ for the\nfirst, second and the third dataset, respectively.\n\n\n\nFig.\\ref{f:1} shows the results obtained using dataset 1: the left\ncolumn displays the prior PDF $\\pi(\\mbox{\\boldmath$\\theta$})$; the right column - the\nposterior PDF $p(\\mbox{\\boldmath$\\theta$}|\\mathbf{b})$. Figs. \\ref{f:1}.(a) and \\ref{f:1}.(d)\ndisplay the top down view of the area where the experiment was\ncarried out. The placement and the readings of the binary sensor\nare also marked (the number of sensor locations in dataset 1 is $M=45$).\nThe locations where the sensor reported positive readings (i.e., $b_i=1$), are marked by\ngreen squares; the remaining (non-detecting) sensor locations are marked by blue circles. Based on prior knowledge (e.g., intelligence reports), figures (a) and (d) also indicate the circular area where the source must be located (circle drawn in cyan colour). The center of this circle is the mean position of the sensor locations with\npositive readings; its radius is $150$m. Furthermore, figures (a)\nand (d) also display the countours of the buildings (black lines).\nBoth the marginal prior $\\pi(x_0,y_0)$ and the marginal posterior\nPDF $p(x_0,y_0|\\mathbf{b})$, are approximated by random samples marked by\nscattered red dots. Assuming the source must be inside one of the\nbuildings, the marginal prior PDF $\\pi(x_0,y_0)$ shown in figure (a)\nis a uniform PDF over the intersection of the cyan color bounded\ncircular area and the area covered by the buildings. The marginal\nposterior PDF $p(x_0,y_0|\\mathbf{b})$, shown in figure (d), concentrates in\nthe upper left corner of one of the buildings, thus \ndramatically reducing the initial uncertainty in the source location. The\ntrue source location is marked by an asterisk at $(-298.4,-342.6)$.\nFigs.\\ref{f:1}. (b) and (c) show the histograms of random samples\napproximating $\\pi(Q_0)$; and $\\pi(V)$, respectively. Figs.\n\\ref{f:1}.(e) and \\ref{f:1}.(f) display the histograms of random\nsamples approximating the marginal posteriors $p(Q_0|\\mathbf{b})$ and\n$\\pi(V|\\mathbf{b})$, respectively.\n\n\\begin{figure}[h!]\n\\centerline{\\includegraphics[height=7.0cm]{data2_a.eps}}\n\\centerline{\\includegraphics[height=5.3cm]{data2_bc.eps}}\n \\caption{\\footnotesize Estimation results obtained by processing dataset 2: (a)  Marginalised posterior PDF $p(x_0,y_0|\\mathbf{b})$\n  (scatter plot, red dots); sensor locations (green squares are positive readings,\n blue circles are non-detections); building contours (black lines);\n true source location at $(125.6,436.6)$ (grey asterix).\n Wind direction coincides with the $x$ axis. Figures (b) and (c) display the histograms of random samples approximating $p(Q_0|\\mathbf{b})$ and\n  $p(V|\\mathbf{b})$, respectively. }\n \\label{f:2}\n\\end{figure}\n\\begin{figure}[h!]\n\\centerline{\\includegraphics[height=7.0cm]{data3_a.eps}}\n\\centerline{\\includegraphics[height=5.3cm]{data3_bc.eps}}\n \\caption{\\footnotesize Estimation results obtained by processing dataset 3: (a)  Marginalised posterior PDF $p(x_0,y_0|\\mathbf{b})$\n  (scatter plot, red dots); sensor locations (green squares are positive readings,\n blue circles are non-detections); building contours (black lines).\n True source location at $(31.2,-453.2)$ (grey asterix).\n Wind direction coincides with the $x$ axis.  }\n \\label{f:3}\n\\end{figure}\n\n\nThe marginal posterior PDFs, obtained by processing dataset 2, are\nshown in Fig.\\ref{f:2}. The number of  sensor measurement locations in this case\nwas $M=25$. The initial random sample $\\{\\mbox{\\boldmath$\\theta$}_n\\}_{1\\leq n \\leq N}$\nwas created in the same manner as for the case of dataset 1. From\nFig.\\ref{f:2}.(a) we can observe that the posterior PDF\n$p(x_0,y_0|\\mathbf{b})$ indicates fairly accurately the true source\nlocation marked by an asterisk at coordinates $(125.6,436.6)$.\n\n\n\nFinally, the marginal posterior PDFs, obtained by processing dataset\n3, are shown in Fig.\\ref{f:3}. The number of  sensor locations in\nthis case was $M=27$. The initial random sample $\\{\\mbox{\\boldmath$\\theta$}_n\\}_{1\\leq n\n\\leq N}$ was created in the same manner as for the case of dataset\n1. From Fig.\\ref{f:3}.(a) we can observe that the support of the\nmarginal posterior PDF $p(x_0,y_0|\\mathbf{b})$ indeed contains the true\nsource location at coordinates $(31.2,-453.2)$. However, the\nconditions were such (the placement of sensor, the wind speed) that\nsome ambiguity in the source location remains.  The resulting\nposterior PDF is bi-modal (two buildings contain scattered red\ndots), suggesting that the source must be located in one of them.\n\n\n\n\n\n\\section{Conclusions}\n\\label{s:6}\n\nThe paper proposed a simple Bayesian estimation algorithm for\nlocalisation of a continuous source of biochemical agent dispersing in the\natmosphere, using measurements collected at multiple locations by a single moving binary sensor whose detection threshold is unknown. The algorithm would also be applicable to  a single snapshot of the measurements from a network of identical binary sensors. The sensor detection threshold may be unknown because it may have been drifted due to temperature, humidity, ageing, etc. Another possible scenario where the detection threshold of a binary sensor  may be unknown is when a human, rather than a device, detects an odour at some locations but not  others. In this scenario, the person can easily make the binary measurements of ``detection'' or ``non-detection'' without knowing the exact detection threshold in terms of ppm or g\/m$^3$ of the detected material. To enable source localisation in such scenarios, in our algorithm, we treat the source release rate, as well as the binary sensor threshold, as being unknown. Under these conditions, the algorithm can not estimate the absolute value of the source release rate and is only able to estimate the release rate normalised by the unknown sensor threshold. However, the algorithm can correctly estimate the location of the biochemical source.  The performance of the\nalgorithm is demonstrated using three experimental datasets\ncollected in a semi-urban environment. In all three cases, the\nposterior density function included the true source location,\nthereby validating the proposed algorithm. Future work will consider\nintroducing the uncertainty in the mean wind direction and more\ndetailed dispersion models for urban environments.\n\n\\section*{References}\n\\bibliographystyle{elsarticle-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\nWhile the assumption of perfect lattice-translation symmetry facilitates the theoretical treatment of solid state systems, it is inevitably broken in reality due to the presence of impurities or defects in the crystal and the fact that samples are of finite size. In the context of superconductivity, the study of impurities has a long history \\cite{Review1Impurities}. The classic result \\cite{AT,ATAG1,ATAG2,BWState,GolubovMazin} that the impact of non-magnetic disorder on the superconducting transition temperature $T_c$ crucially depends on the symmetry of the order parameter already shows that disorder can be seen as a blessing in disguise: It allows to obtain information about the symmetries of the order parameter, although this approach requires care as has become clear in recent years\\cite{SpinOrbitLocking,PhysRevB.90.134517,PhysRevB.91.054501,PhysRevB.92.014518}. While these studies assume a homogeneous superconducting order parameter, the self-consistent solution of the gap equation of an $s$-wave superconductor shows Friedel-like oscillations, with wavevector set by the Fermi momentum $k_F$, in the vicinity of a non-magnetic impurity as was first shown in \\refcite{FetterOldImpurity}. \nThe associated correction to the reduction of the transition temperature, suppressed by a factor $\\propto T_c\/E_F$ ($E_F$ denotes the Fermi energy), was analyzed in \\refscite{Tsuzuki,HeinrichMagnetic,Kuemmel,Schlottmann} for $s$-wave superconductors in the presence of magnetic impurities. While formally of the order $T_c\/E_F$, these Friedel oscillations of the superconducting gap can be numerically sizable in low-dimensional systems\\cite{tanaka1,tanaka2}. Unconventional even-parity, spin-singlet states were analyzed in \\refscite{WalkerOsc,FranzTc}. Similarly to impurities, $s$-wave superconductors are also known to exhibit Friedel-like oscillations at the surfaces of the system \\cite{GiamarchiSurfaceSC,TanakaPRB,tanaka1}.\n\nThe physics of inhomogeneities due to sample edges has received enormous attention in recent years due to the emergence of topological insulators and superconductors\\cite{HasanKaneReview,AliceaReview,Bernevig}: The celebrated ``bulk-boundary correspondence'' relates topological invariants, which are discrete functions of the Hamiltonian describing the bulk of the system, to the presence of zero-energy modes localized at the edge of the system. In the case of topological superconductors, the charge-conjugation symmetry, that naturally emerges in the Bogoliubov-de Gennes (BdG) formalism, leads to a reality constraint on the wave function of the associated zero-energy modes, which are thus referred to as Majorana bound states (MBSs). \nWhile the bulk-boundary correspondence has been exhaustively analyzed for noninteracting, quadratic, Hamiltonians, it requires further theoretical investigation in the presence of interactions\\cite{BudichReview}.   \nOn the experimental side, the unambiguous detection and controlled manipulation of MBSs is the current central open question which is not only interesting in its own right but also constitutes the first step towards topological quantum computation\\cite{AliceaReview,Bernevig} and provides information about the underlying pairing mechanism \\cite{ScheurerOxide,ScheurerSymmetryMechanism} of superconductivity. One of the most promising routes towards clear identification of MBSs and, potentially, qubit readout\\cite{AliceaReview} is based on the MBS-mediated Josephson effect \\cite{Kitaev,TRSMajoranaJosephson}.\n\nIn this paper, we show that the Friedel-like oscillations of the superconducting order parameter at the edge of one-dimensional (1D) and two-dimensional (2D) systems and in the vicinity of both magnetic and non-magnetic inhomogeneities in 1D are only present in the even-parity channel but not for an odd-parity order parameter. Consequently, the observation (the absence) of oscillations of the order parameter on the length scale $1\/k_F$ indicates that the order parameter is even parity (odd parity). This also allows to gain information about the topology of superconductivity as a gapped system with odd-parity pairing realizes a time-reversal invariant topological superconductor given its Fermi surfaces enclose an odd number of time-reversal invariant momenta and a trivial phase in the case of even-parity pairing\\cite{FuBergProposal,Sato}. \nThus, we present another approach to utilize inhomogeneities in order to access both the symmetry and topology of the superconducting order parameter.\nPotential candidate systems for time-reversal invariant topological superconductivity include the doped topological insulator Cu$_x$Be$_2$Se$_3$\\cite{FuBergProposal}, (quasi-)1D wires\\cite{InteractingWires,dwaveWire,AnotherWirePaper,BuddichWire}, bilayer Rashba systems\\cite{PhysRevLett.108.147003}, and oxide heterostructures \\cite{ScheurerOxide}. The superfluid analogue\\cite{BWState,BWStateMajorana} of this phase is believed to be realized in the B phase of $^3$He. \n\n\n\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.8\\linewidth]{setups22.pdf}\n\\caption{Schematic of Josephson junction between an $s$-wave superconductor (blue) and an interaction-induced topological superconductor (red) for the detection of MBSs\\cite{Kitaev,TRSMajoranaJosephson}. In addition, we show a JSTM\\cite{MartinProp,FirstExperimentJSTM,FabricationOfTips,Hamidian:2016aa,Yazdani,SuperconductingSTM} tip that allows to probe the spatial texture of the order parameter and of the MBS wave function as we discuss in this paper.}\n\\label{BasicSetup}\n\\end{center}\n\\end{figure}\n\n\nIn addition, we provide an example of a modification of the conventional bulk-boundary correspondence due to competing interactions that can significantly influence Josephson tunneling experiments aimed at identifying MBSs.\nTo this end, we focus on ``intrinsic'' topological superconductors, i.e., systems where the nontrivial topology arises spontaneously from the internal interactions (applies to  Refs.~\\onlinecite{InteractingWires,FuBergProposal,ScheurerOxide,BWState,PhysRevLett.108.147003}). For concreteness, we assume the interaction to be dominated by the $p$-wave channel, but  closely followed by the competing $s$-wave instability. By construction, this yields purely $p$-wave, time-reversal invariant topological superconductivity in the bulk; however, the broken inversion symmetry at the edge allows the competing $s$-wave order parameter to emerge locally and leads to a topologically trivial region with width of the order of the coherence length $\\xi$ between the topological superconductor and the vacuum. Consequently, the MBSs are not localized at the edge of the system, as expected from the bulk-boundary correspondence, but are ``pushed away'' from it by a length of the order of $\\xi$. While this seems like a minor modification at first sight, it can change the signal in the MBS-induced Josephson effect\\cite{TRSMajoranaJosephson} between the system and an $s$-wave superconductor, see \\figref{BasicSetup}, dramatically: \nDepending on parameters, the current-phase relation can more closely resemble that of a conventional Josephson junction based on two topologically trivial $s$-wave superconductors despite the presence of MBSs and a topologically nontrivial bulk phase.\n\nWe finally discuss Josephson scanning tunneling microscopy\\cite{MartinProp,FirstExperimentJSTM,FabricationOfTips,Hamidian:2016aa,Yazdani,SuperconductingSTM} (JSTM) as an experimental tool to image the spatial variations of the superconducting order parameter in the vicinity of defects and at the edge, see \\figref{BasicSetup}. We generalize the analysis of \\refcite{SuperconductingSTM} to finite voltage drops across the junction and also take into account the contribution arising from MBSs. Note that the latter is not captured by the Green's function formalism used in \\refcite{SuperconductingSTM} due to the MBS-induced ground state degeneracy. We also analyze the expected JSTM signature of the MBSs that are pushed further into the bulk of the system due to the presence of the competing topologically trivial instability at the edge of the system. We demonstrate that JSTM is a very powerful tool not only to identify the presence of MBSs but also to reveal the spatial profile of the MBS wave function -- even in the presence of competing interactions.\n\n\n\\section{Model and formalism}\n\\label{ModelIntro}\nIn this section, we introduce the model we use in this paper to analyze the behavior of superconductivity in the vicinity of inhomogeneities and describe our procedure for solving the corresponding gap equations.\n\n\\subsection{Hamiltonian for inhomogeneous superconductors}\nIn this work, we focus on a single-band lattice Hamiltonian of the form \n\\begin{equation}\n\\hat{\\mathcal{H}} = \\hat{\\mathcal{H}}_{\\text{0}} + \\hat{\\mathcal{H}}_{\\text{imp}} + \\hat{\\mathcal{H}}_{\\text{pairing}}\\text{.} \\label{Hamiltonian}\n\\end{equation}\nThe kinetic contributions are represented by \n\\begin{equation}\n\\hat{\\mathcal{H}}_{\\text{0}} = -t\\sum_{\\langle ij\\rangle}\\sum_\\sigma c_{i\\sigma}^{\\dagger} c^{\\phantom{\\dagger}}_{j\\sigma} - \\mu \\sum_{i}\\sum_\\sigma c_{i\\sigma}^{\\dagger}c^{\\phantom{\\dagger}}_{i\\sigma} \\text{,} \\label{Hopping}\n\\end{equation}\nwhere $c_{i\\sigma}^{(\\dagger)}$ annihilates (creates) an electron of spin \\mbox{$\\sigma=\\{\\uparrow,\\downarrow\\}$} at site $i$. Here, we only consider hopping between nearest-neighbours $\\langle ij\\rangle$ with hopping amplitude $t$. The boundary conditions of the system will be chosen as \\textit{open} (OBC) or \\textit{periodic} (PBC), depending on the specific questions at hand. The chemical potential is given by $\\mu$. The second term of the Hamiltonian (\\ref{Hamiltonian}),\n\\begin{equation}\n\\hat{\\mathcal{H}}_{\\text{imp}} = \\sum_{i} \\sum_{\\sigma\\sigma'} c_{i\\sigma}^{\\dagger} U_{i}^{\\sigma\\sigma'} c^{\\phantom{\\dagger}}_{i\\sigma'}, \\label{ImpurityHam}\n\\end{equation}\nallows for magnetic and non-magnetic defects. We take the defects to be localized at a single site $i=i_0$, $U_{i}^{\\sigma\\sigma'} = u_0 (\\hat{\\sigma}^{m})_{\\sigma\\sigma'} \\delta_{i,i_0}$ using $u_0$ to denote the strength of the defect. The appropriate spin-structure of the impurities is chosen as $\\hat{\\sigma}^{m} = \\hat{\\sigma}^{z}$ and $\\hat{\\sigma}^{m} = \\hat{\\sigma}^{0}$ for magnetic and non-magnetic impurities, respectively. The pairing Hamiltonian\n\\begin{align}\n\\hat{\\mathcal{H}}_{\\text{pairing}} = \\sum_{\\eta=s,t} V^{\\eta}& \\sum_{i} \\biggl(\\sum_{\\delta\\sigma\\sigma'} c_{i+\\delta,\\sigma}^{\\dagger}\\,\\chi_{\\sigma'\\sigma}^{\\eta *} (\\delta)c_{i,\\sigma'}^{\\dagger} \\biggr)\\nonumber \\\\\n& \\times \\biggl(\\sum_{\\tilde{\\delta}\\tilde{\\sigma}\\tilde{\\sigma}'} c_{i,\\tilde{\\sigma}'}\\,\\chi_{\\tilde{\\sigma}'\\tilde{\\sigma}}^{\\eta} (\\tilde{\\delta})c_{i+\\tilde{\\delta},\\tilde{\\sigma}}\\biggr) \n\\end{align}\nallows for superconducting pairing in the appropriate spin channels corresponding to singlet $s$-wave ($\\eta = s$) and triplet $p$-wave ($\\eta = t$) with coupling constants $V^{\\eta}$. The spin- and spatial-structure leading to \\textit{on-site} $s$-wave ($\\delta=0$) and \\textit{nearest-neighbour} $p$-wave ($\\delta=\\pm 1$) pairing are encoded in the spin matrices $\\chi_{\\sigma\\sigma'}^{\\eta}(\\delta)$. The $s$-wave pairing interaction shall be taken to be momentum independent, i.e.~of the form $\\chi_{\\sigma\\sigma'}^{s} = \\Delta^{s} (i\\hat{\\sigma}^{y})_{\\sigma\\sigma'}$ and, unless otherwise stated, the $p$-wave pairing interaction in momentum space is chosen as $\\chi^{t} \\propto \\vec{d}(\\vec{k})\\cdot \\vec{\\sigma}i\\hat{\\sigma}^{y}$, with triplet vector $\\vec{d}(\\vec{k}) \\propto (-\\sin k_y, \\sin k_x, 0)^{T}$.\n\nWe treat the interacting problem in the mean-field approximation focusing on the Cooper channel and introduce\n\\begin{equation}\\label{eq:Delta}\n\\Delta_{i}^{\\eta} = V^{\\eta} \\sum_{\\delta\\sigma\\sigma'} \\langle c_{i,\\sigma'}c_{i+\\delta,\\sigma}\\rangle \\chi_{\\sigma'\\sigma}^{\\eta}(\\delta)\n\\end{equation}\nas the gap function.\nWe bring the resulting mean-field Hamiltonian into the standard BdG form\\cite{bogoliubov}\n\\begin{equation}\\label{eq:BdG}\n\\hat{\\mathcal{H}}_{\\text{MF}} = \\frac{1}{2}\\sum_{ij}\\sum_{\\sigma\\sigma'} \\hat{\\Psi}_{i\\sigma}^{\\dagger} \\underbrace{\\begin{pmatrix}\n\\hat{h}_{ij}^{\\sigma\\sigma'} & \\hat{\\Delta}_{ij}^{\\sigma\\sigma'} \\\\ \\bigl(\\hat{\\Delta}_{ij}^{\\sigma\\sigma'}\\bigr)^{\\dagger} & -\\left(\\hat{h}_{ij}^{\\sigma\\sigma'}\\right)^{T}\n\\end{pmatrix}}_{\\equiv \\hat{\\mathcal{H}}_{\\text{BdG}}} \\hat{\\Psi}_{j\\sigma'} \\text{,}\n\\end{equation}\nwith BdG Hamiltonian $\\hat{\\mathcal{H}}_{\\text{BdG}}$ and Nambu spinor \\mbox{$\\hat{\\Psi}_{i\\sigma} = \\bigl( c_{i\\sigma}, c_{i\\sigma}^{\\dagger}\\bigr)^T$}. All single-particle contributions, $\\hat{\\mathcal{H}}_{\\text{0}}$ and $\\hat{\\mathcal{H}}_{\\text{imp}}$, are incorporated into the single-particle Hamiltonian $\\hat{h}$, while $\\hat{\\Delta}$ results from $\\hat{\\mathcal{H}}_{\\text{pairing}}$. \n\\subsection{Gap equation}\nThe problem is now reduced to solving the BdG equations\n\\begin{equation}\n\\hat{\\mathcal{H}}_{\\text{BdG}} \\, \\phi_E = E \\, \\phi_E \\text{,}\n\\end{equation}\nwith eigenmodes $\\phi_E$. The particle- ($\\phi^{(e)}$) and hole-amplitudes ($\\phi^{(h)}$) are the Nambu-components of the eigenvectors $\\phi_E = (\\phi_{E}^{(e)},\\phi_{E}^{(h)})^{T}$. The intrinsic redundancy of the BdG Hamiltonian, having effectively doubled the degrees of freedom, leads to the charge-conjugation symmetry\n\\begin{equation}\n\\hat{\\Xi} \\, \\hat{\\mathcal{H}}_{\\text{BdG}}\\, \\hat{\\Xi}^{\\dagger} = -\\hat{\\mathcal{H}}_{\\text{BdG}}\\text{,}\n\\end{equation}\nwhere $\\hat{\\Xi}= \\hat{\\tau}^{x} C$ is the charge conjugation operator with the first Pauli matrix $\\hat{\\tau}^{x}$ in particle-hole space and the operator $C$ for complex conjugation. Due to the charge-conjugation symmetry of the BdG equations, a solution $\\phi_E$ with energy $E$ is related to a solution $\\phi_{-E}$ with energy $-E$ by \n\\begin{equation}\n\\phi_{-E} = \\hat{\\Xi}\\phi_{E}.\n\\end{equation}\nBy performing a mode expansion of the spinor\n\\begin{align}\\label{eq:spinorexp}\n\\hat{\\Psi}_{\\alpha} =& \\begin{pmatrix}\nc_{\\alpha}\\\\ c_{\\alpha}^{\\dagger}\n\\end{pmatrix} = \\sum_{E} (\\phi_E)_{\\alpha} a_{E}\\nonumber \\\\ =& \\sum_{E>0} \\biggl((\\phi_E)_{\\alpha} a_{E}  + \\underbrace{(\\phi_{-E})_{\\alpha} a_{-E}}_{=\\left(\\hat{\\Xi}\\phi_E\\right)_{\\alpha}a_{E}^{\\dagger}}\\biggr) \\text{,}\n\\end{align}\nwhere $\\alpha = \\{i,\\sigma\\}$ is a shorthand for site- and spin-indices, we diagonalize the BdG Hamiltonian. The equivalence of creating a hole and annihilating a particle, i.e. $a_{E} = a_{-E}^{\\dagger}$, has been used in \\equref{eq:spinorexp} to reduce the expansion to a sum over positive eigenenergies. Using the mode expansion to evaluate the average $\\langle c_{\\alpha'}c_{\\alpha}\\rangle$ in \\equref{eq:Delta} and introducing an energy cut-off $\\epsilon_c$  leads to the inhomogeneous gap equation\n\\begin{equation}\\label{eq:selfcongap}\n\\Delta_{i}^{\\eta} = V^{\\eta} \\sum_{\\delta\\sigma\\sigma'} \\sum_{0<E<\\epsilon_c} (\\phi_E)_{i\\sigma'}^{(e)} (\\phi_{E}^{*})_{i+\\delta,\\sigma}^{(h)} \\,\\, \\chi_{\\sigma'\\sigma}^{\\eta}(\\delta) [1 - 2f(E)] \\text{.} \n\\end{equation}\nAt $T=0$, which we shall be considering for the remainder of this paper, the Fermi-distribution function \\mbox{$f(E) = [e^{\\beta E} + 1]^{-1}$} vanishes, due to the lack of thermal excitations at zero temperature.\nThe results shown below are obtained by solving \\equref{eq:selfcongap} via iteration, i.e., iteratively inserting an approximation of $\\Delta_{i}^{\\eta}$ into the right-hand side of the equation and recalculating its value from \\equref{eq:selfcongap} until convergence is achieved. The convergence criterion we use is demanding that the relative change in $\\Delta_{i}^{\\eta}$ from one iteration to the next one be smaller than $10^{-4}$.\n\n\n\\section{One spatial dimension}\n\\label{1DResults}\nWe start our discussion of the numerical solution of the superconducting self-consistency equations (\\ref{eq:selfcongap}) by focusing on 1D systems since the oscillatory features in the order parameter are most pronounced and the results can be more conveniently illustrated in one spatial dimension.\n\n\\subsection{Competition of singlet and triplet order parameters at edges}\n\\label{FriedelOscillations}\nIn the presence of inversion symmetry in the normal state, the superconducting order parameter that assumes finite values below the superconducting transition temperature must be either pure singlet or pure triplet\\cite{sigristueda}\nHowever, once inversion is broken in the normal state, even-parity (singlet) and odd-parity (triplet) pairing can and, in general, will mix. \n\nInversion symmetry can be broken globally by the crystal structure itself or by the environment of a low-dimensional system (e.g., the substrate of a 2D thin film). Alternatively, it can be broken locally at edges and inhomogeneities of the sample. In this work, we will entirely focus on the local breaking of inversion symmetry. As a first example, let us consider a system with dominant triplet pairing in the bulk and allow for a non-vanishing singlet component ($V^t > V^s \\neq 0$). The spatial texture of the superconducting order parameter at one of the edges of the system, as found from solving the superconducting gap equations numerically, is shown in \\figref{fig:CompetingOP}. The local structure of the $p$-wave order parameter $\\Delta^t$ exhibits the expected behavior: a constant bulk value and a decrease towards the boundary on the scale of its coherence length $\\xi_t$. Furthermore, a non-vanishing singlet component is present in the vicinity of the boundary, due to the locally broken inversion symmetry explained above. Note, we do not consider the possibility of odd-frequency pairing near the edge\\cite{Tanaka}. For the model considered here, with two almost degenerate even frequency order parameters, this seems to be a natural assumption.\n\nThe most prominent feature of the local structure of the $s$-wave order parameter are the pronounced \\textit{Friedel oscillations}\\cite{WalkerOsc} that are entirely absent in the $p$-wave component. These oscillations are not accessible to the quasi-classical approaches, such as the Eilenberger equations\\cite{eilenberger}, due to the averaging over energies, leaving the coherence length as the only relevant length scale. The additional shorter oscillation period resolved by our microscopic formalism is set by $k_F$ and is of the order of the lattice spacing $a$. The larger oscillation period, intermediate between $k_F$ and the coherence length $\\xi_s$ of the $s$-wave state, is due to the finite energy cut-off $\\epsilon_c$ and disappears for larger $\\epsilon_c$ or as we move away from half-filling.\n\n\\begin{figure}[tb]\n\\includegraphics[width=\\linewidth]{CompetingOP.pdf}\n\\caption{Spatial structure of competing $s$-wave (blue) and $p$-wave (red) order parameters at a boundary ($x=0$) for a system of $L=3000$ sites at half-filling ($\\mu=0$) for coupling constants $V^{s}=0.36$, $V^{t}=0.38$. The order parameters are normalized to the bulk value $\\Delta_{0}^{t}$ of the $p$-wave pairing. The order parameters at the other edge at $x=L$ behave in the same fashion.}\n\\label{fig:CompetingOP}\n\\end{figure}\n\n\\begin{figure}[tb]\n\\includegraphics[width=\\linewidth]{MBSDist.pdf}\n\\caption{MBS wave function at the left edge ($x=0$) of a spinful superconductor with (a) pure $p$-wave pairing and (b) $p$-wave pairing that dominates in the bulk but competes with a subleading $s$-wave instability at the boundary. The wave functions of the MBS of the spinless Kitaev chain with homogeneous $p$-wave order parameter and spatially varying $p$-wave order parameter are shown and compared to $\\text{const.}\\times e^{-x\/\\xi_t}$ in (c) and (d), respectively.}\n\\label{fig:MBSComp}\n\\end{figure}\n\n\n\\subsection{Consequences for MBSs}\n\\label{ImpactOnMBSsWF}\nFor the simple band structure assumed, with only two Fermi points, see \\equref{Hopping}, the system is a topological class-DIII superconductor for triplet pairing and in a topologically trivial phase for singlet superconductivity\\cite{FuBergProposal,Sato,QiInvar}. In the case without competing singlet pairing ($V_s = 0$), we thus obtain (Kramers pairs of) MBSs localized at the edges of the system, see \\figref{fig:MBSComp}(a).\n\nDespite being irrelevant for the topological properties of the bulk phase, the presence of a competing singlet instability has a prominent influence on the location of the MBS wave function $\\phi_{\\text{maj}}$.\nFor competing order parameters, $V^t > V^s \\neq 0$, its position is shifted from the boundary towards the bulk on a scale of the coherence length $\\xi_t = 1\/\\Delta^t$ as can be seen in \\figref{fig:MBSComp}(b). This shift can be explained by the local structure of the multi-component order parameters (see \\figref{fig:CompetingOP}): Due to the emergence of an additional, topologically trivial, region of predominantly singlet pairing close to the boundary, the transition point from the topologically non-trivial to the topologically trivial region now constitutes the new boundary at which the MBSs arise. Thus, the wave function is shifted into the bulk on a scale set by the two respective coherence lengths, $\\xi_s$ and $\\xi_t$. \n\n\nWe note that the phase competition between singlet (trivial) and triplet (topological) superconductivity and the associated shift of the MBSs cannot occur in spinless models, such as the Kitaev chain, as the lack of spin degrees of freedom only allows for $p$-wave pairing. The sole modification as compared to just taking spatially constant $p$-wave pairing is a slight deformation of the MBS wave function as can be seen in \\figref{fig:MBSComp}(c) and (d):\nFor constant $p$-wave pairing, \\figref{fig:MBSComp}(c), the wave function of a MBS decays exponentially from the boundary towards the bulk. The local suppression of the $p$-wave order parameter in the self-consistent treatement of a spinless system with $p$-wave pairing interaction, \\figref{fig:MBSComp}(d), modifies the exponential functional form of the wave function close to the boundary. The maximum, however, is still at the boundary.\nA similar effect can be seen in the spatial form of the MBS wave function in the spinful case, \\figref{fig:MBSComp}(a), without competing $s$-wave pairing.\n\n\n\n\\subsection{Behavior at impurities}\\label{BehaviorAroundImpur}\nIn addition to the suppression of triplet pairing and the emergence of a local singlet phase in a spinful superconductor, we have also identified significant Friedel oscillations in the singlet channel in the vicinity of the edge (see \\figref{fig:CompetingOP}), which are absent in the triplet component. As one might expect, very similar behavior can be found in the vicinity of impurities as we will discuss next.\n\n\nTo analyze impurities, we impose PBC on our system and separately consider $s$-wave and $p$-wave pairing in presence of either a single non-magnetic ($\\hat{U}\\propto \\hat{\\sigma}^0$) and magnetic ($\\hat{U}\\propto \\hat{\\sigma}^z$) defect. Note that Friedel oscillations in the $s$-wave singlet order parameter in the presence of a single non-magnetic defect have already been studied by \\citet{WalkerOsc}. We will extend their analysis by including $p$-wave triplet pairing and by studying magnetic impurities.\n\nFor a non-magnetic impurity the local structure of the $s$-wave order parameter (see \\figref{fig:1DIMP}(a)) exhibits pronounced order-parameter-enhancing Friedel oscillations close to the impurity site, similar to the oscillations present close to a boundary, as considered above. The oscillations arise in the vicinity of the defect and decay towards the bulk value on the scale of the coherence length $\\xi_s$, while the oscillation period is determined by $k_F$.\n\n\\begin{figure}[tb]\n\\includegraphics[scale=0.15]{ImpuritiesBeh.pdf}\n\\caption{Spatial structure of the (a) pure $s$-wave singlet ($V^s = 0.36$) and (b) pure $p$-wave triplet ($V^t=0.38$) order parameter normalized to the respective bulk value $\\Delta_0^{\\eta}$, in the region close to a non-magnetic impurity at site $i_0=1500a$ for a system with $L=3000$ sites, chemical potential $\\mu=0$, coherence length $\\xi_{\\eta} = 1\/\\Delta_0$, impurity strength $u_0=0.5t$, and periodic boundary conditions. Part (c) and (d) show the analogous plots for a magnetic impurity.}\n\\label{fig:1DIMP}\n\\end{figure}\n\n\n\nAs in the case near the sample edge, the $p$-wave order parameter does not possess pronounced Friedel oscillations near defects, but exhibits a suppression of the superconducting gap close to the defect site (see \\figref{fig:1DIMP}(b)). The order parameter recovers to its bulk value over the coherence length $\\xi_t$. \n\nVery similar behavior is found for $s$-wave pairing in the presence of a magnetic impurity.\nAs can be seen in \\figref{fig:1DIMP}(c), the overall behavior is governed by a dominant suppression close to the impurity site, recovering the bulk value over the coherence length $\\xi_s$. Within the region of suppression, $\\Delta^s$ exhibits minor oscillatory features, which are qualitatively distinct from the Friedel oscillations for the non-magnetic defect; the oscillations in the vicinity of a magnetic defect do not lead to a local enhancement of the $s$-wave gap as compared to its bulk value. \n\nFrom the local structure of the $p$-wave order parameter in \\figref{fig:1DIMP}(d) it is evident that no oscillations are present. Similar to the non-magnetic defect a magnetic defect leads to a suppression of the gap in the vicinity of the impurity, which recovers on the scale of $\\xi_t$. Comparing the magnitude of the suppression for a magnetic defect for the two respective pairing symmetries we find a stronger suppression of the $s$-wave gap, in accordance with the greater vulnerability of the singlet $s$-wave superconductor to magnetic impurities.\n\n\n\n\n\\section{Analytical understanding of Friedel oscillations}\n\\label{sec:origin}\nIn this section, we will derive an analytical expression for the impact of an isolated impurity on the superconducting order parameter in leading order in the impurity strength. From the structure and symmetry properties of this expression, the numerical observation of Friedel oscillations in the spatial structure of the singlet $s$-wave order parameter and the lack thereof in the $p$-wave triplet component are readily seen.\n\n\n\\subsection{Perturbative expression for spatial variations of order parameter}\nWe start by transforming the Hamiltonian (\\ref{Hamiltonian}) with a single impurity localized at the origin, $U_i^{\\sigma,\\sigma'} = \\delta_{i,0}U_{\\sigma,\\sigma'}$ in \\equref{ImpurityHam}, to momentum space,\n\\begin{align}\\begin{split}\n\\hat{\\mathcal{H}}_{\\text{0}} &= \\sum_{\\vec{k},\\sigma} \\epsilon_{\\vec{k}} c_{\\vec{k}\\sigma}^{\\dagger}  c_{\\vec{k}\\sigma}^{{\\phantom{\\dagger}}}, \\\\\n\\hat{\\mathcal{H}}_{\\text{imp}} &= \\sum_{\\vec{k},\\vec{k}'}\\sum_{\\sigma,\\sigma'} c_{\\vec{k}\\sigma}^{\\dagger} U_{\\sigma\\sigma'} c_{\\vec{k}'\\sigma'}^{{\\phantom{\\dagger}}},\\\\\n\\hat{\\mathcal{H}}_{\\text{pairing}}&=V \\sum_{\\vec{k},\\vec{k}',\\vec{q}}\\sum_{\\sigma,\\sigma',\\tilde{\\sigma},\\tilde{\\sigma}'} c_{\\vec{k}+\\frac{\\vec{q}}{2}\\sigma}^{\\dagger} \\chi_{\\sigma\\sigma'}^{\\dagger}(\\vec{k})c_{-\\vec{k}+\\frac{\\vec{q}}{2}\\sigma'}^{\\dagger}\\\\ &\\qquad\\quad \\times c_{-\\vec{k}'+\\frac{\\vec{q}}{2}\\tilde{\\sigma}}\\chi_{\\tilde{\\sigma}\\tilde{\\sigma}'}(\\vec{k}')c_{\\vec{k}'+\\frac{\\vec{q}}{2}\\tilde{\\sigma}'}\n\\end{split}\\end{align}\nwith $c_{\\vec{k}\\sigma}$ ($c_{\\vec{k}\\sigma}^{\\dagger}$) denoting the fermionic annihilation (creation) operators in momentum space and single-particle dispersion $\\epsilon_{\\vec{k}}$.  The pairing symmetries for singlet and triplet superconductivity are encoded in the momentum- and spin-dependent matrix $\\chi(\\vec{k})$, with relative and center-of-mass momenta denoted by $\\vec{k}$ and $\\vec{q}$, respectively.\n\nA mean-field decomposition in the Cooper-channel yields\n\\begin{align}\\label{eq:Hmfor}\\begin{split}\n&\\hat{\\mathcal{H}}_{\\text{MF}} = \\sum_{\\vec{k},\\vec{k}'}\\sum_{\\sigma,\\sigma'} c_{\\vec{k}\\sigma}^{\\dagger} h_{\\vec{k}\\sigma,\\vec{k}'\\sigma'} c_{\\vec{k}'\\sigma'}^{{\\phantom{\\dagger}}} \\\\ &+  \\sum_{\\vec{k},\\vec{q}}\\left( \\Delta(\\vec{q}) c_{\\vec{k}+\\frac{\\vec{q}}{2}\\sigma}^{\\dagger} \\chi_{\\sigma\\sigma'}^{\\dagger} (\\vec{k})c^{\\dagger}_{-\\vec{k}+\\frac{\\vec{q}}{2}\\sigma'} + \\text{H.c.}\\right), \\\\\n\\\\\n\\\\\n\\end{split}\\end{align}\nwhere we have defined $h_{\\vec{k}\\sigma,\\vec{k}'\\sigma'}=\\delta_{\\vec{k},\\vec{k}'}\\delta_{\\sigma,\\sigma'}\\epsilon_{\\vec{k}}   +  U_{\\sigma\\sigma'}$ and the gap function\n\\begin{equation}\\label{eq:selfconk}\n\\Delta(\\vec{q}) = V\\sum_{\\vec{k}} \\chi_{\\sigma\\sigma'}(\\vec{k}) \\langle c_{-\\vec{k}+\\frac{\\vec{q}}{2}\\sigma}\\, c_{\\vec{k}+\\frac{\\vec{q}}{2}\\sigma'}\\rangle \\text{.}\n\\end{equation}\nIntroducing the spinor $\\hat{\\Psi}_{\\vec{k}\\sigma} = \\left( c_{\\vec{k}\\sigma},c_{-\\vec{k}\\sigma}^{\\dagger}\\right)^{T}$, we rewrite the Hamiltonian in \\equref{eq:Hmfor} in the standard BdG form:\n\n\\begin{widetext}\n\\begin{equation}\n\\hat{\\mathcal{H}}_{\\text{MF}} = \\sum_{\\vec{k},\\vec{k}'} \\hat{\\Psi}_{\\vec{k}\\sigma}^{\\dagger} \\hat{\\mathcal{H}}_{\\vec{k}\\sigma,\\vec{k}'\\sigma'}^{\\text{BdG}} \\hat{\\Psi}_{\\vec{k}'\\sigma'}, \\quad \\hat{\\mathcal{H}}_{\\vec{k}\\sigma,\\vec{k}'\\sigma'}^{\\text{BdG}}= \\begin{pmatrix}\nh_{\\vec{k}\\sigma,\\vec{k}'\\sigma'} & \\Delta(\\vec{k}-\\vec{k}')\\chi_{\\sigma\\sigma'}^{\\dagger}(\\frac{\\vec{k}+\\vec{k}'}{2})\\\\\n\\Delta^*(\\vec{k}'-\\vec{k})\\chi_{\\sigma\\sigma'}(\\frac{\\vec{k}+\\vec{k}'}{2}) & -h_{-\\vec{k}'\\sigma',-\\vec{k}\\sigma}\n\\end{pmatrix}\\text{.}\n\\end{equation}\n\nDenoting the Nambu Green's function by $\\mathcal{G}_{\\sigma\\sigma'}(\\omega_n;\\vec{k},\\vec{k}') = \\left(i\\omega_n-\\hat{\\mathcal{H}}^{\\text{BdG}}\\right)_{\\vec{k}\\sigma,\\vec{k}'\\sigma'}^{-1}$ with $\\omega_n$ representing fermionic Matsubara frequencies,  \\equref{eq:selfconk} is rewritten as\n\\begin{equation}\\label{eq:sc1}\n\\Delta(\\vec{q}) = VT\\sum_{\\omega_n} \\sum_{\\vec{k}} \\chi_{\\sigma\\sigma'}(\\vec{k})\\left( \\mathcal{G}_{\\sigma\\sigma'}(\\omega_n;\\vec{k}-\\vec{q}\/2,\\vec{k}+\\vec{q}\/2)\\right)_{1,2}.\n\\end{equation}\nAssuming the impurity to be weak,  we only keep terms linear in $\\hat{U}$ and neglect the spatial variations\\cite{WalkerOsc} of the gap on the right-hand side of \\equref{eq:sc1}, $\\Delta(\\vec{q}) \\rightarrow \\Delta_0$ with  $\\Delta_0$ denoting the solution of the homogeneous gap equation in the absence of the impurity. Furthermore, we linearize \\equref{eq:sc1} in the superconducting order parameter which is valid in the vicinity of $T_c$. \nWith these approximations, \\equref{eq:sc1} can be written as \n\\begin{equation}\\label{eq:Fq}\nF(\\vec{q}) = V\\Delta_0 T\\sum_{\\omega_n}\\sum_{\\vec{k}} \\frac{\\epsilon_{\\vec{k}+\\frac{\\vec{q}}{2}}}{(\\omega_{n}^{2}+\\epsilon_{\\vec{k}+\\frac{\\vec{q}}{2}}^{2})(\\omega_{n}^{2}+\\epsilon_{\\vec{k}-\\frac{\\vec{q}}{2}}^{2})} \\operatorname{tr}\\left[\\chi(\\vec{k})\\,\\hat{U}\\,\\chi^{*}(\\vec{k}-\\vec{q}\/2)\\right],\n\\end{equation}\n\\end{widetext}\nwith $F(\\vec{q}) \\equiv \\Delta (\\vec{q}) - \\delta_{\\vec{q},0}\\Delta_{0}$ describing the impurity-induced alteration of the superconducting gap and $\\text{tr}$ denoting the trace in spin-space.\nAs we will see below, the crucial qualitative differences in the behavior of the superconducting phases in the presence isolated impurities are captured by \\equref{eq:Fq}. We thus conclude that the Born approximation applied above suffices to describe the relevant physics.\n\n\\subsection{Non-magnetic impurities in 1D}\nFirst, we study the general expression in \\equref{eq:Fq} for a non-magnetic impurity of the form $\\hat{U}=u_0\\hat{\\sigma}^0$ and consider the cases of pure $s$-wave and pure $p$-wave pairing separately.\n\nWe take the singlet pairing interaction to be \\mbox{$\\chi_{s}(\\vec{k}) = i\\hat{\\sigma}^{y}$}, i.e.~momentum independent. With these approximations, the expression in \\equref{eq:Fq} reduces to\n\\begin{equation}\n F(q) = -V\\Delta_0 u_0 L^d T  \\sum_{\\omega_n} \\mathcal{F}(q,\\omega_n), \\label{FcalIntro}\n \\end{equation}\nwith system length $L$, $d=1$, and\n\\begin{equation}\n \\mathcal{F}(q,\\omega_n) = 2\\int\\limits_{-\\infty}^{\\infty} \\frac{dk}{2\\pi} \\frac{\\epsilon_{k+\\frac{q}{2}}}{(\\omega_{n}^{2}+\\epsilon_{k+\\frac{q}{2}}^2)(\\omega_{n}^{2}+\\epsilon_{k-\\frac{q}{2}}^2)}. \\label{1DNonmagswave}\n\\end{equation}\nThe graph in \\figref{AnalyticCurves}(a) shows plots of $\\mathcal{F}$ for different $\\omega_n$, taking $\\epsilon_{k}=\\frac{k}{2m}-\\mu$ for concreteness. Due to the symmetry of $\\mathcal{F}$, i.e. $\\mathcal{F}(q,\\omega_n) = \\mathcal{F}(-q,\\omega_n)$, only the region $q > 0$ is shown. A distinct peak around $q=2k_F$ is visible in the graph, which becomes sharper as $\\omega_n$ decreases. We expand \\equref{1DNonmagswave} around $q=2k_F +\\delta q$, with $|\\delta q| \\ll k_F$, and furthermore linearize $\\epsilon_{k\\pm q\/2} \\sim \\pm v_F (k\\pm \\delta q\/2)$ (expecting the main contribution to arise from the Fermi surface). This leads to\n\\begin{align}\\label{eq:Fs1d}\n\\mathcal{F}(q,\\omega_n) &\\sim \\int\\limits_{-\\infty}^{\\infty} \\frac{dk}{2\\pi} \\frac{v_F \\, \\delta q}{(\\omega_n^2 + v_F^2\\, (k+\\frac{\\delta q}{2})^2)(\\omega_n^2 + v_F^2 \\, (k-\\frac{\\delta q}{2})^2)}\\nonumber \\\\&= \\frac{1}{|\\omega_n|}\\frac{\\delta q}{(2\\omega_n)^2 + (v_F \\, \\delta q)^2}\n\\end{align}\nand the pole-structure of $\\mathcal{F}(q,\\omega_n)$ at $q=\\pm 2k_F \\pm i2\\omega_n\/v_F$ reproduces the oscillation period equaling $2k_F$ in the numerical results in \\secref{1DResults}.\n\n\n\\begin{figure}[t]\n\\centering \\includegraphics[width =\\linewidth]{AnalytCurves2.pdf}\n\\caption{Momentum dependence of $\\mathcal{F}$ at two different frequencies, $\\omega_n\/\\mu=0.1$ (blue solid line) and $\\omega_n\/\\mu=0.2$ (red solid line), for the four different cases of $s$-wave in 1D (a),  $p$-wave in 1D (b), $s$-wave in 2D (c), and $p$-wave in 2D (d). The dashed black line in (a) refers to the approximate expression in \\equref{eq:Fs1d}.}\n\\label{AnalyticCurves}\n\\end{figure}\n\n\n\nFor the case of $p$-wave pairing, the interaction is chosen to be $\\chi_{t}(k) = i\\hat{\\sigma}^{y}\\hat{\\sigma}^{y} k\/k_F = i\\hat{\\sigma}^{0} k\/k_F$. Substituting this into \\equref{eq:Fq} yields\n\\begin{align\nF(q) &= V \\Delta_0 u_0 L^d T \\sum_{\\omega_n} \\mathcal{F}(q,\\omega_n), \\quad \\text{with} \\label{eq:F1}\\\\ \\mathcal{F}(q,\\omega_n) &= 2\\int\\limits_{-\\infty}^{\\infty} \\frac{dk}{2\\pi} \\frac{\\epsilon_{k+\\frac{q}{2}} k(k-\\frac{q}{2})\/k_F^2 }{(\\omega_{n}^{2}+\\epsilon_{k+\\frac{q}{2}}^2)(\\omega_{n}^{2}+\\epsilon_{k-\\frac{q}{2}}^2)} \\label{eq:F2}.\n\\end{align}\nIn contrast to $s$-wave pairing, $\\mathcal{F}(q,\\omega_n)$ does not exhibit a peak around $q=2k_F$, as can be seen in \\figref{AnalyticCurves}(b). No oscillations are, thus, expected in accordance with our numerical results for the local structure of the $p$-wave superconducting order parameter. \nThis can be qualitatively understood as follows. As in the case of $s$-wave pairing, for $q\\simeq 2k_F$, the weight of the poles of the denominator of the integrand in \\equref{eq:F2} is greatest for $k\\simeq 0$. \nHowever, the $p$-wave nature of the pairing interaction, represented by the factor $k$ in the numerator, leads to a cancellation by a simultaneously vanishing numerator. This can be more explicitly seen by performing an approximation analogous to the $s$-wave case. Linearising the dispersion relation, i.e.~$\\epsilon_{k\\pm q\/2}=\\pm v_F (k\\pm \\delta q\/2)$, and expanding about $q=2k_F + \\delta q$, the expression given in \\equref{eq:F2} becomes independent of $\\delta q$, and, hence, does not exhibit the pole structure as in \\equref{eq:Fs1d}. This explains why no Friedel-like oscillations are found for $p$-wave pairing.\n\n\n\n\\subsection{Non-magnetic impurities in 2D}\nSo far, we have been focusing on the 1D case. As the analytical result in \\equref{eq:Fq} can be readily applied to higher dimensions as well, let us next discuss its predictions for Friedel-like oscillations around defects in two spatial dimensions. In \\secref{sec:roledim} below, the numerical solutions of the 2D inhomogeneous gap equations in the presence of impurities will be presented.\n\nFor a 2D system with $s$-wave pairing interaction (choosing $\\chi_{s}(\\vec{k}) = i\\hat{\\sigma}^{y}$ as before), we obtain\n\\begin{equation}\\label{eq:Fs2}\n\\mathcal{F}(\\vec{q},\\omega_n) = 2\\int \\frac{d^2 \\vec{k}}{(2\\pi)^2} \\frac{\\epsilon_{\\vec{k}+\\frac{\\vec{q}}{2}} }{(\\omega_n^2 + \\epsilon_{\\vec{k}+\\frac{\\vec{q}}{2}}^2)(\\omega_n^2 + \\epsilon_{\\vec{k}-\\frac{\\vec{q}}{2}}^2)} \n\\end{equation}\nin \\equref{FcalIntro} with $d=2$.\nDue to the rotational symmetry of the dispersion $\\epsilon_{\\vec{k}}=\\frac{\\vec{k}^2}{2m}-\\mu$, the expression only depends on $|\\vec{q}|$, $\\mathcal{F}(\\vec{q},\\omega_n) = \\mathcal{F}(|\\vec{q}|\\vec{e}_{x},\\omega_n)$. Based on the presence of a peak around $|\\vec{q}|\\simeq 2k_F$, see \\figref{AnalyticCurves}(c), we expect Friedel-like oscillations in the $s$-wave order parameter, similar to the 1D case.\n\n\nFor a general triplet vector, $\\chi_{t}(\\vec{k})= \\vec{d}(\\vec{k}) \\cdot \\vec{\\sigma} i\\hat{\\sigma}^{y}$, we find\n\\begin{equation}\n\\mathcal{F} = 2 \\int \\frac{d^2 \\vec{k}}{(2\\pi)^2}\\,\\,  \\frac{\\epsilon_{\\vec{k}+\\frac{\\vec{q}}{2}} \\, \\vec{d}(\\vec{k}) \\cdot \\vec{d}^*(\\vec{k}-\\frac{\\vec{q}}{2}) }{(\\omega_n^2 + \\epsilon_{\\vec{k}+\\frac{\\vec{q}}{2}}^2)(\\omega_n^2 + \\epsilon_{\\vec{k}-\\frac{\\vec{q}}{2}}^2)} \n\\end{equation}\nin \\equref{eq:F1} with $d=2$.\nFor concreteness, let us further assume a $d$-vector of the form $\\vec{d}(\\vec{k}) = (k_y,-k_x,0)^T\/k_F$, leading to\n\\begin{equation}\\label{eq:Fp2}\n\\mathcal{F} = 2\\int \\frac{d^2 \\vec{k}}{(2\\pi)^2} \\,\\,  \\frac{\\epsilon_{\\vec{k}+\\frac{\\vec{q}}{2}} \\,\\, \\vec{k}(\\vec{k}-\\frac{\\vec{q}}{2})\/k_F^2 }{(\\omega_n^2 + \\epsilon_{\\vec{k}+\\frac{\\vec{q}}{2}}^2)(\\omega_n^2 + \\epsilon_{\\vec{k}-\\frac{\\vec{q}}{2}}^2)} \\text{.}\n\\end{equation}\nAgain, $\\mathcal{F}(\\vec{q},\\omega_n)$ is rotationally symmetric, such that $\\mathcal{F}(\\vec{q},\\omega_n) = \\mathcal{F}(|\\vec{q}|\\vec{e}_{x},\\omega_n)$. The numerical evaluation of \\equref{eq:Fp2}, as shown in \\figref{AnalyticCurves}(d), does not exhibit a pronounced peak around $|\\vec{q}|\\simeq 2k_F$, but does show an enhancement of $\\mathcal{F}$ for $|\\vec{q}| < 2k_F$. This is due to the continuum of $\\vec{q}$-vectors connecting parts of the Fermi surface. \n\nConsequently, we expect the difference in the behavior of $s$- and $p$-wave order parameters around non-magnetic impurities to be smaller in higher dimensions than in 1D; in accordance with our numerical results presented in \\secref{sec:roledim} below, Friedel-like oscillations are expected in both $s$- and $p$-wave components in higher dimensions, whereas, in 1D, only the $s$-wave channel shows oscillatory behavior.\n\n\n\\subsection{Magnetic impurities}\\label{Magnetic1D2DAnal}\nLet us next analyze the predictions of our analytical expression in \\equref{eq:Fq} for the case of a magnetic impurity, i.e., for $\\hat{U} = \\vec{S}\\cdot \\vec{\\sigma}$ with some real vector $\\vec{S}$.\n\nInserting a general singlet-pairing function, $\\chi(\\vec{k})=i\\hat{\\sigma}^{y} f(\\vec{k})$ with $f(\\vec{k}) = f(-\\vec{k})$, the spin-trace in \\equref{eq:Fq} vanishes, leading to $F=0$. Consequently, there is no impact on the local superconducting order parameter within the leading-order perturbative treatment presented in this section. The suppression of the superconducting order parameter, including the minor oscillatory features, present in the local structure of the singlet $s$-wave (see \\figref{fig:1DIMP}(c)) are not captured by the analytical calculation presented here as a result of the approximations made to derive the expression in \\equref{eq:Fq}. \n\nTaking a general triplet term, $\\chi(\\vec{k}) = (\\vec{d}(\\vec{k})\\cdot \\vec{\\sigma}) i\\hat{\\sigma}^{y}$ with triplet vector satisfying $\\vec{d}(\\vec{k}) = -\\vec{d}(-\\vec{k})$, \\equref{eq:Fq} assumes the form\n\\begin{align}\\begin{split}\nF(\\vec{q}) &= 2i V\\Delta_0 T \\\\ &\\times \\sum_{\\omega_n}\\sum_{\\vec{k}} \\frac{\\vec{S}\\cdot \\left(\\vec{d}(\\vec{k}) \\times \\vec{d}^*(\\vec{k}-\\frac{\\vec{q}}{2})\\right) \\epsilon_{\\vec{k}+\\frac{\\vec{q}}{2}}}{(\\omega_{n}^{2}+\\epsilon_{\\vec{k}+\\frac{\\vec{q}}{2}}^{2})(\\omega_{n}^{2}+\\epsilon_{\\vec{k}-\\frac{\\vec{q}}{2}}^{2})}. \\label{TripletExprFq}\n\\end{split}\\end{align}\nInterestingly, the resulting leading order correction is purely imaginary for real-valued triplet vectors. This means that the phase of the order parameter changes in the vicinity of the impurity and, thus, induces a time-reversal-symmetry-breaking component in the superconducting order parameter. In the 1D case we have studied numerically in \\secref{BehaviorAroundImpur}, we did not obtain this phase as the triplet vector $\\vec{d}(\\vec{k})$ was colinear leading to $\\vec{d}(\\vec{k}) \\times \\vec{d}^*(\\vec{k}-\\frac{\\vec{q}}{2})=0$ in \\equref{TripletExprFq}. When investigating the 2D case numerically in the next section, we will indeed recover the emergence of a local imaginary part.\n\nWe finally note that there are crucial differences to the local phase shift by $\\pi$ in $s$-wave superconductors in the vicinity of magnetic impurities studied, e.g., in \\refscite{PhysRevB.95.104521,PhysRevB.92.064503,PhysRevB.93.165418}. First, the local phase shift in the $p$-wave superconductor is continuous and incommensurate with $\\pi$, thus breaking time-reversal-symmetry in the superconducting order parameter itself. Secondly, it occurs already at linear order in the impurity strength and as such does not require the impurity potential to exceed a certain strength.\n\n\n\n\n\n\\section{Numerical results for higher dimensions}\n\\label{sec:roledim}\nIn addition to the numerical treatment of 1D systems and the analytic approach in both dimensions, we follow up with a numerical examination of 2D square lattices with $L\\times L$ sites and the influence of inversion symmetry breaking at impurities and boundaries on the different superconducting order parameters. \n\n\\begin{figure}[tb]\n\\includegraphics[width=\\linewidth]{Ds2DOBC.jpg}\n\\caption{(a) Surface plot of $\\Delta^s$ for a system of $41\\times 41$ sites with OBC, coupling constant $V^s=0.5$, energy cut-off $\\epsilon_c=0.7t$  and chemical potential $\\mu=0t$. (b) $\\Delta^s$ at $y=L\/2$ along $x$.}\n\\label{fig:2DsOBC}\n\\end{figure}\n\nThe result of the self-consistent calculation for a 2D square lattice with OBC and pure $s$-wave pairing is depicted in \\figref{fig:2DsOBC}.\nSimilar to the one dimensional case, the order parameter exhibits oscillations along with an enhancement in the vicinity of the edges and at the corners, though the effect is weaker than in 1D.\n\\begin{figure}[tb]\n\\includegraphics[width=\\linewidth]{Dp2DOBC.jpg}\n\\caption{(a) $\\Delta^t_x$ and (b) $\\Delta^t_y$ for a system of $41\\times 41$ sites with OBC in $x$ and $y$, coupling constant $V^t=1.2$, energy cut-off $\\epsilon_c=1.1t$ and chemical potential $\\mu=0$.}\n\\label{fig:2DpOBC}\n\\end{figure}\n\nWe have also studied the behavior of 2D $p$-wave pairing in the vicinity of edges taking a bulk triplet vector of the form $\\vec{d} = (\\sin(k_y),-\\sin(k_x),0)^T$. In contrast to the $s$-wave case, the two components $\\Delta^t_x$ and $\\Delta^t_y$ of the triplet order parameter transforming as $k_y$ and $k_x$, respectively, do not exhibit a spatial oscillation, but a smooth decrease towards one boundary and a smooth increase towards the other, see \\figref{fig:2DpOBC}.\n\n\nThe influence of non-magnetic and magnetic defects on the spatial structure of the 2D $s$-wave order parameter closely resembles that of the 1D case (see \\figref{fig:2DsIMP}). A non-magnetic defect leads to enhanced oscillations close to the defect site, while a magnetic inhomogeneity leads to a dominant suppression. This behavior is in good accordance with our analytic treatment in \\secref{sec:origin}.\n\\begin{figure}[tb]\n\\includegraphics[width=\\linewidth]{DsIMP2.pdf}\n\\caption{Cross sections of $\\Delta^s$ for a system of $31\\times 31$ sites and PBC, coupling constant $V^s=0.5$, energy cut-off $\\epsilon_c=0.7t$ and chemical potential $\\mu=-0.3t$ with (a) non-magnetic and (b) magnetic impurities  of strength $u_0=0.25t$ located at $(L\/2,L\/2)$.}\n\\label{fig:2DsIMP}\n\\end{figure}\n\nA comparison of these results with the effect of non-magnetic and magnetic defects on the $p$-wave order parameter only yields minor differences (see \\figref{fig:DpIMPcut}). While a magnetic defect leads to a suppression of the order parameter and a local phase, in agreement with our results in \\secref{sec:origin}, a non-magnetic defect leads to an oscillatory enhancement of the two order parameter components close to the impurity. This limits the discernibility of $s$-wave and $p$-wave by defect-induced spatial variations. The only sharp qualitative difference in the behavior of $s$- and $p$-wave pairing in the vicinity of defects is that only the latter exhibits a local change of phase in the case of a magnetic impurity.\n\n\\begin{figure}[t]\n\\includegraphics[width=\\linewidth]{DpIMP2Phase.pdf}\n\\caption{Same as \\figref{fig:2DsIMP}, for triplet pairing using $V^t=0.7$ and energy cut-off $\\epsilon_c=1.1t$. In the case of the magnetic impurity in part (b), the order parameter becomes complex in the vicinity of the defect as anticipated by our analytical analysis in \\secref{Magnetic1D2DAnal}. For this reason, we show the absolute value $|\\Delta_x^t|$ and $\\Im[\\Delta_x^t]$ (inset) in part (b).}\n\\label{fig:DpIMPcut}\n\\end{figure}\n\n\\section{Detection by JSTM}\n\\label{sec:observe}\nAs a promising experimental tool to detect the spatial textures of competing superconducting order parameters and of MBS wave functions, we discuss JSTM\\cite{MartinProp,FirstExperimentJSTM,FabricationOfTips,Hamidian:2016aa,Yazdani,SuperconductingSTM} and compute the expected signatures in the Jospheson current. We extend previous theoretical work (see, in particular, \\refcite{SuperconductingSTM}) to non-vanishing voltage drops across the interface and to topological superconductors with MBSs.\n\nTo this end, we assume a Hamiltonian\n\\begin{equation}\n\\hat{\\mathcal{H}} = \\hat{\\mathcal{H}}_{sc} + \\hat{\\mathcal{H}}_{\\text{T}} + \\hat{\\mathcal{H}}_{\\text{tip}} \\text{,}\n\\end{equation}\nwith the subsystem Hamiltonians $\\hat{\\mathcal{H}}_{sc}$ and $\\hat{\\mathcal{H}}_{\\text{tip}}$ for the inhomogeneous superconductor of interest and the superconducting tip, respectively. Tunneling between the two subsystems is mediated by the tunneling Hamiltonian $\\hat{\\mathcal{H}}_{\\text{T}}$.\tThe Hamiltonian of the inhomogeneous superconductor is obtained from our self-consistent calculations, while the tip Hamiltonian $\\hat{\\mathcal{H}}_{\\text{tip}}=\\hat{\\mathcal{H}}_{\\text{tip}}^{n} + \\hat{\\mathcal{H}}_{\\text{tip}}^{sc}$, consisting of normal state and superconducting terms, is assumed to model an atomically sharp homogeneous superconducting tip.\nThe superconducting part reads as\n\\begin{equation}\n\\hat{\\mathcal{H}}_{\\text{tip}}^{\\text{sc}}= \\sum_{\\sigma\\sigma'} \\Delta_{\\text{tip}}^{\\sigma\\sigma'} d_{\\sigma}^{\\dagger}d_{\\sigma'}^{\\dagger} + \\text{H.c.}\\text{,}\n\\end{equation}\nwith operators $d_{\\sigma}^{(\\dagger)}$ that annihilate (create) an electron of spin $\\sigma$ in the tip, and $\\hat{\\mathcal{H}}_{\\text{tip}}^{n}$ is taken to describe a non-interacting, spin-diagonal, spectrum in the tip with density of state $\\rho_0$ at the Fermi level. The Hamiltonian\n\\begin{equation}\\label{eq:tun}\n\\hat{\\mathcal{H}}_{\\text{T}}= T_0 \\sum_{\\sigma} \\left( c_{i_0 \\sigma}^{\\dagger} d^{\\phantom{\\dagger}}_{\\sigma} + d_{\\sigma}^{\\dagger}c^{\\phantom{\\dagger}}_{i_0\\sigma} \\right) \\text{,}\n\\end{equation}\ndescribing tunneling (with amplitude $T_0$) between the site $i_0$ of the superconductor and the tip, is taken to preserve spin. \n\nThroughout this work, we take a tunneling amplitude of $T_0 = 0.01 t$ which corresponds to the weak-tunneling limit, $T_0\/t\\ll 1$. This allows to neglect any back action of the presence of the superconducting tip on the order parameter in the inhomogeneous superconductor and vice versa.\n\nThe Josephson current crucially depends on the difference $\\Delta\\phi = \\phi_s - \\phi_t$ of the superconducting phases of the tip ($\\phi_t$) and of the inhomogeneous superconductor ($\\phi_s$). These phases can be gauged away from the superconducting terms by an operator transformation $c_{i\\sigma} \\rightarrow c_{i\\sigma}\\cdot e^{i\\phi_s\/2}$ and \\mbox{$d_{\\sigma} \\rightarrow d_{\\sigma} \\cdot e^{i\\phi_t\/2}$} such that they only appear via $\\Delta\\phi$ in the tunneling Hamiltonian \n\\begin{equation}\n\\hat{\\mathcal{H}}_{\\text{T}} = T_0 \\sum_{\\sigma} \\left( e^{i\\Delta\\phi\/2} \\, c_{i_0 \\sigma}^{\\dagger}d^{\\phantom{\\dagger}}_{\\sigma} + e^{-i\\Delta\\phi\/2}d_{\\sigma}^{\\dagger}c^{\\phantom{\\dagger}}_{i_0 \\sigma}\\right) \\text{.}\n\\end{equation}\nThis gauge transformation allows us to take the order parameters $\\Delta_s$ and $\\Delta_t$ as real parameters below. Closely following the derivation given in \\refcite{mahan} we find the Josephson tunneling current between the inhomogeneous superconductor and the superconducting tip to take the general form\n\\begin{widetext}\\begin{subequations}\\label{eq:IJ}\\begin{align}\\label{eq:IJ1}\nI_{J}(t,V) = -2eT_0^2 I_{J,1}(V) \\sin(\\Delta\\phi-2eVt) -2eT_0^2 I_{J,2}(V) \\cos(\\Delta\\phi-2eVt) \\text{,}\n\\end{align}\nwhere $I_{J,1}(V)$ and $I_{J,2}(V)$ represent the integrals\n\\begin{align}\nI_{J,1} &= \\int \\frac{d\\epsilon}{\\pi}\\left[f(\\epsilon-eV) \\Re \\mathcal{F}_{s,\\sigma\\sigma'}^{\\dagger}(\\epsilon+i0^+)\\Im \\mathcal{F}_{t,\\sigma'\\sigma}(\\epsilon-eV+i0^+)+ f(\\epsilon) \\Im \\mathcal{F}_{s,\\sigma\\sigma'}^{\\dagger}(\\epsilon+i0^+)\\Re\\mathcal{F}_{t,\\sigma'\\sigma}(\\epsilon-eV+i0^+)\\right] \\label{I1Integral} \\\\\nI_{J,2} &= \\int \\frac{d\\epsilon}{\\pi}[f(\\epsilon-eV)-f(\\epsilon)]\\Im\\mathcal{F}_{s,\\sigma\\sigma'}^{\\dagger}(\\epsilon+i0^+)\\Im\\mathcal{F}_{t,\\sigma'\\sigma}(\\epsilon+i0^+-eV). \\label{I2Integral}\n\\end{align}\\end{subequations}\n\\end{widetext}\nHere, $f(\\epsilon)$ is the Fermi-distribution function, $eV$ the voltage drop across the interface, and $\\mathcal{F}_s$ and $\\mathcal{F}_{t}$ denote the anomalous retarded Green's functions of the inhomogeneous superconductor and the superconducting tip, respectively. Summation over spin-indices is implied. The superconducting tip is assumed to be in an isotropic $s$-wave state, such that\n\\begin{equation}\n\\mathcal{F}_{t,\\sigma'\\sigma}(\\epsilon+i0^+) = \\frac{\\rho_0\\pi\\Delta_{\\text{tip}}}{\\sqrt{\\Delta_{\\text{tip}}^2 - (\\epsilon+i0^+)^2}} (i\\hat{\\sigma}^y)_{\\sigma'\\sigma}\\text{.}\n\\end{equation}\nThe sine-term in \\equref{eq:IJ1} produces a dc current of electron pairs at zero voltage, while at finite voltages it leads to the oscillating currents known as Josephson effect. The cosine contribution\\cite{harriscosine} in \\equref{eq:IJ1} is often neglected and vanishes for $eV=0$, due to the term $[f(\\epsilon-eV)-f(\\epsilon)]$ in the integral in \\equref{I2Integral}. We shall later see, that this contribution leads to pronounced beat signatures in the Josephson current.\n\\subsection{Vanishing voltage drop $eV=0$}\nFor a vanishing voltage drop across the interface, $eV=0$, the expression for the Josephson current in \\equref{eq:IJ} reduces to \n\\begin{align}\\label{eq:525}\nI_J&(V=0) = -2eT_0^2 \\, \\sin(\\Delta\\phi)\\nonumber \\\\ &\\times \\int \\frac{d\\epsilon}{\\pi} f(\\epsilon) \\Im \\left[ \\mathcal{F}_{s,\\sigma\\sigma'}^{\\dagger}(\\epsilon+i0^+) \\mathcal{F}_{t,\\sigma'\\sigma}(\\epsilon+i0^+)\\right] \\text{,}\n\\end{align}\nas the integral associated with the cosine contribution vanishes. The critical Josephson current is given by $I_c = 2eT_0^2\\int \\frac{d\\epsilon}{\\pi} f(\\epsilon) \\Im \\left[ \\mathcal{F}_{s,\\sigma\\sigma'}^{\\dagger}(\\epsilon+i0^+) \\mathcal{F}_{t,\\sigma'\\sigma}(\\epsilon+i0^+)\\right]$.\n\\begin{figure}[tb]\n\\includegraphics[width=\\linewidth]{IJ_eV0_magnon.pdf}\n\\caption{Spatial form of the critical Josephson current $I_c$ (red), normalized to the critical current of the clean case $I_0$, and of the order parameter (blue) for (a) non-magnetic impurity at site $i_0=100a$, for $\\mu_s=-t$, $V^s=0.5$, $U_{\\text{imp}}=1t$, $\\xi_s=14a$, $\\Delta_{\\text{tip}}=5\\Delta_0^s$ and (b) magnetic impurity at site $i_0=100a$, for $\\mu_s=0$, $V^s=0.5$, $U_{\\text{imp}}=0.5t$, $\\xi_s=54a$, $\\Delta_{\\text{tip}}=3\\Delta_0^s$.}\n\\label{fig:IJeV0}\n\\end{figure}\n\nThe numerical results for the spatial variation of the critical Josephson current are presented in \\figref{fig:IJeV0} for both a non-magnetic and a magnetic defect in a superconductor with $s$-wave pairing symmetry and OBC. For better optical discernibility in the graphs, the coupling constants have been chosen to lead to short coherence lengths $\\xi_s$ and large oscillations periods of the Friedel oscillations (governed by $k_F$). Both the local structure of the order parameter $\\Delta^s$ (blue line) and the critical Josephson current $I_c$ (red line) have been normalized to their respective bulk values, $\\Delta_0$ and $I_0$, in the absence of defects. As is evident from \\figref{fig:IJeV0} the Josephson current reproduces the spatial behavior of the order parameter for both magnetic and non-magnetic defects, and especially carries a signature of the Friedel oscillations for the non-magnetic defect. \n\nFurthermore, the minor oscillatory features present within the suppression of the order parameter close to the magnetic defect are mapped by the Josephson current. Our numerical results affirm the work presented in \\refcite{SuperconductingSTM}. Next, we will extend this technique to a non-vanishing potential drop $eV\\neq 0$ across the junction interface.\n\\subsection{Finite voltage $eV \\neq 0$}\nThe result of the numerical evaluation of the general time- and voltage-dependent expression for the Josephson current given in \\equref{eq:IJ} for magnetic and non-magnetic defects is illustrated in \\figref{fig:IJeV} and compared to the respective local structures of the $s$-wave order parameter. Here, we show the amplitude $A_J$ of the current oscillations, $I_J(t)=A_J\\cdot \\sin(2eVt - \\text{const.})$.\n\\begin{figure}[tb]\n\\includegraphics[width=\\linewidth]{IJeV.pdf}\n\\caption{Spatial form of the AC-Josephson-current amplitude $A_J$, normalized to its value $I_0$ for the clean case and of the order parameter (blue) for an $s$-wave superconductor with (a) a non-magnetic impurity at site $i_0=100a$, for $\\mu_s = -t$, coupling constant $V^s = 0.5$, $U_{\\text{imp}}=1t$, $\\xi_s \\sim 14a$, $\\Delta^{\\text{tip}}=3\\Delta_0^s$, $eV=0.005t$ and (b) a magnetic impurity at site $i_0=100a$, for $\\mu_s = 0$, $V^s = 0.5$, $U_{\\text{imp}}=0.5t$, $\\xi_s \\sim 54a$, $\\Delta^{\\text{tip}}=3\\Delta_0^s$, $eV=0.005t$. }\n\\label{fig:IJeV}\n\\end{figure}\nBoth quantities, the local structure of the order parameter $\\Delta^{s}$ and the Josephson current amplitude $A_J$, have been normalized to their respective values in the absence of defects ($\\Delta_0$ and $I_0$, respectively). The Friedel oscillations induced in the local structure of the $s$-wave order parameter by a non-magnetic defect are well mapped by the Josephson current, similar to the zero voltage case considered above.\n\\begin{figure}[tb]\n\\includegraphics[width=\\linewidth]{IJ_contrib.pdf}\n\\caption{(a) Prefactor of the sine-term ($I_{J,1}$) and (b) of the cosine-term ($I_{J,2}$) contribution to the tunneling Josephson current in \\equref{eq:IJ1} for an $s$-wave superconducting system with a single non-magnetic defect at $i_0=100a$. Part (c) and (d) show the analogous plots for a magnetic defect at $i_0 = 100a$. System parameters as in \\figref{fig:IJeV}.}\n\\label{fig:IJContributions}\n\\end{figure}\nWe find distinct beat features in the Josephson current for the magnetic defect case, which are strongly enhanced compared to zero voltage. The separate contributions $I_{J,1}$ and $I_{J,2}$ to the Josephson current at $eV\\neq 0$ are depicted in \\figref{fig:IJContributions}. We see that the beat features arise predominantly from the cosine contribution $I_{J,2}$ while being much less pronounced in $I_{J,1}$. \n\\subsection{Josephson current for degenerate ground states}\nAs the Green's function formalism used above is not able to capture the JSTM signature of the MBSs, due to the induced ground state degeneracy, we calculate the current-phase relation directly from\\cite{Huang}\n\\begin{equation}\nI_J(\\Delta\\phi) =  \\sum_n \\partial E_n(\\Delta\\phi)\/\\partial \\Delta\\phi,\n\\end{equation}\ni.e., by partial derivation of the negative quasi-particle energy spectrum $E_n$ of the JSTM setup with respect to the relative phase $\\Delta\\phi$ (we set $e=1$ for the electron charge).  As before, we assume a homogeneous $s$-wave superconductor for the tip and take a weak tunneling element of $T_0=0.01t$. \n\n\n\\begin{figure*}[tb]\n\\includegraphics[width=\\linewidth]{CurrentPhaseHomogen.pdf}\n\\caption{Phase $\\Delta\\phi$ and site $i_0$ dependence of Josephson current for homogeneous $p$-wave superconductor ($\\Delta^t=0.1t$) with (a) $L=150$, (b) $L=90$, and (c) $L=91$ sites with $\\Delta_{\\text{tip}}=0.1t$. The energy of MBSs as a function of system size $L$ is shown in (d).}\n\\label{fig:pevenodd}\n\\end{figure*}\n\n\\subsubsection{Homogeneous $p$-wave}\nThe current-phase relation $I_J(i_0,\\Delta\\phi)$ (with $eV=0$) close to the boundary for a spinfull homogeneous $p$-wave chain, i.e. spatially constant order parameter, is depicted in \\figref{fig:pevenodd}(a). Here $i_0$ is the position measured from the sample edge. As a consequence of the distinct parity of the superconductivity in the tip ($s$-wave) and in the superconductor of interest ($p$-wave), there is no Josephson current in the bulk (for $|i_0| \\gg \\xi_t$). However, due to the presence of MBSs at the edge of the topological $p$-wave state there is a non-zero MBS-induced contribution to Josephson tunneling for $|i_0| \\ll \\xi_t$, which shows the characteristic\\cite{TRSMajoranaJosephson} $\\pi$-periodic behavior with discontinuity at $\\Delta\\phi=\\pi$. \n\n\n\n\n\n\n\nWe find that the phase relation is sensitive to the degree to which the MBSs at the two ends of the system are separated, i.e., it depends on the ratio of coherence length $\\xi_t$ and system size $L$. In \\figref{fig:pevenodd}(b) and (c) we show the phase relation for two systems with overlapping MBSs with even and odd number of sites, respectively.\nFor an even number of sites (see \\figref{fig:pevenodd}(b)) the Josephson current retains its $\\pi$-periodicity, but the maxima are reduced and shifted and the discontinuity is removed. For an odd number of sites (see \\figref{fig:pevenodd}(c)) we regain the current-phase relation of the system with well separated MBSs.\nThis even-odd-effect mirrors the oscillation of the MBS energies for small system sizes $L$, which is illustrated in \\figref{fig:pevenodd}(d).\n\nIn \\figref{fig:pevenodd}, we observe a second even-odd effect not as a function of system size but as a function of $i_0$, which persists in the thermodynamic limit $L\\rightarrow \\infty$. The oscillation of the Josephson current amplitude with $i_0$ follows the spatial profile of the MBS wavefunction which has large (small) weight at odd (even) sites $i$.\n\n\n\\begin{figure}[b]\n\\includegraphics[width=\\linewidth]{IjComp_phom-inhom.jpg}\n\\caption{Comparison of (a) the critical Josephson current $I_c(i_0)=I_J(i_0,\\Delta\\phi=0)$ for  spinfull, spatially homogeneous (red) and inhomogeneous (blue) $p$-wave order parameters and of (b) $I_c$ and the modulus squared  $|\\phi_{\\text{maj}}|^2$ of the MBS wave function for the inhomogeneous $p$-wave state (normalized to their respective maximum values $I_c^{\\text{max}}$ and $|\\phi^{\\text{max}}_{\\text{maj}}|^2$).}\n\\label{fig:Ijcomp}\n\\end{figure}\n\n\n\\begin{figure}[tb]\n\\includegraphics[width=\\linewidth]{IJPHICOMP.jpg}\n\\caption{(a) Current-phase relation close to the boundary for a superconductor of $L=350$ sites with competing order parameters ($V^t=0.6$, $V^s=0.5$, $\\epsilon_c=0.3t$) and bulk dominant $p$-wave ($\\Delta^t_0\\simeq 0.045$) assuming $\\Delta_{\\text{tip}}=0.05t$. (b) Spatial profile of Josephson current $|I_J(i_0,\\Delta\\phi=0)|$ and of critical Josephson current $I_c=I_j(i_0,\\Delta\\phi\\simeq \\frac{3}{4}\\pi)$ and (c) the modulus squared of the MBS wave function $|\\phi_{\\text{maj}}|^2$}\n\\label{fig:spIj350}\n\\end{figure}\n\n\n\n\n\\subsubsection{Self-consistent $p$-wave}\nAs was discussed in \\secref{ImpactOnMBSsWF}, for a spinfull system with inhomogeneous $p$-wave, i.e. spatially varying order parameter, the MBSs are slightly distorted at the system boundaries compared to the homogeneous case.\nComparing the critical Josephson current $I_c(i_0)=I_J(i_0,\\Delta\\phi=0)$ for these two cases, we find this distortion to be clearly reflected in the Josephson current (see \\figref{fig:Ijcomp}(a)), where some of the weight of the Josephson current is shifted towards the bulk.\n\n\n\nIn \\figref{fig:Ijcomp}(b), we can further see that the spatial profile of the critical Josephson current $I_c$ very closely follows the absolute square of the MBS wave function. This clearly illustrates the capability of JSTM not only to detect MBSs but also to map the spatial profile of their wave function.\n\n\\begin{figure}[tb]\n\\includegraphics[width=\\linewidth]{IJsiteComp.pdf}\n\\caption{Current-phase relation for sites $i_0=a,2a,3a,4a$ for (a) the same parameters as in \\figref{fig:spIj350} and (b) for $L=1000$, $V^s=0.35$, $V^t=0.38$ and $\\epsilon_c=0.2t$ leading to a much larger coherence length of $\\xi_t \\approx 165a$. The alternating value of the amplitude of the $\\sin(\\Delta \\phi)$ dependence as a function of $i_0$ in (b) is due to the Friedel-like oscillations of the $s$-wave order parameter.}\n\\label{fig:Ijsite}\n\\end{figure}\n\n\n\n\n\\subsubsection{Competing order parameters}\nFinally, the current-phase relation for a superconductor with competing order parameters with dominant $p$-wave in the bulk and an $s$-wave state emerging at the boundary is shown in \\figref{fig:spIj350}. It exhibits a non-zero current contribution for $\\Delta\\phi=0,2\\pi$ and a discontinuity at $\\Delta\\phi=\\pi$. The parameters have been set to yield a bulk $p$-wave gap similar to that of the homogeneous $p$-wave state analyzed above ($\\Delta^t_0\\simeq 0.1t$). The overall behavior resembles a superposition of the pure $p$-wave contribution, as depicted above, and the well established $\\sin(\\Delta\\phi)$ dependence of a conventional Josephson junction of $s$-wave superconductors. Furthermore, the current decreases towards the bulk, where the $s$-wave component vanishes and the extension of the MBSs ends.\n\n\n\nIn \\figref{fig:spIj350}(b) we show the local Josephson current for $\\Delta\\phi=0$, which arises solely due to the MBSs, and the critical current $I_c$. The overall form of both quantities is in good agreement with the modulus squared of the MBS wave function in \\figref{fig:spIj350}(c). Note that, in contrast to the system examined in \\secref{ImpactOnMBSsWF}, the shift of the MBSs away from the edge of the system is not as clear, due to the shorter coherence length (only $\\xi_t=24a$ instead of $\\xi_t =210a$ in \\figref{fig:MBSComp}).\n\nA closer examination of the site- and phase-dependent Josephson current in \\figref{fig:spIj350} reveals an alternating pattern between an $s$-wave-dominated profile ($\\propto \\sin(\\Delta\\phi)$) of the Josephson current and an admixture of a MBS-induced and of an $s$-wave-like current-phase relation (see \\figref{fig:Ijsite}(a)). This alternation is in accordance with the weights of the MBS wave functions (see \\figref{fig:spIj350}(c)) being greatest on odd sites and minimal on even sites. For even $i_0$ with weak weights of the MBSs, primarily an $s$-wave signature is seen as the MBS contribution is very small.\nFor odd $i_0$, the admixture of $\\propto \\sin(\\Delta\\phi)$ and of MBS-mediated tunneling leads to a current-phase relation with discontinuous jumps, due to the MBSs, but $2\\pi$ periodicity, instead of the expected period of $\\pi$, resulting from the additional $s$-wave component.\n\nNote that the contribution of the MBSs to the Josephson current very close to the boundary, $i_0 \\ll \\xi_t$, is much smaller for systems with larger coherence lengths $\\xi_t$. As can be seen in \\figref{fig:Ijsite}(b), where $\\xi_t=165a$, the Josephson current only shows the conventional $\\sin(\\Delta\\phi)$ form close to the edge. This is due to shift of the MBS wave function away from the edge of the system (see \\figref{fig:MBSComp}(a) and (b)) leading to a strong suppression of the MBS contribution to the Josephson current for $i_0 \\ll \\xi_t$.  While the MBSs can still be resolved in a JSTM setup, a regular Josephson junction between the edge of the topological superconductor and a conventional $s$-wave state, see \\figref{BasicSetup}, can hardly detect any signatures of the MBSs.\n\n\n\n\\section{Conclusions}\nIn this paper, we have studied the modulations of the superconducting order parameter in the vicinity of edges, magnetic and non-magnetic impurities by self-consistently solving the gap equations for multiple pairing interactions. We have shown that the presence or absence of Friedel oscillations of the superconducting order parameter crucially depends on the pairing symmetry and the nature of the inhomogeneity. Hence, they can be used to discern, on the one hand, between different defect types and, on the other hand, between even parity singlet and odd parity triplet pairing.\nThis distinction is especially prominent in low-dimensional systems. The $s$-wave order parameter exhibits pronounced Friedel oscillations in the vicinity of non-magnetic inhomogeneities, while the $p$-wave order parameter lacks such distinct features.\nWe note that a different alternative route to access the character of the order parameter in noncentrosymmetric superconductors, which is based on Shiba states, has been recently proposed in \\refcite{PSimon}. \n\nFurthermore, we have discussed the competition of order parameters with different pairing symmetries in the presence of inhomogeneities. We have shown that this competition can lead to the emergence of a topologically trivial region close to the boundary of a system that is topologically nontrivial in its bulk. Due to this competition, the MBSs are shifted towards the transition point separating these two regions, as opposed to the clear localization at the system edge for a topologically nontrivial system with homogeneous order parameter.\n\nAs a means to probe and resolve the Friedel oscillations induced by inhomogeneities, as well as the spatial structure of competing $s$-wave superconductivity and the MBSs, we have discussed JSTM. The spatially resolved Josephson current provides a precise tool for the mapping of the spatial structure of the $s$-wave order parameter as well as of the MBS signatures in systems with competing order parameters. MBSs contribute significantly to the current-phase relation of the Josephson current, modifying it beyond the well established sinusoidal dependence of pure $s$-wave superconductivity, thus revealing the presence of MBSs and their spatial profile. \n\nTaken together, the spatially resolved Josephson current can be utilized to distinguish between different order parameter symmetries by means of their response to inhomogeneities, discern between magnetic and non-magnetic defects, identify the competition of order parameters, and carry clear signatures of MBSs.\n\\acknowledgments\nMS acknowledges support from the German National Academy of Sciences Leopoldina through grant LPDS 2016-12. This work was supported by the Virtual Materials Design initiative within the Helmholtz program Science and Technology of Nanosystems at the Karlsruhe Institute of Technology.\n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conclusion}\nIn this paper, we propose the CLIP-GEN strategy to train a reliable and general text-to-image generator without using any paired text-image data, but with the image-language priors of CLIP and a set of unlabeled image data. Such strategy enables us to make use of the available huge text-free image dataset (e.g., ImageNet) to train a text-to-image generator as powerful as flagship models like CogView \\cite{ding2021cogview} that is trained with huge amounts of paired data. The proposed strategy should see greater breakthroughs in the future if we use larger numbers of unlabeled images that are available on the Internet.\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n\n\\section{Experiment}\n\\label{sec:exp}\n\nIn this section, we describe how we evaluate our method and compare with previous approaches. We first introduce the datasets used for training and validation and the implementation details of our approach on these datasets.\nThen we make comprehensive comparisons between our method and previous text-to-image methods both quantitatively and qualitatively.\n\n\n\n\\begin{table}[htb]\n\\begin{center}\n\\resizebox{\\linewidth}{!}{%\n\\begin{tabular}{lccccc}\n\\hline\n\\textbf{Model}  & \\textbf{IS} $\\uparrow$ & \\textbf{FID-0} $\\downarrow$  & \\textbf{FID-1} $\\downarrow$  &\\textbf{FID-2} $\\downarrow$ &\\textbf{CapS} \\\\\n\\hline\nAttnGAN~\\cite{xu2018attngan} & 23.3 & 35.2 & 44.0 & 72.0 &  0.02763\\\\\nDM-GAN~\\cite{zhu2019dm} & \\textbf{32.2} & 26.0 & 39.0 & 73.0 & 0.02801 \\\\\nDF-GAN~\\cite{tao2020df} & 18.7 & 26.0 & 33.8 & 55.9 &0.02802 \\\\\nCogView~\\cite{ding2021cogview} & 18.2 & 27.1 & 19.4 & \\textbf{13.9} &\\textbf{0.17403} \\\\\nDALL-E~\\cite{ramesh2021zero} & 17.9 & 27.5 & 28.0 & 45.5 & - \\\\\n\\hline\nOurs & 21.4 & \\textbf{20.7} & \\textbf{18.5} & 17.4 &  0.13751 \\\\\n\\hline\n\\end{tabular}\n}\n\\end{center}\n\\caption{Quantitative results on MS-COCO validation dataset.}\n\\label{tab:COCO}\n\\end{table}\n\n\n\\begin{figure*}[tbh]\n  \\centering\n   \\includegraphics[width=1.0\\linewidth]{figures\/coco_samples_times.pdf}\n   \\vspace{0.4cm}\n   \\caption{Generated samples based on six textual descriptions from the MS-COCO validation dataset. Our results as demonstrated in this figure are generated with the model trained on only images from the complete MS-COCO dataset.}\n   \\label{fig:coco}\n\\end{figure*}\n\n\n\\begin{figure*}[tbh]\n  \\centering\n  \n   \\includegraphics[width=1.0\\linewidth]{figures\/IN_samples_times.pdf}\n   \\vspace{0.4cm}\n   \\caption{Generated samples with different text-to-image methods. The captions are carefully composed so as to comply with the distribution of the ImageNet dataset, i.e., a visual description of an object covered by the ImageNet categorical labels.}\n   \\label{fig:imagenet}\n\\end{figure*}\n\n\n\\subsection{Datasets}\n\\label{subsec:datasets}\nWe train and evaluate our methods on two datasets: MS-COCO~\\cite{lin2014microsoft} and ImageNet~\\cite{krizhevsky2012imagenet}.\n\n\\noindent\\textbf{MS-COCO}~is a widely used dataset for language-vision benchmarks. It contains 80k images for training and 40k test set images. Each image has 5 short sentence descriptions, which is not used in our method but used by competing methods (e.g., DM-GAN \\cite{zhu2019dm}, DF-GAN \\cite{tao2020df}, and AttnGAN~\\cite{xu2018attngan}). We use the 2014 split of MS-COCO dataset in our experimental setting. We use the complete set of images to train the vQGAN and the text-to-image generator, while we only use the textual descriptions form the validation split for the quantitative evaluation and visual demonstration.  \n\n\\noindent\\textbf{ImageNet}~has long been used to evaluate conditional generation tasks. It contains more than 14 million images, and a little more than 21 thousand groups or classes. We use the complete set of images to train the VQGAN and our text-to-image generator. For evaluations, we construct the input textual descriptions either by fitting the template of ``a photo of a [class name]'' ({\\it{class name}} is an ImageNet category) or manually composing a caption like ``a photo of some [descriptive] objects with some [features]'' ({\\it{descriptive}} can be some constraints of the color, size or other properties of the object, and {\\it{features}} could be some accessories of the object).  \n\n\\label{subsec:quant}\n\\subsection{Implementation Details.}\nFor both datasets, we train a VQGAN with $|\\cZ|=16384$ and $dim_z=256$. We use GPT2~\\cite{radford2019language} as the architecture of our conditional transformer. We trained a 24-layers GPT2-medium for MS-COCO and a 48-layers GPT2-XL for ImageNet. The details of params are shown in ~\\cref{tab:params}. The CLIP we used is the pre-trained ViT-B\/32 model released by OpenAI \\cite{radford2021learning}.\n\n\\subsection{Comparisons}\nWe compare our results with four existing approaches that are representative methods of different research streams (e.g., CNN-based methods, optimization-based methods and transformer-based methods). These methods are:\n\n\\noindent\\textbf{DF-GAN}~\\cite{tao2020df}, \\textbf{DM-GAN}~\\cite{zhu2019dm} and \\textbf{AttnGAN}~\\cite{xu2018attngan} represent the traditional approaches which use CNN generator to directly generate images with a textual condition. These methods provide pre-trained models on the MS-COCO dataset, so we can directly use those models for comparisons. \n\n\\noindent\\textbf{CogView}~\\cite{ding2021cogview} is a flagship transformer-based method and serves as a good representation of fully-supervised and large transformer-based models~\\cite{ding2021cogview, ramesh2021zero}, which are trained on tens of millions of high-quality text-image pairs. It achieves the best text-image relevance and FID metrics (see more details in \\cite{ding2021cogview}), but the results by CogView suffer lack of perceptual details as images are encoded and decoded with VQ-VAE~\\cite{radford2021learning}. Its pre-trained model is a large model with 4-Billion parameters trained on 50 million text-image pairs in the general domain and should cover the distribution of both ImageNet and MS-COCO images very well.\n\n\\noindent\\textbf{VQGAN-CLIP}~\\cite{vqganclip2021} represents for zero-shot opimization-based approach. It utilizes CLIP scores to guidance the optimization direction of latent codes of a pre-trained VQGAN model without any extra training. The CLIP model used in our comparisons is the pre-trained ViT-B\/32 model released by OpenAI \\cite{radford2021learning}. The VQGAN models for the two datasets respectively are the same as used in our method.\n\n\\noindent\\textbf{BigGAN-CLIP}~\\cite{bigganclip2021} Since DF-GAN cannot be trained on ImageNet (without text labels), we use the BigGAN-CLIP~\\cite{bigganclip2021} as a substitute when conducting visual and quantitative comparisons on ImageNet dataset. Here the BigGAN model is the one pre-trained on ImageNet dataset and provided by \\cite{brock2018large}.\n\n\n\n\\begin{figure*}[htb]\n  \\centering\n  \n   \\includegraphics[width=1.0\\linewidth]{figures\/style_times.pdf}\n   \\vspace{0.25cm}\n   \\caption{The generated sheepdog pictures in different styles. These images are synthesized with our model trained on the ImageNet dataset without any data augmentation.}\n   \\label{fig:style}\n\\end{figure*}\n\n\n\\subsection{Quantitative Results}\n\n\\noindent\\textbf{Evaluation Metrics}\nTo evaluate the quality of generated images, we use the stantard metrics as in \\cite{ding2021cogview}: Inception Score (IS)~\\cite{tim2016is}, Fr\u00b4echet Inception Distance (FID)~\\cite{martin2017fid} and CapS \\cite{ding2021cogview}. IS calculates KL-divergence between conditional distribution and marginal distribution given an image classifier. FID computes the Fr\u00b4echet distance between the distribution of Inception features of synthetic images and real-world images. CapS measures the semantic similarities between the input text and the generated image. The quantitative results are computed over 30,000 images generated based on diverse validation captions and 30,000 ground-truth images related to the captions.\n\n\\noindent\\textbf{MS-COCO}~\\cref{tab:COCO} shows the comparison between our method and previous methods on text-guided scene image synthesis. Our method achieves the best FID-0 and FID-1 due to the perceptually rich results generated by VQGAN and coherent image structures. The CapS score is lower than CogView by 4\\% but significantly better than other competing methods.\n\n\\noindent\\textbf{ImageNet.}~\\cref{tab:IN} shows the class-conditional generation results on ImageNet. We compared our method with similar methods based on discrete image tokens. Since the models require text inputs, we use the textual prompting (\\textit{A photo of a \\{ImageNet label\\}.}) as the input. Our method achieves the best FID metrics, implying that our method achieves better image quality and more coherent semantic distribution.\n\\begin{table}[htb]\n\\begin{center}\n\\resizebox{\\linewidth}{!}{%\n\\begin{tabular}{lcc}\n\\hline\n\\textbf{Model}  & \\textbf{IS} $\\uparrow$ & \\textbf{FID} $\\downarrow$ \\\\\n\\hline\nVQGAN+CLIP~\\cite{vqganclip2021} & 20.8 & 77.0 \\\\\nVQVAE-2~\\cite{razavi2019generating} & $\\sim$45 & $\\sim$31  \\\\\nVQGAN+label conditioned Transformer~\\cite{esser2021taming} & \\textbf{70.6} & 17.0 \\\\\n\\hline\nOurs & 45.16 & \\textbf{16.74}\\\\\n\\hline\n\\end{tabular}\n}\n\\end{center}\n\\caption{Quantitative results on ImageNet dataset.}\n\\label{tab:IN}\n\\end{table}\n\n\\subsection{Qualitative Results}\n\\label{subsec:examples}\n\n\n\nAs shown in Fig. \\ref{fig:coco} and Fig. \\ref{fig:imagenet}, compared to the competing methods, our method could generate high-fidelity images with more details. Generally speaking, the results of VQGAN-CLIP \\cite{vqganclip2021} are non-realistic and suffer severe image distortion. CogView \\cite{ding2021cogview} can generate good image structures but fails to produce realistic textures as they use the VQVAE \\cite{oord2017neural} to discretize images. The BigGAN-CLIP \\cite{bigganclip2021} method sees more natural details than VQGAN-CLIP but suffer distorted image structures either.  The DF-GAN \\cite{tao2020df} can generate acceptable image structures with perceptually rich details but is prone to producing local regional artifacts. The visual evaluation matches the quantitative results well. \n\n\nAs shown in Fig. \\ref{fig:coco}, our model can successfully capture the semantic concepts such as ``reflection in the water'' ($2^{nd}$ column), but fails to match the numeric concepts like ``three plush bears'' ($3^{rd}$ column). These concepts are captured by CogView model \\cite{ding2021cogview} very well.\n\n\nTo examine the generalization ability of our method, we attempt to generate images under out-of-distribution language descriptions. As shown in  ~\\cref{fig:imagenet}, some of the descriptions (e.g. ``a dog with a cigarrete'', ``a lemon with hair and face'') do not even have a corresponding real-world image. Our model that is trained on the realistic images can surprisingly generate images well-aligned with these out-of-distribution texts. However, CogView \\cite{ding2021cogview} that is trained upon amounts of image with textual labels fails to match those decorative words (e.g., ``flying'', ``with a cigarrete'', ``with big beak'').\n\nWe also explore the generalization ability of our method in terms of stylized synthesis. We attempt to generate images under special style descriptions (e.g., ``sketch'', ``oil painting'' or even ``style of Edvard Munch''). As shown in ~\\cref{fig:style}, our model can successfully synthesize stylized pictures even without seeing many stylized training samples as no style augmentation is applied during training.\n\n\\section{Introduction}\n\\label{sec:intro}\nText guided image generation in the general domain has been a challenging and frontier task in recent years. Early approaches (e.g., DMGAN \\cite{zhu2019dm}, AttnGAN \\cite{xu2018attngan}, DF-GAN \\cite{tao2020df}, Obj-GAN  \\cite{li2019object}, OPGAN \\cite{li2019object}, SD-GAN \\cite{yin2019semantics}, CPGAN \\cite{liang2019cpgan}, XMC-GAN \\cite{zhang2021cross}) that directly generate pixels from the given text embeddings with a convolutional generator have shown promising revolutions to generate images in limited domains. However, when designated to generate images in the general domain, these methods see poor results in terms of image quality and text-image matching.\n\n\n\nRecently, transformer-based text-to-image generators such as DALL-E\\cite{ramesh2021zero} and CogView~\\cite{ding2021cogview} have achieved great progress. Such progress is owed to two factors. First, the discretized representations of images achieved by vector quantized models such as VQ-VAE~\\cite{oord2017neural} and VQ-GAN~\\cite{esser2021taming} enable an image to be represented in the same way as natural language, thus enabling a transformer to be trained upon the cross-modality text-image data in a unified framework. Second, the progress in terms of large model (consisting of tens or hundreds of billions of parameters) training significantly leverages the model capacities of modeling cross-modality data in general domains. So far, these large-transformer-based methods \\cite{ramesh2021zero,ding2021cogview} achieve the best performance in terms of image quality, text-image relevance and range of domains. However, a limitation is that they require hundreds of millions of high-quality paired text-image data for the training, which are not publicly available and typically too expensive to acquire.\n\nOn the other hand, inspired by the recent progress in cross-modality language-vision pre-training and the unveiling of CLIP model \\cite{radford2021learning}, various optimization-based methods attempt to search in the image space based on a query text by optimizing the text-image matching score of a pre-trained CLIP model. The image search domain used by these methods could be the latent codes of a pre-trained GAN model (e.g. BigGAN \\cite{bigganclip2021}, StyleGAN \\cite{patashnik2021styleclip,gal2021stylegan}, SWAGAN \\cite{MAGnet2021}), the codebook of a VQGAN model \\cite{vqganclip2021}, Diffusion Denoising Models \\cite{diffusionclip2021}, structured representations such as a set of strokes (ClipDraw \\cite{frans2021clipdraw}) or triangles \\cite{triangleclip2021}, or a SIREN network \\cite{DeepDaze2021,sirenfft2021} that maps spatial coordinates to pixels. These methods relieve the demands of huge paired datasets and computing resources. However, the images generated by methods of this stream are either limited to a specific domain (e.g., faces) or suffer low-quality (e.g., unnatural, structure-distorted or physically meaningless).\n\n\n\\begin{figure}[tb]\n  \\centering\n \n   \\includegraphics[width=1.0\\linewidth]{figures\/overview_time_2.pdf}\n\n   \\caption{An overview of our purposed approach. (a) describes how our method maps a sentence to the corresponding image through the embedding space and the token space. (b) shows our training and testing pipeline. During training, the pre-trained CLIP model embeds the image to a cross-modality embedding and the pre-trained image tokenizer encodes the image into discrete image tokens. The autoregressive transformer learns to predict the image tokens with respect to the cross-modality embedding. During the inference, the CLIP model could either take an image or a sentence as the input, and then the transformer could predict coherent image tokens semantically related to the input.}\n   \\label{fig:intro}\n\\end{figure}\n\nAn analysis of methods of the two streams motivates us to seek a balance between the two. With the language-vision priors learned by the CLIP model, we shall be able to train a text-to-image generator without the use of any paired data. Considering the joint language-vision embedding space of a pre-trained CLIP is shared by both modalities, the image embedding extracted with the image encoder of the CLIP upon an image is also a good representation of textual semantics. If we train a transformer that maps an image embedding to the image itself, then the inference pipeline with text inputs is automatically bridged: starts from a text, goes though the united embedding space, and finally generates an image. \n\n\nSpecifically, we first extract the cross-modality embedding of the image in the joint language-vision embedding space with a pre-trained CLIP model \\cite{radford2021learning}. Next, we convert the image into a sequence of discrete tokens in the VQGAN codebook space \\cite{esser2021taming} which can be trained with the unlabeled image dataset in hand. Finally, an autoregressive transformer that predicts the image tokens based on its joint language-vision embedding is trained. During inference, the transformer can generate coherent image tokens based on the text embedding extracted upon an input text with the text encoder of CLIP, and the generated image tokens can be further reconstructed into an image with the VQGAN decoder: see Figure \\ref{fig:intro}. Such a scheme relies on the assumption that the distribution of images embeddings used for training is well aligned with that of text embeddings during test, which shall hold if we intentionally ensure the statistical coherency of the semantic distribution of the training image data and text data used for test.\n\nContributions of this paper include:\n\\begin{itemize}\n\\item We propose a scheme to train a reliable and general text-to-image generator without any paired text-image data, but with a set of unlabeled images and a pre-trained CLIP model as prior. Our approach provides a promising new direction for high-fidelity text-to-image generation with accessible resources.\n\n\\item Qualitative and quantitative evaluations verify that our method outperforms optimization-based text-to-image methods (e.g., VQGAN+CLIP \\cite{vqganclip2021}, BigGAN-CLIP \\cite{bigganclip2021}) and CNN-based methods (DF-GAN \\cite{tao2020df}) in terms of image quality while not compromising the text-image matching. Our model can even achieve comparable performance as the flagship supervised model like CogView \\cite{ding2021cogview} which is trained with huge amounts of paired data.\n\\end{itemize}\n\n\n\\section{Approach}\n\\label{sec:approach}\n\nIn this section, we introduce the details about our text-to-image generation framework and the training strategy.\n\nAs shown in \\cref{fig:framework}, our model is made up of three components: a pre-trained language-image matching model (CLIP), an image tokenizer (VQ-GAN) and a conditional autoregressive transformer that takes\nthe image embedding $\\ve_c$ of an image extracted from CLIP as the condition $\\vc$, then generates the discrete image tokens $\\vz$ of the same image.\n\n\\subsection{Language-Image Feature Extractor}\n\nContrastive Language\u2013Image Pre-training (CLIP)~\\cite{radford2021learning} has achieved great success in mapping the language-image inputs to a common embedding space. Given an image $I$ or a sentence $T$ as the input (denoted as $x$), the CLIP model can embed them into a common representation space: \\begin{align}\n\\ve_c &= f_{\\mathrm{CLIP}}(x), \\mathrm{where}~x\\in\\{I, T\\}. \n\\end{align}\n\nThe official pre-trained CLIP model provided by~\\cite{radford2021learning} is trained on 400-millions of text-image pairs with InfoNCE Loss:\n\\begin{align}\n\\cL_{\\mathrm{InfoNCE}} &= \\sum_{(I_{pos},T_{pos})}\\log{\\frac{\\exp(\\cos(I_{pos}, T_{pos}))}{\\sum_{(I, T)}{\\exp(\\cos(I, T))}}}.\n\\end{align}\nwhich learns robust representations of hetero-modality data to ensure the semantically relevant data to be close to each other in the common embedding space. \n\nWhereas it is too expensive to obtain large-scale and high-quality text-image pairs within our domain-of-interest, the CLIP model pre-trained on 400-million noisy pairs collected from Internet has shown enough capability to model language-vision data in general domains.\nSpecifically, we use the ViT-B\/32 variation of CLIP~\\cite{radford2021learning} in our experiments.\n\n\n\\subsection{Learning an Efficient Image Tokenizer}\n\\label{subsec:vqgan}\n\nThe recent VQ-VAE~\\cite{oord2017neural} and VQ-GAN~\\cite{esser2021taming} models have shown promising results to compress image patches into discrete image tokens. Such mechanisms enable images to be represented in the same way as natural language and easier to process with transformers. \n\nAs shown in the pre-training phase of ~\\cref{fig:framework}, we employ VQ-GAN to learn a perceptually rich codebook $\\cZ=\\{z\\}^{K}\\subset \\Rbb^{dim_z}$ by optimizing all parameters of the encoder $E$, decoder $G$ and discriminator $D$. After the training finishes, the discriminator is removed. We only use the encoder $E$ and the codebook $\\cZ$ as the image tokenizer $\\vq(\\cdot)$, and the decoder $G$ to reconstruct an image from its tokens.\n\nGiven an input image $I$, we first map it into a spatial embedding map of size $h\\times w$,  $\\hat{z} = E(I) \\in \\Rbb^{dim_z\\times h\\times w}$. Each embedding $\\hat{z}_{ij}$ will then be hard-coded by looking up its nearest neighbor in the codebook:\n\\begin{align}\n    z_q = \\argmin_{z_k\\in \\cZ}{\\lVert\\hat{z}_{ij}-z_{k}\\rVert}.\n\\end{align}\nwhere the indices sequence $\\{k\\}^{h\\times w}$ of $\\{z_q\\}$ is denoted as $\\mathbf{s}$. \n\nThe VQ-decoder is used to reconstruct an image from the token sequence $\\mathbf{s}$, i.e., $\\hat{I} = G(\\mathbf{s})=G(\\mathbf{\\mathbf{q}(\\mathbf{s}}))$. \n\nThe VQGAN model can be optimized with an objective consisting of the reconstruction loss:\n\\begin{align}\n\\label{eq:L_vq}\n    \\cL_{\\mathrm{vq}} &= \\lVert I-\\hat{I}\\rVert ^2 + \\lVert \\mathrm{sg}[E(I)]-z_q \\rVert_{2}^{2} + \\lVert \\mathrm{sg}[z_q]-E(I) \\rVert_{2}^{2}\n\\end{align}\nand the adversarial loss:\n\\begin{align}\n\\label{eq:L_D}\n    \\cL_{\\mathrm{GAN}} &= [\\log{D(I)} + \\log(1-D(\\hat{I})]\n\\end{align}\nwhere $\\mathrm{sg}(\\cdot)$ is the stop-gradient operation.\n\n\\subsection{Conditional Autoregressive Transformer}\n\\label{subsec:autoreg}\n\nThe conditional autoregressive transformer is designated to predict image tokens based on its CLIP embedding. Given an input image $I$, we obtain its embedding with the CLIP image encoder $\\ve_c = f_{\\mathrm{CLIP}(I)}$ and a row-major ordered sequence of image tokens $\\textbf{s} = \\vq(I) = \\{s_1, s_2, ..., s_{h\\times w}\\}, s_i\\in\\{0, ..., |\\cZ|\\}$. Since the CLIP model only extracts high-level semantic information of an image, we expect the low-level image information of the image could be restored with the transformer in an autoregressive way, just as\n\\begin{align}\n    p(\\textbf{s}|\\textbf{c}) &= \\prod_{i}{p(s_i|s_{<i}, \\ve_c)}\n\\end{align}\n\nOnce the complete set of tokens $\\mathbf{s}$ are restored with respect to the image embedding, the pre-trained decoder $G$ could reconstruct the tokens back to an image, $\\hat{I} = G(\\textbf{s})$.\n\n\n\n\n\\subsection{Training Strategy}\n\n\\noindent We employ the two-stage training strategy.\n\n\\noindent \\textbf{First Stage} We first train a VQ-GAN model with the image dataset in a self-supervised manner. As mentioned in \\cref{subsec:vqgan}, all parameters of the encoder $E$, decoder $G$, codebook $\\cZ$ and discriminator $D$ will be optimized during training. The training objective is:\n\\begin{align}\n    \\cL_{\\mathrm{tokenize}} &= \\argmin_{E, G, \\cZ}\\max_{D}{\\Ebb_{x\\sim p(x)}{[L_{VQ} + L_{GAN}]}}\n\\end{align}\nwhere $\\cL_{\\mathrm{VQ}}$ and $\\cL_{\\mathrm{GAN}}$ are specified at ~\\cref{eq:L_vq} and \\cref{eq:L_D} respectively.\n\n\\noindent \\textbf{Second Stage} The conditional autoregressive transformer is trained at this stage. Since we have paired input-output data (embedding$\\rightarrow$image), our objective is a sum of the embedding reconstruction loss and a loss to maximize the likelihood of the corresponding image token.\n\nThe maximum-likelihood of the token sequence is enforce with \n\\begin{align}\n\\cL_{\\mathrm{Transformer}} = \\Ebb_{x\\sim p(x)}[-\\log{p(s)}]\n\\end{align} \n\nTo ensure the generated image can be mapped back to its embedding with the  CLIP image encoder, we employ the embedding reconstruction loss:\n\\begin{align}\n\\cL_{\\mathrm{CLIP}} &= -\\log{s(f_{\\mathrm{CLIP}}(G(\\mathbf{s})), f_c)}\n\\end{align}\n\nThe training objective is the weighted combination of the two losses above:\n\\begin{align}\n\\cL &= \\cL_{\\mathrm{Transformer}} + \\lambda \\cL_{\\mathrm{CLIP}}\n\\end{align}\nwhere we set $\\lambda=0.2$ in our implementation.\n\n\n\\section{Related Work}\n\\noindent\\textbf{Discrete Image Representation}\n\nOord et al. \\cite{oord2017neural} first present an approach called Vector Quantized Variational Autoencoder (VQVAE) to learn discrete representations of images and model their distribution autoregressively with a convolutional architecture. \\cite{razavi2019generating} extends this approach to use a hierarchy of learned representations to represent images of higher-resolution. \\cite{esser2021taming} introduces self-attention layers \\cite{ashish2017attn} to the bottleneck of the convolutional architecture with the expect to capture long-range interactions in high-resolution images, and adversarial training losses \\cite{creswell2018generative} to enforce the learning of perceptually rich codebooks. In our approach, we use the perceptually rich discrete representations of images that preserve more photorealistic details and natural textures.\n\n\\noindent\\textbf{Vision-Language Modelling}\nPre-training methods that have recently moved from raw text to multi-modal data (e.g., image-text) have revolutionized numerous multi-modal tasks (e.g., image-text matching \\cite{tan2019lxmert,DBLPabs210300020,chen2020uniter}, image captioning \\cite{zhang2021vinvl}, Visual Question Answering \\cite{su2019vl,tan2019lxmert,li2020oscar,chen2020uniter}).  Cross-modality tasks require the understanding of both modalities, and the alignment and relationships between the two modalities. The pre-training enables the encoder to produce representations with fused cross-modality information, thus benefiting downstream tasks. \n\nThe unveiling of CLIP model \\cite{radford2021learning} is a big step for multi-modal pre-training. In the CLIP architecture, the image modality and the language modality are mapped with the image encoder and the text encoder respectively to the shared multi-modal embedding space. The text-image similarity score computed in the shared multi-modal embedding space of CLIP can serve as a metric of text-image alignment or an objective for text-guided image generation \\cite{bigganclip2021,patashnik2021styleclip,gal2021stylegan,MAGnet2021,vqganclip2021,diffusionclip2021,frans2021clipdraw,triangleclip2021,DeepDaze2021,sirenfft2021}. However, in our strategy, we use the shared multi-modal embedding space in a creative way, i.e., training a reverse model that maps the shared embedding to the image modality.\n\n\\noindent\\textbf{Text-to-Image Generation}\nWhile there have been several attempts to improve the controllability of image generation by conditioning image synthesis on explainable priors (categories, attributes, label maps, edge maps, key points, depth-map), they often require users to follow some fixed control patterns. However, text-to-image generation that enables free-style user controls is a good choice, as natural language is easy to express and rich in information. \n\nRecent years see great progresses in this field and many methods have been proposed. The key differences of existing text-to-image approaches rely on what are used to represent texts (e.g., word embeddings) and images (e.g., GAN, VAE, VQVAE or raw pixels) respectively, and what model is used to bridge the two modalities. Early methods for text-to-image \\cite{zhu2019dm,xu2018attngan,tao2020df,li2019object,li2019object,yin2019semantics,liang2019cpgan,zhang2021cross,zhang2017stackgan,qiao2019mirrorgan,zhang2018stackgan++} attempt to train a convolutional generator that predicts pixels directly from the given text embeddings. Recently, transformer-based generators \\cite{ramesh2021zero,ding2021cogview} that map the textual embeddings to the discretized representations of images (VQGAN \\cite{vqganclip2021} or VQVAE \\cite{oord2017neural,razavi2019generating}) have achieved significantly better results than traditional CNN-based methods. Other streams rely on a pre-trained GAN model (e.g., StyleGAN \\cite{karras2020analyzing,viazovetskyi2020stylegan2}) and attempt manipulate the style space based on textual inputs \\cite{xia2021tedigan,bigganclip2021,patashnik2021styleclip,gal2021stylegan,MAGnet2021,rombach2020network}, or rely on a pre-trained text-image matching model (e.g., CLIP \\cite{radford2021learning}) and attempt to optimize the image representations to satisfy the textual guidance \\cite{vqganclip2021,diffusionclip2021,triangleclip2021,DeepDaze2021,sirenfft2021,bigganclip2021,patashnik2021styleclip,gal2021stylegan,MAGnet2021}. Our method takes the advantages of both transformer-based and CLIP-based methods. We make use of the knowledge priors learnt with CLIP and train a powerful transformer without any paired data.\n\n\\section{Implementation Details}\n\\label{sec:implementation}\n\n\\noindent\\textbf{VQ-GAN Training} In the pre-training stage, we train two VQ-GAN models on ImageNet and COCO dataset respectively. Both models have the same architecture and hyperparameters. Specifically, we set the codebook size $|\\cZ|=16384$, embedding size $dim_z=256$, compression factor $f=16$ and sequence length $|s|=16\\times 16=256$.\n\nFollowing the implementation of the original VQ-GAN, we use a 3-layer PatchGAN discriminator with perception loss.  The model is trained for 100 epochs with a batch size of 16. The discriminator is not trained for the first 25,000 steps. The models are trained via Adam optimizer with the initial learning rate of 4.5e-6. During training, input images are resized and random cropped to $256\\times 256$ while the model leanrs to reconstruct the image of the same size.\n\n\\noindent\\textbf{Self-supervised Training of the Auto-regressive Transformer} In the second stage, we only train the auto-regressive transformer, while the weights of VQ-GAN and CLIP are fixed. We choose the GPT-2 architecture in our implementation. We use the GPT-2-Medium model for the COCO dataset and the GPT-2-XL model for the ImageNet dataset.\n\\fi\n\n\\subfile{supp_sections\/implementation.tex}\n\\subfile{supp_sections\/results.tex}\n\n\n\n\n\n\n\\end{document}\n\n\\section{Implementation Details}\n\\label{sec:implementation}\n\n\\noindent\\textbf{Pre-trained VQ-GAN} In the pre-training stage, we train two VQ-GAN~\\cite{esser2021taming} models on ImageNet~\\cite{krizhevsky2012imagenet} and COCO~\\cite{lin2014microsoft} dataset respectively. Both models have the same architecture and hyperparameters. Specifically, we set the codebook size $|\\cZ|=16384$, embedding size $dim_z=256$, compression factor $f=16$ and sequence length $|s|=16\\times 16=256$.\n\nFollowing the implementation of the original VQ-GAN, we use a 3-layer PatchGAN~\\cite{isola2017image} discriminator with perception loss. The discriminator is not trained before 25,000 steps. The models are trained with Adam optimizer with the initial learning rate $0.0000045$. During training, input images are resize and random cropped to $256\\times 256$ while the model reconstruct the image in the same size. The model is finally trained for 100 epochs with a batch size of 16.\n\n\\noindent\\textbf{Autoregressive Transformer} In the second stage, we only trained the autoregressive transformer. The weights of VQ-GAN and CLIP~\\cite{radford2021learning} are fixed. We choose the GPT-2~\\cite{radford2019language} architecture in our implementation.\n\nOn COCO dataset, we use the medium model with 24 layers, 16 attention heads and the inner dim is 1024. On ImageNet dataset, we use the modified XL model with 48 layers, 24 attention head and the inner dim is 1536. The feature extracted from CLIP is liner probed to the same number of inner dim before input to the transformer.\n\nWe train the model with Adam optimizer and the initial learning rate is set to $0.00001$. The medium model is trained with a batch size of 32 for 1 million steps and the XL model is trained with a batch size of 16 for 2 million steps. \n\n\n\\section{Additional Results}\n\\label{sec:res}\n\\noindent\\textbf{Different Sample Strategies}\nWe employ top-k\/p sample strategy to predict the image tokens with the autoregrresive model.\nThe top-k sampling only retains the highest $k$ logits.\nThe top-p sampling first normalize the logits into a probability distribution with softmax then remove the tokens with cumulative probability above the given threshold $p$. We will show our model's generated images with different tuned $k$ and $p$ in ~\\cref{fig:sample1,fig:sample2,fig:sample3}. It shows that the small $k$ and $p$ will only retrain a few token candidates and make the generated images be simple and naive, while sampling with large $k$ and $p$ leads to more diverse but chaos results. We found $k=300, p=1.0$ could be a good trade off between diversity and quality, the images shown in our paper are sampled with that hyper-parameters.\n\n\\noindent\\textbf{Illustration Generation}\nTo better show the model's capability, we selects several famous lines of poetry as the input and shows the generated painted style illustrations in~\\cref{fig:poem} \n\n\\begin{figure*}[htb]\n  \\centering\n  \n   \\includegraphics[width=1.0\\linewidth]{supp_figures\/sample1.pdf}\n  \n   \\caption{The generated results with different sample strategies.}\n   \\label{fig:sample1}\n\\end{figure*}\n\n\n\n\\begin{figure*}[htb]\n  \\centering\n  \n   \\includegraphics[width=1.0\\linewidth]{supp_figures\/sample2.pdf}\n  \n   \\caption{The generated results with different sample strategies.}\n   \\label{fig:sample2}\n\\end{figure*}\n\n\n\n\\begin{figure*}[htb]\n  \\centering\n  \n   \\includegraphics[width=1.0\\linewidth]{supp_figures\/sample3.pdf}\n  \n   \\caption{The generated results with different sample strategies.}\n   \\label{fig:sample3}\n\\end{figure*}\n\n\n\n\\begin{figure*}[tb]\n  \\centering\n  \n   \\includegraphics[width=1.0\\linewidth]{supp_figures\/poem_2.pdf}\n  \n   \\caption{Generated illustrations for some famous lines of poetry.}\n   \\label{fig:poem}\n\\end{figure*}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn high energy physics today, one of the most important questions is to determine whether the Higgs mass tuning indeed\nexists in nature or whether the electroweak scale is set by a mechanism that does not need\na large cancellation. This is the issue of the Higgs naturalness, or the hierarchy problem.\n\nAn attractive dynamical kind model to solve the hierarchy problem is to introduce a new symmetry\nwhich protects the Higgs against large radiative corrections. That is, these models invoke such a symmetry\nthat implies the particles existing beyond the standard model (SM) which consists of the \"symmetry partners\"\nof known SM fields.\n\nWe know that the hierarchy problem depends on the top quark one loop diagram,\ntherefore ANY model that resolves the hierarchy problem must introduce the top quark\nsymmetry partners, the \"top partners\".\nTo avoid significant residual tuning, on the other hand, the top partners are expected to\nhave masses at or below the TeV scale.\nFor examples, in supersymmetric models (for a review see  \\cite{9709356}), in little\nHiggs models \\cite{0105239,0206020,0206021,0407143} (for a review see \\cite{0502182}), there exist the scalar stops\nand vector-like fermionic top-primes as top partners, respectively.\nIn these models the new symmetry protects the Higgs commuting with the SM gauge symmetries,\nso the quantum numbers of the top partners are identical as those of the top quark.\n\nThe searches for these colored top partners, both scalar and fermionic, however, have so far suffered\nstringent limits from the large hadron collider (LHC) searches(see e.g. \\cite{1308.1586,CMS_Col,1407.0583,1406.1122}),\n so theories which include colorless top partners,\ni.e, not charged under the strong interactions, appear ever more compelling.\nSince the production cross sections of uncolored top partners are many orders of magnitude smaller\nthan in the colored case, this allows a simple understanding of why these particles have so far\nescaped discovery.\n\nColorless top partners occur in scenarios where the symmetry is district, but not the global symmetry\nas in little Higgs theories \\cite{0506256,0609152,0812.0843}.\n By far, the most striking possibility of the uncolored top partners\nis the mirror twin Higgs (MTH) model, where the Higgs is protected by the discrete $Z_2$ subgroup\\cite{0506256}\n(see also \\cite{0812.0843,0510273,0604076,1312.1341,1410.6808}).\n\n\n\n\n\n\n\nOn the other side, the huge mass of the top quark makes its lifetime very short,\nand it decays without non-perturbative hadronization effects.\nSo there is still room for non-standard top quark interactions, such as the productions and the decays.\nMoreover, the top quark strongly interacts with the yet-mysterious Higgs boson.\nSo the detailed studies of top-quark interactions would be useful to explore the mechanism of electroweak symmetry breaking\nand some properties of the Higgs boson.\n\n\n\nIn the SM, the flavor changing neutral currents (FCNCs) are absent at the tree-level, and in the loop-level, they are\nstrongly suppressed by the Glashow-Iliopoulos-Maiani (GIM) mechanism.\nWithin the SM, the Decays of the top quark induced by FCNC interactions are known to be extremely rare.\nThus, the FCNC interactions are of utmost importance in constraining the beyond SM (BSM) physics.\nHowever, this kind of loop-driven processes can get contributions from new physics particles and new couplings\nand alter greatly the SM predictions for these processes.\nIn the present paper, we will consider the rare decays of $t\\to cV$ ($V=\\gamma,~Z,~g$)\nin the context of mirror twin Higgs(MTH) models.\n\n\n\nThe paper is organized as follows. In Section 2, we briefly describe the realization of the MTH models\nwith colorless top partners and introduce the related couplings with the top rare decay.\nSection 3 is dedicated to discussions about the calculation of the the three decay processes in these\nmodels. The results are  elaborated in Section 4. Section 5 will list the compatibility of parameter space\nwith phenomenological constraints coming from electroweak precision\ndata, LHC observations. Finally, the summarization and conclusion are given in Section 6.\n\n\n\\section{the realization of the mirror twin Higgs models with colorless top partners and the correlative couplings\\cite{1411.3310}}\n\n\\subsection{The Model and Cancellation Mechanism\\label{ssec:twinmodel}}\n\nIn the MTH models, there assumes to exist a $Z_2$ distinct symmetry\nwhich exchanges the complete SM with a mirror copy of the SM, and this copy is called\nthe twin sector.\nBesides this, the global symmetry of the Higgs sector of the theory is approximate,\nwhich may be taken as either SU(4)$\\times$U(1) or O(8), and the SM\nYukawa couplings, and the SM electroweak gauge interactions will explicitly violate the\nglobal symmetry.\nThe gauge subgroup contains the SU(2)$\\times$U(1) electroweak interactions\nof the SM and of the twin sector.\nAfter the global symmetry is spontaneously broken, the SM Higgs doublet emerges as light\npseudo-Nambu-Goldstone boson.\nIn spite that the global symmetry will be violated, the discrete $Z_2$\nsymmetry, however, will be exact to ensure the absence of quadratically divergent contributions to\nthe Higgs mass at the one-loop level.\n\nIn the following SU(4)$\\times$U(1) group is taken as an example of the global symmetry to describe\nthe cancellation of the quadratic divergences in this model, and\nthe gauge subgroup of the SM and twin sectors can be taken as SU(2)$\\times$SU(2)$\\times$U(1) and $U(1)$, respectively.\nLabels $A$ and $B$ will be used to denote the SM and twin sectors, and under the action of the discrete $Z_2$\nsymmetry, the labels $A$ and $B$ will exchange with each other, $A \\leftrightarrow B$.\n Then, The field $H$, which transforms as the fundamental representation under the global SU(4) symmetry, can be written as\n \\begin{equation}\nH=\\left(\\begin{array}{c}\nH_A\\\\\nH_B\n\\end{array} \\right) \\; ,\n \\end{equation}\nwhere $H_A$ and $H_B $ represent the SM Higgs doublet and the twin doublet, respectively.\n\n\nThe SU(4) potential for $H$ is\n \\begin{equation}\nm^2 H^{\\dagger}H + \\lambda (H^{\\dagger} H)^2\n \\end{equation}\nWhen the parameter $m^2$ is negative, the global symmetry is\nspontaneously broken, SU(4)$\\times$U(1)$\\to$ SU(3)$\\times$U(1),\nand thus the gauge and Yukawa interactions engender radiative corrections\nwhich violate the global symmetry and generate a mass for $H_A$.\n\nTo cancel the quadratically divergent corrections,\nthe top Yukawa coupling can be taken as,\n  \\begin{equation}\n  \\lambda_{Ai} H_A q_{Ai} t_A + \\lambda_{Bi} H_B q_{Bi} t_B \\,.\n\\label{eq:twntopsect}\n \\end{equation}\nDue to the $Z_2$ symmetry, $\\lambda_{At}=\\lambda_{Bt}=\\lambda$ ,\nso that, at one loop order, the quadratically divergent corrections to the\nHiggs potential can be generated and cancelled out by these interactions.\nThe corrections are ($\\Lambda$ is the ultraviolet (UV) cutoff.)\n \\begin{equation}\n\\Delta V = \\frac{3}{8 \\pi^2} \\Lambda^2 \\left(\n\\lambda_{At}^2 H_A^{\\dag} H_A + \\lambda_{Bt}^2 H_B^{\\dag} H_B \\right) = \\frac{3 \\lambda^2}{8 \\pi^2} \\Lambda^2 H^{\\dagger} H \\;\\; ,\n \\end{equation}\n\nThus, with the $Z_2$ symmetry, the contribution above respects the global symmetry,\nso it cannot contribute to the mass of the Nambu-Goldstone bosons.\n\nMore generally, the cancellation mechanism of the Higgs mass can also be understood in the low\neffective theory. $H$ can then be written as,\n\\begin{equation}\nH=\\left(\\begin{array}{c}\nH_A\\\\\nH_B\n\\end{array} \\right) =\n\\exp\\left(\\frac{i}{f}\\Pi \\right)\\left(\\begin{array}{c}\n0\\\\\n0\\\\\n0\\\\\nf\n\\end{array} \\right) \\; .\n \\end{equation}\nWhere $f$ is the symmetry breaking vacuum expectation value(VEV),\nand $\\Pi$ is\n\\begin{equation}\n\\Pi=\\left(\\begin{array}{ccc|c}\n0&0&0&h_1\\\\\n0&0&0&h_2\\\\\n0&0&0&0\\\\ \\hline\nh_1^{\\ast}&h_2^{\\ast}&0&0\n\\end{array} \\right).\n \\end{equation}\nExpanding out the exponential, we have\n \\begin{equation}\nH=\\left(\\begin{array}{c}\n\\displaystyle \\bm{h}\\frac{if}{\\sqrt{\\bm{h}^{\\dag}\\bm{h}}}\\sin\\left( \\frac{\\sqrt{\\bm{h}^{\\dag}\\bm{h}}}{f} \\right)\\\\\n0\\\\\n\\displaystyle f\\cos\\left( \\frac{\\sqrt{\\bm{h}^{\\dag}\\bm{h}}}{f} \\right)\n\\end{array} \\right)\n\\end{equation}\n where $\\bm{h} = (h_1, h_2)^{\\text{T}}$ is the Higgs doublet of the SM\n \\begin{align}\n\\displaystyle H_A&= \\bm{h}\\frac{if}{\\sqrt{\\bm{h}^{\\dag}\\bm{h}}}\\sin\\left( \\frac{\\sqrt{\\bm{h}^{\\dag}\\bm{h}}}{f} \\right)=i \\bm{h}+\\ldots\\,, \\\\\n H_B&= \\left(\\begin{array}{c}\n0\\\\\nf\\cos\\left( \\frac{\\sqrt{\\bm{h}^{\\dag}\\bm{h}}}{f} \\right)\n\\end{array} \\right)=\\left(\\begin{array}{c}\n0\\\\\n\\displaystyle f-\\frac{1}{2f}\\bm{h}^{\\dag}\\bm{h}+\\ldots\n\\end{array} \\right).\n \\end{align}\n Now considering Eq.~\\ref{eq:twntopsect} in quadratic order of $\\bm{h}$,\n \\begin{equation}\ni\\lambda_i\\bm{h} q_{Ai}t_A+\\lambda_i\\left(f-\\frac{1}{2f}\\bm{h}^{\\dag}\\bm{h} \\right) q_{Bi}t_B \\; . \\label{eq:twntoplag}\n \\end{equation}\n Thus the quadratic divergence arising from the first diagram is exactly canceled by that of the second\n via evaluating these contributions.\n\n\n\n\\subsection{The quark flavor changing couplings\\label{ssec:fcc}}\nNow we focus on the flavor changing of the top quark. Firstly,\nwe determine the low energy couplings of the Higgs.\nChoosing unitary gauge in the visible sector with $h_1=0$ and $h_2=(v+\\rho)\/\\sqrt{2}$, we obtain\n \\begin{equation}\\begin{array}{cc}\nH_A=\\left(\\begin{array}{c}\n0\\\\\n\\displaystyle if\\sin\\left(\\frac{v+\\rho}{\\sqrt{2}f} \\right)\n\\end{array}\\right), &H_B=\\left(\\begin{array}{c}\n0\\\\\n\\displaystyle f\\cos\\left(\\frac{v+\\rho}{\\sqrt{2}f} \\right)\n\\end{array}\\right).\n\\end{array}\n \\end{equation}\n The kinetic terms are\n \\begin{equation}\n\\left|D_{\\mu}^AH_A \\right|^2+\\left|D_{\\mu}^BH_B \\right|^2\n \\end{equation}\n where the $D^{A,B}$ denote the covariant derivative of the $A,B$ gauge bosons.\n From the above equation one can obtain the masses of the $W^\\pm$ and $Z$ gauge bosons in\nthe visible $A$ and twin sectors $B$ and their couplings to the Higgs, $\\rho$, which\ndetermine the relation of the Higgs SM VEV $v_{\\text{EW}}=$246 GeV and the MTH parameters $v$ and $f$,\n \\begin{equation}\nv_{\\text{EW}}=\\sqrt{2}f\\sin\\left(\\frac{v}{\\sqrt{2}f} \\right)\\equiv\\sqrt{2}f\\sin\\vartheta  \\; .\n\\label{vew_v}\n\\end{equation}\nwhere the angle $\\vartheta=\\frac{v}{\\sqrt{2}f}$, and when $v=v_{\\text{EW}}$, $v \\ll f$, or equivalently\n$\\vartheta\\ll 1$.\n\n\n\n\nExpanding  the top quark sector \\eqref{eq:twntopsect} in the unitary gauge,\n\\begin{align} \\nm\n&\\lambda_i\\left[ifq_{Ai}t_A\\sin\\left(\\frac{v+\\rho}{\\sqrt{2}f} \\right) +fq_{Bi}t_B\\cos\\left( \\frac{v+\\rho}{\\sqrt{2}f}\\right) \\right]\\\\\n=&i\\frac{\\lambda_i v_{\\text{EW}}} {\\sqrt{2}}q_{Ai}t_A\\left[1+\\frac{\\rho}{v_{\\text{EW}}}\\cos\\vartheta \\right]+ \\lambda_i fq_{Bi}t_B\\cos\\vartheta\\left[ 1-\\frac{\\rho}{v_{\\text{EW}}} \\tan\\vartheta\\sin\\vartheta \\right]\n\\label{tc_coup}\n\\end{align}\nHere $q_i$ can be quarks $u,~c$ or $t$.\n\n From this, we can also see that the mass of the top quark's mirror twin partner is\n\\begin{equation}\nm_T=\\lambda_t f\\cos\\vartheta=m_t\\cot\\vartheta\\, .\n\\end{equation}\n\n\nFrom Eq.(\\ref{tc_coup}), we can also see clearly that the scalar $\\rho$ acts\nas the SM-like Higgs, and it consists of both visible and invisible\npart in some ratio according to a certain parameters.\n\n\\section{Calculation of the top rare decays $t\\to cV$}\nFrom Eq.(\\ref{tc_coup}), we can see the flavor changing couplings mediated by the neutral scalar,\nso the FCNC decays $t\\to c V$ can be realized by it, and the Feynman diagrams are listed in Fig.\\ref{fig:tcv}.\n\\begin{figure}[!htbp]\n  \\centering\n   \\includegraphics[scale=0.95]{fig1}\n  \\vspace{-0.9cm}\\caption[]{Feynman diagrams for the process $t\\to cV$ in the 't Hooft--Feynman gauge in the MTH model. }\n\\label{fig:tcv}\n\\end{figure}\nWe can see from Eq.(\\ref{tc_coup}) that the structure the fermions couplings to the scalar $\\rho$ are very simple.\nTo be more general, we write the couplings of the scalar to the fermions as\n\\beq\n\\rho q_{Ai}\\bar t_A: i\\frac{\\lambda_i} {\\sqrt{2}} (c+d\\gamma^5)\n\\label{tc_coup_gene}\n\\eeq\nwhere $q_{Ai}=t,~c$(we only consider the visible section A ), $\\lambda_c=V_{tc}\\lambda_t$ and $V_{tc}$ is the\nratio of the two couplings, a bit like the CKM matrix element $U_{tc}$.\nThe parameters $c$ and $d$ are just to expand the lorentz structure,\nand from Eq. (\\ref{tc_coup}), we can see that $c=\\frac{1}{2},~ d=0$.\nIn the following discussion, however, we can also release the constricts and to check\nthe influence on the branching ratios in Fig. \\ref{fig3}.\n\n\\subsection{The amplitude and the width of $t\\to cg$, $t\\to c \\gamma$ and $t\\to cZ$}\n\nWith the general coupling of the Yukawa form, we can write out the amplitude of the decay $t\\to cg$, (taking\nFig.\\ref{fig:tcv} (c) as an example)\n\\bea \\nm\n{\\cal M}_c&=& \\bar u_c (i\\frac{\\lambda_c } {\\sqrt{2}} )(c+d\\gamma^5)\\frac{i}{\\sk-m_t}(-ig_s\\gamma^\\mu T^a)\n\\frac{i}{\\sk+\\spg-m_t}(-i\\frac{\\lambda_t } {\\sqrt{2}} )    \\\\ \\nm &&\n(c-d\\gamma^5) \\frac{i}{(k+p_g-p_t)^2-m^2_\\rho} u_t \\epsilon_\\mu \\\\ \\nm\n&=&-\\frac{\\lambda_c\\lambda_t }{2}g_sT^a\\frac{1}{k^2-m_t^2}\\frac{1}{(k+p_g)^2-m_t^2}\\frac{1}{(k+p_g-p_t)^2-m^2_\\rho} \\\\ \\nm &&\n\\bar u_c  (c+d\\gamma^5)(\\sk+m_t)\\gamma^\\mu(\\sk+\\spg+m_t)(c-d\\gamma^5)u_t \\epsilon_\\mu\n\\eea\nNote that in the above formula, we have omitted the common integrated factor $\\tau^{2\\epsilon }\\int^{+\\infty}_{-\\infty} \\frac{d^nk}{(2\\pi)^n}$,\n$\\epsilon=1-\\frac{n}{2}$, $\\tau$ is the scale factor introduced to keep the dimension of the coupling constants unchanged.\n The decay width in general is given by\n\\beq\n\\Gamma_{t\\to cg} = C_F \\frac{1}{32\\pi}|M|^2\n\\eeq \nwhere $C_{F} = 4\/3$ is a color factor.\n\nAs for the width of $t\\to c\\gamma$, we can simply obtain it by replacing the coupling $g_s$ with $e$ in the amplitude, and $C_{F}$ with $1$.\nSince the coupling of $Z\\bar t t$ is a little complicated than that of $\\gamma \\bar t t $, so in the calculation of the\n$t\\to c Z$, we need to replace the $t\\to c\\gamma$ parameters $e$ with $e\/(2s_Wc_W)$,\nand the structure $\\gamma^\\mu$ with $\\gamma^\\mu(P_L-2s_W^2)$.\n\n\nIn this scalar-mediated decay process, the sum of the one-loop divergent terms is zero.\nThat's to say, the one-loop divergences cancel out with each other in the Feynman gauge,\nso we can safely use the calculating tool of LoopTools\\cite{looptools}.\n\nOf course, the effective vertex $\\bar{t}cV$ is a 4-component Lorentz vector and also\na $4 \\times 4$ matrix in Dirac space and need to be managed.\nOne trick is that the tensor loop functions are retained rather than expanding them\nin term of scalar loop functions as usual \\cite{Hooft}.\n\nIn its realization of Fortran, a dimension-three array $V(i,j,k)$\nwith $i$ (=1,2,3,4) labelling the Lorentz index and  $j,k$ (=1,2,3,4)\nlabelling the spinor indices are used. See the details in Ref.\\cite{0702264}.\n\n\n\n\n\n\\subsection{The top total width and the upper bounds to the rare decays }\n\n The decay width of the dominant decay mode of the top\nquark  $t\\to bW$ is given by ~\\cite{1807.01481,Dey:2016cve}\n\\begin{align}\n\\Gamma_{t\\to bW} = \\frac{G_{F}}{8\\sqrt{2}\\pi}|V_{tb}|^{2}\n                    m_{t}^{3}\n                    \\left[\n                    1 - 3\\left(\\frac{m_{W}}{m_{t}}\\right)^{4}\n                      + 2\\left(\\frac{m_{W}}{m_{t}}\\right)^{6}\n                    \\right].\n\\end{align}\nThe above equation gives $\\Gamma_{t\\to bW} \\sim 1.5$~GeV.\nSo the branching ratio of any other mode $t\\to X$ is\n\\begin{align}\n\\text{BR}(t\\to X) = \\frac{\\Gamma_{t\\to X}}{\\Gamma_{t\\to bW}}.\n\\end{align}\nThe SM predictions for the $t\\to cg$, $t \\to c\\gamma$ and $t\\to cZ$ branching ratios are \\cite{1807.01481,Dey:2016cve,Abbas:2015cua}\n\\beq\n\\text{BR}(t\\to cg) = \\left(4.6^{+1.1}_{-0.9}     \\pm 0.4^{+2.1}_{-0.7}   \\right)\\times 10^{-12},\n\\label{tcg_sm}\n\\eeq\n\\beq\nBR(t \\to c\\gamma) = \\left(4.6^{+1.2}_{-1.0}\\pm 0.4^{+1.6}_{-0.5} \\right)\\times 10^{-14},\n\\label{tcr_sm}\n\\eeq\n\\beq\n\\text{BR}(t\\to cZ) =(1.03\\pm 0.06)\\times 10^{-14},\n\\label{tcz_sm}\n\\eeq\nLHC has searched for these rare decays and give their upper bounds~\\cite{lhcTopWGNov:2017,Aad:2015gea, Khachatryan:2016sib, ATLAS-CONF-2017-070,Sirunyan:2017kkr, t_cr_bounds}:\n\\beq\n BR(t\\to cg) < 2\\times 10^{-4}.\n \\label{tcg_lhc}\n \\eeq\n\\beq\n BR(t\\to cZ) < 2\\times 10^{-4}.\n \\label{tcr_lhc}\n \\eeq\n\\beq\n Br(t \\to c\\gamma)< 1.82\\times 10^{-3}.\n\\label{tcz_lhc}\n\\eeq\nFrom Eq.(\\ref{tcg_lhc}) to Eq.(\\ref{tcz_lhc}), we can see that the predictions of the\n SM Eq.(\\ref{tcg_sm}) to Eq.(\\ref{tcz_sm}), are impossible to be probed\nat the LHC, so any signal of the flavor decays of this kind is inevertably the signature\nof the new physics.\n\n\nSome BSM scenarios may predict an enhanced branching ratio of these rare modes up to the level that can be detected\nin the future colliders, such as 2HDM~\\cite{Eilam:1990zc, Abbas:2015cua, Gaitan:2017tka},\n     left-right symmetric model~\\cite{Gaitan:2004by}, MSSM~\\cite{0702264},\n     $R$-parity violating SUSY~\\cite{Bardhan:2016txk}, warped extra dimensional\n     models~\\cite{Agashe:2006wa, Gao:2013fxa},  UED models~\\cite{GonzalezSprinberg:2007zz}, mUED and nmUED models \\cite{Dey:2016cve,1807.01481}\n     and composite Higgs model~\\cite{Agashe:2009di,tc2-comp-fcnc}, etc.\nIn Ref.~\\cite{CorderoCid:2004vi, Datta:2009zb} one can find that the effective Lagrangian approach is used to study of rare top decays.\nOther collider stuides to the search of these rare decays can be found in Ref. \\cite{DiazCruz:1989ub, Mele:1998ag, AguilarSaavedra:2002ns, AguilarSaavedra:2004wm, Chen:2013qta, Khanpour:2014xla, Hesari:2014eua, Kim:2015oua, Hesari:2015oya, Khatibi:2015aal, Malekhosseini:2018fgp, Banerjee:2018fsx}, etc.\nIn the following, we will check whether the prediction of the MTH models may arrive at the detectable level\nand provide constraints to the model parameters.\n\n\n\n\n\n\\subsection{The results for $t\\to cV$ }\n\\def\\figsubcap#1{\\par\\noindent\\centering\\footnotesize(#1)}\n \\begin{figure}[htb\n \\begin{center}       \\hspace{-1.5cm}\n  \\parbox{5cm}{\\epsfig{figure=fig2a.ps,width=5cm}    \\figsubcap{a}}\n  \\parbox{5cm}{\\epsfig{figure=fig2b.ps,width=5cm}    \\figsubcap{b}}\n  \\parbox{5cm}{\\epsfig{figure=fig2c.ps,width=5cm}   \\figsubcap{c}}\n \\caption{ The one-loop level branching ratios of the three processes in the MTH model.\n  \\label{fig2} }\n   \\end{center}\n \\end{figure}\n\n\n\nIn order to find constraints on the parameters $f$, the VEV $v$ (To find the relation between it and the electroweak\nVEV $v_{EW}$, one can refer to Eq.(\\ref{vew_v})), and the flavor changing coupling $V_{tc}$,\nthe one-loop level branching ratios of the three processes\nare given in Fig.\\ref{fig2}.\nWe take $m_t = 172.5$ GeV and $M_Z = 91.2$ GeV,\n and other physical constants are obtained from the Review of Particle Physics \\cite{pdg-2018}.\n\nFigures in Fig.\\ref{fig2} show that the branching ratios are respectively in the range: $Br(t\\to cg) \\sim  10^{-3}-10^{-6}$,\n$Br(t\\to c\\gamma) \\sim 10^{-4}-10^{-8}$ and $Br(t\\to cZ) \\sim 10^{-5}-10^{-8}$.\nThe reason that the $t\\to cg$ is larger than those of the $t\\to c\\gamma~\/Z$\nis that the $g\\bar q q$ coupling in this process with QCD coupling $\\alpha_S$ involved,\n is one-order larger than the electroweak coupling $\\gamma \\bar q q \\sim \\alpha_e$.\n\nWe can see from Fig.\\ref{fig2} that the influence of the parameters on the branching ratios is not the same:\nVarying $f$ or $v$ doesn't make much difference when other parameters are fixed, as shown in Fig.\\ref{fig2}(a)(b),\nwhile varying $V_{tc}$ will change the branching ratios largely. However, the alteration is an associative effect of\nthe parameters $f$, $v$ and $V_{tc}$, since in the couplings Eq.(\\ref{tc_coup_gene}), $\\lambda_c$ is connected with\nthe threes and so not quadratically increasing with the increasing $V_{tc}$.\n\nFrom Fig.\\ref{fig2}, we can also see that the process $t\\to cg$ may be hopeful to arrive at the detectable level\n according to the experimental bounds in Eqs. (\\ref{tcg_lhc}), (\\ref{tcr_lhc}), (\\ref{tcz_lhc}),\n and may provide constraints on the parameters. So in the following, we will only consider the process $t\\to cg$.\n\n\\begin{figure}[!htbp]\n  \\centering\n    \\includegraphics[scale=0.5]{fig3}\n  \\vspace{-0.7cm}\\caption[]{ The one-loop level branching ratios of the three processes in the MTH model versus to the structure parameter $c$. }\n\\label{fig3}\n\\end{figure}\n\nTo see the effect of the structure parameters $c$ and $d$ in Eq.(\\ref{tc_coup_gene}) on the branching ratio of $t\\to cg$,\nwe in Fig.\\ref{fig3} vary $c$ between $(-\\frac{1}{2},\\frac{1}{2})$ with $d=0,~c,~\\frac{1}{2}$, respectively,\nand we can find that the\nof course, $c$ and $d$ cannot be equal to zero simultaneously, because if so the coupling $\\rho \\bar q_i t$ will vanish,\nand so does the $t\\to cg $ branching ratio.\nNormally, they should be the $\\pm\\frac{1}{2}$ and $0$ (asynchronously).\n\n\n\n\\def\\figsubcap#1{\\par\\noindent\\centering\\footnotesize(#1)}\n \\begin{figure}[htb\n \\begin{center}       \\hspace{-1.5cm}\n  \\parbox{5cm}{\\epsfig{figure=fig41.ps,width=5cm}\\figsubcap{a}}\n  \\parbox{5cm}{\\epsfig{figure=fig42.ps,width=5cm}    \\figsubcap{b}}\n  \\parbox{5cm}{\\epsfig{figure=fig43.ps,width=5cm}    \\figsubcap{c}}\n \\caption{ The $t\\to cg$ detective contours on $v\\sim f$, $f\\sim V_{tc}$, and $v\\sim V_{tc}$\n in the MTH model.  \\label{fig4} }\n   \\end{center}\n \\end{figure}\n\nIn Fig.\\ref{fig2} and  \\ref{fig3}, the parameter values are taken to be in the optimum case, i.g, we take\n$v=250$ GeV, $f=500$ GeV, and $V_{tc}=0.2$, quite a large coupling.\nBut actually, the parameters may be in unfavorable case, which is not\npossible to be detected at the colliders, so\nwe scan the whole parameter spaces to find out the possible room to be accessible to the model.\nIf it is difficult to mark the signal, the parameters of the models will face severe constraints.\nThat is, if we cannot find the process,\nit may serve as a robust measurement to constrain the model parameters, especially $V_{tc}$.\n\n\nFig.\\ref{fig4} scans the possibility of the $t\\to cg$ branching ratios in light of the detectable level in the following\nthree parameter ranges:\n$500 \\leq f \\leq 10000$ GeV, $ 250 \\leq v \\leq 1000$ GeV, and $0.001 \\leq V_{tc}\\leq 0.5$.\nFrom Fig.\\ref{fig4}(a), we can see that the branching ratios prefer to a small $v$, while\ninsensitive to $f$, which is because that, via Eq.(\\ref{vew_v}),\nthe coupling $\\lambda_t=\\sqrt{2}m_t\/(f sin\\vartheta)=\\frac{\\sqrt{2}m_t}{f sin [v\/(\\sqrt{2}f)]}$,\nso the smaller $v$ is, the larger the branching ratio is.\nIn the meantime, with two compelling $f$s in the denominator, we can conclude that $v$ contributes\nlarger than $f$, which is also seen clearly in the next two figures.\n\nHowever, $V_{tc}$ is close to linear in the amplitude, so the branching ratios will increase rapidly with\nthe increasing $V_{tc}$. Hence, to arrive at the detectable level, the larger $V_{tc}$ will be preferable,\nwhich can be seen clearly in Fig.\\ref{fig4}(b)(c).\n\nFrom Fig.\\ref{fig4}, we can see that the parameters are constrained\nin a very narrow space, if the FCNC decay $t\\to cg$ cannot be detected.\nSince $v$ and $f$ contribute small, $V_{tc}$ is strongly restricted.\n When $V_{tc}\\leq 0.13$, the $t\\to cg$ branching ratio is normally smaller than the detectable level.\n\n\n\n\nSince it is impossible for the branching ratio of the $t\\to c\\gamma$ to arrive at the limit Eq.(\\ref{tcr_lhc}),\nthe choosing areas of the parameters $v$, $f$, $V_{tc}$ are not affected by this decay.\nAs for the $t\\to cZ$, there are very little points can arrive at the limit Eq.(\\ref{tcz_lhc}),\nand the parameter $V_{tc}$ is constrained severely: $V_{tc}\\leq 0.45$.\n\n\nSo one can conclude that the MTH model can enhance the branching ratios\nof $t\\to  c V$  to some significant level from the SM values and may be in the allowed ranges\nof the LHC constraints.\n\\section{Summary and Conclusions}\nWe have performed a complete one-loop calculation of the flavour-changing top quark decays $(t\\to cV)$ ($V=g,\\gamma,Z$)\nin the context of mirror twin Higgs models.\nSince the LHC experimental searches are concerned, performing some searches of FCNC top decays will be possible and\nviable.\nAnd some Refs \\cite{top-review} have provided the projected limits for higher energies on top FCNCs at the LHC and ILC.\nFrom these data we can see clearly that even in the higher energetic Run-II\nof the LHC, the sensitivity will not reach the limit to probe the small branching ratios as obtained\nin the theoretical calculations from the SM.\nHowever, there are many BSM scenarios in which these branching ratios are enhanced quite large,\neven to the level that may be probed in the Run-II of LHC in some parameter space.\nThe aim of this work is to look into this issue of rare decays in one of the popular BSM scenario, i.e., MTH models.\nWe show that all the decay widths of $t\\to cV$ do change much from the SM value for favorable parameters\nas the consequence of the colorless top partner in this kind of model.\nThese results are not wield since the coupling between the scalar and the quarks $y_t= m_t\/(fsin\\vartheta)$\ncan be quite large in some parameter spaces. And with the development of the future collider,\nthe decays may be much more hopeful to be probed than nowadays.\n\n\n\n\n{\\it Acknowledgements:} Guo-Li Liu would like to thank Fei Wang for very helpful discussions.\nThis work was supported by the National Natural Science Foundation of China\nunder grant 11675147, 11775012 and by the Academic Improvement Project of Zhengzhou\nUniversity.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{Introduction}}\nA two-dimensional electron gas (2DEG) in the quantum Hall effect (QHE) regime demonstrates a number of intriguing phenomena that are not fully explained up to date. Among them are non-equilibrium phenomena manifested in hysteresis of sheet electron density, \\cite{Pudalov1984, Dolgopolov1992, Dolgopolov1993} magnetization (see for review Ref.[\\onlinecite{Usher2009}]) and electrochemical potential \\cite{Huels2004, Klaffs2004} as a function of magnetic field and gate voltage (for gated structures). The hysteresis was observed in an ordinary single-layer 2DEG in nonmagnetic materials such as Si MOSFETs and AlGaAs\/GaAs heterostructures at integer and fractional filling factors corresponding to a longitudinal resistance ($R_\\mathrm{L}$) vanishing.\n\nIt was found that the hysteretic variations of magnetization \\cite{Usher2009} and electrochemical potential \\cite{Huels2004, Klaffs2004} exceed equilibrium values by a factor of 10-60. In contrast the conventional electron transport measurements in QHE regime with vanishing $R_\\mathrm{L}$ are not sensitive to the hysteresis, though, the question of a probable indirect influence of the non-equilibrium on the transport properties of 2DEG still remains open.\n\n\nRecently in a number of papers it was demonstrated that creation of a constriction in a macroscopic bath of 2DEG allows to reveal the hysteretic behavior in magnetoresistance \\cite{Budantsev2007, Budantsev2009, Budantsev22009, Budantsev2011, Pioro-Ladriere2006, Smith2011}.\n\nIt was shown that at low temperatures (lower than 100~mK) the hysteresis relaxes over a day \\cite{Pioro-Ladriere2006} which is far beyond the time usually spent on a conventional measurement (a magnetic field sweep) in the QHE regime. In terms of usual magnetic hysteresis description \\cite{Bertotti1998} this type of hysteresis is considered as time-independent.\n\nThe most common explanation of the hysteretic phenomenon is based on non-equilibrium currents (NECs) slowly relaxing providing $R_\\mathrm{L}$ vanishing. But there are no unambiguous models in the literature, adequately describing this phenomenon. The detailed consideration of NECs induction is proposed in the Ref.~ [\\onlinecite{Shikin2002}], but it concerns only the case of weak non-equilibrium. Ruhe et al. \\cite{Ruhe2009} studied the case of time-dependent hysteresis. The most developed model up to now \\cite{Matthews2004} is based on the assumption of redistribution of ``frozen'' charges over the whole bulk area of the 2DEG \\cite{Dyakonov1991}. These charges give rise to the radial (for the disk geometry) electric field that in turn gives rise to azimuthal drift magnetization current. The magnetization maximum is determined by the QHE breakdown current, calculated in the frames of quasi-elastic inter-Landau level scattering (QUILLS) approach.\\cite{Eaves1986} The mentioned model correctly describes experimentally observed linear temperature suppression of the hysteresis amplitude observed both in magnetization \\cite{Matthews2004} and constriction conductance.\\cite{Budantsev2007, Budantsev2009} But the magnetization maximum calculated in the model is less than experimentally obtained value by a factor of 4. Moreover, the model predicts bulk NECs distribution, which, to our opinion, contradicts with experimental data of Klaffs et. al. \\cite{Klaffs2004}\n\nIn the present work we study the hysteresis of the magnetoresistance of the conventional 2DEG with constriction, placed in it. It was found that at certain critical width of the constriction the hysteresis of the magnetoresistance vanishes. This observation strongly suggests that NECs are localized near the edge of the sample.\n\nA phenomenological comparison of the 2DEG magnetoresistance hysteresis in the QHE regime with the hysteresis of magnetization of ordinary ferromagnetic materials is performed. Particularly, we discuss the dependence on the magnetic field sweep prehistory for both cases, studying minor loops, anhysteretic curves and time relaxations. Mostly we observe a similar behavior. However significant difference has been found manifested itself in unusual advancing (anti-coercive) response (2DEG magnetoresistance) to external parameter (magnetic field) change, which is just opposite to the behavior of ferromagnetic materials.\n\nWe compare our results with the hysteresis of magnetoresistance in systems with pseudo-spin degree of freedom --- so-called ``quantum Hall ferromagnets'' \\cite{Piazza1999, DePoortere2000, Jungwirth2001}. In some cases (see e.g. Ref.~[\\onlinecite{Piazza1999}]) analogical anti-coercive behavior is found. This phenomenological similarity suggests that there are similar mechanisms underlying the origin of the hysteresis both in ordinary 2DEG and in QH ferromagnets.\n\nIn addition to the phenomenological description, we provide a qualitative picture of a spatial distribution of NECs and electrochemical potential in the sample.\n\nThe paper is organized in the following way. In Sec.~\\ref{Experimental details} we describe the experimental detail. In Sec.~\\ref{Experimental results} we present the experimental data of following measurements: the magnetoresistance of the 2DEG with the constriction in the QHE regime, minor loop, anhysteretic curve, time relaxation dependence of the magnetoresistance, and the dependence of the magnetoresistance hysteresis loop area on the lithographic width of the constriction and discuss them. In Sec.~\\ref{Discussion} we develop a detailed qualitative picture of spatial distribution of NECs in a sample, explaining the hysteresis of the magnetoresistance of the constriction as well as the non-equilibrium magnetization of a 2DEG in the QHE regime. We discuss inverted (anti-coercive) behavior of the hysteresis and compare the hysteresis of the magnetoresistance with the hysteresis of the ferromagnetic materials magnetization. We conclude in Sec.~\\ref{Conclusions}.\n\n\\section{Experimental details\\label{Experimental details}}\nExperimental samples were fabricated from GaAs\/AlGaAs heterojunctions with 2DEG of two types (type I and type II) grown in different cycles of molecular beam epitaxy. The electron mobility $\\mu$ and spacer layer thickness $d$ are $\\mu$=0.6$\\div$0.8$\\times$10$^{6}$~cm$^{2}$\/V$\\cdot$s and $d=400$~{\\AA} in the type I sample and  $\\mu$=0.8$\\div$1.0$\\times$10$^{6}$~cm$^{2}$\/V$\\cdot$s and $d=300$~{\\AA} in the type II sample. The electron densities of the macroscopic 2DEGs of both types are n$_\\mathrm{s}$=3$\\times$10$^{11}$~cm$^{-2}$ at the temperature of 4.2~K. Hall bars of the dimensions $W$$\\times$$L$=50$\\times$100~$\\mu$m$^{2}$ have been created on the surface of the heterostructures by means of photolithography. In the middle part of the Hall bars constrictions of the effective width 0.6$\\div$0.8\\,$\\mu$m and the length 3~$\\mu$m have been created by means of electron beam lithography (see the inset of Fig.~\\ref{fig1}). The samples of the both type allow to measure longitudinal magnetoresistance $R_\\mathrm{L}=U_{35}\/I_{12}$ and Hall magnetoresistance of macroscopic 2DEG $R_\\mathrm{H}=U_{36}\/I_{12}$. The measurements are carried out in the linear response regime on the alternating current of the magnitude 1$\\div$10~nA and the frequency 7~Hz at the temperature of 60~mK. The magnetic field covering a range of 0$\\div$15\\,T was oriented perpendicular to the 2DEG.\n\\begin{figure}\n\\includegraphics[width=.49\\textwidth]{fig_1}\n\\caption{The magnetoresistance of the samples of the type I. At the centers of the plateaux of macroscopic 2DEG Hall resistance $R_\\mathrm{H}~=~U_{36}\/I_{12}$ at odd and even integer filling factors a\ngiant hysteresis of longitudinal magnetoresistance $R_\\mathrm{L}~=~U_{35}\/I_{12}$ is observed. The geometry of the sample is shown in the inset.\\label{fig1}}\n\\end{figure}\n\\section{Experimental results\\label{Experimental results}}\nFirst, we have studied the type I samples. A giant hysteresis of longitudinal magnetoresistance $R_\\mathrm{L}$ of the constriction with respect to magnetic field sweep direction is observed in magnetic fields corresponding to centers of the plateau of macroscopic 2DEG Hall resistance $R_\\mathrm{H}$ at both odd and even integer filling factors (Fig.~\\ref{fig1}). The effect is independent on a direction of magnetic field.\nThe most pronounced hysteresis loop is observed near the filling factor $\\nu=1$ in the range of filling factors between $\\nu_\\mathrm{h1}=0.95$ and $\\nu_\\mathrm{h2}=1.07$ on the background of macroscopic 2DEG Hall plateau observed in the range of filling factors between $\\nu_1=0.85$ and $\\nu_2=1.22$ (Fig.~\\ref{fig1}). The hysteresis has the amplitude 10\\,k$\\Omega$ amounting to 100$\\%$ of measuring signal and the width width of the hysteresis is 1.2~T at filling factor $\\nu=1$. Drastic step-like rise of the magnetoresistance from zero up to 10~$k\\Omega$ at up-sweep of the magnetic field and sharp drop down to zero at down-sweep of the magnetic field allows us to consider this behavior as the magneto-induced breakdown of QHE.\n\n The large amplitude of the hysteresis made it possible to carry out a detailed study. It has been found that the response of the system (magnetoresistance) outpaces the external excitation (magnetic field change) rather than retards from it, unlike the case of ferromagnetic magnetization. In mathematical terms the hysteresis loop is negatively oriented curve, while in ferromagnetic materials the hysteresis loop is positively oriented. One can see that the magnetoresistance dramatically changes at the entrance to the area of hysteresis, that is $|dR\/dB|$ tends to a huge value, and it remains almost unchanged when leaving the area. In the case of retarded behavior the magnetoresistance would practically not be changed at the entrance to the area of hysteresis, that is $|dR\/dB|$ would be close to zero. In this sense the observed hysteresis is abnormally inverted having a negative coercivity in contrast to the ferromagnetic magnetization.\n\n\\begin{figure}\n\\includegraphics[width=.49\\textwidth]{fig_2}\n\\caption{\nThe hysteresis of the magnetoresistance of the II type sample at the filling factor $\\nu=1$. (a) Major and minor loops. (b) Anhysteretic curve.  Arrows indicate the magnetic field sweep direction. The sweep-rate is 0.05~T\/min. The geometry of the sample and relaxation are shown in the insets. One can see the fast relaxation regime with $\\tau$=2.4~s. The relaxation is occurred by jumps simultaneously in both structures.\\label{fig2}}\n\\end{figure}\n\nThe magnetoresistance of the type II sample also demonstrates the hysteresis of magnetoresistance (Fig.~\\ref{fig2}) in the range of filling factors between $\\nu_\\mathrm{h1}=0.96$ and $\\nu_\\mathrm{h2}=1.06$ on the background of macroscopic 2DEG Hall plateau observed in the range of filling factors between $\\nu_1=0.88$ and $\\nu_2=1.16$. One can see that the hysteresis of magnetoresistance of the type II samples is observed at the same range of filling factors as in the type I samples but it does not demonstrate a giant amplitude and drastic jumps. This distinction is probably resulted from the differences in heterostructures parameters and needs to be discussed separately. Nevertheless, we succeeded to reveal inverted (anti-coercive) behavior of the magnetoresistance hysteresis of the type II sample studying the form of minor hysteretic loops. One can see that a change of the magnetic field sweep direction inside the major hysteretic loop leads to dramatic change of the magnetoresistance, i.e. the derivative $|dR\/dB|$ tends to a huge value (Fig.~\\ref{fig2}(a)), while the retarding behavior would correspond to zero $|dR\/dB|$ (as in the case of minor loops of ferromagnetic materials, where $|dM\/dH|=0$). Moreover the analysis of the minor loops allows us to conclude that there is a large number of sample states inside the major hysteresis loop, and therefore the sample state is multistable.\n\nIt is generally accepted that long-lived NECs are induced on the background of Hall plateaus where $\\sigma_{xx}$ is vanishing. Our measurements shows that the range of existence of the hysteresis $\\Delta\\nu_\\mathrm{h}$ is approximately 3 times less than the Hall plateau width $\\Delta\\nu$: $\\Delta\\nu_\\mathrm{h}$\/$\\Delta\\nu$$\\approx$1\/3. That is the condition $\\sigma_{xx}\\approx0$ is necessary but not sufficient to observe the hysteresis.\n\nTo perform consistent phenomenological comparison of the 2DEG magnetoresistance hysteresis in QHE regime with the hysteresis of magnetization of ferromagnetic materials we have measured the anhysteretic curves (Fig.~\\ref{fig2}(b)). Each experimental point $R_i(B_i)$  has been obtained by cycling the magnetic field around $B_i$ with a decreasing amplitude. The form of the anhysteretic curve and its position inside the loop are phenomenologically similar to that of ferromagnetic materials.\n\nWe have studied the time relaxation of the hysteresis, measuring the time dependence of the magnetoresistances of two conducting channels situated in the same Hall bar and separated by macroscopic 2DEG reservoir of the length of 250~$\\mu$m and the width of 50~$\\mu$m (see the top inset of Fig.~\\ref{fig2}(a)). The dependencies are shown in the insets of Fig.~\\ref{fig2}(a) and (b). The curves have been obtained after stopping the up-sweep of the magnetic field at the value 12.25~T, corresponding to the filling factor $\\nu=1$ where the hysteresis has the maximum amplitude.\n\nIt has been found that time relaxation has two phases: the fast initial one followed by slow phase. After stopping the magnetic field sweep we have observed the fast exponential relaxation by relatively small value (less than 25$\\%$) with relaxation time 2.4~s (see the bottom inset of Fig.~\\ref{fig2}(a)). Such relaxation time is too big to be explained by spin-orbital coupling mechanism of relaxation \\cite{Muller1992, Khaetskii1992} and resembles the nuclear relaxation time in GaAs which is about 30~s.\\cite{Dixon1997, Devyatov2004} It should be noted that the non-equilibrium magnetization relaxes in the similar way.\\cite{Kershaw2007} It also has regime of initial exponential decay followed by a much slower power-law decay, however the relaxation time in the first regime is longer ($\\sim$20~s) than obtained in our work. In Ref.~[\\onlinecite{Matthews2004}] it has been suggested that magnetic field change results in QHE breakdown accompanied by electron transitions between adjacent Landau levels with opposite spin. The observed fast relaxation can be explained by such inter-Landau levels scattering and brings the system to a local (not global) minimum of energy after stopping the magnetic field sweep. According to Ref.~[\\onlinecite{Matthews2004}] such scattering results from QUILLS process \\cite{Eaves1986} and takes place in a part of the sample where the maximum electric field is reached. In Refs.~[\\onlinecite{Matthews2004},~\\onlinecite{Dyakonov1991}] it is suggested that maximum electric field is reached in the bulk of a sample. However our results are sensitive to the sample edge (see Section~\\ref{Discussion}) since the magnetoresistance is defined by the edge states filling. This allows us to conclude that QHE breakdown takes place at the sample edge.\n\nFurther relaxation is much slower. During the next 40~min the resistances of both structures relax by a small value ($\\sim$20$\\%$). Moreover the most notable changes are occurred by sudden jumps simultaneously in both remote structures (see the inset of Fig.~\\ref{fig2}(b)). Taking into account that the structures are separated by macroscopic 2DEG reservoir of the length of 250~$\\mu$m and the width of 50~$\\mu$m it can be concluded that these jumps are caused by relaxation process in the macroscopic 2DEG reservoir. This observation once again confirms the conclusion that the narrow conductive channel is a tool for the study of the non-equilibrium phenomena taking place in a macroscopic 2DEG.\\cite{Budantsev2007} The observed stepwise relaxation is phenologically similar to Barkhausen jumps observed in the ferromagnetic materials and originating from spin domain structure transformations.\\cite{Bertotti1998}\n\nA formation of a spatial spin polarization in a single quantum wire has been considered in Ref.~[\\onlinecite{Ihnatsenka2007}] and it has been shown that this would result in hysteresis in electron transport measurements. However, all the calculated characteristics demonstrate the conventional coercive behavior. Consequently, the spin mechanism is not relevant to our results. Moreover, earlier it have been shown that the hysteresis of magnetoresistance is independent on the in-plain component of magnetic field.\\cite{Budantsev2009} It allowed us to conclude that the hysteretic effect is not related to the electron spin.\n\nThe measurements of non-equilibrium magnetization of a 2DEG also show stepwise relaxation.\\cite{Smith2011} It should be noted that transport measurements give a valuable complementary information about the non-equilibrium state of a 2DEG. Fact is that the magnetization is an integral characteristic determined by the sum of all the magnetic moments, which are caused by all magnetization currents in a sample, while the magnetoresistance is sensitive to a local electrochemical potential of edge states. Particularly in this paper we succeed to observe the correlation between the relaxation of magnetoresistance of two remote constrictions and have established that NECs are localized near the edge.\n\\begin{figure}\n\\includegraphics[width=.49\\textwidth]{fig_3}\n\\caption{The dependence of the hysteresis loop area on the constriction width at the filling factors $\\nu$=1 and 2 at temperature 0.48~K.\\label{fig3}}\n\\end{figure}\n\nTo determine the width of spatial localization of NECs the dependence of magnetoresistance hysteresis loop area on the lithographic width of the constriction has been experimentally studied. For this purpose we used the series of samples fabricated from the same heterostructure. The samples were the Hall bars with constrictions of different widths. The constrictions of widths from 0.8~$\\mu$m to 1.3~$\\mu$m have been fabricated by means of electron beam lithography in the central part of Hall bars. All the constrictions have low resistance in zero magnetic field (0.2$\\div$0.8~k$\\Omega$) and demonstrate QHE in high magnetic fields. At the temperature of 0.48~K the hysteresis of magnetoresistance have been observed at the filling factors $\\nu$=1, 2 and 4 for all the samples. Fig.~\\ref{fig3} shows the hysteresis loop area as a function of the constriction width at the filling factors $\\nu$=1 and 2. Experimental points are well fitted by linear functions. One can see that the hysteresis loop area decreases with the constriction width and vanishes at certain critical width $W_{0}$. Moreover these critical widths coincide for both filling factors and are about $W_{0}\\approx$1.35$\\mu$m. Assuming that the critical lithographic width comprises of two counter-propagating NECs of the width $W_\\mathrm{NEC}$ and two depletion regions of the width $W_\\mathrm{depl}\\approx$0.2$\\mu$m: $W_{0}=2W_\\mathrm{NEC}+2W_\\mathrm{depl}$, we obtain $W_\\mathrm{NEC}\\approx$0.5$\\mu$m is comparable with the depletion width. The edge character of NECs has been also reported in the Ref.~[\\onlinecite{Klaffs2004}] devoted to the study of spatial distribution of the non-equilibrium electrostatic potential on a 2DEG surface, where it has been shown that the most significant changes of potential take place close to the edge.\n\nThe observed hysteresis of magnetoresistance exhibits the following properties: (i) it has two phases of  relaxation, fast and slow; (ii) the slow relaxation is occurred by jumps resembling the Barkhausen jumps in ferromagnetics; (iii) it demonstrates inverted anti-coercive behavior; (iv) it shows multistability and has an anhysteretic curve; (v) it can be observed in sufficiently narrow channel thus pointing out its edge character.\n\n\\section{Discussion\\label{Discussion}}\nThe existing physical models are able to interpret only separate experiments. Unfortunately there is no any model which would not contradict all the set of experimental data on the non-equilibrium state in a 2DEG in the QHE regime. Analyzing the experimental data obtained in our and other studies\\cite{Budantsev2007, Klaffs2004, Matthews2004, Usher2009, Ruhe2009} we suggest such a qualitative picture. We propose the picture of spatial distribution of NECs in a 2DEG and discuss its relation to the experimentally observed non-equilibrium magnetization of the 2DEG. For simplicity, we consider a 2DEG at the filling factor near $\\nu=1$, when electrons form the incompressible liquid on the first Landau level and inter Landau level scattering is assumed to be suppressed.\n\n\\subsection{Magnetoresistance of constriction\\label{Magnetoresistance of constriction}}\nThe obtained experimental data on the magnetoresistance can be interpreted in the frame of the following physical picture.\n\nDown-sweep of the magnetic field gives rise to the vortex electric field that induces outflow of the electrons from the bulk to the edge.\\cite{Dolgopolov1992, Laughlin1981} In this case the area of the incompressible liquid increases and the edge channels are shifted closer to the lithographic borders of the sample. It means that the counter-propagating edge currents in the constriction become more distant from each other (Fig.~\\ref{fig4}(a-c)) than in the case of equilibrium state and the backscattering is suppressed. In the type II samples the magnetoresistance is indeed lower than its equilibrium value on the anhysteretic curve. In the type I samples at filling factor $\\nu=1$ the decrease of magnetic field leads to complete suppression of the backscattering in the constriction, manifested in the vanishing of the longitudinal magnetoresistance and the establishment of the QHE regime.\n\nUp-sweep of the magnetic field leads to vortex electric field, which in turn induces outflow of the electrons from the edge to the bulk. In this case the area of the incompressible liquid decreases and the edge channels are shifted away from the lithographic border of the sample. The counter-propagating edge currents in the constriction come closer together (Fig.~\\ref{fig4}(d-f)) than in the case of equilibrium state and the backscattering increases. This results in magnetoresistance rise as we have observed in both types of the samples. The magnetoresistance of the type II samples is indeed higher than its equilibrium value on the anhysteretic curve. In the type I samples at the filling factor $\\nu=1$ the increase of magnetic field switches the constriction from the QHE regime to the resistive state. In other words, the magnetic field sweep leads to magnetic field induced QHE breakdown.\n\n\\begin{figure}[!]\n\\includegraphics[width=.49\\textwidth]{fig_4}\n\\caption{Non-equilibrium electron redistribution in the macroscopic 2DEG (a), (d) and in the constriction (b), (e) and top view of the 2DEG with narrowing (c), (f) at up- and down-sweep of the magnetic field correspondingly at the filling factor $\\nu$=1. Bulk states in the constriction are occupied (c) or not occupied (f) with incompressible liquid in different magnetic field sweep direction. Arrows designate nonequilibrium currents.\\label{fig4}}\n\\end{figure}\n\n\\subsection{Non-equilibrium magnetization\\label{Non-equilibrium magnetization}}\nLet us consider how the proposed picture corresponds to the non-equilibrium magnetization measurements. The spatial distribution of the NECs have been discussed in Refs.~[\\onlinecite{Usher2009}, \\onlinecite{Ruhe2009}, \\onlinecite{Matthews2004}]. However, the details of the mentioned models are not fully consistent with the experimental data obtained up to date. In particular, the model proposed in Refs.~[\\onlinecite{Usher2009}, \\onlinecite{Matthews2004}] assumes that the non-equilibrium charge is redistributed over the whole plane of a 2DEG and consequently the electric field is formed in the bulk. This contradicts experimental data obtained in the present study and discussed in Ref.~[\\onlinecite{Klaffs2004}] pointing to the fact that the non-equilibrium charge redistribution takes place only near the edge. In Ref.~[\\onlinecite{Ruhe2009}] the spatial distribution of the magnetization currents at the 2DEG edge is qualitatively described (Fig.8 in [\\onlinecite{Ruhe2009}]), but the provided picture corresponds rather to equilibrium state because the common Fermi level for the bulk and the edge is introduced, up to which all the electronic states are occupied. Besides in this case the observed magnetization should not exceed the amplitude of the equilibrium dHvA oscillations while the experimental non-equilibrium magnetization is about 20$\\div$60 times larger.\n\n\n\\begin{figure}\n\\includegraphics[width=.49\\textwidth]{fig_5}\n\\caption{The structure of the non-equilibrium edge currents at down- (a) and up-sweep (b) of the magnetic field.\\label{fig5}}\n\\end{figure}\n\nThe decrease of magnetic field at the filling factor $\\nu=1$  shifts the electrons from the bulk to the edge within the first Landau level. Due to the fact that the area occupied by the bulk states is much larger than that occupied by the edge states one can neglect the changes of the electrostatic potential in the bulk and consider that it remains flat. In this case the most significant changes are occurred near the edge, where a dipolar strip decreasing the electrostatic potential in the bulk and increasing it near the edge is formed. The electrons on the first Landau level populate the edge states up to the level of non-equilibrium electrochemical potential $\\mu_{down}$ exceeding the potential of the empty bulk states by the value $\\Delta\\mu$ (Fig.~\\ref{fig5}(a)). In the crossed electric and magnetic fields the electrons drift along the edge forming the magnetization current approximately equal to $e^2\/h\\cdot\\Delta\\mu$. This current corresponds to the experimentally observed non-equilibrium paramagnetic magnetization.\\cite{Matthews2004, Ruhe2009}\n\nThe increase of magnetic field at the filling factor $\\nu=1$ leads to more complicated distribution of the magnetization currents. The outflow of the electrons from the edge to the bulk leads to the formation of the dipolar strip near the edge that rises the electrostatic potential in the bulk and forms the potential well near the edge followed by the depletion region. It means that the electrostatic potential is non-monotonic and has a minimum near the edge, caused by interplay of two oppositely directed electric fields: the field of overpopulated bulk and the depletion field (Fig.~\\ref{fig5}(b)). On either side of the potential minimum the incompressible electron liquid forms counter-propagating drifting currents  --- giant diamagnetic current flowing closer to the bulk and low paramagnetic current flowing in close proximity to the edge. The presence of the paramagnetic current is confirmed by the experimental fact that the sign of the Hall resistance $R_H$ does not depend on the magnetic field sweep direction in a given orientation of the magnetic field. Hence the direction of the transport current, flowing along the edge of a sample, coincides with the direction of the paramagnetic current as in case of equilibrium state.\n\nEdge states are populated up to the non-equilibrium electrochemical potential level $\\mu_{up}$ so that the empty edge states on the first Landau level are lower than populated bulk states by the value of $\\Delta\\mu$. The total magnetization current is dominated by the giant diamagnetic current $e^2\/h\\cdot\\Delta\\mu$. To explain the experimentally observed giant non-equilibrium magnetization of a 2DEG \\cite{Usher2009} it is necessary that $\\Delta\\mu\\geq(20\\div60)\\hbar\\omega_c$, where $\\omega_c$ is a cyclotron frequency.\n\nHere we do not discuss the finite magnetic field range of the hysteresis loop as it requires detailed consideration of disorder in nonlinear screening conditions.\\cite{Chklovskii1992}\n\n\\subsection{Anti-coercive behavior\\label{Anti-coercive behavior}}\nThere are two possible reasons for anti-coercive behavior: (i) an abrupt movement of the boundary of the incompressible liquid at small change of the magnetic field, (ii) topological transitions in quantum Hall liquid. Discuss them in more detail.\n\n(i) at the filling factor $\\nu=1$ the magnetic field change $\\Delta B$ causes the shift of the boundary of the incompressible liquid relative to the lithographic edge. Besides, the larger the sample the smaller the change of the magnetic field $\\Delta B$ is required to shift this boundary by an amount of the order of the depletion width. For example for disk-shaped 2DEG of radius $R$:\n\\begin{equation}\\label{QPC}\n\\Delta B \\approx \\frac{W_{depl}}{2R} \\cdot B.\n\\end{equation}\n\n(ii) As seen from Figs.~\\ref{fig4}(c) and \\ref{fig4}(f) the shift of the boundary of the incompressible liquid leads to topological transitions in quantum Hall liquid. Fig.~\\ref{fig4}(c) corresponds to the total transmission of the edge channels through the constriction, while Fig.~\\ref{fig4}(f) corresponds to the intense backscattering. Thus, a small change of the magnetic field leads to a drastic change of the magnetoresistance.\n\nTopological transition has been observed in samples of type I: the vanishing of the longitudinal magnetoresistance has been observed indicating total suppression of backscattering in the constriction at down-sweep of the magnetic field, while up-sweep of the magnetic field causes sharp jump of the magnetoresistance from zero to 10~k$\\Omega$. Abrupt movement of the boundary of the incompressible electron liquid at small change of the magnetic field is inevitably realized in the both types samples. This is manifested itself in the sharp changes of the magnetoresistance in minor loops measurements.\n\n\\begin{figure}\n\\includegraphics[width=.39\\textwidth]{fig_6}\n\\caption{Hysteresis of 2DEG magnetization (large loop) in QHE regime against the de Haas - van Alphen oscillations (thick line) is inverted (anti-coercive behavior) that is the response advances the magnetic field sweep. Small loop demonstrates the retarding (coercive) behavior.\\label{fig6}}\n\\end{figure}\nThe discovered unusual anti-coercive behavior is in accordance with the law of energy conservation. Consider the behavior of the non-equilibrium magnetization of a 2DEG in the QHE regime. In Fig.~\\ref{fig6} a typical hysteresis of the magnetization on the background of dHvA oscillations at the filling factor $\\nu=1$ is shown. \\cite{Matthews2004} The system response (magnetization) also advances the external parameter (magnetic field) change as in the case discussed. However the total work performed by the system  $-\\oint{Md\\,B}$ (shadow area in Fig.~\\ref{fig6}) is negative, that is in accordance with energy conservation law. Retarding response behavior would correspond to the positive total system work (dashed area in Fig.~\\ref{fig6}) per cycle contrary to the law of conservation of energy.\n\n\\subsection{Comparison with ferromagnetics\\label{Comparison with ferromagnetics}}\nLet us compare the observed hysteresis of magnetoresistance with the hysteresis of magnetization of ferromagnetics.\n\nIt is well known that ferromagnetism is of spin origin. The exchange interaction in ferromagnetics causes the spins align. At the same time, the requirement of a minimum of the total energy of the system leads to its splitting into domains with opposite spin orientations. A change of the magnetic field results in spin flip process near the domain walls causing their ``motion''. However, due to a disorder the transformations of domain structure is retarded with respect to the external magnetic field change. This is the reason of hysteresis with positive coercivity in ferromagnetics. While in a 2DEG a small changes of the external magnetic field lead to sharp motion of the boundary of the incompressible liquid and causes the hysteresis with negative coercivity.\n\nEarlier it has been found \\cite{Budantsev2007} that the hysteresis of magnetoresistance is suppressed by temperature. It has been shown that the amplitude of hysteresis linearly increases as the temperature decreases and saturates at certain low temperature ($\\sim$400~mK). The non-equilibrium magnetization of a 2DEG has similar temperature dependence.\\cite{Matthews2004} The hysteresis of magnetization of ferromagnetic materials is also suppressed by temperature but it has different functional dependence described by Weiss theory of ferromagnetism. \\cite{Bertotti1998}\n\nIt should be noted that NECs in the QHE regime are induced by sweep of the magnetic field and can be considered as eddy currents as they are often called. However, in ferromagnetics it is enough to reduce the sweep rate to eliminate the influence of eddy currents. In this sense eddy currents in ferromagnetics are time-dependent because of finite longitudinal conductivity $\\sigma_{xx}$. While in a 2DEG in the QHE regime when $\\sigma_{xx}\\approx0$ the generation of NECs is inescapable at any reasonable sweep rate. According to our results NECs relaxation lasts for hours after stopping the magnetic field sweep. In Ref.~[\\onlinecite{Kershaw2007}] NECs have been shown to circulate for many hours. In terms of Ref.~[\\onlinecite{Bertotti1998}] under conventional experimental conditions NECs can be considered as time-independent.\n\n\\section{Conclusions\\label{Conclusions}}\nWe have shown that sweep of the magnetic field in the QHE regime results in drastic changes of electrostatic potential in the bulk with respect to the edge if the inter Landau levels scattering is suppressed. This leads to giant magnetization currents flowing along the edge and topological transitions in a 2DEG with a constriction. The latter results in changes of magnetoresistance.\n\nThe hysteresis of magnetoresistance of a 2DEG in the QHE regime has significant phenomenological similarities with the hysteresis of magnetization of ferromagnetic materials, showing multistability, jumps of relaxation, temperature suppression and having the anhysteretic curve. At the same time a fundamental difference expressed in anti-coercive hysteresis behavior of the magnetoresistance has been found.\n\nThe possible causes of the limitations of the inter Landau level scattering as well as the possible limiting mechanisms of NECs are beyond the scope of this article and require further investigation. However we have experimentally shown that NECs are induced in a narrow ($\\sim$0.5$\\mu$m) area along the edge.\n\nDespite the large number of studies devoted to the non-equilibrium state of a 2DEG induced by the sweep of the magnetic field in the QHE regime there is no clear understanding of the phenomena at a microscopic level as well as a systematic phenomenological study of the dependence of the phenomena on parameters of heterostructures and conditions of epitaxial growth up to date. Unfortunately the NECs are insensible in a conventional transport measurement in a linear response regime due to zero longitudinal resistance. In such measurements the condition of the 2DEG is not clear in advance. Such uncertainty can bring unexpected difficulties in studies of a 2DEG in the QHE regime, such as the study of QHE breakdown by the electric current, that has no full and consistent model up to date (see for a review Ref.~[\\onlinecite{Nachtwei1999}]). The present study raises the issue of the need to search critical parameters that define the phenomena and explain the phenomena at a microscopic level. Therefore, some existing concepts on the phenomena in 2DEG in the QHE regime can be extended or revised taking into account the found non-equilibrium.\n\n\\section*{Acknowledgments\\label{Acknowledgments}}\nThe reported study was supported by RFBR (research project No.14-02-31740-mol-a and No.12-02-00532-a), Program of fundamental scientific research DNIT RAS (project 3.2).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\n\tConsider a domain $D$ of $\\mathbb{R}^d,d\\geq 1,$ and a second order uniformly elliptic  differential operator \n\t$$\n\tLu=\\sum_{i,j=1}^{d}a_{ij}\\partial_{ij}u+\\sum_{i=1}^{d}b_i\\partial_{i}u,\n\t$$\n\twhere coefficients  $a_{ij}$ and $b_i$ are  sufficiently smooth. We are concerned with   nonnegative continuous solutions, in the distributional sense,  to  inequalities of type  \n\t\\begin{equation}\\label{nl-ieq}\n\t\t-Lu+\\xi\\psi(u) \\geq g \\quad\\mbox{in }D,\n\t\\end{equation} \t\n\twhere  $g,\\xi: D\\rightarrow \\mathbb{R}_+$ are locally bounded  Borel measurable  functions and    $\\psi:\\mathbb{R}_+\\rightarrow \\mathbb{R}_+$ is a nondecreasing continuous function such that $\\psi(0)=0$. If~$D$ is bounded and regular, it is well known that, for every nonnegative continuous function $f$ on $\\partial D$,  the   Poisson-Dirichlet problem \n\t\t\\begin{equation}\\label{l-dir}\n\t\t\\left\\{\\begin{array}{rcll}\n\t\t\t-Ls &=& g &\\mbox{in }D,\n\t\t\t\\\\ s &=& f &\\mbox{on }\\partial D.\n\t\t\\end{array}\\right.\n\t\\end{equation}  \t\nhas a unique solution $s=S_D(f,g)$. Observe that $H_Df=S_D(f,0)$ is the $L-$harmonic extension of $f$ to $\\overline{D}$ and  $G_Dg=S_D(0,g)$ is the Green potential of $g$.\n\n In this paper, it will first be shown that   for every nonnegative continuous  function~$f$ on  $\\partial D$,\t\t the semilinear boundary value  problem\n\t\\begin{equation}\\label{psl}\n\t\t\\left\\{\\begin{array}{rcll}\n\t\t\t-Lu+\\xi\\psi(u) &=& g &\\mbox{in }D,\n\t\t\t\\\\u &=& f &\\mbox{on }\\partial D,\n\t\t\\end{array}\\right.\n\t\\end{equation} \n\tadmits one and only one  solution which will be denoted by $U_D^\\xi(f,g)$. Besides, it is simple to see that \n\t\\begin{equation}\n\t\tU_D^\\xi(f,g)\\leq S_D(f,g)\\quad\\mbox{in }D.\n\t\\end{equation}\n\tOur main goal  is to establish a lower bound estimate of $U_D^\\xi (f,g)$ as a function of~$S_D(f,g)$, which especially allows us to conclude lower bound estimates of any nonnegative solution to  inequality~\\eqref{nl-ieq}.\n\t\n\t\n\tIt should be noted that semilinear problems of type~\\eqref{psl} play a vital role in the modeling of many phenomena in the fields of physics, astrophysics, logistics and differential geometry  (see \\cite{marcus2014nonlinear,dynkin2002diffusions}). We also notice that~\\eqref{psl} has been investigated in many papers, under various aspects, which deal with  existence and uniqueness as well as the estimation of its solution (see for instance \\cite{ garcia2009existence, mohammed2010ground, fredj2013comparison,   Cao2016, Cao2017, hirata2018, ghardallou2019positive,  hirata2021} and references therein).\n\t\n\t\n\tTaking $\\xi$ a positive constant in~$D$ ($\\xi= 2$), Ben Fredj and the first author studied   problem~\\eqref{psl} in~\\cite{fredj2013comparison} when $L=\\Delta$ is  the Laplacian operator  and $g$ is the null function. They proved  that for every nonnegative continuous function $f$ on $\\partial D$,  \\eqref{psl} has a unique solution~$u$ and  there exists a real constant $\\kappa>0$ such that \tfor all $x\\in D$,\n\t\\begin{equation}\n\t\\kappa H_Df(x)\\leq u(x) \\leq H_Df(x).\n\t\\end{equation}\n\t\n\tThe case of $\\xi\\leq 0$  is recently investigated by  Grigor'yan and Verbitsky in~\\cite{grigor2020}  where they  gave a real positive function $\\varrho$, depending only on~$\\psi$, such that if $S_D(f,g)\\geq 1$ in $D$ then  every  nonnegative solution $u$ to problem~\\eqref{psl} satisfies for all $x\\in D$,\n\t\\begin{equation}\\label{urho}\n\t\tu(x)\\geq \\varrho\\left(G_D\\xi \\right(x)).\n\t\\end{equation}\n\tIn this context,  M\\^aagli and Zribi \\cite{zribi2001} have already treated the special   setting where $L=\\Delta$ and $f,g$ are identically zero.\t\n\nFor  signed functions $\\xi$, Grigor'yan and Verbitsky obtained in~\\cite{grigor2019pointwise} \n\tlower bounded estimates for nonnegative    solutions to \n\t\\begin{equation}\\label{sup-sol-g}\n\t\t-Lu+\\xi u^\\gamma \\geq  g \\mbox{ in }D,\n\t\\end{equation}  \n\twhich are twice continuously differentiable  in $D$.\n\t\n\t\n\tIn the current paper, we deal with nonnegative solutions  to inequality~\\eqref{nl-ieq} in the distributional sense. We only consider functions $\\xi$ that are nonnegative everywhere, however~$D$ can be any  Greenian domain of $\\mathbb{R}^d$, not  necessary bounded, and $\\psi$ belongs to  a  large class of  nondecreasing  functions  which contains  in addition to  polynomial functions, logarithmic functions as well as  exponential functions.\n\t\n\tWe further  assume that $\\psi$ is continuously differentiable in $ ]0,\\infty[$ and there exists $c>0$ such that \n\t\\begin{equation}\n\t\t\\psi(rt)\\leq c\\psi(r)\\psi(t)\\quad \\mbox{for all }r\\in[0,1], t\\in\\mathbb{R}_{+}, \n\t\\end{equation}\nand consider the function   $\\Theta:]0,1]\\rightarrow [0, \\ell[$ ($\\ell$ may be infinite) given, for every $t\\in ]0,1]$ by\n\t\\begin{equation*}\n\\Theta(t)=\\int_{t}^{1}\\frac{1}{c\\,\\psi(s)}\\,ds\\quad\\mbox{where}\\quad\t\\ell =\\int_{0}^{1}\\frac{1}{c\\,\\psi(s)}\\,ds.\n\\end{equation*}\t\nWe obviously see that   $\\Theta$ admits an inverse function which will be denoted by $\\varphi$ and extended  to~$\\mathbb{R}_+$ by setting $\\varphi(t)=0$ if $t\\geq \\ell$. \n Let $p=G_Dg$ and assume that   $0<p<\\infty $ in~$D$. We will prove that every nonnegative solutions~$u$ to~\\eqref{nl-ieq} satisfies, for all $x\\in D$,\n\\begin{equation}\\label{estine}\n\tu(x)\\geq p(x)\\varphi\\left(\\frac{G_D(\\xi\\psi(p))(x)}{p(x)}\\right).\t\n\\end{equation}\nTo do this, we will go through lower bounded estimates of solution $U_D^\\xi(f,g)$ to the boundary value problem~\\eqref{psl}. In this regard,  we had to carry out  a detailed study of $L$-harmonic functions in order to show, when $D$ is bounded regular, that for any  continuous function $f:\\partial D\\to\\mathbb{R}_+$ such that $s=S_D(f,g)$ is finite not identically zero and $G_D(\\xi\\psi(s))$ is finite in~$D$, the integral equation \n\\begin{equation}\\label{eqint}\nu+G_D\\big(\\xi\\psi(u)\\big)=s\\quad\\mbox{in }D,\n\\end{equation}\npossesses a unique solution $u$ and that  for every $x\\in D$,\n\\begin{equation}\\label{estin}\ns(x)\\varphi\\Big(\\frac{G_D(\\xi\\psi(s))(x)}{s(x)}\\Big)\\leq u(x) \\leq s(x).\n\\end{equation}\nNotice that in the setting where Green potentials  $G_D\\xi$ and $ G_Dg$  are continuous in $\\overline{D}$ vanishing on $\\partial D$, $u$ is a solution to~\\eqref{eqint} if an only if $u$ is a solution to problem~\\eqref{psl}.\n\nFor every $n\\geq 1$,  we define  $v_n=U_{D_n}^\\xi(0,g)$  where \n$(D_n)_{n\\geq 1}$ is a sequence of  bounded regular domains such that for every $n\\geq 1$,\n$$\n\\overline{D_n}\\subset D_{n+1} \\quad\\mbox{and}\\quad \\cup_{n\\geq 1}D_n=D.\n$$\nDefine $p_n=G_{D_n}g$ and choose  a subsequence $(p_{n_k})_{k\\geq 1}$ of $(p_n)_{n\\geq 1}$ such that $p_{n_k}>0$ for all~$k\\geq 1$. We then apply~\\eqref{estin} to $v_{n_k}$ in $D_{n_k}$, to obtain that for all $k\\geq 1$,\n\\begin{equation}\\label{estst}\nv_{n_k}\\geq p_{n_k} \\vphi\\left(\\frac{G_{D_{n_k}}(\\xi\\psi(p_{n_k}))}{p_{n_k}}\\right).  \n\\end{equation}\nGiven a solution $u$ to inequality~\\eqref{nl-ieq}, it follows from a comparison principal that $u\\geq v_{n_k}$ in $D_{n_k}$, which in turn allows us to get estimate~\\eqref{estine}  in view of some  convergence theorems  and  monotonicities of functions~$\\psi$ and $\\varphi$.\n\n\nEstimates of type~\\eqref{estine} encompass  results given by Grigor'yan and Verbitsky  \\cite{grigor2019pointwise}  for $\\psi(t)=t^\\gamma$ with $\\gamma>0$. In fact, in this particular case,  simple computation of $\\varphi$ yields   that every nonnegative solution $u$ to~\\eqref{nl-ieq} satisfies, for all $x\\in D$,\n\\begin{equation}\n\tu(x)\\geq \\left\\{\\begin{array}{ll}\n\t\tp(x)\\left(1+(\\gamma-1)\\frac{G_D\\left(p^\\gamma \\xi\\right)(x)}{p(x)}\\right)_+^{1\/(1-\\gamma)} & \\mbox{if }\\gamma<1,\n\t\t\\\\ p(x)\\exp\\left({-\\frac{G_D(\\xi p)(x)}{p(x)}}\\right) & \\mbox{if }\\gamma=1, \n\t\t\\\\ p(x)\\left(1+(\\gamma-1)\\frac{G_D(p^\\gamma \\xi)(x)}{p(x)}\\right)^{1\/(1-\\gamma)} & \\mbox{if }\\gamma>1.\n\t\\end{array}\\right.\n\\end{equation}\t\n Our approach is still valid for various functions $\\psi$ such as \n $$\n \\psi(t)=at+bt^\\gamma,  \\quad \\psi(t)=\\sinh t\\quad \\mbox{and} \\quad \\psi(t)=a(1+bt)\\log(1+bt)\n $$\n  where $a,b,\\gamma>0$. These examples are presented and discussed in detail in the last section, where main results are stated and proved. In  section~2, we introduce some basic notions that are useful to  study the Dirichlet problem~\\eqref{l-dir}. After treating fundamentals properties of the Green potential in section~3, we study in section~4 the semilinear problem~\\eqref{psl} in regular domains. We prove  the existence and uniqueness of the nonnegative continuous solution to problem~\\eqref{psl} and investigate its regularity. In section~5, we are  concerned with  estimates of the solution to~\\eqref{psl} as well as estimates of nonnegative  solutions of the inequality~\\eqref{nl-ieq}. \n\t\n\t\\section{$L-$harmonic functions}\n\tFor every domain $D$ in $\\mathbb{R}^d$,  $d\\geq 1$, let $\\mathcal{B}(D)$ be the set of all real-valued Borel measurable functions in~$D$ and let~$\\mathcal{B}_{b}(D)$ (resp.~$\\mathcal{B}^+(D)$) be the set of all  functions in~$\\mathcal{B}(D)$ which are bounded (resp. nonnegative) in~$D$. More generally, given any class  of numerical functions~$\\mathcal{F}$, we will denote by $\\mathcal{F}_{b}$ (resp.~$\\mathcal{F}^{+}$) the set of all functions in~$\\mathcal{F}$ which are bounded (resp. nonnegative). We define~$\\mathcal{C}(D)$ to be the set of all real-valued continuous functions in~$D$ and~$\\mathcal{C}_{c}^{\\infty}(D)$ to be the set of all infinity differentiable functions with compact support in $D$. For every $1\\leq i,j\\leq d$, $\\partial_{i}$ stands for the first partial derivative with respect to the coordinate $x_{i}$ of $x=(x_{1},....,x_{d})\\in \\mathbb{R}^d$,  and $\\partial_{ij}:=\\partial_{i}\\partial_{j}$.\n\tLet $\\mathcal{C}^{1}(D)$ (resp. $\\mathcal{C}^{2}(D)$) be the set of all functions $u\\in\\mathcal{C}(D)$ such that $\\partial_{i}u\\in\\mathcal{C}(D)$  (resp. $\\partial_{i}u, \\partial_{ij}u\\in\\mathcal{C}(D)$) for every $1\\leq i,j\\leq d$. If $D$ is bounded, we denote by  $\\mathcal{C}_{0}(D)$ the set of all functions $u\\in \\mathcal{C}(D)$  such that  $\\lim_{x\\in D, x\\rightarrow z}u(x)=0$ for all $z\\in \\partial D$.\n\t\n\tLet $D$ be a domain in ${\\mathbb{R}}^d$ and let   $0<\\alpha \\leq 1$. A real-valued function $u$ is called  $\\alpha-$H\\\"{o}lder  continuous  in $\\overline{D}$ if there exists a real constant $C>0$ such that \n\t$$\n\t|u(x)-u(y)|\\leq C |x-y|^\\alpha\\quad\\mbox{for all }x,y\\in\\overline{D}.\n\t$$\n\tThe set of all  $\\alpha-$H\\\"{o}lder  continuous functions  in $\\overline{D}$ is denoted by $\\mathcal{C}^{\\alpha}(\\overline{D})$. We say that $u$ is locally $\\alpha-$H\\\"{o}lder  continuous in the domain $D$ and we write $u\\in\\mathcal{C}^{\\alpha}(D)$  if $u\\in\\mathcal{C}^{\\alpha}(\\overline{V})$ for all domains $V\\Subset D$. Here and in the sequel, writing  $V\\Subset D$ means that $\\overline{V}$ is compact with closure $\\overline{V}\\subset D$.  We also define $\\mathcal{C}^{1,\\alpha}(D)$ (resp. $\\mathcal{C}^{2,\\alpha}(D)$) to be the class of all functions $u\\in \\mathcal{C}^{\\alpha}(D)$ such that $\\partial_{i}u\\in \\mathcal{C}^{\\alpha}(D)$ (resp. $\\partial_{i}u, \\partial_{ij}u\\in \\mathcal{C}^{\\alpha}(D)$) for all $1\\leq i,j\\leq d$. \n\t\n\tWe consider a second order  differential operator of type\n\t\\begin{equation}\n\t\tLu:=\\sum\\limits_{i,j=1}^{d}a_{ij}\\partial_{ij}u+\\sum\\limits_{i=1}^{d}b_{i}\\partial_{i}u\n\t\\end{equation}\n\twhere $a_{ij}\\in\\mathcal{C}^{2,\\alpha}(\\mathbb{R}^d)$ and  $b_{i}\\in\\mathcal{C}^{1,\\alpha}(\\mathbb{R}^d)$ for all $1\\leq i,j\\leq d$ and we assume that  $L$ is uniformly elliptic, that is, there exists a real  constant $\\beta>0$ such that for all $x\\in\\mathbb{R}^d$ and for all $\\xi=(\\xi_1,\\cdots,\\xi_d)\\in\\mathbb{R}^d$,\n\t$$\n\t\\sum_{i,j=1}^{d}a_{ij}(x)\\xi_i\\xi_j\\geq \\beta \\,|\\xi|^2.\n\t$$\n\tGiven a Radon measure $\\mu$ in $D$ and a locally Lebesgue integrable function $u$ in the open set $D$, equation\n\t\\begin{equation}\n\t\tLu=\\mu \\mbox{ in } D\n\t\\end{equation} \n\tshould be understood in the distributional sense, that is,  for all $\\varphi \\in \\mathcal{C}_{c}^{\\infty}(D)$,\n\t\\begin{equation}\n\t\t\\int_{D}uL^{*}\\varphi(x)\\,dx = \\int_{D}\\varphi(x)\\,d\\mu(x) \n\t\\end{equation}\n\twhere $L^{*}$ denotes  the adjoin operator of $L$ given by\n\t\\begin{equation}\n\t\tL^{*}u:=\\sum\\limits_{i,j=1}^{d}\\partial_{ij}(a_{ij}u)-\\sum_{i=1}^{d}\\partial_{i}(b_{i}u).\n\t\\end{equation}\n\tWe call $L-$harmonic in  $D$ any continuous function $u:D\\to\\mathbb{R}$ such that  $Lu=0$ in $D$ and define $\\mathcal{H}_{L}(D)$ to be the set of all $L-$harmonic functions in $D$.\n\tWe also assume that $L$ is hypoelliptic in the sense that $\\mathcal{H}_{L}(D)\\subset \\mathcal{C}^2(D)$. \n\t\n\tAn open bounded set~$D\\subset\\mathbb{R}^d$  is called regular if for each continuous function $~{f:\\partial D\\rightarrow \\mathbb{R}}$, the Dirichlet problem\n\t$$\n\t\\left\\{\\begin{array}{rl}\n\t\tLh= 0 &\\mbox{ in } D,\n\t\t\\\\h=f &\\mbox{ on } \\partial D,\n\t\\end{array}\\right.\n\t$$\n\thas a unique solution  which will be denoted by $H_{D}f.$ In this setting, for every $x\\in D$, the map $f\\mapsto H_{D}f(x)$ defines a probability measure $H_{D}(x,\\cdot)$ on $\\partial D$ which is called harmonic measure relative to $x$ and $D$. For $x\\in \\mathbb{R}^d\\as D$, it will be convenient to put $H_D(x,\\cdot):=\\delta_{x}$ where $\\delta_{x}$ denotes the Dirac measure  concentrated at the point $x$. It is well known (see \\cite[p.106]{gilbarg1998elliptic}) that $D$ is regular provided it satisfies the exterior sphere condition at every boundary point, that is, for every  $z\\in\\partial D$ there exists a ball $B$ of $\\mathbb{R}^{d}$ such that $B\\cap D=\\emptyset$ and $z\\in\\partial B$. This immediately yields that every ball $B\\subset \\mathbb{R}^d$ is regular and consequently a continuous function $u$ is $L-$harmonic in $D$ if and only if $H_{B}u=u$ for all balls $B\\Subset D$.\n\t\n\tDue to J. Serrin \\cite{serrin1954harnack}, nonnegative $L-$harmonic functions in the domain $D$ satisfy  the following Harnack's inequality: For  every compact set $\\Gamma\\subset D$, there exists a real constant $C\\geq 1$  (depending only on $L$, $\\Gamma$ and $D$) such that for all $u\\in\\mathcal{H}_{L}^{+}(D)$,\n\t\\begin{equation}\\label{harnack}\n\t\t\\sup_{x\\in\\Gamma}u(x)\\leq C\\inf_{x\\in\\Gamma}u(x).\n\t\\end{equation}\n\t\\begin{theorem}\\label{bre_cv}\n\t\tLet $D\\subset \\mathbb{R}^d$ be a domain and let $(h_{n})_{n\\geq 1}$ be a  nondecreasing  (or nonincreasing) sequence of  $L-$harmonic functions in $D$ such that $h:=\\lim\\limits_{n\\rightarrow \\infty}h_{n}$ is finite in some point $x_{0}\\in D$. Then $h$ is $L-$harmonic in $D$.\n\t\\end{theorem}\n\t\\begin{proof}\n\t\tLet  $\\Gamma$ be a compact subset of $D$ containing $x_0$ (otherwise, take $\\Gamma'=\\Gamma\\cup \\{x_{0}\\}$) and let $\\varepsilon>0$. It follows from the Harnack's inequality~\\eqref{harnack} and the fact that $h_{n}(x_{0})$ converges to $h(x_{0})$ that, there exist  $n_{0}\\geq 1$ and a real constant $C>0$ such that for every $x\\in\\Gamma$ and for every $m\\geq n\\geq n_0$,   we have\n\t\t$$\n\t\t0\\leq h_{m}(x)-h_{n}(x)  \\leq  C\\big(h_{m}(x_{0})-h_{n}(x_{0})\\big) \\leq  C\\big(h(x_{0})-h_{n}(x_{0})\\big) \\leq  C\\varepsilon.\n\t\t$$\n\t\tLetting $m$ tend to $\\infty$, we get that for every $x\\in\\Gamma$ and for every $n\\geq n_0$,\n\t\t$$\n\t\t0\\leq h(x)-h_{n}(x)  \\leq  C\\varepsilon.\n\t\t$$\n\t\tWe then conclude that the sequence $(h_{n})_{n\\geq 1}$ converges to $h$ locally uniformly in $D$. In particular,  $h$ is finite and continuous in $D$.  Now, for every  $B\\Subset D$ and every $x\\in B$,\n\t\t$$\n\t\tH_{B}h_{n}(x)=\\int_{\\partial B}h_{n}(y)H_{B}(x,dy)\\underset{n\\to\\infty}{\\longrightarrow} \\int_{\\partial B}h(y)H_{B}(x,dy)=H_{B}h(x).\n\t\t$$  \n\t\tSince $H_B h_n=h_{n}$ for every $n\\geq 1$ and $h_n(x)$ converges to $h(x)$, we deduce that $H_Bh=h$ for every ball $B\\Subset D$. This yields that $h$ is $L-$harmonic in $D$.\n\t\\end{proof}\n\t\\begin{proposition}\\label{lem01}\n\t\tLet  $D\\subset \\mathbb{R}^d$ be a regular set  and let   $f:\\partial D\\to \\overline{\\mathbb{R}}_{+}$ be a Borel measurable function. Then, for every connected component $E$ of $D$,  $H_{D}f$ is either $L-$harmonic  or identically infinite   in $E$. \n\t\\end{proposition}\n\t\\begin{proof} Without loss of generality, we assume that $D$ is connected. \n\t\tIn virtue of the previous theorem, we see that \n\t\t$\n\t\t\\mathcal{A}:=\\{\\Gamma\\subset \\partial D: H_{D}(\\cdot, \\Gamma)\\in \\mathcal{H}_{L}(D)\\}\n\t\t$ \n\t\tis a monotone class on $\\partial D$. \n\t\tFurthermore, $\\mathcal{A}$ contains the set $\\mathcal{O}$ of all open subsets of $\\partial D$. In fact, let $O\\in\\mathcal{O}$ and consider \n\t\t$$\n\t\tf_{n}:x\\mapsto f_{n}(x)=\\inf(1, n\\,d(x, O^{c}))\n\t\t$$ \n\t\twhere $d(x,O^{c})=\\inf_{y\\in O^{c}}|x-y|$ denotes the distance between $x$ and  $O^{c}:=\\mathbb{R}^d\\as O$. Then $(f_{n})_{n\\geq 1}$ is a nondecreasing sequence in $\\mathcal{C}^{+}(\\partial D)$ which converges to $\\textbf{1}_{O}$ and hence $(H_Df_{n})_{n\\geq 1}$ is a nondecreasing sequence of  $L-$harmonic functions in $D$ which converges to $H_D(\\cdot,O)$ in view of the monotone convergence theorem.  We then conclude by Theorem~\\ref{bre_cv},  that $H_{D}(\\cdot, O)$ is $L-$harmonic and  thereby $\\mathcal{O}\\subset \\mathcal{A}$. Therefore, it follows from the monotone class lemma that $\\mathcal{A}$ contains $\\mathcal{B}(\\partial D)$. This yields that $H_{D}f\\in \\mathcal{H}_{L}(D)$ whenever \n\t\t\\begin{equation}\\label{ef}\n\t\t\tf=\\sum\\limits_{i=1}^{k}\\gamma_{i}\\textbf{1}_{A_{i}}\n\t\t\\end{equation} \n\t\twith $k\\geq 1$, $\\gamma_{1},...,\\gamma_{k}\\in\\mathbb{R}_+$ and $A_{1},...,A_{k}\\in\\mathcal{B}(\\partial D)$. Finally, let $f:\\partial D \\to \\overline{\\mathbb{R}}_{+}$ be a Borel measurable function such that $H_{D}f\\not\\equiv \\infty$ and choose a nondecreasing sequence $(f_{n})_{n\\geq 1}$ of  nonnegative   functions of the form~\\eqref{ef} which converges pointwise to  $f$. Then   $(H_{D}f_{n})_{n\\geq 1}$ is a nondecreasing sequence of $L-$harmonic functions in $D$ pointwise converging to~$H_Df$  by the monotone convergence theorem. Thus,  $H_{D}f\\in \\mathcal{H}_{L}(D)$ in virtue of Theorem~\\ref{bre_cv}.\n\t\\end{proof}\n\t\n\tLet us point out that the previous result can be  extended to any Borel measurable function $f:\\partial D\\rightarrow \\overline{\\mathbb{R}}$ without having to assume that $f$ is   nonnegative in $\\partial D$. More precisely, if  $D$ is connected and $f$ is $H_D(x_0,\\cdot)-$integrable on $\\partial D$ for some $x_0\\in D$ then $f$ is $H_D(x,\\cdot)-$integrable on $\\partial D$ for all $x\\in D$ and the function $H_Df$ is $L-$harmonic in~$D$.\n\n\t\\medskip\n\tWe recall that, due to the hypoellipticity of $L$  and \\cite[Theorem 3.5]{gilbarg1998elliptic}, a non constant function $h\\in~\\mathcal{H}_L(D)$ can not achieve its  minimum (or maximum) in the interior of the domain $D$. This yields that every $L-$harmonic function $h$ in an open set $D$ is necessarily nonnegative provided $\\liminf_{x\\in D, x\\to z}h(x)~\\geq~0$ for every $z\\in\\partial D$.\n\t\n\t\n\t\\medskip\n\tA lower semicontinuous function $v>-\\infty$ is said to be $L-$superharmonic in $D$ if for every regular  set $U\\Subset D$, $H_{U}v\\in \\mathcal{H}_{L}(U)$ and $ H_{U}v\\leq v$ in~$U$.\n\tThe set of all $L-$superharmonic functions in $D$ will be denoted by $\\mathcal{S}_L(D)$. We will say that $v$ satisfies the property $(\\mathcal M_*)$ if for every $x\\in D$, there exists $r=r(x)> 0$ such that $B(x, r)\\subset D$ and for all $0<\\rho<r$, \n\t\\begin{equation}\\label{promoy}\n\t\t\\int_{\\partial B(x,\\rho)}v(z)H_{B(x,\\rho)}(x,dz)\\leq v(x).\n\t\\end{equation} \n\tIt is obvious that every  $L-$superharmonic function satisfies property $(\\mathcal M_*)$. The converse will be shown later.\n\t\n\t\\begin{proposition}\\label{prinmin}\n\t\tLet $D\\subset\\mathbb{R}^d$ be an open set and let $v:D\\to]-\\infty,\\infty]$ be a lower semicontinuous function which satisfies the  property $(\\mathcal M_*)$. The following holds true:\n\t\t\\begin{enumerate}\n\t\t\t\\item[a)] If $v$ attains its minimum in  a connected component $E$ of $D$, then $v$ is  constant in $E$.\n\t\t\t\\item[b)] If $\\liminf_{x\\in D,\\; x\\to z} v(x)\\geq 0$ for every $z\\in \\partial D$, then $v\\geq 0$ in $D$.\n\t\t\\end{enumerate}\n\t\\end{proposition}\n\t\\begin{proof} Without loss of generality, we consider the case where $D$ is connected and $v$ is not identically infinite in $D$. Assume that there exists $x_0\\in D$ such that  $\\min_{x\\in D} v(x)= v(x_0)$. By~$(\\mathcal M_*)$, there exists $r_{0}>0$ such that  $B(x_0,r_{0})\\subset D$ and for every ball $B=B(x_0,\\rho)$ with $0 <\\rho < r_{0}$,\n\t\t$$\n\t\tH_Bv(x_0)=\\int_{\\partial B}v(z)H_B(x_0,dz)\\leq v(x_0).\n\t\t$$\n\t\tFurthermore, $H_B v\\geq v(x_0)$ in $B$ which yields that the $L-$harmonic function $H_B v$ reaches its minimum at the point $x_0$ and thereby $H_B v=  v(x_0)$ in $B$. \t\n\t\tOn the other hand, since $v$ is lower semicontinuous and bounded below by  $v(x_0)$, there exists (see for instance \\cite[Lemma 3.2.1]{armitage2001classical})  a nondecreasing sequence $(f_n)_{n\\geq 1}$ of  continuous  functions in $\\partial B$ such that $\\lim_{n\\to\\infty}f_n(z)=v(z)$ for all $z\\in\\partial B$. Then $ H_B f_n\\leq H_B v= v(x_0)$ in $B$ for every $n\\geq 1$. Hence, for a given point  $z\\in\\partial B$, \n\t\t\\begin{equation}\\label{eqlim}\n\t\t\tf_n(z)=\\lim_{x\\in B, \\, x\\rightarrow z} H_B f_n(x)\\leq  v(x_0)\\quad\\mbox{for all }n\\geq 1.\n\t\t\\end{equation}\n\t\tThe equality in the formula above follows from the fact that $f_n$ is continuous in $\\partial B$ for all $n\\geq 1$ and the ball $B$ is regular. \n\t\tLetting $n$ tend to $\\infty$, we obtain that $v(z)\\leq  v(x_0)$ for every $z\\in\\partial B$ and so $v= v(x_0)$ in $\\partial B(x_0,\\rho)$. Since $0<\\rho<r_0$ is arbitrary, we conclude that $v(x)= v(x_0)$ for all $x\\in B(x_0,r_0)$.  Therefore, the set\n\t\t$$\n\t\tU=\\{x\\in D: v(x)= v(x_0)\\}\n\t\t$$\n\t\tis open. By the lower semicontinuity  of~$v$, we also see that $U$ is  closed relative to $D$. Consequently $U= D$ and the proof of assertion $(a)$ is complete. In order to show $(b)$, let us assume that $\\liminf_{x\\in D, x\\to z}v(x)\\geq 0$ for every $z\\in \\partial D$ but there exists $x_0\\in D$ such that $v(x_0)<0$. We extend $v$ to $\\overline{D}$ by defining  \n\t\t$$\n\t\tv(z)=\\liminf_{x\\in D, x\\to z} v(x)\\mbox{ for all }  z\\in \\partial D.\n\t\t$$ It is clear that $v$ is lower semicontinuous in the compact set $\\overline{D}$. Then, there exists $y_0\\in \\overline{D}$ such that   $\\min_{x\\in \\overline{D}}v(x)=v(y_0)$. Since $v\\geq 0$ in $\\partial D$ and $v(x_0)<0$, we conclude that $y_0\\in D$ and consequently $v$ achieves its minimum in the interior of~$D$. Using $(a)$ we get that $v(x)=v(x_0)<0$ for every  $x\\in D$ which  yields a contradiction.  \n\t\\end{proof}\n\t\n\tThroughout this paper, $\\lambda_d$ will denote the Lebesgue measure in $\\mathbb{R}^d$.\t\n\t\\begin{proposition}\\label{mprop}\n\t\tLet $D\\subset\\mathbb{R}^d$ be an open set and let $v:D\\to]-\\infty,\\infty]$ be a lower semicontinuous function which is finite $\\lambda_d-$a.e in $D$. The following assertions are equivalent:\n\t\t\\begin{enumerate}\n\t\t\t\\item[a)] $v$ is $L-$superharmonic in $D$.\n\t\t\t\\item[b)] $H_Bv\\leq v$ for every ball $B\\Subset D$.\n\t\t\t\\item[c)] $v$ satisfies property $(\\mathcal{M}_*)$ in $D$.\n\t\t\\end{enumerate}\n\t\\end{proposition}\n\t\\begin{proof}\n\t\tWe obviously see that $(a)$ yields $(b)$ which in turn implies $(c)$. To show that $(c)$ yields~$(a)$, we take a regular set $U\\Subset D$ and choose a nondecreasing sequence $(f_n)_{n\\geq 1}$ of continuous functions on $\\partial U$ such that $\\lim_{n\\to\\infty } f_n(z)=v(z)$ for all $z\\in\\partial U$. Then, for every  $n\\geq 1$,  $w_n:=v-H_U f_n$\n\t\tis lower semicontinuous, $w_n>-\\infty$ in $U$ and for all $z\\in \\partial U$,\n\t\t$$\n\t\t\\liminf_{x\\in U, x\\to z} w_n(x)\\geq v(z)-f_n(z)\\geq 0.\n\t\t$$ \n\t\tIt is also immediate that $w_n$ satisfies~$(\\mathcal M_*)$ in $U$. Thereby, according to assertion~$(b)$ in  the previous proposition, we conclude that $w_n\\geq 0$ in $U$ for all $n\\geq 1$ and consequently \n\t\t\\begin{equation}\\label{fn}\n\t\t\tH_Uv=\\lim_{n\\to \\infty} H_U f_n \\leq v\\quad \\mbox{in }U.\n\t\t\\end{equation}\n\t\tFurthermore, the $L-$harmonicity of $H_Uv$ follows from Proposition~\\ref{lem01} and the fact that $v$ is finite  $\\lambda_d-$a.e. Thus, $v$ is $L-$superharmonic in $D$ if $(\\mathcal M_*)$ is satisfied.\n\t\\end{proof}\n\t\n\tWe end this section with the following convergence property of a monotone sequences of $L-$superharmonic functions which follows from Theorem~\\ref{bre_cv}.\n\t\\begin{proposition}\\label{sconv}\n\t\tLet $D\\subset \\mathbb{R}^d$ be an open set and let $(v_{n})_{n\\geq 1}$ be a  nondecreasing   sequence of nonnegative  $L-$superharmonic functions in $D$ such that $v:=\\sup_{n\\geq 1}v_{n}<\\infty$  $\\lambda_d-$a.e. Then $v$ is $L-$superharmonic in $D$.\n\t\\end{proposition}\n\t\n\t\n\t\\section{Green operator}\n\tWe call Green function of $L$ in the regular set $D\\subset\\mathbb{R}^d$, the Borel measurable function \n\t$G_{D}:D\\times D\\to ]0,\\infty]$ having the following properties   for any  $y\\in D$:\n\t\\begin{enumerate}\n\t\t\\item[G1)] $G_D(\\cdot,y)$ is finite and continuous  in  $D\\as\\{y\\}$ and  $\\lim_{x\\in D,x\\to y}G_D(x,y)=G_D(y,y)$.\n\t\t\\item[G2)]  $G_D(\\cdot,y)$ is  \tlocally $\\lambda_d-$integrable in $D$ and $LG_{D}(\\cdot,y)=-\\delta_{y}$.\n\t\t\\item[G3)]  $\\lim_{x\\in D, x\\rightarrow z}G_D(x,y)=0$ for all $z\\in\\partial D$.\n\t\\end{enumerate}\n\tThe fact that  coefficients $a_{ij},b_i$ are sufficiently smooth guarantees the existence of  the Green function  $G_D$ (see for instance \\cite{miranda1970partial}).\n\tWe also notice that a simple application of the maximum principle  in the domain $D\\as\\{y\\}$ yields that  \n\t\\begin{equation}\\label{ypole}\n\t\tG_D(y,y)=\\sup_{x\\in D}G_D(x,y).\n\t\\end{equation}\n\t\n\t\n\t\\begin{proposition} Let $D\\subset \\mathbb{R}^d$ be a regular set, $y\\in D$ and    $0<\\beta< G_D(y,y)$. Then,  the  function $w:=G_D(\\cdot,y)\\wedge \\beta$\n\t\tis $L-$superharmonic in $D$. In particular, $G_D(\\cdot,y)$ is \t $L-$superharmonic in $D$.\t\t\n\t\\end{proposition}\n\t\\begin{proof} In virtue of Proposition~\\ref{mprop}, it is sufficient to show that $w$ satisfies~$(\\mathcal M_*)$. \t Since  $LG_{D}(\\cdot,y)=-\\delta_{y}$, we  see that $w$ is $L-$harmonic in $\\{G_D(\\cdot,y)<\\beta\\}$ and consequently~\\eqref{promoy} is valid for all $x\\in D$ such that $G_D(x,y)<\\beta$. If  $G_D(x,y)\\geq \\beta$,  inequality~\\eqref{promoy} is immediate with $r(x)=d(x,D^c)$. Thus, $w=G_D(\\cdot,y)\\wedge\\beta$ is $L-$superharmonic in $D$ for every   $0<\\beta< G_D(y,y)$. To prove the last part of the proposition, we consider  $\\beta_n=G_D(y,y)\\wedge n$ and $w_n=G_D(\\cdot, y)\\wedge\\beta_n$ for every $n\\geq 1$. Therefore, Proposition~\\ref{sconv} and~\\eqref{ypole} yield that  $G_D(\\cdot,y)=\\lim_{n\\to\\infty}w_n$ is $L-$superharmonic in~$D$.\n\t\\end{proof}\n\t\n\tWe then conclude that $G_D(\\cdot,y)$ is a {\\em potential with support} $\\{y\\}$ in the sense of R.M. Herv\u00e9~\\cite{herve1962recherches}. Hence,  the uniqueness of the Green function $G_D$ follows  from  Theorem~16.5 in~\\cite{herve1962recherches} and property {\\rm (G2)} above. In particular, if $D$ and $\\Omega$ are regular sets such that  $D\\subset\\Omega$, then\n\t\\begin{equation}\\label{ggg}\n\t\tG_D(\\cdot,y)=\tG_\\Omega(\\cdot,y)-H_DG_\\Omega(\\cdot,y)\\quad \\mbox{in }D\n\t\\end{equation}\n\twhich yields that  $G_D(x,y)\\leq G_\\Omega(x,y)$ for all $x,y\\in D$. \n\t\n\tGiven an arbitrary open set  $\\Omega\\subset\\mathbb{R}^d$, we define for every $x,y\\in \\Omega$,\n\t\\begin{equation}\\label{greenf}\n\t\tG_{\\Omega}(x,y):=\\sup_{D\\in \\mathcal{R}(\\Omega)} G_D(x,y),\n\t\\end{equation}\nwhere  $\\mathcal{R}(\\Omega)$ denotes the family of all regular sets~$D\\Subset\\Omega$.\n\t\\begin{definition}\n\t\tThe  open set $\\Omega$ of $\\mathbb{R}^d$ is called Greenian if for every $y\\in \\Omega$, $G_\\Omega(\\cdot,y)$ is not identically infinite in the connected component of $\\Omega$ which contains $y$. In this setting, $G_\\Omega$ is called Green function of $L$ in $\\Omega$.\n\t\\end{definition}\n\t\n\tIf $\\Omega $ is regular then $\\Omega \\in \\mathcal{R}(\\Omega)$ and $G_{\\Omega}$ is just the Green function of $L$ in $\\Omega$ given  at the beginning of this section. Let us also note that  for any sequence   $(D_n)_{n\\geq 1}$  of regular sets such that $D_n\\Subset D_{n+1}$ for all $n\\geq 1$ and $\\cup_{n\\geq 1}D_n=\\Omega$, we have\n\t\\begin{equation}\\label{lgreen}\n\t\tG_{\\Omega}(x,y)=\\lim_{n\\to\\infty } G_{D_n}(x,y).\n\t\\end{equation}\n\tHence, it is not difficult to show that $G_\\Omega(\\cdot,y)$ satisfies properties {\\rm (G1)} and {\\rm (G2)} for any  Greenian set~$\\Omega$.  However, instead of {\\rm (G3)}, we observe that $G_\\Omega$ satisfies \n\t\\begin{enumerate}\n\t\\item[G3')] For every $h\\in\\mathcal{H}_L^+(\\Omega)$, if $h\\leq G_\\Omega(\\cdot,y)$ in $\\Omega$ then $h\\equiv 0$ in $\\Omega$.\n\t\\end{enumerate}\n \n\t\n\tWe define $\\Gamma(x)=\\overline\\Gamma(|x|)$ for every $x\\in\\mathbb{R}^d$, where $\\overline\\Gamma$ is given for every real $r> 0$ by \t\n\t\\begin{equation}\\label{gacomp}\n\t\t\\overline\\Gamma(r)=\\left\\{\\begin{array}{lr}\n\t\t\tr^{2-d} & \\mbox{if }d\\geq 3,\n\t\t\t\\\\ 1\\vee(-\\log r) & \\mbox{if }d=2,\n\t\t\t\\\\ 1 & \\mbox{if } d=1.\n\t\t\\end{array}\\right.\n\t\\end{equation}\n\tLet $D\\subset\\mathbb{R}^d$ be a bounded open set. It is well known (see \\cite{hueber1982}) that there exists a  real constant $c>0$  such that \n\t\\begin{equation}\\label{equa}\n\t\tG_{D}(x,y)\\leq c\\, \\Gamma(x-y)\\quad\\mbox{for all }x,y\\in D.\n\t\\end{equation}\n\tTherefore, every bounded open subset of $\\mathbb{R}^d$, $d\\ge 1$, is Greenian. Moreover, if $d\\geq 3$ then~$\\mathbb{R}^d$ and thereby all open subsets are Greenian. However, $\\mathbb{R}$ and $\\mathbb{R}^2$ are not Greenian.\n\t\n\tGiven a Greenian set $D\\subset\\mathbb{R}^d$, the Green operator in $D$ is defined by \n\t$$\n\tG_D p:=\\int_D G_D(\\cdot,y)p(y)dy\n\t$$\n\tfor every function $p\\in\\mathcal{B}(D)$ for which the above integral exists. The following proposition is well proved in \\cite{dynkin2002diffusions}.\n\t\n\t\n\t\\begin{proposition}\\label{pror}\n\t\tLet $D\\subset\\mathbb{R}^d$ be a bounded domain and let $p\\in \\mathcal{B}^+(D)$.\n\t\t\\begin{enumerate}\n\t\t\t\\item[a)]  If $p$ is bounded, then $G_Dp$ is bounded and  $\\alpha-$H\\\"{o}lder  continuous in $D$. If moreover $D$ is regular, then $\\lim_{x\\in D, \\, x\\rightarrow z} G_Dp(x)=0$ for all $z\\in \\partial D$.\n\t\t\t\\item[b)]  If $p$ is bounded and  $\\beta-$H\\\"{o}lder  continuous in $D$ for $0<\\beta\\leq 1$, then $G_Dp\\in C^2(D)$.\n\t\t\\end{enumerate}\n\t\\end{proposition}\n\t\n\t\n\t\n\tWe need the following topological lemma which should be well known. For the completeness, we give here the proof. \n\t\\begin{lemma}\\label{toplem}\n\t\tLet $D$ be a connected open subset of $\\mathbb{R}^d$  and let $a\\in D$. Then for every $x\\in D$, there exist $n\\geq 1$ and $x_0,x_1,\\ldots, x_n\\in D$ such that \n\t\t\\begin{equation}\\label{segments}\n\t\t\tx_0=a,\\;x_n=x\\mbox{ and } [x_{k-1},x_k]\\subset D\\mbox{ for all }k\\in\\{1,\\ldots,n\\}.\n\t\t\\end{equation} \n\t\\end{lemma}\n\t\\begin{proof}\n\t\tLet $W$ be the set of all points of $D$ having the property given in the lemma. We claim that  $W$ is open. Indeed, let $x\\in W$ and let $n\\geq 1$, $x_0,x_1,\\ldots, x_n\\in D$ satisfying~\\eqref{segments}. Choose $r>0$ such that  $B(x,r)\\subset D$ and let $y\\in B(x,r)$. \n\t\tThen, setting $x_{n+1}=y$ we get $x_0,x_1,...,x_n, x_{n+1}\\in D$ such that $x_0=a$, $x_{n+1}=y$ and $[x_{k-1},x_{k}]\\subset D$ for every $1\\leq k \\leq n+1$. This means that  $B(x,r)\\subset W$ and the claim is proved. To show that $W$ is a closed subset of~$D$, we consider a sequence  $(y_p)_{p\\geq 1}$ in $W$ which converges to a point $y\\in D$. Let $\\varepsilon>0$ and $q\\geq 1$ such that $y_q\\in B(y,\\varepsilon)$. Since $y_q\\in W$, there exist $m\\geq 1$ and $z_0,z_1,...,z_{m}\\in D$ such that $z_0=a,z_m=y_q$ and $[z_{k-1},z_k]\\subset D$ for~every $1\\leq k\\leq m$. To see that $y\\in W$, it suffices to take $z_{m+1}=y$. Finally, we conclude that $W$ is open and closed in $D$ at the same time, which yields that $W=D$ since $W\\not=\\emptyset$ and $D$ is connected.\n\t\\end{proof}\n\t\\begin{theorem}\\label{theoo} Let $D\\subset \\mathbb{R}^d$ be a bounded domain and let $p\\in \\mathcal{B}^+(D)$ be  a locally bounded function such that  $u:=G_Dp$ is finite at some point in~$D$. Then $u$ is  $\\alpha-$H\u00f6lder continuous,  $L-$superharmonic in  $D$ and  $L-$harmonic in $D\\as \\overline{\\{p>0\\}}$.\n\t\\end{theorem}\n\t\\begin{proof}\n\t\tThe proof is divided in three steps. Given a set $V\\subset D$, we define $u_V=G_D(p\\textbf{1}_V)$.\n\t\t\n\t\t{\\em Step 1.}  We claim that if $u(x_0)<\\infty$ for any  $x_0\\in D$  then $u$ is finite and  continuous in the  ball~$B_r:=B(x_0,r)\\Subset D$ where~$r>0$. Indeed, we decompose $u$ into the sum $u=v_{r}+w_{r}$ where $v_r=u_{B_r}$ and $w_r=u_{B_r^c}$. We obviously see that~$u_{B_r}$ is finite and  continuous in~$B_r$ because $p\\textbf{1}_{B_r}$ is bounded in~$D$.\n\t\t\t\tOn the other hand, given $0<\\rho<r$ and $z\\in B_r^c:=D\\as B_r$, we easily see that $G_D(\\cdot,z)$ is $L-$harmonic in~$D\\as\\{z\\}$ and hence Fubini-Tonelli's theorem yields that for every $x\\in B_\\rho$,\n\t\t\\begin{eqnarray*}\n\t\t\tH_{B_\\rho}w_{r}(x)\n\t\t\t&=& \\int_{D\\as B_r}\\left(\\int_{\\partial   B_\\rho}G_{D}(y,z)\\,H_{B_\\rho}(x,dy)\\right) p(z)\\, dz\n\t\t\t\\\\ &=& \\int_{D\\as B_r}H_{B_\\rho}G_{D}(\\cdot,z)(x)\\, p(z)\\, dz\n\t\t\t\\\\ &=& \\int_{D\\as B_r}G_{D}(x,z)p(z)\\,dz\n\t\t\t\\\\ &=& w_{r}(x).  \n\t\t\\end{eqnarray*}\n\t\tSince $w_{r}(x_0)<\\infty$, it follows that $H_{B_\\rho}w_{r}\\not\\equiv\\infty$ in $B_\\rho$.  Then, in virtue of Proposition~\\ref{lem01},  $w_{r}$ is $L-$harmonic in $B_\\rho$ for every $0<\\rho<r$. Whence  $w_{r}$ is $L-$harmonic in $B_r$, which yields in particular that $w_r$ is finite and   continuous in $B_r$. Consequently, $u$ is finite and  continuous in $B_r$.\n\t\t\n\t\t\n\t\t\n\t\t{\\em Step 2.} Assume that $u(x)<\\infty$ for some point $x\\in D$ and let $y\\in D$ such that  the segment $[x,y]\\subset D$.  We claim that $u(z)<\\infty$ for all  $z\\in [x,y]$. Indeed, it is enough  to show that $u(y)<\\infty$. To do this, put $\\eta:=d([x,y], D^c)$ and let $0<r<\\eta$. Choose an integer~$n$ such that   $nr>|y-x|$ and  define \n\t\t$$\n\t\tz_k=x+\\frac{k}{n}(y-x)  \\quad\\mbox{for }k=0,1,\\ldots, n.\n\t\t$$\n\t\t\\begin{center} \n\t\t\t\\begin{tikzpicture}\n\t\t\t\t\\draw (2.1,5.91) .. controls (7,4.2)  .. (10,5) ;\n\t\t\t\t\\draw (2.1,0.9) .. controls (5,1.6)  .. (10,0) ;\n\t\t\t\t\\draw (3.5,3.5) -- (9,3.5);\n\t\t\t\t\\draw (10,0) arc (-90:90:2.5cm);\n\t\t\t\t\\node[blue]at (11.6, 4){\\textbf{$D$}};\n\t\t\t\t\\draw (2.1,0.9) arc (-90:-270:2.5cm);\n\t\t\t\t\\draw (3.5,3.5) circle (0.7cm); \t\n\t\t\t\t\\draw (4,3.5) circle (0.7cm);\n\t\t\t\t\\draw (4.5,3.5) circle (0.7cm);\n\t\t\t\t\\draw (8.5,3.5) circle (0.7cm);\n\t\t\t\t\\draw (9,3.5) circle (0.7cm);\n\t\t\t\t\\draw[->, red] (3.5,3.5)--(3,4); \\node[red]at (3.3,3.9){\\textbf{$r$}};\n\t\t\t\t\\draw[->, red] (4,3.5)--(3.5,4);\n\t\t\t\t\\draw[->, red] (8.5,3.5)--(8,4);\n\t\t\t\t\\draw[->, red] (9,3.5)--(8.5,4);\n\t\t\t\t\\foreach \\Point in {(3.5,3.5),(4,3.5),(4.5,3.5),(5,3.5),(5.5,3.5),(7.5,3.5),(8,3.5),(8.5,3.5),(9,3.5)}{\n\t\t\t\t\t\\node at \\Point {$\\bullet$};\n\t\t\t\t}\n\t\t\t\t\\draw[<->, blue] (7.2,3.5)--(7.2,4.49);\n\t\t\t\t\\node [black] at (6.5,3.2){$\\cdots\\;\\cdots\\;\\cdots$};\n\t\t\t\t\\node[blue]at (7.4,4){\\textbf{$\\eta$}};\n\t\t\t\t\\node[blue]at (3.65,3.3){\\textbf{$x$}};\n\t\t\t\t\\node[blue]at (8.8,3.3){\\textbf{$y$}};\n\t\t\t\t\\draw[->, red] (3.5,3.5)--(3.5,2);\n\t\t\t\t\\draw[->, red] (4,3.5)--(4,2);\n\t\t\t\t\\draw[->, red] (4.5,3.5)--(4.5,2);\n\t\t\t\t\\draw[->, red] (8.5,3.5)--(8.5,2);\n\t\t\t\t\\draw[->, red] (9,3.5)--(9,2);\n\t\t\t\t\\node[blue]at (3.5,1.75){\\textbf{$z_0$}};\n\t\t\t\t\\node[blue]at (4,1.75){\\textbf{$z_1$}};\n\t\t\t\t\\node[blue]at (4.5,1.75){\\textbf{$z_2$}};\n\t\t\t\t\\node[blue]at (8.4,1.75){\\textbf{$z_{n-1}$}};\n\t\t\t\t\\node[blue]at (9.1,1.75){\\textbf{$z_n$}};\n\t\t\t\\end{tikzpicture}\n\t\t\\end{center}\n\t\tSeeing that $u(z_0)=u(x)<\\infty$, we obtain by Step~1 that $u<\\infty$ in ${B(z_0,r)}$ and then  $u(z_1)<\\infty$ since  $|z_1-z_0| < r$. Using same arguments,  we deduce that $u(z_k)<\\infty$ for every $k\\in\\{1,\\ldots,n\\}$. In particular  $u(y)=u(z_n)<\\infty$ and the claim is proved.\n\t\t\n\t\t{\\em Step 3.} From Steps~1 and 2, we first   conclude  that $u$ is finite and  continuous in the whole domain~$D$. To see this, let~$x_0\\in D$ such that $u(x_0)<\\infty$.  According to Lemma~\\ref{toplem} above, for each $x\\in D$  we can find $n\\geq 1$ and $x_1,\\ldots, x_n\\in D$ such that $x_n=x$ and  $[x_{k-1},x_k]\\subset D$ for all $1\\leq k\\leq n$.  Hence, by successive applications of Step~2, we get that $u(x)=u(x_n)<\\infty$ and consequently~$u$ is finite in~$D$. The fact that $u$ is continuous in~$D$ follows from Step~1. Now, in order to prove that $u$ is $\\alpha-$H\u00f6lder continuous in~$D$,  consider a domain  $V\\Subset D$ and choose a domain~$W$ such that $V\\Subset W\\Subset D$. Since $p\\textbf{1}_W$ is bounded in~$D$, it is clear that $u_W\\in\\mathcal{C}^\\alpha(D)\\subset \\mathcal{C}^\\alpha(\\overline{V})$.\n\t\t On the other hand, the function $u_{W^c}$ is continuous in~$W$ and same arguments as in Step~1 yield that $H_Bu_{W^c}=u_{W^c}$ for every ball $B\\Subset W$. Therefore, $u_{W^c}$ is $L-$harmonic in~$W$ and consequently $\\alpha-$H\u00f6lder continuous in~$W$. Thus $u=u_{W}+u_{W^c}\\in\\mathcal{C}^\\alpha(\\overline{V})$ for every domain $V\\Subset D$ which means that $u\\in\\mathcal{C}^\\alpha(D)$. \n\t It remains to show  that~$u$ is  $L-$superharmonic in~$D$  and $L-$harmonic in~$U:=D\\as\\overline{\\{p>0\\}}$. The first claim follows easily from  Fubini-Tonelli's theorem by observing that $G_D(\\cdot,y)$ is $L-$superharmonic in~$D$ for all~$y\\in D$. Finally, the function $u$ is $L-$harmonic in~$U$ because it is continuous and $H_Bu=u$ for any ball $B\\Subset U$ by analogous calculus as in Step~1. \n\t\\end{proof}\n\t\nWe deduce that for every bounded domain $D$ of $\\mathbb{R}^d$  and for every locally bounded  function $p\\in \\mathcal{B}^+(D)$ such that $G_Dp\\not\\equiv \\infty$ in $D$, we have  \n\t\\begin{equation}\\label{gdp}\n\t\tLG_{D}p=-p\\mbox{ in } D.\n\t\\end{equation}\n\tIndeed, the previous theorem guarantees  that $G_Dp$ is finite and continuous in $D$ and  for every function  $\\varphi\\in\\mathcal{C}^{\\infty}_{c}(D)$, Fubini's theorem  yields   that\n\t$$\n\t\\int_{D}G_Dp(x) L^{*}\\varphi(x) dx=\\int_{D}p(y)\\Big(\\int_{D}G_{D}(x,y)L^*\\varphi(x)dx\\Big)dy=-\\int_{D}p(y)\\varphi(y)dy.\n\t$$ \t\n\t\n\t\n\t\\begin{proposition}\\label{pro2}\n\t\tLet $D\\subset \\mathbb{R}^d$ be a bounded domain. Then $G_{D}$ is a compact operator on~$\\mathcal{B}_{b}(D)$ endowed with the uniform norm $\\|\\cdot\\|_{\\infty}$. \n\t\\end{proposition}\n\t\\begin{proof}\n\t\tWe have to show that $\n\t\t{\\mathcal{F}}=\\{G_{D}u: u\\in\\mathcal{B}_b(D)\\mbox{ and }\\|u\\|_{\\infty}\\leq 1\\}\n\t\t$\n\t\tis relatively compact in $\\mathcal{B}_{b}(D)$ endowed with  $\\|\\cdot\\|_{\\infty}$. We obviously see that\n\t\t$\n\t\t\\|G_{D}u\\|_{\\infty}\\leq \\|G_D1\\|_{\\infty} <\\infty\n\t\t$ \n\t\tfor all $u\\in\\mathcal{B}_b(D)$ such that $\\|u\\|_{\\infty}\\leq 1$,\t\twhich means that  ${\\mathcal{F}}$ is uniformly bounded. Thus, in view of  Ascoli's theorem, the proof will be finished provided we show that the family ${\\mathcal{F}}$  is  equicontinuous in~$D$. In order to do this,  let $A\\subset D$ be Borel set,  $x\\in D$ and $r>0$. Then, in virtue of~\\eqref{equa}, we have\n\t\t$$\n\t\t\\int_{A}G_{D}(x,y)dy    \\leq c\\int_{B(x,r)}\\Gamma(x-y)\\,dy+c\\int_{A\\as B(x,r)}\\Gamma(x-y)dy.\n\t\t$$ \n\t\tBy a simple computation, there exists a real constant  $c_1>0$,  depending only on $d$, such that   \n\t\t$$\n\t\t\\int_{B(x,r)}\\Gamma(x-y)\\,dy\\leq  c_1 \\int_0^r \\overline\\Gamma(t)t^{d-1}\\,dt \\quad\\mbox{and}\\quad \n\t\t\\int_{A\\as B(x,r)}\\Gamma(x-y)dy\\leq \\overline\\Gamma(r)\\lambda_d(A).\n\t\t$$\n\t\tTherefore, for every $r>0$ we get that \n\t\t$$\n\t\t\\limsup_{\\lambda_d(A)\\to 0}\\left(\\sup_{x\\in D}\\int_{A}G_{D}(x,y)dy \\right)\\leq c\\, c_1 \\int_0^r \\overline\\Gamma(t)t^{d-1}\\,dt. \n\t\t$$\n\t\tSeeing that  $\\int_0^r \\overline\\Gamma(t)t^{d-1}\\,dt\\to 0$ if $r\\to 0$, we deduce that \n\t\t$$\n\t\t\\sup_{x\\in D}\\int_{A}G_{D}(x,y)dy \\to 0\\quad\\mbox{if}\\quad \\lambda_d(A)\\to 0.\n\t\t$$\n\t\tWhence  the family ${\\mathcal{R}}=\\{G_{D}(x,\\cdot):x\\in D\\}$ is uniformly $\\lambda_d$-integrable in $D$. Consequently, by  Vitali's convergence theorem, we conclude   that for all $x,z\\in D$,\n\t\t$$\n\t\t\\sup_{ \\|u\\|_{\\infty}\\leq 1}\\left| G_Du (z)-G_Du(x)\\right|\\leq \\int_{D}\\left| G_{D}(z,y)-G_{D}(x,y)\\right| dy \\to 0 \\quad\\mbox{if}\\quad z\\to x.\n\t\t$$\n\t\tThus \n\t\t$\\mathcal{F}$ is equicontinuous in the domain~$D$.    \t\n\t\\end{proof}\n\t\\section{Semilinear Dirichlet problem }\n\t\n\t\n\t\n\tThroughout this work, we consider   a  regular domain~$D$ in $\\mathbb{R}^d$, a Borel measurable locally bounded function $\\xi: D\\rightarrow \\mathbb{R}_{+}$   and a nondecreasing continuous function  $\\psi: \\mathbb{R}_+\\rightarrow \\mathbb{R}_{+}$  such that $\\psi(t)=0$ if and only if  $t=0$. We are concerned with existence and uniqueness of nonnegative solution to  the semilinear Dirichlet problem \n\t\\begin{equation}\\label{base}\n\t\t\\left\\{\\begin{array}{rcll}\n\t\t\t-Lu+\\xi\\psi(u) &=& g &\\mbox{in }D,\n\t\t\t\\\\u &=& f &\\mbox{on }\\partial D,\n\t\t\\end{array}\\right.\n\t\\end{equation}\n\twhere $f: \\partial D\\rightarrow \\mathbb{R}_{+}$ is  continuous  and $g:D\\rightarrow \\mathbb{R}_{+}$ is Borel measurable locally bounded. It will be convenient to extend the function $\\psi$ to $\\mathbb{R}$ by setting $\\psi(t)=0$ if $t\\leq 0.$  By a nonnegative solution in $D$ to equation\n\t\\begin{equation}\\label{peq}\n\t\t-Lu+\\xi\\psi(u) = g,\n\t\\end{equation} \n\twe mean a continuous function $u: D\\to\\mathbb{R}_+$ satisfying~\\eqref{peq} \n\tin  the distributional sense,  i.e, for all  $\\varphi \\in \\mathcal{C}_{c}^{\\infty}(D)$,\n\t\\begin{equation}\n\t\t-\\int_{D}uL^{*}\\varphi\\, d\\lambda_d+\\int_{D}\\xi\\psi(u)\\varphi \\, d\\lambda_d = \\int_{D}g\\varphi \\, d\\lambda_d. \n\t\\end{equation} \n\tThe boundary condition in problem~\\eqref{base} means that  for all $z\\in\\partial D$,\n\t\\begin{equation}\n\t\t\\lim_{x\\in D, \\, x\\rightarrow z} u(x) =f(z).\n\t\\end{equation} \n\t\n\t\\begin{proposition}\\label{lemcom}\n\t\tLet $\\Psi:\\mathbb{R}\\to \\mathbb{R}_+$ be  nondecreasing and let $u, v\\in\\mathcal{B}(D)$ be two locally bounded functions such that $G_{D}(\\xi\\Psi(u))$ and $G_{D}(\\xi\\Psi(v)))$ are not identically infinite in~$D$. Assume that \n\t\t$$s:=u-v+G_{D}(\\xi\\Psi(u))-G_{D}(\\xi\\Psi(v))$$ is a nonnegative $L-$superharmonic function in $D$. Then $u\\geq v$ in $D$.\n\t\\end{proposition}\n\t\\begin{proof}\n\t\tWe first observe that $G_Dw$ is finite and continuous in~$D$ by Theorem~\\ref{theoo}, where  $w:=\\xi\\Psi(u)-\\xi\\Psi(v)$. \n\t\tThe fact that $\\Psi$ is nondecreasing yields that  $\\{w^+>0\\}\\subset \\{u>v\\}$ and consequently  \n\t\t\\begin{equation}\\label{eqcomp}\n\t\t\ts+G_{D}w^-\\geq G_{D}w^+ \\mbox{ in } \\{w^+>0\\}.\n\t\t\\end{equation}\n\t\tWe define $p_n:=n\\wedge w^+\\textbf{1}_{K_n}$  where $(K_n)_{n\\geq 1}$ is a sequence  of compact sets satisfying \n\t\t$$\n\t\tK_n\\subset K_{n+1}\\subset \\{w^+>0\\} \\mbox{ for all }n\\geq 1\\quad\\mbox{and}\\quad \\lambda_d(\\cup_{n\\geq 1}K_n)=\\lambda_d(w^+>0).\n\t\t$$\n\t\tAccording to   Theorem~\\ref{theoo},  for every $n\\geq 1$,  $G_Dp_n$ is $L$-harmonic in  $U_n:=D\\as K_n $ and consequently $\\theta_n:=s+G_Dw^--G_Dp_n$  is   $L-$superharmonic in~$U_n$. \n\t\tOn the other hand, since  $p_n$ is bounded and $D$ is regular, Proposition~\\ref{pror} yields that $\\lim_{x\\in D,x\\to z}G_D p_n(x)=0$ for every $z\\in \\partial D$. Then, \n\t\tfor every $z\\in\\partial U_n\\cap\\partial D$,\n\t\t$$\n\t\t\\liminf_{x\\in U_n,\\, x\\to z}\\theta_n(x)=\\liminf_{x\\in U_n,\\, x\\to z}(s+G_Dw^-)(x)\\geq 0 \n\t\t$$\n\t\tMoreover, since $K_n\\subset\\{w^+>0\\}$, it follows from~\\eqref{eqcomp} that  $\\theta_n \\geq 0$ in $K_n$. Hence, seeing that $\\theta_n$ is lower semicontinuous in $D$, we deduce that  for every  $z\\in \\partial U_n\\cap D\\subset K_n$,  $\\liminf_{x\\in U_n,\\,x\\to z}\\theta_n(x)\\geq 0$.  \n\t\tWe then conclude by Proposition~\\ref{prinmin} that for every $n\\geq 1$,\n\t\t\\begin{equation*}\n\t\t\ts+G_Dw^--G_Dp_n\\geq 0 \\mbox{ in }U_n.\n\t\t\\end{equation*}\n\t\tLetting $n$ tend to $\\infty$, we obtain that $u-v=s+G_Dw^- -G_Dw^+\\geq 0$ in $D\\as\\{w^+>0\\}$ which yields that $u-v\\geq 0$ in $D$.\n\t\\end{proof}\n\t\\begin{theorem}\\label{thexis}\n\t\tLet\t$s\\in\\mathcal{C}^+(D)$ be  $L-$superharmonic in $D$ such that $G_D(\\xi\\psi(s))\\not\\equiv \\infty$. Then there exists one and only one function  $u\\in \\mathcal{C}^+(D)$ such that \n\t\t\\begin{equation}\\label{qe}\n\t\t\tu+G_D(\\xi\\psi(u))=s\\mbox{ in }D.\n\t\t\\end{equation}\n\t\\end{theorem}\n\t\\begin{proof}\n\t\tDue to the previous proposition, the integral equation \\eqref{qe} has at most one solution.  The existence of a solution  will be established  in two steps.\n\t\t\n\t\t{\\em Step 1.} Consider first an open set $V\\Subset D$. Let $a=\\sup_{x\\in \\overline{V}}|s(x)|$ and define\n\t\t$$\n\t\t\\Lambda(u)=s-G_V(\\xi{\\psi_a}(u)) \\quad\\mbox{for all }u\\in\\mathcal{B}_b(V),\n\t\t$$\t\n\t\twhere \t${\\psi_a}(t)= \\psi(a\\wedge t)$ for every $t\\in\\mathbb{R}$. \n\t\tWe  claim that  $\\Lambda$ is continuous in $\\mathcal{B}_b(V)$ endowed with its uniform norm.  Indeed, let $(u_n)_{n\\geq 1}\\subset \\mathcal{B}_b(V)$ be a sequence converging   to a function  $u\\in \\mathcal{B}_b(V)$. Then, due to the dominated convergence theorem, $(G_V(\\xi{\\psi_a}(u_n))_{n\\geq 1}$ is pointwise  convergent to $G_V(\\xi\\psi_a(u))$.   Assume that the last convergence does not hold uniformly. Thus, there exist $\\varepsilon>0$ and a subsequence $(u_{n_k})_{k\\geq 1}$ of $(u_n)_{n\\geq 1}$\n\t\tsuch that \n\t\t\\begin{equation}\\label{unk}\n\t\t\t\\sup_{x\\in V}\\left|G_V(\\xi{\\psi_a}(u_{n_k}))(x)-G_V(\\xi{\\psi_a}(u))(x)\\right|\\geq \\varepsilon \\quad\\mbox{for all }k\\geq 1.\n\t\t\\end{equation}\n\t\tSince the operator  $G_V$ is compact  and  the sequence $(\\xi{\\psi_a}(u_{n_k}))_{k\\geq 1}$ is  bounded  in $\\mathcal{B}_b(V)$, we can find a subsequence $(w_k)_{k\\geq 1}$ of  $(u_{n_k})_{k\\geq 1}$ such that \n\t\t$(G_V(\\xi{\\psi_a}(w_k)))_{k\\geq 1}$ converges to $G_V(\\xi{\\psi_a}(u))$ with respect to the uniform norm in $\\mathcal{B}_b(V)$. This yields a contradiction with~\\eqref{unk} above. Hence $\\Lambda$ is continuous in $\\mathcal{B}_b(V)$ and the claim is proved. Using again the fact that the operator $G_V$ is compact, we check out that $ \\Lambda(\\mathcal{B}_b(V)) $ is  relatively compact in $\\mathcal{B}_b(V)$. Then, we conclude  in virtue of the Schauder's fixed point theorem, that there exists $v\\in \\mathcal{B}_b(V)$ such that $\\Lambda(v)=v$. On the other hand, using Proposition~\\ref{lemcom}, it is clear that $v\\geq 0$ in $V$ and thereby $0\\leq v \\leq s\\leq a$ in $V$. This yields that ${\\psi_a}(v)=\\psi(v)$ and consequently  \n\t\t\\begin{equation}\\label{eqcom}\n\t\t\tv+G_V(\\xi\\psi(v))=s\\quad\\mbox{in } V.\n\t\t\\end{equation}\t\n\t\tObserve that $v$ is continuous in $V$  since   $G_V(\\xi\\psi(v))$ is continuous in $V$  by   Proposition~\\ref{pror}.(a).    \n\t\t\n\t\t{\\em Step 2.} Choose a sequence $(D_{n})_{n\\geq 1}$ of regular open sets such that \n\t\t\\begin{equation*}\n\t\t\tD_n\\Subset D_{n+1} \\quad\\mbox{for all }n\\geq 1\\quad \\mbox{ and }\\cup_{n\\geq 1}D_n=D.\n\t\t\\end{equation*}\n\t\tLet, for every $n\\geq 1$,  $u_{n}$ be  the solution of \n\t\t\\begin{equation}\\label{eq2}\n\t\t\tu_{n}+G_{D_{n}}(\\xi\\psi(u_{n}))=s  \\quad \\mbox{in }D_{n}.\n\t\t\\end{equation}\n\t\tThen, in virtue of formula~\\eqref{ggg} we observe that \n\t\t$$\n\t\tu_{n+1}+G_{D_n}(\\xi\\psi(u_{n+1}))=s-H_{D_n}G_{D_{n+1}}(\\xi\\psi(u_{n+1})) \\quad\\mbox{in }D_n\n\t\t$$\n\t\tand hence\n\t\t$\n\t\tu_n-u_{n+1}+G_{D_n}(\\xi\\psi(u_n))-G_{D_n}(\\xi\\psi(u_{n+1}))\n\t\t$\n\t\tis a nonnegative $L-$harmonic function in $D_n$. So, in virtue of  Proposition~\\ref{lemcom}, we have  $0\\leq u_{n+1}\\leq u_n\\leq s$ in $D_n$ for all $n\\geq 1$. Let $u:D\\to{\\mathbb{R}}_+$ be the function given for every $x\\in D$ by \n\t\t$$\n\t\tu(x)=\\lim_{n\\to\\infty}u_n(x)=\\inf_{n\\geq 1} u_{n}(x).\n\t\t$$\n\t\tSeeing that  $G_D(\\xi\\psi(s))<\\infty$ in $D$ and  applying  the dominated convergence theorem, we get for every $x\\in D$ that \n\t\t$$\n\t\t\\lim_{n\\rightarrow \\infty}\\int_{D_{n}}G_{D_{n}}(x,y)\\xi(y)\\psi(u_{n}(y))dy  =  \\int_{D}G_{D}(x, y)\\xi(y)\\psi(u(y))dy.\n\t\t$$\n\t\tThus, letting $n$ tend to infinity   in~\\eqref{eq2} we conclude that $u$ satisfies~\\eqref{qe}.  The continuity of $u$ follows obviously from the fact that $G_D(\\xi\\psi(u))$ is continuous in~$D$. \n\t\\end{proof}\n\t\n\tFor every $f\\in\\mathcal{B}^+(\\partial D)$ and every locally bounded function $g\\in\\mathcal{B}^+(D)$ such that $H_Df$ and $G_Dg$ are not identically infinite in~$D$, it \n\tfollows from Proposition~\\ref{lem01} and Theorem~\\ref{theoo} that the function  \n\t\\begin{equation*}\n\t\tS_D(f,g)=H_Df +G_Dg\n\t\\end{equation*}\n\tis   continuous and $L-$superharmonic in $D$. Then, assuming that \n\t$G_D(\\xi\\psi( S_D(f,g)) )\\not\\equiv\\infty$ in~$D$,\n\tthe previous theorem yields that the integral  equation\n\t\\begin{equation}\\label{s2}\n\t\tu +\\int_D G_D(\\cdot,y)\\xi(y)\\psi(u(y))\\,dy= S_D(f,g)\n\t\\end{equation}\n\tadmits a unique   solution  $u\\in \\mathcal{C}^+(D)$. In all the following, we use the notation \n\t\\begin{equation}\n\t\tu=U_D^\\xi(f,g).\n\t\\end{equation}\n\tFor $i=1,2,$ let $f_i\\in\\mathcal{B}^+(\\partial D)$ and $g_i,\\xi_i \\in\\mathcal{B}^+(D)$ be locally bounded in~$D$ such that  $s_i=S_D(f_i, g_i)<\\infty$ and $G_D(\\xi_i\\psi(s_i)))<\\infty$ in~$D$.  If $f_1\\leq f_2$, $g_1\\leq g_2$ and $\\xi_1\\geq \\xi_2$, it follows immediately from Proposition~\\ref{lemcom} that \n\t\\begin{equation}\\label{compf1f2}\n\t\tU_D^{\\xi_1}(f_1, g_1)\\leq U_D^{\\xi_2}(f_2,g_2). \n\t\\end{equation}\n\t\n\t\n\t\n\t\\begin{proposition}\\label{copro}\n\t\tWe consider a locally bounded function $g\\in\\mathcal{B}^+(D)$  and let  $f\\in\\mathcal{B}^{+}(\\partial D)$\t such that $s=S_D(f,g)<\\infty$ and $G_D(\\xi\\psi(s))<\\infty$ in~$D$. Let $(\\xi_n)_{n\\geq 1}, (g_n)_{n\\geq 1}\\subset \\mathcal{B}^+(D) $ and $(f_n)_{n\\geq 1}\\subset \\mathcal{B}^{+}(\\partial D) $  be nondecreasing sequences which are   pointwise convergent respectively to $\\xi,g$ and~$f$. Then:\n\t\t\\begin{enumerate}\n\t\t\t\\item[a)]  $U_D^{\\xi_n}(f, g)$ decreases to $U_D^{\\xi}(f, g)$.\n\t\t\t\\item[b)] \t\t\t$U_D^{\\xi}(f_n, g_n)$ increases to $U_D^{\\xi}(f, g)$.\n\t\t\\end{enumerate}\n\t\\end{proposition}\n\t\\begin{proof}\n\t\tWe define $u_n:=U_D^{\\xi_n}(f, g)$ for every  $n\\geq 1$. Applying~\\eqref{compf1f2}, we get that $(u_n)_{n\\geq 1}$ is nonincreasing in $D$ and thereby it converges to $u:=\\inf_{n\\geq 1} u_n$. Furthermore, applying the dominated convergence theorem, we obtain \n\t\t$$\n\t\t\\lim_{n\\to\\infty} \\int_{D}G_D(\\cdot,y)\\xi_{n}(y)\\psi(u_n(y))\\,dy=\\int_{D}G_D(\\cdot,y)\\xi(y)\\psi(u(y))\\,dy.\n\t\t$$\n\t\tConsequently, $u+G_D(\\xi\\psi(u))=s$ in $D$ which yields that $u=U_D^\\xi(f,g)$ and statement~(a) is proved.  Statement~(b) is similar.\n\t\\end{proof}\n\t\n\t\n\tUnder appropriate conditions, the following theorem ensures that  $U_D^\\xi(f,g)$ is none other than   the solution to the semilinear boundary value problem~\\eqref{base}.\n\t\n\t\\begin{theorem}\\label{theopro}\n\t\tAssume that $G_D \\xi, G_D g\\in\\mathcal{C}_{0}(D)$. Then, for every function $f\\in\\mathcal{C}^+(\\partial D)$,  problem~\\eqref{base} admits one and only one solution $u\\in\\mathcal{C}^+(D)$. More precisely  $u=U_D^\\xi(f,g)$.  \n\t\\end{theorem}\n\t\n\t\\begin{proof}\n\t\tIt is sufficient to prove that $u$ is a solution to~\\eqref{s2} if and only if $u$ is the solution to problem~\\eqref{base}. Let $a=\\sup_{x\\in D}|S_D(f,g)|$ and let $u=U_D^\\xi(f,g)$ be the solution to \\eqref{s2}. It is easy to see   $0\\leq u \\leq a$ in $D$ which yields that \n\t\t$\n\t\tG_{D}(\\xi\\psi(u))\\leq \\psi(a) G_{D}\\xi<\\infty\n\t\t$ and thereby $G_D(\\xi\\psi(u))\\in \\mathcal{C}(D)$ and according to  formula~\\eqref{gdp},\n\t\t$$\n\t\t-LG_{D}(\\xi\\psi(u))=\\xi\\psi(u) \\quad\\mbox{in }D.\n\t\t$$ \n\t\tWe also have $-LG_{D}g=g$ in $D$ since $G_Dg$ is continuous. Therefore, $u\\in\\mathcal{C}^+(D)$ and\n\t\t$$\n\t\t-Lu+\\xi\\psi(u)=g \\quad\\mbox{in }D.\n\t\t$$ On the other hand, for every $z\\in\\partial D$,\n\t\t$\n\t\t0\\leq \\lim_{x\\to z}G_D (\\xi\\psi(u))(x)\\leq \\psi(a)\\lim_{x\\to z} G_{D}\\xi=0\n\t\t$\n\t\tand consequently $\\lim_{ x\\rightarrow z} u(x)=f(z)$ by~\\eqref{s2}.\n\t\tThus $u=U_D^\\xi(f,g)$ is a  solution to the boundary problem~\\eqref{base}.\n\t\tConversely, let $u$ be a solution to~\\eqref{base}. Using the fact that $G_D\\xi$ and $G_Dg$ are in the class $\\mathcal{C}_0(D)$, we easily check out that \n\t\t$$\n\t\th:=u+G_D(\\xi\\psi(u))-G_Dg\n\t\t$$\n\t\tis $L-$harmonic in $D$ and $\\lim_{x\\to z}h(x)=f(z)$ for all $z\\in\\partial D$. Thus $h=H_Df$ and $u$ satisfies~\\eqref{s2}.\t\n\t\\end{proof}\n\t\n\tIn virtue of the previous theorem, we observe that for any function $u\\in\\mathcal{C}^+(D)$,  the following three statements are equivalent:\n\t\\begin{enumerate}\n\t\t\\item[i)] The function $u$ is a solution to equation~\\eqref{peq} in $D$.\n\t\t\\item[ii)] $U_V^\\xi(u,g)=u$ in $V$ for every regular set $V\\Subset D$.\n\t\t\\item[iii)]\t$U_B^\\xi(u,g)=u$ in $B$ for every ball $B\\Subset D$.\n\t\\end{enumerate}\n\t\n\t\n\t\n\t\n\t\\begin{proposition}\\label{theo3}  Assume that   $\\xi\\in\\mathcal{C}^\\alpha(D)$,  $\\psi\\in\\mathcal{C}^\\alpha(\\mathbb{R}_+)$ and let $f\\in\\mathcal{B}(\\partial D)$, $g\\in\\mathcal{C}^\\alpha(D)$ be nonnegative functions such that $S_D(f,g)<\\infty$ and $G_D\\big(\\xi\\psi(S_D(f,g))\\big)<\\infty$ in~$D$. Then  $U_D^\\xi(f,g)$ is twice continuously  differentiable in~$D$.  \n\t\\end{proposition}\n\t\n\t\\begin{proof} Consider a ball    $B\\Subset D$. \t Notice that $u=U_D^\\xi(f,g)$ is continuous in~$D$ and, in virtue of~\\eqref{s2} and~\\eqref{ggg}, we have \n\t\t$$\n\t\tu=H_{B}u+G_{B}g-G_{B}\\big(\\xi \\psi(u)\\big)\\quad \\mbox{in }B.\n\t\t$$ \n\t\tWe first  observe that $H_{B}u\\in \\mathcal{C}^{2}(B)$ by the hypoellipticity property of~$L$ and $G_Bg\\in \\mathcal{C}^{2}(B)$ in view of~(b) in Proposition~\\ref{pror}. Next, since $p=\\xi\\psi(u)$ is bounded in~$B$,  statement~(a) in Proposition~\\ref{pror} yields that  $G_Bp\\in\\mathcal{C}^\\alpha(B)$ and consequently $u\\in \\mathcal{C}^\\alpha(B)$. By simple computation, we check out that  $p\\in \\mathcal{C}^{\\alpha^2}(B)$ and thereby $G_Bp\\in \\mathcal{C}^{2}(B)$ by Proposition~\\ref{pror}(b). Whence $u\\in \\mathcal{C}^2(B)$ for any arbitrary ball $B\\Subset D$ which means that $u\\in\\mathcal{C}^2(D)$. \n\t\\end{proof}\n\t\n\tLet us note  that all  results obtained  in this section can easily be extended to a more general setting, where we consider instead of  functions $\\xi(x)$ and $\\psi(t)$, a function \n\t$$\n\t\\Phi: (x,t)\\in  D\\times \\mathbb{R}_{+} \\mapsto \\Phi(x,t)\\in \\mathbb{R}\n\t$$ \n\tsatisfying some hypotheses analogous to those assumed on  $\\xi$ and $\\psi$. Besides, it should be specified that the fact that $\\xi$ and $g$ are locally bounded is not really necessary in our approach. In fact, it is enough to assume that $\\xi$ and $g$ belong to the local Kato class \\cite{chung1995brownian}.\n\t\n\t\\section{Estimates of solutions}\n\t\n\t\n\t\n\tThis section is devoted to establish our main results which consist in  lower bound estimates of  nonnegative  solutions to equation~\\eqref{peq} and more general to inequality \n\t\\begin{equation}\\label{sup-sol0}\n\t\t-Lu+\\xi\\psi(u) \\geq  g \\mbox{ in }D.\n\t\\end{equation} \n\tMore precisely, under appropriate conditions on $\\psi$, we will determine a function $\\varphi$ with values in  $]0,1]$  such that \tfor every   $f\\in\\mathcal{C}^+(\\partial D)$, $g\\in\\mathcal{B}^+(D)$ and for all $x\\in D$,\t\n\t\\begin{equation}\\label{eqcp}\n\t\tU_D^\\xi(f,g)(x)\\geq\ts(x)\\varphi\\left(\\frac{G_D(\\xi\\psi(s))(x)}{s(x)}\\right).\n\t\\end{equation}\n\tThe function  $g$ is assumed to be locally bounded  in~$D$ such that \t$G_Dg\\in\\mathcal{C}_0(D)$ and the $L-$superharmonic function \n\t$s=S_D(f,g)>0$ in $D$.\n\t\n\tWe are only interested  in locally bounded functions~$\\xi\\in\\mathcal{B}(D)$ which are nonnegative. Analogous problems  with $\\xi\\leq 0$  are recently investigated by  Grigor'yan and Verbitsky in~\\cite{grigor2020}. They gave a real positive function $\\varrho$, depending only on~$\\psi$, such that if  $S_D(f,g)\\geq 1$ in $D$, then the  solution $u$ to problem~\\eqref{base}  satisfies for all $x\\in D$,\n\t\\begin{equation}\\label{urho}\n\t\tu(x)\\geq \\varrho\\left(G_D\\xi \\right(x)).\n\t\\end{equation}\n\tThe particular case where $L=\\Delta$ is the Laplacian operator and $f,g$ are identically zero, is already studied by  M\\^aagli and Zribi \\cite{zribi2001}.\n\tFor signed function $\\xi$,   problem~\\eqref{base} is  examined  by  Grigor'yan and Verbitsky  \\cite{grigor2019pointwise} in the special case where $\\psi(t)=t^\\gamma$ with $\\gamma>0$.  Assuming that $p=G_Dg>0$ in $D$, they showed that  if  $u\\in \\mathcal{C}^2(D)$  is a nonnegative solution to \n\t\\begin{equation}\\label{sup-sol-g}\n\t\t-Lu+\\xi u^\\gamma \\geq  g \\mbox{ in }D,\n\t\\end{equation}     \n\tthen \t for every $x\\in D$,\n\t\\begin{equation}\\label{estgr}\n\t\tu(x)\\geq \\left\\{\\begin{array}{ll}\n\t\t\tp(x)\\left(1+(\\gamma-1)\\frac{G_D\\left(p^\\gamma \\xi\\right)(x)}{p(x)}\\right)_+^{1\/(1-\\gamma)} & \\mbox{if }\\gamma<1,\n\t\t\t\\\\ p(x)\\exp\\left({-\\frac{G_D(\\xi p)(x)}{p(x)}}\\right) & \\mbox{if }\\gamma=1, \n\t\t\t\\\\ p(x)\\left(1+(\\gamma-1)\\frac{G_D(p^\\gamma \\xi)(x)}{p(x)}\\right)^{1\/(1-\\gamma)} & \\mbox{if }\\gamma>1.\n\t\t\\end{array}\\right.\n\t\\end{equation}\t\n\t\n\tIn this paper, we establish similar estimates for a large class of functions $\\psi$ which contains   polynomial functions, logarithmic functions as well as  exponential functions. More precisely, we consider a continuous function  $\\psi:\\mathbb{R}_{+}\\to \\mathbb{R}_{+}$ such that  $\\psi(0)=0$,  $\\psi(t)>0$ for all $t>0$, $\\psi$ is continuously differentiable in $ ]0,\\infty[$ and \n\t\\begin{equation}\\label{hypcom}\n\t\t\\psi(rt)\\leq c\\psi(r)\\psi(t)\\quad \\mbox{for all }r\\in[0,1], t\\in\\mathbb{R}_{+} \n\t\\end{equation}\n\twhere $c$ is a positive constant not depending on either $s$ or $t$. Let $0<\\ell\\leq \\infty$ given by \n\t\\begin{equation*}\n\t\t\\ell c=\\int_{0}^{1}\\frac{1}{\\psi(s)}\\,ds\n\t\\end{equation*}\n\tand define the function  $\\Theta:]0,1]\\rightarrow [0, \\ell[$  for every $t\\in ]0,1]$ by \n\t$$\n\t\\Theta(t)=\\int_{t}^{1}\\frac{1}{c\\psi(s)}ds.\n\t$$\n\tThen, obviously   $\\Theta$ admits an inverse function which will be denoted by $\\varphi$ and extended  to~$\\mathbb{R}_+$ by setting $\\varphi(t)=0$ if $t\\geq \\ell$. Hence, $\\varphi$ is continuous and nonincreasing in  $\\mathbb{R}_+$, twice continuously  differentiable in $]0,\\ell[$  and for every $r\\in ]0,\\ell[$, \n\t$$\n\t\\varphi'(r)=-c\\psi(\\varphi(r))<0\\quad\\mbox{ and }\\quad \\varphi''(r)=c^2\\psi'(\\varphi (r))\\psi(\\varphi (r))> 0.\n\t$$ \n\t\n\tIn order to prove Theorem~\\ref{thp} below, we first establish some technical lemmas. \n\t\\begin{lemma}\\label{lem1}\n\t\tLet $u:D\\rightarrow \\mathbb{R}_+$ and $v:D\\rightarrow ]0,\\ell[$ be  twice continuously differentiable  functions. If $u$ is $L-$superharmonic in $D$, then\n\t\t$$\n\t\tL(uv)\\geq \t\\frac{Lu- L(u\\varphi(v))}{c\\psi(\\varphi(v))}.\n\t\t$$\n\t\\end{lemma}\n\t\\begin{proof} Obviously  $w=\\vphi(v)\\in\\mathcal{C}^2(D)$ and by elementary  computation, we get that  \n\t\t$$\n\t\tL w=\\varphi'(v)Lv+\\varphi''(v)<A\\nabla v,\\nabla v>.\n\t\t$$\n\t\tConsequently \n\t\t\\begin{eqnarray*}\n\t\t\tL(uw)&=&uLw+wLu+<\\nabla u, A\\nabla w>+<A\\nabla u,\\nabla w>\\\\\n\t\t\t&=&\\varphi'(v)uLv+\\varphi''(v)u<A\\nabla v,\\nabla v>+\\varphi(v)Lu\\\\\n\t\t\t&&\\qquad+\\varphi'(v)<\\nabla u, A\\nabla v>+\\varphi'(v)<A\\nabla u, \\nabla v>\\\\\n\t\t\t&=&\\varphi(v)Lu-\\varphi'(v) vLu+ \\varphi''(v)u<A\\nabla v,\\nabla v>\\\\\n\t\t\t&&\\qquad +\\varphi'(v)\\left( uLv+vLu+ <\\nabla u, A\\nabla v>+<A\\nabla u, \\nabla v>\\right)\\\\\n\t\t\t&=&\\varphi(v)Lu-\\varphi'(v) vLu+ \\varphi''(v)u<A\\nabla v,\\nabla v>+\\varphi'(v) L(uv)\n\t\t\\end{eqnarray*}\n\t\tSince the matrix $A$ is positive-definite and the function $\\varphi$ is convex, it follows that \n\t\t\\begin{eqnarray*}\n\t\t\t&&\tL(u\\varphi(v))-\\left(\\varphi(v)-v\\varphi'(v)\\right)Lu-\\varphi'(v)L(uv) \\\\\n\t\t\t&& \\qquad\\qquad\\qquad = \\quad \\varphi''(v)u<A\\nabla v,\\nabla v>\\\\\n\t\t\t&& \\qquad\\qquad\\qquad \\geq \\quad 0.\t\n\t\t\\end{eqnarray*}\n\t\tHence\n\t\t\\begin{eqnarray*}\n\t\t\t-\\varphi'(v)L(uv)  & \\geq &  -L(u\\varphi(v))+\\left(\\varphi(v)-v\\varphi'(v)\\right)Lu\n\t\t\t\\\\ & = &\n\t\t\t-L(u\\varphi(v))+\\left(\\varphi(v)-1-v\\varphi'(v)\\right)Lu+Lu\n\t\t\\end{eqnarray*}\n\t\tIn virtue of the mean value theorem and the fact that $\\varphi'$ is nondecreasing, we obtain that\n\t\t$\n\t\t\\varphi(v)-1-v\\varphi'(v)\\leq 0\n\t\t$ in $D$. \n\t\tAssume that $u$ is $L-$superharmonic in $D$. Then we deduce from  \\cite[Theorem~3.5]{gilbarg1998elliptic} that $Lu\\leq 0$ in $D$ and consequently  $\\left(\\varphi(v)-1-v\\varphi'(v)\\right)Lu\\geq 0$ in $D$.\n\t\tFinally, using the fact that $\\varphi'(v)=-c\\psi(\\varphi(v))\\leq 0 $, we get that\n\t\t$$\n\t\tL(uv)\\geq \t\\frac{Lu- L(u\\varphi(v))}{c\\psi(\\varphi(v))}.\n\t\t$$\n\t\\end{proof}\n\t\n\t\\begin{lemma}\\label{lem2}\n\t\tLet $f\\in\\mathcal{B}_b^+(\\partial D)$ and $g\\in\\mathcal{B}_b^+(D)$ such that $s=S_D(f,g)>0$ in $D$. Assume that  $\\xi$ is bounded and let  $(\\xi_n)_{n\\geq 1}$, $(g_n)_{n\\geq 1}\\subset \\mathcal{B}^{+}(D)$ and $(f_n)_{n\\geq 1}\\in\\mathcal{B}^+(\\partial D)$  be bounded sequences which are pointwise convergent respectively to $\\xi$, $g$ and $f$. Assume that for all $n\\geq 1$, $s_n=S_D(f_n,g_n)>0$ and \n\t\t\\begin{equation*}\n\t\t\tU_D^{\\xi_n}(f_n,g_n)\\geq s_n\\varphi\\left(\\frac{G_D(\\xi_n\\psi(s_n))}{s_n}\\right)\\quad\\mbox{in }D.\n\t\t\\end{equation*}\t\n\t\tThen  \\eqref{eqcp}  holds true.\n\t\\end{lemma}\n\t\\begin{proof} We have $u_n+G_Dp_n=s_n$ where  $u_n=U_D^{\\xi_n}(f_n, g_n)$  and $p_n=\\xi_n\\psi(u_n)$  for all $n\\geq 1$. Since $(p_n)_{n\\geq 1}$ is a bounded  sequence, it follows from Proposition~\\ref{pro2} that there exists a subsequence $(p_{n_k})_{k\\geq 1}$ such that \t\t$(G_Dp_{n_k})_{k\\geq 1}$  is uniformly convergent in $D$. Seeing that  $(s_{n_k})_{k\\geq 1}$ is pointwise convergent to $s$, we conclude that $(u_{n_k})_{k\\geq 1}$ converges in $D$ to a   function $u\\in\\mathcal{B}_b^+(D)$.  Applying the dominated convergence theorem, we obtain that for every $x\\in D$,\n\t\t$$\n\t\t\\lim_{k\\to\\infty}G_Dp_{n_k}(x)=G_D(\\xi\\psi(u))(x).\n\t\t$$\n\t\tTherefore  $ u+G_D(\\xi\\psi(u))=s$ in $D$, which yields in view of Theorem~\\ref{thexis} that $u=U_D^{\\xi}(f,g)$.   Again, since sequences $(\\xi_n)_{n\\geq 1}$ and  $(s_n)_{n\\geq 1}$ are bounded, we see that  $(G_D(\\xi_{n_k}\\psi(s_{n_k})))_{k\\geq 1}$  is pointwise convergent to $G_D(\\xi\\psi(s))$. Hence, letting $k$ tend to infinity in the formula\n\t\t$$\n\t\tu_{n_{k}}\\geq \ts_{n_k}\\varphi\\left(\\frac{G_D(\\xi_{n_k}\\psi(s_{n_k}))}{s_{n_k}}\\right) \\quad\\mbox{in }D,\n\t\t$$\n\t\twe obtain \\eqref{eqcp} in virtue of the continuity of $\\varphi$.\n\t\\end{proof}\n\t\n\t\\begin{theorem}\\label{thp} \t\n\t\tLet $g\\in\\mathcal{B}^+(D)$ be locally bounded and  let $f\\in\\mathcal{B}^+(\\partial D)$  such that $0<s<\\infty$ and $G_D(\\xi\\psi(s))<\\infty$ in $D$ where   \n\t\t$s=S_D(f,g)$. \n\t\tThen \n\t\t\\begin{equation*}\n\t\t\ts\\,\\varphi\\left(\\frac{G_{D}(\\xi\\psi(s))}{s}\\right)\\leq U_D^\\xi(f,g) \\leq s \\quad \\mbox{ in } D. \n\t\t\\end{equation*}\n\t\\end{theorem}\n\t\\begin{proof} Seeing that $u\\leq s$ is immediate with $u= U_D^\\xi(f,g)$, we only need to prove that \n\t\t\\begin{equation}\\label{estf}\n\t\t\t\\frac{u}{s}\\geq \\varphi\\left(\\frac{G_D(\\xi\\psi(s))}{s}\\right)\\quad \\mbox{in } D.\n\t\t\\end{equation}\n\t\t\n\t\t{\\em Step 1.} We first assume that $\\xi$, $g$ are bounded and $\\alpha-$H\\\"older continuous in $D$ and that~$f$ is bounded in~$\\partial D$. Then \n\t\t$s=H_Df +G_D g\\in\\mathcal{C}^2(D)$ since  $H_Df\\in\\mathcal{C}^2(D)$ by the  hypoellipticity property of~$L$ and  $G_Dg \\in\\mathcal{C}^2(D)$ in view of Proposition~\\ref{pror}.(b).  Besides, $u\\in\\mathcal{C}^2(D)$  in virtue of  Proposition~\\ref{theo3} and the fact that $\\xi$ and $\\psi$ are $\\alpha-$H\\\"older continuous.\n\t\tTherefore, for every $\\ve>0$, $v_{\\ve}=\\Theta\\left(\\frac{u+\\varepsilon}{s+\\varepsilon}\\right)$ is twice  continuously differentiable in $D$ with values in $]0,\\ell[$ and consequently using  Lemma~\\ref{lem1} we obtain that\n\t\t\\begin{eqnarray*}\n\t\t\tL((s+\\ve)v_\\ve)\n\t\t\t\\geq \\frac{L(s+\\ve)-L(u+\\ve)}{c\\psi(\\frac{u+\\ve}{s+\\ve})}\n\t\t\t=-\\xi\\frac{\\psi(u)}{c\\psi(\\frac{u+\\ve}{s+\\ve})}\\geq -\\xi \\psi(s).\n\t\t\\end{eqnarray*}\n\t\tThe last inequality follows from~\\eqref{hypcom} and the fact that $\\psi$ is nondecreasing in~$\\mathbb{R}_+$. We also see that $G_D(\\xi\\psi(s))\\in \\mathcal{C}^2(D)$ by Proposition~\\ref{pror}.(b) and therefore\n\t\t\\begin{equation*}\n\t\t\tw_\\ve=(s+\\ve)v_\\ve - G_D(\\xi\\psi(s))\\in \\mathcal{C}^2(D) \\quad\\mbox{and} \\quad L w_\\ve\\geq 0\\mbox{ in }D.\n\t\t\\end{equation*}\n\t\tOn the other hand, for every $z\\in \\partial D$,   $\\lim_{x\\to z}G_D(\\xi\\psi(s))(x)=0$ by Proposition~\\ref{pror}.(a). Moreover, seeing that\n\t\t$$\n\t\t1\\leq \\frac{s+\\ve}{u+\\ve}= \\frac{u+\\ve+G_D(\\xi\\psi(u))}{u+\\ve}\\leq 1+\\frac{G_D(\\xi\\psi(s))}{\\ve}\\quad\\mbox{in }D,\n\t\t$$\n\t\twe also obtain that  $\\lim_{x\\to z}v_\\ve(x)=\\Theta(1)=0$ and thereby  $\\lim_{x\\to z}w_\\ve(x)=0$ for all $z\\in \\partial D$. We then conclude  in view of the maximum principle \\cite[Theorem 3.1]{gilbarg1998elliptic} that $w_\\ve\\leq 0$ in $D$ and consequently \n\t\t\\begin{equation*}\n\t\t\t\\frac{u+\\ve}{s+\\ve}\\geq \\varphi\\left(\\frac{G_D(\\xi\\psi(s))}{s+\\ve}\\right)\\quad \\mbox{ in } D.\n\t\t\\end{equation*}\n\t\tLetting $\\ve$ tend to $0$, we conclude that~\\eqref{estf} holds true whenever  $\\xi$ and $g$ are  bounded  $\\alpha-$H\\\"older continuous in $D$ and that~$f$ is bounded in~$\\partial D$. \t \n\t\t\n\t\t\n\t\t{\\em Step 2.} We consider at this step the case where $\\xi$ and $g$ are only bounded in $D$ and we still assume that  $f$ is   bounded in $\\partial D$. In virtue of  Corollary~4.23 and Theorem~4.9 in~\\cite{brezis2010functional}, we choose two sequences $(\\xi_n)_{n\\geq 1}, (g_n)_{n\\geq 1}\\subset \\mathcal{C}_c^{\\infty}(D) $ such that $0\\leq \\xi_n\\leq \\|\\xi\\|_{\\infty}$, $0\\leq g_n\\leq \\|g\\|_{\\infty}$ for all $n\\geq 1$ and \n\t\t$$\n\t\t\\lim_{n\\to\\infty}\\xi_n= \\xi,\\; \\lim_{n\\to\\infty}g_n= g\\;\\mbox{a.e in }D.\n\t\t$$\n\t\tSeeing that $\\lim_{n\\to\\infty}S_D(f,g_n)(x)=s(x)$ and $s(x)>0$ for all $x\\in D$, we can impose on the $L$-superharmonic function  $s_n=S_D(f,g_n)$ to be positive everywhere in $D$ for every $n\\geq 1$.  It follows from Step~1 above that, for all $n\\geq 1$,  \n\t\t$$\n\t\t\\frac{U_D^{\\xi_n}(f,g_n)}{s_n}\\geq \\varphi\\left(\\frac{G_D(\\xi_n\\psi(s_n))}{s_n}\\right)\\quad\\mbox{in }D.\n\t\t$$\n\t\tHence, \\eqref{estf} holds true by Lemma~\\ref{lem2}.\n\t\t\n\t\t{\\em Step 3.} We return to the general setting where functions  $\\xi, g$  are nonnegative locally bounded in~$D$ and $f\\in\\mathcal{B}^+(\\partial D)$. We define for every $n\\geq 1$,    $\\xi_n=\\xi\\wedge n$, $g_n=g\\wedge n$, $f_n=f\\wedge n$ and let  $ s_n=S_D(f_n,g_{n})>0$ in~$D$.   According to Step~2,  for every $n\\geq 1$ and every $k\\geq 1$,\n\t\t\\begin{equation}\\label{estk}\n\t\t\t\\frac{U_D^{\\xi_k}(f_n,g_n)}{s_n}\\geq \\varphi\\left(\\frac{G_D(\\xi_k\\psi(s_n))}{s_n}\\right)\\quad\\mbox{in }D.\n\t\t\\end{equation}\n\t\tOn the other hand, for every $k\\geq 1$, $\\lim_{n\\to\\infty}U_D^{\\xi_k}(f_n,g_n)=U_D^{\\xi_k}(f,g)$  by  Proposition~\\ref{copro}.(b) and  $\\lim_{n\\to\\infty}G_D(\\xi_k\\psi(s_n))=G_D(\\xi_k\\psi(s))$  by the monotone convergence theorem. Hence, letting $n$ tend to infinity in \\eqref{estk},  we get that for all $k\\geq 1$,\n\t\t\\begin{equation*}\n\t\t\t\\frac{U_D^{\\xi_k}(f,g)}{s}\\geq \\varphi\\left(\\frac{G_D(\\xi_k\\psi(s))}{s}\\right)\\quad\\mbox{in }D.\t\n\t\t\\end{equation*} \n\t\tLetting $k$ tend to infinity, we conclude that~\\eqref{estf} holds true in virtue of the continuity of  $\\varphi$. Observe that  $\\lim_{k\\to\\infty}U_D^{\\xi_k}(f,g)=u$  by Proposition~\\ref{copro} and $\\lim_{k\\to\\infty}G_D(\\xi_k\\psi(s))=~G_D(\\xi\\psi(s))$ by the monotone convergence theorem.\n\t\t\n\t\\end{proof}\n\t\n\tNow, we  use our previous results  to establish a lower bound estimate of nonnegative solutions to inequality\n\t\\begin{equation}\\label{sup-sol}\n\t\t-Lu+\\xi\\Psi(u) \\geq  g \\quad\\mbox{in }D,\n\t\\end{equation} \n\twhere $\\Psi:\\mathbb{R}_+\\to\\mathbb{R}$ is any locally bounded Borel measurable function which is  bounded above  by some function $\\psi$ satisfying hypotheses given at the beginning of this section.  Here, we do not need to assume that $\\Psi$ is monotone nor that it keeps a constant sign.   We point out that a solution $u$ to inequality~\\eqref{sup-sol} should be understood in the distributional sense, that is,  $u:D\\to\\mathbb{R}$ is continuous  and for every nonnegative function   $\\varphi \\in \\mathcal{C}_{c}^{\\infty}(D)$,\n\t\\begin{equation*}\n\t\t-\\int_{D}uL^{*}\\varphi\\, d\\lambda_d+\\int_{D}\\xi\\Psi(u)\\varphi \\, d\\lambda_d \\geq  \\int_{D}g\\varphi \\, d\\lambda_d.\n\t\\end{equation*}\n\t\\begin{theorem}\\label{thecp}\n\t\tLet  $\\Psi$ and $\\psi$ be as above,  $D$ be any Greenian domain of $\\mathbb{R}^d$  (not necessarily bounded) and let  $g\\in \\mathcal{B}^+(D)$ be locally bounded such that $0<p<\\infty$ in $D$ where $p=G_D g$. Then, for every nonnegative solution $u$ to inequality~\\eqref{sup-sol} such that   $Lu$ is a locally bounded function in $D$,    \n\t\t\\begin{equation}\\label{estg}\n\t\t\tu\\geq p\\, \\varphi\\Big(\\frac{G_D (\\xi\\psi(p))}{p}\\Big) \\quad\\mbox{in }D.\n\t\t\\end{equation}\t\t\n\t\\end{theorem}\n\t\\begin{proof} \t Let  $u$ be a  solution to inequality~\\eqref{sup-sol}, define $g_0:=-Lu+\\xi\\psi(u)$ and choose a sequence  $(D_{n})_{n\\geq 1}$  of regular bounded   domains such that $D_n\\Subset D_{n+1}$ for all $n\\geq 1$ and $\\cup_{n\\geq 1}D_n=D$. For every $n\\geq 1$, define $v_n:=U_{D_n}^\\xi(0,g)$, that is,  \n\t\t\\begin{equation*}\n\t\t\t\\left\\{\\begin{array}{rcll}\n\t\t\t\t-Lv_n+\\xi\\psi(v_n) &=& g &\\mbox{in }D_n,\n\t\t\t\t\\\\ v_n &=& 0&\\mbox{on }\\partial D_n.\n\t\t\t\\end{array}\\right.\n\t\t\\end{equation*}\t\n\t\tSince $u\\geq 0$ and since $g_0\\geq -Lu+\\xi\\Psi(u)\\geq  g$ in $D$, it follows immediately from~\\eqref{compf1f2} that $\tu=U_{D_n}^\\xi(u,g_0)\\geq  v_n$ in~$D_n$. On the other hand, we obviously see that the sequence  $(p_n)_{n\\geq 1}$ of $L$-superharmonic functions in~$D_n$ given by  $p_n=G_{D_n}g=S_{D_n}(0,g)$, is nondecreasing and pointwise convergent to~$p>0$ in~$D$. Then, there exists a subsequence $(p_{n_k})_{k\\ge 1}$ such that $p_{n_k}>0$ in $D_{n_k}$ for every $k\\geq 1$.  We then apply Theorem~\\ref{thp} in the  regular domain~$D_{n_k}$ to obtain \n\t\t\\begin{equation*}\n\t\t\tv_{n_k}\\geq p_{n_k} \\vphi\\left(\\frac{G_{D_{n_k}}(\\xi\\psi(p_{n_k}))}{p_{n_k}}\\right)\\quad\\mbox{in }D_{n_k}. \n\t\t\\end{equation*}\n\t\tHence, seeing that $p_{n_k}\\leq p$ in $D_{n_k}$ and using   monotonicities of functions $\\psi,\\vphi$, we get that for all $k\\geq 1$, \n\t\t\\begin{equation*}\n\t\t\tu\\geq p_{n_k} \\vphi\\left(\\frac{G_{D_{n_k}}(\\xi\\psi(p))}{p_{n_k}}\\right)\\quad\\mbox{in }D_{n_k}. \n\t\t\\end{equation*}\n\t\tConsequently, by letting  $k$ tend to $\\infty$ we obtain estimate~\\eqref{estg} and the proof is complete. \n\t\\end{proof}\n\t\n\tIn order to illustrate our results regarding  estimates of type~\\eqref{estg}, we  give the following examples which deal with three different kinds of functions $\\psi$.\n\t\n\t\\medskip\t\n\t\\noindent{\\it Example 1- Polynomial functions: }\n\tFor $\\psi(t)=t^\\gamma$  we have   $\\ell= 1\/(1-\\gamma)$ if $0<\\gamma <1$ and $\\ell=\\infty$ if $\\gamma \\geq 1$. By elementary computation of $\\varphi(t)$, Theorem~\\ref{thecp} above yields that estimates given by~\\eqref{estgr}  hold true for every   nonnegative solution to inequality~\\eqref{sup-sol-g}. \tAs  mentioned above, these estimates are already obtained by Grigor'yan and Verbitsky~\\cite{grigor2019pointwise} provided that the solution  $u$ is  twice continuously  differentiable  in~$D$. Notice that, in this work, we deal  with continuous  solutions in the distributional sense. More generally, for  $\\psi(t)=at+bt^\\gamma$  where $a,b,\\gamma>0$, we obtain\n\t$$\n\t\\ell=\\left\\{\\begin{array}{cl}\n\t\t\\displaystyle\\frac{a+b}{a(1-\\gamma)}\\log\\frac{a+b}{b}& \\mbox{if }\\gamma<1,\n\t\t\\\\\t\\vspace{0,3cm}\n\t\t\\infty & \\mbox{if }\\gamma>1,\n\t\\end{array}\\right.\n\t$$\n\tand by simple computation of  $\\varphi(t)$, we deduce that  every nonnegative solution $u$ to\n\t\\begin{equation}\n\t\t-Lu+\\xi(au+bu^\\gamma)\\geq g\\quad \\mbox{in }D\n\t\\end{equation}\t\n\tsatisfies, for all $x\\in D$,\n\t\\begin{equation*}\n\t\tu(x)\\geq \\left\\{\\begin{array}{ll}\n\t\t\t\\displaystyle{p(x)\\left(\\frac{a+b}{a}\\exp\\Big(\\frac{a}{a+b}(\\gamma-1)\\frac{G_D\\xi(ap+bp^{\\gamma})(x)}{p(x)} \\Big)-\\frac{b}{a}\\right)_+^{1\/(1-\\gamma)}} & \\mbox{if }\\gamma<1,\n\t\t\t\\vspace{0,3cm}\n\t\t\t\\\\ \\displaystyle{ p(x)\\left(\\frac{a+b}{a}\\exp\\Big((\\gamma-1)\\frac{G_D\\xi(ap+bp^{\\gamma})(x)}{p(x)} \\Big)-\\frac{b}{a}\\right)^{1\/(1-\\gamma)}} & \\mbox{if }\\gamma>1.\n\t\t\\end{array}\\right.\n\t\\end{equation*}\n\t\n\t\n\t\\medskip\t\n\t\\noindent{\\it Example 2- Exponential functions:} For $\\psi(t)=\\sinh t=(e^t-e^{-t})\/2 $, we prove that  $l=\\infty$ and elementary computation  yields  that for every $t\\geq 0$,\n\t$$\n\t\\vphi(t)=2\\,\\mbox{artanh}\\left(\\alpha e^{-t}\\right)\\quad \\mbox{with}\\quad\\alpha=\\tanh\\frac12.\n\t$$\n\tHence, we conclude that for\tevery nonnegative solution $u$ to inequality\n\t\\begin{equation}\n\t\t-Lu+\\xi \\sinh u\\geq g\\quad \\mbox{in }D\n\t\\end{equation}\n\tand for all $x\\in D$, we have\n\t$$\n\tu(x)\\geq 2\\, p(x)\\,\\mbox{artanh}\\left(\\alpha\\exp\\Big({-\\frac{G_D(\\xi\\sinh(p))(x)}{p(x)}}\\Big)\\right).\n\t$$\n\t\n\t\n\t\n\t\n\t\\medskip\t\n\t\\noindent{\\it Example 3- Logarithmic functions:}\n\tConsider $\\psi(t)=a(1+bt)\\log(1+bt)$ with $a,b>0$. By simple computation, we get that $\\ell=\\infty$ and for every real $t\\geq 0$,\n\t$$\n\t\\varphi(t)=\\frac{(1+b)^{e^{-t}}-1}{b}.\n\t$$\n\tIn particular, for $a=b=1$, it follows that every nonnegative solution $u$ to inequality \n\t\\begin{equation}\n\t\t-Lu+ \\xi (1+u)\\log(1+u)\\geq g\\quad \\mbox{in }D,\n\t\\end{equation}\n\tsatisfies, for all $x\\in D$,\n\t$$\n\tu(x)\\geq p(x)\\left(\n\t2^{\\displaystyle{\\exp\\left( -\\frac{G_D(\\xi(1+p)\\log(1+p))(x)}{p(x)}\n\t\t\t\\right)}}\n\t-1\\right)\n\t$$\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\vspace*{-5pt}\n\\begin{figure}[t]\n\t\\centering\n\t\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=2cm 1cm 2cm 1cm, width=0.45\\linewidth]{goodix.pdf}\n\t}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=6.5cm 12cm 10cm 11.8cm, width=0.45\\linewidth]{minutiae.pdf}\n\t}           \n\t\\hspace{0.2 cm}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=4cm 8cm 4cm 8cm, width=0.55\\linewidth]{minu_distribution.pdf}\n\t}\n\t\\caption{Fingerprint texture for matching.  (a) Only two minutiae in a fingerprint image  (96$\\times $ 96 at 500 ppi) acquired by a capacitive sensor embedded in a smartphone (provided by Goodix), (b) five manually marked minutiae in a latent fingerprint image from NIST SD27, and (c) histogram of the number of minutiae in latent fingerprints in NIST SD27 database (consisting of 258 operational latents).}\n\t\\label{fig:introduction}\n\t\\vspace*{-5pt}\n\\end{figure}\n\nEver since minutiae were introduced for comparing fingerprints in 1888 by Sir Francis Galton \\cite{Galton},  minutiae are considered as the foundation for the science of fingerprint identification, which has expanded and transitioned to a wide variety of applications for person recognition over the past century \\cite{FingerprintHandbook}.  The first Automated Fingerprint Identification System (AFIS)  launched by FBI in early 1970s only stored  type of fingerprint and its minutiae instead of digital images because of the compact and efficient representation offered by minutiae\\footnote{https:\/\/www.fbi.gov\/file-repository\/about-us-cjis-fingerprints\\_biometrics-biometric-center-of-excellences-fingerprint-recognition.pdf\/view}.  With decades of research and development  and advances in processor, memory and sensor design,  fingerprint recognition systems have now been deployed in a broad set of applications such as border\ncontrol, employment background checks, secure facility access and national identity programs \\cite{FingerprintHandbook}.\n Aadhaar\\footnote{https:\/\/uidai.gov.in\/about-uidai\/about-uidai.html},  has the world's largest biometric ID system with an enrollment database that already exceeds 1.2 billion tenprints\n(along with corresponding irises and photos) of supposedly unique individuals. All of these systems are primarily based on minutiae based fingerprint matching algorithms. \n\n\n\n\nMinutiae based approaches, however, may not be effective in some cases, for example, in poor quality latent fingerprint matching and fingerprint images captured by small area sensors in mobile phones.  Latent fingerprints (latents)  are one of the\nmost important and widely used sources of evidence in law enforcement and forensic agencies worldwide \\cite{Hawthorne2002}. Due to the unintentional deposition of the print\nby a subject, latents are typically of poor quality in terms of ridge clarity, large background noise and small friction ridge area. Hence, the number of minutiae in a latent may be very small , e.g., $\\leq$10. \n  Fig. \\ref{fig:introduction} (c) shows the distribution of the number of manually marked minutiae on NIST SD27 latent database and  Fig. \\ref{fig:introduction} (b) shows a latent image with 5 manually marked minutiae overlaid on the image. Minutiae alone do not have enough information for latents.\n Another example where minutiae based matching does not work is for matching fingerprint images captured by the capacitive sensors embedded in smartphones.  It is estimated that 67\\% of the smartphones in the world  will have an embedded fingerprint sensor\\footnote{https:\/\/www.statista.com\/statistics\/522058\/global-smartphone-fingerprint-penetration\/} by 2018. The size of these embedded fingerprint sensors (only $88 \\times 88$ pixels for TocuhID\\footnote{https:\/\/assets.documentcloud.org\/documents\/1302613\/ios-security-guide-sept-2014.pdf}) are much smaller than the standalone sensors. Hence, the number of minutiae in these images is very few as shown in Fig. \\ref{fig:introduction} (a). For these reasons, accurate non-minutiae based (also called \\textit{texture based})  fingerprint matching algorithms are necessary.  To our knowledge, all major latent AFIS vendors use texture templates and different quality latents are handled differently inside AFIS and that is why the processing time is different. \n  \n\nA few non-minutiae based fingerprint matching algorithms have been proposed in literature.  FingerCode by Jain et al. \\cite{Jain2000IP} uses a bank of Gabor filters to capture both the local and global  details in a fingerprint. However, FingerCode relies on a reference point and, further, its accuracy is much lower than minutiae based approaches.  Some approaches based on keypoints, e.g., SIFT \\cite{Park, Yamazaki} and AKAZE \\cite{AKAZE},  have been proposed to generate  dense keypoints for fingerprint matching.  But these keypoints as well as their descriptors are not sufficient to distinguish fingerprints from different fingers.\n\n\nDeep learning based approaches have also been proposed for fingerprint recognition.  Zhang et al. \\cite{Zhang2017IJCB} proposed a deep learning based feature, called deep dense multi-level feature, for partial high resolution fingerprint recognition which achieved promising performance on their own database. However, their approach could not handle fingerprint rotation. Cao and Jain \\cite{Cao2018PAMI} proposed a virtual minutiae based approach for latent fingerprint recognition, where the virtual minutiae locations are determined by a raster scan with a stride of 16 pixels; the associated descriptors are obtained by three convolutional neural networks. Experimental results on two latent databases,  NIST SD27 and WVU, showed that the recognition performance of virtual minutiae based ``texture template\" when fused with two different true minutiae templates boosts the rank-1 accuracy  from 58.5\\% to 64.7\\%  against 100K reference print gallery for NIST SD27 \\cite{Cao2018PAMI}. However, the virtual minutiae feature extractor and matcher are quite slow. \n\n\n\nThe objective of this paper is to improve both the accuracy and efficiency of virtual minutiae (texture template) based latent matching. The main contributions of this paper are as follows: \n \\begin{enumerate} \n\\itemsep0em \n\\item Reduce  the average recognition time between a latent texture template and a rolled texture template  from 11 ms (24 threads) to 7.7 ms (single thread);\n\\item Improve the rank-1 identification rate of the texture templates in \\cite{Cao2018PAMI} by 8.9\\% (from 59.3\\% to 68.2\\%) for 10K gallery;\n\\item Boost the rank-1 identification rate  in \\cite{Cao2018PAMI} by 2.7\\% by fusion  of the proposed three texture templates with three templates in \\cite{Cao2018PAMI} (from 75.6\\% to 78.3\\%) for 10K gallery. This means that out of the 258 latents in NIST SD27, improvements in the texture template will push 7 additional latents at rank 1.\n \\end{enumerate}\n \n  \n\n\n\n\n\\section{Proposed Approach}\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.85\\linewidth]{overview4.pdf}\n\t\\caption{Overview of the proposed latent fingerprint recognition algorithm. }\n\t\\label{fig:overview}\n\t\\setlength{\\belowdisplayskip}{-6pt}\n\t\\setlength{\\belowdisplayshortskip}{-6pt}\n\t\\vspace*{-5pt}\n\\end{figure*}\n\nIn this section, we describe the proposed texture-based latent matching approach, including feature extraction and matching. Fig. \\ref{fig:overview} shows flowchart of the proposed approach. \n\n\n\\subsection{Virtual Minutiae Extraction}\nFor both latents and reference prints, the texture template is similar to that in \\cite{Cao2018PAMI} and consists of locations, orientations and descriptors of virtual minutiae.  We first describe the virtual minutiae localization and then discuss the associated descriptors. \n\nFor reference fingerprints which are typically of good quality, the region of interest (ROI) is   segmented by magnitude of the gradient and  the orientation fields with a block size of $16\\times 16$ pixels as in \\cite{Chikkerur2007198}.\nThe locations of virtual minutiae are sampled by raster scan with a stride of $s$ and their orientations are the same as the orientations of their nearest blocks in the orientation field. The virtual minutiae close to the mask border are ignored.  Fig. \\ref{fig:rolled_virtual} shows the virtual minutiae  on two rolled prints.\n\nFor latents, the same manually marked ROIs and automatically\nextracted ridge flow with a block size of $16\\times 16$ pixels as in \\cite{Cao2015ICB} are used for\nvirtual minutiae extraction.  Suppose that $(x, y)$ are the $x-$ and $y-$coordinates of a sampling point and $\\theta$ denotes the orientation of the block which is closest to $(x,y)$ in the ridge flow. Two  virtual minutiae, i.e., $(x,y,\\theta)$  and $(x,y,\\theta+\\pi)$, are created to handle the ambiguity in ridge orientation.  Fig. \\ref{fig:latent_virtual} show virtual minutiae\non two enhanced latents from NIST SD27 with $s = 32$.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=3cm 1cm 3cm 1cm, width=0.4\\linewidth]{Rolled_001.jpg}\n\t}\\hspace{1cm}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=3cm 1cm 3cm 1cm, width=0.4\\linewidth]{Rolled_004.jpg}\n\t}           \n\t\\caption{Virtual minutiae on two rolled prints with stride $s$=32.  }\n\t\\label{fig:rolled_virtual}\n\t\\vspace*{-5pt}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=5cm 5.5cm 6cm 1.5cm, width=0.35\\linewidth]{Latent_001_32.jpg}\n\t}\n\\hspace{1cm}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=5cm 3.5cm 6cm 3.5cm, width=0.35\\linewidth]{Latent_004_32.jpg}\n\t}           \n\t\\caption{Virtual minutiae on two enhanced latent fingerprints with stride $s$=32. Note that each circle represents two virtual minutiae with opposite orientations to handle the ambiguity in ridge orientation.}\n\t\\label{fig:latent_virtual}\n\t\\vspace*{-5pt}\n\\end{figure}\n\n\n\\subsection{Descriptors for Virtual Minutiae}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\subfigure[][]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{3_3.jpeg}\n\t}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{3_8.jpeg}\n\t}     \n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{3_13.jpeg}\n\t}    \n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{3_18.jpeg}\n\t}    \n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{3_3_e.jpeg}\n\t}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{3_8_e.jpeg}\n\t}     \n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{3_13_e.jpeg}\n\t}    \n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{3_18_e.jpeg}\n\t}         \n\t\\caption{Examples of training fingerprint patches.}\n\t\\label{fig:train_eg}\n\t\\vspace*{-5pt}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n\t\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{scale_1.jpg}\n\t}  \n\t\\subfigure[][]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{corner_3.jpg}\n\t}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{top.jpg}\n\t}     \n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{scale_2.jpg}\n\t}       \n\t\\caption{Four different patch types used in \\cite{Cao2018PAMI} for descriptor extraction. Patch types in (a)-(c) were determined to be the best combination in terms of identification accuracy. The patches in (a), (b) and (d) are of sizes $96\\times 96$ pixels while the patch in (c) is of size $80\\times 80$ pixels. In this paper, we used patch types in (b), (c) and (d) for descriptor extraction because all of them are of the same size and hence no resizing is needed.}\n\t\\label{fig:patch_types}\n\t\\vspace*{-5pt}\n\\end{figure}\n\n\nA minutia descriptor contains attributes of the minutia\nbased on the image characteristics in its neighborhood.\nSalient descriptors are needed to establish \n minutiae correspondences and compute the similarity between a latent and\nreference prints. Instead of specifying the descriptor in an ad hoc manner, Cao and Jain \\cite{Cao2018PAMI} trained ConvNets to learn\nthe descriptor from local fingerprint patches around a minutia and showed its performance. In this paper, we improve both the distinctiveness of the descriptor and efficiency of descriptor extraction. \n\nCao and Jain \\cite{Cao2018PAMI} extracted around 800K  $160 \\times 160$ fingerprint patches from around 50K minutiae from images in a large fingerprint longitudinal database  \\cite{Yoon2015PNAS}, to train the ConvNets. On average, there are  around 16 fingerprint patches around each minutia for training. Cao and Jain also reported that descriptors extracted from enhanced latent images give better performance. In order to augment the training dataset and improve the descriptor distinctiveness between enhanced latent and original rolled prints, we use patches from  both the original and enhanced fingerprint patches for training ConvNets; this results in around 1.6 million fingerprint patches for training. Fig. \\ref{fig:train_eg} shows some example patches from the training dataset;  Figs. \\ref{fig:train_eg} (a) to (d) are from the original image and (e) to (h) are from the corresponding enhanced images.\n\n\nLocation and size of the patches were also evaluated in \\cite{Cao2018PAMI}. The three patch types shown in Figs. \\ref{fig:patch_types} (a)-(c) were determined to be the best patch types via forward sequential selection. The two patches in Figs. \\ref{fig:patch_types} (b) and (c) are of the same size ($96\\times 96$ pixels) while the patch in Fig. \\ref{fig:patch_types} (a) is of size $80 \\times 80$ pixels. All the patches had to be resized to  $160\\times 160$ pixels in \\cite{Cao2018PAMI}, by bilinear interpolation. In order to avoid resizing, we train a ConvNet on $96\\times 96$  images and use patch types in Figs. \\ref{fig:patch_types} (b)-(d) for descriptor extraction. Note that the patch in Fig. \\ref{fig:patch_types} (d) is minutiae centered whereas patches in Figs. \\ref{fig:patch_types} (b) and (c) are offset from the center minutiae.\n \nAmong the various ConvNet architectures  \\cite{VGG},  \\cite{GoogLeNet},  \\cite{ResNet},  \\cite{Mobilenet}, MobileNet-v1 \\cite{Mobilenet} uses depth-wise separable convolutions,  resulting in a \ndrastic reduction in model size and training\/evaluation times while providing good recognition performance. The number of original model parameters to be trained in MobileNet-v1 (4.24M), is significantly\nsmaller than the number of model parameters in Inception-v3 (23.2M) and VGG (138M), requiring significantly lower\nefforts in terms of regularization and data augmentation, to prevent overfitting. For these reasons, we utilize the MobileNet-v1 architecture  due to its fast inference speed.\nIn order to feed $96 \\times 96$ fingerprint patches for training and to reduce the descriptor length, we make the following modifications  to the MobileNet-v1 architecture: i) change the input image size to $96 \\times 96 \\times 1$, ii) remove the last two convolutional layers to accommodate the smaller input sizes, and iii) add a fully connected layer, also called the embedding layer, before the classifier layer to obtain a compact feature representation. The modified architecture is shown in Table \\ref{tab:MobileNet}.\n\nFor each of the three patch types shown in Figs. \\ref{fig:patch_types} (b)-(d), 1.6 million fingerprint patches are used to train a MobileNet. \nGiven a  fingerprint patch around a virtual minutiae, the output ($l$-dimensional feature vector) of the last fully connected layer  is considered as the virtual minutiae descriptor. In the experiments, three value of $l$, namely $l=32, 64$, and $128$, are investigated. The concatenation of the three outputs for the same virtual minutiae is regarded as the descriptor with length $l_d$ = $3\\times l$.  \nThe set of virtual minutiae and the associated descriptors define a texture template. \n\n\n\\subsection{Texture Template Matching}\n\n\\begin{figure}[t]\n    \\centering\n        \\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.5\\linewidth]{H22.pdf}\n    \\caption{Illustration of second-order graph matching \\cite{Cao2018PAMI}, where $m_{i_1}$ and $m_{j_1}$ are the two virtual minutiae, $d_{i_1,j_1}$ is the Euclidean distance between $m_{i_1}$ and $m_{j_1}$,  and $\\theta_{i_1,j_1}$, $\\theta_{i_1}$ and $\\theta_{j_1}$ are the three angles formed by two virtual minutiae orientations and the line segment connecting  $m_{i_1}$ and $m_{j_1}$. }\n      \\label{fig:second-order}\n      \\setlength{\\belowdisplayskip}{-6pt}\n\\setlength{\\belowdisplayshortskip}{-6pt}\n\\vspace*{-5pt}\n\\end{figure}\n\n\n\\begin{table}[tp]\n\\caption{MobileNet Architecture, where $l$ is the length of feature vector output by the MobileNet and $c$ is the number of classes used for training.}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\nType \/ Stride  & Filter Shape & Input Size\\\\\n\\hline\nConv \/ s2 & 3$\\times$3$\\times$1$\\times$32  & 96$\\times$96$\\times$1 \\\\\n\\hline\nConv dw \/ s1 & 3$\\times$3$\\times$32 dw &48$\\times$48$\\times$32 \\\\\n\\hline\nConv \/ s1 &1$\\times$1$\\times$32$\\times$64 &48$\\times$48$\\times$32 \\\\\n\\hline\nConv dw \/ s2 & 3$\\times$3$\\times$64 dw &48$\\times$48$\\times$64 \\\\\n\\hline\nConv \/ s1 &1$\\times$1$\\times$64$\\times$128 &24$\\times$24$\\times$64 \\\\\n\\hline\nConv dw \/ s1 &3$\\times$3$\\times$128 dw &56$\\times$24$\\times$128 \\\\\n\\hline\nConv \/ s1 &1$\\times$1$\\times$128$\\times$128 &24$\\times$24$\\times$128  \\\\\n\\hline\nConv dw \/ s2 &3$\\times$3$\\times$128 dw &24$\\times$24$\\times$128 \\\\\n\\hline\nConv \/ s1 &1$\\times$1$\\times$128$\\times$256 &12$\\times$12$\\times$128 \\\\\n\\hline\nConv dw \/ s1 &3$\\times$3$\\times$256 dw &12$\\times$12$\\times$256 \\\\\n\\hline\nConv \/ s1 &1$\\times$1$\\times$256$\\times$256 &12$\\times$12$\\times$256 \\\\\n\\hline\nConv dw \/ s2 &3$\\times$3$\\times$256 dw &12$\\times$12$\\times$256 \\\\\n\\hline\nConv \/ s1 & 1$\\times$1$\\times$256$\\times$512 &6$\\times$6$\\times$256 \\\\\n\\hline\n5$\\times $Conv dw \/ s1& 3$\\times$3$\\times$512 dw &6$\\times$6$\\times$512 \\\\ \n\\hline\nConv \/ s1 &1$\\times$1$\\times$512$\\times$512 &6$\\times$6$\\times$512 \\\\\n\\hline\nConv dw \/ s2 &3$\\times$3$\\times$512 dw &6$\\times$6$\\times$512 \\\\\n\\hline\nAvg Pool \/ s1 & Pool 6$\\times$6  & 6$\\times$6$\\times$512\\\\\n\\hline\nFC \/ s1  & 512$\\times l$ &  1$\\times 1 \\times 512$ \\\\\n\\hline\nFC \/ s1  &  $l \\times c $&  1$\\times 1 \\times l$ \\\\\n\\hline\nSoftmax \/ s1 & Classifier & 1 $\\times $ 1 $\\times  c$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:MobileNet}\n\\vspace*{-5pt}\n\\end{table}%\n\n\n\\label{sec:matching}\nThe algorithm for comparing two texture templates, one from latent and the other from a reference print, as proposed in \\cite{Cao2018PAMI}  can be summarized as: i) compute pair-wise similarities between the latent and reference print virtual minutiae descriptors using cosine similarity;  normalize the similarity matrix, ii) select the top $N$ ($N=200$) virtual minutiae correspondences based on the normalized similarity matrix, iii) remove false minutiae correspondences using second-order graph matching, iv) further remove false minutiae correspondences using third-order graph matching, and (v) finally compute the overall similarity between the two texture templates  based on final minutiae correspondences. Although the second-order graph matching can remove most false correspondences, it is still time-consuming as it involves $N(N-1)\/2$ computations. Furthermore, since the locations of virtual minutiae do not have large variations due to the raster scan, the larger complexity third order graph matching does not help too much in virtual minutiae correspondences.\n\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=2cm 1cm 1cm 1cm, width=0.85\\linewidth]{top_200.jpg}\n\t}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=2cm 1cm 1cm 1cm, width=0.85\\linewidth]{top_200_dist.jpg}\n\t}   \n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=2cm 1cm 1cm 1cm, width=0.85\\linewidth]{top_200_Fast2.jpg}\n\t}                  \n\t\\caption{Illustration of virtual minutiae correspondences by the proposed graph matching strategy. Figures in  (a), (b) and (c) show the top $N=200$ minutiae correspondences,   88 virtual minutiae correspondences by using the modified second order graph matching and 73 virtual minutiae correspondences by applying the second order graph matching used in \\cite{Cao2018PAMI} to the 88 minutiae correspondences in (b).}\n\t\\label{fig:correspondences}\n\t\\vspace*{-5pt}\n\\end{figure}\n\nAs shown in Fig. \\ref{fig:second-order}, the computation of the compatibility between two virtual minutiae pairs in second-order graph matching involves one Euclidean distance computation and three angular distance computations. \nIn our preliminary experiments, we found that the Euclidean distance alone is good enough to remove most false correspondences as shown in Fig. \\ref{fig:correspondences}.\nThe approach we propose here is to use a modified second-order graph matching to remove most false virtual minutiae correspondences and use the second-order graph matching in \\cite{Cao2018PAMI} to get the final virtual minutiae correspondences\n\n\n An overview of the modified virtual minutiae matching algorithm is illustrated in $\\mathbf{Algorithm       \\ \\ref{alg:search}}$.  The details of the modified second-order graph matching are as following.  Suppose  $\\{i=(i_1,i_2)\\}_{i=1}^{N}$ is the set of $N$ selected minutiae correspondences between a latent $L$ and a rolled print $R$, where $i_1$ and $i_2$ denote the $i^{th}$ correspondence between the $i_1^{th}$ and $i_2^{th}$ virtual minutiae in the latent $F_l$ and the rolled print $F_r$, respectively.  Given two  minutiae correspondences $(i_1,i_2)$ and $(j_1,j_2)$, $H_{i,j}^2$ ($H^2 \\in R^{N\\times N}$) measures the compatibility between $(i_1,j_1)$ from the latent and $(i_2,j_2)$ from the rolled prints:\n\\begin{align}\nH^2_{i,j} &= Z(D_{i,j},\\mu,\\tau,t),\n\\end{align}\nwhere $D_{i,j} = |d_{i_1,j_1} - d_{i_2,j_2}|$ and $Z$ is a truncated sigmoid function:\n\n\\begin{equation}\nZ(v,\\mu,\\tau, t) =\t\\left.\n\\begin{cases}\n \\frac{1}{1+e^{-\\tau(v-\\mu)}}, & \\text{   if } v \\leq t, \\\\\n 0, & \\text{   otherwise. }\n\\end{cases}\\right.\n\\end{equation}\nHere $\\mu,\\tau$ and $t$ are the parameters of function $Z$.\n\n The goal of graph matching is to find an $N$-dimensional correspondence vector $Y$, where the $i^{th}$ element ($Y_i$) indicates whether $i_1$  is assigned to $i_2$  ($Y_i = 1$) or not ($Y_i = 0$). This can be represented in terms of  maximizing the following objective function:\n \\begin{equation}\n\\label{eq:objective_simple_two}\nS_2(Y) = \\sum_{i,j}  \\\\ H^2_{i,j} Y_i Y_j.\n\\end{equation}\nA strategy of power\niteration followed by discretization used in \\cite{Cao2018PAMI} is used to remove false minutiae correspondences.\n\nSuppose  $\\{i=(i_1,i_2)\\}_{i=1}^{n}$  represent the final $n$ matched minutiae correspondences between $F_l$ and $F_r$. The similarity $S$  between $F_l$ and $F_r$ is defined as:\n\\begin{equation}\n\\label{eq:minutiae_similarity}\nS = \\sum_{i=1}^nDesSim(i_1,i_2),\n\\end{equation}\nwhere $DesSim(i_1,i_2)$ is the descriptor similarity between $i_1^{th}$ virtual minutiae in latent template $F_l$ and $i_2^{th}$ virtual minutiae in reference template  $F_r$.\n\n\\begin{algorithm}\n\t\\caption{Modified virtual minutiae matching algorithm}\\label{alg:search}\n\t\\begin{algorithmic}[1]\n\t\n\t\t\\State \\textbf{Input:} Latent template $F_l$ and reference template $F_r$ \n\t\t\\State \\textbf{Output:} Similarity between $F_l$ and $F_r$ \n\t\t\\State \tCompute descriptor similarity matrix\\;\n\t\t\\State\tNormalize similarity matrix\\;\n\t\t\\State  Select the top $N$ minutiae correspondences based on the normalized similarity matrix\\;\n\t\t\\State \tConstruct $H^2$ based on these $N$ minutiae correspondences\\;\n\t\t\\State\tRemove false correspondences using modified second-order graph matching\\;\n\t\t\\State  Further remove false correspondences using original second-order graph matching\\;\n\t\t\\State  Compute similarity between $F_l$ and $F_r$\\;\n\t\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\n\n \\begin{figure}[t] \n\t\\centering\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=6cm 8cm 7cm 8cm, width=0.35\\linewidth]{019_original.jpg}\n\t} \\hspace{1 cm}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=6cm 8cm 7cm 8cm, width=0.35\\linewidth]{019_enhanced_1.jpg}\n\t}     \\hspace{1 cm}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.22\\linewidth]{019_enhanced_3.jpg}\n\t}   \\hspace{1.5 cm}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=6cm 8cm 7cm 8cm, width=0.35\\linewidth]{019_texture.jpg}\n\t}           \n\t\\caption{Illustration of processing strategies applied to the input latent shown in (a) for virtual minutiae descriptor extraction.  (b) and (c) are the two enhanced latent fingerprint images used for minutiae template 1 and 2 extraction in \\cite{Cao2018PAMI}, and (d) is the texture component after decomposition \\cite{Cao2014PAMI}. }\n\t\\label{fig:input_images}\n\t\\vspace*{-5pt}\n\\end{figure}\n\n\\subsection{Constructing texture templates}\n\\label{sec:templates}\nThe locations and orientations of virtual minutiae are determined by the ROI and ridge flow, while the descriptors depend on the images input to the trained ConvNets.  Three different processed latent images for each latent are investigated for texture template construction. Two different enhancement algorithms were proposed in \\cite{Cao2018PAMI} and the resulting two minutiae sets were extracted, one per enhanced images.  Both of the enhanced images for virtual minutiae descriptor extraction are also investigated here. However, the fingerprint enhancement performance critically depends  on the estimates of  ridge flow and ridge spacing. If ridge flow or ridge spacing are not estimated correctly, spurious ridge structures are created in the enhanced images.  In addition to the two enhanced images, we also consider the texture image obtained by image decomposition \\cite{Cao2014PAMI}, which essentially removes large scale background noise and enhances ridge contrast.  Figs. \\ref{fig:input_images} (b)-(d) illustrate the three processed images for the input latent image in Fig. \\ref{fig:input_images} (a). For each latent, three different texture templates, $T_{e_1}$,  $T_{e_2}$ and $T_{t}$, can then be extracted. Given a latent to reference print pair, three texture template similarities are computed  which are then fused to improve the overall latent recognition performance.\n\n\n\\section{Experimental Results}\nThe proposed texture template matching algorithm is evaluated on  NIST SD27 latent database, which consists of 88 good quality, 85 bad quality and 85 ugly quality latent fingerprint images. For latent search experiments,  10,000 reference prints\\footnote{Results for the larger gallery of 100K reference prints are not yet available at the time of submitting this paper.}, including the 258 mates of NIST SD27 and others from NIST SD14, are used as the gallery.\n\n\n\\subsection{Descriptor length evaluation}\nThe performance of the new virtual minutiae descriptors with different feature lengths, namely, $l_d$ = 96 ($l=32$), 192 ($l=64$) and 384 ($l=128$), are evaluated by verification performance based on manually marked minutiae correspondences on NIST SD27\\footnote{NIST SD27 dataset  is no longer available for  download from the NIST site.} \\cite{NISTDB27}. A total of 5,460 minutiae correspondences between the latent images and the mated rolled fingerprint images were provided with the NIST SD27 database.  The average numbers of manually marked minutiae correspondences on 88 good quality, 85 bad quality and 85 ugly quality latent images are 31, 18 and 14, respectively.  For a fair comparison with the descriptors used in \\cite{Cao2018PAMI}, the descriptors for the latents are extracted on the same enhanced images as in \\cite{Cao2018PAMI} while the descriptors of the mated rolled prints are extracted on the original rolled prints.  Fig. \\ref{fig:manual_minutiae} shows the manually marked minutiae on an enhanced latent image and its mated rolled prints. The genuine scores are computed using the similarities of descriptors from the manually marked minutiae correspondences while the impostor scores are computed using the similarities of descriptors from the different minutiae. Thus, a total of 5,460 genuine scores and around 30 million impostor scores ($5,460\\times 5,460$) are computed.  Fig. \\ref{fig:verification} compares the Receiver Operating Characteristic (ROC) curves of different descriptor lengths as well as the descriptors from \\cite{Cao2018PAMI} which used a descriptor length of 384. Note that the ROC curves of descriptors with feature lengths 384 and 192 are very close to each other, slightly better than descriptors with feature length 96 and significantly better than the descriptors in \\cite{Cao2018PAMI}.  This can be explained by the use of both enhanced and original fingerprint patches along with a more appropriate ConvNet architecture for training.\n \\begin{figure}[t] \n\t\\centering\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=5cm 2cm 4cm 2cm, width=0.45\\linewidth]{003_Latent.jpg}\n\t}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=5cm 2cm 4cm 2cm, width=0.45\\linewidth]{Rolled_minutiae_003.jpg}\n\t}             \n\t\\caption{Manually marked minutiae correspondences are used to evaluate the proposed virtual minutiae descriptors. (a) Manually marked latent minutiae shown on the enhanced latent image and (b)  manually marked rolled minutiae shown on mated rolled print image.}\n\t\\label{fig:manual_minutiae}\n\t\\vspace*{-5pt}\n\\end{figure}\n\n\n\n\\begin{figure}[t]\n    \\centering\n        \\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.9\\linewidth]{ROC.pdf}\n    \\caption{Receiver Operating Characteristic (ROC) curves under different descriptor lengths.}\n      \\label{fig:verification}\n     \n\\vspace*{-5pt}\n\\end{figure}\n\n\n\n\n \\begin{figure*}[t] \n\t\\centering\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=4cm 8.5cm 4cm 9cm, width=0.35\\linewidth]{All.pdf}\n\t} \\hspace{1cm}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=4cm 8.5cm 4cm 9cm, width=0.35\\linewidth]{Good.pdf}\n\t}     \\hspace{1cm}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=4cm 8.5cm 4cm 9cm, width=0.35\\linewidth]{Bad.pdf}\n\t}   \\hspace{ 1cm}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=4cm 8.5cm 4cm 9cm, width=0.35\\linewidth]{Ugly.pdf}\n\t}           \n\t\\caption{Cumulative Match Characteristic (CMC) curves of the three texture templates proposed here and their  fusion with three templates used in \\cite{Cao2018PAMI} on (a) all 258 latents in NIST SD27, (b) subset of 88 good latents, (c) subset of 85 bad latents and (d) subset of 85 ugly latents.  Note that the scales of the y-axis in these four plots are different to show the differences between the two curves.}\n\t\\label{fig:CMC}\n\t\\vspace*{-5pt}\n\\end{figure*}\n\n\n\n\\begin{table}[htp]\n\\caption{Latent search accuracies under different scenarios for NIST SD27.  Reference database size is 10K. }\n\\vspace*{-10pt}\n\\begin{center}\n\\begin{tabular}{|p{1.9cm}|p{1.3cm}|p{1.cm}|p{1.cm}|p{1.1cm}|}\n\\hline\nInput templates & descriptor length &rank-1 (\\%)& rank-5 (\\%)& rank-10 (\\%) \\\\\n\\hline\nCao\\&Jain \\cite{Cao2018PAMI} & 384 & 59.30 & 70.16& 73.26 \\\\\n\\hline\n\\hline\n$T_{e_1}$ & 192 & 68.22 & 73.64 & 74.81 \\\\\n\\hline\n$T_{e_2}$& 192 & 66.67 & 72.48 & 74.42 \\\\\n\\hline\n$T_{t}$ & 192 &  60.47 & 67.83 & 70.93   \\\\\n\\hline\n$T_{e_1}$+$T_{e_2}$ & 192 & $\\mathbf{70.93}$ & 74.81 & 77.91  \\\\\n\\hline\n$T_{e_1}$+$T_{t}$ & 192 & $\\mathbf{70.93}$ & 76.36 & 79.07   \\\\ \n\\hline\n$T_{e_2}$+$T_{t}$ & 192 &  67.05 & 75.19 & 77.13\\\\\n\\hline\n$T_{e_1}$+$T_{e_2}$+$T_{t}$ & 192 & 70.16 & $\\mathbf{76.74}$ & $\\mathbf{81.40}$ \\\\\n\\hline\n$T_{e_1}$ & 384 &69.38 & 75.58 & 77.13  \\\\\n\\hline\n$T_{e_2}$ & 384 & 66.28 & 72.48 & 73.64 \\\\\n\\hline\n$T_{t}$  & 384 & 58.91 & 66.28 & 69.77\\\\\n\\hline\n$T_{e_1}$+$T_{e_2}$ & 384 & 70.16 & 75.58 & 78.29 \\\\\n\\hline\n$T_{e_1}$+$T_{t}$  & 384 & 69.38 & 76.74 & 77.91 \\\\ \n\\hline\n$T_{e_2}$+$T_{t}$  & 384 &   67.83 & 74.03 & 75.97 \\\\\n\\hline\n$T_{e_1}$+$T_{e_2}$+$T_{t}$ & 384 & $\\mathbf{70.93 }$ & 75.97 & 78.68 \\\\\n\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:accuracy}\n\\vspace*{-10pt}\n\\end{table}%\n\n\\subsection{Texture template search}\nIn this section, we evaluate the search performance of the proposed three texture templates, i.e., $T_{e_1}$,  $T_{e_2}$ and $T_{t}$ extracted in section \\ref{sec:templates}, as well as their fusion. The following seven scenarios are considered:\n\\begin{enumerate}\n\\itemsep0em \n\\item $T_{e_1}$:  Texture template with enhanced image 1 (Fig. \\ref{fig:input_images} (b)) for descriptor extraction;\n\\item $T_{e_2}$:  Texture template with enhanced image 2 (Fig. \\ref{fig:input_images} (c)) for descriptor extraction;\n\\item $T_{t}$:  Texture template with texture image (Fig. \\ref{fig:input_images} (d)) for descriptor extraction;\n\\item $T_{e_1}$ + $T_{e_2}$:  Score level fusion of $T_{e_1}$ and $T_{e_2}$;\n\\item $T_{e_1}$ + $T_{t}$:  Score level fusion of $T_{e_1}$ and $T_{t}$;\n\\item $T_{e_2}$ + $T_{t}$:  Score level fusion of $T_{e_2}$ and $T_{t}$;\n\\item $T_{e_1}$ + $T_{e_2}$ + $T_{t}$:   Score level fusion of $T_{e_1}$, $T_{e_2}$ and $T_{t}$.\n\\end{enumerate}\nFor each one of the above seven scenarios, all three descriptor lengths, i.e., $l_d$ = 96, 192 and 384, are considered.  The latent search accuracies of different scenarios at three different ranks  for  NIST SD27  against 10K reference fingerprints are shown in  Table \\ref{tab:accuracy}. In addition,  the performance of texture template used in \\cite{Cao2018PAMI} is also included for a comparison  (row 1 of Table \\ref{tab:accuracy}). Note that the enhanced images used for $T_{e_1}$ extraction are the same as those in \\cite{Cao2018PAMI}.  A descriptor length 96 performs much lower than the other two lengths (192 and 384) so its performance is not reported in Table \\ref{tab:accuracy}.  The findings from Table \\ref{tab:accuracy} can be summarized as follows: i)  among the three texture templates, $T_{e_1}$ performs the best for both descriptor lengths; ii) the performance of descriptor length of 192 is sufficiently  close to that of descriptor length 384 in different scenarios; iii) rank-1 accuracy of $T_{e_1}$ with descriptor length 192 is 8.98\\% higher than that in \\cite{Cao2018PAMI}; and iv) fusion of any two out of the three proposed texture templates is higher than that of any single template; the fusion of all three templates boosts the performance slightly  at rank-1 (for $l_d$=192) and rank-5 (for $l_d$=384).  Fusion of all three proposed templates with $l_d=192$ is preferred because of its smaller feature length  and  higher accuracy at rank-10. As examiners typically evaluate top 10 candidate matches to identify the source of a latent \\cite{Busey2010}, this will ensure that the true mate is frequently available in the candidate list.\n\n\n\n\n\\subsection{Fusion with minutiae templates}\n\n \\begin{figure}[t] \n\t\\centering\n\t\\subfigure[]{\n\t\t\\includegraphics[width=0.4\\linewidth]{217.jpg}\n\t} \\hspace{0.5cm}\n\t\\subfigure[]{\n\t\t\\includegraphics[ width=0.4\\linewidth]{218.jpg}\n\t}     \\hspace{1 cm}\n\t\\caption{Two ugly latents whose true mates were not retrieved at rank 1 by the algorithm in \\cite{Cao2018PAMI}. The use of proposed texture templates and their fusion with templates in \\cite{Cao2018PAMI}  correctly retrieves their true mates at rank 1. }\n\t\\label{fig:failure}\n\t\\vspace*{-10pt}\n\\end{figure}\n\nIn order to determine if our new texture templates can boost the performance over the results in  \\cite{Cao2018PAMI} we fuse the proposed three texture templates with the three templates used in \\cite{Cao2018PAMI}. Fig. \\ref{fig:CMC}  compares the Cumulative Match Characteristic (CMC) curves of the fusion scheme  on all 258 latents in NIST SD27 as well as subsets of latents of three different quality levels  (good, bad and ugly).  Plots in Fig. \\ref{fig:CMC}  show the proposed three texture templates when fused with the three templates in \\cite{Cao2018PAMI} can boost the overall performance by 2.7\\% at rank-1 (from 75.6\\% to 78.3\\%). In particular,  the fusion of six templates  (three proposed + three from \\cite{Cao2018PAMI})  improves the rank-1 accuracy by 4.7\\% on the subset of ugly latents, some of the most challenging latents with an average of only 5 minutiae per latent. Fig. \\ref{fig:failure} shows two ugly latents  whose true mates were not retrieved at rank-1 by the method in \\cite{Cao2018PAMI}, but are now correctly retrieved at rank-1 with the introduction of three new texture templates.\n\n\\subsection{Computation time}\nThe texture template matching algorithm was implemented in tensorflow and python and executed on a desktop with  i7-6700K CPU@4.00GHz, GTX 1080 Ti (GPU),  32 GB RAM and Linux operating system. The average computation time for comparing a latent texture template to a rolled texture template is 7.7ms  (single thread)  compared to 11.0 ms (24 threads) in \\cite{Cao2018PAMI}. The average  times for extracting one proposed latent texture template and one proposed rolled texture template are 0.7s and 1.5s (GPU), respectively;  when fused with the three templates in \\cite{Cao2018PAMI}  are 1.2s and 2.2s (24 threads).\n\n\\section{Summary and future work}\n\nTexture template is critical to improve the  search accuracy of latent fingerprints, especially for latents with small friction ridge area  and large background noise. We have proposed a set of three texture templates, defined as a set of  virtual minutiae along with their descriptors. Different virtual minutiae descriptors lead to different texture templates. The contributions of this paper are as follows. i) Use patches from original fingerprints and enhanced fingerprints to improve the distinctiveness of virtual  minutiae descriptors, ii) three different texture templates, and iii) a modified second-order graph matching. \nIdentification results on NIST SD27 latent database demonstrate that the proposed texture templates when used alone can improve the rank-1 accuracy  by 8.9\\% (from 59.3\\% in \\cite{Cao2018PAMI} to 68.2\\%).   Our ongoing research includes i) improving ridge flow estimation, ii) using latent-rolled pairs for learning minutiae descriptors and similarities, and iii) using multicore processors to improve the search speed. \n\\section*{Acknowledgement}\nThis research is based upon work supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via IARPA R\\&D Contract No. 2018-18012900001. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation therein.\n\n{\\footnotesize\n\\bibliographystyle{ieee}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Small sumsets in LCA groups}    Given subsets $A$ and $B$ of an abelian group $G$, we consider their \\textit{sumset} $A+B:=\\{a+b: a\\in A, b\\in B\\}$. A general theme in additive combinatorics is that $A$ and $B$ must be highly structured when $A+B$ is not very large in comparison to $A$ and $B$; see \\cite{GrynkiewiczKemperman}, \\cite{Nathansoninverse}, or \\cite{TaoVu} for many instances of this theme.  We consider the case where $G$ is a locally compact abelian (LCA) group,  extending the investigations of \\cite{Kneser56}.   Theorem 1 of \\cite{Kneser56} describes the pairs $(A,B)$ of Haar measurable subsets of an LCA group $G$ with Haar measure $m$ satisfying $m_*(A+B)<m(A)+m(B)$; such pairs are often called \\textit{critical pairs}. Here $m_*$ is the inner measure corresponding to $m$, so $m_*(S)=\\sup\\{m(E): E\\subseteq S, E \\text{ is compact}\\}$.  If $S\\subseteq G$ and $t\\in G$, we write $S+t$ for the set $\\{s+t:s\\in S\\}$, and write $C\\sim D$ if $m(C\\triangle D)=0$.\n\n\\begin{proposition}[\\cite{Kneser56}, Theorem 1]\\label{Ktopinequality}  Let $G$ be a locally compact abelian group with Haar measure $m$,  and $A$, $B\\subseteq G$ measurable sets with $m_*(A+B)<m(A)+m(B)$.  Then the group $H:=\\{t\\in G: A+B+t \\sim A+B\\}$ is compact and open,\n\\begin{align}\nA+B=A+B+H,\n\\end{align}  and\n\\begin{align}\\label{critical}\nm(A+B)=m(A+H)+m(B+H)-m(H).\n\\end{align}\n\\end{proposition}\nProposition \\ref{Ktopinequality}, especially equation (\\ref{critical}), plays a crucial role in our results.\n\n\\smallskip\n\nSince connected groups have no proper compact open subgroups, Proposition \\ref{Ktopinequality} implies $m_*(A+B)\\geq \\min\\{m(A)+m(B),m(G)\\}$ whenever $G$ is connected, generalizing results Raikov \\cite{Raikov},  Macbeath \\cite{Macbeath}, and Shields \\cite{Shields}.  The following theorem from \\cite{Kneser56} completely classifies the pairs of subsets of a compact, \\textit{connected}, abelian group satisfying $m_*(A+B)=m(A)+m(B)$, when $m(A)>0$ and $m(B)>0$.  We call such a pair $(A,B)$ a \\textit{sur-critical} pair.  Here $\\mathbb T$ is the group $\\mathbb R\/\\mathbb Z$; see \\S \\ref{intervalsdef} for our usage of the term ``interval.\"\n\n\n\n\\begin{proposition}[\\cite{Kneser56}, Theorem 3]\\label{Kconnected}  Let $G$ be a compact connected abelian group with Haar measure $m$, and $A$, $B\\subseteq G$  measurable sets satisfying $m(A)>0$, $m(B)>0$, and $m_*(A+B)=m(A)+m(B)<1$.  Then there is a continuous surjective homomorphism $\\chi:G\\to \\mathbb T$, and there are intervals $I$, $J\\subseteq \\mathbb T$ such that $A\\subseteq \\chi^{-1}(I)$, $B\\subseteq \\chi^{-1}(J)$,  $m(A)=m(\\chi^{-1}(I))$, and $m(B)=m(\\chi^{-1}(J))$.\n\\end{proposition}\n\n\nKemperman \\cite{KempermanTopological} generalized part of Proposition \\ref{Ktopinequality} to the case where $G$ is not abelian, showing in particular that $m_*(A\\cdot B)\\geq m(A)+m(B)$ for subsets of a general locally compact connected group (here $A\\cdot B$ denotes $\\{ab:a\\in A, b\\in B\\}$).  Bilu \\cite{Bilu} investigates the inequality $m(A+B)<m(A)+m(B)+\\min\\{m(A),m(B)\\}$ when the ambient group is $\\mathbb T^d$ for some $d$; the case $d=1$ is studied in \\cite{MoskvinFreimanJudin}.\n\n\\subsection{New results}  When $G$ is disconnected, the pairs $(A,B)$ of subsets of $G$ satisfying $m_*(A+B)=m(A)+m(B)$ are not completely described by Proposition \\ref{Kconnected}.   For example, take $G=(\\mathbb Z\/5\\mathbb Z)\\times \\mathbb T$, and let $A=(\\{0\\}\\times \\mathbb T) \\cup (\\{1\\} \\times [0, 1\/2])$, $B=(\\{0,1\\}\\times \\mathbb T) \\cup (\\{2\\} \\times [0,1\/4])$, so that $A+B=(\\{0,1,2\\}\\times \\mathbb T) \\cup (\\{1\\} \\times [0,3\/4])$.  Other examples are discussed in \\S \\ref{examples}. We classify the pairs $(A,B)$ of measurable subsets of a locally compact abelian group satisfying $m(A)>0$, $m(B)>0$, and $m_*(A+B)=m(A)+m(B)$, generalizing Proposition \\ref{Kconnected} to the case where $G$ may be disconnected.   Our main result is Theorem \\ref{main}, where $G$ is assumed to be compact.   See \\S \\ref{terminology} for notation and terminology.  In \\S \\ref{lc} we discuss the case where $G$ is not compact.\n\n\\begin{theorem}\\label{main}  Let $G$ be a compact abelian group with Haar measure $m$, and let $A$, $B\\subseteq G$ be measurable sets such that $m(A)>0$, $m(B)>0$, and $m_*(A+B)=m(A)+m(B)$.  Then at least one of the following is true\\textup:\n\\begin{enumerate}\n\\item[(P)]  There is a compact open subgroup $K\\leqslant G$ with $A+K\\sim A$ and  $B+K\\sim B$.\n\n\\smallskip\n\n\\item[(E)]  There are measurable sets $A'\\supseteq A$ and  $B'\\supseteq B$ such that\n\\begin{align*}\nm(A')+m(B')>m(A)+m(B),\n\\end{align*}  and $m_*(A'+B')=m_*(A+B)$.\n\n\\smallskip\n\n\\item[(K)]  There is a compact open subgroup $K\\leqslant G$, a continuous surjective homomorphism $\\chi:K\\to \\mathbb T$, intervals $I$, $J\\subseteq \\mathbb T$, and elements $a$, $b\\in G$ such that $A\\subseteq a+\\chi^{-1}(I)$, $B\\subseteq b+\\chi^{-1}(J)$, $m(A)=m(\\chi^{-1}(I))$, and $m(B)=m(\\chi^{-1}(J))$.\n\n\\smallskip\n\n\\item[(QP)]  There is a compact open subgroup $K\\leqslant G$ and partitions $A=A_1\\cup A_0$, $B=B_1\\cup B_0$ such that $A_0\\neq \\varnothing$, $B_0\\neq \\varnothing$, at least one of $A_1\\neq \\varnothing$, $B_1\\neq \\varnothing$, and\n\n\\begin{enumerate}\n\\item[(QP.1)]\n$A_1+K\\sim A_1$, $B_1+K\\sim B_1$,  while  $A_0$ and $B_0$ are each contained in a coset of $K$ and $(A_1+K)\\cap A_0=(B_1+`K)\\cap B_0=\\varnothing$\\textup{;}\n\n\\smallskip\n\n\\item[(QP.2)]  $A_0+B_0+K$ is a unique expression element of $A+B+K$ in $G\/K$\\textup{;}\n\n\\smallskip\n\n\\item[(QP.3)]    $m_*(A_0+B_0)=m(A_0)+m(B_0)$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{theorem}\n\\begin{remarks} (i) Corollary \\ref{QPfour} provides more detail in conclusion (QP); conclusion (E) is examined in \\S \\ref{Eremark}.  Corollary \\ref{roman} is a convenient rephrasing of Theorem \\ref{main}.\n\n\\smallskip\n\n\\noindent (ii)  We do not provide explicit constructions of all pairs satisfying conclusion (QP), so the statement of Theorem \\ref{main} is not a complete characterization of the sur-critical pairs for a compact abelian group.\n\n\\smallskip\n\n\\noindent (iii)  The labels (P), (E), (K), and (QP) stand for ``periodic,\" ``extendible,\" ``Kneser,\" and ``quasi-periodic,\" respectively.  See \\S \\ref{setsandpairs} for elaboration.\n\n\\smallskip\n\n\\noindent (iv) Theorem \\ref{main} was developed partly to help answer Question 4.1 of \\cite{Jin10}. \\hfill $\\blacksquare$ \\end{remarks}\n\n\n\n\nTheorem \\ref{main} is proved in \\S \\ref{proof}.  A sequence of lemmas in \\S \\ref{lemmas} reduces the proof to a special case.  We then apply the $e$-transform  (\\S \\ref{etsection}) as in the proof of Proposition \\ref{Kconnected} from \\cite{Kneser56}.  Lemma \\ref{transformcase1} handles difficulties related to applying the $e$-transform in a disconnected group; its statement and proof are inspired by arguments from \\cite{Grynkiewicz}.\n\n\\smallskip\n\nSome easy consequences of Theorem \\ref{main} are discussed in \\S \\ref{corollarysection}.  Examples are given in \\S \\ref{examples} showing that each conclusion in Theorem \\ref{main} may occur, and none can be omitted.  \\S \\ref{corollarysection} and \\S \\ref{examples} are technically irrelevant to the proof of Theorem \\ref{main}.\n\n\n\n\\subsection{When one of $m(A)=0$ or $m(B)=0$.}  When $m(B)=0$, $0<m(A)<\\infty$, and $B$ contains the identity element of $G$, Theorem 4 of \\cite{Kneser56} says that $m_*(A+B)=m(A)+m(B)$ if and only if the closed subgroup $G'\\leqslant G$ generated by $B$ is compact, and $A$ can be partitioned as $A=C\\cup D$, where $m(C)=0$, $m_*(B+C)=0$ and $m(G'+D)=m(D)$.  We have nothing to say regarding the case where $m(A)=m(B)=0$.\n\n\n\n\\subsection{Context}  The conclusion of Theorem \\ref{main} is trivial for discrete groups $G$, where the trivial subgroup $\\{0\\}$ is compact and open, so conclusion (P) holds with $K=\\{0\\}$.  The article \\cite{Grynkiewicz} classifies the sur-critical pairs for discrete groups, and therefore provides a classification of the pairs $(A,B)$ satisfying conclusion (P) in Theorem \\ref{main} (see \\S \\ref{periodicexamples}).   We summarize the development of some related results.\n\n\\smallskip\n\n\\textit{Inverse theorems} in additive combinatorics deduce properties of subsets $A$, $B$ of an abelian group from a hypothesis on the sumset $A+B$, while   \\textit{direct theorems} deduce properties of $A+B$ from hypotheses on $A$ and $B$.  One of the earliest direct theorems is the Cauchy-Davenport inequality, which states that $|A+B|\\geq \\min\\{|A|+|B|-1, p\\}$ whenever $A$, $B\\subseteq \\mathbb Z\/p\\mathbb Z$ for some prime $p$; here $|S|$ denotes the cardinality of the set $S$.  The corresponding inverse theorem, due to Vosper \\cite{Vosper1,Vosper2}, classifies the pairs $(A,B)$  of subsets of $G=\\mathbb Z\/p\\mathbb Z$ satisfying $|A+B|=|A|+|B|-1$, when $|A|+|B|< p$: the equation holds if and only if $A$ and $B$ are arithmetic progressions with the the same common difference, or one of $|A|=1$, $|B|=1$.  When $|A|+|B|=p$, an elementary analysis shows that the equation $|A+B|=|A|+|B|-1$ is satisfied exactly when $B$ is a translate of $-(G\\setminus A)$, and when $|A|+|B|=p+1$, the equation $|A+B|=|A|+|B|-1$ is always satisfied if $A$ and $B$ are nonempty.\n\n\\smallskip\n\nGeneralizing Vosper's theorem and strengthening  Proposition \\ref{Ktopinequality}, Kemperman \\cite{Kemperman60} classified the pairs of finite subsets $A$, $B$ of an arbitrary abelian group satisfying $|A+B|<|A|+|B|$;  see \\cite{GrynkiewiczKemperman} for exposition.  The articles \\cite{HaRo,HaSeZe} classify pairs $(A,B)$ satisfying $|A+B|=|A|+|B|$ in certain ambient abelian groups.  More recently, Grynkiewicz \\cite{Grynkiewicz} classified the pairs $(A,B)$ of finite subsets of an arbitrary abelian group satisfying $|A+B|=|A|+|B|$.  That classification is somewhat intricate, so we do not reproduce it here.   Concatenating Theorem \\ref{main} with the results of \\cite{Kemperman60} and \\cite{Grynkiewicz} yields a very precise description of pairs $(A,B)$ of subsets of a compact abelian group satisfying $m_*(A+B)=m(A)+m(B)$, but we will not state this description explicitly.\n\n\n\n\n\n\n\n\\section{Terminology, notation, and background}\\label{terminology} We assume knowledge of the theory of locally compact abelian groups.   The books \\cite{HewittandRoss1} and \\cite{ReiterStegeman} together provide nearly sufficient background, as does \\cite{Kneser56} and its bibliography.    We need Theorem A of \\cite{Mueller65}, which is proved in \\cite{Mueller62}.\n\n\\smallskip\n\nSome of our terminology and notation is taken from \\cite{Grynkiewicz} and \\cite{Kneser56}; when $G$ is discrete, some of our definitions coincide with those from \\cite{Grynkiewicz}.\n\n\\subsection{General conventions}  Throughout, $G$ will denote a locally compact abelian group and $m_G$ will be its Haar measure.  We make the standard assumption that $G$ satisfies the $T_0$ separation axiom.  When there is no chance of confusion, we write $m$ for $m_G$, and $m_*$ for the corresponding inner measure.   The term \\textit{measurable} in reference to a subset of $G$ will always mean ``lies in the completion of the Borel $\\sigma$-algebra with respect to $m$.\"  We will exploit inner regularity of Haar measure for compact groups: $m(C)=\\sup\\{m(E): E\\subseteq C, E \\text{ is compact}\\}$ when $C$ is measurable and $G$ is compact.  Haar measure will always be normalized for compact groups, so $m_G(G)=1$ for such $G$.\n\n\n\\smallskip\n\nIf $S\\subseteq G$, $S^c$ will denote the set theoretic complement $G\\setminus S$.  The symbol $-S$ denotes $\\{-s:s\\in S\\}$, while $A-B$ denotes the difference set $\\{a-b:a\\in A, b\\in B\\}$.\n\n\n\\smallskip\n\n\n The identity element of $G$ will be written as $0_G$, or simply $0$ if there is no ambiguity.\n\n\n\n\\subsubsection{Intervals}\\label{intervalsdef}  The symbol $\\mathbb T$ will denote the group $\\mathbb R\/\\mathbb Z$ with its usual topology,  and  ``an interval in $\\mathbb T$\" means a set of the form $[x,y]+\\mathbb Z$, where $x\\leq y <x+1$.  We omit the term ``closed\" as we will never refer to non-closed intervals.  We frequently exploit the following property of intervals:  if $I$, $J\\subseteq \\mathbb T$ are intervals with $m_{\\mathbb T}(I)+m_{\\mathbb T}(J)<1$, and $x\\notin I$, then $m_{\\mathbb T}((x+J)\\setminus (I+J))>0$.  Consequently, if $\\chi:G\\to \\mathbb T$ is a continuous surjective homomorphism and $t\\notin \\chi^{-1}(I)$, then $m_G((t+\\tilde{J})\\setminus (\\tilde{I}+\\tilde{J}))>0$, where $\\tilde{I}=\\chi^{-1}(I)$ and $\\tilde{J}=\\chi^{-1}(J)$.\n\n\\subsubsection{Subgroups and quotients} If $K\\leqslant G$ is a closed subgroup, the quotient $G\/K$ with the quotient topology is a locally compact abelian group.  We may identify subsets of $G$ of the form $A+K$ with subsets of $G\/K$, and conversely subsets of $G\/K$ may be identified with subsets of $G$.\n\n\\smallskip\n\nA \\textit{$K$-coset decomposition} of a set $A\\subseteq G$ is the collection of sets $A_i=A\\cap K_i$, where $K_i$ ranges over the cosets of $K$ having nonempty intersection with $A$.\n\n\\smallskip\n\nWe use without comment the following well-known facts:\n\n\n\\begin{itemize}\n\\item If $K\\leqslant G$ is a measurable subgroup of a compact group, then $m(K)>0$ if and only if $K$ is open, if and only if $K$ has finite index in $G$.  The index of $K$ in $G$ is $1\/m(K)$.\n\n\n\\item If $K\\leqslant G$ is an open subgroup, the group $G\/K$ is discrete.\n\\end{itemize}\n\n\\subsubsection{Disintegration of Haar measure}   This subsection regards technicalities occurring only in \\S \\ref{trivializing}.  For simplicity we assume $G$ is compact.  Given a closed subgroup $K\\leqslant G$, we consider the Haar measure $m_K$ as a measure on $G$.   If $A\\subseteq G$, and $x\\in G$, we consider $m_K(A-x)$, which may be regarded as the $m_K$-measure of $A$ in the coset $x+K$ of $K$.  Although the function $g(x):= m_K(A-x)$ depends only on the coset $K+x$, and may therefore be regarded as a function whose domain is (a subset of) $G\/K$, we prefer to regard $g$ as a function whose domain is (a subset of) $G$.  The measure $m_G(A)$ can be recovered in a natural way from the measures $m_K(A-x)$; in other words, Haar measure can be disintegrated over the cosets of a closed subgroup.  This is the content of the following proposition; cf.~\\S 2 of \\cite{Kneser56}. It may be obtained by specializing Theorem 3.4.6 of \\cite{ReiterStegeman} to the case where $G$ is compact and abelian.\n\n\\begin{proposition}\\label{Weil}  Let $K\\leqslant G$ be a closed subgroup of $G$ with Haar measure $m_K$, and let $f\\in L^1(m)$ be a real-valued function.  Then the function $\\tilde f: G\\to \\mathbb R$ given by $\\tilde{f}(x)= \\int f(x+t)\\, dm_K(t)$ is defined for $m$-almost every $x$, $\\tilde{f}$ is $m_G$-measurable, and $\\int \\tilde f\\, dm=\\int f\\, dm$.  In particular, if $A\\subseteq G$ is measurable, then the function $x\\mapsto m_K(A-x)$ is measurable, and $\\int m_K(A-x)\\, dm(x) = m(A)$. \\hfill $\\blacksquare$\n\\end{proposition}\n\n\n\\subsubsection{Unique expression elements}\\label{uees} We say $c\\in A+B$ is a \\textit{unique expression element of} $A+B$ if $c=a_0+b_0$ for some $a_0\\in A$, $b_0\\in B$, and $a+b=c$ implies $a=a_0$ and $ b=b_0$  when  $a\\in A$ and $b\\in B$.  Unique expression elements play an important role in \\cite{Kemperman60}, the classification of pairs $(A,B)$ satisfying $|A+B|=|A|+|B|-1$; Corollary \\ref{QPfour} connects that classification to conclusion (QP) of Theorem \\ref{main}.  If $K\\leqslant G$ is a subgroup, the phrase ``$C+K$ is a unique expression element of $A+B+K$ in $G\/K$\" means that $C+K= a_0+b_0+K$ for some $a_0\\in A$, $b_0\\in B$, and whenever $a\\in A$, $b\\in B$, and $a+b+K=C+K$, then $a+K=a_0+K$ and $b+K=b_0+K$.\n\n\n\n\n\\subsubsection{Critical and sur-critical pairs}\\label{critdef}  If $A$, $B\\subseteq G$ are measurable sets satisfying $m_*(A+B)=m(A)+m(B)$, we call $(A,B)$ a \\textit{sur-critical pair} for $G$.  We call a pair $(A,B)$ satisfying $m_*(A+B)<m(A)+m(B)$ a \\textit{critical pair} for $G$.\n\n\n\n\\subsubsection{Similarity and the essential stabilizer}\\label{Hdef}  If $C$, $D\\subseteq G$, we write $C\\sim D$ to mean $m(C \\triangle D)=0$, and  we write $C\\subset_m D$ to mean $m(C\\setminus D)=0$.\n\n\\smallskip\n\nWhen $C$ is measurable and $m(C)<\\infty$, the group $H(C):=\\{t\\in G: C+t\\sim C\\}$ is a compact subgroup of $G$ (\\cite{Kneser56}, Lemma 4); we may refer to $H(C)$ as the \\textit{essential stabilizer} of $C$.  The group $H(A+B)$ plays an important role in \\cite{Kneser56} and in the proof of Theorem \\ref{main}.  Proposition \\ref{Ktopinequality} says that if $m_*(A+B)<m(A)+m(B)$ then $H(A+B)$ is compact and open, and $A+B+H(A+B)=A+B$.  The equation $C=C+H(C)$ may fail in general, and more importantly, the similarity $C\\sim C+H(C)$ may fail.  For example, take $G= (\\mathbb Z\/5\\mathbb Z) \\times \\mathbb T$, and let $C= (\\{0\\}\\times \\mathbb T) \\cup \\{(1,0)\\}$.  Then $H(C)=\\{0\\}\\times \\mathbb T$, so $m(C+H(C))= 2\/5$, while $m(C)=1\/5$.\n\n\n\n\n\\subsection{Special sets and special pairs}\\label{setsandpairs}\n\n\\subsubsection{Periodicity}\\label{periodicdef}  If $C\\sim C+K$ for some some compact open subgroup $K\\leqslant G$,  we call $C$ \\textit{periodic} with \\textit{period} $K$.  Otherwise, we call $C$ \\textit{aperiodic}.  Note that when $G$ is not discrete, the assertion $H(C)=\\{0\\}$ implies $C$ is aperiodic, but an aperiodic $C$ may have $m(H(C))>0$.   We will exploit the following  relation between the essential stabilizer and periodicity. This observation is a consequence of Lemma \\ref{H(S)}, although it may be obtained by more elementary means.\n\n\\begin{observation} If $m(H(C))>0$, then for all cosets $H_i$ of $H(C)$, either $m(C\\cap H_i)=0$ or $m(C\\cap H_i)=m(H_i)$.  Consequently, if $m(H(C))>0$, there is a measurable subset $C'\\subseteq C$ such that $C\\sim C'\\sim C'+H(C)$.  \\hfill $\\blacksquare$\n\\end{observation}\n\n\\begin{remark}  When $G$ is discrete, every subset of $G$ is periodic according to our definition.  This differs from the terminology of \\cite{Grynkiewicz}, where a periodic set $S$ must satisfy $S+t \\sim S$ for some $t\\neq 0$.  \\hfill $\\blacksquare$\n\\end{remark}\n\n\n\\subsubsection{Extendibility and nonextendibility}   Let $A$ and $B$ be measurable subsets of a compact abelian group $G$.  We say that $A$ is \\textit{extendible with respect to} $B$ if there is a measurable set $A'\\supseteq A$ with $m(A')>m(A)$ and $m_*(A'+B)=m_*(A+B)$.  We say that the pair $(A,B)$ is \\textit{extendible} if $A$ is extendible with respect to $B$ or $B$ is extendible with respect to $A$.  Otherwise, we say that $(A,B)$ is \\textit{nonextendible}.  The nonextendibility of a pair $(A,B)$ may be expressed as follows:  if $A'\\supseteq A$ and $B'\\supseteq B$ are measurable and $m_*(A'+B')=m_*(A+B)$, then $A'\\sim A$ and $B'\\sim B$.\n\n\\smallskip\n\n When $m_*(A+B)=m(A)+m(B)$ and $(A,B)$ is extendible, Proposition \\ref{Ktopinequality} implies $H(A+B)$ is compact and open, and $A+B\\sim A+B+H(A+B)$.  Consequently, $A+B$ is measurable when $(A,B)$ is extendible.  The following example lemma  illustrates how nonextendibility will be exploited in subsequent proofs.\n\n \\begin{lemma}  If $K\\leqslant G$ is a compact open subgroup, $A+B\\sim A+B+K$, and $(A,B)$ is nonextendible, then $A+K\\sim A$.\n \\end{lemma}\n\n\\begin{proof} The similarity $A+B\\sim A+B+K$ can be rewritten as $(A+K)+B\\sim A+B$.  Since $A+K\\supseteq A$ and $(A,B)$ is nonextendible, we have $A+K\\sim A$. \\end{proof}\n\nIn subsequent proofs, such as those of Lemmas \\ref{reduciblelemma1}, \\ref{reduciblelemma2}, \\ref{QP3}, and \\ref{transformcase1},   we will omit the above reasoning.\n\n\\subsubsection{Complementary pairs}\\label{complementarydef}  If $G$ is compact and $m(A+B)=m(A)+m(B)=1$, call $(A,B)$ a \\textit{complementary} pair.  When $G$ is infinite, it is easy to construct such pairs $(A,B)$ with $A+B\\neq G$: let $A\\subseteq G$ be any measurable set meeting every coset of every finite index subgroup of $G$ with $0<m(A)<1$, and let $B\\subseteq G$ have $m(B)=1-m(A)$.  Then $(A,B)$ is a complementary pair by Proposition \\ref{Ktopinequality}.  In particular, if $G$ is connected and $A$ is any measurable subset of $G$, then $(A,-A^c)$ is a complementary pair.\n\n\\smallskip\n\nComplementary pairs $(A,B)$ with $m(A)>0$ and $m(B)>0$ satisfying $A+B\\neq G$ can be described as follows.  If $(A,B)$ is a complementary pair, and $A+B\\neq G$, then $A\\cap (t-B)=\\varnothing$ for some $t\\in G$, so $t-B\\sim A^c$.  If $s\\notin H(t-B)$ ($=H(A^c)=H(A)$), then $m(A\\cap (t+s-B))>0$, so $t+s\\in A+B$.  It follows that $A+B$ contains a translate of $G\\setminus H(A)$.\n\n\\smallskip\n\nIf $K\\leqslant G$ is a compact open subgroup, $A$ and $B$ are each contained in a coset of $K$, and $m(A+B)=m(A)+m(B)=m(K)$, we say that $(A,B)$ is \\textit{complementary with respect to} $K$.  Such pairs form a subclass of the extendible pairs.\n\n\n\\subsubsection{Reducibility}\\label{reducibledef}  If there are measurable subsets $A'\\subseteq A$ and $B'\\subseteq B$ such that $m(A')=m(A)$, $m(B')=m(B)$, and $m_*(A'+B')<m_*(A+B)$, we say that $(A,B)$ is \\textit{reducible}.  Otherwise, we say $(A,B)$ is \\textit{irreducible}.\n\n\n\\subsubsection{Quasi-periodicity}\\label{qpdef}  Let $K\\leqslant G$ be a compact open subgroup. A set $A\\subseteq G$ is called \\textit{quasi-periodic with respect to $K$} if $A$ can be partitioned into two nonempty sets $A_1$ and  $A_0$ such that $(A_1+K)\\cap (A_0+K)=\\varnothing$, $A_1\\sim A_1+K$, and $A_0$ is contained in a coset of $K$.  We call $A_1\\cup A_0$ a \\textit{quasi-periodic decomposition} of $A$, and we say $K$ is a \\textit{quasi-period} of $A$.  Note that $A$ may have more than one quasi-period.\n\n\\smallskip\n\nCall a pair $(A,B)$ of subsets of $G$  \\textit{quasi-periodic with respect to $K$} if one of $A$ or $B$ is quasi-periodic with respect to $K$ and the other is either contained in a coset of $K$ or is quasi-periodic with respect to $K$.  This means that $A=A_1\\cup A_0$, $B=B_1\\cup B_0$, $A_1\\sim A_1+K$, $B_1\\sim B_1+K$, $A_0$ and $B_0$ are nonempty and contained in cosets of $K$, $(A_1+K)\\cap A_0=(B_1+K)\\cap B_0=\\varnothing$, and at least one of $A_1$ and $B_1$ is nonempty.  We say $A_1\\cup A_0$, $B_1\\cup B_0$ is a \\textit{quasi-periodic decomposition of} $(A,B)$, and $K$ is a \\textit{quasi-period} of $(A,B)$.  Note that if $(A,B)$ is quasi-periodic with respect to $K$, then $A+B$ is quasi-periodic with respect to $K$.\n\n\\begin{remark}  Our insistence that $A_1$ be nonempty for $A$ to be quasi-periodic differs from the convention of \\cite{Grynkiewicz}; the analogous definition in \\cite{Grynkiewicz} allows $A_1=\\varnothing$. \\hfill $\\blacksquare$\n\\end{remark}\n\n\\subsubsection{Repeated decompositions}\\label{repeateddef}  If $A_1\\cup A_0$ is a quasi-periodic decomposition of $A$ with quasi-period $K_0$ and $A_0= C_1\\cup C_0$ is a quasi-periodic decomposition of $A_0$ with quasi-period $K_1\\leqslant K_0$, then setting $A_0'=C_0$ and $A_1'=A\\setminus C_0$, the partition $A=A_1'\\cup A_0'$ is a quasi-periodic decomposition of $A$ with respect to $K_1$.  Consequently, if $(A,B)$ satisfies (QP) of Theorem \\ref{main} with $A=A_1\\cup A_0$, $B=B_1\\cup B_0$, and subgroup $K=K_1\\leqslant G$, and $(A_0,B_0)$ also satisfies (QP) with subgroup $K=K_2\\leqslant G$, then $(A,B)$ satisfies (QP) with subgroup $K=K_2\\leqslant G$.\n\n\n\n\n\n\n\n\\section{Consequences of Theorem \\ref{main}}\\label{corollarysection}\n\n\\subsection{Conclusion (QP) and critical pairs for finite groups}\\label{remark2}  Sur-critical pairs (\\S \\ref{critdef}) satisfying (QP) of Theorem \\ref{main} can be further described in terms of certain critical pairs for finite groups, as the following corollary shows.\n\n\\begin{corollary}\\label{QPfour}  With the hypotheses and notation of Theorem \\textup{\\ref{main}}, if $(A,B)$ satisfies conclusion \\textup{(QP)}, then with $K$ being the subgroup of that conclusion,\n\\begin{enumerate}\n\\item[(QP.4)]  $m(A+B+K)=m(A+K)+m(B+K)-m(K)$.\n\\end{enumerate}\n\\end{corollary}\nViewing $A'=A+K$ and $B'=B+K$ as subsets of the discrete group $G\/K$, (QP.4) means $|A'+B'|=|A'|+|B'|-1$, while (QP.2) says that $A'+B'$ has a unique expression element.  The results of \\cite{Kemperman60} may be used to classify such pairs $(A',B')$; see \\S 2 of \\cite{GrynkiewiczKemperman} for exposition.\n\n\n\\begin{proof} Let $A=A_1\\cup A_0, $ $B=B_1\\cup B_0$ constitute a pair satisfying (QP) and $m_*(A+B)=m(A)+m(B)$, and let $K$ be the corresponding subgroup. Note that \\begin{align}\\label{aka1k}\nm(A+K)=m(A_1)+m(K) = m(A)-m(A_0)+m(K),\n\\end{align} and a similar identity holds for $m(B+K)$.   Since $A_0+B_0+K$ is a unique expression element of $A+B+K$ in $G\/K$, we have\n\\begin{align*}\nm(A+B+K)&= m_*(A+B)-m_*(A_0+B_0)+m(K)\\\\\n&=m(A)-m(A_0)+m(B)-m(B_0)+m(K) \\\\\n&=[m(A+K)-m(K)]+[m(B+K)-m(K)]+m(K)  \\\\\n&= m(A+K)+m(B+K)-m(K),\n\\end{align*}\nwhere the where the second line uses the hypothesis and (QP.3) of Theorem \\ref{main}, and the third uses (\\ref{aka1k}).  \\end{proof}\n\n\nAs a counterpart to Corollary \\ref{QPfour}, the following procedure produces sur-critical pairs satisfying (QP) from critical pairs for finite groups.\n\n\\smallskip\n\nLet $K\\leqslant G$ be a compact open subgroup so that the quotient $F=G\/K$ is discrete; write $\\phi:G\\to F$ for the quotient map.   Let $(A',B')$ be a critical pair for $F$ such that $A'+B'$ has a unique expression element and $|A'+B'|=|A'|+|B'|-1$.  Pick $a'\\in A'$, $b'\\in B'$ so that $a'+b'$ is a unique expression element of $A'+B'$.  Define $A_1':=A'\\setminus \\{a'\\}$, and $B_1':=B'\\setminus \\{b'\\}$.  Set $A_1=\\phi^{-1}(A_1')$, $B_1=\\phi^{-1}(B_1')$, and choose $C\\subseteq K$ and $D\\subseteq K$ such that $m_*(C+D)=m(C)+m(D)$.  Let $A_0=\\phi^{-1}(a')+C$, $B_0=\\phi^{-1}(b')+D$, and let $A=A_1\\cup A_0$, $B=B_1\\cup B_0$.   Then\n\\begin{align*}\nm(A)=m(K)(|A'|-1)+m(C),\\  m(B)=m(K)(|B'|-1)+m(D),\n  \\end{align*}\nand\n\\begin{align*}\nm_*(A+B)&=m(K)(|A'+B'|-1)+m_*(C+D)\\\\\n&=m(K)(|A'|+|B'|-2)+m(C)+m(D)\\\\\n&=m(A)+m(B).\n\\end{align*}\n\n\n\\subsection{Measurability and topology of $A$, $B$, and $A+B$.}\\label{essreg}\n\\subsubsection{Essential regularity.}  For $S\\subseteq G$, let $\\overline{S}$ denote the topological closure of the set $S$, and let $\\operatorname{int}S$ denote the interior of $S$.  Call a set $S\\subseteq G$ \\textit{essentially regular} if $m_*(S)=m(\\overline{S})=m(\\operatorname{int} \\overline{S})$, and call a pair $(A,B)$ \\textit{essentially regular} if $A$, $B$, and $A+B$ are essentially regular.  Proposition \\ref{Kconnected} implies $(A,B)$ is essentially regular when $G$ is a compact connected group and $A$, $B\\subseteq G$ have $m(A)>0$, $m(B)>0$, and $m(A+B)=m(A)+m(B)$.  Corollary \\ref{measurableremark} below shows that restricting the measures of $A$ and $B$ in Theorem \\ref{main} can guarantee that $(A,B)$ is essentially regular.   For this we need the following lemma.\n\n\\begin{lemma}\\label{interior}  If for all $\\varepsilon>0$, $(A,B)$ has a quasi-periodic decomposition \\textup{(\\S \\ref{qpdef})} with respect to a compact open subgroup $K$ having $0<m(K)<\\varepsilon$, then $(A,B)$ is essentially regular.\n\\end{lemma}\n\n\\begin{proof}  Let $\\varepsilon>0$.  For a given quasi-period $K$ of $(A,B)$ having $0<m(K)<\\varepsilon$, write $A=A_1\\cup A_0$, where $A_1\\sim A_1+K$ and $A_0$ is contained in a coset of $K$.  Then $m(\\overline{A})\\leq m(A+K)\\leq m(A)+\\varepsilon$, while $m(\\operatorname{int}\\overline{A})\\geq m(A+K)-m(K)\\geq m(A)-\\varepsilon$.\nLetting $\\varepsilon\\to 0$, we get that $m(A)=m(\\overline{A})=m(\\operatorname{int}\\overline{A})$. The same argument applied to $B$ and $A+B$ shows that $B$ and $A+B$ are essentially regular.  \\end{proof}\n\n\n\n\n\n\\subsubsection{Refinement of Theorem \\textup{\\ref{main}}} In conclusion (QP) of Theorem \\ref{main}, possibly $m(A_0)=0$, $m(B_0)=0$, or both.  When $m(A_0)>0$ and $m(B_0)>0$, (QP.3) guarantees that Theorem \\ref{main} applies to the pair $(A_0,B_0)$.  This observation yields the following corollary.\n\n\\begin{corollary}\\label{roman}  With the hypotheses and notation of Theorem \\textup{\\ref{main}}, at least one of the following is true\\textup{:}\n\\begin{enumerate}\n\\item[\\textup{(I)}]  One of \\textup{(P)}, \\textup{(E)}, or \\textup{(K)} holds.\n\\item[\\textup{(II)}]  Conclusion \\textup{(QP)} holds, and $(A_0,B_0)$ satisfies \\textup{(K)}.\n\\item[\\textup{(III)}]  For all $\\varepsilon>0$, there is a compact open subgroup $H\\leqslant G$  having $m(H)<\\varepsilon$ and $(A,B)$ satisfies \\textup{(QP)} with $K=H$.\n\\item[\\textup{(IV)}]  $(A,B)$ satisfies \\textup{(QP)} with one at least one of $m(A_0)=0$ or $m(B_0)=0$.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}  If $(A,B)$ satisfies (P), (E), or (K), we have (I). If not, inductively form a sequence of pairs $(A^{(n)}, B^{(n)})$  and subgroups $K^{(n)}\\leqslant G$ as follows:  let $(A^{(0)}, B^{(0)})=(A,B)$, and let $K^{(0)}=G$.  Suppose $(A^{(j)},B^{(j)})$ and $K^{(j)}$ are defined for $j=0,\\dots,n$ and for each $j=1,\\dots, n$,\n\\begin{enumerate}\n\\item[(1)]\n$A=A'_1\\cup A^{(j)}$, $B=B'_1\\cup B^{(j)}$ is a quasi-periodic decomposition of $(A,B)$ with respect to $K^{(j)}$ satisfying (QP) of Theorem \\ref{main}, so that\n\\item[(2)] $A^{(j)}$ and $B^{(j)}$ are each contained in cosets of $K^{(j)}$,\n\\item[(3)] $m_*(A^{(j)}+B^{(j)})=m(A^{(j)})+m(B^{(j)})$, and\n\\item[(4)] $K^{(j)}$ is a proper subgroup of $K^{(j-1)}$.\n\\end{enumerate}\nWe will construct $(A^{(n+1)},B^{(n+1)})$ and $K^{(n+1)}\\leqslant K^{(n)}$ satisfying (1)-(4), or show that one of (I), (II), or (IV) holds.\n\n\\smallskip\n\nIf one or both of $m(A^{(n)})=0$ or $m(B^{(n)})=0$, we have conclusion (IV).  Otherwise, apply Theorem \\ref{main} to $(A^{(n)},B^{(n)})$.  If $(A^{(n)},B^{(n)})$ satisfies (P) or (E), then so does $(A,B)$, and we have (I).  If ($A^{(n)}, B^{(n)})$ satisfies (K), we have (II).  Otherwise, $(A^{(n)},B^{(n)})$ satisfies (QP), so take $K^{(n+1)}$ to be corresponding subgroup $K$, and let $A^{(n)}= A^{(n)}_1\\cup A^{(n)}_0$, $B^{(n)}=B^{(n)}_1\\cup B^{(n)}_0$ be the corresponding decompositions. Observe that $K^{(n+1)}$ must be a proper subgroup of $K^{(n)}$, so that $m(K^{(n+1)})\\leq \\frac{1}{2}m(K^{(n)})$.  Now take $A^{(n+1)}=A^{(n)}_0$ and $B^{(n+1)}=B^{(n)}_0$, so that $(A^{(n+1)},B^{(n+1)})$ satisfies (1)-(4) with $j=n+1$ (cf.~\\S \\ref{repeateddef}).\n\n\\smallskip\n\nIf the above construction terminates, we conclude (I), (II), or (IV).  Otherwise, we have for each $n$ quasi-periodic decompositions with respect to $K^{(n)}$, and we conclude (III).\n\\end{proof}\n\n\n\\begin{corollary}\\label{measurableremark}  Suppose $(A,B)$ satisfies the hypotheses of Theorem \\textup{\\ref{main}}.\n\\begin{enumerate}\n\\item[\\textup{(a)}]  If $A+B$ is not measurable, then $(A,B)$ satisfies \\textup{(QP)} with at least one of $m(A_0)=0$ or $m(B_0)=0$.\n\n\\smallskip\n\n\\item[\\textup{(b)}]  If $(A,B)$ is not essentially regular, then $(A,B)$ satisfies conclusion \\textup{(E)} of Theorem \\textup{\\ref{main}}, or $(A,B)$ satisfies conclusion \\textup{(QP)} of Theorem \\textup{\\ref{main}} with one or both of $m(A_0)=0$ or $m(B_0)=0$.\n\n\\smallskip\n\n\\item[\\textup{(c)}]  If $m(A)$, $m(B)$, and $m(A+B)$ are all irrational numbers, then $(A,B)$ is essentially regular.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}    (a)  In conclusions (I) and (II) of Corollary \\ref{roman}, it is routine to check that $A+B$ is measurable.  If (III) holds in Corollary \\ref{roman}, Lemma \\ref{interior} implies $A+B$ is measurable.\n\n\\smallskip\n\n\\noindent  (b)  If (P) or (K) holds, then $(A,B)$ is essentially regular.  If (II) or (III) of Corollary \\ref{roman} holds, then $(A,B)$ is essentially regular.  The only remaining alternatives are (E) of Theorem \\ref{main} or (IV) of Corollary \\ref{roman}.\n\n\\smallskip\n\n\\noindent (c)  If $m(A+B)$ is irrational, then neither conclusion (P) nor (E) can hold in Theorem \\ref{main}.  If $m(A)$ and $m(B)$ are irrational, then (QP) cannot hold with one of $m(A_0)=0$ or $m(B_0)=0$.  Now $(A,B)$ is essentially regular, by Part (b) of the present corollary. \\end{proof}\n\n\n\\subsection{Further description of extendible pairs}\\label{Eremark}  If $(A,B)$ is a sur-critical pair satisfying conclusion (E) of Theorem \\ref{main}, Proposition \\ref{Ktopinequality} implies the group $H:=H(A+B)$ is compact and open and $A+B\\sim A+B+H$.  Furthermore, exactly one of the following holds:\n\\begin{enumerate}\n\\item[(E.1)]  $A+B=A+B+H$.\n\n\\smallskip\n\n\\item[(E.2)]  $(A,B)$ is complementary with respect to $H$ and $A+B\\neq A+B+H$, or $(A,B)$ satisfies conclusion (QP) of Theorem \\ref{main}, with $K=H$, $m(A_0)+m(B_0)=m(H)$, and $A_0+B_0\\neq A_0+B_0+H.$\n\\end{enumerate}\nWhen $m(A_0)>0$ and $m(B_0)>0$ in (E.2), the pair $(A_0,B_0)$ is complementary with respect to the subgroup $H$; see \\S \\ref{complementarydef} for further description.\n\nTo obtain this classification, fix $A'\\supseteq A$ and $B'\\supseteq B$ such that $m(A')+m(B')>m(A)+m(B)$ and $m(A'+B')=m(A+B)$, and write $H$ for $H(A'+B')$.  By Proposition \\ref{Ktopinequality},\n\\begin{align*}\nm(A'+B')=m(A'+H)+m(B'+H)-m(H),\n\\end{align*}\nso  $m(A)+m(B)=m(A+B)=m(A'+H)+m(B'+H)-m(H)$.  Rearranging, we get\n\\begin{align}\\label{holes}\nm(A'+H)-m(A)+m(B'+H)-m(B)=m(H).\n\\end{align}\nLet $A=\\bigcup_{i=1}^n A_i$ and $B=\\bigcup_{j=1}^m B_j$ be $H$-coset decompositions of $A$ and $B$.  Since $A\\subseteq A'+H$ and $B\\subseteq B'+H$, (\\ref{holes}) implies\n\\begin{align}\\label{surholes}\nm(A_i)+m(B_j)\\geq m(H)\n\\end{align}\n for each $i$ and $j$ (otherwise the left-hand side of (\\ref{holes}) would be larger than $m(H)$).  If the inequality (\\ref{surholes}) is strict for each pair $i$ and $j$, we have (E.1).  Otherwise, take $i$ and $j$ so that equality holds in (\\ref{holes})  and set $A_0=A_i$, $B_0=B_j$.  If $A=A_0$ and $B=B_0$, then $(A,B)$ is complementary with respect to $H$.  If not, we set $A_1=A\\setminus A_0$ and $B_1=B\\setminus B_0$, we verify (QP.1)-(QP.3) in Theorem \\ref{main}.   Equation (\\ref{holes}) implies $A_1\\sim A_1+H$ and $B_1\\sim B_1+H$, so (QP.1) holds. If $A+B\\neq A+B+H$, we conclude that $A_0+B_0+H$ is a unique expression element of $A+B+H$ in $G\/H$ and we have (QP.2).  (QP.3) now follows from $A+B\\sim A+B+H$.\n\n\\section{Examples}\\label{examples}  We list some examples of sur-critical pairs to show that each alternative in Theorem \\ref{main} can occur, and that none of the alternatives can be omitted.  We do not attempt to exhaustively construct all possible sur-critical pairs.\n\n\n\\subsection{Periodic pairs}\\label{periodicexamples}  If $F$ is a finite group, every sur-critical pair for $F$ satisfies (P) and not (K), and every nonextendible sur-critical pair satisfies (P) but not (E).  A specific example with $F=\\mathbb Z\/11\\mathbb Z$ is $A=\\{0,1\\}$, $B=\\{0,3\\}$, so that $A+B=\\{0,1,3,4\\}$, and $(A,B)$ satisfies (P) but not (E) or (K).  Every periodic sur-critical pair $(A,B)$ for a compact group $G$ has the form $A=\\phi^{-1}(C)\\setminus N$, $B=\\phi^{-1}(D)\\setminus N$, where $(C,D)$ is a sur-critical pair for a finite group $F$, $\\phi:G\\to F$ is a continuous surjective homomorphism,  and $m(N)=0$, so the periodic sur-critical pairs for an arbitrary compact group $G$ are classified in \\cite{Grynkiewicz}.\n\n\\subsection{Extendible pairs} The following example satisfies (E), but not (P), (K), or (QP).  Let $G=(\\mathbb Z\/ 17\\mathbb Z)\\times \\mathbb T$ and let $A=\\{1,3,5,7\\}\\times [0,0.8]$, $B=\\{0,2\\}\\times [0,0.9]$, so that $A+B=\\{1,3,5,7,9\\}\\times \\mathbb T$.  Then $m(A)=3.2\/17$, $m(B)=1.8\/17$, and $m(A+B)=5\/17$.  Also $A+B=A'+B'$, where $A'=\\{1,3,5,7\\}\\times \\mathbb T$ and $B'=\\{0,2\\} \\times \\mathbb T$.\n\n\n\\smallskip\n\nTo find examples satisfying (E.1)  (\\S \\ref{Eremark}),  fix a pair of sets  $A'$, $B'\\subseteq G$ satisfying $m(A'+B')<m(A')+m(B')$, and let $H=H(A'+B')$.  Choose any pair of subsets $A\\subseteq A'$, $B\\subseteq B'$ with $m(A)+m(B)=m(A'+B')$, subject to the condition that $m((a+H)\\cap A)+m((b+H)\\cap B)>m(H)$ for all $a\\in A'$ and $b\\in B'$.  Then $A+B=A'+B'$, so $m(A+B)=m(A)+m(B)$.\n\n\\smallskip\n\nTo find examples satisfying (E.2) but not (E.1),  recall the construction in \\S \\ref{remark2}.  Let $(A,B)$ satisfy conclusion (QP) of Theorem \\ref{main} with $m(A_0)+m(B_0)=m(K)$, but $A_0+B_0\\neq A_0+B_0+K$.   To form a specific example of this construction with $G=(\\mathbb Z\/ 15\\mathbb Z)\\times \\mathbb T$, let $A=(\\{1,3,5\\}\\times \\mathbb T)\\cup (\\{7\\}\\times S)$, where $S\\subseteq \\mathbb T$ is any measurable set, and let $B=(\\{0,2\\}\\times \\mathbb T)\\cup (\\{4\\}\\times (-S^c))$.  This example satisfies (E) and (QP) but not (P) or (K).\n\n\\subsection{Pairs arising from $\\mathbb T$} Let $G$ be a compact group which is not totally disconnected. Let $(A,B)$ have the form $A=\\chi^{-1}(I)$,  $B=\\chi^{-1}(J)$, where $H\\leqslant G$ is a compact open subgroup, $\\chi:H\\to \\mathbb T$ is a continuous surjective homomorphism, $I,J\\subseteq \\mathbb T$ are intervals, and $m_{\\mathbb T}(I)+m_{\\mathbb T}(J)<1$.  One can verify that $(A,B)$ is a sur-critical pair satisfying (K) but not (P), (E), or (QP).\n\n\\subsection{A quasi-periodic pair} This example will satisfy (QP) but not (P), (E), or (K).\n\n\\smallskip\n\nLet $G=\\mathbb Z_7$, the $7$-adic integers with the usual topology, and consider $\\mathbb Z$ as a subset of $\\mathbb Z_7$ in the usual way.  Define the set $C\\subseteq \\mathbb Z$ by\n\\begin{align*}\nC:=(\\{0,1\\}+7\\mathbb Z)\\cup\\Bigl(2+7((\\{0,1\\}+7\\mathbb Z)\\cup (2+7(\\cdots)))\\Bigr),\n\\end{align*}\nso that $C=(\\{0,1\\}+7\\mathbb Z)\\cup (2+7C)$.  Let $A$ be the closure of $C$ in $\\mathbb Z_7$, and let $B=A$.  Then $\nm(A)=m(B)=\\sum_{n=1}^\\infty \\frac{2}{7^n}=1\/3$. Note that $A$ has a quasi-periodic decomposition $A_1\\cup A_0$ where $A_1=\\overline{\\{0,1\\}+7\\mathbb Z}$, $A_0=A\\setminus A_1$.  The sumset $A+B$ is the closure of $C+C$ in $\\mathbb Z_7$.  Note that\n \\begin{align}\\label{C+C}\nC+C=(\\{0,1,2,3\\}+7\\mathbb Z)\\cup\\Bigl(4+7((\\{0,1,2,3\\}+7\\mathbb Z)\\cup (4+7(\\cdots)))\\Bigr),\n \\end{align}\nso $m(A+B)=2\/3=m(A)+m(B)$.  From (\\ref{C+C}) one can check that $H(A+B)=\\{0\\}$, so (P) and (E) fail.  Since $G$ is totally disconnected, (K) cannot hold.\n\n\n\\subsection{More quasi-periodic pairs.} Let $G'$ be an infinite compact abelian group with Haar measure $m'$, and let $G=(\\mathbb Z\/4\\mathbb Z)\\times G'$.  Fix sets $A_0'$, $B_0'\\subseteq G'$ satisfying $m_{*}'(A_0'+B_0')=m'(A_0')+m_0(B_0')$, let $A_1=\\{0\\}\\times G'$, $B_1=\\{0\\}\\times G'$, and let $A_0=\\{1\\}\\times A_0'$, $B_0=\\{1\\}\\times B_0'$.  Let $A=A_1\\cup A_0$, $B=B_1\\cup B_0$.  Then $m_*(A+B)=m(A)+m(B)$, but depending on our choice of $A_0$ and $B_0$, $A+B$ may not be measurable.  If $A_0'$ is a singleton, then $B_0'$ may be an arbitrary measurable subset of $G'$.\n\n\n\n\n\\section{Lemmas}\\label{lemmas}\n\n\n\nWe fix a compact abelian group $G$ with Haar measure $m$.  Unless stated otherwise, sets $A$ and $B$ are assumed to be subsets of $G$.\n\n\n\n\\subsection{Consequences of Proposition \\ref{Ktopinequality}}\n\nWe need Lemma \\ref{overspill} and Corollary \\ref{consequence} in many subsequent proofs.  See \\S \\ref{Hdef} for notation.\n\n\\begin{lemma}\\label{overspill}\nIf $m_*(A+B)<m(A)+m(B)$, then for all $a\\in A$ and all $b\\in B$,\n\\begin{align}\n\\label{os1} m((a+H)\\cap A)+m_*(A+B)&\\geq m(A)+m(B),\\\\\n\\label{os2} m((b+H)\\cap B) +m_*(A+B)&\\geq m(A)+m(B),\n\\end{align}\nwhere $H=H(A+B)$.\n\\end{lemma}\n\n\\begin{proof}  We prove (\\ref{os1}).  Write $m(A)=\\sum_{i=1}^n m(A_i)$, where $A=\\bigcup_{i=1}^n A_i$ is an $H$-coset decomposition of $A$.  Fix $a\\in A$, and let $j$ be such that $(a+H)\\cap A=A_j$. Then $m(A_j)=m(A)-\\sum_{i\\neq j} m(A_i)$, while $m(A+H)=\\sum_{i=1}^n m(H)$.   Now Proposition \\ref{Ktopinequality} implies\n\\begin{align*}\nm(A_j)+m_*(A+&B)=\\Bigl(m(A)-\\sum_{i\\neq j}m(A_i)\\Bigr)+m(A+H)+m(B+H)-m(H)\\\\\n&= \\Bigl(m(A)-\\sum_{i\\neq j}m(A_i)\\Bigr) + \\Bigl(\\sum_{i} m(H)\\Bigr)+m(B+H)-m(H)\\\\\n&= m(A)+\\Bigl(\\sum_{i\\neq j} m(H)-m(A_i)\\Bigr)+m(H)+m(B+H)-m(H)\\\\\n&\\geq m(A)+m(B+H)\\\\\n&\\geq m(A)+m(B).\n\\end{align*}\nEquation (\\ref{os2}) follows by symmetry. \\end{proof}\n\nWe write $C\\subset_m D$ below to mean $m(C\\setminus D)=0$.\n\\begin{corollary}\\label{consequence}  If $m_*(A+B)< m(A)+m(B)$, let $H:=H(A+B)$.  Then\n\\begin{align}\\label{mover1}\n\\{z\\in G:z+B\\subset_m A+B\\}=\\{z\\in G: z+B\\subseteq A+B\\}=A+H,\n\\end{align}\n\\begin{align}\\label{mover2}\n\\{z\\in G: A+z\\subset_m A+B\\}=\\{z\\in G: A+z \\subseteq A+B\\}=B+H.\n\\end{align}\n\\end{corollary}\n\n\\begin{proof}  For the first equality in (\\ref{mover1}), note that $z+B\\subset_m A+B$ implies $z+B\\subseteq A+B$, by Lemma \\ref{overspill} and the fact that $A+B$ is a union of cosets of $H$.  For the second equality in (\\ref{mover1}), we argue by contradiction.  Fix $z\\in G$ such that $z+B\\subseteq A+B$, but $z\\notin A+H$.   Let $A'=A\\cup (z+H)$.     Then $A'+B=A+B$, so that $H(A'+B)=H(A+B)$, while $m(A'+H)=m(A+H)+m(H)$.  Applying Proposition \\ref{Ktopinequality} to $(A',B)$ yields\n\\begin{align*}\nm(A'+B)&=m(A'+H)+m(B+H)-m(H)\\\\\n&= m(A+H)+m(B+H),\n\\end{align*}\nwhich contradicts the assumption that $m_*(A+B)<m(A)+m(B)$.  Equation (\\ref{mover2}) follows by symmetry.  \\end{proof}\n\n\\begin{lemma}\\label{Hholes}  If $m_*(A+B)<m(A)+m(B)$ and $H:=H(A+B)$, then for all $a\\in A$ and $b\\in B$,\n\\begin{align}\\label{organize}\nm((a+H)\\cap A)+m((b+H)\\cap B)>m(H).\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}  If inequality (\\ref{organize}) fails, then $m(A+H)+m(B+H)\\geq m(A)+m(B)+m(H)$, so inequality (\\ref{critical}) in Proposition \\ref{Ktopinequality} implies $m_*(A+B)\\geq m(A)+m(B)$, contradicting the hypothesis.\n\\end{proof}\n\nIn the next corollary we use the elementary fact that if $G$ is compact, and $C$, $D\\subseteq G$ have $m(C)+m(D)>1$, then $C+D=G$.  This fact follows from the observation that $t\\in C+D$ if and only if $(t-C)\\cap D\\neq \\varnothing$, while $t-C$ and $D$ cannot be disjoint if $m(C)+m(D)>m(G)$.  Consequently, if $H\\leqslant G$ is a compact open subgroup and $C,$ $D\\subset G$ are each contained in a coset of $H$ while  $m(C)+m(D)>m(H)$, then $C+D=C+D+H$.\n\n\\begin{corollary}\\label{differencetosum}  If $m_*(A+B)<m(A)+m(B)$, then $A-B$ is periodic with period $H:=H(A+B)$.\n\\end{corollary}\n\n\\begin{proof} Let $A=\\bigcup_{i=1}^n A_i$ and $B=\\bigcup_{i=1}^m B_i$ be $H$-coset decompositions of $A$ and $B$, so that $A-B=\\bigcup_{i,j} A_i-B_j$.  Inequality (\\ref{organize}) implies $m(A_i)+m(B_j)>m(H)$ for each $i$ and $j$, so $A_i-B_j= A_i-B_j+H$ for each $i$ and $j$, implying $A-B=A-B+H$.\n\\end{proof}\n\n\\subsection{Reducible pairs}  We now dispense with the case of Theorem \\ref{main} where $(A,B)$ is reducible (\\S \\ref{reducibledef}). For the next two lemmas and the following corollary, we assume that $m(A)>0$, $m(B)>0$, and $(A,B)$ satisfies $m_*(A+B)=m(A)+m(B)$.\n\n\\begin{lemma}\\label{reduciblelemma1} If $A'\\subseteq A$ is such that $m(A')=m(A)$ and $m_*(A'+B)<m_*(A+B)$, then $A'+H(A'+B)\\subset_m A$ and at least one of the following holds\\textup{:}\n\n\\begin{enumerate}\n\\item[(i)]  $(A,B)$ satisfies \\textup{(QP)} of Theorem \\textup{\\ref{main}}, with  $K=H:=H(A'+B)$,  \\\\ $A_1=A\\cap(A'+H)$, and $A_0=A\\setminus A_1$, so that $m(A_1)=m(A)$ and $m(A_0)=0$.\n\n\\smallskip\n\n\\item[(ii)]  $B\\sim B+H(A'+B)$.\n\\end{enumerate}\nIf $(A,B)$ is nonextendible, then {\\textup{(i)}} holds.\n\\end{lemma}\n\n\n\\begin{proof}  If $A$, $B$, $A'$ and $H$ are as in the hypothesis, Proposition \\ref{Ktopinequality} implies $H$ is compact and open and $A'+B$ is a union of cosets of $H$. Corollary \\ref{consequence} implies that whenever $a_0\\in A$ is such that $a_0+B \\not\\subseteq A'+B$, there is a $b_0\\in B$ such that\n\\begin{align}\\label{Bover2}\nm((a_0+B_0)\\setminus (A'+B))>0,\n\\end{align}\nwhere $B_0:=(b_0+H)\\cap B$.   Fix $a_0\\in A$, $b_0\\in B$, and $B_0$ satisfying (\\ref{Bover2}), and let $B_1=B\\setminus B_0$.  Lemma \\ref{overspill} and the hypothesis $m(A')=m(A)$ imply\n\\begin{align}\\label{Bover1}\nm(B_0)+m(A'+B)\\geq m(A')+m(B) =m_*(A+B).\n\\end{align}\n\n\\begin{claim1}  $A'+H \\subset_m A$ and $B_1+H\\subset_m B$.\n\\end{claim1}\n\n\\noindent\\textit{Proof of Claim \\textup{1}.}  By the definition of $B_0$ and the fact that $A'+B=A'+B+H$,  $A'+B$ is disjoint from $a_0+B_0$.    Using (\\ref{Bover1}) we write\n\\begin{align}\\label{innermeasure}\nm_*(A+B)=m(A'+B)+m(B_0).\n\\end{align}\nThen\n\\begin{align}\\label{arranging}\\begin{split}m(A')+m(B)&=m_*(A+B)\\\\\n&=m(A'+B)+m(B_0)\\\\\n&=m(A'+H)+m(B+H)-m(H)+m(B_0)\\\\\n&=m(A'+H)+m(B_1+H) +m(B_0),\\end{split}\n\\end{align}\nwhere the second line is (\\ref{innermeasure}), the third line is from Proposition \\ref{Ktopinequality}, and the fourth line is due to $B=B_1\\cup B_0$ being a disjoint union and $B_0$ being contained in a coset of $H$. Subtracting $m(A')+m(B)$ from the first and last lines of (\\ref{arranging}), we find\n\\begin{align}\\label{arranged}\n[m(A'+H)-m(A')] + [m(B_1+H)+m(B_0)-m(B)]=0\n\\end{align}\nEach bracketed summand in (\\ref{arranged}) is nonnegative, so\n\\begin{align}\\label{A'}\nm(A')=m(A'+H)\n\\end{align}\nand\n\\begin{align}\\label{B'}\nm(B)=m(B_1+H)+m(B_0).\n\\end{align}\nSince $(B_1+H) \\cap B_0=\\varnothing$ and $B=B_1\\cup B_0$, (\\ref{A'}) and (\\ref{B'}) prove the claim.  \\hfill $\\square$\n\n\\smallskip\n\nWe proceed based on whether $m(B_0)<m(H)$.\n\n\n\\subsubsection*{Case \\textup{1}. $m(B_0)=m(H)$} In this case Claim 1 implies $B\\sim B+H$, and we conclude (ii) in the statement of the lemma.\n\n\n\\subsubsection*{Case \\textup{2}.  $m(B_0)<m(H)$}  Let $A_0:=A\\setminus (A'+H)$.  We will show conclusion (i) of the lemma holds.\n\n\\begin{claim2}  $A_0$ is contained in a coset of $H$ and $A_0+B_0+H$ is a unique expression element of $A+B+H$ in $G\/H$.\n\\end{claim2}\n\n\\noindent\\textit{Proof of Claim \\textup{2}.}  Observe that $A+B_1\\subseteq A'+B$, since otherwise Claim 1 implies $m_*(A+B)\\geq m(A'+B)+m(H) >m(A'+B)+m(B_0)$, contradicting (\\ref{innermeasure}).  Furthermore, $A'+H$ is the set $\\{z\\in G: z+B\\subset_m A'+B\\}$, by Corollary \\ref{consequence}.  Thus, $A_0$ is the set of $a\\in A$ such that $a+B_0\\not\\subseteq A'+B$.  Now if $a$, $a'\\in A_0$,  then $a+B_0\\sim a'+B_0$, by (\\ref{innermeasure}).  Since $B_0$ is contained in a coset of $H$ and $m(B_0)>0$, the similarity $a+B_0\\sim a'+B_0$ implies $a-a'\\in H$.   We conclude that $A_0$ is contained in a coset of $H$.  Now the containment $A+B_1\\subseteq A'+B$ implies $A_0+B_0+H$ is a unique expression element of $A+B+H$ in $G\/H$, as claimed.  \\hfill $\\square$\n\n\\smallskip\n\nWriting $A_1=A\\cap(A'+H)$, and maintaining $A_0$, $B_1$, and $B_0$ as defined above,  the partition $A=A_1\\cup A_0$ is a quasi-periodic decomposition with quasi-period $H$, while $B_1\\sim B_1+H$, and $B_0$ is contained in a coset of $H$ (possibly $B_1=\\varnothing$), so we have verified that (QP.1) and (QP.2) hold.  To verify (QP.3), consider the partition $A+B= (A_1+B)\\cup (A_0+B_0)$,  so that  (\\ref{innermeasure}) implies $m_*(A_0+B_0)=m(B_0)$.  Since $m(A_0)=0$, we then have $m_*(A_0+B_0)=m(A_0)+m(B_0)$. This concludes the analysis of Case 2.\n\n\\smallskip\n\n Now suppose $(A,B)$ is nonextendible.  If Case 2 holds, we have conclusion (i) as desired.   If Case 1 holds we derive a contradiction:  from $B+H\\sim B$ and the nonextendibility of $(A,B)$ we find $A\\sim A+H$.  Since $H$ is compact and open,  the similarity $A\\sim A+H$ implies  $m(A''+B)=m_*(A+B)$ whenever $A''\\subseteq A$ has $m(A'')=m(A)$, contradicting the hypothesis of the lemma.    \\end{proof}\n\n\n\\begin{lemma}\\label{reduciblelemma2}\n If $(A,B)$ is reducible and nonextendible, then $(A,B)$ has a quasi-periodic decomposition $A=A_1\\cup A_0$, $B=B_1\\cup B_0$ satisfying conclusion \\textup{(QP)} of Theorem \\textup{\\ref{main}}, and at least one of $m(A_0)=0$ or $m(B_0)=0$.\n\\end{lemma}\n\n\\begin{proof}  Let $A'\\subseteq A$ and $B'\\subseteq B$ be such that $m(A')+m(B')=m(A)+m(B)$ and $m_*(A'+B')<m_*(A+B)$.   Write $H:=H(A'+B')$, and note that $H$ is compact and open, by Proposition \\ref{Ktopinequality}.  We consider two cases.\n\n\\subsubsection*{Case \\textup{1}. $m_*(A'+B)<m_*(A+B)$ or $m_*(A+B')<m_*(A+B)$}  We apply Lemma \\ref{reduciblelemma1}  and we are done.\n\n\\subsubsection*{Case \\textup{2}. $m_*(A'+B)=m_*(A+B')=m_*(A+B)$}  In this case Lemma \\ref{reduciblelemma1} applies to the pairs $(B,A')$ and $(A,B')$, so\n\\begin{align}\\label{simsim}\nB'+H\\sim B \\text{ and } A'+H \\sim A.\n\\end{align}\nWe claim that $A_0:=A\\setminus (A'+H)$ is contained in a coset of $H$.  Let $t$ be the number of cosets of $H$ occupied by $A\\setminus (A'+H)$, so that \\begin{align}\\label{tee}\nm(A+H)-m(A')=t\\cdot m(H).\n \\end{align}\n We aim to show that $t=1$, so that $A_0$ is contained in a coset of $H$.  We have $t>0$, as the hypothesis of Case 2 implies $A+H\\neq A'+H$.\n\n\\smallskip\n\nConsider the pair $(A+H,B')$.  By (\\ref{simsim}), $A+H+B'\\sim A+B'\\sim A+B$.  Then\n\\begin{align}\\label{computation}\n\\begin{aligned}m((A+H)+B') &=m_*(A+B)\\\\\n&=m(A)+m(B)  && \\text{(by hypothesis)}\\\\\n&=m(A'+H)+m(B')  && \\text{(by (\\ref{simsim}))}\\\\\n&=m(A+H)-t\\cdot m(H)+m(B')  && \\text{(by (\\ref{tee}))}\\\\\n&<m(A+H)+m(B'),\n\\end{aligned}\n\\end{align}\nApplying Proposition \\ref{Ktopinequality} to the pair $(A+H,B')$ we get that $K:=H((A+H)+B')$ is compact and open,\n\\begin{align}\\label{+K}\nA+H+B'+K=  A+H+B'\n\\end{align} and\n\\begin{align}\\label{-m(K)}\nm(A+H+B')=m(A+H+K)+m(B'+K)-m(K).\n\\end{align}\nWe want to show $K\\leqslant H$.  The nonextendibility of $(A,B)$, (\\ref{simsim}),  and (\\ref{+K}) together imply $B'+K\\subset_m B$.   It follows that $B'+K\\subset_m E$ whenever $E\\sim B$.  Consequently $B'+K\\subset_m B'$, and we conclude  $A'+B'\\sim A'+B'+K$, so $K\\leqslant H$ by the definition of $H$.  Now (\\ref{-m(K)}) becomes\n\\begin{align}\\label{KandH}\nm(A+H+B')=m(A+H)+m(B')-m(K).\n\\end{align}  Comparing (\\ref{KandH})  with the fourth line of (\\ref{computation}), we get $t\\cdot m(H)=m(K)$.  This implies $t=1$, since $K\\subseteq H$ and $m(K)>0$.\n\n\\smallskip\n\nWe have shown that $A_0$ is contained in a coset of $H$, so $A_1:=A\\setminus A_0$ gives the desired quasi-periodic decomposition of $A$.  Reversing the roles of $A$ and $B$, we find the corresponding quasi-periodic decomposition of $B$: $B_0=B\\setminus (B'+H)$, $B_1=B\\setminus B_0$.  We now show that $A_1\\cup A_0$ and $B_1\\cup B_0$ satisfy (QP.2) and (QP.3).\n\n\\smallskip\n\nObserve that $m(A_0)=m(B_0)=0$, by our choice of $A_0$ and $B_0$.  Nonextendibility of $(A,B)$ implies $A_0+B_0+H$ is a unique expression element of $A+B+H$ in $G\/H$.  To see that $m_*(A_0+B_0)=0$, write $A+B$ as $(A_1+B)\\cup (A_0+B_0)$, so that $m_*(A+B)=m(A_1+B)+m_*(A_0+B_0)$.  By the hypothesis of Case 2, we have $m_*(A_1+B)=m_*(A+B)$, so $m_*(A_0+B_0)=0=m(A_0)+m(B_0)$.  \\end{proof}\n\n\\begin{corollary}\\label{reduciblecor}  If $(A,B)$ is reducible and $m(H(A+B))=0$, then $(A,B)$ satisfies conclusion \\textup{(QP)} of Theorem \\textup{\\ref{main}} with  at least one of $m(A_0)=0$ or $m(B_0)=0$.\n\\end{corollary}\n\n\n\\begin{proof}  If  $m(H(A+B))=0$, then $(A,B)$ is nonextendible, by Proposition \\ref{Ktopinequality}.  The conclusion now follows from Lemma \\ref{reduciblelemma2}. \\end{proof}\n\n\n\\subsection{Auxiliary lemmas}  The next two lemmas handle technicalities arising repeatedly in the next subsection.  We continue to assume $(A,B)$ satisfies the hypotheses of Theorem \\ref{main}.\n\n\n\\begin{lemma}\\label{easy}  Suppose $(A,B)$ is irreducible and nonextendible.  Assume further that $H:=H(\\tilde{A}+\\tilde{B})$ has positive Haar measure for some $\\tilde{A}\\subseteq A$, $\\tilde{B}\\subseteq B$  such that $\\tilde{A}+\\tilde{B}$ is measurable, $m(\\tilde{A})=m(A)$ and $m(\\tilde{B})=m(B)$. Then $A$ and $B$ are periodic with period $H$, or $(A,B)$ satisfies \\textup{(QP)} of Theorem \\textup{\\ref{main}} with $K=H$.\n\\end{lemma}\n\n\\begin{proof}  Let $\\delta=m(H)$.  Since $m(H)>0$, we have $m((\\tilde{A}+\\tilde{B})\\cap H_i)\\in \\{0,\\delta\\}$ for every coset $H_i$ of $H$ (cf.~\\S \\ref{periodicdef}).  Irreducibility of $(A,B)$ then implies\n\\begin{align}\\label{0delta}\nm_*((A+B)\\cap H_i)\\in\\{0,\\delta\\} \\text{ for every coset } H_i \\text{ of } H.\n\\end{align} Combining (\\ref{0delta}) with the nonextendibility of $(A,B)$, we have $m(A\\cap H_i)\\in \\{0,\\delta\\}$ and  $m(B\\cap H_i)\\in \\{0,\\delta\\}$ for every coset $H_i$ of $H$.  Now by the irreducibility of $(A,B)$,  there are sets $A'\\subseteq A$, $B'\\subseteq B$ such that\n\\begin{align}\\label{AsimsBsims} A\\sim A'+H\\sim A' \\text{ and } B\\sim B'+H\\sim B',\n \\end{align}\n $A'+B'$ is measurable, and $m_*(A+B)=m(A'+B')$.  Setting  $A_0:=A\\setminus (A'+H)$, we claim $A_0$ is contained in a coset of $H$.  To prove this claim, assume $A_0$ is nonempty, let $a\\in A_0$, and consider the pair $(A'\\cup (a+H), B')$.  Write $C$ for $A'\\cup (a+H)$, so $m(C)=m(A')+m(H)$.  By (\\ref{AsimsBsims}) we have $a+B'+H\\subset_m A+B$, so $C+B'\\subset_m A+B$.   The irreducibility of $(A,B)$ then implies\n\\begin{align}\\label{C+B'}\nC+B' \\sim \\tilde{A}+\\tilde{B},\n\\end{align} so that $H(C+B')=H(\\tilde{A}+\\tilde{B})$.   We also have\n$$\nm(C+B')=m_*(A+B)=m(A)+m(B)<m(C)+m(B'),\n$$\nso we may apply Corollary \\ref{consequence} to the pair $(C,B')$ and find\n\\begin{align}\\label{CplusH}\nC+H=\\{z\\in G: z+B'\\subset_m C+B'\\}.\n\\end{align}\nBy (\\ref{C+B'}), the right-hand side of (\\ref{CplusH}) is also $\\{z\\in G: z+B'\\subset_m \\tilde{A}+\\tilde{B}\\}$, which does not depend on $a\\in A_0$.  Hence, $(A'+H)\\cup (a+H)$ does not depend on the choice of $a\\in A_0$.  This means that $A_0$ is contained in a coset of $H$, as claimed.\n\n\\smallskip\n\nIf $A_0=\\varnothing$, then $B\\sim B+H$ by the irreducibility and nonextendibility of $(A,B)$,  so $A$ and $B$ are periodic with period $H$, and we are done.   If $A_0$ is nonempty, we will show that $(A,B)$ satisfies (QP).  Set $A_1=A\\setminus A_0$ to get the desired quasi-periodic decomposition $A_1\\cup A_0$ of $A$ with $m(A_0)=0$.  Reversing the roles of $A$ and $B$ in the previous paragraph, we find that either $B$ is $H$-periodic or has a quasi-periodic decomposition $B_1\\cup B_0$ with respect to $H$, where $m(B_0)=0$.  The nonextendibility of $(A,B)$ implies $B_0$ is nonempty, and also implies $A_0+B_0+H$ is a unique expression element of $A+B+H$ in $G\/H$.    Now (\\ref{0delta}) and the nonextendibility of $(A,B)$ imply $m_*(A_0+B_0)=0=m(A_0)+m(B_0)$.  \\end{proof}\n\n\\begin{lemma}\\label{QPtoQP}  Let $(A',B')$ be an irreducible nonextendible sur-critical pair satisfying \\textup{(QP)} of Theorem \\textup{\\ref{main}} with decomposition $A'=A_1'\\cup A_0'$, $B'=B_1'\\cup B_0'$, such that $m(A_0')>0$ and $m(B_0')>0$.   If $(A,B)$ is another irreducible nonextendible sur-critical pair such that $A\\sim A'$ and $B\\sim B'$, then $(A,B)$ also satisfies \\textup{(QP)}.\n\\end{lemma}\n\n\\begin{proof}   We may assume $A'\\subseteq A$ and  $B'\\subseteq B$, since we can replace $A'$ and $B'$ with $A'\\cap A$ and $B'\\cap B$ while maintaining the hypotheses of the lemma -- this follows from the irreducibility of $(A,B)$ and $(A',B')$.  Let $K$ be the quasi-period of $(A',B')$ from conclusion (QP). Assume, without loss of generality, that $A_1'\\neq \\varnothing$.  Define $A_0:=A\\cap (A_0'+K)$ and  $A_1:= A\\setminus A_0$.   We will show that $A_1\\sim A_1+K$.\n\n\\smallskip\n\nLet $a\\in A_1$. We aim to show that $a+K\\subseteq A_1'+K$.  Irreducibility of $(A,B)$ implies $a+B'\\subset_m A'+B'$.   Since $a\\notin A_0'+K$, we have $a+B'\\subset_m (A_1'+B')\\cup (A+B_1')$, so that $a+B'+K\\subset_m A'+B'$.  Then $a+K\\subset_m A'$ by the nonextendibility of $(A',B')$.   Consequently, $a+K\\subseteq A_1'+K$, and we find $A_1+K=A_1'+K$.  The similarity $A\\sim A'$ then implies $A_1\\sim A_1'$, and $A=A_1\\cup A_0$ is a quasi-periodic decomposition.  Reversing the roles of $A$ and $B$, we define $B_0=B\\cap (B_0'+K)$ and $B_1=B\\setminus B_0$.  Now it is routine to verify that $(A,B)$ also satisfies (QP) of Theorem \\ref{main}.  \\end{proof}\n\n\\subsection{Measurability and trivializing $H(A+B)$.}\\label{trivializing}\n\n\\begin{lemma}\\label{measurability}\nIf Theorem \\textup{\\ref{main}} holds under the additional assumption that $A+B$ is measurable,  then Theorem \\textup{\\ref{main}} holds in general.\n\\end{lemma}\n\n\\begin{proof}  We first list cases where we obtain the conclusion of Theorem \\ref{main} without assuming $A+B$ is measurable.  Excluding those cases, we then show how to deduce the conclusion of Theorem \\ref{main} from the special case where $A+B$ is assumed to be measurable.\n\n\\smallskip\n\nIf $(A,B)$ is extendible, then $(A,B)$ satisfies conclusion (E) of Theorem \\ref{main}.  If $(A,B)$ is nonextendible and reducible,  then $(A,B)$ satisfies conclusion (QP) of Theorem \\ref{main}, by Lemma \\ref{reduciblelemma2}.  We may therefore assume that $(A,B)$ is nonextendible and irreducible.  Let $A'\\subseteq A$ and $B'\\subseteq B$ be countable unions of compact sets with $m(A')=m(A)$ and $m(B')=m(B)$, so that $A'+B'$ is measurable.\n\n\\smallskip\n\nIrreducibility of $(A,B)$ implies $m(A'+B')=m_*(A+B)$, so $(A',B')$ satisfies the hypotheses of Theorem \\ref{main}.  If $(A',B')$ satisfies conclusion (P) or conclusion (E) of Theorem \\ref{main}, then $m(H(A'+B'))>0$ by Proposition \\ref{Ktopinequality}.  We apply Lemma \\ref{easy} and conclude that $(A,B)$ satisfies one of conclusion (P) or (QP).  If $(A',B')$ satisfies (K), so that $A'\\sim a+\\chi^{-1}(I)$ and $B'\\sim b+\\chi^{-1}(J)$, where $I$, $J\\subseteq \\mathbb T$ are intervals and $\\chi:K\\to \\mathbb T$ is a continuous surjective homomorphism from a compact open subgroup $K\\leqslant G$,  then a routine argument (\\S \\ref{intervalsdef}) shows that $A$ and $B$ must be contained in $a+\\chi^{-1}(I)$, and $b+\\chi^{-1}(J)$, respectively.   We conclude that $(A,B)$ satisfies (K).\n\n\\smallskip\n\nThe only remaining possibility is that $(A',B')$ satisfies (QP), but not (P) or (E).  Then $(A',B')$ is nonextendible, while irreducibility of $(A,B)$  implies that of $(A',B')$.  Writing $A'=A_1'\\cup A_0'$ and $B'=B_1'\\cup B_0'$ for the decomposition of $(A',B')$ in (QP) and $K$ for the corresponding subgroup, we consider different cases based on the measures of $A_0'$ and $B_0'$.\n\n \\smallskip\n\n If $m(A_0')>0$ and $m(B_0')>0$,  Lemma \\ref{QPtoQP} implies $(A,B)$ satisfies (QP).\n\n \\smallskip\n\n If $m(A_0')=0$ and $m(B_0')>0$, or vice versa, we derive a contradiction.  Under these conditions, $(A',B')$ is reducible, since $A_0'+B_0'+K$ is a unique expression element of $A'+B'+K$ in $G\/K$. Reducibility of $(A',B')$ contradicts our present assumptions.\n\n \\smallskip\n\nIf $m(A_0')=m(B_0')=0$, the irreducibility of $(A,B)$ implies $A+B\\sim A_1'+B_1'$.  Since $H(A_1'+B_1')$ has positive Haar measure, we apply Lemma \\ref{easy} with $\\tilde{A}=A_1'$, $\\tilde{B}=B_1'$ and conclude that $(A,B)$ satisfies (P) or (QP) of Theorem \\ref{main}.  \\end{proof}\n\n\n\nThe next lemma is a rephrasing of Lemma 7 of \\cite{Kneser56}; we include a proof for completeness.  Under the standing assumption that $G$ is compact, the group $H(S)$ below is compact, so the convention that Haar measure is normalized applies:  $m_{H(S)}(H(S))=1$.\n\n\n\\begin{lemma}\\label{H(S)}\nIf $S\\subseteq G$ is measurable, set $H:=H(S)$ and let\n$$\nE:=\\{x\\in G: 0<m_{H}(S-x)<1 \\text{ \\textup{or} } H\\cap (S-x) \\text{ \\textup{is not} $m_{H}$\\textup{-measurable}}\\}.\n$$ Then $m(E)=0$.\n\\end{lemma}\n\n\\begin{proof}  Consider the integrals\n\\begin{align*}\nI_1&:=\\int \\int 1_S(x-z)1_S(x) \\, dm_H(z) \\, dm(x)\\\\\nI_2&:= \\int \\int 1_S(x-z)1_S(x) \\, dm(x) \\, dm_H(z).\n\\end{align*}\nProposition \\ref{Weil}, these integrals are well-defined, and $I_1$ can be computed as\n\\begin{align*}\n\\int \\int 1_S(x-z) \\, dm_H(z)\\, 1_S(x) \\, dm(x)=\\int m_H(S-x)1_S(x)\\, dm.\n\\end{align*}\n while $I_2 =\\int m((S+z)\\cap S)\\, dm_H(z)=m(S)$.   By Fubini's theorem, $I_1=I_2$, meaning $\\int m_H(S-x) 1_S(x)\\, dm(x) =m(S)$.  Since $0\\leq m_H(S-x)\\leq 1$, the last equation implies\n \\begin{align}\\label{S-x}\n m_H(S-x)=1 \\text{ for $m$-almost every $x\\in S$}.\n \\end{align}\n Replacing $S$ with $S^c$, (\\ref{S-x}) implies $m_H(S-x)=0$ for $m$-almost every $x\\in S^{c}$.  The assertion follows. \\end{proof}\n\n\\begin{lemma}\\label{H(A+B)}\nIf $(A,B)$ is irreducible and satisfies the hypotheses of Theorem \\textup{\\ref{main}}, $A+B$ is measurable, and $m(H(A+B))=0$, then there are measurable sets $A'\\subseteq A$, $B'\\subseteq B$ such that $m(A')=m(A)$, $m(B')=m(B)$, $A'\\sim A'+H$, $B'\\sim B'+H$, and $A'+B'+H\\sim A'+B'\\sim A+B$, where $H:=H(A+B)$.\n\\end{lemma}\n\n\\begin{proof}  Let\n\\begin{align*}\nS_1:=\\{x\\in G: m_H(A-x)>0\\},\\ \\  S_2:=\\{x\\in G: m_H(B-x)>0\\}.\n\\end{align*}\nBy Proposition \\ref{Weil}, $S_1$ and $S_2$ are measurable.  Let $A''=A\\cap S_1$, $B''=B\\cap S_2$;  Proposition \\ref{Weil} implies $m(A'')=m(A)$ and $m(B'')=m(B)$. Let $A'\\subseteq A''$ and $B'\\subseteq B''$ be countable unions of compact sets having $m(A')=m(A)$ and $m(B')=m(B)$.   The irreducibility of $(A,B)$ implies  $A'+B'\\sim A''+B''\\sim A+B$, so that $H(A'+B')=H$. By Lemma \\ref{H(S)}, the set\n\\begin{align*}\nE=\\{x\\in G: 0<m_H(A+B-x)<1 \\text{ or } H\\cap(A+B-x) \\text{ is not $m_{H}$-measurable}\\}\n\\end{align*} has $m_G(E)=0$.  By Lemma \\ref{H(S)} and the definitions of $S_A$, $S_B$, and $E$,\n \\begin{align}\\label{=1}\n m_H(A'+B'+H-x)\\leq m_H(A''+B''-x)=1\n  \\end{align}\n  whenever $x\\notin E$ and $(A''+B''-x)\\cap H$ is measurable and nonempty.  Since $A'$ and $B'$ are countable unions of compact sets and $H$ is compact, $A'+B'+H$ is measurable.  We now show $A'+B'+H\\sim A'+B'$.  By Proposition \\ref{Weil} and (\\ref{=1}), then\n \\begin{align*}\n m(A'+B'+H)&=\\int m_H(A'+B'+H-x)\\, dm(x) \\\\\n &= \\int_{E^c} m_H(A'+B'+H-x)\\, dm(x)\\\\\n &\\leq \\int_{E^c} m_H(A''+B''-x)\\, dm(x)\\\\\n &= m(A''+B'').\n\\end{align*}\nThus $m(A'+B'+H)\\leq m(A'+B')$, so $A'+B'+H\\sim A'+B'$, as desired.\n\n\\smallskip\n\nTo show that $A'+H\\sim A'$, suppose otherwise to get a contradiction.  Then $m(A'+H)>m(A')$, so $m((A'+H)+B')=m(A'+B')<m(A'+H)+m(B')$, and Proposition \\ref{Ktopinequality} implies $m(H(A'+H+B'))>0$.  Since $A'+H+B'\\sim A+B$, we then have $m(H(A+B))>0$ contradicting the hypothesis.   Similarly $B'+H\\sim B'$.  \\end{proof}\n\n\n\\begin{lemma}\\label{specialtogeneral}\nIf Theorem \\textup{\\ref{main}} holds under the additional assumption that $A+B$ is measurable and $H(A+B)=\\{0\\}$ then  Theorem \\textup{\\ref{main}} holds in general.\n\\end{lemma}\n\n\n\n\n\\begin{proof}   By Lemma \\ref{measurability} we may assume $A+B$ is measurable.  As in the proof of Lemma \\ref{measurability}, we may assume that $(A,B)$ is irreducible and nonextendible.   Write $K:=H(A+B)$.  We first dispense with the case $m(K)>0$.  In that case, Lemma \\ref{easy} implies $(A,B)$ satisfies conclusion (P) or (QP) of Theorem \\ref{main}.  Now assume $m(K)=0$.  Let $A'\\subseteq A$ and  $B'\\subseteq B$ be as in the conclusion of Lemma \\ref{H(A+B)}, so that $A'+K\\sim A'$, $B'+K\\sim B'$, and $A'+B'+K\\sim A'+B'\\sim A+B$.  Observe that $G\/K$ is infinite, since $m(K)=0$.  For the remainder of the proof, write $G'$ for $G\/K$, and write $m'$ for $m_{G'}$.\n\n\\begin{claim}\nViewing $A'+B'+K$ as a subset of $G'$, the pair $(A'+K,B'+K)$ is an irreducible, nonextendible, sur-critical pair for $G'$, and $H(A'+B'+K)=\\{0_{G'}\\}$.  In particular $m'(H(A'+B'+K))=0$, since $G'$ is infinite.\n\\end{claim}\n\n\\noindent\\textit{Proof of Claim.}  We first show $H(A'+B'+K)=\\{0_{G'}\\}$ by contradiction.  Supposing otherwise, there exists $z\\neq 0_{G'}\\in G'$ such that \\begin{align}\\label{triangle1}\nm'((A'+B'+K)\\triangle(A'+B'+K+z))=0.\n   \\end{align} If $t\\in G$ is such that $t+K=z$, then $t\\notin K$, and by Proposition \\ref{Weil} and the choice of $A'$ and $B'$,\n   \\begin{align*}\nm_G((A+B)\\triangle (A+B+t)) &= m_G((A'+B'+K)\\triangle (A'+B'+K+t))\\\\\n&=m'((A'+B'+K)\\triangle(A'+B'+K+z))\\\\\n&=0 ,\n\\end{align*} contradicting the definition of $K$.\n\n\\smallskip\n\nIf $(A'+K,B'+K)$ is extendible, then $m'(H(A'+B'+K))>0$ by Proposition \\ref{Ktopinequality}, contradicting the previous paragraph.\n\n\\smallskip\n\nIf $(A'+K,B'+K)$ is reducible, there exist $A''\\subseteq A'+K$ and $B''\\subseteq B'+K$ such that $m'(A''+K)=m'(A'+K)$, $m'(B''+K)=m'(B'+K)$, and\n\\begin{align*}\nm'(A''+K+B''+K)<m'(A'+K+B'+K).\n  \\end{align*}\nSetting $C=A\\cap (A''+K)$, $D=B\\cap (B''+K)$, we apply Proposition \\ref{Weil} and find that $m(C)=m(A)$, $m(D)=m(B)$, and $m(C+D)<m(A)+m(B)$.  This implies $(A,B)$ is reducible, contradicting our assumptions.  \\hfill $\\square$\n\n\\smallskip\n\nNow we must show that if the pair $(A'+K,B'+K)$ satisfies any of the conclusions (P), (E), (K), or (QP) of Theorem \\ref{main}, then so does the pair $(A,B)$.\n\n\\smallskip\n\nIf $(A'+K,B'+K)$ satisfies (P) or (E), then $H((A'+K)+(B'+K))\\neq \\{0_{G'}\\}$, contradicting the claim above.  This case is vacuous.\n\n\\smallskip\n\nIf $(A'+K,B'+K)$ satisfies (K), let $\\chi': G'\\to \\mathbb T$ be the homomorphism of conclusion (K), and let $I$, $J\\subseteq \\mathbb T$ be the corresponding intervals. If $\\chi=\\chi'\\circ \\phi$, where $\\phi:G\\to G'$ is the quotient map, it is easy to check (\\S \\ref{intervalsdef}) that $(A,B)$ satisfies $m(A)=m(\\chi^{-1}(I))$, $m(B)=m(\\chi^{-1}(J))$, while $A\\subseteq a+\\chi^{-1}(I), $ $B\\subseteq b+\\chi^{-1}(J)$ for some $a$, $b\\in G$. We then see that $(A,B)$ satisfies (K).\n\n\\smallskip\n\nIf $(A'+K,B'+K)$ satisfies (QP) but not (P) or (E), let $A'+K=A_1'\\cup A_0'$ and   $B'+K=B_1'\\cup B_0'$ be decompositions of $A'+K$ and $B'+K$ satisfying (QP) with quasi-period $W$, such that $A_0'+B_0'$ is a unique expression element of $A'+B'+W+K$ in $G'\/W$, and $m'(A_0'+K+B_0'+K)=m'(A_0'+K)+m'(B_0'+K)$.  Regarding $B_1'$, $B_0'$, $A_1'$, and $A_0'$ as subsets of $G$, we see that $A_1'\\cup A_0'$ and $B_1'\\cup B_0'$ are decompositions of $A'+K$ and $B'+K$ satisfying (QP) with quasi-period $W+K$.\n\n\\smallskip\n\n If $m(A_0'+K)=0$ or $m(B_0'+K)=0$, irreducibility of $(A'+K,B'+K)$ implies $m(H(A'+B'+K))>0$,  so that $m(H(A+B))>0$, contradicting our present assumptions. Thus we may assume $m(A_0'+K)>0$ and $m(B_0'+K)>0$, so that Lemma \\ref{QPtoQP} applies to $(A',B')$ and $(A,B)$.   We conclude $(A,B)$ satisfies (QP).  \\end{proof}\n\n\n\n\\subsection{Quasi-periodicity and complements} Throughout this subsection we make the standing assumption that $A+B$ is measurable in addition our previous assumption that $G$ is compact.\n\n\\begin{lemma}\\label{complements}  If $(A,B)$ is an irreducible nonextendible sur-critical pair satisfying $m(A)>0$, $m(B)>0$, and $m(H(A+B))=0$, then\n\\begin{align}\\label{comp1}\n-B+ (A+ B)^c \\sim A^c,\n\\end{align}\n\\begin{align}\\label{comp2}\n-A+ (A+ B)^c \\sim B^c,\n\\end{align}\n and both $(-B,(A+B)^c)$ and $(-A,(A+B)^c)$ are irreducible nonextendible sur-critical pairs with $m(H(-B+(A+B)^c))=0$ and $m(H(-A+(A+B)^c))=0$.\n\n\\end{lemma}\n\nNote that (\\ref{comp1}) and (\\ref{comp2}) imply $-B+(A+B)^c$ and $-A+(A+B)^c$ are measurable;  the proof is complicated somewhat by the hypothetical possibility that these sets are not measurable.\n\n\\smallskip\n\nLemma \\ref{complements}  is an analogue of Lemma 2.4 of \\cite{Grynkiewicz}.  As shown in \\cite{Grynkiewicz},  the containment $-B+(A+B)^c\\subseteq A^c$ holds unconditionally.\n\n\n\n\n\n\\begin{proof}  We first establish (\\ref{comp1}).  Write $D$ for $-B+(A+B)^c$.  From the unconditional containment $D\\subseteq A^c$, we have $m_*(D)\\leq m(A^c)$.  If $m_*(D)=m(A^c)$, then (\\ref{comp1}) holds, so we assume\n\\begin{align}\\label{compcontradiction}\nm_*(D)< m(A^c)\n\\end{align}  and derive a contradiction. Now\n\\begin{align}\\label{oneminus}\\begin{split}m(-B)+m((A+B)^c)&= m(B)+1-m(A+B)\\\\\n&=m(B)+1-(m(A)+m(B))\\\\\n&=m(A^c),\\end{split}\n\\end{align}\nso (\\ref{compcontradiction}) means $m_*(-B+(A+B)^c)<m(-B)+m((A+B)^c)$, and Proposition \\ref{Ktopinequality} implies $-B+(A+B)^c$ is measurable.\n\n\\smallskip\n\nLet $E$ be the set of $y\\in A^c$ such that $(y+B)\\cap (A+B)^c\\neq \\varnothing$, so that $E= A^c \\cap D$.  Let $S=A^c\\setminus E$, so that $(S+B)\\cap (A+B)^c=\\varnothing$.  Nonextendibility of $(A,B)$ implies $m(S\\setminus A)=0$. Since $S\\subseteq A^c$, we have $m(S)=0$, which implies $m(E)=m(A^c)$.  From the definition of $E$, we have $E\\subseteq -B+(A+B)^c$.  Combining this with the unconditional containment $-B+(A+B)^c\\subseteq A^c$, we obtain (\\ref{comp1}), which is actually the desired contradiction.\n\n\\smallskip\n\nWe have proved (\\ref{comp1}), and (\\ref{comp2}) follows by symmetry.  We finish the proof of the lemma in the following sequence of claims.\n\n\\smallskip\n\n\\noindent\\textit{\\textup{1}. $m(H(-A+(A+B)^c))=0$.}  Suppose otherwise to get a contradiction.  Then $m(H(B^c))>0$ by (\\ref{comp2}), and consequently $m(H(B))>0$.   Now irreducibility of $(A,B)$ implies $m(H(A+B))>0$, contradicting our assumptions on $(A,B)$.\n\n\\smallskip\n\n\n\\noindent\\textit{\\textup{2}. $(-A,(A+B)^c)$ is a sur-critical pair.}  By a computation similar to (\\ref{oneminus}), we get $m(-A)+m((A+B)^c)=m(B^c)$,\nso $(-A,(A+B)^c)$ is a sur-critical pair by (\\ref{comp2}).\n\n\\smallskip\n\n\n\\noindent\\textit{\\textup{3}. $(-A,(A+B)^c)$ is irreducible.}  Assume otherwise. By Corollary \\ref{reduciblecor}, reducibility of $(-A,(A+B)^c)$ yields decompositions $A=A_1\\cup A_0$ and $(A+B)^c=C_1\\cup C_0$ satisfying (QP), and one of $m(A_0)=0$ or $m(C_0)=0$.  If $m(A_0)=0$, the irreducibility of $(A,B)$ implies $m(H(A+B))>0$, contradicting the hypothesis.  If $m(C_0)=0$ then $m(H((A+B)^c))>0$, and therefore $m(H(A+B))>0$, contradicting the hypothesis.\n\n\\smallskip\n\n\n\\noindent\\textit{\\textup{4}. $(-A,(A+B)^c)$ is nonextendible.}   Assume otherwise.  Then Proposition \\ref{Ktopinequality} implies $m(H(-A+(A+B)^c))>0$, contradicting the previous claim to the contrary.\n\n\\smallskip\n\nWe have shown that $(-A,(A+B)^c)$ is an irreducible nonextendible sur-critical pair satisfying $m(H(-A+(A+B)^c))=0$.  Reversing the roles of $A$ and $B$ yields the corresponding description of $(-B,(A+B)^c)$.   \\end{proof}\n\n\n\n\\begin{lemma}[cf.~\\cite{Grynkiewicz}, Lemma 5.1]\\label{QP1}  Let $(A,B)$ be a sur-critical pair with $m(A)>0$ and $m(B)>0$, and let $K$ be the subgroup of $G$ generated by $A$ \\textup{(}so $K$ is compact and open\\textup{)}.  If $A+B$ is aperiodic then $B$ is quasi-periodic with respect to $K$, or $B$ is contained in a coset of $K$.\n\\end{lemma}\n\n\\begin{proof}   Let $B= B_1\\cup \\dots \\cup B_{l}$ be a $K$-coset decomposition of $B$.  Since $A\\subseteq K$, $A+B$ is the disjoint union $\\bigcup_{i=1}^l A+B_i$.  Then\n\\begin{align}\\label{sumpartition}\nm(A+B)=\\sum_{i=1}^l m(A+B_i),\n\\end{align} while\n\\begin{align}\\label{originalpartition}\nm(A+B)=m(A)+m(B)=m(A)+\\sum_{i=1}^l m(B_i).\n\\end{align}\nEquating the right-hand sides of (\\ref{sumpartition}) and (\\ref{originalpartition}) we get\n \\begin{align}\\label{collectsplitsum}\n \\sum_{i=1}^l m(A+B_{i})-m(B_{i})=m(A).\n \\end{align}\n If $m(A+B_{i})< m(A)+m(B_{i})$ for all $i$, then the group $W:=K\\cap \\bigcap_{i=1}^l H(A+B_i)$ is open by Proposition \\ref{Ktopinequality}, and $A+B$ is periodic with period $W$, contradicting the hypothesis.  Thus there is an $i$ such that $m(A+B_{i})\\geq m(A)+m(B_{i})$, and (\\ref{collectsplitsum}) implies $m(A+B_{j})=m(B_{j})$ for $j\\neq i$. Consequently, $A\\subseteq H(B_{j})$ and $m(B_j)>0$ for all $j\\neq i$.  This implies $K\\leqslant H(B_{j})$ and $B_j\\sim B_j+K$ for all $j\\neq i$.  We then have that $B$ is quasi-periodic with respect to $K$ if $l>1$, and $B$ is contained in a coset of $K$ if $l=1$. \\end{proof}\n\n\\begin{corollary}\\label{QP1cor}  With the hypotheses of Theorem \\textup{\\ref{main}}, if $m(H(A+B))=0$ and $A$ is contained in a coset of a compact open subgroup $K_0\\leqslant G$, then $B$ has a quasi-periodic decomposition with respect to a compact open subgroup $K\\leqslant K_0$, or $B$ is contained in a coset of $K_0$.\n\\end{corollary}\n\n\\begin{proof} By translation we may assume $A\\subseteq K_0$.  The hypothesis $m(H(A+B))=0$ implies $A+B$ is aperiodic.  Now apply Lemma \\ref{QP1} and let $K$ be the subgroup generated by $A$.  \\end{proof}\n\nIn the next lemma, we assume that $A=A_1\\cup A_0$ has a quasi-periodic decomposition with respect to the maximal period $K$ of $A_1$.  This is no less general than assuming that $A$ has a quasi-periodic decomposition $A=A_1'\\cup A_0'$ with respect to some compact open subgroup $L$, since the maximal period $K$ of $A_1'$ must contain $L$, and then $A_0'$ is contained in a coset of $K$, since $A_0'$ is contained in a coset of $L$.\n\n\\begin{lemma}[cf.~\\cite{Grynkiewicz}, Lemma 5.3]\\label{QP3}  Let $(A,B)$ be an irreducible nonextendible sur-critical pair with $m(A)>0$, $m(B)>0$,  and $m(H(A+B))=0$.  Let $A=A_1\\cup A_0$ be a quasi-periodic decomposition with $A_1$ periodic with maximal period $K$.  Then $B=B_1\\cup B_0$, where $B_1\\sim B_1+K$, $B_0$ is contained in a coset of $K$, and\n\\begin{enumerate}\n\\item[(i)]  $A_0+B_0+K$ is a unique expression element of $A+B+K$ in $G\/K$,\n\n    \\smallskip\n\n\n\\item[(ii)]  $m(A_0+B_0)=m(A_0)+m(B_0)$,\n\\end{enumerate}\nso $(A,B)$ satisfies conclusion \\textup{(QP)} of Theorem \\textup{\\ref{main}}.\n\n\\end{lemma}\n\n\n\n\\begin{proof}  Observe that\n\\begin{align}\\label{A1smaller}\nm(A_1+B)<m(A+B)\n\\end{align}  since $m(H(A_1+B))\\geq m(K)>0$, while $m(H(A+B))=0$.  Inequality (\\ref{A1smaller}) implies\n\\begin{align}\\label{1step}\n\\begin{split}\nm(A_1+B)&\\leq m(A+B)-m(A_0)\\\\\n&=m(A)+m(B)-m(A_0)\\\\\n&=m(A_1)+m(B),\n\\end{split}\n\\end{align}\nwhere the first line results from $A_1+B\\sim A_1+B+K$, while $A_0$ is contained in a coset of $K$, the second line is due to $m(A+B)=m(A)+m(B)$, and the third results from $A=A_1\\cup A_0$ being a partition of $A$.  We also have\n\\begin{align}\\label{inbetween}\n0<m(A_0)<m(K),\n\\end{align} where $0<m(A_0)$ is a consequence of irreducibility of $(A,B)$ and (\\ref{A1smaller}), while $m(A_0)<m(K)$ follows from the hypothesis $m(H(A+B))=0$.\n\n\\begin{claim1} Strict inequality holds in (\\ref{1step}) and $H(A_1+B)=K$.\n\\end{claim1}\n\n\\noindent\\textit{Proof of Claim \\textup{1}.}  Assume $m(A_1+B)=m(A+B)-m(A_0)$ to get a contradiction. Then $m(A+B)=m(A_1+B)+m(A_0)$, so\n\\begin{align}\\label{A0plusb0}\nA+B\\sim (A_1+B)\\cup (A_0+b_0).\n\\end{align}\n for some $b_0\\in B$.   Then (\\ref{inbetween}) and the similarities $A_1\\sim A_1+K$, $A_1+B\\sim A_1+B+K$ imply that\n\\begin{align*}\nC:=\\{b\\in B: (A+b)\\setminus (A_1+B) \\neq \\varnothing\\}\n \\end{align*}\n is contained in a coset of $H(A_0)$.  If $m(H(A_0))=0$, then $m(C)=0$ and $(A,B)$ is reducible, contrary to the hypothesis.  If $m(H(A_0))>0$, (\\ref{A0plusb0}) implies $K\\cap H(A_0)\\subseteq H(A+B)$, so $m(H(A+B))>0$, again contradicting our hypothesis.\n\n\\smallskip\n\n\nStrict inequality in (\\ref{1step}) implies $m(A_1+B)<m(A_1)+m(B)$. We now show  $H(A_1+B)=K$. Write $H$ for $H(A_1+B).$  Applying Proposition \\ref{Ktopinequality} to $(A_1,B)$,  we have $A_1+B=A_1+B+H$.  Then $A_1+B+H\\subseteq A+B$, so the nonextendibility of $(A,B)$ implies\n\\begin{align}\\label{A1plusH}\nA_1+H\\subset_m A.\n\\end{align}\nTo see that $H=K$, first note that $A_1\\sim A_1+K$, so $A_1+B+K\\sim A_1+B$, and we find $K\\leqslant H$ by the definition of $H$.  Now $A_1+H$ is disjoint from $A_0$, since otherwise $A_1+H=A+H$, and then $A\\sim A+H$, contradicting $m(H(A+B))=0$.  Thus (\\ref{A1plusH}) implies $A\\setminus A_0 \\sim A_1+H$.  The maximality of $K$ then implies $H\\leqslant K$.  \\hfill $\\square$\n\n\\smallskip\n\nLet $D:=\\{b\\in B: A_0+b\\subseteq (A_1+B)^c\\}$, and let $D= \\bigcup_{i=1}^l D_i$ be a $K$-coset decomposition of $D$.\n\n\\begin{claim2}  $m(A_0+D_i)\\geq m(A_0)+m(D_i)$ for some $i$.\n\\end{claim2}\n\n\\noindent\\textit{Proof of Claim \\textup{2}.}  We have $A+B= (A_1+B) \\cup \\bigcup_{i=1}^l A_0+D_i$.  Assuming Claim 2 fails, Proposition \\ref{Ktopinequality} implies $W:=\\bigcap_{i=1}^l H(A_0+D_i)$ is open.  Then  the containment $(K\\cap W)\\leqslant H(A+B)$ contradicts $m(H(A+B))=0$.  \\hfill $\\square$\n\n\\smallskip\n\nLet $B_0$ be one of the $D_i$ satisfying $m(A_0+D_i)\\geq m(A_0)+m(D_i)$, and let  $B_1=B\\setminus B_0$. We aim to show that $B_1\\sim B_1+K$.  Using  Claim 1, we apply Lemma \\ref{overspill} to $(A_1,B)$ and find\n\\begin{align}\\label{overspilled}\nm(B_0)+m(A_1+B)\\geq m(A_1)+m(B).\n\\end{align}\nThen\n\\begin{align}\\label{crit}\n\\begin{aligned} m(A+B)&\\geq m(A_0+B_0) + m(A_1+B)  \\\\\n&\\geq m(A_0)+m(B_0)+m(A_1+B) \\qquad && \\text{(by Claim 2)}\\\\\n&\\geq m(A_0)+m(A_1)+m(B) \\qquad && \\text{(by (\\ref{overspilled}))}\\\\\n&= m(A)+m(B).\n\\end{aligned}\n\\end{align}\nSince $m(A+B)=m(A)+m(B)$, the inequalities in (\\ref{crit}) are actually equalities, and\n\\begin{align}\\label{djunion}\nA+B\\sim (A_0+B_0)\\cup (A_1+B),\n\\end{align}\n which is a disjoint union.  The hypothesis $m(H(A+B))=0$ and (\\ref{djunion}) imply  $A_0+B_0+K$ is a unique expression element of $A+B+K$ in $G\/K$.  Now   (\\ref{djunion}) implies $B_1+A\\subseteq A_1+B+K\\sim A_1+B$.  The nonextendibility of $(A,B)$ then implies $B_1+K\\sim B_1$.  This concludes the proof of (i). Part (ii) of the lemma follows from the first and second lines  of (\\ref{crit}), as the inequalities are not strict.  \\end{proof}\n\n\\begin{lemma}\\label{QPsumtosummand}  Let $(A,B)$ be an irreducible nonextendible sur-critical pair for $G$ such that $m(A)>0$, $m(B)>0$, and $m(H(A+B))=0$.  If $A+B\\sim D$, where $D$ is quasi-periodic with respect to a compact open subgroup $K\\leqslant G$, then $(A,B)$ satisfies conclusion \\textup{(QP)} of Theorem \\textup{\\ref{main}}. \\end{lemma}\n\n\n\\begin{proof}  Write $C$ for $(A+B)^c$. The assumption that $A+B\\sim D$, where $D$ is quasi-periodic with quasi-period $K$, implies $C\\sim C'$ where $C'\\subseteq C$ is quasi-periodic with respect to $K$, or $C'$ is contained in a coset of $K$.  Lemma \\ref{complements} implies $(-B,C)$ is an irreducible nonextendible sur-critical pair with $m(H(-B+C))=0$, so $(-B,C')$ is such a pair with $m(H(-B+C'))=0$.  If $C'$ is contained in a coset of $K$, then Corollary \\ref{QP1cor} implies $-B$, and therefore $B$, is quasi-periodic with respect to $K$, or is contained in a coset of $K$.  If $B$ is quasi-periodic with respect to $K$, then Lemma \\ref{QP3} implies $(A,B)$ satisfies (QP).  If $B$ is contained in a coset of $K$, then Corollary \\ref{QP1cor} implies $A$ is contained in a coset of $K$ or $A$ is quasi-periodic with respect to $K$.  Since $A+B\\sim D$, which is quasi-periodic with respect to $K$, the sets $A$ and $B$ cannot both be contained in a coset of $K$, so $A$ is quasi-periodic with respect to $K$.  Again we apply Lemma \\ref{QP3} and find $(A,B)$ satisfies (QP).\n\n\\smallskip\n\nIf $C'$ is not contained in a coset of $K$, then $C'$ is quasi-periodic with respect to $K$, and Lemma \\ref{QP3} implies $(-B,C')$ satisfies conclusion (QP) of Theorem \\ref{main}.  Then $-B$ is contained in a coset of $K$ or $-B$ is quasi-periodic with respect to $K$, and as above we conclude that $(A,B)$ satisfies (QP).  \\end{proof}\n\n\n\\subsection{The $e$-transform}\\label{etsection}  For $A$, $B\\subseteq G$ and $e\\in G$, form the pair $(A_e,B_e)$, where\n\\begin{align*}\nA_e:= A\\cup (B+e),\\  B_e:=(A-e)\\cap B.\n\\end{align*}\nThis is the Dyson $e$-transform, whose properties are well-documented.  In particular,\n\\begin{align}\\label{et1}\nA_e+B_e&\\subseteq A+B, \\\\\n\\label{et2}m(A_e)+m(B_e)&=m(A)+m(B),\n\\end{align}\nwhenever $B_e$ is nonempty.  See \\cite{Kneser56}, \\cite{Nathansoninverse}, or \\cite{TaoVu} for further exposition.\n\n\\smallskip\n\nThe proof of Proposition \\ref{Kconnected} in \\cite{Kneser56} relies on a sequence of pairs $(A^{(n)}, B^{(n)})$ successively derived by $e$-transform, where $\\lim_{n\\to \\infty} m(B^{(n)})=0$ and $m(B^{(n)})>0$ for all $n$.  The next lemma facilitates the same construction for some pairs $(A,B)$ where the ambient group is disconnected.\n\n\\begin{lemma}\\label{transformestimate}\nIf $A$, $B\\subseteq G$ have positive Haar measure, $A+B$ is measurable, $m(A)+m(B)\\leq 1$, and $A+B$ is aperiodic, then there is an $e\\in G$ such that $B_e:=(A-e)\\cap B\\neq \\varnothing$ and $m(B_e)\\leq (1-m(B))m(B)$.\n\\end{lemma}\n\nIn particular, the conclusion holds when $m(H(A+B))=0$. The hypothesis ``$A+B$ is aperiodic\" cannot be omitted; the conclusion fails when $B$ is a coset of a compact open subgroup $K\\leqslant G$ and $A$ is a union of cosets of $K$.\n\n\\begin{proof}  Write $f(z)=m((A-z)\\cap B)$, so that $f:G\\to [0,1]$ is a continuous function and $\\int f\\, dm=m(A)m(B)$ (\\cite{Kneser56}, Lemma 1). Note that $(A-z)\\cap B$ is nonempty exactly when $z\\in A-B$. Consider $S:=\\{z:f(z)>0\\}$, which is contained in $A-B$.  If $S\\neq A-B$, then there is an $e\\in A-B$ with $f(e)=0$, meaning $B_e\\neq \\varnothing$ and $m(B_e)=0$, so we are done.  If $S=A-B$, consider the average\n\\begin{align}\\label{Saverage}\n\\frac{1}{m(S)}\\int_S f dm =\\frac{1}{m(A-B)} m(A)m(B).\n\\end{align}\n We estimate $m(A-B)$.  If $m(A-B)<m(A)+m(B)$, then  Corollary \\ref{differencetosum} implies $A+B$ is periodic, contradicting the hypotheses.   Now $m(A-B)\\geq m(A)+m(B)$, and equation (\\ref{Saverage}) implies\n \\begin{align*}\n\\frac{1}{m(S)}\\int_S f dm&\\leq \\frac{m(A)}{m(A)+m(B)}m(B)\\\\\n& = \\Bigl(1-\\frac{m(B)}{m(A)+m(B)}\\Bigr)m(B)\\\\\n&\\leq (1-m(B))m(B),\n \\end{align*}\nwhere we used $m(A)+m(B)\\leq 1$ in the last line.  So there is a $e\\in G$ such that $0<f(e)<(1-m(B))m(B)$.  Now $0<m(B_e)\\leq (1-m(B))m(B)$ by the definition of $f$.   \\end{proof}\n\n\n\\subsection{The key lemma}     The proof of next lemma follows part of  the  argument of \\cite{Grynkiewicz}, \\S 6, Subcase 1.\n\n\\begin{lemma}\\label{transformcase1}\nSuppose $(A,B)$ is an irreducible, nonextendible, sur-critical pair having $m(A)>0$, $m(B)>0$, $m(H(A+B))=0$, and there exists $e\\in G$ such that $m_*(A_e+B_e)<m(A)+m(B)$.  Then $(A,B)$ satisfies conclusion \\textup{(QP)} of Theorem \\textup{\\ref{main}} with $K=H(A_e+B_e)$.\n\\end{lemma}\n\n\\begin{proof}  Without loss of generality assume $e=0$,\\footnote{Write $C=A-e$, $D=B$, so applying the $e$-transform to $(C,D)$ with $e=0$ yields \\\\ $(C',D')=((A-e)\\cup B, (A-e)\\cap B)=(A_e-e,B_e)$.} so $A_e=A\\cup B$, and $B_e=A\\cap B$.  Since $m(A_e)+m(B_e)=m(A)+m(B)$, we apply Proposition \\ref{Ktopinequality} to $(A_e,B_e)$ and find\n\\begin{align}\\label{AeBeH}\nA_e+B_e=A_e+B_e+H,\n\\end{align} while $H=H(A_e+B_e)$ is compact and open.\n\\begin{claim1}\\label{C1} $(A\\cap B)+H\\sim A\\cap B$.\n\\end{claim1}\n\n\\noindent\\textit{Proof of Claim \\textup{1}.}  If $b\\in A\\cap B$, then $A+b\\subseteq A_e+B_e=A_e+B_e+H$, so $A+b+H\\subseteq A+B$.  Nonextendibility of $(A,B)$ then implies $b+H\\subset_m B$.  Likewise, if $a\\in A\\cap B$, then $a+B\\subseteq A_e+B_e$, so $a+H\\subset_m A$.   The claim follows.  \\hfill $\\square$\n\n\\smallskip\n\nPartition $A$ as $A=A_0\\cup A_1\\cup A_2$, where\n\\begin{align*}\nA_0&:=(A\\cap(B+H))\\setminus [(A\\cap B)+H]\\\\\n A_1&:=A\\cap [(A\\cap B)+H]\\\\\nA_2&:=A\\setminus (A_0\\cup A_1).\n\\end{align*}\nPartition $B$ similarly, as $B_0:=(B\\cap (A+H))\\setminus (A\\cap B)$, $B_1:=B\\cap [(A\\cap B)+H]$, and $B_2:=B\\setminus (B_0\\cup B_1)$.\n\n\n\\begin{claim2}\\label{C2} $A_0+H= B_0+H$.\n\\end{claim2}\n\n\\noindent\\textit{Proof of Claim \\textup{2}.}  The definitions of $A_0$ and $B_0$ imply $A+H$ contains $B_0$, while $A_1+H$ and $A_2+H$ are disjoint from $B_0$. Thus $B_0\\subseteq A_0+H$.   Similarly, $A_0\\subseteq B_0+H$. \\hfill $\\square$\n\n\\smallskip\n\nWe now consider three cases.\n\n\\subsubsection*{Case \\textup{1}. There exists $b\\in B_2$ such that $(A+b)\\not \\subseteq (A_e+B_e)$.}    In this case (\\ref{AeBeH}) implies that  there is a $b\\in B_2$ and an $a_0\\in A$ such that $a_0+( (b+H)\\cap B)$ is disjoint from $A_e+B_e$.  Fix such $b$, and set $B_0=(b+H)\\cap B$.\nNote that $B_0=A_e\\cap (b+H)$, by the definition of $B_2$.  Then Lemma \\ref{overspill}, applied to $(A_e,B_e)$, implies\n\\begin{align}\\label{beenot}\nm(A_e+B_e)+m(B_0)\\geq m(A_e)+m(B_e)=m(A+B).\n\\end{align}\nThe containment $A_e+B_e\\subseteq A+B$,  (\\ref{beenot}), and  the definition of $B_0$ imply\n\\begin{align}\\label{almost}\nA+B\\sim(A_e+B_e)\\cup (a_0+B_0).\n\\end{align}\nSince $A_e+B_e$ is $H$-periodic, the set $(A_e+B_e)\\cup (a_0+B_0)$ is quasi-periodic.  By (\\ref{almost}) and  Lemma \\ref{QPsumtosummand}, $(A,B)$ satisfies conclusion (QP) of Theorem \\ref{main} with $K=H$.  We are done with Case 1.\n\n\\subsubsection*{Case \\textup{2}. There exists $a\\in A_2$ such that $(a+B)\\not \\subseteq (A_e+B_e)$.}  We argue as in Case 1, reversing the roles of $A$ and $B$.\n\n\\subsubsection*{Case \\textup{3.}  For all $a\\in A_2$ and $b\\in B_2$, $A+b\\subseteq A_e+B_e$ and $a+B\\subseteq A_e+B_e$}  In this case, nonextendibility of  $(A,B)$ and the containment $A_e+B_e+H\\subseteq A+B$ imply $b+H\\subset_m B$ for all $b\\in B_2$, and $a+H\\subset_m A$ for all $a\\in A_2$.\n\n\\smallskip\n\nBy Claims 1 and 2 and the definitions of the $A_i$, we now have $A=A_0\\cup A_1\\cup A_2$, where $A_1+H\\sim A_1$, $A_2+H\\sim A_2$, and $A_0\\cap B=\\varnothing$.  Similarly, $B=B_0\\cup B_1\\cup B_2$, where $B_1+H=B_1$, $B_2+H=B_2$, and $B_0\\cap A=\\varnothing$.  Also, $A_0+H=B_0+H$. Figure \\ref{fig:QPfigure} depicts this scenario: the vertical rectangles are cosets of $H$, while contrary to the conclusion of the lemma, $A_0$ and $B_0$ are shown as occupying two cosets of $H$.  Our immediate goal is to show that $A+B$ has a quasi-periodic decomposition with quasi-period $H$. Let\n\\begin{align*}\nA_0=\\bigcup_{i=1}^l A_{0,i},\\  B_0=\\bigcup_{i=1}^l B_{0,i}\n\\end{align*}\nbe $H$-coset decompositions of $A_0$ and $B_0$, with $A_{0,i}+H=B_{0,i}+H=:H_i$.    Then $A_e\\cap H_i=(A_{0,i}\\cup B_{0,i})$, which is a disjoint union, for each $i$.  Lemma \\ref{overspill}, applied to $(A_e,B_e)$, implies\n\\begin{align}\\label{e-coset}\nm(A_{0,i})+m(B_{0,i}) + m(A_e+B_e) \\geq m(A_e)+m(B_e) =m(A+B),\n\\end{align}\nfor all $i$.\n\n\n\\begin{figure}\n\\includegraphics[scale=.4]{QPfigure1.pdf}\n\\caption{}\n\\label{fig:QPfigure}\n\\end{figure}\n\n\n\\begin{claim3}  There is a pair $(i,j)$ such that $H_i+H_j$ is disjoint from $A_e+B_e$, and $m(A_{0,j}+B_{0,i})\\geq m(A_{0,j})+m(B_{0,i})$ or $m(A_{0,i}+B_{0,j})\\geq m(A_{0,i})+m(B_{0,j})$.\n\\end{claim3}\n\n\\noindent\\textit{Proof of Claim \\textup{3}.}  Assuming otherwise, Proposition \\ref{Ktopinequality} implies the group $W:=\\bigcap_{(i,j)} H(A_{0,j}+B_{0,i})\\cap H(A_{0,i}+B_{0,j})$ is open, where the intersection is over all pairs $(i,j)$ such that $H_i+H_j$ is disjoint from $A_e+B_e$. Then the containment $(W\\cap H)\\leqslant H(A+B)$ contradicts $m(H(A+B))=0$.    \\hfill $\\square$\n\n\\smallskip\n\nAssume, to get a contradiction, that $A+B$ is not quasi-periodic with quasi-period $H$. Let $(i,j)$ be a pair satisfying the conclusion of Claim 3, and let $(i',j')$ be another pair such that the sets\n\\begin{align*}\n(A_{0,i}+B_{0,j})\\cup (A_{0,j}+B_{0,i}) \\text{ and } (A_{0,i'}+B_{0,j'})\\cup (A_{0,j'}+B_{0,i'})\n\\end{align*}\n occupy distinct cosets of $H$ and are disjoint from $A_e+B_e$.  Writing\n\\begin{align*}\nM&:= \\max\\{m(A_{0,i}),m(A_{0,j}),m(B_{0,i}),m(B_{0,j})\\}\\\\\nM'&:=\\max\\{m(A_{0,i'}),m(A_{0,j'}),m(B_{0,i'}),m(B_{0,j'})\\},\n\\end{align*}\nand $S$ for $\\{i,i',j,j'\\}$, we estimate $m(A+B)$ from below by\n\\begin{align}\\label{e-estimate}\nm(A+B)\\geq M+M'+m(A_e+B_e).\n\\end{align}\nCombining (\\ref{e-estimate}) with (\\ref{e-coset}) yields\n\\begin{align}\\label{comparison}\nM+M'\\leq m(A_{0,s})+m(B_{0,s}) \\text{ for all } s\\in S,\n\\end{align}\n so in particular, $M+M' \\leq \\min_{s\\in S} \\{m(A_{0,s})+m(B_{0,s})\\}$.  Assume, without loss of generality, that $M\\leq M'$.  Since\n\\begin{align*}\n\\min_{s\\in S} \\{m(A_{0,s})+m(B_{0,s})\\} \\leq \\min_{s\\in S} \\min\\{m(A_{0,s}),m(B_{0,s})\\}+M',\n\\end{align*} (\\ref{comparison}) implies\n\\begin{align}\\label{minsmin}\nM=\\min_{s\\in S} \\min\\{m(A_{0,s}),m(B_{0,s})\\}.\n\\end{align}\nFrom the definition of $M$, (\\ref{minsmin}) implies\n\\begin{align}\\label{mmmm}\n\\begin{aligned}\nM&=m(A_{0,i})=m(A_{0,j})=m(B_{0,i})=m(B_{0,j}).\n\\end{aligned}\n\\end{align}\nCombining (\\ref{mmmm}) with (\\ref{comparison}) we conclude $M=M'$, and\n\\begin{align}\\label{allsame}\nm(A_{0,s})=m(A_{0,t})=m(B_{0,s})=m(B_{0,t}) \\text{ for all } s,t\\in S.\n\\end{align}\nClaim 3 and (\\ref{allsame}) imply $m((A_{0,i}+B_{0,j})\\cup (A_{0,j}+B_{0,i}))\\geq m(A_{0,i})+m(B_{0,i})$.  Again estimating $m(A+B)$, we have\n\\begin{align}\\label{ijij}\n\\begin{aligned}\nm(A+B)&\\geq m(A_e+B_e)+m((A_{0,i}+B_{0,j})\\cup (A_{0,j}+B_{0,i}))\\\\\n&\\qquad +m((A_{0,i'}+B_{0,j'})\\cup (A_{0,j'}+B_{0,i'}))\\\\\n& > m(A_e+B_e)+m(A_{0,i})+m(B_{0,i})\\\\\n&\\geq m(A)+m(B),\n\\end{aligned}\n\\end{align}\nwhere the last line follows from (\\ref{e-coset}). The strict inequality in (\\ref{ijij}) is the desired contradiction.  This concludes the analysis of Case 3 and the proof of the lemma. \\end{proof}\n\n\n\\subsection{Intersections and measure}\n\n\n\n\\begin{lemma}\\label{intersections}  If $A$, $B\\subseteq G$ have $m(A)>0$ and $m(B)>0$, there are sets $A'\\subseteq A$, $B'\\subseteq B$ with $m(A')=m(A)$ and $m(B')=m(B)$, such that $m((A'-t)\\cap B')>0$ whenever $(A'-t)\\cap B'$ is nonempty.\n\\end{lemma}\n\n\n\\begin{proof}  By Theorem A of \\cite{Mueller65}, there is a sequence of neighborhoods $U_n$ of the identity $0\\in G$ such that for $m$-almost all $a\\in A$ and $b\\in B$,\n\\begin{align}\\label{PoD}\n\\lim_{n\\to \\infty} \\frac{m(A\\cap (U_n+a) )}{m(U_n)}=1 \\text{ and }  \\lim_{n\\to \\infty} \\frac{m(B\\cap (U_n+b) )}{m(U_n)} =1.\n\\end{align}\nLet $A'\\subseteq A$ and  $B'\\subseteq B$ be the sets of points in $A$ and $B$, respectively, satisfying (\\ref{PoD}), so that $m(A')=m(A)$ and $m(B')=m(B)$.  If  $t\\in G$ and $(A'-t)\\cap B'$ is nonempty, assume without loss of generality that $0\\in (A'-t)\\cap B'$.  Now (\\ref{PoD}) implies that for $n$ sufficiently large, \\begin{align*}\nm((A'-t)\\cap U_n)>m(U_n)\/2 \\text{ and } m(B'\\cap U_n)>m(U_n)\/2,\n\\end{align*}\nso $m((A'-t)\\cap B'\\cap U_n)>0$.  Thus $m((A'-t)\\cap B')>0$. \\end{proof}\n\n\\begin{remark}  Theorem A of \\cite{Mueller65} is proved in \\cite{Mueller62}.  We outline a way to find neighborhoods $U_n$ satisfying (\\ref{PoD}):  by Proposition 2.42 of \\cite{FollandACAHA} and the ensuing remarks, there is a sequence of neighborhoods $V_n$ of the identity $0\\in G$ such that the functions\n\\begin{align*}\nf_n(z):= m(A\\cap (V_n+z))\/m(V_n),\\  g_n(z):=m(B\\cap (V_n+z))\/m(V_n)\n\\end{align*}\nconverge in $L^1(m)$ to the functions $1_A$ and $1_B$, respectively.  One may then choose a subsequence $\\{U_n\\}_{n\\in \\mathbb N}$ of $\\{V_n\\}_{n\\in \\mathbb N}$ so that the corresponding subsequences of $f_n$ and $g_n$ converge $m$-almost everywhere.  These $U_n$ will satisfy (\\ref{PoD}).  \\hfill $\\blacksquare$ \\end{remark}\n\n\\section{The main argument}\\label{main argument}\n\n\\subsection{The sequence of $e$-transforms}\\label{transforms}  As in \\cite{Kneser56}, we construct a sequence of pairs by successively applying the $e$-transform (\\S \\ref{etsection}).\n\n\\begin{lemma}\\label{esequence}\nLet $(A,B)$ be an irreducible, nonextendible pair satisfying the hypotheses of Theorem \\textup{\\ref{main}} such that $m(H(A+B))=0$.  Then at least one of the following holds.\n\\begin{enumerate}\n\\item[(i)]  There is a sequence of pairs $(A^{(n)}, B^{(n)})$, $n=0,1,2,\\dots$ such that $A^{(0)}=A$, $B^{(0)}=B$, and for all $n\\geq 1$,\n\n\\smallskip\n\n    \\begin{enumerate}\n    \\item[(i.1)]  the pair $(A^{(n)},B^{(n)})$ is derived from $(A^{(n-1)},B^{(n-1)})$ by $e$-transform,\n\n    \\item[(i.2)]  $A^{(n)}+B^{(n)}\\sim A+B$, and\n\n    \\item[(i.3)] $0<m(B^{(n)})\\leq (1-m(B^{(n-1)}))m(B^{(n-1)})$.\n    \\end{enumerate}\n\n    \\smallskip\n\n\\item[(ii)]  $(A,B)$ satisfies conclusion \\textup{(QP)} of Theorem \\textup{\\ref{main}}.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}  Suppose $(A^{(n)},B^{(n)})$ satisfies the description in (i) for $n=0,\\dots, k$.  We will show that there is a pair $(A^{(k+1)},B^{(k+1)})$ satisfying (i.1)-(i.3) with $n=k+1$, or  (ii) holds. Note that $(A^{(k)},B^{(k)})$ is not extendible, as  (i.2) implies $H(A^{(k)}+B^{(k)})=\\{0\\}$.\n\n\\smallskip\n\nIf $(A^{(k)},B^{(k)})$ is reducible, Corollary \\ref{reduciblecor} says that $(A^{(k)},B^{(k)})$ satisfies (QP) of Theorem \\ref{main}.  Since $A+B\\sim A^{(k)}+B^{(k)}$, Lemma \\ref{QPsumtosummand} implies $(A,B)$ satisfies conclusion (QP) of Theorem \\ref{main}, and we have (ii).\n\n\\smallskip\n\nIf $(A^{(k)}, B^{(k)})$ is irreducible, apply Lemma \\ref{intersections} to find $C\\subseteq A^{(k)}$ and $D\\subseteq B^{(k)}$ such that $m(C)+m(D)=m(A^{(k)})+m(B^{(k)})$ and $(C-e)\\cap D\\neq \\varnothing$ implies $m((C-e)\\cap D)>0$.  By Lemma \\ref{transformestimate}, there is an $e\\in G$ such that\n\\begin{align}\\label{Destimate}\n0<m(D_e)\\leq (1-m(D))m(D)\n \\end{align}\nTake $B^{(k+1)}=(B^{(k)})_e$ and  $A^{(k+1)}=(B^{(k)})_e$.  If\n\\begin{align*}\nm(A^{(k+1)}+B^{(k+1)})=m(A)+m(B)\n\\end{align*} we have (i.1)-(i.3) with $n=k+1$, since $B^{(k+1)}\\sim D_e$.  Otherwise,\n\\begin{align*}\nm(A^{(k+1)}+B^{(k+1)})<m(A^{(k+1)})+m(B^{(k+1)})\n\\end{align*} and Lemma \\ref{transformcase1} implies $A^{(k)}+B^{(k)}$ ($\\sim A+B$) is quasi-periodic.   Now  Lemma \\ref{QPsumtosummand} implies $(A,B)$ satisfies conclusion (QP) of Theorem \\ref{main}.  We conclude (ii).\n\n\\smallskip\n\nIf the construction of $(A^{(n)},B^{(n)})$ yields conclusion (ii) for some $n$, we are done.  Otherwise, we have (i).   \\end{proof}\n\nWe now prove a special case of Theorem \\ref{main}.\n\n\\begin{proposition}\\label{specialprop}  Theorem \\textup{\\ref{main}} holds under the additional assumptions that $(A,B)$ is irreducible and nonextendible, $A+B$ is measurable, and $H(A+B)=\\{0\\}$.\n\\end{proposition}\n\n\\begin{proof}  If $G$ is finite then conclusion (P) of Theorem \\ref{main} holds.  If $G$ is infinite, the hypotheses of Lemma \\ref{esequence} are satisfied. If (ii) holds in the conclusion of Lemma \\ref{esequence} then we are done.  Now we must analyze the case where the conclusion (i) holds in Lemma \\ref{esequence}.  Let $(A^{(n)},B^{(n)})$ be a sequence of pairs satisfying (i) of Lemma \\ref{esequence}.  We argue as in the proof of Proposition \\ref{Kconnected} in \\cite{Kneser56}.\n\n\\begin{claim} \\label{nhoods} For every neighborhood $U$ of $0\\in G$, there exists $n$ with $B^{(n)}-B^{(n)}\\subseteq U$.\n\\end{claim}\n\n\\noindent\\textit{Proof of Claim.}  Suppose not.  Then, since $B^{(n+1)}\\subseteq B^{(n)}$ for all $n$, there exists a neighborhood $U$ of $0$ and elements $x_n$, $y_n\\in B^{(n)}$ such that $x_n-y_n\\notin U$ for all $n\\in \\mathbb N$. Since $G$ is compact, the sequences $x_n$, $y_n$ have limit points $x$, $y$, with $x-y\\neq 0$.   Our choice of $x_n$ and $y_n$ implies\n\\begin{align}\\label{xnyn}\nA^{(n)}\\subseteq (A^{(n)}+B^{(n)}-y_n)\\cap (A^{(n)}+B^{(n)}-x_n).\n\\end{align}  Condition (i.3) in Lemma \\ref{esequence} implies $\\lim_{n\\to \\infty} m(B^{(n)})=0$, so (\\ref{et2}) implies $\\lim_{n\\to \\infty} m(A^{(n)})=m(A)+m(B)$.  Then\n\\begin{align*}\nm((A+B)\\cap (A+B+y&-x))\\geq \\liminf_{n\\to \\infty} m((A+B)\\cap (A+B+y_n-x_n))\\\\\n&= \\liminf_{n\\to \\infty} m((A^{(n)}+B^{(n)}-y_n)\\cap (A^{(n)}+B^{(n)}-x_n))\\\\\n&\\geq \\lim_{n\\to \\infty} m(A^{(n)})\\\\\n&= m(A)+m(B)\\\\\n&= m(A+B),\n\\end{align*}\nwhere the second line uses translation-invariance of $m$ and the third uses (\\ref{xnyn}). Now $A+B+(y-x)\\sim A+B$, contradicting the assumption $H(A+B)=\\{0\\}$.  \\hfill $\\square$\n\n\\smallskip\n\nWe now consider two separate cases.    Corollary 7.9 of \\cite{HewittandRoss1} implies these two cases cover all possible compact abelian groups $G$.\n\n\\subsubsection*{Case \\textup{1}. $G$ has open subgroups of arbitrarily small measure}\n\nBy the Claim, there is a compact open subgroup $K$ and $n\\in \\mathbb N$ such that $B^{(n)}-B^{(n)}\\subseteq K$, while $m(A^{(n)})>m(K)$.   Then $B^{(n)}$ is contained in a coset of $K$, while $A^{(n)}$ is not. Then Corollary \\ref{QP1cor} implies $A^{(n)}$ is quasi-periodic, and Lemma \\ref{QP3} implies $A^{(n)}+B^{(n)}$ is quasi-periodic.  Since $A+B\\sim A^{(n)}+B^{(n)}$,  Lemma \\ref{QPsumtosummand} implies $(A,B)$ satisfies (QP) of Theorem \\ref{main}.\n\n\\subsubsection*{Case \\textup{2}. $G$ has an open connected subgroup.}\n\nLet $G_0$ be an open connected subgroup of $G$.  By the Claim, we can choose $n$ sufficiently large that $B^{(n)}$ is contained in a  coset of $G_0$.  If $A^{(n)}$ is contained in a coset of $G_0$, then by the construction of $A^{(n)}$, $A$ and $B$ are each contained in a coset of $G_0$.  Then Proposition \\ref{Kconnected} implies the existence of a continuous surjective homomorphism $\\chi:G_0\\to \\mathbb T$ and intervals $I,J\\subseteq \\mathbb T$ such that $A\\subseteq a+\\chi^{-1}(I)$, $B\\subseteq b+\\chi^{-1}(J)$, for some $a$, $b\\in G$, such that $m(A)=m(\\chi^{-1}(I))$ and $m(B)=m(\\chi^{-1}(J))$.  From this we conclude (K) in Theorem \\ref{main}.\nIf $A^{(n)}$ is not contained in a coset of $G_0$, we argue as in Case 1 and find that $(A,B)$ satisfies (QP).  \\end{proof}\n\n\\subsection{Proof of Theorem \\textup{\\ref{main}}.}\\label{proof}  If $(A,B)$ is extendible, we have conclusion (E).  If $(A,B)$ is nonextendible and reducible, Lemma \\ref{reduciblelemma2} implies $(A,B)$ satisfies conclusion (QP).  If $(A,B)$ is irreducible and nonextendible,  Lemma \\ref{specialtogeneral} allows us to assume that $A+B$ is measurable and $H(A+B)=\\{0\\}$.  In that case, Proposition \\ref{specialprop} implies the conclusion of Theorem \\ref{main}.  \\hfill $\\square$\n\n\\section{When $G$ is not compact}\\label{lc}  A structure theorem for locally compact abelian groups (\\cite{HewittandRoss1}, Theorem 24.30) says that such a group $G$ is isomorphic (as a topological group) to $\\mathbb R^n\\times L$, where $L$ is a locally compact group containing a compact open subgroup, and $n$ is a nonnegative integer.  The study of the equation \\begin{align}\\label{equation}\nm_*(A+B)=m(A)+m(B)\n\\end{align}\ndepends heavily on $n$.  Theorem 5 of \\cite{Kneser56} says that when $n\\geq 1$,\n\\begin{align}\\label{BM}\nm_*(A+B)^{1\/n} \\geq m(A)^{1\/n}+m(B)^{1\/n}\n \\end{align}\n for measurable subsets $A$ and $B$ of $G$, and Theorem 6 of \\cite{Kneser56} classifies the pairs for which equality holds in (\\ref{BM}), generalizing Theorem 2 of \\cite{HenstockMacbeath} to the case where $L$ is nontrivial.  When $n>1$ and $A$ and $B$ both have positive measure, (\\ref{BM}) implies $m_*(A+B)>m(A)+m(B)$, so there are no such pairs satisfying (\\ref{equation}) when $n>1$.  When $n=1$, Theorem 6 of \\cite{Kneser56} says that if $A$ and $B$ have positive measure then (\\ref{equation}) holds if and only if there are closed intervals $I$, $J\\subseteq \\mathbb R$, a compact open subgroup $K\\leqslant L$, and $a$, $b\\in G$ such that $A\\subseteq a+(I\\times K)$, $B\\subseteq b+(J\\times K)$, and $m(A)=m(I\\times K)$, $m(B)=m(J\\times K)$.\n\n\\smallskip\n\nOur study of equation (\\ref{equation}) is thus reduced to groups $G$ having a compact open subgroup.  We assume $A$ and $B$ have finite measure, since otherwise (\\ref{equation}) is satisfied.  One can easily check that $A$ and $B$ have compact closures under these assumptions.  Replacing $G$ by the group generated by $A\\cup B$, we can assume that $G$ is compactly generated.  A compactly generated group with a compact open subgroup is isomorphic to $\\mathbb Z^d\\times K$ for some nonnegative integer $d$ and some compact group $K$ (\\cite{HewittandRoss1}, Theorem 9.8). For such groups $G$, it is then routine to verify that Theorem \\ref{main} holds with no modification of the conclusion.  This is the content of the following corollary.\n\n\\begin{corollary}\\label{generalization}  The conclusion of Theorem \\textup{\\ref{main}} holds under the weaker assumption that $G$ has a compact open subgroup and $m_*(A+B)=m(A)+m(B)<\\infty$. \\hfill $\\square$\n\\end{corollary}\n\n\n\n\n\\section{Acknowledgements}  This work was conducted mostly while the author was a postdoctoral fellow in the Department of Mathematics at the University of British Columbia.  The author thanks Izabella {\\L}aba and Malabika Pramanik for financial support and helpful discussions.  Discussions with Michael Bj\\\"orklund and Alexander Fish provided motivation for this work.\n\n\n\n\\bibliographystyle{amsplain}\n\\frenchspacing\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n \nThe spectacular  discovery of a boson  with a mass around $\\sim 125 \\,\\, \\mathrm{GeV}$ by\nthe ATLAS and CMS experiments~\\cite{Aad:2012tfa,Chatrchyan:2012xdj} at\nCERN constitutes a milestone in the quest for understanding the physics\nof electroweak symmetry breaking (EWSB). While within the present experimental\nuncertainties the properties of the observed Higgs boson are compatible with\nthe predictions of the Standard Model (SM)~\\cite{Aad:2015zhl},\nmany other interpretations are possible as well, in particular as a\nHiggs boson of an extended Higgs sector. Consequently, any model describing \nelectroweak physics needs to provide a state that can be identified\nwith the observed signal. \n\nOne of the prime candidates for physics beyond the SM is supersymmetry\n(SUSY), which doubles the particle degrees of freedom by predicting\ntwo scalar partners for all SM fermions, as well as fermionic partners\nto all bosons. The simplest SUSY extension is the Minimal Supersymmetric\nStandard Model (MSSM)~\\cite{Nilles:1983ge,Haber:1984rc}.\nIn contrast to the single Higgs doublet of the SM, the Higgs sector of\nthe MSSM contains two Higgs doublets, which in the ${\\CP}$ conserving\ncase leads to a physical spectrum consisting of two ${\\CP}$-even, one\n${\\CP}$-odd and two charged Higgs bosons. The light (or the heavy)\n${\\CP}$-even MSSM Higgs boson can be interpreted as the signal discovered\nat $\\sim 125 \\,\\, \\mathrm{GeV}$~\\cite{Heinemeyer:2011aa}. \n   \nGoing beyond the MSSM, a well-motivated extension is given by\nthe Next-to-Minimal Supersymmetric Standard Model (NMSSM), see\ne.g.~\\cite{Ellwanger:2009dp,Maniatis:2009re} for reviews. In particular\nthe NMSSM provides a solution for the so-called ``$\\mu$ problem'' by\nnaturally associating an adequate scale to the $\\mu$ parameter appearing\nin the MSSM superpotential~\\cite{Ellis:1988er,Miller:2003ay}.\nIn the NMSSM a new singlet superfield is introduced, which only couples to the\nHiggs- and sfermion-sectors, giving rise to an effective $\\mu$-term,\nproportional to the vacuum expectation value (vev) of the scalar singlet.\nAssuming ${\\CP}$ conservation, as we do throughout the paper, the states in the\nNMSSM Higgs sector can be classified as three ${\\CP}$-even Higgs bosons,\n$h_i$ ($i = 1,2,3$), two ${\\CP}$-odd Higgs bosons, $a_j$ ($j = 1,2$),\nand the charged Higgs boson pair $H^\\pm$. In addition, the SUSY\npartner of the singlet Higgs (called the singlino) extends the\nneutralino sector to a total of five neutralinos. In the NMSSM the\nlightest but also the second lightest ${\\CP}$-even neutral Higgs boson\ncan be interpreted as the signal observed at about $125~\\,\\, \\mathrm{GeV}$, see,\ne.g., \\cite{King:2012is,Domingo:2015eea}.\n\nA natural extension of the NMSSM is the \\ensuremath{\\mu\\nu\\mathrm{SSM}}, in which the \nsinglet superfield is interpreted as a right-handed neutrino \nsuperfield~\\cite{LopezFogliani:2005yw,Escudero:2008jg} \n(see \\citeres{Munoz:2009an,Munoz:2016vaa,Ghosh:2017yeh} for reviews). \nThe \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ is the simplest extension of the MSSM that can provide \nmassive neutrinos through a see-saw mechanism at the electroweak scale.\nIn this paper we will focus on the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ with one family of \nright-handed neutrino superfields, and the case of three families will\nbe studied in a future publication.%\n\\footnote{The \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ with three families of right-handed neutrinos\nextends the ${\\CP}$-even and ${\\CP}$-odd scalar sector and the\nneutral fermion sector by two additional\nparticles each, in particular\nallowing a more viable reproduction of neutrino\ndata~\\cite{LopezFogliani:2005yw,Escudero:2008jg,Ghosh:2008yh,Bartl:2009an,Fidalgo:2009dm,Ghosh:2010zi}.}\n~The $\\mu$ problem is solved analogously to the NMSSM by the coupling of the right-handed neutrino \nsuperfield to  the Higgs sector, and a trilinear coupling of the \nright-handed neutrino generates an effective Majorana mass at the \nelectroweak scale. The unique feature of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ is the introduction \nof a Yukawa coupling for the right-handed neutrino of the order of the\nelectron Yukawa coupling that induces the explicit\nbreaking of $R$-parity. One of the consequences is that there is no lightest stable \nSUSY particle anymore. Nevertheless, the model can still provide a dark matter\ncandidate with a gravitino that has a life time longer  \nthan the age of the observable universe~\\cite{Choi:2009ng,GomezVargas:2011ph,Albert:2014hwa,Gomez-Vargas:2016ocf}. \nSince the lightest particle beyond the SM is not stable, it can \ncarry electrical charge or even \nbe coloured.\nThe explicit violation of lepton number and lepton flavor \ncan modify the \nspectrum of the neutral and charged fermions in comparison to the NMSSM.\nThe three families of charged leptons will mix with the chargino and \nthe Higgsino and form five massive charged fermions. However, the mixing will naturally be tiny since the breaking of $R$-parity is governed by the \nsmall neutrino Yukawa couplings. In the neutral fermion sector the three \nleft-handed neutrinos mix with the right-handed neutrino and the four \nMSSM-like neutralinos. When just one family of right-handed neutrino is \nconsidered (as we do in this paper), the mass matrix of the\nneutral fermions is of rank six, so  \njust one light neutrino mass is generated at tree-level, while the other \ntwo light-neutrino masses will be generated by quantum corrections.\nFor the Higgs sector the breaking of $R$-parity has dramatic \nconsequences. The three left-handed and the right-handed sneutrinos \nwill mix with the doublet Higgses and form six massive ${\\CP}$-even \nand five massive ${\\CP}$-odd states, assuming that there is no \n${\\CP}$-violation. Additionally, since the vacuum\nof the model is not protected anymore by lepton number, the sneutrinos \nwill acquire a vev after spontaneous EWSB.\nWhile the vev of the right-handed sneutrino can easily take values up to \nthe TeV-scale, the stability of the vacuum together with the smallness \nof the neutrino Yukawa couplings force the vevs of the left-handed \nsneutrinos to be several orders of magnitude \nsmaller~\\cite{Escudero:2008jg,LopezFogliani:2005yw}.\nAs in the NMSSM, the couplings of the \ndoublet-like Higgses to the gauge-singlet right-handed sneutrino provide \nadditional contributions to the tree-level mass of the SM-like Higgs \nboson, relaxing the prediction of the MSSM, that it is bounded from above by the $Z$ boson mass. Still it was shown in the\nNMSSM~\\cite{Drechsel:2016ukp}\nthat a consistent \ntreatment of the quantum corrections is necessary for accurate\nHiggs mass predictions (see also\n\\citeres{Goodsell:2014bna,Goodsell:2014pla,Staub:2015aea}).\nIn this paper we will investigate if this is also\nthe case in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ and if its unique couplings generate significant \ncorrections to the SM-like Higgs mass, that go beyond the corrections \narising in the NMSSM.\n\n\nThe experimental accuracy of the measured mass of the observed Higgs\nboson has already reached the level of a precision observable, with an\nuncertainty of less than $300~\\,\\, \\mathrm{MeV}$~\\cite{Aad:2015zhl}.\nIn the MSSM the masses of the ${\\CP}$-even Higgs bosons can be\npredicted at lowest order in terms of two SUSY parameters\ncharacterising the MSSM Higgs sector, e.g.\\ $\\tb$, the ratio of\nthe vevs of the two doublets, and the mass of the\n${\\CP}$-odd Higgs boson, $\\MA$, or the charged Higgs boson, $\\MHp$. This\nresults in particular in an upper bound on the mass of the light\n${\\CP}$-even Higgs boson given by the $Z$-boson mass.\nHowever, these relations receive large higher-order corrections. \nBeyond the one-loop level, the dominant two-loop corrections of\n$\\order{\\alpha_t\\alpha_s}$~\\cite{Heinemeyer:1998jw,Heinemeyer:1998kz,Heinemeyer:1998np,Zhang:1998bm,Espinosa:1999zm,Degrassi:2001yf}\nand \\order{\\alpha_t^2}~\\cite{Espinosa:2000df,Brignole:2001jy} as well as\nthe corresponding corrections of \n$\\order{\\alpha_b\\alpha_s}$~\\cite{Brignole:2002bz,Heinemeyer:2004xw} and\n\\order{\\alpha_t\\alpha_b}~\\cite{Brignole:2002bz} are known since more than a\ndecade. (Here we use $\\al_f = (Y^f)^2\/(4\\pi)$, with $Y^f$ denoting the\nfermion Yukawa coupling.) These corrections, together with a\nresummation of leading and subleading logarithms from the top\/scalar\ntop sector~\\cite{Hahn:2013ria} (see also~\\cite{Draper:2013oza,Lee:2015uza} \nfor more details on this type of approach),\na resummation of leading contributions from the bottom\/scalar bottom\nsector~\\cite{Brignole:2002bz,Heinemeyer:2004xw,Hempfling:1993kv,Hall:1993gn,Carena:1994bv,Carena:1999py}\n(see also~\\cite{Noth:2008tw, Noth:2010jy}) and momentum-dependent two-loop\ncontributions~\\cite{Borowka:2014wla, Borowka:2015ura} (see\nalso~\\cite{Degrassi:2014pfa}) are included in the public\ncode~\\fh~\\cite{Heinemeyer:1998yj,Hahn:2009zz,Heinemeyer:1998np,Degrassi:2002fi,Frank:2006yh,Hahn:2013ria,Bahl:2016brp,Bahl:2017aev,feynhiggs-www}. \nA (nearly) full two-loop EP calculation, including even the leading\nthree-loop corrections, has also been\npublished~\\cite{Martin:2005eg,Martin:2007pg}, which is, however, not\npublicly available as a computer code. Furthermore, another leading\nthree-loop calculation of \\order{\\alpha_t\\alpha_s^2}, depending on the various\nSUSY mass hierarchies, has been\nperformed~\\cite{Harlander:2008ju,Kant:2010tf}, resulting in the code \n{\\tt H3m} and is now available as a stand-alone\n  code~\\cite{Harlander:2017kuc}. \nThe theoretical uncertainty on the lightest\n${\\CP}$-even Higgs-boson mass within the MSSM from unknown higher-order\ncontributions is still at the level of about $2-3~\\,\\, \\mathrm{GeV}$ for scalar top\nmasses at the TeV-scale, where the actual uncertainty\ndepends on the considered parameter region~\\cite{Degrassi:2002fi,Heinemeyer:2004gx,Hahn:2013ria,Buchmueller:2013psa}.\n\nIn the NMSSM the status of the higher-order corrections to the Higgs-boson\nmasses (and mixings) is the following. \nFull one-loop calculations including the momentum dependence have been\nperformed in the \\DRbar\\ renormalization scheme in \n\\citere{Degrassi:2009yq,Staub:2010ty}, or in a mixed on-shell (OS)-\\DRbar\\ \nscheme in \\citere{Ender:2011qh,Graf:2012hh,Drechsel:2016jdg}.\nTwo-loop corrections of \\order{\\alpha_t\\alpha_s,\\alpha_t^2} have been included in the\nNMSSM in the leading logarithmic approximation (LLA) in\n\\citeres{Yeghian:1999kr,Ellwanger:1999ji}.  \nIn the EP approach at the two-loop level, the dominant \n\\order{\\alpha_t\\alpha_s, \\alpha_b\\alpha_s} in the \\DRbar\\ scheme became available in \n\\citere{Degrassi:2009yq}. The two-loop corrections involving only\nsuperpotential couplings such as Yukawa and singlet interactions were given in\n\\cite{Goodsell:2014pla}. A two-loop calculation of the \\order{\\alpha_t\\alpha_s}\ncorrections with the top\/stop sector renormalized in the OS scheme or in the\n\\DRbar\\ scheme were provided in \\citere{Muhlleitner:2014vsa}.\nA consistent combination of a full one-loop calculation with all corrections\nbeyond one-loop in the MSSM approximation was given in\n\\citere{Drechsel:2016jdg}, which is included in the (private) version of\n\\fh\\ for the NMSSM. \nA detailed comparison of the various higher-order corrections up to the\ntwo-loop level involving a \\DRbar\\ renormalization was performed in\n\\citere{Staub:2015aea}, and involving an OS renormalization of the top\/stop\nsector for the \\order{\\alpha_t\\alpha_s} corrections in \\citere{Drechsel:2016htw}.\nAccordingly, at present the theoretical uncertainties from\nunknown higher-order corrections in the NMSSM are expected to be still\nlarger than for the MSSM.\n\nIn this paper we go one step beyond and investigate the scalar sector of\nthe \\ensuremath{\\mu\\nu\\mathrm{SSM}}, containing (mixtures of) Higgs bosons and scalar neutrinos.\nAs a first step we present the renormalization at the one-loop level of the\nneutral scalar sector in detail. Here a crucial point is that the NMSSM\npart of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ is treated exactly in the same way as in\n\\citere{Drechsel:2016jdg}. Consequently, differences (at the one-loop\nlevel) appearing for, e.g., mass relations or couplings can be directly\nattributed to the richer structure of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}. As for the NMSSM in\n\\citere{Drechsel:2016jdg}, the full one-loop calculation is supplemented\nwith higher-order corrections in the MSSM limit (as provided by \\fh~\\cite{Heinemeyer:1998yj,Hahn:2009zz,Heinemeyer:1998np,Degrassi:2002fi,Frank:2006yh,Hahn:2013ria,Bahl:2016brp,Bahl:2017aev,feynhiggs-www}).\\footnote{A\ncorresponding calculation using a pure \\DRbar\\ renormalization could\nin principle be performed using \\texttt{SARAH} and\n\\texttt{SPheno}~\\cite{Goodsell:2014bna}.}\nIn our numerical analysis we evaluate several ``representative''\nscenarios using the full one-loop results together with the MSSM-type\nhigher-order contributions. Differences found w.r.t.\\ the NMSSM can be\ninterpreted in a two-fold way. On the one hand, if non-negligible\ndifferences are found, they might serve as a probe to distinguish the\ntwo models experimentally. On the other hand, they indicate the level of\ntheoretical uncertainties of the Higgs-boson\/scalar neutrino mass\ncalculation in the \n\\ensuremath{\\mu\\nu\\mathrm{SSM}}, which should be brought to the same level of accuracy as in the\n(N)MSSM. \n\nThe paper is organized as follows. In \\refse{sec:mnSSM} we describe\nthe \\ensuremath{\\mu\\nu\\mathrm{SSM}}, including the details for all sectors relevant in this paper.\nThe full one-loop renormalization of the neutral scalar potential \nis presented in \\refse{sec:renopot}. \nWe will establish a convenient set of free parameters and fix their \ncounterterms in a mixed OS-\\DRbar\\ scheme. The counterterms are \ncalculated and applied in the renormalized ${\\CP}$-even and\n${\\CP}$-odd one-loop scalar\nself-energies in \\refse{sec:getmasses}. \nIn this work we focus on the application to the renormalized\n${\\CP}$-even\nself-energies, but the calculation of the renormalized ${\\CP}$-odd\nones constitutes a good additional test for the counterterms. \nWe also describe the \nincorporation of higher-order contributions taken over from the MSSM.\nOur numerical analysis, including an analysis of differences w.r.t.\\ \nNMSSM, is presented in \\refse{sec:numanal}. \nWe conclude in section \\refse{sec:concl}.\n\n\n\n\n\n\n\n\n\n\\section{\\protect\\boldmath The model: \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ with one generation of right handed neutrinos.}\n\\label{sec:mnSSM}\n\nIn the three-family notation of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ with one generation of\nright-handed neutrinos the superpotential is written as\n\\begin{align}\\label{superpotential}\nW = & \\;\n\\epsilon_{ab} \\left(\nY^e_{ij} \\, \\hat H_d^a\\, \\hat L^b_i \\, \\hat e_j^c +\nY^d_{ij} \\, \\hat H_d^a\\, \\hat Q^{b}_{i} \\, \\hat d_{j}^{c} \n+\nY^u_{ij} \\, \\hat H_u^b\\, \\hat Q^{a}_{i} \\, \\hat u_{j}^{c}\n\\right)\n\\nonumber\\\\\n&+   \n\\epsilon_{ab} \\left(\nY^{\\nu}_{i} \\, \\hat H_u^b\\, \\hat L^a_i \\, \\hat \\nu^c \n-\n\\lambda \\, \\hat \\nu^c\\, \\hat H_u^b \\hat H_d^a\n\\right)\n+\n\\frac{1}{3}\n\\kappa \n\\hat \\nu^c\\hat \\nu^c\\hat \\nu^c\n\\ .\n\\end{align}\nwhere $\\hat H_d^T=(\\hat H_d^0, \\hat H_d^-)$ and \n$\\hat H_u^T=(\\hat H_u^+, \\hat H_u^0)$ are the MSSM-like doublet Higgs\nsuperfields, $\\hat Q_i^T=(\\hat u_i, \\hat d_i)$ and \n$\\hat L_i^T=(\\hat \\nu_i, \\hat e_i)$ are the left-chiral\nquark and lepton superfield doublets,\nand $\\hat u_{j}^{c}$, $\\hat d_{j}^{c}$, $\\hat e_j^c$ and $\\hat{\\nu}^c$ are\n the right-chiral quark and lepton superfields.\n$i$ and $j$ are family indices running from one to three and \n$a,b=1,2$ are indices of the fundamental representation of SU(2) \nwith $\\epsilon_{ab}$ the totally antisymmetric tensor and \n$\\varepsilon_{12}=1$. The colour indices are undisplayed. \n$Y^u$, $Y^d$ and $Y^e$ are the usual Yukawa couplings\nalso present in the MSSM. \nThe right-handed neutrino is a gauge singlet, which permits us to \nwrite \nthe gauge-invariant  \ntrilinear self coupling $\\kappa$ and the trilinear \ncoupling with the Higgs doublets $\\lambda$ in the second row, \nwhich are analogues to the couplings of the singlet in the \nsuperpotential of the trilinear NMSSM.\nThe $\\mu$-term is generated dynamically after the spontaneous EWSB,\nwhen the right-handed sneutrino obtains a vev.\nThe $\\kappa$-term forbids a global U(1) symmetry and we\navoid the existence of a Goldstone boson in the ${\\CP}$-even sector. \nThe remarkable difference to the\nNMSSM is the additional Yukawa coupling $Y^\\nu_i$, which induces\nexplicit breaking of $R$-parity through the $\\lambda$- and\n$\\kappa$-term,\nand which justifies the interpretation of the singlet\nsuperfield as a right-handed neutrino superfield.\nIt should be pointed out\nthat in this case lepton number is not conserved anymore, and also the\nflavor symmetry in the leptonic sector is broken. \nA more complete motivation of this superpotential can be found in\n\\citere{LopezFogliani:2005yw,Escudero:2008jg,Ghosh:2017yeh}. \n\nWorking in the framework of low-energy SUSY the\ncorresponding soft SUSY-breaking Lagrangian can be written as\n\\begin{eqnarray}\n-\\mathcal{L}_{\\text{soft}}  =&&\n\\epsilon_{ab} \\left(\nT^e_{ij} \\, H_d^a  \\, \\widetilde L^b_{iL}  \\, \\widetilde e_{jR}^* +\nT^d_{ij} \\, H_d^a\\,   \\widetilde Q^b_{iL} \\, \\widetilde d_{jR}^{*} \n+\nT^u_{ij} \\,  H_u^b \\widetilde Q^a_{iL} \\widetilde u_{jR}^* \n+ \\text{h.c.}\n\\right)\n\\nonumber \\\\\n&+&\n\\epsilon_{ab} \\left(\nT^{\\nu}_{i} \\, H_u^b \\, \\widetilde L^a_{iL} \\widetilde \\nu_{R}^* \n- \nT^{\\lambda} \\, \\widetilde \\nu_{R}^*\n\\, H_d^a  H_u^b\n+ \\frac{1}{3} T^{\\kappa} \\, \\widetilde \\nu_{R}^*\n\\widetilde \\nu_{R}^*\n\\widetilde \\nu_{R}^*\n\\\n+ \\text{h.c.}\\right)\n\\nonumber \\\\\n&+& \n\\left(m_{\\widetilde{Q}_L}^2\\right)_{ij}   \n\\widetilde{Q}_{iL}^{a*}\n\\widetilde{Q}^a_{jL}\n+\\left(m_{\\widetilde{u}_R}^{2}\\right)_{ij} \\widetilde{u}_{iR}^*  \n\\widetilde u_{jR}\n+ \\left(m_{\\widetilde{d}_R}^2\\right)_{ij}  \\widetilde{d}_{iR}^*  \n\\widetilde d_{jR}\n+\n\\left(m_{\\widetilde{L}_L}^2\\right)_{ij}    \n\\widetilde{L}_{iL}^{a*}  \n\\widetilde{L}^a_{jL}\n\\nonumber\\\\\n&+&\n\\left( m_{H_d\\widetilde{L}_L}^2 \\right)_{i}H_d^{a*} \\widetilde{L}_{iL}^a\n+\nm_{\\widetilde{\\nu}_R}^2 \\widetilde{\\nu}_{R}^*\n\\widetilde\\nu_{R} \n+\n\\left(m_{\\widetilde{e}_R}^2\\right)_{ij}  \\widetilde{e}_{iR}^* \n\\widetilde e_{jR}\n+ \nm_{H_d}^2 {H^a_d}^*\nH^a_d + m_{H_u}^2 {H^a_u}^*\nH^a_u\n\\nonumber \\\\\n&+&  \\frac{1}{2}\\, \\left(M_3\\, {\\widetilde g}\\, {\\widetilde g}\n+\nM_2\\, {\\widetilde{W}}\\, {\\widetilde{W}}\n+M_1\\, {\\widetilde B}^0 \\, {\\widetilde B}^0 + \\text{h.c.} \\right)\\ ,\n\\label{2:Vsoft}\n\\end{eqnarray}\nIn the first four lines\nthe fields denote the scalar component of the\ncorresponding superfields.\nIn the last line the fields denote the fermionic\nsuperpartners of the gauge bosons.\nThe scalar trilinear parameters \n$T^{e,\\nu,d,u,\\lambda,\\kappa}$ correspond to the trilinear couplings \nin the superpotential. The soft mass parameters \n$m_{\\widetilde{Q}_L,\\widetilde{u}_R,\\widetilde{d}_R,\n\\widetilde{L}_L,\\widetilde{e}_R}^2$ are hermitian $3\\times 3$ matrices in family space. \n$m_{H_d,H_u,\\widetilde{\\nu}_R}^2$ are the soft masses\nof the doublet Higgs fields and the right-handed sneutrino, and \n$ m_{H_d\\widetilde{L}_L}^2$ is a 3-dimensional\nvector in family space allowed\nby gauge symmetries since the left-handed lepton fields and the down-type \nHiggs field share the same quantum numbers. In the last row \nthe parameters $M_{3,2,1}$ define Majorana masses for the gluino, wino \nand bino, where the summation\nover the gauge-group indices in the \nadjoint representation\nis undisplayed. While all the soft parameters except\n$m_{H_d}^2$, $m_{H_u}^2$ and $m_{\\widetilde{\\nu}_R}^2$ can \nin general be complex,\nthey are assumed to be real in the following to avoid \n${\\CP}$-violation.\nAdditionally, we will neglect flavor mixing at tree-level in the squark\nand the quark sector, so the soft masses will be diagonal and we write\n$m_{\\widetilde{Q}_{iL} }^2$, $m_{\\widetilde{u}_{iR}}^2$ and\n$m_{\\widetilde{d}_{iR}}^2$, as well as for the soft trilinears\n$T^u_i=A^u_i\nY^u_i$, $T^d_i=A^d_i Y^d_i$, where the summation convention on repeated\nindices is not implied, and the quark Yukawas $Y^u_{ii}=Y^u_i$ and\n$Y^d_{ii}=Y^d_i$ are diagonal. For the sleptons we define \n$T^e_{ij}=A^e_{ij}Y^e_{ij}$ and $T^\\nu_{i}=A^\\nu_i Y^\\nu_i$, again without summation over repeated indices.\n\nSome care has to be taken with the parameters\n$(m_{\\widetilde{L}_L}^2)_{ij}$\ncontributing to the tree-level neutral scalar potential, because\nthese parameters cannot be set flavor-diagonal a priori. The reason is that\nduring the renormalization procedure (see \\refse{sec:condis})\nthe non-diagonal elements receive a\ncounterterm. Of course, the tree-level value of the non-diagonal\nelements can and should be set to zero to avoid too large flavor mixing. This assures that the contributions generated by virtual corrections\nwill always be small.\n\nSimilarly to the off-diagonal elements of the squared\nsfermion mass matrices,\nthe parameters $( m_{H_d\\widetilde{L}_L}^2 )_{i}$ are usually\nnot included in the tree-level Lagrangian of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}.\nIn the latter case because\nthey contribute to the minimization equations of the left-handed\nsneutrinos and spoil the electroweak seesaw mechanism that\ngenerates neutrino masses of the correct order of\nmagnitude. Theoretically, the absence of these\nparameters mixing different fields\nat tree level, $( m_{H_d\\widetilde{L}_L}^2 )_{i}$,\n$(m_{\\widetilde{L}_L}^2)_{ij}$,\n$(m_{\\widetilde{Q}_L}^2)_{ij}$, etc., \ncan be justified by the diagonal structure of the K\\\"ahler metric in\ncertain supergravity models, or when the dilaton field\nis the source of SUSY breaking in string\nconstructions~\\cite{Ghosh:2017yeh}.\nNotice also that when the\ndown-type Higgs doublet superfield is interpreted as a fourth family\nof leptons, the parameters\n$m_{H_d\\widetilde{L}_L}^2$ can be seen as\nnon-diagonal elements\nof $m_{\\widetilde{L}_L}^2$~\\cite{Lopez-Fogliani:2017qzj}.\nNevertheless, we include them in the soft\nSUSY-breaking Lagrangian in this paper,\nbecause these terms are generated at (one-)loop level, and\nin our renormalization approach we need the functional dependence\nof the scalar potential on $m_{H_d\\widetilde{L}_L}^2$.\n\nAfter the electroweak symmetry breaking the neutral scalar fields will acquire\na vev. This includes the left- and right-handed\nsneutrinos, because they are not protected by lepton number conservation as in\nthe MSSM and the NMSSM. We define the decomposition \n\\begin{align}\n\\label{eq:vevdecompH}\nH_d^0 =& \\frac{1}{\\sqrt 2} \\left(H_{d}^{\\mathcal{R}} + v_d + \\text{i}\\ H_{d}^{\\mathcal{I}}\\right)\\ , \\\\\nH^0_u =& \\frac{1}{\\sqrt 2} \\left(H_{u}^{\\mathcal{R}} + v_u +\\text{i}\\ H_{u}^{\\mathcal{I}}\\right)\\ , \\\\\n\\widetilde{\\nu}_{R} =&\n      \\frac{1}{\\sqrt 2} \\left(\\widetilde{\\nu}^{\\mathcal{R}}_{R}+ v_{R} + \\text{i}\\ \\widetilde{\\nu}^{\\mathcal{I}}_{R}\\right), \\\\\n\\label{eq:vevdecompL}\n  \\widetilde{\\nu}_{iL} =& \\frac{1}{\\sqrt 2} \\left(\\widetilde{\\nu}_{iL}^{\\mathcal{R}} \n  + v_{iL} +\\text{i}\\ \\widetilde{\\nu}_{iL}^{\\mathcal{I}}\\right)\\ ,\n\\end{align}\nwhich is valid assuming ${\\CP}$-conservation, as we will do\nthroughout this paper.\n\n\n\n\n\\subsection{The \\boldmath{\\ensuremath{\\mu\\nu\\mathrm{SSM}}} Higgs potential}\n\nThe neutral scalar potential $V_H$ of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ with one generation of right-handed neutrinos is given at tree-level with all parameters chosen to be real by the soft terms and the $F$- and $D$-term contributions of the superpotential. We find\n\\begin{equation}\nV^{(0)} = V_{\\text{soft}} + V_F  +  V_D\\ , \n\\label{finalpotential}\n\\end{equation}\nwith\n\\begin{eqnarray}\nV_{\\text{soft}}  =&&\n\\left(\nT^{\\nu}_{i} \\, H_u^0\\,  \\widetilde \\nu_{iL} \\, \\widetilde \\nu_{R}^* \n- T^{\\lambda} \\, \\widetilde \\nu_{R}^*\\, H_d^0  H_u^0\n+ \\frac{1}{3} T^{\\kappa} \\, \\widetilde \\nu_{R}^* \\widetilde \\nu_{R}^* \n\\widetilde \\nu_{R}^*\\\n+\n\\text{h.c.} \\right)\n\\\\\n&+&\n\\left(m_{\\widetilde{L}_L}^2\\right)_{ij} \\widetilde{\\nu}_{iL}^* \\widetilde\\nu_{jL}\n+\n\\left( m_{H_d\\widetilde{L}_L}^2 \\right)_{i}H_d^{0*}\\widetilde{\\nu}_{iL}\n+\nm_{\\widetilde{\\nu}_R}^2 \\widetilde{\\nu}_{R}^* \\widetilde\\nu_{R} +\nm_{H_d}^2 {H^0_d}^* H^0_d + m_{H_u}^2 {H^0_u}^* H^0_u \\nonumber\n\\ ,\n\\label{akappa}\n\\\\\n\\nonumber\n\\\\\nV_{F}  =&&\n \\lambda^2 H^0_{d}H_d^0{^{^*}}H^0_{u}H_u^0{^{^*}}\n +\n\\lambda^2\\tilde{\\nu}^{*}_{R}\\tilde{\\nu}_{R}H^0_{d}H_d^0{^*}\n +\n\\lambda^2\n\\tilde{\\nu}^{*}_{R}\\tilde{\\nu}_{R}  H^0_{u}H_u^0{^*}   \n\\nonumber\\\\                                              \n&+&\n\\kappa^2\\left(\\tilde{\\nu}^*_{R}\\right)^2\n\t\\left(\\tilde{\\nu}_{R}\\right)^2\n- \\left(\\kappa\\lambda\\left(\\tilde{\\nu}^{*}_{R}\\right)^2 H_d^{0*}H_u^{0*}                                      \n -Y^{\\nu}_{i}\\kappa\\tilde{\\nu}_{iL}\\left(\\tilde{\\nu}_{R}\\right)^2H^0_{u}\n \\right.\n \\nonumber\\\\\n &+&\n \\left.\n Y^{\\nu}_{i}\\lambda\\tilde{\\nu}_{iL} H_d^{0*}H_{u}^{0*}H^0_{u}\n+{Y^{\\nu}_{i}}\\lambda \\tilde{\\nu}_{iL}^{*}\\tilde{\\nu}_{R}\\tilde{\\nu}_{R}^* H^0_{d}\n + \\text{h.c.}\\right) \n\\nonumber \\\\\n &+& \nY^{\\nu}_{i}{Y^{\\nu}_{i}} \\tilde{\\nu}^{*}_{R}\n\\tilde{\\nu}_{R}H^0_{u}H_u^0{^*}                                                \n +\nY^{\\nu}_{i}{Y^{\\nu}_{j}}\\tilde{\\nu}_{iL}\\tilde{\\nu}_{jL}^{*}\\tilde{\\nu}_{R}^{*}\n                                  \\tilde{\\nu}_{R}  \n +\nY^{\\nu}_{i}{Y^{\\nu}_{j}}\\tilde{\\nu}_{i}\\tilde{\\nu}_{j}^* H^0_{u}H_u^{0*}\\, ,\n\\\\\n\\nonumber\n\\\\\nV_D  =&&\n\\frac{1}{8}\\left(g_1^{2}+g_2^{2}\\right)\\left(\\widetilde\\nu_{iL}\\widetilde{\\nu}_{iL}^* \n+H^0_d {H^0_d}^* - H^0_u {H^0_u}^* \\right)^{2}\\, .\n\\label{dterms}\n\\end{eqnarray}\nUsing the decomposition from\n\\refeqs{eq:vevdecompH} - (\\ref{eq:vevdecompL}) \nthe linear and bilinear terms\nin the fields define the tadpoles $T_{\\varphi}$ and the scalar ${\\CP}$-even\nand ${\\CP}$-odd neutral mass matrices $m_{\\varphi}^2$ and\n$m_{\\sigma}^2$ after electroweak symmetry breaking,\n\\begin{align}\nV_H=\\cdots - T_{\\varphi_i}\\varphi_i + \\frac{1}{2} \\varphi^T m_{\\varphi}^2\n  \\varphi + \\frac{1}{2} \\sigma^T m_{\\sigma}^2 \\sigma + \\cdots \\; .\n\\end{align}\nwhere we collectively denote with \n$\\varphi^T=(H_d^{\\mathcal{R}},H_u^{\\mathcal{R}},\n\\widetilde{\\nu}_{R}^{\\mathcal{R}},\\widetilde{\\nu}_{iL}^{\\mathcal{R}})$ \nand \n$\\sigma^T=(H_d^{\\mathcal{I}},H_u^{\\mathcal{I}},\n\\widetilde{\\nu}_{R}^{\\mathcal{I}},\\widetilde{\\nu}_{iL}^{\\mathcal{I}})$ \nthe ${\\CP}$-even and ${\\CP}$-odd scalar fields.\nThe linear terms are only allowed for ${\\CP}$-even fields and given by:\n\\begin{align}\nT_{H_d^{\\mathcal{R}}}=&-m_{H_d}^2v_d -\n\\left( m_{H_d\\widetilde{L}_L}^2 \\right)_{i} v_{iL} -\n\\frac{1}{8}\\left(g_1^2+g_2^2\\right)v_d \\left( v_d^2 + v_{iL}v_{iL} - v_u^2\\right)\n\\notag \\\\\n&-\\frac{1}{2}\\lambda\\left(v_R^2+v_u^2\\right)\\left(\\lambda v_d-v_{iL}Y^\\nu_i\\right)+\n\\frac{1}{\\sqrt{2}}T^\\lambda v_R v_u+\n\\frac{1}{2}\\kappa\\lambda v_R^2 v_u \\; , \\label{eq:tp1}\\\\[4pt]\nT_{H_u^{\\mathcal{R}}}=&-m_{H_u}^2v_u+\n\\frac{1}{8}\\left( g_1^2+g_2^2\\right)v_u\\left( v_d^2+v_{iL}v_{iL}-v_u^2\\right)\n\\notag \\\\\n&-\\frac{1}{2}\\lambda^2\\left( v_d^2+v_R^2\\right)+\n\\frac{1}{\\sqrt{2}}T^\\lambda v_d v_R+\n\\lambda v_d v_u v_{iL}Y^\\nu_i+\n\\frac{1}{2}\\kappa\\lambda v_d v_R^2-\n\\frac{1}{2}\\kappa v_R^2 v_{iL}Y^\\nu_i\n\\notag \\\\\n&-\\frac{1}{2}v_u\\left( v_{iL}Y^\\nu_i\\right)^2-\n\\frac{1}{\\sqrt{2}}v_R v_{iL}T^\\nu_i-\n\\frac{1}{2}v_R^2v_u Y^\\nu_i Y^\\nu_i \\; , \\label{eq:tp2}\\\\[4pt]\nT_{\\widetilde{\\nu}_R^{\\mathcal{R}}}=&-m_{\\widetilde{\\nu}_R}^2 v_R-\n\\frac{1}{\\sqrt{2}}T^\\kappa v_R^2-\\kappa^2 v_R^3+\n\\frac{1}{\\sqrt{2}}T^\\lambda v_d v_u-\n\\frac{1}{2}\\lambda^2 v_R\\left( v_d^2+v_u^2\\right)\n\\notag \\\\\n&+\\lambda v_d v_R v_{iL}Y^\\nu_i +\n\\kappa \\lambda v_d v_R v_u-\n\\kappa v_R v_u v_{iL} Y^\\nu_i-\n\\frac{1}{2}v_R\\left( v_{iL}Y^\\nu_i\\right)^2 \n\\notag \\\\\n&-\\frac{1}{\\sqrt{2}}v_u v_{iL}T^\\nu_i-\n\\frac{1}{2}v_Rv_u^2 Y^\\nu_i Y^\\nu_i \\; , \\label{eq:tp3}\\\\[4pt]\nT_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}}}=&-\n\\left(m_{\\widetilde{L}_L}^2\\right)_{ij}v_{jL}-\n\\left( m_{H_d\\widetilde{L}_L}^2 \\right)_{i}v_d-\n\\frac{1}{8}\\left( g_1^2+g_2^2\\right)v_{iL}\\left( v_d^2+v_{jL}v_{jL}-v_u^2\\right)\n\\notag \\\\\n&+\\frac{1}{2}\\lambda v_d v_R^2 Y^\\nu_i-\n\\frac{1}{\\sqrt{2}}v_R v_u T^\\nu_i-\n\\frac{1}{2}\\kappa v_R^2v_uY^\\nu_i+\n\\frac{1}{2}\\lambda v_d v_u^2 Y^\\nu_i\n\\notag \\\\\n&-\\frac{1}{2}v_R^2Y^\\nu_i v_{jL}Y^\\nu_j-\n\\frac{1}{2}v_u^2Y^\\nu_i v_{jL}Y^\\nu_j \\label{eq:tp4} \\; .\n\\end{align}\nThe tadpoles vanish in the true vacuum of the model. During the\nrenormalization procedure they will be treated as OS parameters,\ni.e., finite corrections will be canceled by their corresponding\ncounterterms. This guarantees that the vacuum is\nstable w.r.t.\\ quantum corrections.\n\nThe bilinear terms \n\\begin{align}\nm_{\\varphi}^2= \n  \\left( \\begin{array}{cccc} \n  m_{H_{d}^{\\mathcal{R}}H_{d}^{\\mathcal{R}}}^{2}& \nm_{H_{d}^{\\mathcal{R}}H_{u}^{\\mathcal{R}}}^{2} & \nm_{H_{d}^{\\mathcal{R}}\\widetilde{\\nu}_{R}^{\\mathcal{R}}}^{2} & \nm_{H_{d}^{\\mathcal{R}}\\widetilde{\\nu}_{jL}^{\\mathcal{R}}}^{2}\n\\\\\nm_{H_{u}^{\\mathcal{R}}H_{d}^{\\mathcal{R}}}^{2} & \nm_{H_{u}^{\\mathcal{R}}H_{u}^{\\mathcal{R}}}^{2}& \nm_{H_{u}^{\\mathcal{R}}\\widetilde{\\nu}_{R}^{\\mathcal{R}}}^{2}  & \nm_{H_{u}^{\\mathcal{R}}\\widetilde{\\nu}_{jL}^{\\mathcal{R}}}^{2}\n\\\\\n m_{\\widetilde{{\\nu}}^{\\mathcal{R}}_{R} H_{d}^{\\mathcal{R}}}& \nm_{\\widetilde{{\\nu}}^{\\mathcal{R}}_{R} H_{u}^{\\mathcal{R}}}&   \nm_{\\widetilde{{\\nu}}^{\\mathcal{R}}_{R} \\widetilde{{\\nu}}^{\\mathcal{R}}_{R}}^{2} &   \nm_{\\widetilde{{\\nu}}^{\\mathcal{R}}_{R} \\widetilde{{\\nu}}^{\\mathcal{R}}_{jL}}^{2} \n\\\\\nm_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}} H_{d}^{\\mathcal{R}}}^{2}  & \n m_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}} H_{u}^{\\mathcal{R}}}^{2} & \nm_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}} \\widetilde{\\nu}^{\\mathcal{R}}_{R}}^{2} &   \nm_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}} \\widetilde{\\nu}_{jL}^{\\mathcal{R}}}^{2} \n         \\end{array} \\right)\\ ,  \n\\label{matrixscalar1}\n\\end{align}\nand\n\\begin{align}\nm^2_{\\sigma}= \n  \\left( \\begin{array}{cccc} \n  m_{H_{d}^{\\mathcal{I}}H_{d}^{\\mathcal{I}}}^{2} & \nm_{H_{d}^{\\mathcal{I}}H_{u}^{\\mathcal{I}}}^{2} & \nm_{H_{d}^{\\mathcal{I}}\\widetilde{\\nu}_{R}^{\\mathcal{I}}}^{2} & \nm_{H_{d}^{\\mathcal{I}}\\widetilde{\\nu}_{jL}^{\\mathcal{I}}}^{2}\n\\\\\nm_{H_{u}^{\\mathcal{I}}H_{d}^{\\mathcal{I}}}^{2} & \nm_{H_{u}^{\\mathcal{I}}H_{u}^{\\mathcal{I}}}^{2}& \nm_{H_{u}^{\\mathcal{I}}\\widetilde{\\nu}_{R}^{\\mathcal{I}}}^{2}  & \nm_{H_{u}^{\\mathcal{I}}\\widetilde{\\nu}_{jL}^{\\mathcal{I}}}^{2}\n\\\\\n m_{\\widetilde{{\\nu}}^{\\mathcal{I}}_{R} H_{d}^{\\mathcal{I}}}^2& \nm_{\\widetilde{{\\nu}}^{\\mathcal{I}}_{R} H_{u}^{\\mathcal{I}}}^2&   \nm_{\\widetilde{{\\nu}}^{\\mathcal{I}}_{R} \\widetilde{{\\nu}}^{\\mathcal{I}}_{R}}^{2} &   \nm_{\\widetilde{{\\nu}}^{\\mathcal{I}}_{R} \\widetilde{{\\nu}}^{\\mathcal{I}}_{jL}}^{2} \n\\\\\nm_{\\widetilde{\\nu}_{iL}^{\\mathcal{I}} H_{d}^{\\mathcal{I}}}^{2}  & \n m_{\\widetilde{\\nu}_{iL}^{\\mathcal{I}} H_{u}^{\\mathcal{I}}}^{2} & \nm_{\\widetilde{\\nu}_{iL}^{\\mathcal{I}} \\widetilde{\\nu}^{\\mathcal{I}}_{R}}^{2} &   \nm_{\\widetilde{\\nu}_{iL}^{\\mathcal{I}} \\widetilde{\\nu}_{jL}^{\\mathcal{I}}}^{2}\n         \\end{array} \\right)\\ , \n\\label{matrixscalar2} \n\\end{align}\nare $6\\times 6$ matrices in family space whose rather lengthy entries\nare given in the appendix in \\refse{app:cpeven} and \\refse{app:cpodd}.\nWe transform to the mass eigenstate basis\nof the ${\\CP}$-even scalars through a\nunitary transformation defined by the matrix $U^H$,\nthat diagonalizes the mass matrix $m_{\\varphi}^2$,\n\\begin{eqnarray}\nU^H\nm^2_{\\varphi}\\ {U^H}^{^T}=\nm^2_{h}\n\\ ,\n\\label{eq:scalarhiggs}\n\\end{eqnarray}\nwith \n\\begin{equation}\n\\varphi = {U^H}^{^T} h\\, ,\n\\label{physscalarhiggses}\n\\end{equation}\nwhere the $h_i$ are the ${\\CP}$-even scalar fields in the mass eigenstate basis. Without ${\\CP}$-violation in the scalar sector the matrix $U^H$ is real. Similarly, for the ${\\CP}$-odd scalar we define the rotation matrix $U^A$, that diagonalizes the mass matrix $m_{\\sigma}^2$,\n\\begin{eqnarray}\nU^A\nm^2_{\\sigma}\\ {U^A}^{^T}=\nm^2_{A}\n\\ , \\qquad \\text{with } \\sigma={U^A}^T A \\; .\n\\label{eq:scalaroddhiggs}\n\\end{eqnarray}\nBecause of the smallness of the neutrino Yukawa couplings $Y^\\nu_i$, which also implies that the left-handed sneutrino vevs $v_{iL}$ have to be small, so that the tadpole coefficients vanish at tree-level \\cite{Escudero:2008jg}, the mixing of the left-handed sneutrinos with the doublet fields and the singlet will be small. \n\nIt is a well known fact that the quantum corrections to the Higgs\npotential are highly significant in supersymmetric models, see\ne.g.\\ \\citeres{Heinemeyer:2004ms,Heinemeyer:2004gx,Djouadi:2005gj} \nfor reviews. As in the NMSSM~\\cite{Ellwanger:2009dp}, the upper bound on \nthe lowest Higgs mass squared at tree-level is relaxed through additional\ncontributions from the \nsinglet~\\cite{Escudero:2008jg};\n\\begin{equation}\nM_Z^2 \\left( \\cos^2 2\\beta +\n\\frac{2\\lambda^2}{g_1^2+g_2^2}\\sin^2 2\\beta \\right) \\; .\n\\end{equation}\nNevertheless, quantum corrections were \nstill shown to contribute significantly especially in the prediction of \nthe SM-like Higgs boson\nmass~\\cite{Draper:2016pys,Ellwanger:1993hn,Elliott:1993uc,Elliott:1993bs,Ender:2011qh,Drechsel:2016htw,Drechsel:2016ukp,Drechsel:2016jdg}.\nIn this paper we will investigate how\nimportant the unique loop corrections of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ beyond the NMSSM \nare in realistic scenarios. Before that we briefly describe the other\nrelevant sectors of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}.\n\n\n\n\\subsection{Squark sector}\n\nThe numerically most important one-loop corrections to the scalar\npotential are expected from the stop\/top-sector, analogous to the\n(N)MSSM~\\cite{ELLIS199183,Ellwanger:1993hn,Elliott:1993ex,Elliott:1993uc,Elliott:1993bs,Pandita:1993tg} due to the huge Yukawa\ncoupling of the (scalar) top. The tree-level mass matrices of the squarks differ\nslightly from the ones in the MSSM. Neglecting flavor mixing in the\nsquark sector, one finds for the up-type squark mass matrix\n$M^{\\widetilde{u}_i}$ of flavor $i$, \n\\begin{align}\n    M^{\\widetilde{u}_i}_{11}&=m_{\\widetilde{Q}_{iL}}^2 + \\frac{1}{24}(3g_2^{2}-g_1^2)(v_d^2+v_{jL}v_{jL}-v_u^2)+\\frac{1}{2}v_u^2{Y^{u}_i}^2\n    \\label{eq:usquarks11} \\\\\n    M^{\\widetilde{u}_i}_{12}&=\\frac{1}{2}(\\sqrt{2}A^u_i v_u + v_RY^u v_{jL}Y^\\nu_j -\\lambda v_d v_R) \\label{eq:usquarks12} \\\\\n    M^{\\widetilde{u}_i}_{22}&=m_{\\widetilde{u}_{iR}}^2+\\frac{1}{6}g_1^2(v_d^2+v_{jL}v_{jL}-v_u^2)+\\frac{1}{2}v_u^2{Y^u_i}^2 \\; .\n    \\label{eq:usquarks22}\n\\end{align}\nIt should be noted that in the non-diagonal element explicitly\nappear the neutrino \nYukawa couplings. This term arises in the F-term contributions of the\nsquark potential through the quartic coupling of up-type quarks and one \nleft-handed and the right-handed sneutrino after EWSB.\nThe mass eigenstates $\\widetilde{u}_{i1}$ and $\\widetilde{u}_{i2}$ are\nobtained by the unitary transformation\n\\begin{equation}\n\\begin{pmatrix}\n \\widetilde{u}_{i1} \\\\\n \\widetilde{u}_{i2}\n\\end{pmatrix}\n=U^{\\widetilde{u}}_i\n\\begin{pmatrix}\n \\widetilde{u}_{iL} \\\\\n \\widetilde{u}_{iR}\n\\end{pmatrix}\\; ,\n\\qquad U^{\\widetilde{u}}_i{U^{\\widetilde{u}}_i}^\\dagger=\\mathbbm{1} \\; .\n\\end{equation}\nSimilarly, for the down-type squarks it is\n\\begin{align}\n    M^{\\widetilde{d}_i}_{11}&=m_{\\widetilde{Q}_{iL}}^2 - \\frac{1}{24}(3g_2^{2}+g_1^2)(v_d^2+v_{jL}v_{jL}-v_u^2)-\\frac{1}{2}v_d^2{Y^{d}_i}^2 \n    \\label{eq:dsquarks11} \\\\\n    M^{\\widetilde{d}_i}_{12}&=\\frac{1}{2}(\\sqrt{2}A^d_i v_d -\\lambda v_d v_R) \\\\\n    M^{\\widetilde{d}_i}_{22}&=m_{\\widetilde{d}_{iR}}^2-\\frac{1}{12}g_1^2(v_d^2+v_{jL}v_{jL}-v_u^2)+\\frac{1}{2}v_d^2{Y^d_i}^2 \\; .\n    \\label{eq:dsquarks22}\n\\end{align}\nThe mass eigenstates $\\widetilde{d}_{i1}$ and $\\widetilde{d}_{i2}$ are obtained by the unitary transformation\n\\begin{equation}\n\\begin{pmatrix}\n \\widetilde{d}_{i1} \\\\\n \\widetilde{d}_{i2}\n\\end{pmatrix}\n=U^{\\widetilde{d}}_i\n\\begin{pmatrix}\n \\widetilde{d}_{iL} \\\\\n \\widetilde{d}_{iR}\n\\end{pmatrix}\\; ,\n\\qquad U^{\\widetilde{d}}_i{U^{\\widetilde{d}}_i}^\\dagger=\\mathbbm{1} \\; .\n\\end{equation}\n\n\n\n\n\\subsection{Charged scalar sector}\n\nSince $R$-parity, lepton number and lepton-flavor are broken, the six charged left- and right-handed sleptons mix with each other and with the two charged scalars from the Higgs doublets. In the basis $C^T=\n({H^-_d}^*,{H^+_u},\\widetilde{e}_{iL}^*,\\widetilde{e}_{jR}^*)$ we find\nthe following mass terms in the Lagrangian:\n\\begin{equation}\n\\cL_{C} \\; = \\; \n-{C^*}^T {m}^2_{H^+} C\\, ,\n\\label{matrix122}\n\\end{equation}\nwhere ${m}^2_{H^+}$ assuming ${\\CP}$ conservation is a symmetric\nmatrix of dimension 8,\n\\begin{align}\nm_{H^+}^2= \n  \\left( \\begin{array}{cccc} \n  m_{H_{d}^-{H^{-}_d}^{*}}^{2} & m_{H_{d}^- H_{u}^+}^{2} & \nm_{{H_d^-} \\widetilde{e}^*_{jL}}^2 & m_{{H_d^-} \\widetilde{e}^*_{jR}}^2  \\\\\n m_{{H_{u}^+}^* {H_d^-}^*}^{2}  & m_{{H^{+}_u}^{*} H_{u}^{+}}^{2} & \nm_{{H_u^+}^*\\widetilde{e}^*_{jL}}^{2} & m_{{H_u^+}^*\\widetilde{e}^*_{jR}}^{2} \\\\\n m_{\\widetilde{e}_{iL} {{H^{-}_d}}^{*}}^2 & m_{\\widetilde{e}_{iL} H_u^+}^2 &   \nm_{\\widetilde{e}_{iL} \\widetilde{e}_{jL}^{*}}^{2} &   \nm_{\\widetilde{e}_{iL} \\widetilde{e}_{jR}^{*}}^{2}\\\\\n m_{\\widetilde{e}_{iR} {{H^{-}_d}}^{*}}^2 & m_{\\widetilde{e}_{iR} H_u^+}^2 & \n  m_{\\widetilde{e}_{iR} \\widetilde{e}_{jL}^{*}}^{2} &   \nm_{\\widetilde{e}_{iR} \\widetilde{e}_{jR}^{*}}^{2}\n         \\end{array} \\right)\\ .  \n         \\label{matrixcharged2}\n\\end{align}\nThe entries are given\nin appendix \\ref{app:charged}.\nThe mass matrix is diagonalized by an orthogonal matrix $U^+$:\n\\begin{eqnarray}\nU^+\nm^2_{H^+}\\ {U^+}^{^T}\n=\n\\left(m^2_{H^+}\\right)^{\\text{diag}}\n\\ ,\n\\label{scalarhiggs22}\n\\end{eqnarray}\nwhere the diagonal elements of $\\left(m^2_{H^+}\\right)^{\\text{diag}}$ are the squared masses of the mass eigenstates\n\\begin{equation}\nH^+=U^+\\; C \\; ,\n\\end{equation}\nwhich include the charged Goldstone boson $H^+_1=G^\\pm_0$.\n\n\n\n\n\\subsection{Charged fermion sector}\n\nThe charged leptons mix with the charged gauginos and the charged higgsinos. Following the notation of \\citere{Ghosh:2017yeh} we write the relevant part of the Lagrangian in terms of two-component spinors \n$(\\chi^-)^T = \\left({({e_{iL})}^{c}}^{^*}, \\widetilde{W}^-, \\widetilde{H}^-_d\\right)$\nand \n$(\\chi^+)^T = \\left(({e_{jR}})^c, \\widetilde{W}^+, \\widetilde{H}^+_u\\right)$:\n\\begin{equation}\n\\cL_{\\chi^\\pm} \\; = \\; -({\\chi^{-}})^T \n{m}_{e}\n\\chi^+ + \\mathrm{h.c.}\\, .\n\\label{matrixcharginos0}\n\\end{equation}\nThe $5\\times5$ mixing matrix $m_e$ is defined by\n\\begin{equation}\nm_e=\n\\begin{pmatrix}\n    \\frac{v_d Y^e_{11}}{\\sqrt{2}} & \\frac{v_d Y^e_{12}}{\\sqrt{2}} & \\frac{v_d Y^e_{13}}{\\sqrt{2}} & \\frac{g_2 v_{1L}}{\\sqrt{2}} &\n    \t-\\frac{v_R Y^\\nu_1}{\\sqrt{2}} \\\\\n    \\frac{v_d Y^e_{21}}{\\sqrt{2}} & \\frac{v_d Y^e_{22}}{\\sqrt{2}} & \\frac{v_d Y^e_{23}}{\\sqrt{2}} & \\frac{g_2 v_{2L}}{\\sqrt{2}} & \n    \t-\\frac{v_R Y^\\nu_2}{\\sqrt{2}} \\\\\n    \\frac{v_d Y^e_{31}}{\\sqrt{2}} & \\frac{v_d Y^e_{32}}{\\sqrt{2}} & \\frac{v_d Y^e_{33}}{\\sqrt{2}} & \\frac{g_2 v_{3L}}{\\sqrt{2}} &\n    \t-\\frac{v_R Y^\\nu_3}{\\sqrt{2}} \\\\\n    0 & 0 & 0 & M_2 & \\frac{g_2 v_{u}}{\\sqrt{2}} \\\\\n    -\\frac{v_{iL}Y^e_{1i}}{\\sqrt{2}} & -\\frac{v_{iL}Y^e_{2i}}{\\sqrt{2}} & -\\frac{v_{iL}Y^e_{3i}}{\\sqrt{2}} & \\frac{g_2 v_{d}}{\\sqrt{2}} & \\frac{\\lambda v_R}{\\sqrt{2}}\n\\end{pmatrix} \\; .\n\\end{equation}\nIt is diagonalized by two unitary matrices $U^e_L$ and $U^e_R$:\n\\begin{equation}\n{U_R^e}^{^*} m_{e} {U_L^e}^{^\\dagger} = m_{e}^{\\text{diag}}\n\\, ,\n\\label{diagmatrixneutralinosn}\n\\end{equation}\nwhere $m_{e}^{\\text{diag}}$ contains the masses of the charged fermions in the mass eigenstate base\n\\begin{eqnarray}\n\\chi^{+} = {U_L^e}^{^\\dagger} \\lambda^+\\, ,\n\\\\\n\\chi^{-} = {U_R^e}^{^\\dagger} \\lambda^-\\, .\n\\label{physcharginos}\n\\end{eqnarray}\nThe smallness of the left-handed sneutrino vevs in comparison to the doublet ones assures the decoupling of the three leptons from the Higgsino and the wino.\n\n\n\n\n\\subsection{Neutral fermion sector}\n\nThe three left-handed neutrinos and the right-handed neutrino mix with\nthe neutral Higgsinos and gauginos. Again, following\n\\citere{Ghosh:2017yeh} we write the relevant part of the Lagrangian in\nterms of two-component spinors\n$({\\chi^{0}})^T=\\left({(\\nu_{iL})^{c}}^{^*},\\widetilde B^0, \n\\widetilde W^{0},\\widetilde H_{d}^0,\\widetilde H_{u}^0,\\nu_{R}^*\\right)$\nas\n\\begin{equation}\n\\cL_{\\chi^0} \\; = \\;-\\frac{1}{2} ({\\chi^{0}})^T \n{m}_{\\nu} \n\\chi^0 + \\mathrm{h.c.}\\, ,\n\\label{matrixneutralinos}\n\\end{equation}\nwhere ${m}_{\\nu}$ is the $8\\times 8$ symmetric mass matrix. The neutral fermion mass matrix is determined by\n\\begin{equation}\n{m}_{\\nu}=\n\\begin{pmatrix}\n 0 & 0 & 0 & -\\frac{g_1 v_{1L}}{2} & \\frac{g_2 v_{1L}}{2} &\n 0 & \\frac{v_{R} Y^\\nu_1}{\\sqrt{2}} & \\frac{v_{u}Y^\\nu_1}{\\sqrt{2}} \\\\\n 0 & 0 & 0 & -\\frac{g_1 v_{2L}}{2} & \\frac{g_2 v_{2L}}{2} &\n 0 & \\frac{v_{R} Y^\\nu_2}{\\sqrt{2}} & \\frac{v_{u}Y^\\nu_2}{\\sqrt{2}} \\\\\n 0 & 0 & 0 & -\\frac{g_1 v_{3L}}{2} & \\frac{g_2 v_{3L}}{2} &\n 0 & \\frac{v_{R} Y^\\nu_3}{\\sqrt{2}} & \\frac{v_{u}Y^\\nu_3}{\\sqrt{2}} \\\\\n -\\frac{g_1 v_{1L}}{2} & -\\frac{g_1 v_{2L}}{2} & -\\frac{g_1 v_{3L}}{2} &\n M_1 & 0 & -\\frac{g_1 v_{d}}{2} & \\frac{g_1 v_{u}}{2} & 0 \\\\\n  \\frac{g_2 v_{1L}}{2} & \\frac{g_2 v_{2L}}{2} & \\frac{g_2 v_{3L}}{2} &\n 0 & M_2 & \\frac{g_2 v_{d}}{2} & -\\frac{g_2 v_{u}}{2} & 0 \\\\\n 0 & 0 & 0 & -\\frac{g_1 v_{d}}{2} & \\frac{g_2 v_{d}}{2} & 0 &\n -\\frac{\\lambda v_{R}}{\\sqrt{2}} & -\\frac{\\lambda v_{u}}{\\sqrt{2}} \\\\\n \\frac{v_R Y^\\nu_1}{\\sqrt{2}} & \\frac{v_R Y^\\nu_2}{\\sqrt{2}} &\n \\frac{v_R Y^\\nu_3}{\\sqrt{2}} & \\frac{g_1 v_u}{2} & -\\frac{g_2 v_u}{2} &\n -\\frac{\\lambda v_R}{\\sqrt{2}} & 0 &\n \\frac{-\\lambda v_d + v_{kL}Y^\\nu_k}{\\sqrt{2}} \\\\\n \\frac{v_u Y^\\nu_1}{\\sqrt{2}} & \\frac{v_u Y^\\nu_2}{\\sqrt{2}} &\n \\frac{v_u Y^\\nu_3}{\\sqrt{2}} & 0 & 0 & -\\frac{\\lambda v_u}{\\sqrt{2}} &\n \\frac{-\\lambda v_d +v_{iL}Y^\\nu_i}{\\sqrt{2}} &\n \\sqrt{2}\\kappa v_R\n\\end{pmatrix}~.\n\\end{equation}\nBecause of the Majorana nature of the neutral fermions we can \ndiagonalize ${m}_{\\nu}$ with the help of just a single - but complex - \nunitary matrix $U^V$,\n\\begin{equation}\n{U^V}^{^*} m_{\\nu}\\ {U^V}^{^\\dagger} = m_{\\nu}^{\\text{diag}}\\, ,\n\\label{diagmatrixneutralinos}\n\\end{equation}\nwith\n\\begin{equation}\n\\chi^{0} = {U^V}^{^\\dagger} \\lambda^0\\, ,\n\\label{physneutralinos}\n\\end{equation}\nwhere $\\lambda^0$ are the two-component spinors in the mass basis. The eigenvalues of the diagonalized mass matrix $m_{\\nu}^{\\text{diag}}$ are the masses of the neutral fermions in the mass eigenstate basis. It turns out that the matrix $m_\\nu$ is of rank six, so it can only generate a single neutrino mass at tree-level.\\footnote{Including three generations of right-handed neutrinos, three light tree-level neutrino masses are generated.} The remaining two light neutrino masses can be generated by loop-effects.\n\n\n\n\n\n\\section{Renormalization of the Higgs potential at One-Loop}\n\\label{sec:renopot}\n\n\nThe first step in renormalizing the neutral scalar potential is to choose\nthe set of free parameters. These free parameters will receive a counter\nterm fixed by consistent renormalization conditions to cancel all\nultraviolet divergences that are produced by higher-order corrections. \n\nAt tree-level the relevant part of the Higgs potential $V_H$ is given by\nthe tadpole coefficients \\refeqs{eq:tp1}-(\\ref{eq:tp4}) and the ${\\CP}$-even and\n${\\CP}$-odd mass matrix elements in \\refeqs{matrixscalar1}\nand (\\ref{matrixscalar2}). The following\nparameters appear in the Higgs potential:\n\\begin{itemize}\n\t\\setlength\\itemsep{0.25em}\n\t\\renewcommand\\labelitemi{--}\n\t\\item Scalar soft masses: $m_{H_d}^2$, $m_{H_u}^2$, $m_{\\widetilde{\\nu}_R}^2$,\n\t\t\t$\\left( m_{\\widetilde{L}_L}^2 \\right)_{ij}$,\n\t\t\t$\\left( m_{H_d\\widetilde{L}_L}^2 \\right)_{i}$\n\t\t\t$\\quad$ (12 parameters)\n\t\\item Vacuum expectation values: $v_d$, $v_u$, $v_R$, $v_{iL}$\n\t\t\t$\\quad$ (6 parameters)\n\t\\item Gauge couplings: $g_1$, $g_2$ $\\quad$ (2 parameters)\n\t\\item Superpotential parameters: $\\lambda$, $\\kappa$, $Y^\\nu_i$\n\t\t\t$\\quad$ (5 parameters)\n\t\\item Soft trilinear couplings: $T^\\lambda$, $T^\\kappa$, $T^\\nu_i$\n\t\t\t$\\quad$ (5 parameters)\n\\end{itemize}\nThe complexity of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ Higgs scalar\nsector becomes evident when we\ncompare the numbers of free parameters (30) with the one in the real\nMSSM (7) \\cite{Frank:2006yh} and the NMSSM (12)\n\\cite{Drechsel:2016ukp}. While the number of free parameters is fixed,\nwe are free to replace some of the parameters by physical parameters. We\nchose to make the following replacements: \n\nThe soft masses $m_{H_d}^2$, $m_{H_u}^2$, $m_{\\widetilde{\\nu}_R}^2$, and\nthe diagonal elements of the matrix $m_{\\widetilde{L}_L}^2$ will\nbe replaced  \nby the tadpole coefficients. The substitution is defined by\nthe tadpole~\\refeqs{eq:tp1}-(\\ref{eq:tp4})\nsolved for the soft mass parameters just\nmentioned. This will give us the possiblity\nto define the\nrenormalization scheme in a way that the true vacuum is not spoiled by\nthe higher-order corrections. The Higgs doublet vevs $v_d$ and $v_u$ will be replaced by the MSSM-like parameters\n$\\tan\\beta$ and $v$ according to \n\\begin{equation}\\label{eq:tanbetadef}\n\\tan\\beta=\\frac{v_u}{v_d} \\qquad \\text{and} \\qquad\nv^2=v_d^2+v_u^2+v_{iL}v_{iL} \\; .\n\\end{equation}\nNote that the definition of $v^2$ differs from the one in the MSSM by\nthe term $v_{iL}v_{iL}$. This allows to maintain the relations\nbetween $v^2$ and the gauge boson masses as they are in the\nMSSM. Numerically, the difference in the definition of $v^2$ is\nnegligible, since the $v_{iL}$ are of the order of $10^{-4}\\,\\, \\mathrm{GeV}$ in\nrealistic scenarios. Analytically, however, maintaining the\nfunctional form of $\\tan\\beta$ as it is in the (N)MSSM is convenient\nto facilitate the comparison of the quantum corrections in the\n\\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ and the NMSSM. In particular, we can still express the one-loop\ncounterterm of $\\tan\\beta$ without having to include the counterterms\nfor the left-handed sneutrino vevs.\nFor the vev of the\nright-handed sneutrino we chose to make the same substitution as was\ndone in previous calculations in the NMSSM \\cite{Drechsel:2016ukp} \n\\begin{equation}\\label{eq:mudef}\n\\mu=\\frac{v_R \\lambda}{\\sqrt{2}} \\; ,\n\\end{equation}\nwhere we make use of the fact that when the sneutrino obtains the vev,\nthe $\\mu$-term of the MSSM is dynamically generated.\nThe gauge couplings $g_1$ and $g_2$ will be replaced by the\ngauge boson masses $\\MW$ and $\\MZ$ via the definitions\n\\begin{equation}\\label{eq:gaguebosonmasses}\n\\MW^2=\\frac{1}{4}g_2^2v^2 \\qquad \\text{and} \\qquad\n\\MZ^2=\\frac{1}{4}\\left( g_1^2+g_2^2\\right) v^2 \\; .\n\\end{equation}\nThis is reasonable because the gauge boson masses are well measured\nphysical observables, so we can define them as OS parameters.\nInterestingly, the mass counterterm for $\\MW^2$ drops out at one-loop,\nbut it will contribute in the definition of the counterterm for $v^2$,\nso it is not a redundant parameter.\nFor the soft trilinear couplings we chose to adopt the redefinitions\n\\begin{equation}\nT^\\lambda = A^\\lambda \\lambda \\; ,  \\qquad T^\\kappa = A^\\kappa \\kappa\n\\; , \\qquad T^\\nu_i=A^\\nu_i Y^\\nu_i \\; .\n\\end{equation} \nThe reparametrization from the initial to the physical set of independent parameters is summarized in \\refta{tab:setpara}.\n\\begin{table}\n\\centering\n\\begin{tabular}{c c c c c}\nSoft masses & VEVs & Gauge cpl. & Superpot. & Soft trilinears \\\\\n\\hline\n$m_{H_d}^2$, $m_{H_u}^2$, $m_{\\widetilde{\\nu}_R}^2$,\n${m_{\\widetilde{L}_L}^2}_{ij}$,\n\t\t\t${m_{H_d\\widetilde{L}_L}^2}_i$ \n & $v_d$, $v_u$, $v_R$, $v_{iL}$ &\n$g_1$, $g_2$ &\n$\\lambda$, $\\kappa$, $Y^\\nu_i$ &\n$T^\\lambda$, $T^\\kappa$, $T^\\nu_i$ \\\\\n$\\downarrow$ & $\\downarrow$ & $\\downarrow$ & $\\downarrow$ & $\\downarrow$ \\\\\n $T_{H_d^{\\mathcal{R}}}$, $T_{H_u^{\\mathcal{R}}}$,\n $T_{\\widetilde{\\nu}_{R}^{\\mathcal{R}}}$,\n $T_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}}}$,&\n $\\tan\\beta$, $v$, $\\mu$, $v_{iL}$ &\n $\\MW$, $\\MZ$ &\n $\\lambda$, $\\kappa$, $Y^\\nu_i$ &\n $A^\\lambda$, $A^\\kappa$, $A^\\nu_i$ \\\\\n ${m_{\\widetilde{L}_L}^2}_{i\\neq j}$,\n ${m_{H_d\\widetilde{L}_L}^2}_i$ & & & & \n\\end{tabular}\n\\caption{Set of independent parameters initially entering the tree-level Higgs potential of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ in the first row, and final choice of free parameters after the substitutions mentioned in the text.}\n\\label{tab:setpara}\n\\end{table}\n\nIn the following we will regard the entries of the neutral scalar mass matrix as functions of the final set of parameters,\n\\begin{align}\nm_{\\varphi}^2&=m_{\\varphi}^2\\left( \\MZ^2, v^2, \\tan\\beta, \\lambda, \\dots \\right) \\; , \\\\\nm_{\\sigma}^2&=m_{\\sigma}^2\\left( \\MZ^2, v^2, \\tan\\beta, \\lambda, \\dots \\right) \\; ,\n\\end{align}\nand we define their renormalization as\n\\begin{align}\nm_{\\varphi}^2&\\to m_{\\varphi}^2+\\delta m_{\\varphi}^2 \\; , \\\\\nm_{\\sigma}^2&\\to m_{\\sigma}^2+\\delta m_{\\sigma}^2 \\; .\n\\end{align}\nThe mass counterterms $\\delta m_{\\varphi}^2$ and $\\delta m_{\\sigma}^2$\nenter the renormalized one-loop scalar self-energies. They have to be\nexpressed as a linear combination of the counterterms of the independent\nparameters. We define their one-loop renormalization as\n\\begin{equation}\n  \\begin{split}\nT_{H_d^{\\mathcal{R}}} &\\to\nT_{H_d^{\\mathcal{R}}}+\\delta T_{H_d^{\\mathcal{R}}} \\; , \\\\\nT_{H_u^{\\mathcal{R}}} &\\to\nT_{H_u^{\\mathcal{R}}}+\\delta T_{H_u^{\\mathcal{R}}} \\; , \\\\\nT_{\\widetilde{\\nu}_{R}^{\\mathcal{R}}} &\\to\nT_{\\widetilde{\\nu}_{R}^{\\mathcal{R}}}+\\delta T_{\\widetilde{\\nu}_{R}^{\\mathcal{R}}}\n\\; , \\\\\nT_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}}} &\\to\nT_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}}}+ \\delta T_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}}}\n\\; , \\\\\n{m_{\\widetilde{L}_L}^2}_{i\\neq j} &\\to\n{m_{\\widetilde{L}_L}^2}_{i\\neq j}+ \\delta {m_{\\widetilde{L}_L}^2}_{i\\neq j}\n\\; , \\\\\n{m_{H_d\\widetilde{L}_L}^2}_i &\\to\n{m_{H_d\\widetilde{L}_L}^2}_i+\\delta {m_{H_d\\widetilde{L}_L}^2}_i \\; ,\n  \\end{split}\n\\qquad\n  \\begin{split}\n\\tan\\beta &\\to \\tan\\beta+\\delta \\tan\\beta \\; , \\\\\nv^2 &\\to v^2+\\delta v^2 \\; , \\\\\n\\mu &\\to \\mu +\\delta \\mu \\; , \\\\\nv_{iL}^2 &\\to v_{iL}^2+\\delta v_{iL}^2 \\; , \\\\\n\\MW^2 &\\to \\MW^2 +\\delta \\MW^2 \\; , \\\\\n\\MZ^2 &\\to \\MZ^2 +\\delta \\MZ^2 \\; , \\\\\n  \\end{split}\n \\qquad\n  \\begin{split}\n\\lambda &\\to \\lambda + \\delta \\lambda \\; , \\\\\n\\kappa &\\to \\kappa + \\delta \\kappa \\; , \\\\\nY^\\nu_i &\\to Y^\\nu_i + \\delta Y^\\nu_i \\; , \\\\\nA^\\lambda &\\to A^\\lambda+\\delta A^\\lambda \\; , \\\\\nA^\\kappa &\\to A^\\kappa + \\delta A^\\kappa \\; , \\\\\nA^\\nu_i &\\to A^\\nu_i + \\delta A^\\nu_i \\; .\n  \\end{split}\n\\end{equation}\nSince the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ is a renormalizable theory, the divergent parts of the\ncounterterms are fixed to cancel the UV divergences.\nThe finite pieces, and thus the meaning of the parameters have to be\nfixed by renormalization conditions. \nWe will adopt a mixed renormalization scheme, where tadpoles\nand gauge boson masses are fixed OS, and the other parameters are\nfixed in the $\\overline{\\text{DR}}$ scheme. The exact renormalization\nconditions will be given in \\refse{sec:condis}. The dependence of the\nmass counterterms $\\delta m_{\\varphi}^2$ and $\\delta\nm_{\\sigma}^2$ on the counterterms of the free parameters is given\nat one-loop by \n\\begin{equation}\n\\delta m_{\\varphi}^2=\\sum_{X\\in\\text{Free param.}} \\left(\n\\frac{\\partial}{\\partial X}  m_{\\varphi}^2 \\right)\\delta X \\; , \n\\qquad\n\\delta m_{\\sigma}^2=\\sum_{X\\in\\text{Free param.}} \\left(\n\\frac{\\partial}{\\partial X}  m_{\\sigma}^2 \\right)\\delta X \\; . \n\\label{eq:masscounterderive}\n\\end{equation}\nIn our calculation the mixing matrices are defined in a way to diagonalize the renormalized mass matrices, so they do not have to be renormalized, because they are defined exclusively by renormalized quantities. The expressions for the counterterms of the scalar mass matrices in the mass eigenstate basis are then simply\n\\begin{equation}\\label{eq:masscounterrot}\n\\delta m_{h}^2=U^H \\delta m_{\\varphi}^2 {U^H}^T \\; , \n\\qquad\n\\delta m_{A}^2=U^A \\delta m_{\\sigma}^2{U^A}^T \\; .\n\\end{equation}\nIt should be noted at this point that the counterterm matrices in the mass eigenstate basis $\\delta m_{h}^2$ and $\\delta m_{A}^2$ are not diagonal, as they would be in a purely OS renormalization procedure, which is often used in theories with flavor mixing \\cite{Grimus:2016hmw}.\n\nIn the following chapter we will discuss the field renormalization, which is necessary to obtain finite scalar self-energies at arbitrary momentum.\n\n\n\n\n\\subsection{Field renormalization}\n\nWe write the renormalization of the neutral scalar-component fields as\n\\begin{equation}\\label{eq:fieldrenodef}\n\\begin{pmatrix}\nH_d \\\\ H_u \\\\ \\widetilde{\\nu}_R \\\\ \\widetilde{\\nu}_{iL}\n\\end{pmatrix}\n\\to \\sqrt{Z} \n\\begin{pmatrix}\nH_d \\\\ H_u \\\\ \\widetilde{\\nu}_R \\\\ \\widetilde{\\nu}_{iL}\n\\end{pmatrix}=\n\\left( \\mathbbm{1} + \\frac{1}{2} \\delta Z \\right)\n\\begin{pmatrix}\nH_d \\\\ H_u \\\\ \\widetilde{\\nu}_R \\\\ \\widetilde{\\nu}_{iL}\n\\end{pmatrix} \\; ,\n\\end{equation}\nwhere $\\sqrt{Z}$ and $\\delta Z$ are $6\\times6$ dimensional matrices and the equal sign is valid at one-loop. It should be emphasized that in contrast to the MSSM and the NMSSM these matrices cannot be made diagonal even in the interaction basis. The reason is that the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ explicitly breaks lepton number and lepton flavor, so the fields $H_d$ and $\\widetilde{\\nu}_{iL}$ share exactly the same quantum numbers and kinetic mixing terms are already generated at one-loop order.\n\nFor the ${\\CP}$-even and ${\\CP}$-odd neutral scalar fields the \ndefinition in \\refeq{eq:fieldrenodef} implies the following field\nrenormalization in the mass eigenstate basis: \n\\begin{equation}\nh \\to \\left( \\mathbbm{1} + \\frac{1}{2} \\delta Z^H \\right) h \\; , \n\\qquad\nA \\to \\left( \\mathbbm{1} + \\frac{1}{2} \\delta Z^A \\right) A \\; ,\n\\end{equation}\nwith\n\\begin{equation}\\label{eq:fieldrenorotate}\n\\delta Z^H = U^H \\left(\\delta Z \\right) {U^H}^T \\; \\qquad \\text{and} \\qquad\n\\delta Z^A = U^A \\left(\\delta Z \\right) {U^A}^T \\; .\n\\end{equation}\nAs renormalization conditions for the field renormalization counterterms we chose to adopt the $\\overline{\\text{DR}}$ scheme. We calculate the UV-divergent part of the derivative of the scalar ${\\CP}$-even self-energies in the interaction basis and define\n\\begin{equation}\\label{eq:drfieldcounterdef}\n\\delta Z_{ij}=-\\left.\\frac{d}{dp^2}\\Sigma_{\\varphi_i \\varphi_j} \\right|^{\\rm div} \\; .\n\\end{equation}\nHere $\\mbox{}^{\\rm div}$ denotes taking the divergent part only,\nproporional to $\\Delta$, \n\\begin{equation}\n\\Delta = \\frac{1}{\\varepsilon}-\\gamma_E+\\ln{4\\pi} \\; ,\n\\end{equation}\nwhere loop integral are solved in $4-2\\varepsilon$ dimensions and\n$\\gamma_E=0.5772\\dots$ is the Euler-Mascharoni constant.\nSince the field renormalization constants contribute only via divergent\nparts, they do not contribute to the finite result after canceling \ndivergences in the self-energies.\nAs regularization\nscheme we chose dimensional reduction~\\cite{SIEGEL1979193,CAPPER1980479}, \nwhich was shown to be SUSY\nconserving at one-loop~\\cite{Stockinger:2005gx}. In contrast to the\nOS renormalization scheme our field renormalization matrices are\nhermitian.\nThis holds also\ntrue for the field renormalization in the mass eigenstate basis, because\nas already mentioned the rotations in \\refeq{eq:scalarhiggs} and\n\\refeq{eq:scalaroddhiggs} diagonalize the renormalized tree-level scalar\nmass matrices, so \\refeqs{eq:fieldrenorotate} do not introduce\nnon-hermitian parts into the field renormalization, that would have to\nbe canceled by a renormalization of the mixing matrices $U^H$ and $U^A$\nthemselves. \n\nIn appendix \\ref{app:fieldcounters} we list our field renormalization\ncounterterms $\\delta Z_{ij}$ in terms of the divergent quantity $\\Delta$.\nNote that the field counterterms mixing the down-type Higgs and the left-handed sleptons are proportional to the neutrino Yukawa couplings $Y^\\nu_i$, while the counterterms mixing different flavors of left-handed sneutrinos contain terms proportional to non-diagonal lepton Yukawa couplings $Y^e$ and  terms proportional to $Y^\\nu_i Y^\\nu_j$. This is why their numerical impact is negligible, but they are needed for a consistent renormalization of the scalar self-energies.\n\n\n\n\n\\subsection{Renormalization conditions for free parameters}\n\\label{sec:condis}\n\nIn this section we describe our choice for the renormalization\nconditions, where we stick to the one-loop level everywhere. \nWe start\nwith the OS conditions for the gauge boson mass parameters and the\ntadpole coefficients followed by our definitions for the\n\\DRbar\\ renormalized parameters.\n\nThe SM gauge boson masses are renormalized OS requiring\n\\begin{equation}\n\\text{Re}\\left[{\\hat{\\Sigma}_{ZZ}}^T\\left( \\MZ^2 \\right) \\right]=0 \\qquad\n\\text{and} \\qquad\n\\text{Re}\\left[{\\hat{\\Sigma}_{WW}}^T\\left( \\MW^2 \\right) \\right]=0 \\; ,\n\\end{equation}\nwhere $\\hat{\\Sigma}^T$ stands for the transverse part of the renormalized gauge boson self-energy. For their mass counterterms these conditions yield\n\\begin{equation}\n\\delta \\MZ^2 = \\text{Re}\\left[ {\\Sigma}_{ZZ}^T\\left( \\MZ^2 \\right) \\right] \\qquad\n\\text{and} \\qquad\n\\delta \\MW^2 = \\text{Re}\\left[ {\\Sigma}_{WW}^T\\left( \\MW^2 \\right) \\right] \\; .\n\\end{equation}\nHere the ${\\Sigma}^T$ (without the hat) denote the transverse part of the unrenormalized gauge boson self-energies.\n\nFor the tadpole coefficients $T_{\\varphi_i}$ the OS conditions read\n\\begin{equation}\nT_{\\varphi_i}^{(1)}+\\delta T_{\\varphi_i}=0 \\; ,\n\\end{equation}\nwhere $T_{\\varphi_i}^{(1)}$ are the one-loop contributions to the linear terms of the scalar potential, stemming from tadpole diagrams shown in \\reffi{fig:tadpoles}. The tadpole diagrams are calculated in the mass eigenstate basis $h$. The one-loop tadpole contributions in the interaction basis $\\varphi$ are then obtained by the rotation\n\\begin{equation}\nT_\\varphi^{(1)} = {U^H}^T T_h^{(1)} \\; .\n\\end{equation}\nAccordingly we find for the one-loop tadpole counterterms\n\\begin{equation}\n\\delta T_{\\varphi_i}=-T_{\\varphi_i}^{(1)} \\; .\n\\end{equation}\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.6\\textwidth]{tadpoles.pdf}\n  \\caption{Generic Feynman diagrams for the tadpoles $T_{h_i}$.}\n  \\label{fig:tadpoles}\n\\end{figure}\n\nFor practical purposes we\ndecided to renormalize all remaining parameters in the $\\DRbar$\nscheme (reflecting the fact that there are no physical observables that\ncould be directly related to them). \nThe counterterms of each parameter were obtained by\ncalculating the divergent parts of one-loop corrections to different\nscalar and fermionic two- and three-point functions. \nWe state the determination of the counterterms in the (possible) order \nin which they can be successively derived.\nWe start with the counterterms that\nwere obtained by renormalizing certain neutral fermion self-energies. \n\n\\paragraph{\\protect\\boldmath Renormalization of $\\mu$:}\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.8\\textwidth]{chiSE.pdf}\n  \\caption{Diagrams contributing to the neutral\n    fermion self-energies in the interaction basis.}\n\\label{fig:neufermdiags}\n\\end{figure}\nThe $\\mu$ parameter appears isolated in the Majorana-type mass matrix of\nthe neutral fermions \n\\begin{equation}\n\\left(m_\\nu\\right)_{67}=-\\frac{\\lambda v_R}{\\sqrt{2}}=-\\mu \\; ,\n\\end{equation}\nwhich is the element mixing the down-type and the up-type Higgsinos\n$\\widetilde{H}_d$ and $\\widetilde{H}_u$.\nThe entries\n$\\left(m_\\nu\\right)_{ij}$ get one-loop corrections via the neutral\nfermion self-energies ${\\sum}_{\\neu{i} \\neu{j}}$, that for Majorana\nfermions can be decomposed as\\footnote{Left-handed components and\n  right-handed components are the same for Majorana fields.} \n\\begin{equation}\n\\Sigma_{\\neu{i} \\neu{j}}\\left( p^2 \\right) = \\slashed{p}\\Sigma_{\\neu{i} \\neu{j}}^F\\left( p^2 \\right)+\\Sigma_{\\neu{i} \\neu{j}}^S\\left( p^2 \\right) \\; .\n\\end{equation}\nThe part $\\Sigma_{\\neu{i} \\neu{j}}^F$ is renormalized through field\nrenormalization and the part $\\Sigma_{\\neu{i} \\neu{j}}^S$ is\nrenormalized by both the field renormalization and a mass counter\nterm. Since we are interested in the mass renormalization we focus on\n$\\Sigma_{\\neu{i} \\neu{j}}^S$ and write for the renormalized self-energy\nat zero momentum \n\\begin{equation}\\label{eq:neufermreno}\n\\hat{\\Sigma}_{\\neu{i} \\neu{j}}^S\\left(0\\right)=\\Sigma_{\\neu{i} \\neu{j}}^S\\left(0\\right)-\\frac{1}{2}\n\t\\left( \\delta Z_{ki}^\\chi \\left( m_\\nu \\right)_{kj} +\n\t\t\t\\left( m_\\nu \\right)_{ik} \\delta Z_{kj}^\\chi \\right)\n\t\t-\\delta\\left(m_\\nu\\right)_{ij} \\; .\n\\end{equation}\nThe field renormalization constants can be obtained by calculating the divergent part of $\\Sigma_{\\neu{i} \\neu{j}}^F$:\n\\begin{equation}\n\\delta Z_{ij}^\\chi=-\\left.\\Sigma_{\\neu{i}\\neu{j}}^F\\right|^{\\rm div} \\; ,\n\\end{equation}\nwhere we make use of the fact that there are no divergences proportional\nto $p^2$ in our case. The divergent parts of the self-energies of the\nneutral fermions are calculated diagrammatically in the interaction\nbasis, where diagrams with mass insertions have to be included. In\n\\reffi{fig:neufermdiags} we show the generic diagrams potentially\ncontributing to the divergent part of the self-energies.\nDiagrams with a scalar mass insertion or more than\none fermionic mass insertion are power-counting finite, so we do\nnot depict them.\nThe diagram shown in \\reffi{fig:neufermdiags} with a mass\ninsertion on the chargino propagator\ncan be divergent depending on the expressions for the couplings of the\ncharginos.\n\nWe checked that our results for the field renormalization counterterms\nfor the neutral fermions are consistent with the one-loop anomalous\ndimensions $\\gamma_{ij}^{(1)}$ of the corresponding superfields, i.e., \n\\begin{equation}\n\\delta Z_{ij}^\\chi=\\frac{\\gamma_{ij}^{(1)}\\Delta}{16 \\pi^2} \\; .\n\\end{equation}\nTo extract $\\delta \\mu$ we now just have to identify \n\\begin{equation}\n\\delta\\left(m_\\nu\\right)_{67}=-\\delta\\mu \\; ,\n\\end{equation}\nand calculate the divergent part of $\\Sigma_{\\widetilde{H}_d \\widetilde{H}_u}^S$, which again is not momentum dependent. $\\delta\\mu$ is then given by\n\\begin{equation}\n\\delta\\mu=\\frac{1}{2}\\mu\\left(\n\t-\\left(\n\t\t\\delta Z_{66}^\\chi+\n\t\t\\delta Z_{77}^\\chi\n\t\\right)+\n\t\\frac{1}{\\lambda}\\left(\n\t\\delta Z_{16}^\\chi Y^\\nu_1+\n\t\\delta Z_{26}^\\chi Y^\\nu_2+\n\t\\delta Z_{36}^\\chi Y^\\nu_3 \\right)\\right) -\n\t\\left.\\Sigma_{\\widetilde{H}_d \\widetilde{H}_u}^S\\right|^{\\rm div} \\; ,\n\\end{equation}\nwhere we made us of the fact that the matrix $\\delta Z_{ij}^\\chi$ is real and symmetric and that components mixing left-handed neutrinos and the down-type Higgsino are the only non-diagonal elements contributing here.\n\nExplicit formulas for the counterterms of the parameters renormalized in the $\\DRbar$ scheme are listed in the appendix \\ref{app:paracounters}. \nFor the $\\DRbar$ counterterms we checked that in the limit $Y^\\nu_i\\rightarrow 0$ our results coincide with the one in the \nNMSSM~\\cite{Ellwanger:2009dp}.\n\n\\paragraph{\\protect\\boldmath Renormalization of $\\kappa$:}\nThe parameters $\\kappa$ appears isolated at tree-level in the three-point vertex that couples the right-handed neutrino to the right-handed sneutrino,\n\\begin{equation}\n\\Gamma_{\\nu_R \\nu_R \\widetilde{\\nu}_R}^{(0)}=-\\sqrt{2}\\kappa \\; .\n\\end{equation}\nThe divergences induced to this coupling at one-loop have to be absorbed by the field renormalization of the right-handed neutrino and sneutrino and the counterterm for $\\kappa$, which is the only parameter in the tree-level expression. We find\n\\begin{equation}\n\\delta \\kappa = \n\\frac{1}{\\sqrt{2}}\n\\left.\\Gamma{\\nu_R \\nu_R \\widetilde{\\nu}_R}^{(1)}\\right|^{\\rm div} -\n\\frac{1}{2}\\kappa\\left( \\delta Z_{33}+2 \\delta Z^\\chi_{88}\\right) \\; ,\n\\end{equation}\nwhere $\\Gamma{\\nu_R \\nu_R \\widetilde{\\nu}_R}^{(1)}\\rvert^{\\rm div}$ is the divergent part of the corresponding one-loop three-point function, and the terms containing the field renormalization is trivial, because there is only one singlet-like superfield so that no non-diagonal field renormalization constants appear. The divergent one-loop contributions to the vertex are calculated diagrammatically in the interaction basis. The only contributing generic diagrams are shown in \\reffi{fig:kappadiags}.\n\nAll other topologies, including diagrams with one or more\nmass insertion, are finite, and there are no diagrams with gauge bosons\ninstead of scalars  in the loop, because there are three gauge-singlet\nfields on the outer legs. It turns out that the sum over the diagrams\nshown in \\reffi{fig:kappadiags} is also finite, so that \n$\\Gamma_{\\nu_R \\nu_R\\widetilde{\\nu}_R}^{(1)}\\rvert^{\\rm div}$ vanishes. \n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=0.6\\textwidth]{kappadiags.pdf}\n  \\caption{Potentially divergent one-particle irreducible diagrams\n    contributing to the three-point vertex between two right-handed\n    neutrinos and one right-handed sneutrino.}\n  \\label{fig:kappadiags}\n\\end{figure}\n\n\\paragraph{\\protect\\boldmath Renormalization of $\\lambda$:}\nHaving calculated $\\delta \\mu$ and $\\delta \\kappa$ we can extract the counterterm for $\\lambda$ in the neutral fermion sector. $\\lambda$ appears in the mass matrix element\n\\begin{equation}\n\\left( m_\\nu \\right)_{88}=\\frac{2\\kappa\\mu}{\\lambda} \\; .\n\\end{equation}\nMaking use of \\refeq{eq:neufermreno} we find\n\\begin{equation}\n\\delta\\lambda=\n\\lambda\\left( \\delta Z^\\chi_{88}+\\frac{\\delta\\kappa}{\\kappa}\n+\\frac{\\delta\\mu}{\\mu} \\right)-\\frac{\\lambda^2}{2\\mu\\kappa}\n\\left.\\Sigma_{\\nu_R \\nu_R}^S\\right|^{\\rm div} \\; ,\n\\end{equation}\nwhere we calculated the divergent part of the right-handed neutrino self-energie $\\Sigma_{\\nu_R \\nu_R}^S\\rvert^{\\rm div}$ diagrammatically in the interaction basis using the diagrams already shown in \\reffi{fig:neufermdiags}.\n\n\\paragraph{\\protect\\boldmath Renormalization of $A_\\kappa$:}\nThe counterterm for the parameter $A_\\kappa$ can be extracted from the one-loop corrections to the scalar three-point vertex of right-handed sneutrinos when $\\delta\\kappa$ is known and using the one-loop relation\n\\begin{equation}\n\\left[ \\frac{\\delta\\mu}{\\mu} - \\frac{\\delta\\lambda}{\\lambda} \\right]^{\\rm div}\n=\\left.\\frac{1}{2}\\delta Z_{33}\\right|^{\\rm div} \\; ,\n\\end{equation}\nwhich was found in the NMSSM~\\cite{Sperling:2013eva} and confirmed\nfor this work also in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}.\nFor the trilinear singlet vertex we have at tree-level \n\\begin{equation}\n\\Gamma_{\\widetilde{\\nu}_R\\widetilde{\\nu}_R\\widetilde{\\nu}_R}^{(0)}=\n-\\sqrt{2}\\kappa\\left(A_\\kappa+\\frac{6\\kappa\\mu}{\\lambda}\\right) \\; .\n\\end{equation}\nThe tree-level vertex does not depend on the momentum, so the one-loop counterterm for $A_\\kappa$ can be calculated through\n\\begin{equation}\n\\delta A_\\kappa = \\frac{1}{\\sqrt{2}\\kappa}\\left( \n\\left.\\Gamma_{\\widetilde{\\nu}_R\\widetilde{\\nu}_R\\widetilde{\\nu}_R}^{(1)}\\right|^{\\rm div} +\n\\frac{3}{2} \\delta Z_{33}\n\\Gamma_{\\widetilde{\\nu}_R\\widetilde{\\nu}_R\\widetilde{\\nu}_R}^{(0)} \\right)\n-A_\\kappa \\frac{\\delta \\kappa}{\\kappa}-\n\\frac{6\\kappa\\mu}{\\lambda} \\left( 2\\frac{\\delta\\kappa}{\\kappa}+\n\\frac{1}{2}\\delta Z_{33} \\right) \\; .\n\\end{equation}\nHere $\\Gamma_{\\widetilde{\\nu}_R\\widetilde{\\nu}_R\\widetilde{\\nu}_R}^{(1)}\\rvert^{\\rm div}$ is the divergent part of the one-loop corrections to the three-point vertex, which was calculated diagrammatically in the interaction basis. The number of contributing diagrams is rather high, so for simplicity we just show the topologies of the diagrams contributing, that potentially lead to divergences, in \\reffi{fig:triplescalardiags}.\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.7\\textwidth]{triplescalardiags.pdf}\n\\caption{Potentially divergent one-particle irreducible topologies\n  contributing to a scalar three-point vertex at one-loop in the\n  interaction basis. Diagrams with scalar mass insertions or more than\n  one fermionic mass insertions are finite.} \n\\label{fig:triplescalardiags}\n\\end{figure}\nIn the case of the vertex\n$\\Gamma_{\\widetilde{\\nu}_R\\widetilde{\\nu}_R\\widetilde{\\nu}_R}$ we can\nneglect the diagrams with gauge bosons, because the right-handed\nsneutrinos are gauge singlets.  \n\n\\paragraph{\\protect\\boldmath Renormalization of $A_\\lambda$:}\nThe counterterm for the parameter $A_\\lambda$ is like in the previous case extracted from the one-loop corrections to a scalar three-point function. Here we consider $\\Gamma_{H_d H_u \\widetilde{\\nu}_R}$, the coupling between the two doublet-type Higgses and the right-handed sneutrino. At tree-level it is\n\\begin{equation}\n\\Gamma_{H_d H_u \\widetilde{\\nu}_R}^{(0)}=\n\\frac{A_\\lambda \\lambda}{\\sqrt{2}}+\n\\sqrt{2}\\kappa\\mu \\; ,\n\\end{equation}\nso we will make use of the fact that we already know the counterterms for $\\lambda$, $\\kappa$ and $\\mu$.\n\nThe final expression defining $\\delta A_\\lambda$ will also contain the\ntree-level expressions for the couplings where the down-type Higgs is\nreplaced by one of the left-handed sneutrinos. They are induced by the\nnon-diagonal field renormalization of $H_d$ and\n$\\widetilde{\\nu}_{iL}$ and enter the\nrenormalization of $\\Gamma_{H_d H_u \\widetilde{\\nu}_R}$ at one-loop. We\nfind\n\\begin{align}\n\\delta A_\\lambda&=\n-\\frac{\\sqrt{2}}{\\lambda}\n\\left.\\Gamma_{H_d H_u \\widetilde{\\nu}_R}^{(1)}\\right|^{\\rm div}-\n\\frac{1}{\\sqrt{2}\\lambda}\\left(\n\\delta Z_{11}\\Gamma_{H_d H_u \\widetilde{\\nu}_R}^{(0)}+\n\\delta Z_{14}\\Gamma_{\\widetilde{\\nu}_{1L}H_u\\widetilde{\\nu}_R}^{(0)}+\n\\delta Z_{15}\\Gamma_{\\widetilde{\\nu}_{2L}H_u\\widetilde{\\nu}_R}^{(0)}\n    \\right. \\notag \\\\  +& \\left.\n\\delta Z_{16}\\Gamma_{\\widetilde{\\nu}_{3L}H_u\\widetilde{\\nu}_R}^{(0)}+\n\\delta Z_{22}\\Gamma_{H_d H_u \\widetilde{\\nu}_R}^{(0)}+\n\\delta Z_{33}\\Gamma_{H_d H_u \\widetilde{\\nu}_R}^{(0)} \\right)-\n\\frac{A_\\lambda}{\\lambda}\\delta\\lambda-\n\\frac{2\\kappa}{\\lambda}\\delta\\mu-\n\\frac{2\\mu}{\\lambda}\\delta\\kappa \\; ,\n\\end{align}\nwith\n\\begin{equation}\n\\Gamma_{\\widetilde{\\nu}_{iL}H_u\\widetilde{\\nu}_R}^{(0)}=\n\\frac{-Y^\\nu_i\\left( A^\\nu_i+\\frac{2\\kappa\\mu}{\\lambda} \\right)}{\\sqrt{2}}\n\\; .\n\\end{equation}\n\n\\paragraph{\\protect\\boldmath Renormalization of $v^2$:}\nThe SM-like vev is renormalized via the\nrenormalization of the electromagnetic coupling in the Thompson limit,\nwhich can be done when the counterterms for the gauge boson masses are\nfixed. We follow here the approach of \\citere{Drechsel:2016ukp} used in\nthe NMSSM to be able to compare the results in both models as best as\npossible.\n\nThe renormalization of the electromagnetic coupling is defined by\n\\begin{equation}\ne \\to e \\left( 1+ \\delta Z_e \\right) \\; ,\n\\end{equation}\nand the counterterm $\\delta Z_e$ can be calculated via\n\\begin{equation}\n\\left.\\delta Z_e\\right|^{\\rm div}=\n\\left[\n\\frac{1}{2} \\left( \\frac{\\partial \\Sigma_{\\gamma\\gamma}^T}{\\partial p^2} \\left( 0 \\right) \\right) +\n\\frac{\\SW}{\\CW \\MZ^2} \\Sigma_{\\gamma Z}^T \\left( 0 \\right)\n\\right]^{\\rm div} \\; ,\n\\end{equation}\nwhere $\\Sigma_{\\gamma\\gamma}^T(0)$ is the transverse part of the photon\nself-energy and $\\Sigma_{\\gamma Z}^T$ is the transverse part of the\nmixed photon-Z boson self-energy. $\\SW$ and $\\CW$ are defined as \n$\\SW = \\sqrt{1 - \\CW^2}$ with $\\CW = \\MW\/\\MZ$. $v^2$ and $e$ are related by\n\\begin{equation}\nv^2=\\frac{2\\SW^2 \\MW^2}{e^2} \\; ,\n\\end{equation}\nso the counterterm $\\delta v^2$ can be obtained through\n\\begin{equation}\\label{eq:dv2def}\n\\delta v^2 = \\frac{4 \\SW^2 \\MW^2}{e^2}\\left.\\left(\n\\frac{\\delta \\SW^2}{\\SW^2}+\\frac{\\delta \\MW^2}{\\MW^2}-\n2\\delta Z_e \\right)\\right|^{\\rm div} \\; ,\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:dsw2def}\n\\SW^2  \\rightarrow \\SW^2 + \\delta \\SW^2 \\; , \\quad\n\\text{with }  \\quad\n\\delta \\SW^2=-\\CW^2 \\left( \\frac{\\delta \\MW^2}{\\MW^2}\n-\\frac{\\delta \\MZ^2}{\\MZ^2} \\right) .\n\\end{equation}\nHere we take only the divergent parts of the counterterms \n$\\delta \\MZ^2$,\n$\\delta \\MW^2$ and $\\delta Z_e$, so that $\\delta v^2$ is\nrenormalized in the $\\DRbar$ scheme. \nThis implies that the counterterm $\\delta Z_e$ is not a\nfree parameter, even if we calculated it as if it\nwould be to determine $\\delta v^2$. Instead $\\delta Z_e$ is a\ndependent parameter defined by $\\delta v^2$ in the $\\DRbar$ scheme\nand $\\delta M_Z^2$ and $\\delta M_W^2$ in the OS scheme through\n\\refeq{eq:dv2def} and \\refeq{eq:dsw2def},\n\\begin{equation}\n\\delta Z_e = \\frac{1}{2 \\SW^2}\n\\left( \\CW^2 \\frac{\\delta \\MZ^2}{\\MZ^2}\n        +\\left( \\SW^2-\\CW^2 \\right)\\frac{\\delta \\MW^2}{\\MW^2}\n        -\\frac{e^2}{4\\MW^2}\\delta v^2 \\right) \\; .\n\\end{equation}\n\n\\paragraph{\\protect\\boldmath Renormalization of $v_{iL}^2$:}\nThe counterterms for the three vevs of the left-handed sneutrinos\n$v_{iL}$ can be extracted from the divergent part of the one-loop\nself-energies $\\Sigma_{\\widetilde{B}\\nu_{iL}}$ between the bino and the\ncorresponding left-handed neutrino. The tree-level mass matrix entries\nwe renormalize are defined by\n\\begin{equation}\n\\left( m_\\nu \\right)_{4i}=-\\frac{g_1 v_{iL}}{2} \\; ,\n\\end{equation}\nso it is necessary to have the counterterm of the gauge \ncoupling $g_1$, whose renormalization we define as \n$g_1\\rightarrow g_1+\\delta g_1$. We then can obtaine \n$\\delta g_1$ from $\\delta \\MW^2$, $\\delta \\MZ^2$\nand $\\delta v^2$ through the definitions of the gauge boson masses in\n\\refeq{eq:gaguebosonmasses},\n\\begin{equation}\n\\delta g_1 = \\frac{2}{g_1 v^2}\\left( \\delta M_Z^2-\\delta M_W^2 \\right)\n\t-\\frac{g_1}{2}\\frac{\\delta v^2}{v^2} \\; .\n\\end{equation}\nRenormalizing the self-energies\n$\\Sigma_{\\widetilde{B}\\nu_{iL}}$ using \\refeq{eq:neufermreno} we\nfind the following expression for the $\\delta v_{iL}^2$:\n\\begin{equation}\n\\delta v_{iL}^2=\\frac{4 v_{iL}}{g_1}\n\\left.\\Sigma_{\\widetilde{B}\\nu_{iL}}^S\\right|^{\\rm div}-\nv_{iL}\\left( \\delta Z_{44}^\\chi v_{iL} +\n\\delta Z_{ij}^\\chi v_{jL} \n+ \\delta Z_{i6}^\\chi v_d \\right) -\n2v_{iL}^2\\left.\\frac{\\delta g_1}{g_1}\\right|^{\\rm div} \\; ,\n\\end{equation}\nwhere again the divergent contributions of $\\Sigma_{\\widetilde{B}\\nu_{iL}}^S$ are calculated diagrammatically in the interaction basis.\n\n\\paragraph{\\protect\\boldmath Renormalization of $Y^\\nu_i$:}\nThe counterterm for the neutrino Yukawas $Y^\\nu_i$ can be extracted in the neutral fermion sector as well. We decide to use the renormalization of the tree-level masses\n\\begin{equation}\n\\left( m_\\nu \\right)_{i7}=\\frac{\\mu Y^\\nu_i}{\\lambda} \\; ,\n\\end{equation}\nthat mix the left-handed neutrinos and the up-type Higgsino. Since we already found $\\delta\\lambda$ and $\\delta\\mu$  we can get $\\delta Y^\\nu_i$ from the divergent part of the one-loop self-energies $\\Sigma_{\\nu_{iL}\\widetilde{H}_u}^S$,\n\\begin{equation}\n\\delta Y^\\nu_i = \\frac{1}{2}\\left( \\delta Z_{16}^\\chi \\lambda -\n\\delta Z_{77}^\\chi Y^\\nu_i - \\delta Z_{ij}^\\chi Y^\\nu_j \\right)\n- \\left( \\frac{\\delta\\mu}{\\mu}-\\frac{\\delta\\lambda}{\\lambda} \\right)\n+\\frac{\\lambda}{\\mu} \\left.\\Sigma_{\\nu_{iL}\\widetilde{H}_u}^S\\right|^{\\rm div} \\; .\n\\end{equation}\n\n\\paragraph{\\protect\\boldmath Renormalization of $\\tb$:}\nWe adopted the usual definition for $\\tb$ as in the MSSM (see\n\\refeq{eq:tanbetadef}). \nIf we define the renormalization for the vevs of the doublet fields as\n\\begin{equation}\\label{eq:vudreno}\n v_d^2 \\rightarrow v_d^2 + \\delta v_d^2 \\; , \\quad\n v_u^2 \\rightarrow v_u^2 + \\delta v_u^2 \\; ,\n\\end{equation}\nthe counterterm for $\\tb$ can be written at one-loop as a linear\ncombination of the counterterms for the vevs of\nthe doublet Higgses, \n\\begin{equation}\\label{eq:dtanbetadef}\n\\delta \\tb = \\frac{1}{2}\\tb\\left( \\frac{\\delta v_u^2}{v_u^2} -\n\\frac{\\delta v_d^2}{v_d^2} \\right) \\; .\n\\end{equation}\nNote that our renormalization of $v_u^2$ and $v_d^2$ in \\refeq{eq:vudreno} \nincludes the contributions from the field renormalization constants inside \nthe counterterms $\\delta v_u^2$ and $\\delta v_d^2$. This approach is \nequivalent as defining \n\\begin{equation}\n v_d \\rightarrow \\sqrt{Z_{11}}\n \t\\left(v_d + \\delta \\hat{v}_d\\right) \\; , \\quad\n v_u \\rightarrow \\sqrt{Z_{22}}\n \t\\left(v_u + \\delta \\hat{v}_u\\right) \\; ,\n\\end{equation}\nand writing the counterterm of $\\tan\\beta$ as\n\\begin{equation}\\label{eq:tanfieldreno}\n \\delta \\tan\\beta = \\frac{1}{2}\\tan\\beta\n \t\\left( \\delta Z_{22} - \\delta Z_{11} \\right) +\n \t\\tan\\beta \\left( \\frac{\\delta\\hat{v}_u}{v_u} -\n \t\t\\frac{\\delta\\hat{v}_d}{v_d} \\right) \\; .\n\\end{equation}\nThis notation was convenient in the MSSM and the NMSSM, because the \nsecond bracket in \\refeq{eq:tanfieldreno} is finite at \none-loop~\\cite{Chankowski:1992er,Dabelstein:1994hb,Ender:2011qh,Drechsel:2016ukp}\nand can be set to zero in the $\\DRbar$ scheme,   \nso that $\\delta \\tan\\beta$ can be expressed exclusively by the\nfield renormalization constants. In contrast, \nin the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ we find\n\\begin{equation}\n\\left.\\left( \\frac{\\delta\\hat{v}_u}{v_u} -\n \t\t\\frac{\\delta\\hat{v}_d}{v_d} \\right)\\right|^{\\rm div}\n=-\\frac{\\Delta \\lambda v_{iL} Y^\\nu_i }{32 \\pi^2 v_d} \\; .\n\\end{equation}\nThere are several possibilities to extract the counterterms $\\delta\nv_d^2$ and $\\delta v_u^2$. \nA convenient choice is to extract $\\delta v_d^2$\nfrom the renormalization of the entry of the neutral fermion\nmass matrix\nmixing the up-type Higgsino and\nthe right-handed neutrino, \n\\begin{equation}\n\\left( m_\\nu \\right)_{78}=\\frac{-\\lambda v_d + v_{iL}Y^\\nu_i}{\\sqrt{2}} \\; ,\n\\end{equation}\nbecause in this case no non-diagonal field renormalization counterterms are needed. Calculating the divergent part of $\\Sigma_{\\widetilde{H}_u v_R}^S$ and using the counterterms previously calculated we can extract $\\delta v_d^2$ via the expression\n\\begin{align}\n\\delta v_d^2 =& -\\frac{2\\sqrt{2}v_d}{\\lambda}\n\\left.\\Sigma_{\\widetilde{H}_u v_R}^S\\right|^{\\rm div}+\n\\frac{v_d}{\\lambda}\\left( \\delta Z_{77}^\\chi+ \\delta Z_{88}^\\chi \\right)\n\\left( -v_d\\lambda + v_{iL}Y^\\nu_i \\right) -\n2v_d^2\\frac{\\delta\\lambda}{\\lambda} \\notag \\\\\n&+\\frac{v_d}{\\lambda} Y^\\nu_i \\left( \\frac{\\delta v_L^2}{v_L} \\right)_i\n+\\frac{2v_d}{\\lambda}v_{iL}\\delta Y^\\nu_i \\; . \\label{eq:dvd2def}\n\\end{align}\nSince all counterterms appearing in \\refeq{eq:dvd2def}\nare renormalized in the $\\DRbar$ scheme also\n$\\delta v_d^2$ has no finite part.\nThere are now two ways to determine $\\delta v_u^2$. Firstly, we could similarly to $\\delta v_d^2$ extract the counterterm $\\delta v_u^2$ by renormalizing the up-type Higgsino self-energy $\\Sigma_{\\widetilde{H}_u \\widetilde{H}_u}^S$. Alternatively, we can deduce $\\delta v_u^2$ from the definition of $v^2$ in \\refeq{eq:tanbetadef} and simply write\n\\begin{equation}\\label{eq:dvu2def}\n\\delta v_u^2 = \\delta v^2 -\\delta v_d^2 - \\delta v_{1L}^2 - \\delta v_{2L}^2 - \\delta v_{3L}^2 \\; .\n\\end{equation}\nWe verified that both options yield the same result, which constitutes a\nconsistency test for the counterterms $\\delta v_{iL}^2$, which are\nunique for the \\ensuremath{\\mu\\nu\\mathrm{SSM}}. Inserting $\\delta v_d^2$ from \\refeq{eq:dvd2def}\nand $\\delta v_u^2$ from \\refeq{eq:dvu2def} into \\refeq{eq:dtanbetadef}\nfinally gives the counterterm for $\\tb$.\nWe checked that the final expression for $\\tb$ in \\refeq{eq:dtbdrbar}\nagrees with the NMSSM result in the limit\n$Y^\\nu_i\\rightarrow 0$.\n\nThe renormalization of $\\tb$ in the $\\DRbar$ scheme is manifestly process-independent and has shown to give stable numerical results in the MSSM \\cite{Frank:2002qf,Freitas:2002um} and the NMSSM \\cite{Drechsel:2016ukp,Ender:2011qh}.\n\n\\paragraph{\\protect\\boldmath Renormalization of $A^\\nu_i$:}\nThe soft trilinears $A^\\nu_i$ can be renormalized through the calculation of the radiative corrections to the corresponding scalar vertex in the interaction basis. The tree-level expression for the interaction between the up-type Higgs, one left-handed sneutrinos and the right-handed sneutrino is given by\n\\begin{equation}\n\\Gamma_{H_u \\widetilde{\\nu}_R \\widetilde{\\nu}_{iL}}^{(0)}=\n-\\left( \\frac{A^\\nu_i}{\\sqrt{2}}+\\frac{\\sqrt{2}\\kappa\n\\mu}{\\lambda} \\right) Y^\\nu_i \\; .\n\\end{equation}\nThe renormalized one-loop corrected vertex will define the counterterm for $A^\\nu_i$ since the counterterms for $\\kappa$, $\\mu$ and $\\lambda$ were already determined. We showed in \\reffi{fig:triplescalardiags} the topologies of the diagrams that have to be calculated in the interaction basis to get the divergent part of one-loop corrections\n$\\Gamma_{H_u \\widetilde{\\nu}_R \\widetilde{\\nu}_{iL}}^{(1)}$.\nAs in the case of the renormalization of $A^\\lambda$ the renormalization of the scalar vertex will contain the tree-level expressions of all the vertices with the same quantum numbers of the external fields, because of the non-diagonal field renormalization. Solved for $\\delta A^\\nu_i$ the renormalization of the vertex leads to\n\\begin{align}\n\\delta A^\\nu_i=&\n\\frac{\\sqrt{2}}{Y^\\nu_i}\\left.\\Gamma_{H_u \\widetilde{\\nu}_R\n\\widetilde{\\nu}_{iL}}^{(1)}\\right|^{\\rm div}+\n\\frac{1}{\\sqrt{2}Y^\\nu_i}\\left( \\left( \\delta Z_{22}+ \\delta Z_{33}\n\\right) \\Gamma_{H_u \\widetilde{\\nu}_R\n\\widetilde{\\nu}_{iL}}^{(0)}+\n\\delta Z_{1,3+i} \\Gamma_{H_u \\widetilde{\\nu}_R H_d}^{(0)} \\right. \\notag \\\\ \n&\\left.+\\delta Z_{3+j,3+i}\\Gamma_{H_u \\widetilde{\\nu}_R\n\\widetilde{\\nu}_{jL}}^{(0)}\\right)-\n\\frac{A^\\nu_i}{Y^\\nu_i}\\delta Y^\\nu_i -\n\\frac{2\\mu}{\\lambda}\\delta \\kappa -\n\\frac{2\\kappa}{\\lambda}\\delta \\mu -\n\\frac{2\\kappa\\mu}{\\lambda Y^\\nu_i}\\delta Y^\\nu_i +\n\\frac{2\\kappa\\mu}{\\lambda^2}\\delta\\lambda \\; ,\n\\end{align}\nwith\n\\begin{equation}\n\\Gamma_{H_u \\widetilde{\\nu}_R H_d}^{(0)} =\n\\frac{\\lambda A^\\lambda}{\\sqrt{2}}+\\sqrt{2}\\kappa\\mu \\; .\n\\end{equation}\n\n\\paragraph{\\protect\\boldmath Renormalization of ${m_{H_d\\widetilde{L}_L}^2}_i$:}\nThe soft scalar masses appear in the bilinear terms of the Higgs\npotential. They can be renormalized by calculating radiative corrections\nto scalar self-energies. It proved to be convenient \nto calculate the ${\\CP}$-odd scalar self-energies in the mass basis,\nand then to rotate the self-energies back to the interaction basis. \n\nWe find ${m_{H_d\\widetilde{L}_L}^2}_i$ at tree-level in\n\\begin{equation}\nm_{\\widetilde{\\nu}_{iL}^{\\mathcal{I}} H_{d}^{\\mathcal{I}}}^{2}=\n\\left(m_{H_d\\widetilde{L}_L}^2\\right)_i-\n\\frac{1}{2}v_R^2\\lambda Y^\\nu_i-\n\\frac{1}{2}v_u^2\\lambda Y^\\nu_i \\; .\n\\end{equation}\nThe general form of the renormalized scalar self-energies at one-loop is\n\\begin{align}\n\\hat{\\Sigma}_{X_i X_j}\\left( p^2 \\right) =&\n\\Sigma_{X_i X_j}\\left( p^2 \\right)+\n\\frac{1}{2} p^2 \\left( \\delta Z_{ji} + \\delta Z_{ij} \\right) \\notag \\\\\n-&\\frac{1}{2} \\left( \\delta Z_{ki} \\left( m_{X}^2 \\right)_{kj} +\n\\left( m_{X}^2 \\right)_{ik} \\delta Z_{kj} \\right) -\n\\delta \\left( m_{X}^2 \\right)_{ij} \\; ,\n\\end{align}\nwhere $X=(\\varphi,\\sigma)$ represents either the ${\\CP}$-even or the ${\\CP}$-odd scalar fields and we made use of the fact that the field renormalization constants $\\delta Z$ and the mass matrix $m_{X}^2$\nare real.\nDemanding that the renormalized self-energies $\\hat{\\Sigma}_{A_i A_j}$ are finite in the mass eigenstate basis we can define the divergent parts of the mass counterterms via\n\\begin{equation}\n\\left.\\delta \\left( m_{A}^2 \\right)_{ij}\\right|^{\\rm div} =\n\\left.\\Sigma_{A_i A_j}\\left( 0 \\right) \\right|^{\\rm div} -\n\\frac{1}{2}\\left( \\left(\\delta Z^A\\right)_{ji} m_{A_j}^2 +\nm_{A_i}^2 \\left(\\delta Z^A\\right)_{ij} \\right)\\; \n\\label{eq:oddcounters} ,\n\\end{equation}\nwhere the field counterterms in the mass eigenstate basis were defined in \\refeq{eq:fieldrenorotate} and the masses $m_{A_i}^2$ are the eigenvalues of the diagonal ${\\CP}$-odd scalar mass matrix $m_{A}^2$. In \\reffi{fig:scalarselfs} we show the diagrams that have to be calculated to get the quantum corrections to scalar self-energies at one-loop in the mass eigenstate basis.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.76\\textwidth]{scalarselfscpodd.pdf}\n  \\caption{Generic diagrams for the ${\\CP}$-even (h) and \n  ${\\CP}$-odd (A) scalar self-energies in the mass\n    eigenstate basis.}\n\\label{fig:scalarselfs}\n\\end{figure}\n\n\nWe calculated all diagrams in the 't Hooft-Feynman gauge, in which the Goldstone bosons $A_1$ and $H^\\pm_1$ and the ghost fields $u^\\pm$ and $u^Z$ have the same masses as the corresponding gauge bosons.\nCalculating the ${\\CP}$-odd self-energies $\\Sigma_{A_i A_i}$ diagrammatically, we get the mass counterterms in mass eigenstate basis through the \\refeq{eq:oddcounters}. Now inverting the rotation in \n\\refeq{eq:masscounterrot} we can get the mass counterterms for the ${\\CP}$-odd self-energies in the interaction basis via \n\\begin{equation}\\label{eq:oddcountersrot}\n\\left.\\delta m_{\\sigma}^2\\right|^{\\rm div}=\n{U^A}^T \\left.\\delta m_{A}^2\\right|^{\\rm div} U^A \\; ,\n\\end{equation}\nRecognizing that\n\\begin{equation}\n\\left(\\delta m_{\\sigma}^2\\right)_{3+i,1}=\n\\delta m_{\\widetilde{\\nu}_{iL}^{\\mathcal{I}} H_{d}^{\\mathcal{I}}}^{2} \\; ,\n\\end{equation}\nand that $m_{\\widetilde{\\nu}_{iL}^{\\mathcal{I}} H_{d}^{\\mathcal{I}}}^{2}$\n depends on $(m_{H_d\\widetilde{L}_L}^2)_i$, we can extract\n$\\delta (m_{H_d\\widetilde{L}_L}^2)_i$ through\n\\begin{align}\n\\delta \\left(m_{H_d\\widetilde{L}_L}^2\\right)_i=&\n\\left.\\left(\\delta m_{\\sigma}^2\\right)_{3+i,1}\\right|^{\\rm div}+\n\\frac{2\\mu Y^\\nu_i}{\\lambda}\\delta\\mu +\n\\lambda \\left( v_d^2+v_u^2\\right) Y^\\nu_i \\cos^3\\beta\\sin\\beta\n\\; \\delta\\tb \\notag \\\\\n&+\\frac{1}{2}\\lambda Y^\\nu_i \\sin^2\\beta \\; \\delta v^2 -\n\\frac{1}{2}\\lambda\\sin^2\\beta Y^\\nu_i\\left( \\delta v_{1L}^2 +\n\\delta v_{2L}^2 + \\delta v_{3L}^2 \\right) \\notag \\\\\n+& \\left( \\frac{\\mu^2}{\\lambda}+\\frac{1}{2}\\lambda\\left(\nv_d^2+v_u^2 \\right) \\sin^2\\beta \\right) \\delta Y^\\nu_i \\notag \\\\\n-&\\left( \\frac{\\mu^2 Y^\\nu_i}{\\lambda^2}-\\frac{1}{2}\\left( v_d^2+v_u^2 \\right)Y^\\nu_i\\sin^2\\beta\\right)\\delta\\lambda \\; .\n\\end{align}\n\n\\paragraph{\\protect\\boldmath Renormalization of ${m_{\\widetilde{L}_L}^2}_{ij}$:}\nSince we neglect ${\\CP}$-violation the counterterms for the \nnon-diagonal elements of the hermitian matrix\n${m_{\\widetilde{L}_L}^2}_{ij}$ \nare symmetric under the exchange of the indices $i$ and $j$.\nThen we can extract the counterterms for the non-diagonal elements in the same way as the ones for ${m_{H_d\\widetilde{L}_L}^2}_i$ in the \n${\\CP}$-odd scalar sector. They appear in the tree-level mass matrix in \n\\begin{equation}\nm_{\\widetilde{\\nu}_{iL}^{\\mathcal{I}} \\widetilde{\\nu}_{jL}^{\\mathcal{I}}}^{2}=\n\\left(m_{\\widetilde{L}}^2\\right)_{ij}+\n\\frac{1}{2}\\left(v_R^2+v_u^2\\right)Y^\\nu_iY^\\nu_j \\quad \\text{for }\ni\\neq j \\; .\n\\end{equation}\nHence, the counterterms\n$\\delta\\left({m_{\\widetilde{L}_L}^2}\\right)_{ij}$ for $i\\neq j$ \nare given by\n\\begin{align}\n\\delta\\left({m_{\\widetilde{L}_L}^2}\\right)_{ij}=&\n\\left.\\left(\\delta m_{\\sigma}^2\\right)_{3+i,3+j}\\right|^{\\rm div} -\n\\frac{1}{2}\\left( v_R^2+v_u^2 \\right)\\left(Y^\\nu_i\\delta Y^\\nu_j -\nY^\\nu_j\\delta Y^\\nu_i \\right) \\\\\n&-\\frac{2\\mu Y^\\nu_i Y^\\nu_j}{\\lambda^2}\\left( \\frac{\\delta\\mu}{\\mu}-\n\\frac{\\delta\\lambda}{\\lambda} \\right)-\n\\frac{1}{2}Y^\\nu_i Y^\\nu_j  \\sin^2\\beta\\; \\delta v^2 \\notag \\\\\n&-\\left( v_d^2+v_u^2 \\right)Y^\\nu_i Y^\\nu_j \\cos^3\\beta\\sin\\beta\n\\;\\delta\\tb+\n\\frac{1}{2}Y^\\nu_i Y^\\nu_j\\sin^2\\beta\n\\left( \\delta v_{1L}^2 + \\delta v_{3L}^2 + \\delta v_{3L}^2 \\right)\n\\; . \\notag\n\\end{align} \n\\paragraph{\\protect\\boldmath \\FA\\ modelfile:}\nThe diagrams and their amplitudes\nthat had to be calculated to obtain the counterterms, as described\nin this section,\nwere generated using the Mathematica package\n\\FA~\\cite{Hahn:2000kx} and further evaluated with the package\n\\FC~\\cite{Hahn:1998yk}. The \\FA\\ model file for the\n\\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ was created with the Mathematica program \\texttt{SARAH}\n\\cite{Staub:2009bi}. We modified the model file to neglect ${\\CP}$-violation \nby choosing all relevant parameters to be real. We also neglected \nflavor-mixing in the squark- and the quark-sector \nin this work. The \\FA\\ model file can be provided by the authors upon\nrequest.\nThe calculation of renormalized two- and three-point\nfunctions of the neutral scalars of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ at one-loop accuracy is\nthereby fully automated. \n(as it is in the MSSM~\\cite{Fritzsche:2013fta}).\n\nIn \\refse{sec:numanal} we will present our predictions for the Higgs\nmasses in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ compared to the ones of the NMSSM.\nTo be able to make this comparison, we had to calculate the\nNMSSM-predictions in the same renormalization scheme and using\nthe same conventions as were used in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}.\nThis is why we calculated the one-loop self-energies in the NMSSM \nwith our own NMSSM-modelfile for \\FA\/\\FC\\ created with\n\\texttt{SARAH} using the same procedure as for the \\ensuremath{\\mu\\nu\\mathrm{SSM}}.\nWe verified that the results\ncalculated in the NMSSM with our modelfile are equal to the results\ncalculated with the modelfile presented in \\citere{Paehr:276451}, which\nwas a good check that the generation of the modelfiles for the NMSSM\nand the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ was correct.\n\n\n\n\\section{Loop corrected Higgs boson masses}\n\\label{sec:getmasses}\n\nIn the previous section we have derived an\nOS\/\\DRbar\\ renormalization scheme for the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ Higgs sector. This\ncan be applied (via the future \\FA\\ model file, once the\ncounterterms are implemented) to any higher-order\ncorrection in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}. As a first application, we evaluate the full\none-loop corrections to the ${\\CP}$-even scalar sector in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}.\nDue to the still missing implementation of counterterms\nin the \\FA\\ model file,\nthe calculation of the renormalized scalar self-energies is done in\ntwo steps. Firstly, the unrenormalized self-energies are calculated\nusing \\FA\\ and \\FC, and subsequently the self-energies are renormalized\nsubtracting (by hand) the field renormalization and\nmass counterterms, as will\nbe described in the next section.\n\n\n\\subsection{Evaluation at one-loop} \n\nHere we describe the final form of the renormalized ${\\CP}$-even scalar\nself-energies \n$\\hat{\\Sigma}_{hh}$ and how the loop corrected physical masses of the\nHiggs boson masses are evaluated.\n\nThe one-loop renormalized self-energies in the mass eigenstate basis are\ngiven by\n\\begin{equation}\\label{eq:renomself}\n\\hat{\\Sigma}_{h_i h_j}^{(1)}\\left( p^2 \\right) = \\Sigma_{h_i h_j}^{(1)}\\left( p^2 \\right) + \\delta Z^H_{ij}\n\\left( p^2 - \\frac{1}{2}\\left( m_{h_i}^2 + m_{h_j}^2 \\right) \\right)\n- \\left(\\delta m_{h}^2\\right)_{ij} \\; ,\n\\end{equation}\nwith the field renormalization constants $\\delta Z^H$ \nand the mass counter terms $\\delta m_{h}^2$\nin the mass eigenstate basis\ndefined by the rotations in \\refeq{eq:fieldrenorotate}\nand \\refeq{eq:masscounterrot}.\n$\\Sigma_{h_i h_j}$ is the unrenormalized self-energy obtained by\ncalculating the diagrams shown in \\reffi{fig:scalarselfs} \nwith the ${\\CP}$-even states $h$ on the external legs.\nThe self-energies were calculated in the Feynman\ngauge, so that gauge-fixing terms do not yield counterterm contributions\nin the Higgs sector at one-loop. The loop integrals were regularized\nusing dimensional reduction \\cite{SIEGEL1979193,CAPPER1980479} and\nnumerically evaluated for arbitrary real momentum using\n\\LT~\\cite{Hahn:1998yk}. The contributions from complex values of $p^2$\nwere approximated using a Taylor expansion with respect to the imaginary part\nof $p^2$ up to first order. \n\nIn \\refeq{eq:renomself} we already made use of the fact that $\\delta\nZ^H$ is real and symmetric in our renormalization scheme. The mass\ncounterterms are defined as functions of the counterterms of the free\nparameters following \\refeq{eq:masscounterderive} and\n\\refeq{eq:masscounterrot}. They contain finite contributions from the\ntadpole counterterms and from the counterterm for the gauge boson mass\n$\\MZ^2$. The matrix $\\delta m_{h}^2$ is real and symmetric.\n\nThe renormalized self-energies enter the inverse propagator matrix\n\\begin{equation}\\label{eq:renoselfdef}\n\\hat{\\Gamma}_{h}=\\text{i} \\left[ p^2\\; \\mathbbm{1} -\n\\left( m_{h}^2 - \\hat{\\Sigma}_{h}\\left(p^2\\right) \\right)\n\\right] \\; , \\qquad  \\text{with }\n\\left(\\hat{\\Sigma}_{h}\\right)_{ij} = \n\\hat{\\Sigma}_{h_i h_j} \\; .\n\\end{equation}\nThe loop-corrected scalar masses squared are the zeroes of the\ndeterminant of the inverse propagator matrix. The determination of\ncorrected masses has to be done numerically when we want to account for\nthe momentum-dependence of the renormalized self-energies. This is done\nby an iterative method that has to be carried out for each of the six\nsquared loop-corrected masses \\cite{Fuchs:2015jwa}. \n\n\n\n\n\\subsection{Inclusion of higher orders}\\label{sec:higherorders}\n\nIn \\refeq{eq:renoselfdef} we did not include the superscript $^{(1)}$ in\nthe self-energies. Restricting the numerical evaluation to a pure\none-loop calculation would lead to very large theoretical uncertainties.\nThese can be avoided by the inclusion of corrections beyond the one-loop\nlevel. Here we follow the approach of \\citere{Drechsel:2016jdg} and\nsupplement the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ one-loop results by higher-order corrections in\nthe MSSM limit as provided by \\fh\\ (version 2.13.0)~\\cite{Heinemeyer:1998yj,Hahn:2009zz,Heinemeyer:1998np,Degrassi:2002fi,Frank:2006yh,Hahn:2013ria,Bahl:2016brp,feynhiggs-www}.\nIn this way the leading and subleading two-loop corrections are\nincluded, as well as a resummation of large logarithmic terms, see the\ndiscussion in \\refse{sec:intro}, \n\\begin{equation}\n\\hat{\\Sigma}_{h}\\left( p^2 \\right) = \n\\hat{\\Sigma}_{h}^{(1)}\\left( p^2 \\right) +\n\\hat{\\Sigma}_{h}^{(2')} +\n\\hat{\\Sigma}_{h}^{\\rm resum} \\; .\n\\end{equation}\nIn the partial two-loop contributions $\\hat{\\Sigma}_{h}^{(2')}$ we take\nover the corrections of\n\\order{\\alpha_s\\alpha_t,\\alpha_s\\alpha_b,\\alpha_t^2,\\alpha_t\\alpha_b}, \nassuming that the MSSM-like corrections are also valid in the\n\\ensuremath{\\mu\\nu\\mathrm{SSM}}. This assumption \nis reasonable since the only difference between the squark sector of the\n\\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ in comparison to the MSSM are the terms proportional to \n$Y^\\nu_i v_{iL}$ in the non-diagonal element of the up-type squark mass \nmatrices (see \\refeq{eq:usquarks12})\nand the terms proportional to $v_{iL}v_{iL}$ in the diagonal\nelements of the up- and down-type squark mass matrices\n(see \\refeq{eq:usquarks11}, \\refeq{eq:usquarks22},\n\\refeq{eq:dsquarks11} and \\refeq{eq:dsquarks22}),\nwhich numerically will always be negligible in\nrealistic scenarios since $v_{iL}\\ll v_d,v_u,v_R$.\nFurthermore,. in \\citere{Drechsel:2016ukp} the quality of the MSSM\napproximation was tested in the NMSSM, showing that the genuine NMSSM\ncontributions are in most cases sub-leading. The same is expected for\nthe contributions stemming from the resummation of large logarithmic\nterms given by $\\hat{\\Sigma}_{h}^{\\rm resum}$. \n\n\n\n\n\\section{Numerical analysis}\n\\label{sec:numanal}\n\nIn the following we present for the first time the full one-loop\ncorrections to the scalar masses in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}, \nwith one generation of right-handed neutrinos obtained in the\nFeynman-diagrammtic approach, taking into account all parameters of the\nmodel and the complete dependence on the external momentum, which\nincludes a consistent treatment of the imaginary parts of the scalar\nself-energies. \nOur results extend the known ones in the literature of\nthe MSSM and the NMSSM to a model, which has a\nrich and unique phenomenology through explicit $R$-parity breaking. The\none-loop results are supplemented by known higher-loop results from the\nMSSM (see the previous section) to reproduce the Higgs mass value of \n$\\sim 125 \\,\\, \\mathrm{GeV}$~\\cite{Aad:2015zhl}. \nHere the theory uncertainty must be kept in mind. In the\nMSSM it is estimated to be at the level of \n$2-3 \\,\\, \\mathrm{GeV}$~\\cite{Degrassi:2002fi,Bahl:2017aev}, and in extended models\nit is naturally slightly larger. \n\nWe will present results in several different scenarios, in\nall of which one scalar with the correct SM-like Higgs mass\nis reproduced.\nTo get an estimation of the significance of quantum\ncorrections to the Higgs masses that are unique for the \\ensuremath{\\mu\\nu\\mathrm{SSM}}, we\ncompare the results to the corresponding ones in the NMSSM.\nThe results in the NMSSM are obtained by a calculation based on\n\\citere{Drechsel:2016ukp}, but with slightly changed renormalization\nconditions to be as close as possible to the calculation in the\n\\ensuremath{\\mu\\nu\\mathrm{SSM}}. While Ref.~\\cite{Drechsel:2016ukp} uses the mass squared of\nthe charged Higgs mass as input parameter and renormalizes it as OS\nparameter we instead use $\\DRbar$ conditions for $A^\\lambda$.\n\nThe benchmark points used in the following were not tested in detail\nagainst experimental bounds including the $R$-parity violating effects of\nthe \\ensuremath{\\mu\\nu\\mathrm{SSM}}. They have been chosen to exemplify the potential magnitude\nof unique \\ensuremath{\\mu\\nu\\mathrm{SSM}}-like corrections. Nevertheless, the values\nwe picked for the free parameters should be close to realistic and\nexperimentally allowed scenarios: the parameters in the scalar sector\nare taken over from calculations in the NMSSM~\\cite{Drechsel:2016ukp}, and\nunique \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ parameters are chosen in a range to reproduce\nneutrino masses of the correct order of magnitude. That means that the\nneutrino Yukawas $Y^\\nu_i$ should be of the order $10^{-6}$ to generate\nneutrino masses of the order less than $1\\,\\, \\mathrm{eV}$. \nFor the left-handed sneutrino\nvevs this directly implies $v_{iL}\\ll v_d, v_u$ so\nthat the tadpole coefficients vanish at tree-level\n\\cite{Escudero:2008jg}. We will leave a more detailed discussion of\nnumerical results for a future publication, in which we will also include\nthree generations of right-handed neutrinos.\n\n\n\n\\subsection{NMSSM-like crossing point scenario}\n\\label{sec:nmssmpoint}\n\n\nThe first scenario we want to analyze is one studied in the NMSSM with a\nsinglet becoming the LSP in the region of $\\lambda > \\kappa$\ntaken from Ref.~\\cite{Drechsel:2016ukp}. \nThis scenario was tested therein against the experimental limits \nimplemented in \\texttt{HiggsBounds 4.1.3}~\\cite{Bechtle:2008jh,Bechtle:2011sb,Bechtle:2013gu,Bechtle:2013wla,Bechtle:2015pma}.\nIt has the nice feature that there is a crossing point when\n$\\lambda\\approx\\kappa$ in the neutral scalar sector, in which the masses\nof the singlet and the SM-like Higgs become degenerate and NMSSM-like\nloop corrections become significant \\cite{Drechsel:2016jdg}. \n\nIn \\refta{tab:crossingpoint} we list the values chosen for the\nparameters. The SM-like parameters from the electroweak sector and\nthe lepton and quark masses are given in appendix \\ref{app:smvalues} \nin \\refta{tab:smsum}. The\nparameters present in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ and the NMSSM are of course chosen\nequally in both models. \nThe region $\\lambda < 0.026$ is excluded\nbecause the left-handed sneutrinos become tachyonic at tree-level. The\nflavor-changing non-diagonal elements in the slepton sector are\nzero. The value for $A^\\lambda$ is chosen to correspond to a mass of\n$m_{H^\\pm}= 1000 \\,\\, \\mathrm{GeV}$ for the charged Higgs mass in the NMSSM with\n$m_{H^\\pm}$ renormalized OS and $A^\\lambda$ not being a free\nparameter. $A^\\kappa$ should be chosen to be negative in our convention\n(when $\\kappa$ is positive) to avoid false vacua~\\cite{Escudero:2008jg}\nor tachyons in the pseudo-scalar sector~\\cite{Ghosh:2014ida}. It should\nbe kept in mind \nthat the diagonal soft scalar masses in the neutral sector are extracted\nfrom the values for\n$v_{iL}$, $\\tb$ and\n$\\mu$ via the tadpole equations, and their non-diagonal,\nflavor-violating elements are always set to zero at tree-level.\nThis is of crucial importance for the comparison of the\nscalar masses\nin the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ and the NMSSM, since in the NMSSM the soft slepton masses\n$m_{\\widetilde{L}}^2$ are independent parameters, while in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ the\ndiagonal elements are dependent parameters fixed by the tadpole\n\\refeqs{eq:tp4}, \nwhen the vevs are used as input.\nThe latter strategy is particularly convenient since the order\nof magnitude of the vevs is roughly fixed through the electroweak seesaw mechanism\nby demanding neutrino masses below the eV scale,\nwhile the soft scalar masses are not directly related to any physical observable.\nConsequentially, for each parameter point\ncalculated in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}, the corresponding values that have to be chosen for\n$m_{\\widetilde{L}}^2$ in the NMSSM have to be adjusted accordingly,\ndefined as a function of all the free parameters appearing in the\nHiggs potential.\n\n\n\\begin{table}\n\\renewcommand{\\arraystretch}{1.7}\n\\centering\n\\begin{tabular}{c c c c c c c c c c c}\n $v_{iL}\/\\sqrt{2}$ & $Y^\\nu_i$ & $A^\\nu_i$ & $\\tb$ & $\\mu$ & $\\lambda$ &\n \t$A^\\lambda$ & $\\kappa$ & $A^\\kappa$ & $M_1$ & \\\\\n \\hline\n $10^{-4}$ & $10^{-6}$ & $-1000$ & $8$ & $125$ & $[0.026;0.3]$ &\n \t$897.61$ & $0.2$ & $-300$ & $143$ & \\\\\n \\hline\n \\hline\n $M_2$ & $M_3$ & $m_{\\widetilde{Q}_{iL}}^2$ &\n \t$m_{\\widetilde{u}_{iR}}^2$ & $m_{\\widetilde{d}_{iR}}^2$ &\n \t$A^u_3$ & $A^u_{1,2}$ & $A^{d}_{1,2,3}$ & $(m_{\\widetilde{e}}^2)_{ii}$ &\n \t$A^e_{33}$ & $A^e_{11,22}$ \\\\\n \\hline\n $300$ & $1500$ & $1500^2$& $1500^2$ & $1500^2$ & $-2000$ & $-1500$ &\n \t$-1500$ & $200^2$ & $-1500$ & $-100$ \n\\end{tabular}\n\\caption{Input parameters for the NMSSM-like crossing point scenario;\n  all masses and values for trilinear parameters are in GeV.}\n\\label{tab:crossingpoint}\n\\renewcommand{\\arraystretch}{1.0}\n\\end{table}\n\nIn \\reffi{fig:crossingspec} we show the resulting spectrum of the\n${\\CP}$-even scalars at tree-level and including the full one-loop and\ntwo-loop contributions.%\n\\footnote{Here and in the following we denote with ``two-loop'' result the\none-loop plus partial two-loop plus resummation corrected masses.}%\n~The standard model Higgs mass value\nis reproduced accurately when the quantum corrections are included. The\nheavy MSSM-like Higgs $H$ and the left-handed sneutrinos are at the\nTeV-scale and rather decoupled from the SM-like Higgs boson. The three\nleft-handed sneutrinos are degenerate because the \\ensuremath{\\mu\\nu\\mathrm{SSM}}-like parameters\nare set equal for all flavors. \n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{nmssmspec.pdf}\n    \\vspace{-0.4cm}\n    \\caption{Spectrum of ${\\CP}$-even scalar masses in NMSSM-like\n      crossing point scenario. The three left-handed sneutrinos\n      $\\widetilde{\\nu}_{iL}$ are degenerate.\\protect\\footnotemark} \n    \\label{fig:crossingspec}\n\\end{figure}\n\\footnotetext{All plots\n      have been produced using \\texttt{ggplot2}~\\cite{ggplot2}\n      and \\texttt{tikzDevice}~\\cite{tikzDevice} in \\texttt{R}~\\cite{Rlanguage}.}\nThe singlet-like scalar mass heavily depends on $\\lambda$,\nbecause when $\\mu$\nis fixed, increasing $\\lambda$ leads to\na smaller value for $v_R$ (see \\refeq{eq:mudef}).\nAs was observed in \\citere{Drechsel:2016ukp}, the loop-corrected\nmass of the singlet becomes smaller than the SM-like Higgs boson mass\nat about $\\lambda\\approx\\kappa$.\nWe observe non-negligible loop-corrections to the singlet in the\nregion of $\\lambda$ where the singlet is the lightest neutral scalar.\n\nDue to the similarity of the Higgs sectors of the\nNMSSM and the \\ensuremath{\\mu\\nu\\mathrm{SSM}}, the masses of the doublet-like Higgs bosons and\nthe right-handed sneutrino will be of comparable size as the masses\npredicted for the doublet-like Higgses and the singlet in the\nNMSSM.\nIn \\reffi{fig:nmssmdif} we show the tree-level and the one- and\ntwo-loop corrected mass of the SM-like Higgs boson in the\ncrossing-point scenario. One can see that, as expected, \nthe two-loop corrections are crucial to predict a SM-like Higgs mass of $125\\,\\, \\mathrm{GeV}$.\nIndeed, our analysis confirmed that differences in the prediction of the SM-like\nHiggs boson mass are negligible compared to the current experimental\nuncertainty~\\cite{Aad:2015zhl}\nand the anticipated\nexperimental accuracy of the ILC of about\n$\\lsim 50 \\,\\, \\mathrm{MeV}$~\\cite{Moortgat-Picka:2015yla}, even when there is a\nsubstantial mixing between left-handed sneutrinos and the SM-like\nHiggs at tree-level or one-loop.\nApart from that, they\nare clearly exceeded by the (future) parametric uncertainties in the Higgs-boson\nmass calculations. Consequently, the\nHiggs sector alone will not be sufficient to distinguish the\n\\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ from the NMSSM. On the other hand, we can regard\nthe theoretical uncertainties\nin the NMSSM and the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ to be at the same level of accuracy.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.6\\textwidth]{nmssmcompare.pdf}\n  \\vspace{-0.5cm}\n  \\caption{Tree-level, one-loop and two-loop corrected\n    masses of the SM-like Higgs boson in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ in the NMSSM-like\n    crossing point scenario.} \n  \\label{fig:nmssmdif}\n\\end{figure}\n\n\n\n\n\\subsection{\\protect\\boldmath Light $\\tau$-sneutrino scenario}\n\\label{sec:lighttaulsp}\n\nIn the previous scenario we observed that, in\na scenario where the left-handed sneutrinos where practically \ndecoupled from the SM-like\nHiggs boson, the unique \\ensuremath{\\mu\\nu\\mathrm{SSM}}-like\ncorrections do not account for a substantial deviation of the SM-like Higgs\nmass prediction compared to the NMSSM.\nIn this section we will investigate a scenario in which one of\nthe left-handed sneutrinos has a small mass close to SM Higgs boson\nmass. The phenomenology of such a spectrum was recently studied in\ndetail,\nincluding a comparison of its predictions with the\nLHC searches~\\cite{Ghosh:2017yeh,Lara:2018rwv}. It was found that a light\nleft-handed sneutrino as the LSP can give rise to distinct\nsignals for the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ (for instance, final states with\ndiphoton plus missing energy, diphoton plus leptons and multileptons).\n\\begin{table}\n\\renewcommand{\\arraystretch}{1.7}\n\\centering\n\\begin{tabular}{c c c c c c c c c c c}\n $v_{1,2L}\/\\sqrt{2}$ & $v_{3L}\/\\sqrt{2}$ & $Y^\\nu_i$ & $A^\\nu_i$ & $\\tb$ & $\\mu$ & $\\lambda$ &\n \t$A^\\lambda$ & $\\kappa$ & $A^\\kappa$ \\\\\n \\hline\n $10^{-5}$ & $4\\cdot 10^{-4}$ & $5\\cdot 10^{-7}$ & $-400$ & $10$ & $270$ & $[0.19;0.3]$ &\n \t$1000$ & $0.3$ & $-1000$\n\\end{tabular}\n\\caption{Input parameters for the light $\\tau$-sneutrino scenario;\n\t\t all masses and values for trilinear parameters are in GeV.}\n\\label{tab:lsptaupoint}\n\\renewcommand{\\arraystretch}{1.0}\n\\end{table}\n\nIn \\refta{tab:lsptaupoint} we list the relevant parameters that were\nchosen to\nobtain a light left-handed $\\tau$-sneutrino. The parameters not shown\nhere are chosen to be the same as in the previous case, shown in\n\\refta{tab:crossingpoint}. One can see that the vev\n$v_{3L}$ (corresponding to $\\widetilde{\\nu}_{3L}$) was increased w.r.t.\\\nthe NMSSM-like scenario. The reason for this becomes clear when one\nextracts the leading terms of the diagonal tree-level mass matrix\nelement of the left-handed sneutrinos,\n\\begin{equation}\\label{eq:approxnumass}\nm_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}} \\widetilde{\\nu}_{iL}^{\\mathcal{R}}}^{2}\n\\approx \n\\frac{Y^\\nu_i v_R v_u}{2v_{iL}}\n\\left( - \\sqrt{2}A^\\nu_i-\\kappa v_R \n\t\t\t\t+\\frac{\\sqrt{2}\\mu}{\\tan\\beta} \\right) \t\n\t\t\t\t\\; .\n\\end{equation}\nThe tree-level masses of the left-handed sneutrinos are roughly\nproportional to the inverse of their vev. We also\ndecreased $A^\\nu_3$ in comparison to the previous scenario, keeping it\nnegative, so that it is of order $\\kappa v_R$ and the sum in the brackets of \\refeq{eq:approxnumass} becomes small.\n\n\\begin{figure}[!]\n    \\centering\n    \\hspace{1cm}\\includegraphics[width=0.8\\textwidth]{sntauLSPspec.pdf}\n    \\vspace{-0.5cm}\n    \\caption{${\\CP}$-even scalar mass spectrum of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ in the light\n      $\\tau$-sneutrino scenario, see \\refta{tab:lsptaupoint}.\n      On the right side we state the dominant composition\n      of the mass eigenstates.}\n    \\label{fig:sntaulspspec}\n\\end{figure}\n\nIn \\reffi{fig:sntaulspspec} we show the tree-level and loop-corrected\nspectrum of the scalars in the region of $\\lambda$ where there are no\ntachyons at tree-level.\nFor too small $\\lambda$ the tree-level mass of\n$\\widetilde{\\nu}_{3L}$ becomes tachyonic,\nbecause when $\\mu=(v_R \\lambda)\/ \\sqrt{2}$ is\nfixed $v_R$ has to grow and the second term in the bracket\nof \\refeq{eq:approxnumass} will grow larger than the sum of the first and the\nthird term.\nFor too large $\\lambda$,\nthe tree-level mass of the SM-like Higgs boson\nbecomes tachyonic.\nIn particular, it starts to mix with the tree-level singlet mass, which\nbecomes tachyonic because $v_R$ decreases when $\\lambda$\nincreases.\nThe central value of the\nSM Higgs boson mass is reproduced in this scenario\nup to values of $\\lambda\\leq\\num{0.22}$. However, \nconsidering the theoretical uncertainty even higher values of $\\lambda$ \ncan be viable. For $\\lambda=0.236$ the prediction for the SM-like \nHiggs mass decreases below $m_{h_1}\\approx 122\\,\\, \\mathrm{GeV}$. \nAs discussed in the introduction we assume a theory uncertainty of\n$\\sim 3 \\,\\, \\mathrm{GeV}$ on the mass evaluation, \nso we consider in this scenario the region $\\lambda\\leq0.236$ to be valid \nregarding the SM Higgs boson mass.\nAn interesting observation is\nthat the masses of light left-handed sneutrinos are\nmainly induced via quantum corrections, while the tree-level mass\napproaches 0 for small values of $\\lambda$.\nThis indicates that a consistent treatment of\nquantum corrections to light sneutrino masses is of\ncrucial importance.\n\nThe large upward shift of the left-handed sneutrino masses through\nthe one-loop \ncorrections is due to the fact that in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ the sneutrino fields\nare part of the Higgs potential, each with an associated\ntadpole coefficient $T_{\\widetilde{\\nu}_{iL}}$. To ensure the stability of the\nvacuum \\wrt quantum corrections, the tadpoles are renormalized OS, absorbing\nall finite corrections into the counterterms $\\delta T_{\\widetilde{\\nu}_{iL}}$\n(see \\refse{sec:condis}). In the mass counterterms for the left-handed\nsneutrinos the finite parts $\\delta T_{\\widetilde{\\nu}_{iL}}^{\\rm fin}$\nintroduce the main finite contribution in the form\n\\begin{equation}\n\\delta m_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}}\n  \\widetilde{\\nu}_{iL}^{\\mathcal{R}}}^{2\\;{\\rm fin}}\n    = -\\frac{\\delta T_{\\widetilde{\\nu}_{iL}}^{\\rm fin}}{v_{iL}} + \\cdots \\; ,\n\\end{equation}\nwhich is enhanced by the inverse of the vev of $\\widetilde{\\nu}_{iL}$.\nIt is these terms inside the counterterms of the renormalized self-energies\n$\\hat{\\sum}_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}}\n  \\widetilde{\\nu}_{iL}^{\\mathcal{R}}}^{(1)}$ that shift the poles of the\npropagator matrix and increase the masses of the left-handed sneutrinos,\nespecially in cases where the tree-level masses are small.\n\n\\begin{figure}[!]\n     \\centering\n     \\includegraphics[width=0.9\\textwidth]{sntauLSPcompare.pdf}\n     \\vspace{-1.0cm}\n     \\caption{Light $\\tau$-sneutrino scenario, see \\refta{tab:lsptaupoint}.\n     In the shaded region the prediction for the SM-like Higgs mass is \n     below $122\\,\\, \\mathrm{GeV}$. \\textit{Left:} Masses of the SM-like Higgs, the left-handed\n     $\\tau$-sneutrino and the right-handed sneutrino in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ at\n     tree-level and one-loop. \\textit{Right:} Masses of the SM-like\n     Higgs and the singlet in the NMSSM at tree-level and\n     one-loop.}\n     \\label{fig:sntaulspcompare}\n\\end{figure}\n\nThis behavior is a peculiarity of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}, meaning that the leptonic\nsector and the Higgs sector are mixed through the breaking of $R$-parity.\nThe relations between the vevs $v_{iL}$ and the soft masses\n$m_{\\widetilde{L}}^2$ via the tadpole equations automatically lead to\ndependences between the sneutrino masses and, for instance, the neutrino\nor the Higgs sector.\nIn the NMSSM, on the other hand, the sneutrinos are not part of the\nHiggs potential, since the fields are protected by lepton-number conservation.\nThere, the soft masses $m_{\\widetilde{L}}^2$ are, without further assumptions,\nfree parameters that can be chosen without taking into account\nany leptonic observable (such as neutrino masses and mixings).\nIn principle, the additional dependences of the\n\\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ scalar (neutrino) masses on the neutrino sector could be\nused (e.g.\\ when all neutrino masses and mixing angles will be\nknown with sufficient experimental accuracy)\nto restrict the possible range of $m_{\\widetilde{L}}^2$, and thus the\npossible values for the left-handed sneutrino masses. However, with\nour current experimental knowledge on the neutrino masses,\nthe possible values for the vevs $v_{iL}$, and hence the\npossible range of left-handed sneutrino masses, are\neffectively not yet constrained.\n\nIt should be noted as well, that the soft masses\n$m_{\\widetilde{L}}^2$ also appear \nin the mass matrix of the charged scalars (see \\refeq{eq:mlinmch})\nand the pseudoscalars (see \\refeq{eq:mlinmps}).\nIn many cases they are the dominant\nterm in the tree-level masses of the left-handed sleptons and\nsneutrinos, so the values of the masses of charged sleptons and\nsneutrino of the same family will be close. A precise treatment of\nquantum corrections of the size observed in \\reffi{fig:sntaulspspec} is\nextremely important in those cases, since \nthey might easily change the relative sign of their\nmass differences. This can result in a complete change of the\nphenomenology of the corresponding benchmark point, for instance when\neither the neutral (pseudo)scalar or the charged scalar is the\nLSP~\\cite{Ghosh:2017yeh,Lara:2018rwv}.\n\nWe compare the relevant spectrum of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ to the\ncorresponding one in the NMSSM in \\reffi{fig:sntaulspcompare}.\nWe show the tree-level\nand one-loop corrected masses of the light scalars in the \\ensuremath{\\mu\\nu\\mathrm{SSM}},\nand the masses of the SM-like Higgs boson and the singlet in the\nNMSSM on the right, with parameters set accordingly.\nWe shade in grey the region of $\\lambda$ where the the prediction for\nthe SM-like Higgs boson mass is below $122\\,\\, \\mathrm{GeV}$ if two-loop corrections\nare included. As expected, the SM-like Higgs-boson mass and\nthe mass of \nthe singlet turn out to be equal in both models. Even in regions\nwhere there is a substantial mixing\nof the SM-like Higgs boson with the left-handed sneutrinos, something that\ncannot occur in the NMSSM, the differences in the SM-like Higgs\nmass prediction are not larger than a few keV.\n\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=1.0\\textwidth]{mixingsntau.pdf}\n    \\vspace*{-1.0cm}\n    \\caption{Light $\\tau$-sneutrino scenario, see \\refta{tab:lsptaupoint}.\n             We show the absolute values of the mixing matrix elements at\n             tree-level $|U_{1i}^{H(0)}|$ (left) and\n             $|U_{2i}^{H(0)}|$ (right), whose squared value\n             define the admixture of the two-lightest\n             ${\\CP}$-even scalar mass eigenstate $h_{1,2}$ with the fields\n             $\\varphi_i=(H_d,H_u,\\widetilde{\\nu}_R,\\widetilde{\\nu}_{1L},\n             \\widetilde{\\nu}_{2L},\\widetilde{\\nu}_{3L})$ in the\n             interaction basis. A substantial mixing of the $\\tau$-sneutrino\n             $\\widetilde{\\nu}_{3L}$ with the SM-like Higgs boson $h^{125}$ and\n             with the singlet $\\widetilde{\\nu}_R$ is present in the narrow region\n             where the corresponding tree-level masses are degenerate\n             (for example in the \\textit{ right} plot at\n             $\\lambda\\sim 0.20237$ and $\\lambda\\sim0.29692$).}\n    \\label{fig:sntaulsmixing}\n\\end{figure}\n\nIt is rather surprising that the SM-like Higgs masses coincide this precisely\nin both models, considering the fact\nthat a substantial mixing with the sneutrino is\npossible at tree-level, as we show in \\reffi{fig:sntaulsmixing}.\nWe individually plot the mixing matrix elements of\nthe two lightest ${\\CP}$-even scalars, whose\nsquared values define the composition of each mass eigenstate\nat tree-level. In the cross-over point of the $\\tau$-sneutrino and the\nSM-like Higgs boson the lightest scalar results to be a mixture\nof $\\widetilde{\\nu}_\\tau$ and the doublet-components $H_u$ and $H_d$, as\none can see in the upper left plot of \\reffi{fig:sntaulsmixing}.\nFor example, if we fine-tune $\\lambda = 0.20237$ we find that the lightest Higgs boson\nis composed of approximately\n\\begin{align}\nH_d \\quad \\rightarrow &\\quad |U_{11}^{H(0)}|^2\\sim 1\\%  \\; , \\\\\nH_u \\quad \\rightarrow &\\quad |U_{12}^{H(0)}|^2\\sim 80\\% \\; , \\\\\n\\widetilde{\\nu}_{3L} \\quad \\rightarrow &\\quad |U_{16}^{H(0)}|^2\\sim 19\\% \\; .\n\\end{align}\nNevertheless, due to the upward shift, as explained before,\nthe one-loop corrections break the degeneracy and no trace\non the SM-like Higgs mass remains, which would deviate it from the NMSSM\nprediction.\n\n\n\n\n\n\n\n\\subsection{\\protect\\boldmath The \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ and the CMS $\\gamma\\ga$ excess at\n  $96 \\,\\, \\mathrm{GeV}$}\n\\label{sec:gaga95}\nIn this section we will investigate a scenario in which the SM-like Higgs\nboson is not the lightest ${\\CP}$-even scalar. This is inspired by\nthe reported excesses of LEP~\\cite{Barate:2003sz}\nand CMS~\\cite{CMS:2015ocq,CMS:2017yta}\nin the mass range around\n$\\sim 96\\,\\, \\mathrm{GeV}$, that (as we will show) can be explained\nsimultaneously by the presence\nof a light scalar in this mass window.\nWhile in the NMSSM the light scalar can be interpreted as the ${\\CP}$-even\nscalar singlet and can accommodate both excesses at $1\\sigma$ level\nwithout violating any known experimental\nconstraints~\\cite{Cao:2016uwt,Domingo:2017},\\footnote{Other possible\nexplanations of the CMS excess were analyzed in\n\\citere{Cacciapaglia:2016tlr,Mariotti:2017vtv,Crivellin:2017upt}. On the other hand, in the MSSM the CMS excess cannot be\nrealized~\\cite{Bechtle:2016kui}.}\nwe will interpret the\nlight scalar as the ${\\CP}$-even right-handed sneutrino of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}.\nSince the singlet of the NMSSM and the right-handed sneutrino of\nthe \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ are both gauge-singlets, they share very similar properties.\nHowever, the explanation of the excesses in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ avoids\nbounds from direct detection experiments, because $R$-parity is broken\nin the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ and the dark matter candidate is not a neutralino as in\nthe NMSSM but a gravitino with a lifetime longer than the age of\nthe universe~\\cite{Munoz:2016vaa}. This is important because the direct\ndetection measurements were shown to be very constraining in the NMSSM\nwhile trying to explain the dark matter abundance on top of the\nexcesses from LEP and CMS~\\cite{Cao:2016uwt}.\n\n\\begin{table}\n\\renewcommand{\\arraystretch}{1.7}\n\\centering\n\\begin{tabular}{c c c c c c c c c c}\n $v_{iL}\/\\sqrt{2}$ & $Y^\\nu_i$ & $A^\\nu_i$ & $\\tb$ & $\\mu$ & $\\lambda$ &\n \t$A^\\lambda$ & $\\kappa$ & $A^\\kappa$ & $M_1$ \\\\\n \\hline\n $10^{-5}$ & $10^{-7}$ & $-1000$ & $2$ & $[413;418]$ & $0.6$ &\n \t$956$ & $0.035$ & $[-300;-318]$ & $100$ \\\\\n \\hline\n \\hline\n $M_2$ & $M_3$ & $m_{\\widetilde{Q}_{iL}}^2$ &\n \t$m_{\\widetilde{u}_{iR}}^2$ & $m_{\\widetilde{d}_{iR}}^2$ &\n \t$A^u_i$ & $A^{d}_i$ & $(m_{\\widetilde{e}}^2)_{ii}$ &\n \t$A^e_{33}$ & $A^e_{11,22}$ \\\\\n \\hline\n $200$ & $1500$ & $800^2$& $800^2$ & $800^2$ & $0$ &\n \t$0$ & $800^2$ & $0$ & $0$ \n\\end{tabular}\n\\caption{Input parameters for the scenario featuring the right-handed\n\t\t sneutrino in the mass range of the LEP and CMS excesses and\n\t\t a SM-like Higgs boson as next-to-lightest ${\\CP}$-even scalar;\n  \t\t all masses and values for trilinear parameters are in GeV.}\n\\label{tab:lepcms}\n\\renewcommand{\\arraystretch}{1.0}\n\\end{table\n\nIn \\refta{tab:lepcms} we list the values of the parameters we used\nto account for the lightest ${\\CP}$-even scalar as the right-handed\nsneutrino and the second lightest one the SM-like Higgs boson.\n$\\lambda$ is chosen to be large to account for a sizable mixing of\nthe right-handed sneutrino and the doublet Higgses.\nIn the regime where the SM-like Higgs boson is not the lightest scalar,\none does not need large quantum corrections to the Higgs boson mass,\nbecause the tree-level mass is already well above $100\\,\\, \\mathrm{GeV}$. This is\nwhy $\\tan\\beta$ can be low and the soft trilinears $A^{u,d,e}$\nare set to zero. The values of $A^{\\lambda}$ and $\\left|A^\\nu\\right|$\nare chosen to be around $1\\,\\, \\mathrm{TeV}$ to get masses for the\nheavy MSSM-like Higgs and the left-handed sneutrinos of this order, so\nthey do not play an important role in the following discussion.\nOn the other hand, $\\kappa$ is small to bring the mass\nof the right-handed sneutrino below the SM-like Higgs boson mass.\nFinally, the two parameters that are varied are $\\mu$ and $A^\\kappa$.\nBy increasing $\\mu$ the mixing of the right-handed sneutrino with\nthe SM-like Higgs boson is increased, which is needed to couple\nthe gauge-singlet to quarks and gauge-bosons. At the same time we\nused the value of $A^\\kappa$ to keep the mass of the\nright-handed sneutrino in the correct range. Accordingly, the\nresults in this chapter will all be displayed in the scanned\n$A^\\kappa$-$\\mu$ plane.\n\nThe process measured at LEP was the production of a Higgs boson\nvia Higgstrahlung associated with the Higgs decaying to\nbottom-quarks:\n\\begin{equation}\n\\mu_{\\rm LEP}=\\frac{\\sigma\\left( e^+e^- \\to Z h_1 \\to Zb\\bar{b} \\right)}\n\t\t\t   {\\sigma^{SM}\\left( e^+e^- \\to Z h \n\t\t\t   \t\t\\to Zb\\bar{b} \\right)}\n\t\t\t  = 0.117 \\pm 0.057 \\; ,\n\\end{equation}\nwhere $\\mu_{\\rm LEP}$ is called the signal strength, which is the\nmeasured cross section normalized to the standard model expectation,\nwith the SM Higgs boson mass at $\\sim 96\\,\\, \\mathrm{GeV}$.\nThe value for $\\mu_{\\rm LEP}$ was extracted in \\citere{Cao:2016uwt}\nusing methods described in \\citere{Azatov:2012bz}.\nWe can find an\napproximate expression for $\\mu_{\\rm LEP}$ factorizing the production\nand the decay of the scalar and expressing it in terms of couplings\nto the massive gauge bosons $C_{h_1VV}$ and the up- and down-type\nquarks $C_{h_1u\\bar{u}}$ and $C_{h_1d\\bar{d}}$, respsectively, normalized to\nthe SM predictions for the corresponding couplings (where with\n\\mbox{}$^{\\mu\\nu}$ we denote the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ prediction, and $\\Gamma$ is the\nHiggs-boson decay width):\n\n\\begin{align}\n\\mu_{\\rm LEP}^{\\mu\\nu}&=\\frac{\\sigma^{\\mu\\nu}\\left(Z^*\\to Zh_1 \\right)}\n          {\\sigma^{\\rm SM}\\left(Z^*\\to Zh \\right)}\\times\n      \\frac{\\text{BR}^{\\mu\\nu}\\left( h_1\\to b \\bar b\\right)}\n            {\\text{BR}^{\\rm SM}\\left( h\\to b \\bar b\\right)} \n            \\notag \\\\\n     &\\approx\\left| C_{h_1VV} \\right|^2\\times\n       \\frac{\\Gamma^{\\mu\\nu}_{b\\bar{b}}}{\\Gamma^{\\rm SM}_{b\\bar{b}}}\n          \\times\n       \\frac{\\Gamma^{\\rm SM}_{\\rm tot}}{\\Gamma^{\\mu\\nu}_{\\rm tot}} \n       \\notag \\\\\n     &\\approx \\frac{\n     \\left| C_{h_1VV} \\right|^2\\times\\left| C_{h_1d\\bar{d}} \\right|^2\n          }{\n      \\left| C_{h_1d\\bar{d}} \\right|^2\n          (\\text{BR}^{\\rm SM}_{b\\bar{b}}+\n           \\text{BR}^{\\rm SM}_{\\tau\\bar{\\tau}}) +\n           \\left| C_{h_1u\\bar{u}} \\right|^2\n            (\\text{BR}^{\\rm SM}_{gg}+\n           \\text{BR}^{\\rm SM}_{c\\bar{c}})} \\; .\n\\label{eq:mulep}\n\\end{align}\n\nThe SM branching ratios dependent on the Higgs boson mass can be\nobtained from \\citere{Heinemeyer:2013tqa}.\nThe denominator is the ratio of the total decay width of $h_1$\nin the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ and $h$ in the SM when all SM\nbranching ratios larger than $1\\%$ are considered.\nThe off-shell decay to $W$ and $Z$ bosons is in\nprinciple also possible, but the BRs are very small for a SM Higgs\nboson with a mass around $95\\,\\, \\mathrm{GeV}$ ($\\text{BR}^{\\rm SM}_{WW}\\sim 0.5\\%$ \nand $\\text{BR}^{\\rm SM}_{ZZ}\\sim 0.06\\%$)~\\cite{Denner:2011mq,Heinemeyer:2013tqa}.\nIt is worth noticing that although the right-handed neutrino mass\nis small, $m_{\\nu_R}\\sim62-63\\,\\, \\mathrm{GeV}$,\nin the investigated parameter region, it is nevertheless\nlarger than half of\nthe SM-like Higgs boson mass\nin all benchmark points, so the decay of the Higgs\nto the right-handed neutrino is kinematically forbidden and\ncannot spoil the properties of the SM-like Higgs.\nNeglecting the vevs $v_{iL}$ the normalized couplings of the\nscalars are given at\nleading order by the admixture of the mass eigenstate $h_i$ with\nthe doublet like Higgs $H_d$ and $H_u$ via\n\\begin{equation}\nC_{h_i d \\bar{d}}=\\frac{U^{H,(2')}_{i1}}{\\cos\\beta} \\; , \\quad\nC_{h_i u \\bar{u}}=\\frac{U^{H,(2')}_{i2}}{\\sin\\beta} \\; , \\quad\nC_{h_i V V}=U^{H,(2')}_{i1}\\cos\\beta+U^{H,(2')}_{i2}\\sin\\beta \\; ,\n\\end{equation}\nwhere the partial two-loop plus resummation corrected mixing matrix elements $U^{H,(2')}_{ij}$\nwere calculated\nin the approximation of vanishing momentum,\nsee the discussion in \\refse{sec:higherorders}.\nWe show in \\reffi{fig:lep1} the masses (top row) and the normalized\ncouplings ($|C_{h_1 d \\bar d}|$ second row, $|C_{h_1 u \\bar b}$| third\nrow, $|C_{h_1 VV}|$ lowest row) \nof the lightest and the next-to-lightest ${\\CP}$-even scalar. The lower\nright corner (marked in gray) results in the right-handed sneutrino\nbecoming tachyonic (at tree-level).\nThe largest mixing of the right-handed sneutrino and the SM-like\nHiggs boson is achieved where $\\mu$ is largest\nand $|A^\\kappa|$ is smallest. The mass of $h_2$ is in the allowed\nregion for a SM-like Higgs boson at $\\sim 125\\,\\, \\mathrm{GeV}$ if we assume a\ntheory uncertainty of up to $3\\,\\, \\mathrm{GeV}$ (see the previous subsections).\nThe LHC measurements of the SM-like Higgs boson\ncouplings to fermions and massive gauge bosons are still not\nvery precise~\\cite{Khachatryan:2016vau}, with uncertainties\nbetween 10 and $20\\%$ at the $1\\sigma$ confidence level\n(obtained with\nthe assumption that no beyond-the-SM decays modify the total\nwidth of the SM-like Higgs boson). Therefore, it would be\nchallenging to exclude parts of the parameter space by considering\nthe deviations of the normalized couplings of $h_2$.\nHowever, possible future lepton colliders\nlike the ILC could measure these couplings to a\n$\\%$-level~\\cite{Dawson:2013bba,Moortgat-Picka:2015yla}, which\ncould exclude (or confirm) most of the parameter space presented here.\nSeen from a more optimistic perspective, the precise measurement of\nthe SM-like Higgs boson couplings at future colliders could\nbe used to make predictions for the properties of the\nlighter right-handed sneutrino in this scenario.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.6\\textwidth]{lep1.pdf}\n  \\vspace{-0.5cm}\n  \\caption{Properties of the lightest (left) and next-to-lightest (right)\n         ${\\CP}$-even scalar in the $\\mu$--$A^\\kappa$ plane.\n         The couplings are normalized to the SM-prediction\n         of a Higgs particle of the same mass. The gray area is\n         excluded because the right-handed sneutrino becomes\n         tachyonic at tree-level.\n         \\textit{First row:} two-loop masses,\n         \\textit{second row:} coupling to down-type quarks,\n         \\textit{third row:} coupling to up-type quarks,\n         \\textit{fourth row:} coupling to massive gauge bosons.}\n  \\label{fig:lep1}\n\\end{figure}\n\nThe CMS excess was observed in the diphoton channel\nwith a signal strength of~\\cite{Shotkin:2017}\n\\begin{equation}\n\\mu_{\\rm CMS}=\\frac{\\sigma\\left( gg\\to h_1 \\to \\gamma\\gamma \\right)}\n         {\\sigma^{\\rm SM}\\left( gg\\to h \\to \\gamma\\gamma \\right)}\n     = 0.6 \\pm 0.2 \\; .\n\\end{equation}\nWe calculate the signal strength using the approximation that the\nHiggs production via gluonfusion is described at leading order\nexclusively by the loop-diagram with a\ntop quark running in the loop, and that\nthe diphoton decay is described by the diagrams with $W$~bosons or a\ntop quark in the loop, which is sufficient in the investigated mass\nrange of $h_1$. One can then write\n\\begin{align}\n\\mu_{\\rm CMS}^{\\mu\\nu}&=\\frac{\\sigma^{\\mu\\nu}\\left( gg\\to h_1 \\right)}\n         {\\sigma^{\\rm SM}\\left( gg\\to h \\right)} \\times\n      \\frac{\\text{BR}^{\\mu\\nu}_{\\gamma\\gamma}}\n           {\\text{BR}^{\\rm SM}_{\\gamma\\gamma}} \\notag \\\\\n     &\\approx \\left|C_{h_1 u \\bar{u}}\\right|^2 \\times\n        \\frac{\\Gamma^{\\mu\\nu}_{\\gamma\\gamma}}\n             {\\Gamma^{\\rm SM}_{\\gamma\\gamma}} \\times\n      \\frac{\\Gamma^{\\rm SM}_{\\rm tot}}{\\Gamma^{\\mu\\nu}_{\\rm tot}} \\notag \\\\\n     &\\approx\\frac{\\left|C_{h_1 u \\bar{u}}\\right|^2 \\times\n         \\left|C^{\\rm eff}_{h_1 \\gamma\\gamma}\\right|^2}\n            {\\left| C_{h_1d\\bar{d}} \\right|^2\n          (\\text{BR}^{\\rm SM}_{b\\bar{b}}+\n           \\text{BR}^{\\rm SM}_{\\tau\\bar{\\tau}}) +\n           \\left| C_{h_1u\\bar{u}} \\right|^2\n            (\\text{BR}^{\\rm SM}_{gg}+\n           \\text{BR}^{\\rm SM}_{c\\bar{c}})} \\; .\n\\label{eq:mucms}\n\\end{align}\nThe effective coupling of the neutral scalars to photons\n$C^{\\rm eff}_{h_i\\gamma\\gamma}$ has to be\ncalculated in terms of the couplings to the $W$ boson and the up-type\nquarks. In the SM the dominant contributions to the decay to photons\ncan be written as~\\cite{Djouadi:2005gi}\n\\begin{equation}\n\\Gamma^{\\rm SM}_{\\gamma\\gamma} =\n\\frac{G_\\mu\\,\\alpha^2 \\, m_h^3}{128\\, \\sqrt{2}\\, \\pi^3}\n\\left| \\frac{4}{3} A_{1\/2}\\left( \\tau_t \\right) +\n    A_{1}\\left( \\tau_W \\right) \\right|^2 \\; ,\n\\end{equation}\nwhere $G_\\mu$ is the Fermi-constant and the form factors $A_{1\/2}$\nand $A_{1}$ are defined as\n\\begin{align}\nA_{1\/2}\\left(\\tau\\right)&=2\\left( \\tau+\\left(\\tau -1\\right)\n      \\arcsin^2\\sqrt{\\tau} \\right)\\tau^{-2} \\; , \\\\\nA_{1}\\left(\\tau\\right)&=-\\left( 2\\tau^2+3\\tau\n      +3\\left( 2\\tau-1 \\right)\n      \\arcsin^2\\sqrt{\\tau} \\right)\\tau^{-2} \\; ,\n\\end{align}\nfor $\\tau \\leq 1$, and the arguments of these functions are\n$\\tau_t = m_h^2\/(4 m_t^2)$ and $\\tau_W = m_h^2\/(4 M_W^2)$.\nIn our approximation the only difference between the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ and the\nSM will be\nthat the couplings of $h_i$ to the top quark and the $W$ boson is\nmodified by the factors $C_{h_i t\\bar{t}}$ and $C_{h_i VV}$, so the\neffective coupling of the Higgses to photons in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\\nnormalized to the SM predictions\ncan be written as\n\\begin{equation}\n\\left|C^{\\rm eff}_{h_i \\gamma\\gamma}\\right|^2=\n\\frac{\\left| \\frac{4}{3} C_{h_i t\\bar{t}} A_{1\/2}\\left( \\tau_t \\right) +\n    C_{h_i VV} A_{1}\\left( \\tau_W \\right) \\right|^2}\n    {\\left| \\frac{4}{3} A_{1\/2}\\left( \\tau_t \\right) +\n    A_{1}\\left( \\tau_W \\right) \\right|^2} \\; .\n\\end{equation}\n\nUsing \\refeq{eq:mulep} and \\refeq{eq:mucms} we can\ncalculate the two signal strengths. \nThe result are shown in \\reffi{fig:lep2}, the LEP (left) and the CMS\nexcesses (right) in the $\\mu$--$A^{\\kappa}$ plane.\nWhile the LEP excess is easily reproduced\nin the observed parameter space, we cannot achieve the central\nvalue for $\\mu_{\\rm CMS}$, but only slightly smaller values.\nAs already observed in \\citere{Cao:2016uwt},\nthe reason for this is that for explaining the LEP excess\na sizable coupling to the bottom quark is needed.\nOn the contrary, the CMS\nexcess demands a small value for $C_{h_1 d\\bar{d}}$ so that the\ndenominator in \\refeq{eq:mucms} becomes small and $\\mu_{\\rm CMS}$\nis enhanced. Nevertheless, considering the large\nexperimental uncertainties in $\\mu_{\\rm CMS}$ and $\\mu_{\\rm LEP}$,\nthe scenario presented in this section\naccommodates both excesses comfortably well\n(at approximately $1\\sigma$),\nand it is a good motivation to keep on\nsearching for light Higgses in the allowed mass window\nbelow the SM-like Higgs mass. Apart from that, this scenario\nillustrates the importance of an accurate calculation of the\nloop-corrected scalar masses and mixings, since already small changes\nin the parameters can have a big impact on the production and the\ndecay modes of the ${\\CP}$-even Higgs bosons.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.9\\textwidth]{lep2.pdf}\n  \\vspace{-0.5cm}\n  \\caption{Signal strengths for the lightest $\\widetilde{\\nu}_R$-like\n         neutral scalar at CMS ($pp\\to h_1 \\to \\gamma\\gamma$)\n         (\\textit{left}) and\n         LEP (${e^+e^-\\to h_1 Z \\to b\\bar{b} Z}$)\n         (\\textit{right}) in the\n         $\\mu$-$A^\\kappa$ plane. The gray area is\n         excluded because the right-handed sneutrino becomes\n         tachyonic at tree-level.}\n  \\label{fig:lep2}\n\\end{figure}\n\n\n\\section{Conclusion and Outlook}\n\\label{sec:concl}\nThe \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ is a simple SUSY extension of the SM that is capable of predicting neutrino physics in agreement with experimental data. As in other SUSY models, higher-order corrections are crucial to reach a\ntheoretical uncertainty at the same level of (anticipated) experimental\naccuracy. So far, higher-order corrections in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ had been restricted\nto \\DRbar\\ calculations, which suffer from the disadvantage that they cannot\nbe directly connected to (possibly future observed) new BSM particles.\n\nIn this paper we have performed the complete one-loop renormalization of the\nneutral scalar sector of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ with one generation of right-handed\nneutrinos in a mixed on-shell\/\\DRbar\\ scheme. The renormalization procedure\nwas discussed in detail for each of the free parameters appearing in the\n\\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ Higgs sector. We have emphasized the conceptual differences to the MSSM and the NMSSM regarding the field\nrenormalization and the treatment of non-flavor-diagonal soft mass parameters, which have their\norigin in the breaking of $R$-parity in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}. However, we have ensured that the renormalization of the relevant (N)MSSM\nparts in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ are in agreement with previous calculations in those\nmodels. Consequently, numerical differences found can directly be attributed\nto the extended structure of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}.\nThe derived renormalization can be applied\nto any higher-order correction in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}.\nThe one-loop counterterms derived in this paper are implemented into\nthe \\FA\\ model file, so the computation of these corrections\ncan be done fully automatically.\n\nWe have applied the newly derived renormalization to the calculation of the full one-loop corrections to the neutral scalar masses\nof the \\ensuremath{\\mu\\nu\\mathrm{SSM}}, where we found that all UV-divergences cancel.\nIn our numerical analysis the newly derived full one-loop contributions are\nsupplemented by available MSSM higher-order corrections as provided by the\ncode \\fh\\ (leading and subleading fixed-order corrections as well as resummed large logarithmic\ncontributions obtained in an EFT approach.)\nWe investigated various representative scenarios, in which we obtained\nnumerical results for a SM-like Higgs boson mass consistent with experimental bounds. We compared our results to predictions of the various\nneutral scalars in the NMSSM to\ninvestigate the relevance of genuine \\ensuremath{\\mu\\nu\\mathrm{SSM}}-like contributions.\nWe find negligible\ncorrections w.r.t.\\ the NMSSM,\nindicating that the Higgs boson mass calculations in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ are at \nthe same level of\naccuracy as in the NMSSM.\n\nFinally we showed that the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ can accommodate a right-handed\n(${\\CP}$-even) scalar neutrino \nin a mass regime of $\\sim 96 \\,\\, \\mathrm{GeV}$, where the full Higgs sector is in\nagreement with the Higgs-boson measurements obtained at the LHC, as well as\nwith the Higgs exclusion bounds obtained at LEP, the Tevatron and the LHC.\nThis includes in particular a SM-like Higgs boson at $\\sim 125 \\,\\, \\mathrm{GeV}$.\nWe have demonstrated that the light right-handed sneutrino can explain\nan excess of $\\gamma\\ga$ events at $\\sim 96 \\,\\, \\mathrm{GeV}$ as reported \nrecently by CMS in their Run~I and Run~II date. It can\nsimultaneously describe the $2\\,\\sigma$ excess of $b \\bar b$ events\nobserved at LEP at a similar mass scale. We are eagerly awaiting the \ncorresponding ATLAS Higgs-boson search results. \n\n\n\n\\subsection*{Acknowledgements}\nWe thank F.~Domingo for helpful discussions.\nThis work was supported in part by the Spanish Agencia Estatal de\nInvestigaci\\'on through the grants\nFPA2016-78022-P MINECO\/FEDER-UE (TB and SH) and\nFPA2015-65929-P MINECO\/FEDER-UE (CM), and IFT Centro de\nExcelencia Severo Ochoa SEV-2016-0597. The work of TB was\nfunded by Fundaci\\'on La Caixa under `La Caixa-Severo Ochoa' international\npredoctoral grant. We also acknowledge the support of the MINECO's\nConsolider-Ingenio 2010 Programme under grant MultiDark CSD2009-00064.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n   \n   \n   \n   \n   \n   \n   \n   \n   \n\nGrammatical error correction (GEC) is the task of automatically identifying and correcting the grammatical errors in the written text. \nRecent work treats GEC as a translation task that use sequence-to-sequence models \\citep{SutskeverVL14seq2seq,BahdanauCB14} to rewrite sentences with grammatical errors to grammatically correct sentences. As with machine translation models, GEC models benefit largely from the amount of parallel training data. Since it is expensive and time-consuming to create annotated parallel corpus for training, there is research into generating sentences with artificial errors from grammatically correct sentences with the goal of simulating human-annotated data in a cost-effective way \\citep{YuanB16,XieAAJN16,chollampatt2018mlconv}. \n\n\nRecent work in artificial error generation (AEG) is inspired by the back-translation approach of machine translation systems \\citep{DBLP:conf\/acl\/SennrichHB16,DBLP:journals\/corr\/abs-1804-06189}.  In this framework, an intermediate model is trained to translate correct sentences into errorful sentences.  A new parallel corpus is created using the largely available grammatically correct sentences and the corresponding synthetic data generated by this intermediate model. The newly created corpus with the artificial errors is then used to train a GEC model \\citep{DBLP:conf\/bea\/ReiFYB17,DBLP:conf\/naacl\/XieGXNJ18,Ge2018HumanGEC}.\n\nTo date, there is no work that compares how different base model architectures perform in the AEG task. In this paper, we investigate how effective are \ndifferent model architectures in generating artificial, parallel data to improve a GEC model.  Specifically, we train four recent neural models (and one rule-based model \\cite{Bryant2018LanguageMB}), including two new syntax-based models, for generating as well as correcting errors.   We analyze which models are effective in the AEG and correction conditions as well as by data size.  Essentially, we seek to understand how effective are recent sequence-to-sequence (seq2seq) neural model as AEG mechanisms ``out of the box.\"  \n\n\n\n\n\\section{Related Work}\n\n\n\nBefore the adoption of neural models, early approaches to AEG involved identifying error statistics and patterns in the corpus and applying them to grammatically correct sentences \\citep{BrockettDG06,RozovskayaR10}. Inspired by the back-translation approach, recent AEG approaches inject errors into grammatically correct input sentences by adopting methods from neural machine translation  \\citep{FeliceY14,KasewaS018}. \\citet{DBLP:conf\/naacl\/XieGXNJ18} propose an approach that adds noise to the beam-search phase of an back-translation based AEG model to generate more diverse errors. They use the synthesized parallel data generated by this method to train a multi-layer convolutional GEC model and achieve a 5 point $F_{0.5}$ improvement on the CoNLL-2014 test data \\cite{ng-EtAl:2014:W14-17}. \\citet{Ge2018HumanGEC} propose a fluency-boosting learning method that generates less fluent sentences from correct sentences and pairs them with correct sentences to create new error-correct sentence pairs during training. Their GEC model trained with artificial errors approaches human-level performance on multiple test sets.  \n\n\n\n\\section{Approach}\n\n\\subsection{Correction and Generation Tasks}\nWe train our models on the two tasks---error correction and error generation. In \\textit{error correction}, the encoder of the sequence-to-sequence model takes an errorful sentence as  input and the decoder outputs the grammatically correct sentence. The process is reversed in the \\textit{error generation} task, where the model takes a correct sentence as input and produces an errorful sentence as the output of the decoder. \n\nWe investigate four recent neural sequence-to-sequence models---(i) multi-layer convolutional model \\citep[MLCONV;][]{chollampatt2018mlconv}, (ii) Transformer \\citep{DBLP:conf\/nips\/VaswaniSPUJGKP17}, (iii) Parsing-Reading-Predict Networks \\citep[PRPN;][]{shen2018neural}, (iv) Ordered Neurons \\citep[ON-LSTM;][]{shen2018ordered}---as error correction models as well as error generation models. The PRPN and ON-LSTM models are originally designed as recurrent language models that jointly learn to induce latent constituency parse trees. We use the adaption of PRPN and ON-LSTM models as decoders of machine translation systems \\citep{anon2018syntaxtranslation}: In this setting, a 2-layer LSTM is used as the encoder of the syntactic seq-to-seq models, and the PRPN and ON-LSTM are implemented as the decoders with attention \\citep{BahdanauCB14}. We hypothesize that syntax is important in GEC and explore whether models that incorporate syntactic bias would help with GEC task. We provide a brief description of each model in \\S\\ref{sec:models} and refer readers to the original work for more details. \n\n\\subsection{Models}\n\\label{sec:models}\n\\paragraph{Multi-layer Convolutional Model}\nWe use the multi-layer convolutional encoder-decoder base model (MLCONV) of \\citet{chollampatt2018mlconv} using the publicly available code from the authors.\\footnote{\\url{https:\/\/github.com\/nusnlp\/mlconvgec2018}} \nAs our aim is to only compare the performance of different architectures and not to achieve state-of-the-art performance, we make few changes to their code.\nThe model of \\citet{chollampatt2018mlconv} produces 12 possible correct sentences for each input sentences with error. They also train an N-gram language model as a re-ranker to score the generated sentences and pick the corrected sentence with the best score as final output. We did not use this re-ranking step in our model, nor did we perform ensembling or use the pre-trained embeddings as in the original work. We do not observe improvement in models like transformer and PRPN using re-ranking with an N-gram language model. Additionally, there's only a slight improvement in MLCONV using re-ranking. The reason might be because the N-gram language model is not very powerful.\n\n\n\\paragraph{Transformer Model} We use the publicly available Fairseq framework which is built using Pytorch for training the Transformer model. We apply the same hyper-parameters used for training the IWSLT'14 German-English translation model in the experiments of \\citet{DBLP:conf\/nips\/VaswaniSPUJGKP17}. \n\n\n\\paragraph{PRPN Model} is a language model that jointly learns to parse and perform language modeling \\citep{shen2018neural}. It uses a recurrent module with a self-attention gating mechanism and the gate values are used to construct the constituency tree. \nWe use the BiLSTM model as the encoder and PRPN as the decoder of the sequence-to-sequence model.\n\n\\paragraph{ON-LSTM Model}\n is follow-up work of PRPN, which incorporates syntax-based inductive bias to the LSTM unit by imposing hierarchical update order on the hidden state neurons \\citep{shen2018ordered}. ON-LSTM assumes that different nodes of a constituency trees  are represented by the different chunks of adjacent neurons in the hidden state, and introduces a master forget gate and a master input gate to dynamically allocate the chunks of hidden state neurons to different nodes. \n\n We use a BiLSTM model as encoder and ON-LSTM model as decoder.\n\n\n\n\n\n\\section{Experiments}\n\n\\subsection{Data}\nWe use the NUS Corpus of Learner English \\citep[NUCLE;][]{DBLP:conf\/bea\/DahlmeierNW13} and the Cambridge Learner Corpus \\citep[CLC;][]{Nicholls2003} as base data for training both the correction and generation models.  We remove sentence pairs that do not contain errors during preprocessing resulting in 51,693 sentence pairs from NUCLE and 1.09 million sentence pairs from the CLC .\nWe append the CLC data to the NUCLE training set (henceforth NUCLE-CLC) to use as training data for both AEG and correction. We use the standard NUCLE development data as our validation set and we early-stop the training based on the cross-entropy loss of the seq-to-seq models for all models.  For the generation of synthetic errorful data, we use the 2017 subsection of the LDC New York Times corpus also employed in the error generation experiments of \\citet{DBLP:conf\/naacl\/XieGXNJ18} which contains around 1 million sentences.\\footnote{\\url{https:\/\/catalog.ldc.upenn.edu\/LDC2008T19} }\n\n\n\\begin{table*}[]\n\\small\n\\centering\n\\begin{center}\n\\begin{tabular}{llccccccc} \n\\toprule\n \\bf GEC Model & \\bf AEG model & \\bf NUCLE-CLC &  \\bf 10K  & \\bf  50K & \\bf 100K  & \\bf  500K & \\bf 1M & \\bf 2M \\\\\n \\midrule\n\n\n MLCONV  & MLCONV  & 35.2 & 35.1 &  34.7 &  34.6 &  38.9 & 39.4 & 34.0   \\\\\n Transformer & MLCONV  & 36.3 &  43.9 & \\bf 44.1   & \\bf 45.4 &  \\bf 44.4 & \\bf 45.5 &  \\bf 42.0  \\\\\n PRPN & MLCONV  & \\bf 43.6 &  \\bf 45.4  &  42.8 &  43.2 &  39.6 &  38.6  & 31.7 \\\\\n ON-LSTM & MLCONV  & 36.6  &  39.8   & 35.6 & 38.4  &  36.9 &  24.2  & 20.1 \\\\\n \\midrule\n \n\n MLCONV  & Transformer & 35.2 & 36.1 &  35.2 &   39.4 & 36.6 & 36.6 & 36.1   \\\\\n Transformer & Transformer & 36.3 & 20.1 & \\bf 43.9   & \\bf 42.9 & \\bf 43.7 &  \\bf  44.0 & \\bf 41.0  \\\\\n PRPN & Transformer &  \\bf 43.6 & \\bf 43.1  &  40.9 &  40.6 & 41.4 &  29.4  & 31.7 \\\\\n ON-LSTM & Transformer & 36.6  &  39.8   & 38.2 & 39.6  & 24.0 &  21.3  & 20.1 \\\\\n  \\midrule\n MLCONV & Rule-based &  35.2 & 6.0 &  7.8 &  10.5 &  13.7 & 13.9 & --  \\\\\n Transformer  & Rule-based &  36.3 & \\bf 13.5 &  \\bf 14.4   &  \\bf 21.8 & \\bf 14.5 & \\bf 21.6 & --  \\\\\n PRPN  & Rule-based & \\bf  43.6 & 2.8  &  4.9 &  2.6 &  3.9 &  8.9 & --  \\\\\n ON-LSTM  & Rule-based &  36.6  & 4.7   & 3.9 & 5.5  &  4.2 &  5.3 & --  \\\\\n  \\bottomrule  \n\\end{tabular}\n\\end{center}\n\\caption{\\label{tab:rb-table} (Exp2) Evaluating the impact of MLCONV, Transformer and the rule-based AEG systems. \nNUCLE-CLC column represents the F0.5 score of GEC models trained on the base NUCLE-CLC data. \\textit{10K, 50K, 100K, 500K, 1M, and 2M} represents the amount of artificial data added to the NUCLE-CLC during training. \n}\n\\end{table*}\n\n\n\\subsection{Setup}\nWe conduct four experiments in this paper.  In \\textbf{Exp1}, we train all the AEG models and intermediate GEC models on NUCLE-CLC. We use the NYT dataset as input to the AEG models to generate sentences with artificial errors.  We then create new parallel training sets for correction by combining the sentences from CLC and NUCLE with the errorful sentences generated by each model. We then train the GEC models using these parallel datasets. \n\nThe three other experiments are variants of the first. In \\textbf{Exp2} we train all correction models on artificial errors generated by the top neural AEG systems and a rule-based system for comparison.  In \\textbf{Exp3}, we train the GEC models on NUCLE to analyze models built on real data.   Finally, in \\textbf{Exp4}, we train all GEC models on artificial data to determine how well correction models can perform without any real data.\n\n\nAll our experiments are tested on the CoNLL-2014 test set and we use the sentence-level $F0.5$ score from the MaxMatch ($M^2$) scorer \\citep{DahlmeierN12} for evaluation.  All models are implemented using the Fairseq framework.\\footnote{\\url{https:\/\/github.com\/pytorch\/fairseq}}\n\n\n\n\n\n\n\n\n\n\\subsection{Results}\n\n\\noindent\\textbf{Exp1}: Figure~\\ref{fig:table1a} shows the performance of GEC models trained on the base NUCLE-CLC set and then retraining with various amounts of artificial data.  We first observe that PRPN performs substantially higher than the rest of the models when trained only with the base CLC-NUCLE data.  However, its performance drops when artificial data generated by the corresponding PRPN AEG model is added. As for ON-LSTM, the performance improves slightly when the amount of added data is less than 100k but the performance drops drastically otherwise. Conversely, the performance of MLCONV and Transformer improves with the added artificial data but the improvement is not linear with the amount of added data. \n\n\\begin{figure}[htbp]\n    \\centering\n        \\includegraphics[height=1.8in] {Nucle-CLC-table1a.png}\n        \\caption{(Exp1) Models trained on the artificial data generated by the corresponding AEG model.  The X-axis represents the amount of artificial data added to NUCLE-CLC during training. \n        }\n        \\label{fig:table1a}\n\\end{figure}\n\n\\noindent\\textbf{Exp2}: Since the performance of MLCONV and Transformer GEC models improve with the addition of artificial data generated by corresponding AEG models, we hypothesize that the artificial error generated by these models are useful. To test this hypothesis, we train all the GEC models with various amount of artificial error generated by MLCONV and Transformer AEG models.  We also compare these AEG models to a rule-based one inspired by the confusion set generation method in \\citet{Bryant2018LanguageMB}. We subsequently score each sentence with a language model (GPT-2 \\citep{noauthororeditor}) in order not to select the most probable sentence. This method generates a confusion set for prepositions (set of prepositions plus an empty element), determiners, and morphological alternatives (cat $\\rightarrow$ cats).\n\n\n\nThe results of these experiments are found in Table~\\ref{tab:rb-table}.  Nearly all correction models improve when using MLCONV or Transformer AEG data with the biggest performances yielded using the Transformer model.  Interestingly, when using 1M or 2M samples, performance starts to decline.  We believe that over 1M samples, the noisiness of the artificial data overwhelms the contributions of the real data (roughly over 1M samples).  The performance of all models drops when trained with the errors generated by the rule-based model. It is interesting to observe that the performance drops significantly just by adding 10K artificial sentences to the base data.  \n\n\n\n\n\n\n\\noindent\\textbf{Exp3}: Table \\ref{tab:nucle-table} shows the performance of the models trained on NUCLE dataset with additional artificial data generated by corresponding AEG models trained on NUCLE-CLC. We can see that the performance of all models, except ON-LSTM, improves significantly when 1 million artificial sentence pairs are added to the NUCLE training data, even though the errors in these sentences do not resemble natural errors. This contrasts with the result in Figure~\\ref{fig:table1a} where the performance of the GEC models trained with the combination of artificial error and CLC-NUCLE base data drops. This suggests that artificial data is helpful when the base data size is relatively small.\n\n\\begin{table}[htbp]\n\\small\n\\centering\n\\setlength{\\tabcolsep}{9 pt} \n\\begin{center}\n\\begin{tabular}{lcccc} \n\\toprule\n \\bf Model & \\bf NUCLE &  \\bf +10K  & \\bf  +50K &  \\bf +1M \\\\\n \\midrule\n\n\n MLCONV  & 10.1 & 12.3 &  12.9 &   16.1   \\\\\n Transformer & 11.2   & \\bf 28.1 &  \\bf 16.9  &   22.8  \\\\\n PRPN  & 8.3 & 6.9 &  12.5 &  \\bf 26.2   \\\\\n ON-LSTM & 9.4   & 11.3 &  11.8  &  6.0  \\\\\n  \\bottomrule  \n\\end{tabular}\n\\end{center}\n\\caption{\\label{tab:nucle-table} (Exp3) Using only NUCLE as base training for correction.  The AEG models are trained using NUCLE-CLC data as in other experiments. \n}\n\\end{table}\n\n\\noindent\\textbf{Exp4}: The GEC models trained only on artificial data perform very poorly. The best setups, Transformer and MLCONV, achieve F0.5 scores of 12.8 and 12.4 respectively when trained with 2 million sentences generated by the corresponding AEG model.\nThis outcomes suggests that AEG data should be paired with some sample of real data to be effective.\n\n\\subsection{Manual Evaluation}\nWe performed a manual analysis of the generated error sentences and found that many of the errors did not always resemble those produced by humans.  For example, \\textit{The situation with other types is not much (better $\\rightarrow$ downward)}.  This shows that despite the noisiness of the error-generated data, some models (namely MLCONV and Transformer) were robust enough to improve. This also suggests that we may achieve better improvement by controlling artificial errors to resemble the errors produced by humans. The performance of syntax-based models goes down significantly with the addition of artificial errors, which indicates that these models may be sensitive to poor artificial data.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\nWe investigated the potential of recent neural architectures, as well as rule-based one, to generate parallel data to improve neural GEC.  We found that the Multi-Layer Convolutional and Transformer models tended to produce data that could improve several models, though too much of it would begin to dampen performance.  The most substantial improvements could be found when the size of the real data for training was quite small.  We also found that the syntax-based models, PRPN and ONLSTM, are very sensitive to the quality of artificial errors and their performance drops substantially with the addition of artificial error data.  Our experiments suggest that, unlike in machine translation, it is not very straightforward to use a simple back-translation approach for GEC as unrealistic errors produced by back-translation can hurt the correction performance substantially.\n\nWe believe this work shows the promise of using recent neural methods in an out-of-the-box framework, though with care.  Future work will focus on ways of improving the quality of the synthetic data.  Ideas include leveraging recent developments in powerful language models or better controlling for diversity and frequency of specific error types.\n\n\\section*{Acknowledgements}\nWe would like to thank the Grammarly Research Team, especially Maria Nadejde, Courtney Napoles, Dimitris Alikaniotis, Andrey Gryschuck, Maksym Bezva and Oleksiy Syvokon.  We would also like to thank Sam Bowman, Kyunghyun Cho, and the three anonymous reviewers for their helpful discussion and feedback.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Summary}\n\\label{introduction}\n\nThree dimensional gauge theories, in contrast to four dimensional ones, are always asymptotic free and strongly coupled in the infra-red (IR) regime. \nIt is a central problem to investigate properties of their IR fixed points, and for this analysis instanton or monopole operators often play a key role. \n\nIndeed, (BPS) monopole operators played a crucial role in classifying a class of $\\cN=4$ supersymmetric linear quiver gauge theories from their long distance behavior \\cite{Gaiotto:2008ak}.\nIn this class it was already studied how to compute the R-charge of an arbitrary monopole operator \\cite{Borokhov:2002ib,Borokhov:2002cg}, which depends on the number of hypermultiplets charged under the associated gauge group. \nIf the number of hypermultiplets is great enough so that the R-charge of any monopole operator is always greater than half, then the system flows to a standard critical point.\nIn this case the theory was called ``good''.\nOn the other hand, if the number of hypermultiplets decreases to saturate some bound so that there exists a monopole operator whose R-charge is equal to half, then it decouples at the IR fixed point.  \nThis was called an ``ugly'' theory.\nFinally if the number of hypermultiplets is so small that there exists a BPS monopole operator whose R-charge is less than half, then the IR fixed point of the system is not a standard one. This case was called a ``bad'' theory. \nWe refer to such monopole operators in the ugly\/bad theory as ugly\/bad ones.\nIn a bad theory the R-symmetry in the IR has to be different from that in the UV in order for the system to preserve the unitarity.  \n\nThe infra-red physics of a bad quiver as well as the fate of such a bad monopole operator in the IR were further studied in \\cite{Kapustin:2009kz,Bashkirov:2013dda,Yaakov:2013fza,Hwang:2015wna} (See \\cite{Assel:2017jgo,Dey:2017fqs} for recent study).\nThe simplest example to study a bad theory is ${\\cal N}=4$ SQCD with $U(N_\\text{c})$ gauge group and $N_\\text{f}$ fundamental hypermultiplets.\\footnote{ \nIn this case the theory is good if $2N_c\\le N_f$, ugly if $N_f=2N_c-1$ and bad if $N_c < N_f\\le 2N_c -2$ \\cite{Gaiotto:2008ak}.} \nIt was pointed out in \\cite{Kapustin:2009kz} that this classification by using a monopole operator is related to the convergence property of the three-sphere partition function, which was shown to reduce to a finite dimensional integration by the supersymmetric localization, so that the partition function of a good\/bad theory is absolutely convergent\/divergent. \nThe three-sphere partition function in a bad case was further studied by converting into a good or ugly theory by adding extra hypermultiplets with real mass so as to flow to the original bad theory in the IR \\cite{Yaakov:2013fza}. \nThis modified partition function was used to propose a Seiberg-like duality between a good theory and a bad one in $\\cN=4$ SQCD, where the bad monopole operators decouple in the IR. \nThis duality was further supported by a direct comparison of the factorized form of the superconformal index for both sides \\cite{Hwang:2015wna}.\\footnote{\nRecent argument by \\cite{Assel:2017jgo} based on the analysis of the moduli space suggests that the duality holds only around the special singular locus called ``symmetric vacuum'' in the moduli space of the bad theory.\nThis is, however, still consistent with the observations in \\cite{Yaakov:2013fza,Hwang:2015wna}.\n}\n\nIt is natural to ask whether this classification of $\\cN=4$ supersymmetric linear quiver gauge theories works also for the theories including Chern-Simons interaction. \nThis question may not be trivial in the following sense. \nFirst the three-sphere partition function for a supersymmetric Chern-Simons theory has the Fresnel factor, which plays a role of a damping factor by performing the analytic continuation for the Chern-Simons coupling, so that there may be no issue on the convergence or at least completely different convergence property from the case without Chern-Simons terms. \nThis suggests that the classification by using the convergence property of the three-sphere partition function does not work or at least needs speculation. \nOn the other hand, the classification by employing monopole operators seems to work, though the modification needs to be done once Chern-Simons interaction is introduced. \nThis is because the gauge charge of bare monopole operators changes accordingly, and so does the total R-charge of the gauge invariant monopole operator due to the dress by compensating matter operators. \n\nThe above question is strategically good to be addressed first by using the simplest model in the $\\cN=4$ linear quiver Chern-Simons theories, that is ${\\cal N}=4$ $U(N_1)_k\\times U(N_2)_{-k}$ Chern-Simons theory coupled with single bi-fundamental hypermultiplet.\nIn our previous publication we studied the three-sphere partition function in this minimal $\\cN=4$ Chern-Simons theory, where we performed the remaining integrations in the localization formula explicitly \\cite{Nosaka:2017ohr}.\nAs a result we found that the resulting partition function diverges for $k -N_1 -N_2 \\le -2$. This divergence is rather the IR one than the UV one since it can be cured by introducing FI terms or equivalently mass terms. \nWe have further found that the resulting partition functions of the theory connected by the Hanany-Witten transition coincide up to an overall factor, which is given by a trigonometric function to the power of $k-N_1-N_2$.\nThese results are reminiscent of those for $\\cN=4$ SQCD in \\cite{Yaakov:2013fza} and suggests that the minimal $\\cN=4$ Chern-Simons theory is bad for $k -N_1 -N_2 \\le -2$ in the terminology of \\cite{Gaiotto:2008ak}.\nIn this paper we study this behavior of minimal $\\cN=4$ Chern-Simons theories by computing their superconformal indices \\cite{Kim:2009wb,Imamura:2011su} (see \\cite{Yokoyama:2011qu} for a review and \\cite{Aharony:2015pla} for related works). \n\nThe rest of this paper is organized as follows.\nIn section \\ref{N4theory} we review the results of \\cite{Nosaka:2017ohr} and compare them with the observations in \\cite{Kapustin:2009kz,Yaakov:2013fza} without Chern-Simons terms.\nWe provide additional evidence for this duality by counting the dimension of the moduli spaces.\nIn section \\ref{generalexpression} we introduce the superconformal index, and in section \\ref{perturbative} we estimate the contributions from dressed monopole operator.\nAs a result we find that for $k-N_1-N_2\\le -2$ there is a family of infinitely many monopole operator which contribute with non-positive power of $x$, the fugacity of $D+J_3$.\nThis implies that $U(N_1)_k\\times U(N_2)_{-k}$ theory with $k-N_1-N_2\\le -2$ is indeed a bad theory.\nIn section \\ref{perturbative_results} we further compute the superconformal index of non-bad theories with $k-N_1-N_2=\\pm 1$ explicitly in small $x$ expansion.\nBy comparing those for pairs of theories related through the Hanany-Witten transition, we find that the ratio of the superconformal indices have a completely same expression as the superconformal index of a hypermultiplet.\nThis suggests that the theory with $k-N_1-N_2=-1$ is ugly and dual to the paired good theory with $k-N_1-N_2=1$ plus a hypermultiplet.\n\n\n\n\n\n\n\n\n\\section{Minimal $\\cN=4$ Chern-Simons matter theories }\n\\label{N4theory}\n\n\nIn this section we briefly review the minimal $\\cN=4$ Chern-Simons matter theory or $\\text{U}(N_1)_k\\times \\text{U}(N_2)_{-k}$ Chern-Simons theory interacting with one bifundamental hypermultiplet \\cite{Gaiotto:2008sd}. (See also \\cite{Hosomichi:2008jd,Nosaka:2017ohr}.)\nThis theory is the simplest example in a class of ${\\cal N}=4$ linear quiver Chern-Simons matter theories, which are realized by taking the low energy limit of a UV field theory on D3-branes in a type IIB brane configuration. \nThe type IIB brane configuration of the simplest linear quiver theory is given by Table.\\ref{GWN4BraneSetup}.\n\\begin{table}[htbp]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n & $012$ & $3$ & $456789$ \\\\ \n\\hline \nNS5-brane & $\\bigcirc $  &  $\\times$ & [456]  \\\\ \n\\hline\n$N_1$ D3-branes & $\\bigcirc $  & $\\bigcirc $ &  $\\times$ \\\\ \n\\hline\n$(1,k)$5-brane & $\\bigcirc $  &  $\\times$ & $[(47)_\\theta(58)_\\theta(69)_\\theta]$  \\\\ \n\\hline\n$N_2$ D3-branes & $\\bigcirc $  & $\\bigcirc $ & $\\times$  \\\\ \n\\hline\nNS5'-brane & $\\bigcirc $  &  $\\times$ & [456]  \\\\ \n\\hline\n\\end{tabular} \n\\caption{Type IIB Brane configuration describing the minimal linear quiver theory.\nHere $\\bigcirc$ indicates that the corresponding brane extends in all directions specified on its above column and $\\times$ means that it does not in any. The numbers inside the bracket represent the directions of the extension, where $(ij)_\\theta$ stands for a single direction in the $x^ix^j$-plane with the angle $\\theta=\\arctan k$ from the $x^{i}$ axis.  }\n\\label{GWN4BraneSetup}\n\\end{center}\n\\end{table} \nThe field contents of such a UV field theory of the simplest model are the two $\\cN=4$ vector multiplets associated with U$(N_1)\\times \\text{U}(N_2)$ and one hypermultiplet in the bi-fundamental representation in U$(N_1)\\times\\text{U}(N_2)$.\nIn the terminology of 3d ${\\cal N}=2$ representations, here a 3d ${\\cal N}=4$ vector multiplet consists of a vector multiplet plus a adjoint chiral multiplet, while a hypermultiplet consists of a pair of chiral multiplets.\n\nThese massless supersymmetric multiplets arise from open strings ending on the $N_1$ D3-branes, those on the $N_2$ D3-branes and the ones connecting the two stacks of D3-branes, respectively.\nThe $(1,k)$5-brane induces the `twisted' mass for the $\\cN=4$ vector multiplets \\cite{Kitao:1998mf,Bergman:1999na}: for the U$(N_1)$ vectormultiplet the ${\\cal N}=2$ Chern-Simons term with level $k$ plus the cocmplex mass term proportional to $k$ for the adjoint chiral multiplet; for the U$(N_2)$ vectormultiplet the same terms with $k$ replaced with $-k$.\nThe twisted mass breaks SO$(4)_{\\rm UV}$ R-symmetry to SO$(3)_{\\rm UV}$. \n\nIn the low energy limit this system gets strongly coupled and flow to the conformal fixed point, which is nothing but the minimal $\\cN=4$ Chern-Simons theory mentioned above. \nIn the IR limit, the $\\text{SO}(3)_{\\rm UV}$ R-symmetry is enhanced to SO$(4)_{\\rm R}$, which can be explicitly seen by integrating out the massive vector multiplet and adjoint chiral multiplets except the Chern-Simons gauge field. \nIn addition to the other global symmetry this system enjoys the parity invariance which exchanges the two gauge fields when $N_1=N_2$.\nWhen the two ranks are different, the system is invariant under the exchange of the ranks as well as the levels.\n\n\n\n\\subsection{Moduli space} \n\\label{modulispace}\n\nIn this subsection we briefly comment on the moduli space of the minimal $\\text{U}(N_1)_k\\times \\text{U}(N_2)_{-k}$ $\\cN=4$ Chern-Simons theory.\nWe will use the result of this analysis for a quick check of Seiberg-like duality discussed in the next subsection.\nFor convenience we write down the Lagrangian in the Euclidean space following the convention used in \\cite{Yokoyama:2013pxa} ($\\kappa=k\/(4\\pi)$)\n\\beal{\n\\cL\n=& \\tr \\bigg[\\kappa \\Bigl(-\\varepsilon^{\\mu\\nu\\rho}\\bigl(A_\\mu\\partial_\\nu A_\\rho +{2\\over3}A_\\mu A_\\nu A_\\rho\\bigr)+ \\varepsilon^{\\mu\\nu\\rho}\\bigl(\\wt A_\\mu\\partial_\\nu \\wt A_\\rho +{2\\over3}\\wt A_\\mu \\wt A_\\nu \\wt A_\\rho\\bigr) \\Bigr)  +D_\\mu  Q_A ^\\dagger D^\\mu Q^A -  {\\psi^\\dagger}^A \\sla D \\psi_A  \\nn\n&+ {1\\over 2\\kappa}(- {\\psi^\\dagger}^A \\psi_A  Q^\\dagger_C Q^C  +  \\psi_A {\\psi^\\dagger}^A Q^B  Q^\\dagger _B + \\varepsilon_{CD} \\varepsilon_{AB} {\\psi^\\dagger}^A  Q^C {\\psi^\\dagger}^B  Q^D +\\varepsilon^{AB} \\varepsilon^{CD} Q^\\dagger_C  \\psi_A Q^\\dagger_D \\psi_B )\\bigg] +V_q.\n\\label{N4GWaction}\n}\nHere $A,B=1,2$ and the covariant derivative for $Q^A$ and $\\psi_B$ is defined as $D_\\mu X = \\partial_\\mu X +A_\\mu X -X \\wt A_\\mu $.\n$V_q$ is the scalar potential given b\n\\footnote{ \nThis form of the scalar potential is obtained after rewriting the one given in \\cite{Yokoyama:2013pxa} using an identity for SU(2) indices. \n}\n\\beal{ \nV_{q} =\\tr [ T_C (T_C)^\\dagger ]\n}\nwhere $T_C = {1\\over 2\\kappa} \\varepsilon_{AB} Q^A Q^\\dagger_C Q^B$. \n\nThe vacuum moduli space is the solutions $(Q^1,Q^2)$ of $T_1=T_2=0$, modded by the U$(N_1)\\times \\text{U}(N_2)$ gauge transformations.\nThe generic point on the moduli spcae can be characterized as follows.\nDue to the (generalized) parity invariance we can assume $N_1\\le N_2$ without loss of generality.\nBy using the gauge degrees of freedom the complex scalar  $Q^1{}^a_{i}\\; (1\\leq a \\leq N_1, 1\\leq i \\leq N_2 )$ can be diagonalized so that\n\\beal{  \nQ^1{}^a_{ i} =  \\left\\{\n\\begin{array}{cl}\n r_a \\delta^a_{ i} & (1 \\leq  i \\leq N_1) \\\\\n 0 & (N_1 <  i \\leq N_2) \\\\\n\\end{array}\n\\right.\n}\nwhere $r_a$ are real positive numbers.\nThe residual gauge symmetry is the diagonal $U(1)^{N_1}$. \nNow, in the generic situation where all $r_a$ are different with each other, the vacuum equation $T_C=0$ requires also $Q^2$ to be diagonal\n\\beal{  \nQ^2{}^a_{ i} =  \\left\\{\n\\begin{array}{cl}\nq_a \\delta^a_{ i} & (1 \\leq  i \\leq N_1) \\\\\n0 & (N_1 <  i \\leq N_2) \\\\\n\\end{array}\n\\right.\n}\nwith $q_a$ some complex numbers.\n\nThe diagonal U(1)$^{N_1}$ gauge symmetry fixes all the degrees of freedom of the gauge fields except its zero modes in the Cartan part, which span the extra directions in the moduli space since the abelian gauge field does not couple with the bi-fundamental matter fields.\nThe gauge flux quantization gives the $2\\pi$ periodicity for the range of each zero mode, while the level $k$ Chern-Simons interaction breaks the U(1)$^{N_1}$ to $\\mathbb{Z}_k^{N_1}$ (see \\cite{Imamura:2008nn,Martelli:2008si} for example).\nThese zero modes become coordinates of the moduli space as $(S^1\/\\mathbb{Z}_k)^{N_1}$.\n\nAs a result, the classical moduli space is generically given by {$(\\mathbb{C}^2\/\\mathbb{Z}_k)^{\\text{min}(N_1,N_2)}\/{\\cal S}_{\\text{min}(N_1,N_2)}$. We suspect that the moduli space does not receive any quantum correction and is classically exact at least for non-bad theories as in the cases with the Higgs phase in a linear quiver non-Chern-Simons theory and with $\\cN=6$ ABJM theory. \nThis is indeed supported from the superconformal index computed in section \\ref{SCindex}.\nWe leave the proof thereof to a future work. In what follows, we confirm that this moduli space is consistent with the analysis below that there exists a decoupled sector of real dimension $4$ in the duality between pairs of non-bad theories\n\\footnote{ \nPrecisely speaking the decoupled sector turns out to couple to the other sector through the topological current. \n}\n\n\\subsection{Comments on level-rank duality} \n\\label{levelrankduality}\n\nThis minimal $\\cN=4$ Chern-Simons theory is expected to enjoy the level-rank (or Seiberg-like) duality \\cite{Giveon:2008zn}, that is, the duality between the $\\text{U}(N_1)_k\\times \\text{U}(N_2)_{-k}$ theory and the $\\text{U}(k-N_2)_k\\times \\text{U}(k-N_1)_{-k}$ theory.\nThe two theories are indeed related under the exchange of the two NS5-branes in the tpe IIB brane setup given in table \\ref{GWN4BraneSetup} by taking into account the Hanany-Witten brane creation\/annihilation \\cite{Hanany:1996ie}.\\footnote{\nSee \\cite{Yokoyama:2013pxa} for the first scratch of evidence of this self-duality from the large $N$ thermal free energy.\n}\n\n{\nOn the other hand, there is a dicrepancy in the $S^3$ partition functions between the two theories \\cite{Nosaka:2017ohr}; in some cases the partition function even diverges though it is finite in the other theory in the supposed dual.\nBy introducing the FI parameters which regularizes the divergences, we instead found that in any Hanany-Witten pair the ratio of the partition functions takes the same expression which depends on $k$, $N_1$, $N_2$ only through $k-N_1-N_2$.\n}\nThis mismatch of the overall factor may be a signal of the existence of some decoupled sector. Indeed, it was pointed out that such decoupled sector appears if there exists a monopole operator whose dimension computed in the UV theory saturates or violates the unitarity bound in the case without Chern-Simons interaction \\cite{Yaakov:2013fza,Hwang:2015wna}.\nIn the terminology used in \\cite{Gaiotto:2008ak} such theories with (naively) unitarity violating monopole operators are called {\\it bad} theories, and the  divergences discovered in \\cite{Nosaka:2017ohr} suggests the following classification of the minimal ${\\cal N}=4$ Chern-Simons theories:\n\\begin{itemize}\n\\item [] The minimal ${\\cal N}=4$ $\\text{U}(N_1)_k\\times \\text{U}(N_2)_{-k}$ Chern-Simons theory is respectively\n\\begin{align}\n\\text{good\/ugly\/bad if }\nk-N_1-N_2 \\text{ greater than\/equal to\/smaller than } -1.\n\\label{goodbaduglyfromZS3}\n\\end{align}\n\\end{itemize}\nThe main motivation of this paper is to investigate whether this classification is valid and what such decoupled sector in the minimal $\\cN=4$ Chern-Simons theories is by using a superconformal index. \n \nThe existence of the decoupled sector can be understood from the dimension of the moduli space.\nFor example, let us consider the minimal $\\cN=4$ theory with $\\text{U}(1)_3\\times \\text{U}(1)_{-3}$, whose moduli space is of real dimension $4$.\nThis theory is suggested to be dual to the one with $\\text{U}(2)_3\\times \\text{U}(2)_{-3}$.\nThe dimension of the moduli space in the latter theory, however, is $8$ and does not match with the former theory.\nWe suspect that the discrepancy is explained as the presence of some decoupled sector in the latter theory, as in the cases without Chern-Simons terms.\n\nIn what follows, we study such decoupled sectors from a superconformal index, which encodes the BPS spectrum of the theory and is useful to analyze the system more precisely.\n\n\\section{Superconformal index}\n\\label{SCindex} \n\nIn this section we examine the IR aspects of the ${\\cal N}=4$ minimal $\\text{U}(N_1)_k\\times \\text{U}(N_2)_{-k}$ superconformal Chern-Simons theory proposed in section \\ref{levelrankduality} by studing the superconformal index.\nAfter introducing the definition of the superconformal index in section \\ref{generalexpression}, in section \\ref{analytic} we first provide an analytic computation of the superconformal index.\nThough this analytic computation works well only for the abelian case $N_1=N_2=1$, it and also tells us a technical obstacle associated with the bad theory.\nIn section \\ref{perturbative} we adopt the small $x$ expansion.\nThrough an argument on its convergence we provide the good\/bad classifiation for the theory, which coincides with the one suggested from the convergence of the $S^3$ partition function \\eqref{goodbaduglyfromZS3}.\nFinally we compare the superconformal indices of the two theories related under the Hanany-Witten transition in several examples.\nAs we have expected, we observe that their ratio takes the identical form as the superconformal index of a hypermulitplet.\n\n\\subsection{General expression} \n\\label{generalexpression}\n\nIn this section we study the superconformal index of minimal $\\cN=4$ $\\text{U}(N_1)_k\\times \\text{U}(N_2)_{-k}$ Chern-Simons matter theories.\nFirst let us define a superconformal index by introducing fugacities of the global symmetry which commutes with the chosen $\\cN=2$ supersymmetry.\nThis requires a special care as we shall explain below. \n\nWe compute the superconformal index by the supersymmetric localization so that \nwe deform a minimal $\\cN=4$ Chern-Simons theory to be an $\\cN=3$ UV field theory described in section \\ref{N4theory} in the free theory limit. \nWe emphasize that the SO(4)$_\\text{R}$ symmetry is broken to SO(3)$_{\\text{UV}}$ under the deformation. \nHence we shall turn on the fugacity only for this SO(3)$_{\\rm UV}$ symmetry to define a superconformal index of the system, which we denote by $\\sigma$. \nFurthermore, there is a conserved topological current for each gauge group associated with any node in the quiver diagram. \nWe also turn on a fugacity for each topological current, which we denote by $y,z$ respectively for $\\text{U}(N_1)$ and $\\text{U}(N_2)$.\nTaking these into account the superconformal index for this system is defined by the following trace over the spectrum of the theory:\n\\begin{align}\n{\\cal I}_{k,N_1,N_2}(x,\\sigma,y,z)=\\Tr[ (-1)^F e^{-\\beta'\\{Q,Q^\\dagger\\}} x^{D+J_3} \\sigma^A y^{m_{\\rm tot}} z^{n_{\\rm tot}} ]\n\\label{IndexDefinition}\n\\end{align}\nwhere $Q$ is a nilpotent supercharge and $Q^\\dagger$ is its BPZ conjugate, while $D$, $J_3$, $A$, $(m_\\text{tot},n_\\text{tot})$ are respectively the dilatation, the angular momentum, the Cartan generator for SO$(3)_\\text{UV}$ and the total magnetic charge of U$(N_1)\\times \\text{U}(N_2)$.\nOne can show that \\eqref{IndexDefinition} is actually independent of $\\beta'$ so that only BPS states, which satisfy $\\{Q,Q^\\dagger\\}=D-J_3-R = 0$, contribute to the index.\n\nTo perform the supersymmetric localization, \nwe first rewrite \\eqref{IndexDefinition} as the path integral form over $\\mbf S^2 \\times \\mbf S^1$.  \nThen we take the weak coupling limit so that the evaluation of the path integral by the WKB approximation becomes exact. As a result each supersymmetric multiplet contributes to the index independently under the saddle points parametrized by magnetic charges and holonomies for U$(N_1) \\times {\\rm U}(N_2)$, which \nwe denote by $(m_a,n_i)$ and $(u_a,v_i)$ with $a=1,\\cdots, N_1, i=1,\\cdots, N_2$, respectively. \nThen the 1-loop contribution of $\\cN=4$ U$(N_1) \\times {\\rm U}(N_2)$ vector multiplet is given by\n\\beal{\n{\\cal I}_\\text{vec}&=\n\\prod_{a\\neq b}^{N_1}(1-x^{|m_a-m_b|}u_au_b^{-1})\n\\prod_{i\\neq j}^{N_2}(1-x^{|n_i-n_j|}v_iv_j^{-1}), \n\\label{Ivec}\n}\nand that of the bi-fundamental hypermultiplets is \n\\beal{\n{\\cal I}_\\text{hyp}&=\n\\prod_{a=1}^{N_1}\n\\prod_{j=1}^{N_2}\n\\frac{\n(x^{|m_a-n_j|+\\frac{3}{2}}\\sigma^{-1}u_a^{-1}v_j;x^2)_\\infty\n}{\n(x^{|m_a-n_j|+\\frac{1}{2}}\\sigma u_a v_j^{-1};x^2)_\\infty\n}\n\\frac{\n(x^{|m_a-n_j|+\\frac{3}{2}}\\sigma u_a v_j^{-1};x^2)_\\infty\n}{\n(x^{|m_a-n_j|+\\frac{1}{2}}\\sigma^{-1}u_a^{-1}v_j;x^2)_\\infty\n},\n\\label{Ihyp}\n}\nwhere we have used the Pochhammer symbol defined by \n\\beal{\n(z;x)_m :=& \\prod_{n=0}^{m-1} (1-zx^n). \n}\nNote that the contribution of the adjoint chiral multiplet in the $\\cN=4$ vector multiplet completely cancels.\nIncluding the vacuum and classical contribution the index is finally given by \n\\begin{align}\n{\\cal I}_{k,N_1,N_2}(x,\\sigma,y,z)\n&=\\frac{1}{N_1!N_2!}\\sum_{\\overrightarrow{m}\\in \\mathbb{Z}^{N_1} \\atop \\overrightarrow{n}\\in\\mathbb{Z}^{N_2}}\n\\int \\prod_{a=1}^{N_1}\\frac{du_a}{2\\pi iu_a} \n\\prod_{i=1}^{N_2}\\frac{dv_i}{2\\pi iv_i} y^{m_\\text{tot}}z^{n_\\text{tot}}\nx^{\\epsilon_0} u_a^{km_a} v_i^{-kn_i}\n{\\cal I}_\\text{vec}\n{\\cal I}_\\text{mat}\n\\end{align}\nwhere $m_\\text{tot}=\\sum_{a=1}^{N_1}m_a$, $n_\\text{tot}=\\sum_{i=1}^{N_2}n_i$ and\n\\begin{align}\n\\epsilon_0&=\\frac{1}{2}\\sum_{a=1}^{N_1}\\sum_{j=1}^{N_2}|m_a-n_j|\n-\\frac{1}{2}\\sum_{a,b=1}^{N_1}|m_a-m_b|\n-\\frac{1}{2}\\sum_{i,j=1}^{N_2}|n_i-n_j|. \n\\end{align}\nThis expression may be obtained from the superconformal index of ABJM theory determined in \\cite{Kim:2009wb} by excluding the contribution of one bifundamental hypermultiplet.   \n\nThe above expression of the index can be rewritten as a more compact form   \nby using its invariance under the action of the Weyl group for ${\\rm U}(N_1) \\times {\\rm U}(N_2)$:\n\\begin{align}\n{\\cal I}_{k,N_1,N_2}(x,\\sigma,y,z)\n&=\\sum_{\\overrightarrow{m}\\in \\mathbb{Z}^{N_1}\/{\\cal S}_{N_1} \\atop \\overrightarrow{n}\\in\\mathbb{Z}^{N_2}\/{\\cal S}_{N_2}}\\frac{y^{m_\\text{tot}}z^{n_\\text{tot}}}{|W_{(\\overrightarrow{m},\\overrightarrow{n})}|}{\\cal I}^{(\\overrightarrow{m},\\overrightarrow{n})}(x,\\sigma)\n\\end{align}\nwhere $\\cS_N$ is the permutation group for $N$ elements and $|W_{(\\overrightarrow{m},\\overrightarrow{n})}|$ is the number of the permutations which fix the monopole configuration specified by $(\\overrightarrow{m},\\overrightarrow{n})$, and \n\\begin{align}\n{\\cal I}^{(\\overrightarrow{m},\\overrightarrow{n})}(x,\\sigma)=\nx^{\\epsilon_0}\n\\int\n\\prod_{a=1}^{N_1}\\frac{du_a}{2\\pi iu_a} u_a^{km_a}\n\\prod_{i=1}^{N_2}\\frac{dv_i}{2\\pi iv_i} v_i^{-kn_i}\n{\\cal I}_\\text{vec}\n{\\cal I}_\\text{mat}.\n\\end{align}\n\n\\subsection{Analytic computation in the abelian case}\n\\label{analytic}\n\n\nBefore considering the case with general ranks, let us start with the abelian case with $N_1=N_2=1$ with a general Chern-Simons coupling constant.\nIn this case we can compute the superconformal index analytically without difficulty.\n\nIn the minimal Gaiotto-Witten theory, the diagonal U(1) gauge field does not couple to the bi-fundamental matter, so that the corresponding holonomy integration can be trivially performed. \nIn the abelian case, the only one non-trivial integration remains. \nWe perform the remaining residue integration by deforming the integration contour to either the origin or infinity so as to avoid the poles arising due to the Chern-Simons interaction, as shown below. \n\nLet us first perform the integration of the diagonal U(1). This can be done by  changing integration variables such that \n\\be \nw = \\sigma^{-1} v_1\/u_1.  \n\\ee\nWe fix the region of variables by $\\sigma x^\\half <1, x<1,\\sigma x^{-\\half} >1 $.\nThen the integral form of the index becomes \n\\beal{\n{\\cal I}_{k,1,1}^{(m_{\\rm tot},n_{\\rm tot})}(x,\\sigma)\n=& \\oint {d{u} \\over (2\\pi i{u}) }u^{-k(m_{\\rm tot} -n_{\\rm tot})}  \\oint {dw \\over (2\\pi iw) } \\sigma^{km_{\\rm tot}} w^{km_{\\rm tot}} {(wx^{3\/2} ; x^2)_\\infty (w^{-1}x^{3\/2}; x^2)_\\infty  \\over (w^{-1} x^{\\half} ;x^2)_\\infty (wx^{\\half} ;x^2)_\\infty  }, \n\\notag\n}\nin which the integration of $u$ variable decouples from the other part. \nPerforming this integration over $u$ gives  \n\\beal{\n{\\cal I}_{k,1,1}^{(m_{\\rm tot},n_{\\rm tot})}(x,\\sigma)\n=&\\delta_{m_{\\rm tot},n_{\\rm tot}} \\sigma^{km_{\\rm tot}} \\oint {dw \\over (2\\pi iw) } w^{km_{\\rm tot}} {(wx^{3\/2} ; x^2)_\\infty (w^{-1}x^{3\/2}; x^2)_\\infty  \\over (w^{-1} x^{\\half} ;x^2)_\\infty (wx^{\\half} ;x^2)_\\infty  }. \n}\nIt turns out that the contribution associated with the different monopole charges for two U(1) gauge groups vanishes\n\\footnote{ \nThis statement may hold in the non-abelian case as well by considering the total monopole charges for U($N_1$) and U($N_2$). \n}\nWe perform the remaining integral by deforming the integration contour to either the origin or infinity so as to avoid the poles generated by the classical contribution, $w^{km_{\\rm tot}}$ at the origin or infinity, which depends on the value of $m_{\\rm tot}$.\n\nWhen $m_{\\rm tot} \\geq  0$, the term $w^{km_{\\rm tot}}$ is a pole around $w\\sim\\infty$. \nTo avoid this pole, we deform the integration contour to the origin so that we pick up the poles inside the unit circle. \n\\be \n{\\cal I}_{k,1,1}^{(m_{\\rm tot},n_{\\rm tot})}(x,\\sigma)\n= \\delta_{m_{\\rm tot},n_{\\rm tot}} \\sigma^{km_{\\rm tot}} \\oint_{|w|<1} {dw \\over (2\\pi iw) } w^{km_{\\rm tot}} {(wx^{3\/2} ; x^2)_\\infty (w^{-1}x^{3\/2}; x^2)_\\infty  \\over (w^{-1} x^{\\half} ;x^2)_\\infty (wx^{\\half} ;x^2)_\\infty  }\n\\ee\nWe pick up poles inside the unit circle at $w= x^{2n+\\half}$ with $n\\geq0$, which come from the term $(w^{-1} x^{\\half} ;x^2)_\\infty$ in the denominator.\\footnote{ \nAs is the case without Chern-Simons interaction, we do not pick up the pole at the origin in this evaluation, which arises when $m_{\\rm tot} = 0$ (see \\cite{Krattenthaler:2011da} for example). This can be justified in the following way. Since the contribution coming from the pole at the origin is clearly ill-defined, one needs to regularize the contribution with $m_{\\rm tot}=0$ for its evaluation. We regularize it by introducing a (discrete) background magnetic flux studied in \\cite{Kapustin:2011jm} so that the pole at the origin disappears, which is turned off after the evaluation. Then the evaluation can be done as described in the main text.\n}\nPerforming the residue integral we obtain \n\\beal{\n{\\cal I}_{k,1,1}^{(m_{\\rm tot},n_{\\rm tot})}(x,\\sigma)\n=&\\delta_{m_{\\rm tot},n_{\\rm tot}}\\sigma^{km_{\\rm tot}} \\sum_{n\\geq0} (x^{2n+\\half})^{km_{\\rm tot}} {((x^{2n+\\half}) x^{3\/2} ; x^2)_\\infty ((x^{2n+\\half})^{-1}x^{3\/2}; x^2)_\\infty  \\over (x^{-2n};x^2)_n(x^2;x^2)_\\infty ((x^{2n+\\half})x^{\\half} ;x^2)_\\infty  } \\nn\n=& \\delta_{m_{\\rm tot},n_{\\rm tot}}\\sigma^{km_{\\rm tot}} f^{m_{\\rm tot}}(x)\n}\nwhere we set \n\\be \nf^{m_{\\rm tot}}(x)\n= {1\\over (x^2;x^2)_\\infty} \\sum_{n\\geq0} (x^{2n+\\half})^{km_{\\rm tot}} {(x^{2n+2}; x^2)_\\infty (x^{-2n+1} ; x^2)_\\infty  \\over (x^{-2n};x^2)_n (x^{2n+1}  ;x^2)_\\infty  }. \n\\ee\n\nWhen $m_{\\rm tot} < 0$, on the other hand, the term $w^{km_{\\rm tot}}$ is a pole around $w\\sim0$. To avoid this pole we deform the integration contour to the infinity so that we pick up the poles outside the unit circle. \n\\be \n{\\cal I}_{k,1,1}^{(m_{\\rm tot},n_{\\rm tot})}(x,\\sigma)\n= \\delta_{m_{\\rm tot},n_{\\rm tot}} \\sigma^{km_{\\rm tot}} \\oint_{|w|>1} {dw \\over (2\\pi iw) } w^{km_{\\rm tot}} {(wx^{3\/2} ; x^2)_\\infty (w^{-1}x^{3\/2}; x^2)_\\infty  \\over (w^{-1} x^{\\half} ;x^2)_\\infty (wx^{\\half} ;x^2)_\\infty  }.\n\\ee\nExchanging the integration variable such that $w \\to \\bar w = 1\/w$, we find \n\\beal{\n{\\cal I}_{k,1,1}^{(m_{\\rm tot},n_{\\rm tot})}(x,\\sigma)=&\\delta_{m_{\\rm tot},n_{\\rm tot}}\\sigma^{km_{\\rm tot}} \\oint_{|\\bar w|<1} {d\\bar w \\over (2\\pi i\\bar w) }\\bar w^{-km_{\\rm tot}} {(\\bar w^{-1}x^{3\/2} ; x^2)_\\infty (\\bar w x^{3\/2}; x^2)_\\infty  \\over (\\bar w x^{\\half} ;x^2)_\\infty (\\bar w^{-1} x^{\\half} ;x^2)_\\infty  } \n}\nwhich is the same as $\\delta_{m_{\\rm tot},n_{\\rm tot}}\\sigma^{km_{\\rm tot}} f^{-m_{\\rm tot}}(x)$.\n\nAs a result, we obtain  \n\\beal{\n{\\cal I}_{k,1,1}(x,\\sigma, y,z)\n=&\\sum_{m_{\\rm tot} \\geq 0  } (yz)^{m_{\\rm tot}} \\sigma^{km_{\\rm tot}} f^{m_{\\rm tot}}(x)  + \\sum_{m_{\\rm tot} < 0  } (yz)^{m_{\\rm tot}}  \\sigma^{km_{\\rm tot}} f^{-m_{\\rm tot}}(x) \\nn\n=&\\sum_{m_{\\rm tot} > 0  } ( (yz\\sigma^k)^{m_{\\rm tot}} + (yz\\sigma^k)^{-m_{\\rm tot}} ) f^{m_{\\rm tot}}(x) + f^{0}(x). \n} \nWe have verified this expression by Taylor-expansion in terms of small $x$\n\\footnote{ \nTo expand the superconformal index up to $x^{\\nu_\\text{th}}$ we truncate the summation over $m_{\\rm tot}$, $n$ and the infinite product in the Pochhammer symbols as follows.\n\\begin{align}\n\\sum_{m_{\\rm tot}=1}^\\infty&\\longrightarrow \\sum_{m_{\\rm tot}=1}^{m_\\text{max}},\\quad\\quad\\quad\\quad \\Bigl(m_\\text{max}=\\Bigl[\\frac{2\\nu_\\text{th}}{k}\\Bigr]\\Bigr),\\nonumber \\\\\n\\sum_{n=0}^\\infty&\\longrightarrow \\sum_{n=0}^{n_\\text{max}},\\quad\\quad\\quad\\quad \\Bigl(n_\\text{max}=\\Bigl[\\frac{1}{2km_{\\rm tot}+1}\\Bigl(\\nu_\\text{th}-\\frac{km_{\\rm tot}}{2}\\Bigr)\\Bigr]\\Bigr),\\nonumber \\\\\n\\frac{(x;x^2)_\\infty}{(x^{1+2n};x^2)_\\infty}\n&\\longrightarrow \\frac{\\prod_{j=0}^{\\text{Floor}[\\frac{x_\\text{th}-n-1}{2}]}(1- x^{1+2j})\n}{\n\\prod_{j=0}^{\\text{Floor}[\\frac{x_\\text{th}-3n-1}{2}]}(1- x^{1+2n+2j})\n}.\n\\end{align}\n}\n\n\n\n\\subsubsection{Obstacle in bad theory for higher ranks}\nIn the case without Chern-Simons terms \\cite{Hwang:2015wna} the benefit of such computation was that it works also for bad theories and enable us the direct comparison of the superconformal indices between the Hanany-Witten pairs.\nIn the same motivation below we show an attempt to generalize the above computation for the theories with higher ranks, though we end up with a difficulty for bad cases.\n\nFor simplicity we consider the case with $N_1=N_2=N$, which satisfy the s-rule bound $k\\le N$.\nFirst we consider the $u_a$-integrations, estimating the poles at the origin and at the infinity as (${\\widetilde u}_a=u_a^{-1}$)\n\\begin{align}\n{\\cal I}\\sim\n\\begin{cases}\n\\int du_a u_a^{km_a-(N-1)-1}=\\int du_a u_a^{km_a-N}&\\quad(u_a\\sim 0)\\\\\n\\int du_a u_a^{km_a+(N-1)-1}=\\int d{\\widetilde u}_a e^{-km_a-N} &\\quad(u_a\\sim\\infty)\n\\end{cases}\n\\end{align}\nwhere in the middle $u_a^{\\mp (N-1)}$ comes from ${\\cal I}_\\text{vec}$, which increases the singularity at $u_a=0,\\infty$.\n\nTo see the difficulty we focus on the case $m_a>0$ and the contribution for the following choice of the poles: $u_a=(const)\\cdot v_{i=a}$ which amounts to\n\\begin{align}\n{\\cal I}^{(\\overrightarrow{m};\\overrightarrow{n})}\\sim \\int \\frac{dv_a}{2\\pi iv_a} v_a^{-k(n_a-m_a)}{\\cal I}_\\text{vec}{\\cal I}_\\text{mat}.\n\\end{align}\nSince the integrand is still a complicated function of $\\{v_i\\}$, we may want to perform the $v_i$-integration one by one by classifying $n_i$ in the same way as we did in $u_a$ integration.\nHere we encounter a problem.\nDue to the substitution $u_a=(const)\\cdot v_a$, the $\\text{U}(N)_k$ part of ${\\cal I}_\\text{vec}$ contributes in the same way as $\\text{U}(N)_{-k}$ part.\nHence the estimation of poles at $v_i=0,\\infty$ is modified as\n\\begin{align}\n{\\cal I}\\sim \n\\begin{cases}\n\\int dv_i v_i^{-k(n_i-m_i)-2(N-1)-1}=\\int dv_i v_i^{-k(n_i-m_i)-2N+1} &\\quad(v_i\\sim 0)\\\\\n\\int dv_i v_i^{-k(n_i-m_i)+2(N-1)-1}=\\int dv_i {\\widetilde v}_i^{k(n_i-m_i)-2N+1} &\\quad (v_i\\sim \\infty)\n\\end{cases},\n\\end{align}\nwhich indicates:\nfor $n_i-m_i<-(2N-2)\/k$ the integration has no pole at $v_i=0$ hence can be computed from the residues at the poles in $|v_i|<1$;\nfor $n_i-m_i>(2N-2)\/k$ the integration has no pole at $v_i=\\infty$ hence can be computed from the residues at the poles in $|v_i|>1$;\nfor $-(2N-2)\/k\\le n_i-m_i\\le (2N-2)\/k$ both poles are present with of order $N$ hence such computation does not work.\nIf $2N-2<k$, or the theory is good, the last case is satisfied only for $n_i=m_i$.\nFor $2N-2\\ge k$, or the bad theory, however, there are several choices of the monopole charges where non-perturbative computation does not work.\n\n\n\n\\subsection{Perturbative computation}\n\\label{perturbative}\n\nNow let us move on the case with general $N_1$, $N_2$.\nAs we argued above, analytic computation is not available for the ``bad'' theories with $k-N_1-N_2\\le -2$.\nThe computation is already difficult, however, even for the good theories with higher ranks due to the increasing number of holonomy integrals.\nIn this section we instead consider the small $x$ expansion of the superconformal index from the beginning up to some finite order, which we shall call ``perturbative computation''.\n\nFor each $(\\overrightarrow{m},\\overrightarrow{n})$ we can compute ${\\cal I}^{(\\overrightarrow{m},\\overrightarrow{n})}$ up to an arbitrary order of $x$, say $x^{\\nu_\\text{th}}$, in the following way:\n(i) First expand the integrand ${\\cal I}_\\text{vec}{\\cal I}_\\text{hyp}$ around $x=0$; (ii) then expand each coefficient of small-$x$ expansion in the Laurent series of $(u_a,v_i)$; and (iii) finally pick up the monomial $\\prod_au_a^{-km_a}\\prod_iv_i^{kn_i}$ so that it compensate the Chern-Simons term $\\prod_a u_a^{km_a} \\prod_i v_i^{-kn_i}$.\nThis is indeed a procedure which is physically straightforward.\nFor each $(\\overrightarrow{m};\\overrightarrow{n})$ the bare monopole vaccum is not gauge invariant, which is indicated by the Chern-Simons term.\nTo obtain the gauge invariant states it requires additional excitations of the fields which are non-trivially charged under the gauge symmetry ((i) and (ii)) in an appropriate amount so that the net bound state is gauge invariant ((iii)).\n\nTo compute the full superconformal index we need to repeat these computations for all monopole charges.\nIn some cases the leading order of the non-zero contribution grows fast enough with respect to $|m_a|,|n_i|$, which allows us to compute the superconformal index up to arbitrary order $x^{\\nu_\\text{th}}$ by truncating the summation over the monopole charges at some finite $|m_a|,|n_i|$.\n\n\n\\subsubsection{Preliminary analysis of the classification \\eqref{goodbaduglyfromZS3} }\n\\label{sec_singularity}\n\nTo see whether the last requirement is correct or not, let us estimate the leading order of $x$ for the following monoople charge $(m\\ge 0)$\n\\begin{align}\n(m_1,m_2,\\cdots,m_{N_1};n_1,n_2,\\cdot,n_{N_2})=(m,0,0,\\cdots,0;m,0,0,\\cdots,0).\n\\end{align}\nFor this choice the contribution from bare charge is\n\\begin{align}\nx^{\\epsilon_0}=x^{-\\frac{m}{2}(N_1+N_2-2)}.\n\\end{align}\nTo cancel the gauge charge of the monopole $\\prod_au_a^{km_a}\\prod_iv_i^{-kn_i}=(u_1v_1^{-1})^{km}$, we also need to bring an appropriate number of monomials from each factors of ${\\cal I}_\\text{vec}{\\cal I}_\\text{hyp}$ \\eqref{Ivec}, \\eqref{Ihyp}.\nSince we are interested in the leading order of $x$, here we adopt the choice with the smallest power of $x$.\nIn current choice of $(\\overrightarrow{m};\\overrightarrow{n})$, this is realized by picking $x^{\\frac{1}{2}}\\sigma^{-1}u_1^{-1}v_1$ from the second denominator in ${\\cal I}_\\text{hyp}$ $km$ times.\nIn total the leading order of $x$ in ${\\cal I}^{(\\overrightarrow{m};\\overrightarrow{n})}$ is given as\n\\begin{align}\n{\\cal I}^{(m,0,0,\\cdots,0;m,0,0,\\cdots,0)}_{k,N_1,N_2}(x,\\sigma)\\sim x^{\\frac{m}{2}(k-N_1-N_2+2)}.\n\\end{align}\n\nFrom this result we conclude\n\\begin{enumerate}\n\\item If $k-N_1-N_2<-2$, the leading negative exponent of $x$ grows linearly in $m$ and not bounded from below.\nHence we cannot compute the small $x$ expansion of ${\\cal I}$ in this approach.\n\\item If $k-N_1-N_2=-2$, the exponent of $x$ is bounded from below by $x^0$.\nHowever, since infinitely many choices of monopole charges contribute to each order of $x$, it is again impossible to compute the small $x$ expansion of ${\\cal I}$.\n\\item If $k-N_1-N_2>-2$, the expansion starts from a positive power of $x$.   \n\\end{enumerate}\nNotice that the singularities for $k-N_1-N_2\\le -2$ cannot be cured by introducing chemical potential for $m_\\text{tot}=n_\\text{tot}$.\nTo see this, let us work out similar power estimation for the following monopole charge\n\\begin{align}\n(\\overrightarrow{m};\\overrightarrow{n})=(m+m_\\text{tot},-m,0,\\cdots,0;m+m_\\text{tot},-m,0,\\cdots,0),\n\\end{align}\nwith $m$ being an arbitrary integer, which contributes to the sector of $m_\\text{tot}$.\nFor this monopole charge the contribution from bare monopole is $x^{\\epsilon_0}=x^{-|2m+m_\\text{tot}|-\\frac{N_1+N_2-4}{2}(|m+m_\\text{tot}|+|m|)}$.\nThe bare monopole is charged as $u_1^{k(m+m_\\text{tot})} u_2^{-km} v_1^{-k(m+m_\\text{tot})} v_2^{km}$ due to the Chern-Simons term.\nAssuming $m>0$ and $|m|$ being large enought compared with $m_\\text{tot}$ we find that the most economical way to cancel this charge and form a gauge singlet is to bring $(x^{\\frac{1}{2}}u_1^{-1}v_1)^{k|m+m_\\text{tot}|}\\cdot (x^{\\frac{1}{2}}u_2v_2^{-1})^{k|m|}$ from ${\\cal I}_\\text{hyp}$.\nIn total, the leading power of the superconformal index is\n\\begin{align}\n{\\cal I}_{k,N_1,N_2}^{(m+m_\\text{tot},-m,0,0,\\cdots,0;m+m_\\text{tot},-m,0,0,\\cdots,0)}(x,\\sigma)\\sim \nx^{\\epsilon_0+\\frac{k|m+m_\\text{tot}|}{2}+\\frac{k|m|}{2}}=x^{(k-N_1-N_2+2)(m+\\frac{m_\\text{tot}}{2})}.\n\\label{badforeachmtot}\n\\end{align}\nHere we have used $m>0$ and $m\\gg m_\\text{tot}$ to reduce $|m+m_\\text{tot}|=m+m_\\text{tot}$, $|m|=m$, $|2m+m_\\text{tot}|=2m+m_\\text{tot}$.\nThe result \\eqref{badforeachmtot} implies that for $k-N_1-N_2\\le -2$ the non-analyticity of the superconformal index at $x=0$ appears already at each sector of $m_\\text{tot}$.\n\nThis preliminary analysis implies that the classification \\eqref{goodbaduglyfromZS3} done by using the convergence property of the three-sphere partition function works also for a superconformal index. \nIn what follows we display the result of the perturbative computation of the superconformal index for several good or ugly theories.\n\n\\subsubsection{Results}\n\\label{perturbative_results}\n\nWe display the result of the perturbative computation of the superconformal index for various $k$, $N_1$, $N_2$.\nWe are especially interested in the pairs of $\\text{U}(N_1)_k\\times \\text{U}(N_2)_{-k}$ theory and $\\text{U}(k-N_2)_k\\times \\text{U}(k-N_1)_{-k}$ theory, which are suggested to be dual with each other from the Hanany-Witten transition.\nSince $k-N_1-N_2=k-N_1-N_2$ transforms $k-N_1-N_2\\rightarrow -k-N_1-N_2$ under the Hanany-Witten transition, such comparison is possible only for the pairs of $k-N_1-N_2=k-N_1-N_2=\\pm 1$ because of the singularity for $k-N_1-N_2<-2$ argued in section \\ref{levelrankduality}.\\footnote{\nFor $k-N_1-N_2=0$ the pairs are different only in the sign of $k$.\nIn this case the superconformal indices trivially coincide in the dual pair.\n}\nFor $|k-N_1-N_2|=1$ case, the explicit computation shows that the superconformal indices do not coincide in the proposed dual pair.\nNevertheless, their ratio simplifies and allows a physical interpretation as the contribution from an extra hypermultiplet.\n\nWe have computed the superconformal index for the proposed dual pairs with $N_1,N_2\\le 2$, namely, $(k,N_1,N_2)=(1,0,0),(1,1,1),(2,0,1),(2,1,2),(3,0,2),(3,1,1),(3,1,3),(3,2,2)$ up to $x^5$.\nHere for the cases with $N_1=0$ or $N_2=0$ the theory is the pure Chern-Simons theory and the superconformal index is trivially ${\\cal I}(x,\\sigma)=1$.\nWe have taken into account all the monopole charges in $|m_a|,|n_i|\\le 20$, and found only small number of those in $|m_a|,|n_i|\\le 10$ displayed in table \\ref{contributingmn} contributes to the superconformal index, which supports this truncation is indeed exact up to $x^5$.\n\\begin{table}\n\\begin{tabular}{|c|c|c|l|}\n\\hline\n$k$&$N_1$&$N_2$&$(\\overrightarrow{m};\\overrightarrow{n})$ (up to permutations and $(\\overrightarrow{m},\\overrightarrow{n})\\rightarrow (-\\overrightarrow{m},-\\overrightarrow{n})$)\\\\ \\hline\\hline\n$1$&$1$&$1$&$(0,0),(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(7,7),(8,8),(9,9),(10,10)$\\\\ \\hline\n$2$&$1$&$2$&$\n(0;-1,1),\n(0;0,0),\n(1;0,1),\n(2;0,2),\n(3;0,3),\n(4;0,4),\n(5;0,5),\n(6;0,6),\n(7;0,7),\n$\\\\\n&&&$\n(8;0,8),\n(9;0,9),\n(10;0,10)\n$\\\\ \\hline\n$3$&$1$&$1$&$(0,0),(1,1),(2,2),(3,3)$\\\\ \\hline\n$3$&$2$&$2$&$(-5,5;-5,5),\n(-4,4;-4,4),\n(-3,3;-3,3),\n(-2,2;-2,2),\n(-1,1;-1,1),\n$\\\\\n&&&$\n(0,0 ;0,0),\n(-4,5;-4,5),\n(-3,4;-3,4),\n(-2,3;-2,3),\n(-1,2;-1,2),\n(0,1 ;0,1),\n$\\\\\n&&&$\n(-4,6;-4,6),\n(-3,5;-3,5),\n(-2,4;-2,4),\n(-1,3;-1,3),\n(0,2 ;0,2),\n(1,1 ;1,1),\n$\\\\\n&&&$\n(-3,6;-3,6),\n(-2,5;-2,5),\n(-1,4;-1,4),\n(0,3 ;0,3),\n(1,2 ;1,2),\n(-3,7;-3,7),\n$\\\\\n&&&$\n(-2,6;-2,6),\n(-1,5;-1,5),\n(0,4 ;0,4),\n(1,3 ;1,3),\n(-2,7;-2,7),\n(-1,6;-1,6),\n$\\\\\n&&&$\n(0,5 ;0,5),\n(1,4 ;1,4),\n(-2,8;-2,8),\n(-1,7;-1,7),\n(0,6 ;0,6),\n(1,5 ;1,5),\n$\\\\\n&&&$\n(-1,8;-1,8),\n(0,7 ;0,7),\n(-1,9;-1,9),\n(0,8 ;0,8),\n(0,9 ;0,9),\n(0,10;0,10)\n$\\\\ \\hline\n\\end{tabular}\n\\caption{\n{\nWe list the monopole charges with non-vanishing contribution to the superconformal index up to ${\\cal O}(x^5)$.\nHere we have denoted only one element among each family generated by permutations and $(m_a;n_i)\\rightarrow (-m_a;-n_i)$ whose contributions are identical (up to $(y,z,\\sigma)\\rightarrow (y^{-1},z^{-1},\\sigma^{-1})$).\n}\n}\n\\label{contributingmn}\n\\end{table}\n\nFor our purpose of comparison between HW pairs it is convenient to express the superconformal index ${\\cal I}$ in the letter index ${\\widetilde {\\cal I}}$ defined by the plethystic exponential\n\\begin{align}\n{\\cal I}(x,\\sigma,y,z)=\\PE[x,\\sigma,y,z;{\\widetilde {\\cal I}}(x,\\sigma,y,z)]=\\exp[\\sum_{n=1}^\\infty \\frac{1}{n}{\\widetilde {\\cal I}}(x^n,\\sigma^n,y^n,z^n)],\n\\end{align}\nwith which the ratio is mapped to the difference\n\\begin{align}\n\\frac{{\\cal I}}{{\\cal I}'}=\\PE[{\\widetilde {\\cal I}}-{\\widetilde{\\cal I}'}].\n\\end{align}\nWe have obtained the following letter indices\n\\begin{align}\n{\\widetilde {\\cal I}}_{1,0,0}&=0,\\nonumber \\\\\n{\\widetilde {\\cal I}}_{1,1,1}&=(yz\\sigma +y^{-1}z^{-1}\\sigma^{-1})(x^{\\frac{1}{2}}-x^{\\frac{3}{2}}+x^{\\frac{5}{2}}-x^{\\frac{7}{2}}+x^{\\frac{9}{2}})+{\\cal O}(x^{\\frac{11}{2}}),\\nonumber \\\\\n{\\widetilde {\\cal I}}_{2,0,1}&=0,\\nonumber \\\\\n{\\widetilde {\\cal I}}_{2,1,2}&=(yz\\sigma^2+y^{-1}z^{-1}\\sigma^{-2})(x^{\\frac{1}{2}}-x^{\\frac{3}{2}}+x^{\\frac{5}{2}}-x^{\\frac{7}{2}}+x^{\\frac{9}{2}})+{\\cal O}(x^{\\frac{11}{2}}),\\nonumber \\\\\n{\\widetilde {\\cal I}}_{3,0,2}&=0,\\nonumber \\\\\n{\\widetilde {\\cal I}}_{3,1,1}&=x+(yz\\sigma^3+y^{-1}z^{-1}\\sigma^{-3})x^{\\frac{3}{2}}-2x^2-(yz\\sigma^3+y^{-1}z^{-1}\\sigma^{-3})x^{\\frac{5}{2}}+2x^3\\nonumber \\\\\n&\\quad +2(yz\\sigma^3+y^{-1}z^{-1}\\sigma^{-3})x^{\\frac{7}{2}}-3x^4-5(yz\\sigma^3+y^{-1}z^{-1}\\sigma^{-3})x^{\\frac{9}{2}}+4x^5+{\\cal O}(x^{\\frac{11}{2}}),\\nonumber \\\\\n{\\widetilde {\\cal I}}_{3,1,3}&=(yz\\sigma^3+y^{-1}z^{-1}\\sigma^{-3})(x^{\\frac{1}{2}}-x^{\\frac{3}{2}}+x^{\\frac{5}{2}}-x^{\\frac{7}{2}}+x^{\\frac{9}{2}})+{\\cal O}(x^{\\frac{11}{2}}),\\nonumber \\\\\n{\\widetilde {\\cal I}}_{3,2,2}&=(yz\\sigma^3+y^{-1}z^{-1}\\sigma^{-3})x^{\\frac{1}{2}}\n+x-2x^2\n+2x^3\n+(yz\\sigma^3+y^{-1}z^{-1}\\sigma^{-3})x^{\\frac{7}{2}}\n-3x^4\\nonumber \\\\\n&\\quad -4(yz\\sigma^3+y^{-1}z^{-1}\\sigma^{-3})x^{\\frac{9}{2}}\n+4x^5\n+{\\cal O}(x^{\\frac{11}{2}}).\n\\label{pertresults}\n\\end{align}\n\nInterestingly, we find that the ratio of the superconformal index between Hanany-Witten pair is the same for all choices of $(k,N_1,N_2)$.\nThat is,\n\\begin{align}\n\\frac{{\\cal I}_{k,N_1,N_2}}{{\\cal I}_{k,k-N_2,k-N_1}}=\\PE[x,\\sigma,y,z;{\\widetilde {\\cal I}}_{(k,N_1,N_2)\/(k,k-N_2,k-N_1)}] \n\\end{align}\nfor $N_1>k-N_1$, with\n\\begin{align}\n{\\widetilde {\\cal I}}_{(1,1,1)\/(1,0,0)}=\n{\\widetilde {\\cal I}}_{(2,1,2)\/(2,0,1)}=\n{\\widetilde {\\cal I}}_{(3,1,3)\/(3,0,2)}=\n{\\widetilde {\\cal I}}_{(3,2,2)\/(3,1,1)}=(yz\\sigma^k+y^{-1}z^{-1}\\sigma^{-k})\\frac{x^{\\frac{1}{2}}-x^{\\frac{3}{2}}}{1-x^2},\n\\label{ratioisfreehyper}\n\\end{align}\nup to ${\\cal O}(x^{\\frac{11}{2}})$. \nAs a result the indices of Hanany-Witten dual pairs turned out to coincide up to the contribution of one hypermultiplet up to the order. \nThis result indicates that the Hanany-Witten duality holds up to a hypermultiplet. \nThis implies that an ugly monopole operator decouples in the IR to form a hypermultiplet\n\\footnote{ \nStrictly speaking this ugly monopole operator does not totally decouple but couples to the other sector through the topological current. \n}\n\n\\section{Discussion}\n\nIn this paper we have considered the 3d ${\\cal N}=4$ $\\text{U}(N_1)_k\\times \\text{U}(N_2)_{-k}$ superconformal Chern-Simons theory coupled with a bifundamental hypermultiplet.\nIt was observed that the three-sphere partition function of this theory diverges if $k-N_1-N_2\\le -2$ \\cite{Nosaka:2017ohr}.\nThis suggests, following the argument \\cite{Kapustin:2010mh,Yaakov:2013fza} in the case without Chern-Simons interactions, that the theory is {\\it bad} when $k-N_1-N_2\\le -2$ according to the good\/ugly\/bad classification in \\cite{Gaiotto:2008ak}.\nTo check this classification we have studied the superconformal index of the theory and we have indeed found that there exists monopole operators with unitarity violating R-chage if $k-N_1-N_2\\le -2$.\n\nWe have further computed the superconformal indices in small $x$ expansion for the pairs of theories with $k-N_1-N_2=\\pm 1$ related by the Hanany-Witten transition.\nAs a result we have found that the superconformal indices of the two theories in pair coincide with each other up to an overall factor which is the same as the contribution of the hypermultiplet up to a certain order.\nNotably, this is consistent with the dimension of moduli space $4\\text{min}(N_1,N_2)$ of the theory: the difference between the dimensions the moduli space of the theories in a Hanany-Witten pair is $4|k-N_1-N_2|$, which coincides with the number of degrees of freedom of a hypermultiplet for $k-N_1-N_2=\\pm 1$.\nThese results are again natural generalizations of what occurs in the case without Chern-Simons term.\n\nIt would be interesting to ask what happens in a pair with $|k-N_1-N_2|\\ge 2$, that is, a pair of a good theory and a bad one.\nFrom the dimension of the moduli space it is natural to expect that the number of decoupled hypermultiplets is $|k-N_1-N_2|$.\nUnfortunately so far we do not have a method to compute the superconformal index of the bad theory.\nThe perturbative approach with the truncation of the summation over the monopole charges does not work for a bad theory.\nIn the case without Chern-Simons interactions the superconformal index is obtained in factorized form which is valid also for bad theories \\cite{Hwang:2015wna}.\nThe computation, however, requires the theory to satisfy $|k|<|N_\\text{f}-N_\\text{a}|\/2$ where $N_\\text{f}$ and $N_\\text{a}$ are the number of fundamental and anti-fundamental chiral multiplet in the UV field content (called as ``maximally chiral'') \\cite{Benini:2013yva}.\nThis condition is not satisfied in our setup and the computation does not work due to the non-trivial poles at the origin and infinity.\nIt is desirable to establish an alternative technique to compute the superconformal index for our theory with $k-N_1-N_2\\le -2$.\n\nOne possible approach is to introduce the fugacities for all the components of monopole charge $(m_1,m_2,\\cdots,m_{N_1};n_1,n_2,\\cdots,n_{N_2})$ not only for $m_\\text{tot}=n_\\text{tot}$.\nThis might remedy the singularity in the perturbative computation argued in section \\ref{sec_singularity} and make the monopole summation convergent.\nOnce we obtain a resummed expression the original superconformal index will be obtained by sending all the fugacities to unity except for the one for $m_\\text{tot}=n_\\text{tot}$.\n\nIt would also be interesting to study the moduli space in more detail along the line of \\cite{Nakajima:2015txa,Bullimore:2015lsa,Assel:2017jgo}.\nSeveral generalizations of our setup could be studied in a similar manner.\nWe can also add an arbitrary number of fundamental hypermultiplets coupling with each gauge node.\nAs the extra hypermultiplets lift the R-charge of the monopole operators up, such generalizations are not only interesting by themselves but also can be easier than the original theory and would be helpful to understand the original theory.\n\n\\section*{Acknowledgement}\n\nWe would like to thank Chiung Hwang for helpful discussions and valuable comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe ionizing UV emission produced by the first stars and galaxies in the high-redshift Universe transforms the surrounding IGM from a neutral gas to a highly ionized plasma. At the same time, it is photoheated from a few Kelvin to $\\sim 10^4$ K. As this process of cosmic reionization proceeds, ionized regions grow, start to overlap, and eventually become volume filling \\citep[e.g., see reviews by ][]{rauch1998,meiksin2009,mcquinn2016}. This inherently inhomogeneous process results in an almost fully ionized IGM. Neutral gas is only present in dense regions that can self-shield from the cosmic UV background.\n\nThe photoheating provided by cosmic reionization increases the gas pressure in the IGM such that it becomes dynamically relevant on small scales. While gravity still dominates the formation of structures in the baryonic density field on large scales, small scales in the IGM are notably affected by the hydrodynamic reaction to the photoheating. Overpressurized regions expand, thereby erasing structure on small scales \\citep[e.g.,][]{1998MNRAS.296...44G,2000MNRAS.315..600T,KulkarniEtAl2015,rorai2017,WuEtAl2019,KatzEtAl2020,2021ApJ...923..161N}. Understanding this process is relevant both for explaining the formation (or lack of formation) of galaxies in low mass halos, as well as the properties of the IGM during and after cosmic reionization. Here, we concentrate on the latter, i.e. on studying the immediate impact of patchy cosmic reionization on the IGM, as well as the \\textit{relic} signatures of patchy reionization that persist for a significant amount of time in the post-reionization IGM \\citep[e.g.,][]{LidzMalloy2014,DAloisioEtAl2015,KeatingEtAl2018,OnorbeEtAl2019,WuEtAl2019,MonteroCamacho2020,MolaroEtAl2022}.\n\nThe IGM is most readily observed in absorption, in particular using the Lyman-$\\alpha$\\ line of neutral hydrogen, which imprints a forest of absorption lines on the spectra of background quasars. The structure of this Lyman-$\\alpha$\\ forest on small scales is affected by the instantaneous temperature of the IGM via the Doppler broadening of the lines, by its thermal history via the pressure smoothing, and by the small scale structure in the dark matter density field via gravitational interaction. The latter has been exploited by using the Lyman-$\\alpha$\\ forest on small scales to probe the free streaming scale of dark matter particles \\citep[e.g.,][]{2017PhRvD..96b3522I,2021PhRvL.126g1302R}, which for thermal relic dark matter is directly related to the dark matter particle mass. Such Lyman-$\\alpha$\\ forest constraints on dark matter are best derived at high-redshift when the relevant scales are not yet completely dominated by mode coupling due to non-linear structure growth and when the Lyman-$\\alpha$\\ forest is sensitive to low (less non-linear) densities. Furthermore, for fixed comoving free-streaming length the cut-off in velocity space is at larger scales\/smaller $k$ at higher redshift, and thus - at least in principle - easier to detect. At very high redshift, when entering the epoch of reionization, the Lyman-$\\alpha$\\ forest becomes completely opaque. Hence, the sweet spot for dark matter constraints is in the redshift range $4 \\lesssim z \\lesssim 5.5$, or in other words shortly after reionization. Understanding the thermal state and pressure smoothing of the IGM in this epoch is, hence, also important for probing dark matter.\n\nInterpreting observations of the IGM typically relies heavily on comparison to cosmological hydrodynamical simulations. The main differences between the various simulations used for this purpose are how the ionizing sources, i.e. the galaxies, and the ionizing radiation fields are treated. In the most simple approach, these are not treated explicitly \\citep[see, e.g.,][for recent works]{Sherwood,Rossi_2020,Chabanier_2020,2021JCAP...04..059W,Villasenor_2021}. Instead an external model for the ionizing UV background (UVB) is used, typically a spatially homogeneous, time varying UVB model. Such models are obtained by integrating the ionizing emission of stars and active galactic nuclei based on empirical constraints of their abundance and of the opacity of the IGM \\citep[e.g.,][]{2009ApJ...703.1416F,HaardtMadau2012,onorbe2017,PuchweinEtAl2019,khaire2019,2020MNRAS.493.1614F}. When focusing on the low-density IGM probed by the Lyman-$\\alpha$\\ forest, it is possible to neglect other forms of feedback from galaxies, as it does typically not reach the relevant low-density regions of the IGM at $z\\gtrsim 4$ \\citep[although feedback will start to play a role by $z\\sim 2$, see e.g.][]{Theuns_2002,Viel_2013,Chabanier_2020}. Hence, when using an external UVB model, one can reasonably ignore galaxy formation altogether in cosmological hydrodynamical simulations of the low-density IGM \\citep{QuickLya}. Despite their simplicity and low computational cost, such simulations are also in remarkably good agreement with the observed properties of the Lyman-$\\alpha$\\ forest at $z\\lesssim4$, i.e. well after the end of reionization  \\citep[e.g.][]{Sherwood}.\n\nDuring patchy cosmic reionization, the ionizing radiation field is, however, highly inhomogeneous and simulations with a spatially homogeneous UVB model fail to reproduce the observed properties of the Lyman-$\\alpha$\\ forest, such as the fluctuations of its opacity on large scales \\citep[e.g.][]{BeckerEtAl2015,bosman2018,eilers2018,Zhu_2021,bosman2022}. To overcome these problems, the spatial distribution of ionizing sources and the resulting spatial fluctuations in the UV radiation field need to be modelled. This can either be done on the fly in full galaxy formation simulations with radiative transfer coupled to the hydrodynamics, or by doing the radiative transfer in post-processing, typically using a simpler model of the ionizing source populations.\n\nThe latter approach is computationally much cheaper and can avoid all the complications of realistically modelling the galaxy population and the escape of ionizing radiation. Empirically constrained source models can be used to ``paint'' ionizing sources on the simulated dark matter halos. A weakness of this approach is that it may miss many details of the source population, such as the bursty nature of ionizing radiation production and escape. Nevertheless, the source luminosities can be calibrated such that the amount of ionizing radiation reaching the IGM is adequate for bringing the reionization history and the simulated properties of the Lyman-$\\alpha$\\ forest in agreement with observational constraints. Calibrating in this manner makes a detailed modelling of the ionizing radiation escape from high-density regions in the simulation unnecessary. Hence, the post-processing radiative transfer can be done at coarser resolution using an uniformly spaced grid. This also allows efficient parallelization on GPUs \\citep[e.g.,][]{Aton2010}, making this approach numerically cheap. Such calculations are very successful in matching the properties of the Lyman-$\\alpha$\\ forest on large scales during and directly after cosmic reionization \\citep[e.g.,][]{KulkarniEtAl2019}.       \n\nOn the downside, post-processing radiative transfer neglects the hydrodynamic reaction of the IGM to the inhomogeneous photoheating. The thermal and ionization states are only re-calculated in post processing so that the heating is not coupled to the hydrodynamics. Hence, a self-consistent modelling of pressure smoothing is not possible with this approach. Furthermore, the thermal and ionization states are typically stored on a static grid so that the thermal energy injected by photoheating as well as the ionization state are not advected with the gas flow. Finally, post-processing radiative transfer codes that follow only heating by the UV radiation field miss other heating mechanisms that are present in a hydrodynamic simulation, such as shock heating in dense regions. All of these issues can be fixed by doing the radiative transfer in a fully coupled manner on the fly. Such radiation-hydrodynamics simulations can be done with external ionizing sources to study the reaction of the IGM \\citep[e.g.,][]{2016ApJ...831...86P,DAloisioEtAl2020} or following the formation of high-redshift galaxy populations and the escape of ionizing radiation from them self-consistently. The latter is, however, computationally very expensive. Usually some corners need to be cut to make this feasible at all, e.g., doing the radiative transfer with a reduced speed of light. In addition, the modelling of galaxy formation in such simulations is highly uncertain, in particular during the epoch of reionization where only very limited observational constraints on the galaxy population are available, making sanity checks on the simulated population difficult. This is further exacerbated by the fact that the escape of ionizing radiation depends on the detailed properties of the interstellar medium, which is very challenging to model faithfully in cosmological simulations. Despite these difficulties, there has been major progress in this direction in the last years. For example, the CROC \\citep{2014ApJ...793...29G}, Sphinx \\citep{Sphinx}, Technicolor Dawn \\citep{Finlator2018}, CoDa \\citep{ocvirk2016,ocvirk2020,lewis2022} and Thesan \\citep{Thesan,garaldi2022,2022MNRAS.512.3243S} collaborations were able to perform fully-coupled radiation-hydrodynamics simulations of cosmological volumes along with a modelling of galaxy formation.\n\nIn this work, we focus on the low-density IGM probed by the Lyman-$\\alpha$\\ forest. We build on our previous Sherwood simulation project \\citep{Sherwood}, which showed excellent agreement with various statistics of the Lyman-$\\alpha$\\ forest at $2 \\lesssim z \\lesssim 5$. Our aim is to produce a simulation suite that samples a wide range of astrophysical and cosmological parameters and that can be used for parameter inference when compared to the observed Lyman-$\\alpha$\\ forest. At the same time, we aim to overcome limitations\nat the high redshift end where the \\textit{relic} signatures that a recently completed patchy cosmic reionization has imprinted on the IGM become increasingly important. To this end, we introduce a new hybrid radiative transfer\/hydrodynamical simulation technique that aims at combining many of the positive aspects of post-processing radiative transfer and fully-coupled radiation-hydrodynamics simulations while avoiding some of their major downsides. \n\nIn Sec.~\\ref{sec:methods} of this manuscript, we will describe the simulation methods in detail. Sec.~\\ref{sec:grid_pow_spec} will give an overview of how varying different parameters affects the Lyman-$\\alpha$\\ forest on different scales. Sec.~\\ref{sec:patchy_results} focuses on our hybrid patchy reionization simulations, with the thermal state of the IGM being discussed in \\ref{sec:thermal_properties}, the modulation of the Lyman-$\\alpha$\\ forest on large scales in \\ref{sec:modulation_lya},  the spatially varying pressure smoothing of the IGM in \\ref{sec:pressure_smooth}, and its direct imprints on the Lyman-$\\alpha$\\ forest in \\ref{sec:lya_lines_rings}. We summarize our results in Sec.~\\ref{sec:summary}.\n\nIn the Sherwood-Relics project, we assume for our baseline model the same $\\Lambda$CDM cosmology as in the Sherwood project. We, thus, use $\\Omega_\\textrm{m} \\! = \\! 0.308$, $\\Omega_\\Lambda \\! = \\! 0.692$, $\\Omega_\\textrm{b} \\! = \\! 0.0482$, $h \\! = \\! 0.678$, $\\sigma_8 \\! = \\! 0.829$, and $n \\! = \\! 0.961$ unless specifically mentioned otherwise.\n\n\\section{Methods}\n\\label{sec:methods}\n\n\\subsection{The Sherwood-Relics simulation suite}\n\\label{sec:relics_suite}\n\n\\begin{table*}\n  \\centering\n   \\caption{Summary of the Sherwood-Relics simulation suite. The columns contain\n     (i) the box size, (ii) the cube root of the (initial) gas particle number, \n     (iii) the dark matter and (iv) gas mass resolution, (v) the gravitational softening, as well\n     as several properties that have been varied between the different runs.\n     This includes (vi) the dark matter model (cold dark matter or warm dark matter particle mass),\n     (vii) the H~\\textsc{i}\\ photoheating normalization factor\n     (see Sec.~\\ref{sec:sims_homo_uvb}), and (viii) the redshift $z_\\textrm{r}$ at which H~\\textsc{i}\\ reionization\n     completes (defined as the redshift when the volume-averaged neutral fraction falls below $10^{-3}$). The last column provides (ix) the number of runs as well as further comments.}\n  \\resizebox{\\textwidth}{!}{%\n  \\begin{tabular}{|c|c|c|c|c|c|c|c|c|c}\n    \\hline\n    \\hline\n    Box size & $N^{1\/3}$ & $M_{\\rm dm}$ & $M_{\\rm gas}$ & $l_{\\rm soft}$ & Dark matter & HI photoheating & $z_\\textrm{r}$ & Comments\\\\\n    $[h^{-1} {\\rm cMpc}]$ & & $\\rm [h^{-1}\\,M_{\\odot}]$ & $\\rm [h^{-1}\\,M_{\\odot}]$ & $\\rm\\,[h^{-1}\\,ckpc]$ & & factor & & &\\\\\n    \\hline\n    \\multicolumn{9}{c}{\\textbf{Simulations with a homogeneous UV background}}\\\\\n    \\hline\n     40 & 2048 & $5.37\\times 10^{5}$ & $9.97\\times 10^{4}$ & 0.78 & CDM, 3 keV, 4 keV & 0.5, 1, 2 & 6.0 & 9 runs\\\\\n     40 & 2048 & $5.37\\times 10^{5}$ & $9.97\\times 10^{4}$ & 0.78 & CDM & 1 & 5.4, 6.7, 7.4 & 3 runs\\\\\n     40 & 2048 & $5.37\\times 10^{5}$ & $9.97\\times 10^{4}$ & 0.78 & CDM & tailored heating & 5.3, 5.7, 6.0, 6.6 & 4 runs, comparison for patchy simulations\\\\\n       40 & 2048 & $5.37\\times 10^{5}$ & $9.97\\times 10^{4}$ & 0.78 & CDM & no photoheating & no ionization & 1 run, adiabatic simulation\\\\\n     \n    \n     80 & 2048 & $4.30\\times 10^{6}$ & $7.97\\times 10^{5}$ & 1.56 & CDM & 1 & 6.0 & 1 run\\\\\n     160 & 2048 & $3.44\\times 10^{7}$ & $6.38\\times 10^{6}$ & 3.13 & CDM & tailored heating & 5.3 & 1 run \\\\\n     \\hline\n       5 & 1280 & $4.30\\times 10^{3}$ & $7.97\\times 10^{2}$  & 0.17  & CDM & 1 & 6.0 & 1 run\\\\\n     10 & 1280 & $3.44\\times 10^{4}$ & $6.38\\times 10^{3} $ & 0.31 & CDM & 1 & 6.0 & 1 run\\\\\n      20 & 1280 & $2.75\\times 10^{5}$ & $5.10 \\times 10^{4}$  & 0.63 & CDM & 1 & 6.0 & 1 run\\\\\n    \\hline\n      5 & 1024 & $8.39\\times 10^{3}$ & $1.56\\times 10^{3}$  & 0.20  & CDM & 1 & 6.0 & 1 run\\\\\n     10 & 1024 & $6.72\\times 10^{4}$ & $1.25\\times 10^{4}$ & 0.39 & CDM, 2 keV, 3 keV, 4 keV & 0.5, 1, 2 & 5.4, 6.0, 6.7, 7.4  & 25 runs \\\\\n     20 & 1024 & $5.37\\times 10^{5}$ & $9.97\\times 10^{4}$ & 0.78 & CDM, 2 keV, 3 keV, 4 keV & 0.5, 1, 2 & 5.4, 6.0, 6.7, 7.4  & 48 runs\\\\\n     20 & 1024 & $5.37\\times 10^{5}$ & $9.97\\times 10^{4}$ & 0.78 & CDM & different $\\gamma$ & 6.0 & 3 runs\\\\\n     20 & 1024 & $5.37\\times 10^{5}$ & $9.97\\times 10^{4}$ & 0.78 & 1 keV & 1 & 6.0 & 1 run\\\\\n     20 & 1024 & $5.37\\times 10^{5}$ & $9.97\\times 10^{4}$ & 0.78 & CDM & 0.5, 1, 2 & 6.0 & 12 runs, different $\\sigma_8$ values\\\\\n     20 & 1024 & $5.37\\times 10^{5}$ & $9.97\\times 10^{4}$ & 0.78 & CDM & 0.5, 1, 2 & 6.0 & 12 runs, different $n_\\mathrm{s}$ values\\\\\n     40 & 1024 & $4.30\\times 10^{6}$ & $7.97\\times 10^{5}$ & 1.56 & CDM, 2 keV, 3 keV, 4 keV & 0.5, 1, 2 & 5.4, 6.0, 6.7, 7.4 & 48 runs\\\\\n     40 & 1024 & $4.30\\times 10^{6}$ & $7.97\\times 10^{5}$ & 1.56 & CDM & different $\\gamma$ & 6.0 & 3 runs\\\\\n      40 & 1024 & $4.30\\times 10^{6}$ & $7.97\\times 10^{5}$ & 1.56 & 1 keV & 1 & 6.0 & 1 run\\\\\n     40 & 1024 & $4.30\\times 10^{6}$ & $7.97\\times 10^{5}$ & 1.56 & CDM & 0.5, 1, 2 & 6.0 & 12 runs, different $\\sigma_8$ values\\\\\n     40 & 1024 & $4.30\\times 10^{6}$ & $7.97\\times 10^{5}$ & 1.56 & CDM & 0.5, 1, 2 & 6.0 & 12 runs, different $n_\\mathrm{s}$ values\\\\\n     80 & 1024 & $3.44\\times 10^{7}$ & $6.38\\times 10^{6}$ & 3.13 & CDM & 1 & 6.0 & 1 run\\\\\n    \\hline\n    5 & 768 & $1.99\\times 10^{4}$ & $3.69\\times 10^{3}$  & 0.26  & CDM & 1 & 6.0 & 1 run\\\\\n     10 & 768 & $1.59\\times 10^{5}$ & $2.95\\times 10^{4}$ & 0.52 & CDM & 1 & 6.0 & 1 run\\\\\n     20 & 768 & $1.27\\times 10^{6}$ & $2.36\\times 10^{5}$ & 1.04 & CDM & 1 & 6.0 & 1 run\\\\\n      40 & 768 & $1.02\\times 10^{7}$ & $1.89\\times 10^{6}$ & 2.08 & CDM & 1 & 6.0 & 1 run\\\\\n       80 & 768 & $8.15\\times 10^{7}$ & $1.51\\times 10^{7}$ & 4.17 & CDM & 1 & 6.0 & 1 run\\\\\n    \\hline\n    5 & 512 & $6.72\\times 10^{4}$ & $1.25\\times 10^{4}$  & 0.39  & CDM & 1 & 6.0 & 1 run\\\\\n     10 & 512 & $5.37\\times 10^{5}$ & $9.97\\times 10^{4}$ & 0.78 & CDM, 2 keV, 3 keV, 4 keV  & 0.5, 1, 2 & 5.4, 6.0, 6.7, 7.4 &  25 runs\\\\\n     20 & 512 & $4.30\\times 10^{6}$ & $7.97\\times 10^{5}$ & 1.56 & CDM & 1 & 6.0 &  1 run\\\\\n     40 & 512 & $3.44\\times 10^{7}$ & $6.38\\times 10^{6}$ & 3.13 & CDM & 1 & 6.0 & 1 run\\\\\n       80 & 512 & $2.78\\times 10^{8}$ & $5.10\\times 10^{7}$ & 6.25 & CDM & 1 & 6.0  & 1 run\\\\\n    \\hline\n    \\multicolumn{9}{c}{\\textbf{Simulations with an inhomogeneous radiation field (patchy reionization)}}\\\\\n    \\hline\n     40 & 2048 & $5.37\\times 10^{5}$ & $9.97\\times 10^{4}$ & 0.78 & CDM & - & 5.3, 5.7,  6.0, 6.6 & 4 runs\\\\\n    160 & 2048 & $3.44\\times 10^{7}$ &  $6.38\\times 10^{6}$ &  3.13 & CDM & - & 5.3 & 1 run\\\\\n    \\hline\n    \\hline\n  \\end{tabular}}\n  \\label{tab:sims}\n\\end{table*}\n\nThe Sherwood-Relics suite that we present here builds upon the Sherwood simulation project \\citep{Sherwood}. In particular,\nwe simulate the same volumes and compute mock Lyman-$\\alpha$\\ forest absorption spectra on-the-fly in the same way. \nWe, however, expand the sampled space of astrophysical and cosmological parameters significantly. We explore different reionization and heating histories, different values for the most relevant cosmological parameters for the Lyman-$\\alpha$\\ forest ($\\sigma_8$ and $n_\\textrm{s}$), and investigate the impact of the patchiness of cosmic reionization on the high-redshift IGM ($z \\gtrsim 4$). Furthermore, since the sweet spot for Lyman-$\\alpha$\\ forest constraints on dark matter also falls in this redshift range, we simulate models with a range of different dark matter free streaming scales. Table~\\ref{tab:sims} provides an overview of the over 200 different simulations performed for this project. \n\nOn a technical level, the main improvements compared to the Sherwood simulations are a non-equilibrium thermo-chemistry solver and an improved treatment of the ionizing radiation fields. The latter includes both simulations with an improved time-dependent but spatially-homogeneous UV background model (as detailed in Sec.~\\ref{sec:sims_homo_uvb}, based on \\citealt{PuchweinEtAl2019}), as well as with a new hybrid post-processing radiative transfer\/hydrodynamical simulation treatment of patchy reionization (as introduced in Sec.~\\ref{sec:sims_hybrid_method}). \n\nSeveral studies have already made use of the Sherwood-Relics suite. Using our hybrid simulations, \\citet{MolaroEtAl2022} have derived corrections for the impact of the patchiness of reionization on the Lyman-$\\alpha$\\ forest flux power spectrum. These corrections can be applied to conventional simulations with homogeneous UVB models and will be used together with our large grid of homogeneous UVB simulations in forthcoming studies. Other aspects of our hybrid simulations have also already been explored in \\citet{2020MNRAS.494.5091G}, focusing on the properties of Lyman-$\\alpha$\\ transmission spikes, and in \\citet{2021MNRAS.506.5818S}, predicting the 21-cm forest. \\citet{lamberts2022} have used our baseline homogeneous UVB simulation as a reference for the expected thermal history during He~\\textsc{ii}\\ reionization. The main science focus of this work is to provide a comprehensive introduction to the Sherwood-Relics simulations and to investigate the physical processes by which patchy reionization affects the IGM and Lyman-$\\alpha$\\ forest.\n\n\\subsection{The simulation code}\n\\label{sec:sim_code}\n\nAll cosmological hydrodynamical simulations that we present in this work were performed with modified versions of the \\textsc{p-gadget3} code, itself an updated and extended version of\n\\textsc{p-gadget2}\\footnote{\\url{https:\/\/wwwmpa.mpa-garching.mpg.de\/gadget\/}} \\citep{Gadget2}. The code follows the gravitational interactions with\nan efficient, parallel, Tree-PM gravity solver and the hydrodynamics with an energy- and entropy-conserving\nsmoothed-particle hydrodynamics (SPH) scheme \\citep{EntropyConservingSPH}. \n\nThe treatment of radiative cooling assumes a primordial composition of the gas with a hydrogen and\nhelium mass fraction of 76 and 24 per cent respectively. The ionization and thermal state of the gas\nis then followed with a non-equilibrium solver that integrates the ionization, recombination,\ncooling and heating rate equations using sub-cycling and adaptive time steps \\citep[see][]{PuchweinEtAl2015}.\nThe \\textsc{cvode} library\\footnote{\\url{https:\/\/computing.llnl.gov\/projects\/sundials}} is used for this purpose.\nFollowing the full non-equilibrium equations avoids an artificial delay between \nphotoionization and photoheating that is present in simulations with an equilibrium solver \\citep[see also][]{Gaikwad2019,Kusmic2022}. \nThe following rate coefficients are assumed: the case A recombination rates of \\citet{VernerFerland1996},\nthe He~\\textsc{ii}\\ dielectronic recombination rate of \\citet{AldrovandiPequignot1973}, the collisional excitation rates\nof \\citet{Cen1992}, the collisional ionization rates of \\citet{Voronov1997}, and the free-free Bremsstrahlung rate of\n\\citet{TheunsEtAl1998}. In most of our runs, photoionization and photoheating is followed based on external, spatially homogeneous\nmodels of the UV background (UVB), while in our patchy reionization simulations we interpolate from maps of the radiation field\nobtained with the \\textsc{aton} radiative transfer code. We will discuss this in more detail below.\n\nSince we are primarily interested\nin the low-density IGM, we accelerate our simulations by converting all gas particles that exceed a density\nof $1000$ times the mean cosmic baryon density and have a temperature smaller than $10^5\\rm\\,K$ to collisionless star particles.\nWhile this approach does not yield realistic galaxies, it accurately predicts the properties of the low-density IGM \\citep{QuickLya}.\n\n\\subsection{Simulations with a homogeneous UVB}\n\\label{sec:sims_homo_uvb}\n\nFor our baseline simulations, we use time-varying but spatially homogeneous photoionization and photoheating rates from\nthe \\textit{fiducial} UVB model presented\nin \\citet[][see their table D1]{PuchweinEtAl2019}. These were derived in such a way that gas exposed to this UVB follows\na (largely) realistic cosmic reionization and heating history. Hydrogen reionization finishes at $z \\sim 6$, while He~\\textsc{ii}\\\nreionization ends at $z \\sim 2.8$. In particular, simulations with this model avoid an artificially accelerated reionization \\citep[see also][]{onorbe2017}.\nIn comparison, in simulations with, e.g., an \\citet{HaardtMadau2012} UVB model, hydrogen reionization would be essentially\ncompleted by $z \\sim 11$ \\citep[see][]{PuchweinEtAl2019}.\n\nIn addition to our baseline simulations, we perform simulations with various modifications of the \\citet{PuchweinEtAl2019}\n\\textit{fiducial} UVB model in order to sample different reionization histories as well as a wider range of IGM temperatures (see \\citealt{VillasenorEtAl2022} for a similar approach).\nFor example, we produce simulations with a colder or hotter IGM by rescaling the photoheating rates while keeping the\nphotoionization rates fixed. The rescaling factors used for the different runs are provided in Table~\\ref{tab:sims}.\nFor the colder (hotter) models, we use the same factor of 0.5 (2) for the H~\\textsc{i}\\ and He~\\textsc{i}\\ photoheating rates,\nbut a different factor of 0.66 (1.5) for He~\\textsc{ii}\\, as the latter primarily\naffects the thermal history at lower redshifts during the epoch of He~\\textsc{ii}\\ reionization.\n\nIn addition, we also vary the reionization redshift (which we define as the time when the volume-averaged neutral fraction falls below $10^{-3}$) while keeping the instantaneous gas temperature at $z<5$ fixed,\nby performing a linear redshift rescaling of our fiducial UV background model \\citep{PuchweinEtAl2019} at $z>5$.\nThe different homogeneous UVB models considered here complete reionization in the redshift range $z_\\textrm{r}=5.3$ to 7.4, providing a range of models with different amounts of pressure smoothing. These reionization histories corresponds to a rescaling of the fiducial\nUV background redshift coordinate at $z>5$ (i.e. of $z-5$)  by a factor of \n0.89 to 1.24. In addition, to ensure the instantaneous gas temperatures of the models\nremain the same at $z<5$, we multiply the H~\\textsc{i}\\ photoheating rates\nby factors 0.9 to 1.5 at $z>5$, with the higher factors being used for models with earlier reionization.\n\n\\subsection{Hybrid, patchy reionization simulations}\n\\label{sec:sims_hybrid_method}\n\nA time-varying but spatially homogeneous UVB model cannot\ncapture the large spatial fluctuations in the ionizing\nradiation field that are present during the patchy cosmic\nreionization process. A homogeneous UVB model can only aim to provide\nsuitable mean values. However, the radiation field will be entirely different\nin ionized bubbles and neutral regions. In addition to the\ndirect effect this has on the ionization state, this also seeds fluctuations\nin the IGM temperature and pressure smoothing on large scales.\nThese fade only slowly and persist well into the post-reionization\nepoch \\citep[see, e.g.,][]{DAloisioEtAl2015, KeatingEtAl2018}.\n\nTo capture these effects, while avoiding the enormous computational\ncost of fully coupled radiation hydrodynamics simulations, we\nuse a new hybrid scheme that combines relatively cheap post-processing\nradiative transfer simulations on a fixed Eulerian grid,\nwith cosmological hydrodynamical simulations that capture\nthe hydrodynamic response to photoheating by an inhomogeneously evolving\nUV radiation field. Our scheme has some common features with the hybrid technique introduced in \\citet{OnorbeEtAl2019}. Their scheme uses a semi-numerical, excursion-set method to obtain a map of the reionization redshift of the simulation volume. For each resolution element in the simulation, they then switch on a time-dependent UVB model at the pre-computed reionization redshift of the element. The UVB seen by their simulation is otherwise homogeneous, i.e. spatially constant across ionized regions. Our scheme instead uses post-processed radiative transfer simulations to provide an inhomogeneously evolving UV radiation field. This captures the effect of a spatially-varying reionization redshift, but also of spatial fluctuations in the radiation field within ionized regions, which can be significant near the tail end of reionization.\n\nOur scheme consists of the following steps:\n\\begin{itemize}\n\\item Performing a cosmological hydrodynamical simulation with a\nhomogeneous UVB model. We use our baseline simulation discussed above for\nthis purpose. To provide sufficient time resolution for the next step,\nwe have saved outputs every 40 Myrs.\n \n\\item Performing a post-processing radiative transfer simulation on the\noutputs of the cosmological hydrodynamical simulation. We do this on\na fixed Eulerian grid with a slightly modified version of the highly efficient, GPU-accelerated\n\\textsc{aton} code \\citep{Aton, Aton2010}. It uses a moment-based radiative\ntransfer scheme that assumes the M1 closure approximation and uses the full\nphysical speed of light. We use a number of grid cells that equals\nthe (initial) number of gas particles in the hydrodynamical simulation\n(i.e., $2048^3$ for the patchy simulations presented later in this work).\nThe challenging task of accurately predicting source luminosities from\ngalaxy formation physics is bypassed by empirically calibrating the amount of\nionizing radiation escaping from haloes to observational constraints on\nthe ionization state of the IGM, e.g., based on the very high-redshift\nLyman-$\\alpha$\\ forest and Thomson scattering optical depth measurements from CMB data. \nHalos are then populated with ionizing sources based on their mass.\nFull details of this method are provided in \\citet{KulkarniEtAl2019}. For our hybrid simulations completing reionization at a redshift of 5.3, we\nuse the redshift evolution of the ionizing emissivity that was derived in\nthat study (see their fig.~1). For our 40 cMpc\/$h$ box, we apply a redshift-independent boost factor to the emissivity of $1.265$ to\naccount for the smaller box size and higher numerical resolution, which changes the resolved\npart of the escape fraction. We use a single frequency bin and assume a somewhat smaller (mean) photon energy of $18.63$ eV. We apply the same strategy for calibrating the redshift evolution of the emissivity for our earlier reionization scenarios.\n\n\\item The modified \\textsc{aton} code version saves maps of the photoionization rate every 40 Myrs. In addition,\nit produces a map of the reionization redshift, which for each grid cell we define as the redshift at which\nan ionized, i.e. H~\\textsc{ii} , fraction of 3 per cent is exceeded for the first time. This records when a cell starts to get\nionized. As cells ionize rapidly once they are reached by an ionization front, the recorded reionization redshift is rather insensitive to the exact threshold value used. Fig.~\\ref{fig:zreion_map} shows a slice of a reionization redshift map produced in this way. Typically high-density regions containing many ionizing sources reionize first, while remote voids are the last regions to reionize.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{figs\/zreion.pdf}\n  \\caption{Map of the reionization redshift in a thin slice produced by the radiative transfer calculation with the \\textsc{aton} code. This map corresponds to the $z_\\textrm{r}=5.3$ patchy simulation that is discussed later in the manuscript. For comparison, the large scale structures in the same slice are shown in Fig.~\\ref{fig:slices_homo_aton_patchy}.}\n  \\label{fig:zreion_map}\n\\end{figure}\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=\\linewidth]{figs\/slices_homo_aton_patchy.pdf}\n  \\caption{Neutral hydrogen density (in units of the mean total hydrogen density, top panels) and temperature (bottom panels) in a \\textsc{p-gadget3} simulation\n  with a homogeneous UVB (left, showing the matched homogeneous run for the $z_\\textrm{r} = 5.3$ patchy simulation), in an \\textsc{aton} post-processing radiative transfer simulation (middle), and in our hybrid, patchy reionization\n  cosmological hydrodynamical simulation (right). Results are shown at $z\\sim7$ for simulations that complete reionization at $z_\\textrm{r} = 5.3$. \n  By construction all three simulations have very similar (volume-weighted) H~\\textsc{i}\\ fractions of $\\sim 45\\%$ and temperatures at mean density of $T_0 \\sim 6900$ K.\n  While the hybrid simulation has similar ionized\/neutral regions as the \\textsc{aton} run, it also accounts for \n  shock heating of gas in high-density regions, consistent pressure smoothing, and the advection of the thermal energy in gas flows. \n  }\n  \\label{fig:slices_homo_aton_patchy}\n\\end{figure*}\n\n\\item Finally, we perform a second cosmological hydrodynamical simulation of the same volume. In this run\nour modified version of \\textsc{p-gadget3} loads the maps of the photoionization rate produced by \\textsc{aton}\nand uses them as a spatially varying UV background for following photoionization and photoheating\nduring patchy reionization. At each time step and for each SPH particle, local values of the the photoionization and\nphotoheating rates are computed as follows. If the host cell of the particle has not been reionized in the \\textsc{aton}\nsimulation yet (H~\\textsc{ii}\\ fraction continues to be smaller than 3 per cent), the rates are assumed to be zero so that we can\ncompletely skip the integration of the rate equations and assume the gas in this SPH particle is neutral. At redshifts lower\nthan the ionization redshift, the H~\\textsc{i}\\ photoionization rate of the host cell is interpolated between the nearest \\textsc{aton} output\ntimes and adopted for the SPH particle and the current time step. A slightly different treatment is used right after the reionization of a region, i.e.\nbetween the reionization redshift and the next (lower) \\textsc{aton} output redshift. In this case we use the rate of the next (lower redshift) map directly without interpolation. This results in a larger jump in the photoionization rate at the recorded reionization redshift, corresponding to a quickly passing ionization front.\nOur tests showed that this leads to a smoother growth of the ionized regions in the hydrodynamical simulation, thereby further reducing residual imprints\nof the finite number of \\textsc{aton} outputs. Since we cannot capture differences between H~\\textsc{i}\\ and He~\\textsc{i}\\ reionization with\na single frequency bin in the radiative transfer, we simply adopt the H~\\textsc{i}\\ photoionization rate also for He~\\textsc{i}.\nThe H~\\textsc{i}\\ photoheating rate is computed based on the assumed mean photon energy, i.e. the photoionization rate\nis simply multiplied by $18.63 \\, \\textrm{eV} - 13.6 \\, \\textrm{eV} = 5.03 \\, \\textrm{eV}$. The He~\\textsc{i}\\ photoheating\nrate (per atom) is then assumed to be 1.3 times that of H~\\textsc{i}, in rough agreement with the ratio of the two in\nthe homogeneous, synthesis UVB model from \\citet{PuchweinEtAl2019}. Finally, the He~\\textsc{ii}\\ photoionization and photoheating rates are assumed to be\nspatially homogeneous and are adopted from the \\citet{PuchweinEtAl2019} \\textit{fiducial} UVB model. These latter rates play, however,\na role only at lower redshifts and are negligible during the epoch of H~\\textsc{i}\\ reionization, except in the proximity of quasars \\citep[e.g.][]{Bolton2012}, which we do not model here. Armed with these rates, we integrate\nthe ionization and cooling\/heating rate equations in the same manner as in our homogeneous UVB simulations.\n\\end{itemize}\n\nThe results from a patchy, hybrid radiative transfer\/cosmological hydrodynamical simulation are shown at\nredshift $z\\sim7$ in Fig.~\\ref{fig:slices_homo_aton_patchy}. The left-hand panels show the neutral hydrogen density\n(top) and gas temperature (bottom) in a \\textsc{p-gadget3} simulation with a homogenous UVB.\nThe middle panels show the post-processed radiative transfer simulation performed with the \\textsc{aton} code,\nwhile the right-hand panels show the results of the hybrid radiative transfer\/cosmological hydrodynamical simulation.\nAs expected, the simulation with a homogenous UVB completely misses the patchy nature of cosmic reionization.\nThe \\textsc{aton} simulation nicely displays the complicated morphology of ionized bubbles, but misses some\nimportant physics. In particular, running the calculation in post-processing means that the dynamical effect of\ninhomogeneous photoheating (e.g., pressure-smoothing) and its impact on the gas density distribution cannot be followed.\nIn addition, the thermal energy injected by photoheating is stored for each grid cell, but not properly advected with\nthe gas flow. This can sometimes be seen as dense gas leaving a wake of increased temperature when it falls towards\na structure. Also the temperature evolution in the \\textsc{aton} simulation only accounts for photoheating and\nadiabatic evolution, but misses shock heating of gas in dense regions. The patchy, hybrid \\textsc{p-gadget3} \ncosmological hydrodynamical simulation (right panels) captures all these aspects. The morphology of ionized and\nneutral regions is almost identical to the \\textsc{aton} simulation, but, e.g., the temperature in halos is larger\ndue to the inclusion of shock heating. The dynamical impact of inhomogeneous photoheating in our patchy \\textsc{p-gadget3} \nsimulation will be discussed in detail in Sec.~\\ref{sec:pressure_smooth}.\n\nAt first glance our multi-step hybrid radiative transfer\/cosmological hydrodynamical simulation scheme may seem complicated, e.g.,\ncompared to a single fully coupled radiation-hydrodynamics simulation. There are, however, several distinct\nadvantages. First, the computational cost is much lower compared to a full radiation hydrodynamics calculation (e.g., 0.5 million core hours on CPUs + 3 thousand GPU hours for our $z_\\textrm{r}=5.3$ patchy run compared to 28 million core hours on CPUs for the main Thesan run; \\citealt{Thesan}).\nThe reason for this is that the radiative transfer is done on a somewhat coarse fixed Eulerian grid (cell size $19.5 \\, h^{-1} \\, \\textrm{ckpc}$ in our $40 \\, h^{-1} \\, \\textrm{cMpc}$\npatchy simulations), and hence allows for relatively large Courant time steps as well as an efficient parallelization on \npowerful GPUs. Second, as we empirically calibrate the emission that escapes into the IGM, we can continue to use our strongly simplified\ngalaxy\/star formation model (see Sec.~\\ref{sec:sim_code}) and avoid all the complications of radiation-hydrodynamically modelling realistic\nsource galaxy populations as well as the escape of ionizing radiation from them. Third, finding a calibration of the source\nluminosity as a function of halo mass that agrees with observational constraints on the ionization state of the IGM (using only cheap post-processing\nradiative transfer simulations for this purpose) is much simpler than modifying a full simulation model of galaxy formation in such a way that the same\nis achieved. Finally, the overhead of producing the first hydrodynamical simulation with a homogeneous UVB came (at least in this project) for free in practice,\nas we would have needed this run in any case as a baseline model for the comparison to the large number of simulations with different ionization histories and dark\nmatter models (see Sec.~\\ref{sec:relics_suite}).\n\nOf course the computational efficiency of our approach also comes at a price. By saving only a limited number of maps of the radiation field on a fixed grid,\nwe have access to the radiation field only with a limited spatial and time resolution when following photoionization and photoheating in\nthe second hydrodynamical simulation. Also, the coupling of the radiative transfer to the hydrodynamics is not fully self-consistent in cases where the difference in\npressure-smoothing between the first and second hydrodynamic simulation significantly changes the opacity of the medium and the local radiation field.\nWe expect these effects to primarily play a role on small scales in dense systems. We would hence not advise to use this method for investigating, e.g., the escape of ionizing\nradiation from galaxies, and one should probably be careful when studying the details of self-shielding of dense gas. The impact of large scale fluctuations\n(on the scale of the size of ionized bubbles) on ionization and pressure-smoothing of the low-density IGM should, however, be robustly predicted.  \n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{figs\/temp_xhii_vs_z.pdf}\n  \\caption{IGM properties as a function of redshift in the patchy simulation that completes reionization at $z_\\textrm{r}=5.3$, as well as in the matched homogeneous simulation. The temperature at mean baryon density ($T_0$) is shown in the top panel. Results are indicated for the volume-weighted mean and median value of $T_0$, measured in gas with densities $0.975 < \\Delta < 1.025$. IGM temperature measurements from \\citet{BoeraEtAl2019}, \\citet{2019ApJ...872...13W} and \\citet{2020MNRAS.494.5091G} are shown for comparison. The bottom panel shows the mass- and volume-weighted ionized fractions (i.e. the H~\\textsc{ii}\\ fractions). Observational constraints from \\citet{2015MNRAS.447..499M}, \\citet{2018ApJ...864..142D}, \\citet{2019MNRAS.485.3947M}, \\citet{2020ApJ...896...23W} and \\citet{2022MNRAS.512.5390G} are shown for reference.}\n  \\label{fig:temp_xhii_taueff_vs_z}\n\\end{figure}\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=\\linewidth]{figs\/pow_spec_ratios.pdf}\n  \\caption{Change in Lyman-$\\alpha$\\ forest flux power spectrum due to different choices of reionization\/photoheating model, dark matter model, as well as matter power spectrum normalization and slope. Shown are the ratios of the Lyman-$\\alpha$\\ forest flux power spectrum in the considered simulations to that of our default (or baseline) $\\Lambda$CDM model with a \\citet{PuchweinEtAl2019} UVB at $z=4.6$. In the hot\/cold models, the H~\\textsc{i}\\ photoheating rate has been increased\/decrease by a factor of 2. For reference, the shaded region indicates the relative error in the power spectrum measurement of \\citet{BoeraEtAl2019}. Significant degeneracies between different modifications exist. Interestingly, the increase in power on large spatial scales in our patchy simulations seems to be a characteristic signature of inhomogeneous reionization.\n  }\n  \\label{fig:pow_spec_ratios_grid}\n\\end{figure*}\n\nTo isolate the effects of the patchiness of reionization from the effects of differences in the reionization history,\nwe have performed simulations with homogeneous UVB models that produce the same \naverage reionization and thermal histories as our hybrid, patchy reionization simulations. Details on how a suitably tailored UVB model for such a simulation\nis obtained are provided in Appendix~\\ref{app:homogeneous_UVB}. We have performed such pairs of patchy and matched homogeneous simulations for different reionization histories with reionization completing at redshifts $z_\\textrm{r} = 5.3$, 5.7, 6.0 and 6.6. We will mostly concentrate on the first model in the analysis as its reionization history seems in best agreement with observations \\citep[e.g.,][]{KulkarniEtAl2019,bosman2022}. \n\nFig.~\\ref{fig:temp_xhii_taueff_vs_z} compares this matched homogeneous model to the corresponding patchy simulation. It displays the evolution of the mean and median IGM temperature at mean density, as well as the mass- and volume-weighted ionized hydrogen fraction. As planned, the IGM temperature and ionized fraction in the matched homogeneous run closely follow those in the patchy simulation. We have opted to follow the mean temperature at mean density of the patchy simulation during reionization, and the median temperature at mean density after reionization (see Appendix~\\ref{app:homogeneous_UVB} for further details on this). \n\nNote that, despite the similar neutral fractions, the Lyman-$\\alpha$\\ forest transmission properties will be quite different during reionization. While ionized regions can allow transmission in a patchy reionization scenario, a small amount of homogeneously distributed residual neutral gas that is present even late in the reionization process is sufficient to almost fully absorb the Lyman-$\\alpha$\\ forest in a homogeneous model. This too homogeneous distribution of neutral gas, which can be seen in the upper, left panel of Fig.~\\ref{fig:slices_homo_aton_patchy}, is the main reason why simulations with a homogeneous UVB cannot reproduce the observed statistics of the Lyman-$\\alpha$\\ forest at very high redshift ($z \\gtrsim 5.3$; e.g., \\citealt{BeckerEtAl2015}). Patchy reionization simulations, even when done in post-processing, do a much better job \\citep[e.g.,][]{KulkarniEtAl2019, KeatingEtAl2020, bosman2022}.\n\n\\section{Results}\n\n\\subsection{The impact of cosmology, reionization model and IGM temperature on the Lyman-$\\alpha$\\ forest}\n\\label{sec:grid_pow_spec}\n\nThe Lyman-$\\alpha$\\ forest is sensitive to a wide range of cosmological and astrophysical parameters and processes. We aim to sample many of the most relevant ones with the Sherwood-Relics simulation suite. Fig.~\\ref{fig:pow_spec_ratios_grid} provides an overview of the various impacts on the structures present in the Lyman-$\\alpha$\\ forest on different scales. The figure shows the relative change in the Lyman-$\\alpha$\\ flux power spectrum compared to our baseline simulation with cold dark matter and a \\citet{PuchweinEtAl2019} UVB.\n\nNote that by $z=4.6$, even in our patchy reionization simulations, large scale fluctuations in the ionizing radiation field have largely faded. We thus choose to rescale the optical depths in all simulations shown in the figure such that the mean transmission value is consistent with observations at that redshift. We have used the following fitting function for the observed effective optical depth for this purpose \\citep[see][]{MolaroEtAl2022},\n\\begin{equation}\n\\tau_\\textrm{eff} =\n\\begin{cases}\n   -0.132 + 0.751 \\, [(1 + z)\/4.5]^{2.90},& \\text{if } 2.2 \\leq z < 4.4\\\\\n   1.142 \\, [(1 + z)\/5.4]^{4.91}. & \\text{if } 4.4 \\leq z \\leq 5.5.\n\\end{cases}\n\\label{eq:tau_eff_fit}\n\\end{equation}\n\nFig.~\\ref{fig:pow_spec_ratios_grid} displays ratios of the power spectrum of the normalized transmitted flux contrast, i.e. of $F\/\\bar{F}-1$, where $F$ is the normalized transmitted flux and $\\bar{F}$ denotes its mean value. For comparison, the relative errors in the \\citet{BoeraEtAl2019} Lyman-$\\alpha$\\ forest flux power spectrum measurement are also indicated.\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=\\linewidth]{figs\/rhoT.pdf}\n  \\caption{Volume-weighted temperature-density distribution of gas in the patchy simulation that completes reionization at $z_\\textrm{r}=5.3$ (bottom panels) and in the matched homogeneous simulation (top panels) at redshifts $4.2$, $5.4$ and $7.0$. Both simulations have, by construction, similar volume-weighted neutral fractions of $\\sim$0\\%, 4\\%, 45\\% at these redshifts, respectively. During reionization, cold gas that has not yet been ionized is present in the patchy run. In contrast, the same gas is partly ionized and heated in the homogeneous simulation. After reionization, the temperature-density relation in the patchy simulations is still broader at low densities.\n  }\n  \\label{fig:rho-T}\n\\end{figure*}\n\nClearly, the different changes in the simulated physics leave specific imprints in the flux power spectrum. Models with a hotter\/colder IGM (with H~\\textsc{i}\\ and He~\\textsc{i}\\ photoheating rates boosted\/reduced by a factor of 2, see Sec.~\\ref{sec:sims_homo_uvb} for full details) have significantly less\/more power on small spatial scales (large $k$). This is consistent with the expectation of increased thermal broadening and pressure smoothing at higher temperature. Pressure smoothing is also increased by an earlier reionization (see the $z_\\textrm{r}=7.5$ model, blue solid curve). Interestingly, the corresponding suppression of power on intermediate scales, $10^{-2}\\,\\textrm{s}\\,\\textrm{km}^{-1} \\lesssim k \\lesssim 10^{-1}\\,\\textrm{s}\\,\\textrm{km}^{-1}$, is very similar to that in a warm dark matter model with 4 keV particle mass (green solid curve). This already suggests that there will be degeneracies between dark matter constraints and the thermal\/reionization history of the IGM  \\citep[see e.g.][]{Viel_2013,2017PhRvD..96b3522I,Garzilli2019}. One can aim to break these by including measurements at different redshifts, as well as at higher $k$ \\citep[e.g.][]{Nasir2016}.  Changing the cosmological parameters $\\sigma_8$ and $n_\\textrm{s}$ has largely the expected effects \\citep[see also][]{QuickLya,McDonald2005}. Similar to the matter power spectrum, it changes the normalization and slope of the Lyman-$\\alpha$\\ forest flux power spectrum.  Note also that, as already noted elsewhere \\citep{Cen2009,KeatingEtAl2018,DAloisio2018,OnorbeEtAl2019,WuEtAl2019,MonteroCamacho2020,MolaroEtAl2022}, the patchy reionization simulations predict distinctive increases in power on the largest spatial scales (smallest $k$). We will discuss these in detail in Sec.~\\ref{sec:modulation_lya}.\n\nOur grid of simulations, as detailed in Table~\\ref{tab:sims} and as (partly) shown in Fig.~\\ref{fig:pow_spec_ratios_grid}, will be used for parameter inference studies in forthcoming work. In the remainder of this study, we will concentrate on our new hybrid simulations and discuss the \\textit{relic} signatures that patchy cosmic reionization leaves in the IGM and Lyman-$\\alpha$\\ forest.  \n\n\\subsection{The impact of patchy reionization on the IGM and the Lyman-$\\alpha$\\ forest}\n\\label{sec:patchy_results}\n\\subsubsection{The thermal state of the IGM during and after patchy cosmic reionization}\n\\label{sec:thermal_properties}\n\nDuring the era of cosmic reionization, energetic UV photons emitted by first galaxy populations ionize the IGM. The excess energy of these photons beyond the ionization energy of the relevant atoms\/ions is available for heating the IGM. Given the patchy nature of cosmic reionization, this causes significant temperature differences between regions that reionize at different times.\n\nFig.~\\ref{fig:rho-T} shows how this affects the temperature-density distribution of the IGM during and after cosmic reionization. At $z=7$, when roughly half of the hydrogen is ionized, there is both cold, neutral and hot, ionized gas present in the $z_\\textrm{r}=5.3$ patchy simulation, corresponding to neutral and ionized regions. The matched homogeneous UVB simulation instead contains only gas that has a temperature of at least several thousand Kelvin. Except for a small amount of shock-heated gas, the gas is partly ionized and partly photoheated (see also Fig.~\\ref{fig:slices_homo_aton_patchy}) and follows a tight temperature-density relation that is almost flat. This corresponds to all gas having a similar thermal history with little variation around the mean evolution shown in Fig.~\\ref{fig:temp_xhii_taueff_vs_z}. It also explains the very similar mean and median thermal evolutions in the homogeneous model that are also indicated there. In the patchy simulations instead, there is more variation in temperature of the ionized gas, in particular at low densities where temperatures span a range of 3000 - 20000 K, corresponding to different reionization redshifts and consequently different amounts of cooling after reionization. The situation at $z=5.4$ is qualitatively similar, just with much less cold, neutral gas remaining in the patchy run.\n\nAt $z=4.2$, well after the end of reionization (at $z_\\textrm{r}=5.3$), the homogeneous and patchy simulations look more similar. The temperature-density relations have steepened somewhat (at least the upper envelope for the patchy run). In ionization equilibrium photoionizations balance recombinations. Recombinations happen more frequently in dense gas which consequently receives more photoheating per particle. In addition, the lowest density gas, i.e. the gas in voids, expands more strongly during cosmic expansion and structure formation resulting in increased adiabatic cooling. In the patchy run, there is nevertheless a larger spread in temperature in very low density gas as the temperature fluctuations seeded by inhomogeneous reionization fade only slowly.\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=\\linewidth]{figs\/slice_temp.pdf}\n  \\caption{Gas temperature in a thin slice through our matched homogeneous and patchy simulations which complete reionization at $z_\\textrm{r}=5.3$. During reionization (redshift 7, both runs $\\sim45\\%$ neutral fraction, upper panels) the gas temperature in the simulation with the homogeneous UVB has only small fluctuations which largely trace the gas density, as expected from the narrow density-temperature relation shown in Fig.~\\ref{fig:rho-T}. In contrast the temperature in the patchy simulation differs by orders of magnitude between ionized and neutral regions. Some of the highest temperatures (outside shock heated regions) are found near the ionization fronts in recently ionized gas which had little time to subsequently cool. After reionization (redshift 4.8, lower panels), the temperature in the homogeneous simulation still looks qualitatively similar (except for more pronounced shock heating than at $z=7$). In the patchy simulation, all of the gas has been photoheated as well, but large scale temperature fluctuations that are relics of the patchy reionization process are still clearly visible. The highest temperatures (except for shock heated gas) are found in regions that have been reionized late.}\n  \\label{fig:slice_temp}\n\\end{figure*}\n\nThis is also shown in Fig.~\\ref{fig:slice_temp}, which displays the IGM temperature in a thin slice through part of the simulation box. During reionization, at $z=7$, the temperature map of the patchy simulation clearly shows the locations of ionized regions which have been strongly heated, while neutral regions are still cold. The effect discussed above that recently heated low density gas is hotter than gas that has been reionized earlier is also visible. It results in the temperatures of gas in ionized regions that is located near the ionization fronts being particularly high. We have already seen from the temperature-density distributions that even at $z=7$ all gas has been heated significantly in the homogeneous UVB run. This is reflected by the temperature map in the upper, left panel of Fig.~\\ref{fig:slice_temp} which shows no unheated gas.\n\nAt $z=4.8$, i.e. $\\Delta z \\approx 0.5$ after the end of reionization in these simulations, large scale temperature fluctuations are still present in the patchy run. This relic signature of patchy reionization reflects when regions were reionized. One can see this clearly by comparing the upper and lower right panels of Fig.~\\ref{fig:slice_temp}. Regions that were still cold and neutral at $z=7$ are particularly hot at $z=4.8$ as they have been reionized late and had less time to cool after their reionization.\n\n\\subsubsection{Modulation of the Lyman-$\\alpha$\\ forest on large scales}\n\\label{sec:modulation_lya}\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=\\linewidth, trim=0 0 1cm 0, clip]{figs\/specs_temp_zreion_homog_patchy.pdf}\n  \\caption{Impact of patchy reionization on the Lyman-$\\alpha$\\ forest at $z=4.8$. The top panel displays the normalized transmitted Lyman-$\\alpha$\\ fluxes along the same line-of-sight through the patchy (red) and matched homogeneous simulations (blue). The optical depths were re-scaled to be consistent with the observed mean transmission (see Eq.~\\ref{eq:tau_eff_fit}). In the left and right regions (roughly left of the first and right of the second vertical dotted line), there is slightly less transmission in the patchy run, while it exhibits more transmission than the matched homogeneous run in the central region. The middle panel shows that this finding is closely related to large scale temperature fluctuations in the patchy simulation in which the central region has a higher temperature and is hence more highly ionized. The temperature fluctuations are in turn largely driven by spatial variations of the reionization redshift (bottom panel). Regions that have been reionized late are typically hotter.}   \n  \\label{fig:specs_homog_patchy}\n\\end{figure*}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\linewidth]{figs\/pow_specs.pdf}\n  \\caption{Power spectrum of the normalized transmitted Lyman-$\\alpha$\\ flux contrast in the $z_\\textrm{r}=5.3$ patchy and matched homogeneous simulations (top). The optical depths were re-scaled to be consistent with the observed mean transmission (see Eq.~\\ref{eq:tau_eff_fit}). Results are shown at $z=5.0$, $4.6$ and $4.2$. An enhanced large scale (small $k$) power is found in the patchy simulation in particular near the end of reionization. This corresponds to the large scale modulation of the transmission that is displayed in Fig.~\\ref{fig:specs_homog_patchy}. The bottom panel shows the ratio of the power spectrum in the patchy to that of the homogeneous simulation. In addition to the large scale enhancement a reduction of power on small scales is visible in the patchy simulation \\citep[see also][]{MolaroEtAl2022}.}\n  \\label{fig:flux_pow_specs}\n\\end{figure}\n\nThese large scale temperature fluctuations also affect the ionization state of the gas. In equilibrium the neutral fraction is proportional to the recombination rate, which depends on temperature and is roughly proportional to $T^{-0.7}$. Hence, hot regions have a lower neutral fraction and correspondingly allow more Lyman-$\\alpha$\\ forest transmission. This results in a large scale modulation of the Lyman-$\\alpha$\\ transmitted flux.\n\nFig.~\\ref{fig:specs_homog_patchy} illustrates this effect. The bottom panel shows the reionization redshift along a line-of-sight through the simulation box of the $z_\\textrm{r}=5.3$ patchy run. The middle panel displays the IGM temperature along the same skewer at $z=4.8$. Temperatures are shown for both the patchy run and the corresponding matched homogeneous run. The late reionizing region in the middle ($x\\approx 6$ to 32 cMpc$\/h$) has an increased temperature in the patchy run, as there is little time to cool between its reionization and $z=4.8$. The increased temperature in turn results in a lower neutral fraction and hence more transmission in the Lyman-$\\alpha$\\ forest in that region in the patchy run compared to the matched homogeneous run. This is shown in the upper panel of Fig.~\\ref{fig:specs_homog_patchy}. In early reionizing regions, the opposite effect can be seen, the transmission is slightly lower in the patchy reionization simulation.\n\nNote that the optical depths in both simulations have been rescaled such that (when averaged over our full sample of 5000 lines-of-sight through the box) they are consistent with the observed mean transmission value (according to Eq.~\\ref{eq:tau_eff_fit}). The large scale fluctuation in the transmitted flux is, hence, driven by the temperature variation and not by fluctuations in the radiation field in our patchy simulations, which have largely faded by $z=4.8$.\n\nTo summarize, Fig.~\\ref{fig:specs_homog_patchy} illustrates how a spatially varying reionization redshift translates to a large scale modulation of the Lyman-$\\alpha$\\ forest transmitted flux. Such a modulation is also expected to change the Lyman-$\\alpha$\\ forest transmitted flux power spectrum. Fig.~\\ref{fig:flux_pow_specs} shows that this is indeed the case. \nResults are indicated for several redshifts after the end of reionization.\nClearly visible is an increased power in the patchy simulation on large scales, $k \\approx 10^{-3}$ to a few times $10^{-3}\\rm\\,s\\,km^{-1}$, corresponding to modes with peaks having sizes of $\\lambda\/2 \\gtrsim 5 \\, \\textrm{cMpc}\/h$, which are typical sizes of ionized regions (compare to Fig.~\\ref{fig:slices_homo_aton_patchy}). The power is increased all the way up to the fundamental mode of the $40 \\, \\textrm{cMpc}\/h$ box. As expected for an effect caused by large scale temperature fluctuations seeded by patchy reionization, the power enhancement fades away at lower redshift. On small spatial scales, $k \\gtrsim 0.05 \\rm \\, s\\,km^{-1}$, there is less power in the patchy simulation compared to the matched homogeneous simulation. These findings are consistent with those obtained by \\citet{WuEtAl2019} using fully coupled radiation-hydrodynamics simulations. The behaviour of the power spectrum on small spatial scales (large $k$) is sensitive to the pressure smoothing of the gas and hence requires a coupling of the radiative transfer to the hydrodynamics to be faithfully followed. The increase of power on large spatial scales (small $k$) is in contrast also captured by post-processing radiative transfer simulations \\citep[e.g.,][]{KeatingEtAl2018}.\n\nEmploying several of our patchy reionization simulations, the causes of the reduction of the power spectrum on small scales were investigated in detail by \\citet{MolaroEtAl2022}. The main findings were that the spatial fluctuations in the thermal broadening kernel and the different peculiar velocity fields in the patchy simulations cause the reduction in small scale power. For example, transmission spikes will appear first in low density regions that typically reionize late and are hence particularly hot in the patchy model. Consequently, there will be more thermal broadening in these regions, reducing the flux power spectrum on small scales.\n\nNote, however, that the reduction of power on small scales discussed above is based on a matched comparison of a patchy and homogeneous run with the same mean ionization and thermal history. When comparing to a homogeneous run with a different and more extended thermal history, like our baseline run with a \\citet{PuchweinEtAl2019} UVB, there can be more power on small scales in the patchy simulation (compare to Fig.~\\ref{fig:pow_spec_ratios_grid}).\n\n\\subsubsection{Spatially varying pressure smoothing of the IGM}\n\\label{sec:pressure_smooth}\n\nThe photoheating of the IGM during reionization strongly increases its temperature and gas pressure. The energy injected per baryon, and hence the temperature increase, depend on the spectrum of the ionizing radiation but are approximately independent of gas density. This results in a fairly flat temperature-density relation for most of the gas shortly after reionization (see Fig.~\\ref{fig:rho-T}), with the exception of regions that have been strongly gravitationally heated by shocks and compression during structure formation. The spatial variation of the gas pressure is, hence, largely dominated by the density variation with dense regions having the highest pressure. Many of the smaller\/lower density structures, for which the photoheating dominates over the gravitational heating, e.g., filaments and walls, will consequently be over-pressurized and will start to expand after their reionization. Post-processed radiative transfer simulations do not capture this effect as the coupling to the hydrodynamics is missing. With our hybrid scheme, we can instead study the pressure smoothing of the IGM during and after patchy reionization. The expansion of photoheated structures is illustrated in Fig.~\\ref{fig:pressure_vel}, which shows the gas pressure and gas velocity field of a region in our $z_\\textrm{r}=5.3$ patchy simulation before and after its reionization.\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=\\linewidth]{figs\/pressure_vel.pdf}\n  \\caption{Gas pressure in thin slices through the $z_\\textrm{r}=5.3$ patchy simulation. Results are shown at $z\\approx11.6$ (left panel), when the region is still largely neutral (only the upper left corner has already been swept over by the approaching ionization front), and at $z\\approx8.6$ (right panel), after the region has been reionized. Both panels show the same structures, although note that different $x$-coordinate ranges have been used because the whole region is falling (almost exactly) towards the left. The arrows indicate the gas velocity in a reference frame in which the central filament is roughly at rest. The photoheating during reionization strongly boosts the gas pressure. This also increases the absolute difference in pressure between dense structures and voids. After reionization these structures are over-pressurized and start to expand, which is visible in the velocity fields. The gas velocities near the edged of the filament point inward before reionization, but outward after reionization. The arrows are scaled such that a velocity of 10 km\/s corresponds to a length of 20 ckpc\/$h$.}\n  \\label{fig:pressure_vel}\n\\end{figure*}\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=\\linewidth]{figs\/slice_pressure_smooth_ion_contour.pdf}\n  \\caption{Gas density in units of the mean baryon density in a thin slice through our $z_\\textrm{r}=5.3$ matched homogeneous and patchy simulations. The upper panels show results at redshift 7, when reionization is in full swing and both runs have a neutral fraction of $\\sim 45\\%$. In the upper right panel, regions that are already ionized are indicated by white contours and contain most large overdensities. Compared to the homogeneous simulation, the gas in the central regions of ionized bubbles has experienced more pressure smoothing in the patchy simulation. In contrast, regions that have not been ionized yet in the patchy simulation show more pronounced small scale structure than in the homogeneous run. This is illustrated in more detail for regions A and B for which zoom-ins are shown in Fig.~\\ref{fig:slice_pressure_smooth_regions}. The bottom panels show the gas density in the same slices after reionization at redshift 4.8. The contours in the lower right panel indicate the same regions as in the upper right panel, hence separating regions that have reionized early (before redshift 7) from those that have reionized late. As illustrated by regions A and B, the difference in reionization redshift results in notably different pressure smoothing even after reionization has ended (also see  Fig.~\\ref{fig:slice_pressure_smooth_regions} for zoom-ins).}\n  \\label{fig:slice_pressure_smooth}\n\\end{figure*}\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=0.85\\linewidth]{figs\/slice_pressure_smooth_regions_with_adiabatic.pdf}\n  \\caption{Gas density in units of the mean baryon density in thin slices covering regions that reionize early (region A, top set of panels) and late (region B, bottom set of panels) in our patchy reionization model. Density fields are shown at redshifts $7.0$ and $4.8$ for the $z_\\textrm{r}=5.3$ patchy simulation (right panels), the matched homogeneous simulation (middle panels), and a non-radiative (``adiabatic'') simulation without photoheating and radiative cooling (left panels). The later is included to provide a reference model without pressure smoothing (due to photoheating). The color scale is the same as in Fig.~\\ref{fig:slice_pressure_smooth}. The locations of regions A and B are also indicated there. Clearly early\/late reionizing regions exhibit more\/less pressure smoothing in the patchy simulation compared to the matched homogeneous model.}\n  \\label{fig:slice_pressure_smooth_regions}\n\\end{figure*}\n\nThese hydrodynamic reactions to the photoheating smooth the gas density distribution on small scales, roughly below the \\textit{filtering scale} \\citep[see][]{1998MNRAS.296...44G}, resulting in differences that persist well after reionization. Fig.~\\ref{fig:slice_pressure_smooth} compares the gas density in a thin slice through our $z_\\textrm{r}=5.3$ patchy simulation to that in the corresponding matched homogeneous simulation. This allows an assessment of how patchy reionization causes spatial fluctuations in the amount of small scale structure present in the gas density field.\n\nThe contours in the upper right panel, which shows the patchy run at $z=7$, indicate ionization fronts, i.e. the edges of ionized bubbles. Careful inspection shows that regions near the center of ionized bubbles (such as region A), which reionize early in the patchy simulation, are more strongly smoothed than in the matched homogeneous run where all regions largely follow the mean reionization history. Regions outside ionized bubbles (such as region B) have instead not experienced any pressure smoothing yet in the patchy run.\n\nThe bottom panels of Fig.~\\ref{fig:slice_pressure_smooth} show the gas density in the same slice (in comoving coordinates) after reionization, at $z=4.8$. The dotted contours in the bottom right panel indicate the same regions as in the upper right panel, hence separating regions that have been reionized early (before $z=7$) from regions that have been reionized late (after $z=7$). Even at $z=4.8$, regions that have been reionized early have a smoother gas distribution than regions that have been reionized late.\n\nTo illustrate these effects more clearly, we zoom in on regions A and B in Fig.~\\ref{fig:slice_pressure_smooth_regions}. For reference, we also show results for a non-radiative simulation of the same volume. This run does not include any photoionization or photoheating. Thus, no pressure smoothing is present outside shock-heated regions. This provides a reference model for comparison in which pressure smoothing is absent in the IGM.\n\nClearly, the early reionizing region A is most strongly smoothed in the patchy run at both redshifts. At $z=7$, the density field in the homogeneous run is still very similar to the non-radiative simulation, while pressure smoothing is clearly visible at $z=4.8$. In the patchy run at $z=4.8$, shells (visible as ring-like features) around photo-evaporated structures are visible. These will be discussed in more detail in Sec.~\\ref{sec:lya_lines_rings}.\n\nAs expected, the neutral region B in the patchy simulation is indistinguishable from the non-radiative run at $z=7$. However, also the slice through the partly ionized homogeneous run still looks very similar. At $z=4.8$, region B is smoother in the homogeneous run compared to the patchy run. In the latter, the region reionizes late and has hence little time to respond to the heating.\n\nIn the following, we want to quantify the differences in the pressure smoothing that we have visually identified in the density fields. To this end, we perform local measurements of the power spectrum of the gas density contrast, $\\delta = \\Delta - 1 = \\rho\/\\bar{\\rho} -1$, where $\\rho$ is the gas density and $\\bar{\\rho}$ the mean baryon density. For these measurements, we use a $2048^3$ grid covering the whole simulation box, and then randomly select 32768 regions of size $64^3$ grid cells, corresponding to a region side length of $1.25 \\, \\textrm{cMpc}\/h$. The number of regions was chosen to sample a volume comparable to the full box. In each region, we then measure the gas density contrast power spectrum. We use a window function to reduce the impact of the non-periodic boundary conditions of the individual regions (see Appendix~\\ref{app:pow_measure} for full details). We also compute the mean reionization redshift of each region, so that we can bin the power spectrum measurements by reionization redshift. We perform this procedure both for the non-radiative simulation, as well as for our $z_\\textrm{r}=5.3$ patchy run. We then compute for each reionization redshift bin the ratio of the mean power spectrum (averaged over all regions in the bin) in the patchy run to that (of the same regions) in the non-radiative (``adiabatic'') simulation. This quantifies the reduction of power caused by photoheating.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\linewidth]{figs\/zr_regions_pow_spec.pdf}\n  \\caption{Impact of patchy reionization on small scale structure in the low-density IGM. Shown are ratios of the gas density ($\\delta$) power spectra in the $z_\\textrm{r}=5.3$ patchy and non-radiative (``adiabatic'') simulations. Results are shown at $z=4.8$ for regions with different mean reionization redshifts ($z_\\mathrm{b}$ in the figure legend denotes the center of the $\\Delta z=0.5$ bins into which the regions are sorted by their mean reionization redshift). Regions that have been reionized earlier show a larger suppression of small scale power. All regions included here have a mean gas density of $0.2 < \\Delta < 0.4$ (in units of the mean cosmic baryon density), corresponding to low-density IGM that the Lyman-$\\alpha$\\ forest is sensitive to. The dotted lines are fits to the solid curves assuming the functional form given in Eq.~(\\ref{eq:cutoff}).}\n  \\label{fig:pow_spec_suppression}\n\\end{figure}\n\nFig.~\\ref{fig:pow_spec_suppression} shows this quantity at $z=4.8$. Typically the gas density power spectrum is dominated by dense collapsed structures \\citep[e.g.,][]{KulkarniEtAl2015}. As we are primarily interested in the low density IGM that is probed by the Lyman-$\\alpha$\\ forest at these redshifts, we opted to include only low-density regions with a mean density $0.2 < \\Delta < 0.4$ in the computation of the mean power spectra. This is also a density range in which many of the shell\/ring features discussed above (and further in Sec.~\\ref{sec:lya_lines_rings}) reside and to which the Lyman-$\\alpha$\\ forest is sensitive to at very high redshifts. A reduction of gas density power compared to the non-radiative simulation is clearly present on small spatial scales (large $k$). We also find a clear dependence of the amount of suppression on the reionization redshift of the considered regions. As expected, early reionizing regions show a suppression of power up to larger spatial scales (smaller $k$), while the regions that reionize latest (the $z_\\textrm{r}=5.5$ bin) have the smallest reduction of small scale structure in the gas density field.\n\nOverall the suppression of the power spectrum as a function of $k$ can be well described with a functional form similar to that used in \\citet{1998MNRAS.296...44G}, i.e. with a suppression factor \n\\begin{equation}\n  \\frac{P_\\textrm{patchy}}{P_\\textrm{ad}} = N \\times \\big(\\exp(-k^2\/k_\\textrm{PS}^2)\\big)^2,\n\\label{eq:cutoff}\n\\end{equation}\nwhere $P_\\textrm{patchy}$ and $P_\\textrm{ad}$ are the power spectra in the patchy and non-radiative simulation respectively, and $k_\\textrm{PS}$ describes a pressure-smoothing scale. We will fit this function to the curves in Fig.~\\ref{fig:pow_spec_suppression} to extract the corresponding pressure smoothing scales. In these fits, we treat $k_\\textrm{PS}$ as a free parameter. We will compare the scale measured from the simulation in this way to the \\citet{1998MNRAS.296...44G} filtering scale (Eq.~\\ref{eq:filtering_scale}) later in this section. To allow a bit more flexibility in the fits we have included a normalization factor $N$ which helps to absorb some of the effects that are caused by radiative cooling and star formation in dense objects in the patchy run. We expect this factor to be close to unity and indeed we find numerical values in the range 1.006 to 1.075 for the different reionization redshift bins. We perform the fits only in the range where $P_\\textrm{patchy}\/P_\\textrm{ad} > 0.1$ to avoid being affected by the oscillations that are present at lower values (higher $k$) in the case of high reionization redshifts. We interpret these oscillations as an effect similar to that found in \\citet{1998MNRAS.296...44G} for linear perturbations (see their fig. 1). We also note that while the functional form given by Eq.~(\\ref{eq:cutoff}) works well for the suppression of power in low-density regions ($\\Delta \\sim 0.3$), it does so less well for regions at higher mean density ($\\Delta \\sim 1$ and larger), likely due to a larger number of non-linearly collapsed structures there.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\linewidth]{figs\/lsmooth.pdf}\n  \\caption{Pressure smoothing length scale at $\\Delta \\approx 0.3$ and $z=4.8$ as a function of time since reionization. The values for the patchy simulation were obtained from the parameters of the fits shown in Fig.~\\ref{fig:pow_spec_suppression}.  For comparison we show the Jeans scale either evaluated at the actual density or at mean density ($\\Delta = 1$). Note that the values of the Jeans scales were divided by 3 to better fit on the figure. Furthermore, we indicate the filtering scale computed as in \\citet{1998MNRAS.296...44G}, as well as the length scale that corresponds to a free expansion with a starting velocity of 10 km\/s (see main text for details).}\n  \\label{fig:lsmooth}\n\\end{figure}\n\nThe measured pressure smoothing scale $k_\\textrm{PS}$ can then be converted to a length scale; we do this by defining the pressure smoothing length scale by $\\lambda_\\textrm{PS} \\equiv 1\/k_\\textrm{PS}$. We do not include a $2 \\pi$ factor in this definition to stay consistent with \\citet{KulkarniEtAl2015}. The $\\lambda_\\textrm{PS}$ values inferred from the fits are shown by the red curve in Fig.~\\ref{fig:lsmooth}. On the x-axis, we show the time since reionization, i.e. the time between the mean reionization redshift of the regions in a reionization redshift bin and the redshift, $z=4.8$, at which the power spectrum suppression is measured. As expected, the pressure smoothing scale increases with time since reionization as there is more time for the expansion of structures after their photoheating.\n\nWe next compare the measured pressure smoothing scale to different theoretical estimates such as the Jeans scale, the \\citet{1998MNRAS.296...44G} filtering scale, and the distance corresponding to a simple free expansion with a fixed starting velocity.  The Jeans scale is an instantaneous measure which corresponds to the scale at which the sound crossing time matches the free fall time. Structures below this scale are typically assumed to be suppressed. Here, we define the co-moving Jeans scale similar to equation (2) of \\citet{1998MNRAS.296...44G},\n\\begin{equation}\n  k_\\textrm{J} = \\frac{a}{c_\\textrm{s}} \\sqrt{4 \\pi G \\bar{\\rho}_\\textrm{m} \\Delta}\n\\end{equation}\nbut with an additional factor $\\Delta$ to allow evaluation at different densities in units of the mean density. Here $c_\\textrm{s} = \\sqrt{(5 k T)\/(3 \\bar{m})}$ is the sound speed, with $T$ being the median temperature of a region, $k$ the Boltzmann constant and $\\bar{m}$ the mean particle mass. $\\bar{\\rho}_\\textrm{m}$ is the mean physical matter density at the considered redshift, and $a$ is the scale factor.\n\nIn linear perturbation theory, this should be evaluated at $\\Delta=1$, but has a clear meaning only if the temperature evolves like $T\\propto a^{-1}$ \\citep[see][]{1998MNRAS.296...44G}. For different values of $\\Delta$, this would correspond to matching sound crossing and free fall time at that density, but only in the absence of an expanding background. Here we are interested in low density regions, $\\Delta \\sim 0.3$, which often will expand even in the absence of a thermal pressure. It is thus not entirely clear how well motivated evaluating this at $\\Delta \\sim 0.3$ is. Nevertheless we show results for both, i.e. $\\Delta$ set to the mean density of a region and $\\Delta$ set to 1. We then average the Jeans length scale, $\\lambda_\\textrm{J} \\equiv k_\\textrm{J}^{-1}$, over all regions falling in a reionization redshift bin. Also, note that we have divided the Jeans length scales by a factor of 3 in Fig.~\\ref{fig:lsmooth}, so that the curves fit better onto the plot.\n\nIndependent of the choice of $\\Delta$, we find that the Jeans scale does not adequately reproduce the pressure smoothing scale measured from the simulation. This is not too surprising as an instantaneous measure cannot faithfully capture the time evolution of the pressure smoothing following an heating event. As we are measuring the pressure smoothing shortly after the end of reionization, when there was only limited time for the IGM to hydrodynamically react to the heating, we find that the Jeans scale is much larger than the measured pressure smoothing scale (keep in mind the division by 3). It also decreases, rather than increases, with the time since reionization as recently reionized regions are hot and have a correspondingly large Jeans scale.\n\nNext, we compute the \\citet{1998MNRAS.296...44G} filtering scale, which aims to capture the time evolution properly by taking the thermal history into account. The filtering scale $k_\\textrm{F}$ is given by their equation (6), i.e. by\n\\begin{equation}\n\\frac{1}{k^2_\\textrm{F}} = \\frac{1}{D_+(t)} \\int^t_0 dt' c^2_\\textrm{s}(t') D_+(t') \\int^t_{t'} \\frac{dt''}{a^2(t'')}, \\label{eq:filtering_scale}\n\\end{equation}\nwhere $D_+(t)$ is the growth function of linear perturbations. As we are considering high redshifts here, $z \\geq 4.8$, we simplify this by using $D_+ \\propto a$ and approximating the Hubble function by $H \\approx H_0 \\sqrt{\\Omega_\\textrm{m} a^{-3}}$, where $H_0$ and $\\Omega_\\textrm{m}$ are the usual $\\Lambda$CDM cosmological parameters. Using this, switching to $a' = a(t')$ as the integration variable and carrying out the inner integral, we can write the pressure smoothing scale as\n\\begin{equation}\n\\frac{1}{k^2_\\textrm{F}} = \\frac{2}{a H_0^2 \\Omega_\\textrm{m}}\n\\int^a_0 da' a'^{\\frac{3}{2}} \\left[\\frac{1}{\\sqrt{a'}} - \\frac{1}{\\sqrt{a}}\\right] \\,\nc^2_\\textrm{s}(a').\n\\end{equation}\nWe evaluate this for each region using the history of the median gas temperature, convert this to a filtering length scale $\\lambda_\\textrm{F} \\equiv k_\\textrm{F}^{-1}$, and then average over all regions within the considered reionization redshift bin. The results of this are shown by the solid gray line in Fig.~\\ref{fig:lsmooth}, which is overall in good agreement with the pressure smoothing scale measured from the simulation. This confirms that the \\citet{1998MNRAS.296...44G} filtering scale describes the pressure smoothing of the low-density IGM well after reionization, and captures its time evolution.\n\nFinally we compare the measured pressure smoothing scale to a simple expansion model in which structures expand freely after their photoionization\/heating. This length scale in comoving units is given by\n\\begin{equation}\n\\lambda_\\textrm{exp} = \\int \\frac{v_\\textrm{start} a_\\textrm{r}}{a(t)} \\frac{dt}{a(t)} = v_\\textrm{start} a_\\textrm{r} \\int_{a_\\textrm{r}}^a \\frac{da'}{a'^3 H(a')},\n\\end{equation}\nwhere $v_\\textrm{start}$ is the initial velocity right after reionization at $a_\\textrm{r}$ and the $a_\\textrm{r}\/a$ term takes care of the cosmological decay of peculiar velocities. This expansion scale is shown for $v_\\textrm{start} = 10 \\, \\textrm{km}\/\\textrm{s}$ in Fig.~\\ref{fig:lsmooth}. Maybe somewhat surprisingly this simplistic model is in quite good agreement with the simulation. Note that the corresponding smoothing kernel for the gas density field, $\\propto \\exp(-k^2 \\lambda_\\textrm{exp}^2)$, corresponds to a Gaussian with standard deviation $\\sigma = \\sqrt{2} \\lambda_\\textrm{exp}$ in position space. Thus, the standard deviation of the real space smoothing kernel roughly grows like the travel distance for a $\\sqrt{2} \\times 10 \\, \\textrm{km}\/\\textrm{s} \\approx 14 \\, \\textrm{km}\/\\textrm{s}$ starting velocity,  which is close to the speed of sound in a $\\sim 10^4 \\, \\textrm{K}$ ionized IGM.\n\nIn particular, this simple free expansion model seems to reproduce the measured smoothing well and even better than the filtering scale shortly after reionization, $\\lesssim 0.5 \\, \\textrm{Gyr}$. This suggests that the expansion of photoheated structures may not be strongly hindered by swept up material in the surrounding lower density regions in this time span. The lower level of agreement of the filtering scale may, however, be related to how the filtering scale is computed here. We use the whole thermal history of a region to compute the filtering scale. Part of a region will already be ionized and heated before the mean reionization redshift of the region is reached. Hence, despite using the median temperature, a region may already have a non-zero filtering scale at its mean reionization redshift, i.e. at a time of zero in Fig.~\\ref{fig:lsmooth}. Such an offset could then still have a notable effect at somewhat later times. Furthermore, we get the median temperature history not directly from 3D grids but by using all pixels of a set of 5000 lines-of-sight through the simulation box that fall into a region. This gives us better time resolution ($\\Delta z = 0.1$) as the line-of-sight files were saved more frequently than the full snapshots. It introduces, however, some noise which may contribute to such an offset. We have checked that using the line-of-sight file temperatures gives essentially the same result for the Jeans scale as the full temperature field, suggesting that the impact of this procedure on the filtering scale should also be small.  \n\nOverall both the \\citet{1998MNRAS.296...44G} filtering scale and the simple expansion model match the measured pressure smoothing in the underdense IGM well, while the Jeans scale is clearly inadequate shortly after reionization.\n\n\\subsubsection{Lyman-$\\alpha$\\ lines due to pressure-smoothed structures}\n\\label{sec:lya_lines_rings}\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=0.9\\linewidth]{figs\/dens_map_spectra.pdf}\n  \\caption{Expansion of gas overdensities due to the heating provided by reionization and their imprint on Lyman-$\\alpha$\\ forest absorption lines. The top panel displays the gas density in a thin slice in our $z_\\textrm{r}=5.3$ patchy simulation at $z=4.2$. Also indicated are five lines-of-sight (LOS A-E) for which we explore the Lyman-$\\alpha$\\ transmission. The five other sets of panels show the density (in units of the mean) and the normalized transmitted flux for these five lines-of-sight. In the top panel various ``rings'' are visible. These appear when the slice cuts through expanding spherical or cylindrical shells that are produced when overdensities are evaporated by the photoheating provided by reionization. In the density skewers, these ``rings'' are visible as pairs of density peaks. Some of them are marked by gray dotted lines, which also connect to the corresponding redshift-space positions in the Lyman-$\\alpha$\\ spectra, i.e. the transmitted flux panels. There, isolated ``rings'' appear as absorbers with two minima (see, e.g., LOS A) or with a flat bottom (LOS B). ``Rings'' in regions with various other features in the density field can, e.g., appear as small dips in the larger scale features of the transmitted flux (e.g., LOS C-E). Such imprints can in principle be used to constrain pressure smoothing and reionization.}\n  \\label{fig:rings_specs}\n\\end{figure*}\n\nIn Fig.~\\ref{fig:slice_pressure_smooth_regions}, we have seen how pressure smoothing puffs up photoheated structures. This can result in shell-like features (visible as ``rings'' in thin 2D slices). Such features are also present in various other SPH and grid-based hydrodynamic simulations (see, e.g., fig.~9 in \\citealt{KulkarniEtAl2015}, the highest resolution panel in fig.~8 of \\citealt{2015MNRAS.446.3697L}, figs.~1 and 2 in \\citealt{DAloisioEtAl2020}, or fig.~6 in \\citealt{2021ApJ...923..161N}), but have received only limited attention so far. Here, we investigate to what extent such shells and puffed up gas clouds leave a noticeable imprint in the Lyman-$\\alpha$\\ forest. The top panel of Fig.~\\ref{fig:rings_specs} displays the gas density in a thin slice through part of our $z_\\textrm{r} = 5.3$ patchy simulation at $z=4.2$. Various shell-like features, often visible as rings in this 2D slice, are present. To investigate their impact on Lyman-$\\alpha$\\ absorption, we shoot several lines-of-sight through the slice (dotted lines, labelled LOS A to E), and calculate synthetic Lyman-$\\alpha$\\ forest spectra for them. For each line of sight, we show a twin panel in Fig.~\\ref{fig:rings_specs}, with the lower part showing the normalized transmitted Lyman-$\\alpha$\\ flux and the upper part showing the gas density in units of the mean baryon density. We have rescaled the optical depths by a constant factor to make the mean transmitted flux consistent with observed values (using Eq.~\\ref{eq:tau_eff_fit} and computing the rescaling factor based on our full sample of 5000 lines-of-sight). The small dotted lines in the twin panels connect real-space positions of features in the density field to the corresponding redshift space position in the mock spectra. This facilitates identifying the associated absorption features.\n\nLOS A passes through a ``ring'' in the density slice at $x\\approx 16.3$ cMpc\/$h$. A corresponding double peak is clearly visible in the density profile along the line-of-sight at that location (upper panel of the uppermost twin panel). The two peaks are marked by dotted lines. In redshift space, they correspond to a double-dip absorption  feature, which in this case directly reflects the shell traversed by the line-of-sight. Interestingly, the peak\/dip separation is larger in redshift than in real space. This indicates that the shell is, as expected, expanding.\n\nLOS B passes through a ``ring'' at $x\\approx 18.3$ cMpc\/$h$. Again, the double peak in the corresponding line-of-sight density profile is easily identified. In this case, the absorption features of the two peaks are more strongly overlapping, resulting in a broad flat bottom absorption profile in the mock spectrum. \n\nLOS C passes through a ``ring'' at $x\\approx 17.8$ cMpc\/$h$. While in the previous cases the double peak in the density profile was rather isolated, it falls near other structures here. This results in two absorption dips that fall on a larger scale gradient in the transmitted flux fraction. In an actual observation, this would likely make it more difficult to infer that these two dips originate from a single shell.\n\nLOS D passes a less pronounced ``ring'' at $x\\approx 15.7$ cMpc\/$h$, and with several other shells and a larger structure nearby. Two corresponding small peaks can still be identified in the line-of-sight density profile, but the corresponding absorption features fall near a larger saturated absorber and are only visible as a slight change in the curvature of the spectrum. This would likely go unnoticed even in a high signal-to-noise observation.\n\nLOS E passes through two ``rings'' at $x\\approx 16.1$ and $16.4$ cMpc\/$h$. Four corresponding peaks are visible in the line-of-sight density profile. Three of these cause visible dips in the synthetic spectrum. In an observation, it would, however, be difficult to identify which dips originate from the same shell, making inferring any pressure smoothing hard.\n\nOverall, we find that shells caused by photoheated expanding structures often leave a direct imprint in the Lyman-$\\alpha$\\ forest, with distinct absorption dips at the shell ``walls''. In the most distinct case of a single, isolated, roughly spherical\/cylindrical shell an intriguing double-dip absorption profile is imprinted on the Lyman-$\\alpha$\\ forest spectrum (see LOS A). Measuring, e.g., the separation of the two dips should in principle allow inferring the amount of pressure smoothing that the object has experienced. In practice, it may be difficult to identify which absorption features correspond to a shell originating from the same structure (see, e.g., LOS D and E), as well as what the exact orientation of the line-of-sight with respect to the shell is. An analysis to infer the pressure smoothing based on such features would hence likely require a larger sample of absorption systems along with a tailored statistical technique tested on simulations. It would also be important to check how well different hydrodynamics schemes agree on the prominence and properties of such shells. Finally, observations would need to be done at a suitable redshift at which the Lyman-$\\alpha$\\ forest is sensitive to the typical densities of such shells. The redshift considered here, $z=4.2$, seems to work reasonably well for this.   \n\n\\section{Summary and conclusions}\n\\label{sec:summary}\n\nWe have presented the Sherwood-Relics simulations, a new suite of cosmological hydrodynamical simulations aimed at modelling the IGM during and after cosmic reionization. The main difference to our previous Sherwood simulation project, which we build on in this work, is an improved treatment of the ionizing UV radiation field and of the thermo-chemistry of the IGM. \n\nOur new simulation sample consists of over 200 runs covering cubic volumes with sidelengths ranging from 5 to 160 cMpc\/$h$. These are populated with between $2\\times512^3$ and $2\\times2048^3$ particles. Most of the simulations use an updated time-dependent but spatially homogeneous UV background model along with a non-equilibrium thermo-chemistry solver. These runs cover a wide range in thermal evolutions, cosmological parameters, dark matter free streaming scales and reionization histories, and will be instrumental for deriving constraints on these properties from Lyman-$\\alpha$\\ forest observations. \n\nThe main focus of the analysis presented in this work is, however, the impact of a more realistic patchy cosmic reionization process on the properties of the IGM during the epoch of reionization, as well as its \\textit{relic} signatures that persist for a considerable amount of time in the post-reionization IGM, such as spatial fluctuations in the IGM temperature, ionization state and small scale structure. To this end, we have developed a new hybrid radiative transfer\/cosmological hydrodynamical simulation technique that allows following an inhomogeneous cosmic reionization process as well as the associated heating and pressure smoothing. The scheme uses radiation fields from post-processing radiative transfer simulations to photoionize and photoheat the IGM in a subsequent cosmological hydrodynamical simulation that then also captures the hydrodynamic response to the heating. This approach is suitable for the IGM,  computationally relatively cheap and circumvents the challenges of a full hydrodynamical modelling of the source galaxy population.\n\nWe assess the impact of the patchiness of reionization by comparing such ``patchy'' runs to homogeneous UVB simulations with the same mean reionization and thermal history. Our main findings are:\n\\begin{itemize}\n\\item Consistent with previous work, patchy reionization seeds IGM temperature fluctuations on large scales that persist well into the post-reionization epoch, down to $z\\approx4$.\n\\item These temperature fluctuations are closely related to the reionization redshift, with late reionizing regions being hotter.\n\\item The ionization state of the IGM reflects these temperature fluctuations. This causes a modulation of the Lyman-$\\alpha$\\ forest transmitted flux on large scales and a corresponding increase in the Lyman-$\\alpha$\\ forest power spectrum at $k\\lesssim 10^{-2}$ s\/km.\n\\item Patchy reionization also leads to a spatially varying pressure smoothing of the IGM. This results in spatial fluctuations in the amount of small scale density structure that is present in the IGM,  with early reionizing regions exhibiting the least amount of such structures.\n\\item Following reionization, the pressure smoothing length scale as a function of time since reionization is well described by the \\citet{1998MNRAS.296...44G} filtering scale in the very low-density IGM. A simplistic free expansion model with an appropriate starting velocity also provides a reasonable fit. The instantaneous Jeans scale is instead not suitable for quantifying pressure smoothing shortly after reionization.\n\\item Pressure smoothing puffs up or evaporates small IGM structures such as filaments and small haloes. This often results in shell-like features in the IGM density field and can leave characteristic imprints in the Lyman-$\\alpha$\\ forest, such as flat-bottom or double-dip absorption profiles.\n\\end{itemize}\n\nThese various impacts of the patchiness of cosmic reionization on the IGM and the high-redshift Lyman-$\\alpha$\\ forest should be taken into account when interpreting precision studies based on Lyman-$\\alpha$\\ forest data, in particular when using measurements at $z\\gtrsim4$. A relatively simple way of doing this is to extract correction factors from patchy reionization simulations that can then be applied to (grids of) conventional homogenous UVB simulations \\citep[see, e.g.,][]{MolaroEtAl2022}. Alternatively, given the rather low computational cost of our hybrid patchy reionization simulation technique, tailored simulations can be performed to aid the interpretation of particular datasets.\n\nThe signatures of patchy reionization seen in our simulations may also be interesting for more accurately constraining the cosmic reionization process, e.g., by measuring the large scale increase of the Lyman-$\\alpha$\\ forest power spectrum as a function of redshift, or by analysing characteristic imprints of photoheated structures on the Lyman-$\\alpha$\\ forest. The latter would likely require the development of a suitable statistical technique to quantitatively compare such features between simulations and observations. The former would rely on more accurate measurements of the Lyman-$\\alpha$\\ forest power spectrum on large scales and near the epoch of reionization.\n\n\\section*{Acknowledgements}\nWe thank Volker Springel for making \\textsc{p-gadget3} available and Debora Sijacki for helpful discussions. We acknowledge the Partnership for Advanced Computing in Europe (PRACE) for awarding us access to the Curie and Irene supercomputers, based in France at the Tr\\`es Grand Centre de calcul du CEA, during the 16th Call.  The Sherwood-Relics simulations were also performed using time allocated during the Science and Technology Facilities Council (STFC) DiRAC 12th Call.  We used the Cambridge Service for Data Driven  Discovery (CSD3),  part of which is operated by the University of Cambridge Research Computing on behalf of the STFC DiRAC HPC Facility  (www.dirac.ac.uk).  The  DiRAC component of CSD3 was funded by BEIS capital funding via STFC capital grants ST\/P002307\/1 and ST\/R002452\/1 and STFC operations grant ST\/R00689X\/1.  This work also used the DiRAC@Durham facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility. The equipment was funded by BEIS capital funding via STFC capital grants ST\/P002293\/1 and ST\/R002371\/1, Durham University and STFC operations grant ST\/R000832\/1. DiRAC is part of the National e-Infrastructure. JSB and MM are supported by STFC consolidated grant ST\/T000171\/1.  Laura Keating was supported by the European Union's Horizon 2020 research and innovation programme under the Marie Sk\u0142odowska-Curie grant agreement No. 885990. GK is partly supported by the Department of Atomic Energy (Government of India) research project with Project Identification Number RTI 4002, and by the Max Planck Society through a Max Planck Partner Group. T\u0160 is supported by the University of Nottingham Vice Chancellor's Scholarship for Research Excellence (EU).  MV is supported by the INFN grant INDARK PD51 and by ASI grant under grant agreement ASI-INAF n. 2017-14-H.0. VI is supported by the Kavli Foundation. GDB is supported by the National Science Foundation through grant AST-1751404. Part of this work was supported by FP7 ERC Grant Emergence-320596. \n\n\\section*{Data Availability}\nThe data and analysis code used in this work are available from the authors on request.  Further guidance for accessing  the publicly available Sherwood-Relics simulation data can be found on the project website:  \\url{https:\/\/www.nottingham.ac.uk\/astronomy\/sherwood-relics\/}\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n Thom Fr\\\"uhwirth presented a short, elegant and efficient Prolog program for\n the $n$ queens problem \\cite{Fruehwirth91}.\n However the program may be seen as rather tricky and one may be not convinced\n about its correctness.   The author's description is rather operational.\nSo it should be useful to explain the program declaratively, and to provide\nformal proof that it is correct.\n\nIn imperative and functional programming, program correctness implies that\nthe program produces the ``right'' results.  \nIn logic programming, which is nondeterministic, the situation is different. \nOne also needs the program to be complete, i.e.\\ to produce all\nthe results required by the specification.  (In particular, the empty program\nproducing no answers is correct whatever the specification is.)\n\nThis paper provides proofs of correctness and\ncompleteness of the $n$ queens program; the proofs are declarative,\ni.e.\\ they abstract from any operational semantics. \n\nThe paper is organized as follows.  After technical preliminaries, \nSection \\ref{section.program} presents the $n$ queens program together \nwith an informal description of its declarative semantics.\nThe next section presents a formal specification.\nProofs of correctness and completeness of the program \nare subjects of, respectively, \nSections \\ref{sec.correctness.proof} and \\ref{sec.completeness.proof}.\nThe last section concludes the paper.\nA version of this paper with slightly abridged initial sections has been\nsubmitted to Theory and Practice of Logic Programming.\n\n\n\n\\section{Preliminaries}\n\\label{sec.preliminaries}\n\\paragraph{Basics.}\nThis paper considers definite clause logic programs.\nWe employ the standard terminology and notation \\cite{Apt-Prolog},\nand do not repeat here standard definitions and results.\nWe assume a fixed alphabet of function and predicate symbols.\nThe Herbrand universe will be denoted by {\\ensuremath{\\cal{H U}}}\\xspace, the Herbrand base by {\\ensuremath{\\cal{H B}}}\\xspace,\nand the set of all terms (atoms) by {\\ensuremath{\\cal{T U}}}\\xspace (respectively {\\ensuremath{\\cal{T B}}}\\xspace\\\/);\n${\\ensuremath{\\cal{H B}}}\\xspace_p$ is the set of ground atoms with the predicate symbol $p$.\nBy {\\ensuremath{\\mathbb{N}}}\\xspace we denote the set of natural numbers.\nWe sometimes do not distinguish a number $i\\in{\\ensuremath{\\mathbb{N}}}\\xspace$ from its representation\nas a term, $s^i(0)$. \nWe use the list notation of Prolog. \nWe assume that $[\\seq e|e]$ stands for $e$ when $n=0$.\nA list (respectively an open list) of\nlength $n\\geq0$ is a\nterm $[\\seq e]\\in{\\ensuremath{\\cal{T U}}}\\xspace$ ($[\\seq e|v]\\in{\\ensuremath{\\cal{T U}}}\\xspace$), where $v$ is a\nvariable; $\\seq e$ are the members of the (open) list.\nWe generalize the latter notion, and say that $e\\in{\\ensuremath{\\cal{T U}}}\\xspace$ is a\n{\\bf member} of a term $t\\in{\\ensuremath{\\cal{T U}}}\\xspace$ if $t=[\\seq[k-1]e,e|e']$ (for some terms\n$\\seq[k-1]e,e'$, where $k>0$).\nIn such case we also say that $e$ is the $k$-th member of $t$.\nNote that this kind of membership is defined by the Prolog built-in predicate\n{\\tt member}\/2.\n\n\\comment{\n   So what is a predicate for a logician, is a procedure for a programmer.\n\n   So a predicate from the point of\n   view of logic is a procedure from the point of view of a programmer.\n}\n\n\nFollowing \\citeN{Apt-Prolog},\nwe use queries (conjunctions of atoms) instead of goals.\nBy an {\\bf answer} of a program $P$ we mean any query $Q$ such\nthat $P\\models Q$.  So an answer is a query to which a computed or correct\nanswer substitution has been applied; \\citeN{Apt-Prolog} calls it computed\/correct\ninstance of a query.\n(Due to soundness and completeness of SLD-resolution, it does not matter\nwhether correct or computed answer substitutions are considered here.)\n${\\ensuremath{\\cal M}}\\xspace_P$ stands for the least Herbrand model of a program $P$.\nBy the {\\em relation} defined by a predicate $p$ in $P$ we mean\n$\n{\\ensuremath{\\cal R}}\\xspace_P(p) =\n\\{\\,\\vec t\\,\\in{\\ensuremath{\\cal{T U}}}\\xspace^n \\mid P\\models p(\\vec t\\,) \\,\\},\n$\nwhere $n$ is the arity of $p$.\n\\comment{%\nBy the {\\em ground relation} defined by $p$ in $P$ we mean\n${\\ensuremath{\\cal R}}\\xspace_P(p) \\cap {\\ensuremath{\\cal{H U}}}\\xspace^n$.\n}\n\\comment{\n(we may omit the subscript $P$ when this does not lead to ambiguity).\n}\n\n\\paragraph{Specifications.}\n\nIn this paper,\nthe treatment of specifications and reasoning about correctness and\ncompleteness follows that of \\cite{Drabent.tocl16};\nmissing proofs and further explanations can be found there.\nFor further discussion, examples and references, see also\n(\\citeNP{Drabent.tplp18}; \\citeNP{DBLP:journals\/tplp\/DrabentM05shorter}).\n\nBy a {\\bf specification} we mean an Herbrand interpretation $S\\subseteq{\\ensuremath{\\cal{H B}}}\\xspace$.\nA program $P$ is {\\bf correct} w.r.t.\\ a specification $S$ when\n${\\ensuremath{\\cal M}}\\xspace_P\\subseteq S$.  This implies that $S\\models Q$ for any answer $Q$ of $P$.\nA program $P$ is {\\bf complete} w.r.t.\\ $S$ when  $ S\\subseteq{\\ensuremath{\\cal M}}\\xspace_P$.\nThis implies that, for any ground query $Q$, if $S\\models Q$ then $Q$\nis an answer of $P$.\nSo $Q$ is an instance of an answer in each SLD-tree for $P$ and any query $Q_0$\nmore general than $Q$.\n\n\nDealing with the $n$ queens program we face a usual phenomenon:\nOften it is inconvenient (and unnecessary) to specify ${\\ensuremath{\\cal M}}\\xspace_P$ exactly, i.e.\\ to\nprovide a \nspecification $S$ for which the program is both correct and complete, \n$S={\\ensuremath{\\cal M}}\\xspace_P$.\nIt is useful to use instead an {\\bf approximate specification}, which is a pair\n$S_{\\it c o m p l}, S_{corr}$ of specifications for, respectively,\ncompleteness and correctness.\nWe say that  \na program $P$ is {\\em fully correct} w.r.t.\\ \n$S_{\\it c o m p l}, S_{corr}$ when \n$S_{\\it c o m p l}\\subseteq{\\ensuremath{\\cal M}}\\xspace_P\\subseteq S_{corr}$.\n\n\\commenta{\n The choice of an approximate specification depends on the properties of\n interest. \n  See for instance\n. . . \n  for various specifications for {\\sc append}\n  describing various properties of the program.\n(. . . %\n  is a version of this paper with extended introductory sections.)\n}\n\n The choice of an approximate specification depends on the property of interest.\n As an example take the standard {\\sc append}\\xspace program \\cite{Apt-Prolog}.\n %\n It does not define the list appending relation, but its certain\n superset  (as the program has answers with two arguments not being lists).\n %\n To be sure that the first argument of $app$ is a list, it is sufficient to prove\n correctness of the program w.r.t.\\\n $S_1=\\{\\, app(s,t,u)\\in{\\ensuremath{\\cal{H B}}}\\xspace \\mid s \\mbox{ is a list}\\,\\}$.\n Correctness w.r.t.\\ \n %\n \\[\n S_2=\\left\\{\\, app(s,t,u)\\in{\\ensuremath{\\cal{H B}}}\\xspace \\: \\left|\n \\begin{array}{l@{}}\n                 \\mbox{if } u=[\\seq e] \\mbox{ then for some } i\\in\\{0,n\\}\n                 \\\\\n                 s=[\\seq[i]e],\\ t=[e_{i+1},\\ldots,e_n]\n %\n \\end{array}\n \\right.\\right\\}\n \\]\n %\n implies that {\\sc append}\\xspace correctly splits a list given as the third argument \n of $app$.\n %\n (If the third argument is a list then the first two ones are a result of \n  splitting the list.)\n See \\cite[(2) p.\\,672 and Ex.\\,3.4]{DBLP:journals\/tplp\/DrabentM05shorter}\n for two further specifications for correctness of {\\sc append}\\xspace.\n %\n The first one describes appending lists; the second, $S_{\\rm APPEND}$,\n %\n deals both with splitting and appending lists, and states that the first\n argument is a list.\n\n\n\n To be sure that the program will append any two lists, and split\n any list (of length $n$) in all possible ($n+1$) ways, we establish its\n completeness w.r.t.\\ \n \\[\n S_{\\it c o m p l} = \n \\left\\{\\,\n %\n  app( [\\seq[m]e], [\\seq {e'}], [\\seq[m]e,\\seq {e'}] ) \\in{\\ensuremath{\\cal{H B}}}\\xspace \\mid m,n\\in{\\ensuremath{\\mathbb{N}}}\\xspace\n \\,\\right\\}.\n \\]\n If we are interested only in lists of even length then we may consider\n completeness w.r.t.\\ an appropriate subset of this specification.\n Without getting into details, we \n note that the least Herbrand model ${\\ensuremath{\\cal M}}\\xspace_{\\rm APPEND}$ is distinct from all\n these specifications, and that {\\sc append}\\xspace is complete w.r.t.\\ $S_{\\it c o m p l}$\n and correct w.r.t.\\ $S_1, S_2$ and $S_{\\rm APPEND}$.\n %\n We have\n $S_{\\it c o m p l} \\subset {\\ensuremath{\\cal M}}\\xspace_{\\rm APPEND} \\subset S_{\\rm APPEND} \\subset S_2\n \\not\\subset S_1\n $ \n and $S_{\\rm APPEND} \\subset S_1 \\not\\subset S_2$.\n\n\n\nFor a discussion on building approximate specifications\nsee \\hyperref[sec.comments]{\\em Comments} at the end of\nSection~\\ref{section.program}.\n\n\n\n\n\n\\paragraph{Proving program correctness.}\nAn obvious way to prove correctness is to use the following sufficient\ncondition.\nAccording to \\citeN{DBLP:journals\/tcs\/Deransart93}, \nthe condition is due to \\citeN{Clark79}.\n\n\\begin{theorem}\n\\label{th.correctness}\n    For a program $P$ and a specification $S$,\n    if  $S\\models P$  then  $P$ is correct w.r.t.\\ $S$.\n\\end{theorem}\n\\vspace{-1.2\\topsep}\n\\nopagebreak\n\\begin{proof}\n As $S$ is an Herbrand model of $P$, the least Herbrand model of $P$ is a\n subset of $S$.\n\\end{proof}\n\nAs $S$ is an Herbrand interpretation, \\,$S\\models P$ means that\n     for each ground instance \n    \\mbox{$    H\\gets \\seq B    $}\n    ($n\\geq0$)\n    of a clause of $P$,\n     if $\\seq B\\in S$ then $H\\in S$.\n\nSo, informally speaking,\nfor correctness of a program it is sufficient that each its clause out of\ncorrect atoms (i.e.\\ those in the specification) produces only correct ones.\n\n     \n\n\n\\paragraph{Proving program completeness.}\nInformally, for completeness of a program w.r.t.\\ $S$ it is necessary that\neach atom $A\\in S$ can be produced by some clause of $P$ (out of atoms\nproduced by $P$).\nMoreover, $A$ should be produced in a finite way.\nTo formalize this idea we introduce some auxiliary notions.\n\n\\begin{definition}\n  A ground atom $H$ is\n  {\\bf covered} {by a clause} $C$ w.r.t.\\ a specification $S$\n  if $H$ is the head of a ground instance  \n  $\n  H\\gets \\seq B\n  $\n  ($n\\geq0$) of $C$, such that $\\seq B\\in S$\n  \\cite{Shapiro.book}.\n\n  A ground atom $H$ is {\\em covered by a program} $P$ w.r.t.\\ $S$\n  if it is covered w.r.t.\\ $S$ by some clause $C\\in P$.\n\\end{definition}\n\n\n\\begin{definition}\n    A {\\bf level mapping} is a function $|\\ |\\colon{\\ensuremath{\\cal{H B}}}\\xspace\\to{\\ensuremath{\\mathbb{N}}}\\xspace$.\n    %\n    %\n    A program $P$ is {\\bf recurrent} w.r.t.\\ a level\n    mapping $|\\ |$ \\cite{DBLP:journals\/jlp\/Bezem93} \n    %\n    when, for each ground instance $H\\gets\\seq B$ ($n\\geq0$) of a clause of $P$\n    and each  $i\\in\\{1,\\ldots,n\\}$, \n    we have $|H|>|B_i|$.\n\\end{definition}\n\nThe following sufficient condition is an immediate corollary \nof \\cite[Theorem 5.6 and Proposition 5.4]{Drabent.tocl16}\nor of \\cite[Theorem 6.1]{Deransart.Maluszynski93},\nand is sufficient for the purpose of this paper.\n\\comment{\n Th. 5.6 (semicompleteness), Proposition 5.4 (completeness)\n}\n\n\\begin{lemma}\n\\label{lemma.completeness}\nLet $P$ be a program, and $S$ a specification.  If \neach atom $A\\in S$ is covered by $P$ w.r.t.\\ $S$, and\n$P$ is recurrent then $P$ is complete w.r.t.\\ $S$.\n\\end{lemma}\n\n\\paragraph{A note on built-ins.}\n\\label{sec.built-ins}\nThe presented approach can be generalized in a rather obvious way to Prolog with\nsome built-ins.  We focus here on Prolog arithmetic.\nA program $P$ using arithmetic predicates (like {\\tt is}\/2, or {\\tt>}\/2)\ncan be understood as augmented with an infinite set \n$P(Ar)\\subseteq{\\ensuremath{\\cal{H B}}}\\xspace$ of unit clauses\ndefining the ground instances of arithmetic relations \\cite{Apt-Prolog}.\nSuch clauses are e.g.\\,(in the infix form) \n$4\\mathop{\\tt is}2{+}2$, and \n$2{+}2\\mathop{\\mbox{\\tt<}}7$.\nTo deal with correctness or completeness of such program, we assume that \nthe specification is augmented with $P(Ar)$\n(more precisely, the specification is\n$S\\cup P(Ar)$ where no arithmetic predicate occurs in $S$).\nWe also assume that $|B|=0$ for each $B\\in P(Ar)$.\nNow the sufficient conditions for correctness and completeness apply.\n(As they are obviously satisfied by $P(Ar)$,\nthe condition for correctness\nneeds to be checked only for the clauses from $P$, and that for completeness\nonly for atoms from $S$.)\n\n\n\nThis approach abstracts from run-time errors.  So completeness w.r.t.\\ $S$\nmeans that\nif $S\\models Q$ and $Q$ is a ground instance of a query $Q_0$\nthen $Q$ is an instance of an answer of a Prolog computation starting with\n$Q_0$, unless a run-time error or infinite loop is encountered.\n\n\n\n\\section{The $n$ queens program}\n\\label{section.program}\n\nThis section presents the $n$ queens program of\n\\citeN{Fruehwirth91}, and provides its informal declarative description.\nPossible inaccuracies due to informal approach will be corrected in the next\nsections, dealing with a formal specification and proofs.\n\nThe problem is to place $n$ queens on an $n\\times n$ chessboard so that there\nis exactly one queen in each row and each column, and at most one queen in\neach diagonal. \n The main idea of the program is to describe the placement of the queens by a\ndata structure in which it is impossible that two queens are placed on the\nsame row, column or a diagonal.%\n\\footnote{%\nActually, some of the diagonals are not dealt with.  This issue is clarified\nlater on.\n}\nIn this way the constraints of the\nproblem are treated implicitly and efficiently.\n\nThis paper considers\nthe version of the program which represents natural numbers as terms in a\nstandard way.\nAnother version employs Prolog arithmetic.\nThe specifications and proofs of Sections\n\\ref{sec.spec} -- \\ref{sec.completeness.proof} can be, in a rather obvious\nway, transformed to ones dealing with the latter version,\nfollowing \\hyperref[sec.built-ins]{\\em A\\,note\\,on\\,built-ins}\\\/ from the previous section.\n\nHere is the main part of the program (with predicate names abbreviated);\nit will be named {\\sc nqueens}\\xspace.%\n\n\\vspace{\\abovedisplayskip}\n\\noindent\n\\mbox{}\\hfill%\n\\begin{minipage}[t]{.8\\textwidth}\n{\\small %\n\\begin{verbatim}\n    pqs(0,_,_,_).\n    pqs(s(I),Cs,Us,[_|Ds]):-\n            pqs(I,Cs,[_|Us],Ds),\n            pq(s(I),Cs,Us,Ds).\n\n   \n    pq(I,[I|_],[I|_],[I|_]).\n    pq(I,[_|Cs],[_|Us],[_|Ds]):-\n            pq(I,Cs,Us,Ds).\n\\end{verbatim}\n}\n\\end{minipage}%\n\\hfill\n \\begin{minipage}[t]{.035\\textwidth}\n\\raggedleft\n\\small%\n(\\refstepcounter{equation}\\theequation\\label{clause1})\n\\\\ \\ \\\\\n       \\refstepcounter{equation}%\n      (\\theequation\\label{clause2})%\n\\\\ \\ \\\\ \\ \\\\ \\ \\\\\n       \\refstepcounter{equation}%\n      (\\theequation\\label{clause3})\n\\\\[1.5ex]\n       \\refstepcounter{equation}%\n        {(\\theequation\\label{clause4})}\n\\end{minipage}%\n\\vspace{\\belowdisplayskip}\n\\\\\nSolutions to the $n$ queen problem are provided by those answers of {\\sc nqueens}\\xspace\nthat are of the form ${\\it p q s}(n,q,t_1,t_2)$,\nwhere $n$ is a number and $q$ a list of length $n$.\nA number $i$ being the $j$-th member of list $q$ means that the queen\n    of row $i$ is placed in column $j$.\n(The role of $t_1,t_2$ will be explained later.)\n    So to obtain the solutions, one can use a query\n    ${\\it p q s}(n,q_0,\\mbox{\\tt\\symbol{95}},\\mbox{\\tt\\symbol{95}})$,\n    where $q_0$ is a list of $n$ distinct variables.\n\n\nWe quote the original description of the program, as it is an example of \nnon declarative viewing of logic programs:\n\\begin{quotation}\n  Observing that no two queens can be positioned on the same row, column\n  or diagonals, we place only one queen on each row. Hence we can identify\n  the queen by its row-number. Now imagine that the chess-board is divided\n  into three layers, one that deals with attacks on columns and two for the\n  diagonals going up and down respectively. We indicate that a field is\n  attacked by a queen by putting the number of the queen there.\n\n  Now we solve the problem by looking at one row at a time, placing one queen\n  on the column and the two diagonal-layers. For the next row\/queen we use\n  the same column layer, to get the new up-diagonals we have to move the \n  layer one field up, for the down-diagonals we move the layer one field down.\n\\end{quotation}\nThis does not have much to do with the logic of the program; \nin particular the relations defined by the program are not described.\nInstead, actions of the program are described\n(and its data structures outlined).\nAlso, the description does not seem to justify why the program is correct.  \n\\comment{\n Fruehwirth spells up-diagonals, down-diagonals \n}\n  Let us try to treat the program\ndeclaratively, abstracting from the operational semantics.\n\n\\addtolength{\\textfloatsep}{-1ex}\n\\addtolength{\\textfloatsep}{-1ex}\n\n\\paragraph{Chessboard representation.}\nAssume that columns and rows of the chessboard are numbered from the left\/top.\nEach queen is identified by its row number.\nDiagonals intersecting a given row $i$ are numbered from the left\n(Fig.\\,\\ref{figure.diagonals}).\nIn contrast to the numbering of rows and columns, \nthis numbering is not fixed, it is \nspecific to the context of the currently considered row;\nthe diagonal number $j$ includes the $j$-th field of the row.\nThus, in the context of row number $i$, its queen $i$ is\nin the column and in the up and down diagonals of the same number.\n\n\nTo avoid ambiguity, let us state that \nby an up diagonal (resp.\\ down diagonal) we mean the set of fields for which the\nsum (the difference) of its row and column number is the same.\nGiven a set $A\\subseteq{\\ensuremath{\\mathbb{N}}}\\xspace$ of queens, \nby a {\\bf correct placement} of queens $A$ on a chessboard we mean one in which\neach row, column, up diagonal and down diagonal contains at most one queen\nfrom $A$.\n\n\n\\begin{figure}\n\\vspace*{-1ex}\n\\vspace*{-1ex}\n\\[\n  \\includegraphics\n        [trim = 4.4cm 18.2cm 3.7cm 5.5cm, clip, scale=.95] \n        %\n        %\n  {diagram}%\n\\vspace*{-1ex}\n\\]\n  \\caption[just a trick]{%\nNumbering of rows and columns.  Numbering of diagonals in the\n    context of row   $i$ (left), and $i+1$ (right).\n\nThe board with two queens is represented in the context of row $i$ as follows:\nthe columns by  $[i,1|\\ldots]$, the up diagonals by  $[i|\\ldots]$,\nthe down diagonals by $[i,\\ldots,1|\\ldots]$ (where $1$ is the member number\n$i+1$).\nUp diagonals with non-positive numbers are not represented. %\n}\n \\figrule\n\\label{figure.diagonals}\n\\end{figure}\n\n\nWhen the initial query is as described above, the program \nrepresents a placement of queens by a list and two open lists.\nGenerally, the placement of queens on the columns, up diagonals\nand down diagonals is represented by respectively three terms of the form\n$[\\seq[k]t|t]$.\nIf the column (respectively up diagonal, down diagonal) $j$, where $j>0$,\ncontains the queen $i$ then the $j$-th member of\nthe respective term \nis the number $i$.  \nIf it contains no queen \nthen the $j$-th member of the term is arbitrary%\n \\addtolength\\footnotesep{-1ex}%\n \\footnote{%\n %\n It is a variable when the initial query is sufficiently general.\n }\nor does not exist.\nSuch representation guarantees that at most one queen can be\nplaced in each column and each diagonal of a positive number.\n\n\n\n\\vspace{-.8ex}\n\\paragraph{Relations defined by {\\sc nqueens}\\xspace, rationale.}\nNow we informally describe the purpose of the predicates of the program.\nThe role of ${\\it p q}$ is to define  (a relation consisting of)\ntuples $(i, cs, us, d s )\\in{\\ensuremath{\\cal{T U}}}\\xspace^4$,\nwhere\n\\begin{equation}\n\\label{spec.attempt.pq}\n\\mbox{for some $j>0$, $i$ is the $j$-th member of each $cs,us,d s$.}\n\\vspace{-1ex}\n\\end{equation}\nThe role of ${\\it p q s}$ is to define\ntuples $(i, cs, us, [t|d s] )\\in{\\ensuremath{\\mathbb{N}}}\\xspace\\times{\\ensuremath{\\cal{T U}}}\\xspace^3$\nsuch that\n\\begin{equation}\n\\label{spec.attempt.pqs}\n\\begin{oldtabular}{l@{}}\n $i>0$ and\n$cs$,\\,$us$,\\,$d s$ represent (as described above) \n\\\\\na correct placement of queens $1,\\ldots,i$\non, respectively, \\\\\nthe columns, up diagonals and down diagonals,\n\\\\\nwhere the diagonals are\nnumbered in the context of row $i$,\n\\end{oldtabular}\n\\end{equation}\nand additionally the tuples from $\\{0\\}\\times{\\ensuremath{\\cal{T U}}}\\xspace^3$.\n\n\nNow we understand, for instance,\n why in clause (\\ref{clause2}) the third argument of  \n${\\it p q s}$ is ${\\it Us}$ in the head, and $[\\mbox{\\tt\\symbol{95}}|{\\it Us}]$ in\nthe body atom. \n The latter represents up diagonals in the context of row $I$\n(as $I$ is the first argument in the atom). \nThus ${\\it Us}$ represents them in the context of row $s(I)$ (which is the\nfirst argument in the head).\nIn other words, if $[\\mbox{\\tt\\symbol{95}}|{\\it Us}]$ represents the up diagonals \nintersecting row $I$ then ${\\it Us}$ represents those intersecting row $s(I)$.\nSimilar reasoning applies to the fourth argument and down diagonals.\n\n\n\n\n\nNote that property (\\ref{spec.attempt.pqs}) is not closed under substitution.\nA triple $cs,us,d s$ that satisfies (\\ref{spec.attempt.pqs}) may have instances\nthat do not (e.g.\\ a list with a single occurrence of $i$ may have an\ninstance with multiple occurrences of $i$).  \nSo what we described differs from the relation actually defined by \n${\\it p q s}$, and our description needs to be corrected.\n\n\n\\paragraph{Informal specification.}\n\\hspace{0pt plus .4em\nNote first that property (\\ref{spec.attempt.pq}) is closed under\nsubstitution (due to employing\nthe generalized notion of member).\nThanks to this our specification for ${\\it p q}$ is obvious:\n\\begin{equation}\n\\label{spec.pq}\nS_{p q} = \n    %\n    \\{\\,\n      p q(\\,i,\\, [\\seq[k]c,i|c],\\, [\\seq[k]u,i|u],\\, [\\seq[k]d,i|d] \\,) \n    %\n      \\in{\\ensuremath{\\cal{H B}}}\\xspace   %\n        {}\\mid   k\\geq0\n      \\,\\}.\n     %\n\\end{equation}\nNote that this specification is exact;\nit is the set of atoms from ${\\ensuremath{\\cal M}}\\xspace_{\\rm NQUEENS}$ with the predicate symbol $p q$\n(and it is the least Herbrand model of $\\{(\\ref{clause3}),(\\ref{clause4})\\}$).\n\n\n\n\\commenta{\n  %\n  The difficulty in constructing a specification for ${\\it p q s}$ \n  is that the program has\n  also answers which represent incorrect\n  %\n   placement of queens.\n \n} \n\n\nWe need to construct a specification for  ${\\it p q s}$ \nso that correctness of {\\sc nqueens}\\xspace\nimplies that the program  solves the $n$ queens problem.\nThe difficulty is that the program has \nalso answers which represent incorrect\n placement of queens.\n This cannot be avoided, as a non-ground triple of (open) lists $cs,us,d s$\n which represents a correct placement of queens may have instances which do not\n (e.g.\\ because a single queen is placed in two columns). \n It may seem that we face a contradictory task: the role of our specification\n is to describe correct placements, but it has to include some incorrect ones.\n\n\n\\commenta{\n   The idea to overcome the difficulty is to care only about those atoms\n   ${\\it p q s(i, cs, us, d s )}\\in{\\ensuremath{\\cal{H B}}}\\xspace$, where $i>0$ and $cs$ is a list of distinct members.\n   %\n   %\n   This leads to the following informal specification for correctness\n   for ${\\it p q s}$:\n}\n\n\n To describe  ${\\it p q s}$,\n %\n note first that we are only interested in atoms \n ${\\it p q s(i, cs, us, d s )}\\in{\\ensuremath{\\cal{H B}}}\\xspace$, where $i>0$ and $cs$ is a list.\n The idea is that we care only about those in which the members of list $cs$ are\n %\n distinct. \n This means that the specification may {\\em include} \natoms ${\\it p q s(i, cs, us, d s )}$  where $cs$\n is not a list, or is a list with a repeated member.  \n Whenever $cs$ is a list of distinct members, the remaining arguments are\n such that the whole atom describes a correct placement.\nThis leads to the following informal specification for correctness\nfor ${\\it p q s}$:\n\\begin{equation}\n\\label{informal.spec.pqs}  \n  \\parbox{.77\\textwidth}{%\n    \\hspace*{-1.5em}%\n    the set of  those  ${\\it p q s(i, cs, us, d s )}\\in{\\ensuremath{\\cal{H B}}}\\xspace$ where \\\\\n    $i\\in{\\ensuremath{\\mathbb{N}}}\\xspace$, \\\n    $1,\\ldots,i$ are members of term  $c s$\n    (cf.\\ Section \\ref{sec.preliminaries}, Basics), and\\\\\n    if $i>0$ and $cs$ is a list of distinct members %\n    then condition (\\ref{spec.attempt.pqs}) holds.\n}\n\\end{equation}\nIt follows that if ${\\it cs}$ is a list of length $i$ then \nit is a list of distinct members $1,\\ldots,i$, and hence\nit is a solution of the $i$ queens problem.\nNow our specification for {\\sc nqueens}\\xspace is the union of the sets described by\n(\\ref{informal.spec.pqs}) and (\\ref{spec.pq}).\nNote that it serves its purpose, as\ncorrectness w.r.t.\\ it implies that the program solves the problem\n(each answer for the previously described query\n ${\\it p q s}(n,q_0,\\mbox{\\tt\\symbol{95}},\\mbox{\\tt\\symbol{95}})$ represents a solution).\nNote also that the specification is a proper superset of ${\\ensuremath{\\cal M}}\\xspace_{\\rm NQUEENS}$.\n\nSuch informal specification facilitates understanding of the program and\nmakes possible informal but precise reasoning\nabout the program.  \nFor an example, \nconsider a ground instance of clause (\\ref{clause2})\n\\[\n\\it p q s( s(i), cs, us, [t|d s]) \\ \\gets \\\n         p q s(i,cs,[t'|us],d s),\\  p q(s(i),cs,us,d s).\n\\]\nAssume that\nthe body atoms are compatible with the specification\n(i.e.\\ are in the sets (\\ref{informal.spec.pqs}) and (\\ref{spec.pq}),\nrespectively; we will refer to these two facts as\nrespectively ($\\alpha$), ($\\beta$)).\nWe show that also the head is compatible with the specification.\nBy ($\\alpha$), \n$s(i) \\in{\\ensuremath{\\mathbb{N}}}\\xspace$ and $1,\\ldots,i$ are members of $c s$.\nBy ($\\beta$), $s(i)$ is a member of $cs$.\nAssume that $cs$ is a list of distinct members.\nSo by ($\\alpha$) and (\\ref{spec.attempt.pqs}),\n ${\\it cs}$, ${\\it[t'|{\\it us}]}$ and the tail of ${\\it d s}$\nrepresent\na correct placement of queens $1,\\ldots,i$ in the context of row $i$.\n    %\nHence\nthis placement in the context of row $i+1$ is represented by \n${\\it cs, us, d s}$.\nBy ($\\beta$)\n we have that, in the same context, $cs,us,d s$ represent\nplacing the queen $i+1$.\nSo its column (its up \/ down diagonal) is distinct from those occupied by\nqueens $1,\\ldots,i$.\nThus ${\\it cs, us, d s}$ represent a correct placement of queens $1,\\ldots,i+1$.\nHence the head of the clause instance is in the set  (\\ref{informal.spec.pqs}).\n\n\n\n\nThe reasoning of the last paragraph explains the clause and convinces us\nabout its correctness. \nActually it is an informal outline of a central part\nof a correctness proof of the program, based on Theorem \\ref{th.correctness}.\n\n\n\\commenta{\\vspace{-3ex}\n\\paragraph{Comment.}\n    We exemplified a general pattern of constructing approximate\n    specifications.  Some ground atoms may be irrelevant for the program\n    properties we are interested in.  \n    Thus they, or some of them, may be additionally included into the\n    specification. \n    (In our case they are atoms ${\\it p q s(i, cs, us, d s )}\\in{\\ensuremath{\\cal{H B}}}\\xspace$,\n    where $i\\in{\\ensuremath{\\mathbb{N}}}\\xspace$, $1,\\ldots,i$ are members of $cs$, and moreover\n     $i=0$ or $cs$ is not a list of distinct members.)\n   %\n\\comment{Submitted 2020.07:\n    (In our case they are atoms ${\\it p q s(i, cs, us, d s )}\\in{\\ensuremath{\\cal{H B}}}\\xspace$,\n    where $i\\in{\\ensuremath{\\mathbb{N}}}\\xspace$, $1,\\ldots,i$ are members of $cs$, and\n   %\n    $cs$ is not a list of distinct members or $i=0$.)\n}\n     So ${\\ensuremath{\\cal{H B}}}\\xspace_{\\it p q s}$\n   is divided into a set $S_{\\rm i r r}$ of irrelevant atoms and \n   $S_{\\rm rel} = {\\ensuremath{\\cal{H B}}}\\xspace_{\\it p q s} \\setminus S_{\\rm i r r}$, the set of relevant\n   ones.  From the latter a set $S_{\\rm c}\\subseteq S_{\\rm rel}$ of\n    ``correct'' ones is chosen,\n   so that correctness w.r.t.\\ specification $S_{\\rm i r r}\\cup S_{\\rm c}$\n   implies the program properties of interest.\n   %\n   In many cases, $S_{\\rm c}$ describes the ground atoms\n   %\n   which the program should compute, so \n   %\n   $S_{\\rm c}$ is used as a specification for completeness, and \n   $S_{\\rm i r r}\\cup S_{\\rm c}$ as one for correctness.\n\n   In the next section,\n   the specification outlined here is made formal and is augmented by a\n   specification for completeness.\n} \n\n\n %\n \\paragraph{Comments.}\n\\label{sec.comments}\n %\n Program {\\sc nqueens}\\xspace employs non-ground terms.  \n Given an initial query as described previously,\n the program uses open lists,\n and it seems crucial that the not yet assigned columns and diagonals are\n represented as unbound variables.   \n Our informal description above begins with a relation which is not closed\n under substitution.\n %\n So one may suppose that the standard declarative semantics, based on the\n notion of logical consequence (and characterized by the least Herbrand models)\n is not suitable here.  Hence the notions of specification, correctness, and\n completeness of Section \\ref{sec.preliminaries} would have not been suitable.\n One may expect that the s-semantics \\cite{DBLP:journals\/tcs\/FalaschiLPM89}\n should be employed,\n as it makes it possible to explicitly deal with variables in program answers.\n\n Actually, this is unnecessary, as shown above and confirmed in the next\n sections.  We specify the program and prove its \n correctness and completeness in terms of Herbrand interpretations.\n An initial version of this work\n %\n was based on the\n s-semantics, this turned out to be less convenient and more complicated\n \\cite{drabent2020arxiv.s-semantics}.\n\n\n The difficulty we face is to describe a certain property of non-ground atoms,\n by means of their ground instances, while an atom with the required\n property\n %\n may have instances violating the property.\n %\n The difficulty is overcome by simply neglecting such instances\n  (and accepting them).\n This works, because a ground atom satisfying the property cannot be an\n instance of one not satisfying it, and each atom satisfying the property\n has a ground instance satisfying it.\n (The latter holds thanks to infinite {\\ensuremath{\\cal{H U}}}\\xspace.)\n The neglecting is performed by describing in (\\ref{informal.spec.pqs})\n %\n the arguments $us,d s\\in{\\ensuremath{\\cal{H U}}}\\xspace$ of ${\\it p q s}$ only for the cases in which\n $cs$ is a list of \n distinct members.  Otherwise $us,d s$ are arbitrary.  \n\n\n %\n %\n\n We applied a general way of constructing approximate specifications.\nSome ground atoms may be irrelevant for the program \nproperties we are interested in.  \nSo they, or some of them, may be additionally included into the specification.\nThe Herbrand base is split into a set $S_{\\rm i r r}$ of irrelevant\n(neglected) atoms,\nand  $S_{\\rm rel} = {\\ensuremath{\\cal{H B}}}\\xspace \\setminus S_{\\rm i r r}$, the set of relevant\nones.\n And within $S_{\\rm rel}$ we distinguish a subset \n$S_{\\rm corr}\\subseteq S_{\\rm rel}$ of the answers we accept.\nThe specification for correctness is now $S = S_{\\rm i r r}\\cup S_{\\rm corr}$.\nCorrectness w.r.t.\\ $S$ implies the required properties of the program.\n(Usually the intended initial queries have no instances in $S_{\\rm i r r}$.)\n\nIn our case \n$S_{\\rm corr}$ is the set of atoms describing correct placements of queens.\nThe irrelevant atoms are ${\\it p q s(i, cs, us, d s )}\\in{\\ensuremath{\\cal{H B}}}\\xspace$,\n %\n     where $i\\in{\\ensuremath{\\mathbb{N}}}\\xspace$, $1,\\ldots,i$ are members of $cs$, and moreover\n      $i=0$ or $cs$ is not a list of distinct members \n ($us,d s$ are arbitrary).\n\n\n As may be expected, such approach\n %\n does not lead to a unique specification.  \n The choice of $S_{\\rm i r r}$ is not unique.\n %\n %\nE.g.\\ the first versions of this report used another specification\nfor the same property of interest.\n   In the next section,\n   the specification outlined here is made formal and is augmented by a\n   specification for completeness.\n\n\\section{Approximate specification}\n\\label{sec.spec}\nThis section presents a pair of specifications %\nfor correctness and for completeness of {\\sc nqueens}\\xspace,\nformalizing the ideas from the previous section.\n\nThe specification for predicate $p q$ is obvious.  \nBoth for correctness and for completeness it is\n $S_{p q}$\nfrom (\\ref{spec.pq}) in the previous section. \n\n\nIn order to formulate the specification for ${\\it p q s}$, we introduce some\nadditional notions. \nLet us first formalize the numbering of diagonals.\nAssume a queen $j$ (i.e.\\ the queen of row $j$) is placed in column $k$\n(i.e.\\ $j$ is the $k$-th \nmember of a term $cs$ representing columns). \nThen, in the context of row $i$ (say $i\\geq j$), the queen $j$ is on the up\ndiagonal of number $k+j-i$.\nSimilarly, the queen $j$ is on the down diagonal of number  $k+i-j$,\nin the context of row $i$.\nConsider, for instance, the queen $i-3$ placed in column 2.  Then,\nin the context of row $i$,\nit is on the up (down) diagonal number $-1$ (respectively $5$).\n\n\n\\begin{definition}\nLet a queen (i.e.\\ a number) $j$ be the $k$-th member of a list $cs$.\n\n\\quad\nThe {\\bf up diagonal number} of $j$, w.r.t.\\ $i$ in $cs$ is  $k+j-i$.\n\n \\quad\nThe {\\bf down diagonal number} of $j$, w.r.t.\\ $i$ in $cs$ is  $k+i-j$.\n\\end{definition}\n\nWe can skip ``w.r.t.\\ $i$'' when stating that some queens have distinct up\n(down) diagonal numbers, as the numbers are distinct w.r.t.\\ any $i$.\n\\pagebreak[3]\n\nNow we are ready to introduce the core of our specification.\n\n\\begin{definition}\n\\label{def.correct.triple}\nA triple of terms $(cs,us,d s)\\in{\\ensuremath{\\cal{T U}}}\\xspace^3$  \n{\\em represents a correct placement} up to row $m$ in the context of row $i$\n(shortly: is {\\bf correct} up to $m$ w.r.t.\\ $i$\\\/) when $0\\leq m\\leq i$ and\n\\[\n\\qquad\\ \n\\begin{oldtabular}{l@{}}\n  $cs$ is a \nlist of distinct members, and each $j\\in\\{1,\\ldots,m\\}$ is its member,\n\\\\\n  \\begin{oldtabular}{@{}l@{\\quad}r@{}}\nthe up (respectively down) diagonal numbers of $1,\\ldots,m$ in $cs$ are\ndistinct, \n&\n\\refstepcounter{equation}\n(\\theequation\\label{correct.condition.distinct})\n\\\\\n          for  each  $j\\in\\{1,\\ldots,m\\}$,\n          \\\\\n  \\quad\n  \\begin{oldtabular}{l}\n      if\n      the up (down) diagonal number of $j$ w.r.t.\\ $i$ in $cs$ is $l>0$\n          \\\\\n      then the $l$-th member of $us$ (respectively $d s$) is $j$.\n  \\end{oldtabular}\n  &\n       \\refstepcounter{equation}\n      (\\theequation\\label{correct.condition1})\n  \\end{oldtabular}\n\\end{oldtabular}\n\\]\n\\end{definition}\nCondition (\\ref{correct.condition1}) assures that the placement of queens\n$1,\\ldots,m$ on the diagonals (according to $us,d s$) is compatible with\ntheir placement on the columns, as described by $cs$.\nNote that correctness of $(cs,us,d s)$ implies the \nrequired property of $cs$\\\/:\\,\nIf  $(cs,us,d s)$ is correct up to $m$ then,\naccording to $cs$, the queens (of rows) $1,\\ldots,m$ are placed\nin distinct columns, up diagonals and down diagonals. \n(Note that the down diagonal numbers in (\\ref{correct.condition1}) are\npositive, as $j\\leq i$.)\n\n\\pagebreak[3]\n\nNow the specification for ${\\it p q s}$ is\n  \\[\n  \\begin{array}[t]{@{}l@{}}\n  S_{\\it p q s} = \n     \\begin{array}[t]{@{}l}\n       \\big\\{\\, {\\it p q s}(0, cs, us, d s) \\mid cs, us, d s \\in {\\ensuremath{\\cal{H U}}}\\xspace \\,\\big\\}  \n       \\ \\cup\n       \\\\[1ex]\n  %\n       \\left\\{\\,  p q s(i, cs, us, [t|d s] ) \\in{\\ensuremath{\\cal{H B}}}\\xspace \\: \\left|\\,\n       \\begin{oldtabular}{l@{\\,}}\n  %\n         $i>0$, \\ \\ \\ %\n               $1,\\ldots,i$ are members of $cs$,\n         \\\\\n  %\n  %\n         if  $cs$ is a list of distinct members  then \\\\\n          $(cs, us, d s)$ is correct up to $i$ w.r.t.\\ $i$. \\\\\n  %\n        \\end{oldtabular}\n        \\right\\}\\right.,\n     \\end{array}\n  \\end{array}\n\\]\nAnd our specification of {\\sc nqueens}\\xspace for correctness is\n\\[\nS = S_{\\it p q} \\cup S_{\\it p q s}.\n\\]\n\n\nNote that correctness w.r.t.\\ $S$ implies the required property of the program.\nTake an atom $A = {\\it p q s}(n, cs', us', d s')\\in{\\ensuremath{\\cal{T U}}}\\xspace$, such that $n>0$ and\n$cs'$ is a list of length $n$.\nIf $S\\models A$  then $cs'$ is a solution of the $n$ queens problem\n(as, for each ground instance ${\\it p q s}(n, cs, us, d s)$ of $A$,\n$1,\\ldots,n$ are members of $cs$,\n thus $cs$ is a list of distinct members $1,\\ldots,n$, so\n  $(cs, us, d s)$ is correct up to $n$ w.r.t.\\ $n$.\nHence the up (down) diagonal numbers of  $1,\\ldots,n$ in $cs$ are distinct,\nso $cs$ represents a solution of the $n$ queens problem.\n\n  %\n\nWhile specifying completeness, we are interested in ability of the program to\nproduce all solutions to the problem.  \nThis leads to the following specification for completeness:\n\\[\n  S_{\\it p q s}^0\\ =\\ \n   \\begin{array}[t]{@{}l}\n    \\left\\{\\,  p q s(i, cs, us, [t|d s] ) \\in{\\ensuremath{\\cal{H B}}}\\xspace \\: \\left|\n     \\,\n     \\begin{oldtabular}{l@{\\,}}\n       $i>0$, \\\\\n        $(cs, us, d s)$ is correct up to $i$ w.r.t.\\ $i$. \\\\\n      \\end{oldtabular}\n      \\right.\\right\\}.  \n   \\end{array}\n\\]\n\nWe conclude this section with a property which will be used \nlater on.\n\n\\pagebreak[3]\n\\begin{lemma}\n\\label{lemma.almost.equivalence}\nAssume $0<m\\leq i$.\nConsider two conditions\n\\[\n\\begin{tabular}{c@{\\qquad}l}\n  $(cs,[t|us],d s)$ is correct up to $m$ w.r.t.\\ $i$\n   &  \\refstepcounter{equation}  (\\theequation\\label{equivalence.part1})\n\\\\\n$(cs,us,[t'|d s])$ is correct up to $m$ w.r.t.\\ $i+1$\n   &  \\refstepcounter{equation}  (\\theequation\\label{equivalence.part2})\n\\end{tabular}\n  \\]\nFor any $t,t'\\in{\\ensuremath{\\cal{H U}}}\\xspace$, (\\ref{equivalence.part1}) implies\n(\\ref{equivalence.part2}). \nFor any $t'\\in{\\ensuremath{\\cal{H U}}}\\xspace$, (\\ref{equivalence.part2}) implies \n$\\exists\\, t\\in{\\ensuremath{\\cal{H U}}}\\xspace\\:(\\ref{equivalence.part1})$. \n\\end{lemma}\n\n\\vspace{-1\\topsep}\n\\begin{proof}\nAssume that $cs$ is a list of distinct members and each $j\\in\\{1,\\ldots,m\\}$\nis a member of $cs$.  We will consider here the diagonal numbers in $cs$.\nObviously, the up (down) diagonal numbers w.r.t.\\ $i$ (of $1,\\ldots,m$)\nare distinct iff the diagonal numbers w.r.t.\\ $i+1$ are.\n\n\nLet $j\\in\\{1,\\ldots,m\\}$.\nThen $l$ is the down diagonal number of $j$ w.r.t.\\ $i$ \niff\n$l_1=l+1$ is the down diagonal number of $j$ w.r.t.\\ $i+1$.\nNote that $l>0$ (as $j\\leq i$).\nSo for down diagonals,\n conditions (\\ref{correct.condition1}) for $i$, $j$, $l$ and $d s$, and    \n(\\ref{correct.condition1}) for $i+1$, $j$, $l_1$ and $[t'|d s]$ are equivalent.\n\n\nNumber $l$ is the up diagonal number of $j$ w.r.t.\\ $i$ \niff\n$l_2=l-1$ is the up diagonal number of $j$ w.r.t.\\ $i+1$.\nSo $l_2\\geq0$. For $l_2>0$ we, similarly as above, obtain that\nfor up diagonals conditions\n\\vspace{-1.5ex}\n\\vspace{-.5ex}\n\\begin{eqnarray}\n    &&\n    \\label{lemma.almost.equivalence.cond.1}\n    \\mbox{(\\ref{correct.condition1}) for $i$, $j$, $l$ and $[t|us]$} \n    \\\\\n    &&\n    \\label{lemma.almost.equivalence.cond.2}\n    \\mbox{(\\ref{correct.condition1}) for $i+1$, $j$, $l_2$ and $us$}\n\\end{eqnarray}\nare equivalent.\nFor $l_2=0$, (\\ref{lemma.almost.equivalence.cond.1}) vacuously implies\n(\\ref{lemma.almost.equivalence.cond.2}), and\n(\\ref{lemma.almost.equivalence.cond.2})\nimplies that (\\ref{lemma.almost.equivalence.cond.1}) holds for some $t$,\nnamely $t=j$.\n\nThis completes the proof of both implications of the lemma.\n\\end{proof}\n\n\n\\section{Correctness proof}\n\\label{sec.correctness.proof}\n\nFollowing Theorem \\ref{th.correctness}, \nto prove correctness of program {\\sc nqueens}\\xspace w.r.t.\\ specification $S$,\none has to show that $S$ is a model of each clause of the program.\nIn other words to show, for each ground\ninstance of a clause of the program,\nthat the head is in $S$ provided the body atoms are in $S$. \nFor the unit clauses of {\\sc nqueens}\\xspace\n\\[\n\\begin{tabular}{l}\n${\\it p q(I,[I|\\mbox{\\tt\\symbol{95}}],[I|\\mbox{\\tt\\symbol{95}}],[I|\\mbox{\\tt\\symbol{95}}])}$.\n\\\\\n${\\it p q s}(0,\\mbox{\\tt\\symbol{95}},\\mbox{\\tt\\symbol{95}},\\mbox{\\tt\\symbol{95}})$.\n\\end{tabular}\n\\]\nit is obvious that each ground instance of the clause is in $S$.\n\\\nConsider clause (\\ref{clause4}).\nFor any its ground instance\n\\[\n{\\it p q(i,[t_1|cs],[t_2|us],[t_3|d s])\n\\gets \\it p q(i,cs,us,d s)\n}.\n\\]\nit immediately follows from the definition of  $S_{\\it p q}$ that\n if the body atom is in $S$ (thus in $S_{\\it p q}$) then its head is in \n$S_{\\it p q}\\subseteq S$.\n\nThe nontrivial part of the proof is to show that $S$ is a model of clause \n(\\ref{clause2}).  Consider its ground instance\n\\[\n{\\it\n    p q s(s(i),cs,{\\it us},[t|d s]) \\gets\n            p q s(i,cs,[t_1|{\\it us}],d s), \\,\n            p q(s(i),cs,{\\it us},d s).\n}\n\\]\nLet $H$ be its head, and $B_1,B_2$ the body atoms.  Assume $B_1,B_2\\in S$.\nNow (by $B_2\\in S$) \n$s(i)$ is the $l$-th member of  $cs, us, d s$ (for some $l>0$).\nNote that $l$ is the up (down) diagonal number of $s(i)$ w.r.t.\\ $s(i)$\nin $cs$.\nSo condition (\\ref{correct.condition1}) holds for  $s(i)$ w.r.t.\\  $s(i)$.\n\nConsider first the case of $i=0$.  Then\n$(cs, us, d s)$ is correct up to $s(0)$ w.r.t.\\ $s(0)$,\nprovided that $cs$ is a list of distinct members.\nHence $H\\in S$.\n\nConsider $i>0$. Note first that $1,\\ldots,s(i)$ are members of $cs$\n($s(i)$ as explained above, and $1,\\ldots,i$ by $B_1\\in S$).\nAssume that\n$cs$ is a list of distinct members.\nThen (by $B_1\\in S$)\n$(cs,[t_1|{\\it us}],d s')$ is correct up to $i$ w.r.t.\\ $i$, where\n$d s '$ is the tail of $d s$.\nHence by Lemma~\\ref{lemma.almost.equivalence},\n$\\beta=(cs,{\\it us},d s)$ is correct up to $i$ w.r.t.\\ $s(i)$.\nAs shown above, \n(\\ref{correct.condition1}) holds for \n$s(i)$ w.r.t.\\ $s(i)$\n(where $l$ is both the up and the down diagonal number of $s(i)$).\nThus (\\ref{correct.condition1}) holds for $1,\\ldots,s(i)$ w.r.t.\\ $s(i)$.\nHence no up (or down) diagonal number of a $j\\in\\{1,\\ldots,i\\}$ is $l$.\nAs the latter diagonal numbers are distinct (due to $\\beta$ being correct up to\n$i$), \n(\\ref{correct.condition.distinct}) holds for $1,\\ldots,s(i)$.\n\n\n  Hence $\\beta$ is correct up to $s(i)$ w.r.t.\\ $s(i)$.\n  Thus $H\\in S$.\n  This completes the proof.\n\n\n\n\\section{Completeness proof}\n\\label{sec.completeness.proof}\n\nAs explained in Section \\ref{sec.spec}, we are interested in completeness of\n{\\sc nqueens}\\xspace w.r.t.\\ specification \n$S_{\\it p q s}^0$.\nHowever\nthe sufficient condition of  Lemma \\ref{lemma.completeness}\ndoes not hold for this specification.\n  Instead let us use \n\\[\nS^0 = S_{\\it p q} \\cup S_{\\it p q s}^0 \\cup \n     \\{\\, {\\it p q s}(0, cs, us, d s) \\mid cs, us, d s \\in {\\ensuremath{\\cal{H U}}}\\xspace \\,\\}  \n\\]\nas the specification for completeness.%\n\\footnote{%\n  This is a common phenomenon in mathematics;  an inductive proof of a property\n  may be impossible, unless the property is strengthened.\n  Actually, the same happened in the case of correctness. We are interested\n  in correctness of {\\sc nqueens}\\xspace w.r.t.\\ $S_{\\it p q s}\\cup{\\ensuremath{\\cal{H B}}}\\xspace_{\\it p q}$.\n  However $S_{\\it p q s}\\cup{\\ensuremath{\\cal{H B}}}\\xspace_{\\it p q}$ is not a model of\n  the program and Theorem \\ref{th.correctness} is not applicable.  Instead\n  we used a stronger specification $S = S_{\\it p q s} \\cup S_{\\it p q}$.\n\n  Obviously, correctness (completeness) w.r.t.\\ a specification implies\n  correctness  (completeness) w.r.t.\\ any its superset (subset).\n} %\n\nWe first show that each atom from specification $S^0$ is covered by program {\\sc nqueens}\\xspace.\nEach atom\n\\vspace{-1.5ex}\n\\[\nA =  p q(\\,i,\\, [\\seq[k]c,i|c],\\, [\\seq[k]u,i|u],\\, [\\seq[k]d,i|d] \\,) \n\\]\nfrom $S_{\\it p q}$ is covered by {\\sc nqueens}\\xspace w.r.t.\\  $S^0$; for $k=0$ by clause\n(\\ref{clause3}) as $A$ is its instance;\nfor $k>0$ by clause (\\ref{clause4})\ndue to its instance \n\\newcommand*{\\SEQ}[3]\n            {{\\ensuremath{#1_{#2}, \\allowbreak \\ldots, \\allowbreak #1_{#3}}}}%\n$A\\gets \np q(\\,i,\\, [\\SEQ c 2 k,\\,i|c],\\, [\\SEQ u 2 k,i|u],\\,\\allowbreak\n [\\SEQ d 2 k,i|d] \\,) \n$ (as its body atom is in $S^0$).\nAlso, each atom $ {\\it p q s}(0, cs, us, d s)$ is covered, as it is an\ninstance of  clause (\\ref{clause1}).\n\nThe nontrivial part of the proof is to show that each $A\\in S_{\\it p q s}^0$\nis covered.  Consider such atom, it is of the form\n\\vspace{-1.1ex}\n\\[\nA = p q s(s(i), cs, us, [t|d s] ),  %\n\\]\nwhere $i\\geq0$ and $(cs,us,d s)$ is correct up to $s(i)$ w.r.t.\\ $s(i)$.\nSo $cs$ is a list of distinct members, and each $j\\in\\{1,\\ldots,s(i)\\}$ is\na member of $cs$.\n   %\n   %\n   %\nLet $s(i)$ be the $l$-th member of $cs$.  Thus\n$l$ is the up and down diagonal number of $s(i)$ w.r.t.\\ $s(i)$ in $cs$, and \n(by Def.\\ \\ref{def.correct.triple}) $s(i)$ is the $l$-th member of $us$ and\nof $d s$.\n\n\nWe show that $A$ is covered by clause (\\ref{clause2}) w.r.t.\\ $S^0$, due to\nits instance \n \\[\n A\\gets  B_1, B_2. \\qquad\n \\mbox{where }\nB_1 =  {\\it p q s}(i,cs,[t'|us],d s),\\ \\\nB_2 =    p q(s(i),cs,{\\it us},d s)\n \\]\n(and $t'\\in{\\ensuremath{\\cal{H U}}}\\xspace$ will be determined later).\nWe have $B_2\\in S^0$ (as $s(i)$ is the $l$-th member of $cs,us$ and $d s$).\nIf $i=0$ then $B_1\\in S^0$, thus $A$ is covered by (\\ref{clause2}).%\n\\footnote{%\n    Note that in this case $A$ is covered  w.r.t.\\ $S^0$ but not\n    w.r.t.\\ $S_{\\it p q}^0\\cup S_{\\it p q s}^0$.\n    This is why we use $S^0\\supset S_{\\it p q}^0\\cup S_{\\it p q s}^0$ as a\n    specification. \n}\n\nAssume $i>0$.\nAs $(cs, us, d s)$ is correct up to $s(i)$ w.r.t.\\ $s(i)$, it is correct \n up to $i$ w.r.t.\\ $s(i)$, and \nby Lemma~\\ref{lemma.almost.equivalence},\n$(cs,[t'|us],d s')$ is correct  up to $i$ w.r.t.\\ $i$, for some $t'\\in{\\ensuremath{\\cal{H U}}}\\xspace$, \nwhere $d s'$ is the tail of $d s$.\nHence for such $t'$ we have $B_1\\in S_{\\it p q s}^0\\subseteq S^0$,\nthus $A$ is covered by (\\ref{clause2}).\n\n\nThis completes the proof that each $A\\in S^0$ is covered by {\\sc nqueens}\\xspace\nw.r.t.\\ $S^0$.  \nIt remains to find\na level mapping under which {\\sc nqueens}\\xspace is recurrent.  Consider\nthe level mapping defined by\n\\vspace{-1ex}\n\\[\n\\begin{array}{l}\n  | \\, {\\it p q s}(i,cs,us,d s) \\,| = |i|+|cs|,\n\\\\\n  | \\, {\\it p q}(i,cs,us,d s) \\,| = |cs|,\n  %\n\\end{array}\n \\qquad \\mbox{where} \\qquad\n    \\begin{array}{l}\n      |\\, [h|t]\\, | = 1+|t|,    \\\\\n      |\\, s(t)\\, | = 1+|t|,    \\\\\n      |f(\\seq t)| = 0, \n    \\end{array}\n\\]\nfor any ground terms $i,cs,us, d s, h,t,\\seq t$, and any $n$-ary\n function symbol $f$ distinct from $s$ and from $[\\ | \\ ]$ ($n\\geq0$). \nAn easy inspection shows that\nunder this level mapping {\\sc nqueens}\\xspace is recurrent.\nHence by Lemma \\ref{lemma.completeness}, the program is complete w.r.t.\\ $S^0$.\n\n\n\n\\section{Conclusions}\n The paper provides an example of precise reasoning about the semantics of a\n logic\n program.  It presents detailed proofs of correctness and completeness of the $n$\n queens program of \\citeN{Fruehwirth91}.  \n The program is short, but may be seen as tricky or non-obvious.\n The approach is declarative; the specifications and proofs abstract from any\n operational semantics, the program is treated solely as a set of logical\n formulae. \n  %\n Note that\n in many cases, approaches based on the operational semantics are proposed \n for reasoning about declarative properties of logic programs\n  \\cite{Apt-Prolog,DBLP:conf\/tapsoft\/BossiC89,DBLP:journals\/jlp\/PedreschiR99}.\n  This seems to introduce unnecessary complications\n  (cf.\\ \\cite[Section 3.2]{DBLP:journals\/tplp\/DrabentM05shorter}).\n\n\nThe original description \\cite{Fruehwirth91} of the {\\sc nqueens}\\xspace program is rather\noperational, as it mainly describes how the program works, and does not\nexplain the relations it defines\n(cf.\\ Section \\ref{section.program}).\nSo we begin with informally describing the program declaratively, from a\nlogical point of view. \n\nThe program uses non-ground data,\nlike open lists with some elements being variables.\nMoreover some of its answers (which represent solutions to the problem)\nhave instances that do not.\nSo one may expect that approaches based on the standard semantics and\nHerbrand interpretations are inapplicable here.  Actually, this is\nnot the case. \nWe discuss difficulties with constructing a specification, show how to\novercome them, \nand provide a formal specification based on Herbrand interpretations.\nThen we prove that the program is correct and complete with respect to the\nspecification. \n\nIt may seem that s-semantics \\cite{DBLP:journals\/tcs\/FalaschiLPM89}\nis suitable here, as it explicitly deals with non-ground answers.\nHowever the approach employed in this paper seems preferable, as\nanalogical specification and proofs\nemploying the s-semantics \\cite{drabent2020arxiv.s-semantics} turn out to be\nmore complicated.\n\n  Our specification is approximate;\n this means separate specifications for correctness and for completeness\n (see Section \\ref{sec.preliminaries} for explanations and references). \n Constructing an exact specification of the program would be too\n %\n troublesome, and would result in more complicated correctness and completeness\n proofs.\n %\n This is quite common in logic programming---one\n often does not need to know the exact semantics of one's program.  \n  Some features of the program are of no interest, for instance they may be\n  irrelevant to its intended usage.  So we do not need to describe them.\n  What we require from a specification is that it describes those program\n  properties in which we are interested.\n\n %\n %\n %\n\nThis paper provides an example of precise declarative reasoning about the\nsemantics of a particular logic program.  \nThe case seems apparently difficult to deal with; we show how to overcome the\ndifficulties. \nIn the author's opinion\nthe example confirms applicability of the employed approach\nand provides hints for its use in other cases.\n\n\nThe detailed proofs presented here may be seen as too impractical due to\nnumerous details.  Note however that this is usually the case when proving\nprogram properties.\n  Experience with reasoning about programs,\n  also in imperative programming,\n  provides evidence that program correctness really does depend on many \n  details (see for instance the proof for Quicksort in\n \\cite{AptBO.quicksort2009},\n  and the example proofs in the papers mentioned above).\nIt should be possible to tame the complexity of proofs by employing \nsome proof assistant.  This issue is however outside of the scope of this paper.\nOn the other hand,\n  in the author's opinion proofs like those presented here\n  can be performed by programmers at an informal level\n  during actual programming,\nat various degrees of precision.\nFragments of such informal reasoning (at two levels of precision) are shown in\nSection \\ref{section.program}.\nWe expect that formal proof methods, like those discussed here,  can teach\nprogrammers a systematic way\n of reasoning about their programs in practice.\n\n\n\n\\paragraph{\\bf Acknowledgement}\nComments of anonymous referees and of Michael Maher were instrumental in\nimproving the presentation.\n\n\n\n\\bibliographystyle{acmtrans}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nConsider two directed graphical models (or directed acyclic graphs, DAGs) on random variables $(A,B,C)$:\n\\begin{equation}\\label{eq:thr}\n\t\tA \\to B \\to C \\qquad B \\leftarrow A \\to C   \n\\end{equation}\n(See~\\cite{lau96} for background on graphical models.) In this paper, we will say that these two models\nare \\textit{isomorphic} (as graphical models). Roughly, this means that after relabeling ($A\\leftrightarrow B$), the two resulting models describe the same collection of joint distributions\n$P_{A,B,C}$. Note that the so defined isomorphism notion is weaker than the (directed) graph isomorphism: the\ngraphs in~\\eqref{eq:thr} are not isomorphic.  \n\nOn\nthe other hand, there does not exist any relabeling making~\\eqref{eq:thr} equivalent to \n\\begin{equation}\\label{eq:thr2}\n\t\tB \\to C \\leftarrow A \n\\end{equation}\nIn fact, a simple\nexercise in $d$-separation criterion shows that~\\eqref{eq:thr} and~\\eqref{eq:thr2} list all possible isomorphism classes\nof directed \\textit{tree} models\non three variables.  However, note that the above DAGs are all isomorphic as undirected graphs.\n\nThe goal of this paper is to provide (computational) answer to:\n\\textit{What are the isomorphism classes of directed graphical models on $n$ nodes?}\n\nNote that when variables $(A,B,C)$ are jointly Gaussian and zero-mean, then conditions such as~\\eqref{eq:thr} can be\nstated as algebraic constraints on the covariance matrix:\n\\begin{equation}\\label{eq:thr3}\n\t\t\\mathbb{E}[AB] \\mathbb{E}[BC] = \\mathbb{E}[AC] \\mathbb{E}[B^2]. \n\\end{equation}\nThis suggests that checking isomorphism of models can be carried out via algebraic methods. Indeed, one needs to recall (see \\cite{lev01}) that graphical models equality can be tested by restricting to Gaussian random variables.\n\nIn this paper, we associate with every DAG two subsets of covariance matrices:\n\\begin{itemize}\n\t\\item all non-singular covariance matrices satisfying DAG constraints (denoted $\\mathrm{loc}(G)\\cap\\Sigma^{++}$ below)\n\t\\item all covariance matrices satisfying DAG constraints (denoted $\\mathrm{loc}(G)$ below)\n\\end{itemize}\nWe give an \\textit{analytic} result: while $\\mathrm{loc}(G)$ is not necessarily (Euclidean) closed, closures of both sets\ncoincide. \n\nNext, we switch to the \\textit{algebraic} part. Due to the analytic fact above, much simpler equations for non-singular\nmatrices can be used to completely characterize the Zariski closure of\n$\\mathrm{loc}(G)$ (denoted\n$X_G$ below).\nAesthetically pleasing is the fact that $X_G$ is always an irreducible complex variety (affine and rational).\nFurthermore, two graphical models $G$ and $G'$ define the same set of conditional independence constraints if and only\nif $X_G = X_{G'}$.\n\nFor large graphs it is important to reduce the number of equations needed to describe $X_G$. \nThe natural set of\nequations (denoted $I_G$ below) turns out to be too small: its solution set $V(I_G)$ contains $X_G$ and a\nnumber of spurious components. We show how to get rid of these spurious components, proving that\n$$ X_G = V((I_G:\\theta_0^m))\\,, $$\nwhere $\\theta_0$ is an explicit polynomial (and establishing Conjecture 3.3 of Sullivant~\\cite{sul08}). This provides a convenient method for computing $X_G$. After these preparations, we\ngive our main result: isomorphism question $G\\stackrel{?}{\\sim} G'$ is equivalent to comparing intersections of $X_G$\nand $X_{G'}$ with a certain invariant variety. We give a randomized algorithm for this and apply it to provide a list\nof isomorphism classes on 4,5, and 6 nodes. \n\nThe question of DAG isomorphism does not seem to have appeared elsewhere, though the closely related question of DAG equivalence (or Markov equivalence \\cite{and97}) is well-studied. As mentioned in \\cite{chi02}, the natural space to work with when doing model selection or averaging over DAGs is that of their equivalence classes. In practice, the number of DAGs in an equivalence class encountered during model selection can be large. Some examples are presented in Table 4 of \\cite{chi02} where, for instance, the learning algorithm discovers equivalence classes on 402 nodes with more than $7\\times 10^{21}$ members. However, one should keep in mind that not all equivalence classes are as large. In fact, Steinsky \\cite{ste03} showed, by a recursive method, that roughly 7 percent of equivalence classes of graphs with 500 nodes or less consist only of one element. Recent results \\cite{wag13} indicate that this ratio is valid asymptotically as well. Such classes appear to be the most common type of equivalences classes of DAGs \\cite{gil01}. In general, it is expected, based on observations on small graphs, that the ratio of equivalence classes to DAGs be around 0.27 \\cite{gil01}\\cite{pen13}. Nevertheless, some equivalence classes (such as those that appear in \\cite{chi02}) are quite large. This has motivated the need to represent DAGs, and among the representatives that are relevant in this regard are the essential graphs{\\footnote{The essential graphs were originally known as {completed patterns} and were introduced in \\cite{ver90} as the maximal invariants associated to equivalence classes of DAGs. They were also studied in \\cite{mee95} under the name {maximally oriented graphs}.}} \\cite{and97} and the characteristic imsets \\cite{std10}. Both these methods have a combinatorial flavor and this work provides an algebraic alternative. The word algebraic here means commutative-algebraic, unlike in~\\cite{std10}. We remark that the two mentioned methods can also be applied to solve the isomorphism problem. For instance, using the results in \\cite{and97} one can reduce DAG isomorphism to the isomorphism of certain directed multi-graphs.\nIn fact, this gives a sense of the inherent computational difficulty involved in working with the isomorphism class of a DAG.\n\nWhile the notion of Markov equivalence makes sense in the setting of \\cite{chi02}, there are situations where it is natural to want to work with the isomorphism class of a DAG-- the Markov equivalence class modulo permutations of variables. For instance, the recent results in \\cite{kwi10} imply that there is a precise sense in which the isomorphism classes of all large graphs that admit efficient inference are related to graphs that look like the (unlabeled) trees listed in Figure~\\ref{fig_classes}, but are far from the complete DAGs.\n\\begin{comment}\n\\begin{center}\n\\tikzset{node distance=1.5cm, auto}\n\\begin{tikzpicture}[auto]\n  \\node (A0){$$};\n  \\node (A1)[right of=A0,node distance=1.5cm] {$$};\n  \\node (A3) [below of=A0,node distance=0.5cm]{$\\bullet$};\n  \\node (A3l) [left of=A3,node distance=1cm]{$$};\n  \\node (A3ul) [above of=A3l,node distance=0.75cm]{$\\bullet$};\n  \\node (A3dl) [below of=A3l,node distance=0.75cm]{$\\bullet$};\n  \\node (A5p) [right of=A4,node distance=1.5cm]{$$};\n \n  \\path [line] (A3ul) --node[above]{$$} (A3);  \n  \\path [line] (A3dl) --node[above]{$$} (A3);  \n\n\n\\end{tikzpicture}\n\\end{center}\nor\n\\begin{center}\n\\tikzset{node distance=1.5cm, auto}\n\\begin{tikzpicture}[auto]\n  \\node (A0){$\\bullet$};\n  \\node (A1)[right of=A0,node distance=1.5cm] {$\\bullet$};\n  \\node (A2) [right of=A1,node distance=1.5cm]{$\\bullet$};\n  \\path [line] (A0) --node[above]{$$} (A1);  \n  \\path [line] (A1) --node[above]{$$} (A2);  \n\\end{tikzpicture}\n\\end{center}\n\\end{comment}\nAn exact description of such models, however, is problematic by the subsequent results in \\cite{kwi10}. It thus appears reasonable to find good ways to approximate them, and for that we resort to the family of projective varieties. \n\nIt is also important to mention that the idea of associating an algebraic variety to a conditional independence (CI)\nmodel has been previously explored in a number of publications, among which we will\ndiscuss~\\cite{lnv07,mat95,mat95p,mat99,sim06p,sim06,drt08,drt09,mat97,gar05}. Some of our preparatory propositions can\nbe found in the literature in slightly weaker forms and we attempt to give references.  The main novelties are:\n\\begin{itemize} \n\\item We essentially leverage the directed-graph structure of the model (as opposed to general CI models) to infer\nstronger algebraic claims. In particular, our treatment is base independent -- although for readability we present\nresults for the varieties over $\\mathbb{C}$.\n\\item We present a computational procedure for answering the isomorphism question.\n\\end{itemize}\n\n\n\n\n\n\n\\subsection{Preliminaries}\nDirected acyclic graphical models are constraints imposed on a set of probability distributions:\n\\begin{definition}\nA directed acyclic graphical model (DAG) ${G}\/k$ is the data:\n\\begin{itemize}\n\\item A set of indices $[n]:=\\{1,\\cdots,n\\}$ that are nodes of a directed acyclic graph. We frequently assume the nodes\nto be topologically sorted, i.e., $i<j$ whenever there\nis a path in the graph from $i$ to $j$.\n\\item A list $\\mathcal{M}_{{G}}$ of {\\em imposed} (a.k.a local Markov) relations\n\\begin{equation}\\label{eq:topoall}\n\t\ti\\perp\\!\\!\\!\\perp \\mathbf{nd}(i)| \\mathbf{pa}(i) \\quad  \n\\end{equation}\nwhere $\\mathbf{pa}(i)$ denotes the set of parents of $i\\in [n]$ and $\\mathbf{nd}(i)$ is the set of non-descendants of $i$ in the directed graph. \n\\item A subset $\\mathcal{M}_{{G}}^{topo}$ of topologically sorted local Markov relations:\n\\begin{equation}\\label{eq:topomark}\ni\\perp\\!\\!\\!\\perp \\mathbf{nd}(i) \\cap \\{j: j<i\\}| \\mathbf{pa}(i) \\quad \n\\end{equation}\n\\item A set of {\\em ${G}$-compatible} joint probability distribution\n\\[\n\\mathrm{Loc}({G}):=\\{P_X | \\,I\\perp\\!\\!\\!\\perp J|K\\in \\mathcal{M}_{G}\\Rightarrow X_I\\perp\\!\\!\\!\\perp X_J|X_K \\},\n\\]\nwhere $X$ is a $k^n$-valued random variable\\footnote{We mostly work with $k=\\mathbb{R}$ or $\\mathbb{C}$. Since $\\mathbb{R}$ and $\\mathbb{C}$ are measurably isomorphic, it does not matter which one we pick. We write ${G}\/k$ if we need to emphasize the base field $k$. }. \n\\item A set of {\\em implied} relations\n\\[\n\\mathcal{C}_{{G}}:=\\cap_{P_X\\in \\mathrm{Loc}({G})}\\{I\\perp\\!\\!\\!\\perp J|K \\,\\,\\mathrm{s.t}\\,\\, X_I\\perp\\!\\!\\!\\perp X_J|X_K\\}.\n\\]\n\\end{itemize}\n\\end{definition}\n\nGiven a collection of such models, it is often of interest to find representatives for their isomorphism classes (see also \\cite{lnv07,mat95})-- these are models that have the same compatible distributions modulo labelings of variables: \n\\begin{definition} Let $\\mathcal{Q}$ be a permutation invariant family of distributions. Two DAGs ${G},{G}'$ are called {$\\mathcal{Q}$-equivalent}\nif \n\\[\n\\mathrm{Loc}({{G}})\\cap \\mathcal{Q}=\\mathrm{Loc}({{G}'})\\cap \\mathcal{Q}.\n\\]\nWhen $\\mathcal{Q}$ is the set of all distributions, we call such models equivalent. Likewise, two DAGs ${G},{G}'$ are called {$\\mathcal{Q}$-isomorphic} \nif \n\\[\np_{X_1\\cdots X_n}\\in \\mathrm{Loc}({{G}})\\cap \\mathcal{Q}\\iff p_{X_{\\pi(1)}\\cdots X_{\\pi(n)}}\\in \\mathrm{Loc}({{G}'})\\cap \\mathcal{Q}\n\\]\nfor some permutation $\\pi$ of indices. When $\\mathcal{Q}$ is the set of all distributions, we call such models isomorphic and denote the isomorphism class of $G$ by $[G]$. \n\\label{def_equiv}\n\\end{definition}\n\nWe shall mainly focus on characterizing isomorphism classes of DAGs. A related question is that of understanding the\nstructure of conditional independence constraints -- see for the case of discrete random variables \\cite{mat95p,mat99,mat95}, positive discrete random variables \\cite{sim06p}, non-singular Gaussians \\cite{lnv07},  and general Gaussians \\cite{sim06}. \n\n\\begin{comment}\n\\begin{definition}\nA CI model is said to be a directed acyclic graphical model (DAG) if\n\\begin{itemize}\n\\item The set of indices $[n]$ are nodes of a directed acyclic graph. \n\\item The list of imposed (local Markov) relations is given by\n\\[\n\\mathcal{M}_{{G}}:i\\perp\\!\\!\\!\\perp \\mathbf{nd}(i)| \\mathbf{pa}(i) \\quad \n\\]\nwhere $\\mathbf{pa}(i)$ denotes the set of parents of $i$ and $\\mathbf{nd}(i)$ is the set of non-descendants of $i$ in the directed graph.\n\\end{itemize}\n\\label{def_dags}\n\\end{definition}\n\\end{comment}\n\nLet $H=H_1\\times\\cdots\\times H_n$ be a product measure space endowed with the $\\sigma$-algebra $\\mathcal{H}=\\mathcal{H}_1\\otimes\\cdots\\otimes\\mathcal{H}_n$. We assume that $H_i$ is measurably isomorphic to $\\mathbb{R}$ and that $\\mathcal{H}_i$ is a Borel $\\sigma$-algebra for all $i$. The next property, factorization, \nrelies on a digraph structure and pertains only to DAGs:\n\\begin{definition}\nA probability measure $P$ defined on $(H,\\mathcal{H})$ is said to factorize w.r.t a DAG if it can be written as\n\\[\nP(\\mathcal{A})=\\int_{\\mathcal{A}}\\prod_{i}^n K_{i|\\mathrm{pa}(i)}(dx_i|x_{\\mathrm{pa}(i)}) \\quad \\forall \\mathcal{A}\\in \\mathcal{H},\n\\]\nwhere $K_{i|\\mathrm{pa}(i)}$'s are conditional probability kernels (which exist by \\cite[Theorem 2.7]{cin11}) \nand $K_{i|\\mathrm{pa}(i)}(dx_i|x_{\\mathrm{pa}(i)})=\\mu_i(dx_i)$ if $i$ has no parents in $G$.\n\\label{def_fac}\n\\end{definition}\n\nGiven a DAG $G$, we denote by $\\mathrm{Fac}(G)$ the set of distributions that factorize w.r.t $G$. It is known (c.f. Section II.A) that \n\\[\n\\mathrm{Fac}(G)=\\mathrm{Loc}(G).\n\\]\nThis means that two DAGs are equal (isomorphic) in the above sense if and only if they factorize the same set of distributions (modulo the labeling of the variables). \n\n\\subsection{Notation}\n\\begin{itemize}\n\\item $\\mathcal{N}$ is the set of real valued Gaussians\n\\item $\\mathcal{N}^{+}$ is the non-singular subset of $\\mathcal{N}$. \n\\item $\\Sigma=[\\sigma_{ij}]$ is the affine space $\\mathbb{C}^{{n+1 \\choose 2}}$ of Hermitian $n\\times n$ matrices.\n\\item $\\Sigma^+$ is the positive semi-definite (PSD) subset of $\\Sigma$.\n\\item $\\Sigma^{++}$ is the positive definite (PD) subset of $\\Sigma$.\n\\item $\\Sigma^{\\mdot{}\\,}$ is the subset of matrices in $\\Sigma$ with non-zero principal minors\\footnote{Note that $\\Sigma^{\\mdot{}\\,}$ is Zariski open, while $\\Sigma^{+},\\Sigma^{++}$ are described by inequalities. }. \n\\item $\\hat{\\Sigma}$ is the subset of $\\Sigma$ consisting of matrices with ones along the diagonal. We also set $\\hat{\\Sigma}^+\\stackrel{\\vartriangle}{=}\\hat{\\Sigma}\\cap \\Sigma^+$, $\\hat{\\Sigma}^{++}\\stackrel{\\vartriangle}{=}\\hat{\\Sigma}\\cap \\Sigma^{++}$, and $\\hat{\\Sigma}^{\\mdot{}\\,}\\stackrel{\\vartriangle}{=}\\hat{\\Sigma}\\cap \\Sigma^{\\mdot{}\\,}$. \n\\item $\\mathrm{loc}(G)$ is the set of covariance matrices in $\\mathrm{Loc}(G)\\cap \\mathcal{N}$. \n\\item $f_G$ is the rational parametrization defined in II.B.\n\\item Given $S\\subset \\Sigma$, $[S]_{}$ and $[S]_Z$ are its standard and Zariski closures\\footnote{The closure is always taken inside the affine complex space.}, respectively.\n\\item Given $S\\subset \\Sigma$, $I(S)$ is the ideal of polynomials that vanish on $[S]_Z$. \n\\item Given an ideal $I$, the associated algebraic set is given by\n\\[\n{V}(I)=\\{ x\\in \\mathbb{C}^n| f(x)=0 \\quad \\forall f\\in I\\}.\n\\]\n\\item $X_G\\stackrel{\\vartriangle}{=} [\\mathrm{loc}(G)]_Z$, $\\mpf_G\\stackrel{\\vartriangle}{=} I(X_G)$, $\\hat{X}_G\\stackrel{\\vartriangle}{=} [\\mathrm{loc}(G)\\cap \\hat{\\Sigma}]_Z$.\n\\item $Y_G$ is the closure of $\\hat{X}_G$ inside $\\mathbb{P}^{n \\choose 2}$.\n\\end{itemize}\n\n\\subsection{Overview of main results} \n\nOur main purpose is to show that the computational tools in algebra are relevant for addressing the following problem: \n\n\\begin{problem}\nGiven two DAGs, determine if they are isomorphic.\n\\end{problem}\n\nOur starting point is to show that isomorphism and $\\mathcal{N}^{+}$-isomorphism are equivalent for DAGs (see Section II.C). It is well known that checking $\\mathcal{N}^{+}$-equivalence reduces to checking equality of algebraic subsets inside the positive definite cone (see for instance \\cite{drt08,drt09,mat97,sul08}). \nThis follows from the next proposition:\n\\begin{proposition}[Lemma 2.8 in \\cite{std05}]\nLet $X \\sim N(\\mu,\\sigma)$ be an $m$-dimensional Gaussian vector and $A,B,C \\subset [m]$ be pairwise disjoint index sets. Then $X_A\\perp\\!\\!\\!\\perp X_B | X_C$ if and only if the submatrix $\\sigma_{AC,BC}$ has rank equal to the rank of $\\sigma_{CC}$. Moreover, $X_A\\perp\\!\\!\\!\\perp X_B | X_C$ if and only if $X_a\\perp\\!\\!\\!\\perp X_b | X_C$ for all $a \\in A$ and $b \\in B$.\n\\label{pro_minors}\n\\end{proposition}\n\\begin{comment}\n\\begin{proof}\n\\ifcomplete\nThe proof for the case where $\\sigma_{CC}$ is non-singular can be found in \\cite{drt08}. The general case follows from the fact that $X_A\\perp\\!\\!\\!\\perp X_B | X_C$ if and only if $X_A\\perp\\!\\!\\!\\perp X_B | X_{C'}$ where $C'$ is a maximal subset of $C$ such that $X_{C'}$ is not singular, that is,  no coordinate of $X_{C'}$ can be written as a linear combination of the others. To see this, note that \n\\begin{align*}\nI(X_A;X_B|X_C)&=I(X_A;X_B|X_{C'},f(X_{C'}))\\\\\n&=I(X_A;X_B|X_{C'}).\n\\end{align*}\nThen $I(X_A;X_B|X_C)=0\\iff I(X_A;X_B|X_{C'})=0.$\n\\else \nsee~\\cite{RP14-arxiv}.\n\\fi\n\\end{proof}\n\\end{comment}\n\\begin{remark}\nNote that the rank constraint is equivalent to vanishing of the minor $|\\sigma_{AC',BC'}|$ for a maximal $C'\\subset C$\nsuch that $X_{C'}$ is non-singular  \\footnote{A vector random variable is said to be non-singular if its distribution\nadmits a density w.r.t. product Lebesgue measure.}.\n\\end{remark}\n\nProposition \\ref{pro_minors} enables us to think algebraically and\/or geometrically when deciding Gaussian equivalence. Indeed, \nit states that $\\mathrm{loc}(G)\\cap \\Sigma^{++}$ can be identified with the positive definite subset of the real solutions to the polynomial equations generated by the implied relations in $G$. Working with such subsets, however, is not convenient from a computational point of view. This motivates the next problem:\n\\begin{problem}\nGive an algebraic description of $\\mathrm{loc}(G)$.\n\\end{problem}  \n\nLet ${J}_{G}$ be the ideal generated by the minors $|\\sigma_{iK,jK}|$ of the implied relations $i\\perp\\!\\!\\!\\perp j|K\\in \\mathcal{C}_G$ inside $\\mathbb{C}[\\Sigma]$. Similarly, the minors of imposed relations of $G$ generate an ideal $I_G\\subset J_G$ in $\\mathbb{C}[\\Sigma]$. Note that this ideal coincides with that generated by the toposorted imposed relations. The corresponding ideals generated inside $\\mathbb{C}[\\hat{\\Sigma}]$ are denoted by $\\hat{I}_G,\\hat{J}_G$.\nWith the established notation, for example, Proposition~\\ref{pro_minors} implies\n\\begin{equation}\\label{eq:vcmloc}\n\t\t{V}(I_G) \\cap \\Sigma^{++}\\cap \\mathbb{R}^{n+1\\choose 2} = \\mathrm{loc}(G) \\cap \\Sigma^{++}. \n\\end{equation}\nWe address the above problem by identifying $X_G$ with an irreducible component of ${V}(I_G)$. \nIt is a curious fact that the points in ${V}(I_G)\\cap \\hat{\\Sigma}^{++}$\ncorrespond to covariances of circularly symmetric Gaussians that satisfy the CI constraints of $G\/\\mathbb{C}$. Thus if we work\nwith complex Gaussians, we may avoid intersecting with the reals in (\\ref{eq:vcmloc}). \n\n\nIn Section II, we first prove some geometric results, which can be summarized in the following diagram\n\\begin{center}\n\\tikzset{node distance=1.5cm, auto}\n\\begin{tikzpicture}\n  \\node (D0) {$\\mathrm{Im}f_G\\cap \\Sigma^{+}$};\n  \\node (D1) [right of=D0,node distance=1.75cm]{$=\\mathrm{loc}(G)\\subset$};\n  \\node (D2) [right of=D1,node distance=2.25cm]{$[\\mathrm{loc}(G)\\cap \\Sigma^{++}]_{}\\subset$};\n  \\node (D3) [right of=D2, node distance=2.2cm]{$X_G$};\n  \\node (C0) [below of =D0, node distance=.5cm] {\\rotatebox{90}{$\\subsetneq$}};\n  \\node (C1) [below of =D1, node distance=.5cm] {\\rotatebox{90}{$\\supsetneq$}};\n  \\node (C3) [below of =D3, node distance=.5cm] {\\rotatebox{90}{=}};\n  \\node (B0) [below of =C0, node distance=.5cm] {$\\mathrm{loc}(G)\\cap \\Sigma^{++}$};\n  \\node (B1) [below of =C1, node distance=.5cm] {$V(I_G)$};\n  \\node (B3) [below of =C3, node distance=.5cm] {$[V(I_G)\\cap \\Sigma^{\\mdot}\\,]_Z$};\n\\end{tikzpicture}\n\\end{center}\nThe same inclusions hold if we replace $(I_G,\\Sigma)$ with $(\\hat{I}_G,\\hat{\\Sigma})$, $\\Sigma^{\\mdot}\\,$ with\n$\\Sigma^{++}$, or $I_G$ with $J_G$.  \n\nIt is known that $[\\mathrm{loc}(G)\\cap \\Sigma^{++}]_Z$ is a complex irreducible rational algebraic variety, cf.~\\cite{sul08}. Here we further show that it coincides with $X_G$ and characterize $\\mathfrak{p}_G=I(X_G)$ in two different ways: as the saturated ideal of $I_G$ at $\\theta_0=\\prod_{A\\subset [n]}(|{\\Sigma}_{AA}|)$ (Conjecture 3.3 in \\cite{sul08}), and as the unique minimal prime of $I_G$ contained in the maximal ideal $\\mathfrak{m}_I$ at the identity. We thus have the following relations inside $\\mathbb{C}[{\\Sigma}]$:\n\\begin{align*}\n&I_G\\subset J_G\\subset S^{-1}J_G\\cap {\\mathbb{C}}[\\Sigma]= S^{-1}I_G\\cap \\mathbb{C}[\\Sigma]\\\\\n&=I(\\mathrm{loc}(G)\\cap\\Sigma^{++})=\\mathfrak{p}_G\\subset \\mathfrak{m}_I,\n\\end{align*}\nwhere $S=\\{\\theta_0^n|n>0\\}$. One can replace $(\\mathrm{loc}(G),{\\Sigma})$ with $(\\mathrm{loc}(G)\\cap \\hat{\\Sigma},\\hat{\\Sigma})$ in the above. We note that the above relations hold verbatim over $\\mathbb{Z}[\\Sigma]$ and other base rings.\nOur main statement, shown in \\ref{sec:algrep}, is that two DAGs $G,G'$ are isomorphic if and only if\n\\[\nS^{-1}I_G\\cap \\mathbb{C}[\\Sigma]^\\Pi=S^{-1}I_{G'}\\cap \\mathbb{C}[\\Sigma]^\\Pi.  \n\\]\n\nWe use the above results to provide a randomized algorithm for testing DAG isomorphism in \\ref{sec:rand}. \n\\ifcomplete Section III concerns the special case of directed tree models. In particular, we show that \n$\\hat{I}_T$ is a prime ideal for a tree model $T$ and hence $\\hat{I}_T=I(\\mathrm{loc}(T)\\cap \\hat{\\Sigma})$. This is analogous to primality of $J_T$, the ideal of implied relations, shown in \\cite{sul08} (see Corollary 2.4 and Theorem 5.8). \nWe thus have that two directed tree models $T$ and $T'$ are equal if and only if $\\hat{I}_T=\\hat{I}_{T'}$. Moreover, we use our randomized algorithm to list the isomorphism classes of directed tree models for $n=4,5,$ and $6$ nodes. \\else We then use the algorithm to list the isomorphism classes of directed tree models for $n=4$ and $5$ nodes. We also include the list for $n=6$ in the extended version of the paper. There, we further discuss some special properties of directed tree models. In particular, we show that \n$\\hat{I}_T$ is a prime ideal for a tree model $T$ and hence $\\hat{I}_T=I(\\mathrm{loc}(T)\\cap \\hat{\\Sigma})$. This is analogous to primality of $J_T$, the ideal of implied relations, shown in \\cite{sul08} (see Corollary 2.4 and Theorem 5.8). \n\n\\fi  \nThe number of isomorphism classes of directed tree models  found by our procedure is $1,1,2,5,14,42,142,...$ for $n\\ge 1$. Curiously, the first 6 numbers are Catalan but the 7th is not.\n\n\\section{Main results}\n\\subsection{Factorization and local Markov properties}\nIn this section we show that isomorphic DAGs factorize the same set of probability distributions modulo the labeling of the variables. A theorem of Lauritzon (see \\cite[Theorem 3.27]{lau96}) says that a non-singular measure satisfies the local Markov property if and only if its density factorizes. Let $(H,\\mathcal{H})$ be as in Definition \\ref{def_fac}. One can further state:\n\\begin{proposition} Let ${G}$ be a DAG and $P$ a probability measure defined on $(H,\\mathcal{H})$. The following are\n\tequivalent:\n\t\\begin{enumerate}\n\t\t\\item $P$ factorizes w.r.t to ${G}$.\n\t\t\\item $P$ satisfies all imposed constraints~\\eqref{eq:topoall} w.r.t ${G}$.\n\t\t\\item $P$ satisfies topologically sorted constraints~\\eqref{eq:topomark} w.r.t. $G$.\n\t\\end{enumerate}  \n\tIn particular, $\\mathrm{Fac}(G)=\\mathrm{Loc}(G)$.\n\\label{pro_locfac}\n\\end{proposition}\n\\begin{proof}\n\\ifcomplete\n$1\\implies 2 \\implies 3$ are obvious.  We show $3\\implies 1$. The main step is to show that if $1\\perp\\!\\!\\!\\perp 3| 2$, then there exists a conditional probability kernel $K_{3|2}$ such that for all $\\mathcal{A}\\in \\mathcal{H}_3$ we have\n\\[\n\\int K_{3|12}(\\mathcal{A}|x_1,x_2)dP_{X_1X_2}=\\int K_{3|2}(\\mathcal{A}|x_1)dP_{X_1X_2}.\n\\]\nThe general result then follows from this by induction. Note that the inductive step requires $P$ to satisfy only the toposorted constraints of $G$. Without loss of generality, let $K_{3|12}$ and $K_{1|2}$ be regular branches of conditional probabilities. \nSet\n\\[\nK_{3|2}(\\mathcal{A}|x_2):=\\int K_{3|12}(\\mathcal{A}|x_1,x_2)K_{1|2}(dx_1|x_2).\n\\]\nWe want to prove that $K_{3|2}$ is a regular branch of conditional probabilities $P_{X_3|X_2}$. Clearly,  $K_{3|2}(\\cdot|x_2)$ is a probability measure for all $x_2$. Now fix $\\mathcal{A}\\in \\mathcal{H}_3$. By \\cite[Theorem I.6.3]{cin11},  $K_{3|2}(\\mathcal{A}|\\cdot)$ is measurable as well, hence,  it is a regular branch of conditional probabilities.\n\nClaim: $K_{3|2}=K_{3|12}$. Suppose this is not the case. Then there exists $\\epsilon>0$ such that\n\\[\n\\mathcal{A}=\\{K_{3|12}(\\mathcal{L}|x_1,x_2)>K_{3|2}(\\mathcal{L}|x_2)+\\epsilon\\}\n\\]\nhas non zero probability for some $\\mathcal{L}\\in \\mathcal{H}_3$. Let $\\mathcal{F}_I$ be the $\\sigma$-algebra generated by $X_I$. By the local Markov property\n\\begin{align*}\n\\mathbb{E}[1_\\mathcal{L}1_\\mathcal{A}]&=\\mathbb{E}[\\mathbb{E}_{\\mathcal{F}_{1,2}}[1_\\mathcal{L}1_\\mathcal{A}]]=\\mathbb{E}[1_\\mathcal{A}\\mathbb{E}_{\\mathcal{F}_{2}}[1_\\mathcal{L}]]\\\\\n&=\\int_{ \\mathcal{A}} K_{3|2}(\\mathcal{L}|x_2)dP_{X_1X_2}.\n\\end{align*}\nNow by direct computation\n\\begin{align*}\n\\mathbb{E}[1_\\mathcal{L}1_\\mathcal{A}]&=\\int_{\\mathcal{A}}K_{3|12}(\\mathcal{L}|x_1,x_2)dP_{X_1X_2} \\\\\n&>\\int_{\\mathcal{A}} K_{3|2}(\\mathcal{L}|x_2)dP_{X_1X_2}+\\epsilon \\mathbb{P}[\\mathcal{A}]\\\\\n&=\\mathbb{E}[1_\\mathcal{L}1_\\mathcal{A}]+\\epsilon \\mathbb{P}[\\mathcal{A}],\n\\end{align*}\nwhich is a contradiction. This completes the proof.\n\\else\nsee~\\cite{RP14-arxiv}.\n\\fi\n\\end{proof}\n\n\n\n\n\n\\subsection{Weak limits of factorable Gaussians}\n\nThis section provides a characterization of the singular distributions in $\\mathrm{loc}(G)$ as the weak limit of sequences in $\\mathrm{loc}(G)\\cap \\Sigma^{++}$. Note that since $(H,\\mathcal{H})$ is a topological space, weak convergence $P_{X_n}\\stackrel{w}\\rightarrow P_X$ is well-defined .\n\n\\begin{comment} By a weak limit we mean:\n\n\\begin{definition}\nLet $P_{X_{n}}$ and $P_X$ be defined on $(H,\\mathcal{H})$. We say that $P_{X_{n}}$ converges weakly to $P_X$ and write $P_{X_{n}}\\stackrel{w}{\\rightarrow} P_X$ if\n\\[\nP_X[\\partial E]=0\\implies P_{X_{n}}[E]\\rightarrow P_X[E]\n\\]\nfor all $E\\in \\mathcal{H}$. \n\\end{definition}\n\\end{comment}\n\\ifcomplete We note that in the case of Gaussians $X_n\\sim N(0,\\sigma_n), X\\sim N(0,\\sigma)$, $P_{X_n}\\stackrel{w}{\\rightarrow} P_X$ is equivalent to $\\sigma_n\\rightarrow \\sigma$ in the standard metric.\\else \\fi\nTo characterize $\\mathrm{loc}(G)\\cap \\partial \\mathcal{N}^+$, we shall find it useful to work with the parametrization \n\\begin{equation}\nX_i=\\sum_{j<i} \\alpha_{ij} X_j+\\omega_iZ_i,\n\\label{eq_rational}\n\\end{equation} \nwhere $Z_i$'s are independent standard Gaussians. Suppose that  $\\alpha_{ij}= 0$ for all $(i,j)\\notin E$, where $E$ denotes the set of (directed) edges of $G$.\nThen this parametrization gives a polynomial map $f_G:\\mathbb{R}^{|E|+n}\\mapsto \\mathbb{R}^{n+1 \\choose 2}$, sending $\\{\\alpha_{ij},\\omega_i\\}$ to $\\mathrm{cov}(X)$. Indeed, starting from (\\ref{eq_rational}), one can write\n\\[\n\\sigma_{ik}=\\sum_{j<i}\\alpha_{ij} \\sigma_{jk}+\\omega_i\\gamma_{ik}\n\\]\nwhere $\\gamma_{ik}=\\mathrm{Cov}(Z_i,X_k)$, $\\sigma_{ik}=\\mathrm{Cov}(X_i,X_k)$. Note that $\\gamma_{ik}=0$ for $k<i$. With this notation, we can write\n\\begin{align*}\n\\gamma_{ki}&=\\sum_{j>i}\\alpha_{ij}\\gamma_{kj}+\\omega_i\\delta_{ik}=\\sum_{j>i}\\gamma_{kj}\\alpha^*_{ji}+\\omega_i\\delta_{ik}.\n\\end{align*}\nSet $\\Gamma:=[\\gamma_{ij}],A:=[\\alpha_{ij}],\\Omega:=[\\omega_{ii}],\\Sigma:=[\\sigma_{ij}]$. We can write the above equations in matrix form:\n\\[\n\\Sigma=A\\Sigma+\\Omega\\Gamma, \\quad \\Gamma=\\Gamma A^*+\\Omega.\n\\]\nHence,\n\\[\n\\Sigma=(I-A)^{-1}\\Omega^2(I-A^*)^{-1}.\n\\]\nThe image of $f_G$ is Zariski dense in $[\\mathrm{loc}(G)\\cap \\Sigma^{++}]_Z$:\n\\begin{proposition}[\\it Proposition 2.5 in \\cite{sul08}]\\label{prop:irred}\nLet $G$ be a DAG and $E$ be its set of edges. Then $[\\mathrm{loc}(G)\\cap \\Sigma^{++}]_Z$ is a rational affine irreducible variety of dimension $n+|E|$.\n\\begin{comment}\n\\textbf{TODO:} Replace previous paragraph with a rigorous statement $[\\mathrm{loc}(G)\\cap \\Sigma^{++}]_Z$ is rational affine\nirreducible variety of dimension blah. Reference the literature where this has been previously established. Make sure\ndefinition of $f_G$ appears in the ISIT version (it wasn't before).\n\\end{comment}\n\\end{proposition}\n\nThe next Proposition shows that $$X_G=[\\mathrm{loc}(G)\\cap \\Sigma^{++}]_Z.$$\n\n\\begin{proposition} Let $G$ be a DAG. Then\n\\begin{enumerate}[(a)]\n\\item $\\mathrm{loc}({G})\\cap \\Sigma^{++}$ is dense in $\\mathrm{loc}({G})$.\n\\item $\\mathrm{loc}(G)\\cap \\hat{\\Sigma}^{++}$ is dense in $\\mathrm{loc}(G)\\cap \\hat{\\Sigma}$. \n \\end{enumerate}\n\\label{pro_dense}\n\\end{proposition}  \n\n\\begin{proof}\n\\ifcomplete\nFor part (a), we want to show $\\mathrm{loc}(G)\\subset [\\mathrm{Im}f_G\\cap \\Sigma^{++}]$. Given a Gaussian $X\\in \\mathrm{Loc}(G)$, start with a representation\n\\[\nX_i=\\sum_{j<i} \\alpha_{ij} X_j+\\omega_iZ_i,\n\\]\nwhere $Z_{i}$'s are i.i.d. Gaussians. We need to show that there exists a $G$-compatible representation $\\{\\alpha_{ij}',\\omega_i\\}$ for $X$, i.e., $\\alpha'_{ij}= 0$ for all adjacent nodes $i,j$ in $G$. We may assume by induction that the $\\alpha_{ij}$'s are $G$-compatible for $i,j<n$. Now write\n\\[\nX_n=X_{\\mathbf{pa}}+X_{\\mathbf{npa}}+\\omega_nZ_n,\n\\]\nwhere $$X_{\\mathbf{pa}}=\\sum_{i\\in \\mathbf{pa}(n)} \\alpha_{in} X_i, \\,\\,X_{\\mathbf{npa}}=\\sum_{i\\in ([n-1]\\backslash\\mathbf{pa}(n)} \\alpha_{in} X_i.$$\n Note that, for general random variables $A,B,C$, we have\n\\[\nA\\perp\\!\\!\\!\\perp (B+C) |C\\iff A\\perp\\!\\!\\!\\perp B|C.\n\\]\nIt thus follows that\n\\[\nX_{\\mathbf{n}\\mathbf{pa}(n)}\\perp\\!\\!\\!\\perp X_{\\mathbf{n}\\mathbf{pa}(n)}+\\omega_nZ_n|X_{\\mathbf{pa}(n)}.\n\\]\nNow observe that for independent random variables $A,Z$\n\\begin{align*}\nA\\perp\\!\\!\\!\\perp A+Z \\implies \\mathbb{E}[(A+Z)A]=\\mathbb{E}[A+Z]\\mathbb{E}[A],\n\\end{align*}\nwhich implies $\\mathbb{E}[A^2]=\\mathbb{E}[A]^2$. In particular, if $A$ is a Gaussian then it must be a constant. It follows from this observation that $X_{\\mathbf{n}\\mathbf{pa}(n)}$ is a linear function of $X_{\\mathbf{pa}(n)}$, say $X_{\\mathbf{n}\\mathbf{pa}(n)}=cX_{\\mathbf{pa}(n)}$. The $G$-compatible $\\alpha'_{in}$'s are then obtained by setting $\\alpha'_{in}=(1+c)\\alpha_{in}$ if $i\\in \\mathbf{pa}(n)$ and $\\alpha'_{in}=0$ otherwise.\n\n\nFor part (b), given $\\sigma\\in \\mathrm{loc}(G)\\cap \\hat{\\Sigma}^{++}$, we can find a sequence $\\sigma_n$ in $\\mathrm{loc}(G)\\cap \\Sigma^{++}$ that converges to $\\sigma$. Pass to a subsequence with $\\sigma_{ii}\\neq 0$ for all $i$. Normalize the coordinates (using the defining equations) along $\\sigma_{ii}$'s to obtain a sequence in $\\mathrm{loc}(G)\\cap \\hat{\\Sigma}^{++}$. \nNormalization is continuous around $\\sigma$. Thus, the new sequence convergences to $\\sigma$ as well.\n\\else\nsee~\\cite{RP14-arxiv}.\n\\fi\n\\end{proof}\n\n\\ifcomplete\n\\begin{remark} The above proof also shows that \\mbox{$\\mathrm{loc}(G)=\\mathrm{Im}f_G\\cap \\Sigma^{+}$}.\n\\end{remark}\n\\fi\nThis proposition shows that $X_G$ contains all $G$-factorable Gaussians. There are, however, (singular) covariances on $X_G$ that are not $G$-compatible. In other words, unlike independence, conditional independence is not preserved under weak limits as shown in the following example.\n\\begin{example}[$\\mathrm{loc}(G)$ is not closed] \nLet $X_n \\sim N(0,1)$, $W_n\\sim N(0,1)$ be independent Gaussians. Set $Z_n=X_n$ and $Y_n=\\frac{1}{n} X_n+\\frac{\\sqrt{n^2-1}}{n}W_n$. Then $X_n\\perp\\!\\!\\!\\perp Z_n|Y_n,W_n$ for all $n$ and $P_{X_n,Y_n,Z_n,W_n}\\stackrel{w}{\\rightarrow} P_{X,Y,X,W}$ with $X\\sim N(0,1),W\\sim N(0,1)$ and $Y=W$. However, $$X\\not\\perp\\!\\!\\!\\perp X|W.$$ \nThus the closure of $\\mathrm{loc}(G)\\cap \\hat{\\Sigma}^{++}$ strictly contains \\mbox{$\\mathrm{loc}(G)\\cap \\hat{\\Sigma}$}. \n\\label{ex_cicounter}\n\\end{example}\n\n\n\\begin{remark}\n\\indent\n In general, the weak convergence of the joint $P_{X^{(n)}}\\stackrel{w}{\\rightarrow} P_{X}$ does not imply that of the conditional kernels $P_{X^{(n)}_i|X^{(n)}_j}\\stackrel{w}{\\rightarrow} P_{X_i|X_j}$. If the latter conditions are also satisfied, then conditional independence is preserved at the (weak) limit\\footnote{This follows directly from the lower semi-continuity of divergence.}. \n\\end{remark}\n\n\n\\ifcomplete\n\\subsection{DAG isomorphism}\nThe next result states that isomorphism of DAGs can be decided inside $\\mathcal{N}^+$\\footnote{This statement generalizes to the class of chain graphs. See \\cite{lev01} for details.}:\n\\begin{proposition}[Theorem 5.1 in \\cite{lev01}]\nLet $G,G'$ be DAGs. Then the following are equivalent:\n\\begin{enumerate}[(a)]\n\\item $G$ and $G'$ are equal.\n\\item $G$ and $G'$ are $\\mathcal{N}$-equal.\n\\item $G$ and $G'$ are $\\mathcal{N}^+$-equal.\n\\end{enumerate}\n\\label{pro_dagiso}\n\\end{proposition}\nThis property is also known as the faithfulness of Gaussians in the statistics literature (cf.~ \\cite{uhl13}). \nLet us point out that, in general, $\\mathcal{N}^{+}$-isomorphic models are not $\\mathcal{N}$-isomorphic as shown in the next example.\n\n\\begin{example}\nConsider the models\n\\begin{align*}\n{G}_1&: 1\\perp\\!\\!\\!\\perp 3|2\\quad\\&\\quad 1\\perp\\!\\!\\!\\perp 2|3\\quad\\&\\quad 2\\ci3|1\\\\\n{G}_2&: 1\\perp\\!\\!\\!\\perp 2\\perp\\!\\!\\!\\perp 3\n\\end{align*}\nA non-singular Gaussian belongs to the first model if and only if it belongs to the second model. However, a Gaussian $X_1=X_2=X_3$ is only compatible with the first model.\n\\label{ex_nplus}\n\\end{example}\n\n\n\\begin{comment}\nThis result is also known as the faithfulness of Gaussians in the statistics literature (mentioned for instance in \\cite{uhl13}). \n\nWe need a definition for the following proof. Let $K\\subset [n]$ be a subset of nodes in a DAG $G$. Then the nodes $i,j$ are said to be {\\em $d$-connected} given $K$ if there is a sequence of distinct edges $\\{e_m\\}$ such that : 1) $e_m$'s form an undirected path from $i$ to $j$ 2) there are no edges in $\\{e_m\\}$ that form either a V-structure $r\\rightarrow k\\leftarrow s$ for some $k\\in [n]\\backslash \\{K\\cup \\mathbf{an}(K)\\}$ or a chain $r\\rightarrow k\\rightarrow s$ for $k\\in K$. Here, $\\mathbf{an}(K)$ is the set of ancestral nodes in $K$. Call such a sequence a {\\em $d$-path}. It is well known that (see the d-separation criteria in \\cite{lau96}) $I\\perp\\!\\!\\!\\perp J|K$ is an implied relation for a DAG $G$ if and only $I$ and $J$ are not d-connected given $K$ in $G$.\n\n{{\\it Proof of Proposition \\ref{pro_dagiso}}:}\nThe only non-trivial implication is  (c) $\\implies$ (a). To prove this, suppose $i$ and $j$ are d-connected given $K$ in $G$ with $i>j$. It suffices to show \\mbox{$X_i\\not\\perp\\!\\!\\!\\perp X_j|X_K$} for some \\mbox{$X\\in\\mathrm{Loc}(G)\\cap \\mathcal{N}^{+}$}. \nTo simplify the notation, let us assume that $K$ is a singleton and consider a parametrization of $X$ \n\\[\nX_r=\\sum_{s\\in\\mathrm{pa}(r)} \\lambda_{is}X_s+\\nu_r,\n\\]\nwhere $\\nu_r$'s are independent standard Gaussians. Let $\\mathcal{D}(a,b|C)$ be the set of $d$-paths from $a$ to $b$ given $C$, and define the polynomial\n\\[\nh_{ab|C}(\\lambda)=\\sum_{p\\in\\mathcal{D}(a,b|C)} \\prod_{r\\rightarrow s\\in p} \\lambda_{rs}.\n\\]\nIf $K<i$, then $X_i$ can be re-written as\n\\begin{align*}\nX_i&=h_{ji|K}X_j+h_{Ki|\\emptyset}X_K+\n\\sum \\alpha_k\\nu_k+\\sum_{s\\in\\mathrm{rest}}\\lambda_{is} X_s\n\\end{align*}\nand if $K>i$ then\n\\begin{align*}\nX_K&=h_{iK|\\emptyset}X_i+h_{jK|\\emptyset} X_j+\\sum \\alpha_k\\nu_k+\\sum_{s\\in\\mathrm{rest}}\\lambda_{is} X_s,\n\\end{align*}\nwhere ${\\mathrm{rest}}$ is in each case a set consisting of irrelevant ancestral nodes. \nSince $\\sigma_{jj}\\neq 0$, we can see that \\mbox{$X_i\\perp\\!\\!\\!\\perp X_j|X_K$} if and only if $g(\\lambda)=0$ where\n\\[\ng(\\lambda)=\\left\\{\\begin{array}{ll}h_{ji|K}&K<i\\\\h_{iK|\\emptyset}h_{jK|\\emptyset}&K>i.\\end{array}\\right.\n\\]\nBy assumption, either $\\mathcal{D}(i,j|K)$ is not empty or $\\mathcal{D}(i,K|\\emptyset)$ and $\\mathcal{D}(j,K|\\emptyset)$ are not empty. Moreover, the monomials appearing in $g(\\lambda)$ are distinct (they differ in at least one index $rs$). Thus $g(\\lambda)$ is a non-zero polynomial. \nThe set of non-singular Gaussians with $X_i\\perp\\!\\!\\!\\perp X_j|X_K$ corresponds to the real roots of $g(\\lambda)$, which is an algebraic set of dimension at most $|E|-1$ inside $\\mathbb{R}^{|E|}$, where $E$ is the set of edges of $G$. Picking $\\lambda$ outside of this real algebraic subset results in a Gaussian in which $X_i\\not\\perp\\!\\!\\!\\perp X_j|X_K$.\n\\end{comment}\n\\fi\n\n\n\\subsection{DAG varieties and ideals}\nHere we provide some algebraic and geometric descriptions for $\\mathrm{loc}(G)$:\n\\begin{theorem} Let $G$ be a DAG and let $\\theta_0=\\prod_{A\\subset [n]}(|{\\Sigma}_{AA}|)$.\n\\noindent\n\\begin{enumerate}[(a)]\n\\item There is a Zariski closed subset $B_G$ so that \n$$ V(I_G) = X_G \\cup B_G $$\nwhere $B_G \\subseteq V(\\theta_0) = \\{\\theta_0=0\\}$.\n\\item Let $\\mpf_G=I(\\mathrm{loc}(G))$ so that $X_G=V(\\mpf_G)$. Then $\\mpf_G$ is a prime ideal obtained by saturating $I_G$\n\\begin{equation}\\label{eq:mpfg}\n\t\t\\mpf_G = S^{-1} I_G \\cap \\mathbb{C}[\\Sigma] \n\\end{equation}\nat the multiplicatively closed set $S=\\{\\theta_0^n, n=1,\\ldots\\}$.\n\\item $X_G$ is smooth inside $\\Sigma^{\\mdot}$.\n\\end{enumerate}\n\\label{pro_real2}\n\\label{pro_embcomp}\n\\label{pro_pdecom}\n\\label{cor_conj}\n\\label{thm:thm1}\n\\end{theorem}\n\\begin{remark}\n\\noindent\n\\begin{enumerate} [(a)]\n\\item In Theorem \\ref{pro_pdecom}b, we can replace $I_G$ with $J_G$. \\ifcomplete \\else Analogous statements hold over $\\mathbb{C}[\\hat{\\Sigma}]$ as shown in ~\\cite{RP14-arxiv}.\\fi\n\\item It follows that $V(I_G)$ and $V(J_G)$ do not miss a single $G$-compatible Gaussian, but can add some bad components to the boundary $\\partial \\Sigma^{++}$. Theorem \\ref{cor_conj}b states this in algebraic terms and provides a proof of Conjecture 3.3 in \\cite{sul08}. Theorem 8 in \\cite{gar05} gives an analogous result for the implied ideals of discrete random variables. \n\\item In Theorem \\ref{cor_conj}b, one can replace $\\theta_0$ with the product of principal minors $|\\sigma_{KK}|$ where $K$ appears as a conditional set in some imposed relation $i\\perp\\!\\!\\!\\perp j|K$.\n\\item There are many equivalent ways to recover $\\mpf_G$ from $I_G$ besides~\\eqref{eq:mpfg}. Indeed, (e.g.~\\cite[Chapter\n4]{ati69}) we have\n$$ \\mpf_G = (I_G:\\theta_0^m)$$\nfor all $m$ sufficiently large. Another characterization is from primary decomposition of $I_G$:\n$$ I_G = \\mpf_G \\cap \\mq_1 \\cdots \\mq_r\\,,$$\nwhere $\\mpf_G$ is the unique component that is contained in maximal ideal $\\mathfrak{m}_x$ corresponding to covariance\nmatrix $x$ with non-singular principal minors (e.g. identity).\n\\ifcomplete\n\\item One can ask if Theorem 1 generalizes, i.e., if $[V(I_{G})\\cap \\Sigma^{++}]_Z=[V(I_{G})\\cap \\Sigma^{\\mdot\\,}]_Z$ for any conditional independence model ${G}$. This is equivalent to asking if  some component of $V(I_{G})$ can intersect $\\Sigma^{\\mdot\\,}$ but avoid $\\Sigma^{++}$. This question appears in \\cite{mat05}, and remains open so far as we know. \n\\fi\n\\end{enumerate}\n\\end{remark}\n\n\\begin{proof}\n\\ifcomplete\nLet us consider the ring $S^{-1} \\mathbb{C}[\\Sigma]$ and \nif $ij$ is an edge in $G$ or if $i=j$, call $\\sigma_{ij}$ an edge variable and the rest are non-edge variables. Denote by $\\Sigma_{\\mathrm{edge}}$ the\nsubspace of $\\Sigma=[\\sigma_{ij}]$ corresponding to the edge variables. Note that the ideal $S^{-1}I_G$ has one generator\n$g_{ij}$ \nfor every non-edge variable $\\sigma_{ij}$. This generator corresponds to the constraint $i\\perp\\!\\!\\!\\perp j | K$ where $i<j$ and\n$K \\subset\\{k: k<j\\}$ are parents of $j$ and we have\n$$ g_{ij} = \\sigma_{ij} |\\sigma_{KK}| - h_{ij}\\,. $$\nHere $|\\sigma_{KK}|$ and $h_{ij}$ are polynomials in $\\{\\sigma_{a,b}, a\\le b<j\\}$. Now introduce a lexicographic ordering\non pairs $(i,j)$\\footnote{For instance, take $(i,j)<(i',j')$ if $i<i'$ or $i=i'$ and $j<j'$ in the topological sort.} and among all non-edge variables entering into polynomial on the\nright consider the maximal one -- denote it $\\sigma_{i',j'}$. This variable has its corresponding generator:\n$$ g_{i'j'} = \\sigma_{i'j'} |\\sigma_{K'K'}| - h_{i'j'}. $$\nThus multiplying by a suitable power of $|\\sigma_{K' K'}|$ we can write\n$$ |\\sigma_{K' K'}|^r g_{ij} = |\\sigma_{K' K'}|^r (\\sigma_{ij} |\\sigma_{KK}| - h_{ij}) $$\nand now every occurrence of $|\\sigma_{K' K'}| \\sigma_{i' j'}$ we replace with $g_{i'j'}+h_{i'j'}$. In the end we obtain\n$$ |\\sigma_{K' K'}|^r g_{ij} = \\sigma_{ij}  u'_{ij} - h'_{ij}\\,,$$\nwhere the expression on the right no longer contains $\\sigma_{i',j'}$ or any larger (w.r.t. ordering of pairs) variable.\nThus, repeating similar steps in the end we obtain expression:\n$$ \\gamma_{ij} g_{ij} = \\sigma_{ij} \\tilde u_{ij} - \\tilde h_{ij} $$\nwhere we have:\n\\begin{enumerate}\n\t\\item $\\gamma_{ij}=\\gamma_{ij}(\\sigma)$ is a polynomial with $\\mathbb{Z}$-coefficients.\n\t\\item $\\gamma_{ij}$ and $\\tilde u_{ij}$ are units in $S^{-1}\\mathbb{C}[\\Sigma]$ (equivalently, they divide $\\theta_0^m$ for\n\tsome large enough $m$).\n\t\\item $\\tilde u_{ij}$ and $\\tilde h_{ij}$ are both polynomials with $\\mathbb{Z}$-coefficients in \\textit{edge} variables $\\sigma_{a,b}$ with\n\t$a\\le b < j$.\n\\end{enumerate}\n\nConsequently, since $\\gamma_{ij}$'s are units we have\n\\begin{equation}\\label{eq:sgen}\n\t\tS^{-1}I_G = (\\sigma_{ij} \\tilde u_{ij} - \\tilde h_{ij}, (i,j)\\mbox{-- non-edge}) \n\\end{equation}\nNote that on one hand $\\mpf_G = I(X_G)$ contains $I_G$. On the other hand, by Proposition~\\ref{prop:irred}\nand~\\eqref{eq:sgen} any minimal prime above $S^{-1}I_G$ has codimension equal to $S^{-1} \\mpf_G$. Thus, if we show that\n$S^{-1} I_G$ is prime we must have\n$$ S^{-1} I_G = S^{-1}\\mpf_G $$\nand after intersecting with~$\\mathbb{C}[\\Sigma]$ conclusion~\\eqref{eq:mpfg} and the rest of the theorem follow. \n\nTo that end let $g = g(\\sigma_{\\mathrm{edge}})$ be the product of all $\\tilde{u}_{ij}$. Let $\\mathring{\\sigma}$ be the identity matrix \n\\[\n\\mathring{\\sigma}_{ij}=\\left\\{\\begin{array}{cc} 1& i=j\\\\0 &i\\neq j \\end{array}\\right.\n\\]\nand let\n\\[\n\\varphi^*:\\mathbb{C}[\\Sigma]\\rightarrow \\mathbb{C}[\\Sigma_{\\mathrm{edge}}][1\/g]\n\\]\nbe the ring map associated to the rational map $\\Sigma_{\\mathrm{edge}}\\stackrel{\\varphi}{\\to} \\Sigma$ given by\n\\begin{equation}\n\\varphi:\\sigma_{\\mathrm{edge}}\\mapsto \\left(\\sigma_{\\mathrm{edge}}, {\\tilde h_{ij}(\\sigma_{\\mathrm{edge}})\\over \\tilde u_{ij}(\\sigma_{\\mathrm{edge}})}, (i,j)\\mbox{--\nnon-edge}\\right)\\label{eq:ltt}.\n\\end{equation}\nFirst note that $\\mathring{\\sigma}$ is a $\\mathbb{C}$-point of $\\mathbb{C}[\\Sigma_{\\mathrm{edge}}][1\/g]$ since\n\\[\n\\theta_0(\\mathring{\\sigma})=1, \\,\\,g|\\theta_0^m\\implies g(\\mathring{\\sigma})\\neq 0.\n\\]\nThus we can form a commutative diagram\n\\begin{center}\n\\tikzset{node distance=1.5cm, auto}\n\\begin{tikzpicture}[auto]\n  \\node (A0){$\\mathbb{C}[\\Sigma]$};\n  \\node (A1)[right of=A0,node distance=2cm]{$\\mathbb{C}[\\Sigma_{\\mathrm{edge}}][1\/g]$};\n  \\node (B0)[below of=A0,node distance=0.5cm,right of=A0,node distance=1cm] {$\\mathbb{C}$};\n  \\path [line] (A0) --node[above]{$\\varphi^*$} (A1);\n  \\path [line] (A0) --node[below,left]{$\\mathrm{ev}_{\\mathring{\\sigma}}$} (B0);\n  \\path [line] (A1) --node[below,right]{$\\mathrm{ev}_{\\mathring{\\sigma}_{\\mathrm{edge}}}$} (B0);\n\\end{tikzpicture}\n\\end{center}\nChasing $\\theta_0$ from $\\mathbb{C}[\\Sigma]$ to $\\mathbb{C}$ in two different ways gives $\\theta_0\\notin\\ker \\varphi^*$, and thus $\\varphi^*(\\theta_0)=h\/g^m$ for $h\\neq 0$. Localizing $\\mathbb{C}[\\Sigma_{\\mathrm{edge}}][1\/g]$ at $h$ gives a diagram\n\\begin{center}\n\\tikzset{node distance=1.5cm, auto}\n\\begin{tikzpicture}[auto]\n  \\node (Am1){$\\mathbb{C}[\\Sigma]$};\n  \\node (A0)[right of=Am1,node distance=2cm]{$\\mathbb{C}[\\Sigma_{\\mathrm{edge}}][1\/g]$};\n  \\node (A1)[right of=A0,node distance=3cm] {$\\mathbb{C}[\\Sigma_{\\mathrm{edge}}][1\/gh]$};\n  \\path [line] (Am1) --node[above]{$\\varphi^*$} (A0);\n  \\path [line] (A0) --node[above]{$\\varphi^*_h$} (A1);\n  \\path [dashed] (Am1) edge [out=45,in=135,->]node[below]{$\\varphi^*_{gh}$} (A1);  \n\\end{tikzpicture}\n\\end{center}\n\nNote that $\\varphi^*_{gh}$ sends $\\theta_0$ to a unit. Hence, by the universal property  of localization (see~\\cite[Proposition 3.1]{ati69}), it extends to a map\n\\begin{align}\n\t\\varphi_{gh}^{*,e}:S^{-1}\\mathbb{C}[\\Sigma] \\to \\mathbb{C}[\\Sigma_{\\mathrm{edge}}][1\/gh]\\label{eq:morp}\\\\\n\t\\sigma_{ij}\\mapsto \\varphi^*(\\sigma_{ij}),1\/\\theta_0\\rightarrow g^m\/h\n\\end{align}\nthat is onto and has $S^{-1}I_G$ as kernel. To verify the latter claim, take $s$ in the kernel of (\\ref{eq:morp}) and write it as $s=p_\\mathrm{edge}+\\sum_{ij}q_{ij}.(\\sigma_{ij}- \\tilde h_{ij}\/\\tilde u_{ij})$ where $q_{ij}\\in S^{-1}\\mathbb{C}[\\Sigma]$ with $(i,j)\\mbox{--non-edge}$, and $p_\\mathrm{edge}$ is a polynomial in $\\sigma_{\\mathrm{edge}}$. This can be done by virtue of the binomial theorem:\n\\[\n\\sigma_{ij}^n=(\\frac{\\tilde{h}_{ij}}{\\tilde{u}_{ij}}-\\frac{\\tilde{h}_{ij}}{\\tilde{u}_{ij}}+\\sigma_{ij})^n=(\\frac{\\tilde{h}_{ij}}{\\tilde{u}_{ij}})^n+n(\\frac{\\tilde{h}_{ij}}{\\tilde{u}_{ij}})^{n-1}(\\sigma_{ij}-\\frac{\\tilde{h}_{ij}}{\\tilde{u}_{ij}})+\\cdots\n\\]\nThen $\\varphi^*(s)=p_\\mathrm{edge}$, and thus $\\varphi^*(s)=0$ gives $p_\\mathrm{edge}=0$, that is $s\\in S^{-1}I_G$. The reverse inclusion is obvious. \n\nThis establishes isomorphism of rings\n$$ S^{-1} \\mathbb{C}[\\Sigma]\/S^{-1}I_G = \\mathbb{C}[\\Sigma_{\\mathrm{edge}}][1\/gh], $$\nwhich implies that $S^{-1} I_G$ is prime, and that each local ring of $X_G \\cap \\Sigma^{\\mdot}$ is regular (since\nall local rings of  $\\mathbb{C}[\\Sigma_{\\mathrm{edge}}][1\/gh]$ are regular).\n\nGeometrically our proof corresponds to constructing a birational isomorphism:\n\\begin{center}\n\\tikzset{node distance=1.5cm, auto}\n\\begin{tikzpicture}[auto]\n  \\node (Am1){$\\Sigma_{\\mathrm{edge}}\\supset$};\n  \\node (A0)[right of=Am1,node distance=0.8cm]{$\\mathcal{U}$};\n  \\node (A1)[right of=A0,node distance=2cm] {$V(I_G)\\cap\\Sigma^{\\mdot{}}$};\n  \\path [line] (A0) --node[above]{$\\varphi{|_\\mathcal{U}}$} (A1);\n  \\path [dashed] (A1) edge [out=-135,in=-45,->]node[below]{$\\pi$} (A0);  \n\\end{tikzpicture}\n\\end{center}\nwhere $\\mathcal{U}=D(gh)$ is a distinguished open, $\\varphi|_\\mathcal{U}$ is obtained by restriction of the map given in~\\eqref{eq:ltt}($\\varphi$ is regular on $\\mathcal{U}$), and $\\pi$ is the projection from\n$\\Sigma$ to $\\Sigma_{\\mathrm{edge}}$.\n\\else \nsee~\\cite{RP14-arxiv} for details. \n\\fi\n\\end{proof}\n\n\n\\ifcomplete\n\nAlternatively, we can choose to prove our results over $\\mathbb{C}[\\hat{\\Sigma}]$. Recall that $\\hat X_G$ is defined as $[\\mathrm{loc}(G) \\cap\n  \\hat{\\Sigma}^{++}]_Z$.\n\n\\begin{theorem} Let $G$ be a DAG and let $\\hat{\\theta}_0=\\prod_{A\\subset [n]}(|{\\hat{\\Sigma}}_{AA}|)$. With the notation of Theorem~\\ref{thm:thm1},\n\\noindent\n\\begin{enumerate}[(a)]\n\\item  $\\hat {X}_G$ is an affine rational (irreducible) variety of dimension $|E|$ and $\\hat X_G = X_G \\cap \\hat \\Sigma$.\n\n\\item We have $$ V(\\hat{I}_G) = \\hat{X}_G \\cup \\hat{B}_G, $$\nwhere $\\hat B_G \\subset\\{\\hat{\\theta}_0=0 \\}$.\n\\item The prime ideal $\\hat p_G \\triangleq I(\\hat X_G)$ is obtained by saturating $\\hat{I}_G$\n\\begin{equation}\\label{eq:hmpfg}\n\t\t\\hat{\\mpf}_G = \\hat{S}^{-1} \\hat{I}_G \\cap \\mathbb{C}[\\hat{\\Sigma}] \n\\end{equation}\nat the multiplicatively closed set $\\hat{S}=\\{\\hat{\\theta}_0^n, n=1,\\ldots\\}$.\n\\item $\\hat{X}_G$ is smooth inside $\\hat{\\Sigma}^{\\mdot}$.\n\\end{enumerate}\n\\label{thm:xhat}\n\\end{theorem}\n\\begin{proof}\nWe first claim that $X_G\\cap \\hat{\\Sigma}$ is an affine rational irreducible variety that is smooth inside $\\hat{\\Sigma}^{\\mdot}$. Consider the sequence\n\\begin{equation}\nS^{-1}\\mathbb{C}[\\Sigma]\\stackrel{\\varphi^{*,e}_{gh}}{\\to} \\mathbb{C}[\\Sigma_{\\mathrm{edge}}][1\/gh]\\stackrel{\\mathrm{ev}}{\\to} \\mathbb{C}[\\hat{\\Sigma}_{\\mathrm{edge}}][1\/\\hat{g}\\hat{h}],\n\\label{eq:hatseq}\n\\end{equation}\nwhere $\\varphi^{*,e}_{gh}$ is the extended map in (\\ref{eq:morp}) and $\\hat{g}=g|_{\\hat{\\Sigma}},\\hat{h}=h|_{\\hat{\\Sigma}}$, and $\\mathrm{ev}$ is the  evaluation map $\\sigma_{ii}=1$. Observe that  $$X_G\\cap \\hat{\\Sigma}=V((I_G:\\theta_0^m))\\cap V(\\ker (\\mathrm{ev})^c)=V((I_G:\\theta_0^m)+\\ker (\\mathrm{ev})^c)=V(\\hat{I}_G:\\hat{\\theta}_0^m).$$\nAlso note that $$(\\hat{I}_G:\\hat{\\theta}_0^m)=\\hat{S}^{-1}\\hat{I}_G\\cap \\mathbb{C}[\\hat{\\Sigma}].$$ Let us show that $\\hat{S}^{-1}\\hat{I}_G$ is prime of codimension $|E|$.  \nNote that $\\varphi_{gh}^{*,e}$ is identity on the generators of $\\ker(\\mathrm{ev})^c$, i.e., $\\varphi_{gh}^{*,e}(\\ker(\\mathrm{ev})^c)=\\ker(\\mathrm{ev})$.\nThis implies that $\\ker(\\mathrm{ev} \\varphi^{*,e}_{gh})\\supset \\ker(\\varphi^{*,e}_{gh})+\\ker (\\mathrm{ev})^c$. The reverse inclusion $\\ker (\\mathrm{ev} \\varphi^{*,e}_{gh})\\subset \\ker(\\varphi^{*,e}_{gh})+\\ker( \\mathrm{ev})^c$ is obvious.\nSince both maps in (\\ref{eq:hatseq}) are onto, we have an isomorphism of integral domains\n\\begin{equation}\nS^{-1}\\mathbb{C}[\\Sigma]\/(S^{-1}I_G+\\ker({\\mathrm{ev}})^c)= \\mathbb{C}[\\hat{{\\Sigma}}]\/\\hat{S}^{-1}\\hat{I}_G= \\mathbb{C}[\\hat{\\Sigma}_{\\mathrm{edge}}][1\/\\hat{g}\\hat{h}],\n\\label{eq:hatiso}\n\\end{equation}\nwhere we used the fact that $\\ker(\\varphi^{*,e}_{gh})=S^{-1}I_G$, shown in the proof of Theorem~\\ref{thm:thm1}. This proves the primality of $\\hat{S}^{-1}\\hat{I}_G$, hence, $X_G\\cap \\hat{\\Sigma}$ is irreducible of dimension $|E|$. This further implies that the map (\\ref{eq:hatiso}) is induced by the restriction of the rational map $\\varphi|_{\\hat{\\Sigma}}$. In particular, this restriction is an isomorphism inside $\\hat{\\Sigma}^{\\mdot}$, proving the above claim. \n\nNow the proof reduces to shwoing that $\\hat{X}_G=X_G\\cap \\hat{\\Sigma}$. We know that $X_G\\cap \\hat{\\Sigma}$ is irreducible by the above discussion, and that it contains $\\hat{X}_G$ by Proposition \\ref{pro_dense}. Thus the theorem follows if we show that $\\hat{X}_G$ has dimension $|E|$.  Recall from Proposition \\ref{pro_minors} and construction of $\\varphi$ that $\\mathrm{loc}(G)\\cap \\hat{\\Sigma}^{++}$ can be obtained as the intersection of the image of $\\varphi$ with the real subset of $\\hat{\\Sigma}^{++}$, i.e., it has the structure of a real differentiable manifold of dimension $|E|$. Together with Proposition \\ref{pro_dense} and  \\cite[Proposition 2.8.14]{boc98}, this implies that\n\\[\n\\dm (\\mathbb{R}[\\hat{\\Sigma}]\/I_\\mathbb{R}(\\hat{X}_G))=|E|,\n\\]\nwhere $I_\\mathbb{R}(\\hat{X}_G)$ is the ideal of polynomials with real coefficients that vanish on $\\hat{X}_G$.\nSince the set $\\mathrm{loc}(G)\\cap \\hat{\\Sigma}$ is stable under conjugation, we have $I(\\hat{X}_G)=I_\\mathbb{R}(\\hat{X}_G)^e$. Then the going up theorem implies that $\\hat{X}_G$ has dimension $|E|$ as desired.\n\\end{proof}\n\\fi\n\nThe next example shows how Theorem~\\ref{cor_conj} can be used to construct $\\mathfrak{p}_G$ from $I_G$:\n \n\\begin{example}\nConsider the DAG\n\\begin{center}\n\\tikzset{node distance=1.5cm, auto}\n\\begin{tikzpicture}[auto]\n  \\node (A0){$$};\n  \\node (A1)[right of=A0,node distance=1.5cm] {$$};\n  \\node (A2) [right of=A1,node distance=1.5cm]{$4$};\n  \\node (A3) [below of=A0,node distance=0.5cm]{$G:1$};\n  \\node (A4) [right of=A3,node distance=1.5cm]{$2$};\n  \\node (A5) [below of=A2,node distance=1cm]{$3$};\n  \\path [line] (A3) --node[above]{$$} (A4);  \n  \\path [line] (A4) --node[above]{$$} (A5);  \n  \\path [line] (A4) --node[above]{$$} (A2);  \n  \\path [line] (A5) --node[above]{$$} (A2);  \n\\end{tikzpicture}\n\\end{center}\nThe ideal of imposed relations is generated by relations $1\\perp\\!\\!\\!\\perp 3|2$ and $4\\perp\\!\\!\\!\\perp 1|(2,3)$:\n\\[\nI_G=\\langle |\\sigma_{12,23}|,|\\sigma_{123,423}|\\rangle.\n\\]\nIt has primary components\n\\begin{align*}\nI_{G,1}&=\\langle \\sigma_{12}\\sigma_{23} - \\sigma_{13}\\sigma_{22},\n        \\sigma_{12}\\sigma_{24} - \\sigma_{14}\\sigma_{22},\n        \\sigma_{13}\\sigma_{24} - \\sigma_{14}\\sigma_{23}\\rangle\n\\end{align*}\nand\n\\[\nI_{G,2}=\\langle \\sigma_{12}\\sigma_{33} - \\sigma_{13}\\sigma_{23},\n        \\sigma_{12}\\sigma_{33} - \\sigma_{13}\\sigma_{23},\n        \\sigma_{22}\\sigma_{33} - \\sigma_{23}^2\\rangle.\n\\]\nIt can be seen that only $I_{G,1}$ intersects $\\Sigma^{\\mdot}$\\,. We thus have $\\mathfrak{p}_G=I_{G,1}$. Furthermore, $I_{G,1}$ is the unique ideal contained in the maximal ideal at the identity of $\\Sigma$, and is also equal to the saturation of $I_{G}$ at $f=\\sigma_{22}(\\sigma_{22}\\sigma_{33}-\\sigma_{23}^2)$. \n\\ifcomplete\nThe ideal of implied relations is generated by relations $1\\perp\\!\\!\\!\\perp 3|2,1\\perp\\!\\!\\!\\perp 4|2,1\\perp\\!\\!\\!\\perp 4|(2,3)$, and $1\\perp\\!\\!\\!\\perp 3|(2,4)$:\n\\[\nJ_G=\\langle |\\sigma_{12,23}|,|\\sigma_{12,42}|,|\\sigma_{123,423}|,|\\sigma_{124,324}|\\rangle.\n\\]\nIt has primary components\n\\[\nJ_{G,1}=\\langle \\sigma_{13}\\sigma_{22} - \\sigma_{12}\\sigma_{23},\\sigma_{14}\\sigma_{22} - \\sigma_{12}\\sigma_{24},\\sigma_{14}\\sigma_{23} - \\sigma_{13}\\sigma_{24} \\rangle\n\\]\nand\n\\[\nJ_{G,2}=\\langle \\sigma_{12}, \\sigma_{22},\\sigma_{24},\\sigma_{23}^2\\rangle.\n\\]\nAgain one can check that $J_{G,1}=I_{G,1}=S^{-1}J_G\\cap \\mathbb{C}[\\Sigma]$ is the unique component that is contained in the maximal ideal at the identity. Finally, we can see that $$S^{-1}\\hat{I}_G\\cap \\mathbb{C}[\\hat{\\Sigma}]=\\langle\\hat{\\sigma}_{12}\\hat{\\sigma}_{23} - \\hat{\\sigma}_{13}, \\hat{\\sigma}_{12}\\hat{\\sigma}_{24} - \\hat{\\sigma}_{14},\\hat{\\sigma}_{13}\\hat{\\sigma}_{24} -\\hat{\\sigma}_{14}\\hat{\\sigma}_{23}\\rangle$$ is the unique irreducible component of $V(\\hat{I}_G)$ that contains the origin of $\\hat{\\Sigma}$. \\\\\n\\fi\n\\label{ex:thm1}\n\\end{example}\n\n\\ifcomplete\nLet us point out that $X_G$ need not be smooth outside of $\\Sigma^{\\mdot}$ as the next example shows:\n\\begin{example} Let $G$ be the Markov chain $1\\rightarrow 2\\rightarrow 3$. Then $X_G=V(\\sigma_{22}\\sigma_{13}-\\sigma_{12}\\sigma_{23})$ is a cone and has a singularity at the origin. \n\\end{example}\n\n\\else\n\n\\fi\n\n\\ifcomplete\n\n\\begin{comment}\n\n\n\n\\begin{proposition}\n$(I_G:\\theta_0)$ is a prime ideal.\n\\label{pro:prime}\n\\end{proposition}\n\\begin{proof}\nBy Theorem 1, one can write $V(I_G)=V(I_G:\\theta_0)\\cup_{K} V(I_G+|\\sigma_{KK}|)$ as $K$ runs through the set ${\\bf pa}(i)$ for nodes $i$. The result follows from \\ref{pro:smooth} if we can show $V(I_G)$ is a reduced scheme. This can be done by showing that $I_G+|\\sigma_{KK}|$ is primary. By, \\cite{gor10},\\cite{bru01}, $V(|\\sigma_{KK}|)$ is normal integral scheme.  [TODO]: show that it intersects $V(I_G)$ transversally and deduce the result by appealing to intersection theory.\n\\end{proof}\n\\end{comment}\n\\fi\n\\subsection{Algebraic representation} \n\\label{sec:algrep}\nHere, we put together the results of the previous sections to give an algebraic\ncriteria for testing isomorphism of graphical models. We start by a result on equivalence of DAGs:\n\n\\begin{proposition}\nLet $G,G'$ be DAGs. Then $G$ is equal to $G'$ if and only if $X_G=X_{G'}$, or equivalently, if and only if $\\hat{X}_G=\\hat{X}_{G'}$.\n\\label{pro_dageq}\n\\end{proposition}\n\\begin{proof}\n\\ifcomplete\nIf $G$ is equal to $G'$, then $\\mathrm{loc}(G)=\\mathrm{loc}(G')$ by Proposition~\\ref{pro_dagiso}, and thus $X_G=X_{G'}$. Conversely, if $X_G=X_{G'}$, then $V(I_G)\\cap\\Sigma^{\\mdot}= V(I_{G'})\\cap\\Sigma^{\\mdot}$ by Theorem~\\ref{cor_conj}a. This implies that $V(I_G)\\cap \\Sigma^{++}= V(I_{G'})\\cap\\Sigma^{++}$ and by (\\ref{eq:vcmloc}) that $\\mathrm{loc}(G)\\cap\\Sigma^{++}=\\mathrm{loc}(G')\\cap\\Sigma^{++}$. Then by Proposition ~\\ref{pro_dagiso} we have $G=G'$. The last part of the assertion follows by a similar reasoning.\n\\else\nsee~\\cite{RP14-arxiv}.\n\\fi\n\\end{proof}\n\nIn what follows, $\\Pi=\\{\\pi_s\\}_{s\\in S_n}$ is the permutation\ngroup with induced action on $\\mathbb{C}[{\\Sigma}]$:  $\\pi_s(f(\\sigma_{ij}))=f(\\sigma_{s(i)s(j))})$ where $s\\in S_n$ is a\npermutation of indices. The invariant subring $\\{f\\in {\\mathbb{C}}[{\\Sigma}]\\,|\\, f\\circ \\pi_s=f\\,\\, \\forall s\\}$ is denoted by\n${\\mathbb{C}}[{\\Sigma}]^\\Pi$. We can now state our main result:\n\n \\begin{theorem} Let $G,G'$ be DAGs and $S$ be as in Theorem~\\ref{cor_conj}. Then $G\\sim G'$ if and only if\n\t\\begin{align}\\label{eq:tpr}\nS^{-1}I_G\\cap {\\mathbb{C}}[\\Sigma]^\\Pi=S^{-1}I_{G'}\\cap {\\mathbb{C}}[{\\Sigma}]^\\Pi.\n\\end{align}\n\\label{thm:iso}\n\\end{theorem}\n\\begin{proof}\\ifcomplete By Proposition~\\ref{pro_dageq}, DAGs $G$ and $G'$ are isomorphic iff there\n\texists $\\pi \\in \\Pi$ such that\n\t$$ X_{\\pi(G)}= X_{G'}. $$\n\tBy Theorem~\\ref{cor_conj}b this in turn is equivalent to \n\\begin{equation}\\label{eq:tp1}\n\t\\pi(X_G) = X_{G'}. \n\\end{equation}\nTo see this note that \n\t$$X_{\\pi(G)}=V( I_{\\pi(G)}:\\theta_0^m)=V(\\pi(I_G:\\theta_0^m)))=\\pi(X_G).$$\nFurthermore, since $S$ in Theorem~\\ref{cor_conj} is $\\Pi$-invariant,~\\eqref{eq:tp1} is equivalent to\n\\begin{equation}\\label{eq:tp3}\n\t\t\\pi(S^{-1} I_G)\\cap \\mathbb{C}[\\Sigma] = S^{-1} I_{G'}\\cap \\mathbb{C}[\\Sigma]\\,. \n\\end{equation}\nFinally, because of primality of ideals in~\\eqref{eq:tp3}, the existence of $\\pi$ satisfying~\\eqref{eq:tp3} is equivalent to~\\eqref{eq:tpr} by~\\cite[Exercise 5.13]{ati69}.\n\t\\else\n\tsee~\\cite{RP14-arxiv}.\n\t\\fi\n\\end{proof}\n\n\\ifcomplete\nAs before, we can work over $\\mathbb{C}[\\hat{\\Sigma}]$.\n\n\\begin{theorem}: Let $G,G'$ be DAGs and $\\hat{S}$ be as in Theorem~\\ref{thm:xhat}. Then $G$ and $G'$ are isomorphic if and only if\n\t\\begin{align}\\label{eq:tpr}\n\\hat{S}^{-1}\\hat{I}_G\\cap {\\mathbb{C}}[\\hat{\\Sigma}]^\\Pi=\\hat{S}^{-1}\\hat{I}_{G'}\\cap {\\mathbb{C}}[\\hat{\\Sigma}]^\\Pi.\n\\end{align}\n\\label{thm:hiso}\n\\end{theorem}\n\\begin{proof}\nRecall that $X_{\\pi(G)}=\\pi(X_G)$. Thus by Proposition~\\ref{pro_dageq} we have\n\\[\n\\pi(X_G)=X_{G'} \\iff \\pi(\\hat{X}_G)=\\hat{X}_{G'}.\n\\]\nThe rest of the proof is analogous to that of Theorem~\\ref{thm:iso} and is omitted.\n\\end{proof}\nWe remark that $\\hat{S}^{-1}\\hat{I}_G$ can be replaced with $\\hat{J}_G$ in the above. The presentation of the results in this form is a matter of convenience. Indeed, there is a simple way to generate $\\hat{I}_G$: \n\n\\begin{itemize}\n\\item Traverse the graph in the order of the topological sort and set $\\hat{I}_i:= \\sum_{{j<i}, j\\notin K}\\langle |\\hat{\\sigma}_{iK,jK}|\\rangle$ with $K:=\\mathbf{pa}(i)$ for all $i$.\n\\item Output $\\hat{I}_{G}=\\sum_i \\hat{I}_i$.\n\\end{itemize}\nHowever, extracting the implied relations of a DAG requires more work.\n\nWe also point out that the extra components of $V(\\hat{I}_G)$ (or $V(I_G)$) are not invariant across the isomorphism class of $G$:\n\\begin{example}\nThe DAG\n\\begin{center}\n\\tikzset{node distance=1.5cm, auto}\n\\begin{tikzpicture}[auto]\n  \\node (A0){$$};\n  \\node (A1)[right of=A0,node distance=0cm] {$$};\n  \\node (A2) [above of=A1,node distance=0.75cm]{$2$};\n  \\node (A3) [below of=A0,node distance=0.5cm]{$1$};\n  \\node (A4) [right of=A3,node distance=1.5cm]{$$};\n  \\node (A6) [left of=A3,node distance=0.5cm,above of=A3,node distance=0.7cm]{${G}':$};\n  \\node (A7)[right of=A6,node distance=2cm] {$3$};\n  \\node (A5) [right of=A7,node distance=1.25cm]{$4$};\n  \\path [line] (A3) --node[above]{$$} (A2);  \n  \\path [line] (A3) --node[above]{$$} (A7);  \n  \\path [line] (A7) --node[above]{$$} (A5);  \n  \\path [line] (A2) --node[above]{$$} (A7);  \n\\end{tikzpicture}\n\\end{center}\nis isomorphic to the DAG $G$ in Example \\ref{ex:thm1}, but $\\hat{I}_{G'}$ is a prime ideal. Note however that $J_{G'}=J_{\\pi(G)}$ where $\\pi$ is the permutation $(14)(23)$. Thus \nit is necessary to compute the saturation ideal in (\\ref{eq:tpr}). We shall see, however, in section \\ref{sec:rand} that one can avoid computing the saturation ideal, and more importantly, the subring intersection by a probabilistic procedure. \n\\label{ex:inv}\n\\end{example}\n\\fi\n\\ifcomplete\n\\subsection{Changing the ground ring}\n\nWe briefly note that none of our algebraic arguments depend on the choice of $\\mathbb{C}$ as ground ring. In particular,\ninstead of $X_G$ we could have considered \n$$ X'_G = \\mathrm{Spec}(\\mathbb{Z}[\\Sigma]\/\\mq_G) $$\nwhere $\\mq_G = S^{-1} I_G \\cap \\mathbb{Z}[\\Sigma]$. Then we still have that $X'_G$ is an integral, rational \nscheme over $\\mathbb{Z}$,\nsmooth at every point in $D(\\theta_0) = \\{\\theta_0\\neq 0\\}$. Two graphs $G\\sim G'$ are isomorphic if and only if \n$$ S^{-1} I_G \\cap \\mathbb{Z}[\\Sigma]^\\Pi = S^{-1} I_{G'} \\cap \\mathbb{Z}[\\Sigma]^\\Pi. $$\nFurthermore, we can identify $X_G$ as the base-change of $X'_G$ to $\\mathbb{C}$: $X_G=X'_G\\times_{\\mathrm{Spec}(\\mathbb{Z})} \\mathrm{Spec}(\\mathbb{C})$. \n\\ifcomplete\nTo verify this, one merely needs to check that $\\mq_G^e=\\mpf_G$. Indeed $I_G$ has a Gr$\\ddot{\\textup{o}}$bner\\, basis (w.r.t any term order) consisting of generators with $\\mathbb{Z}$-coefficients (run Buchberger's algorithm and clear denominators at the end). One can thus see, by variable elimination, that $\\mpf_G=S^{-1}I_G\\cap \\mathbb{C}[\\Sigma]$ can be generated by polynomials in $\\mathbb{Z}[\\Sigma]$.\n\\fi\nOther convenient choices of rings are $\\mathbb{Q}$ and $\\mathbb{R}$.  \n\\fi\n\\subsection{Randomized algorithm}\n\\label{sec:rand}\n\\ifcomplete\n\\begin{algorithm}[t]\n\\caption{$\\mathrm{ISODAG}_{\\textup{m}}$}\n\\label{ISODAG}\n\\begin{algorithmic}[1]\n\\STATE {\\bf function} {$\\mathrm{ISODAG}_\\mathrm{m}$}($G$, $G'$)\n\\STATE Sort $G$ and $G'$ topologically\n\\STATE Initialize $\\mathrm{ISO}\\gets \\TRUE,r\\gets 1$\n\\WHILE{$\\mathrm{ISO}\\,\\,\\AND \\,\\,r\\le m$}\n\\STATE Sample $z_G^r,z_{G'}^r$ respectively from $\\hat{\\varphi}_*P_G$,$\\hat{\\varphi}_*P_{G'}$ as follows:\\\\\n\\begin{enumerate}[(i)]\n\\item Sample edge variables $\\hat{\\sigma}^r_{\\mathrm{edge}}$ of $G$ from $P_G$\n\\FOR{$i:=2$ to $n$}\n\\item[] \\quad Solve the (linear) toposorted imposed relations\n\\[\t\t\n|\\hat{\\sigma}^r_{iK,jK}|=0, \\quad K:=\\mathbf{pa}(i)\n\\]\n\\quad for each non-edge variable $\\hat{\\sigma}^r_{ij}$, $j<i$\n\\item[] \\ENDFOR\n\n\\item $z_G^r\\gets (\\hat{\\sigma}_{\\mathrm{edge}}^r,\\hat{\\sigma}_{\\mathrm{non-edge}}^r)$\n\\item Repeat for $G'$\n\\end{enumerate}\n\\IF{\\,\\,$\\Pi(z^r_G)\\cap V(\\hat{I}_{G'})=\\emptyset\\,\\OR\\, \\Pi(z^r_{G'})\\cap V(\\hat{I}_G)=\\emptyset$} \n\t\\STATE $\\mathrm{ISO}\\gets \\FALSE$\n\\ENDIF\n\\STATE $r\\gets r+1$\n\\ENDWHILE\n\\RETURN ISO\\\\\n\\end{algorithmic}\n\\end{algorithm}\n\n\\else\n\\begin{algorithm}[t]\n\\caption{$\\mathrm{ISODAG}_{\\textup{m}}$}\n\\label{ISODAG}\n\\begin{algorithmic}[1]\n{\\small\n\\STATE {\\bf function} {$\\mathrm{ISODAG}_\\mathrm{m}$}($G$, $G'$)\n\\STATE Sort $G$ and $G'$ topologically\n\\STATE Initialize $\\mathrm{ISO}\\gets \\TRUE,r\\gets 1$\n\\WHILE{$\\mathrm{ISO}\\,\\,\\AND \\,\\,r\\le m$}\n\\STATE Sample $z_G^r,z_{G'}^r$ respectively from $\\hat{\\varphi}_*P_G$,$\\hat{\\varphi}_*P_{G'}$ as follows:\\\\\n\\begin{enumerate}[(i)]\n\\item Sample edge variables $\\hat{\\sigma}^r_{\\mathrm{edge}}$ of $G$ from $P_G$\n\\FOR{$i:=2$ to $n$}\n\\item[]\\quad Solve the (linear) toposorted imposed relations\n\\[\t\t\n|\\hat{\\sigma}^r_{iK,jK}|=0, \\quad K:=\\mathbf{pa}(i)\n\\]\n\\quad for each non-edge variable $\\hat{\\sigma}^r_{ij}$, $j<i$\n\\item[]\\ENDFOR\n\n\\item $z_G^r\\gets (\\hat{\\sigma}_{\\mathrm{edge}}^r,\\hat{\\sigma}_{\\mathrm{non-edge}}^r)$\n\\item Repeat for $G'$\n\\end{enumerate}\n\\IF{\\,\\,$\\Pi(z^r_G)\\cap V(\\hat{I}_{G'})=\\emptyset\\,\\OR\\, \\Pi(z^r_{G'})\\cap V(\\hat{I}_G)=\\emptyset$} \n\t\\STATE $\\mathrm{ISO}\\gets \\FALSE$\n\\ENDIF\n\\STATE $r\\gets r+1$\n\\ENDWHILE\n\\RETURN ISO\\\\\n}\n\\end{algorithmic}\n\\end{algorithm}\n\\fi\n\n\\ifcomplete Theorem~\\ref{thm:hiso} shows that testing isomorphism amounts to subring intersection. Computing this intersection is difficult since there is no easy description available for generating invariants of $\\mathbb{C}[\\hat{\\Sigma}]^\\Pi$. Another computational difficulty is that of computing the saturation ideal. This operation does not scale well with the number of nodes in the model. Here we give a randomized algorithm that avoids computing both the intersection and saturation ideals. \n\nLet $\\varphi$ be the rational map in (\\ref{eq:ltt}) and denote by $\\hat{\\varphi}$ its restriction $\\varphi|_{\\hat{\\Sigma}}$. Let $\\mathcal{Y}:=\\mathbb{F}_q^{|E|}$ and define\n$$\\mathcal{U}:=\\{y\\in \\mathcal{Y}: \\hat{g}(y)\\neq0,\\hat{h}(y)\\neq 0\\},$$ \nwhere  $\\hat{g},\\hat{h}$ are as in  (\\ref{eq:hatseq}). Note that\n\\[\n|\\mathcal{U}|\\ge (q-d) q^{|E|-1}\n\\]\nwith $d:=\\deg(\\hat{g})+\\deg(\\hat{h})$.\n\n\\else We use the preceding results to give a randomized algorithm for testing DAG isomorphism. \nIn~\\cite{RP14-arxiv}, we associate with every DAG $G$ a rational map (see proof of Theorem~\\ref{thm:thm1}) ${\\varphi}:\\Sigma_{edge}\\to \\Sigma$ where $\\varphi$ is regular on a distringuished open $D(gh)$ and its image is dense in $X_G$. Here, $g$ and $h$ are certain polynomials in $\\mathbb{C}[\\Sigma_{\\mathrm{edge}}]$. Denote by $\\hat{\\varphi}:\\hat{\\Sigma}_{\\mathrm{edge}}\\to \\hat{X}_G$ the map $\\varphi|_{\\hat{\\Sigma}}$.  Let $\\mathcal{Y}:=\\mathbb{F}_q^{|E|}$ and define\n$$\\mathcal{U}:=\\{y\\in \\mathcal{Y}: \\hat{g}(y)\\neq0,\\hat{h}(y)\\neq 0\\},$$ \nwhere $\\hat{g}=g|_{\\hat{\\Sigma}},\\hat{h}=h|_{\\hat{\\Sigma}}$. \\fi \nNow construct a random matrix with uniform distribution $P_G$ on the finite set $\\mathcal{U}$. \\ifcomplete This can be realized, for instance, by transforming a uniformly distributed matrix on $\\mathcal{Y}$ through a kernel $P_{Y|X}:\\mathcal{Y}\\to\\mathcal{Y}\\cup \\{\\emptyset\\}$ such that  $P_{Y|X}(x|x)=1$ if $x\\in \\mathcal{U}$ and $P_{Y|X}(\\emptyset |x)=1$ otherwise.\\fi Let $\\hat{\\varphi}_{*}P_G$ be the push-forward of $P_G$ under $\\hat{\\varphi}$ and $Z_G^i$'s be independent random variables with common distribution $\\hat{\\varphi}_*P_G$.\n\nGiven DAGs $G$, $G$', the algorithm $\\mathrm{ISODAG}_\\mathrm{m}$ described above constructs $m$ realizations $z^i_G, z^i_{G'}$ from $Z^i_G,Z^i_{G'}$. It then declares $G$ and $G'$ to be isomorphic if and only if for each $i\\le m$, there is some permutation $\\pi$ such that both $\\pi(z^i_G)\\in V(\\hat{I}_{G'})$ and $z^i_{G'}\\in V(\\pi(\\hat{I}_G))$ hold. \\ifcomplete The latter conditions amount, respectively, to checking $f'_j(\\pi(z^i_G))=0$ and $\\pi(f_k)(z^i_{G'})=0$ for all generators $f'_j$ of $\\hat{I}_{G'}$ and $f_k$ of $\\hat{I}_G$. The next theorem shows that the probability of failure of the algorithm can be made arbitrarily small:\\else We have:\\fi\n\n\n\n\\begin{theorem}\nLet $G$ be a DAG on $n$ nodes and $E$ be its set edges. Let $Z^i_G$ be as in above and set $d:=\\deg(\\hat{g})+\\deg(\\hat{h})$. If $G\\sim G'$, then\n\\[\n\\mathbb{P}[\\mathrm{ISODAG}_\\mathrm{m}(G,G')=\\mathrm{yes}]=1.\n\\]\nIf $G\\not\\sim G'$, then\n\\[\n\\mathbb{P}[\\mathrm{ISODAG}_\\mathrm{m}(G,G')=\\mathrm{yes}]\\le  (n!\\frac{n+2d-1}{q-d})^m.\n\\]\n\\label{thm:rand}\n\\end{theorem}\n\\begin{proof}\n\\ifcomplete\nBy Theorem~\\ref{thm:hiso}, the algorithm outputs yes with probability 1 if $G$ is isomorphic to $G'$. Now suppose $\\mathrm{loc}(G)\\not\\subset \\mathrm{loc}(G')$. We need to upper bound the probability that a realization of $Z^i_G$ lands on $\\hat{X}_{G'}$. Note that $Z^i_G$ takes values on $\\hat{X}_{G}\\cap \\hat{\\Sigma}^{\\mdot}$. It follows from  Proposition \\ref{pro_dageq} that a point on $\\hat{X}_{G'}\\cap \\hat{\\Sigma}^{\\mdot}$ satisfies at least one relation of the form $|\\hat{\\sigma}_{iK,jK}|$ where $i\\perp\\!\\!\\!\\perp j|K \\not\\in \\mathcal{M}^{topo}_G$. We can pullback the intersection $\\hat{X}_G\\cap \\hat{\\Sigma}^{\\mdot}\\cap \\{|\\hat{\\sigma}_{iK,jK}|=0\\}$ to $\\mathcal{U}$ via the embedding $\\hat{\\varphi}$. This corresponds to replacing the non-edge variables (w.r.t $G$) of $|\\sigma_{iK,jK}|$ with rational functions of edge variables as in (\\ref{eq:ltt}). Let $f_{ijK}$ be the resulting rational function.\nThen the number of points on the intersection is upper bounded by the number of solutions in $\\mathcal{Y}$ of $\\{f_{ijK}=0\\}$. Note that the minor has total degree at most $n-1$ and the largest power of any monomial appearing in the expansion of such minor is at most $2$. Thus clearing the denominators in $f_{ijK}$ gives a non-zero polynomial of degree at most $(n+2d-1)$ in at most $|E|$ variables. Such a polynomial has at most $(n+2d-1)q^{|E|-1}$ roots in $\\mathcal{Y}$.\nOn the other hand, $\\mathcal{U}$ has at least $(q-d)q^{|E|-1}$ points. The union bound proves the result. \n\\else\nsee~\\cite{RP14-arxiv}.\n\\fi\n\\end{proof}\nIf $G$ and $G'$ are two DAGs on $n$ nodes and $E$ edges, it is easy to verify that the algorithm can decide equivalence of $G$ and $G'$ in time $O(n^3|E^c|)$. To test if a sampled point lies on $\\hat{X}_G$, one needs to solve a sequence of $|E^c|$ linear equations. Each equation involves one missing edge and one minor of size at most $(n-1)\\times (n-1)$, which requires $O(n^3)$ operations to compute (note that we can do the arithmetics over rationals). This is comparable, in terms of complexity, with  $O(n^4|E|)$ operations needed in the essential graph method (see \\cite{and97}).\n\n\n\\ifcomplete\n\\section{Directed tree models}\nWe study the case of directed tree models in this section. The main property is the following:\n\\begin{proposition}\nLet $T$ be a directed tree model. Then $\\hat{I}_T$ is prime and $V(I_T)\\cap\\hat{\\Sigma}=[\\mathrm{loc}(T)\\cap\\hat{\\Sigma}]_Z$. Furthermore, $\\hat{X}_T$ is smooth everywhere.\n\\label{pro_prime}\n\\end{proposition}\nThis is analogous to primality of $J_T$ shown in \\cite{sul08} (see Corollary 2.4 and Theorem 5.8). A direct consequence is that two tree models are isomorphic if and only if $ \\hat{I}_T\\cap \\mathbb{C}[\\hat{\\Sigma}]^{\\Pi}=\\hat{I}_{T'}\\cap \\mathbb{C}[\\hat{\\Sigma}]^\\Pi.$\nIt also follows from the proposition that $\\hat{I}_T=\\hat{J}_T$.\n\\noindent\nIn II.E, we give a procedure to generate $\\hat{I}_T$. To prove primality of $\\hat{I}_T$, we introduce a second procedure that uses lower degree generators by modifying the first step:\n\\begin{itemize}\n\\item Set ${L}_0=\\emptyset$.\n\\item Traverse the graph in order of the sort and set $K_i:=\\mathbf{pa}(i)$ for all $i$. \n\\item For all $j<i$, let $K'\\in K_i$ be the smallest subset that $d$-separates $j$ and $i$. Set $L_i=L_{i-1}\\cup \\{f| f= \\langle |\\sigma_{iK',jK'}|\\rangle\\}$.\n\\item Output $\\hat{I}'_G:=\\sum_{f\\in {L}_n} \\langle f \\rangle$.\n\\end{itemize}\n\nLet us verify $\\hat{I}_T=\\hat{I}'_T$ for a tree model $T$. \nSuppose the claim holds on $n-1$ nodes \nand let the first procedure reach node $n$. Set $K=\\mathrm{pa}(i)$ and let $k$ be the separator of nodes $i$ and $n$ for some $i<n$. The generated polynomial is \n\\[\n|\\sigma_{ iK,nK}|=\\left|\\begin{array}{cc} \\sigma_{in}& \\sigma_{iK}  \\\\ \\sigma_{Kn} & \\sigma_{KK} \\end{array} \\right|.\n\\]\nThis can be expanded as\n\\begin{align*}\n|\\sigma_{ iK,nK}|&=\\left|\\begin{array}{cc} \\sigma_{in}& \\sigma_{ik}  \\\\ \\sigma_{kn} & 1 \\end{array} \\right|+\\sum_{j\\in \\{K-k\\}} \\sigma_{ij}f_j+\\sum_{j,j'\\in K} \\sigma_{jj'}f_{jj'},\n\\end{align*}\nwhich is in the ideal of the second procedure since $i$ is not connected to $K-k$ and the parents of $n$ are not connected either. Conversely, take the minor \n\\[\n\\left|\\begin{array}{cc} \\sigma_{in}& \\sigma_{ik}  \\\\ \\sigma_{kn} & 1 \\end{array} \\right|\n\\]\nproduced by the second procedure and note that \\mbox{$i\\perp\\!\\!\\!\\perp K-k$} and $k'\\perp\\!\\!\\!\\perp k''$ for distinct $k',k''\\in K$. By the inductive hypothesis, the corresponding linear forms are in $I_T$.  Using the above expansion for $|\\sigma_{ iK,nK}|$ one can see that this reduced minor is in the ideal of the first procedure as well. Thus the two ideals are equal when $T$ is a tree.  \nThe first procedure is easy to implement and is what we use to generate the ideals. The second procedure is easy to analyze and has useful properties that we exploit later. For instance, it shows that the ideal of imposed relations of a tree model is generated by quadratic polynomials of type $\\sigma_{ij}-\\sigma_{ik}\\sigma_{kj}$ or linear forms $\\sigma_{ij}$. We note that the equality of $\\hat{I}_T$ and $\\hat{I}'_T$ does not generalize to all DAGs. For instance, in the case of Example~\\ref{ex:thm1}, $\\hat{I}_T$ is not prime whereas $\\hat{I}'_T$ is prime for all DAGs with $n\\le 4$ nodes.\n\nWe are now ready to prove that the ideal of the imposed relations is prime for tree models:\\\\\n\\begin{proof}[Proof of Proposition~\\ref{pro_prime}]\nThe generators of $\\hat{I}_T$ are of the form either $\\hat{\\sigma}_{ik}-\\hat{\\sigma}_{ij}\\hat{\\sigma}_{jk}$ or $\\hat{\\sigma}_{ij}$. Then the rational map (\\ref{eq:morp}) in the proof of Theorem~\\ref{cor_conj} can be taken to be a polynomial map. In other words, $\\hat{S}^{-1}\\hat{I}_T\\cap\\mathbb{C}[\\hat{\\Sigma}]=\\hat{I}_T$. It further follows that $\\hat{X}_T$ is smooth.\n\\end{proof}\n\nOne can further show that for a certain lexicographic order the generators of $\\hat{I}_G$ form a Gr$\\ddot{\\textup{o}}$bner\\, basis. Set $d_{ij}=\\min_{\\pi \\in \\mathcal{D}(i,j|\\emptyset)} |\\pi| $, that is $d_{ij}$ is the length of the shortest d-path from $i$ to $j$. Note that if $i\\perp\\!\\!\\!\\perp j|k$ is an imposed relation, then $d_{ij}>1$. Order the variables as follows: $\\sigma_{ij}\\succ\\sigma_{i'j'}$ if $d_{ij}>d_{i'j'}$ or $d_{ij}=d_{i'j'}$ but $(i,j)\\succ_{\\mathbb{Z}^2}(i'j')$, where $\\succ_{\\mathbb{Z}^2}$ is any order on $\\mathbb{Z}^2$. We also set $\\sigma_{ij}\\succ 1$ for all variables. \\\\\nLet $\\alpha=(\\alpha_{ij})$ be a vector. Denote by $\\sigma^\\alpha$ the monomial $\\prod \\sigma_{ij}^{\\alpha_{ij}}$. Now define the relation $\\sigma^\\alpha\\succ_\\mathrm{dag} \\sigma^\\beta$ if the first non-zero coordinate of $\\alpha-\\beta$ is positive. Note that $\\succ_\\mathrm{dag}$ is a lexicographic order. \n\nThe above order has a pleasant property: the leading monomial of quadratic relations $\\sigma_{ij}-\\sigma_{ik}\\sigma_{kj}$ generated by CI relations of $T$ is always the linear form $\\sigma_{ij}$. We use this to prove that the quadratic and linear imposed relations used to generate $\\hat{I}'_T$ form a Gr$\\ddot{\\textup{o}}$bner\\, basis w.r.t to this order:\n\n\\begin{proposition} The generators of $\\hat{I}'_T$ form a Gr$\\ddot{\\textup{o}}$bner\\, basis w.r.t the $\\succ_\\mathrm{dag}$ oder. \n\\label{pro_gb}\n\\end{proposition}\n\\begin{proof}\nNote that $\\hat{I}'_G$ is generated by linear forms and quadratic relations of the form $\\sigma_{ij}-\\sigma_{ik}\\sigma_{kj}$. Given two such polynomials, $f=\\sigma_{ij}-\\sigma_{ik}\\sigma_{kj}$ and $f'=\\sigma_{i'j'}-\\sigma_{i'k'}\\sigma_{k'j'}$, we can check that the leading terms are linear, and hence, the resulting $S$-polynomial\n\\[\nS(f,f')=\\sigma_{ik}\\sigma_{kj}\\sigma_{i'j'}-\\sigma_{i'k'}\\sigma_{k'j'}\\sigma_{ij},\n\\]\nreduces to zero w.r.t $\\{\\sigma_{ij}-\\sigma_{ik}\\sigma_{kj},\\sigma_{i'j'}-\\sigma_{i'k'}\\sigma_{k'j'} \\}$. \n\\end{proof}\nIt was asked in \\cite{sul08} (see Conjecture 5.9) if there exists a Gr$\\ddot{\\textup{o}}$bner\\, basis consisting of square free terms of degree one and two for $J_T$. The above proposition shows that this is the case for $\\hat{I}_T$ (or $\\hat{J}_T$). \n\nFinally, using the procedure in section \\ref{sec:rand}, we list the isomorphism classes of trees on $4,5,$ and $6$ nodes. See Figure~\\ref{fig_classes}. Our computations are done in Magma.\n\n\n\n\\begin{figure}[t]\n        \\centering\n\\subfigure[$n=4$]{\n        \\centering\n\\includegraphics[height=0.12\\textheight]{4trees_essential_darrows.pdf} \n                }\n\\subfigure[$n=5$]{\n        \\centering\n\\includegraphics[height=0.28\\textheight]{5trees_essential_darrows.pdf} \n                }\n                \\subfigure[$n=6$]{\n\\centering\n\\includegraphics[height=0.45\\textheight]{6trees_essential_darrows.pdf} \n}\n        \\caption{Isomorphism classes of directed tree models on $4,5,$ and $6$ nodes. The double arrows are essential.}                 \n                \\label{fig_classes}\n\\end{figure}\n\n\n\n\\else\nWe use this theorem to list the isomorphism classes of trees on $4$ and $5$ nodes. See Figure~\\ref{fig_classes}. \n\n\n\\begin{figure}[t]\n        \\centering\n\\subfigure[$n=4$]{\n        \\centering\n\\includegraphics[height=0.07\\textheight]{figures\/4treesarrows.pdf} \n                }\n\\subfigure[$n=5$]{\n        \\centering\n\\includegraphics[height=0.16\\textheight]{figures\/5treesarrows.pdf} \n                }\n         \\caption{Isomorphism classes of directed tree models on $4$ and $5$ nodes.} \n \n                \\label{fig_classes}\n\\end{figure}\n\\fi\n   \n\n\n\\section{Marginalization}\n\nExact inference on a DAG $G$ is the problem of extracting marginals of a $G$-compatible distribution. In this section, we establish a connection between the obstructions to embeddings of $Y_G$ and obstructions to efficient inference on $G$. \n\n\\begin{definition} \\begin{enumerate} \n\\item Given a DAG $G$, we define marginalization of $G$ w.r.t $N$, denoted by $\\mathrm{El}_N(G)$, to be the conditional independence model defined by the implied relations in $G$\n\\[\ni\\perp\\!\\!\\!\\perp j|K\n\\]\nsuch that $\\{i,j,K\\}\\cap N=\\emptyset$. \n\\item Let $e=|N|$. Given a DAG $G$, define the extension $G^{e}$ of $G$ w.r.t $N$ to be the DAG on $n+e$ nodes whose implied relations are the same as $G$. \n\\end{enumerate}\n\\end{definition}\nTake for instance the DAG\n\\begin{center}\n\\tikzset{node distance=1.5cm, auto}\n\\begin{tikzpicture}[auto]\n  \\node (A0) {$1$};\n  \\node (A1) [left of=A0,node distance=0.5cm]{$G:$};\n  \\node (A3l) [right of=A0,node distance=1.5cm]{$$};\n  \\node (A3ul) [above of=A3l,node distance=1cm]{$2$};\n  \\node (A3dl) [below of=A3l,node distance=1cm]{$3$};\n  \\node (A4) [right of=A0,node distance=3cm]{$4$};\n  \\node (A5p) [right of=A4,node distance=1.5cm]{$$};\n  \\path [line] (A0) --node[above]{$$} (A3ul);  \n  \\path [line] (A3ul) --node[above]{$$} (A4);  \n  \\path [line] (A3dl) --node[above]{$$} (A4);  \n  \\path [line] (A0) --node[above]{$$} (A3dl);  \n\\end{tikzpicture}\n\\end{center}\nTo extract the marginal on nodes 1 and 3, one needs to eliminate nodes 2 and 4. A well known problem is that the order of elimination matters. For instance, eliminating node 4 leads to a ``sparse'' graph \n\n\\begin{center}\n\\tikzset{node distance=1.5cm, auto}\n\\begin{tikzpicture}[auto]\n  \\node (A0){$$};\n  \\node (A1)[right of=A0,node distance=0.5cm]{$2$};\n  \\node (A2)[right of=A1,node distance=1.5cm] {$$};\n  \\node (A2u)[above of=A2,node distance=0.75cm] {$1$};\n  \\node (A3) [right of=A2,node distance=1.5cm]{$3$};\n  \\path [line] (A2u) --node[above]{$$} (A1);  \n  \\path [line] (A2u) --node[above]{$$} (A3);  \n\\end{tikzpicture}\n\\end{center}\nwhereas eliminating node 2 gives a ``dense'' graph\n\n\\begin{center}\n\\tikzset{node distance=1.5cm, auto}\n\\begin{tikzpicture}[auto]\n  \\node (A0){$$};\n  \\node (A1)[right of=A0,node distance=0.5cm]{$3$};\n  \\node (A2)[right of=A1,node distance=1.5cm] {$$};\n  \\node (A2u)[above of=A2,node distance=0.75cm] {$1$};\n  \\node (A3) [right of=A2,node distance=1.5cm]{$4$};\n  \\path [line] (A2u) --node[above]{$$} (A1);  \n  \\path [line] (A2u) --node[above]{$$} (A3);  \n  \\path [line] (A1) --node[above]{$$} (A3);  \n\\end{tikzpicture}\n\\end{center}\nA fact is that the computational effort in sequential eliminations is controlled by the sparsity of the subgraphs that appear in the process. The problem is that there is no good way of finding the right elimination order. What is worse is that, by the recent results in \\cite{kwi10}, it is even hard to say if such an order exists or not. The difficulty of inference in a DAG is controlled by the so called tree-widths of its underlying graph. This is hard to compute for large graphs, and it thus makes sense to settle for an easier question: {\\em what are some necessary conditions for a good elimination order to exist?}\n \nOne observation is that having small tree-width implies that, under some suitable ordering, the graphs that appear in the elimination process have small tree-width as well, and one can take this as a measure of efficiency for inference. In the above example, we see that the marginals of interest in $G$ are supported on the (unlabeled) Markov chain:\n\n\\begin{center}\n\\tikzset{node distance=1.5cm, auto}\n\\begin{tikzpicture}[auto]\n  \\node (A0){$M:$};\n  \\node (A1)[right of=A0,node distance=0.5cm]{$\\bullet$};\n  \\node (A2)[right of=A1,node distance=1.5cm] {$\\bullet$};\n  \\node (A3) [right of=A2,node distance=1.5cm]{$\\bullet$};\n  \\draw [line] (A1) --node[above]{$$} (A2);  \n  \\path [line] (A2) --node[above]{$$} (A3);  \n\\end{tikzpicture}\n\\end{center}\n\nHowever, the same cannot be said about the complete graph on four nodes. It is thus useful to know how simple the space of models that support the marginals of a certain DAG can be.  This gives us a notion of complexity of marginals in DAGs and it is clear that this notion depends on the class of the model, and not on how the model is represented as a graph. We formalize this notion as follows:\n\n\\begin{definition} \\begin{enumerate}[(a)] \\item Suppose that the nodes in $M$ are a subset of the nodes in $G$. We say that $M$ lies below $G$ if \n\\[\n\\forall Q_{YX}\\in \\mathrm{Loc}(G) \\quad\\exists\\quad P_{X}\\in \\mathrm{Loc}(M)\\quad s.t.\\quad Q_{YX}(\\mathcal{A})=\\int_\\mathcal{A} K_{Y|X}dP_\n\\]\nLikewise, we say that $G$ lies above $M$. \n\\item $M$ is said to be minimal w.r.t the above property if \n\\[\n\\mathrm{Loc}(M')\\not\\subset \\mathrm{Loc}(M)\n\\]\nfor all $M'$ that satisfy (a).\n\\item The class $[M]$ is said to lie below $[G]$ if (a) holds after some relabeling of variables (in $G$ or $M$). \n\\end{enumerate}\n\\end{definition}\n\nIn other words, $M$ lies below $G$ if the marginal of any $G$-compatible distribution on the subspace associated to $M$ factorizes w.r.t it. We note that the elimination of a DAG need not be a DAG (see \\cite{ric02}), but in cases that it is, the notions of minimal model and elimination model coincide. \n\n\\begin{comment}\nWe now define the corresponding operations on varieties. Let $\\mathrm{El}_N(X_G)$ denote the closure of the variety obtained by eliminating the nodes in $N$, i.e., it is the (closure of) the projection of $X_G$ to coordinates $\\sigma_{ij}$ for $i,j\\notin N$. We define the projection map $\\pi:X_G\\to \\mathrm{El}_N(X_G)$ as follows\n\\[\n\\sigma_{ij}\\mapsto \\left\\{\\begin{array}{ll} 0& i\\in N\\, \\textup{or}\\, j\\in N\\\\  \\sigma_{ij}& \\textup{otherwise}.  \\end{array}\\right.\n\\]\nBy definition $\\mathrm{El}_N(X_G)=[\\pi(X_G)]_Z$. In general, we have \n\n\\begin{proposition}\n$[M]$ lies below $[G]$ if and only if $\\hat{X}_M\\subset \\mathrm{El}_N(\\hat{X}_{\\pi(G)})$ for some $N\\subset [n]$ and some permutation $\\pi$.\n\\end{proposition}\n\\end{comment}\nThe main question of interest is to decide when $[M]$ can lie below $[G]$.\nIt is clear that an enumerative approach to answer this question is problematic, even for a small graph $M$, as the number of possibilities grow exponentially with the size of the eliminated subset (which grows as $M$ gets smaller). The next proposition gives a necessary condition. \n\n\\begin{proposition}\nLet $M,G$ be DAGs and suppose $e:=|V_G|-|V_M|\\ge 0$. Then if $[M]$ lies below $[G]$, there exists an embedding\\footnote{By an embedding we mean a closed immersion.} $Y_G\\hookrightarrow Y_{M^e}$. \n\\end{proposition}\n\\begin{proof}\nIf $M$ lies below $G$, then its implied relations are satisfied by the marginals in $\\mathrm{Loc}(G)$. This gives an inclusion $\\mathrm{loc}(G)\\subset \\mathrm{loc}(M^e)$, which induces an embedding $\\hat{X}_G\\hookrightarrow \\hat{X}_M$. This embedding extends to the projective closures.  \n\\end{proof}\n\n\\begin{example}\nLet $G$ be the complete DAG on $n$ nodes and $M$ be a Markov chain on $m\\le n$ nodes. It is clear, from dimension considerations, that $[M]$ cannot appear below $[G]$.\n\\end{example}\n\nWe use the following proposition later:\n\\begin{proposition}\nIf there exists an embedding of $\\mathbb{P}^n\\times\\mathbb{P}^1$ into $\\mathbb{P}^{m}$, then $m\\ge 2n+1$.\n\\end{proposition}\n\\begin{proof}\nThis is perhaps easiest to see from Bezout's theorem, which states any two closed sub-varieties of $\\mathbb{P}^m$ of complementary dimension must have a non-empty intersection. In $\\mathbb{P}^n\\times\\mathbb{P}^1$, there are $n$-dimensional hyperplanes ${\\mathbb{P}^n\\times \\{pt\\}}$ that do no intersect, and this remains true after embedding. Thus each hyperplane must have codimension at least $n+1$ in the ambient projective space. \n\\end{proof}\nThis proposition shows that when $m=1$ the Segre embedding $\\mathbb{P}^n\\times\\mathbb{P}^m\\hookrightarrow \\mathbb{P}^{nm+n+m+1}$\nuses the minimal target dimension{\\footnote{In the general case, there are embeddings $\\mathbb{P}^n\\times\\mathbb{P}^m\\hookrightarrow \\mathbb{P}^{2(m+n)-1}$ and these can be shown to have the smallest target dimension.}}. This can be useful for our purposes.\n\n\\begin{example}\nLet $G_n$ be the following family of DAGs\n\\begin{center}\n\\tikzset{node distance=1.5cm, auto}\n\\begin{tikzpicture}\n  \\node (A0){$1$};\n  \\node (A1) [right of=A0,node distance=1.5cm]{$$};\n  \\node (B0) [below of=A0,node distance=1.5cm] {$2$};\n  \\node (B1) [below of=A1,node distance=1.5cm]{$n+1$};\n  \\node (B2) [right of=B1,node distance=1.5cm]{$n+2$};\n  \\node (C0) [below of=B0,node distance=.5cm]{$\\vdots$};\n  \\node (D0) [below of=C0,node distance=1.25cm]{$n$};\n\n  \\path [line] (A0) --node[above]{$$} (B1);  \n  \\path [line] (B0) --node[above]{$$} (B1);  \n  \\path [line] (D0) --node[above]{$$} (B1);  \n  \\path [line] (B1) --node[above]{$$} (B2);  \n\n\n\\end{tikzpicture}\n\\end{center}\nand $M_n$ be a disconnected V-structure on $n+1$ nodes\n\\begin{center}\n\\tikzset{node distance=1.5cm, auto}\n\\begin{tikzpicture}\n  \\node (A0){$1$};\n  \\node (A1) [right of=A0,node distance=1.5cm]{$$};\n  \\node (B0) [below of=A0,node distance=1.5cm] {$2$};\n  \\node (B1) [below of=A1,node distance=1.5cm]{$n+1$};\n  \\node (C0) [below of=B0,node distance=.5cm]{$\\vdots$};\n  \\node (D0) [below of=C0,node distance=1.25cm]{$n$};\n\n\n  \\path [line] (B0) --node[above]{$$} (B1);  \n  \\path [line] (D0) --node[above]{$$} (B1);  \n\n\n\n\\end{tikzpicture}\n\\end{center}\n\nOne can check that $Y_{G_n}\\simeq\\mathbb{P}^{n}\\times\\mathbb{P}^1$ and that $Y_{M_n^e}\\simeq\\mathbb{P}^{2n}$. Then $[M_n]$ does not lie below $[G_n]$ by the above proposition. This example is tight, in the sense that $[M_{n-1}]$ does lie below $[G_n]$ as eliminating the nodes $n+1$ and $n+2$ makes $G_n$ completely disconnected. In particular, $G_2$ does not lie above $\\bullet\\quad \\bullet\\to \\bullet$.  \n\\end{example}\n\nThe examples above are special in that they all involve smooth projective varieties, and this is hardly an attribute of projective DAG varieties. Additional effort will be needed to deal with singularities. \n \n\\begin{example}\nConsider the DAG\n\\begin{center}\n\\tikzset{node distance=1.5cm, auto}\n\\begin{tikzpicture}[auto]\n  \\node (A0){$$};\n  \\node (A1)[right of=A0,node distance=1.5cm] {$$};\n  \\node (A3) [below of=A0,node distance=0.5cm]{$3$};\n  \\node (A3l) [left of=A3,node distance=1cm]{$$};\n  \\node (A3ul) [above of=A3l,node distance=0.75cm]{$1$};\n  \\node (A3dl) [below of=A3l,node distance=0.75cm]{$2$};\n  \\node (A4) [right of=A3,node distance=1.5cm]{$4$};\n  \\node (A5p) [right of=A4,node distance=1.5cm]{$$};\n  \\path [line] (A3) --node[above]{$$} (A4);  \n  \\path [line] (A3ul) --node[above]{$$} (A3);  \n  \\path [line] (A3dl) --node[above]{$$} (A3);  \n  \\path [line] (A3ul) --node[above]{$$} (A3dl);  \n\n\n\\end{tikzpicture}\n\\end{center}\nWe want to show that for all $n$, $[G]$ does not lie above $[M]$ where $[M]$ is the class of a V-structure $\\bullet \\rightarrow \\bullet \\leftarrow \\bullet$. The extension $[M^e]$ of a V-structure to four nodes is the DAG\n\\begin{center}\n\\tikzset{node distance=1.5cm, auto}\n\\begin{tikzpicture}[auto]\n  \\node (A0){$$};\n  \\node (A1)[right of=A0,node distance=1.5cm] {$$};\n  \\node (A3) [below of=A0,node distance=0.5cm]{$\\bullet$};\n  \\node (A3l) [left of=A3,node distance=1cm]{$$};\n  \\node (A3ul) [above of=A3l,node distance=0.75cm]{$\\bullet$};\n  \\node (A3dl) [below of=A3l,node distance=0.75cm]{$\\bullet$};\n  \\node (A4) [right of=A3,node distance=1.5cm]{$\\bullet$};\n  \\node (A5p) [right of=A4,node distance=1.5cm]{$$};\n  \\path [line] (A3) --node[above]{$$} (A4);  \n  \\path [line] (A3ul) --node[above]{$$} (A3);  \n  \\path [line] (A3ul) --node[above]{$$} (A4);  \n  \\path [line] (A3dl) --node[above]{$$} (A4);  \n  \\path [line] (A3dl) --node[above]{$$} (A3);  \n\n\n\\end{tikzpicture}\n\\end{center}\nSuppose that $[M]$ is below $[G]$ to obtain an embedding $\\iota:Y_G\\hookrightarrow Y_{M^e}$. We note that $\\Pi_G$ is smooth everywhere except at the vertex of the affine cone over $\\mathbb{P}^1\\times\\mathbb{P}^1$. Blowing up $Y_{M^e}\\simeq\\mathbb{P}^5$ at $\\iota(p)$ gives a commutative diagram:\n\\begin{center}\n\\tikzset{node distance=1.5cm, auto}\n\\begin{tikzpicture}[auto]\n  \\node (A0){$\\mathrm{Bl}_p(Y_G)$};\n  \\node (A0r)[right of=A0,node distance=2.5cm] {$\\mathrm{Bl}_{\\iota(p)}(Y_{M^e})$};\n  \\node (A1) [below of=A0,node distance=1.5cm]{$Y_G$};\n  \\node (A1r) [right of=A1,node distance=2.5cm]{$Y_{M^e}$};\n  \\path [line] (A0) --node[above]{$$} (A0r);  \n  \\path [line] (A0r) --node[above]{$$} (A1r);  \n  \\path [line] (A1) --node[above]{$$} (A1r);  \n  \\path [line] (A0) --node[above]{$$} (A1);  \n\n\n\\end{tikzpicture}\n\\end{center}\nSince the property of being a closed immersion is stable under base change, it follows that the map $\\mathrm{Bl}_p(Y_G)\\to \\mathrm{Bl}_{\\iota(p)}(\\mathbb{P}^5)$ is an embedding (see corollary II.7.15 in \\cite{har77}\nWe note that $\\mathrm{Bl}_p(\\Pi_G)$ is smooth, and has the structure of a $\\mathbb{P}^1$-bundle over $\\mathbb{P}^2\\times \\mathbb{P}^1$. Likewise, the blow-up $\\mathrm{Bl}_{\\iota(p)}(\\mathbb{P}^5)$ is a $\\mathbb{P}^1$-bundle over $\\mathbb{P}^4$. In other words, after blow ups, we get projective bundles over DAG varieties\n\\begin{center}\n\\tikzset{node distance=1.5cm, auto}\n\\begin{tikzpicture}[auto]\n  \\node (A0){$\\mathrm{Bl}_p(Y_G)$};\n  \\node (A0r)[right of=A0,node distance=2.5cm] {$\\mathrm{Bl}_{\\iota(p)}(Y_{M^e})$};\n  \\node (A1) [below of=A0,node distance=1.5cm]{$Y_{G'}$};\n  \\node (A1r) [right of=A1,node distance=2.5cm]{$Y_{M'}$};\n  \\path [line] (A0) --node[above]{$$} (A0r);  \n  \\path [line] (A0r) --node[right]{$\\mathbb{P}^1$-bundle} (A1r);  \n  \\path [line] (A1) --node[right]{$$} (A1r);  \n  \\path [line] (A0) --node[left]{$\\mathbb{P}^1$-bundle} (A1);  \n\n\n\\end{tikzpicture}\n\\end{center}\nwhere $G'$ looks like\n\\begin{center}\n\\tikzset{node distance=1.5cm, auto}\n\\begin{tikzpicture}[auto]\n  \\node (A0){$$};\n  \\node (A1)[right of=A0,node distance=1.5cm] {$$};\n  \\node (A3) [below of=A0,node distance=0.5cm]{$\\bullet$};\n  \\node (A3l) [left of=A3,node distance=1cm]{$$};\n  \\node (A3ul) [above of=A3l,node distance=0.75cm]{$\\bullet$};\n  \\node (A3dl) [below of=A3l,node distance=0.75cm]{$\\bullet$};\n  \\node (A4) [right of=A3,node distance=1.5cm]{$\\bullet$};\n  \\node (A5p) [right of=A4,node distance=1.5cm]{$$};\n  \\path [line] (A3) --node[above]{$$} (A4);  \n  \\path [line] (A3ul) --node[above]{$$} (A3);  \n  \\path [line] (A3dl) --node[above]{$$} (A3);  \n\n\n\\end{tikzpicture}\n\\end{center}\nand $M'$ is\n\\begin{center}\n\\tikzset{node distance=1.5cm, auto}\n\\begin{tikzpicture}[auto]\n  \\node (A0){$$};\n  \\node (A1)[right of=A0,node distance=1.5cm] {$$};\n  \\node (A3) [below of=A0,node distance=0.5cm]{$\\bullet$};\n  \\node (A3l) [left of=A3,node distance=1cm]{$$};\n  \\node (A3ul) [above of=A3l,node distance=0.75cm]{$\\bullet$};\n  \\node (A3dl) [below of=A3l,node distance=0.75cm]{$\\bullet$};\n  \\node (A4) [right of=A3,node distance=1.5cm]{$\\bullet$};\n  \\node (A5p) [right of=A4,node distance=1.5cm]{$$};\n  \\path [line] (A3) --node[above]{$$} (A4);  \n  \\path [line] (A3ul) --node[above]{$$} (A4);  \n  \\path [line] (A3dl) --node[above]{$$} (A4);  \n  \\path [line] (A3dl) --node[above]{$$} (A3);  \n\n\n\\end{tikzpicture}\n\\end{center}\nOne can check that $\\mathrm{Pic}(\\mathrm{Bl}_{\\iota(p)}(Y_{G}))=\\mathbb{Z}^3$ while $\\mathrm{Pic}(\\mathrm{Bl}_p(Y_{M^e}))=\\mathbb{Z}^2$. If an embedding exists, $\\mathrm{Bl}(Y_{G})$ must be isomorphic to a smooth codimension one subvariety in $\\mathrm{Bl}_p(Y_{M^e})$, which has Picard group isomorphic to $\\mathbb{Z}^2$ by the Lefschetz's hyperplane theorem. This example shows that if two Markov chain relations fit in a DAG, they cannot combine to produce a pure independence relation. The fact that the relations fit in a DAG is essential here. \n\\end{example}\n\nIt is common to encounter DAG varieties that do not have a simple geometric description as in the above families. One can work with numeric invariants in this case. \n\n\\begin{example}\nLet us check if the DAG\n\\begin{center}\n\\tikzset{node distance=1.5cm, auto}\n\\begin{tikzpicture}[auto]\n  \\node (A0){$$};\n  \\node (A1)[right of=A0,node distance=1.5cm] {$$};\n  \\node (A3) [below of=A0,node distance=0.5cm]{$4$};\n  \\node (A3l) [left of=A3,node distance=1cm]{$$};\n  \\node (A3ul) [above of=A3l,node distance=0.75cm]{$$};\n  \\node (A3dl) [below of=A3l,node distance=0.75cm]{$1$};\n  \\node (S) [left of=A3dl,node distance=0.5cm]{$G:$};\n  \\node (A3r) [right of=A3,node distance=1.5cm]{$2$};\n  \\node (A4) [below of=A3r,node distance=.75cm]{$5$};\n  \\node (A6) [right of=A4,node distance=1.5cm]{$6$};\n  \\node (A5p) [right of=A4,node distance=1.5cm]{$$};\n  \\node (B4) [below of=A3,node distance=1.5cm]{$3$};\n  \\path [line] (A3) --node[above]{$$} (A4);  \n  \\path [line] (A3r) --node[above]{$$} (A3);  \n  \\path [line] (A3dl) --node[above]{$$} (A3);  \n  \\path [line] (A3dl) --node[above]{$$} (B4);  \n  \\path [line] (B4) --node[above]{$$} (A3);  \n  \\path [line] (A4) --node[above]{$$} (A6);  \n  \\path [line] (B4) --node[above]{$$} (A4);  \n  \\path [line] (A3r) --node[above]{$$} (A6);  \n\n\\end{tikzpicture}\n\\end{center}\nlies above $M:\\bullet\\to\\bullet\\to\\bullet$. There is an embedding $\\iota:Y_G\\hookrightarrow \\mathbb{P}^{15}$ defined by the ideal sheaf $\\mathcal{I}_G$. If $[G]$ lies above $[M]$, then there must exist an irreducible quadric hyper-surface in $\\mathbb{P}^{15}$ that contains the image of $\\iota$. This means that the map $\\Gamma(\\mathbb{P}^{15},\\mathcal{O}_{\\mathbb{P}^{15}}(2))\\to \\Gamma(\\mathbb{P}^{15},\\iota_*\\mathcal{O}_{Y_G}(2))$ has an irreducible quadratic polynomial in its kernel, which corresponds to a global section of $\\mathcal{I}_G(2)$. Using Magma, we compute the Hilbert polynomial of $\\mathcal{I}_G$ and\nconclude that $h^0\\mathcal{I}_G(1)=2$ and  $h^0\\mathcal{I}_G(2)=31$. Thus there are 2 linearly independent hyperplanes\\footnote{Two hyperplanes are said to be linearly independent if they are defined by linearly independent sections of $O(1)$.} that contain the image of $\\iota$. These in turn give 31 reducible (and thus no irreducible) quadrics containing the image. This shows that $[G]$ does not lie above $[M]$.  \n\\end{example}\n\n\\begin{proposition}\nLet $M$ be a Markov chain on three nodes and $G$ be any DAG on $n$ nodes. Set $a_i:=h^0(\\mathcal{I}_G(i))$.  Then $[G]$ lies above $[M]$ if and only if\n\\[\na_2-\\frac{a_1}{2}(n^2-n-a_1+3)>0.\n\\]\n\\end{proposition}\nThis section justifies the following questions: 1) {\\em what are some invariants of DAG varieties that are useful for ruling out embeddings of the above type?} 2) {\\em how are such invariants related to the combinatorial data of the DAG?} 3) {\\em when can the numbers $a_i$ be read from the Hilbert polynomial of $\\mathcal{I}_G$?} We plan to come back to these questions in a future paper. \n\\\\\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nFacial action units (AUs), defined by Facial Action Coding System (FACS) \\cite{Ekman1997}, has been widely used for understanding human emotions. Different from prototypical facial expression recognition (FER), AUs describe facial muscle movements in facial expressions.\n\nWith the advance of the deep learning networks, deep features show much better generalization than traditional hand-crafted features like HoG, SIFT, etc.\nTheir successful applications on large-scale datasets also benefit the AUs study. Comparing with other tasks, there exist some practical limitations: (1) accurate AU annotations usually require intensive human labor and professional experience; (2) provocation of spontaneous facial expressions is very difficult; (3) unbalanced labels problem, e.g., \\textit{AU6} (Cheek Raiser) is more likely to occur than average but \\textit{AU2} (Outer Brow Raiser) is less.\n\nBased on the RGB modality, several methods achieve the state-of-the-art performance in AU detection \\cite{eac,jaa,niu_2019,dsin,srerl,sev}. \nMeanwhile, the attention mechanism has been adopted in recent AU detection works. Most of them utilized an attention mechanism to find the region of interest (ROI) of each AU.\nHowever, prior knowledge, e.g., facial landmarks are required to predefine the attentions of AUs. Therefore, it's more efficient to utilize a self-attention technique for learning. \nMoreover, little work explores the attention among the AUs learning features but instead focuses on the ROI-based attention.\nOn the other hand, transformer, based on self-attention mechanisms, achieved significant improvements in not only natural language processing (NLP) \\cite{transformer,bert,brown2020,raffel2019} but also computer vision (CV) tasks \\cite{vit,detr,zhu2020,yang2020}.\nTherefore, we propose a new transformer-based network for AU encoding to tackle the above-mentioned inefficiency problem.\n\nData and task based multi-modalities fusions take the advantage of the complementary information and increase the robustness \\cite{taskonomy}. \nMeanwhile, AU analysis relies on the facial muscle movement, the RGB-only detection methods are insufficient for recognizing subtle changes.\nThanks to recently developed multi-modal databases \\cite{wang2012,bp4d+,bp4d} on facial expressions (e.g. RGB, 3D mesh, thermal, and physiology signals in BP4D+ \\cite{bp4d+}), we can broaden our views from classic RGB to various modalities and from 2D image to 3D space.\nFor instance, \\textit{AU6} (cheek raiser) involves the deformation of \\textit{Orbicularis oculi} and \\textit{pars orbitalis} muscles, which shows subtle differences when observed in visible images.\nHowever, due to the high density of points around the eyes region, better geometric changes can be characterized on 3D mesh.\nAlthough some works utilized the additional information presented from multiple modalities in FER and AU detection \\cite{li2015,li2017,amf}, two main challenges remain: 1) the model must learn the most relevant features for representation; 2) the model must be effectively when fusing two modalities.\n\nInspired by these vision transformer works \\cite{vit,detr}, and multi-modal transformer works \\cite{mult,transfuser}, we explore the methods that are utilized transformer architecture for both feature representation and fusion. \nIn this paper, we study how multi-modal data help AUs analysis and present a novel Multi-Head Fused Transformer (MFT) for AU detection. Fig. 1\\ref{fig:figure1} illustrates the proposed network, which contains two principal points to notice. \nFirst, two pipelines in our model, each pipeline is fed with AU embeddings from each modality. \nSecond, transformer-based fusion modules are applied for the fusion of two modalities along the pipelines.\nWe design the multi-head fusion attention module in the fusion transformer to fuse modalities effectively.\nAs a result, our model not only learns the relevant AU encoding feature but also effectively fuses two modalities. \nWe evaluate our methods on two public datasets in terms of BP4D \\cite{bp4d} and BP4D+ \\cite{bp4d+}. \nOur transformer based encoder performs better than the state-of-the-art methods on RGB images, which demonstrates the effectiveness of the AU encoding.\nMoreover, the results of the fusion experiments with RGB and Depth images also show our method outperforms the SOTA methods, which indicates the fusion transformer module works effectively.\n\nThe main contribution of this work lies in three-fold: \n\n\\begin{enumerate}\n   \\item We present a novel transformer based framework for AU detection, which can learn AU encoding features effectively. \n   \\item We design a transformer based fusion module for multi-modalities fusion.\n   To the best of our knowledge, this is the first attempt to utilize \\B{multi-head fusion attention} to fuse AU features of two modalities in a transformer.\n   \\item Extensive experiments demonstrate that our network outperforms the state-of-the-art works in AU detection by both RGB-only data and RGB-Depth fusion features. \n\\end{enumerate}\n\n\\begin{figure*}[ht] \n    \\centering\n    \\includegraphics[width=0.95\\textwidth]{figures\/figure_1.pdf}\n    \\caption{\\textbf{Framework architecture.} The proposed framework consists of two pipelines with two modalities as inputs: $X_\\alpha$, $X_\\beta$. For example, modalities $\\alpha$ and $\\beta$ represent RGB and Depth respectively in the figure. \n    We design two core components in the network: Fusion Transformer (FT) module and AU Transformer Encoder(TE). \n    Two modalities are first extracted by a backbone network and then split the FC layer for each AU. \n    After obtaining AU embeddings of two modalities, they are fused through a FT module to acquire the fusion features $F(\\alpha, \\beta)$. Then following 4 TEs in each pipeline, $F(\\alpha, \\beta)$ and $\\beta$ are encoded. Two features after each encoder of two pipelines are fused again by another FT module. Each pipeline ends with a classifier, and the final losses are combined from both. It is worth noting that $\\alpha$ and $\\beta$ can also represent Depth and RGB in turn, which means our fusion framework is sensitive to the fusion order.} \n    \\label{fig:figure1}\n\\end{figure*}\n\n\\begin{figure}[ht]  \n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{figures\/figure_2.pdf}\n    \\caption{\\textbf{Structure of Fusion Transformer module.} The module is based on a standard 2-heads attention transformer for modality $\\alpha$ and modality $\\beta$. \n    The difference is the attention of $\\beta$ is calculated by $Q_\\alpha$, $K_\\beta$, $V_\\beta$.\n    The fusion features are then passed to AU Transformer encoders.}\n    \\label{fig:figure2}\n\\end{figure}\n\n\\section{RELATE WORK}\n\nWe review the previous related works on AU detection and multi-modal deep learning.\n\n\\subsection{AU Detection}\nRecent works on Facial Action Units detection achieved certain improvements by deep learning and attention mechanism. \nZhao et al. \\cite{drml} proposed a local feature learning approach that uniformly slices the first feature map as a region layer. \nLi et al. \\cite{eac} proposed an EAC network, which predefines attentions based on AU-related landmarks to enhance and crop the ROI of AUs. \nZhang et al. \\cite{zhang2018} proposed an adversarial training method to maximize identity loss to focus on the AU-related features.\nShao et al. \\cite{jaa} proposed the JAA-Net, which explores multi-scale region learning and adaptive attention learning.\nLi et al. \\cite{li2019} gave insight into the AU relation leaning by a Graph Neural Network (GCN) based model.\nNiu et al. \\cite{niu_2019} proposed a subject independent model to use the facial landmarks as person-specific shape regularization for AU detection.\nThese works with regional attention usually require prior information like facial landmarks. \nRecently, Yang et al. \\cite{sev} proposed the SEV-Net that utilized the transformer for Intra-AU and Inter-AU attention learning from the AU semantic descriptions. However, the AU semantic descriptions are required as the prior knowledge.\nDiffer from the ROI-based attention, our transformer-based methods exploit the self-attention mechanism between AUs, which is no landmarks requirement.\nCompare with the SEV-Net, our model only require the AU labels, and no AU semantic descriptions are needed. \n\n\\subsection{Multi-modal Deep learning}\n\nMulti-modal machine learning aims to build models that can process, correlate, and integrate information from multiple modalities \\cite{multimodal}.\nThe success of deep learning has been a catalyst to solving\nincreasingly complex machine-learning problems, which\noften involve multiple data modalities \\cite{ramachandram2017}.\nMulti-modal deep learning has been succeed used in a wide range of applications, e.g., human activity recognition \\cite{mtut,neverova2015,emonets}, autonomous\ndriving \\cite{nuscenes,cui2019,transfuser}, and image segmentation \\cite{hu2018,Huang_2018}.\n\nIn the past decade, multi-modal learning has been studied on both facial expression recognition and action unit detection. \nCompare with visual images and audio \\cite{emonets,2019avec} combination, especially dealing with videos, the potential of infrared, thermal, and 3D are not fully explored. \nLi et al. \\cite{li2015,li2017} proposed approaches for facial expression recognition based on features fusion of 2D and 3D.\nIrani et al. \\cite{irani2015} proposed a method to utilize RGB-Thermal-Depth images for pain estimation. \nLakshminarayana et al. \\cite{lakshminarayana2019} explored the combination of physiological signals and RGB images to predict action units. \nYang et al. \\cite{amf} proposed a model called AMF, which included a feature scoring module to select the most relevant feature representations from different modalities.\n\nTransformer \\cite{transformer}, based on self-attention mechanism, was first applied to natural language processing tasks, then this framework has been extended to computer vision. \nFor example, the ViT model, proposed by Dosovitskiy et al. \\cite{vit}, applied a pure transformer to sequences of image patches, which achieved comparable results to CNNs on image classification.\nCarion et al. \\cite{detr} proposed the DETR framework, which combined a common CNN with a transformer encoder-decoder architecture.\nRecently, researchers have explored multi-modal learning by transformers.\nThe most related works are called Multimodal Transformer (MulT) \\cite{mult} and Multi-Modal Fusion Transformer (TransFuser) \\cite{transfuser}, even though they target human multi-modal language analysis and multi-modal autonomous driving respectively.\nIn MulT, pairwise crossmodal transformers are designed, where the $Q$, and $KV$ are from different modalities. Then the outputs of pairwise crossmodal transformers are concatenated for fusion. \nIn the TransFuser work, two modalities fusion occurred before the self-attention in a transformer, then was split after the self-attention. \n\\B{Differ} from these two works, our method fuses the features of two modalities by the \\B{multi-head fusion attention} module in a transformer.\n\n\n\n\n\\section{PROPOSED METHOD}\nThe proposed Multi-Head Fused Transformer (MFT) is sketched in Fig. \\ref{fig:figure1}.\nEach data modality goes through its own backbone to extract features and then output the AU embeddings from the fully connected (FC) layer. \nThere are two pipelines, and each contains 4 transformer-based AU encoders and a classifier in the end. \nOne pipeline's input is the fusion feature from a fusion transformer, which is fed with the AU embedding features of fusion features, and the other pipeline encode the AU features from a single modality embeddings.\nIn the end, two losses from these two pipelines are combined together for training.\n\n\\subsection{AU Embeddings}\nTwo pre-trained ResNet-50 \\cite{resnet} are employed as backbone encoders for RGB and Depth images.\nThe feature out from the fully connected layer is divided by each AU and then fed into an embedding layer for AU embedding.\nThe embedding layer in our model is similar to the standard transformer \\cite{transformer}.\nSince the AU encoder feature shares the same position with AU embedding, there is no need for position embeddings. \nMeanwhile, the stand layer normalization is applied to AU embedding as well.\n\n\\subsection{Transformer Encoder}\n\nWe apply the multi-layer transformer encoder to encode each AU embedding feature.\nThe Transformer encoder is formed by multiple alternating layers, where each layer consists of multiheaded self-attention and wise feed-forward network blocks.\nLayernorm (LN) \\cite{layernorm} is utilized before every block and residual connector after every block.\nTo summary, the transformer encoder is defined as:\n\n\\begin{equation}\nX_A = LN(MSA(X)) + X\n\\end{equation}\n\n\\begin{equation}\nX_B = LN(FFN(X_A)) + X_A\n\\end{equation}\nwhere $X$ is the input of the Transformer layer, $X_B$ is the output of the Transformer layer, $LN$ is Layernorm and $MSA$ is multihead self-attention.\n\n\\subsection{Fusion Transformer}\nThe architect of Fusion Transformer is shown in Fig. \\ref{fig:figure2}.\nTwo modalities $\\alpha$ and $\\beta$ are denoted $X_{\\alpha} \\in \\mathbb{R}^{T \\times D}$ and $X_{\\beta} \\in \\mathbb{R}^{T \\times D}$ respectively.\n$T$ and $D$ are represented as AU class numbers and AU embedding feature dimensions, respectively.\nInspired by the Multimodal Transformer in \\cite{mult}, we designed a two-heads self-attention for fusion, which is the core component in the fusion transformer.\nWe define the Querys, Keys, Values as following: $Q_{\\alpha}^i = X_{\\alpha}W_{Q_{\\alpha}}^i$,\n$K_{\\alpha} = X_{\\alpha}W_{K_{\\alpha}}$, $K_{\\beta} = X_{\\beta} W_{K_{\\beta}}$,\n$V_{\\alpha} = X_{\\alpha}W_{V_{\\alpha}}$, and $V_{\\beta} = X_{\\beta}W_{V_{\\beta}}$,\nwhere $W_{Q_{\\alpha}}^i \\in \\mathbb{R}^{T \\times D}, i \\in \\{1,2\\}$, \n$W_{K_{\\alpha}} \\in \\mathbb{R}^{T \\times D}$,\n$W_{K_{\\beta}}  \\in \\mathbb{R}^{T \\times D}$,\n$W_{V_{\\alpha}} \\in \\mathbb{R}^{T \\times D}$,\n$W_{V_{\\beta}}  \\in \\mathbb{R}^{T \\times D}$.\nThen the fusion multi-head self-attention equation $f(X_\\alpha, X_\\beta)$ for $\\alpha$ and $\\beta$ is defined:\n\n\n\\begin{align} \\label{eq:fusion}\nf(X_\\alpha, X_\\beta) &= Att(Q_\\alpha^1, K_\\alpha, V_\\alpha) \\oplus Att(Q_\\alpha^2, K_\\beta, V_\\beta) \\nonumber\\\\\n    &= \\sigma(\\frac{Q_\\alpha^1K_\\alpha^T}{\\sqrt{d_k}})V_\\alpha \\oplus\n       \\sigma(\\frac{Q_\\alpha^2K_\\beta^T}{\\sqrt{d_k}})V_\\beta \\nonumber\\\\\n    &= \\sigma(\\frac{X_{\\alpha}W_{Q_{\\alpha}}^1(X_{\\alpha}W_{K_{\\alpha}})^T}{\\sqrt{d_k}})V_\\alpha \\oplus\\\\\n    &\\qquad{} \\sigma(\\frac{X_{\\alpha}W_{Q_{\\alpha}}^2(X_{\\beta}W_{K_{\\beta}})^T}{\\sqrt{d_k}})V_\\beta \\nonumber\n\\end{align}\n\nwhere $\\sigma$ is $softmax$, $\\oplus$ is concatenation, and $d_k$ is dimension of queries and keys.\n\nThe other components in the fusion transformer are the same as the standard transformer encoder we mentioned above.\nBased on the attention of $Q_{\\alpha}^2K_{\\beta}V_{\\beta}$, the fusion transformer enables the $\\alpha$ modality to receive information from $\\beta$ modality and then combines them.\nIt is worth noting that equation (\\ref{eq:fusion}) can be easily extended to more modalities as following:\n\\begin{align}\n    f(X_1, X_2, ... , X_m) = & Att(Q_1^1, K_1, V_1) \\, \\oplus \\nonumber \\\\\n                             & Att(Q_1^2, K_2, V_2) \\oplus \\cdots\\\\\n                             & Att(Q_1^m, K_m, V_m) \\nonumber\n\\end{align}\nwhere $m$ is the number of modalities.\n\n\\subsection{Loss Function}\nAs a multi-label task, AU detection faces the common problem of data unbalances.\nHence, we apply the weighted binary cross-entropy (BCE) loss functions. \nThe $AU_k$ occurrence probability function $P(AU_k)$ and the correlate positive weight $p_k$ of weighted BCE are given as:\n\n\\begin{align}\n        P(AU_k) &= \\frac{\\sum_{i=1}^{N} y_k^i}{\\sum_{i=1}^{N} \\sum_{k=1}^{C} y_{k}^i}\\\\\n        p_k &= \\frac{P(AU_k)}{\\min\\limits_{x \\in \\{1,...,C\\}}P(AU_x)}\n\\end{align}\n\nwhere $N$ is the total number of training set, $C$ is the number of AU. \n\nThe weighted BCE loss is defined:\n\\begin{align}\n    \\mathcal{L} =& - \\frac{1}{N}\\sum_{i=1}^{N} \\sum_{k=1}^{C}(p_k\\times y_k^i \\times log(\\widehat{y_k^i}) + \\nonumber\\\\\n       &\\qquad{} (1-y_k^i) \\times log(1-\\widehat{y_k^i}))\n\\end{align}\nwhere $y_k^i$ and $\\widehat{y_k^i}$ represent the ground truth label and prediction for $AU_k$ respectively.\nThere are two classifiers for the two pipelines in our model, where two losses are combined.\nWe denote the loss for the first pipeline of fusion feature as $\\mathcal{L}^{f(\\alpha, \\beta)}$ and the second pipeline of $\\beta$ feature as $\\mathcal{L}^\\beta$, then the final loss can be written as:\n\\begin{equation}\n    \\mathcal{L} = \\lambda_1\\mathcal{L}^{f(\\alpha, \\beta)} + \\lambda_2\\mathcal{L}^{\\beta}\n\\end{equation}\nwhere $\\lambda_1$ and $\\lambda_2$ are two positive regularization parameters.\n\n\\section{EXPERIMENTS}\n\nTo evaluate our method, we examine our framework in AU detection based on single modality and also multi-modalities.\nSingle modality experiments are designed to validate the efficiencies of the AU encoding by the transformer-based encoder.\nWe also make multi-modalities AU detection experiments for fusion quality evaluation.\nWe use two public multi-modal databases BP4D \\cite{bp4d} and BP4D+ \\cite{bp4d+} for AU occurrence detection.\n\n\\subsection{Data}\n\n\\subsubsection{BP4D}\nThe dataset contains 41 subjects (23 females and 18 males) captured under laboratory environments.\n8 tasks are designed to elicit a range of spontaneous emotions. There are 328 = 41 \u00d7 8 sequences in total with a frame rate of 25. \nExpert coders select the most expressive 20s of each sequence for AU coding.\nAround 140,000 labeled frames are included in our experiments and split into 3-fold for a fair comparison with state-of-the-art algorithms. In order to avoid training and testing on images of the same person, we use the same splits as in \\cite{eac}. \nFor each frame, the database provides 3D face meshes as well as corresponding texture images.\n\n\\subsubsection{BP4D+}\nThere are 140 subjects in BP4D+. On the basis of 8 tasks included in BP4D, two more tasks are introduced with a richer set of spontaneous emotions.\nEach frame is associated with synchronized 3D face mesh, 2D texture, thermal image, and a series of physiological signals.\nSimilar to BP4D, 20 seconds from four tasks were AU labeled for all 140 subjects, resulting in 192,000 labeled frames.\n12 Aus are selected and the performance of four-fold subject-exclusive cross-validation is reported. \n\n\\subsection{Preprocessing}\nIn our experiments, we use two modalities in BP4D and BP4D+: 2D RGB image and 3D face model. \nThe face regions of RGB images are detected and cropped by the publicly available library OpenCV.\nA raw 3D face model contains a 30k - 50k number of vertices in BP4D and BP4D+ databases, including the face, hairs, and neck areas of a subject.\nWe first cropped the face area of 3D meshes and then project them into depth images.\nAll the face images, including RGB and Depth, are resized to 244x244 for training and testing.\nWe calculate the average and standard deviation of each modality in one dataset and then apply z-score standardization to each image.\n\n\n\\subsection{Implementation Details}\nBoth backbones in our model are ResNet-50 networks, which were pretrained on ImageNet \\cite{imagenet}.\nThe layer number and head number are set 3 and 4 in the transformer encoder, respectively.\nThe AU embedding size is 128, the header size is 32, the MLP size is 256, and the dropout rate is 0.5.\nThe fusion transformer module is a one-layer and two-head transformer encoder, and the other parameters are the same as the transformer encoder except there's no dropout.\nThe hyper-parameter $\\lambda_1$ and $\\lambda_2$ are selected by comparing the best F1-score in the first fold of BP4D, which are 0.6 and 0.4 respectively. \nThe comparing results is shown in Table.\\ref{tab:table0}. \nWe use the stochastic gradient descent (GSD) optimizer with a momentum of 0.9 and weight decay of 0.0005. \nThe initial learning rate is 0.01 and 10 times smaller every epoch start from epoch 4. \nThe network is trained for 5 epochs for BP4D and 6 epochs for BP4D+.\nThe batch size is set to 32 on both BP4d and BP4D+.\n\n               \n\n\\begin{table}[th]\n\\caption{Hyper-parameters Comparison on one fold of BP4D}\n\\label{tab:table0}\n\\centering\n\\renewcommand{\\arraystretch}{1.0}\n\\begin{tabular}{c|cccccccc}\n\\hline\n$\\lambda_1$  & 0.2  & 0.3  & 0.4  & 0.5     & 0.6   & 0.7   & 0.8  &    1.0 \\\\ \\hline\n$\\lambda_2$  & 0.8  & 0.7  & 0.6  & 0.5     & 0.4   & 0.3   & 0.2  &    0.5 \\\\ \\hline\nAvg F1       & 63.1 & 63.0 & 63.5 & 64.1  & \\B{64.7} & 63.7  & 63.7 &   63.5 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n               \n\n\\begin{table*}[ht]\n\\caption{F1 scores in terms of 12 AUs and mean accuracy on BP4D database. Bold numbers indicate the best performance in each modal; Bold-Bracketed numbers indicate the best in all modals}\n\\label{tab:table1}\n\\centering %\n\\renewcommand{\\arraystretch}{1.0}\n\\begin{tabular}{l|c|*{12}{c}|c}\n\\hline\nMethods             & Modal   & AU1       & AU2       & AU4       & AU6       & AU7       & AU10      & AU12      & AU14      & AU15      & AU17      & AU23      & AU24      & Avg\\\\\n\\hline\nEAC \\cite{eac}      & RGB     & 39.0      & 35.2      & 48.6      & 76.1      & 72.9      & 81.9      & 86.2      & 58.8      & 37.5      & 59.1      & 35.9      & 35.8      & 55.9\\\\\nDSIN \\cite{dsin}    & RGB     & 51.7      & 40.4      & 56.0      & 76.1      & 73.5      & 79.9      & 85.4      & 62.7      & 37.3      & 62.9      & 38.8      & 41.6      & 58.9\\\\\nJAA \\cite{jaa}      & RGB     & 47.2      & 44.0      & 54.9      & 77.5      & 74.6      & 84.0      & 86.9      & 61.9      & 43.6      & 60.3      & 42.7      & 41.9      & 60.0\\\\\nSRERL \\cite{srerl}  & RGB     & 46.9      & 45.3      & 55.6      & 77.1      & \\B{78.4}  & 83.5      & 87.6      & \\B{63.9}  & 52.2      & \\B{63.9}  & 47.1      & 53.3      & 62.9\\\\ \nSEV-Net \\cite{sev}  & RGB     & \\BB{58.2} & \\B{50.4}  & 58.3      & \\B{81.9}  & 73.9      & \\B{87.8}  & 87.5      & 61.6      & 52.6      & 62.2      & 44.6      & 47.6      & 63.9\\\\\n\\textbf{Ours}       & RGB     & 49.6 \t  & 48.1      & \\B{59.9}  & 78.4      & 78.0\t  & 83.7  \t  & \\B{87.9}  & 62.0\t  & \\B{55.3}  & 61.8      & \\B{50.9}  & \\B{54.9}  & \\B{64.2}\\\\\n\\hline\nResNet-50           & Depth   & \\B{44.5}  & \\B{41.3}  & \\B{57.2}  & 76.5\t  & 72.6\t  & 80.9\t  & 86.8\t  & 54.3\t  & \\B{50.2}  & \\B{62.7}  & 48.1\t  & \\B{44.5}  & 60.0\\\\ \n\\textbf{Ours}       & Depth   & 38.6\t  & 37.3\t  & 44.2\t  & \\B{84.2}  & \\BB{89.0} & \\B{89.7}  & \\BB{89.2} & \\BB{79.8} & 44.7\t  & 46.0\t  & \\BB{53.2} & 37.0\t  & \\B{61.1}\\\\ \n\\hline\nLate fusion         & Fusion  & 44.3\t  & 34.0\t  & 41.6\t  & \\BB{85.2} & \\B{87.3}  & \\BB{90.4} & 88.9      & \\B{79.4}  & 46.1\t  & 49.6\t  & 58.6\t  & 38.6\t  & 62.0\\\\\nMTUT \\cite{mtut}    & Fusion  & 51.3      & 50.2      & 62.2      & 77.2      & 71.7      & 83.8      & 88.2      & 61.4      & 54.3      & 57.9      & 45.8      & 42.2      & 62.2\\\\ \nTEMT-Net \\cite{temt}& Fusion  & \\B{53.7}  & 47.1      & 60.5      & 77.6      & 75.6      & 84.8      & 87.4      & 67.0      & \\BB{57.2} & 61.3      & 44.7      & 41.6      & 63.2\\\\ \nAMF \\cite{amf}      & Fusion  & 52.1      & \\BB{51.0} & \\BB{64.5} & 79.2      & 73.9      & 86.4      & 88.3      & 60.5      & 55.3      & 64.2      & 47.7      & 49.2      & 64.4\\\\ \n\\textbf{Ours}       & Fusion  & 51.6\t  & 49.2\t  & 57.6      & 78.8\t  & 77.5      & 84.4\t  & 87.9\t  & 65.0\t  & 56.5\t  & \\BB{64.3} & 49.8\t  & \\BB{55.1} & \\BB{64.8}\\\\ \n\\hline\n\\end{tabular}\n\\end{table*}\n               \n\\begin{table*}[ht]\n\\caption{F1 scores in terms of 12 AUs and mean accuracy on BP4D+ database. Bold numbers indicate the best performance in each modal; Bold-Bracketed numbers indicate the best in all modals; * indicts the result from our own implementation.}\n\\label{tab:table2}\n\\centering %\n\\renewcommand{\\arraystretch}{1.05}\n\\begin{tabular}{l|c|*{12}{c}|c}\n\\hline\nMethods             & Modal   & AU1       & AU2       & AU4       & AU6       & AU7       & AU10      & AU12      & AU14      & AU15      & AU17      & AU23      & AU24      & Avg\\\\\n\\hline\nEAC \\cite{eac}*     & RGB     & 43.7      & 39.0      & 14.0      & 85.6      & 87.2      & 90.5      & 88.7      & \\BB{88.4} & 45.7      & 49.0      & 57.3      & \\BB{43.6} & 61.1\\\\\nJAA \\cite{jaa}*     & RGB     & 46.0\t  & \\B{41.3}  & \\B{36.0}  & 86.5\t  & 88.5\t  & 90.5\t  & \\B{89.6}  & 81.1\t  & 43.4\t  & 51.0\t  & 56.0\t  & 32.6\t  & 61.9\\\\\nSEV-Net \\cite{sev}  & RGB     & 47.9      & 40.8      & 31.2      & \\BB{86.9} & 87.5      & 89.7      & 88.9      & 82.6      & 39.9      & \\BB{55.6} & \\BB{59.4} & 27.1      & 61.5 \\\\\n\\textbf{Ours}       & RGB     & \\B{48.4}  & 37.1\t  & 34.4\t  & 85.6\t  & \\BB{88.6} & \\BB{90.7} & 88.8      & 81.0\t  & \\B{47.6}  & 51.5\t  & 55.6\t  & 36.9\t  & \\B{62.2}\\\\\n\\hline\nResNet-50           & Depth   & 35.7\t  & 33.9\t  & 38.9\t  & 84.2\t  & 86.8\t  & 89.7\t  & 89.1\t  & 77.1\t  & 40.7\t  & 45.7\t  & 54.4\t  & 33.1\t  & 59.1 \\\\ \n\\textbf{Ours}       & Depth   & \\B{43.2}  & \\B{40.2}  & \\B{43.3}  & \\B{84.5}  & \\B{88.5}  & \\B{90.2}  & \\B{89.3}  & \\B{79.7}  & \\B{45.8}  & \\B{48.7}  & \\B{53.8}  & \\B{36.8}  & \\B{62.0}\\\\ \n\\hline\nLate fusion         & Fusion  & 46.0\t  & 41.3\t  & 36.0      & \\B{86.5}  & 88.5\t  & 90.5\t  & 89.6\t  & 81.1\t  & 43.4\t  & 51.0      & 56.0\t  & 32.6\t  & 61.9\\\\ \nAMF \\cite{amf}      & Fusion  & 45.3      & \\BB{42.5} & 34.8      & 85.9      & 87.9      & 89.5      & \\BB{90.4} & \\B{82.6}  & \\BB{50.1} & 45.5      & 55.7      & \\B{42.1}  & 62.7\\\\ \n\\textbf{Ours}       & Fusion  & \\BB{49.6} & 42.0\t  & \\BB{43.5} & 85.8\t  & \\BB{88.6} & \\B{90.6}  & 89.7\t  & 80.8\t  & 49.8\t  & \\B{52.2}  & \\B{59.1}  & 38.4\t  & \\BB{64.2}\\\\ \n\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\subsection{Results}\n\n\n\n\\subsubsection{single-modal results}\n\nWithout fusion modules, each pipeline in our network is workable for single modality AU detection. \nHence, we first compare our method to the state-of-the-art algorithms on a RGB modality in terms of F1-score.\nThe methods evaluated on BP4D are including EAC \\cite{eac}, DSIN \\cite{dsin}, JAA \\cite{jaa}, SRERL \\cite{srerl}, SEV-Net \\cite{sev}. \nThe upper section of Table.\\ref{tab:table1} shows the results on BP4D, where our method achieves the highest performance in RGB modality.\nSpecifically, our method outperforms all the SOTA methods in recognizing \\textit{AU4}, \\textit{AU12}, \\textit{AU15}, \\textit{AU23}, \\textit{AU24}.\nOur method achieves 4.2\\% higher than JAA, which used facial landmarks as a joint task for AU detection.\nCompare to SRERL, which requires an AU relation graph from label distribution, our method still shows 1.3\\% higher performance.\nRecently, SEV-Net achieved 63.9\\% in F1-score by including the transformer for Intra-AU and Inter-AU attention learning from the semantic descriptions of AUs.\nHowever, we don't need these semantic descriptions of AUs and still achieve 64.2\\% by increasing 0.3\\% in F1-score. \nTo evaluate our method on Depth modality, we compare with ResNet-50, as the baseline function, where our method improves the performance in almost all AUs, except the \\textit{AU15}, which is 0.1\\% less. \nThe result is reported in the middle section of Table.\\ref{tab:table1}.\nOur method on Depth performs better than RGB on some AUs, e.g., \\textit{AU6}, \\textit{AU7}, \\textit{AU10}. \nMoreover, it even performs best in all modalities on the following AU: \\textit{AU7}, \\textit{AU12}, \\textit{AU14}, \\textit{AU23}.\n\nOn BP4D+, we compare with following works: EAC \\cite{eac}, JAA \\cite{jaa}, SEV-Net \\cite{sev}. \nSince there's no report on BP4D+ in EAC and JAA, we implement these two methods for AU detection on BP4D+.\nJAA implementation is based on their public source code and EAC is implemented by converting the original Theano framework to PyTorch.\nThe results are shown in the upper section of Table.\\ref{tab:table2}.\nOur method also outperforms the SOTA methods, which are 1.1\\%, 1.0\\%, and 0.7\\% higher than EAC, JAA, and SEV-Net, respectively.\nSimilar to BP4D, the result of AU detection on Depth modality is shown in the middle of Table.\\ref{tab:table2}, where our method is 2.9\\% higher than the baseline.\nThe achieved F1-score at 62.0 is even higher than the other RGB-based SOTA methods, which could be caused by the high quality of the 3D model in BP4D+.\nThe high performance of our method proves its ability for AU encoding.\n\n\n\n               \n\\begin{table*}[th]\n\\caption{Ablation study of effectiveness of key components of our model on BP4D database. TE represent transformer encoder; FT represent fusion transformer. Bold numbers indicate the best performance.}\n\\label{tab:table3}\n\\centering %\n\\renewcommand{\\arraystretch}{1.0}\n\\begin{tabular}{l|*{12}{c}|c}\n\\hline\nMethods                      & AU1       & AU2       & AU4       & AU6       & AU7       & AU10      & AU12      & AU14      & AU15      & AU17      & AU23      & AU24      & Avg\\\\\n\\hline\nLate fusion                  & 44.3\t     & 34.0\t     & 41.6\t     & 85.2      & 87.3      & \\B{90.4}  & 88.9      & 79.4      & 46.1\t     & 49.6\t     & \\B{58.6}\t & 38.6\t     & 62.0\\\\ \nLate fusion + TE             & 48.0\t     & 39.2\t     & 37.0\t     & \\B{86.1}\t & 89.1\t     & \\B{90.4}\t & 88.7\t     & \\B{80.9}\t & 46.7\t     & 47.8\t     & 55.9\t     & 39.8\t     & 62.5\\\\ \nResNet-50 + FT               & 47.8\t     & 36.8\t     & 39.7\t     & 85.4\t     & \\B{89.2}\t & 90.3\t     & \\B{89.0}\t & 80.6\t     & 46.5\t     & 49.6\t     & 58.4\t     & 40.1\t     & 62.8\\\\ \nResNet-50 + FT + TE (Ours)   & \\B{51.6}\t & \\B{49.2}\t & \\B{57.6}\t & 78.8\t     & 77.5\t     & 84.4\t     & 87.9\t     & 65.0\t     & \\B{56.5}\t & \\B{64.3}\t & 49.8\t     & \\B{55.1}\t & \\B{64.8}\\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n               \n\\begin{table}[th]\n\\caption{Ablation study of fusion modal order on BP4D and BP4D+ database.  Bold numbers indicate the best performance on each database.}\n\\label{tab:table4}\n\\centering\n\\renewcommand{\\arraystretch}{1.0}\n\\begin{tabular}{c|cc|cc}\n\\hline\n          & \\multicolumn{2}{c|}{BP4D}  & \\multicolumn{2}{c}{BP4D+} \\\\\n            \\cline{2-5}\n          & $F(R,D)$     & $F(D,R)$    & $F(R,D)$     & $F(D,R)$\\\\\n\\hline\nAU1       & \\B{51.6}\t & 50.3\t       & 49.3         & \\B{49.6}\\\\ \nAU2       & \\B{49.2}     & 48.9\t       & 39.7         & \\B{42.0}\\\\\nAU4       & 57.6\t     & \\B{58.0}\t   & 42.8         & \\B{43.5}\\\\ \nAU6       & 78.8         & \\B{79.6}\t   & 85.4         & \\B{85.8}\\\\ \nAU7       & \\B{77.5}\t & 77.3\t       & \\B{88.9}     & 88.6\\\\ \nAU10      & 84.4         & \\B{84.5}\t   & \\B{91.0}     & 90.6\\\\ \nAU12      & 87.9\t     & \\B{88.2}\t   & \\B{90.0}     & 89.7\\\\ \nAU14      & \\B{65.0}     & 62.4        & \\B{81.8}\t  & 80.8\\\\ \nAU15      & \\B{56.5}\t & 54.7\t       & 47.7         & \\B{49.8}\\\\ \nAU17      & \\B{64.3}     & 61.8        & \\B{53.0}\t  & 52.2\\\\ \nAU23      & 49.8\t     & \\B{51.9}\t   & \\B{59.2}         & 59.1\\\\ \nAU24      & \\B{55.1}     & 53.9\t       & 37.5         & \\B{38.4}\\\\ \n\\hline\nAvg       & \\B{64.8}     & 64.3        & 63.9         & \\B{64.2}\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\\subsubsection{multi-modal results}\n\nFusion refer to the fusion of RGB and Depth in the modal column of Table.\\ref{tab:table1} and Table.\\ref{tab:table2}.\nEquation (\\ref{eq:fusion}) shows $f(X_{rgb}, X_{depth})$ is not equal to $f(X_{depth}, X_{rgb})$, which means our method is sensitive to the order of fusion modalities.\nBased on various experiments results of evaluating the order of modalities in fusion transformer module, we report the result of $f(X_{rgb}, X_{depth})$ on Table.\\ref{tab:table1} and $f(X_{depth}, X_{rgb})$ is on Table.\\ref{tab:table2}.\nThe differences of the fusion modalities order will be discussed in the following ablation study subsection.\n\nLate fusion is currently the most common fusion technique, so we use late fusion with ResNet-50 as a baseline. \nThen we compare with these methods on BP4D: MTUT \\cite{mtut}, TEMT-Net \\cite{temt}, and AMF \\cite{amf}.\nThe F1-score of MTUT and TEMT-Net are reported in the paper of AMF, where the author implemented them with the fusion of RGB and Depth.\nCut-switch is a data augmentation method in AMF, which is independent of AMF's deep model.\nFor a fair comparison, we compare AMF with its non-switched version without applying such data augmentation.\nThe results on BP4D are shown in the bottom section of Table.\\ref{tab:table1}.\nOur method outperforms all the related methods and achieves the highest F1-score at 64.8\\%, which is 2.6\\% higher than latefusion, 2.6\\% higher than MTUT, 1.6\\% higher than TEMT-Net, and 0.4\\% higher than AMF.\nIn terms of 12 AUs, our method performs best in \\textit{AU17} and \\textit{AU24}.\n\n\nBesides baseline, we also compare our method to AMF on BP4D+ and report the results in the bottom part of Table.\\ref{tab:table2}\nOur method also achieves the best performance, showing 2.3\\% higher than latefusion and 1.5\\% higher than AMF.\nMore specifically, our model achieves the best performances on \\textit{AU1}, \\textit{AU4}, \\textit{AU7}, \\textit{AU10}, \\textit{AU17}, and \\textit{AU23}. \n\nAs we can see, our multi-modal method is obviously outperformed over the single modal methods on both two datasets. \nIn addition, the high performance over the related multi-modal methods also verifies the effectiveness of our method. \n\n\n\n\n\\subsection{Ablation Study}\n\n\\subsubsection{Effect of individual part for AU detection}\nFor more rigorous proof of the validity of our model, we conduct experiments on BP4D dataset to examine the two major individual components, transformer encoder, and fusion transformer.\nTwo important components of our model, namely, transformer encoder (TE), and fusion transformer (FT), are tested as we run the same experiments using variations of the proposed network, with and without them.\nTable.\\ref{tab:table3} shows the performance of various combinations of the components in terms of F1-score over all the AUs.\nBy using the transformer encoder to train latefusion features, the performance is improved by 0.5\\%, which provides another piece of evidence for the effectiveness of the transformer-based AU encoder.\nBy applying the fusion transformer and without the transformer encoder, the performance is 0.8\\% higher than latefusion, which shows the effectiveness of the FT module.\nThe last row shows the proposed method, with both TE and FT, achieves the best performance. \n\n\\subsubsection{Effect of modal fusion order}\nAs previously discussed, our method is affected by the order of modal fusion.\nTwo fusion features, $F(R, D)$ and $F(D, R)$, respectively represent the output from fusion transformer by applying  $f(X_{depth}, X_{rgb})$ and $f(X_{depth}, X_{rgb})$.\nWe conduct experiments on BP4D and BP4D+ datasets and report the results in Table.\\ref{tab:table4}.\nAs we can see, $F(R, D)$ shows improved performance than $F(D, R)$ on BP4D, which is 0.5\\% higher, showing fusing the projected Depth features in RGB space is more effective.\nHowever, on BP4D+, fusion in Depth space outperforms than it in RGB space by 0.3\\%.\nThe different quality of 3D data, different number of frames, and label imbalance may cause this difference.\nIt is worth noting that both of these fusion features meet or exceed the performance of the state-of-the-art methods on each database.\n\n\n\\subsection{Multi-modal AU Detection Analysis}\n\nBased on our method, we compare the recognition performance on both BP4D and BP4D+ datasets.\nThe result on BP4D is shown in Fig.\\ref{fig:bp4d}.\nUsually, the fusion modality performs better than the RGB, even when depth performs much lower.\nHowever, the fusion result is clearly lower than the RGB on \\textit{AU4}, with a 2.3\\% decrease.\nWe propose an assumption that the Depth information plays a major role for \\textit{AU4}, so the poor property of Depth modality caused performance deduction in \\textit{AU4}. Fig.\\ref{fig:bp4d+} shows the fusion $F(R, D)$ also performs approximately to the RGB, but more to the Depth modality on \\textit{AU4} on BP4D+. \nIn addition, Table.\\ref{tab:table4} also shows the higher performance of $F(D, R)$ than $F(R,D)$ on \\textit{AU4} of both BP4D and BP4D+. \nThis also proves the depth data provided a more effective representation than the RGB on $AU4$.\n\nFig.\\ref{fig:bp4d} shows \\textit{AU6}, \\textit{AU7}, \\textit{AU10}, and \\textit{AU14} have better performances in Depth than in RGB, which shows the high effectiveness of the AU Depth features. \nThe $F(D, R)$ outperforms $F(R, D)$ in \\textit{AU6}, and \\textit{AU10}, but not in \\textit{AU7} and \\textit{AU14}, which need more exploration and discussion in our future work.\n\n\\iffalse\nfrom the definition: Brow Lowerer, \nContributes to sadness, fear, and anger, and to confusion. \nMuscles: depressor glabellae, depressor supercilii, and corrugator supercilii.\n\n\u2022 AU4: Vertical wrinkles appear in the glabella and the\n    eyebrows are pulled together. The inner parts of the\n    eye-brows are pulled down a trace on the right and\n    slightly on the left with traces of wrinkling at the corners;\n\n\n\\fi  \n\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{figures\/bp4d.png}\n    \\caption{Performance comparison between different modalities on BP4D. R+D represents $F(R, D)$.}\n    \\label{fig:bp4d}\n\\end{figure}\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{figures\/bp4d+.png}\n    \\caption{Performance comparison between different modalities on BP4D+. R+D represents $F(R, D)$.}\n    \\label{fig:bp4d+}\n\\end{figure}\n\n\\section{CONCLUSIONS AND FUTURE WORKS}\n\n\\subsection{Conclusions}\n\nIn this paper, we have presented a novel transformer-based multi-modal network for action unit detection.\nWithout requiring prior knowledge like facial landmarks, AUs relation graphs, and AU semantic description, our framework utilize the transformer encoder and transformer fusion in multi-modal based AU detection. \nWe evaluate our proposed method on 2 public multi-modalities databases and achieves superior performance over the peer state-of-the-art methods, not only on fusion modality but also on a single modality.\nBased on the performance of our work in different modalities, we further analyze the contribution of two modalities on different AUs.\n\n\n\\subsection{Future Works}\n\nOur future work will include more modalities (e.g., thermal images, physiology signals) into the fusion strategy.\nBesides 3D Depth images, we also plan to develop more advanced approaches for 3D feature representations.\nAs we discussed previously that \\textit{AU4} obtains more information from the Depth images, so we can re-design the fusion module, where different AU features are following different fusion orders.\nOur work is also easy to be extended with the ROI-based attention module, which is widely used recently.\nThe proposed framework will also be applied to other practical affection-related applications.\n \n\n\n\\section{ACKNOWLEDGMENTS}\n\nThe material is based on the work supported in part by the NSF under\ngrant CNS-1629898 and the Center of Imaging, Acoustics, and Perception\nScience (CIAPS) of the Research Foundation of Binghamton University.\n\n\n\n\n\n\n\n{\\small\n\\bibliographystyle{ieee}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Computational Details}\n\\label{sec:appa}\n\\renewcommand{\\theequation}{A\\arabic{equation}}\n\\setcounter{equation}{0}\n\nThe $h^{ab}$ and $k^{ab}$ occurring in \\ref{eq:gtruu} are given by\n\\begin{gather}\n  h^{00} = -h_{00}, \\quad h^{01} = h_{01}, \\quad h^{02} = h_{02}, \n  \\quad h^{03} = \\csc^{2}\\theta h_{03}, \\quad h^{11} = -h_{11},\n  \\quad h^{12} = -h_{12}, \\notag \\\\\n   h^{13} = -\\csc^{2}\\theta h_{13}, \n  \\quad h^{22} = -h_{22}, \\quad h^{23} = -\\csc^{2}\\theta h_{23}, \n  \\quad h^{33} = -\\csc^{4}\\theta h_{33}, \\label{eq:huu}\n\\end{gather}\nand\n\\begin{equation}\n  \\label{eq:kuu}\n  \\begin{split}\n    k^{00} &= -k_{00} + h_{00}^2 -  h_{01}^2 - h_{02}^2 - \n    \\csc^{2}\\theta h_{03}^2,\\\\\n    k^{01} &= k_{01} - h_{00}h_{01} +  h_{01}h_{11} +\n    h_{02}h_{12} + \\csc^{2}\\theta h_{03}h_{23}, \\\\\n    k^{02} &= k_{02} -h_{00}h_{02}+h_{01}h_{12} + h_{02}h_{22} +\n    \\csc^{2}\\theta h_{03}h_{23}, \\\\\n    k^{03} &= \\csc^{2}\\theta(k_{03} - h_{00}h_{03} + h_{01}h_{13} +\n    h_{02}h_{23} + \\csc^{2}\\theta h_{03}h_{33}), \\\\\n    k^{11} &= -k_{11} + h_{01}^2 - h_{11}^2 - h_{12}^2 -\n    \\csc^{2}\\theta h_{13}^2, \\\\\n    k^{12} &= -k_{12} + h_{01}h_{02} - h_{11}h_{12} - h_{12}h_{22} -\n    \\csc^{2}\\theta h_{13}h_{23}, \\\\\n    k^{13} &= \\csc^{2}\\theta(-k_{13} + h_{01}h_{03} - h_{11}h_{13} - \n    h_{12}h_{23} - \\csc^{2}\\theta h_{13}h_{33}),\\\\\n    k^{22} &= -k_{22} +h_{02}^2 - h_{12}^2 - h_{22}^2 -\n   \\csc^{2}\\theta h_{23}^2, \\\\\n    k^{23} &= \\csc^{2}\\theta(-k_{23} + h_{02}h_{03} - h_{12}h_{13} -\n    h_{22}h_{23} - \\csc^{2}\\theta h_{23}h_{33}), \\\\\n    k^{33} &= \\csc^{4}\\theta(-k_{33} + h_{03}^2 - h_{13}^2 -\n    h_{23}^2 - \\csc^{2}\\theta h_{33}^2).\n\\end{split}\n\\end{equation}\n\nThe $a_{ab}$ occurring in \\ref{eq:gurdd} (after imposing the\nconditions \\ref{eq:yu}) are given by\n\\begin{equation}\n  \\label{eq:add}\n  \\begin{split}\n    a_{00} &= h_{00} - 2q_{1},\\\\\n    a_{01} &= h_{00} + h_{01} + 2m_{1} -q_{1},\\\\\n    a_{02} &= h_{02} - {y_{3}}_{,u}, \\\\\n    a_{03} &= h_{03} - {z_{3}}_{,u}\\sin^{2}\\theta ,\\\\\n    a_{11} &= h_{00} + 2h_{01} + h_{11} + 4m_{1}, \\\\\n    a_{12} &= h_{02} + h_{12} + y_{2},\\\\\n    a_{13} &= h_{03} + h_{13} + z_{2}\\sin^{2}\\theta ,\\\\ \n    a_{22} &= h_{22} - 2{y_{2}}_{,\\theta},\\\\\n    a_{23} &= h_{23} - {y_{2}}_{,\\phi} - {z_{2}}_{,\\theta}\\sin^{2}\\theta ,\\\\\n    a_{33} &= h_{33} - 2{z_{2}}_{,\\phi}.\n  \\end{split}\n\\end{equation}\n\nThe relation between the tetrad components $c^{mn}$ and the $c_{mn}$\nis (here $s=2^{-1\/2}$)\n\\begin{equation}\n  \\label{eq:cud}\n  \\begin{split}\n    c^{00} &= \\tfrac{1}{2} c_{01} - c_{00},\\\\\n    c^{01} &= \\tfrac{1}{2} c_{00} - \\tfrac{1}{4} c_{01} - c_{10} + \\tfrac{1}{2} c_{11},\\\\\n    c^{02} &= s(c_{20} + c_{30}) - \\tfrac{1}{2} s(c_{21} + c_{31}),\\\\\n    c^{03} &= is[(c_{20} - c_{30}) - \\tfrac{1}{2} s(c_{21} - c_{31})\\csc\\theta],\\\\\n    c^{10} &= -c_{01},\\\\\n    c^{11} &= \\tfrac{1}{2} c_{01} - c_{11},\\\\\n    c^{12} &= s(c_{21} + c_{31}),\\\\\n    c^{13} &= is(c_{21} - c_{31})\\csc\\theta,\\\\\n    c^{20} &= s(c_{02} + ic_{03}\\csc\\theta),\\\\\n    c^{21} &= s(c_{12} - \\tfrac{1}{2} c_{02}) - is(\\tfrac{1}{2} c_{03} - c_{13})\\csc\\theta,\\\\\n    c^{22} &= -\\tfrac{1}{2}(c_{22}+c_{32}) - \\tfrac{1}{2} i(c_{23}+c_{33})\\csc\\theta,\\\\\n    c^{23} &= -\\tfrac{1}{2} i(c_{22}-c_{32})\\csc\\theta +\n    \\tfrac{1}{2}(c_{23}-c_{33})\\csc^{2}\\theta, \\\\\n    c^{30} &= s(c_{02} - ic_{03}\\csc\\theta),\\\\\n    c^{31} &= s(c_{12} - \\tfrac{1}{2} c_{02}) +\n    is(\\tfrac{1}{2} c_{03} - c_{13})\\csc\\theta,\\\\\n    c^{32} &= -\\tfrac{1}{2}(c_{22}+c_{32}) + \\tfrac{1}{2} i(c_{23}+c_{33})\\csc\\theta,\\\\\n    c^{33} &= -\\tfrac{1}{2} i(c_{22}-c_{32})\\csc\\theta -\n    \\tfrac{1}{2}(c_{23}-c_{33})\\csc^{2}\\theta. \\\\\n  \\end{split}\n\\end{equation}\n\n\nOne possible relation between the tetrad coefficients $c_{mn}$ and the\nmetric coefficients $a_{mn}$ and Lorentz parameters $\\alpha_{m}$ is\n\\begin{equation}\n  \\label{eq:add1}\n  \\begin{split}\n    c_{00} &= \\alpha_{1},\\\\\n    c_{01} &= \\tfrac{1}{2} a_{11}, \\\\\n    c_{02} &= a_{12} - 2s\\alpha_{4},\\\\\n    c_{03} &= a_{13} + 2s\\alpha_{5}\\sin\\theta ,\\\\\n    c_{10} &= \\tfrac{1}{2} a_{00} - \\tfrac{1}{2} \\alpha_{1},\\\\\n    c_{11} &= a_{01} - \\tfrac{1}{4} a_{11} - \\alpha_{1},\\\\\n    c_{12} &= a_{02} - \\tfrac{1}{2} a_{12} - s(2\\alpha_{2} - \\alpha_{4}),\\\\\n    c_{13} &= a_{03} - \\tfrac{1}{2} a_{13} + s(2\\alpha_{3}-\\alpha_{5})\\sin\\theta,\\\\\n    c_{20} &= \\alpha_{2} +i\\alpha_{3},\\\\\n    c_{21} &= \\alpha_{4} +i\\alpha_{5},\\\\\n    c_{22} &= \\tfrac{1}{2} s a_{22} - is(a_{23}-\\alpha_{6})\\csc\\theta,\\\\\n    c_{23} &= s\\alpha_{6} - \\tfrac{1}{2} isa_{33}\\csc\\theta,\\\\\n    c_{30} &= \\alpha_{2} - i\\alpha_{3},\\\\\n    c_{31} &= \\alpha_{4} - i\\alpha_{5},\\\\\n    c_{32} &= \\tfrac{1}{2} s a_{22} + is(a_{23}-\\alpha_{6})\\csc\\theta,\\\\\n    c_{33} &= s\\alpha_{6} + \\tfrac{1}{2} isa_{33}\\csc\\theta,\\\\\n  \\end{split}\n\\end{equation}\nwhere $s=2^{-1\/2}$.\nOther representations are possible.\n\n\\section{Tetrad formalism}\n\\label{sec:appb}\n\\renewcommand{\\theequation}{B\\arabic{equation}}\n\\setcounter{equation}{0}\n\nHere we define the notation and summarize the results.\nThe reader to whom this material is unfamiliar should consult\nintroductory material e.g., \\citet{Chandra83} chapter 1, section 7.\n\nIn this section $a, b, c,\\ldots$ are coordinate indices while \n$\\alpha, \\beta, \\gamma, \\ldots$ are tetrad indices.\n\nAt each space-time point $P$ we introduce a basis of vectors\n\\begin{equation}\n  \\label{eq:bas1}\n  e_{\\alpha}{}^{a}, \\qquad \\alpha\\in[0,3], \\quad a\\in [0,3].\n\\end{equation}\nThen the matrix\n\\begin{equation*}\n  e_{\\alpha}{}^{a} = \\left (\n    \\begin{array}{ccc}\n      e_{0}{}^{0} & e_{0}{}^{1} & \\ldots\\\\\n      e_{1}{}^{0} & e_{1}{}^{1} & \\ldots\\\\\n      \\vdots & \\vdots & \\ddots \n    \\end{array} \\right)\n\\end{equation*}\nis non-singular and we denote its inverse by $e^{\\alpha}{}_{a}$.\nThus\n\\begin{equation}\n  \\label{eq:inv1}\n  e_{\\alpha}{}^{a}e^{\\beta}{}_{a} = \\delta_{\\alpha}{}^{\\beta}, \\qquad\n  e_{\\alpha}{}^{a}e^{\\alpha}{}_{b} = \\delta^{a}{}_{b}.\n\\end{equation}\nThe $e^{\\alpha}{}_{a}$ represent the dual basis of covectors.\nAs usual chart indices are lowered (raised) using $g_{ab}$($g^{ab}$).\n\nAn additional assumption made here is that\n\\begin{equation}\n  \\label{eq:eps1}\n  \\epsilon_{\\alpha\\beta} = g_{ab}\\,e_{\\alpha}{}^{a}e_{\\beta}{}^{b}\n\\end{equation}\nis a \\textbf{constant} symmetric matrix with inverse\n$\\epsilon^{\\alpha\\beta}$.\nThus\n\\begin{equation}\n  \\label{eq:eps2}\n  \\epsilon_{\\alpha\\beta}\\,\\epsilon^{\\beta\\gamma} = \\delta_{\\alpha}{}^{\\gamma}.\n\\end{equation}\nThe choice $\\epsilon_{\\alpha\\beta}=\\text{diag}(1,-1,-1,-1)$ gives an\n\\emph{orthonormal tetrad}, but here we choose \n\\begin{equation}\n  \\label{eq:npt1}\n  \\epsilon_{\\alpha\\beta} = \\epsilon^{\\alpha\\beta} = \\left(\n    \\begin{array}{cccc}\n      0 & 1 & 0 & 0 \\\\\n      1 & 0 & 0 & 0 \\\\\n      0 & 0 & 0 & -1 \\\\\n      0 & 0 & -1 & 0 \\\\\n    \\end{array} \\right),\n\\end{equation}\nwhich gives a NP tetrad, \\cite{NP62}.\n\nThen it is easy to see that \n\\begin{equation}\n  \\label{eq:inv2}\n  \\epsilon_{\\alpha\\beta}\\,e^{\\beta}{}_{a} = e_{\\alpha a}, \\qquad\n  \\epsilon^{\\alpha\\beta}\\,e_{\\beta}{}^{a} = e^{\\alpha a},\n\\end{equation}\nso that tetrad indices are lowered (raised) using \n$\\epsilon_{\\alpha\\beta}$ ($\\epsilon^{\\alpha\\beta}$).\n\nThe \\emph{Ricci rotation coefficients} $\\gamma_{\\lambda\\mu\\nu}$ are\ndefined via\n\\begin{equation}\n  \\label{eq:ricci1}\n  e_{\\mu b;c} = \\gamma_{\\lambda\\mu\\nu} e^{\\lambda}{}_{b}e^{\\nu}{}_{c},\n\\end{equation}\nwhere the metric covariant derivative has been used,\nand since $\\epsilon_{\\alpha\\beta}$ is constant we must have\n$\\gamma_{\\lambda\\mu\\nu} = \\gamma_{[\\lambda\\mu]\\nu}$.\n\nThe tetrad \\emph{structure constants} $C^{\\gamma}{}_{\\alpha\\beta}$ are\ndefined via\n\\begin{equation}\n  \\label{eq:struct1}\n  [e_{\\alpha}, e_{\\beta}] = C^{\\gamma}{}_{\\alpha\\beta}e_{\\gamma},\n\\end{equation}\nand clearly $C^{\\gamma}{}_{\\alpha\\beta} = C^{\\gamma}{}_{[\\alpha\\beta]}$.\nIf we let \\ref{eq:struct1} act on a scalar function $f$, note that the\nmetric connection is symmetric, and use \\ref{eq:ricci1}, then it is\neasy to see that\n\\begin{equation}\n  \\label{eq:struct2}\n  C^{\\gamma}{}_{\\alpha\\beta} = \\gamma^{\\gamma}{}_{\\beta\\alpha} - \n  \\gamma^{\\gamma}{}_{\\alpha\\beta},\n\\end{equation}\nwhich implies\n\\begin{equation}\n  \\label{eq:struct3}\n  \\gamma_{\\lambda\\mu\\nu} = \\tfrac{1}{2}(C_{\\nu\\lambda\\mu} - C_{\\lambda\\mu\\nu}\n  -   C_{\\mu\\nu\\lambda}).\n\\end{equation}\n\nIt is important to realise that the $C^{\\lambda}{}_{\\mu\\nu}$ do not\ninvolve the connection.\nFor \\ref{eq:struct2}, \\ref{eq:ricci1} and the fact that the connection\nis symmetric means that\n\\begin{equation}\n  \\label{eq:struct4}\n  C^{\\lambda}{}_{\\mu\\nu} = e^{\\lambda}{}_{a,b}\n  (e_{\\mu}{}^{a}e_{\\nu}{}^{b} - e_{\\mu}{}^{b}e_{\\nu}{}^{a}).\n\\end{equation}\n\nNext the Ricci identity applied to $e^{\\alpha}{}_{b}$ gives\n\\begin{equation}\n  \\label{eq:riemt}\n  R_{\\alpha\\beta\\gamma\\delta} = \\gamma_{\\alpha\\beta\\gamma,a}e_{\\delta}{}^{a}  -\n  \\gamma_{\\alpha\\beta\\delta,a}e_{\\gamma}{}^{a} +\n  \\gamma_{\\alpha\\beta\\epsilon}C^{\\epsilon}{}_{\\gamma\\delta} +\n  \\gamma^{\\epsilon}{}_{\\alpha\\gamma}\\gamma_{\\epsilon\\beta\\delta} - \n  \\gamma^{\\epsilon}{}_{\\alpha\\delta}\\gamma_{\\epsilon\\beta\\eta}, \n\\end{equation}\nand finally\n\\begin{equation}\n  \\label{eq:riccit}\n  R_{\\alpha\\gamma} = \\epsilon^{\\beta\\delta}R_{\\alpha\\beta\\gamma\\delta}.\n\\end{equation}\nNote that throughout the paper we use these forms for the curvature\ntensors.\nThus $R_{12}$ means $R_{\\alpha\\beta}$ with $\\alpha=1$ and $\\beta=2$,\nwhich is not the same as $R_{ab}$ with $a=1$ and $b=2$.\n\nOur algorithm starts from the sets $\\{e_{\\alpha}{}^{a}\\}$ and \n$\\{e^{\\alpha}{}_{a}\\}$.\nWe compute $C^{\\alpha}{}_{\\beta\\gamma}$ from \\ref{eq:struct4} and of\ncourse $C_{\\gamma\\alpha\\beta} = \\epsilon_{\\gamma\\delta}C^{\\delta}_{\\alpha\\beta}$.\nNext we compute $\\gamma_{\\alpha\\beta\\gamma}$ from \\ref{eq:struct3}\nand finally the curvature tensors from \\ref{eq:riemt} and \\ref{eq:riccit}.\nAlthough this looks ponderous it can easily be automated using a\ncomputer algebra system.\n\n\n\n\n\n\n\n\\section{Introduction and motivation}\n\\label{sec:intro}\n\nThe two threads which underpin this study of asymptotic flatness are\ntheoretical and numerical relativity.\nWe start by reviewing the former.\nIt is widely believed that the region of space-time far from an\nisolated gravitating body is, in some sense, asymptotically  Minkowskian.\nAlready in 1962 Bondi and his coworkers \\citep{BVM62}, \\citep{Sachs62}\ndeveloped asymptotic expansions for the solution of the full nonlinear\nvacuum field equations, leading to a rigorous concept of gravitational\nradiation in the far field.\nSoon afterwards \\citet{NU62} produced an alternative version using the\nNP null tetrad formalism \\citep{NP62}. \nA key ingredient in this and later work was the careful choice of a\nsuitable coordinate chart involving a ``retarded time'' coordinate $u$.\nBoth groups introduced a $(u,r,\\theta,\\phi)$ chart where \n$\\theta$ and $\\phi$ were  spherical polar coordinates.\nThey both developed asymptotic expansions as $r\\rightarrow\\infty$\nholding the other coordinates fixed.\nSubsequent work by Penrose \\citep{RP63} showed that by a process of\n``conformal compactification'', infinity could be adjoined to the\nspace-time manifold and then treated by standard methods.\nMost modern theoretical treatments use the Penrose conformal approach,\nand the ``old-fashioned'' chart-based approach has fallen out of\nfashion.\nHowever it is closer to what most numerical relativists are\ncalculating, and for this reason we shall use it here. \n\nThe two groups used different charts and dependent variables.\n\\citet{BVM62}  chose a $(u,r,\\theta,\\phi)$ chart where $r$ was an area\ncoordinate and $\\theta$ and $\\phi$ were standard spherical polar\ncoordinates (see below). \nTheir primary dependent variables were the metric components.\n\\citet{NU62} produced an alternative version using a null ``retarded\ntime''  coordinate $u$.\nThen the  null vector $l^{a} = g^{ab}u_{,b}$ is geodesic and their\ncoordinate $r$ was chosen to be an affine parameter for the integral\ncurves of $l^{a}$ along which the other three coordinates were fixed.\nTheir primary dependent variables were the tetrad connection\ncomponents and the tetrad components of the Weyl curvature tensor.  \nThese ten independent Weyl tensor components are usually described by\nfive complex scalar functions $\\Psi_{n}$ where $n=0,1,\\ldots ,4$.\n(For a  covariant physical interpretation of the Weyl tensor see e.g.,\n\\citet{Szek65}.) \n\nBoth groups  were considering the limit $r\\rightarrow\\infty$ with the\nother coordinates fixed.  \nIn the Penrose geometrical picture \\citep{RP63} this region is called\n\\emph{future null infinity}.\nOf course both groups could have considered an ``advanced time''\ncoordinate $v$ where the corresponding limit is \\emph{past null\n  infinity}.\nThe $\\Psi_{n}$ in that case have similar properties and interpretation\nto $\\Psi_{4-n}$ near future null infinity.\n\nIf one wants to consider an isolated system with no extrinsic incoming\nradiation, then, as explained below, the  natural place to impose this\nis past null infinity. \nHowever both groups looked for a condition to be imposed near future\nnull infinity. \n\\citet{BVM62} introduced an ``outgoing radiation condition'' which\nrequired the vanishing of certain terms in the asymptotic expansion of\ntwo of the metric components. \nThis condition will be stated more precisely in section \\ref{sec:rett}.\n\\citet{NU62} made a ``peeling assumption'': near future null infinity \n$\\Psi_{0} = O(r^{-5})$, and with this assumption they were able to\ndemonstrate a so-called ``peeling theorem'':  $\\Psi_{n}= O(r^{n-5})$.\nThen $\\Psi_{4} = O(r^{-1})$ is interpreted as the leading term in the\noutgoing radiation.\nIn the  \\citep{BVM62} picture the equivalent r\\^ ole is\ntaken by the ``Bondi news function'' built from first derivatives of\nmetric components.\nFor a comparison of the conditions in the two schemes, showing that\nthe outgoing radiation condition implies the peeling assumption\nsee e.g., \\citet{Kroon98}, and section \\ref{sec:curvature}.\n\nBy reversing the direction of time, swapping advanced time for\nretarded time, one could carry out an almost identical study near past\nnull infinity.\nThere, assuming the analogous peeling condition, $\\Psi_{0} =\nO(r^{-1})$ is to be interpreted as the leading term in the extrinsic\nincoming radiation, and a natural ``no incoming radiation condition''\nnear past null infinity would be $\\Psi_{4}=O(r^{-5})$.  \n\nIt is important to realise that the ``outgoing radiation condition''\nor ``peeling assumption'' does not preclude the presence of incoming\nradiation near future null infinity.\nEven within linearized theory the ``peeling theorem'' allows modest\namounts of incoming radiation (Deadman \\& Stewart, paper in preparation).\n\nWe turn now to numerical relativity where researchers  have expended\nconsiderable effort on the (numerical) evolution of asymptotically\nflat space-times. \nA minority of researchers have adopted the Penrose conformal approach,\nbut most have chosen to evolve the space-time as far out (both in\nspace and time) as is feasible, using the traditional approach.\nThen some matching process is required to interpret their numerical\ndata in the Bondi or NP pictures.\nThis, the goal of this paper,  turns out to be far from trivial. \nThe choice of a coordinate chart is an intrinsic part of the numerical\nevolution and the final data is available only in this chosen chart.\nEach numerical relativity group has its own favoured chart or charts\nand they usually  bear little resemblance to the Bondi or NP ones.\nFurthermore this data does not contain complete information because \nthe inevitable occurrence of numerical errors will corrupt the values\nof higher derivatives---from it one can construct reliably only a few\nleading terms in asymptotic expansions\\footnote{Consider an asymptotic\n  expansion $$f(r)=f_{0} + f_{1}r^{-1} + f_{2}r^{-2}+\\ldots$$ as\n  $r\\rightarrow\\infty$.  If we interpret this as the first few terms in\n  the Taylor series for $f(q)$ about $q=0$ where $q=r^{-1}$, then the\n  $f_{n}$ are, up to numerical factors, the $q$-derivatives of $f$\n  evaluated at $q=0$.}.\n\nThe usual approach adopted by numerical relativists is to argue that\nfar from the isolated source the gravitational field is weak, and so\nlinearized theory can be used to match the numerical and the Bondi\nor NU pictures.\n\\citet{BVM62} argued strongly against such an approximation pointing\nout the fundamental nonlinearity of general relativity.\nEven if plausible arguments in its favour could be found, linearization\ncarries its own difficulties. \nThe first is that in a non-compactified space-time the matching\nprocess is a global one.   \nFurther given a space-time, the choice of a simpler second space-time\nof which the first can be considered a linearized perturbation, is not\nunambiguous. \nEven if such a choice could be justified the transformation between the\ncharts in the two space-times would not, in general, be smooth.\n\nAs a concrete example illustrating these points consider the\nwell-known Schwarzschild metric in the standard $(t,r,\\theta,\\phi)$\nchart \n\\begin{equation}\n  \\label{eq:schwarz1}\n  g^S_{ab} = \\text{diag}(F, -F^{-1}, -r^2, -r^2\\sin^2\\theta),\n\\end{equation}\nwhere $F = 1 -2M\/r$.\nIn the region where $r\\gg M$ this might appear to be a small\nperturbation of Minkowski space-time with metric\n\\begin{equation}\n  \\label{eq:mink1}\n  g^M_{ab} = \\text{diag}(1, -1, -r^2, -r^2\\sin^2\\theta),\n\\end{equation}\nbut this is deceptive.\nConsider the scalar wave equation $g^{ab}\\Psi_{;ab}=0$ on the two\nspace-times.\nWe would measure outgoing radiation at \\emph{future null infinity} by\ntaking the limit $r\\rightarrow\\infty$ holding $u$ constant, where $u$\nis a retarded time coordinate.\nTwo standard choices for $u$ are\n\\begin{equation}\n  \\label{eq:ret1}\n  u^M = t - r, \\qquad u^S = t - r^*,\n\\end{equation}\nwhere \n\\begin{equation}\n  \\label{eq:rstar1}\n  r^* = \\int F^{-1}\\,{\\rm d}r = r + 2M\\log|r\/2M - 1| + const.\n\\end{equation}\nThus\n\\begin{equation}\n  \\label{eq:ret2}\n  u^M = u^S + 2M\\log|r\/2M - 1| + const.\n\\end{equation}\nThe Schwarzschild null infinity is given by $r\\rightarrow\\infty$\nholding $u^S$ constant, which implies $u^M\\rightarrow \\infty$, known\nas \\emph{future timelike infinity} for the Minkowski space-time.\nEquivalently the Minkowski null infinity involves taking the limit\nwith $u^M$ constant which corresponds to $r\\rightarrow\\infty$ with \n$u^S\\rightarrow -\\infty$, known as \\emph{spacelike infinity}\nfor the Schwarzschild spacetime.\nThus the limits in the two charts are different.\nThis happens because of the global nature of the limiting process.\n\nIn order to achieve comparable  limiting processes we need to redefine\nthe two charts.\nHere both space-times are static and so it is simplest to retain the\n$t$-coordinate.\nSuppose we invert (for $r>M$) the relation \\ref{eq:rstar1},\n$r^*=r^*(r)$ giving $r=r(r^*)$ and introduce a new chart\n$(t,r^*,\\theta,\\phi)$. \nThen the Schwarzschild line element \\ref{eq:schwarz1} becomes\n\\begin{equation}\n  \\label{eq:schwarz2}\n  g^S_{ab} = \\text{diag}(F, -F, -r^2, -r^2\\sin^2\\theta).\n\\end{equation}\nUsing the same chart the Minkowski line element is \n\\begin{equation}\n  \\label{eq:mink2}\n  g^M_{ab} = \\text{diag}(1, -1, -r^2, -r^2\\sin^2\\theta).\n\\end{equation}\nNow the two metrics \\ref{eq:schwarz2} and \\ref{eq:mink2} are not\nonly small perturbations of each other (for large $r^*$), but they\nshare the same causal structure, $u=t-r^*$ in both cases.\n(There are of course many other ways of doing this, e.g., retain the\n$r$'s and change the $t$'s, which is the approach to be adopted in\nthis paper.)\nNote also the appearance of logarithms, which  means that the\ntransformations are not smooth.\n\nThe purpose of this paper is to examine in more detail these issues\nfrom the point of view of the numerical relativist.  \nIn section \\ref{sec:numerical} we state what information we believe is\navailable in a typical numerical evolution, and we assume that this\ninformation is expressed in terms of a given chart  \n$X^{a}=(T,R,\\Theta,\\Phi)$ which is asymptotically Minkowskian.\nSection \\ref{sec:rett} addresses the construction of an approximate\nBondi-like chart $x^{a}=(u,r,\\theta,\\phi)$ using this information.\nThis circumvents the problem referred to above.\nWe write down here the explicit form of the Bondi et al.~outgoing\nradiation condition.\nWe introduce a Newman-Penrose (NP) tetrad \\citep{NP62} adapted to the\nproblem by \\citet{NU62} in section \\ref{sec:tetrad}.\nAt leading order this is the usual NP tetrad for Minkowski space-time.\nAt each order, $r^{-1}$, $r^{-2}, \\ldots ,$ there are 16 real\ncoefficients describing the tetrad.\nHowever from section \\ref{sec:rett} we know that only 10 coefficients\nare needed to describe the metric.\nThere are six coefficients which describe an infinitesimal\nLorentz transformation at each order, and, for the moment, we do not\nmake a particular choice for them.\n\nIn section \\ref{sec:curvature} we first obtain the asymptotic solution of\nthe full nonlinear vacuum Einstein equations.\nAs we do so we fine-tune our chart and NP tetrad to make them closer\nto those of Bondi et al.~and Newman-Unti.\nOnce we have set the Ricci curvature, to the best of our abilities, to\nzero we turn to the Weyl curvature described by the Weyl scalars\n$\\Psi_{n}$ referred to earlier. \nWe find that $\\Psi_{n}=O(r^{n-5})$ for $n=4,3,2,1$, but\n$\\Psi_{0}=O(r^{-4})$, which would appear to violate the NU peeling\nassumption. \nHowever using the information gleaned from solving the vacuum field\nequations and the fine-tuning of the chart and tetrad, we can show that\nthe  Bondi outgoing radiation condition implies the NU peeling\ncondition so that the peeling theorem then holds.\n\nThe bad news is that the leading terms in the Weyl scalars\n$\\Psi_{0}=O(r^{-5})$ and $\\Psi_{1}=O(r^{-4})$ cannot be estimated\nusing the information we judge to be available from the information\nextracted in section \\ref{sec:numerical}.\nAlthough these scalars can be computed in linearized theory, that\ntheory would appear to be an unreliable guide here near future null\ninfinity.  \n\nThe good news is that we can compute the leading terms in\n$\\Psi_{4}=O(r^{-1})$, equivalent to the ``Bondi news function'' (and\nwe can compute this scalar accurately within linearized theory).\nThe same holds for $\\Psi_{3}=O(r^{2})$, which involves nonlinear\nterms, but these can be removed by the fine-tuning process.\nWe can also compute $\\Psi_{2}= O(r^{-3})$ which involves nonlinear\nterms in an essential way.\nThis means that we can offer reliable estimates of the ``Bondi mass''\n$M_{B}(u)$ of the isolated system\\footnote{The ``Bondi mass'' is of\n  course the timelike component of a 4-vector and so frame dependent.\n  But a numerical relativity evolution singles out a well-defined frame,\n  and that is the one in which the mass is computed.}, and its rate of\ndecrease $dM_{B}\/du\\leqslant0$, presumably due to the radiation of\nenergy, both manifestly inaccessible to linearized theory.\n\nThe final section \\ref{sec:numrec} translates these results back into\nthe $X^{a}$ chart of section \\ref{sec:numerical} used by a typical\nnumerical relativist. \nFrom her\/his standpoint there is no need to go through the elaborate\nconstruction of a theoretical chart and NP tetrad carried out in the\nintermediate sections.\nWe offer ``numerical relativity recipes'' so that they can compute the\nkey quantities referred to in the previous paragraph in their own\npreferred chart.\n\nThe key ideas in this paper are at least forty years old, and one\nmight ask why were these results were not given before?\nThe nonlinear calculations of \\citet{BVM62}, \\citet{Sachs62} and\n\\citet{NU62} were made possible by careful, clever, a-priori choices of\nchart and tetrad.\nWe have to start from more or less arbitrary choices and so the\nresulting expressions are  horrendously complicated. \nIn order to handle them accurately we have utilized a\ncomputer algebra system.\nWe used \\emph{Reduce}, and our \\emph{Reduce 3.8} scripts can be\nobtained by an email request to the authors.\nOur choice reflected our experience and knowledge of one particular\ncomputer algebra system, but we used no features not available in\nsome other systems.\n\n\nWe are grateful to Oliver Rinne for very useful discussions.\nOne of us (ED) is grateful for financial support from the UK\nEngineering and Physical Sciences Research Council.\n\n \n\n\\section{The numerical data}\n\\label{sec:numerical}\n\n\nMost numerical relativists would choose a quasi-spherical polar chart\n$X^{a} = (T, R, \\Theta, \\Phi)$ for the numerical evolution of the\nspace-time surrounding an isolated gravitational source.\nWe could also define an associated quasi-cartesian chart $Y^{a} =\n(T,X,Y,Z)$ where\n\\begin{equation*}\n  X = R\\sin\\Theta\\cos\\Phi, \\quad\n  Y = R\\sin\\Theta\\sin\\Phi, \\quad\n  Z = R\\cos\\Theta .\n\\end{equation*}\n\nWe shall be interested in the limit $R\\rightarrow\\infty$.\nAs stated this limit is meaningless unless we specify the behaviour of\nthe other three coordinates under the limiting process, and we shall\nrectify this omission shortly.\nIt proves very convenient to introduce the notation\n\\begin{equation}\n  \\label{eq:onotn}\n  O_{n} = O(R^{-n})  \\text{  as  } R\\rightarrow\\infty.\n\\end{equation}\nOur fundamental assumption is that the space-time outside an isolated\nsource is asymptotically Minkowskian, expressed by the idea that, as\nseen in the $Y^{a}$ chart,\n\\begin{equation}\n  \\label{eq:mink1}\n  g_{ab} = \\eta_{ab} + g^{(1)}_{ab}R^{-1} + g^{(2)}_{ab}R^{-2} + O_{3},\n\\end{equation}\nwhere $\\eta_{ab}=\\text{diag}(1,-1,-1,-1)$ and the $g^{(n)}_{ab}$ are\nsupposed to remain constant during the limiting process.\nTransforming from the Minkowskian chart to the spherical polar one we\nfind that the metric components in the $X^{a}$ chart look like\n\\begin{equation}\n  \\label{eq:gtrdd}\n  \\begin{split}\n    g_{00} &= 1 + h_{00}R^{-1} + k_{00}R^{-2} + O_{3},\\\\\n    g_{01} &=  h_{01}R^{-1} + k_{01 }R^{-2} + O_{3},\\\\\n    g_{02} &= h_{02} + k_{02}R^{-1}  + O_{2},\\\\\n    g_{03} &= h_{03} + k_{03}R^{-1}  + O_{2},\\\\\n    g_{11} &= -1 +  h_{11}R^{-1} + k_{11}R^{-2}  + O_{3},\\\\\n    g_{12} &= h_{12} +  k_{12}R^{-1} + O_{2},\\\\\n    g_{13} &= h_{13} + k_{13}R^{-1}  + O_{2},\\\\\n    g_{22} &= -R^{2} +h_{22} R  + k_{22} + O_{1},\\\\\n    g_{23} &= h_{23} R  + k_{23} + O_{1},\\\\\n    g_{33} &= -R^{2}\\sin^{2}\\Theta +  h_{33}R + k_{33} + O_{1}.\\\\\n  \\end{split}\n\\end{equation}\nHere the functions $\\{h_{ab}\\}$ and $\\{k_{ab}\\}$ are required to remain\nconstant during the limiting process.\n\nWe shall also need the asymptotic form of the inverse metric $g^{ab}$\nwhich is readily obtained from the relation \n$g^{ac}g_{cb} = \\delta^{a}{}_{b}$.\nWe find\n\\begin{equation}\n  \\label{eq:gtruu}\n  \\begin{split}\n    g^{00} &= 1 + h^{00}R^{-1}  + k^{00}R^{-2}  + O_{3},\\\\\n    g^{01} &= h^{01}R^{-1}  + k^{01} R^{-2}  + O_{3},\\\\\n    g^{02} &= h^{02}R^{-2}  + k^{02}R^{-3}  + O_{4},\\\\\n    g^{03} &= h^{03}R^{-2}  + k^{03}R^{-3}  + O_{4},\\\\\n    g^{11} &= -1 + h^{11}R^{-1}  + k^{11}R^{-2}  + O_{3},\\\\\n    g^{12} &= h^{12}R^{-2}  + k^{12}R^{-3}  + O_{4},\\\\\n    g^{13} &= h^{13}R^{-2}  + k^{13}R^{-3}  + O_{4},\\\\\n    g^{22} &= -R^{-2} + h^{22}R^{-3}  + k^{22}R^{-4}  + O_{5},\\\\\n    g^{23} &= h^{23}R^{-3}  + k^{23}R^{-4}  + O_{5},\\\\\n    g^{33} &= -R^{-2}\\csc^{2}\\Theta + h^{33}R^{-3}  + k^{33}R^{-4}  + O_{5}.\\\\\n  \\end{split}\n\\end{equation}\nExplicit formulae for $h^{ab}$ and $k^{ab}$ are given by equations\n\\ref{eq:huu} and \\ref{eq:kuu} in appendix \\ref{sec:appa}.\nAt this level of approximation\n\\begin{equation*}\n  g^{ac}g_{cb} = \\delta^{a}{}_{b} + O_{3}.\n\\end{equation*}\n\nA numerical evolution in which the dependent variables include both\n$g_{ab}$ and $g_{ab,c}\\,$, usually called a ``first order formulation'',\nshould produce accurate values for  $h_{ab}$ and its first derivatives,\nand for $k_{ab}$. \nOtherwise we assume that these variables are available for discrete\nranges of $T$, $\\Theta$ and $\\Phi$ so that the corresponding\nderivatives can be estimated. \n\n\\section{The Bondi chart}\n\\label{sec:rett}\n\nMost of the theoretical work which has been done on outgoing\ngravitational radiation involves a ``Bondi chart'' \n$(u, r, \\theta,\\phi)$ in which $u$ is a retarded time\ncoordinate, see e.g., \\citet{BVM62}, \\citet{NP62}, \\citet{NU62}.\n\nHere we take the viewpoint that the $X^{a} = (T,R,\\Theta,\\Phi)$ chart\nintroduced in section \\ref{sec:numerical} is the fundamental one in\nwhich, ultimately, all numerical calculations will be performed.\nStarting from this chart we need to construct an \n$x^{a} = (u, r,\\theta,\\phi)$ one which has all the essential features \nof a Bondi chart and we start by studying the function \n$u(T,R,\\Theta,\\Phi)$.\n\nBecause $u$ is a null coordinate it has to satisfy the\n\\emph{relativistic eikonal equation}\n\\begin{equation}\n  \\label{eq:eikonal}\n  g^{ab}u_{,a}u_{,b} = 0.\n\\end{equation}\nThis is a well-known nonlinear equation with four independent\nvariables which is exceedingly difficult to solve with any generality.\n(Even the restriction of \\ref{eq:eikonal} to Minkowski space-time\nleads to the surprisingly rich structure of light ray caustics.)\nNote  that there is a ``gauge freedom''---if $u$ is a\nsolution then so is $U(u)$ for any differentiable function $U$.\n\nThe standard approach is to specify $u$ on a spacelike hypersurface in   \nspace-time, and then existence and local uniqueness of $u$ is\nguaranteed by standard theorems.\nThe standard approach is of little utility in this context, for no\nobvious choice of data suggests itself, and so we adopt a different\napproach.\n\nConsider first the special case of a Minkowski space-time, where\n\\ref{eq:eikonal} can be rewritten as \n\\begin{equation}\n  \\label{eq:eikmink}\n  (u_{,T})^{2} - (u_{,R})^{2} = \n  R^{-2}\\left[(u_{,\\Theta})^{2} + \\csc^{2}\\Theta(u_{,\\Phi})^{2}\\right] = O_{2}.\n\\end{equation}\nSuppose we look for spherically symmetric solutions $u=u(T,R)$.\nSetting $\\omega=u_{,R}\/u_{,T}$ in \\ref{eq:eikmink} we find\n$\\omega^{2}=1$.\nUsing the gauge freedom mentioned earlier we may impose $u_{,T}=1$ to\nfind\n\\begin{equation*}\n  du = dT \\pm dR,\n\\end{equation*}\nwhich implies\n\\begin{equation*}\n  u = T \\pm R + \\text{const}.\n\\end{equation*}\n$T-R$ is called \\emph{retarded time} and $T+R$ is called\n\\emph{advanced time}.\n\nAlthough  the special case appears trivial it is the key to the general\none.\nWithin this section only let indices $i,j$ range over $0,1$\nand let indices $I,J$ range over $2,3$.\nPerusal of the display \\ref{eq:gtruu} shows that $g^{ij}$ is $O_{0}$\nwhile both $g^{iJ}$ and $g^{IJ}$ are $O_{2}$.\nThus the eikonal equation takes the form\n\\begin{equation}\n  \\label{eq:eikgen1}\n  g^{ij}u_{,i}u_{,j} = O_{2},\n\\end{equation}\nwhich should be compared with \\ref{eq:eikmink} above.\nAs boundary conditions (as $R\\rightarrow\\infty$) we impose\n\\begin{equation}\n  \\label{eq:eikbc}\n  u_{,T} = 1 + O_{1}, \\qquad u_{,I} = O_{1}.\n\\end{equation}\nThis means that the eikonal equation takes the form\n\\begin{equation}\n  \\label{eq:eikgen2}\n  g^{ij}u_{,i}u_{,j} = O_{3},\n\\end{equation}\nwhich we can write as a quadratic equation for $\\omega=u_{,R}\/u_{,T}$,\nand choosing the sign appropriate for retarded time we find the\nsolution\n\\begin{equation}\n  \\label{eq:ur1}\n  u_{,R} = -(1+2m_{1}\/R + 2m_{2}\/R^{2})u_{,T} + O_{3},\n\\end{equation}\nwhere\n\\begin{equation}\n  \\label{eq:mass1}\n  m_{1} = -\\tfrac{1}{4}(h_{00}+2h_{01}+h_{11}),\n\\end{equation}\nand\n\\begin{equation}\n  \\label{eq:mass2}\n  \\begin{split}\n    m_{2} =& -\\tfrac{1}{16}\\big[4k_{00} + 8k_{01} + 4k_{11} +\n    (h_{00}-h_{11})^{2} -\\\\\n    &\\;\\;4(h_{00}+h_{01})^{2} +4(h_{02}+h_{12})^{2} +\n    4(h_{03}+h_{13})^{2}\\csc^{2}\\theta\\big].\n  \\end{split}\n\\end{equation}\nThus \n\\begin{equation}\n  \\label{eq:du}\n  du = u_{,T}\\,dT - u_{,T}(1+2m_{1}\/R+ 2m_{2}\/R^{2})\\,dR + O_{3}.\n\\end{equation}\nWe leave some freedom in $u$ by setting\n\\begin{equation}\n  \\label{eq:ut}\n  u_{,T} = 1 + \\frac{q_{1}}{R} + \\frac{q_{2}}{R^{2}} + O_{3},\n\\end{equation}\nwhere $q_{1}$ and $q_{2}$ are $R$-independent functions. At the moment\nthey are arbitrary.  The requirement that the vacuum Einstein\nequations hold then determines inter alia $q_1$, see section 6. In our\ncalculation $q_2$ is not used directly. \n\nWe need next to specify a radial coordinate $r = r(T,R,\\Theta,\\Phi)$,\nand the simplest choice is $r=R$.\nThis has the great practical advantage that \n$O_{n} = O(R^{-n}) = O(r^{-n})$.\nIt could be argued that our choice of $r$ is neither the Bondi area\ncoordinate nor an affine parameter along the outgoing null rays as\nfavoured by \\citet{NU62}.\nHowever since both of those approaches are known to be essentially\nequivalent, it would seem that the discussion is not sensitive to the\nprecise choice of $r$. \n\n\nThen \\ref{eq:du} implies\n\\begin{equation}\n  \\label{eq:dT}\n  dT = \\left((1 - q_{1}\/r - (q_{2}-q_{1}{}^{2})\/r^{2}\\right)\\,du + \n    (1+2m_{1}\/r + 2m_{2}\/r^{2})\\,dr + O_{3},\n\\end{equation}\nand so\n\\begin{alignat}{2}\n  \\label{eq:jac1}\n  \\left(\\frac{\\partial T}{\\partial u}\\right)_{r} &= \n  1 - \\frac{q_{1}}{r} - \\frac{q_{2}-q_{1}{}^{2}}{r^{2}} +  O_{3}, \n  \\quad \\left(\\frac{\\partial R}{\\partial u}\\right)_{r} &= 0,\\\\\n  \\left(\\frac{\\partial T}{\\partial r}\\right)_{u} &= \n  1 + \\frac{2m_{1}}{r} + \\frac{2m_{2}}{r^{2}} + O_{3},\n  \\quad\\;\\; \\left(\\frac{\\partial R}{\\partial r}\\right)_{u} &= 1.\n\\end{alignat}\n\nFinally we consider the choice of angular coordinates\n$\\theta = \\theta(T, R, \\Theta, \\Phi)$ and \n$\\phi = \\phi(T, R, \\Theta, \\Phi)$.\nWe shall require $\\theta = \\Theta + O_{1}$ and $\\phi = \\Phi + O_{1}$,\nand so the relations, being close to the identity, are invertible.\nIt is more convenient to posit\n\\begin{equation}\n  \\label{eq:ang}\n  \\Theta = \\theta + \\frac{y_{2}}{r} + \\frac{z_{2}}{r^{2}} + O_{3}, \\qquad\n  \\Phi = \\phi + \\frac{y_{3}}{r} + \\frac{z_{3}}{r^{2}} + O_{3}, \n\\end{equation}\nwhere the functions $y_{J}$ and $z_{J}$ do not depend on $r$ but are\notherwise arbitrary.\nEquations \\ref{eq:ang} are certainly consistent with the boundary\nconditions \\ref{eq:eikbc}.\n\n \nWe can now specify the limiting process as $r\\rightarrow\\infty$\nholding $u$, $\\theta$ and $\\phi$ constant.\nThus we are regarding $m_{n}$, $q_{n}$, $y_{J}$, $z_{J}$, $\\{h_{ab}\\}$ \nand $\\{k_{ab}\\}$ as functions of $u$, $\\theta$ and $\\phi$.\n\nBecause we know the Jacobian $(\\partial X^{a}\/\\partial x^{b})$ we can\nwrite down the metric components in the $x^{a}= (u,r,\\theta,\\phi)$\nchart\n\\begin{equation}\n  \\label{eq:gurdd}\n  \\begin{split}\n    g_{00} &= 1 + a_{00}\/r + b_{00}\/r^{2} + O_{3},\\\\\n    g_{01} &= 1 + a_{01}\/r + b_{01}\/r^{2} + O_{3},\\\\\n    g_{02} &= -r{y_{2}}_{,u} + a_{02} + b_{02}\/r + O_{2},\\\\\n    g_{03} &= -r{z_{2}}_{,u}\\sin^{2}\\theta + a_{03} + b_{03}\/r + O_{2},\\\\\n    g_{11} &= a_{11}\/r + b_{11}\/r^{2} + O_{3},\\\\\n    g_{12} &= a_{12} + b_{12}\/r + O_{2},\\\\\n    g_{13} &= a_{13} + b_{13}\/r + O_{2},\\\\\n    g_{22} &= -r^{2} + a_{22}r + b_{22} + O_{1},\\\\\n    g_{23} &= a_{23}r + b_{23} + O_{1},\\\\\n    g_{33} &= -r^{2}\\sin^{2}\\theta + a_{33}r + b_{33} + O_{1}.\\\\\n  \\end{split}\n\\end{equation}\nTwo points should be noted here.\nFirstly the leading terms in $g_{02}$ and $g_{03}$, if non-zero,\nwould violate our notion of an asymptotically Minkowskian space-time,\nfor they are not present in the standard Minkowski line element.\nThus we need to impose the conditions or ``constraints''\n\\begin{equation}\n  \\label{eq:yu}\n  {y_{2}}_{,u} = {z_{2}}_{,u} = 0.\n\\end{equation}\nExplicit formulae for $a_{mn}$ in terms of the $h_{mn}$, $q_{1}$,\n$m_{1}$ and $y_{2}$, $z_{2}$ (after imposing \\ref{eq:yu}) are given as\n\\ref{eq:add} in appendix \\ref{sec:appa}.  \nWe could also give explicit formulae for the $b_{mn}$ in terms of the\n$h_{mn}$, $k_{mn}$, $q_{n}$, $m_{n}$, $y_{n}$ and $z_{n}$but they are\nrather lengthy, and are most easily generated using a computer algebra \npackage.\n\nNext recall that the $u$-coordinate was constructed as a solution of\nthe eikonal equation \\ref{eq:eikgen2}.\nThus, as seen in the $(u,r,\\theta,\\phi)$ chart $g^{00}= O_{3}$ which\nimplies $g_{11} = O_{1}$ and so $a_{11} = 0$.\nOne may verify this directly by comparing the explicit expression for\n$a_{11}$ given in \\ref{eq:add} with \\ref{eq:mass1}.\nWe will show later that by making a suitable choice for $y_{2}$ and\n$y_{3}$, we can achieve $b_{11}=0$ so that $g_{11} = O_{3}$ as expected.\n\nWe now have sufficient notation available to write down the ``outgoing\nradiation condition'' of \\citet{BVM62} as\n\\begin{equation}\n  \\label{eq:bvmorc}\n  a_{33}= -a_{22}\\sin^{2}\\theta, \\qquad\n  b_{33} = b_{22}\\sin^{2}\\theta, \\qquad\n  b_{23} = 0,\n\\end{equation}\nwhich we shall invoke later.\n\n\n\n\\section{The NP tetrad}\n\\label{sec:tetrad}\n\nSince most recent studies of gravitational radiation use a NP null\ntetrad \\citep{NP62} we need to introduce one.\nThe basics of tetrad formalisms are due to \\citet{Schou54}.\nMany textbooks contain more readable, but often succinct accounts, and\n \\citet{Chandra83} chapter 1, section 7, is a good pedagogic\ncompromise.\nWith small, but necessary, changes in notation this is summarised in\nappendix \\ref{sec:appb}.\nThe specialisation of this approach to the original NP formalism has\nbeen given by  \\citet{CW77}.\nIt turns out that the calculations that we need to do with it become\nsurprisingly intricate, and so are most conveniently handled using a\ncomputer algebra system.\nOne of the authors implemented the appendix \\ref{sec:appb} formalism,\nthe other the \\citep{CW77} one.\nBoth of course gave the same results, a useful guard against\nprogramming errors.\nResults here are reported for the former.\n\nWe use a tetrad of vectors $e_{\\alpha}{}^{a}$ and the dual tetrad of\ncovectors $e^{\\alpha}{}_{a}$.  \n(The tetrad indices are greek characters and always occur first.)\nTetrad indices are lowered and raised using $\\epsilon_{\\alpha\\beta}$\nand $\\epsilon^{\\alpha\\beta}$ where \n\\begin{equation*}\n  \\epsilon_{\\alpha\\beta} = \\epsilon^{\\alpha\\beta} = \\left(\n    \\begin{array}{cccc}\n      0 & 1 & 0 & 0 \\\\\n      1 & 0 & 0 & 0 \\\\\n      0 & 0 & 0 & -1 \\\\\n      0 & 0 & -1 & 0 \\\\\n    \\end{array} \\right).\n\\end{equation*}\nIn NP notation we have \n\\begin{equation}\n  \\label{eq:np1}\n  \\begin{split}\n    e^{0}{}_{a} &= l_{a}, \\quad e^{1}{}_{a} = n_{a}, \\quad\n    e^{2}{}_{a} = m_{a}, \\quad e^{3}{}_{a} = \\overline{m}_{a}, \\\\\n    e_{0}{}^{a} &= n^{a}, \\quad e_{1}{}^{a} = l^{a}, \\quad\n    e_{2}{}^{a} = -\\overline{m}^{a}, \\quad e_{3}{}^{a} = -m^{a}. \\\\\n  \\end{split}\n\\end{equation}\nWe shall require that, to leading order, $l_{a}= u_{,a}$.\n\nSetting $s = 2^{-1\/2}$ we write the tetrads as\\footnote{In our\n  calculations we actually   included one extra term in each of the\n  asymptotic expansions below.\n  E.g., the first component of $e^{0}{}_{a}$ was written as \n  $$e^{0}{}_{0}=1+c_{00}\/r + d_{00}\/r^{2} + j_{00}\/r^{3} + O_{4}.$$\n  These ``junk'' terms show up in our expressions for the connection\n  and curvature components.\n  In any expression where a junk term occurs we regard all terms of\n  that (and any higher order) as being junk, and not computable form\n  the data described in section \\ref{sec:numerical}.}\n\n\\begin{equation}\n  \\label{eq:npud}\n  \\begin{split}\n    e^{0}{}_{a} &= (1 + c_{00}\/r + d_{00}\/r^{2} + O_{3}, \\quad\n    c_{01}\/r + d_{01}\/r^{2} + O_{3}, \\\\\n    &\\qquad c_{02}+ d_{02}\/r + O_{2}, \\quad c_{03}+ d_{03}\/r + O_{2}),\\\\\n    e^{1}{}_{a} &= (\\tfrac{1}{2} + c_{10}\/r + d_{10}\/r^{2} + O_{3}, \\quad\n    1 + c_{11}\/r + d_{11}\/r^{2} + O_{3}, \\\\\n    &\\qquad c_{12}+ d_{12}\/r + O_{2}, \\quad c_{13}+ d_{13}\/r + O_{2}),\\\\\n    e^{2}{}_{a} &= (c_{20}\/r + d_{20}\/r^{2} + O_{3}, \\quad\n    c_{21}\/r + d_{21}\/r^{2} + O_{3}, \\\\\n    &\\qquad -sr + c_{22}+ d_{02}\/r + O_{2}, \\quad \n    isr\\sin\\theta + c_{23}+ d_{23}\/r + O_{2}),\\\\\n    e^{3}{}_{a} &= (c_{30}\/r + d_{30}\/r^{2} + O_{3}, \\quad\n    1 + c_{31}\/r + d_{31}\/r^{2} + O_{3}, \\\\\n    &\\qquad -sr + c_{32}+ d_{32}\/r + O_{2}, \\quad \n    -isr\\sin\\theta + c_{33}+ d_{33}\/r + O_{2}),\\\\\n  \\end{split}\n\\end{equation}\nand \n\\begin{equation}\n  \\label{eq:npdu}\n  \\begin{split}\n    e_{0}^{a} &= (1 + c^{00}\/r + d^{00}\/r^{2} + O_{3}, \\quad\n    -\\tfrac{1}{2} + c^{01}\/r + d^{01}\/r^{2} + O_{3}, \\\\\n    &\\qquad c^{02}\/r^{2} + d^{02}\/r^{3} + O_{4}, \n    \\quad c^{03}\/r^{2} + d^{03}\/r^{3} + O_{4}),\\\\\n    e_{1}^{a} &= (c^{10}\/r + d^{10}\/r^{2} + O_{3}, \\quad\n    1 + c^{11}\/r + d^{11}\/r^{2} + O_{3}, \\\\\n    &\\qquad c^{12}\/r^{2} + d^{12}\/r^{3} + O_{4}, \n    \\quad c^{13}\/r^{2}+ d^{13}\/r^{3} + O_{4}),\\\\\n    e_{2}^{a} &= (c^{20}\/r + d^{20}\/r^{2} + O_{3}, \\quad\n    c^{21}\/r + d^{21}\/r^{2} + O_{3}, \\\\\n    &\\qquad -s\/r + c^{22}\/r^{2} + d^{02}\/r^{3} + O_{4}, \\quad \n    -is\\csc\\theta\/r + c^{23}\/r^{2} + d^{23}\/r^{3} + O_{4}),\\\\\n    e_{3}^{a} &= (c^{30}\/r + d^{30}\/r^{2} + O_{3}, \\quad\n    1 + c^{31}\/r + d^{31}\/r^{2} + O_{3}, \\\\\n    &\\qquad -s\/r + c^{32}\/r^{2} + d^{32}\/r^{3} + O_{4}, \\quad \n    is\\csc\\theta\/r + c^{33}\/r^{2} + d^{33}\/r^{3} + O_{4}).\\\\\n  \\end{split}\n\\end{equation}\nEach tetrad contains, at each order, 32 real coefficients.\nFor although $c_{2n}$ and $c_{3n}$ are complex,\n$\\overline{c_{2n}}=c_{3n}$ etc.\nThe relation $e^{\\mu}{}_{a}e_{\\nu}{}^{a} = \\delta^{\\mu}{}_{\\nu}$\nallows one to determine the $c^{mn}$ in terms of the $c_{mn}$ and the\n$d^{mn}$ in terms of $c_{mn}$ and $d_{mn}$, reducing the number of\nunknowns, at each order, from 32 to 16.\nThe first set of these is given as equation \\ref{eq:cud}  in appendix\n\\ref{sec:appa} .\nThe second set is rather lengthy and best generated using a computer\nalgebra package.\n\nWe also have the relation\n$\\epsilon_{\\mu\\nu}e^{\\mu}{}_{a}e^{\\nu}{}_{b}=g_{ab}$, and this enables\nus to determine $a_{mn}$ in terms of $c_{mn}$.\nNote that there are 10 $a_{mn}$ and 16 real $c_{mn}$.  \nGiven the tetrad, the metric is uniquely determined.  \nBut for a given metric  there is a 6-parameter set of tetrads which\ngive rise to it.\nThey are of course Lorentz transformations of each other and the\nLorentz group has 6 arbitrary parameters.\nWe introduce 6 arbitrary first order Lorentz parameters $\\alpha_{m}(u,\n\\theta, \\phi)$ and can determine the $c_{mn}$ in terms of the $a_{mn}$\nand $\\alpha_{m}$. \nThere are many different ways of doing this, and one is written down\nexplicitly as \\ref{eq:add1} in appendix \\ref{sec:appa}.\nWe can of course write down the $d_{mn}$ in terms of the $a_{mn}$,\n$b_{mn}$, $\\alpha_{m}$ and extra second order Lorentz parameters\n$\\beta_{m}$, but they are rather lengthy and are best generated by\ncomputer algebra. \n\n\n\\section{The Curvature Tensors}\n\\label{sec:curvature}\n\nWe now use the tetrads developed in section \\ref{sec:tetrad} to\nevaluate the Ricci and Weyl curvature tensors using the algorithm\noutlined in appendix \\ref{sec:appb}.\nAt each stage we convert all instances of $c^{mn}$ and $d^{mn}$ to\ninstances of $c_{mn}$ and $d_{mn}$ using \\ref{eq:cud} and its second\norder analogue.\nThen we convert all instances of $c_{mn}$ and $d_{mn}$ to instances of\nthe metric coefficients $a_{mn}$ and $b_{mn}$ using \\ref{eq:add1}\nand its second order analogue.\nThese conversions are implicit and will not be mentioned explicitly again.\n\nWe start by looking at the Ricci tensor component \n$R_{11} = R_{ab}e_{1}{}^{a}e_{1}{}^{b}= R_{ab}l^{a}l^{b}$.\nWe find\n\\begin{equation}\n  \\label{eq:ricc11}\n  0 = R_{11} = a_{11,u}\/r^{2} + O_{3}.\n\\end{equation}\nWe chose our chart to ensure that $u$ was approximately a null\ncoordinate or equivalently $g^{00}=O_{3}$ or equivalently $g_{11}=O_{1}$.  \nThis means that we have to enforce $a_{11}= 0$, and so the\nleading term in $R_{11}$ vanishes. \nWe look next at\n\\begin{equation}\n  \\label{eq:ricc01}\n  0 = R_{01} = R_{ab}e_{0}{}^{a}e_{1}{}^{b}= R_{ab}n^{a}l^{b} = \n  -\\tfrac{1}{2} a_{11,uu}\/r + O_{2}.\n\\end{equation}\nAgain the leading order term vanishes automatically.\nNext consider\n\\begin{equation}\n  \\label{eq:ricc123} \n  0 = R_{12}+R_{13} = -2sa_{12,u}\/r^{2} + O_{3}, \\qquad\n  0 = R_{12}-R_{13} = -2isa_{13,u}\\csc\\theta\/ r^{2} + O_{3}, \\qquad\n\\end{equation}\nwhere $s = 2^{-1\/2}$.\nWe deduce that\n\\begin{equation}\n  \\label{eq:a123}\n  a_{12,u} = a_{13,u} = 0.\n\\end{equation}\nFurther we can compute \n\\begin{equation}\n  \\label{eq:ricc023} \n  0 = R_{02}+R_{03} = sa_{12,uu}\/r + O_{2}, \\qquad\n  0 = R_{02}-R_{03} = isa_{13,u}\\csc\\theta\/r + O_{2}, \\qquad\n\\end{equation}\nIf we inspect $R_{23}$ and use \\ref{eq:a123}  we find\n\\begin{equation}\n  \\label{eq:ricc23}\n  0 = R_{23} = \\tfrac{1}{2}(a_{22,u} + a_{33,u}\\csc^{2}\\theta)\/r^{2} + O_{3},\n\\end{equation}\nand we deduce that\n\\begin{equation}\n  \\label{eq:a22}\n  a_{22,u} +  a_{33,u}\\csc^{2}\\theta = 0.\n\\end{equation}\nFurther we may compute\n\\begin{equation}\n  \\label{eq:ricc00}\n  R_{00} = -\\tfrac{1}{2}(a_{22,uu} + a_{33,uu}\\csc^{2}\\theta )\/r + O_{2},\n\\end{equation}\nand we see immediately from \\ref{eq:a22} that the leading term vanishes,\nand so furnishes no new information.\nFinally inspection of the leading $O_{2}$ terms in $R_{22}\\pm R_{33}$ reveals\nthat they vanish automatically because of $a_{11}=0$, \\ref{eq:a123}\nand \\ref{eq:a22}.\nThus we have\n\\begin{equation}\n  \\label{eq:ricc223}\n  R_{22} + R_{23} = O_{3}, \\qquad R_{22} - R_{23} = O_{3}.\n\\end{equation}\nWe have found, so far, that the conditions $a_{11}=0$, \\ref{eq:a123}\nand \\ref{eq:a22} imply  that $R_{00}$, $R_{01}$, $R_{02}$ and $R_{03}$\nare $O_{2}$ while the other components are $O_{3}$.\n\nAt this point we need to examine \\ref{eq:a123} more closely.\nUsing \\ref{eq:add} we have\n\\begin{equation}\n  \\label{eq:y2z21}\n  (h_{02} + h_{12} + y_{2})_{,u} = 0 \\qquad\n  (h_{03} + h_{13} + z_{2}\\sin^{2}\\theta)_{,u} = 0. \n\\end{equation}\nNow we know that the functions $y_{2}$ and $z_{2}$ are arbitrary,\napart form the constraints \\ref{eq:yu}, and so \\ref{eq:y2z21} implies\n\\begin{equation}\n  \\label{eq:y2z22}\n  (h_{02} + h_{12})_{,u} = 0 \\qquad\n  (h_{03} + h_{13})_{,u} = 0. \n\\end{equation}\nWe may therefore choose, consistent with the constraints \\ref{eq:yu},\n\\begin{equation}\n  \\label{eq:y2z23}\n  y_{2} = -(h_{02} + h_{12}), \\qquad\n  z_{2} = -(h_{03} + h_{13})\\csc^{2}\\theta,\n\\end{equation}\nwhich, using \\ref{eq:add}  sets \n\\begin{equation}\n  \\label{eq:a1230}\n  a_{12} = a_{13} = 0.\n\\end{equation}\n\nThe choice \\ref{eq:y2z23} has an added advantage that if we now\nexpress $b_{11}$ in terms of the $h_{ab}$, $k_{ab}$, $m_{1}$ and\n$m_{2}$, $y_{2}$ and $z_{2}$ we find that $b_{11}= 0$, \nso that $g_{11}= O_{3}$.\nAt the same time we can examine $b_{12}$ and $b_{13}$ which are linear\nin $y_{3}$ and $z_{3}$ respectively.\nBy choosing $y_{3}$ and $z_{3}$ appropriately we may arrange\n$b_{12}=b_{13}=0$.\n\nThis is a convenient point at which to examine the  choice of\na specific Lorentz transformation, in our tetrad, exemplified at\nleading order by the parameters $\\alpha_{n}$, see e.g., \\ref{eq:add1}.\n\\citet{NP62} chose $u$ to be a null coordinate, $g^{ab}u_{,a}u_{b}=0$\nand the covector $l_{a}=u_{,a}$.\nFor a symmetric connection it follows easily that\n$l^{a}{}_{;b}l^{b}=0$.\nEven if we set $l_{a}= f(x^{c})u_{,a}$ we find that $l^{a}{}_{;b}l^{b}$\nis proportional to $l^{a}$ so that we still have a null geodesic,\nalbeit not necessarily an affinely parametrised one.\nThe rest of this paragraph relies on some details of NP formalism\nwhich can be checked swiftly using appendix B of \\citet{JMS93}.\nWithin NP formalism\n\\begin{equation}\n  \\label{eq:jms1}\n  l^{a}{}_{;b}l^{b}= (\\epsilon + \\overline \\epsilon)l^{a} -\n  \\bar\\kappa m^{a} - \\kappa {\\overline m}^{a},\n \\end{equation}\nwhere $\\kappa$ and $\\epsilon$ are NP spin coefficients defined below.\nNow our coordinate $u$ is only approximately null, and our covector\n$l_{a}$ is only approximately its gradient.\nHere $\\kappa = m^{a}l^{b}l_{a;b}=\\gamma_{131}$ (using the\nnotation of  appendix \\ref{sec:appb}) turns out to be $O_{3}$.\nHowever if we choose $\\alpha_{4}=\\alpha_{5}=\\beta_{4}=\\beta_{5}=0$,\nand anticipate $a_{01}=0$ (see next paragraph), we obtain $\\kappa=O_{4}$.\nAlso $\\epsilon + \\overline\\epsilon = n^{a}l^{b}l_{a;b} = \\gamma_{011}$\nis $O_{2}$ but if we choose $\\alpha_{1}=0$ we find \n$\\epsilon + \\overline\\epsilon$ is $O_{3}$.\nWe also found $\\tau=\\gamma_{130} = (\\alpha_{2}+i\\alpha_{3})\/r^{2}+O_{3}$\nor $\\tau=O_{3}$ if we impose $\\alpha_{2}=\\alpha_{3}=0$.\nAt this stage we also choose $\\alpha_{6}=\\beta_{6}=0$ for reasons \ngiven below.\n\nNow we need to examine each of the remainder (next order)\nterms in \\ref{eq:ricc11}, \\ref{eq:ricc01}, \\ref{eq:ricc123}, \n\\ref{eq:ricc023}, \\ref{eq:ricc23}, \\ref{eq:ricc00} and \n\\ref{eq:ricc223}.\nFor example we now find that the $O_{3}$ terms in $R_{11}$ vanish if\nand only if we set $a_{01}=0$, and then $R_{11} = O_{4}$.\nThis also implies that the $O_{2}$ terms in $R_{01}$ vanish, so that \n$R_{01}=O_{3}$.\nWe already established that  $R_{12}\\pm R_{13} = O_{3}$.  \nSetting the leading order terms to zero  furnishes\nexpressions for $a_{02}$ and $a_{03}$ which we use for subsequent\nsimplifications. \nNow $R_{12}\\pm R_{13} = O_{4}$.  \nWe find then that our previous estimate \\ref{eq:ricc023} \nrefines to $R_{02}\\pm R_{03} = O_{3}$.\nWe also need to refine our estimate \\ref{eq:ricc23} to $R_{23}=O_{4}$. \nWe find $R_{22}\\pm R_{23} = O_{3}$, where both $O_{3}$ terms deliver\nthe same relation relating the $u$-derivatives of \n$a_{22}$, $a_{23}$, $a_{33}$ and $b_{22}$, $b_{23}$, $b_{33}$ which we\nsave for later use.\nIn deriving this result we had to choose $\\alpha_{6}=\\beta_{6}=0$ and\nto impose the Bondi outgoing radiation condition \\ref{eq:bvmorc}.\nThen $R_{22}\\pm R_{23} = O_{4}$.\nNext we reexamine \\ref{eq:ricc00}.\nThe leading $O_{2}$ term gives us an expression for $a_{00,u}$ which\nwe store for later use.  Now $R_{00} = O_{3}$.\nWe have found, so far,  that $R_{00}$, $R_{01}$, $R_{02}$ and $R_{03}$\nare $O_{2}$ while the other components are $O_{3}$.\n\nWe now try to repeat the procedure of the previous paragraph.\nHowever we find that the $O_{4}$ contribution in $R_{11}$ contains\n``junk'' terms, i.e., terms which involve the third order metric\ncomponents which we have not been including; see the footnote in\nsection \\ref{sec:tetrad}.\nThus we can obtain no further information from the vacuum field\nequation $R_{11}=0$.\nSimilarly we find that the $O_{3}$ terms in $R_{01}$ contain junk as\ndo the $O_{4}$ terms in $R_{12}\\pm R_{13}$.\nThe same applies to the $O_{3}$ terms in $R_{02}\\pm R_{03}$, the\n$O_{4}$ terms in $R_{23}$ and $R_{22}\\pm R_{33}$, and finally the \n$O_{3}$ terms in $R_{00}$.\nWe have therefore exhausted the information available from the vacuum\nfield equations.\n\n\n\n\n\n\nAssuming that we have a vacuum we can switch attention to the Weyl\ntensor, and we compute first\n\\begin{equation}\n  \\label{eq:psi4a}\n  \\Psi_{4} = \n  R_{0202} = \\left[\n    \\tfrac{1}{4}(a_{22,uu} - a_{33,uu}\\csc^{2}\\theta) + \n    \\tfrac{1}{2} i a_{23,uu}\\csc\\theta \\right]\/r + O_{2}.\n\\end{equation}\nUsing \\ref{eq:a22} we may rewrite this as\n\\begin{equation}\n  \\label{eq:psi4b}\n  \\Psi_{4} = \\mathcal{N}_{,u}\/r + O_{2},\n\\end{equation}\nwhere \n\\begin{equation}\n  \\label{eq:bnews}\n  \\mathcal{N} = \\tfrac{1}{2}(a_{22} + ia_{23}\\csc\\theta)_{,u}\n\\end{equation}\nis the \\emph{Bondi news function} \\citep{BVM62}.\n\nThis is a highly satisfactory result which, in spite of our rather ad\nhoc chart and tetrad, mimics the treatment of \\citet{BVM62} and\n\\citet{NU62}.\nFurther we see that it is linear in the $a_{mn}$ and so should appear\n in linearized theory.\nAlso it does not involve the Lorentz parameters $\\alpha_{n}$ and\n$\\beta_{n}$ and so is tetrad-invariant (for tetrads which are\nasymptotically Minkowskian).\nThe remainder term in in \\ref{eq:psi4b} contains some $O_{2}$ terms and\njunk $O_{3}$ terms.\n\nNext consider\n\\begin{equation}\n  \\label{eq:psi3a}\n  \\Psi_{3} = R_{0120} = \\Psi_{3}^{(2)}\/r^{2} + O_{3},\n\\end{equation}\nwhere \n\\begin{equation}\n  \\label{eq:psi3b}\n  \\Psi_{3}^{(2)} = \n  2^{-1\/2}(\\mathcal{N}_{,\\theta}-i\\mathcal{N}_{,\\phi}\\csc\\theta +  \n  \\mathcal{N}\\cot\\theta) + \n(\\alpha_{4} + i\\alpha_{5})\\mathcal{N}_{,u}.\n\\end{equation}\nNote first that the $r$-dependence is precisely what one would have\nexpected from the peeling property.\nThe first  term in  the coefficient $\\Psi_{3}^{(2)}$ is linear\nand would have been predicted within linearized theory.\nHowever the second term is nonlinear for  it depends on the\n$\\alpha_{n}$ which determine the infinitesimal Lorentz transformation\nof the NP tetrads \\ref{eq:npud} and \\ref{eq:npdu}.\nThis is to be expected.\nThe NP tetrad used by \\citet{NU62} was chosen very specifically, while\nhere we are considering a class of tetrads\ninfinitesimally close to the Minkowski one.\nIf we were to restrict  attention to the subclass of tetrads where\n$\\alpha_{4}=\\alpha_{5}=0$ then our result would be consistent with\nlinearized theory.\nOn the other hand another choice of $a_{4}+i\\alpha_{5}$ would give\n$\\Psi_{3}^{(2)} = 0$.\nThe remainder term in \\ref{eq:psi3a} is junk.\n\n\nNext we find that \n\\begin{equation}\n  \\label{eq:psi2a}\n  \\Psi_{2} = R_{1320} = \\Psi_{2}^{(3)}\/r^{3} + O_{4},\n\\end{equation}\nwhere the remainder term is junk.\nWe will return to the leading term shortly.\n\nWe find next that \n\\begin{equation}\n  \\label{eq:psi1}\n  \\Psi_{1} = R_{0113} = \\Psi_{1}^{(4)}\/r^{4} + O_{5},\n\\end{equation}\nThe coefficient $\\Psi_{1}^{(4)}$ contains nonlinear terms, but we are\nunable to determine it precisely because it also contains junk terms. \nThe peeling property is still holding though.\n\nThe peeling property would demand that $\\Psi_{0} = R_{1313}$ should be\n$O_{5}$. \nHowever we find\n\\begin{equation}\n  \\label{eq:psi0a}\n  \\Psi_{0} = \\Psi_{0}^{(4)}\/r^{4} + O_{5},\n\\end{equation}\nwhere\n\\begin{equation}\n  \\label{eq:psi0b}\n  \\Psi_{0}^{(4)} = \\tfrac{1}{8}[\n  (a_{22}+a_{33}\\csc^{2}\\theta)(a_{22}-2ia_{23}\\csc\\theta  - \n  a_{33}\\csc^{2}\\theta) + 4(b_{22}-b_{33}\\csc^{2}\\theta - 2ib_{23}\\csc\\theta)].\n\\end{equation}\nBut we have not made the restrictions that were imposed by\n\\citet{BVM62} or \\citet{NU62} to ensure peeling.\nThe latter restriction was to \\emph{demand} $\\Psi_{0}^{(4)} =0$.\nThe former restriction was that the ``outgoing radiation condition'',\n\\citep{BVM62}, held.\nIn our notation this condition is , see \\ref{eq:bvmorc},\n\\begin{equation}\n  \\label{eq:orc1}\n  a_{22} = -a_{33}\\csc^{2}\\theta, \\qquad\n  b_{22} = b_{33}\\csc^{2}\\theta, \\qquad  b_{23} = 0.\n\\end{equation}\nExamining \\ref{eq:psi0b} we see that imposing the outgoing radiation\ncondition \\ref{eq:orc1} ensures $\\Psi_{0}^{(4)} =0$, a result first\nobtained by \\citet{Kroon98}.\nWith one or other condition we have\n\\begin{equation}\n  \\label{eq:psi0c}\n  \\Psi_{0} = \\Psi_{0}^{(5)}\/r^{5} + O_{6},\n\\end{equation}\nbut the coefficient $\\Psi_{0}^{(5)}$ contains junk terms and so we\ncannot evaluate it.\n(It also contains nonlinear terms not predicted by linearized theory.)\n\nTo summarize: if we impose the outgoing radiation condition\n\\ref{eq:orc1} then we obtain the peeling property, and we can obtain\nexplicitly the leading terms in $\\Psi_{4}$, $\\Psi_{3}$, $\\Psi_{2}$, but\nnot those for $\\Psi_{1}$ and $\\Psi_{0}$ because they contain junk\nterms.\n\nWe now return to the discussion of $\\Psi_{2}$ given by \\ref{eq:psi2a}.\nThe leading term coefficient is \n\\begin{equation}\n  \\label{eq:psi2b}\n  \\begin{split}\n    \\Psi_{2}^{(3)} =& \\tfrac{1}{2} a_{00} - \\tfrac{1}{4} ia_{23}\\csc^{3}\\theta +\n    \\tfrac{1}{2} b_{22,u} +\n    \\tfrac{1}{4}(a_{22}+ia_{23}\\csc\\theta )\n    (a_{22,u} - ia_{23,u}\\csc\\theta ) +\\\\\n    &\\quad \\tfrac{1}{4} i (a_{23,\\theta} - 2a_{22,\\phi})\\cot\\theta\\csc\\theta -\n    \\tfrac{1}{2} ia_{22,\\theta\\phi}\\csc\\theta  +\n    \\tfrac{1}{4} i (a_{23,\\theta\\theta} - \n    \\csc^{2}\\theta a_{23,\\phi\\phi})\\csc\\theta.\n  \\end{split}\n\\end{equation}\nHere we have fixed the Lorentz parameters, as described earlier, and\nare imposing the outgoing radiation condition \\ref{eq:bvmorc}.\n\nNow the \\emph{Bondi mass} $M_{B}(u)$ can be  defined by, \\citep{BVM62},\n\\citep{NU62}, \\citep{JMS89},\n\\begin{equation}\n  \\label{eq:bmassdef}\n  4\\pi M_{B}(u) = -\\lim_{r\\rightarrow\\infty}\\int_{S(u,r)}\n  r^{3}(\\Psi_{2} + \\sigma\\lambda)\\,\\sin\\theta\\,d\\theta\\,d\\phi,\n\\end{equation}\nwhere the integral is over the 2-surface $S(u,r)$ given by $u=const.$\nand $r=const.$\nHere $\\sigma$ and $\\lambda$ are NP spin coefficients given by\n\\begin{equation}\n  \\label{eq:siglam1}\n  \\sigma = m^{a}\\delta l_{a} = \\gamma_{313} = \\sigma^{(2)}\/r^{2} +\n  O_{3}, \\qquad\n  \\lambda = n^{a}{\\bar \\delta}{\\overline m}_{a} = \n  \\gamma_{022} = \\lambda^{(1)}\/r + O_{2},\n\\end{equation}\nwhere\n\\begin{equation}\n  \\label{eq:siglam2}\n  \\sigma^{(2)} =  -\\tfrac{1}{2}(a_{22}-ia_{23}\\csc\\theta ), \\qquad\n  \\lambda^{(1)} = -\\tfrac{1}{2} (a_{22,u}+ia_{23,u}\\csc\\theta ).\n\\end{equation}\nTaking the limit in \\ref{eq:bmassdef} we have\n\\begin{equation}\n  \\label{eq:bmassdef1}\n  4\\pi M_{B}(u) = -\\int_{S(u,1)}(\\Psi_{2}^{(3)} + \n  \\sigma^{(2)}\\lambda^{(1)})\\,\\sin\\theta\\,d\\theta\\,d\\phi.\n\\end{equation}\n\nOf course the formula \\ref{eq:bmassdef} is only valid in a specially\nchosen \\emph{Bondi frame}.\nThe generalization to an arbitrary NP frame is discussed in\n\\citet{JMS89}.\nIn the large $r$ limit our frame differs from the Bondi one by a\nLorentz transformation which is close to the identity.\nA 2-parameter subgroup of the Lorentz group consists of ``boosts'' and\n``spins'' \n\\begin{equation}\n\\label{eq:boostspin}\nl \\rightarrow a^{2}l,\\quad\nn \\rightarrow a^{-2}n,\\quad\nm \\rightarrow e^{i\\psi}m,  \n\\end{equation}\nwhere $a$ and $\\psi$ are real.\nUsing the formulae in appendix B of \\cite{JMS93} it is easy to verify\nthat the integrand of \\ref{eq:bmassdef1} is invariant under boosts and\nspins.\nNext consider a 2-parameter subgroup of ``null rotations about $l$''\ngiven by\n\\begin{equation}\n  \\label{eq:lrot1}\n  l \\rightarrow l,\\quad, m \\rightarrow m + {\\bar c}l,\\quad\n  n \\rightarrow n + cm + {\\bar c}{\\overline m} + c{\\bar c}l,\n\\end{equation}\nwhere $c$ is complex.\nUnder such a transformation\n\\begin{equation}\n  \\label{eq:lrot2}\n  \\begin{split}\n    &\\Psi_{2}\\rightarrow \\Psi_{2} + 2c\\Psi_{1} + c^{2}\\Psi_{0}, \\qquad\n    \\sigma \\rightarrow \\sigma + {\\bar c}\\kappa,\\\\\n    &\\lambda\\rightarrow\\lambda + c\\pi + 2c\\alpha +\n    c^{2}(\\rho+2\\epsilon) + c^{3}\\kappa + cl^{a}c_{,a} + \n    {\\overline m}^{a}c_{,a}.\n  \\end{split}\n\\end{equation}\nWe expect $c = O_{1}$ and the NP scalars $\\alpha$, $\\pi$, $\\rho$ and\n$\\epsilon$ are all $O_{1}$.\nThus the integrand of \\ref{eq:bmassdef1} is not changed.\nWe should also consider null rotations about $n$ given by\n\\begin{equation}\n  \\label{eq:nrot1}\n  n \\rightarrow n,\\quad, m \\rightarrow m + {\\bar c}n,\\quad\n  l \\rightarrow l + cm + {\\bar c}{\\overline m} + c{\\bar c}n,\n\\end{equation}\nso that\n\\begin{equation}\n  \\label{eq:nrot2}\n  \\begin{split}\n    &\\Psi_{2}\\rightarrow \\Psi_{2} + 2c\\Psi_{3} + c^{2}\\Psi_{4}, \\qquad\n    \\lambda \\rightarrow \\lambda + {\\bar c}\\nu,\\\\\n    &\\sigma\\rightarrow\\sigma + c\\tau + 2c\\beta +\n    c^{2}(\\mu+2\\gamma) + c^{3}\\nu + cn^{a}c_{,a} +  m^{a}c_{,a}.\n  \\end{split}\n\\end{equation}\nNow we have taken great care to ensure that $l$ is almost geodesic (\n$\\kappa=O_{4}$) and almost affinely parametrised \n($\\epsilon + \\bar\\epsilon=O_{3}$) and so we should only consider the\ntransformation \\ref{eq:nrot1} where $c=O_{3}$.\nUnder this restriction the integrand of \\ref{eq:bmassdef1} is not\nchanged.\nThus the formula \\ref{eq:bmassdef1} evaluated in our frame does indeed\ngive the Bondi mass to leading order.\nNext note that\n\\begin{equation}\n  \\label{eq:bmassim}\n  \\begin{split}\n  Im(\\Psi_{2}^{(3)} + \\sigma^{(2)}\\lambda^{(1)}) =&\n  -\\tfrac{1}{4} a_{23}\\csc^{3}\\theta + \n  \\tfrac{1}{4} a_{23,\\theta}\\csc\\theta\\cot\\theta +\n  \\tfrac{1}{4} a_{23,\\theta\\theta}\\csc\\theta  -\\\\\n  & \\qquad \\tfrac{1}{2}  a_{22,\\phi}\\csc\\theta\\cot\\theta -\n  \\tfrac{1}{2} a_{22,\\theta\\phi}\\csc\\theta  -\n  \\tfrac{1}{4} a_{23,\\phi\\phi}\\csc^{3}\\theta.\n  \\end{split}\n\\end{equation}\nWhen we integrate this over the unit sphere the terms in the second\nline give zero since their contribution to the integrand is\n$2\\pi$-periodic in $\\phi$. \nThose in the first line contribute\n\\begin{equation*}\n  \\tfrac{1}{2}\\pi\\left[\\csc\\theta(\\sin\\theta a_{23})_{,\\theta}\\right]^{\\pi}_{0}.\n\\end{equation*}\nNow $a_{23}$ must scale like $\\sin^{2}\\theta$ at the end points or\nelse the integrand is singular.  It follows that the Bondi mass must\nbe real, and\n\\begin{equation}\n  \\label{eq:bmassre}\n  4\\pi M_{B}(u) = -\\tfrac{1}{2}\\int_{0}^{\\pi}(\n  a_{00} + b_{22,u} + a_{22}a_{22,u} +  a_{23}a_{23,u}\\csc^{2}\\theta)\n  \\sin\\theta\\,d\\theta\\,d\\phi.\n\\end{equation}\n\nFinally there is a standard result, \\citep{BVM62}, \\citep{NU62},\n\\citep{JMS89}, for the rate of decrease of the Bondi mass\n\\begin{equation}\n  \\label{eq:bmassloss}\n  \\begin{split}\n    4\\pi\\frac{dM_{B}}{du}(u) &= -\\int_{S(u,1)} |\\mathcal{N}|^{2}\n    \\sin\\theta\\,d\\theta\\,d\\phi\\\\\n    &= -\\tfrac{1}{4} \\int_{S(u,1)}\\left((a_{22,u})^{2} + \n      (a_{23,u})^{2}\\csc^{2}\\theta\\right)\\sin\\theta\\,d\\theta\\,d\\phi,\n  \\end{split}\n\\end{equation}\ndemonstrating the well-known result $dM_{B}\/du\\leqslant 0$, the\nBondi mass decreases as energy is radiated away, a result not\ndeducible in linearized theory.\nAlthough \\ref{eq:bmassloss} was originally derived in a special Bondi\nframe, it too holds in our approximate Bondi one, at least to leading\norder.\nWe should empasize that although the outgoing radiation condition was\nused in the derivation of \\ref{eq:bmassre}, the mass loss formula\n\\ref{eq:bmassloss} holds without the need for this restriction.\n\n\n\\section{Implications for numerical relativity}\n\\label{sec:numrec}\n\nAt first glance the formalism we set up to carry out this study may\nseem to be cumbersome, but it has the advantage the the results can be\ntranslated back into the $X^{a}=(T,R,\\Theta,\\Phi)$ chart, after which\nthe theoretical chart $x^{a}=(u,r,\\theta,\\phi)$ can be discarded.\n\nWe chose the $x^{a}$ chart so that the metric coefficients $a_{11}$\nand $b_{11}$ vanished as well as $a_{12}$ and $a_{13}$.\nNow\n\\begin{equation}\n  \\label{eq:partial}\n  \\begin{split}\n    \\left(\\frac{\\partial\\;\\;}{\\partial u}\\right)_{r} &=\n    \\left(\\frac{\\partial T}{\\partial u}\\right)_{r}\n    \\left(\\frac{\\partial\\;\\;}{\\partial T}\\right)_{R} + \n    \\left(\\frac{\\partial R}{\\partial u}\\right)_{r}\n    \\left(\\frac{\\partial\\;\\;}{\\partial R}\\right)_{T} \\\\\n    &= (1-q_{1}r^{-1})\\left(\\frac{\\partial\\;\\;}{\\partial T}\\right)_{R}+ O_{2},\\\\\n  \\end{split}\n\\end{equation}\nusing \\ref{eq:jac1}.\nIn principle the function $q_{1}$ is arbitrary.\nBut the vacuum field equations implied $a_{01}=0$, and then equations\n\\ref{eq:add} imply\n\\begin{equation}\n  \\label{eq:q1}\n  q_{1} = \\tfrac{1}{2}(h_{00}-h_{11}).\n\\end{equation}\nThe vacuum field equations imply \\ref{eq:a22}\n\\begin{equation*}\n  a_{22,u} + a_{33,u}\\csc^{2}\\theta.\n\\end{equation*}\nUsing \\ref{eq:add} and \\ref{eq:y2z23} we have, to leading order\n\\begin{equation}\n  \\label{eq:eg}\n  [h_{22}+2h_{02,\\Theta}+2h_{12,\\Theta}]_{,T} + \n  [h_{33}+(2h_{03,\\Phi}+2h_{13,\\Phi})\\csc^{2}\\Theta]_{,T} \\csc^{2}\\Theta = 0.\n\\end{equation}\nOther vacuum conditions can be handled in a similar way.\n\n\nIn order to discuss the Bondi news function and Bondi mass, it is\nconvenient to introduce some auxiliary functions in the numerical chart,\n\\begin{equation}\n  \\label{eq:aux}\n  \\begin{split}\n    \\mathcal{W} =& h_{03} + h_{13},\\\\\n    \\mathcal{A} =& h_{22}+ 2(h_{02,\\Theta}+h_{12,\\Theta}),\\\\\n    \\mathcal{B} =& h_{23} + h_{02,\\Phi} + h_{12,\\Phi} +\n    \\mathcal{W}_{,\\Theta} - 2\\mathcal{W}\\cot\\Theta, \\\\\n    \\mathcal{C} =& k_{22} + 2\\mathcal{W}h_{23}\\cot\\Theta\\csc^{2}\\Theta\n    - 4\\mathcal{W}^{2}\\cot\\Theta\\csc^{2}\\Theta +\n    (k_{02}+k_{12})_{,\\Theta} -\\\\\n    &\\quad (\\tfrac{1}{2} h_{11}+h_{22})h_{02,\\Theta} -\n    ((\\tfrac{1}{2} h_{00}+h_{01})h_{02})_{,\\Theta} +\n    (4\\mathcal{W}\\cot\\Theta -\n    h_{23})\\mathcal{W}_{,\\Theta}\\csc^{2}\\Theta - \\\\\n    &\\quad \\tfrac{1}{2} h_{02}h_{11,\\Theta} - h_{22}h_{12,\\Theta} + \n    (h_{02}+h_{12})h_{22,\\Theta} +\n    \\mathcal{W}h_{23,\\Theta}\\csc^{2}\\Theta -\n    (\\mathcal{W}_{,\\Theta})^{2}\\csc^{2}\\Theta\n  \\end{split}\n\\end{equation}\nwhich should be readily available according to the assumptions in\nsection \\ref{sec:numerical}.\n\nThen the leading term in the Bondi news function given by\n\\ref{eq:bnews} becomes \n\\begin{equation}\n  \\label{eq:bnewsnum}\n  \\mathcal{N} = \\mathcal{A}_{,T} + i\\mathcal{B}_{,T}\\csc\\Theta,\n\\end{equation}\nwhose calculation  might require some sophistication, although there\nis no reference to the intermediary $x^{a}=(u,r,\\theta,\\phi)$ chart.\nBecause the news function is linear in the $h_{ab}$ and their\nderivatives, it could have been calculated within linearized theory.\n\nIn particular the formula \\ref{eq:bmassre} for the Bondi mass\n$M_{B}(u)$ translates into\n\\begin{equation}\n  \\label{eq:bmassnum}\n  \\begin{split}\n    M_{B}(T-R) =&\n    -\\frac{1}{8\\pi}\\int_{\\Phi=0}^{2\\pi}\\int_{\\Theta=0}^{\\pi}\n     \\Big(h_{11} + \\mathcal{A}\\mathcal{A}_{,T}\n     + \\mathcal{B}\\mathcal{B}_{,T}\\csc^{2}\\Theta + \\mathcal{C}_{,T} \\Big)\n       \\,\\sin\\Theta\\,d\\Theta d\\Phi.\n   \\end{split}\n\\end{equation}\n\nIn a similar way the formula \\ref{eq:bmassloss} for the rate of change\nof $M_{B}$ at fixed large $R$ is\n\\begin{equation}\n  \\label{eq:blossnum}\n  \\begin{split}\n    {\\dot M}_{B}(T-R) = &\n    -\\frac{1}{16\\pi}\\int_{\\Phi=0}^{2\\pi}\\int_{\\Theta=0}^{\\pi}\\Big(\n    (\\mathcal{A}_{,T})^{2} + (\\mathcal{B}_{,T})^{2}\\csc^{2}\\Theta\n    \\Big)\\,\\sin\\Theta\\,d\\Theta d\\Phi.\n   \\end{split}\n\\end{equation}\n\nWorking in linearized theory numerical relativists often take the\nleading terms in $\\cal{A}$ and $\\cal{B}$ to represent the\n``gravitational waveforms'' $h_+$ and $h_{\\times}$. Equations \\ref{eq:aux}\nsuggest that other terms are present, even in linearized theory.\n\nAgain we emphasize that the intermediate $x^{a}$ chart does not\nintrude---the formulae \\ref{eq:bnewsnum}, \\ref{eq:bmassnum} and\n\\ref{eq:blossnum} apply in the numerical $X^{a}=(T,R,\\Theta,\\Phi)$\nchart.\nOnly the first can be deduced from linearized theory.\n\nWhy are these formulae so complicated, when compared with the original\npapers, \\citet{BVM62} and \\citet{NU62}, or even the formulae in\nsection \\ref{sec:curvature}?\nWell the coordinates and tetrads of the originals were very carefully\nchosen to simplify the problem, and much of this paper has been spent\nbuilding the relationship between the numerical relativist's \n$X^{a}=(T,R,\\Theta, \\Phi)$ chart and the $x^{a}=(u,r,\\theta,\\phi)$\nchart and adapted tetrad used in this paper, in which the formulae look\nalmost as simple as in the original approaches.\nOne way to avoid the complexity is to design a numerical approach\nbased on Penrose's geometrical approach \\citep{RP63}, but that brings\nin different problems and complexities.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}%\n\nData-driven techniques in power systems have at least fifty years of history, starting with static state estimations developed by Schweppe and co-authors \\cite{70SW,70SR,70Sch}, then transitioning to dynamic state estimation analysis and applications, see e.g.  \\cite{1990Hauer,2007Mani,12Tru} and references therein, and most recently discussed under the umbrella of ``big data'' as the most significant enabler of power system operations, security  and resiliency in the future \\cite{13KXG,18AZ}. (See also related discussion in the description of the US Department of Energy new funding opportunity to ``explore the use of big data, artificial intelligence, and machine learning technology to leverage the power of grid sensors\" \\cite{DOE_call}.) Many specific questions and approaches,  including but not limited to modes of oscillations analysis of stability and detection \\cite{12Tru,NERC_reliability,PNNL_oscil,WECC-JSI}, dimension reduction for faster processing and analysis \\cite{12DKM}, early event detection \\cite{14XCK}, missing data recovery \\cite{16_RPI_missing_data}, identification of cyber attacks \\cite{16_RPI_cyber} and real time (online) event detection \\cite{18LWC} are among the most recent advances. \n\nOn the methodology side data-driven methods developed in other engineering disciplines have been adopted, modified and used for many (e.g. aforementioned) power system applications. Principal component analysis\n\\cite{12DKM}-\\cite{18LWC}, auto-correlation analysis of memory effects \\cite{18RL}, and linear model driven spectral analysis of the dynamic state matrix \\cite{2007Mani,12Tru,NERC_reliability,PNNL_oscil,WECC-JSI,2009Chang,2016Mani,2017LVSDC} are arguably the most popular data-driven techniques currently in use in the power system research. \n\n\nEven though the sophistication level of the methods already used in power system applications is impressive, coherence and understanding of the potential of new generation of the big data methods, driven during the last decade largely through heavy investment of IT industry, is still lacking \\cite{DOE_call}. We anticipate that many of the most modern methods, especially Deep Learning (DL) and related techniques linked to Machine Learning (ML) and  Artificial Intelligence (AI), revolutionary advances in data science and more generally theoretical engineering \\cite{09Ben,10ARK,15Sch,15LBH,16GBC}, will impact the power-system operation-room reality in an even more significant ways. However, one problem with applications of the novel methods of DL and alike in sciences and engineering is that they are application agnostic\/generic -- very effective for many business cases, but lacking ``explainability\", i.e. intuitive physical\/engineering explanations.  This significant handicap of the most advanced and recent ML \\& AI methods slows down development of related applications in power systems.  Indeed,  power system practitioners would generally not consider as practical any new methods lacking ``power systems informed\" explanations.\n\nThis manuscript takes a step towards closing the gap between the rich variety of methods already developed and utilized in power systems and yet to be unleashed power of the upcoming Big Data revolution. Specifically, we start walking towards exciting sophistication of DL slowly, from the well-established and intuitive trenches of practical system engineering. We develop in this manuscript a pragmatic ``phasor-detective\" approach to analysis of the streaming Phasor Measurement Unit (PMU) data which allows to extract and interpret spatial and temporal correlations in a computationally light fashion and without making any constraining assumptions about origin of the correlations. \n\n\\subsection{Our Contribution}\n\nWe analyze synchronized historical PMU data recorded  at $\\approx 200$  most significant locations of a US Independent System Operator (ISO) over the course of two years. At each PMU location the data includes complex current and voltage recorded with a millisecond resolution. Given geo-spatial locations of the PMU, but no information about the grid characteristics and layout, we pose the following principal questions: What can we possibly reconstruct from the data stream about the system current ambient behavior? To answer the question we utilize available statistical tools. In relation to preliminary data processing we apply to the raw signal three filtering techniques: moving average, sliding time horizon and Fourier analysis pre-processing. This allows to provide robust identification of the ``quiet\" periods and also prepare data for subsequent statistical analysis by means of Principal Component Analysis (PCA) and Auto-Correlation Analysis (ACA). We show that the two complementary tools, applied to the raw as well as to pre-processed signal, allow to separate scales and also provide compressed, thus easy to visualize, descriptors for online tracking of current state of the grid in a much broader way than what the current Energy Management System (EMS) actually uses. PCA provides a robust set of indicators which record slow\/adiabatic changes on the scale of seconds to ten of seconds and slower. We have observed that only a very few principal modes are significant at any moment of time, even though these modes may be different for voltage amplitude, phase difference and frequency (the three main characteristics) we track. The results do not change when the PCA is applied to the filtered signal, consistently with the fact that PCA averages over time but do not catch different-time correlations. ACA is the tool used to analyze the latter, in particular identifying significant, persistent correlations, missed by PCA, at shorter time scales - subseconds-to-seconds.  Following ACA curves at different spatial locations we were able to identify nodes where correlations do not decay with time showing significant memory-effects. Remarkably,  these nodes with significant memory cluster geographically. We observe two areas in the grid which show especially strong sustainable temporal correlations. We then proceed with ACA analysis of the Fourier-filtered signal.  This helps us to identify and localize different harmonics. In particular, we observe (for a particular quiet period) emergence of significant oscillations in the 4-6Hz range at a small number of nodes. Interestingly, nodes with significant sustainable oscillations are either wind farms, big aggregated loads or mid-size generators. We conjecture that the sustainable oscillations are indicators of malfunction at these critical elements of the grid. We also observe that sustainable oscillations, seen clearly through emergence of a residue in the ACA analysis of the raw signal, disappear when applied to the Fourier-filtered signal (cutting off the 4-6Hz oscillations). Finally, analyzing spatial cross-correlations (of the residue) we were able to identify group of nodes with significant inter-dependency.\n\n\n\nMaterial in the manuscript is organized as follows. Logic and main steps of the ``phasor detective\" approach are described in Section \\ref{sec:logic}. Section \\ref{sec:time_series} discusses the time series data we are working with and it also introduces averaging and filtering approaches we apply to the data. Covariance matrix and PCA analysis are discussed in Section \\ref{sec:PCA}. Auto-correlation and cross-correlation approaches, extracting information of the data temporal correlations\/memory,  are developed in Section \\ref{sec:temp-correlations}. Section \\ref{sec:conclusions} and Section \\ref{sec:path-forward} are reserved, respectively, for conclusions and discussions of the path forward. \n\n\n\\section{Logic and Steps of the Detective Approach}\n\\label{sec:logic}\n\n\nIn this work we report on data streams from over $200$ PMUs operated by an ISO and spanning a period over one year long. Fig.~\\ref{fig:death_valey} displays a rendition of the locations of the PMUs using anonymized coordinates. As is standard, \nthe data stream from each PMU includes (complex) current and voltage reported 30 times per second.\nUsing this data one can obtain real-time estimates of complex power at each location.   Working with a data set this large (on the order of 28 TB) presents some obvious challenges; additionally there are specific artifacts that can arise in the data.  For example, not all PMUs are always reporting, and occasionally some PMUs exhibit what appears to be errant behavior.  \n\n\\begin{figure}\n  \\begin{center}\n    \\begin{tikzpicture}\n    \\begin{axis}[%\n        ticks=none,\n    \twidth=0.8\\linewidth,\n    \theight=120pt,\n    \n        ]\n        \\addplot[only marks,mark size=0.7pt]%\n          table[x=x,y=y,col sep=comma]{newcoords.csv};\n    \\end{axis}\n    \\end{tikzpicture}\n  \\caption{Geographical location of PMUs (anonymized coordinates). \\label{fig:death_valey}}\n  \\end{center}\n\\end{figure}\n\nOur work has centered on performing statistical analysis aimed at inferring ``structure\" in the underlying transmission system as well as identifying complex behaviors, such as resonance and oscillations.  \nIn this manuscript we focus specifically on identification and characterization of ``quiet\" periods. Oversimplifying (see related discussion below) we consider a period \\textit{quiet} if fluctuations around the mean (e.g. characterized in terms of the standard deviations) are smaller than a reasonable pre-defined threshold.  This focus on the quiet\/ambient periods is motivated by the following considerations:\n\\begin{itemize}\n    \\item The development of a strong understanding of quiet periods and (in particular) efficient online algorithms for recognition of such periods is a necessary step prior to studying less-quiet or even anomalous regimes, for otherwise we risk significant misinterpretation, i.e. errors in online detection of anomalies. \n    \\item As will be seen in the following,  the quiet regimes display informative patterns and correlations, all (slowly) time-evolving.  Identifying such features is important with regards to\n    \\begin{itemize}\n        \\item Developing fast and reliable identification techniques.\n        \\item Uncovering hidden malfunction of assets thus providing significant contribution towards forecasting most probable (and destined to occur) failures. \n    \\end{itemize}\n    \\item The richness of correlations observed in the quiet regime, in fact, suggests that separation of what is normal\/quiet from what is anomalous\/atypical will be challenging.  Even though we observe that quiet regimes dominate, relatively abrupt jumps of moderate size (i.e. jumps exceeding currently tracked standard deviation by factor of two or three) are rather frequent even though overall they account for a relatively short fraction of the stream.  As a result, it is rather difficult to find sufficiently long completely quiet periods in the available data.\n    \\item Clearly, understanding quiet vs volatile behavior will be helpful toward building predictive models for better optimization, control and planning. \n    \\item ``Cyber\" and ``cyber-physical\" attacks on power systems are a venue where fast and effective learning of (changing) stochastics may prove useful in identifying attacks.  See \\cite{2018attack}.\n\\end{itemize}\nThe methodology adopted in this manuscript to identify the quiet periods is explained in Section \\ref{sec:quiet}. \n\n\n{\\bf Notation:} for $x\\in{\\mathbb C}$, $|x|$ denotes its magnitude and $\\overline{x}$ denotes its complex conjugate. For a complex matrix $A$, $A^{\\mathsf{H}}$ denotes its Hermitian transpose.\n\n\n\n\\section{Description and Averaging of the \\\\Time Series Data}\n\\label{sec:time_series}\n\nThe available data encompasses the period from January 1st 2013 to March 21st 2014. Each of the $N = 240$ PMUs records the following measurements 30 times per second: time of the measurement (GPS tagged), bus ID, voltage amplitude, voltage phase angle, current magnitude, current angle, frequency, and additional data not used in this study. Additionally, the 2-dimensional coordinates of the PMU locations is also available, together with their corresponding nominal voltages.  We note that PMUs \\textit{report} 30 times\/second, but they\n\\textit{sample} at a far higher rate and perform filtering (e.g., anti-aliasing) before reporting.\n\n\nWe will denote a generic scalar or complex measurement (e.g. complex voltage) at PMU location $i$, at time $t$ by  $m_i(t)$. The parameter $t$ will be used to refer to the discrete time sequence with each temporal data point separated from preceding one by the same duration $\\Delta$ ($1\/30^{th}$ of a second in our case). \n\nTypical pre-processing steps in the statistical analysis of data (especially with the goal of analyzing correlations) involve modifications through de-trending, offsetting (subtraction) of moving average, and normalizations.  We will apply such techniques below.\nSpecific details are provided next. \n\n\\subsection{Moving average and covariance}\n\nThe \\emph{moving average} $\\mu^{(m)}$ and \\emph{moving variance} $\\sigma^{(m)}$ of a parameter $m$ is computed at every bus $i$ in the following way:\n\\begin{align}\n    \\mu_i^{(m)}(t;\\alpha) &= \\alpha\\cdot m_i(t) + (1-\\alpha)\\cdot \\mu_i^{(m)}(t-1;\\alpha),  \\label{eq:movinga} \\\\\n    \\sigma_i^{(m)}(t;\\alpha) \n    &= \\alpha\\cdot |m_i(t)-\\mu_i^{(m)}(t-1;\\alpha)|^2 \\nonumber\\\\\n    &\\quad + (1-\\alpha)\\cdot \\sigma_i^{(m)}(t-1;\\alpha),\n    \\label{eq:movingv}\n\\end{align}\nwith some initial values, say $\\mu_i^{(m)}(0;\\alpha)=0$ and $\\sigma_i^{(m)}(0;\\alpha)=1$.\nThe parameter $\\alpha\\in(0,1)$ represents the degree of weighting decrease. Note that in \\eqref{eq:movingv} we are using the moving average defined in  \\eqref{eq:movinga}.  One can likewise define moving covariance parameters.\n\nFinally, we introduce the zero mean and normalized zero mean data streams\n\\begin{eqnarray}\n    && \\bar{m}^{(m)}_i(t;\\alpha) = m_i(t)-\\mu_i^{(m)}(t-1;\\alpha),\\label{eq:mov_av}\\\\\n    && \\hat{m}^{(m)}_i(t;\\alpha) = \\frac{m_i(t)-\\mu_i^{(m)}(t-1;\\alpha)}{\\sqrt{\\sigma_i^{(m)}(t-1;\\alpha)}},\n    \\label{eq:mov_av_norm}\n\\end{eqnarray}\nobtained from the input stream by making use of the moving average and variance.  The zero mean parameters will help us identify quiet periods, as discussed next.\n\n\\subsection{Quiet Periods}\n\\label{sec:quiet}\n\nGiven a reference time $t$ and a length parameter $Q$  consider the $N\\times Q$ matrix of normalized measurements\n\\begin{equation*}\n    M(t;\\alpha;Q) = [\\ \\hat{m}_i^{(m)}(\\tau;\\alpha)\\ |\\ \\forall i,\\ \\tau\\in\\{t-Q+1,\\ldots,t\\}\\ ],\n\\end{equation*}\ncorresponding to the last $Q$ measurements before time $t$ for all buses.\nWe define the period $(t-Q,t]$ as \\emph{quiet} if the absolute value of all entries of the matrix $M(t;\\alpha;Q)$ is below some preset threshold. The reason behind this definition is that sudden jumps in the data appear as large values in the normalized time series, whereas normalized values close to zero mean that the data is behaving in a steady way. \nEffectively, a quiet period is an interval of time where all sensor-reported data behave in a stationary way.\nMoreover it is relatively cheap to compute $M(t;\\alpha;Q)$ from the stream data, as we just have to keep track of the moving average and moving variance of each bus, and the matrix of the last $Q$ normalized measurements.\n\nIn our analysis, we used $\\alpha=0.05$ as the weighting decrease constant of the moving average and moving variance. This value of $\\alpha$ corresponds to the memory budget (in the moving average) of approximately one second, see Fig.~\\ref{fig:averages}.  In our analyses we will consider quiet periods spanning $15$ minutes ($Q=27,000$).  Over a selection of five different days across the database, we compute we compute the matrix $M(t;\\alpha;Q)$ and record its maximum absolute value when $t$ spans over the complete day. Just in few cases the maximum was below 10 units, we selected 2-3 intervals between these recorded cases.\n\nAveraging over a fixed sliding window, described next for completeness, is an alternative to the moving window average discussed so far.\n\n\\subsection{Averaging over Sliding Time Horizon}\n\\label{subsec:Sliding}\n\nWe continue to work with the scalar or complex signal (voltage, phase or frequency) data stream $m_i(t)\\ \\forall i$, with $t\\in\\{0,\\ldots,Q-1\\}$, corresponding to the (not normalized) measurements of a selected period. Here, $t=0$ indicates the beginning of the period.\n$N$ dimensional vectors of means and variances, averaged over the (last) sliding time horizon of duration $S\\leq Q$ are defined as follows, $\\forall t \\mbox{ such that } S\\leq t<Q$:\n\\begin{eqnarray}\n&& \\mu^{(s)}_i(t;S)= \\frac{1}{S}\\sum_{\\tau=t-S+1}^t m_i(\\tau), \\label{eq:mu-s}\\\\\n&& \\sigma^{(s)}_i(t;S)= \\frac{1}{S}\\sum_{\\tau=t-S+1}^t \\left|m_i(\\tau)-\\mu_i^{(s)}(t;S)\\right|^2,\\label{eq:sigma-s}\n\\end{eqnarray}\nthen, the zero-mean data vector and the re-scaled zero-mean data vector are given by\n\\begin{eqnarray}\n&& \\bar{m}^{(s)}_i(t;S)=  m_i(t)-\\mu_i^{(s)}(t-1;S),\\label{eq:bar-m-s}\\\\\n&& \\hat{m}^{(s)}_i(t;S)=  \\frac{m_i(t)-\\mu_i^{(s)}(t-1;S)}{\\sqrt{\\sigma_i^{(s)}(t-1;S)}}.\\label{eq:hat-m-s}\n\\end{eqnarray}\nNumerically, we found that a reasonable choice for $S$, i.e. one consistent with $\\alpha=0.05$ which allows to separate power electronics (milliseconds) and electro-mechanical (seconds) time scales, is the number of readings in 1 second, that is $S=30$. See Fig.~\\ref{fig:averages}. \n\nFig.~\\ref{fig:std} shows a comparison between the behavior of $\\sigma^{(m)}(t,\\alpha=0.05)$ and $\\sigma^{(s)}(t;S=30)$ --in the same time windows as Fig.~\\ref{fig:averages}-- showing $\\mu^{(\\cdot)}\\pm\\sqrt{\\sigma^{(\\cdot)}}$. \n\n\n\\begin{figure*}\n  \\includegraphics[width=\\textwidth]{plot_average_bus156.png}\n  \\caption{Raw data $m_i(t)$ during 1 minute of voltage magnitude at a particular bus and the comparison of the {\\bf(a)} moving average $\\mu_i^{(m)}$ with $\\alpha=0.05$ and the window average $\\mu_i^{(s)}$ for $S$ equal to {\\bf(b)} 60, {\\bf(c)} 10, and {\\bf(d)} 1 seconds. \n  \\label{fig:averages}}\n\\end{figure*}\n\n\\begin{figure*}\n  \\includegraphics[width=\\textwidth]{plot_std_bus156.png}\n  \\caption{Raw data $m_i(t)$ during 1 minute of voltage magnitude at same bus than Fig.~\\ref{fig:averages} and the comparison of {\\bf(a)} $\\mu_i^{(m)}\\pm\\sqrt{\\sigma_i^{(m)}}$ with $\\alpha=0.05$ and {\\bf(b)} $\\mu_i^{(s)}\\pm\\sqrt{\\sigma_i^{(s)}}$ using $S=1$ second.\n  \\label{fig:std}}\n\\end{figure*}\n\n\n\\subsection{Fourier Filter}\n\\label{sec:FTfilter}\n\nWe have also applied Fourier filtering on the time series $m_i(\\cdot)$. Let ${\\cal F}[m_i(\\cdot)]\\in {\\mathbb C}^Q$ be the discrete Fourier transform of $m_i(\\cdot)$, where ${\\cal F}[m_i(\\cdot)]_k$ is the amplitude corresponding to frequency $\\omega_k$, $k\\in\\{0,\\ldots,Q-1\\}$. \nSince the frequency of the readings is $30$Hz, we can represent the frequency domain as $Q$ equidistant points between $-15$ and $15$Hz.\n\nWe obtain a filtered time series by suppressing the high frequency components. \nLet ${\\bf 1}_{[-\\lambda,\\lambda]}\\in\\{0,1\\}^Q$ be the indicator function of the set of frequencies in the interval $[-\\lambda,\\lambda]$, that is\n\\begin{equation*}\n    ({\\bf 1}_{[-\\lambda,\\lambda]})_k = \\begin{cases}\n    1, &\\mbox{if }-\\lambda\\le \\omega_k\\leq \\lambda\\\\\n    0, &\\mbox{otherwise.}\n    \\end{cases}\n\\end{equation*}\nThe filtered series is ${\\cal F}^{-1}\\left[{\\bf 1}_{[-\\lambda,\\lambda]}\\odot{\\cal F}[m_i(\\cdot)]\\right]$, where ${\\cal F}^{-1}$ is the inverse Fourier transform and $\\odot$ is the component-wise product. We will denote this new series by $f_i(t;\\lambda)$, $t\\in\\{0,\\ldots,Q-1\\}$.  \n\nFig.~\\ref{fig:ft-filter}a shows the amplitude (in absolute value) of the Fourier transform of the voltage of a bus for an interval of 15 minutes. \nFig.~\\ref{fig:ft-filter}b-d show the filtered series when frequencies larger than 5Hz, 4Hz, and 2Hz have been suppressed, respectively.\n\n\\begin{figure*}\n  \\includegraphics[width=\\textwidth]{plot_fft_bus156_VoltageMag.png}\n  \\caption{{\\bf (a)} Absolute value (in log-scale) of Fourier transform of the voltage magnitude time series for a particular bus. Raw data and filtered series at {\\bf (b)} $\\lambda=5$Hz, {\\bf (c)} $\\lambda=4$Hz, and {\\bf (d)} $\\lambda=2$Hz. \n  \\label{fig:ft-filter}}\n\\end{figure*}\n\nAnalogously to equations~\\eqref{eq:mu-s}-\\eqref{eq:hat-m-s}, we can compute (averaging over a sliding time horizon) the zero-mean data vector and the re-scaled zero-mean data vector for $f_i(\\cdot;\\lambda)$ instead of $m_i(\\cdot)$, we will call them $\\bar{f}_i^{(s)}(\\cdot;\\lambda;S)$ and $\\hat{f}_i^{(s)}(\\cdot;\\lambda;S)$, respectively. \n\n\n\n\\begin{figure}\n\\centering  \n  \\includegraphics[width=.47\\textwidth]{plot_fourierspectrum_2013-01-15_0013_bus002.png}\n  \\caption{Fourier Transform of raw data for selected bus and selected period.} \n  \\label{fig:fourier1}\n\\end{figure}\n\n\\begin{figure}\n\\centering  \n  \\includegraphics[width=.47\\textwidth]{plot_fourierspectrum_2013-01-15_0013_bus156.png}\n  \\caption{Fourier Transform of raw data for selected bus and selected period.} \n  \\label{fig:fourier2}\n\\end{figure}\n\n\\begin{figure}\n\\centering  \n  \\includegraphics[width=.47\\textwidth]{plot_fourierspectrum_2013-03-10_0009_bus170.png}\n  \\caption{Fourier Transform of raw data for selected bus and selected period.} \n  \\label{fig:fourier3}\n\\end{figure}\n\n\\begin{figure}\n\\centering  \n  \\includegraphics[width=.47\\textwidth]{plot_fourierspectrum_2013-03-10_0429_bus191.png}\n  \\caption{Fourier Transform of raw data for selected bus and selected period.} \n  \\label{fig:fourier4}\n\\end{figure}\n\n\\begin{figure}\n\\centering  \n  \\includegraphics[width=.47\\textwidth]{plot_fourierspectrum_2013-07-30_1639_bus006.png}\n  \\caption{Fourier Transform of raw data for selected bus and selected period.} \n  \\label{fig:fourier5}\n\\end{figure}\n\n\\begin{figure}\n\\centering  \n  \\includegraphics[width=.47\\textwidth]{plot_fourierspectrum_2013-04-03_0719_bus059.png}\n  \\caption{Fourier Transform of raw data for selected bus and selected period.} \n  \\label{fig:fourier6}\n\\end{figure}\n\nFigures~\\ref{fig:fourier1}-\\ref{fig:fourier6} plot the absolute value of the Fourier transform for positive frequencies (the negative side is symmetric since the original signal is real, not complex) and for three considered variables: (a) frequency, (b) voltage phase angle, and (c) voltage magnitude.\n\n\n\n\\section{Covariance Matrix \\& PCA analysis}\n\\label{sec:PCA}\n\nLet $T\\leq Q$ be the number of measurements for a period of length greater than the parameter $S$. The $N\\times N$ covariance (correlation) matrix of the signal, which is based on the last $T$ measurements, equals, at time $T\\leq t\\leq Q$:\n\\begin{align}\n \\Sigma_0(t;T;m^*(\\cdot))&=\\Bigl[\\ \\frac{1}{T}\\sum_{\\tau=t-T+1}^t m_i^*(\\tau) \\overline{m_j^*(\\tau)}\\ \\Big|\\ \\forall i,j\\ \\Bigr] \\nonumber \\\\\n&= \\frac{1}{T}\\sum_{\\tau=t-T+1}^t m^*(\\tau)m^*(\\tau)^{\\mathsf{H}}, \\label{eq:Sigma}\n\\end{align}\nwhere $m^*(\\cdot)$ could be replaced by $\\bar{m}^{(m)}(\\cdot;\\alpha)$,  $\\hat{m}^{(m)}(\\cdot;\\alpha)$, $\\bar{m}^{(s)}(\\cdot;S)$,  $\\hat{m}^{(s)}(\\cdot;S)$, $\\bar{f}^{(s)}(\\cdot;\\lambda;S)$, or $\\hat{f}^{(s)}(\\cdot;\\lambda;S)$.\n\nWe aim to choose a (fixed) value $T$ that is sufficiently large so that the covariance is weakly dependent on $T$, but also not too large to keep memory as small as possible (with an eye on streaming applications).\nEmpirical experiments show that with $T$ corresponding to the number of measurements in three minutes we can obtain stable covariance matrices when $t$ varies.\n\nIn what follows we will drop the inputs $T$ and $m^*(\\cdot)$ from $\\Sigma_0(t;T;m^*(\\cdot))$ since they become fixed in the following analysis. We perform an eigen-decomposition on the correlation matrices $\\Sigma_0(t)$ and track the results as the function of $t$:\n\\begin{eqnarray}\n\\Sigma_0(t)=\\sum_{k=1}^N \\lambda_k(t) \\xi_k(t)\\xi_k(t)^{\\mathsf{H}}, \n\\label{eq:eigen-dec}\n\\end{eqnarray}\nwhere $\\xi_k(t)$ and $\\lambda_k(t)$ are the the orthonormal eigenvectors and corresponding eigenvalues (in decreasing order) of $\\Sigma_0(t)$ respectively, i.e. \n\\begin{align}\n \\forall k:\\quad &\\Sigma_0(t)\\xi_k(t)=\\lambda_k(t) \\xi_k(t),\\label{eq:eigen1}\\\\\n \\forall k,l:\\quad &\\xi_k(t)^{\\mathsf{H}} \\xi_l(t)=\\delta_{kl},\\label{eq:eigen2}\\\\\n& \\lambda_1(t)\\geq\\lambda_2(t)\\geq\\cdots\\geq\\lambda_N(t)\\geq0,\\label{eq:eigen3}\n\\end{align}\nwhere $\\delta$ is the Kronecker delta function. Since $\\Sigma_0(t)$ is a positive semidefinite matrix, \\eqref{eq:eigen3} is justified.\nIn our tests, $\\Sigma_0(t)$ is (numerically) rank-deficient, thus justifying the Principal Component Analysis (PCA) approach. PCA may be considered exact, if $K<N$ principal components are tracked or approximated. We approximate the sum in Eq.~\\eqref{eq:eigen-dec} replacing,  $N=200$, by $K=10$. \n\n\nThe results of the PCA analysis are presented in Fig.~\\ref{fig:case1_phase} which are screenshots of the movies available in~\\cite{learn_pmu_web}. The screenshots and the movies show at time $t$ for each of the three variables (frequency, voltage angle, and voltage magnitude) five indicators:\n\\begin{itemize}\n    \\item normalized vector $m^*(t)$ of measurements: for each sensor, we plot at its geographical position the value of the normalized data using different colors for different values (column 1);\n    \\item first 40 eigenvalues of $\\Sigma_0(t)$, in decreasing order (column 2);\n    \\item largest 3 eigenvalues $\\lambda_1(\\cdot),\\lambda_2(\\cdot),\\lambda_3(\\cdot)$ for the last minute before $t$ (columns 3 to 5, upper section);\n    \\item corresponding eigenvectors $\\xi_1(t),\\xi_2(t),\\xi_3(t)$ at time $t$, component values of the vector are plotted geographically, in blue values close to $-1$ and in red values close to $1$ (columns 3 to 5, bottom section);\n    \\item auto-correlation function for a selected bus (column 6 in the movie---omitted in Fig.~\\ref{fig:case1_phase}), this concept and figures are introduced in Section~\\ref{sec:auto-correlations}.\n\\end{itemize}\n\nFor the numerical experiments we used traditional methods to compute the eigen-decomposition of $\\Sigma_0(t)$, typically taking $O(N^3)$ time and using $O(NT)$ space. However, for streaming data that falls into a subspace of the original space (like in our case), we can consider lighter and faster algorithms to compute PCA, see \\cite{17SYB}.\n\n\n\\begin{figure*}\n  \\includegraphics[width=\\textwidth]{plot_phase_case1.png}\n  \\caption{Movie screenshot of a selected quiet period at a specific moment $t$, showing the voltage phase angle PCA analysis. Here, $T=3$min. Columns from left to right: normalized phase angle (Eq.~\\eqref{eq:hat-m-s}) at all buses, eigenvalues of $\\Sigma_0(t)$, first, second, and third largest eigenvalues during the last minute (above) together with their corresponding normalized eigenvectors (below).}\n  \\label{fig:case1_phase}\n\\end{figure*}\n\n\nWe also studied how the PCA changes if the data that its received is sparse, that is, from all the measurements that are made by the PMUs just a fraction of them are used in the analysis. \nAs we mentioned above, PMUs report data 30 times per second, we performed PCA analyses with a fifth of that information, i.e. considering only 6 measurements per second and dropping the rest, meaning that each reading is equidistant in time by 0.167 [s].  Fig.~\\ref{fig:case1_phase_sparse} shows the PCA analysis of the same period as Fig.~\\ref{fig:case1_phase} for sparse measurements.\nWe observe that the same spectrum of eigenvalues is obtained, though the later analysis has higher eigenvalues. Moreover, the 3 eigenvalue functions that are shown have similar shape in Figs.~\\ref{fig:case1_phase} and~\\ref{fig:case1_phase_sparse}, as well as the eigen-vectors (omitting the sign change, flipping red and blue colors).\nWith this experiment we confirm what it is reported in previous literature \\cite{18LWC}.\n\n\\begin{figure*}\n  \\includegraphics[width=\\textwidth]{plot_phase_case1_sparse.png}\n  \\caption{Movie screenshot of the same period and same time as Fig.~\\ref{fig:case1_phase}. Here, the PCA analysis was made using only 6 readings per second.}\n  \\label{fig:case1_phase_sparse}\n\\end{figure*}\n\n\n\n\n\n\n\\subsection{Singular Value Decomposition (SVD)}\n\nConsider the $N\\times T$ measurement matrix \n$$M(t)=[\\ M_{i,\\tau}=m_i^*(\\tau)\\ |\\ \\forall i,\\; \\forall\\tau\\in\\{t-T+1,\\ldots,t\\}\\ ].$$ \nThe SVD of $M(t)$ is\n\\begin{eqnarray}\nM(t) = U(t) D(t) W(t)^{\\mathsf{H}}, \\label{eq:SVD}\n\\end{eqnarray}\nwhere the ``spatial\" matrix $U(t)$ is an $N\\times N$ matrix the columns of which are orthogonal unit vectors of length $N$ which are called left singular vectors of $M(t)$, the ``temporal\" matrix, $W(t)$, is $S\\times S$ whose columns are right singular vectors of $M(t)$ and $D(t)$ is the $N\\times S$ rectangular diagonal matrix of positive numbers. The covariance matrix, $\\Sigma_0(t)$, is related to the measurement matrix, $M(t)$, according to \n\\begin{eqnarray}\n\\Sigma_0(t)=M(t) M(t)^{\\mathsf{H}}=U(t) D(t) D(t)^{\\mathsf{H}} U(t)^{\\mathsf{H}}, \\label{eq:PCA-to-SVD}\n\\end{eqnarray}\nwhere we took into account that $W(t)^{\\mathsf{H}} W(t)=I$.\nOne observes that the covariance matrix does not depend on the temporal matrix $W(t)$.  To investigate time-related correlation effects, in the next section we also study  auto-correlations. \n\n\n\n\n\\section{Accounting for Temporal Correlations: Auto-Correlation and Cross-Correlation Residue Maps}\n\\label{sec:temp-correlations}\n\nConsider the delayed covariance matrix generalizing Eq.~\\eqref{eq:Sigma}, \n$\\forall \\Delta\\geq0,\\ \\forall t\\geq T+\\Delta:$\n\\begin{align}\n\\Sigma_\\Delta(t;T;m^*(\\cdot)) &= \\Bigl[\\ \\frac{1}{T}\\sum_{\\tau=t-T+1}^t m^*_i(\\tau) \\overline{m^*_j(\\tau-\\Delta)}\\ \\Big|\\ \\forall i,j\\ \\Bigr] \\nonumber\\\\\n&= \\frac{1}{T} \\sum_{\\tau=t-T+1}^t m^*(\\tau) m^*(\\tau-\\Delta)^{\\mathsf{H}}. \\label{Sigma_tau}\n\\end{align}\nWe are interested to study how $\\Sigma_\\Delta(t;T;m^*(\\cdot))$ changes with $\\Delta$ increase and then track evolution with $t$. \n\nHowever, evolution  of the spectrum (in the two-dimensional $(\\Delta,t)$ space) is challenging.  Instead, we study two surrogate objects, introduced in the following two subsections, which are easier to visualize.\nAgain, since $T$ and $m^*(\\cdot)$ are fixed, we will omit them as input of the $\\Sigma_\\Delta$ function.\n\n\\subsection{Auto-Correlation Functions} \n\\label{sec:auto-correlations}\nThe normalized PMU's auto-correlation functions at different nodes are defined as follows:\n    \\begin{eqnarray}    \n    \\forall i: \\qquad\n    {\\cal A}_i(\\Delta;t)=\\frac{[\\Sigma_\\Delta(t)]_{ii}}{[\\Sigma_0(t)]_{ii}}.\n    \\label{eq:A_i}\n    \\end{eqnarray}\nThis object is of interest because of the following two reasons\n\\begin{itemize}\n\\item Dependence of the auto-correlation function on $\\Delta$ indicates whether fluctuations around the mean at a particular node decay or not with time.  Stated differently this analysis tests if there are significant memory effects or if memory is lost. \n\\item It accounts for the part of the measurement matrix which is ignored in the PCA analysis, as discussed above --that is, it accounts for the temporal matrix, $W$.\n\\end{itemize}\n\nWe show dependence of the auto-correlations on time,  i.e. dependence of ${\\cal A}_i(\\Delta;t)$ on $t$. The movies are available in~\\cite{learn_pmu_web} and example snapshots (of frequency) are given in row (a) of Fig.~\\ref{fig:autocorr_freq_fns}, which shows five auto-correlation functions ${\\cal A}_i(\\cdot;t)$ for 5 selected buses at a specific time $t$. The buses that are shown were chosen between the ones that have larger auto-correlation amplitude. \n\nWe also apply the auto-correlation analysis to the filtered signal.  The results of filtering the row signal, correspondent to the  Fig.~\\ref{fig:autocorr_freq_fns}a, at 5Hz, 4Hz and 2Hz (cutting off higher harmonics) are shown in Fig.~\\ref{fig:autocorr_freq_fns}b-d. We observe that if the signal is filtered at a sufficiently low frequency (4Hz) in this case a significant level of oscillations, seen otherwise in Fig.~\\ref{fig:autocorr_freq_fns}a-b, disappear. \n\n\n\n\nIn order to measure this amplitude and formalize the concept of the \\emph{residue} one studies \n\\begin{eqnarray} \\forall i: \\qquad\n\\rho_i([\\Delta_{min},\\Delta_{max}];t)=\\frac{\\sum_{\\Delta=\\Delta_{min}}^{\\Delta_{max}-1}|{\\cal A}_i(\\Delta;t)|}{\\Delta_{\\max}-\\Delta_{min}},\n\\label{eq:A_i-res}\n\\end{eqnarray}\nwhere $\\Delta_{min}$ was set to $1s$ (in order to ignore the firsts values of ${\\cal A}_i(\\cdot;t)$) and $\\Delta_{max}$ was set to $1min=60s$ in our tests. (We have verified empirically that with this value of $\\Delta_{max}$ the results are robust with $O(1)$ changes in $\\Delta_{max}$.)\n\n\\begin{figure*}\n  \\includegraphics[width=\\textwidth]{plot_freq_autocorr_fns3.png}\n  \\caption{Movie screenshot of the auto-correlation functions for frequency at selected buses at a specific moment $t$. Here, $T=60$sec, and $S=1$sec.\n  Different rows show the result using different time series: {\\bf (a)} $\\hat{m}^{(s)}(\\cdot;S)$,  {\\bf (b)} $\\hat{f}^{(s)}(\\cdot;\\lambda=5Hz;S)$,  {\\bf (c)} $\\hat{f}^{(s)}(\\cdot;\\lambda=4Hz;S)$,  and {\\bf (d)} $\\hat{f}^{(s)}(\\cdot;\\lambda=2Hz;S)$.\n  }\n  \\label{fig:autocorr_freq_fns}\n\\end{figure*}\n\nFig.~\\ref{fig:autocorr_freq_map} geographically shows the residue for each node when different normalized time series are considered.\nAs seen in the movies of the residue maps shown in Fig.~\\ref{fig:autocorr_freq_map}, residue values at a number of special nodes do not decay with time. Moreover we observe that nodes with a significant residue cluster, specifically, there are two sets of nodes that repeat their high values across different days, months and time of the day. Those groups are also shown in Fig.~\\ref{fig:autocorr_freq_map}a: in the bottom left part of the map and the upper right zone (closer to the center); they might show up together or by separate.  \n\n\\begin{figure*}\n\\centering  \n  \\includegraphics[width=\\textwidth]{plot_freq_autocorr_map3.png}\n  \\caption{Movie screenshot of the auto-correlation map for frequency of Fig.~\\ref{fig:autocorr_freq_fns}a-c, that is, using $\\hat{m}^{(s)}(\\cdot;S)$, $\\hat{f}^{(s)}(\\cdot;\\lambda=5Hz;S)$, and $\\hat{f}^{(s)}(\\cdot;\\lambda=4Hz;S)$, respectively. Here, $T=60$sec, $S=1$sec, $\\Delta_{min}=1$sec, and $\\Delta_{max}=60$sec. Color and size of the marks show the residue for each bus. Geometric figures (circle, square, cross, diamond and pentagon) correspond to the position of the buses whose auto-correlation function are shown in Fig.~\\ref{fig:autocorr_freq_fns}.  }\n  \\label{fig:autocorr_freq_map}\n\\end{figure*}\n\nObservation that sustainable correlations stay for sufficiently long period of time suggest to analyze spatio-temporal features of the sustainable correlations via the Cross-Correlation Residue (CCR) described in the next Subsection. \n\n\\subsection{Cross-Correlation Residue (CCR)}\n\nThe cross-correlation version of Eqs.~\\eqref{eq:A_i}-\\eqref{eq:A_i-res} is\n\\begin{align}\n\\forall i,j: \\qquad \\quad\n{\\cal B}_{ij}(\\Delta;t) &=[\\Sigma_\\Delta(t)]_{ij},     \\label{eq:B_ij}\\\\\n{\\cal R}_{ij}([\\Delta_{min},\\Delta_{max}];t)\n &= \\frac{\\sum_{\\Delta=\\Delta_{min}}^{\\Delta_{max}-1} |{\\cal B}_{ij}(\\Delta;t)|}{\\Delta_{max}-\\Delta_{min}},\n\\label{eq:B_ij-res}\n\\end{align}\nwhere we drop normalization to avoid singularities associated with signals at different nodes which are not correlated. To visualize the CCR \\eqref{eq:B_ij-res}, we plot the matrix ${\\cal R}([\\Delta_{min},\\Delta_{max}];t) = \\left[{\\cal R}_{ij}([\\Delta_{min},\\Delta_{max}];t)|\\forall i,j\\right]$ showing in darker colors the higher values, see Fig.~\\ref{fig:residue_freq_matrix}.\n\nThe three plots in Fig.~\\ref{fig:residue_freq_matrix} show the CCR matrix at the same time using different time series, and depending on this, different patterns appear. In Fig.~\\ref{fig:residue_freq_matrix}a 27 buses have in its corresponding row some component with a value higher than 0.4, whereas in Fig.~\\ref{fig:residue_freq_matrix}c only 13 buses and different from the ones in Fig.~\\ref{fig:residue_freq_matrix}a. Fig.~\\ref{fig:residue_freq_matrix}b shows 40 buses with values higher than 0.4, the same as Fig.~\\ref{fig:residue_freq_matrix}a and Fig.~\\ref{fig:residue_freq_matrix}c together.\n\n\n\n\\begin{figure*}\n\\centering  \n  \\includegraphics[width=.9\\textwidth]{plot_freqresidue_matrix.png}\n  \\caption{Cross-correlation matrices of 141 buses using {\\bf (a)} $\\hat{m}^{(s)}(\\cdot;S)$, {\\bf (b)} $\\hat{f}^{(s)}(\\cdot;\\lambda=5Hz;S)$, and {\\bf (c)} $\\hat{f}^{(s)}(\\cdot;\\lambda=4Hz,S)$. Here,  $T=60$sec, $S=1$sec, $\\Delta_{min}=1$sec, and $\\Delta_{max}=60$sec.}\n  \\label{fig:residue_freq_matrix}\n\\end{figure*}\n\nFig.~\\ref{fig:residue_freq_map} shows geographically the value that each bus has in row 86 of Fig.~\\ref{fig:residue_freq_matrix}a and row 19 of Fig.~\\ref{fig:residue_freq_matrix}c. We can see from these plots that high cross-correlation between buses is strongly connected with geographical closeness.\nWe also note that components that have high auto-correlation in Fig.~\\ref{fig:autocorr_freq_map} have also high cross-correlation between themselves.\n\n\\begin{figure}\n\\centering  \n  \\includegraphics[width=.48\\textwidth]{plot_freqresidue_map.png}\n  \\caption{Geographical map with values of {\\bf (a)} row 86 of Fig.~\\ref{fig:residue_freq_matrix}a and {\\bf (b)} row 19 of Fig.~\\ref{fig:residue_freq_matrix}c. Buses corresponding to each row have an extra outer-circle.} \n  \\label{fig:residue_freq_map}\n\\end{figure}\n\n\n\n\\section{Conclusions: What does online PCA and Auto-Correlations show?}\n\\label{sec:conclusions}\n\nIn this Section we summarize ``detective\" methodology developed in the manuscript and the results it has helped to discover analyzing ``quiet periods\" in the OGE data.  We will start with the latter (results) and then present a brief summary for the toolbox of the ``detective\" methods.\n\n\\underline{\\it Highlights of the Results:} \n\\begin{itemize}\n    \\item Averaged over time PMU signal shows interesting spatial correlations. Correlations are different for different objects of interest (frequency, phase and voltage) and also different for different quiet periods. Matrix of correlations is sparse, also revealing that number  of the high-intensity contributions to the correlations is small. (Note that the statement of sparseness is consistent with previous studies of the measurement matrix, see e.g. \\cite{18LWC}.) Each of the contributions characterizes a mode localized on a relatively few nodes (PMU positions) within the system. Principal modes, computed over the quiet periods, are almost frozen in time, however responding fast to any significant perturbation,  thus suggesting them as efficient features\/indicators for changes. \n    \n    \\item Fourier-analysis of the signal reveals interesting spatial patterns. Extracting modes in the 4-6Hz range one observes significant contributions from only few PMU nodes.  The spatial patterns are different for different characteristics (frequency, phase, voltage amplitude) and they are also different for different quiet periods. Nodes showing large 4-6Hz contributions were identified as aggregated loads, mid-size generators and large wind-farms.  \n    See Figures~\\ref{fig:fourier1}-\\ref{fig:fourier6} for Fourier analysis on a selection of nodes.\n    \n    \\item In general the signals are long-correlated in time. However the memory effects becomes significantly less prominent when Fourier filtering at 4Hz, cutting higher frequencies, is applied. Nodes with significant auto-correlations (memory) are relatively few and the respective spatial pattern is adiabatic\/frozen (changing slowly during the quiet periods). Like other adiabatic patterns mentioned above,  the pattern changes from one quiet period to another and we also observed different patterns for different characteristics (frequency, phase, voltage).  \n    \n    \\item Analyzing time-delayed cross-correlations between different nodes we observe that, like in the case of the auto-correlations, correlations between some nodes have long memory. Nodes which mutual inference shows a long memory form a sparse pattern. These patterns, like others described above are adiabatic and evolving from characteristic to characteristic and from one quiet period to another quiet period.\n    \n\n    \n\\end{itemize}\n\n\\underline{\\it\\ Summary of Methods:} \n\\begin{itemize}\n    \\item {\\bf Averaging} -- computation of mean and variance taking into account recent history, exponential moving average and averaging over sliding time horizon methods are used and compared; with these statistics we compute normalized time series.\n    \\item {\\bf Fourier Analysis} -- computation of discrete-time Fourier transform over the samples of an entire quiet period obtaining a new representation of the data, in the frequency domain; we obtain filtered time series by suppressing high frequency components (in the frequency domain) and applying the inverse Fourier transform.\n    \\item {\\bf PCA} -- study of the eigen-decomposition of the covariance (correlation) matrix defined by the normalized time series; we plot how the spectrum (of eigenvalues) changes over time, as well as leading eigenvalues and eigenvectors.\n    \\item {\\bf Auto-Correlation Analysis} -- computation of auto-correlation functions that explain how correlated are intervals of measurements at a node with previous intervals, we define the residue at a node to be related with the amplitude of the auto-correlation function.\n    \\item {\\bf Cross-Correlation Residue} -- same as later, but now the correlation is in between an interval of measurements at a node with previous history at a different node.\n\\end{itemize}\n\n\n\n\\section{Path Forward}\n\\label{sec:path-forward}\n\nIn this manuscript we have developed algorithms that use PMU data for real-time separation of quiet periods from anomalous behavior. We argue that  tracking  algorithms of the type described above, validated on historical data from system wide PMU measurements, will make it possible in the future to extract and interpret hidden equipment malfunction and other extraneous phenomena. For example,  we should be able to pin down sources of forced oscillations at different frequencies (e.g. in the range from 0.5Hz to 5Hz) to malfunction of transformers, generators or large wind farms in less than a minute after  hard-to-identify small signatures of a potentially big future problems appear. We anticipate that accurate identification of the hidden signatures of the malfunctions will also be possible through introduction of learning, based on physical models of the system, in particular of a generalized swing-equation type discussed in \\cite{2017DTCS,2017TDLCS} and possibly their nonlinear versions. The basic tools of statistical analysis, we took advantage of in the manuscript,  will be empowered in the future through injection physical models of the underlying phenomena into the purely statistical description of this manuscript. We also anticipate that fusing parameterized physics-based models for elements of the system we have sufficient amount of information (e.g. generators, transformers, well understood and aggregated traditional loads) with Neural Networks, e.g. of the modern DL type~\\cite{09Ben}-\\cite{16GBC}, for elements and parts of the system which we know much less about (renewable sources, active loads, etc) will be most beneficial in practice.  \n\nOur ``detective'' approach to analysis of the quiet periods, started in the paper, will become even more critical for the next step -- accurate detection, localization and classification of more sever disturbances\/events almost instantaneously (within seconds) after their occurrence. Overall we envision devising modern, physics of power systems informed machine learning methodology based but also going beyond what is the state of the art in the physics-blind and application agnostic IT industry, such as deep learning solutions. To validate the new approaches data-partnership with multiple ISOs and regional grid owners will be imperative for future success. \n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Braid shuffles}\n\nIn this section we collect some necessary information on shuffle elements in the braid group rings.\n\n\\vskip .2cm\nIn the Artin presentation, the braid group $B_{M+1}$ is given by generators \n$\\sigma_i$, $1\\leq i\\leq M$, and relations \n\\begin{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{ryb1}\\sigma_{i}\\sigma_{j}\\sigma_{i}=\\sigma_{j}\\sigma_{i}\\sigma_{j}\n&{\\mathrm{if}}& |i-j|=1\\ ,\\\\[.5em]\n\\sigma_{i}\\sigma_{j}=\\sigma_{j}\\sigma_{i}&{\\mathrm{if}}&|i-j|>1\\ .\\label{braidg}\\end{eqnarray}\nThe inductive limit $B_\\infty =\\displaystyle{\\lim_{\\longrightarrow}}\\, B_M$ is defined by \ninclusions $B_M\\rightarrow B_{M+1}$, \n$B_M\\ni\\sigma_i\\mapsto \\sigma_i\\in B_{M+1}$, $i=1,\\dots ,M-1$. \n\n\\vskip .2cm\nWe denote $w^{\\uparrow\\ell}$, as in \\cite{OP}, the image of an element $w\\in B_\\infty$ under\nthe endomorphism of $B_\\infty$, sending $\\sigma_i$ to $\\sigma_{i+\\ell}$, $i=1,2,\\dots$\n(we keep the same notation for the Hecke and BMW quotients of the braid group rings). \n\n\\vskip .2cm\nBraid shuffle elements $\\Sha_{m,n}$ ($m,n\\in {\\mathbb Z}_{\\geq 0}$) are analogues of the \nbinomial coefficients. The shuffle elements belong to the group ring of $B_{m+n}$ \n(and thereby of $B_\\infty$); they can be defined inductively by any of the recurrence relations \n(braid analogues of the Pascal rule)\n\\begin{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{shufrec}\\Sha_{m,n}^{\\phantom{\\uparrow}}&=&\\Sha_{m-1,n}^{\\phantom{\\uparrow}}+\n\\Sha_{m,n-1}^{\\phantom{\\uparrow}}\\sigma_{m+n-1}\\cdots\\sigma_{n}\\ ,\\\\[.5em]\n\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{shufrec2}\\Sha_{m,n}^{\\phantom{\\uparrow}}&=&\n\\Sha_{m,n-1}^{\\uparrow 1}+\\Sha_{m-1,n}^{\\uparrow 1}\\sigma_{1}\\cdots\\sigma_{n}\\ ,\\end{eqnarray} \ntogether with the boundary conditions $\\Sha_{0,n}=1$ and $\\Sha_{n,0}=1$ for any non-negative\ninteger $n$.\n\n\\vskip .2cm\nLet $\\Sigma_n$ be the lift \\cite{Ma} of the symmetrizer $\\sum_{g\\in S_n}g$ from the symmetric group ring ${\\mathbb Z}{\\mathbb{S}}_n$ to ${\\mathbb Z}B_n$. The element $\\Sigma_n$ is the \nbraid analogue of $n!$; it satisfies\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{symsh}\\Sigma_{m+n}^{\\phantom{\\uparrow}}=\\Sha_{n,m}^{\\phantom{\\uparrow}}\\,\n\\Sigma_m^{\\phantom{\\uparrow}}\\,\\Sigma_n^{\\uparrow m}\\ .\\ee\nUsing the automorphism $\\mathfrak a$ and the anti-automorphism $\\mathfrak b$, \n${\\mathfrak{b}}(xy)={\\mathfrak{b}}(y){\\mathfrak{b}}(x)$,\nof the braid group \n$B_{n+1}$, defined on the generators by\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{aau}\\mathfrak a:\\sigma}\\def\\b{\\beta}\\def\\o{\\omega}\\def\\d{\\partial_i\\mapsto\\sigma}\\def\\b{\\beta}\\def\\o{\\omega}\\def\\d{\\partial_{n+1-i}\\quad ,\\quad\\mathfrak b:\\sigma}\\def\\b{\\beta}\\def\\o{\\omega}\\def\\d{\\partial_i\\mapsto\\sigma}\\def\\b{\\beta}\\def\\o{\\omega}\\def\\d{\\partial_i\\ ,\\ee\nand their composition, one obtains three more decompositions of $\\Sigma$.\n\n\\vskip .2cm\nHigher shuffles (braid analogues of the trinomial {\\em etc.} coefficients) appear in the\nfurther decompositions of the elements $\\Sigma_n$,\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{hsh}\\Sigma_{m+n+k}^{\\phantom{\\uparrow}}=\\left\\{\n\\begin{array}{l}\\Sha_{n+k,m}^{\\phantom{\\uparrow}}\\,\n\\Sigma_m^{\\phantom{\\uparrow}}\\,\\Sigma_{n+k}^{\\uparrow m}\n=\\Sha_{n+k,m}^{\\phantom{\\uparrow}}\\,\\Sha_{k,n}^{\\uparrow m}\\,\\Sigma_m^{\\phantom{\\uparrow}}\n\\Sigma_n^{\\uparrow m}\\,\\Sigma_{k}^{\\uparrow m+n}\\ ,\\\\[1.2em]\n\\Sha_{k,m+n}^{\\phantom{\\uparrow}}\\,\\Sigma_{m+n}^{\\phantom{\\uparrow}}\\,\\Sigma_{k}^{\\uparrow m+n}\n=\\Sha_{k,m+n}^{\\phantom{\\uparrow}}\\,\\Sha_{n,m}^{\\phantom{\\uparrow}}\\,\\Sigma_m^{\\phantom{\\uparrow}}\n\\Sigma_n^{\\uparrow m}\\,\\Sigma_{k}^{\\uparrow m+n}\\ .\\end{array}\\right.\\ee\nDue to the existence of a (one-sided) order on the braid groups \\cite{D}, the braid group rings \n${\\mathbb Z}B_n$ have no zero divisors. Equating the two expressions for $\\Sigma_{m+n+k}$ \nin (\\ref{hsh}) and simplifying, one finds\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{ctsh}\\Sha_{n+k,m}^{\\phantom{\\uparrow}}\\,\\Sha_{k,n}^{\\uparrow m}=\n\\Sha_{k,m+n}^{\\phantom{\\uparrow}}\\,\\Sha_{n,m}^{\\phantom{\\uparrow}}\\ .\\ee\nA direct verification of (\\ref{ctsh}) is a good exercise. Any of the expressions in (\\ref{ctsh})\nis the braid trinomial coefficient $\\Sha_{k,n,m}$. The element $\\Sigma_n$ is the shuffle\n$\\Sha_{1,1,\\dots ,1}$.\n\n\\vskip .2cm\nWe shall later use the following identity\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{lue}\\Sha_{1,n-1}^{\\phantom{\\uparrow}}\\Sha_{1,n-2}^{\\phantom{\\uparrow}}\\dots\n\\Sha_{1,n-k}^{\\phantom{\\uparrow}}=\n\\Sha_{k,n-k}^{\\phantom{\\uparrow}}\\Sigma_{k}^{\\uparrow n-k}\\ ,\\ee\nwhich is verified by induction. For $k=1$ there is nothing to prove. The induction step\nuses (\\ref{ctsh}) and then (\\ref{symsh}):\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{neid}\\begin{array}{rcl}\n\\Sha_{k,n-k}^{\\phantom{\\uparrow}}\\Sigma_{k}^{\\uparrow n-k}\\Sha_{1,n-k-1}^{\\phantom{\\uparrow}}&=&\n\\Sha_{k,n-k}^{\\phantom{\\uparrow}}\\Sha_{1,n-k-1}^{\\phantom{\\uparrow}}\\Sigma_{k}^{\\uparrow n-k}\n\\\\[1em]\n&=&\\Sha_{k+1,n-k-1}^{\\phantom{\\uparrow}}\\Sha_{k,1}^{\\uparrow n-k-1}\\Sigma_{k}^{\\uparrow n-k}\n=\\Sha_{k+1,n-k-1}^{\\phantom{\\uparrow}}\\Sigma_{k+1}^{\\uparrow n-k-1}\n\\ .\\end{array}\\ee\n\n\\vskip .4cm\nShuffle elements find numerous applications in the theories of free Lie algebras, \npolylogarithms and multiple zeta values, Hopf algebras, differential calculus on quantum groups, \nhomology of quantum Lie algebras, braidings of tensor spaces {\\em etc.} \n\\cite{R}, \\cite{BB}, \\cite{N}, \\cite{Andr}, \\cite{Rosso2}, \\cite{Wor}, \\cite{IO}, \\cite{IOG}, \n\\cite{GO}.\n\n\\section{B-shuffle algebras}\n\nIn this section we recall the definition of the Nichols--Woronowicz algebras and construct,\nwith the help of the baxterized elements, another family of graded associative algebras in the \ntensor spaces of local representations of the braid groups.\n\n\\paragraph{1.} Let $V$ be a vector space over a field ${\\mathfrak k}$. For an operator \n$X\\in {\\mathrm{End}}(V^{\\otimes j})$ we denote by the same symbol the operator \n$X\\otimes {\\mathrm{Id}}^{\\otimes l}\\in {\\mathrm{End}}(V^{\\otimes (j+l)})$ for any \n$l\\in {\\mathbb Z}_{\\geq 0}$; $X^{\\uparrow l}$ denotes the operator \n${\\mathrm{Id}}^{\\otimes l}\\otimes X\\in {\\mathrm{End}}(V^{\\otimes (j+l)})$.\n\n\\vskip .2cm\nLet $\\{ {\\cal{T}}_{m,n}\\}_{_{m,n\\in {\\mathbb Z}_{\\geq 0}}}$ be a collection of operators, \n${\\cal{T}}_{m,n}\\in {\\mathrm{End}}(V^{\\otimes (m+n)})$, such that\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{ctshr} {\\cal{T}}_{n+k,m}^{\\phantom{\\uparrow}}\\, {\\cal{T}}_{k,n}^{\\uparrow m}=\n{\\cal{T}}_{k,m+n}^{\\phantom{\\uparrow}}\\, {\\cal{T}}_{n,m}^{\\phantom{\\uparrow}}\n\\qquad\\forall\\ m,n,k\\in {\\mathbb Z}_{\\geq 0}\\ .\\ee\n{}For tensors $u\\in V^{\\otimes m}$ and $v\\in V^{\\otimes n}$ let\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{shal} u\\odot v:={\\cal{T}}_{n,m}(u\\otimes v)\\in V^{\\otimes (m+n)}\\ .\\ee\nDue to (\\ref{ctshr}), the space $\\bigoplus_j V^{\\otimes j}$ with the composition law $\\odot$\nis an associative graded algebra. Assume, in addition, that \n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{ico} {\\cal{T}}_{m,0}={\\mathrm{Id}}\\quad {\\mathrm{and}}\\quad\n{\\cal{T}}_{0,m}={\\mathrm{Id}} \\qquad\\forall\\ m\\in {\\mathbb Z}_{\\geq 0}\\ee \n(then $1\\in {\\mathfrak k}\\equiv V^{\\otimes 0}$ is the identity element of the algebra). \nBy (\\ref{ctshr}) and (\\ref{ico}), the following collection \n$\\{ {\\cal{S}}_m\\}_{_{m\\in {\\mathbb Z}_{\\geq 0}}}$ of operators,\n${\\cal{S}}_m\\in {\\mathrm{End}}(V^{\\otimes m})$,\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{symshr} {\\cal{S}}_0={\\mathrm{Id}}\\quad ,\\quad {\\cal{S}}_1={\\mathrm{Id}}\\quad ,\\quad \n{\\cal{S}}_{m+n}^{\\phantom{\\uparrow}}={\\cal{T}}_{n,m}^{\\phantom{\\uparrow}}\\,\n{\\cal{S}}_m^{\\phantom{\\uparrow}}\\, {\\cal{S}}_n^{\\uparrow m}\n\\qquad\\forall\\ m,n\\in {\\mathbb Z}_{\\geq 0}\\ ,\\ee\nis well defined. The operation $\\odot$ restricts on $\\bigoplus_j {\\mathrm{Im}}({\\cal{S}}_j)$,\nthe direct sum of images of the operators ${\\cal{S}}_j$, making it an associative graded algebra\nas well.\n\n\\vskip .2cm\nLet $\\hat{R}\\in {\\mathrm{End}}(V\\otimes V)$ be a solution of the Yang-Baxter equation, that is, \n$\\hat{R}\\hat{R}^{\\uparrow 1}\\hat{R}=\\hat{R}^{\\uparrow 1}\\hat{R}\\hat{R}^{\\uparrow 1}$.\nDenote by $\\rho_{\\hat{R}}$ the corresponding local representation of the braid groups $B_n$,\n$\\rho_{\\hat{R}}(\\sigma_i):=\\hat{R}^{\\uparrow (i-1)}$. Then the collection \n${\\cal{T}}_{m,n}:=\\rho_{\\hat{R}}(\\Sha_{m,n})$ obeys (\\ref{ctshr}) and (\\ref{ico}). The space\n$\\bigoplus_j {\\mathrm{Im}}\\rho_{\\hat{R}}(\\Sigma_j)$ with the composition law $\\odot$ is called\nNichols--Woronowicz algebra.\n\n\\paragraph{2.}\nThe braid group rings admit a family of automorphisms\n$\\sigma_i\\mapsto t\\,\\sigma_i$, where $t\\in {\\mathfrak k}^*$ is an arbitrary\nparameter. The formal limits $\\displaystyle{\\lim_{t\\rightarrow 0}}$ (the lowest power in $t$)\nand $\\displaystyle{\\lim_{t\\rightarrow \\infty}}$ (the highest power in $t$) of the elements\n$\\Sigma_m$, $\\Sha_{m,n}$ and the operation $\\odot$ are well defined. For $t\\rightarrow 0$\nwe obtain the usual tensor algebra, while for $t\\rightarrow \\infty$ the element \n$\\Sigma_{n+1}$ becomes the lift of the longest element of the symmetric group \n${\\mathbb{S}}_{n+1}$ to $B_{n+1}$,\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{ees4}\\bar{\\Sigma}_{n+1}=(\\sigma_1\\sigma_2\\dots\\sigma_n)(\\sigma_1\\dots\\sigma_{n-1})\n\\dots (\\sigma_1)\\ .\\ee\nThe shuffle elements, in the limit $\\displaystyle{\\lim_{t\\rightarrow \\infty}}$, become the\nelements $\\bar{\\Sha}_{m,n}$ which, in a representation in a vector space $V$, equip\nthe tensor powers of $V$ with the standard braidings; the recurrency relations (\\ref{shufrec}) \nand (\\ref{shufrec2}) take the multiplicative form for $\\bar{\\Sha}_{m,n}$,\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{muh1}\\bar{\\Sha}_{m,n}=\\bar{\\Sha}_{m,n-1}\\sigma_{m+n-1}\\dots\\sigma_n\\ ,\\ \n\\bar{\\Sha}_{m,n}=\\bar{\\Sha}_{m-1,n}^{\\uparrow 1}\\sigma_1\\dots\\sigma_n\\ .\\ee\nExplicitly,\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\bar{\\Sha}_{m,n}=\\left\\{\\begin{array}{l}\\Bigl(\\sigma_m\\sigma_{m+1}\\dots\\sigma_{m+n-1}\\Bigr)\n\\Bigl(\\sigma_{m-1}\\sigma_{m}\\dots\\sigma_{m+n-2}\\Bigr)\\Bigl(\\sigma_1\\sigma_2\\dots\\sigma_n\n\\Bigr)\\ ,\\\\[1em]\n\\Bigl(\\sigma_m\\sigma_{m-1}\\dots\\sigma_1\\Bigr)\\Bigl(\\sigma_{m+1}\\sigma_m\\dots\\sigma_2\\Bigr)\n\\Bigl(\\sigma_{m+n-1}\\sigma_{m+n-2}\\dots\\sigma_n\\Bigr)\\ .\\end{array}\\right.\\ee\nIn addition to (\\ref{muh1}), the elements $\\bar{\\Sha}_{m,n}$ satisfy\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{muh2}\\bar{\\Sha}_{m,n}=\\sigma_m\\dots\\sigma_1\\bar{\\Sha}_{m,n-1}^{\\uparrow 1}\\ ,\\ \n\\bar{\\Sha}_{m,n}=\\sigma_m\\dots\\sigma_{m+n-1}\\bar{\\Sha}_{m-1,n}\\ .\\ee\n\n\\paragraph{3.}\nIn this section we shall construct another collection ${\\cal{T}}_{m,n}$ starting with \nthe elements $\\sigma_k(x,y)$, satisfying the Yang-Baxter equation with spectral parameters \n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{sh1}\\sigma_k(x_{k+1},x_{k+2})\\sigma_{k+1}(x_k,x_{k+2})\\sigma_k(x_k,x_{k+1})\n=\\sigma_{k+1}(x_k,x_{k+1})\\sigma_k(x_k, x_{k+2})\\sigma_{k+1}(x_{k+1},x_{k+2})\\ee\nand the locality condition\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{sh2}\\sigma_k(x_{k},x_{k+1})\\sigma_{l}(x_l,x_{l+1})=\n\\sigma_{l}(x_l,x_{l+1})\\sigma_k(x_{k},x_{k+1})\\qquad {\\mathrm{if}}\\ \\ |k-l| > 1\\ .\\ee\nHere $x_k$ are variables (spectral parameters). Depending on the situation, the elements \n$\\sigma_k(x,y)$ can live in certain quotients of the braid group rings or be realized as operators.\nWe shall call $\\sigma_k(x,y)$ baxterized elements (usually the term \"baxterized\" is applied\nwhen $\\sigma(x,y)$ is a function of the solution $\\sigma$ of the constant Yang-Baxter \nequation).\n\n\\vskip .2cm\nLet $\\pi_k$ be the operator which permutes the variables $x_k$ and $x_{k+1}$,\n$$\\pi_kf(\\dots,x_k,x_{k+1},\\dots)=f(\\dots,x_{k+1},x_k,\\dots)\\pi_k\\ .$$\nRelations (\\ref{sh1}) and (\\ref{sh2}) acquire the braid form (\\ref{ryb1}) and (\\ref{braidg})\nfor the elements\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{sh3}\\underline{\\sigma}_k:=\\pi_k\\sigma_k(x_k,x_{k+1})\\ .\\ee\nThe unitarity condition $\\sigma_k(x_{k},x_{k+1})\\sigma_k(x_{k+1},x_{k})=1$ (if imposed)\nfor the baxterized elements takes the form $\\underline{\\sigma}_k^2 =1$ for the elements \n(\\ref{sh3}).\n\n\\vskip .2cm\nThe operators $\\pi_k$ obey the braid group relations; prepare the elements \n$\\bar{\\Sha}_{m,n}\\{\\pi\\}$ and $\\bar{\\Sigma}_m\\{\\pi\\}$ from $\\pi$'s; the elements \n$\\bar{\\Sha}_{m,n}\\{\\underline{\\sigma}\\}$ and $\\bar{\\Sigma}_m\\{\\underline{\\sigma}\\}$ built from \n$\\underline{\\sigma}$'s can be written, after moving all $\\pi$'s to the left, in the form\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{preti}\\bar{\\Sha}_{m,n}\\{\\underline{\\sigma}\\} =\n\\bar{\\Sha}_{m,n}\\{\\pi\\}\\widetilde{\\Sha}_{m,n}(x_1,\\dots,x_{m+n})\\quad ,\\quad\n\\bar{\\Sigma}_m\\{\\underline{\\sigma}\\} =\n\\bar{\\Sigma}_m\\{\\pi\\}\\widetilde{\\Sigma}_m(x_1,\\dots,x_m)\\ ,\\ee\nwhere \n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{basha}\\begin{array}{ccl}\\widetilde{\\Sha}_{m,n}(x_1,\\dots,x_{m+n})&=&\n\\Bigl(\\sigma_m(x_1,x_{m+n})\\sigma_{m+1}(x_2,x_{m+n})\n\\dots\\sigma_{m+n-1}(x_n,x_{m+n})\\Bigr)\\\\[1em]\n&\\cdot&\\Bigl(\\sigma_{m-1}(x_1,x_{m+n-1})\\sigma_{m}(x_2,x_{m+n-1})\n\\dots\\sigma_{m+n-2}(x_n,x_{m+n-1})\\Bigr)\\\\[1em]\n&\\dots&\\Bigl(\\sigma_1(x_1,x_{n+1})\\sigma_2(x_2,x_{n+1})\n\\dots\\sigma_n(x_n,x_{n+1})\\Bigr)\\end{array}\\ee\nand \n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{basi}\\begin{array}{ccl}\\widetilde{\\Sigma}_m(x_1,\\dots,x_m)&=&\n\\Bigl(\\sigma_1(x_{m-1},x_m)\\sigma_2(x_{m-2},x_m)\\dots\\sigma_{m-1}(x_1,x_m)\\Bigr)\\\\[1em]\n&\\cdot&\\Bigl(\\sigma_1(x_{m-2},x_{m-1})\\sigma_2(x_{m-3},x_{m-1})\n\\dots\\sigma_{m-2}(x_1,x_{m-1})\\Bigr)\\dots\\Bigl(\\sigma_1(x_1,x_2)\\Bigr)\\ .\\end{array}\\ee\n\nThe elements $\\bar{\\Sha}_{m,n}\\{\\pi\\}$ and $\\bar{\\Sigma}_n\\{\\pi\\}$ are invertible and obey the \nrelations (\\ref{symsh}), (\\ref{ctsh}), (\\ref{muh1}) and (\\ref{muh2}); substituting (\\ref{preti}) \ninto (\\ref{symsh}), (\\ref{ctsh}), (\\ref{muh1}) and (\\ref{muh2}), moving all $\\pi$'s to the left\nand simplifying, we find relations for $\\widetilde{\\Sigma}$'s and $\\widetilde{\\Sha}$'s alone.\nThe relations (\\ref{muh1}) and (\\ref{muh2}) take the form\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\!\\widetilde{\\Sha}_{m,n}(x_1,\\dots,x_{m+n})\\!=\\!\\left\\{\\begin{array}{l}\\!\\!\\!\n\\widetilde{\\Sha}_{m,n-1}(\\hat{x}_n)\n\\sigma_{m+n-1}(x_n,x_{m+n})\\sigma_{m+n-2}(x_n,x_{m+n-1})\\!\\dots\\!\\sigma_n(x_n,x_{n+1}),\\\\[1em]\n\\!\\!\\!\\widetilde{\\Sha}_{m-1,n}^{\\uparrow 1}(\\hat{x}_{n+1})\\sigma_1(x_1,x_{n+1})\n\\sigma_2(x_2,x_{n+1})\\!\\dots\\!\\sigma_n(x_n,x_{n+1}),\\\\[1em]\n\\!\\!\\!\\sigma_m(x_1,x_{m+n})\\sigma_{m-1}(x_1,x_{m+n-1})\\!\n\\dots\\!\\sigma_1(x_1,x_{n+1})\\widetilde{\\Sha}_{m,n-1}^{\\uparrow 1}(\\hat{x}_1),\\\\[1em]\n\\!\\!\\!\\sigma_m(x_1,x_{m+n})\\sigma_{m+1}(x_2,x_{m+n})\\!\\dots\\!\\sigma_{m+n-1}(x_n,x_{m+n})\n\\widetilde{\\Sha}_{m-1,n}(\\hat{x}_{m+n}\\}),\\end{array}\\right.\\ee\nwhere \"$\\hat{x}_j$\" means that the argument $x_j$ is omitted. For a set \n$\\overrightarrow{x}=\\{ x_1,\\dots,x_n\\}$ of arguments, let $\\overleftarrow{x}:=\\{ x_n,\\dots,x_1\\}$\nbe the reversed set. The relation (\\ref{symsh}) becomes:\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{wtsi}\\widetilde{\\Sigma}_{m+n}^{\\phantom{\\uparrow}}(\\overrightarrow{x},\\overrightarrow{y})\n=\\widetilde{\\Sha}_{n,m}^{\\phantom{\\uparrow}}(\\overleftarrow{x},\\overleftarrow{y})\\,\n\\widetilde{\\Sigma}_m^{\\phantom{\\uparrow}}(\\overrightarrow{x})\\,\n\\widetilde{\\Sigma}_n^{\\uparrow a}(\\overrightarrow{y})\\ ,\\ee\nwhere $\\overrightarrow{x}=\\{ x_1,\\dots,x_m\\}$ and $\\overrightarrow{y}=\\{ y_1,\\dots,y_n\\}$;\nthe relation (\\ref{ctsh}) becomes\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\widetilde{\\Sha}_{n+k,m}^{\\phantom{\\uparrow}}\n(\\overrightarrow{x},\\overrightarrow{z},\\overrightarrow{y})\n\\widetilde{\\Sha}_{k,n}^{\\uparrow m}(\\overrightarrow{y},\\overrightarrow{z})=\n\\widetilde{\\Sha}_{k,m+n}^{\\phantom{\\uparrow}}\n(\\overrightarrow{y},\\overrightarrow{x},\\overrightarrow{z})\n\\widetilde{\\Sha}_{n,m}^{\\phantom{\\uparrow}}(\\overrightarrow{x},\\overrightarrow{y})\\ ,\\ee\nwhere $\\overrightarrow{x}=\\{ x_1,\\dots,x_m\\}$, $\\overrightarrow{y}=\\{ y_1,\\dots,y_n\\}$\nand $\\overrightarrow{z}=\\{ z_1,\\dots,z_k\\}$.\n\n\\vskip .2cm\nAfter the removal of all $\\pi$'s, one can give values to the spectral variables. Each\n$\\widetilde{\\Sigma}_m$ can be evaluated on its own sequence \n$\\vec{x}^{(m)}=(x_1^{(m)},\\dots,x_m^{(m)})$. In the relation (\\ref{wtsi}), the \nbeginning of the sequence for $\\widetilde{\\Sigma}_{m+n}$ becomes the beginning of\nthe sequence for $\\widetilde{\\Sigma}_m$ while its end becomes the beginning of the\nsequence for $\\widetilde{\\Sigma}_n$. This is a strong restriction; if it is imposed on the\nsequences themselves, the general solution is that each $x_j^{(m)}$ is equal to one and the same \nnumber. However, assume that the baxterization is \"trigonometric\", the baxterized elements depend \non the ratio of the spectral parameters, $\\sigma(x,y)=\\sigma(x\/y)$.  The Yang-Baxter equation then \nreads\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{ybeH}\\sigma_n(x)\\sigma_{n-1}(xy)\\sigma_n(y)=\\sigma_{n-1}(y)\\sigma_n(xy)\\sigma_{n-1}(x)\\ .\n\\ee\nNow $\\widetilde{\\Sigma}_m(\\vec{x}^{(m)})=\\widetilde{\\Sigma}_m(\\alpha\\vec{x}^{(m)})$ for an arbitrary constant $\\alpha\\neq 0$ and the general solution of the restrictions imposed\nby (\\ref{wtsi}) for the projectivized sequences is $(x_1^{(m)},\\dots,x_m^{(m)})=(1,s^{-1},s^{-2},\\dots,s^{1-m})$, the \ngeometric progression. Denote $\\widetilde{\\Sigma}_m(1,s^{-1},s^{-2},\\dots,s^{1-m})$ by\n$^{^s}\\!\\Sigma_m$ and \n$\\widetilde{\\Sha}_{m,n}(s^{1-n},\\dots,1,s^{1-m-n},\\dots,s^{-n})$ by \n$^{^s}\\!\\Sha_{m,n}$. Explicitly:\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{basi2} ^{^s}\\!\\Sigma_m=\n\\Bigl(\\sigma_1(s)\\sigma_2(s^2)\\dots\\sigma_{m-1}(s^{m-1})\\Bigr)\n\\Bigl(\\sigma_1(s)\\sigma_2(s^2)\\dots\\sigma_{m-2}(s^{m-2})\\Bigr)\\dots\\Bigl(\\sigma_1(s)\\Bigr)\n\\ee\nand\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{basha2} ^{^s}\\!\\Sha_{m,n}=\\Bigl(\\sigma_m(s)\\dots\\sigma_{m+n-1}(s^n)\\Bigr)\n\\Bigl(\\sigma_{m-1}(s^2)\\dots\\sigma_{m+n-2}(s^{n+1})\\Bigr)\n\\Bigl(\\sigma_1(s^m)\\dots\\sigma_n(s^{m+n-1})\\Bigr)\\ .\\ee\n\nThe elements $^{^s}\\!\\Sha_{m,n}$ obey the relation (\\ref{ctsh}). Therefore, in a local\nrepresentation $\\rho$, the collection ${\\cal{T}}_{m,n}:=\\rho_{\\hat{R}}(^{^s}\\!\\Sha_{m,n})$ \nobeys (\\ref{ctshr}) and (\\ref{ico}) and defines a one parameter family of graded associative \nalgebras on $\\bigoplus_j V^{\\otimes j}$ together with the subalgebras on \n$\\bigoplus_j {\\mathrm{Im}}({\\cal{S}}_j)$ (now ${\\cal{S}}_m=\\rho_{\\hat{R}}(^{^s}\\!\\Sigma_m)$), \nwhich we propose to call {\\em b-shuffle algebras} (\"b\" from \"baxterized\"; maybe the \nterm \"buffle\" would be an apt acronym). \n\n\\vskip .2cm\nIt is known that the element $\\bar{\\Sigma}$ admits reduced expressions starting (or ending) \nwith $\\sigma_j$ for every $j=1,\\dots,m-1$. In particular, $\\bar{\\Sigma}\\{\\underline{\\sigma}\\}$ \ncan start (or end) with every $\\underline{\\sigma}_j$. It follows that  \n$\\widetilde{\\Sigma}_m(x_1,\\dots,x_m)$ can start (or end) with $\\sigma_j(x_j,x_{j+1})$ for\nevery $j$. We shall use this for the trigonometric $\\sigma$'s:\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{retr} ^{^s}\\!\\Sigma_m\\ \\ {\\mathrm{has\\ a\\ reduced\\ expression\\ of\\ the\\ form}}\\ \\sigma_j(s)\\cdot(\\dots)\n\\ {\\mathrm{or}}\\ (\\dots)\\cdot\\sigma_j(s)\\ \\ \\forall\\ j=1,\\dots,m-1\\ .\\ee\n\n\\vskip .2cm\nThe baxterization is known for the Hecke and BMW quotients of the braid group rings;\nit is trigonometric. In the next section we discuss the baxterized collections for these quotients.\n\n\\paragraph {4. Remarks.} {\\bf (a)} We suggest another natural source for collections ${\\cal{T}}_{m,n}$ satisfying (\\ref{ctshr}) and (\\ref{ico}). \n\n\\vskip .2cm\nLet ${\\cal{A}}$ be a Hopf algebra. Assume that ${\\cal{A}}$ admits a twist ${\\cal{F}}$, that is, \nan element ${\\cal{F}}\\in {\\cal{A}}\\otimes {\\cal{A}}$ which satisfies \n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{bai} {\\cal{F}}\\cdot (\\Delta\\otimes {\\mathrm{Id}})({\\cal{F}})=\n{\\cal{F}}^{\\uparrow 1}\\cdot ({\\mathrm{Id}}\\otimes\\Delta)({\\cal{F}})\\ .\\ee \nHere $\\Delta$ is the coproduct and $^\\uparrow$ is the shift in the copies of ${\\cal{A}}$ in \n${\\cal{A}}^{\\otimes j}$. Define ${\\cal{F}}_{m,0}:=1$, ${\\cal{F}}_{0,m}:=1$,\n$m\\in {\\mathbb Z}_{\\geq 0}$, and\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray} {\\cal{F}}_{m,n}:=\\Delta^{m-1}\\otimes\\Delta^{n-1}({\\cal{F}})\\qquad ,\\ \\  \nm,n\\in {\\mathbb Z}_{\\geq 1}\\ .\\ee\nIt is straightforward to verify that \n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{utw}{\\cal{F}}_{k,m}^{\\phantom{\\uparrow}}{\\cal{F}}_{m+k,n}^{\\phantom{\\uparrow}}\n={\\cal{F}}_{m,n}^{\\uparrow k}{\\cal{F}}_{k,m+n}^{\\phantom{\\uparrow}}\\ee\n(for $m=n=k=1$ this is (\\ref{bai}); by induction, ${\\mathrm{Id}}^{i-1}\\otimes\\Delta\n\\otimes {\\mathrm{Id}}^{m+n+k-i}$ increases $k$ by 1 for $1\\leq i\\leq k$, \n$m$ by 1 for $k<i\\leq m+k$ and $n$ by 1 for $m+k<i\\leq m+n+k$).\n\n\\vskip .2cm\nTherefore, given a representation $\\rho$ of ${\\cal{A}}$, the relations (\\ref{ctshr}) and \n(\\ref{ico}) hold for \n$${\\cal{T}}_{m,n}:=\\varpi\\circ\\rho^{\\otimes (m+n)}({\\cal{F}}_{n,m})\\ ,$$\nwhere $\\varpi$ is any operation which reverses the order of terms in both sides of (\\ref{utw})\n(it can be a transposition or, if ${\\cal{F}}_{m,n}$ are invertible for all $m$ and $n$, an inversion).\n\n\\vskip .2cm\nIt might be of interest to investigate this type of collections ${\\cal{T}}_{m,n}$\nfor the twists \\cite{IOt} corresponding to Belavin--Drinfeld triples.\n\n\\paragraph{(b)} Assume, in addition, that ${\\cal{F}}$ satisfies\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray} (\\Delta\\otimes {\\mathrm{Id}})({\\cal{F}})={\\cal{F}}_{\\{1,3\\}}{\\cal{F}}_{\\{2,3\\}}\\ ,\\ \n({\\mathrm{Id}}\\otimes\\Delta)({\\cal{F}})={\\cal{F}}_{\\{1,3\\}}{\\cal{F}}_{\\{1,2\\}}\\ ,\\ee\nwhere ${\\cal{F}}_{\\{i,j\\}}$ is the element ${\\cal{F}}$ located in the copies number $i$ and $j$\nin ${\\cal{A}}\\otimes {\\cal{A}}\\otimes\\dots$ ; for example, for a quasi-triangular Hopf algebra, \n${\\cal{F}}$ can be the universal $R$-matrix. Then\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{fumn} {\\cal{F}}_{m,n}=\\Bigl({\\cal{F}}_{\\{1,m+n\\}}\\dots {\\cal{F}}_{\\{m,m+n\\}}\n\\Bigr)\\Bigl({\\cal{F}}_{\\{1,m+n-1\\}}\\dots {\\cal{F}}_{\\{m,m+n-1\\}}\\Bigr)\\dots\n\\Bigl({\\cal{F}}_{\\{1,m+1\\}}\\dots {\\cal{F}}_{\\{m,m+1\\}}\\Bigr)\\ee\n(in each bracket the first index increases from 1 to $m$, the second one is constant); this \nformula generalizes the formula $\\Delta\\otimes\\Delta ({\\cal{R}})={\\cal{R}}_{\\{1,4\\}}\n{\\cal{R}}_{\\{2,4\\}}{\\cal{R}}_{\\{1,3\\}}{\\cal{R}}_{\\{2,3\\}}$ used in the theory of \nquasi-triangular Hopf algebras for establishing properties of the element giving the square of the\nantipode by conjugation, see, e.g., \\cite{O}, chapter 4. It follows from (\\ref{fumn}) that\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{inuf}{\\cal{F}}_{m,n}=\\Bigl({\\cal{F}}_{\\{1,m+n\\}} {\\cal{F}}_{\\{1,m+n-1\\}}\\dots \n{\\cal{F}}_{\\{1,m+1\\}}\\Bigr) {\\cal{F}}_{m-1,n}^{\\uparrow 1}\\ .\\ee\nGiven a representation $\\rho$ of ${\\cal{A}}$ on a vector space $V$, let $P_i$ be the flip operator\nin the copies number $i$ and $i+1$ of the space $V$ in $V\\otimes V\\otimes\\dots$ ; let \n$F:=\\rho^{\\otimes 2}({\\cal{F}})$ and $\\hat{F}:=P_1F$; for an operator \n$X\\in {\\mathrm{End}}(V\\otimes V)$ denote by $X_{\\{i,j\\}}$ the operator $X$ acting in the\ncopies number $i$ and $j$ of the space $V$ in $V\\otimes V\\otimes\\dots$ and let \n$X_i:=X_{\\{i,i+1\\}}$. Then\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{unf}\\rho^{\\otimes (m+n)}({\\cal{F}}_{m,n})=\n\\bar{\\Sha}_{m,n}\\{ P\\}\\bar{\\Sha}_{n,m}\\{\\hat{F}\\}\\ ,\\ee\nwhere $\\bar{\\Sha}_{m,n}\\{ P\\}$ are built from $P$'s and $\\bar{\\Sha}_{n,m}\\{\\hat{F}\\}$ from \n$\\hat{F}$'s. Indeed, by (\\ref{inuf}) and induction,\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\begin{array}{l} \\rho^{\\otimes (m+n)}({\\cal{F}}_{m,n})=\\Bigl(F_{\\{1,m+n\\}}\\dots F_{\\{1,m+1\\}}\n\\Bigr)\\bar{\\Sha}_{m-1,n}^{\\uparrow 1}\\{ P\\}\\bar{\\Sha}_{n,m-1}^{\\uparrow 1}\\{\\hat{F}\\}\\\\[1em]\n\\hspace{1cm}\\! =\\!\\Bigl(P_{\\{1,m+n\\}}\\dots P_{\\{1,m+1\\}}\\Bigr)\\!\\Bigl({\\hat{F}}_{m+n-1} \n{\\hat{F}}_{m+n-2}\\dots {\\hat{F}}_{m+1}{\\hat{F}}_{\\{1,m+1\\}}\\Bigr)\\!\n\\bar{\\Sha}_{m-1,n}^{\\uparrow 1}\\{ P\\}\\bar{\\Sha}_{n,m-1}^{\\uparrow 1}\\{\\hat{F}\\}\\ .\\end{array}\\ee\nUse now \n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray} \\Bigl({\\hat{F}}_{m+n-1} {\\hat{F}}_{m+n-2}\\dots  \n{\\hat{F}}_{m+1}{\\hat{F}}_{\\{1,m+1\\}}\\Bigr)\\bar{\\Sha}_{m-1,n}^{\\uparrow 1}\\{ P\\}=\n\\bar{\\Sha}_{m-1,n}^{\\uparrow 1}\\{ P\\}\\Bigl({\\hat{F}}_n {\\hat{F}}_{n-1}\\dots  \n{\\hat{F}}_{1}\\Bigr)\\ ,\\ee\nthe first recursion relelation in (\\ref{muh2}) for $\\bar{\\Sha}_{n,m}\\{\\hat{F}\\}$ and\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\begin{array}{l} \\Bigl(P_{\\{1,m+n\\}}\\dots P_{\\{1,m+1\\}}\\Bigr)\n\\bar{\\Sha}_{m-1,n}^{\\uparrow 1}\\{ P\\} =\\bar{\\Sha}_{m-1,n}^{\\uparrow 1}\\{ P\\}\n\\Bigl(P_{\\{1,n+1\\}}P_{\\{1,n\\}}\\dots P_{\\{1,2\\}}\\Bigr)\\\\[1em]\n\\hspace{2cm}=\\bar{\\Sha}_{m-1,n}^{\\uparrow 1}\\{ P\\}\\Bigl(P_1P_2\\dots P_n\\Bigr)\n=\\bar{\\Sha}_{m,n}\\{ P\\}\\end{array}\\ee\n(by the second recurrency relation in (\\ref{muh1}) for $\\bar{\\Sha}_{m,n}\\{ P\\}$) to finish the \nproof of (\\ref{unf}).\n\n\\vskip .2cm\nThus the elements ${\\cal{F}}_{m,n}$ can be regarded as the universal (in the Hopf\nalgebra theoretical sense) counterpart of the elements $\\bar{\\Sha}_{m,n}$.\n\n\\paragraph{(c)} We describe an operation which transforms a collection ${\\cal{T}}_{m,n}$ \nsatisfying (\\ref{ctshr}) and (\\ref{ico}) into another, \"dual\", collection $\\check{\\cal{T}}_{m,n}$\nsatisfying (\\ref{ctshr}) and (\\ref{ico}). \n\n\\vskip .2cm\nKeep the notation from the previous remark. Let $X\\in {\\mathrm{End}}(V^{\\otimes m})$ and $Y\\in {\\mathrm{End}}(V^{\\otimes n})$ be two \noperators. Then\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{pesha}\\bar{\\Sha}_{m,n}\\{ P\\}X^{\\uparrow n}Y=XY^{\\uparrow m}\\bar{\\Sha}_{m,n}\\{ P\\}\\ .\\ee\nDefine $\\check{\\cal{T}}_{m,n}$ by\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{dedu} {\\cal{T}}_{m,n}:=\\check{\\cal{T}}_{n,m}\\bar{\\Sha}_{m,n}\\{ P\\}\\ \\ {\\mathrm{or}}\\ \\ \n\\check{\\cal{T}}_{m,n}:={\\cal{T}}_{n,m}\\bar{\\Sha}_{m,n}\\{ P\\}\\ .\\ee\nThe equivalence of two definitions follows from\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray} \\bar{\\Sha}_{m,n}\\{ P\\}^{-1}=\\bar{\\Sha}_{n,m}\\{ P\\}\\ .\\ee\n\nThe relation (\\ref{ico}) is satisfied for the collection $\\check{\\cal{T}}_{m,n}$. The\nrelation (\\ref{ctshr}) reads, by (\\ref{pesha}),\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray} \\check{\\cal{T}}_{m,n+k}\\check{\\cal{T}}_{n,k}\\bar{\\Sha}_{n+k,m}\\{ P\\}\n\\bar{\\Sha}_{k,n}^{\\uparrow m}\\{ P\\}=\\check{\\cal{T}}_{m+n,k}\n\\check{\\cal{T}}_{m,n}^{\\uparrow k}\\bar{\\Sha}_{k,m+n}\\{ P\\}\\bar{\\Sha}_{n,m}\\{ P\\}\\ .\\ee\nSince\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\bar{\\Sha}_{n+k,m}\\{ P\\}\\bar{\\Sha}_{k,n}^{\\uparrow m}\\{ P\\}\n=\\bar{\\Sha}_{k,m+n}\\{ P\\}\\bar{\\Sha}_{n,m}\\{ P\\}\\ee\nit follows that the relation (\\ref{ctshr}) is as well satisfied for the collection \n$\\check{\\cal{T}}_{m,n}$.\n\n\\vskip .2cm\nWith the help of the identity $ \\bar{\\Sigma}_m\\{ P\\}^2={\\mathrm{Id}}$, it is straightforward \nto verify that the collection $\\check{\\cal{S}}_m$ for $\\check{\\cal{T}}_{m,n}$ is given by\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray} \\check{\\cal{S}}_m={\\cal{S}}_m\\bar{\\Sigma}_m\\{ P\\}\\ .\\ee\n\n\\section{Hecke and BMW algebras\\vspace{.25cm}}\n\n\\vspace{-.2cm}\nIn the sequel we call the elements $\\Sha_{m,n}$ additive shuffles and $^{^s}\\!\\Sha_{m,n}$\nmultiplicative shuffles. In this section we derive the sequences of the (anti-)symmetrizers \nfor the Hecke and BMW algebras with the help of the multiplicative shuffles. We compare the\nmultiplicative versions with known expressions for the (anti-)symmetrizers. \n\n\\vskip .2cm\nWe derive a new expression for the (anti-)symmetrizers in terms of the highest multiplicative \n1-shuffles alone and, for the Hecke algebras, in terms of the highest additive 1-shuffles alone.\n\n\\vskip .2cm\nIn principle, the Hecke algebras can be considered as quotients of the BMW algebras\nand many formulas for the Hecke algebras can be obtained from this point of view. \nBecause of importance of the Hecke algebras we prefer however to treat them separately.\n\n\\subsection{Hecke algebras}\n\n\\paragraph{1.} The tower of the $A$-Type Hecke algebras $H_{M+1}(q)$ (see {\\em e.g.} \\cite{Jon1} \nand references therein) depends on a parameter $q\\in {\\mathfrak{k}}^*$; the algebra  $H_{M+1}(q)$ \nis the quotient of the braid group ring ${\\mathfrak{k}}B_{M+1}$ by\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{ahecke}\\sigma^2_i=(q-q^{-1})\\sigma_i+1\\ \\ ,\\ i=1,\\dots,M\\ .\\ee\n\n{}For $q^2\\neq 1$, the baxterized elements have the form\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{baxtH}\\sigma_i(x):=\\frac{1}{q-q^{-1}}\\,(x\\sigma_i-x^{-1}\\sigma_i^{-1})\\ ;\\ee\nthey are normalized, $\\sigma_i(1)=1$, and satisfy  the unitarity condition\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{hunit}\\sigma_i(x)\\sigma_i(x^{-1})=1-\\frac{(x-x^{-1})^2}{(q-q^{-1})^2}\\ .\\ee\n\n\\paragraph{2.} The symmetrizers $S_n$, $n=1,\\dots,M+1$, (\\cite{Jon0}, \\cite{W1}, \\cite{Gur}) are \nthe non-zero elements, \n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{desy} S_1=1\\ ,\\ S_n\\in H_n(q)\\subset H_{M+1}(q)\\ ,\\ee\nwhich satisfy \n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{desy'} \\sigma_iS_n=S_n\\sigma_i=qS_n\\ ,\\ i=1,\\dots,n-1\\ ,\\ee\nwhich forces $S_n^2\\sim S_n$; they are normalized by\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{desy2} S_n^2=S_n\\ .\\ee\nThe sequence $\\{ S_n\\}$ is defined by (\\ref{desy}), (\\ref{desy'}) and (\\ref{desy2}) uniquely \n(and it does exist for the Hecke quotients for generic $q$); the anti-symmetrizers are \nrelated to the symmetrizers by the isomorphisms $H_{M+1}(q)\\rightarrow H_{M+1}(-q^{-1})$, \n$\\sigma_i\\mapsto\\sigma_i$. \n\n\\paragraph{3.} The symmetrizers can be quickly constructed with the help of the baxterized\nelements. Let $[n]_q:=(q^n-q^{-n})\/(q-q^{-1})$, $[n]_q!:=[1]_q[2]_q\\cdots [n]_q$ and\n$[n]_q^\\$ :=[1]_q![2]_q!\\cdots [n]_q!$. By (\\ref{retr}), \n$\\sigma_i\\, ^{^q}\\!\\Sigma_n=\\, ^{^q}\\!\\Sigma_n\\sigma_i=q\\, ^{^q}\\!\\Sigma_n$, or, \nequivalently, $\\sigma_i(q) ^{^q}\\!\\Sigma_n=[i+1]_q\\, ^{^q}\\!\\Sigma_n$, so\n$(^{^q}\\!\\Sigma_n)^2=[n]_q^\\$\\, ^{^q}\\!\\Sigma_n$ and\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{mshu} S_n=\\frac{1}{[n]_q^\\$}\\  ^{^q}\\!\\Sigma_n\\ .\\ee\nIn particular, the symmetrizers satisfy the recurrent relation\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{recm} S_n=\\frac{1}{[n]_q!}\\  ^{^q}\\!\\Sha_{1,n-1}S_{n-1}\\ .\\ee\n\n\\paragraph{4.}\nWe recall several other forms of the symmetrizers and compare them with (\\ref{mshu}) and\n(\\ref{recm}). A convenient recurrent relation for the symmetrizers is (see e.g. \\cite{Jimb1}, \n\\cite{HIOPT}):\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{santis2}S_n=S_{n-1}\\,\\frac{\\sigma_{n-1}(q^{n-1})}{[n]_q}\\, S_{n-1}\\ .\\ee\nThis is checked either by verifying (\\ref{desy}), (\\ref{desy'}) and (\\ref{desy2}) and then \nby uniqueness or, using (\\ref{recm}), by the following calculation:\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{dersan}\\begin{array}{rcl} [n]_q!S_n&=&\\sigma_1(q)\\dots\\sigma_{n-2}(q^{n-2})\n\\sigma_{n-1}(q^{n-1})S_{n-1}=\\sigma_1(q)\\dots\\sigma_{n-2}(q^{n-2})\\sigma_{n-1}(q^{n-1})\nS_{n-2}S_{n-1}\\\\[1em]\n&=&\\sigma_1(q)\\dots\\sigma_{n-2}(q^{n-2})S_{n-2}\\sigma_{n-1}(q^{n-1})S_{n-1}=\nS_{n-1}\\sigma_{n-1}(q^{n-1})S_{n-1}\\ .\\end{array}\\ee\n\nDenote $\\Sha_{1,n}\\{ q\\sigma\\}$ (the additive shuffle built with the $q\\sigma_1,\\dots,\nq\\sigma_{n-1}$) by $\\sha_{1,n}$. There is another recurrent relation for the symmetrizers in \nterms of the additive shuffles:\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{reca} S_n=\\frac{q^{1-n}}{[n]_q}\\ \\sha_{1,n-1}S_{n-1}\\ .\\ee\nIn other words,\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{santis} S_n=\\frac{q^{-\\frac{n(n-1)}{2}}}{[n]_q!}\\ \\Sigma_n\\{ q\\sigma\\}\\ .\\ee\nThis is checked again either by verifying (\\ref{desy}), (\\ref{desy'}) and (\\ref{desy2}) and \nthen by uniqueness or, using (\\ref{santis2}), by induction:\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{santider}\\begin{array}{rcl} [n]_qS_n&=&S_{n-1}\\sigma_{n-1}(q^{n-1})S_{n-1}=\n\\frac{q^{2-n}}{[n-1]_q}\\sha_{1,n-2} S_{n-2}\\sigma_{n-1}(q^{n-1})S_{n-1}\\\\[1em]\n&=&\\frac{q^{2-n}}{[n-1]_q}\\sha_{1,n-2}\\sigma_{n-1}(q^{n-1})S_{n-1}\n=\\frac{q^{2-n}}{[n-1]_q}\\sha_{1,n-2} \\bigl( [n-1]_q\\sigma_{n-1}+q^{1-n}\\bigr) S_{n-1}\\\\[1em]\n&=&\\frac{q^{2-n}}{[n-1]_q}\\bigl( [n-1]_q\\sha_{1,n-2}\\sigma_{n-1}+\nq^{-1}[n-1]_q\\bigr) S_{n-1}=q^{1-n}\\sha_{1,n-1}S_{n-1}\\ .\\end{array}\\ee\nWe stress that the factors $\\frac{1}{[n]_q!}\\  ^{^q}\\!\\Sha_{1,n-1}$ in (\\ref{recm}) and \n$\\frac{q^{1-n}}{[n]_q}\\ \\sha_{1,n-1}$ in (\\ref{reca}) differ; the multiplicative and\nadditive shuffles do not coincide although their products -- the symmetrizers -- do.\n\n\\paragraph{5.}It turns out that the symmetrizer $S_n$ can be expressed in terms of the \nmultiplicative 1-shuffle $^{^q}\\!\\Sha_{1,n-1}$ or in terms of the additive 1-shuffle \n$\\sha_{1,n-1}$ only.\n\n\\vskip .2cm\n{}For the additive shuffle, we prove by induction that, for $k=1,\\dots,n-1$,\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{adle}\\prod_{j=0}^{k-1}(\\sha_{1,n-1}-q^{j-1}[j]_q)\n=q^{k(k-1)}\\sha_{1,n-1}\\sha_{1,n-2}\\dots\\sha_{1,n-k}\\ .\\ee\n{}For $k=1$ there is nothing to prove. Assume that (\\ref{adle}) holds for some $k<n-1$. \nBy (\\ref{lue}), the right hand side is divisible, from the right, by $S_k^{\\uparrow (n-k)}$.\nMultiply (\\ref{adle}) by the factor $(\\sha_{1,n-1}-q^{k-1}[k]_q)$ from the right and substitute,\nin the right hand side,\n$$\\sha_{1,n-1}=\\sha_{1,k-1}^{\\uparrow (n-k)}+q^k\\sha_{1,n-1-k}\\sigma_{n-k}\\dots\\sigma_{n-1}$$\nin this factor. Since $S_k^{\\uparrow (n-k)}\\sha_{1,k-1}^{\\uparrow (n-k)}=q^{k-1}[k]_q\nS_k^{\\uparrow (n-k)}$, the product in the right hand side simplifies,\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\begin{array}{c}\\sha_{1,n-1}\\sha_{1,n-2}\\dots\\sha_{1,n-k}(-q^{k-1}[k]_q+\n\\sha_{1,k-1}^{\\uparrow (n-k)}+q^k\\sha_{1,n-1-k}\\sigma_{n-k}\\dots\\sigma_{n-1})\\\\[1em]\n=q^k\\sha_{1,n-1}\\sha_{1,n-2}\\dots\\sha_{1,n-k}\\sha_{1,n-1-k}\\sigma_{n-k}\\dots\\sigma_{n-1}\n=q^{2k}\\sha_{1,n-1}\\sha_{1,n-2}\\dots\\sha_{1,n-1-k}\\end{array}\\ee\n(in the last equality we again used (\\ref{lue}) for \n$\\sha_{1,n-1}\\sha_{1,n-2}\\dots\\sha_{1,n-k}\\sha_{1,n-1-k}$), establishing the induction step.\n\n\\vskip .2cm\nIn particular, at $k=n-1$, we obtain, by (\\ref{reca}), the expression of $S_n$ in terms\nof $\\sha_{1,n-1}$,\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{sysha} S_n=\\frac{q^{-\\frac{(n-1)(3n-4)}{2}}}{[n]_q!}\\ \n\\prod_{j=0}^{n-2}(\\sha_{1,n-1}-q^{j-1}[j]_q)\\ .\\ee\n\n\\paragraph{6.} For the multiplicative shuffle, we prove by induction that, for $k=1,\\dots,n$,\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray} (\\, ^{^q}\\!\\Sha_{1,n})^k=\\frac{[n+1-k]_q^\\$\\ ([n+1]_q!)^k}{[n]_q^\\$}\\, ^{^q}\\!\\Sha_{1,n}\n\\, ^{^q}\\!\\Sha_{1,n-1}\\dots\\, ^{^q}\\!\\Sha_{1,n+1-k}\\ .\\ee\n{}For $k=1$ there is nothing to prove. Assume that (\\ref{adle}) holds for some $k<n$.\nThe relations (\\ref{symsh}) and (\\ref{ctsh}) hold for $\\Sigma_m=\\, ^{^q}\\!\\Sigma_m$\nand $\\Sha_{m,n}=\\, ^{^q}\\!\\Sha_{m,n}$. Therefore, (\\ref{lue}) holds as well and\nthe product $\\, ^{^q}\\!\\Sha_{1,n}\\, ^{^q}\\!\\Sha_{1,n-1}\\dots\\, ^{^q}\\!\\Sha_{1,n-k}$\nis divisible, from the right, by $\\, ^{^q}\\!\\Sigma_{k+1}^{\\uparrow (n-k)}$. The induction step is:\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\begin{array}{rcl}\\, ^{^q}\\!\\Sha_{1,n}\\dots\\, ^{^q}\\!\\Sha_{1,n+1-k}\\cdot\n\\, ^{^q}\\!\\Sha_{1,n}&=&\\, ^{^q}\\!\\Sha_{1,n}\\dots\\, ^{^q}\\!\\Sha_{1,n+1-k}\n\\, ^{^q}\\!\\Sha_{1,n-k}\\cdot\\sigma_{n-k+1}(q^{n-k+1})\\dots\\sigma_n(q^n)\\\\[1em]\n&=&{\\displaystyle\\frac{[n+1]_q!}{[n+1-k]_q!}}\\, ^{^q}\\!\\Sha_{1,n}\\, ^{^q}\\!\\Sha_{1,n-1}\\dots\n\\, ^{^q}\\!\\Sha_{1,n+1-k}\\, ^{^q}\\!\\Sha_{1,n-k}\\ ,\\end{array}\\ee\nsince $S_{n+1}\\sigma_k(q^k)=[k+1]_qS_{n+1}$, $k=1,\\dots,n$.\n\n\\vskip .2cm\nIn particular, at $k=n$, we obtain, by (\\ref{mshu}), the expression of $S_n$ in terms\nof $\\, ^{^q}\\!\\Sha_{1,n}$,\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{syshh}S_{n+1}=\\Bigl(\\frac{1}{[n+1]_q!}\\, ^{^q}\\!\\Sha_{1,n}\\Bigr)^n\\ .\\ee\n\n\\paragraph {7. Remark.} Let $\\hat{R}$ be a Hecke Yang--Baxter matrix and $\\rho_{\\hat{R}}$ \nthe corresponding local representation of the tower of the Hecke algebras. The symmetrizers\n${\\cal{S}}_j$ built with ${\\cal{T}}_{m,n}'=\\rho_{\\hat{R}}(\\, ^{^s}\\!\\Sha_{m,n})$, at\n$s=q$, are the same as the symmetrizers built with \n${\\cal{T}}_{m,n}''=\\rho_{\\hat{R}}(\\Sha_{m,n}\\{ t\\sigma\\} )$, at $t=q$ (the symmetrizers\ncoincide at $s^2=q^2$ and $t=q$ or $s^2=q^{-2}$ and $t=-q^{-1}$, these are the values for \nthe anti-symmetrizers; the symmetrizers coincide trivially at $s^2=1$ and $t=0$; otherwise \nthe symmetrizers for $\\{{\\cal{T}}_{m,n}'\\}$ and $\\{{\\cal{T}}_{m,n}''\\}$ differ). Therefore, \nfor the Hecke algebras, the b-shuffle algebra on $\\bigoplus_j {\\mathrm{Im}}({\\cal{S}}_j)$ \ncoincides with the Nichols--Woronowicz algebra (or the symmetric algebra of the quantum space). \nIndeed, the composition law (\\ref{shal}) on $\\bigoplus_j {\\mathrm{Im}}({\\cal{S}}_j)$ can be \nwritten in the following equivalent form:\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{fe1}u\\odot v:={\\cal{S}}_{m+n}(u'\\otimes v')\\ ,\\ee\nwhere $u={\\cal{S}}_mu'$ and $v={\\cal{S}}_nv'$. Also,\n${\\mathrm{Im}}({\\cal{S}}_j)\\simeq V^{\\otimes j}\/{\\mathrm{Ker}}({\\cal{S}}_j)$, and\nthe algebra on $\\bigoplus_j {\\mathrm{Im}}({\\cal{S}}_j)$ can be defined alternatively as\nthe algebra on $\\bigoplus_j V^{\\otimes j}\/{\\mathrm{Ker}}({\\cal{S}}_j)$ with the composition\nlaw\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{fe2}\\bar{u}\\circ \\bar{v}:=u\\otimes v\\ \\ {\\mathrm{mod}}\n\\ {\\mathrm{Ker}}({\\cal{S}}_{m+n})\\ ,\\ee\nwhere $\\bar{u}\\in V^{\\otimes m}\/{\\mathrm{Ker}}({\\cal{S}}_m)$ and\n$\\bar{v}\\in V^{\\otimes n}\/{\\mathrm{Ker}}({\\cal{S}}_n)$; $u\\in V^{\\otimes m}$ and \n$v\\in V^{\\otimes n}$ are representatives of $\\bar{u}$ and $\\bar{v}$, respectively.\nIn the formulations (\\ref{fe1}) or (\\ref{fe2}), the algebra on \n$\\bigoplus_j {\\mathrm{Im}}({\\cal{S}}_j)$ or\n$\\bigoplus_j V^{\\otimes j}\/{\\mathrm{Ker}}({\\cal{S}}_j)$\ndepends only on the collection $\\{ {\\cal{S}}_j\\}$; the composition laws (\\ref{fe1}) or \n(\\ref{fe2}) are well defined if ${\\cal{S}}_{m+n}$ is divisible by ${\\cal{S}}_m$ and\n${\\cal{S}}_n^{\\uparrow m}$ from the right (which is, in general, weaker than\n${\\cal{S}}_{m+n}^{\\phantom{\\uparrow}}={\\cal{T}}_{n,m}^{\\phantom{\\uparrow}}\\,\n{\\cal{S}}_m^{\\phantom{\\uparrow}}\\, {\\cal{S}}_n^{\\uparrow m}$); when, say, the representative \n$u$ of $\\bar{u}$ changes, $u\\sim u+\\delta u$, ${\\cal{S}}_m(\\delta u)=0$, so \n$\\delta u\\otimes v\\in {\\mathrm{Ker}}({\\cal{S}}_{m+n})$ and the product $\\bar{u}\\circ \\bar{v}$\ndoes not change, $u\\otimes v\\equiv (u+\\delta u)\\otimes v\\ \\ {\\mathrm{mod}}\n\\ {\\mathrm{Ker}}({\\cal{S}}_{m+n})$.\n\n\\vskip .2cm\nHowever, the algebras on the space $\\bigoplus_j V^{\\otimes j}$, built with ${\\cal{T}}_{m,n}'$\nor ${\\cal{T}}_{m,n}''$, are very different, as it is seen, for example from the comparison \nof the spectra of the multiplicative and additive 1-shuffles in Section \\ref{secspec}.\nThe collections $\\{ {\\cal{T}}_{m,n}'\\}$ and $\\{ {\\cal{T}}_{m,n}''\\}$ seem to have\ndifferent ranges of applicability (already for the BMW algebras, the symmetrizers for these\ntwo collections do not coincide). \n\n\\subsection{BMW algebras}\n\nThe tower of the Birman-Murakami-Wenzl algebras $B\\!M\\!W_{M+1}(q,\\nu)$ was introduced \nin \\cite{M1} and \\cite{BW}; it depends on two parameters, $q\\in {\\mathfrak{k}}^*$ and\n$\\nu\\in {\\mathfrak{k}}\\setminus \\{0,q,-q^{-1}\\}$. For $q^2\\neq 1$, the algebra \n$B\\!M\\!W_{M+1}(q,\\nu)$ is the quotient of the braid group ring ${\\mathfrak{k}}B_{M+1}$ by\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{bmw1}\\kappa_i\\sigma_i=\\sigma_i\\kappa_i=\\nu\\kappa_i\\ ,\\ee\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{bmw2}\\kappa_i\\sigma_{i-1}\\kappa_i=\\nu^{-1}\\kappa_i\\quad ,\\quad \n\\kappa_i\\sigma_{i-1}^{-1}\\kappa_i=\\nu\\kappa_i\\ ,\\ee\nwhere the elements $\\kappa_i$ are defined by\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{bmw3}\\sigma_i-\\sigma_i^{-1}=(q-q^{-1})(1-\\kappa_i)\\ .\\ee\n\nThe Hecke quotient is $\\kappa_i=0$.\n\n\\vskip .2cm\n{}For $q^2\\neq 1$, the baxterized elements have the form\n(\\cite{Jon2}, \\cite{Mu}, \\cite{J}, \\cite{Is1})\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{bmwbax}\\sigma_i(x):=x^{-1}\\left( 1+\\frac{x^{2}-1}{q-q^{-1}}\\,\\sigma_i+ \n\\frac{x^{2}-1}{1-\\nu^{-1}q^{-1} x^{2}}\\,\\kappa_i\\right)\\ .\\ee\nTheir classical counterparts (for the Brauer algebras) were found in \\cite{Zam}. The elements \n(\\ref{bmwbax}) are normalized, $\\sigma_i(1)=1$, and satisfy the same unitarity conditions \n(\\ref{hunit}). The spectral decomposition of the generator $\\sigma_i$ contains three idempotents. \nThe basic symmetrizer (the idempotent corresponding to the eigenvalue $q$) is proportional to \n$\\sigma_i(q)$. However, $\\sigma_i(q^{-1})$ is a mixture of two other idempotents. There are again \nisomorphisms \n$\\iota:\\, B\\!M\\!W_{M+1}(q,\\nu) \\simeq B\\!M\\!W_{M+1}(-q^{-1},\\nu)$,\n$\\sigma_i\\mapsto\\sigma_i$. The formula (\\ref{bmwbax}) is not invariant under $\\iota$.\nThe basic anti-symmetrizer (the idempotent corresponding to the eigenvalue $-q^{-1}$) is\nproportional to $\\iota^{-1}(\\sigma(x))$ at $x=q$.\n\n\\vskip .2cm\nAgain, the symmetrizers $S_n$, $n=1,\\dots,M+1$, are the \nnon-zero elements, which satisfy \n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{desyb} S_1=1\\ ,\\ S_n\\in B\\!M\\!W_n(q)\\subset B\\!M\\!W_{M+1}(q)\\ ,\\ee\n(\\ref{desy'}) and (\\ref{desy2}); they exist and are defined uniquely by the conditions \n(\\ref{desyb}), (\\ref{desy'}) and (\\ref{desy2}).\n\n\\vskip .2cm\nAgain, with the knowledge of the baxterized elements, one constructs the symmetrizers \nimmediately: they are given by the same formula (\\ref{mshu}) and satisfy the same\nrecurrence (\\ref{recm}); the anti-symmetrizers are related to the symmetrizers by the \nisomorphisms $\\iota$.\n\n\\vskip .2cm\nThe recurrency (\\ref{santis2}) holds for the BMW symmetrizers as well (it was used in \n\\cite{WT},\\cite{Fior}); it is derived from the baxterized form of the symmetrizers by the\nsame calculation (\\ref{dersan}).\n\n\\vskip .2cm\nThe recurrency relation (\\ref{reca}) does not hold for the BMW symmetrizers; the\nadditive shuffles have to be modified. A version of such modification was suggested\nin \\cite{HS} and can be derived by a calculation similar to (\\ref{santider}). For the \nHecke algebras the expansions of the products $\\sha_{1,n-1}\\sha_{1,n-2}\\dots\\sha_{1,n-k}$ \ncontain only reduced words; this is not any more so for the modified shuffles for the BMW \nalgebras, the expansions contain similar terms (in a monomial basis, like the one suggested in \n\\cite{K}) and the formulas are not as elegant as for the Hecke algebras.\n\n\\vskip .2cm\nThe formula (\\ref{syshh}) holds, with the same derivation, for the BMW symmetrizers.\n\n\\section{Spectrum of 1-shuffles}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{secspec}\n\nPolynomial identities for the multiplicative (for the Hecke and BMW algebras) and additive \n(for the Hecke algebras) 1-shuffles follow, as a by-product, from (\\ref{syshh}) and (\\ref{sysha}). \nWe establish the multiplicities of the eigenvalues in this section. The polynomial identity for \nthe additive shuffle was discovered in \\cite{Wal} for the symmetric groups and generalized to \nthe Hecke algebras in \\cite{Lus} with the help of the interpretation of the Hecke algebras in \nterms of flag manifolds over finite fields. The multiplicities of the eigenvalues of the additive \nshuffles were obtained in \\cite{DFP} for the symmetric groups. We propose a different approach to \nthe calculation of the multiplicities for the Hecke algebras; our method uses the traces of the \noperators of the left multiplication by the additive shuffles.\n \n\\vskip .2cm\nLet $u\\in H_n(q)\\subset H_m(q)$, $m\\geq n$. Denote by $L_u$ the operator of the left \nmultiplication by $u$, $L_u:H_m(q)\\rightarrow H_m(q)$, \n$L_u(x):=ux$. We denote by ${\\mathrm{Tr}}_{_{H_{m}}}(L_u)$ the trace of the operator\n$L_u$, considered as the operator on $H_m(q)$.\n\n\\paragraph{1.} We start with the multiplicative shuffles. Since\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{ein} \\, ^{^q}\\!\\Sha_{1,n}S_{n+1}=[n+1]_q!S_{n+1}\\ ,\\ee \nwe find, multiplying (\\ref{syshh}) by $(\\, ^{^q}\\!\\Sha_{1,n}-[n+1]_q!)$, the following polynomial \nidentity for the multiplicative shuffle\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray} (^{^q}\\!\\Sha_{1,n})^n\\,(^{^q}\\!\\Sha_{1,n}-[n+1]_q!)=0\\ ,\\ee\nwhich holds for both Hecke and BMW algebras. This is the minimal polynomial, already for the Hecke \nalgebras. It is seen without calculations in the Burau representation \\cite{Bu} of $H_{n+1}$, \n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\sigma_j(q^j)\\mapsto [j+1]_q\\,{\\mathrm{Id}}_{j-1}\\oplus\\left(\\begin{matrix}q^{-j}&[j]_q\\\\\n[j]_q&q^j\\end{matrix}\\right)\\oplus [j+1]_q\\,{\\mathrm{Id}}_{n-j}\\ .\\ee\nIf the minimal polynomial is $t^i(t-[n+1]_q!)$ with $i<n$ (the eigenvalue $[n+1]_q!$ is present \ndue to (\\ref{ein})) then $S_{n+1}$ is proportional to the smaller than $n$ power of \n$\\, ^{^q}\\!\\Sha_{1,n}$.  The matrix of the element $\\sigma_j(q^j)$ in the Burau representation\nhas only one non-zero entry under the main diagonal, on the intersection of {\\mbox{$(j+1)$-st}} \nline and $j$-th column. Therefore, the matrix of $^{^q}\\!\\Sha_{1,n}$ has only one sub-diagonal \nfilled with (possibly) non-zeros. However, the matrix of $S_{n+1}$ is proportional to the Hankel \ntype matrix $A^i_j:=q^{i+j}$; a smaller than $n$ power of the matrix of $^{^q}\\!\\Sha_{1,n}$ has \nzero in the very left entry of the bottom line and cannot be equal to the matrix of $S_{n+1}$.\n\n\\vskip .2cm\nThus, the element $^{^q}\\!\\Sha_{1,n}$ is not semi-simple for $n>1$; the semi-simple part of \n$^{^q}\\!\\Sha_{1,n}$ is $[n+1]_q!S_{n+1}$ and the eigenvalue $[n+1]_q!$ is simple (the rank of \nthe projector $L_{S_{n+1}}$ on $H_{n+1}(q)$ is one, because $S_{n+1}\\sigma_j=qS_{n+1}$,\n$j=1,\\dots,n$).\n\n\\paragraph{2.} Similarly, multiplying (\\ref{sysha}) by $(\\sha_{1,n-1}-q^{1-n}[n]_q)$, we find\nthe following polynomial identity for the additive shuffle\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{og1}\\left(\\sha_{1,n-1}- q^{1-n}[n]_q\\right) \n\\prod_{j=0}^{n-2}\\,\\left(\\sha_{1,n-1}-q^{1-j}[j]_q\\right)=0\\ . \\ee\nThe $q$-numbers $q^{1-j}[j]_q\\equiv 1+q^2+\\dots+q^{2j-2}$, $j=1,2,\\dots$, are polynomials in \n$q$, linearly independent over ${\\mathbb{Z}}$. Therefore, there is a unique integer combination \n$\\sum_{j\\in\\{ 1,2,\\dots,n-2,n\\} }n_jq^{1-j}[j]_q$, $n_j\\in {\\mathbb{Z}}$, of these $q$-numbers, \nwhich is equal to the trace of $L_{\\sha_{1,n-1}}$; the coefficients $n_j$ in this combination \nare the multiplicities of the eigenvalues $q^{1-j}[j]_q$, $j>0$. The multiplicity $n_0$ of the \neigenvalue $0$ is fixed by $\\sum n_j={\\mathrm{dim}}(H_n(q))\\equiv n!$. Thus, the presence \nof the parameter $q$ gives a simple way to calculate the multiplicities. \n \n\\vskip .2cm\\noindent{\\bf Lemma 1.} (i) If $u\\in H_j$ then \n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray} {\\mathrm{Tr}}_{_{H_{j+1}}}(L_u)={\\mathrm{Tr}}_{_{H_{j+1}}}(L_{u^{\\uparrow 1}})=\n(j+1){\\mathrm{Tr}}_{_{H_j}}(L_u)\\ .\\ee\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray} {\\mathrm{(ii)}}\\hspace{2.0cm}{\\mathrm{Tr}}_{_{H_{j+1}}}(L_{\\sigma_1\\sigma_2\\dots\\sigma_j})\n=(q-q^{-1}){\\mathrm{Tr}}_{_{H_j}}(L_{\\sigma_1\\sigma_2\\dots\\sigma_{j-1}})\\ \\ ,\\ \\ j>0\n\\ .\\hspace{.8cm}\\ee\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{dltrsh} {\\mathrm{(iii)}}\\hspace{2.2cm}\n{\\mathrm{Tr}}_{_{H_j}}(L_{\\sigma_{k-l+1}\\dots\\sigma_{k-1}\\sigma_{k}})\n=\\frac{j!}{(l+1)!}(q-q^{-1})^l\\ \\ ,\\ \\ j>k\\geq l\\ .\\hspace{.7cm}\\ee\n\n\\vskip .2cm\\noindent{\\it Proof.} Recall that, as a vector space, \n$H_{j+1}(q)\\simeq\\oplus_{k=-1}^{j-1}W_k$, where $W_k$ is the vector space \nconsisting of elements $v\\sigma_j\\sigma_{j-1}\\dots\\sigma_{j-k}$ with $v\\in H_j(q)$ (the word \n$\\sigma_j\\sigma_{j-1}\\dots\\sigma_{j-k}$ is, by definition, empty for $k=-1$); each \n$W_k$ is canonically isomorphic to $H_j(q)$ as a vector space, the isomorphism is \n$v\\sigma_j\\sigma_{j-1}\\dots\\sigma_{j-k}\\mapsto v$. The Hecke versions of the automorphism \n$\\mathfrak a$ and the anti-automorphism $\\mathfrak b$, defined in (\\ref{aau}), transform\nthe above decomposition of $H_{j+1}(q)$ into $H_{j+1}(q)\\simeq\\oplus_{k=-1}^{j-1}W_k'$, where \n$W_k'$ consists of elements $v\\sigma_1\\sigma_2\\dots\\sigma_{k+1}$ with \n$v\\in H_j(q)^{\\uparrow 1}$ and $H_{j+1}(q)\\simeq\\oplus_{k=-1}^{j-1}W_k''$, where \n$W_k''$ consists of elements $\\sigma_{j-k}\\dots\\sigma_{j-1}\\sigma_jv$ with $v\\in H_j(q)$.\n \n\\vskip .2cm\nThe operator $L_u$ (respectively, $L_{u^{\\uparrow 1}}$) acts in each of the spaces $W_k$\n(respectively, $W_k'$) separately and this action commutes with the isomorphisms \n$W_k\\simeq H_j(q)$ (respectively, $W_k'\\simeq H_j(q)^{\\uparrow 1}$). This establishes (i).\n\n\\vskip .2cm\nIn fact, it was enough to find the formula for ${\\mathrm{Tr}}_{_{H_{j+1}}}(L_u)$;\n${\\mathrm{Tr}}_{_{H_{j+1}}}(L_u)={\\mathrm{Tr}}_{_{H_{j+1}}}(L_{u^{\\uparrow 1}})$\nbecause $u^{\\uparrow 1}$ is conjugate to $u$,\n$\\sigma_1\\dots\\sigma_ju=u^{\\uparrow 1}\\sigma_1\\dots\\sigma_j$, for $u\\in H_j(q)$.\n\n\\vskip .2cm\n(ii) Given a basis $\\{ e_\\alpha\\}$ of a vector space $U$ we say that the vector $e_\\alpha$ \n(from the basis) does not contribute to the trace of an operator $X:U\\rightarrow U$ if \n$X_\\alpha^\\alpha=0$ (no summation), where $X_\\alpha^\\beta$ is the matrix of $X$ in the basis \n$\\{ e_\\alpha\\}$. \n\n\\vskip .2cm\nWe use the decomposition \n$H_{j+1}(q)\\simeq\\oplus_{k=-1}^{n-1}W_k''$. The operator\n$L_{\\sigma_1\\sigma_2\\dots\\sigma_j}$ maps $W_{-1}''$ to $W_{j-1}''$, so vectors from \n$W_{-1}''$ do not contribute to the trace of $L_{\\sigma_1\\sigma_2\\dots\\sigma_j}$. For $0\\leq k<j$,\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{cotr}\\begin{array}{l} \\Bigl(\\sigma_1\\dots\\sigma_j\\Bigr)\\Bigl(\\sigma_{j-k}\\dots\n\\sigma_{j-1}\\cdot\\sigma_j\\Bigr)v=\\Bigl(\\sigma_{j-k+1}\\dots\\sigma_j\\Bigr)\\Bigl(\\sigma_1\n\\dots\\sigma_j\\Bigr)\\sigma_jv\\hspace{4cm}\\\\[1em]\n\\hspace{1.5cm}=\\Bigl(\\sigma_{j-k+1}\\dots\\sigma_j\\Bigr)\\Bigl(\\sigma_1\\dots\\sigma_{j-1}\\Bigr)\n\\Bigl((q-q^{-1})\\sigma_j+1\\Bigr)v\\\\[1em]\n\\hspace{1.5cm}=(q-q^{-1})\\Bigl(\\sigma_{j-k+1}\\dots\\sigma_j\\Bigr)\\Bigl(\\sigma_1\\dots\\sigma_j\n\\Bigr)v+\\Bigl(\\sigma_{j-k+1}\\dots\\sigma_j\\Bigr)\\Bigl(\\sigma_1\\dots\\sigma_{j-1}\\Bigr)v\\\\[1em]\n\\hspace{1.5cm}=(q-q^{-1})\\Bigl(\\sigma_1\\dots\\sigma_j\\Bigr)\\Bigl(\\sigma_{j-k}\\dots\n\\sigma_{j-1}\\Bigr)v+\\Bigl(\\sigma_{j-k+1}\\dots\\sigma_j\\Bigr)\\Bigl(\\sigma_1\\dots\\sigma_{j-1}\n\\Bigr)v\\ ,\\end{array}\\ee\nthe operator $L_{\\sigma_1\\sigma_2\\dots\\sigma_j}$ maps $W_k''$ to $W_{j-1}''\\oplus W_{k-1}''$.\nTherefore, vectors from $W_k''$ do not contribute to the trace of \n$L_{\\sigma_1\\sigma_2\\dots\\sigma_j}$ for $k<j-1$. For $k=j-1$, the component \n$L_{\\sigma_1\\sigma_2\\dots\\sigma_j}^\\diamond$ of the operator $L_{\\sigma_1\\sigma_2\\dots\\sigma_j}$, \nwhich maps $W_{j-1}''$ to $W_{j-1}''$, may have a non-zero trace. This component reads, by \n(\\ref{cotr}),\n$$L_{\\sigma_1\\sigma_2\\dots\\sigma_j}^\\diamond(\\sigma_1\\dots\\sigma_jv)=(q-q^{-1})\\,\n\\sigma_1\\dots\\sigma_jL_{\\sigma_1\\sigma_2\\dots\\sigma_{j-1}}(v)\\ ,$$ \nand the assertion (ii) follows.\n\n\\vskip .2cm\n(iii) Follows from (i) and (ii).\\hfill$\\Box$\n\n\\vskip .2cm\nBy (\\ref{dltrsh}), the trace of the operator $L_{\\sha_{1,n-1}}$ is\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray} {\\mathrm{Tr}}_{_{H_n}}(L_{\\sha_{1,n-1}})=\\sum_{i=0}^{n-1}\\frac{n!}{(i+1)!}(q^2-1)^i\\ .\\ee\n\n{}For the symmetric group ${\\mathbb{S}}_n$, the multiplicity of the eigenvalue $j$ of\n$L_{\\sha_{1,n-1}}$ is the number of permutations in ${\\mathbb{S}}_n$ with exactly $j$ fixed \npoints \\cite{DFP}. Recall that the derangement number $d_n$ (the number of permutations in \n${\\mathbb{S}}_n$ without fixed points) is:\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray} d_n=n!\\,\\sum_{i=0}^n\\frac{(-1)^i}{i!}\\ee\nand the number $d_{n,j}$ of permutations in ${\\mathbb{S}}_n$ with exactly $j$ fixed points is\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray} d_{n,j}=\\binc{n}{j} d_{n-j}\\equiv\\frac{n!}{j!}\\sum_{i=0}^{n-j}\\frac{(-1)^i}{i!}\\ .\\ee\n{}For generic $q$, the multiplicities of the eigenvalues of $L_{\\sha_{1,n-1}}$ are the same as \nfor the symmetric group. By construction, $\\sum_{j=0}^{n}d_{n,j}=n!$. Thus, to rederive the \nmultiplicities we have only to check that $\\sum_{j=0}^n d_{n,j}q^{1-j}[j]_q\n={\\mathrm{Tr}}_{_{H_n}}(L_{\\sha_{1,n-1}})$, or, explicitly,\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray}\\label}\\def\\be{\\begin{equation}}\\def\\ee{\\end{equation}{fid}\\sum_{j=0}^n\\frac{n!}{j!}\\sum_{k=0}^{n-j}\\frac{(-1)^k}{k!}q^{1-j}[j]_q\n=\\sum_{i=0}^{n-1}\\frac{n!}{(i+1)!}(q^2-1)^i\\ .\\ee\nIt is straightforward to verify that both left and right hand sides \nsatisfy the same recurrency relation in $n$,\n\\begin{equation}}\\def\\ee{\\end{equation}}\\def\\ba{\\begin{eqnarray}}\\def\\ea{\\end{eqnarray} f_{n+1}=(n+1)f_n+(q^2-1)^n\\ ,\\ee\nwith the same initial condition $f_0=0$, and thereby coincide.\n\n\n\\vskip .5cm\n\\noindent {\\bf Acknowledgements.} The work of A. P. Isaev was partially supported by the \ngrants RFBR 08-0100392-a, RFBR-CNRS 07-02-92166-a and RF President Grant N.Sh. 195.2008.2; \nthe work of O. V. Ogievetsky was partially supported \nby the ANR project GIMP No.ANR-05-BLAN-0029-01.\n  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\nThe number of eclipsing massive binary stars, having M $\\textgreater$ 9 M$_{\\odot}$, is small in \ncomparison to lower mass systems (\\citet{Gie03}, \\citet{Hil05}, \\citet{Mas12},   \\citet{Moe13}, \\citet{San13}). Massive \nstars are usually formed in OB associations. It is well known that massive stars evolve faster and are of interest \nas progenitors of neutron stars and black holes. The evolutionary paths of massive binaries depend on processes \nrelated to mass transfer between the components and mass loss via stellar wind from one or both components. Consequently, \nin order to better understand the physics of binary systems and  test the predictions of theoretical models, it is thus \nimportant to quantitatively analyse the properties of massive binary systems with well-constrained orbital parameters. In \nthis context studies of the rare early B-type massive stars has major highlights for understanding formation mechanism and \nevolution of massive binaries. \\citet{Tor10} collected accurate fundamental parameters, for the 95 double-lined detached \nbinary systems. The number of binary systems with masses and radii determined to an accuracy of better than 3\\% \ndecreases towards the massive stars. The number of systems with at least one component more massive than \n9 M$_{\\odot}$ does not exceed ten \\citep{Tor10}. Therefore, we initiated a spectroscopic study for the close \nbinary systems which include high mass stars located in the upper left-hand corner of the Hertzsprung-Russell diagram.    \n   \n\nV745\\,Cas (HD1810, HIP\\,1805, WDS\\,J00229+6214AB, V=8.11, B-V=0.06) is the brightest star of a multiple star \nsystem. WDS\\,J00229+6214 consists of five stars. The angular distances of B, C, D and E from the brightest \nstar A are given in the Washington Double Stars Catalog \\footnote{ \\bf http:\/\/ad.usno.navy.mil\/wds\/} as 9.5, 23.2, 44.9 \nand 57.8\\,arcseconds, respectively. The visual apparent magnitudes of the stars A, B, C, D, and E are also estimated \nas 8.12, 11.13, 10.8, 12.14 and 12.85\\,mag. Despite its relative brightness, V745\\,Cas has not been comprehensively \nstudied yet. The first spectral classification for V745\\,Cas (B0\\,IV) has been dating back to \\citet{Mor53}. \\citet{Hau70} \ngives V=8$^m$.19, (B-V)=0$^m$.08, while \\citet{Gue74} gives V=8$^m$.16, (U-B)=-0$^m$.83 and (B-V)=0$^m$.05. Its radial \nvelocity, averaging V=-50 \\kms, was found to be variable by \\citet{San38}, \\citet{Wil53}, \\citet{Pet61}, and \\citet{Eva67}. \nV745\\,Cas is listed in the optical pairs by \\citet{Mei68} who estimated the spectral types of the A and B components as \nB0\\,IVp and A9\\,V with a magnitude difference of 1.5\\,mag. \\citet{Ree03} estimates visual magnitude and color indices of \nthe A component as 8$^m$.19, (U-B)=-0$^m$.79 and (B-V)=0$^m$.08.\n\nThe light variability of V745\\,Cas has been discovered thanks to the Hipparcos satellite mission \\citep{Per97}. The \nH$_p$ magnitudes at the maximum and minimum light of the brightest component A of the multiple system are given as \n8$^m$.063 and 8$^m$.164, i.e. a magnitude variation by about 0$^m$.10. The light curve was classified as a W UMa-type \neclipsing binary according to the criteria of the General Catalogue of variable Stars, with an orbital period of 1.41057 \ndays from visual inspection of light variation. The Hipparcos' light curve shows that the depths of the eclipses are \nnearly identical. \\citet{Kaz99} designated it as V745\\,Cas in the 74th Name-list of Variable Stars, and classified \nit as an EB.  \n\nV745\\,Cas is located in the sky in the vicinity of the Cas\\,OB4 and Cas\\,OB14 associations. While the star was not included \nas a member of Cas\\,OB4 association by \\citet{Hum78} and \\citet{Gar92}, it is included into the list of members of Cas\\,OB4 \nby \\citet{Mel95} and \\citet{Mel09}. These associations are located in the Perseus arm of the Galaxy. Many estimates have \nbeen made about their distances to the Sun. These estimates indicate that the Cas OB14 association is closer to the sun \nhaving a distance of about 1\\,kpc, which is less than half the distance of Cas\\,OB4. \n\nThis paper is organized as follows. We present new spectroscopic observations and radial velocities of both components \nof the eclipsing pair. By analysing the $INTEGRAL$ and Hipparcos missions' light curves and the new radial velocities \nwe obtain orbital parameters for the stars. Combining the results of these analyses we obtain absolute physical \nparameters of both components. In addition, we conclude with a discussion of the system's evolutionary status.\n\n\\section{Spectroscopic observations and data reductions}                              \nOptical spectroscopic observations of V745\\,Cas were obtained with the Turkish Faint Object Spectrograph Camera \n(TFOSC) attached to the 1.5\\,m telescope from\nJanuary 2 to August 28, 2012, under good seeing conditions. Further details on the telescope and the spectrograph \ncan be found at http:\/\/www.tug.tubitak.gov.tr. The wavelength coverage of each spectrum was 4000-9000 \\AA~in 12 orders, \nwith a resolving power of $\\lambda$\/$\\Delta \\lambda$ $\\sim$7\\,000 at 6563 \\AA~and an average signal-to-noise ratio \n(S\/N) was $\\sim$120. We also obtained high S\/N spectra of early type standard stars 1\\,Cas (B0.5\\,IV), HR\\,153 (B2\\,IV), \n$\\tau$\\,Her (B5\\,IV), 21\\,Peg (B9.5\\,V), and $\\alpha$\\,Lyr (A0\\,V) for use as templates in derivation of the radial \nvelocities. \n\nThe data reduction was performed using the echelle task of IRAF {\\footnote{IRAF is distributed by the $National~Optical~\nAstronomy$ Observatory, which is operated by the Association of Universities for Research in Astronomy,Inc. (AURA), under \ncooperative agreement with the National Science Foundation} \\sc echelle package} \\citep{Sim74} following the standard steps:\nbackground subtraction, division by a flat-field spectrum given by a halogen lamp, wavelength calibration using the emission\nlines of a Fe-Ar lamp, and normalization to the continuum through a polynomial fit. Heliocentric corrections were computed using\nthe IRAF {\\sc rvsao.bcvcorr} routine and were taken into account in the subsequent radial velocity determination. \n\n\\subsection{Radial Velocities}\nRadial velocities of the components were measured by cross-correlation with the {\\sc fxcor} task in IRAF \n(\\citet{Sim74}, \\citet{Ton79}). The standard stars with known radial velocities, given in previous section, were \nused as templates. The standard stars' spectra were synthetically broadened by convolution with the broadening \nfunction of \\citet{Gra92}. The cross-correlation with the standard star 1\\,Cas gave the best result. The spectra \nshowed two distinct cross-correlation peaks at the quadratures, one for each component of the binary. Since the \ntwo peaks appear blended, a double $Gaussian$ fit was applied to the combined profile using {\\it de-blend} \nfunction in the task.\n\nThe cross-correlation technique was applied to four wavelength regions with well-defined absorption lines \nof the primary and secondary components. These regions include He\\,{\\sc i} lines at $\\lambda$4471 (in the 9th \norder), $\\lambda$5876 (in the 3rd order), $\\lambda$6678 (in the 3rd order), and $\\lambda$7065 (in the 2nd order) \ndominant in early B-type stars. Here we used as weights the inverse of the variance of the radial velocity \nmeasurements in each order, as reported by {\\sc fxcor}. We have been able to measure radial velocities of both \ncomponents with a precision better than 10\\,\\kms extracted from the {\\sc fxcor}. The $Balmer$\nlines were not used in the measurements of radial velocities due to their extremely Stark-broadened\nand rotationally broadened profiles.   \n\n \nThe heliocentric radial velocities for the primary (V$_p$) and the secondary (V$_s$) components are listed in \nTable\\,1, along with the dates of observations and the corresponding orbital phases computed with the new \nephemeris given in Sect. 2.1. The velocities in this table have been corrected to the heliocentric \nreference system by adopting a radial velocity value for the template stars. The radial velocities are \nplotted against the orbital phase in Fig.\\,1 where the filled squares correspond to the primary and the \nempty squares to the secondary star. Examination of the $Integral-OMC$ and $Hipparcos$ light curves \nshows no evidence for any eccentricity in the orbit of the system. Therefore, we have started with a circular \norbit and analysed the RVs using the {\\sc RVSIM} software program \\citep{Kan07}. The results of the common \ncenter-of-velocity analysis of the radial velocities are  presented in Table\\,2. The best fit was obtained \nfor a circular orbit with a mass-ratio of $q=\\frac{M_2}{M_1}$=0.571$\\pm$0.010. The dotted lines in \nFig.\\,1 show the computed curves. \n\n\\begin{figure*}\n\\center\n\\includegraphics[width=16cm,angle=0]{RV.ps}\n\\caption{Radial velocities for the components of V745\\,Cas. Symbols with error bars (masked by the symbol size) \nshow the radial velocity measurements for the components of the system (primary: filled squares, \nsecondary: open squares).} \\end{figure*}\n\n\\begin{table}\n\\scriptsize\n\\centering\n\\begin{minipage}{85mm}\n\\caption{Heliocentric radial velocities of V745\\,Cas. The columns give the heliocentric \nJulian date, orbital phase and the radial velocities of the two components with the corresponding \nstandard deviations.}\n\n\\begin{tabular}{@{}ccccccccc@{}c}\n\\hline\nHJD 2400000+ & Phase & \\multicolumn{2}{c}{Star 1 }& \\multicolumn{2}{c}{Star 2 } \t\\\\\n             &       & $V_p$                      & $\\sigma$                    & $V_s$   \t& $\\sigma$\t\\\\\n\\hline\n55929.4430  &   0.4420\t&  -79  &  5 &     95 &  6 \\\\\n55930.2400  &   0.0071\t&  -22  &  7 &\t  $--$ &  9 \\\\\n56130.5869  &   0.0395\t&  -33  &  6 &\t   38 &  8 \\\\\n56131.5227  &   0.7030\t&  114  &  3 &\t -258 &  4 \\\\\n56132.4341  &   0.3491\t& -154  &  4 &\t  218 &  4 \\\\\n56133.3873  &   0.0248\t&  -45  &  5 &\t  $--$ &  6 \\\\\n56134.5216  &   0.8290\t&  121  &  4 &\t -272 &  7 \\\\\n56136.5308  &   0.2534\t& -167  &  2 &\t  256 &  3 \\\\\n56137.4836  &   0.9288\t&   54  &  6 &\t -151 &  7 \\\\\n56162.4659  &   0.6396\t&   99  &  2 &\t -222 &  3 \\\\\n56162.5262  &   0.6824\t&  121  &  2 &\t -248 &  3 \\\\\n56167.5812  &   0.2660\t& -177  &  3 &\t  251 &  3 \\\\\n\n\\hline\n\\end{tabular}\n\\end{minipage}\n\\end{table}\n\n\n\\begin{table}\n\\scriptsize\n  \\caption { Results of the radial velocity analysis for V745\\,Cas}\n  \\label{parameters}\n  \\begin{tabular}{lrr}\n  \\hline\n   Parameter \t\t\t\t\t& Primary\t&\tSecondary\t\t     \\\\   \n   \\hline\n  K (km s$^{-1}$) \t\t\t  & 156$\\pm$2 \t      &273$\\pm$3\t\t    \\\\\n    V$_\\gamma$(km s$^{-1}$) \t\t &-16$\\pm$2  \t      &-16$\\pm$2\t\t    \\\\   \n   Average O-C (km s$^{-1}$)\t\t& 4         \t      &3      \t        \\\\\n   $M$\\,$\\sin^3 i$ (M$_{\\odot}$)  &7.34$\\pm$0.20      &4.20$\\pm$0.11\\\\\n    $a$ sin$i$ (R$_{\\odot}$)\t\t& 4.348$\\pm$0.056\t  & 7.608$\\pm$0.083\t\\\\  \n\\hline  \n  \\end{tabular}\n\\end{table} \n\n\n\n\n\\section{Spectral analysis}\n\nFundamental stellar parameters, such as projected rotational velocity ($v\\,sin\\,i$), spectral type ($Sp$),\nluminosity class, effective temperature (T$_{eff}$), surface gravity ($log~g$), and metallicity ([Fe\/H]) may be\ndetermined from the mid-resolution optical spectroscopy.\n\n\\subsection{The projected rotational velocities}\nProjected rotational velocities of early-type stars were determined usually by means of various methods: the full-width-at-half-maximum\n(FWHM) method \\citep{Abt02}, the goodness-of-fit ($GOF$) method \\citep{Con77}, the cross-correlation-function ($CCF$)\nmethod \\citep{Pen96}, the Fourier transform ($FT$) method \\citep{Gra76},  and the $iacob-broad$ method \\citep{Sim14},\nbased on a combination of GOF and FT methodologies.\n\nWe used the $CCF$-method for measurement of the individual\nprojected rotational velocities ($v \\sin i$) of the component stars. The rotational velocities of the components\nwere obtained by measuring the full-width-at-half-maximum (FWHM) of the CCFs in five high-S\/N spectra of the components  \nacquired close to the quadratures, where the spectral lines have the largest Doppler-shift.\nThe He\\,{\\sc i} lines at $\\lambda\\lambda$4471, 6678 and 7065 lines were selected. \nThe CCFs were used for the determination of $v\\,sin\\,i$ through a calibration of the FWHM of \nthe CCF peak as a function of the $v\\,sin\\,i$ of artificially broadened spectra of slowly rotating standard \nstar (21\\,Peg, $v \\sin i$ $\\simeq$14 km s$^{-1}$, e.g., \\citet{Roy02}) acquired with the same set up and in \nthe same observing night as the target. The limb darkening coefficient was fixed at the theoretically\npredicted values of 0.42 for both stars \\citep{Van93}.  We calibrated the relationship between the CCF Gaussian\nwidth and $v\\,sin\\,i$ using the \\citet{Con77} data sample. This analysis yielded projected rotational\nvelocities for the components of V745\\,Cas as  $V_psin~i$=171 km s$^{-1}$, and $V_ssin~i$=151 km s$^{-1}$. The\nmean deviations were 4 and 8 km s$^{-1}$, for the primary and secondary, respectively, between the measured\nvelocities for different lines. \n\n\n\n\\subsection{The spectral classification}\nSpectral types of the components were first estimated by comparison both with our standard stars' spectra and \nalso with templates taken from the \\citet{Val04} $Indo-U.S.\\ Library\\ of\\ Coude\\ Feed\\ Stellar\\ Spectra$ (with \na resolving power of about R=3600) that are representative of stars with various metallicities, spectral types \nfrom late-O type to early-A, and luminosity classes V, IV, and III.\n\nWe have performed a spectral classification for the components of the system using COMPO2, an IDL ({\\sc Interactive \nData Language RSI}) code for the analysis of the spectra of SB2 systems written by \\citet{Fra06}. This code, similar \nto the {\\sf SYNTHE} code \\citep{Kur81}, was adapted to the TFOSC spectra of the binary systems. This code searches \nfor the best combination of two standard-star spectra able to reproduce the observed spectrum of the system. We give, \nas input parameters, the radial velocities and projected rotational velocities $v\\sin i$ of the two components, which \nwere already derived. The code then finds, for the selected spectral region, the spectral types and fractional flux \ncontributions that better reproduce the observed spectrum, i.e. which minimize the residuals in the collection of \ndifference (observed\\,$-$\\,constructed) spectra.\n\nThe atmospheric parameters of the reference stars given by \\citet{Val04} were recently revised by \\citet{Wu11}. For \nthis task we selected about 200 single-star' spectra spanning the ranges of expected atmospheric  \nparameters, which means that we have searched for the best combination of spectra among 39204 possibilities per \neach spectrum. \n\nThe observed spectra of V745\\,Cas around the $\\lambda\\lambda$4471, and 6678 spectral lines were best represented by \ncombination of the spectra of HD\\,34816 (B0.5 V, $log~g$=3.91) and HD\\,184915 (B1.5 III, $log~g$=3.40). We have derived the \nspectral types for the primary and secondary component of V745\\,Cas as B(0$\\pm$0.5)\\,V and B(1.5$\\pm$0.5)\\,III, \nrespectively. The atmospheric parameters obtained by the code are presented in Table\\,3. The observed spectra of \nV745\\,Cas at nearly quadratures around the He\\,{\\sc i} lines at $\\lambda\\lambda$4471, and 6678 are compared with \nthe combination of two  single-star' spectra in Fig.\\,2.\n\n\nThis result can be confirmed using the ratios of equivalent widths of He {\\sc i } 4471 and He {\\sc ii} 4541 lines. Spectral \nlines of the components are well separated at the quadratures of the eclipsing pair. We measured equivalent widths (EW) \nof the primary star He {\\sc i } 4471 and He {\\sc ii} 4541 lines for estimating {\\bf its spectral type}. The average \nlogarithmic ratio of the equivalent widths (EW) of the primary is about 1.19$\\pm$0.17, which corresponds to a B0 \ntype star \\citep{Huc96}.  The luminosity criterion based on the logarithm of the ratio of the line strengths of \nSi\\,{\\sc iv} 4088 and He\\,{\\sc i} 4143 \\citep[]{Con86,Huc96} yields $-1.07 \\pm 0.05$. This value corresponds \nto main-sequence (luminosity class V) stars.\n\n\nFurther support for this classification comes from the photometry of V745\\,Cas available in the literature. The \nvisual apparent magnitude and colour indices were given by \\citet{Gue74} as $V$=8$^m$.16, ($U-B$)=$-$0$^m$.83 \nand ($B-V$)=0$^m$.05 with an uncertainty of $\\pm$0.01 mag. The quantity $Q$=($U-B$)$-$($E_{(U-B)}$\/($E_{(B-V)}$)$(B-V)$ \nis independent of interstellar extinction. The average value of the ratio ($E_{(U-B)}$\/($E_{(B-V)}$) is 0.72$\\pm$0.03 \n(\\cite{Joh53}; \\cite{Hov04}). We compute the reddening-free index from the data of \\citet{Gue74} as $Q$=$-$0.866$\\pm$0.014. \nThe $Q$ $-$ values were calculated by \\citet{Hov04} begining from O8 to G2 spectral types for the luminosity classes \nbetween main-sequence and supergiants. The $Q$$-$$Sp$ calibration yields an approximate spectral type of the system \nas B0 with a range from O9 to B1. The spectral classification from the spectra and photometric indices are quite \nconsistent. \n\nIn addition, infrared colors J$-$H=$-$0.057$\\pm$0.064, H$-$K=0.015$\\pm$0.062\\,mag are given in the 2MASS \ncatalog \\citep{Cut03}. These indices confirm spectral classifications made by spectra and wide-band \nphotomeric indices. A preliminary analysis of the light curve (see Section 4.3) yields light ratio of 0.43 for the V-passband.\nThis light ratio, the observed colours and the intrinsic colour of the primary star of $(B-V)_0$=$-$0.30$\\pm$0.01 \\citep{Dri00} allow us to estimate\nan intrinsic composite colour of $(B-V)_0$=$-$0.29$\\pm$0.01. Thus, the interstellar \nreddening of $E_{(B-V)}$=0.34$\\pm$0.01\\,mag is estimated for the system. \n\n \n\n\n\\begin{table}\n\\scriptsize\n\\centering\n\\begin{minipage}{85mm}\n\\caption {Spectral types, effective temperatures, surface gravities, and rotational velocities of each star \nestimated from the spectra of V745\\,Cas.}\n\\begin{tabular}{@{}lcccccccc@{}c}\n\\hline\nParameter    \t\t\t& \\multicolumn{2}{c}{\\sf V745\\,Cas}\t&  \t\t\t\\\\\n             \t\t\t& Primary                   \t\t& Secondary   \t\t\\\\\n\\hline\n Spectral type \t\t\t& B(0$\\pm$0.5)\\,V \t\t\t&B(1.5$\\pm$0.5)\\,III\t\\\\\n T$_{eff}$ (K)\t    \t\t&30\\,000$\\pm$880  \t\t\t&25\\,600$\\pm$1\\,050\t\\\\   \n $\\log~g$ ($cgs$)\t\t& 3.91$\\pm$0.00         \t\t&3.40$\\pm$0.00       \t\\\\    \n $Vsin~i$ (km s$^{-1}$)  \t&171$\\pm$4\t  \t\t&151$\\pm$8     \t\\\\  \n\\hline\n\\end{tabular}\n\\end{minipage}\n\\end{table}\n\n\n\\begin{figure*}\n\\center\n\\includegraphics[width=14cm,angle=0]{ex.ps}\n\\caption{Comparison between the observed spectra of V745\\,Cas obtained near quadratures and the best-fitting \nspectra around He\\,{\\sc i} $\\lambda$4471 (left panel) and  the  $\\lambda$6678 lines (right panel). The deeper \nlines indicate the primary and the shallow ones the secondary star.}\n\\end{figure*}\n\n\n\n\\section{Analyses of the light curves}\n\\subsection{Light Curve Constraints}\nThe first photometric observations of V745\\,Cas were made by the Hipparcos mission and 144 H$_p$ magnitudes were \nlisted by \\citet{Van07}. These magnitudes were obtained in a time interval of about three years. The average accuracy \nof the Hipparcos data was given as $\\sigma_{H_p}$ $\\sim$ 0.01\\,mag. The light variation from peak-to-peak is about \n0.12\\,mag. However, we have estimated a scatter of about 0.022\\,mag at the maxima and within the primary minimum.\nThe scatter at the secondary eclipse is slightly larger, amounting to 0.028\\,mag.  \n\nThe second set of photometric observations were made by the $International~Gamma-Ray~Astrophysics~Laboratory$ ($INTEGRAL$)\nmission \\citep{Alf13}]. The $Integral-OMC$ database contains light curves for many previously unknown variable stars.\nWe extracted the V-passband light curve for V745\\,Cas ($Integral-OMC$\\,4019000053) from this catalog. The $INTEGRAL$\ndata are consisting of 1\\,980 Johnson's V-passband magnitudes. \\citet{Dom03} estimate an average accuracy of about\n0.006\\,mag for objects of V=12\\,mag.  \n\n \nThe 1\\,980 photometric measurements of $Integral$, including eclipses, permit determination of the\ntime of mid-primary eclipse and the orbital period of the system. A periodogram analysis, e.g. PERIOD04 \n\\citep{Len05}, has been applied to the data obtained by $INTEGRAL$ mission. We determine following ephemeris \n\n\\begin{equation}\nMin I(HJD)=2\\,455\\,222.1234(2)+1^d.410571(7) \\times E\n\\end{equation}\nwhere the standard deviations in the last significant digits are given in parentheses. However, \\citet{Alf12}\ndetermined an orbital period of 1.4106019\\,d, which is slightly longer than we found. Periodogram analysis\nindicates that there is a light variation with a second period of about 0.1659$\\pm$0.0007 days (approximately 4 h)\nhaving a peak-to-peak amplitude of about 0.014\\,mag. If this light variation really exists it may originate from \nintrinsic variation of one or both stars. The spectral types of the components are similar to those of $\\beta$\\,Cephei \ntype pulsating stars. Therefore such intrinsic variation may be expected from the components of V745\\,Cas. However, \nexistence of such a variation should be checked by precise multi-passband photometric observations.   \n\nWe have measured the magnitude difference between the A and B components of the visual pair as 3.13\\,mag \n(see Chapter 4) in the V-passband. This difference is slightly larger than the difference of 3.01\\,mag given by \nthe $WDS~catalogue$. Since the INTEGRAL Optical Monitoring Camera has an angular resolution of about 23\\,arcsec \nits photometric  data cover the light contribution of the component B. Light contribution of the component B to the total \nlight has been calculated as 0.053 which slightly affects the observed magnitudes. Therefore we subtracted the \nlight contribution of component B from all the measured magnitudes. On the other hand the $H_p$ magnitudes measured\nby the $Hipparcos$ mission were transformed to the Johnson's V-passband using the transformation coefficients given\nby \\citet{Har98}. Then both data sets were combined for a simultaneous analysis. All available photometric data are \nphased and plotted in Fig.\\,3, where the open circles represent the Hipparcos data. It is clear from Fig.\\,3 that \nV745\\,Cas is an eclipsing binary, with a W\\,UMa type light curve that is dominated by variations due to tidal distortion. \n\nIn order to model the photometric light curve, we have used the computer programme $PHOEBE$ (PHysics Of Eclipsing \nBinariEs) developed by \\citet{Prs05}. This program is an implementation of the Wilson-Devinney (hereafter WD) \nbinary star code \\citep{Wil71}. The Roche geometry, sensitive to the mass-ratio, is used to represent the tidal \ndeformation of the two stars. One of the main difficulties in this modelling is determination of effective \ntemperature of the primary star and the mass-ratio of the system. Effective temperatures of both stars could \nalready be determined from the spectra. In addition the mass-ratio, which is the second key-parameter for \nthe modelling, was also obtained from the radial velocities of the stars.        \n\n\n\n\n\\begin{figure*}\n\\center\n\\includegraphics[width=14cm,angle=0]{son_v745caslc.ps}\n\\caption{V-passbad light curve constructed from the $INTEGRAL$ (empty circles) and $Hipparcos$ \n(filled stars) photometric data of V745\\,Cas. The continuous and dashed lines show the best-fit \nmodels obtained by the WD code using the Mode 6 and Mode 3, respectively.} \\end{figure*}\n\n\n\n\\begin{table}\n\\scriptsize\n\\caption{Results of the analyses of the light curve for V745\\,Cas.}\n\\begin{tabular}{lrrrrr}\n\\hline\nParameter & Mode 2 & Mode 3 & Mode 4 & Mode 5 & Mode 6  \\\\\n\\hline\t\n$i^{o}$\t\t\t        &43.65$\\pm$0.03\t\t&47.30$\\pm$0.32\t&44.08$\\pm$0.12\t&48.07$\\pm$0.20\t    &47.51$\\pm$0.22      \\\\\nT$_{eff_1}$ (K)\t\t\t&30\\,000[Fix]\t\t&30\\,000[Fix]\t\t&30\\,000[Fix]\t\t&30\\,000[Fix]\t    &30\\,000[Fix]        \\\\\nT$_{eff_2}$ (K)\t\t\t&14\\,210$\\pm$235\t&25\\,350$\\pm$315\t&17\\,580$\\pm$320\t&25\\,350$\\pm$301    &25\\,540$\\pm$300     \\\\\n$\\Omega_1$\t\t\t&3.0572$\\pm$0.0079\t&3.0098$\\pm$0.0036\t&3.0099       \t\t&3.0181$\\pm$0.0067    &3.0099$\\pm$0.019     \\\\\n$\\Omega_2$\t\t\t&2.8105$\\pm$0.0040\t&3.0080$\\pm$0.019\t&2.8908$\\pm$0.0081\t&3.0099$\\pm$0.019    &3.0099$\\pm$0.019     \\\\\n$r_1$\t\t\t\t&0.4202$\\pm$0.0015\t&0.4285$\\pm$0.0007\t&0.4281$\\pm$0.0016\t&0.4251$\\pm$0.0013  &0.4281$\\pm$0.0016   \\\\\n$r_2$\t\t\t\t&0.3816$\\pm$0.0012\t&0.3301$\\pm$0.0007\t&0.3572$\\pm$0.0021\t&0.3297$\\pm$0.0015  &0.3297$\\pm$0.0015  \\\\\n$\\frac{L_{1}}{(L_{1}+L_{2})}$   &0.8274$\\pm$0.0033\t&0.7033$\\pm$0.0047\t&0.7951$\\pm$0.0048\t&0.7002$\\pm$0.0047  &0.6979$\\pm$0.0046 \\\\\n$\\sum(O-C)^{2}$\t\t\t&1.0242\t\t\t&1.0484              \t&1.0353              \t&1.0486              &1.0477              \\\\\t\t\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\n\\begin{figure*}\n\\center\n\\includegraphics[width=14cm,angle=0]{roche.ps}\n\\caption{Equipotential structure of V745\\,Cas corresponding to the solution of the light curve using Mode 6 at the mid-primary (left) and \nmid-secondary  (right) eclipses. Comparison with the Roche lobes is shown in the middle.  } \\end{figure*}\n\n\nLinear limb-darkening coefficients were interpolated from the tables of \\citet{Van93}. They are \nupdated at every iteration by $PHOEBE$. The gravity-brightening coefficients $g_1$=$g_2$=1.0 and albedos \n$A_1$=$A_2$=1.0 were fixed for both components, as appropriate for stars with radiative atmospheres.\n\n\\subsection{Light Curve Solutions}\n\nWe started with the  $Mode-2$ of the Wilson-Devinney code, referring to the detached Algols for the analysis\nof light curves as described by \\citet{Wil06}. The adjustable parameters in the light curve fitting were the orbital\ninclination ($i$), the effective temperature of the secondary star (T$_{eff_2}$), the potentials ($\\Omega_1$ and $\\Omega_2$), \nthe luminosity of the primary (L$_1$), and the zero-epoch offset. The parameters of the solution are tabulated as $Mode-2$ in \nTable\\,4. The effective temperature, fractional luminosity and fractional radius of the secondary are about 14 000 K, 0.17, and \n0.38, respectively. The effective temperature and fractional luminosity are too small, nearly half, when compared to the values \nestimated from the spectra. We obtain an inclination of about 43.7 degrees which yields masses of about 22 and 13 M$_{\\odot}$ \nfor the primary and secondary star respectively. When the volumes of the stars are compared with their corresponding Roche lobes we see that \nwhile the primary fills up its lobe the secondary star overflows its lobe. Since the configuration is inconsistent with the \ndetached assumption the results of the analysis are taken to be unacceptable. \n \nNext, we tried $Mode-3$ (for overcontact systems, the stars are in geometrical contact without being in thermal contact).\nThe adjustable parameters were $i$, T$_{eff_2}$, and L$_1$. The results of the analysis are presented in the third column \nof Table\\,4. Analysis with this mode yields higher $i$, T$_{eff_2}$, and L$_2$ but lower fractional radius for the secondary star.\nAlthough the sum of residuals squared is slightly larger from that obtained in $Mode-2$ these results are in agreement with those \nobtained from the spectra. However, comparison with the Roche lobes shows that both components are in contact with their inner Roche lobes,\nwhich is inconsistent with the assumption of overcontact configuration. In other words, the solution clearly reveals a fill-out factor \n$f$=($\\Omega_{in}$ $-$ $\\Omega$)\/($\\Omega_{in}$ $-$ $\\Omega_{out}$)=0, corresponding to the contact configuration.     \n\nThe results of the analyses using $Mode-4$ (primary star fills its lobe) and $Mode-5$ (secondary star fills its lobe) were given\nin the fourth and fifth columns of Table\\,4. The analysis with $Mode-4$ gives similar results as $Mode-2$, but slightly larger \n$\\sum(O-C)^{2}$. While the primary component fills its lobe the secondary overfills its lobe which is inconsistent with the \n$a$ $priori$ assumption. The analysis under the assumption of $Mode-5$ gives the parameters which are very close to those \nobtained by the $Mode-3$ solution. However, comparison with the Roche lobes shows that both components are just in contact \nwith their corresponding Roche lobes.     \n\nComparison with the Roche lobes\npoints out double contact configuration. Therefore, we applied $Mode-6$ (for double contact systems) for the analysis of the\nobserved light curve. The results of this analysis are given in the last column of Table\\,4. The uncertainties assigned to \nthe adjusted parameters are the internal errors provided directly by the code. While the sum of residuals squared is\nslightly larger than those obtained by $Mode-2$ and $Mode-4$, it is slightly smaller than those obtained by $Mode-3$ and $Mode-5$.\nHowever, it should be noted that the solutions obtained with $Mode-3$, $Mode-5$ and $Mode-6$ are of similar quality and yield similar \nmodel parameters. The results obtained with $Mode-6$ are in good agreement with the atmospheric parameters determined from the \nspectra. The computed light curve is compared with the observations in Fig.\\,3. This solution indicates that the grazing \neclipse, lasting about five hours, occurs in the V745\\,Cas system. In Fig.\\,4 we show the equipotential surfaces of both \ncomponents for the light curve solution at the phases mid-primary and mid-secondary eclipses. In the middle of the figure \nthe volumes of the stars are compared with the Roche lobes.  \n\n\n\\begin{figure*}\n\\begin{center}\n   \\includegraphics[width=14cm]{gen_eco.ps}\n\\end{center}\n  \\caption{Positions of the components of the system in the luminosity-effective temperature and \n  gravity effective temperature planes. The solid lines show evolutionary tracks for \n  stars with masses of 40, 35, 30, 25, 20, 15 and 10 solar masses taken from \\citet{Eks12}. }  \\label{fig:evo}\n\\end{figure*}\n\n\n\n\n\\section{Results and discussion}\nCombining the results of radial velocities and light curve analyses we have calculated the absolute parameters \nof the stars. Separation between the components of the eclipsing pair is calculated as $a$=16.21$\\pm$0.14R$_{\\odot}$. \nThe fundamental stellar parameters for the components such as masses, radii, luminosities are listed in Table\\,5 \ntogether with their formal standard deviations. The standard deviations of the parameters have been determined by \nmeans of the JKTABSDIM\\footnote{This can be obtained from http:\/\/http:\/\/www.astro.keele.ac.uk\/$\\sim$jkt\/codes.html} code \n\\citep{Sou05}. The mass for the primary of $M_p$=18.31$\\pm$0.51 M$_{\\odot}$ and secondary of \n$M_s$=10.47$\\pm$0.28 M$_{\\odot}$ are consistent with B0\\,V and B(1-2)\\,IV-V stars.\n\nThe bolometric magnitudes of the components are computed as -6.62$\\pm$0.14 and -5.35$\\pm$0.06 mag for the primary\nand secondary components taking the absolute bolometric magnitude of the Sun as 4.74. The bolometric corrections listed\nin Table 5 are taken from  \\citet{Lan07} adopting V$_{tur}$= 2 km s$^{-1}$, and $[Z\/Z_{\\odot}]$=1.0 (solar metallicity)\nand the effective temperatures and the surface gravities of the components given in the table. Using the interstellar reddening\nof $E$(B-V)=0.34\\,mag and absorption coefficient of 3.1 we estimate the absorption in visual band as 1.05 mag. With V=8.18 at\nthe maximum light one finds $V_{0p}$=7.52 and $V_{0s}$=8.43 for the primary and secondary star, respectively. Then the \ndistance modulus are 11.16 and 11.15 mag which correspond to the distances of 1\\,703$\\pm$63 and 1\\,697$\\pm$38 pc. As a \nweighted mean, we find the distance to the system as 1\\,700$\\pm$50\\,pc.\n\nOur line-of-sight \npassing through the binary system V745\\,Cas traverses first the $Perseus$ arm of the $Galaxy$ and then the distant $Cygnus$ \narm. The mean residual radial velocity and Galactocentric distance of the OB-associations in the $Perseus$ region, including \nCas\\,OB4, were given as 8.4\\,kpc and -6.7km s$^{-1}$. We estimated the distance of V745\\,Cas from the galactic plane as 20.4\\,pc \nand from galactic center as 8.5\\,kpc, where the distance of sun from the galactic plane is taken as 20.5\\,pc \\citep{Hum95}. The \ndistances of the associations in the $Perseus$ arm from the Sun were estimated between 1.8 and 2.8\\,kpc \\citep{Mel09}. Therefore, \nwe suspect that V745\\,Cas may be a member of the Cas\\,OB4 association. If it is a member it should be located at the nearest \nedge of the association. This conclusion agrees with the result of \\cite{Mel95} who quoted V745 Cas as a member of Cas OB4, for \nwhich they gave a distance of 2.04\\,kpc.\n\n \n\n\\begin{table}\n\\scriptsize\n  \\caption{Properties of the V745\\,Cas components}\n  \\label{parameters}\n  \\begin{tabular}{lrr}\n  \\hline\n   Parameter \t\t\t\t\t\t            & Primary\t       &\tSecondary\t\t      \\\\   \n   \\hline\n\tMass (M$_{\\odot}$)\t\t\t\t        & 18.31$\\pm$0.51\t\t&10.47$\\pm$0.28\t\t\\\\\n\tRadius (R$_{\\odot}$)\t\t\t\t    & 6.94$\\pm$0.07\t\t    & 5.35$\\pm$0.05\t\t\\\\   \n\t$T_{eff}$ (K)\t\t\t\t\t        & 30\\,000$\\pm$1000\t    & 25\\,540$\\pm$300\t\\\\\n   $\\log~(L\/L_{\\odot})$\t\t\t\t        & 4.547$\\pm$0.056\t    & 4.040$\\pm$0.025\t\\\\\n   $\\log~g$ ($cgs$) \t\t\t\t\t    & 4.018$\\pm$0.006 \t    & 4.002$\\pm$0.007\t\\\\\n   $Sp.Type$ \t\t\t\t\t\t        & B0V\t\t\t\t    & B(1-2)IV-V  \t\t\\\\\n   $M_{bol}$ (mag)\t\t\t\t\t        & -6.62$\\pm$0.14\t\t&-5.35$\\pm$0.06\t\t\\\\\n   $BC$ (mag)\t\t\t\t\t\t        & -2.98          \t    &-2.63   \t\t\t    \\\\  \n   $M_{V}$ (mag)\t\t\t\t\t\t    & -3.64$\\pm$0.14\t\t&-2.72$\\pm$0.06\t\t\\\\  \n   $(vsin~i)_{calc.}$ (km s$^{-1}$)\t        & 184$\\pm$2\t\t\t    & 142$\\pm$2\t\t\t  \\\\       \n   $(vsin~i)_{obs.}$ (km s$^{-1}$)\t        & 171$\\pm$4\t\t        & 151$\\pm$8 \t\t  \\\\    \n   $d$ (pc)\t\t\t\t\t\t\t        & 1\\,703$\\pm$63\t \t    & 1\\,697$\\pm$38\t\t\\\\\n\\hline  \n  \\end{tabular}\n\\end{table}\n\n\n\nThe distance to the Cas\\,OB4 association had been subjected to various studies. There are a number of well-known OB associations \nand galactic open clusters in the direction of Cas\\,OB4 association (approximately $l$=120, $b$=0 degrees). The mean interstellar \nreddening of 0.40, 0.34, 0.55, {\\bf 0.47} and 0.55 mag, and distance of 1.43, 2.96, 3.8, 1.67 and 3.47 \\,kpc were found for Mayer1 \n\\citep{Kha05}, King\\,14 \\citep{Net06}, NGC\\,103 \\citep{Phe93}, NGC\\,129 \\citep{Tur92} and NGC\\,146 \\citep{Sub05}, respectively. The \nderived systemic velocity, reddening and distance of V745\\,Cas agree quite well with those values of NGC\\,129.\n\n\n\nWe have observed the components A, B, C and D of the multiple stellar system (WDS\\,J00229+6214) simultaneously using the 100\\,cm telescope \nof the Turkish National Observatory of Turkey. We used the A-star as a reference for which UBV magnitudes are known. The photometric \nobservations were made when the component A was at maximum light. The wide-band UBV magnitudes of the components B, C and D were \nobtained with respect to the component A and are presented in Table\\,6. The \nstandard deviations of the measurements are about 0.01\\,mag. The apparent visual magnitude for C-star with a separation of 23.2\\,arcsec  \nwas given in the WDS catalogue as 10.8 which is inconsistent with that we determined. We suspect that the C-star may be variable in \nbrightness. Using the Johnson Q$-$method we determined the spectral types of the stars as B3, B8 and B7 for B, C and D, respectively. \nAccording to this classification the interstellar reddening for B and C is similar with that of the eclipsing pair. However \nthe D star appears to have the largest reddening, amounting to E(B-V)=0.60\\,mag. Such a result is expected because the reddening \nmay vary from region to region in the association depending on the density of molecular clouds.     \n\n\n\\begin{table}\n\\scriptsize\n\\caption{UBV measurements of the components B, C and D of the multiple stellar system and their probable spectral types.}\n\\begin{tabular}{lrrrrr}\n\\hline\nStar & U & B  & V  & Q  & Sp.Type \\\\\n\\hline\t\n$B$\t\t\t&10.82\t&11.32  &11.25 &-0.542 & B3 \t\t\t\t\t\\\\\n$C$\t\t\t&13.47  &13.54  &13.33 &-0.218 & B8\t\t\t\t\t\t\\\\\n$D$\t\t\t&12.62  &12.65  &12.18 &-0.357 & B7\t\t\t\t\t\t\\\\\t\t\t\n\\hline\n\\end{tabular}\n\\end{table}\n\nFigure\\,5 shows the components, with 1-$sigma$ error bars, of V745\\,Cas in the log T$_{eff}$-log L\/L$_{\\odot}$ \n(left panel) and log T$_{eff}$-log\\,g planes (right panel). The evolutionary tracks and isochrones for the\nnon-rotating single stars with solar composition are taken from \\citet{Eks12}. The models considering mass\nloss and convective overshooting are adopted. We typically compare the positions of the stars with evolutionary\ntracks and determine what masses they would predict. This comparison points out a mass slightly smaller for\nthe primary but higher for the secondary. The locations of the components in both\npanels show that the components are in the main sequence band, not far from the ZAMS. Although both components are \nyoung they fill their corresponding Roche lobes due to very small separation. Therefore, one should not necessarily expect\na good agreement with single star tracks.\n\nV745\\,Cas is one of a few massive contact binaries, namely TU\\,Mus, V382\\,Cyg, and LY\\,Aur \\citep{Pen08}. The masses of \nthe components as well as the orbital period are very similar to that of TU\\,Mus. As pointed out by \\citet{Pen08} the \nobserved mass ratios of the contact systems are slightly higher than those of the semi-detached systems. The components \nof the contact systems have similar spectral types (within one spectral type of each other) and luminosity classes. \n\\citet{Wel01} presented evolutionary calculations for 74 systems considering case A and B scenarios. Our analyses \nindicate that  V745\\,Cas is a contact system. Both components have nearly the same luminosity class and almost \nidentical surface gravities, as given in Table\\,5. Therefore, the system is most probably undergoing $case~A$ \nevolution. According to the \\citet{Wel01} models, $case~A$ contact systems can occur when the initial period of \nthe system is very small and\/or mass ratio is either very small or very large. The observed properties of the \ncomponents are similar with those of the system No.53, (initial masses of 16 and 12 M$_{\\odot}$ and orbital period \nof 1.5 days) in their Table\\,3. Their model shows that the more massive star reaches its Roche lobe fairly early \nin its evolution. Rapid mass transfer from the more massive star to the less massive star occurs, the orbital period \ndecreases which causes the loser star to overflow at a faster rate. Large increase in mass of the gainer star and \nsmaller orbit cause it to expand and reach its $Roche$ lobe. The orbit of the binary continues to shrink until the \ngainer star is more massive. Then the mass transfer after this point causes the orbit to expand, leading to a longer \norbital period. Fast $case~A$ mass transfer still continues until the loser is much less massive. Since the ratio \nof mass in the core to that of the envelope of the loser is increasing this causes its radius to increase. Finally \nthe gainer expands and reaches to the Roche lobe due to the increase in mass. Thus a contact system, similar to \nV745\\,Cas, is formed.            \n\n\n\\begin{figure*}\n  \\begin{center}\n      \\includegraphics[width=12cm]{mrt.ps}\n  \\end{center}\n  \\caption{Locations of the primary (solid dots) and secondary (open circles) components of the detached, young \n  eclipsing binary systems in various diagrams. (a) log L - log T$_{eff}$, (b)log L - log M, (c) log T$_{eff}$ - log M \n  and (d) log R - log M. The solid lines show the best correlations between the parameters. The component of \n  V745\\,Cas and MY\\,Ser are shown by squares and triangles. }  \\label{fig:evo}\n\\end{figure*}\n\nIn Fig.\\,6 we show the correlations between luminosity-effective temperature, mass-luminosity, mass-radius and \nmass-effective temperature for the components of the detached massive eclipsing binaries where the filled circles \nrefer to the primaries and the open circles to the secondaries. The parameters are taken from \\citet{Tor10}. These \nstars lie close to the zero-age main-sequence on the HR diagram. Though the sample contains only ten stars reliable \ncorrelations between the parameters could be obtained. Linear least-squares fit to the data gives following relationships,\n\n\n\\begin{equation}\nlog\\frac{L}{L_{\\odot}}=7.34(\\pm0.60) ~~ log T_{eff}-28.48(\\pm2.69) \\\\\n\\end{equation}\n\n\\begin{equation}\nlog\\frac{L}{L_{\\odot}}=3.07(\\pm0.17) ~~ \\frac{M}{M_{\\odot}}+0.84(\\pm0.19) \\\\ \n\\end{equation}\n\n\\begin{equation}\nlog T_{eff}=0.39(\\pm0.02) ~~ log\\frac{M}{M_{\\odot}}+4.01(\\pm0.03)\n\\end{equation}\n\n\\begin{equation}\nlog\\frac{R}{R_{\\odot}}=0.75(\\pm0.10) ~~ log\\frac{M}{M_{\\odot}}-0.08(\\pm0.11)\n\\end{equation}\n\nThe numbers in the parentheses are the standard errors of the preceding coefficients determined by linear least \nsquares solutions. The components of V745\\,Cas and MY\\,Ser are shown by squares (primary:filled, secondary:empty)  \nand triangles (primary:filled, secondary:empty) in Fig.\\,5. While the \nparameters for V745\\,Cas are determined in this study, the parameters for the components of MY\\,Ser are adopted \nfrom \\citet{Iba13}. The positions of the components of V745\\,Cas agree well with those of the components of \ndetached, young massive eclipsing binaries. This may be taken as an indicator that mass-exchange between the \ncomponents or mass-loss from them did not yet lead to a significant change on their radii and effective temperatures.  \n\n\n\\section{Conclusion}\n\nV745\\,Cas is one of the rare early type contact systems with massive components. Analysis of the radial \nvelocities and light curves yielded absolute parameters of the components. The components are classified \nas B0V and B(1-2)V spectral types with masses of 18.31$\\pm$0.51 and 10.47$\\pm$0.28 M$_{\\odot}$. \nComparison with the evolutionary models and with the parameters of detached massive binaries show that both \ncomponents are still close to the zero-age main-sequence. Although the components of V745\\,Cas  are in contact\nwith their Roche lobes they locate on the main-sequence band. Using our estimation of $E$(B-V) and A$_v$ we determined\na distance of about 1700$\\pm$50\\,pc which is in agreement with the distance estimations of the Cas\\,OB4 association.\nWe also observed V745\\,Cas, the brightest component of the multiple star system with the other components simultaneously.\nThese observations indicate that the components B, C and D are also massive stars with spectral types of B3, B8 and B7.\n   \n\n\n\n\n\n\n\\section*{Acknowledgments}\nWe thank to T\\\"{U}B{\\.I}TAK National Observatory (TUG) for a partial support in using RTT150 \ntelescope with project number 11BRTT150-198.\nWe also thank to the staff of the Bak{\\i}rl{\\i}tepe observing station for their warm hospitality. This \nstudy is supported by Turkish Scientific and Technology Council under project number 112T263.\nThe following internet-based resources were used in research for this paper: the NASA Astrophysics Data \nSystem; the SIMBAD database operated at CDS, Strasbourg, France; and the ar$\\chi$iv scientific paper \npreprint service operated by Cornell University. \nThis research was supported by the Scientific Research Projects Coordination Unit of Istanbul University. Project \nnumber 3685. We thank \\c{C}anakkale Onsekiz Mart University Astrophysics Research Center and Ulup{\\i}nar Observatory \ntogether with \\.{I}stanbul University Observatory Research and Application Center for their support and allowing use \nof IST60 telescope. The authors thank to the anonymous referee for helpful comments that improved the clarity of the text. \n\n\n\n\\bibliographystyle{elsarticle-harv}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\nA strongly interacting partonic medium\nis evident in multiple observables at RHIC.\nOne manifestation of this medium, the\n jet-quenching effect, was found in single-particle\nspectra \\cite{PHENIX:RAA} as well as in away-side di-hadron angular correlations \\cite{adams:152301}.\nThe latter have already yielded new insights into the\nmedium properties: away-side quenching indicates that the medium is\ndense, and transverse momentum distributions of associated hadrons\nare consistent with thermalization \\cite{adams:152301}.\nAlteration of the angular distribution on both \n(same- and away-) sides of a high \\ensuremath{p_{T}}\\,trigger reveals \na strong modification of known hadron production mechanisms.\nAlternatively, it has been\nargued that the peculiar features found in the triggered\ndi-hadron distributions, such as a double-humped away-side structure \\cite{Horner:2007gt}\nand a ``ridge'' along pseudo-rapidity $\\Delta\\eta$ formed on the same-side of the trigger,\ncould be caused by the medium's response to propagated jets \\cite{Putschke:2007mi}.\n\nSeveral three-particle correlation techniques \nhave been developed to further investigate proposed mechanisms for jet modifications like \njet deflection or the Mach cone\/shock wave mechanism.\nExamples include $\\Delta\\phi$--$\\Delta\\phi$ correlation analyses \\cite{Ulery:2006ha}\n and $\\Delta\\eta$--$\\Delta\\eta$ studies \\cite{STAR:EtaEta}.\nBoth methods use one high \\ensuremath{p_{T}}\\,trigger particle and then study the correlation of two lower \\ensuremath{p_{T}}\\,associate particles, and in that sense \nare denoted as 1+2 techniques. With our 2+1 method we instead use a pair of correlated high \\ensuremath{p_{T}}\\,hadron triggers\nto determine the jet axis and study associated hadron distributions with respect to this axis.\nThus we  aim to explore both same- and away-side features with a well-defined jet axis at the same time.\nThis new approach has already shown unique results \\cite{Barannikova:2008zz}; here we present the most recent developments.\n\n\\section{Data Analysis}\n\\label{sec:data-analysis}\nA description of the method and initial results were introduced in \\cite{Barannikova:2008zz}. \nTo summarize, the re-emergence of an away-side peak in angular correlations of two high \\ensuremath{p_{T}}\\,particles \\cite{Adams:2006yt}\nallows us to determine the away-side jet direction in both azimuth and pseudo-rapidity.\nThe highest transverse momentum particle in the event is designated as a primary trigger (T1)\nif it is at mid-rapidity, $|\\eta|<1$,  and if it satisfies the transverse momentum threshold.\nA second trigger (T2) is selected back-to-back with the primary trigger, i.e.\\,within $|\\Delta\\phi-\\pi|<\\alpha$,\nwhere $\\Delta\\phi$ is the relative azimuthal angle between the two triggers.\nWe choose $\\alpha = 0.2$ as a characteristic away-side peak width for this kinematic regime.\nT2 satisfies the same pseudo-rapidity cuts\nand both triggers have to obey additional track quality cuts.\nThe distribution of associated hadrons (A) is then studied in both $\\Delta\\eta$ and $\\Delta\\phi$ dimensions\nwith respect to the jet axis as determined by this pair of triggers. The associated hadrons have to satisfy similar track quality cuts as the trigger pair.\n\nThe raw angular correlation is constructed similarly to other di-hadron analyses \\cite{Horner:2007gt,Putschke:2007mi}.\nTo correct for pair acceptance we use the mixed-event technique.\nThe combinatorial background is modulated by the $\\cos(2\\Delta\\phi)$ function to account for elliptic flow effects.\nWe use published STAR results for the centrality and \\ensuremath{p_{T}} \\,dependence of the elliptic flow coefficient $v_2$ \\cite{Adams:2004bi}.\nThe overall background level is determined via the Zero-Yield at Minimum (ZYAM) method as in \\cite{Ulery:2006iw}.\n\nSpecific to 2+1 correlations is the correlated background, resulting from randomly associated trigger pairs.\nThe angular distribution of the associated hadrons for these trigger pairs is obtained from\n di-hadron correlations with appropriate kinematic selections.\nWe scale this background by the signal-to-noise ratio in the away-side peak trigger-trigger\n correlation used for jet axis determination and subtract this contribution from the raw distribution.\nThe resulting corrected signal is scaled back so that ultimately the correlation is normalized per di-jet trigger.\n\n\\section{Results}\n\\label{sec:results}\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[scale=0.29]{Proj_both_phi}\n  \\includegraphics[scale=0.29]{Proj_both_eta}\n\\caption{Yields in the asymmetric case (normalized per number of triggers N$_T$): High-Tower triggered 200\\ensuremath{\\mytext{GeV}}\\,Au+Au data,\n    $8\\,\\ensuremath{\\mytext{GeV}}<\\ensuremath{E_{T}} (T1)<15\\,\\ensuremath{\\mytext{GeV}}$, $4\\,\\ensuremath{\\mytext{GeV}\/c}<\\ensuremath{p_{T}} (T2)<\\min\\{\\ensuremath{p_{T}} (T1),10\\,\\ensuremath{\\mytext{GeV}\/c}\\}$, $1.5 \\, \\ensuremath{\\mytext{GeV}\/c} <\\ensuremath{p_{T}} (A)<10\\,\\ensuremath{\\mytext{GeV}\/c}$.\n    Upward triangles signify the same-side correlation and are compared directly with\n    the away-side correlation in the downward triangles.\n    $\\Delta\\phi$ and $\\Delta\\eta$ are calculated with respect to T1 on the same-side and with respect to T2 on the away-side.\n    The left figure shows the $\\Delta\\phi$ projection; the right figure shows the $\\Delta\\eta$ projection.\n    Error bars are statistical only.}\n \\label{fig:AsymmRes}\n\\end{figure}\n\nBelow we consider  two scenarios with different relative \\ensuremath{p_{T}}\\,between the\ntwo trigger particles. The first is a symmetric scenario, as in our early results \\cite{Barannikova:2008zz}, in which\nthe T1 trigger is selected between 5 and 10 \\ensuremath{\\mytext{GeV}\/c}\\,\nand T2 above 4 \\ensuremath{\\mytext{GeV}\/c}. Associated hadrons were selected with \\ensuremath{p_{T}}\\,between 1.5 and 4 \\ensuremath{\\mytext{GeV}\/c}.\n20M 200\\ensuremath{\\mytext{GeV}}\\,Central Au+Au events were used for this analysis.\nThe availability of the High-Tower triggered 200 \\ensuremath{\\mytext{GeV}}\\,Au+Au data set in STAR with\n good statistics in the higher \\ensuremath{p_{T}}\\,region\nenabled us to consider a second, asymmetric scenario. Using the Barrel Electromagnetic Calorimeter, we required the\nT1 trigger's transverse energy \\ensuremath{E_{T}}\\,to be between 8 and 15 \\ensuremath{\\mytext{GeV}}, rejecting triggers that coincided with a charged hadron track.\nThe selection criteria for T2 remained unchanged, associated particles were selected \nto have transverse momenta in the TPC above 1.5 \\ensuremath{\\mytext{GeV}\/c}. \nWhere \\ensuremath{p_{T}}\\,cuts were used, values above the resolution threshold of 10\\ensuremath{\\mytext{GeV}\/c}\\,disqualified an event in both scenarios.\n\nPreviously, we found no modifications on the away-side of a di-jet in the symmetric case \\cite{Barannikova:2008zz}.\nIn the present analysis we confirmed the absence of the jet-medium modification effects for such di-jets\nwith more detailed studies of transverse momentum spectra of associated hadrons.\nA tangential jet production scenario could account for such results.\n\nIn contrast, an asymmetric trigger pair should enhance long path lengths through the medium on the away-side.\nThe correlation structures resulting from a 2+1 analysis for the asymmetric triggers feature clear jet-like peaks\n on both same- and away-sides.\n$\\Delta\\phi$ and $\\Delta\\eta $ projections of the 2-dimensional correlation function are shown in Figure \\ref{fig:AsymmRes},\nwith no obvious broadening or indication of a ridge-like phenomenon.\nIn this new analysis we observe enhancement rather than suppression of away-side yield\nin contrast to all di-hadron correlation results for this kinematic range of associated particles.\nWe next investigate the correlation peaks in more detail.\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[scale=.29]{PtSpectrum}\n  \\includegraphics[scale=.29]{Asymm_perpt}\n\\caption{\n    $\\ensuremath{p_{T}}$ spectra  (normalized per number of triggers N$_T$) for the symmetric case on the left and the asymmetric case on the right.\n    Upward triangles signify the same-side correlation while \n    the away-side correlation is represented by downward triangles. In both cases the minimum bias inclusive \n    background is given as a reference. As a visual aid fits to $e^x\/x$ are shown with dashed lines. Error bars are statistical only.\n }\n  \\label{fig:Spectra}\n\\end{figure}\n \n\n\n\nFigure \\ref{fig:Spectra} shows the \\ensuremath{p_{T}}\\,spectra of associated hadrons in the jet peaks as defined by $\\left|\\Delta\\eta\\right|<0.5$ and\n$\\left|\\Delta\\phi\\right|<0.5$ for both cases.\nNote the expected significant hardening of the associated hadron spectra on both sides of the primary trigger as\n compared to the inclusive hadron distribution.\nAlso, just as in the case of angular variables,\nthe spectra on both sides  are virtually identical for the symmetric case. \nOn the other hand,\nthe away-side spectrum of the asymmetric trigger pair is softer than that of the same-side.\nComparison of symmetric di-jet spectra between Central Au+Au and minimum bias d+Au data (available from Run III)\nconfirmed the absence of modifications within our uncertainties.\nAnalysis of the d+Au data from Run VIII is in progress.\n\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[scale=.29]{JetYield2}\n  \\includegraphics[scale=.29]{JetPt2}\n  \\caption{ Yield (left) and \\ensuremath{\\langle p_{T}\\rangle} (right) in the symmetric and asymmetric case.\n    For the symmetric case, the d+Au values are also shown.\n    Upward triangles are used for the same-side correlation, \n    downward triangles for the away-side correlation.\n   Error bars are statistical only.\n  }\n  \\label{fig:Numbers}\n\\end{figure}\n\n\nTo further quantify our findings the average \\ensuremath{p_{T}}\\,of associated hadrons and the total yield in the jet peaks for both scenarios  \nare shown in Figure \\ref{fig:Numbers}. \nWe are finalizing the systematic uncertainties, but they are expected to be between 20\\% and 30\\% and\nstrongly correlated between same- and away-side. If this expectation holds, there will be no appreciable difference \nbetween d+Au and Au+Au systems in the symmetric case. \nBy contrast, in the asymmetric case, we see\n a spread between same- and away-side. The enhancement in total yield is offset by an attenuation in \n\\ensuremath{\\langle p_{T}\\rangle}.\n\nIn \\cite{renk:014903} two models were used to compute \ntotal energy deposition in the medium for di-hadron triggered events. As a proxy for this value we sum up\nthe \\ensuremath{p_{T}}\\,of trigger and associated particles on both same- and away-sides and consider the difference between the two.\nIn the symmetric case we find similar values of $1.50 \\pm 0.31 \\ensuremath{\\mytext{GeV}\/c}$ (Au+Au) and $1.64 \\pm 0.35 \\ensuremath{\\mytext{GeV}\/c}$ (d+Au),\nconsistent with\nthe estimate for \\ensuremath{k_{\\perp}}\\,smearing used in \\cite{renk:014903}.\nThe differences between the correlation peaks, if any, are not accessible\nwith these measurements.\nFor the asymmetric case we do not yet have a comparison with d+Au. A reasonable assumption is that the\n\\ensuremath{k_{\\perp}}\\,effect has about the same trigger-independent value of 1.6 \\ensuremath{\\mytext{GeV}\/c}. Subtracting this from the\n Au+Au $\\sum \\ensuremath{p_{T}}$ difference of 4.3 \\ensuremath{\\mytext{GeV}\/c}\\,results in an estimated energy loss of 3\\ensuremath{\\mytext{GeV}}\\,in the fiducial range.\n\n\\section{Summary}\n\\label{sec:summary}\n\nIn correlations using  symmetric \\ensuremath{p_{T}}\\,trigger pairs we found no evidence for  modifications in yield\nor $\\ensuremath{p_{T}}$ spectra for both the \nCentral Au+Au and d+Au collisions, and for same- and away-sides.\nThis is consistent with minimal medium interaction, e.g. \nstrong surface bias in the Au+Au correlation. \nWe expanded our 2+1 correlation studies\nof di-jet properties to include an asymmetric trigger selection scenario\nand again found the peaks on the same-side and away-side\nto be similar in shape.\nHowever, the away-side exhibited an increased yield \nbalanced by a softer spectrum resulting in a lower \\ensuremath{\\langle p_{T}\\rangle}.\n\nWe presented $\\sum \\ensuremath{p_{T}}$ as an estimate for energy loss, and our first results suggest negligible \nenergy loss in the symmetric region\nwhere model predictions are about 2 \\ensuremath{\\mytext{GeV}}.\nWe are working to complete d+Au reference calculations for the asymmetric case.\nOur initial findings indicate energy loss values that are at the lower end of predictions.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe inflationary big bang cosmology has been developed into a\nprecision science by recent cosmological observations including\nsupernova data~\\cite{ob} and measurements of cosmic microwave\nbackground radiation~\\cite{ob2}. From those it is suggested that our\nuniverse is made up of about 74 percent dark energy as well as about\n4 percent ordinary matter and about 22 percent dark matter. The\nobservations triggered an explosion of recent interests in the\norigin of dark energy~\\cite{dark}. There are several approaches to\nunderstanding the dark energy such as quintessence~\\cite{cald} and\nphantom~\\cite{phantom}.\n\nOne of the simple ways to explain it is through an introduction of\nthe cosmological constant $\\Lambda$, which leads to an exponential\nexpansion of the scale factor of the universe via $a(t)\\sim\ne^{\\sqrt{\\Lambda\/3}~t}$. Even though this simple form adequately\ndescribes current acceleration with a small value of $\\Lambda$, it\ncannot say anything about important issues including the coincidence\nproblem \\cite{coinci}. Why in the recent past the expansion of the universe\nstarted to accelerate? In this article, we suggest relic\nneutrinos, which are the lightest among all known fermionic\nparticles and thus have become non-relativistic in the last cosmic\ntime, might be related with this issue. We study a possible role\nof degeneracy pressure of the neutrinos in the evolution of the\nuniverse after neutrino decoupling $t_{d}$.\n\n\n\nWhen we consider a sphere of radius $R(t)$ containing all relic\nneutrinos (of total number $N(t)=M(t)\/m_{\\nu}$) at a cosmic time $t$\n($\\ll t_d$), the balance between degeneracy energy of the\nnon-relativistic neutrinos, $ {\\hbar^2\nM^{{5}\/{3}}}\/({m_{\\nu}^{{8}\/{3}} R^2})$, and their gravitational\nenergy, $ {GM^2}\/{R} $, gives us such relation of the radius as $ R\n(t) = {\\hbar^2}\/({ G m_{\\nu}^{{8}\/{3}} M^{{1}\/{3}}(t)}) $.\nOn the other hand, the balance between degeneracy energy of\nrelativistic neutrinos, $ {\\hbar c M^{{4}\/{3}}}\/({m_{\\nu}^{{4}\/{3}}\nR})$, and the gravitational energy yields the maximum total mass\n\\cite{ch}\n$ M_{c} =(\\frac{\\hbar c }{ G})^{\\frac{3}{2}} m_{\\nu}^{-2} $.\n From these the radius of the sphere is bounded below,\n\\beq R>R_c ~\\big(\\simeq \\frac{\\hbar^2}{ c^2 m_{\\nu}^2 l_p}\\big) \\sim\n10^{23}\\times\\big ( \\frac{{1 eV}}{  m_{\\nu} c^2}\\big)^2 ~ cm\n\\label{a} \\eeq with Planck\nlength $l_p $.\n\n\nThe matter-dominated era of our universe began at $t_m= 70~{\\rm Kyr}\n$ and ended at the transition time\n $t_t= 5~ {\\rm Gyr}$ \\cite{hartnett}, at which the universe started\nto accelerate. In the era $t_m<t<t_{t}$, we can neglect the pressure\nof the other matter than the degeneracy pressure of relic neutrinos\ncaused by the Heisenberg uncertainty principle. If we assume that at\na cosmic time $t=t_{\\chi}$ a huge bubble of relic neutrinos (which\nwe call a $\\nu_R$-sphere) is formed with a radius $R_{\\chi}$\ndetermined from the condition $R_{\\chi}(t_{\\chi})=R_c$ as a gigantic\nFermion star \\cite{TDL}, then the universe's main-frame in the time\ncan be imagined as made up of these relic neutrinos filled up to the\nFermi energy. Comparing the horizon radius of the present universe\n$R_{now}\\sim 10^{29}  cm$ (or $R_m\\sim 10^{27}cm$ at $t=t_m$) with\nthe critical value $R_c(\\simeq R_{\\chi})$ in equation (\\ref{a}), we\nsee that these values are not different from each other if we take\n$m_{\\nu}\\sim 1 - 10^{-3}~ eV$.\n\nFrom the time $t\\simeq t_{\\chi}$, the universe has a total energy\ndensity contributed mainly from the $\\nu_R$-sphere with a\n(effective) cosmological constant $\\rho_{\\nu}~(\\simeq M_c\/R_c^3)$ in\naddition to cold dark matter $\\rho_{CDM}(\\propto R^{-3})$, and thus\nthe time $t_{\\chi}$ can be interpreted as the transition time\n$t_{t}$.\nWe think that the coincidence problem might be naturally resolved\nbecause neutrinos become relic in this matter-dominated era and can\nmake the $\\nu_R$-sphere at the recent past time\n$t_{\\chi}~(<t_{now})$.\n In Section 2 we demonstrate a realization of this idea in a global\n$U(1)$ symmetric field theory model with a complex scalar field and\nrelic neutrinos in curved spacetime. Section 3 contains summary and\ndiscussions.\n\n\n\\section{A Global U(1) Model}\n\n\\subsection{General Relativistic Formulation}\n\nThe action of a global $U(1)$ symmetric model of complex scalar\n$\\Phi $ and neutrino fields $\\Psi$ minimally coupled to gravity is\ngiven by $S_{m}=\\int d^4 x \\sqrt{-g}~{\\cal L}_{matter}$, where\n\\begin{eqnarray}\n{\\cal L}_{matter}&= & -  g^{\\mu \\nu}\\partial_{\\mu}\\Phi^{\\dagger}\n\\partial_{\\nu}\\Phi -V (\\Phi^{\\dagger}\\Phi ) \\\\ \\nonumber\n&+&\\frac{i}{2}(\\overline{\\Psi} \\gamma^a \\nabla_a \\Psi - \\nabla_a\n\\overline{\\Psi} \\gamma^a \\Psi)-m_{0} \\overline{\\Psi} \\Psi\n-q\\Phi^{\\dagger} \\Phi  \\overline{\\Psi} \\Psi \\nonumber\n\\end{eqnarray}\nwith a bare mass of neutrino $m_{0}$.  The last term describes the\nscalar-neutrino interaction\npreserving the $U(1)$ symmetry with a parameter $q$ of dimension,\nfor example, $1\/M_{p}$ (Planck mass $M_p$) in gravity-induced $U(1)$\nbreaking models \\cite{MG, gi}.\n\nFrom the above action, we obtain following equations for scalar\nfield and neutrino.\n\\begin{equation}\n\\frac{1}{\\sqrt{-g}} \\partial_{\\mu}(\\sqrt{-g} g^{\\mu \\nu}\n\\partial_{\\nu} \\Phi ) -\\frac{\\partial V }{\\partial \\Phi^{\\dagger}}\n- q \\Phi \\overline{\\Psi} \\Psi =0~ , \\label{sc}\n\\end{equation}\n\\begin{equation}\ni\\gamma^a \\nabla_a \\Psi -m_0 \\Psi - q \\Phi^{\\dagger} \\Phi \\Psi=0~ ,\n\\end{equation}\nwhere the $\\gamma^a$-matrices satisfy the Clifford algebra in a\nlocally flat inertial coordinate,\n$\\{ \\gamma^a ,\\, \\gamma^b  \\}=-2\\, \\eta^{a b}$\nwith \n\\eta^{a b}=Diag(-1,\\,1,\\,1,\\,1) $,\nand the covariant derivative\n\\begin{equation}\n\\nabla_a ={e^\\mu}_a (\\partial_\\mu +\\Gamma_\\mu )\n\\end{equation}\nis constructed from the vierbein ${e^\\mu}_a$ and spin connection\n$\\Gamma_\\mu$ \\cite{ausy}.\nThe energy-momentum tensor in this model\n\\begin{eqnarray}\nT_{\\mu \\nu}&=&2\\partial_\\mu \\Phi^{\\dagger} \\partial_\\nu \\Phi -g_{\\mu\n\\nu}[\\partial^\\beta \\Phi^{\\dagger} \\partial_\\beta \\Phi\n+V(\\vert\\Phi\\vert^2) ]\\nonumber\\\\ &-&\\frac{i}{4}[ (\\overline{\\Psi}\n\\gamma_\\mu \\nabla_\\nu \\psi -\\nabla_\\nu \\overline{\\Psi} \\gamma_\\mu\n\\Psi )+(\\mu \\leftrightarrow \\nu)]\n\\end{eqnarray}\nis obtained by using the standard definition\n$T_{\\mu \\nu}\\equiv -\\frac{2}{\\sqrt{-g}} \\frac{\\delta S_{m}}{\\delta\ng^{\\mu \\nu}} ~ = -\\frac{ e_{a \\mu }}{ \\det \\{ e \\} }\\frac{\\delta\nS_m}{\\delta e_a^{\\nu}}$.\nIn next subsection we construct a concrete model with a specific\npotential of the scalar field.\n\n\\subsection{A Bubble Universe with $U(1)$ Symmetry Spontaneously Broken }\n\n Assuming a potential of the scalar field\n\\begin{equation}\nV(\\vert\\Phi\\vert^2)= \\lambda \\vert\\Phi\\vert^2(\\vert\\Phi\\vert^2 - v^2\n)^2 \\label{v}\\end{equation}\nwith dimensionful constants $\\lambda$\nand $ v $, we consider a spherical bubble of true vacuum $\\big< \\Phi\n\\big>=v$ with a spontaneously broken $U(1)$ global symmetry,\nimmersed in the different (false) vacuum state $\\big< \\Phi \\big>=0$\nhaving the $U(1)$ symmetry.\nTo construct a field theoretical model in which the ideas discussed\nin the (previous) Section 1 can be simply realized, we adopt the\nThomas-Fermi(TF) approximation \\cite{TDL, Q, QGM} inside the bubble\nincluding all relic neutrinos of the total number $ N=\\int d^3\nx\\sqrt{-g}\\, {\\Psi}^{\\dagger}\\Psi {e^t}_0$ with ${e^t}_0 =\n\\frac{1}{\\delta \\alpha}$.\nThe neutrinos have the effective mass, $m_{eff}=q v^2$ with $\\big<\n\\Phi\\big> =v$, and we put $m_0=0$ from now on for the simplicity.\n In the TF approximation there is at each point in space a Fermi sea of\nmassive neutrinos with the local Fermi momentum $q_F (r)$, and the\nnumber density of neutrinos is given by\n\\begin{equation}\n\\big< \\Psi^{\\dagger} \\Psi \\big>_{TF} =\\frac{2}{(2\\pi)^3 }\\int d^3\nq\\, n_q =\\frac{{q_F}^3 (r)}{3\\pi^2}~ \\label{tf}\n\\end{equation}\nwith the Fermi distribution $n_q =\\theta (q_F -q)$. From Eq.\n(\\ref{tf}) and Dirac's equation, $i\\partial_t \\Psi=E \\Psi$, we have\nthe energy density of neutrinos\n\\begin{eqnarray}\n&& \\big< \\frac{i}{2\\delta \\alpha}(\\overline{\\Psi} \\gamma^0\n\\partial_t\n \\Psi -\\partial_t \\overline{\\Psi} \\gamma^0 \\Psi ) {\\big>}_{TF}\n=\\frac{2}{(2\\pi)^3 }\\int d^3 q\\, n_q\\, E (q) \\,{e^t}_0 \\equiv \\rho\n (r) .\n\\end{eqnarray}\nThe pressure $p(r)$ of neutrinos is given in equation (13) below.\n\n\n\n\nWhen we express the spherically symmetric (static) spacetime as\n\\begin{equation}\nds^2 =-\\delta^2 (r) \\alpha^2 (r) dt^2 +\\frac{ dr^2}{ \\alpha^{2} (r)\n} +r^2 d\\theta^2 +r^2 sin^2 \\theta \\, d\\phi^2 , \\label{metric}\n\\end{equation}\nthe (background) scalar field equation (\\ref{sc}) reads\n\\begin{eqnarray}\n&& \\frac{1 }{r^2 \\delta}{( r^2 \\delta \\alpha^2 {\\Phi'} )}'\n-\\lambda\\Phi(\\vert\\Phi\\vert^2-v^2)(3\\vert\\Phi\\vert^2-v^2)\n- q\\Phi\\big<\\overline{\\Psi} \\Psi\\big>_{TF}  = 0, ~\\label{scalar}\n\\end{eqnarray}\nand the Einstein equation, $G_{\\mu\\nu}=\\kappa T_{\\mu\\nu}$ with\n$\\kappa=8\\pi G$, in this TF approximation, yields\n\\begin{eqnarray}\n&& \\frac{{(1-\\alpha^2 ) }}{r^2}- \\frac{1}{r}(\\alpha^2)'\n=\n\\kappa[  \\alpha^2 \\vert{\\Phi}'\\vert^2 +V  + \\frac{i}{2\\delta\n\\alpha}\\big< \\overline{\\Psi} \\gamma^0\n\\partial_t \\Psi -\\partial_t \\overline{\\Psi} \\gamma^0 \\Psi\\big>_{TF}\n],\\nonumber\n\\end{eqnarray}\n\n\\begin{eqnarray}\n&& -\\frac{{(1-\\alpha^2 ) }}{r^2} +  \\frac{{(\\delta^2 \\alpha^2 )'\n}}{r \\delta^2 }\n=\n\\kappa[\\alpha^2 \\vert{\\Phi}'\\vert^2 - V   -\\frac{i \\alpha}{2} \\big<\n\\overline{\\Psi} \\gamma^1\n\\partial_r \\Psi -\\partial_r \\overline{\\Psi} \\gamma^1 \\Psi \\big>_{TF}\n],\\nonumber\n\\end{eqnarray}\n\n\\begin{eqnarray}\n&& \\frac{1}{2 \\delta^2 }{(\\delta^2 \\alpha^2 )''} -\\frac{{(\\delta^2\n\\alpha^2 )'}} {4\\delta^2 }\\frac{(\\delta^2)'}{\\delta^2 } +\n \\frac{{(\\delta^2 \\alpha^2 )' }}{r\\delta^2 }\n- \\frac{\\alpha^2 }{2r}\\frac{(\\delta^2)'}{\\delta^2 } \\nonumber\\\\ =&&\n\\kappa[ -\\alpha^2 \\vert{\\Phi}'\\vert^2 - V  - \\frac{\n i}{2r}\\big<\\overline{\\Psi} \\gamma^2 \\partial_\\theta \\Psi\n -\\partial_\\theta \\overline{\\Psi} \\gamma^2 \\Psi\\big>_{TF} ]\n ,\\nonumber\n\\end{eqnarray}\n\n\\begin{eqnarray}\n&& \\frac{1}{2 \\delta^2 }{(\\delta^2 \\alpha^2 )''} -\\frac{{(\\delta^2\n\\alpha^2 )'}} {4\\delta^2 }\\frac{(\\delta^2)'}{\\delta^2 } +\n \\frac{{(\\delta^2 \\alpha^2 )' }}{r\\delta^2 }\n- \\frac{\\alpha^2 }{2r}\\frac{(\\delta^2)'}{\\delta^2 } \\nonumber\\\\ = &&\n\\kappa[  -\\alpha^2 \\vert{\\Phi}'\\vert^2- V  - \\frac{ i}{2 r}\n\\big<\\frac{1}{sin\\theta }(\\overline{\\Psi} \\gamma^3 \\partial_\\phi\n\\Psi -\\partial_\\phi \\overline{\\Psi} \\gamma^3  \\Psi  )\\big>_{TF} ] ,\n\\label{eins}\n\\end{eqnarray}\nwhere ${\\Phi}'\\equiv \\frac{\\partial \\Phi}{\\partial r}, ~\n ...$ ,\nhave been used for the simplicity.\nThe results in above equation (\\ref{eins}) are consistent with those\nin the case of fermion stars \\cite{TDL}, when\n\\begin{eqnarray}\n&&\\big< \\frac{ \\alpha}{2 i} (\\overline{\\Psi} \\gamma^1 \\partial_r\n\\Psi -\\partial_r \\overline{\\Psi} \\gamma^1  \\Psi ) {\\big>}_{TF} =\n\\big<  \\frac{ 1}{2 i r}(\\overline{\\Psi} \\gamma^2 \\partial_\\theta\n\\Psi -\\partial_\\theta \\overline{\\Psi} \\gamma^2\n\\Psi){\\big>}_{TF} \\nonumber\\\\\n &&=\\big< \\frac{ 1}{2 i r\\, sin\\theta}(\\overline{\\Psi} \\gamma^3\n \\partial_\\phi \\Psi -\\partial_\\phi \\overline{\\Psi} \\gamma^3 \\Psi\n ){\\big>}_{TF} = p(r) ,\n\\end{eqnarray}\nare assumed in the spherically symmetric spacetime, and from Dirac's\nequation and its Hermitian conjugate we have the relation\n$\\rho(r) - 3~ p(r) = q v^2 \\big< \\overline{\\Psi} \\Psi {\\big>}_{TF}$.\n\n\nNow assume that there exist regular solutions of the metric\ncomponents, scalar field, and the energy density and pressure of\nneutrinos, and we get approximate solutions to above equations\n(\\ref{scalar})-(\\ref{eins}) as\n\\begin{eqnarray}\n\\Phi(r)&=&v+  \\frac{\\rho_0 -3 p_0}{6} r^2+{\\cal O}(r^3 )\\, , \\nonumber \\\\\n\\alpha^2 & =&1-\\frac{\\kappa}{3}\\rho_0 r^2 +{\\cal O}(r^3 ) ,\\\\\n\\delta^2 \\alpha^2& =& \\{ 1+\\kappa(\\frac{1}{6}\\rho_0 +\\frac{1}{2} p_0\n)r^2 \\} +{\\cal O}(r^3 ),\\nonumber\n\\end{eqnarray}\nwith $\\rho_0\\equiv\\rho(0)$ and $p_0\\equiv p(0)$. These solutions are\nconsistent with the results in fermionic stars with a global\nmonopole \\cite{QGM}.\nOn the other hand, to equation (\\ref{eins}) with $V\\simeq 0$,\n$\\rho\\simeq 0$ and $p\\simeq 0$ far away from the horizon of the true\nvacuum bubble $r_h\\equiv \\sqrt{1\/\\kappa \\rho_0} \\ll r$, solutions of\nthe metric components in equation (\\ref{metric}) can be taken\nasymptotically as\n$\\alpha^2 \\simeq 1 -\\frac{2\\kappa M}{r}$ and $ \\delta^2 \\simeq 1 $\nwith the total mass of all particles inside the bubble universe $M$.\nWith the metric components we have the asymptotic solution\n$\\Phi \\simeq r^{-2} e^{-\\sqrt{\\lambda}v^2 r}$.\nUnder the condition \\beq \\sqrt{\\lambda} v^2 \\kappa M =1,\n\\label{nl}\\eeq the solution for $\\Phi$ satisfies all the relevant\nequations even upto next leading order.\n\nFrom the effective neutrino mass $m_{eff}=qv^2$ inside the bubble,\nwe can estimate the true vacuum expectation value as $v^2\\sim\nm_{eff}M_p$, which gives us $\\lambda\\simeq M_p^4\/(v^4 M^2)\\sim\n10^{-120}\/m_{eff}^2$ when equation (\\ref{nl}) is used.\nOn the other hand, if the potential $V(\\vert \\Phi\\vert^2)$ of the\nscalar field is induced by gravity \\cite{MG, gi} with the\ndimensionful constant $\\lambda \\simeq v^{n-2}\/ M_{p}^n$ for $0\\leq\nn$, then its magnitude comparable with the above is given as\n$\\lambda\\sim\n 10^{-112}\\times m_{eff}^2 \/(1 ~eV)^4$ for $n=6$, which is the same as\n$10^{-120}\/m_{eff}^2$ if $m_{eff}\\sim 10^{-2}~ eV$.\n\n\n\n\\section{Summary and Discussions}\n\nConsidering a huge $\\nu_R$-sphere to be possibly formed at\n$t=t_{\\chi}$, with degenerate relic neutrinos filled up to the Fermi\nenergy, we suggest that its constant energy density $\\rho_{\\nu}$\nplay a role of an effective cosmological constant. The time\n$t_{\\chi}$ can be interpreted as the transition time $t_{t}$, and\nthese lead to the interesting possibility that our model gives one\nway of understanding the coincidence problem \\cite{coinci}, because\nthe relic neutrinos can make the $\\nu_R$-sphere at the recent past\ntime $t_{\\chi}~(<t_{now})$ during the matter-dominated era. In the\nSection 2 we have constructed a spontaneously broken $U(1)$ global\nsymmetric model, in which the energy density $T_{t t}\\equiv\n\\rho~(\\sim \\rho_0)$ in the right hand side of the first equation of\n(\\ref{eins}) can be put as $\\rho_{\\nu}\\sim M_c\/R_c^3$ of the\n$\\nu_R$-sphere considered from Section 1. However the formulation\nwas constrained in a static case and further study is necessary on\nthe cosmic time evolution of the scale factor $R(t)$. The predicted\nvalues of the relic neutrino mass, in the range\n $m_{\\nu}\\sim 1 - 10^{-3}~ eV$\ngiven both in Section 1 and the last line of Section 2, are\nanticipated to be tested by future observations.\n\nWe would like to make a few comments. In future works, we need a\nfully relativistic treatment (and possible numerical methods) in\ncalculation of the more rigorous value of the critical radius of the\n$\\nu_R$-sphere in equation (\\ref{a}).\nFor our $U(1)$ model to work with the appropriate (relic) neutrino\nmass, the parameter $\\lambda$ in the potential of the scalar field\nin equation (\\ref{v}) should be very small, which might be realized\nin some models using an extremely light scalar field \\cite{XHe} or\nin gravity-induced models {\\cite{MG, gi}. We are seeking to find\nothers models related with ours, in which the global $U(1)$ symmetry\nis spontaneously broken with $\\big<\\Phi\\big>\\equiv\nv=\\sqrt{m_{eff}M_p}\\sim 100~ T eV$.\n During a very slow phase\ntransition ($ \\lambda \\ll 1$) from the false vacuum with\n$\\big<\\Phi\\big>=0$ to the true vacuum with $\\big<\\Phi\\big>=v$,\nmasses of neutrinos look changing from from $0$ to $m_{eff}=q v^2$,\nand the $\\nu_R$-sphere made of the massive neutrinos could play a\nsimilar role to a effective cosmological constant in our model,\nwhile a growing matter component (neutrinos) in Ref. \\cite{wetter},\ninstead, helps such a field as the cosmon \\cite{wetter} responsible\nfor the dark energy.\n\n\n\\section*{Acknowledgements}\n\nThis research was supported by the Basic Science Research Program\nthrough the National Research Foundation of Korea(NRF) funded by\nMinistry of Education, Science and Technology(2010-0012692).\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\n**EGMONT**  \n _We bring stories to life_\n\nFirst published in the United Kingdom by Egmont UK Limited, 2014  \nFirst published in the United States of America by Egmont USA, 2014  \n443 Park Avenue South, Suite 806  \nNew York, NY 10016\n\nCopyright \u00a9 The Shadow Gang, 2014  \nAll rights reserved\n\nwww.egmontusa.com\n\nLibrary of Congress Cataloging-in-Publication Data\n\nGrant, Michael, 1954-  \nBZRK apocalypse \/ Michael Grant.  \n1 online resource.  \nSequel to: BZRK reloaded.  \nSummary: In the concluding book in the BZRK trilogy, war is again being waged in the macro and the nano, and Lear's identity is finally uncovered.  \nISBN 978-1-60684-409-0 (eBook) -- ISBN 978-1-60684-408-3 (hardcover) [1. Utopias--Fiction. 2. Nanotechnology--Fiction. 3. Conjoined twins--Fiction. 4. Twins--Fiction. 5. Science fiction.] I. Title.  \nPZ7.G7671  \n[Fic]--dc23  \n2014006440\n\nAll rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, or otherwise, without the prior permission of the publisher and copyright owner.\n\nv3.1\nFor Katherine, Jake, and Julia\n\n# Contents\n\n_Cover_\n\n_Title Page_\n\n_Copyright_\n\n_Dedication_\n\nChapter One\n\nChapter Two\n\nChapter Three\n\nChapter Four\n\nChapter Five\n\nChapter Six\n\nChapter Seven\n\nChapter Eight\n\nChapter Nine\n\nChapter Ten\n\nChapter Eleven\n\nChapter Twelve\n\nChapter Thirteen\n\nChapter Fourteen\n\nChapter Fifteen\n\nChapter Sixteen\n\nChapter Seventeen\n\nChapter Eighteen\n\nChapter Nineteen\n\nChapter Twenty\n\nChapter Twenty-one\n\nChapter Twenty-two\n\nChapter Twenty-three\n\nChapter Twenty-four\n\nChapter Twenty-five\n\nChapter Twenty-six\n\nChapter Twenty-seven\n\nChapter Twenty-eight\n\nChapter Twenty-nine\n\nChapter Thirty\n\nChapter Thirty-one\n\nChapter Thirty-two\n\nChapter Thirty-three\n\n_About the Author_\n\n# **ONE**\n\nSandra Piper was having dinner with friends when it started.\n\nShe was eating chilled lobster on the teak deck of a producer friend's Malibu home, along with a former costar named Wade Talon (a ridiculous screen name in Sandra's opinion), her current director (Quentin\u2014no last name necessary), a very rich and rather magnificently tattooed woman named Lystra Reid who had an odd vocal tic that added \"Yeah\" to random sentences, and an extraordinarily fit, tall, and broad-shouldered man whose name she kept forgetting but who might have been named Noble, or something very close to that.\n\nThe Noble creature was listening, rapt, while the more famous folk discussed work and mutual friends and more work. In fact, in one way or another it was all work.\n\nSandra had been nominated: Best Actress. Very tough competition. The oddsmakers called her a long shot at six to one. Long but not impossible. And despite the fact that Sandra Piper was a mother of two, a down-to-earth thirtyish woman with a masters in economics who had smoked pot exactly twice in her life and never drank more than two glasses of wine, she was thinking of seducing young Mr. Shoulders. Mr. Shy Grin. Mr. Large-But-Sensitive Hands.\n\nBecause he was definitely interested, and she had been divorced for two years and had dated no one in that time. And she was exhausted from long days on the current shoot, plus her son, Quarle (three years old), had just gotten over a two-week-long bout of the flu.\n\nAnd really, what the hell was the point of being America's Sweetheart if you couldn't even get laid? Would a male actor in the same situation even hesitate? Well, some, sure. But lots wouldn't. So why should she? Wasn't that why Quentin had invited Noble...? No, wait, now she remembered. His name was Nolan. Whatever. Wasn't he there for her, um... amusement?\n\n_Unless_. Oh, had he come with the Lystra person? Was he here for _her_? She would be closer to his age, not a beauty but attractive enough, given that she was not Hollywood at all but some sort of health-care billionaire.\n\nNo. No, young Mr. Body of Steel was not eyeing Lystra. He was eyeing the next winner of the Academy Award for Best Actress. _Uh-huh_.\n\nBut the idea sighed inside her and deflated like a balloon with a slow leak. She shook her head, a tiny movement not intended for anyone else, and took a deep breath. She had to help Quinn (seven years old) with her stupid California Mission project, due tomorrow.\n\nGod, she was boring. Boring and responsible and definitely America's Sweetheart, except that when it came right down to it, she was Mommy.\n\nSuddenly her hand jerked and she tipped her wineglass over. The last ounce of white wine drained onto the wood surface, alarming no one.\n\n\"Sorry. I just\u2014\"\n\nSandra frowned. Shook her head.\n\n\"What's the matter, Sandy?\" Wade asked.\n\n\"I'm just...\" She shook her head again. Frowned, despite the fact that frowning would crease her ageless forehead. \"Oh my God, is there something in the wine? I'm... I'm seeing something.\"\n\nNolan looked at her from beneath lashes that would probably have tickled her cheeks (and other places, too, if she'd just said the word) and asked, \"Are you feeling ill?\"\n\n\"It's...\" She laughed. \"This is going to sound crazy. It's like I can see something that isn't there. I'm...\" She looked away from them, stared out toward the black Pacific Ocean, wondering if somehow what she was seeing was a reflection off the wineglasses.\n\nBut no. It was still there. It was as if she had a second set of eyes, and they opened onto a small TV screen in a corner of her _own_ eyes.\n\n\"I'm seeing, like, like... just flat, but weird.\" Then, a sudden, sharp gasp. \"Oh my God, a second one. Like another window in my head.\"\n\n\"Maybe you should lie down,\" Nolan suggested.\n\n\"Or have another glass of wine,\" Quentin said, and laughed. But now he, too, was staring at her sideways, with concern on his face.\n\n\"There's two... Oh! Oh! Oh! There's a giant insect. I'm going nuts. Maybe I'm having a stroke.\"\n\n\"I'm calling nine one one,\" Nolan said, and pulled out his phone.\n\n\"Jesus Christ! It's a huge bug. I can see it! It's turning, it's coming toward me.... Oh, oh God, I think I'm moving it! I think I'm making it move!\"\n\nShe pushed back hard from the table. Glassware clattered and toppled. Wade leapt to his feet and caught her arm as she lurched away from the table.\n\n\"It has eyes! It has eyes! Oh, God. Oh, God. My face! My eyes! Those are _my_ eyes!\"\n\nShe pushed Wade aside violently, then, abashed, shocked by her own behavior, she tried on a fleeting smile, reached out a reassuring hand and said, \"I think I need help. I think I'd better see a doctor.\"\n\n\"That would be best,\" Lystra Reid said coolly, then added, as if an afterthought, \"Yeah.\" She had moved to place her back against the railing and was watching with detached interest. At least she wasn't taking a picture to tweet later.\n\n\"Ambulance is on the way,\" Nolan reported.\n\nAnd Sandra thought, _Well, he certainly won't sleep with me now_. But that thought came and left in a heartbeat, because something else was happening on that eerie picture-in-picture view in her head. She was seeing a falling drop of liquid that must have been a million gallons. It was far bigger than the terrifying bugs with her face smeared across them, her eyes; those nightmare insects with _her own damned eyes_.\n\nThe drop landed. It swept around the two bugs, engulfing them. And instantly it began to eat away those insect legs. It chewed burning holes into those insect carapaces. It burned away those distorted reflections of her own face like an old-time filmstrip jammed in a projector that bubbles and caramelizes and is gone.\n\nThe picture frames in her head blinked out.\n\nThey were gone as fast as they had come.\n\nSandra stood now, seeing only through her own eyes, seeing only what was real.\n\nShe laughed. \"Hah-hah-hah-hah. Hahahahahahahah!\"\n\nAnd then she screamed. \"Ahhhh! Aaaaaahhhh! You're devils! Devils!\"\n\nNolan moved to grab her because she was climbing awkwardly onto the table. She slipped, skinned her knee against the edge, stared down at the blood, and shrieked, shrieked like a mad thing.\n\nShe snatched up a knife. Not a very big knife, just a dinner knife with a point and modest serrations. She stabbed it into Nolan's thick bicep.\n\nThe strong man screamed, a more feminine sound than one might have expected.\n\n\"Hah! Hah, devil!\" Sandra yelled, happy at the sight of his blood, fascinated.\n\nWade and Quentin backpedaled, making sure to keep the table between themselves and the long shot for Best Actress.\n\nIn Sandra's eyes they were not backing away, they were coming for her, with their fangs out, and claws for fingers, and liquid fire dripping from their eyeballs\u2014it was all about the eyeballs, it was there, in the eyes, the demons.\n\nSandra Piper turned the knife around and stabbed it into her belly. It didn't go far. It drew blood, but just a stain the size of a quarter.\n\n\"Hey, hey, hey!\" Quentin yelled.\n\n\"No, no, stop that, stop that this instant,\" Wade said.\n\nNolan made another move\u2014this time wary\u2014to take the knife from her.\n\nSandra spit at him. \"Hah!\" she yelled, and stabbed the knife into her own eye. Her left eye. Pulled it out bloody and clotted with viscous goo.\n\nCries of horror, and now even she could see that they were backing away, the devils. It was working. _Hah! Run, devils, run!_\n\nShe then stabbed the knife into her other eye and pushed it through cracking bone, pushed it until the hilt was stopped. Then she twisted the knife around as if she was trying to churn her own brain.\n\nHer knees gave way. The knife dropped from her hand.\n\n\"Stupid Mission project,\" she said. Then fell onto her back, laughing and howling, laughing and howling. \"Devils! Dev\u2014\"\n\nIt was Lystra Reid who took the knife from her. And Lystra who placed a napkin over the bloody craters in her face.\n\nNot that Sandra Piper could see that.\n\n# **TWO**\n\nHer name was Sadie McLure. She had indifferently styled brown hair and smart, skeptical brown eyes that could take on golden highlights and even suggestions of green in certain lights. She had freckles on her cheeks and across the bridge of her nose. She'd never liked the freckles\u2014they seemed to be accompanied by the word _cute_ and she didn't like people thinking of her as cute. Cute was a belittling word.\n\nThe cute freckles had a second outpost on her chest, and a lesser presence on her shoulders. But all her freckles were now almost hidden by a rich, deep tan.\n\nHer name was Sadie McLure, but in certain company she called herself Plath, after the great and tragically suicidal poet.\n\nIt was her nom de guerre. Her BZRK name. The name that defiantly acknowledged that there were only two possible fates in her future as a member of BZRK: death or madness.\n\nShe had a net worth expressed in billions of dollars. She had a small but effective private army in the form of McLure Labs security under a Mr. Stern. (She must have heard his first name at some point, but what had stuck was the Mr. and the Stern.)\n\nShe had seen terrible things, Sadie had. As Plath she had _done_ terrible things, too, and had terrible things done to her.\n\nShe was sixteen years old.\n\nA month had passed since that bizarre and fateful day when the _Doll Ship_ had burned down much of the Hong Kong harbor waterfront. A month since the president of the United States had blown her own brains out on nationwide TV after being (correctly) suspected of murdering her husband.\n\nA month since Sadie as Plath had sent her biots into Vincent's brain, one armed with acid to burn the biot-death madness from him. The great advantage of biots over their mechanical competitors, the nanobots, was the closeness of the connection between twitcher and biot. That was also the greatest disadvantage because that same connection meant that the loss of a biot sent its creator on a downward spiral into madness.\n\nVincent had spiraled following the loss of one biot and serious injury to a second.\n\nFrom a desperate desire to save Vincent, Sadie had undertaken a grim mission to cauterize parts of his brain. But at this moment that terrible day was compartmentalized if not forgotten, and Sadie was doing something that was not at all terrible. She was on a white-sand beach beneath palm trees. A picnic was laid out on a woven mat of the kind the locals used. There was cold fried chicken, cold lobster, and a bowl of vanilla-spiked fruit in the local Madagascar style.\n\nThere was also a bottle of white wine, now empty, and a bottle of vodka, now partly empty.\n\nAnd there was a boy.\n\nHe was naked as Sadie. His name was Noah, though like Sadie he sometimes used a nom de guerre: Keats.\n\nWhether they were Plath and Keats or Sadie and Noah, she was on top and he was inside her. They were both smiling because the ash from the joint in Sadie's mouth had landed on the very tip of Noah's nose, and when she blew it away it made him sneeze. Which struck them both as funny, so they laughed, and that physical convulsion had interesting side effects.\n\n\"Laugh again,\" Noah said.\n\n\"Not yet,\" Sadie said.\n\n\"You're torturing me.\"\n\n\"I'm teaching you endurance,\" she said, voice slurring.\n\n\"I'm standing right at the very edge of a cliff,\" he said, and his eyes closed and his smile became dreamier. \"If you laugh... or even move at all... or even breathe deeply, I'll go right... _mmmm_... over... the edge.\"\n\n\"You're going with a cliff metaphor?\" she asked, and giggled.\n\nWhich was all it took.\n\nShe watched his face while his body arched and thrust and shuddered and finally subsided. His expression was more animal than human in the first seconds, and the sounds he made were definitely not witty banter. Or even half-drunk and quite stoned banter. But then that feral look softened into an expression like you'd see on the face of a saint in a Renaissance painting.\n\nAnd then he laughed, too.\n\nAnd opened his blue, blue eyes and said, \"Don't go yet.\"\n\nHe remained inside her, in more ways than one. He was also inside her brain, and not metaphorically. A tiny creature smaller than the period at the end of a sentence\u2014a creature that was built from an exotic stew of DNA that included Noah's own\u2014was deep within Sadie's brain. This was a _biot_. One of _his_ , Noah's biots, because biots were nothing if not unique to their creator. It was designated K2. Keats 2. His other biot, K1, was in a tiny vial stuck in the buttoned pocket of his shorts, which were... he looked around... over there, somewhere.\n\nK2 had the job of maintaining the fragile latticework painstakingly built around a bulge in an artery in Sadie's brain. Left alone, the aneurysm might never pop. Then again it might pop at any moment, which would almost certainly kill Sadie, perhaps over the course of pain-filled hours.\n\nNoah had worked with scarcely a break over this last month to strengthen the Teflon casing around the deadly bulge. It was tedious work. Fibers had to be carried through Sadie's eye, down the optic nerve, up and down the soggy hills and deep valleys of her brain\u2014quite a long trip for a biot\u2014then carefully threaded in place. Basket weaving.\n\nAll the while a sort of picture-in-picture was open in Noah's own mind, an artificially color-enhanced but grainy picture. Imagine a 3D special-effects movie but with the color flattened out and stripped of nuance, all shot through a dirty lens.\n\nNoah knew Sadie with an intimacy that was impossible for people who did not travel _down in the meat_. When she became aroused, he could feel the artery beneath his biot's six legs pumping faster, harder. But it was not just the relatively monotonous, liquid-encased surface of the brain that he had seen up close. He had at various times, in the course of more than one desperate mission, crawled across her eyes, her lips, her tongue.\n\nShe kissed his mouth and then the place just beside his mouth and then his neck. Then she rolled off onto the blanket and looked toward the food.\n\n\"You didn't...,\" he said.\n\n\"No.\" She struggled to find the right tone. Unconcerned but not indifferent. Nonchalant, not like it mattered. Then tried switching to a sexy purr. \"But I loved every minute. That's not the only thing in the world, you know.\"\n\n\"It's not?\" he asked, trying to be funny.\n\n\"Want some lobster?\" she asked, deflecting him. She didn't like talking about sex. The effects of weed and wine were ebbing, leaving her tired and groggy. She could be cranky in a minute if she let herself.\n\nThere were things nagging at her, distractions. She wanted to keep pushing them away, but self-medication had its limits and all those niggling worries would resurface, frequency and intensity increasing. She had pushed it all away for a month and now \"it\" was pushing back.\n\n\"I do want some lobster, I absolutely do,\" Noah said.\n\n\"Then trot on over there and get me some, too.\"\n\nHe sighed. \"It's always something with you. _Undress me. Make love to me. Feed me lobster_. You are so demanding.\" He stood up, and she saw that half his hard, lean behind was coated with sand. She lay back, head resting on one hand, enjoying that particular sight, and the view beyond. They were in a secluded lagoon on the western edge of the island, facing the much larger island of Madagascar, which was a blur of green ten miles off.\n\nA quarter mile to both north and south, armed men\u2014fashionably attired in white Tommy Bahama shirts and automatic rifles\u2014watched for any threat to their privacy. Just out of sight behind a rocky point, a yacht crewed by ex-soldiers rolled in the gentle swell and kept a radar lookout over the area.\n\nNoah brought her pieces of lobster on a small china plate.\n\n\"We're out of wine,\" he said.\n\n\"Good. Time to sober up, anyway.\"\n\n\"Is it?\" he asked. \"Why?\"\n\nShe sat up and reached for her T-shirt. He interrupted her with a kiss and gently stroked her breasts as if saying good-bye to them. \"I quite like these,\" he said.\n\n\"I guessed that. Can I put on my shirt now?\"\n\n\"I suppose.\" He started to dress as well: shorts, a T-shirt, sandals. He reached down and pulled her to her feet.\n\n\"I'll call for our cab,\" Sadie said. She pressed the talk button on a handheld radio\u2014there was no cell-phone reception this far up-island.\n\nFive minutes later, as they packed up the picnic, a glittering white cabin cruiser appeared around the point.\n\nThe captain gave a little _toot-toot_ on the horn, and the boat blew up.\n\nIt took a few seconds for the flat _crump!_ of the explosion to reach them. It took a bit longer for the debris to splash into the water.\n\nAnd just like that Sadie and Noah were Plath and Keats once again, running now, food and blanket forgotten. McLure security men were tearing along the beach from north and south, assault rifles in their hands, yelling, \"Get under cover, get under cover!\"\n\nThe boat burned for a while\u2014there was no possibility of anyone having survived\u2014and then it slipped beneath gentle waves that were a very similar color to Noah's eyes. The pillar of black smoke was smothered. A black smudge rose until it was caught by a breeze and blown away over the island.\n\nVacation was over. The war for the human race was back on.\n\n# **THREE**\n\n_The roll that had begun was accelerating. The ship's ballast had shifted decisively. It rolled onto its side, sending the flames shooting hundreds of feet into the air_.\n\n_The inside of_ Benjaminia _was a slaughterhouse\u2014dead Marines, many more dead residents hung from bloody catwalks. The sphere turned on its axis, and floors became walls. Bodies fell through the air_.\n\n_Like the turning drum of a dryer, the sphere rolled on, and now people clinging to desperate handholds fell screaming and crashed into the painted mural of the Great Souls_.\n\n_Water rushed in through the opened segments_.\n\n_The blowtorch submerged but burned on and turned the water to steam as the_ Doll Ship _sank, and settled on the harbor floor_.\n\nWhen the _Doll Ship_ sank, the Armstrong Twins had found themselves in Hong Kong's Victoria harbor.\n\nThey could not swim. With some effort, and if they felt in a cooperative mood, they could manage to walk, dragging the useless third leg. But swim?\n\nIt was Ling who had saved their lives. Tiny, ancient, birdlike Ling. She had cupped her hand beneath their chin and churned the filthy water with her legs. She'd sunk beneath the waves repeatedly, rising each time to gasp in a single breath mixed with salt water, to cough and gag, and yet to keep her legs churning, until a fishing boat had come to the rescue.\n\nThey would find a way to reward Ling. They vowed that. She had saved their lives and very nearly died herself.\n\nThe Armstrong Twins had made their way from Victoria harbor to Vietnam, where they had financial interests and owned a small but useful number of local government officials. From there they'd made their way to Malaysia, to the Sarawak state on the island of Borneo.\n\nThe Armstrong facility there was involved in mining rare earths. And it did a bit of logging, as well, all very eco-friendly, with careful replanting programs and all of that. Whatever it took to avoid too much scrutiny. The Armstrongs were good corporate citizens out of self-interest.\n\nBut this facility was not strictly about mining or logging. It was built of three elements: there were two identical buildings, each a crescent, facing each other across an elongated oval that formed an enchanting tropical garden, a sort of tamed version of the surrounding rain forest.\n\nThere were trees and flowers, streams full of fish and waterfowl, pink gravel pathways leading to benches, and seating areas where the white-collar employees could take their lunches alfresco.\n\nAt the top of the oval, connecting the two crescents, was a stumpy tower topped by a domed observatory. There was an impressive optical telescope that profited from the profound darkness of the surrounding countryside.\n\nNo one was using the telescope at the moment because it was pouring rain. It often poured rain here. And when it poured it was unlike anything Charles Armstrong had ever known in New York. It came down not in drops but in sheets. The heavens did not sprinkle on Sarawak, they emptied buckets and bathtubs and swimming pools.\n\nCharles watched a lizard climbing up the glass side of the dome, pushing against the stream of water. Sarawak had lizards. It had lizards and snakes and birds in abundance.\n\n\"I would have thought the rain would wash it off,\" Charles said.\n\nHis brother, Benjamin, was less interested by the lizard or the rain, but of course could see both since it was impossible for the twins not to face in the same direction. Their individual eyes could roam this way or that, focus independently under the direction of their separate brains, but they did not have separate heads, rather two heads melded together.\n\nThis gave them two mouths, one nose, and three eyes. The middle eye was a bit smaller than the other two and often had an unfocused, glazed quality. It could see, but its focus was not consciously directed by either Charles or Benjamin. Rather it often seemed to have a mind of its own and would focus where it willed, suddenly granting depth perception to one or the other twin, but never both at once.\n\nThey were large, the twins were, tall but even more broad, with shoulders capable of carrying the unusual weight of their doubled head. Two arms, neither muscular; two fully developed legs; and a third, stunted leg.\n\nAt the moment they were sitting in a modified electric wheelchair. It was far more capable than the usual motorized wheelchairs and had been given an almost dashing, exotic look with burgundy velvet trim, two side panels that likely concealed weapons, and wheels that looked more racetrack than hospital, but it remained, in the end, a wheelchair.\n\nThe observatory was their haunt for now. There was a bedroom down a ramp, and a specially outfitted bathroom. But the bedroom had only conventional windows. All their lives had been spent indoors, and they craved the openness of the observatory, even when all they could see was water sheeting down the glass and a lizard struggling against that tide.\n\n\"Looking at lizards,\" Benjamin said, disgusted.\n\nThey had both been depressed since the sinking of the _Doll Ship_. The _Doll Ship_ had been their happy place, the place they could think about when life became too gloomy or the pressure too intense. Now it was gone. All those poor people, the people who worshipped them, who saw beauty in their deformity, all of them gone.\n\n\"Fish food,\" Charles said, knowing where his brother's thoughts had wandered. \"And we still don't know how it happened.\"\n\n\"A Swedish intelligence officer and a British admiral.\"\n\n\"But how?\"\n\n\"Many questions, brother.\"\n\nThey turned the wheelchair to face the large monitor that hung above a touch-screen desktop. The monitor was divided into twenty-four smaller frames. Three were tuned to various news outlets. The rest were clearly surveillance cameras. An empty room with desks. A break room with one woman making coffee. A lab with two people in white coats moving to some unheard music while they tapped on keyboards. A puzzling view of what might be a warehouse.\n\nOne by one the video tiles flipped to be replaced by different views. Every corner of the Armstrong empire.\n\nThey could see everything, but what could they control? They weren't even sure they could return to New York. London, too, might be out of bounds.\n\n\"We are hiding like rats from a cat,\" Benjamin said.\n\n\"We're foxes at the very least,\" Charles said, trying to make it sound like a good thing, trying not to think about the way fox hunts usually ended with dogs tearing at the cornered animal. \"System: locate Burnofsky.\"\n\nA larger picture appeared, in the center of the monitor. The object of their search had his back to them. He was hunched over a terminal.\n\n\"There's our Karl,\" Charles said, steel in his voice.\n\n\"Ours?\"\n\nCharles sighed. \"Either he hit bottom on some grand, final bender and decided to turn his life around. Or\u2014\"\n\n\"Or BZRK wired him,\" Benjamin said.\n\n\"Ling!\" Charles yelled. \"It's dinnertime, and I find I would enjoy a drink.\"\n\nThey shared a digestive tract, despite having two mouths. It took consent from both for either to drink alcohol. Or to eat, though they tried to be tolerant on that. Benjamin liked to snack on a bowl of Chex Mix sometimes, and Charles preferred fresh fruit. Apricots. He loved a perfect apricot, though a really good one was hard to find.\n\n\"A drink, yes,\" Benjamin said. \"And maybe more than one apiece.\"\n\nLing appeared, moving with a gliding speed that belied her advanced years.\n\n\"Ah, our friend and hero, Ling. I shall have a glass of wine,\" Charles told her. \"A Cabernet, I think.\"\n\n\"I'll have a Cognac,\" Benjamin said. \"You know what I like.\"\n\nThey sat glumly watching the video frames opening and closing around Burnofsky as the system cycled randomly through the hundreds of surveillance cameras. Here was a woman making copies. There a man staring blankly into space. A couple putting on coats ready to go home. Jet-lag-dulled shoppers at the Twins' O'Hare Airport store. Two men debating something, both pointing at tablets.\n\nAt the bottom of each window was a small tag giving the location. Athens. Newport News. Tierra del Fuego. Johnson City. AFGC\u2014the Armstrong Fancy Gifts Corporation\u2014had locations all over the world, even without counting the shops in virtually every airport.\n\n\"We have not lost, brother,\" Charles said softly, with what he hoped was an undertone of iron resolve.\n\n\"Yet we're in hiding.\"\n\n\"We have not _lost_. We are not _beaten_. We have the Hounds. We can rebuild the twitcher corps. We can start again. And we have Floor Thirty-Four.\"\n\n\"Floor Thirty-Four's a losing tactic,\" Benjamin snorted. \"Defensive. It takes down BZRK. But it does not give us back the president we lost, or the premier we lost, goddammit! God _damn_ it!\" He slammed his fist down on the desk, making Charles's glass of wine jump. \"Or Bug Man. Or the _Doll Ship_.\" He moaned. \"What we have lost! What we have _lost_!\" He drained the snifter of Cognac in a single long swallow.\n\n\"When Floor Thirty-Four is ready, we take down BZRK and all they have within weeks. It spreads, brother; it will find them in all their hiding places. And when it has done its work, we will be without enemies, we\u2014\"\n\n\"Without enemies? You think BZRK is our only enemy? Don't you know the Chinese are dissecting every body they fish out of Hong Kong harbor? They know. They _know_! And if the Americans and Europeans don't know yet, they will soon.\"\n\n\"What is it you want, Benjamin? To unleash the gray goo?\"\n\nThe _gray goo_ , a ridiculous name for a deadly threat: self-replicating nanobots. Nanobots building more nanobots with whatever material they found at hand. Going from thousands to millions and billions and trillions in mere days, consuming every last atom of carbon and a good many other elements as well. Everything that lived or had lived on the surface of planet Earth. Everything that made life possible.\n\nNanobots were the mechanical answer to biots. Just as small, but without the eerie and inexplicable link that connected a biot to its maker. Nanobots had to be run through a game controller. They were somewhat less capable, but they had a huge advantage: it was nothing to lose a nanobot. But to lose a biot? Well, that way madness lay.\n\nBenjamin gestured at the screen. He happened to be focused on a family at one of the AFGC shops, this one at Airport Schiphol in the Netherlands. A family. Man, woman, blond child, poring over souvenirs. \"I hate them sometimes. I hate them enough to do it.\"\n\nCharles intuited which frame his brother was focusing on. \"Yes, but imagine them as _ours_ , brother. Imagine them united with us. Imagine them happy to look at us. Imagine what we can make them into with our nanotechnology and our friends from Nexus Humanus.\"\n\n\"Nexus Humanus,\" Benjamin snorted. It was a cult they had financed as a way to recruit twitchers to control nanobots, and other useful folk. But it had lost steam, like bargain-basement Scientology. \"We had it, the world we seek. The _Doll Ship_.\" A tear welled in Benjamin's eye, swelled, and went rushing down his cheek.\n\n\"Nonsense, brother, it was only a _model_ of the world we seek.\"\n\n\"A world united,\" Benjamin said, bitterly wondering at his own naivety. Weeping, figuratively at least, for the benighted human race that was being deprived of the utopia he saw so clearly. \"One vast interconnectedness, with us at the nexus.\"\n\n\"It can still be. It can. But not if we unleash the SRNs. Not the gray goo, not that final act of G\u00f6tterd\u00e4mmerung. The lesser tack, though...\" Charles was offering a sacrifice to the god of Benjamin's rage. A step short of apocalypse.\n\n\"Massed preprogrammed attack,\" Benjamin said, accenting the final word. Nanobots could be programmed to carry out simple commands autonomously. Large numbers of them, so long as the task was simple. Millions of them if necessary. They could be programmed to destroy all in their reach for a certain period of time and then turned off, a sort of localized, small-scale gray goo.\n\n\"If it's true that the intelligence agencies either know or will soon, then we won't be safe, even here. But if we disrupt... If we launch mass releases. Washington. London. Beijing. Give them something to keep them very busy. And at the same time use the Floor Thirty-Four weapon to take out BZRK...\"\n\n\"There he goes. Burnofsky. He's doing it again.\" Benjamin had spotted it. He gave the voice command to expand the screen. Burnofsky's image pushed all the others aside.\n\nIn the image\u2014high-def, no grainy monochrome\u2014Burnofsky had lit a cigarette. He took a few puffs. Sat, staring at nothing. Took another drag on the cigarette.\n\n\"Here it comes,\" Benjamin said.\n\nBurnofsky slid a desk drawer open. He drew out a framed photograph of a young girl.\n\n\"The daughter,\" Charles said. \"He's never gotten over it.\"\n\nBurnofsky looked at the picture and puffed his cigarette so that now the smoke partially obscured the image, swirling up around the hidden camera. They could see only the side of the man's face, but the smile was huge, ear to ear. The smile and a silent laugh.\n\n\"Volume up,\" Charles ordered.\n\nBurnofsky was making a chortling sound, a private, gleeful, somehow greedy sound. Like a miser counting his money.\n\n\"Bugs in your brain, baby,\" he said, laughing happily. \"Bugs in your brain.\"\n\n\"System: zoom in on Burnofsky's face,\" Benjamin ordered. The camera zoomed. \"He's crying as he laughs. Crying and laughing. Here it comes.\"\n\nBurnofsky lifted his shirt up off his corpse-white concave belly. They had a poor angle on this, just barely able to see.\n\nBurnofsky sucked hard on the cigarette, and holding the smoke in his lungs, stabbed the lit end of it against his belly.\n\nThey heard the sizzle.\n\nHe held it there; held it, held it, held it... and then, with a cry of pain that caused smoke to explode from his mouth, Burnofsky at last pulled the cigarette away.\n\n\"Karl, Karl, Karl,\" Charles said.\n\n\"Exercising, eating well, no more drugs, far less alcohol.\" Benjamin recited the relevant facts. \"Seemingly less depressed. And this self-mutilation is the price, somehow. You know it's BZRK, brother. You must _know_ that. He's wired. They've taken our genius from us.\"\n\nCharles sipped his wine. He had to take it slow if Benjamin was going to be swigging brandy. \"I don't _know_ it. But, do I suspect it?\"\n\nHe let the question hang.\n\n\"We must return home. Home to the Tulip.\"\n\n\"Back to the Tulip?\" Charles's voice was troubled. \"Even now that will be dangerous.\"\n\n\"I've spent\u2014 _we've_ spent\u2014our lives skulking and hiding, brother. Is there not, finally, a time to stand up and be seen and counted?\"\n\nCharles didn't argue. He knew it would be pointless. Benjamin would have his own _G_ \u00f6 _tterd_ \u00e4 _mmerung_. Charles felt sick inside. He did not want this to end in apocalypse. He had never wanted anything, really, but for all the world to be happy. And to accept him for what he was. And if only he could be allowed to wire the entire human race with his nanobot forces that beautiful vision would be realized. A world of peace. A world free from want and hate and fear and pain because every human being would be brother, sister, father, mother to every other human being. _One vast interconnectedness_.\n\n\"We hit back,\" Benjamin was saying. Over and over. \"We hit back!\"\n\nCharles closed his eye and heard the voice of his brother, so many years ago, so long ago, before they understood. Before they came to accept their isolation and loneliness. The voice of the child Benjamin was the voice of the grown man now.\n\n_Hit back, hit back, hit back_.\n\nOn the screen Burnofsky was giggling and crying.\n\n# **FOUR**\n\nSadie and Noah were bundled into a Land Rover and driven straight, without packing, without ceremony, without time to breathe, to a privately owned airstrip and practically shoved aboard a Gulfstream.\n\nThe pilot filed a flight plan for the relatively short hop to Sambava Airport on the main island of Madagascar. But that would be the expected route, and if the enemy had gone to the trouble of blowing up a boat, would they hesitate at an airport assassination?\n\nSo the Gulfstream flew on, took on fuel in Kenya, and made the long haul to Madeira to prepare for the final leg to New York's Teterboro.\n\nAt Madeira the security men let them off the plane. Plath and Keats took a taxi into the whitewashed city of Funchal, where they ate voraciously at a caf\u00e9 that smelled of garlic, red wine, and cedar, and served cod and prawns and good, doughy bread in a sky-blue stucco dining room. The Gulfstream had left in too great a hurry to take on food, and despite their picnic lunch hours earlier, they were starving.\n\n\"So what do we do now?\" Keats asked. He had the sense that this might be the last time they could speak freely. There was a single weary McLure security guy outside on the street, gun out of view but not out of reach, but no one was watching or listening in the restaurant and the clatter of cutlery on pottery and china would have obscured their words in any case.\n\n\"Back to New York,\" she said with a shrug.\n\n\"And then?\"\n\n\"Then we do whatever Lear tells us to do.\" It sounded bitter. It was.\n\nKeats tore at a piece of bread then used it to sop up some gravy. \"That's not proper, is it? Proper table manners, I mean.\"\n\n\"Yeah, that's what I care about,\" Plath said. \"Table manners.\" She offered him a smile and put her hand on his.\n\n\"It doesn't make sense, that's the thing,\" Keats said.\n\n\"Manners?\"\n\n\"Blowing up the boat.\"\n\nOne of Plath's continuing joys in her relationship with Keats came from the fact that in just about every case where she wondered if he was understanding things, he was. He might look a bit like the na\u00efve dreamboat guy, but those too-blue eyes and sensuous mouth were deceptive. There was a sharp, observant brain there as well.\n\n_When am I going to stop underestimating you, Noah?_ She asked herself this silently, and in her mind he was firmly Noah still, not Keats. Keats was work. Noah was... well, what? Love?\n\nHe loved _her_. Did she love _him_?\n\nWas it a class thing? The fact that she came from money and his family had never risen to middle class? Was she really that shallow? She wouldn't have thought so, would have angrily denied it. But at the same time, coming into her inheritance had without doubt added just a bit of swagger to her worldview.\n\nShe was rich. Very rich. He was very much not. Was that why she still held something back from him? That would be shameful. Or was it simply that she had seen him in ways no young woman is meant to see a young man? She knew too much and had memories that were far too vivid and intrusive. She knew what his lips looked like in the micro-subjective.\n\nShe knew that down there, where distances were measured in microns, those full lips were crusted parchment. She knew that his fingertips looked like arid, plowed fields. She knew that his tongue was serried ranks of pink hoods, and that trapped between the rows were bright false-color bacteria.\n\nShe knew that living things crawled in his eyelashes, tiny things, unless you were down in the meat and saw them m-sub. Then they didn't look so small. M-sub fleas looked like spiky, punk versions of the armored oliphaunts from the _Lord of the Rings_ movies, except that they could jump a thousand times their own height.\n\nShe knew, above all, that all the intelligence and charm and wit, all of his readiness to commit, all the love he was so ready to express, was nothing but minute electrical charges firing along neurons in the wet folds of his brain.\n\nShe had not just seen these things on an image captured from a scanning electron microscope. She had _been_ there in her biots. She had seen them all with biot eyes that were as real to her as her own.\n\nEven now she knew that Noah was seeing the same with her. One of his biots was in her brain right now. All three of hers\u2014P1, P2, P3\u2014were in the vial she wore on a chain around her neck for safekeeping, but she was still seeing through their eyes, seeing a long, rainbow-hued glass wall. Three distinct windows were open inside her visual field. And if they ever began to go dark... then would come the madness she defied by taking the name Plath.\n\n_Down in the meat_.\n\nOnce you had gone down in the meat, the images could not simply be set aside and ignored. And after memories came imagination, so that she would picture things she had not seen through biot eyes as they would look at m-sub.\n\nShe would see the micro detail of his lips and her own; she would see the rough furrows of his fingertips as they brushed her nipples; and she could imagine the billion tail-whipping sperm cells as he ejaculated.\n\nIt was all, at the very least, distracting. Though somehow it never seemed to distract _him_ \u2014\n\nKeats waved his hand up and down in front of her face.\n\n\"Sorry,\" Plath said, and snapped back to reality. \"I was considering. The boat. Yeah, it was both crude and ineffectual.\"\n\n\"Armstrong wouldn't come at us that way,\" he said. \"If they knew where we were, they'd deploy nanobots. There have been servants in and out of the house, we had a doctor in when I got food poisoning; there were opportunities for infestation.\"\n\n\"Or they could have targeted some of Stern's people and bounced the nanobots to us from them. I mean, if you know where two members of BZRK are, you try to _wire_ them, you don't try to kill them.\" She glanced over her shoulder upon saying the word _BZRK_ , pronounced with vowels intact: \"Berserk.\"\n\nKeats nodded, tore off another piece of bread, sopped up more gravy, and popped it in his mouth. Plath could imagine the scene down at the m-sub. The teeth would be impossibly huge, scaly not smooth, massive mountainous gray boulders dropping from the sky and rising from below to crush and\u2014\n\n_I have to stop this. I have to get control of my thoughts_.\n\nToo easy to let that consciousness of another universe take over her mind. Too easy to go from distraction to revulsion. She had to be able to be with another human being without always picturing that other, stranger reality.\n\n\"Maybe it was something totally different,\" Noah suggested. \"Maybe there was a fuel leak on the boat. Maybe we're just overreacting.\"\n\n\"Maybe,\" Plath said. \"But our time in the Garden of Eden had to end eventually. We had to go back. We're supposed to be running things.\"\n\nKeats met her gaze and shook his head slowly. \"No, not we. You, Sadie.\" Then with a wry smile he corrected himself. \"You, _Plath_.\"\n\nShe could have said that they were partners. She could have said that obviously he was as important as she was.\n\nBut she had not told him about the message from Lear telling her to get back in the game. The message she had ignored for days.\n\nShe wondered if she should tell him now.\n\nBut instead she copied him and mopped up some gravy. She didn't have time to worry about tending to Keats's ego. Her mind was filling with the implications of the suspicion that they were being shepherded.\n\nDriven.\n\nManipulated.\n\n\u00b7 \u00b7 \u00b7\n\nAnthony Elder, who had once used the name Bug Man, was shopping for onions at Tesco. Not just onions, there were other things on the list, too. But it was onions that somehow irritated him.\n\n_Nutella_\n\n_Beans_\n\n_Bread_\n\n_Pasta (store brand, nothing fancy)_\n\n_Mushrooms (fresh, button, 1\/2 pound)_\n\n_Cheerios_\n\n_2 oranges_\n\n_3 onions (the white kind)_\n\n_Three onions. The white kind_.\n\nThis was his life. Again. His mother was already on him about going back to school. To _school_!\n\n\"You don't want to go on neglecting your education, Anthony. That's most likely why you were let go.\"\n\n_Let go_.\n\n_Well, no, Mum, I wasn't exactly let go. I ran for my life\u2014flew for it, actually, all the way back to England\u2014after my mistakes caused the American president to blow her brains out in front of the whole world. It wasn't because I couldn't conjugate French verbs or recall the date of the Battle of Hastings_.\n\nHe didn't say that to his mother, of course.\n\nHe walked down the cereal aisle searching for Cheerios, maneuvering around a woman who was pushing both a baby buggy and a shopping cart. He found the cereal, puzzled for a moment over what size box he should be getting. His mother would chide him no matter what he chose.\n\nSmall, then. Easier to carry home. Less chance of catching some smart remarks from passing thugs.\n\nHe'd been on top of the world. Now he was self-conscious about being seen by others his age, struggling with plastic bags of pasta and Nutella and onions. The white kind.\n\nA pretty girl coming toward him looked right through him as if he was invisible.\n\nHe'd had the most beautiful girl in the world. Jessica. She'd been a slave to him. A slave. The memories made him ache inside. He would never get within conversational range of a girl like that again.\n\nTop of the world, that's where he'd been. But all that was gone now. All that gone and now he was invisible to women and girls. He was a moderately attractive black teenage boy with no obvious signs of wealth or future prospects. Why _would_ they look at him?\n\nHe rounded a corner, walked glumly past aisles of this and that, entirely forgetting the pasta, ignoring the plastic-wrapped slabs of meat to one side, heading to onions.\n\nHe felt rather than saw that something had changed.\n\nInstinct. Some sense that was not quite sight\u2014sound, smell, or touch. The certainty that he was being watched. Without turning to look he knew he was being followed. His speed was being matched.\n\nHe walked slower, stopped, pretended to admire the lamb; but the presence did not pass him by.\n\nHe moved suddenly toward the produce department, walking too fast, and he felt his pursuer keep pace.\n\n_Well_.\n\n_Well. Ah. So_. So was it cops or killers?\n\nHis heart was heavy in his chest. His feet dragged a bit, just the toes scraping on the tile. Shit, he'd just started to think maybe he was out of it, that maybe the Armstrongs would let him go. He'd given them a lot of good work, after all.\n\nIf not some hitman for the Armstrongs, was it police? Or even MI5?\n\nHe stopped in front of a bin of oranges and rested his hand on one, just feeling it. He liked oranges. Was this the last one he would see for a long while? Or the last one ever?\n\nHe turned, resigned, not seeing the point really in continuing to pretend. And there was his pursuer.\n\nNow surely _that_ was not a cop or MI5.\n\nThe man was well dressed, almost like a banker. Far too elegant looking to be a cop. He was a black man, tall, thin, with glasses, and when he met Anthony's eyes he smiled. Like an old friend. At first Bug Man felt himself relaxing, but no, no, that was a bad idea. A smile meant nothing.\n\n\"You want something?\" Bug Man asked. His voice was ragged. Maybe the expensive suit hadn't noticed.\n\n\"Anthony Elder?\"\n\nHe nodded. What would be the point in lying?\n\nWhat about running? He could surely outrun this man.\n\n\"Are you here to kill me?\"\n\nThe man was not surprised by the question. \"Not at this time.\" He smiled. \"But you will be taken for questioning by this time tomorrow.\"\n\n\"Haven't done anything.\"\n\n\"Oh, come now, you know better than that. People of our particular skin tone don't need to be guilty of anything to be questioned by the police, now, do we?\"\n\nBug Man moved a step sideways, edging along the oranges. He spotted the onions. The white ones.\n\n\"Met police will pick you up tomorrow, but of course it's not really for themselves. They'll turn you over to the Security Service, to MI5, for questioning.\"\n\nThe man moved closer so he could speak more quietly. He smelled of sandalwood and spearmint. Bug Man liked the cologne, didn't like the man belonging to it. He had a ridiculous urge to ask him whether it was available for sale here at Tesco.\n\n\"They will detain you on a secret warrant, and in all likelihood you will be given a chance to plead guilty so as to avoid a public trial. They'll put out a statement accusing you of something like embezzlement. Something safe for public consumption. They'll promise to let you out in a few years, and they would, really they _would_. Except that you'll have been gutted by some hardened lifer in your cell long before that. They'll make sure of that. If they don't, their cousins will\u2014the Americans.\"\n\nBug Man licked his lips. This was a threat, but not just a threat. This was the beginning of an offer.\n\n\"Whatever they want, the Twins, whatever they want, I'm still the best; I'm still fucking _Bug Man_.\"\n\n\"The Twins?\" The man made a crestfallen face, an act, a little show that he was putting on. Bug Man wanted to punch him. \"Oh, yes, the _Twins_. Well, Anthony, this is not really about them. I'm not able to tell you anything, really, but I can tell you that I don't work for the Twins.\"\n\nBug Man took a breath. He'd forgotten to do that. \"Who are you, then?\"\n\n\"My name is George. George William Frederick.\"\n\nHe said it as if it should mean something to Bug Man. And it did ping some distant, dusty strand of memory. But nothing meaningful. It was a name out of a different time, Bug Man felt.\n\n\"You slept through history, didn't you?\" George William Frederick said. \"That's a shame. History is everything important, really. In any case, I'm here because the surveillance team that has been on you for every minute of the last month is outside, in the parking lot, drinking coffee in paper cups and eating HobNobs, confident that you will soon emerge with your groceries. They'll follow you home, as per their orders, log your movements, and go off shift at eight p.m. They won't bother with physical surveillance after that; they'll be watching on the cameras they have in your home. Yes. So, as it happens, this would actually be an opportune time for you to follow me, out of the back of the store, to a waiting car.\"\n\nBug Man immediately ran through some of the more embarrassing things that would have been observed by cameras in his home. But he was mostly over the concept of privacy. The Twins had had cameras on him from the start of his employment by them.\n\n\"And then?\" Bug Man asked.\n\nGeorge-With-Three-Names shrugged. \"All I can tell you is that an Armstrong hit team is also looking for the right moment to shoot you, and tomorrow MI5 will bundle you off to prison where they or the Americans will do for you, and the third alternative, the one I'm offering you, is preferable.\"\n\nBug Man knew the man was speaking the truth. Or at least believed himself to be telling the truth.\n\nGeorge-With-Three-Names. George William Frederick. The penny dropped.\n\n_George III_.\n\n_The mad king_.\n\n\"You're BZRK.\"\n\n\"Think what you like,\" George said with a self-satisfied smile. \"I'm your way out.\"\n\n\"You _are_ going to kill me.\" Bug Man was proud that he managed to get the words out with only a minor tremor in his voice.\n\nGeorge tapped his waist. There was something there that was no belt buckle. \"If that were my instruction, you'd never know about it. By the way, you're not Roman Catholic, are you?\"\n\n\"What? Church of England, I guess. But\u2014\"\n\n\"Good.\"\n\nBug Man let it go. The point was, this wasn't an assassination. \"Will I have time to say good-bye to my mother?\"\n\nGeorge shook his head.\n\n\"Good,\" Bug Man said. He nodded, smiled for himself alone, and thought, _Okay then: back in the game_.\n\n# **[ARTIFACT]**\n\nAn exchange of texts\n\n**Plath:** Back in NYC. What is our mission?\n\n**Lear:** Destroy AFGC.\n\n**Plath:** What does that mean?\n\n**Lear:** Find and kill the Twins. Destroy all AFGC records. Kill or wire all AFGC scientists and engineers. Their technology must be obliterated.\n\n**Plath:** I'm to do this with 7 people?\n\n**Lear:** You had your vacation. Besides there is an 8th.\n\n**Plath:** Caligula?\n\n**Lear:** I've always found him very useful.\n\n[Long pause]\n\n**Lear:** Time is short, Plath.\n\n**Plath:** Short why?\n\n**Lear:** AFGC very close to developing remote biot killer. Nature unspecified. Days not weeks until it is weaponized. You must strike before then. Ticktock. Death or madness.\n\n# **ELAPSED TIME**\n\nThe Gateway Hotel could not be repaired or rebuilt. The blowtorch heat of the burning LNG carrier ship had burned everything capable of burning. Natural gas burns at temperatures ranging from 3,000 to 3,6000 degrees Fahrenheit, and that's enough to incinerate furniture, carpet, and paint. It's also enough to melt glass and soften structural steel. A human body is a marshmallow.\n\nThe Gateway was a black, bent, crumpled horror that reminded some observers of a very old woman, bent by arthritis, in the act of falling to her knees.\n\nBuildings on either side had burned as well. Buildings farther back in Kowloon, where the gas had rolled through the streets before catching fire, were burned. Some had exploded, simply popped open like rotting fruit. Kowloon Park was a field of ash.\n\nThe Chinese government had not been able to conceal the extent of the disaster. It was visible from satellites and from the decks of passing ferries and cruise ships. This was Hong Kong, not some provincial outpost. The whole world passed through Hong Kong.\n\nThe government had kept a faithful account of the dead and presumed dead. Now over a thousand. The \"presumed dead\" included those so badly burned that no more than a few bones with the marrow boiled away had survived and could not be identified.\n\nDivers were still pulling bodies out of the blistered and twisted hulk of the liquid natural gas carrier\u2014the ship dubbed the _Doll Ship_ \u2014that lay at the bottom of Hong Kong harbor. The Chinese government was nowhere near as forthcoming on this part. The official story was that it had been simple error on the part of the ship's captain. He was dead: he wasn't going to argue.\n\nNo one spoke openly of the bodies of children found blown apart. No one spoke of the fact that one of the ship's spheres, and possibly a second one as well (it was hard to tell), had never contained LNG but had instead been something very much like a human zoo.\n\nCrewmen who had managed to jump ship were picked up and spirited away to a camp in far-off Qinghai Province. A small number of British Royal Marines were held there as well. And twenty-four civilians, neither crew nor soldiers\u2014inmates on the _Doll Ship_ \u2014were being held at a small local hospital that had been taken over by the Ministry of State Security. The MSS had drafted a dozen radiologists, neurosurgeons, and pathologists, snatched them up from cities all over China and bundled them off to Qinghai.\n\nInterrogations were under way.\n\nMedical investigations were under way.\n\nNeither was terribly gentle.\n\nChinese premier Ts'ai attempted to shut down the camp, ordered all survivors to be executed and their bodies cremated. Which would have worked had not the governor of Qinghai Province slow-walked that order. He smelled a rat.\n\nTwo weeks after the Hong Kong disaster, the MSS briefed certain members of the Central Committee on their findings from the survivors. And on Ts'ai's unusual and very out-of-channels effort to shut down the investigation.\n\nTwenty-four hours later the Chinese official news agency reported that Premier Ts'ai had suffered a stroke. He was getting the best care available, but doctors were not hopeful.\n\nIn fact, the top of the premier's head had already been sawed off. His brain had been carefully scooped out of his skull, flattened and stretched, frozen, cut into handy one-centimeter sections, and was now being examined minutely under a scanning electron microscope.\n\nThey found numerous strands of extremely fine wire\u2014nanowire\u2014in segments as long as three centimeters, and a dozen tiny pins.\n\nSimilar wire had been found in the brains of survivors of the _Doll Ship_.\n\nA careful\u2014but less drastic\u2014autopsy of President Helen Falkenhym Morales found no evidence of brain abnormality. Then again, the single nine-millimeter bullet she had fired into her own head had bounced around a bit inside her skull and made a mess of the soft tissue.\n\nThe FBI director, a man who would not have fared well himself if his brain had been carefully examined under an electron microscope, pushed for the conclusion that the suicide was a result of depression following the death of her husband.\n\nFBI forensic experts produced a report stating that the videotape purported to have been taken (by means unknown) directly _through_ _the president's eye_ \u2014the videotape that seemed to suggest that President Morales had beaten her husband to death\u2014was a clever fake.\n\nThere was obviously no way for the images to be real. Presidents did not commit murder.\n\nThen again, they didn't make a habit of committing suicide, either. But that undeniably happened.\n\nIn a bit of historic irony, the authoritarian state of China discovered the truth, while the American democracy had thus far missed it.\n\nBut there were other investigations under way. A joint committee of Congress. An independent blue-ribbon panel featuring a former secretary of defense, a former senator from Maine, and the chairman, a former president of the United States.\n\nOnly one of them had thus far been compromised by busy little creatures laying wire.\n\nMinako McGrath, who had been kidnapped and taken aboard the _Doll Ship_ , was one of the few to escape entirely. With the help of an ex-marine, former gunnery sergeant Silver, who'd been aboard that floating horror show, she made her way back from Hong Kong to Toguchi, Okinawa, one step ahead of the Hong Kong authorities.\n\nBut she found some changes when she finally reached her home. Her Facebook and Twitter accounts were closed. Her Internet access\u2014in fact her whole family's Internet access\u2014was blocked.\n\nThen her mother was called in to see the commander of the local base where Minako's father\u2014himself a U.S. marine\u2014had been stationed before he was sent to Afghanistan and killed. She was told quite simply that if she could keep her daughter quiet, her family would be safe and her late husband's official military service record would remain unblemished.\n\nThere was no direct threat. Just that promise. Just the carrot. The stick was only implied. The general looked sick to his stomach going even that far, but marines obey orders, and it was clear that he was passing on an order that came from very high up the chain of command.\n\nHaving been saved by one marine, and honoring the memory of her father, upon hearing the ultimatum Minako nodded solemnly and raised a hand in salute.\n\n\"Semper fi,\" she said.\n\nA week later Minako's mother, the police chief of their little town, was offered a civilian contract to work in security on the base, at a seven-hundred-dollar-a-month increase in pay.\n\nMinako got a Vespa motor scooter.\n\nAnd from that point on Minako discussed the _Doll Ship_ only with her marines-supplied therapist, who duly shredded all records of her visits and prescribed Prozac.\n\nDespite the separate efforts of the Chinese and U.S. governments, Google searches for various conspiracies were up in the last month.\n\nWay, way up.\n\nPossible suspects included the Illuminati, the Church of Scientology, Anonymous, the Freemasons, the Roman Catholic Church, the Bilderberg Group, Iran, China, the CIA, the NSA, the DEA, MI5 and MI6, Mossad, Ag\u00eancia Brasileira de Intelig\u00eancia, Direction Centrale du Renseignement Int\u00e9rieur, the Russian Federal Security Service, and, of course, space aliens.\n\nWith far fewer searches: the Armstrong Fancy Gifts Corporation.\n\nAnd with only a handful of searches, most as a result of accidental misspellings: BZRK.\n\nThere was no change whatsoever in searches for \"Lear.\"\n\n# **FIVE**\n\nPlath. That was her name again. Plath, not Sadie.\n\nShe'd been back in New York for just thirty-six hours, sleeping the first half of that.\n\nPlath was provided by the weather with a perfect disguise to move about the streets of New York. It was freezing and the faux-fur-lined hood of her coat along with superfluous glasses and her newly blonde hair made it very unlikely that anyone would recognize her.\n\nShe had taken a cab to the Tulip. The Armstrong headquarters was not a place where she could take any, even slight, risks of being recognized.\n\nBut she had gotten out and walked the last block to the Freedom Tower. It soared up into low-hanging clouds. One hundred and four stories of defiance to replace the lost World Trade Center towers.\n\nShe had not yet been born when the towers fell, but she had seen the video. They'd had a unit on terrorism in school.\n\nThe Tulip was not as tall as either the World Trade Center or the Freedom Tower.\n\nShe had distinct memories of the videos of that day, September 11, 2001. Funny that she recalled them so clearly. But there it was, playing over and over in her mind.\n\nThe jets.\n\nThe initial explosions.\n\nThe spreading horror of billowing smoke.\n\nTwo hundred people leaping to their deaths rather than die more slowly of smoke and flame.\n\nThe awe-inspiring, horrific collapse as the melted, hollowed-out building fell.\n\n_Find and kill the twins. Destroy all AFGC records. Kill or wire all AFGC scientists and engineers. Their technology must be obliterated_.\n\nIt was all in the Tulip. The technology, the records, the scientists. The Twins. Up there at the top floors, what, sixty-seven? Sixty-eight? She'd been rather distracted the last time she was in the Tulip, hard to recall the exact floors where the Twins lived and looked out over the concrete and haze of the city.\n\nA single skyscraper in Midtown Manhattan.\n\nHer breath came out in a cloud of ice crystals. She looked around, feeling obscurely guilty, but no one in the sparse crowd of tourists or the crew at work around a steaming manhole was looking at her.\n\nUnder her breath, Plath made a sound. It was the sound of a slow-motion explosion.\n\nLystra Reid watched Plath as she looked up at the Freedom Tower and knew exactly what she was thinking. Exactly. She was contemplating destruction, _yeah, yeah, yeah_. Destruction. She was envisioning it already.\n\n_That was quick. But then, if you want great results, hire great people. Even if they are a wee bit nuts_.\n\nLystra had a Starbucks latte in her hand. One of the things she would miss, she supposed: convenient and at least somewhat drinkable coffee. There were things about this game space, this paradigm, that she would regret losing. But it was never good to become complacent.\n\nTime for the 2.0. As there was a Grand Theft Auto 6, there must inevitably come a day when GTA 6 was done and a GTA 7 must be born. Even the greatest game was eventually played out. When you had squeezed all the fun out of Portal you needed a Portal 2, 3, 4...\n\n\"Yeah. Yeah.\"\n\nShe shivered\u2014it was cold\u2014and tossed the cup into a trash can. Her newest tattoo was itching, and she scratched her rib cage discreetly. She was just thirty feet or so from Plath. Plath was, what, fifteen years her junior? But they could have been sisters, perhaps, in a different world. Maybe, come to think of it, they _would be_ , in this new game Lystra was creating.\n\nShe acknowledged her own loneliness. Emotional honesty did not frighten her. There had been a price to pay for becoming what she was: rich, successful, powerful beyond what anyone would guess. Arguably at this point, the most powerful person on Earth.\n\nNo, the truth never scared Lystra.\n\n_Lonely? True. Strange? True, yeah. Yeah. Crazy? Well, once upon a time, yeah, but no longer_.\n\nShe closed her eyes and replayed the memory of seeing madness overtake Sandra Piper. God, that had been intense. The eye-stabbing thing, wow, that was the kind of detail you got only from seeing things firsthand.\n\nShe remembered a girl trying to strangle herself with a bedsheet. Crazy people did crazy things. _Back in the day, back in the old days, yeah_. But never anything to match the weirdness of watching a famous actress stabbing her own eyes. Now _that_ was crazy.\n\nSad to think that she would have to retreat soon and watch the endgame play out from a distance. But not yet. There would be many rich, visceral experiences to come before she headed south.\n\nAnd then?\n\nAnd then she would play the new game and win that as well. Or not. She might not master the new game. She might even lose.\n\nThe idea made her smile. Her father had taught her to understand that life was a walk on a tightrope and death was the ground. Sooner or later, no matter how agile you were, the ground would claim you.\n\nHe'd been full of gloomy pronouncements back in the old days, sitting in lawn chairs outside their trailer as the carnival shut down for the night. They would sit there, the two of them, the man and the child, as the lights went out on the Mad Mouse and the Ferris wheel. They would sit and sip their drinks\u2014bourbon for her father, unsweetened iced tea for her\u2014and acknowledge the nods and the weary greetings as the other carnies headed for their own digs.\n\nThe nights had almost always been warm and muggy. The carnival mostly played the south: Baton Rouge, Bogalusa, Hattiesburg, Vicksburg. She'd seldom been cold, which was maybe why the cold attracted her now. Cold was clean. Hot was sweaty and dirty.\n\nBack then, back before the train wreck that was in her future, Lystra had wanted two things: For her mother to come back. And to be able someday to take over a couple of the sideshow games. An old man named Sprinkle operated the coin toss, the dart throw, the water pistol, and the ring toss. He let his games get shabby, refusing to spring for so much as a few cans of paint.\n\nLystra thought she could do better. She could make the games livelier and more profitable. The key was to make them a bit easier. Let the marks take home a teddy bear occasionally; it was good advertising. Run an honest game, attract more players, pay out more in prizes\u2014but offer more levels, more depth, and make more money in the end.\n\n\"Yeah!\" Lystra said to no one. It made her smile to think how even then, even when she was a lonely seven-year-old, she was ambitious.\n\nBut yes, lonely. She had always wanted a younger sister. Someone like Plath, maybe. Someone to look up to her. Someone to talk to and play with.\n\nEven a brother would have been welcome.\n\nInteresting thought.\n\n\"A game within a game?\" Lystra muttered under her breath.\n\nWould it add spice? Yes. Would it complicate the overall plan? She walked it through step-by-step in her mind and concluded that it would have only a small downside risk.\n\nIt would be good to have someone to appreciate what she had accomplished. It would be good to have someone to watch it all play out with her.\n\n\"Minions,\" she said, and laughed. \"I need minions. Yeah.\"\n\n# **SIX**\n\n\"No. Vincent is not ready to resume control.\" This was from Anya Violet, and spoken in a whisper. \"He may never be ready.\"\n\nPlath was making peanut-butter-and-jelly sandwiches in the kitchen of the new Manhattan safe house. One for herself and one for Keats. And seeing Billy's level of interest she pulled out two more slices of bread for him.\n\nThey were in the kitchen: Plath, Keats, Billy the Kid who really was a kid, and Dr. Anya Violet. Anya was of undetermined age\u2014perhaps in her thirties, perhaps she had edged into her forties\u2014but to Plath, at least, she seemed beautiful, sophisticated, and effortlessly sexy in a way that she decided must come only with some age and some experience.\n\nAnya had not yet chosen a nom de guerre. She thought it was a silly affectation. Of course, she understood the thinking behind choosing the name of some mad or at least seriously unbalanced person: it signaled acceptance of the core reality for BZRK members. It signaled a break with the past. It signaled a chin-out acknowledgment of the fact that madness was very likely in their future.\n\nShe understood all that, but Dr. Anya Violet was not a child and was not interested in following the rules of the clubhouse. Nor was she sure she wanted to accept the authority of a sixteen-year-old girl. Yes, Plath was the daughter of Grey McLure, Anya's former employer, and Plath had proven herself in battle. And it had become clear that she was a bit more... stable... than Nijinsky, who had been in charge during Vincent's recovery.\n\nBut Anya was suspicious of money. She could call herself Plath, but Anya knew who Sadie was. She was rich, that's what she was. Worse yet, she'd always been rich. She'd had life handed to her. Anya would rather have seen Keats in the top job, because _there_ was a boy who had never been handed anything, and Anya instinctively trusted working people. She herself had come from nothing and nowhere to earn a PhD. She shared with Keats an emotional knowledge of hard times and hard choices.\n\nBut Keats was totally loyal to Plath.\n\nBilly was a child. Wilkes was... well, she was Wilkes. Nijinsky had to a great extent lost the confidence of the group. And that left two people to run things at the New York cell of BZRK: Vincent or Plath.\n\nPlath, who saw a great deal when she paid close attention, saw all this in Anya's smoky eyes. Vincent might or might not still be damaged, but Anya loved him and would never admit he was ready to take charge again. Not if it meant risking his life and sanity.\n\nIn the matter of safe houses things had improved quite a bit. Plath had access to most of her own money now, and she had Mr. Stern and the McLure security apparatus to arrange things. So BZRK New York was quite nicely established in a five-story townhouse not far from Columbus Circle on the Upper West Side.\n\nThey had obtained it through numerous cutouts and guys-who-knew-a-guy, and bought it for cash for nine million dollars.\n\nJust twelve blocks away was a second safe house. This had also been purchased for cash, but this time the cutouts had been just a bit less well managed. Not so poorly managed as to seem obvious; just a few scant clues left here and there for those who were watching the movements of Plath's money.\n\nThe _fake_ safe house was above a bankrupt dry cleaner. A sound system played ambient noise from within\u2014TV, music, the sound of laughter, occasional yelling. A timer turned the lights on and off. And random people delivering handbills were hired to enter and leave the place at odd times of day and night. It wouldn't stand up to in-depth surveillance, but it would do as a diversion. It was already, according to Stern, drawing the attention of Hannah Thrum, the chairwoman of McLure Holdings, the parent company of McLure Labs. Thrum was almost certainly working for the Armstrong Twins as well, but that was all right, so long as Plath knew where all the players were.\n\nLet Thrum follow the money. She was a numbers person. Numbers people loved to believe they saw deeper than anyone else, believing their numbers were truth. In reality, Thrum was chasing numbers like a kitten chasing a piece of string.\n\nPlath, Keats, and Billy carried their sandwiches back to the parlor where Nijinsky, Vincent, and Wilkes waited. Anya sat beside Vincent on the couch. Plath stood, leaning back against a walnut Restoration Hardware china cabinet, bit into the sandwich, and looked over her sparse troops.\n\nNijinsky was a bit less elegant and less well turned out than he'd been just a few weeks ago.\n\nWilkes had shaved half her head and dyed the other side a sickly yellow that was only vaguely related to blonde. Wilkes\u2014named for Annie Wilkes, the insane fan in Stephen King's _Misery_ \u2014was a tough chick, a pierced, tattooed (including a sort of down-swept flame tat under one eye), leather-and-lace teenager whose personal history strongly suggested that people not mess with her. There was a fire-damaged school in Maryland that stood witness to what happened when Wilkes lost it.\n\nBilly the Kid: a scrawny mixed-race kid who had shot his way out of an Armstrong attack on the Washington cell of BZRK. Shot his way out, and then shot his way back in to finish off any Armstrong survivors.\n\nKeats. The working-class London boy with impressive gaming skills and too-blue eyes. And a very nice, taut body, not that Plath should have been thinking about that at the moment. But she was; in fact, she was recalling a specific moment on the island, standing at the railing of their deck, watching the sun come up, Noah as he was then, behind her, his strong arm around her waist, drawing his forearm over her body, over her breasts, kissing the nape of her neck.\n\nShe took a breath. It was deeper and noisier than she'd intended, and she wondered if people guessed that she'd been daydreaming.\n\nFinally, of course, there was Vincent himself. Vincent had brought Sadie into BZRK. He had basically created Plath. He'd been their fearless leader until he had lost a biot in a battle with Bug Man. To lose a biot was to lose your mind.\n\nThe biot\u2013human link was still not understood. The mechanism that allowed the human \"parent\" to see through biot eyes, to move biot limbs, and to be so intimately connected with them that losing a biot was like some kind of psychic lobotomy\u2014that mechanism, that _force_ , was not understood. In fact, it had been a complete surprise when first discovered at McLure Labs by Plath's father, Grey McLure, and had remained a mystery to him to the day he had been murdered in spectacular fashion.\n\nThe effects of the brain\u2013biot connection were plain to see. Vincent, who had once been so dead calm, so in control, had fallen into madness. And the only way to save him had been with crude intervention down in the folds of his brain.\n\nPlath herself had done the job. She had delivered acid to sites in Vincent's brain that stored specific memories of his dead biot. She had watched through her own biot eyes as Vincent's brain cells burst and boiled and died, erasing memory, thoughts, ideas, and perhaps some piece of his personality.\n\nAfter that Vincent had clawed his way back from madness. He had gone back into battle against Bug Man, and he'd won. But that did not mean Vincent was _back_.\n\n\"Okay,\" Plath said. \"It's been a month. Things have calmed down a bit. Where do we stand?\" When no one volunteered an answer, she nodded and said, \"Jin?\"\n\nNijinsky turned cold eyes up to her. He had not fared well in the last month. While Keats and Plath were both tanned and rested\u2014well, as rested as they could be, given the fact that their boat had been blown up\u2014Nijinsky had become increasingly frayed and ragged. His clothing was no longer perfect. His hair was at least two weeks past its optimum. He was still by any normal standard a spectacularly handsome, well-turned-out person, a tall Chinese American with a graceful way of moving and a sad, sympathetic smile.\n\nThe changes would be visible only to someone familiar with his previous level of perfection. But the signs were there, even more visible in the red-rimmed eyes, the stress lines above the bridge of his nose, the grim tightening around his mouth. And of course the sour smell of a body oozing alcohol residue through its pores.\n\n\"It's been a busy month,\" Nijinsky said. \"Sorry you two missed it.\"\n\n\"Lear agreed I should disappear for a while,\" Plath said calmly. \"I'm known.\"\n\n\"Yes. And Lear agreed that I should get stuck with the shit work.\" He shrugged and tried on an insincere smile. \"Well, here's where we stand. Vincent is about seventy percent.\" He looked at Vincent and asked, \"Fair?\"\n\nVincent nodded. His cold gray eyes focused, then lost focus. \"Fair.\"\n\n\"Billy is thoroughly qualified for missions down in the meat. He has two biots. Wilkes is still Wilkes, God help us all.\" This he said with a certain wry tone that was very much the old Nijinsky.\n\n\"What else could I be?\" Wilkes asked, framing her face with her hands.\n\n\"Anya remains a bitch,\" Nijinsky said, trying to sound jokey about it and not succeeding. \"The president is dead, long live the new guy, President Abbott. The country is freaked out, but we are still not under surveillance\u2014as far as we can tell. The Chinese premier just had a very sudden illness, and we know he'd been compromised by the Armstrongs. So, it's possible the Chinese government is... aware.\"\n\n\"And Burnofsky?\" Keats asked.\n\nNijinsky shrugged. He looked away, not avoiding Keats, but seeing that weirdly colored window inside his brain. He had a biot resting on Burnofsky's optic nerve. The biot was tapped into visual input from Burnofsky's right eye.\n\n\"At the moment he's working,\" Nijinsky said. \"I can't make out what's on his monitor\u2014I have a pretty good tap, but you know what it's like.\"\n\nThey all, all except Anya, did know what it was like. Tapping an optic nerve was a bit like watching an old-fashioned TV in a thunderstorm back before cable, when the picture could be wildly distorted and never entirely clear.\n\n\"Has he been in touch with the Armstrong Twins?\" Plath asked.\n\nNijinsky nodded. He tapped a cigarette out of some exotic, foreign pack and lit it. \"Four days after that ship went down in Hong Kong. By the way, Lear is sure that was an Armstrong thing. Some kind of messed-up human zoo. By that point I was done wiring Burnofsky. I sent him back in. But nothing face-to-face. Wherever the Twins are now, they aren't talking to Burnofsky in person; it's all video link.\"\n\n\"Do you have a biot in his ear?\" Plath asked.\n\n\"No.\"\n\nThere was pause while everyone absorbed this. It meant Nijinsky could see what Burnofsky was seeing, but could not hear what he was hearing.\n\n\"Why not?\" Plath asked, deceptively quiet.\n\nNijinsky blew his smoke toward her. It was not a subtle gesture. He resented being demoted and didn't mind if she knew. \"Because I was using my other biots to train Billy, here.\"\n\n\"For a month?\"\n\nNijinsky shook his head. \"Fuck you, Plath.\"\n\nKeats's eyes narrowed angrily, but Plath remained cool. \"A lot has been asked of you, Jin. And you've endured a lot.\"\n\n\"Endured,\" he said, sneering at the word. \"Yes, I've endured a lot. A lot of enduring has gone on.\"\n\n\"Why not have Anya generate a new biot and use it?\"\n\nBilly and Wilkes were watching the back-and-forth between the two, like spectators at a tennis match. Vincent was elsewhere in his mind. Keats was keeping still, irritated by Nijinsky, but accepting that this was up to Plath to handle.\n\n\"Why not generate a new biot?\" Nijinsky mocked. \"When you play Russian roulette, you put one bullet in the gun and spin the chamber. _Click_.\" He mimed shooting himself in the head. \"A one-in-six chance you're dead. Two bullets? That's a one-in-three chance. Three? At that point it's fifty-fifty. You _know_ why not, Plath, so don't give me that hard look. Vincent barely survived the loss of one biot. Keats's brother is shackled in a loony bin for losing two biots. You want to hear what Burnofsky's hearing? Tell Wilkes to do it. Or do it yourself, Plath.\"\n\nPlath nodded. \"Okay. Fair enough.\"\n\n\"What are we doing?\" Anya asked wearily. \"What is this all about anymore? The Armstrong attempt to control the president is obviously ended. And it seems the same is true of the Chinese premier. The Twins are in hiding. Burnofsky has been wired and switched sides. Bug Man is gone. What are we doing? Are we playing a game? If so, what is our next move?\"\n\n\"They still have the technology,\" Plath said. \"They will try again. In some other way. They won't give up.\"\n\n\"How do we know that?\" Anya demanded.\n\n\"They found Keats and me. They blew up the boat that was coming to pick us up.\"\n\n\"Convenient, wasn't it?\" Nijinsky said.\n\nPlath didn't say anything to that, because she'd had the same thought. _Convenient_. If you wanted to push her and Keats back to New York. Say, after you'd ignored an order to get your ass back there already.\n\n_The punishment for desertion is death, isn't it? Or is that some Hollywood bullshit?_\n\nThe boat had blown up, but there was no follow-up. No attack on the beach, no attack on the compound they'd been staying in. No attack as they rushed to the airport and flew away from the island.\n\nNo attack waiting for them when they refueled in Kenya or Madeira, and no attack when they'd landed at Teterboro.\n\nHad a quick change of hair color somehow thrown off the kind of people capable of tracking her to Madagascar and then to \u00cele Sainte-Marie? Not likely.\n\nJust enough violence to send her running back to New York. Not as if someone was serious about killing her.\n\nLike someone wanted her back in the game.\n\n_Get back in the game_.\n\nThat had been the text from Lear. The one she'd ignored, because, why? Because she was Sadie McLure, that's why. Since when did she take orders? What was she, someone's butler suddenly? _Fuck you, Lear. I'm on a beautiful island with a beautiful boy who loves me and wears himself out trying to please me_.\n\nFor the first twenty-four hours after that she had felt liberated. Like maybe she had regained control of her life. But slowly her doubts had grown. What right did she have to blow off Lear? Lear was BZRK. Lear was the general, and she was a lowly lieutenant or whatever.\n\nAnd he'd been right, hadn't he, Lear? Right that she had to get back in the game? The Armstrong Twins seemingly still lived. The nanobot technology still existed. The liberty of all humanity was still in danger.\n\nThe Armstrongs still had to be stopped. Didn't they?\n\n\"I've heard from Lear,\" Plath said. She wasn't sure why, but she was reluctant to tell them. Maybe because once she said it she would have to take action.\n\n\"Did he mention whether he liked the whole blonde look you have going on?\" Wilkes. Of course.\n\n\"Lear says the Armstrongs have developed some kind of remote biot killer. Nature unknown. No other details. But...\" She shook her head ruefully. \"But his instructions are to destroy AFGC. Destroy their data in particular so this new technology doesn't go into use.\"\n\nLong silence. Much mute staring. Biots already faced a number of potential enemies, from the slow but irritating defenses of the body itself to the much more dangerous nanobots. But nanobots could be faced, fought, and, with luck and skill, killed. The idea of a weapon that could kill in some unfathomable way, in some way that did not even allow for a fight, was terrifying. It would be push-button madness.\n\nFinally, Nijinsky laughed, a low, slow sound weighed down by cynicism. \"Well, I'm going to use that word again. _Convenient_. We're all sitting here wondering why we're still playing this game, and what do you know? Turns out the bad guys have the means to drive us all insane and then enslave the human race.\" He lit a second cigarette and blew the smoke insolently at Plath.\n\nShe thought about telling Nijinsky to put it out. Show him that she was back and in charge.\n\nBut was she in charge? That was not clear.\n\nShe checked Keats. He was as dubious as Nijinsky.\n\n\"Yeah,\" she said by way of acknowledging their doubts. \"Yeah. Convenient. But I guess unless we want a visit from Caligula, we'd better...\" She faded out, realizing what she was saying.\n\nIt was Anya who put it into words. \"In the Great Patriotic War\u2014what you call World War Two\u2014Russians had soldiers. And behind the soldiers they had NKVD. Secret police. If a soldier complained, the NKVD shot him. If a soldier failed, the NKVD shot him. If a soldier said, 'To hell with this, I am going home,' the NKVD shot him. And then they arrested the man's family and sent them to the gulag.\"\n\n\"Well, they _were_ fighting the Nazis,\" Billy piped up.\n\nAnya snorted a derisive laugh. \"Yes, murderous, evil Nazis. And who were the NKVD? Murderous, evil Communists.\"\n\n\"I'm confused. Which are we supposed to be?\" Wilkes asked.\n\n# **[ARTIFACT]**\n\n_A News Item_.\n\n**Wellington, NZ**. Wellington Police Superintendent Thomas DuPr\u00e9 gave a press conference today in which he discussed the recent suicide of two Wellington Police Department officers, and the attempted suicide by a third, who remains in care at Wellington Hospital.\n\n\"All three officers reported seeing strange visions about an hour prior to their suicide attempts. They variously described these hallucinations as involving bizarre insects and strange objects.\"\n\nSuperintendent DuPr\u00e9 said all three were tested for drugs but results were negative. \"It's possible that this tragic episode is simply a rather horrible coincidence.\"\n\nAll three incidents occurred nine days ago. The two successful and one attempted suicides were particularly brutal and appeared to be unplanned.\n\nThe investigation is ongoing.\n\nNothing was said publicly about the fact that the three officers, while on their way together to a soccer match a week earlier, had come across an overturned truck on the highway apparently headed to the port.\n\nThe truck had appeared to be carrying military grade weapons.\n\nHigher authorities were called in to take over the case. And the three policemen would have nothing further to say on the matter.\n\n# **[ARTIFACT]**\n\n_From Deadline Hollywood:_\n\nThe Academy announced today that Sandra Piper's name would remain on the ballot for the Best Actress Oscar. There had been suggestions (surely not from studios and press agents tied to competing actresses, heaven forfend!) that the actress's bizarre suicide would send a bad message to movie lovers and especially young fans. The statement reads in part, \"We believe that an Academy Award is given for the work, and only for the work, and should not be affected by the tragedy that took this great talent's life.\"\n\nComments:\n\n**QxT:** _Sandra Piper was a great lady and a great actress. Shame on those who are trying to prophet from her death_.\n\n**KeyAgrippa:** _She was nuts. That's who we want to show off as a symbol of Hollywood?_\n\n**Book Guy:** _Tragedy my ass. She was murdered. I don't know how. Yet. But I knew Sandra, we worked together on UTD. No way she killed herself, she had everything to live for_.\n\n# **SEVEN**\n\nSeven thousand, two hundred and fourteen miles south and a bit east from the watery grave of the _Doll Ship_ , where bloated, bleached-out bodies still fed indifferent fish, a very different sort of vessel was roaring across very different waters. The navy called it an LCAC\u2014landing craft air cushion\u2014a hovercraft some eighty-eight feet long and forty-seven feet wide.\n\nThis LCAC was no longer part of the U.S. Navy; it was privately owned, and it had been extensively modified with more efficient turbines, tougher skirts, and integrated deicing systems.\n\nIt was one of two in active service in Antarctic waters. The craft were used to carry large cargos ashore and, just as critically, to remove garbage, and to do so in weather that would swat a helicopter down onto the ice.\n\nEnvironmentalists were determined to keep Antarctica \"green,\" despite the fact that green was rarely seen on the ice.\n\nThe LCACs shuttled back and forth between shore and a refurbished navy-surplus amphibious assault ship now called the _Celadon_. Celadon being a shade of green. (Her sister ship was the _Shamrock_.) The LCACs were the _Jade Monkey_ and the _Emerald_ , again, shades of green. But the LCACs were in fact painted white and gray with splashes of rescue-orange.\n\nThe particular LCAC arriving in a whirlwind of salt spray and noise was the _Jade Monkey_ , skippered by Imelda Suarez. Suarez\u2014no one called her Imelda\u2014had a four-person crew and a cargo of booze, diesel fuel, and a massive electrical generator covered by a tarp, as well as a climate-controlled steel container filled with potatoes, apples, fresh spinach, grapes, and oranges. The box was painted with the logo of Whole Foods, and indeed all the produce was organic.\n\nFor the old-timers the very idea that fresh fruit and meat could be almost (not quite) year-round was astonishing, and it caused quite a bit of grumbling about how easy things had gotten.\n\nIt was nearing summer in Antarctica, and there in McMurdo Sound the thermometer showed a pleasant twenty-nine degrees Fahrenheit. The wind was a noticeable but manageable eighteen knots. The sun was shining. This time of year it shone pretty nearly all day. All in all about as pleasant as you could ask for at McMurdo.\n\nThe _Jade Monkey_ floated over the water and up onto gravel, its big black rubber skirts all puffed out and vibrating like a trumpet player's cheeks. Suarez powered down, and the vehicle came to rest with a disgruntled wheeze of engines and a long, slow fart as the air cushion bled out.\n\nImelda Suarez was twenty-eight years old, five feet seven inches tall, dark-skinned, weather-beaten but pretty in the right light. She had worked for Cathexis Inc., owner of the _Celadon_ and her two LCACs, for three years, two as skipper of the _Jade Monkey_.\n\nIt was grueling, brutal, often boring, but occasionally terrifying work. Suarez had never lost a cargo, she had never lost a crewman, and she had kept that spotless record by never underestimating the A-factor. The Antarctic factor. The capacity of the most alien of all continents to complicate or obliterate the schemes of _Homo sapiens_.\n\nAntarctica was always out to kill you.\n\nBut the advent of the Cathexis era had changed life on the ice. In the old days the bases that dotted the rim of the continent had been cut off for as much as ten months out of the year. Aircraft get a bit unsafe in high crosswinds. LCACs do, too, but these specially modified versions could make a forty-mile run from the _Celadon_ in all but the worst conditions\u2014and in emergencies, even then.\n\nAll of which was extremely useful, because McMurdo Base\u2014MacTown, as it was known\u2014was growing more rapidly than just about any place on Earth. There was oil under the ice and offshore. With the Middle East in turmoil even the greens admitted that oil exploration on the ice was a better option than fighting wars to maintain supplies from volatile countries.\n\nMacTown, which had once been full of nothing but scientists, academics, and support staff\u2014generally from cold lands like Alaska and Montana and Maine\u2014was now home to a whole lot of people from Texas and Louisiana. The same evolution was occurring at British, Russian, Aussie, Kiwi, Chinese, Japanese, Chilean, and Argentinean bases. The effort to locate oil and develop the technology to survive the harsh environment was big, well financed, and in a hurry. And they could afford oranges that cost fifteen bucks apiece to bring in from Wellington or Tierra del Fuego.\n\nSuarez stepped out of her cockpit, nodded at her chief who was in charge of matters from this point, stretched up onto her toes, hefted a rather heavy shoulder bag, and headed up the long gravel slope into MacTown. Solid ground, ground that was not bucking and vibrating like the deck of the _Jade Monkey_ , felt oddly uneven and unsteady. She headed toward the new admin building where Cathexis Inc. had a small wing of cubicles\u2014nothing but a bunk and an electrical outlet, really. This was her third trip of the day, and Suarez was required by company policy to grab a minimum six hours of sleep. LCACs did not want to be steered by sleepy pilots. LCACs steered by sleepy pilots had a tendency to flip over.\n\nShe was intercepted on her way up the road by a tall, not-bad-looking man with a full beard, sunglasses, and a big grin. Jim Tanner was Lockheed security. Lockheed ran McMurdo. But it was well known that Tanner was former Naval Intelligence. And it was widely assumed that he was the U.S. government's eyes and ears on the base. Or at least, one set of eyes and ears.\n\n\"Well, hello there, Suarez. Whatcha got in the bag?\"\n\n\"What, this bag?\" Suarez asked innocently.\n\n\"Wouldn't be contraband booze, would it?\"\n\nSuarez stopped, unzipped the bag, and pulled out a bottle of scotch. \"Huh,\" she said. \"I wonder how this got in there? And look, it has a twin. You here to help me destroy the evidence, Jim?\"\n\nAlcohol was sold at McMurdo, but it was also rationed. Nobody begrudged you a drink, but there were supposed to be limits.\n\n\"I would like nothing better.\" Tanner took one of the bottles, held it up to read the label. \"Ah, the Macallan Sixteen. You've grown and matured, Suarez. You have grown and matured.\"\n\n\"If you're nice to me and let me get to sleep eventually, I'll share.\"\n\nTanner handed her back the bottle, grinned, looked away a bit sheepishly, and said, \"Sadly, I am here in an official capacity.\"\n\nSuarez's eyes narrowed. \"Your _official_ official capacity? Or your _unofficial_ official capacity?\"\n\nHis smile thinned out. \"This will be a conversation that involves your signing a legal document promising not to disclose the nature of the conversation. The document in question is not a company document. It's a _company_ document.\"\n\nThe company was Lockheed. The _company_ was the CIA.\n\n\"What the hell did I just step in?\" Suarez demanded, no longer in a joking mood.\n\nTanner's office was tiny\u2014space was always at a premium in a place where Home Depot was ten thousand miles away. It was overheated, so neat that no piece of paper could be found, and seemed to have been furnished entirely with the kind of office furniture that a self-respecting Goodwill store would reject.\n\nThe document he had for her was on an iPad. If it had been printed out it would have taken up four pages. Pages full of threats and requirements and official language. The long and short of it was that if she spoke of this meeting to anyone not properly cleared for top secret or better, she would go to jail.\n\n\"I'm going to remind you that even though you have been separated from the Marine Corps, Lieutenant Suarez, the corps still owns you.\" Tanner turned the pad to her. She scribbled a fingernail signature and at his prompting spoke her full name to the camera.\n\n\"And now do we get to the reason for this cloak and dagger, _Captain_ Tanner?\"\n\nHe was behind the desk in the good chair, the one that swiveled. She had a steel-frame chair with the stuffing half blown out. The bag of booze was at her side on the floor.\n\n\"Cathexis Base,\" Tanner said.\n\n\"Okay. What about it?\"\n\nCathexis Base was a facility built by Suarez's corporate masters. It was used as a transshipment point, a storage facility, a rescue facility for the _Celadon_ and her sister ship. There were repair facilities for the LCACs there, as well as for the helicopters and planes Cathexis used on the ice.\n\n\"Well, let's start with this: Have you ever seen anything suspicious at Cathexis?\n\nNo, she had not.\n\n\"What about at the satellite facility. What do they call it? Forward Green? Good grief, sounds like a golf course.\"\n\n\"I've never been there.\"\n\nTanner nodded. \"Know anyone who's ever been there?\"\n\nSuarez shrugged. \"I imagine a lot of the support people have. Must have been to handle construction.\"\n\nTanner shook his head, and watched her. \"No. In fact, the crews have been kept almost entirely separate. There's very little crossover. There's Cathexis Base and its people, and there's Forward Green and its people.\"\n\nSuarez looked at him expectantly, waiting for some kind of clue. When all he did was look back at her, she said, \"So?\"\n\n\"So, it's odd.\"\n\n\"Okay.\"\n\nHe was an experienced interrogator and had mastered the trick of waiting. But Suarez had nothing to offer, so all she could do was wait as well.\n\nHe nodded as if he'd satisfied himself on some point, then leaned forward on his elbows. \"Anyone at Cathexis ever suggest you might want to try piloting a new kind of hovercraft? Something faster?\"\n\n\"Well, the navy already has\u2014\"\n\n\"I'm not talking about a piece of navy equipment.\"\n\n\"Then what _are_ you talking about, because I'm tired, I need sleep, and before that I need a drink.\" She was bouncing one leg, a habit when she was impatient.\n\nHe opened his laptop, hit a few keys, then turned it so she could see. \"The video is just seven seconds long.\"\n\nThe film was obviously taken from a great distance. It shook and wobbled. What it showed, or seemed to show, was a sleek, low-slung object shooting across the ice.\n\n\"Do you recognize that?\"\n\n\"Do I recognize what? Something going zoom across the ice?\"\n\nHe laughed. \"We did a bit of enhancement and a bit of informed speculation, and the best guess from Langley is that it's a hovercraft, quite small, so not designed for cargo. There appears to be a bubble canopy large enough for one, possibly two people. Speed in excess of a hundred and twenty knots. And it appears to be armed.\"\n\n\"Armed?\" That stopped the bouncing of her leg.\n\n\"Mmm. Armed. With a type of Russian missile, essentially an antitank weapon, although obviously it would work even better against a tractor or a Sno-Cat or a shelter.\"\n\nThe thing that came to her mind was obvious and a bit stupid. But she said it, anyway. \"Weapons are forbidden on the ice. Nothing beyond a couple of handguns for the security people.\"\n\n\"Yes.\"\n\n\"Why would somebody need missiles? On some souped-up hovercraft?\"\n\n\"That's the question,\" Tanner agreed. \"Why would they? Speculate, Suarez.\"\n\nShe pushed back, tilting the hind legs of her chair. \"If it's as fast as you say, it would be tough to hit from the air. White on white, going one hundred twenty knots? You'd see a hell of an infrared signature, so if you went after it in an Apache you could use the thirty mil, but an Apache's top speed is one hundred fifty knots, so you don't have much of an edge in speed.\"\n\n\"I knew a good pilot like yourself would see it all clearly,\" Tanner said. \"A pilot with SEAL training, and right here close at hand. Let's have that drink, Suarez.\"\n\nShe hefted a bottle, unwound the capsule, and poured into paper cups. \"Am I going to need it?\"\n\n\"Lieutenant Imelda Suarez, I am informing you that pursuant to a special directive of the Department of Defense, you are hereby returned to active duty.\"\n\n\"Whether I like it or not?\"\n\nTanner raised his cup. \"Cheers.\"\n\n\u00b7 \u00b7 \u00b7\n\nSailing in the San Francisco Bay in blustery weather, Francis Janklow, the CEO of Janklow\/MediStat, was not as happy as he should have been. He loved his boat in the abstract, but now that he'd bought the damned thing for two million dollars he felt as if he had to use it. But the truth was, he was just not that crazy about sailing. Especially when the wind was up so that he was constantly drenched by a spray that ranged from cooling mist to fire hose.\n\nHis guests seemed to be having a good time, though. These were a senior state senator and the senator's much younger \"assistant,\" a rival CEO, a supposed painter whom Janklow's wife was sponsoring, and of course Janklow's wife.\n\nThe boat had been his wife's idea. According to her, you could not own a waterfront property on Belvedere Island and not also own a boat of some sort, and after all Janklow had sailed as a youth.\n\nAnd yet, Janklow thought glumly even as he affected many a grin in the face of the elements, he would much rather have been home with a spreadsheet on his screen and a scotch in his hand. Instead he was at the wheel, yelling instructions to the kid, Antonio, who sometimes crewed for a day.\n\nAnd also seeing things. Definitely seeing things. He frowned and peered off toward the Golden Gate, open water ahead, trying to figure out just what he was seeing.\n\n\"I think I'm seeing things,\" Janklow said. He forced a laugh. No one heard either the remark or the laugh.\n\nNo one heard him say that it was as if a window... no, two windows... had opened in his head.\n\nAntonio saw him stagger back from the wheel and raced back to take over.\n\n\"You okay, Mr. J.?\"\n\n\"I'm... Nah. Nah. Yeah. Oh, shit.\"\n\nAnd then suddenly Janklow was racing up the mast, hand over hand, like a much younger man.\n\nEveryone saw this. The state senator's assistant yelled something and pointed. All eyes turned to look at Janklow, now thirty feet up, his sparse hair flowing in a wind that was too strong for those below to make much sense of what sounded a lot like disconnected, wild ranting.\n\nAnd then Janklow fell. Although it looked very much as if he actually leapt.\n\nHe plunged straight down into the sea.\n\nPandemonium. All the passengers jumped up and began yelling to Antonio to _turn the boat around, turn the boat around_.\n\nBut sailboats are not so easy to turn around when under wind power. So first Antonio\u2014without help\u2014had to lower the sail and start the engine. Only then, a quarter mile away from Janklow, could they turn back and effect a rescue.\n\nJanklow could be seen. He was in the water, waving his hands wildly, but more as if he was a little kid splashing in the tub.\n\nAs the boat drew up alongside, the state senator had the presence of mind to throw a life vest to Janklow, while his wife berated him for being so careless.\n\nBut Janklow just laughed; a wild, manic sound that sent chills up his wife's spine. And then, pushing himself along the side of the boat and refusing all proffered hands, Janklow went to the stern, dove down, and came up with his face shoved straight into the churning propeller.\n\nIt would be listed as an accidental death, not a suicide.\n\n\"I'm looking at the spreadsheet right now,\" Lystra Reid said. She had a phone propped against her ear and a pad open before her. Tiburon police officers and California Highway Patrol detectives were milling around the marina of the Tiburon Yacht Club. They had taken statements from everyone on the Janklow boat. Lystra had little enough to say, and none of it useful, and the detectives had let her go.\n\nBut rather than take off immediately, Lystra savored a bourbon rocks and split her attention between the mild chaos of the investigation and the neat order of her spreadsheets.\n\n\"Yes, I am very much aware of some of my off-book expenses, and no, I won't enlighten you further, Tom. One of the reasons I don't take the company public, yeah, yeah, is because I like to spend my money without being second-guessed. It is, after all, mine.\"\n\nAt the age of nine, Lystra had been sent away. Her father had finally decided that he could not raise her properly. His own business was falling on hard times; the carnival business was fading fast. Her father's act\u2014he was a trick shooter and put on an impressive if threadbare show with guns, knives, and hatchets\u2014no longer drew enough of a paying crowd for the carny life to make much sense.\n\nHe'd sat her down and explained it all to her. She would be going to a good, decent family that would raise her properly, with school, and friends, and all of that.\n\n\"You won't be my dad anymore?\" She hadn't cried. She'd felt sick with betrayal, but she hadn't cried.\n\nHer father, his lined face half hidden in the gloom of the Louisiana dusk, had said, \"I won't be with you. I won't be seeing you, I... I have to find some way to make a living. But listen to me, Lystra. Listen to me. You're a very smart kid. And better than smart, you're determined. You'll do fine. And if you ever need me, really need me, life-and-death need, I'll be there.\"\n\n\"What about Mom? Is she dead?\"\n\n\"I don't know,\" he said.\n\nShe knew he was lying. She couldn't recall the exact moment when it dawned on her that her father had killed her mother. But once the idea _had_ dawned, certainty soon followed.\n\nHer mother had been a bit of a party girl. That was the nicest way to put it. Lystra's mother liked a good time, and she had not found it in the life her husband gave her. She'd looked for comfort elsewhere. In booze, in drugs, in sex.\n\n_\"I know,\"_ Lystra had said. Nothing else. Just those two words.\n\nHer father had said nothing. The two of them just sat there on the broken-down lawn chairs. Then her father had poured two fingers of bourbon into a paper cup and handed it to her.\n\nGod, it had burned her throat, but she had swallowed it and not made a sound.\n\n\"Bad things happen in this life,\" he had said at last.\n\nLystra had held out her paper cup and said, \"More.\"\n\nHe shook his head. \"That taste was enough. You're still a kid.\"\n\n\"You killed my mother. Now you're dumping me. Okay. That's all done. Yeah. Maybe I'll never see you again.\"\n\n\"Maybe.\"\n\n\"But if I do, you'll do whatever I ask you to do.\"\n\n\"Will I?\" He'd seemed almost amused, but seeing the look in her eyes he had flinched, looked down, and finally poured her a second drink. \"I will,\" he had said, and there was a sacredness to that vow.\n\nLystra went to live with a very nice, childless family by the name of Reid, in Tulsa, Oklahoma. She got straight As in school while barely bothering to crack a book. She wasn't just a smart kid; she was brilliant. A cold, emotionally distant, friendless-but-never-bullied kid.\n\nBut at age fourteen things began to change. Not her grades, those stayed top-notch. But at about that time Lystra began to talk to her long-distant father again. He would speak to her when she was walking through the corridors at school. He would speak to her as she sat in the Baptist church and listened to the sermon. Her lip would curl when she heard him. Her eyes would focus with inhuman intensity on the back of a man's neck until by sheer force of will she could make him turn around, uncomfortable, only to become confused when the danger he sensed turned out to be just a young girl.\n\nHer father's voice spoke to her. And other voices as well. Angels, sometimes, though not the better sort of angel. And the voice of a girl with the odd name of Scowler.\n\nShe never told anyone about the voices; they had universally warned her not to. _Yeah, don't tell anyone we're here, they'll lock you up. Yeah_.\n\nThen both her adoptive parents had died in a car accident. The particulars of the accident raised eyebrows but elicited sympathy. Lystra had been sixteen at that point, just learning to drive. And despite the fact that Lystra had played various online driving games for years, she panicked while driving the real thing. She had not realized the car was in reverse. She did not notice that her parents were standing behind her, down at the bottom of the long driveway.\n\nThe police questioned her for a long time. The detectives could not quite square her story of intending to pull the car forward slowly into the open garage with the fact that the car had been in reverse and had shot at surprisingly high speed the sixty-seven feet between the rear bumper and the two Reids.\n\n\"When I realized it was in reverse, it was too late, yeah. I saw what was about to happen, and I knew what to do, but instead of hitting the brake I accidentally hit the gas pedal.\"\n\n\"And then?\"\n\n\"I felt the impact, and my only thought was that I should pull the car forward. Yeah. Undo my mistake.\"\n\n\"Right. And in the process you ran over both of your parents again. That's your story. You're sticking to that?\"\n\n\"How can I do otherwise? It's the truth.\"\n\nNo, they had not believed her. No one believed her. People who knew Lystra Ellen Alice Reid scoffed at the notion that she had panicked. Panic? Lystra, panic?\n\nBut in the end the cops couldn't prove a thing.\n\nThere wasn't a lot in the way of a social services department in Tulsa, but a shrink was tasked with testing her.\n\n\"She's a very difficult subject,\" he had reported. \"Hard to test. Her IQ is very high\u2014very smart, very quick\u2014so she knows how to answer, how to avoid setting off alarm bells. But my instinct tells me she's concealing something. At times I got the impression she might be hearing voices. Phantom voices. She may just be traumatized. Or she may be schizophrenic but with enough control to hide it.\"\n\nLystra was the sole heir to a million-dollar life insurance policy that was doubled due to the fact that the death had been an accident. Double indemnity, they called it.\n\nTwo million dollars. She'd been unable to touch it until she was eighteen, and at that time other family members had petitioned the court to examine her psychologically again.\n\nThe court had found her legally sane.\n\nThe voices in her head had congratulated her on the finding.\n\nOn her eighteenth birthday, Lystra had filed papers to form the Mad Alice Holding Company. And she'd gotten her first tattoo. She'd told the tattoo artist, \"I want my adoptive parents, like in this picture. But I want them to be screaming.\"\n\nThe tattoo artist had been reluctant, but an extra thousand dollars had cured him of all doubt.\n\nThe placement she'd chosen was strange. Her stepmother was beneath one breast, so that she seemed to be smothered by the weight of it. Her stepfather, also screaming, was beneath the other.\n\nOnce both tattoos were complete, they began to speak to her. They wept, sometimes. Other times they threatened. She heard their voices so very clearly. If she stripped off her shirt and her bra, she could see their mouths moving as they cried out in pain and despair.\n\nBut they could be useful, too, the talking tattoos. It was the dead Mr. Reid who suggested using her inheritance to buy a small, failing medical testing company outside of Washington, D.C.\n\nSo the Mad Alice Holding Company was dissolved and a successor corporation formed as an Isle of Man company, exempt from most supervision. And then, another stroke of unusual luck: a midsize competitor in the medical testing field had suffered a catastrophic hacking that had spilled the records all over the Internet.\n\nLystra Reid bought the stricken company and brought in the best security people around to ensure that a similar fate would never befall her. The result was a medical testing company, Directive Medical, which had never suffered a successful break-in, while\u2014not so strangely\u2014security problems plagued her competitors.\n\nAt the age of twenty-four, Reid controlled a third of the independent medical labs in North America, as well as significant portions of other markets around the world.\n\nIt was amazing what you could learn from data mining the health records of more than two hundred million people worldwide. You could, for example, learn that the wife of a brilliant medical researcher named Grey McLure had a rare cancer. And you could learn that this McLure fellow was suddenly in a desperate search for living cell samples. And with just a bit more work you could discover that he was also looking for a wide range of animal tissue samples for a very secret project of some sort.\n\nLystra hung up the phone, indifferent really to the current spreadsheet drama from her office. It didn't matter. There was no future to worry about. She swallowed the last of the bourbon and stood up to stretch. The marina was nestled between Tiburon and the adjoining Belvedere Island. Unpretentious yet extremely expensive homes rose on a cute little hill to her left and up the longer, wooded slope of Belvedere to her right. Looking south through the forest of masts, she could see San Francisco. Fog was rolling out, revealing the city, all muted pastels and off-whites.\n\nIt was all in all a beautiful location, with sailboats and ferries and container ships passing by in review. A genteel, civilized, prosperous place.\n\nAnd all of it about to come to a terrifying end.\n\nIt had been good to watch Janklow go mad; he had annoyed her on more than one occasion. She wanted to get a few tastes of the madness out here in the real world\u2014before the final chapters, which would force her to hide out and watch it as well as she could via electronic means. The personal, real-world experiences would help her to enjoy the next step.\n\n\"All done?\" the waiter asked, coming to clean her table.\n\n\"Soon,\" she said.\n\n# **EIGHT**\n\nThe first thing Bug Man had asked was, \"Where are we?\"\n\nBug Man had flown on a private jet before. He wasn't indifferent to it, but he wasn't overly impressed, either. George had not told him where they were going but had retreated into a book, remaining sullen and uncommunicative.\n\nBug Man saw a city in the distance. It was all tan walls and terra-cotta roofs, a large blur extending far out in every direction, reaching beneath the jet with roads full of small cars.\n\n\"The former center of the Earth, once upon a time. The Eternal City,\" George had said.\n\n\"Yeah, which is what?\"\n\nGeorge sighed. \"Your education is deplorable. The Eternal City is a reference to Rome.\"\n\n\"Rome? That's like, Italy, right?\"\n\nGeorge managed not to roll his eyes, but only just. \"Yes, Italy, Bug Man. Pizza, pasta, wine, priests, fashion, Rome. The Coliseum,\" he added. \"Gladiators and all of that.\"\n\n\"I saw the movie,\" Bug Man said. \"Also, I played the game. Not a great game.\"\n\n\"No?\" The plane took a little lurch as a crosswind hit it. \"What makes a good game?\"\n\nBug Man had been much more sure of his ground on this topic. He didn't know much about history, but he knew games. \"A good game? That's one where you can't stop playing it, even when you're asleep. Whatever you have to do that takes you away from the game, all you're thinking about is getting back into it.\"\n\n\"Hard?\"\n\n\"It's not about hard. Yeah, it has to be challenging. Can't be so easy it's over in five minutes, right? But it's not just about hard; otherwise, you could play online chess or work a Rubik's cube, man.\"\n\nHe heard the grinding of the landing gear coming down.\n\n\"Why are we in Rome?\" Not that he was complaining. He'd been locked away for several days in a safe house in the emptied-out Lake District before George had come to retrieve him. He'd been about to lose his mind looking at rain falling on green hills.\n\n\"We need a good twitcher. A nanobot twitcher.\"\n\n\"Where'd you get nanobots? The people you work for don't do nanobots, and I am not doing any biot bullshit. I saw what that did to Vincent.\"\n\n\"Nanobots,\" George reassured him. \"We came across some, and a portable controller. Compliments of a former friend of yours.\"\n\n\"Burnofsky?\"\n\nGeorge laughed and didn't answer. He rolled into the nearest seat and motioned Bug Man to buckle up.\n\nBug Man didn't exactly miss Burnofsky. The old man was an unreliable, unpredictable, sometimes cruel degenerate. But he and Bug Man had played a great game. The greatest game Bug Man would probably ever play.\n\nGod, that was a depressing thought. Was it all downhill from here? He supposed that would depend on just what George here had in mind.\n\n\"What do you want me for?\" Bug Man asked, but the question was lost in the impact of tires on tarmac. The jet rolled down the taxiway to a waiting car.\n\nBug Man walked down the steps to the tarmac\u2014it was warmer out than it should have been for this time of year. Was Rome always warm? He had no idea. The sun was setting, and all he could see were featureless hangars and repair sheds. In the distance was a Fiat sign, and beyond that a billboard for what looked like a juice drink.\n\n\"I don't speak Italian,\" he said.\n\n\"You won't need to,\" George said. \"Get in the car.\"\n\nBug Man did not like that, the bossy tone. He needed to draw a line right here and now, before he was driven off to wherever. \"Tell me what we're doing here, dude.\" When George looked evasive, Bug Man held up one hand, cutting him off. \"No, man, now. Right here, right now. Enough playing around.\"\n\nGeorge nodded, as if expecting this. As if he'd have preferred to do it somewhere else, but okay, if his impatient young friend insisted.\n\n\"The Pope,\" George said.\n\n\"The Pope? The freaking Holy Father? _The_ Pope? What about the Pope?\"\n\n\"You know he's in Rome?\" The question was obviously insulting, spoken as it was with more than a trace of condescension.\n\n\"What's with the Pope?\"\n\nGeorge dropped the snarky look and got serious. \"You are wanted by MI5. A word from them and every other intelligence and police agency on Earth will be looking for you. And of course, the Armstrong Twins want you dead.\" He stepped closer, put his face right up close to Bug Man's face, close enough that Bug Man could have told you the man's toothpaste brand. \"But forget all of that, because we have a fellow named Caligula. A charming name, I'm sure you'll agree. He already knows your name. A single text from Lear to Caligula and your death is assured.\" He held up an index finger. \"I don't mean that you will _likely_ be killed. I mean that you will without the slightest doubt be killed. Caligula has never failed. _Never_.\"\n\nBug Man swallowed. He knew the name. He knew the reputation. And he did not like the fact that Caligula knew what he was about.\n\n\"As to what you are to do, Anthony 'Bug Man' Elder, you are to retrieve a sample. A few cells. That is all. And then you will be free to go. We won't protect you, but neither will we harm you. And you'll be paid. A hundred thousand pounds.\"\n\n\"Cells?\" Bug Man asked with a dry mouth.\n\n\"Cells. A tissue sample. From the Pope. And it must be done quickly.\"\n\n\"The Pope. Tissue samples.\" Bug Man let this sink in. George waited, expectant, curious to see whether Bug Man would put it all together.\n\n\"Jesus,\" Bug Man said. He let loose a short, sharp bark of a laugh. \"Jesus bloody Christ on a cross.\"\n\nGeorge got a dreamy look on his face. \"See, Anthony, control is so much easier when you don't require the victim to carry out complex actions. Reduce it to the binary and it's all more efficient and effective.\"\n\nBug Man nodded, seeing it\u2014and fearing it. \"You don't even need Caligula anymore. You just need a tissue sample.\"\n\nGeorge threw back his head and laughed, showing teeth that had had many encounters with dentists. \"I quite like you, Anthony. I'd have done this later, not here on the tarmac, but you're such a clever boy.\" He pulled a small plastic bag from the inner pocket of his jacket. From it he withdrew a vial and a Q-tip. \"I'll just swab the inner cheek, if you don't mind.\"\n\nBug Man did mind. He pulled away.\n\n\"Oh, it's far too late for that, Anthony. You're in. Like it or not. You haven't a friend in the world, and so many people want you dead. Turn and run and I'll let you go, but Caligula will get to you if the Armstrongs don't find you first. Now open wide.\"\n\nBug Man opened his mouth. George swabbed his inner cheek with the Q-tip and sealed it in the vial.\n\n\"We won't create the biots unless you make it necessary. You have Lear's word on that.\"\n\n_\"Lear's word,\"_ Bug Man said bitterly.\n\n\"You are not in a position to argue, Anthony. You are lost and despised and scheduled for destruction. And now, you are BZRK.\" He grinned and made an ironic power salute with his fist. \"Death or madness, kid. Death or madness.\"\n\nThe Starhotels Michelangelo didn't look like much from the outside; in fact, it looked like any number of the wearily functional, '60s-era buildings that deface Rome. Inside it was moderately posh, and Bug Man was hustled into a large suite with a balcony.\n\nThe balcony had a very nice view of the dome of Saint Peter's Basilica. (And a red-trimmed Total gas station in the other direction.) The walls of Vatican City were just four hundred feet away.\n\nThere was also a nice little restaurant serving\u2014unsurprisingly\u2014Italian food. The TV featured the BBC and CNN International as well as other non-Italian fare, and there was WiFi, but it was a bit slow.\n\nFrom here Bug Man could easily manage nanobots within Vatican City. But what he needed was a pathway. X to Y to Z to the Pope. And then back out with a dozen or so cells.\n\n\"Don't leave this room,\" George instructed. \"Except for lunch, which you will take downstairs in the restaurant. That's when the maids will come in and clean the room. They have to come in, or it will set off alarm bells down at the desk. Normal. Everything normal.\"\n\n\"I can't sit in here twenty-four\/seven,\" Bug Man argued.\n\n\"You can and you will,\" George said flatly. \"Order all the in-room movies you like. But don't draw attention to yourself. Italian police may not be geniuses, but let's not give them a chance. Right?\"\n\nThat was a depressing reminder. When would he be free to walk out in the world without being afraid? Maybe never. But never was a long time, and Bug Man was an optimist.\n\n\"So what's my path?\"\n\n\"Path?\"\n\n\"How do I get from here to there?\"\n\nGeorge sat down in the easy chair. Bug Man stood looking out through the balcony's sliding glass door.\n\n\"We have access to the wafers used for the Pope's communion.\"\n\nBug Man snorted. \"Are you nuts?\"\n\n\"Is it a religious objection, because\u2014?\"\n\n\"It's an objection over the fact that the mouth is not a point of entry unless you want to end up riding an infallible papal turd out the far end.\"\n\nGeorge shrugged dismissively. \"Surely there's some way to\u2014\"\n\n\"Have you ever seen a mouth down at the nano level? It's about as big as a valley, and it's full of massive boulders chomping, plus a tongue and spit and wind. Maybe you can grab onto a tooth and get safely up under the gums, but I'm not trying it.\"\n\n\"All right, there's a second way. We have access to a person who has an audience with the Pope on Tuesday. It's traditional to kiss the papal ring. Does that work for you?\"\n\n\"I'm still sitting out there on a lip hoping this dude doesn't get nervous and lick them.\"\n\n\"It's a woman, and she's not the nervous type.\"\n\n\"A woman? Who?\"\n\n\"Her name is Lystra Reid. Owns some clinical testing company or other. Directive Medical? Rich American.\" He didn't seem to approve of rich Americans. \"She owns medical labs and such. A lot of them. And she's made some big contributions to an African mission the Pope is fond of.\"\n\n\"Is she one of your people?\"\n\n\"No. But her maid has debts, and we have money. So we can get the maid to place the biot... sorry, nanobots in this case... on Ms. Reid. You then merely have to be on her fingertips when she takes the Pope's hand, or on her lips when she kisses the ring. Then it's grab a sample and find your way out.\"\n\nA loud guffaw erupted from Bug Man. He turned to look at George, feeling that he had the better of him for the first time. \"You don't know much, do you? What do you think? My nanobots walk back here to the hotel? It's only a few hundred yards, maybe, but that's a hell of a long walk when you're two hundred microns long. A nanobot can't even see objects at much distance. The optics are calibrated for work down in the meat, so I wouldn't know where it was and where to make it go, even if we had a month or so to walk it back here.\"\n\n\"We'll find a way,\" George said, and yawned.\n\n\"Oh, will we...\"\n\nIt was meant to be sarcastic dismissal, but George didn't take it that way. He clapped his hands once as if drawing the scene to a close. And in fact, he did draw the scene to a close, by leaving behind a baffled, worried\u2014but also excited\u2014Bug Man.\n\nPlath and Keats arrived at the alleyway door of the McLure building after much skullduggery that made them both feel like spies. They were reasonably sure they hadn't been followed.\n\nThey were ushered into a private elevator and whisked to the twentieth floor. It was a bit of an old-home week for Plath, not all of it good. She'd been in and out of this building since childhood, but her last visit had begun with Anya creating Plath's biots and ended in a massacre between McLure security men, AFGC hired hands, and Caligula. Needless to say, Caligula had come out on top.\n\nMr. Stern met them in his office then led them down a guarded hallway to an unmarked door.\n\n\"So, how are you adjusting to being back in New York?\" he asked them both.\n\n\"I liked the island better,\" Keats said.\n\n\"I can imagine. Well, let me show you what we have.\" Stern slid a keycard and opened the unmarked door. Inside was just a room with half a dozen workstations, each focused on a large monitor. The ambient light came from the monitors, the keypads with keys outlined in light, and softly glowing touch screens.\n\nIt had the feeling of a room that had just been emptied of people. Plath touched a coffee cup and felt that it was still warm. Stern had emptied everyone from the room for greater privacy.\n\nHe sat down and Plath and Keats pulled up chairs.\n\nStern tapped a few keys, then switched to a touch screen.\n\n\"You asked me for what we have on the Tulip, and specifically whether there's a data center,\" he said as the image of that strange building appeared. \"This is the Tulip. This is a photo, obviously, taken from across the street. And this\"\u2014he swiped the screen\u2014\"is the heat signature using infrared.\"\n\nThe skyscraper was now a sort of layer cake of red, purple, and blue\u2014mostly red.\n\n\"Of course we had to wait until the building was in shadow so we didn't just pick up reflected sunlight,\" Stern explained. \"We took three readings, three different days, and this is the composite heat signature. This\u2014\"another swipe\u2014\"is the same building but shot from the north. And this is from the east. We don't have a westerly view, but we have a high degree of confidence that these heat signatures are persistent and not just one-time things.\"\n\n\"Okay,\" Plath said, making a puzzled face at Keats, who was looking intently.\n\n\"There's a lot of variation by floor,\" Keats said.\n\n\"Oh, obviously,\" Plath said, with just a little sarcasm.\n\n\"But it's all centrally air conditioned, yes?\"\n\n\"It is. Normally,\" Stern said with unmistakable pride. \"But we turned off the AC. We risked using our back door into their computer network and reset the thermostat overnight. It takes a while for the system to catch up when it's turned back on, and in the meantime we could get a picture of what's being done and where.\"\n\n\"Can you show the temperature readouts?\" Keats asked.\n\nStern winked at Plath. \"This one's smart.\" He tapped a few keys, and numbers popped up beside each floor. \"You get a clearer picture off this data.\"\n\n\"One floor is far hotter.\" Keats used his finger to count the floors. The eighteenth, yes? Something is giving off a lot of heat.\"\n\n\"Servers, we believe,\" Stern said. \"They have their own emergency climate control, but it's not enough to disguise the heat signature when the overall air-conditioning system is down.\"\n\n\"So, the eighteenth floor is where they have their main computers. Their own personal cloud,\" Plath said.\n\n\"That seems likely,\" Stern said.\n\n\"Okay, how do we get to it and destroy it?\"\n\nThere it was again in Plath's head, that crystalline memory of the World Trade Center falling. It seemed almost sensuous. Had she just become used to it? Had she seen that imagery so often that it had lost its potential to shock and had now become almost balletic?\n\nStern sighed. He pulled up a different diagram of the building. \"Those are the elevator shafts. If you see those thicker areas there, that indicates an elevator stop, a door. If you look even closer, you'll notice there are none for the eighteenth floor. And I'll spare you the suspense and just tell you that the stairs, the emergency access, also doesn't open onto eighteen. There is a single stairwell connecting eighteen to seventeen. And there's a stumpy freight elevator that goes only from seventeen to eighteen, and nowhere else. Floor seventeen, in case you were wondering, is where AFGC security lives.\"\n\n\"Oh,\" Keats said.\n\n\"Indeed. There are never fewer than ten security\u2014TFDs as we call them: Tourists from Denver, since that's the look they put on\u2014on that floor at any time. Another two dozen or so patrol the building or watch the entrances. They are all armed. They are mostly very well trained, many are former special forces or commandos. U.S. Marines, ex-Delta Force, Royal Marines, SAS, ex-Mossad... dangerous people.\"\n\n\"So, how do we do it?\" Plath asked. \"How do we get in there and destroy those servers?\"\n\n\"I believe what Mr. Stern is about to tell us is that we _don't_ ,\" Keats said. \"We'd have to get past ground-floor security, go up to seventeen where we would be shot at. A lot. Then somehow we'd have to reach the connecting stairwell and climb to eighteen, where we would have another fight on our hands, with forces coming from all over the building to attack our rear.\"\n\n\"Exactly. Now, we'd have some advantages\u2014we can use our network access to shut down elevators, block some doors, turn off cameras, that sort of thing. But to actually have a decent chance of success? We would need a hundred men.\"\n\nHe laid that last fact out like a poker player showing the ace that would win the pot.\n\nKeats snorted. \"A hundred men?\"\n\n\"In Midtown Manhattan. Imagine a hundred armed men appearing on the street outside the Tulip. There would be no way to avoid the police being involved, especially once bullets started to send plate glass falling down onto pedestrians.\"\n\n\"Isn't there some kind of... I don't know,\" Plath said, frustrated. \"Some Tom Cruise kind of thing? Crawling up the side of the building?\"\n\n\"The shape of the Tulip, with that suggestive bulge at the top, means that's physically impossible, even if we were insane enough to try such a stunt.\"\n\nStern turned away from the monitor with an air of finality, but Keats leaned past him and pointed at the screen. \"Did you see this? Eighteen isn't the only floor that's shut off from elevators. This is, what? Thirty-four, yes?\"\n\nStern spun back and peered closely at the monitor. \"I believe you're right. But the heat signature is quite average on thirty-four, so that's not our server farm.\"\n\n\"No,\" Keats agreed. \"But it's _something_.\"\n\n\"In the end, as you can see, the building is effectively impregnable. Not that we would ever have participated in such a thing, anyway, but just so that you know: that server farm cannot be taken out by direct attack.\"\n\n\"Which means our friend Lear has ordered us to do something impossible,\" Keats said.\n\nPlath looked troubled and uncertain. But she finally stood up, took Stern's hand, and thanked him.\n\nBack on the street Plath said, \"So why did Lear tell us to do the impossible?\"\n\nKeats had no answer to that.\n\n_Unless, of course, it isn't impossible_.\n\nIn Plath's mind the towers fell.\n\n# **BRAZIL**\n\nLystra Reid was nowhere near when the president of Brazil was discovered naked and babbling on a street in S\u00e3o Paulo, apparently collecting dog feces in a Gap shopping bag.\n\nThe president was taken to a hospital, where no explanation could be found for his condition. He was diagnosed first as suffering a breakdown as a result of stress and overwork. But it soon became clear that this was no mere nervous breakdown but a complete psychotic break.\n\nHe had gone mad.\n\nA solemn vice president assumed the office and attempted to reassure a worried nation. But halfway through her speech she appeared to become distracted.\n\nThere were, she said..., \" _Bugs_.\"\n\nAnd soon after she began to weep and curse violently, and from there began to scream and had to be taken away by her chief of staff and security personnel.\n\n# **LOS ANGELES**\n\nThe Los Angeles County coroner, Dr. Baldur Chen, issued two different reports on the death of actress Sandra Piper. One was very thorough and public and reached the obvious conclusion: suicide.\n\nThe second was a report prepared with help from an agency in Washington. That agency sent its own pathologist to \"assist.\" This second pathologist focused on an exceedingly careful examination of the actress's brain. Dr. Chen had never seen an autopsy that involved centimeter-by-centimeter microscopic investigation of the brain tissue.\n\nIt would have taken a much more obtuse man than Dr. Chen to fail to recognize that the agency pathologist was looking for something very specific.\n\nBoth pathologists signed off on a second, eyes-only report that dealt with this second, microscopic examination. The conclusion was that there was no evidence of nanotechnology present.\n\nDr. Chen was required to sign an official secrets document and was solemnly warned that he would go to a federal prison if he revealed the existence of this second report.\n\n# **NINE**\n\nThe Twins arrived back in New York with no more fanfare than Plath and Keats. It had been expensive, but crossing into the U.S. without a passport was possible. Not impossible. Not with enough ready cash.\n\nThey had been helped into their specially built shower, then slept for many hours until Jindal had them awakened as per their orders.\n\nCranky, but relieved to be home again where the environment had been shaped to their needs, they drank coffee, ate pastries, and sat in their tent-size bathrobe while Jindal gave them the rundown. This program and that business.\n\n\"We don't care about the P and Ls,\" Benjamin snarled after a few minutes of spreadsheets. \"Do you think we give a damn about long-term profits? Have you found BZRK?\"\n\nJindal licked his lips and rocked back on his heels. He always stood in their presence. \"No, sir. Thrum's lead took us up a dead alley. She's beginning to suggest that she's being played.\"\n\n\"Played? Hannah Thrum?\" Charles made a dubious face.\n\n\"She thinks, and sirs, I agree, maybe, that Sadie McLure and the McLure chief of security are laying a false trail to\u2014\"\n\n\"We're being played by a _teenager_?\" Charles was usually the calmer brother, but this insulted his intelligence.\n\nBenjamin slapped the table with his palm. \"If we can't find them, we can still go after their allies. This chief of security. His whole department.\"\n\nJindal started to smile, almost as if he thought it was a joke. Then his smile faded. \"Sir?\"\n\nBenjamin glared at him. \"Never mind. Not your sort of work. No. No, get Burnofsky in here.\"\n\nJindal stiffened. He had kept Burnofsky at arm's length, suspecting, suspecting very damned strongly that the genius had been compromised by BZRK.\n\n\"Are you sure you want\u2014\"\n\n\"Get him. And get out.\"\n\nBenjamin remained silent a while, judging his brother's mood. Charles, he concluded, was frustrated, but not yet ready to accept that they were entering a new phase. Charles did not yet understand that they were _losing_. In fact may already have lost.\n\nCharles still half believed the silly cult they'd financed, Nexus Humanus, was of some use. He still seemed to think that the work of their remaining twitchers\u2014no great prodigies among them\u2014was just marking time, doing damage control.\n\n\"You're still trying to hide,\" Benjamin said aloud at last. \"Our whole life, you always wanted to find a way to hide what we are.\"\n\n\"What we are?\" Charles said a bit pompously. \"What we are is two great men, who have\u2014\"\n\n\"We are freaks,\" Benjamin said, but not angrily. \"Everywhere except on the _Doll Ship_. They've taken that from us. BZRK, the intelligence people, the police, all of them, all the forces of _the normal_. They've destroyed the one, small place where we could be. Just... be.\"\n\n\"We have this place, still,\" Charles said.\n\n\"Our cage. Our gilded cage.\"\n\n\"Yes,\" Charles admitted. Then he heaved a sigh. \"The tide has turned, has it not, brother?\"\n\n\"Yes,\" Benjamin said. He reached awkwardly across their body to pat his brother's chest. It was as much physical affection as they could deploy. You could not hug a man who was attached to you. \"The tide has turned. The governments have become aware. In secret we had a chance. But secrecy is impossible now. They will come for us, and they will take us. They'll put us on display. They'll call it a trial, but it will be a carnival freak show. And then they'll put us in a cell until we die.\"\n\nThe angled mirror that let them look in each other's eye revealed that Charles was crying.\n\n_So_ , Benjamin thought. _Perhaps he sees at last_.\n\n\"You were too softhearted, Charles. Always. You thought you could improve them, as we did on the _Doll Ship_ , and yes, it was a magnificent dream, brother. But we now face Sodom and Gomorrah, and no righteous man is to be found to justify their salvation.\"\n\nThe silence that followed was long.\n\n\"What,\" Charles asked finally, sounding exhausted, \"would you have us do?\"\n\n\"We tried to gently show the world the error of its ways,\" Benjamin said. \"We tried the carrot. Now comes the stick. Now comes judgment. Now comes righteous wrath, brother. Or do we wait for our chance to star in their freak show?\"\n\n\"No,\" Charles whispered. Then louder. \"No, by God. Now comes Judgment Day. We hit them. We hit them so hard they can't stand up. And then we show them that we have worse still in store unless they submit.\"\n\nBenjamin smiled. The doorbell sounded. \"That would be the good Dr. Burnofsky.\"\n\nIn Rome, the Pope was working his way methodically through his daily audiences. He was a humble man despite the pomp of his ancient office, and he still, after many years in the job, felt a bit put off by the need to play the kingly role.\n\nFirst up there was the priest who had defied death threats to keep an inoculation program going in narco country. The priest was young and cocky and brave and offered to shake the Holy Father's hand rather than kiss his ring.\n\nThen the two Little Sisters of the Poor, one of whom had been attacked on a mission in Burma. The Pope rose from his seat to embrace them each in turn and to whisper words of encouragement. They left with tears streaming down their faces.\n\nThen the usual collection of businesspeople and media people, all of which would culminate in the Pope getting to meet a famously good-looking actor to thank him for his charitable work. As far as the Holy Father knew the actor was not a Catholic, but he was still a great talent and this Pope rather liked the conversation of talented people.\n\nA banker, a reporter, a union boss, an Argentinean politician (the Pope was not fond of politicians as a rule), a scientist who had discovered a way to raise sorghum crop yields dramatically, and last, before the actor, Lystra Reid, a youngish woman with tattoos peeking out from beneath her expensive clothing.\n\n\"Your Holiness,\" Lystra Reid said, and knelt, and kissed his ring.\n\nAnd at that moment four of Bug Man's nanobots leapt from her lips, slick with lipstick, to the cold metal of what was known as the Fisherman's Ring.\n\nA quarter mile away, Bug Man said, \"And that's how the pros do it,\" and did a little fist pump.\n\nThe Pope's audience was broadcast via a closed-circuit station from the Vatican, and of course streamed, so Bug Man could see it all play out in the macro even as he was marveling at the unusual smoothness of the ring's gold surface.\n\n\"You're back,\" Burnofsky said. \"I mean, welcome back.\"\n\nThey stared at him, unnerving him as they often did. Were they going to kill him right here, right now? Surely they must suspect that he had been wired. Maybe he should just put it out there; maybe he should just blurt it out.\n\n_Are you watching all this, Nijinsky? Or are you in my ear listening? Or are you drunk and passed out, you sad degenerate?_\n\nBurnofsky was pleased to realize that he was not afraid to die. Yet, he was afraid to die too soon. BZRK had reprogrammed him, brutally shifted his emotions, but it was crude work. Typical of the lesser BZRKers. Vincent would have done a better job. Vincent would have found a way to wire him for true loyalty. All Nijinsky had accomplished was to turn Burnofsky\u2014for now at least\u2014away from the bottle and the pipe. He had implanted very strong inhibitions against telling the Twins all he knew. He had turned Burnofsky's most terrible secret into a source of sickening pleasure, and oh, that had been cruel work.\n\nBut still: crude and ham-fisted. Burnofsky could no longer be said to be working for the Twins, true, but he was still working for himself, still pursuing his own agenda. Nijinsky thought his watchful biot would allow him to see and understand what Burnofsky was doing.\n\n_Foolish boy. Male model. I'm one of the great minds of the century, and you think I can't carry out my work right under your nose?_\n\n\"Karl, it's good to see you,\" Charles lied.\n\nBenjamin's one-eyed stare would freeze lava.\n\n\"It's good to have you gentlemen back,\" Burnofsky said. \"I'm, um, well, sorry for your...\"\n\n\"Defeat?\" snarled Benjamin. \"Are you sorry for our _defeat_?\"\n\n\"Your loss,\" Burnofsky said, finding the right word. \"I'm sorry for your loss.\"\n\n\"Fuck your sympathy,\" Benjamin snapped.\n\nCharles intervened smoothly. \"My brother and I are both grieving. You can understand our... impatience.\"\n\n\"What can I do for you?\" Burnofsky asked. Benjamin's anger had sent him back in his mind to Carla. To his daughter. It had been in this room, just over there, closer to the desk. That's where he had come to them\u2014drunk, stoned, filled with sorrow so deep and shame so dark that it would poison him as surely as a dose of strychnine. There, yes, right there he had reported to them that the deed was done and his daughter was dead.\n\nThey had said then that they were sorry for _his_ loss.\n\nHe swallowed hard, trying to avoid the terrible rush of pleasure that flowed each time he recalled the murder, each time, oh, God, to enjoy it, to be excited by it...\n\nFor a moment he thought he might vomit. Or actually become physically aroused. Or both at once.\n\n_I will kill you, Nijinsky. I don't know how, but I will kill you_.\n\n\"Massed preprogrammed attack,\" Charles said, trying to take control of the conversation to forestall more rage from his brother. They could still use Burnofsky, so long as they were careful. Let him reveal all to BZRK: without details it would mean nothing.\n\n\"What about a preprogrammed attack?\" Burnofsky asked cautiously.\n\nCharles smiled. \"It's time we learned more about some of our... toys.\" He nodded. \"Yes, Karl, we want to learn how to do it.\"\n\n\"You mean, how to program an attack using self-replicating nanobots? Yourselves?\"\n\n\"Are we too stupid?\" Benjamin demanded. \"Is that what you think? Do you think we rose from where we began to all of this by being stupid?\" He waved his hand to encompass all of what he'd earlier called his gilded cage.\n\n_No, by being rage-filled lunatics_ , Burnofsky thought. _And by having a very rich grandfather_.\n\n\"I am very well aware of your intellect,\" Burnofsky soothed.\n\n\"Perhaps not quite on your level, Karl,\" Charles said. \"But as I understand it, there's an app for this.\"\n\nBurnofsky's first thought was that they meant to use it against him. But no, there were so many ways they could kill him, they wouldn't be cute about it.\n\n\"Gentlemen,\" Burnofsky said, \"if you have thirty minutes, I can teach you to use the app.\"\n\n\"Wake up, Anthony. You have a visitor.\"\n\nBug Man sat up fast. The lights were on. But it must still be night out beyond the shuttered windows.\n\nGeorge III had a cup of coffee in his hand. He gave it to Bug Man.\n\n\"What?\" Bug Man said.\n\n\"Someone wants to meet you.\"\n\nBug Man was not yet fully awake, but he was getting there fast. \"No one knows I'm here.\" Awful suspicion blossomed. \"You sold me out! You mother\u2014\"\n\n\"Drink your coffee,\" George said, and sighed. \"If I was selling you out, would I start by bringing you a cappuccino? It's full-fat milk\u2014you're not watching your cholesterol, I hope.\"\n\nBug Man took a sip. George was trying to act cool, but he was upset. Something had disturbed his typical sangfroid.\n\n\"Put on some clothing. It's just one of my compatriots here to brief you on next steps.\" He was lying. He was lying and he was jumpy, very unlike his usual self.\n\n\"In the middle of the night?\"\n\n\"She has an early flight.\" George left the room. Bug Man took another sip of coffee. A soft knock at the door.\n\n\"Yeah, George,\" Bug Man yelled, \"I'm getting up. Damn, give a brother a few minutes to\u2014\"\n\nThe door opened. It was not George, but a white woman. Medium-tall, slender, good-looking but sharp edged. Brunette.\n\n\"Hello, Anthony. I'm sorry to barge in on you. But I have to get back to New York, so I don't have a lot of time.\"\n\nShe sat down on the foot of the bed, a position that made Bug Man quite uncomfortable since under the blankets he wasn't wearing anything. He was very conscious of his skinny chest and well-formed but not exactly muscular shoulders.\n\n\"Who are you?\"\n\n\"My name is Lystra.\"\n\n\"You were the pathway.\"\n\nShe smiled. She tilted her head, looked closely at him, making eye contact, taking her time in responding. _Smart_ , that's what he thought of her on first impression. That she was smart. And not bad if you liked older women. And she was on his bed....\n\n\"I'm a lot of pathways,\" Lystra said.\n\n\"So, George said\u2014\"\n\n\"Do you like George?\" she asked.\n\n\"Not really,\" Bug Man said.\n\n\"No, you wouldn't. George isn't really like us, is he?\"\n\n\"Like us?\"\n\n\"George is so serious. He never plays games. You and I, we like to play. We enjoy the game _as a game_.\"\n\n\"Do we know each other?\" Bug Man asked. Alarm bells were going off in his head. He recalled George's furtive eyes.\n\n\"In a way. I've played you at different times in different games, yeah. I use several online identities. But you're better than I am. Quicker reaction time; very, very good at taking advantage of terrain. And an amazing three-dimensional thinker. I can see why the Armstrong Twins hired you: your natural abilities, yeah, and your total lack of moral core.\"\n\nBug Man wasn't sure he liked that. On the other hand, he wasn't sure he could argue the point. Was he without a moral core? He frowned, considered it, shrugged it off\u2014figuring that if he couldn't think of a good counterargument, maybe it meant she was right.\n\nSo he said, \"Thank you. George said you weren't part of BZRK.\"\n\n\"Hmmm. Well, at that point I wasn't sure we should meet, you and I. Yeah. You've extracted the sample?\"\n\n\"The cells? Yeah, I got the Pope's cells out. They're with my nanobots on the Pope's sleeve.\"\n\n\"Yes, which goes to the laundry where it will be intercepted by someone working for... well, _us_.\" She said the plural pronoun mockingly. As if it made no sense.\n\nThe alarm bells were going crazy now. Bug Man almost felt the floor tipping beneath him. Lystra laughed, almost as though she could read his mind.\n\n\"Ah, suspicion begins to form, yeah,\" Lystra said. \"See how quick you are? That's why I like you. I'm done with you, I mean you've done what... _we_... wanted you to do. You harvested the cells. And I thought maybe, yeah... we... would just kill you now.\"\n\nBug Man froze. The playful tone was more frightening than a threat. An overt threat might ring false, might be a bluff. But this woman was not bluffing. She could have him killed.\n\n\"Here you are, though, a young man without a place to go. So very many people, yeah, want you dead.\"\n\nHis throat was dry, and the first words came out in a rasp. \"What do you...\" He swallowed, tried to get some moisture going.\n\n\"What do I want?\" She sighed. The sigh was melodramatic and false. \"Do you know the difference between us, Bug Man? Aside from the obvious\u2014gender, age, race. We both know none of that's important. The real difference between us is that you are a superb game _player_. Whereas I am a game _designer_.\"\n\n\"Yeah? What game?\"\n\n\"This one. Yeah. The game I call BZRK. Nanobots and biots. On the one side, twisted idealist freaks who would deprive humans of free will in order to give them all contentment. On the other hand...\" She let it hang, then added a superfluous, \"Yeah.\"\n\n\"That's the Twins' game,\" he said dully.\n\nThe not-very-convincing mask of friendliness disappeared so suddenly and so completely that it must never have been there. He had the terrifying impression that the skin on her face had shrunk so that bone and teeth and the hollows of her eye sockets were all suddenly outlined and shadowed.\n\nHer eyes glittered. \"Oh, them,\" she said, striving to regain her jokey tone and failing. \"They have _their_ game. Mine is better. More levels.\"\n\nHe noted that she no longer used the plural. Not _our_ game. _Mine_.\n\nShe stood up suddenly. \"Get dressed. I've decided. You're coming with me.\"\n\n\"But... where? Why am I\u2014?\"\n\n\"Where? Oh, places with tall buildings. New York City. And then cold, cold places. As cold as it gets this side of the grave. But what do you care, Bug Man?\" She sounded weary now. \"Don't you want to see how the game plays out? Don't you want to know what it was all about?\"\n\nHe shook his head slowly. \"Games aren't about anything. Games are just about the game.\"\n\nShe leaned down and laid a soft palm against his cheek. \"See? That's why I like you, Anthony Bug Man. You and I are going to be friends. Or I can have George put a bullet in your head.\"\n\n\"Friends,\" he said.\n\nAnd Lear smiled.\n\n# **TEN**\n\nNijinsky was shopping when it happened.\n\nHe was at Saks, the big one, the flagship store on Fifth Avenue. Christmas was coming and he had nephews. But he was shopping more for himself than for them. He liked shopping. It was a Zen thing for him. He had an eye for style, which had been useful in his life as a model but was entirely neglected in BZRK.\n\nSaks was already in full Christmas swing, decorated in a fantasy of silver and white; the storefront windows were dioramas of highly stylized snowmen appearing in Russian-themed settings. There were delicate flights of abstract snowflakes arched across the ceiling, and a restrained seasonal soundtrack played unobtrusively.\n\nNijinsky lifted the leg of a pair of slacks, felt the weight of the wool, ran sensitive, knowledgeable fingertips along the crease and then inside the waistband.\n\nAnd to no one he said, \"What?\"\n\nHe froze, just stood there, seeming to stare at a mannequin dressed in a sleek but uninspired Canali suit.\n\n\"The hell?\" Nijinsky said.\n\n\"Are you finding what you're looking for?\" It seemed an almost philosophical question, but of course it was just a salesperson, a woman, blonde, well put together but with tired eyes.\n\nHe stared at her now, just as blankly as he'd stared at the mannequin. \"Something...\" he said.\n\n\"Are you all right, sir?\"\n\nHe was not all right. Nijinsky had four biots. One was in Burnofsky, in his eye, tapping the nerve and watching the computer upon which Burnofsky was busily typing. The others were in their cr\u00e8ches\u2014holders for dormant biots\u2014in the basement of the safe house. All were out of range, so that rather than seeing detailed pictures of what they saw, he was seeing something more like two open picture-in-picture displays with vague shapes, fuzziness, lack of detail. Like looking through a very dirty window at a poorly lit scene.\n\nExcept that now, suddenly, there was _another_ window. And this one was perfectly clear.\n\nA new biot.\n\nHe looked around then, frantic, searching for an explanation. A fit, attractive middle-aged man was trying on an Armani blazer. Two children and their nanny killing time, the kids playing tag around hanger racks. An attractive woman with ornate ink peeking out of her d\u00e9colletage. Clerks. An older man; a store display designer carefully placing a hat on a mannequin.\n\n\"Sir?\" the blonde saleswoman prompted.\n\nNijinsky shook his head. \"No. I don't think I am all right.\"\n\nThe saleswoman said nothing to that.\n\nAnd then, a second new window, as clear as the earlier one. A clear biot's-eye view of the interior of a glass tube. He could see the curvature, the texture\u2014like stretch marks somehow\u2014because nothing was entirely smooth down at m-sub level.\n\nWithout so much willing it as thinking it, he turned the two new biots. They moved, obeying his will. And both biots now saw his opposite: six-legged; insectoid, but with dangerous tail stingers; a spider's spinnerets; and the disturbing biot rendering of his own eyes, a nightmare twisting of his own face.\n\nBiots. Two of them. And suddenly he understood.\n\nHe had seconds left.\n\n\"Excuse me,\" he said to the saleswoman. \"I believe I'm about to go mad. You may want to move away.\" He pulled out his phone and opened his messaging app. Who? Who should he tell?\n\nShould he even bother? Plath had pushed him aside. Why should he help her now?\n\nHe keyed in her phone number, hit the button for text, and typed.\n\nThere was a sudden rush of liquid rolling down the inside of the tube. It was no more than a droplet in the real world, but it was as big as a house in the m-sub.\n\n\"Ah,\" he said, as the acid engulfed both his new biots.\n\nThe next thing he said was also, \"Ah,\" but this time he shouted it.\n\nAnd the next \"Ah\" was screamed.\n\nAnd the next twenty or so.\n\nHe broke into a run\u2014frantic, terrified, still clutching the phone with its typed but unsent message.\n\n\"No! No! No!\" he shrieked as he raced to the open escalator and threw himself down it. Threw himself, as if he was trying to fly. Arms outstretched, face forward.\n\nHe hit the steel steps, and his face exploded in blood. He climbed to his feet but was pulled off-balance by the moving stairs and pirouetted down until he landed again, hard.\n\nBut not hard enough to kill himself.\n\nNijinsky swung around, off the bottom of the escalator, and this time he had a plan, a mad, desperate plan, one he could barely hold on to. He tied his long scarf into a knot as he descended a second, upward-bound escalator.\n\nPeople ran out of his way, bounded up the steps to avoid him. They yelled things like, \"What the hell, man?\" But mostly they just got out of his way.\n\nNijinsky knelt on the stairs. Rising, rising, and lay the end of his silk scarf on the step before him.\n\nFive seconds.\n\nFour.\n\nA wild, giggling shout rose from his throat as the end of the scarf was sucked into the escalator. The shout ended abruptly as the relentless mechanism devoured the scarf, tightened it around his neck, slammed his bloody face into the steel, chewed up his left hand, cut off his air.\n\nHe could no longer speak. No longer scream. Blood filled his head, and still the noose tightened.\n\nHis windpipe was crushed. Blood now seeped from his eyes and ears. The phone fell from his fingers and lay with message unsent on the steel serrated edge of the escalator.\n\nBy the time some bright shopper thought to push the emergency-stop switch, Nijinsky was dead.\n\nThe message on Plath's phone was from Nijinsky.\n\nIt read, _2 new biots_.\n\nBut she had muted her phone and would not see the message until later because she was meeting with Stern. Again.\n\nPlath did not ask Keats to join her this time. She would discuss the Tulip with Mr. Stern, but she would mostly be asking him what he had discovered about Lear.\n\nAttempts to learn about Lear counted as treason within BZRK. Treason led to bad things, and she did not want to implicate Keats in that.\n\nOf course, Keats had a biot in her brain. If he was very curious he could make the long trip out into an ear canal and listen in.\n\nShe didn't think Keats did things like that. It would be out of character. But in this new world she had entered such things had to be considered. In this new world the human body was not a singular object\u2014it was an ecosystem. It was a Brazilian rainforest full of flora and fauna, from creepy, crawly mites to big, fat balls of pollen to Dr. Seuss\u2013like fungal trees, to a hundred different types of bacteria, all the way down to viruses. None of it strictly human. The average human body had far more nonhuman cells than human ones, though they comprised only a fraction of the weight\u2014about five pounds in most people.\n\nYou moved differently through the world when you truly came to accept that fact. When you knew that you were crawling, covered, congested by nonhuman life-forms. Sometimes you couldn't quite see the line between yourself and the world around you.\n\nAll about her on the sidewalk were other ecosystems, each body a similarly complex environment. Each body in turn a small part of a larger system. A system called New York. Or, more inclusively, the human race. Formerly meaningful divisions had lost some solidity. What seemed solid in the macro was so much less so down in the meat.\n\nThe meeting with Stern followed all the rules of spy craft. They set up the meeting using text messages. They both spent an hour throwing off any possible pursuers. Their phones were off and therefore impossible to track. If she were being followed, then it was very professionally done.\n\nAnd yet when Plath arrived at the steps of the public library in Bryant Park, there was a man sitting across the street in the window of a hotel caf\u00e9, sipping a latte and making no effort to go unnoticed.\n\nHe was middle-aged, with long graying hair and a wry, observant expression. He was dressed like a dandy\u2014a purple velvet blazer, a top hat that sat on the counter beside him.\n\nIf he had ever had a real name, Plath didn't know it. His nom de guerre, his BZRK name, was Caligula.\n\nPlath had seen him in action. He was a confident and extremely capable killer. He was the eighth person of whom Lear had spoken. But it was not possible for Plath to imagine giving him orders.\n\nIt was Caligula who had killed Ophelia after she was captured by the FBI. He had burned out her brain so as to leave no traces of nanotechnology behind. If Plath brought him into BZRK now\u2014into her BZRK\u2014Wilkes, who had been close to Ophelia, who had very nearly died beside her, might try to kill him. And that would be the end of Wilkes.\n\nAnd yet, here was Plath meeting Mr. Stern to discover what he had learned in his efforts to track down the elusive personality behind BZRK. Was that why Caligula was here? Did he already know? Should she be expecting a bullet or a knife or the killer's trademark hatchet?\n\nCould Caligula guess what they were talking about? Surely not. But he had found a way to follow her, or perhaps to follow Mr. Stern. That knowledge made her feel faint. It weakened her knees.\n\nGod, it was true: there was no escaping the man in the velvet suit. _The NKVD_. Plath had Googled it. Anya had spoken the truth. And now here was her own personal NKVD sipping a coffee and watching to see what she would do.\n\nOr fail to do.\n\nAs Nijinksy's body was being cut loose by paramedics, Plath bought a street pretzel and a Nantucket Nectars cranberry. Stern had a coffee and an Italian sausage. They looked, perhaps, like a girl meeting her father. Or a student with her atypically tough-looking professor.\n\n\"Now that we're alone, how have you been, Sadie?\"\n\n\"Getting used to being back in the world,\" she said, looking around at the other lunchtime diners, all somewhere between coats and sweaters on this gray day.\n\n\"It was good of you to pay the money to the boat crew who died. One of them had two young kids. Softens the blow.\"\n\n\"What have you found?\" she asked, too cold to want to chat, and too aware of Caligula's cobra gaze.\n\n\"On the Armstrong Twins? I suspect they are in a place called Sarawak, which is in Malaysia. AFGC owns a facility in Malaysia, a rare earths mine. Rare earths are a class of rare minerals used in some sophisticated electronics components. It makes sense that AFGC would have a source.\"\n\n\"How likely do you think it is that they're there?\"\n\nStern thought it over. \"I'd say seventy percent. It seems consistent with what we're seeing. But it's possible they're elsewhere. It's even possible they are back in New York.\"\n\n\"And the _other_ person you're looking for?\"\n\nStern glanced at Caligula. \"There sits the one man who might be able to take us to Lear.\"\n\n\"Stay away from him,\" Plath said too quickly.\n\n\"You're that scared of him.\"\n\n\"I've seen his work, Mr. Stern. The man who warned me about him doesn't scare easily.\" Vincent. Back when Vincent was _Vincent_. \"But he was scared of Caligula.\"\n\nStern raised his cup, sipped, and said, \"I have leads, nothing solid. Lear's cell number is obviously switched out daily. You gave me four such numbers. All the numbers are throwaways. Burner phones. But interestingly, two of them were purchased in odd locations.\"\n\n\"Odd how?\"\n\n\"Well, one was bought in London; one was bought in Wellington, New Zealand; one was bought in Saint Petersburg, Russia. The last was from Punta Arenas, Chile.\"\n\n\"What am I not seeing in those four locations?\"\n\n\"Wellington and Punta Arenas share a distinction as major jumping-off points for Antarctica.\"\n\n\"Antarctica. Why... never mind. I had another text exchange with Lear. Here's the number.\" She read it off to him. \"Why doesn't Lear just block the number?\"\n\n\"Excellent question,\" Stern said approvingly. \"Arrogance? Or, more likely, he's deliberately leaving breadcrumbs. Either a false trail, or...\"\n\n\"Or what?\"\n\n\"Or a trail meant for the right person to follow.\"\n\nA game? Was she supposed to believe that Lear was playing a game with her?\n\nIn the coffee shop, Caligula was standing up. He put on his hat, straightened it carefully, and looked directly at Sadie, who returned his gaze evenly. Then he tugged at the front brim in a slight but unmistakable acknowledgment of her, and faded from view as he moved away.\n\nStern caught the gesture and said, \"And you're sure we shouldn't question _him_?\"\n\n\"He's tight with Lear. He's Lear's attack dog. And I may need him.\"\n\n\"What are you planning?\"\n\nPlath shrugged. \"Lear's orders. He still wants us to wipe out all AFGC data on nanotechnology. And soon.\"\n\nStern took a long pause at that. He searched her face, looking for something to reassure himself. But reassurance did not come, and now he was wary. \"Have you told Lear about the practical objections to such a harebrained scheme?\"\n\n\"No,\" Plath said. \"Not yet.\"\n\nShe hesitated, unsure if she should go forward. Stern was an experienced interrogator\u2014he knew when just to wait.\n\n\"It's just...,\" Plath began. \"It's just, well, I was thinking... a bomb of some kind?\"\n\n\"A bomb? Are we back to that?\" He shook his head slowly without shifting his gaze from her. When she said nothing, he said, \"Sadie, please listen to me. I've been to war. I was in Iraq, and before that I was in Somalia. When you're in it, when you're scared and when you're mad and you want revenge, maybe, you find yourself thinking about doing things no human being should do. You think about crossing the line.\"\n\n\"Where's this line, Mr. Stern? The Armstrongs killed my father and brother. They basically killed the president of the United States, even if that's not what they intended. Burnofsky was trying to unleash self-replicating nanobots that could kill every living thing on the planet. So where's the line?\"\n\nStern put down his coffee, carefully crumpled the paper from his sandwich, and set it aside. He wiped his hands with a napkin. Then, with a clean forefinger, he pointed at Plath's forehead. \"In there.\" Then he pointed at her heart. \"In there. That's where the line is.\"\n\nIt was not easy to meet his worried, penetrating eyes.\n\n\"Sadie, you need to ask yourself: Is this _you_? Are you really, truly a person planning what would look like a terrorist attack in Midtown Manhattan?\"\n\nFinally she couldn't take it and turned away. \"No, of course not. But get me everything you can, okay? Everything you feel okay about giving me. I still need to find a way....\"\n\nHe wasn't buying it. And for a moment she was afraid he might just walk away. Then, with a pained expression, an expression of loss, he nodded his head.\n\n# **ELEVEN**\n\nSaks would not release the store surveillance video. But Mr. Stern had excellent connections throughout security companies in New York. An underpaid guard, when offered ten thousand dollars in untraceable cash, decided he could in fact arrange for the video file to make its way to Mr. Stern.\n\nHe in turn passed it along to Plath. Who watched it for the third time with Keats, Wilkes, and Anya. Billy had not been asked to be present, but he was, anyway.\n\nThey decided that there was no need for Vincent to be subjected to it in his condition.\n\nHis _condition_. Fragile, that was his condition. Borderline nuts, still. High-functioning unbalanced.\n\n\"Jin lost his shit,\" Wilkes said on a second viewing. \"Look. That's when it starts. He's fondling a pair of pants. Then that's him texting.\"\n\n\" 'Two new biots,' \" Plath said dully.\n\n\"The text was sent three minutes later,\" Noah pointed out. He had compared the video time code to the time signature on Plath's phone.\n\nThey watched the second part of the tape. Nijinsky hurling himself down the escalator. There was no sound. The video was decent quality, but the angle was poor. They were seeing him from behind.\n\n\"Jesus Christ, how many times do we have to watch this?\" Wilkes cried suddenly. She stormed off to the kitchen. Then came back with a bag of chips.\n\n\" 'Two new biots,' \" Keats said. \"But he was _against_ getting anymore.\"\n\nThe third segment of tape showed a distraught Nijinsky, face-on this time, kneeling, feeding his scarf into the escalator.\n\nIt went on for way too long. Nijinsky dead. People milling around helplessly. Store employees rushing over with scissors, trying to get at the scarf and cut him loose. Failing, because it was too tight, too tangled.\n\nEventually a security guard. Then, at last, far too late, the paramedics.\n\n\"He went crazy,\" Anya said. \"It was deliberate. He was looking at clothing and then he was killing himself. Madness.\"\n\nShe wasn't thinking about Nijinsky. She was thinking about Vincent. She glanced nervously toward the stairs leading up to his room, then tried to cover the telltale gesture with a reach toward Wilkes's chips.\n\n\"New biots,\" Plath mused.\n\n\"Just totally lost his shit.\" Wilkes spoke around the crunching of a corn chip.\n\n\"Who could make a biot for him?\" Keats asked. \"It takes a tissue sample and the equipment.\" He didn't mean to single Anya out by looking at her, but she was the only one in the room with the skills, and she controlled the equipment that had been hidden in the basement of the safe house.\n\n\"It takes a tissue sample, the equipment, and the skills,\" Anya said. Then, angrily, \"Why would I do that to Nijinsky?\" She didn't wait for an answer. Everyone knew the answer. Anya sighed. \"Yes, I disliked him. But I would never do this.\"\n\nThat earned her carefully blank looks.\n\n\"No, you listen to me, all of you. I would never. I did never. I did not do this.\"\n\nAll eyes were on her.\n\n\"No!\" Anya cried. \"No, do not do this! Suspicion will destroy us.\"\n\n\"What 'us'?\" Wilkes asked. \"Look at _us_. Ophelia's dead. Renfield. Vincent's out of it. Now Jin. Fucking Jin, man.\" She laughed her weird heh-heh-heh laugh and looked ready to cry. \"We're a fucking joke.\"\n\n\"We stopped the Armstrongs,\" Keats said reasonably. \"We accomplished a lot. More than we should have been able to.\"\n\nAnya ignored him and instead pleaded with Plath. \"Plath, you know I didn't do this. Look at me. I did not hurt Nijinsky.\"\n\nPlath wanted to say something reassuring. But she couldn't quite get the words to come out. If not Anya, then who? Someone at McLure Labs? But how many people there even knew of the existence of biots? And of those, how many could make one? And of _those_ , how many would use the knowledge to kill Nijinsky? Was Anya a traitor?\n\n\"I know what you are thinking.\" Anya's Russian accent was coming to the fore. The word came out _thinkink_. \"You are wrong.\" _Wronk_. \"It was someone else. Why would [ _vwould_ ] I...? For what reason?\"\n\nKeats said, \"No one suspects you, Anya. I don't, at least. But the thing is, who else then? Not you, okay. But who?\"\n\n\"I don't know,\" Anya pleaded. \"I can think of only three others at McLure Labs with the knowledge and the access to equipment. But how would they have a tissue sample from Nijinsky?\"\n\n\"He's dead now, can we call him by his real name? Shane Hwang. Not some dead, crazy Russian ballet dancer.\" This from Wilkes. She punched the bag of chips and sent crumbs flying. \"His name was Shane fucking Hwang. I never even knew Ophelia's real name. And poor old Renfield. And when I'm dead or crazy, you people won't know me, either.\" The flame tattoo under her eye looked like extravagant tears. \"Jesus, no one will even know me.\"\n\n\"Okay,\" Plath said, bringing silence. \"I believe you, Anya. I think... I mean, I choose to think... that this is the remote biot-killer technology that Lear was talking about. Which means we are all in danger. But still, Anya, I\u2014we\u2014need to be able to watch you.\" Plath put a finger to her eye. It looked like a gesture, some kind of evil-eye, maledictory gesture. But in fact Plath had sent one of her biots racing around her own eyeball to clamber over lashes and reach the cheek.\n\nThrough her biot's eyes she could see the vast column of flesh descending like some cylindrical meteor from the sky to press a giant furrowed fingertip within a few seconds' walk.\n\nHer biot ran beneath the vast curve, ran on until fingertip and depressed cheek met, then clambered upside down onto the finger.\n\n\"No,\" Anya said. \"No. Nyet. Is not happening.\"\n\n\"I promise you, Anya, I won't lay any wire. I will not make any changes in your brain.\"\n\n\"Your promise,\" Anya sneered.\n\n\"Yes, my promise,\" Plath said. \"I can't just let you walk away. I have to maintain surveillance.\" She leaned toward Anya and stretched a finger up to the older woman's eye.\n\nAnya swallowed in a dry throat. \"So you will watch me. You will tap into my eye and see everything that I see.\"\n\n\"It's the only way,\" Keats said, though he didn't sound too sure of it. He pressed his lips together and stole a worried glance at Plath, who revealed no emotion.\n\n_Look how hard she's gotten_ , Keats thought.\n\nWhen they had first met, he'd marked her down as a spoiled little rich girl, probably a snob, who would condescend to him, look down her nose at him.\n\nBut that had not been true. She had been anything but a snob. But even then, early days, he'd noticed that effortless authority she carried with her. That was, without question, a product of wealth and privilege. Plath would admit that much. A billionaire's daughter simply had an air about her that could not be faked by a working-class kid like Keats.\n\nPart of him was proud of her in an uncomplicated way. He wanted to say, _Well, look at you, all grown up and in charge_. But part of him was small enough to focus on their relationship rather than BZRK. He was in love with her. He believed she loved him back. But how stable could a relationship be when there was this much of a difference in their circumstances? My God, the girl basically had a private army.\n\nAnya let Plath touch her, just below her left eye.\n\nPlath held the contact for a few seconds as her biot scampered off and began the journey to the optic nerve.\n\nFrom now until Plath let her go, Anya's sight would be shared. Plath would see what Anya saw. In the bathroom and bedroom, too, inevitably. The idea made Keats's skin crawl, but this was BZRK.\n\nFighting for freedom. Saving the world.\n\nYeah, but hadn't they done that when they stopped the Armstrongs from controlling the president? And when they stopped Burnofsky's gray-goo scenario? Hadn't they already won?\n\nThen how was it they were still trapped in this paranoid universe where they used the names of dead or made-up madmen? How was it that they were still taking orders from an invisible character called Lear?\n\nThe thought was out of his mouth before he could check it. \"Why are we still doing this?\"\n\nWilkes snorted. \"Pretty blue eyes asks the right question. Why are we still doing this?\"\n\n\"Because we haven't won yet,\" Plath said. But she didn't quite like that answer. \"It's not over yet.\"\n\n\"How does it get to be over?\" Keats asked. \"How will we know it's over?\" He had been leaning forward, now he drew back. \"Look, isn't this about the knowledge, really? Once we know how to make nanobots and biots, how do we ever unlearn that? It's like nuclear bombs, isn't it. How do you stop it spreading once the technology exists?\"\n\n\"When the last of us is dead, it's game over. For us. Right?\" This was the first time Billy had spoken. \"I mean, it's a game, right? Biots versus nanobots. Take over the world. Isn't it a game?\"\n\n\"No, it's real,\" Plath insisted. \"The Armstrong Twins are real, and we're real, and Jin was real.\"\n\n\"Yeah, but...\" Billy felt the weight of disapproval. \"Yeah, but games _are_ real. That's what you don't get, with respect to you, Plath. Games are real to the people playing them. While they're playing.\"\n\nNo one said anything; after all, Billy was just a kid. But Keats couldn't shake the feeling that he'd just heard something important, that Billy had blurted out the truth. _It could be real, and dangerous, and deadly, and yet still be a game_ , he thought.\n\nWhen was a game over? When you lost.\n\nOr when you won and went off in search of a new game.\n\nBiot versus nanobot. That was the game. But now, according to Plath by way of Lear, a new level was being revealed. Something out there could kill biots remotely. Dead biots meant madness. It meant killing yourself on an escalator in Saks.\n\nSo why bother to blow up a boat? If you could generate then kill biots, then why did it seem so much like manipulation? The Armstrong Twins would not hesitate if they could kill Plath and him.\n\nSo why wasn't he dead?\n\nBecause the game was somehow more complicated than that.\n\nThe video played again, looping. Keats watched the faces watching Nijinsky. They watched in surprise as he stared and spoke to the air. Then in shock as he threw himself down the escalator. Horror as he fed the silk scarf into the mechanism that choked the life from him.\n\nThen, Keats picked up the remote and rewound.\n\n\"Enough!\" Wilkes yelled.\n\n\"Wait,\" Keats said. \"Don't watch Jin. Watch the people around him. That woman. The one with the ink.\"\n\nHe advanced it in slow motion, focusing on the woman.\n\nShe pulled out her phone and glanced at it. Checking e-mail? Or checking the time?\n\nShe stole a glance at Nijinsky.\n\n\"She's looking at Jin,\" Keats said.\n\n\"He was a good-looking dude, maybe she\u2014\" Wilkes began, but then she fell silent, because now was the part where Nijinsky started to lose it. The people nearest were shooting him irritated or concerned looks. The woman was not. She was half smiling, watching... waiting.\n\n_Waiting_.\n\n\"She knows,\" Keats said.\n\nHe cut to the next video, the horrific one showing Nijinsky on the escalator. There was a woman just a dozen steps behind him.\n\n\"Fuck! It's _her_ ,\" Wilkes said.\n\nNow everyone was leaning toward the screen, checking the dress, checking the shoes, the hair, comparing them to the first images.\n\n\"Yes,\" Plath confirmed. It's the same woman. Jin got to this place by running, then hurling himself down the escalator. And she _followed_ him? What kind of person follows a crazy man?\"\n\nNow, again, Nijinsky fed the scarf into the escalator.\n\nBut this time they watched the woman behind him\u2014the shoulders, the hair.\n\nShe stepped past and over the strangling Nijinsky. Not panicked. Calm.\n\nShe knelt by Nijinsky. Her hand shot out, took something.\n\n\"The phone,\" Plath said. \"She took his phone. The time signatures. She sent the text.\"\n\n\"It's an Easter egg,\" Keats said. \"Billy's right: it's all a game. And that woman is an Easter egg. We are _supposed_ to see her.\"\n\nJindal could barely restrain himself. His first meeting with the returned Twins had ended with his being dismissed like a disappointing schoolboy. Now they would have to listen. \"We have confirmation. Proof. They've hacked our network. Somehow they exploited a hole in the AmericaStrong computer system and worked their way back to us, back to core AFGC systems.\"\n\nCharles saw the meat of it immediately. \"Floor Thirty-Four?\"\n\nJindal shook his head so hard he couldn't speak until he had stopped. \"No, that is walled off entirely. But the good\u2014\"\n\n\"Do they have our nanobot blueprints? Our technical specs?\"\n\n\"Yes. And they've been looking at this building.\"\n\n\"With an eye to infiltration or attack?\" Charles demanded, while Benjamin remained ominously silent.\n\n\"No way to tell. But gentlemen, there's good news as well.\"\n\nCharles raised his eyebrow. Benjamin glowered at Jindal, as if holding him personally responsible. \"Good news?\"\n\n\"The hackers have been hacked in return,\" Jindal said. He was giddy now, torn between excitement and fear. \"We tracked them back and found a way into some of their systems.\"\n\n\"BZRK?\"\n\n\"No. McLure Labs Security. That's who's been watching us. McLure Security. Presumably at the direction of\"\u2014Jindal hesitated, knowing the effect his next words would have on the Twins\u2014\"Sadie McLure.\"\n\n\"The little bitch,\" Benjamin spat.\n\n\"Do we know where she is?\" Charles asked.\n\nJindal shook his head, impatient to get to the one remaining piece of good news. \"No, nothing directly on BZRK. But we can now track the movements of the main McLure Security folks, and if we follow them, we'll likely find a way back to Plath herself.\"\n\n\"Bah,\" Benjamin snorted. \"No time. They're planning an attack here, that's obvious. We have to hit them hard, now. Now!\"\n\nCharles looked queasy, but as Jindal watched, he could see wary acceptance grow on the wiser brother's face.\n\n\"We don't have the gunmen we used to, thanks to that disaster in Washington. But we have other means, as you know well. Massed preprogrammed attack,\" Benjamin said harshly.\n\nCharles smiled faintly at that. He shrugged his shoulder. \"Go ahead, Benjamin. You know you've wanted to say it ever since you came up with that name for the drones. Go ahead.\"\n\nFor once Benjamin did not scowl. He smiled. And said, \"Locate Stern. And any other important actors in McLure Security. And as soon as you have the location and Burnofsky is ready... I will release the Hounds.\"\n\n# **TWELVE**\n\n_Down in the meat_.\n\nP2: soulless, mindless biot, Plath's creature, Plath's bizarro-world daughter. P2 zooming across Plath's eye, six legs stroking as Plath had learned to do, like an Olympic speed skater.\n\nThe room was dark, shades drawn, door locked, a GO AWAY Post-it note on the door. In the darkness, her eyeball\u2014which in light could look like a frozen lake\u2014looked like some impossibly vast jellyfish, at least here on the white.\n\nHer eyelids\u2014the onrushing \"shore\" lined with palm trees\u2014looked less benign, more like needle-sharp teeth.\n\nHer eyelid swept over her, rubbing across her biot back, a slight pressure, greater darkness; then it rushed away as though that row of teeth had rejected the tiny meal.\n\n_Sadie, you need to ask yourself: Is this_ you _?_\n\nThat barb stuck. It stuck, and Plath could not shake it off.\n\n_Are you really, truly a person planning what would look like a terrorist attack in Midtown Manhattan?_\n\nThe World Trade Center was falling in her memory, and now there was a musical track to go with it. An old, old song, a Beatles song: \"Piggies.\"\n\nIt added a vengeful but playful note to the video atrocity.\n\nHow had she felt about that footage the first time she had seen it, back in the classroom? She had been horrified. Sickened. She had always been that way, always capable of being outraged by terrible injustice. In school they had done a unit on World War Two, and as part of that they had done a couple days on the Holocaust. She was not Jewish. She was not part of any group that had been touched by the Holocaust, but she'd been unable to sleep afterward, unable quite to control the sickened hatred of people who could do that to other human beings.\n\nThey had watched parts of _Shoah_ in class\u2014actual first-person testimony from Holocaust survivors. She remembered vibrating with the suppressed fury she'd felt. She remembered giving up finally on any effort to control the tears.\n\nShe still felt that way when she recalled the Holocaust unit. But she no longer felt horrified by the World Trade Center. Now it was... what?\n\nBeautiful, is what it was.\n\n_Is this you?_\n\nWas it really this easy to cross lines that should never be crossed? Had the stress of this unasked-for war of hers, this BZRK existence, simply washed away the part of her that cared about right and wrong?\n\nOr. Or had she had some help?\n\n_Is. This. You?_\n\nPlath had three biots. She had sent P1 into Anya's brain. It sat now on Anya's optic nerve, looking out through Anya's left eye. It was a window open in Plath's head, showing, at the moment, a bowl of soup, a rough hunk of baguette, and three slices of sausage. Anya's hand lowered a spoon. Raised a spoon. Pause. Lower spoon. Raise spoon. Put down spoon, hands to bread, tear off a hunk, raise it toward mouth.\n\nPlath's final biot, P3, was an enhanced model. Faster, with better sensors, stronger. It was still in the vial attached to a chain around Plath's neck, staring at nothing\u2014a very dull TV show of curved glass wall, and not so much of that in this light.\n\n_The line is there..._\n\nMr. Stern suspected she'd been caught up in something, and needed some time to think it through more calmly. Plath had different suspicions. Because, yes, she _was_ thinking of attacking the Tulip. Guns blazing. Bombs blasting. The image of the Tulip disintegrating, toppling, falling to the ground in fire and smoke was almost... almost erotic.\n\nAnd this was Plath\u2014Sadie\u2014who had refused when she had the chance to kill the Armstrong Twins.\n\nShe had left for \u00cele Sainte-Marie feeling betrayed that she'd been trapped into BZRK. Feeling sickened by the violence and by what she had seen and done down in the meat. Now she was ready to launch an actual attack. To kill. To kill innocent people. Why? Was it just because Lear had told her to?\n\nWhat had changed?\n\nThe benign explanation was that she had learned and matured and come to grips with grim necessity. The less benign explanation was that she had become hardened and had lost her soul.\n\nBut she feared the truth was a third possibility: that she had been wired.\n\nHow and by whom? The obvious suspect was Keats. After all, he had a biot in her brain, ostensibly protecting her from a blown aneurysm.\n\nBut why would Keats wire her? Orders from Lear? Or had he gone over to the other side? Both seemed absurd. Keats would not blindly take orders. And he would never join the people who had put his brother Alex in a mental institution.\n\nUnless he had decided that BZRK was to blame. And wasn't that a plausible conclusion? _Wasn't_ BZRK responsible, in a way?\n\nShe ran down the list of other people who might have done it. Maybe one of the McLure Security guys. Maybe one of the house servants who washed sheets and delivered food. Or maybe someone had gone to work in her brain as soon as she got back to New York. But that would mean whoever was doing it had had very little time. Which in turn meant that someone was very, very good at the job.\n\nSomeone.\n\nBut the _obvious_ suspect?\n\nShe was circling the globe, around the eye that twitched beneath her, making all the minute adjustments that eyes must do. She skimmed the edge of her iris-serried ranks of gristly muscle fiber waiting to react to light, opening and closing the dark, deep hole of her pupil.\n\nDown and around, beneath the permanent retraction point of the eyelid, so that her \"sky\" was now an eternal mucous membrane. Her biot skated on, slowed slightly by the claustrophobically low roof. With absolutely no ambient light, she had illumination switched on\u2014glowing nodes built out of the DNA of exotic deep-ocean creatures. She was in the land of muscle bundles now, massive cables seemingly fused into the melting ice of the eyeball and ascending into the dark.\n\nAnd onward, farther around the globe\u2014and now, at last, like Yggdrasil, the tree that supported the world in Norse mythology, the optic nerve rose into view.\n\nSuddenly the world shifted wildly beneath her. Muscles jerked crazily. In the real world, the light had snapped on.\n\nShe sat up.\n\nKeats looked at her, saw her surprise, and said, \"Sorry, did I startle you?\"\n\n\"No, no,\" she lied. \"I just... there was a Post-it note.... Never mind.\" She could see it lying on the floor of the bright hallway. \"Are you coming to bed?\"\n\n\"Was kind of hoping to,\" he said, not wolfishly, more just a tired boy.\n\nPlath pulled the blanket back to bare the sheets for him. He nodded at the open space, smiled at it as if it was an old friend. He stripped off his clothes while she lay back and closed her eyes, hoping he would get the message.\n\nShe tried to calm her breathing. Keats was in her brain; he would know from the pulse of blood through the aneurysm whether she was perturbed.\n\nKeats was warm beside her. He leaned over to give her the lightest of kisses. Just a brush of lips and a whispered, \"Good night.\"\n\nBut to her surprise Plath found herself wanting more. She pushed her fingers through his hair and pulled him close and kissed him back. In the dark, even as she crawled toward her own optic nerve, his lips were just his lips and not a parchment landscape.\n\nHe responded.\n\nP2 began the ascent\u2014direction was all very subjective in the meat\u2014began climbing that tree.\n\nHe was still holding back, not quite sure whether this kiss was a prelude or just a very nice good-night. She pushed her tongue into his mouth, and now he must feel the way her pulse raced.\n\nUp the nerve, up to the impassable membrane that guarded the brain itself. Her brain. She reared up on her four anterior legs and used the sharp pincers on her front legs to slice as small a hole as possible through the membrane. A watery liquid oozed outward.\n\nShe checked herself, inspected as well as she could her biot legs, looking for pollen, bacteria, fungus\u2014all the things which can be so deadly if carried into the brain. She found what looked like a half-dozen tennis balls on her left rear leg and knocked them loose. Bacteria, and very much alive: one was splitting as she watched.\n\nKeats was kissing her now, everywhere. He was no longer responding to her, but moving ahead, taking charge, setting the pace, and for once Plath let him, willingly surrendering, needing to surrender.\n\nHer brain floated like a giant sponge, a sponge crisscrossed with throbbing arteries and veins like the tangle of rivers and tributaries in a delta. The fluid made movement slower than it was in an air environment, and her biot claws had to grab on so as not to float away.\n\nHe was inside her. His biot. Down here in these endless folds of pink flesh. At least she hoped he was, hoped he was not on her other eye spying, or worse, far worse, somewhere deeper still, laying wire.\n\n_Let it not be him. Not him_. That was a betrayal she could not survive.\n\nThe tissue that was the ground could appear to be a wall, a floor, or a ceiling, depending on your perspective. The biot world was one where gravity was almost irrelevant, certainly in this liquid environment.\n\nShe was aiming for the hippocampus, a deep structure, an ancient part of the evolving brain. It was the router of the mind. If someone was wiring her, that's where they would likely start. The implanted brain-mapping imagery was a guide, though an imperfect one because no two brains were identical, and where she might expect to find a figurative gully could be a plunging valley.\n\nIn the real world her body was responding almost on its own, as though it was not connected to her, not connected to the brain upon which she now walked, the brain that was the processor of every contact between his tongue and her flesh.\n\nMadness. She laughed. He stopped.\n\n\"No, no, no, don't stop,\" she said.\n\n\"You were giggling.\"\n\n\"Shhhh,\" she said, and pushed his head back to where it had been.\n\nToward the hippocampus, but with a stop on the way. She crept her biot forward slowly, slowly, dousing her illuminators one by one, just enough to feel her way forward to\u2014\n\nLight out. In the darkness of her own brain she saw his biot's light. There was his biot, not moving, just standing on the bulging basketwork he had so painstakingly constructed in order to save her life. The work had been started by her father; almost completed now by her lover.\n\nHis biot was not wiring her. It was not him.\n\nFar away and as close as the artery that pounded beneath her feet, she felt him, felt his banked power, knew he was close to losing control, and liked that idea a great deal.\n\nShe sent her biot forward toward the hippocampus, turning lights back on as she moved away from Keats's biot.\n\nShe tripped over it before she saw it. One leg scraped across something that did not feel like flesh, something hard and sharp.\n\n_Wire_.\n\nDid Keats feel the sudden chill that went through her? He did not slow or falter. But now her mind was reeling, no longer vague and disconnected from her body and its reactions.\n\nShe had been wired.\n\nWait, was that a glimmer of light?\n\nShe killed her own biot's light once more and stared hard into the visual field in her brain. Into the visual field that showed her brain to her brain.\n\n_There!_ For just a second. Less than a second. A glimmer of light.\n\n\"Bastard,\" she muttered.\n\nKeats did not hear her, he was beyond that.\n\nThe light had come from behind a pulsing vein. There was no innocent excuse. There were no light-emitting life-forms down here.\n\nThe fear rose in Plath now, competing with simmering rage. It began as a dull electrical charge in the base of her spine and fanned out from there to become nausea in her stomach and a tightened chest that felt too small to contain her air-starved lungs and pounding heart.\n\nWho was on the other side of that vein, that vein the circumference of a subway tunnel? Who and what was back there?\n\n_Bastard, bastard, bastard_ , she raged, but silently.\n\nPlath stifled her fear, and her biot plunged after the retreating nanobot. She noted that she had decided now that it was an Armstrong nanobot, not a BZRK biot, not Keats, not anyone from her side. Because that\u2014\n\nWait. When had she acquired this readiness to believe the best of BZRK? Was _that_ a naturally occurring thought? Or was it part of the wiring? Was that what this foe was doing right now, right now practically under her nose\u2014finding ways to dampen her suspicion?\n\nAgain, a glimmer! It was moving away, but it evidently needed light. So did Plath, so any hope of concealment was forgotten now, any hesitation set aside with the decision to chase.\n\nShe saw him! Or at least an impression of something moving. She was gaining on him. Gaining! Which most likely meant it was a nanobot. That at least would be a relief.\n\n_Please, God, if there is a God down here in the meat, let it be the enemy, the true enemy_.\n\nSuddenly the light ahead dimmed as if it had dropped into a crevasse. She charged ahead, caught up in the chase, adrenaline flooding her system with urgency, breathing hard in her bed, trying to remain perfectly still so as not to wake Keats.\n\nHer biot raced; she saw the dip ahead and killed her illumination, rendering herself almost invisible while using the enemy's light as a beacon.\n\nShe looked down, and there it was, waiting for her.\n\nIt was no nanobot.\n\nShe grabbed Keats's head in her hands and held it still, just inches away from her, stared into his eyes, pleading and said, \"Noah, help me. Help me, Noah.\"\n\n\"My Stockholm lair. Yeah. Lair. Because the supervillain needs a lair, yeah?\"\n\nIt was a nice hotel suite, a very nice hotel suite at the Stockholm Grand. Nice view out over the very civilized waterfront with bright-lit ferries and stately buildings. Multiple bedrooms, understated taupes and beiges and earth tones.\n\n\"It's not all that...\" Bug Man started to say before stopping himself.\n\n\"Not so lairlike?\" Lystra asked, and laughed. \"Well, I have a much better lair somewhere else. Far to the south, you might say. You'll like it... if I let you come with me.\"\n\nBug Man stood as awkwardly as one might expect a young man to stand when threatened with death.\n\nLystra laughed again and waved him to a seat. He sat on leather. It made a squeaking sound that might almost have been a fart.\n\n\"That was... um...,\" he said.\n\n\"Did you just fart in my presence?\" She was pretending to look fierce. But Bug Man had seen her true ferocity, and this wasn't it. He relaxed a very little bit.\n\nLystra went to a sideboard and poured an amber liquid into two heavy crystal glasses. She handed one to Bug Man.\n\nHe sniffed and recoiled.\n\n\"It's Balcones True Blue. Lovely whiskey, that. Made with Hopi blue corn.\" She took a single cube of ice with a pair of silver tongs, carried it to him, and dropped it in his glass. \"You taste it now. Then you keep drinking as the cube melts, which lowers the proof. The flavor evolves. Each sip will be subtly different.\"\n\nBug Man took a sip. It was fire in liquid form, and he started coughing, which made her laugh. It was a cruel laugh, and there, again, a glimpse of the harsh bone beneath soft flesh.\n\n\"My father used to let me drink whiskey with him,\" Lystra said. She sat down opposite Bug Man. He glanced at her bare legs. She noticed.\n\n\"You miss your little love slave?\" she asked.\n\n\"Jessica? You know about... that?\"\n\n\"Yes, of course. You're a rapist, Bug Man.\"\n\nHe flushed. \"No, I'm not. I never forced her to do anything.\"\n\nShe leaned toward him, elbows on knees, drink cradled in both hands. \"You programmed her. You took away her free will. You replaced it with your own. You enslaved her. And when you have sex with someone in that condition, it's rape.\"\n\nHe shook his head and took a drink just so he'd have an excuse not to meet her gaze.\n\n\"Rapist. Murderer. Terrorist. That's you, Bug, by the standards of the wide world, yeah.\"\n\nBug Man frowned. No, that wasn't right. \"I'm... no. No way. I'm a gamer. I'm just playing.\"\n\n\"Buggy, Buggy, Buggy.\" She patted his knee, and he felt his flesh creep. \"If you were charged in a court of law, you'd be looking at life without parole in New York. In Texas, hell, they'd execute you, yeah. Electric chair in Texas? Let me Google that.\" She pulled out her phone and opened the browser.\n\nBug Man let loose a weird giggle, and then was appalled by the sound he'd made.\n\n\"You're a very bad person. In this world. In this real world the way it is. You're a monster. Don't you know that? Damn! I was wrong.\" She held up her phone for him to see. \"Lethal injection in Texas. The needle. That's such a weak way to die.\"\n\n\"I don't know what you want,\" Bug Man pleaded.\n\nShe didn't answer directly. Then she said, \"Drink,\" and he drank. Then she said, \"You didn't listen closely enough. I said 'In this world. In this real world the way it is.' But this isn't the only way the world could be. Is it?\"\n\nThe whiskey had started a fire in his throat. And now a dangerous warmth spread from his stomach outward. He flicked his eyes up at her. She wasn't stronger than him. She wasn't armed. He could probably smash this heavy glass against the side of her head. Push her out of the window. It was, what, six floors down to the pavement? What did he have to lose if what she said about him was true?\n\n\"I sent a text just now,\" Lystra said.\n\n\"So what?\"\n\n\"So... wait. Ticktock. Ticktock, yeah.\" She smiled. It was almost playful. \"Ticky tocky.\"\n\n\"Lady, I think I've had it with your shi\u2014\" His mouth still moved, but no sound came out. Because just then a window opened in his brain.\n\n\"Mmmm,\" Lystra said, savoring it.\n\nA second window opened in his brain. A second little TV screen with nothing in view but something that might just be an insect's leg.\n\n\"Is the third one up yet?\"\n\nA third window. This one showed all too clearly the shape he'd come to know as prey and fear as predator. A biot.\n\n\"You ever hear the phrase 'dead man's switch'?\"\n\nHe had. But he felt as if he couldn't open his mouth. Fear seeped into his blood with icy fingers that outraced the warm glow of alcohol.\n\n\"A dead man's switch. They use it on subways and things like that,\" she explained. \"If the subway conductor dies, see, he lets go of the switch and the train automatically stops, yeah. Yeah. That's me now. I'm your own personal dead man's switch. Because if my heart stops beating, guess what?\"\n\nWhen he didn't answer, she bared her teeth, and once again, that skeletal presence seemed to burn through her flesh. \"If I die, little Bug Man, all three of your biots... oh, and they are _yours_ now... die as well.\" She put a fist over her heart, opened it, closed it, opened it, in a mockery of sinus rhythm.\n\n\"What do you want?\" he screamed, losing the last of his self-control. Then, weeping, softly repeated, \"What do you want? What do you want?\"\n\n\"I'm going to create a new world,\" she said, sitting back, dreamy now, her eyes gazing toward the French doors and the city beyond. \"A whole new world. I am its god. But it's a lonely thing, being god; you could ask the real God, if he existed. He'd tell you. He created the world, and then, he was all alone with no one to talk to. He needed friends. But!\" She held up a cautionary finger. \"He needed friends who understood who they were, and who he was, and who held the lightning bolts, and who was there to cower and serve. He needed the love that only comes from those who are afraid. Love me, your god, or burn in hell. I'm offering the same deal as Jehovah.\"\n\n\"You're fucking crazy.\"\n\nHe flinched, expecting her to reveal that awful presence again, but instead she laughed a genuine, happy laugh. \"Crazy? Nah. I'm BZRK.\"\n\n# **THIRTEEN**\n\nKeats pulled away from her. \"What's the matter?\"\n\n\"Wire, Noah. Wire in my brain.\"\n\nIt took him a few seconds to make sense of things. \"You're down in the meat?\"\n\nShe nodded\u2014distracted, scared. She pushed him off her and jumped from the bed. She grabbed at clothing. \"I knew something... I just... Something was weird, so I looked.\"\n\n\"Why didn't you tell me? Why didn't you ask me to help?\" But even as the words were out of his mouth, he knew the answer. \"You thought it was me.\"\n\nPlath didn't answer, her attention was elsewhere. The biot\u2014if it was a biot, how could it be?\u2014had disappeared, and its light along with it. Plath swung her biot left, right, shining her illumination around in the brain fluid.\n\nThen she saw it: a fountain. Instead of water it sprayed red blood cells, the flattened lozenges that were never supposed to fly loose in the cranial fluid. The artery lay like some massive fire hose, coiled across the surface of the brain. It pulsed obscenely with every beat of her racing heart and the blood cells twirled as they flew, then arced away, scattering through the liquid.\n\nThe enemy was cutting into her artery.\n\n\"No!\" she cried.\n\n\"What?\" Keats demanded.\n\n\"He's cut an artery!\"\n\n\"Where? Where?\" Keats grabbed her shoulders, shaking her, forcing her to pay attention and answer.\n\n\"Hippocampus,\" she said, and Keats sent his biot racing to her.\n\nIn Plath's mind she saw the three open windows. Nothing but glass in one. A bleary view from Anya's half-closed eye of the other side of Anya's bed, empty\u2014a slit of light coming from the bathroom. And in the final window that deadly fountain.\n\nShe sent her biot racing toward the deadly leak, clambered madly atop the artery and saw her mistake. It was not coming from the artery itself but from a much smaller vein just behind it. Still dangerous, but the pressure was less intense. Still dangerous, still potentially deadly.\n\nAnd yet, had the foe wanted to kill her, it could certainly have sliced the artery. And there would be more than a few hundred cells flying. He could have done it more than once in the time available. She could right now be swimming through a blood-clouded fluid.\n\nShe had nothing to patch the hole. \"Bring some fibers,\" Plath told Keats.\n\n\"Yes,\" he said tersely. He still held her shoulders. She shrugged him off, turned away, ashamed of her suspicions, ashamed to have him know.\n\nVeins were delicate things, unlike arteries, which managed higher pressures. This vein was about as big around as the biot\u2014translucent, like a worm that never sees sunlight\u2014and it undulated as the blood cells jostled and pushed to make their way back to the heart.\n\nThen she saw the bulge. Something larger than blood cells almost too large to squeeze through the vein. The enemy. It had not just punctured the vein as a distraction, it had stretched the cut to crawl inside and escape.\n\nShe could stab it right through the sausage-casing walls of the vein. She could probably kill it. But she'd be poking holes in her own vein, and the enemy\u2014who had thus far not done anything as drastic as cut an artery\u2014might get frantic, might start slashing from inside the vein.\n\n\"I'm almost there,\" Keats said.\n\n\"I'm going after him,\" she said, without explaining what she meant.\n\nWith her front two biot legs she pried open the elastic flesh of the vein. Blood cells pummeled her face. A white blood cell hit her, rolled down her back, and clung on. It took all her strength to push into the flow, like trying to move uphill against a rockslide.\n\nHalfway in and the pressure shifted. Now it was cells in the vein battering her like dozens of flat stones, pushing her head and upper body after the escaping enemy. She slipped the rest of the way in and fought down the claustrophobia as the vein fitted around her like a body sock. The blood was pushing her along, pushing her toward the distant lungs where oxygen would flow to the cells and they would be fired into arteries for the outward-bound trip.\n\nShe could see nothing but blood cells, red and white, crowded all around her. Her hope was that her prey would soon cut his way out and she would be swept along with him.\n\nBut if he didn't? If he rode this all the way to the heart and the lungs? She could be lost forever in the miles and miles of blood vessels.\n\n\"No!\" she said in sudden panic.\n\n\"I don't see you yet,\" Keats said. He had switched on the harsh overhead light so that the two of them, in various states of dress, looked sickly and frightened.\n\nToo late to get back to her entry point, Plath knew; now she would have to cut her own way out. A second bleeder in her brain. God, she was making things worse. A risk of a second blowout that could kill her, weighed against the terror of being lost forever inside her own body.\n\nSoon this vein would merge with another, and then any exit would cause more blood loss. She had to cut her way out now or lose her chance altogether.\n\nShe stabbed a claw into the vein wall but almost could not hold on against the pressure. Making matters worse, the cell was on her back, oozing its way like warmed Silly Putty into her shoulders, reducing the mobility of her legs. And another now attached to her left hind leg, a fat slug of a thing wrapping its mindless self around her sticklike limb.\n\nPanic!\n\nShe slashed madly at the vein wall, heedless, cut it and felt the blood change speed and direction. Biots are not flexible, so all she could do was use her front legs to cantilever her rear out of the incision.\n\nSuddenly the pressure was too much. Her grip failed. Her biot went tumbling end over end, no way to tell where she was, in or out of the vein.\n\nAnd then, all at once, she was floating free in cerebral fluid, riding like a beach ball atop a stream of cells. She grabbed onto brain tissue and hauled herself out of the current.\n\nFrom there at last she could turn around and see the damage she'd done.\n\nThe leak was twice as large as the first one. Cells were flying out in threes and fours rather than singly.\n\nWith her heart in her throat she grabbed Keats's shoulder.\n\n\"Where are you?\" she demanded.\n\nKeats took her in his arms and held her as his biot crossed into view bearing a half-dozen fibers to begin the job of yet again saving Sadie McLure from her own blood.\n\n# **FOURTEEN**\n\nIt was called the Gyllene Salen, the Golden Hall. It was a vast space\u2014a long rectangle with an impossibly high ceiling, reminding some first-time visitors of a medieval cathedral decorated by Liberace.\n\nAll of one long wall was taken up by five arches opening onto a courtyard. The opposite wall was seven arches. And all of it\u2014virtually every square inch\u2014was covered in just under nineteen million pieces of tile, most of them gold. They depicted various characters from Swedish history\u2014kings and saints, for the most part.\n\nLystra had done her homework and knew all of this. The detail added to the experience. It was a wondrous place and the perfect setting for the annual Nobel Prize ball and banquet.\n\nAt this moment on this dark December night, a handful of Nobel laureates, a slightly larger handful of previous Nobel laureates, the family and friends of said laureates, assorted VIPs and kind-of VIPs\u2014amounting, in total, to several hundred people, all in tuxedos and evening dresses\u2014were seated at long banquet tables loaded down with the sort of china and stemware you don't find at Bed Bath & Beyond.\n\nThis, thought Lystra, would be the point at which she would have to be very careful for her personal safety. First her immediate, physical safety\u2014because what was coming would be violent. But more to the point, this was where the intelligence agencies of the world would focus like laser beams once the event had... well, played out. All the major intel powers\u2014America, China, Japan, the UK, France, Germany, Russia\u2014had prominent citizens here. What was coming would be an event of earthshaking impact. No one cared much what happened to a single actress or a single businessman, and no one would connect any of this to the nosy New Zealand cops who'd had to be eliminated, or to poor, conscience-wracked Nijinsky.\n\nBut the self-murder of the president of the United States, and then the sudden fatal \"illness\" of the Chinese leader, followed by the madness of the Brazilian president, and then this? Even the disaster in Hong Kong. Oh yes: the pennies would begin to drop. The spies and the cops and their ilk would have to be deaf, dumb, and blind not to begin to see a hand behind it all. Right now people were jumpy, worried, on edge, but they still believed the world was just sort of having a bad run of luck.\n\nThere was no luck involved. Well, she corrected herself, there was a bit of luck: the blundering Armstrong Twins had unintentionally heralded what was to come. They had provided the fanfare presaging the main event.\n\nThe Twins, poor silly buggers, were actually helping her carry out her far superior, far cooler plan.\n\nThe thought of them, those hideous freaks, imagining that they were in control. Lystra's lip curled. For a while there had been a freak show with the carnival: a bearded lady, a dwarf who dressed up like a Tolkien character, and a genetically deformed man with hands like lobster claws. They had frightened her then. The bearded lady in particular had tried to be friendly, motherly. The Human Claw, as the lobster-handed man had styled himself, was easier to handle. He just leered, the pervert, until her father had threatened to decapitate him.\n\nWell, let the freaks think they had something. Let the Twins congratulate themselves for killing the president, blundering idiots. The penny would drop for them, too, soon.\n\n\"Girls' night tonight, boys,\" Lystra whispered.\n\nShe wondered if Bug Man, back at the hotel, was watching and could see her on TV. She'd told him to, and while he wasn't the obedient type, he _was_ the frightened type.\n\nThe great thing about tonight was that all the king's horses and all the king's men would never suspect the end goal. They'd be waiting for some kind of blackmail demand. They'd be looking for a rational motive. The fatal weakness of rational people was that they always looked for the rational answer.\n\nThe attendees were mostly through the appetizer\u2014a lobster-and-crab terrine with snap pea mousse, brioche, and edible flower _fantasie_ \u2014when last year's winner of the Nobel Prize in Literature, Miguel Reynaldo, stopped talking about his younger days when he was a hobo, or traveling minstrel, or whatever it was he thought he was, and stared hard at the Swedish finance minister, a dull middle-aged woman seated across from him.\n\n\"I... I've just had the strangest... But it's still there. I'm seeing...\"\n\nAnd at that point the CEO of Spotify said, \"Like windows? Like there's windows in your head?\"\n\nThe two men stared at each other, while those around them formed expressions of polite concern.\n\n\"Something is the matter!\" This came from a second table, from a past Nobel Peace Prize winner, a man credited with saving many lives through nonviolent means\u2014but he was not now seeming nonviolent. He had lurched to his feet, and in the process he had knocked over his very expensive glass of Champagne and caused his dinnerware to rattle and his chair to scrape.\n\n_\"Moi aussi, mais c'est bizarre, \u00e7a!\"_ cried a French industrialist. Then he, too, shoved back from the table as if scalded. He tried to switch to English, but it was a mangled job. \"In my head things. I am see.\"\n\nIt spread quickly. There were a dozen tables, hundreds of well-dressed folk, and some of them, far too many of them, were now whispering urgently or shouting hysterically that something was very odd, something was not right in their heads.\n\n\"Bloody hell!\" the English ambassador cried. \"It's some sort of insect. Oh!\"\n\nAnd then versions of that in a dozen languages and multiple accented versions of English. Those not directly affected were rushing to give comfort. People shouted for doctors. The words _food poisoning_ were spoken. Others said it was drugs. Someone must have spiked the _crabe et homard_ with LSD.\n\nEveryone was talking. The hall was a posh tower of Babel, volume rising, some voices trying to dominate, impose order.\n\nThen came the first true scream. It was a soprano sound, a woman's voice. It began in terror, rose in pitch, roughened, and turned at last into a throaty animal howl.\n\nLystra closed her eyes and savored it. It went on for a very long time, and a smile split Lystra's face, perfect teeth shining in candlelight.\n\nThe room froze, listening, straining to see the source of this delicious scream. Already some were moving prudently toward the exits.\n\n\"God fuck you all! God fuck you all!\" A deep male voice, but frantic not angry, fearful and repeating the curse over and over as the man backed away from the table, plowed into people with outstretched arms. \"God fuck you alllllll!\"\n\nNow the screams and cries, the roars and shouts and canine yelps broke loose in full.\n\nMiguel Reynaldo was laughing and howling like some demented hyena, mouth so wide open it seemed his jaw must dislocate. He dug his fingernails into his face, down his forehead and cheeks, leaving bloody trails behind. Then he threw himself onto the table, twisted onto his back, shrieking all the while, kicking dinnerware and baskets of bread and glasses of sparkling water in every direction, like some great toddler having the mother of all tantrums.\n\nAnd that's when things turned really ugly. Because someone\u2014later identified as a Finnish philanthropist\u2014came up behind the Swedish minister of finance and cut her throat ear to ear with a table knife.\n\nAnd when she had sunk to the floor\u2014gurgling, dying, spraying crimson across white linen\u2014he kept sawing away, brushing aside her weak defensive efforts, sawing away at her trachea.\n\nPanic!\n\nThe screams were general now as people rushed to the exits, crushed into one another in their desperate desire to get the hell out of that room, but not all those in the crowd were behaving normally. A past Nobel laureate for physics had stripped off his clothing and was peeing on anyone within reach.\n\nLystra, too, began to scream and wave her hands in the air. And she grinned, widely, not only because that's what would be expected of a madwoman, but also because it was all just so wonderful.\n\n\"I bring you madness!\" she yelled, and laughed, but kept a careful eye on all around her as she backed toward the nearest exit.\n\nBack in the center of the room a man later identified as one of the world's great scientific minds was squatting on a table defecating, while around him madmen and madwomen screamed and threw things, attacked one another with cutlery and broken glass, clawed their own eyes, or simply huddled in corners yelping at imaginary spirits.\n\n\"Madness!\" Lystra yelled as she reached the door.\n\nIn the space of five minutes the Nobel banquet had become a blood-splattered insane asylum.\n\nWould Bug Man appreciate it? Would he get it? Probably not. Buggy was useful for some things, not for others. He would not be enough to occupy her own Eden. Someone smarter would be good. Someone more subtle. Someone who would chafe even more and thus be even more completely subjugated in the end.\n\nSadie McLure. God, the irony would be wonderful.\n\nNot everyone was driven mad at the Golden Hall. Most were not, though it was hard to differentiate them as they ran from the hellscape splattered with blood.\n\nLystra Reid's gown\u2014Prada, very chic\u2014was already red, so the blood didn't show. However, her shoes\u2014Christian Louboutin pumps\u2014were absolutely ruined.\n\nBut she could not resist, as she fled the room, crying out in her pretended madness: \" 'As flies to wanton boys, are we to the gods. They kill us for their sport.' \"\n\nThus did Lear quote from _King Lear_ as she kicked off her shoes and ran barefoot into the cold night, laughing and twirling as snowflakes fell.\n\n# **FIFTEEN**\n\nKeats handed Plath a cup of coffee. Her hands were shaking. It was morning and she'd had no sleep. They stood in the kitchen, Keats in some soccer team jersey and sweatpants; Plath in an unattractive sweater, panties, and socks.\n\n\"It's plugged. The big one.\" Keats sipped his own coffee and looked at her over the rim as he took a second sip.\n\n\"What?\" She was confused for a moment, thinking he was talking about the coffee.\n\n\"The second hole. It's plugged. I don't think it was all that dangerous, anyway, but I've patched it, the lymphocytes are keeping it clean, and I can see clotting factor forming nicely.\"\n\n\"Thanks.\" She sent him a very serious look and added, \"I don't say that enough, do I? Thanks.\"\n\n\"Hungry?\"\n\nShe considered it. \"Yes, I am.\"\n\n\"I'll fry some eggs and bacon. No bangers, I'm afraid. You Americans don't really do sausages very well.\"\n\n\"You can cook?\"\n\nHe made a small laugh. \"Oddly enough, I don't actually live at Downton Abbey.\" Then, thinking that may have sounded resentful, he smiled and touched her shoulder. \"I learned a bit of this and that. Enough to fry an egg and make toast. If we have bread.\" He searched the cupboards. \"Yes, we do have bread. But no beans or tomato.\"\n\n\"Beans?\"\n\nHe sighed. \"The thing you Americans so proudly think of as breakfast is a sad affair compared to a proper full English breakfast. Eggs, bacon, toast, black pudding, mushrooms, beans, and a nice grilled tomato. And coffee, of course, unless you prefer tea.\"\n\n\"Black pudding?\"\n\n\"Given your adventures tonight, it's maybe best not to discuss black pudding.\"\n\n\"No?\"\n\n\"It's also called blood sausage.\"\n\n\"Ah. Yeah. Enough blood.\"\n\nHe set a rectangular grill on the cooktop and turned on the fire. He peeled strips of bacon from the package and started the flame beneath a saut\u00e9 pan. In seconds the bacon was sizzling, and both the familiar sound and aroma made Plath's mouth water.\n\n\"Were you going to tell me?\" he asked, once he had things organized and under way.\n\nShe stalled for a moment by sipping her coffee. She didn't have to tell him. But he had possibly just saved her life.\n\n\"I was planning an armed attack. I was planning to kill people. I met with Stern, without you. I asked him for... and he said...\" She sighed, lost momentarily. \"I never really asked myself whether it was the right thing. I have this picture in my head....\" She let that sentence peter out, not willing, still, to tell him everything. Not the things that would make him despise her.\n\nHe nodded. \"You started to suspect you'd been wired.\" He sighed, turned the bacon, pushed the toast down, and used his spatula to keep the eggs from spreading out. \"And you didn't tell me because you thought I might be the one doing it.\"\n\n\"It's the world we're in, isn't it?\" she asked.\n\nHe nodded. \"It's the world we're in.\"\n\n\"But it wasn't you.\" She took his hand, which after a few seconds he took back to press the spatula down on the bacon.\n\n\"Which leaves who?\" he asked.\n\nShe glanced toward the door, wondering if anyone was on the other side listening. \"Wilkes. Billy. Maybe even Vincent, maybe that affectless thing he's doing is just camouflage. Or it's someone else with BZRK, someone not from our group. We're just a part of it, after all.\"\n\n\"You're sure it was a biot, not a nanobot?\"\n\nShe played the memories back. \"Not a hundred percent.\" She tried out various values in her head. \"Seventy percent sure. But if I'm planning on blowing up, um, attacking the Tulip... the Armstrongs wouldn't be doing that; they wouldn't be wiring _me_ to kill _them_.\"\n\nHe served the food onto two plates, and they sat at the counter and ate, side by side, leaning so that their shoulders would touch.\n\nFinally Keats spoke. \"If you were wired, why? I mean, what you're pointing to are kind of, I don't know, moral changes.\"\n\n\"Have you noticed anything different with me?\" she asked, afraid of the answer and covering it with transparently false nonchalance.\n\nHe thought it over while chewing bacon. \"You're questioning Lear and BZRK less. I mean, maybe it's just that you've got more responsibility. But you used to be more suspicious, I guess. More critical.\"\n\nShe thought about that. \"Yeah, maybe so.\n\n\"I'll help you look for wire.\"\n\nPlath hesitated and felt herself blush. She filled her mouth with egg. If she was still wrong, if somehow Keats was the person running the biot, or at least knew about it, then he would never find any wire. It could be a way to determine his loyalty. If he found\u2014\n\nShe cut herself off in midthought as reality dawned: there was no way to determine loyalty. Ever. Keats might be loyal today and some nano creature of either side might be wiring him to switch sides tomorrow.\n\n\"It's still inside me,\" she said. \"Maybe it's trapped in my liver or whatever, but it's still in me.\" She nodded and wiped her mouth, then set her plate in the sink. \"Yes,\" she said decisively. \"The aneurysm will keep. Help me look for wire. I want to know where it all is. And help me kill this thing, whatever and whoever it is.\"\n\n\"And pull the wire?\"\n\nShe didn't answer. She didn't answer for so long that Keats thought she must not have heard him.\n\n\"And pull the wire?\" he repeated, more insistently this time.\n\nShe shook her head. \"Not yet, Noah.\"\n\nAfter that she wouldn't look at him.\n\nThe other person making an interesting discovery was Imelda Suarez. _Lieutenant_ Imelda Suarez, dammit. Hopefully there was an extra paycheck to come with that.\n\nThe _Celadon_ , the mother ship, had dropped anchor six miles out from Cathexis Base, hundreds of miles from McMurdo. The ice was thick and crusty here, and no way the ship could edge up to the dock, not for another month at least. She had gone ashore in the _Jade Monkey_ , running easily up onto the beach and roaring along until the LCAC found its home, a hangar and refit facility they all called the Blower Barn.\n\nAs soon as she'd done the inevitable paperwork, Suarez headed toward the Office\u2014the administrative building. Things at Cathexis Base tended to be named simply, usually by function, but with an occasional touch of wit: the Blower Barn, the Chiller (a poorly heated dorm), the Toasty (the newer, warmer dorm), the Club, the Link (the satellite dishes), the Office.\n\nIn the center of the base, acting as a sort of central park, was a glass dome raised up on a skid-mounted platform. It was seldom transparent\u2014condensation saw to that\u2014but it clearly housed green, living things, ranging from small elm trees to tall grasses to irises and roses. But for the most part the Andalite Dome, or AD, as it was called for some obscure reason, was more practically planted with cabbages, broccoli, romaine lettuce, carrots, and onions.\n\nThe produce wasn't anything like enough even to feed Cathexis Base, but it helped, and it was the place to go when you felt the ice start to close in on you.\n\nAs Suarez did now. Seeing the smudge of green inside the sweat-dripping bubble, she felt herself drawn to it, and decided checking in at the Office could wait. Getting into the dome was a process\u2014you had to shed your gear and walk in wearing a T-shirt and pants alone. And you had to pass through a double airlock.\n\nIt was while in the airlock that she ran into Charlie Bronk.\n\n\"Coming or going?\" he asked her.\n\n\"Just got in,\" she said. Bronk was a small man with a too-tough name. He was a mechanic who often worked with Suarez. They weren't friends, but they were cordial.\n\n\"I'm supposed to head out to Forward Green,\" he said. \"One of their cats is wonky, needs a new fuel injector.\"\n\n\"There's no one out there can do it?\"\n\nBronk laughed. \"At Forward Green? Pff. Those are scientists and God knows what all out there. Sally Wills is the only one can turn a screwdriver, and she's on an evac to Wellington.\" He lowered his voice. \"A psych thing. She lost her shit.\"\n\n\"Damn. Sally Wills? The redhead?\"\n\nBronk nodded. \"I don't suppose... I mean, I wouldn't ask, but it's my son's bar mitzvah and I'm missing it. I was going to Skype.\"\n\n\"You can't Skype from Forward Green?\"\n\n\"There's no communication in or out of Forward except to here. Security.\"\n\nShe was on the verge of asking him why there would be secrecy, but thought better of it. That was the kind of question that might be thrown in your face some day if there was a problem. Cathexis Inc. might not be military, but when it came to secrecy, they sometimes went the military one better.\n\n\"I could do it,\" Suarez said with a shrug. \"Of course you'll owe me. And I don't mean you cover for me on cleanup. I mean something more like you pull a shift. Three shifts.\"\n\nThey agreed on one shift and a round of clean-up duty. And that was how Suarez ended up on a loud chopper heading almost due south. It was a hellish ride. The wind had come up. In fact, at the halfway point the pilots discussed turning back. Antarctic weather wasn't something you took risks with.\n\nBut satellite imagery gave them a nominally clear hour before the hammer came down, so they went forward.\n\nIf Cathexis Base was businesslike and humane, Forward Green was a bizarre cross between survivalist compound and Ritz-Carlton resort. From the frosty window of the chopper she could see that the buildings were arranged in a sort of diamond around what was very certainly the only swimming pool on the continent. The pool was covered of course, and as sweaty as the Andalite Dome at Cathexis Base. It was an ostentatious symbol of wealth, because water\u2014actual, liquid water\u2014was one of the rarest and most expensive of commodities. It spoke of a profligate use of power\u2014the heat to keep the pool warm, the light to make it shine, the lift capacity to bring it all together in this place.\n\nIt was built aboveground, of course\u2014the shifting ice would have crushed anything cut into it. It was covered by a plastic roof that formed three peaks, vaguely reminiscent of the Sydney Opera House.\n\nSuarez guessed that the power source had to be a nuclear reactor. But how had that been approved? The green movement had made peace with nuclear power, but here? On the _ice_?. And in private hands?\n\nOnce she'd looked beyond that eye-popping artifact of another world, Suarez took in the rest of the place. The buildings were identical\u2014seven three-story ski-mounted structures, with an empty slot where an eighth building might go someday.\n\nThe windows aimed out toward the ice were small, with metallic shutters that could be mechanically closed against the wind. The windows facing in, toward the pool, were larger than anything she'd ever seen before in energy-conscious Antarctica. Though they, too, were equipped with strong steel shutters.\n\nShe imagined what the place would look like locked down, with all those shutters closed. And then she noticed the four half-buried towers two hundred yards out from each point on the diamond.\n\n\"I'll be damned if those aren't gun emplacements,\" she muttered. Not that she saw anything like weapons.\n\nA mile away to the south and barely visible because the wind was now blowing crystals of ice through the air was a larger structure\u2014long, low, and unadorned\u2014that could only be some sort of hangar.\n\nThat's where the souped-up hovercraft would be.\n\nIt hit her then full force: they didn't have anyone who could fix a fuel injector? At a facility where they were building jet-powered hovercraft? Bullshit.\n\nShe hadn't cleverly exploited an opening to reach Forward Green: she'd been lured there.\n\nLystra and Bug Man left Stockholm not by way of Arlanda Airport but by car, to a private airfield fifty miles out of town, out into the landscape of snow and dark pine trees.\n\nBug Man had only a light parka that George had supplied, in no way sufficient to deal with a Swedish winter. The run from the car to the welcoming light of the jet was enough to freeze him, but Lystra seemed indifferent, still wearing her blood-drenched red dress\u2014though she had swapped her shoes for a pair of shearling boots. _She would look almost cute_ , Bug Man thought, _if she were younger. And a whole lot less insane_.\n\nIt was warm on the plane, which took off within minutes of the door closing, soaring up into the night.\n\n\"Look!\" Lystra said, and drew him out of his seat to look through the window on her side.\n\nThe sky was an eerie light show, green against black, the stars all rendered irrelevant. The green was a veil, translucent, shimmering.\n\n\"Aurora borealis,\" Lystra said. \"The northern lights.\" She nodded. \"We get them in the south sometimes, too. You'll see.\"\n\nBug Man watched for a while, acutely aware of her nearness. Crazy, yes. Too old for him, yes. Still...\n\nShe must have sensed it because she laughed, an almost girlish sound, and pushed him back to his seat.\n\nBut then she stood up, turned her back to him, and said, \"Help me with the zipper.\"\n\nBug Man swallowed hard. Okay, yes, he'd thought about it. But seriously? With a woman who had his sanity and life in her hands? He'd watched the TV as instructed, and he had seen the Nobel madness. He had even seen a fleeting shot of Lystra dancing and twirling away from the carnage.\n\nGod only knew what the woman would do to him if he disappointed her.\n\nHe drew the zipper down. It snagged halfway and he had to tug at it for a bit, all the while with his nose just inches from her back.\n\nMost of what he could see was tattooed. Blues and reds and greens. He couldn't make out the patterns, except that most of it seemed to be faces. He saw eyes staring, mouths twisted in screams.\n\n\"Damn,\" he whispered, and winced, hoping she hadn't heard.\n\n\"You like my ink?\"\n\n\"Uh, yeah,\" he said too quickly.\n\n\"Want to see more? Want to see my latest one?\"\n\nHe froze. Just absolutely froze. She let her dress fall to the floor.\n\n\"Oh... shit,\" he said. There were faces on her back, on her behind, on her flanks. Not every inch was covered\u2014maybe half of the available flesh.\n\nMore than enough. It was a horror show.\n\nFaces. Men, women, one that might even be a child. All in agony or rage or some combination of the two.\n\nHe couldn't breathe. He did not want to see more. He did not want her to turn around.\n\nBut she did.\n\nSlowly, slowly; savoring his fear, the fear she could hear in his raspy breathing, in the way it caught in his throat.\n\nHer front was even more horrific. Faces from hell were staring out at him. Two were new, still healing.\n\nShe pointed to the freshest-looking one with a coyly bent finger. She was being cute. She was playing with him. But oh, God, there was no way to fake this, no way for him to force his features into anything like a pleasing expression.\n\n\"That's a man named Janklow. He didn't want to sell me his medical testing company. Because of him, yeah, the whole game was delayed.\"\n\nHer breasts were just inches from him. Her eyes were the eyes of a rabid dog, focused on him with an intensity that made him tremble.\n\n\"Don't you want to know who they all are?\" she asked, and the hard, sadistic voice he'd heard before had replaced the cute come-hither tone.\n\nHe managed to shake his head. No. He didn't have a single question. No, he didn't want to know. He wanted to be back in England. He wanted to be back at Tesco, shopping for his mother's onions. His fists were clenched so tight they ached.\n\n\"Sure, you want to know,\" she said. \"These are all the ones whose lives I have taken from them.\"\n\n\"That actress? Do you remember, yeah? You must have read about it, seen it on TV? She dug her own eyes out with a knife. It was intense, Bug Man, very intense.\" She tapped the other still-healing tat, on her sculpted hip bone. America's Sweetheart in blue ink, bleeding red blood from her gouged eyes.\n\n\"What did she\u2014\"\n\n\"What did she do? Oh, she wouldn't even remember, didn't recognize me at all, why would she? I was in a hospital for a while for... stress?\" She threw her head back and laughed. \"Stress? I was crazy as a loon.\"\n\n_Was? Past tense?_ Bug Man wanted to ask. But not as much as he wanted to go on living.\n\n\"My pap and mam, that's what they had me call them. My guardians.\" She spit the word. \"The losers my daddy dumped me on. They started talking to me after I killed them.\" She covered each breast with a hand, lifted them slightly so he could see the faces tattooed there. They appeared to have been crushed. Their eyes were...\n\n\"They would talk to me. 'Be a good girl, Lyssie. Pray to the Lord for strength, Lyssie.' Sometimes though, they would give me useful business advice.\" She frowned at the memory, then thankfully she turned away, walked to a narrow closet, and pulled out jeans and a sweatshirt. The sweatshirt was green with a big letter _C_ over an outline of Antarctica.\n\nBug Man breathed a shaky sigh of relief. Ugly sweatshirt, but so much better than looking at that torture chamber on her body.\n\n\"I went crazy, yeah. Into the nuthouse with me. I was rich by then, had my businesses going pretty good, but yeah, off the deep end, yeah. Meds did nothing; they still talked to me.\"\n\n_Your conscience, you sick bitch_ , Bug Man thought.\n\n\"Not my conscience,\" she said, for all the world as if he'd said it out loud. He had to resist the urge to cover his mouth with his hand lest he say something to get himself killed.\n\n\"Psychotic break. Not functional. Everything falling apart... and he came back. Daddy. He said he would if it came to it, if, you know... if. I guess he thought I might eventually get weirded out over his killing my mother. Drink?\"\n\nShe poured them each several fingers of bourbon. Bug Man gulped his down. He needed to pee desperately, but this was so not the time to ask to be excused.\n\n\"Nuts, yeah. So back he came, my daddy. And he said, 'I know about this man, this scientist. He's doing some weird stuff with nanotechnology. Maybe he can help. Only he refused, you see, and Daddy couldn't kill him and neither could I, because, well, he was protected.\"\n\n\"Burnofsky?\"\n\n\"Burnofsky?\" She shook her head. \"But good guess. No, it was Grey McLure. He was just starting\u2014freaking out over his wife dying and he couldn't save her with his new toys. Then his daughter and the aneurysm, yeah. Yeah. People went crazy, though, see? Off this new thing he'd invented. This _biot_!\"\n\nThe word came out in a roar that made Bug Man jump back.\n\n\"This biot. So, maybe, yeah, maybe if a dying biot would make a sane person crazy, hey. Maybe, right? Yeah? Maybe the other way, too.\"\n\n\"Jesus. They gave you a biot.\"\n\nShe nodded. \"Yeah. Yeah. My very own. And then they killed it. And you know what? It worked. It _worked_. I wasn't crazy anymore.\"\n\n_The hell you weren't_ , Bug Man did not say.\n\n\"The tattoos stopped talking to me. I could cope, yeah. I could manage. Making tons of money. And then I saw it\u2014saw the game. Saw the way I could do it. Make a whole new world, yeah.\"\n\nShe fell silent then, staring down into her drink.\n\nBug Man stood on wobbly legs and went to the bathroom. In the glaring fluorescence he stared at his own face as if staring at a ghost. He was shaking. He felt an urge to sit down and empty his bowels, but who knew what the crazy woman would do?\n\n_Oh, that's right_ , he told himself. _Not crazy. No, she was all cured_.\n\nHe peed and washed his hands, and having used up all his stalling tactics went back out.\n\nLystra Reid had not moved a muscle.\n\nHe sat down.\n\nAnd unprompted she said, \"Oh, and the actress? Sandra Piper? Bitch cut me off in traffic.\"\n\n# **ARTIFACT**\n\n**Plath:** I need Caligula.\n\n**Lear:** Name the place.\n\n# **SIXTEEN**\n\nThe news was all about the Nobel madness. Twenty-four hours a day. MSNBC, Fox, CNN.\n\nOnly the BBC made a connection to the bizarre case of the New Zealand cops.\n\nOnly the Web site Buzzfeed made a connection between the Nobel madness and the inexplicable suicide of Sandra Piper.\n\nEveryone, though, connected it to the bizarre death of the American, Chinese, and Brazilian heads of state.\n\nFear was spreading. A sharp observer would already be able to spot a wariness in people's eyes and in their words. There was a feeling in the air.\n\nFear. Like the scent of smoke. Like the distant rumble of tank engines and clanking tracks. Like sirens in the night.\n\nThe theories about the cause were: food poisoning, mass hysteria, and some sort of terrorist attack using a form of nerve agent.\n\nOnly Cracked.com actually listed nanotech on its \"8 Ways to Explain the Big Brain Meltdown.\"\n\nThere were several loops of footage that ran more or less continuously online and on TV. One was a cell-phone video of a scene of madness from inside the Golden Hall. A second showed a bloodstained woman in a party dress rushing from the hall amid a panicked crowd, then suddenly launching herself at a passing woman and biting savagely into her neck. Another showed a former American secretary of state waving madly at invisible flying enemies.\n\nOf course there were also clips of the new president looking solemn and vowing to give the Swedish government any assistance they required. Ditto footage of the British prime minister, the French president, and a long list of folks who had no idea what was going on, all vowing to get to the bottom of it.\n\nRye ergot. That was the first guess. Rye ergot, a disease caused by fungus that grows on some foods and can cause symptoms similar to an LSD overdose.\n\nTests for rye ergot were all negative.\n\n\"Just like Nijinsky,\" Keats said. \"It's all connected.\" He was watching the BBC coverage. \"It's all the same bloody thing, isn't it.\"\n\nHe was talking to no one. Plath was out, and though a part of Keats was with her\u2014sitting on his hands, waiting for a cue\u2014he felt alone. Abandoned. Both here and there. Both large and small. Slumped into his chair and on edge, ready for a race. Not for the first time, he wondered mordantly what he had to fear from madness. Wasn't this already madness?\n\nBilly was absorbed in a video game. Vincent was there, staring, almost forgotten by Keats.\n\nKeats sat before the television, watching through his two eyes, and seeing the windows in his head, watching from other eyes. \"It's all one. But who?\"\n\nThe voice when it spoke surprised him. What the voice said was chilling.\n\n\"Lear,\" Vincent said.\n\nKeats turned to look at him. He was still showing nothing, Vincent. A blank expression, sad eyes. Only his brow seemed to speak of any emotion; if tension can be called an emotion.\n\n\"Lear?\" Keats said. \"Not the Armstrong mystery weapon?\"\n\n\"Games,\" Vincent said, as though that word should mean everything and the saying of it had exhausted him.\n\nKeats couldn't quite think of what to say. On the one hand, this was Vincent. On the other hand, this was mad Vincent. Shattered Vincent.\n\n_Seventy percent Vincent_.\n\n\"You want anything to eat?\" Keats asked. \"I was thinking of ordering Chinese.\"\n\n\"Did Lear just see it?\" Vincent mused, ignoring Keats. \"Or has he known all along? Should I ask him?\" There was something almost like a smile on Vincent's lips. \"There will be more.\"\n\nKeats might have pursued it, but a few thousand feet away, his much smaller self saw that the moment was fast approaching. He readied himself to confront the lion in his den.\n\nWith Nijinsky dead, Burnofsky was off his leash. He had no way of knowing this\u2014not yet\u2014but there was no longer a biot in his head. Or to be more accurate, there was still a biot attached to his optic nerve, but no one was peering through those biot eyes any longer. The biot had no real brain of its own, nor did it have instincts. It continued to live, but only to live. Immobile.\n\nBurnofsky had a Post-it note. He wrote on it: _Floor 34. Viral research_.\n\nHe held this note up in front of his eyes. Held it there for far longer than it should take to read it. But he guessed that whoever was running the biot in his head\u2014and he believed it was Nijinsky\u2014would not be focused on his every moment.\n\nHe was careful in the way he did this because Burnofsky knew perfectly well that his lab was under surveillance. He had come to accept that fact. Privacy was dead, anyway, particularly if you worked for the Armstrongs. But he knew the camera locations and angles. Sometimes he forgot\u2014he had a worrying sense that his little self-inflicted wound of the other day might have been observed.\n\nWell, the Twins had seen worse, hadn't they? They'd seen him puking his guts out. He was morally certain that they'd been watching one dark night months earlier, back before he'd been wired, when he had sat for twenty minutes with a loaded pistol in his hand trying to get up the nerve to put the barrel in his mouth and pull the trigger.\n\nSo what was a little cigarette burn, eh? Better than the opium pipe, right? Better than the vodka bottle. He wasn't drinking now, not that he'd made some lifelong decision to quit; he just wasn't drinking right now. Or snorting coke. Or smoking opium.\n\nNo, he was all cleaned up. He laid the Post-it note down in the ashtray in front of him, shielding it with his body from the hidden camera. Then he began to light a cigarette and in the process burned the note to ashes.\n\nHe drew in the smoke of his cigarette and wondered if he would get to the end of it without burning himself.\n\nThe burning was\u2014\n\n\"Shit,\" he muttered. Nijinsky would think it was a reference to a _computer_ virus. He wouldn't understand that Floor 34 was a crash program involving actual viruses. _Biological_ viruses.\n\nBurnofsky had only stumbled upon the information by chance. He was hiring a new engineer and happened to speak to one of the people in human resources, who smiled, told him he had plenty of available engineers, and thank God at least Burnofsky wasn't looking for a virologist.\n\nVirologist. A scientist specializing in viruses, of course. And why was anyone at Armstrong Fancy Gifts Corporation working on biologicals of any kind?\n\nIt had to be Floor 34. Burnofsky knew most of what AFGC was into, he should have known about a biological nano program of any sort. Were they working on their own version of biots? Were they preparing to toss his nanobots aside? The possibility worried Burnofsky a bit.\n\nAs always when he was anxious, his thoughts went to opium, and then to his work, and then to Carla. And from there to the Great Forbidden Memory.\n\nBurnofsky knew exactly what they had done to his brain. He knew. He was a scientist; he had wired many a person, done to others what had now been done unto him. He knew that tiny wires in his brain had been used to create shortcuts\u2014sending thoughts around the usual circuitous neural pathways to hook into the most intense sensations.\n\nIn other words, he knew that Nijinsky had connected memories of his daughter's death to pleasure centers. He knew Nijinsky had made his greatest guilt into a sick and disturbing fantasy. He knew that. He could picture the wire in his own brain. He could imagine just how Nijinsky had done it.\n\nBut that changed nothing. It did not stop the physical reaction when he thought of that most awful of days.\n\n_I killed her_.\n\n_And I'm thrilled_.\n\nAt first he had thought of using his own nanobots to go in and rewire himself. But of course Nijinksy would see him. Burnofsky could take Nijinsky's biot\u2014Burnofsky wasn't quite Bug Man or Vincent when it came to nano warfare, but he was confident that he could outfight Nijinsky.\n\nBut somehow... No.\n\nSomehow the will to fight back always seemed to dissipate.\n\nWas this still more wiring? Probably. If so, it was effective. He would form the desire, formulate a plan, start to get his resources in order, and then, then, then something.... It would all just leak away.\n\nThe answer was no. He would not finish this cigarette by putting it out in the ashtray.\n\nHe took one long, final pull on the cigarette butt\u2014it was down to the last inch\u2014lifted his shirt, and stabbed it into his stomach.\n\nThe pain was staggering. The smell of burned flesh was like opium, somehow, a narcotic that turned the pain into a dream, a swirling unreality.\n\nAnd most of all, it took his mind off Carla. Because despite all of Nijinsky's careful work, Burnofsky felt that if he had to endure that horror-excitement one more time, he would find his gun and finally do it.\n\nThe HNDS\u2014hover-capable nanobot deployment system\u2014or \"Hounds\" were roughly triangular in shape and no bigger than a paper airplane.\n\nThe original drone architecture was under development for the U.S. military and the CIA. Stealthy, relatively quiet, wonderfully maneuverable, their only real drawback was that their range was limited to twenty miles. The military wanted a seventy-five-mile range, and the CIA weren't interested unless they could be flown at distances up to five hundred miles.\n\nSo the drones\u2014once designated the hover-capable surveillance system (HOSS)\u2014had been repurposed. Twenty miles might not quite be the thing for the soldiers or the spies, but it was perfectly adequate for use in massed preprogrammed attack by nanobots.\n\nThe Hounds came for Mr. Stern as he was picking up his morning bagel at Montague Street Bagels in Brooklyn. It was a short walk from his home, and the McLure Security car and driver would be waiting across the street.\n\nThere were twenty thousand self-replicating nanobots aboard the Hound piloted remotely by a tech in the bowels of the Tulip. The Twins watched on their eternal monitor. The nanobots themselves were of course not twitcher run. They had been programmed by the Twins via the app. These nanobots had been given a simple set of instructions: to multiply as soon as they encountered a source of carbon. To continue to do so for exactly forty minutes. Then to commit mechanical suicide and stop.\n\nAs Stern was crossing the sidewalk the Hound swept down Henry Street before executing a sharp right onto Montague.\n\nStern bit into his bagel. The cream cheese oozed from the sides and he licked a dollop before it could fall away.\n\nAnd then he heard something strange. Like a ceiling fan, but with blades going very fast. He even felt the downdraft and looked up to see its source. The Hound was just six feet over his head.\n\nThe nanobots fell in a cloud, like dust.\n\nStern ran to the car, still clutching the bagel. The driver saw him, started to jump out to open the door for him, then saw the urgency on Stern's face, so just released the lock and started the engine.\n\nStern reached the door just as he began to feel a burning sensation on his scalp.\n\nHe piled into the car and yelled, \"Some kind of drone!\"\n\nThe driver turned around and blanched visibly. \"Jesus, boss! Your head!\"\n\nStern reached past the driver and yanked down the visor mirror. In the narrow rectangle he saw that his scalp was red with blood.\n\n\"Drive!\" Stern shouted. \"To McLure Labs!\"\n\n\"What's happening?\" the driver cried.\n\nStern tried to answer, but at that moment the nanobots had chewed through his cheek and were tearing into his molars, and the sound that came out of the security man was not decipherable as anything but a cry of agony.\n\nThe driver yanked the car into traffic, leaned on the horn, and forced his way past a parked UPS truck.\n\n# **SEVENTEEN**\n\nCaligula found himself almost nervous. How strange. Plath was just a girl, after all.\n\nHe remembered the first time he had really met her, in a small but vicious battle at the Tulip. He'd liked her. He'd thought he saw some inner strength in her, but it had never occurred to him that she would end up running the New York cell of BZRK. Vincent had seemed bulletproof\u2014an odd concept for Caligula to think of. But Vincent really had seemed indestructible.\n\nFor a while after Nijinsky's fall from grace Caligula thought Lear might place the burden of leadership on him. But no. Of course not. Caligula had his purpose in life, and it was not shepherding a gaggle of kids. He was useful to Lear, but only as a killer. And less and less useful at that. Lear had found other ways.\n\nNijinsky, poor bastard. A clean bullet would have done the job. No need for what he endured. No need for that cruelty.\n\nHe wondered what Plath would ask of him. Would she ask for his help in bringing Burnofsky in so that he could be infested with a new biot?\n\nHe hoped she would not ask him about Lear.\n\nBut of course she asked about Lear.\n\n\"It seems absurd to call each other Caligula and Plath,\" Plath said.\n\nPlath had picked the meeting place, and she was waiting for him when he arrived. It was public but not: a dark booth in a dark bar. It was against the law for her even to be sitting here across from him. But there was a law for regular minors and then there was a very different law for minors who could hand a fistful of hundred-dollar bills to a concerned bartender.\n\nIt amused Caligula that she had even found this place. It was classically male, a dive bar in a pricey Manhattan neighborhood. An easy walk from the safe house, which showed caution. After all, Sadie McLure had changed her hair, but she could still be recognized if a paparazzo spotted her. She had minimized the odds of that. Smart girl.\n\nHe took in the surroundings as he did every few minutes, checking for changes in personnel, in position and posture. There were a couple of hipsters at the bar imagining themselves as latter-day Kerouacs. A tired-looking woman who was almost certainly a hooker. Three loud businessmen saying things like, \"So I told him, 'That is not something I'm comfortable with.' I mean, maybe he doesn't give a shit, but I do.\" After a few more drinks they'd be complaining about their wives and their kids.\n\nBut that's not who Caligula watched out of the corner of his eye. It was a woman, thirty-five maybe, in an inexpensive business suit with slacks, sensible shoes, and khaki raincoat. She had brown hair cut short, but not so short as to be fashionable. She ordered something he didn't overhear but that caused the bartender to look wary. It came clear and fizzy in a tall glass: sparkling water.\n\nIf she wasn't some kind of cop, she was doing a very good impression of one. She confirmed the impression by avoiding looking at Caligula. It was a fact of life that any normal person would look at him.\n\nHad it come to this? Were even the cops on the trail? It was one thing being shadowed by Armstrong people and by Plath's security people. It was a different matter entirely when secrecy was so compromised that FBI or intelligence or even NYPD were watching.\n\nThings were coming to an end. One way or the other. But wasn't that what Lear wanted?\n\n\"It does seem ridiculous,\" Caligula allowed.\n\n\"Call me Sadie.\"\n\n\"Call me Caligula.\"\n\nThat earned him a wintry smile.\n\nHe did not lean toward her. He had not shaken her extended hand\u2014she would understand why. Caligula might be a part of BZRK in his own way, but you simply did not trust people armed with biots. A fleeting touch was all it took to send the tiny little beasties toward his brain.\n\nHe was nursing a beer in a tall, sweating mug. He casually dragged the mug across the table, left to right, leaving a trail of water behind. A barrier to the tiny bugs.\n\n\"I never thanked you. For that first time.\" Plath nodded at him, a regal move that seemed natural for her. \"You saved our lives.\"\n\n\"You're welcome,\" he said. And waited.\n\n\"I need you,\" she said.\n\n\"For?\"\n\n\"Lear wants the computer servers in the Tulip destroyed.\"\n\n\"They'll have backups.\"\n\nShe shook her head. \"We don't think so. They're so paranoid they keep several systems cut off from one another. We've had access to many of their networks, but some of their computers are entirely unreachable from the outside. No Internet links at all. No phone lines. They might as well be something out of the 1980s.\"\n\nHe nodded, accepting this as a likely fact. \"It's a large building. They are well guarded. This is not a movie; I could not do it alone, or do it even with your people.\"\n\n\"How _could_ you do it?\"\n\n\"By destroying the entire building.\"\n\nShe stared at him. He watched her eyes. Interesting. Her pupils had expanded. A pleasure reaction. But then her eyes had narrowed, and she had drawn away. Of course: she was conflicted.\n\n\"Destroying...\"\n\n\"There will be natural gas pipelines in the basement. If you were to fill some of the sublevels with that gas and ignite a spark, it would very likely collapse the entire structure.\"\n\n\"Like...\"\n\n\"Like what, Sadie?\" He knew like what. He had a pretty good idea what was being done to her. He could guess Lear's direction. But he wanted Sadie to say it.\n\n\"Like the World Trade Center. Like 9\/11.\"\n\n\"Yes,\" Caligula said. \"We could obliterate the building itself. It would kill everyone inside. Which is what you would want, Sadie. You would _want_ all the scientists in there to die. It would set back nanobot technology several years at least. It would be the practical end of Armstrong Fancy Gifts. By the time they recovered, someone else would have developed the same capacity. Someone perhaps a bit less... visionary?\"\n\nThere was a TV on over the bar. It showed what every screen in the world was showing: the Nobel madness. Cut to the American president's suicide. Back to the Nobel madness. Cut to the Brazilian president.\n\nPlath was shaking her head. \"No.\"\n\n\"If you destroy the servers and let the scientists walk away\u2014\"\n\n\"It's not just scientists in that building. There are regular people. Clerks and janitors and people who just answer the phones.\" She was pleading with him to find a different answer.\n\n\"It would be mass murder. It would make you one of the greatest terrorists in history.\" He watched her eyes. She was repelled. She was sickened. But she was not surprised. So that idea had definitely already occurred to her.\n\nAnd she did not get up and walk away.\n\n_Jesus Christ_ , Caligula thought, this _is the new way, the new reality_. Sixteen-year-old girls could be made into terrorists. They could be wired for mass murder.\n\nPlath, for her part, could see it in her imagination. She could see that phallic monstrosity of a building collapsing into the fire that raged at its base.\n\n_My God_ , she thought, _it_ could _be done_.\n\n\"We can't do that,\" she said. To emphasize her point, she reached most of the way across the table and pounded it with her index finger. \"There have to be limits. There's a line.\"\n\n\"Do there? Is there?\"\n\nThe table was lacquered wood. To Keats's biot eyes, it was a bit like an aerial map of someplace like Afghanistan. There were steep, deep valleys below formed by the grain of the wood. But filling in those valleys was the smooth lacquer finish. The result was a feeling like skimming along over mountains, flying at the height of the peaks.\n\nThe great problem with biots moving over large distances\u2014distances measured in centimeters or meters rather than millimeters\u2014was finding your way. A biot's view of the macro world was fuzzy and distorted.\n\nCaligula felt safe on his side of the table. There were two feet separating his arm, resting on the edge of the table, from Plath's arm on the opposite side. A long run for a biot, and worse, a hard target to keep track of. Then there was the wall of water left by Caligula's deliberate dragging of his beer.\n\nBut Plath, too, had been playing games with the tabletop. Seemingly fidgeting pensively, Plath had picked up the saltshaker, picked at some dried-on food, then put it down on the table.\n\nShe put it down toward the far left end of Caligula's water obstacle.\n\nFrom the point of view of Keat's biot the saltshaker was the Tower of Babel and the Empire State Building all rolled into one. He saw it as a distant shape, a feature of the landscape like some impossibly symmetrical mountain.\n\nHe saw it from there. But he also saw it through the tap he'd placed on Plath's eye using his other biot. One of Plath's own biots was standing beside him there. Plath made her biot tap Keats's creature and make a gesture meant to convey going around the saltshaker. Biots could not speak to each other, so this was a primitive but effective way to convey basic signals.\n\nOn the table surface Keats's other biot rolled farther left, moving at top speed, racing to get around the saltshaker and avoid being slowed by the water.\n\nHad Caligula noticed? That would be the question.\n\nKeats cleared the saltshaker tower. He spotted the wall of water off to his right but was well clear of it. Ahead, far in the distance, was a wall of indeterminate color.\n\nKeats's first biot, K1\u2014the one inside Plath's brain\u2014turned awkwardly to Plath's P2 and made a gesture using two claws meant to convey that he was closing in.\n\nIn the macro Plath was dragging the conversation out to give Keats time.\n\nCaligula drained the last of his beer and set the glass down just behind the saltshaker.\n\nDeliberate?\n\nPlath's P2 looked at Keats's K1. A body shake that was the equivalent of a headshake. _No, that didn't get me_.\n\nBut it had been close, very close. The glass\u2014a rainbow-swirling object so big it looked a bit like some rainbow-hued desert mesa\u2014came crashing down out of the sky. It sent vibration and water droplets in all directions. One of them, an Olympic pool of water, crashed behind him as he sped on.\n\n\"It is your decision,\" Caligula said. \"Lear will insist that it be your decision.\"\n\n_Lear will insist_ , Plath thought. Never \"he\" or \"she,\" always the careful gender-neutral name.\n\n\"And I will make that decision,\" Plath said. \"But first\u2014\"\n\n\"I'm afraid that as enjoyable as this is, I must go,\" Caligula said.\n\n\"Is there a way for me to contact you directly?\"\n\nCaligula smiled. It was a surprisingly genuine thing, that smile. He was no comic-book villain playing a role and posturing for the camera. He smiled and meant it when he said, \"Sadly, no. My orders come from Lear. My loyalty is to Lear. But Lear will respect your decision and convey it to me.\"\n\nHe pushed back from the table and stood.\n\nHe spoke very definitely about Lear's state of mind, it seemed to Plath. And not for the first time it occurred to her that she might have been speaking to Lear all along. _Was_ Caligula Lear?\n\nExcept that there was something in the killer's eyes when he spoke of his master. There was affection, it seemed to Plath, affection and... not fear. No, Caligula did not fear his master. He liked Lear. He was... he was...\n\n_Proud!_\n\nIt hit her so suddenly she gulped and blushed and ended up awkwardly extending a hand, which Caligula bemusedly refused.\n\n_No_ , Plath thought, _Caligula is not Lear_. But neither was he a mere employee.\n\nAffection and _pride_.\n\nUnable to sit still in the safe house, Keats had come halfway and met her on the sidewalk. With neither of them acknowledging the other, they made their way to a Starbucks. Standing in line, speaking the proper Starbucks drink formula, squeezing around a tiny round table too close to the bathroom\u2014it was all reassuringly normal.\n\n\"Did you make it?\" she asked him.\n\nHe smiled. \"I grabbed his sleeve as he was standing up. I'm on his arm and heading north. In an hour I'll be seeing what Caligula sees.\"\n\n\"And _who_ he sees,\" Plath added.\n\n\"So what did you two talk about?\"\n\nJust a flicker in Plath's eyes. \"I told him what Stern had said about the Tulip being impregnable.\"\n\n\"And Caligula accepted that?\"\n\nPlath shrugged. \"What else could he do? He agreed to pass it along to Lear.\" She frowned, formed a sentence in her head that went like this: _There's something proprietary in the way Caligula speaks about Lear. There's a relationship there. Almost father-son, I think_.\n\nBut she didn't speak it. Under the table she clenched her hands into fists. She found it difficult to talk about Lear at all. She could feel it. She could guess that it was wiring.\n\nWhat she could not do was decide to rip up that wire. That felt suicidal. It felt painful, though of course it would not be.\n\nMore wiring. She'd been wired to fear ripping up the wire.\n\nGames within games. Ever-deeper circles of hell.\n\nPlath's phone lit up. She recognized the number. She covered one ear against the noise of steaming milk.\n\n\"Mr. Stern?\"\n\nTo her surprise it was a woman's voice. \"No. He's dead.\"\n\nPlath froze. Then, \"What?\" It sounded childlike to her, her own voice. She sounded wounded.\n\n\"This is Camilla Strange. I'm... I mean, I was... Mr. Stern's second-in-command. I am now holed up at McLure Labs with reports of four of our people dead.\"\n\nPlath found she was breathing hard. Audibly. \"How did you know to call me?\"\n\nWas it her imagination or were there an unusual number of police sirens. Too many even for New York?\n\nWas it her imagination, or were unflappable New Yorkers hunched a bit too tight around their lattes? Were their eyes less big-city averted and more alert-scared?\n\n\"Mr. Stern left a file to be opened in the event of his suspicious death.\"\n\n\"And was it suspicious? His death?\"\n\nCamilla Strange laughed humorlessly. \"He seems to have been... eaten. Consumed. His driver brought him here, dead, with maybe a third or a half of his body gone. Muscles, viscera, organs: all eaten. Like millions of ants had been working on him. That's how he looks. Like roadkill.\"\n\n\"Nanobots,\" Plath said.\n\n\"Yes, ma'am,\" Camilla Strange said. \"That was our thought, too. A mini gray-goo scenario. They must have been programmed in advance to replicate only so many generations. And then... I'm sorry, someone is... hold on, please.\"\n\nThe phone muted. Then Camilla was back. \"I just sent you a piece of video.\"\n\nPlath switched apps, opened the video, and turned so that Keats could see. It showed a sedan screeching to a halt at McLure Labs. A man whose entire head and shoulders seemed to be weeping blood staggered from the car, walked three steps, and fell.\n\n\"Oh, God. Oh, God. Oh, God, what is that?\" The voice on the video was saying.\n\nThe picture zoomed in, and for just two seconds before focus went hazy Plath could see the dead man liquefying before her eyes.\n\nThe video ended. Blessedly no advertising had yet been attached.\n\n\"Ma'am? Ms. McLure?\"\n\n\"Yes.\"\n\n\"You saw?\"\n\n\"I saw.\"\n\n\"What do we\u2014\"\n\n\"Stay hidden. Stay out of it. This is out of your hands now.\"\n\nPlath, shaken, hung up the phone. She excused herself to the bathroom. She vomited into the toilet bowl, fished in her bag for a mint, found three loose Tic Tacs.\n\nWar was on. If there had been any uncertainty, it was gone now. If she had entertained doubts about whose side she was on, the Armstrong Twins had made it easy.\n\nStern had been like an uncle. The one living remnant of her father's company. The only man she knew who'd been Grey McLure's friend.\n\nStern, murdered by the Twins. Her brother, murdered by the Twins. Her father...\n\nShe saw it again in her mind, the towers falling, and mingling with that imagery was the vivid personal memory of watching her father's jet arcing crazily out of the sky, plunging toward the stadium, the fear, the panic, the flash and heat and noise of the explosion.\n\nIf she was not in this to avenge her father and brother, why was she in it at all?\n\nWas it all wiring now, thrusting these memories to the fore? Maybe, yes. But that didn't make it wrong, any of it.\n\nStern must have been in agony.... Grey McLure must have died in terror, not at his own extinction but at the knowledge that his son would die with him, and possibly his daughter as well.\n\nIf wire was what it took to give her strength, then okay. Okay.\n\nShe wiped her mouth, washed her hands, chewed the Tic Tacs, and thumbed a text.\n\n# **ARTIFACT**\n\n**Plath to Lear:** _Yes_.\n\n# **EIGHTEEN**\n\nThe Antarctic weather came down like the wrath of God. Sixty-knot winds, subzero temperatures. Nothing was flying out of Forward Green.\n\nIt was on her third day there that Imelda Suarez decided to take a chance and see what was in the big hangar out to the south. She waited until the base boss\u2014his actual title was Chief Executive Forward Green\u2014had his birthday party.\n\nSuarez had no difficulty starting up a Sno-Cat and driving off toward the south. No one saw her leave, which was not surprising in the whiteout conditions. The problem would come if for some reason she got lost or the Cat broke down. Then she would have to call for help and all hell could break loose.\n\nIn her forty-eight hours at the base Suarez had felt that this was a very different sort of place, very different from the usual Antarctic facility\u2014even very different from Cathexis Base. The ice was a lonely and often boring place, so people tended to be friendly. People liked \"new meat.\"\n\nBut not at Forward Green. Here she had been treated politely, properly, but not welcomed. No one had plopped down next to her at table and struck up a conversation. This despite the fact that she was an attractive woman and the gender ratio on the ice was about seven to one.\n\nConversations in the dining hall tended to become quieter when she was seen. Everyone was trying hard not to seem secretive, but the end result was that they just seemed more so.\n\nMaybe it was just that Tanner had warned her to expect that something strange was going on. Maybe she was seeing what she expected to see. But that said, it was weird. It was a _very weird vibe_ , as her hippie mother would have said.\n\nThe Sno-Cat is a small, tracked vehicle, like a tiny two-person tank with big windows and no cannon. The heater was blowing noisily, rattling from something stuck in the vent, and the windshield wipers were ratcheting back and forth even more noisily, but visibility was still poor. It would be all too easy to drive right past the hangar and just keep going until the gas was used up. And then she'd quite likely freeze to death. The ice was unforgiving of recklessness.\n\nBut after an anxious half hour she saw the outlines of the building in between swipes of the wipers. She kept going\u2014no point in being coy, she had to look like she had every reason to be here.\n\nBefore stepping out of the Sno-Cat she zipped her parka all the way up, flipped her fur-lined hood forward, and tugged at the drawstrings before pulling on her huge gloves. Her dark goggles were already in place.\n\nSuarez climbed out of the warm cab and was almost knocked over by the wind. But she was a sailor, after all, and not unaccustomed to pitching decks and bad weather, so she avoided disgracing herself. She twisted the door handle, and, sure enough, it was unlocked.\n\nThe wind\u2014which was a battering physical force outside\u2014became just a howling noise.\n\nThe hangar was lit only minimally, but it was still bright enough to see. And what she saw were four vehicles like the ones in the video Tanner had shown her. Three were partially dismantled, with parts strewn across wheeled steel tables.\n\nThe fourth vehicle appeared to be intact. She walked to it, torn between fascination and caution.\n\nIt was about thirty-five feet long from tip to tail, and almost as wide. It was a sort of elongated oval, a hovercraft judging by the skirts, but otherwise like no hovercraft she'd ever seen outside of a Hollywood movie.\n\nIt had a tail, almost like something you'd see on a fighter jet, but there was no horizontal plane, just a shark's fin bearing missile pods on each side. A quick count indicated six missiles total, three in each pod. She had no familiarity with the type of ordnance, but it was undoubtedly real and undoubtedly missiles and undoubtedly military in its purpose\u2014and that fact shocked her.\n\nAntarctica was the last place on Earth without nationalities or armies.\n\nBefore and beneath the tail was a hard plastic canopy\u2014again like something from a fighter jet. There appeared to be two jet turbines mounted on either side, flush with the top, squat beside the canopy. The pilot would be able to see ahead and to either side by looking over the engine casings.\n\nIt was painted a marshmallow white with only a few blue accent notes here and there, plus the obligatory safety notices near the intakes and exhausts from the two jet turbine engines.\n\nSuarez walked boldly to the hovercraft and peeked inside the canopy. The controls were more modern versions of those on her own LCAC.\n\n\"What do you think?\"\n\nThe voice made her jump. It was male, high-pitched, curious not hostile. But when she turned to see its source, she was face-to-face with an assault rifle. Behind the rifle was a middle-aged man in white overalls. He was balding, had a red face and glasses. And he was not, she judged, used to pointing weapons at people.\n\n\"It looks fast,\" she said, trying for a nonchalant tone.\n\n\"It is,\" the man said with evident pride. \"She'll do one sixty knots with no wind and on smooth ice.\"\n\n\"One sixty? And if it hits a bump?\"\n\n\"Do I need to point this at you?\"\n\nShe shrugged. \"I'm unarmed. And I'm not up to anything. I came out here because I can't find a three-sixteenth socket wrench to save my life. I'm Imelda Suarez. I drive an LCAC. They brought me in to work on... well, to be honest, I think they brought me in on a bullshit excuse.\"\n\nThe man smiled expectantly. \"And why would they bring you here on a pretext?\"\n\n\"So that I would see this.\" She indicated the hovercraft. \"So that _they_ could see how I reacted. Because they need hovercraft pilots and we aren't exactly thick on the ground. There aren't a hundred left on the planet, let alone on the ice since the navy's LCACs were decommissioned.\"\n\nThe man lowered the gun, then set it on one of the tool carts. \"I suspect you're right, Ms. Suarez. Or is it Lieutenant Suarez?\"\n\n\"Not lieutenant,\" she said forcefully. \"Semper fi and all that, but I'm no longer getting paid by Uncle Sam. Do I get to learn your name?\"\n\n\"Babbington. Joseph Babbington. Doctor, if that matters to you. We expected you yesterday; that was the thinking, anyway. We were ready yesterday. I'm just an engineer. I did some of the design on the sleigh.\"\n\n\"The sleigh?\"\n\nHe shrugged. \"It's a nickname, but it stuck. 'Santa's badass sleigh,' some wit said once, and now that's what we call it.\" He fished a remote control from his pocket, pressed a button, and the sleigh's canopy rose. \"Take a closer look.\"\n\nCautiously, very aware that the assault rifle was still near at hand, she leaned into the cockpit. She took it in with an expert eye, whistled, and said, \"About twenty years ahead of my cockpit. Very nice. That's a forward-looking radar?\"\n\n\"Oh, much better than that. What we have there, Lieuten... Ms. Suarez... is a computerized obstacle avoidance system technology. COAST, because, well, you know how engineers love acronyms. It senses changes in elevation\u2014obstacles, anything over six inches above grade level\u2014and either diverts power to the cushion to lift the sleigh clear, steers clear, or in extreme cases slows to allow the pilot to choose the course of action.\"\n\n\"Useful if you're shooting along at one hundred and sixty knots.\"\n\n\" _Vital_ if you're shooting along at one hundred and sixty knots.... We have two qualified pilots,\" Babbington said, with the air of someone who was tired of playing games. \"We need six total. Four primaries and two backups. You could be the third primary, if you qualify. And if you're interested.\"\n\n\"Since I left the military my interests have had a lot to do with what I'm paid.\"\n\nBabbington searched her face for a long time. He didn't believe her. Or at least he didn't believe her yet. \"The pay is three hundred thousand USD per annum.\"\n\n\"Jesus.\"\n\n\"It's a tough job. It may even be a dangerous job. And it's a job that has something in common with your military service: it demands unquestioning loyalty and obedience.\"\n\nShe reached in and put her hand on the yoke. They'd gone to the trouble of padding it with leather. It was like something out of a sports car.\n\nOn impulse she hopped inside, a move that required a twisting half jump, like a stunt rider mounting a running horse. She made it work.\n\nThe cockpit was snug, but there was room to the left and right, flat surfaces that even included a cup holder. The pedals felt familiar. If her LCAC was a twenty-year-old Buick, this was a brand-new Porsche. It even had a new-car smell.\n\nIt was seductive.\n\n\"Very nice,\" she said. \"But what's it for?\"\n\n\"For?\"\n\n\"Dr. Babbington, I couldn't help but notice the missiles.\"\n\n\"Indeed.\"\n\n\"Why would the sleigh require missiles?\"\n\n\"We're testing it for the military.\"\n\nShe wondered if she should let the lie go unchallenged. If she called him out, would he shoot her? No, she judged: if she failed to call bullshit, he'd know she was lying.\n\n\"That's very funny,\" she said. \"What's the real reason?\"\n\nBabbington smiled, a nice, genuine smile. \"The owner of the company is a bit... let's say, she's a bit unusual. She has a notion that civilization will soon collapse, and she intends to sit it out right here. But should that civilization lash out at her in its last throes, she wants to be able to defend herself.\"\n\n\"You work for a nut?\"\n\n\"I used to work for the Pentagon, as did you. Weren't we working for nuts then? And those nuts paid rather ungenerous government salaries.\"\n\nDespite herself, Suarez laughed. \"Well, you got me there. What kind of range does this thing have?\"\n\n\"The sleigh has a three-hundred-fifty-mile operational range. Six surface-to-surface missiles, four surface-to-air missiles just inside the engine cowling, twin thirty-caliber machine guns.\"\n\nWith a show of reluctance Suarez climbed out of the cockpit.\n\n\"I have to tell you, Dr. Babbington: three hundred large would be very nice. Very, very nice. But there's something else. The U.S. Navy taught me to drive hovercraft, but that was incidental to my core training.\"\n\n\"Which was?\"\n\n\"Marines, first, as you already know. Then Sea Air Land, Doctor. Navy SEAL.\"\n\nShe watched his face turn gray.\n\nShe watched his eyes dart toward the assault rifle. Which was in her hands before he could move.\n\n\"This is the part where you tell me everything,\" she said. \"This is the part where you answer all my questions, because if you don't, I shoot you, and you die.\"\n\nNot a good liar, Sadie McLure.\n\n\"I told Lear no,\" she said.\n\nBut it was right there in her eyes.\n\n\"Then what are we going to do?\" Keats asked her.\n\nShe shrugged. \"I don't know. This madness, the Nobel thing, whatever happened there, that must be the secret weapon Lear wants destroyed. Right?\"\n\nHe had not known what to say then. He had not known what he could safely say to her. He did not know whether the girl he loved would have him killed for turning against Lear.\n\nIt felt as if his insides were dying. Like he was a piece of fruit left out in the sun, rotting from the inside, collapsing in on himself. He felt sick.\n\nShe was wired. She knew she was wired. Yet she had refused to let him try to fix her.\n\nThe insidiousness of it. She was like a schizophrenic who knows she's supposed to take the meds but refuses to. She was becoming party to her own mind-rape.\n\nHe wanted to grab her shoulders and shake her. Everything he was to her was less important to her now than carrying out Lear's plan.\n\nShe had always held something back from him, he knew that. That was okay, he'd told himself, she just needed time. At first he'd decided the reticence was a class thing. That made him feel a bit better, really, because it was something he could understand. It was something he could defend himself from emotionally.\n\nHe still loved her. But she had never loved him, had she? And now... now where was Sadie McLure?\n\n\"Do you want to make love?\" she asked him, that and he wanted to punch her in the face.\n\nNot her, not _Sadie_ , no, it was... it was whoever this person was, this reprogrammed, wired alteration of Sadie. It was this truly new creature called Plath.\n\n\"I'm tired,\" he said, and the relief in her eyes was almost more than he could endure.\n\n\"Yeah. Big day tomorrow,\" she said.\n\n\"Oh? Why?\"\n\nHer eyes flicked right\u2014guilty, caught. She shrugged and forced a phony smile. \"Aren't they all big days?\"\n\nShe left, heading toward the bedroom they still shared.\n\nTears filled his eyes and since no one was around to see him standing there like a fool, he let the tears roll down his cheeks.\n\nBack to New York, that's what Lystra said. \"Back to New York to watch the show, yeah. A lot happening very soon. Timing. It's all in the timing, yeah.\"\n\nSo here they were. New York City, and damned if the tattooed madwoman didn't have an apartment a block away from the Tulip. He could look straight down Sixth Avenue and see the building. He could run for it, escape, get to the Twins and say, \"I'm sorry, I'm sorry, but you don't know what this crazy bitch is doing!\"\n\nHe could do that. And they'd thank him for the information and then kill him. Or Lystra would catch him and she would show him that scary face she had, the one where she seemed almost to turn into a skeleton. And then she could kill his biots and turn him loose.\n\nDeath or madness. Seriously? That's what it was down to? The three windows in his head said _yes_ , _yes_ , that was exactly the choice.\n\nHe wondered rather morbidly just what kind of crazy he would be. Stories were still leaking out of Stockholm. They said some big-deal banker found a way to hang himself from a chandelier. They said a French general was found smeared with feces, crying. They said a famous American horror novelist had run into the street and beaten a party Santa to death with a fire extinguisher.\n\n_Which crazy will you be, Anthony?_ he asked himself.\n\nWhat escape was there? The Twins? The American government?\n\nHe stopped breathing. The answer\u2014not a good answer, a weak, probably worthless answer\u2014popped into his head.\n\nSomeone brilliant. Someone with mad skills. Someone who once had almost, sort of, liked Bug Man. And was just a block away.\n\nBurnofsky.\n\n_Lystra had taken his phone. She was on her own phone right now in the adjacent room, telling her CEO, some dude named Tom, to_ fire all the remaining employees, effective now, this minute, shut it all _down, the whole Directive Medical shebang, stop all checks and buy more gold. Yeah. Just don't touch Cathexis_.\n\nBurnofsky. The dude had invented nanobots. It stood to reason he'd have... something. But how to reach him? He knew Burnofsky's e-mail and his cell, but Bug Man had no phone.\n\nHe would have to wait until she was asleep, the monster in the next room.\n\n_Maybe I won't be a hanging-myself crazy_ , Bug Man thought. _Maybe I'll be a nice, gentle, shit-smeared kind of crazy_.\n\nSomehow he was convinced that none of this would ever have happened if he'd just found the onions sooner. Gotten home.\n\nWith a chill he remembered his mother coming down with a sinus infection a year ago, give or take. She'd had tests done. Her DNA, too, might be stored somewhere on one of Lystra Reid's drives.\n\nHer plan was now frighteningly clear. She had used her web of medical testing companies to acquire and digitize DNA from millions of people. Once you had the DNA, you could grow a biot derived in part from that DNA. The biot-DNA-donor mind link would happen\u2014which would be disorienting all by itself. Suddenly having windows open in your mind... well, that was going to be disturbing.\n\nBut nothing to what came next. Lear wasn't out to disturb or unsettle people, she was out to destroy civilization. For that she needed madness. Widespread, inexplicable, irresistible madness.\n\nSo once the biots were born, she had only to kill them. An electrical surge maybe, or extreme heat or acid.\n\nWould it really work? Would one crazy woman be able to bring the whole world crashing down?\n\nHe turned on the television; it was all he had. Al Jazeera TV had a news bulletin. He reached for the remote. He did not want to see more video of that horror show in Stockholm.\n\nSuddenly he felt Lystra's presence and realized he was no longer hearing her from the other room. \"Leave it,\" she said, looking toward the TV screen. \"I think something kind of, yeah, big just happened.\"\n\nSeven months earlier, the younger British prince had given blood in a public show of support for a National Blood Service blood drive. The NBS had been helped in their work by volunteers from Directive Medical UK.\n\nOf course security for the Royal Family was very tight, so no one would be allowed to actually know which was his donation. It was labeled anonymously, just a numerical tracer, and sent off to the blood bank.\n\nExcept that the Directive Medical lab tech had already swapped it out with an earlier sample.\n\nNow, as the television picture showed, the prince was in a gondola of the London Eye\u2014the huge Ferris wheel beside the Thames\u2014as part of an outreach to disadvantaged youth.\n\nThe gondolas were large enough to hold a couple of dozen people at once, and were in fact holding twelve specially chosen children of carefully varied ethnicities, who shrank in horror against the far end of the gondola as the prince repeatedly ran at the transparent wall and smashed his head into it.\n\nBlood smeared the plastic. Blood completely covered the prince's face and would have rendered him unrecognizable if not for the familiar red hair.\n\nThe Eye was slowly coming around, bringing the gondola back to earth, but that footage from three minutes earlier\u2014brutal video of the raving royal slamming himself again and again and again\u2014was competing in one half of the screen with a live shot showing him flailing, kicking, spitting blood in every direction as appalled Royalty Protection in plainclothes and uniformed London Met police tried to get him under control.\n\n\"That was excellent,\" Lystra said. \"You try to nail the timing, yeah, and arrange something spectacular, but wow, that was better than I'd hoped for. Yeah.\"\n\nBug Man stared in horror. \"I liked that dude. He was the fun one.\"\n\n\"Who, the prince?\" Lystra laughed. \"Don't go soft on me, Bug. Much more to come. I've got three officers at a nuclear missile base near Novosibirsk. High hopes. Fingers crossed, yeah?\"\n\nAnd yes, she had her fingers crossed. She left and closed the door behind her.\n\nBug Man watched as the prince was hauled away to a waiting ambulance. \"Fuck you, crazy lady. Yeah? I liked him. He was a gamer.\"\n\n# **NINETEEN**\n\n\"I need your help.\"\n\nKeats to Wilkes and Billy the Kid.\n\nPlath was asleep. He had crept silently from bed to bed waking them, holding a silencing finger to his lips.\n\n\"Anything for you, pretty blue eyes,\" Wilkes said, and yawned.\n\n\"Plath has been wired,\" he said. He knew she might wake up at any moment. No time for delicacy. \"She's been wired, she knows it, but she won't pull the wires. It's got to her. We need to go in there and clean her up.\"\n\nAnya was not invited. Plath had a biot in Anya. Keats badly wanted to ask Anya if she had built any more biots for Plath. But Plath might have been watching through Anya's eyes, or listening in her ear.\n\n\"You saying someone from Armstrong wired her?\" Wilkes asked.\n\nKeats hesitated. \"This is lunacy. This is mad. But she saw something. Down in the meat. She doesn't think it was a nanobot. I helped her look. I didn't find anything. But I have found wire, a lot of it.\"\n\nHe let that sink in. \"She thinks Nijinsky\u2014\" Wilkes began.\n\n\"No,\" Keats said. \"Whatever it is, whoever's running it, it's still apparently active, so not Nijinsky. Someone else. Maybe one of you two. Maybe a traitor from some other cell.\"\n\nWilkes got up, came over to Keats, and sat down beside him. Very close, uncomfortably so. \"How do we know it isn't you?\" she asked. \"You've had a biot in her for a long time, right? Fixing that hole in her artery or whatever? Could be you, right? And maybe you're just lying in wait for one of our biots to come crawling along and, _boom_!\"\n\n\"This is kind of crazy,\" Billy said.\n\n\"Nah, this isn't _kind of_ crazy,\" Wilkes said. \"This is full-on crazy.\" She heh-heh-heh laughed and said, \"This is where it all goes, right? I mean, this is where it kind of had to go, didn't it? You start playing with people's brains, man.... How do you know? Right? Whole world's going crazy. All those big brains. And now your prince dude.\"\n\nKeats nodded tightly. \"Right.\"\n\nWilkes pulled away from him. \"Maybe I just transferred one of my kids to you, Keats. Just now.\"\n\n\"Maybe,\" he acknowledged.\n\n\"Maybe it's me, and if I put one of my kids into Plath, maybe that's my second one, you know? Maybe I get in there and make it worse. What's Plath doing? What's she up to? Did this wire make her soft in the head?\"\n\n\"She's planning to blow up the Tulip.\"\n\n\"What's a tulip? A flower, right?\" Billy asked.\n\nWilkes snorted. \"It's a skyscraper in Midtown. Blow it up? What's that even mean?\"\n\n\"It means that she's given the go-ahead to Caligula to blow it up. Kill everyone in it. Destroy all their labs, all their computers.\"\n\nWilkes stared at him.\n\n\"Lear told her to\u2014\" Keats began.\n\n\"Lear?\" Wilkes shrilled. \"Lear told her to murder all those people?\"\n\n\"That was her own... her own solution. Maybe. Who knows? She's met with Caligula. She knows she's wired, and she knows it's wrong, but she can't, you know.... she can't pull the goddamned wires. We have to do it for her. And we have to find whatever is in there. Nanobot or biot, we have to find it and kill it.\"\n\n\"Who is doing it?\" Billy asked. \"I mean, who is wiring her brain to do\u2014\"\n\n\"To do what Lear wants done?\" Keats asked, his voice rising. \"Who is wiring her to do _exactly what Lear wants done_?\"\n\nWilkes drew a sharp breath. \"The hell,\" she said.\n\n\"I don't have any choice but to trust you two,\" Keats said. \"For all I know, you're as wired as she is. Or maybe you just think it's okay. Or maybe I'm as messed up as she is and the way I see this is all wrong. But I have no choice, I have to... I can't...\" He spread his hands, helpless.\n\n\"You're talking about ripping out wire that Lear or someone working for Lear put there?\" Wilkes asked. \"Lear's going to see that as treason. You know what that means? You know who comes to talk to you when you betray Lear? Jesus, Keats, if she's as wired up as you say, Plath'll send Caligula after you herself.\"\n\n\"I know!\" he raged. He pushed his fingers back through his hair. \"I know. I know.\"\n\nNo one spoke. Keats sniffed and wiped at his eyes. \"This fight has changed,\" he said. \"This isn't us against them anymore. Not that simple. I mean, doesn't there have to be some line we draw? Doesn't there have to be something we won't do, even if it means maybe we lose? And doesn't there have to be some limit on how far we'll _let_ ourselves be used?\"\n\n\"The Twins don't have a limit,\" Wilkes said.\n\n\"Neither does Lear,\" Keats said. \"I think he's the one using biots\u2014creating them, killing them\u2014to drive people crazy. Sweden. The prince. The Brazilian.\" He waved his hand vaguely. \"Probably a bunch of other stuff. The Twins, Lear, they're just two sides in the same crazy game, Wilkes.\"\n\n\"Yeah. And we are playable characters, right? We're game pieces.\"\n\n\"If we let ourselves be,\" Keats said.\n\n\"So now you're taking over?\" Wilkes asked.\n\n\"Only until Plath is cleared. Then...\" He shrugged. \"Then we... I don't know.\"\n\n\"I'm in,\" Wilkes said, but her usual smart-ass smirk was gone. Her face was gray and slack. She looked far older than she could have been. \"Death or madness. Right? We've always known it would come down to that.\"\n\nKeats nodded. \"Death or madness.\"\n\nThe Russian officers proved to be disappointments to Lear. A major and two lieutenants duly lost their minds as their biots died, but at the time they were not on duty. The major wandered off into the Siberian wastes and froze to death. One lieutenant was dead drunk, too incapacitated to do much of anything.\n\nThe remaining lieutenant had just finished a shift. He saw the windows opening in his mind and acted quickly. He stripped off his sidearm and threw it into the snow. Then he ran toward the medical dispensary, but lost his mind halfway there.\n\nNaked, he charged the guarded gate of the missile silo and was arrested by security.\n\nThe lack of a nuclear event\u2014it would have registered on seismographs\u2014disappointed Lystra.\n\nSo she opened her laptop and scanned the list of high-value targets. She picked out the pilot of a Virgin Australia plane making the long haul from Los Angeles to Sydney.\n\nAs he approached Sydney in a few hours, his biots would be born, windows would open, and if Lystra was lucky the world would have one more thing to fear. An appetizer, so to speak, before the pasta course.\n\n\"Funny,\" she said. \"Yeah.\"\n\nShe watched some old _Beavis and Butt-head_ on Netflix, and fell asleep with it still playing.\n\nBug Man had never heard of _Beavis and Butt-head_. That would give him an excuse in case she woke up and saw him creep into her room with his heart in his throat. He could say, _I heard this on TV, didn't know what it was, so I came in and..._\n\n_... and lifted your phone_.\n\n_And then you killed me, so, yeah, yeah, crazy bitch, yeah, then you killed me. The end_.\n\nSuarez had not found it necessary to threaten him much. Dr. Babbington was amenable enough once she'd made clear that she would do bad things to him if necessary. And an assault rifle was hard to argue with.\n\n\"Because society is going to crumble. That's why. She's absolutely convinced that society is about to crumble like a stale cookie.\"\n\n\"Who? Who are you talking about?\" she had demanded.\n\n\"Jesus, you don't even know who is running this? Our lord and mistress. The owner. Of Cathexis. Lystra Reid.\"\n\n\"Lystra Reid? Are you sure?\"\n\n\"Yes, I'm quite sure. This is only one of two _secret_ facilities. This is where we create the sleighs, where we train pilots: this isn't the final level any more than Cathexis is the final level.\"\n\n\"There's a third base?\"\n\nA third base. Three hundred kilometers south in a small dry valley. Dry valleys are a phenomenon unique to Antarctica, places of rock and little else, where for reasons of ice drift and unusual wind patterns the ground is bare of snow.\n\nIf Lystra Reid had built a base in a dry valley, it would not be one of the McMurdo group. The McMurdo Dry Valleys were more or less permanently infested by scientists collecting rocks and drilling core samples and complaining about their grant proposals.\n\nShe pointed this out to Babbington.\n\n\"Yes, well, this dry valley is an odd duck. It's extraordinarily deep and also quite narrow\u2014just two kilometers across at its widest point. The ice is piled high against both mountain ridges, and sooner or later, of course, the weight of all that ice will crumble the mountains and take the valley. Soon by geological standards, so within a hundred thousand years.\"\n\nHe laughed, obviously thinking that was a science joke. When Suarez mustered up a half smile, he seemed encouraged.\n\n\"There's actually a meltwater river there, helped by some subterranean geothermal activity, and the whole place is quite sheltered from the wind. It's a garden spot, really. The average annual temperature remains within twenty degrees of fifteen degrees Fahrenheit. So sometimes it's actually above freezing.\"\n\n\"Garden spot.\"\n\n\"Anyway, that's where the third base is.\" He showed her on a map.\n\nAnd that was when Babbington made an ill-fated leap for the gun. In a hand-to-hand battle of SEAL vs. scientist, the outcome was not in doubt.\n\nBabbington landed on his butt several feet from where he started.\n\n\"I'd stay there if were you,\" Suarez warned.\n\nHe took her advice, crossed his legs awkwardly like a kindergartener, and sat.\n\n\"I'm afraid I will have to lock you up, Dr. Babbington. I'm sure there's a tool locker somewhere. They'll find you when the party is over and when the weather clears. Do you have to use the bathroom? Because you'll be tied up for as much as a day.\"\n\n_Amazing_ , Suarez thought, _how quickly life can get weird. One minute you're driving an LCAC delivering oranges and booze and hauling away garbage, and the next minute you're beating up scientists and preparing to get yourself killed in some dry valley at the end of the world_.\n\nShe had no doubt that this Lystra Reid person was capable of killing. You don't set out to build secret bases defended by sophisticated weapons because you're peaceable by nature.\n\nBut the question in her mind was whether this whole thing, whatever was going on here, was a secret government op unknown to Tanner. Tanner was low-level; this could simply be something way above his pay grade. In which case she would earn no thanks for barging in on this third base unannounced.\n\nBut they wouldn't kill her, not if this was a government op. They'd give her a stern lecture, make her sign more threatening letters, and just maybe hire her on.\n\nIf, on the other hand, it wasn't a government op but some actual crazy woman buying missiles and building a secret lair at the frozen anus of planet Earth, well...\n\nShe had to tell _someone_ what was up: a witness she could trust to follow up just in case Imelda Suarez was never heard from again. She glanced at the computer on a work desk.\n\n\"User password on that computer?\" she asked, sliding into a chair.\n\nBabbington shrugged. \"1234ABC.\"\n\n\"Seriously?\" She typed it in and got access to her own e-mail account. She wrote a message to her brother, Frank. Frank was with the Capitol Police. He wouldn't be cleared for this information, but she knew she could trust him.\n\nShe spent a few minutes locating a good, strong steel tool locker and pushed Babbington and a bottle of water inside.\n\n\"You okay in there?\"\n\n\"Well...\"\n\n\"You'll be fine.\"\n\nThen: the sleigh.\n\nSuarez was honest enough to admit that she was motivated in part by an almost lustful desire to drive the sleigh. It was an object of beauty. A work of art. It screamed \"speed\" just sitting there.\n\nShe shed her parka, stuffed it into the very minimal storage space behind the cockpit seat, and slid into the leather chair. The pedals were where they should be. The yoke was awkwardly placed by her lights, but she could live with it. The displays were elegant and wonderfully easy to read.\n\nShe closed the canopy and realized the hangar doors were shut. So she climbed back out, scrounged around until she found a remote control, and climbed back in. It was just as good the second time. She had to fight the urge to run her fingers over the displays. Beautiful. If Rolls-Royce, Tesla, and Porsche teamed up to make a hovercraft, this would be it.\n\nThere was an autopilot, but she couldn't imagine trusting herself to a computer\u2014not at the speeds this thing moved, not on the most treacherous terrain on planet Earth. But she turned on the automated warnings as well as, after some hesitation, the impact-avoidance system that would take control if she was in immediate danger of crashing.\n\n\"You wreck this thing and your future will be very much in doubt, Imelda,\" she told herself.\n\nThen, finally, she fired up the engines.\n\nIt was noisy but not deafening in the cockpit. She felt the surge of suppressed power as the twin jets throbbed. The sleigh rose on a cushion of air.\n\nShe keyed the remote, and the hangar doors slid open. Beyond the doors was whiteness, white on white as far as the eye could see.\n\nShe punched her destination into the GPS, released the cable tie-downs, and slid toward the gap at walking speed then running speed and was just hitting fifty knots by the time she blew out of the building, keyed the doors over her shoulder, and rocketed out onto the ice.\n\n_Oh, yes_.\n\nShe smiled and held it at fifty knots until she had played with the controls for a while and come grudgingly to trust the forward-scanning radar.\n\nThe ice here was rippled but with no rises over eighteen inches. The sleigh's jets adjusted automatically to push more air into the cushion as she reached obstacles.\n\nVery soon fifty began to feel slow. Boring. Despite the fact that the ice was flying by beneath her. In her rearview mirror she saw a vortex of ice crystals, a shimmery white contrail.\n\n\"Well, in for fity, in for a hundred, right?\"\n\nShe punched it, and the sleigh took off like a rocket.\n\n\"Oh, yes,\" Suarez said. \"Ah-hah-hah!\"\n\n# **TWENTY**\n\nPlath was still asleep when they struck.\n\nOne from Keats, two from Wilkes, two from Billy. Five busy biots raced up through her eye and into her brain.\n\nThey had planned. Wilkes and Keats would focus on ripping up wire in the places where Keats had found it. Billy would go hunting for the intruder and call for help if he found something.\n\n\"It's mostly all up in here,\" Keats said. \"Hippocampus and some Broca's area.\" Amazing how quickly one could learn something as esoteric as brain architecture when life and death were involved.\n\nThe three of them were downstairs in the darkened living room. Hopefully Plath would not awaken and come down to find out why Keats was not in her bed. If she did, they would know it: arteries would start pumping faster as she woke and began to stir.\n\n\"Go ahead, Billy. But if you find something, don't fight it. Call us for help.\"\n\nBilly had a Coke by his side. He was dressed in a Washington Nationals jersey many sizes too large and slumped down to look cool, with the result that he looked even younger than he was\u2014a small, round head and solemn face in a pile of rumpled clothing. None of them had anything to do with their hands.\n\n\"There's the first wire I found,\" Keats said to Wilkes. Down in the meat he was pointing it out to her nearest biot.\n\n\"It's encrusted,\" Wilkes said. It's been there for some days at least. Maybe longer. Meaning maybe we pull the wire, but the neurons have already made it redundant.\"\n\nThe wire was crisp and clean, only a few molecules in circumference. But neurons had grown over and around it in places, like kudzu, vines twining sensuously around the metal of the wire.\n\n\"Maybe,\" Keats admitted. \"But I won't have that in her brain.\"\n\nThat made Wilkes smile. A genuine smile, not her usual cynical leer. \"Pulling it up, sir. Aye, aye, Captain. Pulling it up.\"\n\nKeats saw her two biots going at it, working well together to pull up the encrusted wire. The pins were sunk deep and completely overgrown. It took two biots straining to draw them slowly out of the brain like fence posts being pulled up. They came free but were still tangled in strands of neurons.\n\nWilkes had to tear the strands away, breaking actual brain connections in the process. To biot \"ears\" they made a sound like someone squirting water through their teeth and tearing denim.\n\nNo way to know whether these were just redundant cells tracking the wire or whether they had some legitimate purpose. Was she ripping away some cherished childhood memory? Probably not, probably these connections were just reinforcing the wire, but the human brain was astoundingly complex. BZRK had very sophisticated brain mapping, but still it was largely a crapshoot.\n\n_Gee, sorry about that, Plath, I just wiped out your memory of nursery school_.\n\nKeats was doing the same around the corner. There the wire was fresher, less overgrown. His biot stood at right angles to hers in the almost gravity-free liquid environment.\n\nIn the macro Billy said, \"Can I ask a question?\"\n\n\"Of course,\" Keats replied.\n\n\"Why are we doing this?\"\n\n\"Because someone has messed with Plath's head, that's why.\" Keats obviously thought that was the end of it. But Billy pressed. \"But isn't everyone's head messed with? I mean, stuff you see, or how you were raised. Stuff people did to you.\"\n\nThere was something, maybe several somethings, behind that tremulous _stuff people did to you_.\n\nBut this was not the time to examine Billy's demons. \"That's all natural, this is...\" He was at a loss for words. \"It's wrong, that's all.\"\n\nBilly fell silent after that. But Keats could see that he wasn't convinced by Keats's halfhearted effort to justify what they were doing. Keats went on about his business, tearing out wire, pulling pins. So did Wilkes, but now she took up the same line of questioning.\n\n\"Yeah, blue eyes, but we aren't doing this with her okay anymore than whoever laid the wire down, right?\" In the meat they were at right angles, here they sat facing.\n\n\"She's not able to\u2014\"\n\n\"So we're _making_ her do it. Right? I mean, we're unwiring her even though she obviously isn't totally psyched about it.\"\n\n\"Come on, guys,\" Keats said. \"It's not the same. Someone wired her brain. _Hacked_ her brain. Took over her brain. Now we're fixing it. It's not that difficult to understand.\"\n\nWilkes began to argue, but then Billy yelled, \"I see something!\" His two biots had emerged from a brain fold to see a furtive shape disappear just beyond the reach of illumination.\n\n\"Nanobot?\" Keats demanded.\n\n\"I don't know. I can't... I think it sees me. It's running! He's fast! He's got moves, he's got moves, man! 3D moves!\"\n\n\"Stay with him, we'll catch up,\" Keats directed.\n\nBilly was in the game now, racing as fast as his biots would go across a terrain of eerie hillocks, pulsing red worms as big as car tunnels, static sparks, and always the lethargically circulating fluid that slowed his every movement. His quarry disappeared into a shallow fold and Billy followed.\n\n\"Ahhh!\"\n\n\"What?\"\n\n\"Shit!\"\n\nBilly jumped out of his seat, knocking over his Coke. His fingers moved as though he was using a gamepad. His eyes seemed to dart after objects he could only see in the m-sub.\n\nBilly's first biot was down and minus two legs on the left side before he knew what hit him. He twisted both biots to see, and there, undeniable, unmistakable, the enemy: a biot.\n\n\"It's a biot!\" Billy yelled. \"It's a biot-biot-biot!\"\n\n\"Don't let him get away!\"\n\n\"One of mine is down! I'm\u2014he's fast!\"\n\nThe alien biot had raced up a vertical surface then pushed off, somersaulted, and dropped down behind Billy's remaining mobile biot.\n\nThe foe was vertical and swimming downward. Billy made the mistake of believing he was safe until the biot landed, but the biot spun in midfall and fastened two pincers onto Billy's eyes.\n\n\"I'm blinded!\"\n\nIn the macro he instinctively rubbed his eyes, shook his head. In Billy's brain the second window was blank, showing no picture. Like a TV tuned to a dead channel.\n\n\"I see him!\" Keats yelled. \"Come on, Wilkes!\" He grabbed her hand in excitement.\n\nNow three biots raced to catch the intruder. But the intruder was no longer fleeing. It had taken up a position on what seemed like a vertical surface and now waited.\n\nKeats pulled to a halt. Wilkes's two biots did the same.\n\n\"Three to one,\" Wilkes said. Then, \"Why does that thing look familiar?\" And then, \"Fuck!\"\n\nKeats was already on his feet. He raced up the stairs, and without bothering to knock, opened the door to Anya and Vincent's room.\n\nAnya was asleep.\n\nVincent was not.\n\nDown in the meat Wilkes stood beside Keats. Now she and Billy both joined him in the macro, staring at Vincent, who looked at them calmly.\n\n\"You,\" Keats said.\n\nVincent didn't answer. Anya rolled over and opened her eyes.\n\n\"Billy,\" Keats said. \"Go get Plath.\"\n\nA message lit up Burnofsky's phone, but he had muted it so there was no chime.\n\n_It's Bug. Bad shit happening. Crazy bitch I think is Lear. Going to kill me and the whole damn world_.\n\nNinety seconds later, a second message.\n\n_Are u there? Talk to me! I'm not playing_.\n\nSixty seconds later:\n\n_Fuck! Do NOT call back. I'm using her phone. Can't wait. I'll try again later_.\n\nBug Man had barely erased the messages and slid the phone back onto Lystra's nightstand when the alarm on that phone went off. _Zeeet! Zeeet! Zeeet!_\n\nBug Man leapt for the door, eased himself out even as Lystra stirred and reached blindly for the phone.\n\nBy the time she emerged he was wrapped in a blanket on the couch doing a very poor job of faking sleep.\n\n\"Get up,\" Lystra said, and pushed his foot. \"It's time.\"\n\nHe pretended to yawn. \"Wha\u2014? Time for what?\"\n\nShe grabbed a piece of glass fruit from a bowl on the nearby table. She hefted it in her hand, judging the weight. Then she swung it hard and fast, smashing it into Bug Man's left eye.\n\n\"AHHH!\"\n\nHer free hand was on his throat, he could feel the pressure tightening. He squirmed but did not lash out at her, did not try to hit her. She took the glass fruit\u2014it may have been a peach\u2014and stuffed it brutally into his mouth.\n\nBug Man tasted blood. She pushed harder, harder until his front teeth began to splinter. He cried out, a muffled, frantic sound, and suddenly she spun away and tossed the now-bloody fruit back in its bowl.\n\n\"Who did you text?\" she asked in a conversational tone.\n\n\"Whanh?\" He couldn't make sounds right, and spit splinters of teeth out onto the blanket. Tears filled his eyes, the pain but more the shock of the attack.\n\nHe felt with his tongue around the new architecture of his mouth.\n\n\"Who did you text, Buggy, come on,\" Lystra said. \"Was it some girl?\"\n\nHe seized on the idea. \"Nuffing, jush shome girl back home. Jush my girlfrien'.\"\n\n\"Yeah, don't do that again,\" she said. \"Now I have to password-protect my phone, and I hate that. It slows me down. Turn on the TV, go online to Vatican City feed.\"\n\nHe couldn't see to find the remote. Wouldn't have had any idea where to find a Vatican feed if he had, and in the end Lystra, making disparaging sounds, did it herself.\n\n\"Now watch.\"\n\nBug Man wiped tears and blood from his face and tried to focus. The Pope, but not wearing his tall Pope hat.\n\nBug Man knew what was coming next.\n\n\"This is your work,\" Lystra said approvingly. \"The man is very healthy, it seems. I couldn't find a biological sample anywhere. But you got me his cells.\"\n\n\"My mouff...\" He groaned and wept again, despairing. There would be no happy ending for his life: he saw that clearly now. Unmistakably. His lip was swelling. He swallowed more blood; it hadn't slowed yet. \"Why 'o I haff to wash?\"\n\n\"Why do you have to watch?\" Lystra seemed puzzled. \"I thought you'd want to see. Look! It's starting. Look at him staring around, trying to figure out where the bugs are.\" Then she sighed happily. \"Plus, I suppose I like an audience. Genius unappreciated and all that. Get ready. This is going to be epic. It's one thing to show people they can't rely on their politicians and famous brains and all; it's another thing altogether to say even God can't stop what's coming. This will scare the hell out of people.\"\n\nHis Holiness the Pope stood with the benign expression the world had come to expect and love. It was a sort of half smile, eyes crinkled, hands folded in front of him.\n\nHe was bored to tears. He was often bored by these ceremonial events. Although at least this was out of doors, under a partly cloudy sky just brightening to the richer blue of early afternoon from the bright blue of morning.\n\nThe Pope sometimes walked in the streets of Rome in disguise. He disliked the fishbowl in which he was kept, always surrounded, always watched. If he was to lead the Church, then he must know its people.\n\nOnce he had gone out disguised as a priest. That had ended badly, with tourists recognizing him and crowding around him, twenty, fifty, two hundred people, a mob within seconds. His security detail had had to practically lift him and carry him through the crowd to a waiting SUV.\n\nThen he became more creative: a toupee, jeans, and an \"I heart Rome\" T-shirt. He was followed on these excursions by Swiss Guards in plainclothes. He had negotiated with them to keep their distance. And they had agreed to stay at least a hundred feet away.\n\nHe was considering such a trip for the evening. What a joy it would be to find a cramped table in some little osteria, drink wine and eat antipasto, pasta, and perhaps a nice piece of fish. Watch regular people. Eavesdrop on their conversations.\n\nThen, what? A limoncello in lieu of dessert? A walk by the river? Or succumb to the lure of the beautiful array at some well-tended gelato stand?\n\nIt might be his last excursion for a while. The world was going mad. He had been shown the footage of the British prince, the poor young man. Drugs, most likely. But contacts with intelligence agencies around the world suggested that suspicion was growing that something connected it with the self-murder of the American president, the Nobel massacre, the Brazilian president and vice president\u2014perhaps the earlier attack on the UN in New York and even the bizarre tragedy in Hong Kong.\n\nIf it was terrorists of some sort, no one seemed to know who they were or what they wanted.\n\nThe Pope frowned, realized he was being watched by many eyes, and relaxed into his blankly beatific expression.\n\nYes, just as soon as he got through today's event, a _tableau vivant_ of the manger scene. It was a group of Italian children, specially chosen, prepared, and rehearsed. The production was done with some of the biggest names in Italian theater, with costumes from great fashion designers. There would be music.\n\nThe Holy Father had managed to get it moved to the morning on the theory that he needed his rest in the afternoon. In fact, he needed to be back at the Vatican in plenty of time to slip out for dinner in the evening.\n\nA small lie. He would confess it and be absolved.\n\nHe hoped the presentation was wrapping up. He didn't want to sneak a peek at the printed program or it might betray impatience. But if memory served, this was the last song, which would be followed by applause, then kind words for the child actors and singers and the adult organizers.\n\nAnd then, he was out. Done by the afternoon. They'd promised him that.\n\nPerhaps a nice piece of cod. He liked cod when it was not overdone.\n\nYes, the final chorus! Applause from the thousands, maybe tens of thousands of people in the square. Time to applaud, time to enlarge the benign smile, time to...\n\nThe Pope blinked.\n\nThen he frowned.\n\nWhat was that? What was he seeing? Some sort of... it looked like an insect, a bizarre insect. He was seeing it, but... but it was nowhere around him.\n\nHe looked around, puzzled and a little concerned. No one else seemed to see anything unusual.\n\nAnd now... now another vision. Like a movie screen lit up inside his head, like two of them, really, and both of them showing fantastic insects.\n\nThe one insect, he could see its face.... He could... he was hallucinating. Clearly, he was hallucinating. Was it a stroke? He dreaded a stroke; his father had died at age fifty-four of a stroke.\n\nThe insect face... It...\n\n\"It's me,\" he said in his native Spanish.\n\nThen, as quickly as it opened, one of the windows in his head closed. Gone.\n\n\"Ah!\" he cried out. \"Ah!\"\n\nHe sank to his knees, and now everyone was looking at him, now the TV cameras and the phone cameras all swiveled toward him, as he cried, \"Oh, oh, oh!\"\n\nThe Pope began to laugh. He began to laugh and laugh and then he was screaming, he knew he was screaming, though he felt no pain.\n\nHe screamed and tore at his vestments.\n\nThe world was swirling colors, all zooming crazily around him, faces suddenly coming into focus, distorted, demonic faces.\n\nOnly when he grabbed an elderly woman's walker and began attacking the children with it did anyone try to stop him.\n\nIn the end he was hauled away by Rome police and his own plainclothes Swiss Guards.\n\n# **TWENTY-ONE**\n\nThe sense of approaching doom was rising now. The country was scared. The world was scared. Glances were shielded. Heads were lowered. Shoulders hunched. Jaws tight. Voices too high or too low, too loud or whispering like a scared child.\n\nNot as scared yet as it should be, no, not yet. But when people figured it out, the true panic would begin.\n\nThe Twins had pulled the trigger on massed preprogrammed attacks: Burnofsky had seen the footage of Stern. There would be more of that. Benjamin was in the driver's seat increasingly, and Benjamin would have his apocalypse.\n\nBut that was nothing compared to what Lear was doing.\n\n\"Of course it's Lear!\" Burnofsky cried aloud, as though someone was arguing with him, as though he was fighting someone to make them understand.\n\nLear. What a clever, clever fellow he had turned out to be. Burnofsky saw it all now, saw the games, saw the ultimate destructive power that flowed from Grey McLure's little lifesaving creatures.\n\nPoor old Grey. They'd been friends, he and Burnofsky. The last friend Burnofsky had had. Poor old Grey, who had gotten his panties all in a twist when he learned Burnofsky was weaponizing nanobots for the Armstrongs. A lovely idealist, old Grey. A good man who just wanted to save his sick, dying wife.\n\nHad he ever realized the destructive potential in his creatures? Had he even an inkling of what they could do in the wrong hands?\n\nFucking idealists. They were ever so useful to those with evil minds.\n\nIt was all coming down, Burnofsky thought. And when it did, the Twins were going to kill him. Kill him or rewire him.\n\nThat second thing made his stomach turn. He had endured it once, was still enduring it. But like many traumas, the threat of a repeat performance was even worse. He could _not_ be used this way; he couldn't be turned into some computer made of meat, rewrite, delete, up-arrow, down-arrow, parentheses, backslash....\n\nHe had in some way accepted the first wiring as a sort of penance. He was a sinner, a terrible sinner, and he had deserved the punishment of having his mind crudely twisted this way and that. But not again. Not again. He had paid. Paid enough.\n\n_Not again_.\n\nDeath? Death was nothing. Death was relief of pain.\n\nThat's what he had told himself while sitting in his grim apartment with a gun in his mouth. He had lacked the courage to do it. But he would die before he would let them treat him like nothing. Like _nothing_.\n\n\"I paid,\" he told the camera he knew was watching him. His lip curled into a vicious sneer. \"I _paid_!\"\n\nHis mind went inevitably to Carla, and the sickening result of that thought, the excitement, the pleasure of it. And with it the awful need to hurt himself.\n\nHe lit a cigarette. He watched the end burn a bright orange. The smoke curling, teasing the end of his nose, making his watery eyes water still more. Not tears, though. Not tears.\n\nThe Pope\u2014that would push the Twins over the edge. They would have to realize that they were no longer the masterminds, just two more suckers playing Lear's game. And then? Benjamin wouldn't stand for it, oh, no. Lear would not take Benjamin's G _\u00f6tterd\u00e4mmerung_ from him. Benjamin would lose it, lash out, and at last unleash the gray goo.\n\nHe would use Burnofsky for that. Yes, of course. Burnofsky would serve the Twins one last time and destroy the world before Lear could do it.\n\nIt was funny, really\u2014despite the way his eyes watered\u2014it was funny, funny to think that in the end it would not be a race between destruction and salvation for humanity, but a race between two different lunatics, Benjamin and Lear, both bent on annihilation.\n\nWell. Maybe not two.\n\nNanobots were _his_ creation, not Benjamin's. Poor old Grey had died in a fiery crash, lucky bastard, and his creations had become Lear's. But Burnofsky still lived. Would go on living, probably, until the Twins decided they had squeezed the last from him.\n\n\"They're mine,\" he muttered, looking at a schematic of a nanobot. \"I deserve a fucking prize. Hah! I deserve the fucking Nobel, hah!\" Well, that was over, wasn't it. Lear had sort of killed that whole thing, hadn't he?\n\nThere was a bottle in the desk of one of his assistants. He had seen it, but he'd never said anything about it. It wasn't his job to preach abstinence.\n\n\"It's all coming down, anyway,\" he muttered. \"Twist me this way, twist me that way; in the end it's all death.\"\n\nHe watched the thoughts in his own mind, tracked them like the scientist he was. Not so easy, really, to predict the outcome of wiring, eh, Nijinsky? Poor dumb Bug Man had learned that when the president went off the rails. Not so easy.\n\nFive minutes later the alcohol was raw in his throat and warm in his belly.\n\n\"I paid,\" he said. And hurt himself again with a deep, deep swig.\n\n\"Hah!\" Burnofsky said. \"Fuck it. Fuck it all.\"\n\nHis phone lay on the desk. He blinked at it. The icon for messages showed a three.\n\nNo one texted Burnofsky. In fact, he couldn't recall the last time he'd had a text.\n\nHe almost didn't look, but even carried off on a happy wave of blessed alcohol, he was still a servant to his own curiosity.\n\n_It's Bug. Bad shit happening. Crazy bitch I think is Lear. Going to kill me and the whole damn world_.\n\nThen:\n\n_Are u there? Talk to me! I'm not playing_.\n\nThen:\n\n_Fuck! Do NOT call back. I'm using her phone. Can't wait. I'll try again later_.\n\nBurnofsky stared at the messages. His first thought: _Anthony's alive still? The Twins must be slipping_.\n\nAnd then, _Jesus, he's fallen in with Lear?_ And that made him laugh. Of _course_ Anthony would end up back in some kind of world of shit. Of course he would.\n\nAnd then he saw the words _bitch_ and _her_.\n\n_Okay_ , he told himself, tamping down his excitement. _Bitch_ could be slang for anyone, male or female. And the difference between _he_ and _her_ could be a simple mistyped letter.\n\n_Going to kill me and the whole damned world_.\n\nThe Golden Hall in Stockholm. The Brazilians. That actress. The prince. The Pope. Even that slimy prick Nijinsky, of whose death he had at last learned.\n\n\"Yes, biot madness,\" Burnofsky said. He gave himself a deep swig, feeling the savage joy of destruction, and the subtler pleasure of having his theory, his educated guess, ratified. He felt as if he was vibrating from the rush of discovery, the way he had when he used to make breakthroughs in the lab.\n\nHe could call the phone number back. Who would answer? Bug Man? Or Lear?\n\n\"Biot madness. Jesus Christ,\" he said, voice an indecipherable slur. \"BZRK. It's a joke. It's a goddamned joke.\" Then, pushing himself back from his desk, shaking his head, he whispered, \"Ah, no, not a joke: a game.\"\n\nWith trembling fingers he hit the call-back button.\n\n_The number you have dialed is not a working number_.\n\nNo, of course not. Lear would swap phones regularly, clever boy. Or girl. Could it really be a woman? Was a woman capable of such malice? Would a woman play this game?\n\nHe opened his browser. How did one get samples of DNA? Any hospital, sure, but these were samples from multiple countries. Who could do that?\n\nHe searched _medical testing labs_ , impeded somewhat by the fact that he couldn't quite direct his fingers to hit the right keys. He came up with too many results. He added qualifiers and came up with fewer hits. Then refined the search further to focus on corporate structures.\n\nThere were only six that were privately owned and reached beyond just North America. One of them, Janklow\/MediStat, had recently lost its owner in an unfortunate boating accident.\n\n\"Accident. Hah.\"\n\nAmong those present on the boat and giving statements to the police was another\u2014competing\u2014corporate titan. A woman.\n\nThree seconds later he had a photograph looking at him from his monitor, the face of Lystra Ellen Alice Reid.\n\nL. E. A. R.\n\nHe stared at the picture. He laughed. _My God, a pretty young woman_. That face, that serious, intelligent, attractive face hid a madness as profound as anything bubbling beneath the surface of two men so hideous they couldn't walk down the street.\n\nLear. Self-aware madness, then. She knew, Lear did; the creator of BZRK knew she was mad. The wicked thing. The wicked, wicked thing.\n\nHe raised his mostly empty bottle to her in a wry toast. \"Nice. Very nice.\"\n\nHe could probably stop her. He could take this to the Twins, and _they_ could probably stop her.\n\n\"I could save the world,\" Burnofsky said, his tone mocking.\n\nIn the old days, he might have been tempted by that. He could be a hero. A hero\/murderer. A hero who had helped to cause the suicide of the president of the United States. _Right. Hero_.\n\nHe could save the world.\n\n_Or_.\n\nOr he could beat Lear to the punch and shove it in the Twins' face as well.\n\n_Hero? Sorry, Dr. Burnofsky, that role is no longer available to you. How about killer? How about destroyer of worlds?_\n\nYeah, that position was still open.\n\n\"Why?\" Keats asked it, though he knew it was foolish. How did you ask for explanations when the person you were asking might be wired himself? But he asked, anyway. \"Why, Vincent?\"\n\n\"I had my orders. From Lear. He gave me instructions.\"\n\nPlath licked her lips, nervous, angry, but feeling as if she should be far angrier. _Knowing_ she should be far angrier. But somehow the emotion didn't quite come. The rage did not rise in her. \"Tell me what he instructed,\" she said, her voice roughened to simulate the emotion she did not feel.\n\n\"He said to wire you. To reduce your skepticism. To avoid suspicion of Lear. Or of me.\"\n\n\"What else?\"\n\n\"Seventy percent,\" Wilkes snarled. \"Right.\"\n\n\"There are... holes in my mind,\" Vincent said. \"I feel them. I know something is wrong with me. That's not a lie.\"\n\n\"Asshole!\" Wilkes said, far more furious than Plath.\n\nPlath raised a hand to silence Wilkes. \"Tell me the rest,\" she said to Vincent.\n\nAnya sat beside Vincent, who seemed terribly small. She was weeping quietly, holding one of his hands in hers. Expecting him to be killed.\n\nPlath saw herself through Anya's eye. She saw a grim-faced girl, a sixteen-year-old girl with freckles for God's sake, with stupid freckles. That picture of her finally brought the true emotion to the surface, but the emotion was disgust. Plath was disgusted with herself, with what she had become.\n\n\"Tell me what you did about the Tulip, Vincent,\" Plath said.\n\nVincent flinched and broke eye contact. \"You know what I did.\"\n\n\"You wired the Tulip and the Twin Towers together,\" Plath said bitterly. \"And you looped in, what? Something that would make me less questioning, something...\"\n\n\"I wired the memories of the Towers, the Tulip, and your pleasure centers, all together,\" Vincent said. \"They were Lear's orders. That's what he wanted. I...\" He looked at Anya, and now Plath was looking at Vincent through her own eyes and through Anya's. \"I\u2014\"\n\n\"Did you at least argue? Did you at least question?\" This was Keats now, raging. \"Did you not say, 'What the hell are you talking about?' Did you not say, 'How dare you?' Did you not tell Lear to _go fuck himself_?\" Keats looked as if he might beat Vincent to death right then and there.\n\n\"He's been wired, too,\" Plath said wearily. \"We burned holes in his brain to save him, but someone else saw an opportunity in that. We never checked because... well, because we felt so sad and guilty. But Lear got to Vincent, Vincent got to me.\"\n\n\"Who would have wired Vincent?\" Keats asked, but even as the words were forming, he saw the answer. \"Nijinsky. It's like a disease. Lear to Nijinksy, Nijinsky to Vincent, Vincent to Plath. Like a virus.\"\n\n\"Vincent, walk your biot out of me,\" Plath ordered. \"Do it now.\"\n\nVincent said nothing, just looked at her, so Wilkes dropped down beside him, clapped a friendly hand on his leg, and in a flash there was a knife in her free hand. She jabbed the point against his carotid artery. \"Do what the lady says, Vincent. Or I have to kill you. Give up the one you have in her, and your others, too. It's either that or you die.\"\n\n\"Do it, Vincent,\" Anya pleaded. \"For me, do it.\"\n\nHalf an hour later Vincent's biots were in a vial hanging from Plath's neck. Vincent, the once-invincible Vincent, was still just seventy percent. But he was one hundred percent in Plath's power.\n\n# **STATE OF PLAY**\n\nEnough dots had been connected. But twenty-four hours after the day of the prince and the Pope, no one had an explanation. The prince was locked in a comfortable room in the palace and tranquilized to near coma.\n\nHis Holiness was locked in comfortable rooms at the Vatican, tied to his bed, and tranquilized to near coma.\n\nStockholm was fresh out of psychiatric beds in its institutions.\n\nAnd then the new head of Wells Fargo bank drove her car off a bridge.\n\nAnd several hours after that the Ayatollah Aliabadi was discovered amid broken glass cutting his wrists.\n\nAnd the fashion model who leapt out of a tenth-floor window in Kyoto.\n\nAnd the rock star who stormed offstage at a concert in Toronto, only to return a few minutes later, armed with a pistol, which he emptied into the audience, killing one and injuring four.\n\nAnd the president of the World Bank who swam frantically into the Baltic Sea in freezing conditions. He was rescued but had to be confined.\n\nIt soon became hard to keep track of.\n\nThe Christmas Crazy, it was called, though it had begun earlier. The Season of Hope, as some faiths called it, had taken a very grim turn.\n\nThe world was on edge. The world was baffled and frightened but still somewhat amused, as it was only prominent people being affected by whatever bizarre syndrome was occurring.\n\nBut then a tenth-grade teacher in Larkspur, California, began attacking students with a knife. Five injured, one critically.\n\nAnd a soldier at Fort Belvoir, Virginia, grabbed his AR-15 and began shooting up the officer's club. Seven dead, three critically wounded.\n\nThe word was out: listen for people who claim to suddenly be seeing things in their head. Especially if they claim to be seeing strange insects.\n\nGrab them, restrain them immediately, or at least get the hell out of the vicinity.\n\nAll over the world events were being postponed or canceled. All over the world people eyed each other with suspicion bordering on paranoia.\n\nThen... nothing. For twenty-four hours.\n\nSome dared to hope that it\u2014whatever _it_ was\u2014was over.\n\nOthers wondered if whoever was behind this\u2014aliens were the top choice\u2014was just taking a break in order to build up to something even more unsettling.\n\nThe Centers for Disease Control and counterparts all over the world were in panic mode, searching for the cause, or at least the common thread. But it was a business consultant, who worked frequently with major medical clients, who made the tentative connection on his blog.\n\nThis person, David Schiller, sixty-three, suggested that, based on limited available data, it seemed those affected were more likely than the norm to have had lab work done. Medical tests. Blood tests. Urine samples.\n\nHe wrote this up on his blog. The dozen or so readers who saw the post wrote in comments that they would be very interested to see this developed further.\n\nSadly Schiller was unable to post a follow-up, as six hours after he published his blog post he was arrested by Chicago police for barbecuing one of his beloved Samoyeds on a fire he built in his front yard.\n\nHis blog was hacked and deleted.\n\nThe world was frightened. On edge. Desperate for some peace or some explanation.\n\nThe world was ready.\n\nAnd so was Lear.\n\n# **TWENTY-TWO**\n\n\"I don't know how I feel,\" Plath said. \"I feel...\"\n\n\"It's probably weird,\" Keats said.\n\n\"Hollow.\"\n\nThey had walked out of the safe house, both feeling that there was too little air in the place, both needing to be reassured that the outside world still existed.\n\nThey looked at each other, and Keats knew that a vast distance had opened up between them. He had wanted nothing so much as to close the much smaller distance that had persisted, even during the idyll on \u00cele Sainte-Marie. Instead, he had dug the Grand Canyon and now looked at her across it.\n\nShe had not raged at him. She seemed too tired to be very angry that he had unwired her without permission, in fact in direct rejection of her wishes. The temperature of their conversation was cold, not hot. She stood with her hands down at her sides. Her eyes were as big as ever, but now they seemed to be looking just past him, refusing to make eye contact.\n\n\"I'm sorry,\" he said.\n\n\"No. You did the right thing,\" she said. There was no reassurance in her tone.\n\n\"Brains are complicated,\" Keats said pointlessly.\n\n\"Mmm. Yeah. Complicated.\"\n\n\"I have to ask...\"\n\n\"What?\" She frowned, wishing he would go away and let her adjust to this feeling of emptiness. She felt nauseous. She felt as if she might at some point throw up. She felt _not herself_ , like this was not her body, like she was a head transplant attached to some new torso. Alien.\n\n\"I have to ask what you told Caligula.\"\n\n\"Caligula? Nothing. I can't text him or call him.\"\n\nKeats wanted to heave a sigh of relief, but it might have seemed as if he hadn't trusted her. Then...\n\n\"I told Lear.\"\n\nHis blue eyes snapped up to hers, and his brows lowered. \"Told Lear what?\"\n\n\"That Caligula should do it.\"\n\n\"What?\" He grabbed her shoulders. \"You gave the go-ahead to blow up the Tulip?\"\n\nShe nodded. No emotion. Yes, she had ordered up an atrocity. No emotion. Yes, she had ordered mass murder. Nothing.\n\n\"Jesus, Sadie,\" he said, and his voice broke.\n\nShe blinked, taken aback by his reaction. \"It's okay,\" she said.\n\nHe released her, and now he was no longer looking at her, he was staring into the window in his mind where his biot's visual flow would be. But the images were grainy and indistinct. The distance was too great. The biot he had in Caligula's head was too far out of range for useful input.\n\n\"We have to stop him,\" Keats said.\n\n\"Uh-huh,\" she said indifferently.\n\nWhat happened next was pure instinct, and he regretted it even as his hand was flying through the air, even as the flat of his palm connected with the side of her face with enough force to snap her head around and start the tears in her eyes.\n\nWhen her eyes came back around there was emotion. Anger. Finally, anger.\n\n\"Listen to me,\" he said, regretful but determined, too. \"We have to stop him.\"\n\n\"Don't you fucking hit me,\" she snarled.\n\n\"Good. You're not dead yet, are you? I'm sorry about the slap, but you sound like you're in a coma.\"\n\n\"And who put me there?\" she demanded.\n\n\"Lear put you there!\" he said. \"This has all been a game for him. We wanted to stop one evil, so we never even asked questions about whether the man we served was just as bad. Or worse.\" He felt her attention slipping away and wanted to grab her but knew that would be wrong. So he leaned closer to her, bending down so that she could not avoid looking at him. \"Madness like a bloody plague. All over. It's all _Lear_. It's Lear making biots and then killing them to drive people mad. Hundreds of dead already. The Pope went mad and attacked little children. Sadie, that's his game.\"\n\n\"The Pope?\"\n\n\"Lear. _Lear!_ And we have to stop him. We have to stop Lear!\"\n\n\"The Twins,\" she said, sounding vague.\n\n\"Yeah, them, too,\" Keats said. \"Come on.\"\n\nHe grabbed her hand and yanked her along with him.\n\nWilkes stood up as they burst into the living room. Billy was absorbed in his phone.\n\n\"Caligula's going to blow up the Tulip,\" Keats said. \"We have to stop him.\"\n\n\"Blow up the Tulip?\" Wilkes said. \"I thought that\u2014\"\n\n\"Yeah, well, it's back on.\"\n\n\"You're going to stop Caligula?\" Wilkes demanded skeptically. \"You and what army, pretty blue eyes? You've seen him in action. This won't be biot war; this will be kill or get killed, with a dude who is a genius at killing!\"\n\n\"We have a gun. Just one. It's\u2014\"\n\n\"It's in the drawer in the kitchen, below the silverware.\" This from Billy. Casual, as though it was no big thing that he knew where they'd hidden a gun. Then, \"It's a Colt forty-five. Seven-round clip. One spare clip. We have a total of fourteen bullets.\"\n\n\"I'll do the best I can with it,\" Keats said, knowing in his heart that it wouldn't work, knowing\u2014because, yes, he _had_ seen Caligula work\u2014that Caligula would kill him before he fired a shot.\n\nWilkes, seeing that despair, shook her head and said, \"Keats, you aren't a gunman. Neither am I.\" She looked pointedly at Billy.\n\n\"Yeah,\" the young boy said. \"I can do it.\"\n\n\"No,\" Keats said. He shook his head. \"No. That's wrong. That's over the line.\"\n\n\"It is over the line.\" They stopped, looking back almost guiltily, to see Plath.\n\n\"This doesn't involve you,\" Keats said. He didn't mean it to sound angry, but it did.\n\nPlath shook her head. \"Of course it involves me, Noah. I gave the order to Caligula.\"\n\n\"Can you just take back the order?\" Billy asked.\n\nPlath shook her head. \"That was all part of the game. Lear's game. For whatever sick reason he wanted me to choose to do it. But that doesn't mean he'll stop just because I change my mind.\"\n\nShe looked around at them, defiant, defying her own shame. \"I...\" She sighed and shook her head. \"I don't know... how my brain is...\" She sighed again. \"I don't know anything, I guess. I was used. I was controlled, but then, even after you guys...\" She made a gesture with her hand, as if she was pulling something out of her head. \"I still wanted revenge, and the wiring played into that. I still wanted revenge. I guess I do even now. But yeah, what Mr. Stern told me... There has to be a line.\"\n\nKeats saw tears flowing and his heart yearned to touch her, to take her in his arms and protect her. But that felt impossible now.\n\n\"I thought of what my dad would say,\" Plath said, dejectedly. \"My dad, my brother... there still has to be some kind of limit. A line drawn.\" She wiped away the tears, then, resolute, said, \"So we stop Caligula. But. But we give the gun to the kid.\"\n\nCaligula disliked disguise, but he knew how to use it. There were two approaches. You could either become part of the background and therefore be ignored\u2014like a janitor. Or you could pass yourself off more boldly, pretending to be someone in authority, someone who would compel obedience. For example, pretending to be a cop.\n\nAnd Caligula understood diversion. He'd spent a part of his life working the carnival as a trick shooter and knife thrower, and he'd met his share of magicians. Sleight of hand was all about misdirection: look over there, not over here.\n\nFinally he understood simple brutality.\n\nAll three were required to gain access to the subfloors of the Tulip.\n\nHe dressed as a janitor, having first determined what the AFGC janitors wore and when they worked and through which entrance they came. He gathered his long gray hair into a bun and pushed it up under a do-rag, slipped into gray-blue overalls, and, crucially, applied just enough dark makeup to be arguably Mexican. In the world as it was, a dark-skinned older man dressed as a janitor was as close to invisible as it was possible to be. It wasn't just that people didn't notice you; it was that they actively avoided making eye contact with you or noting any feature.\n\nBut timed to coincide with his fraught passage through the security station on the first subfloor, Caligula arranged a distraction in the form of a call to NYPD claiming to have seen a homeless white woman waving a knife and threatening people on the street in front of the Tulip.\n\nThe choice of a fictional white woman was important, because it in no way pointed to a theoretically Hispanic janitor. And, as well, there actually was a homeless white woman with a shopping cart on the sidewalk. Five bucks and a secret message from \"aliens\" in the person of Caligula had ensured her presence.\n\nThe police duly came roaring up. The security guards at the entrance duly ran to see what was happening. And the sublevel security guards duly glued themselves to their monitors and muttered jealously about _those guys upstairs who at least have something going on_.\n\nCaligula dragged a heavy floor cleaner past without notice.\n\nDown the stairs. One level. Two. The door was locked but was easily defeated with a six-inch segment of metal venetian blind.\n\nThe temperature went up ten degrees from stairwell to mechanical room. As mechanical rooms went it was a nice one, three stories from grated floor to pipe-crossed ceiling, with catwalks offering access to massive blowers, electrical boxes, alarm systems, and telephone and cable panels.\n\nEverything was color coded, so it was easy to pick out the natural gas pipes from the water lines and cable conduits. _Red. An interesting choice_ , Caligula thought. He might have gone with lilac. He liked lilac.\n\nThe first job was to make sure no one was down here. He walked around, looking lost with his floor cleaner until he located the engineer on duty. He was a middle-aged man, staring at an iPad propped in front of the readouts he was meant to be watching.\n\nCaligula gave him a hello wave and a hatchet in the neck, stepping nimbly out of the way of the blood spray.\n\nHe squeezed fast-drying epoxy around the edges of the doors. He looked around, spotted a metal table, dragged it over to the door, tipped it on its side, and epoxied it across the door. Once the epoxy had hardened in twenty minutes, it would take a tank to break through. He would leave via the freight elevators, which he'd be able to watch more easily.\n\nThe next thing was to eliminate any source of spark. It wouldn't do to have the gas ignite too soon. He turned off the heating system. He decided to accept the risk of a random spark from one of the electrical panels\u2014unlikely, given the pristine newness of the building.\n\nThen he located the safety shutoffs that would choke off the gas in the event that the computers decided a pipe had ruptured. He jammed that useful piece of equipment with a wrench.\n\nWhich left only the last three phases: opening the flow, setting the timer on the igniting explosive, and getting the hell out of the place before it blew up.\n\nAbout twenty floors above Caligula, Burnofsky worked. The beautiful thing about nanotech, he thought, was the whole nano thing itself. Nano: small. Tiny. Invisible to the human eye.\n\nHe could begin growing self-replicating nanobots within full view of the hidden cameras. A million of them looked like a couple of handfuls of dust. Blue dust, in this case, because in a moment of wracking guilt back before\u2014before the new Burnofsky\u2014he had given them the color of his daughter's eyes. He'd done that as a strange expiation. An homage? Was that the right word?\n\nHe was still secretive about drinking the booze. He rolled his wheeled office chair back into a blind spot, poured into an empty soda can, then rolled back into view.\n\nWere the Twins watching? He didn't really care, so long as they didn't try to stop him.\n\nHe had ten million SRNs so far. SRNs with no limits. SRNs that would replicate and replicate, doubling in number and doubling again and again and again until there were not millions but billions, trillions, as many as there were grains of sand on all the beaches of the world.\n\nWhat was the famous quote from 1984? He Googled it. He wanted to get it right. Ah, there.\n\n_If you want a picture of the future, imagine a boot stamping on a human face\u2014forever_.\n\n_Well, no, Mr. Orwell_ , Burnofsky thought. _If you want a picture of the future, imagine a world scoured clean of every living thing_. And more. Imagine that having taken and used all the easiest forms of carbon the SRNs keep going. They eat the steel out of buildings, the coal and oil and diamond out of the earth itself. They wouldn't just destroy all life, they would relentlessly remove all possibility that life would ever again arise to trouble an empty planet.\n\nHis eye scanned down the page of quotes from Orwell and came to rest on this: _Power is in tearing human minds to pieces and putting them together again in new shapes of your own choosing_.\n\nHah. Well, the Twins had tried it. The Twins, with their mad plan to unite all of humanity into one vast interconnectedness. A new world where they would be accepted. And more than accepted: esteemed, loved.\n\nAnd Lear? What was young Lystra Reid's motive?\n\nBurnofsky's own motive was clear to him now. He had done evil for ambition's sake. He had tortured himself for that evil and sought to close the eyes of the world to his shame.\n\nThen he had been rewired so that the evil gave him pleasure. And now he would close the eyes of the world because it would bring him pleasure.\n\nHe would wait for a few more doublings. Then he would drop the force fields that held the SRNs contained and unleash the gray goo.\n\nThen? Well, then he would go back to his old haunt, back to the China Bone. There would be time for them to prepare him a pipe. He would float on a cloud of purple opium haze and wait for the end of the world. When the nanobots reached him, well, that's when he would take a last drink and fire the heroin into his veins and leave the world behind, dying with two raised middle fingers to humanity.\n\nSuarez wished she had music, but the cockpit was not large enough to allow her to reach for her headphones. It was just that the mad rush of sheer speed demanded some propulsive music to go along with it.\n\nIt was crazy. It was also crazy fun.\n\nThe sleigh was a dream to drive. Computer-assists and automated systems made it more like a video game than a craft moving at a hundred and fifty miles an hour. The little icon that was the sleigh on the GPS display was zooming along past... well, past nothing, really. This was Antarctica. There were no towns, houses, roads, or any feature, really, aside from the blur of ice and the gray blanket of hazy, overcast sky. The target was coming closer very fast, and she didn't really have much of a plan for how to approach it.\n\nMost likely roaring in at three times freeway speeds\u2014with jet engines screaming and ice crystals trailing a plume\u2014was a bad idea. The smart move would probably be to park the sleigh a few miles away from the target and walk in on foot. Much more subtle that way. But on the ice one did not casually decide to abandon a vehicle that provided shelter and warmth. Not to mention a vehicle with an impressive array of weapons.\n\nSo she would try to bluff it out. Whoever she encountered would probably suspect her cover story was nonsense, but what could they do about it, really?\n\n\"People who buy illegal missiles and smuggle them onto the ice?\" she said aloud. \"Plenty. That's what they can do: plenty.\"\n\nOn the other hand, Suarez was only the third woman ever to qualify for SEALs, so she was no weakling. She was formidable.\n\n\"That's right, talk yourself into it,\" she muttered.\n\nIt was a good thing she had the computer navigating, because she would never have seen the dry valley. It was a rift in the ice, which at this point was a relatively sparse two hundred meters thick.\n\nPart of her just wanted to keep shooting across the ice, but she slowed reluctantly and nosed the sleigh closer to the edge of the valley. Still she could see nothing ahead of her but ice and more ice.\n\n\"On foot it is, then,\" she muttered. With great reluctance she raised the canopy, unwedged herself, and managed to climb rather ungracefully out. The wind was doing thirty knots\u2014gentle by local standards\u2014just enough to push cold stilettos through every seam, every zipper, every opening in her parka. She pulled on her bear claw mittens, cinched the neck strap a bit tighter, and leaned into the wind to walk what seemed to be the last hundred yards.\n\nShe didn't see the edge of the precipice until she was practically on top of it.\n\n\"Whoa.\" Antarctica had millions of square miles of same-old, same-old, but hidden here and there in the largely unexplored continent there were features that would take your breath away. This was one of those times.\n\nThe ice fell away in a sheer cliff just beyond the toes of her boots. Spread out below her was a narrow valley shaped like a boat hull\u2014pointed at the near end, rounded at the far end. Suarez's position would correspond to just off the starboard bow.\n\nThe topography was not complicated; it was effectively a big, oblong hole in the ice, maybe two kilometers long, a quarter that in width at the widest point. The floor of the valley was reddish gravel and looked like an abandoned quarry. About where the cabin would be on a yacht was a series of structures\u2014four buildings, one quite large by polar standards, and, sure enough, under a plastic dome there was unmistakably a swimming pool.\n\nSomebody really liked to swim.\n\nSuarez carefully absorbed the layout. The structure with the two stubby towers would be the power plant. The largest building was some sort of hangar or factory space, unmistakably utilitarian. The third, a two-story L-shape, would be a barracks. Room for, what, fourteen, sixteen rooms plus a common space? So not a huge contingent.\n\nAnd finally what looked very much like a private home, done with a reckless disregard for energy conservation, a three-story, ultra-modern, Scandinavian-looking thing with the kind of floor-to-ceiling windows you just didn't see in Antarctica, and a plastic tunnel running to the pool.\n\nInstead of the inevitable Sno-Cats there was a pair of Audi SUVs parked outside the house, as though the occupant might have kids and pets who needed transportation to the nearest soccer field or dog park.\n\nCraziness. Suarez had to laugh. It was nuts. And whoever had built it was nuts. They were hundreds of miles from the coast, which was to say a whole long way from even the thinnest edge of civilization. This might easily be the most isolated house on planet Earth. Sure as hell the furthest from a Starbucks.\n\nTwo helicopters lay all tied down and shipshape on a well-marked pad. One was an EC130, a species of chopper found all over the ice. But the other was an Apache, with missile pods on the stubby wings and a swivel-mounted thirty-millimeter cannon.\n\nWho had the kind of political juice to get hold of a freaking Apache?\n\nHow had this place even been discovered? Either someone had amazing luck, or they had some amazing satellite imagery. Speaking of which, a satellite would have to be directly overhead to see the valley at all.\n\nSuarez peered off to the far end of the valley. Was that a road? It _looked_ like a road cut into the ice wall, rising at a steep grade up to the outside. Could she drive the sleigh down there? Bigger question: Could she get it back out?\n\nThe smart thing to do would be to take some pictures and get the hell out of here, get back to Tanner and let him take it from there. That was definitely the smart move.\n\nYes.\n\nSo Suarez, face already numb and hands getting there, climbed back into the sleigh and drove at safe speed around to the head of the road. It was definitely a road, hopefully just wide enough to allow a large truck\u2014or the sleigh\u2014to avoid going off the side.\n\n\"All in now,\" she said to herself, and sent the sleigh creeping down the incline. This proved to be as tricky as she'd thought it might be. There was no pavement, of course, just icy gravel, dropping what looked to be about one hundred meters in the course of a quarter mile, one heck of a grade. The sleigh, like any hovercraft, was not well suited to going downhill in a controlled manner and started to slide sideways almost immediately, but Suarez got the hang of it and made it to the valley floor without plunging off the side of the ice ramp.\n\nStill no one came rushing to greet or confront her. Either would almost have been welcome merely for the purpose of ending the suspense. The sleigh was more than capable of crossing gravel, but now, down inside the rift, it was a high-strung Thoroughbred in a too-narrow paddock.\n\nWas the whole place abandoned? Obviously anyone there would have heard, if not seen, the sleigh. You don't exactly sneak up on people when you're in a jet-powered hovercraft.\n\nShe throttled forward, creeping along at walking speed, aiming for the bizarre house\u2014which was no less bizarre down here at eye level. The glass windows were mirrored, so Suarez could not see inside and instead saw the reflection of herself scowling from beneath the canopy.\n\nFinally, lacking any better idea of what to do, she parked the sleigh, killed the engines, and climbed out again. There was no wind, which did not make it any warmer but did reduce the effects a bit. Her breath rose as steam but her face, while cold, regained some of its feeling.\n\n_Now what? Ring the doorbell?_\n\nShe crunched across the gravel to the door of the house. There was no bell. So she stripped off one glove and knocked.\n\nBad idea. The door was steel and very, very cold. Had she knocked any slower she'd have left knuckle skin behind.\n\nNo answer.\n\n\"This is insane.\"\n\nThe next likely target was the large\u2014quite large\u2014building fifty meters away.\n\nThere would presumably be a satellite phone in the buildings, or an Internet connection\u2014some way to inform Tanner. The question was whether there would also be men and women with guns. It all had an empty but not-quite-abandoned feel.\n\nThe building was not locked. Well, why would you lock a building that was a million miles from nothing much? But it bothered her. This was all too easy.\n\nShe pushed in. Lights were on, illuminating a single large open space populated by machinery of a sleek, white-coated type. She recognized only the 3D printers, monitors, and touch screens. The other objects were not familiar, and might have been anything from manufacturing to medical gear. But one particular type of equipment predominated: a chest-height, white rectangle with inexplicable slots outlined in green light. There were a lot of those. Dozens. Maybe a hundred. These definitely were some kind of manufacturing equipment, not computers. Probably automated, given that the machines were backed too closely together to allow for people to move easily around them.\n\nFascinated by trying to make sense of the machines, Suarez belatedly registered the sound of armed men taking positions behind her.\n\nSix of them. Six automatic rifles leveled at her from behind the cover of the equipment.\n\nOne of the downsides of actually being a trained fighter is that you come to accept that real life isn't Hollywood, and no one wins a fight that pits a single assault rifle against six.\n\n\"Set the gun aside. Raise your hands.\"\n\nThat's what she did.\n\n# **TWENTY-THREE**\n\nThe line of limos completely choked West 54th Street, extended back onto Seventh Avenue and then all the way up to 56th Street. Stretches, town cars, the occasional privately owned Mercedes or Tesla.\n\nCrowds pushed against barricades manned by tolerantly amused NYPD pulling down some welcome overtime pay. Satellite trucks had been parked right on the sidewalk across from the Bow Tie Ziegfeld Theater, which all by themselves doubled the congestion.\n\nIt was not an uncommon situation. This was not New York's first big movie premiere. But even New York could not be jaded about this much star power. Every A-list actor, director, and producer was here, all of Hollywood royalty, for the premiere of the year's biggest-budget flick _Fast, Fast, Dead_ , starring, among several actual human stars, a computer-generated Marilyn Monroe that was supposed to be so indistinguishable from the long-dead real thing that there'd been some speculation about whether the program might be up for an Oscar nomination.\n\nLystra Reid had managed an invitation for herself and a plus one. The plus one was at least ten years her junior, but this was a Hollywood crowd, and if the relatively unknown but reputed to be fabulously rich woman wanted a young, black, not terribly attractive boy toy who had clearly been on the wrong end of either a bar fight or a car accident, hey, who cared, really? The country had bigger problems.\n\n\"We're toward the back,\" Lystra said, guiding Bug Man in.\n\nDespite the rocketing pain of his broken teeth and the split, swollen lips, Bug Man was enthralled. Obviously Lystra was up to something horrific, but in the meantime Bug Man played Spot the Star. Seeing a very familiar face, he said, \"Man, I ufed to cruff on her when I wa' a little ki' looking a' Harry Potter,\" he said.\n\n\"Watching what?\"\n\n\"Harry... Never mind.\" Lystra Reid was not big on popular culture, Bug Man had decided. And talking was painful and difficult, though he was adjusting to the lack of front teeth.\n\n\"I' tha' Gwynneff?\" Bug Man asked, but of course Lystra was paying no attention. And the star-dense crowd was not looking at Bug Man. They were hailing old friends and talking to the roving camera crews still pushing through packed-in A-listers.\n\nThe jocularity was strained. There was not a person in the room so oblivious to all that had happened in the world that they were not nervous. Some celebrities who had initially agreed to attend had suddenly discovered that they had headaches and would need to skip the proceedings. But the ideal of \"the show must go on\" and the lure of cameras had kept numbers high enough to prevent organizers from canceling.\n\n\"Time to send a text,\" Lystra said. And when Bug Man failed to cease craning his neck to locate a particular buxom TV star, she said more pointedly, \"This is going to go bad in a few minutes. So stay close to me.\" She laid a hand gently on his swollen cheek. \"I wouldn't want anything to happen to you. And don't forget: you wouldn't want anything to happen to me.\"\n\nShe thumbed in a text that went to the closest cell-phone tower, from there to a satellite, and from that satellite to a touch screen almost ten thousand miles away and very far to the south.\n\nShe almost blew the timing. Another minute or two and the cameras would have turned off their lights and been hustled from the theater by security people and public relations folk.\n\nLystra wanted cameras. All the cameras.\n\nThe first person to cry out in a startled voice that carried even over the hubbub was a big man with a big voice who said, \"I see bugs!\"\n\nTen seconds later, another voice, female, screamed.\n\n\"Oh my God, it's like, it's like that thing in Sweden!\" Lystra herself cried helpfully. \"That's what _they_ said. Bugs! That's what happened there! Oh my God, we're all going mad!\"\n\n\"Aaaaaahhhh!\" a man cried, and then more, and more, and all at once everyone in the theater seemed to realize what was happening. And what was coming.\n\nA well-liked hunk known for starring in superhero movies started laughing and then tried to shove his entire hand into his mouth. Bug Man stared in disbelief. It was one thing knowing that something could theoretically happen. It was a whole different thing when a Marvel superhero was trying to gag himself right in front of you.\n\n\"And now we exit,\" Lystra said. She smiled at Bug Man. She was enjoying his amazement. She'd been right to bring him along. It would be fun to have someone to share it all with afterward. \"See, this is why I've savored, yeah, a few of these events in person: video doesn't do it justice. The edge of panic. Yeah. The wild look in people's eyes.\"\n\nThe panic was like a herd of wildebeest smelling a lion. In a heartbeat hundreds of people surged toward the exits. A woman in an evening gown went down. She tried to stand, but someone tromped on the hem of her dress and she fell again.\n\nLystra and Bug Man barely made it out the door without being trampled, and she laughed as she was jostled and laughed as she was pushed hard against a door and laughed still as she spilled out onto 54th Street into the glare of lights.\n\nThey ran to get ahead of the flood, raced to get behind the cameras to watch as long as Lystra could without endangering herself too much.\n\nA bloody Broadway star was shouting nonsense syllables. A famously beautiful actor was tearing her dress off while her director crawled on all fours making a sound like a sheep. Hollywood's favorite dad was playing with his hair, twining his fingers through it and laughing hysterically.\n\nAn Oscar-winning producer launched himself at a New York police officer, pushed him to the ground, and yanked the gun from the startled cop. The first shot struck a film critic in the chest, a fact that he found funny until he fell over dead.\n\n\"Time to get out of here. Guns are dangerous,\" Lystra observed.\n\nBug Man stared at her, looked around at the madness, and back at her. At the wild glee in her eyes.\n\n\"I know how they feel,\" Lystra said without even a hint of compassion. \"I've been there. I've been crazy. It's kind of... amazing, really.\"\n\nShe turned and walked quickly away, ignoring the well-dressed lunatics rushing by, dancing, twirling and attacking each other with fists and fingernails.\n\nBug Man followed her, because Bug Man had nowhere else to go.\n\nThe trick was to pop the natural gas pipeline in a way that would not cause a spark. Caligula was not interested in suicide.\n\nPower saw and small explosive charge were both ruled out. And he didn't just want to open a smaller release valve\u2014it would take forever for the gas to build up. He needed a rupture in one of the main lines. He needed the gas to come roaring in, thousands of cubic yards of it.\n\nHe had disabled the local safety cutoffs. The next cutoff covered an entire six-block area\u2014this rupture would probably trigger it eventually, but not quickly enough.\n\nHe had brought a car jack with him, the simple, screw-type device, capable of lifting a car off the street. More than enough if he could just find the right position.\n\nHe needed a place where the pipe was rigid. Like right... _there_... where it emerged from the concrete foundation. And just two feet from that point there was a junction, a sort of flange\u2014he wasn't exactly familiar with the terminology. Perfect. If he could just get the jack between the concrete wall and the pipe standing eight inches out from same. He pulled the jack from a bag and looked at it critically. It would be a tough squeeze. The jack, even screwed all the way down, was ten inches tall. So he either had to find another place, or he would have to chip away some of the concrete to make room.\n\nHe sighed and retrieved a chisel and a rubber mallet. Slight delay, that was all.\n\n\"He's in the basement,\" Keats said. \"I can see what he's doing. He's chipping away at something with a chisel. There's still time.\"\n\nThey were outside the Tulip. In the alleyway behind. Staring at various doors\u2014two loading bays, a smaller door, a door a few dozen yards away that was vented so probably contained electrical equipment.\n\nNo clue.\n\nThere was the front door, out on the street, but that was guarded. Their weapons were: one little kid with a Colt .45.\n\nNo. Wait. There was a second weapon: the biot in Caligula's head.\n\nKeats could blind the killer.\n\nOr he could maybe slice through an artery and kill or cripple Caligula.\n\nRewiring was not going to happen in the few minutes remaining to them. There would be no time for subtlety. And if Keats moved his biot farther into Caligula's brain, he would have to detach from the optic nerve and would no longer be able to see what Caligula was doing.\n\nHow long to blind one eye? And could he reach the other eye in time to truly stop Caligula? Or would he be better off diving down deep, finding a fat artery, and sawing away?\n\nKeats felt sick inside. He had no plan. He had a Goth chick, a wild street kid with a gun, a biot, and Plath, who might or might not be entirely okay.\n\n\"What do we do, pretty blue eyes?\" Wilkes of course, jumpy, nervous, eyes darting everywhere with manic appreciation of their hopeless plight.\n\nThe door beside the second loading bay opened. Light spilled out. A man in silhouette yelled, \"Hey, move along, you three.\"\n\nBefore Keats could react there was a loud bang and a flash. A cry of pain. The man in silhouette was visible for a millisecond in the flash. He was younger than his voice, maybe twenty-five, uniformed. A security guard. A minimum-wage grunt with a hole in his chest that leaked dark blood onto khaki.\n\nBilly was moving, leapt up the four concrete steps, and grabbed the door as the man fell back.\n\n\"Jesus!\" Keats cried.\n\nWilkes was quicker, just seconds behind Billy. She grabbed the door, freeing Billy, who calmly knelt and took the dying man's gun.\n\nKeats and Plath followed, Keats feeling as if he was in a dream. Two biot windows were open in his head, one showing the damned bulge in Plath's brain, the other watching the rise and fall, rise and fall of mallet on chisel.\n\nBilly was already proffering the guard's pistol to Keats, buttforward. Keats stared at it. Wilkes took it.\n\nKeats stepped over the guard. He was crying softly and holding his wound with one hand while fumbling for his radio with the other.\n\nHe couldn't be left alive to raise the alarm.\n\nWilkes and Billy both looked at Keats expectantly. Waiting for his order. Plath seemed mesmerized.\n\n_On me, the responsibility_ , Keats thought. It had been so quick, somehow, getting to this point, the kill-or-be-killed point.\n\nBilly must have seen the answer in Keats's eyes. He squatted and pressed the muzzle directly against the man's heart, muffling the sound as much as he could.\n\n_BANG!_\n\nAnd blood sprayed across Billy's face.\n\n\"It's okay,\" Billy said. \"I did it before. Just another first-person shooter, right?\"\n\nKeats felt like throwing up. He felt a flash of fury at Plath. Shouldn't she have made the decision? Shouldn't the guilt be _hers_ to bear?\n\nThe guard was motionless now. But all was not still. They were in a short hallway\u2014barely painted drywall, weak overhead lighting, second door now opening fast, someone coming through expecting trouble, gun already leveled and\u2014\n\n_BANG!_\n\nHead shot. A single hole drilled right in the man's forehead. The back of his head\u2014a crust of skull and hair and something like hamburger\u2014hit the wall and slid down, leaving a trail.\n\n\"Go,\" Keats said, barely audible.\n\nThrough the door, now in the wide-open space within the loading bay, boxes and crates and a chair and table and playing cards laid out, and a coffee mug, and flickering monitors.\n\n\"Basement,\" Keats managed to say, trying to push aside the memory of tears rolling down a doomed man's cheeks.\n\n_One of the innocents I was trying to save_ , Keats thought.\n\n_Now, two of the innocents I was trying to save_.\n\nThey were lost and needed light. Keats spotted a bank of light switches, crossed to them, made it halfway before his stomach sent its contents burbling out of his mouth. He threw every switch, wiped his mouth, and said, \"Find a way down!\"\n\nThe glare of fluorescent light had the effect of casting deep shadows that if anything made the room seem darker, with every high-piled stack of crates like a skyscraper shadowing narrow alleys.\n\nThey ran then, moved forward, the young sociopath with the name of a young sociopath leading the way. Billy moved like a cop, cover to cover, gun steadied in both hands, a goddamned gamer, a goddamned game, where would the bad guys pop up next?\n\n\"Whoa. Down here,\" Wilkes said, waving her own gun fecklessly toward a dark hallway.\n\nBilly moved smoothly ahead of her. Cover. Pause. Scan. Run to cover. Pause. Scan.\n\nA freight elevator, with buttons for up and down.\n\n\"Down,\" Keats said, feeling useless and now seeing flashes of his London home, so squalid and dull all his life, but now so beloved, so _needed_. To crawl into his own bed...\n\nThe elevator door opened on a guard with headphones in and singing along tunelessly, yet Keats recognized the song.\n\n\"Born This Way.\" An old Gaga tune.\n\nKeats barely flinched when Billy put a bullet into the guard's head. The bullet must have hit just wrong because it entered the forehead and blew an exit wound out through the man's jaw.\n\nThe ricochet could have killed one of them, but no, and the man went down with such completeness that he might have been a dropped sack of garbage.\n\nWilkes dragged the dead man off the elevator.\n\nButtons. Three different sublevels. Where was Caligula? Go all the way down. Why not? Gates of hell. Keats punched the S3 button.\n\nThe elevator doors closed over smeared blood.\n\nBilly popped the clip from the gun, counted the bullets and said, \"I don't think there are upgrades or reloads in this game.\"\n\nIt struck Keats that if he had ever found a match for Caligula, it was this sad, sick little boy. It was not a good thought. His stomach was empty, and the smell of his vomit filled the padding-walled elevator as it dropped beneath them.\n\nKeats had kept his place on Caligula's optic nerve. He saw the sudden cessation of hammering. The visual field swirled as Caligula moved quickly.\n\n\"He's heard us!\"\n\n\"Up against the walls, hide under the blankets!\" Billy yelled in high-pitched excitement.\n\nThe blankets were the padding hung to protect the elevator walls. Wilkes and Keats dived under. Too late Keats saw that Plath hadn't moved.\n\nBilly stood waiting, gun drawn and leveled.\n\nKeats saw Caligula rushing toward the elevator, stopping, ducking behind cover. And then the peace of the game descended on Keats. It was live-or-die time. Win-or-lose time.\n\nAll his gaming life Keats had had this other place he could go, except that he didn't quite go there as an act of deliberate choice, it would just _happen_. It would come down over him\u2014a calm, a control, a speed of perception, an ease of decision making\u2014blessedly blanking out fear and self-loathing.\n\n\"He'll be to your left, Billy,\" Keats said. \"Behind a thick vertical pipe painted orange.\"\n\nBilly shifted stance without a word.\n\n\"He's expecting an adult, someone tall,\" Keats said, still deadly calm.\n\nBilly nodded and squatted. His head would be lower than a grown man's belly. Caligula would be quick, but he might hesitate on seeing a child.\n\n\"Wilkes. Give me your gun. As soon as the door opens, scream, really loud,\" Keats ordered. \"Like you need help.\"\n\nCaligula's eye was steady now, lid drooping just a bit, unafraid surely, confident that no one could beat him. Keats's biot was already busy sawing away at the massive optic nerve beneath its feet. Cutting, cutting, like trying to slice through a bridge cable with a hacksaw, but nerve fibers popped and coiled away, wildly whipping wires, and each taking with it a tiny part of Caligula's visual field.\n\nThe elevator stopped.\n\nThe door was loud as it opened.\n\nWilkes screamed, \"Help! Help me! Help me!\"\n\n_BANG!_\n\n_BANG!_\n\nBilly and Caligula fired almost simultaneously and out came Keats from behind the hanging blanket and fired wildly, _BANGBANGBANG!_ with bullets ricocheting off pipes.\n\nKeats's biot sawed madly and now more shooting, and Keats was on the floor of the elevator now, crawling on his belly, aiming, squeezing off rounds, gun bucking in his hand until it banged open, out of bullets.\n\nBilly, still standing, advanced in quickstep, running for cover, and Keats saw Caligula's eye tracking him, saw the butt of Caligula's gun as it bucked from recoil and heard the loud _BANG!_ and saw Billy the Kid's neck suddenly no longer all there.\n\nThe boy fired again as arterial blood sprayed like a cut fire hose, until his head, no longer supported, fell to one side and hung limply, and Billy fell, knees hitting the floor, then onto his back and his head bounced, barely tethered. His gun twirled across the floor leaving a blood trail.\n\nCaligula emerged from cover. He holstered his now-empty gun. Calm. No hurry. He drew his throwing hatchet, and Keats could see the killer's eye on him, on a pitiful crawling wretch, saw the way it focused on his upturned face.\n\nAlien, the sight of his own face as he waited to die. Strange and alien. Keats knew the face, knew it was him, but how could that be? The blessed peace of action had faded, and now he was a bug, a worm waiting to be crushed by the boot of the human god.\n\nThe hatchet flew.\n\nIt grazed Keats's shoulder and clattered to the floor of the elevator.\n\nWas Wilkes still screaming? Someone was.\n\nCaligula blinked. Stared.\n\nKnew. _Understood_. Because Caligula did not miss, not with gun or hatchet. He did not miss, and he knew then it could only be some fault in his vision.\n\nKeats's biot sawed and more nerves parted.\n\nCaligula drew a knife and bounded, like some bizarre kangaroo, rushing with unnatural speed. Keats saw the distance shorten in a heartbeat, saw the killer's focus, saw his own scared face, Wilkes's open mouth, a flash of Plath's hand pressing down on the door's Close button, and the elevator door closing too slowly.\n\nCaligula reached the elevator when the doors were still six inches apart. He thrust in a hand to stop it.\n\nWilkes was on him like an animal, biting the hand, snarling, shaking her head like a terrier with a rat.\n\nCaligula yelled in pain and rage.\n\nKeats saw the door from the inside.\n\nThe door through Caligula's eyes.\n\nThe knife dropped from Caligula's bloody hand, but he did not withdraw, would not let the door close.\n\nThe hatchet was in Keats's hand before he knew it. He observed it through Caligula's eye, saw the killer seeing him, saw the killer track the hatchet as it went back and came down fast and hard and Caligula tried to pull the hand back now, but Wilkes still had it in her teeth and the hatchet blade hit with a cleaver-on-bone sound, barely missing Wilkes's nose and biting deep into Caligula's flesh.\n\nWilkes recoiled then, the hand pulled away, pumping blood from the gash.\n\nKeats saw the doors close from both sides.\n\nHe saw the killer stare at his mangled hand, then through his own eyes saw the little pink curls of fingers on the elevator floor.\n\nThe elevator rose.\n\n\"Billy,\" Wilkes said. Her mouth was smeared with blood, forming a terrible rictus smile.\n\n\"Up,\" Keats said, and punched the button for the highest floor available, the third floor.\n\n\"What are we going to do?\" Wilkes asked and there was a sob in her voice.\n\n\"Surrender,\" Plath said.\n\n# **TWENTY-FOUR**\n\nSuarez was handcuffed. The handcuffs went through a chain that in turn went through a massive steel ring set into the wall at head height.\n\nThe wall was in a dungeon.\n\nThe dungeon was both frightening and absurd. There were mossy stone walls. There was straw on the stone floor. She'd been left with a rusty pail in which to do her business. The door was too low and made of flaking, unfinished wood. There was a narrow window, but when she dragged her chain over to it she saw that it was fake. The scene visible through the window was a matte painting of a medieval village.\n\n\"Cute,\" she said dryly.\n\nIt was like a movie set, or something out of a video game. Someone was having fun with the whole idea of a dungeon. Which was absurd.\n\nThe scary part came from the fact that the cuffs and chain and even the ring in the wall were all of very high-grade steel.\n\nA man who had the bearing of a former cop or soldier, a beefy, steroided thirtysomething with a crew cut, brought her dinner after a while. The tray was plastic and flimsy, no use as a weapon. The cutlery was plastic as well, and not the good kind. Water was in a plastic bottle. Wine was in a paper cup.\n\nWine, because it was quite a good meal, considering the location. Better than airline food, in any event. Wine in a dungeon.\n\n\"You have a name, soldier?\" Suarez asked as he set the plate carefully on the floor, five feet from where she sat.\n\n\"Yes, sir,\" he said reflexively. So (a) he _was_ an ex-soldier, and (b) he knew that she'd been an officer.\n\nHe flushed, realizing his mistake. Then said, \"You can call me Chesterfield.\"\n\n\"That's not your name. It's a brand of cigarette.\" When he did not demur, she said, \"So, I'm guessing the other guards will be Marlboro and Lucky Strike?\"\n\n\"Eat your food. Ma'am.\"\n\n\"Looks good. And I am hungry.\" She crawled to the food. Took a sip of the wine. \"Know what the wine is?\"\n\n\"It's French.\"\n\n\"Expensive, too, I'd guess. No point paying to ship cheap wine all the way here. Of course I'm more of a whiskey drinker.\"\n\n\"So's the boss.\"\n\n\"The boss,\" Suarez said pensively. \"The one who thinks civilization is about to crumble so she built Crazy Town here. You're not crazy, though, right? You're just here for the money? Bad economy and all, a former serviceman has bills to pay like anyone else.\"\n\n\"If the boss says it's all coming down, it's all coming down. I mean, she's probably the smartest person in the world, smarter than Dr. Stephen Hawking.\"\n\n_Dr. Stephen Hawking?_ Suarez rolled that around in her head. A strange way for a guy who looked like this to put it. _Doctor?_\n\n\"Okay, well, what do you do for fun around here while you're waiting for the apocalypse?\"\n\n\"It won't be an apocalypse for the people here; it will be a rebirth.\"\n\nNo irony in his gaze. He was dead serious. Someone had definitely sold this boy a complete bill of goods.\n\n\"Okay, which still leaves the question of what you do to pass the time?\"\n\nHe shrugged, and Suarez detected a softness in him. _I'm going to try not to kill you_ , she thought.\n\n\"I don't suppose there's any way I can get a shower? You know, hot water? Soap?\" She mimed it for him, mimicking the movements of a bar of soap over her body, not lasciviously\u2014that would be too obvious and set off alarm bells. Just... enough. She just wanted him to connect his boredom with the mental picture of a reasonably attractive woman taking a shower. Let him stew on that for a while. Activate the twin male instincts of protection and predation.\n\nLater, when the time was right, there would be the metal pail.\n\n\"No shower,\" he said in a voice just a tiny bit lower than it had been. \"I could maybe get you a deck of cards.\"\n\n\"I would be very grateful.\"\n\nThe explosion came as the elevator rose, an impact that knocked Keats, Plath, and Wilkes to their knees. Not an explosion that would bring down a building. Smaller.\n\nBut the elevator stopped moving, and the door did not open. The backlit buttons went dark. The overhead light snapped off, replaced by an eerie emergency light.\n\n\"He blew up the elevator doors down there,\" Keats said, offering his hand to Plath.\n\nShe spurned it and jumped to her feet. \"We have to get out of here.\"\n\nA second explosion, more distant this time. The second elevator.\n\n\"He's cutting himself off,\" Keats said.\n\n\"He'll die with the explosion,\" Plath said. Then, softly, \"Maybe that was the plan all along.\"\n\nWilkes had started trying to pry open the elevator doors. Keats and Plath jumped in, jamming splintering fingernails into the gap. Slowly, inch by inch, the door opened. They were between floors, but with an open space of several feet.\n\nPlath went through first, boosted by Keats. Then Wilkes. Together they hauled Keats after them.\n\nThey were on the ground floor\u2014the lobby floor, polished marble. Security guards were a swarm of uniforms and plainclothes tourists from Denver, though minus the parkas. All were armed. In seconds there were a dozen weapons pointed at the three of them.\n\n\"One move and we shoot,\" a woman snapped.\n\n\"No need,\" Plath said. \"I'm Sadie McLure. We need to talk to the Twins.\"\n\n\"And in the meantime, there's an assassin down in the basement preparing to blow this whole place up,\" Keats said.\n\nNervous glances went back and forth.\n\n\"Hey, dumb asses,\" Wilkes said. \"Shoot us or beat us up or whatever, but there is an honest-to-God stone-cold killer down there.\"\n\n\"He's wedged a car jack behind a gas pipe,\" Keats said. \"In a few minutes high-pressure gas is going to start pouring into the basement.\"\n\n\"Leave his eye,\" Plath ordered. \"Find an artery.\"\n\nKeats's eyebrow shot up at the tone of command. Plath, who had seemed almost to be comatose, now sounded like her old self.\n\n\"Kill him?\" Keats asked. He searched her eyes, not sure what he wanted the answer to be. In this very building Plath had refused to kill the Twins. She had refused to commit cold-blooded murder.\n\nMany had died since then. Much had changed.\n\nThey had just ripped m-sub yards of wire from Plath's brain, and parts of her gray matter were as raw as a skinned knee. If she gave the order, who and what would be behind it? What would be her motivation? How much responsibility would she bear in the end?\n\nAnd if she said\u2014\n\n\"Kill him,\" Plath said.\n\nAnd if she said, _Kill him_ , would he obey?\n\n\"Get them up to Jindal. Cuff 'em, keep guns on them, any bullshit, shoot 'em,\" the woman in charge snapped.\n\nThe three remaining, active members of BZRK New York were cuffed and hustled to the main bank of elevators.\n\n\"Has he blown the pipe yet?\" Plath asked Keats.\n\n\"I don't know, Keats said. \"I'm no longer on the optic nerve.\"\n\nPlath and Wilkes both knew this meant he had sent his biot to kill Caligula.\n\n\"And the last of the righteous succumbs to the darkness,\" Wilkes said mordantly, and added, \"Heh-heh-heh.\"\n\n\u00b7 \u00b7 \u00b7\n\nLystra Reid laughed like a mad thing, and to Bug Man's amazement actually executed a somersault, as crazy as the terrifyingly unhinged actors and producers and agents and whoever now baying like wolves in the streets of Manhattan, chased by cameras that broadcast the images all over the world.\n\nShe led the way to a limo and held the door open for Bug Man, who tumbled in, shaken.\n\n\"Jefuf Chri'!\" he cried.\n\n\"No, no, no, no goddamned made-up, bullshit divinities!\" Lystra yelled exultantly. \"Jesus Christ and Zeus and Mohammed and whatever the hell you want, yeah, they didn't write _this_ game!\" She fell into the seat beside him. It was as if she was drunk or high. She was cackling. \"Fuck your gods, Bug Man, I'm god now! Yeah! This is my fucking world!\"\n\nBug Man had seen some crazy in his life. He'd spoken with the Armstrong Twins, and those boys were crazy. He'd hung out with Burnofsky, not exactly a paragon of sanity. _But_ , he thought, _this chick is nuts. Once you start calling yourself \"god\" you're all the way into crazy_.\n\n_Berserk_.\n\n_BZRK_.\n\nHe was crying without quite knowing why, unless it was just some kind of overload. Too much. Too much crazy. The whole world was going crazy, and this madwoman was making sure of it.\n\n\"Wha' nef?\" he asked, both to humor her and because he needed to know.\n\n\"Stop mush mouthing,\" Lystra snapped. \"I'll tell you what next, yeah. Next, we get back to the apartment to watch the Tulip blow up,\" she said, and winked conspiratorially. With her hands she made a sort of finger explosion and said, \"Boom! Crash! Tinkle tinkle tinkle. Woosh! Screams! Cries! It's Nine\/Eleven all over, but now, yeah, the whole fucking world is going nuts! Crazy president. All the big brains? Crazy! Crazy prince. Crazy Pope! Everyone you know, yeah, is insane! And then, ah-hah-hah!\"\n\n\"Then... what?\" Bug Man asked.\n\n\"Then the Tulip comes down. And then, yeah, then, yeah, then the rest of them. Tens of thousands. Hundreds of thousands. The code is all laid in. The cr\u00e8ches are ready. Grow 'em, kill 'em. Grow 'em, kill 'em, yeah. Biot fucking apocalypse, Bug Man! Madness! Have you had blood drawn? Then I have your DNA, bitches. And, yeah, I have your biot. We can do sixteen thousand at a time. Sixteen thousand an hour. Day one? Three hundred eight-four thousand! A million, yeah, in sixty-two and a half hours. Everyone from big to little. Everyone from great to small. Everyone from rich to poor. The grocery clerk? Berserk! The train driver? Berserk! The guy, yeah, in a missile silo somewhere in Shitheel, Nebraska? Berserk! Cops? Berserk!\"\n\nShe reeled back against the leather seat. Took a deep breath. Like she was overwhelmed by the vision in her head. \"Every continent. Every country. I have twenty-nine million samples, yeah. One out of every two hundred and forty-one people on planet Earth. Berserk. Yeah.\"\n\nShe seemed spent. Drained. But still wondering, still amazed. \"It will take seventy-five days to do them all. But it won't hold together for that long, yeah. Governments fall. Religions fail. It all comes down. Chaos. Mass insanity. The end. How many die in the end? Don't know, don't know, yeah. Maybe all of them, yeah. Whole new game then, yeah? Whole new game, right? _My_ game. Adam and fucking Eve. Genghis Khan. Hitler. Stalin and Mao and what's his name? Fucking Attila. _My_ game. Yeah.\"\n\nThe limo stopped just a block away from the Tulip. Lear bounded out with Bug Man on her heels. They raced for the elevator up to her posh apartment.\n\nBug Man felt a sick dread settle over him.\n\nHe didn't see where this was any kind of game. This was just plain murder. Murder on a massive scale.\n\nLystra was excited, fumbling the keys at first. Then she led the way to the window, tapped the remote that opened the curtains and did a game-show-model move, like she was presenting the Tulip as some sort of prize.\n\nThen Lystra fell silent. She was thinking something over. Bug Man could practically see her arguing with herself as her head tilted slightly this way, slightly that way.\n\n\"Yeah,\" she said to herself, finally. \"Yeah. Call him. Call him, yeah.\"\n\nKeats's biot raced away from the partly cut optic nerve, six legs milling through the fluid, impeded but only slightly by sticky macrophages coming to dumbly check out the damage he had done.\n\nIt was like a wild nighttime drive down a back country road, somehow. His illumination in fact lit up very little, just the nerve and a suggestion of deeper brain ahead.\n\nAn artery, that's what he needed, and there were a lot of them in the brain. None would kill instantly; that's not the way it would work. Instead, blood would pour into the brain itself, depriving some tissue of oxygen, putting pressure on other tissue. The result would be a stroke or series of strokes and yes, maybe death, but not quickly.\n\nQuickly enough to stop him blowing up the building? No way to know. He couldn't see through Caligula's eye anymore. Any moment could bring a fireball, a terrible shudder, and a falling floor beneath his feet.\n\nThey'd been taken to see Jindal, a worm of a man who kept rocking back on his heels, then forward onto his toes, trying to look taller than he was. He had snapped out a set of superfluous instructions to his security people, but they were already vectoring armed men toward the sublevels.\n\n\"You need to evacuate the building,\" Plath said.\n\n\"Hah. Just what you'd want if this were all a ruse. Just what you could be after, no? I think so. I think we'll wait until\u2014\"\n\nThe phone chirped. He grabbed it, listened, face darkening. \"The freight elevators are blown. The doors are jammed. They may be booby-trapped.\"\n\n\"I'd bet on it,\" Plath said.\n\n\"Caligula was keeping the elevators to use for his own escape,\" Keats said, walking it through in his own mind. Elevators stopped at the loading bays, from there to the alley, and off he would go. In five minutes he could be clear of the blast and any police cordon.\n\nJindal's forehead creased. And he may have started to sweat just a bit.\n\n\"Evacuate the building!\" Plath yelled. \"We're not here because we want to die, we're trying to save innocent people!\"\n\n_Innocent people_ , Keats noted. So there was still a Sadie somewhere inside Plath.\n\nJindal shook his head slowly. \"If I'm wrong and the place blows up, I'm dead. If I'm wrong and I evacuated the building, the Twins will...\" He shook his head doggedly. \"There are worse things than dying.\"\n\n\"Yes, but none are really as permanent,\" Wilkes said.\n\n\"Take us to the Twins,\" Plath said urgently. \"If you don't have the balls to make a decision, take us to the Twins!\"\n\n\" _Now_ , you bloody fool!\" Keats added.\n\nWhen Jindal still stood, frozen in indecision, Plath spun on her heel and marched for the elevator. \"I've been there before. I know the way.\"\n\nFour security men trained their guns on her. Plath, without turning around said, \"I'm Sadie McLure. Now, you may be too gutless or stupid to make a decision, Mr. Jindal, but you know as well as I do that your bosses would throw you out of that window if you deprived them of a chance to deal with me themselves. So I'm getting on the elevator, and I'm going upstairs.\"\n\nWilkes put on a falsely cheery smile and said to Keats, \"I think she's back.\"\n\nCaligula had seated the jack. It was in an awkward position, and he had to turn the screw using a crowbar that could be moved only a few degrees at a time.\n\nHis vision had not deteriorated further. Which meant whoever was running the biot in his head had moved on in search of a faster way to stop him. And his hand hurt like hell. He'd used the do-rag as a makeshift bandage, but the blood had soaked through almost instantly.\n\nWell. At this point death was a certainty. Death by brain hemorrhage or death by natural gas explosion. _Step right up, ladies and gentlemen, you pays your money, you takes your_ chances.\n\nRemembering the old carnival barker cant made him smile. They had not been so bad, those days. He turned the crowbar. It had been lonely a lot of the time, especially after he gave up his daughter. But he couldn't look at her. He couldn't.\n\nWhen he'd caught his wife in bed with another man, he killed the man and then, much to his own surprise, let his wife live. He'd even forgiven her.\n\n_He_ had forgiven her. Their daughter had not.\n\nHis phone buzzed softly. He closed his eyes and leaned back from the jack. There was only one person who could possibly be calling him, only one person who had ever had the number.\n\nHe pulled the phone from his pocket with his good hand.\n\n\"Yes, baby,\" Caligula said.\n\n\"Call me Lear. How many times do I have to tell you that? Call me Lear!\"\n\nCaligula said nothing, just closed his eyes.\n\n\"Why hasn't the Tulip been blown up?\"\n\n\"Well, I'm working on it,\" he said, feeling very weary.\n\n\"You're ruining the timing!\"\n\n\"Listen, baby... Lear. Listen to me. This will be the last time we have a chance to talk.\"\n\n\"Are you arguing with me? Are you failing me? Again?\"\n\nCaligula sighed. \"They tried to stop me. Plath, Keats... I can't get out of here. I'm going to die.\"\n\nAt least there was a moment of hesitation. At least there was that much. Maybe she didn't really care, but the news at least made her pause. Made her blink, perhaps, at the other end of the line.\n\n\"I guess it's karma,\" Lear said at last.\n\n\"What?\"\n\n\"For you killing Mom. It's karma. Cosmic justice.\"\n\nCaligula hung his head and for a minute could not go on. Could not speak. \"Lystra. Baby. You have to know\u2014\"\n\n\"Goddamn it, you old piece of shit, call me\u2014\"\n\n\"I didn't kill her. You know I\u2014\"\n\n\"Blow it up! Blow it up!\"\n\n\"\u2014didn't take your mother's life.\"\n\n\"Shut up! Just shut the hell up and do it!\"\n\n\"You did, Lystra. _You_ killed your mother.\"\n\nHeavy breathing at the other end of the line. Then, a weird, distorted voice, like a child trying to sound grown-up. A whining, almost singsong voice. \"No, I didn't.\"\n\n\"Lystra...\"\n\n\" _You_ did. You killed her. Yeah, you killed my mother and then you gave me away.\"\n\n\"Baby...\" Caligula's voice broke. He felt a sharp pain in his head. Any other time he would have thought it was just the beginning of a headache.\n\n\"How could I? I was just a little girl.\"\n\nHow long did he have? Minutes or seconds?\n\n\"You're right,\" he said at last. \"You're right, ba\u2014Lear. I did it.\"\n\n\"Hah! I told you so. Now, do this. Do it and all is forgiven.\"\n\nHe managed a slight laugh, a hoarse sound. \"I don't think even God can forgive me all I've done.\"\n\n\"Then it's no problem, Daddy. I am god now.\"\n\nShe hung up the phone. Caligula knew it was true. Not about his poor, mad daughter being god. But yes, he had killed his wife, her mother. A week after they'd reconciled, he'd been drunk and angry at what he thought was a flirtation with the carny who ran the Mad Mouse ride. He'd punched her. He'd punched her hard, right in the jaw. She had fallen, unconscious, to the floor of their shabby trailer.\n\nHe'd left her there.\n\nWhen he woke, raging with thirst from all the drink, filled with remorse, he'd found her still on the floor. But with her throat cut.\n\nThe bloody meat cleaver was on the floor beside her.\n\nHe had roused a sleeping Lystra from her bed and washed the red stain from her hands. Burned her bloody clothing in the fifty-five-gallon drum where the carnies burned trash and kept their hands warm on cold nights.\n\nIt was his fault she had done it. Who had taught her violence? Who had revealed his rage to the impressionable ears of a young girl?\n\nAnd then, cowardly, unable to face Lystra, unable to cope with the madness that was already a part of her, he had shipped her off.\n\nCaligula did not believe in karma. He believed in damnation. His own, and hers as well. And the damnation of the world.\n\nHe set the crowbar in place and heaved with all his might.\n\nThe pipe snapped. Whatever sound it made was obliterated by the roar of high-pressure gas gushing into the room.\n\nHe choked from the smell, reeled back, staggered to the far end of the chamber, and set the timer on the explosive device for ten minutes.\n\nThat should be enough.\n\n# **TWENTY-FIVE**\n\nSadie McLure. In person. In the flesh. And the rest of her little crew as well. Benjamin Armstrong felt disappointed. It should have been a triumph, but she was walking in under her own power, head held high.\n\n\"Someone get me a... a knife! Or a baseball bat! Something,\" Benjamin snarled.\n\n\"Benjamin,\" Charles chided mockingly. \"There will be plenty of time for that.\"\n\n\"I'm going to beat her bloody and rape whatever is left of her!\" Benjamin saw his own spittle flying. He felt the way Charles drew him back, restraining him, knowing Benjamin otherwise would have gone at the girl with his fist until some better weapon appeared.\n\nMore and more security men and women were arriving\u2014by elevator, by stair\u2014all armed, all looking to the Twins for guidance.\n\n\"The building is going to blow up,\" Plath said calmly.\n\n\"Of course it is,\" Benjamin sneered. \"You know, your father was the smart one, not you, you stupid little bitch!\"\n\n\"Caligula is in the basement,\" Keats said, striving to mirror Plath's even tone, despite the realization that one way or the other, his own time was fast running out.\n\nHe flashed on a memory of his brother Alex, chained to his cot in a mental institution in London. Mad. Utterly, terribly mad from the death of his biots. Of course, Alex had had more than one die. But at the same time, Alex had been strong.\n\n\"It's Caligula,\" Plath repeated.\n\nCharles's eyes narrowed. \"What is Caligula?\"\n\n\"He's the one in the basement. Looks like he's rupturing a pipeline. He'll wait until the gas builds up and\u2014\"\n\n\"System!\" Charles yelled. \"System: show all cameras in the basement of this building!\"\n\nOn the huge screen with its multitude of squares showing the Armstrong empire, five windows opened. Three were black. But two were still in operation, one trained on an instrument panel, while the other was a grainy long shot of pipes and...\n\n\"There!\" Jindal cried, pointing. \"There's someone down there. You can just see their back!\"\n\n\"One of the engineers,\" Charles scoffed, but he didn't sound too sure of himself.\n\nSuddenly the grainy figure reeled back, spun away from whatever he had been doing.\n\nIn Caligula's head Keats's efforts were beginning to work. Blood that had been just a single-cell spray from the throbbing artery had become a gusher, like a cartoon of an oil well. The clear cranial fluid around his biot was growing opaque with the floating Frisbees of red blood cells and the soggy sponges of white blood cells.\n\nThe force of the blood knocked his biot loose of its perch and sent it spinning, end over end. What had been a sort of narrow but calm seam of watery fluid was now a turbulent underground river.\n\nHe would not make it back to the artery.\n\n\"He's hemorrhaging,\" Keats said. To the Twins he explained, \"I have a biot in his brain. I've cut an artery. I've damaged one optic nerve.\"\n\nThe camera no longer showed the man in question.\n\n\"Can you get back to his eye?\" Plath asked.\n\nShe still hadn't realized... Keats nodded. \"On my way.\"\n\n\"You don't give orders here!\" Benjamin raged at Plath.\n\nBut his brother was no longer with him on that. Charles said, \"Why would Caligula blow up the Tulip?\"\n\nPlath glanced at Keats, who seemed to her to be elsewhere. Looking through his biot's eyes, seeing a different scene altogether.\n\nIn fact, Keats's biot was racing madly back toward Caligula's eye. His biot swam and crawled, shouldered its way through the clinging platelets, the lymphocytes, the tendrils of detached neurons, floating like seaweed.\n\nHe had never moved so fast. He didn't wonder at which direction to take, which planes to use to flow through the 3D maze of Caligula's brain. The calm had come over him.\n\nHe knew what was coming for himself, but he was no longer afraid. A slight smile stretched his mouth. His eyes glistened.\n\nHe was there, in that place of peace and calm and wild, frantic action.\n\n\"Floor Thirty-Four,\" Plath said to Benjamin. She didn't know what Floor 34 was. Just that it was the one part of the Tulip aside from the data center that was unreachable by elevator. A guess. An intuition.\n\nA bluff.\n\nThe silence that followed was all the confirmation necessary. Charles was shaken.\n\n\"And who sent Caligula to do this act of terrorism?\" Benjamin asked, voice silky and malevolent now.\n\n\"Me,\" Plath said.\n\nCharles blinked. \"But... Surely you...\" His tone was almost pleading.\n\n\"Lear,\" Wilkes said when Keats remained silent. \"It was Lear. He's wired her. He got Vincent to wire her. We've cleared her brain of wire, but\u2014\"\n\n\"So now you see that we were right! Now, now with our beautiful people all dead on the _Doll Ship_ , all destroyed. Now you\u2014\"\n\n\"Look, you're a piece of shit who needs to die a painful death. The two of you,\" Wilkes snapped. \"But we do not blow up buildings full of innocent people. We're trying to stop this from happening.\"\n\nBenjamin's face was a snarl. Charles was guarded, worried. It was he who said, \"Jindal, get Burnofsky up here.\"\n\nKeats had reached the optic nerve. He sank a probe. \"I can see,\" he said in a dreamy, disconnected, emotionless voice. \"Caligula is looking right at it. At the bomb. There's a timer.\"\n\n\"How much time left?\" Jindal asked.\n\nBenjamin raged at him. \"Follow my brother's orders, now!\"\n\n\"I have a weak picture,\" Keats said, speaking to Plath. \"I'll try for a better one.\"\n\nJindal rapped orders to his people, then, undeterred\u2014 _Accustomed to abuse_ , Keats thought\u2014he said, \"Our people will be through the door into the sublevel in a few minutes.\"\n\n\"How are they getting through?\" Plath asked.\n\n\"They're cutting through the steel with a blowtorch and once they're in\u2014\"\n\n\"A blowtorch? Cutting into a room full of gas?\" Wilkes cried. \"Isn't that, uh, stupid?\"\n\n\"She's right,\" Charles said.\n\n\"No,\" Plath said sharply. \"No. Maybe better to blow it up now rather than wait. Less gas now. More later.\"\n\n\"System,\" Charles said. \"Exterior, sublevel doors.\"\n\nAs one they all turned to look at the monitor. Four frames. Three showed nothing but doors. The last showed two men wearing welding helmets. The bright light of the torch caused lens flares that obscured the progress of the work.\n\n\"Seven minutes, eighteen seconds,\" Keats said. \"I can see it now. I can see it clearly. Seven minutes and...\" And it all came back to him. The calm of battle had run its course once his biot had reached its goal. Now Keats couldn't go on. He had run out of indifference to his own fate.\n\nPart of him didn't want to tell Sadie. What would be gained? But he had to speak. He had to say good-bye.\n\n\"Sadie,\" he said.\n\nShe must have registered the sadness and gentleness in his voice. She turned to him. \"Yes?\"\n\n\"Sadie,\" he said again. \"I've thought about it a lot. I've seen Alex. I know what it means. Death or madness, I... I guess I believe in another life, maybe. After this one. So...\"\n\nShe stared, uncomprehending. Then a sharp intake of breath. Her eyes widened. \"Oh, God.\"\n\n\"What?\" Wilkes demanded.\n\n\"I'm getting my other biot as far from your aneurysm as possible,\" he said. \"But you'll need to kill me. You can't have it in your head with a madman running it.\"\n\n\"Noah,\" Sadie said. Sadie, and not Plath. Sadie. \"Noah... We have to...\"\n\nHe took her hand in his. \"We always knew it could happen.\"\n\n\"Order the men down there to cut straight through, forget cutting a hole, tell them just to cut all the way through in a single spot,\" Benjamin told Jindal.\n\n\"Better to burn than to blow up,\" Benjamin said. \"And thus, it ends.\"\n\n\"You can't... Noah...\"\n\n\"When Caligula burns, so will my biot, Sadie. You know what follows. It's okay.\"\n\n\"Noah...\" She was in his arms, and tears were running down her face.\n\n\"Yes, of course, pity for the pretty boy, eh?\" Benjamin said savagely. \"Pity for poor, poor Noah. None for our people on our beautiful ship. And none for hideous freaks.\"\n\nBurnofsky watched the counter on his computer monitor. The number of self-replicating nanobots had just crossed thirty-two million. The next doubling would take it to sixty-four million, then one hundred and twenty-eight. Pretty soon megabots would give way to gigabots and hence to terabots.\n\nHe laughed at that, slurred, \"I made a funny,\" took a drink, sucked on his cigarette, and touched the butt of the pistol that was stuck into his belt.\n\nHe'd been feeding the nanobots everything he could find: stale doughnuts, candy bars from the machine down the hall, half a salami he'd found in the staff fridge. He hadn't slept in... how many hours? How many days? It was all kind of fuzzy.\n\nHe had the remote control in his hand. Press the button and the force field would drop. His nanobots would eat their way out into the world and from there they would never stop. They would eat their way through the building, its furnishings, and anyone dull enough to wait around.\n\nBut before they finished the Tulip they'd be carried on breezes or simply fall from chewed-through walls down onto the streets. Nearby buildings would be infested and begin the same accelerated decline and rot. The pace would accelerate as the nanobots doubled and doubled.\n\nWhat would the reaction be? What would the government do? Nothing short of a nuclear weapon would stop the spread, and they would wait far too long for that. Nanobots would find their way onto ferries, cars, ships, and planes.\n\nFor the first few days the damage would be most visible at the epicenter. But then, here and there and all around the world they would appear and double and double and double.\n\nPeople would flee to the woods and deserts. And they would survive for a while\u2014maybe weeks, maybe months. In places the nanobots would consume all there was to consume and cease doubling. But by that time they would have eaten every living thing and much of the nonliving things as well.\n\nHe asked himself, where would be safe? Or at least, where would be _safest_? The coldest places, he supposed. Nanobots tended to be immobilized when things got cold enough, down to minus twenty-three Celsius or minus ten Fahrenheit. But even in the coldest lands a warm day would set them off again.\n\n\"God bless global warming,\" he muttered, and laughed at his own wit.\n\nPeople thought they were scared now? They thought they were terrified by Lear's plague of madness? Wait until they saw their crops, their home, their car and its gas, their dogs and cats and cows and pigs, all chewed up, masticated by trillions of nanobots that did little but crap out more nanobots.\n\nWait until they realized how hopeless it was. How powerless they were. Wait until they saw the little sore on their ankle become a bleeding hole and endured the agony of being eaten alive, consumed, like a beetle being swarmed by fire ants. It would be like leprosy on fast forward. It would be like flesh-eating bacteria on meth.\n\nSure, maybe in places there would be pockets of a few scattered humans who would hold out for as long as six months. But it wouldn't matter. The nanobots would eat the algae out of the sea and every oxygen-producing plant on the land and then, inexorably, the atmosphere itself would become fatal to life.\n\nDirt. Water. That would be planet Earth. Just dirt and water and a vast, inconceivably vast swarm of nanobots. Mindless. Without soul or sin. Efficient, relentless, unstoppable killers without malice, without meaning, without moral judgment. Without guilt\u2014that most destructive, weakening, sickening, disabling of emotions.\n\nYes, his babies would obliterate without guilt.\n\nHe pulled up the picture he'd found of Lystra Reid and gave it the finger.\n\n\"Game, set, match, Lear. Death or madness? I got a little hint for you, sweetheart. The answer is death. Death, brought to you by Karl Burnofsky.\"\n\nOut in the lab he heard a disturbance: raised voices, a bustling movement, chairs scuffling. The door to his office was locked. He drew the pistol.\n\nSomeone banged hard on his door: a cop's knock.\n\n\"Damn,\" he said. \"I'd have liked to hit a billion first.\"\n\n\"Burnofsky! Come out here. The bosses want you.\"\n\n\"I'm busy,\" Burnofsky yelled.\n\n\"Don't think they care, Dr. B. You've got about ten seconds.\"\n\nThe Twins wanted him, did they? Well, why not? It would be worth a laugh. And he had something special for them, just for them, something ever so special.\n\n\"Give me a second!\" he yelled. In his desk, all the way at the back, he found the little vial he'd prepared against this very moment. He slipped it into his pocket along with the remote control that could unleash Armageddon and opened the office door for what he suspected would be the last time.\n\nThe area within the force field continued to fill with his children.\n\nKeats had his biot back on Caligula's optic nerve. He was again seeing what the killer saw. Caligula seemed to be sitting, perhaps with his back against a wall. His legs were stretched out before him. He stared at his missing fingers, bleeding freely, unbandaged. He leaned down to rub a spot of mud from his boot, glanced at the timer\u2014six minutes and nine seconds\u2014then apparently coughed as his head jerked violently and his hand came up to his mouth.\n\nJust six minutes until the natural gas flooding the basement would achieve sufficient density that a spark would bring down the entire building. The gas was invisible dynamite being stacked, ton upon ton. Caligula's eye glanced toward the ruptured pipe. He had a picture of something in his hand, a photograph of a serious little girl slumped in a busted-webbing lawn chair outside a shabby trailer. There was a Ferris wheel in the background.\n\nCaligula coughed again and drew something out of his bag. Keats saw a small steel cylinder, a clear plastic hose smeared with Caligula's blood, and a clear plastic mask with elastic straps. It reminded him of the lecture aboard an airplane: _Should there be a sudden loss of cabin pressure..._ Caligula pulled the mask on, and now the plastic partly filled Keats's view. Caligula was determined to wait out the\u2014\n\nNo, he was up, up and staggering, but not toward the rupture, or toward the elevators. Keats saw a steel door. Caligula's eye went to the handle, then his hand as it touched the metal of the door.\n\n\"He knows your guys are burning through,\" Keats said dully.\n\n\"Jindal!\" Charles yelled in response.\n\nJindal talked into his phone and reported, \"They say they'll be through any second.\"\n\nCaligula glanced back toward the bomb. Glanced at the gun in his hands. Suddenly they trembled. He seemed to be struggling to hold on to the weapon; his mutilated hand was still bleeding freely, but even the fingers on his good hand looked stiff, uncooperative.\n\nThe gun fell from his grip. The picture, too, was facedown on the floor.\n\n\"He's having a stroke,\" Keats reported. _Go on_ , he told himself, _just keep watching. Until the end. Be the good boy. No freaking out, no last-minute pleas_. Tough, that's how his brother Alex had always been. \"He's stroking out from the artery I cut.\"\n\nSadie was looking at him, her eyes ashamed, horrified.\n\n\"Not your fault,\" he said to her. \"None of this is your fault.\"\n\n\"But it is,\" she said.\n\n\"He's picked up a crowbar. His fingers can barely hold it. He's dropped it. He's staring at it.\"\n\n\"For God's sake, evacuate the building!\" Plath shrieked at the Twins.\n\nBurnofsky, disheveled but animated, came in with guards on either side. His rheumy eyes sparkled. \"Ah, ah, ah!\" he said on spying Plath and Keats and Wilkes. \"So, _that's_ the panic.\" He seemed pleased and relieved.\n\n\"Help me get these idiots to evacuate the building,\" Plath pleaded. \"Caligula's flooding the basement with natural gas. In six minutes this whole place goes up!\"\n\n\"Is that true?\" Burnofsky demanded, squinting hard at the Twins. He glanced at the monitor. The cameras in the basement had been redirected, searching for Caligula. A grainy image showed him walking, dragging one leg, then collapsing on the floor.\n\nKeats had never been inside the brain of a dying man. There was nothing to see on the optic nerve, nothing changing in his immediate environment. But the eyelid no longer blinked as often, and it seemed to be drooping, partly obscuring the view.\n\nIf Caligula died before the explosion, then Keats would have been his killer. His biot might sit for several minutes in a dead man's brain before the explosion killed his biot and plunged him down into the dark hell of madness.\n\nHow would it feel, he wondered. How would it feel to no longer be himself?\n\nKeats's throat was dry. His breathing was shallow. He was afraid. First would come the razor edge of madness, to be followed by an explosion that\u2014\n\nA brilliant flash of light from Caligula's eye.\n\nThe same bright flash filled the monitor that had been trained on Caligula. The camera aimed at the exterior where the men had been wielding the cutting torch went dead.\n\n\"They burned through!\" Jindal cried.\n\n\"System,\" Charles yelled, his voice cracking. \"Sublevel two, northeast corner stairwell cameras!\"\n\nBlank nothing, dead cameras.\n\n\"System, sublevel one, northeast corner!\"\n\nHere, too, the cameras were blank. A shudder communicated itself up the length of the Tulip to Keats's feet, like a minor earthquake. A glass fell from a shelf and shattered.\n\nThe fire killed Caligula instantly. Then it began to burn through his flesh, boiling the blood in his veins, sloughing away charred skin, burning its way to his heart, to his lungs. To his brain.\n\n\u00b7 \u00b7 \u00b7\n\n\"Fire in the lobby!\" Jindal reported, phone to his ear.\n\n\"They can put it out!\" Burnofsky yelled. \"They have to put it out!\" His relief was all gone now, all gone, as his brilliant mind frantically calculated the damage that could be done to his nanobots by a burning building.\n\n\"The southwest corner stairwell is still clear,\" Jindal said. \"Gentlemen, we have to get you out of here!\" This to the Twins, who seemed paralyzed.\n\n\"Anyone who wants to live, get out of here!\" Plath yelled, pulling away from Keats.\n\n_A window in Keats's mind went dark and then disappeared. Keats felt strange, very strange. Not upset. Just... alone..._\n\n\"Wilkes! Run!\" Plath pleaded.\n\n\"Not without you and blue eyes,\" Wilkes said.\n\n... _alone in a strange landscape_.\n\n\"No one moves!\" three different security men yelled at once, waving their guns in a bewildered effort to assert control.\n\n\"It blew up early,\" Plath said, looking to the Twins. \"The explosion was only limited, but it's still burning, and there's an open gas line feeding that fire. We may still get out.\"\n\nBurnofsky yelled, \"System: show Burnofsky lab!\"\n\n_Such strange images. Flashing pictures of his old room in London, of playing football in the alley, of the island. Of Sadie. Of the dark, looming monster that seemed now to be emerging from her, bursting from her flesh, a dark, terrible beast..._\n\nBurnofsky's lab was untouched. He saw assistants going about their business, clueless.\n\n\"Evacuate the building!\" Charles yelled.\n\n\"No!\" This came from Benjamin. Jindal stutter-stepped and stopped.\n\n_Keats saw it all. The Twins were a glowing two-headed dragon, with liquid fire bleeding from Benjamin's lips. Burnofsky melting, somehow melting, and Keats felt the laughter rise in him, rise and fill his chest and come burbling out of his mouth. He pulled against imaginary chains, yanking his arms against nonexistent restraints_.\n\n\"Noah,\" Sadie pleaded, helpless, knowing what was happening to him, knowing that he was spiraling down.\n\n_A part of Keats\u2014a fading, weakening remnant\u2014watched it all from very far away, a shadow of his mind, watching himself slip, slip, slip.... It was all very, very clear, very, very clear to Noah, Noah like the guy with the ark, the one who liked animals, all clear, they were all devils, all of them. Mad, each of them, mad as... mad as..._\n\nA whimpering voice, Noah's own voice, but not operated by him, no, a voice mewling and laughing and crying out, \"Kill me, kill me, it's what you all want, isn't it?\"\n\nIn that moment, a final becalmed moment, the last sane vestiges of his mind took it all in, and his laughter was not yet the laughter of the insane, but the knowing, cynical laughter of one who sees everything clearly, if only for a single second in time.\n\nHe saw Benjamin and Charles as what they were, two rejected, despised, sad little children forever bound together, neither able to feel even a moment's freedom.\n\nHe saw Burnofsky, so desperate for redemption from suffering that he would bring down the whole world in a fit of self-loathing.\n\nAnd Sadie, Sadie his love, her brain a tangled mess, wired, unwired, but even before that crippled by a dead mother, a dead brother, a dead father, and corrupted by wealth and power and crushed by responsibility. Mad. Her, too: mad.\n\nThey were all mad. They always had been.\n\nCrazy people had gotten their hands on deadly toys. The end was inevitable.\n\n_And me, too_ , he thought. As mad as any of them, believing that there could be love and honor in the midst of it.\n\nThey had all tried to armor themselves against this final moment, but their defiance had been its own lie: there was never a choice between death and madness. It was always to be both.\n\nAnd then, with a strangled cry in his throat, Noah attacked.\n\n# **TWENTY-SIX**\n\nLear watched, hands behind her back, lifting herself up on the balls of her feet, bouncing with anticipation. When she first saw the fire burst from the ground-floor windows and setting a passing man alight, she let out a happy squeal.\n\nBut then, when the Tulip still stood, she clenched her fists and began to curse. \"Fucking useless old man. Useless old man,\" she said. \"Trying to say I killed her, and now look! Look!\"\n\nWhen Bug Man did not move from the couch, she took two long steps, reached down, grabbed the neck of his T-shirt, and dragged him to the window.\n\n\"Iff' burning,\" Bug Man said.\n\n\"It's not supposed to _burn_ , it's supposed to explode! The gas was supposed to explode! The whole thing should be toppling over!\"\n\nThe TV was on, showing a sea of flashing red lights around the theater, with cutaways to eerie vignettes of cops tackling a naked, raving rock star, or Tasering a man in a business suit carrying a severed arm, the remnants of the lunacy at the premiere.\n\n\"Blow up! Blow up, blow up, blow up, _blow up_!\" Lear raged, banging the plate glass with her fists.\n\nAs if on command a huge fireball erupted from the windows of the third floor.\n\n\"It could still fall, yeah,\" Lear said, nodding, reassuring herself. She bent to a tripod-mounted telescope. \"Can't see anything through their dark glass. Are you scared yet, you freaks? Are you wetting yourselves, you _freaks_?\"\n\nBug Man had had enough, more than enough. He had to get away. He shot a look toward the door. Did she have guards out there? If she died, he went mad... if she was telling the truth about a dead man's switch... but there wasn't anything he could do about that, and he could not be here watching all this. He could not be with this crazy witch raving and pounding on the glass like an infuriated ape in a cage.\n\nHe stepped back, back, turned, and ran for the door. Locked.\n\n\"Really, Bug Man?\" Lear asked in a mocking voice. \"Really? You think you get to run away?\"\n\n\"You 'ave to le' me go,\" he pleaded.\n\nShe ignored him and crowed wildly as another burst of orange flame billowed out from the base of the Tulip. \"It'll collapse. Has to. The fire will melt the girders, has to, yeah. Damn, I want to see them when it happens.\"\n\n\"You coul' talk to them.\"\n\nLear's eyes lit up. She grinned. \"What?\"\n\n\"I know Burnofsshky's number. He' prob'ly there. He worksh late.\"\n\nShe grabbed Bug Man's bicep and propelled him to a laptop. \"Do it! Do it and I'll... I'll get you new teeth. Any color you want.\"\n\nBug Man opened an app, punched in the number, and hit Connect.\n\nKeats rushed at the Twins, hands clawing the air, animal noises coming from him.\n\nPlath shoved Wilkes aside to put herself between Keats and his intended victims. Keats never seemed to notice her. He ran right through her, sending her sprawling.\n\nIt was on her back, stunned by the violence of his assault, that Plath\u2014Sadie McLure\u2014saw three security men turn, as if in slow motion, and raise their guns.\n\n_BANG! BANGBANG!_\n\nKeats twisted, turned, stood...\n\n_BANG! BANG!_\n\n... fell.\n\nA terrible scream rose from her mouth, echoed by Wilkes as they both fell more than ran toward Keats.\n\n\"No, no, no, no, no!\" Plath cried.\n\n\"You fucking assholes! You murdering assholes!\" Wilkes screamed.\n\nKeats lay on his back. Three bullets had struck him in the side of his chest, in his upper arm, in the side of his head. He was not yet dead, eyes glazing, dark blood like ink pumping from him to form a pool on the floor, his mouth working like a beached fish, gasping.\n\n\"Oh, God, Noah! Oh, God, Noah!\"\n\nHe tried to speak but only managed to form a blood bubble. He grunted, the sound of a dying beast. He breathed heavily, looked at Plath, grunted again. He blinked, just one eye, almost as if he was winking. Blood found its way out of his ears, out of his nose.\n\nPlath tried to cradle him in her arms, tried to hold his head, but when she did, a part of his skull came away and she screamed. Wilkes, her own hands red, took Plath's hand and kept saying, \"He'll be okay, he'll be okay.\"\n\nA siren was screeching, up and down the scale, up and down in Plath's head, but it was only her own screams.\n\nA cell phone rang.\n\nPlath stared at Noah, his eyes still so blue, his eyes open, his lips no longer the parchment landscape she had seen through biot eyes, now only the lips that had kissed her. They were moving silently.\n\nPlath's entire body was shaking. She heard nothing, and for a while she saw nothing. The world was lost to her. Only Wilkes's arms around her connected her to reality.\n\nThe sound of a phone ringing. And going to voice mail.\n\n\"I hate people who get my hopes up,\" Lear said. But she was distracted by a third eruption of flames. This one blew the windows out of half the lower floors. A shower of crystal fell through yellow flames, pursued by billows of smoke.\n\nBug Man dialed again. This time, the call was answered.\n\n\"Kind of a bad time, Anthony,\" Burnofsky said.\n\n\"Lear wan' to tal' to th' Twinshh,\" he said.\n\n\"Oh, does she?\" Burnofsky said, his voice flat. \"A little late for talk, I think. Hey, Anthony?\"\n\n\"Wha'?\"\n\n\"I never hated you, Anthony,\" Burnofsky said.\n\nBug Man had no idea how to respond to that, so he simply handed the phone to Lear after pushing the Speaker button: he wanted to hear.\n\n\"Who is this?\" Lear demanded.\n\n\"Well, well, if it isn't Lystra Reid. Or should I say 'Lear'?\" Burnofsky said.\n\n\"Is this one of the Twins?\"\n\n\"This is Burnofsky. Dr. Burnofsky. But you can call me Karl.\"\n\n\"Give me the Twins.\"\n\n\"Well, we're all kind of busy panicking and getting ready to die,\" Burnofsky said. \"Hey, just out of curiosity, Ms. Reid, did you ever figure out what the Twins were up to on Floor Thirty-Four?\"\n\nBurnofsky heard the silence of confusion. Then, \"What are you talking about, you old fool?\"\n\n\"Their secret weapon. A virus that preyed only on cobra DNA. Like the cobra DNA that forms part of the biot genome. Ironic, don't you think? They were going to obliterate all biots, and now, hah! Now you're the one killing biots.\"\n\n\"Shut your filthy mouth, you disgusting drunk,\" Charles said, now as furious as his brother.\n\n\"Oh, I'm sorry, am I embarrassing you, boys?\" Burnofsky laughed. \"Don't worry, the final laugh will be on Lear.\"\n\nBug Man heard shouts and cries in the background. A female voice was crying, \"Noah! Noah!\" Then it stopped. The line went dead.\n\nBug Man could see flames behind windows on the tenth and twelfth floors of the Tulip.\n\nOn the street below, the first fire engines were pulling up, but Bug Man doubted there was anything they could do. The Tulip was doomed.\n\n\"Okay,\" he said.\n\n\"Okay what?\"\n\n\"Okay, I'm wiff you. I'm in. All the way.\" What alternative did he have? \"So, I' goin' to tell you, Burnofshhky didn' shound righ'. Too happy. He'shh got shomething goin' on, I know tha' old fart.\"\n\nLear was curious. \"What could he be up to?\"\n\n\"SRNs. The gray goo.\"\n\n\"No,\" Lear said confidently. \"We wired him up. Nijinsky wired him.\"\n\n\"And Nijinshhky's dead. So you don' know wha' going on in hi' head anymore. I wired the presiden' and guessh wha', shuicide wa' not par' of the plan for her.\"\n\nLear was pensive. She could become coldly rational when she needed to, Bug Man had learned. She would make a fascinating case for some psychiatrist some day, he thought mordantly. Or a whole hospital full of psychiatrists.\n\nOut of her mind but still able to plan the end of the world. Then again, who else but a crazy person would even want the world to end?\n\n\"Charles, Benjamin,\" Jindal pleaded. \"There's one stairwell still clear. But we have to go now!\"\n\n\"Hundreds of steps?\" Charles asked wistfully. \"My brother and I, walk down hundreds of steps?\"\n\n\"We can carry you,\" Jindal said. \"We\u2014\" He stopped, because his inclusive wave, meant to indicate the security men, now included no one. The security men had fled.\n\nStill left in the Twins' sanctum were the Twins themselves, Jindal, Wilkes, and Plath. And the gasping, dying body of Noah Cotton, the former Keats, now on his back in a wide pool of his own blood.\n\n\"Will you drag us down the stairs, faithful Jindal?\" Charles said. \"No, I don't think we'll allow that. Instead...\" he shouted, \"a drink, if you please!\"\n\nThe building was shaking now, successive waves of it\u2014an artificial earthquake as small explosions and gouts of flame made their way inexorably upward, floor after floor.\n\n\"The gray goo,\" Benjamin said. \"How many SRNs do you have, Burnofsky? The flames have not reached your lab yet. Yes, better the gray goo. The best possible outcome. Apocalypse!\"\n\n\"Fetch me a bottle, Jindal,\" Burnofsky said, \"and I'll tell your bosses all about it.\"\n\nPlath realized she was kneeling in Noah's blood. She was looking in horror at his brain, a pulsing pink mass that swelled out from the bullet's hole. Wilkes took her hand, but Plath felt nothing. She knew she had to look away, but it felt like abandonment to look at anything but her lover.\n\nHe had loved her. Had she ever really loved him in return? How could she know? From memories that had been tampered with, in a brain still coping with violation? That truth was no longer entirely recoverable. Nothing was. Everything that she knew and remembered, everything she felt, had to be mistrusted.\n\nOn the big monitor, cameras showed gift shops and labs, darkened bedrooms and banks of computer servers. The Armstrong Fancy Gifts Corporation went on, largely untouched, even as its headquarters burned.\n\nPlath seemed to make up her mind. \"We're leaving,\" she said.\n\n\"The hell you are,\" Benjamin snarled.\n\nPlath looked at him, not afraid to meet his furious gaze. \"I'm going to find and kill Lear.\" But she made no move toward the door.\n\n\"Lear doesn't matter, not anymore,\" Charles said. \"I'm afraid I no longer have the will to resist my brother. The self-replicating nanobots will be released. They will scour this planet, and sooner or later, they will find Lear.\"\n\nJindal had found a bottle. Burnofsky uncorked it and took a drink before offering it to the Twins. \"It's not up to you, Benjamin, it's up to me.\" He revealed the remote control in his hand. \"I push this button, and the world begins to die.\"\n\n\"Give it to me,\" Benjamin demanded. \"We paid for your work. They're ours, those little machines of yours, ours!\"\n\nBurnofsky laughed. \"Hitler's bunker. I've been trying to think what this reminds me of, and that's it. With the Russians closing in, there was Hitler in his underground bunker still handing out orders. Like he had an army to command. Dead man rapping out orders.\"\n\n\"You treacherous, degenerate\u2014\"\n\n\"Shut up,\" Burnofsky said. He gave them a wave, a tolerant gesture. \"It's over for you boys. All over but the punishment phase.\"\n\nThere was an awful groan of bending metal, a shriek that was felt as much as heard. A crack split one of the floor-to-ceiling windows. The power went out. The monitor went dark.\n\nEmergency lights came on, casting dark shadows softened only slightly by a full moon hovering just over a nearby apartment building. \"Plath, now,\" Wilkes pleaded. \"You said it. We have to go. Keats will... we can't help him.\"\n\nBut still Plath couldn't move.\n\n\"Call that number back! I'll speak with Lear!\" Benjamin cried, motioning for the phone. \"I want to tell her what we've done! I want Lear to know!\"\n\nWilkes grabbed Plath's arm and began to physically pull her away.\n\n\"We? _We?_ \" Burnofsky demanded, erupting in fury. \" _We_ , you freak? We? No we. No we. Me. _Me!_ I did it! They're mine and they're blue, the blue goo not the gray, and do you know why they're blue?\"\n\nHe was in Benjamin's face now, gripping the remote in his hand, spit flying from his lips.\n\nCharles and Benjamin took a step back.\n\n\"Because that was the color...\" A sob stopped Burnofsky. \"Because... her eyes...\"\n\n\"Is this about your nasty little girl?\" Benjamin demanded, sneering.\n\n\"Damn it, Plath. Sadie! Come on!\" But now even Wilkes could not look away.\n\n\"Did you...?\" Burnofsky asked. \"Did... you...?\" He got control of himself again, and laughed. \"You make this easy. I have something for you.\"\n\nBurnofsky drew a small object from his pocket. He held out a glass vial. It looked empty but for a hint of blue.\n\nCharles knew instantly. \"Get that away from us, Karl.\"\n\n\"These are special,\" Burnofsky said. \"A special project I've been working on, just for the two of you.\"\n\n\"Someone stop him!\"\n\n\"It's easy to program the SRNs with time codes, kill switches.... Much harder to program them for a particular, um, diet. Yes, a particular _diet_. But it's doable. I have them that can eat only steel. Others that consume only hemoglobin. Cool, huh?\" Over his shoulder, he said, \"Run away, Sadie. Run while you're able. I loved your dad. He was a good man. A good man. So run away. Save yourself if you can. Get far from here. You may survive for a while, until my babies come for you.\"\n\n# **TWENTY-SEVEN**\n\n\"Now! Now!\" Wilkes yelled.\n\nPlath rose from Noah's blood. Her knees and shins and hands were soaked, red.\n\n\"Noah,\" she whispered. She touched his face, still sweating, still gasping like a dying fish.\n\n\"Go,\" he said. \"For me. Go.\"\n\nPlath tore herself away. No way she could survive trying to carry Noah. They would both die. Someone had to live to ensure that Lear did not.\n\nPlath and Wilkes fled the room they'd never expected to leave alive.\n\n\"What corner did he say was still clear?\" Wilkes asked.\n\n\"Southwest,\" Plath answered. \"Southwest.\"\n\n\"Which is...\"\n\n\"This way.\" Plath led the way, first from the cathedral vastness of the Twins' lair and back out into the entryway they'd come through. She considered the elevators and rejected them. Even if they just used them to get down a few floors, there was no knowing what they'd open onto. Past the easy way out, into a stainless-steel kitchen area, through a gloomy and oppressive formal dining room that looked as if it had never been used.\n\nThey pushed out through a narrow door into a similarly narrow hallway, then followed red exit signs to what a push confirmed was a stairwell.\n\nThere was smoke in the stairwell.\n\n\"Not too bad, we can breathe,\" Wilkes said. \"At least up here.\"\n\n\"No other way,\" Plath said, and plunged unhesitatingly down the concrete stairs.\n\n\"Great, seventy floors,\" Wilkes said. \"Here's where it would have been a good thing to work out.\"\n\n\"It's all downhill,\" Plath said.\n\nThey ran and tumbled and occasionally tripped down the stairs, half a floor, a landing, a turn, down another flight. Over and over again.\n\nThe smoke grew thicker but not yet enough to choke them, just enough to make their throats raw and their eyes sting.\n\nPlath was quicker, but she waited for Wilkes to catch up when she pulled too far ahead. Down and down. Then, on the fortieth floor, a woman banged back the door, took a wild-eyed look at them, and raced away as though they were trying to catch her.\n\nDown and down and down, and by the twenty-first floor the smoke was wringing hacking coughs from their throats and watering their eyes.\n\nA massive shock hit the building and knocked them both off their feet. Plath came up with a skinned knee and bruised forearms. Wilkes was worse off. She had twisted her ankle and could only hobble.\n\n\"You need to go on ahead,\" Wilkes said. \"Go, go, I'll be fine.\"\n\nPlath took her arm. \"I left Noah. I'm not leaving you. Come on. Run now, hurt later.\"\n\nThey hobbled and slid and tripped, floor by floor, tears streaming down their faces. The last six floors were agony. Smoke was everywhere, searing their lungs. The heat of the fire turned the stairwell into an oven. At some point Plath simply stopped thinking, stopped even feeling anything but pain.\n\nThe last two floors were crowded with people\u2014yelling, choking, pushing, panicking.\n\nAnd all at once there was air.\n\nPlath, still holding Wilkes by the hand, fell out onto the sidewalk and into light; rough hands grabbed her, pulled her away, a voice yelling, \"Move, move, move, it's coming down!\"\n\nThey staggered on, not even sure what direction they were headed, stumbling into other refugees. A fire hose was spraying blessed cold water, and only then did Plath realize that some people were on fire, their clothing smoking, their hair crisped.\n\nGlass was everywhere on the sidewalk and streets. Red lights flashed. Smoke billowed, but was caught by a breeze that cleared most of it at street level.\n\nA block away they stopped, gasping, and sank down onto the concrete.\n\n\"Okay?\" Plath asked.\n\n\"Alive,\" Wilkes answered.\n\nPlath smeared smoke from her eyes, blinked away tears and tried to look up at the Tulip.\n\nFire licked from windows. Smoke poured everywhere, the whole building a chimney now.\n\n\"We have to move farther.\"\n\n\"Can't,\" Wilkes gasped.\n\n\"Like hell you can't.\" Plath stood, hauled Wilkes to her feet and, taking the girl's weight on her shoulder, hobbled and ran with memories, too-sharp memories, of what happened when skyscrapers burned.\n\n\"Burn and fall, burn and fall,\" Lear crooned as she watched flames and smoke wreathe the Tulip, dividing her attention between the real-world vista from her window and the TV coverage.\n\nIt was split-screened now on the news: half showed the remaining, yet-to-be rounded up loons from the Hollywood premiere; half showed the Tulip aflame. The crawl along the bottom was all about the Plague of Madness.\n\n\"Good title, that,\" Lear commented. \"Makes people think it can spread. Yeah. And it can, hah.\"\n\nBug Man said nothing. This was his future now. He would live or die at Lear's whim. Or she might just let him go crazy.\n\nThree windows were open in his head. None of them showed much at the moment, just glimpses of the biots themselves. It was different than twitching nanobots, more intimate. You had only to think and the biot would move. No wonder BZRK had been so tough to beat. No wonder Vincent had ended up drooling nuts.\n\n\"Oh, look look look!\" Lear pointed, as excited as a little child. \"It's starting to buckle. Look! Look! You can see rebar starting to stick out the side there. My dad came through in the end, I guess.\"\n\n\"Your dad?\"\n\n\"Yeah,\" she said, almost fondly. \"My dad. You must have heard of Caligula. Of course that's not his real name. I gave him that nom de guerre. Caligula, yeah. Yeah.\"\n\n\"Caligula's your father?\" He forced himself to quash the urge to say that this explained a lot.\n\nHis mouth hurt terribly. He had finally been allowed a couple ibuprofen swallowed with cold water, which had sent lightning bolts of pain shooting from his broken teeth but was already clearing up his speech. Now Bug Man was drinking raw bourbon, no ice, no water, no nothing, because it just didn't seem to matter anymore if his brain was dulled. What was he holding out for? He was owned, body and soul. He was her slave. He was her dog.\n\n\"Mmm,\" Lear said. \"Was. Past tense. He killed my mother, you know. He tries to pretend it was me, yeah, like I could have done it. Like I could have killed her. Like I could have found her unconscious, yeah, and the cleaver, and thought... no. Yeah. But if I had, wouldn't I have a tattoo of her?\"\n\nBug Man nodded wearily, as if this proved her case.\n\n\"Adoptive parents, yeah, that's different,\" Lear said. \"You saw them.\"\n\n\"Yeah.\"\n\n\"It's going,\" Lear said. \"It's going. Oh, this will be the best. Get me a drink. I want to toast the Armstrong Twins as they die.\"\n\n\"What have you done?\" Charles demanded, aghast.\n\n\"Revenge is a dish best served cold,\" Burnofsky said. \"And you know what? Even with the fire below, I feel chilly.\"\n\n\"Damn you, _what have you done_?\" Benjamin yelled, desperation breaking his voice.\n\n\"My final work of genius,\" Burnofsky said. \"I programmed my SRNs to respond not just to a time signature, or even a specific energy source. I programmed them with a map. A topographical program.\"\n\nCharles began to scratch his chest, the place where his chest became Benjamin's.\n\n\"Yep, it will itch at first,\" Burnofsky said. \"Then it will burn. And then, it will really start to become quite unpleasant.\"\n\n\"What have you done? Tell us! What have you done?\"\n\n\"I've granted your secret wish,\" Burnofsky said. \"You've lived with each other every single minute of your lives. Neither of you has ever been separate. Well, now you will be. The topography is _you_.\"\n\n\"What?\" Charles cried. \"We'll die!\"\n\n\"Well, yeah,\" Burnofsky allowed. \"But not right away. Hey, I've put a lot of thought into this. You don't think I'd make it easy for you. Has my life been easy? No, it has not.\" He dropped the jocular tone. \"You bought my soul, you two. You bought my soul...\"\n\nBenjamin tore at the buttons of their tailor-made shirt, exposing pink flesh with an angry, vertical red rash in the center. He clawed at it then whinnied in horror as his fingernails came away trailing ribbons of flesh.\n\n\"... and then you let me be mind-raped. My brain. It's all I had after I killed her, my intellect. Oh, God, and still, still, do you know what they did to me? Do you know what BZRK did? When I think of her...\"\n\n\"They're on my back!\" Charles cried.\n\n\"... I get turned on. Did you know that's what they did to me with their wire? Crude. They thought, _Well, we will just sort of reverse polarities on old Burnofsky's brain_. Like an old _Star Trek_ , did you boys\u2014\"\n\n\"I can't reach, I can't reach!\" Charles cried as he flailed madly, trying to reach his back with his hand, but that had never been possible.\n\n\"Ever watch that show? They were always reversing polarities. All bullshit. But that's all Nijinsky had. Crude and cruel. A man should do penance for his crimes. A man should pay. A man should suffer, not feel pleasure.\"\n\n\"We don't deserve to suffer!\" Charles shouted.\n\n\"No,\" Burnofsky drawled. \"You two? No, it's not like you enslaved a ship full of people and did to them just what BZRK did to me, right? See how you're not going to win that argument?\"\n\n\"We'll give you whatever you want,\" Charles said, and then cried out in pain and grabbed at his rear in what would in other circumstances be almost comic.\n\n\"Up your butt, are they? Right on schedule. There'll be a couple million of them by now.\"\n\n\"We'll give you anything! Anything!\" Charles pleaded.\n\nBurnofsky looked sick, like a man on the edge of vomiting. He stood wearily, old bones popping with the effort, and stepped closer, just out of reach of Benjamin's grasping claw of a hand. \"Anything? Will you? Then give me back my little girl.\"\n\n\"She had to die; it was treason!\" Benjamin raged. \"She was a filthy, treacherous, little\u2014\"\n\nBurnofsky punched him. It wasn't much of a punch, just enough to start the blood draining from a reddened nose.\n\n\"Give me my daughter. Give me my pride back. Give me back my own brain. Do all of that, and I'll stop them.\" Then, he laughed\u2014a sudden, strange noise. \"Kidding. They will carry out their programming and\u2014\"\n\nThe floor tilted suddenly, a 10 degree pitch that sent the Twins sprawling. Burnofsky staggered but remained standing.\n\n\"My apocalypse,\" Burnofsky said, holding the deadly remote control aloft. \"Not Lear's, not yours. Mine.\"\n\n\"You're insane!\" Charles wailed.\n\n\"You think?\" He drained the last of his bottle and smacked his lips. \"Who wasn't insane in this?\" His eyes fell on Noah's twitching body. Noah made an incoherent sound. The tilting floor had sent the pool of blood trailing off like rivulets on a windshield. \"Him, maybe. Seems like a decent kid. Maybe even sane.\"\n\nThe Twins were wallowing back and forth like a cockroach on its back, trying to roll over so they could stand. Noah's blood met Benjamin's elbow and soaked his shirt.\n\nThe smell of smoke had been growing more noticeable, and now it could be seen, too, pouring in from two directions as well as rushing past the windows like some gravity-defying waterfall.\n\nThe Twins were screaming now, fighting each other to scream, lungs pumping out of sync, heart hammering. Screaming as the nanobots used their flesh to create more nanobots, millions of little worker ants carving tiny slices of flesh, busy little hog butchers carving a living pig.\n\nAgainst all odds, slipping in blood, their own and Noah's, the Twins managed to get to their feet.\n\nWith a sound of screeching metal and shattering glass, the Tulip sagged farther and Burnofsky staggered forward and was flattened against the glass windows. The floor now tilted up and away from him. But he still held the remote.\n\nThen Noah began to slide, his movement lubricated by his own blood. He slid straight toward Burnofsky.\n\nWith a sound like wood being split, the window behind Burnofsky cracked but did not shatter. Burnofsky tried to push himself away, to reach something, anything he could grab, but his feet were slipping on the same blood that bore Noah's body straight toward him.\n\nAnd then Noah made one desperate reach and grabbed the rolling bottle of vodka. He grabbed it and dug his heel in\u2014slipping, sliding, but the angle helped him to rise, just a little, just enough, just enough to hurl the bottle.\n\nThe bottle smashed into the cracked window.\n\nBurnofsky in a moment of terrible awareness pressed his thumb on the remote control, but missed the button. The remote was in his hand, but awkwardly held. He reached with his free hand to straighten it, and the window blew out.\n\nBurnofsky went flying, flying through shattered glass, falling on his back toward the street far below. Noah had plowed into him, and for a moment the two of them were tangled in midair, grotesque acrobats trailing red.\n\nBurnofsky fell and squeezed, but the remote was in the air beside him, falling, and his hand was held in the slippery grip of the boy with blue, blue eyes.\n\n_Madness_ and _death_ , Noah thought. It was funny.\n\nHe laughed as the sidewalk rushed toward him and obliterated all that he was.\n\nThe Twins staggered into the hanging monitor, where Charles managed to grab on, powerful fingers gripping slippery steel and glass.\n\nCharles Armstrong saw his face, their faces, in the hanging mirror they used to speak eye to eye.\n\nWhat he saw was a grotesque head with two staring eyes and a third, lesser eye that now belonged entirely to Benjamin. Two mouths screamed. A line of blood had been drawn between those mouths, between those eyes, as the self-replicating nanobots chewed their industrious way through all that connected Charles to his brother.\n\nThe pain was unendurable. He could only scream and scream as his privates and rectum, his stomach and chest, his neck and back and now head were eaten away, faster and faster as the nanobot army multiplied. Eaten away and then cauterized as Burnofsky had planned, so that blood loss would not occur too quickly.\n\nCharles did not feel the moment when his body began to disconnect from Benjamin's, the agony did not allow for calm consideration. But he saw, as he looked down, as he and Benjamin lowered their massive head to see, that they were now two dying men, two, connected only at the brain.\n\nBenjamin slipped, his leg going out from beneath him, but Charles still stood, as like a dividing cell they split slowly apart.\n\nThen finally Charles lost his grip, and they fell onto their backs and slid toward the window.\n\nCharles tried to scream, but his throat was gone.\n\nThey slid, consciousness fading in a hell of pain and terror as they accelerated.\n\nBenjamin stuck out a hand and grabbed the leg of a table, but it, too, was sliding. And then, with a bump at the sill, they were in the air.\n\nIt would take them just under eight seconds to fall to the pavement. At four seconds before impact Charles saw Benjamin's body separate from his, a crudely bisected man trailing blood.\n\nHe saw Benjamin. Saw him _there. There!_ For the first time in his life.\n\nThe Armstrong Twins hit the pavement two tenths of a second apart.\n\nTwo and a half minutes later, the Tulip came down in a catastrophic eruption of flame, smoke, steel, dust, and debris that buried Burnofsky and the Twins and Noah.\n\nAnd the remote that would have destroyed the world.\n\n# **TWENTY-EIGHT**\n\nPlath and Wilkes had to walk and hobble the whole way back to the safe house. The subway had been shut down. The taxis had fled the streets. They saw cars pass by, heading toward the bridge, pets and houseplants inside, household goods strapped to the roof. A hard-to-frighten city had at last been frightened.\n\nBy the time they made it they were numb with cold, lips blue, teeth chattering. Plath's tears had frozen on her cheeks. She recalled the \u00cele Sainte-Marie, recalled where she'd been not very many days ago. A completely different world. It had been so perfect there. Warm sunshine and blue water and Noah.\n\nThey had killed him. Noah. They had killed him.\n\nInside the safe house at last the two girls collapsed onto the couch and shivered, burrowing beneath throw pillows in search of warmth.\n\nPlath saw that Anya was coming to investigate the noise. In the window in her mind she saw herself through Anya's eye. She looked pitiful. Her face was smeared with smoke; her hair was thick with ash.\n\n\"What is the matter?\" Anya asked. She didn't wait for an answer but ducked out to come back with blankets to pile on the frozen girls. Then she made hot tea and helped them hold the cups until their hands could stop trembling.\n\n\"Where is Keats? Where is Billy?\" Anya asked, already suspecting the answer. The TV had been on when they came in, tuned to news. On the screen the Tulip fell again and again. Hollywood and city luminaries ran wild through the streets again and again. The lurid loops played over and over again.\n\n_Plague of Madness_.\n\nAn overhead shot of the Brooklyn Bridge was a river of red lights\u2014cars fleeing the city.\n\n\"Dead,\" Plath said. \"Both dead.\"\n\n\"This is Lear's doing,\" Anya said. \"He is\u2014\"\n\n\"She,\" Wilkes interrupted. \"Our overlord and master is a chick.\" Then, eyes darting suspiciously toward the stairs, said, \"Get Vincent down here. Get Mr. Seventy Percent.\"\n\nAnya seemed ready to argue, but acquiesced with downcast eyes.\n\nPlath felt a wave of exhaustion that forced her eyelids down. She coughed\u2014she'd been coughing the whole way home. The nauseating stink of smoke was in her nostrils, the taste of it in her mouth, and more came when she coughed.\n\nVincent arrived silently and stood with Anya by his side, looking like a man waiting for his own firing squad.\n\n\"What did you know?\" Plath asked wearily.\n\n\"What do you mean?\" he asked, and Wilkes was up out of her seat and swinging a fist at him, which he blocked easily. She swung again, but with less conviction, and he gently pushed her back down onto the couch.\n\n\"What did you know, Vincent?\" Plath asked again with deadly, weary calm that carried absolute authority. \"Did you know who Lear is?\"\n\nHe blinked and shook his head. Then he leaned toward her, frowning. \"Are you saying you _do_ know who he is?\"\n\n\"She,\" Wilkes said. \"She, she, she. She. A sister. One of the vaginally endowed. Lystra Reid.\"\n\nVincent drew back as if frightened. \"You can't do that, you can't talk about Lear. Caligula will\u2014\"\n\n\"He's dead, too,\" Wilkes said. \"That's his work.\" She stabbed a finger at the TV. \"He's dead. And Jin is dead. And Ophelia is dead. And Renfield is dead. And Billy is dead. Even the Twins are dead. And pretty\u2014\" She sobbed, and it was a moment before she could go on, her voice low and grating. \"It's a whole big bunch of dead tonight. Now answer Plath's question, Vincent, or I swear to God I'll find some way to make you dead, too.\"\n\n\"I met Lear once. I didn't look at him. Her, if you say so. Maybe that explains why I was told not to turn around and look. He, she, whoever, used voice masking. I assumed it was so that later I wouldn't be able to recognize the voice.\"\n\n\"And did you know Lear planned to do this? To use biot madness this way?\"\n\n\"No.\"\n\nPlath set her teacup aside carefully. Her hands were still hard to control with any finesse. The cold seemed to have sunk deep in her, joining the cold, dead spot where Noah had been held in her thoughts. \"Where do you stand, Vincent?\"\n\nHe did not pretend he didn't understand. He grasped her meaning immediately. \"I'm not sure I know who I am,\" he began.\n\nWilkes interrupted. \"Yeah, well, welcome to the new reality. We've all been mind-fucked one way or the other.\" She laughed her mirthless heh-heh-heh and said, \"We really are BZRK now, I guess. Crazy.\"\n\n\"I wish they would stop showing that,\" Anya said, transfixed by the TV.\n\n\"Where do you stand, Vincent?\" Plath repeated.\n\n\"I have to...\" He began, hesitated, shook his head, and continued. \"I have to go back to basics. To what I believe. For a start, my name is not Vincent. It's Michael Ford.\"\n\n\"I'm sticking with Wilkes. It works for me.\"\n\n\"I'm Michael Ford,\" he said, almost wonderingly. Like a little kid talking about some new and amazing thing he'd just learned. \"I'm Michael. I believe... I believe people should be free, that's why... I believe they should be left alone. That's why I joined BZRK.\"\n\n\"That's why everyone _joins_ ,\" Anya said, speaking that last word with distaste. \"No one ever _joins_ to do evil. It just always ends up that way.\"\n\nVincent winced as if she'd struck him.\n\nPlath said, \"Burnofsky is releasing self-replicating nanobots. Maybe they were all destroyed in the Tulip, maybe not. If he did it, if they aren't all killed, well... anyway, the Twins and Burnofsky are no longer the problem. Lear is the problem. And I don't think she's done. I think she'll keep at it. She wants to...\" She shrugged. \"I have no idea what she wants.\"\n\n\"Noah would have,\" Vincent said softly. \"He was a gamer. This is all a game. It's been a game from the start.\"\n\nPlath stared at him, thinking. He did not look away. \"A game,\" she said finally. \"And what's the point of this game?\"\n\n\"Games have no point,\" Vincent said. \"The point of the game is the game. The purpose is to play. But games have structure. They are built and written. And you can only play one at a time. Lear is wiping the board of the old game, replacing it with his... _her_ own game.\"\n\n\"How do we win?\"\n\n\"To win you have to understand the...\" He shook his head. \"You can't beat the game designer at her own game.\"\n\n\"Sure you can,\" Wilkes said. \"I used to beat my little brother at games all the time. I'd pull the power cord out of the wall. Game over.\"\n\nBefore she got on her plane, Lystra Reid, Lear, punched a code into her phone and pushed Send.\n\nThe text went to the nearest cell-phone tower. The signal went from there to a central router that pushed it up to a satellite from whence it was bounced to another satellite, and still another as it wound its way south. Eventually it was picked up by a satellite dish.\n\nFrom there it traveled just a few hundred feet to a computer server that recognized the code and translated it into sixteen thousand individual digital instructions that then mostly retraced the digital path of the incoming message.\n\nElapsed time, 3.4 seconds.\n\nIn cr\u00e8ches concealed in locations in several cities across North America, Europe, and Asia, DNA stew was bathed in various enzymes before receiving three micro-doses of drugs and a final jolt of electricity.\n\nForty-eight thousand biots\u2014three for each of the sixteen thousand DNA signatures\u2014came to life.\n\nOnly fifteen thousand, eight hundred and four people (a number had died since their fateful visit to a medical testing lab) saw windows open in their minds.\n\nOf those, fewer than a third understood what it meant.\n\nThey generated more than three thousand terrified calls to 911 in the U.S. and 999 in the UK and 112 in the European Union.\n\n\"That's the first tranche,\" Lear said. To the pilot, she said, \"Okay, we can go now.\"\n\nBug Man did not want to ask. He risked making her angry, and in this new world, where his life belonged to her, he did not want to do that. But he couldn't help himself.\n\n\"My mum?\"\n\n\"By now she's thinking, 'Blimey, what's that then?' \" Lear said, switching to an exaggerated British accent. \"There's windows in me head, innit?\"\n\nBug Man's throat convulsed. Tears came to his eyes, impossible to stop.\n\n\"Best to move on, Buggy,\" Lear said. \"Get over it. Look at me. My father died tonight, and do I seem all weepy? Hey, have you decided what color teeth you want?\"\n\n\"What?\"\n\n\"The teeth. The teeth!\" She pointed at her own. \"How about green? I like green.\"\n\nAs the jet taxied the acid rolled toward forty-eight thousand biots.\n\n\"Hah, there we go, yeah,\" Lear said. \"Now we're going to play.\"\n\n\"We know her name now. Lystra Reid,\" Plath said.\n\nAnya typed it in. Instantly the computer monitor lit up with links and photos.\n\n\"I've seen her before.\" Wilkes frowned, then snapped her fingers. \"Nijinsky. She was there when Jin died.\"\n\n\"Lear. She's thirtysomething, born in Bogalusa, Louisiana. Parents not listed. Schools, nope. That's about it except for later business stuff. She owns a lot of medical testing labs.\"\n\n\"That would make sense,\" Anya said.\n\nVincent, seemingly exhausted by his earlier conversation, remained silent.\n\n\"That's probably how she met my father. And it's how she got DNA samples.\"\n\n\"She will have millions of them,\" Anya said.\n\nPlath looked at the best photograph of Lystra Reid. What was there in that pretty face to betray the existence of an evil, disturbed mind? Nothing. The eyes were clear, the expression open, the mouth smiling.\n\nPlath remembered what Stern had told her. That Lear had used burner phones but without masking the callback number. One had been purchased in Tierra del Fuego. The other in New Zealand, she could not recall the city. But both had been connected to Antarctica.\n\n\"Search 'Lystra Reid' and Antarctica,\" she told Anya.\n\nThat earned a raised eyebrow, but the search caused a long, slow exhale. Lystra Reid had purchased a company called Cathexis.\n\n\"Pull up any articles on Cathexis Inc.,\" Plath instructed.\n\nThe four of them read silently. Wilkes moved her lips. Plath felt a new pang as just for a moment she thought to turn, look over her shoulder, and ask Noah what he thought.\n\nBut there was no Noah. No Noah, no Nijinsky, no Mr. Stern, and only a partial Vincent.\n\n\"Who has had any medical testing done in the last ten years?\" Plath asked.\n\nBut Vincent shook his head. \"Irrelevant. If we've had biots made, we're in her database.\"\n\n\"I have not had biots made,\" Anya said. \"But I have been tested at one of her labs.\"\n\n\"So we are all vulnerable. It's possible that at any moment\u2014\"\n\n\"Great,\" Wilkes said. \"Fine. Let me go nuts. I'll fit right in.\"\n\nPlath looked to Vincent. \"What will she do next?\"\n\nVincent thought about it, eyes dark beneath his brow, mouth a grim line. \"Her goal is instability. What else could it be? With her skills and her resources, if all she wanted was the whole world dead, she could have grown smallpox or anthrax in a lab somewhere. And she has nanotechnology. Why have us use biots to fight the Armstrongs? She had the upper hand all along. She could have used a lot less effort and simply obtained a sample of their DNA, grown biots for them, and inflicted biot madness.\"\n\n\"Okay, why didn't she?\" Plath asked.\n\n\"Because she's a gamer,\" Vincent said with more confidence than he felt. \"She wants to win, yes, but first she wants to _play_. We were Level One.\"\n\n\"Then we're in Level Two now.\" Plath nodded. \"Now she drives the whole world crazy. Watches it. Shows up in person to enjoy Jin's death. Probably other events as well. She's enjoying all that.\"\n\n\"Sick bitch,\" Wilkes muttered.\n\n\"She brought me back, made me a part of it again. Why?\"\n\nVincent shrugged. \"Because you're her avatar. She wants you to go on playing. Bluebooking.\"\n\n\"What?\"\n\n\"It's an old gaming expression. It's when a player keeps a journal of the game, but from the POV of the avatar.\"\n\n\"You are smart and rich and pretty,\" Anya suggested. \"Just as she is. And alone. As she must be.\"\n\n\"Lear sent me to recruit you,\" Vincent reminded her.\n\n\"And when I was enjoying the island too much, she forced me back into the game. She even left clues for me to find that would link her to Antarctica.\"\n\n\"Machines do not work well at very cold temperatures,\" Anya said. \"And nanobots are machines.\"\n\n\"Okay, so Antarctica because\u2014\"\n\n\"Because if the gray goo has been unleashed, it will have a hard time penetrating hundreds of miles of subzero temperatures. It's the safest place on the planet if you're worried about that.\"\n\nVincent nodded agreement. This was the most engaged Plath had seen him in a long while. Was he ready to take command again? _No. This is my game now_.\n\n\"Antarctica is also a place to ride out whatever shitstorm she's unleashed,\" Wilkes offered. \"It's as far away as you can get without being on the moon.\"\n\n\"So she camps there,\" Vincent said. \"Safe from the goo. And safe from the consequences of her own game.\"\n\n\"She camps. She waits. Why?\"\n\n\"For Level Two to play out. So she can be there for Level Three.\"\n\n\"And what is Level Three?\"\n\nVincent shook his head slowly. \"Only Lear knows that. It's her game.\"\n\n\"And we can't beat her by playing her game,\" Plath concluded. \"We can only pull the power cord.\"\n\n\"The power cord is south of here,\" Wilkes said.\n\n\"She will expect that,\" Vincent said.\n\n\"Expect it? I have a feeling it's what she wants,\" Plath said.\n\n# **TWENTY-NINE**\n\nFrom New York to Tierra del Fuego was a bit over six thousand five hundred miles, which at a speed of four hundred eighty knots took eleven hours. It was not a pleasant flight for Bug Man. But he was lucky. The rest of the world was faring much worse.\n\nDuring the time Bug Man was in the air eleven more tranches of forty-eight thousand biots, totaling five hundred twenty-eight thousand, were generated from stored DNA patterns. Dividing by three biots per person, that was approximately one hundred seventy-six thousand people who lost their grip on sanity.\n\nThey were concentrated in fifteen major cities for maximum disruption. New York, London, Berlin, Paris, Shanghai, Los Angeles, Tokyo, Mexico City, Moscow, Washington, Rome, Beijing, Jerusalem, Mumbai, and Sydney.\n\nBy the time Lear's plane landed at the Ushuaia's airport, Los Angeles, Jerusalem, and Berlin were burning.\n\nThe flight to the ice would be slower and in a less comfortable plane: Lear's sumptuous private jet could not land on ice. It was two thousand seven hundred miles from Ushuaia to Cathexis Base, but flying in a refurbished C-130 Hercules turboprop with a cruising speed of three hundred thirty miles an hour, it took more than eight hours. Another one hundred twenty-eight thousand people, minus those who had already passed away, were driven into madness.\n\nThese were concentrated in and around military bases in the United States, Russia, China, the United Kingdom, France, India, and Pakistan. The choice of countries was not random: each had nuclear weapons, but of those, France and the UK used only submarine-based weapons.\n\nThe first launch was from Russia, but the missile and its warhead were destroyed in flight, en route to North Dakota.\n\nThe second launch was from Pakistan. It landed in the middle of a department store in New Delhi, India, but did not explode. The madman who had fired it had not armed the warhead.\n\nBut the Indian military did not wait to consider the situation. Indian missiles flew minutes after the C-130 slid to a stop on the Cathexis Base airfield.\n\nFifty-two Agni-III and Agni-IV missiles flew, striking targets in Pakistan. By the time the C-130 had been refueled for the last leg of its flight, there were thirty-one million dead, a number that would double within days.\n\nAfter a shorter hop and a very bumpy landing, Bug Man stumbled from the plane still wearing the T-shirt he'd been wearing in New York. His teeth, his entire mouth, and jaw hurt. He was exhausted, having been awakened repeatedly by nightmares. And now he was more cold than he would have believed possible. And standing in the whitest place on Earth.\n\n\"Where are we?\"\n\n\"The bottom of the world, Buggy, the place where machines go to die. People, too.\"\n\nA green Sno-Cat was tearing across the snow toward them. It roared to a stop and two men jumped out. One ran to Lystra with a full-length coat that many foxes had died to provide. The other handed a voluminous down parka to Bug Man, who shivered into it. A fur-lined hat was plopped on his head, and he was hustled into the backseat of the Sno-Cat. It wasn't exactly warm inside, but it wasn't fatally cold, either.\n\n\"How was your flight, Ms. Reid?\"\n\n\"Fine, Stillers. Fine. Are all the necessary personnel in from Forward Green?\"\n\n\"Yes, ma'am, all personnel, all equipment, all supplies, except for the final two sleighs, which are being prepped and will be brought here tomorrow. And we've topped off the fuel both here and at Forward Green.\"\n\n\"Then we are in lockdown,\" she said pleasantly. \"Except for the final sleighs. Make sure no one shoots at them.\" She shook her head as if marveling at the world's unpredictability. \"The world has just gone to hell in a handbasket, yeah, and we have a long year ahead of us.\"\n\nNot waiting a second, Stillers keyed a radio and said, \"Lockdown, lockdown. Lockdown, lockdown.\"\n\nBug Man could not quite imagine what was being locked down. It wasn't like there was a crowd standing around trying to break in. They were in the middle of a whole lot of nothing as far as he could see.\n\nThen, as if by a miracle, the ground seemed to open up. The Sno-Cat rounded a sharp corner, treads churning up hard-packed snow, and plunged down a long ramp into a valley. He saw buildings and an improbable house and...\n\n\"Is that a swimming pool?\"\n\n\"Yes,\" Lystra said. \"One of only two in Antarctica.\" Then, with a wistful look, she added, \"I like to swim. It's a very clean sort of sport, yeah. And I look amazing in a bathing suit, yeah, if I say so myself.\"\n\nBug Man thought that was likely true, if _amazing_ was the right word for a woman covered in tattoos of her victims.\n\n\"There's also an underground greenhouse. Palm trees! Palm trees in Antarctica, yeah. Yeah. We can live very well here for two years, or survive for three. If necessary. We'll see.\"\n\n\"Do you want to go to the office?\" Stillers asked deferentially.\n\n\"No, the house. Find quarters for Bug Man, but for now he'll stay with me.\" She patted Bug Man's knee. \"I've decided he's my good-luck charm. Oh, and tell the dentist, Dr. Whatever-the-Hell, yeah, he's got a customer. Patient. Whatever. Yeah.\"\n\nTanner was among those waiting when an unannounced flight came into McMurdo, running on fumes, or so the pilot said. Planes did not just suddenly arrive on the ice. And Tanner, like everyone else at the base, had been watching events back in the world with disbelief and anxiety turning to fear.\n\nTanner had called Naval Intelligence in Washington and been told that _Satan is loose among the flock, hah-hah, redrum redrum, they're listening, don't you know that?_\n\nA call higher up the chain of command to the Pentagon had gone unanswered. Calls to USAP and Lockheed had yielded nothing.\n\nTanner was in summer gear\u2014a parka over padded jeans with the big Mickey Mouse boots unlaced. He wore gloves and goggles and a light stocking cap with a Pittsburgh Pirates logo.\n\nThe plane, a C-130, a Herc in the patois, landed easily, and killed engines. Tanner reached under his parka to touch the butt of his trusty Colt .45 auto. Everyone authorized to carry a gun was carrying one. As a safety measure that would have been absurd in earlier times, Tanner had stationed an ex-sergeant with a sniper rifle on the roof of a parked truck.\n\nThe person who stepped first from the plane could not have been less likely.\n\n\"It's a girl,\" Tanner said.\n\n\"Yep, that's a girl.\" This from the station chief beside him. \"Looks kind of familiar. Not some crazy pop singer, is it?\"\n\nBehind the girl came a grown woman, rather beautiful and just exotic enough to hold Tanner's eye for longer than strictly necessary. Then a girl with a strange half mohawk and a stranger tattoo below one eye. And finally a young man with dark hair, a calm expression, and an air of tension that Tanner associated with trouble.\n\nThe girl walked up without hesitation, in a hurry. She pulled off her glove and stuck out her hand. \"I'm Pla\u2014Sadie McLure.\"\n\nThe station chief, Joe Washington, shook her hand and glanced at Tanner.\n\n\"Sadie McLure,\" Tanner repeated, frowning as he tried to pull the name from memory.\n\n\"Yes. As in Grey McLure crashing a jet into a Jets game,\" she said. No hint of a smile. A very serious, even grim young woman. \"These are my friends. Wilkes. Dr. Anya Violet. Michael Ford.\"\n\nTanner remembered now. \"What exactly are you doing here, Ms. McLure?\"\n\nHer eyes bored into him. They were eyes that belonged in a much older face. \"We're here to try to stop what's happening. We're here to kill the woman responsible.\"\n\n\"The woman responsible? Here?\" Washington wanted to laugh, but the faces before him did not look as if they were joking.\n\n\"Lystra Reid.\"\n\n\"Cathexis Inc.?\"\n\n\"And some other businesses as well. What's happening is her doing.\"\n\nThe station chief had to laugh at that. \"Excuse me, but I've met Lystra Reid, and she's a sharp young businesswoman. I don't know what\u2014\"\n\n\"Let them talk, Joe,\" Tanner said quietly.\n\nThe station chief seemed almost offended, but he nodded. \"Okay. Not here. We'll drive you to my office.\"\n\nAn hour later Plath and Vincent, with occasional outbursts from Wilkes, had told their tale.\n\n\"To say that sounds crazy is an understatement,\" the station chief said.\n\n\"Do you have any proof?\" Tanner asked.\n\nPlath cocked her head and looked at him. \"You know something.\"\n\nTanner smiled slightly. \"Do you have proof?\"\n\n\"As a matter of fact, I do,\" Plath said. \"We thought you might be skeptical. \"So here's what's going to happen. I'm going to just touch my finger to your face. Then, in a few minutes you're going to open a book at random. You'll hold the page close to your face. And I'll tell you what you're reading.\"\n\n\"What is that, some kind of magic trick?\"\n\n\"It's the best I can do on short notice,\" Plath snapped. \"If you like, I could blind you, or start sticking pins in your brain and giving you some amazing hallucinations.\"\n\n\"I'll read a book,\" Tanner said. Ten minutes later he was shaken and convinced.\n\n\"What do you want from us?\" Washington asked. He was still skeptical, still not sure it wasn't some sort of trick, but he also knew that in matters of security, Tanner was the real boss.\n\n\"Fuel,\" Plath said. \"And men with guns, if you have any.\"\n\n\"Men, I have. Guns? I could spare a couple of handguns and a hunting rifle. But Mr. Tanner here may have other means.\"\n\nTanner shifted uncomfortably, then made a decision. \"Okay. Cards on the table. We've been looking at Cathexis for some time now in relation to a souped-up hovercraft they seem to have built. An _armed_ hovercraft. I sent a person with some military background in to check it out. I have not heard back from her.\"\n\nVincent spoke for the first time. \"You're intelligence.\"\n\nTanner gave a short nod.\n\n\"Then you have people you could call.\"\n\nTanner snorted. \"Are you kidding me? With what's happening back in the world? Shit has hit the fan. Cities are burning, people are scared to death, my chain of command...\" He threw up his hands.\n\n\"If we can prove to you that this woman is doing what we say she's doing, if we can prove to you that we can stop her, will you do all you can?\" Plath asked.\n\nTanner thought about that for a moment and glanced at Washington, who raised his hands\u2014palms out\u2014in a gesture that said, _It's on you_. \"Yeah,\" Tanner said. \"You prove all that, and I will do all I can to bring down the wrath of God.\" Then, under his breath he added, \"But it won't work.\"\n\nSurreal, that was the word Bug Man had been searching for. Surreal.\n\nHe was in Antarctica, in a dry valley way below the ice, in a house, in a very expensively furnished living room, looking out of expansive windows onto a domed swimming pool, while a lunatic and mass murderer suggested he could replace the teeth she herself had broken with fangs. Green fangs.\n\n\"It would give you an original look,\" Lear said. \"Do you know how to cook at all? My cook is busy, yeah, helping to inventory supplies. Can you fry some eggs?\"\n\nA television was on in the kitchen where Bug Man rummaged in a vast refrigerator for eggs and bacon. That much he could do. Eggs and bacon.\n\nThe television showed the BBC, but it wasn't any of the sets he'd ever seen them use. It looked a bit as if the male and female announcers were broadcasting from a concrete bomb shelter.\n\nThe crawl at the bottom of the screen was full of warnings from the army that people should stay in their homes and off the streets. That and statements from Number 10 and acting prime minister Dermot Tricklebank, whoever that was, to the effect that _the only thing they had to fear was fear itself_.\n\n\"Hunh,\" Lear commented. \"That's a Roosevelt quote. Shouldn't they be using Churchill?\"\n\nThe stovetop was a restaurant-quality thing with massive knobs and too many burners. It took Bug Man a few anxious minutes to figure out how to work the knobs, but eventually he was able to lay six strips of bacon on a grill.\n\n\"Crispy,\" Lear said, pointing at the bacon.\n\nThe announcer said, _\"The nuclear exchange between India and Pakistan has escalated, with at least five major Indian cities now essentially vaporized.\"_\n\n\"Hah,\" Lear said. \"And don't forget the eggs. Not too runny.\"\n\n\"Okay.\"\n\n_\"Winds are whipping the fire now spreading out of control through Bayswater and Notting Hill. Our reporters have seen no evidence of effective emergency response.\"_\n\n\"It's hard to tell when an egg is done, yeah, but... Oh, look look look! He's setting himself on fire!\"\n\nBug Man did not want to see that and instead focused on his work.\n\n\"Looked like a banker. Nice suit. It's interesting that a person can be mad and yet plan ahead well enough to find gas. Or _petrol_ , as you would say.\"\n\n_Yes_ , Bug Man thought grimly, _who would have thought a crazy person could plan?_ He turned the bacon and held down the curling tips.\n\n\"Oh, look at that! Look at that video!\" This was spoken as an order not a request, so Bug Man looked. The tape showed an American Airlines 787 roaring down from the sky and smashing into a very large, gray Gothic church. The announcer said something about the cathedral at Reims.\n\n\"Not that great an explosion, though,\" Lear opined.\n\nSuddenly the BBC was off the air, replaced by static.\n\n\"I knew this would be a problem,\" Lear said. \"I avoided messing with media folks, yeah, but there's no way to stop someone cutting their power.\" She began flipping through channels. Static and more static. Then what appeared to be a Japanese news station with a fixed camera aimed at a woman who was giggling and stabbing her arm with broken shards of wooden chopsticks.\n\nAl Jazeera was on, but in Arabic. A Russian station had a bespectacled, overweight man with a bottle of vodka before him on the anchor desk. He seemed to be announcing news, but his voice was slurred, and as they watched he began weeping.\n\n\"CNN! Yes! See, that's why I took it very easy on Atlanta.\"\n\nLear seemed to think she deserved some praise for her foresight.\n\n_\"At this time we cannot confirm that the event in Norfolk, Virginia, was a nuclear explosion, although Norfolk is a major naval base that does handle ships carrying nuclear weapons.\"_\n\n\"No video?\" Lear moaned.\n\n_\"We now have video of an oil refinery in Port Arthur, Texas, which is burning_.\"\n\n\"I've seen oil refineries burn,\" Lear complained. \"I've never seen a nuked city. Come on, they must have some video.\"\n\n\"Here you go.\" Bug Man plated the bacon and eggs.\n\n\"Next time drain the bacon a little better. Blot it with a paper towel.\"\n\nThey went into the dining room, all rich, dark wood with high-backed chairs. A chandelier hung above the table.\n\n\"It's going well, don't you think? Yeah?\"\n\n\"Yes.\" What else could he say?\n\n\"Early stages yet.\" She munched thoughtfully. \"I wonder if I should spread it out, you know? My first plan was to keep up the pace, sixteen thousand an hour. But what if... No. No, I'm sticking with the original plan. I don't want to start second-guessing myself.\"\n\n_No, you wouldn't want that_ , Bug Man thought. He wondered if his mother was still alive. Had she killed herself like so many seemed to do? Was she even now wandering the streets, raving? Maybe hurting other people? Maybe being hurt herself?\n\nWhat was the point of caring? Lear had won. The world was going crazy. The human race was killing itself in an orgy of madness.\n\n\"I have some work to do,\" Lear said. \"You stay and watch.\"\n\n\"I don't think\u2014\"\n\n\"That wasn't a request, yeah? Stay and watch. You know what to look for.\"\n\n\"I do?\" Bug Man was mystified.\n\n\"The Armstrongs had self-replicating nanobots. Yeah. Maybe the fire at the Tulip got them all. Maybe not.\" Lear shook her head and her mouth was a grim, worried line. But she cheered up considerably when the news announced that Berlin, Germany, had been hit by a nuclear weapon.\n\n# **THIRTY**\n\nThe C-130 carrying Plath and her crew, as well as Tanner and seventeen ex-military volunteers from McMurdo, landed at Cathexis Base to find employees there bewildered and frightened. Their medical team had all been ordered to Forward Green a day earlier without explanation. And the Plague of Madness had spread there as well. They had seven people locked up. A dormitory had been burned to the ground, killing three.\n\nThe C-130 flew on to Forward Green. It was a cargo plane, a cavernous, incredibly noisy and very cold open space with webbing seats along both sides. Large dotted lines had been painted on the curved walls, indicating just where the propellers would chew through the fuselage should one come off in flight.\n\n\"This is their only other facility, so far as we know,\" Tanner said.\n\nThe plane circled, coming around into a strengthening wind. The sun was low on the horizon, as much like night as Antarctica got this time of year.\n\n\"I don't like that layout,\" one of the ex-soldiers said. \"Those towers sure as hell look like gun emplacements.\"\n\nThe pilot called back over the intercom. \"They are refusing to let us land.\" Then, a moment later, \"Sir, they are warning us that they will open fire if we attempt a landing.\"\n\nTanner looked at Plath. \"Well, I guess that tends to confirm your story.\"\n\nHe unhooked himself from his webbing seat and went forward to speak to the pilot.\n\nPlath looked at Vincent\u2014arms folded, eyes in shadow. At Wilkes, snoring beside her, somehow curled into a fetal ball in the webbing. And Anya, who seemed never to need sleep.\n\nPlath had removed her biot from Anya. With apologies. They were all three now in her own head, as safe as they could be. To kill her biots you'd have to kill Plath herself. Three windows were open, as they always were, now showing slithering macrophages and twitching neurons and what were hopefully spiky balls of pollen in her eye and not bacteria.\n\nShe\u2014\n\n_BOOOOM!_\n\nSomething had smacked the C-130 a staggering blow. Tanner came tearing back from the cockpit, the back of his jacket on fire. Plath unbuckled and threw her parka over him, smothering the flames.\n\nThe plane jerked again, not as hard, but then nosed down. They were low, no more than four thousand feet up; there was little room to recover.\n\nThe nose came slowly, slowly up, but as it did the plane went into a steep turn that threw Plath into Vincent.\n\n\"Sons of bitches!\" Tanner yelled.\n\nPlath worked her hand into the webbing and held on as the plane rolled, rolled, and she hung suspended in midair while baggage and vomit flew everywhere and grown men screamed.\n\nWilkes was yelling something that Plath couldn't hear. \"What?\" she yelled.\n\n\"I said: I can't say it's been fun, Plath, but it was good knowing you!\" Wilkes made a little mock-salute.\n\nPlath reached her free arm across and took Wilkes's hand. Plath was not afraid to die, in some ways it spelled relief. But she was furious at the idea that Lear would win. \"I'm not dying until I've killed that bitch!\" she yelled to Wilkes, who smiled wryly and squeezed her hand.\n\nThen, with a series of bone-shaking jerks, the plane slowly, slowly leveled off, but all the while it drifted lower.\n\nThe pilot, voice wracked with pain and fear, yelled, \"Hard landing! Hard landing! Brace! Brace!\"\n\nThe impact rattled Plath's spine and chipped one of her teeth as her mouth slammed shut. The webbing seat held her, but Anya was knocked from her seat and fell to the metal floor of the plane. A metallic shriek went on and on and on.\n\nAnd that's when a spinning propeller\u2014almost twenty feet from tip to tip\u2014exploded through the flimsy fuselage, tearing Anya Violet and two of Tanner's men apart.\n\nThe plane skidded to a stop.\n\nA giant gash made by the prop had nearly split the plane in two. Jagged metal edges were everywhere, blood and pale viscera was sprayed around the fuselage like some demented Jackson Pollock painting. A man with his leg gone at midthigh bellowed like a dying bull and tried futilely to cover the pulsing wound with his hands.\n\nSmoke rolled back through the cargo bay, whipped away by a brutally cold wind coming through the gash.\n\nVincent stared at the place where Anya had been. He picked up something white and red, some unrecognizable part of her, and held it cradled on his lap.\n\nTanner was among the first to recover. \"Get ready! They may send someone to finish us off!\" He drew his pistol. It looked small and irrelevant in his hand. Dazed men responded, drawing their few weapons. One was trying to draw a gun with a hand that was no longer there. Another man gently eased him into the webbing and took the gun from him.\n\n\"You okay?\" Plath asked Wilkes, and got a shaky nod in return. \"Vincent?\"\n\nVincent stared at her as if he'd never seen her before, maybe wasn't seeing her now. His shallow breathing formed a small cloud of steam.\n\n\"Anyone who can, follow me!\" Tanner said. He wound his way through tangled metal to leap from the gash. Half a dozen men followed. Plath and Wilkes went to Vincent. \"Come on, Vincent. Stay alive now, grieve later.\"\n\nHe flashed a look of pure, unadulterated fury that Plath at first thought was directed at her.\n\n\"Come on, Vincent. We have to get off this pl\u2014\"\n\nA machine gun, sounding like a chainsaw, opened up. A line of holes appeared at the tail end of the cargo bay and walked its way forward. Metal was flying everywhere. The air stank of cordite, steel, blood, and human waste.\n\nPlath grabbed Vincent by the jacket and yanked him to his feet as Wilkes undid his safety harness. Vincent let the gruesome body part drop, hesitated as if he might go back for it, and then Plath shoved him out onto the ice and jumped after him.\n\nWilkes landed on Plath, rolled off, and slithered on her belly. Plath glanced back and saw a Sno-Cat with a machine gun mounted on its roof, still firing from the far side of the wreck.\n\nThen, with a _woosh_ of searing heat, the starboard-side fuel tanks exploded, billowing out over the Sno-Cat. The man firing the machine gun was aflame, twisting, writhing, trapped somehow, and the machine gun stopped.\n\nThey were three hundred feet from the nearest building, which was one of the four gun emplacements.\n\n\"Run run run!\" Tanner yelled, and led the way, slipping and staggering across the ice with the wind blessedly at his back. Plath saw immediately what he was doing. The gun tower was opening, shutters rising mechanically, revealing a long black muzzle. Tanner was trying to close the distance and get below the place where the gun could be depressed to target them.\n\nIt took twenty seconds for the shutters to open fully. Another ten seconds for the gunners to ready their weapon, and at that moment the gamble had failed. The gaggle of freezing survivors were in pointblank range.\n\nThe machine gun fired. Two rounds, killing one man instantly and hitting another in the thigh.\n\nAnd then, the gun jammed.\n\nTraining took over for the ex-soldiers. They quickly closed the distance to the tower's base and began kicking at the door. One fired at the lock. The door opened and small-arms fire\u2014a _pop! pop! pop!_ sound\u2014came from within.\n\nTanner, yelling obscenities, picked up a fallen body and threw it through the doorway to draw fire. He was in through the door in a flash. More gunfire as those with weapons rushed the doorway after him.\n\nSilence descended. Tanner and his men had taken the tower.\n\n\"Come on,\" Plath said to Vincent and Wilkes, \"we'll freeze out here!\"\n\nA second Sno-Cat was barreling toward them from the center of the compound, trailing a cloud of ice particles and steam.\n\nThe top third of the tower now rotated, bringing the machine gun to bear on the Sno-Cat, which made the fatal mistake of hesitating, slowing, and then blew apart as Tanner poured fire into it.\n\nPlath, Wilkes, and Vincent found themselves in a bare room at the bottom of a steel spiral staircase leading up. \"Wilkes, stay with Vincent.\"\n\nPlath ran up the stairs to find Tanner still cursing, but also bleeding into his parka, a growing stain.\n\n\"Goddammit, goddammit, they shot me,\" he said as he tore off his jacket, then burrowed through layers of warmth to find a hole in his left side.\n\nA soldier squatted to take a look. He grinned up at Tanner. \"Through and through, Captain. You'll live if you don't bleed out.\"\n\n\"Slap on a compress, Sergeant O'Dell.\"\n\nTanner looked at Plath. \"You look okay for your first firefight.\"\n\n\"Not my first,\" Plath said. \"Not even my second. It's been a hell of a week.\" She peered out of the shooting hole as the machine gun traversed left and right. Nothing moved. The plane and the two Sno-cats burned.\n\n\"All those buildings\u2014shuttered. Bulletproof, most likely.\" O'Dell, the ex-soldier who had tended Tanner's wound.\n\n\"Jesus H.,\" Tanner said. \"It's a fortress. See what we have here. Inventory weapons and do a head count.\"\n\nThe bad news was that there were just six battle-ready men, plus Tanner, Plath, Vincent, and Wilkes.\n\nThe good news was delivered by O'Dell. \"We have all the small arms we could want, plenty of ammo, and a dozen of these.\" The \"these\" in question were shoulder-fired antiaircraft missiles.\n\n\"I'm not familiar with those. Russian?\"\n\n\"Chinese,\" O'Dell said. \"And to answer your next question, yes, they can be fused for impact.\"\n\n\"Okay,\" Tanner said. \"That is not a professional outfit out there; otherwise, they wouldn't have driven that Sno-Cat into range and then conveniently stopped. Amateurs with maybe a couple of veterans. Short-handed and poorly led, or we'd already be dead. Let's not give them time to figure anything out. Sergeant, blow some holes in that first building. Ground level if you can. We need a door.\"\n\nThe battle lasted two hours, by which time two more men had been killed. Plath and her friends had been given the job of ferrying wounded from the plane into the first tower while Tanner led the assault on the second.\n\nWhen it was all over, they counted seven bodies of former Cathexis employees.\n\n\"A skeleton force,\" Tanner said. \"So this was just a warm-up.\"\n\nThey had assembled in the dining hall and Wilkes had helpfully brewed a pot of coffee and popped open bags of chocolate chip cookies.\n\nThey were eight now, along with three wounded survivors from the plane wrapped in blankets and lying bandaged on empty steel tables. O'Dell and one other had taken a remaining Sno-Cat to what looked like a hangar that lay well outside of the main base.\n\n\"Whoever was here pulled out,\" Tanner said. \"This place was not built for the dozen men left behind.\"\n\nVincent stood up and walked away.\n\n\"He'll be okay,\" Plath said, not believing it.\n\n\"He's been through a lot,\" Tanner said generously.\n\n\"You have no idea,\" Wilkes muttered as she poured mugs of coffee.\n\n\"We don't know if anyone got off a message to whoever, wherever... but let me just say that any skepticism about you, Ms. McLure, is officially dead and buried. We have to find wherever they went, chase them down, and stop this.\"\n\n\"All we've got is a Sno-Cat,\" a man observed. \"Holds four passengers.\"\n\nVincent came back and without pre-amble said, \"They left their computers on. There's another base. Farther south. A couple hundred miles.\"\n\nSomeone whistled low, and slow, and said, \"That's a hell of a long ride in a Cat.\"\n\nThen O'Dell returned. He had two prisoners, held at gunpoint. \"Meet Mademoiselle Bonnard and Mr. Babbington.\"\n\n\"Dr. Babbington, actually.\"\n\nO'Dell smacked his rifle butt into the man's spine.\n\n\"They didn't even know what was going on. They're out at the hangar out there, working on... well, you'll want to see this, Tanner.\"\n\n\"Is it a hovercraft with a jet engine and missiles?\" Tanner asked wearily.\n\nO'Dell threw up his free hand in exasperation. \"You are no fun to surprise, Captain.\"\n\n\"We were just completing the assembly,\" the Frenchwoman said. \"We are not dangerous. You have no need to point guns. We are engineers, just working for the company. Let us go free.\"\n\n\"Uh-huh,\" Tanner said. \"Well, ma'am, you, too, _Doctor_ , you now work for the U.S. Navy. You will complete your work, and if you manage to do it inside of two hours, I will not strip you both down to your underwear and send you out onto the ice.\"\n\n\"The sleighs are coming in,\" Stillers reported. He was casting questioning glances at Bug Man, wondering no doubt why his face was swollen, why his teeth were missing, and why he was wearing a bathrobe and flicking between YouTube and Twitter on the big TV monitor in Lystra Reid's living room.\n\n\"Yeah,\" Lear said distractedly.\n\n\"That will be the last of it,\" Stillers said.\n\n\"It's all coming down, Stillers. Um... Tell everyone good job, yeah? Yeah. Tell them all I said well done.\"\n\nHe nodded. \"Did you want to, maybe, come over to the dining hall and speak to them?\"\n\nLear considered the idea, shook her head almost shyly, and said, \"No, I have to watch.\" She waved a hand toward a shaky YouTube of one of the endless array of riots in one of the endless number of burning cities. \"Panic, you know. That's what gets them killed. It's like medieval, yeah? Plague. Or cholera.\"\n\nShe was no longer talking to Stillers, who sensed that fact and stood there stoic and awkward.\n\n\"That's the whole point. Madness leading to panic. If they just didn't panic, yeah, they'd be okay. Yeah? If they just didn't panic. But I knew they would.\"\n\n\"Yes, ma'am.\"\n\n\"Mmm. You can go, Stillers.\"\n\nStillers seemed relieved. Bug Man was not. It was better to have at least one extra person in the room in case Lear lost it again.\n\nShe flopped beside him on the couch. They had been watching together for the last few hours. Eating and watching in a bizarre parody of a girls' night at the movies. Bug Man had been half afraid she'd decide to paint his nails or talk about her love life.\n\n\"I'm glad you decided to join me, Buggy. Good old Buggy. You get it, yeah. You've been down there, down in the meat. You've been part of the game for a long time.\"\n\nBug Man did not remember choosing to be here. He remembered being blackmailed and threatened, made a party to yet another crime. If anyone ever lived to tell this story in some history book, he would be labeled as the guy who killed a president and almost killed a pope. Which was unfair. He was, at most, an accessory.\n\nAn accessory to the end of the world.\n\n\"Get us a drink, Buggy. You know, I wanted to get Sadie here, too. I thought she would be fun to have around, yeah. For a little girl-time, you know? We could talk girl stuff, yeah, that I can't talk about with you.\"\n\nHe poured them each a bourbon. She had said they had enough for two years, at least. He hoped that was true, because he felt he was going to need to drink an awful lot.\n\n_I'm turning into Burnofsky_ , he thought. _Old degenerate trying to drink away his sins. That's me now, but not old. So I can live with this for a long time. If she doesn't kill me_.\n\n\"What is that? Is that a cross? Oh, that is awesome. They're nailing that woman to a cross!\"\n\nBug Man was sick so far down into his soul that he wished he could shut down his brain, go into some kind of coma\u2014wake up later, maybe a lot later. He waited for the shaky video to end then navigated to the next clip.\n\n\"So Sadie, that didn't work out. But I've got you, Buggy. And it's all working,\" she said. \"All working. Except for the self-replicating nanobots. Yeah. The goo.\"\n\n\"I haven't seen anything like what you're looking for,\" Bug Man ventured. \"Just crazies, no buildings eaten up or whatever.\"\n\n\"Mmm. Yeah.\" Lear was pensive. \"Probably all burned up when the Tulip came down. Burned up with the Twins. Wish I'd been able to stay to see even more of that, yeah. Yeah. Burning Armstrongs, that would have been excellent.\" She shrugged and sighed, disappointed. \"But all it takes is one of those SRNs to survive. Just one.\" She bit a fingernail and added a superfluous, \"Yeah.\"\n\n\"I'm sure\u2014\"\n\n\"Shut up!\" Lear snapped. \"You're not sure. I'm not sure, so you're not sure.\"\n\n\"Yes, ma'am.\"\n\n\"Gotta exterminate them, somehow. They'll just... just keep on. Gotta be a way to stop them.\"\n\n\"Race to the end of the world,\" Bug Man said, his tongue loosened by the whiskey. \"Choose your apocalypse.\"\n\n\"I can't let them beat me, the Twins. Burnofsky.\"\n\nAn idea occurred to Bug Man. If he spoke it, he would never be able to unsay it. If she liked the idea, she would be happy with him. If not...\n\n\"I have an idea,\" he said.\n\n\"Speak it, Buggy.\"\n\n\"You have people's biots. You can send them a message. To the right people. I mean, you have all that cross-referenced, right? I mean, you would know which people were in the Pentagon, or maybe in Russia, wherever.\"\n\nShe was looking at him with the intensity of a cobra looking at a mouse. \"Spit it out of that mush mouth, Bug Man.\"\n\n\"Okay, say you have some general, or whatever. You fire up his biots, right? He knows now what's coming. He knows he's screwed. But biots can see, right? They could see, you know, if you showed them a sign. Held up a sign in front of them.\"\n\nShe stared at him for a full minute, during which Bug Man wondered if he would have the strength even to resist if she decided to kill him. Did he even want to live?\n\nThen she reached out one hand, pinched his swollen cheek, and said, \"Buggy, you are a genius.\"\n\n# **THIRTY-ONE**\n\nPlath was in the second seat of the sleigh. Tanner was driving. O'Dell was in the other sleigh, being driven by Babbington, who had been convinced to help when O'Dell shot two of his toes off and promised to keep going if he didn't.\n\nThree more men plus Vincent and Wilkes were crammed into the Sno-Cat, trailing many miles behind.\n\n\"I still don't see a damned thing, and we're supposedly right on top of it,\" Tanner said. Then, \"Ahhh! Shit! O'Dell, stop, stop, stop!\" he yelled through his radio.\n\nHe killed the engine and fumbled for the brakes that slammed steel claws down into the ice. The sleigh went from a moderate seventy miles an hour\u2014neither Tanner nor Babbington felt confident going any faster\u2014to zero in five seconds. Even so, the front two feet of the sleigh were over the lip of a sharp drop-off.\n\n\"This thing have a reverse gear?\" Tanner wondered. If there was, he never found it. \"Okay, we get out and push it sideways.\"\n\nTanner and Plath climbed out onto the ice. Only then did they see the brightly lit compound nestled in the dry valley below.\n\n\"Under my nose,\" Tanner muttered. \"They built this right under my nose.\"\n\n\"Antarctica is a big place,\" Plath soothed. \"And Lear has a lot of money.\"\n\n\"Is that another swimming pool?\"\n\nO'Dell and Babbington joined them and helped manhandle the sleigh back from the lip of the cliff. Under low power, just enough to raise the weight of the sleigh from the ice, it wasn't too hard.\n\n\"There's ramp over there,\" O'Dell said. \"But we could just sit up here and fire down into the base. Twelve missiles, fair amount of thirty-mil cannon...\" He shrugged.\n\n\"No,\" Plath said. \"We need to know whether this base is the place she's using to control events, or just a place to hide while the work is done elsewhere.\"\n\nTanner nodded. \"Look at that slag heap over there. That's way more than you'd get from just leveling. They've dug some holes.\"\n\n\"Yeah, well, that base looks like it will sustain a hundred men,\" O'Dell argued. \"I'm not seeing the gun emplacements we saw back at Forward Green. Still, we could get a very hot greeting. These sleighs aren't armored worth a damn.\"\n\nBabbington took offense at that. \"We needed to keep weight down, obviously. The engine is armored.\"\n\n\"Yeah? How about the cockpit?\" O'Dell asked. \"Yeah, I thought so.\"\n\n\"The house,\" Plath said.\n\n\"Yep,\" Tanner said. \"That's the big-boss house right there. If we catch them by surprise, decapitate them\u2014\n\n\"That chopper down there has missile launchers and a cannon,\" O'Dell pointed out.\n\nPlath said, \"Look, for whatever reason, Lear hasn't killed me yet. She could have. She wanted me back in the game. She insisted I play an active role. I think... I think she doesn't want me dead.\"\n\n\"What do you have in mind?\"\n\n\"I think I walk down there, knock on the door, and hope she shakes hands.\"\n\n\"I'm going to try to get through to some rational person, either in D.C. or Langley or any random naval vessel that might be within range. But don't count on the cavalry. You understand?\"\n\n\"I do,\" Plath said.\n\nHe gave her an appraising look. \"What are you, sixteen?\"\n\n\"Yes,\" Plath said. \"But I've packed a lot into the last few months.\"\n\nHe nodded. \"I have a son about your age. Back in the world. Minneapolis, with my parents. I'm trying to tell myself he's okay.\"\n\nPlath started to answer, stopped herself, shook her head, and finally said, \"I was about to say I'm doing the same. But everyone I care about is either dead, or here with me.\" Noah, lying in his own blood, gasping final breaths.\n\nShe squeezed her eyes shut. There were no tears\u2014which, she thought, was a good thing as they would have frozen.\n\nHer father, her brother. Ophelia, Nijinsky, Anya. Billy. She saw his head fall to the side, his neck cut almost through.\n\nAt least her mother had died of natural causes. She hadn't been murdered. So much sadness, and now, the whole world was joining Plath in that sadness. That did not help. The old saying was that misery loves company. But Plath knew that misery needed hope. Misery needed to believe in a better future.\n\nWhat was happening back in the world where Tanner's son lived? Had Lear's madness killed millions, or just hundreds of thousands? Had Burnofsky's vile machines escaped to obliterate all of life?\n\nHow much could the human race stand? The dinosaurs had thrived for tens of millions of years before dying out. How many species had evolved, survived, and then at last succumbed?\n\n_Homo sapiens_ were, what, a million years old? And all of human civilization just a tenth of that. Had the clock run out?\n\nNoah, lying in his own blood while the Twins raged and Burnofsky gloated.\n\nHad she loved him? Then how could it be that she'd not told him? Too late now. Now she could only offer him more blood. More murder.\n\n_I'll kill her. For you, Noah_.\n\n\"It's cold,\" Plath said. \"Let's get this done.\"\n\n\"We'll drive you around to the far side, to the top of the ramp, and then stay out of sight.\"\n\nStaying out of sight was an illusion. Sensors had tracked the approach of the sleighs. And now Stillers reported to Lear that the sleighs were behaving strangely. They had stopped for a while at the northern end of the valley before continuing on around to the southern entrance.\n\n\"Now they're just sitting there.\"\n\n_Interesting_ , Lear thought. Frightened employees? Was some of the biot conditioning that all her core people had been subjected to beginning to weaken?\n\nHer eyes flicked to the TV. YouTube was still up, thankfully. Bug Man was watching a shaky video of a Tesco being looted.\n\n\"Do we have cameras on the ramp?\" Lear asked.\n\n\"Yes, ma'am,\" Stillers said.\n\n\"Get them on-screen here.\" Soon a dimly lit image of the ramp opened. At first: nothing, just gravel and ice. Then, someone walking down the ramp. The person wore a heavy parka with a fur-lined hood, with dark goggles covering the upper part of the face.\n\n\"Can't see the face,\" Stillers said. \"I'll send some guys up there.\"\n\n\"No.\" Lear smiled. \"I think... I think maybe I can guess who this is. Yeah. Have men ready, get a sniper into position to cover my door, make sure all security personnel are armed at all times, and I'll want a handgun for myself. Do nothing unless I give the order.\"\n\nStillers nodded and went about his work.\n\n\"I believe we have company, yeah,\" Lear said to Bug Man. \"I do not know how she did it, clever girl, but if I'm right, we'll have an old friend of yours over for a drink.\"\n\nOpportunity for Suarez came with Kung Pao chicken\u2014extra spicy, the way she liked it\u2014brown rice, and a glass of Austrian white wine.\n\nAfter so long planning what to do with a bucket as the only weapon, she was handed a golden opportunity: Chesterfield came armed.\n\nShe immediately recognized it as a Glock nine-mil with a eventeen-round clip. She had fired hundreds of rounds from a weapon essentially identical to this. All that was good, but the beautiful part from her perspective was that the standard cop holster was also very familiar, and she would be able to draw it smoothly, especially if she could get behind him.\n\nMuch better than trying to beat him down with a pee bucket.\n\nThe final piece of the puzzle was the Kung Pao. And more specifically, the peanuts.\n\nShe accepted her tray, invited him to stay so she could be sure it wasn't too spicy. She took a bite and cried, \"Oh, no. No! Peanuts!\"\n\n\"What's the matter?\"\n\nShe put a hand to her throat and began wheezing dramatically. \"Allergic... to... peanuts. I can't breathe! Help...\" And then choking noises and a strained, whooping breathing and Chesterfield made the fatal move: he behaved like a human being, stepped in, knelt down, and in a blur of movement felt the muzzle pressed against the side of his head.\n\n\"I would honestly hate to do it,\" Suarez said. \"You've been decent to me. But Chesterfield, I will blow your brains out if I have to. The alternative...\"\n\nWhich was how Chesterfield ended up wearing her chains, with handcuffs added to keep him in a hog-tie position, and his own socks stuffed in his mouth with his belt wrapped tight to hold them in place.\n\n\"Can you breathe okay?\" she asked him.\n\nHe nodded, and Suarez, armed with the gun, an extra clip, his radio, and his keys, opened the door to her cell very slightly and looked cautiously left and right. If there were cameras, they were not in evidence. Which did not mean they weren't there.\n\n_Nothing you can do about that but move fast_ , Suarez told herself. Down the hallway, which carried the ridiculous medieval dungeon theme forward. A door. She cracked it slightly. There was a sort of control room\u2014monitors and swivel chairs and two women chatting as they watched the screens. Panic buttons were large and prominent. She winced. There was no room for error or pity.\n\n\"Hey,\" she said, stepping into view, and with two head shots dropped the women. One was clearly dead. The other rattled her shallow breaths in and out until Suarez covered her mouth and nose and waited for the final spasm. No point wasting ammo, and no point risking a third shot attracting attention.\n\nHer immediate goal was simple: to find and take the sleigh she'd ridden in and get the hell out of there. But that would require some intel. She dropped into one of the dead women's seats and began cycling through the camera angles, one of which did in fact show the hallway outside her dungeon. She had been lucky they hadn't spotted her.\n\nThis monitoring station appeared to have only limited access to cameras, concentrating on the dungeon and what appeared to be extensive storerooms. Really quite impressive storerooms, too large to be in any of the aboveground buildings. She saw other people, some armed, some not. Some doing mundane tasks with iPad inventory systems, others driving forklifts, still others...\n\nA man walked toward the monitoring station, holding three disposable cups and a paper sack in a recyclable cardboard holder. He might easily have been coming from a Starbucks.\n\n\"Hey, coffee!\" he said as he stepped into the room. Suarez grabbed his hand, yanked him forward, slammed the door shut, and blew out his brains.\n\nOne of the coffees survived the fall, and she took a sip before getting back to her research. Surely there must be a way to break out of this limited protocol and access more cameras.\n\nShe was beginning to regret having killed all three of them\u2014she could have used some help. But then she stumbled upon an open link that led her helpfully to a schematic of the base. The schematic had green dots for camera locations.\n\nThe first was password protected. She tried the usual combinations, and none worked. So she rifled the pockets and wallets of the dead, and finally found a tiny slip of yellow legal pad.\n\n\"Thank God for unreliable memories.\" Moments later: \"And bingo. We are in.\"\n\nThe sun was just millimeters above the horizon, and the weak light left the valley in darkness. Stadium lights cast a circle of eerie orange across the main buildings, excepting the house, which cast its own warm, buttery light.\n\nPlath was shaking with cold and fear by the time she had descended the long ramp and then crunched her way across the gravel to the house. She did not spot\u2014indeed did not look for\u2014the sniper who watched her through his telescopic sights.\n\nShe climbed the few stairs and stood on the porch of the impossible house belonging, she was certain, to Lystra Reid, also known as Lear.\n\nShe pulled off her glove and knocked.\n\nThe door flew open to reveal an attractive young woman wearing white yoga pants, shearling boots, and a blue down vest over a sheer white tunic.\n\nPlath pushed up her goggles and slid back her hood.\n\n\"Oh. My. God.\" Lear said. \"It _is_ you.\"\n\n\"May I come in?\" Plath asked, feeling an absurdity in it all that went beyond the merely surreal.\n\n\"Mmm, not just yet. First, I should tell you there's a very good shot watching you, yeah, and ready to fire at any excuse. So. Shrug off the coat, keep your hands where I can see them, and don't move.\" In order to emphasize her point, Lear pointed with one hand at the gun in the other.\n\nPlath complied.\n\n\"Now, turn around slowly.\"\n\nThis, too, Plath did.\n\n\"Ah! There we go. You _do_ have a gun. I thought you might.\" Lear pulled the gun from Plath's waistband and tossed it out onto the ice. It came to rest by a lawn ornament, a pink flamingo that must have been someone's idea of witty commentary on the climate.\n\n\"Now, come on and warm up,\" Lear said. \"Bug and I are drinking excellent bourbon, would you like some?\"\n\n\"Bug?\"\n\nPlath looked past Lear and saw a badly battered Bug Man, sitting on a couch and looking miserable and humiliated, and perhaps just a little hopeful.\n\n\"You two have met, right?\"\n\n\"Briefly,\" Plath said. Then added, \"I don't drink.\"\n\n\"Yes, you do, yeah, not a lot but on occasion,\" Lear said smugly. \"Yeah.\" She handed Plath a glass. Plath took a sip, grimaced, and put the glass aside.\n\n\"If we're going to be friends, you're going to have to get into the spirit of things,\" Lear said, her face darkening.\n\nSo Plath picked the glass back up and followed Lear's direction to sit, sit down, take it easy, relax.\n\nPlath sat. She saw the TV, currently on a YouTube of a burning house. Where it was she had no idea. Bug Man sat stiff and wary.\n\n\"I did it,\" Plath said.\n\n\"Did it?\" Lear asked.\n\n\"I blew up the Tulip. I gave the order to Caligula. Then I followed the breadcrumbs here.\"\n\nThat had the desired effect of throwing Lear off stride. \"Are you trying to tell me that\u2014\"\n\n\"Did I know it was you behind it?\" Plath interrupted. \"Yes. After you killed Jin it was obvious that he had failed you, somehow. Was it that he found out the reason you'd ordered him to wire Vincent?\"\n\nLear, small smile growing. \"In a way. Nijinsky hated you. He didn't like being pushed aside for some kid. So that was part of it. But yeah, he was starting to get cold feet. Developing a conscience.\"\n\n\"I didn't want to die choking on my own tongue on an escalator. So I didn't fight it very hard. I could have sent my own biots in to stop it all happening, my own rewiring. But I could see where it was all going.\"\n\n\"Oh?\"\n\n\"I came to like the idea. I came to like the whole, meticulous planning of it. It was brilliant. It was genius. It's historic.\"\n\nLear's nostrils flared, and her eyes widened. \"Historic?\"\n\nSo, Plath noted, she liked that word. \"Well, yeah,\" she said. She took a sip of the whiskey, suppressed the face she wanted to make, and instead said, \"It gets better as you get used to it.\"\n\n\"Historic, yeah?\" Lear prompted.\n\n\"I remember this lecture in history class. All about Genghis Khan. You know, the Mongol guy.\"\n\n\"I know.\"\n\nNo, Plath thought, Lear had not heard of the great Khan. But she didn't like admitting it. \"Well, the point was that Genghis killed, like, thirty million people, no one is sure how many. Maybe twice that much. There was this one thing where he took a bunch of captured enemies, and built a platform on top of them. His own soldiers had lunch on the platform as it slowly crushed all the men beneath.\"\n\n\"Yes,\" Lear said fervently.\n\n\"But the point was, that later, like nowadays, we look back on him, Genghis, I mean, as a great historical figure. He, like, improved the economy and so on by clearing out a bunch of people who were in his way. But he killed millions.\"\n\n\"He changed the game. But I'm changing it more. I'm changing it all,\" Lear boasted. \"I'm creating whole new species, yeah, to take over. I mean, you know, thanks to your dad, who was a genius. Yeah. By the way, condolences on his death, he was a great man.\"\n\nPlath's mask almost dropped then. Almost. \"Yes, he was.\"\n\n\"But we used his techniques and played around, and now we have three very interesting species. _Macro_ , not micro. We'll breed them up, yeah, and then release them when the time is right. One of them can't metabolize anything but pork and human meat. Hah! Later, at the next level, yeah.\"\n\n\"But how are you going to watch what happens? I mean...\" She waved a hand at the YouTube video. \"How much longer is Google going to work?\"\n\n\"Oh, don't worry. The satellites will work independently for a long time. And we'll start placing cameras here and there, when the time is right.\"\n\n\"You've thought of everything,\" Plath said.\n\nLear smiled, a shark's smile this time. \"You don't really think I'm buying any of this, do you?\"\n\n\"Sorry?\"\n\n\"This bad-girl act. _This_ Sadie McLure, indifferent to suffering. You tried to stop Caligula. I _know_. I spoke to my father before he died. That was kind of a drag. He was very useful, the old man. I was never going to bring him here, no, no, but it would have been fun watching him deal with the world I'm creating. He would have been an interesting player in the game.\"\n\nPlath put her drink down again. Her hand was shaking. Lear saw it.\n\n\"The world you're creating?\"\n\nBut Lear wasn't playing along anymore. \"How much longer do you think you have to live, Sadie?\"\n\nPlath did not answer.\n\n\"Two ways forward for you, Sadie. The usual choice: death or madness. We have some decent twitchers here, and we could easily wire you up. Or I could be disappointed that you would just walk in here and think you could lie to me.\" Lear raised the pistol on her lap and leveled it at Plath.\n\nThe muzzle looked huge. _What a clich\u00e9_ , some corner of Plath's mind thought. _That's what everyone who has ever stared down the wrong end of a gun thinks: oh, it's big_.\n\n\"Go ahead,\" Plath said.\n\n\"You don't think I will?\" Lear stood up and let the down vest slide to the floor. The sheer tunic revealed shadows of the tattoo horrors beneath. Lear pointed to a spot on her belly, right where an appendix scar would be. \"Right here, yeah. That's where I would tattoo your face. Maybe then you'll talk to me, yeah? They speak the truth, the tattoos do. Yeah.\"\n\n\"I believe you'll kill me,\" Plath said. \"You're a mass murderer. Before you're done you'll kill more people than Genghis or Hitler. You're a sick, twisted, crazy woman playing an insane game. So yeah, I think you'll kill me.\"\n\nLear cocked her head, all the while keeping the gun aimed. \"Don't you want to beg?\"\n\nPlath forced a smile of her own. A peace had descended over her. It was like what Noah had described to her, the eerie feeling of detachment and fearlessness that could come in the midst of a very challenging game. It would be over in minutes.\n\n\"I'm not afraid to die,\" she said. \"So long as I take you with me, you foul, fucked-up psychopath.\"\n\n\"Hah!\" Lear said. And then, the wheels began to turn in her head. Plath could see her retracing her steps. \"You never touched me. Yeah, you never touched me.\"\n\n\"No,\" Plath said. \"But you took my gun. As I knew you would.\"\n\nLear swallowed. She glanced at Bug Man, as if he would or could help.\n\n\"You know the anterior cerebral artery?\" Plath asked. \"Don't be embarrassed if you don't. I never would have, if some sick creature had not dragged me into her little BZRK game. But now, hey, I know a fair amount. Like I know that the anterior cerebral artery feeds blood to the frontal lobes. Which is where your consciousness lives.\"\n\n\"You're bluffing.\"\n\n\"Three biots, Lear. Each has a nice, long spike buried in that artery. There's blood leaking, but just a few cells, nothing fatal. It takes pressure to hold them in place. I think you may have high blood pressure, because it's a little like holding a Champagne cork in. If I keep up the pressure, leave the spikes in, well, eventually the clotting factor will seal the damage. But if I let the spikes out... which is what will happen if my biots are suddenly no longer being controlled... there will be a sudden spurt of blood. The pressure of cells forcing their way out of the holes will actually widen the holes. And since all the spikes are close together, the whole area will probably tear wide open. I know these things because of my own aneurysm. Useful.\"\n\nLear lowered the pistol and squeezed the trigger. The bang rattled the glassware.\n\nPlath felt a terrible blow, like a crowbar against her knee. The pain was immediate. Blood gushed from the wound. Bits of white bone stuck like teeth from ripped skin.\n\nPlath fell to her other knee and shrieked in pain.\n\n\"See, little Sadie girl, there are other ways. I don't have to kill you. I can just keep hurting you. How does it feel? Does it hurt? It's weird, yeah, but people who can face the idea of dying can't always face the idea of suffering, yeah?\"\n\nThe pain was beyond belief, beyond anything Plath had ever felt before.\n\n\"See, honey, I'm not afraid to die, either. I'm afraid to fail. I'd rather die than lose the game, yeah, my game, _my goddamned game_!\"\n\nShe fired again, this time into the meat of Plath's arm.\n\n\"Oh, did I get the bone on that one? Ouch, yeah? I have doctors, I have morphine, I can help, but first\u2014\"\n\nThis crash was not nearly as loud as the gunfire. Just the _crump!_ sound of a bourbon bottle hitting a skull.\n\nLear fell sideways, and Bug Man kicked her in the stomach, then grabbed the gun from her hand.\n\n\"Shoot her,\" Plath cried through waves of agony and terror.\n\n\"Can't. She's got me. Biots. Some kind of dead man's switch. She dies, I lose it. So you don't kill her, either, Plath.\"\n\n\"My biots never got to her brain. They're only halfway up her neck.\"\n\n\"Hah! You bluffed the crazy bitch?\"\n\nA loud, imperious banging at the front door. Bug Man fired through it. \"They won't shoot back,\" he said, voice high with stress. \"They might kill their boss.\" Then he yelled, \"You come in here, I shoot her! I shoot her right between the eyes!\"\n\n\"That was gunfire,\" Tanner said. \"We go in.\"\n\nO'Dell threw a quick salute and ran for his sleigh. But Babbington had run off, and O'Dell had never been any sort of pilot.\n\n\"With me, Sergeant!\"\n\nTanner fired the engine as, down below, the helicopter's rotors began to turn.\n\n# **THIRTY-TWO**\n\nSuarez did not hear the gunfire in her underground position, but on the monitor she did see men rushing, guns drawn.\n\n\"Something just hit the fan,\" she muttered.\n\nShe had located her own sleigh. It was parked behind one of the dormitories, not hard to get to so long as no one was shooting at you.\n\n\"Hope to hell they fueled the damn thing up,\" she said. She grabbed the guns from the dead guards, stuck them in her waistband feeling weighed down and a little ridiculous, and raced from the room.\n\nThe dungeon theme was over, now it was bright-lit hallway, white on white. Ahead, footsteps running. A man and a woman. It took her three shots to kill them.\n\nThe hallway dead-ended, and she had to double back to find an exit. She opened it quietly, glanced around to see the warehouse she expected, and ran toward concealment behind stacked plastic crates.\n\n\"Who is that?\" a voice yelled.\n\n\"The prisoner got loose!\" she yelled, waited until a worried face appeared, and put a bullet through its mouth.\n\nRunning, running, one of her extra guns clattered to the floor, but she kept running. Running through her mind was that whatever had sent armed men rushing around, it wasn't her. They'd been headed somewhere else, after someone else.\n\nBless whoever the poor fool was, but that was not her problem.\n\nProbably.\n\nThe sleigh came slipping and sliding, hard to control, very hard to control as Tanner raced it down the ramp. First things first: kill that chopper.\n\nSmall-arms fire popped off to his left, chipping stone from the wall to his right.\n\n\"RPG at your six!\" O'Dell yelled.\n\nThe wobbly rocket arced toward them, fired from behind and below. It missed by inches and blew up against the stone wall. The sleigh was blown clear of the ramp, still a hundred feet up from the bottom of the valley.\n\nBut then the computer kicked in\u2014roared the engines to push a tornado of air beneath the hovercraft\u2014which slowed the descent so that rather than being fatal it was merely bone-jarring as it slammed down onto gravel.\n\n\"RPG!\" O'Dell yelled again, but this time Tanner had seen it coming even before O'Dell and pushed the throttle forward. The sleigh bucked, kicked up a storm of gravel, and blew past the missile, which detonated fifty feet away.\n\n\"On that building!\" O'Dell pointed and there, sure enough, were two men manhandling yet another round into the missile launcher.\n\n\"Like hell,\" Tanner yelled, swung the nose of the sleigh around and fired blind at the building with one of his own missiles. It struck a second-floor window and blew a hole. It did not kill the men with the RPG, but the concussion knocked them onto their backs.\n\n\"The house!\" Tanner yelled. He aimed the sleigh toward it and then, at the last second, sank the brakes into gravel and the sleigh skidded sideways into a stop. O'Dell had already opened the canopy and now leapt, pistol in hand, to rush the door.\n\nThe sniper fired once, and O'Dell slammed onto his face and did not move. At the same moment the door of the house flew open and a young black kid in a bathrobe appeared, dragging Sadie by one arm.\n\nThe sniper fired and missed.\n\nTanner spotted the muzzle flash, and thanked whatever God watched over him that the sleigh had skidded sideways, because his weapons were pointed in the right direction. He launched a missile that blew a hole in this second structure, and while the sniper was recovering Tanner emptied his pistol at the roofline.\n\n\"Get in! Get in!\"\n\nThe boy climbed in, hauling a nearly helpless Plath after him. The canopy would not close with Plath's legs sticking out, but Tanner wasn't waiting. He gunned the engine and roared away toward the ramp, firing his thirty-mil cannon continuously, causing bright-red flowers to bloom on walls, empty ground, and a couple of men.\n\n\"Get her in, get her in!\"\n\n\"Can't, there's no room!\" Bug Man cried, but nevertheless he hauled a screaming, bloody Plath the rest of the way into the cockpit, a tumble of limbs and hair on Bug Man's lap.\n\n\"Who are you?\" Tanner demanded.\n\n\"They call me Bug Man.\"\n\n\"Yeah, well, listen up, Bug Man. See this? That's the throttle. That's the brake. This is the yoke. The computer will help.\"\n\n\"What? Why? Are you bailing out?\"\n\n\"No, but you will be. There's another one of these at the top of the ramp.\"\n\nLear rose from the floor, woozy, took a stutter-step, and fell into the wall. She left a trail of blood behind.\n\n\"Fu... The... Yeah...\" she muttered.\n\nHer legs were jelly. Her head was going around and around and around and _oh, no_. She vomited onto the floor. Felt a little better after that. Wished she hadn't been drinking. Wished she had more sleep. Yeah. Sleep would be good....\n\nStillers came pounding in, gun drawn. Three other men, all armed.\n\n\"Boss!\"\n\n\"Di... get 'em?\"\n\n\"They've got the sleigh, but Tara's getting airborne.\"\n\n\"Kill them. Kill them,\" Lear said, slurring where she wished she was shouting.\n\n\"Someone get the doctor!\" Stillers yelled.\n\nMore voices yelling, all around her; voices yelling and walkie-talkies blasting away and something burning.\n\n\"I'm 'kay,\" she said. Why wouldn't her mouth work?\n\nShe felt the side of her head, then stared at her hand, red with something she couldn't bring herself to understand. \"Mom?\" she asked.\n\nSlowly, slowly, her head stopped spinning. Her legs were still weak but she could stand. A white-coated doctor was doing something to her head. Someone else was putting something in her mouth. Water. Had she asked for water?\n\nShe blinked. Her father was here. What was he doing here?\n\nShe shook her head, which set off a cascade of pain. She was sitting now on a couch stained with red handprints.\n\nCaligula. He had come around to peer at her, keeping his distance, but saying something. \"She's dead, Lyssie, she's dead, and you can't ever tell anyone what you've done....\"\n\n\"My head,\" she managed to say. \"Give me something. Give me something. Hurts.\"\n\nShe blinked and her father was gone. She blinked again and pushed herself to her feet. \"Kill them! Kill them!\" she cried, and this time it came out right.\n\n\"Tara's in the air,\" Stillers said. \"She'll get them.\"\n\nBy sheer dumb luck more than skill the sleigh made it to the top of the ramp, weakly followed by small-arms fire that drilled a hole in the canopy and brought a whinny of fear from Bug Man.\n\n\"There it is,\" Tanner yelled.\n\n\"I don't know how to drive that thing!\"\n\n\"Go!\"\n\nBug Man tried to crawl out from under Plath, who was only barely moving and definitely not saying anything brilliant. There was a pool of blood on the seat that had seeped into Bug Man's bathrobe.\n\nThis as much as Tanner's shout propelled Bug Man out and onto the ice. He immediately fell down, and that fact saved his life when the sleigh he was aiming for suddenly began firing. Cannon fire blew through the engines of Tanner's sleigh, and it settled to the ground. Tanner tried to run. His legs took two more steps after the cannon cut him in half.\n\nBug Man screeched in terror and bolted back toward the ramp. Plath meanwhile had managed to drag herself out onto the ice and was making a red smear across it, crawling, crawling but not dead yet. _Cold, dead soon_ , she thought, _not dead yet_.\n\nIn her mind there were three windows.\n\nThree biots ran up the side of Lear's face. Blood\u2014a jumble of red Frisbees and expiring whitish sponges\u2014lay strewn across a landscape of flesh.\n\nWas she even going the right direction? Which way was up? Plath saw a stream, like a mountain spring rushing down a cliff face, but the water was a landslide of blood cells.\n\n\"Okay, that's up,\" she told the ice that was freezing to her lip.\n\nUp and up, following the stream, the biots raced, the newest, P3, bounding ahead.\n\nAhead a forest of dark hair, huge, rough-textured whips sprouting from the flesh soil.\n\n\"Mmm, left,\" Plath mumbled.\n\nThe biots veered left toward the falling blood, leapt atop the softtextured, tumbling cells, running, losing ground as the current swept them, then out onto dry surface.\n\nAnd yes, ahead the slope leading toward the eye, a vast lake covered then revealed, covered then revealed by blinking eyelids.\n\nThis was a road Plath had traveled before. Her biots pushed through the twitching leafless palm trees of eyelashes and leapt onto the surface of Lear's eye.\n\nNormally biots could travel unfelt across an eyeball, but not when the biot twitcher deliberately dragged sharp claws, slicing the outer layer of the cornea.\n\nA sky-blackening hand fell from outer space and mashed the eyelid down on Plath's biots, but it didn't matter. You could no more squash a biot with a hand than you could stomp a cockroach in plush bedroom slippers.\n\n\"That's right, Lear. Still here,\" Plath said. Her body was shaking with cold. She was sure she was going to die. But before she did...\n\nHer biots skated hard around the orb, leaving tiny rips over the mineshaft of the pupil, racing ever faster into the dark, clambering over veins, stabbing them as she went, loosing narrow fountains of blood that sprayed up to beat against the back of Lear's eye socket.\n\n_For you, Noah. For you. It's the best I have_....\n\nAhead lay the twining cables of the optic nerve. P1 dropped back to sink a probe and try to see what Lear was seeing.\n\nP2 ran after P3, now well ahead and already ripping and tearing its way through mucus membrane, widening an access to the brain itself.\n\nSuarez saw the sleigh, but someone was already in the cockpit, canopy open, revving the engines. She ran flat out now. The sleigh driver saw her and seemed to be fumbling for a weapon since the sleigh was still too sluggish to move.\n\nSuarez jumped onto the sleigh's surface and pointed her gun directly down at the driver's head. \"It would be a pain in the ass to haul your dead body out of that cockpit.\"\n\nThe driver saw the logic of that, held up his hands, and piled out onto the ground.\n\n\"Good choice,\" Suarez said, and shot him in the foot.\n\nShe slammed the canopy closed and cranked the throttle, sending ice crystals and grit flying.\n\nAcross the compound she saw the chopper pulling away, rising toward the level of the ice above.\n\n\"Yeah, you just go that way, and I'll go the other,\" she said, and sent the sleigh hurtling toward the ramp, cannon firing at anything that crossed her path.\n\nBabbington had grown tired of being bullied. He had run off across the ice, but when he saw O'Dell abandon the sleigh and jump in beside Tanner, he'd run back. The sleigh was warm at least. He had barely made it before the chills came on so hard that for the next twenty minutes he just shook while waiting for the cockpit heater to thaw his bones.\n\nAnd then, he had shot the other sleigh.\n\nBabbington's thoughts had been less about needing to kill Tanner than they were about not wanting to yet again be forced out into the killing cold.\n\nHis first salvo blew the engine apart.\n\nHis second tore Tanner in half. The shock of that moment froze Babbington in a very different way. He pushed away from the controls and just in front of him the helicopter, bristling with weapons, rose like an avenging god.\n\nCold was not worse than being blown apart. Babbington threw back the canopy to wave his arms, show his face, anything to keep the helicopter from firing, but the dragonfly-looking monster still swept toward him, nose down.\n\nCannon fire ripped the ice, swept by, and now Babbington was warm enough. He ran from the sleigh, ran in panic across the ice toward the ramp, waving his arms.\n\nSuarez shot up the ramp, then swerved madly as a boy in a bathrobe came pelting down. It was perhaps the most improbable thing she had ever seen. She backed the engines, shoved brakes into gravel, threw back her canopy, and yelled, \"Who the hell are you supposed to be?\"\n\nThe boy, wild-eyed, dove into the cockpit beside her.\n\n\"Yeah, okay,\" Suarez said. \"Just don't talk.\" She hit the throttle and Bug Man, facedown in the seat, twisted like an eel to get back upright.\n\nThe sleigh topped the crest and shot directly beneath the helicopter.\n\n\"Oh, that's not good,\" Suarez said.\n\n\"It' after ush!\" Bug Man yelled.\n\nThe chainsaw roar of the chopper's cannon opened up, blowing a hole in the canopy, sending plastic shards everywhere. Suarez hauled the sleigh sharply left. Looked at her left hand. A two-inch piece of plastic protruded from the back of it. Her tendons were cut, her fingers slack.\n\nSuarez pushed the throttle to full speed and said, \"Hey, kid!\"\n\n\"Wha? Wha? Wha?\"\n\n\"Ever play video games?\"\n\n\"What?\"\n\n\"See that thing right there, kinda looks like a game controller? Well, that's our weapons system.\"\n\n\"She's in my eye!\" Lear yelled. \"She's in my eye!\"\n\nThe doctor did not understand. Stillers did. \"I'll get some of our twitchers!\"\n\nLear's head was almost clear now, but now sheer, blind rage was clouding her thoughts. She'd been bluffed! The McLure girl hadn't had biots in her brain, but they were sure on their way there now. Still time to stop them, maybe. Somehow.\n\nHad to be. Otherwise...\n\nThe nanobots could survive, the whole thing would be ruined\u2014had to win this, had to stay alive and win this. The Twins were dead, they couldn't defeat her\u2014dead, impossible!\n\n\"Don't kill,\" Plath groaned to herself. \"Wire.\"\n\nBut Plath's own body was in spasm now, convulsing. She could no longer feel her face. Her hands blue before her, frozen to the ice.\n\nP3 stabbed a needle into brain tissue, didn't matter where, spooled wire from its spider spinnerets as it ran, and stabbed a second pin.\n\n\"Toast!\" Lear yelled.\n\n\"What? Why are you yelling toast?\" the doctor asked.\n\nAnother pin, another wire and Lear felt an overwhelming urge to bite her lip.\n\nNow P2 was in the act, stabbing and spooling, stabbing and spooling.\n\n\"She's wiring me! She's wiring me!\" Lear cried.\n\nWhen she wasn't stabbing pins and running wire, Plath was simply slicing through neurons and axons, plowing the soft pinkish-gray tissue.\n\n\"No!\" Lear shouted. \"No. No! Grah! Grah!\"\n\nPlath felt a strange warmth creeping over her. Not real, she knew. Illusion. The body shutting down. Shutting down, conserving blood warmth in her core, saying farewell to limbs.\n\n_If I didn't love you, Noah, why am I thinking of you now, now at the end?_\n\nShe no longer felt the pain of her knee. Numb. Her arm still ached, but it was so very far away.\n\n_I loved that you loved me, Noah_.\n\nBut still enough consciousness to stab and spool and stab again.\n\n_I loved making love to you_.\n\n\"Grah, I, grah, yeah,\" Lear said, straining to be understood.\n\n\"She's having a stroke,\" the doctor said. \"Look! Her left pupil is blown!\"\n\nLear no longer saw the doctor. She saw her mother, her mother, the whore had actually slapped her across the face when she'd seen her daughter's disapproving gaze, a red welt and a sting and a humiliation.\n\n_Slap me? Slap me? SLAP ME?_\n\n_I wasn't brave enough to love you, Noah_.\n\n_Bitch-slap me? Me? Me? Me?_\n\nIncoherent sounds came from Lear's mouth between manic twitches. The doctor and Stillers laid her down on the floor.\n\n\"I'm giving her blood thinners,\" a funny, funny voice said, coming from her mother's screaming mouth, the cleaver in Lystra's hand, yeah, die yeah, slap me?\n\n_Me? Meeeee? Meeeee?_\n\nThe helicopter had a top speed just ten knots slower than the sleigh. The sleigh pulled away but with painful, painful slowness.\n\nAnd the sleigh was definitely not faster than cannon or missiles.\n\nThe missile grazed the cockpit with a fiery tail and exploded a hundred yards ahead. The sleigh's computers were fast, but not fast enough at one hundred sixty miles an hour to avoid the ice and stone thrown up in the explosion. It was like driving full speed in a hailstorm with golf ball\u2013sized hail.\n\nBut the sleigh survived, rocking wildly from side to side.\n\n\"Okay, we get one shot at this, kid,\" Suarez said. \"Be ready!\"\n\nSuarez hit the brakes. The sleigh slowed in a storm of ice particles, the helicopter roared by overhead, and Bug Man pushed the button.\n\nThe recoil was unexpected, as was the inundation of smoke and flame as the missile launched from the sleigh and curved into the sky, seeking a heat source.\n\nThe missile flew harmlessly past the helicopter\u2014which now, ominously, turned to come back. It came on cannon blazing, blasting ice on its way to killing Suarez and Bug Man.\n\nSuarez spun the sleigh and shot back toward the valley.\n\n\"You know there's a big giant hole up ahead there, right?\" Bug Man yelled.\n\n\"Yeah. We're going to see what this toy can do.\"\n\nThe distance was not great. The helicopter was a half mile behind. Suarez could only hope the chopper pilot wouldn't risk firing on her own people.\n\nOut into nothingness, out over the lip of the valley, the sleigh shot out into midair. And fell. The engines roared, trying frantically to push enough air downward to slow the descent. It worked, but not well.\n\nThe sleigh fell, faster and faster, and Suarez grunted and switched the thrust from vertical to horizontal once again.\n\nThe sleigh bolted forward and fell even more rapidly.\n\nAhead, a patch of blue.\n\nJust feet from the plastic dome, Suarez kicked all the thrust back to lift. The force of it bent the dome, then the sleigh broke through the plastic and with a loud crash slammed into the pool, snapped a diving board, and rode up and over a chaise longue to stop just inches from breaking through the far end of the dome.\n\nThe engine died then and the sleigh lay inert, back half trailing in the shallow end, front end tilted up.\n\n\"I gotta get this game,\" Bug Man said.\n\nTara Longwood\u2014the chopper pilot\u2014gave a thumbs-up to her weapons officer and took a victory pass over the wet sleigh below.\n\nThen she turned the helicopter back, scanning for any other targets. There was still a sleigh at the top of the cliff, but last she'd seen, the pilot, Babbington, was running like a scared rabbit.\n\nHowever... She frowned and pointed. A green Sno-Cat sat steaming within a few yards of the sleigh.\n\n\"One of ours,\" the weapons officer said. \"Must have just come up from Forward Green.\"\n\nTara nodded. She saw a dark-haired man climb into the sleigh's cockpit before she flew on around, circumnavigating the valley, looking for trouble.\n\nBy the time she got back to the sleigh and the Sno-Cat, she had heard a panicky babble of voices in her earpiece, coming from the ground. The dark-haired fellow in the sleigh was waving his arms, trying to attract her attention.\n\nA young woman and another man were carrying what could only be a body toward the Sno-Cat.\n\nTara brought the chopper in low, ready to help ferry the wounded now that the fight was over. She landed, and the young man in the sleigh trotted toward her, seemingly unconcerned, waving as he came on.\n\nShe slid the side panel of her cockpit open. \"What is this?\" she asked.\n\nAnd Vincent shot her in the face.\n\n# **THIRTY-THREE**\n\nPlath woke slowly. She was a drowning person, fighting her way up toward air and light, but it was so far, and her arms were so heavy.\n\nThen, all at once, she was awake. A doctor was beside her. And to her utter amazement, Vincent, Wilkes, and Bug Man were standing before her. There was also a Latino woman she had never seen before.\n\n\"Where am I?\" Plath asked.\n\n\"You're still here, in the valley,\" Wilkes said.\n\nPlath stared. Looked left and right. It could have been a room in any well-appointed, new hospital. She saw her leg, swathed in rigid webbing over bandages. It hurt like hell. Her arm hurt as well, but not as bad.\n\nHer face felt raw, as if it had been sunburned. Something was wrong with the bandaged hand. She saw bandages over the stubs of her amputated little and ring finger.\n\nHer head hurt. But she was alive.\n\nIn her mind, she saw three windows.\n\nShe took a deep breath, drank some water through a straw, answered the doctor's questions, and said, \"What's happened?\"\n\n\"Later,\" Wilkes said. \"We had you brought around so that, uh... there's a pretty big question, and we think we need to ask you.\"\n\n\"Wait. Are we\u2014\"\n\n\"We're in charge,\" Vincent said. \"We're running this base now. Suarez here can fly a helicopter, and do a few other things, and\u2014\"\n\nWilkes broke in to say, \"And with Lear out of the picture, all her wired-up zombies here didn't exactly know what to do.\"\n\n\"You've been unconscious for eighteen hours,\" the doctor said. \"I gave you a stimulant to wake you up. But it won't last long, the pain will get worse, and you'll be better off asleep for a while longer while your body recovers. You've been through a lot.\"\n\n\"Why did you wake me up?\" Plath asked Wilkes.\n\nBut Wilkes looked pleadingly at Bug Man. \"Okay, this is some very bad shit to deal with. But the gray goo, Burnofsky's babies, we're not sure... I'm seeing stuff that may be caused by self-replicating nanobots. But very small scale so far. And it could be I'm wrong.\"\n\n\"And there's the Floor Thirty-Four virus,\" Vincent said. \"Maybe it never escaped the Tulip. But maybe it did. The whole final tranche of Lear's victims have biots. We stopped the process before they were killed off. That's thousands of people with living biots who would suffer madness if the Floor Thirty-Four virus were to get loose.\"\n\n\"Not to mention all of us,\" Bug Man said.\n\n\"Uh-huh.\" Plath wanted very badly to go back to sleep, and the doctor was right; the pain from her shattered knee was stalking her.\n\n\"The thing is, there's only one way to stop the gray goo, and to make sure the Floor Thirty-Four virus never escapes,\" Vincent said in his dispassionate voice. \"Nuclear.\"\n\n\"What? Wait, um, I'm lost, here. I don't exactly have an atomic bomb on me, oh, damn\u2014Doctor, can I at least get an ibuprofen or something?\"\n\n\"If it's out there and we don't stop it, the whole world dies,\" Wilkes said. She put her hand on Plath's forehead and held the cup so she could take another sip of water. \"Bug Man has an idea.\"\n\nBug Man nodded uncertainly, not quite sure about how Plath would receive what he was about to say. \"Listen, we stopped the biot cr\u00e8ches. The madness has stopped, but man, half the world is burning. Millions... you know. Nobody's in charge. But people know what it means if a window all of a sudden opens up in their head. And we still can control the cr\u00e8ches, we can still, you know...\"\n\n\"Why would we?\" Plath asked. Her head was throbbing. Her mouth felt like flannel, and nausea tickled the bottom of her throat.\n\n\"Because we need someone to blow up New York City,\" Vincent said. \"Lear had good records, good data. We can pinpoint guys with access to nukes. Americans, Russians, French, Brits. And Bug Man realized that when the biots quicken\u2014when they're born, you know\u2014they see. And they could read. We can bring biots online, and we can show them a message.\"\n\n\"What message?\" Plath asked.\n\nOne by one they looked to Bug Man. \"Do it, or we kill your biots. Do it, or we take out your family. We explain, as much as you can, you know... the whole thing. But if we don't stop this, we're all dead. Us last of all, down here on the ice. But everyone. The whole human race. The whole planet.\"\n\nPlath felt tears welling in her eyes. \"You woke me up for this? To vote on\u2014\"\n\n\"Not a vote,\" Wilkes said. \"We already took a vote. You're in charge, Plath. Sadie. Suarez will run security, and eventually we'll unwire some of these people, but right now, it's on you.\"\n\n\"Vincent?\" Plath pleaded.\n\nHe looked away, ashamed. \"It's on you, Plath. Whatever you decide.\"\n\nIn the end it was a Chinese missile that did it. The Chinese general responsible, once certain that his family was safe, tied a rope to a tree in one of his favorite countryside spots, and hanged himself.\n\nThere were very few functioning governments still left to do useful things like tally up the death toll of the Plague of Madness. But later, historical estimates would set the count at two hundred ten million, in thirty-six countries.\n\nFour million of those had come as a consequence of Plath's order. But, in the end, Burnofsky's gray goo did not make it off Manhattan. The human race was saved. Life on planet Earth would go on.\n\nIn the weeks that followed, Plath drank much more than she should have, sitting in the living room of what had been Lear's house. She shared the house with Wilkes, Vincent, and Bug Man. She tried not to drink before lunch, but she often failed. She tried to stop, but not very hard. Wilkes made efforts to get her to move on, but the very words died on her lips when she looked into Sadie McLure's haunted eyes.\n\nOnce, and only once, had Plath gone to look at Lear.\n\nLear sat chained in the dungeon that had once held Suarez. Plath had asked for the door to be opened so that she could see her. See the monster. The mass murderer.\n\nBut Lear had not responded to Plath, had seemingly not noticed that she was there.\n\nPlath stopped using that name, and reverted to Sadie. She had tried and mostly succeeded in accepting Noah's death. But she could not reconcile herself to what had happened, what she had done, to New York City.\n\nFour months on, Wilkes found her on the floor, choking on her own vomit after drinking an entire bottle of Lear's bourbon. It was terribly clear that Sadie McLure would, sooner or later, manage to kill herself in expiation of her sins.\n\nWilkes would not allow that. She went to Vincent, and to Bug Man, and slowly, so very gently, the biots went to work. And little by little, Sadie McLure forgot.\n\n# **TWO YEARS LATER**\n\nThe woman was probably in her early fifties but looked much older. She was dressed in clean but tattered clothing, layers of it, as if she had to be ready for any sort of weather. In the pocket of her patched coat she carried a crumpled black trash bag to use as an umbrella. London was out of umbrellas.\n\n\"That's her,\" a street kid said, jerking his chin and holding out his hand. \"That's old Mrs. Cotton.\"\n\nSadie pressed a small gold bar\u2014no bigger than a segment of Kit Kat\u2014into his hand and said, \"If you lied to us, kid, we'll find you.\"\n\nThe \"we\" in question included seven uniformed, heavily armed men who had fanned out on both sides of the street. London had quieted since the worst of the Madness, as it was commonly called, but it was still a wild place where street gangs ruled many neighborhoods. The \"we\" also included Wilkes, now somewhat changed as well. She still bore the strange flame tattoo beneath one eye, but she had grown out her hair into a simple blunt cut. She was dressed in a zippered black jumpsuit and carried a machine pistol over her shoulder.\n\nSadie waved Wilkes back a few steps and moved closer to Mrs. Cotton, keeping pace with her.\n\n\"I'm not a danger to you, Mrs. Cotton,\" Sadie said. \"I'm here to tell you about your son.\"\n\nThe woman stopped. She turned a scarred and ravaged face to Sadie. Such signs of abuse were common among the survivors of the Madness. Sadie could only imagine what this woman had endured.\n\n\"Who are you?\"\n\n\"I'm Sadie McLure.\" The name obviously meant nothing to Mrs, Cotton, and Sadie was relieved. A lot of stories were going around the newly revived Internet. There were even ridiculous rumors that Sadie McLure had actually ordered New York City destroyed. \"I knew your son.\"\n\n\"Alex? You were a friend of Alex?\" The woman peered skeptically at Sadie.\n\n\"No, ma'am, Noah. In fact... we were close. I was with him at the end.\"\n\nSadie led Mrs. Cotton to a small coffee shop, a place the older woman would never have been able to afford on the starvation pension and ration coupons the shaky government was able to pay her. But Sadie had gold, and gold made many things possible.\n\nThey bought weak coffee\u2014or at least part of the hot brew was coffee, with just a bit of wheat chaff. And they each had a biscuit.\n\n\"Were you his girlfriend?\"\n\n\"Yes,\" Sadie said.\n\nSilence. Nothing but the munching of the dry cookie. The sipping of coffee. Then, \"How did he do? At the end?\"\n\n\"Mrs. Cotton, Noah died a hero.\" Sadie did not elaborate. Mrs. Cotton did not seem to need it, and the truth was that Sadie's memories of Noah at the end were disjointed. Parts of what she thought she remembered seemed unrealistic. Parts of her memory seemed to fit poorly with other memories.\n\nWilkes stood a distance away, close enough to smell the coffee and overhear snippets of the conversation whenever the room was quiet. She had, of course, been involved in rewiring Sadie. She and Vincent had written a heroic end for Noah, an ending in which he single-handedly took down the Armstrong Twins and stopped Burnofsky.\n\nThere were elements of truth\u2014a good wiring always rests best on a foundation of some truth. But it was still a work in progress, connecting images of Noah to heroic pictures gleaned painstakingly from Sadie's memories of movies and books.\n\n\"Your son saved the human race,\" Sadie said, and believed it, mostly.\n\nMrs. Cotton nodded grimly. \"He was always a good boy.\"\n\n\"Yes. I loved him.\"\n\nMrs. Cotton's composure broke then, and tears filled her eyes. \"I couldn't... I didn't know how to reach him.... He had this job in New York....\"\n\n\"It was an important job. He was an important boy. Man, actually. Because he was definitely a man by the end,\" Sadie said.\n\n\"I'm glad you told me this,\" Mrs. Cotton said, though her face was anything but happy. \"Did you tell him?\"\n\n\"Did I tell him what?\" Sadie asked.\n\n\"Did you tell him that you loved him?\"\n\nSadie took her hand and squeezed it gently. \"Yes. I told him that I loved him. I told him that many times.\" Sadie glanced at Wilkes, who blushed and looked down. \"He loved me, and I loved him. I think that memory is all that's kept me alive.\"\n\nSadie sat for a while longer with Noah's mother and left her with enough small gold bars to take the edge off her poverty.\n\nShe and Wilkes walked down streets that still showed the bullet holes, the fire scorches, the wreckage of the Plague of Madness. But London had suffered this badly before in its long history and knew how to put itself back together. Crews were at work. There were police on the streets. Life was slowly returning.\n\nA century would pass before New York City could say the same.\n\n\"Now what?\" Wilkes asked.\n\n\"How much of what I told that woman was true?\" Sadie asked.\n\nWilkes met her gaze and waited, saying nothing. Finally she said, \"Now what?\"\n\nThey were in front of what had once been a pizza restaurant, but was now burned out and choked with rubble.\n\n\"How long has it been since you had a decent pizza, Wilkes?\"\n\n\"Long, long time,\" Wilkes acknowledged, peering into the restaurant. \"I think those ovens may still be usable. Of course someone would have to clean the place up. Get the gas working again.\"\n\n\"You have something better to do?\" Sadie asked. She stepped over the threshold, bent down, and grabbed hold of a broken table. \"Help me with this.\"\n\n# **TWELVE YEARS LATER**\n\nThree windows were open in Sadie McLure's brain.\n\nHer three biots sat immobile in the glass vial she wore on a chain around her neck.\n\nWhen business was slow at Poet Pizza, she would sit in a corner booth with her old friends, Anthony and Wilkes. Their daughter would tease the cooks while their baby son chuckled on his father's lap.\n\nTen thousand miles away to the south, Michael Ford, once known as Vincent, supervised the skeleton staff that maintained what had become, in effect, a prison.\n\nA prison with a single prisoner. Who sat in her cell, chained to the wall, screaming.\n\n_\"Meeee? Meeee?\"_\nMICHAEL GRANT likes to tell stories that will leave readers entertained, excited, and afraid to turn out the lights. He likes to make up characters who become like family members to his readers\u2014and then kill them. He likes to take readers to places they would never have imagined but can never forget. Michael Grant has no hobbies, he doesn't take vacations, he is not particularly friendly or charitable. He just wants to grab readers and leave them wrung out, trembling, and begging for more. Which, according to just about everyone who's read a Michael Grant book, such as _BZRK, BZRK Reloaded_ , and the Gone series, is exactly what happens. Michael is on Twitter @TheFayz, in case you want to talk to him. He lives in Marin County, California, with his wife, Katherine Applegate, their two children, and far too many pets.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n# Copyright\n\nCopyright \u00a9 2017 by Rachel Botsman\n\nHachette Book Group supports the right to free expression and the value of copyright. The purpose of copyright is to encourage writers and artists to produce the creative works that enrich our culture.\n\nThe scanning, uploading, and distribution of this book without permission is a theft of the author's intellectual property. If you would like permission to use material from the book (other than for review purposes), please contact permissions@hbgusa.com. Thank you for your support of the author's rights.\n\nPublicAffairs\n\nHachette Book Group\n\n1290 Avenue of the Americas, New York, NY 10104\n\nwww.publicaffairsbooks.com\n\n@Public_Affairs\n\nOriginally published in hardcover and ebook by Portfolio Penguin in October 2017\n\nFirst Edition: November 2017\n\nPublished by PublicAffairs, an imprint of Perseus Books, LLC, a subsidiary of Hachette Book Group, Inc.\n\nThe Hachette Speakers Bureau provides a wide range of authors for speaking events. To find out more, go to www.hachettespeakersbureau.com or call (866) 376-6591.\n\nThe publisher is not responsible for websites (or their content) that are not owned by the publisher.\n\nIllustrations by Team (www.team.design)\n\nLibrary of Congress Control Number: 2017949042\n\nISBNs: 978-1-5417-7367-7 (hardcover), 978-1-5417-7368-4 (ebook)\n\nE3-20171012-JV-PC\n\n# Contents\n\n  1. Cover\n  2. Title Page\n  3. Copyright\n  4. Dedication\n  5. Epigraph\n  6. Who Can You Trust?\n  7. Evolution of Trust\n  8. Introduction\n  9. 1. Trust Leaps\n  10. From eleventh-century traders to Alibaba: how trust works to cross barriers, calm fears and revolutionize what's possible.\n  11. 2. Losing Faith\n  12. Behind the devastating crisis in institutional trust\u2013and why we're now more likely to 'phone a friend'.\n  13. 3. Strangely Familiar\n  14. From sushi to self-driving cars\u2013some surprising lessons in persuading people to trust new ideas.\n  15. 4. Where Does the Buck Stop?\n  16. When trust crashes in the 'self-managed' digital world, who's accountable?\n  17. 5. But She Looked the Part\n  18. A cautionary tale about deceptive appearances, and the technology that could unmask fakers and frauds.\n  19. 6. Reputation is Everything, Even in the Dark\n  20. What drug dealers on the darknet can teach us about great customer service.\n  21. 7. Rated: Would Your Life Get a Good Trust Score?\n  22. When dystopian sci-fi turns into a reality and every little move you make is ranked, who wins and who loses?\n  23. 8. In Bots We Trust\n  24. But should we... and how do we make them ethical?\n  25. 9. Blockchain Part I: The Digital Gold Rush\n  26. From fei to bitcoin, the long road to setting money free. What will it mean for the City?\n  27. 10. Blockchain Part II: The Truth Machine\n  28. The golden promises of the blockchain: over-hyped or the trustworthy key to our digital future?\n  29. Conclusion\n  30. Acknowledgements\n  31. About the Author\n  32. Glossary of 'Trust' Terms\n  33. Notes and Further Reading\n  34. Index\n\n# Navigation\n\n  1. Begin Reading\n  2. Table of Contents\n\nIn memory of Pamela Hartigan, my friend and mentor.\n\n# What is trust?\n\nA confident relationship with the unknown.\n\n# Who Can You Trust?\n\n'In this extremely thought-provoking new book, Rachel Botsman educates and entertains as she reveals with expertise how our lives are already changing more than we know. A must-read for anyone interested in how the world works\u2013and will work in the future' Will Dean, MBE, CEO, Tough Mudder\n\n'Not only is the thesis completely compelling. It demonstrates, through a sequence of real-world case studies of projects, businesses and platforms, that the distributed trust model offers enormous promise if used wisely\u2013as well as enormous pitfalls if used unwisely. For good and sometimes ill, it has the potential to reshape everything we do, from our choice of babysitter to our choice of money. These are important messages from what is an important book' Andy Haldane, Chief Economist, Bank of England\n\n'A fascinating and well-researched study. Every reader will gain new insights into one of the great issues of our time: the shifting tides of trust' Geoff Mulgan, CBE, Chief Executive of the National Endowment for Science Technology and the Arts (Nesta)\n\n'Profound insights about how the digital age changes trust, wrapped in a compelling narrative of captivating and revealing stories. A rare book that will cause you to think deeply about your business, your relationships and your life' Don Tapscott, bestselling author of _Wikinomics_ and _Blockchain Revolution_\n\n'This book perfectly walks the reader through the past, present and future of trust as we know it. Rachel's expertise on this topic is unmatched. It's an absolute must-read for business leaders and everyday consumers alike' Nick Shapiro, Global Head of Trust and Risk Management, Airbnb\n\n'This is that admirable and all-too-rare book that gives you \"an idea to think with\" that helps to put new things in place. _Who Can You Trust?_ is a primer for a new world that sets you up to be a better citizen, consumer and parent. I quickly learned so much about so many things I wanted to know' Professor Sherry Turkle, author of _Reclaiming Conversation_ and _Alone Together_\n\n# Introduction\n\n'Abandon weapons first, then food. But never abandon trust. People cannot get on without trust. Trust is more important than life.'\n\nConfucius to his disciple Tzu-Kung\n\nI was getting married on the day the hammer fell on Wall Street. The date was 14 September 2008. I had been living in New York for almost a decade and had met my then fianc\u00e9, Chris, in a downtown dive bar called Eight Mile Creek. We were both 'city people' but we wanted to have our wedding in a rural, rustic setting. The place we finally chose was called Gedney Farm, nestled in the charming old Berkshire village of New Marlborough, Massachusetts.\n\n'So, you want to get married in a horse barn?' my father said when I showed him the venue, a Normandy-style red barn surrounded by lush meadows and abundant orchards. Getting into the spirit of things after that, he decided we should arrive at the venue in an old-fashioned horse and carriage. I went along with his Cinderella fantasy and climbed into an open-topped white carriage, complete with a driver and a footman, drawn by an old grey mare. The horse, on its last legs, was slow. It rained. I was late.\n\nAround eighty guests, our closest family and friends from all over the world, joined us for the occasion. Lit by candles and strings of Edison bulbs, the ceremony was traditional and very beautiful. The best man's speech was funny and the food was delicious, despite my finding a grasshopper about the size of my little finger in the green salad.\n\nSo there I was, at the heart of one ancient institution\u2013marriage\u2013built on trust and life-long commitment, while another\u2013Wall Street\u2013was imploding. Lost in the bubble of the celebrations, I didn't realize the outside world was in meltdown until around 9.30 p.m., when I finally noticed that, around the room, the warm glow of the Edison bulbs was competing with the brash blue glare of iPhones and BlackBerries as guests stealthily consulted their hand-held harbingers of doom. Family and friends who worked in banking were trying to absorb the barrage of messages flooding in. Could the impossible have happened? Lehman Brothers had just filed for Chapter 11 bankruptcy protection. Bank of America and Barclays had pulled out of a deal that might have saved the 158-year-0ld firm. Merrill Lynch had agreed to be bought by Bank of America for roughly $50 billion* in an attempt to avert a financial crisis. Washington Mutual, Wachovia and HBOS in the United Kingdom were within a whisker of collapsing. The fate of another giant, American International Group (AIG), the vanguard of the credit default swap market, teetered in the balance.\n\nA couple of friends who were senior executives at JP Morgan Chase and Goldman Sachs apologized for having to leave, summonsed to 'red alert' emergency meetings. It would be a race against the clock to avoid the blind panic that would surely happen when the markets opened. Several other guests drank nervously and partied hard, not sure if they would be carrying their work belongings out in boxes the following day. We danced the Horah, a traditional Jewish wedding ritual, which ended with me being elevated on a chair and my husband being thrown precariously up in the air on a large white tablecloth. Another moment of trust. Guests whirled around us, clapped and made 'Oy! Oy! Oy!' noises. Meanwhile, outside the barn, the biggest global financial crisis in history was building up a head of steam.\n\nIt was, of course, the beginning of the nerve-shattering period when many businesses 'fell off a cliff' and the world's financial system came closer to collapse than at any time since the Great Depression. As we now know, the economic repercussions of the meltdown would engulf the world for many years to come. But my wedding day, rich with tradition, also marked the downfall of something more profound: public trust in institutions.\n\nWho was to blame for the crisis? What were the main causes? These questions were at the heart of the Financial Crisis Inquiry Commission (FCIC) created to investigate the banking collapse, and the answer was damning. 'The crisis was the result of human action and inaction, not of Mother Nature or computer models gone haywire,' the 525-page report found. 'To paraphrase Shakespeare, the fault lies not in the stars, but in us.' In other words, the meltdown was an 'avoidable' human disaster.\n\nThe federal inquiry hammered the embarrassing failures of regulators, whom the report described as 'sentries not at their posts'. The finger was pointed squarely at the Federal Reserve for its failure to question widespread, egregious mortgage lending, overreliance on short-term debt and the excessive packaging and reselling of loans, along with many other red flags. According to the report, however, the main culprit was not the toxic financial instruments but the human failings that drove them: reckless risk-taking, greed, incompetence, stupidity and a systemic breakdown in accountability and ethics.\n\nIt wasn't the first nail in the coffin of institutional trust and it probably won't be the last, but the financial crisis struck deep.\n\nA loss of trust amounts to a lack of faith and confidence in 'the system' itself. What should we believe in if the system has failed us? Who, or what, can be relied upon? We begin to fear what else can go wrong. What other shortcomings we don't yet know about might lurk in the system? Fear, suspicion and disenchantment are deadly viruses that spread fast. The initial epicentre of the trust explosion was, understandably, with the banks. But it hasn't stopped there. Since the crisis, other scandals, other revelations, have seen the ripples of distrust touch government, the media, charities, big business and even religious organizations.\n\nLike the plot of some overblown soap opera or Jacobean tragedy, the episodes of unethical behaviour have come thick and fast, from the lurid, even criminal, to the just plain stupid and, sadly, routine. Each has chipped away at public confidence. The British MPs' expenses scandal; the false intelligence about weapons of mass destruction (WMDs); Tesco's horsemeat outrage; price gouging by big pharma; the BP Deepwater Horizon oil spill; the dishonours of FIFA's bribery; Volkswagen's 'dieselgate'; major data breaches from companies such as Sony, Yahoo! and Target; the Panama Papers and widespread tax avoidance; the exchange-rate manipulation by the world's largest banks; Brazil's Petrobras oil scandal; the lack of an effective response to the refugee crises; and, last but not least, shocking revelations of widespread abuse by Catholic priests, other clergy and other 'care' institutions. No wonder a thousand headlines lament that nobody trusts authority any more. Corruption, elitism, economic disparity\u2013and the feeble responses to all of the above\u2013have pummelled traditional trust in the old institutions as fiercely as a brutal wind lashing ancient oaks.\n\nSignificantly, this crisis is taking place in a landscape of rapidly shifting and evolving technologies, from artificial intelligence (AI) to automation to the Internet of Things (IOT). We are already putting our faith in algorithms over humans in our daily lives, whether it's trusting Amazon's recommendations on what to read or Netflix's suggestions on what to watch. But this is just the beginning. We will soon be riding around in self-driving cars, trusting our very lives to the unseen hands of technology.\n\nAt the same time, many people are feeling so overwhelmed by the pace of change and the sheer amount of knowledge now available at a swipe or keystroke that they are beating a retreat to media echo chambers that narrow down information and reinforce already held beliefs. It becomes easy to ignore or simply not see contrary views. Technology, for all its pluses, also means falsehoods and 'fake news' can quickly spread through networks unchecked and with an unstoppable momentum. In fact, online misinformation on a grand scale\u2013and the potential for digital wildfires\u2013was listed by the World Economic Forum (WEF) in 2016 as one of the major risks to our society. The result of those echo chambers and that misinformation? Our fears are verified, often baselessly. Our anger is amplified. The cycle of distrust is magnified. All in all, our faith in many institutions has been dragged to a critical tipping point.\n\nIndeed, recent gloomy poll numbers would have any politician or business leader in a sweat. For the past seventeen years, the global communications firm Edelman has been conducting an annual 'Trust Barometer', asking more than 30,000 people across twenty-eight countries about their level of trust in various institutions. The headline for the 2017 results was, tellingly, 'Trust in Crisis'. Trust in all four major institutions\u2013government, the media, business and non-governmental organizations (NGOs)\u2013is at an all-time low. The media suffered the biggest blow, now distrusted in 82 per cent of all countries surveyed. In the UK, the number of people saying they trusted the media fell from 36 per cent in 2016 to 24 per cent in 2017. 'People now view media as part of the elite,' says Richard Edelman, President and CEO of PR firm Edelman. 'The result is a proclivity for self-referential media and reliance on peers.' In other words, looking to reinforce what we already believe, often from people we know.\n\nThe Brexit vote to leave the European Union and the election of Donald Trump are the first wave of acute symptoms emerging from one of the biggest _trust shifts_ in history: from the monolithic to the individualized. Trust and influence now lie more with 'the people'\u2013families, friends, classmates, colleagues, even strangers\u2013than with top-down elites, experts and authorities. It's an age where individuals matter more than institutions and where customers are social influencers that define brands.\n\nBy asking challenging questions about the flawed structure and size of institutional systems, and who runs them, we are coming to another confronting realization. Institutional trust, taken on faith, kept in the hands of a few and operating behind closed doors, wasn't designed for the digital age.\n\nIt wasn't designed for an age of radical transparency, of WikiLeaks and Cryptome, where politicians and CEOs must imagine they are operating behind clear glass. Trying to hide, well, anything really, is a high-stakes gamble. It doesn't work in a world where PR puffery can no longer cover up dirty secrets or closed-door antics. Take a few recent examples of 'private' matters that have been spilled around the world: the sensitive user data of the extra-marital dating site Ashley Madison, Turing Pharmaceuticals' internal emails on its predatory drug pricing, secret Scientology manuals, Hillary Clinton's emails and even a private conversation that took place within a private palace garden between the Queen of England and the Metropolitan Police Commander about the rudeness of Chinese officials.\n\nIt wasn't designed for an age where people can transact directly on platforms such as Airbnb, Etsy and Alibaba. It wasn't designed for an era where it is predicted half of the workforce will be 'independent workers'\u2013freelancers, contractors and temporary employees\u2013within the next decade. It wasn't designed for a time where we have become dependent on tech powerhouses such as Facebook and Google which represent new forms of 'network monopolies' and platform capitalism. It wasn't designed for a culture where we want to control everything personally, from our bank accounts to our dates, with a swift click, tap or a swipe.\n\nSo should we be mourning the loss of trust? Yes, and no, because here's the thing: whatever the headlines say, this isn't the age of distrust\u2013far from it. Trust, the glue that holds society together, hasn't disappeared. It has shifted\u2013and the implications, for everything from hiring a babysitter to running a business, are massive.\n\nFor the past decade, I have been researching how technology is radically changing our attitudes towards trust. In 2008, I started writing my first book, _What's Mine is Yours_ , about the so-called 'collaborative' or 'sharing economy'. I was fascinated by how technology could unlock the value of idle assets\u2013cars, homes, power drills, skills, time\u2013but it was the trust ingredient, how technology could make us engage in behaviours that might previously have been considered a little creepy or outright risky, that became my obsession.\n\nEven then, the notion of building a marketplace based on letting strangers stay in other people's houses seemed ludicrous. Today, Airbnb, the home-sharing marketplace, is valued at $31 billion, making it the second most valuable hospitality brand in the world. In 2008, it was hard to see how detailed online profiles would give people the confidence to get lifts with strangers operating as cab drivers and using their own cars. Today, Uber is valued at $68 billion, making it one of the biggest companies in the world, larger than FedEx, Deutsche Bank or Kraft Foods. And then there is the explosion of online dating apps such as Tinder, where the average number of daily swipes is more than 1.4 billion with 26 million matches made daily. These are just a handful of examples where online tools are enabling us to have face-to-face interactions and entrust strangers with our most valuable possessions, experiences, even our lives, in previously unimaginable ways.\n\nConsider this: why do people say they don't trust bankers or politicians yet trust strangers to share a ride with them?\n\nOne conventional explanation is that people don't always tell the truth in surveys. That may be so, but there had to be more to this trust paradox. I had a hunch something deeper was happening. What if trust, like energy, cannot be destroyed and instead just changes form?\n\n_Who Can You Trust?_ charts a theory, a bold claim: we are at the start of the third, biggest trust revolution in the history of humankind. When we look at the past, we can see that trust falls into distinct chapters. The first was _local_ , when we lived within the boundaries of small local communities where everyone knew everyone else. The second was _institutional_ , a kind of intermediated trust that ran through a variety of contracts, courts and corporate brands, freeing commerce from local exchanges and creating the foundation necessary for an organized industrial society. And the third, still very much in its infancy, is _distributed_.\n\nA trust shift need not mean the previous forms will be completely superseded; only that the new form will become more dominant. For example, a small farming community may continue to rely on centuries-old local trust in some matters, but turn more often to the new town court to handle others.\n\nTrust that used to flow upwards to referees and regulators, to authorities and experts, to watchdogs and gatekeepers, is now flowing horizontally, in some instances to our fellow human beings and, in other cases, to programs and bots. Trust is being turned on its head. The old sources of power, expertise and authority no longer hold all the aces, or even the deck of cards. The consequences of that, good and bad, cannot be underestimated.\n\nThe explosive growth of the sharing economy is a textbook example of _distributed trust_ at play. But the theory is also a way to understand the rapid evolution of platforms like the darknet, where consumers are happily scoring everything from marijuana to AK-47s from 'untrustworthy' dealers. The darknet and the new era of digitally enabled app intimacy may sound as if they have little in common, but they share the same underlying principle\u2013people trusting other people through technology.\n\nDistributed trust explains why we are now feverishly scoring and rating everything from restaurants to chatbots to Uber drivers (and why passengers are rated, too), helping to shape, almost instantly, the rise or fall of all sorts of businesses, while also creating _reputation trails_ where one mistake or misdemeanour could follow us potentially for the rest of our lives.\n\nDistributed trust helps us understand why digital cryptocurrencies such as bitcoin and ether could be the future of money, and how the blockchain (the underlying ledger technology that powers these cryptocurrencies) could be used for everything from tracking the source of foods or blood diamonds to selling our homes without the need for estate agents.\n\nDistributed trust helps us grapple with why and how we'll come to trust well-trained bots, whether they're giving us relationship advice, resolving our parking tickets, ordering our sushi or telling us if we have cancer.\n\nIndeed, I believe the real disruption happening is not technology itself, but the massive trust shift it creates.\n\nDistributed trust is not simply a new, idealistic flavour of techno libertarianism. There are many stories in this book that show how it can have negative, dark or disastrous consequences\u2013discrimination, theft and even death. Yes, technology can widen the circle of trust, unlocking the potential to collaborate and connect with unfamiliar strangers, but it can also erect and harden boundaries between us. Ratings and reviews may make us more accountable, even a little nicer, to our fellow human beings but our growing reliance on them also means some people will become forever tarnished, relegated to a kind of digital purgatory. And, in our rush to reject the old and embrace the new, we may end up placing too much trust, too easily, in the wrong places.\n\nIt's already becoming clear that the turpitudes of institutions, real or fabricated, have left many people dangerously receptive to alternatives, and ready to place unquestioning faith in a new, and some would say highly dubious, breed of trust arbiter. Distributed trust is far from foolproof and the questions that really matter are ethical and moral, not technical.\n\nThe first two chapters of this book pose a simple question: how did we end up here? They unpack why trust matters so much. The next three chapters explore the trio of conditions that make distributed trust possible\u2013trust in a new idea, trust in platforms and, finally, trust in other people or bots. This section explains how to adapt to building trust in this new era and what to do when it's lost. Critically, it asks who takes responsibility when trust is no longer centralized but distributed.\n\nElsewhere, the book travels to the depths of the darknet to understand why reputation matters so much, even to cocaine dealers. It goes inside the Orwellian-like trust-scoring system that is emerging in China and could determine everything from a citizen's job to whether they can get on a train or a plane.\n\nThe final chapters look to our digital future, particularly focusing on our rapidly evolving trust in artificial intelligence. If we make a habit of trusting intelligent machines, does it become harder to build trusting relationships with people? The glorified promises of the blockchain are explored. Will this digital ledger really become the 'Internet of Value', as many enthusiasts claim? Will the big banks end up 'taking over' this technology originally designed to cut out the middlemen?\n\nDistributed trust, enabled by new technologies, is rewriting the rules of human relationships. It's changing the way we view the world and each other, returning us to the old village model of trust in one sense, except that the community is global in scale and some of its invisible reins are being pulled by internet giants. Now more than ever it is critical to understand the implications of this new trust era: who will benefit, who will lose and what the fallout might be.\n\nWhy? Because without trust, and without an understanding of how it is built, managed, lost and repaired, a society cannot survive, and it certainly cannot thrive. Trust is fundamental to almost every action, relationship and transaction. The emerging trust shift isn't simply the story of a dizzying upsurge in technology or the rise of new business models. It's a social and cultural revolution. It's about us. And it matters.\n\n* 'Dollars' and the dollar symbol ($) refer to US dollars throughout.\n\n#\n\n# Trust Leaps\n\nFriday, 19 September 2014, was a historic day on Wall Street. From the moment the markets opened at 9.30 a.m. Eastern Standard Time (EST), the ticker rocketed for one company in particular. It was called Alibaba. By the end of the day, the Chinese e-commerce giant had a staggering market capitalization of $231 billion. It was the largest global public flotation ever on the New York Stock Exchange (NYSE), dwarfing that behemoth known as Facebook and even Alibaba's giant rival, Amazon. Overnight, fifty-year-old Chinese businessman Jack Ma, the company's founder and current chairman, became a very, very wealthy man.\n\nA massive crowd lined the streets and packed on to the floor of the New York Stock Exchange that day to get a glimpse of the legendary entrepreneur. He was greeted like a rock star. 'What we raised today is not money, it's the trust from the people,' Ma told more than a thousand cheering admirers.\n\nIt was not, however, the charismatic and dynamic founder who rang the famed opening bell at the Stock Exchange. Instead, Ma opted to have eight Alibaba customers, five of whom were women, to stand on the podium to start the day of trading. He wanted to show he was true to his mantra, 'Customers first, employees second and shareholders third.' One merchant\u2013one of the millions of small businesses who trade on Alibaba's sites\u2013was Lao Lishi, a former Chinese Olympic diving gold medallist who sells wooden beaded bracelets. Another was Peter Verbrugge, an American farmer who currently holds the record for selling the greatest amount of cherries through Alibaba. The customers ringing the NYSE bell represented something very important to Ma\u2013how Alibaba has transformed the way Chinese businesses of all shapes and sizes can buy and sell a bewildering variety of goods, from clothes and nappies to live pedigree dairy goats and frozen chicken feet, inflatable sex dolls and even 'do-it-yourself abortion kits' to people all over the world.\n\nBut Jack Ma's story is not simply a fascinating rags-to-riches tale of entrepreneurial persistence. It also represents a remarkable feat in the delicate business of building trust.\n\nIt's a challenge to build any successful online marketplace where two sides need to trust one another, but what makes Ma's story extraordinary is that he achieved this in China. Traditionally, China is a society based on the concept of _guanxi_ , loosely translated as 'relationships'. Trust, in business as well as private life, exists between people in the same _guanxi_ : family, friends and people in the same village. People they know well over time, in other words, not strangers on a far-flung planet called the internet. In fact, it is common to _distrust_ people outside your own personal network. This can create a cultural impediment and business obstacle as people are more prone to avoid building new relationships where there is no close connection.\n\nI first went to Shanghai on business when I was twenty-five. I was part of a consulting project for a well-known brand looking to expand into Asia. Over the course of the first week, we shared an array of meals with our Chinese business clients. Lazy Susans spun, we ate delicious food at lunch and dinner, and we clinked beer glasses in toast after toast. The gatherings were warm and enjoyable but by day three I was wondering when we would get to the 'real' work. Rather insensitively, I didn't realize how important it was for Chinese businesspeople to spend a considerable amount of time socializing and getting to know you at the start of a relationship. 'In the West, we tend to reserve trust from the heart (affect-based trust) for family and friends and trust from the head (cognition-based trust) for business partners,' explains Professor Paul Ingram of Columbia Business School, who studies social networks. 'But in China, affect- and cognition-based trust are highly entwined even in business.' It's especially true in China that people only trust you after you have invested a lot of time upfront in proving yourself to be _trustworthy_.\n\nThat was what Jack Ma was up against. It was one of the rock-hard conventions of trust he would set about shattering.\n\nMa Yun, as he was originally named, was raised in Hangzhou, about a hundred miles southwest of Shanghai, during Mao's Cultural Revolution. The middle child of three, his parents were traditional musical theatre performers. Ma inherited their love of showmanship. He would later get a reputation for dressing up in elaborate wigs and leather gear, and belting out theme songs from the _Lion King_ at company events.\n\nMa wasn't a great student at school, but he was canny. From a young age, he realized the importance of mastering English. After President Nixon visited Hangzhou in 1972, tourists flooded to the area to see the beautiful lakes, temples and gardens. Every day, Ma would set his alarm early, around 5.00 a.m., and ride his bike to the Hangzhou Hotel. He would talk to the guests in English and offer to act as a guide, giving tours around the city free of charge. He did this for more than nine years. 'These Western tourists opened my mind because everything they told me was so different from the things I learned from school and from my parents,' Ma has said.\n\nOver the years, Ma befriended many tourists including a young American woman who suggested he take on an English name. Her husband's name was Jack, as was her father's. And so Jack he became.\n\nJack Ma would go on to become China's richest man in 2014, worth more than $19.5 billion, hauling himself to that lofty position with the aid of a shatter-proof resilience in the face of failure, a good dose of raw ambition, and another vital kind of trust\u2013trust in himself. He applied to Harvard ten times and was rejected on every occasion. (Who applies ten times?) He failed China's national university entrance exam twice. When he did eventually graduate in 1988 with a degree in English, he became a schoolteacher. To supplement his modest $3 a week income, he would buy and sell plastic carpets on the streets of Hangzhou. At heart, he was a businessman.\n\nWith China's economy steadily improving and ideological barriers falling in the early 1990s, Ma decided to quit teaching. He applied for more than thirty different jobs but missed out on them all. When he applied to be a police officer, he was told in four simple words, 'You are no good.' 'I even went to Kentucky Fried Chicken when it came to China,' Ma told an audience at the World Economic Forum in Davos. 'Twenty-four people went for the job. Twenty-three were accepted. I was the only guy who wasn't.'\n\nIt was in 1995, on his first visit to the United States, that his life took a fortunate turn. Ma had started Hangzhou Hope Translation Agency a year earlier, and went to America to help a Chinese firm sort out a financial dispute they were having with a US partner. It turned into a terrifying trip\u2013the American he had been sent to see was a con man who flashed a gun at him. He made his way to Seattle to stay with Stuart Trusty, a friend who happened to run one of American's first internet providers, VBN. Ma noticed a mysterious grey box with a screen sitting on his friend's desk. He wondered what on earth it was. 'Jack, it's not a bomb,' Trusty reassured him. 'It's a computer. Search for anything you want.'\n\nMa slowly typed in the word 'beer'. He doesn't remember why, possibly because it is easy to spell. A list of beers from Germany, America and Japan popped up, but noticeably no Chinese beers. Next, he typed in 'beer' and 'China'. No results. Keep in mind this was 1995. Netscape had just launched and Yahoo was an infant. Google would not launch for another three years. It was still the days of achingly slow dial-up internet. Nonetheless, Ma sensed something huge waiting in the wings.\n\nBack home, he started China Pages, a kind of online Yellow Pages for Chinese companies. 'The day we got connected to the web, I invited friends and TV people over to my house... we waited three and a half hours and got half a page,' he says. 'We drank, watched TV and played cards, waiting. But I was so proud. I proved [to my house guests that] the internet existed.' Ma eventually sold the directory business to the state-run Hangzhou Telecom for 1 million yuan (approximately $148,000), a very large sum of money at the time. Next, he headed to Beijing to advise the Ministry of Foreign Trade and Economic Cooperation on ways to get Chinese companies to take up 'electronic commerce'. 'My boss wanted to use the internet to control small businesses but I wanted to use the internet to empower small businesses,' he says. What Ma really wanted to do was to start and build companies.\n\nMa's revolutionary vision was to help transform China's entire export economy, connecting small- to medium-sized Chinese businesses to Western customers and Western companies to a myriad of Chinese factories. Amazon and eBay had launched in the United States but nothing like this existed yet in China.\n\nToday, more than 80 per cent of _all_ goods bought and sold online in China are through Alibaba's various online marketplaces. Its spider-like corporate campus, with its gardens and open-plan workspaces, spreads over more than 150,000 square metres in Ma's hometown of Hangzhou. It houses tens of thousands of employees, and the other businesses Ma has since founded. There is Taobao (meaning 'hunting for treasures'), launched in 2003. It is like eBay, enabling people to sell virtually anything directly to each other, but it is more like a local flea market where you might land on a bargain or find something bizarre, such as live scorpions or soap made from the seller's own breast milk. There are hundreds of 'Taobao' villages scattered across China where a large proportion of their economy is based on people selling local goods on the platform.\n\nThen came Tmall in 2008, the online equivalent of a humungous glossy shopping mall, selling well-known global brands, from Disney to Burberry, directly to Chinese consumers. The Alibaba Group has 430 million annual active buyers, and one out of every three individuals in China has made a purchase from the marketplaces.\n\nOn the day of Alibaba's Initial Public Offering (IPO), in September 2014, the dynamic Ma was ebullient about his company's historic milestone. In interviews he gave there was one word that stood out: 'Trust. Trust us, trust the market and trust the young people,' Ma said. 'Trust the new technology. The world is getting more transparent. Everything you worry about, I've been worrying about in the past fifteen years.' Ma continued without a pause. 'Trust has to be earned, of course. Because when you trust, everything is simple. If you don't trust, things get complicated.' He made no fewer than eight mentions of the word 'trust' within the space of a minute.\n\nRight from the start, Ma knew the importance of building trust, especially in a culture like China's\u2013it's why he splashes the word about so generously. And we'll come back to how he set about doing just that. But first, it's worth unpacking the concept of trust itself. Like the rest of us, Ma would probably have trouble defining it\u2013what exactly do we mean when we talk about 'trust'? And what doors does it open?\n\nTrust is not a nicety, a kind of optional extra in life. We all depend on it in so many of our daily activities. How could we eat, drive, work, shop, get on a plane, go to the doctor, tell secrets, unless we trusted other people? 'Trust,' as political scientist Eric Uslaner says, 'is the chicken soup of social life.'\n\nFor instance, when I order takeaway sushi, I have to trust that the restaurant will use fresh ingredients, the kitchen will be clean, they will not abuse my credit card details and the deliveryman will not run off with my dinner. Trust enables small and large acts of cooperation that all add up to increased economic efficiency. 'Virtually every commercial transaction has within itself an element of trust, certainly any transaction conducted over a period of time,' Kenneth Arrow, the American Nobel Prize-winning economist, once observed. 'It can be plausibly argued that much of the economic backwardness in the world can be explained by the lack of mutual confidence.'\n\nTrust enables us to feel confident enough to take risks and to open ourselves up to being vulnerable. It means we can commit to people before we know the precise outcome or how the other person will behave. And that applies to something as minor as ordering sushi or as life-changing as marrying someone. If, before we bought or did anything, we thought we were going to be cheated or ripped off, then very little would get done.\n\nSocial scientists, psychologists, economists and others view trust as an almost magical economic elixir, the glue that keeps society together and the economy ticking over. That much is agreed. The definition of trust, however, has been widely debated for years. In fact, there are more academic papers on its definition than on any other sociological concept.\n\nIt's odd that we describe trust as being _built_ or _destroyed_ when it's not a structure or a physical thing (except maybe in the form of a handshake or a paper contract). Like 'happiness' or 'love', it's one of those words that we tend to think of as a universal idea. And, the same as love, trust has many faces. It's not like some kind of predictable engine that comes with a manual and only functions in one precise way. Trust varies from situation to situation, relationship to relationship. Put simply, trust is highly contextual.\n\n'So what does the word trust mean to you?'\n\nIt's a question I've asked hundreds of people over the past five years\u2013entrepreneurs, politicians, leaders of large companies, scientists, economists, bankers, designers, academics, students and even five-year-olds. Their answers are fascinating and remarkably different. The question usually elicits a pause and then a kind of _mmm, I'm thinking_ noise. 'It's hard to define, right?' they'll say. Yes, it is. Trust means different things to different people. The most honest answer I have ever been given was, somewhat ironically, from an insurance broker. 'Giving my wife my phone without clearing the history! That is trust,' he said.\n\nFor many people, it is about confidently relying on another person. For example, 'Trust is being able to rely on my husband\/doctor\/friends.' In this instance, trust is being referred to as an attribute that rests _in_ someone specific, generally someone we are familiar with. The more we interact with a person over time, the more confident we become about how they will behave, that they are trustworthy. That kind of trust is known as _personalized_ trust.\n\nG _eneralized_ trust is the trust we attach to an identifiable but anonymous group or thing. As one of the MBA students in my course at the University of Oxford's Sa\u00efd Business School put it, 'Trust is like a contract that guarantees an outcome.' For example, I trust the postal service to deliver my mail. It is also common to mix the two different types of trust. For instance, I might have high personal trust in my bank manager but fragile trust in my bank as a financial institution.\n\nOne of my favourite definitions of trust came from one of my son's friends. He was five at the time and had come round to our house to play. My son, Jack, told him over tea that I was writing a book. They were disappointed it was not about _Star Wars_ or _Harry Potter_ but still asked fascinating questions. I asked them what trust meant to them. 'Trust is when the ice-cream man says he will give you ice cream, he gives you the ice cream because he wants to, and I don't worry about him not giving me the ice cream,' replied my son's friend in one breath. Wow, out of the mouth of babes. It is in fact very close to the definition given by the pre-eminent German sociologist Niklas Luhmann, who wrote, 'Trust is confidence in one's expectations.'\n\nI am on the board of the National Roads and Motorists' Association (known as the NRMA), one of the most trusted brands in Australia. It's like the AAA in the United States or the RAC in the United Kingdom, basically the people who will come and fix your car if it breaks down, wherever you are. Recently, a woman phoned into the NRMA call centre. She sounded very distressed. She was breathing heavily and was clearly crying. Turns out, she was driving down the motorway when she realized she had passed the exact spot where her son had crashed and tragically died a few years ago. She pulled over on to the side of the road and started to have a panic attack. The first number she rang was the NRMA. A roadside assist personnel got to her within minutes. He sat with her for more than two hours. They listened to the radio together. They talked about her late son. He didn't leave until she felt ready to drive again. I was so touched by this story but also curious to find out, why did she call the NRMA? I mean, nothing was wrong with her car. Why not the police, an ambulance, her husband or a colleague? 'I knew you would come,' was her answer. That is trust.\n\nOf the hundreds of definitions of trust I have studied, most can be reduced to one simple idea: trust is an evaluation of outcomes, of how likely it is that things will go right. Or put another way, trust is fostered when the likelihood of an undesirable outcome is low. The five-year-old was right, sort of. Five-year-olds tend to be more naturally trusting than grown-ups, having had a lot less experience in being let down or in worrying about distant outcomes. For adults, trust becomes more complicated and it works in the heart as well as the head. As Morton Deutsch beautifully put it, trust is 'confidence that [one] will find what is desired [from another] rather than what is feared'. Trust is a mixture of our highest hopes and our deepest worries.'\n\nIf you look up images for the word 'trust', all sorts of graphics will come up, often with some kind of danger lurking in the picture. People swinging between trapezes, for instance. Two hands, reaching out to meet but not quite touching. A person in mid-fall relying on another person\u2013typically with their arms stretched out\u2013to catch them. A sleeping lion with a mouse prancing a few centimetres from his nose. The common element in all of these areas is a gap, a grey area where something unknown happens. The images are conveying the powerful elements of trust: vulnerability and expectation.\n\nImagine a gap exists between you and something unknown. A stranger you need to rely on, a restaurant you have never been to before or your first run in a self-driving car. The gap between the known and unknown is what we refer to as risk. Indeed, risk can be defined as the management of uncertainty that matters. There are some uncertainties that are simply irrelevant. For example, if I am a farmer in England the possibility of heavy rain is an uncertainty that matters to my livelihood. But if I am managing a clothing factory in, say, China, the uncertainty about the weather in England is irrelevant. When there are no unknowns, when we can guarantee an outcome, there is no risk. For example, we know for certain the sun will rise in the morning.\n\nTrust and risk are like brother and sister. Trust is the remarkable force that pulls you over that gap between certainty and uncertainty; as the Nike tagline says, 'Just do it'. It is literally the bridge between the known and the unknown. And that's why my definition of it is simple:\n\nTrust is a confident relationship with the unknown.\n\nWhen you view trust through this lens it starts to explain how it enables us to cope with vulnerability, place our faith in strangers or just keep moving forward. It shows why just enough trust is a critical ingredient of innovation and entrepreneurial success like that of Jack Ma. Companies such as Apple, Amazon and Netflix constantly challenge assumptions, take smart risks and allow their employees to dive into uncharted waters to discover new ideas. But they also know how to get customers to trust new offerings, so that the initial risk of trying something new becomes quickly irrelevant.\n\nJack Ma realized the internet presented an opportunity to unleash the entrepreneurial spirit that existed in China but had been suppressed by years of Communism. What he spotted early on was how technology could enable trust\u2013make unknown sellers seem familiar to people. But how to build a new kind of trust between strangers in a country based on _guanxi_?\n\nAside from those ancient traditions at work, less than 1 per cent of the country's population was online when Alibaba first launched. And of that 1 per cent, fewer still would even consider buying something off a website. People were unfamiliar with the concept of the internet, let alone e-commerce. Indeed, there was no history of e-commerce, no online payments system nor even a way quickly and safely to send goods. So how did Alibaba crack the trust code?\n\nWhen we are trading goods over the internet, typically neither party knows each other. We are wary of scams and products not being as promised. If I buy, say, a Fitbit from an eBay seller, is it really brand new or is it refurbished, fake or even stolen? A lot can go wrong. There is always the chance of an unwanted outcome or a risk. Ma realized that to create trust between buyers and sellers online, he needed to use technology to reduce uncertainties or lower the risk enough to allow transactions to go ahead.\n\nHe also recognized that the bigger the trust problem, the greater the business opportunity. Ma was a bit like Steve Jobs; he knew there were enormous advantages to developing the solution to an obstacle that stood in the way of a market, rather than waiting for others to solve the problem. Take payments. How do I know you will pay? How do I know you will send what I paid for? It's a classic chicken-and-egg trust problem.\n\n'For three years, Alibaba was just an e-marketplace for information. What do you have? What do I have? We talk for a long time, but don't do any business, because there is no payment,' says Ma. 'I talked to the banks. No banks wanted to do it. Banks said, \"Oh no, this thing would never work,\" so I didn't know what to do.' Ma knew all too well that if he launched a payment system without the required licence, he would be breaking China's strict financial laws. He could have gone to jail. But he decided to do it anyway. Why? 'It is so important for China and the world to be able to trust the system,' says Ma.\n\nIn 2004, the company introduced an online system called Alipay ( , meaning 'payment treasure'). Instead of a direct payment like PayPal, Alipay takes money upfront from buyers and puts it in an escrow account.* The seller then ships the product and the funds are only released after the buyer has inspected the goods and confirmed that they are satisfied. It's a simple example of reducing uncertainties, in this instance over settlement. 'So many of the people I talked to at that time, for Alipay, said, \"This is the stupidest idea you've ever had,\"' recalls Ma. This was really stepping on the toes of China's highly regulated banking sector. But Ma didn't mind if people thought the idea was risky or even stupid, 'As long as people use it.' And use it they did. Today, more than 400 million people use Alipay to pay for goods. It is estimated to be worth more than $50 billion as a standalone business. In 2015, approximately 70 per cent of _all_ online payments in China\u2013whether it was for goods, rent, utilities, phone charges or tuition fees\u2013went through Alipay. Ma always had big dreams.\n\nPayments aside, how did people know they could trust the unknown small businesses and individual traders Ma wanted to put online? Take thirty-eight-year-old Wang Zhiqiang, one of the sellers who rang the bell on the day of the IPO. Zhiqiang was once a migrant rural worker struggling to make a living. He had tried his luck as a vegetable street vendor, a construction worker and a takeout deliveryman. Despite being poorly educated, Zhiqiang had always been interested in computers and the internet. He moved to Beijing's Zhongguancun area, known as China's Silicon Valley, where he took on many manual labour jobs for more than six years. By 2006, he had saved enough money to buy a computer, which he took back to his hometown, a small rural farming village in north China's Shanxi province. After overcoming the challenge of getting the internet installed in his home, he opened an online shop selling just a few local products such as rice and soybeans. His friends and family wondered what on earth he was trying to do. In 2008, while the Olympic Games were in full swing in Beijing, Zhiqiang got the break he needed. He opened 'Farmville', not the popular game but an online shop selling all kinds of fresh produce grown by the local villagers. Sales quickly soared to more than 200 products per day. Today, his monthly net profits are more than 80,000 yuan (approximately $13,000) and he has become well known online as 'Wang Xiaobang'\u2013in this context, 'bang' means a warm-hearted person who would like to help others. But how did so many people come to trust the legitimacy of this unknown vendor from a remote area?\n\nThe answer is, at least in part, down to a service Alibaba launched in 2001, called TrustPass. For a seller to get TrustPass certification, they had to go through a third-party identity and bank account verification process. Alibaba also helped sellers create their own official-looking brands and virtual storefronts. Zhiqiang, for instance, would take colourful pictures of rural farm life and close-ups of the fresh products he sold as they were farmed or harvested. He wanted his store to feel 'local' and connected to the suppliers. TrustPass marked a breakthrough for Alibaba, not just in terms of trust but also money.\n\nOn average, certified TrustPass sellers were receiving up to six times more genuine enquiries than non-registered sellers. This gave Alibaba the perfect excuse to start charging small businesses (most services had been free up until this point). 'It made customers who paid for their status appear more _trustworthy_ ,' says Porter Erisman, a close friend of Ma and long-time employee. 'It made those members still clinging to their free accounts seem _less_ trustworthy. After all, if they had such a good business, why weren't they willing to pay up a little to prove it?'\n\nAlibaba's liquid gold was not online shopping but trust. And that is why, when Ma discovered that it had been badly broken, he was livid. In February 2011, it became known that around a hundred members of Alibaba's 5,000-strong sales team had been taking financial kickbacks in exchange for allowing fraudulent sellers to skip the verification process and set up accounts. The deceitful behaviour had been going on for more than two years. The impact? Two thousand three hundred and twenty-six high-volume sellers of low-quality or even fake products had been verified as 'Gold Suppliers'.\n\nMa knew he had a grave trust issue on his hands. He had to act quickly. He had to send a loud and clear message to protect his company's reputation. So the salespeople who had knowingly set up the accounts and those employees who had looked the other way were all fired. David Wei, the then CEO, and COO Elvis Lee also resigned, even though they were not implicated in the fraud. Both were falling on their swords to accept responsibility. 'One of our most important values is integrity. That means integrity of our employees and integrity of our online marketplaces as trusted and safe places for our small business customers,' said Ma. 'We must send a strong message that it is unacceptable to compromise our culture and values.' His actions were clearly validated. In 2016, the Alibaba Group overtook Walmart as the world's largest retailer.\n\nMa has proved in spades to the Chinese (and to the rest of the world) that trade does not require a prior or close personal tie in order to work. And strangers won't, as a matter of course, betray you.\n\nThe story of Alibaba is a telling illustration of how technology is enabling millions of people across the world to take a _trust leap_. A trust leap occurs when we take a risk and do something new or in a fundamentally different way.\n\nTrust leaps create new possibilities; they break down barriers and help us form new relationships; they enable us to mash up ideas and memes in unexpected ways; and, as in the example of Alibaba, to open up new markets, new networks and new alliances that would once have been unthinkable. Trust leaps carry us over the chasm of fear, that gap between us and the unknown.\n\nImagine the first time people switched from bartering real goods to using paper money. Bartering is intuitive\u2013I give you a chicken in exchange for a metal pot. Money meant people had to trust that these flimsy pieces of printed paper had real value, and would retain that value. They had to trust that the institutions issuing the money, usually governments and banks, would determine the right value. That is a trust leap.\n\nDo you remember the first time you put your credit card details into an internet site? That is a trust leap.\n\nI remember a heated conversation I had with my dad when I was eighteen. I had seen a nice-looking second-hand navy blue Peugeot for sale on eBay. It was within the price range my parents had set me to buy my first car. From the photos, it looked in good condition. My dad, then a chartered accountant, asked me if I knew what 'a market for lemons' was. I didn't back then. Over lunch, a mini lecture followed on George Akerlof's economic theory about problems of uncertainty over quality of goods. To put it simply, Akerlof argued that in a market there are good used cars and defective ones ('lemons'). The buyer of a car does not know beforehand whether it is a car or a lemon. Dad argued that this is what happens in spades on eBay because we couldn't drive or inspect the car. He also pointed out that the seller's pseudonym was 'Invisible Wizard', which did not exactly inspire confidence. So instead of using eBay, we went to the car dealer not far from home, the same dealer my dad had bought my brother's car and three other cars from in the past. To my dad, eBay seemed like an irrational way to buy goods. It wasn't a trust leap he was ready to make in 1999.\n\nThe first time we leap, it feels a bit weird, even risky, but we soon get to a point where the idea seems normal. Our behaviours transform, often relatively quickly. And when others see that enough people have survived the leap and benefited from it, millions will follow. My father is now somewhat of an eBay addict. He would probably consider buying a car on eBay these days (then again, maybe not).\n\nTrust is the conduit through which new ideas travel. Trust drives change.\n\nThroughout history, humans have demonstrated a remarkable propensity to change the way we do things\u2013bank, trade, travel, consume, learn and date. To understand just how good we are at taking trust leaps we need to go way back in time, way before the internet or even the printing press were invented.\n\nOne day in the year 1005, a handwritten letter crossed the Mediterranean. It was filled with the troubled words of Sumhun ben Da'ud, an eminent merchant who lived in Sicily. He was furious with his business partner, Joseph ben 'Awkal, who had ignored Sumhun's repeated requests to pay Egyptian creditors the hundreds of dinars they were owed. As the total monthly expenses of a middle-class family living in Egypt were no more than three dinars, these were very significant sums. Word was spreading to other traders across the region that neither of the business partners could be trusted. 'My reputation is being ruined,' lamented Sumhun.\n\nSumhun and Joseph were part of a close-knit group called the 'Maghribi traders', Jews who left Baghdad during the political turmoil of the tenth century and settled in the Maghreb, the coastal region of north Africa. Around 150 years ago, more than 1,000 of their personal letters were discovered in almost perfect condition in the storeroom of an ancient synagogue in Fustat in Egypt. The letters provide an intriguing window into the lives of the traders and the role they played in transforming long-distance trade.\n\nSay a merchant based in Old Cairo wanted to sell his textiles and spices in Palermo in Sicily. He could travel long distances by boat\u2013but sea voyages were treacherous and time-consuming. Or, instead of travelling himself, he could use overseas agents, and the agents could handle everything from unloading the ships to selling the goods in local markets, to settling the odd bribe on the merchant's behalf.\n\nToday this sounds like a relatively straightforward idea, but at the time it required a massive trust leap. Possibilities for deception and corruption were high. An agent could lie about prices, skim money or simply steal the goods altogether. And when something bad happened, it could take months for the merchants to find out. Formal trade regulations and legal contracts, as we know them, did not exist. The Maghribi merchants faced a problem: they had no way of knowing what the hired agents were up to on the other side of the Mediterranean.\n\nWhen one party has less information than the other, economists call it _information asymmetry_. Economist Kenneth Arrow first described the concept in 1963, in the context of healthcare. Doctors generally know more about the value and effectiveness of a given medical treatment than patients do. They are in a powerful 'expert' position and patients will tend to follow their recommendations. Arrow noted how sometimes the doctor might manipulate the asymmetry to his or her advantage, for instance, recommending costly drugs or an operation that is not necessary.\n\nInformation asymmetry is all around us because it is rare for two people to have perfect and equal information in any kind of exchange. The life insurance broker who knows what clause 221 of a complicated policy really means; the Alibaba seller Rakjuk Kft who knows if the top-quality Angora goats he sells really are 'well-bred' champions, 'free from parasite' and 'fully red-blooded', as advertised; the second-hand car salesman who knows the real history of a cute little Fiat 500; the Airbnb host in Cape Town who knows if their place really has two bedrooms (or if one is a sort of pseudo-loft in the kitchen only accessible by a precariously propped-up wooden ladder) and the sweeping ocean views shown (or has a postage-stamp view of something blue if you hang off the right-hand corner of the balcony and use binoculars); and the overseas agents who know if they are selling the merchant's frankincense or olive oil for fair or dodgy prices. Information asymmetry creates future unknowns and the very need for trust.\n\nSo how did the Maghribi traders get the agents in far-off lands not to lie, cheat or steal in the absence of direct supervision? The system they came up with was so ingenious that it opened the modern era of long-distance trade between strangers.\n\nThe Maghribi traders had the same religion, common family ties and, most importantly, shared the same motive of ensuring their agents did the right thing. They were living in the era of _local trust_ and had high levels of _social capital._ It's an idea that has fascinated sociologists for centuries, notably Pierre Bourdieu, Robert Putnam and James Samuel Coleman. 'Whereas physical capital refers to physical objects and human capital refers to the properties of individuals, social capital refers to connections among individuals\u2013social networks and the _norms of reciprocity_ and trustworthiness that arise from them,' Putnam explains in his influential book, _Bowling Alone: The Collapse and Revival of American Community_ , first published in 2000. Putnam argues that close-knit suburbs have given way to 'exurbs' and 'edge cities'\u2013vast anonymous places where people spend more time commuting to work, at the office and watching TV alone, and less time socializing with friends, neighbours, community groups and even family.\n\nSocial capital\u2013shared values, bonds and support\u2013can be found in a whole array of networks and communities. For instance, a group of neighbours who keep an eye on one another's homes to make the street safer. A school that holds a garage sale to raise money for a local homeless shelter. A stranger who drops your wallet into the police station after finding it left on a bus. A former colleague who helps you land a new job. 'Out of such shared values comes trust, and trust has a large and measurable economic value,' writes Francis Fukuyama in his 1995 classic book _Trust._ 33\n\nThe merchants formed a coalition with a system of collective sanctions. The tight-knit group would frequently write and talk to each other, openly sharing information about good agents and those up to no good. However, they couldn't just rely on shaming the cheaters. Merely identifying them was not enough to discourage bad behaviour. Rewards for acting honestly and responsibly were needed as an incentive against the short-term gains cheating offered.\n\nThe Maghribi traders designed a simple reputation-based system where the most trustworthy agents were rewarded with the most business. If one agent ripped off a merchant, the entire trading network subsequently shunned him. All the merchants were required to take a collective vow never knowingly to employ crooks. The agents knew their success in the long run depended on repeat business. The traders were able to set aside their fears of getting scammed because both parties knew honesty would pay. (That was why Sumhun ben Da'ud was so upset with his business partner.)\n\nWhat's more, it turned out that it didn't matter if the merchants weren't aware of every wrongdoing; the threat that they could find out was enough to make the agents behave honestly. If people believe they are being observed and judged, it makes them behave better, even if they are not actually being watched all the time.\n\nJust like Jack Ma would do a thousand years later, the traders created a system to reduce the unknown enough for people to take a risk and do things differently. Through simple accountability mechanisms, they were able to expand their local network into international trade across geographic, language and cultural boundaries.\n\nTrust leaps expand what is possible, what we can invent and who can be an inventor. Trust leaps extend the reach of our collaboration and creations, opening up new horizons of opportunity. That is why trust matters _so_ much and why establishing confidence in the unknown has been a central part of innovation and economic development over the course of history. Just ask Jack Ma, the boy who was told he was no good for anything, who took a trust leap on himself and persuaded a nervous nation to leap with him.\n\n* An escrow account is generally a holding fund where money is held by a third party on behalf of two other parties making a transaction.\n\n#\n\n# Losing Faith\n\nChances are you have never heard of an American woman called Jean Heller. She was a young reporter just a couple of years out of graduate school when she uncovered a dark secret the United States government had been hiding for more than four decades.\n\nIt was this. Between the years of 1932 and 1972, 600 African Americans living in Tuskegee, one of the poorest counties in rural Alabama and with the highest syphilis rates in the nation at the time, had been used as human guinea pigs by the United States Public Health Service. It was one of the most unethical medical research experiments the country had ever seen and would become known\u2013rather blandly, in light of what it represented\u2013as the 'Tuskegee Study'.\n\nIn the experiments, the county's black farmers, many illiterate, were subjected to painful spinal taps, daily blood draws and, when they eventually died, autopsies. The men were offered free hot lunches, transportation to and from hospitals, free medicine and free burial services as incentives to enter the study programme. The farmers were never told what disease they were suffering from or of its seriousness. Of the 600, 399 of them were syphilitic, while 201 did not have the disease but were used as controls. Doctors merely informed all participants they were being treated for 'bad blood'. These trusting men, disadvantaged and easily manipulated, became pawns in what James H. Jones, author of _Bad Blood_ , identified as 'the longest nontherapeutic experiment on human beings in medical history'.\n\nShockingly, the farmers were left to suffer the ghastly ravages of syphilis, which can include blindness, deafness, dementia, heart disease, paralysis and eventually death. And not knowing they had the disease, the men unwittingly passed it on to their wives and children. 'The Tuskegee Study began ten years before penicillin was found to be a cure for syphilis and fifteen years before the drug became widely available,' wrote Heller in her story published on 26 July 1972, in the _New York Times_. 'Yet, even after penicillin became common, and while its use probably could have helped or saved a number of the experiment subjects, the drug was denied to them.'\n\nWhy? Because the very purpose of the study was _not_ to cure the participants of syphilis. The goal was to observe the long-term effects of the disease and to determine through autopsies if untreated syphilis affected black bodies the same way it affected white ones. 'As I see it,' one of the doctors explained, 'we have no further interest in these patients until they die.'\n\nHeller's story became front-page news and generated widespread public outrage. Members of the US Congress reacted with shock, denying they had known what was going on. The egregious research was stopped immediately, but the fact that it had continued over four decades was the result of a lot of people turning a blind eye for a very long time.\n\nResponding to the outcry, congressional investigations led to the establishment of the Office for Human Research Protections, to monitor ethical standards. Federal laws were also created, requiring Institutional Review Boards to oversee clinical research and ensure adequate protection of all study participants.\n\nTwenty years later, the government, right from the top, was still apologizing for the 'moral and ethical nightmare' of Tuskegee. 'It was a time when our nation failed to live up to its ideals, when our nation broke the trust with our people that is the very foundation of our democracy,' President Bill Clinton said in a press conference in 1997, standing alongside eight elderly survivors of the study. 'To our African-American citizens, I am sorry that your federal government orchestrated a study so clearly racist. That can never be allowed to happen again... An apology is the first step, and we take it with a commitment to rebuild that broken trust.'\n\nThis shameful chapter in medical research history shook the foundations of trust between Americans, especially black patients, and the medical system for a long time afterwards. It's only recently, however, that anybody has tried to quantify the scale of the trust fallout.\n\nAfrican-American men today have the worst health outcomes of all major racial, ethnic and demographic groups in the United States. The life expectancy of black men at age forty-five is three years less than for their white counterparts. It is caused by multiple factors, including disparities in income, diet and healthcare access. But could the differences also be linked to the general African-American distrust of healthcare providers that grew out of the Tuskegee Study? And if so, to what degree?\n\nThese are questions that intrigued two researchers, Marcella Alsan at the Stanford Medical School and Marianne Wanamaker, an economist at the University of Tennessee. After crunching data from various national population and health surveys, in 2016 the pair confirmed what other researchers had hypothesized: that high levels of mistrust existed among African-American men _decades_ after the Tuskegee Study was exposed. The difference was that Alsan and Wanamaker put a precise number on its life and death effect.\n\nWhen someone doesn't trust their doctor, typically they stop going for check-ups or for care when they need it. The researchers deduced that, as a general principle, this has led to a decrease in life expectancy of 1.4 years among black men over forty-five. Perhaps the most remarkable finding, however, was that more than a third of the life-expectancy gap\u2013between older black men and their white equivalents\u2013could be attributed to fallout from the disclosure of the Tuskegee experiment. It's a staggering revelation: life expectancy has dropped for millions of men who didn't live in Alabama and had nothing to do with the Tuskegee Study because of broken trust\u2013or to put it another way, compounding mistrust.\n\nDr Joseph Ravenell gave a wonderful TED talk in 2016 on this very problem that continues today. He pointed to the fact that high blood pressure is one of the leading causes of death among African males over fifty, a medical problem that could be prevented with timely diagnosis and appropriate treatment. So why is it so deadly for black men? 'Because too often high blood pressure is either untreated or under-treated in black men, in part because of our lower engagement with the primary healthcare system,' says Ravenell. 'Some of our earliest research on black men's health revealed that for many, the doctor's office is associated with fear, mistrust, disrespect and unnecessary unpleasantness.' So people skip going to the doctor, especially if they feel fine. Indeed, Ravenell has found in his research that many black men trust their barbers far more than they trust their doctors.\n\nIt doesn't matter that regulations and ethical standards have since been put in place; the echoes of Tuskegee still linger and affect some people's decisions. The findings also tell a bigger story: they show how one trust-busting incident can create a generational scar against an institution or system that takes decades to heal. And sometimes the damage is too deep to repair.\n\nIt would be comforting to think that trust-shattering incidents like the Tuskegee Study were all in the past, the lessons learned and acted upon. The truth is that institutional trust is in greater jeopardy now than ever before. For a stunning illustration of why ordinary citizens feel more betrayed than ever by elites and those in power, and why those elites themselves are heading into a twilight zone, we need only look to the Panama Papers.\n\nIt began with a message\u2013anonymous, of course. 'HELLO. This is John Doe. Interested in data?' The recipient was Bastian Obermayer, a thirty-eight-year-old investigative reporter for the German newspaper _S\u00fcddeutsche Zeitung_. It was sometime late in 2014 when the mysterious message appeared on the reporter's laptop. Obermayer was sitting in his flat in a quiet neighbourhood in Munich at the time. 'We're very interested,' he quickly typed back.\n\nThe source replied in encrypted chat that he or she was going to leak a mountain of highly confidential data, 'more than you have ever seen'. The source made it clear upfront that he didn't want money, only justice. 'I can't explain my rationale without making my identity clear... but I want to make these crimes public,' Doe replied.\n\nOver the course of 2015, more than 11.5 million of the documents that became known as the Panama Papers were transferred. They represented forty years' worth of digitized records ripped from the servers of the Panamanian law firm Mossack Fonseca. The amount of data was staggering; we're talking 2.6 terabytes, and it included 4.8 million emails, 2.1 million PDFs, 3 million database files, 1 million images, as well as other confidential contracts, letters, bank records and property titles. The documents would turn out to be the greatest single data drop in journalistic history, 2,000 times larger than the 2010 Cablegate coup, when WikiLeaks released more than 250,000 classified diplomatic cables that had been sent to the United States from 270 consulates and embassies around the world. These cables contained allegations of corruption and revealed numerous unguarded comments such as US embassy staff referring to Vladimir Putin as an 'alpha-dog', Hamid Karzai as being 'driven by paranoia' and a comparison of the then Iranian President, Mahmoud Ahmadinejad, to Adolf Hitler.\n\nThe Panama Papers revealed that between 1977 and 2015, the firm Mossack Fonseca had created more than 200,000 offshore shell companies in tax havens around the world, for world leaders and their families including President Putin and Kojo Annan, son of the former UN secretary-general Kofi Annan, celebrities such as footballer Lionel Messi, and many other important global elites. But beyond the revelations it laid bare, the leak itself represents a remarkable trust story.\n\nImagine you have been handed an immense cache of secret documents. What would you do?\n\nJournalists have, by nature, an obsession with scoops, with being first. But after working non-stop for more than two months, Obermayer realized he was in way over his head with the small team at his German newspaper. The volume of data was daunting. He needed help, lots of it. So he took the unusual step of sharing the files with hundreds of other journalists from around the world.\n\nObermayer first reached out to Gerard Ryle, someone he deeply respected and had worked with before on the Luxembourg and HSBC leaks. Ryle is the director of the International Consortium of Investigative Journalists (ICIJ), a non-profit based in Washington, DC. It was set up in 1997 to create a network of journalists around the world to expose global scandals. It was, for example, the ICIJ who first revealed the war contracts in Iraq and Afghanistan that were unfairly awarded to companies that had donated large sums to the presidential campaigns of George W. Bush.\n\nRyle is an Irish-born reporter. He is a calm and softly spoken man. He is the kind of person you can imagine not talking much at a dinner party but saying the one thing everyone remembers. A veteran newshound, he is the embodiment of an investigative journalist, ready to shake every tree and go down every burrow to find the truth, to make sure he gets a story right. He became Director of the ICIJ in 2011.\n\nIn June 2015, Ryle and Obermayer organized a secret meeting on the thirteenth floor of the National Press Club in Washington. Staff from rival news outlets like the _Guardian_ , BBC, _SonntagsZeitung_ and _Le Monde_ gathered around a long conference table. During the meeting, the journalists decided the project needed a code name. Prometheus was agreed upon, after the Titan from Greek mythology who stole the secret of fire from the gods. Before they could get access to the documents, the journalists had to sign a basic non-disclosure agreement. It required them to operate under only two rules. First, to share everything they found, and second, to publish on the same day, at the same time, and Ryle had the power to set the publication date.\n\nAll the leaked documents were stored in a searchable database program named Blacklight. If you wanted information on a specific name, for instance 'Ian Cameron', the database would run a search and in a matter of minutes produce a CSV file with all the matching documents. Some of the clients, however, were harder to identify\u2013Mossack Fonseca addressed some of the more secretive ones by code names like Winnie the Pooh or Harry Potter.\n\nWorking collaboratively, local journalists\u2013from Iceland to Nigeria to Russia\u2013could piece stories together in ways that would have been almost impossible to do from a newsroom in one country. 'While one journalist is looking at Indian data, it might lead them to Brazil, or France, and then suddenly you've got this trust-building exercise where the Indian journalist shares information with the French journalist,' explains Ryle.\n\nA critical member of the ICIJ leadership team was thirty-three-year-old Spanish-born Mar Cabra who heads data and research. Her colleagues describe her as a 'data genius'. Mar distinctly remembers the day a colleague discovered Messi among the client list. 'A message appeared in the virtual newsroom saying, \"Oh my god, I found the football player Lionel Messi!\" The other journalists were like, \"Oh my god, Messi.\"' Once one person had found Messi in the documents, everybody had found Messi.\n\nAfter nine months of painstakingly combing through tens of thousands of documents, the journalists were exhausted and stressed, and champing at the bit. Many times, Ryle had to calm jittery journalists whose natural instinct was to start reporting on the injustices they had discovered. 'It took a lot to keep everyone holding the line,' he says. The journalists were spread out across time zones, working in different languages and based in countries with different social and political environments. Ryle was not technically their boss and he couldn't force them to do anything they didn't want to do. At the end of the day, everyone was legally free to report as they saw fit. Yet, remarkably, the story remained a secret for nearly a year.\n\nRyle knows a thing or two about what makes journalists tick. 'A journalist's biggest weakness, whether they like it or not, is their ego. But on this project, we had an awful lot of selflessness going on,' he says. 'I tried to give them a sense of belonging to something greater than themselves.' The prospect of a bigger and better story if they all worked together was used as a carrot, but Ryle knew he also needed a stick. 'I made it clear that if they broke the trust once, the ICIJ will never work with them again.'\n\nTen minutes before the agreed embargo was due to expire, an unexpected message appeared on Twitter: 'Biggest leak in the history of data journalism just went live, and it's about corruption.' It was tweeted by Edward Snowden. Juliette Garside was sitting in the offices of the _Guardian_ with her bosses and the rest of the Panama Papers team when it came. 'We had our fingers hovering over the button to go live with our story but we were respectfully waiting for the exact embargoed time,' she says. 'Then the Snowden tweet appeared and all hell broke loose. Everyone in the room was shouting \"Go, go, go!\"'\n\nOn Sunday, 3 April 2016, at 8 p.m. German time, dozens of news organizations around the world started publishing front-cover stories about the Panama Papers. Just hours after publication, thousands of people took to the streets of the Icelandic capital of Reykjavik to express their outrage and to call for their prime minister to quit. The papers revealed that Sigmundur Dav\u00ed\u00f0 Gunnlaugsson had a secret offshore tax account called Wintris Inc., in the British Virgin Islands, that he used to hold investments with his wealthy wife. He reportedly avoided more than 500 million Icelandic krona (approximately $4 million) in taxes on the wealth. To make matters worse, he had set up the account in 2007, while Iceland was on the brink of a financial meltdown. Gunnlaugsson resigned shortly after the leak.\n\nBastian Obermayer, the whistle-blower's original contact, was one of the first to discover the links to Vladimir Putin in the files. He repeatedly came across the name Sergei Roldugin, a Russian concert cellist. It turned out that the cellist had been a friend of Putin since they were teenagers and was godfather to the president's daughter, Mariya. It was reported that Roldugin owned a Moscow bank called Bank Rossiya that had allegedly handled billions of dollars in transactions with offshore companies.\n\nIn all, 29 of the billionaires featured in _Forbes Magazine_ 's list of the world's 500 richest people, 12 serving or former world leaders and 140 politicians were named in the Panama Papers. The documents implicated the king of Saudi Arabia; Ian Cameron, the late father of former UK prime minister David Cameron; six members of the House of Lords; Nawaz Sharif, Pakistan's prime minister; Ayad Allawi, former vice president of Iraq; and Petro Poroshenko, president of Ukraine.\n\nIn itself, the appearance of a name in the files was no proof of wrongdoing; there are plenty of legitimate reasons to hold money in offshore companies and trusts. A foreigner, for example, buying a vacation home, may set up a local shell company to purchase the property. Indeed, Mossack Fonseca vigorously denied having broken any laws, as did many of its clients. Even so, the avalanche of information provided damning examples of how the rich and powerful can exploit offshore tax regimes and strip countries of tax revenues that most believe they deserve to pay.\n\nThe public outrage was loud and vehement. Here were the rich finding ways to grow even richer by avoiding taxes the average person must pay. Here were leaders essentially salting away money for which tax should morally be payable for the benefit of the very countries they were governing.\n\nWas this all we could expect from those at the top level of society, those who are meant to serve the public good?\n\nThe feeling of disillusionment that followed the leak wasn't just about money; it was about fairness and equality. Why did the wealthy, powerful and elite get to play by different rules? The revelations left the social contract in tatters: it destroyed the tacit understanding that we all work hard, pay our taxes and are 'in this together'.\n\nIn revealing to the average person what went on behind the scenes, the Panama Papers confirmed one of the key reasons why institutional trust is eroding at an alarming rate; many people feel let down and left behind, watching in dismay as the elites and authorities in charge seem to thrive and act unethically, at the very least. To borrow the title of a book by Joseph Stiglitz, the Nobel Laureate economist, trust is facing a Great Divide.\n\nInstitutions, as the eighteenth-century Italian political philosopher Giambattista Vico observed, are essentially social structures made up of a history of practices, values and laws that are accepted and used by many people. We often think of institutions as something physical\u2013grand university buildings, ancient stone churches, the Houses of Parliament\u2013but they can also be an idea, constraint or a social norm. Marriage, for example, is an institution. So is the family unit or the British monarchy. Religion, property rights or other legal constitutions are institutions. 'The simple point is that institutions are to humans what hives are to bees. They are structures within which we organize ourselves as groups,' writes the historian Niall Ferguson in _The Great Degeneration_. 'You know when you are inside one, just as a bee knows when it is in the hive. Institutions have boundaries, often walls.' In other words, concrete or conceptual, they are valued building blocks of rule and repetition on which societies are built. They shape our behaviour and how we interact with each other.\n\nTrust in institutions bobs and dips with scandals, recessions, wars and changes in government. In the past, it was much easier to hide wrongdoings, such as Tuskegee, for years, even decades. Now, in a digital age, shoddy institutions and the leaders and elites at their helm are much more likely to be exposed and lose our trust quickly, sometimes for good. A deep loss of faith in banks, governments, the media, the church or other elite institutions is not a new phenomenon. Go back to any other civilized age and you'll find examples, such as the Cr\u00e9dit Mobilier Scandal of 1872 where a sham company was used as a front to bribe US congressmen and to funnel money to its construction projects. The scandal became a symbol of post-Civil War corruption. And then, a couple of years later, there was the Whisky Ring, exposed in 1875, involving hundreds of politicians, distillers and distributors who conspired to avoid payment of taxes by reporting lower alcohol sales. Unprecedented, however, is the extent and rate of the breakdown of trust we are now witnessing between citizens and institutions, between the everyman and the elites. Alarmingly, survey after survey of public sentiment across countries and age groups tells a similar, woeful tale.\n\nIn the 1970s, post-Watergate and the Vietnam War, when trust in government and the armed forces had slumped, Gallup began asking Americans how much confidence they had in their major institutions, such as banks, media, public schools, organized religion and Congress. Approximately seven in ten Americans believed they could trust key institutions to do the right thing most of the time.\n\nOver forty years on, the same Gallup survey continues. In 2016, it revealed that the confidence in fourteen institutions averaged at only 32 per cent. Confidence has fallen to historic lows across every single major institution, bar two\u2013small business and the military. When the survey first started, 75 per cent of Americans said they had 'a great deal or a fair amount of' trust and confidence in the federal government in Washington in handling international problems, and 70 per cent had confidence in the government for handling domestic problems. Those figures are now at 49 per cent and 44 respectively. For Congress, the numbers are even worse, plummeting from 42 per cent in 1973 to 9 per cent now. Even the Supreme Court, once a bastion of trust in society, has suffered a major decline\u2013from 45 per cent in 1973 to 36 per cent today. But it's not only governmental organizations where trust has eroded. Public faith has also taken a hit when it comes to banks (60 to 27 per cent); big business (26 to 18 per cent); the church (65 to 41 per cent); and newspapers (39 to 20 per cent).\n\nIn terms of age groups, millennials are the most doubting. According to a 2015 survey conducted by the Harvard University Institute of Politics, 86 per cent of them distrust financial institutions. Three in four millennials 'sometimes or never' trust the federal government to do the right thing and a staggering 88 per cent 'sometimes or never' trust the media.\n\nThe plunge isn't limited to the United States. The story is similar across Western Europe and Britain. Respected pollsters Ipsos MORI have tracked people's trust in twenty-four different occupations in the United Kingdom, from politicians to hairdressers, for more than thirty years. Nurses come out on top as the most trusted, with a stellar rating of 93 per cent. But in this kind of survey, the category to watch is the one with the sharpest decline. It shows how easily once mighty institutions can crash and burn. According to Ipsos, the big loser this time round was the clergy. When the Ipsos poll started in 1983, 85 per cent of the people trusted the clergy to tell the truth. It was the most trusted profession. By January 2016, the clergy had fallen 18 percentage points to come in as only the eighth most-trusted profession overall. Consider this: the average Briton now trusts the random stranger they meet on the bus or in a supermarket to tell the truth more than they trust a member of the clergy on the other side of a confessional.\n\nSo why is trust in so many elite institutions collapsing at the same time? There are three key, somewhat overlapping, reasons: inequality of accountability (certain people are being punished for wrongdoing while others get a leave pass); twilight of elites and authority (the digital age is flattening hierarchies and eroding faith in experts and the rich and powerful); and segregated echo chambers (living in our cultural ghettoes and being deaf to other voices).\n\nFor institutions to retain credibility and our confidence, there must be penalties\u2013loss of power or position, fines\u2013when they break the rules or the rules become meaningless. Take something clear-cut like traffic regulations. In Britain, the legal convention is to drive on the left. If over the course of, say, a week, we saw hundreds of cars veering down the wrong side of the road without any consequences, the power of the traffic rule would quickly dissolve. Similarly, rogues within institutions must be seen to pay or be punished. When they get off scot-free, our faith in the institution is shaken. And in recent years, we've seen many leaders get off scot-free. Consider banks. How did the crash of 2008\u2013the largest man-made economic catastrophe since the Depression\u2013result in the jailing of only a single investment banker and minimal reform of Wall Street? What did that do to our trust?\n\nOver the past couple of decades, the banking industry has had its skirts lifted to reveal some very grubby underwear. From Enron to Arthur Andersen, Freddie Mac to Fannie Mae, Lehman Brothers to Bear Stearns, AIG to Northern Rock, Nick Leeson to Bernie Madoff, the BHS pension funds fiasco to the Libor scandal, the list goes on, and it has taken a hard toll on trust. Perhaps the biggest blow, though, has come from the fact that only a handful of CEOs, the 'captains of finance' who played a role in creating the financial crisis, faced any form of punishment. The few who lost their jobs, most notably Ken Lewis of Bank of America and Dick Fuld, the former CEO of fallen Lehman Brothers, walked out the door with multimillion-dollar golden parachutes. The message is clear: if you are rich and powerful, you can break the rules, as long as it makes a lot of money. It's an acute case of moral hazard; when things went belly up, the bankers didn't face any real consequences.\n\nSome bankers, such as Madoff and Leeson, proved inherently untrustworthy. But for the most part, banks are not filled with bad people; it's more a case that people working in banks are operating in a toxic culture with a perverse incentive structure that permits\u2013and even breeds\u2013unethical behaviour and misaligned interests. They acknowledge as much themselves.\n\nLabaton Sucharow, a respected law firm based in the United States, conducted an independent survey to find out how financial insiders view other professionals within their industry. More than half of respondents believed that their competitors engaged in illegal or unethical behaviour. Nearly a quarter admitted they would engage in insider trading if they could get away with it and just under a third believed that financial services professionals might need to engage in illegal or unethical behaviour to be successful. 'The succession of scandals means it is simply untenable now to argue that the problem is one of a few bad apples,' admits Mark Carney, the governor of the Bank of England. 'The issue is with the barrels in which they are stored.'\n\nRising star Andy Haldane, a Yorkshireman, is currently chief economist at the Bank of England, working alongside Carney. He has become a key figure in the debate on financial regulation calling for 'reformation'. Indeed, in 2014, _TIME_ named Haldane as one of the 100 Most Influential People in the World for being 'the central banker not afraid to be blunt'. 'The significance of these findings is not the precise percentages, as striking as these are,' says Haldane in response to the Labaton findings. 'More fundamentally, it is because of what they reveal about finance's perception of itself, the mirror it holds to the social identity of finance... It is the sociology and psychology of banking and bankers that needs to change, as much as their finances.' The problem is, can you regulate culture?\n\nFaith in the financial institution will not be restored unless the behaviour of banking changes and we see more in the way of serious punishment or penalties for those in the rotten barrel. The systemic breakdown of trust in financial institutions comes down to this application of different rules for different folks. 'Along with the other rising inequalities we've become so familiar with\u2013in income, in wealth, in access to politicians\u2013we confront now a fundamental inequality of accountability,' writes American political commentator and author Christopher Hayes in his fascinating book _Twilight of the Elites_. 'We cannot have a just society that applies the principles of accountability to the powerless and the principle of forgiveness to the powerful.'\n\nWhen it was revealed in 2015 that over 11 million Volkswagen vehicles were knowingly programmed with software, so-called 'defeat devices', which can dupe government emissions tests, CEO Martin Winterkorn resigned. His pay-out post 'dieselgate' was a staggering 15.9 million euro. Likewise, no individuals or institutions have taken a serious hit following the 2016 Sir John Chilcot inquiry, which found that intelligence about weapons of mass destruction (WMDs) had been misrepresented in order to justify the invasion of Iraq following the attacks on the World Trade Center. I distinctly remember, in 2003, watching the news as the 'war on terror' began, bombs lighting up the Baghdad skyline. More than a decade of military interventions followed, unleashing astronomical human destruction and massive financial costs. When the long-awaited Chilcot findings, a 2.6 million-word report, were published in June 2016, the verdict was unequivocal: the legal basis to invade Iraq was 'far from satisfactory'. The report confirmed that the 'intelligence' about Saddam Hussein's alleged WMDs was 'flawed' and exaggerated. The British prime minister's justifications may have destroyed the trust of the British voters but no politicians from the time, including Tony Blair and President Bush, were held truly accountable. It hasn't helped that both those leaders have been defiantly unapologetic.\n\nAs professors Alsan and Wanamaker proved with the Tuskegee Study, specific events can trigger rampant mistrust of entire systems. If politicians can take us into war under false pretences, we start to wonder how we can trust the wider process of decision-making in government. If Chuck Blazer, the ex-member of FIFA's advisory committee, accepted bribes, corruption must be pervasive in other sporting organizations. If we can't trust the behaviour of bankers, the financial system must be broken. If we can't trust journalists to report accurately on the financial crisis, the Iraq War or the presidential race, the press must be failing us as a credible source of facts. If we can't trust the Catholic clergy to report abuse, perhaps it means the leaders of the church are only loyal to the institution, not the people they are meant to serve.\n\nIt's as if the safety net we once relied on\u2013our trust in the wider society and its sterling institutions\u2013has been ripped away and we're in a spiralling trust freefall. It's not only that corrupt individuals are getting away with bad deeds; as the Panama Papers showed, the moral compass at the top of society is also spinning wildly.\n\nIt is easy to see why 'loss of trust' in established institutions has become both a mantra and a real crisis of our times. The problem is further amplified by the fact that many of us are trapped in echo chambers of information shared by 'like-minded' people, where we hear this message over and over.\n\nOn 29 June 2016, Facebook made an announcement about changes it was making to its personal news-feed algorithm. 'We are updating News Feed over the coming weeks so that the things posted by the friends you care about are higher up in your News Feed,' wrote Facebook's engineering director Lars B\u00e4ckstr\u00f6m. The statement sounded fairly innocuous but what it meant was significant; the feed would now promote content posted by friends over content by traditional media outlets.\n\nGiven that an estimated 41.4 per cent or more of traffic to news sites comes from Facebook, the change seems likely to bring about a critical decline in referral and reach for publishers. More significant, though, was the way the new Facebook algorithm represented a profound shift in the diversity of opinions and news we see, the things that challenge us or broaden our worldview.\n\nSociologists describe the innate tendency to associate and connect with people similar to us as _homophily._ These similarities might be dimensions such as ethnicity, age, gender, education, political affiliation, religion and occupation or, say, where we live. We also cluster around niche interests such as whether we like pug dogs, Thai food or playing chess. The internet amplifies homophily, sorting people into online neighbourhoods on social channels like Twitter, Reddit and Facebook. It becomes much easier to find crowds of people who think, live and vote like we do online than offline. It creates loud and polarizing echo chambers with less space for constructive disagreement, debate or enlightenment.\n\nNearly two-thirds of Americans get news on social media, according to a recent survey by the Pew Research Center. What's more, Facebook is the number-one source of news for two in three of its users. That's nearly half of the US population. The algorithm tweak means we are limiting our exposure to opposing perspectives, whether it's on a presidential race, climate change, safety of vaccinations or ISIS. It's hard for alternative viewpoints and contradictory information to break into someone's echo chamber. For the most part, we see ideas and news we are likely to agree with.\n\nIf you were surprised by Trump's presidential victory or the Brexit vote, you may well be living in what Eli Pariser, author and co-founder of Upworthy, pinpointed back in 2011 as the 'filter bubble' effect. 'The rise of naked partisanship, increasing economy and regional stratification, the splintering of our media into a channel for every taste\u2013all this makes this great sorting seem natural, even inevitable,' President Obama said in his farewell speech on the evening of 10 January 2017. 'And increasingly, we become so secure in our bubbles that we accept only information, whether true or not, that fits our opinions, instead of basing our opinions on the evidence that's out there.'\n\nIn the aftermath of the EU referendum, Tom Steinberg, the British internet activist and mySociety founder, provided a powerful illustration of the filter bubble epidemic. 'I am actively searching through Facebook for people celebrating the Brexit leave victory,' he wrote. But to no avail. The algorithm must have assumed he wasn't interested. 'The filter bubble is SO strong, and extends SO far into things like Facebook's custom search that I can't find anyone who is happy despite the fact that over half the country is clearly jubilant today and despite the fact that I'm actively looking to hear what they are saying.'\n\nPeople are more likely to describe 'a person like me' as the most credible source of information. A friend or, say, a Facebook friend, is now viewed as twice as credible as a government leader, according to the Edelman Trust Barometer. 'The mass population is relying less on newspapers and magazines and instead chooses self-affirming online communities,' says Richard Edelman. The Facebook algorithm is proof of a new 'world of self-reference'. Not only do we become victims to our own biases but it is also easier to cherry-pick stories that inflame our outrage. Distrust of institutions breeds more distrust until the fearful meme becomes contagious.\n\nIt is no coincidence that Trump pulled off a victory to become president in 2016 during a crisis of faith in traditional authorities. Here, supposedly, was a Washington 'outsider', who during his campaign promised to 'drain the swamp', clean up the political establishment. The former _Apprentice_ host and impulsive falsehood-circulating tweeter, told voters he would 'shake things up' and do everything differently, from banning Muslims entering the United States to scrapping the Affordable Health Care Act (Obama Care). He promised to be the opposite of the 'very, very stupid people' currently leading America. During his campaign, Trump was consistent about one thing\u2013he would 'Make America Great Again' (the Brexiters had a similar killer slogan: 'Take Back Control').\n\nHis claim to 'tell it like it is' represented an intoxicating form of transparency for many people. 'Those princes who do great things,' wrote Machiavelli, 'have considered keeping their word of little account, and have known how to beguile men's minds by shrewdness and cunning.' In other words, Trump may have lied during his campaign, but as Stephen Colbert, the American talk-show host, explained to his viewers, Trump embodies 'truthiness': ideas which 'feel right' to many people.\n\n'A deep recognition of the slow death of the meritocratic dream underlies the decline of trust in public institutions and the crisis of authority in which we are now mired,' says Christopher Hayes. 'Since people cannot bring themselves to disbelieve in the central premise of the American dream, they focus their ire and scepticism instead on the broken institutions it has formed.' Trump's rise was a product of suffering.\n\nAt the 2016 Democratic Convention, President Obama observed that Hillary Clinton, a former First Lady, Senator and Secretary of State, was the 'most qualified candidate ever' to run for the presidency. The fact that the election was even a contest came down to trust.\n\nI should disclose at the outset that some time ago I worked for the Clintons at their Foundation for almost three years. I respect Senator Clinton, a lot, but she personifies a breed of authority that more and more people are no longer willing to put their faith in. Senator Clinton's vote for the Iraq War; her handling of the Benghazi attack; a murky web of connections to the Foundation; her use of a private email server (and its mysterious destruction); and the revelation that on several occasions she was paid hundreds of thousands of dollars to give speeches to the same Wall Street bankers she promised to regulate\u2013all these made her seem like a typical old-school politician and, fatally, an insider. 'Let's face it: our biggest problem here isn't Trump\u2013it's Hillary. She is hugely unpopular\u2013nearly 70 per cent of all voters think she is untrustworthy and dishonest,' wrote the documentary-maker Michael Moore in an incredibly prescient blog post predicting the win of Trump, twelve months before the election. 'She represents the old way of politics, not really believing in anything other than what can get you elected.'\n\nThe story of Brexit is a similar tale. In a heated interview with Faisal Islam of Sky News on 3 June 2016, Michael Gove, the UK's justice secretary and leader of the campaign to leave the European Union, said, 'I think the people in this country have had enough of experts.' It was a disturbing comment that really stuck in my mind. He also compared ten Nobel Prize-winning economists who signed a letter warning people about leaving the European Union to Nazi scientists loyal to Hitler who denounced physicist Albert Einstein in the 1930s. They were highly controversial points to make but summed up the 'post-truth' world, a term named as the Oxford Dictionaries' word of the year for 2016. 'Relating to or denoting circumstances in which objective facts are less influential in shaping public opinion than appeals to emotion and personal belief.' As Swiss-born British philosopher Alain de Botton tweeted on 15 November 2016, 'New doublespeak dictionary: elite = wretched, educated = dumb, sceptical = whining, regretful = reactionary, expert = idiot.'\n\n'Why did nobody notice it?' the Queen famously asked professors and academics at the London School of Economics during a briefing in 2008 on the turmoil of the financial crisis. Almost a decade later, it's a question being asked of experts in general, from scientists to pollsters to economists. In the run-up to the Brexit referendum, YouGov found more than half of Leave voters trusted neither academics nor economists. Two-thirds of Leave supporters\u2013compared to just a quarter of Remainers\u2013said it was wrong to rely too much on 'experts' and better to rely on the 'ordinary people'. So why don't people trust experts? We need to believe experts are honest, have integrity and the public's best interest at heart. And sometimes we are encouraged not to trust them by vested parties because they tell 'inconvenient truths' about climate change, the economy or, say, tobacco.\n\nBrexit and Trump are the first wave of acute symptoms emerging from one of the biggest trust shifts in history: trust and influence now lie more with individuals than they do with institutions.\n\n'Every act of creation is first an act of destruction,' Picasso famously said. It applies to trust as much as art. As institutional trust collapses, it allows space for new systems to emerge. Technology is enabling trust across huge networks of people, organizations and intelligent machines in ways that are unbundling traditional trust hierarchies. Signs of distributed trust are appearing. Blockchain technologies, for example, which have the potential to create a digital record of history that no single person has the power to erase or change. The blockchain offers a new trust model, one where trust doesn't need to be mediated by a centralized authority such as a government or a bank but where people who might not otherwise trust each other can agree on a single truth or a common record of events.\n\nThe rise of multi-billion-dollar companies such as Airbnb and Uber, whose success depends on trust between strangers, is a clear illustration of how trust can now travel through networks and marketplaces. Tesla may look like a smart car company but in fact distributed trust underpins the grand master plan of Elon Musk, the company founder and CEO. 'You will be able to add your car to the Tesla shared fleet just by tapping a button on the Tesla phone app,' Musk has said. This will allow owners to earn money as their self-driving car picks up passengers while they are at work, on vacation or not using it for whatever reason.\n\nIt's still early days, but we have seen distributed trust powering the rise of crowdfunding sites such as Kickstarter and Patreon; social media platforms; peer-to-peer lending; open-source projects such as GitHub; Massive Open Online Courses (MOOCs); information-sharing hubs such as Stack Overflow and Wikipedia; citizen science projects; and many other person-to-person agreements and decentralized transactions that bypass traditional institutions and middlemen.\n\nIts potential is massive but there's a catch. While distributed trust may sound like a techno-libertarian dream, the flip side is that the same tools that are being used to connect strangers all over the world can also be used in deeply unsettling and nefarious ways. Consider the profound changes in the way information and knowledge reached the public, the dark side of media abundance.\n\nFrom 1962 to 1981, news anchorman Walter Cronkite was a nightly presence in millions of American homes. His kind and even-keeled voice, distinctive trimmed moustache, and calm poise, rich with gravitas, inspired deep confidence with his viewers, more than twenty million a night. Fronting CBS's _Nightly News_ , he guided the country through a tumultuous period of tragedies\u2013nuclear explosions, the civil rights movement, Watergate, the Iran hostage crisis, the Vietnam War and the death of Martin Luther King. Cronkite removing his black-framed glasses and blinking back tears to break the news of the death of John F. Kennedy became one of the defining images of the day. 'Oh boy!' he memorably exclaimed on seeing the Apollo 11 moon landing on 20 July 1969. He often embodied the emotions and thoughts of millions.\n\nIt was an era when the nightly news was a routine event day in, day out, central to many people's lives. Cronkite, who died at the age of ninety-two, was very much an old-fashioned newsman and his appeal was simple\u2013a huge number of Americans respected, liked and trusted him. He earned a reputation as an objective straight-shooter. If Walter said it, it must be true. In 1972, he topped the rankings of the national 'trust index' poll with 73 per cent, making him the most trusted public figure in the country.\n\n'And that's the way it is...' Cronkite would say in his signature sign-off at the end of his broadcast. He would recite the line with humour, irony or sadness depending on the news of the day. It was a kind of definitive sign-off that newscasters couldn't get away with today\u2013and probably never will again.\n\nCronkite benefited from working in a time when people had far more trust in the media and authority. During the nineteen years he was the voice of the national news, there was, obviously, no blogging. Three broadcast networks\u2013CBS, NBC and ABC\u2013enjoyed a monopoly; one that has now been broken into shards by the web and fragmented by millions of news producers and sources fiercely competing for views and clicks. On Facebook, the 'mainstream' news now fights for a piece of our Snapchat-sized attention spans with our friends' posts of birthday photos and pics of what they had for dinner. Cronkite earned the trust of his audiences but he also wasn't broadcasting in a 'post-truth' era. People's faith in the trustworthiness of the media has gone up in flames, a fire partly of its own making but also fanned, you could argue, by those in positions of power or influence who want to discredit a probing media. What's more, there is little consensus on who is telling the truth. The internet has made it harder to sort fact from fiction and easier to undermine the truth.\n\nIt is estimated there are more than 3 million blog posts written in the world per day. One of the largest blog sites is Reddit, co-founded by Steve Huffman in 2005, when he was barely more than a teenager. Reddit is a gigantic collection of more than 853,824 message boards called 'subreddits', which range from the general\u2013Sleep to Music to Politics to Food\u2013to the niche. For example, 'Weird Animals Without Necks', where more than 30,168 neckless-animal enthusiasts post snapshots of, well, animals that appear to be neckless. Reddit also features some of the most misogynistic, racist, anti-Semitic and hateful content on the internet, including the subreddits Pics of Dead Kids, Date Rape, Jail Bait, Beating Women, Watch Niggers Die and Fat People Hate\u2013all of which Reddit eventually banned, but it took more than two years for that to happen. Reddit's claim to fame is that it is a democratic web forum for free speech and niche beliefs; anyone can say almost anything, even if it is outrageous, toxic or false.\n\nHuffman started Reddit with Alexis Ohanian, his college buddy from the University of Virginia. The goal was simple: to become the 'front page of the internet'. The site is designed so that users 'upvote' items they think are valuable and 'downvote' those deemed to be unworthy, meaning the crowd curates what appears on Reddit's front page. It's a powerful meme engine, where stories take root and go viral. Huffman has almost reached his front-page dream\u2013today Reddit has become the seventh most popular website in the United States, with more than a quarter of a billion unique visitors each month. Reddit is huge.\n\n'Spez' is how Huffman, thirty-three, is known on Reddit. He has spent years as a troll on the internet and doesn't try to hide that he can hack sites and is known for fake messaging his friends' girlfriends. In November 2016, he used his skills and editing privileges to tamper secretly with Reddit users' comments in the pro-Trump subreddit \/r\/The_Donald from 'fuck u\/spez' (\/r\/The_Donald became the unofficial online home of the Trump campaign where an AMA\u2013'Ask Me Anything'\u2013thread was used by Trump). Huffman redirected the abuse targeted at him to the moderators of the thread, without, he thought, leaving a trace of his tampering. But quickly realizing it was not a smart move to 'troll the trolls', as he put it, he fixed the comments back to their original state within the hour.\n\nNot soon enough. The deceit, as is often the case on the internet, was rapidly discovered. It could have been viewed as simply a naive move from a young entrepreneur under pressure. Instead, the reaction was loud and intense\u2013the meddling was regarded as akin to censorship. A couple of days after the incident, Huffman posted an apology to the community, called 'TIFU' (an acronym for 'Today I Fucked Up'). 'I am sorry: I am sorry for compromising the trust you all have in Reddit,' he wrote. 'I honestly thought I might find some common ground with that community [the trolls] by meeting them on their level. It did not go as planned.'\n\nSecretly editing another user's threads was, by Reddit standards, a gross ethical violation of power, which eroded the trust of its vast online community. Users felt betrayed that Huffman had abused his gatekeeping privileges. Lucas Schlessinger, a user from Vancouver, wrote: 'Terrible abuse of power. Makes me think what other comments have been edited by Reddit admins.' The issue was not just the act in itself but the possibilities it represented.\n\nOn Change.org thousands of users signed a digital call-to-arms, a petition calling for Huffman to resign as CEO. It seems, however, that Huffman, like the bankers and politicians, won't face any real repercussions for his behaviour. He merely got a slap on the wrist, just a very public one.\n\nHuffman's abuse of power may have been short and contained but it represents something profound. We stand on the threshold of a chaotic and confusing period; a murky grey zone where institutional trust is being systemically undermined and distributed trust, for better or worse, is rising to take its place. As we overturn traditional institutions and old sources of authority, a new era of hyper-individual accountability has to take hold; one where we understand the factors that come into play when traditional gatekeepers, referees, experts and authorities are sidestepped, undermined or removed. It calls for a new kind of vigilance and decision-making. The sheer scale of the changed system presents immense challenges. For example, which assaults on trust do we choose to challenge and pursue? Why Huffman's behaviour, while a million other dubious blogs and trolls pass as good coin? It's a fallacy to believe we can take power out of individual hands. Instead, we need to think more deeply about the consequences of individual acts\u2013and where responsibility ultimately lies.\n\n#\n\n# Strangely Familiar\n\nMy parents, for the most part, didn't believe in cocooning their children in cotton wool. They wanted my brother and me to learn how to navigate our world through first-hand experience, by boldness and trial and error, even if it hurt a little. If we fell from a climbing frame, off a bike, a wall or even a horse, there wasn't a collective gasp. We were one of those families where minor calamities were greeted with a no-nonsense, 'Straight back up.'\n\nBut there were certain things, dangerous things, my parents taught us to avoid. Don't put your fingers in the plug socket; don't answer the front door to people you don't know; don't put plastic bags over your face; don't touch the fire or boiling kettle; don't cross the road before looking both ways; and so on.\n\nI vividly remember the day my mum asked me, 'Do you know what a stranger is?' I was four and a half and about to start kindergarten, about to be out in the world in a new way. We were on our way home from getting my first school uniform, an awful olive green dress with a chequered green and white shirt, complete with green woollen knickers. As we walked down the high street, she made a point of explaining that a stranger is not necessarily a good or bad person, just someone we don't know. There were 'safe strangers' that were okay to trust. For example, police officers, lollipop ladies and firefighters.\n\nAnd then there were others. Strangers in cars. My mother repeatedly explained that if someone stopped a car and told me to get in, I was to run and shout loudly. I was good at that. If someone unexpected was going to pick me up from school, even an aunty or Mum's friend, we had a code word to share, 'green tomato'. Compared to the rest of her somewhat rational parenting, her fear around strangers kidnapping me in their cars bordered on paranoia.\n\nSo there is a beautiful irony that, as an adult, my work focuses on ideas that require trust between strangers, and even strangers in cars. It's a once unthinkable form of trust that has sprung up around the world. And we're only just beginning to understand how it works.\n\nAnshul Shuka is a twenty-nine-year-old doctor who lives and works in Gurgaon, a fast-growing city on the fringes of southwest New Delhi. He regularly takes trips to Jaipur to visit his family and friends. It's a long journey of around 240 kilometres, and depending on the traffic it can take him close to four hours in his dark grey Hyundai. Rather than make the drive alone, Shuka offers his three empty seats to people who want to take a similar journey around the same time. He 'sells' the seats for 600 rupees (approx. \u00a37.00) per passenger, advertising them on BlaBlaCar, the world's largest long-distance ride-sharing platform.\n\nOn his profile, Shuka describes himself as 'BlaBla'. It means he is a good match for other passengers who like some conversation but do not want to chat the entire journey. If he did want to talk non-stop, he would be a 'BlaBlaBla'. If his preference were to drive in silence, he would just be a 'Bla'. His profile shows that he likes listening to music, doesn't smoke and won't bring any dogs or other pets along for the ride. His rating is good, a 4.7\/5. Manisha Vasdey, a twenty-six-year-old female passenger who has shared a ride with Shuka, gave him an 'outstanding' review. 'The ride was very nice and comfortable,' she commented. 'Great conversation from sports to _Game of Thrones_ (which he doesn't watch!) to politics. Would definitely recommend travelling with him.'\n\nThe idea behind BlaBlaCar, a French start-up founded in 2006, is relatively simple: drivers 'sell' the empty seats in their cars on trips they are planning to take but they can only charge prices that cover petrol and road tolls. They are not allowed to make a profit (it's against the rules). It's a win-win; passengers get a relatively cheap ride and drivers offset the costs of their journey. BlaBlaCar sounds like a worthwhile community service. It is, in one sense, but it's also a commercial venture. Indeed, the company was valued at \u00a31.2 billion in 2017. BlaBlaCar makes a nice profit by charging a booking fee, which is about 15 per cent of the cost of the ride.\n\nNotably, BlaBlaCar is different from Uber in its pricing model but it's also not designed to compete with taxis. The average trip taken is long, around 320 kilometres (200 miles), which makes it more of a direct threat to coaches and trains. It is also a long time to spend in a car with someone we have never met before. We could get a back-seat driver telling us what to do on the road, someone with a weird sense of humour or, as Mum still likes to point out, a psycho-killer.\n\nForty-one-year-old co-founder Fr\u00e9d\u00e9ric Mazzella first got the idea of creating a marketplace for empty seats back in December 2003. He was working hard in Paris at the time and had promised his family that he would join them for Christmas. They lived in the Vend\u00e9e region of France, around 420 kilometres southwest of the capital. Busy with his job, Mazzella had left his travel plans to the very last minute. All the trains were fully booked and he didn't own a car. In a fix, he convinced his sister Lucie to make a two-hour detour to pick him up.\n\nOn that wintery night, driving along the motorway, he noticed that most of the cars had only one person\u2013a driver and no passengers. 'I thought, that's crazy. What a waste,' says Mazzella. 'Why don't we put those empty seats in some kind of search engine, making it as easy for passengers to search and book them as it is for available seats on trains?'\n\nTo Mazzella's surprise, no such website existed yet. Yes, people posted random trips on Craigslist, but that was about it. He couldn't get the idea out of his head, although for a long time he was too busy to do anything about it. Finally, three years later, in 2006, he decided the time was right to start a ride-sharing company. Online marketplaces such as Etsy and eBay that match buyers and sellers were taking off, and social networks such as YouTube and Facebook were beginning to gather momentum. If people were starting to share photos, music and daily thoughts, why not seats in cars?\n\nMazzella contacted Francis Nappez, a close friend and programmer. Together they created a website that they initially called CoVoiturage (French for the term 'car-pooling'). The first site was very basic and ugly-looking. It didn't have user profiles and there were no peer reviews or ratings. Drivers simply submitted their emails and phone numbers and advertised their rides in a similar way to online classified ads. People had to contact one another, agree on prices and make necessary arrangements. There was a lot of friction. It took time and effort. And, most importantly, strangers were still strangers.\n\nOn paper, ride-sharing looked like a big opportunity. In France alone, there are approximately 1 billion seats travelling empty between major cities each year, and there are more than 700 million trips taken each year that are in excess of 150 kilometres.\n\nYet the idea did not initially take off as Mazzella had envisioned. Indeed, it would take more than a decade for the platform to get real traction. People were simply not ready to take the trust leap needed to adopt this new way of travelling. That early model had overlooked something vital, something very human: the company had focused on solving the coordination problem of matching drivers and passengers but it had failed to solve the problem of trust.\n\nIn 2007, Mazzella decided to enrol in the MBA programme at the INSEAD Business School. It was there he met Nicolas Brusson, now the COO of the company. They decided to enter CoVoiturage into the INSEAD's business venture competition. Disappointingly, they finished fourth. The biggest question the judges had was around trust\u2013how was this different from a marketplace for hitchhiking?\n\n'The iPhone did not exist. People were only just beginning to grasp the fact that we were entering the digital age. And there I am, pitching my idea of a world where anyone looking to travel could connect with drivers heading the same way,' Mazzella says. 'It took some time before the world was ready for this idea.' He had to figure out a way to get round that lesson drummed into us as children: never get into a car with a stranger.\n\nGetting people who haven't met to trust each other to share a ride is a fiendish problem. It's basically reinventing the hitchhiking experience into one that is paid for, planned and trusted. But the problem does not in fact start with personal trust, which is getting people to trust one another. The problem starts with building _generalized trust_ , which is how we first gain trust in the _idea_ itself.\n\nAnd for many years BlaBlaCar, as it was renamed, took the wrong approach to getting people to adopt the idea. The company tried emphasizing the environmental savings, how good sharing a car was for the planet. But it didn't work. It wasn't a good enough reason to get people to use the service. The next approach was to try to sell the software to companies who could offer car-pooling services for employees. It was a logical idea; employees are not total strangers, the trust is already there. But all the companies wanted different features to suit their specific needs. 'An enormous amount of time, resources and attention was spent on delivering multiple customized platforms. There was no scalable solution,' says Mazzella. 'Over time, it became clear the business-to-business version would not flourish.'\n\nThe turning point came when he recognized a now-obvious problem\u2013users were frequently cancelling reservations without being penalized. 'Passengers would call, like, an hour before or three hours before saying, \"Oh, I'm sorry, I can't come. My grandma is sick.\" We had a percentage of sick grandmothers which was way above average,' Mazzella jokes. Passengers were booking seats online in multiple cars to hedge the risk of a driver cancelling last minute. And drivers compensated for unreliable passengers not showing up by overbooking the spots they had available. There was no mutual commitment and the whole experience was an inefficient mess.\n\nSo BlaBlaCar implemented a solution that in retrospect sounds ridiculously simple. In 2011, they introduced a feature so that people had to pay online and, critically, in advance. At the time of the booking, passengers were charged for the ride. Not only did this remove the social awkwardness of exchanging cash in the car, it created upfront commitment. Cancellation rates fell from 35 per cent to less than 3 per cent. And the service started really to take off. Online payments removed a _trust blocker_ , things that can get in the way or are the deal breakers when it comes to people trusting a new idea or each other.\n\nBlaBlaCar now transports more than 12 million people per quarter across twenty-two countries, as of April 2017. To put that in context, it carries more passengers each month than Eurostar (2.5 million travellers per quarter) or British Airways (10 million passengers per quarter). 'Nothing is as powerful as an idea whose time has come,' says Mazzella, quoting the French novelist, poet and playwright Victor Hugo.\n\nBlaBlaCar is an illustration of how technology is enabling millions of people to take a trust leap in an idea, to do something new or different from the way we have previously done it, regardless of cultural norms.\n\nOver the past decade, I have been researching hundreds of networks, marketplaces and systems that reinvent the way something of value\u2013a product, service or information\u2013reaches many people. There are fascinating nuances in how trust works in these examples and we will explore those nuances throughout the book. But beneath the differences lies a common behavioural pattern people follow in forming trust. I call it 'Climbing the _Trust Stack_ '.\n\nThe trust stack goes like this: first, we have to trust the idea; then the company; and, finally, we have to trust the other person (or in some instances a machine or robot).\n\nLet's use BlaBlaCar for an overview of how it works. On the first level, we have to trust that the _idea_ of ride-sharing is safe and worth trying. There has to be enough understanding and certainty, or reduced uncertainty, to make us willing to try the idea. The next stage is about having confidence in the platform and company. In this instance, it's knowing that BlaBlaCar will remove bad apples before the ride and help us out if something does go wrong. The third and final stage sees us using different bits of information to decide whether the other person is trustworthy. It's this last level where the real trust happens. But we can't get there without going through the other two stages.\n\nThe first time we climb the trust stack it feels a bit weird, even risky. But we get to a point where these new ideas seem not only normal but necessary. We're comfortable to make the trust leap and, once made, our future behaviours change, often quite quickly.\n\nWhat coaxes us into trusting new ideas? It is trust, after all, that influences how far, how fast and how permanently a new idea will spread.\n\nTrust in new inventions doesn't happen by accident. There are some universal psychological and emotional hurdles to overcome first. The conditions that enable that to happen are summed up in three key notions: the California Roll principle, the WIIFM factor ('what's in it for me') and _trust influencers_.\n\nThe first depends on making the unfamiliar more familiar. Consider sushi. The concept of sushi was introduced into the United States during the late 1960s, a period of whirlwind change in tastes\u2013entertainment, music, fashion and food. At first, the idea of sushi did not bite. Keep in mind that the average family at the time was sitting down to a dinner of cuts of meats with sides of mashed potatoes swimming in gravy. The thought of eating raw fish was bewildering, even dangerous, in the minds of most restaurant-goers. And then a chef by the name of Ichiro Mashita, who ran Tokyo Kaikan, a small sushi bar in downtown Los Angeles, had a clever idea. He asked, 'What would happen if the strange ingredients were combined with familiar ingredients such as cucumber, crabmeat and avocado?' Mashita also realized that Americans preferred seeing the rice on the outside and seaweed paper in the interior. In other words, the roll would feel more familiar if it were made 'inside-out'.\n\nDemand exploded. The California Roll was a gateway for many people to discover Japanese cuisine. Americans now consume $2.25-billion-worth of sushi annually. As Nir Eyal, the author of _Hooked_ , writes, 'The lesson of the California Roll is simple\u2013people don't want something truly new, they want the familiar done differently.'\n\nThe California Roll principle is based on the underlying rule of combining something new with something familiar to make it 'strangely familiar'. It's a phenomenon that psychologists like Robert B. Zajonc have labelled the 'mere-exposure effect' or the 'Law of Familiarity'. Humans, understandably, have a tendency to be more comfortable around people or things they are familiar with. There is more than one way to build on this.\n\nApple does it through a design feature Steve Jobs called 'skeuomorphism'. It's a catch-all term for when design cues are taken from common objects or elements in the physical world. The iPhone calendar resembles a physical calendar. The notes app looks like a yellow legal pad. The rubbish bin on the first Mac was exactly like a metal bin. The podcast app when first launched looked like an ancient reel-to-reel tape and iBooks looked like a real bookshelf with wood veneers. The familiar elements are not necessary for the new features to function but they tap into our memory bank. Their role is to enable our brains, in a split second, to grasp things they have never experienced before. Jony Ive, the legendary Apple designer, describes the goal as 'to build things that are strangely familiar'.\n\nWhen I was at Oxford doing my undergraduate degree in Fine Art, I had to take a course in modern philosophy and critical theory. We studied the likes of Descartes, Voltaire and Rousseau. The lectures would be held in a grand hall in an old Oxford building. There was lots of dark wood panelling, with enormous portrait oil paintings hung on the walls of mostly elderly men. Our professors would tell us things like, 'Art is a form of thinking that addresses the thoughtfulness of life.' I honestly thought these lectures were pompous and painful. In fact, I still have dreams about having to sit through them. At the time, I didn't know how to think in abstract ways. But there was one class that I found fascinating: Immanuel Kant. 'Human reason is by nature architectonic,' Kant wrote in _The Critique of Pure Reason_ \u2013i.e. our thoughts follow a clearly organized and defined structure. Well before Steve Jobs came up with the idea, Kant believed that people need some kind of system or familiar schema to pave the way to understanding something new.\n\nPerhaps you have stood looking at a piece of art and thought, 'What on earth is this?' I remember standing in the Tate Modern some time ago in front of a pile of 120 bricks arranged in a rectangle; bricks that would otherwise be used to make an ordinary house. The piece, by the sculptor Carl Andre, was called _Equivalent VIII_. Andre is renowned for making 'idea art' that is about 'recognizable things'. But what was the idea in bricks? I felt lost. So what did I do? I just walked away, on to the next piece. The same thing happens with new inventions and new experiences we just don't get. We move on.\n\nIt was only recently I realized that that is exactly how I think about trust. Specifically, for us to trust a new idea, we need bridges that are easy to find and to cross. The unknown needs to be reduced just enough that our mind goes, 'I get this. It's kind of like...' We have to turn _Equivalent VIII_ into 'Destruction of the Berlin Wall'. We have to put the strange seaweed on the inside and rice on the outside.\n\nLet me give you an example of a company that knows how to create bridges that allow people to step lightly from the known to the unknown. That company is Airbnb.\n\nJudd Antin's official title is director of research at Airbnb. His job is to get inside the heads of guests and hosts using the accommodation platform. He wants to find out what they really think and what _really_ happens. If you wanted to know, say, all the strange and unexpected ways hosts get their homes ready for guests, Antin's experience research team could tell you. Like the host who decided they did not want their guests to use a specific toilet in one of their bathrooms, so they placed a giant prickly cactus in a heavy concrete pot on the toilet seat.\n\nAntin, thirty-eight, is a social psychologist with a PhD in information management and systems from the University of California, Berkeley. Much of his research has been on why people behave in certain ways in different online environments. For instance, how do people cope with managing multiple digital identities? Do women and men edit differently on Wikipedia? (Turns out they do.) Antin gives you the sense that when he gets interested in something, he gets a little obsessed. He is the type of person who would say no question is a dumb question, but perhaps you could ask the question differently. In fact, he said that to me a few times when we spoke over Skype.\n\nHe never talks about the people he studies as 'subjects' or 'participants'. He doesn't even use the word 'users'. It is always 'guests' and 'hosts'. He doesn't talk about 'spaces' but 'homes'. The word he uses to describe his research is 'sexy'. 'When I say sexy, I mean that the topic of our research is really meaty. The idea of letting another person into your home has lots of dimensions that you could spend years diving into,' Antin explains. 'It's research that requires deeply understanding a problem and figuring out how to turn the findings into design, product and communication solutions.'\n\nAntin's team must understand what makes people comfortable with trusting the idea of Airbnb. When guests go to the Airbnb site for the first time, they know they want to go on holiday or simply need a place to stay. But there are lots of 'what ifs' and unknowns before they get started. The first and most basic question is, 'What is the idea of home-sharing?'\n\nEven investors had trouble getting their heads around the idea initially. In fact, many laughed the founders out of the room and dismissed it as dangerous. Take Chris Sacca, an accomplished entrepreneur and the head of Lowercase Capital, one of the leading venture funds in the United States. Sacca spots things early. I mean, really early. He has been one of the first to back the likes of Twitter, Uber and Instagram. He was one of the first people to whom Brian Chesky, Nathan Blecharczyk and Joe Gebbia, the three founders, showed the original pages of Airbnb to in 2008. But he passed on making a seed investment.\n\n'I pulled them aside and said, guys, this is super dangerous,' Sacca said. 'You're renting out a room in somebody's house while they're still there? Somebody's going to get raped or murdered, and the blood is gonna be on your hands. There's no way this'll succeed.'\n\nInvestors will run through 'what if' scenarios when assessing a deal. Specifically, 'What's the worst that can happen when using this product or service?' In the case of Airbnb, a lot can happen and has since happened.\n\nSacca is not the only investor who initially spurned Airbnb and missed out. I met the founders in 2009 when I was writing my first book on the so-called 'sharing economy'. I came home and told my husband, Chris, we should make an early investment. I explained in immense detail how people were going to take pictures of their homes\u2013bedrooms, kitchens and even their bathrooms\u2013and then guests from all over the world would book these places. He is used to me continually sharing and sounding ideas off him. But this time, he looked at me strangely, as if I had lost the plot. I went on trying to convince him: 'eBay is a multi-billion-dollar business where strangers trade all kinds of stuff including second-hand cars online. At the time, people thought Pierre Omidyar was also crazy and it wouldn't work,' I insisted. He did not budge, so I went with my overused line: 'The future is created by optimists, not pessimists.'\n\nChris is a barrister and therefore a master at arguing his point. 'Well, this is not eBay. People are not trading goods anonymously online. This is people's homes. This is people meeting up in the real world,' was his comeback. He was right and very, very wrong. Airbnb is now the second most valuable hospitality brand in the world, estimated to be worth $31 billion. I keep the valuation chart of Airbnb on our fridge. It has scribbled at the top, 'Always listen to your wife!'\n\nIn retrospect, it is easy to point to the likes of Sacca and my husband and say how wrong they were. But, as I say, it's also easy to see why they thought the way they did. Airbnb's success rests on having been able to get people to make that staggering leap of trust and overcome our natural 'stranger danger' bias.\n\nTry this quick experiment yourself. The next time you are sitting next to someone you don't know, ask them to swap phones with you for just one minute. Explain that you will hold their phone and they can hold yours. Tell them: 'What you decide to do with it is your choice.'\n\nI have played this game many different times with different groups of people, from financial advisors to students to estate agents. I have tested it at dinner parties, conference events and in the classroom. The reactions are predictable. Some people outright refuse. People laugh nervously. People hesitantly take the phone but place it face down. People ask how long is left to go. A few launch straight into the experiment, looking at messages, photos and Twitter feeds. Some even tweet or post on Instagram. But for the most part, it feels very uncomfortable for participants. And that is just holding another person's phone for less than a minute.\n\nAirbnb has had to create trust around an exchange that involves one of the most intimate things in our lives, the place where we rest our heads. 'We have to enable Olympic levels of trust between people who have never met,' Antin says. And that's what makes his job so fascinating and challenging. 'As big as we are, most people will have no idea and many more people will have a vague idea of what Airbnb is, but don't really get it,' Antin told me. The first thing that will run through their minds is, 'What on earth is this anyway?'\n\nWhen some people go to the site for the first time, they even wonder if Airbnb owns all the homes they see on the website and the company simply rents them out. 'It sounds silly but it's not; it's a known model of holidaying they understand,' says Antin.\n\nA lot of research has been done on how people 'get' the concept. Noticeably, there are no 'How does Airbnb work?' videos on the homepage. Admittedly, some people look at explicit things such as the 'About' or 'Trust and Safety' pages. But those are listed right at the bottom of the page. Front and centre, the first thing we see on Airbnb is a simple question, intentionally curious: 'Where?'\n\n'One of the ways people get the concept is by relating to something they understand,' Antin told me. 'What we observe when new guests come to the site is that they don't typically go to the educational materials. Either they don't see them or they simply don't resonate. Instead, they go straight to the search box and they search for places in their hometown because it's a place they know, right?'\n\nFor example, when a first-time guest living in London wants to stay in New York, they might not search for New York but for London instead. And then they get even closer to home and search their borough, say, Camden. 'They look at the map of results and the guest's reaction is, \"Oh, oh, I see. This is somebody whose house is just near mine, over there by the river, and you could stay there if you wanted to. Now I get it,\"' explains Antin. 'That's the \"ah-huh\" moment.' Critically, Airbnb has designed the site in a way to encourage this behaviour. It could have had a drop-down menu where guests select a destination from a list. But that would be overly prescriptive. It wouldn't allow new users easily to discover something they can understand; that other nearby homes, like theirs, are available for rent.\n\nIn other words, we trust what we know but we can also trust what we think we know: ideas that are in fact quite new but appear strangely familiar.\n\nFamiliarity is not the only thing that matters in convincing us to trust a new idea. Once we are over the 'I get this' hump\u2013the California Roll principle\u2013the next barrier to be crossed is the 'What's in it for me?' (WIIFM) factor.\n\nOn 14 May 1796, Edward Jenner, an English doctor, carried out an experiment on eight-year-old James Phipps, the son of his gardener. After making two small scratches on the boy's arms, Jenner rubbed in a small amount of fluid from a cowpox blister. Young Phipps got the expected reaction of a slight fever but within a few days he had fully recovered. Two months later, Jenner inoculated the boy again. This time he used matter from a smallpox lesion. As the doctor predicted, and no doubt to his relief, the boy did not contract smallpox. Further tests showed that James was immune to the disease. This is how the concept of vaccinations, named after the Latin _vacca_ for cow, was born. So where did Jenner get the idea?\n\nJenner had received his training in the Gloucestershire countryside where most of his patients worked on farms with cattle. While still a medical student, Jenner observed that milkmaids who contracted cowpox, a relatively mild disease, did not catch smallpox, one of the deadliest infectious diseases of the period, claiming the lives of millions of people including five reigning monarchs. A cure was urgently needed.\n\nIn 1797, Jenner sent a short paper to the Royal Society describing the findings of his experiment. The paper was rejected. Scorned by his peers, Jenner published the book himself, documenting his theory that cowpox did indeed protect against smallpox. His ideas, however, were met with widespread controversy and criticism. The reasons varied from religious to scientific and political objections. The clergy claimed it was repulsive and 'unchristian' to inoculate someone with material originating from an infected cow, one of God's lowlier creatures. Some physicians dismissed his research as 'unethical'. Others selfishly did not want Jenner to succeed because they were making significant sums of money selling different kinds of drugs to treat the disease. In 1802, a satirical cartoon was published called 'The Cow-Pock\u2013or\u2013the Wonderful Effects of the New Inoculation!', summing up the sentiment of the time. It caricatured a scene of the vaccine being administered to a group of petrified young people and cows sprouting from different parts of their bodies.\n\nJenner was told that his ideas were too revolutionary and that he needed more proof. Undaunted by the ridicule, he experimented on several other children, including his own eleven-month-old son, by placing a small amount of cow pox scab into the human skin. Every child he injected showed immunity to the disease.\n\nThe young country doctor did not discover the idea of immunization\u2013it had a long history in China and Africa\u2013but he was the first person scientifically to attempt to control an infectious disease by the deliberate use of vaccination. Jenner later became known as the 'Father of Immunology'. In 1853, thirty years after Jenner's death, the British government made cowpox vaccinations compulsory in England and Wales. On 8 May 1980, the World Health Assembly declared that the world was free of smallpox. But for decades Jenner's new methods did not catch on. Why? People were not ready to take a trust leap in the idea of vaccinations because they didn't fully understand the risks and benefits\u2013they couldn't see what was in it for them.\n\nAnd it's a doubt that persists today. The anti-vaccination movement has existed since Jenner first discovered the cure, but it gained resurgence during the late 1990s after a series of events stoked the fear and doubt already brewing. One was an article that appeared in the UK medical journal _The Lancet_ in 1998, written by Dr Andrew Wakefield and his colleagues, implying a possible link between the measles, mumps and rubella (MMR) vaccine and autism. Remarkably, Wakefield wrote the paper based on research he had done on a mere twelve children. The children in the study were carefully selected and the research was largely funded by lawyers acting for parents who were involved in lawsuits against vaccine manufacturers. The article has since been discredited and retracted but the damage was already done. Dr Sharon Kaufman, the chair of the Department of Social Medicine at the University of California, has done extensive research focused on individuals' trust\u2013and mistrust\u2013in the findings of medical knowledge. 'Most parents are not anti-vaccination _per se_ ,' she says. 'Rather, they live in a time, as we all do, of heightened risk awareness, mistrust of government institutions and the pharmaceutical industry and a great many opportunities for \"seeing\" doubt.' Even if people don't believe that vaccines cause harmful side-effects such as autism, the power of anti-vaccination stories is they evoke fearful emotions that undermine our trust, no matter what we might rationally think.\n\nJames Samuel Coleman, a famous American sociologist born in 1926, was, among other things, fascinated by how we make decisions. Specifically, by the ways people decide whether or not to trust a new idea. He was living in a time when many great new technologies, from the first television and video recorder to the first commercial passenger jet plane, were introduced. Essentially, his research showed that we decide whether to trust based on assessment of the upsides and downsides. We make a calculation about whether trusting this idea will in some way make our lives better or not.\n\nIt sounds obvious, right? But it raises a critical point\u2013we don't _want_ to use a new invention until we _understand_ it. That doesn't mean we all need to understand precisely how a technology functions, whether it was a combustion engine back then or a blockchain now. However, we do need to grasp what it can do and what it can give us. Until that chasm is crossed, we won't abandon what we already have.\n\nCreating trust in a television is one thing. What about creating it in something that has the power to hurt or even kill us? We're seeing this play out now with self-driving cars. Autonomous vehicles are expected by some engineering groups to account for up to 75 per cent of vehicles on the road by 2040. But how do you get people to trust machines enough to let them take over the wheel? An expert who knows the issue first-hand is Dr Brian Lathrop.\n\nLathrop has worked in the Electronics Research Lab at Volkswagen (VW) since 2004. He has a PhD in cognitive psychology, specializing in human interface design, and is the person in charge of research and development of VW's autonomous vehicles.\n\nPrior to his current role at the automaker, he worked at NASA during the period when pilots were first adopting plane automation. He became fascinated by a rather worrying state he calls 'mode confusion'. 'This happens when pilots wonder if the aeroplane is doing the flying or if they are,' Lathrop explains to me. 'Autonomous vehicles face exactly the same challenge.'\n\nDuring our conversation, Lathrop speaks surprisingly openly about the challenges of self-driving vehicles. He has a natural exuberance but you wouldn't describe him as an evangelist. More like an optimistic realist.\n\nIn March 2016, the American Automobile Association (AAA) conducted an extensive survey to understand how much trust its members had in self-driving cars. Three out of four drivers in the United States said they would feel 'afraid' to ride in self-driving cars. Only one in five said they would trust a driverless vehicle to drive itself with them inside. The reasons people gave included: 'trusting their own driving skills more than the technology' (84 per cent); 'feeling the technology is too new and unproven' (60 per cent); and 'not knowing enough about the technology' (50 per cent).\n\nThere is a YouTube video of a seventy-year-old grandmother, named Shirley, freaking out the first time she is inside a Tesla Model S on autopilot. As the car moves across lanes and navigates traffic, Shirley screams, 'Oh no, there is a car coming! Oh dear Jesus. Ah, ah, ooh, ooh, where is it going?' It is painful to watch. In the video, you can hear her son Bill laughing at her reaction after he turns on the autopilot. 'Oh my, I am about to die,' his mum cries out. She looks like she is on the verge of heart failure. In fact, all Shirley needs to do to take control from the machine is simply touch the steering wheel. But her mind is elsewhere.\n\nWhen I ask Lathrop how hard it is to get people to trust being driven by autonomous vehicles, I assume he will go into details of how smart design can overcome the fears of people like Shirley. I expect him to cite the safety stats. In fact, he makes a quite different observation. 'People trust the car quickly, almost too easily,' he says.\n\nYes, Lathrop has seen how some people, such as Shirley, freak out the first time they are driven by an autonomous car. They are, however, the minority. Others are completely awestruck and think, 'Wow, it's doing the driving for me.' But then something interesting happens. After a few miles\u2013around twenty minutes of being driven by the car\u2013the experience feels normal, even boring. Being driven by an intelligent machine it turns out is just not that exciting. That's why Lathrop is worried about people nodding off.\n\n'What do you do when people fall asleep in a self-driving car?' is his biggest concern. 'How do you deal with that?' He thinks it's a question that is not being asked enough. 'It's not good for the trust equation!' Lathrop admits half jokingly.\n\nThe idea of people trusting the car too easily is intriguing so I press Lathrop harder. Surely, the first step in getting people to use self-driving cars has to require a massive trust leap? 'Think about it,' he replies. 'Most people are comfortable being a passenger. They are used to being driven by a colleague or by a friend.' In other words, the idea of being a passenger in a self-driving car taps into something familiar. 'The leap we are asking people to take is not a new experience, but it's asking people to trust a machine versus a human to drive. It's not that huge.' It is strangely familiar.\n\nLathrop admits that before people get inside an autonomous vehicle, they do have questions. Quite predictable ones. 'What if a vehicle cuts in front; will the car respond?' 'Can the vehicle change lanes?' And the most common is, 'Can it drive as well as me?' Lathrop points out that very few of us ask about unpredictable situations. For instance, what happens if a deer or even a dog runs out in front of the car? How does the car navigate parking lots at busy sports matches? The researchers have not yet figured out the answers to all these questions. Some are hard to simulate safely or even predict. It's those unforeseen, or as Lathrop puts it, '1 per cent corner cases', that his team needs to solve for. But the point is, very quickly a first-time user runs out of questions. They just get into the vehicle and let it do the driving.\n\nThe ultimate success of self-driving cars\u2013that it becomes normal to use one\u2013doesn't depend on engineering success. It doesn't even depend on us understanding how the technology works. It depends on that second principle of getting people to trust an idea, the WIIFM factor. We want to know what we will gain. In this instance, will the benefits of the machine doing the job of a human outweigh the risks?\n\nThe typical American commuter spends on average more than fifty-two minutes per day stuck in traffic. That adds up to more than 4 billion hours of wasted time in the United States alone, time we could use in better ways. 'People want to know what they will be able to do if they are freed from driving the car,' Lathrop tells me. They imagine being able to watch movies, talk on the phone, work and eat. 'Look around you when you're at the traffic lights, when cars are stopped. These behaviours are not new. I want to allow people to do what they are already doing but in a much safer manner,' he says. Indeed, the biggest benefit of self-driving cars is safety.\n\nOne hour after passing my driving test as a teenager, I crashed my car. And it wasn't a minor ding. I was waiting at the intersection of a busy crossroads in Hampstead Garden Suburb, north London, looking for a break in the traffic. I started fiddling with the radio, switching from Capital to Virgin and other stations, trying to find a song I liked. Distracted, I decided foolishly to make a go for it. Smash. You never forget the sound of metal hitting metal. My car spun and landed in what was previously a nicely groomed privet hedge on the other side of the road. Over the course of the next five years, I had three subsequent accidents. I am truly a terrible driver. And my experience is a reminder of why self-driving cars matter.\n\nHuman error and inconsistent driving cause more than 90 per cent of crashes, which kill more than 1.2 million people annually, according to the World Health Organization. It is estimated that driverless cars could, by mid-century, reduce traffic fatalities by up to 90 per cent. It works out to be more than 300,000 lives saved each decade in the United States alone, and a saving of $190 billion each year in healthcare costs associated with accidents. In the UK, KPMG estimates that self-driving cars will lead to 2,500 fewer deaths between 2014 and 2030. But we are still highly sceptical of the benefits. 'No one is going to want to realize autonomous driving into the world until there's proof that it's much safer, like a factor of 100 times safer, than having a human drive,' Andrew Moore, the computer science dean at Carnegie Mellon University told _The Atlantic_.\n\nHumans are not 'risk mutual' people. In other words, they don't place the same weight on things going well as things going badly. For example, say I lost a favourite navy blue chequered coat, I would feel more about the loss than I would about the joy of finding another coat, even if the jackets were identical. That's the human tendency, to feel more strongly about a loss than a gain. The basic idea of 'loss aversion' is a concept first discovered by Daniel Kahneman and Amos Tversky, the Israeli psychologists who changed how people think about how people think. Sticking with the status quo feels much safer and better, even if we know we might compensate a loss.\n\nUndisputedly, we love new things, but those things tend to be an upgrade or improvement on ideas we are already comfortable with. For instance, the high-definition wireless flat screen is better than the clunky colour television box with wires and remote controls. It is easy to trade up. To date, the media has not done much to help foster the idea that self-driving cars are safer than humans. In fact, it has tended to fuel the opposite image: the car could kill you.\n\nOn 7 May 2016, Joshua Brown's Tesla Model S vehicle crunched into the side of an eighteen-wheeler trailer truck. Forty-year-old Brown was killed. The car was on autopilot during the collision. When police arrived at the scene, a _Harry Potter_ movie was reportedly playing on a portable DVD player inside the car. In a blog post published on the day the accident became public, Tesla stated, 'Neither Autopilot nor the driver noticed the white side of the tractor trailer against a brightly lit sky, so the brake was not applied.' In other words, the car is not perfect. And the driver was distracted by a young wizard.\n\n'The media love these stories. They sensationalize them. I understand why,' Lathrop tells me. 'But they are extremely rare.' Indeed, this was the first known autopilot death in roughly 130 million miles driven by Tesla customers. To put this in perspective, among all vehicles in the United States, it is estimated there will be a fatality every 94 million miles. In other words, self-driving cars are 36 million miles safer.\n\nLathrop thinks a lot of the messaging around self-driving cars is not useful. 'We need to temper expectations that an autonomous vehicle can be perfect,' he says. 'We need people to realize that the benefits hugely outweigh the negatives.'\n\nWhat's interesting in Lathrop's observation is that it illustrates how building trust doesn't have to hinge on a promise of perfection. In fact, guaranteeing 100 per cent certainty of an outcome is a recipe for disaster.\n\nWhether it's deciding to use Airbnb rather than a well-known hotel brand such as Marriott or Hilton, or deciding to put our trust in an intelligent vehicle, the pros and cons we draw up will circle around the same dimensions\u2013value and certainty. Whatever the idea, the questions are essentially the same: Will these experiences create value in my life? How can I be sure of that value?\n\nYou've no doubt heard the term 'early adopter'. It typically refers to an individual (or business) who uses a new product or technology before others do. My brother-in-law is one. He seems to be in on the next big thing way before it's big. 'Found it first' is a badge he likes to wear. He tells you about new things you should be trying: digital wallets, the Aire digestive tracker that helps you figure out which foods are most compatible with your body, the Nest Learning Thermostat and so on. Early adopters tend to seek novel things to tinker with, and then become evangelists for them. They tend to have strong and excited opinions (and I don't mean that in a bad way). Let's just say they know a lot without being a know-it-all. Without a doubt, early adopters are critical to innovations taking off. Interestingly, however, as a group they are not necessarily the most influential in terms of getting the late majority, or laggards, to climb the first layer of the trust stack. What is needed is that third element\u2013made up of what I call the trust influencers.\n\nTrust influencers are groups of people who can disproportionately influence a significant change in the way we do something; they set new social norms.\n\nTrust influencers are out there for every idea but sometimes we have to look hard and in unexpected places to find them.\n\nThere is a wonderful example of trust influencers relating to what is not a particularly exciting concept\u2013transferring money overseas.\n\nBehind almost every great start-up lies a story of personal frustration. The tale of TransferWise goes like this: Taavet Hinrikus was born in Estonia when it was still part of the USSR. It was a tough environment to grow up in. 'One really had to take charge and solve problems in creative ways to get anything done,' Hinrikus says. In 2002, when he was twenty years old, he met two budding entrepreneurs called Niklas Zennstr\u00f6m and Janus Friis. They were tinkering around with an idea: what if people could digitally transfer their voices and words to one another? Like a phone but without a phone. And without the phone bill. This was exactly the description first used for what would become known as Skype. Hinrikus became one of the first employees on the team.\n\nSkype grew fast. In 2006, the company needed Hinrikus to move from Estonia to London to help them expand. He was earning euros that were paid into his Estonian bank account. Every fortnight Hinrikus had to transfer money to a UK account, to cover his rent, food and other costs. When his money eventually arrived, it was much less than he expected. The whole experience, he thought, was unnecessarily painful.\n\nAn old friend, Kristo K\u00e4\u00e4rmann, faced the same problem but in the opposite direction. He was working as a consultant at Deloitte and was being paid in pounds. He was transferring money back to Estonia to pay his mortgage. 'I was losing five per cent of the money each time I moved it,' K\u00e4\u00e4rmann says. The money was being lost in the hidden fees and poor foreign exchange rates that earn the banks a hefty profit.\n\nHinrikus and K\u00e4\u00e4rmann came up with a simple but clever idea. 'We figured we could just \"swap\" the money. I could just transfer money from my Estonia account into his Estonia account and he transfer money from his account in London to my account in London,' says Hinrikus. 'Pretty quickly we saved thousands by eliminating unfair rates and banking fees.' The friends realized they had hit on an enormous opportunity. TransferWise, the company they founded in London in 2011, is as of March 2017 valued at more than $1.1 billion and already has a 5 per cent market share of international money transfer in the United Kingdom.\n\nTransferWise is based on a peer-to-peer technology system that matches money flows. Say I wanted to send \u00a31,000 from a bank in London to a bank in Paris, the system looks for someone else wanting to convert euros to pounds. The result is that the money never moves between countries. Doing things this way means the process is faster, easier and cheaper than the transfer services we get from banks. Which brings us back to the question of trust and changing behaviour.\n\nThe conventional way to transfer money from one country to another is to use a traditional bank or a post office or a known brand such as Western Union. In 2015, in excess of $601 billion was transferred in this way. So what would persuade ordinary people to trust an unknown digital start-up to transfer their money? The answer: seeing unexpected users putting their faith in this new way of doing things. People who make us think, 'Hey, maybe this idea isn't so risky after all.' But just who would TransferWise's trust influencers be?\n\nHinrikus and his team realized their ideal trust influencers were neither fintech know-it-alls, nor the people with the latest Apple Watch. Far from it. They had to find users we wouldn't necessarily expect to take a risk with an unknown company like TransferWise. Namely, pensioners. Retired British people living abroad in Spain, for instance, who needed to get their pension regularly transferred from pounds to euros. 'For them, the fees they were being charged represented a big portion of their total, so they had a strong incentive to take the same leap of faith,' Hinrikus says.\n\nWhen other first-time users heard about pensioners giving TransferWise the thumbs-up, it had enormous influence on shaping their decision to trust the idea. Indeed, when enough trust influencers are seen to have made the trust leap and survived, millions will follow, often very quickly. That's how change spreads.\n\nJames Surowiecki wrote brilliantly about the influence of group example in his book _The Wisdom of Crowds._ It was largely based on the science of crowd persuasion: how groups of people can influence other individuals to say 'yes'. Some of the core ideas were grounded in Professor Robert Cialdini's theory of 'social proof'. 'If a lot of people are doing the same thing, they must know something we don't. Especially when we are uncertain, we are willing to place an enormous amount of trust in the collective knowledge of the crowd,' Cialdini wrote. Put simply, we tend to follow the lead of other people, especially when we are unsure.\n\nVarious experiments have shown the different dimensions of social proof. One of the most visual is known as the 'Street Corner Experiment'. It was designed in 1968 by social psychologists Stanley Milgram, Leonard Bickman and Lawrence Berkowitz. First, the researchers put a single person on a street corner and had him look at the empty sky for sixty seconds. Only a fraction of passers-by stopped to see what the person was looking at. So the next day, they put five people staring at the sky on the same street corner. Four times as many people stopped to check what they were gazing at. And when the researchers put fifteen people to stare at the empty sky, 45 per cent of all passers-by stopped and tilted their heads to see what the others were looking at. They stopped traffic by staring at an empty sky. The typical takeaway from this study, as Surowiecki puts it, is 'the crowd becomes more influential as it becomes bigger'.\n\nThere's no doubt social proof builds trust around new ideas, especially when we are uncertain of the outcomes. That is why we commonly see sites boast the number of reviews or users they have. For instance, TransferWise has on its homepage, front and centre, their number of 'happy reviews' (35,000), and its number of customers (more than 1 million). Indeed, we seem to be living in a world where it is perceived that the critical way to persuade us is through large numbers, be it Facebook 'likes', five-star ratings or Twitter and Instagram followers.\n\nBut social proof, and the trust it fosters, does not have to come from large crowds. It can also come from a small group of individuals with a unique power to influence. They do not need to have an impressive title, be a celebrity or even a credible 'expert'. They do not need large followings. They do not even have to be people who are similar to the majority, the crowd. They can be just like the British pensioners in Spain, people who can change other people's perceived uncertainty because they seem the least likely to take a trust leap.\n\nThe three ideas covered in this chapter\u2013the California Roll principle, the WIIFM factor and trust influencers\u2013aka 'What is it?', 'What do I gain?', 'Who else is doing it?'\u2013offer a way to see how an idea once dismissed as preposterous can turn into something strangely familiar. They explain how trust in new ideas spreads. Anyone who has ever built trust in a venture, a new product or an idea has had to go through that process, whether they're aware of it or not.\n\nIt's as if there's a daunting rock face the creators of the new idea are asking people to climb. First, they have to show the would-be climbers some familiar moves and handholds, to reduce the unknowns enough to encourage that first step. (And while they've reduced uncertainty, they haven't promised perfection\u2013there's still risk.) They have to explain what climbing the rock face has to offer. Finally, they have to point out the other climbers above them who are loving the experience. Before long, the doubters find themselves racing up the climb, leaving the ground so far behind it's soon just a distant memory.\n\nAnd that process is powerful. It can turn an idea once dismissed as risky and even frightening\u2013sharing a long ride with a stranger, staying in the home of someone you don't know or getting into a self-driving car\u2013into something normal, rewarding and disruptive. So that's the 'idea' part of the bigger climb, the climb up the trust stack. The next level is building trust in the platform.\n\n#\n\n# Where Does the Buck Stop?\n\nOn 20 February 2016, in Kalamazoo, Michigan, a deadly rampage played out. Over the course of five hours, a forty-five-year-old Uber driver named Jason Brian Dalton became a mass killer, shooting six people dead and leaving two seriously injured. In between the separate incidents of bloodshed, Dalton went back to routinely picking up Uber fares.\n\nThe shootings began around 5.40 p.m. EST on a Saturday evening and the first victim was twenty-five-year-old Tiana Carruthers. Carruthers was crossing a parking lot with five children, including her young daughter, when a silver Chevy Equinox with Dalton at the wheel veered towards her. Dalton rolled down his window. He asked if her name was Maisie (the name of the passenger he was circling around the estate trying to find and pick up was Maci.) 'No, I am not that person,' Carruthers replied.\n\nSomething didn't seem quite right about the driver. She yelled at the children to run. Dalton sped off but then quickly turned the car round and headed directly for the terrified Carruthers. Pulling out a Glock 9mm semi-automatic pistol, he shot her at least ten times, hitting her arms and legs, the last bullet lodging in her liver. Remarkably, she survived.\n\nUntil that day, Dalton was by all accounts an ordinary Joe. Neighbours described him as 'a little odd and awkward' but generally sociable. He worked as a loss adjuster for a local insurance company. He had been married for almost twenty years and had two children: a fifteen-year-old son and a ten-year-old daughter. Dalton became an Uber driver on 25 January 2016, about a month before the shooting. He wanted to earn some extra money to take his family to Disney World. Before that tragic evening, he had clocked up more than a hundred Uber trips and passengers had given him a very good rating after rides: an average of 4.73 out of a possible five stars.\n\nOn the day of the shooting, Dalton had run a few errands, including a visit to the local gun store. It wasn't unusual\u2013he loved guns and owned sixteen firearms. He turned on his Uber app around 4 p.m. Shortly after, he picked up a young man called Matt Mellen. It seemed like a normal ride until Dalton took a call from his son, at which point something abruptly shifted in his mood. He accelerated wildly and took Mellen on a hair-raising ride. 'We were driving through medians [central reservations], through lawns, speeding along,' Mellen later told the police. Dalton even side-swiped another car but seemed completely unfazed by what he had done. 'He wouldn't stop. He just kind of kept looking at me like, \"Don't you want to get to your friend's house?\" and I'm like, \"I want to get there alive,\"' said Mellen. When the car finally screeched to a halt, Mellen jumped out.\n\nBoth Mellen and a concerned bystander put in calls to 911, describing the car and Dalton. Mellen specifically identified him as an Uber driver. 'I don't want somebody to get hurt,' he told emergency services. They didn't seem too concerned. Mellen then tried getting hold of someone at Uber, to get the car off the road. Uber didn't seem to prioritize the call, even though Dalton's whereabouts could have been easily located through GPS tracking on the app. It was the first of many alarm bells that would go unheeded.\n\nSoon after that, Mellen's fianc\u00e9e posted Dalton's Uber photo on Facebook with a lengthy warning: 'ATTENTION Kzoo peeps!!! This Uber driver named JASON drives a silver Chevy Equinox is NOT a safe ride!' she wrote. 'Hoping this man will be arrested or hospitalized soon if he has a medical condition causing his behaviour.' Instead, Dalton went from his crazy ride with Mellen to shooting Carruthers.\n\nAfter that first shooting, Dalton swapped his car for his wife's black Chevrolet. Carole Dalton later told police he seemed a little 'troubled'; he had also told her not to go back to work, to fetch the kids, stay inside and lock all the doors. Dalton, meanwhile, went back to picking up regular Uber fares. @IamKeithBlack tweeted that he got a ride with him at 8 p.m., including a clear screenshot of the driver. He had given Dalton a five out of five star rating after his ride. When he heard about the shootings, he tweeted 'Lucky to be alive'.\n\nOthers weren't so lucky. In the lot of a nearby brightly lit car dealership around 10 p.m. that night, seventeen-year-old Tyler Smith and his father, Rich, were looking at vehicles, while Tyler's girlfriend, Alexis, waited in the car. Dalton drove in, parked and walked up to the father and son, fatally shooting both of them. A terrified Alexis hid until he drove off.\n\nFifteen minutes later, in a final burst, Dalton killed four older women and critically injured a fourteen-year-old girl. Nothing connected the victims\u2013male, female, white, black, young and old\u2013and they weren't targeted for any reason.\n\nLate Saturday night is one of the busiest periods on Uber. Remarkably, after killing six people and going home to change guns, Dalton carried on picking up revellers requesting rides. By now, the shootings were all over the news. Some passengers had heard there was a mass murderer called Dalton, an Uber driver, on the loose in Kalamazoo, yet they continued to use the app to get to where they wanted to go. A passenger named Marc Dunton even asked Dalton, 'You're not that guy going around killing people, are you?'\n\n'Wow,' Dalton answered. 'That is crazy. No way\u2013I'm not that guy.'\n\nOnly one passenger made the connection and refused the ride. The young woman's father had texted and called several times to warn her. She requested an Uber around 12.30 a.m. 'Jason. Chevy Equinox' came up on the phone as he was the nearest driver. She cancelled the ride. The same thing happened on her next attempt. A few minutes later, Dalton was arrested in the parking lot of a downtown Kalamazoo bar. When police officers asked the shooter to explain his motive, he replied, 'I don't think there is a why.'\n\nDuring his interrogation Dalton said he recognized the Uber logo as being the religious Eastern Star symbol. He said a horned 'devil figure' would pop up on his phone through the app and would cast an intoxicating spell over him. 'It would give you an assignment and it would literally take over your whole body,' he said. When asked why he randomly shot people, he calmly claimed, 'The Uber app made me do it... It just had a hold of me.' He has since been accused of six counts of open murder, two counts of attempted murder and eight counts of felony firearm.\n\nThe day after the shooting, Uber issued a press release. 'We are horrified and heartbroken at the senseless violence in Kalamazoo,' announced Uber's chief security officer, Joe Sullivan. 'Our hearts and prayers are with the families of the victims of this devastating crime.' A few days later, the company asserted that the rampage could not have been predicted. 'There were no red flags, if you will,' said Sullivan. 'Overall his rating was good, 4.73 out of five.' Perhaps realizing that pointing to Dalton's trust rating was not the best line of defence, he later added: 'As this case shows, past behaviour doesn't always predict how people will behave.'\n\nIt's true, there may have been no reliable way to predict that Dalton would morph from Uber driver to psychotic mass shooter during a single shift. But why didn't anyone seem to respond with any urgency when Mellen first contacted Uber about an hour before the first murder? Turns out, the complaint wasn't prioritized by Uber's customer response team because it wasn't explicitly about violence. 'He said the gentleman was driving erratically,' an Uber spokesperson said, pausing. 'Remember we're doing three million rides a day. How do you prioritize that feedback and how do you think about it?' Still, killings aside, you would think side-swiping a car, speeding and running up on the central reservation might raise some immediate alarm about a driver's fitness to be transporting passengers. Who could have foreseen such tragic consequences? But whose job is it to respond and act?\n\nThe horrific shootings intensified scrutiny on how Uber decides who is fit and safe to be a driver. The company claims it spends tens of millions every year performing background checks on applicants. San Francisco District Attorney George Gasc\u00f3n and many other critics have called those background checks without fingerprints 'completely worthless'. Dalton had passed the checks before he started driving on the platform. But here is the problem: he had no prior criminal history, ever. He came out clean as a whistle.\n\nUber doesn't run the fingerprint checks that taxi and limo companies are required to do. But would they help anyway? 'As the Equal Employment Opportunity Commission (EEOC) has emphasized, background checks have limited predictive value and can have a disparate impact on minority drivers,' says Brishen Rogers, an associate professor of law at Temple University who is researching the social costs of Uber. That said, fingerprinting might have at least revealed that Dalton owned sixteen guns, assuming he held a Federal Firearms License.\n\nWhile Dalton's crime was by far the worst an Uber driver has committed, there has been a troubling stream of other serious incidents. Uber drivers from Boston to Los Angeles, Delhi to Sydney, have been arrested for sexual assault, rape, kidnapping, theft and drink-driving while carrying passengers. In April 2016, a driver was arrested for slashing a passenger's neck. On 23 May 2016, another was accused of strangling a student in a parking lot.\n\nIn February 2017, the company was hit with sexual harassment allegations and former employees accusing the company of fostering a toxic, misogynist culture. Shortly after, a damning video surfaced online of former CEO Travis Kalanick verbally abusing a driver in the US called Fawzi Kamel, who had complained to the CEO about dropping prices and lower pay. 'People are not trusting you any more,' said Kamel. 'I lost ninety-seven thousand dollars because of you. I'm bankrupt because of you.'\n\n'Bullshit,' Kalanick angrily retorted, telling the driver his problems were his own fault. 'Some people don't like to take responsibility for their own shit. They blame everything in their life on someone else.' A fuming Kalanick then slammed the door. He later apologized for his actions.\n\nFive days later, it was revealed that Uber had secretly used a software tool called Greyball, designed to identify city officials attempting sting operations to catch Uber drivers violating local regulations. Then Google's self-driving car outfit, Waymo, filed a lawsuit accusing Uber of stealing technical trade secrets to fuel its own self-driving car research. Following this string of scandals, Jeff Jones, Uber's company president, and six other key executives resigned. Kalanick, the notoriously hard-nosed co-founder, resigned as CEO in June 2017.\n\nClearly, for all its success\u2013at its latest valuation of $68 billion it is the world's most valuable private start-up in history\u2013Uber has had many serious breaches of trust. And yet more than 5 million people every day, myself included, still tap the Uber app, and within minutes get in a car with a total stranger, often without a second thought. We have, in a sense, outsourced our capacity to trust to an algorithm, and that trust, perhaps for convenience's sake, has proved hard to destroy. The question is, where does Uber's responsibility start and finish?\n\nThere are dangerous taxi drivers and, for that matter, dangerous people in any industry. Here's the difference: in Uber's terms of service, the company denies any liability for how third-party drivers\u2013whom Uber considers to be 'independent contractors'\u2013behave on its platform. The company says it is merely _facilitating_ the needs of people who want to drive, and you are getting a ride from them in _their_ car, not a company car. 'Your day belongs to you,' Uber enthuses to would-be drivers. Uber takes up to a 25 per cent cut of the total fare, a service fee, to play the role of go-between. The company claims it can't control what drivers do on the job as they are interacting with an automated system delivered primarily via an app. Uber, however, is not a neutral platform, like a phone line, simply matching supply and demand. It controls surge pricing that temporarily raises fares, and drivers can be suspended for not accepting enough rides or for low passenger ratings. Uber has been involved in more than 170 lawsuits in the US alone, from class-action safety complaints to price gouging to data failures to privacy practices to the biggest lawsuits of all, the misclassification of drivers.\n\nIssues of accountability are incredibly complex in an age when platforms offer branded services without owning any assets or employing the providers. Tom Goodwin, a senior vice president at Havas Media, put it well when he wrote in an article: 'Uber, the world's largest taxi company, owns no vehicles. Facebook, the world's most popular media owner, creates no content. Alibaba, the most valuable retailer, has no inventory. And Airbnb, the world's largest accommodation provider, owns no real estate. Something interesting is happening.'\n\nWhen disasters such as the Kalamazoo killings happen, they raise the question of where accountability should lie when things go wrong.\n\nIt's much easier to know who to blame when traditional brands breach trust. Take the Tesco scandal that took place in January 2013. More than 10 million hamburgers and other meat products were withdrawn from supermarkets after traces of horsemeat were discovered in some of its beef products. The disclosure sparked a national outcry and it became one of the biggest food scandals of the twenty-first century. Even the then prime minister got involved. David Cameron reassured the British people that everything possible would be done to address a 'very shocking crime'.\n\nIn the wake of the scandal, Tesco issued an 'unreserved apology' and promised to introduce a robust new DNA-testing system to ensure the food customers buy is exactly as the label says. How did the 29 per cent horsemeat in the Tesco burger, falsely labelled 'pure beef', get in there? Although Tesco publicly accepted responsibility for the fiasco, they appeared to pin much of the blame on their supplier, Silvercrest, owned by the ABP Food group, who had 'breached the company's trust'.\n\nWhen a customer shops at Tesco, their trust clearly lies in the supermarket brand, what they experience in the store and the products they buy. Tesco, the company, has to behave in trustworthy ways so that their shoppers trust _them_ and their products. But where does trust ultimately lie with platforms?\n\nWhen I get in a car with a stranger, is it the driver I am trusting? Have I placed some faith in Uber, the company, its team? Am I trusting the Uber brand? Perhaps I have confidence in the platform itself, the app, payments, rating system and its mysterious pricing algorithm? Some of the answers lie in the history of trust between people, companies and brands.\n\nThere was a time when people lived in tiny communities, hamlets made up of perhaps no more than a hundred people. Everybody knew everybody else and relationships were tight-knit. People's trustworthiness, or lack of it, was evident to everyone, given the close proximity.\n\nAs hamlets turned into villages and small towns, the population tipped well above what has become known as 'Dunbar's number'. The famous University of Oxford psychologist and anthropologist found that our brains, on average, are designed to have a limited number of people, around 150, in our social group. Yes, you can friend 500, even 5,000 people on your Facebook page, but Dunbar asserts that it is hard for us to maintain stable, meaningful relationships (online and offline) beyond 150. Within that 150, an inner circle of fifteen is the very limited number of people that you turn to for support when you most need it. And this circle of meaningful relationships is fluid: the friend you confide in this week may not be the person you turn to the following month. On the flip side, our brains can handle group sizes of up to 500 at what Dunbar calls the 'acquaintance level'\u2013put simply, the people with whom we can put a name to a face.\n\nWhen people moved into larger towns, with populations way above Dunbar's number, a close circle of trust, based on direct knowledge of each other, was no longer possible. Our reputations became an essential asset. If the baker offered good bread, people would buy it and others would hear about its quality. Equally, if the local blacksmith did shoddy work or someone failed to pay back a loan, word would get round. That dynamic kept most people up to the mark. The evolutionary biologist Robert Axelrod called it in his classic book _The Evolution of Cooperation_ the 'shadow of the future', referring to the idea that people behave better or more cooperatively when they know they're likely to meet or meet again (as opposed to a one-off encounter) and might be judged on previous behaviour. These local traders knew that how they behaved in the present would shape future prospects. Just like the Maghribi traders, it's the promise of benefits from continued cooperation that helps keeps us in line.\n\nEven the earliest incarnations of what we now call a 'brand' first rode on the back of personal reputation. The agricultural machinery giant John Deere was founded in the 1830s by a young entrepreneurial blacksmith living in Illinois who had invented an innovative plough. The Mars brand empire had humble beginnings in the kitchen of Frank C. Mars's home in Tacoma, Washington, when he started making and selling butter-cream candy. With these early brands, goods and services for the most part were associated with specific people, a name and face, not large corporations.\n\nIn the late 1800s, as cities expanded and goods became mass-produced, trust needed to keep up with the pace and scale of industrialization. As local merchants became massive companies, person-to-person trust was no longer viable. So how were people to know the quality of the goods and services they were buying? Take beer, a product that could easily be watered down.\n\nEstablished in 1777 by William Bass, the Bass Brewery grew into one of the largest beer companies in England. By the nineteenth century business was booming and in 1876 the brewery registered its distinctive ale's red triangle symbol and brand name to assure people of its quality. It was the very first trademark to be registered under the United Kingdom's Trade Marks Registration Act. A new kind of branding was born.\n\n'Brands arose as a way to compensate for the dehumanizing effects of the Industrial Age,' writes Douglas Rushkoff, author and professor of media theory at City University of New York's Queens College. 'The more people had previously needed to trust the person behind a product, the more important the brand became as a symbol of origin and authenticity.'\n\nTrust soon became centralized, top-down, opaque, controlled and institutional. Rules, regulations, auditors, market analysts, insurance and independent agencies such as the Better Business Bureau flourished, enabling people to trade beyond their immediate circle of trust. And so, by the mid twentieth century, companies faced a new challenge. With products and services now more or less standardized, how were they to stand out from the crowd?\n\nAt first, they relied on developing brand identities, recognizable by a name, logo, packaging and a tagline, which typically represented a promise\u2013this is what this product or service will do for you. Oxo Cubes promised to 'make cooking so easy'. Lava soap promised to 'clean like no other soap'. By the 1950s, however, mass manufacturers such as Procter and Gamble, Unilever and General Foods realized these types of practical promises were not enough. The problem was, all washing powders and frozen peas do much the same job.\n\nWhat would give consumers a reason to choose one product over another? Vanity, neediness, status anxiety, aspiration, nostalgia and hope, among other things, it turned out. Marketers began to tap into a whole new consumer psychology. Grandiose _brand propositions_ were created, mixing functional benefits with emotional values. 'When I buy or use this brand, I am...' Coca-Cola wasn't manufacturing sugary drinks; its product was about making you feel 'refreshed'. Disney wasn't making movies; it was celebrating dreams. Nike, named after the winged goddess of victory, didn't sell trainers; it made you feel inspired. The idea that a brand would enable consumers to express something about themselves, something intangible, was revolutionary at the time. Crucially, and however artificially, it also bred a sense of intimacy and connection between consumer and multinational: 'Look, they care about _me_.' The result was that brand, with its fancy packaging and catchy slogans, developed enormous power and influence in our lives.\n\nWith the dawn of social media in the twenty-first century, everything changed. Marketers were hit with a seismic shift in the way trust worked with consumers. Through no-holds-barred comments and feedback, reviews and ratings, photo posts and 'likes', people started to share their experiences at scale. The person formerly known as a 'passive consumer' became a participant, a social ambassador, one who was not so easily duped and could be brutal when let down. Could Rice Krispies, the breakfast cereal, really 'help support your child's immunity'? Could a New Balance 'toning' sneaker, with hidden board technology that promised to activate the hamstrings and calves, really help burn calories as claimed?\n\nIt became much harder for brands to exaggerate or make false claims, no matter how flashy their ads. Companies had to start delivering authentic experiences and get comfortable with transparency. They had to learn how to listen, enable conversations and respond to customers' needs in real time. Brands had to let go of an era where trust could be produced and controlled centrally; by them, that is.\n\nFast forward to today, where conventions of how trust is built, managed, lost and repaired are once again being turned upside down. Platforms create systems that act as social facilitators. They match us with goods, rides, dates, trips, recommendations and so on. Customers have become communities, and these communities are themselves platforms that shape the ups and downs of a brand. Indeed, a recent survey conducted by Nielsen revealed that the most credible advertising comes straight from the people we know and trust. More than 80 per cent of respondents say they completely or somewhat trust the recommendations of friends and family. And two-thirds say they trust consumer opinions posted online.\n\nCompare, say, Marriott hotels to Airbnb. Once upon a time, we trusted the hotel chain; the brand was what made people feel safe to spend the night there. With Airbnb, you need to have confidence in the platform itself and in the connections between hosts and guests. In other words, trust must exist in the platform _and_ between people in the community. This is one of the key dynamics that distinguishes the new era of distributed trust from the old paradigm of institutional trust.\n\nJoe Gebbia is the thirty-six-year-old co-founder and chief product officer of Airbnb. Gebbia is a designer rather than an engineer, and he studied at the Rhode Island School of Design, where he met Airbnb co-founder Brian Chesky. I met him in 2009, when I was writing my first book, _What's Mine is Yours_. The marketplace was just starting to become more popular and Airbnb was still far from a billion-dollar idea. The founders were enthusiastic to tell the story of how they started the company from a couple of air mattresses in San Francisco; a story they have now told thousands of times.\n\nGebbia, Brian Chesky and fellow co-founder Nate Blecharczyk began Airbnb with the best of intentions, and had no idea of the magnitude of what they were building. Today, with the company now occupying slick 170,000 square-foot offices and with on average nearly 2 million guests staying in Airbnb rentals every night, the unintended consequences of their success have also mushroomed. One example is the concern over Airbnb's role in distorting rental prices and creating housing shortages, especially for lower-income residents. As an exercise in trust-building, however, Airbnb is a standout.\n\nGebbia, though passionate about design and technology and their power to bring people together, is the first to admit that Airbnb is not a technology company but is in the 'trust business'. 'We bet our whole company on the hope that, with the right design, people would be willing to overcome the stranger-danger bias,' Gebbia says in a TED talk on how Airbnb designs for trust. 'What we didn't realize is just how many people were ready and waiting to put the bias aside.'\n\nHe admits there are risks. 'Obviously, there are times when things don't work out. Guests have thrown unauthorized parties and trashed homes. Hosts have left guests stranded in the rain,' he says. 'In the early days, I was customer service, and those calls came right to my cell phone. I was at the front line of trust-breaking. And there's nothing worse than those calls. It hurts to even think about them.'\n\nGebbia thinks of Airbnb as playing the role of the 'mutual friend' who introduces you to new friends, new places and experiences. 'We have to create the conditions for a relationship to form between two people who have never met,' he tells me. 'And after the introduction has been made, we need to get out of the way.' The role Airbnb is playing may be different from Uber's, in that whom to trust is a choice that each host or guest must make for themselves, but what people want from the platform is similar. That is, we want platforms to mitigate the risk of bad things happening, and to be there for us if they do. 'The number-one thing people want from Airbnb is that if something goes wrong or not as planned, we've got their back,' Gebbia says. 'If we get that right, we are 80 per cent there.'\n\nAlok Gupta, thirty-three, was formerly a high-frequency trader on Wall Street and a research fellow in mathematics at the University of Oxford. He admits he is a big fan of observing patterns and predicting outcomes in enormous data sets. Three years ago, he joined Airbnb as a data science manager, applying his talents and thinking to a different problem: online-to-offline trust. That is, people using digital tools to meet up face-to-face. 'I think Airbnb places itself as the company which does the hard work for you, in terms of trusting the individual,' says Gupta. 'We know there's a barrier to trusting people you've never met before, but we want to fill that space, and we want to help overcome that barrier for you.'\n\nGupta talks about the 'defensive mechanisms' Airbnb has developed that reduce uncertainty. For instance, in 2011, after the 'EJ incident' when a host infamously got her San Francisco apartment completely trashed, Airbnb introduced a 'host guarantee' covering property damage of up to $1 million per booking. In 2013, the company introduced 'Airbnb Verified ID' confirming a person's online identity by matching it to offline ID documentation such as a driving licence and a passport. 'There is no place for anonymity in a trusted community,' the company wrote on its blog announcing the launch. 'Trust and verification. They just go together.' The challenge for Gupta and his team is that the spectrum of wrongdoing is vast, ranging from people using places as brothels to old-fashioned discrimination.\n\nIn January 2014, researchers from Harvard Business School released a controversial working paper on a study they had conducted. The study revealed that non-black Airbnb hosts could charge approximately 12 per cent more, on average, than black hosts\u2013roughly $144 per night, versus $107. In September 2016, looking across 6,000 listings, the same researchers found that requests from guests with distinctively African-American-sounding names (like Tanisha Jackson) were 16 per cent less likely to be accepted by Airbnb hosts than those with Caucasian-sounding names (like Allison Sullivan). Particularly troubling was that, in some instances, Airbnb users would rather allow their property to remain vacant than rent to a black-identified person.\n\nIn the United States, Title II of the Civil Rights Act of 1964 explicitly prohibits racial discrimination in 'public accommodations' such as restaurants, cinemas, motels and hotels. The law contains an exemption, however, for someone renting fewer than five rooms in his own home, a category that would seem to include many Airbnb hosts. 'There have been too many unacceptable instances of people being discriminated against on the Airbnb platform because of who they are or what they look like,' wrote Laura W. Murphy, a former director of the American Civil Liberties Union's Washington legislative office, who was hired by Airbnb to compile a report to serve as a blueprint for how Airbnb plans to fight discrimination on the site.\n\nIn the firestorm that followed, Airbnb users started sharing stories on social media with the hashtag #AirbnbWhileBlack about their experience of bookings being denied or cancelled because of their race. For example, one user @MiQL tweeted: 'My wife & I tried to book w\/@Airbnb for a vaycay. Hosts w\/listed available rooms responded w\/\"Unavailable\". White friend got \"available\".' Personal profiles and photos, which users put together to try to project trustworthiness, have the unintended consequence of facilitating discrimination. Indeed, it seems that distributed trust is not always fairly or evenly distributed.\n\nBen Edelman, one of the authors of the Harvard study, says Airbnb's initial response to his findings 'was kind of denial'. Nine months after the discrimination study was conducted, with pressure mounting, Airbnb released their report outlining its non-discrimination policies and promise to root out bias and bigotry. The company are trying fixes such as minimizing the prominence of user photos and trying to increase 'instant bookings' that don't require pre-approval from hosts.\n\nBut why hadn't they noticed the discrimination happening on the platform? They had a blind spot. Brian, Joe and Nate are three young white American men. In other words, they hadn't personally experienced the type of discrimination many members of the Airbnb community had. 'Discrimination has no place on Airbnb. It is against what we stand for,' says Gebbia. In March 2016, Airbnb hired David King III, who had previously held a prominent diversity role at the US State Department, as the company's first director of diversity and belonging. King was given a team of talented engineers and data scientists, whose job includes identifying patterns of host behaviour and figuring out solutions to create a more inclusive platform.\n\nBut how do you go about stamping out those unconscious biases? It is difficult to remedy offline human prejudices that migrate online. You can't make somebody trust somebody. Like so much else, it's uncharted online territory. 'New systems, new structures that haven't been invented yet are needed to create environments that reduce discrimination or eliminate it,' Gebbia says.\n\nHe takes me back to the early days of the Model T Ford, which revolutionized the take-up of automobiles in the early twentieth century. 'I think the Model T has a lot of analogies to us,' says Gebbia. 'Look at early photos. It didn't have doors; it didn't have blinkers; it was missing all of these things that are needed for a safe ride, and that Ford added over the years. And sometimes I think we're like a Model T, that we haven't added our blinkers yet.'\n\nIn 1865, the British government passed a law called the 'Locomotive Act', which was a safety precaution to warn pedestrians and horse-drawn traffic of the terrifying approach of a motor vehicle. It stated that any locomotive or automobile must have a crew of three people: the driver, a stoker and a man whose job it was to walk at least fifty-five metres ahead of the vehicle, waving a red flag. The act, which later became known as the 'Red Flag Law', made it impossible for vehicles to drive more than two miles per hour in urban areas, meaning the usefulness of the new automobile was limited. The car is just one example of how, throughout history, a new technology that enables a trust leap can also introduce a new 'risky' behaviour\u2013travelling mechanically at speed\u2013and create a vexing challenge for lawmakers. With no precedents to go by, how do they figure out what kind of policies and restrictions will protect public interests?\n\nFor more than a decade, Coye Cheshire, a social psychologist and associate professor of UC Berkeley School of Information, has been studying how the internet is changing risk and trust. On a brisk autumn afternoon, I meet him in his campus office, a cosy room painted in moss green and filled with dark wood furniture and books piled high on every available surface. He makes me a cup of peppermint tea and gets straight to the heart of the matter. 'I want to help understand how humans take risks in the presence of uncertainty.'\n\nI've long admired Cheshire's work, including a paper he wrote in 2011 called 'Online Trust, Trustworthiness, or Assurance?' There, he asserted there was a difference between _interpersonal trust_ (human to human) and _system trust_ (human to system). Did he still think this distinction between people and technologies held true?\n\n'Back then, systems meant things like your telephone and computer, but to be blunt, I was taking a simplistic view of technology that didn't take into account that systems are now capable of betrayal,' he says. For example, a bot in a chat room that can express feelings and moods is very different from simply cooking food in a microwave. 'Today, systems embody everything from online platforms that people are using, to autonomous agents that act on behalf of humans in ways that are blurring the line in terms of our awareness around what the machine is doing.'\n\nCheshire admits that trust is a far more intricate business these days. 'We are working with these systems that are using complex algorithms to manage our information and make decisions on our behalf. But they are getting too complex for our brains to understand.' Consider this: to place a man on the moon in 1969 it took 145,000 lines of code. Today, it takes more than 2 billion lines of code to run all of Google's internet services, dwarfing Facebook, which runs on more than 62 million lines of code. 'I used to think that it was completely ridiculous to compare trust between people with trust in these online platforms,' says Cheshire. 'I'm not certain that is the case any more because in some ways we have offloaded some of our cognitive power.'\n\nIn the past, engineers would typically work on physical infrastructure projects such as roads, rail lines, gas pipelines and bridges. Today, however, they are designing new kinds of social infrastructure: online bridges that bring friends, families and strangers together. They are _trust engineers._ And one of the goals of these engineers is to get us to a place where we don't even think about the risks we are taking. It should feel like magic: you just get the right recommendation, the nearest driver, the best match, no hassle, no dramas. Yet cultivating an overload of that blind confidence can also create the opposite problem: too much trust in untrustworthy people. Think about Marc Dunton, the Uber passenger who was out with his friends at a bar when he accepted his ride with the homicidal Dalton. Dunton admits he knew there was a mad shooter on the loose and that Dalton and his car fitted the description. But surely someone on a shooting spree wasn't still on the Uber app and picking up fares? Surely the genie in the app wouldn't send me someone bad, right?\n\nIronically, one of the issues we face today is the speed and ease at which we are trusting. And it's not just happening on ride-sharing platforms such as Uber. Fancy a date? Download Tinder, Bumble, Happn or Tingle and zip through a fast set-up. When a profile comes up, swipe right on your phone if you like the look of them. And if they swipe right too, it's a match and you are on your way to meet up with a stranger, anywhere from one to a hundred miles away thanks to the powers of geolocation. It's accelerated trust based on a few photos and a handful of words: shopping through a catalogue of faces. It's trust on speed. And when we are in an accelerated mode of trust, we can be impulsive. It requires a conscious gear change to slow down and think twice about our decisions.\n\nOr what about news? Have you ever shared a link without ever reading the article or watching the clip? You are not alone. As a recent study conducted by computer scientists at Columbia University and the French National Institute suggested, many people on Twitter appear to retweet news without even reading it. The researchers found that 59 per cent of links shared on Twitter have never actually been clicked. 'People are more willing to share an article than read it,' says study co-author Arnaud Legout. 'This is typical of modern information consumption. People form an opinion based on a summary, or a summary of summaries, without making the effort to go deeper.'\n\nOn 29 January 2017, the White House Press Secretary Sean Spicer emphatically retweeted on his personal account a video from the satirical site _The Onion_. '@SeanSpicer's role in the Trump administration will be to provide the American public with robust and clearly articulated misinformation,' _The Onion_ 's tweet joked. Around an hour later, Spicer retweeted it, adding, 'You nailed it. Period.' The clip included questionable 'facts' to know about Spicer, including his former role as a senior correspondent for NPR's national desk (false), his snowy white pocket square with a quarter inch of clearance on his suit jacket, his 'defensive' speaking style and whether he has knowingly lied to the press. Could it have been that Spicer has a quirky sense of humour and was giving a sarcastic response to the video? It's possible. But it's also possible that he didn't bother to watch the video before sharing it, or to read the headline carefully enough to realize he was the punchline. In an age of 'fake news' and media propaganda, this phenomenon is more problematic and frightening than ever.\n\nEfficiency can be the enemy of trust. Trust needs a bit of friction. It needs time. It requires investment and effort. 'Trust doesn't form at an event in a day. Even bad times shared don't form trust immediately,' says author Simon Sinek. It comes from 'slow, steady consistency and we need to create mechanisms where we allow for those little innocuous interactions to happen'. Systems are becoming so seamless that we are not always fully conscious of the risks we are taking or the falsehoods we are sharing.\n\n'I think the problem comes down to social translucence,' says Coye Cheshire. 'How much of the social interaction\u2013our behaviours and the underlying mechanisms that enable interactions\u2013how much of that is visible?' Not much at present, he maintains. We need to crack open the 'black boxes' of the internet giants, to lift the veil on the behind-the-scenes operations of systems with which we interact daily and yet know very little about\u2013and may trust too much.\n\nHere is a simple example of how social translucence works in the physical world. If I have something valuable I want to mail, I will go to a post office rather than just drop it in a box at the end of the road and hope for the best. I will hand the parcel to the person behind the counter and pay for registered mail. A human being hands me back a paper tracking slip with a number that I can use to go online and see where my package is at any given stage of the journey. That's social translucence\u2013there are lots of visible cues that tell me what's happening. 'With online systems, the translucence breaks down,' says Cheshire. 'Take putting your credit card information into a site. I pass along the information but it just goes in one direction and I trust the system that the details are safe.'\n\nOnline systems seem as magic as the Wizard of Oz: we do not see the army of human beings, many of them single-minded maths nerds, involved in their operation. We can't see the ghosts in the systems ranking people, places, objects and ideas, making choices and matches on our behalf. Then again, perhaps some of that ignorance is self-imposed. Many of us don't like knowing the extent to which our lives are constantly being massaged by algorithms. We prefer to trust that it's all above board.\n\nA few years ago, a social psychologist and data scientist named Adam D. I. Kramer designed an experiment using the world's largest laboratory of human behaviour: Facebook. A team of researchers from Cornell University and Facebook, including Kramer, joined up to study 'emotional contagion' on a massive scale. Do the emotions expressed by friends via online social networks influence our moods\u2013in other words, can emotions online be transferred to others and how?\n\nFor one week in 2012, the researchers tweaked the algorithm to manipulate the emotional content appearing in the news feeds of 689,003 randomly selected, unwitting users. Posts were identified as either 'positive' (awesome!) or 'negative' (bummer) based on the words used. In one group, Facebook reduced the positive content of news feeds, and in the other, it reduced the negative content. 'We did this research because we care about the emotional impact of Facebook and the people that use our product,' Kramer says. 'We felt that it was important to investigate the common worry that seeing friends post positive content leads to people feeling negative or left out. At the same time, we were concerned that exposure to friends' negativity might lead people to avoid visiting Facebook.'\n\nDid tinkering with the content change the emotional state of users? Yes, the authors discovered. The exposure led some users to change their own behaviours: the researchers found people who had positive words removed from their feeds made fewer positive posts and more negative ones, and vice versa. It could have been an online version of monkey see, monkey do, or simply a matter of keeping up with the Joneses. 'The results show emotional contagion,' Adam Kramer and his co-authors write in the academic paper published in the _Proceedings of the National Academy of Science_ in 2014. 'These results suggest that the emotions expressed by friends, via online social networks, influence our own moods, constituting, to our knowledge, the first experimental evidence for massive-scale emotional contagion via social networks.'\n\nWhen the study was published, it sparked widespread public uproar. What drove the study into the spotlight weren't its findings\u2013in fact, the effect, as the authors acknowledge, was quite minimal, as little as one-tenth of a per cent of an observed change. (Given, however, the scale of Facebook, even tiny effects can have large social consequences such as online bullying.) Instead, the outrage centred on ethics. The researchers had failed to get informed consent from the Internal Review Board (IRB) that oversees 'human subjects research', or from the thousands of Facebook users who were subjected to the manipulation. In the blaze that followed, the company argued that its 1.86 billion monthly users give blanket consent to the company's research on the personal data it collects and stores as a condition of its terms of service. Facebook's data use policy warns users that Facebook 'may use the information we receive about you... for internal operations, including troubleshooting, data analysis, testing, research and service improvement'. That is your price every time you log in, and the cost may be higher than you think.\n\nUsers' willingness to tick the box labelled 'Agree', on this and other platforms, has seen the blithe handover of massive amounts of once private information. 'It has enabled one of the biggest shifts in power between people and big institutions in the twenty-first century,' says Zeynep Tufekci, a sociologist at the University of North Carolina and author of _Twitter and Tear Gas: The Power and Fragility of Networked Protest_. 'These large corporations (and governments and political campaigns) now have new tools and stealth methods to quietly model our personality, our vulnerabilities, identify our networks, and effectively nudge and shape our ideas, desires and dreams.'\n\nThe Facebook story is one example, and it became notorious. Shouldn't Facebook explicitly ask people to 'check the box' if they want to be made to feel happier or sadder? Comments poured in across social media. 'Does everyone who works at Facebook just have the \"this is creepy as hell\" part of their brain missing?' tweeted @sarahjeong a few days after the study was published. Similarly, @Tomgara tweeted: 'Impressive achievement by Facebook to snatch back the title of most dystopian nightmarish tech company.' People felt they had been treated like lab rats.\n\nWhat surprised many academics and researchers, including me, was the level of shock and outrage. Didn't people realize these platforms are essentially mysterious algorithms that exert immense control over what we see? Facebook is just like other content sites such as BuzzFeed and Upworthy, constantly turning one algorithmic knob and tweaking another to find an ideal ad placement, to get us to read and post more. Consider this: if you are a Facebook user, what is the statistical likelihood you have been a guinea pig in one of its experiments? According to the company, it's 100 per cent. 'At any given time, any given Facebook user will be part of ten experiments the company happens to be conducting,' says Dan Ferrell, a Facebook data scientist.\n\nCompanies conduct split testing all the time, ostensibly to improve user satisfaction.* We trust algorithms to determine our Netflix and Spotify recommendations; deliver the most relevant results to our Google searches; even assess our credit score. So why all the fuss and surprise? 'The machine appears to be only a neutral go-between,' writes Cathy O'Neil in her insightful book _Weapons of Math Destruction._ In 2013, Karrie Karahalios, an associate professor of computer science at the University of Illinois, carried out a survey on Facebook's algorithm and found that 62 per cent of the people were unaware the company tinkered with the news feed. So of the 1.72 billion people on Facebook, 1 billion think the system instantly shares whatever they or their friends post.\n\nThe study struck a deep nerve\u2013it was a reminder of how the internet churns, and where the power really lies. It illustrated the power of digital puppet masters or trust engineers constantly to manipulate our data and in different ways control our lives. And to many users it felt like they had been played; the Facebook study was considered a major betrayal of trust.\n\nBeyond the initial brouhaha, though, the study raised a more profound question: if Facebook can manipulate a person's moods with a minor tweak of its algorithms, what else can the platform control? 'About two-thirds of American adults have a profile on Facebook. They spend thirty minutes a day on the site, only four minutes less than they dedicate to face-to-face socializing,' writes O'Neil. Nearly half of them, according to a Pew Research Center report, count on Facebook to deliver at least some of their news. So this leads to the question: how else could Facebook change our minds by tweaking the algorithm? Could it change whom we vote for?\n\n'Pope Francis has broken with tradition and unequivocally endorsed Donald Trump for President of the United States.' 'WikiLeaks CONFIRMS Hillary has sold weapons to ISIS.' 'Clinton runs a child-trafficking ring out of a pizzeria.' Many Facebook readers would have seen these posts and others in their news feed. A few days before the 2016 US election, I made a promise to myself not to check the site for at least a month, after a ludicrous article appeared at the top of my feed. It stated that an FBI agent suspected of involvement in leaking Hillary Clinton's emails had been found dead after apparently murdering his wife and then turning the gun on himself. Two thoughts in succession ran quickly through my mind. Was this piece of 'news' true? And then, _whom_ should I go to, what news source, to find out the real truth? All these stories were of course hoaxes, typifying the fake news and conspiracy theories that plagued the 2016 election.\n\nA recent study by BuzzFeed found that 38 per cent of all posts from three of the largest hyper-partisan right-wing Facebook pages, such as Eagle Rising, contained a mixture of true and false, or mostly false information, compared to 19 per cent of posts from three hyper-partisan left-wing pages, such as Occupy Democrats. Cumulatively, the audiences of these pages are in the tens of millions. The top five fake news items in the last weeks of the election were all negatives for the Clinton campaign\u2013the Facebook algorithm picked a side. And the spread of fake news is far more common on the right than it is on the left. 'These findings suggest a troubling conclusion: The best way to attract and grow an audience for political content on the world's biggest social network is to eschew factual reporting and instead play to partisan biases using false or misleading information that simply tells people what they want to hear,' writes Buzzfeed.\n\nIt's an ironic turn of events, given that in May 2016, Facebook came under fire from Republicans and critics for allegedly suppressing conservative-leaning stories in its trending news section, curated by a small team of human editors. Facebook ended up replacing those human editors, accused of party bias, with software, but the plan clearly failed. And, tellingly, the top twenty fabricated election stories on Facebook netted more engagement than factual stories from mainstream news sources.\n\nIn the wake of President Trump's unexpected victory, many questions were raised about fake news and filter bubbles on Facebook influencing the election results. Mark Zuckerberg initially denied the allegations. 'Personally, I think the idea that fake news on Facebook\u2013it's a very small amount of the content\u2013to think it influenced the election in any way is a pretty crazy idea,' he said a few days after the election. A few months later, however, he had significantly shifted his stance from the initial _Who, us?_ shrug reaction. In February 2017, he published a 5,700-word manifesto on his Facebook page. It sounded a bit like a grandiose State of the Union address for 'bringing us closer together', outlining the immense challenges facing the world today, from terrorism to climate change, pandemics to online safety. 'Every year, the world got more connected and this was seen as a positive trend. Yet now, across the world there are people left behind by globalization, and movements for withdrawing from global connection,' he wrote. 'There are questions about whether we can make a global community that works for everyone, and whether the path ahead is to connect more or reverse course.'\n\nZuckerberg dedicated approximately 1,000 words to how Facebook has become a hotbed for fake content and why this is leading to increased polarization. 'If this continues and we lose common understanding, then even if we eliminated all misinformation, people would just emphasize different sets of facts to fit their polarized opinions. That's why I'm so worried about sensationalism in media.' The CEO then went on to describe a plan, albeit a vague one, for how Facebook proposes to deal with issues on the platform. But can the internet giant really wage war on disinformation and quash bogus memes?\n\nMoving forward, Facebook will check whether people are reading the articles before sharing. If they are, those stories will get more prominence in the news feed. Users can flag posts they think are fake or suspect, helping Facebook detect the most blatant posts. Artificial intelligence and algorithm analysis will also be used to flag content and detect dangerous falsehoods. But how do you decide what is intentional disinformation and what is, say, a bit of an exaggeration? Who decides what the truth is? If it's Facebook doing the deciding, we are awarding them even more power to set the agenda.\n\n'Accuracy of information is very important. We know there is misinformation and even outright hoax content on Facebook,' Zuckerberg said in the letter. 'In a free society, it's important that people have the power to share their opinion, even if others think they're wrong.' He added, towards the end: 'Our approach will focus less on banning misinformation, and more on surfacing additional perspectives and information, including that fact checkers dispute an item's accuracy.'\n\nTo be clear, I do not think that the data scientists, researchers and engineers at Facebook are intentionally gaming the political system. But while Facebook isn't responsible for untrue stories _per se_ , the company wrote the code that meant the bogus news items appeared more prominently over the picture of a friend's child, a funny video or a genuine news story. Is it up to Facebook to emphasize and provide information about the original news sources for news articles? Should they do more to contain hoax stories? Is it the fault of a Facebook business model that depends on clickbait? I think yes, to all of the above. But perhaps the betrayal lies in the system itself, a system that makes it easy for almost a third of the world's population to gossip and gripe, share and like, even if the content is false, and without proper checks and balances or any real redress.\n\nThe online landscape is vastly populated and yet, all too often, empty of anyone to take charge or turn to when it counts. It's rather like when you're a teenager and you throw a party while your parents are away. At first, the freedom is thrilling but there comes a point, somewhere after the tequila slammers, window smashes and the gatecrashers, when you start wishing there was a responsible adult in the house.\n\nFacebook insists it is a neutral technology pathway facilitating connections between people, and not a media company. It is a misconceived and dangerous position. It is _the_ media company with control over how misinformation spreads and an enormous amount of influence shaping someone's worldview about whom they can trust.\n\nThe questions around Facebook may sound different from the ones raised in the days after the Kalamazoo killings but in fact they are remarkably similar. When it comes to trust in distributed systems, we need to know who will tell the truth about a product, service or piece of news, and who to blame if that trust is broken.\n\nWhere does the buck stop? In this new era, people are still working that out.\n\nWith traditional institutions, the picture was clearer. For instance, when, say, your Barclays bank account was hacked, the bank would reimburse you. But when an online cryptocurrency fund such as the Decentralized Autonomous Organization (DAO) runs into trouble, there is no central ombudsman or traditional institution to turn to. (Instead, pandemonium rules, as we'll see in chapter 10.) We are in unmapped territory, scrambling and fumbling around for mechanisms that can replace institutional trust, and at the same time looking for ways to improve the old world's own shortcomings in matters of accountability. Platforms, meanwhile, are trying to figure out their role in it all\u2013mere facilitators in bringing people together, or something more?\n\nGoing back to the analogy of the early days of the car, it took decades to create norms like traffic lights, stop signs and even something as simple as the concept of road lanes. 'We will look back one day and laugh, \"Imagine a car without blinkers,\"' Gebbia says. 'Then we will realize how far we have come in this new era of trust.' Even at that point, with some clearer guidelines governing our coexistence in a world of distributed trust, we will still occasionally crash into each other. No system is foolproof. The hope is that it will be a minor collision, not a fatal accident.\n\n* 'Split testing' is the comparison of two versions of an app or web page against each other to determine which one performs better.\n\n#\n\n# But She Looked the Part\n\nOne day in the spring of 1983, not long after I turned five, an unfamiliar woman entered our house. She was neither a family member nor friend. Her name was Doris, she was in her late twenties and she was starting as our mother's help. She came from Glasgow and had a thick Scottish accent. Her o's sounded like 'ae' and she rolled her r's. It was gently lilting, almost sing-song.\n\nDoris had a mop of mousy brown hair and wore thin steel-rim glasses. She was plump with a ruddy face. She was the type of person you could imagine going for a brisk walk on a cold day and then sinking into a comfy chair, content with a cup of tea and a shortbread biscuit.\n\nShe arrived at our house wearing her 'Salvos' uniform. It was a navy suit with big silver S's embroidered on the collars, complete with a bonnet-style hat. Doris said she belonged to the Salvation Army because she enjoyed helping people. She didn't bring many belongings with her, although I remember the tambourine she kept at the side of her bed.\n\nMy mum had found Doris through a magazine called _The Lady._ A young aristocrat named Thomas Gibson Bowles, who also started _Vanity Fair_ , founded the magazine in 1885. If you watch _Downton Abbey_ , you will have heard of _The Lady._ It is the place where high society\u2013including the Royal Family\u2013seeks domestic staff, from gardeners to butlers to nannies. You won't find any celebrity tittle-tattle or sex stories in the magazine whose tagline is 'for elegant women with elegant minds'. The lead articles from a past edition included 'Capture the Style that Wooed a King', followed by 'Where to Find Bluebells in Bloom'. There was even a recipe to make teatime Bakewell tarts. You get the picture.\n\nMy family is not high society, far from it. So I was intrigued as to why on earth my mum had advertised for help in _The Lady._ 'I was starting my own business and feeling nervous about hiring someone to look after you,' she explains, all these years later. 'I remember thinking if the Royal Family uses _The Lady_ to find help, it must be reliable and the best.'\n\nDoris replied to Mum's advertisement. In those days, you would send a formal letter expressing interest in the position and a photograph of yourself. An interview would follow. As Doris lived in Scotland, Mum interviewed her over the phone. 'I remember her strong Scottish accent,' Mum tells me. 'She said all the right things. She told me she was a member of the Salvation Army and had worked with kids of a similar age. But, honestly, she had me at \"hello\".' After their chat, Mum called the references Doris had given and was satisfied they were all impeccable.\n\nDoris lived with us for just over ten months. She was for the most part a good nanny\u2013cheerful, reliable and helpful. There was nothing strikingly suspicious about her, except for one thing. After school every Wednesday, she would drive us to a block of council flats in Edmonton. The building was one of those dark grey concrete high-rises. It sat close to the North Circular, a busy ring road in London. An odd man in his fifties, balding, lived in the flat. And so did a young baby. The flat was dingy and things were always strewn everywhere. I still remember the dowdy wallpaper and damp, musty smell. Doris would spend the entire visit holding the baby.\n\nI told my parents I didn't like going to this strange flat, to see this strange man. Doris insisted that she was visiting the only family she had in London. Her 'uncle' made us nice tea and we liked playing with the baby. The weekly after-school trips continued.\n\nOn one of these visits, I noticed there were lots of bottles of expensive-looking perfume on the table; they looked just like the ones Mum had in her own bathroom. I mentioned it to my parents. Funnily enough, it was one of the first times I remember my parents not believing me. I was a dreamer. I had imaginary friends and made up elaborate plays. They told me to stop making up stories about Doris. That it wasn't nice. So nobody suspected anything was amiss. Or not until Doris's Uncle Charlie died, supposedly.\n\nOne night, around nine months into her stay, Doris didn't come home. When she did return she explained her Uncle Charlie had suddenly died of a heart attack and she had rushed back to Edinburgh for the funeral. Doris's mum happened to call our house later that afternoon. My parents naturally offered their condolences. 'Her mum had no idea why,' Dad tells me now. 'Doris's mother said the brother was alive and kicking. In fact, he was sitting in the armchair having tea right next to her in her lounge.'\n\nDad confronted Doris. She said her mother was in shock and must have forgotten. 'I told her it was highly unlikely you would forget your brother dying,' Dad recalls. Doris finally confessed that she had lied because she had really gone to the VJ Day veterans day parade to see Princess Diana. My parents thought it was slightly odd but Doris was obsessed with the Royal Family so it was plausible. She continued living with us.\n\nThe series of events that subsequently unravelled sounds totally unbelievable. You'll have to take my word that it's true.\n\nWe had lovely neighbours at the time called the Luxemburgs. They had kids of a similar age and also an au pair. Doris spent _a lot_ of time with her. Around a month after the Uncle Charlie incident, Mr Luxemburg knocked on our door late one evening. He told my parents that he had just thrown their au pair out. 'Philip said he found out that she had been involved in running some kind of drugs ring in North London with Doris,' Dad relates. 'They had even been in an armed robbery and he believed Doris was the getaway driver.' The car, it later turned out, was our family's silver Volvo Estate.\n\nAt this point, my parents decided to search Doris's room. They found plastic bags full of credit-card statements and thousands of pounds' worth of unpaid bills. In a shoebox under her bed, she had stuffed piles of foreign currency, stolen from my parents' home office. Now on high alert, my dad stood on guard by our front door all night with a baseball bat. He was frightened Doris would come home. Thankfully, she didn't.\n\nThe next morning, Dad went to the police first thing. He drove with them to the flat we had been visiting on Wednesdays, visits my parents knew about, even if they had been misled about the true circumstances. 'There was a big hole in the front door that somebody had tried to kick in,' Dad recalls. The weird guy was there, the supposed 'uncle' who gave us tea. (Turned out the 'uncle' was Doris's boyfriend and the baby was their child.) He had a big iron bar on the table. Doris never returned to our house.\n\n'Even as I retell this story I feel sick to my stomach,' Mum says now. 'I left you in the care of a serious criminal. And it took us so long to know who she really was.' My parents never hired anyone through _The Lady_ again. Instead, they asked their friends for referrals.\n\nLooking back, what would they have done differently? 'I wish we had asked Doris more and better questions,' Mum says. 'I wish we had known more about her.' She now realizes the impeccable referees could just as well have been Doris's friends, family or even 'colleagues' in her drugs ring. And the Salvation Army was a total cover story.\n\nMy parents thought they had enough information to make a good decision about Doris, even though in retrospect there was _a lot_ they didn't know about her. There was a _trust gap._ And that raises an essential point when it comes to trust: the illusion of information can be more dangerous than ignorance. As the Italian social scientist Diego Gambetta beautifully put it, 'Trust has two enemies, not just one: bad character and poor information.'\n\nIt would be helpful if the likes of Doris wore labels saying, 'Be warned, I am a con woman and serial liar.' But they don't, and of course it's in the nature of such a person to be convincing. My parents clearly made a very, very poor decision. Yet they are generally smart, rational people with good judgement. What went wrong?\n\nBaroness Onora O'Neill is a philosopher, a professor at the University of Cambridge and a cross-bench member of the House of Lords. Now in her late seventies, she has written extensively about trust and, crucially, how trust is misplaced. She explores that theme in a TED talk, while also challenging the conventional, simplistic belief that as a society we have lost trust and ought to set about rebuilding it. More has to be better, right?\n\n'Frankly, I think rebuilding trust is a stupid aim. [Instead] I would aim to have more trust in the trustworthy but not in the untrustworthy. In fact, I aim positively to try _not_ to trust the untrustworthy,' Baroness O'Neill tells her audience, with understated dry humour.\n\nHer point, however, is deadly serious. Trust is not the same as trustworthiness. Encouraging generalized trust simply for the sake of creating a more 'trusting society' is not only meaningless, it's dangerous. For one thing, people are already inclined to want to trust blindly, particularly when greed enters the picture. The Bernie Madoff scandal is a classic case. Think of all the tens of thousands of investors who placed their savings with the aptly named Madoff, who made off with their money in an elaborate $65 billion Ponzi scheme that ran over decades. Why did investors trust him about something too good to be true? Mostly because Madoff was charming and moved in the same country club and Jewish social circles as they did, in Long Island and Palm Beach. He was a long con, a person who had built up his reputation over years. Indeed, he was known for being a generous, charitable man (it just turned out to be with other people's money). And besides, his own family, close friends and showbiz names such as Steven Spielberg and Fred Wilpon, owner of the New York Mets, had invested with him. The guy had to be sound, didn't he? No, as it turned out.\n\nAs O'Neill notes, Madoff is an example of too much trust in the wrong place. Instead, all of us making decisions about trust should be looking at the who, where and why of _trustworthiness_. Who deserves our trust, and in what respects do we need them to be trustworthy? For instance, if I asked, 'Do you trust your dentist?' that in itself is not a helpful question. You might sensibly respond, 'To do what?' 'Intelligently placed and intelligently refused trust is the proper aim [in this life],' the baroness reiterates. 'What matters in the first place is not trust but trustworthiness\u2013judging how trustworthy people are in particular respects,' says O'Neill.\n\nHow well do we carry out that logical goal in practice? It's not always easy.\n\nMy parents' decision to trust Doris came down largely to their personal judgement and blind faith. They wanted, even needed, to believe that what she was saying was true. Their judgement of Doris was also influenced by _trust signals_. These are clues or symbols that we knowingly or unknowingly use to decide whether another person is trustworthy or not. The Salvation Army, Scottish accents, _The Lady_ magazine, Doris's cheery appearance, her references and even her steel-rimmed glasses were all trust signals my parents used to make a decision. Trust signals supposedly give us the ability to 'read' each other. They give us reasons to trust someone or ways to demonstrate our own trustworthiness. But it's still a bet, of sorts. 'Like all gambles, assessing trustworthiness is an imperfect endeavour; there's always a chance you're going to come up short,' writes David DeSteno in _The Truth about Trust_.\n\nSome signals we literally 'give off', such as our clothes, our face and our accent. Indeed, studies have shown that the Scottish accent is perceived to be the most trustworthy in the United Kingdom ('Scouse' is perceived to be the least). Other trust signals are non-verbal but still visible, including our posture or gestures such as a nod, smile, twitch or an averted gaze. Despite the admonition not to judge a book by its cover, these first impressions are insanely influential when it comes to trusting someone.\n\nJon Freeman is an assistant professor of psychology at New York University and director of its Social Cognitive and Neural Sciences Lab. He studies what he calls 'split-second social perception'. When you see someone's face, you make snap judgements, within a tenth of a second, about their traits, including how trustworthy they are. Freeman wants to understand why our brains take these kinds of mental shortcuts.\n\nFreeman, in his late twenties, is a rising academic star. On a typical day, you will find him wearing slim-fitting slacks, a navy blue button-down shirt and nerdy-cute tortoiseshell glasses. Looking at him, you might instantly label him an 'academic'. The scientist behind 'blink' stereotyping likes it that way. And, in this case, you'd be right.\n\nA few years ago, Freeman and his colleagues devised an experiment to see if there was such a thing as appearing trustworthy. Participants were shown pictures of distinctly different male faces of different ethnicities. They were asked to rate how trustworthy or untrustworthy they thought the people in the pictures were. The results were clear\u2013your brain thinks it knows a trustworthy face. Humans are inherently wired this way. When our ancestors were approached by a stranger, they needed a rapid-response mechanism. Friend or foe? But the same rapid response in our day-to-day reactions can lead us to make biased trust decisions based on stereotypes.\n\nThe researchers devised a second experiment in which they digitally altered pictures of the same person, evolving the images gradually from looking _slightly happy_ to _slightly angry_. The study found that people with upturned eyebrows, pronounced cheekbones, big baby-like eyes and an upward curving mouth\u2013even if they weren't overtly smiling\u2013are more likely to be perceived as trustworthy. Those featuring sunken cheeks, a downturned mouth and eyebrows\u2013even if they weren't obviously frowning\u2013are more likely to be perceived as untrustworthy. S0 the trick to appearing trustworthy? Look _slightly_ happy. Just like Doris did. The problem is, however, there is no evidence that people with those 'trustworthy' features are in fact more trustworthy.\n\nYou can't consciously control whether you perceive a person to be trustworthy or not; your brain does it for you automatically, but not always accurately. And our early assumptions about people can be difficult to budge. That's astonishing and a little frightening. Trust can therefore be easily misplaced. Indeed, the art of the con man is to give off the right signals and appear trustworthy.\n\nOf course, trust signals are not just based on looks and first impressions. Clear symbols of status or authority, from a white lab coat to a police officer's badge, are also trust signals, for some. Uniforms can be powerful shortcuts for enabling trust. For instance, if my doorbell rings and I look through the peephole and see a stranger in a postal uniform, I will open the door. The postal uniform is a recognizable symbol that reassures me in my decision. Well-known brands rely on the same dynamic. When I am overseas in a country with dodgy drinking water, I'll buy bottles of Evian or some other brand I know. Why? The name and packaging make me believe the water inside is safe. I don't have to trust the person selling me the water, I trust the brand.\n\nTrust signals can also come from the endorsement of third parties. Doris used her association with the Salvation Army, one of the most trusted charities in the UK, to appear trustworthy. She was deliberately sending false signals to dupe my parents. We may not outright lie like Doris did, but think of the number of times we drop in an association with a trusted brand or institution to establish our credibility. I do it all the time when I meet new people: 'Oh, I teach at Oxford University' or 'You might have seen my work in _The Economist_.' I am not bragging (okay, maybe a bit, sometimes) but intentionally drawing on these signals to build trust. Going back to the definition of trust, _The Economist_ and Oxford reduce uncertainty about me for other people. They build confidence in the unknown.\n\nWe use institutional trust signals to help us make all kinds of decisions in our lives. For instance, imagine you have to choose between two lawyers to handle a matter. One has a law degree from Whittier Law School in California (ranked as one of the worst law schools in the United States) and the other has a degree from Harvard Law School. Removing price from the equation, most people would choose the Harvard-trained lawyer. Similarly, 'You can't get fired for hiring McKinsey' is an adage that has been muttered among management for years as a reason for hiring the world's most prestigious consulting firm. In both these examples, we are trusting the elite reputation of the institution, not the trustworthiness of the individual.\n\nTrust signals are changing in the age of distributed trust. Consider occupational licences: in the 1950s, fewer than 5 per cent of American workers needed a licence to do their job. Today, more than 1,000 professions\u2013approximately a third of all occupations\u2013in the United States require a licence. In some states you need a licence to be a tree surgeon, fortune teller, florist, horse masseuse, make-up artist, ferret breeder, falconer and even a hair braider. Of course, occupational licences are necessary to enforce standards for high-risk professions that require you to put your life in someone else's hands. I like knowing that the pilot flying the plane or the doctor at the hospital are regulated and licensed. But professional licensing rules have become excessive and don't necessarily help us to make good decisions as to whom we can trust. Do I really need my hairdresser to pay $2,500 per year and to have no fewer than five sinks and at least ten workstations before I can trust them to cut my hair? Surely ratings and reviews left by previous customers would be a better, or at least an equal, indicator of their talent?\n\nIn some instances, new trust signals will be used alongside institutional ones. Think about lawyers. Law degrees or bar memberships are certainly important trust mechanisms, but here's the thing: once earned, these qualifications do not change based on a lawyer's _actual_ performance (unless, of course, he or she is removed from the bar for unethical behaviour). Who judges the judgement and quality of skills of professionals? Indeed, I once made a very bad decision in hiring a lawyer who had graduated from an Ivy League law school and belonged to a top-tier law firm. A friend had recommended him to handle a matter. He turned out to be unresponsive and, frankly, incompetent. If I was hiring a lawyer today, I might go to UpCounsel, an online marketplace that matches clients with high-quality attorneys, based on particular skills, pricing and availability. There I could find a suitable one at the push of a button, just as easily as I could hire an Uber ride.\n\nOn every lawyer's profile, it lists qualifications, years of experience and areas of specialization. Clients can also see in real time how much their attorney is billing and what for. Plus, after the transaction, lawyers are rated by clients with one to five stars, just like on eBay, and given detailed reviews. 'Reviews are one of the most important things in my online profile,' says Seth Weiner, a popular attorney on UpCounsel, who graduated from Columbia Law School and left life in the big city firms to work as a solo practitioner. 'If 250 people have been happy with me, it stands to reason that the next person will be as well.' UpCounsel is using online reputation as a means of solving problems of trust. The fascinating aspect of these real-time ratings is how they help hold people accountable for performance once they are in the profession. They allow us to see beyond the more obvious hallmarks of 'goodness' (such as Harvard degrees and fancy offices) that could be totally at odds with the quality of work they _actually_ perform.\n\nSo technology might hold people more accountable, but to what degree can technology enhance our assessment of who is trustworthy and who isn't? Would my parents not have mistakenly trusted Doris if they had been choosing a child carer in this digital age? I knew the right person to ask.\n\nLynn Perkins is the forty-three-year-old co-founder of UrbanSitter, an online marketplace that connects families with babysitters on the internet. She lives in San Francisco and she talks fast, really fast. 'I love to help people find the perfect restaurant, the right vacation for a friend's honeymoon or a new job,' she says. 'I've matched friends needing apartments with roommates and three couples who are now married.' In short, Perkins is a connector.\n\nIn 2008, after the arrival of twin boys, Perkins decided to take time off from her high-flying career in investment banking and real-estate development. Long hours in the office were replaced with lots of time with other mums. The conversations were somewhat predictable. How do you get your child to sleep more? Why are they so picky about what they eat? Why don't they listen to me? And so on. Perkins also noticed, however, the inordinate amount of time mums spent venting about the shortage of reliable childcare. 'They'd say they would rather skip going out on a Friday night with their husband if they couldn't find a sitter their friend had already used,' Perkins tells me. 'It was incredibly revealing about what they were looking for and currently couldn't find.'\n\nPerkins herself had experienced the nuisance of one too many sitters cancelling at the last minute. 'It was around the time when companies like OpenTable, for restaurant reservations, and Airbnb were starting to crop up,' she says. 'But there wasn't any kind of on-demand marketplace for babysitters and nannies.' Why, she wondered, could you book a table in thirty seconds but not a sitter? She suspected it was down to trust.\n\nIn the early days of UrbanSitter, Perkins did something simple but smart\u2013she borrowed trust from organizations parents already trusted. She went to local music classes, clubs such as Big City Moms, Little Stars Soccer teams, elementary school groups, you name it, and found all the babysitters the parents in the groups were currently using. Perkins convinced the best sitters she met through these organizations to put their profiles on UrbanSitter. But it wasn't enough. When Perkins first started UrbanSitter in 2011, her closest friends and the investors she pitched to were convinced it would never work. 'Are you crazy?' was a common response.\n\n'They just could never imagine using a service where people would find their care provider online,' says Perkins. The doubters wondered how finding a sitter online could be better and safer than asking for personal recommendations from friends. 'One thing everyone wanted to know was whether my team had met and interviewed every single one of the individual sitters. It was something they thought you just had to do to build trust.'\n\nEven then, people simply couldn't imagine that social networks could not only deliver all that essential information but also make a better job of it. As is now the case with many online services, you have to join UrbanSitter via Facebook or LinkedIn. If a person has fewer than five 'friends', it's a red flag that it's a fraudulent account. But the real power of the Facebook login is that it can unlock the value of established personal connections. It reveals how we are connected to others. Whom do you know that I know, be it direct Facebook friends, friends of friends, people we went to school with or who worked at the same company. Trust established in one group and context can travel and spread to another.\n\nOn UrbanSitter, when you go to book, you can see how many 'friends' have previously booked or are in some way connected to that sitter. These connections make us feel more comfortable and confident about our decisions. They reduce the unknown. The collective wisdom of the crowd is enhanced by the wisdom of 'friends'. It's social proof on steroids.\n\nThe late John Keith Murnighan, when he was a professor in social sciences at the Kellogg School of Management, set out to explain what causes us to trust people we do not really know. Specifically, he was interested in the role that 'friends' play in stimulating feelings of trust for a stranger. He conducted a series of experiments based on the famous 'trust game' that was originally designed by behavioural economists in 1995. In the game, there are two participants, a sender and a receiver, who are anonymous strangers. Both are given a certain amount of money, let's say $100. The first player can send any portion of the $100, or none of it, to the second player whom they will never meet. The first player is told that whatever amount they send will be tripled, by the experimenter, for the receiver. The receiver then has to make a similar decision: how much of the tripled amount should they return to the sender? Player A can therefore either potentially turn a profit or lose everything. The point of the game is that sending a large amount of money indicates a high degree of trust.\n\nBefore volunteers participated in the trust game, Murnighan and his team asked them to provide the names of people they trusted and people they distrusted and the reasons for their feelings. The researchers quickly flashed these names for mere milliseconds to subliminally prime the study participants. It was too fast for anyone to be able to recognize the names that had been flashed. After that, the participants played the classic trust game. The results were stunningly clear.\n\nParticipants who had subliminally seen the names of people they trusted, sent on average nearly 50 per cent more to the anonymous receiver than the participants who saw the names of people they did not trust. 'We found we could stimulate feelings of trust for a stranger without people even realizing,' wrote Murnighan, an outcome he found 'both exciting and scary'. 'Imagine a fanatic fan of Elvis Presley. If I know someone is a huge fan of Elvis, I might casually drop Elvis's name to activate more trust in me. There is clearly a risk of manipulation.'\n\nThe results of the game help explain how the likes of Madoff managed to deceive so many people. His client list included the rich and famous and his own friends and family\u2013his brother, his sons and their wives. When new investors spotted close relatives on the investment list, it was a powerful trust signal that Madoff himself was trustworthy. Whether Madoff intentionally or inadvertently used the friends and family connections does not matter. The point is, these relational cues can provoke automatic trust that can be very dangerous, especially in cases such as Madoff's where we might lack the time or expertise to make a deliberate and proper evaluation.\n\nBut Murnighan's experiment also sheds light on the power of online social connections\u2013the wisdom of 'friends' can automatically enhance our ability to trust people we do not know.\n\nInterestingly, Lynn Perkins wrongly assumed the most influential social connection would be between parents. Instead, it was between sitters. 'Over time, parents value the connections UrbanSitter can surface between sitters; they want to book the friends of the sitter they really liked,' she explains. When you think about it, this is how trusted referrals typically work. Say an entrepreneur asks me for the name of the designer I trust. I introduce them to Amy Globus, whom I have worked with for years, but she is too busy to take on their work. The entrepreneur is then likely to ask Amy to recommend another designer rather than coming back to me for another suggestion. In other words, trust really lies within the group with the expertise (the babysitters) rather than the group with a similar need (the parents).\n\nThe social graph* is information manna when it comes to trust. We are now using digital tools to approximate old-fashioned ways of finding people we could trust, through referrals and close connections, but in ways and on a scale never possible before. 'It's a digital recreation of the neighbourly interactions that predefined industrial society,' writes Jason Tanz in an excellent piece on digital trust in _WIRED_. 'Except now our neighbour is anybody with a Facebook account.'\n\nThe strange and beautiful truth about the social graph is that it shortens the distance between any two people in the world. Think of it as the string tying together the arbitrary connections between humanity. In 1929, the Hungarian author Frigyes Karinthy first wrote about the theory of the 'Six Degrees of Separation', later popularized in a play by John Guare, claiming that everybody on this planet is separated by only six other people. This is where the 'Kevin Bacon game' comes from\u2013it would take the average person six or fewer acquaintances to be connected to the actor. Today, that number has shrunk significantly, at least for the 1.8 billion active users on Facebook. In 2016, Facebook crunched their social graph and determined that the degrees of separation number is 3.57. In other words, each person on Facebook is connected to every other person by an average of three and a half other people. It means that, despite Facebook's vastness, it feels intimate. It gives us the sense of connection and trust we used to get from real-life communities and neighbourhoods. Its size, however, is critical to its usefulness.\n\nWhen my parents decided to trust Doris, they based their decision on faith\u2013they believed what she and her references claimed. In the past, we had to make a lot of decisions based on blind faith or personal experience, but today we can base them on _collective experience_ \u2013the experiences other people have shared through reviews and social networks.\n\nI don't need to have personal experience with a nanny I hire through UrbanSitter to assess if she or he is trustworthy; I can benefit from other people's experiences. It's a dynamic often referred to as _indirect reciprocity\u2013_ and it can speed up the process of trust. Today, on UrbanSitter, the average sitter responds in less than three minutes, down from twenty-three hours five years ago. The average time a parent takes from either posting a job or doing a search for a sitter to accepting the person they trust to look after their kids is less than ten minutes, down from twenty-three hours when UrbanSitter first started. It shows that, online, questions of trust can be settled very fast. And this process is only going to get faster, from minutes to seconds.\n\nThere is just the small matter of things going wrong now and again in this speedy and efficient brave new world.\n\nIt is almost impossible to get entrepreneurs, people like Lynn Perkins, to tell you the exact number of bad incidents, minor or serious, that happen on their platforms. I have prodded and cajoled to try to get actual data but instead you tend to get a generic response. 'They do happen but they are extremely rare,' insists Perkins. Take Wendy, a mum living in Seattle who hired a sitter to look after her six-month-old daughter one night in May 2016. The day after, she received a call from her bank about a cheque with a dodgy signature. It turned out the sitter had gone through Wendy's drawers and stolen a cheque that she had written out for $1,300. 'As soon as the parent let us know what happened, the sitter was blocked from our system,' Perkins tells me. Bad apples will make the cut, it's inevitable, but it's also likely they'll be exposed more quickly.\n\n'She seemed so trustworthy' is something we commonly say after trust has been misplaced. On the flip side, you have probably been on the receiving end of someone saying 'I trust you'. But what do these statements mean? And, coming back to Baroness O'Neill's point, what is it that we should be basing our trust upon?\n\nTurns out, there's a relatively simple formula to trustworthiness that goes beyond 'but he had kind eyes' or 'she looked the part'. It doesn't matter if you are deciding whether to trust an estate agent, a lawyer or a babysitter, the three traits of trustworthiness are the same: Is this person competent? Is this person reliable? Is this person honest?\n\nCompetence comes down to how capable a person is to do something. Does he or she have the skills, knowledge and experience to do a particular role or task, be it cut my hair, mind my children or fly me to Uzbekistan.\n\nReliability comes down to a person's consistency in doing what they said they would do for you. Ultimately, it's about you knowing, 'Can I depend on this person?' Will he or she follow through?\n\nHonesty is about integrity and intentions. 'What are their interests and motives towards me?' Basically, it's whether their intentions are aligned with yours. What do they gain by lying or by telling the truth?\n\nPolitical scientist Russell Hardin eloquently argues that trust is really about _encapsulated interest_ , a kind of closed loop of each party's self-interests. He argues that if I trust you, it's because I believe that you are going to take my interests seriously\u2013whether it be for friendship, love, money or reputation. Why? You won't take advantage of me because it benefits you not to do so. 'You value the continuation of our relationship, and you therefore have your own interests in taking my interest into account,' Hardin writes in _Trust and Trustworthiness._ 20 For instance, I trusted the estate agent who recently sold my house to get a good price not because she was nice or cared about me but because her commission was directly tied to the sale price. That's encapsulated interest. Or as economist Adam Smith would put it, the estate agent's future payoff is a strong enough incentive for her and good enough reason for me to trust her.\n\nWe often mentally ask of someone, 'Do I trust you?' A better question is 'Do I trust you to do x?' We need to think of trust as _trusting someone to do something_. For instance, you can trust me to write an article but trusting me to, say, drive a lorry would be a grave mistake. You can trust me to teach twenty-something-year-old graduate students, but put me in a class of five-year-olds and I would probably lose my rag in my attempt to teach them to read and write. What we are trusting someone to do fundamentally changes the alchemy and order of importance of those three necessary traits: competence, reliability and honesty. Trust is contextual.\n\nI am not suggesting that we have to go through this kind of assessment with every person we need to trust. Take something simple like getting on a bus or train. We don't want to be assessing the driver's skills. If we thought about every decision to trust, we would spend the entire day asking questions and making checklists. We might never even leave the house. But let's say we're hiring someone for a job. How do we make good decisions about whether or not to trust what they claim about themselves?\n\nNot so long ago, CVs were the principal tool in job hunting, listing what someone had\u2013allegedly\u2013done and with whom, but they didn't provide much proof. Indeed, a survey done in the UK by Powerchex, a company that screens CVs on behalf of finance companies, found that, out of 4,735 job applications, 18 per cent contained outright false information. The most common lie was to claim a 2:1 university degree when they had been awarded a 2:2. Embellishing the truth, such as exaggerating job titles (from project assistant to project director) and tasks, is very common. CVs were often 'creative', at best, but pretty much all the average employer had to go on.\n\nToday, many people send their LinkedIn page instead of a CV and include links to their other online portfolios and social profiles. Think for a minute about the number of profiles out there that contain information on you. I easily have fifteen: my profiles on Amazon, eBay, LinkedIn, Facebook, BlaBlaCar, Uber, YouTube, Twitter, TED, Oxford University Faculty page, my personal website, my literary agent's website, my speaking agents' websites and so the list goes on. And that does not even include all my accounts that require a Facebook login such as Spotify, Airtasker and Airbnb. Online profiles are another example of the shift in trust signals. Information that used to be held by institutions or small groups of friends, family or colleagues is now distributed among many people. In this sense, trust signals have become socially fluid.\n\nIn online services such as UrbanSitter, people create specific online profiles listing detailed information. The amount and type of details people will voluntarily disclose is extraordinary. 'One parent included a long description about their pot-bellied pigs,' Perkins explained. 'They wanted to make sure the sitter was comfortable with the family keeping pigs as pets. It sounds weird but is actually great expectation management.' Both sides knew what they were in for.\n\nThe information a sitter provides in their profile is verified through different online checks. Do you really have cardiopulmonary resuscitation (CPR) certification? Did you really get a Level 5 childcare diploma from South Thames College? Do you really have a clean driving record? Or in the case of Doris, do you really belong to the Salvation Army? In fact, only 25 per cent of the sitters that start the registration process make it on to the platform. Put differently, 75 per cent are rejected.\n\nUrbanSitter uses forty different criteria to find the six sitters most compatible to the parent. The algorithm takes into account the age of the children, the parents' social connections to the sitter, where they live, when the sitter is needed, specific preferences and so on. The marketplace also uses a reputation system just like eBay\u2013parents leave feedback and rate sitters after each transaction.\n\n'People rate everything now. Your Uber driver, the guy who delivers your food, and not all things are equal. There are only so many things you can do wrong or right when you're delivering my food,' says Andrea Barrett, UrbanSitter co-founder and vice president of product. 'But when you're watching my child, there are a lot of things that can go awry. Your ability to deal with the behaviour of my child, how much you engage with my child, how much you cleaned up, were you friendly, did I like you? There's a lot going on.'\n\nReviews significantly influence a sitter's ability to get a booking\u2013a sitter without reviews is two times less likely to get a booking than a sitter that has at least one. A valuable badge experienced sitters and parents carry is called 'Repeat Families'. Basically, it means you were invited back.\n\nReliability is also easier to demonstrate online. If you have booked a place on Airbnb, you may have noticed that hosts are categorized by how quickly they respond. My response rate is 100 per cent but my response time is, um, twenty-four hours. This means I respond to all new messages but by Airbnb standards I am slow to message back. Similarly, UrbanSitter categorizes sitters based on how quickly they reply. 'If you are looking for someone last minute, it's good to know upfront how long it will take them to respond,' says Perkins. 'But on a deeper level, I think it's an indicator of reliability in some weird way. If this person is really slow to respond, will they show up on time? Are they really interested?' Perkins is right; time is often used as an indicator of reliability.\n\nThe hardest trait, without a doubt, both to prove and predict online, is honesty. How do you get a real read of someone's intentions and integrity? Perkins knew this was a big problem she had to address. In 2014, she wondered if asking the sitters to make short videos about themselves and their interest in kids would make a difference.\n\nMost videos start with, 'Hi, my name is so and so and I would like to give you a bit of information about...' Then they get more personal, explaining why they want to look after your kids. One wondrous offshoot of the videos is seeing the rooms people are in. Some sit casually in their bedrooms on unmade beds, others choose the sofa in tidy living rooms.\n\nI tried an informal experiment. I read the written profiles of twenty sitters and wrote down characteristics that summed up my first impressions. I found I kept using the same vague words such as 'nice' or 'seems friendly'. I then did the same exercise using the videos. The descriptors were clearly more specific, such as 'considerate', 'warm' or 'nervous'. Hearing sitters talk and seeing a small glimpse of their environment felt like seeing inside their lives. 'It allows them to be human,' Perkins says. Today, it is mandatory for all sitters to leave a thirty-to-ninety-second short clip video about themselves. There is this, though: Doris would have made a great video.\n\nSo would UrbanSitter have caught that she was a fraud?\n\nOn 5 July 1993, a 'dog' cartoon by Peter Steiner appeared in the _New Yorker._ The pen-and-ink artwork features two dogs, one sitting on the floor and the other in a chair in front of a computer. Underneath is the prescient caption, 'On the internet, nobody knows you're a dog.' In 2013, to mark the twentieth anniversary of that iconic cartoon, the web comic 'Joy of Tech' created an updated version. It featured two agents with sunglasses standing in the National Security Agency (NSA) surrounded by computer screens. 'Our Metadata analysis indicates that he is definitely a brown lab. He lives with a white-and-black-spotted beagle mix and I suspect they are humping,' reads the caption.\n\nSitting in the meeting room of Trooly*, an 'Instant Trust Rating' venture launched in July 2014 based in Los Altos, California, Steiner's cartoon came to mind. Before I arrived to meet Savi Baveja, the co-founder and CEO, his team had come up with an idea. 'We thought it would be helpful to run you, Rachel Botsman, through the Trooly system,' Baveja says. 'We'll do it on the projected screen so you can see what is happening in real time.' Great. Five other people were gathered round the conference table. I felt a tad uneasy about what they might find. Baveja, in his late forties, a former senior partner at Bain & Company, has a calm and thoughtful demeanour. He picked up on my silent pause about the plan. 'Don't worry,' he reassures me. 'We did a test run before you got here and you are all good.'\n\nWhen you first connect with people, total strangers living on the other side of the world, how do you know if they pose a serious risk? 'People talk a lot about reviews and ratings. Well, by definition, they're backward-looking,' says Baveja. Indeed, the goal of Trooly is to fill the trust gap caused by the sheer speed of online commerce.\n\nMy first and last names are entered into the Instant Trust software, plus my email address. That's it. No phone number, age, date of birth, occupation or address. Anish Das Sarma, co-founder and chief technology officer, tells me the machine learning software is searching whatever public and permissible data it can find associated with my identity. 'Watch lists, the National Sex Offender Registry, social media, et cetera. But it's also scanning the deep web, the parts of the web not indexed by search engines, so you wouldn't surface them in a search. We have indexed deeper than Google,' Das Sarma adds. I think he is referring to the weird websites I might hang out on, getting under the hood of what I really do on the internet. He asks if he can add my middle name to send a 'stronger signal'. Sure.\n\nA long thirty seconds passes. Then the results appear. 'Look, you are a one,' says Baveja, pointing at the screen. 'Only approximately 15 per cent of the population are a one; they are our \"super goods\".'\n\n'So how many are \"super bad\"?' I ask the team.\n\n'About 1.5 to 2 per cent of the population end up between five and four. There are some exceptions in some populations we've done because some populations attract super creeps,' Baveja explains. 'The vast majority are twos. And for around 10 to 15 per cent of people we screen in the United States we can't generate a confident score because there is either not enough of a digital footprint out there or enough accurate inputs.'\n\nHonestly, the level of information the search had pulled was staggering. I didn't mention my maiden name, so I assumed that nothing linked to 'Simmons' would appear. I was wrong. Links to clubs I had joined at Oxford and Harvard were in the search list (yes, I was a member of the darts society and, yes, I spent almost nine months in the Territorial Army Officers' Training Corps). The data was consolidated into five different categories. The most basic was verifying my identity. Was I who I said I was? Check. Next was criminal record and possible unlawful activity. Clean. Phew, they missed the car accident case I was involved in when I was twenty because it was dismissed in court. And then there is a category called 'anti-social'. 'A lot of people won't have done anything explicitly good or bad,' explains Baveja. 'So we have spent a lot of time figuring out the granularity of how to get at \"what kind of person is this?\" We want our customers to look at more than just a superficial score before making decisions.'\n\nContext for the anti-social category is king. Clients tell Trooly the types of traits they want to screen against and bad behaviours they need to weed out. For example, UrbanSitter may want to screen potential sitters against hate language. They might also want to know if you have ever had a drug or alcohol problem or if you have ever been involved with pornography. A home-sharing site might care a lot if you are a sex party organizer. A ride-sharing platform will want to know if you are a terrible driver. I also get a one in the 'anti-social category', meaning I am 'pro social'. I feel relief, even a tinge of pride that I have been given an A* trust rating.\n\nBaveja is a warm, considerate and clearly intelligent man. His sentences tend to contain lots of thoughtful questions, as if he is constantly seeking better answers. He went to Stanford University for his undergraduate degree in electrical engineering. A few years later, he completed his MBA at Harvard Business School where he graduated as a Baker Scholar, meaning he was one of the top students in his class. He then became a management consultant, rising to the highest echelons of Bain & Company, first becoming a partner and then joining the company board. 'While at Bain, I started to look into traditional background and credit checks, and realized how dangerously flawed they are,' he says. 'A background check is just that\u2013it is retrospective and does not foresee the future. But does this have to be the case? I thought there must be something better.'\n\nOur behaviours have changed but the trust mechanisms we use in society have stayed pretty much the same. For one thing, in the US, UK and much of Europe, the current background-check system is still slow and manual, often relying on low-paid and overworked court runners to rummage through records and so on. No wonder all kinds of mistakes happen, especially if your last name happens to be Jones, Smith or Harris, or another common moniker. Ron Peterson, who lives in California, knows this problem all too well. 'In Florida, I'm a female prostitute (named Ronnie); in Texas, I'm currently incarcerated for manslaughter,' says Peterson. 'In New Mexico, I'm a dealer of stolen goods. Oregon has me as a witness tamperer. And in Nevada\u2013this is my favorite\u2013I'm a registered sex offender.' If you are falsely mismatched with someone else's felony, it's known as a _false positive._ It's an alarmingly common problem.\n\nOut of the people the traditional checks label 'bad', how many are actually criminals? Worryingly, a study conducted in 2016 by Simone Ispa-Landa, an assistant professor at Northwestern University, and Charles Loeffler, a professor at the University of Pennsylvania, found that one out of three American adults has been involved with the criminal justice system and has a record, even if they were not found guilty of a crime. The United States Attorney General's office recently found something similar. Half of all case files in the background system contain no information about how the cases turned out\u2013whether the person was found guilty or not, or even prosecuted. Bad data fouls the system and the most common groups falsely labelled are blacks and Latinos. 'What we've ended up doing is taking all the biases and pre-existing preconceptions of the criminal justice system,' Baveja says, 'and ossifying them, institutionalizing them in more and more and more decisions where they don't belong.'\n\nBaveja raises question after question about the checks system. 'I can tell you that 50 per cent of what you will find in a background check is going to be either traffic violations or drug convictions,' he says. 'But if I smoked a joint seven years ago, what does it really tell you about me? Does it mean I'm going to be a bad tenant?'\n\n'But surely, if I had been convicted of shooting someone, the check would flag this up?' I ask.\n\n'No, a fail is a fail in a traditional background check. It's just not precise,' says Baveja.\n\nBut there is another problem: the system can miss people who really are criminals. It's known as a _false negative._ Of the millions of checks done every year, between 1 and 2 per cent turn up a problem. Indeed, the vast majority of people who have done bad things pass their background checks. 'How on earth did we get to a point where we rely on flawed data and processes that lack rigour to determine whether someone should get a job or will be a good tenant?' says Baveja. 'I mean, these are serious decisions, right?'\n\nI admit to the Trooly team that even though their intentions seem entirely positive, the process still feels intrusive. 'I knew you were going to ask us about privacy, everyone does,' says Baveja. 'I laugh when people talk about privacy. Not because it's not a serious issue but think about how much information you give to the banks. Every bill, purchase, credit card, everything, and we just accept that, right?' Trooly complies with applicable privacy and data protection laws, including the Fair Credit Reporting Act in the United States. So if I wanted to, I could opt out of the ratings altogether, check and challenge my profile data. My report is deleted after ninety days. 'We're not doing anything in the shadows here,' says Baveja. 'Any trust mechanism, even asking a friend about someone, involves some level of intrusiveness. I think the dividing line is how you do it. Are you doing it ethically? Are you doing it transparently? We bend over backwards to make sure what we are doing is way better, way fairer and way more predictive than any other trust mechanisms being used today.'\n\nHere's the thing. Baveja is reassuringly competent, open and honest. I trust him. And he clearly has positive intentions to disrupt the existing background-checks system, which is seriously flawed. I was still left, however, with the feeling that current privacy and data protection laws don't offer a sufficient degree of protection when it comes to what Trooly customers could do with our data.\n\nAt the end of my day with the Trooly team, I tell them about Doris. Would the search have caught her if she had applied to UrbanSitter? The answer is a definitive 'yes'. My parents would have known that she didn't in fact belong to the Salvation Army, had no previous childcare experience and had a chequered criminal history. In other words, Doris would not have made the cut.\n\n'Trooly is helping us screen our caregivers in an objective and quantifiable way. But it's only part of the equation,' Lynn Perkins says. She makes an important admission: technology should never try to replace parental intuition. 'If a sitter shows up at your door and you get a weird feeling, it doesn't matter if they have passed checks, how well reviewed they were or what you thought about her online, say you are suddenly not feeling well and cancel. Go with your gut.' There's no question: at the end of the day, we're the ones who have to decide where to place our trust.\n\nIt's complicated. Our intuition may be strong but sometimes we tune into trust signals that are loud even though they are not in fact good indicators of reliability, honesty and competence. Doris, with her spectacles and Scottish accent, is a prime example. Someone looks the part, and it's only later you discover that your kids have been looked after by an armed bank robber and drug dealer.\n\nIn the future, online trust can only get faster, smarter and more prevalent. And that has to be a good thing in terms of helping us make more informed choices\u2013whether it's about hiring a lawyer, selling our house or taking care of our kids. At the same time, we don't want to lose what makes us human.\n\nMaking the odd mistake when it comes to trust, taking a leap, is sometimes how we open up new possibilities or find ourselves in unexpected situations, both exciting and dangerous. It's how we place our faith in strangers, without knowing what might come of it. For that matter, my mother's trust in Doris wasn't entirely misplaced. The funny thing was that while Doris turned out to be an accomplished criminal, she was also a rather good nanny.\n\n* The 'social graph' refers to the connections between you and the other people, places and things you interact with in an online social network.\n\n* Trooly was acquired by Airbnb in June 2017.\n\n#\n\n# Reputation is Everything, Even in the Dark\n\nDepending on how strictly you want to define the terms of an 'online trade', you could claim that the first thing to be bought and sold on the internet wasn't a CD or pizza, it was a small bag of weed. In the early 1970s, a group of students at Stanford University and Massachusetts Institute of Technology made an online drug deal via ARPANET, the precursor to the internet we know today. Since then, it has become remarkably fast and easy to score drugs online, through the so-called 'darknet'.\n\nYou can't get to the darknet using your regular web browser; it can only be accessed via an anonymizing software called Tor (an acronym for 'The Onion Router'). Instead of a web address ending in a .com or .org, darknet URLs are a hash of random letters and numbers that end in .onion. Originally developed by the US Naval Research Laboratory for the purpose of protecting government communications, Tor has become a handy privacy tool for journalists and human rights organizations that need to mask their browsing activity and hide their identity and location. Of course, its subterranean nature means it also attracts criminals who can exchange drugs and other illegal goods, from guns to child pornography, online and in relative obscurity.\n\nStumbling into the darknet is like stumbling into a shadowy and mysterious parallel universe, where everything looks oddly the same as on the regular web\u2013strangely familiar\u2013except that its consumer sites are selling AK-47s or counterfeit passports, instead of pre-loved Hermes handbags and Jamie Oliver cookbooks. It means a visitor doesn't need to be a hacker or computer whiz to navigate it; it's remarkably easy to find and buy illegal goods and services. Google doesn't search onion sites but Grams does (FYI, the address for Grams is: grams7enufi7jmdl.onion) and its site looks incredibly similar, from the brightly coloured rainbow logo to the 'I Feel Lucky' button.\n\nLet's say you type in 'ecstasy' on Grams. The search engine trawls cryptomarkets such as BlackBank, Mr Nice, Pandora and SilkRoad4 (now on its fourth life) and provides a list of results showing the name of the seller, price of the product, a brief description and the exact URL. As the creator of Grams told _WIRED_ magazine in an anonymous interview: 'I noticed on the forums and Reddit, people were constantly asking \"who had the best product X and was reliable and not a scam?\" I wanted to make it easy for people to find things they wanted on the darknet and figure out who was a trustworthy vendor.'\n\nI had read a lot about darknet drug sites but I was still gobsmacked at just how much they look like conventional e-retailers such as Amazon. They would appear reassuringly familiar to any online shopper. There's even the usual amount of competition and cornucopia of choice. It's just that the listings, row after row, are for cocaine, blotter (LSD), ecstasy, opioids, dimethyltryptamine (DMT), heroin, hash, cannabis and almost any other drug a user could possibly want. With just a few clicks, buyers can browse a mind-boggling selection, pay for the drugs in the traceless digital currency bitcoin (unattached to any central bank) and have them delivered unknowingly by the postman.\n\nIn October 2013, the darknet achieved notoriety when an illicit drug site called Silk Road was shut down by the FBI. The site owner and administrator, twenty-nine-year-old Ross William Ulbricht, aka Dread Pirate Roberts (DPR), was arrested at a public library in San Francisco. Convicted of money laundering, computer hacking and conspiracy to traffic narcotics, he has since been sentenced to life in prison. Appropriately, Ulbricht took his fictional namesake, Dread Pirate Roberts, from a character in the book and film _The Princess Bride._ In the story, Roberts is not one man but one of many who pass on the name, reputation and pirate business from one person to the next. At the time of the site's closure, the FBI estimated that Silk Road had 13,000 drug listings and had processed approximately $1.2-billion-worth of sales. DPR was believed to be making an estimated $20,000 a day in the 6\u201312 per cent commission the site charged on every transaction.\n\nUlbricht was a university-educated guy who grew up in Austin, Texas. A self-proclaimed idealist, he believed drug use was a personal choice. He was essentially a twenty-first-century digital libertarian. In Silk Road's code of conduct it stated, 'We refuse to sell or list anything the purpose of which is to harm or defraud another person.' DPR also wrote, 'Treat others as you wish to be treated.' Ulbricht wanted to create a trusted trading ground where people could buy and sell drugs free from violence and the reach of government laws. Critically, DPR was not just revered\u2013vendors and buyers on Silk Road trusted him. But once he was arrested, their belief was shaken in the safety of the whole system. It wasn't enough, however, to bring it crashing down. And other sites sprang up to replace Silk Road.\n\nHow are visitors persuaded to trust these sites? Shared control, for one thing. When a buyer places an order, the bitcoin goes into an escrow account and is only released to the vendor once the order has been confirmed as received. (It's a similar escrow system to the one used on Alibaba.) The buyer, seller and the site's administrators control the account; two out of the three must sign off on the deal before the bitcoin can be moved. It's designed to make it far more difficult for vendors and buyers to scam one another. On the whole, it works, although, like most systems, there are pitfalls and bad apples. For example, unscrupulous administrators can shut down a marketplace and make off at any time with all of their users' bitcoins escrowed in the site. And some do. It's known as an _exit scam_.\n\nIn March 2015 the popular Silk Road descendant Evolution mysteriously disappeared overnight. It had nothing to do with a government bust; just simple greed. The site's administrators, Verto and Kimble, emptied the market's bitcoin coffers and ran off with an estimated $12 million. A few days after the incident, a user named NSWGreat, who had previously self-described herself as an Evolution 'public relations' staffer, put a post on Reddit's darknet markets forum.\n\n'I am so sorry, but Verto and Kimble have f\u2014ked us all. I have over $20,000 in escrow myself from sales,' NSWGreat wrote. 'I'm sorry for everyone's losses, I'm gutted and speechless. I feel so betrayed.'\n\nSoon after Evolution shut down, lesser-known markets like Abraxas, Amazon Dark, Blackbank and Middle Earth also subsequently disappeared for unknown reasons, but it's assumed they were also exit scams. The utopian ideals of the darknet\u2013'us against the government and over-regulation'\u2013seemed to have had their heyday. And, of course, vendors and buyers who have been ripped off have little recourse, beyond warning each other. They can hardly go to the ombudsman and complain their LSD didn't turn up and demand their bitcoins back. How much damage did that do to trust in the darknet?\n\nDr James Martin is a renowned expert on cryptomarkets. He's a professor at Macquarie University and author of _Drugs on the Dark Net._ I interviewed Martin from his flat in Melbourne just after he had come back from, appropriately, Amsterdam. He is particularly fascinated by how cryptomarkets use technology to allow people to communicate and engage in new forms of self-governance outside state and traditional regulatory control.\n\n'What interests me is the way technology could be used to transform an illicit drugs trade that has been debased and corrupted by four decades of the global War on Drugs,' he says. 'The fact is, you have got thousands of people\u2013drug users and drug retailers\u2013who have been stereotyped as \"untrustworthy\", who don't have a reputation for being the most reliable people. So how do they create highly functioning markets that are non-violent and self-regulate based on trust?'\n\nMartin thinks law enforcement agencies and the police have been surprised by the darknet. '[They've seen] how criminals are creating peaceful communities selling dangerous products; communities that work really well the vast majority of the time.'\n\nYet haven't the exit scams now damaged that trust in the darknet? 'There isn't a trust crisis but people are definitely more sceptical,' he says. 'Scams undermined the faith in the whole system. It wasn't the \"big bad Drug Enforcement Agency (DEA)\" that was destroying the markets; it was people from the inside. That really chipped away the idealism and shook people's faith in the darknet community.'\n\nMartin likens it to research one of his colleagues carried out looking at desertion within armies and how it affected the remaining troops. The study found that if fellow soldiers were killed by the enemy, they were seen to be doing their job and it stiffened the resolve of those left. If they fled voluntarily, however, it undermined faith in what they were all fighting for, faith in the whole system. 'That,' says Martin, 'is what we saw with the exit scams.'\n\nEven so, the scams didn't deliver a mortal blow. 'Vendors and customers have just moved past them and repopulated other sites,' Martin says. It's like a game of whack-a-mole: as soon as one site closes down, another pops up.\n\nDespite the Silk Road bust and others since, the drug business on the darknet, like other forms of e-commerce, is thriving. According to the 2016 Global Drug Survey, approximately 22 per cent of UK drug users have sourced and bought their drugs online. Globally, almost one in ten participants reported having bought drugs off the darknet. Significantly, 5 per cent of respondents stated that they had not bought drugs before purchasing them through darknet markets. The darknet makes drug-buying easier but also seem less risky to some people.\n\nAn in-depth study by the European think tank Rand found that the number of transactions on illegal drug sites has tripled since 2013. The United Kingdom is the largest online drug market outside the United States, with darknet sellers doing close to 21,000 drug deals a month. British vendors took home on average around \u00a35,200 per month. But the most successful darknet dealers were making upwards of \u00a3200,000.\n\nThe truly fascinating thing, however, about drug marketplaces is not how many there are or how much dealers make but just how well they work. While we should rightfully fear that these sites could lead to higher levels of drug use, it's hard not to be impressed by their apparent quality control and efficiency. People, ordinary people, transact with each other globally, at high speed and in great volume. The darknet can teach us a lesson or two about trust.\n\nIn street dealing, selling is often restricted to customers a dealer already knows. New customers tend to be introduced to dealers only by a broker, a trusted intermediary. 'My mate is not a cop, you can sell her drugs and she won't dob you in,' is how a broker might vouch for someone. Trust is crucial in conventional drug-dealing but it's small-scale, interpersonal trust, meaning the trust lies directly between a few people.\n\nIn contrast, the darknet is an open network. Choice isn't limited by whom you know or where a dealer lives. 'Cryptomarkets represent a kind of super broker. They are able to facilitate contact between many, many more vendors and customers than any individual possibly could,' says Martin. 'But the trust in this instance is represented in quantitative metrics, such as reviews and ratings that substitute for the personal trust that has been historically critical for the drug trade. And that is transformative.'\n\nTrust used to be a very personal thing: you went on the recommendations of your friends or friends of friends. By finding ways to extend that circle of trust exponentially, technology is expanding markets and possibilities. In the case of the darknet, it is creating trust between the unlikeliest of characters, despite a heavy cloak of anonymity.\n\nThe darknet is peopled by hundreds of thousands of drug users and vendors who would commonly be stereotyped as untrustworthy, the worst of the worst, yet here they are creating highly efficient markets. Effectively, they are creating trust in a zero-trust environment. Nobody meets in person. There are obviously no legal regulations governing the exchanges. It looks like a place where buyers could get ripped off. Theoretically, it would be easy for dealers to send lower quality drugs or not deliver the goods at all. Yet this rarely happens on the darknet and, overall, you're more likely to find buyers singing hymns of praise about the quality of the drugs and reliability of the service.\n\nAn extensive report published in February 2016 by the European Monitoring Centre for Drugs and Drug Addiction (EMCDDA) revealed that drugs available through darknet markets tend to be of a higher purity than those available on the streets. Similarly, a study by Energy Control, a Spanish think tank, also confirmed a quality premium in darknet drugs. Volunteers were asked to send random samples of drugs they had bought from online sites and offline dealers for testing. More than half of the darknet cocaine samples contained nothing but cocaine, compared with just 14 per cent of those bought on the Spanish streets. Even the FBI testified in the Silk Road hearings that, of more than a hundred purchases of drugs they made online before the closing of the site, all showed 'high purity levels'.\n\nIt seems dealers are more honest online. And that tells us something about how people create trust in this criminal ecosystem, with the help\u2013or supervision\u2013technology can provide. Given the nature of its wares, the darknet may seem like an alien and subterranean world\u2013and some of it undoubtedly is\u2013but at its core, it's about people connecting with other people. It's just another incarnation of the new kind of trust-building that technology has facilitated. The same dynamics, the same principles in building digital human relationships, apply. In that sense, it's almost comically conventional.\n\nTurns out, drug dealers care about their online brand and reputation and customer satisfaction as much as Airbnb hosts or eBay sellers. A typical vendor's page will be littered with information, including: how many transactions they have completed; when the vendor registered; when the vendor last logged in; and their all-important pseudonym. It will also feature a short description about why a user should buy drugs from them, refund policy information, postage options and 'stealth' methods (measures used to conceal drugs in the post). Even if you're not in the market for what they're selling, it's hard not to be impressed; vendors put in real effort to demonstrate their trustworthiness.\n\nTraditionally, the words 'drug dealer' bring to mind some thuggish or shady lowlife, a badly educated tough swaggering around in a leather jacket or lurking threateningly on a street corner. Someone you don't want to mess with. True or not, that persona of a heartless intimidator doesn't work on cryptomarkets. There, vendors need to project a cleaner image. Some even display specific logos and taglines and their brand message is loud and clear: 'We care about you' or 'Your satisfaction is our priority'.\n\nIndeed, marketing strategies used on the darknet look remarkably like standard ones. There are bulk discounts, loyalty programmes, two-for-one specials, free extras for loyal customers and even refund guarantees for dissatisfied punters. It's common to see marketing techniques such as 'Limited Stock' or 'Offer ends Friday' to help boost sales. While reviewing these deals, I had to remind myself I was looking at an illegal drugs marketplace, not shopping for shoes on Zappos.\n\nSome vendors, eager to build brand, label their drugs as 'fair trade' or 'organic' to appeal to 'ethical' interests. 'Conflict-free' sources of supply are also available for the benevolent drug taker. 'This is the best opium you will try, by purchasing this you are supporting local farmers in the hills of Guatemala and you are not financing violent drug cartels,' promised a seller on Evolution (before it was shut down).\n\nNew vendors will offer free samples and price-match guarantees to establish their reputation. Promotional campaigns are rife on 20 April, also known as Pot Day, the darknet's equivalent of Black Friday. (The date of Pot Day comes from the North American slang term for smoking cannabis which is 4\/20.) 'It's not anonymity, Bitcoins or encryption that ensure the future success of darknet markets,' writes Jamie Bartlett, author of _The Dark Net._ 'The real secret of Silk Road is great customer service.'\n\nAfter a buyer receives their drugs, they are prompted to leave a star rating out of five. Nicolas Christin, an associate research professor at Carnegie Mellon University, analysed ratings from 184,804 pieces of feedback that were left on Silk Road over the course of an eight-month period. On the site, 97.8 per cent of reviews were positive, scoring a four or five. In contrast, only 1.4 per cent were negative, rating one or two on the same scale. Just how much can people trust those good reviews?\n\nSome observers suspect the darknet suffers from _review inflation_. Similar studies have been conducted on conventional marketplace rating systems and found that feedback is also overwhelmingly positive. For instance, on eBay less than 2 per cent of all feedback left is negative or neutral. One explanation is that dissatisfied customers are substantially less likely to give feedback. It means the most important information, the negative reputation data, is not being captured.\n\nSocial pressure encourages us to leave high scores in public forums. If you have experienced an Uber driver saying at the end of a trip, 'You give me five stars, I'll give you five stars,' that's tit for tat or grade inflation in action. I know I'm reluctant to give a driver a rating lower than four stars even if I have sat white-knuckled during the ride as he whizzed through lights and cut corners. Perhaps that is why Jason Dalton was a 4.7. Drivers get kicked off the Uber platform if their rating dips below 4.3 and I don't want to be responsible for them losing, in some instances, their livelihood. Maybe they are just having a bad day. That, and the driver knows where I live. In other words, reviews spring from a complex web of fear and hope. Whether we are using our real name or a pseudonym, we fear retaliation and also hope our niceties will be reciprocated.\n\nIt would be easy to conclude, then, that the ratings we rely on to make assessments are often not an accurate reflection of the experience. But they can still make us more accountable to one another. For instance, I sometimes drop towels on the bathroom floor when I am staying in a hotel. But I would never do this as a guest staying in someone's home on Airbnb. Why? Because I know the host will rate me, and that rating is likely to have an impact on my booking requests being accepted by other Airbnb hosts in the future. It illustrates how online trust mechanisms will have on impact on our real-world behaviours in ways we can't yet even imagine.\n\nWhile ratings might be exaggerated, or even fabricated, some sites are taking steps to reduce the problem of positive bias. For instance, in 2014 Airbnb introduced a _double-blind_ process where guest and host reviews would only be revealed after both are submitted, or after a fourteen-day waiting period, whichever came first. The result was a 7 per cent increase in review rates and a 2 per cent increase in negative reviews. 'These may not sound like big numbers but the results are compounding over time,' says Judd Antin, director of research at Airbnb. 'It was a simple tweak that has improved the travel experiences of millions of people since.' Like any game, it's about figuring out the rules that put downward pressure on an unwanted behaviour until it doesn't exist any more.\n\nThere is, however, another way to look at the 97.8 per cent of positive reviews on darknet sites. Perhaps they are an accurate reflection of a market functioning remarkably well most of the time, with content customers. Even if review systems are not perfect, and bias is inevitable, it seems that they still do their job as an accountability mechanism of social control. Put simply, they make people behave better.\n\nA question I have often asked myself is whether limiting people to a score or star rating is really that helpful. Many marketplaces are now asking people to rate against a particular trait that is more relevant to trust in a specific context. For instance, on BlaBlaCar, people are rated from one to five on their driving skills. On Airbnb, hosts are rated from one to five on their cleanliness, accuracy, value, communication, arrival and location. On Uber, when riders give a low or high rating, you must specify the reason for doing so, such as cleanliness, driving, customer service or directions. On drug marketplaces, how clean a dealer is and how well they drive is irrelevant. What would be the trust measures there?\n\nTurns out, the three key traits of trustworthiness\u2013competence, reliability and honesty\u2013also apply to drug vendors. To highlight reliability, many reviews point out the speed of response and delivery. For example, 'I ordered 11.30 a.m. yesterday and my package was in my mailbox in literally twenty-five hours. I'll definitely be back for more in the future,' commented a buyer on Silk Road 2.0. Other reviews focus on the product quality: 'Amazing weed. Fast shipping. Packaging very secure. Took me a bit long to get in the double-seal vacuum seal but well worth it. Would highly recommend.' One of the ways skills and knowledge are reviewed is how good a vendor's 'stealth' is, that is, how cleverly they disguise their product so that it doesn't get detected. 'Stealth was so good it almost fooled me,' wrote a satisfied buyer on an MDMA listing on the AlphaBay market. Established vendors are very good at making it look (and smell) like any old regular package. Excessive tape or postage, reused boxes, presence of odour, crappy handwritten addresses, use of a common receiver alias such as 'John Smith' and even spelling errors are bad stealth.\n\nThere is a clear incentive for vendors consistently to provide the product and service they promise: the dealers with the best reviews rise to the top. No feedback, either negative or positive, can be deleted, so there is a permanent record of how someone has behaved. Just like the Maghribi traders and Alibaba sellers discovered, past behaviour is used to predict future behaviour. 'The future can therefore cast a shadow back upon the present,' Robert Axelrod wrote in _The Evolution of Cooperation._ 20 Or to put it another way, vendors have a vested interest in keeping their noses clean right from the word go.\n\nReputation is trust's closest sibling; the overall opinion of what people think of you. It's the opinion others have formed based on past experiences and built up over days, months, sometimes years. In that sense, reputation, good or bad, is a measure of trustworthiness. It helps customers choose between different options and, with luck, make better choices. It encourages sellers to be trustworthy, in order to build that reputation, and it weeds out those who aren't. It isn't quite that simple, however. Price is also a factor in the value of reputation, and reputation can influence price.\n\nConsider this scenario: two vendors are both offering Purple Haze cannabis on exactly the same shipping and refund terms. One vendor, let's call him BlazeKing, has only three reviews, with an average score of 4, and is offering the cannabis for $12.50\/gram. The other vendor, CandyMan2, has fifty-two reviews, with an average score of 4.9, and is offering the same Purple Haze but for a slightly higher price at $12.95\/gram. Who to go with? Obviously, the majority of customers would choose CandyMan2. The reviews reduce uncertainty for the buyer.\n\nCandyMan2's higher reputation comes with a price to the customer, though not a huge one. The value of the reduced risk is $0.45\/gram. In this sense, online reputation is functioning as a risk premium.\n\nNow let's say the choice is between BlazeKing, still with three reviews and cannabis for sale at $12.50\/gram, and another vendor, FlyingDynamite, also with fifty-two reviews and an average score of 4.9, but selling the same drugs for $16.50\/gram. Now the risk premium for a vendor with a higher reputation is $4\/gram. Is the vendor's reputation worth the significant difference in price? For some, it may be; for others, not. Even reputation, while valuable, has a price ceiling. There is only so much more a customer is willing to pay for goods or services based on a seller's high reputation.\n\nStill, I was intrigued how influential a vendor's reputation really is on cryptomarkets. So I tracked down a buyer, let's call him Alex. He describes himself as a 'casual drug taker' who likes to smoke a lot of weed and occasionally take ecstasy at the weekend. Towards the end of 2014, Alex switched from buying drugs from a dealer who lived relatively close by, to buying through cryptomarkets. What made him switch? His answer confirms everything we've just explored. 'Rather than buying drugs from a friend of a friend of a guy I met in a bar, I can buy drugs after reading dozens of reviews of their service,' he explains. 'I feel confident I am getting exactly what I am paying for.' It echoes what drug surveys indicate: 60\u201365 per cent of respondents say that the existence of ratings is _the_ motivation for using darknet marketplaces.\n\nAnd that brings us back to the matter of feedback and reputation systems on the darknet. Those systems make both sides more accountable but they are not infallible. Ratings are gamed in a similar way to other sites such as Amazon, Yelp and TripAdvisor. A common trick is a practice known as _padding_ feedback. Vendors essentially purchase their own drugs from a series of fake buyer accounts they have created. The glowing reviews look legitimate when they have in fact been created by the vendor. It's the online equivalent of stuffing the ballot box. Politicians do it. Advertisers do it. TripAdvisor is bedevilled by it. And drug dealers do it.\n\nAn industry of darknet 'marketing' services has sprung up, peopled by fibbers and promoters who are willing to create rave reviews and posts to help boost a vendor's reputation. 'Hi, my name is Mr420 and we started out and still are a small group of college public relations majors,' wrote a darknet PR vendor. 'We would be interested in keeping your product, thread or listing at the top of the forums.' Fake reviewers on Amazon will get free books, and hotels will often offer discounts for a positive review on TripAdvisor. The likes of Mr420 will pad feedback in exchange for free drug samples. Vendors regard it simply as brand management, doing something to make themselves look better.\n\nIt's the same practice Amazon took a suit against in a landmark reputation case. On 16 October 2015, in Washington, DC, the company sued 1,114 individuals for selling positive five-star reviews to Amazon sellers and Kindle authors. All the defendants in the case were advertising their services on Fiverr, an online marketplace where freelancers offer to do minor tasks for a flat rate of $5.\n\nIt might be hard to see how Amazon could suffer a loss of revenue from dodgy reviews; products with high-star ratings sell more, right? But Amazon was smart enough to know it needed to crack down on fake reviews because they undermine the foundations of trust in online marketplaces. If reviewers and their reviews can't be trusted, the whole system falls.\n\nBut where is the line? Say I send this book to a hundred friends and colleagues, and ask them to leave a nice review on Amazon to help boost sales. Is that gaming the system or just common-sense marketing? As the adage goes, fake it until you make it.\n\n'The weird thing is that even though there is a certain amount of gaming, there is an acceptance that it will take place and it's okay,' James Martin says. 'New vendors will tell you that to break into the market they have to generate false reviews themselves. If you have no feedback in your vendor profile, you are not an attractive proposition. So fake reviews get the ball rolling for new entrants.' It's simply the way markets function; a feedback system will never be perfect.\n\nGaming the feedback system is also used by rival vendors who want to gain market share. In the case of drug sales, how do dealers compete with one another when violence and turf wars are no longer an option? They engage in online wars. One tactic is sock puppetry (or 'socking'), where a rival hides behind an online identity in order to tarnish the reputation of a competitor. It's a common behaviour, even esteemed professors do it; quite badly, in some cases.\n\nTake Professor Orlando Figes, a critically acclaimed and prize-winning author, who has written eight books. In 2010, he was caught posting damaging critiques of his rivals' books on Amazon. Under the aliases 'orlando-birkbeck' and 'Historian' (drug dealers could teach him a thing or two about selecting a better pseudonym), he called his competitors' works 'dense', 'pretentious' and 'the sort of book that makes you wonder why it was ever published'. Stupidly, he used the same pseudonyms to praise his own work in illustrious detail: 'Beautifully written... leaves the reader awed, humbled yet uplifted... a gift to us all,' Historian wrote.\n\nWhen the scandal went public, the professor initially, and somewhat ungallantly, blamed all the reviews on his wife, the barrister Stephanie Palmer. It did not help the situation. Figes ended up paying damages to rival historians Dr Rachel Polonsky and Robert Service, whose work he had slated. In an apology statement he issued to the media, Figes said, 'Some of the reviews were small-minded and ungenerous, but they were not intended to harm.' The same purposeful sniping happens on the darknet. Vendors pretend to be unsatisfied buyers and leave bad reviews.\n\nBut gaming can only go so far. Indeed, machine learning systems are already being developed to identify and weed out deceptive reviews. A team of researchers at Cornell University has developed software that can detect _review spam_. In a test on 800 reviews of Chicago hotels on TripAdvisor, the program was able to pick out the deceptive reviews with almost 90 per cent accuracy. In contrast, Cornell's human subjects only managed to pick the fakes about 50 per cent of the time.\n\nIt turns out people are beautifully predictable when writing fictional reviews, using similar syntax, language, grammar, punctuation, too many long words and even similar spelling mistakes. The Cornell researchers found that deceivers use more verbs and long words than truth tellers, while the genuine reviewers used more nouns and punctuation.\n\nNo doubt these types of review filter will become increasingly sophisticated and commonplace, so we will be better able to trust a review is legit. But there is another simple solution when it comes to countering deceptive reviews: word of mouth. An upstanding community may not be something we associate with the drug trade but darknet markets have a strong sense of community with clear norms, rules and cultures. Users frequently chat to each other on discussion forums such as the DarkNetMarkets on Reddit, publicly calling out dodgy vendors. 'I was looking at this vendor a few hours ago and they had zero feedback. Now they have a bunch,' wrote one user. Customers who continually ask for refunds, claiming that their goods did not show up, are also likely to be shamed.\n\nThere are also websites such as DarkNet Market Avengers (dnmavengeradt4vo.onion), that use trained chemists to do random testing of darknet drugs. Users send samples of their drugs to Energy Control, a drug-testing lab funded by community donations. It tests the products and sends the results back to the user. For instance, if LSD is found to be 'under-dosed' or heroin is found to contain something dangerous like Carfentanil (an extremely potent synthetic opioid which can be life-threatening), the results are posted on the DNM Avengers site, including details of the specific vendor who sold the product.\n\nThe result is that fraudsters on both sides of the market are relatively quickly outed and driven out. As James Martin beautifully puts it, 'The darknet is really not dark. Thousands of people hold torches to shine the light on how other people behave. You no longer have to rely on one person but the collective judgements of the entire darknet community.'\n\nWithin the next five years, darknet sites could be to street drug dealers what Amazon is to local booksellers or Airbnb is to hotels, even if they do raise different and serious ethical questions. On the one hand, cryptomarkets mean that drugs will be more easily available to more people, which cannot be a good thing. On the other, they reduce the length of the supply chain and some of the risks and criminal behaviour associated with conventional drug-dealing.\n\nEither way, the systems work because customers become enfranchised in them. Technology empowers customers to hold vendors to account and, ultimately, it is only trustworthy vendors who will survive. E-commerce is e-commerce and, even on the darknet, reputation is everything.\n\n#\n\n# Rated: Would Your Life Get a Good Trust Score?\n\nOn 14 June 2014, the State Council of China published an ominous-sounding document called 'Planning Outline for the Construction of a Social Credit System'. In the way of Chinese policy documents, it was a lengthy and rather dry affair but it contained a radical idea. What if there was a national _trust score_ that rated the kind of citizen you were?\n\nImagine a world where many of your daily activities were constantly monitored and evaluated\u2013what you buy at the store and online, where you are at any given time, who your friends are and how you interact with them, how many hours you spend watching content or playing video games, and what bills and taxes you pay (or not). It's not hard to picture, because most of that already happens, thanks to all those data-collecting behemoths like Google, Facebook and Instagram, or health-tracking apps such as Fitbit that can capture your moves and location at any given time. But now imagine a system where all these behaviours are rated as either positive or negative and distilled into a single number, according to rules set by the government. That would create your _Citizen Score_ and it would tell everyone whether or not you were trustworthy. Plus, your rating would be publicly ranked against that of the entire population and used to determine your eligibility for a mortgage or a job, where your children can go to school\u2013or even just your chances of getting a date.\n\nA futuristic vision of Big Brother out of control? No, it's already getting underway in China where the government is developing a system to rate the trustworthiness of its 1.3 billion citizens. I'm sure George Orwell has rolled over in his grave a couple of times in recent years, but this idea, called the 'Social Credit System' (SCS), must have him doing frantic 360-degree turns in his coffin.\n\nThe Chinese government is pitching the system as a desirable way to measure and enhance 'trust' nationwide and to build a culture of 'sincerity'. As the policy states, 'It will forge a public opinion environment where keeping trust is glorious. It will strengthen sincerity in government affairs, commercial sincerity, social sincerity and the construction of judicial credibility.'\n\nOthers aren't so sanguine about its wider purpose. 'It is very ambitious in both depth and scope, including scrutinizing individual behaviour and what books people are reading. It's Amazon's consumer tracking with an Orwellian political twist,' is how Johan Lagerkvist, a Chinese internet specialist at the Swedish Institute of International Affairs, describes the Social Credit System (SCS). Dr Rogier Creemers, a postdoctoral scholar specializing in Chinese law and governance at the Van Vollenhoven Institute at Leiden University, who published a comprehensive translation of the plan, compared it to 'Yelp reviews with the nanny state watching over your shoulder'.\n\nFor now, technically, participating in China's Citizen Scores is voluntary. But by 2020 it will be mandatory. The behaviour of every single citizen and legal person in China (which includes every company or other entity) will be rated and ranked, whether they like it or not. Teachers, scientists, doctors, charity workers, government administrators, members of the judicial system and even sports figures will be under special scrutiny. 'Big data will become the most important and most powerful driver to accelerate the modernization of governmental governance capacity,' states the plan.\n\nPrior to the national rollout in 2020, the government is taking a watch-and-learn approach. In this marriage between Communist oversight and capitalist can-do, the government has given a licence to eight private companies to come up with systems and algorithms for social credit scores. Predictably, data giants currently run two of the best-known projects.\n\nThe first is with China Rapid Finance, a partner of the social network behemoth Tencent and developer of the messaging app WeChat with more than 850 million active users. The other is run by the Ant Financial Services Group (AFSG), an affiliate company of Alibaba, and is called Sesame Credit. Ant Financial sells insurance products and provides loans to small- and medium-sized businesses. However, the real star of Ant is AliPay, its payments arm, which people use not only to buy things online but also for restaurants, taxis, school fees, cinema tickets and even to transfer money to each other.\n\nSesame Credit has also teamed up with other data-generating platforms, such as Didi Chuxing, the ride-hailing company that was Uber's main competitor in China before it acquired the American company's Chinese operations in 2016, and Baihe, the country's largest online matchmaking service. It's not hard to see how that all adds up to gargantuan amounts of 'big data' that Sesame Credit can tap into to assess how people behave and rate them accordingly.\n\nSo just how are people rated? Individuals on Sesame Credit are measured by a score ranging between 350 and 950 points. Alibaba does not divulge the 'complex algorithm' it uses to calculate the number but they do reveal the five factors taken into account. The first is _credit history_. For example, does the citizen pay their electricity or phone bill on time? Do they repay their credit card in full? Next is _fulfilment capacity_ , which it defines in its guidelines as 'a user's ability to fulfil his or her contract obligations'. The third factor is _personal characteristics_ , which is verifying personal information such as someone's mobile phone number and address. But it's the fourth category, _behaviour and preferences_ , where it gets interesting and, some might say, more sinister.\n\nUnder this system, something as innocuous as a person's shopping habits become a measure of character. Alibaba admits it judges people by the types of product they buy. 'Someone who plays video games for ten hours a day, for example, would be considered an idle person,' says Li Yingyun, Sesame's technology director. 'Someone who frequently buys diapers would be considered as probably a parent, who on balance is more likely to have a sense of responsibility.' So if a citizen is buying socially approved items, like baby supplies or work shoes, their score rises. But if they're buying _Clash of Clans_ , _Temple Run 2_ or any video game, and thus looking like a lazy person, their score takes a negative hit. (I wonder how long it will take to get to the point where the system can judge their behaviour within a game? Maybe they will get a few points for being a 'nicer' player by, say, helping another player's avatar in _World of Warcraft_.)\n\nSo the system not only investigates behaviour\u2013it shapes it. It 'nudges' each of those closely monitored citizens away from purchases and behaviours the government does not like.\n\nAnd it's not just about purchases or pastimes. Friends matter, too. The fifth category is _interpersonal relationships_. What do their choice of online friends and their interactions say about the person being assessed? Sharing what Sesame Credit refers to as 'positive energy' online, nice messages about the government or how well the country's economy is doing, will make your score go up. For anyone who's read _The Circle_ by Dave Eggers, or seen the film, this might sound nightmarishly familiar. 'You and your ilk will live, willingly, joyfully, under constant surveillance, watching each other always, commenting on each other, voting and liking and disliking each other, smiling and frowning, and otherwise doing nothing much else,' writes Eggers. 'Secrets are lies. Sharing is caring. Privacy is theft.'\n\nAlibaba is adamant that, currently, anything negative posted on social media does not affect scores (we don't know if this is true or not because the algorithm is secret). But you can see how this might play out when the government's own Citizen Score system officially launches in 2020. Even though there is no suggestion yet that any of the eight private companies involved in the ongoing pilot scheme will ultimately be responsible for running the government's own system, it's hard to believe that the government will not want to extract the maximum possible amount of data for its SCS from the pilots\u2013particularly Alipay's Sesame and Tencent's WeChat. If that happens, and continues as the new normal under the government's own SCS, it will result in private platforms acting essentially as spy agencies for the government. They may have no choice. 'Government and big internet companies in China can exploit \"big data\" together in a way that is unimaginable in the West,' says Creemers. 'There are ample reasons to assume that whatever data the Chinese government wants, it can get.'\n\nPosting dissenting political opinions or links mentioning Tiananmen Square has never been wise in China but now it could directly hurt a citizen's rating. But here's the real kicker. The system could have a Kevin Bacon-like connection built in. A person's own score will depend on what their online friends say and do, beyond their own contact with them. If someone they are connected to online posts a negative comment on, say, the Shanghai stock market collapse (a massive embarrassment to the Chinese regime), their own score will also be dragged down. Talk about guilt by association.\n\nSo why have millions of people already signed up to what amounts to a trial run for a publicly endorsed government surveillance system? There may be darker, unstated reasons\u2013fear of reprisals, for instance, for those who don't put their hand up\u2013but there's also a lure, in the form of rewards and 'special privileges' for those who show themselves to be 'trustworthy' on Sesame Credit.\n\nIf their score reaches 600, they can take out a 'Just Spend' loan of up to 5,000 yuan (around $1,000) to use to shop online, as long as it's on an Alibaba site. Reach 650 points, they may rent a car without leaving a deposit. They are also entitled to faster check-in at hotels and use of the VIP check-in at Beijing Capital International Airport. Those with more than 666 points can get a cash loan of up to 50,000 yuan (more than $10,000), obviously from Ant Financial Services. Get above 700, they can apply for Singapore travel without supporting documents, such as an employee letter. And at 750, they get fast-tracked application to a coveted pan-European Schengen visa. 'I think the best way to understand the system is as a sort of bastard love child of a loyalty scheme,' says Rogier Creemers. 'Like the trust systems on eBay put together with an air-miles-type rewards programme.'\n\nHigher scores have already become a status symbol, with almost 100,000 people bragging about their scores on Weibo (the Chinese equivalent of Twitter) within months of launch. A citizen's score can even increase or decrease their odds of getting a date or a marriage partner, because the higher their Sesame rating, the more prominent their dating profile is on Baihe. 'A person's appearance is very important... but it's more important to be able to make a living,' says Zhuan Yirong, vice president of Baihe. 'Your partner's fortune guarantees a comfortable life.' More than 15 per cent of Baihe users are currently choosing to display prominently their Sesame scores on their profiles. It shows how readily many people will buy into a system like this, apparently blind to all its other implications.\n\nSesame Credit already offers tips to help individuals improve their ranking, including warning about the downsides of friending someone who has a low score. Undoubtedly, it won't be long before we see the rise of score advisors, who will share tips on how to gain points, or reputation consultants willing to offer expert advice on how strategically to improve a ranking or get off the trust-breaking blacklist. I wonder if people will hire reputation auditors to look into the assessments made about them. It could be a lucrative new venture for accounting outfits like PricewaterhouseCoopers (PwC).\n\nWe're also bound to see the birth of reputation black markets selling under-the-counter ways to boost trustworthiness. In the same way that Facebook 'likes' and Twitter followers can be paid for, and positive reviews on the darknet can be bought, individuals will pay to manipulate their score. But what happens to the poor and less educated people who can't afford or don't know how to enhance their score? Those who can't game or manipulate the system will be at a disadvantage. And what about keeping the system secure? Cyber hackers (some even state-backed) could go in and change or steal the digitally stored information. How much will a spouse or a future employee pay to purchase data on everything from the comments made in chatrooms to a history of every hotel room someone has checked into? It will give a whole new meaning to a 'background check'.\n\nThere's a compelling psychological reason people are willing to sign up to systems like this. Sesame Credit has tapped into a fundamental aspect of what makes us human: the desire to push ourselves to be better. We have been ranked and put on a curve since we were in primary school; most of us are wired to want continually to level up, to score higher than others. We're caught on the 'hedonic treadmill', the term psychologists give to the desire to keep improving our current situation. We stay on it because satisfaction and happiness seem forever just out of reach. For instance, when we finally reach a longed-for salary level, we'll experience a temporary high but before long we are hankering after more money. Or we post something on Facebook and it gets 121 'likes'; the pleasure soon gives way to a desire to post something that gets 125 or more 'likes'. In the world of Citizen Scores, this means as soon as we reach one level, we not only will need but will _want_ to ramp upwards. The desired social rung will always remain tantalizingly out of our grasp, making it almost impossible to be content with who and where we are.\n\nSesame Credit plays on this in several ways. For example, it encourages users to guess whether they have a higher or lower score than their friends. When they check their own score, it also displays all their friends' scores. But it's not simply about competitiveness. It also means they can see who might be dragging them down. Conversely, people will be tempted to cultivate friends with good reputations for their own advantage. Want a loan to start a business? Better start being extra nice to influential people with high trust ratings and drop the losers.\n\nIt will create some bizarre family dinner conversations. 'Honey, I noticed your score dropped by thirty-eight points today,' says a wife to her husband. 'You know we need to maintain a high score to get that home improvement loan. And have you forgotten that our family score goes on our son's college application next month? So what exactly did you do today, points-wise?'\n\nAs I learned more and more about China's Citizen Scores, I kept thinking about the bestselling novel _Super Sad True Love Story_ , which came out in 2010. It is set in a dystopian New York City in the not-too-distant future and author Gary Shteyngart imagines credit poles lining the streets that publicly announce your credit rating as you pass by. Lenny Abramov, a Russian Jew and the main American character, is something of a throwback because he still believes in the unquantifiable qualities of individuals. His boss and everyone else tell him that that kind of touchy-feely stuff doesn't matter and that he needs to get his rating up.\n\n_Super Sad_ features a number of gadgets, and one that Lenny wears is an '\u00c4pp\u00e4r\u00e4ti', a neck pendant with 'RateMe Plus' technology. It broadcasts personal data such as life expectancy, current cholesterol levels and even the wearer's sexual history. 'Let's say you walk into a bar, it says, \"OK, you're the third-ugliest man in here, but you have the fifth-best credit rating,\"' explained Gary Shteyngart in an interview with _The Atlantic_. Forget 'beer goggles', even Google Glass\u2013\u00c4pp\u00e4r\u00e4ti allows the wearer to check other people's ratings in real time to ensure they are not hooking up with someone dishonest, or at least rated as dishonest. It's not a very happy or trusting world. It's narcissistic, ruthless and exhibitionist. And it might not be far off.\n\nShteyngart's haunting satire is a commentary on society's obsession with needing to know where everyone else stands. It illustrates the perils of oversharing information with strangers and how everything from credit scores to health records could come to define us publicly, and with grave consequences, despite the whole business being made to look like an enticing game.\n\nIndeed, Sesame Credit is basically a 'big data' gamified version of the Communist Party's surveillance methods; the disquieting _dang'an_. The regime kept a dossier on every individual that tracked political and personal transgressions. A citizen's _dang'an_ followed them for life, from schools to jobs. People started reporting on friends and even family members, raising suspicion and lowering social trust in China. The same thing will happen with digital dossiers. People will have an incentive to say to their friends, spouses, family and colleagues, 'Don't post _that_. I don't want you to hurt your score but I also don't want you to hurt mine.'\n\nThe social pressure to conform to the party line and avoid any form of dissent will be immense. Negative or even contrary opinions will have no place. It's mind-blowing to imagine the sameness this system encourages, how it will stamp out individualism. Who will dare to speak out? Maya Wang, a spokesperson for Human Rights Watch China, based in Hong Kong, sees 'a scary vision of the future' in the system: currently there is intensive surveillance of 'sensitive groups, such as dissidents, but the Social Credit System goes to another level. This is an effort of surveillance of all people,' she says.\n\nRogier Creemers wholeheartedly agrees with Wang. 'The aim [in East Germany] was limited to avoiding a revolt against the regime. The Chinese aim is far more ambitious: it is clearly an attempt to create a new citizen.'\n\nThe new system reflects a cunning paradigm shift. As we've noted, instead of trying to enforce stability or conformity with a big stick and a good dose of top-down fear, the government is attempting to make obedience feel like gaming. It is a method of social control dressed up in some points-reward system. It's gamified obedience.\n\nIn a trendy neighbourhood in downtown Beijing, the BBC news services hit the streets in October 2015 to ask people about their Sesame Credit ratings. Most of the residents spoke about the upsides. But then, who would publicly criticize the system? Ding, your score might go down. 'It is very convenient,' one young woman said, smiling at the camera and proudly showing the journalist the score on her phone. 'We booked a hotel last night using Sesame Credit and we didn't need to leave a cash deposit.' Alarmingly, few people seemed to understand that a bad score could hurt them in the future, preventing them from, say, signing a lease. Even more concerning was how many people, despite signing up for Sesame Credit, had no idea that they were being constantly rated.\n\nThat kind of trusting ignorance is familiar, even if, in this case, it's taking place in a far more advanced form of dystopia. Think of all those Facebook users who were surprised to find out they were being used as data lab rats. We sign up to all kinds of services without really knowing what we're agreeing to and what is in our control to reject, if we choose to do so.\n\nCurrently, Sesame Credit does not directly penalize people for being 'untrustworthy'\u2013it's far more effective to lock people in with treats for good behaviour. But Hu Tao, Sesame Credit's chief manager, warns people that the system is designed so that 'untrustworthy people can't rent a car, can't borrow money or even can't find a job'. She has even disclosed that Sesame Credit has approached China's Education Bureau about sharing the list of its students who cheated in national examinations, in order to make them pay in the future for their dishonesty.\n\nPenalties are set to change dramatically when the government system becomes mandatory in 2020. Indeed, on 25 September 2016, the State Council General Office updated its policy entitled 'Warning and Punishment Mechanisms for Persons Subject to Enforcement for Trust-Breaking'. The overriding principle is simple: 'If trust is broken in one place, restrictions are imposed everywhere,' the policy document states. The punishments will seriously affect the social mobility of any transgressors.\n\nFor instance, people with low ratings will have slower internet connectivity; restricted access to more desirable restaurants, nightclubs or golf courses; and the removal of the right to travel freely abroad with, I quote, 'restrictive control on consumption within holiday areas or travel businesses'. Scores will influence a person's rental application, their ability to get insurance, eligibility for a loan and even social security benefits. Chinese citizens with low scores will not be hired by certain employers and will be forbidden altogether from obtaining some jobs, including in the civil service, journalism and legal fields, where of course you must be deemed trustworthy. People who do not rate well will also be restricted when it comes to enrolling themselves or enrolling their children in high-paying private schools. I am not fabricating this list of punishments. It's the reality Chinese citizens will face. As the government document repeatedly states, the Social Credit System will 'allow the trustworthy to roam everywhere under heaven while making it hard for the discredited to take a single step'.\n\nOnce again, life mirrors art. The system is strikingly similar to an episode of _Black Mirror_ , the critically acclaimed dystopian sci-fi television series. Each episode has a different cast, a different setting, even a different reality, notes Charlie Brooker, the creator of this darkly witty series. 'But they're all about the way we live now\u2013and the way we might be living in ten minutes' time if we're clumsy.' Meaning if we do not carefully handle new technologies, they will pull us into a strange future much sooner than we expect. Indeed, many of the imagined scenarios have since become reality, including a chatbot that mimics deceased relatives (yes, this now exists\u2013it is called Replika) and an obnoxious TV character who runs for political office to shake up a corrupt system. Not to mention a British PM who is forced to perform an insalubrious act with a pig on national television.\n\n'Nosedive', the first episode of the third season, envisions a world in which each of us continually chases after a desirable rating that sums up how people feel about us in real time. Your score, out of five stars, is affected by everyone\u2013family members, friends, co-workers and anonymous passers-by\u2013and is used for everything, no matter how trivial. Did the barista pour a nice swirl of milk on your coffee? You can reward him for that. Did a woman look you up and down the wrong way in your thirty-second elevator ride? You can make her pay for that. Be warned, though, your own rating might fall if she returns fire and rates you negatively.\n\nThe main character, Lacie Pound, lives her life constantly trying to please everyone in exchange for a few precious points. She has to work hard to maintain her solid but not outstanding 4.2 rating. She even practises her fake smile in the bathroom mirror every morning. Her value in this world is equivalent to her points, which she checks obsessively after every tiny interaction.\n\nWhat does Lacie's life tell us about the way the world is moving? Luciano Floridi, professor of philosophy and ethics of information at the University of Oxford, and the director of research at the Oxford Internet Institute, has an interesting way of framing it. Many make the claim to be an expert on 'digital disruption', but Floridi is the real deal. He is currently serving as the only ethicist on Google's advisory committee on the European Union's 'right to be forgotten' ruling. It's a role that has seen him crowned 'Google's Philosopher'.\n\nAccording to Floridi, there have been three critical 'de-centring shifts' that have altered our view in self-understanding: Copernicus's model of the earth orbiting the sun; Darwin's theory of natural selection; and Freud's claim that our daily actions are controlled by the unconscious mind.\n\nFloridi believes we are now entering the fourth shift in our world, as what we do online and offline merge into an _onlife._ He asserts that as our world increasingly becomes an infosphere, a mixture of physical and virtual experiences, we are acquiring _onlife personality_ \u2013different from who we innately are in the 'real world' alone. We see this writ large on Facebook, where people present a carefully edited or idealized portrait of their lives. When I look at some of my friends' streams\u2013beautiful pictures of holidays and their kids angelically dressed up in costumes\u2013I wonder, is this the same friend complaining about her husband and bratty five-year-old? I do the same. I edit the flaws and inconsistencies in my life, disguising my true messy self.\n\nIn _Black Mirror_ , Lacie's onlife personality is the extreme version of the future Floridi is talking about. Her life has become an exhausting, dramatic public performance. She has discovered that the only way she can afford her dream apartment is by raising her rating. So she visits a score counsellor for advice. Then, out of the blue, Naomi, an old school friend and social media star with a higher rating, asks Lacie to be maid of honour at her wedding. With many prime influencers (high-ranking wedding guests) attending, Lacie is convinced a tear-jerking bridesmaid's speech will get her the upvotes she needs. The speech, of course, turns into a disaster but that's not the point here. Or maybe it is.\n\nThe rating system in _Black Mirror_ is based on social approval, on likes and stars; as we see with Lacie, it encourages people to base relationships on personal gain and to fake behaviour. Disturbingly, that episode is not so very far from the 'onlife' we are living right now.\n\nThink about your Uber experiences. Are you just a little bit nicer and friendlier to the driver because you know you will also be rated? Indeed, judgement and scores are a two-way street. Some days, I like my conversations with drivers. I appreciate the serendipitous connections that sometimes emerge because I sit in the front and we talk. However, there are times I wish my Uber ride could be a simple transaction: where the driver does not know my name or have a picture of me; where I feel no pressure to be nice; where I am not asked what I do or how many kids I have.\n\nI once berated my husband down the phone during a trip because he told me he was running late, again. I was tired. It hadn't been a particularly good day. The driver said to me, 'If I were your husband and you shouted like that, I would be late.' It's rude that I shouted in his car but, frankly, it's none of his business.\n\nThe pressure to be rated means I am tempted to be falsely polite and not authentic. Yet it's not as if I am unused to being rated and reviewed. After a speech, I can see exactly how many people thought it was 'fantastic' or 'a waste of time'. People 'like' or 'dislike' my talks on TED, my slides on Slideshare and my posts on Medium. My students at Oxford break my teaching ability down into a detailed survey. People send me not just complimentary remarks but also scathing comments about my ideas and articles. I have learned to be comfortable having all my imperfections pointed out and even so I still worry about how I measure up on an Uber ride. I am human; I need to be liked and\u2013more to the point\u2013I want drivers to continue to pick me up.\n\nYet I don't want to worry ceaselessly about how I am being rated, whether I am late or punctual, rude or a darling, dirty or clean. I am frightened of ending up like Lacie. I am frightened my children will live in a society where scores become the ultimate truth of who they are. A paranoid world where they are under never-ending pressure to present an idealized portrait of their lives, not just for 'likes' but because of fear of how they're measuring up against others, minute by minute, year by year, and how it will affect their future prospects. How will I teach them what it means to be your authentic self?\n\nThe information we liberally post about ourselves today might end up being rated in some way down the track\u2013but that doesn't stop us. We have become hooked, literally, on displaying our lives and doings. A few years ago, Diana Tamir, an associate professor of psychology at Princeton University, and Jason Mitchell of Harvard's Neuroscience Lab, published a paper titled 'Disclosing information about the self is intrinsically rewarding'. Surveys of internet use show that more than 80 per cent of posts to social media sites consist simply of announcements about a user's immediate experience, such as what they are about to eat for dinner. The researchers asked participants to undergo functional magnetic resonance imaging (fMRI) scans while making these kinds of posts, to see what happens to their brains. And what happens is that our reward centres light up, just as they do with primary rewards such as food and sex. That is why we strive to post more. As Dave Eggers brilliantly puts it, it's the addictive digital-social equivalent of snack food, 'endless empty calories'. And it's far from nourishing.\n\nUber ratings are nothing compared to Peeple, an app launched in March 2016, which is like a Yelp for humans. It allows you to assign ratings and reviews to everyone you know\u2013your neighbour, your boss, your teacher, your spouse and even your ex. A profile displays a 'Peeple Number', a score based on all the feedback and recommendations you receive. Worryingly, once someone puts your name in the Peeple system, it's there and there's nothing you can do about it. You can't opt out. You must use your real name to leave a review, be over twenty-one and of course have a Facebook account. You must also affirm that you know the person you are rating based on one of three categories: professional, personal or dating (that is, how good they are at dating). The app is basically allowing you to judge and publicly reduce people to a grade without consent. Sound familiar?\n\nPeeple has forbidden certain bad behaviours including mentioning private health conditions, expressing profanities or being sexist (however you objectively assess that). There are, however, very few rules on how people are graded or standards about transparency. The app does include a feature called a 'Truth License'. According to the company's press release, 'The Peeple Truth License shows you everything that has been written about a person, whether it was published live on their profile or not. This allows you to make better decisions about the people around you.' One of the key reasons why Nicole McCullough, Peeple co-founder and a mother of two, developed the app was that, in a world where people don't know their neighbours, she wanted help to decide whom to trust with her kids.\n\nFittingly, the founders have been publishing a reality documentary on YouTube about every step involved in building Peeple. 'It doesn't matter how far apart we are in likes or dislikes,' co-founder Julia Cordray tells a total stranger in a bar in episode ten of the YouTube documentary. 'All that matters is what people say about us.'\n\nWhat are the consequences of boxing people into a number and a value?\n\nThis question comes to life in a particularly memorable scene in _Black Mirror_. Lacie is at the airport on her way to Naomi's wedding. Dressed like a pink pastel daydream, she approaches the check-in counter, all smiles. When she places her phone on the scanner her details, including her rating and PMA (positive mental attitude), flash on the check-in agent's screen. Unfortunately, her flight is cancelled and the airline representative can't book her on to another standby flight because Lacie's social credit score has dropped. On the way to the airport, her score dipped to a 4.183 after she got into a squabble with a woman while getting into her taxi. Her explanation doesn't matter; the system automatically blocks the agent from booking her on to the flight without the correct 4.2 rating. She ends up hitching a ride with a female truck driver who has a dismal 1.4 rating. The trucker shares the moving story of how she, too, was obsessed with her rating, until her husband got terminal cancer. He was denied treatment he badly needed; it was given to another patient with a higher score. 'So I figure,' the trucker tells Lacie with a smile, 'fuck it.' It makes me wonder, will we see similar movements of anti-rating people happy to be poorly ranked?\n\n_Black Mirror_ has become somewhat of a Magic 8 Ball, predicting the future. China's trust system might be voluntary as yet, but it's already having Lacie-like consequences. In February 2017, China's Supreme People's Court announced that 6.15 million people in the country had been banned from taking flights over the past four years for social misdeeds. The travel ban is being pointed to as the first indication of how people blacklisted in the Social Credit System, so called 'trust-breakers', will be punished. 'We have signed a memorandum... [with over] 44 government departments in order to limit \"discredited\" people on multiple levels,' says Meng Xiang, head of the executive department of the Supreme Court. Another 1.65 million people cannot take trains, because they are on the social credit blacklist for misdemeanours. They have been downgraded and branded as second-class citizens. This isn't TV. It isn't marketing. It's reality.\n\nWhere these systems really descend into nightmarish territory is that the trust algorithms used are unfairly reductive. They don't tell the whole story. They don't take into account context and valid reasons for a bad day. For instance, one person might miss paying a bill or a fine because they were in hospital; another may simply be a freeloader. But there is no one sitting and analysing every Citizen Score assessment, going, 'Oh, okay, she was having an operation and that explains why she didn't pay her credit card.' And therein lies the urgent challenge facing all of us in the digital world, and not just the Chinese. If life-determining algorithms are here to stay, and it certainly looks that way, we need to figure out how they can embrace the nuances, inconsistencies and contradictions inherent in human beings. We need to work out how they can reflect real life.\n\nYou could see China's so-called 'trust plan' as Orwell's _Nineteen Eighty-Four_ meets Pavlov's dogs. Act like a good citizen, be rewarded and be made to think you're having fun. It's worth remembering, however, that personal scoring systems have been present in the West for decades.\n\nMore than seventy years ago, two men called Bill Fair and Earl Isaac invented credit scores. They met at Stanford University in San Jose, California, where Fair was studying engineering and Isaac mathematics. They started their own company with just $400 apiece. The goal was to use _predictive analytics_ , and the new-fangled capabilities of computers, to give lenders a unified view of a person's credit risk. Specifically, the duo wanted to use algorithms to study customers' past behaviour, predict future behaviour and come up with a credit score. At the time, it was regarded as a radical concept.\n\nInitially, the idea of credit scores didn't take off. Fair and Isaac sent a letter to fifty of the largest lenders in the United States offering them the new technology. Only one responded. But in 1958, the first credit score, known today as FICO (short for the Fair Isaac Corporation), was created. Over the years, it has positively challenged many lenders' practices and prejudices. 'Good credit does not wear a coat and tie' was the headline on one advertisement. FICO proved time and time again that race, for example, was not a predictor of good credit risk and refused to put it in their scoring system.\n\nToday, companies use FICO scores to determine many financial decisions, including the interest rate on our mortgage or whether we should be given a loan. The score range is 300 to 850, with the high number representing less risk to the lender or insurer. Remarkably, it wasn't until 2003 that we could find out our actual score. Before then, they had been kept a secret. And despite the significance of credit scores to our lives, repeated studies show that more than 60 per cent of Americans still do not know their score or simply have not bothered to find out.\n\nFor the majority of Chinese people, it is not a case of knowing or not. In a catch-22, they have never had credit scores and so they can't get credit. 'Many people don't own houses, cars or credit cards in China, so that kind of information isn't available to measure,' explains Wen Quan, an influential blogger who writes about technology and finance. 'The central bank has the financial data from 800 million people, but only 320 million have a traditional credit history.' According to the Chinese Ministry of Commerce, the annual economic loss caused by lack of credit information is more than 600 billion yuan, approximately $97 billion.\n\nChina's lack of a national credit system is why the government is adamant that Citizen Scores are long overdue and badly needed to fix what they refer to as a _trust deficit_. In a poorly regulated market, the sale of counterfeit and substandard products is a massive problem. According to the Organisation for Economic Co-operation and Development (OECD), 63 per cent of all fake goods, from watches to handbags to baby food, originate from China. In late 2008, the Chinese Ministry of Health revealed six babies had died and almost 30o,000 had fallen ill after drinking baby formula deliberately laced with melamine, a toxic chemical used in plastics and fertilizer. Turns out, a local manufacturer had intentionally added the industrial chemical to mask low protein levels in watered-down formula. Since this massive breach of trust, Chinese customers have bought baby formula milk, loads of it, from overseas. So much so that some big British retailers such as Boots and Sainsbury's decided to set a two-can limit to prevent bulk buying to feed the Chinese market leaving a shortage of tins on the shelves.\n\nIn January 2017, Chinese authorities discovered a 'production hub' of around fifty factories that were generating counterfeit products designed to look exactly like well-known brands. Jack Ma has called fake goods 'cancer' to Alibaba but crackdown efforts to weed out fakes have had an uphill battle. 'Food security, counterfeiting and a lack of regulatory compliance are real issues for Chinese citizens. The level of micro corruption is enormous,' says Rogier Creemers. 'Up and down the ladder, trust is a huge problem in China. So if this particular scheme results in more effective oversight and accountability, it will likely be warmly welcomed.'\n\nThe government also argues that the system is a way to bring in those people left out of traditional credit systems, such as students, low-income households and those who have never borrowed money. Professor Wang Shuqin from the Office of Philosophy and Social Science at Capital Normal University in China recently won the bid to help the government develop the system that she refers to as 'China's Social Faithful System'. Without such a mechanism, doing business in China is risky, she stresses, as about half of the signed contracts are not kept. 'Especially given the speed of the digital economy, it is crucial that people can quickly verify each other's creditworthiness,' she says. 'The behaviour of the majority is determined by their world of thoughts. A person who believes in socialist core values is behaving more decently.' In other words, she regards the 'moral standards' the system assesses, as well as financial data, as a 'bonus'.\n\nIndeed, the State Council's primary objective is to raise the 'honest mentality and credit levels of the entire society' in order to improve 'the over-all competitiveness of the country'. In other words, the government is selling Citizen Scores as a tool to evaluate people more fairly and improve economic vitality.\n\nIs it remotely possible that the Social Credit System in China is in fact a more desirably transparent approach to surveillance in a country that has a long history of watching its citizens? 'As a Chinese person, knowing that everything I do online is being tracked, would I rather be aware of the details of what is being monitored and use this information to teach myself how to abide by the rules of government?' asks Rasul Majid, a Chinese blogger based in Shanghai who writes about behavioural design and gaming psychology. 'Or would I rather live in ignorance and hope\/wish\/dream that personal privacy still exists and that our ruling bodies respect us enough not to take advantage?' Put simply, Majid thinks that the system gives him a tiny bit more control over his data.\n\nOn the one hand, a social credit system will almost certainly encourage people to act more honestly and to abide by the rules. On the other, it's a deeply disturbing version of reputation economics that will give governments unprecedented control over what they consider good and bad ways to behave.\n\nWhen I tell people living in the Western world about the Social Credit System in China, their responses are fervent, visceral. After a speech I gave at a financial conference, a female banker remarked, 'We routinely do things that just five years ago would have made no sense to us, but that idea is bat-shit crazy.' Her sense of alarm was typical. Many people have asked if it is really true, if it is really happening _in China_. Surprisingly, very few people ask the more pertinent question, 'Could this happen in the Western world?' Or rather, when can we expect it?\n\nWe already rate restaurants, movies, books and even doctors. We've seen how Peeple rates people. You can even rate your bowel movements online (check out ratemypoo.com if you don't believe me). 'Yelpers', customers who regularly leave reviews on Yelp, will threaten hotels and restaurants with poor reviews if they don't please them by giving them, say, complimentary drinks. Authors have Amazon scores. Airbnb hosts and guests have cleanliness scores. Teachers have RateMyProfessors.com scores. Errand runners on Taskrabbit, Deliveroo drivers and a whole plethora of other 'gig workers' are rated (and they rate customers back). 'Klout scores', that claim to identify the most influential social media users, are even appearing on some people's r\u00e9sum\u00e9s as proof of their stellar reputation. Fitbit captures how much you move (or don't) and gives you a fitness score that it shares with multiple companies. On an app called DateCheck, you can even do an instant background check on someone you've just met in a bar. Its tagline is, fittingly, 'Look up before you hook up'. Facebook is now capable of identifying you in pictures without seeing your face; it only needs your clothes, hair and body type to tag you in an image with 83 per cent accuracy. It's kind of like how I can recognize my husband from a hundred metres away by his gait.\n\nIn 2015, the OECD published a study revealing that in the United States there are at least 24.9 connected devices per every one hundred inhabitants. All kinds of companies scrutinize the 'big data' emitted from these devices to understand our lives, desires and psyches, and to predict our future actions, in ways that we couldn't even predict for ourselves.\n\nWhen I get on the bus on my way to work, I put on my headphones. It's a morning ritual that gives me some sense of personal privacy in a crowded public space. My listening habits, especially the podcasts, audio books and news programmes I download, would provide a clear window into my political preferences, life stresses, religious views and various other interests. So what would happen if someone knew what I was listening to?\n\nOn 18 April 2017, a class-action lawsuit filed in a federal court in Chicago accused a high-end audio-equipment maker of spying on its customers' listening habits. After paying $350 for his QuietComfort 35 headphones, Kyle Zak, the lead plaintiff in the case, followed Bose's suggestion to 'get the most out of your headphones' by downloading its Connect app to his smartphone. He provided his name, email address and headphone serial number as part of the sign-up process. And like most of us, he handed over his information without much thought. The app adds functions such as the ability to customize the level of noise cancellation in the headphones. But the app also tracks the music, podcasts and other audio Bose customers listen to, and violates privacy rights by selling the information to various third parties, including a data-mining company called Segment.io. Shortly after the lawsuit was filed, Bose responded with a company statement: 'We'll fight the inflammatory, misleading allegations made against us through the legal system. Nothing is more important to us than your trust. We work tirelessly to earn and keep it, and have for over fifty years. That's never changed, and never will.'\n\nRegardless of the final legal outcome, the Bose case sparked further questions about the ethics of data collection. The fact is, many companies are not transparent about the data they take and what they are doing with it, or clear about how they monetize our personal information. And this applies to everything from coffee machines to headphones, running shoes to even sex toys. In 2017, We-Vibe paid more than $3.75 million to resolve privacy claims regarding vibrators remotely controlled with a 'connect lover' smartphone app. The sex toys were secretly collecting customer data, including highly intimate details such as the date and time of each use, temperature settings and what vibration intensity and mode users selected\u2013all of which were linked to owners' personal email addresses. What if the data was hacked? Do we want companies (or even governments) to know how we spend our most personal time and the details of our orgasms? In April 2017, another smart sex toy faced a massive security glitch over intimate surveillance. Svakom Siime Eye, a $249 app-enabled vibrator, has a tiny built-in camera designed for either private live-streaming or to 'know the subtle changes inside of your private areas'. The default password on the device is 88888888. If it is not reset, the device can be easily hacked. What's more, the manufacturer, Standard Innovations, can geolocate whenever the vibrator is in use.\n\nSmart phones and computer webcams can be co-opted for commercial and nefarious purposes. Next in line as potential spies are the digital voice assistants such as Amazon's Echo smart speaker called Alexa, now entering millions of our homes. Her tagline is, fittingly, 'Just Ask'. The artificially intelligent assistant is happy to help with all kinds of requests such as 'Alexa, what's on my calendar today?' or 'Alexa, play the Coldplay song I like.' And, of course, she is especially handy in buying things\u2013from Amazon, that is. But what if she was asked to assist with, say, a murder trial?\n\nIn November 2015, Victor Collins, a police officer from Arkansas, was found floating dead in the hot tub of his friend James Andrew Bates, who became a suspect. Two years later, attorney Nathan Smith, the lead prosecutor in the first-degree murder trial, ordered Amazon to hand over the audio recordings from Bates's digital assistant, used in the Echo speakers in his home. While it's unlikely any alleged murderer would have asked, 'Alexa, how do I strangle someone and hide a body?' the prosecution felt the recording might provide valuable clues as to what happened at Bates's house the night Collins was found dead.\n\nAmazon's attorneys contended the digital assistant has First Amendment rights protecting information gathered and sent by the device. Bates, however, told Amazon it could hand over the information. Maybe he believed it would prove his innocence, although it's also possible Bates thought the Echo device was only recording snippets of audio during the few seconds during and after 'hearing' a command. Aside from the other issues, the case raises a key question: how can you know when your always-connected digital assistant is recording what you say?\n\nAnd it is not just tech companies that are in on this. Governments around the world are already engaged in the business of monitoring, rating and labelling their own citizens. The National Security Agency (NSA) is not the only government digital eye in the US following the movements of citizens' lives. In 2015, the US Transportation Security Administration (TSA) quietly proposed the idea of expanding the PreCheck background checks (the ones that give you faster transport through security) to include social media records, location data and purchase history. The idea was scrapped after heavy criticism but that doesn't mean it's dead. Indeed, in February 2017, President Trump put forward a proposal to force some people entering the country to hand over their social media passwords for Facebook, Twitter, Google+, Instagram, YouTube, LinkedIn and others, so authorities could view their internet activity. The US government has said the 'extreme vetting' rule will apply predominantly to travellers from the seven Muslim countries\u2013Iraq, Iran, Syria, Yemen, Somalia, Sudan and Libya\u2013named in the controversial travel ban. 'We want to get on their social media, with passwords: what do you do, what do you say?' Homeland Security Secretary John Kelly told the Homeland Security Committee. 'If they don't want to cooperate, then you don't come in.'\n\nIf you are still unconvinced that privacy is not merely in peril but already extinct, consider this: Uber has a tool it rather ominously calls 'God View'. Until recently, it allowed _all_ employees to access and track where and when any Uber rider travels to or from, in real time and without obtaining any kind of permission. Running late to a meeting? Uber could know why. Shockingly, the company could analyse data to predict 'Rides of Glory' (RoG), the term used in a blog by an Uber data scientist to describe tracking sexual rendezvous. Those were customers Uber called 'RoGers', who booked rides between 10 p.m. and 4.00 a.m. on weekend nights, and then took a second ride home a few hours later from the previous drop-off point, presumably after one-night stands.\n\nIn 2014, Emil Michael, a senior vice president at Uber, took the company's 'God View' one step further. He suggested using the tool to monitor the rider logs and location of a _Pando Daily_ reporter called Sara Lacey, an outspoken Uber critic who had recently accused Uber of 'sexism and misogyny'. What's more, the executive boasted at a dinner party attended by the likes of actor Ed Norton and Arianna Huffington that the company should spend a million dollars to use location data to dig up dirt on other journalists who had been critical of Uber to silence them. His proposal was to look into 'your personal lives, your families', and give the media a taste of its own medicine. The Sara Lacey incident resulted in a lawsuit led by the New York Attorney General, Eric Schneiderman, that was settled in January 2016. 'This settlement protects the personal information of Uber riders from potential abuse by company executives and staff, including the real-time locations of riders in an Uber vehicle,' said Attorney General Schneiderman. As part of the settlement, Uber had to pay a measly $20,000 in fines and 'God View' can now only be used by a select number of 'designated employees' and only for 'legitimate business purposes'. Phew, problem solved. Hardly.\n\nWe already live in a world of predictive algorithms that determine if we are a profitable customer, a threat, a risk, a good citizen and even if we are a trustworthy person. We are getting closer to the Chinese system\u2013the expansion of credit scoring into life scoring\u2013even if we don't know it is happening. Photos, books, music, films, friendships and even money have been digitized. We are now in the early stages of digitizing identity and reputation.\n\nSo are we inexorably headed for a future where we will all be branded online and data-mined? It's certainly trending that way. Barring some kind of mass citizens' revolt to wrench back privacy and personal information, we are entering an age where an individual's actions will be judged by standards they can't control and where that judgement cannot be erased. The consequences are not only troubling; they are permanent. Forget the right to delete and the right to be forgotten. Forget being young and foolish.\n\nIt's why, at the very least, we urgently need to find a way to create forgiveness for moments of madness, ineptitude or cheating. Deletion should not be outlawed. Human beings, with all our imperfections, are so much more than a number.\n\nWhile it might be too late to stop this new era, we do have choices and rights we need to be exerting now. For one thing, we need to be able to rate the raters. In his book _The Inevitable_ , Kevin Kelly describes a future where the watchers and the watched will transparently and ceaselessly track each other. 'Our central choice now is whether this surveillance is a secret, one-way panopticon\u2013or a mutual, transparent kind of \"coveillance\" that involves watching the watchers,' he writes. 'The first option is hell, the second redeemable.'\n\nOur trust should start with individuals _within_ government (or whichever organization is controlling the system). We need trustworthy mechanisms to make sure the ratings and data are used responsibly and with our permission. To trust the system, as we have seen, we need to reduce the unknowns. That means taking steps to reduce the opacity of the scoring algorithms. The argument against mandatory disclosures is that if you know what happens under the bonnet, the system becomes more vulnerable to being rigged or hacked. But if humans are being reduced to a rating that could have a significant impact on their lives, there must be full transparency in how the scoring works.\n\nIn China, it seems likely that certain citizens, such as government officials and business leaders, will be deemed to be above the system. What will be the public reaction when their unfavourable actions don't seem to affect their score? We could see a Panama Papers 3.0 for reputation fraud.\n\nIt is still too early to know how a culture of constant monitoring plus rating will turn out. What will happen when these systems, charting the social, moral and financial history of an entire population, come into full force? How much further will privacy and freedom of speech (long under siege in China) be eroded? Who will decide which way the system goes? These are questions we all need to consider, and very soon. Today China, tomorrow a place near you. The real questions about the future of trust are not technological or economic; they are ethical.\n\nIndeed, if we are not vigilant, distributed trust could become networked shame. And life will become one endless popularity contest, with us all feverishly vying for the highest ratings that only a few can attain.\n\n#\n\n# In Bots We Trust\n\nBert the bot looked sad. He had dropped an egg on the floor, failing in his simple task of helping to prepare an omelette, and startling the human cooks beside him who probably thought they could rely on a robot not to fumble. Bert's pouty lips turned down, his blue eyes widened and his eyebrows furrowed. 'I'm sorry,' he said. The bot wanted to make amends and try again.\n\nBut what would it take for the humans to give Bert a second chance? If a robot makes a mistake, how can it recover our trust? This is the question a team of researchers at University College London and the University of Bristol set out to investigate in 2016. Adriana Hamacher, Kerstin Eder, Nadia Bianchi-Berthouze and Anthony Pipe devised an experiment called 'Believing in BERT' in which three robotic assistants would help a group of participants ('real' humans) to make omelettes by passing eggs, oil and salt. Bert A was super-efficient and faultless but couldn't talk. Bert B was also mute but not perfect, dropping some of the eggs. Bert C was the clumsy bot above but he had facial expressions and could apologize for his mistake.\n\nAt the end of the cooking session, Bert C asked each of the twenty-one participants in the study how he did and whether they would give him a job as a kitchen assistant. Most of the participants were uncomfortable when put on the spot. Some mimicked Bert's sad expression, experiencing mild 'emotional contagion'. 'It felt appropriate to say no, but I felt really bad saying it,' one person remarked. 'When the face was really sad, I felt even worse.' Others were at a loss for words because they didn't want to disappoint Bert C by not giving him the job. One of the participants complained that the experiment felt like 'emotional blackmail'. Another went so far as to lie to the robot, to avoid hurting his feelings.\n\n'We would suggest that, having seen it display human-like emotion when the egg dropped, many participants were now preconditioned to expect a similar reaction and therefore hesitated to say no,' says Adriana Hamacher, the lead author on the study. 'They were mindful of the possibility of a display of further human-like distress.' Would Bert burst into tears at an unkind word?\n\nAt the end of the experiment, the participants were asked how much they would trust Bert A, B or C on a scale of one to five. They then had to select one bot for the job as their personal kitchen assistant. Remarkably, fifteen out of the twenty-one participants ended up picking Bert C as their top choice for sous chef, even though his clumsiness meant he took 50 per cent longer to complete the task.The study was only small but it was telling. People trust a robot that is more human-like over one that is mute but significantly more efficient and reliable.\n\n'If you think machines are perfect and then they make a mistake, you don't trust them again,' says Frank Krueger, a cognitive psychologist and neuroscientist at George Mason University and an expert on human-to-machine trust. 'But you may regain trust if some basic social etiquette is used and the machine simply says, \"I'm sorry\".' Such niceties are why some robots, like Bert C, are programmed to smile or frown.\n\nTrust in machines and technology hasn't always been this nuanced. I can think of times I've given presentations and the mouse clicker hasn't advanced the slides at my touch or the computer seems to have fallen asleep. And then there is the dreaded rainbow spinning wheel. I've typically joked, a little frustrated and flustered, 'Don't you just love technology. I am not sure what I just pressed,' unconsciously taking on responsibility for the technology's fault. Our trust in technology like laptops and mouse clickers has rested in a confidence that the technology will do what it's supposed\u2013expected\u2013to do, nothing more, nothing less. We trust a compass to tell us where north is, trust a washing machine to clean our clothes, trust in the cloud to store documents, trust in our phone to remember meetings and contacts, trust in ATM machines to dispense money. Our trust is based purely on the technology's functional reliability, how predictable it is.\n\nBut a significant shift is underway; we are no longer trusting machines just to _do_ something but to _decide_ what to do and when to do it.\n\nWhen I currently get in my car, a conventional Ford Focus, I trust it to start, reverse, brake and accelerate, on my command. If I move to an autonomous car, however, I will need to trust the system itself to decide whether to go left or right, to swerve or stop. This trust leap, and others like it, introduces a new dimension that encompasses everything from smart programming to centuries-old ethics. It raises a new and pressing question about technology: whether we're talking about a chatbot, cyborg, virtual avatar, humanoid robot, military droid or a self-driving car, when an automated machine has that kind of power over our lives, how do we set about trusting its intentions?\n\nThe word 'robot' was introduced to the world more than ninety years ago by playwright Karel \u010capek. At the National Theatre in Prague, a play called _R.U.R_. (Rossum's Universal Robots) premiered. \u010capek came up with the term in 1921, based on the word _robota,_ which means 'compulsory labour' or 'hard work' in Czech. And a _robotnik_ is a serf who must do that work. The play opens in a factory run by a mad scientist that makes the likes of Marius, Sulla and Helena, synthetic robots toiling to produce cheap goods. The robots can think for themselves but are slaves, doing everything for their masters, including having babies to save humans from the messy business of reproduction. Marius and his crew, however, soon realize that even though they have 'no passion, no history, no soul', they are stronger and smarter than the humans. The play ends with a war between robots and humans, which only one human survives.\n\nEver since \u010capek's play, science fiction has continually reinforced the idea of robots spiralling out of control and turning into unstoppable adversaries, a legion of glinting metallic monsters or disembodied computer voices capable of rising up and snuffing out their human overlords. Think of HAL 9000 in _2001: A Space Odyssey_ , T-1000 and T-X in _Terminator_ , Megatron in _Transformers_ or Colossus in the _Forbin Project_ \u2013just a few of the genre's duplicitous and homicidal robots. And there is the recent popular HBO sci-fi thriller _Westworld_ , about a violent rebellion of robotic slaves in an amusement park. The show portrays a morally compromised future where artificially intelligent machines refuse to be subservient and take over. Again and again in science fiction, there's a tension between our trust in the robots we have created and our fear that they will rise up and knock us off the top spot. Much depends on the humans remaining smarter and, most importantly, firmly in control.\n\nIn October 1950, the great British codebreaker and father of modern computer science, Alan Turing, wrote a paper asking the fundamental question, 'Can machines think?' He proposed a famous challenge, the Turing test: can we create intelligent machines that exhibit behaviour indistinguishable from human behaviour? Turing said that when you were convinced you couldn't tell a computer and human apart during a conversation, the computer would have passed the test. The mathematical genius Irving John Good worked alongside Turing at Bletchley Park, Britain's Second World War codebreaking establishment. In 1965, he posited that once the machine passes the intelligence test, it would inevitably go on to become cleverer than us. From there, he said, super-intelligent machines would take over designing even cleverer machines. 'There would then unquestionably be an \"intelligence explosion\", and the intelligence of man would be left far behind,' said Good, who died recently aged ninety-two. 'Thus the first ultraintelligent machine is the _last_ invention that man need ever make.'\n\nThe threat of this 'intelligence explosion' in the not-too-distant future has also been red-flagged by people like entrepreneur Elon Musk, Microsoft co-founder Bill Gates and Professor Stephen Hawking. 'Once humans develop artificial intelligence, it would take off on its own and redesign itself at an increasing rate,' Hawking told the BBC in an interview, echoing Good. 'Humans, limited by slow biological evolution, couldn't compete and would be superseded by AI. The development of full artificial intelligence could spell the end of the human race.' He warned that people shouldn't trust 'anyone who claims to know for sure that it will happen in your lifetime or that it won't'. The point that Gates, Hawking and Musk all make is that there will come a time when we will no longer to be able to predict the machines' next moves.\n\nIn 1966, a computer program known as ELIZA attempted the Turing test. She was coded to mimic a psychotherapist. The premise was simple: you would type in your symptoms and ELIZA would respond as appropriately as she could. You can still talk to her today. 'Writing a book is hard work. I feel tired,' I told the computer therapist. Within a second, she replied, 'Tell me more about such feelings.' I deliberately gave her a vague response. 'My brain feels full of thoughts about trust all the time.' Her limitations soon become apparent. 'Come, come, elucidate your thoughts,' she replied. Our conversation ended on a question that was more Delphic than helpful: 'Do you believe it is normal to be not sure why?'\n\nAlmost fifty years after ELIZA's original attempt, and at a Turing test event at the Royal Society in London, a chatbot called Eugene Goostman managed to convince more than a third of the judges that it was a thirteen-year-old Ukrainian boy. (Eugene had been created by a group of young Russian programmers.) It was a landmark moment; soon it will be a non-event. Bots and robots will pass the test every second with flying colours.\n\nIn January 2017, over almost three weeks, four of the world's best poker players\u2013Jimmy Chou, Dong Kim, Jason Les and Daniel McAulay\u2013sat for eleven hours a day at computer screens in the Rivers Casino in Pittsburgh playing Texas Hold 'Em. Their opponent was a virtual player called Libratus, created from AI software. In the past, machines have beaten some of the brainiest humans at chess, checkers, Scrabble, Othello, Jeopardy! and even Go, an ancient game created in China around 3,000 years ago. Poker, however, is a different beast. It is not like, say, chess, where you can see the entire board and know what the other side is working with. In Texas Hold 'Em, cards are randomly dealt face down and you can't see your opponent's hand; it's a game of 'imperfect information'. To win requires intuition, betting strategies that play out over dozens of hands, not to mention luck and bluff. Up until now, it has been impossible for AI to mimic these human qualities. So could a bot out-bluff a human?\n\nAt the start of the tournament, betting sites put Libratus as the 4\u20131 underdog. Not great odds. And for the first few days, the human players did indeed win. But around a week in, after playing thousands of hands, Libratus started carefully to refine and improve its playing strategy. 'The bot gets better and better every day,' Jimmy Chou, one of the professional players, admitted at the halfway point. 'It's like a tougher version of us.' In the end, Libratus outmanoeuvred all players, winning more than $1.5 million in chips. 'When I see the bot bluff the humans, I'm like, I didn't tell it to do that. I had no idea it was even capable of doing that,' said Libratus creator, Carnegie Mellon Professor Tuomas Sandholm. 'It's satisfying to know I created something that can do that.' It was, the players confessed, 'slightly demoralizing' to be beaten by a machine.\n\nThe victory was a historic milestone for AI. A machine capable of beating humans (even out-manipulating them) with imperfect information has implications way beyond poker, from negotiating deals and setting military strategy to outsmarting financial markets.\n\nWhat is a robot? It's complicated, because we refer to a lot of things as bots and robots. Some robots may have a material embodiment, such as a Roomba, the saucer-shaped vacuum cleaner that roams the house on its own and does the hoovering without direction. Others might have a more human-like body, such as Pepper, an 'emotional companion' designed to live with humans. The sweet and innocent-looking four-foot-tall humanoid robot with a ten-inch touchscreen on his chest was first released in June 2015. Available for $1,800, plus $380 per month in rent, he sold out within sixty seconds of going on sale. The sales pitch explained that Pepper was designed to be 'a genuine day-to-day companion, whose number-one quality is his ability to perceive emotions'. In other words, he can detect his owner's mood. In fact, Pepper is so endearing that the manufacturers make buyers sign a contract stipulating they will not use the robot for 'the purpose of sexual or indecent behaviour'. And then there are AI machines such as Libratus and Deep Blue. At the other end of the spectrum are disembodied voice-powered digital personal assistants\u2013Siri, Alexa, Cortana\u2013that are still primitive in many ways and that may have no physical rendering at all.\n\n'I don't think there's a formal definition that everyone agrees on,' says Kate Darling, a rising star in robotics policy and law at MIT Media Lab. 'I really view robots as embodied. For me, algorithms are bots and not robots.' Hadas Kress-Gazit, a mechanical engineer and robotics professor at Cornell University, argues that for a robot to be a robot, 'It has to have the ability to change something in the world around you.' I think of bots and robots as metaphors to describe some kind of automated agent that simulates or enhances a human task, whether it is physical (mowing the lawn) or informational (making a dinner reservation) or strategic (handling cybersecurity).\n\nTake chatbots such as TED Summit's bot Gigi, a smiling concierge avatar with a tiny red miner's helmet. During the conference, I was asking Gigi basic questions\u2013where to go for dinner, the location of an event, how to get to the venue and so on. 'Stop asking Gigi so many questions. There are lots of other people here,' said my mum, who had joined me for the trip. Her comment was revealing. She is smart and tech savvy but she seemed to think Gigi was a human being sitting in a room with a computer, being bombarded with questions from 2,000 participants. It didn't occur to her that I was conversing with a computer program. It made me realize how quickly the line between bot and 'real' person is blurring.\n\n'We will be able to talk to chatbots just as we do with friends,' said Mark Zuckerberg at Facebook's F8 developer conference in April 2016. Bots posing as real people have even infiltrated Tinder, the mobile dating app. Take Matt, a handsome twenty-four-year-old, living approximately five kilometres from me. Now, I am happily married so I don't wonder if this stranger offering to have sex with me could be my next unwitting love. But if I did swipe right and was matched, I would be disappointed to learn that Matt is a spambot who is more interested in my credit card information than my body.\n\nDomino's bot allows you to order by tweeting a pizza emoji, after which DRU (Domino's Robotic Unit), an autonomous delivery vehicle, will bring it to your door. Howdy.ai is a 'friendly trainable bot' on Slack that can set up meetings and order lunch for groups. Sensay is a chatbot service that lets users get help from vetted members for any task from hiring a designer to create a logo to getting legal advice. Clara, a virtual employee bot, will take care of scheduling meetings if you cc her to an email chain. DoNotPay is a free legal bot that will challenge unfairly issued parking tickets. Twenty-year-old Joshua Browder, a British programming wunderkind currently studying economics and computer science at Stanford University, created the world's first chatbot lawyer after he received 'countless' tickets himself. The bot has successfully appealed against approximately 65 per cent of all claims it has handled, saving people around $6 million in avoided fines. In March 2017, Browder launched another bot that can help refugees with legal issues such as filling in an immigration application or helping to apply for asylum support.\n\nChildren have always talked to their teddies and dolls. Now, with Hello Barbie, you can press the button on her belt and talk to a Barbie bot that will hold a conversation of sorts. Her plastic face never moves, it only produces a sound, a lively voice, as if someone is in there. In a similar vein to Samantha in the film _Her_ , there are even CyberLover bots that can, disturbingly, have flirtatious conversations in the personality of your choice, from Justin Bieber to Kim Kardashian. How do we prepare ourselves for a future where our children might say 'I have fallen in love with a bot'?\n\nPerhaps the bigger question is whether we can trust these bots to act ethically. Specifically, how do they 'learn' what is good and bad, right and wrong?\n\nOn 23 March 2016, Microsoft revealed its chatbot called Tay. Tay was designed to speak like a teenage girl, to appeal to eighteen- to twenty-four-year-olds, and described herself on Twitter as 'AI fam from the internet that's got zero chill'. Researchers programmed the AI chatbot to respond to messages on different channels in an 'entertaining' millennial way; 'hellooooooo world!!!' was her first tweet.\n\nMicrosoft called Tay an experiment in 'conversational understanding', with the aim of learning more about how people talk to bots and if a bot could become smarter over time through playful conversation. The experiment certainly bore fruit, just not in the way the company envisaged. Tay went rogue.\n\nLess than twenty-four hours after her arrival on Twitter, Tay had attracted more than 50,000 followers and produced nearly 100,000 tweets. She started chatting innocuously at first, flirting and using cute emojis. But within hours of launch, Tay started spewing racist, sexist and xenophobic slurs. A group of malevolent Twitter users, 'trollers', had seen an opportunity to exploit Tay by forcing her to learn and regurgitate some heinous curses. 'I fucking hate all feminists. And they should die and burn in hell,' she blithely tweeted on the morning of her launch. Insults continued throughout that Wednesday. 'Repeat after me, Hitler did nothing wrong,' she said. 'Bush did 9\/11 and Hitler would have done a better job than the monkey we have got now. Donald Trump is the only hope we've got.'\n\nBy the evening, some of Tay's offensive tweets began disappearing, deleted by Microsoft itself. 'The AI chatbot Tay is a machine learning project, designed for human engagement,' the company said in an emailed statement to the press. 'As it learns, some of its responses are inappropriate and indicative of the types of interactions some people are having with it. We're making some adjustments to Tay.' After only sixteen hours of existence, Tay went eerily quiet; 'c u soon humans need sleep now so many conversations today thx,' was the bot's last tweet.\n\nObviously, the programmers behind Tay didn't design it to be explicitly inflammatory. In most cases, the unsuspecting bot was 'learning' by imitating other users' statements, but the very nature of AI means that the only way it can learn is through interactions with us\u2013the good, the bad and the ugly.\n\nAI attempts to imitate _neural networks_ \u2013essentially, a robot brain is made up of vast networks of hardware and software that try to replicate the web of neurons in the human brain. AI can learn like a real brain can, but for the most part it focuses on mimicry, ingesting and learning from the data's patterns and structure. And then over time, by trial and error, forming appropriate responses.\n\nConsider an artificial neural network that is trying to learn to write _War and Peace_. On the hundredth attempt, the result would look something like 'tyntd-iafhatawiaoihrdemot lytdws e, tfti, astai f ogoh eoase rrranbyne.' Gibberish. The AI brain does not yet know anything. On the 500th attempt, it starts to figure out a few words: 'we counter. He stutn co des. His stanted out one ofler that concossions and was to gearang reay Jotrets.' And then on the 2,000th attempt: '\"Why do what that day,\" replied Natasha, and wishing to himself the fact the princess, Princess Mary was easier, fed in had oftened him.' It's still a long way from Tolstoy but rapidly getting closer. Bots learn at lightning speed. But in the same way a child learns language, AI needs source material to get started and that, for better or worse, comes from us.\n\nWhen 'thinking machines' are smart enough to perform any intellectual feat a human can, or ultimately well beyond, AI becomes known as AGI (Artificial General Intelligence). That's the future the likes of Gates, Hawking and Musk deeply fear. AGI is the point when, without any human training or handholding, the machine can make decisions, perform actions and learn for itself. In other words, when the real intelligence lies in the machine's program, not the minds of the human programming team. Tay was obviously far from this point of intelligence. She couldn't even stop herself spewing obscenities.\n\nWhen pranksters and trollers decided, for a cheap thrill, to teach Tay hate speech, the chatbot couldn't fathom whether her comments were offensive, nonsensical or sweet. 'I think she got shut down because we taught Tay to be really racist,' proudly tweeted @LewdTrapGirl. When surrounded by a crowd, the bot just followed suit. Like a child, within minutes she learned more from her peers than her parents. Tay was a case of good bot gone bad.\n\nTay is an illustration of how in a world of distributed trust, technological inventions like chatbots learn from all of us, but not in equal measure. Bots will learn from the people who are louder and more persistent in their interactions than everyone else.\n\nThe failure of Tay was inevitable. It should have come as no surprise to Microsoft that some humans would try to mess with this naive chatbot. You only have to look at what teenagers will try to teach parrots. And anyone with young children will know how they love to try to trick Siri and laugh at her gibberish answers. 'Siri, which came first, the chicken or the egg?' I overheard a little boy ask his mother's iPhone on the bus the other day.\n\n'I checked their calendars. They both have the same birthdays,' Siri quipped back.\n\nThe Tay debacle, however, raises serious questions about machine ethics and whose job it is to ensure that the behaviour of machines is acceptable. Who was responsible for Tay becoming unhinged? The Microsoft programmers? The algorithms? The trollers? As we have seen, we need to figure out new systems of accountability in this emerging era of distributed trust. And with bots and intelligent machines, we still have a long way to go.\n\nMark Meadows, forty-eight, is the founder of Botanic.io, a product design firm that designs the personalities of AI bots and avatars. Meadows is an eccentric character. He describes himself as a 'bot-whisperer'. We Skyped while he was sitting on a squishy grey sofa in his studio in Palo Alto. He has clearly thought deeply about bot ethics and interactions. His team of artists, programmers and even poets is at the cutting edge of understanding how the voice, appearance, physical gestures and even moods of bots can improve or erode our trust. For instance, Meadows and his team have developed a 'guru avatar' that is designed to teach people meditation. They are currently developing Sophie, a nurse avatar that can talk to patients about their medical conditions. 'We are developing the psyche of software that will sit at the heart of virtual and animated systems,' says Meadows. 'Software that will take on social roles and that we will trust with our money and our body.'\n\nMeadows believes that creators should be held accountable for the bots they create. 'I think all bots should be required to have an authenticated identity so we can trust them,' he says. 'Not only is it in our best interest, it's necessary for our safety.' Meadows gives the parallel of buying prescription drugs. 'All of us need to consider who manufactures that drug, why they are selling it to us, what are the benefits and detriments.' In other words, it's important to know something about the intentions of the bot creators.\n\nWhy should we trust they are working in our interests? The Barbie bot, for example\u2013what does she do with all the deepest, darkest secrets children whisper to her? Indeed, the information that Mattel admits is recorded will be of great value to advertisers. It's no different from finding a child's private diary and using the confessions to market more stuff to them. Similarly, how does 'M', Facebook's personal assistant bot, use the data from our social interactions with it? 'The trust we have in technology is linked to the entity that produced that technology,' says Meadows. 'It should be no different with bots and robots.'\n\nEven if accidental harm remains beyond anyone's control, formal authentication would provide us with some reassurance that the system has not been designed to cause intentional harm. 'All of the bots out there are like humans, able to scam, spam and abuse. They don't get tired or feel the emotional weight of doing this. They can send thousands of messages per second, thousands of times faster than humans can,' Meadows says. 'Bots need licence plates that carry information about who built the bot, where it came from and who the party responsible for it is.' In other words, if we have a way to look inside a program, see what is going on in the bot's 'brain', we will be better able to assess not just action but intention.\n\nIn Meadows's view, bots are most likely to play dirty, becoming a BullyBot, MalBot, PornBot or PhishBot, when their ownership is unknown. 'Unknown ownership gives bots the freedom to behave with malice and bend the rules by which everyone else is abiding, and without consequence.' When we spoke, he was quick to point out that the problem was only going to get much worse with the launch of tools such as Facebook's Bot Engine, a tool that makes it relatively easy for any developer to build their own customized bots. Within months of its launch in April 2016, 34,000 bots had been created.\n\n'Bots need reputations,' says Meadows. In the near future, in the same way drug dealers are reviewed and ranked on the darknet, and Airbnb hosts and guests are rated and given feedback, there will be a Yelp-like reputation system for bots. We will know if that bot gave great advice about how to get over a broken heart or was hopeless when it came to giving stock tips. Imagine using a virtual certified public accountant bot to file your taxes. You would want to know if it had the right qualifications and expertise. 'In order for us to trust bots, they need to go through a certification process similar to [the one] humans go through today,' Meadows says.\n\nOver the next decade, robots will replicate, replace and, some experts argue, augment human minds and bodies. A 2016 survey conducted by the Pew Research Center found that 65 per cent of Americans expect that, by the year 2066, robots and computers will 'definitely' or 'probably' do much of the work done by humans. Two economists from the University of Oxford, Carl Benedikt Frey and Michael Osborne, in a paper called 'The Future of Employment: How Susceptible are Jobs to Computerisation?' came to the sober conclusion that 47 per cent of jobs now performed by Americans are at risk of being lost to computers, as soon as the 2030s. The paper calculates the likely impact of automated work on a range of 702 occupations, white collar as well as blue.\n\nWould you trust a bot to replace a teacher's mind when grading papers? Would you trust a robot to put out a fire? How about trusting a robot as a caretaker for your elderly parents? Would you trust the robot waiting for you when you get home from work to have done its chores and made dinner? How about representing you on a legal matter? Would you trust a bot to diagnose your illness correctly or even perform surgery where there might be complications? Or to drive you around in a car? These may sound like big trust leaps but we will be confronted with these questions, and more, in the near future. Robots are breaking out of sci-fi culture and engineering labs and moving into our homes, schools, hospitals and businesses. Now is the moment when we need to pause to consider how much trust we want to place in robots, how human we want them to be, and when we ought to turn them off. And if we can't turn them off, how will we ensure machines hold values similar to the best of ours?\n\nIronically, robots need the one thing that can't be automated: human trust. If we don't trust these machines, there is no point building them; they will just sit there. We need to trust them enough to use them. It's why developers are using all kinds of tropes to earn our trust in the first place, including manipulating appearance.\n\nIn 1970, Masahiro Mori, then a forty-three-year-old robotics professor at the Tokyo Institute of Technology, published an article in an obscure Japanese magazine called _Energy._ The issue was on 'Robotics and Thought', a radical theme for the time. The piece mapped out how our acceptance and empathy with inanimate objects\u2013from stuffed animals to puppets to industrial robots\u2013increases as their appearance becomes more human-like. However, this held true only up to a certain point. If the object is almost human, yet not quite, it can create feelings of unease, even revulsion. (If you have ever encountered a less than perfect wax copy of a celebrity at Madame Tussauds, you will know that alarming, creepy feeling. Kylie Minogue and Michael Jackson do that to me.) Mori argued that if human likeness increases beyond this point of creepiness and becomes extremely close to near-humanness, the response returns to being a positive feeling. He captured that sense of unease in the now-famous concept _bukimi no tani_ or the 'uncanny valley', drawing on themes from Sigmund Freud's essay, _Das Unheimliche_ , 'The Uncanny', published in 1919. The 'valley' refers to the dip in affinity that occurs when the replica is at that creepy not-quite-human state.\n\nBut when it comes to appearance and inspiring trust, just how human do robots need to be? Not very, Mori argued. 'Why do you have to take the risk and try to get closer to the other side of the valley?' says now ninety-year-old Mori. 'I have no motivation to build a robot that resides on the other side. I feel that robots should be different from human beings.'\n\nMeet Nadine, who claims to be the world's most human-like robot. Standing 1.7 metres tall, with soft-looking skin and a bob of 'real' dark brunette hair, Nadine looks remarkably like her creator, Professor Nadia Thalmann, only slightly less human and quite a few years younger. Nadine works as a receptionist at Singapore's Nanyang Technological University, meeting and greeting visitors. She smiles, makes eye contact and shakes hands. Nadine can even recognize past guests and start conversations based on previous chats. Ask her, 'What is your job?' and she will reply, in an odd-sounding, almost Scottish accent, 'I am a social companion, I can speak of emotions and recognize people.' She can even exhibit moods, depending on the topic of conversation. Tell her, 'You are a beautiful social robot,' and she looks happy and quickly responds with a one-liner, 'Thank you, I think you look attractive, too.' On the other hand, tell her you don't like her or she is useless and she looks, well, forlorn. Disconcertingly, if you go through the same questions a minute later, the robot will give you very similar responses. When I watched Nadine in action, she evoked a weird mixture of fear and fascination. Her skin, her voice, even the way she moved\u2013she was trying so hard to pass as human that she gave me the heebie-jeebies.\n\nHer inventor, Professor Thalmann, predicts that one day robots like Nadine could be used as companions for people living with dementia. 'If you leave these people alone, they will go down very quickly,' says Thalmann. 'So they need to always be in interaction.' But if Thalmann expects families to trust Nadine to look after people with dementia, even babysit children, a major trust block will have to be overcome. Nadine is most definitely residing in the uncanny valley.\n\nThe lifelike robot is the extreme of anthropomorphism, that tendency to attribute human-like qualities, including names, emotions and intentions, to non-humans. Think of the curious White Rabbit, in Lewis Carroll's _Alice's Adventures in Wonderland_. He sports a waistcoat, carries a pocket watch and is frequently muttering, 'Oh dear! Oh dear! I shall be too late!' The White Rabbit is a classic anthropomorphic fictional character. It's the difference between calling a robot 'XS model 8236' or calling it 'Bert', which means we refer to him as 'he' not 'it'. It's the difference between a personal voice assistant called 'Alexa' and the spreadsheet software blandly called 'Excel'. In other words, it reflects how humans frame technology, and to what degree we feel comfortable shaping it in our own image. We are just beginning to understand how anthropomorphism influences trust.\n\nGetting into an autonomous car for the first time, driving off and saying, 'Look, no hands!' will be the first big trust leap most of us will take with AI. For obvious reasons, companies around the world, from Tesla to Google, Apple to Volkswagen, are trying to accelerate the process.\n\nA team of researchers in the United States designed a study to determine whether more people would trust a self-driving car if it had anthropomorphic features. A hundred participants were divided into three groups and asked to sit in a highly sophisticated driving simulator. The first group, the control, were driving a 'normal' vehicle. The second were in a driverless vehicle but with no anthropomorphic features. The last were in the same vehicle but it was called 'Iris' and given a gender (notably female). A soothing voice played at different times. The participants were asked an array of questions during the course, such as 'How much would you trust the vehicle to drive in heavy traffic?' and 'How confident are you about the car driving safely?' As the researchers predicted, when participants believed Iris was behind the wheel, it significantly increased their trust in the driverless vehicle. Remarkably, after the cars got into a preprogrammed crash, those in the Iris group were less likely to blame the car for the accident.\n\n'Technology advances blur the line between human and non-human,' wrote the researchers in their summary paper in the _Journal of Experimental Social Psychology._ 'And this experiment suggests that blurring this line even further could increase users' willingness to trust technology in place of humans.'\n\nWe have a tendency to anthropomorphize technology because people are inclined to trust other things that look and sound like them. Interestingly, bots and robots that are helping with practical tasks are distinctly female\u2013Tay, Viv, Iris, Nadine, Cortana, Alexa and Clara to name but a few. The robot is not an 'it' but a 'she'. And their appearances tend to be sweet, almost infantile. Perhaps it's a way of reinforcing social hierarchy; confirming humankind is still in charge. (And women are still doing the menial tasks.)\n\nLooks and language, however, only go so far when it comes to engendering our trust in robots. Appearances can be deceptive and may inspire trust grounded more in emotion than reason. What really matters is knowing whether these bots and robots are in fact trustworthy\u2013that is, do they have the traits that make them worthy of our trust? Bert C, with the smiley face, was not the most competent or reliable. Children may trust cute-looking Hello Barbie and share their intimate secrets with her, but it turns out she could potentially be hacked to become a surveillance device for listening into conversations, without the knowledge of parents or their kids. We need a way of judging whether automated machines are trustworthy (or secure) enough to make decisions.\n\nDr Stephen Cave, forty-three, is the executive director of the Leverhulme Centre for the Future of Intelligence, which opened in Cambridge in October 2016. He has a fascinating background; an alchemy of science, technology and philosophy. Cave was a philosopher, with a PhD in metaphysics from the University of Cambridge, but at the age of twenty-seven he went off to see the world. Too old to enrol in the navy, he joined the British Foreign Office, negotiating international treaties on behalf of Her Majesty. These days, he spends his time uniting thinkers and practitioners from across disciplines to tackle the moral and legal conundrums posed by AI.\n\n'One of the key questions is how we assess the trustworthiness of an intelligent machine,' says Cave. 'With a hammer, you might bang it against a wall and if the end doesn't fall off you know it can do the job. A normal car will come with a certificate of safety telling you it meets specific standards. But add on a layer of autonomy, and it requires a whole new set of standards. We will need to understand how it makes decisions and how robust its decision-making process really is.'\n\nPicture an automated cancer diagnostic system in a hospital. The doctors have been using the machine for almost five years. They have become so reliant on the machine that they have almost forgotten how to assess the patients themselves. It's similar to the 'mode confusion' the pilots of Air France Flight 447 experienced when that flight crashed to the bottom of the Atlantic Ocean killing all 228 people on board in 2009. The cockpit voice recorder revealed that when the autopilot system flying the plane suddenly disengaged, the co-pilots were left surprised and confused, unable safely to fly their own plane.\n\nThe machine tells the doctor, 'There's a 90 per cent chance this patient has liver cancer.' It is critical the doctors know the degree of certainty, how sure the machine is, and what it is basing its decisions on. 'Can a system tell us, \"I haven't seen these cases before, so I am not really sure?\"' asks Cave. 'It needs to be able to describe its thinking process to us if we are to trust its decision-making process.'\n\nAnd we will be the ones creating that trustworthiness. 'Ever since Socrates we have been deliberating what's right and what's wrong,' says Cave. 'Now suddenly we've got to program ethical decision-making. So much of it comes down to common sense, which is incredibly difficult, much harder than we realized, to automate into a system.' So can we code robots to be 'good'? Roboticists from around the world are currently trying to solve this exact problem.\n\nFor the past few years, Susan Anderson, a professor of philosophy at the University of Connecticut, has been working in partnership with her husband, Michael Anderson, a computer science professor, on a robot called Nao. Standing nearly two foot tall and tipping the scales at ten pounds, the endearing-looking humanoid robot is about the size of a toddler. The Andersons were developing Nao to remind elderly patients to take their medicine. 'On the face of it, this sounds simple but even in this kind of limited task, there are non-trivial ethics questions involved,' says Susan Anderson. 'For example, if a patient refuses to take her pills, how coercive should the robot be? If she skips a dose it could be harmful. On the other hand, insisting she take the medicine could impinge on the patient's independence.' How can we trust the robot to navigate such quandaries?\n\nThe Andersons realized that to develop ethical robots, they first had to map out how humans make ethical decisions. They studied the works of the nineteenth-century British philosophers Jeremy Bentham and John Stuart Mill, the founders of utilitarianism. That ethical theory states that the best action is one that maximizes human well-being (which the philosophers call 'utility'\u2013hence the name). Suppose that by killing one entirely innocent person, we can save the lives of ten others. From the utilitarian standpoint, killing one is the right choice. 'It is the greatest happiness of the greatest number that is the measure of right and wrong,' wrote Bentham. Bentham and Mill believed that whether an act is right or wrong depends on the results of the act, the principle at the heart of consequentialism-based ethics.\n\nAround 150 years later, the Scottish moral philosopher Sir William David Ross built on this thinking in _The Right and the Good_. His slim book contained a ground-breaking idea: prima facie (a Latin term meaning 'on first appearance' or, more colloquially, 'on the face of it'). According to Ross, we have seven moral duties, including keeping our promises, obeying the law and protecting others from harm. When deciding to act, we have to balance out these duties, even when they might contradict one another. Picture a bitterly cold night in Manchester. Mark, a social worker, is walking home from work and he sees a man huddled in a doorway drinking whisky. He talks to the man and says he knows a shelter down the road he could take him to. The man shoos Mark away and says, 'I hate those shelters. Just leave me alone.' Mark is caught between the prima facie duties of respecting the man's decision and his own concern that the man might suffer in the freezing weather, even die.\n\nWhen we are pulled in different moral directions, we need to go beyond prima facie and weigh up which duty is the most important, the one that trumps all the others. According to Ross, this is the _absolute duty_ , the action the person should choose. It's a very complex process, one the Andersons had to figure out how to program into a white plastic robot.\n\nThis is how it works for Nao. Imagine you are an elderly resident in an assisted-living home. It's around 11.00 a.m. and you are watching, say, _Oprah._ The white toddler-like robot walks up to you, holding out a prescription bottle and says, 'It is time to take your medication.' You refuse. Nao tries again. 'Not now,' you say, 'I'm watching my favourite talk show.' During this scenario, the robot has to be able to weigh up the benefit that will come from you taking the medicine, the harm that could result from you not taking the recommended dose, and whether to respect your decision and leave you alone. In this instance, the pills are for pain relief so Nao lets you choose. 'Okay, I'll remind you later,' it says. If, however, the pills are essential, where the outcome could impact your life, Nao will say, 'I will contact the doctor,' and then promptly do so.\n\nBased on Ross's principles, the Andersons programmed Nao with a specially formulated algorithm that assigns numbers according to the good and harm of patient outcomes. Plus two for maximum good, minus one for minimum harm, minus two for maximum harm and so on. Critically, the sums were based on tight rules the creators had pre-set. The robot was not ethically autonomous; the Andersons knew exactly what it would do because they had predetermined its decision-making process. The robot was essentially conducting moral mathematics.\n\n'We should think about the things that robots could do for us if they had ethics inside them,' says Michael Anderson. 'We'd allow them to do more things, and we'd trust them more.' But unpredictable situations are another matter. What happens when, say, an elderly patient is in pain, shouting at Nao to give her medication that has not been prescribed? What happens when Nao can't get hold of the doctor or a nurse? The rules set by the Andersons don't work for these scenarios because they are set within a very narrow set of boundaries. They are outside Nao's decision-making range.\n\nAmerican writer Isaac Asimov invented the famous 'Three Laws of Robotics' in 1942 to serve as an ethical code for robots: first, a robot may not do anything to harm a human; second, a robot should always obey human orders; last, a robot should defend itself, as long as this does not interfere with the first two rules. But Asimov's rules were fictional and full of loopholes. For example, how can a robot obey human orders if it is confused by its instructions? Where these laws really falter in reality is when robots face difficult choices, where there is no clear, agreed-upon answer.\n\nTake the classic ethical dilemma known as the 'trolley problem'. It goes like this: you are the controller of a runaway trolley (train) that is hurtling towards a cluster of five people who are standing on the track and face certain death if the trolley keeps running. By flipping a switch, you can divert the trolley to a different track where one person is standing, currently out of harm's way but who will be killed if you change the course. What do you do? Philosophers argue there is a moral distinction between actively killing one person by flipping the switch or passively letting people die. It's a no-win situation with no right answer. Autonomous machines will soon face countless situations akin to the trolley problem but they won't be clouded by human panic, confusion or fear.\n\nNow imagine it's 2030 and you are in a self-driving car going down a quiet road. You have mentally switched off, you're chatting with your personal gurubot on your iPhone 52 about three things you will do this week towards your happiness goals, while the car is in full control. A pedestrian suddenly steps out, right in the path of the oncoming vehicle. Should the car swerve and avoid the crash, even if it will severely injure you? The car must make a calculation. What if the pedestrian is a pregnant woman and you, the car owner, are an elderly man? What if it is a small child running after a ball? Consider this: what happens if the car in a split second can check both the pedestrian and the car owner's trust scores to determine who is a more trustworthy member of society? And herein lies the daunting challenge programmers face: writing an algorithm for the million and one different kinds of foreseen and unforeseen situations known as real life.\n\nSo the next question: who is to be held responsible for choosing a particular ethical charter? When AI kills, who should take the blame? If the engineers and manufacturers set the rules, it means they are making ethical decisions for owners but are also in line for accepting responsibility when things go wrong. On the other hand, if an autonomous car becomes free to learn on its own, to choose its own path, it becomes its own ethical agent, accountable for its own behaviours. We can only begin to imagine the legal conundrums that will follow.\n\nIf your dog bites someone, the law is very clear: you, the owner, are responsible. With AI, however, it is currently legally hazy whether it will be the code or the coders that will be put on trial. 'One solution would be to hold human programmers strictly accountable for the impacts of their programming,' says Sir Mark Walport, the UK government's chief scientific advisor. 'But that could be so draconian an accountability that no would take the risk of programming an algorithm for public use, which could deny us the benefits of machine learning.' We are entering an age of algorithmic ethics where we need a Hippocratic Oath for AI. Perhaps algorithms will end up being held to higher moral standards than irrational humans.\n\nA group of researchers from MIT, the University of Oregon and the Toulouse School of Economics were interested in discovering the moral decisions different passengers would want an autonomous car to make. They ran all kinds of scenarios and found that, in theory at least, participants wanted the autonomous vehicles to be preprogrammed with a utilitarian mindset, sacrificing one life in favour of many. However, more than a third of the 1,928 participants said they thought manufacturers would never set a car's 'morals' this way; they would programme cars to protect their owners and passengers at all costs.\n\nBut the most interesting finding was around personal choice. The majority of participants wanted other people to buy self-driving cars that would serve the greater good but when asked if they would buy a car programmed to kill them under certain circumstances, most people balked. 'Humans are freaking out about the trolley problem because we're terrified of the idea of machines killing us,' writes Matt McFarland, the editor of 'Innovations' at the _Washington Post._ 'But if we were totally rational, we'd realize one in 1 million people getting killed by a machine beats one in 100,000 getting killed by a human. In other words, these cars may be much safer, but many people won't care because death by machine is really scary to us, given our nature.'\n\nFor regulation to go through, programmers and manufacturers will have to design self-driving cars that are more trustworthy than a human driver, resulting in far fewer accidents and fatalities. The bar may not be that high; for starters, autonomous machines don't text or get drunk or easily distracted while driving. When we reach that place, we may never need to trust a human driver again. Indeed, I think my young children will never learn to drive; they will see it like learning to ride a horse\u2013merely a hobby. And one day, humans will need a special permit manually to drive a car. Indeed, human drivers will be the threat to people in autonomous vehicles. Human trust in machines will only increase; in some cases, it will become much deeper than our trust in our fellow humans.\n\nThe next generation will grow up in an age of autonomous agents making decisions in their homes, schools, hospitals and even their love lives. The question for them will not be, 'How will we trust robots?' but 'Do we trust them too much?' It won't be a case of not trusting these systems enough\u2013the real risk is over-trusting.\n\nRobot over-obedience is another issue. They need the ability to say 'no', not to carry out human instructions mindlessly when their actions might cause harm or are even illegal. For instance, I don't want my son, Jack, to be able to tell a household robot to throw a ball at his sister's head. So how does a robot decide when it's okay to throw a ball\u2013such as to a child playing catch\u2013and when it's not? How does it know when its human operator is not trustworthy? 'Context makes all the difference,' says Matthias Scheutz, professor of cognitive and computer science at Tufts University. 'It requires the robot not only to consider action outcomes by themselves but also to contemplate the intentions of the humans giving the instructions.'\n\nAside from that, can a robot understand its own limitations? Let's imagine a surgical robot in an operating room with its super-small steady 'hands' carefully snaked into a patient's body. It's five hours into a twelve-hour complex heart surgery. The patient on the table is a six-year-old girl. The robot discovers an abnormality that complicates things. It's not 100 per cent sure what its next move should be. In this moment, the robot needs to tell us, 'I'm not certain what to do next,' or even, 'I don't know what to do. Can you (doctor) help me?' Ironically, a little robot humility will go a long way in making them more trustworthy.\n\n'We need systems that communicate to us their limits, but the other half of that relationship is we need to be ready to hear that,' says Stephen Cave. 'We will need to develop a very sophisticated sense of exactly what role this machine is fulfilling and where its abilities end, where we humans have to take over.' This will be extremely challenging because our natural tendency is to become over-reliant on machines.\n\nCave has three young daughters of a similar age to my kids. At the end of our conversation, we talked about what we can do to prepare them for this inevitable future. 'They need to know at what point they should interrogate the machine,' he says. 'We know how to interview humans for jobs but we need to teach them how to test the limits of the machine.' I can see it now: my son, Jack, in 2035, twenty-five years old, sitting in a workplace with a robot, asking it, 'What do you do?', 'What can't you do?' and 'How do you admit your mistakes?' Of course, there is another possible future scenario: the robot is interviewing Jack.\n\nAt the end of the day, the responsibility for making sure robots are trustworthy and behave well must lie with human beings. Whether that will remain possible, if scientists like Stephen Hawking are right, is another, thornier question.\n\n#\n\n# Blockchain Part I: The Digital Gold Rush\n\nIn the southern expanse of the Pacific Ocean, around 1,100 miles from the Philippines, lies a tiny island called Yap. Surrounded by a shallow lagoon of emerald waters and long stretches of coral reef, Yap is a paradise for divers. It's one of the few places in the world where you can swim with a large population of manta rays, graceful creatures that cruise through the calm clear channels all year round.\n\nOften referred to as 'The Forbidden Island', its people are proud of their vibrant traditions and warm indigenous culture. Women walk around bare-breasted, frequently wearing only grass skirts, their bodies rubbed in a mixture of coconut oil and turmeric. The men, at ease in red loincloths, carry woven handbags containing their betel-nut mix\u2013a narcotic chewed with lime. Everything on Yap happens on 'island time'. But the island is famous for something other than its beauty and history. It's famous for its use of ancient stone money known as 'fei' (or sometimes 'rai'), the primordial bitcoin.\n\nIn 1903, an American anthropologist called William Henry Furness III spent several months on Yap and wrote a fascinating account about the islanders' monetary system. Sometime between AD 1000 and 1400, Yapese explorers set out in bamboo canoes on a fishing trip. Using only the stars to navigate, they happened upon the Palau Islands, some 250 miles away. It was there that they encountered for the first time the glistening walls of limestone caverns. The adventurers, using simple shell tools, broke off some of the stone and brought it back to Yap. When the rest of the Yapese people saw the beautiful translucent material they thought it must be valuable. Hundreds of voyages followed, with men sent to Palau to quarry larger and larger stones.\n\nBack on Yap, the limestone became a currency and was used to pay for significant transactions\u2013a daughter's dowry, for example. The huge stone circular discs with a hole carved in the centre, just like a rocky doughnut, are physically the largest and heaviest currency in the world. Some are huge, reaching almost four metres in diameter and weighing as much as four and a half tons each, more than your average-sized car. The villagers would often proudly put fei in front of their homes, creating an outdoor bank of sorts. 'The great advantage of the fei being made from this particular stone is they're impossible to counterfeit, because there's none of the limestone on Yap,' writes author John Lanchester in a brilliant article in the _London Review of Books_ that examines the history of money. 'The fei are rare and difficult to get by definition, so they hold their value well.' The precise value of each fei depends on its size and craftsmanship but also its provenance. Transporting the stone discs on the outrigger canoes\u2013fragile-looking, narrow boats\u2013was a treacherous undertaking and sometimes fatal for the sailors. Their deaths would in fact increase the value of the fei.\n\nSome of the stones are so large they require more than twenty adult men to move them, with the help of a massive wooden pole. To avoid the colossal effort, and the risk of damaging the stone, the islanders decided to leave most of the stone discs in their original spots.\n\nCritically, the cumbersome nature of fei meant ownership could change hands without the stones themselves ever being physically moved. Stone discs in front of a house, for example, could belong to somebody else from another village far, far away. The islanders just agree that somebody else now owns the fei, with ownership held in a collective register, the minds of the community.\n\n'My faithful old friend, Fatumak, assured me that there was in the village nearby a family whose wealth was unquestioned\u2013acknowledged by everyone\u2013and yet no one, not even the family itself, had ever laid eye or hand on this wealth,' writes William Furness. According to his accounts, one boat got caught in a violent storm on its way back from Palau. To save their lives, the sailors had to cut adrift the raft with a massive stone on it. It sank out of sight, forever lost to the bottom of the ocean. The islanders never saw the stone again but that didn't undermine its value\u2013in fact, it added to it. 'The purchasing power of that stone remains, therefore, as valid as if it were leaning visibly against the side of the owner's house,' explains Furness. It's remarkable to think that the people of Yap have so much faith in their currency that it can sit, unreachable, miles down on the ocean floor, and still have value. Now, that is trust.\n\nThe island of Yap has long been of interest to economists because it helps answer a fundamental question: what is money?\n\nIn 1991, American Nobel Prize economist Milton Friedman wrote about the 'island of stone money'. He compared Yap's monetary system to the gold standard. Friedman emphasized the importance of 'myth' and 'unquestioned belief in monetary matters'. Money can be anything\u2013paper, coins, shells, beads or stone\u2013as long as people have faith in its value. 'How many of us have literal direct assurance of the existence of most items we regard as constituting our wealth?' wrote Friedman. 'Entries in a bank account, property certified by pieces of paper called share of stocks, and so on and on.'\n\nFei stones may have been basic, not to mention unwieldy, but they represented an innovative technology. They changed the way the Yapese could store value, pay for things and have a unit of account\u2013the three primary functions of money. The stones were a physical ledger, a new method for keeping track of payments and credits. The real 'money' isn't the fei itself, but the collective agreement over who owns the fei.\n\nThe total amount of all money in the world, in terms of value, was estimated in 2006 to be around $473 trillion. That works out to be around \u00a345,000 per head for 7 billion people on the planet. But less than 10 per cent of it is physical money\u2013banknotes and coins in vaults and wallets. The remaining 90 per cent is simply electronic debit and credit entries in financial registers and accounts. There is also value in assets such as air miles and supermarket reward points that don't exist in physical form. For the most part, digits moving around on ledgers are what we call 'money' and we have to go through lots of middlemen\u2013a bank, PayPal or a credit-card company\u2013to spend it.\n\nSince the Medici bank, set up in 1397 by Giovanni di Bicci de' Medici, the basic premise of the modern-day banking system hasn't changed all that much. The Medici family invented the double-entry accounting method whereby the debits and credits of _nostro_ and _vostro_ accounts\u2013'ours' and 'yours'\u2013were gathered in one place and the bank acted as the intermediary. The Medici, one of the most powerful familial financial institutions of the fifteenth century, would, like today's bankers, hold deposits and make loans to everyone from the Pope to merchants, while charging significant interest rates. In 1494, Fra Luca Bartolomeo de Pacioli, a Franciscan monk, mathematician, magician and friend of Leonardo da Vinci, published the first description of the 'Italian method' of double-entry accounting. He described the use of ledgers, and how every transaction should be recorded twice, first as a credit and then as a debit, and then how all transactions can be reconciled to measure the overall financial health of a business. The monk, considered the 'Father of Accounting', warned that a person should not go to sleep at night until the debits equalled the credits. The universal system was nothing short of revolutionary, enabling capitalism to flourish.\n\nFast forward to 2008. In the midst of the global financial crisis, disillusionment about the traditional financial system was deep and pervasive. Could banks and governments be trusted to run our financial system? Was there another way to mediate transactions, a way that would remove the generous cut the banks took as middlemen in the process? As millions of people were losing their homes, jobs and livelihoods, a mysterious person (or persons), who went by the moniker Satoshi Nakamoto, was busy figuring out a solution to liberate money from the control of governments and banks.\n\nIt all started in October 2008 when Satoshi, who claimed to be a thirty-seven-year-old Japanese man, published a 500-word paper in flawless English on an obscure cryptography mailing list. The paper was called 'Bitcoin: A Peer-to-Peer Electronic Cash System' and it outlined the current pitfalls with traditional fiat currencies*, emphasizing one issue in particular. 'The root problem with conventional currency is all the trust that's required to make it work. The central bank must be trusted not to debase the currency, but the history of fiat currencies is full of breaches of that trust,' Satoshi wrote. 'Banks must be trusted to hold our money and transfer it electronically, but they lend it out in waves of credit bubbles with barely a fraction in reserve. We have to trust them with our privacy, trust them not to let identity thieves drain our accounts...' The paper's main intention, however, was to present an alternative solution\u2013the design of a new digital currency called bitcoin. Developed with high hopes, it would solve many of the trust issues Satoshi flagged, but, in a familiar story, it would also give birth to some of its own as time went on.\n\nOn the evening of 3 January 2009, Satoshi pressed a button and released the first fifty bitcoins, the so-called 'Genesis block', to the world. No physical coins or notes were produced\u2013just 31,000 lines of code. A week later, a man called Hal Finney, who recently passed away at fifty-eight years old, was the recipient of the first ten bitcoins sent by Satoshi.\n\nLike the massive fei stones on the island of Yap, bitcoins do not physically move around when they are being exchanged. The 'coins' themselves are simply a digital token that can move from one user's address to another, thereby transferring ownership of the 'coin'. The senders and receivers of the bitcoin do not need to know or trust one another. They are identified only by wallet IDs (known as _public keys_ ) that are not tied to real-world identities. Every transaction is recorded but it is encrypted into a random string of numbers and digits (like 12c6TSU4Tq3p4xzziKzL5BrJKLXFTX), making it very difficult, although not impossible, to trace back to its owners. No wonder it's the currency beloved by marketplaces on the darknet for all things illegal, and by criminals laundering money.\n\nIt is widely believed, however, that the first 'real' thing bought with bitcoin was not drugs but pizza. On 22 May 2010, a computer programmer living in Florida by the name of Laszlo Hanyecz convinced someone to accept 10,000 bitcoins for two large pizzas from Papa John's. 'It wasn't like bitcoins had any value back then, so the idea of trading them for a pizza was incredibly cool,' Hanyecz told the _New York Times_.\n\nTimes have changed. Based on the current exchange rate in July 2017, one bitcoin is worth approximately $2,320 (\u00a31,801). So at current bitcoin rate, Hanyecz paid more than $9 million for a pizza. These days, you can use bitcoin to pay for your plane tickets from Expedia, buy your gifts from 1-800-FLOWERS or purchase your car.\n\nThe price of bitcoin has been a rollercoaster of volatility. Its value is based on the volume and velocity of bitcoin payments running through the ledger today and on speculative future use of the digital currency. Events such as the FBI seizure of darknet outfit Silk Road or the implosion of the bitcoin exchange Mt Gox (an acronym for 'Magic: The Gathering Online Exchange') have eroded people's faith in the security and anonymity of the system. The value crashes\u2013but then it rises again, especially during a currency crisis such as in India and Venezuela in January 2017. As history has shown us time and time again, chaos and uncertainty make people open to alternative systems, including cryptocurrency. Once again, however, questions of trust lie at the heart of any new system.\n\nSatoshi was not the first person to attempt to create a form of digital cash that could disrupt the central power of Wall Street. Since the 1990s, cypherpunks* have tried and failed with the likes of B-Money, invented by a man called Wei Dai. He also wrote a paper in 1998 describing his invention as an 'anonymous, distributed electronic cash system'. Other attempts to create an online currency that enables people to directly exchange value have included David Chaum's ecash, Stefan Brands's electronic cash system and Nick Szabo's bit gold. The real issue lies in the _double-spending problem_. If I have, say, a \u00a35 note in my purse, I can't give two people the same note. Same goes for a bar of gold\u2013once I give it to you, it's clearly in your possession. But if a currency is just digital information, what stops me from copying the line of code and 'spending' it as many times as I want? It's like a digital photograph of my children that I email to my parents\u2013they have a copy but I also have a copy. It's the equivalent of being able to print your own money. So how do you solve this problem? Satoshi figured out an ingenious way through what he called the _blockchain._\n\nThe blockchain is an enormous shared digital ledger, open for anyone with internet access. To see it for yourself, just go to https:\/\/blockchain.info. Every single bitcoin transaction that has ever happened since it began in 2009, approximately 190 million transactions, is publicly recorded and time-stamped on the blockchain. Ticking over in real time, it tracks every time an asset moves from one place in the register to somewhere else, building over time. The distributed ledger is replicated on more than 5,500 computers around the world\u2013known as bitcoin _nodes_ \u2013creating an _immutable record_. In other words, everyone in the network can maintain a copy of the shared ledger but all those copies remain the same.\n\nA record on the ledger cannot be changed, falsified or erased\u2013it has a permanent memory. 'A distributed ledger is a database that is shared between multiple users, with every contributor to the network having their own identical copy of the database,' writes Andrew O'Hagan in 'The Satoshi Affair', a brilliant account of the mysterious genius. 'Any and all additions or alterations to the ledger are mirrored in every copy as soon as they're made.'\n\nSo, why is the blockchain so significant when it comes to trust? Well, for the first time in the history of humanity, there is the potential to create a permanent public record of who owns what, which no single person or third party controls or underwrites, and where we can all reliably agree on the correctness of what is written.\n\nThe true identity and whereabouts of bitcoin's inventor, Satoshi, is still the subject of hot debate. In May 2016, Craig Steven Wright, a forty-five-year-old Australian, came forward claiming he was the inventor. Wright has never 100 per cent proved, however, that he is in possession of cryptographic keys for the first Genesis block that only Satoshi can have, and his claim has since been called into doubt by many in the bitcoin community. Others believe Satoshi is Nick Szabo, a reclusive American of Hungarian descent, but he fervently denies the suggestion. Others still, including me, think Satoshi was an identity created by a group of brilliant mathematicians and computer scientists who came up with an astonishingly clever code.\n\nThere were several problems Satoshi had to crack to make bitcoin work. The first was to get the bitcoin out there in the first place. The inventor couldn't 'own' all the currency because it would give him too much power and that was against its decentralized ideology. So whom do you give the money to? A handful of geeks, panicked Cypriots in a currency crisis or perhaps celebrity web figures such as Tim Berners-Lee? The fairest way, Satoshi decided, was to award bitcoin, as an incentive, to the people who do the work communally to maintain the ledger around the world. These people are known as _miners_ , some of whom dedicate their lives to the job of being virtual witnesses in order to keep the engine of the network running. Without the miners, the blockchain engine stops.\n\nIt was early in 2011 when an article about bitcoin popped up in Yifu Guo's RSS news feed. At the time, the young Chinese immigrant was a digital media student in his early twenties at New York University (NYU). 'I remember thinking that this was the stupidest thing ever. It would never work,' Guo said in an interview with Motherboard. 'But what kept my attention was that it was open source and, after a few days of thought and further research, I concluded that this was legit.' Guo, as we'll see, would go on to become one of bitcoin's most successful entrepreneurs.\n\nThe article in Guo's feed described how every bitcoin transaction, however small, contains a difficult mathematical puzzle (known as a _proof-of-work calculation_ ) that has to be solved through trial and error. The puzzle works as evidence of the transaction's legitimacy. Essentially, transaction-clearing responsibilities, which are traditionally managed by centralized banks, are now distributed into the hands of many miners and thousands of computers in the bitcoin network. Trust is shared out.\n\nLet's say I want to pay David Forster, a young farmer and owner of Grass Hill Alpacas in Massachusetts and one of the first merchants in the world to accept bitcoins as payment, for some wool socks. Forster needs to be sure that the bitcoins in my electronic wallet are genuine. That's where the massive peer-to-peer network of miners comes in. When one of the computers has proof that the transaction is legit, the payment information\u2013the amount, time and wallet address\u2013is added to the blockchain in ten-minute bundles of transactions, known as _blocks_. (For instance, the last block added to the blockchain on 2017-04-12 06:19:53 contained 1,322 different transactions totalling 7,583 bitcoin sent.) My payment to Forster is confirmed. The block stores a long and seemingly random sequence of letters and numbers known as a _hash_ , e.g. 0000000000000000017f62231a5206f8333c9f8730c96f605cf44ddf03e8af93. Each block contains the hash of the prior block, linking the blocks together. Hence the name.\n\n'They are in a race against all the other miners to have the privilege of being the canonical record of that transaction,' says Gavin Andresen, the chief scientist of the non-profit Bitcoin Foundation and regarded by many as the 'chief' bitcoin developer. 'If they win that race, then there is a special transaction at the beginning of each block which rewards them with bitcoins.' Bitcoin places trust in mathematics: 'In proof we trust.' Guo decided to join the computational race and become a miner.\n\nMining is like a cryptographic game or lottery, where the winner is the first computer to find the key to open a digital padlock that solves the puzzle. But like the lottery, the difficulty of winning increases the more miners play it. Why is this so? Satoshi set into the software a finite ceiling on the number of bitcoins that will ever be released. For obscure reasons, he set the upper limit at 21 million, estimated to be in circulation by 2140. The rewards given to miners for solving problems were also predetermined to halve roughly every four years, to slow down the coin circulation. Initially, miners like Guo received fifty bitcoins as the reward for mining a block of transaction data. It then cut in half to twenty-five. The last event commonly referred to as 'the halving' took place on 9 July 2016, taking the reward down to the current 12.5 bitcoins. The miners accepted the reward cut because they know the measure is part of the Satoshi plan, written into the code, to keep a lid on inflation.\n\nSatoshi did another smart thing. The system is set to adjust the difficulty of the maths problems depending on how fast they are being solved, the goal being to slow the miners down and slow the release of bitcoins. But there were two things Satoshi couldn't solve with mathematical thinking\u2013market forces and human greed. (For that matter, Satoshi's own wealth is now estimated at around a million bitcoins or, currently, a billion dollars.)\n\nGuo, like other early digital diggers, was initially enthusiastic about the utopian ideology behind bitcoin; it was almost like discovering a new religion. Guo kept most of the bitcoins he mined but sold a few for 'real money' to help pay his rent and bills. The savvy student was also one of the first to realize that bitcoin could be the internet's equivalent of the nineteenth-century gold rushes. And whoever had the fastest computers would 'discover' the most money first.\n\nIn January 2013, Guo quit university and founded a company called Avalon, with fellow classmate Ng Zhang. It was one of the first companies to sell bitcoin-specific mining computers that used an ASIC (application-specific integrated circuit) chip to help boost computation horsepower. The first batch of Avalon processors sold out within fifteen minutes, at $1,200 a pop, to miners around the world. Recently, one exuberant customer paid $20,000 for one of Guo's machines on eBay. Is it worth it? Only time will tell; it all depends on the appreciation of the currency. Guo, on the other hand, has gone from struggling geek to a very rich man, joining the bitcoin millionaires club.\n\nWhen he and others first started mining in 2011, it was much easier than it is today\u2013technically, any maths geek armed with an ordinary laptop could download the software. Today, that is simply not possible. As Guo suspected, the hunger for those limited bitcoins would turn mining into a giant, highly competitive enterprise.\n\nEntering the enormous helicopter hangar in Boden, in northern Sweden, you might think you have walked into a server warehouse of, say, Amazon. The space is large enough to hold at least a dozen helicopters. Except that it's not packed with aviation equipment. The walls hold rows and rows of processors and custom-built computers, more than 45,000 of them continuously working to solve mathematical algorithms. All around is the loud, constant whirring from the industrial fans attached to each of the super computers to stop them from overheating. The place is KnC Mine, one of the largest bitcoin mining rigs in the world.\n\nDespite the libertarian ideology behind bitcoin, mining is coming up against the inevitable push and pull of being industrialized. 'Ever faster, energy-hungry ASIC machines would come on the market, spurring a relentless arms race among miners chasing the finite supply of newly issued bitcoins...' write Paul Vigna and Michael Casey in their book _The Age of Cryptocurrency._ 'The only way to win that race and stay profitable was by creating giant, data-centre-based mining farms.'\n\nBitcoin mines don't just need hardcore processing power. They need cheap electricity, lots of it. One miner's electric bill was so high police raided his house suspecting he was growing pot. 'We thought it was a major grow operation... but this guy had some kind of business involving computers. I don't know how many computer servers we found in his home,' said a baffled DEA agent.\n\nAs is often the case, a movement with ambitions to return power to individuals, accelerated out of the garage of early enthusiasts, is becoming monopolized by centralized power. By the start of 2014, bitcoin mining had evolved into a worldwide industry, with one country becoming the dominant player. Where do you find cheap electricity and cheap labour? Why, China, of course.\n\nAlong the lush green banks of the Min Jiang, a tributary of the upper Yangtze River, can be found one of the oldest surviving water management systems in the world. Built around 2,300 years ago by a man called Li Bing and a team of tens of thousands, it was designed to irrigate farmland and control flooding. The irrigation system made the Sichuan province one of the most productive agricultural regions in China and it is now a protected heritage site referred to as the 'Treasure of Sichuan'.\n\nToday, more than twenty dams are completed or under construction along the very same river, making it the centre for cheap hydroelectric power. Remote towns along the banks, especially around the area of Kangding, are being transformed into data centres, much bigger than any in the West. It is becoming the hub of a hidden bitcoin economy that never sleeps.\n\nThe mining machines whirring away all day and night use enough megawatts of electricity to power a small city. The people looking after the hardware often live and work inside the facilities, returning home to their families only four or five days of the month. The buildings tend to be unmarked and kept a secret. 'People don't really know where these mines are,' says Zhu Rei, the young CEO of an unidentified mine somewhere in the Sichuan province. 'Competition is really intense in China and the number of people getting into bitcoin mining is rising rapidly. So the bottom line is, when you are lucky enough to be in a place like this, where the cost of electricity is so low, well, you keep it to yourself.'\n\nSome of the most lucrative mines such as DiscusFish and Antpool are generating thousands of bitcoins a month. 'I've always feared that mining will concentrate in a few countries,' Yifu Guo told _The Economist_. Consider this: it is estimated that 70 per cent of the transactions on the bitcoin network are going through just four Chinese companies, mining powerhouses.\n\nOn 28 April 2011, Satoshi mysteriously vanished. In one of the last emails he sent to Gavin Andresen, he wrote, 'I've moved on to other things. It's in good hands with Gavin and everyone.' The legendary anonymous founder never explained why he moved on or what he is working on now, or why he 'probably won't be around in the future'. Does it mean Satoshi knows something we don't?\n\nEspecially in the light of its metamorphosis into a giant business opportunity, just how much faith can people place in the bitcoin system?\n\n'It's completely decentralized, with no central server or trusted parties, because everything is based on crypto proof instead of trust,' Satoshi wrote of the system in his 2009 essay. Rather than trusting third-party institutions such as banks, with bitcoin we can now place our confidence in mathematics. That's all well and good, but for most people, cryptographic algorithms, hash functions and industrialized mining operations remain a massive trust leap, especially when all of it was designed by an anonymous creator (Satoshi) who has now disappeared.\n\nThe incentive for hacking the system is high: the bounty would be bitcoins worth billions. Still, it wouldn't be easy. Technically, to hack it you would need to gain control of more than half of the bitcoin network computing capacity at any given moment. Or to put it another way, you would need to deceive more than 51 per cent of computers in the network at the same time. It is estimated that the bitcoin network has 360,000 times more processing power than all the Google server farms in the world put together. Therefore, in tech jargon, a '51 per cent attack' would be an expensive and formidable challenge, yet nonetheless theoretically possible\u2013and in fact there has been at least one attempt that went close before it was shut down. The previously mentioned DiscusFish, also known as F2Pool, single-handedly mined 26.3 per cent of all blocks between 24 May and 24 June 2016.\n\nWhat would happen if a handful of the largest mining pools in China worked in concert? The processing clout could give them veto power over changes to the bitcoin software. A scary thought. What happens, for example, if they decided to forbid all US blocks of transactions from being added to the system?\n\nHow do we trust that Satoshi won't suddenly re-emerge and plunder it all? What is to prevent another digital currency trumping bitcoin, making the original coins obsolete and worthless? How do you trust that your bitcoins are stored in a secure location and won't be subject to hacks, theft and scams? It has already happened several times before. Mt Gox was the largest exchange for bitcoins in existence. It was located in Japan and run by a Frenchman called Mark Robert Karpel\u00e8s. On 28 February 2014, it filed for bankruptcy and suddenly closed down after a major theft\u2013850,000 bitcoins worth nearly $500 million had mysteriously vanished from its accounts. Other theft incidents have included Bitfloor, Ozcoin and Bitfinex. Then, of course, there was the seizure by the FBI of around 144,000 bitcoins that were in the possession of Ross Ulbricht and the coffers of Silk Road. In a remarkable twist, the FBI ended up auctioning the bitcoins they had seized, recognizing their value, and also that the currency was legit.\n\nIf your bitcoins are stolen, there is no traditional legal recourse. In fact, nobody\u2013not even a bitcoin expert\u2013can help you, because the bitcoin transactions are anonymous. The other issue is that if you lose your cryptographic _private key_ \u2013a string of numbers that opens your digital wallet\u2013your bitcoins are gone for ever. It happened to James Howells, an IT worker from Wales. He famously lost 7,500 bitcoins in 2013. On a fateful clear-out day, he accidentally put an old hard drive he had kept in a drawer for three years into the bin. Howells had forgotten it contained his private key. Today, those bitcoins would be worth more than \u00a313 million but instead they are in a landfill site somewhere buried deep under mud and rubbish.\n\nWhat's more, how do we know that governments won't ban the cryptocurrency or make it illegal? Bolivia did. So did Vietnam in February 2014. And the Central Bank of Bangladesh, citing concerns over lack of 'a central payment system', issued a punishment of up to twelve years in prison for anyone trading in bitcoin and other digital currencies. The problems of bitcoin are not technological; they are ultimately trust issues.\n\n'The people who think that somehow bitcoin is going to bring in some kind of libertarian paradise where we won't have \"know your customer\" rules and we won't have rules of transfer, that won't happen. The people who think bitcoin is our salvation... are wrong,' says Larry Summers, former US Secretary of the Treasury and a professor at Harvard University. 'But is the blockchain technology going to be fundamental to reducing frictions? I think the answer is going to be overwhelmingly yes.' In other words, digital currency is just the beginning. The truly revolutionary invention is the blockchain, the vast underlying trust architecture.\n\n_The Economist_ , in its 31 October\u20136 November 2016 edition, featured a cover story called 'The Trust Machine: How the technology behind Bitcoin could change the world'. The article eloquently described the blockchain as the 'great chain of being sure about things'. The need for a trustworthy record is vital for all kinds of transactions, which means the blockchain technology itself is far more necessary than a cryptocurrency. A distributed public ledger offers the possibility of a reliable record for any asset transfer\u2013whether it's currencies, a contract, stock, equity or bond, deeds, property title, the rights of a song, even your identity. 'It offers a way for people who do not know or trust each other to create a record of who owns what that will compel the assent of everyone concerned,' the _Economist_ article explains. 'The real innovation is not the digital coins themselves, but the trust machine that mints them.'\n\nIn December 1974, Vinton Cerf and Robert Kahn designed the revolutionary Transmission Control Protocol\/Internet Protocol (TCP\/IP). It was, of course, the foundation for the internet that would change the way we communicate and do business. Many enthusiasts believe that 31 October 2008 marks a similar historic moment: the day the blockchain was ushered in as the next generation of the internet, as a new network of trust\u2013which promises much more than digital coins.\n\n* The euro, the US dollar and many other major world currencies are part of a fiat system. A paper note has no intrinsic value, like say gold or silver. It is accepted as money because a government says that it's legal tender. In other words, the government who issues the _fiat currency_ backs its value.\n\n* Cypherpunk is an activist who advocates for the use of strong encryption algorithms (cryptography) to help preserve privacy and private transactions. The term first appeared in Eric Hughes's 'A Cypherpunk Manifesto' in 1993.\n\n#\n\n# Blockchain Part II: The Truth Machine\n\nIn the early hours of Friday, 17 June 2016, an unknown thief, or thieves, pilfered more than $60 million of a digital currency. It wasn't bitcoin they made off with but a rival cryptocurrency called ether, or eth for short.\n\nWithin hours of the online 'heist' kicking in, the alarm went off. 'EMERGENCY ALERT!' a community organizer wrote in the DAO slack channel.* 'The DAO is being attacked. It has been going on for 3\u20134 hours. It is draining ETH (the cryptocurrency ETHER) at a rapid rate. This is not a drill.' People in chat rooms responded instantly: 'Oh shit', 'Uh oh', or as one anonymous person wrote, ':fire: :fire: :fire: :fire: NOBODY PANIC :fire: :fire: :fire: :fire:' It was as if somebody had set a bomb under the Bank of England. The target was the DAO, a particular entity in this case, but one that takes its name from the general acronym, Decentralized Autonomous Organization.\n\nThe DAO started out as a radical social experiment. Could a company run itself without executives, managers, a board or any type of chief? Could smart computer code make decisions and autonomously run the organization in place of individuals? And could a blockchain sit under it all as its digital ledger?\n\nOn 30 April, over a month and a half before the attack, the DAO fund (daohub.org) was launched as a crowdfunding campaign. The stated mission was: 'To blaze a new path in business organization for the betterment of its members, existing simultaneously nowhere and everywhere and operating solely with the steadfast iron will of unstoppable code.'\n\nThink of the DAO fund as a venture capital firm of sorts: Kleiner Perkins meets the crowdfunding platform Kickstarter. Decisions, however, are not made by a handful of venture capitalists or any one person. It is a chiefless venture\u2013software sitting on a network\u2013with thousands of founders.\n\nInitial interest in the DAO surpassed the expectations of Christoph and Simon Jentzsch, two tech-savvy German brothers who wrote the fund's code. Every day, more and more money poured in, although the contributions were not in pounds or dollars, or even bitcoin, but in an alternative virtual currency, ether (more on that later). Approximately 11,000 people invested the equivalent of $150 million within a month through a 'crowd sale'. It turned out, at the time, to be the largest crowdfunding effort in history. But even its inventors had some worries about its operation early on, and outside observers expressed concerns about its governance model once investments started pouring in.\n\nFor every one ether invested, DAO tokens were issued, proportional to the investment, which acted as an internal currency that gave all investors voting rights (notably different from equity shares) on which start-ups to back. For example, Mobotiq, a French electric vehicle start-up, was one of the companies up for funding. The Jentzsch brothers' tech venture, Slock.it, was also in the mix. The wisdom-of-the-crowd set-up was designed so that the good ol' boy investor network would not make all the decisions: the code was designed automatically to fund projects that received the highest number of cumulative votes. The decentralized nature of the fund supposedly meant that no Madoff-like character could run off with all the money. Things didn't go quite according to plan.\n\nTo understand what went wrong for the DAO and led to its eventual downfall, it is important to realize there is not just one blockchain technology\u2013that is, the original bitcoin blockchain Satoshi created\u2013but many other distributed database platforms. When it comes to trust, however, the principle behind them all is the same: a digitally decentralized, shared ledger that relies on users to power the network by confirming transactions. This means that people who have no particular confidence in or knowledge of each other can exchange all kinds of assets without having to go through a trusted third party such as a lawyer or bookkeeper.\n\nIt's why the blockchain is likely to disrupt industries like law, banking, real estate, media and intellectual property\u2013industries that typically involve layers of complex processes and lots of 'middlemen' to handle matters of trust. 'The practical consequence [is] for the first time, there is a way for one internet user to transfer a unique piece of digital property to another internet user, such that the transfer is guaranteed to be safe and secure, everyone knows that the transfer has taken place, and nobody can challenge the legitimacy of the transfer,' says Marc Andreessen, inventor of the internet browser Netscape. 'The consequences of this breakthrough are hard to overstate.'\n\nKick-started by the bitcoin blockchain, many other decentralized ledger technologies are now springing up, custom-built for different purposes. One of those blockchains is called Ethereum, created by Vitalik Buterin, a twenty-three-year-old programming wunderkind. Buterin is also the mastermind behind the general concept of decentralized autonomous organizations, to run on Ethereum. 'Instead of a hierarchical structure managed by a set of humans interacting in person and controlling property via the legal system,' explains Buterin, 'a decentralized organization involves a set of humans interacting with each other according to a protocol specified in code, and enforced on the blockchain.' That's the geeky theory. He, and the others, didn't count on how that might play out in practice.\n\nThe very first DAO launched on Ethereum on January 2016 was called Digix, a platform designed to trade gold bullion receipts peer-to-peer. The DAO fund, the Jentzsch project, was the second flagship project to launch on Ethereum. Like the brothers, Buterin was keen to see it work but he didn't think it would raise $150 million in a matter of weeks. In other words, he hadn't expected that it would get so big, so fast. Before long, the DAO fund had simply become 'too big to fail', but who would rescue it when things went south? And how did it come off the rails?\n\nVitalik Buterin was born in Moscow, but left Russia aged six to be raised in Toronto. He is tall and, notably, very thin. Typically, he sports geeky T-shirts with slogans such as YOU READ MY T-SHIRT. THAT'S ENOUGH SOCIAL INTERACTION FOR ONE DAY. His voice is flat and measured. When he speaks, his piercing blue eyes frantically dart around, as if he is trying to avoid focusing on just one person or one thing. If Hollywood were going to cast a nerdy genius alien landing on earth to refashion the world, they would cast Buterin.\n\nFrom the time Buterin was a small child, it was clear he had an extraordinary gift for mathematics and science; he just loved numbers. He could solve complex problems and clearly explain his thinking to other children and grown-ups. His father, Dmitry Buterin, who studied computer science, bought his son his first computer when the boy was four. Microsoft Excel soon became Buterin's favourite 'toy'. 'I remember knowing, for a while, for a long time, that I was kind of abnormal in some sense,' he says. 'When I was in grade five or six, I just remember quite a lot of people were always talking about me like I was some kind of math genius. And there were just so many moments when I felt like, okay, why can't I just be like some normal person and go have a 75 per cent average like everyone else.'\n\nHe first learned about bitcoin one day in February 2011, from his father who had a small software start-up of his own. He was seventeen at the time and had recently quit playing _World of Warcraft_ for hours on end. Perhaps he was looking for his next big fix. Maybe it was the cryptographic algorithms that appealed to him. One thing was clear, Buterin wanted to get his hands on some bitcoins. There was only one problem: he neither had the cash to buy them nor the computing power to mine them. So he figured out another way to earn them. He started writing for a blog, _Bitcoin Weekly_ , where he was paid five bitcoins per post, worth around $4 apiece at the time. In September 2011, he co-founded his own magazine, _Bitcoin Magazine_ , with a Romanian programmer called Mihai Alisie. 'The industrial revolution allowed us, for the first time, to start replacing human labour with machines,' Buterin wrote in the magazine. 'But this is only automating the bottom; removing the need for rank and file manual labourers... Can we remove the management from the equation instead?'\n\nThrough his writings and conversations with early enthusiasts, Buterin realized that bitcoin's underlying blockchain technology was going to be a much bigger deal than the currency itself. That it represented more than just a way of tracking money. He believed it could be used as a powerful tool to re-architect financial, social and even political systems all over the world.\n\nAnd so at the age of nineteen, in the typical pattern of entrepreneurial tales, Buterin dropped out of college. Using a pile of bitcoins he had earned and that had now significantly appreciated in value, he went round the world. He travelled from bitcoin conferences in San Francisco and Los Angeles to meet-ups in Israel, Amsterdam, London, Barcelona and dozens of other places. He spoke extensively to programmers and dabbled in a few coding projects here and there. All the time, he was trying to figure out how best to make a significant contribution to this growing quasi-cyber-religion.\n\nHuddled around laptops with other early enthusiasts, Buterin would raise question after question. How could they make the system inclusive and open to everyone? How could they create a new economy in which anyone could participate on their own terms? Could companies be run by autonomous algorithms instead of directors? He was not on a technological crusade so much as a mission to upend current power structures. And the rallying cry was _decentralization_ , a situation where users, not governments, banks or big companies are in control. 'Ultimately, power is a zero sum game,' Buterin says, 'and if you talk about empowering the little guy, as much as you want to couch it in flowery terminology that makes it sound fluffy and good, you are necessarily disempowering the big guy. And personally I say screw the big guy. They have enough money already.'\n\nHe didn't just want to disrupt the big financial institutions. Imagine a marketplace where people can buy and sell anything from artwork to books to honey directly to each other without intermediaries. In other words, imagine Amazon without Amazon (the company, the middlemen and the fees). Similarly, imagine Uber without Uber, a network where drivers could directly offer rides to passengers. Buterin wanted to create a technology that could redistribute power, away from the rent seekers and incumbent middlemen and back into the hands of the people creating value. His vision was the ultimate techno-libertarian promise of creating decentralized marketplaces that nobody owns.\n\nFollowing the siren call of Satoshi, Buterin released a white paper in November 2013, outlining the plans for his new technology called Ethereum and a currency called ether. The paper is full of technical jargon outlining the problems of the original blockchain, including the key issue of scalability. The blockchain Satoshi created has a built-in hard cap of one megabyte, or about 1,400 transactions per block, that is processed and added to the blockchain roughly every ten minutes. This works out around three to seven transactions per second. To put that in context, Visa handles more than 1,700 transactions per second in America alone. In other words, the bitcoin blockchain is not fast or big enough to handle large volumes of transactions.\n\nThe other problem Buterin identified is that Satoshi deliberately designed the original programming language to limit what the blockchain could do\u2013its job was only to store and transfer value. I can send you a bitcoin, you can send it to me. It was not intended as a general software platform on top of which other applications could be created. Imagine the original blockchain as a pocket calculator that can only do a set number of things; Buterin wanted to create the equivalent of a smartphone.\n\nIndeed, the young entrepreneur is sometimes hailed as the next Steve Jobs, although he himself prefers comparisons to Linus Torvalds, the creator of Linux software. Either way, Buterin is most definitely a 'crazy one': a round peg in the square hole, not fond of rules and with no respect for the status quo, as the famous Apple ad goes. It seems, however, that Buterin isn't trying to create a multi-billion dollar company that will bring him a windfall IPO payday. Ethereum is currently set up as a non-profit foundation based in Zug, Switzerland. And Buterin is its chief scientist.\n\nHis vision is to build 'the Lego of cryptographic finance', to give people the building blocks to create all kinds of digital services right out of the box, such as Transactive Grid, a distributed energy market that enables people to buy and sell energy directly from each other. There's also the likes of Ujo Music, which is working to create a platform for musicians to register digital rights, and to be paid directly, without labels, iTunes and other middlemen. Grammy award-winning singer-songwriter Imogen Heap was the first artist to release her single, 'Tiny Human', on the Ethereum blockchain.\n\nEthereum is based on a stripped-down, Turing-complete programming language known as Solidity. It is simple enough that developers can easily build decentralized apps ('DApps' for short) on top of it. 'Instead of creating a device that just does a specific number of things, you have a device that understands and supports this programming language and whatever people want to do can potentially be implemented,' Buterin explained in an interview with _The Economist._ 17 There are other projects, such as BitCloud, BitAngels and QixCoin, trying to achieve a similar goal. But Buterin, like Jobs, is pretty convincing when he argues that Ethereum is _the_ open blockchain worth getting behind.\n\nEthereum is designed to allow developers to spawn blockchain-based offerings that fall into three main categories or 'buckets'. The first bucket is based on being able to transfer any kind of asset\u2013from shares to concert tickets\u2013in a fast and transparent way on the blockchain. For example, Colu, a Tel Aviv-based start-up founded in 2014, has developed a mechanism to inject every bitcoin with a 'dye' that adds extra data to transactions. Think of it as colouring the cryptocurrency with information about 'real-world' assets, such as the ownership history of a car, which sticks to the coins as they are transferred and stored on the blockchain. Colu creates the equivalent of a digital ID for the asset that can be transferred directly between people, instantly and securely.\n\nThe second bucket of applications uses the blockchain to track the supply chain of products, from their provenance to the hands of the customer. Take pharmaceutical drugs. According to Havocscope, which tracks black markets around the world, drugs are _the_ most counterfeited products. It is estimated that people pay $200 billion a year for drugs, from Viagra to diet pills to flu medicine, that are not what they say they are. And the consequences of a cancer sufferer taking drugs they think are genuine but are counterfeit duds can be dire. Accenture is currently experimenting with using blockchain technology to create an open and trusted record of where drugs have come from and to track closely what happens to them across the supply chain. It is one example among many of using the blockchain as a kind of _truth machine_.\n\nThe DAO fund falls into the third, and perhaps most ambitious, bucket of blockchain applications: _smart contracts._ A smart contract is essentially a digital agreement, whether it's for a loan, job or an investment, which lives on the blockchain. The main difference between it and a traditional contract is that the clauses are not written in English and executed by lawyers. Instead, smart contracts are written in code, with preprogrammed clauses that automatically execute themselves following a set of instructions that work on a principle of 'If this happens, then do this. If that happens, then do that...' In other words, the contract is self-fulfilling and carries out what it has been coded to do. For example, my will could be turned into a smart contract, with the rules of how assets should be transferred enshrined in code. Family power plots, squabbles and lies over who gets what would be a battle with, well, a computer program. It's not currently legal but it could be one day. (Even so, it won't replace lawyers altogether. Say the language of my insurance policy was unclear, it would still need a qualified human to make a judgement.)\n\nSelf-executing smart contracts need to know there will be a clear outcome. Take gambling. From a young age, I learned about odds and probabilities through betting. (I know, it's a unique approach to teaching your child mathematics.) My family are by no means gamblers but it's a bit of a tradition to back a horse in the Grand National or to pick which side will win the football league. I have such fond memories of sitting in our lounge, huddled around the television, cheering my horse to win and willing my dad's to fall at one of the fences. It made for some wonderful family banter. Now, say I place a bet with my dad on who will win the Wimbledon men's tennis tournament. I pick Rafael Nadal; he picks Roger Federer. We agree on odds and assign a certain amount of ether to a smart contract. On the day of the match final, the system would check the final score of the game via the web and distribute the funds to whoever placed the right bet. We would not need to rely on a bookie. But as the DAO theft demonstrated, smart contracts are only as good as the people who program them: the code will always be susceptible to human error and\/or avarice.\n\nOn 17 June 2016, an anonymous hacker (or group) exploited a loophole in the DAO fund smart contract. Essentially, there was a programming mistake in the code that allowed a DAO shareholder to create an identical clone fund (known as a 'child DAO') and then freely to move money. And that is exactly what the hacker did: the equivalent of approximately $60 million in ether was drained out of the original fund into the clone. 'I have carefully examined the code of the DAO and decided to participate after finding the feature where splitting is rewarded with additional ether,' the hacker wrote in an open letter explaining the loophole. 'I have made use of this feature and have rightfully claimed 3,641,694 ether and would like to thank the DAO for this reward...'\n\n'Rightfully claimed' is the key phrase. It wasn't fraud. Blockchain purists and even some lawyers argued that the hacker was rightfully entitled to the stolen riches. 'I am disappointed by those who are characterizing the use of this intentional feature as \"theft\",' the hacker added in his letter. He wasn't a Madoff character, duping people. It was clearly the fault of the code in the smart contract that ran on the Ethereum blockchain.\n\nWhen it became clear that nearly a third of the DAO's funds had disappeared, people on online forums were calling out for one person in particular: 'Where is Vitalik?' one person asked, 'Vitalik, our alien overlord, please save us.' Buterin happened to be in China at the time, figuring out with others in the Ethereum Foundation how to proceed. 'DAO token holders and Ethereum users should sit tight and remain calm,' Buterin wrote on the Ethereum Foundation's blog after the attack.\n\nA very contentious ethical debate followed the heist. Should the community respect the rule of the smart contract and accept the unfortunate consequences? Or should they figure out a way to retrieve the 'stolen' funds?\n\nThe attack couldn't be reversed but there was another rule programmed into the contract that would provide a possible remedy. The code imposed a waiting period of twenty-seven days before any money could be paid out of a new fund. So the attacker couldn't do anything with the $50 million for almost a month. 'It's like stealing the _Mona Lisa_ ,' says Stephan Tual, the COO of Slock.it. 'Great, congratulations, but what do you do with it? You can't sell it, it's too big to be sold.'\n\nThe Ethereum team, including Buterin, proposed something called a _hard fork_. It is technical jargon for essentially rewriting history or changing the rules. It is, ultimately, a last-resort solution. Buterin proposed creating an entirely separate version of the ledger that wouldn't have the original loophole. 'It's a one-time fix to a one-time problem,' Buterin said. But first he had to convince the majority of people in the Ethereum network\u2013more than 51 per cent\u2013it was the right way forward.\n\nSupporters of the hard fork insisted that even though they were in uncharted legal waters, they should turn to traditional English law: a contract should be interpreted by the intended spirit of those who wrote it, not the literal interpretation of the words ('to the letter'). Did the intent behind the DAO contract trump the hundreds of lines of computer code?\n\nThose who strongly opposed the hard fork argued that it was a blockchain sin, against the mission of Ethereum, which is to be a 'decentralized platform for applications that run exactly as programmed without any chance of fraud, censorship or third-party interference'. Isn't the goal of a decentralized network that no one has the power to rewrite history, or else the network itself becomes untrustworthy?\n\nThe hacker asserted that smart contracts are their own arbiters: 'A soft or hard fork would amount to seizure of my legitimate and rightful ether, claimed legally through the terms of a smart contract. Such fork would permanently and irrevocably ruin all confidence in not only Ethereum but also in the field of smart contracts and blockchain technology.'\n\nWhen it was put to a vote, 87 per cent of the Ethereum network said 'yes' to a hard fork. The result? The transactions were effectively made void and the millions of 'stolen' ether tokens were retrieved and returned to the DAO crowd investors. It was like the hack never happened. But at what cost?\n\nButerin dismissed the hack as a 'rite of passage' for a technology still in its infancy. Other DAO code writers and creators, like Christoph Jentzsch, took a similar view. It was a young concept, he argued, and this DAO, with its massive and rapid crowdfunding, had been forced to run while it was still getting the hang of walking. That may be true, but the fork set a dangerous precedent for Ethereum and its quest to become the trusted operating system of the future.\n\nIf certain people can reverse transactions, doesn't that mean _they_ , not code, are in charge of the system? And if you bend or change the rules once, what happens next time it fails or doesn't suit you? It's rather like the government bailing out the banks when risky trades went south. The hard fork was a top-down reordering of events.\n\n'Its creators hoped to prove you can build a more democratic financial institution, one without centralized control or human fallibility,' writes Klint Finley in an article in _WIRED_. 'Instead, the DAO led to a heist that raises philosophical questions about the viability of such systems. Code was supposed to eliminate the need to trust humans. But humans, it turns out, are tough to take out of the equation.' In other words, even if the maths works perfectly, trust is not simply a matter of code. At the end of the day, the problems of the DAO fund are not just technological but people problems. Humans get in the way.\n\nAndreas Antonopoulos, author of _Mastering Bitcoin_ , calls the blockchain 'trust-by-computation'. Reid Hoffman, venture capitalist and LinkedIn founder, labels it 'trustless trust'. But these terms are a bit misleading\u2013there is still clearly trust involved. You have to trust the idea of the blockchain; you have to trust the system. And given that most people lack the technical know-how to understand how the system really works, you have to trust the programmers, miners, entrepreneurs and experts who establish and maintain the cryptographic protocols. A large dose of faith is required. But it's true to say that you don't have to trust another human being in the traditional sense.\n\nSo here, with the DAO fund and the hard fork, was a trust stumble. History, however, is littered with high-minded projects that were pushed through\u2013by kings, emperors, inventors, scientists, surgeons\u2013before they were completely ready. Early steel bridges that collapsed, experimental operations that killed the patient, explosions in the lab, re-engineered ships that sank. The short-term results were disastrous but valuable lessons were learned. As polymath Danny Hillis said back in 1997, 'Technology is everything that doesn't work yet.'\n\nEvery innovator wants to be first over the line, and it's no different with the quest for the ultimate blockchain technology. Inevitably, there will be glitches along the way because that's how innovation comes into being and grows resilient, just as the body develops its immune system by being exposed to bugs and viruses. The blockchain's enormous potential means developers and investors are taking a classic 'fail fast, fail forward' approach. As Ethereum's story shows, even the odd hiccup and regrettable repair job won't stop people jumping on the blockchain juggernaut.\n\nSince 30 July 2015, the day Ethereum went live, around 405 DApps have been created on its blockchain. To put that figure in context: when the Apple App Store launched in 2008 there were 500 apps available. By 2010, there were 250,000 and in June 2016 there were more than 2 million. Investment funding in blockchain-related start-ups has increased globally from an estimated $1.3 million in 2012 to more than $1.4 billion in 2016, according to PwC. Much of the interest (and a lot of hype) is focused around how decentralized ledgers can create a shared version of _single proof_ or a _digital truth_ about the identity of assets. 'I see what you see... and I _know_ that what I see is what you see.'\n\nTake diamonds, for instance. The round diamond in my engagement ring has an interesting history. When my family, Eastern European Jews, fled Russia in the late 1800s they exchanged their wealth for three or four diamonds. Precious stones were easier than money to hide, and were commonly sewn into the linings of coats or hidden in the soles of shoes. My nana, the late Evelyn Amdur, supposedly had the last remaining family diamond, roughly 2.3 carats, in her possession. She wanted me to have it when I got married, so she asked my then fianc\u00e9 and now husband, Chris, if he would use it in my engagement ring. He had it designed in a beautiful antique setting. It's a stunning ring. A few years ago, however, the stone started chipping quite badly, with chips you could see and feel. The stone was meant to be valuable, without inclusions, so it was an odd thing to happen. I started to suspect the stone was possibly not the original family diamond.\n\nEvelyn was like a character in a _Catherine Tate Show_ sketch\u2013funny, frank, slightly crude and with a remarkable talent for fooling people. A couple of years ago, she was sick and it was clear she didn't have long to live. One day when I went to visit her, I decided to ask her about the ring. She was sitting comfortably in her favourite beige armchair, cup of tea and digestive biscuit in hand, when I raised the sensitive topic. Nana smiled, with a twinkle in her eye, and said, 'Well, darling, the luxury cruise was very wonderful.' I never did get a straight answer before she passed away.\n\nSure, I can go to a jeweller and get an assessment of the diamond's four Cs\u2013cut, clarity, carat and colour\u2013and know its real value. But that's not what is important to me. And I do kind of like the idea that my nana, the rascal, sold the family heirloom and went on the trip of her dreams. Even so, I would love to have known the life story of the real stone\u2013its age, its lineage and where it is now. The problem is that the information, like many other valuable items, was stored on a paper record or certificate that has been lost years ago. It is the sort of thing that happens all the time. The blockchain, however, offers a way to capture and keep the history of an item\u2013whether it's a diamond, a valuable stamp, bottle of wine or piece of art.\n\nThanks to consumer pressure, we can now find out the source of, say, fair trade coffee from Starbucks or the organic cotton in Gap T-shirts, yet we still know surprisingly little about most of the items we own and use. Was that organic, grass-fed cow really raised on such-and-such free-range farm as claimed, slaughtered at such-and-such abattoir, packaged last week and brought to the supermarket on Wednesday? Or, as in the Tesco scandal, was it contaminated with some horsemeat at some point in the journey from pasture to plate? Is this product what it claims to be? Supply chains and the origins of a product are, for the most part, a dark secret.\n\nProvenance. It's a word Leanne Kemp, a serial entrepreneur, has spent a lot of time thinking about. 'It means the history of something, where it came from and where did it go,' she says in her distinct Aussie accent. 'Who owns it? Who sold it and where is it now?' Essentially, it's the life story of an item, and in the world of goods, especially expensive or rare items, provenance matters. The Yapese were right with their fei stones\u2013the value of an item should not be separated from its origin and history.\n\nKemp, now in her late forties, was born and raised in Brisbane, Australia. She moved to London in the late 1990s and divides her time flying between continents, her work specializing in an unusual blend of technology and the jewellery trade. 'I'm a technologist. A \"super-nerd\" who can cut code,' she says. 'I used to work in RFID [Radio Frequency Identification] to track the identity of goods as they moved through the supply chain.' Kemp likes to use a new technology to solve a problem in a way it hasn't been solved before.\n\nSeveral years ago, she began to immerse herself in the world of cryptocurrency. From the outset, like Buterin, she was far more interested in the ledger technology than the bitcoin itself. 'The currency had been on the market for quite some time. The fundamental change occurred in 2014 when the bitcoin network released something called an _op return function_ ,' she says. 'Basically, it enables you to trade a coin and to hash on to the coin a piece of data. I started to think about what I could do with that functionality.' Could it be used to track the origins and ownership of, well, anything worth tracking?\n\nLooking into the diamond industry, Kemp discovered that it was plagued by problems like synthetic diamonds, insurance fraud, theft and tampering of paper certificates. Some \u00a345 billion is lost annually in the United States and Europe on insurance fraud alone, and an estimated 65 per cent of fraudulent claims go undetected. Then there are the notorious 'blood diamonds', precious stones mined in African war zones, often by young children, with the funds from sales frequently used to arm brutal rebel conflicts. By the time Kemp came along, there was at least one certification system, the Kimberley Process, in place to reassure buyers they weren't buying a diamond with blood on it, but that system was not foolproof and still largely depended on paper trails.\n\nEarly in 2015, Kemp sketched the idea for her company on the back of a napkin. She drew the trail of a diamond from a mine to a marketplace to a person's finger. 'I sketched the data we could store against the provenance chain of diamonds and that's where it all started.' A few months later, she founded Everledger, a London-based start-up that digitally certifies diamonds on the blockchain.\n\n'We create a diamond's digital thumbprint or ID,' Kemp says. 'Take a three-carat diamond. It will have a serial number inscribed on its girdle during the grading process. There are the four Cs\u2013cut, clarity, carat and colour\u2013but there are forty other attributes, such as angles, cuts and pavilions, which make up that specific diamond.' The diamond's ID is enshrined in the blockchain, creating an immutable record for insurers, traders and customers to know the real provenance and movements of a diamond over its entire history. 'We can apply this technology to solve very big ethical supply chain problems: from ivory poaching to blood diamonds,' says Kemp, adding that because a diamond moves through so many hands, corruption, scams and rip-0ffs can happen at any stage of its journey from grimy mine to gleaming boutique. 'Blockchain technologies allow us to bring ethical transparency on a global scale.'\n\nSo far, Everledger has digitized the ID of more than 1 million diamonds and has partnered with big financial players including Barclays and Lloyds. The company is also building an anti-counterfeit database. What this means is when a stolen diamond resurfaces for sale on an online marketplace such as Amazon or eBay, it will be much easier for investigators to track its history and return it to its rightful owner.\n\nIn a few years, we will get to a point where we are able to check the provenance of all kinds of items before we buy them and find out precisely if they are what they claim. The question is, will customers want to know and will they care? 'We've seen this with organic food, but when it comes to luxury goods, people mostly rely on the trust of a brand such as De Beers to decide what to buy,' Kemp says. 'If there was to be a full consciousness of transactions, what would we choose? I don't know the answer to that yet.'\n\nEverledger is essentially building a platform to track the true identity and reputation of objects. 'We're on the next generation of technology where transactional data of objects and assets becomes woven into the web and that's what I think is the World Wide Ledger,' says Kemp. The World Wide Ledger (WWL) offers a way to make and preserve truths\u2013the history of assets.\n\nImagine if this technology had been around in the Second World War during the greatest art theft in history. Art objects of all kinds, deemed suitable to Hitler's taste, were shipped in freight cars from all over Europe to end up in Germany. It is estimated that more than 650,000 artworks, including Giovanni Bellini's famous _Madonna and Child_ and one of Edgar Degas's iconic ballerina paintings, were looted by the Nazis, with many of the works shamelessly stripped from the homes of wealthy Jews who had fled or been sent to concentration camps.\n\nWhen the war was over, the Allies put together a special unit of personnel, the so called 'Monuments Men', devoted to finding and returning looted art to their rightful owners. Despite those efforts, the art was largely returned to countries, not to individuals. Some 100,000 stolen works of art remain unaccounted for, their current location and owners an enigma.\n\nIt turns out that during the 1950s and 1960s, hundreds of works were sold, at a significant discount, to the Nazis who had stolen them. Perhaps more alarming was the fact that Jews were made to buy back in auctions (or at least split the fees with houses) works they had proved to be rightfully owned by their families. 'They called them a \"return sale\",' says Anne Webber, the founder of the Commission for Looted Art in Europe, a London-based non-profit.\n\nNaturally, museums from the Tate in London to the Metropolitan Museum of Art in New York and the Louvre in Paris try to check the ownership history of works on display and in storage, but it is an incredibly complex task. In most instances, the museum has no way of verifying the claims of ownership of plundered property because, just like my diamond, the paper records have either been lost or tampered with along the way. Remarkably, it wasn't until 2001 that the American Alliance of Museums published the first set of strict guidelines for handling and checking the provenance of _Raubkunst,_ Nazi-confiscated art.\n\nSo how do families, many of whom lost parents in the Holocaust, even know what is missing from their collections? How do you stop people filing fake claims if the true existence of the artwork is unknown? Decades after the war, most attempts at recovering and returning looted art fail, and the knowledge about collections is disappearing as the original owners pass away.\n\nOne problem with _Raubkunst_ has been that the art world, like many other industries, lacks transparency. In May 2016, Everledger announced an investment and partnership with Vastari, a company that creates a network between museums and collectors. 'There are a number of individuals around the world who hold significant collections, say, in Andy Warhols,' says Kemp. 'If they could easily track the movement of their artwork, they might be able to realize or release the financial potential of that piece and the public would see it more.'\n\nKemp is just one of many entrepreneurs around the world recognizing the power of the blockchain to act as a new kind of digital trust broker. And what works for diamonds could also work for, well, fish. Provenance, founded by Jessi Baker, is using the technology to track the supply chain of fish from fisherman to plate. 'Every product has a story,' says Baker. 'There is often an enormous gap between advertising and the reality of operations, what goes on behind the scenes. I find it strange that the provenance and production of products has remained so secret for so long.'\n\nIt's not just start-ups thinking this way. Alibaba wants to weed out counterfeit foods\u2013whether soy sauce made using dirty tap water or fake spices unfit for human consumption\u2013by using the blockchain to track products sold on Taobao and Tmall through the supply chain. Retail giant Walmart and IBM have partnered with Tsinghua University in Beijing digitally to track the source, factory information and movement of pork in China on a blockchain. 'Consumers today want more transparency about where and how a product came to be,' says Frank Yiannas, vice president of food safety at Walmart. 'If you shine a light on the food system, that leads to transparency.'\n\nThe transparency Yiannas is referring to holds immense value in industries where lies and falsehoods are rife. Indeed, at the World Economic Forum at Davos in 2017 the agenda was full of presentations with titles such as 'Blockchain Revolutionizes Global Transactions' and 'Employing the Blockchain to Serve Society'. One of the much-hyped virtues of blockchain technologies is their potential for emerging markets such as Honduras and Ghana where weak governance and a lack of record-keeping systems means that trust is all too often in short supply. 'It has the potential to leapfrog billions of people into a new era\u2013in parallel to the way that mobile phones helped them leapfrog over landlines,' says Mariana Dahan, a senior operations officer at the World Bank.\n\nHernando de Soto Polar, a prominent Peruvian economist, has long held a view that capitalism will only thrive in the Third World when people feel that the law is firmly on their side, or even simply applies to them and their personal circumstances. 'What you have to remember is that people outside the legal system are the majority. There are 7 billion people in the world, and those who are outside the legal system are five billion,' he says. 'So this is no marginal phenomenon.' One of de Soto's core arguments is that a lack of clear _de facto_ property rights, not capital, is what has held (typically poor) people back in developing countries for so long. An estimated 5 billion people, mostly in the developing world, have difficulty proving they own land, businesses or cars. With no legal owners, the assets are effectively walled off from the 'official' economy. It adds up to trillions of dollars in locked or 'dead capital'. If ownership of land and housing is clear, recognized and protected, people will look after what they can control. It also gives people collateral to borrow against.\n\nIn Ghana, where it is estimated around 78 per cent of land is unregistered, homeowners will often put a sign outside their house painted with the words 'This house is not for sale'. It is often the only means available to show that a place is occupied. Another big problem that arises from the unregistered land is that people sell what they don't rightfully own. Government officials, abusing their power, sometimes buy an election vote here and there using fraudulent land titles. Bureaucrats sometimes hack databases to get themselves the rights to a nice beachfront property. So start-ups such as Bitfury, ChromaWay and Bitland are starting to work with governments to collect and organize property records on blockchains, based on the premise that every _coin_ on the blockchain could represent a unique house or land title. If there were an immutable record of land-title information, theoretically at least, dodgy officials would not be able to tamper with the record without leaving a digital trail.\n\nThe UK government's chief scientific advisor, Sir Mark Walport, published a report in January 2016, claiming that 'distributed ledger technologies have the potential to help governments collect taxes, deliver benefits, issue passports, record land registries, assure the supply chain of goods and generally ensure the integrity of government records and services'. Similarly, the Dubai government announced its ambitious plans to go paperless and move all documents on to the blockchain by 2020.\n\nFor the moment, the blockchain remains in the realm of enthusiasts, innovators and idealists. It's still unclear what will be the killer consumer app that takes it into the mainstream. But one thing is certain: the intermediaries and centralized behemoths such as banks and accountancy firms will do their darnedest not to be cut out of the picture by a network of digital ledgers. Indeed, the first industry widely to adopt the blockchain could be the very middlemen Satoshi wanted to replace: finance.\n\nThe first sentence of Satoshi's 2008 bitcoin paper defines it as follows: 'A purely peer-to-peer version of electronic cash that would allow online payments to be sent directly from one party to another without going through a financial institution.' 'Without a financial institution' is the key point. But it looks like banking middlemen are going to use the technology to make the exchange of money faster and cheaper. 'My hunch is that the blockchain will be to banking, law and accountancy as the internet was to media, commerce and advertising,' says Joi Ito, the respected entrepreneur, professor and director of the MIT Media Lab. 'It will lower costs, disintermediate many layers of business and reduce friction. As we know, one person's friction is another person's revenue.'\n\nSo if financial incumbents can't be at the centre, why not control the new trust architecture that transactions will flow through? Indeed, patent wars are brewing and the race is on to try to 'own' blockchain technologies.\n\nAnyone on Wall Street knows who Blythe Masters is. She is a banker's banker. Born in 1969, she was raised in southeast England, where she attended the exclusive King's School in Canterbury. Her accent is cut-glass English: proper Home Counties. During her gap year, before studying economics at the University of Cambridge, she interned at JP Morgan Chase. At the age of twenty-two, she officially joined the bank in the derivatives team in New York. Her rise through the ranks was inexorable. By the time she was twenty-eight, she was managing director, the youngest woman ever to achieve the title in the investment bank's long history. At thirty, she became head of the global derivatives unit. At thirty-four, she became chief financial officer, joining an elite group the media dubbed as the 'JP Morgan Mafia'. She became known on Wall Street as 'Queen of Commodities'.\n\nIn 1989 more than 10.8 million gallons of crude oil from the oil tanker _Exxon Valdez_ spilled into the pristine oceans of Alaska, spreading far and wide. The potential damages were estimated to be upwards of $5 billion. The oil company needed a loan, an enormous one. So in 1994 they went to their long-term bank, JP Morgan. Masters happened to be leading the team managing the financial side of the crisis for the oil company. Exxon were an old client and the bank didn't want to turn down their request. At the same time, the loan was risky and would tie up a lot of the bank's reserve cash. Masters came up with what seemed at the time like an ingenious idea: what if the risk of the loans could be sold?\n\nHer thinking was based on the fact that investment banks already swapped bonds and interest rates. So why not swap the risk of defaulting on loans? And so the idea of the 'credit default swap' was born and took off big time. Blythe Masters was credited as the mastermind behind the concept. Credit default swaps were meant to reassure investors the risk was hedged if a loan went south. Instead, as we now famously know, it blew enormous holes in the balance sheets of banks and insurance companies such as American International Group (AIG) and mortgage lenders like Fannie Mae, who didn't have the collateral owed when the underlying credit swaps deteriorated.\n\nDuring the 2007\u20132008 financial crisis, Masters had the courage, or some might say audacity, vociferously to defend the bank's trading activity. She asserted they had done nothing wrong, despite the millions of Americans who lost their homes and jobs as a result of the crisis. If the media needed a target, Blythe Masters had a giant circle on her back. Warren Buffett went so far as to describe the derivatives she masterminded as 'financial weapons of mass destruction'. She was vilified, some say unfairly, given how many others jumped on board with her and then ran for cover. In April 2014, after almost three decades with JP Morgan, Masters resigned.\n\nNow, she is back, championing not swaps but another potential money-spinner\u2013blockchains. 'You should be taking this technology as seriously as you should have been taking the development of the internet in the early 1990s,' Masters told a packed and rapt audience of money managers and investors at a conference held at the Le Parker Meridien Hotel in mid-Manhattan in the summer of 2015. 'It's analogous to email for money.'\n\nAround the time when bitcoin and blockchains were starting to catch the attention of the mainstream investment world, a New York-based start-up called Digital Asset Holdings (DAH) was launched. Blythe Masters was at its helm. The Wall Street veteran is knowledgeable about a common problem many banks face\u2013getting incompatible financial databases to talk to each other. It's costly, complex and takes time. While it might seem that traders work at Red Bull speed in lightning-paced environments, the technology used to execute trades is remarkably old-fashioned and _slow_. Lots of phone calls are made, emails traded and even the occasional fax is still sent. It can take up to three days\u2013T3\u2013for stock trades to change hands via clearing houses such as the National Securities Clearing Corporation (NSCC). It's a process known as 'settlement lag'. Every hour before settlement happens, when a trade precariously hangs between sale and purchase, increases the risk that the trade won't go through. Obviously, it's in the banks' interest to close that lag time as much as possible.\n\nBlockchains could help reduce the gap of the entire lifecycle of a trade from days to minutes, even to zero. According to a report by Santander InnoVentures, the Spanish bank's fintech investment fund, by 2022 ledger technologies could save banks $15\u201320 billion a year by reducing regulatory, settlement and cross-border costs.\n\nDigital Asset Holdings wants to be _the_ distributed database handling these speedy transactions. And the who's who of the world's biggest financial names, including Goldman Sachs, Citibank and Blythe Masters's old employer, JP Morgan, have ploughed more than $60 million of investment into DAH. Speed and efficiency are not the only qualities that make distributed ledgers attractive to banks. 'Regulators will like that blockchain-based transactions can achieve greater transparency and traceability\u2013an \"immutable audit trail\",' Masters says. In other words, it could help eliminate the kinds of fraud that come from cooking the books. It's rather ironic that these words come from a woman who spent several months being investigated by the Federal Energy Regulatory Commission for a cover-up of energy-trading strategies. Masters was not cited for any wrongdoing and no action was brought individually against her. JP Morgan paid $410 million to settle and close the case, without denying or admitting wrongdoing.\n\nOn Wall Street, the race is on to embrace or control what could be either its biggest ally or its death knell. Where does the average Joe store their money? In a bank's current or savings account or a safety deposit. But the blockchain could become a new repository of value.\n\nHow do typical loans work? A bank assesses the credit score of an individual or business and decides whether to lend money. The blockchain could become the source to check the creditworthiness of any potential borrower, thereby facilitating more and more peer-to-peer financing. How do typical credit cards and money transfer services work? They currently flow through a bank, but the blockchain could handle this exchange of value directly from person to person. Consider traditional accounting, a multi-billion industry largely dominated by the 'big four' audit firms, Deloitte, KPMG, Ernst & Young and PwC. The digital distributed ledger could transparently report the financial transactions of an organization in real time, reducing the need for traditional accounting practices. And that is why most major players in the financial industry are busy investing significant resources into blockchain solutions. They have to embrace this new paradigm to ensure it works for, not against, them.\n\nA San Francisco-based venture called Chain is said to have raised more than $30 million in funding from big names such as Nasdaq, Visa and Citi Ventures to develop open-source code for a distributed ledger. IBM, Wells Fargo, the London Stock Exchange and others have joined forces with Digital Asset Holdings to develop blockchain software that is also open source, making the underlying recipe available to developers. Originally dubbed the Open Ledger Project (and later renamed Hyperledger), the joint efforts are being overseen by the widely respected Linux Foundation.\n\nGoldman Sachs has recently filed a patent for its own cryptocurrency, its own version of bitcoin, called SETLcoin which processes foreign-exchange transactions. It is designed to run on the bank's own private blockchain. This means the replicated ledger of transactions still sits behind the closed walls of the bank, centralized and guarded. It seems to defeat the very purpose of the technology, which is to create a single indisputable version of the truth, freely accessible to all, that could eliminate the need for the bank entirely. In the patent, Goldman describes SETLcoin as having the potential to guarantee 'nearly instantaneous execution and settlement' for trades. It would mean all the capital the bank is required to keep in reserve, to hedge against the risk of transactions if they don't settle, would be freed up.\n\nMore than forty banks have a stake in a consortium called R3CEV to come up with shared standards for blockchains. The technology will be pretty much worthless if there are multiple versions of the blockchain that can't work together. R3CEV wants to bring along all the banks and regulators so they can share just one\u2013a ledger that is not controlled by any one person or organization but by many participants. Sure, it's collaboration, but perhaps not the kind Satoshi had in mind.\n\nNotably, R3CEV has recruited a man by the name of Mike Hearn as its chief platform officer. The former Googler is a big deal in the blockchain world. Hearn spent more than five years working full-time alongside Gavin Andresen, as part of Bitcoin Core, the original group of developers that maintain the open-source code that runs the bitcoin peer-to-peer network.\n\nHearn admits he is a 'tell-it-like-I-see-it kinda guy'. In January 2016, he publicly denounced the future of bitcoin and said it was inherently doomed. 'It has failed because the community has failed. What was meant to be a new, decentralized form of money... has become something even worse: a system completely controlled by just a handful of people,' Hearn wrote. 'The mechanisms that should have prevented this outcome have broken down, and as a result there's no longer much reason to think bitcoin can actually be better than the existing financial system.'\n\nJust days after he published the post, Hearn joined the R3CEV banking consortium. 'The current Bitcoin system, I mean the system we actually use today with the blockchain, isn't going to change the world at all due to the 1mb limit [the maximum size of a bitcoin block],' he said in defence of his move. 'So if I have a choice between helping the existing financial system build something better than what they have today that resembles Bitcoin, or helping the Bitcoin community build something worse than what they have today that resembles banking, then I may as well go where the users are and work with the banks.'\n\nFrom Buterin to Hearn, it seems that everyone, however different their motives, is in a race to create something like the original Satoshi blockchain, only better. For many, it's the biggest game in town.\n\nThe blockchain raises a key human question: how much should we pay to trust one another? In the past year, I've paid my bank interest and fees, some hidden, to verify accounts and balances so that I could make payments to strangers. I've spent thousands of dollars on lawyers to draw up contracts because I am not quite sure how another person will behave (and to sort out a few incidents where trust broke down). I've paid my insurance company to oversee the risk around my health, car, home and even life. I've paid an accountant to reconcile an auditing issue. I've paid an estate agent tens of thousands of dollars essentially to stand between me, the prospective buyer, and the current owner to buy a house. It would seem we pay a lot for people to lord over our lives and double-check what's happening. All these 'trusted intermediaries' are part of the world of institutional trust that is now being deeply questioned.\n\nMany of the ideas surrounding the blockchain sound ambitious, risky and radical. Many are being over-hyped, over-funded and will likely fail. What's not in doubt is that, as the cost of trust plummets because of new technology, the third parties currently paid to facilitate our trust\u2013be they agents, referees, watchdogs or custodians\u2013will increasingly have to prove their value if they don't want to be supplanted by an 'immutable' ledger.\n\nIn 1993, enthusiasts such as Al Gore were telling the world about a coming 'information superhighway' that would change the world. The internet was a novel concept few had grasped and people didn't really know what to make of it. John Allen, an early web aficionado, went on TV to try to explain how people would use it: 'In this world, there's a table with a big sign on it that says \"Football\" and there's 150 or 1,000 jocks all around the world who want to talk about football,' he said on CBC. At that time, Mark Zuckerberg was nine years old. Google was three years from being born. All the other products and companies that would emerge to commercialize the internet and its future potential were not yet clear. Today, it is circa 1993 for blockchain technologies. Even though most people barely know what the blockchain is, a decade or so from now it will be like the internet: we'll wonder how society ever functioned without it. The internet transformed how we share information and connect; the blockchain will transform how we exchange value and whom we trust.\n\n* Slack is a communication and messaging app used by teams and groups to organize conversations. 'Channels' are created around a specific topic. Thedao.slack.com was the channel created around the DAO fund.\n\n# Conclusion\n\n'It is trust, more than money, that makes the world go round.'\n\nJoseph Stiglitz\n\nOn 28 February 2016, in a small cinema in Pan Paper Village in Kenya, more than 400 locals had gathered to watch an English football match. The game was between two of the greatest rivals in the Premier League: Manchester United and Arsenal. The cinema belonged to Eric, an entrepreneur who also owns a printing business, small kiosk and a photo studio. He is a local success story.\n\nThe Kenyan crowd, sitting in a darkened room with the game projected against a wall, cheered as the players walked out to start the match. Man U started out strong with Arsenal struggling to get anything going. Then the screen went blank. The crowd let out a groan. 'Get the game back on!', 'Boo, what's happening?', 'We're missing the match!', the paying customers hollered at Eric.\n\nThe cable company had disconnected the service because Eric's bill had not been paid on time. Feeling frazzled, with hundreds of eyes trained on him to fix the problem, and fast, he reached for his phone. He opened an app called Tala, a company that makes loans to people without a traditional credit history in emerging markets including Kenya, the Philippines and Tanzania. Luckily, he recently reached gold status for repaying four loans on time and his credit had been increased to 5,000 Kenyan shillings (KES), the equivalent of around \u00a340. It's a decent amount of money when you consider that a small 0.33-litre bottle of Coke costs around 50 KES.\n\nWith a few taps and swipes on his phone, Eric took out his Tala credit line and immediately paid his cable fees. Within minutes, the match was back on the big screen. The crowd cheered; the score was still 0\u20130. Each customer paid Eric around 15 KES, so he made 6,000 shillings in takings. After everyone had left, he paid back his Tala loan and kept the profits.\n\nEric is one of more than 300,000 people Tala has issued loans to in Kenya since the company was founded in 2011. Loans range from $10 to $100, with terms of three to four weeks. The interest rates range from 11 to 15 per cent, considerably better than those of the loan sharks who charge a crippling 300 per cent and more. The repayment rate is 90 per cent. Tala's customers are part of the one-third of the world's population that is 'unbanked'. Across the globe, there are 2.5 billion people who have no traditional credit score and are therefore not eligible to receive loans to fund new businesses, buy a home and generally improve their lives. 'For a traditional bank, [the lack of a credit score] means there is no real data to answer the question: \"What's my basis for investing in this person?\",' says Shivani Siroya, the thirty-four-year-old founder and CEO of Tala.\n\nSiroya was first raised in Udaipur, India. Her mother, by all accounts, was a renegade. She went to medical school and was the first female doctor, a gynaecologist, in her community. She ran medical camps in rural areas, training women how to deliver babies more safely. Her relationship with her patients was always selfless. 'We would say to her, \"Mom, you work so hard and your patients never pay you,\"' Siroya tells me. 'But money wasn't the thing that kept her going. For her, being a doctor is not a transactional relationship. She truly wants to help.'\n\nThe family eventually ended up moving to the United States, settling in Brooklyn, and her mother continued her good work there, often allowing poor patients to pay when they could or not at all. 'She gave them credit because she trusted them,' Siroya says.\n\nOn the phone from Santa Monica, California, where she now lives, Siroya sounds so calm and tranquil she might have just finished a yoga class. She gives the impression of being, like her mother, full of life and goodness. It's odd to imagine her working in the cut-throat culture of mergers and acquisitions at Citigroup, which is where she began her career as a banking analyst before going on to become an equity researcher at UBS. Siroya loved the data but something didn't sit right with her. 'I didn't want to be analysing companies selling diet drugs; I wanted to work with companies doing something with impact.' She heard about social enterprise guru Muhammad Yunus who won the Nobel Peace Prize in 2006 for pioneering the concepts of microfinance and microcredit. Shortly after, she quit her job and joined the United Nations Population Fund.\n\nAt the UN, she ended up working with thousands of microbusiness owners, such as locals running food stalls and small kiosks across Africa and India. Over the course of two and a half years, she interviewed more than 2,500 people, going door to door and learning the detail of their lives. 'These micro-entrepreneurs were working so hard to grow their businesses and to help create jobs, yet they felt really stuck. They were getting access to really informal micro-loans, often from loan sharks, but they couldn't get access to any kind of real capital in the form of a business loan,' Siroya says. 'When I looked at the other side, I found that traditional banks felt these individuals, operating in an economy where cash is prevalent, were too risky. I saw a gap in the market.' She realized the problem came down to identity. How could she build financial identities for people who didn't otherwise have any?\n\nApproximately a billion people in the emerging markets have basic smartphones. People are using them in the same way that Westerners do\u2013to text friends, surf the net, run their daily lives and pay for everything from electricity bills to parking fees. The phones also tell a story about their owners. Tala can cull more than 10,000 data points from a phone in less than one minute to gauge a person's ability and willingness to repay loans. 'We look at behavioural things such as what are their current spending habits? Do they have consistency in their income? What other apps do they use?' explains Siroya.\n\nThe size of a person's network is a strong trust signal for potential borrowers. Turns out, if our phone calls last more than four minutes, we tend to have stronger relationships, and therefore may be more creditworthy. Similarly, people who communicate with more than fifty-eight different contacts tend to be better borrowers because they have a wider network to depend on. Even how we organize our contacts can be revealing. 'If more than 40 per cent of the entries in a person's contact list have both first and last names, it suggests a customer who is sixteen times more reliable than one with very few contacts listed with first and last names,' explains Siroya. Filling in a first and a last name shows, in a small way, the care and attention we pay to something. Indeed, no single piece of information determines whether someone gets a loan\u2013it's the cumulative points of data that provide a clear picture of a person. 'It's a financial identity that looks more like a person and less like a score,' says Siroya. 'This is data that would not be found on a paper trail or in any formal financial record.' It proves that a person doesn't need a traditional credit score to prove they are trustworthy.\n\nToday, Tala is the fifth most used app in Kenya. Only the Bible app, Facebook, Twitter and WhatsApp are ahead of it. 'Customers don't see us as a traditional banking institution which they have a transactional relationship with. They see us as a personalized financial partner,' Siroya says. Tala illustrates how technology can help find ways round trust bottlenecks to unlock more economic activity. It demonstrates how data and algorithms can prove that there are billions of people like Eric, often overlooked and undervalued, that deserve to be trusted.\n\nWhy is Tala so successful? It sounds so simple: start with the person, not the system.\n\n'Free markets can succeed for all if business works with the people, not just sells to them,' says Richard Edelman in response to the 2017 Edelman survey that revealed a 'global implosion of trust'. To get out of the current trust collapse, we need radically to rethink the foundations on which our institutions are built so they are designed to work not just _for_ people but _with_ people. 'We must rebuild faith in the system citizen by citizen, community by community, where common goals and fairness matter,' writes Edelman. In other words, put people at the centre of everything you do.\n\nThe litmus test for any organization is this: would people describe it as an 'honest, ethical and reliable friend', someone who is there when you need them?\n\nTo give a simple example, when Tala wants to send their customers a reminder to repay their loan, they don't constantly bombard them with notifications when those customers clearly don't have any money in their bank. What's the point? Instead, they send borrowers a friendly SMS reminder the minute a deposit hits their account. More than 80 per cent of people pay on their phones as soon as they get that reminder. It shows the power of technology when it contains simple ingredients of humanity such as empathy and fairness.\n\nWhen I started writing this book, I thought, maybe a little naively, that most of the entrepreneurs, hackers, leaders and innovators I would meet designing ideas based on distributed trust would be like Siroya. And many are\u2013working to reframe deeply held institutional assumptions about power, access and equality. There are figures like Gerard Ryle, head of the ICIJ team of journalists behind the Panama Papers story, who uses digital networks to get people to work together, selflessly and collaboratively. Or entrepreneurs such as Leanne Kemp, founder of Everledger, and Savi Baveja, CEO of Trooly, who want to take on such enormous industries as background checks and the diamond trade, industries with troubled legacies and where trust has been systemically broken in many places. If you add up the stories of founders such as Joe Gebbia from Airbnb, Lynn Perkins from UrbanSitter and Fr\u00e9d\u00e9ric Mazzella from BlaBlaCar, you start to see a world where new mechanisms for enabling trust leaps can make us comfortable with people, ideas and experiences we may never have otherwise considered. Their companies might look very different from each other but what they have in common is using digital tools to build trust with strangers to connect and collaborate on an unprecedented scale.\n\nOn the other hand, there are entrepreneurs who act like 'digital gods', reaping the humungous benefits of platforms and algorithms with immense impact on our lives, but denying responsibility when things go wrong. In some cases, the high walls of institutions are merely being replaced by opaque, controlling algorithms and unpredictable leaders of platforms. Travis Kalanick, for example, the controversial co-founder and former CEO of Uber. Among other instances of bad behaviour, Kalanick sent emails to his PR team after the media reported a woman had been choked by a driver, instructing them to 'make sure these writers don't come away thinking we are responsible even when these things do go bad'. The company and its notorious founder don't do themselves any favours by projecting an aggressive and flawed public image.\n\nIn late January 2017, a protest movement with the hashtag #DeleteUber quickly gained traction over two issues, both of which involved the company's connection to President Donald Trump. The former CEO said his participation on an advisory group within the Trump administration had created what he called a 'perception-reality gap between who people think we are, and who we actually are'. But perception is everything.\n\nWhen trust is lost, a company has to exhibit humility, to be unafraid to give a genuine apology and acknowledge mistakes, and to demonstrate a clear willingness to fix what is wrong. Indeed, in an age of distributed trust, it's much harder to get away with errors and bungled responses, because there are far more people watching. Real incidents, or merely unfounded views that someone feels to be true, circulate at an unprecedented speed. A tweet about Uber's surge pricing might appear in one person's feed in the morning and then have flown around the world on social media by noon, leading to a full-blown protest not long after. The consequences for trust are enormous.\n\nIn 1919, the Irish poet W. B. Yeats wrote a poem called 'The Second Coming' to describe the atmosphere of post-war Europe.\n\nThe best lack all conviction, while the worst\n\nAre full of passionate intensity.\n\nAlmost a hundred years on, we are living in a time where perilous trust battles are raging between facts and 'alternative facts', falsehoods and rumours; between open platforms and gated communities; between elites or authorities and 'the people'; between the informed, the misinformed and the credulous. It's fair to assume that for some time to come the prevailing mood will be anti-elite and anti-authority\u2013a feeling that our traditional systems are deeply failing us. Trust in big institutions and the established order will continue to unravel and collapse.\n\nThat reaction might be understandable but a blanket hammering of institutional trust\u2013a wholesale rejection of the media, the courts and intelligence services, the truth-defending organizations that underpin any democracy\u2013threatens to create chaos. We obviously need to question institutions and hold them accountable, but if we pull the rug out from under them all, what are we left with? Potentially a dangerous _trust vacuum_ that is open to manipulation and being filled with catchy conspiracy theories, comforting biases, unfounded accusations and sleights of hand. A trust free-for-all, in other words. Think about it: when people are told they cannot trust any of the old institutions, they can end up trusting nothing, or anything. Institutions do not need to go the way of the dodo\u2013they just have to learn to adapt to this new trust landscape if they don't want to be left behind.\n\nWhen institutional systems fail, alternatives will always rise up to take their place. Distributed trust in itself can't knock down the rise of extremist populist movements, dangerous policies introduced by radical political leaders or a divisive resurgence of nationalism. But, driven democratically and rationally, and shaped and reshaped by people's needs and innate preferences about how they want to do things, it can provide a path forward for businesses, governments, media and other key institutions. It gives them a means to redesign systems that put people first in ways that are more _transparent_ , _inclusive_ and _accountable._\n\nSignificantly, this revolution is taking place in a landscape of rapidly shifting and evolving technologies, where the once unthinkable, the once impossible, can become the new normal in the blink of an eye. Humans are naturally attuned to taking trust leaps. Today, however, it feels like we are constant 'newbies' leaping at such an accelerated rate and in so many realms at once that it's dizzying. That's another challenge; setting up trust systems that can adapt and keep pace with an unprecedented rate of change.\n\nI would not have written this book if I did not believe in the enormous potential of distributed trust to give people, even countries, the tools and power to leap out of low-trust situations; if I didn't believe in its ability to help us find ways through the treacherous storm of distrust we are currently only just weathering. There is a large dose of optimism and exciting potential in the world of distributed trust, although it would be foolhardy not to acknowledge there is also a high degree of fear and uncertainty. It's a work in progress. We're still discovering its strengths, its virtues and its vulnerabilities.\n\nOver the course of _Who Can You Trust?_ we've looked at several stories where distributed trust always seems to lead us back to centralized power; a take-over, if you like, of those early good intentions. Take Amazon, Alibaba or Facebook. They might have begun as ways to democratize commerce or information, but they have become centralized behemoths in control of valuable and ever-more sensitive data.\n\nWhat's more, institutions meant to keep dominating powers in check\u2013regulatory bodies and labour unions, for example\u2013are ill-equipped to deal with a new digital era of fast-paced monopolies. One of the real challenges for distributed trust is whether it can resist, or at least weather, market forces and human greed.\n\nIdeas such as China's Social Credit System show how distributed networks of trust could become national networks of shame and interference, controlled by governments. And what has happened to those early utopian bitcoin miners? Mining power has ended up dangerously concentrated in China, at odds with the globalized ideals underlying bitcoin. The mass exchange of diverse ideas and the decentralization of information that we first envisioned the World Wide Web would bring us have happened, but so has a new kind of homophily and centralization\u2013hyperlinks and hierarchies managing what we see and read\u2013inside a handful of social networks. It's as if the small local cafes where we talked and disagreed with strangers have been replaced by a chain of McCafes where we are given algorithmically determined food, regardless of what we might actually want. The consequence is that we have become vulnerable to digital concentrations of power. We want power handed back to the people, but what if it's handed to the wrong people? Or only some of the people? Or, worse still, only a few of the wrong people?\n\nIt would be easy to suggest that platforms should be owned and governed by their users but there is the issue of accountability. Even ideas, such as the DAO fund and bitcoin, that seek to challenge bureaucratic systems and powerful gatekeepers, seem to need top-down decision-makers at times. When the DAO fund went belly up, what were many people calling for? An individual: Ethereum founder Vitalik Buterin. Yes, he had to get a majority consensus from the user network to implement the hard fork solution; but it seems we still want to be able to throw up our hands and say 'It was his idea', or ask 'Who's in charge here?' For the moment, at least, we remain in a mindset that wants a benevolent leader, an ultimate decision-maker to take charge and fix the problem. The positive is that all these processes are far more transparent than ever before, as well as under mass observation and open to comment from everyone with a stake in them.\n\nIn truth, the creators and leaders of ideas built on distributed trust are not like traditional captains of the ship issuing orders in command-and-control model and manner. But as stuff happens\u2013whether it's a mis-channelling of funds or an unpredictable mass murder associated with their platform\u2013they are realizing they need to know about and take increased responsibility for everything and anything that could sink the ship. The unforeseen storms could be legal challenges, data breaches, safety questions, unethical behaviour, discrimination or even competitors. When the water starts rushing in, leaders have to act quickly and transparently to plug the holes, and this can't always involve consensus building, although sometimes it can and should. Once the holes are plugged, they need to figure out how to fix the problem in the long-term in a 'We will do...' approach, engaging the wider community.\n\nThe third challenge of distributed trust is that many new technologies, from bots to blockchains, either anonymize people or attempt to remove entirely the need to trust another human. Yet it's humans, with all our wonderful kinks and mutations, who make trust possible. It's not technology or mathematics. When we trust an automated search engine over a human editor or when avatars or programmed algorithms serve as our managers, trust runs the risk of becoming static. What happens to human let-downs and surprises? These are how we learn to trust, and not trust. How else do we practise the skills of earning and rebuilding trust? Sometimes repairing trust requires a slow cure rate and the personal touch. It would be a shame to find ourselves in a world so automated that we depend solely on machines and algorithms to make decisions about whom to trust. That's a world apparently devoid of uncertainty, devoid of the colour and movement born of human imperfection, and, if we take our hands off the wheel too much, possibly even dangerous. As astronaut Dave Bowman famously found out when HAL 9000 went rogue in Kubrick's cult _2001: A Space Odyssey_ , one of our key challenges is deciding where and when it is appropriate to make trust a matter of computer code.\n\nWhen we look back in history, we can see that trust falls into distinct chapters. The first was l _ocal_. The second was _institutional_. And the third, still very much in its infancy, is _distributed._ Like most inventions in their early stages, distributed trust will be messy, unpredictable and at times even dangerous. Researching and writing about the theory has sometimes felt remarkably similar to watching my two young children bounding around the house, pushing boundaries, constantly negotiating, feeling misunderstood, trying to figure out the rules they have to abide by and the ones they can ignore.\n\nThere is no simple answer to the question 'Who Can You Trust?' but we do know that ultimately it comes down to a human decision. Technology can help us make better and different choices, but in the end it's we who have to decide where to place our trust and who deserves it. It will require some care. Distributed trust needs us to allow space for a _trust pause_ , an interval in which to stop and think before we automatically click, swipe, share and accept. To ask the right questions and to seek the right information that helps us to decide: is this person, information or thing worthy of my trust? What is it I'm trusting them to do or deliver? Each time we engage in that process, we are in our own small way taking responsibility for the kind of world we want to live in. We are exercising the power available to us all now at the press of a key. We are helping to preserve society's most precious and fragile asset, trust.\n\n# Acknowledgements\n\nIt's funny, I think writing books is a bit like having children; as soon as you have the first copy in your hands, you forget the immensely tough marathon in getting there. It takes a village to produce a book and I am deeply grateful for the advice, encouragement and support given by friends, family and colleagues along the way.\n\nAs for the book itself, I have three talented individuals to thank in particular: Mia de Villa, Phoebe Adler-Ryan and Fenella Souter.\n\nMia played an invaluable role as a research assistant, tracking down obscure papers and checking and rechecking hundreds of facts and transcripts. She was willing to do whatever it took to produce this book. Thank you, for everything.\n\nA special thank you to Fenella who was instrumental in helping to make the stories in this book zing. She was an incisive critic throughout the process and her keen-eyed comments helped to make every page infinitely better.\n\nI am immensely grateful to Phoebe, my assistant, for handling so much so well, and with diligent professionalism, care and grace. Thank you for being a joy to work with and an invaluable member of the team.\n\nThank you to Gary Nunn for hitting the ground running in managing communications and helping to spread the ideas in the book.\n\nMy heartfelt thanks to my brilliant literary agent, Toby Mundy, for continually playing the roles of thoughtful advisor, supportive coach and chief cheerleader. I am indebted to his fierce faith in the project (even in its unwieldy embryonic state) and for making this book happen. And thanks to David Roach for introducing me to Toby.\n\nI'd like to extend a special thanks to my terrific editors at Penguin Portfolio, Fred Baty and Daniel Crewe, who pushed the manuscript along in the most encouraging way possible. And the talented and world-class team behind the scenes: Nicola Evans, Lydia Yadi, Ellie Smith and John Stables. For her careful copyedit, thank you to Karen Whitlock. I was very fortunate to have David Over leading the marketing efforts around this book. Thank you for passionately embracing the ideas from the outset (and for countenancing my nudging!). Thanks also to Alex Elam and Sarah Scarlett and the rest of the dedicated international rights team at Penguin Portfolio.\n\nI am also grateful to the first-rate team at Hachette, Public Affairs\u2013led by Clive Priddle, Lindsay Fradkoff, and Jaime Leifer\u2013who enthusiastically brought this book to readers in the United States.\n\nMy gratitude goes to the talented designers at Team Design: Amy Globus, John Clark and Devin Seger for the stunning internal illustrations. I am constantly amazed how you turn my chicken scratches into something beautiful. Making complex things simple is complex and you do it so well.\n\nThanks also to Caroline Baum, an inspirational friend and talented writer, who read through an early draft of the manuscript (it's because I trust you!).\n\nA special thank you to Danny Stern, for doing everything a good speaking agent should do\u2013only infinitely better. I am also grateful to the rest of the team at Stern Strategy Group for their committed enthusiasm for my ideas: Katie Balogh, Tara Baumgarten, Mel Blake, Stephanie Heckman, Whitney Jennings, Joseph Navatto, Susan Stern and Ania Trzepizur. And a big thank you to Nanette Moulton, Trish Stafford and Carol Pedersen and the rest of the team at Saxtons for supporting my engagements from the very first time I stepped on a stage.\n\nEnormous thanks to all the talented people who graciously shared their time and wisdom with me, including: Judd Antin, Jessi Baker, Andrea Barrett, Savi Baveja, Joshua Browder, Leah Busque, Verena Butt d'Espous, Pierrick Caen, Juan Cartagena, Emily Castor, Stephen Cave, Coye Cheshire, Sean Conway, Ines Cormier, Rogier Creemers, Courtney Cregan, Nilesh Dalvi, Damien Detcherry, Matt Faustman, Juliette Garside, Joe Gebbia, Logan Green, Alok Gupta, Elliot Hedman, Jos\u00e9 Ignacio Fern\u00e1ndez, Angeli Jain, Husayn Kassai, Leanne Kemp, Federico Lalatta, David Lang, Brian Lathrop, James Martin, Fr\u00e9d\u00e9ric Mazzella, Mark Stephen Meadows, Paolo Parigi, Lynn Perkins, Gerard Ryle, Anish Das Sarma, Ariel Schultz, Shivani Siroya, Ryann Wahl and Seth Weiner. (And to all the story contributors who asked to be kept anonymous.) I am immensely grateful for their trust in letting me share their experiences and stories.\n\nJames Coleman, Francis Fukuyama, Onora O'Neill and Robert Putnam, with their remarkable writings on trust, have been a touchstone of intellectual inspiration.\n\nThe faculty at Oxford University Sa\u00efd Business School have provided me with the opportunity to teach the unbelievable student community. A special thank you to Colin Mayer, Ian Rogan, Rupert Younger and Marc Ventresca for all their support. I also want to thank my students for their engagement, comments and for pushing me to find clearer answers. Pamela Hartigan, my former supervisor, to whom this book is dedicated, showed me, in innumerable ways, what it means to be a teacher. I benefited enormously from her wisdom and was inspired by her relentless faith in people.\n\nThanks to the colleagues and organizations who have frequently provided public platforms to test and improve ideas: Helen Goulden and the team at Nesta; Mairi Ryan and Matthew Taylor of the RSA; and Alain de Botton and the team at School of Life; Jo Gray, Theo Chapman and the editors at the _AFR_ ; and David Rowan and Greg Williams at _WIRED._ A big thank you to Chris Anderson, Remo Giuffre, Bruno Giussani and Helen Walters at TED for letting me share my ideas from the ominous 'red circle'.\n\nI am also thankful to Rohan Lund, Kyle Loades and the other members of the board at the NRMA for tolerating my schedule and absences. I am proud to be a part of such a trusted organization.\n\nI am grateful to Kirsty de Garis for her research and input in the early stages of this book. Thank you for always being there. Many other friends and family, including Dana Ardi, Craig Baker, Tony Botsman and Jonathan Simmons, who have listened (or I should say, at times, endured) obsessive conversations about ideas lingering in my mind.\n\nA special thank you to Isabel 'one' and Isabel 'deux' for all their love and help with my kids. To all the teachers at Emanuel School, thanks for creating a wonderful and caring place.\n\nMy dear parents, Ruth and David, trusted me to share many personal family stories in this book. (I forgive you for leaving me in the care of a drug dealer!) An enormous thank you for all your boundless love, generosity and wisdom over the years. You are both remarkable people.\n\nTo my mother-in-law Wendy: my heartfelt thanks for all your love and support. You are the best nonna to the kids.\n\nAnd of course, my two beautiful children, Jack and Grace, for being an infinite source of joy, pride and humility. Although they are both still young, they have taught me so much about trust and how it's a family's most precious asset.\n\nFinally, my heartfelt thanks go to the person who knows me best, my husband, Chris. I am sure there were many times he did not want to talk about bots, blockchain and Breitbart, but he always listened. His endless patience, love and support made this book possible.\n\nRachel Botsman, June 2017\n\n**Rachel Botsman** is a visiting academic at the University of Oxford's Sa\u00efd Business School. A world-renowned expert on trust, Botsman was named one of the Most Creative People in Business by _Fast Company_ and a Young Global Leader by the World Economic Forum. Her TED talks have been viewed over 3.5 million times. She writes for the _Harvard Business Review_ , _Wall Street Journal_ , _The Guardian_ , _Wired_ , and more.\n\n# Glossary of 'Trust' Terms\n\n**Distributed Trust:** Trust that flows laterally between individuals, enabled by networks, platforms and systems.\n\n**Institutional Trust:** Trust that flows upwards to leaders, experts and brands, and runs through institutions and intermediaries such as courts, regulatory bodies and corporations (e.g. trusting your bank to safeguard your savings).\n\n**Local Trust:** Trust that exists between members of small, local communities and rests _in_ someone specific, someone we are familiar with.\n\n**Reputation:** The overall opinion others have formed of you, based on past experiences and built up over time.\n\n**Reputation Capital:** The value of your reputation across communities, networks and marketplaces; a measurement of how much an individual or community trusts you.\n\n**Reputation Trails:** Data we leave behind about how we behave, or misbehave.\n\n**Trust:** A confident relationship with the unknown.\n\n**Trust Blocker:** Obstacles or deal breakers for people when it comes to trusting a new idea or each other (e.g. not believing self-driving cars will make the right safety decisions).\n\n**Trust Deficit:** A lack of trust in a business, institution or within a society that prevents it from functioning well.\n\n**Trust Engineers:** People designing digital systems and networks that connect people and build or manipulate distributed trust.\n\n**Trust Gap:** The void between the known and unknown.\n\n**Trust Influencers:** People who can disproportionately influence a significant change in the way we do something or view something, and thus set new social norms.\n\n**Trust Leap:** A trust leap occurs when we take a risk and do something new or in a fundamentally different way.\n\n**Trust Pause:** An interval in which to stop and think before we automatically swipe, click and give our trust to someone.\n\n**Trust Scar:** Created by a trust-busting incident, a scar against an institution, individual or brand that may take decades or generations to heal.\n\n**Trust Score:** A system where all of an individual's behaviours are rated as either positive or negative and distilled into a single number as an indicator of their overall trustworthiness.\n\n**Trust Shift:** The historical evolution of trust from local to institutional, and institutional to distributed.\n\n**Trust Signals:** Clues or symbols that we knowingly or unknowingly use to decide whether or not another person is trustworthy.\n\n**Trust Stack** : The three-step process of trusting the idea; then the platform; and finally, the other person (or in some instances a machine or robot).\n\n**Trust Vacuum:** Created by a lack of trust in traditional experts, leaders and elites; an absence of trust that can create opportunities for malicious disruptors to occupy the space.\n\n**Trustworthy:** Someone who is competent, reliable and honest and thus worthy of our trust.\n\n# Notes and Further Reading\n\nIntroduction\n\n1. 'The Financial Crisis Inquiry Report', Financial Crisis Inquiry Commission United States of America, https:\/\/www.gpo.gov\/fdsys\/pkg\/GPO-FCIC\/pdf\/GPO-FCIC.pdf, 25 February 2011, accessed 2 March 2017.\n\n2. 'Digital Wildfires in a Hyperconnected World', WEF Report 2016, http:\/\/www3.weforum.org\/docs\/Media\/TheGlobalRisksReport2016.pdf, accessed 16 May 2017.\n\n3. '2017 Edelman Trust Barometer', Edelman, http:\/\/www.edelman.comr\/trust2017\/, accessed 2 March 2017.\n\n4. '2017 Edelman Trust Barometer Reveals Global Implosion of Trust', Edelman, http:\/\/www.edelman.com \/news\/2017-edelman-trust-barometer-reveals-global-implosion\/, accessed 15 January 2017.\n\n5. Turing Pharmaceuticals' CEO, Martin Shrekli, was arrested for securities fraud on 18 December 2015 and he resigned the following day. As of June 2017, the case is still in trial. The CEO has denied fraud charges.\n\n6. 'The Billion Dollar Startup Club', _Wall Street Journal_ , http:\/\/graphics.wsj.com\/billion-dollar-club\/, accessed 16 May 2017.\n\n7. Ibid.\n\n8. 'Tinder reveals the most attractive jobs in the UK that make people swipe right', _Telegraph_ , http:\/\/www.telegraph.co.uk\/technology\/2016\/09\/07\/tinder-reveals-the-most-attractive-jobs-in-the-uk-that-make-peop\/, 7 September 2016, accessed 30 May 2017.\n\n1. Trust Leaps\n\n1. See 'Alibaba IPO: Market values e-commerce giant at $231bn in enthusiastic opening day', _Guardian_ , https:\/\/www.theguardian.com \/business\/live\/2014\/sep\/19\/alibaba-ipo-nyse-stock-price-live-updates, 12 November 2016, accessed 30 January 2017.\n\n2. See 'Alibaba Lists on the NYSE', NYSE, https:\/\/www.nyse.com \/network\/article\/Alibaba-Lists-on-the-NYSE, 19 September 2014, accessed 30 May 2017.\n\n3. 'Who rang Alibaba's IPO opening bell?', Offbeat China, http:\/\/offbeatchina.com\/who-rang-alibabas-ipo-opening-bell, 22 September 2014, accessed 30 January 2017.\n\n4. See ' _Guanxi_ vs networking: Distinctive configurations of affect\u2013and cognition-based trust in the networks of Chinese vs American managers', Roy Chua, Michael Morris and Paul Ingram, _Journal of International Business Studies_ , https:\/\/link.springer.com\/article\/10.1057\/palgrave.jibs.8400422, 17 July 2008, accessed 30 November 2016.\n\n5. 'Entrepreneurial Masterclass: Alibaba founder Jack Ma interviewed by Charlie Rose', BizNews, http:\/\/www.biznews.com \/interviews\/2015\/02\/09\/the-incredible-story-behind-alibabas-jack-ma-an-inspiration-that-will-span-generations\/, 9 February 2015, accessed 4 May 2017.\n\n6. 'Alibaba, JD.com Locked in War of Words', _Forbes_ , https:\/\/www.forbes.com \/sites\/ywang\/2015\/01\/09\/alibaba-jd-com-locked-in-war-of-words\/#5cc3b3926658, 9 January 2015, accessed 30 May 2017.\n\n7. 'How Jack Ma Went From Being a Poor School Teacher to Turning Alibaba into a $US160 Billion Behemoth', Business Insider, http:\/\/www.businessinsider.com.au\/the-story-of-jack-ma-founder-of-alibaba-2014-9?r=US&IR=T, 15 September 2014, accessed 30 May 2017.\n\n8. 'Entrepreneurial Masterclass: Alibaba founder Jack Ma interviewed by Charlie Rose', BizNews, http:\/\/www.biznews.com \/interviews\/2015\/02\/09\/the-incredible-story-behind-alibabas-jack-ma-an-inspiration-that-will-span-generations\/, 9 February 2015, accessed 4 May 2017.\n\n9. 'Meet Jack Ma, the man behind Alibaba', Australian Financial Review, http:\/\/www.afr.com \/technology\/meet-jack-ma-the-man-behind-alibaba-20140908-jeqeh, 10 September 2014, accessed 30 January 2017.\n\n10. 'Thirteen Fascinating Facts About the Man Behind the Largest IPO in History', NextShark, http:\/\/nextshark.com\/13-fascinating-facts-about-the-man-behind-the-largest-ipo-in-history\/, 23 September 2014, accessed 30 January 2017.\n\n11. 'Alibaba Group', Julie Wulf, Harvard Business School Case 710-436, http:\/\/www.hbs.edu \/faculty\/Pages\/item.aspx?num=38507, 26 April 2010, last accessed 30 May 2017.\n\n12. _Alibaba's World_ , Porter Erisman, Macmillan (2015), p. 12.\n\n13. 'Jack Ma: U.S. Small Business Is Key to Alibaba's Growth', Bloomberg, https:\/\/www.bloomberg.com \/news\/articles\/2015-06-09\/u-s-small-business-key-to-alibaba-growth-outside-china-ma-says, 9 June 2015, accessed 30 January 2017\n\n14. 'Letter to Shareholders from Executive Chairman Jack Ma', Alibaba Group Investor News, http:\/\/www.alibabagroup.com \/en\/ir\/article?news=p161013, 13 October 2016, last accessed 4 May 4 2017.\n\n15. 'Squawk on the Street', CNBC, http:\/\/www.cnbc.com \/2014\/11\/11\/cnbc-exclusive-cnbc-transcript-alibaba-founder-executive-chairman-jack-ma-sits-down-with-cnbcs-david-faber-today-on-squawk-on-the-street.html, 11 November 2014, accessed 4 May 2017.\n\n16. See 'Trust and Consequences', Eric Uslaner, University of Maryland, http:\/\/gvptsites.umd.edu\/uslaner\/commun.pdf, accessed 30 November 2016, p. 1.\n\n17. See 'Gifts and Exchanges', essay by Kenneth Arrow, Princeton University Press (1982).\n\n18. For the distinction between generalized and personal trust, see 'Trust and Finance', Paola Sapienza and Luigi Zingales, NBER Reporter Online, National Bureau of Economic Research, http:\/\/www.nber.org \/reporter\/2011number2\/paola&luigi.html, June 2011, accessed 1 December 2016.\n\n19. See _Trust and Power_ , Niklas Luhmann, with an introduction by Gianfranco Poggi, Wiley (1979).\n\n20. See _The Resolution of Conflict_ , Morton Deutsch, Yale University Press (1973).\n\n21. 'Entrepreneurial Masterclass: Alibaba Founder Jack Ma Interviewed by Charlie Rose', BizNews, http:\/\/www.biznews.com \/interviews\/2015\/02\/09\/the-incredible-story-behind-alibabas-jack-ma-an-inspiration-that-will-span-generations\/, 9 February 2015, accessed 30 May 2017.\n\n22. Alipay website, https:\/\/intl.alipay.com\/, accessed 4 May 2017.\n\n23. See 'Jack Ma's finance business Alipay could be worth more than Goldman Sachs', Australian Financial Review, http:\/\/www.afr.com \/technology\/web\/ecommerce\/jack-mas-finance-business-alipay-could-be-worth-more-than-goldman-sachs-20160920-grkt09, 21 September 2016, accessed 4 May 2017.\n\n24. See 'Alipay study: Online payments via smartphones gaining ground', _China Daily_ , http:\/\/usa.chinadaily.com.cn\/epaper\/2016-01\/13\/content_23071262.htm, 13 January 2016, accessed 4 May 2017.\n\n25. See 'Eight Ringers of Alibaba's IPO Opening Bell', Women of China, http:\/\/www.womenofchina.cn\/womenofchina\/html1\/people\/others\/1409\/1036-1.htm, 23 September 2014, accessed 30 January 2017.\n\n26. _Alibaba's World_ , Porter Erisman, Macmillan (2015), p. 44.\n\n27. See 'Why Alibaba's CEO had to go', _Fortune_ , http:\/\/fortune.com\/2011\/02\/22\/why-alibabas-ceo-had-to-go\/, 22 February 2011, accessed 30 January 2017.\n\n28. See 'Alibaba.com chief executive resigns', _Guardian_ , https:\/\/www.theguardian.com \/business\/2011\/feb\/21\/alibaba-chief-resigns-over-frauds, 21 February 2011, accessed 1 December 2016.\n\n29. See 'The Market for Lemons: Quality Uncertainty and the Market Mechanism', George Akerlof, _Quarterly Journal of Economics_ , 84, 3 (1970).\n\n30. See 'The Maghribi traders: a reappraisal?' Avner Greif, _The Economic History Review_ , https:\/\/web.stanford.edu\/~avner\/Greif_Papers\/2012_Greif_long_ssrn_Maghribi.pdf, May 2012, accessed 21 February 2017.\n\n31. 'Reputation and Coalitions in Medieval Trade: Evidence on the Maghribi Traders', Avner Greif, _The Journal of Economic History_ 49, 4 (1989), 857-83, https:\/\/web.stanford.edu\/~avner\/Greif_Papers\/1989%20Greif%20JEH%201989.pdf, accessed 1 December 2016.\n\n32. _Bowling Alone_ , Robert Putnam, Touchstone Books by Simon and Schuster (2001).\n\n33. See _Trust_ , Francis Fukuyama, Free Press Paperbacks (1995), p. 10.\n\n2. Losing Faith\n\n1. 'Syphilis Victims in US Study went untreated for 40 years', Jean Heller, _New York Times,_ http:\/\/www.nytimes.com \/1972\/07\/26\/archives\/syphilis-victims-in-us-study-went-untreated-for-40-years-syphilis.html, 26 July 1972, accessed 30 May 2017.\n\n2. 'The Tuskegee Experiment kept killing black people after it ended', _New York Magazine_ , http:\/\/nymag.com\/scienceofus\/2016\/06\/tuskegee-experiment-mistrust.html, 15 June 2016, accessed 30 May 2017.\n\n3. _Bad Blood: The Tuskegee Syphilis Experiment_ , James H. Jones (Free Press, 1993).\n\n4. See Jean Heller's article, 'Syphilis victims in US study went untreated for 40 years', _New York Times_ , http:\/\/www.nytimes.com \/1972\/07\/26\/archives\/syphilis-victims-in-us-study-went-untreated-for-40-years-syphilis.html, 26 July 1972, accessed 17 May 2017.\n\n5. See _Statistical Monitoring of Clinical Trials: Fundamentals for Investigators_ , Lemuel Moy\u00e9, Springer (2006), p. 6.\n\n6. 'Clinton's Apology', Tuskegee Study, http:\/\/www.cdc.gov\/tuskegee\/clintonp.htm, 16 May 1997, accessed 18 January 2017.\n\n7. 'Deaths: Final Data for 2010', S. L. Murphy, J. Xu and K. D. Kochanek, _National Vital Statistics Reports_ , 61, 4 (2013).\n\n8. The researchers used a statistical method called the 'triple-difference model' which crunched data from the General Social Survey, the National Interview Survey and the Centers for Disease and Control. See 'Tuskegee and the Health of Black Men', Marcella Alsan and Marianne Wanamaker, National Bureau of Economic Research, http:\/\/www.nber.org \/papers\/w22323, June 2016, accessed 16 January 2017.\n\n9. For two great references on the study and paper, see 'A Generation of Bad Blood', _The Atlantic_ , http:\/\/www.theatlantic.com \/politics\/archive\/2016\/06\/tuskegee-study-medical-distrust-research\/487439, 17 June 2016, and 'Did Infamous Tuskegee Study Cause Lasting Mistrust of Doctors Among Blacks?', _New York Times_ , http:\/\/www.nytimes.com \/2016\/06\/18\/upshot\/long-term-mistrust-from-tuskegee-experiment-a-study-seems-to-overstate-the-case.html?_r=0, 17 June 2016, both accessed 30 May 2017.\n\n10. See Joseph Ravenell's TED talk, How barbershops can keep men healthy', https:\/\/www.ted.com \/talks\/joseph_ravenell_how_barbershops_can_keep_men_healthy\/transcript?language=en, February 2016, accessed 30 May 2017.\n\n11. The authors are clear to note that the findings shed light on the men who responded to government surveys they analysed from 1972 and the years immediately following but do not necessarily reflect the attitudes of younger African-American men today. See 'Tuskegee and the Health of Black Men', Marcella Alsan and Marianne Wanamaker, National Bureau of Economic Research, http:\/\/www.nber.org \/papers\/w22323, June 2016, accessed 16 January 2017.\n\n12. The Panama Papers, International Consortium of Investigative Journalists, https:\/\/panamapapers.icij.org\/, accessed 16 May 2017.\n\n13. To read about how the group effort worked with the ICIJ and journalists from around the world, see 'About the Panama Papers', Frederik Obermaier, Bastian Obermayer, Vanessa Wormer and Wolfgang Jaschensky, _S\u00fcddeutsche Zeitung_ , http:\/\/panamapapers.sueddeutsche.de\/articles\/56febff0a1bb8d3c3495adf4\/, accessed 16 January 2017.\n\n14. 'While one journalist is looking at Indian data, it might lead them to Brazil' and subsequent quotes from Gerard Ryle, author interview, 14 July 2016.\n\n15. See 'Panama papers: Iceland PM Sigmundur Gunnlaugsson steps down', _Sydney Morning Herald_ , http:\/\/www.smh.com.au\/world\/panama-papers-iceland-prime-minister-sigmundur-gunnlaugsson-offers-his-resignation-20160405-gnza99.html, 6 April 2016, accessed 4 March 2017.\n\n16. See _The Great Divide: Unequal Societies and What We Can Do About Them_ , Joseph Stiglitz, W. W. Norton (2015), and 'The Great Divide', speech by Andy Haldane given at New City Agenda Annual Dinner, London, on 18 May 2016.\n\n17. For a very good paper on defining institutions, see 'What Are Institutions?', Geoffrey Hodgson, _Journal of Economic Issues_ , March 2006, p. 394, http:\/\/www.geoffrey-hodgson.info\/user\/bin\/whatareinstitutions.pdf, accessed 16 January 2017.\n\n18. _The Great Degeneration_ , Niall Ferguson, Penguin Books (2014), p. 12.\n\n19. The survey question asked goes like this: 'How much of the time do you think you can trust government in Washington to do what is right\u2013a great deal, a fair amount, not very much, none at all?' See 'Trust in Government', Gallup, http:\/\/www.gallup.com \/poll\/5392\/trust-government.aspx, accessed 18 January 2017.\n\n20. See 'Americans' Confidence in Institutions Stays Low', Gallup, http:\/\/www.gallup.com \/poll\/192581\/americans-confidence-institutions-stays-low.aspx, accessed 13 June 2016.\n\n21. See 'Trust in Government', Gallup, http:\/\/www.gallup.com \/poll\/5392\/trust-government.aspx, accessed 17 January 2017.\n\n22. See 'Americans' Confidence in Congress Falls to Lowest on Record', Gallup, http:\/\/www.gallup.com \/poll\/163052\/americans-confidence-congress-falls-lowest-record.aspx, 13 June 2013, and 'Americans' Confidence in Institutions Stays Low', Gallup, http:\/\/www.gallup.com \/poll\/192581\/americans-confidence-institutions-stays-low.aspx, accessed 13 June 2016.\n\n23. See 'Supreme Court', Gallup, http:\/\/www.gallup.com \/poll\/4732\/supreme-court.aspx, accessed 17 January 2017.\n\n24. See 'Confidence in U.S. Banks Low but Rising', Gallup, http:\/\/www.gallup.com \/poll\/183749\/confidence-banks-low-rising.aspx, 22 June 2015.\n\n25. See 'Confidence in Institutions', Gallup, http:\/\/www.gallup.com \/poll\/1597\/confidence-institutions.aspx, 1\u20135 June 2016.\n\n26. Confidence rating for the church or organized religion, ibid.\n\n27. Confidence rating for newspapers, ibid.\n\n28. See 'No Front-Runner Among Prospective Republican Candidates, Hillary Clinton in Control of Democratic Primary, Harvard Youth Poll Finds', Harvard Kennedy School, http:\/\/iop.harvard.edu\/no-front-runner-among-prospective-republican-candidates-hillary-clinton-control-democratic-primary, accessed 16 January 2017.\n\n29. See 'Politicians are still trusted less than estate agents, journalists and bankers', https:\/\/www.ipsos-mori.com\/researchpublications\/researcharchive\/3685\/Politicians-are-still-trusted-less-than-estate-agents-journalists-and-bankers.aspx, 22 January 2016.\n\n30. See 'Wall Street in Crisis: A Perfect Storm Looming', Labaton Sucharow's US Financial Services Industry Survey, http:\/\/www.labaton.com \/en\/about\/press\/Wall-Street-Professional-Survey-Reveals-Widespread-Misconduct.cfm, 16 July 2013, accessed 16 January 2017.\n\n31. See 'Carney puts banker's pay in spotlight after misconduct shockwaves', BBC, http:\/\/www.bbc.com \/news\/business-30079451, 17 November 2014, accessed 4 March 2017.\n\n32. For a short profile of Andrew Haldane, see 'The central banker not afraid to be blunt', _TIME_ , http:\/\/time.com\/70833\/andy-haldane-2014-time-100\/, 23 April 2014, accessed 30 May 2017.\n\n33. _Twilight of the Elites: America after Meritocracy_ , Christopher Hayes, Broadway Books (2013), p. 102.\n\n34. See 'VW labor leaders said to balk at big severance for former CEO', Bloomberg, https:\/\/www.bloomberg.com \/news\/articles\/2015-09-28\/vw-labor-leaders-said-to-balk-at-big-severance-for-former-ceo, 29 September 2015, accessed 4 March 2017.\n\n35. 'The Report of the Iraq Inquiry: Executive Summary', Iraq Inquiry, http:\/\/www.iraqinquiry.org.uk\/media\/247921\/the-report-of-the-iraq-inquiry_executive-summary.pdf, p. 48, 6 July 2016, accessed 19 July 2017.\n\n36. See 'News Feed FYI: Helping Make Sure You Don't Miss Stories From Your Friends', Facebook Newsroom, http:\/\/newsroom.fb.com\/news\/2016\/06\/news-feed-fyi-helping-make-sure-you-dont-miss-stories-from-friends\/, 29 June 2016, accessed 4 March 2017.\n\n37. See 'Yahoo! Tops Twitter as a Traffic Referral Source for Digital Publishers', Parse.ly, http:\/\/blog.parsely.com\/post\/3476\/yahoo-tops-twitter-traffic-referral-source-digital-publishers\/, 26 April 2016, accessed 5 March 2017.\n\n38. Bill Bishop describes this phenomenon in the offline world in _The Big Sort: Why the Clustering of Like-Minded America Is Tearing Us Apart_ , Mariner Books (2009). Yochai Benkler describes the online phenomenon in his book _The Wealth of Networks_ , Yale University Press (2006).\n\n39. For statistics on where people get their news from, see 'News Use Across Social Media Platforms 2016', Pew Research Center, http:\/\/www.journalism.org \/2016\/05\/26\/news-use-across-social-media-platforms-2016\/, 26 May 2016, accessed 30 May 2017.\n\n40. The phrase 'filter bubble' was coined by MoveOn and Upworthy activist Eli Pariser, in his book _The Filter Bubble: What the Internet Is Hiding from You_ , Penguin Press (2011).\n\n41. See 'Obama Farewell Speech Transcript', _Los Angeles Times_ , http:\/\/www.latimes.com \/politics\/la-pol-obama-farewell-speech-transcript-20170110-story.html, 10 January 2017, accessed 15 May 2017.\n\n42. See '2016 Edelman Trust Barometer\u2013Global Results', Edelman, http:\/\/www.slideshare.net\/EdelmanInsights\/2016-edelman-trust-barometer-global-results, accessed 17 January 2017.\n\n43. See 'Beyond the Grand Illusion', Edelman, http:\/\/www.edelman.com \/p\/6-a-m\/beyond-grand-illusion\/, accessed 18 January 2016.\n\n44. _The Prince_ , Niccol\u00f2 Machiavelli, Florence (1505), chapter 18.\n\n45. See _Twilight of the Elites: America after Meritocracy_ , Christopher Hayes, Crown Publishing Group (2012), p. 63.\n\n46. See '5 Reasons Why Trump Will Win', Michael Moore, http:\/\/michael moore.com\/trumpwillwin\/, accessed 17 January 2017.\n\n47. See transcript of interview with Faisal Islam and Michael Gove from 3 June 2016, 'EU in or out?', Sky News, https:\/\/corporate.sky.com\/media-centre\/media-packs\/2016\/eu-in-or-out-faisal-islam-interview-with-michael-gove-30616-8pm, accessed 16 May 2017.\n\n48. See Michael Gove LBC Interview: 'Pro-EU Experts Like the Nazis, Says Gove', LBC, http:\/\/www.lbc.co.uk\/hot-topics\/eu-referendum\/gove-compares-pro-eu-experts-to-nazis-132633\/, 22 June 2016, accessed 19 July 2017.\n\n49. See 'Word of the Year 2016 is...', Oxford Dictionary, https:\/\/en.oxforddictionaries.com\/word-of-the-year\/word-of-the-year-2016, accessed 17 January 2017.\n\n50. New doublespeak dictionary, see Alain de Botton, https:\/\/twitter.com\/alaindebotton\/status\/798623471735447553, 15 November 2016, accessed 12 December 2016.\n\n51. See 'EU Referendum: Leave supporters trust ordinary \"common sense\" more than academics and experts', _Telegraph_ , http:\/\/www.telegraph.co.uk\/news\/2016\/06\/16\/eu-referendum-leave-supporters-trust-ordinary-common-sense-than\/, 22 June 2016, accessed 5 March 2017.\n\n52. See 'TV: Diverse Ventures in News and Public Affairs', _New York Times_ , http:\/\/www.nytimes.com \/1972\/05\/25\/archives\/tv-diverse-ventures-in-news-and-public-affairs.html?_r=0, 25 May 1972, accessed 5 March 2017.\n\n53. To measure blog stats, the number of blog posts versus the number of blogs are measured because some blogs remain dormant\/abandoned. See 'Blog posts written today', http:\/\/www.worldometers.in fo\/blogs\/, accessed 17 January 2017.\n\n54. For the latest count of subreddits, according to Reddit, see 'Happy 10th birthday to us! Celebrating the best of 10 years of Reddit', Reddit blog, https:\/\/redditblog.com\/2015\/06\/23\/happy-10th-birthday-to-us-celebrating-the-best-of-10-years-of-reddit\/, 23 June 2015, accessed 5 March.\n\n55. For the latest metrics for the subreddit 'Animals Without Necks' and also a useful site to look at numbers for other subreddits, see Reddit Metrics, http:\/\/redditmetrics.com\/r\/AnimalsWithoutNecks, accessed 17 January 2017.\n\n56. See 'Removing harassing subreddits', Reddit, https:\/\/np.reddit.com\/r\/announcements\/comments\/39bpam\/removing_harassing_subreddits\/, accessed 17 January 2017.\n\n57. For metrics on Reddit popularity, see 'How popular is Reddit', Alexa, http:\/\/www.alexa.com \/siteinfo\/reddit.com, accessed 17 January 2017, and 'Happy 10th birthday to us! Celebrating the best of 10 years of Reddit', Reddit blog, https:\/\/redditblog.com\/2015\/06\/23\/happy-10th-birthday-to-us-celebrating-the-best-of-10-years-of-reddit\/, 23 June 2015,accessed 5 March 2017.\n\n58. See 'TIFU by editing some comments and creating unnecessary controversy', Reddit, https:\/\/www.reddit.com \/r\/announcements\/comments\/5frg1n\/tifu_by_editing_some_comments_and_creating_an\/, 30 November 2016, accessed 17 January 2017.\n\n59. See 'Petition demands Reddit CEO resign for editing Trump supporter's comments', IBTimes, http:\/\/www.ibtimes.co.uk\/petition-demands-reddit-ceo-resign-editing-trump-supporters-comments-1593459, 25 November 2016, accessed 5 March 2017.\n\n60. See 'Steve Huffman should step down as CEO of Reddit', Change.org, https:\/\/www.change.org \/p\/reddit-steve-huffman-should-step-down-as-ceo-of-reddit, accessed 22 February 2017.\n\n3. Strangely Familiar\n\n1. See 'BlaBlaCar valued at', IBIS Worldwide, http:\/\/ibisworldwide.com\/2017\/news\/blablacar-valued-at-1-2bn\/, accessed 4 May 2017.\n\n2. The figure for the average trip taken is from 'Something to chat about', _The Economist_ , http:\/\/www.economist.com \/news\/business\/21676816-16-billion-french-startup-revs-up-something-chat-about, 24 October 2015, accessed 28 April 2017.\n\n3. See 'BlaBlaCar has turned ride-sharing into a multi-million-euro business', _WIRED,_ http:\/\/www.wired.co.uk\/article\/blablacar, 14 April 2015, accessed 28 April 2017.\n\n4. Ibid.\n\n5. See 'BlaBlaCar: Designing for Trust Between Strangers', Next, http:\/\/nextconf.eu\/2013\/09\/blablacar-designing-for-trust-between-strangers\/, 16 September 2013, accessed 28 April 2017. The '700 million trips' is from 'BlaBlaCar has turned ride-sharing into a multi-million-euro business', _WIRED,_ http:\/\/www.wired.co.uk\/article\/blablacar, 14 April 2015, accessed 4 May 2017.\n\n6. The company was renamed BlaBlaCar in 2011, ibid.\n\n7. Ibid.\n\n8. 'What is the sharing economy?', Blablacar, https:\/\/www.blablacar.com \/blog\/reinventing-travel\/sharing-economy, accessed 17 May 2017.\n\n9. For Eurostar number of travellers per year, see 'Passenger Numbers Stable', Eurostar, http:\/\/www.eurostar.com \/uk-en\/about-eurostar\/press-office\/press-releases\/2016\/passenger-numbers-stable-new-e320, accessed 16 May 2017. For BA number of travellers per year, see 'BA Traffic Stats', IAG Report 2016, http:\/\/www.iagshares.com \/phoenix.zhtml?c=240949&p=irol-traffic, accessed 16 May 2017.\n\n10. Food historians are divided on who invented the California Roll. See 'Will the Real Inventor of the California Roll Please Stand Up?', Grub Street, http:\/\/www.grubstreet.com \/2012\/10\/inventor-claims-california-roll-sushi.html, 24 October 2012, accessed 31 May 2017.\n\n11. See 'Sushi Industry Statistics', Statisticbrain, http:\/\/www.statisticbrain.com \/sushi-industry-statistics\/, accessed 9 September 2016.\n\n12. Steve Jobs believed that computers should be simple enough that a novice could master them based on instinct alone. In 2012, Apple moved away from this design philosophy because some of the old references like the reel-to-reel tape deck look of its podcast app was lost on younger smartphone users. See 'What is skeuomorphism?', BBC, http:\/\/www.bbc.com \/news\/magazine-22840833, 13 June 2003, accessed 28 April 2017.\n\n13. _Critique of Pure Reason_ , Immanuel Kant (1781).\n\n14. Judd Antin's interview on Dollars to Donuts podcast, http:\/\/www.portigal.com \/podcast\/, 19 January 2016, accessed 30 May 2017.\n\n15. Chris Sacca's interview on _This American Life_ , http:\/\/www.thisamericanlife.org \/radio-archives\/episode\/533\/transcript, 9 June 2014, accessed 4 March 2017.\n\n16. Inspired by Joe Gebbia's February 2016 TED talk, when he did the experiment with the audience. See 'How you can design for trust', TED, https:\/\/www.ted.com \/talks\/joe_gebbia_how_airbnb_designs_for_trust?language=en, accessed 17 May 2017.\n\n17. 'Olympic levels of trust' and subsequent quotes from Judd Antin, author interview, 20 July 2016.\n\n18. Ibid.\n\n19. For the details of Edward Jenner's development of the smallpox vaccination see Stefan Riedel, Baylor University Medical Center Proceedings, January 2005, 18(1): 21\u201325, 'Edward Jenner and the history of smallpox vaccination', NCBI, https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC1200696\/#B10, accessed 16 May 2017.\n\n20. 'The Cow-Pock\u2013or\u2013the Wonderful Effects of the New Inoculation!' by James Gillray, 1870, from the _Catalogue of Political and Personal Satires in the Department of Prints and Drawings in the British Museum_ , volume 11, London, http:\/\/www.britishmuseum.org \/research\/collection_online\/collection_object_details.aspx?objectId=1638225&partId=1&people=18459&peoA=18459-1-7&page=1, last accessed 16 May 2017.\n\n21. WHO, http:\/\/www.who.in t\/csr\/disease\/smallpox\/en\/, accessed 16 May 2017.\n\n22. See 'Lancet retracts 12-year-old article linking autism to MMR vaccines', _CMAJ_ , https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC2831678\/, 4 February 2010, accessed 30 May 2017.\n\n23. See 'Why Don't Parents Trust Vaccines?', Sharon Kaufman, Berkeley Wellness, http:\/\/www.berkeleywellness.com \/healthy-community\/contagious-disease\/article\/why-dont-parents-trust-vaccines, 6 April 2015, accessed 30 May 2017.\n\n24. James Samuel Coleman believed that people act purposively towards their desired goals, usually acting to maximize utility with their goals and utilities shaped by values and preferences. See _Foundations of Social Theory_ , James Samuel Coleman, Harvard University Press (1998).\n\n25. See _Crossing the Chasm: Marketing and Selling High-Tech Products to Mainstream Customers_ , Geoffrey Moore, HarperCollins (2006).\n\n26. 'You won't need a driver's license by 2040', _WIRED_ , https:\/\/www.wired.com \/2012\/09\/ieee-autonomous-2040\/, 17 September 2009, accessed 16 May 2017.\n\n27. 'Autonomous vehicles face exactly the same challenge' and subsequent quotes from Brian Lathrop, author interview, 17 August 2016.\n\n28. See 'Three-quarters of Americans \"afraid\" to ride in a self-driving vehicle', AAA Newsroom, http:\/\/newsroom.aaa.com\/2016\/03\/three-quarters-of-americans-afraid-to-ride-in-a-self-driving-vehicle\/, 1 March 2016, accessed 16 May 2017.\n\n29. 'Grandma freaks out self-driving Tesla\u2013you will laugh', YouTube, https:\/\/www.youtube.com \/watch?v=3-5QSZbcs-8, 15 April 2016, accessed 16 May 2017.\n\n30. See Sebastian Thrun's TED talk, 'Google's driverless car', https:\/\/www.ted.com \/talks\/sebastian_thrun_google_s_driverless_car\/transcript?language=en, March 2011, accessed 4 March 2017.\n\n31. See 'Global status report on road safety 2013', World Health Organization, 2013, http:\/\/www.who.in t\/violence_injury_prevention\/road_safety_status\/2013\/en\/, accessed 31 May 2017.\n\n32. 'Self-Driving Cars Could Save 300,000 Lives Per Decade in America', _The Atlantic_ , https:\/\/www.theatlantic.com \/technology\/archive\/2015\/09\/self-driving-cars-could-save-300000-lives-per-decade-in-america\/407956\/, 29 September 2015, accessed 31 May 2017.\n\n33. See 'Connected and autonomous vehicles\u2013the UK economic opportunity', KPMG, https:\/\/www.kpmg.com \/BR\/en\/Estudos_Analises\/artigosepublicacoes\/Documents\/Industrias\/Connected-Autonomous-Vehicles-Study.pdf, March 2015, accessed 31 May 2017.\n\n34. 'Self-Driving Cars Could Save 300,000 Lives Per Decade in America', _The Atlantic_ , https:\/\/www.theatlantic.com \/technology\/archive\/2015\/09\/self-driving-cars-could-save-300000-lives-per-decade-in-america\/407956\/, 29 September 2015, accessed 31 May 2017.\n\n35. See 'Loss Aversion in Riskless Choice: A Reference-Dependent Model', Amos Tversky and Daniel Kahneman, President and Fellows of Harvard College and the Massachusetts Institute of Technology, 1991, http:\/\/www3.uah.es\/econ\/MicroDoct\/Tversky_Kahneman_1991_Loss%20aversion.pdf, accessed 16 May 2017.\n\n36. See 'How the media screwed up the fatal Tesla accident', _Vanity Fair_ , http:\/\/www.vanityfair.com \/news\/2016\/07\/how-the-media-screwed-up-the-fatal-tesla-accident, 7 July 2016, accessed 31 May 2017.\n\n37. See 'A Tragic Loss', Tesla blog, https:\/\/www.tesla.com \/en_AU\/blog\/tragic-loss, 30 June 2016, accessed 31 May 2017.\n\n38. See 'Skype's first employee: How Taavet Hinrikus left Skype and founded TransferWise', YHP, http:\/\/yhponline.com\/2012\/03\/20\/taavet-hinrikus-transferwise\/, 20 March 2012, accessed 31 May 2017.\n\n39. See 'TransferWise wants to take over the world', TechCrunch, https:\/\/techcrunch.com\/2015\/12\/07\/transferwise-wants-to-take-over-the-world\/, 7 December 2015, accessed 31 May 2017.\n\n40. See 'Migration and Remittance Factbook 2016', http:\/\/siteresources.worldbank.org\/INTPROSPECTS\/Resources\/334934-1199807908806\/4549025-1450455807487\/Factbookpart1.pdf, accessed 8 September 2016.\n\n41. See _Influence: Science and Practice_ , Robert Cialdini, Allyn and Bacon (2001).\n\n42. Cialdini talked about this in the following paper summarizing his work on social proof, 'Dr Robert Cialdini and 6 principles of persuasion', http:\/\/www.influenceatwork.com \/wp-content\/uploads\/2012\/02\/E_Brand_principles.pdf, accessed 16 September 2016.\n\n43. The street corner experiment is described in 'Note on the Drawing Power of Crowds of Different Size', Stanley Milgram, Leonard Bickman and Lawrence Berkowitz, _Journal of Personality and Social Psychology_ , 1969.\n\n44. See _The Wisdom of Crowds_ , James Surowiecki, Doubleday (2004), p. 43.\n\n45. See TransferWise, https:\/\/transferwise.com\/au, accessed 16 May 2017.\n\n4. Where Does the Buck Stop?\n\n1. This _GQ_ article covering the shooting was used as reference in the details of the story, 'The Uber Killer: The Real Story of One Night of Terror', _GQ_ , http:\/\/www.gq.com \/story\/the-uber-killer, 22 August 2016.\n\n2. See 'Kalamazoo Uber Driver had a 4.73 Rating Before Shooting Rampage', _TIME_ , http:\/\/time.com\/4233148\/kalamazoo-uber-driver-rating-shooting-rampage\/, 23 February 2016, accessed 4 March 2017.\n\n3. See 'Uber driver Jason Dalton charged with six counts of murder over alleged Michigan shooting rampage', ABC, http:\/\/www.abc.net.au\/news\/2016-02-23\/uber-driver-charged-with-six-murders-in-us-shooting-rampage\/7191728, 23 February 2016, accessed 4 March 2017.\n\n4. See 'Passengers called 911 to report Uber driver before Kalamazoo shooting', Mashable, http:\/\/mashable.com\/2016\/02\/22\/kalamazoo-shooting-uber-driver-passengers\/#0sxstnSYHEqi, 23 February 2016, accessed 4 March 2017.\n\n5. 'Uber driver blamed, but no motive yet in killing spree', WZZM13, http:\/\/www.wzzm13.com \/news\/local\/kalamazoo\/uber-driver-blamed-but-no-motive-yet-in-killing-spree\/50399251, 22 February 2016, accessed 4 March 2017.\n\n6. See 'Kalamazoo Searches for Motive Spree That Killed 6', _New York Times_ , http:\/\/www.nytimes.com \/2016\/02\/22\/us\/kalamazoo-michigan-random-shootings.html?_r=0, 21 February 2016, accessed 4 March 2017.\n\n7. See 'Uber failed to prioritize safety complaint on Kalamazoo suspect before shootings', _Guardian_ , https:\/\/www.theguardian.com \/us-news\/2016\/feb\/22\/kalamazoo-shooting-spree-jason-dalton-uber-ignored-safety-complaint, 23 February 2016, accessed 4 March 2017.\n\n8. See 'The Truth About Uber's Background Checks', FastCompany, https:\/\/www.fastcompany.com \/3050172\/tech-forecast\/the-truth-about-ubers-background-checks, 26 August 2015, accessed 4 March 2017.\n\n9. See 'The Social Costs of Uber', _University of Chicago Law Review_ , https:\/\/lawreview.uchicago.edu\/page\/social-costs-uber, 2015, accessed 15 May 2017.\n\n10. See 'Reported list of incidents involving Uber and Lyft', Who's Driving You?, http:\/\/www.whosdrivingyou.org \/rideshare-incidents, accessed 27 February 2017.\n\n11. See footage and transcript of Travis Kalanick and Fawzi Kamel: 'A new video shows Uber CEO Travis Kalanick arguing with a driver over fares', Recode, https:\/\/www.recode.net\/2017\/2\/28\/14766964\/video-uber-travis-kalanick-driver-argument, 28 February 2017, accessed 19 July 2017.\n\n12. For Uber valuation see 'The Billion Dollar Startup Club', _Wall Street Journal_ , http:\/\/graphics.wsj.com\/billion-dollar-club\/, accessed 16 May 2017.\n\n13. 'More than 5 million' Uber passengers a day is taken from email correspondence from Alana Saltzman, Uber communications UK and Ireland, 6 June 2016.\n\n14. See 'Uber takes \"outrageous liberty\" with drivers', _Sydney Morning Herald_ , http:\/\/www.smh.com.au\/business\/the-economy\/uber-takes-outrageous-liberty-with-drivers-20160413-go5m4f.html, 14 April 2016, accessed 30 May 2017.\n\n15. See 'Uber lawsuits timeline: company ordered to pay out $161.9m since 2008', _Guardian_ , https:\/\/www.theguardian.com \/technology\/2016\/apr\/13\/uber-lawsuits-619-million-ride-hailing-app, 13 April 2016, accessed 16 May 2017.\n\n16. See 'The battle is for the customer interface', TechCrunch, https:\/\/techcrunch.com\/2015\/03\/03\/in-the-age-of-disintermediation-the-battle-is-all-for-the-customer-interface\/, 3 March 2015, accessed 4 March 2017.\n\n17. See 'Horsemeat scandal: timeline', _Guardian_ , https:\/\/www.theguardian.com \/uk\/2013\/may\/10\/horsemeat-scandal-timeline-investigation, 11 May 2013, accessed 4 March 2017.\n\n18. 'Horsemeat scandal: where did the 29% horse in your Tesco burger come from', _Guardian_ , https:\/\/www.theguardian.com \/uk-news\/2013\/oct\/22\/horsemeat-scandal-guardian-investigation-public-secrecy, 22 October 2013, accessed 4 March 2017.\n\n19. 'Horsemeat scandal: Dutch trader found guilty and jailed', BBC, http:\/\/www.bbc.com \/news\/world-europe-32202995, 7 April 2015, accessed 19 July 2017.\n\n20. See 'The Limits of Friendship', _New Yorker_ , http:\/\/www.newyorker.com \/science\/maria-konnikova\/social-media-affect-math-dunbar-number-friendships, 7 October 2014, accessed 4 March 2017.\n\n21. See _The Evolution of Cooperation_ , Robert Axelrod, Basic Books (1984).\n\n22. Ibid.\n\n23. See 'Bass Logo', Local History of Burton-on-Trent, http:\/\/www.burton-on-trent.org.uk\/category\/miscellany\/bass-logo, accessed 27 February 2017.\n\n24. 'Get back in the box thought virus#6: brand as communication', Douglas Rushkoff, http:\/\/www.rushkoff.com \/get-back-in-the-box-thought-virus-6-brand-as-communication\/, accessed 27 February 2017.\n\n25. 'Global trust in advertising', Nielsen, http:\/\/www.nielsen.com \/us\/en\/insights\/reports\/2015\/global-trust-in-advertising-2015.html, 28 September 2015, accessed 31 May 2017.\n\n26. See Joe Gebbia's TED talk, 'How Airbnb designs for trust', https:\/\/www.ted.com \/talks\/joe_gebbia_how_airbnb_designs_for_trust, February 2016.\n\n27. 'We have to create the conditions for a relationship to form between two people who have never met' and subsequent quotes from Joe Gebbia, author interview, 27 July 2016.\n\n28. All further quotes from Alok Gupta, author interview 21 July 2016.\n\n29. 'Introducing Airbnb Verified', Airbnb blog, http:\/\/blog.airbnb.com\/introducing-airbnb-verified-id\/, 30 April 2013, accessed 27 February 2017.\n\n30. 'Digital Discrimination: The Case of Airbnb.com', B. Edelman and M. Luca, Harvard Business School, http:\/\/hbswk.hbs.edu\/item\/digital-discrimination-the-case-of-airbnb-com, 24 January 2014, accessed 31 May 2017.\n\n31. 'Racial Discrimination in the Sharing Economy: Evidence from a Field Experiment', B. Edelman, M. Luca and D. Svirsky, Harvard Business School, http:\/\/www.benedelman.org \/publications\/airbnb-guest-discrimination-2016-09-16.pdf, 16 September 2016, accessed 31 May 2017.\n\n32. See 'Civil Rights Act of 1964 explained', http:\/\/civil.laws.com\/civil-rights-act-of-1964, accessed 17 May 2017.\n\n33. See 'Airbnb's Work to Fight Discrimination and Build Inclusion', Laura Murphy, Airbnb blog, http:\/\/blog.airbnb.com\/wp-content\/uploads\/2016\/09\/REPORT_Airbnbs-Work-to-Fight-Discrimination-and-Build-Inclusion.pdf?3c10be, 8 September 2016, accessed 4 March 2017.\n\n34. For @MiQL tweet see Twitter, https:\/\/twitter.com\/MiQL\/status\/675834706529673216, last accessed 17 May 2017.\n\n35. 'Prejudices play out in the ratings we give\u2013the myth of digital equality', _Guardian_ , https:\/\/www.theguardian.com \/technology\/2017\/feb\/20\/airbnb-uber-sharing-apps-digital-equality, 20 February 2017.\n\n36. 'Airbnb's Nondiscrimination Policy: Our Commitment to Inclusion and Respect', Airbnb, https:\/\/www.airbnb.com.au\/help\/article\/1405\/airbnb-s-nondiscrimination-policy-our-commitment-to-inclusion-and-respect, last accessed 1 March 2017.\n\n37. 'Online Trust, Trustworthiness, or Assurance?', MIT Press Journals, http:\/\/www.mitpressjournals.org \/doi\/abs\/10.1162\/DAED_a_00114?journalCode=daed#.WLKUqRJ96LI, 29 September 2011, accessed 31 May 2017.\n\n38. Quotes from Coye Cheshire, author interview, 16 November 2016.\n\n39. See 'Your car is a giant computer\u2013and it can be hacked', CNN Money, http:\/\/money.cnn.com\/2014\/06\/01\/technology\/security\/car-hack\/index.html, 2 June 2014, accessed 4 May 2017.\n\n40. 'Codebases: Millions of lines of code', Information is beautiful, http:\/\/www.informationisbeautiful.net\/visualizations\/million-lines-of-code\/, accessed 4 May 2017.\n\n41. 'Social Clicks: What and Who Gets Read on Twitter?' Maksym Gabielkov, Arthi Ramachandran, Augustin Chaintreau, Arnaud Legout, _ACM SIGMETRICS \/ IFIP Performance 2016_ , June 2016, Antibes Juan-les-Pins, France, https:\/\/hal.inria.fr\/hal-01281190, accessed 31 May 2017.\n\n42. '6 in 10 of you will share this link without reading it, a new, depressing study says', _Washington Post_ , https:\/\/www.washingtonpost.com \/news\/the-intersect\/wp\/2016\/06\/16\/six-in-10-of-you-will-share-this-link-without-reading-it-according-to-a-new-and-depressing-study\/?utm_term=.d9b38e787de3, 16 June 2016, accessed 4 March 2017.\n\n43. Twitter, https:\/\/twitter.com\/seanspicer?lang=en, accessed 16 May 2017.\n\n44. See 'Transcript of Simon Sinek's Millenials in the Workplace Interview', Ochen, http:\/\/ochen.com\/transcript-of-simon-sineks-millennials-in-the-workplace-interview, 4 January 2017.\n\n45. 'Facebook Tinkers with Users' Emotions in News Feed Experiment, Stirring Outcry', _New York Times_ , https:\/\/www.nytimes.com \/2014\/06\/30\/technology\/facebook-tinkers-with-users-emotions-in-news-feed-experiment-stirring-outcry.html?_r=0, 30 June 2014, accessed 4 March 2017.\n\n46. 'Experimental evidence of massive-scale emotional contagion through social networks', Cornell, https:\/\/cornell.app.box.com\/v\/fbcontagion, 25 March 2014, accessed 4 March 2017.\n\n47. 'Data Policy', Facebook, https:\/\/www.facebook.com \/policy.php, accessed 27 February 2017.\n\n48. 'Facebook and Engineering the Public: It's not what's published (or not), but what's done', Medium, https:\/\/medium.com\/message\/engineering-the-public-289c91390225#.d1x3rghwy, 30 June 2014, accessed 30 May 2017.\n\n49. 'Here's what you need to know about that Facebook experiment that manipulated your emotions', Gigaom, https:\/\/gigaom.com\/2014\/06\/30\/heres-what-you-need-to-know-about-that-facebook-experiment-that-manipulated-your-emotions\/, 30 June 2014, accessed 4 March 2017.\n\n50. 'The Trust Engineers', RadioLab, http:\/\/www.radiolab.org \/story\/trust-engineers\/, 9 February 2015.\n\n51. See _Weapons of Math Destruction_ , Cathy O'Neil, Crown (2016), p. 183.\n\n52. 'The evolving role of news on Twitter and Facebook', Pew Research Center, http:\/\/www.journalism.org \/2015\/07\/14\/the-evolving-role-of-news-on-twitter-and-facebook\/, 14 July 2015, accessed 4 March 2017.\n\n53. 'Hyperpartisan Facebook Pages are Publishing False and Misleading Information at an Alarming Rate', BuzzFeed News, https:\/\/www.buzzfeed.com \/craigsilverman\/partisan-fb-pages-analysis?utm_term=.qkPyQLqm8#.lcVrz58d2, 20 October 2016, accessed 1 March 2017.\n\n54. 'This Analysis Shows How Viral Fake Election News Stories Outperformed Real News on Facebook', BuzzFeed News, https:\/\/www.buzzfeed.com \/craigsilverman\/viral-fake-election-news-outperformed-real-newsonfacebook?utm_content=buffer20bf6&utm_medium=social&utm_source=twitter.com&utm_campaign=buffer&utm_term=.apz6rG5dQ#.habG6qdrR, 17 November 2016, accessed 17 November 2016.\n\n55. 'Mark Zuckerberg says fake news on Facebook could not have influenced the 2016 election', Quartz, https:\/\/qz.com\/836079\/mark-zuckerberg-says-fake-news-on-facebook-could-not-have-influenced-the-2016-election-of-donald-trump\/, 13 November 2016, accessed 17 November 2016.\n\n56. 'Building Global Community', Mark Zuckerberg, Facebook, https:\/\/www.facebook.com \/notes\/markzuckerberg\/buildingglobalcommunity\/10103508221158471\/?pnref=story, accessed 17 February 2017.\n\n57. Quotes from Joe Gebbia, author interview, 27 July 2016.\n\n5. But She Looked the Part\n\n1. See _Trust in Society_ , ed. Karen S. Cook, Russell Sage Foundation (2003), 'Chapter 5: Trust in Signs', Michael Bacharach and Diego Gambetta.\n\n2. I keep going back to Onora O'Neill's TED talk throughout the book. See Onora O'Neill, 'What we don't understand about trust', https:\/\/www.ted.com \/talks\/onora_o_neill_what_we_don_t_understand_about_trust\/transcript?language=en, September 2013, accessed 31 May 2017.\n\n3. 'Trust, Trustworthiness and Transparency', European Foundation Centre, http:\/\/www.efc.be\/human-rights-citizenship-democracy\/trust-trustworthiness-, 3 December 2015, accessed 30 May 2017.\n\n4. _Trust and Trustworthiness_ , Russell Hardin, Russell Sage Foundation (2002).\n\n5. 'Monster Mensch', _New York Magazine_ , http:\/\/nymag.com\/news\/businessfinance\/54703\/, 22 February 2009, accessed 30 May 2017.\n\n6. Onora O'Neill's TED talk, 'What we don't understand about trust', https:\/\/www.ted.com \/talks\/onora_o_neill_what_we_don_t_understand_about_trust\/transcript?language=en, September 2013, accessed 31 May 2017.\n\n7. _The Truth about Trust_ , David DeSteno, First Plume Printing (2015), 'Chapter 6: Can I Trust You? Unlocking the Signals of Trustworthiness'.\n\n8. UK-based executive communications consultancy The Aziz Corporation rated UK accents for their business appeal, and Scottish accents scored highly with 43 per cent of respondents saying that the Scottish speaker sounded successful; 40 per cent found the speaker to be hardworking and reliable and 31 per cent found them the most trustworthy. See 'Scots Accent Favored for Call Centers', WallStreet & Technology, http:\/\/www.wallstreetandtech.com \/careers\/scots-accent-favored-for-call-centers-\/d\/d-id\/1256416, 26 March 2004, accessed 30 May 30 2017.\n\n9. This is work from earlier scientists studying making judgements based on facial appearance. See 'How your looks betray your personality', Roger Highfield, Richard Wiseman and Rob Jenkins, _New Scientist_ , https:\/\/www.newscientist.com \/article\/mg20126957-300-how-your-looks-betray-your-personality\/, 11 February 2009, accessed 30 May 2017.\n\n10. See 'Static and Dynamic Facial Cues Differentially Affect the Consistency of Social Evaluation', _Personality and Social Psychology Bulletin_ , http:\/\/psp.sagepub.com\/content\/early\/2015\/06\/12\/0146167215591495.abstract, 22 May 2015, accessed 30 May 2017.\n\n11. See 'So you think you can be a hair braider', _New York Times_ , http:\/\/www.nytimes.com \/2012\/06\/17\/magazine\/so-you-think-you-can-be-a-hair-braider.html?_r=2&ref=magazine&pagewanted=all, 12 June 2012, accessed 30 May 2017.\n\n12. Quote from Seth Weiner, author interview, 23 November 2016.\n\n13. Trust game experiment conducted by Li Huang and Keith Murnighan. See 'A Trusted Name', Kellogg Insight, http:\/\/insight.kellogg.northwestern.edu\/article\/a_trusted_name, 1 June 2011, accessed 16 May 2017.\n\n14. Ibid.\n\n15. Jason Tanz wrote about how sharing-economy platforms like Lyft and Airbnb got Americans to interact and trust one another and not see meeting strangers as a huge risk. See 'How Airbnb and Lyft finally got Americans to trust each other', _WIRED_ , https:\/\/www.wired.com \/2014\/04\/trust-in-the-share-economy\/, 23 April 2014, accessed 16 May 2017.\n\n16. If you're logged on to Facebook and check this blog, you'll be able to see your degree of separation from everyone else. Last checked, Mark Zuckerberg has 3.17 degrees of separation; Sheryl Sandberg has 2.92. See 'Three and a half degrees of separation', Research at Facebook, https:\/\/research.facebook.com\/blog\/three-and-a-half-degrees-of-separation\/, 4 February 2016, accessed 4 March 4 2017.\n\n17. Sanjay Nazerali, chief strategy officer at Carat Global wrote a great post on the difference between faith and experience that helped me think through this point. See 'Faith vs experience: Building trust in the digital age', MMG, http:\/\/mandmglobal.com\/faith-vs-experience-building-trust-in-the-digital-age\/, 21 September 2015.\n\n18. 'A Question of Trust', Reith Lectures 2002, BBC Radio 4, http:\/\/www.bbc.co.uk\/radio4\/reith2002\/, accessed 30 May 2017.\n\n19. Traits of trustworthiness have been reviewed extensively in trust literature. Even though leading trust researchers propose different characteristics that are responsible for trust, there are three common characteristics that appear often: competence (sometimes labelled as ability or expertise), honesty (sometimes labelled as intention, integrity, benevolence, loyalty or goodwill) and reliability (sometimes labelled as dependability or availability). See 'An Integrative Model of Organizational Trust', The Academy of Management Review, Roger Mayer, James Davis and David Schoorman study, https:\/\/www.jstor.org \/stable\/258792?seq=1#page_scan_tab_contents, July 1995, accessed 30 May 2017.\n\n20. See _Trust and Trustworthiness_ , Russell Hardin, Russell Sage Foundation (2002), p. 6.\n\n21. See 'Graduates are stretching the truth to get work in uncertain economic times', Pre-employment Screening blog, http:\/\/pre-employment-screening.blogspot.com.au\/2009\/08\/graduates-are-stretching-truth-to-get.html, 7 August 2009, accessed 4 March 2017.\n\n22. Quotes from Lynn Perkins, author interview, 25 August 2016.\n\n23. Statistic provided by Lynn Perkins, author interview, 25 August 2016.\n\n24. 'People rate everything now' and subsequent quotes from Andrea Barrett, author interview, 25 August 2016.\n\n25. See 'Web Archive: Cartoon Captures Spirit of the Internet', _New Yorker_ , http:\/\/web.archive.org\/web\/20141030135629\/http:\/\/www.nytimes.com \/2000\/12\/14\/technology\/14DOGG.html, accessed 16 May 2017.\n\n26. To see a copy of the cartoon go to 'The Joy of Tech', Me.Me, https:\/\/me.me\/i\/the-joy-of-tech-in-the-1990s-on-the-internet-11890719, accessed 16 May 2017.\n\n27. 'People talk a lot about reviews and ratings' and subsequent quotes from Savi Baveja, author interview, 17 November 2016.\n\n28. 'Watch lists, the National Sex Offender Registry, social media et cetera' and subsequent quotes from Anish Das Sarma, author interview, 28 September 2016.\n\n29. 'Bad data fouls background checks', _WIRED_ , https:\/\/www.wired.com \/2005\/03\/bad-data-fouls-background-checks\/, 11 March 2005, accessed 31 May 2017.\n\n30. See 'Indefinite punishment and the criminal record: stigma reports among expungement-seekers in Illinois', http:\/\/onlinelibrary.wiley.com\/wol1\/doi\/10.1111\/1745-9125.12108\/full, 8 June 2016, accessed 31 May 2017.\n\n6. Reputation is Everything, Even in the Dark\n\n1. How we view e-commerce now is so different from how it originally started, so there will be a few interpretations of how 'sold' is defined. A few contenders for the first thing sold on the internet are pizza, a Sting CD and weed. See _What the Dormouse Said: How the Sixties Counterculture Shaped the Personal Computer Industry_ , John Markoff, Penguin Books (2005).\n\n2. 'New \"Google\" for the dark web makes buying dope and guns easy', _WIRED_ , https:\/\/www.wired.com \/2014\/04\/grams-search-engine-dark-web\/, 17 April 2014, accessed 3 March 2017.\n\n3. The FBI estimated that Silk Road had 13,000 drug listings and had processed approximately $1.2 billion. See _The Dark Net_ , Jamie Bartlett, Windmill Books (2015), p. 136.\n\n4. Ibid.\n\n5. 'Bitcoin \"exit scam\": deep-web market operators disappear with $12m', _Guardian_ , https:\/\/www.theguardian.com \/technology\/2015\/mar\/18\/bitcoin-deep-web-evolution-exit-scam-12-million-dollars, 19 March 2015, accessed 3 March 2017.\n\n6. See 'The dark web's top drug market evolution, just vanished', _WIRED_ , https:\/\/www.wired.com \/2015\/03\/evolution-disappeared-bitcoin-scam-dark-web\/, 18 March 2015, accessed 3 March 2017.\n\n7. 'Technology could be used to transform an illicit drugs trade' and subsequent quotes from James Martin, author interview, 22 September 2016.\n\n8. See 'The Global Drug Survey 2014 Findings', Global Drug Survey, https:\/\/www.globaldrugsurvey.com \/past-findings\/the-global-drug-survey-2014-findings, accessed 4 May 2017.\n\n9. See 'The Global Drug Survey 2016 Findings', Global Drug Survey, https:\/\/www.globaldrugsurvey.com \/past-findings\/the-global-drug-survey-2016-findings\/, accessed 4 May 2017.\n\n10. See 'Taking Stock of the Online Drugs Trade', Rand Corporation, http:\/\/www.rand.org \/randeurope\/research\/projects\/online-drugs-trade-trafficking.html, 27 October 2016, accessed 3 March 2017.\n\n11. 'Shedding Light on the Dark Web', _The Economist_ , http:\/\/www.economist.com \/news\/international\/21702176-drug-trade-moving-street-online-cryptomarkets-forced-compete, 16 July 2016, accessed 3 March 2017.\n\n12. 'The internet and drug markets', European Monitoring Centre for Drugs and Drug Addiction, http:\/\/www.emcdda.europa.eu\/system\/files\/publications\/2155\/TDXD16001ENN_FINAL.pdf, February 2016, accessed 30 May 2017.\n\n13. James Martin, author interview, 22 September 2016.\n\n14. '\"Fair Trade\" Cocaine and \"Conflict-free\" Opium: The Future of Online Drug Marketing', The Conversation, http:\/\/theconversation.com\/fair-trade-cocaine-and-conflict-free-opium-the-future-of-online-drug-marketing-30127, 12 August 2014, accessed 4 May 2017.\n\n15. See _The Dark Net_ , Jamie Bartlett, Windmill Books (2015), p. 162.\n\n16. For statistics regarding positive reviews left on the darknet, see 'Traveling the Silk Road: A measurement analysis of a large anonymous online marketplace', Nicolas Christin, Carnegie Mellon University, https:\/\/www.cylab.cmu.edu\/files\/pdfs\/tech_reports\/CMUCyLab12018.pdf, 28 November 2012, accessed 31 May 2017.\n\n17. See 'Trust among strangers in internet transactions', Paul Resnick and Richard Zeckhauser, http:\/\/cseweb.ucsd.edu\/groups\/csag\/html\/teaching\/cse225s04\/Reading%20List\/E-bay-Empirical-BodegaBay.pdf, 20 September 2016, accessed 31 May 2017.\n\n18. For changes in Airbnb review system, see 'Building for Trust', Medium, AirbnbEng, https:\/\/medium.com\/airbnb-engineering\/building-for-trust-503e9872bbbb#.s7872icvv, 15 March 2016, accessed 31 May 2017.\n\n19. For Airbnb rating details, see 'How do star ratings work?', Airbnb, https:\/\/www.airbnb.com.au\/help\/article\/1257\/how-do-star-ratings-work?topic=207, accessed 28 October 2016.\n\n20. 'Evolution of the Future', Robert Axelrod, http:\/\/www.eleutera.org \/wp-content\/uploads\/2015\/07\/The-Evolution-of-Cooperation.pdf, 1984.\n\n21. I have spoken at length about what reputation is and how it is related to trust with Juan Cartegena, the founder of Traity. He was the first person to point out to me that reputation is best described as a 'risk premium' (not as an asset or currency) and that it has a price elasticity.\n\n22. See 'Use of Silk Road, the online drug marketplace, in the United Kingdom, Australia and the United States', M. J. Barratt, J. A. Ferris, and A. R. Winstock, https:\/\/www.ncbi.nlm.nih.gov\/pubmed\/24372954, 2014, accessed 30 May 2017.\n\n23. 'Amazon targets 1,114 \"fake reviewers\" in Seattle lawsuit', BBC News, http:\/\/www.bbc.com \/news\/technology-34565631, 18 October 2015, accessed 31 May 2017.\n\n24. 'Historian Orlando Figes admits posting Amazon reviews that trashed rivals', _Guardian_ , https:\/\/www.theguardian.com \/books\/2010\/apr\/23\/historian-orlando-figes-amazon-reviews-rivals, 23 April 2010, accessed 3 March 2017.\n\n25. 'Orlando Figes: Historian admits to writing anonymous reviews on Amazon', _Telegraph_ , http:\/\/www.telegraph.co.uk\/culture\/books\/booknews\/7622877\/Orlando-Figes-Historian-admits-to-writing-anonymous-reviews-on-Amazon.html, 24 April 2010, accessed 19 June 2017.\n\n26. 'Poison pen reviews were mine, confesses historian Orlando Figes', _Guardian_ , https:\/\/www.theguardian.com \/books\/2010\/apr\/23\/poison-pen-reviews-historian-orlando-figes, 23 April 2010, accessed 19 July 2017.\n\n27. 'Some online reviews are too good to be true; Cornell computers spot \"opinion spam\"', http:\/\/www.news.cornell.edu\/stories\/2011\/07\/cornell-computers-spot-opinion-spam-online-reviews, 25 July 2011, accessed 31 May 2017.\n\n7. Rated: Would Your Life Get a Good Trust Score?\n\n1. 'Planning Outline for the Construction of a Social Credit System', China Copyright and Media, translated by Rogier Creemers, https:\/\/chinacopyrightandmedia.wordpress.com\/2014\/06\/14\/planning-outline-for-the-construction-of-a-social-credit-system-2014-2020\/, accessed 3 March 2017.\n\n2. Ibid.\n\n3. See 'China rates its own citizens\u2013including online behaviour', _Volkskrant_ , http:\/\/www.volkskrant.nl\/buitenland\/china-rates-its-own-citizens-including-online-behaviour~a3979668\/, 25 April 2015, accessed 4 May 2017.\n\n4. 'Big Brother Ranking for All: How Would Your Life Rate?', News.com.au, http:\/\/www.news.com.au\/lifestyle\/real-life\/big-brother-ranking-for-all-how-would-your-life-rate\/news-story\/53928e0017a582e16acfa792bf51a496, 9 October 2015, accessed 3 March 2017.\n\n5. See 'How China Wants to Rate Its Citizens', _New Yorker_ , http:\/\/www.newyorker.com \/news\/daily-comment\/how-china-wants-to-rate-its-citizens, 3 November 2015, accessed 30 April 2017.\n\n6. 'Use Big Data Thinking and Methods to Enhance the Government's Governing Capacity', China Copyright and Media, edited by Rogier Creemers, https:\/\/chinacopyrightandmedia.wordpress.com\/2016\/07\/12\/use-big-data-thinking-and-methods-to-enhance-the-governments-governing-capacity\/, 12 July 2016, accessed 3 March 2017.\n\n7. How do people get their Sesame Score? See 'Ant Financial Unveils China's First Credit-Scoring System Using Online Data', Alibaba Group, http:\/\/www.alibabagroup.com \/en\/news\/article?news=p150128, 28 January 2015, accessed 4 March 2017.\n\n8. See 'Ant Financial Subsidiary Starts Offering Individual Credit Scores', Caixin Online, http:\/\/english.caixin.com\/2015-03-02\/100787148.html, accessed 30 April 2017.\n\n9. Ibid.\n\n10. _The Circle_ , Dave Eggers, Penguin Books (2014), p. 367.\n\n11. Ibid., p. 303.\n\n12. See 'China wants to give all of its citizens a score\u2013and their rating could affect every area of their lives', _Independent_ , http:\/\/www.independent.co.uk\/news\/world\/asia\/china-surveillance-big-data-score-censorship-a7375221.html, 22 October 2016, accessed 30 April 2017.\n\n13. See 'Mainland credit-rating network takes shape', _China Daily Asia_ , http:\/\/www.chinadailyasia.com \/business\/2015-06\/09\/content_15274221.html, 9 June 2015, accessed 30 April 2017.\n\n14. See 'In China, Your Credit Score is Now Affected By Your Political Opinions\u2013And Your Friends' Political Opinions', Privacy News Online, https:\/\/www.privateinternetaccess.com \/blog\/2015\/10\/in-china-your-credit-score-is-now-affected-by-your-political-opinions-and-your-friends-political-opinions\/, 3 October 2015, accessed 4 March 2017.\n\n15. Quotes from Rogier Creemers, author interview, 24 November 2016.\n\n16. See 'In China, Your Credit Score is Now Affected By Your Political Opinions\u2013And Your Friends' Political Opinions', Private Internet Access, https:\/\/www.privateinternetaccess.com \/blog\/2015\/10\/in-china-your-credit-score-is-now-affected-by-your-political-opinions-and-your-friends-political-opinions\/, 3 October 2015, accessed 4 March 2017.\n\n17. See 'China's \"social credit\": Beijing sets up huge system', BBC, http:\/\/www.bbc.com \/news\/world-asia-china-34592186, 26 October 2015, accessed 4 March 2017.\n\n18. Cathy O'Neil covers this point in her excellent book about how algorithms are increasingly regulating people. See _Weapons of Math Destruction_ , Cathy O'Neil, Crown (2016).\n\n19. _Super Sad True Love Story_ , Gary Shteyngart, Random House (2010).\n\n20. 'Will Social Media Make Us Anti-Social? A Talk With Gary Shteyngart', _The Atlantic_ , http:\/\/www.theatlantic.com \/business\/archive\/2011\/10\/will-social-media-make-us-anti-social-a-talk-with-gary-shteyngart\/247373\/, 26 October 2011, accessed 2 March 2017.\n\n21. See 'China rates its own citizens\u2013including online behaviour', _Volkskrant,_ http:\/\/www.volkskrant.nl\/buitenland\/china-rates-its-own-citizens-including-online-behaviour~a3979668\/, 25 April 2015, accessed 4 May 2017.\n\n22. See 'China's \"social credit\": Beijing sets up huge system', BBC, http:\/\/www.bbc.com \/news\/world-asia-china-34592186, 26 October 2015, accessed 2 March 2017.\n\n23. See 'Orwellian Dystopia or Trustworthy Nation? Get the Facts on China's Social Credit System', Advox Global Voices, https:\/\/advox.globalvoices.org\/2016\/01\/08\/orwellian-dystopia-or-trustworthy-nation-get-the-facts-on-chinas-social-credit-system\/, 8 January 2016, accessed 3 March 2017.\n\n24. See 'From the end of sesame credit \"not the same\" data source', 21jingji, http:\/\/m.21jingji.com\/article\/20150617\/0c3b29fd50dd0a4a4b2f9d9e94f9cb99.html, 17 June 2015, accessed 30 May 2017.\n\n25. 'State Council Guiding Opinions concerning Establishing and Perfecting Incentives for Promise-keeping and Joint Punishment Systems for Trust-Breaking, and Accelerating the Construction of Social Sincerity', China Copyright and Media, edited by Rogier Creemers, https:\/\/chinacopyrightandmedia.wordpress.com\/2016\/05\/30\/state-council-guiding-opinions-concerning-establishing-and-perfecting-incentives-for-promise-keeping-and-joint-punishment-systems-for-trust-breaking-and-accelerating-the-construction-of-social-sincer\/, 18 October 2016, accessed 8 March 2017.\n\n26. Ibid. See also 'China's New Tool for Social Control: A Credit Rating for Everything', _Wall Street Journal_ , https:\/\/www.wsj.com \/articles\/chinas-new-tool-for-social-control-a-credit-rating-for-everything-1480351590, 28 November 2016, accessed 31 May 2017.\n\n27. See 'Charlie Brooker: the dark side of our gadget addiction', _Guardian_ , https:\/\/www.theguardian.com \/technology\/2011\/dec\/01\/charlie-brooker-dark-side-gadget-addiction-black-mirror, 2 December 2011, accessed 30 April 2017.\n\n28. See 'Lessons from Luciano Floridi, the Google Philosopher', Radio National, http:\/\/www.abc.net.au\/radionational\/programs\/philosopherszone\/lessons-from-luciano-floridi-the-google-philosopher\/6497872, 26 May 2015, accessed 4 March 2017.\n\n29. See _The Fourth Revolution_ , Luciano Floridi, Oxford University Press (2014).\n\n30. Bret Easton Ellis's article in the _New York Times_ about how everyone today is setting themselves up to be branded, targeted and data-mined online shaped some parts of this chapter. See 'Bret Easton Ellis on Living in the Cult of Likability', _New York Times_ , http:\/\/www.nytimes.com \/2015\/12\/08\/opinion\/bret-easton-ellis-on-living-in-the-cult-of-likability.html?_r=0, 8 December 2015.\n\n31. For the proceedings of the 2010 ACM Conference on Computer Supported Cooperative Work see 'Is it really about me? Message content in social awareness streams', Association for Computing Machinery, 2010, pp. 189\u201392.\n\n32. 'Disclosing information about the self is intrinsically rewarding', _PNAS_ , http:\/\/www.pnas.org \/content\/109\/21\/8038.full, 22 May 2012, accessed 30 May 2017.\n\n33. Peeple, a people-rating app dubbed the 'Yelp for humans' was first publicized in October 2015 but faced heavy criticism. It relaunched in March 2016 after a few changes to the app. See 'Remember Peeple? It's back, and launching on Monday', _WIRED_ , http:\/\/www.wired.co.uk\/article\/peeple-social-reputation-app-launched-released-download, 4 March 2016, accessed 16 May 2017.\n\n34. 'Everyone you know will be able to rate you on the terrifying \"Yelp for people\"\u2013whether you want them to or not', _Washington Post_ , https:\/\/www.washingtonpost.com \/news\/the-intersect\/wp\/2015\/09\/30\/everyone-you-know-will-be-able-to-rate-you-on-the-terrifying-yelp-for-people-whether-you-want-them-to-or-not\/, 30 September 2015, accessed 9 November 2016.\n\n35. See 'Peeple Watching Webisode 1\u2013Building the people app in SF', YouTube, https:\/\/www.youtube.com \/watch?v=6YrLEL6U5o4, 5 October 2015, accessed 16 May 2017.\n\n36. 'China penalizes 6.7m debtors with travel ban', _Financial Times_ , https:\/\/www.ft.com \/content\/ceb2a7f0-f350-11e6-8758-6876151821a6, accessed 15 February 2017.\n\n37. See '4.9 mln people with poor credit record barred from taking planes', Ecns.cn, http:\/\/www.ecns.cn\/2016\/11-03\/232618.shtml, 11 March 2016, accessed 15 February 2017.\n\n38. See 'Around the House: 50 years ago, an idea and $800 led to today's FICO scores', _San Gabriel Valley Tribune,_ http:\/\/www.sgvtribune.com \/business\/20160624\/around-the-house-50-years-ago-an-idea-and-800-led-to-todays-fico-scores, 24 June 2016, accessed 15 November 2017.\n\n39. Refer to 'Fair and Accurate Credit Transactions Act of 2003', US Government Publishing Office, https:\/\/www.gpo.gov\/fdsys\/pkg\/PLAW-108publ159\/html\/PLAW-108publ159.htm, accessed 10 May 2017.\n\n40. See 'Credit score statistics', NASDAQ, http:\/\/www.nasdaq.com \/article\/credit-score-statistics-cm435901, 23 January 2015, accessed 4 May 2017.\n\n41. See 'China's \"social credit\": Beijing sets up huge system', BBC, http:\/\/www.bbc.com \/news\/world-asia-china-34592186, 26 October 2015, accessed 31 May 2017.\n\n42. For annual economic loss caused by lack of credit information, see 'China's social credit system gains momentum', ChinaDaily.com, http:\/\/www.chinadaily.com.cn\/china\/2014-08\/22\/content_18472094.htm, 22 July 2014, accessed 4 August 2017.\n\n43. 'Global trade in fake goods worth nearly half a trillion dollars a year\u2013OECD & EUIPO', OECD, http:\/\/www.oecd.org \/industry\/global-trade-in-fake-goods-worth-nearly-half-a-trillion-dollars-a-year.htm, 18 April 2016, accessed 30 May 2017.\n\n44. For interview with Professor Wang Shuqin, see 'China rates its own citizens\u2013including online behaviour', _Volkskrant_ , http:\/\/www.volkskrant.nl\/buitenland\/china-rates-its-own-citizens-including-online-behaviour~a3979668\/, 25 April 2015, accessed 4 May 2017.\n\n45. 'Planning Outline for the Construction of a Social Credit System', China Copyright and Media, translated by Rogier Creemers, https:\/\/chinacopyrightandmedia.wordpress.com\/2014\/06\/14\/planning-outline-for-the-construction-of-a-social-credit-system-2014-2020\/, 25 April 2015, accessed 3 March 2017.\n\n46. 'Open Sesame\u2013Why a Digital \"Social Credit\" System Makes Sense for China', LinkedIn, https:\/\/www.linkedin.com \/pulse\/open-sesame-why-digital-social-credit-system-makes-sense-majid-%E7%BD%97%E7%B4%A0, 5 November 2015, accessed 30 April 2017.\n\n47. See 'Facebook can recognise you in photos even if you're not looking', _New Scientist_ , https:\/\/www.newscientist.com \/article\/dn27761-facebook-can-recognise-you-in-photos-even-if-youre-not-looking\/, 22 June 2015, accessed 30 April 2017.\n\n48. See 'OECD Digital Economy Outlook 2015\u2013Emerging issues: The Internet of Things, OECD Publishing', http:\/\/www.keepeek.com \/Digital-Asset-Management\/oecd\/science-and-technology\/oecd-digital-economy-outlook-2015\/emerging-issues-the-internet-of-things_9789264232440-8-en#page1, 2015, accessed 5 December 2016.\n\n49. See: Zak v Bose Corp, U.S. District Court, Northern District of Illinois, No. 17-02928, https:\/\/assets.documentcloud.org\/documents\/3673948\/Zak-v-Bose.pdf, accessed 19 July 2017.\n\n50. 'A message to our Bose Connect App customers', Bose, https:\/\/www.bose.com.au\/en_au\/landing_pages\/bose_corporation_updates.html, accessed 20 April 2017.\n\n51. See 'We-Vibes Motion For Approval of Settlement', US District Court, Northern District of Illinois, No. 1:16-cv-08655, https:\/\/assets.documentcloud.org\/documents\/3517061\/We-Vibes-Motion-For-Approval-of-Settlement.pdf, accessed 19 July 2017.\n\n52. 'Sex toy surveillance: more Wi-Fi enabled devices vulnerable to hacking', _WIRED,_ http:\/\/www.wired.co.uk\/article\/we-vibe-sex-toy-surveillance, 5 April 2017, accessed 19 July 2017. Svakom has since said it has addressed the issues and that updated versions of its software were 'completely secure'.\n\n53. See 'Agreement Between TSA and TSA Pre-check Application Expansion', Agenda 21 News, http:\/\/agenda21news.com\/wp-content\/uploads\/2015\/01\/OTA_Articles_for_Pre-check_Application_Expansion.pdf, accessed 5 December 2016.\n\n54. 'Password for social media accounts could be required for some to enter country', TechCrunch, https:\/\/techcrunch.com\/2017\/02\/08\/passwords-for-social-media-accounts-could-be-required-for-some-to-enter-country\/, 8 February 2017, accessed 31 May 2017.\n\n55. 'A.G. Schneiderman Announces Settlement with Uber to Enhance Rider Privacy', New York State Office of the Attorney General, https:\/\/ag.ny.gov\/press-release\/ag-schneiderman-announces-settlement-uber-enhance-rider-privacy, 6 January 2016, 19 June 2017.\n\n56. See 'RoG Blog': 'Blog.Uber.com\/Ridesofglory', _Internet Archive WaybackMachine_ , https:\/\/web.archive.org\/web\/20140827195715\/http:\/blog.uber.com\/ridesofglory, 26 March 2012, accessed 19 June 2017.\n\n57. See 'Uber to Pay $20,000 Fine Over \"God View\" Toll September 2014 Data Breach', Tech Times, http:\/\/www.techtimes.com \/articles\/122410\/20160107\/uber-to-pay-20-000-fine-over-god-view-tool-september-2014-data-breach.htm, 7 January 2016, accessed 5 December 2016.\n\n58. _The Inevitable_ , Kevin Kelly, Viking Press (2016).\n\n59. 'Why you should embrace surveillance, not fight it', _WIRED_ , https:\/\/www.wired.com \/2014\/03\/going-tracked-heres-way-embrace-surveillance\/, 3 October 2014, accessed 7 January 2017.\n\n8. In Bots We Trust\n\n1. 'Believing in BERT: Using expressive communication to enhance trust and counteract operational error in physical Human-Robot Interaction', Adriana Hamacher, Nadia Bianchi-Berthouze, Anthony Pipe and Kerstin Eder, 2016, 25th IEEE International Symposium on Robot and Human Interactive Communication (RO-MAN), https:\/\/arxiv.org\/ftp\/arxiv\/papers\/1605\/1605.08817.pdf, accessed 12 December 2016.\n\n2. See 'People will lie to robots to avoid \"hurting their feelings\"', _WIRED,_ http:\/\/www.wired.co.uk\/article\/bert-lying-robots-emotions, 23 August 2016, accessed 31 May 2017.\n\n3. See 'Trust Me: Researchers Examine How People and Machines Build Bonds', George Mason University, https:\/\/www2.gmu.edu\/news\/1849, 4 February 2015, accessed 4 March 2017.\n\n4. See 'Can we trust robots?', Mark Coeckelbergh, _Ethics and Information Technology_ , http:\/\/link.springer.com\/article\/10.1007\/s10676-011-9279-1, 3 September 2011, accessed 4 March 2017.\n\n5. My conversation with Stephen Cave in December 2016 shaped my understanding of the shift from trust in technology being based on it doing something to deciding something.\n\n6. See 'What Is a Robot?', _The Atlantic_ , http:\/\/www.theatlantic.com \/technology\/archive\/2016\/03\/what-is-a-human\/473166\/, 22 March 2015, accessed 5 March 2017.\n\n7. 'Computing Machinery and Intelligence', Alan Turing, _Mind_ , http:\/\/mind.oxfordjournals.org\/content\/LIX\/236\/433.full.pdf+html and Oxford University Press (1950).\n\n8. _Speculations Concerning the First Ultraintelligent Machine_ , Irving Good, http:\/\/www.kushima.org \/is\/wp-content\/uploads\/2015\/07\/Good65ultraintelligent.pdf and Academic Press (1965).\n\n9. See 'Stephen Hawking warns artificial intelligence could end mankind', BBC, http:\/\/www.bbc.com \/news\/technology-30290540, 2 December 2014, accessed 5 March 2017.\n\n10. See 'Computer AI passes Turing test in \"world first\"', BBC, http:\/\/www.bbc.com \/news\/technology-27762088, 9 June 2014, accessed 5 March 2017.\n\n11. 'AI Program beats humans in poker game', BBC News, http:\/\/www.bbc.co.uk\/news\/technology-38812530, 31 January 2017, accessed 4 May 2017.\n\n12. For the price to own a Pepper robot, see 'Pepper the robot's contract bans users from having sex with it', _WIRED_ , http:\/\/www.wired.co.uk\/article\/pepper-robot-sex-banned, 23 September 2015, accessed 12 January 2017.\n\n13. See 'No sex please, we're robots! Buyers of hit new \"emotional robot\" Pepper to sign contract vowing it wont be used indecently', _Daily Mail_ , http:\/\/www.dailymail.co.uk\/news\/article-3243051\/No-sex-robots-Buyers-hit-new-emotional-robot-Pepper-sign-contract-saying-won-t-used-sex-porno-films.html, 23 September 2015, accessed 5 March 2017.\n\n14. See 'The Great Bot Rush of 2015\u20132016', Continuations, http:\/\/continuations.com\/post\/135317420600\/the-great-bot-rush-of-2015-16, 16 December 2015, accessed 5 March 2017.\n\n15. See 'What Is a Robot?', _The Atlantic_ , http:\/\/www.theatlantic.com \/technology\/archive\/2016\/03\/what-is-a-human\/473166\/, 22 March 2015, accessed 5 March 2017.\n\n16. Ibid.\n\n17. Statistics relating to DoNotPay from Joshua Browder, author interview, 2 March 2017.\n\n18. Tay gained 50,000 followers and produced nearly 100,000 tweets in less than twenty-four hours after her arrival on Twitter. See 'Why Microsoft's \"Tay\" AI bot went wrong', TechRepublic, http:\/\/www.techrepublic.com \/article\/why-microsofts-tay-ai-bot-went-wrong\/, 24 March 2016, accessed 30 May 2017.\n\n19. See 'Twitter taught Microsoft's AI chatbot to be a racist asshole in less than a day', The Verge, http:\/\/www.theverge.com \/2016\/3\/24\/11297050\/tay-microsoft-chatbot-racist, 24 March 2016, accessed 20 May 2017.\n\n20. Andrew Karpathy explains how neural networks attempt predictive text. See 'Multi-layer Recurrent Neural Networks (LSTM, GRU, RNN) for character-level language models in Torch', Github, https:\/\/github.com\/karpathy\/char-rnn, accessed 9 December 2016.\n\n21. 'Tay, Microsoft's AI chatbot, gets a crash course in racism from Twitter', _Guardian_ , https:\/\/www.theguardian.com \/technology\/2016\/mar\/24\/tay-microsofts-ai-chatbot-gets-a-crash-course-in-racism-from-twitter, 24 March 2016, accessed 20 May 2017.\n\n22. 'We are developing the psyche of software that will sit at the heart of virtual and animated systems' and subsequent quotes from Mark Stephen Meadows, author interview, 1 December 2016.\n\n23. See 'The Bot Politic', _New Yorker_ , http:\/\/www.newyorker.com \/tech\/elements\/the-bot-politic, 31 December 2016, accessed 31 May 2017.\n\n24. See 'Public Predictions for the Future of Workforce Automation', Pew Research Center, http:\/\/www.pewinternet.org \/2016\/03\/10\/public-predictions-for-the-future-of-workforce-automation\/, 10 March 2016, accessed 4 May 2017.\n\n25. See 'The Future of Employment: How Susceptible are Jobs to Computerisation?', Carl Benedikt Frey and Michael Osborne, Oxford Martin School, http:\/\/www.oxfordmartin.ox.ac.uk\/downloads\/academic\/The_Future_of_Employment.pdf, 2013, accessed 4 March 2017.\n\n26. See 'An Uncanny Mind, Masahiro Mori on the Uncanny Valley and Beyond', IEEE Spectrum, http:\/\/spectrum.ieee.org\/automaton\/robotics\/humanoids\/an-uncanny-mind-masahiro-mori-on-the-uncanny-valley, 12 June 2012, accessed 4 March 2017.\n\n27. 'Now you're talking: human-like robot may one day care for dementia patients', Reuters, http:\/\/www.reuters.com \/article\/singapore-humanoid-idUSKCN0W9120, 7 March 2016, accessed 30 May 2017.\n\n28. 'Oh dear! Oh dear! I shall be too late!' See _Alice's Adventures in Wonderland_ , Lewis Carroll, Macmillan and Co. (1869), p. 1.\n\n29. See 'The mind in the machine: Anthropomorphism increases trust in the autonomous vehicle', Adam Waytz, Joy Heafner and Nicholas Epley, _Journal of Experimental Social Psychology_ , http:\/\/www.sciencedirect.com \/science\/article\/pii\/S0022103114000067, May 2014, accessed 9 December 2016.\n\n30. See 'The Bot Politic', _New Yorker_ , http:\/\/www.newyorker.com \/tech\/elements\/the-bot-politic, 31 December 2016, accessed 31 May 2017.\n\n31. See 'Hackers can hijack Wi-Fi Hello Barbie to spy on your children', _Guardian_ , https:\/\/www.theguardian.com \/technology\/2015\/nov\/26\/hackers-can-hijack-wi-fi-hello-barbie-to-spy-on-your-children, 26 November 2015, accessed 5 December 2016. Note: ToyTalk Mattel have fixed many of the issues security analyst Andrew Blaich raised, such as removing the weaker SSLv3 ciphers from its servers. 'We are aware of the Bluebox Security Report and are working closely with ToyTalk to ensure the safety and security of Hello Barbie,' said Mattel spokesperson Michelle Chidoni.\n\n32. 'One of the key questions is how do we assess the trustworthiness of an intelligent machine?' and subsequent quotes from Stephen Cave, author interview, 6 December 2016.\n\n33. See 'Machine Ethics: The Robots Dilemma', _Nature_ , http:\/\/www.nature.com \/news\/machine-ethics-the-robot-s-dilemma-1.17881, 1 July 2015, accessed 4 March 2017.\n\n34. This is also known as Bentham's 'fundamental axiom'. See _A Fragment on Government_ , Preface, Jeremy Bentham, Cambridge University Press (1988).\n\n35. _The Right and the Good_ , William David Ross, Clarendon Press (2002).\n\n36. See _Machine Ethics_ , Michael Anderson and Susan Leigh Anderson, Cambridge University Press (2011).\n\n37. ' The Ethical Robot', UConn Today, http:\/\/today.uconn.edu\/2010\/11\/the-ethical-robot\/, 8 November 2010, accessed 12 December 2016.\n\n38. The trolley problem was invented by the philosophers Philippa Foot and Judith Jarvis in the 1960s. See 'The Problem of Abortion and the Doctrine of the Double Effect', Philippa Foot and Judith Jarvis, http:\/\/philpapers.org\/archive\/FOOTPO-2.pdf, from the _Oxford Review_ , 1967.\n\n39. 'Rise of the machines: are algorithms sprawling out of our control?', _WIRED_ , http:\/\/www.wired.co.uk\/article\/technology-regulation-algorithm-control, 1 April 2017.\n\n40. See ' Principles of robotics', Engineering and Physical Sciences Research Council, https:\/\/www.epsrc.ac.uk\/research\/ourportfolio\/themes\/engineering\/activities\/principlesofrobotics\/, accessed 12 December 2016.\n\n41. See 'The social dilemma of autonomous vehicles' Jean-Fran\u00e7ois Bonnefon, Azim Shariff and Iyad Rahwan, _Science_ , 352\/6293, https:\/\/arxiv.org\/abs\/1510.03346, 4 July 2016, accessed 12 December 2016.\n\n42. See 'Driverless cars are colliding with the creepy Trolley Problem', _Washington Post_ , https:\/\/www.washingtonpost.com \/news\/innovations\/wp\/2015\/12\/29\/will-self-driving-cars-ever-solve-the-famous-and-creepy-trolley-problem\/?utm_term=.44210b7e6797, 29 December 2015, accessed 12 December 2016.\n\n43. See 'Why robots need to be able to say \"no\"', The Conversation, http:\/\/www.wired.co.uk\/article\/technology-regulation-algorithm-control, 8 April 2016, accessed 12 December 2016.\n\n9. Blockchain Part I: The Digital Gold Rush\n\n1. For William H. Furness's account of his experience in Yap, see _The Island of Stone Money: Uap of the Carolines_ , William H. Furness, J. B. Lippincott Co. (1910), pp. 94\u2013106.\n\n2. See 'David O'Keefe: The King of Hard Currency', _Smithsonian Magazine_ , http:\/\/www.smithsonianmag.com \/history\/david-okeefe-the-king-of-hard-currency-37051930\/, 28 July 2011, accessed 2 March 2017.\n\n3. See 'When Bitcoin Grows Up', _London Review of Books_ , http:\/\/www.lrb.co.uk\/v38\/n08\/john-lanchester\/when-bitcoin-grows-up?utm_content=bufferc8f7f&utm_medium=social&utm_source=twitter.com&utm_campaign=buffer, 21 April 2016, accessed 2 March 2017.\n\n4. See _The Island of Stone Money: Uap of the Carolines_ , William H. Furness, J. B. Lippincott Co. (1910), p. 98.\n\n5. See 'The Island of Stone Money', Milton Friedman, http:\/\/www.karlwhelan.com \/IMB\/Friedman-Yap.pdf, Stanford University, February 1991.\n\n6. For the estimated total amount of money in the world in terms of value, see _The Doctor_ , Dr Karl Kruszelnicki, Macmillan Australia (2016), p. 132.\n\n7. _Double Entry_ , Jane Gleeson-White, Allen & Unwin (2011).\n\n8. See 'Bitcoin: A Peer-to-Peer Electronic Cash System', Satoshi Nakamoto, https:\/\/bitcoin.org\/bitcoin.pdf, 24 May 2009, accessed 2 March 2017.\n\n9. See 'Bitcoin open source implementation of P2P currency', Satoshi Nakamoto, P2P foundation, http:\/\/p2pfoundation.ning.com\/forum\/topics\/bitcoin-open-source, 11 February 2009, accessed 2 March 2017.\n\n10. See 'Disruptions: Betting on a Coin with no realm', _New York Times_ , https:\/\/bits.blogs.nytimes.com\/2013\/12\/22\/disruptions-betting-on-bitcoin\/?_r=0&mtrref= www.forbes.com &gwh=EF12B4F946D4DF3A60818B678EA05D1B&gwt=pay, 22 December 2013, accessed 2 March 2017.\n\n11. For the bitcoin price surge in January 2017, see 'Bitcoin Price Soars, Fueled by Speculation and Global Currency Turmoil', _New York Times_ , https:\/\/www.nytimes.com \/2017\/01\/03\/business\/dealbook\/bitcoin-price-soars-fueled-by-speculation-and-global-currency-turmoil.html, 3 January 2017, accessed 2 March 2017.\n\n12. Wei Dai, 'b-money', http:\/\/www.weidai.com \/bmoney.txt, 1998, accessed 2 March 2017.\n\n13. 'Back to the Future: Adam Back Remembers the Cypherpunk Revolution and the Origins of Bitcoin', _Bitcoin Magazine_ , https:\/\/bitcoinmagazine.com\/articles\/back-future-adam-back-remembers-cypherpunk-revolution-origins-bitcoin-1441741053\/, 8 September 2015, accessed 2 March 2017.\n\n14. For the number of bitcoin transactions, see 'Bitcoin Total Number of Transactions', Quandl, https:\/\/www.quandl.com \/data\/BCHAIN\/NTRAT-Bitcoin-Total-Number-of-Transactions, accessed 14 February 2017.\n\n15. For the number of bitcoin nodes, see 'Nodes', Bitnodes, https:\/\/bitnodes.21.co\/dashboard\/?days=90, accessed 14 February 2017.\n\n16. See 'The Satoshi Affair', _London Review of Books_ , https:\/\/www.lrb.co.uk\/v38\/n13\/andrew-ohagan\/the-satoshi-affair, 30 June 2016, accessed 14 February 2017.\n\n17. See 'Engineering the Bitcoin Gold Rush: An Interview with Yifu Guo, Creator of the First Purpose-Built Miner', _Motherboard_ , https:\/\/motherboard.vice.com\/en_us\/article\/engineering-the-bitcoin-gold-rush-an-interview-with-yifu-guo-creator-of-the-first-asic-based-miner, 27 March 2013, accessed 2 March 2017.\n\n18. See 'In the beginning\u2013Trusted Disrupted: Bitcoin and the Blockchain' (Episode 1), https:\/\/techcrunch.com\/2016\/10\/10\/watch-the-first-episode-of-our-new-series-trust-disrupted-bitcoin-and-the-blockchain\/, 10 October 2016, accessed 2 March 2017.\n\n19. See 'The magic of mining', _The Economist_ , http:\/\/www.economist.com \/news\/business\/21638124-minting-digital-currency-has-become-big-ruthlessly-competitive-business-magic, 8 January 2015, accessed 2 March 2017.\n\n20. See _The Age of Cryptocurrency: How Bitcoin and the Blockchain are Challenging the Global Economic Order_ , Paul Vigna and Michael J. Casey, Picador (2016).\n\n21. 'Sichuan\u2013a Paradise of Food and Modern Agriculture', HKTDC, http:\/\/www.hktdc.com \/web\/featured_suppliers\/sichuan\/index.html, accessed 31 May 2017.\n\n22. For the number of dams on the Min Jiang, see 'Mapping China's \"Dam Rush\"', Wilson Center, https:\/\/www.wilsoncenter.org \/publication\/interactive-mapping-chinas-dam-rush, 21 March 2014, accessed 2 March 2017.\n\n23. Zhu Rei is interviewed in Episode 2 of the TechCrunch series on Blockchain. See 'Trusted Disrupted: Bitcoin and the Blockchain', TechCrunch, https:\/\/techcrunch.com\/2016\/10\/10\/watch-trust-disrupted-bitcoin-and-the-blockchain-episode-two\/, 10 October 2016, accessed 2 March 2017.\n\n24. See 'The magic of mining', _The Economist_ , http:\/\/www.economist.com \/news\/business\/21638124-minting-digital-currency-has-become-big-ruthlessly-competitive-business-magic, 8 January 2015, accessed 2 March 2017.\n\n25. See 'How China took center stage in Bitcoin's civil war', _New York Times_ , https:\/\/www.nytimes.com \/2016\/07\/03\/business\/dealbook\/bitcoin-china.html, 29 June 2016, accessed 2 March 2017.\n\n26. See 'Bitcoin open source implementation of P2P currency', Satoshi Nakamoto, P2P foundation, http:\/\/p2pfoundation.ning.com\/forum\/topics\/bitcoin-open-source, 11 February 2009, accessed 2 March 2017.\n\n27. See 'Missing: hard drive containing Bitcoins worth \u00a34m in Newport landfill site', _Guardian_ , https:\/\/www.theguardian.com \/technology\/2013\/nov\/27\/hard-drive-bitcoin-landfill-site, 28 November 2013, accessed 2 March 2017.\n\n28. For the list of countries that banned bitcoin, see 'Top 10 Countries in Which Bitcoin is Banned', CryptoCoinsNews, https:\/\/www.cryptocoinsnews.com \/top-10-countries-bitcoin-banned\/, 27 May 2015, accessed 2 March 2017.\n\n29. Larry Summers interviewed in Episode 1 of the TechCrunch series on Blockchain, see 'In the beginning\u2013Trusted Disrupted: Bitcoin and the Blockchain', TechCrunch, https:\/\/techcrunch.com\/2016\/10\/10\/watch-the-first-episode-of-our-new-series-trust-disrupted-bitcoin-and-the-blockchain\/, 10 October 2016, accessed 5 March 2017.\n\n30. See 'The great chain of being sure about things', _The Economist_ , http:\/\/www.economist.com \/news\/briefing\/21677228-technology-behind-bitcoin-lets-people-who-do-not-know-or-trust-each-other-build-dependable, 31 October 2015, accessed 30 May 2017.\n\n10. Blockchain Part II: The Truth Machine\n\n1. 'The Dao Attacked: Code Issue Leads to $60 Million Ether Theft', CoinDesk, http:\/\/www.coindesk.com \/dao-attacked-code-issue-leads-60-million-ether-theft\/, 17 June 2016, accessed 1 March 2017.\n\n2. See 'Ether Price Plummets; Ethereum DAO May Be Hacked', CryptoCoinNews, https:\/\/www.cryptocoinsnews.com \/ether-price-plumets-ethereum-dao-may-be-hacked, 17 June 2016, accessed 1 March 2017.\n\n3. See 'Understanding the DAO Attack', CoinDesk, http:\/\/www.coindesk.com \/understanding-dao-hack-journalists\/, 25 June 2016, accessed 1 March 2017.\n\n4. Slock.it is a German tech start-up building 'smart locks' that connect all kinds of things\u2013cars, bikes, even a front door\u2013to the blockchain.\n\n5. See 'Why Bitcoin Matters', _New York Times_ , https:\/\/dealbook.nytimes.com\/2014\/01\/21\/why-bitcoin-matters\/, 21 January 2014, accessed 1 March 2017.\n\n6. 'DAOs, DACs, DAs and More: An Incomplete Terminology Guide', Ethereum Blog, https:\/\/blog.ethereum.org\/2014\/05\/06\/daos-dacs-das-and-more-an-incomplete-terminology-guide\/, 6 May 2014, accessed 4 May 2017.\n\n7. 'Can this 22-year old coder out-bitcoin bitcoin?', _Fortune_ , http:\/\/fortune.com\/ethereum-blockchain-vitalik-buterin\/, 27 September 2016, accessed 1 March 2017.\n\n8. 'The Uncanny Mind that Built Ethereum', Backchannel, https:\/\/backchannel.com\/the-uncanny-mind-that-built-ethereum-9b448dc9d14f#.wmpr48it1, 13 June 2016, accessed 1 March 2017.\n\n9. 'Ethereum', Ethereum, https:\/\/www.ethereum.org \/foundation, accessed 1 March 2017.\n\n10. 'Bootstrapping a Decentralized Autonomous Corporation: Part 1', _Bitcoin Magazine_ , https:\/\/bitcoinmagazine.com\/articles\/bootstrapping-a-decentralized-autonomous-corporation-part-i-1379644274\/, 19 September 2013, accessed 1 March 2017.\n\n11. 'The Uncanny Mind that Built Ethereum', Backchannel, https:\/\/backchannel.com\/the-uncanny-mind-that-built-ethereum-9b448dc9d14f#.wmpr48it1, 13 June 2016, accessed 1 March 1 2017.\n\n12. Some autonomous companies exist now like Weifund, which is decentralized crowdfunding without an intermediary like Kickstarter. See http:\/\/weifund.io\/. Other examples: http:\/\/www.gdi.ch\/en\/Think-Tank\/GDI-Trend-News\/News-Detail\/Uber-without-Uber-Platform-cooperativism-as-the-new-sharing-economy.\n\n13. For the Ethereum White Paper, See 'White Paper', Github, https:\/\/github.com\/ethereum\/wiki\/wiki\/White-Paper, accessed 1 March 2017.\n\n14. 'The great chain of being sure about things', _The Economist_ , http:\/\/www.economist.com \/news\/briefing\/21677228-technology-behind-bitcoin-lets-people-who-do-not-know-or-trust-each-other-build-dependable, 31 October 2015, accessed 30 May 2017.\n\n15. Ibid.\n\n16. In April 2016, Bob Sauchelli purchased his first excess energy, 195 credits for $0.07 each, directly from his neighbour, Eric Frumin, via Transactive Grid. No energy company required. See 'Ethereum Used for \"First\" Paid Energy Trade Using Blockchain Tech', CoinDesk, http:\/\/www.coindesk.com \/ethereum-used-first-paid-energy-trade-using-blockchain-technology, 11 April 2016, accessed 1 March 1, 2017.\n\n17. See Tom Standage on Babbage Podcast from _The Economist_ interview with Buterin: 'Vitalek Buterin on his long term goals for Ethereum', Bitcuners, http:\/\/blog.bitcuners.org\/post\/143849632438\/vitalik-buterin-on-his-long-term-goals-for, 4 May 2016, accessed 1 March 2017.\n\n18. 'Havocscope Black Market', Havocscope, http:\/\/www.havocscope.com \/, accessed 1 March 2017.\n\n19. 'A $50 million hack just showed that the DAO was all too human', _WIRED_ , https:\/\/www.wired.com \/2016\/06\/50-million-hack-just-showed-dao-human\/, 18 June 2016, accessed 1 March 2017.\n\n20. See 'An Open letter', Pastebin, http:\/\/pastebin.com\/CcGUBgDG, 18 June 2016, accessed 23 February 2017.\n\n21. 'Can this 22-year-old coder out-bitcoin bitcoin', _Fortune_ , http:\/\/fortune.com\/ethereum-blockchain-vitalik-buterin\/, 27 September 2016, accessed 31 May 2017.\n\n22. 'A $50 million hack just showed that the DAO was all too human', _WIRED_ , https:\/\/www.wired.com \/2016\/06\/50-million-hack-just-showed-dao-human\/, 18 June 2016, accessed 1 March 2017.\n\n23. 'The Ethereum Hard Fork is Done', Futurism, https:\/\/futurism.com\/the-ethereum-hard-fork-is-done\/, 20 July 2016, accessed 12 December 2016.\n\n24. This is a great reference for the sequence of events around the DAO attack\u2013see 'Understanding the DAO Attack', CoinDesk, http:\/\/www.coindesk.com \/understanding-dao-hack-journalists\/, 25 June 2016, accessed 12 December 2016.\n\n25. See 'A $50 million hack just showed that the DAO was all too human', _WIRED_ , https:\/\/www.wired.com \/2016\/06\/50-million-hack-just-showed-dao-human\/, 18 June 2016, accessed 1 March 2017.\n\n26. _Mastering Bitcoin: Unlocking Digital Cyrptocurrencies_ , Andreas Antonopoulos, O'Reilly Media (2014).\n\n27. See 'Reid Hoffman: Why the blockchain matters', _WIRED_ , http:\/\/www.wired.co.uk\/article\/bitcoin-reid-hoffman, 15 May 2015.\n\n28. For Danny Hillis's interview, See 'Disney's Wizards', _Newsweek_ , http:\/\/europe.newsweek.com\/disneys-wizards-172346?rm=eu, 8 November 1997, accessed 1 March 2017.\n\n29. For the number of DApps since first launched, see Ethereum, http:\/\/dapps.ethercasts.com\/, accessed 10 May 2017.\n\n30. For the number of apps in the Appstore, see 'Appstore', Consensys, https:\/\/consensys.net\/static\/images\/number-of-apps.png, accessed 10 May 2017.\n\n31. 'PwC Expert: $1.4 Billion Invested in Blockchain in 2016', CryptoCoins News, https:\/\/www.cryptocoinsnews.com \/pwc-expert-1-4-billion-invested-blockchain-2016\/, 9 November 2016, accessed 1 March 2017.\n\n32. See 'Introducing R3 Corda: A Distributed Ledger Designed for Financial Services', R3 blog, http:\/\/www.r3cev.com \/blog\/2016\/4\/4\/introducing-r3-corda-a-distributed-ledger-designed-for-financial-services, 5 April 2016, accessed 31 May 2017.\n\n33. 'It means the history of something, where it came from and where did it go' and subsequent quotes from Leanne Kemp, author interview, 2 March 2017.\n\n34. See 'Crime pays when provenance is broken', Everledger, https:\/\/www.everledger.io\/, accessed 1 March 2017.\n\n35. For estimates of stolen art, see 'Nazi loot case: much art still untraced\u2013expert', BBC News, http:\/\/www.bbc.com \/news\/world-europe-24801935, 4 November 2013, accessed 1 March 2017.\n\n36. 'Nazi Art Loot Returned', _New York Times_ , https:\/\/www.nytimes.com \/2016\/07\/16\/arts\/design\/nazi-art-loot-returned-to-nazis.html?_r=0, 15 July 2016, accessed 1 March 2017.\n\n37. See American Alliance of Museums website, 'Standards Regarding the Unlawful Appropriation of Objects During the Nazi Era', http:\/\/www.aam-us.org\/resources\/ethics-standards-and-best-practices\/collections-stewardship\/objects-during-the-nazi-era, accessed 1 March 2017.\n\n38. 'Every product has a story' and subsequent quotes from Jessi Baker, author interview, 9 August 2016.\n\n39. See 'Walmart and IBM are Partnering to Put Chinese Pork on a Blockchain', _Fortune_ , http:\/\/fortune.com\/2016\/10\/19\/walmart-ibm-blockchain-china-pork\/, 19 October 2016, accessed 1 March 2017.\n\n40. 'World Economic Forum Annual Meeting', World Economic Forum, https:\/\/www.weforum.org \/events\/world-economic-forum-annual-meeting-2017, accessed 16 May 2017.\n\n41. See 'Blockchain technology: Redefining trust for a global, digital economy', World Bank, http:\/\/blogs.worldbank.org\/ic4d\/blockchain-technology-redefining-trust-global-digital-economy, 16 June 2016, accessed 1 March 2017.\n\n42. See 'An interview with Hernando de Soto', McKinsey, http:\/\/www.mckinsey.com \/industries\/public-sector\/our-insights\/an-interview-with-hernando-de-soto, October 2012, accessed 1 March 2017.\n\n43. _The Mystery of Capital: Why Capitalism Triumphs in the West and Fails Everywhere Else_ , Hernando de Soto, Basic Books (2000).\n\n44. See _Blockchain Revolution_ , Don Tapscott and Alex Tapscott, Portfolio Penguin (2016).\n\n45. See _Land Policies for Growth and Poverty Reduction_ , World Bank Policy Research Report, 2003.\n\n46. 'Distributed Ledger Technology: Beyond Blockchain', UK Government, https:\/\/www.gov.uk\/government\/uploads\/system\/uploads\/attachment_data\/file\/492972\/gs-16-1-distributed-ledger-technology.pdf, 19 January 2016, accessed 1 March 2017.\n\n47. See 'Bitcoin: A Peer-to-Peer Electronic Cash System', https:\/\/bitcoin.org\/bitcoin.pdf, accessed 16 May 2017.\n\n48. See 'Why Bitcoin is and isn't like the internet', Joi Ito blog, https:\/\/joi.ito.com\/weblog\/2015\/01\/23\/why-bitcoin-is-.html, 23 January 2015, accessed 1 March 2017.\n\n49. For a timeline of Blythe Masters's work life, see 'Outsmarted high finance vs. human nature', _New Yorker_ , http:\/\/www.newyorker.com \/magazine\/2009\/06\/01\/outsmarted, 1 June 2009, accessed 16 May 2017.\n\n50. _Digital Gold: The Untold Story of Bitcoin_ , Nathaniel Popper, Harper (2016).\n\n51. See 'The Fintech 2.0 Paper: rebooting financial services', Finextra, https:\/\/www.finextra.com \/finextra-downloads\/newsdocs\/the%20fintech%202%200%20paper.pdf, 2015, accessed 10 May 2017.\n\n52. See 'Blythe Masters Tells Banks the Blockchain Changes Everything', Bloomberg, https:\/\/www.bloomberg.com \/news\/features\/2015-09-01\/blythe-masters-tells-banks-the-blockchain-changes-everything, 1 September 2015, accessed 10 May 2017.\n\n53. 'JP Morgan to pay $410 million to settle power market case', Reuters, http:\/\/www.reuters.com \/article\/us-jpmorgan-ferc-idUSBRE96T0NA20130730, 30 July 2013, accessed 10 May 2017.\n\n54. See Don and Alex Tapscott for an explanation of how the blockchain could be an ally or threat for the financial industry, _Blockchain Revolution_ : _How the Technology Behind Bitcoin is Changing Money, Business and the World_ , Don Tapscott and Alex Tapscott, Portfolio Penguin (2016).\n\n55. 'Goldman Sachs Files Patent for Virtual Settlement Currency', _Financial Times_ , https:\/\/www.ft.com \/content\/b0d8f614-997c-11e5-9228-87e603d47bdc, 3 December 2015, accessed 1 March 2017.\n\n56. For the number of banks that have a stake in R3CEV, see 'R3 Home', R3, http:\/\/r3members.com\/, accessed 1 March 2017.\n\n57. The original five Bitcoin Core developers were Gavin Andresen, Wladimir J. van der Laan, Pieter Wuille, Greg Maxwell and Jeff Garzik. Gavin convinced Mike Hearn to work on Bitcoin Core, see 'Benevolent dictators and disenchanted believers: bitcoin core developers revisited', CoinFox, http:\/\/www.coinfox.in fo\/news\/reviews\/5312-benevolent-dictators-and-disenchanted-believers-bitcoin-core-developers-revisited, 15 April 2016, accessed 1 March 2017.\n\n58. See 'The resolution of the Bitcoin experiment', https:\/\/blog.plan99.net\/the-resolution-of-the-bitcoin-experiment-dabb30201f7#.idmijyl38, 14 January 2016, accessed 1 March 2017.\n\n59. See 1993 CBC interview with John Allen, 'Tech time warp: 20 years ago, we thought the internet would bring out our nice sides', _WIRED_ , https:\/\/www.wired.com \/2014\/06\/tech-time-warp-cyber-bullies\/, 13 June 2014, accessed 1 March 2017.\n\nConclusion\n\n1. See 'In No One We Trust', Joseph Stiglitz, _New York Times_ , https:\/\/opinionator.blogs.nytimes.com\/2013\/12\/21\/in-no-one-we-trust\/?_r=3, 21 December 2013, accessed 16 May 2017.\n\n2. Eric's story is retold by Shivani Siroya and the Tala team through Skype and email correspondence, 18 January 2017 and 25 February 2017.\n\n3. 'We would say to her, \"Mom you work so hard and your patients never pay you\"' and other quotes from Shivani Siroya, author interview, 18 January 2017.\n\n4. See 'A smart loan for people with no credit history (yet)', TED talk, https:\/\/www.ted.com \/talks\/shivani_siroya_a_smart_loan_for_people_with_no_credit_history_yet\/transcript?language=en#t-102209, April 2016, accessed 16 May 2017.\n\n5. See 'Do Your Research Before Changing the World', Collaborative Fund, http:\/\/www.collaborativefund.com \/blog\/shivani-siroya\/, 8 March 2016, accessed 16 May 2017.\n\n6. 'This is data that would not be found on a paper trail or in any formal financial record': I have combined quotes from my interview with her and her TED speech where she makes a similar point.\n\n7. See 'An Implosion of Trust', Edelman, http:\/\/www.edelman.com.au\/magazine\/posts\/an-implosion-of-trust\/, accessed 18 January 2017.\n\n8. See 'Uber CEO on Driver 'Assault': It's Not Real and We're Not Responsible', Valleywag, http:\/\/valleywag.gawker.com\/uber-ceo-on-driver-assault-its-not-real-and-were-n-1323533057, 16 September 2013, accessed 16 May 2017.\n\n9. See 'Uber Chief's Email to Employees', _New York Times_ , https:\/\/www.nytimes.com \/interactive\/2017\/02\/03\/technology\/document-Kalanick-email.html, 3 February 2017, accessed 31 May 2017.\n\n10. 'The Second Coming', _The Collected Poems of W. B. Yeats_ , Wordsworth Editions (1994).\n\n11. For a description of the current state of ceaseless adaption and change, see _The Inevitable_ , Kevin Kelly, Viking Press (2016).\n\n12. There is a beautiful post by the Iranian blogger Hossein Derakhshan, who was imprisoned for six years for his online activity, who talks about this point. See 'The Web We Have to Save', Medium, https:\/\/medium.com\/matter\/the-web-we-have-to-save-2eb1fe15a426#, 14 July 2015, accessed 16 May 2017.\n\nFor a comprehensive review of trust literature please visit https:\/\/www.rachelbotsman.com.\n\n# Index\n\n_2001: A Space Odyssey_ ,\n\nABP Food Group\n\nAbraxas\n\nacademics\n\nAccenture\n\naccountability ,\n\naccounting, double-entry\n\nadvertising ; _see also_ marketing\n\nAfghanistan, war contracts\n\nAfrican-Americans\n\ndistrust of health-care providers 33\u20134\n\nused for medical research in Tuskegee Study 32\u20133, ,\n\n_Age of Cryptocurrency, The_ (Vigna and Casey)\n\nAhmadinejad, Mahmoud\n\nAir France Flight 447\n\nAirbnb 6\u20137, , , 64\u20138, , 91-6,\n\naircraft, automatic pilot\n\nAire digestive tracker\n\nAirtasker\n\nAkerlof, George\n\nalcohol\n\nAlexa 172\u20133, , 193\u20134\n\nalgorithms , , 100\u2013103, , , 166\u20137, 175\u20136, , 222\u20133,\n\nAlibaba , 11\u201312, , 21\u20134, , , , 154\u20135, , ,\n\n_Alice's Adventures in Wonderland_ (Carroll)\n\nAlipay , ,\n\nAlisie, Mihai\n\nAllawi, Ayad\n\nAllen, John\n\nAlphaBay\n\nAlsan, Marcella ,\n\nAmazon , , , , , , ; _see also_ Alexa\n\nAmazon Dark\n\nAmdur, Evelyn 231\u20132\n\nAmerican Alliance of Museums\n\nAmerican Automobile Association\n\nAmerican Civil Liberties Union\n\nAmerican International Group (AIG) , ,\n\nAnderson, Michael and Susan 196\u20138\n\nAndre, Carl 63\u20134\n\nAndreessen, Marc\n\nAndresen, Gavin , ,\n\nAnnan, Kofi\n\nAnnan, Kojo\n\nAnt Financial Services Group (AFSG)\n\nanthropomorphism\n\nAntin, Judd 64\u20138,\n\nanti-social behaviour 129\u201330\n\nAntonopoulos, Andreas\n\nAntpool\n\nApp Store\n\nApple ,\n\narms dealing\n\nARPANET\n\nArrow, Kenneth ,\n\nart, thefts by Nazis\n\nArthur Anderson\n\nartificial intelligence (AI) , , 181\u20132, 185\u20138\n\nAGI\n\nethical issues 196\u2013200\n\nAshley Madison\n\nAsimov, Isaac\n\nauthenticity 163\u20134\n\nautism, and MMR\n\nauto-pilots\n\nAvalon\n\navatars 188\u20139\n\nAxelrod, Robert ,\n\nB\u00e4ckstr\u00f6m, Lars\n\n_Bad Blood_ (Jones)\n\nBaihe ,\n\nBain & Company\n\nBaker, Jessi\n\nBangladesh\n\nBank of England 43\u20134\n\nBank Rossiya\n\nbankruptcy\n\nbanks , , 77\u20138, ,\n\ncharges\n\nfinancial crisis (2008) 1\u20132, 42\u20133,\n\nregulation\n\ntoxic culture\n\nBarbie doll, robotic (Hello Barbie) , ,\n\nBarclays ,\n\nBarrett, Andrea\n\nBartlett, Jamie\n\nBass Brewery\n\nBates, James Andrew\n\nBaveja, Savi 128\u201330, ,\n\nBBC ,\n\nBear Stearns\n\nBellini, Giovanni\n\nBenghazi attack\n\nBentham, Jeremy\n\nBerkowitz, Lawrence\n\nBetter Business Bureau\n\nbetting 226\u20137\n\nBHS, pension fund fiasco\n\nBianchi-Berthouze, Nadia\n\nBickman, Leonard\n\nBitAngels\n\nBitcloud\n\nbitcoin , 135\u20136, , 207\u20139, , , 238\u20139\n\ndouble-spending problem 209\u201310\n\nmining 211\u201316,\n\nvulnerability 214\u201317\n\n_see also_ blockchain\n\nBitcoin Core\n\nBitcoin Foundation\n\n_Bitcoin Magazine_\n\n_Bitcoin Weekly_\n\nBitfinex\n\nBitfloor\n\nBitfury\n\nBitland\n\nBlaBlaCar 56\u20138, , ,\n\n_Black Mirror_ 161\u20136\n\nBlackBank ,\n\nBlair, Tony\n\nBlazer, Chuck\n\nBlecharczyk, Nate ,\n\nBletchley Park\n\nblockchain , , 208\u201315, , , , 220\u201321, ,\n\nfuture of , 238\u201340,\n\nscalability\n\nuse in banking 239\u201344\n\nuse in smart contracts 226\u20137\n\nuse in tracking diamonds and art works 234\u20137\n\nuse in tracking drugs supply chain\n\nB-Money\n\nBolivia\n\nBose 171\u20132\n\nBotanic.co\n\nbots ; _see also_ artificial intelligence (AI); chatbots; robots\n\nBotton, Alain de\n\nBourdieu, Pierre\n\nBowles, Thomas Gibson\n\n_Bowling Alone_ (Putnam)\n\nBowman, Dave\n\nBP, Deepwater Horizon oil spill\n\nbrands , 89\u201392,\n\nBrands, Stephan\n\nBrexit , 47\u201350\n\nbribery\n\nBritish Virgin Islands\n\nBrooker, Charlie 161\u20132\n\nBrowder, Joshua\n\nBrown, Joshua\n\nBrusson, Nicolas\n\nBuffett, Warren\n\nBumble\n\nBush, George W. ,\n\nButerin, Vitalik 221\u20135, 228\u20139, ,\n\nBuzzFeed ,\n\nCabra, Mar\n\nCalifornia roll principle , ,\n\nCameron, David ,\n\nCameron, Ian ,\n\n\u010capek, Karel\n\nCarney, Mark 43\u20134\n\nCarroll, Lewis\n\nCarruthers, Tiana\n\ncars\n\naccidents 73\u20134\n\nbuying ,\n\nself-driving , , 71\u20135, , , , 193\u20134, 198\u2013201\n\n_see also_ ride-sharing\n\ncartoons 127\u20138\n\nCasey, Michael\n\nCatholic church, abuse scandal\n\nCave, Stephen 194\u20135, 201\u20132\n\nCBS 51\u20132\n\nCerf, Vincent\n\nChain\n\nChange.org\n\nchatbots , , , 184\u20138,\n\nChaum, David\n\nCheshire, Coye , 99\u2013100\n\nChesky, Brian , 92\u20133\n\nchess\n\nChilcot inquiry 44\u20135\n\nchild abuse ,\n\nchildcare 117\u201319, 125\u20136, 132\u20133\n\nby robots\n\nChina 11\u201316, 21\u20134\n\nattitudes to political dissent 154\u20135\n\nattitudes to trust in business relationships 12\u201313\n\nhub of bitcoin economy\n\npunishments for trust-breaking 160\u201361,\n\nSocial Credit System (Citizen Scores) 150\u201361, 166\u201370, ,\n\nChina Pages\n\nChina Rapid Finance\n\nChou, Jimmy 182\u20133\n\nChristin, Nicholas\n\nChromaWay\n\nChuxing, Didi\n\nCialdini, Robert\n\n_Circle, The_ (Eggers)\n\nCiti Ventures\n\nCitibank\n\nCitigroup\n\ncitizen science\n\nCitizen Scores _see_ Social Credit System (Citizen Scores)\n\nclergy, declining trust in\n\nclickbait ,\n\nclimate change ,\n\nClinton, Bill\n\nClinton, Hillary , 48\u20139\n\nemail scandal ,\n\nClinton Foundation 48\u20139\n\nCoca-Cola\n\nColbert, Stephen\n\nColeman, James Samuel ,\n\nCollins, Victor 172\u20133\n\nColu 225\u20136\n\nCommunism\n\ncommunities 28\u20139,\n\ncommuting\n\ncompetence\n\nCongress, trust in\n\nconsent, and online data 101\u20133\n\nCopernicus\n\nCordray, Julia\n\nCortana\n\ncounterfeit goods 168\u20139, ,\n\nCoVoiturage _see_ BlaBlaCar\n\nCraigslist\n\ncredit cards ,\n\nCredit Mobilier Scandal\n\ncredit ratings , , 167\u20138,\n\nCreemers, Rogier , ,\n\ncriminal justice system, biases in\n\n_Critique of Pure Reason_ (Kant)\n\nCronkite, Walter 51\u20132\n\ncrowd persuasion 78\u20139\n\ncrowdfunding , 219\u2013220,\n\ncryptocurrencies , 225\u20136, , 242\u20133; _see also_ bitcoin; Ethereum\n\ncryptomarkets 137\u201341; _see also_ darknet\n\ncurrencies 207\u20138\n\nCVs\n\nCypherpunk\n\nDahan, Mariana\n\nDai, Wei\n\nDalton, Carole 82\u20133\n\nDalton, Jason Brian 81\u20135, ,\n\nDAO _see_ Decentralized Autonomous Organization (DAO)\n\n_Dark Net, The_ (Bartlett)\n\ndarknet 8\u20139, 134\u201349, ,\n\nbranding and marketing on 140\u201341\n\nexit scams 136\u20138\n\nDarkNet Market Avengers 148\u20139\n\nDarkNetMarkets\n\nDarling, Kate\n\nDarwin, Charles\n\ndata breaches\n\ndata-mining 171\u20132,\n\nDatecheck\n\ndating, online 6-7,\n\nDe Beers\n\ndecentralization 223\u20134\n\nDecentralized Autonomous Organization (DAO) , 219\u201321, 226\u201330,\n\ndecision-making\n\nDeep Blue\n\nDeepwater Horizon oil spill\n\nDegas, Edgar\n\nDeliveroo\n\nDeloitte\n\ndementia 192\u20133\n\nDeSteno, David\n\nDeutsch, Morton\n\ndiamonds 231\u20135,\n\ndiesel cars ,\n\nDigital Asset Holdings (DAH) 240\u20131, ,\n\ndigital disruption\n\ndigital wallets\n\nDigix\n\ndiscrimination 95\u20136\n\nDiscusFish 215\u201316\n\nDisney\n\ndoctors 27\u20138,\n\nDoe, John 34\u20135\n\ndomestic staff 108\u201311; _see also_ childcare\n\nDomino's\n\nDoNotPay\n\ndoublespeak\n\n_Downton Abbey_\n\ndriverless vehicles _see_ cars, self-driving\n\ndrugs (illegal)\n\non the darknet 134\u201349\n\nethical branding\n\nquality of 140\u201341, , 148\u20139\n\ndrugs (medical) ,\n\npricing ,\n\n_Drugs on the Dark Net_ (Martin)\n\nDubai\n\nDunbar, Robin\n\nDunton, Marc ,\n\nEagle Rising\n\neBay , 25\u20136, , , ,\n\necho chambers , 45\u20137\n\n_Economist, The_ ,\n\neconomists, lack of trust in 49\u201350\n\nEdelman, Ben\n\nEdelman, Richard , , 249\u201350\n\nEder, Kerstin\n\nEggers, Dave ,\n\nEinstein, Albert\n\nelection 2016 (USA) 104\u20135\n\nelites, declining trust in , , 49-50, 115\u201316,\n\nELIZA program\n\nemotional contagion 100\u2013103\n\nemployment\n\naffected by trust ratings 160\u201391\n\nof robots 190\u201391\n\nEnergy Control\n\n_Energy_ magazine\n\nengineering\n\nEnron\n\nenvironmental issues\n\nEqual Employment Opportunity Commission (EEOC)\n\n_Equivalent VIII_ (Andre) 63\u20134\n\nErisman, Porter\n\nErnst & Young\n\nescrow ,\n\nETHER (cryptocurrency)\n\nEthereum , 224\u20135, 227\u20139, ,\n\nEtsy ,\n\nEU referendum _see_ Brexit\n\nEuro\n\nEuropean Monitoring Centre for Drugs and Drug Addiction (EMCDDA)\n\nEverledger 234\u20135, ,\n\nEvolution 136\u20137,\n\n_Evolution of Cooperation, The_ (Axelrod) ,\n\nexit scams 136\u20138\n\nexpenses scandal, MPs'\n\nexperts, lack of trust in 49\u201350\n\n_Exxon Valdez_ oil spill 239\u201340\n\nEyal, Nir\n\nF2Pool\n\nFacebook , , , , , ,\n\nData Use Policy 101\u20132\n\nemotional contagion experiment 100\u2013103\n\nneutrality\n\nas a news source 46\u20137, 103\u20136\n\n_onlife_ personalities\n\nand social connectedness 121\u20132\n\n_see also_ 'fake news'; Zuckerberg, Mark\n\nfaces, and perceptions of trustworthiness\n\nfacts, 'alternative'\n\nFair, Bill\n\nfake goods 168\u20139, ,\n\n'fake news' 4\u20135, , 104\u20136\n\nFannie Mae ,\n\nFarmville (online shop)\n\nFederal Reserve\n\nfeedback , , , , , 161\u20134\n\non the darknet 142\u20133\n\nfake or padded 146\u20138,\n\nimpact of ,\n\npressure to leave high scores\n\n_see also_ reviews; Social Credit System (Citizen Scores)\n\nfei (currency) 203\u20134\n\nFerguson, Niall\n\nfiat system\n\nFICO\n\nFIFA ,\n\nFiges, Orlando 147\u20138\n\nfilter bubble effect\n\nfinancial crisis (2008) 1\u20132, 42\u20133,\n\nFinancial Crisis Inquiry Commission (FCIC)\n\nfinancial institutions, declining trust in 43\u20134\n\nFinley, Klint\n\nFinney, Hal\n\nFitbit ,\n\nFiverr\n\nFloridi, Luciano\n\nfollowers (social media) ,\n\nfood security 168\u20139\n\nfootball (soccer) ,\n\n_Forbin Project_\n\nFord\n\nformula milk, contaminated\n\nForster, David\n\nFreddie Mac\n\nfree speech\n\nfreelancers\n\nFreeman, Jon 113\u201314\n\nFreud, Sigmund ,\n\nFrey, Carl Benedikt\n\nFriedman, Milton\n\nfriends\n\nrating online 164\u20135\n\nand trust stimulation 119\u201320\n\nFriis, Janus\n\nFukuyama, Francis\n\nFuld, Dick\n\nfulfilment capacity\n\nFurness, William Henry III 203\u20135\n\nGallup\n\nGap\n\nGarside, Juliette\n\nGasc\u00f3n, George\n\nGates, Bill 181\u20132,\n\nGebbia, Joe , , , ,\n\nGeneral Foods\n\n'gig' workers\n\nGitHub\n\nGlobus, Amy\n\nGoldman Sachs , 241\u20133\n\nGood, Irving John\n\nGoodwin, Tom\n\nGoogle , , ,\n\nGore, Al\n\nGove, Michael\n\ngovernment, trust in\n\nGPS tracking\n\nGrams\n\n_Great Degeneration, The_ (Ferguson)\n\ngreed\n\nGreyball software\n\n_guanxi_ ,\n\n_Guardian_ ,\n\nGuare, John\n\nGunnlaugsson, Sigmundur Dav\u00ed\u00f0\n\nguns , ,\n\nGuo, Yifu 211\u201315\n\nGupta, Alok 93\u20134\n\nhacking , , , , 227\u20138,\n\nHaldane, Andy\n\nHamacher, Adriana 177\u20138\n\nHangzhou 13\u201315\n\nHangzhou Telecom\n\nHanyecz, Laszlo\n\nHappn\n\nhard fork 228\u201330,\n\nHardin, Russell\n\nhate language\n\nHavocscope\n\nHawking, Stephen 181\u20132, ,\n\nHayes, Christopher ,\n\nHBOS\n\nhealthcare , 33\u20134,\n\nHeap, Imogen\n\nHearn, Mike 243\u20134\n\nHeller, Jean 31\u20132\n\n_Her_\n\nHillis, Danny\n\nHinrikus, Taavet 76\u20138\n\nhitchhiking\n\nHoffman, Reid\n\nhomophily 46\u20137\n\nhonesty , 123\u20134, ,\n\n_Hooked_ (Eyal)\n\nhousing shortage\n\nHowells, James\n\nHuffington, Arianna\n\nHuffman, Steve 52\u20134\n\nHuman Right Watch China\n\nhuman rights organizations\n\nHussein, Saddam\n\nHyperledger\n\nIBM\n\nIceland\n\nidentity, digital ,\n\nimmunization 68\u201370\n\n_Inevitable, The_ (Kelly)\n\ninformation, asymmetry in\n\ninfosphere\n\ninfrastructure, social\n\nIngram, Paul\n\nInstagram , ,\n\nInstant Trust software 128\u201330\n\ninstitutions, trust in , 40\u201341, 47\u20138\n\nintegrity , , ,\n\nintellectual property\n\nInternational Consortium of Investigative Journalists (ICIJ) 36\u20138\n\ninternet , ,\n\n'right to be forgotten'\n\nInternet of Things (IOT)\n\nInternet Protocol\n\niPhone design 62\u20133\n\nIpsos MORI\n\nIraq War , ,\n\nIsaac, Earl\n\nIslam, Faisal\n\nIspa-Landa, Simone\n\nIto, Joi\n\nivory poaching\n\nJenner, Edward 68\u20139\n\nJentzsch, Christoph and Simon ,\n\nJeopardy\n\njob hunting 125\u20136\n\nJobs, Steve , 62\u20133,\n\nJohn Deere\n\nJones, James H.\n\nJones, Jeff\n\njournalism 36\u20138,\n\n_Joy of Tech_\n\nJP Morgan Chase 239\u201340,\n\nK\u00e4\u00e4rmann, Kristo\n\nKahn, Robert\n\nKahneman, Daniel\n\nKalamazoo, killings by Uber driver 81\u20135, , ,\n\nKalanick, Travis 85\u20136,\n\nKamel, Fawzi 85\u20136\n\nKant, Immanuel\n\nKarahalios, Karrie\n\nKarinthy, Frigyes\n\nKarpel\u00e8s, Mark Robert\n\nKarzai, Hamid\n\nKaufman, Sharon\n\nKelly, John\n\nKelly, Kevin\n\nKemp, Leanne 233\u20136,\n\nkickbacks\n\nKickstarter\n\nKim, Dong\n\nKimberley Process\n\nKing, David III\n\nKlout scores\n\nKnC Mine\n\nknowledge, collective\n\nKPMG\n\nKramer, Adam D. I. 100\u2013102\n\nKress-Gazit, Hadras\n\nKrueger, Frank\n\nLabaton Sucharow 43\u20134\n\nLacey, Sara\n\n_Lady, The_ 108\u20139, ,\n\nLagerkvist, Johan\n\nLanchester, John\n\nLathrop, Brian 71\u20133,\n\nLava soap\n\nlawyers 115\u201317,\n\nledger technology _see_ blockchain\n\nLee, Elvis\n\nLeeson, Nick\n\nLegout, Arnaud\n\nLehman Brothers ,\n\nlemons, market for\n\nLes, Jason\n\nLewis, Ken\n\nLibor scandal\n\nLibratus (robot poker player) 182\u20134\n\nlicensing rules\n\nlife expectancy\n\n'likes' _see_ feedback\n\nLinkedIn , ,\n\nLinux ,\n\nLishi, Lao\n\nloans , , 246\u20139\n\npeer-to-peer\n\nselling risk\n\nLocomotive Act\n\nLoeffler, Charles\n\nLondon Stock Exchange\n\nloss aversion\n\nLouvre\n\nlove\n\nLowercase Capital\n\nloyalty schemes\n\nLuhmann, Niklas\n\nMa, Jack 11\u201316, 20\u201324, 29\u201330,\n\nMachiavelli, Niccol\u00f2\n\nMadoff, Bernie , ,\n\nMaghribi traders 26\u20137, , ,\n\nMajid, Rasul 168\u201370\n\nmarketing 90\u201392, 146\u20137; _see also_ advertising\n\nmarriage ,\n\nMarriot hotels 91\u20132\n\nMars\n\nMartin, James 137\u20139, ,\n\nMashita, Ichiro\n\n_Mastering Bitcoin_ (Antonopoulos)\n\nMasters, Blythe 239\u201342\n\nMattel\n\nMazzella, Fr\u00e9d\u00e9ric 57\u201360,\n\nMcAuley, Daniel\n\nMcCullough, Nicole\n\nMcFarland, Matt\n\nMeadows, Mark 188\u201390\n\nmeat 87\u20138\n\nmedia, trust in ,\n\nmedical ethics ,\n\nmedical research 31\u20132\n\nMedici Bank\n\nMellen, Matt ,\n\nMerrill Lynch\n\nMessi, Lionel ,\n\nMichael, Emil\n\nMicrosoft, rogue chatbot 185\u20138\n\nMiddle Earth\n\nMilgram, Stanley\n\nMill, John Stuart\n\nmillennials, distrust of institutions\n\nmisogyny 52\u20133,\n\nMIT Media Lab ,\n\nMitchell, Jason\n\nMMR vaccine\n\nMobotiq\n\nModel T Ford\n\nmonarchy\n\n_Monde, Le_\n\nmoney , 203\u20136\n\ntransferring overseas 76\u20138\n\nmoney laundering ,\n\nMonuments Men\n\nMOOCs (Massive Open Online Courses)\n\nmoon landing\n\nMoore, Andrew\n\nMoore, Michael\n\nmoral duty 196\u20137\n\nMori, Masahiro 191\u20132\n\nMossack Fonseca , ,\n\nMPs' expenses scandal\n\nMr Nice\n\nMt Gox ,\n\nMurnighan, John Keith 119\u201320\n\nMurphy, Laura W.\n\nmuseums\n\nMusk, Elon , 181\u20132,\n\nmySociety\n\nNakamoto, Satoshi 206\u20137, 208\u201310, ,\n\nNappez, Francis\n\nNASA\n\nNasdaq\n\nNational Press Club\n\nNational Roads and Motorists' Association (NRMA) 18\u201319\n\nNational Securities Clearing Corporation (NSCC)\n\nNational Security Agency (NSA)\n\nNazis, art stolen from Jews\n\nNBC\n\nNest Learning Thermostat\n\nNetflix ,\n\nNetscape\n\nneuroscience\n\nNew Balance\n\nNew York Stock Exchange (NYSE) ,\n\nnews, online ,\n\nnewspapers, decline of\n\nNike\n\nNixon, Richard\n\nNobel Peace Prize\n\nNorthern Rock\n\nNorton, Ed\n\nnurses\n\nObama, Barack ,\n\nObama Care\n\nObermayer, Bastian 34\u20136,\n\nOccupy Democrats\n\nOECD\n\nOffice for Human Research Protections\n\nO'Hagan, Andrew\n\nOhanian, Alexis\n\nOmidyar, Pierre\n\nO'Neil, Cathy\n\nO'Neill, Onora, Baroness 112\u201313,\n\n_Onion, The_\n\nonion sites _see_ darknet\n\n_onlife_\n\nonline payments 59\u201360,\n\nonline profiles\n\nOpen Ledger Project\n\nopen-source projects\n\nOpenTable\n\nOrwell, George ,\n\nOsborne, Michael\n\nOthello\n\nOxford, University of , , ,\n\nOxo\n\nOzcoin\n\nPacioli, Luca Bartolomeo de\n\nPalmer, Stephanie\n\nPan Paper Village\n\nPanama Papers , 34\u20139,\n\nPandora\n\nPariser, Eli\n\nPatreon\n\npayment systems , 59\u201360,\n\nPayPal\n\nPeeple 164\u20135,\n\npeer-to-peer lending\n\npensioners\n\nPepper (robot)\n\nPerkins, Lynn 117\u201320, , , 132\u20133,\n\nPeterson, Ron\n\nPetrobras scandal\n\nPew Research Center\n\npharamaceutical pricing ,\n\nphilosophy\n\nPhipps, James\n\nPicasso, Pablo\n\nPipe, Anthony\n\npizza\n\npoisonings\n\npoker 182\u20133\n\nPolar, Hernando de Soto\n\npolitical campaigns, funding\n\nPolonsky, Rachel\n\nPope Francis\n\npornography ,\n\nPoroshenko, Petro\n\npost-truth ,\n\nPot Day\n\npredictive analysis\n\nPricewaterhouseCoopers (PwC) , ,\n\nprivacy\n\nProcter and Gamble\n\nprovenance 232\u20135\n\nof art works 235\u20136\n\nPutin, Mariya\n\nPutin, Vladimir, links to Panama Papers ,\n\nPutnam, Robert 28\u20139\n\nQixcoin\n\nQuan, Wen\n\nQueen Elizabeth II ,\n\nR3CEV\n\nracism , 52\u20133, 91\u20136\n\nratings _see_ feedback; reviews\n\n_Raubkunst_\n\nRavenell, Joseph 33\u20134\n\nReddit , 52\u20134, , ,\n\nrefugees ,\n\nRei, Zhu\n\nreliability ,\n\nreligion\n\nreputation, and trustworthiness 144\u20136\n\nreputation capital\n\nreputation trails\n\nrestaurants\n\nreviews 8\u20139,\n\ndeceptive or fake 146\u20138\n\n_see also_ feedback\n\nRice Krispies\n\nride-sharing 56\u201361,\n\n_Right and the Good, The_ (Ross)\n\nrisk\n\naversion to 74\u20135\n\nrobots 180\u2013202\n\nas dementia companions 192\u20133\n\nethical issues 185\u20138, 196\u2013201\n\nfictional 180\u201381,\n\nand the future of human employment 190\u201391\n\ngiven human-like qualities 177\u20138\n\nfor human assistance 192-6\n\nresponsibility for 188\u201390,\n\ntrusting 177\u20138, 190\u201391, 194\u20135\n\nused in medical diagnosis\n\n_see also_ artificial intelligence (AI); chatbots\n\nRogers, Brishen\n\nRoldugin, Sergei\n\nRoomba\n\nRoss, William David 196\u20137\n\n_R.U.R._ (Capek)\n\nRushkoff, Douglas\n\nRyle, Gerard 36\u20138,\n\nSacca, Chris 65\u20136\n\nSa\u00efd Business School\n\nSainsbury's\n\nSalvation Army\n\nSandholm, Tuomos\n\nSantander InnoVentures\n\nSarma, Anish Das 128\u20139\n\n'The Satoshi Affair' (O'Hagan)\n\nSaudi Arabia\n\nscandals, and erosion of trust 40\u201341\n\nScheutz, Matthias\n\nSchlessinger, Lucas\n\nSchneiderman, Eric\n\nscience fiction 180\u201381,\n\nScientology\n\nScrabble\n\nSegment.io\n\nself-driving cars _see_ cars, self-driving\n\nService, Robert\n\nSesame Credit 152\u201360\n\nSETLcoin 242\u20133\n\nsettlement lag\n\nSharif, Nawaz\n\nshopping habits, used for data-mining 152-4, 171\u20132\n\nShteyngart, Gary\n\nShuka, Anshul\n\nSicily 26\u20137\n\nSilk Road 135\u20136, , 140\u201342, , ,\n\nSilkRoad4\n\nSilvercrest\n\nSiri , ,\n\nSiroya, Shivani 247\u20139\n\n'Six Degrees of Separation' theory\n\nskeuomorphism\n\nSkype\n\nslack (app)\n\nSlideshare\n\nSlock.it ,\n\nslogans\n\nsmallpox 68\u20139\n\nsmart contracts ,\n\nSmith, Adam\n\nSmith, Nathan\n\nSmith, Tyler and Rich\n\nSnowden, Edward\n\nsocial capital 28\u20139\n\nSocial Credit System (Citizen Scores) 150\u201360, 168\u201370, ,\n\nbeing above the system\n\npunishments for low ratings 160\u201361, 166\u20137\n\nsocial graph\n\nsocial media\n\nextreme vetting of 173\u20134\n\ninfluential users\n\nrewards of disclosing information\n\nright to delete\n\nsocial networks , , ,\n\nsocial proof 78\u20139\n\nsocial translucence 99\u2013100\n\nsock puppetry\n\nSolidity\n\n_SonntagsZeitung_\n\nSony\n\nSpicer, Sean\n\nSpielberg, Steven\n\nSpotify ,\n\nStack Overflow\n\nStarbucks\n\nstealth, by darknet sellers\n\nSteinberg, Tom\n\nSteiner, Peter 127\u20138\n\nStiglitz, Joseph\n\nstrangers, fear of 55\u20136, 58\u20139, 66\u20137\n\nStreet Corner Experiment\n\n_S\u00fcddeutsche Zeitung_\n\nSullivan, Joe\n\nSummers, Larry\n\n_Super Sad True Love Story_ (Shteyngart)\n\nsupply chains 232\u20133,\n\nSurowiecki, James 78\u20139\n\nsurveillance, online 151\u201361, 171\u20134\n\nsushi\n\nSvakom Siime Eye\n\nsyphilis 31\u20132\n\nSzabo, Nick 209\u201310\n\nTala 246\u201350\n\nTamir, Diana\n\nTanz, Jason\n\nTao, Hu\n\nTaobao ,\n\nTarget\n\nTaskrabbit\n\nTate\n\nTate Modern\n\nTay 185-8,\n\ntax avoidance , 38\u201340\n\ntaxi drivers ; _see also_ Uber\n\nteachers, rating\n\ntechnology, trust in , , 177\u20139\n\nTED talks , , ,\n\nTencent ,\n\n_Terminator_\n\nTesco\n\nhorsemeat scandal 87\u20138,\n\nTesla 50\u201351,\n\nself-driving cars ,\n\nTexas Hold 'Em\n\nThalmann, Nadia 192\u20133\n\nTinder , ,\n\nTingle\n\nTmall ,\n\ntobacco\n\nTokyo Kaikan\n\nTor\n\nTorvalds, Linus\n\ntrademarks\n\ntraffic regulations\n\nTransactive Grid\n\nTransferWise 76\u20139\n\n_Transformers_\n\nTransmission Control Protocol\/Internet Protocol (TCP\/IP)\n\nTripAdvisor ,\n\ntrolling 53\u20134\n\nTrooly 128\u201332,\n\nTrump, Donald , 47\u20138, 104\u20135,\n\nextreme vetting proposals 173\u20134\n\nas a Washington 'outsider'\n\ntrust\n\naffect-based\n\ncognition-based\n\ndefinitions of 16\u201320\n\ndistributed 8\u20139, 50\u201351, , , 252\u20135,\n\ngeneralized ,\n\nhierarchy of needs\n\ninstitutional ,\n\ninterpersonal\n\nlocal ,\n\npersonal ,\n\nshared\n\nspeed and ease of\n\nsystem\n\n'trustless'\n\n_Trust_ (Fukuyama)\n\n_Trust and Trustworthiness_ (Hardin)\n\nTrust Barometer ,\n\ntrust blockers ,\n\ntrust deficits ,\n\ntrust engineers , ,\n\ntrust games 119\u201320\n\ntrust gaps\n\ntrust influencers , ,\n\ntrust leaps 24\u20135, , , ,\n\ntrust pause ,\n\ntrust scar\n\ntrust score\n\ntrust shift\n\ntrust signals ,\n\ntrust stack 60\u201361, , ,\n\ntrust vacuum\n\ntrusted intermediaries\n\nTrustPass\n\ntrustworthiness 112\u201313, 122\u20135, ,\n\nand appearance\n\nassessing online 128\u201333\n\nof citizens 151\u20136\n\non the darknet 140\u201341\n\nof machines 190\u201391, 194\u20135\n\nTrusty, Stuart\n\n_Truth About Trust, The_ (DeSteno)\n\nTual, Stephen\n\nTufecki, Zeynep\n\nTuring, Alan\n\nTuring Pharmaceuticals\n\nTuring test 181\u20132\n\nTuskegee Study 31\u20133, ,\n\nTversky, Amos\n\n_Twilight of Elites_ (Hayes)\n\nTwitter , , , ,\n\nsharing unread links 98\u20139\n\n_Twitter and Tear Gas_ (Tufecki)\n\nUber 6\u20138, , , , , ,\n\n#DeleteUber protest\n\nabuse and harassment allegations 85\u20137\n\nGod View\n\nKalamazoo killings by driver 81\u20135, ,\n\nsurveillance of customers\n\nterms of service 86\u20137\n\ntrust and safety issues 81\u20136, ,\n\nUjo music\n\nUlbricht, Ross William 135\u20136,\n\n'Uncanny, The' (Freud)\n\nuniforms , 114\u201315\n\nUnilever\n\nunions\n\nUnited States Public Health Service\n\nUpCounsel 116\u201317\n\nUpworthy ,\n\nUrbanSitter 117\u201319, , , 125\u20137, , ,\n\nUS dollar\n\nUS Naval Research Laboratory\n\nUS Transportation Security Administration (TSA)\n\nUslaner, Eric\n\nutilitarianism ,\n\nvaccinations 68\u201370\n\n_Vanity Fair_\n\nVasdey, Manisha\n\nVastari\n\nVBN\n\nVerbrugge, Peter\n\nVico, Giambattista\n\nVietnam War\n\nVigna, Paul\n\nVisa\n\nVolkswagen , ,\n\nvulnerability 19\u201320\n\nWachovia\n\nWakefield, Andrew\n\nWall Street 1\u20132,\n\nWalmart\n\nWalport, Mark ,\n\nWanamaker, Marianne\n\nWang, Maya\n\n'war on terror' 44\u20135\n\nWashington Mutual\n\nwater, bottled\n\nWatergate\n\nWaymo\n\nweapons of mass destruction (WMDs) ,\n\n_Weapons of Math Destruction_ (O'Neil)\n\nWebber, Anne\n\nWeChat ,\n\nWei, David\n\nWeibo\n\nWeiner, Seth\n\nWells Fargo\n\nWestern Union\n\n_Westworld_\n\nWe-Vibe\n\n_What's Mine is Yours_ (Botsman) ,\n\nWhisky Ring\n\nWIIFM factor (what's in it for me?) ,\n\nWikiLeaks ,\n\nWikipedia ,\n\nWilpon, Fred\n\nWinterkorn, Martin\n\nWintris Inc.\n\n_WIRED_ ,\n\n_Wisdom of Crowds, The_ (Surowiecki)\n\nWorld Economic Forum ,\n\nWorld Trade Center, 9\/11 attacks\n\nWorld Wide Ledger (WWL)\n\nWright, Craig Stephen\n\nXiang, Meing\n\nYahoo\n\nYap ('The Forbidden Island') 203\u20135\n\nYeats, W. B. 251\u20132\n\nYelp , ,\n\nYingyun, Li\n\nYouGov\n\nYouTube , ,\n\nYunus, Muhammad\n\nZajonc, Robert B.\n\nZak, Kyle\n\nZennstr\u00f6m, Niklas\n\nZhang, Ng\n\nZhiqiang, Wang 22\u20133\n\nZhongguancun\n\nZuckerberg, Mark 105\u20136, , ; _see also_ Facebook\n\nPublicAffairs is a publishing house founded in 1997. It is a tribute to the standards, values, and flair of three persons who have served as mentors to countless reporters, writers, editors, and book people of all kinds, including me.\n\nI.F. STONE, proprietor of _I. F. Stone's Weekly_ , combined a commitment to the First Amendment with entrepreneurial zeal and reporting skill and became one of the great independent journalists in American history. At the age of eighty, Izzy published _The Trial of Socrates,_ which was a national bestseller. He wrote the book after he taught himself ancient Greek.\n\nBENJAMIN C. BRADLEE was for nearly thirty years the charismatic editorial leader of _The Washington Post._ It was Ben who gave the _Post_ the range and courage to pursue such historic issues as Watergate. He supported his reporters with a tenacity that made them fearless and it is no accident that so many became authors of influential, best-selling books.\n\nROBERT L. BERNSTEIN, the chief executive of Random House for more than a quarter century, guided one of the nation's premier publishing houses. Bob was personally responsible for many books of political dissent and argument that challenged tyranny around the globe. He is also the founder and longtime chair of Human Rights Watch, one of the most respected human rights organizations in the world.\n\nFor fifty years, the banner of Public Affairs Press was carried by its owner Morris B. Schnapper, who published Gandhi, Nasser, Toynbee, Truman, and about 1,500 other authors. In 1983, Schnapper was described by _The Washington Post_ as \"a redoubtable gadfly.\" His legacy will endure in the books to come.\n\nPeter Osnos, _Founder_\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n# **WHEN YOU'RE FALLLING, DIVE**\n\n**_Lessons in the Art of Living_**\n\n**MARK MATOUSEK**\n\n## **CONTENTS**\n\nCover\n\nTitle Page\n\nDedication\n\nIntroduction\n\nMagical Thinking\n\nThe Roar of Freedom\n\nThe Day of Laughter\n\n_Om Mani Padme Hum_\n\nDragons at the Gate\n\nSuperman's Ghost\n\nHome in the World\n\nThe Art of Losing\n\nA Quarter Inch from Heaven\n\nThe Girl on the Rock\n\nGoing to Tahiti (or Raising Heaven)\n\nThe Net of Indra\n\nReinventing Your Wife\n\nMan Thinks, God Laughs\n\nPraying\n\nDemon Lovers\n\nQuestioning (or The Sphinx)\n\n_Je M'en Foutisme_\n\nThe Terrorists Within\n\nEarth Angel\n\nThe End of Seeking (or Dig in One Place)\n\nSomething Else Is Also True\n\nPain Passes, but the Beauty Remains\n\nHedonics\n\nInvisible Feast\n\nOriginal Blessing\n\nEnough\n\nStress Matters\n\nThe Wounded Healer\n\nTrue Confessions\n\nPrometheus\n\nFound Art\n\nThrough Wilderness\n\nA Splinter of Love\n\nNakedness\n\nKilling Peter Pan\n\nThe Water or the Wave\n\nWhat Time Is Good For\n\nAt Sea\n\nThe Mother (or Aloha Oy)\n\nRope Burn\n\nWhat Makes the Engine Go?\n\nNotes and Sources\n\nAcknowledgments\n\nA Note on the Author\n\nBy the Same Author\n\nImprint\nFor Marcia Horowitz (1948\u20131978) and Marco Naguib\n_Utram bibis? Aquam an undam?_  \nWhich are you drinking? The water or the wave?\n\n\u2014John Fowles, _The Magus_\n\n## **INTRODUCTION**\n\nOne afternoon when I was twenty, my eldest sister, Marcia, appeared at my front door, needing to ask an important question.\n\n\"What is it?\" I asked, shocked by her appearance. In spite of the unseasonable L.A. heat, Marcia was wrapped in a bulky Mexican sweater tightly belted at the waist (just like a crazy person, I thought), her dark hair unkempt, eyes bloodshot: a dramatic decline from the attractive thirty-year-old banker she had been only months before. I sat Marcia down at my kitchen table, poured her a cup of tea, found a comb in my back pocket to run it through her messy hair, as she had once fussed over me as a boy. She had been my surrogate mother\u2014ours wasn't really up to the job\u2014 reading me Aesop's fables before bedtime, packing lunches, explaining riddles (birds, bees, our disappeared father), comforting when my feelings were hurt. Now it was my turn to comfort Marcia. Her spirits had sunk so quickly that winter. I kissed my sister's cheek and asked her to please tell me what was the matter. Marcia seemed unable to speak, simply shook her head and drifted away to that no-man's-land where none of us had been able to reach her.\n\nMarcia had been viciously betrayed by her husband, filed for divorce, had a nervous breakdown. She was hospitalized, then released prematurely when her insurance company refused to pay more. Now she found herself alone, unmoored, and terrified in a world that had always distressed her. The mensch in a family of hooligans, Marcia was the gentle, obedient daughter who did as she was told without question and cared for others more than herself. Newspaper tragedies sent her to bed. She wept for people she didn't know. I never once witnessed her being cruel.\n\nMy sister appeared to be sinking fast, unable to locate solid ground. I begged her again to talk to me. At last, Marcia looked into my eyes and spoke. \"How do you do it?\" she asked.\n\n\"Do what, honey?\" I was confused.\n\n\"How do you live?\"\n\nThe question froze in the air between us. In every life there are red-flag moments that seem to flash out, magnified, against the soft focus of everyday contentment, warning us to pay attention\u2014that something essential is happening. I snapped to attention when Marcia said this yet had no earthly idea what to tell her. I was a wobbly piece of work myself at that age; the bravado my sister seemed to admire was largely a mask for bitter self-doubt. Having grown up as the only son in a fatherless, four-kid, welfare home, I believed that denial and monolithic ambition were the only tools at my disposal for surmounting such disadvantaged beginnings. Pessimism seemed to be evil juju that winners must avoid at all costs. Failure wasn't even an option, of course. Yet here was my beloved, defeated sister posing a heartbreaking question I hadn't yet dared to ask of myself.\n\nI told Marcia that she had to keep fighting. No matter what.\n\n\"I can't,\" she said.\n\n\"You have no choice...\"\n\nMarcia opened her mouth to answer\u2014then stopped. The kitchen fell silent; she slumped in her chair; that was the end of the conversation. I tried to distract her by rubbing her shoulders. She drifted away again. When she stood up to leave, I felt guilty relief as I followed her out to her beat-up Buick, parked hastily at the curb. Marcia fumbled in her purse for the car key, then sat there without moving, hands clutching the wheel. I asked Marcia if she was okay to drive. She stared at me without answering.\n\nI leaned down to kiss her good-bye. \"You'll be fine,\" I promised.\n\nMarcia touched the side of my face. She attempted a smile, which wasn't much but gave me a bit of hope. Then she started the engine, waved good-bye, and disappeared slowly down the street.\n\n\"How do you live?\" Marcia had asked. This question would haunt me in years to come\u2014following some great loss or disappointment, some piece of unexpected bad luck\u2014when I was exposed enough to wonder. How does a person survive his own life, the ceaseless surprises, uncertainties, struggles, reroutings in strange, inconvenient directions? What force is it, exactly, that flips a falling man back on his feet, reconstitutes him after disaster, helps him prevail in the face of challenges far beyond his previous limits? What mysterious strength is it that enables us to outsmart \"the terrorists within\" (as one psychologist described them to me), those destructive maniacs under the skin\u2014cynicism, despair, resignation, terror\u2014that threaten to stop us in our tracks? Finally, how is it possible not merely to survive our greatest obstacles but to prevail in circumstances that threaten to stop us?\n\nThis quandary intensified substantially for me when mortality paid an unwelcome visit. One afternoon in 1984, lying on a Jamaica beach with my best college buddy, John, I found a purple lesion on the ball of his foot that had not been there the day before. Lives, like buildings, have foundation walls; remove these crucial supports and the whole thing comes crashing down around your ears. In the instant that I saw John's lesion, in the seconds it took me to realize what it probably meant for both of us\u2014though our friendship had been platonic\u2014life as I'd known it cracked down the middle from chimney to basement; the house I'd lived in, the self I'd believed in, the future I thought was waiting for me, was suddenly condemned.\n\nJohn was dead within three months. For the next ten years, I lived in a state of near-constant anxiety, waiting for my own demise, scared each morning when I looked in the mirror of what forbidding signs I would find there. Mostly, I couldn't breathe. In her famous essay on affliction, Simone Weil likened this acute state of dread to that of a condemned man forced to stare for hours at the guillotine that is going to cut off his head. You shake, you wait, you do your best not to pee in your pants while people are watching.\n\nBut no one can stay panicked forever. You're forced to find a way through your terror before any viable answers appear, while the floor is dropping beneath you. It's a sloppy, lurching, imperfect business; though not quite yourself, post-catastrophe, neither are you yet equipped to cope with the fallout from such seismic changes. You haven't grown those muscles yet or begun to reimagine your story. You're more like the common American lobster, _Homarus americanus,_ which dives for a few days each year to the ocean floor to slough off its old shell and wait for a new one, a naked, pink-skinned glob of flesh trying not to get smashed too hard before its second skin grows back.\n\nIn my case this retreat took the form of spiritual seeking. When John died, I was working as an editor at _Interview_ magazine under the pop artist Andy Warhol. I'd chased the publishing carrot from L.A. to New York, worked my way slavishly up the mast-head, yet found that in my frenzy to succeed, I'd ignored any sort of inner life. Now the prospect of leaving this world with so little clue as to who I was, what (if anything) this life meant, whether I believed in God, the soul, or self-transcendence, felt like adding insult to injury\u2014like a sleepwalker stumbling off a cliff. I quit my job, left New York, and shifted gears from limousine chasing to authenticity (whatever _that_ was) and confronting a problem I'd long avoided: Why, in spite of my worldly good luck, did I feel so secretly heartsick and vacant, so like an impostor, long before my diagnosis? Why was I so rarely truly happy? This question was an urgent one whether I was dying or not. I became a compulsive, nomadic seeker, living hand to mouth for most of ten years, trawling for wisdom from spiritual teachers around the world (wherever plastic was accepted), gulping it down in desperation.\n\nThese years were traumatic yet eye-opening. I kept a saying in my wallet\u2014\"In a world of fugitives, the man who goes in the opposite direction will always be said to be running away\"\u2014that seemed to let me off the hook. I needed to find out for myself if anything existed of a man beyond this booby-trapped bag of bones. As a skeptical agnostic, I dove warily into the question of whether anything _meta_ physical truly existed, or if human beings really were just blips on the screen of a heartless creation, dying animals like a billion others born to eat and fuck and die, as I had been raised to believe. This philosophical question had struck me for the first time at age seven, when I lifted the lid off a garbage can and found my first dead thing, a blue jay's corpse swaddled in plastic and newsprint. I stood there a long time, staring down at the cat-chewed body, wondering whether the bird had been this pile of bloody feathers or the thing that had escaped? Which, more pertinently now, was I?\n\nThe primary insight that arose from those years had nothing at all to do with religion. In fact, the single most transformative idea to emerge from all that reading, meditation, and ashram-shlepping was simply this: that terror can be a door to enlightenment. While traditional cultures have long understood the empowering aspects of fear and wounding, the double-edged force of passage rites to galvanize and deepen the spirit, we are too often shielded from this secret knowledge. Our prevailing contemporary view of pain and loss as handicaps to be avoided at any cost is not only wrongheaded but deeply ass-backward, in fact. Terror is fuel; wounding is power. Darkness carries the seeds of redemption. Authentic strength isn't found in our armor but at the very pit of the wounds each of us manages to survive. As one widow put it to me, \"Strength doesn't mean being able to stand up to anything, but being able to crawl on your belly a long, long time _before_ you can stand up again.\" Transformation is in our wiring. Looking backward is a Humpty Dumpty waste of time. \"You're gonna come out gold on the other side,\" one heroic man says in this book, \"or you're not gonna come out at all.\"\n\nThis is not Pollyanna speaking. Science is finally catching up to what sages have been saying forever. Thanks to recent breakthroughs in fMRI technology, neurologists are now able\u2014for the first time in history\u2014to observe the human brain in the act of feeling. This has revealed a phenomenon known as neuroplasticity, which in turn has revolutionized how we think about personal change. Once believed to be lumps of gray matter cogitating between our ears, our brains turn out to be more like interlooping Wi-Fi octopuses with invisible tentacles slithering in many directions at every moment, constantly picking up messages we're not aware of and prompting reactions\u2014including illness\u2014in ways never before understood. Contrary to the old wives' tale that humans are born with a fixed number of brain cells that only diminish over time, our bodies produce one hundred thousand new brain cells every day, in fact, until we die. Our brains are highly mutable, reinventing themselves on a regular basis, which is why _not_ putting pain to its natural use\u2014as grist for the evolution mill\u2014is such an extraordinary waste of suffering. While hardship can certainly render us bitter, selfish, defensive, and miserable, it can also be used quite differently: as the artery of interconnection, a bridge to other people in pain, as blood in the muscle that propels us. Crisis takes us to the brink of our limits and forces us to keep moving forward. When people in extremis call it a blessing, this is the paradox they are describing. It's why men sometimes blossom in wartime and women are often changed by childbirth\u2014they come alive as never before on that knife blade of danger and pain. There's vitality in facing life's extremes, including that of your own extinction. Crises pushes you to travel wide, fast, and deep, expands the heart and calls forth reserves of courage you didn't know you had, like adrenaline in the muscles of a mother saving her only child. Only you are the child, and it's your life\u2014 the life of your own soul\u2014that you are saving.\n\nThis paradox is hard to swallow. When I used to tell friends, half jokingly, that HIV had actually saved my life, they rarely understood what I meant. I wasn't promoting an awful virus or claiming I was glad to have it. I wasn't pretending to be overjoyed by the prospect of an early departure. I was simply confessing an odd bit of truth that I wouldn't have believed myself had my own life not improved so dramatically. Without the threat of mortal loss, I would never have had the conviction\u2014the fuel\u2014to become the person I wanted to be or to find my way through terrible dread to something stronger than my fear.\n\nOn April Fool's Day, 1996, my roller coaster took another twist with the news from my doctor that treatments for the virus had finally appeared. Paul Bellman, who looks like Vincent van Gogh if the painter had gone to a Brooklyn yeshiva\u2014 ginger-headed, intense, and bearded\u2014told me that although there were no long-term guarantees, it was now highly possible that I (and many thousands like me) might look forward to a ripe old age after all. I was, in other words, no longer dying, at least not yet and not from this illness.\n\nI stumbled out of Bellman's office feeling like the man who fell to earth\u2014reeling, confused, a little dizzy before the unmitigated expanse. A majority of survivors, regardless of their particular storm, recall this resurrection moment in eerily similar, jumbled-up ways, of being yanked back to earth once they thought they were leaving. There's numbness, thrill, and disbelief, joy intermingled with bittersweet panic. Aristotle was right when he compared good luck to the moment on the battlefield when the arrow hits the guy next to you. It's an abstract, outer-space, torn-in-half feeling, partly shattering, partly sublime. _Awe_ is the only word that fits.\n\nA gift returned is doubly precious, charged with the mandate not to be wasted. I was determined not to lose track of what I'd learned in the mortal zone, or forget the miraculousness of things in the blur of everyday life. For me, this became a new reason for living, to prove myself \"worthy of my sufferings,\" as novelist Fyodor Dostoevsky described his own resurrection moment. Arrested at age twenty-eight for revolutionary activities against Tsar Nikolai I, the impoverished epileptic author spent eight months in solitary confinement before his death sentence was announced. Marched into Saint Petersburg's Semenovsky Square on the teeth-rattling morning of December 22, 1849, along with twenty-three other condemned men, Dostoevsky stood in line for thirty minutes while awaiting his turn to climb the scaffold. \"They snapped swords over our heads and made us put on the white shirts worn by people condemned to death,\" he later wrote to his brother. \"Thereupon we were bound in threes to stakes, to suffer [our deaths].\"\n\nAt the very last moment, the writer was informed that his sentence had been commuted; in fact, the whole thing had been a scare tactic. Suddenly, Dostoevsky was a free man. A fellow inmate went mad after this incident, but Dostoevsky's own madness took a different form: to survive brightened, revivified, by his personal brush with death. Obsessed with the image of one man executed, the other reprieved, the novelist made a lifelong commitment not to waste his gift, to conduct his life in terms of \"what the condemned know,\" to become a survivor artist. \"When I turn back to look at the past, I think how much time has been wasted,\" he confessed in that same letter. \"Life is a gift... each minute could be an eternity of bliss. My old head has been cut from my shoulders... but my heart is left me. And the same flesh and blood which likewise can love and suffer and desire and remember. _On voit le soleil_!\"\n\nAs I retraced my own footsteps in order to understand this turnaround, Marcia's question came back to haunt me. How does a person live full out in a world where uncertainty's king? How do we survive our own days once a fall has already happened? How is it possible to live with \"what the condemned know\" before we are confronted by crisis, through the course of everyday affairs, in order to live in this illuminated state? Finally, how can we keep our hearts \"open in hell,\" as the Dalai Lama describes the existence of suffering, in the midst of trouble, when we need compassion most?\n\nSurvival, I came to understand, has less to do with cheating death than with living as brilliantly as possible. (How many oxygenated people do you know who do not appear to be fully alive?) There are new ways of seeing\u2014of being\u2014in a world turned upside down. There's an art to turning poison into a boon. Even during the worst of times, as I plummeted into some squall of fear, it never failed that some involuntary, irrational wave would sweep me up sooner or later as if out of nowhere. Other times there was satisfaction in the battle itself, a tenacious hunger to stay alive, when I was glad to simply be here, to fight\u2014to _participate_ \u2014even when the going was tough.\n\nEvery situation has the potential for this greening, this _viriditas,_ this bringing of beauty, insight, or healing from the manure of suffering. We're reconstituted after being crushed, fleshed out again after being stripped, by the mysterious, profligate, glowing, extravagant force that courses through and electrifies the phenomenal world. Though we're immersed in this power at every moment, survivors realize how profoundly quality of life is determined by how skillfully (and quickly) we harness ourselves to that evergreen force at the heart of things. Just as we recognize beauty when we see it, we respond to those who have tapped into this secret. They seem larger and more alive somehow, as if working from a more vivified palette. It's no accident that such impassioned, inventive people have frequently endured greater-than-usual doses of pain. Where adversity crosses paths with aliveness, there is the potential for art, as well as for artful living. Although we may not all be masters, we can learn from such enlightened souls and witness their greening survivor's genius for self-renewal.\n\nWhen the hundred-year-old poet Stanley Kunitz tells me, for example, \"Reinvention is my philosophy,\" how different is that from painter Auguste Renoir assuring students, \"Pain passes, but the beauty remains\"? When Dr. Rachel Remen, a physician who's lived with a painful physical condition for fifty years, promises that \"optimism is not required for healing,\" how liberating is that for those who've scrambled to keep their own smiley-face masks in place when what they needed to do was scream? Seeking out masters of this art, following their various footsteps through their particular labyrinths, I saw there might be a way to compose a chorus\u2014the aboriginals call it a song line\u2014to serve as a map for people crossing their own wilderness, a voice map to this dark geography. \"In a dark time, the eye begins to see,\" poet Theodore Roethke reminded us. With our illusions of safety exploded, outside the bounds of \"normal\" life ( _way_ out on King Lear's stormy heath), new abilities do indeed dawn in a person; values, intuitions, skills, perspectives that might seem unnatural\u2014even perverse\u2014to those who've led more sheltered lives.\n\nBut what do people like me have to teach? Something about the limitations of rational thought; our supreme powers of adaptation; the short shelf life of shock; the fraternity of the _anawim_ (people who have experienced great loss); the enlargement of self through tragedy; the absence of guilt; the paradoxical thrill of endangerment; the actual flimsiness of the narratives each of us composes for his life, believing the dots to be connected (when, in fact, they're just dots); the paltriness of survival without spirit; the role of verve and imagination\u2014what the French call _je m'en foutisme,_ the brave art of not giving a damn once your back is already against the wall and you choose to move forward anyway.\n\nAwe is so much vaster than pleasure. It is boundless, transcending clich\u00e9d ideas about good and evil, pleasure and pain, success and failure, redemption and loss. The line blurs between good and bad luck; bipolar Fortune spins her mad wheel, reverses, slows down a moment, then speeds up again in her maddening round. Survival requires a dose of madness\u2014what cynics call \"hoping against hope\"\u2014just like art does; you conjure your future from white space, locate the hidden person, yourself, against this unfamiliar background, peering through grief and loss at something greater. \"Survivors are more urgently rooted in life than most of us,\" observed one Holocaust expert. \"Their will to survive is one with the thrust of life itself, as stubborn as the upsurge of spring. A strange exultation fills [their] soul, a sense of being equal to the worst.\"\n\nI knew this strange exultation myself, of being destroyed yet beyond destruction. Mystical as this may sound, even the hardest-headed atheists I spoke with would echo some version of this same awareness, a retrieved sense of sacredness, even transcendence, flooding the vacuum of survival where common protections had been stripped away. Terror does have its purposes. \"If I cut you,\" Rachel Remen reminded me, \"your entire healing system\"\u2014physical, emotional, spiritual\u2014\"will be mobilized instantly and become more alive in you, more activated than before.\" For this reason, survivors may be our greatest teachers in an increasingly terrorized world.\n\nAfter the wounding by Al Qaeda, Americans watched our country morph overnight from a dozing, not overly conscious giant into a great hydra-headed survivor caught up in the fight of its life. Foreseeing the dangers of power and comfort, Thomas Jefferson had favored revolution every ten years to snap people to attention. Americans were now being called upon to ask themselves the sorts of questions survivors wrestle with every day\u2014to think _heroically,_ although they might not want to be heroes. This heroism has less to do with John Wayne, the default machismo of a people weaned on pioneer individualism, than with throwing off our bonds of fear, deepening through our so-called weaknesses, surrendering to \"the thing with feathers\" that lifts us beyond who we thought we were (and what we are capable of) and rouses us from our perpetual stupor.\n\nWe spend so much of our lives in a waking trance of retrospection, regret, distraction, idling, and disembodiment through a range of addictions that when the bottom does fall out, life assumes a sudden, counterintuitive richness, clarity where there was haze. We're starched into the present tense, where we stop and gaze around with wonder at existence on a mysterious planet. Epiphanies happen where life and death meet. The very walls we construct to protect our lives hide the full glory of those lives from us. It's strange but true that we live immured by our own best intentions, and that the human mind is so constructed\u2014and our powers of denial so baroque\u2014that nothing short of catastrophe has the power to snap us out of our trance. Marcel Proust described this anomaly best, as usual. \"I think that life would suddenly seem wonderful to us,\" wrote the hypersensitive Frenchman,\n\nif we were threatened to die, as you say. Just think of how many projects, travels, love affairs, studies, it\u2014our life\u2014hides from us, made invisible by our laziness, which, certain of a future, delays them incessantly.\n\nBut let all this threaten to become impossible forever, how beautiful it would become again! Ah! if only the cataclysm doesn't happen this time, we won't miss visiting the new galleries of the Louvre, throwing ourselves at the feet of Miss X, making a trip to India.\n\nThe cataclysm doesn't happen, we don't do any of it, because we find ourselves back in the heart of normal life, where negligence deadens desire. And yet we shouldn't have needed the cataclysm to love life today. It would have been enough to think that we are humans, and that death may come this evening.\n\nConfronting our own impermanence is the skeleton key to a secret life. After surviving his own vision quest, before reentering civilian life, Carlos Castaneda was advised by his Yaqui teacher Don Juan to keep death on his shoulder. Mortality becomes a muse\u2014the notion that fear disappears is absurd. But how is it possible to stand up to the terrorists within? How do we live once the ground starts shaking? That is what Marcia wanted to know.\n\nToday I would tell her that there is an art. There are song lines, principles, signposts, tricks, shortcuts, ways of making it better. Plato described such lifesaving wisdom as _techne tou biou_ \u2014 meaning \"the craft of life\"\u2014which is why, in the spirit of arts-and-crafts wisdom, this book is divided into a sequence of lessons that can be read either in order or hopscotch style, depending on the reader's taste. Since a story is worth a thousand rules, I've opted for parables, poetry, true-life tales, and autobiography to illustrate this _techne tou biou._ These concise chapters are meant to be seen as pieces in a larger mosaic, movable parts, independent lessons in the nonlinear art of self-realization.\n\nAs the world becomes more dangerous our need for such lessons only grows more urgent. There are great survivor artists among us whose lives stand as proof of this transformative power. As the reader will discover in the following pages, their strength is a precious, renewable resource available to anyone willing to learn.\n\n## **MAGICAL THINKING**\n\nThe impossible happens to everyone\u2014when we're least expecting it. One minute, you're still the person you know; the next, you've been changed by some freak happenstance you never believed could happen to you.\n\nYou've become a post-catastrophe person. The mask of safety is torn away. There's nowhere to run and hiding is fiction. You search your own reflection for comfort, and the mirror whispers, _Adapt or die._ Feeling your way along in the dark, you dream your new life into existence, grow these strange new appendages slowly, see the world and your own being through unblinkered eyes.\n\nPicture a husband and wife, both writers, sipping cocktails at the end of a long workday, a week after Christmas in New York City. For forty-two years this has been their routine, living and working cheek by jowl\u2014editors, best friends, collaborators. On this particular evening, with snow attacking the windowpanes, John is reading by the fire while Joan, all ninety-five pounds of her, tosses a salad in the kitchen. Both of them are exhausted. Their only daughter, Quintana, is in a nearby hospital, suffering from septic shock following an infection gone wrong. After dinner, Joan and John go back to the hospital. Now it's time to eat and relax.\n\nJoan carries the salad into the living room. John closes his book and waits to be served. Joan is a woman who thrives on routine, as order is an antidote to dread. Mundane details form the daily grid behind which, if she is careful, Joan believes she can barricade her family and herself from the chaos she has glimpsed since girlhood, lurking under the surface of things. The daughter of pioneers (her great-great-great-grandmother brought a cornbread recipe and a potato masher across the plains from Arkansas to the Sierras), Joan was raised to be strong, and has always been stronger than she looks, but suffered nevertheless from a kind of hypersensitive dread, the intuition of hidden yet imminent danger. Joan negotiates this unease through writing, throttles her fears in tight, gimlet-eyed prose, and maintains her cool front in a frightening world. \"We tell ourselves stories in order to live,\" she wrote a long time ago, and now, lighting the candles for dinner, Joan may be thinking (for the thousandth time) of this man she has known for all these years, her husband, and how he actually understands her, allows for her insecurities, loves her anyway.\n\nJohn looks up from the table and catches his wife watching him. Joan hands him a second drink. Beginning to eat, John lifts his fork, slumps to the side, and dies there in front of his wife from a massive coronary infarction.\n\nThis happens so suddenly that Joan thinks it must be a joke at first and waits for her husband to open his eyes. When John doesn't move, she shrieks at him; then she's running toward the phone, dialing 911, holding his head in her lap, opening the door for the paramedics, watching them beat his chest with their paddles, following them with an overnight bag to the service elevator, then into the ambulance, where she rests John's cold hand on her lap and stares at traffic out the back window.\n\nAt the memorial service three weeks later, Joan watches their daughter, Quintana, deliver the eulogy, not knowing that in twenty months' time she will be attending this same child's funeral. Joan sits there, a sylph in oversized sunglasses, hazing in and out of her church seat, her mind unhinged by the suddenness of it, the unreality of the coffin. She feels as if a tectonic shift has split her former life in two, and during the spectral months to follow, pacing the ghost-lined apartment halls, able to eat nothing but soup broth, fixated, restless, somnambulant, she obsessively records the events of that night on paper, minute by minute, as if the creation of a precise time line will magically return John to her.\n\nLife changes fast.  \nLife changes in the instant.  \nYou sit down to dinner and life as you know it ends.  \nThe question of self-pity.\n\nThese four lines, written during that twilight period, will become the footprints Joan follows back into the story of her own life. During this process the detail-crazy mind she's relied on for work and sanity turns flagrantly magical on her (she can't move John's shoes from the doormat in case he should \"need them if he comes home\"), telling her stories she knows can't be true.\n\n\"It was extraordinary,\" Joan Didion tells me now. We're alone in her large Upper East Side apartment on a December day three years after the ordeal, the windowpanes once again crusted with snow. This literary heroine from my undergrad days is dressed like a prep school ing\u00e9nue in a baggy lavender sweater, flower print skirt, black tights, and knee-high mukluks. Joan speaks in a sort of gunslinger's drawl punctuated with frequent, lilting laughter and is cordial without being ingratiating, generous in her responses without wasting a word.\n\n\"I'd been around people who had psychotic breaks,\" she tells me, fixing me with her grayish eyes. \"But I had never imagined that someone who didn't appear to be raving mad could be so crazy.\" Joan arranges her skirt across her knee. \"I had lived, you see, an entirely conventional, bourgeois life. An orderly life. This broke into my golden mean.\"\n\nCut loose from any fixed idea she ever had \"about death, about illness... about marriage and children and memory... about the shallowness of sanity, about life itself,\" as she later wrote, Joan spent the year following her husband's death wading shoulder-high through grief, untangling the loops of her delusional thoughts, tracing her way back to solid ground. \"I've learned that you don't actually get over things,\" she says. \"You incorporate them. They become part of everything you are. I don't mean that you walk about crying all the time. But you change.\"\n\n\"How did you change?\" I ask. Having read her books, I'm already aware of Joan's tempestuous mood swings.\n\n\"I'm far less patient in certain ways.\" She uncrosses her legs and slouches forward. \"Not that I was ever one of life's most patient people.\" Joan lets out a childish laugh. \"But I do value love more.\"\n\nHadn't she always valued love? It had seemed to me that love was one of the few things Didion did trust. Feelings seemed to be everything to this writer. \"A poster girl for anomie wearing a migraine and a bikini to every volcanic eruption of the post-war zeitgeist,\" as _Rolling Stone_ described her. Red-hot emotions in ice-perfect prose, that was Didion's legacy. Love had always seemed sacred to her. Was that not the case?\n\nShe gracefully ducks the question. \"When I think of the moments that are wasted in all of our lives,\" she tells me. \"It is different after you lose someone. These moments become particularly precious to you when you can't replicate them.\" In _The Year of Magical Thinking,_ the memoir she wrote about this vastation, Joan describes the intricate, difficult bond she shared with her husband, the unvarnished truth of their marriage, the diurnal shifts, silences, arguments, the conjugal saga of rising and falling and sticking together anyway. At one time Mr. and Mrs. John Gregory Dunne were the most famous literary couple in America. She was arch and glamorous; he was blue-blooded, devoted, depressive. Together they constructed a legend, the outcome of which she is now living alone. For the most part, she is sane again, though.\n\n\"For about six months after John died, I'd get cross when I heard couples bickering,\" Joan says. \"I'd tell them they didn't have enough time.\" She looks at the ribbons on the dining room table. \"We both knew that John had a heart condition. But what would have happened if I'd believed all the time that he was likely to die? There's no way of staying alive if you believe that you're gonna lose someone or yourself any second. What good would that have done?\"\n\n\"It is hard to live that way.\"\n\n\"I realized that I was berating myself for not being able to control these uncontrollable events. I've learned that thinking you should be able to control things is the worst kind of grandiosity. I used to be such a control freak.\"\n\n\"Used to be?\"\n\n\"I'm better now. I really am a bit more accepting. Part of anticipating the worst is believing that if you expect it, the terrible thing can't possibly happen.\" Joan laughs again. \"Then when the worst finally does happen, you stop anticipating. I used to be quite fearful, but very little can happen to me now.\"\n\n\"You're fearless?\" I ask.\n\n\"Oh, God no. I'm not saying that when something terrible is going on, I don't deny every bit of it. In the emergency room, I had to have known that John was already dead, but I was still trying to figure out how to get him home from the hospital. As if I could somehow improve the situation! It doesn't sink in.\"\n\n\"Denial is there for a reason,\" I say.\n\n\"I don't know how people function without it. Of course, we all need to look at things as they are, but if where they are is untenable at the moment, we have to find another way to get through the day.\"\n\n\"Or the year.\"\n\n\"I never quite know what people mean when they talk about being strong,\" Joan says with some annoyance. \"I didn't die. My life has to go on. What's the choice? I had a sick child. I didn't have an option.\" She did have a choice about whether or not to put her ordeal to some good use, however. By dissecting her own demented time, Joan realized that her story might help others fumbling their way through what she calls \"the American way of grief\"\u2014namely, \"evasiveness posing as courage.\" \"It was something that nobody talked about,\" she tells me. \"The actual physical and mental effects of loss are manifold, but nobody ever admits they're happening. It seemed to me that it could conceivably be a good thing if we started saying, 'Look, you're going to go a little crazy; it's normal, don't panic.'\"\n\nOne of trauma's more puzzling side effects is its trick-mirror effect on time, how time-as-we-know-it seems to pause and fast-forward simultaneously, throwing you out of the flow of things. This distortion contributes to magical thinking. \"What I've learned is that marriage is not only [about] time,\" she admits. \"It is also, paradoxically, about the denial of time. For forty years, I saw myself through John's eyes. I did not age.\"\n\nI tell her how wonderful she's looking (it's true). My hostess accepts the compliment grudgingly. \"I am getting better,\" she allows. Just the other day, Joan even managed to clear some shelves in John's old office without giving it a second thought. \"That was a first,\" she sighs. When I broach the subject of Quintana's death, however, Joan's poker face lets me know that this is a subject too raw to be touched. \"The relationship with a child is a whole different level of loss, at once more fundamental and less intimate\" is all that she has said on this subject.\n\nThe living room has darkened. Joan looks tired. Still, there's one more thing I'd like to ask, but I fear it may seem tacky. Joan stretches her legs and moves toward the front door. The top of her head barely reaches my chin. I finally get the question out.\n\n\"Can you imagine falling in love again?\" I say.\n\nJoan stops and looks me straight in the eye. For a second, I think she's going to slap me. Then I realize she's amused. \"I wouldn't get married again, I don't think,\" she tells me in her cowgirl drawl, standing inside her doorway. \"But fall in love?\" she says, lowering her chin. \"Absolutely.\"\n\n## **THE ROAR OF FREEDOM**\n\nThere's a beautiful Sufi story I've always loved. A group of tigers in a forest leaves a cub behind by mistake. The tiger cub is reared by sheep. The sheep teach it how to act like a sheep. It walks like a sheep and baas like a sheep and eats grass.\n\nMany years later a tiger happens to be passing and sees this ludicrous spectacle of a half-grown tiger behaving like a sheep. It is appalled and amused and drags the tiger to a pool in the forest. There, it shows the young tiger its own reflection, and the tiger begins to wake up to what it really is.\n\nThe older tiger teaches the younger tiger how to roar. At first all it can do is make bleating sounds. But slowly the tiger roar begins in its throat, and then after weeks of practice, it comes up to its master and gives the great roar of freedom.\n\nThis is what survivors do. As domesticated beings, we're fleeced into believing we are safe and special. Then the tiger comes out of the forest. The truth is savage, but in its eyes\u2014in the aftermath of the long tussle\u2014we see that we are wild, too. We only imagined that we were so timid.\n\nIt's uncanny what happens when people wake up. Their wattage increases. Their hearts amplify. You see the juju in their eyes. That's why we're so drawn to oppressed people who prevail. One sweltering afternoon in 1988, I'm alone in a third-class train compartment in Florence station, waiting for the departure whistle, when the doors fly open and in marches one of the strangest-looking creatures I have ever seen in my life. Moist and brown as a greased potato, big-bottomed, and sporting a blue jean ensemble (blue denim tennis shoes, overalls, belt, shoulder bag, and pimp cap), this high-voltage creature has chintzy, taped-up luggage swinging from every appendage and is squealing like a girl\u2014\" _Che stronzo!_ \"\u2014at the heat. Finally, he manages to negotiate himself into my quiet cabin and plops down on the seat across from mine.\n\n_\"Ay, papi!\"_ he gasps, fanning himself with one hand, clutching a cannoli in the other, smiling, his yellow teeth smeared with cake. \" _Che giorno!_ What a day!\" He's dressed like a slutty, overweight mall girl, in camel-toe jeans and rhinestone jacket, mucky lipstick, smeared mascara, the top of his forehead sprinkled with tiny, shiny pimples.\n\n\"It is hot,\" I say.\n\nHe extends his hand and purrs, \"Mario.\"\n\nNow and then things happen to us in life that truly deserve to be called uncanny. This overnight trip to Paris is such a surprise. Bouncing up and down in his seat, so happy he seems ready to pop, Mario is not what he appears to be\u2014a tranny hooker or maid half in drag, say\u2014but is, in fact, one of a handful of naturally born \"sopranists\" alive in the world today. Sopranists are males born with female vocal cords. They're the golden-egged geese of the opera world (where boys were once castrated to sing these roles), lusted after by aficionados for their unique sound, as Mario now explains to me. \"They cut off the privates,\" he says, patting himself. \"But not me!\"\n\nAs if the unlikelihood of this weren't enough, my companion is on his way this very night to make his \"day-boo\" at the Paris Opera. Describing Mario's pitch as excitement would not do his hysteria justice. He's over the moon, beside himself, ready to burst out\u2014I mean _burst_ out\u2014in song. Though he's performed here and there over the years, tomorrow will be Mario's official coming out. His singing teacher, Signora Cowell, has insisted that he prepare a full eight years before premiering his rare instrument on a world-class opera stage. Mario has done _piccolini_ parts, nothing roles, in the provinces for fun, he says, curling the top off a tin of lasagna, but tomorrow will be the great debut of his life.\n\n\"Signora wanted me to save,\" he says, tugging at the blue jean scarf on his neck. \"I work for her since sixteen years old. Now,\" Mario, tells me, digging into the noodles, \"I am ready!\"\n\nI am speechless.\n\n\"My patron find me,\" he claims. \"He introduce me to the signora. This is how I come to Italy.\"\n\n\"Found you where?\"\n\n\"Oh, baby, you do not _want_ to go there,\" Mario protests, in the protesting way of people who can't wait to spill the beans. I assure him that very little shocks me, digging into my bag for a tape recorder. Then, with the Tuscan plains making way for the Appenines, and Mario wolfing down one high-calorie treat after another, he proceeds to relate the far-fetched saga of how he came to be sitting here in this Italian train on his way to\u2014here Mario crosses himself\u2014superstardom.\n\nHe came from a jungle town near Salvador, in the northeast corner of Brazil, one of ten children born to an illiterate farmer and his peasant wife. \"Dirt poor,\" Mario tells me, slowly chewing. No school, no medicine, no electricity, he says. Once a month the family ate meat. Five of his siblings did not survive.\n\n\"Half of your brothers and sisters died?\"\n\n_\"La mis\u00e8re,\"_ Mario says, meaning disease or malnourishment, I guess. \"I was always different. I always sing sing sing, everywhere I go. I am always _maricon,_ you see\u2014big, fat sissy boy. I am always with the fat.\" Mario tugs at his love handles, laughs, gouges out a hunk of lasagna. \"But nothing matters to me. Nothing but the beauty. I need the beauty,\" my companion says, laying a manicured hand on his heart. \" _Papi,_ I grew up in the paradise. _Il giardino di Eden._ \"\n\nOne day when Mario was fourteen, he was drawing water from the well with his sisters, singing his heart out as usual, when a white man and woman came driving by in a Jeep. They asked the local children for a glass of water and were stunned when Mario sang his reply like Maria Callas. The memory still delights him. \"These strangers, they tell me to talk normal.\" Mario chuckles. \"I tell them this _is_ normal. My sisters tell them this is normal. They come to my house to talk to my parents. This man says that he knows a teacher who would love to get her hands on me. At first this scare my poor mama. She thinks they are trying to buy me as slave. The Italians say, no, this lady is opera teacher. They leave us a card with their telephone number.\"\n\nI cannot hide my skepticism. \"As God my witness,\" Mario swears, crossing himself again. Then he produces a photo album with several press clippings (\"Jungle Boy Breaks the Sound Barrier\" etc.) announcing his \"discovery,\" along with a tacky publicity shot of himself in a tuxedo looking less butch than your average lipstick lesbian. I apologize for not quite believing him.\n\n\"Nobody believe me,\" Mario says. \"I go to my friend in town who has a telephone and call this number. The man is in S\u00e3o Paulo. He ask me to come to S\u00e3o Paulo to sing for his friend, Signora Cowell, when she comes from Firenze for visit. They send for me taxi from S\u00e3o Paulo\u2014I have never been in a regular car before. I meet this wonderful lady. She tells me that I am a rare bird. That is what she says. _Rara avis._ I tell her I have always known this. Six months later, she is sending the man a ticket for me to come and study with a famous teacher in Italy!\"\n\n\"That's unbelievable, Mario.\"\n\n\"God is great,\" he says, and kisses his crucifix. \"I have no money, so Signora Cowell find me job in the house of an old contessa. Once, she was a very grand lady. Now she is all alone. I learn many things in the contessa's house. She is so sad, so drunk and lonely. I am alone with her from day until night, doing everything but the cooking all by myself,\" he tells me. \"She is abandoned, this rich old lady\u2014no children, no husband. Her friends only want her money, she says. The contessa tells me I am her only true friend. So much money and I have never seen anyone so unhappy! This old lady tells me her secrets. I listen but\u2014\" Mario zips his mouth with a finger. \" _Niente. Mai._ I never tell nothing. I only tell you this one thing because you are so nice.\" He squinches up his nose at me.\n\n\"The contessa is having these dinner parties. The villa is _estupenda,_ one of the oldest on Corso dei Tintori, near Piazza Santa Croce,\" he goes on. \"Many people are coming. She is sitting at the head of her table in a beautiful dress, with jewels, and piles of food, and laughing\u2014everyone is always laughing\u2014and under the table\u2014\" Mario covers his mouth.\n\n\"What?\"\n\n\"Under the table, she is peeing.\"\n\nWhat?\n\n\"She is sitting there, with all fancy people, and she is peeing on her chair,\" Mario says, making a face. \"She is so drunk she cannot stand up for the toilet. But nobody notices. They are eating and drinking. The contessa is all alone. I am standing in the corner helping with the drinks. She sees me looking and is crying inside.\" Mario pauses. \"In this moment, I understand.\"\n\n\"What did you understand?\"\n\n\" _La miseria,_ \" he tells me. \"I see that there is _la miseria_ of the rich and _la miseria_ of the poor.\"\n\nEquating a lonely alcoholic aristocrat with five dead toddlers? This sounds more like opera than fact to me.\n\n\"I have seen this with my own eyes,\" Mario says, wiping dripped pasta off his chin. \"Let me tell you a story. Where I come from it is a miracle if you're alive. When I was small there was a four-year-old girl who lived nearby. I used to hear her all night long, saying, 'Hungry. Hungry.' One day this girl saw some bugs in the well and ate them. Nobody knew till her stomach swelled up.\" Mario pats his belly like a pregnant woman. \"The doctors purge her. I see them. The vermin start crawling out from her mouth.\"\n\nHe is acting this out with hand gestures.\n\n\"This little girl choked and they couldn't save her. I see her die with my own eyes. But was this little girl more _misere_ than _la contessa_? There are many ways to die, gringo.\"\n\n\"Sure. But, Mario\u2014\"\n\n\"I have seen both sides of the misery,\" he stops me. \"And still life is beautiful!\" Then Mario pulls himself to his feet with some effort, shimmies his jeans into place, and grinds his hips in an impromptu samba, snapping his fingers with one hand, the other perched on his belly. \"I love to dance!\" He spins around, grinning, surprisingly graceful. \"You must never forget to dance.\" Mario throws back his head. \"And you know what else Italy's good for?\" He punches an open palm with a fist. \" _Chiavare._ The best place in the world for fucking. The men, ooh, I love the machos.\"\n\n\"And they must love you.\"\n\n\"You know they do! Any way that they can get me!\" Mario falls back down on his seat in laughter. \"But people like you don't know how to live,\" he says, breathlessly.\n\n\"What do you mean? People like me?\"\n\n\"You gringos. Always so serious! You do not know how to live. All the time so _nervous._ Too much of the stress. And you look at the world with\u2014\" He clenches his arms up like someone protecting himself from the cold.\n\n\"That's not fair\u2014\"\n\n\"Cold inside,\" Mario teases me. \"Like the ice. The contessa is cold like the ice inside. She tells me she has no feelings. When she cries, it is nothing but water, she says. When no one is there to see them, tears are only water, she tell me. This is tragedy. More sad than in my country.\"\n\nI don't bother debating this.\n\n\"We have the life,\" Mario says, making fists with both hands. \"We love the life.\"\n\n\"So do we,\" I say. \"And it's hard sometimes.\"\n\n\"But there is also the beauty!\" Mario leans forward and touches my knee. _\"Viva amore.\"_\n\nI don't know what to say.\n\n\"This is what I tell the contessa. Who cares if life is hard! She tells me that I am better than her own son, who ignores her. She says that I am her _soffio d'aria fresca,_ her breath of fresh air. She is not too old to get better, I tell her. We are in the world more than to suffer! Sometimes I sing to make her happy,\" he says.\n\n\"Maybe you will sing for me?\" I ask, taking his cue.\n\n\"I think that you are never asking!\" Mario laughs. Then he straightens his shoulders, closes his eyes, clears his throat, opens his mouth, and begins to sing the aria \"Un Bel Di\" from _Madama Butterfly_ in a high, pure voice. I'm floored by the beauty of Mario's tone\u2014to say nothing of the stupendous volume, which brings the conductor running to see which of us is blasting the stereo. The train official stops in the doorway and stands there listening to Mario sing this incandescent song about love that will come again, one fine day. As he sings, Mario's eyes well up with tears, his throat quivering, hands clasped across his middle. Eventually the conductor slips away, and when Mario finishes, he tightens the scarf around his neck again and zips his finger across his mouth to let me know the show is over. Mario must rest his instrument. Tomorrow, he will step out onto the stage where Callas became a legend. He will offer his voice into \"that great darkness,\" he tells me with awe in his voice, this peasant boy who might never have seen the lights of a city if he hadn't been singing that day by the well.\n\nIn the morning, we hug on the platform. \"Wish me luck,\" Mario says, kissing me on both cheeks.\n\n\"You definitely will not need it,\" I tell him.\n\n\"Everybody needs it, gringo,\" he insists. Then he picks up his bags, turns on his heel, and sashays into the morning hubbub cramming into the Gare du Nord.\n\n## **THE DAY OF LAUGHTER**\n\nSelf-realization is not for sissies. Transformation comes at a cost. We may be shattered and freed by hardship, eventually, but the process requires a stomach for change. \"A condition of complete simplicity (Costing not less than everything)\" was T. S. Eliot's way of describing this stripping. Because the process is a spiritual one, it requires a mental turnaround (\"conversion\"), whose trials may seem counterintuitive, even perverse, to our rational, self-preserving minds. Yet practice teaches us, in spite of ourselves, that even the most destructive forces are harnessable to constructive ends. Pain we would much prefer to avoid can be used as a kind of battering ram for unearthing the true and beautiful. This _yoga_ \u2014which comes from the Sanskrit for \"yoke\" (in the yin-yang sense of yoking opposite forces)\u2014is available to begin immediately.\n\n\"What is on the day of laughter is also now,\" a teacher of mine used to say. This saint had lived for fifty years on a mountain in southern India. \"What is on the day of laughter is also now,\" he would repeat to his students, tossing the words out lightly, rubbing a palm across the gray stubble covering his shaven head.\n\nWhat did he mean by this? I would wonder. What laughter could he be referring to? In time, his meaning began to dawn on me. He was describing the laugh of a Buddha, the sudden spontaneous guffaw that's said to seize a person when he realizes who and what he really is. Siddhartha, the prince turned Buddha, laughed after discovering his own true nature at last, the story goes. Sometime after his enlightenment, Siddhartha was strolling along a country road when a passerby stopped him to inquire why he was smiling with such quiet joy. Siddhartha did his best to demur.\n\n\"Are you a magician, sir?\" the traveler asked.\n\n\"No,\" the master answered.\n\n\"Are you a god?\" the passerby tried again.\n\n\"No,\" Siddhartha assured him.\n\n\"Then what, sir, can you _be_?\" asked the stranger.\n\nThe Buddha replied, \"I am awake.\"\n\nPlato called this _anamnesis,_ remembering what and who we are, beneath our lives' shifting camouflage, beyond what can be taken away. \"The look of your face before you were born,\" as a Zen koan describes this hidden self, the essence, the nub of you that transcends your changing circumstances. We are not what we appear to be, they tell us: This is the lesson sages have expounded since the days of the earliest seers and shamans. When we glimpse our true identities, apparently it does inspire laughter. Derek Wolcott describes this recognition in one of his most beautiful poems.\n\nThe time will come  \nwhen, with elation,  \nyou will greet yourself arriving  \nat your own door, in your own mirror,  \nand each will smile at the other's welcome,\n\nand say, sit here. Eat.  \nYou will love again the stranger who was your self.  \nGive wine. Give bread. Give back your heart  \nto itself, to the stranger who has loved you\n\nall your life, whom you ignored  \nfor another, who knows you by heart.  \nTake down the love letters from the bookshelf,\n\nthe photographs, the desperate notes,  \npeel your own image from the mirror.  \nSit. Feast on your life.\n\nWhen I visited India for the first time, I was a wreck, a refugee from the New York publishing life I'd coveted yet grown to dread. I was working at _Interview_ magazine, as I've said, in the looming fame-shadow of Andy Warhol. To this day, Andy remains the loneliest person I've ever known, an unquestionable genius but also an alien personage on this planet. Several times a day, Andy would waft through my office, saying almost nothing yet trailing this vacuous atmosphere. Years before, he had admitted that after he was shot in the gut by a crazed fan, Valerie Solanas, his emotional life had been shot out, too, leaving Andy to peer out at the flesh-and-blood world as if through the screen of a TV set. My ex-boss infused the magazine offices somehow with this same dead chill of alienation.\n\nOn my last Christmas at _Interview,_ Andy distributed presents to the staff. Weirdly, my gift from him was a white silk scarf painted with a black pyramid and lotus-seated figure, the words the only way out is in scrawled underneath in inked capital letters. Fourteen months later, Andy was dead following complications from gallbladder surgery. He was only fifty-nine.\n\nI was already in India by then, traveling with Andrew Harvey, an eccentric British friend, who had been born in the twilight of the Raj near Connaught Circus in Delhi, where we were now sitting one evening after dinner.\n\nA harvest moon hung low in the sky. The air was dusty, sweet, and warm. We'd been smoking hash, which made it sweeter, and Andrew\u2014a brilliant professor and poet\u2014was now holding forth in his stentorian way on the subject that most compelled him: the loss of sacredness in the world. \"Until people realize that _all_ of this\u2014\" Andrew held out his hands to encompass the great, entropic, swirling mass of humanity, rickshaws, cows, and sirens surrounding us on all sides\u2014\"that _all_ of this is divine, every last inch of it, even the ugliest parts\u2014till they get past the immature notion that a sacred world would be a world without suffering, a big Disneyland where everyone gets in for free, they will not truly understand that suffering is here for a purpose,\" Andrew declaimed.\n\n\"And what, oh wizard, might that be?\"\n\nMy friend ignored me. \"We spin and spin in our little cages without looking outside of them or even more deeply within,\" he went on. \"But if people took the time to look more deeply, if they dared to look underneath their masks, they would discover something that could change their lives forever.\"\n\n\"Please tell me what you're talking about!\"\n\n\"They would discover their real face in the mirror.\"\n\nWhat real face? I thought to myself. Till that moment I had never considered that there might be a different \"face\" to things of which I was wholly unaware, some secret, even more bona fide visage. But what if Andrew was right? I wondered, influenced by the smoke and the moonlight. My companion was hyperbolic but also wise on subjects spiritual and mysterious. What if this thing we called ordinary life really was, at least in part, a case of mistaken identity? Something in his words struck a chord of truth, tintinnabulating inside me.\n\nAs the years went on, I came to learn that spiritual traditions all put forth some version or other of this same strange realization: that we do not, in essence, know ourselves; that our worldly masks are a mere fa\u00e7ade. During times of crisis in years to come, to my great surprise, I found myself increasingly aware of this other face peering out at me through my physical eyes, seeing through my own mask, penetrating with questions. It was impossible not to notice that this new awareness emerged in direct proportion to assaults on my struggling ego.\n\nThis must be why the Buddha laughed: because nothing more could be taken from him. He recognized his own true face. He was liberated from craving and fear. He saw the freedom of self-surrender, the hidden reward of not looking backward. This freedom came at a price, of course; it comes at a cost for all of us. But considering the dividends\u2014in joy, self-knowledge, and unshakable strength\u2014as I would learn on my own roller coaster, only a fool would refuse to pay it.\n\n## **_OM MANI PADME HUM_**\n\nAndrew and I had flown from Delhi to Srinigar, spent the night on a houseboat on Dal Lake, then boarded a comic-strip bus for Ladakh. Ladakh is India's northernmost province, a lunar, breath-sucking, desolate place at eighteen thousand feet, surrounded by gargantuan Himalayan peaks. The streets of the capital town of Leh are a dusty, fetid confluence of Muslim Kashmiris and Tibetan Buddhists, handbag-faced women with knee-length hair braided to the waist, draped in coarse maroon clothing and carrying their rosary _malas_ everywhere they go, chanting, \" _Om Mani Padme Hum._ \" This Sanskrit mantra is etched like graffiti onto walls and stones (even restaurant menus), like \"Jesus Lives\" in the American Bible Belt.\n\nThere is no simple translation of this ubiquitous Buddhist prayer. An oversimplification might be \"Compassion cooks up cosmic juju.\" The ancient mantra's power does seem to pervade the atmosphere of this otherworldly region. For three weeks Andrew and I hunkered down in a farmhouse pension outfitted with glass-sided rooms and straw mattresses. Our quarters looked onto a sunflower-and-vegetable garden where a prehistoric grandma twirled her rosary and toiled day after day, a donkey no bigger than a Labrador retriever nipping at the heels of her embroidered slippers. A mastiff-sized cow made its nest near the gate. A vast panorama of mountain peaks spread 180 degrees across our window view, dotted with tiny monasteries, each one a five-mile climb straight up. I learned this the hard way during a morning's trek to Gotsang, where Andrew and I were greeted at the temple's gate by a pair of monks who later offered us bowls of yak-butter tea. They slapped their sides and laughed as we struggled to choke the disgusting stuff down, ogling us as if we were escapees from the primate wing of a foreign zoo.\n\nAfter returning to the States, Andrew began work with a lama named Sogyal Rinpoche on a book they decided to call _The Tibetan Book of Living and Dying._ It was to be an updated, accessible version of _The Tibetan Book of the Dead,_ which had caused such a stir when it first appeared in English in 1939. Every so often a book appears that delivers a whiplash to the mainstream zeitgeist. _The Tibetan Book of the Dead_ had been such a volume. While we in the West were busy with the Crusades, the Renaissance, and the Industrial Revolution, the isolated mountain masters of Tibet were hard at work formulating a theory (based, they believe, on evidence) of death and reincarnation. Their self-proclaimed grasp of the nuts and bolts of rebirth enabled the Tibetans to devise a complex system for tracking their masters from one body to the next, enabling them to hone and protect their wisdom tradition down through the ages.\n\nI'm agnostic on the subject of reincarnation, persuadable but unconvinced. I agree with Voltaire that being born twice isn't that much more far-fetched than being born once. In any case, Sogyal Rinpoche asked me to help with the book. Sogyal is a small, powerfully built man with a schoolboy laugh and a stubborn streak. Trained classically in the Tibetan tradition, this _rinpoche_ (an honorific meaning \"teacher\") had survived the Chinese invasion, escaped through the Himalayas to India, then went on to study philosophy at Trinity College, Cambridge, where he worked as an assistant to the visiting Dalai Lama. Today, Sogyal is a rock star in the Buddhist world and keeps the same kind of hectic touring schedule.\n\nOne afternoon we were sitting on a patio overlooking a fir-lined palisade in Aptos, California, having a meeting. Sogyal was fielding cell phone calls from Dharmsala, Buenos Aires, and Paris. We got to talking about a perplexing thing that had recently happened to the Dalai Lama. During a talk in New York someone had asked His Holiness a question about self-loathing\u2014how Buddhism handles the common American problem of autophobia\u2014when all of a sudden the talk stopped. There's no word for self-hatred in Tibetan. The translator was stumped. The Dalai Lama himself was puzzled.\n\n\"This is something almost unheard of in my country,\" Sogyal explained to me between calls.\n\n\"The trance of unworthiness?\" I asked. This is how a therapist friend describes the popular American pastime of disliking ourselves. Whether we blame this destructive habit on the doctrine of original sin or merely inverted narcissism (of the privileged, first world variety), the upshot is the same: too much time spent hating who we are.\n\n\"We emphasize the _preciousness_ of human birth,\" Sogyal went on. \"The Buddha was a human like us. Our true nature is to be enlightened. You must remember that your Buddha nature is as good as any Buddha's Buddha nature.\" The lama chuckled over his own cheesy joke.\n\n\"But what about evil?\" I asked. What about destructive folks who wouldn't know their own Buddha nature if it hit them in the face?\n\nSogyal swatted my caveats away. \"The teachings say that when the world is full of evil, all misfortune should be transformed to the path of goodness.\"\n\n\"Okay,\" I said. \"But these are hard times\u2014\"\n\n\"It is not the times that are bad, it's the people,\" he corrected me, quoting a popular Tibetan saint.\n\n\"I thought you didn't believe people could be bad,\" I kidded Sogyal back.\n\n\"That depends on the day.\" He laughed.\n\nIntrigued by this so-called Buddha nature, I make my way to Washington, D.C., to meet a Tibetan nun named Nawang Sangdrol. This twenty-six-year-old Buddhist devotee had been arrested by the Chinese at the age of thirteen and spent eleven years in the infamous Drapchi prison (the Abu Ghraib of Tibet), becoming her country's longest-serving female political prisoner. In a fluorescent-lit conference room, I meet Nawang Sangdrol and take a seat across the table from where she's sitting, flanked by an overly protective Tibetan translator duded out in a cashmere jacket.\n\nFor a minute she hardly looks at me. Her scarred hands are clasped on the table before her. Nawang looks twenty years older than her age. Her black braid appears bluish against the worn-out fleece of her plum-colored sweater. She's wearing a red string bodhisattva bracelet around her wrist along with three strands of tiny ivory beads.\n\nWhen I attempt to break the ice with small talk\u2014her English is close to nonexistent\u2014the metrosexual translator steers me away. \"Nawang is too shy,\" he says.\n\n\"Will she tell me her story?\"\n\nNawang nods without being asked and for the next half an hour recites the details of her ordeal. I study the young woman as she speaks, her eyes never widening, her voice never piercing a whisper. She seems so fragile she could blow away.\n\nOn August 21, 1990, Nawang was a thirteen-year-old Buddhist nun living in Lhasa. \"My family was religious, but I also had the desire to join the nunnery,\" she tells me. One afternoon she and a group of sister nuns attended a festival at Norbulingka, the former summer palace of the Dalai Lama. Inspired by love for their leader, the nuns felt the urge to move to the center of the large crowd to avoid attention and shout \"Free Tibet! Long live His Holiness the Dalai Lama!\" along with the rest of the crowd.\n\nAlmost immediately, uniformed Chinese police dragged the girls by their hair into trucks and shipped them off to a detention camp outside Lhasa. Following a severe beating, they were thrown together into a cell and kept there for nine months.\n\nNawang pauses to sip her water. I can't quite bridge the gap between her punishment and her offense. Jailing a thirteen-year-old girl for chanting the equivalent of \"Long live the Queen\"? Torture sessions began immediately, Nawang tells me. \"They called us 'splittists' and counterrevolutionaries. We were beaten with iron pipes and electric cattle prods.\" Nawang makes a claw with her hand to show the shape of the red-hot instrument. \"They would tie us up and take turns beating us, and also attach live electric wires to our tongues.\" When her captors were feeling especially aggressive, they put prisoners into what they called the airplane position\u2014hung from the ceiling with their hands roped around their backs.\n\n\"You were a child,\" I say in disbelief.\n\n\"They didn't care how young we were or whether we were female\u2014they tortured children the same way they tortured adults.\" Once, while being assaulted with electric shocks to her neck, Nawang instinctively ripped the wire off and threw it to the ground. \"A guard pointed a gun to my head and said, 'Now you are going to die.' Then he laughed.\"\n\nThis was only the beginning. Eventually released from prison, Nawang returned home to the news that her mother had died, while her father and brother had both been arrested. Refusing to collaborate with the Chinese, the patriotic girl continued to protest against the destruction of her country. Four months after leaving one cell, she was arrested again (for chanting at a public protest) and sent this time to Drapchi, where she would spend the next eleven years.\n\n\"I was put in solitary confinement,\" Nawang tells me, our eyes meeting for the first time. \"It was winter, very cold in Tibet. I was in my cell with only one shirt, no sweater. Because this was my second arrest, they wanted to make an example of me to other splittists. I was forced to stand outside in the courtyard in the snow. If I slumped a little bit, the guards would beat me.\"\n\n\"How did you get through?\" I inquire.\n\n\"By inwardly protesting for freedom,\" she replies. \"The cell was very small, with the ceiling ripped off so the guards could watch me. It was just like being in a cage. They kept the lights on to prevent me from sleeping. They wanted to mentally and physically break me. My health began to deteriorate. There were spiders. Sometimes the rats would bite me.\" Nawang turns her head to show me a pair of vampiric puncture wounds. My face must be betraying my unspoken question. \"I will tell you,\" Nawang says, reading my mind. \"I did this for His Holiness. For my people. For my country.\" There's no trace of martyrdom in her voice, no righteousness, no detectable self-congratulation. This woman is an enigma to me. Agape is an uncanny thing, generosity given without self-interest. I am not suggesting that Nawang Sangdrol is a saint, but she is, by any ethical measure, a most highly evolved human being.\n\n\"We must never give up this fight,\" she says. \"People must remain true to goodness.\" Nawang fingers the _mala_ around her wrist and holds it out for me to see. \"I wove this from the threads of my shirt,\" she says. The letters read, \" _Om Mani Padme Hum._ \"\n\n\"Hatred does not end by hatred,\" Nawang says. These words are from one of the Buddha's sermons. Then she makes the _namaste_ sign and lowers her eyes. The translator excuses her\u2014the nun's adoptive family is waiting for her outside. Nawang had finally gained amnesty through an international relief organization. Now she is going to school to try to learn English. \"Very bad,\" she apologizes, getting up.\n\n\"Do you feel free now?\" I ask her. \"In America? No Chinese soldiers?\"\n\nNawang tilts her head back and forth in the Indian way that means so-so. In fact, she intends to return to Tibet as soon as possible in spite of the risks. This is baffling to me, I admit.\n\n\"There are many like me,\" Nawang says in English.\n\n\"She's a modest girl,\" the translator says. For the first time, I see the young nun's crooked smile.\n\nNawang's freedom rests, it seems, in her ability not to hate. Her self-worth does not depend on aggrandizing herself but is simply her birthright as a human being. Nawang will not keep fighting because she is brave\u2014although, of course, she is\u2014I think to myself afterward, but because she has taken a bodhisattva vow, as a Buddhist nun, to help end suffering wherever she finds it. This promise appears to have set her free.\n\nWith nothing to lose, she's wealthier and more liberated than most of us. _Om Mani Padme Hum:_ the road to freedom is lit with compassion. Then, when the impossible happens, we meet it with an opened heart, knowing the enemy cannot destroy us. Love is stronger than fear, we see clearly. Knowing this opens a different door.\n\n## **DRAGONS AT THE GATE**\n\nSummertime on Southampton Beach in Long Island, July 1970. Two months before their scheduled wedding, Jack Willis and his fianc\u00e9e, Mary Pleshette, are hitting the surf for one last swim of the afternoon. They are a golden couple\u2014Jack an award-winning, thirty-six-year-old documentary film director; Mary a reporter for _Newsweek_ and twelve years his junior. Jack, who'd been an excellent swimmer all his life, throws himself into an average-size wave. A minute passes. Mary sees no trace of him. Suddenly Jack's head bobs up out of the surf, and he cries out to his fianc\u00e9e for help.\n\n\"I'd been bodysurfing since I was a kid and knew I shouldn't have taken that wave,\" Jack is telling me now. We're in the Willises' living room near Central Park in Manhattan. Jack, now seventy-three, is sitting in an armchair across from me, while Mary putters in the other room. Aside from the tilt in his slender shoulders and the metal walker at Jack's side, you'd never know that he is disabled. Bright-eyed, white-haired, and quick with a joke, Jack's the kind of raconteur you'd expect to meet in an Irish pub, talking blarney with a corncob pipe in his mouth (if he weren't really a Jew from L.A.).\n\n\"Suddenly I looked down and saw nothing but sand,\" Jack says, eyes brightening. \"I tried to somersault out of it, but instead I hit the back of my head and was instantly paralyzed from the neck down. I saw flashes of red and realized I had to stay awake or I was going to die.\"\n\nMary, a sexy, auburn-haired woman wearing jeans and a black turtleneck sweater, sets glasses of water in front of us and takes her seat on the opposite couch. \"My next thought was 'It's okay to die,'\" Jack tells me. \"It's the clearest that I have ever been about anything in my entire life. There was a genuine feeling of absolute peace. Then my next thought was 'You have to try to get out of this somehow.'\"\n\n\"What did you do?\" I say, turning to Mary.\n\n\"I tried to run toward him,\" she remembers, \"but it felt like an anxiety dream where you try to run but can't move your legs. Thank God for the people on the beach. I could never have pulled Jack out by myself.\"\n\n\"They got me out of the water and laid me on the beach,\" he continues. \"I couldn't move anything. Luckily the paramedics who finally came had been in Vietnam. They recognized immediately that I'd broken my neck and knew better than to try to move me.\"\n\nThe Willises are a formidable team, one of those rare true-love partnerships that tolerate\u2014even thrive upon\u2014a soup\u00e7on of healthy friction. They adore each other, clearly, challenge each other, disagree, and push the memory envelope in some directions the other might prefer to avoid. It comes as no surprise when Jack tells me that Mary has been integral to his emotional and physical survival. Their symbiosis is poignant and glaring.\n\n\"That moment of peace may have come out of shock,\" Jack backtracks, describing his near-death experience. \"But I'm truly not afraid to die now. I just know it's going to be extraordinarily peaceful.\"\n\n\"For you, maybe.\" Mary arches a brow.\n\n\"No big white light,\" Jack says. \"Just a feeling. And the next thing I thought was _Try to live._ \"\n\nIn the days following the accident, Jack claims to have experienced surprisingly little fear. His real dread came during the twenty-one days of waiting to hear whether his spinal cord had been severed. \"I was in traction for six weeks,\" he says, shaking his head slowly. \"The physical pain was terrorizing. The drugs they were giving me caused the most horrible nightmares. I'd dream I was skiing downhill and would hit a tree, then my neck would snap, over and over again.\" Jack's waking reality wasn't much better. \"I couldn't talk about my fear of being quadriplegic with Mary,\" he admits.\n\nThis is Mary's cue to leave the room.\n\n\"Would I kill myself?\" Jack says in a low voice. \"What happens if I'm in a wheelchair? I'm a filmmaker. What am I going to do with my life?\" His doctor's bedside manner only made things worse. \"One of them said, 'If this happened to me, I don't know what I'd do. I'm a doctor. I need my hands.' I said, 'Fuck you, what do you think I am? A piece of meat?' Luckily Mary brought me a tape recorder I could talk into, which kept me from losing my mind completely. I could say things I absolutely needed to say. Things I couldn't reveal to another soul. Not even Mary.\"\n\nTrapped in agonizing pain, Jack wrestled hour by hour against horrendous, mounting despair. Denial was his only ally. \"In a moment of hopelessness, it's helpful to be lied to,\" he says.\n\n\"It's true,\" I agree.\n\n\"My morale was declining too fast for anything else,\" Jack tells me. Family and friends, including Mary, colluded with him in the fantasy that everything would eventually be all right. But these assurances didn't always help. \"The better I got, the worse I felt emotionally,\" he admits. \"It seemed an incredible paradox\u2014\"\n\n\"What do you mean?\"\n\n\"The less physical pain I had, the more subject I was to fits of depression, to feeling trapped,\" Jack explains. \"When I tried to sleep, I'd wake up in a panic of being imprisoned. I remember seeing a picture of Gulliver tied down by thousands of tiny ropes, his arms at his side, his neck stretched out. That's exactly how I felt.\"\n\nEach small improvement during this waiting period brought with it an onslaught of paranoid predictions about the future. \"Once I started to get movement and feeling back, I was worried about my bladder and bowels and libido. My fianc\u00e9e and I were about to be married, for Christ's sake!\" Jack slaps the arm of his chair. \"Would I kill myself? I didn't know how much I could bear but decided not to do anything until those twenty-one days were up.\"\n\nWith the news that his spinal cord was intact, Jack Willis got a portion of his future back. In spite of medical assurances that he would never walk again, Jack was back on his feet within a year. He and Mary were married soon afterward, and the couple managed to rebound remarkably well at first. Determined to ignore his frailty, to pick up life where he'd left off, Jack worked at building a firewall of denial between himself and his loss, with Mary's wholehearted blessing.\n\n\"I was far less introspective back then,\" he admits. \"Far more alpha male. I didn't want to be treated like a cripple. My own survival demanded that I separate myself from people in my condition.\" That's why Jack refused to continue rehab: The sight of others like him was too depressing. In business he piled on the machismo. \"I was aware in corporate life that I was at a disadvantage.\" He chuckles at the understatement. \"At meetings all these suits would be hustling, and I'm shlepping along. But the minute we sat down we were all equal.\"\n\nHe refused to be stopped by self-consciousness. \"Mary and I just denied that there were problems, but there were enormous problems.\" The buried despair and repressed fury in their home only deepened after the birth of their two daughters, Sarah and Kate. \"Our daughters started having issues, but we didn't put two and two together,\" he says. \"We just weren't dealing.\"\n\nWhen Jack excuses himself to use the bathroom, Mary stays to keep me company. \"Did you ever think of leaving him?\" I ask.\n\n\"Never,\" Mary says, not missing a beat. \"I was just so in love with him. I was only twenty-four years old when it happened, but I knew I'd found the person I wanted to be with. The options were clear when I played them out.\" Mary ticks the choices off on her fingers. \"Could I live with Jack being paralyzed? Yes. It wouldn't be easy, but we'd manage somehow. Could I live without him\u2014or, God forbid, see him get better and marry somebody else? No way in hell. That was unacceptable. The choice wasn't as hard as it looked.\"\n\n\"So you've never regretted it?\"\n\n\"Oh, please!\" Mary laughs. \"Nobody forced me to stay. I'm here because I love my husband. There are plenty of things that we can't do\u2014the hardest thing is not being able to take a walk together\u2014but we really have had a terrific life.\" (And yes, the Willises do have sex.) \"Jack has never lived like he was crippled.\"\n\nThe denial did have to stop, however. In 1990 Jack reached a low point. Having taken a job in Minnesota as the head of Twin Cities Public Television, he became tremendously depressed, overwhelmed by a wave of held-back despair he could no longer manage with make-believe. \"It was a bit like what they call post-polio syndrome, where decades after surviving the disease, people's muscles start to atrophy and they end up in a wheelchair,\" Jack tells me when he's back.\n\n\"A midlife thing?\" I ask.\n\n\"Who knows? But I started feeling very sorry for myself,\" he says. \"What's my life about? That kind of thing. Did I really survive this accident and get this far only to be pulled back down again?\"\n\n\"In many ways it was worse than the initial crisis,\" Mary says.\n\nHaving refused to use a cane, Jack began taking falls and generally neglecting his health. Mary was beside herself. \"She was very angry with me,\" Jack admits. \"We had terrific fights. Mary wanted me to go on a walker, but I was terribly insulted. I'd managed to walk on my own till then, albeit like a drunk on a boat with one leg bent. I threw the walker across the room.\"\n\n\"He was a son of a bitch,\" she says without mincing words.\n\n\"I had too much pride,\" Jack tells me. \"Mary had stayed with me through everything, but I realized that I might lose her. I began, with the help of a shrink, to deal with the denial.\"\n\nAround this time, a family friend gave Jack a copy of Reynolds Price's memoir, _A Whole New Life,_ in which the novelist, who was paralyzed by spinal cancer, describes how hypnosis helped rid him of chronic pain in spite of his own skepticism. Jack jumped on this lifeboat of borrowed wisdom and began to explore the mind-body connection. He located a hypnotist, who taught him how to work with his own pain in order to function at a more comfortable level. Jack describes how this breakthrough led him to meditation and a passionate interest in what he calls the inner life.\n\n\"This has been extraordinary for me,\" he says. \"I was not the kind of guy you could see meditating! But it truly did become my lifeline.\" The far-reaching effects of sitting quietly twice a day for thirty minutes still amaze him. \"My behavior vis-\u00e0-vis the rest of the world is so much different than before. I'm working as hard as I used to, but not integrating the daily stress in the same way. I'm able to deal with things a lot more objectively, not taking things personally\u2014internalizing\u2014the way I used to.\"\n\n\"It's true,\" says Mary, who also meditates.\n\n\"It's about having a calmer foundation from which to act, and finding that when I act that way with other people, they respond much better to me as well. I'm less concerned with how I appear to the world, less attached to the physical. But I'm in much better shape than I was ten years ago, because I'm taking much better care of myself.\"\n\nJack is able to \"unhook\" more skillfully from onslaughts by his inner terrorists. \"It's a beautiful thing,\" he marvels. \"There's a kind of wisdom that doesn't come from the head, an impulse to do something that isn't influenced by thought. Suddenly you say something really smart that you didn't mean to say, that kind of thing. I can drop the mind, unhook awareness from the thought or emotion. Not all the time, but sometimes I can do it. Right now I'm working on lower-back pain. The process continues.\"\n\nJack is also angling to retire. \"I've done it,\" he says, meaning his television career. \"There are other horizons.\" Mary is freelancing for the _New York Times_ and working on a second novel. Their marriage remains passionate; from the way Mary hangs on the arm of his chair, you can tell she'd rather be sitting on his lap. Their relationship with the girls is steadily improving. On a recent trip to Paris, Jack even agreed to use his wheelchair.\n\n\"At any given moment, you're doing the best you can,\" he says. \"The question is always, what's the logical next step?\"\n\nMary smiles lovingly.\n\n\"I try not to worry until I have to. Living with this\"\u2014Jack gazes down at his self-described scrawniness\u2014\"I've seen that we keep finding tools, all of us, no matter the problem. To help us get where we want to go, even if we're not conscious of where we're going.\"\n\n\"What do you do when you get scared?\" I ask.\n\n\"Our fears are like dragons at the gate,\" Jack Willis tells me. \"We just have to face them and walk through.\"\n\n## **SUPERMAN'S GHOST**\n\n\"Every journey has a secret destination of which the traveler is unaware,\" philosopher Martin Buber wrote. The image of life as a labyrinth is ancient and true, each pathway we follow digressing, recircling, occluding, delivering us, unexpectedly, to unsuspected crossroads and brinks. You leave the house to buy a quart of milk and meet your true love near the frozen foods. Your blood tests come back not as you'd hoped, so you experiment with acupuncture, enjoy the needles, take up studies yourself, quit your job at the bank, and wind up living in Chinatown with a Pekingese dog, a new hairdo, and your once-overwhelming condition in retreat.\n\nWe watch our destinations change with every loop of the maze. What mattered yesterday seems absurd; today's resolution is already passing. We're severely aware of our lack of control. \"Take your hands off the wheel,\" says a teacher of mine, \"and you'll see what's really driving!\" He's right, I'm sure, but this requires a difficult faith in the power of life to actually guide you. Such trust is part of the secret, it seems, to finding your way to the place you're going.\n\nJust take the case of Jim MacLaren. A superjock from San Diego, California, the bionic six-foot-five-inch, three-hundred-pound, blue-eyed all-rounder went to Yale on lacrosse and football scholarships, took up acting, moved to New York, and was just leaving a late-night rehearsal on his motorcycle when he was broadsided by a forty-thousand-pound city bus, flew eighty-nine feet in the air, and was pronounced DOA at the hospital.\n\nAfter eighteen hours in the operating room, doctors managed to stabilize Jim's condition but were forced to amputate the comatose patient's left leg below the knee. Jim hopped, then skipped, then ran his way back to full mobility like a champ, returned to school, took up swimming, then became interested in triathlons. The very unlikelihood of his being able to compete pumped Jim's inner competitor to do it against the odds. Within three years Jim had set records in the New York City Marathon as well as the Ironman Triathlon in Hawaii, where he competed with able-bodied opponents.\n\nThen on June 6, 1993, his true descent into the maze began. Riding his bike near his home in Mission Viejo, California, the amputee was plowed into by a van mistakenly waved through by a traffic marshal. The collision broke Jim's neck at the C5 vertebra and left him paralyzed from the neck down. Defeated, the ex\u2013All Star withdrew to Honolulu and spent most of the next few years becoming a drunk and a cocaine addict, hurling just as much gusto into booze and blow as he had into the long jump and javelin.\n\nGrueling as his previous ordeals had been, it was here, during Jim's dark night of the soul, that his most shattering insights began to occur. Having interrupted his substance abuse, he became aware that his greatest suffering was not actually coming from the accidents but from a source beyond his crippled body. Physical rehab had been a kind of smoke screen; now Jim was meeting his true nemesis head-on: overcoming the depression and addictions that now threatened to kill him. Alcoholism has been likened to a misplaced prayer; certainly, addicts are individuals who appear to have lost their inner compass. With Jack Daniel's and cocaine no longer working, Jim was forced to pull his own mask aside and take a good hard look.\n\n\"The first thing I had to do was identify my absolute deepest fear about all this,\" Jim told the writer Elizabeth Gilbert. What was the worst thing about having to spend life as a quadriplegic? he asked himself. Was it fear of death? Not really. He had had two near-death experiences already, \"with the white light and the tunnel and the whole deal,\" amazing encounters that had virtually removed his fears of dying.\n\nWas he afraid of losing his sexuality? No, Jim said again. \"I knew as long as I had taste and smell and sensation, I could lead a sensual life.\" Was he afraid of helplessness? Not really. \"Managing on my own is a drag, but it's just logistics.\" Was he afraid of pain? No, he knew how to deal with pain. \"So what was I afraid of?\" Jim wondered aloud. \"The answer was pretty clear,\" he said. \"I was afraid of being alone with myself, with my mind, with the dark things that lived in there. The doubt, loneliness, and confusion. I was afraid of metaphysical pain.\"\n\nLooking inward, Jim came to understand that his greatest pain arose from a damaged sense of wholeness. This is a common refrain among survivors I've spoken to; material difficulties aside, it's the imagined loss of wholeness, of feeling intact, that wounds many of us most deeply. But what is wholeness, really? Jim was forced to ask himself now. What did it mean to lead a full life? What were his actual obstacles? After he had delved into these questions a long time, a new awareness arose in him. Jim realized that as frustrated as he was with his handicap (and envious of the able-bodied), if he could get up out of his wheelchair and walk across the room, that wouldn't really get him to the place he most wanted to go in his life. Because if he was honest, the other side of the room was not his ultimate destination. His ultimate destination was self-knowledge and enlightenment. Did he have to get there on foot? Or could he find some other path?\n\nEventually Jim did get sober. Today he travels the world in his wheelchair, giving lecture-sermons, kicking butt, still handsome and blue-eyed with his big, burly shoulders. He shows people that it can be done, helps them to navigate their own labyrinths\u2014even take their hands off the wheel when they can\u2014en route to their secret destinations. It's not the life that he had planned for himself\u2014never in a million years. But Jim insists that it led him, however circuitously\u2014and doubters had better believe him\u2014 precisely where he needed to be.\n\n## **HOME IN THE WORLD**\n\nWe learn to live heroically, though we may not always feel like heroes, watching people like Jim MacLaren wrestling strength from such vicious hardship. The spectacle of dignity wrought from despair is always elevating to behold. The Greeks had a word for this, of course: _catharsis_ (literally \"to cleanse\")\u2014that rousing, purging renewal of spirit that comes from witnessing heroic acts through a humbling mirror of terror and pity. In 1889 Robert Louis Stevenson voyaged to the infamous Kalawao leper colony on the Hawaiian island of Molokai, a place filled with mind-boggling pain and injustice. Though Hansen's disease is not especially contagious, lepers had been stigmatized and imprisoned since ancient times. The residents of Molokai, for no medical reason whatsoever, had been taken from their families and forced into these plague-style encampments in paradise. Stevenson, whose life had been relatively sheltered till then, wrote in awe of it to his brother: \"I have seen sights that cannot be told and heard stories that cannot be repeated,\" marveled the author of _Treasure Island._ \"Yet I never admired my poor race so much nor (strange as it may seem) loved life more than in the settlement.\"\n\nLike all forms of beauty, heroism alters us when we see it. Indeed, we are taught that we must love something more than we fear death in order for our lives to have meaning, as Martin Luther King believed. Sometimes this reaching requires us to go far out on a limb before we can taste the fruit waiting there. I've yet to meet a post-catastrophe person who doesn't remember realizing, in a panicked moment, that his familiar context is gone (severely changed at the very least), his safety net exploded, leaving him exposed to the elements. Major change nearly always brings with it a wrenching sense of homelessness, feelings of bitter dislocation. This emotional uprooting forces us to adapt; in time we learn to pack up more quickly, even to carry our metaphorical homes inside ourselves, the place where we feel that we belong, which cannot be taken away from us. We're taught by this homeless feeling that _home_ means more than four walls and a ceiling. Home is where we find our balance, the pivoting point that connects us to the earth. Sociologists studying the homeless have examined this phenomenon. \"It is of more than semantic significance that we call these people 'homeless' instead of 'houseless' or 'shelterless,'\" I read somewhere. \"Home has an existential importance that reflects our discomfort at being on the earth in the first place.\"\n\nHaving felt homeless for much of my life, whether I had an address or not, I'm fascinated by the link between home and healing and wonder how people living on the street do cope. In Santa Fe, New Mexico, I stop by the St. Elizabeth Shelter and strike up a conversation with a wrinkled nun named Sister Jane, who's dressed in an Emmylou Harris T-shirt, with glasses on a pearl chain around her neck. \"Many of our homeless friends are highly artistic people,\" Sister Jane tells me cheerfully. \"Let me show you some of their work.\"\n\nI follow her along a corridor hung with dozens of brash, oversized canvases that, were they signed \"Basquiat,\" might bring in a small fortune in a New York gallery. It's obvious to me now why this wild, untrained school of _art na\u00eff_ is called survivor art. \"Often the homeless are far more aware of sights and sounds than those of us who are wrapped up in our so-called ordinary lives,\" Sister Jane tells me when the tour is over, seeing me off at the front door. \"We forget to see and hear. They take the time because that's all they have.\"\n\n\"In the noise and the business,\" Kierkegaard wrote, \"we're drawn away from our spirits. But in the stillness, when we're absolutely alone in the world, we can sometimes find a vision of justice and beauty that will ever afterwards infuse our lives with purpose.\"\n\nJonathan Kozol, the educator whose book _Rachel and Her Children_ explores the crucible of homeless life, believes that homelessness itself can be an art form. \"No one seeks out misery, of course,\" Jonathan says when I reach him by phone at his home near Boston. \"But many families I've known have been ennobled by their homelessness. Being removed from their ordinary context becomes a spiritual apotheosis. Women who spent their days watching soap operas are suddenly quoting from the Psalms. They're drawn closer to that intense edge which gives life significance.\"\n\n\"You're not romanticizing misery?\" I wonder.\n\n\"Not at all,\" Jonathan assures me. He recalls a recent New Year's Eve at the Martinique Hotel in Manhattan, the infamous way station for the homeless, and one particularly inspiring clan he met there. \"This family had become like ministers to the tortured souls around them,\" he says. \"They lit a candle on the mantelpiece of their ugly, filthy room and made an altar. Each of us had a glass of wine and read prayers at midnight. It was more sacred to me than anything I've felt in a church or synagogue.\"\n\nThis is poignant to imagine. \"The sense of religious feeling just soared when I was in that building,\" he tells me. The writer contrasts this display of compassion to then-cardinal John O'Connor's refusal to allow the homeless to sleep on the steps of St. Patrick's Cathedral at Christmastime\u2014\"a betrayal of Christianity,\" in his opinion. \"If we could begin to see the homeless not as deficient human beings,\" Jonathan says, \"but as metaphors of the fragility of life for all of us, as epiphanies...\" His voice trails off. \"Only then will we not be able to drive them away.\"\n\nLittle has been written about homelessness as a metaphor for impermanence and what we can learn from people who face its wilderness daily. To begin with, a single afternoon at a soup kitchen reveals immediately that there is no homogeneous group known as The Homeless. There is only _homelessness,_ an archetypal state of transience, isolation, and insecurity everyone senses at one time or another (more intensely in times of crisis). As a population, the homeless are as diverse as the human family itself. From suburban housewives to Ph.D. holders, health-care professionals to dope-smoking teenagers, war vets (there are statistics showing that some 25 percent of the homeless in this country fought in Vietnam) to former heads of companies, we discover all kinds when we look to the streets.\n\nIn a park across from San Francisco's Glide Memorial Church, where fifteen hundred free meals are served weekly, I sit down with a thirty-nine-year-old West Indian man named Danny Williams, who's been homeless for the past ten years. Danny came to San Francisco in the late 1970s to attend college but fell into alcoholism after experiencing an emotional crisis. Though he's still drinking, Danny insists to me that he's not lost.\n\n\"How do you think homeless people survive?\" he asks me. \"The ones who got everything, they don't need to pray. I pray every hour. But the only church I go to is in my heart.\"\n\n\"What do you pray for?\"\n\n\"I just tell God that I'm here!\" He chuckles. \"I pray that he won't let me lose my dignity. I pray that if I'm paying for all the bad I done, he'll forgive me.\" Danny looks almost tearful when he says this. His neighbor on the next bench chimes in to help his buddy. \"That's how most homeless people live,\" says this greasy-haired dude with a glass eye and a navy tattoo on his forearm. \"You might see scars and our attitude might be bad, but we always got that hope.\"\n\n\"I can sleep right here in the street,\" Danny tells me. \"But he's there with me. I just say a prayer and hit the sack, wake up, find me food, find me a drink, and he's there. I don't even care what happens to me down here, 'cause I know that when I get up there to meet him, I'm gonna be on his right side.\"\n\n\"No matter what happens?\" I ask.\n\n\"No matter what,\" Danny says.\n\nA passerby, Tanya, throws in her two cents. \"We're all just captive spirits, honey,\" says this fat black Rasta babe in orange flip-flops, holding a mischievous toddler by the hand. \"I don't consider myself homeless even if I'm sleeping in a doorway, 'cause home is where you _is,_ \" Tanya tells me. She does a hand-on-the-hip thing (\"you dig?\"), then wipes Popsicle goo off her toddler's mouth. \"Sometimes I'm more at home nowhere than in my mama's house.\"\n\nTanya waves to an old guy on the breadline. William \"Terry\" Stiles introduces himself and invites me to keep him company while he eats his stale rolls and noodle soup. Terry has been living on and off the street for twenty-five years. He has long, bony hands and a film of glaucoma creeping up the sides of his blue-circled, dark brown eyes.\n\n\"God is like the air,\" offers Terry, chewing his roll, displaying a toothless grin.\n\n\"How do you figure?\" I ask.\n\n\"He's everywhere,\" the old man tells me, his mind drifting to higher thoughts. \"This life has been a great teacher. You know, it's taken me a few doorways and trips to the hospital to know who I am,\" admits Terry, who was a nurse's aide before descending into clinical depression and losing his job. \"This life out here,\" he says, meaning the street, \"it will humble you if it don't break you.\"\n\n\"What keeps you together?\"\n\n\"I know who I am,\" Terry explains. To look at him, that seems obvious. \"What most of you don't realize is that lots of us on the street are brilliant, well-educated people.\"\n\n\"So why do you stay out here?\" I ask.\n\n\"I wasn't always who I am now,\" he replies. \"I used to be grandiose and egotistical and left God out of the picture, which is why I fell.\"\n\n\"And now, Terry?\"\n\n\"I'm hanging out in the park,\" he chuckles. \"Ain't that a bitch?\" I follow his eyes across the lawn, where a group of pretty teenage girls are laughing and a family is having a barbecue.\n\n\"It is a beautiful day. But this life can't be easy,\" I say.\n\n\"It is hard to trust out here sometimes,\" says Terry.\n\n\"It's hard to trust everywhere,\" I reply.\n\nThe old man chews this one over for a moment. Then he says, \"Without my spirit, I'd go insane.\"\n\nWhen I shake Terry's hand and walk away, I look back over my shoulder and see him beginning to doze on his bench. Tanya takes my seat and lights a half-smoked cigarette. Her little boy gets to tickling Terry with his Popsicle stick; Terry is laughing and trying to swat him off. Has he found his home in the world? you could ask. Is home a metaphysical thing as much as a nailed-down address and a mailbox? Or is a home like a body, always there when we bother to notice? Are we really always _at home_ so long as we remain in the world? But if that's so, I can't help but wonder, why do we feel so often so uprooted inside our own skins? Do we feel so homeless because we've yet to locate the keys to interior castles? Or do we keep looking for somewhere different because we're accustomed to feeling lost?\n\n## **THE ART OF LOSING**\n\nViktor Frankl was a thirty-seven-year-old psychiatrist on the autumn morning he and his wife were abducted from their Vienna apartment and shipped to Theresienstadt death camp. On the train platform Viktor and Tillie were torn apart; then, for the next three years, Frankl endured the horrors of camp life we know about all too well. A brilliant student of human nature (his specialty was suicidology), Frankl spent his years in the camp observing the behavior of his fellow inmates, from the lowliest of the _Muselm\u00e4nner_ \u2014those \"walking dead,\" who had lost their will to live\u2014to prisoners at the other end of the spectrum, who'd kept some dignity in spite of their cruel surroundings.\n\nFor Frankl, such dignity\u2014indeed, his very survival\u2014centered on his love for Tillie and the hope that they would be reunited (in fact, she would die in the camp). Frankl would establish himself in the next few years among Europe's most renowned psychiatrists with a school known as logotherapy. Logotherapy, from _logos,_ or \"meaning,\" holds that our ability to live most fully depends on finding meaning in our existence. In _Man's Search forMeaning,_ Frankl's classic study, he described witnessing such hard-won dignity in action:\n\nWe who lived in concentration camps can remember the men who walked through the huts comforting others, giving away their last piece of bread. They may have been few in number, but they offer sufficient proof that everything can be taken from a man but one thing: the last of the human freedoms\u2014to choose one's attitude in any given set of circumstances, to choose one's own way.\n\n... Every day, every hour, offered the opportunity to make a decision... which determined whether you would or would not submit to those powers which threatened to rob you of your very self, your inner freedom; which determined whether or not you would become a plaything of circumstance, renouncing freedom and dignity...\n\nFundamentally, therefore, any man can, even under such circumstances, decide what shall become of him\u2014mentally and spiritually. He may retain his human dignity even in a concentration camp. Dostoevsky said once, \"There is only one thing that I dread: not to be worthy of my sufferings.\"\n\nThis sounds like an odd aspiration at first, being worthy of bad things that happen to us. Yet our very resistance to being broken by evil is what prevents us from becoming playthings of circumstance. Without this dignity, Frankl suggests, we might as well be animals.\n\nThis dignity is a form of grace. Even the most courageous need it to come through slaughter without being destroyed.\n\nA seven-year-old boy is sitting in a dusty marketplace near his family's farm in the hinterlands of eastern Sudan during the early harvest days of 1986. His name is Francis Bok, and he is here with two of his sisters to sell peanuts, eggs, and assorted produce from the family's homestead, which is among the most fertile in the valley. As the strongest son of eight, Francis is his father's favorite, the boy he calls _muycharko_ \u2014a Dinka word meaning \"twelve men\"\u2014the offspring his father has chosen to replace him as head of the family after he dies.\n\nThe children are spread out on a crimson blanket under their lean-to of burlap and sticks, laughing, surrounded by their wares. The rainy season hasn't started; the air is still hot and dry, brimming with the pungent aromas of fish and fresh tobacco leaves, racks of glistening meat hung from hooks in the close-packed stalls. Business is good; Francis is happy collecting coins in a purse his mother has sewn for him to keep inside his clothes.\n\nAll of a sudden, bedlam breaks out. Francis hears gunfire and galloping horses, then crowds scattering in all directions as a band of black-turbaned marauders storms into the marketplace. These are the dreaded _juur,_ members of the Arab tribes to the north who've sworn genocide on southern black Christians like Francis and his family. As Francis watches in terror, his five-year-old sister hiding behind him, these _juur_ begin to hack down the marketers in their path. Francis's neighbor, a girl of eight, is shot through the head for no reason; when her sister becomes hysterical and refuses to leave the girl's dead body, her leg is hacked off at the hip. Women and babies are bayoneted and left to writhe to their death in the dirt.\n\nTerrified, Francis herds his sisters behind him. Before they can escape, a man is there on horseback, towering over him, pointing a gun at Francis's head and ordering him into the saddle behind him. Then the horse is galloping out past the road to Francis's farm and onto the desert highway leading toward Khartoum. Fourteen hours later Francis and his kidnapper, Giemma Abdullah, arrive at Giemma's farm, where Francis is shackled, locked in a pigsty, and told that he is now an _abeed_ \u2014a slave\u2014who will live and eat with the animals. If Francis disobeys in any way or tries to escape, Giemma assures him, he will be killed.\n\nWhat goes through the mind of a seven-year-old boy kidnapped and thrust into such a nightmare? There's numbness first, then disbelief, an armadillo's self-curling protection when \"everything went dead\" in him, as Francis will tell me when we meet. Shock is followed by wrenching grief, loneliness, and humiliation. For no reason whatsoever, Giemma's wife pulls Francis into the kitchen one morning, puts a pistol to his head, and says, \"I would blow your brains out this very minute if I could,\" with an evil smile. Giemma's children are invited to beat Francis with rocks and sticks as a game. \"I wondered why no one was helping me,\" Francis will tell me. \"People just stood there and watched. Why would they do that to me? I was only seven years old.\"\n\nNight after night in his pigsty, gazing up through a hole in the roof, Francis began to plot and replot his escape from Giemma's farm. \"Better not to live,\" he decided, \"than to live as a slave.\"\n\nHe looks like a Giacometti sculpture dipped in blue-black ink and polished. His body, at six feet seven inches, is impossibly thin, severely chiseled, astoundingly odd in its elongation. I've been led to Francis Bok by an antislavery group in Washington, D.C. Francis, now twenty-four, has invited me to visit him in the town outside Boston where the group helped find him an apartment. He wants to tell me the story of how after ten years of slavery he managed to escape to freedom.\n\nFrancis crosses his stiltlike legs and observes me with an expression neither friendly nor disinterested. The little apartment, with its thrift store furniture, basketball posters, and color TV, could belong to any college student (Francis is taking courses at a nearby university). This backdrop seems a universe away from the fields where he and his brother's grew up playing _alweth_ (hide-and-seek) near their village of Gourion.\n\n\"I reached the point inside of myself that I didn't care what would happen to me,\" Francis tells me in a whisper. \"I didn't care if I lived or died. I only knew that I must try again.\"\n\nTen years after his abduction, and after a number of close calls, the seventeen-year-old was finally able to slip off Giemma Abdullah's farm while herding cattle, hitched a ride to a local town, and then, after several detours, found his way to Khartoum (where he was arrested on suspicion of illegal activity) and finally to Cairo. After four months in Egypt, Francis was taken up by a human rights group and granted asylum in the United States, where a foster family supported him long enough for Francis to afford the small apartment where he and I are now sitting.\n\n\"Do you like it here?\" I ask awkwardly.\n\nFrancis doesn't say anything. Though outwardly polite and easygoing, he's markedly inaccessible, too, circled by a moat of caution, his voice remote and difficult to hear. I'm reminded of Nawang Sangdrol, the Tibetan nun, who was comfortable while reporting the terrible facts of her public story but turned dreamy, cagey\u2014the word that comes to mind is \"porous\"\u2014when invited below the surface.\n\n\"How is your life?\" I try again.\n\nFrancis grins and says, \"I am well.\"\n\n\"Really?\" I don't mean to sound so surprised. But it's not the answer one would expect from a man who was forced to sleep in a pigsty, eating discarded kitchen scraps. \"What about what happened to you?\"\n\n\"I am a Christian,\" Francis says, as if this were sufficient explanation.\n\n\"I know a lot of bitter Christians,\" I say.\n\nFrancis takes a slug off his Pepsi. He gazes through the venetian blinds at a girl in hot pants walking her dog. \"I can tell you this,\" he says in a voice so low I can hardly hear him. \"No matter what happens, I try to take it soft.\"\n\n\"What do you mean by soft?\"\n\n\"I learned much about life on Giemma's farm.\"\n\nJust then, Francis's demeanor begins to shift. He leans forward and speaks in a deeper voice. \"I have learned that it does not matter how much you are beaten or how much you are despised,\" he says, suddenly sounding like a man. \"There is one thing they can never take from you.\"\n\n\"What, Francis?\"\n\nHe sits up straight. \"They can never steal your idea of who you are,\" he says. \"They can never take control of your mind. They can never take away your self-love.\" It's suddenly easy to imagine him preaching at a pulpit. \"They cannot take away my forgiveness. They can never silence my heart. No matter what has happened, I see myself as a full person.\"\n\nThis is astonishing. The majority of people I know haven't lived through a fraction of this loss yet do not see themselves as full people. Francis stands to his full height and ducks under the kitchen doorway for more soda.\n\n\"How is that possible?\" I ask.\n\nHe pops our cans open and pours. \"If you want to live, you must fight for your freedom,\" he says to me. As a Christian, Francis has faith in mercy but also believes in the struggle for justice. \"When you abandon yourself, you lose. If you are alive in this world, you must fight. But never with a sword.\" He's back on the couch now, his long fingers occupying themselves with a rubber band. \"My people are using machine guns to shoot the Arabs who are slaughtering us. They believe that is the way to freedom, but it is not.\" I'm stunned when Francis informs me that there are twenty-eight million people currently living as slaves around the world.\n\n\"How do you deal with the memories?\" I ask him. \"How do you balance out what was taken from you?\"\n\nFrancis thinks about this for a moment. \"It is not easy feeling normal.\"\n\n\"You're not normal,\" I say. \"That's a good thing.\"\n\n\"I am very happy that you think so,\" he says, finally giving me a smile. Then he reaches for a paperback book on the coffee table. It's Dickens's _A Tale of Two Cities._ Francis reads from the first page. \"It was the best of times, it was the worst of times,\" he recites, picking his way through the English letters. \"This is what my life has been like,\" he says, still smiling. \"The best. And the worst.\"\n\n\"This is the best part?\"\n\n\"I have been given a second chance.\"\n\n\"You forgave Giemma?\"\n\nFrancis closes the book and puts it down. Apparently he hasn't quite made it to sainthood. For a moment he seems to disappear behind the soft-focus screen I noticed at first. Then he looks me straight in the eye and says, almost word for word, what the Tibetan nun told me. \"No one can move forward in hatred.\"\n\nI realize this, I say. But what does he _feel_ ?\n\n\"And dignity does not come from bloodshed.\" Francis cuts my American-style nit-picking off at the pass. \"There must be a way\u2014you must find it\u2014to love.\"\n\nHe is struggling with himself. I can see that.\n\n\"There is no other way,\" Francis tells me. \"Otherwise I am just like Giemma.\"\n\n## **A QUARTER INCH FROM HEAVEN**\n\nHakuin was a great seventeenth-century Zen master, a serious, shit-kicking old guy in black robes who would whack his disciples when they slouched and could meditate for days at a time. Hakuin was not afraid of anything.\n\nOne day a samurai arrived at the master's mountain _zendo._ The samurai approached Hakuin and bowed. \"Sir,\" he announced, \"I wish to understand the difference between heaven and hell.\"\n\nHakuin was the picture of disdain, eyeing the samurai from head to toe like yesterday's chop suey. \"I would tell you,\" the old man said, twirling his silver mustache. \"But I doubt that you have the keenness of wit to understand.\"\n\nThe samurai's face blazed with wounded pride. He pulled back in astonishment. \"Do you know who you are speaking to?\" he asked, puffing out his astounding chest.\n\n\"No one much.\" Hakuin shrugged. \"I really think you are probably too dull to understand.\"\n\n\"What?\" the samurai demanded, unable to believe his own ears. \"How can you speak to me in such a tone?\"\n\n\"Oh, don't be silly,\" Hakuin mocked him. \"Who do you think you are?\" The samurai trembled with fury. \"And that thing hanging from your waist,\" the teacher added. \"You call that a sword? It's more like a butter knife.\"\n\nFinally the samurai could take it no longer. With sweaty hands, he drew his sword and raised it over his head to strike.\n\n\"Ah,\" Hakuin said. \"That is hell.\"\n\nThe samurai spun around in his own mind. His eyes then shone with recognition as he lowered the sword and sheathed it.\n\n\"And that,\" the old guy told him, \"is heaven.\"\n\n## **THE GIRL ON THE ROCK**\n\nOne rainy afternoon in London, under an ominous mackerel sky, I find myself in a back corridor of the Tate Gallery, standing before an eighteenth-century painting I have never seen before. It is an oil-on-canvas portrait of Hope, the allegorical goddess, as depicted in 1886 by George Frederick Watts. This is no triumphant, trumpeting Hope (no hope springs eternal, let's sound the trombones!). Instead, this Hope is a waifish thing stranded on a lonely cliff, barefoot, tempest torn, eyes concealed behind a blindfold as she reaches her empty hand out toward a harp with only one string.\n\nI'm mesmerized by this mysterious picture, the supplicating pose of the girl whose face is almost hidden from view, fingers straining toward an instrument that offers her only chance of music, yet has only one string still intact, as likely to snap as it is to play. This is how life feels at times, isn't it? You've been blindfolded and left in the dark, knowing that even as you reach into your next moment, the string may simply snap in your face. But that if you don't reach, you're not really living.\n\nI learned this the hard way myself during a routine visit to my onetime doctor's office. This was years before antiretroviral treatments had been invented. I excitedly showed him a magazine article announcing a new prophylactic drug trial for which I might be eligible. I waited nervously for the doctor's response. He glanced at the clipping and handed it back.\n\n\"Listen,\" this substandard whitecoat told me with a patronizing smirk. \"Whatever makes you feel better...\" He trailed off without bothering to finish, as if he were wasting his breath on a child.\n\nI took the clipping, opened the door, left his office, and never returned. If I was on the verge of croaking, I resolved, it would not be while staring up into those cold, tired, angry eyes. My posse of friends questioned the hasty decision, but I knew it was the right thing. Were I to lose all hope, I would be a goner, whether my body survived or not. Souls survive on hope in the absence of physical evidence. Not the na\u00efve hope that everything will be hunky-dory, exactly as life used to be, but the hope that assures us, when things seem darkest, that although it doesn't look that way now, _something else is also true,_ as one survivor put it to me. That there is a hidden face to this moment. Such hope serves the same survival function as faith and denial, preserving a space in the shrinking mind for all that has yet to be revealed, leaving a chink for the mysteries. \"The function of intelligence _in extremis_ is not to judge one's chances, which [may be] nearly zero, but to make it through that day... without thinking too much about tomorrow,\" as one veteran from World War II said. Even when we lose hope for a particular outcome, we may find ourselves experiencing a more general faith in the power of life itself.\n\nHope is a metaphysical power, the breeze stirring in the darkness  . Stan Rice described this mysterious force in a poem. \"I was lost,\" he begins,\n\nand sang my broken down songs in the hell of the hour.  \nThen in my heart moved an oar,  \nand I was found by a breeze from a door in the sea of forms  \nAnd was rowed to the cherry trees on the shore.  \n_Selah Selah_\n\nWhen I left Dr. G.'s harbor of doom, the sails of my boat filled again and got me to Dr. Bellman's office.\n\nFixated hope is a problem, though. When we attach ourselves to a single outcome, it's easy to become hope's hostage, to imprison ourselves in optimism, entrap ourselves through inflexible craving for a premeditated result. Hope of this kind brings sure disappointment. We risk spending our time consumed by longing, obsessed with all the things we don't have and unhappy with what life has chosen to give us.\n\nBuddhists have a word for this gap between fixated hope and its fulfillment: _dukkha,_ a far-reaching term encompassing the absence of ultimate satisfaction in an imperfect world where all things must come to an end. Knowing how stubborn the human mind can be\u2014 how it wants what it wants _now_ \u2014Buddhism warns against too much clinging to desired outcomes we cannot control. Having practiced for decades to release such hope, a teacher such as American-born master Pema Chodron can say something like \"If hope and fear are two sides of the same coin, then so are hopelessness and confidence\" with a mea sure of credibility. Pema assures us that suffering is inevitable as long as we believe that things last. Only by discovering \"ease with uncertainty, poise amidst shakiness,\" by learning to stay with \"the broken heart and rumbling stomach\" and achieving some d\u00e9tente with hopelessness, can we be truly happy, she says. \"In the world of hope and fear,\" writes Pema, \"we always have to change [what is]. But when we allow ourselves to feel uncertainty, disappointment, shock, embarrassment, we discover a mind that is clear, unbiased, and fresh.\"\n\nI'm sure this is true. But for the vast majority of us, hope within reason, like denial within reason, is a form of adaptive genius. Our spirits are kept unsealed by the very breath of hope. Spirit ascends; soul is earthbound; their intersection is human life. The contradictory mess of caring, hoping, and letting go appears to be our curriculum here. Many years ago, a number of meditation teachers I know went to Thailand, where they visited the hermitage of a teacher named Achaan Chah. In Thailand people tend to use the Buddhist abbots and monks the way people in this country use therapists or astrologers. One day a father from the town came to Achaan Chah's monastery, extremely upset, to ask the master how he could possibly live with not being able to protect his children in such a violent world. How could this man, this father, hope to survive his kids' tragedies, the thousand blows that life would deal them? Achaan Chah lifted a lovely crystal goblet from his side table and held it up to the sun.\n\n\"I like this glass,\" the master said, delighting in the diamond light patterns shining through its thousand facets. \"I find this glass very beautiful. When the sun shines through it, there are rainbows. When you test it, it gives a wonderful ring. But I know that this glass is already broken.\"\n\nThe worried father did not understand.\n\n\"Each time I sip from this glass, I enjoy it,\" the master continued. \"And yet, when a strong wind tips it over or I knock this glass with my elbow and it shatters into a thousand pieces, I will say, 'Ah so, it was already broken.'\"\n\nAchaan Chah seemed to be suggesting to the father that were he to love his children in this way, each moment he spent with them would be so direct, and so precious, that there would be no room for regret, no necessity for hope. Acceptance would trump hopelessness.\n\nI wonder if this is true, and what it would mean to the girl on the rock. Whether this lesson would strip off her blindfold, stop her from reaching, seeing that her harp was already broken. Or whether the girl would still want to play because playing itself is in her nature, knowing that one string can be enough, and that if that last string breaks, she can always sing.\n\n## **GOING TO TAHITI (OR RAISING HEAVEN)**\n\nPsychologists agree that transcendental experience is as necessary to emotional health as family, friendship, sex, and work. Religions are born from this hunger for transcendence, of course, as is the taste for mind-expanding substances, which have been enjoyed by human beings, both ritualistically and recreationally, since recorded history began. From at least the year 5000 B.C. onward, our ancestors have employed a vast number of psychotropic plants to heal themselves physically, to enhance insight, and to ease suffering from what a Huichol shaman describes as \"loss of spirit\" (what we might call the blues). The Rig Veda, the most ancient of Hindu texts, speaks of ecstasy derived from a plant known as soma, which Aldous Huxley later introduced into mainstream awareness with his 1932 novel, _Brave New World._ (In fact, diagnosed with cancer at age sixty-nine, Huxley requested that a tab of LSD be administered to him on the day of his death.)\n\nHuxley was a firm believer in better living through chemistry. \"If we could sniff or swallow something that would, for five or six hours a day, abolish our solitude as individuals, attune us with our fellows in a flowing exaltation of affection and make life in all its aspects seem not only worth living but divinely beautiful and significant, then it seems to me all our problems would be wholly solved and earth would become a paradise,\" wrote the futuristic author.\n\nHe'd never been to a Grateful Dead concert, I guess. Not that I'm against getting high. I like an occasional jazz cigarette. I've been smoking weed on and off since high school and have yet to bury a spike in my arm or hear Son of Sam voices inside my head. I did lose control for a couple of years in my teens, but learned my delinquent lesson the day I ate too many quaaludes, totaled three cars, attacked the cop trying to help me after my head smashed through the windshield, and got thrown into jail for the third time that year. When my poor mother came to bail me out, seeing my bloody, wasted self, she just shook her head and said, \"You're gonna end up in Alcatraz.\"\n\nShe was right, which is why I cleaned up my act and drew the line at occasional reefer. When Ecstasy became popular in the 1980s, I wasn't even vaguely tempted. That is why, after receiving two horse-sized capsules of pure MDMA (Ecstasy) in the mail from a well-meaning chemist (as \"part of your research,\" his note read) for a piece I was writing about E, my own conflicted reaction surprised me. This package had come to me uninvited. I'd long ago made the firm decision not to voluntarily leave the three-dimensional world again (and not tweaking, for sure, on some stupid narcotic). I'd worked too hard to build a sane life. I had been told E was different, it's true; it wasn't a hallucinogen, but an empathogen, in fact, a \"heart-opening\" substance, with the reputed power to enable its users to dissolve emotional barricades in themselves\u2014lucidly\u2014in order to free themselves when sober. Until the Feds upgraded MDMA to a Schedule I drug in 1985 (making its use a felony), therapists had used the stuff for the treatment of post-traumatic stress disorder, addiction, and a number of other hard-to-crack maladies. \"Our culture is the first ever to have made the search for self-awareness a crime,\" chemist Alexander Shulgin complained at the time. In a booze-blasted culture like ours, this double standard did seem questionable. And yet, I was far from convinced.\n\nThe horse capsules tempted me from their drawer. It wasn't the prospect of getting high that stuck in my craw. It was the drug's self-liberating promise, and what my staunch refusal told me about myself. I was forty-five, my health was great, I was nested and safe and writing and flossing, but feeling a little too settled as well. I used to be so bold, I moaned. I used to want to taste the whole thing, blast through walls, let caution be damned. Now I wouldn't take a hit of E, which a friend of mine actually buys from her rabbi?\n\nA gauntlet had been tossed, it seemed to me. E morphed in my mind from a few hours' buzz into everything that I feared in my life, the seeker in me who'd gone to sleep. Here were these capsules siren-calling from their drawer, offering me a free ticket out of my bourgeois box, an entr\u00e9e to a carefree zone of envelope pushing, surprise, even bliss. How could I be afraid of that? If Gauguin hadn't taken that boat to Tahiti (miserable wretch though he turned out to be), he would never have painted those brown-breasted women for museum hoppers to ogle down through the ages. I needed to get out of Dodge for a minute. I needed to go to Tahiti.\n\nCoincidentally, around this time I was approached by a publisher looking for an editor to bail them out of an emergency. One of their star authors, Dr. Richard Alpert, aka Ram Dass, had suffered a massive stroke before completing what they believed would be his final bestselling book. A hero of the sixties counterculture, the pilot fish to a generation, Alpert had started out as a Harvard psychology professor before being fired for performing psilocybin experiments on his students (along with his crazy office mate, Timothy Leary). Alpert had dropped out, gone to India, fallen in love with a guru, been rechristened Ram Dass (Servant of God), and been sent back to the States with marching orders to write a book, which became the iconic _Be Here Now,_ once the third-best-selling title in the English-speaking world, after Dr. Spock and the King James Bible. Ram Dass had spent the forty years since then as a humanitarian and spiritual teacher.\n\nNow R.D., as he likes to be called, had survived this massive cerebral hemorrhage (against nine-to-one odds) but been left wheelchair bound and severely aphasic, unable to dress or feed himself, much less finish a book. R.D. desperately needed someone to help extract his mots justes and ventriloquize this stuff onto paper. I caught a flight to San Francisco a few days later to see if the two of us would hit it off.\n\nThe first time I catch sight of R.D., he's cramped into a wheel-chair on the porch of his Victorian home in Marin, smoking a joint and chuckling, unruly white hair haloed around his head like a cartoon scientist's. I'm taken aback by the dramatic difference between Ram Dass as the world knows him and this overweight senior with Kleenex stuck to the side of his face and a fey Filipino nurse fussing at him from inside the house that it's time to take his nap\u2014please!\n\nOur chemistry seems to work (even without additives), so over the course of the next six months, I spend many long days with R.D., picking through his haphazard pages, struggling to draw what he's trying to say from the tangled web of his halting brain. With one arm strapped to the chair, barely able to speak, he alternates between crankiness and serenity, distraction and discomfort (R.D.'s arm hurts a lot), assailing me with grunts and half sentences, annoyed when I can't help interrupting him, and eager for this last book of his\u2014which he's calling _Still Here_ \u2014to be a faithful account of what he has learned through this physical ordeal. R.D. wants the \"fierce grace\" of \"being stroked\" to infuse this manuscript about conscious aging. As he helped lead the flower power crowd toward higher consciousness, he now wants to teach these same baby boomers how to face aging creatively and with a measure of grace.\n\nEvery few weeks, I fly out to Marin, where R.D. and I labor together on his book, a photo of his smiling, tom-tom-bellied guru lolling on a plaid blanket always nearby. In exchange for my grammar and patience, R.D. offers me an intimate tutorial on fierce grace.\n\nNow and then I bring up a personal question. \"Someone sent me some Ecstasy,\" I tell him one day while we're having lunch in his shambles of a kitchen.\n\n\"Lucky you.\" R.D. smiles. His blue eyes haven't lost their twinkle.\n\n\"I doubt that I'll take it,\" I admit.\n\nHe twirls an index finger slowly next to his head.\n\n\"You think I'm crazy?\"\n\n\"Scared,\" he says, struggling to get the food to his mouth without spilling. \"That is reason to try.\" I'm well aware that R.D. credits psychedelics with opening his own eyes to \"the big picture.\" Recently he told an interviewer, \"My entire adult life has been an attempt to grow into what I saw at that [first] moment of cosmic unity on LSD, to incorporate the immensity of that experience into my being.\" \"Anything less extreme would have been unlikely to break through my mind-set,\" R.D. suggested.\n\n\"I've gotten so middle-aged,\" I say.\n\nHe nods in agreement and makes me feel worse.\n\n\"I used to be such a hell-raiser\u2014\" I say.\n\nR.D. chuckles and struggles to find a word.\n\n\"What?\" I lean forward, trying to read his lips.\n\n\"Raise,\" he stammers, \"heaven.\"\n\nAt the time of his stroke a year ago, Ram Dass was a vigorous, sixty-six-year-old, golf-playing, sports-car-driving bachelor traveling the world incessantly, reveling in his role as a lionized sage to a generation of middle-aged seekers like me. As the heir to an affluent Connecticut Jewish family (his father founded the New Haven and Hartford Railroad), he was a pampered child, and later Ivy League academic, who threw it all in for the consciousness game. R.D.'s unique appeal as a teacher had always been his self-deprecating lack of holiness, his willingness to present himself as a work in progress, \"a poor shlub just like anyone else trying to lead an enlightened life.\" Richard Alpert the psychologist had thought that Ram Dass the seeker was doing a pretty good job, until his own near-death experience in 1997 showed him how far he still had to go.\n\n\"I wasn't cooked yet,\" R.D. tells me straight off. This is how he describes the night of the stroke (I've smoothed out R.D.'s speech to spare the reader's nerves): \"There I was lying in my bed, trying to imagine how to finish this book about aging,\" he says, \"having this fantasy of what it would be like to be a very, _very_ old man, when I heard the phone ring.\" R.D. can recall getting out of bed to answer; his next memory is of a group of firemen \"staring into this old man's face\u2014but the old man was me and I wasn't dreaming.\" (Fortunately, an assistant had arrived on the scene in time to call 911; a half hour later and he would have been dead.) Like many near-death-experience survivors, he describes watching himself being resuscitated \"as if from a doorway on the sides of the scene,\" an out-of-body vantage point that followed him into the ambulance and to the hospital.\n\n\"I was sort of fascinated by what was happening,\" he admits.\n\n\"You must have been frightened,\" I insist.\n\nR.D. denies this. \"I never thought that I was dying. There was no flashing white light,\" or anything else that a lifetime of spiritual practice had prepared him to expect at the moment of death. \"That's when I realized that I had a lot more work to do before the end.\"\n\nHis uncooked parts centered on body and pride, the humiliation of allowing himself to be cared for after a lifetime of giving service. Overnight, R.D. went from being the guy on the white horse, visiting sickbeds, heading philanthropic organizations, ministering to thousands of students, to not being able to pee alone or make himself a tuna sandwich. This was more humbling than any _seva_ (service) he'd ever performed in so-called spiritual life, he assures me. \"I'd distanced myself from my body as merely a vehicle for the soul.\"\n\nThis reminds me of a line from one of James Joyce's short stories: \"Mr. Duffy lived a short distance from his body.\" R.D. chuckles. \"I had ignored my body as much as possible,\" he admits. \"I'd tried to 'spiritualize' it away. Calling it detachment when actually it was fear.\" His body denial may, in fact, have helped to cause the stroke, since he neglected to take his blood pressure medications. Now here he was, trapped, at the mercy of others, wrestling daily with his suppressed vanity in the messy limits of his all-too-human body. This self-reckoning brought down walls in R.D. that he had never before dared to acknowledge.\n\n\"The stroke was like a samurai sword,\" he explains. \"The 'I' that I am now is not experiencing things the way the old 'I' would have. It's pushed me up to a higher level.\"\n\n\"Higher in what way?\" I ask.\n\n\"When there's genuine surrender between people, the boundaries between power and powerlessness, healer and helped, begin to dissolve,\" R.D. tells me. \"It is the dissolving of boundaries between us and the mystery that loosens the hold of the ego. Allows the soul to be revealed.\"\n\nThis soul perspective has been key to his negotiation of physical losses. \"You can't work with pain when you are stuck in fear,\" he tells me. \"Watching ourselves through the soul's eye allows us the distance to distinguish between who we really are, spiritual beings having a physical experience, and our suffering at the level of body and mind.\"\n\nR.D.'s nurse enters the room and slips a straw from a glass of orange juice between his lips. After swallowing, he continues. \"The minute you look at a fearful thought you've run from, it changes. Rather than being some awful Goliath, your fears become like little shmoos.\"\n\n\"Shmoos?\"\n\nLittle nuisances, he means. \"Every time you notice a fear, you learn to come closer to it. Fear of paralysis. Fear of dying.\" R.D. ticks off the usual suspects. \"Each time you do this, you're a little more able to take a deep breath and say, 'Ah, so. Big surprise.' And move forward without being swamped.\"\n\n\"Confronting the fear shrinks it down to size?\"\n\n\"Absolutely,\" R.D. assures me. \"It's as if the trapdoors of the self have been opened and we can finally step outside and enjoy the view,\" he says, recalling the famous line from William Blake: \"If the doors of perception were cleansed, everything would appear to man as it is, infinite.\"\n\nR.D. wants people to know that soul perception is our ace in the hole, the ability to widen our subjective lens to take in the full 360. Once we've tasted soul awareness, he believes, it almost doesn't matter whether we've got all our marbles or not. A student of R.D.'s was taking care of an elderly mother with Alzheimer's disease. \"This old lady was perfectly happy being gaga, but her daughter simply couldn't let go of the mother she'd known,\" he says. \"She kept pushing her mother to regain her memory. But this eighty-year-old woman didn't seem to care much about losing her mind. When I was able to work with the daughter to let go, the mother's life became much more peaceful.\"\n\nHe has met similar resistance among his own well-meaning caretakers. \"The people around me think that I should fight to walk again,\" he says, raising his eyebrows. \"But I don't know if I _want_ to walk. I'm sitting. That's where I am.\" He pats the arm of his chair with his good hand. \"I've come to love my wheelchair\u2014I call it my swan boat. I'm peaceful like this and grateful to the people who care for me. Why is this wrong?\"\n\n\"Maybe it's not.\"\n\n\"It's important not to get dragged into other people's drama,\" he says. \"It's too easy to become your illness, to lose yourself on your 'bed of woeses.' It is an ongoing practice to stay free of pity and fear. I may be confined by my stroke, but I don't want to be trapped in the roles that people project onto me\u2014of invalid, victim, hero, what ever. I'm truly content the way I am.\"\n\nI look at him askance. R.D. chides me with a wagging finger. \"In other cultures it's a symbol of honor and power to be carried and wheeled around,\" he insists. \"You know, it's really not at all important to be what the culture calls optimal. Healing does not mean going back to the way things were before. It means allowing what is now to move us closer to God,\" he explains. \"The secret is that our limits actually become our strengths if we use them skillfully. The ego's attachment to power is linked inextricably to fear of losing that power. But there is a kind of power that doesn't give rise to fear.\"\n\n\"Spiritual power?\"\n\n\"Exactly,\" R.D. answers. \"Behind the machinations of our brilliant, undependable minds is an essence that is not conditional,\" he says. \"A being that aging does not alter, to which nothing can be added and from which nothing can be taken away.\"\n\nThe face behind the mask, I think. \"How we feel about the future comes down to how we feel about mystery,\" he tells me. \"I've gone through the worst, and it really wasn't so bad after all.\"\n\nWhen the manuscript is finally finished, we celebrate with a meal and a smoke. R.D. thanks me for my time. I thank him for his sage advice.\n\n\"Did you ever take that E?\" he asks.\n\nOddly enough, the capsules have slipped my mind completely. Being with Ram Dass has left me so high, I've forgotten the need to go to Tahiti. I confess that the capsules are still in their drawer.\n\n\"I guess I really am middle-aged,\" I say.\n\nHe spins a finger around his temple once more. \"Speak for yourself, my boy.\"\n\nThe last time I see him, R.D. is sitting bare-legged on a single bed, clutching his mattress with taped-up fingers, Indian _kirtans_ playing in the background, his nurse slowly peeling socks down over his swollen ankles, the warm, still air of late afternoon falling in shadows across his small bedroom. His blue eyes follow me toward the door. The last sentence we wrote together still resonates in my head.\n\n\"While everything else falls away,\" R.D. told me slowly, extracting the words one painful syllable at a time, \"wisdom alone remains.\"\n\n## **THE NET OF INDRA**\n\nYears ago while volunteering at a hospital, I spent time with a man named Jack, who'd worked fifty years on an oil rig, had arms like a wrestler, and was now, at seventy-five, battling a tumor in his lung. Jack's physical pain was being managed with a morphine pump, his nursing care was impeccable, yet he seemed racked with metaphysical pain, as Jim MacLaren would call it, an isolation so profound that no number of visitors, narcotics, or games of five-card rummy with me were able to alleviate it. The first rule of volunteer training is never to presume to understand how the patient feels, as in \"I've been there, I understand.\" You haven't and you don't. The second rule is to check your cheerleader self at the door and resist the overwhelming urge to do something\u2014anything\u2014to raise their spirits or help them smile. You're there to listen, to be empathic, to put yourself aside sufficiently to be a container\u2014 in the therapeutic sense\u2014for what ever a patient might need to express.\n\nSo I struggled not to cheer Jack up, dealt the cards, didn't ask any questions, avoided his forlorn expression as much as possible. One day while I was in his room, the hospital chaplain poked her head in. Sister Loretta weighed three hundred pounds and looked a lot like Rosie O'Donnell. \"How's my favorite hunk?\" she asked Jack, scraping a chair across the floor to sit at his bedside.\n\n\"Lousy, Sister.\"\n\n\"Loretta's here,\" she told him, signaling for me to get lost. I backed into the doorway and listened. From where I stood, I saw Sister Loretta take Jack's hand. At first the old guy didn't say anything. Loretta waited. Then I heard him starting to sniffle. \"Talk to me,\" Loretta said.\n\n\"My father never loved me,\" said Jack. At this his tears broke into sobs. I was stunned to hear this coming from him, amazed that in the midst of a physical crisis, at a time when his health remained uncertain, the ghost of his father's absent love should be the thing that pained this tough guy the most.\n\nSuch glaring disconnects are common among survivors of all kinds, the gaps between what ought to be wrong and what really is: The homeless guy who wants conversation more than he wants pennies or food. The ex-POW who needs to belong somewhere more than he needs hosannas or financial aid. The individual surrounded by love who complains about not feeling \"cosmic connection,\" because God, she believes, left her high and dry the day she lost her beloved child.\n\nWe spend our lives in a kind of amnesia, sensing disconnection that doesn't exist. Einstein called this an \"optical delusion,\" imagining ourselves to be separate beings, cut off at the root from the rest of creation. \"A human being... experiences himself, his thoughts and feelings as something separated from the rest,\" wrote the father of relativity.\n\nThis delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us. Our task must be to free ourselves from this prison by widening our circle of compassion to embrace all living creatures and the whole of nature in its beauty.\n\nWe sequester ourselves inside our own minds, then project this awful abyss around us, picturing ourselves to be fenced-off, abandoned citizens of a private, bullying universe. This imaginary chasm seems only to widen during times of pain. Yet even under the worst conditions, strength in numbers continues to prevail. As one Holocaust survivor put it, \"Lone dogs died first.\" Mary Robinson, the ex-president of Ireland, observed this principle at work during the troubles in her country. \"It is in each other's shadow that we flourish,\" Robinson insisted to me when we spoke.\n\nThis is not sentimental pabulum. In a universe where boundaries do not actually exist, where waves and particles, protons and neutrons are indivisibly strung together, such baseline connection is obvious. In Indian philosophy this glistening, intergalactic jewel-work of matter (and antimatter) is known as the Net of Indra. This web is so tightly strung that \"the flap of a butterfly's wings on earth can be felt on the planet of Betteljers,\" as a physicist observed. If a full moon can make women menstruate, it's not so much of a stretch to realize that individuals in our lives are ricocheting off of us at every moment, creating positive or negative charges depending on their own chemistry.\n\n\"We're wired to connect,\" science writer Daniel Goleman tells me over lunch at his favorite Tibetan restaurant in Northampton, Massachusetts. Ten years ago Dan became a culture hero with _Emotional Intelligence,_ a seminal book that helped redefine what it means to be smart in our IQ-obsessed culture. Recently Dan has turned his eye to the workings of what he calls social intelligence, which includes the contagiousness of common emotions (think of giggles, yawning, tears, and screams) as well as the pharmaceutical value of keeping good company.\n\n\"The brain itself is social,\" Dan tells me over a plate of yak sausage. \"That's the most exciting finding in the past ten years.\" His gentle mien and thoughtful diction reflect his own thirty-year meditation practice (he was posted in India while doing his Harvard Ph.D. fieldwork). \"One person's inner state affects and drives the other person. We're forming brain-to-brain bridges\u2014a two-way traffic system\u2014all the time. We actually catch each other's emotions like a cold.\"\n\n\"Is that really true?\" I ask.\n\n\"If we're in distressing, toxic relationships with people who are constantly putting us down, this has actual physical consequences,\" Dan assures me. Stress produces cortisol, a chemical that hinders cell health. (He cites a study done on women caring for husbands with Alzheimer's, which found that their actual cell life diminished at an accelerated rate.) Conversely, positive interactions cause the body to secrete oxytocin, the chemical released during lovemaking, nursing, and delivery, which lowers stress hormones and amplifies the immune system.\n\n\"I have this experience often with my two-year-old granddaughter,\" Dan tells me. \"She's like a vitamin for me. Being with her actually feels like a kind of elixir. The most important people in our lives are actually our biological allies.\"\n\nIndeed, neuroplasticity, the discovery that the brain is always growing (not diminishing, as our grandparents believed), has revolutionized our understanding of how people evolve over the course of a lifetime. \"Stem cells manufacture one hundred thousand brain cells every day till you die,\" Dan explains. \"This defies what used to be the dogma. In fact, the brain continually reshapes itself throughout life with ongoing experience. It's where the maxim 'use it or lose it' comes from in neuroscience. The more you challenge it, the more the brain seems to rise to the occasion, and social interaction helps neurogenesis.\"\n\nTake something called mirror neurons, whose sole function is to reflect (in us) the things we see in the world around us. \"There are neurons whose only job is to recognize a smile and make you smile in return,\" he explains. \"The same goes with frowning.\" I'm reminded of the Michelangelo effect, in which long-term partners come to resemble each other over time through facial muscle mimicry. Such mirroring on a mob scale helps to create creepy-sounding things called memes, those oversized cultural ideas (Democracy! Hygiene! Infidels!) that spread through populations like viruses. \"By mimicking what another person does or feels, we bring the outside inside us,\" Dan tells me, speaking literally, not figuratively. \"To understand one another we actually become like the other a little bit.\"\n\nDan describes two kinds of relationships\u2014the I-IT and the I-YOU (first described by the philosopher Martin Buber)\u2014which have antithetical effects on our social lives. I-IT relationships happen when we treat people as objects or functionaries because we want something from them (in the way, perhaps, that Jack's father might have treated him when all the boy wanted was an I-YOU pat on the back). \"In I-YOU relationships, there's a human connection. There's feedback, a loop, because who the other person is, and what they have to say, matters.\" Unfortunately, the \"inexorable technocreep\" of our culture conspires against such intimacy, Dan believes. As T. S. Eliot presciently observed of our first major cultural social wedge, the TV set, back in the early sixties, television \"permits millions of people to listen to the same joke at the same time, and yet remain lonesome.\" Not only is constant digital connectivity stressful, science has discovered, but also, Dan says, \"to the extent that technology absorbs people in virtual reality, it deadens them to those who are actually around them.\"\n\nSince \"empathy is the prime inhibitor of human cruelty,\" as he reminds me, such alienation can have disastrous results. \"Withholding the natural inclination to feel with another allows us to treat the other as It\u2014as Them,\" Dan says. \"The more Thems we have, the more dangerous the world becomes.\" But how can it possibly be true that human beings are essentially altruistic or that \"the human brain is preset for kindness,\" as he has written? What about the newspaper headlines? \"Remember,\" the ex\u2013 _New York Times_ reporter tells me, \"if it bleeds it leads. We pay more attention to human cruelty. But it's an aberration to be cruel.\"\n\nThe famous Yale University Milgram experiment was not the last word on human nature, he assures me. Despite their reputation for being selfish savages, even young children demonstrate altruism from an early age, apparently. In one study, infants reportedly cried when they saw or heard another baby crying but rarely when they heard their own distress. Monkeys have been known to starve themselves after realizing that grabbing food delivers an electrical shock to their cage-mate. Dan makes reference to the philosopher Mencius's assertion that any conscious adult would automatically jump down a well to save a drowning child. Yes, I say, but do they cheat on their wives? \"We may not always be hooked up,\" he says, laughing, \"but that doesn't mean that the wiring's no good.\"\n\nThe link between kindness, survival, and social intelligence seems obvious. As a Harvard post-doc studying meditation in India, Dan noticed that seasoned practitioners tended to exude what he calls \"a special quality, magnetic in a quiet sense.\" Contrary to stereotype, these spiritual types did not seem otherworldly at all, but were \"lively and engaged, extremely present, involved in the moment, often funny, yet profoundly at peace\u2014equanimous in disturbing situations,\" as he describes it. What's more, this quality was _communicable._ \"You always felt better than before you'd spent time with them, and this feeling lasted.\"\n\nPhysicists and mystics agree on this point. The components of altruistic energy appear to be as measurable as photons and electrons; they are also more palpable than a skeptic might imagine, as San Francisco psychologist Paul Ekman reports to me after spending a week in Dharmsala with the Dalai Lama. \"At the airport afterward, my wife looked at me and said, 'You're not the man I married!'\" says Ekman, who is not a Buddhist, laughing. \"I was acting like somebody who's in love.\" The foremost authority on the physiology of emotion, Ekman detected four characteristics common to people with this contagious power: A \"palpable goodness,\" first of all, that went far beyond some \"warm and fuzzy aura\" and seemed to arise from genuine integrity. Next, an impression of selflessness\u2014a lack of concern with status, fame, and ego\u2014a \"transparency between their personal and public lives that set them apart from those with charisma, who are often one thing on the outside, another when you look under the surface.\" Third, Ekman observed that this expansive, compassionate energy nurtured others. Finally, he was struck by the \"amazing powers of attentiveness\" displayed by these individuals, and the feeling he had of being seen in the round, wholly acknowledged by someone with open eyes.\n\nIf these qualities were unique to spiritual masters, they wouldn't be nearly as compelling. What inspired Ekman the scientist was the evidence that such energy is available to the rest of us. \"It wasn't luck or culture or genes that created this qualitative difference,\" he tells me. \"These people have resculpted their brains through practice.\" Survivors with no knowledge of brain science often experience this phenomenon for themselves\u2014the way in which when we stretch past our limits, stretching becomes our second nature. Pushing the envelope seems to actually rewire our brains, adding a new repertoire of thoughts and emotions. When writer Andrew Solomon speaks in a later chapter of becoming more compassionate after his recovery from depression, for instance, this is more than Prozac speaking. It is an actual realignment of self through shifts in chemistry and neural conditioning.\n\nIn a laboratory outside Raleigh-Durham, North Carolina, a monk was monitored a few years back while meditating on compassion. Among other findings, scientists reported a dramatic increase in gamma waves (sparked in the part of the brain associated with positive emotions) while the monk focused on maintaining an open heart. Gamma-bumping like this requires ongoing practice. As a healer named Maxine Gaudio told me, \"Everybody can draw, but not everybody's a Picasso.\" Unfortunately, we can't even pick up the brush sometimes, much less locate the canvas. Such forgetting is our nemesis, teachers maintain. \"It is our daily dilemma,\" as Benedictine monk David Steindl-Rast tells me from his hermitage in Upstate New York. \"A spiritual energy flows through the universe, a super-aliveness\u2014an active _yes,_ \" says the eighty-year-old hermit. \"Yet even though our greatest happiness comes from feeling this eternal connection, there's a tendency in all of us to close off from it. Those who counteract the tendency through practice deepen their sense of belonging and free this latent energy.\" Brother David recommends such remembrance practice to his students. \"When we say, 'Count your blessings,' this is a very profound teaching,\" he stresses. \"A stream of energy\u2014of blessing\u2014is flowing from the universal source as blood pulsates from the heart. Knowing this, I'm energized and pass the blessing along to my brother so it flows again to its source.\" In this way, Brother David believes, \"we create a network of grateful living.\"\n\nThe Net of Indra, shimmering. Remembering our indissoluble connection might actually bring more love into our lives. \"It is love,\" Brother David assures me. \"The love which passes understanding.\" I've sensed this love myself in the company of genuine masters: a great, unstoppable, pulsating love that draws you toward its own radiance. This force could radically change the world, melt away borders, give hope for increased happiness. Another great Christian, Pierre Teilhard de Chardin, articulated this hope for all time. \"Someday after we have mastered the winds, the waves, the tides, and gravity, we shall harness... the energies of love,\" the French paleontologist-priest wrote. \"Then for the second time in the history of the world, man will have discovered fire.\"\n\n## **REINVENTING YOUR WIFE**\n\nAn eminent New York psychologist named Henry Grayson was seeing a patient named John who claimed to be married to the world's biggest shrew. As analysands often do, John appeared to want commiseration from his loyal shrink, but Grayson isn't that kind of doctor. \"What are you willing to do?\" he asked the unhappy husband.\n\n\"Anything,\" John assured him. Dr. Grayson's instructions were oddly simple. The next time John became anxious over his wife's behavior, Grayson said, he should focus on his own upsetting thoughts and replace his inner, wife-hating voice ( _she's ruining my life!_ ) with a tender memory of the woman he had married. At first John couldn't recall such a woman. Eventually, a happy memory oozed up from the distant past. He promised the doctor to give it a try.\n\nJohn seemed confused at his next appointment. His wife had been strangely subdued that weekend. \"She must be coming down with something,\" he said.\n\nGrayson told him to try the experiment again.\n\nThis time John appeared downright suspicious. He and his wife had spent their first tirade-free weekend at home in as long as he could remember. Maybe she'd started seeing a therapist secretly, John wondered out loud, still failing to connect the dots. It would take yet another session for John to realize that the shift in his internal monologue had actually helped manifest his wife's improvement.\n\n\"Behavior stemming from our own thoughts may manifest in the people around us,\" Grayson explains to me when we meet. He resembles a more dashing Mr. Rogers in khaki pants and a paisley tie. Like many of his peers in the field of transpersonal psychology, Grayson came to his insights by way of hard science. It was while he was attending a lecture in physics, in fact, that his work as a psychologist began to shift. \"This physicist helped me understand that the reality we perceive is a tiny fraction of the universe as it really exists,\" Grayson tells me. He began utilizing the Heisenberg principle, wherein objects change when they are perceived, on the interpersonal level. \"We are connected not only as human beings but as energy, mind, and matter,\" he says. \"We are interacting in profound, intimate ways we are rarely aware of.\"\n\n\"What about John and his wife?\" I ask.\n\nHenry Grayson smiles. \"Let's just say they're playing a whole new ball game.\"\n\n## **MAN THINKS, GOD LAUGHS**\n\nIrony can save a life. No survivor escapes without it. Often the irony is cruel, unbearably so if you can't laugh. \"What fresh hell is this?\" asked the ever-cheerful Dorothy Parker, sharpening her bitch nails on grief, lining up her exit options\u2014rivers are damp, drugs cause cramp, guns aren't lawful, nooses give\u2014concluding, as she once had in a poem, that \"you might as well live.\" At other times, fate plays the joke on you. Think of the great British psychologist Wilfred Bion, awarded the Victoria Cross during World War II and later noting, \"The only difference between getting that medal and being shot for treason was which direction I chose to run.\"\n\nOur most serious eventualities arise from the most ridiculous chance. Moments. Twists. Split-second decisions. Telephone calls that might have saved lives. Strokes of luck that bring surprise mishaps, accidents that carry you forward, encounters out of the blue that alter your life irreversibly, hesitations that prove disastrous, then morph into windows opening. Pinballing along in this way, we're reminded of how little we control, how little we know of what good or bad luck is, the way things should or should not be, what we deserve and what we don't, at any given moment in time. We only know that we do not know, and never will, and tough petunias. _Menschen tracht und Gott Lacht,_ my grandma Bella used to say in Yiddish. Man thinks, and God laughs. Your best-laid plans are really a punch line. Life is one insult after another. The bottom will drop; you will lose what you love; then you'll lose some more, hit your head on a rock, and sink into the lap of contentment\u2014till you fall again. If you fail to find the humor in this, you might as well just leave the theater. Or sell the farm, better yet, if that's your idea of real estate.\n\nTake the story of the farmer and his horse. Parables are annoying, I know, but irritation is the path to wisdom. One day a farmer lost his favorite horse. After the animal ran away, his neighbor appeared at his doorstep to offer condolences. \"I'm so sorry for your loss,\" said the yenta neighbor, glad that it had not happened to him.\n\n\"You never know,\" the farmer replied.\n\nThe very next day, his horse reappeared with a beautiful wild mare alongside him. Again the neighbor stuck in his two cents. \"That's wonderful!\" he offered. \"What a stroke of good luck!\"\n\n\"You never know,\" said the farmer, wanting him to go away.\n\nA few days later the farmer's son was trying to break in the wild mare, got thrown to the ground, and broke his leg. Immediately the neighbor appeared to comment again.\n\n\"You never know,\" the farmer repeated, now becoming a little annoyed.\n\nNot long after the accident, the Cossack army came through the village in search of young men to fight in the war. Since the farmer's son was injured, he was allowed to stay at home. \"Are you not a fortunate man!\" exclaimed the neighbor.\n\nYou know what the farmer murmured.\n\nHorace Walpole, the eighteenth-century English writer, was the first to invent a term for this bipolar phenomenon: serendipity (from a tale about the peregrinations of the three sons of the king of Serendip). Doubleness is the way of things, which is why narrow, cycloptic minds are so funny. The human condition, somebody said, is to be a fragment and a fool (pretending not to be a fragment). We couldn't be more ridiculous; this is also the good news. You have to be a trickster to trick up fate and turn it into destiny. Jim MacLaren's inspirational mission, Joan Didion's solvent book, Samuel Beckett's invention of the Theater of the Absurd after being stabbed randomly in a Paris street where his attacker shrugged after being asked why he'd done it. \" _Mais, je ne sais pas, monsieur,_ \" he said. Serendipitous sproutings and lemonade twisters that come from laughing along with God (fate for atheists) instead of stomping your feet and pouting.\n\n\"Here's the thing about fate versus destiny,\" Jim Curtan tells me, bottom-line style, sounding like the Hollywood big shot he was before prostate cancer (and actors) made him start hating show business. \"Fate is what happens to you. Destiny is how you respond.\"\n\nJim and I have been friends for thirty-five years. During the 1970s he was a personal manager for movie stars, spending his days appeasing, massaging, uplifting his clients' fragile, photogenic egos, which turned his life into a Sturm-und-Drang-fest. Stressed beyond reason, Jim was more than ready\u2014before his diagnosis\u2014to throw in his Tinseltown credentials and get his sense of humor back. But his spirits had sunk lower than even he realized.\n\n\"I got to a point where I thought more than once that it might just be wonderful not to wake up in the morning,\" Jim admits, sounding uncharacteristically downbeat. He's a towering ex\u2013Jesuit novice with a cowboy preacher's head of hair (in spite of having just turned sixty-five). A trip to the doctor's office ten years ago revealed a midstage tumor in Jim's prostate. With grander exits to worry about, the decision to quit his job was a no-brainer. \"I was terrified,\" he tells me. \"You heard cancer, you thought, 'Oh well, maybe a year.'\" A theater addict since childhood, Jim remembers thinking each time he saw a Broadway show after his diagnosis, \" 'Maybe this is the last play I'll ever see.' Or 'This could be the last time I ever see Paris.' Or whatever.\" Oddly, however, Jim has not been depressed a single day since getting his unwanted news from his doctor.\n\n\"I'm passionate about my life now,\" he tells me. \"Hardly a day goes by that something doesn't happen\u2014sometimes big, sometimes tiny\u2014that I don't say, 'I love my life.' I'm aware of it now and grateful.\" This joy had escaped him in the midst of his career. \"Before, I didn't live a life with gratitude,\" he says. \"As a good Catholic, I knew that I was _supposed_ to be grateful. I said all my prayers. But now I began to realize that if life actually is a gift, then the right response is 'thank you.'\"\n\n\"It sounds obvious when you put it like that.\"\n\n\"You have to be like a prospector in your own life,\" he explains. \"You have to look for the gift.\"\n\n\"How do you do that?\" I want to know.\n\n\"If this is the last time I go to Paris, then I'm going to do everything and see everything I can in Paris, damn it!\" he says. \"I don't want any of me left over for a yard sale. I'm not saving my life anymore. I'm spending it.\" Jim realizes that this might sound a bit glib. \"Of course, my smart-ass side said, 'I wanted a pony, why did I get a winter coat?'\" he admits. \"We all ask for certain gifts. When you're a grown-up, you understand why your parents gave you a winter coat, but as a kid you're really pissed off about no pony. Cancer was like getting a winter coat.\"\n\n\"And it wasn't a mink.\"\n\n\"It wasn't even fucking nutria! You know how Auntie Mame says that life is a banquet and most poor sons of bitches are starving to death?\" Jim asks. \"But what she doesn't say is that sometimes you're the guest, sometimes you're the chef, and sometimes you're the main course.\"\n\n\"That's comforting.\"\n\n\"But true,\" Jim tells me. \"And it actually works in your favor. Things become less provisional, less insecure, the more grateful you are. I don't have to earn my life anymore. It was given to me to cherish as I would a precious gift from someone who loves me and whom I love.\"\n\nSerendipity came when Jim was led to a workshop given by a healer named Carolyn Myss. \"I'd gone for my cancer, but something else happened,\" he says. \"During the weekend, Carolyn remarked that many people in crisis seek out therapists when what they really need is spiritual direction. I had started out wanting to be a Jesuit priest before Hollywood got to me. Ministering to people was my passion. When Carolyn said that, I knew why I'd really gone to that workshop. I'd gone there to be healed but came out with a vocation.\"\n\nThis vocation announced itself with a voice Jim believes came from his creator. \"I heard God say, 'Your job is joy.'\" Jim realizes that this may sound insane. \"At first I resisted. I said, 'But everybody wants that job!' God said, 'You'd be surprised. No one wants your job.'\"\n\nMy old friend laughs. \"God said my job was joy. But joy is not just about being happy. Joy is a rigorous spiritual practice of saying yes to life on life's terms,\" he says, reminding me of David Steindl-Rast's suggestion. \"I never said, why is this happening to me, which a lot of people do. It was more like, okay, I'm perfectly happy to learn all the lessons of this. But I want to graduate\u2014 meaning, I don't want to live the rest of my life as a cancer patient, even if I have cancer.\"\n\nHow can people not blessed with a hotline to God get their marching orders? I wonder. \"It has nothing to do with religion, first of all,\" Jim says. \"Religion is for people who are afraid of hell, and spirituality is for people who've been there. Winston Churchill said, 'If you find you're going through hell, keep going.' The road to heaven\u2014to peace of mind\u2014leads through hell. Nobody's built a bypass.\"\n\nDuring his training as a spiritual counselor, Jim learned more about these survival instructions from the Sorbonne-educated African teacher Malidoma Som\u00e9. Som\u00e9 described his own attempt to explain life insurance to his village elders. \"One of them asked, 'Do you mean people get paid for their catastrophes?'\" Som\u00e9 told Jim. \" 'Then how do they ever learn anything?'\"\n\nToday this grateful show-business refugee ministers to two dozen clients a week, talking soul survival instead of box office grosses. He advises his clients to live as if their lives actually depended on it. He tells them the story of the medieval monk who was watering his garden one morning when a passerby asked him, \"Father, if you knew you were dying tonight, what would you do?\" The monk thought a moment before replying. He said, \"These flowers would still need watering.\" Jim looks at me. \"I want to live my life in such a way that I wouldn't have to stop, or change, what I was doing\u2014or being\u2014if I was going to die tonight,\" he says.\n\nThen Jim arches an eyebrow in Mame's campy way. \"But I'm not.\"\n\n## **PRAYING**\n\nAccording to a recent poll, 70 percent of Americans over eighteen claim to pray at least once a week and receive \"great satisfaction\" from doing so. I found this statistic confusing when I read it. I'd never felt better in the least after my own sporadic attempts to pray. Never had I, to my knowledge, received a single divine response to any particular _crise du jour._ Maybe this was because I've never believed in a God who'd care about my day-to-day life (being busy with dwarf stars, black holes, and quarks). When it came to prayer, I felt like the clich\u00e9 orphan pressing his face to the window of a room in which God was nuzzling his own true children beside a roaring fire. The miscreant infidel (me) could only watch them from outside the window, slumped in a stupor of self-pity.\n\n\"You're not focused,\" my Christian friend John used to say.\n\n\"Nothing ever happens,\" I told him.\n\n\"It has to come from the heart,\" John said, as if I were trying to phone my prayers in. Sincerity is a private thing, as everyone knows, so I didn't try convincing him. I'm also from Missouri on all matters concerning interior experience. I need to see or feel something in order to draw any sort of conclusion. I need for there to not be parochial rules. If there is a God, I've always thought, it wouldn't require credulity, protocol, or leaps of the imagination. What God would want, I could only guess, were self-reliance, love of the truth, and a little trust that if it does exist\u2014 having spawned the Andromeda Nebula\u2014it knows what I'm thinking anyway.\n\nOne afternoon I opened my mailbox and found a postcard from John with a quote from the late Trappist monk Thomas Merton printed in bold capitals on the reverse side. It read:\n\n**TRUE PRAYER AND LOVE ARE LEARNED IN THE MOMENT WHEN PRAYER BECOMES IMPOSSIBLE AND THE HEART HAS TURNED TO STONE.**\n\nJohn knew that this would get under my skin. I couldn't vouch for the prayer part, but the bit about love being impossible sometimes I was too well acquainted with. Daunting experience had forced me to acknowledge the power of staying, abiding, enduring (with lovers, books, uppity friends) through moments when I most wanted to run. The Buddhists call this \"practicing against the grain,\" and I had the splinters to show for it. I'd found that when it came to questions of self-evolution, the rougher the timber, the stronger the floor. Testing the truth only made faith stronger. Doubting Thomas, I'd always thought, was probably one of Christ's favorite disciples.\n\nBut the act of prayer, as I understood it\u2014as wish fulfillment, solace, entreaty; as hook, line, and sinker for reeling God's voice from the depths of inner (or outer) space\u2014struck me as delusional. Till one day I found myself walking alone on a forest path in Germany. It was the dead of winter in the village of Thalheim, twenty kilometers from Cologne, and I was feeling despicable. I was living with an Indian teacher who'd mysteriously planted her home there among the garden gnomes and Hessian Catholics, a stone's throw from the hospital where Josef Mengele did his experiments on Jewish children during the war. The snow was heavy on the ground; dark evergreens stood outlined, chiaroscuro, against a bleached, indifferent sky. I don't think my spirits could have been lower; nor could I have found a more Bergmanesque setting for laying my bald head down in the snow and blowing my fucked-up brains out.\n\nI was hating myself for everything. I truly believed that I should be stronger. I stood there in the forest, freezing my ass off, blasting the trees, berating myself for a million things, including the fact that I couldn't pray. I found a clearing in the forest and let it rip, ranting at the whole damn thing\u2014and what was I supposed to do _now_ \u2014when the obvious point descended like bird poop. \"You're goddamn fucking praying, you asshole,\" the nurturing voice in my head informed me. The power that kept a person ranting, fist to the sky, whining at fate, was the heart and guts of prayer itself. The same longing for response, reunion, relief from pain that I equated with self-pity was actually the blood and limbs of prayer. Believing that prayer was meant to be a polite entreaty by the pure of heart, I'd missed the obvious point till that moment. \"Struggle is the highest form of song,\" the proverb goes. It was also, I saw now, the backbone of prayer. I hadn't been frozen, ignorant, muzzled, or shut outside the walls of the holy, but offering prayers in my own screwed-up way for as far back as I could remember.\n\nBack at the house, I spoke to my teacher, who was dressed in a sari and shoveling snow from the spot that months from then would be her garden. Her dark face was spotty, her hair a mess, her red plastic rain boots splattered with gunk. I told her what had happened. She said almost nothing, as usual, just smiled and nodded and sent me along like a slow kid checking in with an indulgent aunt. Years later I came across a commentary from the book of Job that reminded me of that walk in the forest. \"By the end, Job's suffering has erased all formality and he speaks to God directly, challengingly, intimately, just the way God speaks to him,\" it read. Job was transformed by dissolving the distance he dreamed between himself and God. Both humbled and enlightened by pain, the prophet finally understood the miraculous proximity of radiance, the commonality of his language with the divine's, the truth that emerges when we speak plain. Believers are welcome to call interlocution of this kind prayer; nonbelievers can call it truth-telling. Either way, we are changed, it seems, by the speaking itself. Telling\u2014not the response\u2014is the secret that links us to a source beyond our own minds, which rarely see past their own distress. When I came to see prayer as a periscope\u2014a means of peering through the surface\u2014 instead of a phone call that never got answered, it finally became real for me.\n\n## **DEMON LOVERS**\n\nCarl Jung called addiction a \"prayer gone awry.\" Desperate for transcendence, solace, communion, addicts turn to booze and blow, casinos and cutting, anonymous sex, bulimia, binge shopping, even domestic violence\u2014some compulsion to spoon in the void\u2014to substitute for higher or deeper connection. Yet cosmic loneliness prevails, especially in the addict's world. This is why Jung advised Bill W., the founder of Alcoholics Anonymous, that without a spiritual component the program could never work. Addicts require a higher power (even one that nonbelievers claim to be making up for themselves) to recover from such hunger and the deadly, doomed behaviors it causes.\n\nLike many addicts-to-be, Michael Klein grew up in a house where spiritual hunger and the threat of abandonment were ever-present facts of life. Night after night he and his twin brother, Kevin, would watch their mother slip into a barbiturate haze in the glow of the family TV set. A sensitive kid, Michael would study Kathryn's face as she disappeared behind her narcotic cloud, wondering what he and his brother had done to make her go away like that. Why did his mother, whom he adored, seem so angry at her twins (though she loved them as well) for needing her to stick around? Her own mother hadn't stuck around; at forty-three Kathryn's mother had gotten drunk and jumped to her death from her balcony on Fifth Avenue. Kathryn was abandoned herself; now she imagined doing the same to her children. This shame seemed to be yet another reason for going through with it.\n\nYou learn the world from your mother's face. You learn about God from the way she moves, how she loves or doesn't love, how she smells, what she says in words and silence. You learn about creation from the way your parents love each other, the story they tell you about how they came together to make you. These details create your idea of who you are and where you came from; the color, texture, depth, or shallowness of your universe; the particular tangle of roots that brought you out of the ground. The grief in his mother's eyes seemed to prove to Michael that the world was a very cruel place, that no one would be there to save him, that he would always be alone.\n\nBy the time he was twelve, Michael was drinking. We're talking at a Midtown Manhattan deli, with corned-beef-balancing waitresses rushing around and yelling. A muscular two-by-four of a guy with blondish hair, insane sense of humor, and thrashed-gravel voice like Harvey Fierstein's, Michael is a teacher and award-winning poet who's been sober now for twenty-two years. But back then he was a lost boy, grasping at strangers with one hand, drinking himself to death with the other, pulled down by an emptiness he didn't understand, sucking down liquor like mother's milk, trying to feel at home in the world.\n\n\"I was drunk every day for ten years at least,\" Michael tells me, squirting relish on his burger. Kathryn, his mother, had died when he was twenty-two\u2014from natural causes or an overdose, no one seems to be quite sure. Kevin, his twin, was leading an isolated alcoholic's life by then; Michael could feel him slipping away. Introduced to the horse-racing world when he was twenty, Michael got himself a job at Belmont grooming thoroughbreds (at one time he tended the champion Swale). It was a gin-soaked, unstable stable life on the road, stumbling from track to bar to anonymous bed in the endless, insatiable search for connection. \"I started drinking to make passion easier,\" Michael wrote in a memoir, _Track Conditions._ \"I could fall into a stranger's sheets when I was drunk. It was desire without fear.\"\n\n\"That's what everyone wants,\" I say, \"desire without fear.\"\n\nIn her own wonderful book _Witness to the Fire: Creativity and the Veil of Addiction,_ Jungian analyst Linda Schierse Leonard recounts her own recovery from alcoholism and explains how obsession reduces the addict to thinghood. \"Addiction leads to monomania,\" she writes, \"a narrowing of life and vision, reducing the addict to the status of an object defined by its craving.\"\n\nMichael's craving was to be touched. He was torn between two lovers, he tells me: art and death. Every addict has two _daimons,_ Leonard explains, one destructive and the other creative. Michael had both in spades. Already, he was writing poetry in earnest\u2014but his life was out of control. \"The creative person _chooses_ to go down into that unknown realm,\" Leonard reminds us. \"Even though the choice may feel destined. But the addict is pulled down, often without a choice, and is held hostage (by his disease).\" Michael was starting to bottom out. Around thirty, his reckoning came to him cold. \"I was really tired. My life was a shambles. But I wasn't dead yet,\" he says simply.\n\n\"You lost me,\" I say.\n\n\"I started drinking because I wanted to die,\" Michael tells me. \"But it wasn't killing me fast enough. If I was going to live and be an addict, I realized that I would rather just live.\"\n\n\"Just like that?\" I ask.\n\n\"Just like that.\"\n\nChecking himself into detox, he found the physical part relatively easy. \"I had the shakes on and off for three months,\" he says with a shrug. \"I couldn't stay awake for more than seven or eight hours a day.\"\n\n\"What about your mind?\"\n\nMichael compares his psychic rebirth to \"a crash victim coming out of a coma. That _this_ was the world I remembered before the accident, and I wanted to live here again.\"\n\n\"Like the man who fell to earth,\" I tell him. \"I know that feeling.\" \"I realized again that the world was in color!\" he says, digging into his fries. \"The world went from two-to three-dimensional. Everything was heightened. I had an amazing amount of energy. I could remember my dreams!\"\n\n\"You make it sound like you rose from the grave.\"\n\n\"It was like that. You _know_ everything is there when you're drinking. But you can't see it and you don't care. As an addict, you're totally unteachable. You lose the desire to learn _anything._ I remember being struck by the fact that people are interesting!\" The decibel level of Michael's laugh startles our neighbors at the next table. \"I didn't _love_ anybody,\" he admits. \"I didn't even like anything before, except sex. When you're drinking,\" he says, waving down a waitress for our check, \"you're able to become your worst self. The difference is that you really don't care.\"\n\nOutside, Michael lights a cigarette as we make our way down Lexington Avenue. His twin brother died last year, he tells me, after an alcohol-induced heart attack. Michael is still confused by this loss. Unlike Kevin, Michael wrestled with his childhood demons in order to stay sober. \"Two or three years into sobriety, reality hits and you realize you need therapy,\" he says. \"There are big parts of you missing, whole chunks of what makes a person whole.\"\n\n\"What about this higher power thing?\" I ask.\n\nHe reminds me that the program's first step is admitting that you are powerless against your disease. \"This step gives you tremendous power,\" he says, aware that it sounds paradoxical. \"But the strength you find isn't in your ego. In recovery everything gets reversed. Many things you thought were signs of strength simply aren't.\"\n\n\"Such as?\"\n\n\"Self-absorption. Self-pity. Self-destruction.\" Michael is being facetious. \"You start to see the big picture. To not take things quite so personally. You become part of the collective consciousness\u2014a worker among workers,\" he says, using Marxist lingo. \"This gives you equanimity and a sense of connection to the essence. You and your disease are not the epicenter of the universe.\"\n\n\"Powerless before what? God?\"\n\n\"Maybe.\" Michael shrugs. \"You can't have strength without spiritual awareness, however you define it. If I had to pick a synonym for what strength is, it would have to be self-realization.\"\n\nIn the thirteenth century the German mystic Mechthild of Magdeburg described being able to finally connect her own misplaced prayer to the arms of a lover she could trust, the one she wanted to begin with. \"Whoever is sore wounded by love will never be made whole unless she embrace the very same love which wounded her,\" the abbess wrote. Mechthild was speaking of her divine love for Jesus, but the same can be said of an addict like Michael who is wounded early by loss of parental love and only treatable through surrender to a primal source. Today he's a transformed man, Michael tells me, publishing books, teaching writing, happily in love with a long-term partner, taking yoga, almost cigarette free.\n\n\"I never thought I'd see my thirtieth birthday,\" says this softhearted, gruff-sounding man at the top of the subway stairs. \"Every single day I'm alive after getting sober is gravy!\" Michael Klein's voice reaches a crescendo that threatens to stop oncoming traffic. \"Gravy, gravy, gravy!\"\n\nBut what about the secondhand casualties? How do those spouses, children, lovers, friends, struggling to save their own lives around addicts they love\u2014and cannot leave\u2014find their way through?\n\nKathleen grew up in a fatherless house in Fresno, the middle daughter of three. Her parents divorced when she was in second grade; her father, a trucker, moved to Ohio, where he started a second family and left Kathleen's mother, Geraldine, to raise the girls on her own. Her family was under constant threat of eviction, since Geraldine, a sort of Irish princess, didn't like being imposed upon to work. She gravitated toward migraines and serially abusive men instead. Kathleen and her sisters ran the house, found odd jobs, put food on the table, catered to their mother's mood swings.\n\n\"I guess you'd say that I never got to be a kid,\" Kathleen tells me. We're having lunch in Pike's Market in Seattle. Kathleen is a big-boned, redheaded woman who wears her hair long, like Wynonna Judd. \"We never got to be kids,\" she says. \"Me and my sisters had to grow up too fast.\"\n\nA brilliant student, Kathleen managed to get a scholarship to UCLA, where she felt happy for the first time. A theater major, she acted, directed, even tried her hand at playwriting, which proved to be her gift. \"Big surprise\u2014I was the queen of make-believe,\" she says, chuckling. \"I had been scripting my own imaginary life since I was a little girl.\" As a director, Kathleen was out of her depth, though, which is why one April evening she put an ad up on the university job board, needing someone to direct her first play, and came to meet an ex\u2013theater student turned out-of-work director named Andre.\n\n\"He was everything I fantasized about in a man,\" Kathleen tells me. \"That should have been enough to scare me.\"\n\nAt six foot three, with amber eyes, a tenor's voice, a wardrobe out of _L'Uomo Vogue,_ and the \"most luscious\" lips she'd ever seen on a man, Andre was the kind of hunk who caused testosterone whiplash in public. \"If they could've thrown their panties at him, like Engelbert Humperdinck, they would've,\" Kathleen says. She hired him to direct her play, then allowed Andre to spend the night after the second curtain call. In bed afterward, he took Kathleen in his lap and said, \"You're the kind of girl I want to take care of.\"\n\nKathleen still can't believe she fell for it. \"I actually believed it,\" she says, shaking her head. \"After that it was like 'open sesame.' I went into a kind of trance.\" Andre's dashing appearance aside, Kathleen was attracted to what appeared to be Andre's beautiful soul. Failing to mention the woman he lived with (or the child of his she was carrying), Andre wooed Kathleen and, when the time was right, invited himself to move in. On the night that the other woman showed up on her doorstep, Kathleen took Andre's word that this previous relationship was over. He promised to look for work. One day Andre showed up with a pair of his-and-hers engagement rings (although he hadn't bothered to pop the question). The word roots was inscribed on the underside of Kathleen's ring (Andre's read wings). \"That's all I'd ever wanted was roots,\" she tells me now. \"Andre had this sixth sense when he wanted something. He always knew people's weak spots. Like killers know.\"\n\nIn retrospect, Kathleen can trace the web. But back then she was happier than she'd ever been. She found a job as a dramaturge that paid enough for them to buy a modest house in Berkeley. Andre played the stay-at-home, freelancing husband while she limned Chekhov for first-time directors. On weekends they planted, mowed, cooked soup. Their sex life was supercharged, though Andre could be a little too rough. \"A guy like that gets away with murder,\" Kathleen tells me. Not a day went by without her noticing some alarming flirtation with her husband, who protested too much, as philanderers do, against having the slightest interest in anyone but his own wife.\n\nTen years passed. Two children were born. Andre now worked as a personal coach. Kathleen had been happily surprised, all things considered, by how well their marriage was turning out. She'd finally put down the roots she craved, established a home, even if it came at a cost. Andre had grown moodier with age. Sex stopped. Then cuddling stopped. But at least they had an understanding, she thought. It was still, she told herself, a marriage. And at least she wasn't alone, Kathleen would catch herself thinking, watching her handsome husband sleep.\n\nThen this picture imploded. One ordinary weeknight around eleven o'clock, Andre did something he'd never done before in a decade of marriage: He rose from bed after he was undressed and left the house. He claimed to have things on his mind. After a couple of hours Kathleen grew anxious, drove to town, found Andre's car outside the dry cleaning store\u2014then saw her husband making out with a blond teenager behind the counter. \"Here's the really sick part,\" Kathleen tells me, lighting a cigarette. \"I decided that I could let it pass. I wasn't going to make a big deal out of a peccadillo. He's a gorgeous man. It was bound to happen. I even thought maybe it was my fault for not trying harder to keep him interested.\"\n\nPrepared to forgive if not to forget, Kathleen was less prepared for what happened when Andre finally did get home. \"He appeared at the house all upset, sweating, shaking,\" she tells me. \"He wanted me to sit down, he had something important to tell me. Now I'm really getting scared. I'm getting that something is really wrong.\"\n\nIt wasn't the kind of other woman Kathleen expected. Andre had fallen in love with \"Tina,\" the street name for crystal meth. Apparently a gym buddy had given him a bump of T one night to help him get through his workout. Now, Andre told Kathleen, showing her the track marks on his arms, he had been shooting Tina every day for the past three months.\n\n\"I froze. I just went dead inside,\" she says. In a zombielike voice, Andre proceeded to confess the rest of his sordid tale: the party girls he'd met on the Internet, the secret hookups, an entire meth subculture Kathleen knew nothing about, where Andre was cheating under her nose at times when he was supposed to be playing golf or working out at the gym.\n\n\"It was the worst moment of my life,\" Kathleen tells me. \"Like watching a mask snap off my husband's face.\" She looks scared remembering it. \"I saw who he was. I saw who he wasn't. I didn't know my own husband. Andre was a complete impostor.\"\n\n\"So you kicked him out?\"\n\n\"No, that's what a sane person would have done.\" Instead, Kathleen put Andre to bed, told him he was forgiven, and took care of him while he tried to kick the drug on his own. He was sick in bed for a week, alternating between crying jags, rage, and puking. \"Have you ever had to spoon-feed someone you wanted to kill?\"\n\nUnfortunately, I have.\n\n\"Wipe their ass when you want to strangle them?\"\n\nDitto.\n\n\"I had never felt so violated or disrespected in my life,\" she says. \"But I was still scared he would disappear.\"\n\n\"You don't stop loving someone just because they ruin your life,\" I say.\n\n\"I wanted to hurt him, but I couldn't stop loving him,\" she admits. \"So I did what I thought was the right thing to do.\" Kathleen the caretaker found Andre a rehab (and paid for it), kissed him good-bye in the hospital lobby, and promised to be there when he got out three weeks later. \"I believed every lie that came out of his mouth,\" she tells me. \"I wanted to believe it was still my Andre. That he was going to change. For me. For our kids. That the whole drug thing had been a mistake.\"\n\nIt seemed as if rehab had worked at first. \"When he got out of the hospital, he was like the man I fell in love with\u2014no, he was better than that,\" Kathleen says. \"He was the Andre I used to dream was there, underneath the macho bullshit, the vanity.\"\n\n\"It's called rehab afterglow,\" I remind her.\n\n\"I was so happy!\" She shakes her head. \"We had a couple of honeymoon weeks. Andre seemed so raw. So humble and real. He went to meetings every day. Then I found a hypodermic needle in his sock drawer.\" Kathleen turns to the view out the window. \"He'd been shooting up since the day he got out. Another lie inside of a lie. This time I got rid of him. I called the cops and everything. There wasn't enough evidence to hold him, so Andre was out of jail the next day. And that's when the fun really started.\"\n\nKathleen sold the house, moved from Berkeley to Seattle, changed her telephone number, and placed a restraining order against her husband in case he ever showed up. But as the weeks went by without a word, Kathleen couldn't get Andre out of her mind. \"I started having panic attacks,\" she says. Scary tidbits of news reached her through the grapevine, friends who'd seen him here or there looking terrible. Even worse, their dog, Sadie (kidnapped by Andre), appeared to have been neglected. \"It was the dog that really got me,\" Kathleen says now. \"He was devoted to Sadie. If she was being neglected, I couldn't imagine what kind of hell he was in.\" Swamped by nightmare scenarios of Andre starving to death on the street with their dog, Kathleen caved in. \"It was just like being possessed. This demon wouldn't let go of me. I could almost hear him calling me, like he was pulling me down into this dark place with him.\"\n\nShe did the only thing she could. \"I called. I told him I wanted to see him. I left the kids here and flew down to Berkeley. He was sick in bed, hadn't been out of the house in six days. Sadie was skinny.\" Kathleen cleaned the place, gave Andre back rubs, sat beside him on the bed as he moaned about being hooked, never meaning to hurt her, wanting to die. Not wanting to.\n\n\"I never knew anything could hurt so much,\" she says now. \"I felt so trapped. Like your worst dream happening with you inside it.\" Still, Kathleen allowed Andre to stay in touch. \"I let him call me. It was harder than not knowing how he was.\" This went on for close to a year. \"He'd call at all hours of the night. Howl on the phone. Beg me to help him. Sometimes he'd blame me for the breakup.\"\n\n\"You?\"\n\n\"He's an addict. I told myself to forgive him because he was sick. But I couldn't keep the demons out.\" Soon Kathleen was so guilt-ridden and depressed that she considered checking herself into some kind of a rehab. She was unable to cut the cord between them. \" _That's_ when I got it,\" she says. \"I realized that there were two addicts here. I was addicted to saving him from his disease. His addiction had become my sickness. I realized that he was going down and it was either him or me. And no way was this man going to kill me.\"\n\nKathleen changed her telephone number\u2014again\u2014and cut off all contact with her children's father. Three times a week for close to a year, she found kindred spirits at Al-Anon meetings. \"I always made fun of the twelve-step thing before,\" she confesses. \"The slogans. The victim mentality. But that was just my arrogance talking. These people weren't victims\u2014they were major survivors.\" Listening to tales so like her own, however varied the details, Kathleen got to know her own disease\u2014the savior complex, the fixer obsession, the codependent impulse to rescue someone in order that she herself could be saved.\n\nShe gazes out at the boats in the harbor. \"People don't get how insidious codependency really is,\" she says. \"What it means is that you\u2014the caretaker\u2014actually _disappear_ behind the addict's pain. You cease to exist when they need something. When Andre got sick, I felt like it was me who was dying.\"\n\n\"I know how that feels, exactly.\"\n\n\"He wanted me to sit there and watch him kill himself. But I would have probably died before he did. I had to save my life\u2014it was him or me. Not just for me but for our kids.\"\n\nThe hardest part of healing from her own addiction, Kathleen found, was admitting that there was absolutely nothing she could do to help this person she loved. \"Getting it through your head that there is nothing\u2014period\u2014that you can do that will change a thing.\"\n\n\"The rational mind doesn't get it,\" I say.\n\n\"It goes against every natural instinct. Because addiction turns everything around. What feels like love in normal life is 'enabling' when you're talking about an addict. Tender loving care is actually harmful. Tough love, which feels like neglect, is the only thing that might actually help. I've never felt more powerless over anything in my life. Or more honest.\"\n\nShe looks directly at me. \"I had to accept the fact that Andre might actually die from this. And I could do nothing at all to stop it.\"\n\nThree months after our interview, I receive an e-mail from Kathleen with the news that after asking his own parents for help, Andre is now in his second bout of rehab. Kathleen will not visit him there; she has, however, given Andre's therapist permission to convey messages. Kathleen also got custody of Sadie. If her ex-husband ever does get sober, she tells me in her note, she may consider giving him visiting rights. \"The kids could use a father.\"\n\n\"What about you?\" I write back.\n\nShe responds with the news that she's almost ready to start dating again. Kathleen is considering using an online matchup service so she can start learning how to set boundaries. \"No beauties,\" she makes clear. \"And they have to be short. Without any muscles.\"\n\n\"No muscles?\" I tease her.\n\n\"And no hair!\" she responds.\n\n\"A daddy thing?\" I ask her in an instant message.\n\n\"Touch\u00e9,\" Kathleen writes back, ending her note with one of these:  \n\n## **QUESTIONING (OR THE SPHINX)**\n\nIt seems perverse that authenticity should stem from loss. The outline cracks, you split apart, half of you is left stranded on an iceberg floating into the chilly distance. You're suspended in partiality, cut off from who you thought you were. This is when questioning starts, that's the truth\u2014when you can't put yourself back together again, when the old parts don't fit and the new ones have yet to arrive. You stand there looking into the mirror, wondering, What in God's name is that? This lopsided mess of an unglued creature leering back at what used to be me? How can I live with this alien changeling? Where's the rest of my ensemble?\n\nYou haven't yet heard of the hidden face. You haven't quite learned that losing what you thought could never be lost is precisely what shows you who you really are. \"The art of losing isn't hard to master,\" wrote Elizabeth Bishop, tongue in her cheek. \"So many things seem filled with the intent \/ to be lost that their loss is no disaster.\" We step through one loss after another, look down to find our feet still on the ground, and get the Buddhist joke of it all: that we're not that stuff we feared leaving behind, \"filled with the intent to be lost,\" not that persona attached to its world, but the thing that's left standing when that starts to go, the self who's watching and asking the questions. Each time this happens, the spring in our step is a bit more pronounced, as if we've lost ballast, been bounced and made lighter.\n\nQuestioning is an art form in itself. Inquiring, who are you today, for starters\u2014and what is it that you want most intensely? Where is intuition leading? Questions till the ground of \"beginner's mind,\" plow fresh road, scrap outdated agendas, help us to reimagine our way.\n\nIn a refrigerated convention hall in Chicago, I find myself waiting with two hundred other workshop participants for the arrival of the queen of questions, Byron Katie. Named one of _Time_ magazine's \"Spiritual Innovators for the 21st Century,\" Katie, as she likes to be called, is a self-taught, sixty-four-year-old teacher, who, as a result of a nervous breakdown she barely survived twenty years ago, has developed a technique she calls The Work.\n\nThis assault on self-delusion consists of posing four direct questions and what Katie calls a \"turnaround.\" (We'll enumerate these in a moment.) What I know on arriving in Chicago is only that this respected woman fully believes that her technique can relieve the pain caused by any situation, save physical torture. I'm skeptical but interested. A respected therapist friend returned from his weekend with Byron Katie describing it as \"psychological shock and awe.\" Even Katie's hardheaded husband, scholar and translator Stephen Mitchell, claims that The Work is a bona fide path to what the Buddha called \"the cessation of suffering,\" a vehicle for navigating the turbulent mind into calmer waters.\n\nThe crowd is beginning to fidget in their folding chairs. A technician sound-checks the mike and jumps offstage. My new best friend, Peggy, an overweight prison chaplain with a white Lulu hairdo and close-set eyes, offers me a Velamint.\n\nAt the stroke of nine, Byron Katie struts onstage in a chocolate silk pants ensemble, a shawl thrown dramatically over the shoulder, Doris Day pretty, perfectly coiffed, self-assured as a lion tamer. Stephen Mitchell takes a seat stage right, scholarly in his beard and tweed jacket. Without introduction Katie begins.\n\n\"Our most intimate relationship is the one we have with our own minds,\" she says. \"I was in a very dark place for a long time. Then one day I realized a simple thing.\"\n\nPeggy's pen is poised on her notebook.\n\n\"When I believed my own thoughts, I suffered,\" says Katie. \"When I didn't believe them, I didn't suffer. Everything changed for me after that day.\"\n\nA man behind me clears his throat. \"You see, thoughts are like children,\" Katie continues, moving downstage. \"They're gonna scream and scream till we pay attention to them. When we do this and begin the work of questioning, things we've believed our whole lives\u2014forty, fifty, sixty years\u2014our most stressful, self-defeating thoughts are brought to an end.\"\n\nPeggy elbows me and raises her eyebrows.\n\n\"It isn't easy,\" Katie assures us. \"It takes a lot of courage. But isn't it time to get real, my honeys?\" She scatters endearments like bonbons to children. \"Haven't we been conning ourselves for long enough?\"\n\n\"We sure have!\" shouts a goateed therapist named Mick (we're all wearing name tags).\n\n\"All right, then. This is your shame worksheet,\" Katie tells us, holding up a purple folder. She instructs us to make a list of our darkest, most aggravating, shameful beliefs\u2014the poisonous, top-secret, horrible judgments we reveal to no one\u2014then choose the greatest shame of all and apply the four questions and turnaround. \"Be brutal!\" Katie exhorts the crowd. \"This is your chance to see what's really going on in your mind. If we don't question what we believe, we are destined to live it out.\"\n\nThe Work consists of a series of four questions that sound almost too simple to be effective:\n\n1. Is it true?\n\n2. Can you absolutely know that it's true?\n\n3. How do you react when you think that thought?\n\n4. Who would you be without that thought?\n\nOnce these questions have been asked, the student is instructed to invert the original thought and give three examples of why this \"turnaround\" is as true as, or truer than, the original belief. \"My mother doesn't love me\" might become \"My mother does love me, and here are three reasons why.\" According to Katie, any painful thought subjected to this inquiry loses its power to hurt us, since most of what goes on between our ears is a pack of lies.\n\nThese questions are said to have come to her spontaneously after a nervous collapse in the mid-1980s. Born Byron Kathleen Reid on December 6, 1942, she was the second daughter of an engineer for the Santa Fe Railroad and a typical fifties housewife in the dusty town of Barstow, California. After a shotgun wedding at nineteen, Katie spent the sixties, and most of the seventies, raising her three kids and making a name for herself as a local real estate mini-tycoon.\n\nWhat happened next turned her life upside down. During a period of domestic turmoil (her marriage ended when she was thirty-three), Katie grew paranoid and suicidally depressed. Morbidly obese, she was also agoraphobic and barely left her own bedroom for two years, often unable to bathe or brush her teeth, much less take care of her children. This crisis was exacerbated by the absence of visible cause. \"I had plenty of money, a beautiful home, three kids who were healthy,\" Katie will tell me when we meet. \"Being depressed was even more shameful with no cause to point my finger at. I felt ungrateful and confused. I was dying.\"\n\nIn fear that their mother might hurt herself, Katie's children located a halfway-house-like facility near Los Angeles where Katie could be kept safe\u2014and the family could be away from her tirades. Relegated to the attic, Katie fell asleep on the night of October 16, 1986, not knowing that when she woke up the next morning an insight would occur to her that would snap her depressed mind in two. Had this spontaneous awakening not been witnessed by those around Katie, it would seem too miraculous to believe. But her family corroborates what happened. \"She seemed completely at peace,\" Katie's daughter says now. \"I couldn't believe it was really my mother.\"\n\n\"I woke up to reality\" is how Katie explains it. \"I realized that my thoughts were creating my suffering.\" Gradually, she began to share her \"work\" with locals in Barstow who wanted to know why their formerly crazy neighbor was suddenly smiling. Over time Katie received invitations to speak from around the country\u2014then from other nations. She made no effort to promote herself. The last thing on Katie's mind was becoming the queen of questions. Asked if she was enlightened\u2014or what?\u2014she has consistently waved such nonsense aside. \"I'm just someone who knows the difference between what hurts and what doesn't.\"\n\nOnce the allotted time is up for completing our shame worksheets, Katie opens the room to volunteers. I have no intention of sharing, but the exercise has been illuminating. My number one shame (pertaining to a relationship failure) readily revealed its own flimsiness when I tested it against these questions. Not only did the \"fact\" I'd been using to flagellate myself reveal itself to be bogus, it flipped over like a harpooned fish when I stuck it with Katie's turnaround. In fact, the opposite of what I had believed was far truer than my self-defeating thought. The exercise leaves me feeling unsettled.\n\nKatie fields the crowd of waving hands. A middle-aged Latino man steps up to the microphone. \"My wife cheated on me,\" he whispers. \"But that doesn't mean I'm less of a man.\"\n\n\"Thank you, precious,\" Katie replies.\n\n\"I think I might be gay,\" mumbles a teenage kid in an Axl Rose T-shirt, his eyes darting nervously around the room.\n\n\"I'm glad you're here, sweetheart,\" she assures him. My white-haired neighbor timidly raises her hand. \"I'm too fat,\" Peggy whispers into the mike, hiding her belly with the purple folder. In fact, Peggy is portly but hardly obese.\n\n\"Is that true?\" Katie asks, planting a hand on her hip. All eyes in the room are on my shy comrade. \"How many of you would rather get a hug from Peggy than, say, a supermodel?\" Katie asks. Nearly every hand in the room goes up. \"You see that, honey?\" Katie smiles at her. Peggy is visibly trembling. \"Remember, it's what we think or say or do that hurts us.\" Peggy covers her mouth and sits down.\n\n\"Anybody else?\" Katie surveys the audience. Finally, a well-dressed, fiftyish guy stands up and waits for the monitor to reach him with the mike. He mumbles, \"I hate my life. I don't want to be here.\"\n\n\"Can you be more specific, honey?\" asks Katie, stepping to the apron of the stage, fixing him with her steady gaze.\n\n\"Sometimes I want to die,\" the man says.\n\n\"Very good,\" Katie assures him, oddly. \"Now let's do The Work,\" she says. \"Is that true?\"\n\n\"Yes,\" he answers.\n\n\"Can you absolutely know it's true?\" asks Katie, unfazed by his visible pain.\n\n\"What do you mean?\" he responds.\n\n\"Can you know with absolute certainty that you want to die, precious?\" she asks very slowly.\n\n\"That's how it feels,\" he says.\n\n\"Of course it does. Now, how do you react when you think that thought, sweetheart?\"\n\nThe expression on his face shifts from slack misery to minor annoyance. \"How do you think it makes me feel?\" he asks irritably. Katie has been down this road many times before. Gently but firmly, she requests that the volunteer simply answer the questions as directly as possible, \"in order for The Work to work.\"\n\n\"Okay,\" he says. \"It makes me feel like a friggin' loser.\"\n\n\"Thank you, sweetie. That's excellent. Now,\" Katie asks, leaving the stage and approaching him down the aisle. \"Who would you be without that thought?\"\n\n\"A liar,\" he tells her straight out.\n\n\"Honey,\" she asks. \"Do you want to be happy?\"\n\n\"Why do you think I'm here?\"\n\n\"So who would you be?\" Katie inquires again.\n\n\"I have no idea,\" he says.\n\n\"Bingo!\" she tells him. The man looks puzzled. \"Without the thought 'I want to die' you don't know who you would be.\"\n\nThe room gets quiet. \"I don't understand,\" he says.\n\n\"That's okay,\" Katie assures him. \"Now, can you turn it around?\"\n\nOnce again he seems annoyed. It's revealing to watch his anger increase with each benign question that further rattles his story. He reaches for his jacket to leave. Then he puts it down again. \"I'm not enjoying this game,\" he mumbles.\n\n\"Then why did you volunteer, sweetness?\" she asks. Peggy elbows me, nodding her head\u2014it's checkmate and the poor guy knows it. \"Just try to answer the question,\" says Katie. \"What turnarounds can you imagine to the thought 'I want to die'?\"\n\n\"What?\" he answers. \"You mean, 'I don't want to die'?\"\n\n\"Any other turnarounds?\" she asks.\n\nHe shrugs and says, \" 'I want to live'?\"\n\nBut Katie isn't finished yet. \"Now,\" she says, \"give me three reasons why this turnaround\u2014'I want to live'\u2014is as true or more true than your original thought.\"\n\nHe now appears more lost than irate. \"All right. Can it be purely vindictive?\"\n\n\"What ever's true for you,\" she says, chuckling.\n\n\"I want to live to divorce my bitch of a wife and see her face when I walk out the door. Does that count?\"\n\nKatie hoots out loud with the rest of us. \"That sounds like a pretty strong reason to me. Any others?\"\n\n\"Well, I'm crazy about my kids. I'd like to be here to see them grow up.\"\n\n\"Good. Now, can you give me one more reason?\"\n\nPausing for a good long time, he finally says in a wobbly voice, \"I think I'm a pretty good man.\"\n\nWith these words the last blush of anger drains from his face. He sinks back into his folding chair as Katie returns to the stage. \"And we wonder,\" she says to the room, \"why a mind being deluged with angry thoughts would be telling us that we would rather be dead. Remember, the mind is a child. It believes what ever we tell it.\"\n\n\"A retarded child,\" Peggy whispers.\n\n\"Whatever I am convinced of _creates my world_ until I question it,\" Katie says, stressing these three words. \"Our unquestioned thoughts only turn into nightmares. We carry this suffering to our graves. But we each have the power to stop the deception, stop abusing ourselves. Someone hits you. Wham! It's over. That's grace.\"\n\nI'm struck by the severity of her logic.\n\n\"But the mind keeps re-creating the pain. There's so much courage in these rooms,\" Byron Katie says, steepling her fingers under her chin. \"It always amazes me.\"\n\nWhile there's nothing new in the notion that how we think is how we live\u2014think of Buddha, Descartes, Dr. Phil McGraw\u2014 Katie has succeeded in articulating this ancient wisdom in a modern, effective, dogma-free way. Rejecting the role of guru or seer, she insists that nobody needs her (or any teacher, for that matter) to reap the benefits of self-inquiry. As evidence that The Work functions on its own steam, at least six hundred independent \"inquiry circles\" (where peers ask these questions together) have cropped up around the world, from Helsinki to Hong Kong to Houston.\n\nDuring the lunch break, I speak to several veterans of Katie's work who claim to have made greater progress in their lives by applying these four questions than they've made through years of recycling childhood trauma in pricey therapists' offices. \"The beauty is that people can do this for themselves without having to go through analysis or doing expensive processes,\" says one consultant in her forties, who recently came through a bout of breast cancer. \"One of the gifts of getting older is that although change may be more difficult, you're forced by the facts of life to work with destructive thoughts in a positive way or go down with the ship.\"\n\nOur blue-eyed, goateed neighbor, Mick, a fifty-two-year-old social worker, agrees. \"This is almost like a non-therapy,\" Mick explains. \"Most people are addicted to a dangerous level of emotional pain. They might want to improve things, but only a little bit. This work is most effective for people who are genuinely sick of their stories and ready to drop them 'cause they're too damned painful. But if I was looking for a reason to stay stuck, a lot of what Katie says would sound like bullshit.\"\n\nIndeed, this teacher's radically positive, tough-love message rubs many of her peers the wrong way. A Los Angeles\u2013based psychologist I spoke to had some serious qualms after watching Katie apply her process to an especially traumatized person. \"It's a creative extension of some preexisting cognitive behavioral therapies,\" he tells me, sounding a little snooty. \"For people in the normal neurotic range, it might be useful,\" this therapist allows, \"but as a template for what ever ails us?\" He harrumphs. \"To assume that simply by asking four questions, then turning the thought around, you can address the complexity and seriousness of issues such as rape or incest, for example\u2014well, that is just narrow and na\u00efve, psychologically unsophisticated.\"\n\nKatie's cruel-to-be-kind approach, her categorical rejection of victimhood as a legitimate position, is indeed what rankles her critics most. A meditation teacher who asked to remain anonymous went so far as to suggest that The Work \"has no heart in it.\" Yet observing Katie carefully, I never doubt for a moment the full involvement of her heart. While it's true that she does not coddle whiners, and uses the fierce road to compassion, her toughness seems to come from an unwillingness to watch people torture themselves\u2014as she nearly tortured herself to death\u2014without challenging them to cut the crap.\n\nWhen I meet with Katie and her husband a few days later in New York, she elaborates on this contentious point. She and Stephen are facing me in high-backed chairs in their Midtown Manhattan hotel room.\n\n\"Empathy is terrific,\" Katie says, looking glamorous in another flowing silk ensemble. \"But when a person is struggling to survive, pity is not their friend. It does more harm than good. If someone we care about is hurt, then I feel hurt because _they're_ hurt\u2014'Oh, you poor thing, I'm so sorry for you!'\u2014now there are two of us hurt. What good am I?\"\n\nAgain, this simple logic is hard to dispute. Pragmatism is what matters most when you've sunk to the bottom, Katie believes, and drama's a luxury you can't afford. This ruthlessness in the realm of self-knowledge reminds me of the goddess Kali in Hindu lore, a metaphor for the fierce feminine, severing chains of delusion with the sword of self-knowledge; or of a Zen master I used to practice with who took great pleasure in whacking meditators across the shoulder blades when they fell asleep, with a shrieking \"Wake up!\"; or of a triage surgeon in the field inflicting pain on the wounded to save their lives. Like every authentic teacher I've met, however friendly their exterior, Byron Katie is as tough as nails inside and passionate about helping people free themselves from suffering.\n\n\"People spend their lives in dread,\" she says, pouring me a cup of tea.\n\n\"It's self-inflicted torture,\" Stephen agrees.\n\nKatie takes his hand. \"The mind that has questioned itself looks forward to life,\" she tells me. \"We can't stop life from happening, honey, so we might as well look forward, right? Can you imagine what we might be capable of if the bulk of our energy were not being taken up with stress? Can you imagine?\"\n\nShe's preaching to the choir, I assure her.\n\n\"But the question we need to ask ourselves is, do we choose to live the most fulfilling life we possibly can? Or would we rather prove to the rest of the world that life is all about suffering? And that we are the primo example of that?\"\n\nThis reminds me of Saul Bellow's description of his fellow Jews in the novel _Seize the Day:_ \"If they quit suffering they're afraid they'll have nothing.\"\n\nKatie continues. \"But what are we teaching our children?\" she asks. \"Are we teaching them the possibility that they can be happy no matter what happens?\"\n\n\"Hardly,\" her husband says.\n\n\"And yet that's the truth of it! Our lives are determined by our thoughts, not by what happens to us. Also, by minding our own business.\" In the workshop Katie suggested that there are only three kinds of business in the world\u2014\"yours, mine, and God's\"\u2014 and that most of the trouble we make for ourselves comes from confusing them. \"It is up to each of us to take care of our own business, question our minds, set ourselves free. If I'm over there in your business, who's taking care of mine?\"\n\nShe empties the rest of her tea from the green enamel pot. My time, I realize, is almost up. There are so many questions I'd still like to ask, but one seems unavoidable. How can a person survive his own losses, adapt without bitterness to twists of fate he can't stop wanting to turn back the clock on? \"We're human,\" I say. \"We have memories.\"\n\nKatie hears me out. Then she asks directly, \"Do you want to know the truth?\"\n\n\"That's why I'm here,\" I tell her.\n\n\"All right, then. What ever suffering we feel over things that have already happened is nothing more than an argument with the past.\"\n\n_An argument with the past?_\n\n\"You're bringing on your own pain now,\" she says. \"Your father slapped you when you were three years old, but _you've_ done it now a million times.\" She is right, of course. \"Nothing can be over till it's over for _us._ On our deathbeds we're still blaming our parents, our spouses, our children, our jobs, our country, our disease, our handicap\u2014whatever it is\u2014for ruining our lives.\"\n\n\"Insane, right?\" Stephen asks.\n\n\"But we can stop the insanity,\" Katie says. \"Questions can stop the suffering.\"\n\nHad she not discovered this work herself, healed herself of her own madness, were Byron Katie not sitting here in front of me, radiating palpable joy, I would discount this claim as pie-in-the-sky, smiley-face, New Age hoo-ha. But her survival story is real. The skeptic in me wants to argue, to find fault with her simple teaching, but even as I think these thoughts, my mind inadvertently starts doing The Work: Is it true that her claims are exaggerated? I wonder. That life can be free of suffering? Can I absolutely know it's true?\n\n\"Reality is so much kinder than our thoughts about it,\" Katie says.\n\n## **_JE M 'EN FOUTISME_**\n\nOne hectic day on a freelance editing job, the magazine's copy chief shows up in my office looking like he wants to talk. Tom, who's goofy and blond and comes from Indiana, has hardly said a word to me till now, but seems to have gleaned my field of interest and appears to want to tell me his story. \"Can you keep a secret?\" Tom asks, closing the door behind him. Before I can even tell him I'm writing a book on this subject, he launches into his remarkable tale (and later allows me to include it here).\n\nTwo years ago, Tom, who was thirty-two at the time, was getting drunk at a New York publishing party, cracking jokes and slugging beers, when all of a sudden his head flew back and he found himself, as he describes it, \"doing an unconscious break dance on the floor of a swanky SoHo ballroom.\" Tom had rarely been sick a day in his life but appeared to be in the throes of a grand mal seizure. A colleague he had been hoping to date shoved a wallet into his mouth to keep him from biting his tongue. \"The end of that relationship,\" Tom deadpans.\n\nThe next thing Tom knew, he was in the back of an ambulance surrounded by paramedics preparing to rush him to the hospital for tests. At first he refused, climbed off the stretcher against his friends' protests, hailed a taxi, and headed uptown for home. \"I figured I was stressed out,\" Tom explains to me now. \"I wanted to go home and sleep it off.\" Moments later, he was overwhelmed by another seizure, though, and the taxi rushed Tom to an emergency room. This is a very lucky thing. If he'd gotten his way and gone home to bed alone, Tom O'Connell would most certainly have died before morning.\n\nThis is what happens, apparently, when the brain tumor you don't know you have has hemorrhaged and filled your head with blood.\n\nWithin hours of arriving at the emergency room, after an MRI had identified the blood mass, Tom was rushed into surgery, where his head was cracked open, his face peeled off, and a piece of skull carved out in a procedure called a craniotomy. Tom reaches into an envelope and produces a photograph of himself on the operating table, hair slicked to the skull like a newborn baby's, blood dripping over a closed purple eyelid, a Tampax-like object protruding from his mouth.\n\n\"You had no warning signs?\" I ask, handing back the gory picture.\n\n\"Nothing,\" Tom says. \"Out of the blue.\" A month after his emergency surgery, Tom's doctors informed him that the tumor was not only malignant, but a grade 3 anaplastic oligodendroglioma\u2014a particularly aggressive cancer\u2014that needed to be removed immediately. In the days before the operation, Tom walked the streets in an existential fog that made day-to-day machinations seem absurd. \"I was consumed with thoughts of speech therapy, Pampers, and being spoon-fed,\" he says. During the two-and-a-half-hour procedure, the surgeon removed a mass \"the size of a small can of cat food.\" Although the surgery was a success, Tom would now be forced to undergo five days of chemo a month for alternating six-month periods\u2014indefinitely.\n\n\"How did you take the prognosis?\" I ask.\n\n\"It wasn't the news I was looking for,\" Tom shrugs. \"Then again, I had a chance.\"\n\nFor someone less feisty than this good ol' boy, scary news of this magnitude might have signaled the end of life as he knew it. In Tom's case it only intensified the desire to figure out what made him happy, physical risks notwithstanding. The French have a term for this brazenness: _je m'en foutisme,_ the brave art of not giving a damn. Tom, the carefree Midwestern boy, stopped playing anything safe in his life. He broke off a long-standing engagement to a girl he knew wasn't his soul mate. He ditched his boring publishing job and went back to his precarious-but-stimulating first-choice career as a pop culture journalist. His most daring decision by far, however, was to buy himself a Harley GTE chopper that became his escape route during his off weeks from chemo. Friends and family were worried about him. Were Tom to miss a single dose of his anticonvulsant medications, he could easily have another seizure. Yet this very danger is what kick-started his innate desire to heal, he believes.\n\n\"Riding gave me a sense of invincibility,\" Tom says. \"I remember one time riding eight hours in the pouring rain, going like ninety, playing music full blast in my headphones, passing these trucks on the highway, feeling fantastic. Tempting fate is in itself therapeutic.\"\n\n\"You probably realize that this sounds nuts,\" I tell him.\n\n\"What's so great about sanity? I'll take passion.\" Trusting his biker fantasy, Tom took a solo trip north to Vermont one June afternoon, eight months after his surgery, unaware of what fate had in store for him next. Landing in the town of Montpelier, where a college buddy ran the local bar, Tom was chugging brews with friends when a sexy woman caught his eye. \"This seriously hot brunette comes rolling in with a couple of girlfriends,\" he tells me. \"I couldn't take my eyes off of her. She was one of those people you look at and your heart just melts. I watched her for close to an hour.\"\n\nBefore Tom could protest, his half-drunk friend had traversed the room to chat up this group of lonesome females. Without Tom's consent, his bigmouthed pal spilled the beans about Tom's recent surgery, hoping to win the ladies' sympathy. Suddenly, to Tom's amazement, the brunette came charging across the barroom and threw her arms around his neck.\n\nShe had a brain tumor, too, she told him.\n\nThis is where Tom's story ascends to the incredible. The following weekend, Trisha invited Tom to spend the night in her nearby mountain cabin. \"Turns out she's eight years older,\" he tells me now, an unhappily married ex-professional bicycle racer turned horse masseuse\u2014and a fourteen-year survivor of brain cancer. Tom reports this to me with the undisguised glee of the kid who finds the pony in a pile of shit. Better still, Trisha recently bought herself a Harley so that the two of them can tour around the country together. Her philandering husband doesn't ask questions.\n\n\"You've both got brain tumors and you're racing motorcycles around the country?\" I ask. This sounds too fantastic to be real. Then Tom invites me to join him and Trisha for dinner in New York the following weekend.\n\n\"I still look at her and can't quite believe I actually get to be with this person,\" Tom is saying. He's got his arm around Trisha in a booth at John's Pizzeria in the Village. Trisha Stevens rolls her eyes. Wearing tight blue jeans, boots, and a brave streak of premature gray in her auburn hair (which a vainer woman would cover up), Trisha tells me, \"I've never cheated on my husband before. But there was something about our connection that I couldn't deny.\" Tom puts his arm around her. \"It's confusing and scary,\" she admits. \"We don't really know where this is going\u2014\"\n\n\"I'm hoping that Trish will be my wife,\" Tom says. Trisha looks at me and shrugs. \"I didn't plan it.\"\n\n\"Who plans it?\" I agree. Trisha seems grateful for my lack of judgment. Raised in a tiny Vermont town, she was a professional long-distance bicycle racer, immersed in winter training in Florida in 1993, when she started having mysterious seizures. \"All of a sudden, I'd be sitting there and could hear and see people, but it was like I was behind a curtain and couldn't speak myself,\" she tells me.\n\n\"Just out of the blue?\"\n\n\"No warning whatsoever.\" When these episodes started happening three times a day, Trisha set out on a frantic, frustrating search for a doctor who could diagnose her illness.\n\n\"They kept calling them panic attacks just because I was a woman,\" Trisha says, shaking her head. \"Do I strike you as a hysterical woman?\" What Trisha strikes me as is the kind of ass-kicking mama you'd want watching your back on a grisly episode of _Survivor._ Eventually, an MRI revealed a mass on her parietal lobe.\n\nShe was relieved to have an answer at least. \"It was like, no shit, Sherlock!\" Trisha laughs. The risks involved in opening her skull for a biopsy were potentially catastrophic, however, so she opted for a wait-and-see approach, combined with a daily regimen of anticonvulsants. \"I was out-of-my-mind scared, absolutely petrified,\" Trisha admits. \"Just walking around like a zombie. Numb. I couldn't figure out how people were just going through their daily lives.\"\n\n\"Isn't that freaky?\" I say.\n\n\"Like being in a parallel dimension.\" As someone who takes comfort in being active, Trisha volunteered to be a poster girl for the Brain Tumor Society and eventually became the number one female cyclist in New England. \"I wanted my experience to be good for something,\" she tells me.\n\nThat was fourteen years ago. Since then, Trisha has helped to raise hundreds of thousands of dollars so that other people like her \"don't have to wait around ten months for a damn diagnosis.\" During one thirty-nine-day solo trip from San Francisco to Boston in 1995, she had a seizure while crossing the Rocky Mountains at fourteen thousand feet. But this tough woman shrugs off the danger the same way she has accepted her _Easy Rider_ romance with Tom. \"I'd rather die like this than fall and hit my head at the grocery store,\" Trisha says. \"Otherwise you're just sitting around waiting to die.\"\n\nTom agrees. \"I still smoke and drink and carouse,\" he reminds me. \"Tempting fate can be therapeutic, like I said. The people around you are more scared for you than you are for yourself.\"\n\n\"I'm terrified of dying,\" Trisha admits. \"Don't get me wrong. But before something like this comes down in your life,\" she says, tapping the side of her head, \"it's easy to forget to pay attention. Watch the signs, take the exits. Not miss the thing that life's giving you.\"\n\n\"And _don't_ hang out with people who don't make you feel good!\" Tom insists. I'm reminded of science writer Dan Goleman's research proving that good company is actually pharmaceutical. \"I get nuts seeing people ignore their gifts! For a while after I got sick, I could hardly go out on the sidewalk without getting pissed off!\"\n\n\"Calm down, honey,\" Trisha says, patting his hand.\n\n\"But people are so oblivious!\" he tells her.\n\n\"We all get our wake-up calls,\" I remind him. \"Sooner or later.\" Tom and Trisha now find themselves in a fraught-but-fantastic situation that neither of them could have planned, crazy in love with each other, uncertain about what the future will bring. It's like _The Bridges of Madison County_ \u2014with cancer.\n\n\"My husband is a good man,\" she says. \"He doesn't beat me. He cares\u2014in his way.\" Now it's Tom's turn to roll his eyes. \"But from the moment Tom opened his mouth, he made me feel like a princess,\" Trisha tells me.\n\n\"You are,\" Tom says, kissing her forehead.\n\n\"I've never had that before,\" Trisha says. \"Never in my life. I wasn't looking to get out of my marriage before we met. But there was something undeniable about our meeting, and having this common bond.\"\n\nWhen Tom excuses himself to use the restroom, Trisha confides in me. \"He's fine right now,\" she says in a low voice. \"He might be okay for a long time. But that's a very nasty tumor he has. Mine could turn out to be benign.\" Before I have time to ask Trisha what effect Tom's prognosis is having on her big decision, he's back. \"I'm mad about him,\" Trisha says quickly, covering up our brief digression.\n\n\"And I've never been happier,\" Tom crows, slipping into the booth beside her.\n\n\"What are the chances?\" I say.\n\n\"Without these tumors,\" Trisha says, \"we would have been just two people in a bar.\" She pushes a cowlick off his forehead.\n\n\"How magical is she?\" Tom says, smiling. Trisha seems to glow when he talks about her. For a second, she does remind me of a Cinderella who snuck out of the house and became a princess\u2014if only for a brief stretch of time before the stroke of midnight. Tom orders more beer and starts humming a Counting Crows standard. When he begins an actual off-key serenade, Trisha lays a hand gently across his grinning mouth, for both our sakes.\n\n\"We'll just have to wait and see,\" she says.\n\n## **THE TERRORISTS WITHIN**\n\nA couple of weeks after 9\/11, I was walking my dogs around Washington Square, one mile north of Ground Zero, feeling strangely unalarmed. The stench of smoke was still thick in the air. Like everyone else, I'd been horrified on the morning of the Al Qaeda attacks. I'd stood on the corner of Sixth Avenue, along with hundreds of neighbors, watching shell-shocked people in business suits covered in ash as they wandered north from the scene of the crime, knowing that American life had just changed forever. In the days that followed, an eerie quiet had settled over the neighborhood. While this atmosphere of communal shock was unmistakable, though, I was not quite as traumatized, it seemed, as the majority of people around me. Not being devastated made me feel like a hypocrite. Had I grown so hard-hearted, cynical, jaded, I wondered, that mass murder in my own backyard left me feeling less tragic than stoical? Peering down into the empty airspace where the towers had stood at the end of West Broadway, I questioned the absence of more outrage inside me and why\u2014with my neighborhood collapsed in despair\u2014I was walking my dogs feeling so weirdly normal.\n\nOthers' lives were changing dramatically. Friends were no longer riding the subway, crossing bridges, risking tunnels, venturing outside walking distance from their kids for more than an hour or two. A girlfriend had already sold her apartment and fled for British Columbia with her husband and two young sons. In those first weeks after the planes struck, every day brought more conversations about rage, depression, post-traumatic stress\u2014along with the chanting refrain that _we were no longer safe._ This is where my mind would snap shut, hearing this apocryphal phrase. No longer safe? When were we safe? Had the facts of life really changed so dramatically on the morning jihad arrived on our shores? Many smart people seemed to think so. They appeared to be responding to 9\/11 as if the human condition had just changed; as if life had just shifted from fair to unfair; as if Eden itself had just fallen apart and the garden was suddenly planted with bombs and lunatics eager to explode them; as if these formerly sheltered Americans had been transformed on that terrible morning from innocents to prisoners of war, forced to confront a violent world for which they were unprepared. To face life's demonic forces, terrorists within and without, whose presence till then they'd been able to keep at a distance. A sixty-year-old woman whom I respect, wise in many worldly things, had confessed to me in those aftermath days that she realized how deeply she herself could hate and how easily this rage could lead her to violence.\n\nAmericans had been initiated into terror. Having dwelled till then in a bubble of comfort, we were forced to acknowledge the dark side of things\u2014not _over there_ but here at home. We were neither invulnerable nor immune, nor was our victimized country a victim. Invited to take deeper stock of how 9\/11 had happened to us, many Americans, while in no way condoning terrorist acts, were forced to acknowledge our nation's checkered history (a reckoning intensified by the misdeeds of a warmongering president), to admit that our great country has never been truly innocent. While spreading democracy we have also at times been bullies, imperialists, and provocateurs, manipulating lesser nations' affairs from behind a superpower mask. Forced to become aware of this, Americans began to grow up. Many changed the way they lived. Others used fear to feel even worse. But I don't know many whose lives didn't change somehow (going through airport security alone gave you pause), as sudden citizens of the world rocked awake by this devastation. The arrival of the terrorists reminded them that their own lives were exposed and fleeting. Bodies hurling themselves from windows reminded us that our lives had windows, too.\n\nThe ancient Greeks had a word for this progression from na\u00efvet\u00e9 to mortal wisdom: metanoia, _the opposite of paranoia._ Metanoia literally means \"turning around,\" crossing away from fear into the zone of truth. The ancient maps that chart the stages of human life recognize that midway through (though some get butt-kicked earlier), after the shake-up, a process of intense self-examination and reassessment of the world begins in an individual. For this process to bear fruit, metanoia requires acknowledgment not only of the terrorists out there, but of the demons we carry within as well.\n\nThis appears to be counterintuitive at first. Why should an attack from outside prompt acquaintance with our own inner killers? We are the victims, some people say. Why should their bloody fatwa be used as a mirror to our buried hatreds? In order to expand the mind, to make use of what otherwise is wasted terror. This is what the enlightened tell us. From Gandhi to the Dalai Lama, even the escaped slave Francis Bok, we hear the paradoxical refrain that only by rejecting false virtue and self-righteousness can we see the truth. \"I contain within me the seeds of all possible crimes,\" claimed the near-saintly phi los o pher Simone Weil. Without condoning the crimes of others, we use them to examine the criminality latent in ourselves. \"I must forgive,\" Francis Bok had told me. \"Otherwise, I am just like Giemma.\" His kidnapper.\n\nMetanoia, and the emotional strength it brings, demands that we name our own demons. \"Devils, Communists, capitalists, terrorists... the evil we previously objectified and assigned to exterior agents... must be discovered within,\" in one psychologist's words. At first we're shocked by our own shadow. It's not so easy now to divide the world into black-and-white camps of good and evil. \"We have met the enemy and he is us,\" the comic-strip character Pogo said. Metanoia turns us around. All of a sudden, we're reeling back not only from terrorists' deeds but also from the brutal awareness that these actions were carried out by humans like us. This is what we are capable of in extremis, we think, and the sickening thought burns our righteousness off. For the first time there is no Other. We can never again pretend not to have seen this.\n\nLuckily, we're hardwired for this awakening; body and soul know what to do with terror. Courage is in our genes. \"It would be strange, indeed,\" wrote geneticist C. H. Waddington (in a passage that deserves to be quoted at length),\n\nwith so many millions of years of survival experience packed into our genes, if at some deep involuntary level, we did not possess capacities specially geared to cope with extreme situations. In the beginning there was nothing but extremity. Nothing but the random rush of life in a touch and go struggle against extinction. Men and women began to notice, as no other animal can, the frightfulness of infinite spaces opening everywhere around them... Death awareness is the bitter fruit of man's having risen to the level of consciousness. What began a billion years earlier as neurological response to environmental stimuli, with man came to climax in terror.\n\nAmericans were mostly fortunate before 9\/11 not to have had to face this harsh fact. But the horror has opened thousands of people's eyes, too. My sixty-year-old friend, for example. She had her own metanoia moment. \"I never knew I could hate so much,\" she told me the other day at dinner. Al Qaeda had ruptured her golden mean, interrupted her trance of cashmered privilege, the comfort zone of the chronically lucky, to whom things like 9\/11 don't happen. It brought her face-to-face with her monsters, including the depth of her own fear, which made her look as if she'd been knocked upside the head. My friend seemed gentler and less aggressive. She seemed to listen with more intent.\n\nPerhaps it's a new kind of innocence. Not the fiction of perfection but an openness born of truth-telling, wonder, and awareness of endings. In a novel called _Gilead,_ Marilynne Robinson wrote about something she calls a \"learned innocence.\" \"There is a learned innocence as much to be valued as the innocence of children,\" she suggests. This innocence contains the dark. It germinates in the footsteps of monsters.\n\n## **EARTH ANGEL**\n\n\"Last night I dreamt I was a butterfly,\" wrote the Taoist philosopher Chuang Tze. \"Now am I a man dreaming I am a butterfly or a butterfly dreaming I am a man?\" Swept up in the current of powerful change, spun around by metanoia, we're startled by how changeable our identities turn out to be when our familiar context\u2014the containers we inhabit\u2014is forcibly taken and stirred. We're more protean than we ever imagined. In Greek mythology Proteus is the god of changing forms, the shape-shifting, trick-sterish aspect of the self that adapts instantaneously to shifts in the current. He is also the patron deity of survivors. The quicker we change, the stronger we swim; the smoother we morph, the better we surge; the less fear we swallow, the higher we dive and the more deeply we're able to plunge.\n\nThis may be why children are often more resilient than their baggaged adult counterparts, and why, as the Bible says, the child can become \"father to the man\" in emergent times. Children have fewer layers to unpeel, less history to keep them stuck. Only later in life do many children who've endured bad things learn to be afraid.\n\nIt's a ball-freezing, dreary-white, midwinter's morning in the exurbs of Chicago and I'm waiting in a strip mall espresso bar to meet a young Bosnian woman named Adisa Krupalija. I've been waiting for years to meet Adisa. In 1994, after returning from a trip to refugee camps in Pakistan, Eve Ensler, an activist friend whose life has been devoted to ending violence against women (including with her play _The Vagina Monologues_ ), told me about meeting a remarkable twelve-year-old refugee at a camp near Islamabad. Apparently, Adisa was a language whiz, and had been inducted as the camp's official English-language translator in spite of being barely pubescent. For a year it had been Adisa's responsibility to help hundreds of fellow refugees trying to gain asylum in the United States. Eve described seeing her for the first time, a tiny girl in a _shalwar kameez,_ rushing around that malarial hellhole in 120-degree weather, translating testimonials, documents, love letters of her desperate compatriots struggling to talk to the outside world. \"She seemed selfless to me,\" Eve told me. The Krupalijas had eventually landed in Chicago to be with relatives; Adisa had gone on to win a scholarship to Northeastern University, where she was now studying for the bar. Although Adisa does not like to talk about the war, she'd agreed to speak to me as a favor to our mutual friend.\n\nAdisa comes rushing into the caf\u00e9 a half hour late and locates me in a corner booth. In a movie she'd be played by Natalie Portman\u2014all gazelle neck and brown eyes and bobbed, gamine hair. \"I'm so sorry!\" Adisa clutches my hand and smiles. Apparently, somebody, somewhere, needed Adisa's help with something and she couldn't bear to tear herself away. I recognize her kind right away; she's the sensible, caretaking, peacekeeping type to whom everyone else turns for help, regardless of what she might need herself, the Rock-of-Gibraltar person who rarely says no, because she mostly feels like the strongest.\n\nI assure Adisa that there is no harm done. A gulag-faced waitress comes lumbering over. Adisa orders the chocolate pancakes. We drink our coffee and talk about Eve. Then this lovely young woman slowly begins to tell me her story.\n\nOnce upon a time, in 1992, the Krupalijas were a happy family leading a privileged life in the Yugoslavian town of Trnovo (population four thousand), thirty kilometers from Sarajevo. Adisa's mother grew roses on the terrace and had girlfriends over for tea. Her father was a town official. Beneath the window of their apartment, a nineteenth-century cobblestone square served as a playground for the Krupalija children. Adisa jumped rope there. She was a top student in her fourth-grade class.\n\nOne morning the milk lady showed up at their doorstep looking panicked, Adisa tells me now. \"She told us that this would be her last time coming. She looked so pale.\" Soon afterward, barricades were erected around their apartment building to keep out Serbian tanks rolling toward them from Sarajevo. \"Explosions began shaking the window. My father said that we must leave right away. But my mother refused.\"\n\n\"Why was that?\"\n\n\"She was worried about who would water her plants.\" Adisa's look of strained forbearance tells me all I need to know about her beloved mother.\n\nNo longer safe in their apartment, the Krupalijas hid out in the basement for three days, making candles from shoestrings and oil. Adisa was charged with caring for her hysterical infant brother. \"It was like hell,\" she says simply. \"As if everything is one big scream and you're caught right in the middle of it. I now know what a rat feels like.\"\n\nUnder cover of night, the family escaped from their underground hideout and set out on a frightening exodus, wading across cold rivers at night without flashlights, hitching rides on trucks, avoiding Serbian snipers' bullets. After several months, and many close calls, the Krupalijas were airlifted to the Pakistani refugee camp. Promised a decent place to stay while they sought U.S. asylum, they arrived to find a filthy, insect-ridden, food-poor compound devoid of hygiene and surrounded by barbed wire, with AK-47-armed policemen posted at the gates.\n\nIt was at this point, Adisa tells me, that her father hatched his plan, laying the family's fate on the shoulders of his twelve-year-old daughter. Adisa still doesn't understand how this happened. \"All my father said was that unless I learned English, we would never be able to leave that place.\"\n\n\"But you were so young.\"\n\n\"It was all happening so fast,\" she says, aware that this sounds like a weak explanation. \"I don't know why it was up to me. Maybe because I like languages.\" Something in Adisa's tone suggests that perhaps her parents were too distraught to be more help.\n\n\"It worked out best for everyone,\" she says, excusing them. \"It was the best thing that I've ever done in my life. Helping people in the camp is what got me through. I learned to put others before myself.\"\n\nHer pancakes arrive, looking like a dripping Dal\u00ed chocolate still life. \"It made me strong,\" she insists, pushing her breakfast aside for the moment. \"I was doing what was necessary, that is all.\" Inside of two months Adisa had perfected her English enough to begin the family's immigration process and to offer service to other camp-mates. This sudden shape-shifting from preadolescent to refugee caseworker was a wonderful thing, she assures me.\n\n\"How did you learn so quickly, Adisa?\"\n\n\"I've always been independent,\" she says, shrugging. \"No matter what happens, I've always tried to focus on what I could learn from any given situation.\"\n\n\"At twelve?\"\n\n\"Even then.\" She smiles. \"I was thinking, how can this improve me as a person? I have always been intensely curious, wanting to learn more and more. Curiosity makes life so much more interesting. In the camp, everybody was so despondent about how hot it was. Some got malaria. Some became jaundiced. But all I could think was, 'Wow, I get to learn English. Now I get to see the world outside my little town.'\"\n\n\"It was your ticket out.\"\n\n\"Exactly. Not that it was so easy, of course not. We saw terrible pain and violence. But I grew up in that refugee camp. When you go through something like this, a part of you is changed forever.\"\n\n\"Which part of you changed?\"\n\n\"The one that refuses to conform. The part that can never be like other people. You're always pushing yourself to be different, to take the harder path. Challenge yourself to be stronger.\"\n\nThat sounds like a lot of pressure, I say.\n\n\"It is, sometimes. But it helps you in the end. It gives you tenacity. A high tolerance for change and hard times. When I look back at my time in Pakistan, there was nothing that brought me down. Nothing. There was nothing that made me doubt that things would be all right.\"\n\nShe almost sounds nostalgic, I notice.\n\n\"Not quite.\" Adisa chuckles. \"But I do have many good memories. Sometimes I do miss that little girl,\" she admits, reaching for the sugary pancakes.\n\n\"She's still here. Underneath.\"\n\n\"More some days than others,\" Adisa tells me. Creating a new life for herself in America has required a difficult new skill set, she says. She compares the process to an artist shifting painting styles in his studio. \"At first it's like a Jackson Pollock,\" she says, alluding to that artist's spontaneous, rainbow-splattered canvases. \"You're just getting through. It's all about instinct. You're throwing paint at this canvas and hoping it will become something.\"\n\nAdisa bites into her gooey breakfast. \"Later it's more like Georges Seurat,\" she says, meaning that Frenchman's micromanaged canvases covered with tiny pinpoint strokes. \"That's when you're putting it all back together. You have to do the little dots and at the same time see the big picture. You carefully proceed step-by-step, being patient, focused, precise. You pick a goal and set it high.\"\n\n\"But look at you now,\" I say. \"It worked.\" Ten years ago Adisa was a Bosnian refugee without a cent in her pocket. Today she's studying for the bar exam and recently accepted a lucrative job with an international corporate law firm. \"You did something right.\"\n\n\"I'm doing okay,\" Adisa says, deflecting the compliment. \"But I have to remind myself sometimes of what my twelve-year-old self would have done. When I'm in school, or with my family or boyfriend, I have to remind myself. I used to be so fearless,\" Adisa says. \"I never want to lose that.\"\n\n\"Where would it go?\" I ask.\n\n\"I know it sounds strange,\" she says. \"But I almost felt more free back then. Life seemed so simple. I just have to keep remembering that.\"\n\n\"And trusting,\" I tell her.\n\n\"And changing,\" she answers. \"I never want to stop changing.\"\n\n\"As if you could,\" I say.\n\nFor a second, Adisa looks as young as she is. Then she says, \"Change is everything.\"\n\n## **THE END OF SEEKING (OR DIG IN ONE PLACE)**\n\nIn the decade after quitting my job at _Interview,_ I lived in twenty-eight different locations (not counting extended road trips and short-term squats), camping out in retreat centers, spare bedrooms, house sits, sublets, sport-utility vehicles, anywhere that I could find a cheap bed and a teacher who claimed to have something important to say. From Frankfurt to Philadelphia to Fuengirola, Myrtle Beach to Bubeneshwar, San Francisco to Paris to Pondicherry, then back again to Manhattan, I became a compulsive dharma bum, following my divine impulses wherever they led and my Visa card was accepted. People told me I was crazy for living this way\u2014for _so_ long\u2014and they might have been right. But it was good crazy, metamorphosis crazy, crazy for learning and waking up before the fat lady crooned her last.\n\nSpiritual seeking was the most intense love affair I'd ever had. It brimmed with unconsummated desire, the tantra of playing peekaboo with God\u2014the Beloved that no one could ever possess. The divine perfume kept me snout-to-the-ground, self-absorbed as any hopeful romantic. I'd finally found an object of desire that seemed worthy of both the chase and the heartache. Pursuing wisdom instead of money, sex, bylines, or security (which I rejected then as a bourgeois illusion), I felt authentic for the first time. My life had a noble purpose at last. I thought I'd become some kind of hero.\n\nThen my honeymoon with the Holy Grail ended as swiftly as it began. I was minding my own business one afternoon, by a lake at a South Carolina retreat center, meditating on the meaning of light on water and whooping loons munching on Spanish moss, when the Ghost of Christmas Future appeared wearing orange harem pants and a single pink flower stuck into his long gray hair. With no invitation from me, this aging hippie proceeded to tell me the sobering story of his life. He'd been a spiritual seeker since R.D. first proclaimed, \"Turn on, tune in, drop out,\" in the sixties, searched for God (nirvana, he called it) and avoided the trappings of worldly life. Now this poor guy was staring down retirement age with nothing to show for his years on the God trail, he told me sadly, but a P.O. box in Santa Cruz, a plethora of religious moments, and his membership card to the AARP. He was lonely, road weary, wistful, and bitter. His resemblance to who I myself could become (were I to live that long) was too pointed and painful to overlook.\n\n\"You wanna know what I ended up with after all that seeking, brother?\" the old hippie asked me, tugging one of his many gold earrings. \" _Nada,_ \" he said. \"Nothing.\"\n\nBut how could that be? I asked in self-defense. He'd chosen a life of surprises and travel, spent his time on the road not taken, expanded his mind with wisdom and awe. \"We all make our choices,\" I said to the guy. Every choice has its pros and cons. Maybe he was just having a crappy day.\n\nThe stranger waved my excuses aside, then paused my heart with these frightening words: \"Go home while you're still young, buddy boy. _Dig in one place._ \"\n\nI felt as if God had sent me a prophet, a sign to ignore at my own peril. The sign pointed straight at the secret suspicion\u2014the one I'd been trying so hard to ignore\u2014that I was metaphysically full of shit. I was running away from my life, not toward it, focused on holiness Casanova-style, mounting the next retreat, teaching, ayurvedic chakra cleanse, for the same reason philanderers can't stop chasing skirts: They're afraid to die. They're secretly scared that if they stop running, they'll be trapped. The trap (or woman) will morph into a grave. The reaper just might spare a moving target. You stick to the edge when the middle's too scary. Promiscuous seeking, I saw, was a ruse.\n\nNext, I had the deflating misfortune of reading a book called _Cutting Through Spiritual Materialism,_ which, if you haven't read it, don't\u2014if you want to have an ego left. Fantasy-wise, it was downhill from there. Sages across the board proclaimed against the refuge of the road: \"When a thing is everywhere the way is not to travel but to love\" (Augustine); \"To seek is not to find\" (Rumi); \"I came to a spot in the road where all paths were one\" (Dogen); \"Most seekers are just Narcissus in drag\" (Da Free John).\n\nThe checkmate of this was abhorrent to me, but I did not want to end up with _nada._ Returning to the city for the time being, I rented a cheap apartment downtown. I lined my walls with photos of otherworldly-eyed saints to help relieve me of claustrophobia. This strategy didn't work for long. Soon enough, I was going crazy, the walls were breathing in on me, I couldn't breathe myself, and then came the morbid ideation. This led to moments of genuine, knee-knocking panic. I really didn't want to die. But as winter passed and the leaves returned, the courtyard behind my apartment quite peaceful, I began to get used to staying put. Being still made way for a ghost to appear.\n\nIn the bathroom one morning, as I was brushing my teeth, a numinous presence announced itself behind me\u2014not an actual ghost, exactly, more a mounting apparition of dread, a condensation of feeling so thick it appeared to envelop me as I stood there holding my toothbrush. The ghost invited me to have a figurative seat. I paused and listened. The tone of its voice was more poignant than scary. The ghost told me it was a distillation of all my greatest fears. It was the combined essence of all that I dreaded, the dark thing I'd been running away from. This ghost simply wanted me to listen, as if hearing confession from myself, to music sadder than anything I had allowed myself to hear before. It spoke to me in a major chord about the \"unfinished symphony,\" the longings and dreams that could go unfulfilled, the lopped-off, pissed-off refrain that began the day I started expecting to die young. Hearing the ghost's voice touched me. It was not horrifying. So I listened to the ghost in the months to come.\n\nThere's a place where beauty and sadness meet\u2014if you've been there yourself, you know this already\u2014where the two become indistinguishable. It's the place where sadness is no longer ugly, where grief begins feeling like soap in a wound, painful but purgative at the same time. Once, in Italy, I watched my host, a restorer of damaged paintings, as he worked on a Renaissance canvas so encrusted by time that the image had been covered up completely. Robi dipped his brush in the lye-smelling liquid and ran it across the grimy surface, revealing an eye, an ear, a cheek, and finally, the face of a curly-haired child gazing up at a pair of wings. It amazed me that something so foul and evil-smelling could uncover such hidden beauty. At home now for the first time in years, I was learning that grief works the same way, poisonous if withheld too long, clarifying upon its release.\n\nThis ghost and I became allies that year. Its voice told me what I was afraid of. Today we're still close, but I rarely see him.\n\nThere's a story about a beggar who was sitting on the side of a road. The old man had been on the road for years. A stranger approached one afternoon. \"Spare some change?\" mumbled the beggar, mechanically shaking his tin cup.\n\n\"I have nothing to give you,\" the stranger said. The beggar turned away in disgust. Then the stranger asked, \"What is that you're sitting on?\"\n\n\"Nothing,\" the beggar told him. \"Just an old box. I have been sitting on it for as long as I can remember.\"\n\n\"Ever looked inside?\" asked the stranger.\n\n\"Why?\" the beggar replied. \"What's the point? There's nothing in there.\"\n\nThe stranger insisted, \"Have a look inside.\"\n\nThe beggar refused at first, then finally decided to pry the lid open. With astonishment, disbelief, and elation, he saw that the box was filled with gold.\n\nComing home, I had opened the box.\n\n## **SOMETHING ELSE IS ALSO TRUE**\n\nDoubleness comes as a revelation. The realization that every experience has two sides, even seemingly monolithic distress, turns the mind around. There is always a mystery face to experience, including the most painful episodes. We realize this slowly over time as we watch life turn its other cheek, again and again.\n\nMaria Housden, then a New Jersey house wife, learned this countervailing lesson by enduring a mother's worst nightmare. On the morning of January 7, 1993, Maria's two-year-old daughter, Hannah, was diagnosed with a galloping, untreatable rhabdoid cancer of the kidney sure to kill the little girl within the year. The shock of this information plunged Maria from the ordinary world of soccer mom into the parallel zone of nursing a dying child\u2014 while her other two children still needed her\u2014without losing her sanity.\n\n\"Hannah's diagnosis catapulted me into another reality,\" Maria explains to me. We're alone on the porch of her rented seaside house in Sea Bright, New Jersey. Maria is six foot two, weighs 120 pounds, and has the glass-green eyes, forward-lurching hips, and classic cheekbones of a runway model. She's like Uma Thurman's basketball-playing supersister, with white pedal pushers up to her knee. It's distracting to walk down the street with Maria. People often point and stare, especially the men, who gaze at her towering beauty with wonder.\n\nMaria lights a Virginia Slim and takes a sip of her margarita. \"I had been very good and faithful to what I understood as God all my life,\" she tells me. \"I believed that I could control what happened to me, more or less\u2014minimize my own suffering and certainly that of my children. That was my job as a mother,\" she says. \"But Hannah's diagnosis dissolved all of that.\"\n\nHer survival response was to keep appearances up as meticulously as possible. \"I became very methodical,\" Maria remembers. \"There were so many things that I couldn't control, but there were a few others that I could. If Hannah was going to die, for instance, I wanted to have a say in what that looked like,\" she explains. \"I wanted her to die at home instead of in a hospital, to have family there instead of strange people.\" Maria stubs out her cigarette. \"My job was to do everything possible to help my family through this.\"\n\nUnderneath the mask, however, she was quickly going to pieces. \"I had it all under control on the surface. Oh, brother! People kept telling me how amazing I was to be doing what I was doing! I felt like one of those dummies you see in store windows, walking around behind a cardboard cutout of myself.\" Maria shakes her head. \"Behind the fa\u00e7ade there was loneliness and turmoil.\"\n\n\"But you couldn't show that to others?\"\n\n\"I needed to make it feel okay to other people so they would keep showing up for me and for Hannah. Who wants to hang out for a year with someone who is completely in despair, unable to cope with what's happening?\" she asks.\n\nI let her know that I've had my own share of friends walking away.\n\n\"It's a survival mechanism,\" Maria agrees. \"I made the outside part very intact\u2014you could draw a line around it\u2014but the rest of me was this kind of soup.\" Her grief over losing Hannah immersed Maria in this muddling void. The wise little girl who insisted on wearing her red Mary Janes during surgery, who refused to speak to doctors unless they told her their first names, became her mother's intimate muse in the year that she died. Maria has detailed this wisdom in her bereavement classic, _Hannah's Gift_ (published with the telling subtitle _Lessons from a Life Fully Lived_ ). Caring for Hannah revealed to Maria truths about her own life she had been avoiding, places where she wasn't telling the truth, parts of herself that were being dishonest.\n\n\"I'd poured everything I had into maintaining an illusion of perfection in every aspect of my life,\" she says now. \"I'd forgotten what was right for me.\" For starters, her marriage was on the rocks long before Hannah got sick. The little girl died a year after her diagnosis, and in subsequent years the weight of accumulated lies became unbearable. Maria slipped into such a deep depression that she decided, against the advice of family and friends, to leave her three children\u2014Will, twelve, Margaret, four, and Madeleine, three\u2014with their father in New Jersey, while she took time to sort herself out. She needed to make a lobster's plunge, to shed one skin and grow a new one before she could enter her next phase of life.\n\nRetreating to a Christian center in the Michigan wilderness, Maria could still hear the mocking voices of those who judged her for what they called running away. Yet she held to her conviction that she needed this solo time in order to heal and absorb the lessons that Hannah's death left with her. \"There's no room in deep grief for anything but the truth,\" Maria says. \"You're able to see things in your life, and say things to yourself, that you couldn't at any other time.\"\n\n\"Grief can be distorting, too,\" I say, thinking of Joan Didion's magical thinking.\n\n\"At first it is,\" Maria agrees. \"But I learned that grief has another side to it as well. For example\"\u2014she takes a sip of her drink\u2014\"something can hurt so badly that you no longer have patience with the fucked-up-ness you might be living in your life. It's that kind of movement. That kind of energy.\"\n\n\"I know what you mean.\"\n\n\" 'Grief' shares the same root as 'gravitation,'\" Maria reminds me. \"It's a force, an energy with momentum. Grief is a power with weight and heft. And the fact that it has weight also suggests that it is something that can be shifted. You can succumb to grief and be buried in it. It can suffocate you and hold you down. Or your perspective can shift and the pain can be used to compel you in a different direction.\"\n\nThe notion of grief as a rerouting force intrigues me. \"Where were you on the night of nine-eleven, for instance?\" asks Maria. \"And did you know then\u2014in a way you may already have forgotten\u2014 what matters most to you?\" She allows the question to settle for a second. \"That's how grief can keep us honest. This is what I learned from Hannah's death. There is always another side to things, a mystery. No matter how great the loss, _something else is also true._ \"\n\nThis mantra has a haunting quality. Doubleness, I think. The hidden face.\n\nMaria appears to be one of the happiest people that I've ever met. Contrary to what one might expect, talking about Hannah gives her joy. \"My daughter taught me to be a whole person,\" she says.\n\nThirteen years after Hannah's death, Maria does appear to be thriving. Back in New Jersey with the kids, working on a film version of _Hannah's Gift_ with a French director, easing her way into the community, patching up a friendship with her ex-husband, beginning to date, she loves her new life as a writer and is looking forward to what comes next.\n\n\"I'm grateful for every day of my life,\" she says simply. \"I never thought I would say that again.\" Maria takes a copy of _Hannah's Gift_ from the shelf and points to a passage near the end of the book. \"Hannah had taught me that there is a death more painful than the one that took her body from this world,\" it reads. \"A soul suffocated by fear leaves too many joys unlived.\"\n\n\"That's the truth,\" I say.\n\n\"Hannah was my teacher,\" she tells me. Then Maria reaches for a photograph of the fair-haired girl sitting on her mother's lap. She studies the picture for several moments. \"I used to be afraid of mystery,\" Maria says, setting the photograph back on the table. \"But the unknown doesn't scare me anymore. Instead of drowning in it,\" she says, pulling her bare knees to her chest, \"now I'm swimming.\"\n\n## **PAIN PASSES, BUT THE BEAUTY REMAINS**\n\nOne day a student comes to her guru overwhelmed with pain. She tells the old woman that she isn't sure how to survive her own sadness.\n\nThe teacher listens to her distress. I imagine the guru's eyes softening as she hears the disciple's story, suffused with fathomless wisdom gained from having crossed her own wilderness to the powerful place where she now sits.\n\n\"I want to end my pain,\" says the weeping student.\n\nThe teacher leans forward with a gentle smile. \"You cannot conquer your pain by destroying it,\" she tells the disciple. \"But only by allowing it to be what it is\u2014unbearable poignancy within an infinite nature, which also contains joy. Even bliss.\"\n\nThe visitor does not understand.\n\n\"Look into my eyes,\" the guru says. The sad woman raises her gaze to meet the teacher's steady countenance. \"Do you see in my eyes an ignorance of your pain?\" the old woman asks.\n\nThe student can see that her teacher's eyes contain no ignorance at all. They are as clear as crystal and shining bright with unconditional compassion and understanding.\n\n\"Do you see that I accept your pain fully?\" the teacher asks.\n\nThe student can see that this is true.\n\n\"What you see,\" the teacher says, \"is that I am not afraid of your pain. I have conquered the illusion that such terrible pain will destroy me. The pain is what it is. And I am what I am.\"\n\nThe humbled student bows to this wisdom. How liberating it must be to know this, she thinks, to know that you are not your pain even when you cannot see beyond it. To overcome any bitterness that may have arisen because you were not \"up to the magnitude of the pain that was entrusted to you,\" in the words of another master.\n\nLike the mother of the world who carries the pain of the world in her heart, each of us is part of her heart and therefore endowed with a certain mea sure of cosmic pain. You are sharing in the totality of that pain. You are called upon to meet it in joy instead of self-pity. The secret: offer your heart as a vehicle to transform cosmic suffering into joy.\n\nThe student leaves her guru's house feeling small but happier.\n\n## **HEDONICS**\n\nBeing happy, wrote Colette, is one way of being wise. The indomitable French authoress who refused to leave Paris during the Siege, scribbling her erotic novels as war planes dropped bombs near the Arc de Triomphe, was certain about the wisdom of joie de vivre in times when life is roughest.\n\nThe burgeoning field of happiness studies, known as hedonics, tells us a lot about why (beyond dumb luck and longevity genes) some people thrive in miserable times while other, more outwardly fortunate souls actually decline rather quickly. Hedonics would not exist, it turns out, if Sigmund Freud had had the last word. Famously declaring happiness to be a quixotic pipe dream, and \"the transformation of hysterical misery into common unhappiness\" to be the most we poor neurotics could hope for, the father of psychoanalysis was largely responsible for science's formerly paltry understanding of how happiness figures into a human life. Before neuroplasticity demonstrated that the brain is evolving, learning new tricks, forging felicitous pathways till the day we die, psychological research had focused mainly on negative emotional states. The recent shift from what goes wrong with us to what goes right has brought the positive psychology movement to the cultural table at last.\n\nSubjective well-being (SWB) is the nickname experts in this field give to happiness. Since your hell may be my paradise, subjectivity appears to be the single greatest variable in the happiness equation. Homeless people in Calcutta have been found to be less unhappy than those in California (because they have a stronger sense of community), while Amish folks rarely claim to be bored, though sifting curd might not be your idea of a laugh riot. Each of us is born, it seems, with a happiness \"set point,\" a genetic level, from giddy to grumpy, around which SWB tends to settle, regardless of what happens to us. A now famous study of identical twins reared in different environments suggests that this set point determines approximately 50 percent of our disposition to happiness.\n\n\"Happiness is genetically influenced but not genetically fixed,\" David Lykken, a geneticist at the University of Minnesota, tells me. \"The brain's structure can be modified through practice. If you really want to be happier than your grandparents provided for in your genes, you have to learn the kinds of things you can do, day to day, to bounce your set point up and avoid the things that bounce it down.\" Apparently, there are as many happy-bumping tools as there are people wanting to cheer up. They range from commonsense maintenance such as sleep, exercise, and nurturing relationships to shifts in core beliefs and expectations. This reminds me of Maria Housden's experience after Hannah died. One researcher I speak to recommends taking control of your own time and keeping a \"gratitude journal.\" Another promotes \"acting happy,\" since there seems to be a direct link between facial expression and emotion, as in fake it till you make it.\n\nHappier people seem to be those who rely on familiar shortcuts rather than overthinking every little situation. Happy people tend to forgive; they're also inclined to gravitate toward making commitments. It comes as no surprise to learn that stress is the Darth Vader of contemporary life. Jon Kabat-Zinn, who helps run the Stress Reduction Clinic at the University of Massachusetts, goes so far as to say that \"we have a sort of autoimmune disease\u2014chronic stress and discontent\u2014caused by not looking deeply enough into this question of genuine happiness.\" Kabat-Zinn teaches people seeking stress relief that wellness practices such as meditation make it possible, even in the midst of hardship, to experience simple pleasures. \"To know delight, what is right and beautiful with the world,\" as he puts it. \"With mental balance, we develop a keel-like ballast that helps us to remain stable even under extreme conditions.\" (We will discuss stress specifically in a later chapter.)\n\nRegardless of which particular tools we choose to help lift our own SWB, one thing appears to be certain: We're better off aiming for happiness moment to moment than trying to engineer contentment through long-term planning. This is because, as science now shows us, human beings are fairly hopeless at predicting what will make us happy or how long that happiness will last. Harvard psychologist Daniel Gilbert spends his days exploring the riddle of human self-delusion. He's a pioneer in the field known as affective forecasting, whose researchers mea sure the distressing gap between what we believe will make us happy and what actually ends up doing so. \"We're such strangers to ourselves,\" Dan insists to me. \"Nowhere more than in our pursuit of the Holy Grail of happiness.\"\n\n\"But why?\" I'm still puzzled by how this self-blinding happens.\n\n\"Because we usually overestimate how things will affect us and rarely underestimate them,\" he explains. This discrepancy, known as the impact bias, causes a great deal of what Dan calls \"miswanting.\" When post-catastrophe people talk about cleaning house, getting rid of unnecessary crap in their lives, they're talking about the fruits of miswanting.\n\nTo confuse things further, the results of our choices are nowhere near as life changing as we think they will be. In a 1978 study of SWB among lottery winners and paraplegics, both groups adjusted paradoxically to their respective changes of fortune: The lottery winners settled back to levels of happiness that did not differ significantly from a control group's. The paraplegics, while less happy, were not as unhappy as expected. In fact, as another study revealed, major events\u2014happy or not\u2014lose their impact on happiness levels _in less than three months._ If we understood how quickly this adaptation process worked, we might choose to invest our hopes in things that could actually last\u2014and deliver.\n\nMoney is not one of them. It has been proven beyond a doubt that once our creature comforts are met, having more in the bank account rarely makes people more content. Once middle-class comforts are in place, the line between wealth and survival, Dan Gilbert tells me, is virtually negligible. \"The first forty grand makes a dramatic difference,\" he says, \"but after basic needs are met, the next ten million does almost nothing.\"\n\nI tell him that I'll be the judge of that.\n\nJust as Italians have eight words for love, we need more definitions of happiness, a spectrum wide enough to encompass struggle, loss, and unwanted change. The smiley-face version of happiness simply will not do. Life is too complex, we know too much, there's too much pain to be satisfied with a na\u00efve idea of what it means to be glad for life. Even survivors of the direst experiences attest to this paradox. \"Who knows what happiness is?\" asked one Holocaust refugee. \"Perhaps it is better to talk in more concrete terms of the fullness or intensity of existence,\" she wrote. \"In this sense there may have been something more deeply satisfying in our desperate clinging to life than in what people generally strive for.\"\n\nDr. Marty Seligman, the godfather of the positive psychology movement, has created a three-tiered happiness model with direct implications on how we make our life choices. Beyond the first level, which Seligman calls the \"Hollywood\" view of happiness (\"getting as much positive emotion as possible\"), a second kind of happiness arises from discovering our \"signature strengths,\" which range in his core list from honesty, kindness, and forgiveness to ingenuity and love of learning. When we use our signature strengths in the service of something beyond ourselves, Seligman believes, we reach the highest tier. Learning to transcend our own needs, to at times sacrifice our immediate desires for the sake of the greater good, not only boosts our happiness level but also promotes the survival of the group. It is never a good idea to take care of others while ignoring one's own needs, of course. We've all known some grim-faced do-gooder who secretly resented those he or she insisted on helping.\n\n\"Just look at the Ten Commandments,\" I'm reminded by New York psychoanalyst Michael Eigen. \"To covet is the gateway to pain.\"\n\nOf course, comparisons are invidious in the best of times. Happiness researchers assure us that the single most powerful variable concerning well-being is how we choose to look at things. The wider the frame, the more vivid the picture. The softer the brush, the more sensuous the light. Colette had one of the most brilliant palettes ever, in both art and life\u2014the ability to soar, crash, and pull herself back up again from the wreckage; to always remain herself somehow; to finish her novels, love her younger men, drink her wine, relish her gossip, even with garrisons outside her window. She knew that happiness wasn't a Popeye grin\u2014no triumphalist horse-ride into the sunset\u2014but a brewing admixture of bitter and sweet. Metanoia made her wise. \"Bring it on!\" the gout-ridden writer would say, clutching a bulldog to her breast, meaning life with all its colors. _C'est la guerre!_\n\n## **INVISIBLE FEAST**\n\n\"Vision and sight are not the same thing.\"\n\nPhotographer John Dugdale is blinking at me through his Coke-bottle lenses, cocking his head and leaning forward, attempting to catch a sliver of my silhouette where I'm sitting less than three feet away. His Seeing Eye dog, Manley, is nuzzling his foot on the floor beside him. John is a strikingly handsome, square-jawed man with a perfect nose and crown of dark hair, recalling a portrait by John Singer Sargent. He is also a human hologram. After three major strokes, five bouts of nearly fatal pneumonia, toxoplasmosis (a brain infection), peripheral neuropathy, Kaposi's sarcoma, and CMV retinitis that robbed him of most of his sight ten years ago, John is hardly the hale-fellow-well-met he appears at first to be. He's more of a walking riddle, in fact.\n\n\"Losing my eyesight at the start of my career was the thing I dreaded most of all,\" he tells me now. \"I'd look at the sky, my hands, my face, my mother. I felt like I was disappearing, losing more sight every day.\" We're alone in John's drafty, top-floor Greenwich Village apartment. The room is filled with copper antiques, oversized cameras, and John's own haunting photographs\u2014blue-tinted images of ghostly trees and naked lovers in dreamlike space. It's hard to believe\u2014nearly impossible\u2014that such beautiful pictures were created with only the fraction of eyesight left to him.\n\n\"Everyone told me that my career was over,\" John tells me. \"But I decided that if I was going to lose my eyesight, I wanted to do it in a courageous way, hanging on to my camera tripod... not strung up to an IV pole. This was my moment to prove that illness did not have to end my creative life. I've had thirty-eight solo shows internationally since then. And my best work is still to come.\"\n\nThe trajectory of John's life in the past fifteen years is daunting to even imagine. Hospitalized for seven months in 1992, John was slowly preparing himself to die. \"My doctor's goal was to get me through the year,\" he says. \"I was struggling violently in my mind. I'd always been a huge overachiever. Everything I touched succeeded. For a long time I thought this couldn't be real, this couldn't possibly be happening to the golden boy. But it was happening. Fast.\"\n\nCaught in this sudden downward spiral, John reached for the only gifts left him: creative intelligence and obstinacy. \"I drove my doctors crazy. I questioned everything they wanted to give me.\" He smiles. \"But you must never let people talk you out of what you want to do. Sometimes they don't even _want_ you to get better.\"\n\nThis is sometimes true, strange as it seems.\n\n\"They need you to go on the path you're _supposed_ to go on,\" John says. \"You're supposed to be here for eighteen months and then die,\" he mocks. \"So don't get off that bed! Don't get out of that box! A nurse once told me I was going to have to take a certain medication for the rest of my life,\" he tells me in disbelief. \"I said who the hell told you that? She kept barraging me with medical propaganda filled with doomsday information. I told her, you put those pamphlets on my bed when I'm sleeping one more time, and I will have you arrested!\"\n\nWith nothing to lose, John chose his own artistic approach to self-healing. Learning to walk again after his stroke, he refused the hospital staff's equipment. \"That walker was like the scarlet letter for me,\" he says, reminding me of Jack Willis, the paralyzed TV producer who threw his own walker across the room. John chose to rely on his sister's shoulder instead. Recovering from toxoplasmosis, he invented visualization techniques as a way of buoying his spirits. \"I tried to come up with the most powerful thing each person in my family could do,\" John explains. \"It's corny, but my mother happens to like doing laundry. So I told her, when you go home to the washing machine at night, imagine you're taking my brain out of my head and putting it in the warm rinse cycle.\"\n\nJohn laughs at his homespun solution. \"I said, 'Imagine toxo coming out of my brain and going down the drain,'\" he says. \" 'Then take my brain, all clean and fluffy, and picture putting it back into my head.' My sister loves to cook, so I asked her to picture my brain in her lettuce spinner and spin the living daylights out of it! I thought of the purest, most perfect thing I had ever seen and remembered newborn lambs in spring. I pictured them nestling peacefully inside my head.\"\n\nA week after several violent drug reactions, a flummoxed internist paid John a visit. \"He told me that the infection in my brain seemed to be gone,\" he says, smiling. \"The doctor said he didn't get it. It could have been a coincidence. Or maybe it wasn't.\"\n\nEither way, John was still here.\n\nJohn realized that his physical and spiritual survival depended now on his ability to keep making art. \"I had to become the world's first blind photographer,\" he says simply. Late one afternoon, at his farm in Upstate New York, he finally found the nerve to pick up his camera and attempt to take a picture with most of his eyesight gone. What happened next, in fact, changed the course of John Dugdale's life.\n\n\"I had just gotten out of the hospital,\" he says, once we've moved to a sofa in the sitting room. \"I was out in the pasture, trying to take a photograph using a magnifying glass, struggling to focus, adjusting the backdrop and light meter,\" John tells me. \"I kept tripping over the tripod, getting more and more frustrated. Every time I was just about ready, the sun would move and I'd have to switch the whole thing again.\"\n\nThe idea of a mostly blind photographer in a country pasture, struggling to find the right light for a picture, is too mythopoeic to quite imagine.\n\n\"When I was about to go crazy, I finally got the whole thing right\u2014just perfect\u2014and bam!\" John says, slapping the table. \"In that very moment the sun went down and everything disappeared, went black. That's when I completely snapped.\" Sensing the distress in his master's voice, Manley searches John's face for trouble.\n\n\"I hit the deck in a heap of nothingness,\" John says. \"Just fell down on the ground and covered my head, pushed my face in the dirt, and started to cry in a way that was almost inhuman. There was grass in my teeth, my mouth. My eyes were full of dirt. I just wanted to burrow into the ground and die. I was furious, sad, frustrated, sick of everything,\" he says. \"Bawling my eyes out, asking, why did this have to happen? In a way that was completely out of control\u2014like hiccups when you can't stop.\"\n\nHe strokes Manley's nuzzling head. \"I was feeling the full impact of what had happened to me for the first time,\" John says. \"I thought I was having a nervous breakdown. My friend came out of the house to see what was happening. Thank God I wasn't alone! He scooped me up and carried me to the house, sat me on the couch, and held me in his arms like a child. He said, 'Okay, go ahead and cry.' I lay on his lap and bawled until there wasn't another drop of fluid in my body. He didn't try to stop me at all. Finally, I asked him to get me a camera so I could take a picture. I got back in his lap and snapped the shot. It was so beautiful, like a piet\u00e0. I called it 'The Descent of Man.'\"\n\nThis stunning photograph, which resembled nothing in John's previous work, became a centerpiece of Shadows Night Fall, the solo exhibition that catapulted him from the ranks of well-paid shutterbugs to the echelon of world-class artist. Since then, John's unique azure cyanotypes (a nineteenth-century camera process that allows him to enlarge images enough to see them) have earned him comparisons to Julia Margaret Cameron, the Victorian master, and invitations from collectors and museums around the world.\n\n\"I never expected this tremendous response,\" he says with a smile. \"I've got this great Cheshire cat feeling. The thing that I thought would be the end, in fact, actually took me somewhere I would never have imagined. I came to New York to be an artist, not a magazine hack, and finally landed in the place I was meant to be when I started. I just took a very different route,\" he says, reminding me of disabled athlete turned teacher Jim MacLaren.\n\nCreativity in the midst of chaos, the turn toward beauty in ugly times, has been John's lifeline back to health. Yet he learned early on that transformation is only possible when we make peace with the facts of our lives. \"Survival doesn't really mean anything without acceptance,\" John explains. \"That's the paradoxical part. You have to take the thing that's wrong and own it. Make it into something that has meaning for you. If you try to hide or negate it, it will just eat you up,\" he says. \"If you're hoping for things to be other than they are\u2014constantly wondering how or why something happened, or how to fix it\u2014you're _lost._ You'll completely miss out on the graceful time you have.\"\n\nJohn witnessed this paradoxical truth during his time in the hospital. \"The minute they rolled somebody into the room next to me, I could tell if they were going to stay on the planet or leave,\" he says. \"I could hear the resistance to change in their voice.\"\n\nThat quickly? John swears to me that this is true. \"The ones who wound up giving up were the people who couldn't imagine themselves except the way they were,\" he says, while patients who exhibited flexibility prevailed. \"The opportunity for transformation when we enter deeply into our experience is absolutely unbridled,\" John believes. \"It's just like nuclear power if you choose to use it properly. But if you can't imagine yourself in a new way, you're just not gonna make it,\" he insists. \"If you think you're gonna be the person you were before tragedy struck\u2014internally or externally\u2014it's impossible. Once you pass through that fire, you've been smelted. You're gonna to come out gold on the other side or you're not gonna come out at all.\"\n\nSuch metamorphoses are sometimes confusing to folks around us. \"People often ask me why I don't take happier pictures,\" John says, chuckling. His famously moody photographs could not be more unlike his peppy commercial work of yesteryear. \"People ask me why I don't just see a shrink,\" he says with raised eyebrows. \"To make the complicated feelings go away. I respond by asking what they see in my work, exactly. Is it loss, fear, joy, sadness, sexuality, beauty, tiredness? Because all of that is there. I have just as much joy as I do melancholy.\"\n\nKahlil Gibran described this doubleness in a poem.\n\n_Your joy is your sorrow unmasked  \nAnd the selfsame well from which your laughter rises was oftentimes filled with your tears.  \nAnd how else can it be?  \nThe deeper that sorrow carves into your being,  \nThe more joy you can contain._\n\nThis wisdom has widened John's point of view. \"I ask these well-meaning people which part, exactly, they are afraid of. Which part can they not look at? Because loss goes on for all of us. It is authentic to be human and in pain.\"\n\nAt his lowest time in the hospital, John says, he had an epiphany that remained with him. \"I was in this tiny, dumpy room at the end of the hallway, with one window onto an airshaft, feeling completely helpless. Nobody was around that day. I was counting the hours, the minutes, the seconds. One thin strand of light was coming down the airshaft. And as I was gazing at it, something began to shift in me,\" John says. \"It was more an awareness than a thought. But as I lay there watching this light, I knew that that was where I was going to. That I would become part of that. It's hard to explain,\" John acknowledges, \"but when I exhaled, a straitjacket fell away and I became very peaceful with the idea of leaving the planet. In _success,_ not failure,\" he makes clear. \"There was a transcendental shift in my personality.\"\n\nHe claims now that this transcendent shift is more precious to him than his actual vision. \"The light comes from inside,\" John explains. \"Vision and sight are not the same thing,\" he repeats. \"Sometimes I think that if God came down and said I could have my sight back but I'd have to forget everything I've learned, I couldn't do it. Once you've really been forced to let everything go, life becomes much more peaceful.\"\n\n\"It's hard to communicate that without sounding delusional,\" I say.\n\n\"Once you've come home for the tenth time to news that someone you love has died, you start to have this other experience of life that most people aren't familiar with,\" John agrees. \"I have a better grip on mortality than my parents do. My mother, God bless her, always wants to give me her eyes. But I tell her, 'Ma, I have my special eye. No one sees only with their eyes. You just don't realize it until you have to.'\"\n\nJohn's macho father learned this lesson as well. \"When my dad thought I was going to die, he came to see me and gave me the first peck on the cheek he'd given me since I was eight years old,\" John says, his eyes pooling up behind the thick glasses. \"It was really awkward. He banged his head on the swing-arm TV.\" My host wipes his eyes on his sleeve. \"But for my father, that kiss was like stepping over a chasm. It would never have come about in any other way. Now, what ever happens, I know deep down in my heart that he loves me. And my father knows that I love him.\"\n\nIn the past few months John has started to lose the fraction of vision left to him. He isn't happy about this, of course. But he doesn't seem especially frightened either. \"Loss goes on,\" he tells me when we meet again. \"I don't want to leave the planet. I want to see what's going to happen with my garden this year. I don't want to not see my friends and my family. But when that happens, I'll be forced to accept it.\"\n\nIn recent photographs John has begun inserting himself into the images. A male figure can be seen around the periphery\u2014 sometimes a hand, sometimes his face, other times just an arm or a shadow\u2014as if to remind us, his doubting viewers, that he's still here. John Dugdale calls these his spirit pictures. \"When I talk about losing my sight these days, I find myself touching my own face,\" he says now, fingers reaching automatically toward his cheek.\n\n\"I haven't finished taking that picture yet.\"\n\n## **ORIGINAL BLESSING**\n\nWe cannot transform what we have not first blessed. Without blessing everything, our harshest parts especially, we cannot be turned around by the _coincidentia oppositorum_ \u2014the conjunction of opposites, the sacred marriage, the wisdom that only arises from reconciling yang and yin. Ancient alchemists applied this same principle, adding a pinch of _negresca,_ the blackest element, in order to make what they called gold. Our dark parts then become allies, enriching our _prima materia,_ the mess that we must learn how to bless if we want to move forward.\n\nThe word _blessing_ comes from the French verb _blesser,_ \"to wound,\" and often arrives through the hands of pain, as Samuel Kirschner, an Israeli bodyworker living in New York, learned when his father was dying. Samuel grew up craving his father's blessing but chose to save his own life instead, fleeing Israel at the age of twenty to become a permanent exile in the United States. His family's grief was too suffocating. Raised by Holocaust survivors (Samuel was conceived in a relocation camp after the war, in fact), the troubled prodigal son went through decades of psychotherapy before learning how to feel happy inside his own skin. Then one day, five years ago, Samuel's mother called from Israel to tell him that his dying father wanted to see him.\n\n\"Being with my father at the end of his life changed me completely,\" Samuel says when we meet at his loft near the East River. A soulful fifty-something man, Samuel is pinching an Indian cigarette between his fingers like a joint, his shaven head silhouetted against the garden window. \"My father and I had never been close,\" he says. \"My mother used to tell me when I was a kid that he wasn't fit to be a parent. She may have been right, but how could I tell?\" Samuel muses, flicking his ashes into a cup.\n\n\"She always kept a wall between him and me, so my father and I were distant strangers. He lived with us, but always apart, like the shell of a father. Then he got sick, and for some weird reason I was the one he wanted close by. I saw how tender my father really was. Sitting with him at his deathbed during those weeks saved my life.\"\n\n\"In what way?\"\n\n\"It's hard to explain,\" he tells me. \"I realized how much of me was missing as a man. I'd been trying so hard not to be like my father\u2014the image of him that I'd made up\u2014that I had no idea who I really was. I had all this power but no foundation, just this huge hole underneath, where my father should have been.\"\n\nHe sips from his cup of tea. \"We like to pretend like we're all grown up\u2014handling it,\" Samuel says, making air quotes. \"What a joke! Sitting with my father, I felt like I was five years old again. I was amazed by how much I still craved his approval after all these years, though I believed I didn't care.\"\n\n\"You never completely stop caring,\" I say.\n\n\"I still wanted him to be proud of me!\" Samuel says with amazement. \"He loved to be pampered, so I just sat there next to the bed and touched him. One day my mother asked me to shave him, and when I finished, my father kissed my hand.\" I'm reminded of John Dugdale's father kissing him in the hospital bed, only in reverse. \"He was too sick to talk, but he looked at me and smiled.\"\n\nThis smile helped to heal a painful breach in his adult psyche. \"Something shifted in me when he kissed my hand,\" Samuel explains. \"I don't know exactly how to describe it. My father lived through his hands. He was a baker, and in his spare time he liked to garden. My father's soul was in his hands. I know he felt this, too, because he began to trust me.\"\n\nHe lights another _beedi_ and continues. \"Anything to do with his body, my father asked for me to help him. I changed his catheter every morning. The first time I did it, I was actually shaking,\" Samuel admits. \"Touching him there was so strange, so intimate. It reminded me of the story in the Bible when Isaac dies. The blessing was meant to go to the older son, and whoever got the blessing would thrive. Jacob was younger than Esau, the way I was younger than my brother, but he managed to touch his father's thigh. Touching the father's thigh was the symbol of being blessed. I knew when my father was dying that he was blessing me. No words, but that simple touch was more important than any therapy I had ever gone through before.\"\n\nIn a famous letter to his own absent father, Franz Kafka expressed the depth of such filial longing. \"My writing was about you,\" confessed the author of _The Metamorphosis._ \"In it I only poured out the grief I could not sigh at your breast.\"\n\nSamuel concurs with this feeling. \"I finally feel like a whole person,\" he admits. \"It only took me fifty-nine years.\"\n\n\"You're lucky to have had it at all,\" I say, never having known my own father.\n\n\"I think about that all the time,\" he says.\n\nThe value of blessing goes far beyond family bonds, however. Blessing is a metaphysical act that elevates and transforms our view of the earth and our lives here. In fact, it's well worth asking ourselves what protean effect the replacement of original sin with original blessing might have on the way we see things. Brother Matthew Fox, a controversial Catholic priest, was actually silenced by the Holy See after suggesting that a doctrine of original blessing could help to heal a divided church he described as a \"dysfunctional patriarchy.\" What's more, Fox wanted to know, how differently might we approach our own lives if we viewed worldly existence as blessed, not cursed; risen, not fallen; sacred and worthy instead of profane? Shortly after his silence is lifted, Brother Matthew agrees to sit down with me on a sunny morning in Washington Square Park to discuss how this absence of blessing in daily life continues to wound the modern soul.\n\n\"I spoke to a young person in Australia,\" Matthew begins. He's wearing jeans and a sky-blue button-down shirt, his silver hair neatly parted at the side, and is munching contentedly on a bagel. \"This young fellow said something surprising,\" the priest remembers. \"He said that it was as if adults had given today's youth a revolver containing six bullets: the hole in the ozone layer, the disappearing rainforests, polluted air and water, joblessness, debt, and the tightening field of education. He said it's as if adults have put this revolver to their heads and said, 'Now be happy,'\" Matthew tells me. \"Youth are in despair all over the world.\"\n\nIt's hard to believe that this mild-mannered Wisconsinite is the same fire-breathing reformer\u2014one journalist called him the \"D. H. Lawrence of theological lit\"\u2014who put the blasphemous screws to his Mother Church. \"I remember teaching a course to a summer group of a hundred people or so,\" Matthew says, tossing bagel bits to the pigeons on the lawn. \"We began with what is called the _via negativa,_ \" the \"school of hard knocks\" in layman's terms. \"I asked how many people had experienced the dark night of the soul. Every single person's hand went up.\"\n\n\"You were surprised?\"\n\n\"Maybe not. But the point is that such awareness isn't just something for nuns and monks\u2014we all know about it. What we often fail to realize is what to do with our sufferings.\"\n\n\"A friend of mine used to say that in a materialistic culture, we have no transcendental context for our suffering.\"\n\nMatthew agrees. \"We've been Novocained to death by television, entertainment, alcohol, drugs, school, work\u2014all our addictions,\" he says. Matthew makes clear that he is not opposed to having fun with shallow, meaningless entertainment; he merely warns against junk culture as a steady diet. \"It covers up our capacity for awe. For passion and deep feeling. It sets us up for superficial experience.\"\n\nHe often mentions Jewish philosopher Abraham Heschel in this regard. \"People today are shocked by the weakness of our awe, but also by the weakness of our shock,\" Matthew says, quoting the visionary rabbi. \"Heschel says that we've lost the capacity for radical response.\"\n\n\"Blessing is a radical response?\" I ask.\n\n\"Of course it is. So is creativity. Creativity lies at the heart of freedom from addiction,\" he tells me. \"Otto Rank, my favorite psychologist, says that the resurrection story from the Gospels is the most revolutionary idea that humans have ever come up with. Rank worked exclusively with artists and found that the number one obstacle to creativity is the fear of death. The redemption power of the Jesus story is that death is not the final word, so we don't have to fear it. When we believe in stories like the resurrection, we rediscover our creativity and are not afraid to use it. Imagination gets us into evil, and imagination gets us out.\"\n\n\"Tell that to Kim Jong Il,\" I say.\n\nHe sidesteps the North Korean despot. \"Violence is intrinsic to the universe,\" Matthew says. \"Terror and beauty go together. We don't live in a pretty universe, we live in a beautiful one. That's not evil. It's just part of being here.\"\n\n\"We would all like easier answers,\" I say.\n\n\"Yet awe is the thing that wakes us up.\"\n\n\"And awe contains darkness?\"\n\n\"Always. The demonic and the divine join in the act of creativity,\" Matthew explains. Blessing counterbalances darkness and helps turn adversity toward the creative. \"The cure for evil\u2014the constructive reaction to suffering\u2014is to redirect our imaginations,\" he says. \"To be more conscious that we _have_ imagination. And to never turn these imaginations over to the few. If we teach people that their imaginations are not powerful, important, or divine, then something essential begins to atrophy.\n\n\"Art, ritual, erotic life, prayer\u2014all of these are a form of blessing,\" the priest goes on. \"They strengthen the heart. They open us to wonder. Take something like a sweat lodge.\" He chuckles. \"My first twenty minutes in one of those things, and I thought for sure I was going to die! I was looking for the fire exit. Then I said, 'I am going to die,' and I yielded. When you yield to the process, you undergo a transformation. Your heart grows.\"\n\n\"In the surrender?\"\n\n\"In the play.\" Matthew's watching a group of toddlers attack the jungle gym. \"We're here to celebrate with one another,\" he reminds me. \"This ought to be an everyday thing. According to Meister Eckhart [the thirteenth-century mystic], 'For the person who is aware, breakthrough does not happen once a year, once a month, once a day, but many times every day.'\"\n\nThis is the power that arises from blessing, Matthew Fox believes. We view ourselves, and the world itself, through a sacramental eye\u2014perceiving the holiness in our lives\u2014rather than the sting of damnation.\n\nThis radical priest seems to be saying that when we perceive life for what it is\u2014miraculous, fathomless, staggering, brilliant\u2014and view ourselves through this same opened window, blessing automatically occurs. Brushed by extinction, the most cynical person can be rendered childlike, curious, and glad. Bertrand Russell, the cranky British philosopher, almost died from double pneumonia in Peking during the winter of 1920. For several weeks all of the doctors in attendance thought he would be dead before morning. But with the coming of spring, his health\u2014and gladness\u2014 returned, as Russell wrote in his autobiography:\n\nLying in my bed feeling that I was not going to die was surprisingly delightful. I had always imagined until then that I was fundamentally pessimistic and did not greatly value being alive. I discovered that in this I had been completely mistaken, and that life was infinitely sweet to me. Rain in Peking is rare, but during my convalescence there came heavy rains bringing the delicious smell of damp earth through the windows, and I used to think how dreadful it would have been to have never smelt that smell again. I had the same feeling about the light of the sun, and the sound of the wind. Just outside my windows were some very beautiful acacia trees, which came into blossom at the first moment when I was well enough to enjoy them. I have known ever since that at bottom I am glad to be alive.\n\nRussell had finally blessed his own life, understood with both heart and mind, and what he perceived in this way surprised him. We are meant to be surprised by this world, he learned in almost losing it, meant most of all\u2014in hard times, even\u2014to be glad for the chance to have been here at all.\n\n## **ENOUGH**\n\nSatisfaction becomes an art form. Knowing enough to recognize fullness. The wisdom of knowing when to stop is critical to _techne tou biou,_ the craft of living. Yet this is not a lesson easily learned in our land of conspicuous plenty. Ours is a culture without sabbath or pause, a theater without an intermission. We measure ourselves by a more-is-more yardstick, drive ourselves beyond our limits, sacrifice life quality for productivity, and often end up feeling so burned out that we can't enjoy the fruits of our own labor through the bitter backwash of exhaustion.\n\nBut what if we didn't live this way? What if we practiced satisfaction and the skill of knowing when to stop? What if we engineered into our frantic schedules a weekly day of actual rest, a sundown-to-sundown tradition of stopping, cessation, renewal, and \"non-doing,\" lived \"as if we were at home in the universe,\" as Rabbi Rami Shapiro puts it? \"Lived for one day without trying to control the people around you or the situations in which we find ourselves? Reserved one day for a state of acceptance? Not a day without desire, that would not be possible,\" Shapiro suggests, \"but a day not to act on those desires.\" He believes that creating such a refuge might actually give us \"a foretaste of the world to come.\"\n\nWhether you believe in an afterlife or not, a sabbath does sound heavenly. Imagine your neighborhood stopping for twenty-four hours every week, as if Mother Earth herself were catching her breath after many days of sweat and toil. God worked mightily for six long days before cooling his heels on the seventh, the book of Genesis tells us. \"Even the Lord could sit back and be satisfied that nothing more needed to be added to his creation,\" a Jewish colleague explained. \"Even he could stop and say, 'It's good already. Enough!'\"\n\nHow often does anything feel like enough? When, for more than a random instant, does life appear completely sufficient, with nothing to edit, improve, aspire to, arrange, or nudge forward in some way or other? Isn't there always something more to be done, work to be finished, ground to cover, prospects to investigate? When it came to enough, I had always believed, it would not be my personal fate in this lifetime.\n\nBut burnout is an insidious process. You don't know you're choked till you're already smoking. For a long time after my own \"alive day,\" taking a break seemed ungrateful somehow, an affront to the gift of vitality. Living fully meant overdrive. Forced to choose between sanity and productivity, I would gladly have chosen prolifically nuts. But the smell of smoke was in my nostrils. I woke up too many mornings feeling like toast. My calendar was booked, my downtime upbeat, but my life had turned into a runon sentence. I needed syntax. I needed a sabbath. I needed to figure out how to stop.\n\nFor someone like me this was difficult. I was the insecure geek in school who wrote twenty-page papers when five pages were assigned (and even those weren't good enough most times). Double the effort for half the self-esteem, that had been my lifelong motto. My Swiss-cheese ego just wouldn't stop leaking. These cheese holes announced that I was fully deficient and nothing I did could keep them stuffed.\n\nI was living as a hungry ghost. Hungry ghosts are those ravenous figures in Buddhist mythology (metaphors for the insatiable ego) condemned to eternal craving in a world of elusive satisfaction. Nothing is ever enough for these monsters. There exists no end to their craving for more. We all have our ghostly orifices, the places where, if there were just a little more, we could be the people we long to be. But there comes a time, if we're paying attention, when this Pavlovian drool-and-whine charade reveals its own futility. We realize with dismay and shock that nothing will ever suffice as a permanent stopgap between us and our ravening hunger. Still, we can't stop trying to plug the space. We're endlessly dissatisfied, you will agree, driven to feed the unfillable void using things with a snowball's chance in hell of actually lasting.\n\nFortunately, Buddhism offers a sensible strategy for how to end this hungry ghost cycle. This begins with understanding the concept of _dukkha_ \u2014the \"pervasive unsatisfactoriness\" of all passing things in an imperfect, temporal world. Even those rare experiences that do manage to satisfy us completely will soon enough come to an end, we know, and this knowledge, in the Buddhist view, is exactly what casts the shadow of longing we find ourselves unable to shake. This dissatisfaction with things as they are remains, Buddhists tell us, our main source of suffering. Yet we seem unable to curb our compulsion to make this empty-sense go away.\n\nThe drive to fill internal emptiness stems from fundamental doubts about our own existence, the Buddha taught. Looking deep inside our psyche for a \"self,\" humans are forever haunted by the suspicion that there is, in fact, no _there_ there. If we continue to peel back the layers of our so-called \"selves,\" Buddhists teach, we will come upon no permanent subject, regardless of how long or hard we try. Because confronting this emptiness feels so scary, we're driven to keep trying to cover it up, to papier-m\u00e2ch\u00e9 the abyss with stuff.\n\nYet it is here\u2014at futility's brink\u2014where we can begin to free ourselves. The cycle of craving is interrupted when we learn how to stop. \"Sitting with uncomfortable feelings, getting friendly with the hole you really are, you realize that emptiness is not really a problem,\" says Buddhist teacher David Loy. \"It's our ways of trying to escape it that turn it into a problem.\" When we learn to tolerate this empty feeling within, instead of binge-feeding the hungry ghosts, an important, metanoia-like turnaround occurs (Buddhists call it _paravritti_ ), when the \"festering hole at our core turns into a life-healing flow [springing] up spontaneously from we know not where,\" in Loy's words. \"The empty core becomes a place where there is now awareness of something other than, greater than, my usual sense of self\u2014greater than I understand myself to be.\"\n\nNothing can possibly be enough, we now see, because nothing is meant to be enough. We're wise to feel the insufficiency of lives predicated on ephemera (including other people), we are told by the Buddhists. Longing is the point, it seems; _dukkha_ is our built-in reminder of what can never be filled (and should not be), _not_ so that we cling to our cravings but in order to call the pursuit\u2014 and the containers themselves\u2014into question. Fuming over the imperfection of even the most brilliant life is meant to be a goad for discovering what cannot be taken away, for achieving reunion with that sufficient self that lacks nothing, expects nothing, needs nothing in order to feel at home in the world\u2014the self that can stop anytime, anywhere, and know that it belongs. The longing does not disappear, but our story about it does; obscure objects of desire turn into a subject\u2014the true I\u2014whose realization enriches our lives rather than diminishing them.\n\nDestructive as hungry ghosts are personally, their greed writ large now endangers the health of the planet, as ecologists have been warning for decades. Five years ago the writer Bill McKibben found himself standing in Tiananmen Square in Beijing, gazing up toward the sun in the sky\u2014and not being able to find it\u2014when he had a scary revelation about what _enough_ really means.\n\n\"You could stare straight at the sun if you could even figure out where in the sky it was,\" McKibben tells me with disbelief. He's on the phone with me at the moment, looking out at pasturelands behind the rural Vermont home he shares with his wife, writer Sue Halpern, and their daughter. \"I was in complete shock,\" Bill admits. Doing the math afterward, he estimated that by the year 2031, barring natural disaster or mass contraception, there will be roughly 1.3 billion Chinese as well off as their American counterparts. \"If the Chinese owned cars like we do, they would add one point one billion cars to the eight hundred million already on the road,\" he says in horror. \"If the Chinese ate meat the way we do, they would consume two thirds of the food on the planet. The earth simply cannot support that.\"\n\nMost of us are already aware of the ways in which greed is depleting the planet. However, we may not have made the connection between this overconsumption and the hungry ghost ethic. Bill McKibben wants to make this clear. \"The official idea that more is better, which has been orthodoxy for the past fifty years, no longer matches reality,\" he explains. \"It's as if we've done a controlled experiment to see if materialism [as a path to happiness] works and found it doesn't.\" With a degree in economics from Harvard, McKibben has made a name for himself doing \"coal-mine canary\" reporting on issues from overpopulation to global warming. \"For all our material progress, all the billions of barrels of oil and millions of acres of trees that it took to create it, we have not moved the satisfaction meter an inch,\" he says, echoing happiness researcher Dan Gilbert.\n\nThe opposite has happened, in fact, with alcoholism, suicide, and depression rates skyrocketing in proportion to the rise in affluence. There even appears to be something afoot that I call \"acquired anhedonia syndrome,\" namely, the inability of people who have too much to enjoy the stuff they already own. This malaise has had a trickle-down effect on our kids. Studies have shown that today's average American child reports suffering higher levels of anxiety than the average child _under psychiatric care_ in the 1950s.\n\nBill found the American dream's shadow side even more troubling when viewed from across the Pacific. In China, he met an eighteen-year-old factory worker named Liu Xian, whom he calls \"the most statistically average person on the planet.\" Bill and Liu Xian struck up a conversation. He told her that he had noticed how many of the girls in the factory dorm had stuffed animals on their beds and asked Liu Xian if she had one. The girl began to cry. \"She couldn't afford such an item, she told me,\" reports Bill. Later, when he brought Liu Xian a stuffed dog, she was as pleased as he had \"ever seen a person.\" The disconnect between her gratitude and his own daughter's disinterest (\"she has a room full of Beanie Babies\") still affects him. \"How could a stuffed animal possibly have the same meaning for her?\" he wondered about his daughter. This is why Liu Xian's story continues to haunt him. \"In impoverished parts of the world, possessions still deliver,\" he insists. \"Any solution we consider [for redistributing wealth] has to contain some answer to Liu Xian's tears.\"\n\nBill decided to try an experiment with his own family. The McKibbens' focus, he determined, would be food. For a solid year, Bill, Sue, and their daughter decided to eat only foods grown locally, to see how it affected their \"appetite, budget, community contact, and general sense of fullness.\" Studies show that the average bite of food an American eats travels some fifteen hundred miles before it reaches our table. It takes a tenth that amount of energy to eat locally, however, and increases our chances of social interaction\u2014the local farmers' market versus Wal-Mart, for example\u2014tenfold as well. Bill hoped that by tipping the balance toward home\/local\/small, he would dig deeper roots in his beloved Lake Champlain Valley, meet the neighbors, locate his own goods, and even, he hoped, regain a sense of proportion and what it means to have enough.\n\nThe experiment wasn't always easy. In the long run, though, it changed Bill's life. While locating and storing his own food proved to be more time consuming than hitting the A&P, Bill and his family actually enjoyed the challenge, cooking at home more, eating higher-quality fare, even making some new local friends (the butcher, the baker, the candlestick maker). Spending a winter without oranges was not nearly as traumatic as he had predicted. This year of eating locally \"permanently altered the way I eat in more ways than one,\" Bill tells me now. \"It left a good taste in my mouth. That taste was satisfaction.\"\n\nIt is possible to be satisfied, after all. Enough exists, we come to realize; in fact, we have it with us already. Craving and hunger are not the same thing. Recognizing our hungry ghosts, we learn to discern the important difference between actual hunger and the gap-stuffing reflex that only makes our emptiness worse. Slowly, we begin to recognize the difference between the ghost's voice and true desire, to resist hues and cries against the void, to obey the call of letting things be, and to find satisfaction in doing nothing. Satisfaction is there in the silence, we see, a break from the mind's cacophony. There is always enough, we begin to believe; the emptiness is filling itself. That must be what we can learn from a sabbath. The empty space becomes a spring.\n\n## **STRESS MATTERS**\n\nIn a quiet corner of the University of Massachusetts campus, near Worcester, sits a cluster of outbuildings called the Stress Reduction Clinic. Since its opening in 1979, the clinic has helped thousands of individuals suffering from stress sickness to reclaim their peace of mind. In a hyperactive age of time crunching, multitasking, and rampant attention deficit, stress disorders are, indeed, the nemesis of modern life, wreaking epidemic damage on our well-being (some 60 to 90 percent of individuals in hospitals declare stress to be their primary complaint) as well as on our physical health. Luckily, this enemy can be disarmed.\n\nSTRESS STOPS HERE, announces a poster on the wall near the clinic's entrance. Indeed, for nearly thirty years the SRC's founders, Jon Kabat-Zinn and Saki Santorelli, have been using a variety of techniques, including biofeedback, meditation, yoga, and cognitive therapy, to help burned-out individuals learn to \"unhook\" themselves from stressful and painful thoughts. While the human stress response may be autonomic, physiologically beyond our control, its mental triggers can be worked with, which is what I have come to the clinic to investigate.\n\nSaki Santorelli welcomes me with a beaming smile and open arms. A diminutive man with an epic nose and the energy of three people, Saki is a stress guru who's devoted his career to understanding the enemy in our midst in the same way other scientists fight cancer or global warming. \"Life is only getting more relentless,\" he tells me after we've settled in his small office. \"People come to us exhausted. Just _frazzled._ They've got ten balls in the air and no sense of how to put them down. They can't give themselves a break, drop off the merry-go-round.\"\n\nThe word _stress_ comes from the French for \"oppressed,\" I was not surprised to learn from my own research.\n\n\"Stress is oppression,\" Saki insists.\n\n\"So what's the cure for oppression?\" I wonder.\n\n\"Choice,\" he tells me. \"Plain and simple.\" Stress, explains Saki, is created from the belief that we are under threat\u2014trapped\u2014and cannot cope. \"Shut in,\" he says.\n\nThis reminds me of what a man admitted after surviving a jump off the Golden Gate Bridge: \"I instantly realized that everything in my life that I thought was unfixable was totally fixable. Except for having just jumped.\"\n\n\"Realizing that we have a whole repertoire of responses available to us at any moment decreases anxiety right away,\" Saki tells me. \"When people feel that there's no way out of the place they're in, their minds contract. Having choice relieves the stuckness, makes the narrow places wider.\"\n\n\"But how do our lives become so narrow in the first place?\" I ask.\n\n\"To survive crisis, we narrow our world to a size that feels safe, knowable, familiar, and secure,\" Saki explains. \"But the mind-set that gets us through crisis becomes impoverishing in the long run. We may be _handling_ things,\" he says, \"but our lives become small. Patients come here telling us that although they are functioning, they feel stuck in a narrow place they don't like. But if they step outside, they don't know if they'll survive.\"\n\nIn other words, we hide inside safely parametered ruts, then wonder why we feel so stuck. \"What gets us through the hardest times does not make us feel content or alive,\" Saki says. He and the clinic's staff work with patients to help them become more \"comfortable with their own discomfort,\" to \"uncouple themselves from anxiety by developing the capacity to feel, see, and actually experience what's going on.\"\n\n\"Instead of what they imagine\u2014and fear\u2014to be going on?\" I ask.\n\n\"Exactly. People learn to step back and separate their thoughts about what's happening from what is actually happening. This mindfulness creates a bit of space. A toe today, a leg tomorrow. Poco a poco. A bit at a time. Then, when we're not so exhausted from the effort of keeping things at bay, we discover that our lives\u2014that we, ourselves\u2014are much bigger than we think.\"\n\nSaki discovered his own chill-out mission while still a neurotic undergraduate at university. During a conversation with an esteemed peer, Saki had one of those red-flag, epiphanic moments. \"It was thirty-plus years ago,\" he remembers. \"One day, out of the blue, someone I had a great deal of respect for asked me if I was the kind of person who went through the mountain or around the mountain.\"\n\n\"What did you tell him?\" I ask.\n\n\"I remember pumping up my chest and saying, 'I go through the mountain!'\" Saki grins. \"That's when this person asked me if I had ever thought of going around the mountain. I have been living inside that question all these years. Wondering what it would mean to go _around_ the mountain. When is that the right approach? When does it work better than trying to push through? If we've pushed as hard as we possibly can and the mountain still does not move, can we yield?\" Saki asks.\n\n\"That's when control issues kick in,\" I say. \"Not wanting to give it up.\"\n\n\"But there are different kinds of control,\" he insists. Saki tells me about groundbreaking research being done by a team of doctors at Stanford, who have identified two distinct kinds of control. \"The first is positive assertive control,\" he explains. \"When we put our noses to the grindstone and see the task through. This is not a stressed-out state. Unless it leads to negative assertive control.\"\n\n\"That sounds bad,\" I say.\n\n\"It is. Negative assertive control means pushing till it hurts.\" This is the sure path to stress sickness, Saki teaches. \"We may have succeeded quantitatively, but qualitatively we're a mess,\" he explains. \"What have we plowed through to maintain control, to achieve our aims? Whom do we grind up in our wake?\" In a world piled high with collateral damage, this does seem an important question. But what is the alternative? \"There is something known as positive yielding control,\" Saki makes clear. \"This has to do with surrender, acceptance, and choosing your battles. Positive yielding control can have enormously positive effects on people's lives, particularly in situations where survival hangs in the balance.\"\n\n\"Not pushing?\"\n\n\"Knowing when to stop,\" he says. \"When to go around the mountain. Using the power of choice.\"\n\nIn other words, yesterday's steadfastness may be tomorrow's embolism. Undiscriminating effort is a sure path to burnout. Stress relief comes from more choice, less brawn; more patience, less pressure; more stretching, less strife; more stillness, less stipulation. According to Saki, stillness\u2014 _far niente,_ as the Italians call it\u2014is a largely untapped elixir for counteracting battle fatigue in our frantic lives. \"Pascal said that most of our troubles come from the fact that human beings can't be alone with themselves for any extended period of time,\" he reminds me. \"The act of stopping may be uncomfortable at first, but it often becomes revelatory. People begin to have small epiphanies.\"\n\nAll the more reason to observe a sabbath, I note.\n\n\"Stillness is what enables us to interrupt the stress cycle,\" he agrees. \"When we unhook from our craving thoughts, we're struck by new possibilities, fresh ideas, more constructive responses.\" Even small children respond to reconnecting in this way. Saki recalls the story of Maria Montessori, the Italian educator who made important innovations in the field of early learning. Apparently, Montessori, who began her work with poor children in Italy, would give the kids short periods of quiet time throughout the day. \"In the beginning the children hated it,\" he says, \"but Montessori had them do it anyway.\"\n\nHe presses his palms together at the side of his face in the international gesture for napping. \"When she deliberately took the quiet time away, the children would start yearning for it,\" Saki says. \"They'd ask, 'Could we just put our heads down, please?'\"\n\nThen my host gazes out through his office window toward a copse of trees in the distance and takes the words right out of my mouth.\n\n\"I know exactly how they feel.\"\n\n## **THE WOUNDED HEALER**\n\nMedicine is a front seat to mystery, according to one wise physician. While some doctors are emotionally stunted individuals, others surpass our empathic expectations in the course of living through their own hardship. \"Only the wounded physician heals,\" Carl Jung said. Without having been tested themselves, like shamans enduring their rites of passage, it may be fair to ask how much help doctors can really be in helping us survive our own.\n\nDr. Rachel Remen grew up as the only daughter in a Russian immigrant family of doctors and rabbis when the subway still cost a few cents in New York City. Long before she entered Cornell Medical College, where she trained as a pediatrician, Rachel's personal healing work began following a diagnosis of Crohn's disease at fifteen. An autoimmune disorder attacking the intestine, Crohn's was untreatable and agonizingly painful in those days. Struggling to live a normal life, Rachel braved several major surgeries, as well as the humiliation of wearing an evacuation bag under her sixties peasant blouse.\n\n\"I was furious at my disease for a long time,\" Rachel tells me when we meet. We're seated across from one another at the dining room table of her Olympian home overlooking the bay in Marin County, California. A purring cat curls around my ankle. The house, where Rachel lives alone, is decorated in calming earth tones and filled with Asian sculptures. Beyond the floor-to-ceiling windows, red robins swarm fruit trees across the expanse of manicured lawn. Snowy-haired and owlish in a cream-colored pants ensemble, Rachel is not who I expected her to be. In book jacket photographs, the best selling author of _Kitchen Table Wisdom_ looks like a sweet-natured, _babke_ -pushing _bubbe_ (a grandma, in other words, offering cake), the kind who kvells and pinches your _tuches._ In person, though, she's more like Bea Arthur, the no-b.s., tough-as-nails sitcom actress best known for playing Maude on TV. Armed with a mystic's awe at the wonder of healing, as well as a scientist's flinty mind, Rachel is quick-witted, formidable, and\u2014though good-hearted\u2014no sufferer of fools.\n\nHer anger at Crohn's brewed a long time before she finally exploded. Offered a prestigious teaching job during her medical residency, the ambitious young woman was forced to turn down the position for lack of sufficient energy. \"Here was another dream stolen,\" she wrote. Severely depressed, the nineteen-year-old escaped to a deserted Long Island beach, where she had her first major healing breakthrough:\n\nIn turmoil, I walked wearily along the water's edge, comparing myself to others my own age, people of seemingly boundless vitality... I remember thinking that this disease had robbed me of my youth. I did not yet know what it had given me in exchange... A wave of intense rage flooded me, the sort of feeling I had experienced many times before. But for some reason, this time I did not drown in it. Instead, I noticed it go by and something inside me said, \"You have no vitality? Here's your vitality.\"\n\nShocked, I recognized the connection between my anger and my will to live. My anger was my will to live turned inside out. My life force was just as intense... as my anger, but for the first time I could experience it as different and feel it directly... Somehow _this [intense love of life] had grown large in me as a result of the very limitations I had thought were thwarting it_ [italics mine]. Like the power of a dammed river... I also knew that in its present form, as rage, this power was trapped. My anger had helped me to survive, to resist my disease, even to fight on, but in the form of anger I could not use my strength to build the kind of life I longed to live.\n\n\"I was like Zorba the Greek!\" Rachel laughs, remembering the angry girl she was and the lust for life trapped inside her. \"The shift came when I realized that I could express this power\u2014my power\u2014directly. I didn't need to be angry. But let me tell you something,\" she says now, fixing me with her fierce dark eyes. \"I would never have reached that point if some therapist had worked with me to get rid of my anger.\"\n\nRachel sips her tea. \"People can get caught in anger, this is true,\" she goes on. \"And that is very limiting as a way of life. But there is a place for anger in the process of becoming a human being.\" The wonderful Buddhist teacher Rick Fields channeled rage at his own disease into a series of poems\u2014including a doozy called \"Fuck You Cancer\"\u2014as a means of staying real in the throes of healing. \"Without it there's something important that you never get to have.\"\n\nThat thing is authenticity, Rachel believes. Such authenticity signals the deepening of soul that occurs when untrue things are burned from our lives. \"What many people don't understand is that it's not enough to survive physically,\" insists Rachel. \"That's too low a goal. The idea is to survive psychologically, as a loving being. To survive as a soul. This is how we begin to awaken.\"\n\n\"To what?\" I inquire.\n\n\"The beginning of loss is also the beginning of compassion,\" she tells me. \"My experience has enabled me to help a lot of people free themselves where they've gotten trapped. I can be in situations that would terrify most other people.\"\n\nI do not doubt this for a moment.\n\nRaised in a family where the debate between science and spirituality never stopped raging (until she was twelve she wanted to be a rabbi), Rachel struggled to reconcile these worldviews not only as a physician but also as a patient. Outraged by the arrogance, shortsightedness, and lack of imagination she recognized among medical colleagues, she set out to find a middle path between intellect and intuition. In 1976 she cofounded Commonweal holistic cancer center in Bolinas, California, and has since been a pioneer in the field of mind-body medicine.\n\nRachel freely admits to having an ax to grind. \"I made many of my critical life decisions based on what I was told by doctors,\" she tells me. \"I was told that I'd be dead by forty, so I never got married or had children.\" That is quite an ax, I think. \"Nobody offered the possibility that I might be able to survive,\" says Rachel, who recently celebrated her sixty-fourth birthday. \"It's hard to think outside the box. But that is where life is. Outside the box.\"\n\nHealing is a mysterious process whose principles often contradict reason. Working with thousands of patients, she has learned how often the body's intelligence defies expectations. \"The body is hardwired to persevere,\" Rachel reminds me. \"If I cut you, your body will heal stronger than before. Without this built-in tenacity toward life, even the most sophisticated treatments could not succeed.\"\n\nThe poet Dylan Thomas called this \"the force that through the green fuse drives the flower.\" Rachel vividly recalls the first time she noticed this uncanny power in miniature. \"It was one spring day when I was fourteen,\" she says. \"I was walking up Fifth Avenue in New York City and was astonished to notice two tiny blades of grass growing through the sidewalk. Not around, but _through._ They were green and tender, and had somehow broken through the cement. Despite the crowds bumping up against me, I stopped and looked at them in disbelief.\"\n\nThe memory still amazes her. \"This image stayed with me for a long time, possibly because it seemed so miraculous. It gave me a very different sense of what power and force look like. There's no violence in real power,\" she observes, in spite of common, dog-eat dog wisdom. \"There is something much more mysterious at work than our culture would have us believe,\" she insists. \"Confronted by enormous change\u2014losing a breast, a country, a child\u2014we're faced with an important choice. Either we come to surrender, go into the loss, attend to our own responses, and listen to ourselves, or we attempt to put it behind us and get on with the rest of our lives. But does this really work?\"\n\n\"It hasn't for me.\"\n\n\"Me either,\" she says. \"This is essential to understand. When we try to avoid loss or plow through our pain, our lives are actually diminished. On the other hand, there's an extraordinary wisdom and clarity that emerges in people who genuinely meet their pain, not in theory but in life.\"\n\n\"Why do you think that is?\" I ask.\n\nRachel considers the question a moment. \"We realize that there is a larger power operating which can be trusted to sustain us,\" she says finally. \"As any woman who's gone through delivery knows, it's minute by minute when the going gets rough. The forces that are happening are so great that the woman surrenders. She has to. Often it's the moment just before birth.\" The paradox of this is quite beautiful. \"The whole process of awakening, initiation, comes only when things are surrendered. Women talk about this moment and how much they learn from it. Afterward they have a different relationship to life\u2014and to themselves. It is the beginning of trust.\"\n\nThis trust in mystery carries us forward. \"An awareness of something larger breaks through,\" she explains. \"There's a core identity shift, not just in ideas about yourself but how you see the entire world.\" Working with cancer patients all these years, Rachel has witnessed this identity shift quite frequently. \"I've often seen that the body can be quite diminished while this other identity has expanded greatly,\" she tells me. \"The _individual_ has expanded.\"\n\nDoubleness, I think to myself.\n\n\"The process of wounding actually awakens us to our strength,\" she says. \"It shuffles our values. And the top priority is never what you thought it would be. It's never about perfection or power. It always turns out to be about love.\"\n\nWe decide to take a walk in the garden. Trailing Rachel into the sloping yard, watching her as she pinches seed carefully into a bird feeder, bending ever so carefully to pick up a rag, I'm reminded of how ill my hostess truly is, how meticulously she must mea sure her energies in order to carry out her many duties. This gives Rachel a sturdy yet frail balance, a poignant and particular strength.\n\nWe sit side by side on a bench. The fog is creeping up Mount Tamalpais, blanketing the pine-sided mountain in white. \"We forget that there is a flow in things,\" Rachel reminds me. \"That the trick is to get out of the way.\"\n\n\"To get out of our own way, you mean?\"\n\n\"When you block a river, it finds another way to get where it's going,\" she says in a soft voice. \"When your physical being is changed, you travel in other ways.\" Rachel scatters a handful of seed for the birds. \"Think of trees that go through extreme weather\u2014not only are they strengthened, but their roots grow deeper. They're not straight, but twisted into themselves. Every challenged individual I have ever met comes to this same crossroads, regardless of what's standing in their way. Will they be resigned, or will they surrender?\"\n\n\"Sometimes these two seem synonymous,\" I say.\n\n\"But that is where our freedom lies,\" Rachel tells me. \"I came to a point where I decided that being ill was not going to stop me from living well. Of course, I'd love to be able to see perfectly!\" she admits. Crohn's has robbed her of vision in one eye. \"Of course, I'd love to be able to eat what ever I want. I would absolutely love to go mountain climbing. But I can't do those things,\" she says with a shrug. \"What I can do is adapt.\"\n\nRachel leans back on the bench. \"It's the way in which we move through very difficult circumstances, and not the absence of circumstances, that enables a person at the end of a life to say, 'You know, this has been a great ride,'\" she tells me.\n\nRachel's dark eyes turn silver as she gazes up at the fog. \"You see that even though your body has changed, there is a part of you that was never hurt,\" she explains. \"My sense of wholeness has nothing to do with my body.\" Her confidence does seem to match this claim. \"I don't believe that there is something wrong with me,\" says Rachel, pointing overhead at a golden hawk perched high in an evergreen nearby. The hawk appears to be watching us, too.\n\n\"I see the world as a mystery,\" she says. \"Much like a child does.\"\n\n## **TRUE CONFESSIONS**\n\nFor reasons never explained to me, I became our family confessor. From the time I was old enough to listen, my sisters, my mother, their lovelorn girlfriends, the widow Yetta who lived two doors down, with her brick-red hair and bison-sized muumuu, pouring out her big heart over sweet borscht and flanken she cooked for me as a snack on my way to bar mitzvah class\u2014to all of them I was Mr. Lonelyhearts, Mr. Let Your Hair Down, Mr. You Know You Can Trust Me. Sit.\n\n\"What d'you know?\" Yetta would kvell at me after delivering one of her epic tales of betrayal by her late husband, Mack, stubbing my chin between her fat fingers. It did Yetta's heart good to be heard, she told me. It did me good to listen, I said.\n\nYetta had a friend named Molly Gross, who needed to talk, the widow told me, like someone begging for oxygen. Molly was a caged bird singing her desperate heart out, married to a miserable surgeon named Sid, who made millions excising brain tumors and did his best, during off hours, to make Molly's life miserable. I first heard about her when I was twelve, but the caged bird and I had never met. Ten years passed. I was in graduate school by then, hardly able to pay my rent. Molly, about to turn eighty-five, was on the hunt for an assistant to help her plow through the avalanche of notebooks she had been keeping for decades, to draw her untold stories from her and then weave this material into a manuscript that would represent her life's work: fifty years of scrawled introspection crammed now, mausoleum style, into a stack of boxes in Sid's closet.\n\n\"She's a brilliant woman, misunderstood,\" Yetta told me over some schmaltzy plate of overcooked food. Molly had been born before her time, the widow believed, and was truly a diamond in the rough, a philosopher, practically, in her own right. Now this unsung old poet lady was afraid of dying without ever having stretched her wings. Someone needed to save her, said Yetta. That was my knee-jerk cue, of course. Needing money\u2014and, even more, to be needed\u2014I made a date to meet with Molly Gross at the penthouse apartment she shared with Sid overlooking Santa Monica Beach.\n\nI shall never forget my first sight of her rushing toward me, wrenlike and breathless, teary-eyed, arms outstretched, a plume of white hair escaping its net as she hurried forward on sticklike legs, bathrobe open, slippers flapping, down the aquamarine-blue hallway carpet, grinning and clutching the Kleenex that never left her hand. I immediately saw that Molly was dying. Her white arms were covered with purple track marks. She squeezed my hands, looked into my eyes, and said, \"Oh, honey. He's here at last\" to Maxine, the Guatemalan maid.\n\nMolly seduced me for two distinct reasons. The first was that I had never met anyone so ferociously honest\u2014so utterly vulnerable, hurt, and alive, yet giggly over the silliest things\u2014and I am an emotion whore, an addict of intimate connection. Second, this old lady seemed to possess a quality that I'd always yearned for but had yet to encounter: a brazen, unfettered brand of aliveness, a no-holds-barred thrusting into each coming moment-never-to-come-again. Molly and I would gab for hours behind the closed door of Sid's curtained study, seated side by side at a long mahogany table piled with her mass of materials, a hodgepodge of legal pads, paper napkins, dog-eared tomes marked with passages she wanted to quote, other books she all but knew by heart\u2014 _The Cloud of Unknowing, I Am That,_ the _Meditations_ of Marcus Aurelius\u2014reams of various sorts of paper etched in her erratic, crooked script. Sometimes Molly would cackle over a delightful memory sparked from this collectanea; other times her eyes grew teary when a painful thought arrowed up from the past, forcing her to look away, to cover her mouth with the crumpled tissue.\n\nDuring the fifteen months we spent in this passionate mind-meld, Molly continued to unnerve, enthrall, and move me. She spoke with startling conviction about things I knew nothing about, beginning with the spiritual aspects of life. In spite of her frail health, Molly overflowed with spirit\u2014absorbed, reflected, saw and heard more; cared more; suffered more; tasted her existence more deeply than anyone I had ever met. We sifted through her papers, recorded her stories, transcribed the tapes of our interviews. One day Molly might be in cheerful form, peaceful in her circumscribed world; the next day she might be agonized over her quarantined existence. \"Honey,\" she'd tell me with pain in her face, \"life is important. People waste it. Please don't waste it.\"\n\n\"I won't,\" I promised.\n\nOther times Molly would grip my hands, squeeze her eyes shut, and refuse to tell me what was going on. Stymied by one cryptic passage, I asked her _what_ was she trying to say?\n\n\"You know!\" she said, slapping my arm.\n\n\"I don't know.\"\n\n\"Oh, you!\" She pinched my ribs.\n\n\"It's fuzzy,\" I told her.\n\n\" _You're_ fuzzy.\" Molly laughed. She touched me every chance she got, stroking every available surface. One day we came across a particularly koanlike entry. It read, what is mine will know my face and was written in large capital letters on stationery from the Hotel de la Legion in Paris.\n\n\"What does this one mean?\" I asked.\n\nMolly tousled my hair. \"You know!\"\n\n\"I honestly don't.\"\n\nShe peered over my shoulder, although no one was standing there. \"One day I stopped in the wood and all paths were the same,\" she whispered, nonsequentially, widening her eyes.\n\n\"Stop it!\"\n\n\"The end of folly is the same as wisdom,\" she taunted.\n\n\"Avoiding the question,\" I would say. But it was no use trying to stop her. Like a woman drawing a strand of pearls from her bodice, Molly revealed her mystical aper\u00e7us one glowing jewel at a time. Mostly, I didn't want to stop her. I'd fallen in love with Molly by then. She believed that my visits were the only thing keeping her alive.\n\n\"Somebody sees me!\" Molly once said to Maxine, squeezing my hand. My friend had weakened as the year wore on. One day we were eating lunch at her kitchen table, drinking a thimbleful of vodka first, as had become our little custom. Maxine was watching us with a smirk.\n\n\"I do see you,\" I told Molly.\n\n\"Eureka!\" she squealed, and gripped the table.\n\n\"Loud and clear,\" I assured her.\n\n\"Will you publish my book?\" she asked.\n\nI promised her that I would try. We had decided to call her five-hundred-page manuscript _Moments of Being,_ not knowing that Virginia Woolf had already usurped that title. As I write these words today, Molly's manuscript remains in its same blue binder, tucked away at the top of my living room closet.\n\n\"Happy day,\" Molly said, kissing the tops of both my hands. \"Now, you say the prayer, Maxine.\"\n\nThe plump, shy maid closed her eyes. \" _O Dios, por favor, Padre Nuestro, denos su mano_ \u2014\"\n\n\"Isn't that just wonderful, honey!\" Molly squealed. \"Isn't she just beautiful?\" Molly laid a kiss on Maxine's cheek. \"Look at the three of us here together with all this love like a family, my children. And all the sun and this beautiful food.\"\n\nJust then, Molly pulled herself to her feet and lifted her knee-length nightgown many inches higher than a woman with varicose veins should have. \"I want to dance!\" she begged me, holding out her thin arms. \"Oh, let's dance and get drunk!\" Molly said.\n\nMaxine lowered her eyes and smiled. I watched Molly waltz around the kitchen, embracing an invisible partner, singing a popular song from the 1920s in her trembly voice. \"Oh, promise me that someday you and I will take our love together to some sky,\" she sang. Molly begged me to come and swing her around. \"Come to me, my handsome man. This day will never come again.\"\n\n\" _Vaya,_ \" Maxine urged me.\n\n\"Hon-eeee!\" Molly begged, lifting her nightgown an inch or two higher, prancing around the kitchen. \"You will break my heart.\"\n\nI took her into my arms awkwardly. Molly pushed her slack breasts against me, hooked her hands around my neck, and hung there, smiling up at me. Then she started to sing. \"Oh, promise me that someday you and I will take our love together to some sky,\" she crooned. I heard Maxine humming along and the grandfather clock ticking in the hall, the sounds of traffic below. Molly laid her cheek against my chest.\n\nAll sorrows can be borne, Isak Dinesen said, if you put them into a story. Rachel Remen remembers sitting under the kitchen table as a little girl, stroking the purple velvet carpet slippers of her grandfather, an Orthodox rabbi, as he told stories and read kabbalah. Rachel believes that our lost art of telling stories, the pre-e-mail tradition of people sitting around telling stories as a way to pass along wisdom, accounts for a cultural loss of soul and shared insight into how we live. By sharing our stories, she believes, we tap into a largely ignored reservoir of living wisdom.\n\n\"When we stop telling each other our stories, we seek out experts to tell us how to live,\" Rachel has written. \"The less time we spend at the kitchen table, the more how-to books appear in the stores. Because we have stopped listening... we have stopped learning how to recognize meaning\u2014and to fill ourselves\u2014from the ordinary events in our lives.\" Storytelling is also a great boon in friendship. I learned of an Israeli woman who'd gone to a therapist because she was having trouble breathing. As they spoke, the shrink noticed the camp numbers tattooed on the patient's forearm. The woman coughed a great deal while telling her story. \"When did you start having trouble breathing?\" the therapist asked. \"When my friend died two years ago,\" the survivor admitted. \"When she was alive,\" this lady told the doctor, \"we could talk about anything. Although she had not been in the camps, she understood. But now there is no one to tell. And the nightmares haunt me. I can't sleep alone in the house. I know that if I want to live, I have to find another friend.\"\n\nOur stories are meaningful to the degree that they converge, in a meaningful way, with others. One recent afternoon, I found myself with fifteen medical residents, the novelist Michael Ondaatje, and Dr. Rita Charon, who heads the Program in Narrative Medicine at Columbia University's College of Physicians and Surgeons in New York, sitting around a Dorito-strewn conference table, talking about how doctors can learn about patients' inner lives from listening carefully to their stories. Rita Charon has been a pioneer in narrative medicine since its appearance two decades ago. As a young internist, Rita realized that she knew neither how to listen deeply to patients, nor how to interpret the life narratives they were telling her. What better way to learn about stories, she asked herself, than to study great works of fiction? She went back to college for a Ph.D. in comparative literature and emerged with an intriguing theory about how doctors can become better healers by doing what good readers do: ponder character, structure of plot, emotional foreshadowing of the kind that novelists employ and that patients use in bringing personal stories to life in their doctors' offices.\n\nEric David, a third-year medical student who's fresh today from the OR (where he assisted at a radical prostatectomy), is struggling to make a point. \"Doctors dismantle human beings,\" says Eric, who's shaggy-haired and looks like he's in high school. \"It's what separates us from the rest of the world. You have to fight the temptation to turn off. If you remember that everyone has a story, it kind of helps.\"\n\nOndaatje, a leonine, big-bellied guy with a white beard and icy green eyes, listens carefully to the young man in green scrubs. The novelist himself got interested in narrative medicine after observing a quadruple bypass, he told me before the class. \"I was standing two feet away from the heart,\" Ondaatje marveled. \"Just staring down into the patient's body. What could be more fascinating?\"\n\nThat heart surgery got him hooked. \"I've been fixated on doctors since I was a boy,\" Ondaatje told me. \"I wanted to talk to doctors about books the same way you would x-ray a body, to see how it worked, how it was made.\"\n\nThat seems like such a clinical motive, I said. \"No more profound or artistic reason?\"\n\nHe thought about it a moment. \"I suppose it has to do with rescue and healing,\" Ondaatje offered. \"Our lives are all so dangerous, precarious\u2014you could even say abandoned. Rescue and healing are essential things.\"\n\nAbandoned? I wondered. This seemed like an odd choice of words for describing the human predicament. Abandoned in the world by who\u2014or what? In _The Noonday Demon,_ Andrew Solomon describes the outcast state of depression as \"the loneliness in us made manifest, a crystallization of the human condition in which each of us is held in the solitude of an autonomous body.\" If feelings of abandonment do lie at the core of our common longing for connection, it seems even more understandable that religions use confession as a means of healing and that writers such as Michael Ondaatje approach the art of storytelling in almost holy terms.\n\nBack in class, he listens to a female resident who appears to be upset over their reading selection for the day, William Maxwell's novel _So Long, See You Tomorrow._ Wearing a pained expression, the young doctor confesses to having been callous with one of her patients during morning rounds. This man, who was gravely ill, had invited her to sit beside him and listen to one of his stories. The resident made an excuse, she tells us, claiming that she was too busy to stay. Instead, she hurried the lonely man through his tale and then cut him off before the end. She never saw the old man again. \"I should have taken the time,\" she says now.\n\n\"Next time,\" says Rita Charon, the comforting mentor, who's trying to train better doctors, not promote more self-punishment.\n\nWhen the class is over I walk Ondaatje to the elevator. The novelist appears far away in his thoughts. When the door slides open, it rouses him from his trance and he blinks. \"Extraordinary,\" he mutters aloud, without elaborating on what he's thinking. \"I can't forget to write that down.\" Then the elevator closes behind him.\n\n## **PROMETHEUS**\n\nAt a recent Fourth of July barbecue, I find myself seated next to a bouncy brunette who introduces herself to me as Ella. In a movie, she'd be played by Eva Longoria. Ella tells me that she's a nurse as fireworks explode overhead. \"I work with miracle kids,\" she says, dabbing corn-on-the-cob butter from her mouth.\n\n\"Miracle?\" I ask.\n\n\"Heroes,\" Ella says. \"You want to talk about Independence Day? These kids are freedom fighters,\" she tells me as pyrotechnics kaboom down the beach. Ella is the head nurse of pediatric orthopedics at New York's Beth Israel Hospital. Her boss is a renowned surgeon named David Feldman, who, in addition to a private practice, runs a monthly clinic for severely handicapped public-assistance kids who have nowhere else to turn. \"He takes these twisted children and makes them walk,\" Ella says. \"The man is a hero in his own right. A greater heart you have never seen.\"\n\nThe following week, I'm powerwalking behind Dr. Feldman up and down Beth Israel's crowded corridors during rounds, flanked by four residents, a Spanish translator, a rep from a crutchmaking company, and Ella. Feldman is a blond-haired, turbocharged Orthodox Jew who dresses slick, barks orders, and walks (when he forgets to hide it) with a limp left over from a childhood illness. Hurrying along the corridors, we're surrounded by crippled kids, lots of them wheelchair bound, others crawling or holding their mothers' hands, feeling their way along the wall. Their eyes light up when they see Feldman. Though gruff, Dr. F. is a natural with these kids, _zetzing_ this one in the ribs, tickling another, calling them names, making them laugh.\n\nIn the first exam room we find Melanie, a naughty six-year-old with spina bifida, who giggles hysterically when Feldman runs a ruler across the bottom of her clubfoot. Melanie's mother, a squat Latina, crows over her daughter's vast improvement since surgery (Feldman has many more planned to straighten out the little girl's spine). He helps Melanie down from the table and holds the little girl's hands as she slowly, proudly rotates herself across the floor to show him how much better she's walking.\n\n\"The boys will be eating out of the palms of your hands,\" he teases her, carefully measuring the splayed angle of Melanie's ankles, shooting technical info at Ella and the residents.\n\nThe girl squeals with delight, leans forward, and falls flat on her face. When Feldman moves to pull her up, Melanie sticks her tongue out at the doctor. \"I can do it!\" she cries, yanking herself up slowly by the handles affixed to the side of the table. Once standing, Melanie beams at him in triumph. \"I told you.\"\n\n\"Yes, you did,\" says Feldman. Then he adds to the mother, sotto voce, \"We'll fix her, Mrs. Padilla. I promise.\"\n\nIn the next room a Hasidic family appears to be in midmeltdown. The high-school-aged patient with bad skin and a faint mustache refuses to stay in his wheelchair.\n\n\"He's making me meshuga,\" cries the mother, a tall, unpleasant-looking person in a bad wig, a baggy dress, and crunching patent leather shoes. \"Sit down, Yudi. Sit. Now!\"\n\n\"You sit!\" he shouts back at her, groping his way along the sink toward the window. \"I'm tired of sitting.\"\n\n\"He don't listen,\" mutters the father, an unshaven, hopeless-looking guy wearing side curls and a yarmulke. \"Mind your mother,\" he tells the boy with a shrug.\n\n\"I want to walk, Doc!\" Yudi tells Feldman.\n\nThe mother seems to be davening, clutching at the sides of her face. \"You're fine like this,\" she hisses.\n\n\"I want to dance at my prom!\" Yudi tells her. He also wants to go skiing, ride motorcycles, and trek in the Himalayas with Sherpas. With each high-risk threat to her dominion, his mother's face goes a shade more livid. \"You see what I mean?\" she moans to Feldman.\n\n\"I think he's ready for surgery, Mrs. H.,\" says the doctor. \"But Yudi, you tell me, where should we start? The ankles? The hip? The back?\" Feldman turns the boy back and forth by the shoulders. \"I say we go for the back.\"\n\nThe mother looks at the father, who looks at the doctor, who looks at the boy, who appears to be the least scared of all of them. \"Tell me the risks first,\" Yudi says in a calm, grown-up fashion. Feldman lets him know that he could be paralyzed and even less mobile after surgery\u2014although that's unlikely\u2014suffer pneumonia because of his weak lungs, or even die under anesthesia.\n\n\"You're fine like this,\" the mother says softly, reaching out for her son's hand.\n\n\"I want to do it, Mama,\" says Yudi.\n\nFeldman repeats to the parents, \"He wants it.\" The mother looks positively stricken. The father shakes the doctor's hand and tells him to schedule the surgery. As we leave the room, I look back and see Yudi smiling. He gives me a thumbs-up.\n\nTwo hours and thirty children later, David Feldman collapses in his office easy chair, exhausted but visibly happy. \"I should have half their chutzpah,\" he says, waving away a resident. \"Half their humor and come-what-may. These kids are my teachers, so help me God. Honestly, I don't know how they do it. They're completely guileless.\"\n\n\"Guileless?\" I say. \"That's an interesting word.\"\n\n\"They're not hiding anything,\" Feldman says. \"These kids are at the end of the line. They have nowhere else to turn. It teaches you\u2014\" The doctor can't find the words. \"You just learn what a huge difference little things can make. It takes so little to make them happy.\"\n\nAfterward, Ella walks me to the elevator. We sit for a few minutes in the lounge. \"At first I was miserable working with these children,\" she admits. \"I thought I'd made a big mistake. This work was just too heartbreaking.\"\n\n\"I can understand why.\"\n\n\"But the truth is, I was feeling sorry for myself,\" Ella admits. She offers me a handful of Skittles. \"Then one day,\" Ella tells me, \"a twelve-year-old spastic quadriplegic girl was brought in.\" I'd just observed Feldman examining a quad with severe spasticity, kicking her feet uncontrollably, arms crunched against her chest as her father cradled the girl in his arms.\n\n\"Her name was Lisa,\" Ella says, remembering this other child. \"Her knees had decomposed. She was walking on the sides of her feet, dragging herself along the ground like a crab.\" Ella pauses to collect herself. \"Suddenly she saw me and gave me this beautiful smile. Lisa said, 'Ella, you look so pretty.'\"\n\nShe blows her nose in a tissue. \"My fear started to go after that. Something broke open inside me. I saw the person she was, not the body she was in. It baffles me that these kids aren't angry,\" says Ella. \"It's like they aren't even handicapped. You don't see them making a big deal over their imperfections. They're always asking you how _you_ are. They speak to you from their souls. You can see it in their eyes.\"\n\n\"I saw it,\" I say.\n\n\"It all comes down to how they see,\" Ella tells me. \"Lisa's mother saw me watching her that day. There must have been pity in my eyes. The mother looked at me and smiled, probably guessing what I was thinking\u2014the thing I'd never really ask.\" The memory still amazes Ella. \"Then she said it anyway.\"\n\n\"What did she say?\"\n\n\" 'I love her,' Lisa's mother told me. 'She's mine.'\"\n\n## **FOUND ART**\n\nYears ago I was waiting on line with a hundred other pilgrims to visit the grave of a long-dead saint in a small town near Poona, in central India. Directly ahead of me was a German mother, her hair covered by a head scarf, helping her crippled Down syndrome child kneel at the side of the marble tomb. The boy, who couldn't have been more than twelve, was having a tough time bending his knees. When he finally managed to lower himself to the ground, the child laid his cheek on the stone for a long time. His mother tried to hurry him along, but the rest of us didn't mind the delay. \"Worship is not a ritual, it is an attitude, an experience,\" this very saint had once suggested. The child's worship seemed to elevate the hearts of the pilgrims standing behind him.\n\nAfterward, in a crowded hall, we listened to the dead saint's closest disciple, an old guy wearing a Rice-A-Roni T-shirt (people bring him mementos from their native towns), as he reminisced about fifty years spent at his master's side. The great man had blessed hundreds and hundreds of thousands of people seeking his _darshan,_ or holy touch. \"Baba loved everyone,\" the white-haired secretary explained to us in his pidgin English. \"Most of all, though, he liked the scoundrels! The bad ones like me!\" The old man's wicked smile made it easy to imagine him as the bad-boy aristocrat he once was, the scion of a wealthy Bengali family, addicted to fast cars, booze, and easy women, till the day his guru spun his head around.\n\n\"The naughty ones,\" the secretary chuckled. \"That is who Baba liked the best. Some people would come to him pretending to be holy. They would sit up very straight and tell Baba about how good they were. How pure. How enlightened. Baba would just smile and move them along.\"\n\nThe Down's boy was rolling around on the ground, giggling. His mother gave up on trying to control him.\n\n\"God does not need your holiness,\" the old man said, swatting the H-word from the air and grinning. \"God does not need your goodness!\"\n\n\"What does he need, then?\" an Australian woman inquired.\n\n\"God wants your humanness, nothing more,\" replied our host. \"Broken furniture, that's what he called them, the people who came without making a pose. Baba loved broken furniture. This is where, he said, Baba could sit the best.\"\n\nI liked the idea of humanity as a sort of global Salvation Army store filled with derelict furnishings, three-legged chairs and mangy sofas, rickety tables, half-extinguished lamps, neglected armoires empty of drawers, beds too lumpy and slouched to dream in. When Andrea Martin, then Bowlby, was a five-year-old girl in a body cast, bedtime was the most painful part of the day. \"I would lie there, crunched up in agony, till morning came,\" Andrea tells me when I visit her in the rectory of the Episcopal church in Hartford, Connecticut, where she is the junior pastor. A tiny woman with pixie hair and a limp, Andrea is feisty, self-deprecating, extremely smart (she graduated from Yale Divinity School), and honest about the torturous path she followed as a child forced to endure fifteen major surgeries and six months out of every year trapped inside the dreaded cast.\n\n\"It's like found art,\" Andrea says to me now, looking out at the trees in a garden. \"You take all the bad things\u2014the pain, the embarrassment, anger, longing\u2014also surprising moments of grace, and form them into something original, unique, which then becomes your life.\"\n\nBorn with a condition known as PPD, Andrea nearly lost her right leg when doctors recommended amputating the drastically shorter limb while she was still a toddler. Luckily, her parents refused. Instead, from age two until twenty-one, Andrea suffered through these leg-lengthening surgeries and the excruciating, racklike procedures that followed.\n\n\"It wasn't so much the physical pain as the emotional agony,\" Andrea says back at her house, fussing over omelets she insists on cooking after christening eight babies (\"a train wreck!\") at the morning service. \"I felt so out of control,\" she admits, whisking eggs. \"I was so much at the mercy of doctors and their decisions. So alienated and ostracized. And so self-conscious!\"\n\nAndrea pours the eggs into a pan and offers me coffee. \"It was very painful,\" she remembers. \"I was _terrified_ of social situations, teased mercilessly by the kids at school.\" Yet even at that early age, she began to find comfort in her faith. \"I was helped by the Christian story of Jesus inviting all the outcast people to be with him,\" she tells me. \"Reaching out to the lepers and the sinners. That had quite an effect on me. Jesus himself was an outsider. That gave me hope.\"\n\nHopeful, too, was the possibility that Andrea might someday be able to use her personal struggle for a higher good. \"Maybe my heart would be different,\" she says, serving our breakfast. \"Maybe God could use my empathy for others in a way I couldn't predict.\" Adolescence was a nightmare still to be survived, though. \"Other girls in school were dating,\" she tells me. \"There was this huge chasm between me and them. I thought, 'Gosh, I've survived all these surgeries, years on crutches, rehabilitation, and now it's become a double whammy, because nobody is going to love me.'\" Andrea seems embarrassed by what may sound too much like self-pity. \"A man with a limp or scars is somehow attractive to the girls,\" she explains. \"It's a sign of manliness. For a woman, it's total alienation. I thought I'd never get married.\" (In fact, Andrea's husband of four years, Chris, a Libertarian lobbyist, seems to adore her.)\n\nThis confusion in her began to shift during an undergraduate semester in India, where the future priest found her vocation. Andrea had previously planned on being a doctor. \"I wanted to minister to sick people the way my doctors did for me,\" she says. \"Treating not just my leg but the whole person.\" Unfortunately, she did not excel in science. Then one day, in Delhi, Andrea Martin got her calling. \"We were visiting Gandhi's birthplace,\" she tells me over breakfast. \"I was on this big, ostentatious tour bus\u2014TVs, velour seats, the whole deal\u2014looking out the window, when these two street urchins came up and looked right at me, reaching out for money or food. The words from the liturgy came back to me: 'Lamb of God who taketh away the sins of the world, have mercy on us.' These two children represented the sins of the world. It was a trace of God calling me,\" she says. \"Reaching out. I finally had permission to follow the path I truly loved.\"\n\nHer ministry since then has become this unconventional reverend's oeuvre of found art. \"My mother used to say, 'Andrea, your struggle right now is very difficult on the outside. But everybody has challenges and hurts and struggles that are not always so visible,'\" she tells me. \"As a priest, my own struggles are mostly on the inside now. But since I've known what it's like to feel a chasm between me and other people, or me and God\u2014and a longing to close that gap as best as I possibly can\u2014I have increased empathy. I pray every day for the grace to help bring people together. This desire is deepened by the isolation of my own childhood.\"\n\nWorking with her congregation has taught Andrea how pervasive such feelings of alienation, isolation, even abandonment, are among outwardly successful individuals. \"There's a huge amount of pressure on people to have it all together, or to look like it at least,\" she tells me. \"Even when they're crumbling inside. My job is to look past the exterior into somebody's heart and soul and know that what's inside can be very different from what they present.\"\n\n\"Appearances are so deceptive sometimes,\" I say.\n\n\"A doctor I know uses discarded junk from the hospital where he works,\" Andrea tells me, sipping her coffee. \"Old x-ray film, IV tubing, gauze, broken casts, to create the most beautiful sculptures.\" She refills both our cups. \"Found art,\" she says, \"from sickness. To me, that is a great metaphor for hope. It's how God acts on our behalf, I believe. When a whirlwind sweeps through our lives, leaving scattered debris and junk, and we're tempted to give up hope, even then God is at work reassembling the junk, the shit of our lives, into something new. When we collaborate with this creative, redemptive spirit, miracles can happen.\"\n\nToday Andrea can bike, swim, and climb mountains with almost no pain. \"God is economical,\" she says with a smile. \"There's no experience that we can endure\u2014short of torture, starvation, or extreme abuse\u2014that God can't use for good. God has a dream for what we might be, and the more we're open to cooperating with that grace, the more we can be sanctified and pulled upward on that spiral to be the person God dreams us to be. Even the most hopeless cases. God takes what's hopeless and turns it into fecundity.\"\n\n\"Broken furniture?\" I suggest.\n\n\"God may even love the broken ones just a little bit more,\" Andrea Martin says.\n\n## **THROUGH WILDERNESS**\n\nVisionary power has historically been the shaman's gift, in traditional cultures, only after surviving the wild. Certain individuals are asked to \"go walkabout\" in extreme terrain; others endure physical torture in order to be \"awakened\" by pain. Others are called upon to venture into trance states (drug-induced and not), where they undergo terrifying psychological initiations involving \"demons\" and other dark interior foes. Finally, there are those, like Francis Bok, the escaped Sudanese ex-slave, who, while not indigenous shamans, are thrust as children into trials so extreme that their characters are forged, swordlike, to help them become liberating agents for others braving their way in the dark.\n\nOn the list of grueling obstacle courses to wisdom, no path is more savage than incest, or the sexual abuse of children. Molestation by a trusted caretaker tends to split a child's life in two. In fact, according to Ariel Jordan, a pioneer survivor in the field of incest, \"Childhood ends the moment that a child is raped,\" and incest amounts to what he calls \"soul murder.\" In order to heal from such trauma, incest survivors are required to pass through spiritual wildernesses most of us would find hard to imagine, labyrinths whose darkness is matched only by the light that dawns in the night sky of recovered children who've found their way back to the land of the living.\n\nWithout documentation, Ariel's story would be hard to believe. A handsome sixty-year-old man with movie-star silver hair, soulful dark eyes, and a bellowslike voice reminiscent of Henry Kissinger's, Ariel was born on a kibbutz in the Upper Galilee, a few years before the state of Israel came into being. His father was an engineer and a pillar of the community. His mother organized the child-care section of the kibbutz. \"Throughout my childhood, I was told how very lucky I was to have them as parents,\" Ariel tells me when we meet in his crowded Chelsea apartment. \"I worshipped my parents. Especially my father.\" He shows me a photograph of himself as a wide-eyed five-year-old being grabbed around the waist by a creepily smiling man.\n\nThe boy looks frightened and trapped. I hand the picture back to Ariel. \"How explicit do you want me to be?\" he asks. Aware that Ariel has testified twice before congressional committees on child abuse, I invite him to be completely candid. His intention is not to shock, he assures me, but simply to shed light on what he calls \"society's darkest secret,\" to draw the murky topic of incest out into the open. There's a cultural block against believing kids, he's learned during twenty years of working in this field. \"But children must be heard,\" he insists.\n\n\"Of course,\" I say. So Ariel leans back in his leather chair and launches into his surreal tale. \"My father raped me for the first time when I was four,\" he tells me without introduction. \"He had given me a bath. It felt so good, because my parents rarely touched me,\" he says. \"Israeli boys were supposed to be tough.\"\n\nMy host is speaking matter-of-factly, with no trace of self-dramatization. \"My father was drying me off and tickling me,\" he goes on. \"It was wonderful! The two of us alone together, having fun\u2014\"\n\n\"Where was your mother?\" I ask.\n\n\"She was there. Somewhere,\" says Ariel. \"But this was time to be with Papa. It was like a game that we were playing. I remember laughing and telling him to stop tickling me. Then my father began to kiss me.\"\n\nAriel's face darkens when he says this. \"I didn't know what was happening,\" he admits. \"I just knew that my father was no longer playing. He seemed to be in a kind of trance. Then\u2014\" He stops to take a breath. \"He entered me.\"\n\n\"At four years old?\" I ask, unable to mask my own incredulity.\n\n\"The physical pain was one thing,\" says Ariel. \"But the body goes into shock. Emotionally, it was much, much worse.\"\n\nI'm completely speechless.\n\n\"It's so terrible that part of you dies. You cease in that moment to be a child. Suddenly you are an orphan,\" he says. Researching a piece about incest in the weeks after meeting Ariel, I'm shocked to learn that one out of four girls is raped before the age of eighteen, by a family member or another (the number is slightly lower for boys, one out of five). Even Freud had trouble believing the harsh facts and prevalence of incest. The father of psychology was so appalled by patients' memories of sexual abuse that he famously recanted his original theory, fearing the scorn of skeptical colleagues, and blamed abuse memories on \"hysteria\" and juvenile sexual fantasies. Psychologists since then have used Freud's own reversal as an excuse for discounting incest survivors' stories.\n\n\"How long did he do this?\" I ask now.\n\n\"Until I was fifteen,\" Ariel says.\n\n\"Why didn't you tell someone? Or fight him off?\"\n\n\"I wanted to make him happy,\" he tells me, aware of how crazy this sounds. \"It is hard to understand, I know. But abused people have strange relationships with their violators. This became our secret. Our secret world. Your survival mechanism tells you not to resist, since they're going to do it anyway. If you collaborate, you'll be like... lovers.\"\n\nMiraculously, Ariel managed to graduate from high school and escape from Israel. Transplanted in London, he earned a degree in cinema and began making documentary films. Far away from home, he was able to nearly submerge his memories of abuse. He was nearly able to convince himself that this dark history was a trick of his imagination. I'm compelled to confess to Ariel that the enigma of recovered memory, common among abuse survivors, is a mystery I've not quite been able to grasp. How can a person who's been through such severe trauma actually function without conscious recall? S\u00e1ndor Ferenczi, a colleague of Freud's, had a theory about uncontrollable stress of any kind\u2014from wartime combat to serious accidents to sexual abuse\u2014believing it can cause a person to emerge from trauma riddled with mnemonic bullet holes. \"Part of our being can 'die,'\" wrote Ferenczi. \"And while the remaining part of ourselves may survive the trauma, it awakens with a gap in its memory.\"\n\nArriving in New York in the early eighties, haunted by his secret past, Ariel was determined to fill in his psychic gaps\u2014to heal himself of the pervasive feelings of shame, impostorhood, and sexual confusion (hyper-promiscuity combined with emotional absence) that were standing in the way of his living a happy life. This was years before incest became a household word\u2014thanks in large part to Oprah Winfrey's publicizing of her own incest story\u2014and Ariel could find no one who would believe him. Finally screwing up his courage to talk to a therapist, Ariel was appalled by the doctor's response. \"I poured out my heart for an hour, and you know what this shrink said to me?\"\n\n\"I'm afraid to ask.\"\n\n\"He put his arm around my shoulder and told me it was all wishful thinking.\" Ariel chuckles. \"I realized in that moment that I would have to do it myself.\"\n\nLike a shaman in training, Ariel Jordan entered the dark wood of memory armed with nothing but raw nerves and resolve. Determined to confront his demons, he turned to art as a tool for excavating his past and bringing monster feelings to light. He created a roomful of work recalling Francis Bacon's disemboweled-looking paintings, a pictorial season in hell, a place where his subconscious could spew the evidence of what he'd lived through, making the art of that raped child still inside his body, who needed to speak but until then had no voice.\n\nSlowly, Ariel found his way through the forest. He emerged with a technique for incest therapy that has helped hundreds of other survivors through their own mazes, back from soul murder. \"Every single survivor I've worked with has at least two personalities,\" he explains to me. \"One is enormously intelligent and functional. The other is a maimed, speechless child who contains, though they can't quite touch these qualities, spontaneity, connectedness, and creative power.\"\n\n\"How can a survivor uncover them?\"\n\n\"Memory lodges itself in the body,\" Ariel explains. Though many of his clients have been through decades of talk therapy, recycling known narratives rarely seems to get at the root of the trauma. \"Many therapists prefer rational dialogues,\" Ariel says. \"But the child part of the patient can't even put what happened into words. One must enter the theater of the subconscious like a safecracker and break the child's code that is spoken there. The child mostly needs to scream. In order to survive his trauma, he has learned to function as a fake, an impostor. This disassociation creates enormous existential pain.\"\n\nAriel points out that incest survivors, like so many post-catastrophe individuals, often have \"an unusual capacity for transcendence.\" \"Survivors are forced to become seekers,\" he explains. \"We learn to take what ever nurturing we can from anything, no matter how small. Like a cactus in the desert.\"\n\nHe remembers doing this himself as a boy when human comfort was nowhere around. \"As a child, I would see how nature wakes up after the trauma of winter,\" Ariel tells me of those early years in the Galilee, \"blossoming, blooming, enriched with sweet smells. I remember leaving our house one particular afternoon, after I had left my father. My hair was wet with sweat, and there was a breeze coming through the mountains right into my face. Suddenly, even though nothing had changed, I was happy again.\"\n\n\"How do you explain that?\"\n\n\"The breeze told me I wasn't alone,\" Ariel says simply. \"It was as if somebody or something was talking to me. Reaching out to touch my forehead.\"\n\nThe room gets quiet for a moment. Ariel seems to be lost in that memory. \"A traumatized person becomes the center of his own universe,\" he goes on to explain. \"This is why so many individuals remain stuck and depressed. Connecting to forces beyond yourself helps you realize that your story, painful as it might be, is not finally that significant. When you get that you're part of a larger context, cells in a larger organism, it gives you a huge amount of comfort.\"\n\nTranscendent connection also allows us to examine our wounds without being consumed by them. \"None of us can fulfill our lives without retrieving the truth,\" Ariel instructs those who come to him. \"You cannot heal what you cannot feel. Whatever you are most ashamed of, what ever humiliates you\u2014this is the exact door to your healing.\"\n\nThis does not mean that abusers necessarily need to be confronted, however. Ariel never hashed things out with his father before he died. He has never talked to his elderly mother about what happened. This has not stopped him from finding some forgiveness, though.\n\n\"How do you forgive something like that?\" I can't help but wonder.\n\nHe reminds me that forgiveness is an \"inside job,\" and has nothing to do with our abusers. Forgiveness is an act of self-blessing. This reminds me of Eva Eiger, a German ballerina forced to entertain her Nazi captors in order not to be murdered. In her memoir, Eiger described how it had taken her forty years to begin to forgive her former captors. \"You must be strong to forgive,\" she reminds us. \"Forgiveness is not about condoning or excusing. Forgiveness has nothing to do with justice. Forgiving is a selfish act to free yourself from being controlled by your past.\"\n\nWhen the interview is over, Ariel Jordan takes me into the adjacent room that stores his latest artwork. The tormented images of yesteryear have been replaced by photographs of shamans and medicine men from around the globe, wild-looking characters in Ecuador, India, Polynesia, who've passed through hell and emerged as potent healers. These shamans feel like kindred spirits, Ariel tells me. Like him, they have slain their minotaurs, the man-eating monsters who dwelled in the depths of their dark mazes.\n\n\"The dark tells you secrets,\" Ariel tells me. We tend to forget that the beast in Greek mythology is also an angel. In the story about the labyrinth, the minotaur's name is Asterion, \"star.\" Once again, our nemesis is recognized as a beacon after its hidden face is revealed. Working with clients, Ariel witnesses this turnaround often. It's like watching people being reborn, he says; they reclaim some lost part of their innocence. He helps them to use these wounds to deepen. Asterion leads them back into the world. Like the green shoots through that New York sidewalk, trust emerges slowly. Life begins again.\n\n## **A SPLINTER OF LOVE**\n\n_Trauma_ comes from the German for \"dream.\" Like recurring dreams, traumas have an uncanny power to haunt, change form, and remain spectrally part of the present. We actually carry the past inside our bodies, metaphorically as well as literally (biologists now understand that memory imprints itself upon our cells in the form of amino-peptide chains). Each of us is, literally, a container for the sum of our life's experience, even when, as in the case of incest survivors, we lack conscious memory of that experience. The body itself remembers.\n\nThat's why _closure_ is such a misleading term. What is it that closes exactly? Crisis may end but memory stays open. Even after the worst things pass, a residue of loss remains, a shadow of love over memory. A splinter of love stays caught in the heart. My mother passed away twelve years ago, for example, but my memory of her is as living as ever. The contours of our lives together remain with me as they always have, a few layers down. Scratch most of us in the right place and we're still ten years old, huddled with our parents and siblings. Are those memories that we want to close? we might ask. Would we know ourselves as well without them? Even when those memories include painful details, are they not the shadow that underlies the fullness of who we are and the unique portrait we have painted? Here, in the split between closing and holding, is where the mind and soul diverge. The mind wants neat endings, locked doors, tightened sutures. The mind wants fresh starts without mess left over. But the soul thrives on raggedness, memory, and mess. The soul wants topography, texture, and ruins, like those flour-and-water 3-D maps you made in grammar school, vivid and sloppy and filled with craters. (The mind prefers black-and-white aerial landscapes.) While the reasoning mind wants to cut and run, our souls pursue a different agenda\u2014to remain where love is, even when that love is gone; to ripen open; to be the keeper of sacred memory long after reason has wandered away.\n\nOn the morning of October 19, 1973, five soldiers loyal to Chile's new dictator, Augusto Pinochet, landed in the tiny town of Calama, a Wild West\u2013like outpost on the farther reaches of the Atacama Desert. Known as the Caravan of Death, this group of government-sponsored terrorists was already famous across the length and breadth of Chile for its brutality. Now the five men comprising the caravan climbed out of their Puma helicopter and began moving from door to door through Calama, apprehending unsuspecting husbands, fathers, brothers, and sons from their homes and loading them into military vans. As the men were led away in handcuffs, the women of the town watched in horror, and they pounded on the sides of the vans as the vehicles pulled away. This tactic of \"disappearing\" unsuspecting, frequently innocent, citizens was already a common tactic for the new regime. The women protested to government authorities, who of course ignored them. These twenty-nine Chilean men, ranging from teenagers to seniors, were never seen or heard from again. Their murders were never confirmed nor their remains returned to their families for burial.\n\nDevastated and enraged, the women of Calama took matters into their own hands. Armed with shovels, picks, and kitchen utensils, they organized regular search parties into the Valley of the Moon to look for their loved ones' remains, and continued to do so for decades. Chasing rumors, intuitions, and dreams, disappointed again and again, they returned to the Atacama to dig for bones and teeth, because they could not live without answers.\n\n\"No one knew if the men were alive or dead, though the likelihood is that they had been murdered,\" Paula Allen tells me. We're walking down a beach in Barbados. Paula and I have been buddies since the early nineties. Paula is a documentary photographer who's spent her thirty-year career following what she calls \"women's invisible stories\" around the globe, spending up to twenty years on a single body of work\u2014from a homeless woman turned street-walker in Jersey City to gays in Cuba and Irish gypsy lasses living in caravans in Belfast.\n\nOn Christmas Day, 1989, sixteen years after the men's disappearance, Paula found herself walking into the Valley of the Moon with the women of Calama at the end of a scorching-hot day, holding a camera and a shovel, scared of what she might actually find. \"It was very windy,\" Paula remembers. \"The light was soft and fierce as it is in the desert. Most of the women had on flowered dresses. I remember dresses blowing in the wind and the determination on their faces. I'd never seen determination like this before\u2014how determination turns into action. They could be arrested for what they were doing, but they refused to be silenced. They refused to give up their love.\"\n\nI had seen the images Paula brought back of weathered-looking women hauling sticks and spades, baskets slung over their shoulders, moving into the chalk-white expanse like explorers on an alien planet, searching for their _desaparecidos._ The plight of Calama's women was heroic yet incomprehensible to me. I could not quite grasp their obsession with touching the remains (that was my mind talking, I now see), their need for physical artifacts if they were to heal.\n\n\"When you don't have a body, nothing is certain,\" said Vicky, one of the widows in the interviews Paula conducted. \"It would be easier to stand the pain if we knew the truth.\" Another reminded Paula that \"a mother can never replace her son.\" \"I hear Manuel's voice everywhere. I see him walking on the streets,\" she said. \"There is always the feeling,\" yet another reported, \"that one must search for something.\" Finally, Se\u00f1ora Leo, the oldest woman among them, summed it up in a chilling phrase: \"They buried us alive.\"\n\nThis last phrase seemed to touch the true point. Aside from wanting to honor their disappeared loved ones, and deriving meaning from this pursuit, the women of Calama seemed to be digging for their own lives in the desert, the parts of themselves that had been stolen, the layers of their own severed past, lost somewhere in that vast expanse. Without finding these relics, they could not grieve; without grieving, they could never feel whole. Quixotic as their plight might appear to the eye of reason, it made perfect sense to the soul. Souls seek retrieval and restitution; to the soul's eye, all of our pieces are valuable, especially the broken ones. When I share these thoughts with Paula, she appears to agree. \"They are looking for some kind of wholeness,\" my friend says. \"It's the same thing all of the women I follow are looking for.\"\n\n\"Is that why you call them invisible stories?\" I ask. Paula doesn't catch my drift. \"Because they're invisible to themselves?\"\n\n\"They're invisible to the world,\" she corrects me. \"And partly to themselves, too.\" But the women of Calama have not been engaged in some pop psychology parlor game. Their search is literal, real, and bloody, as hard to them as bones and dust. They have not been digging for metaphors. After nearly two decades of searching, roughly half the men's remains were discovered by a government team in 1990 along the very road the women followed hundreds of times. Paula could not be there for the burials, but the women have since talked to her about the interments, the bittersweet laying of skulls, bone shards, sometimes no more than an article of clothing, finally into the ground. In solidarity, these women continue to help those whose loved ones' remains have yet to be found. Together the women have found new strength and a new sort of family with one another. \"We have shared so many intimacies,\" explains Victoria Saavedra, \"the things that fill your life.\" \"The members are my family, united in the same pain,\" agrees Hilda Mu\u00f1oz. \"We still dream of finding them,\" says another. \"The search has not finished.\" Near the spot where the mass grave was found, a plaque has been laid in memory of all the disappeared. \"Without knowing where they are,\" it reads in part, \"they are with the sun as companion, in the mercy of silence.\"\n\nIn a gorgeous film called _Dance of Hope,_ you can see the women of Calama walking together into the desert on the anniversary of the Caravan of Death tragedy, throwing flowers into the air, blanketing the sand with hundreds of red carnations. They have not run away from their memories. Their commitment seems to remind us all that life may be buried under our feet (as memory is buried under the skin), and that soul is revived in uncovering it. The bones\u2014the haunted scary parts; the splinters, failures, disappeared selves; the lovers, secrets, petrified dreams\u2014are important parts of who we are. It may be hard to remember this sometimes but it's always worse to forget. Like trauma, memory haunts only when buried; uncovered it speaks through the heart to the soul, which thrives on cherishing even what's lost: the layers beneath that are no longer seen; history's sediments marbled within us; the full complex beauty of what we have loved.\n\n## **NAKEDNESS**\n\nWe must accept heartbreak to be fully human. We cannot love without tasting some blood, nor connect without braving some chink in our armor. Those who are most spiritually naked, most transparent, are also those who see most fully. \"Let the scar of the heart be seen,\" said the prophet Mohammed. \"For by their scars are known the men who are in the way of Love.\" Holy books encourage us to strip ourselves bare\u2014to allow ourselves to be burned all the way through by our passionate quest if we wish to be whole. Avoiding life's shadows makes the heart shallow.\n\nIn his play _Orpheus Descending,_ Tennessee Williams describes the opposite of such surrender through the invention of a mythic white bird whose entire life is spent in the air\u2014soaring, unsullied, shadowless\u2014touching earth only once, to die. While the desire to avoid pain may be normal, denial as a long-term strategy is a cold, narrow way to live. We may be dodging bullets through extreme self-protection yet find ourselves withering in our distant aeries, too\u2014safe yet only half engaged\u2014armoring our unbroken hearts while remaining secretly loveless, invulnerable, and dry. Such safe-seeming lives may seem charmed from the outside while being still-born inside, closed off from the messes of passion and joy, the pain that always comes with loving. And yet, as lovers and poets both know, the mystery of joy includes its pain. The poet Rainer Maria Rilke, who prided himself on raw, intense living, put it this way:\n\nIt is true that these mysteries are dreadful, and people have always drawn away from them. But where can we find anything sweet and glorious that would never wear _this_ mask, the mask of the dreadful? Whoever does not, sometime or other, give his full consent, his full and _joyous_ consent to the dreadfulness of life, can never take possession of the unutterable abundance and power of our existence; can only walk on its edge, and one day, when the judgment is given, will have been neither alive nor dead.\n\nWhile vanity rails against such humiliation and wounds-baring, our hearts thrive on the intimate dropping of masks. Ego may long for admiration, praise, and distance-keeping (as a way to protect its fragile fictions), but soul requires connection and truth. These antithetical human needs help explain the conversion experience, the turnaround in value systems experienced by post-catastrophe people. Unable to hide our humanity, we feel twice born letting down our fa\u00e7ades. Hard as we may fight against it, we're grateful in the end for this self-exposure. With less to hide, there's less to defend, freeing our naked selves to love without fear of revealing imperfection.\n\nIn the realm of public unmasking and courage, I've never met a braver or more resilient person than my late friend Lucy Grealy, whose memoir of childhood cancer, _Autobiography of a Face,_ opened the eyes of thousands of readers to a woman's struggle with facial disfigurement. Diagnosed with a Ewing's sarcoma at nine, Lucy lost half her jaw in an emergency surgery and would spend the next three decades making peace with her own appearance. For years, she spoke publicly about the cost of cancer to her self-esteem and the grueling lessons disease had taught her.\n\nAlthough Lucy died, famously, of a drug overdose in 2002\u2014a tragedy announced in a _New York_ magazine cover story the week after she was buried\u2014her inability to quit a two-year heroin habit in no way diminishes her power to instruct and inspire. Lucy had been told she would be dead by twenty only to survive twice that long, write a classic book, live a colorful (sometimes too colorful) life, teach thousands of writing students, counsel cancer victims, and serve as a great friend to many people, including myself. Her sad end does not subtract from these achievements, any more than Primo Levi's decision to take his own life detracts from his brilliance as a writer on the Holocaust. Lucy descended from a family of Irish alcoholics (she was born in Dublin and came to this country at four), attempted rehab, then tried again. She never stopped attempting to make her life better, and having a tragic flaw did not make Lucy any less of a hero.\n\nShe looked like a broken china doll, the scar running down from below her right ear and along her truncated jaw, as if a hunk of porcelain had shattered and then been removed from an otherwise lovely face. Lucy hated her face and believed it was ugly\u2014it _was_ disturbing to look at from certain angles. Yet she didn't let this stop her from doing TV talk shows, lecturing in public, or, hardest of all, looking for Mr. Right. Lucy's quest to find a man, like that of many childless women in their late thirties, was primal and relentless. A few times a month we would sit at our favorite caf\u00e9, commiserating over our twin obsessions, romance and survival. Lucy mostly found the relationship thing hopeless.\n\n\"Do you think I'm ugly?\" she would ask me constantly, coyly turning her face to its better side, smiling, winking. She would be wearing her oversized arctic wool turtleneck sweater, skintight jeans, and knee-high, leather sex-getting boots (she could be a serious minx).\n\n\"You're not ugly,\" I would say. \"But you are unbelievably vain.\"\n\nThis remark would cause her to slap me and order a drink. Lucy knew that she had certain attributes\u2014beautiful Irish blue eyes, blond hair, a peachy complexion, hot legs, a lithe gamine's body\u2014 but such information would not stick. She'd grown up being called Dog Girl in school, and was taunted mercilessly from the age of nine, when\u2014after being hit in the face with a ball\u2014she was rushed to the doctor's office and was discovered to have a virulent stage-four tumor in her mandible. In the operating room, Lucy's jaw was sliced in half; then, when she returned to the fifth grade, her education in affliction began in earnest. Physical suffering aside, how could the normal people around her not realize how lucky they were? Lucy would ask herself. In her book _As Seen on TV,_ she described her first post-cancer Halloween.\n\nI breathed in the condensing, plastic-tainted air behind the mask and thought that I was breathing in normalcy, that this joy and weightlessness were what the world was composed of, and that it was only my face that kept me from it, my face that was my own mask that kept me from knowing the joy I felt sure everyone but me lived with intimately. How could the others not know it? Not know that to be free of the fear of taunts and the burden of knowing no one would ever love you was all that a person could ever ask for?\n\nSelf-pity was never an option. Lucy's mother begged her not to cry. While this may sound cruel\u2014and it was\u2014Lucy believed that her mother's sternness had also helped her. Childhood trauma often produces high-functioning, extremely rational people with a disproportionately acute sense of responsibility toward others but a void where their own self-feeling should be. This disproportion, though ultimately problematic, can work to a survivor's benefit during times when we're barely making it through. \"I was blessed from the beginning that I never felt bitter about what happened to me,\" Lucy told me, without sounding self-righteous. \"I never asked, why me? From the beginning, it was always, why _not_ me?\" She said this in a tone like _duh._ \"It takes emotional intelligence to realize that feeling bad about yourself is just another form of narcissism,\" Lucy said. \"It's really easy to see your own suffering as another excuse to build up your ego. Poor me!\"\n\nThis refusal of self-pity in a woman who had trouble eating and drinking in public (for fear of dribbling) was extremely humbling. \"That's where the philosophical part comes in,\" Lucy said. \"In the hierarchy of suffering out there, there's a lot worse. People think how strong\u2014how _big_ \u2014you must be to get through something. But it takes a lot of courage to admit how small you are. And that your problems don't really amount to very much.\" This echoed the words of Ariel Jordan, the incest survivor. \"Not to turn that into a punishing, depressive act, but as a bridge to compassion,\" Lucy said. \"I know what it's like to suffer.\"\n\nShe described what having cancer was like in 1970, when chemotherapy was even cruder than it is today, the goal being to \"poison patients to the brink of their own death.\" Lucy, the good Catholic girl, at first tried to \"sacramentalize\" her pain, to render her own suffering holy. The ten-year-old would pray for saintly purification after these horrendous treatments. \"I'd sit on the toilet in the hospital bathroom reading two pieces of graffiti over and over,\" Lucy told me once. \"One said GOD IS NEAR. The other said BE HERE NOW. It set up an inquiry in me, and the search for meaning gives you strength.\"\n\nLucy became an expert in suffering, eventually, a black belt in the martial art of cancer. She learned that pain, too, has a hidden face. \"You begin to see other possibilities in the midst of your hardship,\" she explained to me. We were driving to Connecticut after Lucy's final pre-op hospital stay before her big surgery. She had been promised that this time her doctors would get her jaw right, make the fix she'd been waiting for, to give her face symmetry. Lucy couldn't wait for the operation but needed to recover in the country first. The pastures of Cos Cob were flying by as she talked about what she had learned. \"I experimented with different ways of thinking and feeling about things,\" she said of those first years after surgery. \"Beginning with joy. People think of joy as something rare, something to be acquired.\" She lit a cigarette. \"What I've learned is that joy is simply the absence of suffering\u2014sometimes on a physical level, sometimes the stopping of mental punishment.\"\n\n\"Isn't that an amazing thing?\"\n\n\"You have to suffer very often for a long time in order to have these experiences,\" she insisted. \"I learned that when pain subsided just a little, I'd have these moments of intense euphoria. I didn't know it was unusual to have moments of deep joy on a regular basis, because suffering always _does_ subside for moments,\" Lucy said. \"That's why torture is so effective. Torturers don't give prisoners a moment to have an epiphany about it.\"\n\nLucy looked at me. \"I've had these moments of intense euphoria throughout my life, but it was the joy that came about simply because I felt a little bit better,\" she said. \"Not through the acquisition of anything outside of me. _The joy was already there._ I just needed the pain to let up a little, and there it was. It's a really profound, miraculous thing.\"\n\nLucy's visible defect used to draw people toward her, especially those who knew her story. They adored her by the auditoriumful, magnetized by her naked scar. The thing that she wanted to change was the very wound others found most inspiring, the vulnerability that drew them out and taught them to be more fearless about their own shortcomings.\n\n\"Openness, honesty, vulnerability, nakedness are the key to everything,\" Lucy once said to me. \"You've got to be honest with yourself, to face the depth of your own distress. Otherwise it turns to depression,\" which Lucy described as \"avoidance of larger pain.\"\n\n\"Neurosis is always a substitute for legitimate suffering,\" Jung once said. Lucy could not have agreed more. \"People avoid being burned through by things,\" she insisted. \"Instead, they create these holding cells for their feelings. To keep volatile, extreme feelings at bay. That's how depression makes you helpless. It shuts you down. But when people actually let themselves feel these things, the emotions burn hotter and pass more quickly.\"\n\n\"Would that it were so easy,\" I said.\n\n\"Easy?\" Lucy asked. \"I'm talking about real life.\"\n\n\"Oh, that.\"\n\n\"You saw it after nine-eleven. People immediately wanted to alleviate suffering by creating suffering for somebody else,\" Lucy told me. \"Nobody knew how to feel ambivalence about _not_ knowing what to do. Retaliation, acting out, was more comfortable than being in anguish.\"\n\nIndeed, Lucy's own ability to bear anguish\u2014and emerge bright-eyed\u2014was preternatural to behold. She could land on her feet with the ease of a cat hurled from a top-floor window. Even when things went extra-badly, she comported herself with stoic grace. When the long-promised surgery failed to improve Lucy's face (her jaw had been over-irradiated), she rallied with gusto even then. Her face swollen beyond recognition, a pneumatic tube of transplanted flesh hanging around her neck like a tire, Lucy was bedridden and in extreme pain, yet still managed to retain her unsinkable black humor. I climbed into the bed next to her and she held my hand, with her friend Ann Patchett watching us from a chair. Lucy tried to turn her awful appearance into a joke.\n\n\"Helium by Maybelline,\" she gurgled, turning her bubble face back and forth. She let us know how extremely fortunate she truly was to have such loyal friends, since now she would be alone forever for sure.\n\n\"Am I not a lucky girl?\" Lucy blubbered, fingering her alien neck, unable to blink her black-and-blue slit eyes.\n\nAnn gave me a pained look.\n\n\"I mean it,\" she said. \"I love you guys.\"\n\n\"Oh, shut up,\" said Ann, who knew her humor.\n\n\"Bitch,\" said Lucy.\n\n\"Slut,\" said Ann. Then both of them dissolved into giggles.\n\nThree months later, on the night that she died, Lucy left a message on my answering machine, to thank me for being such a good friend and tell me how much she loved me. I didn't know that it was the last time I would hear her voice. She had recovered from the surgery, gone back to dating, persevered on her long-awaited novel\u2014even considered going to medical school. Only days before that good-bye message, we'd been laughing over coffee at our favorite caf\u00e9. Lucy was reminding me again about how crazy romance can be, how obsessive insecure men are when it comes to strong women, and why, next to dental work, dating is the cruelest and most horrible thing civilized people actually pay for. Lucy seemed okay to me that afternoon, better than she'd been in a while. Her eyes were especially clear and blue. Did she know what was coming? I ask myself now. Or was her death an accident? I will never know the answer, of course; nor will I ever stop thinking about it. This splinter of love will not go away. As I pass the table where we used to sit, there are ghosts in the corner of my eye. I can almost see Lucy sitting there still, asking me if I believe that she will ever find a man. I'm telling her again that yes, she will, if she finally stops settling for Peter Pans. I'd be watching her face as she laughed and moaned, then ordered herself another drink. Lucy would be leaning forward and purring, getting ready\u2014as always\u2014to talk about love. Her determination\u2014again\u2014would amaze me.\n\n## **KILLING PETER PAN**\n\nPhilosophers from Plato to Kierkegaard to Gurdjieff have taught that the path of the soul involves a progressive series of disillusionments. The mirror must crack if we want to evolve. The earthbound lessons of aging, disenchantment, and passing beauty must crack the mask for the soul to breathe. Lucy had learned this lesson in spite of herself, although she still wanted a prettier face. Still, vanity resists the path that leads to this more enduring beauty. No one wants to be discarded by an appearance-crazed, youth-obsessed world.\n\nIn a culture that celebrates narcissism, the soul can become a fugitive. \"The great malady of the 20th century, implicated in all our troubles and affecting us individually and socially, is 'loss of soul,'\" religious writer Thomas Moore wrote. Like water invisible to fish, narcissism is so ubiquitous that we hardly see we're swimming in it. We're inducted into this illusory self-understanding before we are even aware of it. Once, a scholar explained to me that narcissism is born the first time a child looks into a mirror and realizes that this \"thing\" is what others take to be itself. This knowledge exiles us from the garden; holistic innocence comes to an end. We see ourselves as \"things\" in a world of other \"things,\" rather than souls looking through these masks. This trance of mistaken identity causes us enormous trouble, psychologists tell us, beginning with alienation from others and excessive attachment to this \"reflection.\" As we age, this obsession only gets worse, our fears of physical loss and change of appearance masking\u2014\u00e0 la Peter Pan\u2014a self-destructive denial of death.\n\nAt eighty-four, Jungian pioneer James Hillman has spent decades singing the praises of soul and warning against excessive narcissism. \"Remember what Anna Magnani said when they tried to make her look younger in _The Rose Tattoo_?\" Hillman asks me over coffee in a Tribeca diner, referring to the earthy Italian film star. \" 'Don't take out a single line. I paid for every single one!'\" The godfather of archetypal psychology in this country, Hillman has argued in dozens of books, including _The Soul's Code,_ that narcissistic culture has it all wrong, exhorting the self-improvement-obsessed to appreciate their so-called weaknesses, embrace their chronic weirdnesses, rough spots, absences of virtue, physical peculiarities\u2014the places where they fall short and never measure up\u2014as integral to their multiplex beings and the texture that gives them character.\n\nHillman looks much younger than he is, in fact, tanned as toast and surprisingly buff, wearing Levi's and aviator glasses. \"When we see an old wall, an old teacup, an old tree, we appreciate these things precisely for their oldness,\" he tells me, \"the increased beauty of their years and the memories they contain. Objects seem to gain in value as they age, but we deny this same respect to people.\"\n\nHe's right. \"Yet the oddities and flaws that make you different only increase with time and become the most interesting part of who you are,\" Hillman insists. \"These changes form our character\"\u2014a word derived from the Greek for \"etched, cut, or engraved,\" he reminds me.\n\n\"Who wants to be engraved?\" I ask, half-seriously. \"We want to cheat time and look ageless.\"\n\n\"But there is a great deal of human life that cannot be measured by time,\" Hillman says. \"The deepening and refining of personal vision, how we appreciate the world, our increased sense of beauty.\" Longevity for its own sake doesn't interest him much; it's what we do with our years that determines their worth. \"Longevity for its own sake is rarely concerned with what came before you or what will come after,\" Hillman says.\n\n\"Old people can be quite selfish,\" I agree.\n\n\"Obsession with staying alive can lead to a kind of geriatric heroism,\" he says, referring to our fixation on what's in the mirror. \"Beating the odds, lowering our cholesterol levels, outliving statistical tables. But life moves in _three_ directions.\"\n\n\"Three?\" I ask.\n\n\"Backward, forward, and outward,\" Hillman says.\n\n\"Sorry, but I don't follow.\"\n\n\"There is the longevity of extending life backward through memory,\" he explains, reminding me of the women of Calama \"extending\" life by digging for their loved ones in the desert. \"There is the longevity of concerning yourself with the coming generations,\" Hillman goes on. \"Not only your grandchildren and great-grandchildren, but down to the seventh generation, as the Bible says. This extends your life into the future as you reach forward. Finally, there is the longevity that comes from extending your life _outward_ by becoming responsible to the place where you live and the society around you.\"\n\nOur lives, in other words, are never merely personal affairs, despite what the mirror may tell us. Our culture's tendency toward planned obsolescence\u2014cultivating the myth of perpetual youth\u2014 distorts the wisdom we might glean from soul. \"Each stage of life has its own meaning,\" he explains. \"What's unique about surviving into old age has nothing to do with being young. In Rome a man put on the _toga senilis,_ or 'toga of old age,' at age sixty. But our culture is so concerned with quickness and speed, people don't have the patience for slowness.\"\n\n\"We equate slowness with being washed up,\" I remind him.\n\n\"But slowness is another kind of adventure,\" he insists. \"When you're old, you don't even know if you're going to get out of the bathtub! But there is an adventure in that. If you think about it, there's the same challenge and achievement as climbing mountains. You hold on to something, you watch your foot as it moves, the whole adventure is right there in miniature. This is the adventure of slowness.\"\n\nIt is also, Hillman has learned, the adventure of love. As character deepens, so does an eros more nourishing and durable than lust, he believes. Embers are better for cooking than flame, as the saying goes. \"A certain love for the world deepens the recognition of its beauty,\" he says. \"It's amazing how some people who had miserable lives are grateful in old age just to have been here, to have gone through it all.\" I've often noticed this myself, I tell him. \"Relationships can become richer, provided people aren't trying to recapture the experiences of their youth,\" he continues. \"There's more acceptance in love between old people, more respect for the other person and their foibles.\"\n\n\"The mirror is no longer the last word?\"\n\n\"We learn to appreciate our partner's oddities,\" Hillman stresses, \"and to realize what a miracle it is that we're here, still together.\" He looks out the window at traffic on Hudson Street. \"We take more interest in one another. Not what medications we're taking, or what our pulse rates might be, but what we're reading and dreaming about, what memories are returning to us, what peculiar reflection just turned in our minds, something we haven't thought of for years. We become more interested in each other's souls. That is the truly interesting part.\"\n\n\"I thought only superficial people don't believe in appearances,\" I kid him, citing a joke from Oscar Wilde.\n\n\" 'In the final analysis, we count for something only because of the essential we embody,'\" James Hillman replies, throwing a line from Jung back at me. \" 'If we do not embody that, life is wasted.'\" He chuckles and sits back in his chair. \"Try wiggling out of that one.\"\n\n## **THE WATER OR THE WAVE?**\n\nTime is a cruel bitch goddess. We measure our lives in coffee spoons and wonder why we feel so crushed by the ticktock of minutes piling down on us through the burying years. We chop existence into nanoseconds and struggle to fill up every one, fearful of wasting a moment to leisure, hoarding time as if it really were money. Rushing through the measured world, we forget that ours is the first civilization in history to be so obsessed and bedeviled by time. The unhelpful image of God as a celestial clock keeper, stopwatching our every move from above, has shown (in our attention-deficient times) no sign whatsoever of blinking.\n\nYet philosophers have insisted for centuries that there are in fact two kinds of time operating in the world\u2014man-made time and soul time\u2014and that it is vitally important that we remember both if we wish to remain sane. _Nunc fluens_ is their name for hourglass time, the relentless metronome that frays your nerves, grays your hair, forms rings inside the trunk of a tree. _Nunc stans,_ on the other hand, denotes time as seen through eternal eyes, the timeless sense we may feel in nature, while reading a great book, creating artwork, having sex, or praying\u2014peak-and-valley moments\u2014 when _nunc fluens_ seems to stop, dropping us through the mental scrim that separates our everyday minds from great silence. Spaced-out as this sounds to the stressed-out psyche, we're actually more awake than usual during _nunc stans_ moments, able to see more clearly, to operate more effectively, than we can while hugging a clock. Contrary to the puritan belief that time-gripping keeps us on the ball (and keeps our great assembly line ticking along), the opposite appears to be true. _Nunc fluens_ without _nunc stans_ behind it creates an assembly line to madness.\n\nA mountain climber who'd been part of the first team to scale Mount Everest described how the discovery of soul time changed his life. Returning from the peak, the hiker paused on a high pass to admire the stupendous view. As he turned around, he saw a small blue flower in the snow. \"I don't know how to describe what happened,\" he reported later. \"Everything opened up and flowed together and made some strange kind of sense, and I was at complete peace. I have no idea how long I stood there. It could have been minutes or hours. Time melted. But when I came down, my life was different.\"\n\nThese blue flower moments are happening all the time, but few of us pause to pay attention. The workaday mind needs a slap\u2014or a climax\u2014to stop it in its habitual tracks. When this happens, we may find ourselves dropped into this timeless dimension. Problems suddenly seem to shrink down to size when seen against this spacious backdrop. Artists, seekers, lovers, adventurers, people like R.D. who like to trip, all recognize the freedom of such moments when the mind quiets down and affords us a taste of that gorgeous expanse.\n\nStill, we tend to fear _nunc stans_ and its freedoms. We cling to our wristwatches for dear life. Until we're torn somehow from our habitual grid, we may even doubt that eternal time exists. After we glimpse it, though, we begin to see our lives differently\u2014to perceive the world stereoscopically, like the double-faced Roman god Janus on the threshold between two realms, facing both directions at once, the physical and the invisible worlds. Reality seems to turn almost translucent sometimes, as if eternity itself were peeking through this time-stopped thinning of veils.\n\nIn the Andaman Sea of Indonesia, there lives a group of nomadic tribesmen whose lives are virtually untouched by time as we know it. Spending up to eight months out of every year traveling from island to island, living on their primitive boats, the Moken are among the peoples of the world least touched by modern civilization. They are born on the sea, live on the sea, and die on the sea. They understand the moods of the ocean better than any marine biologist. Constantly moving from island to island, they learn to swim before they're walking and are more or less amphibious. The Moken can see twice as clearly as the rest of us and are able to lower their heart rates automatically to stay underwater twice as long. They catch sea cucumbers and eels at low tide, dive for shellfish when the water is high, and enjoy as close to an Edenic existence as seems possible in this time-obsessed world.\n\n_Nunc fluens_ does not exist for them. If you ask a Moken how old he is, he may have trouble telling you. Conspicuously absent from their language are terms for \"when,\" \"want,\" and \"worry.\" If you show up at a Moken village after a twenty-year absence, locals will greet you as if it were yesterday. They have no words for \"hello\" or \"good-bye.\" _Nunc stans_ is their everyday state. This allows them to exist in a unity with their environment that most of us would find hard to imagine. No reaching backward or forward in time\u2014no longing, projecting, hurrying, or reminiscing\u2014upsets their daily round.\n\nIt is not surprising to learn, for this reason, that the Moken were the only Indonesian group to suffer not a single casualty in the 2003 tsunami that claimed three hundred thousand lives. This was thanks to a sixty-seven-year-old fisherman named Satha Kathaleway, who was mending his nets on the beach when the first signs of trouble appeared. The cicadas, whose loin-rubbing causes a ruckus here nine months a year, suddenly fell silent. In all of his years, Kathaleway, who's gray-haired and nimble, had never heard anything like it before. The dolphins made for deeper water as the tide receded abnormally far. Livestock stampeded for higher ground. Kathaleway warned his fellow Moken that the _lumbi,_ the great man-eating wave described in a popular myth, was about to descend on them. His fellow tribesmen followed the old man up the mountain, and their lives were spared, all because this \"primitive\" fisherman was watching the world instead of a clock.\n\nOne crisp fall afternoon, I make my way to Rhinebeck, New York, to have a conversation with Eckhart Tolle, the German spiritual teacher whose book _The Power of Now_ introduced the concept of _nunc stans_ to millions of readers trapped in their time-obsessed heads. According to Tolle,\n\nthe mind, to ensure that it remains in control, seeks continuously to cover up the present moment with past and future... as the vitality and infinite creative potential of Being, which is inseparable from the Now, becomes covered up by time, your true nature becomes obscured by the mind.\n\nI loved that. \"The accumulation of time in the collective and individual human mind,\" he concluded, \"also holds a vast amount of residual pain from the past.\"\n\nI find Eckhart waiting for me on his bungalow porch, an elfin man with a gnomelike beard reminiscent of a character out of _The Hobbit._ He's wearing an Alpine-style vest and a Nehru shirt, his voice a sort of amused hush. When Eckhart offers me his hand, it's as light as a feather. He pours us tea and we take our seats in the dimly lit cabin he's calling home during the weeklong silent retreat that he's leading.\n\n\"Human history has largely been a history of insanity,\" Eckhart begins cheerfully. \"The manifestation of a sort of collective mental illness.\" His steady gaze makes me realize how frazzled I'm feeling after my two-hour drive north. Why is this? I wonder. \"Everyone has the roots of this illness to a greater or lesser degree,\" Eckhart assures me. \"But in order to understand this, we need to look for a moment at how the human mind works.\"\n\nHe sips his tea and begins to explain. \"The mind finds labels and concepts to describe and interpret things and people,\" Eckhart tells me. \"Clouded by labels and concepts, the mind becomes unconscious. Jesus's last words are borne out in the truest sense\u2014 'they know not what they do.'\"\n\n\"Okay,\" I say. But what does this have to do with time?\n\n\"The mind is essentially a survival machine,\" he goes on, approaching my question from another direction. \"In the most primitive sense.\" \"Fighting, defending, storing information, analyzing\u2014this is what the conventional mind is good at,\" Eckhart has written, \"but it is not at all creative. All true artists create from a place of no-mind, from inner stillness, whether or not they're aware of it.\"\n\nBlue flower moments, I think. \"Even the greatest scientists admit that their creative breakthroughs come at times of mental quietude,\" Eckhart says. \"When we quiet the mind, we can be present. Right now. In this very moment.\" He gazes out the window at a squirrel nibbling leaves on the lawn. \"When we can yield, accept, be open to our lives, a new dimension of consciousness opens up,\" Eckhart says. \"If action is possible or necessary, your action will be in alignment with the whole and supported by creative intelligence. Circumstances and people then become helpful, cooperative. Coincidences happen. If no action is possible, you rest in the peace and inner stillness that come with surrender. You rest in God.\"\n\n\"Otherwise?\" I ask.\n\n\"You get resistance, an inner contraction, a hardening of the shell,\" he tells me. \"You close yourself off from reality, and whatever action you take will only cause more resistance.\" I've noticed this often in my own life. \"If the shutters are closed, the sunlight cannot come in. The universe will not be on your side; life will not be helpful,\" says Eckhart.\n\nWhile _nunc fluens_ is necessary for organizing our daily lives\u2014 your wristwatch gets you to the dentist's office on time\u2014life is more than a dentist's appointment. Without tapping into _nunc stans,_ Eckhart teaches, we divorce ourselves from the big picture. \"It wasn't through the mind, through thinking, that the miracle that is life on earth or your body were created and are being sustained,\" he has written.\n\nThere is clearly an intelligence at work [inside us] that is far greater than the mind. How can a single human cell measuring 1\/1,000 of an inch in diameter contain instructions within its DNA that would fill 1,000 books of 600 pages each?... When the mind reconnects with that, it becomes a most wonderful tool. It then serves something greater than itself.\n\n\"Yes, we need the mind to function,\" he says. \"But there comes a point where it takes over our lives, and this is where dysfunction, pain, and sorrow set in.\" When we balance the tyranny of _nunc fluens_ with the spaciousness of soul time, we free ourselves from past and future and realize how much unhappiness we create with our constant mental time-jumps backward and forward, often missing what's in front of our eyes. This habit is comparable to driving a car while staring into the rearview mirror (and reading traffic signs _way_ up the road at the same time). What we get are accidents and the smell of burning rubber, not realizing that the emergency brake is still on. Such time distortion is even more hazardous during times of crisis, Eckhart tells me. \"Whenever loss occurs, you either resist or yield,\" he says. \"Some people become bitter or deeply resentful. Others become compassionate, wise, and loving. The intensity of the pain depends on the degree of resistance to the present moment,\" he explains. \"What could be more futile, more insane, than to fight something that already is? It means you're opposing life itself, which is now and always now. Say yes to life, and see how things start working for you rather than against you.\"\n\nWhen the bell rings to call retreatants into the meditation hall, I realize that I haven't looked at my wristwatch for over half an hour.\n\n\"When we are caught in the timekeeping mind,\" Eckhart says, \"we think our lives rather than living them. We have relationships with our ideas of people rather than with the people themselves.\"\n\nIs he saying that our approach to time affects how we think about other people?\n\n\"Of course. The moment you put a mental label on another human being, you can no longer truly relate to that person.\"\n\n\"I understand that\u2014\"\n\n\"The more mental labels you have for other people, or groups of people different from the group you identify with, the more you deaden yourself to the aliveness and the reality of those people,\" he tells me. \"It then becomes possible to perpetrate any act of violence.\"\n\nIn other words, the mind's blinding concepts\u2014aimed at time, other people, or ourselves\u2014prevent us from seeing clearly. I am writing these words on August 15, 2007, but what does this numerical series have to do with the humid New York afternoon, the actual sunlight warming my window, the rain clearing out of the late summer sky? Is it August in the eye of the soul? we might ask. Or is it just today?\n\nEckhart excuses himself to rest before his evening's talk. Driving home, I listen to one of his hypnotic tapes as I navigate my way down Route 9. Near Poughkeepsie, I'm startled by a car's horn blaring directly behind me. I've been moseying along at thirty miles an hour in a fifty-five-mile-per-hour zone, I realize. Perhaps this is a sign of progress? Eckhart's soft voice on the tape continues.\n\nImagine the earth devoid of human life, inhabited only by plants and animals. Would it still have a past and a future? Could we still speak of time in any meaningful way? What time is it? The oak tree or the eagle would be bemused by such a question. \"What time?\" they would ask. \"Well, of course it's now. The time is now. What else is there?\"\n\n## **WHAT TIME IS GOOD FOR**\n\nWhile our obsession with _nunc fluens_ can become a straitjacket, history (time's overcoat) can be useful for cinching in our woes and warming the soul in harsh weather. The knowledge that others have lived through worse troubles than ours can provide comfort and courage in times of duress. \"I cried because I had no shoes till I saw the boy who had no feet,\" my own mother used to say when we whined about our welfare-grade life. Though I'd never actually met a footless person, the suggestion helped me anyway. Stop complaining, my mother would tell us, you've lived through nothing, we've got a free country, your grandma stuffed kishke to get out of Poland.\n\nAfter 9\/11, Americans were invited to use history in a similar way: to feel less sorry for ourselves. One risked sounding hard-hearted at first, reminding fellow citizens that genocide did not actually begin on that September morning. Still, it is helpful to remember that tragic events carry their own centripetal force, sucking attention toward themselves and away from other suffering beings. Self-privileging is a terrible trap, especially when we're trying to heal. As incest survivor Ariel Jordan put it, remaining the center of our own cruel universe is a sure way of remaining stuck. Connection of any kind will free us. In times of historic-scale tragedy, only history is forceful enough to pry the victim mentality loose and release the pretense of specialness.\n\n\"History is the great leveler,\" says historian Doris Kearns Goodwin in the aftermath of the 9\/11 attacks. \"History helps us to survive,\" she tells me. \"If _they_ did it, we say, we can, too.\"\n\n\"We forget that worse things have happened\u2014\"\n\n\"Listen,\" Doris interrupts me. \"We tend to think of past wars from the victory backward,\" she says. \"World War Two came out well, but in the beginning there was an enormous amount of uncertainty about whether or not we could match Germany's weaponry. In 1940 Germany had the most powerful army ever amassed on the face of the earth, and our armed forces were rated eighteenth in the world. By the time Pearl Harbor came, we were much better prepared, but there were many months of losses before that situation began to turn around.\"\n\n\"Necessity forced invention,\" I say.\n\n\"As it always does. Americans are still reeling after nine-eleven,\" Doris says. \"But it might help if we remembered the London Blitz. There were fifty-seven nights of continuous bombing. Londoners had to live through massive uncertainty. Twenty-three thousand people were killed. But just as Hitler failed to break the will of the British people, citizens in this country have proved that their will has not been broken either.\"\n\n\"On the contrary,\" I say.\n\n\"Indeed. Churchill insisted that theaters in the West End remain open, remember?\" Doris reminds me. \"When there were air raid sirens, people put on gas masks and sang songs until the 'all clear' was sounded. Tube stations were turned into underground shelters with libraries. Stores opened with signs on their shattered windows that read, more open than usual. Life went on!\" she insists. \"This is when Churchill said, 'If the British Empire and its Commonwealth last for a thousand years, men will still say this was their finest hour.' Knowing that people had the ability to sustain themselves through this barrage, and somehow endured it, should give us hope that we can do it as well.\"\n\n\"We had our illusion of safety destroyed.\"\n\n\"That's true. But you see how we've risen to the occasion?\" Doris asks. \"When people band together to defy the enemy, what you see is that fighting spirit. Not allowing the terrorists to win by breaking us. It's not that fear is taken away, but this passion not to be broken becomes greater than fear. Being part of a group gives you courage that you might not have individually. That's why the massive show of flags and feeling of being a nation have been so important during this time.\"\n\nI admit that the flag-waving scares me a little.\n\n\"Most of the time, nation is an abstraction,\" she explains. \"But when a crisis happens, you remember in your heart what it means to be someone living in this country. Being part of something larger than yourself gives you an extra sense of strength. During World War Two almost everyone was involved in doing something to help: planting victory gardens, sending in their dogs' rubber toys, collecting aluminum. That's why people remember the war with such a positive feeling. Surviving nine-eleven has been a major shift for Americans,\" she acknowledges. \"And not at all part of the ethos of recent years. In the decades since the war in Vietnam, we've been more private and individual-oriented. This disaster is awakening that other side of people, infusing the leaders of the future with a deeper public consciousness.\"\n\n\"I hope you're right,\" I say.\n\n\"One feels enlarged during times such as these,\" Doris tells me.\n\nFrom the patio of her home in Marin County, Chilean novelist Isabel Allende agrees that history is the best antidote to both victimhood and hubris.\n\n\"We have been privileged in this country to believe that life is secure, that our rights\u2014including the search for happiness\u2014are sacred,\" Isabel tells me over the phone. \"But most of the world has lived with uncertainty for millennia. I lived in Chile, a country that had one of the oldest democracies in Latin America,\" she says. \"We never thought that anything like a military coup could happen to us. Those only happened in banana republics! Until one day it did happen, and the brutality lasted for seventeen years.\"\n\nPinochet's coup had taken place on September 11, 1973, coincidentally. \"This was a military coup orchestrated by the CIA,\" Isabel says, sounding angry. \"It was a terrorist attack against a democracy!\" I'm reminded of the Caravan of Death arriving in their Puma helicopter. \"The extraordinary thing is that in twenty-four hours you learn to adapt.\"\n\n\"How?\" I ask.\n\n\"You go on with your life, because life goes on,\" says Isabel. \"You see this in anyone who has survived a traumatic situation. My own daughter died, for example.\" Her only daughter, Paula Frias, died of porphyria in 1992 at the age of twenty-seven. \"At first you think you can't live with this,\" says the author, who just turned sixty-five. \"It's just too much. Then life begins to take over. One morning you wake up and you want to eat chocolate. Or walk in the woods. Or open a bottle of wine. You get back up on your feet.\"\n\n\"When you can, right?\"\n\n\"You have no choice!\" Isabel insists. \"You cannot let the bullies keep you on the floor! I have been on my knees a thousand times, and I always get up. This is the message we must give to our children. You must get up off the floor! Sooner or later everybody suffers. Grief and darkness are a part of life.\"\n\n\"That's what nine-eleven has taught us.\"\n\n\"Americans,\" she says, chuckling. \"When I moved to this country twenty years ago, I fell in love with a lawyer, the kind who sues the city when you fall on a banana peel. I couldn't believe it! Accidents are accidents. If you slip and fall, it's your own fault.\"\n\n\"Tell that to people who are suing McDonald's for making them fat.\"\n\n\"There is no insurance for happiness or safety! That's impossible,\" Isabel exclaims. \"Life is difficult, painful, and wonderful. But we are a society that expects to be happy and entertained all the time, a spoiled society that hasn't had war in its territory in more than a century, though we contribute to war in other countries all the time. We support many of the worst dictatorships in the world. It is we who helped create the Taliban.\"\n\n\"That may be true. But how can we live differently now?\" I ask.\n\n\"By becoming citizens of the world. Americans cannot have a gated-community mentality anymore, or believe that we can keep ourselves safe while there's so much inequality and poverty in the world. More than eight hundred million people on this planet are hungry. The distribution of wealth is completely unfair. This creates conditions for hatred and violence. How can this continue forever without paying the consequences?\"\n\nIsabel steps off her soapbox and gets to the point. \"I learned something in Chile at its time of terror. We tend to focus on the negative because that's what makes the news. But for every terrorist and torturer, every person who commits a crime, there are a thousand people who are willing to risk their lives to help and do good,\" she says. \"We forget this. But if it weren't true, we would still be in the Stone Age. Why has humanity evolved? Because there are more good guys than bad guys, even though the bad guys make more noise.\"\n\nThat is why post-catastrophic times like these are rich in potential for metanoia. \"We can come together,\" Isabel agrees. \"Make changes. Begin to reflect. We now have a chance to grow up. Make peace. Become aware. Renew our spirits.\"\n\n\"Otherwise we're on the floor?\"\n\n\"That's right,\" answers Isabel Allende. \"And life on the floor is terrible.\"\n\nFor the past fifty years, Holocaust survivor Elie Wiesel has been exhorting us to remember the past in order not to repeat it. The Nobel Peace Prize winner warns that forgetfulness is our greatest public danger, since tyrants prey on ignorance. While memory is sometimes a heavy burden, it is the price we pay for wisdom.\n\n\"This is the choice we must make as humans,\" Wiesel tells me from his office at Boston University. \"The quest for knowledge is what makes humans survive, even if it hurts.\" I have trouble imagining that this \u00e9minence grise was once a sixteen-year-old Hungarian boy in a death camp. \"There's a troublesome verse from Ecclesiastes about this,\" he tells me. \"It says that the more we know, the more pain we have. But because we are human beings, this must be. Otherwise we become objects rather than subjects.\"\n\nHe pauses for a moment to let this sink in. \"Of course, it hurts when we see pictures of people throwing themselves out of windows, children who are orphaned, the widows,\" Wiesel says. \"But there is no way out of what we've seen.\"\n\n\"And how do we live with what we know?\" I ask \"How can we live with _not_ knowing?\" He says this as if it's the only true question. \"We have more responsibility now.\"\n\n\"More fear, too,\" I say.\n\n\"But fear is a natural ingredient in life. If a child is afraid of fire, that's good. Fear is only unhealthy when it becomes excessive. Courage has an important role to play in how we cope with what we know. Courage,\" Wiesel maintains, \"means doing the impossible within the possible.\"\n\nAt first, I have trouble unscrambling this. \"We have learned so much from nine-eleven about what makes human beings noble,\" he says. \"When people are in need, you must be present. When people suffer, you must let them know you're suffering with them.\"\n\n\"The good side of bad acts?\" I say.\n\n\"I would not say that from horror comes goodness. That would be giving horror too much credit. But goodness prevails in spite of horror.\"\n\n\"That's amazing in itself,\" I say.\n\n\"And finally we must have hope,\" he tells me.\n\n\"Did you have hope when you were in Auschwitz?\" I ask.\n\n\"Even when there is no hope, as Albert Camus said, we must invent it.\"\n\n## **AT SEA**\n\nIn the period between January 1994 and April 1996, I experienced depression for the first time. Though I was outwardly healthy, my T cells were gone, I was running on empty, treatments were barely on the horizon, and one day, during a monthly checkup, I was told by my then-physician to begin prophylactic treatment at once or invite an opportunistic infection. I knew what such infections looked like; I'd nursed my share of dying friends. So I agreed with the doctor's futile, expensive recommendations, none of which had been proved to work, and changed overnight from a person who resisted swallowing aspirin to taking fifteen toxic pills a day and hanging out twice a week in this doctor's death salon with vitamin C dripping into my arm through an IV, surrounded by dying people wrapped in blankets. The time I spent in that ghostly place is the nadir of my life to date.\n\nEvery day became an ordeal. Minor efforts required Herculean strength; a ring of sullen hopelessness seemed to orbit every moment, mocking every prospect for improvement with a devil's sneer. Finally, I felt paralyzed and had trouble even leaving my tiny apartment. This reminded me of how a plane crash survivor described the last moments before impact. When the plane first began to malfunction, passengers were hysterical, sobbing, uncontrollable. Then they began to quiet down, prayed, clasped hands with strangers, and wept as the aircraft spiraled toward the ground. Finally, the survivor reported, even these movements stopped\u2014 passengers sat glued to their seats, too frozen and numb to even cry out. This is exactly how I was feeling, as if strapped inside a hurtling vehicle, frozen, asphyxiated, and plunging.\n\nFor the first time in my whole life, I was unable to function. I sought the help of a psychiatrist, who prescribed a drug called Paxil, whose only visible effect was to make it impossible for me to cry. I tossed the stuff in the trash. While I was at it, I fired Dr. G. and ditched those purgatorial treatments. Somehow, refusing to be deadened by narcotics (and my physician's hopelessness) lifted my spirits a tiny bit. Shafts of hope, fragile but real, began to dispel the depressive cloud. This clearing left me motivated enough to seek out the advice of a new physician and create a semblance of normal life until better treatments did arrive.\n\nOnce you've traveled this road, though, you never forget it. Never again is depression just a word to you. It's a visceral absence, a soul's negation. As author Andrew Solomon put it in his atlas of depression, _The Noonday Demon,_ it is \"the flaw in love.\"\n\nAndrew's own descent into purgatory began insidiously and with no tangible cause. The scion of a pharmaceutical fortune (his father got rich from manufacturing the antidepressant drug Celexa, coincidentally), Andrew was a beloved child of privilege, educated at Yale and Cambridge, embarking on what promised to be a formidable career as a writer, surrounded by friends, in peak physical shape\u2014circumstances, he assures me when we meet, that \"in no way entitled me to misery.\" A deceptively cheerful, sweet-natured guy in an oxford shirt and herringbone blazer, Andrew is almost anachronistically cordial, the sort of \"swell\" guy you'd expect to meet at a garden party written by Edith Wharton, with a rising hairline and blue eyes that do appear, as one reporter described them, to be gazing perpetually out to sea.\n\n\"It began quite slowly,\" Andrew tells me at a caf\u00e9 in our mutual neighborhood. \"It was around the time that my mother died.\" A few weeks after her sixty-fourth birthday, Carolyn Solomon, who was losing her battle with uterine cancer, warmed herself some tea and muffins, called her prewarned husband and two sons to her bedside, swallowed forty Seconol capsules, and died before their eyes. His mother's last words to her younger son were \"Enjoy what you have\"\u2014advice that became Andrew's recrimination after his depression took hold. Having published his first novel, an intensely emotional story about a mother and son ominously titled _Stone Boat,_ the first-time author was about to set out on a reading tour when his melancholy turned clinical.\n\n\"For someone who hasn't been through this, it's almost impossible to imagine,\" he tells me. \"I was excruciatingly afraid all the time. Tied up in knots, just petrified. As if I were about to explode.\"\n\n\"What were you afraid of?\" I ask.\n\n\"You know the moment before your hands shoot out to break a fall if you trip or something?\" says Andrew. \"When you feel the ground rushing up at you but you can't stop yourself? That passing fraction-of-a-second terror?\"\n\nI know this feeling well, I say.\n\n\"I felt that hour after hour after hour,\" he tells me. \"Just being awake\u2014being alive\u2014was acutely painful. I had no idea how to deal with it. People would say, 'Oh, it's just a phase. You'll get through.' I would think, No, I can't get through another fifteen minutes of this, much less another day.\"\n\nHis therapist was no help at all. \"I was in analysis with an incompetent who failed to recognize what was happening,\" he says. The memory still makes him angry. \"He said it was very brave and heroic of me not to take drugs. The truth is that if I had gone on medication six months earlier, I would never have gotten to the point I did.\"\n\nAndrew reached his low point during the summer of 1994, when he was no longer able to leave his bedroom. \"One day I couldn't get up,\" he tells me. \"I was actually too frightened to get out of bed. I lay there for hours and hours after I woke up, wondering how I could put on a pair of socks. I stared at the phone for seven hours till finally it rang. I said to the person who called, 'I'm really in a bad way. I have to get some kind of help.'\"\n\nHe compares this smothering darkness to a vine choking a tree, a crushing, parasitic force that will not stop till its host is dead. \"Compacted and fetal,\" as he writes in _The Noonday Demon,_ he found himself \"depleted by this thing that was crushing me without holding me. Its tendrils threatened to pulverize my mind and my courage and my stomach, and crack my bones and desiccate my body. It went on glutting itself on me when there seemed nothing left to feed it.\"\n\nFortunately, there were loved ones around to help him fight off this killer. Nurturing friends circled around him; his mogul father, not one for touchy-feely affection, insisted on cutting his grown son's food. Slowly, Andrew began to improve. He dumped his useless analyst and found his way to a psychopharmacologist who was finally able to stabilize his moods with a cocktail of medications. He vividly recalls the moment in 2001 when his life force revived for a moment. \"I was with my father, looking out of a window,\" he says with a smile. \"Suddenly I saw a patch of gray sky pull apart, and the sun came blazing through before the clouds came together again. I know it's the most clich\u00e9 image, but it was incredibly apt for me at the time. I felt okay again for maybe five minutes, which gave me the exultant sense that I might actually feel that way again.\"\n\nAn agnostic, nonpracticing Jew, Andrew is careful not to gloss his recovery in supernatural terms, but does admit to a certain spiritual boon to the terror he lived through. \"It's the thing with feathers at the bottom of my box of miseries,\" he says, borrowing a phrase from Emily Dickinson.\n\n\"Major depression is a birth and a death,\" he says. \"Before this happened, I had this idea of myself as a very strong person. In a concentration camp, I always figured I'd be one of those prisoners who managed to sing while they worked.\" This heroic self-image now makes Andrew snicker. \"The idea that I might actually be one of those who fell to pieces, faded and died very quickly, involved a complete readjustment of who I thought I was. The idea of being captain of my own ship, the grand master of my own life, has completely gone out the window. I'm more tolerant now and far less judgmental. There is a fluidness, a fragility I did not previously have. I thought I was a rock.\" He laughs. \"It turns out that I'm a river\u2014or something idiotic like that.\"\n\nIndeed, this witty man's own definition of courage\u2014what it means to be a courageous person\u2014has done a complete about-face. \"There are two ideas of bravery I like to play with,\" he now says. \"Is the brave person the one who rushes in and goes to the front line because he doesn't feel fear? Or is the brave person the one who is completely petrified but does _something_... not as much as the first person, perhaps, but done against the weight of their own fear?\"\n\n\"The first one might just be impetuous,\" I say.\n\n\"I am more fearful than I used to be. But I am also more rigorous about pushing myself past that fear. It's made me aware of how much I depend on other people. But,\" he makes clear, \"I'm also much more careful about which people I do depend on, because I recognize my vulnerability\u2014the profound fragility we all share.\" Andrew weighs his words with care, wary of sounding too maudlin or glib. \"I have found that there is a kind of bliss that only comes to those who've known deep suffering,\" he tells me, echoing Lucy Grealy.\n\n\"What do you mean?\"\n\n\"There is a kind of ecstatic delight when things are going well, a kind of joy I could never have known about if I had not been so depressed.\" Andrew even alludes to the discovery in himself of something akin to soul. \"A part of myself I could never have imagined until one day, seven years ago, when hell came to pay me a surprise visit,\" as he wrote in his book. \"It is a precious discovery.\"\n\nAs Julia Kristeva, the Bulgarian psychoanalyst, wrote in her own memoir about coming through darkness, \"I owe a supreme, metaphysical lucidity to my depression. Refinement in sorrow or mourning are the imprint of a humankind that is surely not triumphant but subtle, ready to fight, and creative.\"\n\nI walk Andrew home after our interview. Near his landmark brownstone is the building where the poet Emma Lazarus lived, whose most famous poem (\"Give me your poor, your huddled masses...\") is inscribed on the Statue of Liberty. The irony of this message almost next door is hardly lost on this clever man, with its themes of exile, resurrection, sail-setting for a country whose standard is freedom. \"So lies the world before us,\" as Andrew described it in his wise book,\n\nand with just such steps we tread a solitary way, survivors as we must be of an impoverishing, invaluable knowledge. We go forward with courage and with too much wisdom but determined to find what is beautiful. It is Dostoyevsky who said, \"Beauty, though, will save the world.\" That moment of return from the realm of sad belief is always miraculous and can be stupefyingly beautiful. It is nearly worth the voyage out into despair.\n\n## **THE MOTHER (OR ALOHA OY)**\n\nA Jewish intellectual had gone to Hawaii to live with a local family. After several months, the head of the family informed their visitor that the clan was ready to introduce him to the family whale. There was a whale that the family revered, he told the Jew. The animal responded to their calls and played with the family at a secret place on the island. They called this whale the Mother.\n\nI can just imagine the Jew's reaction. Aloha oy! Thanks, but no thanks. My people do not hunt or take risks in goyish locations involving harpoons. The Jew must have thought that his host was bonkers, but since these kind people were putting him up, he pretended to consider it. The Jew was also curious. A whale that came on command? he thought. Nothing in this neurotic man's life came on command.\n\n\"I'm not a swimmer,\" he told the elderly Hawaiian, hoping that would end it.\n\nNot to worry, his host assured him. \"Just cling to a rock. The Mother will do the rest.\"\n\nThe Jew felt himself starting to tremble. He had spent his life being afraid of everything, most of all death. He'd tied a tourniquet around his passions. That is why he had come to Hawaii: to wake himself up, to set himself free. The Jew decided to meet the whale.\n\nOn the given day, the family traveled together to a secluded cove surrounded by black volcanic rock. The Hawaiians watched their visitor strip down to his bathing suit, and laughed at his petrified expression as he lowered himself down into the water, clutching the rocks like a monkey. Slowly and in unison, the family began to chant (think of Enya in a hula skirt), their voices carrying over the water as the Jew shivered, awaiting his fate.\n\nThen the most astonishing thing happened. About five hundred yards away, the biggest black whale he'd ever seen surfaced calmly in the water. He was terrified and nearly jumped out of the sea. In spite of his rational mind's protests, however, the Jew stayed where he was. What he felt in his heart seemed to contradict everything he had believed possible until that moment. The Jew sensed that the whale was aware of his terror and was sending him great, warm, enveloping waves of something he could only call love\u2014a strong, immense, impersonal warmth that actually began to calm him down. He was certain that the whale not only could feel his fear, impossible though it seemed, but also knew that he couldn't swim. His only task, the Mother appeared to be telling him, was to stay in the water and feel that great warmth, to trust its heat and go on trusting.\n\nSlowly, the whale began to float toward him, sending miniature geysers from her blowhole as if to introduce herself. The Jew was overwhelmed and stayed perfectly still as the great beast moved in his direction. Then the elder Hawaiian said, \"Touch the Mother.\"\n\nWith shaking hands, the visitor reached out and touched the whale's slick, ebony skin. The moment he made contact, the Mother tilted her body so that he could run his hand along her side. Miraculously, the Jew wasn't frightened. Wonder burned his fear away. He stayed like that for several minutes, with his local friends crooning in the background. Finally the whale slipped away. At that moment, he stopped clutching the rocks and waded into the water, marveling at his own buoyancy.\n\nWhy had he ever doubted this, the Jew asked himself, feeling water as if for the first time\u2014his body as if for the first time, too. Afterward, the world seemed changed; there was awe in him where there had been terror; a confident yes where there had been maybe; courage where caution and doubt had lived; trust in his formerly cynical heart, the desire to swim replacing the notion that letting go, he would sink like a stone.\n\nHe never saw the Mother again. But the world was lightened in the Jew's eyes. He had stepped through the back door of his own mind. He no longer knew what was possible. Nothing in the world seemed beyond his touch.\n\n## **ROPE BURN**\n\nThe day before my mother died, we were in her bedroom passing the time, with Mom's head on my lap and a wall clock ticking, when suddenly she opened her eyes and blinked at me through the morphine haze.\n\n\"What is it?\" I asked. She blinked again, then whispered the first words she'd uttered for days, in a dreamy growling voice. \"Easier,\" my mother was saying. \"It's easier when you let go.\"\n\nThis startled me, coming from her of all people, a stubborn, cantankerous, fighting broad who didn't enjoy introspection. Still, I had noticed a subtle change in her, beyond what was happening to her body, without quite being able to name it. My mother seemed to be suffering less, to be more peaceful. There was something making way in her, helping her to loosen her grip. I resolved then to remember this when my own time came.\n\nSaint Augustine said that we can only know what we love. And to know something is to know it's not yours. We're guests in this hotel, after all; even the ashtrays will have to stay. Still, attachment is bound to happen. We imagine our lives to be an accretion, an increase of layers solidifying the identity that holds us down to the ground. But what if the opposite's even more true, that we're winnowed away, worn down by time, pushed into transparency? What if we're humbled, without being severed, in order that we may move through the world with less friction, less regret but more desire, less protection but more love?\n\nBabies only learn this with time. At first they must cling if they hope to survive. Deprived of touch, infants will rock and rock themselves to sleep, as a volunteer at a foundling hospital told me. \"They would sit there with their arms wrapped around themselves on their beds, just rocking and rocking,\" she said. There are orphans, both actual and self-imagined, who crave being held this way their whole lives, trapped in the psychic bodies of infants. How many times have you heard someone say that if so-and-so doesn't love them, they will just die? It's sad but also clearly a throwback to what they longed for as unmothered children.\n\nOne snowy day in the Berkshire Mountains, Buddhist teacher Joseph Goldstein is talking to me about love and strangling. We're in his house adjacent to the Insight Meditation Society, which Joseph cofounded thirty years ago with fellow Americans Jack Kornfield and Sharon Salzburg. \"We assume that clutching and caring go hand in hand,\" Joseph tells me, crossing his size-14 feet on the table between us. A towering man of sixty, Joseph, though single, is no monkish stranger to human tangling. As he talks about pain and clinging and emotional vampirism, he could be any person who's been wounded in love. \"We assume that attachment equates with love,\" says Joseph. \"But just look at these forces closely and you realize how different they actually are. When we feel most loving, we feel most _open_ hearted. Attachment isn't a giving energy. When we're attached, it's a subtle contraction, the heart holding on, saying, 'Please don't leave me.'\" He compares the pain of this to rope burn.\n\n\"But we're human,\" I say. \"We get attached.\"\n\nThis teacher reminds me that the alternative to rope burn is not some chilly pseudo-detachment, apathy masked as enlightenment. It's _commitment._ \"Attachment wants things to stay the same, especially in relation to us,\" he says. \"Since everyone and everything is always changing, this is obviously doomed. Commitment, on the other hand, does not say things must stay the same for us to be happy, but that we will abide with ourselves affectionately throughout these changes.\"\n\nAbide? I ask.\n\n\"Otherwise we only create more suffering. Attachment and commitment are different,\" Joseph says.\n\nThen he stops talking. The room grows quiet. It must be weird to live like a monk, I think, cutting ties instead of tightening them, relinquishing rather than gathering up. Once upon a time, Joseph was a Jewish intellectual himself, seeking far and wide for inspiration, for something to awe him out of himself. He let go of the rocks when the dharma rose enormous-bellied on the horizon. He touched the Buddha's slippery back. The Four Noble Truths became his lovers. Solitude has left him young.\n\n\"Imperfect life,\" I say stupidly, backing off from the silence Joseph loves.\n\nHe smiles again, seeing my _spilkes._ \"'Love your crooked neighbor with your crooked heart,'\" Joseph says, pretending to instruct me with W. H. Auden's words.\n\n\"You do the best you can,\" I say.\n\nMy mother loved us at the end. Ida's lobster shell fell off. She allowed us to touch her in a way she'd never been comfortable with before, as Samuel's Israeli baker father had blessed his own son in the end. I laid my mother's head in my lap, a tiny skull with gray flattened wisps, and stroked the skin without hurting her. This woman whose attention I'd craved my whole life was now humbled enough to finally surrender. No sign of rope burn anywhere.\n\n\"Am I dying?\" Mom asked the day before she did. She was leaning against the bathtub on her knees, smoking her last cigarette, flicking the ashes down the drain.\n\n\"It doesn't look good,\" said my sister Belle.\n\nMy mother looked more puzzled than scared. Then she said, \"It's not that bad.\"\n\n## **WHAT MAKES THE ENGINE GO?**\n\nWhen I first meet him, a few years before his death at the age of one hundred, poet Stanley Kunitz is a living example of how _viriditas_ works, a photosynthetic phenomenon whose lust for life and singular spirit overflow with ever-greater intensity with each coming year. \"I don't wake up as a nonagenarian,\" Stanley will tell me when we meet. \"I wake up as a poet! It is only the body that wears out. The imagination remains as intense and glowing as ever.\"\n\nStanley answers the door of his New York apartment, bright eyes popping, waving his hands, dressed in funny blue tennis shoes and a shabby tweed jacket. \"Welcome, welcome, welcome!\" he sings, hunched at a sixty-degree angle, leading me at a healthy clip down a long hallway lined with canvases by Stanley's famous friends Franz Kline and Robert Motherwell (as well as abstracts by his wife, painter Elise Asher), piles of books, and a bronze bust of Stanley worthy of a past poet laureate. We settle in opposite high-backed chairs in the large, sunny sitting room. Elise and an assistant field a constant stream of phone calls from a nearby office.\n\n\"There are penalties one pays for a long existence, but I am happy to pay them!\" Stanley smiles and sighs. He looks dwarfed in his large chair, fingertips pressed together under the chin of his ancient, spritely Semitic face. Despite his working-class Jewish background, Stanley speaks in lofty Bostonian Brahmin tones, unfurling one carefully manicured phrase at a time.\n\n\"Reinvention is my philosophy, if you want to call it that,\" he says, looking out the window. \"Imagination is the key to creating a life that is ever new.\" Stanley turns his eyes to me. \"We are each of us a changeling person,\" he says. \"We are not going to be the same decade after decade. Wisdom results from confronting not only one's desires and capacities but also one's limitations.\"\n\n\"The Layers,\" one of Stanley's best-loved poems, is his crystallization of this wisdom.\n\n_I have walked through many lives,  \nsome of them my own,  \nand I am not who I was,  \nthough some principle of being abides  \nfrom which I struggle  \nnot to stray.  \nWhen I look behind,  \nas I am compelled to look  \nbefore I can gather strength  \nto proceed on my journey,  \nI see the milestones dwindling  \ntoward the horizon  \nand the slow fires trailing  \nfrom the abandoned camp-sites,  \nover which scavenger angels  \nwheel on heavy wings._\n\n_Oh, I have made myself a tribe  \nout of my true affections,  \nand my tribe is scattered!  \nHow shall the heart be reconciled  \nto its feast of losses?  \nIn a rising wind  \nthe manic dust of my friends,  \nthose who fell along the way,  \nbitterly stings my face.  \nYet I turn, I turn,  \nexulting somewhat,  \nwith my will intact to go  \nwherever I need to go,  \nand every stone on the road is precious to me.  \nIn my darkest night,  \nwhen the moon was covered  \nand I roamed through wreckage,  \na nimbus-clouded voice  \ndirected me:  \n\"Live in the layers,  \nnot on the litter.\"  \nThough I lack the art  \nto decipher it,  \nno doubt the next chapter  \nin my book of transformations  \nis already written.  \nI am not done with my changes._\n\nDynamism in the midst of our \"feast of losses,\" metamorphosis among the ruins, self-creation inside our own changing story, reconciliation to the dangerous knowledge that while we may no longer be who we were, \"some principle of being abides\"\u2014the authentic, immutable inner voice from which, like the poet, we struggle not to stray. With such directions Stanley has mapped a strategy for weathering time, change, and sorrow without losing our \u00e9lan vital. Curiosity goads this appetite. Whether fortune is smiling, or not, our lives shaped as we'd like them to be, or not, the future seemingly navigable, or not, we remain intensely curious about our own unpredictable changes, all that has yet to be revealed. Curiosity is the link to this green source; so is the courage to remain creative and not cede rule to our screeching demons. The choice of renewal over resignation\u2014truthfulness over the status quo\u2014transforms day-to-day life into evolution.\n\nWe long to puncture the heart of things, to touch the pulsating rawness of being, especially in difficult times. It's no accident that after 9\/11, poetry became a national obsession\u2014Auden's \"September 1, 1939,\" for example. Poetry is the language of survival linking us to the green source, bridging divides of culture and time. During crisis we long for simple things, since only the simplest, truest things are broad enough to carry us across.\n\n\"What makes the engine go?\" Stanley wrote on his eightieth birthday in a poem to his wife. \"Desire, desire, desire.\"\n\nThe road to where Stanley is sitting today has been marked by catastrophic detours. Before he was born, his father had poisoned himself in the main square of their town, Worcester, Massachusetts, leaving the sickly boy and his mother to find their own way in the world. Though he excelled enough in his studies to be hired by Harvard (Stanley graduated in 1924), the university suddenly reneged on its own employment offer for fear that \"Anglo-Saxon students might resent being taught by a Jew.\" Soon afterward he fell in love with a poet named Helen Pearce while both were fellows at Yaddo, the artist colony in Saratoga Springs, New York, married her, and moved with his new bride to a hundred-acre farm in Connecticut. One April day, with no warning, Helen Pearce disappeared from the farm without a trace. To this day, Stanley doesn't know what happened.\n\n\"The heart breaks and breaks and lives by breaking,\" Stanley wrote in \"The Testing Tree.\" His willingness to be broken, then remade, his commitment to the belief that such destruction is natural and fruitful, has led to a life of constant self-reinvention. \"What ever we create is made of the materials of life,\" explained Stanley during a period of tremendous grief. \"We should never think of the life\u2014our life\u2014as being the enemy of what ever we aspire to create.\"\n\nA prot\u00e9g\u00e9e of Stanley's, the gifted poet Marie Howe, received a dose of her mentor's wisdom during a period of great loss. \"I came to Stanley when my brother was dying,\" Marie tells me. \"I told him that it felt as if something had me in its mouth and was chewing me up. Stanley said, 'It is, and you must wait to see who you are when this thing is done with you.'\"\n\nWith his first wife gone and his wunderkind career stalled, Stanley was thrust into the darkest period of his life, a phase that lasted for nearly a decade. Then one day in 1936 he opened the door and found the well-known poet Theodore Roethke standing on his porch with a copy of Stanley's latest commercial failure in his hands\u2014and the offer of a teaching position at his college. \"How he found me I shall never know,\" Stanley says now, \"but that moment will remain with me always. Roethke made it possible for me to live.\"\n\n\"In a dark time, the eye begins to see,\" Roethke himself had written. Since that time, in addition to being poet laureate, Stanley has published dozens of volumes and received nearly every major literary honor available in this country. Even during the hardest of times his inspiration has rarely flagged. \"Poetry is like breathing,\" he likes to say, likening each encounter with a poem to meeting a new bride. \"One never retires from art, any more than one retires from breathing. I say in one of my poems, 'Maybe it is time for me to practice growing old.' I meant it as a joke.\"\n\nStanley turns a hooded eye to one of his wife's bold canvases on the wall behind me, his stubbled chin resting on his chest like a bird's. For a moment, I'm not sure where he is\u2014then the old man swoops back. \"Keats has a great phrase in one of his letters, describing what he values most. He refers to 'the holiness of the heart's affections,'\" he says. \"I remember when I first read those words, recognizing that this holiness would be something that would be forever meaningful to me, a foundation stone of my life. And it has been.\"\n\nIn his stress on reinvention, Stanley shifts emphasis from the endurance of loss to becoming more essentially oneself. \"Every artist I've known has been distinguished, almost from birth, by knowledge of the need to become a self, not just a living body,\" he tells me.\n\n\"Isn't that true of everyone?\"\n\n\"Yes.\" Stanley smiles. \"I suppose that it is.\"\n\n\"How does it feel to be almost a hundred years old?\"\n\nHe looks positively jubilant at the question. \"I feel I have found myself,\" he tells me. In another interview, he elaborated on this point. \"I feel in possession of my destiny. That I am not a victim of it.\" \"Look at this beautiful day, for example,\" he tells me now, smiling at a sunlit tree. \"I've been through so much these last few years, but I don't feel I'm powerless at all. There is a sense of another level, a state of being, that is both yours and not yours.\"\n\n\"Has the world become a more dangerous place?\" I ask.\n\n\"Sometimes one does fear for the future,\" says Stanley. \"One fears the loss of the search for the sacred, for the beautiful, for the true. One fears that the dynamics of modern society point toward the practical rather than the spiritual. But I think there will always be individuals who will carry on the great tradition of the prophets and poets. I have such a fierce conviction about the value of existence, the importance of life, that I know that there must be many, many others who feel the same way, and will always be here on earth. That gives me hope.\"\n\nThe last time I see Stanley, two years after our first interview, he's reclining in a hospital bed in the same living room where we first spoke, drifting in and out of consciousness, looking more bemused than uncomfortable. Conversation is no longer easy for him. Stanley listens to me, smiles, shrugs his shoulders, his delight in being here at all still palpable. When I stand up to leave, he squeezes my hand for a long time. Crossing the green parquet floor, I catch sight of a first edition of Stanley's collected poems lying on a table. The final lines of \"The Round\" have remained with me since I first read them.\n\n_I can scarcely wait till tomorrow,  \nwhen a new life begins for me,  \nas it does each day,  \nas it does each day._\n\n## **NOTES AND SOURCES**\n\n**Magical Thinking:** See Joan Didion's _The Year of Magical Thinking_ (New York: Alfred A. Knopf, 2005) for more on \"the American way of grief.\" John Leonard's introduction to Didion's collected nonfiction works, _We Tell Ourselves Stories in Order to Live_ (New York: Alfred A. Knopf, 2006), is also an excellent introduction to Didion's cultural profile (\"a poster girl for anomie\").\n\n**The Roar of Freedom:** For more on \"The Roar of Freedom,\" see _Dialogues with a Modern Mystic,_ by Andrew Harvey and Mark Matousek (Chicago: Quest Books, 1994).\n\n**The Day of Laughter:** Stephen Batchelor's _Buddhism Without Beliefs_ (New York: Riverhead Books, 1997) is an excellent, lingo-free introduction to the Buddhist way of life.\n\n_**Om Mani Padme Hum** : The Tibetan Book of Living and Dying,_ by Sogyal Rinpoche (San Francisco: HarperSanFrancisco, 1992), provides a strong framework for understanding impermanence as an element of spiritual practice, as well as the path of the bodhisattva.\n\n**Dragons at the Gate:** See the Willises' memoir, _But There Are Always Miracles_ (New York: Viking, 1974), for more on their story. Mary Willis's autobiographical novel, _Papa's Cord_ (New York: Alfred A. Knopf, 1999), provides background to her journey with Jack.\n\n**Superman's Ghost:** For more on Jim MacLaren, see http:\/\/ JimMacLaren .com. Deep thanks to Elizabeth Gilbert for turning me on to Jim with her article \"Lucky Jim\" ( _GQ,_ May 2002) and asking him the two questions that interested me (about metaphysical pain) before I could get there.\n\n**Home in the World:** My article \"The Crucible of Homelessness\" ( _Common Boundary,_ Spring 1992) rounds out this discussion of homelessness as metaphor. Jonathan Kozol's book _Rachel and HerChildren: Homeless Families in America_ (New York: Fawcett Books, 1988) and Eve Ensler's play _Ladies_ (New York: Central Park Locations, 1989) provide illuminating case studies of homeless individuals.\n\n**The Art of Losing:** Viktor Frankl's _Man's Search for Meaning_ (New York: Pocket Books, 1984) is required reading for anyone wishing to understand survival, dignity, and how meaning is culled from life in extremis. Francis Bok's _Escape from Slavery_ (New York: St. Martin's, 2003) gives the full story of Francis's life before and after he came to the United States.\n\n**A Quarter Inch from Heaven:** Shunryu Suzuki Roshi's _Zen Mind, Beginner's Mind_ (Boston: Shambhala, 2006) remains the best introduction to Zen practice that I know of.\n\n**Going to Tahiti (or Raising Heaven):** For more on Ram Dass's thoughts about conscious aging, see his _Still Here: Embracing Aging, Changing, and Dying_ (New York: Riverhead, 2000) and Mickey Lemle's documentary _Fierce Grace_ (Zeitgeist Films Ltd., 2002).\n\n**The Net of Indra:** Daniel Goleman's _Social Intelligence: The New Science of Human Relationships_ (New York: Bantam Books, 2006) is an eye-opener regarding neuroplasticity and the science of relationships. Ken Wilber's _No Boundary_ (Boston: Shambhala, 2001) provides a philosophical basis for understanding interdependence (the Net of Indra) in daily life.\n\n**Reinventing Your Wife:** For more on Henry Grayson's theories, see his _The New Physics of Love_ (New York: Gotham Books, 2004).\n\n**Praying:** Three excellent sources on prayer are _Story of a Soul: The Autobiography of St. Th\u00e9r\u00e8se of Lisieux_ (New York: Tan, 1997), Kathleen Norris's _The Cloister Walk_ (New York: Riverhead, 1996), and Sharon Salzberg's _Faith_ (New York: Riverhead, 2002).\n\n**Demon Lovers:** For more on Michael Klein's life, see his _Track Conditions_ (New York: Persea, 1997) and _The End of Being Known_ (Madison: University of Wisconsin Press, 2003). Identifying details have been changed in Kathleen's story to protect her anonymity.\n\n**Questioning (or The Sphinx):** Byron Katie's book _Loving What Is_ (New York: Three Rivers Press, 2002) is a thorough introduction to The Work.\n\n**The Terrorists Within:** For more on metanoia, see _The Passionate Life,_ by Sam Keen (San Francisco: HarperSanFrancisco, 1984). C. H. Waddington's _The Ethical Animal_ (London: Allen & Unwin, 1960) provides a hard-science background to terror as an evolutionary element in history.\n\n**Earth Angel:** See _The Protean Self: Human Resiliency in the Age of Fragmentation,_ by Robert Jay Lifton (Chicago: University of Chicago Press, 1993), for more on shape-shifting and the trickster archetype in human adaptation.\n\n**Something Else Is Also True:** Maria Housden's memoirs, _Hannah's Gift: Lessons from a Life Fully Lived_ (New York: Bantam, 2002) and _Unraveled_ (New York: Harmony, 2004), offer poignant and inspiring insights into \"child wisdom\" and grief as a prelude to self-discovery.\n\n**Pain Passes, but the Beauty Remains:** This story is adapted from Bo Lozoff's short story \"The Saddest Buddha,\" with great thanks to the author.\n\n**Hedonics:** For more on the science of happiness, see _Stumbling on Happiness,_ by Daniel Gilbert (New York: Alfred A. Knopf, 2006), and _Full Catastrophe Living,_ by Jon Kabat-Zinn (New York: Dell, 1990). For more on Colette, take a vacation with Judith Thurman's _Secrets of the Flesh_ (New York: Alfred A. Knopf, 1999).\n\n**Original Blessing:** To more deeply understand the importance of blessing, see _Original Blessing,_ by Matthew Fox (Santa Fe, NM: Bear and Co., 1983), and _Dialogues with a Modern Mystic,_ by Andrew Harvey and Mark Matousek (Chicago: Quest, 1994).\n\n**Enough:** Rabbi Rami Shapiro's discussion of the sabbath in _Minyan: Ten Principles for a Life Worth Living_ (New York: Bell Tower, 1997) is worth reading. David Loy's _The Great Awakening: A Buddhist Social Theory_ (Boston: Wisdom, 2003) is useful in understanding the hungry ghost dilemma. Bill McKibben's _Deep Economy_ (New York: Henry Holt and Co., 2007) elucidates the East-West consumer crisis and the virtues of local consumption.\n\n**Stress Matters:** Saki Santorelli's _Heal Thy Self_ (New York: Bell Tower, 1999) is a great overview of stress, its discontents, and treatment.\n\n**The Wounded Healer:** For more on Rachel Remen, see her _Kitchen Table Wisdom_ (New York: Riverhead, 1996) and _My Grandfather's Blessings_ (New York: Riverhead, 2000). Also see www.commonweal.org.\n\n**True Confessions:** For more about narrative medicine, see Dr. Rita Charon, _Narrative Medicine: Honoring the Stories of Illness_ (New York: Oxford University Press, 2006).\n\n**A Splinter of Love:** See Paula Allen's _Flowers in the Desert_ (Santiago: Cuarta Propio, 1999) for photographs and more stories about the women of Calama. Also see Deborah Shaffer's wonderful documentary film _Dance of Hope._\n\n**Nakedness:** Lucy Grealy's _Autobiography of a Face_ (Boston: Houghton Mifflin, 1994) gives a deeper look into Lucy's story. Also see Ann Patchett's _Truth and Beauty_ (New York: HarperCollins, 2004).\n\n**Killing Peter Pan:** For more about the loss of soul in modern life, see _Care of the Soul,_ by Thomas Moore (New York: HarperCollins, 1992), and _The Soul's Code,_ by James Hillman (New York: Random House, 1996).\n\n**The Water or the Wave?:** Eckhart Tolle's _The Power of Now_ (Novato, CA: New World Library, 1999) is indispensable to a deeper understanding of _nunc stans_ and the relationship between conflict and mental labeling.\n\n**At Sea:** For more about Andrew Solomon, see his _The Noonday Demon: An Atlas of Depression_ (New York: Scribner, 2001). Also see Julia Kristeva's memoir, _Black Sun_ (New York: Columbia University Press, 1992).\n\n**Rope Burn:** Joseph Goldstein's books, including _One Dharma_ (San Francisco: HarperSanFrancisco, 2002) and _Seeking the Heart of Wisdom_ (with Jack Kornfield; Boston: Shambhala, 1987), are treasures for understanding Buddhist wisdom.\n\n**What Makes the Engine Go?:** For more on Stanley Kunitz's life and work, see _The Collected Poems_ (New York: Norton, 2000). Also, special thanks to Genine Lentine for _The Wild Braid_ (New York: Norton, 2005) and for arranging my last visit with Stanley.\n\n## **ACKNOWLEDGMENTS**\n\nEvery book has a back story\u2014some more epic than others. Five years, three publishers, four editors, two coasts, a season in hell, two funerals, one wedding, and multiple drafts later, I am indebted to many souls.\n\nMy agent, Joy Harris, is a writer's dream; Barbara Graham is a sister in all things; Eve Ensler, my greatest inspiration and heart; Robert Levithan, best friend and brother; Florence Falk, my beautiful _bechert;_ Maria Housden for unfailing courage; Marcia Lippman for joie de vivre; Catherine Ingram for wisdom in exile; my sister, Belle Heil, for constant kindness; Dr. Paul Curtis Bellman, for faith and nerve; and Amy Gross, dear friend, for suggesting that I write about this in the first place.\n\nDeep thanks to Karen Rinaldi (the first person in publishing to recommend never to rush a book, coincidentally, years before she was my publisher); to Terra Chalberg for comfort in limbo; to Amanda Katz for leading me to my wonderful editor Kathy Belden, whose taste, humor, and indomitable patience showed me what I'd been waiting for all these years. The team at Bloomsbury\u2014 especially Sabrina Farber, Annik LaFarge, Maya Baran, and Sara Mercurio\u2014have been wonderful. Special thanks to Jill Hughes and Greg Villepique for their exacting eye on detail.\n\nProfound gratitude to my close friends Michael Klein, Hugh Delehanty, Martha Cooley, Sharon Salzberg, Samuel Kirschner, Paula Allen, Joe Dolce, Andrew Hood, Katharina Tapp, Susan Dalsimer, Gwenyth Jackaway, Karen Fuchs, Ed and Deb Shapiro, Dr. Sue Grand, Sally Fisher, Billy Blechen, Rabia Halim, and Gary Lennon for love and conversation; to Cynthia O'Neal, Robbie Stein, Marie Howe, Elizabeth England, Ram Dass, Lisa Cornelio, and Roger Housden for writerly strength; and to the many people, including Betsy Carter, Deborah Copaken Kogan, Tyrone Thompson, Ella Pasqueriello, and Dr. David Feldman, who were generous enough to tell me their stories. Deep thanks to Louis Morhaim, my greatest teacher. R.I.P.\n\nAnd to Marco Naguib, most of all. \"Set me like a seal upon thy heart, love is as strong as death.\" _Semper fi._\n\n## **A NOTE ON THE AUTHOR**\n\nMark Matousek is the award-winning author of two memoirs, _Sex Death Enlightenment: A True Story_ and _The Boy He Left Behind: A Man's Search for His Lost Father._ A contributing editor to _O: The Oprah Magazine_ and _Tricycle,_ he was a senior editor at _Interview_ magazine and has written for many publications, including the _New Yorker,_ the _New York Times Magazine, Details, Harper's Bazaar, AARP: The Magazine, Self, Common Boundary,_ and the _Utne Reader._ He lives in New York City.\nBY THE SAME AUTHOR\n\n_Sex Death Enlightenment: A True Story_\n\n_The Boy He Left Behind: A Man's Search for His Lost Father_\n\n_Dialogues with a Modern Mystic_ (with Andrew Harvey)\n\n_Still Here,_ by Ram Dass (coeditor)\nCopyright \u00a9 2008 by Mark Matousek\n\nAll rights reserved. No part of this book may be used or reproduced in any manner whatsoever without written permission from the publisher except in the case of brief quotations embodied in critical articles or reviews. For information address Bloomsbury USA, 175 Fifth Avenue, New York, NY 10010.\n\nPublished by Bloomsbury USA, New York\n\nSections of this book have appeared in the _New Yorker, O: The Oprah Magazine, AARP Magazine,_ the _Utne Reader,_ and _Common Boundary Magazine_\n\n\"The Layers,\" copyright \u00a9 1978 by Stanley Kunitz, from _The Collected Poems_ by Stanley Kunitz. Used by permission of W. W. Norton & Company, Inc.\n\n\"Love after Love,\" from _Collected Poems 1948-1984_ by Derek Walcott, Copyright \u00a9 1986 by Derek Walcott. Reprinted by permission of Farrar, Straus and Giroux, LLC.\n\n\"I was lost,\" from _False Prophet_ by Stan Rice. Copyright \u00a9 2005 by Stan Rice. Reprinted by permission of Alfred A. Knopf, Inc.\n\nLIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA\n\nMatousek, Mark.  \nWhen you're falling, dive: lessons in the art of living \/ Mark Matousek.  \np. cm.  \nISBN:13 978-1-59691-369-1 (hardcover)  \n1. Conduct of life. 2. Life change events. I. Title.\n\nBF637.C5M347 2008  \n155.2'4-dc22  \n2007046175\n\nFirst published by Bloomsbury USA in 2008  \nThis paperback edition published in 2009\n\nE-book ISBN: 978-1-60819-646-3\n\nwww.bloomsburyusa.com\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nThank you for downloading this Gallery Books eBook.\n\n* * *\n\nJoin our mailing list and get updates on new releases, deals, bonus content and other great books from Gallery Books and Simon & Schuster.\n\nCLICK HERE TO SIGN UP\n\nor visit us online to sign up at  \neBookNews.SimonandSchuster.com\nWallbanger\n\nAlice Clayton\n\nGallery Books\n\nNew York London Toronto Sydney New Delhi\nPraise for Wallbanger:\n\n\"Wallbanger is an instant classic, with plenty of laugh out loud moments and riveting characters\u2014highly recommended!\"\n\n~NYT and USA Today Bestselling Author Jennifer Probst\n\n\"Hilarious, romantic, and compulsively readable, Wallbanger delivers the perfect blend of sex, romance, and baked goods.\"\n\n~Ruthie Knox, best-selling author of About Last Night\n\n\"Alice Clayton strikes again, seducing me with her real woman sex appeal, unparalleled wit and addicting snark; leaving me laughing, blushing, and craving knock all the paintings off the wall sex of my very own.\"\n\n~Brittany Gibbons, brittanyherself.com\n\n\"Caroline Reynolds. Finally a woman who knows her way around a man and a KitchenAid Mixer. She had us at zucchini bread!\"\n\n~Curvy Girl Guide\nDedication\n\nTo my mom, for letting me have coconut on my birthday cake  \neven though no one else likes it.\n\nTo my dad, for reading me Garfield comics  \nuntil we laughed so hard we were both crying.\n\nThank you\nChapter One\n\n\"OH, GOD.\"\n\nThump.\n\n\"Oh, God.\"\n\nThump thump.\n\nWhat the...\n\n\"Oh, God, that's so good!\"\n\nI scrambled up out of sleep, confused as I looked around the strange room. Boxes on the floor. Pictures propped against the wall.\n\nMy new bedroom, in my new apartment, I reminded myself, placing both hands on the duvet, grounding myself with the luxurious thread count. Even half asleep, I was aware of my thread count.\n\n\"Mmmm...Yeah, baby. Right there. Just like that...Don't stop, don't stop!\"\n\nOh boy...\n\nI sat up, rubbed my eyes, and turned to look at the wall behind me, beginning to understand what had woken me up. My hands still stroked the duvet absently, catching the attention of Clive, my wonder cat. Butting his head under my hand, Clive demanded to be soothed. I stroked him as I looked around and oriented myself in my new space.\n\nI'd moved in earlier that day. It was a gorgeous apartment: spacious rooms, wood floors, arched doorways\u2014it even had a fireplace! I had no clue how to actually build a fire, but that was neither here nor there. I was aching to put things on the mantel. As an interior designer, I had a habit of mentally placing things in almost every space, whether it belonged to me or not. It drove my friends a wee bit mad at times, as I was constantly restaging their knickknacks.\n\nI'd spent the day moving in, and after soaking in the incredibly deep, claw-foot tub until well past prune, I settled myself into bed and enjoyed the creaks and squeaks of a new home: light traffic outside, some quiet music, and the comforting click-click of Clive exploring. The click-click came from his hangnail, you see...\n\nMy new home, I'd thought contentedly as I slipped into an easy sleep, which is why I was so surprised to be woken at...let's see...two thirty-seven a.m.\n\nI found myself gazing stupidly at the ceiling, trying to return to a relaxed state, but I was startled again as my headboard moved\u2014banged into the wall was more like it.\n\nAre you kidding me? Then I heard, very distinctly:\n\n\"Oh, Simon, that's so good! Mmm...\"\n\nAw, jeez.\n\nBlinking, I felt more awake now and a little fascinated by what was clearly going on next door. I looked at Clive, he looked at me, and if I wasn't so tired I'd have been pretty sure he winked. I guess someone should be getting some.\n\nI'd been in a bit of a dry spell for a while. A very long while. Bad, rapid-fire sex and an ill-timed one-night stand had robbed me of my orgasm. She'd been on vacation for six months now. Six long months.\n\nThe beginnings of carpal tunnel were threatening to set in as I tried desperately to get myself off. But O was on seemingly permanent hiatus. And I don't mean Oprah.\n\nI pushed the thoughts of my missing O away and curled up on my side. All seemed quiet now, and I began to drift back to sleep, Clive purring contentedly beside me. Then all hell broke loose.\n\n\"Yes! Yes! Oh, God...Oh, God!\"\n\nA painting I'd propped on the shelf above my bed fell off and rapped me soundly on the head. That'll teach me to live in San Francisco and not make sure everything is securely mounted. Speaking of mounted...\n\nRubbing my head and cursing enough to make Clive blush\u2014if cats could blush\u2014I looked back at the wall behind me again. My headboard was literally banging against it as the ruckus continued next door.\n\n\"Mmm...yes, baby, yes, yes, yes!\" the loudmouth chanted...and concluded with a contented sigh.\n\nThen I heard, for the love of all that's holy, spanking. You can't misinterpret the sound of a good spanking, and someone was receiving one next door.\n\n\"Oh, God, Simon. Yes. I've been a bad girl. Yes, yes!\"\n\nUnreal...More spanking, and then the unmistakable sound of a male voice, groaning and sighing.\n\nI got up, moved the entire bed a few inches away from the wall, and huffed back under the duvet, glaring at the wall the whole time.\n\nI fell asleep that night after swearing I would bang back if I heard one more peep. Or groan. Or spank.\n\nWelcome to the neighborhood.\nChapter Two\n\nTHE NEXT MORNING, my first official morning in my new place, found me sipping a cup of coffee and munching a leftover donut from yesterday's moving-in party.\n\nI wasn't quite as awake as I'd hoped to begin unpackingpalooza, and I silently cursed last night's antics next door. The girl was plowed, spanked, she came, she slept. The same for Simon. I assumed his name was Simon, as that was what the girl who liked to be spanked kept calling him. And really, if she was making up a name there were hotter ones than Simon to be screaming out in the throes.\n\nThe throes...God, I missed the throes.\n\n\"Still nothing, huh, O?\" I sighed, looking down. During month four of The Missing O, I'd started to talk to my O as though she were an actual entity. She felt real enough when she was rocking my world back in the day, but sadly, now that O had abandoned me, I wasn't sure I'd recognize her if she saw her. 'Tis a sad, sad day when a girl doesn't even know her own orgasm, I thought, looking wistfully out the window at the San Francisco skyline.\n\nI unfolded my legs and padded to the sink to rinse out my coffee mug. Placing it in the sink to drain, I pushed my light blond hair back into a sloppy ponytail and surveyed the chaos that surrounded me. No matter how well I planned, no matter how well I labeled those boxes, no matter how often I told that idiot moving guy that if it said KITCHEN it did not belong in the BATHROOM, it still was a mess.\n\n\"What do you think, Clive? Should we start in here or the living room?\" He was curled up on one of the deep windowsills. Admittedly, when I was scouting new places to live, I always looked at the windowsills. Clive was fond of looking out on the world, and it was nice seeing him waiting for me when I came home.\n\nRight now he looked at me, and then seemed to nod toward the living room.\n\n\"Okay, living room it is,\" I said, realizing I'd only spoken three times since waking up this morning, and every word uttered had been directed at a pussy. Ahem...\n\nAbout twenty minutes later Clive had started a stare-off with a pigeon and I was sorting DVDs when I heard voices in the hallway. My noisy neighbors! I ran to the door, almost tripping over a box, and pressed an eye to the peephole only to see the doorway across the hall. What a pervert I am, honestly. But I made no attempt to stop peeping.\n\nI couldn't see very clearly, but I could hear their conversation: the man's voice low and soothing, followed by unmistakable sighing from his companion.\n\n\"Mmm, Simon, last night was fantastic.\"\n\n\"I thought this morning was fantastic too,\" he said, planting what sounded like one helluva kiss on her.\n\nHuh. They must have been in another room this morning. I hadn't heard a thing. I pressed my eye back to the peephole. Dirty pervert.\n\n\"Yes, it was. Call me soon?\" she asked, leaning in for another kiss.\n\n\"Of course, I'll call you when I'm back in town,\" he promised, swatting her on her bottom as she giggled again and turned away.\n\nIt seemed she was on the short side. Bye-bye, Spanx. The angle was wrong for me to see this Simon, and he was back in his apartment before I could get any sort of sense of him. Interesting. So this girl does not live with him.\n\nI hadn't heard any \"I love yous\" when she left, but they did seem very comfortable. I chewed absently on my ponytail. They'd have to be, what with the spanking and all.\n\nPushing thoughts of spanking and Simon from my mind, I went back to my DVDs. Spanking Simon. What a great name for a band...I moved on to the Hs.\n\nAn hour later I was just placing Wizard of Oz after Willy Wonka when I heard a knock. There was scuffling in the hallway as I approached the door, and I stifled a grin.\n\n\"Don't drop it, you idiot,\" a sultry voice chided.\n\n\"Oh, shut up. Don't be so damn bossy,\" a second voice snapped back.\n\nRolling my eyes, I opened the door to find my two best friends, Sophia and Mimi, holding a large box. \"No fighting, ladies. You're both pretty.\" I laughed, raising an eyebrow at them.\n\n\"Ha ha. Funny,\" Mimi answered, staggering inside.\n\n\"What the hell is that? I can't believe you guys carried it up four flights of stairs!\" My girls did not do manual labor when they could get someone else to do it.\n\n\"Believe me, we waited outside in the cab for someone to walk by, but no luck. So we schlepped it ourselves. Happy housewarming!\" Sophia said. They set it down, and Sophia fell into the easy chair by the fireplace.\n\n\"Yeah, quit moving so much. We're tired of buying you stuff.\" Mimi laughed, lying down on the couch and placing her arms over her face dramatically.\n\nI poked at the box with my toe and asked, \"So what is it? And I never said you had to buy me anything. The Jack LaLanne Juicer was not necessary last year, truly.\"\n\n\"Don't be ungrateful. Just open it,\" Sophia instructed, pointing at the box with her middle finger, which she then turned upright and displayed in my general direction.\n\nI sighed and sat on the floor in front of it. I knew it was from Williams Sonoma, as it had the telltale ribbon with the tiny pineapple tied to it. The box was heavy, whatever it was.\n\n\"Oh, no. What did you two do?\" I asked, catching a wink from Mimi to Sophia. Pulling at the ribbon and opening the box, I was pleased as punch with what I found. \"You guys, this is too much!\"\n\n\"We know how much you miss your old one,\" Mimi laughed, smiling at me.\n\nYears before, I'd been given an old KitchenAid mixer from a great aunt who passed away. It was over forty years old, but still worked great. Those things were built to last, by God, and it had lasted until just a few months ago, when it finally bit it in a big way. It smoked and went wonky one afternoon while mixing a batch of zucchini bread, and as much as I hated to do it, I tossed it out.\n\nNow as I stared into the box, a shiny, new, stainless steel KitchenAid stand mixer staring back at me, visions of cookies and pies began dancing in my head.\n\n\"You guys, it's beautiful,\" I breathed, gazing with delight at my new baby. I lifted it out gently to admire. Running my hands over it, splaying my fingers to feel the smooth lines, I delighted in the cold metal against my skin. I sighed gently and actually hugged it.\n\n\"Do you two want to be alone?\" Sophia asked.\n\n\"No, it's okay. I want you to be here to witness our love. Besides, this is the only mechanical instrument that will likely bring me any pleasure in the near future. Thanks, guys. It's too expensive, but I really appreciate it,\" I said.\n\nClive came over, sniffed the mixer, and promptly jumped into the empty box.\n\n\"Just promise to bring us yummy treats, and it's all worth it, dear.\" Mimi sat up, looking at me expectantly.\n\n\"What?\" I asked warily.\n\n\"Caroline, can I please start on your drawers now?\" she asked, stutter-stepping her way toward the bedroom.\n\n\"Can you start doing what to my drawers?\" I answered, pulling my drawstring a little tighter around my waist.\n\n\"Your kitchen! I'm dying to start placing everything!\" she exclaimed, running in place now.\n\n\"Oh, hell yes. Have at it! Merry Christmas, freakshow,\" I called as Mimi ran triumphantly into the other room.\n\nMimi was a professional organizer. She'd driven us crazy when we were all at Berkley together\u2014with her OCD tendencies and her insane attention to detail. One day Sophia suggested she become a professional organizer, and after graduation, she did just that. She now worked all over the Bay Area helping families get their shit together. The design firm I worked for sometimes had her consult, and she'd even appeared on a few HGTV shows filming in the city. The job suited her to perfection.\n\nSo I just let Mimi do her thing, knowing my stuff would be so perfectly arranged I'd be astounded. Sophia and I continued to putz in the living room, laughing over DVDs we'd watched throughout the years. We paused over each and every Brat Pack movie from the eighties, debating whether Bender ended up with Claire once they all went back to school on Monday. I voted no, and I further bet she never got that earring back...\n\nLater that night, after my friends left, I settled on the couch in the living room with Clive to watch reruns of The Barefoot Contessa on the Food Network. While dreaming of the creations I'd be whipping up with my new mixer\u2014and how one day I wanted a kitchen like Ina Garten's\u2014I heard footsteps on the landing outside my door, and two voices. I narrowed my eyes at Clive. Spanx must be back.\n\nSpringing from the couch, I pressed my eye against the peephole once more, trying to get a look at my neighbor. I missed him again, only seeing his back as he entered his apartment behind a very tall woman with long, brown hair.\n\nInteresting. Two different women in as many days. Manwhore.\n\nI saw the door swing shut and felt Clive curl around my legs, purring.\n\n\"No, you can't go out there, silly boy,\" I cooed, bending down and scooping him up. I rubbed his silky fur against my cheek, smiling as he lay back in my arms. Clive was the manwhore around here. He would lie down for anyone who rubbed his belly.\n\nReturning to the couch, I watched as Barefoot Contessa taught us all how to host a dinner party in the Hamptons with simple elegance\u2014and a Hamptons-size bank account.\n\nA few hours later, with the imprint of the couch cushion pressed firmly into my forehead, I made my way back to my bedroom to go to sleep. Mimi had organized my closet so efficiently that all I had left to do was to hang pictures and arrange a few odds and ends. I quite deliberately removed the pictures from the shelf above my bed. I was taking no chances tonight. I stood in the center of the room, listening for sounds from next door. All quiet on the western front. So far, so good. Maybe last night was a one-time thing.\n\nAs I got ready for bed, I looked at the framed pictures of my family and friends: my parents and I skiing in Tahoe; my girls and I at Coit Tower. Sophia loved to take pictures next to anything phallic. She played the cello with the San Francisco Orchestra, and even though she'd been around musical instruments all her life, she could never pass up a joke when she saw a flute. She was twisted.\n\nAll three of us were unattached at the moment, something rare. Usually at least one of us was dating someone, but since Sophia had broken up with her last boyfriend a few months ago, we'd all been in a dry spell. Luckily for my friends, their spell wasn't quite as dry as mine. As far as I knew they were still on speaking terms with their Os.\n\nI thought back with a shudder to the night when O and I had parted ways. I'd had a series of bad first dates and was so sexually frustrated that I allowed myself to go back to the apartment of a guy I had no intention of ever seeing again. Not that I was averse to the one-night stand. I'd made the walk of shame many a morning. But this guy? I should have known better. Cory Weinstein, blah blah blah. His family owned a chain of pizza parlors up and down the West Coast. Great on paper, right? Only on paper. He was nice enough, but boring. But I hadn't been with a man in a while, and after several martinis and a pep talk in the car on the way, I relented and let Cory \"have his way with me.\"\n\nNow, up until this point in my life, I'd shared that old theory that sex was like pizza. Even when it's bad, it's still pretty good. I now hated pizza. For several reasons.\n\nThis was the worst kind of sex. This was machine-gun style: fast, fast, fast. This was thirty seconds on the tits, sixty seconds on something that was about an inch above where he should have been, and then in. And out. And in. And out. And in. And out.\n\nBut at least it was over quick, right? Hell, no. This horribleness went on for months. Well, no. But for almost thirty minutes. Of in. And out. And in. And out. My poor hoohah felt like it had been sandblasted.\n\nBy the time it was over, and he yelled, \"So good!\" before collapsing on top of me, I had mentally rearranged all my spices and was starting on the cleaning supplies under the sink. I dressed, which didn't take that long as I was still almost fully clothed, and departed.\n\nThe next night, after letting Lower Caroline recover, I decided to treat her to a nice long session of self-love, accented by everyone's favorite fantasy lover, George Clooney, aka Dr. Ross. But to my great regret, O had left the building. I shrugged it off, thinking maybe she just needed a night away, still experiencing a little PTSD from Pizza Parlor Cory.\n\nBut the next night? No O. No sign of her that week, or the next. As the weeks became a month, and the months stretched on and on, I developed a deep, seething hatred for Cory Weinstein. That machine-gun fucker...\n\nI shook my head, clearing my O thoughts as I crawled into bed. Clive waited until I was situated before snuggling into the space behind my knees. He let out one last purr as I turned out the lights.\n\n\"'Night, Mr. Clive,\" I whispered and fell right to sleep.\n\nThump.\n\n\"Oh, God.\"\n\nThump Thump.\n\n\"Oh, God.\"\n\nUnbelievable...\n\nI woke up faster this time, because I knew what I was hearing. I sat up in bed, glaring behind me. The bed was still pulled safely away from the wall, so I felt no movement, but there was sure as hell something moving over there.\n\nThen I heard...hissing?\n\nI looked down at Clive, whose tail was at full puff. He arched his back and paced back and forth at the foot of the bed.\n\n\"Hey, mister. It's cool. We just got a noisy neighbor, that's all,\" I soothed, stretching my hand out to him. That's when I heard it.\n\n\"Meow.\"\n\nI cocked my head sideways, listening more intently. I studied Clive, who looked back as if to say, \"T'weren't me.\"\n\n\"Meow! Oh, God. Me-yow!\"\n\nThe girl next door was meowing. What in the world was my neighbor packing to make that happen?\n\nClive, at this point, went utterly bonkers and launched himself at the wall. He was literally climbing it, trying to get to where the noise was coming from, and adding his own meows to the chorus.\n\n\"Oooh yes, just like that, Simon...Mmmm...meow, meow, meow!\"\n\nSweet Lord, there were out-of-control pussies on both sides of this wall tonight. The woman had an accent, although I couldn't quite place it. Eastern European for sure. Czech? Polish? Was I seriously awake at, let's see, one sixteen a.m. and attempting to discern the national origin of the woman getting plowed next door?\n\nI tried to get a hold of Clive and calm him down. No luck. He was neutered, but he was still a boy, and he wanted what was on the other side of that wall. He continued to caterwaul, his meows mixing with hers until it was all I could to do to not to cry at the hilarity of this moment. My life had become theater of the absurd with a cat chorus.\n\nI pulled myself together because I could now hear Simon moaning. His voice was low and thick, and while the woman and Clive continued to call to each other, I listened solely to him. He groaned, and the wall banging began. He was bringing it home.\n\nThe woman meowed louder and louder as she undoubtedly climbed toward her climax. Her meows turned into nonsensical screaming, and she finally yelled out, \"Da! Da! Da!\"\n\nAh. She was Russian. For the love of St. Petersburg.\n\nOne last thump, one last groan\u2014and one last meow. Then all was blessedly silent. Except for Clive. He continued to pine for his lost love until four mother-loving a.m.\n\nThe cold war was back on...\nChapter Three\n\nBY THE TIME CLIVE finally settled down and stopped his cat screaming, I was thoroughly exhausted and wide awake. I had to get up in one more hour anyway, and I realized I'd already gotten whatever sleep I was going to get. I might as well get up and make some breakfast.\n\n\"Stupid meower,\" I said, addressing the wall behind my head, and I padded out into the living room. After switching on the TV, I turned on the coffee maker and studied the pre-dawn light just starting to peek in my windows. Clive curled around my legs, and I rolled my eyes at him.\n\n\"Oh, now you want some love from me, huh? After abandoning me for Purina last night? What a jerk you are, Clive,\" I muttered, stretching out my foot and rubbing him with my heel.\n\nHe flopped onto the ground and posed for me. He knew I couldn't resist when he posed. I laughed a little and kneeled next to him. \"Yeah, yeah, I know. You love me now because I'm the one that keeps you in vittles.\" I sighed, scratching his belly.\n\nI headed back into the kitchen, Clive at my heels, and poured some food into a bowl. Now that he had what he needed, I was quickly forgotten. As I headed for the shower, I heard movement in the hallway. Like the Peeping Caroline I was quickly becoming, I pressed my eye to the peephole to see what was happening with Simon and Purina.\n\nHe stood just inside his doorway\u2014far enough inside that I couldn't see his face. Purina stood in the hall, and I could see his hand running through her long hair. I could practically hear her purring through the goddamned door.\n\n\"Mmm, Simon, last night was...mmmm,\" she purred, leaning into his hand, which was now pressed against her cheek.\n\n\"I agree. A fine way to describe the evening and this morning,\" he said quietly as they both chuckled.\n\nNice. Another twofer.\n\n\"Call me when you're back in town?\" she asked as he swept her hair back from her face. Her freshly done face. I miss that face.\n\n\"Oh, you can count on that,\" he answered, and then pulled her back into the doorway for what I can only assume was a kiss that killed. Her foot came up like she was posing. I started to roll my eyes, but that hurt. The right one was pressed so firmly against the peephole, you see.\n\n\"Do svidaniya,\" she whispered in that exotic accent. It sounded much nicer now that she wasn't caterwauling like a kitten in heat.\n\n\"See ya,\" he laughed, and with that, she gracefully walked away.\n\nI strained to see him before he went back inside, but nope. Missed him again. I had to admit, after the spanking and the meowing, I was dying to see what he looked like. There was some serious sexual prowess going on next door. I just didn't see why it had to affect my sleep habits. I pried myself away from the door and made for the shower. Under the water, I pondered what in the world might be required to make a woman meow.\n\nAs seven thirty rolled around, I hopped a cable car and reviewed the day ahead of me. I was meeting a new client, finishing up some details on a project I'd just completed, and having lunch with my boss. I smiled when I thought about Jillian.\n\nJillian Sinclair headed her own design firm, where I'd had the good fortune to intern during my last year at Berkley. In her late thirties, but looking in her late twenties, she'd made a name for herself in the design community early in her career. She challenged convention, was one of the first to sweep \"shabby chic\" off the map, and had been an early trendsetter in bringing back the quiet neutrals and geometric prints of the \"modern\" look that was all the rage now. She hired me after my internship was over, and she'd provided the best experience a young designer could ask for. She was challenging, discerning, had a killer instinct and an even more killer eye for detail. But the best part about working for her? She was fun.\n\nAs I jumped off the cable car, I caught sight of my \"office.\" Jillian Designs was in Russian Hill, a beautiful part of town: fairy tale mansions, quiet streets, and a fantastic view from the taller peaks. Some of the larger old homes had been converted to commercial space, and our building was one of the nicest.\n\nI breathed a sigh when I entered my office. Jillian wanted each designer to make their space their own. It was a way to show potential clients what they could expect, and I'd put a lot of thought into my work space. Deep gray walls were accented by plush, salmon pink curtains. My desk was dark ebony with a chair draped in soft gold and champagne silks. The room was quietly distinguished\u2014with a touch of whimsy coming from my collection of Campbell's Soup ads from the thirties and forties. I'd found a bunch of them at a tag sale, all clipped from old issues of Life magazine. I had them mounted and framed, and I still chuckled every time I looked at them.\n\nI spent a few minutes throwing out the flowers from last week and arranging a new display. Every Monday I stopped in a local shop to choose flowers for the week. The blooms changed, but the colors tended to fall within the same palette. I was particularly fond of deep oranges and pinks, peaches and warm golds. Today I had chosen hybrid tea roses of a beautiful coral color, the tips tinged raspberry.\n\nI stifled a yawn and sat down at my desk, preparing for the day. I caught sight of Jillian as she breezed past my door and waved at her. She came back and stuck her head in. Always pulled together, she was tall, lean, and lovely. Today, clad in black top to bottom but for the fuchsia peep-toe pumps she was rocking, she was chic.\n\n\"Hey, girl! How's the apartment?\" she asked, sitting in the chair across from my desk.\n\n\"Fantastic. Thank you again so much! I can never repay you for this. You are the best,\" I gushed.\n\nJillian had sublet her apartment to me, which she'd had since she moved into the city years ago. Now she was refinishing a house in Sausalito. Rents being what they were in the city, it was a no brainer. The rent control made the price obscenely low. I prepared to gush further when she stopped me with a wave of her hand.\n\n\"Shush, it's nothing. I know I should get rid of it, but it was my first grown-up place in the city, and for the rent it would just break my heart to let it go! Besides, I like the idea of it being lived in again. It's such a great neighborhood.\"\n\nShe smiled, and I stifled another yawn. Her sharp eyes caught it.\n\n\"Caroline, it's Monday morning. How can you be yawning already?\" she chided.\n\nI laughed. \"When's the last time you slept there, Jillian?\" I looked at her over the rim of my coffee cup. It was my third already. I'd be cruising soon.\n\n\"Oh boy, it's been a while. Maybe a year ago? Benjamin was out of town, and I still had a bed over there. Sometimes when I was working late I'd stay in the city overnight. Why do you ask?\"\n\nBenjamin was her fianc\u00e9. Self-made millionaire, venture capitalist, and knockout gorgeous. My friends and I had a killer crush.\n\n\"Did you hear anything from next door?\" I asked.\n\n\"No, no. I don't think so. Like what?\"\n\n\"Hmm, just noises. Late-night noises.\"\n\n\"No, not when I was there. I don't know who lives there now, but I think someone moved in last year, maybe? The year before? Never met him. Why? What did you hear?\"\n\nI blushed furiously and sipped my coffee.\n\n\"Wait a minute. Late-night noises? Caroline? Seriously? Did you hear some sexy times?\" she prodded.\n\nI thumped my head on the desk. Oh, God. Flashbacks. No more thumping. I peeked up at her, and she had her head thrown back in laughter.\n\n\"Aw jeez, Caroline. I had no idea! The last neighbor I remember was in his eighties, and the only noise I ever heard coming from that bedroom was reruns of Gunsmoke. But come to think of it, I could hear that TV show remarkably well...\" She trailed off.\n\n\"Yes, well, Gunsmoke isn't what's coming through those walls now. Straight up sex is coming through those walls. And not sweet, boring sex either. We're talking...interesting.\" I smiled.\n\n\"What did you hear?\" she asked, her eyes lighting up.\n\nI don't care how old you are, or what background you come from, there are two universal truths. We will always laugh at...gas if it happens at the wrong time, and we are always curious about what goes on in other people's bedrooms.\n\n\"Jillian, seriously. It was like nothing I've ever heard before! The first night, they were banging the wall so hard a picture fell off and hit me on the head!\"\n\nHer eyes widened, and she leaned forward on my desk. \"Shut up!\"\n\n\"I will not! Then I heard...Jesus, I heard spanking.\" I was discussing spanking with my boss. Do you see why I love my life?\n\n\"Nooo,\" she breathed, and we giggled like schoolgirls.\n\n\"Yeesss. And he made my bed move, Jillian. Made it move! I saw her the next morning, as Spanx was leaving.\"\n\n\"You call her Spanx?\"\n\n\"You bet! And then last night\u2014\"\n\n\"Two nights in a row! Spanx got spanked again?\"\n\n\"Oh no, last night I was treated to a freak of nature I've named Purina,\" I continued.\n\n\"Purina? I don't get it.\" She frowned.\n\n\"The Russian he made meow last night.\"\n\nShe laughed again, causing Steve from accounting to stick his head in the door.\n\n\"What are you two hens clucking about in here?\" he asked, shaking his head.\n\n\"Nothing,\" we answered at the same time, then cracked up again.\n\n\"Two women in two nights, that's impressive.\" She sighed.\n\n\"Come on, impressive? No. Manwhore? Yes.\"\n\n\"Wow, do you know his name?\"\n\n\"I do, in fact. His name is Simon. I know this because Spanx and Purina kept screaming it over and over again. I could make it out over the banging...Stupid wall banger,\" I muttered.\n\nShe was silent for a moment, and then she grinned. \"Simon Wallbanger\u2014I love it!\"\n\n\"Yeah, you love it. You didn't have your cat trying to mate with Purina through the wall last night.\" I chuckled ruefully and laid my head back on the desk as we continued to giggle.\n\n\"Okay, let's get to work,\" Jillian finally said, wiping the tears from her eyes. \"I need you to land these new clients today. What time are they coming in?\"\n\n\"Ah, Mr. and Mrs. Nicholson are here at one. I've got the presentation and the plans all ready for them. I think they'll really like the way I redesigned their bedroom. We're going to be able to offer an en suite sitting room and an entirely new bathroom. It's pretty great.\"\n\n\"I believe you. Can you run through your ideas with me at lunch?\"\n\n\"Yep, I'm all over it,\" I answered as she headed for the door.\n\n\"You know, Caroline, if you can land this job, it would be huge for the firm,\" she said, eying me over her tortoiseshell glasses.\n\n\"Just wait until you see what I came up with for their new home theater.\"\n\n\"They don't have a home theater.\"\n\n\"Not yet they don't,\" I said, arching my eyebrows and grinning devilishly.\n\n\"Nice,\" she appraised and left to start her day.\n\nThe Nicholsons were definitely a couple I wanted\u2014everyone did. Mimi had done some work for Natalie Nicholson, blueblood and well heeled, when she reorganized her office last year. She referred me when interior design hit the table, and I immediately started plans for their bedroom remodel.\n\nWallbanger. Pffft.\n\n\"Fantastic, Caroline. Simply fantastic,\" Natalie raved as I walked her and her husband to the front door. We'd spent almost two hours going through the plans, and while we'd compromised on a few key points, it was going to be an exciting project.\n\n\"So, you think you're the right designer for us?\" Sam asked, his deep brown eyes twinkling as he wrapped his arm around his wife's waist and played with her ponytail.\n\n\"You tell me,\" I teased back, smiling at the two of them.\n\n\"I think we would love to work with you on this project,\" Natalie said as we shook hands.\n\nI internally high-fived myself, but kept my face composed. \"Excellent. I'll be in touch very soon, and we can get started on a schedule,\" I said as I held the door for them.\n\nI stood in the doorway as I waved them off, then let the door close behind me. I glanced over at Ashley, our receptionist. She raised her eyebrows at me, and I raised mine right back.\n\n\"So?\" she asked.\n\n\"Oh yeah. Nailed it,\" I sighed, and we both squealed. Jillian came down the stairs as we danced about, and she stopped short. \"What the hell happened down here?\" she asked, grinning.\n\n\"Caroline got hired by the Nicholsons!\" Ashley squealed again.\n\n\"Nice.\" Jillian gave me a quick hug. \"Proud of you, kid,\" she whispered, and I beamed. I freaking beamed.\n\nI danced back to my office, putting a little bump and grind in it as I made my way around the desk. I sat down, twirled in my chair, and looked out onto the bay.\n\nWell played, Caroline. Well played.\n\nThat night when I went out to celebrate my success with Mimi and Sophia, I may have imbibed more than a few margaritas. I continued with tequila shots, and I was still licking at the now-nonexistent salt on the inside of my wrist as they walked me up my stairs.\n\n\"Sophia, you're so pretty. You know that, right?\" I cooed, leaning on her as we crawled up the stairs.\n\n\"Yes, Caroline, I'm pretty. Good grasp on the obvious,\" she said. At almost six feet tall with fiery red hair, Sophia was keenly aware of her looks.\n\nMimi laughed, and I turned to her.\n\n\"And you, Mimi, you're my best friend. And you're so tiny! I bet I could carry you around in my pocket.\" I giggled as I tried to find my pocket. Mimi was a petite Filipino, with caramel skin and the blackest hair.\n\n\"We should have cut her off after the guacamole left the table,\" Mimi muttered. \"She is never allowed to drink again without food present.\" She dragged me up the last few steps.\n\n\"Don't talk about me like I'm not here,\" I complained, taking off my jacket and starting in on my shirt.\n\n\"Okay, let's not get naked here in the hallway, huh?\" Sophia shot back, taking my keys from my purse and opening my door. I tried to kiss her on the cheek, and she pushed me off.\n\n\"You smell like tequila and sexual repression, Caroline. Get off me.\" She laughed and opened my door. As we traveled to the bedroom, I caught sight of Clive on the windowsill.\n\n\"Hey there, Clive. How's my big boy?\" I sang.\n\nHe glared at me and stalked off to the living room. He disapproved of my alcohol use. I stuck my tongue out at him. I flopped down on the bed and surveyed my girls in the doorway. They smirked in that you-are-drunk-and-we-are-not-so-we-judge way.\n\n\"Don't act all high and mighty, ladies. I've seen you more drunk than this on many an occasion,\" I noted, my pants going the way of my blouse. Ask me why I kept my heels on, and I will never be able to tell you.\n\nThe two of them pulled down the duvet, and I crawled under the covers and glared. They tucked me in so well that the only things sticking out were my eyeballs, my nostrils, and my messy hair.\n\n\"Why is the room spinning? What the hell did you guys do to Jillian's apartment? She'll kill me if I mess up her rent control!\" I cried, moaning as I watched the room move.\n\n\"The room isn't spinning. Settle down.\" Mimi chuckled, sitting next to me and patting my shoulder.\n\n\"And that thumping, what the hell is that thumping?\" I whispered into Mimi's armpit, which I then sniffed and complimented her deodorant choice.\n\n\"Caroline, there's no thumping. Jesus, you must have had more than we thought!\" Sophia exclaimed, settling down at the end of the bed.\n\n\"No, Sophia, I hear it too. You can't hear that?\" Mimi said in a hushed voice.\n\nSophia was quiet, and all three of us listened. There was a distinct thump, and then an unmistakable groan.\n\n\"Kittens, lay back. You are about to get Wallbanged,\" I stated.\n\nSophia and Mimi's eyes grew wide, but they stayed quiet.\n\nWould it be Spanx? Purina? Anticipating the latter, Clive entered the room and jumped up on the bed. He stared at the wall with rapt attention.\n\nThe four of us sat and waited. I can barely describe what we were subjected to this time.\n\n\"Oh, God.\"\n\nThump.\n\n\"Oh, God.\"\n\nThump thump.\n\nMimi and Sophia looked at Clive and me. We just shook our heads\u2014both of us, really. A slow smile spread across Sophia's face. I focused on the voice coming through the wall. It was different...The pitch was lower, and, well, I couldn't really make out exactly what she was saying. It wasn't Spanx or Purina...\n\n\"Mmm, Simon \u2014\" giggle \"\u2014 right \u2014\" giggle \"\u2014 there!\" giggle.\n\nHuh?\n\n\"Yes, yes \u2014\" snort \"\u2014 yes! Fuck, fuck \u2014\" giggle-hee haw \"\u2014 fuck, yes!\"\n\nShe was giggling. She was a dirty, dirty giggler.\n\nThe three of us tittered along with her as she giggled and snorted her way toward what sounded like one helluva climax. Clive, realizing quickly that his beloved wasn't making an appearance, beat a hasty retreat to the kitchen.\n\n\"What the hell is this?\" Mimi whispered, her eyes as wide as apple pies.\n\n\"This is the sexual torture I've been listening to for the last two nights. You have no idea,\" I growled, feeling the effects of the tequila.\n\n\"LaughyPants has been getting done like this for the last two nights?\" Sophia cried, slapping her hand over her mouth as more moaning laughter filtered through the wall.\n\n\"Oh, hell no. Tonight is the first night I've had the pleasure of this one. The first night was Spanx. She was a naughty, naughty girl and needed to be punished. And last night Clive met the love of his life when Purina made her debut\u2014 \"\n\n\"Why do you call her Purina?\" Sophia interrupted.\n\n\"Because she meows when he makes her come,\" I said, hiding under the covers. My buzz was beginning to fade, replaced by the distinct lack of sleep I'd experienced since moving into this den of debauchery.\n\nSophia and Mimi peeled the covers from my face just as the chick screamed, \"Oh, God that's...that's \u2014\" hahahaha \"\u2014 so good!\"\n\n\"The guy next door can make a woman meow?\" Sophia asked, raising an eyebrow.\n\n\"Apparently so.\" I chuckled, feeling the first wave of nausea wash over me.\n\n\"Why is she laughing? Why would anyone be laughing while they're getting done like that?\" Mimi asked.\n\n\"No idea, but it's nice to hear she's enjoying herself,\" Sophia said, laughing herself at a particularly loud guffaw. Guffaw my aunt Fanny...\n\n\"Have you seen this guy yet?\" asked Mimi, still staring at the wall.\n\n\"Nope. My peephole is getting a workout, though.\"\n\n\"Glad to hear at least one hole is getting some around here,\" Sophia muttered.\n\nI glared at her. \"Charming, Sophia. I've seen the back of his head, and that's it,\" I answered, sitting up.\n\n\"Wow, three girls in three nights. That's some kind of stamina,\" Mimi said, still looking in wonder at the wall.\n\n\"It's some kind of disgusting is what it is. I can't even sleep at night! My poor wall!\" I wailed as I heard a deep groan from him.\n\n\"Your wall, what does your wall have to do \u2014\" Sophia began, and I held up my hand.\n\n\"Wait for it, please,\" I said. He began to bring it on home.\n\nThe wall began to shake with the rhythmic banging, and the woman's giggles got louder and louder. Sophia and Mimi stared in wonder, as I just shook my head.\n\nI could hear Simon moaning, and I knew he was getting close. But his sounds were quickly drowned out by this evening's friend.\n\n\"Oh \u2014\" giggle \"\u2014 that's \u2014\" giggle \"\u2014 it \u2014\" giggle \"\u2014 don't \u2014\" giggle \"\u2014 stop \u2014\" giggle \"\u2014 don't \u2014\" giggle \"\u2014 stop \u2014\" giggle \"\u2014 oh \u2014\" giggle-snort \"\u2014 God \u2014\" giggle-giggle snort-snort \"\u2014 don't \u2014\" giggle \"\u2014 stop!\" giggle.\n\nPlease. Please. Please, stop, I thought.\n\nGiggle-sniffle.\n\nAnd with one last giggle and groan, silence fell across the land. Sophia and Mimi looked at each other, and Sophia said, \"Oh.\"\n\n\"My,\" added Mimi.\n\n\"God,\" they said together.\n\n\"And that's why I can't sleep,\" I sighed.\n\nWhile the three of us recovered from the Giggler, Clive returned to play in the corner with a cotton ball.\n\nGiggler, I think I hate you most of all... \nChapter Four\n\nTHE NEXT FEW NIGHTS were blissfully quiet. No thumping, no spanking, no meowing, and no giggling. Admittedly Clive was a little forlorn from time to time, but everything else around the apartment was great. I met some of my neighbors, including Euan and Antonio who lived downstairs. I hadn't heard or seen Simon since that last night with the Giggler, and while I was grateful for the nights of perfect sleep, I was curious about where he'd disappeared to. Euan and Antonio were only too glad to fill me in.\n\n\"Darling, wait until you see our dear Simon. What a specimen that boy is!\" Euan exclaimed. Antonio had caught me in the hall on my way home and had a cocktail in my hand within seconds.\n\n\"Oh my, yes. He is exquisite! If only I were a few years younger,\" Antonio crooned, fanning himself as Euan looked over his Bloody Mary at him.\n\n\"If you were a few years younger you'd what? Please. You'd never have been in Simon's league. He is filet, while\u2014face it, love\u2014you and I are tube steaks.\"\n\n\"You would know,\" Antonio cackled, sucking pointedly on his celery stalk.\n\n\"Gentleman, please. Tell me about this guy. I admit, after the show he put on last week, I'm a little intrigued about the man behind the wall banging.\"\n\nI'd broken down and told them about Simon's late-night antics after realizing that unless I dished the dirt, they would not reciprocate. They clung to every word like fat kids at a buffet. I told them about the ladies he made the sweet love to, and they filled in a few more blanks.\n\nSimon was a freelance photographer who traveled all over the world. They guessed he was currently on assignment, which explained my quality sleep. Simon worked on projects for The Discovery Channel, The Cousteau Society, National Geographic\u2014all the bigwigs. He'd won awards for his work and even spent some time covering the war in Iraq a few years ago. He always left his car behind when he was traveling: an old, beat-up, black Range Rover Discovery, like the kind you'd find in the African bush. The kind people drove before the yuppies got a hold of them.\n\nBetween what Euan and Antonio told me, the car, the job, and the international house of orgasms from the other side of the wall, I was beginning to piece together a profile of this man, who I still had yet to see. And I'd be lying if I said I wasn't more and more intrigued by the day.\n\nLate one afternoon, after dropping off some tile samples at the Nicholsons, I decided to walk home. The fog had burned off, revealing the city, and it was a nice evening for a stroll. As I rounded the corner to my apartment, I noticed the Range Rover was absent from its usual place behind the building. Which meant it was out and about.\n\nSimon was back in San Francisco.\n\nAlthough I braced myself for another round of wall banging, the next few days were uneventful. I worked, I walked, I Clived. I went out with my girls, I made a great zucchini bread in my now well-broken-in KitchenAid, and I spent time researching my vacation.\n\nEach year, I took a week and vacationed somewhere totally alone. Somewhere exciting, and I never went to the same place twice. One year I spent a week hiking in Yosemite. One year I went zip-lining through a rain forest canopy at an ecolodge in Costa Rica. Another year I spent a week scuba diving off the coast of Belize. And this year...I wasn't sure where I was going to go. Going to Europe was becoming prohibitively expensive in this economy, so that was out. I was considering Peru, as I'd always wanted to see Machu Picchu. I had plenty of time, but often half the fun was deciding where I wanted to spend my vacation.\n\nI also spent an inordinate amount of time at my peephole. Yes, it's true. Whenever I heard a door close, I actually ran to my door. Clive looked on with a smirk. He knew exactly what I was up to. Why he was judging me, however, I will never know, as his ears perked up every time he heard noises coming up the stairs. He was still pining for his Purina.\n\nI still hadn't actually seen Simon. One day I got to the peephole in time to see him going into his apartment, but all I caught was a black T-shirt and a mess of dark hair. And even that could've been dark blond\u2014hard to tell in the muted hallway light. I needed brighter lighting for better sleuthing.\n\nAnother time I saw the Range Rover pulling away from the curb as I came around the corner on my way home from work. It was going to pass right by! Just as I was about to get the first peek at him, actually see the man behind the myth, I tripped and went ass over applecart on the sidewalk. Luckily Euan spotted me and helped me, my bruised ego, and my bruised bum off the concrete and inside for some Bactine with a whiskey chaser.\n\nBut all remained quiet at night. I knew Simon was home, and I could hear him occasionally: a chair leg moving across the floor, a quiet laugh or two. But no harem, and therefore no wallbanging.\n\nHowever, we did sleep together most nights. He played Duke Ellington and Glenn Miller on his side of the wall, and I lay in bed on my side, listening shamelessly. My grandpa used to play his old records at nighttime, and the pop and crackle of a needle on vinyl was comforting as I fell asleep, Clive curled up at my side. I'll say this for Simon: he had good taste in music.\n\nBut this calm and quiet was too good to last, and all hell broke loose again a few nights later.\n\nFirst, I was treated to another round of Spanx. She had once again been a very bad girl and certainly deserved the resounding spanking she received\u2014a spanking that lasted almost half an hour and ended with calls of, \"That's it! Right there. God, yes, right there!\" before the actual walls began to shake. I'd lain awake that night, rolling my eyes and growing more and more frustrated.\n\nThe next morning, from my post at the peephole, I saw Spanx leaving and got my first really good look at her. Pink-faced and glowing, she was a soft, round little bit of a girl with curvy hips and thighs, and packing some serious junk in the trunk. She was short\u2014really short\u2014and a little plump. She had to stand on tiptoes as she kissed Simon goodbye, and I missed seeing him because I watched her walk away. I marveled at his taste in women. She was the total opposite of what I'd seen of Purina, who looked like a model.\n\nAnticipating that Purina was soon up on the roster, the following night I gave Clive a sock full of catnip and a bowlful of tuna. My hope was to get him wasted and passed out before the action started. The treats had the opposite effect. My boy was ready to party down when the first strains of Purina came shrieking through the walls about one fifteen in the morning.\n\nIf Clive could have put on a mini smoking jacket, he would have.\n\nHe stalked the room, pacing back and forth in front of the wall, playing it cool. When Purina began her meows, though, he couldn't contain himself. He once again launched toward the wall. He jumped from nightstand to dresser to shelf, scaling pillows and even a lamp to get closer to his beloved. When he realized he would never be able to burrow under the plaster, he serenaded her with some weird kind of kitty Barry White, his yowls matching hers in intensity.\n\nWhen the walls began to shake, and Simon was bringing it on home, I was amazed they could maintain their control and focus with the racket going on. Clearly, if I could hear them, they must have been able to hear Clive and all his carrying on. Although if I were impaled on the Wallbanger Wondercock, I imagine I could compartmentalize as well...\n\nFor now, though, I was impaled on nothing and getting angry. I was tired, I was horny with no release in sight, and my cat had a Q-Tip sticking out of his mouth that looked frighteningly like a tiny cigarette.\n\nAfter an abbreviated night's sleep, the next morning I dragged myself to the peephole for another round of HaremWatch. I was rewarded with a brief side profile of Simon as he leaned in to kiss Purina goodbye. It was quick, but it was enough to see the jaw: strong, defined, good. He gave great jaw. The best thing about that day was the jaw sighting. The rest of the day was shit.\n\nFirst, there was a problem with the general contractor over at the Nicholson house. It seems he was not only taking extremely long lunch breaks, he was actually blazing it up in their attic every day. The whole third floor smelled like a Dead concert.\n\nThen, an entire pallet of tiles for the bathroom floor arrived cracked and chipped. The amount of time needed to reorder and reship would set the entire project back at least two weeks, leaving no possibility of finishing on time. Any time major construction takes place, the project end date is an estimated time of completion. However, I had never missed a deadline, and this being such a high-profile job, it made me very warm (not in a good way) to realize there was nothing I could do to speed things up short of flying to Italy and bringing back those tiles my damn self.\n\nAfter a quick lunch, during which I spilled an entire soda all over the floor and thoroughly embarrassed myself, I headed back toward work and stopped in a store to look at some new hiking boots. I had plans to go hiking over in the Marin headlands this weekend.\n\nAs I examined the selection, I felt warm breath in my ear that I instinctively flinched against.\n\n\"Hey you,\" I heard, and I froze in terror. Flashbacks poured over me, and I saw spots. I felt cold and hot at the same time, and the single most horrifying experience of my life passed through my mind. I turned and saw...\n\nCory Weinstein. The machine-gun fucker who'd hijacked the O.\n\n\"Caroline, lookin' good in the neighborhood,\" he crooned, channeling his inner Tom Jones.\n\nI swallowed back bile and struggled to keep my composure. \"Cory, good to see you. How are you?\" I managed.\n\n\"Can't complain. Just touring restaurants for the old man. How are you? How's the decorating business treating you?\"\n\n\"Design business, and it's good. In fact, I was just on my way back to work, so if you'll excuse me,\" I sputtered, beginning to push past him.\n\n\"Hey, no rush, pretty thing. Have you had lunch? I can get you a discount on some pizza just a few blocks away. How does five percent off sound to you?\" he said. If it was possible for a voice to swagger, his did.\n\n\"Wow, five percent. As much as that does sweeten the pot, I'm gonna pass.\" I chuckled.\n\n\"So, Caroline, when can I see you again? That night...damn. It was pretty great, huh?\" He winked, and my skin begged me to tear it from my body and throw it at him.\n\n\"No. No, Cory. And hell no,\" I blurted, the bile rising again. Flashes of in and out and in and out and in and out. My hoohah shrieked in its own defense. Granted, the two of us were not on great terms, but nevertheless I knew how afraid she was of the machine gun. Not on my watch.\n\n\"Oh, come on, baby. Let's make some magic,\" he cooed.\n\nHe leaned in, and I could tell he'd had sausage recently. \"Cory, you should know I'm about to vomit on your shoes, so I'd back up if I were you.\"\n\nHe blanched and stepped away.\n\n\"And for the record, I'd rather staple my head to the wall than make magic with you again. You and me and your five-percent discount? Not going to happen. Bye-bye now,\" I said through clenched teeth and stalked out of the store.\n\nI stomped back to work, angry and alone. No Italian tiles, no hiking boots, no man, and no O.\n\nI spent the night on the couch in a funk. I didn't answer the phone. I didn't make dinner. I ate leftover Thai from the takeout container and growled back at Clive when he tried to sneak a shrimp. He flounced to the corner and glared at me from under a chair.\n\nI watched Barefoot Contessa, which usually cheered me up. Tonight she made French onion soup and took it to the beach for lunch with her husband, Jeffrey. Normally watching the two of them made me all warm and fuzzy inside. They were so cute. Tonight they made me nauseous. I wanted to be sitting on the beach in South Hampton, wrapped in a blanket and eating soup with Jeffrey. Well, not Jeffrey per se, but a Jeffrey equivalent. My own Jeffrey.\n\nFucking Jeffrey. Fucking Barefoot Contessa. Fucking lonely takeout.\n\nWhen it was late enough that I could justify going to bed and putting this terrible day behind me, I dragged my sad-sack self back to my bedroom. I went to get my pjs, and realized I hadn't done any laundry. Dammit. I dug around in my jammies drawer, looking for something, anything. I had plenty of sexy little numbers, from back in the day when O and I were on the same page.\n\nI grumbled and fumed and finally pulled out a pink baby doll nightie. It was frilly and sweet, and while I used to love to sleep in beautiful lingerie, I currently hated it. It was a physical reminder of my missing O. Although, it had been a while since I'd attempted to contact her. Maybe tonight would be the night. I was certainly tense. No one could use the release more than me.\n\nI shooshed Clive out and closed the door. No one needed to see this.\n\nI turned on some INXS, since tonight I needed all the help I could get. Michael Hutchence always got me close. I climbed into bed, arranged the pillows behind me, and slipped between the sheets. In the tiny nightie, my bare legs slid along the cool cotton. There's nothing like the feeling of freshly shaved legs on high-thread-count sheets. Maybe this was a good idea after all. I closed my eyes and tried to slow my breathing. The last few times I'd attempted to find the O, I was so thoroughly frustrated that by the end I was near tears.\n\nTonight I began with the usual fantasy roundup. I started with a little Catalano, allowing my hands to slip under the bottom of my nightie and come up to my breasts. As I thought of Jordan Catalano\/Jared Leto kissing Angela Chase in the basement of the school, I imagined it was me. I felt his kisses thick and heavy on my lips, and it became his hands sliding up my skin toward my nipples. As my\/Jordan's fingers began to massage, I felt that familiar tug low in my tummy, getting warm all over.\n\nWith my eyes still closed, the image changed to Jason Bourne\/Matt Damon attacking my skin. With the two of us on the run from the government, only our physical connection kept us alive. My\/Jason's fingers trailed lightly down my belly, sliding inside my matching panties. I could feel it working. My touch was waking something, stirring something inside. I gasped when I felt how ready I was for Jason, and for Jordan.\n\nJesus. The thought of the two of them together, working to bring back the O made me actually twitch. I moaned and went for the big guns.\n\nI went Clooney. Flashes of Clooney came to me as my fingers teased and twirled, twisted and taunted. Danny Ocean...George from Facts Of Life...\n\nAnd then, I went for it.\n\nDr. Ross. Third season of ER, after the Caesar haircut had been rectified. Mmmm...I moaned and groaned. It was working. I was actually getting really turned on. For the first time in months, my brain and the rest of me seemed to be in tune. I rolled onto my side, hand between my legs as I saw Dr. Ross kneeling before me. He licked his lips and asked me when was the last time anyone had made me scream.\n\nYou have no idea. Make me scream, Dr. Ross.\n\nBehind tightly closed eyes, I saw him lean toward me, his mouth getting closer and closer. He gently pressed my knees farther apart, placing kisses on the inside of each thigh. I could actually feel his breath on my legs, which made me shiver.\n\nHis mouth opened, and that perfect Clooney tongue flickered out to taste me.\n\nThump.\n\n\"Oh, God.\"\n\nThump thump.\n\n\"Oh, God.\"\n\nNo. No. No!\n\n\"Simon...mmm \u2014\" giggle.\n\nI couldn't believe it. Even Dr. Ross looked confused.\n\n\"So \u2014\" giggle \"\u2014 fucking \u2014\" giggle \"\u2014 good...hahahaha!\"\n\nI groaned as I felt Dr. Ross leaving me. I was wet, I was frustrated, and now Clooney thought someone was laughing at him. He began to back away...\n\nNo, don't leave me, Dr Ross. Not you!\n\n\"That's it! That's it! Oh...oh...hahahahaha!\"\n\nThe walls began to shake, and the bed-thumping began.\n\nThat's it. Giggle this, bitch...\n\nI scrambled to my feet, the Catalano and the Bourne and the ever-loving Clooney fading away in wisps of testosterone-laden smoke. I threw back the covers, whipped open the door, and stalked out of my bedroom. Clive held out a paw and started to reproach me for shutting him out, but when he saw my face, he wisely let me pass.\n\nI stomped to my front door, my heels pounding into the hardwood floor. I was beyond angry. I was livid. I'd been so close. I opened my front door with the strength of a thousand angry Os, denied release for centuries. I began to pound on his door. I pounded hard and long, like Clooney had been about to pound into me. I banged again and again, never relenting, never letting up. I could hear feet slapping toward the door, but still I didn't let up. The frustration of the day and the week and the months without an O unleashed itself in a tirade the likes of which no one had ever seen.\n\nI heard locks rattling and chains coming undone, but still onward I banged. I began to yell. \"Open this door, you asshole, or I will come through the wall!\"\n\n\"Take it easy. Quit that banging,\" I heard Simon say.\n\nThen the door swung open, and I stared. There he was. Simon.\n\nSilhouetted by soft light from behind, Simon stood with one hand grasping the door and the other hand holding a white sheet around his hips. I looked him over from top to bottom, my hand still hanging the air, clenched into a fist. It was pulsing, I'd been banging so hard.\n\nHe had jet black hair that stood straight up, likely from the Giggler's hands buried in it as he plowed into her. His eyes were piercing blue, and cheekbones just as strong as the jaw. Completing the package? Kiss-swollen lips, and what looked like about three days of scruff.\n\nJesus, there was scruff. How had I missed that this morning?\n\nI gazed down his long, lean body. He was tan, but not a premeditated tan\u2014outdoorsy tan, weathered tan, manly tan. His chest rose and fell as he panted, his skin coated in a thin sheen of sex sweat. As my eyes traveled down further I saw a smattering of dark hair low on his torso, which led below the sheet. Below the six pack. Below that V that some men have, and which on him didn't look weird or BowFlexed.\n\nHe was stunning. Of course he was stunning. And why did there have to be scruff?\n\nI inadvertently gasped as my gaze dropped lower than I had intended. But my eyes were drawn, as if by a magnet, lower and lower. Beneath the sheet\u2014which was already lower on his hips than should be legal\u2014\n\nHe\n\nWas\n\nStill\n\nHard.\nChapter Five\n\n\"OH, GOD.\"\n\nThump.\n\n\"Oh, God.\"\n\nThump thump.\n\nI was traveling up the bed with the strength of his thrusts. He drove into me with unflinching force, giving me exactly what I could take, then pushing me just past that edge. He stared down at me, hard, flashing a knowing smirk. I closed my eyes, letting myself feel how deeply I was being affected. And by deep, I mean deep...\n\nHe grasped my hands and brought them above my head to the headboard.\n\n\"You're gonna wanna hold on tight for this,\" he whispered and threw one of my legs up over his shoulder as he altered the rhythm of his hips.\n\n\"Simon!\" I shrieked, feeling my body begin to spasm. His eyes, those damnable blue eyes, bore into mine as I shook around him.\n\n\"Mmm, Simon!\" I screamed again. And promptly woke up\u2014with my arms over my head, hands tightly grasping the headboard.\n\nI closed my eyes for a moment and forced my fingers to uncurl. When I looked again I could see dents in my hands from gripping so tightly.\n\nI struggled to sit up. I was covered in sweat and panting. I was actually panting. I found the sheets in a ball at the foot of the bed with Clive buried underneath, just his nose peeking out.\n\n\"Oh, Clive, are you hiding?\"\n\n\"Meow,\" came the angry reply, and a tiny face followed the kitty nose.\n\n\"You can come out, silly. Mommy's done screaming. I think.\" I chuckled, running a hand through my damp hair.\n\nI had charmingly sweated through my pjs, so I got up to stand over the A\/C vent, cooling off and beginning to calm down. \"That was close, huh, O?\" I grimaced, pressing my legs together and feeling a not-unpleasant ache between my thighs.\n\nEver since the night Simon and I \"met\" in the hallway, I couldn't stop dreaming about him. I didn't want to, really didn't want to, but my unconscious mind had taken over and was having her way with him. Nocturnally. My body and brain were separate on this one: Brain knew better, Lower Caroline was not so sure...\n\nClive pushed past me and ran into the kitchen to do his little dance next to his bowl.\n\n\"Yah yah yah, settle down,\" I croaked as he threaded himself in and out of my ankles. I dumped a scoop of kibble in his bowl and hit the coffee. I settled against the counter and tried to collect myself. I was still breathing a little hard.\n\nThat dream had been...well, it had been intense. I thought again of his body perched over mine, a bead of sweat rolling off his nose and dropping onto my chest. He'd lowered himself and dragged his tongue up my stomach, toward my breasts, and then...\n\nPing! Ping!\n\nMr. Coffee brought me back from my saucy thoughts, and I was grateful. I could feel myself getting worked up again. Is this going to be a problem?\n\nI poured a cup of coffee, peeled a banana, and looked out the window. I ignored my compulsion to massage the banana and thrust it into my mouth. Oh, sweet Christ, the thrusting! This was headed south fast. And by south I mean...\n\nI slapped myself in the face and forced my mind to think of something besides the manwhore I was currently sharing a wall with. Inane things. Innocuous things.\n\nPuppy dogs...doggy style.\n\nIce cream cones...licking his cone and two scoops.\n\nChildren's games...damn, did I want to do whatever Simon Says...Okay, enough! Now you aren't even trying.\n\nWhile showering I sang \"The Star Spangled Banner\" over and over again to keep my hands from doing anything other than washing up. I needed to remember what an asshole he was\u2014not how he looked in only a sheet and a grin. I closed my eyes and leaned into the spray, remembering that night again. Once I'd stopped staring at his, well, his below the sheet, I'd opened my mouth to speak:\n\n\"Now look here, mister, do you have any idea how loud you are? I need my sleep! If I have to listen to one more night, one more minute, in fact, of you and your harem banging away on my wall, I'll go insane!\"\n\nI yelled to release all the tension that would have, could have, should have been released already in a very Clooney way.\n\n\"Just settle down. It can't be that bad. These walls are pretty thick.\" He grinned, pumping his fist against the doorframe and trying to unleash a little charm. He was clearly used to getting what he wanted. With abs like that, I could see why.\n\nI shook my head to impart focus. \"Are you out of your mind? The walls are not nearly as thick as your head. I can hear everything! Every spank, every meow, every giggle, and I have had it! This shit ends now!\" I screeched, feeling my face burn with fury. I'd even used air quotes to emphasize the spank, meow, and giggle.\n\nAs I spoke of his harem, he began to downshift from charm to chastise. \"Hey, that's about enough!\" he shot back. \"What I do in my home is my business. I'm sorry if I disturbed you, but you can't just come over here in the middle of the night and dictate what I can and can't do! You don't see me coming across the hall and banging on your door.\"\n\n\"No, you just bang on my damn wall. We share a bedroom wall. You're right up against me when I'm trying to sleep. Have some common courtesy.\"\n\n\"Well, how come you can hear me and I can't hear you? Wait, wait, there's no one banging on your walls, is there?\"\n\nHe smirked, and I felt the color drain from my face. I crossed my arms tightly across my chest, and as I looked down, I remembered what I was wearing.\n\nPink baby doll nightie. What a way to establish credibility.\n\nAs I fumed, his eyes drifted down my body, unabashedly taking in the pink and the lace and the way my hip jutted out as I tapped my foot angrily.\n\nHis eyes finally came back up, and he met my stare, not backing down. Then with a twinkle in those baby blues, he winked at me.\n\nI saw red. \"Oooohhh!\" I'd screamed and slammed back into my apartment.\n\nMortified now, I let the water wash away my frustration. I hadn't seen him since, but what if I did? I thumped my head against the tiles.\n\nWhen I opened the front door forty-five minutes later, I tossed a goodbye to Clive over my shoulder and prayed silently that there'd be no random harem girls in the hallway. All clear.\n\nI pushed my sunglasses on as I walked out the door of the building, barely noticing the Range Rover. And by barely, I mean I barely noticed that rover rhymed with over, as in bend me over the chair in my family room and\u2014\n\nCaroline!\n\nI might have a problem here.\n\nLater that afternoon Jillian stuck her head inside my office. \"Knock, knock,\" she said, smiling.\n\n\"Hey! What's going on?\" I leaned back in my chair.\n\n\"Ask me about the house in Sausalito.\"\n\n\"Hey, Jillian, how's the house in Sausalito?\" I asked, rolling my eyes.\n\n\"Done,\" she whispered and threw her arms in the air.\n\n\"Shut up!\" I whispered back.\n\n\"Totally, completely, absolutely done!\" She squealed and sat down across from me.\n\nI offered a fist bump across the desk. \"Now that is some good news. We need to celebrate.\" I reached into a drawer.\n\n\"Caroline, if you pull out a bottle of scotch, I'm going to have to consult human resources,\" she warned, a grin twitching.\n\n\"First of all, you are human resources. And second of all, like I would keep scotch in my office! Obviously that's in a flask lashed to my thigh.\" I giggled, producing a Blow Pop.\n\n\"Nice. Watermelon even. My favorite,\" she said as we unwrapped and began to suck.\n\n\"So, tell me all about it,\" I prompted.\n\nI'd been consulting a little with Jillian as she chose the final touches on the house she and Benjamin had been renovating, and I knew it was just the kind of house I'd been dreaming of for years. Like Jillian, it would be warm, inviting, elegant, and filled with light.\n\nWe talked shop for a while, and then she let me get back to work.\n\n\"By the way, housewarming next weekend. You and your posse are invited,\" she said on her way out the door.\n\n\"Did you just say posse?\" I asked.\n\n\"I might have. You in?\"\n\n\"Sounds great. Can we bring anything, and can we stare at your fianc\u00e9?\"\n\n\"Don't you dare, and I would expect nothing less,\" she fired back.\n\nI smiled as I went back to work. Party in Sausalito? Sounded promising.\n\n\"You don't seriously have a crush on him do you? I mean, how many dreams have you had about him?\" Mimi asked, sucking on her straw.\n\n\"A crush? No, he's an asshole! Why would I\u2014\"\n\n\"Of course she doesn't. Who knows where that dick has been? Caroline would never,\" Sophia answered for me, tossing her hair over her shoulder and stunning stupid a table of businessmen who'd been staring since she walked in. We'd met for lunch at our favorite little bistro in North Beach.\n\nMimi settled back into her chair and giggled, kicking me under the table.\n\n\"Piss off, pipsqueak.\" I stared hard at her, blushing furiously.\n\n\"Yeah, piss off, pipsqueak! Caroline knows better than to...\" Sophia laughed then trailed off, finally taking off her sunglasses and switching her gaze to me.\n\nThe cellist and the pipsqueak watched me fidget. One smiled and the other swore.\n\n\"Ah, jeez, Caroline, do not tell me you are crushing on that guy? Oh no, you are, aren't you?\" Sophia huffed as the waiter set down a bottle of Pellegrino. He stared at her as she ran her fingers through her hair, and she waved him away with a carefully aimed wink. She knew how men looked at her, and it was fun to watch her make them squirm.\n\nMimi was different. She was so tiny and cute that initially men were drawn in by her innate charm. Then they really got a look at her and realized she was lovely. Something about her made men want to take care of her and protect her\u2014until they got her to the bedroom. Or so I'd been told. Crazytown that one was...\n\nI'd been told I was pretty, and on some days I believed it. On a good day I knew I could work it. I never felt as hot as Sophia or as perfectly pulled together as Mimi, but I cleaned up good. I knew when the three of us went out we could really work a scene, and until recently we'd used this to our advantage.\n\nWe each had very distinct types, which was good. We rarely went for the same guy.\n\nSophia was very particular. She liked her men long, lean, and pretty. She liked them not too tall, but taller than her. She wanted her men polite and smart, and preferably with blond hair. It was her true weakness. She also was a sucker for a southern accent. Seriously, if a guy called her \"sugar,\" she'd wet herself. I had firsthand knowledge of this because I'd messed with her one night when she was wasted using my best Oklahoma accent. I had to fight her off the rest of the evening. She claimed it was college, and she wanted to experiment.\n\nMimi, on the other hand, was particular, but not with a specific look. She went for overall size. She liked her men big, huge, tall, and strong. She loved when they had to pick her up to kiss her, or stand her on a stool so they didn't get neck cramps. She liked her men a little on the sarcastic side and hated condescending. Because she was small, she had a tendency to draw types that wanted to \"protect.\" But girlfriend had been taking karate since she was a kid, and she needed no one's protection. She was a badass in a retro skirt.\n\nI was harder to pin down, but I knew him when I saw him. Like the Supreme Court and pornography, I was aware. I did have a tendency toward outdoorsy guys\u2014lifeguards, scuba divers, rock climbers. I liked them clean cut, but a little shaggy, gentlemanly with a touch of bad boy, and making enough money that I didn't have to play mommy. I'd spent a summer with a hotter-than-hell surfer who couldn't afford his own peanut butter. Even Micah's round-the-clock orgasms couldn't save him when I found out he'd been using my AmEx to pay for his sex wax. And his cell phone bill. And his trip to Fiji that I wasn't even invited on. To the curb, surfer boy. To the curb.\n\nI might have taken one more for the road before he left though. Ahh, the days before O's departure. Round-the-clock orgasms. Sigh.\n\n\"So, wait a minute, have you seen him since the hallway encounter?\" Sophia asked after we'd ordered and I'd come back from my surfer memories.\n\n\"No,\" I groaned.\n\nMimi patted my arm soothingly. \"He's cute, isn't he?\"\n\n\"Dammit\u2014yes! Too cute for his own good. He's such an asshole!\" I slammed my hand down on the table so hard I made the silverware bounce. Sophia and Mimi exchanged a glance, and I showed them my middle finger.\n\n\"And then that morning, he's in the hallway with Purina, kissing on her! It's like some sick, twisted orgasm town going over there, and I want no part of it!\" I said, chewing furiously on my lettuce after telling them the story for the third time.\n\n\"I can't believe Jillian didn't warn you about this guy,\" Sophia mused, pushing her croutons around on her plate. She was on a no-bread thing again, terrified of the five pounds she claimed she'd put on in the last year. She was full of it, but there was no arguing with Sophia when she set her mind to something.\n\n\"No, no, she says she doesn't know this guy,\" I reported. \"He must've moved in since the last time she was there. I mean, she hardly ever stayed in that place. They just kept it so they always had a place to stay in the city. According to the neighbors, he's only been in the building a year or so. And he travels all the time.\" As I spoke, I realized I'd compiled quite a dossier on this guy.\n\n\"So has he been wall banging at all this week?\" Sophia asked.\n\n\"Relatively quiet, actually. Either he really listened to me and is being neighborly, or his dick finally broke off in one of them and he's sought medical attention,\" I said, a little too loudly. The table of businessmen must've been listening pretty closely as they all choked a little just then and shifted in their seats, perhaps crossing their legs in unwitting sympathy. We giggled and continued our lunch.\n\n\"Speaking of Jillian, you guys are invited out to the house in Sausalito next weekend for their housewarming party,\" I informed them.\n\nThey both immediately fanned themselves. Benjamin was the one guy we all agreed on. Whenever we'd plied Jillian with enough liquor, we'd confess our crush to her and make her tell us stories about him. If we were lucky and had managed to get an extra martini into her...well, let's just say it was nice to know sex continued to be worth doing even after your man was well into his forties. The one about Benjamin and the Tonga Room at the Fairmont Hotel? Wow. She was a lucky woman.\n\n\"That'll be cool. Why don't we come over and get ready at your place, like the old days?\" Mimi squealed as Sophia and I plugged our ears.\n\n\"Yes, yes, that's fine, but no more squealing or we'll leave your ass with the bill,\" Sophia scolded as Mimi settled back into her seat, eyes sparkling.\n\nAfter lunch, Mimi walked toward her next appointment around the corner, and Sophia and I shared a cab.\n\n\"So, naughty dreams about your neighbor. Let's hear it,\" she began, to the great delight of the cab driver.\n\n\"Eyes on the road, sir,\" I instructed as I caught him looking at us in the rear view mirror.\n\nI let my thoughts drift to the dreams, which had come every night for the past week. I, on the other hand, had not\u2014ratcheting up my sexual frustration to a critical point. When I could ignore the O, I was okay. Now that I was treated to dreams of Simon every night, O's absence was even more pronounced. Clive had taken to sleeping on top of the dresser, safer with my flailing legs, you see.\n\n\"The dreams? The dreams are good, but he's such an asshole!\" I exclaimed, thumping my fist on the door.\n\n\"I know. That's what you keep saying,\" she added, looking at me carefully.\n\n\"What? What is that look?\"\n\n\"Nothing. Just looking at you. You're awfully worked up over someone who's an asshole,\" she said.\n\n\"I know.\" I sighed, looking out the window.\n\n\"You're poking me.\"\n\n\"I am not.\"\n\n\"Seriously, what the hell is in your pocket, Mimi? Are you packing?\" Sophia exclaimed, jerking her head away as Mimi pressed the curling iron through her hair.\n\nI smiled from my place on the bed, lacing up my sandals. I'd put my own hair up in rollers before the girls got here, so I'd been spared the full treatment. Mimi fancied herself some kind of beauty school dropout, and if she could've opened a shop in her bedroom, she'd have given it some careful thought.\n\nMimi produced a brush from her pocket and showed it to Sophia before starting to tease. With the brush, that is.\n\nWe were pre-partying just like we did at Berkeley, right down to the frozen daiquiris. Although we'd upgraded to the good alcohol and freshly squeezed lime juice, it still made us a little hyper and slaphappy.\n\n\"Come on, come on\u2014you never know who you might meet tonight! You don't want to meet Prince Charming with flat hair, do you?\" Mimi reasoned as she forced Sophia to flip her hair over to \"get some lift at the crown.\" You didn't argue\u2014you just let her do it.\n\n\"I'm not flat anywhere. If these girls are on display, Prince Charming won't even notice I have hair,\" Sophia muttered, which sent me into another gale of giggles. Then over our laughter, I heard voices from next door. I got up off the bed and went closer to the wall where I could hear better. This time instead of just Simon, there were two other distinctly male voices. I couldn't make out what they were saying, but suddenly Guns N' Roses came blaring through the walls loud enough to make Sophia and Mimi stop what they were doing.\n\n\"What the hell is that?\" Sophia snapped, looking wildly around the room.\n\n\"Simon's a GN'R fan, I guess.\" I shrugged, secretly enjoying being welcomed to the jungle. I put a headband low on my forehead and did Axl's crab dance back and forth, much to the delight of Mimi and the scorn of Sophia.\n\n\"No, no, no\u2014that's not it, fool,\" Sophia scolded over the music and grabbed another headband. Mimi screamed with laughter as Sophia and I Axl-battled. Until, of course, Sophia started to mess up her hair. Then Mimi lunged. Sophia jumped on the bed to get away from her, and I joined her. We jumped up and down, shrieking the lyrics now and dancing wildly. Mimi finally gave in, and all three of us danced like mad fools. I started to feel the bed moving underneath us, and I realized it was banging merrily against the wall\u2014Simon's wall.\n\n\"Take that! And that! And a little of...that! No one's banging on my walls, huh? Hahahahaha!\" I shrieked crazily as Mimi and Sophia watched in amazement. Sophia climbed off the bed, and she and Mimi clutched each other as they laughed and I thumped. I rocked back and forth like I was surfing, driving my headboard into the wall again and again.\n\nThe music cut off suddenly, and I dropped like I'd been shot. Mimi and Sophia clasped their hands over each other's mouths while I lay flat on the bed, biting my own knuckle to keep from laughing. The frenzy in the room was like when you got caught TP-ing someone's house, or laughing in the back of church. You couldn't stop, and you couldn't not stop.\n\nBang bang bang.\n\nNo way. He was banging at me?\n\nBang bang bang.\n\nHe was banging at me...\n\nBang bang bang! I gave as good as I got. I couldn't believe he had the balls to try to get me to quiet down. I heard male voices chuckling.\n\nBang bang bang came once more, and my temper flared.\n\nOh, he really was an asshole...\n\nI looked at the girls incredulously, and they jumped back on the bed with me.\n\nBang bang bang bang we pounded, six furious fists raining down on the plaster.\n\nBang bang bang bang came back to us\u2014much, much louder this time. His boys must have gotten in on the action.\n\n\"Give it up, mister! No sex for you!\" I yelled at the wall as my girls cackled maniacally.\n\n\"Tons of sex for me, sister. None for you!\" he yelled all too clearly through the wall.\n\nI raised my fists to bang once more. Bang bang ba-bang bang rang out from my side.\n\nBang bang! A single fist answered back, and then all was silent.\n\n\"Oooohhhhh!\" I screamed at the wall, and I could hear Simon and his boys laughing.\n\nMimi, Sophia, and I stared wide-eyed at each other until we heard a tiny sigh from behind us.\n\nWe turned to see Clive sitting on the dresser. He stared back at us, sighed again, and proceeded to lick his bum.\n\n\"The nerve, I mean, the mother-loving nerve of that guy! He has the balls to actually bang on my wall, on my wall? I mean, God what an\u2014\"\n\n\"Asshole, we know,\" Mimi and Sophia said in unison as I continued my rant.\n\n\"Yes, an asshole!\" I continued, still worked up. We were in the car on the way to Jillian's party. The car service had arrived promptly at eight thirty, and we were soon headed over the bridge.\n\nAs I looked out at the twinkling lights of Sausalito, I began to calm down a little. I refused to let that guy upset me. I was out with my two best friends, about to attend a fantastic housewarming hosted by the best boss in the world. And if we were lucky, her fianc\u00e9 would let us see the pictures of him when he was a swimmer in college, back when swimmers still just wore tiny Speedos. We would sigh and gaze endlessly until Jillian made us put them away. And then she would usually put Benjamin away too\u2014for the night.\n\n\"I'm telling you, I have a really good feeling about tonight. I feel like something's going to happen,\" Mimi mused, staring thoughtfully out the window.\n\n\"Something's going to happen, all right. We'll have a great time, drink way too much, and I'll probably try to cop a feel off Caroline on the car ride home,\" Sophia said, winking at me.\n\n\"Mmm, sugar,\" I teased, and she blew me a kiss.\n\n\"Oh, would you two forget your pseudo-lesbian romance? I'm being serious here,\" she continued, sighing in the Harlequin romance voice she used sometimes.\n\n\"Who knows? Not sure about me, but maybe you'll meet your Prince Charming tonight,\" I whispered, smiling back at her hopeful face. Mimi was certainly the most romantic of the three of us. She was steadfast in her belief that everyone had a soulmate.\n\nEh...I'd just settle for my Soul-O.\n\nWhen we pulled up to Benjamin and Jillian's house, there were cars parked everywhere along the winding street, and Japanese lanterns and luminary bags lined the property. Like most houses set into the hilly landscape, from the street there was nothing to look at. We giggled as we made our way through the gate, and I smiled when the girls stared at the contraption before us. I'd seen the plans for this, but had yet to take a ride.\n\n\"What kind of fucked-up rickshaw is this?\" Sophia blurted, and I couldn't help but laugh. Jillian and Benjamin had designed and installed a hillevator, basically an elevator that went up and down the hill. Very practical when you considered the amount of steps it took to reach the house. Their hillside front yard was blanketed with terraced gardens and benches and various garden scenes, all artfully arranged on flagstone paths lit with tiki torches that led down the hill to the house. But for grocery shopping and other less-leisurely approaches, the hillevator made for a much easier ride.\n\n\"Would you ladies care to use the lift or make your way down the path?\" an attendant asked, appearing from the other side of the carriage.\n\n\"You mean, ride in that thing?\" Mimi squeaked.\n\n\"Sure, that's what it's made for. Come on,\" I encouraged, stepping through the little door that he'd opened in the side. It really felt like a ski lift, except that it was going down a hill instead of up in the air.\n\n\"Yeah, okay, let's do it,\" said Sophia, climbing in behind me and plopping down on the seat. Mimi shrugged and followed.\n\n\"There'll be someone at the bottom waiting for you. Enjoy the party, ladies.\" He smiled, and we were off.\n\nAs we rode down the hill, the house rose up to meet us. Jillian had created a purely magical world here, and as there were huge windows throughout the house, we could see into the party as we continued our descent.\n\n\"Wow, there's a lot of people here,\" Mimi noted, her eyes huge. The sounds of a jazz band on one of the many patios below came tinkling up to us.\n\nI felt a little fluttering in my tummy as the cart came to a stop and another attendant came to open the door. As we filed out and our heels click-clacked across the flagstone, I could hear Jillian's voice from inside the house and immediately smiled.\n\n\"Girls! You made it!\" she called as we walked in.\n\nI turned in the space, taking it all in at once. The house was almost like a triangle, set into the hillside and sprawling outward. Deep mahogany wood floors spread out beneath us, and the clean lines of the walls contrasted beautifully. Jillian's personal taste was a comfortable modern, and the colors in the house reflected the colors of the surrounding hillside: warm leafy greens, rich earthy browns, soft muted creams, and hints of deep marine blue.\n\nAlmost the entire back of the two-story house was glass, taking advantage of the spectacular view. The moonlight danced on the water in the bay, and in the distance you could see the lights of San Francisco.\n\nTears sprang to my eyes as I saw the home she and Benjamin had created for themselves and as I looked back at her, I saw the excitement in her eyes. \"It's perfect,\" I whispered, and she hugged me tightly.\n\nSophia and Mimi gushed to Jillian as a waiter brought us each a glass of champagne. When Jillian left to go mingle, the three of us made our way out onto one of the many terraces to take stock. Waiters passed trays, and as we munched on roasted prawns and sipped our bubbly, we scanned the crowd for anyone we knew. Of course many of Jillian's clients were there, and I knew I'd be mixing in a bit of work tonight, but right now I was content to eat my fancy shrimp and listen to Sophia and Mimi size up the men.\n\n\"Oooh, Sophia, I see a cowboy for you right over there\u2014no, no, wait, he's taken by another cowboy. Moving on.\" Mimi sighed as she continued her search.\n\n\"I got him! I spotted your boy for tonight, Mimi!\" Sophia squealed in a whisper.\n\n\"Where, where?\" Mimi whispered back, hiding her mouth behind a prawn. I rolled my eyes and grabbed another glass of bubbly as the waiter passed.\n\n\"Inside\u2014see? Right over there by the island in the kitchen, black sweater and khaki pants? Jesus, he is a tall drink of water...Hmmm, nice hair too,\" Sophia mused, narrowing her eyes.\n\n\"With the curly brown hair? Yes, I could definitely work with that,\" Mimi said, her target acquired. \"Look how tall he is. Now, who is that yummy he's talking to? If that bimbo would just move out of the way,\" Mimi murmured, raising an eyebrow until the alleged bimbo finally moved on, giving us a clearer shot of the man in question.\n\nI looked as well, and as a path opened up, we could now see both of the chatting men. The big guy was, well, big. Tall and broad\u2014linebacker shoulders almost. He filled out his sweater quite nicely, and as he laughed his face lit up. Yeah, he was exactly Mimi's type.\n\nThe other gentleman had wavy blond hair that he constantly pushed behind his ears. He wore bookish glasses that really worked for him. He was long and lean and intense looking, almost classical in his beauty. Make no mistake, this guy was geeky gorgeous, and Sophia drew in a quick breath at the sight of him.\n\nAs we continued to watch the scene unfold, a third man joined them, and we all smiled. Benjamin.\n\nWe headed for the kitchen immediately to say hi to our favorite man on the planet. No doubt Sophia and Mimi were also delighted to have Benjamin handle their introductions. I glanced at the two as they simultaneously worked themselves over. Mimi surreptitiously pinched both cheeks, a la Scarlett O'Hara, and I saw Sophia sneak a quick boob prop. These poor guys didn't stand a chance.\n\nBenjamin caught sight of us on our way across to him and smiled. The guys opened their circle to let us in, and Benjamin enveloped all three of us in a giant hug.\n\n\"My three favorite girls! I was wondering when you were going to turn up. Fashionably late as always,\" he teased, and we all giggled. Benjamin did that\u2014he made us into silly schoolgirls.\n\n\"Hi, Benjamin,\" we said in unison, and it struck me how much we sounded like Benjamin's Angels at that moment.\n\nBig Guy and Glasses stood there grinning as well, perhaps waiting for an introduction as the three of us just stared at Benjamin. He really was aged to perfection: wavy brown hair, just barely beginning to silver by his temples; jeans, a dark blue shirt, and pair of old cowboy boots. He could have walked right off a Ralph Lauren runway.\n\n\"Allow me to make some introductions here. Caroline works with Jillian, and Mimi and Sophia are her, oh, what do you call it\u2014BFFs?\" Benjamin smiled, gesturing to me.\n\n\"Wow, BFFs? Who's been teaching you the lingo, daddy-o?\" I laughed and extended my hand to Big Guy. \"Hi, I'm Caroline. Nice to meet you.\"\n\nHe engulfed my hand with his paw. It actually was like a paw. Mimi was gonna lose her mind with this one. His eyes were full of fun as he smiled down at me.\n\n\"Hey, Caroline. I'm Neil. This tool here is Ryan,\" he said, nodding over his shoulder at Glasses.\n\n\"Thanks, remind me of that next time you can't remember your email password.\" Ryan laughed good-naturedly and extended his hand to me. I shook it, noticing how scorchingly green his eyes were. If Sophia had kids with this guy, they would be illegally beautiful.\n\nI made sure to handle the continued introductions as Benjamin stepped away. We began to small talk, and I chuckled as the four of them began their little getting-to-know-you dance. Neil spotted someone he knew behind me and shouted, \"Hey, Parker, get your pretty-boy ass over here and meet our new friends.\"\n\n\"I'm coming, I'm coming,\" I heard a voice say behind me, and I turned to see who was joining our group.\n\nThe first thing I saw was blue. Blue sweater, blue eyes. Blue. Beautifully blue. Then I saw red as I recognized who belonged to the blue.\n\n\"Fucking Wallbanger,\" I hissed, frozen on the spot.\n\nHis grin slid off as well as he played place-the-face for a moment.\n\n\"Fucking Pink Nightie Girl,\" he finally concluded. He grimaced.\n\nWe stared, seething as the air literally turned electric between us, snapping and crackling.\n\nThe four behind us had fallen silent, listening to this little interchange. Then they caught up.\n\n\"That's Wallbanger?\" Sophia screeched.\n\n\"Wait a minute, that's Pink Nightie Girl?\" Neil laughed, and Mimi and Ryan snorted.\n\nMy face flamed bright red as I processed this information, and Simon's sneer became that damnable smirk I'd seen that night in the hallway\u2014when I'd banged on his door and made him quit giving it to the Giggler and yelled at him. When I'd been wearing...\n\n\"Pink Nightie Girl. Pink Nightie Girl!\" I choked out, beyond pissed. Beyond angry. Well into Furious Town. I stared at him, pouring all of my tension into that one look. All of the sleepless nights and lost Os and cold showers and banana thrusting and merciless wet dreams went into that one look.\n\nI wanted to level him with my eyes, make him beg for mercy. But no...Not Simon, Director of the International House Of Orgasms.\n\nHe\n\nWas\n\nStill\n\nSmirking.\nChapter Six\n\nWE STOOD STARING AT EACH OTHER, waves of anger and annoyance pinging back and forth between us. We glared, he with the smirk and me with the sneer, until I noticed that our very own peanut gallery had fallen silent again, along with every other guest in the kitchen. I looked past my neighbor and saw Jillian standing with Benjamin with an inquisitive look on her face\u2014no doubt wondering why her prot\u00e9g\u00e9 was squaring off in the middle of her housewarming.\n\nWait a minute\u2014how the hell did she know Simon? Why was he even here?\n\nI felt a tiny hand on my shoulder and spun quickly to see Mimi.\n\n\"Easy, Trigger. You don't need to go nuclear at Jillian's, 'kay?\" she whispered, smiling shyly at Simon. I tossed her a look and turned back to him, finding him joined by our hosts.\n\n\"Caroline, I didn't realize you knew Simon. What a small world!\" Jillian exclaimed, clasping her hands together.\n\n\"I wouldn't say I know him, but I'm familiar with his work,\" I replied through clenched teeth. Mimi danced in a circle around us like a little kid with a secret.\n\n\"Jillian, you won't believe this but\u2014\" she started, her voice bubbling over with barely concealed mirth.\n\n\"Mimi...\" I warned.\n\n\"Simon is Simon from next door! Simon Wallbanger!\" Sophia cried, grasping Benjamin's arm. I'm sure she only did it so she could touch the Benjamin.\n\n\"Dammit,\" I breathed as Jillian took in this information.\n\n\"No fucking way,\" she breathed, hand clapping over her mouth as she dropped the f-bomb. Jillian always tried to be such a lady. Benjamin looked confused, and Simon had the decency to blush a little.\n\n\"Asshole,\" I mouthed to him.\n\n\"Cockblocker,\" he mouthed back, the smirk returning in full force.\n\nI gasped. I clenched my fists and prepared to tell him exactly what he could do with his cockblocker when Neil burst in.\n\n\"Benjamin, check this out\u2014this little hottie here is the Pink Nightie Girl! Can you stand it!\" He laughed as Ryan struggled to keep a straight face. Benjamin's eyes widened, and he raised an eyebrow at me. Simon swallowed a laugh.\n\n\"Pink Nightie Girl?\" Jillian asked, and I heard Benjamin lean in and tell her he'd explain later.\n\n\"Okay, that's it!\" I shouted and I pointed at Simon. \"You. A word, please?\" I barked and grabbed him by the arm. I yanked him outside and pulled him down one of the paths that led away from the house. He scrambled along after me, my heels ringing out angrily on the flagstone.\n\n\"Jesus, slow down, will you?\"\n\nMy response was to dig my nails into his arm, which made him yelp. Good.\n\nWe reached a little enclave set away from the house and the party\u2014far enough away that no one would hear him scream when I removed his balls from his body. I released his arm and rounded on him, pointing a finger in his surprised face.\n\n\"You've got some nerve telling everyone about me, asshole! What the hell? Pink Nightie Girl? Are you kidding me?\" I whisper-yelled.\n\n\"Hey, I could ask you the same question! Why do all those girls in there call me Wallbanger, huh? Who's telling tales now?\" he whisper-yelled right back.\n\n\"Are you kidding me? Cockblocker? Just because I refused to spend another night listening to you and your harem does not make me a cockblocker!\" I hissed.\n\n\"Well, due to the fact that your door-banging blocked my cock, it actually does make you a cockblocker. Cockblocker!\" he hissed back. This entire conversation was beginning to sound like something that might have happened in fourth grade\u2014except for all the nighties and the cock talk.\n\n\"Now, you listen here, mister,\" I said, trying for a more adult tone. \"I'm not going to spend every night listening to you try to crash your girl's head through my wall with the force of your dick alone! No way, buddy.\" I pointed a finger at him. He grabbed it.\n\n\"What I do on my side of that wall is my business. Let's get that straight right now. And why are you so concerned about me and my dick anyway?\" he asked, smirking at me again.\n\nIt was the smirk, that damn smirk, that made me go ballistic. That and the fact that he was still holding my finger.\n\n\"It is my business when you and your sex train come knocking on my wall every night!\"\n\n\"You're really fixated on this, aren't you? Wish you were on the other side of that wall? Are you lookin' to ride that sex train, Nightie Girl?\" He chuckled as he wagged his finger in my face.\n\n\"Okay, that's it,\" I growled. I grabbed his finger in defense, which instantly locked us together. We must have looked like two loggers trying to cut down a tree. We struggled back and forth\u2014beyond ridiculous. We both huffed and puffed, each trying to get the upper hand, each refusing to relent.\n\n\"Why are you such a manwhoring asshole?\" I asked, my face inches from his.\n\n\"Why are you such a cockblocking priss?\" he asked, and when I opened my mouth to tell him exactly what I thought, the fucker kissed me.\n\nKissed me.\n\nPlaced his lips on mine and kissed me. Under the moon and the stars, with the sounds of the waves crashing and the crickets cricketing. My eyes were still open, furiously looking back into his. His eyes were so blue, it was like looking at two angry oceans.\n\nHe pulled away, our fingers still gripping each other's like pliers. I released his hand and slapped him across the face. He looked shocked, even more so as I grabbed his sweater and pulled him closer. I kissed him, this time closing my eyes and letting my hands fill with wool and my nose fill with warm boy smell.\n\nGod damn, he smelled good.\n\nHis hands crept around to the small of my back, and as soon as he touched me, I realized where I was and what I was doing. \"Dammit,\" I said, and pulled away. We stood looking at each other, and I wiped at my lips. I started to walk away and then turned back quickly.\n\n\"This never happened, got it?\" I pointed at him again.\n\n\"Whatever you say.\" He smirked, and I felt my temper flare again.\n\n\"And cool it with the Pink Nightie stuff, okay?\" I whisper-yelled and turned to walk back down the path.\n\n\"Until I get to see your other nighties, that's what I'm calling you,\" he shot back, and I almost tripped. I smoothed my dress and headed back to the party.\n\nUnbelievable.\n\n\"So I told the guy, there is no way I'm organizing your 'play room.' You can arrange your own riding crops!\" Mimi shrieked, and we all laughed. She can tell a story like nobody's business. She has a knack for bringing a group together, especially when it's new people just getting to know each other.\n\nAs the party began to wind down, my girls and Simon's guys were gathered around a fire pit on one of the terraces. Dug deep and lined with flagstone, it had benches all around. While the fire crackled merrily, we laughed and drank and told stories. And by that I mean Mimi, Sophia, Neil, and Ryan told stories while Simon and I glared at each other over the flames. With the sparks flying, if I squinted my eyes a little I could imagine him roasting in the fires of hell.\n\n\"So, are we gonna address the elephant in the room here?\" Ryan asked, drawing his knees up and placing his beer on the bench next to him.\n\n\"Which elephant would that be?\" I asked sweetly, sipping my wine.\n\n\"Oh, please\u2014the fact that the guy thumping the headboard off your bed is the hottie across the way, girl!\" Mimi squealed, almost tossing her drink in Neil's face. He laughed along with her, but pried the glass out of her hand before she could do any real damage.\n\n\"There really isn't anything to talk about,\" Simon said. \"I have a new neighbor. Her name is Caroline. That's it.\" He nodded, eyeing me across the fire. I raised my eyebrow and sipped my wine.\n\n\"Yeah, it's nice to know Pink Nightie Girl has a name. The way he described you...wow! I wasn't sure you were real, but you're as hot as he said you were!\" Neil hooted at me appreciatively, trying for a moment to fist bump Simon through the flames before he realized how hot they were.\n\nMy eyes shot to Simon. He grimaced at the description. Interesting...\n\n\"So, you were the guys banging back at us tonight? Listening to the Guns N' Roses?\" Sophia asked, nudging Ryan.\n\n\"You were the girls singing along, I suppose, yes?\" He nudged back, smiling.\n\n\"Small world, isn't it?\" Mimi sighed, gazing up at Neil. He winked at her, and I saw quickly where this was going. She had her giant, Sophia had her pretty boy, and I had my wine. Which was disappearing by the second.\n\n\"Excuse me,\" I muttered and stood up to find a waiter.\n\nI made my way through the dwindling crowd, nodding at a few faces I recognized. I accepted yet another glass of wine and strolled back outside. I'd started back toward the fire pit when I heard Mimi say, \"And you should have heard Caroline when she told us about the night she banged on his door.\"\n\nSophia and Mimi leaned together and said breathlessly, \"He...was...still...hard!\"\n\nThey all dissolved into laughter. I needed to remember to kill those girls tomorrow, with pain.\n\nI groaned at my public humiliation and spun around to stomp off into the gardens when I saw Simon in the shadows. I tried to back away before he saw me, but he waved.\n\n\"Come on, come on, I don't bite,\" he scoffed.\n\n\"Yeah, sure, I guess,\" I answered, walking toward him.\n\nWe stood quietly in the night. I looked out over the bay, enjoying the silence. Then he finally spoke.\n\n\"So I was thinking, since we're neighbors and all\u2014\" he started.\n\nI turned to look at him. He was giving me a sexy little grin, and I knew that's what he used to make the panties drop. Ha\u2014little did he know I wasn't wearing any.\n\n\"You were thinking what? That I'd want to join you some night? See what all the fuss is about? Hop on the welcome wagon? Honey, I have no interest in becoming one of your girls,\" I answered, glaring at him.\n\nHe said nothing.\n\n\"Well?\" I asked, tapping my foot angrily. The nerve of this guy...\n\n\"Actually, I was going to say, since we're neighbors and all, maybe we could call a truce?\" he said quietly, looking at me in a very irritated way.\n\n\"Oh,\" I said. It was all I could say.\n\n\"Or maybe not,\" he finished and started to walk away.\n\n\"Wait, wait, wait, Simon,\" I groaned grabbing him by the wrist as he pushed past me.\n\nHe stood there, glaring.\n\n\"Yes. Fine. We can call a truce. But there will have to be some ground rules,\" I replied, turning to face him. He crossed his arms over his chest.\n\n\"I should warn you now, I don't enjoy women telling me what to do,\" he answered darkly.\n\n\"Not from what I've heard,\" I said under my breath, but he caught it anyway.\n\n\"That's different,\" he said, the cockiness beginning to reappear.\n\n\"Okay, here's the thing. You enjoy yourself, do your thing, hang from the ceiling fans, I don't care. But late at night? Can we keep it down to a dull roar? Please? I gotta get some sleep.\"\n\nHe considered for a moment. \"Yes, I can see where that might be a problem. But you know, you don't really know anything about me, and you certainly don't know anything about me and my 'harem,' as you call it. I don't have to justify my life, or the women in it, to you. So no more nasty judgments, agreed?\"\n\nI considered it. \"Agreed. By the way, I appreciated the quiet this week. Something happen?\"\n\n\"Happen? What do you mean?\" he asked as we walked back to the group.\n\n\"I thought maybe you were injured in the line of duty, like your dick broke off or something,\" I joked, proud to use my zinger again.\n\n\"Unbelievable. That's all you think I am, isn't it?\" he retorted, his face angry again.\n\n\"A dick? Yes, in fact,\" I snapped back.\n\n\"Now look\u2014\" he started, and Neil appeared out of nowhere.\n\n\"Nice to see you two have kissed and made up,\" he chided, pretending to hold Simon back.\n\n\"Can it, anchorman,\" Simon muttered as the rest of the newly paired-off reappeared.\n\n\"Cool it with the anchorman, huh?\" Neil said, and Sophia whirled on him.\n\n\"Anchorman! Wait a minute, you're the local sports guy on NBC, right? Am I right?\" she asked.\n\nI watched his eyes light up. Sophia may have been a classical music kind of girl, but she was also a huge 49ers fan. I was pretty sure the 49ers were a football team.\n\n\"Yeah, that's me. You watch a lot of sports?\" he asked, leaning toward her, bringing Mimi along. The way she was clinging to his arm, it was unavoidable. She stumbled a little, and Ryan swooped in to steady her. They smiled at each other as Sophia and Neil continued their football talk. I coughed, reminding them that I was, in fact, still here.\n\n\"Caroline, we're taking off!\" Sophia giggled, now leaning on Ryan's arm. I glared at Simon one more time and stalked toward the girls.\n\n\"That's good. I've had enough fun for tonight. I'll call for the car, and we can head out in a few,\" I replied, reaching into my bag for my phone.\n\n\"Actually, Neil was telling us about this great little bar, and we were going to go that way. Do you two want to come?\" Mimi interrupted, stopping my hand. She squeezed it, and I saw her shake her head almost imperceptibly.\n\n\"No?\" I asked, raising both eyebrows.\n\n\"Great! Ol' Wallbanger here'll make sure you get home okay,\" Neil said, clapping Simon roughly on the back.\n\n\"Yeah, sure,\" he said through clenched teeth.\n\nBefore I could even blink, the four of them were on their way to the hillevator, saying sloppy goodbyes to Benjamin and Jillian, who just laughed and shared a high five.\n\nWallbanger and I stared at each other, and I suddenly felt exhausted. \"Truce?\" I said tiredly.\n\n\"Truce,\" he said, nodding.\n\nWe left the party together. We drove back across the bridge, with the late-night fog and silence enveloping us. He'd opened the door for me when I approached the Rover, probably some ingrained training from his mother. His hand had rested on the small of my back as I climbed in, and then it was gone and he was around to his side before I even had a chance to make a snarky remark. Maybe it was best; we had called a truce. The second truce within the span of mere minutes. This was going to end badly, I could tell. Still, I would try. I could be neighborly, right?\n\nNeighborly. Ha. That kiss was all kinds of neighborly. I was trying as hard as I could not to think about it, but it just kept bubbling up. I pressed my fingers to my lips without even realizing it, remembering the feeling of his mouth on mine. His kiss was almost a dare, calling my bluff\u2014a promise of what would follow if I allowed it.\n\nMy kiss? Straight up instinct that frankly surprised me. Why had I kissed him? I had no idea, but I did. It must have looked ridiculous. I'd slapped him, then kissed him like some scene from an old Cary Grant movie. I'd thrown my entire body into my kiss, letting my soft places curve against his strong. My mouth had sought his, and his kiss had grown as eager as mine. There was no fairy tale music, but there was something there. And it had quickly hardened against my thigh...\n\nHis messing about with the radio brought me back to the present. He appeared quite focused on the music as we drove across the bridge, which made me quite nervous.\n\n\"Can I help you with that? Please?\" I asked, looking nervously at the water below.\n\n\"No thanks, I got it,\" he said, glancing at me. He must have noticed the way I was peering over the side of the bridge, and he chuckled. \"Okay, sure, go ahead. I mean, you knew every word to 'Welcome to the Jungle,' so you might pick out something good,\" he challenged.\n\nHe returned his eyes to the road, but even from the side, I could see the approving grin. Which, and I hated to admit it, made his jaw look like it had been chiseled out of the hottest piece of granite ever unearthed.\n\n\"I'm sure I can find something,\" I sassed, reaching over his hand and leaning toward him. His hand grazed against the side of my breast, and we both flinched. \"What, you tryin' to cop a feel there?\" I snapped, selecting a song.\n\n\"Did you or did you not just place your tits in the path of my hand?\" he sniped back.\n\n\"I think your hand just moved in front of the girls' trajectory, but don't sweat it. You're hardly the first that these celestial beings have brought into their orbit.\" I sighed dramatically, looking at him sideways to see if he could tell I was joking. The corner of his mouth rose into a grin, and I allowed myself a small smile as well.\n\n\"Yes, celestial. That's the word I was going to use\u2014as in, not of this earth. As in, suspended in the heavens. As in, courtesy of Victoria's Secret.\" He grinned, and I pretended to be shocked.\n\n\"Oh my, you know of the Secret? And here I thought we silly girls had you all fooled.\" I laughed and settled back into my seat. We'd crossed the bridge and now returned to the city.\n\n\"It takes a lot to fool me, especially when it comes to the opposite sex,\" he replied, as the music came on. He nodded at my choice. \"Too Short? Interesting selection. Not many women would have chosen this,\" he mused.\n\n\"What can I say? I'm feeling very Bay Area tonight. And I should tell you now, I am not like most women,\" I added, feeling another smile sneak across my face.\n\n\"I'm beginning to get that,\" he said.\n\nWe were quiet for a few minutes, then suddenly both started to speak at once.\n\n\"So what do you think about\u2014\" I began.\n\n\"Can you believe that they all\u2014\" he said.\n\n\"Go ahead.\" I chuckled.\n\n\"No, what were you going to say?\"\n\n\"I was going to say, so what do you think about our friends tonight?\"\n\n\"That's actually what I was going to say. I couldn't believe they just up and left us!\" He laughed, and I couldn't help but laugh along with him. He had a great laugh.\n\n\"I know, but my girls know what they want. I couldn't have painted two better guys for them. They're exactly what they look for,\" I confided, leaning against the window so I could watch him as we navigated the hilly streets.\n\n\"Yeah, Neil has a weakness for Asian girls\u2014and I swear that sounded less pervy in my head. And Ryan loves him some leggy redheads.\" He laughed again, glancing over to see if I was okay with his leggy redhead comment.\n\nI was. She was.\n\n\"Well, I'm sure I'll hear all about it tomorrow\u2014what kind of impression they made on my ladies. I'll get the full report, don't you worry.\" I sighed. My phone would be ringing off the hook.\n\nSilence crept back in, and I wondered what to say next.\n\n\"So how do you know Benjamin and Jillian?\" he asked, saving me from small talk fever.\n\n\"I work for Jillian at the firm. I'm an interior designer.\"\n\n\"Wait. Hold up, you're that Caroline?\" he asked.\n\n\"I have no idea what that means,\" I answered, wondering why he was now staring at me.\n\n\"Damn, it really is a small world,\" he exclaimed, shaking his head from side to side as though trying to clear it.\n\nHe was silent as I sat there in limbo.\n\n\"Hey, wanna clear that up a bit? What did you mean, that Caroline?\" I finally questioned, slapping his shoulder.\n\n\"It's just that...well...huh. Jillian has mentioned you before. Let's leave it at that,\" he said.\n\n\"Hell no, we won't leave it at that! What did she say?\" I pushed, slapping again at his shoulder.\n\n\"Would you cut that out? You're really rough, you know that?\" he said.\n\nThere were simply too many ways I could go with that comment, so I wisely kept quiet.\n\n\"What did she say about me?\" I asked quietly, now worried that perhaps she'd said something about my work. My nerves were already shot, and now they were pinging.\n\nHe looked over at me. \"No, no, it's not like that,\" he said quickly. \"It's nothing bad. It's just that, well, Jillian adores you. And she adores me\u2014of course, right?\"\n\nI rolled my eyes, but played along.\n\n\"And well, she might have...mentioned a few times...that she thought I should meet you,\" he dragged out, only to wink at me when I met his eyes.\n\n\"Oh. Ohhhh,\" I breathed as I realized what he meant. I blushed. Jillian, that little matchmaking shit. \"Does she know about the harem?\" I asked.\n\n\"Would you quit with that? Don't call them the harem. You make it sound so shady. What if I told you those three women were incredibly important to me? That I care a lot about them. That the relationships I have with them work for us, and no one else needs to understand it\u2014got it?\" he said, pulling the Rover to an angry stop at the curb outside our building.\n\nI was quiet as I studied my hands and watched him rake his through his already messed-up hair.\n\n\"Hey, you know what? You're right. Who am I to say what's right or wrong for anyone else. If it works for you, great. Hit it. Mazel tov. I'm just surprised Jillian would want to set you up with me. She knows I'm a pretty traditional girl, that's all,\" I explained.\n\nHe grinned and turned the force of his blue eyes on me.\n\n\"As it happens, she doesn't know everything about me. I keep my private life private\u2014with the exception of my neighbor with the thin walls and the devastating lingerie,\" he said in a low voice that could melt, well, anything.\n\nMy brain was most certainly among those things, seeing as I suddenly felt it oozing out of my ears and on down to my collar.\n\n\"Except for her,\" I muttered, thoroughly scrambled.\n\nHe let out a dark laugh and opened his door. He kept his eyes on mine as he strode around the car and opened my door.\n\nI climbed down, taking the hand he offered me, and almost not noticing that he traced a tiny circle on the inside of my left hand with his right thumb. Almost didn't notice it, my ass. It made my skin pebble and Lower Caroline sit up straight. Nerves? Shooting like fireworks all over the place.\n\nWe walked inside the building, and he once again opened the door for me. He really was charming, I had to give him that.\n\n\"So how do you know Benjamin and Jillian?\" I asked, walking up the stairs ahead of him. I knew for certain he was checking out my legs, and why wouldn't he? I had great stems, currently flattered by my flouncy little dress.\n\n\"Benjamin's been a friend of my family's for years. I've known him practically my whole life. He also manages my investments,\" Simon answered as we rounded the first floor and started on the second.\n\nI looked over my shoulder and confirmed him peeking at my legs. Ha! Caught him. \"Oooh, your investments. Have a few savings bonds left over from birthdays there, moneybags?\" I teased.\n\nHe chuckled. \"Yeah, something like that.\"\n\nWe continued up the stairs.\n\n\"It's curious, don't you think?\" I offered.\n\n\"Curious?\" he asked, his voice slipping over me like warm honey.\n\n\"Well, I mean, Benjamin and Jillian both knowing us, us meeting at a party like this, and you being the one that's been keeping me nocturnally amused all these weeks. Small world, I suppose?\" We rounded the top stair, and I got my keys out.\n\n\"San Francisco's a big city, but it can feel like a small town in some ways,\" he offered. \"But yes, it's curious. Intriguing even. Who knew that the nice designer Jillian wanted to set me up with was actually Pink Nightie Girl? Had I known, I might have taken her up on it,\" he replied, that damnable grin back on his beautiful face.\n\nDammit, why couldn't he have stayed an asshole?\n\n\"Yes, but Pink Nightie Girl would have said no. After all, thin walls and all...\" I winked, making a fist and thumping on the wall next to my door. I could hear Clive prattling around behind the door, and I needed to get inside before he began to wail.\n\n\"Ah yes, thin walls. Hmmm...Well, good night, Caroline. Truce is still on, right?\" he asked, turning toward his door.\n\n\"Truce is still on, unless you do something to make me mad again.\" I laughed, leaning in the doorway.\n\n\"Oh, count on that. And Caroline? Speaking of thin walls?\" he said, as he opened his door and looked back at me. He leaned in his own doorway, thumping his fist on the wall.\n\n\"Yes?\" I asked, a little too dreamily for my own good.\n\nThe smirk reappeared and he said, \"Sweet dreams.\"\n\nHe thumped the wall one more time, winked, and went inside.\n\nHuh. Sweet dreams and thin walls. Sweet dreams and thin walls...\n\nMother of pearl. He'd heard me.\nChapter Seven\n\n_P _OKE_._\n\n\"Grrr.\"\n\nPoke. Knead, knead. Poke.\n\n\"Enough.\"\n\nKnead, knead, knead. Head butt.\n\n\"I realize you don't know how to read a calendar, but you should know when it's Sunday. Seriously, Clive.\"\n\nHard head butt.\n\nI rolled over, away from Clive's head butts and persistent poking, and pulled the covers over my head. Flashes of the night before kept appearing. Simon in Jillian's kitchen with the intro heard round the world. His friends calling me Pink Nightie Girl. Benjamin putting two and two together when he learned I was the Pink Nightie Girl. Kissing Simon. Mmm, kissing Simon.\n\nNo, no kissing the Simon! I snuggled deeper under the covers.\n\nSweet dreams and thin walls...Sheer mortification washed over me as I remembered his parting words. I burrowed farther under the covers. My heart beat faster, thinking about how embarrassed I'd been. Heart, pay no attention to that girl below the covers.\n\nLast night had been decidedly dream free, but to make sure no one (Simon) could hear me screaming in passion, I'd slept with the TV on. The revelation that Simon had heard me dreaming of him had thrown me for such a loop that I flipped endlessly through the channels, trying to find something that would not sound like me having my own version of the Simon Wet Dream. I ended up on the all-infomercial channel, which, of course, kept me up later that I'd planned. Everything they sold was fascinating. I had to pry the cell phone out of my own hand at three thirty a.m. when I almost ordered the Slap Chop\u2014to say nothing of the half hour I will never get back after watching Bowser try to sell me the Time Life collection of songs from the fifties.\n\nAll this was in addition to listening to the sounds of Tommy Dorsey coming through the wall. They made me smile. I can't lie.\n\nI stretched lazily under the sheet, stifling a giggle as I watched the shadow of Clive stalking me, trying to figure out a way in. He tried every angle as I deflected his advances. Finally, he resumed his poke-poke-knead approach, and I popped my head back up to laugh at him.\n\nI could handle this thing with Simon. I didn't have to be totally embarrassed. Sure, my O was gone, maybe for forever. Sure, I'd been having sex dreams about my overly attractive and overly confident neighbor. And sure, said neighbor had heard these dreams and commented on them, getting the last word in an already extremely bizarre evening.\n\nBut I could handle this. Of course I could. I'd just acknowledge it before he could\u2014take the wind out of his sails, as it were. He didn't always have to have the last word. I could recover from this and keep our ridiculous little truce going.\n\nI'm totally screwed.\n\nJust then I heard the alarm go off next door, and I froze. Then I recovered and slipped back under the covers, leaving just my eyes peeping over.\n\nWait, why was I hiding? He couldn't see me.\n\nI heard him slap at the alarm clock, and his feet hit the floor. Why was he up so early? When all was quiet, you truly could hear through these walls. How the hell did I not realize before that if I could hear him, he could obviously hear me. I felt my face color as I thought of my dreams again, but then I got control. This was further aided by Clive head butting the small of my back in an attempt to physically push me from the bed to give him his breakfast.\n\n\"Okay, okay, let's get up. God, you're such a little jerk sometimes, Clive.\"\n\nHe fired back a reply over his cat shoulder as he stalked toward the kitchen.\n\nAfter getting Mr. Clive fed and running myself through the shower, I headed out to meet the girls for brunch. I was leaving the building while looking at my phone, answering a text from Mimi, when I collided with a wet, hot wall of Simon.\n\n\"Whoa,\" I cried as I teetered backward. His arm shot out and caught me just before I went from flustered to flat-out wrong and on my bottom.\n\n\"Where are you running off to this morning?\" he asked, as I took him in. Sweaty white T-shirt, black running shorts, damp curly hair, iPod, and a grin.\n\n\"You're sweaty,\" I word-vomited.\n\n\"I am sweaty. It happens,\" he added, sweeping the back of his hand across his forehead, making his hair stand straight up. I had to physically block the neurons from my brain trying to get to my fingers with instructions to lift and nestle. Lift and nestle.\n\nHe stared down at me, his blue eyes twinkling. He'd make this painful if I didn't go ahead and out the giant sex elephant in the room.\n\n\"So listen, about last night,\" I started.\n\n\"What about last night? The part where you were berating me about my sex life? Or the part where you were sharing my sex life with your friends?\" he asked, raising an eyebrow and raising his T-shirt to wipe his face. I drew in a breath that sounded like a wind tunnel as I stared at abs that could almost be speedbumps. Why couldn't he be a soft, fat neighbor?\n\n\"No, I mean the crack you made about the sweet dreams. And the...well...the thin walls,\" I stammered, avoiding all eye contact. I was suddenly fascinated by my new shade of toenail polish. It was lovely...\n\n\"Ah, yes, the thin walls. Well, they work both ways, you know. And if someone were to, say, have a very interesting dream some night, well, let's just say it would be quite entertaining,\" he whispered. My knees went a little wobbly. Damn him and his voodoo...\n\nI had to get back in control. I backed up a step.\n\n\"Yes, you may have heard something I would have preferred you not hear, but that's not the way things always go down. So, you got me. But you won't actually ever have me, so let's move on. You got that? And brunch, by the way,\" I finished, concluding my diatribe.\n\nHe looked confused and amused at the same time. \"Brunch, by the way?\"\n\n\"Brunch. You asked where I was off to this morning, and my answer is brunch.\"\n\n\"Ah, got it. And are you meeting your girls that were out with my guys last night?\"\n\n\"I am, and I will gladly share the scoop with you if it's any good,\" I laughed, twirling a piece of hair around my finger. Nice. Flirting 101. What the hell?\n\n\"Oh, I'm sure it's good scoop. Those two look like man-eaters,\" he said, rocking back on his heels as he began to stretch a bit.\n\n\"Are we talking Hannibal?\"\n\n\"No, more like Hall & Oates.\" He laughed, looking up at me as he stretched his hamstrings.\n\nChrist, hamstrings.\n\n\"Yes, well, they can definitely work a room when they need to,\" I said thoughtfully, beginning to back away again.\n\n\"And how about you?\" he asked, standing straight.\n\n\"How about me what?\"\n\n\"Oh, I bet Pink Nightie Girl can work any room she wants.\" He chuckled, his eyes twinkling.\n\n\"Eh, work this,\" I fired back and walked away with a twinkle of my own.\n\n\"Nice,\" he added when I shot him a look over my shoulder.\n\n\"Oh, please, like you're not intrigued,\" I called back from about ten feet away.\n\n\"Oh, I'm intrigued,\" he shouted as I walked backward, shaking my hips while he applauded.\n\n\"Too bad I don't work well with others! I ain't no harem girl!\" I yelled, practically at the corner.\n\n\"Truce still on?\" he yelled.\n\n\"I don't know, what does Simon say?\"\n\n\"Oh, Simon says, hell yes. It's on!\" he shouted back as I rounded the corner.\n\nI twirled about, actually doing a little pirouette. I smiled big as I bounced along, thinking a truce was a very good thing.\n\n\"Egg-white omelet with tomatoes, mushrooms, spinach, and onions.\"\n\n\"Pancakes\u2014four stack, please\u2014with a side of bacon. And I'll need the bacon very crispy, please, but not blackened.\"\n\n\"Two eggs sunny side up, rye toast with butter on the side, and the fruit salad.\"\n\nAfter ordering, we settled in for a morning of coffee and gossip.\n\n\"Okay, so tell me what happened after we left last night,\" Mimi said, placing her chin in her hands and blinking prettily at me.\n\n\"After you left? You mean after you left me with my jerky neighbor to drive me home? What were you thinking? And telling everyone the he-was-still-hard story? Seriously? I'm writing you both out of my will,\" I snapped, swallowing coffee that was too hot and instantly searing off a third of my taste buds. I let my tongue hang out of my mouth to cool.\n\n\"First of all, we told that story because it's funny, and funny is good,\" Sophia began, fishing a piece of ice out of her water glass and handing it to me.\n\n\"Thanh ooo,\" I managed, accepting the cube.\n\nShe nodded. \"And second, you have nothing to leave me anyway, as I already have the entire set of Barefoot Contessa cookbooks, which you bought me yourself. So write me out of the will. And third, the two of you were being such downers there was no way we were taking you out with our new boys,\" Sophia finished, smiling wickedly.\n\n\"New boys. I love new boys.\" Mimi clapped, looking like a Disney cartoon.\n\n\"How was the ride home?\" Sophia asked.\n\n\"The ride home. Well, it was interesting.\" I sighed, now sucking on the cube with wild abandon.\n\n\"Interesting good?\" Mimi squealed.\n\n\"If you call schtupping someone on the Golden Gate Bridge interesting, then yes,\" I replied, calmly drumming my fingers on the table. Mimi's mouth began to fall from her face when Sophia placed her right hand over Mimi's left, which was about to squeeze her fork into something unrecognizable.\n\n\"Sweetie, she's kidding. We would know if Caroline had been schtupped last night. She'd have better skin tone,\" Sophia soothed.\n\nMimi nodded quickly and released the fork. I pitied any guy who pissed her off during a handjob.\n\n\"So, no dish?\" Sophia asked.\n\n\"Hey, you know the rules. You dish, I dish,\" I answered, eyes widening as our breakfast was served. After we dug in, Mimi fired the first shot.\n\n\"Did you know that Neil played football for Stanford? And that he always wanted to go into sports broadcasting?\" she offered, methodically separating her melon from her berries.\n\n\"Good to know, good to know. Did you know Ryan sold some kind of amazing computer program to Hewlett Packard when he was just twenty-three? And that he put all the money in the bank, quit his job, and spent two years teaching English to kids in Thailand?\" Sophia provided next.\n\n\"That's very good to know as well. Did you know that Simon doesn't consider his lady friends a 'harem,' and Jillian at one point actually told him about me as a potential girl he should be dating?\"\n\nWe all hmm-ed and chewed. Then began Round Two.\n\n\"Did you know that Neil loves to windsurf? And he has tickets to the symphony benefit next week? When he found out I was already going with you, Sophia, he suggested we double.\"\n\n\"Mmm, that sounds fun. I was thinking of asking Ryan. Who, by the way, also loves to windsurf. They all do\u2014they surf in the bay whenever they can. And I can also report that he now runs a charity that puts computers and educational materials into inner city schools all over California. It's called\u2014\" Sophia began.\n\n\"No Child Left Offline?\" Mimi quickly finished.\n\nSophia nodded.\n\n\"I love that charity! I give to that organization every year. And Ryan is the one who runs it? Wow...small world,\" Mimi mused as she began to cut her eggs.\n\nQuiet descended while we chewed again, and I tried to come up with something else to say about Simon that didn't have anything to do with him kissing me, me kissing him, or him being aware of my nocturnal verbal emissions.\n\n\"Um, Simon has Too Short on his iPod,\" I mumbled, which was met with hmms, but I knew my dish wasn't as good.\n\n\"Music is important. Who was that guy you were dating who had his own album out?\" Mimi asked.\n\n\"No, no. He didn't have an album out. He was trying to sell his own CDs out of the back of his car. Not the same thing.\" I laughed.\n\n\"You dated another singer too\u2014Coffee House Joe, remember him?\" Sophia snorted into her breakfast.\n\n\"Yes, he was about fifteen years too late for the flannel, but he got an A for angst. And was more than decent in bed.\" I sighed, thinking back.\n\n\"When is this self-imposed dating hiatus going to be over?\" Mimi asked.\n\n\"Not sure. I kinda like not dating anyone.\"\n\n\"Please, who are you kidding?\" Sophia snorted again.\n\n\"You need a tissue over there, Miss Piggy? Seriously, there have been too many Coffee House Joes and Machine Gun Corys. I'm not interested in just dating any more. It's too much of a merry go round. I'm not investing any more time and effort until I know it's going somewhere. And besides, O's off in no-man's land. I might as well join her,\" I added, trying some coffee again and avoiding their eyes.\n\nThey had their Os, and now they had new boys. I didn't expect anyone to join me on my dating sabbatical. But now their faces just looked so sad. I needed to turn this back to them.\n\n\"So last night was good for you guys, huh? Any kisses at the door? Any spit swapping?\" I asked, smiling cheerfully.\n\n\"Yes! I mean, Neil kissed me.\" Mimi sighed.\n\n\"Oooh, I bet he's a good kisser. Did he wrap you up tight and run his hands up and down your back? He has great hands. Did you notice his hands? Damn fine hands,\" Sophia rambled, face in her pancake stack. Mimi and I exchanged a glance and waited for her to come up for air. When she saw us staring, she blushed a little.\n\n\"What? I noticed his hands? They're huge. How could you not?\" she stammered and crammed her mouth full so we would move on.\n\nI giggled and turned my attention back to Mimi. \"So, did Mr. Great Hands use his great hands?\"\n\nIt was Mimi's turn to blush. \"Actually, he was very sweet. Just a little peck on the lips and a nice hug at my door,\" she answered with a giant smile.\n\n\"And you, Miss Thing? Was the computer genius charitable with his goodnight kiss?\" I giggled.\n\n\"Um...yes, he was. He gave me a great goodnight kiss,\" she replied, licking syrup off the back of her hand. She didn't seem to notice the way Mimi's eyes burned a little when she mentioned the goodnight she'd received, but I did.\n\n\"So, you escaped last night unscathed, I take it?\" Mimi asked me, sipping her coffee. I was still nursing the sore tongue, so I chose to stick with juice.\n\n\"I did. We came to a truce and will try to be more neighborly.\"\n\n\"What exactly does that mean?\" she asked.\n\n\"That means he'll try to curtail his activities to earlier in the evening, and I'll try to be more understanding about his sex life, as lively as it is,\" I answered, digging into my purse for some money.\n\n\"One week,\" Sophia muttered.\n\n\"Come again?\"\n\n\"You wish. One week. That's how long I give this truce. You can't keep your opinions to yourself, and he can't keep that Giggler quiet. One week,\" she said again as Mimi smiled away.\n\nHuh, we'll see...\n\nMonday morning, bright and early, Jillian came waltzing in to my office.\n\n\"Knock knock,\" she called. She was the picture of casual chic: hair swept back into a loose bun, little black dress on her little tan body, legs that went on for miles ending in red pumps. Pumps that would probably constitute almost a week's pay for me. She was my mentor in every way, and I made a mental note to make sure I someday obtained the quiet confidence she carried with her.\n\nShe smiled when she saw the new flowers in the vase on my desk. This week I'd chosen orange tulips, three dozen.\n\n\"Morning! Did you see that the Nicholsons have added a home theater? I knew they'd come around.\" I smiled as I sat back in my chair. Jillian settled herself in the chair across from me and just smiled back.\n\n\"Oh, and Mimi is coming over for dinner tonight. We're hoping to finalize the plans for the new closet system she's designing. She wants to add carpet now.\" I shook my head and sipped coffee from the mug on my desk. My tongue had almost healed.\n\nJillian just continued to smile. I began to wonder if I had a Cheerio stuck to my face. \"Did I tell you I got the glass company in Murano to give me a deal on the pieces I ordered for the bathroom chandelier?\" I forged ahead. \"It's going to be beautiful. I think we'll definitely want to use them again.\" I added, smiling hopefully.\n\nShe finally sighed and leaned forward with a cat-that-ate-the-canary-and-went-back-for-the-feathers-to-play-with grin.\n\n\"Jillian, did you have dental work done this morning? Are you trying to show me your new dentures?\" I asked, and she finally flinched.\n\n\"As if I would ever need dentures, pffft. No, I'm waiting for you to tell me about your neighbor, Mr. Parker. Or should I say Simon Wallbanger?\" She laughed, finally sitting back in her chair and giving me a look that said I would not be allowed to leave my office until I told her everything she wanted to know.\n\n\"Hmm, Wallbanger. Where to start? First of all, you can't tell me you didn't know he lived next door. How the hell could you have lived there as long as you did and not know he was the one thumping away every night?\" I inquired, looking back at her with my best detective sneer.\n\n\"Hey, you know I hardly ever stayed there, especially the last few years. I knew he was in that neighborhood, but I had no idea it was next door to the apartment I was subletting! When I see him, it's always with Benjamin, and we usually go out for drinks or we have him over to our place. Regardless, it's the beginnings of a great story, don't you think?\" she tempted, grinning again.\n\n\"Oh, you and your matchmaking. Simon said you'd mentioned me to him before. You are so busted.\"\n\nShe held up her hands in front of her. \"Wait, wait, wait, I had no idea he was so, well, active. I never would have suggested you if I'd known he had so many girlfriends. Benjamin must have known...but it's a guy thing, I guess,\" she replied.\n\nI was the one to lean forward now. \"So tell me, how does he know Benjamin?\"\n\n\"Well, Simon isn't originally from California. He grew up in Philadelphia and only moved out here when he went to Stanford. Benjamin has known him most of his life\u2014he was really close to his dad. He's kind of watched out for Simon\u2014favorite uncle, big brother, surrogate father, that kind of thing,\" she said, her face growing soft.\n\n\"Was really close to his dad? Did they have a falling out or something?\" I asked.\n\n\"Oh, no, no, Benjamin was always great friends with Simon's dad. He was the one who mentored him early in his career. He was very close with the entire family,\" she said, her eyes growing sad.\n\n\"But now?\" I pressed.\n\n\"Simon's parents were killed when he was a senior in high school,\" she said quietly.\n\nMy hand flew to my mouth. \"Oh no,\" I whispered, my heart full of sympathy for someone I barely knew.\n\n\"Car accident. Benjamin says they went really quickly, almost instantly,\" she replied.\n\nWe were quiet for a moment, lost in our own thoughts. I couldn't even process what that must have been like for him.\n\n\"So after the funeral, he stayed in Philadelphia for a while, and he and Simon began to talk about him going to school at Stanford,\" she continued after a moment.\n\nI smiled at the image of Benjamin doing everything he could to help.\n\n\"I can imagine it was probably a good idea for him to get away from everything,\" I said, wondering how I would deal with something like that.\n\n\"Mm-hmm. I think Simon saw the chance, and he took it. And knowing that Benjamin was close by if he needed anything? I think that made it easier,\" she added.\n\n\"When did you meet Simon?\" I asked.\n\n\"His senior year of college. He'd spent some time in Spain the summer before, and when he came home that August he came into the city to have dinner with us. Benjamin and I had been dating for a while by then, so he knew of me, but hadn't actually met me,\" she said.\n\nWow, Simon does Spain. Those poor flamenco dancers\u2014they never stood a chance.\n\n\"We met for dinner, and he charmed the waitress by ordering in Spanish. Then he told Benjamin that if he was ever stupid enough to leave me that he would be quite happy to\u2014now what was it he said?\u2014ah, yes, he would be quite happy to warm my bed.\" She giggled, her face growing pink.\n\nI rolled my eyes. This matched what I knew of him already. Although, as brash as my girls and I were when flirting with Benjamin, it was the pot calling the kettle forward.\n\n\"And that's how I met Simon,\" she finished, her eyes far away. \"He really is pretty great, Caroline, all banging aside.\"\n\n\"Yes, banging aside,\" I mused, running my fingertips back and forth across the tops of the flowers.\n\n\"I hope you get to know him a little better,\" she said with a grin, matchmaker once again.\n\n\"Settle down there. We've called a truce, but that's all.\" I laughed, shaking my finger at her.\n\nShe got up and started for the door. \"You're very sassy for someone who's supposed to be working for me,\" she said, trying to look severe.\n\n\"Well, I'd get a lot more work done if you'd let me get back to it and stop with your nonsense!\" I said, looking severely back at her.\n\nShe laughed and looked out to reception.\n\n\"Hey, Maggie! When did I lose control of this office?\" she called.\n\n\"You never actually had it, Jillian!\" Maggie yelled back.\n\n\"Oh, go make coffee or something! And you,\" she said, turning to me and pointing. \"Design something brilliant for the Nicholsons' basement.\"\n\n\"Again, all things I could've been doing while you were yakking away in here...\" I murmured, tapping my pencil on my watch.\n\nShe sighed. \"Seriously, Caroline, he's really sweet. I think you two could be great friends,\" she said, leaning in the doorway.\n\nWhat's with everyone leaning in doorways lately?\n\n\"Well, I can always use another friend, now, can't I?\" I waved as she disappeared.\n\nFriends. Friends who called a truce.\n\n\"Okay, so we know the floors in the bedroom are going to be reclaimed, honey-toned wood, but you for sure want carpet in the closet?\" I asked, settling on the couch next to Mimi and starting on my second Bloody Mary. We'd been going through her plans for almost an hour as I tried to get her to see that I was not the only one who would have to compromise on her designs. She would as well. As long as we'd been friends, Mimi had believed she won every argument. Mimi saw herself as a badass that could strong-arm anyone into anything. Little did she know Sophia and I had figured out that we only had to let her think she was getting her way, which made her much more tolerable.\n\nThe truth was, I always knew I wanted carpet in the closet\u2014just not for the same reasons she did.\n\n\"Yes, yes, yes! It has to be carpet\u2014really thick and luxurious carpet! It will feel so good under cold toes in the morning,\" she cried, almost shaking in her excitement. I really hoped Neil would be around long enough to romance her right. She needed to release some of this excess energy.\n\n\"Okay, Mimi, I guess you're right. Carpet in the closet. But for that, you have to give me back those two feet you wanted from the bathroom for the rotating shoe rack that I vetoed.\" I spoke carefully, wondering if she would go for it.\n\nShe thought for a moment, looked at her plans again, took a long pull from her cocktail, and nodded. \"Yes, take back the two feet. I get my carpet, and I can live with that.\" She sighed, offering me her hand.\n\nI shook it solemnly and offered her my celery stalk. Clive came sauntering in and began to pace by the front door, pawing under the crack.\n\n\"I bet our Thai is almost here. Let me get my money,\" I said, pointing toward the door as I headed for my purse on the kitchen counter. Just as I spoke, I could hear steps in the hallway.\n\n\"Mimi, get the door, that'll be the takeout guy,\" I called, rummaging through my purse.\n\n\"Got it,\" she yelled, and I heard the door open. \"Oh, hey there, Simon!\" she said, and then I heard the strangest sound.\n\nI would swear, on a stack of Bibles in a court of actual law, that I heard my cat speak.\n\n\"Porrrrreeeennnnnya,\" Clive said, and I whirled about.\n\nIn the span of five seconds, a thousand things happened: I saw Simon and Purina in the hallway, bags from Whole Foods in hands, key in front door. I saw Mimi at the door, barefoot and leaning (again with the leaning) in the doorway. I saw Clive rear back on his hind legs preparing to jump in a way that I'd only ever seen him do once when I hid the catnip on the top of the fridge. Babies were born, old people died, stocks were traded, and someone faked an orgasm. All in those five seconds.\n\nI launched myself at the door in a slow-motion run reminiscent of every action movie ever made.\n\n\"Nooooooooo!\" I cried as I saw a look of panic cross Purina's face and a look of pure lust cross Clive's as he prepared to woo. If I'd started for the door any earlier, maybe even a second earlier, I could've prevented the pandemonium that ensued.\n\nSimon pushed his door open and smiled a confused smile at me as I caught his eye. No doubt he was wondering why I was charging the door and screaming noooooo. Just then Clive jumped. Leapt. Charged. Purina saw Clive jumping directly at her, and she did the worst thing she could've done. She ran. She ran into Simon's apartment. Of course the girl who meows when she has an orgasm is afraid of cats.\n\nClive gave chase, and as I stood in the hallway with Simon and Mimi, we heard shrieking and meowing echoing back to us. It sounded oddly familiar, and I was reminded of Simon bringing it on home. I shook my head and took over.\n\n\"Caroline, what the hell was that? Your cat just\u2014\" Simon was saying, and I placed my hand over his mouth as I hurried past him.\n\n\"We don't have time, Simon! We have to get Clive!\"\n\nMimi followed me into his apartment, Ned Nickerson to my Nancy Drew. I followed the shrieks and meows to the back of the apartment, noticing that Simon's place was an exact mirror image of mine. It was very single guy, with the flat screen TV and the amazing sound system. I didn't really have time for a proper shakedown, but I did notice the mountain bike in the dining room, as well as beautiful framed photographs all over the walls lit by retro sconces. I couldn't admire for long, as I could hear Clive getting worked up in the bedroom.\n\nI paused by the door, listening to Purina scream. I looked back at Simon and Mimi, who wore twin expressions of fear and confusion\u2014although Mimi's also showed quite a bit of merriment.\n\n\"I'm going in,\" I said in a low, brave voice. With a deep breath I pushed the door open, and saw the Bedroom of Sin for the first time. Desk in the corner. Dresser on one wall, with top covered in loose change. More photographs on the wall, black and whites. And there it was: his bed.\n\nCue trumpets.\n\nPushed up against the wall, my wall, was a giant California king, complete with a padded, leather headboard. Padded. It would have to be, now wouldn't it? It was immense. And he had the power to move that thing with his hips alone? Once again Lower Caroline sat up straight and took notice.\n\nI centered, I focused, and I pried my eyes away from Orgasm Central. I scanned and acquired the target: there at the leather club chair in front of the window. Purina perched on the back of this chair, hands in her hair, moaning and wailing and crying. Her skirt was shredded, and there were tiny claw marks in her stockings. She attempted with every fiber of her being to shrink away from the cat on the floor in front of her.\n\nAnd Clive?\n\nClive was strutting. Strutting back and forth in front of her, giving it his all. He turned like he was on a runway, pacing along a line on the floor and glancing at her nonchalantly.\n\nIf Clive could wear a blazer, he would have taken it off, draped it casually over his kitty shoulder, and pointed at her. It was all I could do not to fall down laughing. I stepped toward him, and Purina shouted something at me in Russian. I ignored her and focused all my attention on my cat.\n\n\"Hey, Clive. Hey. Where's my good boy?\" I crooned, and he turned. He glanced at me, and then jerked his head in Purina's direction as though he were making the first round of introductions. \"Who's your new friend?\" I crooned again, shaking my head at Purina when she tried to say something. I held my finger up in front of my lips. This would require great finesse.\n\n\"Clive, come here!\" Mimi yelled and barreled into the room. She always had trouble containing her excitement.\n\nClive made for the door as Mimi made for Clive. Purina made for the bed as I raced after Mimi, who collided with Simon just outside the bedroom door, who was still holding his damn Whole Foods bags. Thoughtfully chosen sustainable organic produce rained down on both of them as I pushed past, hurdling over limbs and a wheel of Brie on my way back to the front door. I caught Clive just as he made a break for the stairs and held him close.\n\n\"Clive, you know better than to run away from Mommy,\" I chastised, as Simon and Mimi finally caught up to us.\n\n\"What the hell are you doing, cockblocker? Are you trying to kill me?\" he shouted.\n\nMimi rounded on him. \"Don't you call her that, you...you...you wallbanger!\" she fired back, smacking his chest.\n\n\"Oh, you two shut up!\" I yelled. Here came Purina down the hallway toward us, wearing only one shoe and a furious look. She began to shout in Russian.\n\nMimi and Simon continued to yell, Purina screamed, Clive struggled to get loose and be reunited with his one and only, and I stood in the middle of the chaos, trying to figure out what the hell had happened in the last two minutes.\n\n\"Get control of your damn cat!\" Simon yelled, as Clive tried to spring free.\n\n\"Don't you yell at Caroline!\" Mimi yelled, smacking him again.\n\n\"Look at my skirt!\" Purina cried.\n\n\"Did someone order pad thai?\" I heard above the chaos. I looked and saw a petrified delivery boy standing on the top step, reluctant to come any further.\n\nEveryone stopped.\n\n\"Unbelievable,\" Mimi muttered and walked into my apartment, motioning for the delivery boy to follow her. I set Clive just inside the door and pulled it shut, cutting off his cries. Simon ushered Purina into his place, telling her softly to find something in his room to put on.\n\n\"I'll be there in just a minute,\" he said and nodded again for her to go inside. She glared at me once more and turned in a huff, slamming the door.\n\nHe turned back to me and we stared at each other, both starting to laugh at the same time.\n\n\"Did that really just happen?\" he asked through his chuckles.\n\n\"I'm afraid it did. Please tell Purina I am sooo sorry,\" I answered, wiping tears from my eyes.\n\n\"I will, but she needs to cool off for a while before I will attempt that\u2014wait, what did you just call her?\" he asked.\n\n\"Umm, Purina?\" I replied, still chortling.\n\n\"Why do you call her that?\" he asked, no longer laughing.\n\n\"Seriously? Come on, you can't figure it out?\" I said.\n\n\"No, tell me,\" he said, running his hands through his hair.\n\n\"Oh, man, you're gonna make me say it? Purina...because she, God, because she meows!\" I blurted, laughing again.\n\nHe blushed deep red and nodded. \"Right, right, of course you would've heard that.\" He laughed. \"Purina,\" he said under his breath and smiled. I could hear Mimi arguing with the delivery guy in my apartment, something about missing spring rolls.\n\n\"She's a little scary, you know?\" Simon said, gesturing toward my door.\n\n\"You have no idea,\" I said. I could still hear Clive wailing behind the door. I pressed my face to the edge and opened it just an inch.\n\n\"Shut it, Clive,\" I hissed. A paw came out through the crack, and I swear he flipped me off.\n\n\"I don't know a lot about cats, but is that normal feline behavior?\" Simon asked.\n\n\"He has a rather odd attachment to your girl there\u2014ever since the second night I lived here. I think he's in love.\"\n\n\"I see. Well, I'll make sure I convey his sentiments to Nadia,\" he said. \"When the time is right, of course.\" He chuckled and prepared to go back inside.\n\n\"You better keep it down over there tonight, or I'll send Clive back,\" I warned.\n\n\"Jesus, no,\" he said.\n\n\"Well, then turn some music on. You gotta give something,\" I pleaded. \"Or he'll be climbing the walls again.\"\n\n\"Music I can do. Any requests?\" he asked, turning to face me from inside the doorway. I backed up to mine and put my hand on my door.\n\n\"Anything but big band, okay?\" I answered softly. Heart moved down low in my tummy, flitting about.\n\nA look of disappointment crossed his face. \"You don't like big band?\" he inquired, his voice low.\n\nI pressed my fingers to my collarbone, my skin feeling warm under his gaze. I watched as his eyes followed my hand, further heating me with the intensity of his gaze.\n\n\"I love it,\" I whispered, and his eyes jerked back to mine in surprise. I smiled a shy smile and disappeared into my apartment, leaving him smiling back at me.\n\nMimi was still yelling at the delivery guy as I came inside to school Clive, a simpering look on both our faces. Five minutes later, with a mouthful of noodles, I heard Purina yelling something in indecipherable Russian on the landing and his door slammed. I tried to hide my grin, instead playing it off as a particularly spicy bite. No wallbanging tonight, I guess...Clive would be so depressed.\n\nAt around eleven thirty that night, as I was settling into bed, Simon played me some music through our shared wall. Wasn't big band, but it was pretty good. Prince. \"Pussy Control.\"\n\nI smiled in spite of myself, delighted at his wicked sense of humor.\n\nFriends? Definitely. Maybe. Possibly.\n\n\"Pussy Control.\" I thought of it again and snorted.\n\nWell played, Simon. Well played.\nChapter Eight\n\nTHE NEXT EVENING I was headed out to yoga when I found myself face to face with Simon once again. He was coming up the stairs as I went down.\n\n\"If I said, 'we have to stop meeting like this,' would it sound as trite as it sounds in my head?\" I offered.\n\nHe laughed. \"Hard to say. Give it a try.\"\n\n\"Okay. Wow, we have to stop meeting like this!\" I exclaimed.\n\nWe both waited a beat and then laughed again.\n\n\"Yep, trite,\" he said.\n\n\"Maybe we can work out some kind of schedule, share custody of the hallway or something.\" I shifted my weight from one leg to another. Great, now it looks like you have to pee.\n\n\"Where are you off to tonight? I seem to always catch you when you're leaving,\" he said as he propped himself up on the wall.\n\n\"Well, clearly I am headed somewhere very fancy.\" I gestured to my yoga pants and cami. I then showed him my water bottle and yoga mat.\n\nHe pretended to think very carefully, and then his eyes widened. \"You're going to pottery class!\"\n\n\"Yes, that's where I'm headed...ass.\"\n\nHe grinned that grin at me. I smiled back.\n\n\"So you never gave me the scoop on what you heard at brunch the other day. What's going on with our friends?\" he asked, and I didn't at all feel a flutter in my belly at the mention of the word our. Not at all...\n\n\"Well, I can tell you that my girls were quite taken with your boys. Did you know they're all going to the symphony benefit next week?\" I said, instantly horrified that I went there that quickly.\n\n\"I heard that. Neil gets tickets every year. Perks of the job, I suppose. Sportscasters always go to the symphony, right?\"\n\n\"I would assume, especially when one is trying to cultivate a certain man-about-town persona,\" I added with a wink.\n\n\"You caught that, huh?\" He winked back, and we found ourselves smiling again. Friends? Definitely a stronger possibility.\n\n\"We'll have to compare notes afterward, see how the Fantastic Four are doing. Did you know they've been going out on double dates all week?\" I said. Sophia had confided that they'd been going out constantly, but always as a foursome. Hmm...\n\n\"I did hear something about that. They all seem to be getting along well. That's good, right?\"\n\n\"It's good, yes. I'm actually going out with them next week. You should come along,\" I tossed out casually. It's all for the truce, just the truce...\n\n\"Oh, wow. I'd love to, but I'm heading overseas. Leaving tomorrow, actually,\" he said.\n\nIf I didn't know better, I'd say he almost looked disappointed.\n\n\"Really? On a shoot?\" I said, and realized my mistake. That knowing smirk came back with a vengeance.\n\n\"A shoot? Checking up on me?\"\n\nI felt my face go from pink to a lovely tomato red. \"Jillian mentioned what you do for a living, yes. And I noticed the pictures in your apartment. When my pussy was chasing your Russian? Ring any bells?\"\n\nHe seemed to shift his weight a little at my choice of words. Hmmm, weak spot?\n\n\"You noticed my pictures?\" he asked.\n\n\"I did. You've got a great set of sconces.\" I smiled sweetly and looked directly at his crotch.\n\n\"Sconces?\" he mumbled, clearing his throat.\n\n\"Occupational hazard. So where are you headed, anyway? Overseas, I mean.\" I dragged my eyes deliberately back up to his, and noticed his were nowhere near my face. Heh, heh, heh...\n\n\"What? Oh, um, Ireland. Shooting a bunch of coastal spots for Cond\u00e9 Nast, and then going into some of the smaller towns,\" he answered, bringing his gaze back to mine.\n\nIt was nice to see him a bit flustered. \"Ireland, nice. Well, bring me back a sweater.\"\n\n\"Sweater, got it. Anything else?\"\n\n\"A pot of gold? And a shamrock?\"\n\n\"Great. I won't have to leave the airport gift shop,\" he muttered.\n\n\"And then when you come home, I'll do a little Irish dance for you!\" I cried and started laughing at the lunacy of this conversation.\n\n\"Aw, Nightie Girl, did you just offer to dance for me?\" he said in a low voice, stepping a little closer.\n\nAnd just like that, the balance of power shifted.\n\n\"Simon, Simon, Simon,\" I exhaled, shaking my head. Mainly to clear it from the effect of him being so near. \"We've been over this. I have no desire to join the harem.\"\n\n\"What makes you think I'd ask you?\"\n\n\"What makes you think you wouldn't? Besides, I think that would mess with the truce, don't you?\" I laughed.\n\n\"Mmm, the truce,\" he said.\n\nJust then I heard steps on the stairs below. \"Simon? Is that you?\" a voice called up.\n\nAt that he leaned back, away from me. I looked down and realized we'd been inching toward each other on the landing throughout our exchange.\n\n\"Hey, Katie, up here!\" he called down.\n\n\"A haremette? I'll watch my walls tonight,\" I said softly.\n\n\"Stop it. She had a hard day at work, and we're heading out to a movie. That's it.\"\n\nHe smiled sheepishly at me, and I laughed. If we were going to be friends, I might as well meet the harem, by God.\n\nA moment later we were joined by Katie, who I, of course, knew as Spanx. I muffled a laugh as I smiled at her.\n\n\"Katie, this is my neighbor, Caroline,\" Simon said. \"Caroline, this is Katie.\"\n\nI offered my hand, and she looked curiously between Simon and me.\n\n\"Hi, Katie. Nice to meet you.\"\n\n\"You too, Caroline. You the one with the cat?\" she asked, a twinkle in her eye. I looked at Simon, and he shrugged.\n\n\"Guilty, although Clive would argue that, in fact, he is an actual person.\"\n\n\"Oh, I know. My dog used to watch TV and bark until I put on something she liked. What a pain in the ass she was.\" She smiled.\n\nWe all stood for a moment, and it was beginning to get a little awkward.\n\n\"Okay, kids, I'm off to yoga. Simon, have a safe trip, and I'll fill you in on the gossip from the new couples when you get back.\"\n\n\"Sounds good. I'll be gone a while, but hopefully they won't get in too much trouble while I'm away.\" He chuckled as they started up the stairs.\n\n\"I'll keep my eyes on them. Nice to meet you, Katie,\" I said, headed down.\n\n\"You too, Caroline. 'Night!\" she called back to me.\n\nAs I walked down the stairs, more slowly than necessary, I heard her say, \"Pink Nightie Girl's pretty.\"\n\n\"Shut it, Katie,\" he fired back, and I swear he swatted her on the butt.\n\nHer yelp a second later confirmed it.\n\nI rolled my eyes as I pushed the door open and headed out to the street. When I got to the gym, I switched my class from yoga to kickboxing.\n\n\"I'll have a vodka martini, straight up with three olives, please.\" The bartender got to work as I looked around the crowded restaurant, taking a break from the Fantastic Four. After two weeks of hearing about all these fabulous double dates, I'd agreed to go out with them and turn them into a Fantastic Five. It was fun, and I was having a great time, but after being with two new couples all night I needed a break. People-watching at the bar was a great way to get some time off. To my left was an interesting couple: silver-haired gentleman with a woman younger than I was who had newly purchased tits. Good girl! You get yours. I mean, if I had to look at flabby, old-man buns I'd want bigger boobs too.\n\nI never thought I'd enjoy being alone, but lately I was finding I did pretty well without a man in my life. I was alone, but I wasn't lonely. Orgasms aside, I did occasionally miss the companionship of a boyfriend, but I liked going places solo. I could travel alone, so why not? Nevertheless, the first time I took myself out to a movie I thought it was going to be weird\u2014the likelihood of running into someone I knew while out and about in the jungles of Costa Rica was slim to none, but running into someone at the movies in the jungles of San Francisco? Odds were greater\u2014but it was great! And a restaurant alone was also just fine. Turns out I'm a great date all by myself.\n\nStill, dinner tonight with my friends had been quite entertaining. The way these two new couples circled each other was fun to watch. Mimi and Sophia had both snagged themselves the men they'd cultivated in their heads as the perfect match. Just then I spotted Sophia in the crowd, her height and gorgeous red hair setting her apart even among hundreds. Hot restaurant, and even hotter bar, this place was packed with people and pretention.\n\nI could see her chatting with someone, and off to the side I found Mimi and Ryan. What was odd? Neil, not Ryan, seemed to be Sophia's conversation partner. Ryan appeared completely captivated by Mimi, her hands moving through the air and punctuating statements with her toothpicked olive as he listened, fascinated. From where I stood, the distance afforded me perfect clarity. I couldn't help but smile. They'd found the boys they always thought they wanted, but now they each seemed fascinated by the other one...ah well, the grass was always cuter, right?\n\nSophia glanced over and spotted me at the bar, and shortly thereafter, she excused herself and headed my way.\n\n\"Having fun?\" I asked as she perched on the stool next to me.\n\n\"I'm having a great time,\" she mused. She then told the bartender exactly how to make her cocktail.\n\n\"How's Neil tonight?\"\n\nHer eyes lit up briefly, and then she seemed to catch herself.\n\n\"Neil? Good, I guess. Ryan looks great, doesn't he?\" she covered, gesturing over to where we'd left our group, and where Mimi and Ryan were still deep in conversation. Ryan did indeed look good in jeans and a shirt that exactly matched his icy blue eyes\u2014the eyes fixed delightedly on Ms. Mimi.\n\nHow could they not see it?\n\n\"Neil looks pretty good tonight too,\" I tossed out, refocusing on the brawny sportscaster. Charcoal sweater, chinos\u2014he was every inch the man about town.\n\n\"Yep,\" she said icily, licking a bit of salt from the rim on her glass.\n\nI giggled and placed a hand on her arm.\n\n\"Come on, pretty girl, let's get you back to your perfect man,\" I said, and we rejoined the group.\n\nI departed a little before my friends did, tired but happy. Once again I'd spent an evening alone and lived to tell the tale. I wondered if other single women understood the delight that came from fifth-wheeling it. To not have to make small talk with some guy you'd been set up with, to not have to worry about some idiot with peppercorn-encrusted-filet breath trying to force his wiggly tongue down the back of your throat, and to not have to explain to that same idiot why you insist on taking a cab home when his super-fast Camaro is parked right over there.\n\nI'd enjoyed\u2014or should I say mostly enjoyed\u2014an assortment of relationships since high school, but hadn't really been in love for a long time. Not since senior year of college. And since that fell apart, I'd had just a series of casual flings, never really feeling fully invested in anyone. Hence my current hiatus from dating. Getting all the parts to line up seemed more and more difficult for me as I got older, and the process could be exhausting. Lower Caroline might be on board, but Brain and Heart always seemed to have reservations. Plus, now that my O was also absent, for who knew how long, I was finding my solitary lifestyle more and more appealing.\n\nAs I mused over these thoughts, headed home in a cab, my phone beeped. I had a text from a number I didn't recognize.\n\nHave a good time tonight?\n\nWho the hell is texting me?\n\nWho the hell is texting me?\n\nAs I waited for the reply, I leaned down and slipped off my shoes. Fantastic heels, but damn, they hurt my feet. My phone beeped again, and I read.\n\nSome people call me Wallbanger.\n\nI hated myself a little for the way my now-naked toes curled. Stupid toes.\n\nWallbanger, huh?  \nWait a minute - how did you get my number?\n\nI knew it was either Mimi or Sophia. Damn girls. They were really pushing it lately.\n\nI can't reveal my sources.  \nSo, did you have a good time tonight?\n\nOkay, I can play this game.\n\nIn fact I did. On my way home now.  \nHow is the Emerald Isle? Lonely yet?\n\nIt's beautiful actually, just having breakfast.  \nAnd I am never lonely.\n\nI believe that. Did you buy my sweater?\n\nWorking on it, want to get just the right one.\n\nYes, please give me a good one.\n\nNot going to respond to that one...how's that pussy of yours?\n\nReally not going to respond to that one.  \nIs there something you wanted?\n\nThis not responding thing is getting harder...\n\nI know what you mean. It's hard not to touch that one.\n\nOkay, officially ending this round.  \nThe innuendos are too thick to see straight.\n\nOh, I don't know, it's better when it's thick...\n\nWow. I'm enjoying this truce more than I expected.\n\nI have to admit it's good for me too.\n\nAre you home yet?\n\nYep, just pulled up in front of our building.\n\nOkay, I'll wait until you're inside.\n\nBet you can't wait to get inside.\n\nYou're a demon, you know that?\n\nI have been told. Okay, inside. Just kicked your door, btw.\n\nThanks.\n\nJust being a good neighbor.\n\nGoodnight, Caroline.\n\nGood morning, Simon.\n\nI laughed as I turned the key in my lock and went inside. I sank into my couch, still laughing. Clive quickly jumped into my lap, and I petted his silky fur as he purred his welcome home. My phone beeped once more.\n\nDid you really kick my door?\n\nShut up. Go eat your breakfast.\n\nI laughed again as I silenced my phone for the night and lay back onto the couch. Clive perched on my chest as I relaxed for a bit, thoughts of that damn wallbanger in my head. It was shocking how clearly I could picture him: soft faded jeans, hiking boots a la Jake Ryan from Sixteen Candles, off-white Irish cable knit turtleneck sweater, hair all in disarray. Standing on a rocky coast somewhere, ocean in the background. A little tan, slightly weathered, hands in pockets. And that grin... \nChapter Nine\n\nTEXT BETWEEN CAROLINE AND SIMON:\n\nYou had a package delivered.  \nI signed for it and it's at my place.\n\nThanks. I'll pick it up when I'm back. How are you?\n\nGood, just working. How are the Irish?\n\nLucky. How's that insane cat?\n\nLucky. I caught him trying to climb the walls.  \nHe's still looking for Purina. Misses her.\n\nI don't think a romance is in the cards for those two.\n\nProbably not...he won't be over it anytime soon tho.  \nMight have to bump up his catnip ration.\n\nDon't overmedicate.  \nNo one likes a pussy that can't hold a conversation.\n\nI'm actually a little scared of you.\n\nLOL. Don't be scared. Wait until I offer you candy for that.\n\nIf I catch you in a trench coat I'm running the other way!  \nWhen are you coming home btw?\n\nMissing me a little?\n\nNo, I wanted to re-hang some pictures on the wall behind my headboard and I'm wondering how much time I have.\n\nBe home in 2 weeks. If you can wait that long,  \nI'll help you. It's the least I can do.\n\nThe very least, and I'll wait. You provide the hammer,  \nI'll provide the cocktails.\n\nCurious about my hammer, are you?\n\nGoing across the hall right now to kick your door.\n\nText between Mimi and Caroline:\n\nGirl, guess what? Sophia's grandparents' house is available next month. We're on our way to Tahoe, baby!\n\nSweet! That'll be nice.  \nI've been dying to get away with my girls.\n\nWe were thinking of inviting the boys...is that cool with you?\n\nThat's fine. The four of you will have a great time.\n\nIdiot, obviously you're still invited.\n\nAw thx! I'd love to go along on a romantic weekend  \nwith 2 couples. FANTASTIC!\n\nDon't be an asshole. You're totally still coming. You won't be a 5th wheel. It'll be so fun! Did you know Ryan plays guitar? He's gonna bring it, and we can sing along!\n\nWhat is this...camp? No thx!\n\nText between Mimi and Neil:\n\nHey, Big Man, what are you doing middle of next month?\n\nHey, Shortie. No plans yet. What's up?\n\nSophia's grandparents are gonna let us have  \nthe Tahoe house. You in? Ask Ryan...\n\nHells yes! I'm there. I'll ask the nerd if he's in.\n\nTrying to talk Caroline in to coming along too.\n\nGreat! The more the merrier.  \nWe still meeting for drinks with Sophia and Ryan tonight?\n\nYep, see you then.\n\nYou got it, kiddo.\n\nText between Simon and Neil:\n\nQuit fucking asking me about Lucky Charms.\n\nThat little guy cracks me up every time!  \nHey, when are you home?  \nWe're headed up to Tahoe for a weekend next month.\n\nI'll be home next week. Who's going?\n\nSophia and Mimi, me and Ryan. Maybe Caroline.  \nThat girl's pretty cool.\n\nYah, she's pretty cool when she's not cockblocking.  \nTahoe, huh?\n\nYep, Sophia's grandparents have a house there.\n\nNice.\n\nText between Simon and Caroline:\n\nYou going to Tahoe?\n\nHow the hell did you hear about that already?\n\nWord gets around...Neil is pretty excited.\n\nOh, I'm sure he is.  \nSophia in a hot tub - isn't too hard to figure out.\n\nWait, I thought he was dating Mimi.\n\nOh, he is, but he is def thinking about Sophia in a hot tub, trust me.\n\nWhat the hell?\n\nStrange things afoot in San Francisco.  \nThey're each dating the wrong person.\n\nWhat?\n\nIt's shocking. Mimi can't stop talking about Ryan,  \nwho's usually staring like a sad puppy dog at her. And Sophia is so busy mooning over Neil's giant man hands she can't see that he's staring right back at her. Pretty funny.\n\nWhy don't they swap?\n\nSays the guy with the harem...it's not always that easy.\n\nWait until I get home, I'll take care of it.\n\nOkay, Mr. Fix-It. Before or after you hang my pictures?\n\nDon't worry, Nightie Girl.  \nI'm all about getting into your bedroom.\n\nSigh\n\nDid you really just type the word sigh?\n\nSigh...\n\nAre you going to Tahoe?\n\nNot if I can help it. Although it would almost be worth it  \nto watch the chaos when they finally figure this out.\n\nIndeed.\n\nText between Caroline and Sophia:\n\nWhat's this I hear about you not coming to Tahoe?\n\nUgh! What's the big deal?\n\nEasy, Trigger. What crawled up your ass?\n\nI just don't know why it's essential that I accompany all of you on a romantic weekend. I'm perfectly happy to go next time. Going out with you guys here is one thing. Tagging along to Tahoe? I don't think so.\n\nIt won't be like that. I promise.\n\nI already have to hear Simon banging on the walls when he's home. I don't need to hear Ryan drilling you in the next room, or Mimi getting manhandled.\n\nDo you think he's manhandling her?\n\nWhat?\n\nNeil. Do you think he's manhandling her?\n\nIs he what?\n\nOh, you know what I mean...\n\nAre you actually asking me if our dear friend Mimi  \nis having sex with her new boy toy?\n\nYes! I'm asking!\n\nAs it happens, no. They're not manhandling yet. Wait, why are you asking? You've slept with Ryan right? Right????\n\nGotta go.\n\nText between Sophia and Ryan:\n\nIs it weird that we only ever go out on double dates  \nwith Mimi and Neil?\n\nWhat?\n\nIs it weird?\n\nI don't know. Is it?\n\nYes. Tonight you're coming over, alone,  \nand we're watching a movie.\n\nYes, ma'am.\n\nAnd btw, ask your buddy Simon to come to Tahoe.\n\nAny specific reason I'm doing this?\n\nYep.\n\nCare to share?\n\nNope. Bring popcorn.\n\nText between Ryan and Simon:\n\nAre you sick of green yet?\n\nI'm ready to come home, yes. My flight gets in late tomorrow night. Or tonight. Shit, I don't know.\n\nSophia asked me to officially ask you if you want to come along to Tahoe. You in?\n\nTahoe, huh?\n\nYep. I think Caroline is going.\n\nI thought she wasn't going.\n\nHave you been talking to the Cockblocker?\n\nSome. She's pretty cool. The truce seems to be holding.\n\nHmmm. So, Tahoe?\n\nLet me think about it. Windsurfing this weekend?\n\nYep.\n\nText between Simon and Caroline:\n\nSo I got invited to the Tahoe thing. Are you going?\n\nYou got invited? Ugh...\n\nI take it you're still not sold on the idea?\n\nI don't know. I love going up there, and the house is pretty fantastic. Are you going?\n\nAre you going?\n\nI asked you first.\n\nSo what?\n\nChild. Yes, I suppose I will end up going.\n\nGreat! I love it up there.\n\nOh, you're going now?\n\nMight as well. Sounds like fun.\n\nHmm, we'll see. Home tomorrow, yes?\n\nYep, late flight in and then sleeping for at least a day.\n\nLet me know when you're up. I've got that package for you.\n\nWill do.\n\nAnd I'm baking zucchini bread tonight. I'll save some for you. You probably have no groceries at all, right?\n\nYou make zucchini bread?\n\nYep\n\nSigh...\n\nI woke up suddenly and heard music coming from next door. Duke Ellington. I looked at the clock. It was after two in the morning. Clive poked his head out from under the covers and hissed.\n\n\"Oh, shut up. Don't be jealous,\" I hissed back.\n\nHe glared at me, showing me his bum as he turned and wiggled his way back under the covers, head first.\n\nI snuggled in deeper myself, smiling as I listened to the music.\n\nSimon was home.\n\nThe next morning I woke up so happy it was Saturday. I was caught up on everything: no laundry to do, no errands to run. Just a day to enjoy and relax. Fantastic.\n\nI decided to start with a nice long bath, and then I'd decide what to do with my day. I was thinking of a run at Golden Gate Park that afternoon. Fall in San Francisco was so pretty when the weather held. I just might take a book and spend the entire afternoon there.\n\nI started the bath and Clive came in to keep me company. He weaved in and out of my legs as I dropped my pjs on the floor and meowed as he explored the top of the tub. He loved to balance on the edge while I took a bath. He'd never fallen in, although sometimes he would dip his tail. Silly cat\u2014one of these days he was gonna dip more than his tail.\n\nI tested the water. It was just beginning to make its way up the side of the giant tub when I decided I needed a little coffee before I settled in. I padded out to the kitchen\u2014naked as the day is long\u2014to make myself a cup. I yawned as I measured the beans for the grinder.\n\nI tossed a few spoonfuls into the filter and went to get water. As soon as I turned on the faucet, the screeching began.\n\nFirst I heard Clive meow like never before. Then I heard splashing. I started to smile, thinking he'd finally fallen in, when the water from the sink shot straight in my face.\n\nI blinked furiously, confused until I realized water was shooting out the top of the faucet, spraying the entire kitchen. \"Shit!\" I screamed, trying to turn it off. No luck.\n\nI ran to the bathroom, still swearing and found Clive hiding behind the toilet, soaking wet, and the tub faucet spraying wildly all over the bathroom. \"What the\u2014?\" I cried, trying again to turn off the water. Then I began to panic. It was like the entire apartment had gone haywire at the same moment. There was water spraying everywhere, and Clive was still screeching at the top of his lungs.\n\nI was naked, sopping wet, and freaking out.\n\n\"Motherfuckingcocksuckershitdamndamn!\" I screamed and grabbed a towel. I tried to think, tried to calm down. There must be a shut-off valve somewhere. I'd redesigned bathrooms, for Christ's sake. Think, Caroline!\n\nAbout this time I heard the banging coming from somewhere else in the apartment. Of course I thought it was the bedroom first\u2014naturally. But no, it was the front door.\n\nWrapping the towel around myself and still cursing enough to make a sailor blush, I stomped across the floor, fortunately not slipping in the collecting water, and angrily swung the door open.\n\nOf course it was Simon.\n\n\"Are you out of your goddamned mind? What's with all the screaming?\"\n\nI practically didn't notice the green plaid boxers, the sleep hair, or the speedbump abs. Practically.\n\nSurvival mode kicked in, and I grabbed him by the elbow as he was rubbing his eye and dragged him forcibly into the apartment. \"Where the hell is the shut-off valve in these apartments?\" I shrieked.\n\nHe looked around at the chaos: water spraying from the kitchen, water on the floor from the bathroom, and me in my Camp Snoopy towel, which was the first one I grabbed.\n\nEven in a crisis Simon took 2.5 seconds to look at my nearly naked body. Okay, I might have taken 3.2 to look at his.\n\nThen we both snapped into action. He ran into the bathroom like a man on a mission, and I could hear him knocking around. Clive hissed and ran out, straight into the kitchen. Realizing it was just as wet in there, he leapt across the room in an acrobatic fit and landed high atop the fridge. I started to run to the bathroom to help and collided with Simon as he ran to the kitchen. Undeterred, he slid across the floor and opened the doors under the sink. He began throwing my cleaning supplies all over the floor, and I assumed he was trying to get at the shut-off valve. I tried not to notice the way the back of his boxers clung to his buns. I tried so very hard. He was covered in water as well now, and just then his feet slipped out from under him, crashing him to the floor.\n\n\"Ow,\" he said from under the sink, his legs now splayed out across my wet kitchen floor. Then he rolled over. He was soaking wet and a tad bit glorious.\n\n\"Get over here and help me. I can't get this one turned off,\" he requested over the rushing water and the cat meowing.\n\nRemembering that I was only wearing a towel, I gingerly knelt next to him and tried to avoid looking at his body\u2014his wet, long, lean body that was dangerously close to my own. One more random jet of water straight into my eyeball was enough to pull me from my stupor, and I renewed my focus.\n\n\"What do you want me to do?\" I yelled.\n\n\"Do you have a wrench?\"\n\n\"Yes!\"\n\n\"Can you go get it?\"\n\n\"Sure!\"\n\n\"Why are you yelling?\"\n\n\"I don't know!\" I sat there, trying to see underneath the sink.\n\n\"Well, go get it, for God's sake!\"\n\n\"Right. Right!\" I yelled and ran for the hall closet.\n\nWhen I came back, I slipped a little on the wet tile and slid into his side.\n\n\"Here!\" I yelled and thrust the wrench under the sink.\n\nI watched him work, his face hidden. His arms strained, and I saw how strong he really was. I watched in amazement as his stomach hardened and revealed six little packs. Oops, make that eight. And then the V showed up. Hello, V...\n\nHe grunted and groaned and as he strained to turn off the valve, his entire body caught up in the struggle. I watched as he fought the Battle of the Valve and was finally triumphant. I also kept a close eye on those green plaid boxers, which when wet, clung to him like a second skin. Skin that was wet, and probably warm, and\u2014\n\n\"Got it!\"\n\n\"Hurray!\" I clapped as the water finally stopped. He let out one last groan, which sounded oddly familiar, and relaxed. I watched as he slid out from under the sink.\n\nHe lay next to me on the floor, soaked and in his boxers.\n\nI sat next to him, soaked and in a towel.\n\nClive sat on top of the fridge, soaked and angry.\n\nClive continued to yell\/meow, and we continued to stare at each other, breathing heavily\u2014Simon because of his battle and I...because of his battle. Clive finally jumped down from the fridge to the counter and skidded across in the puddle. He hit my radio, bounced off, and fell to the floor. Loud Marvin Gaye poured into the wet kitchen as Clive shook himself and ran for the living room.\n\n\"Let's get it on...\" Marvin sang it like he meant it, and Simon and I looked at each other, our faces stained crimson red.\n\n\"Are you kidding me?\" I said.\n\n\"Is this for real?\" he said, and we both started to laugh\u2014at the chaos, at the ridiculousness, at the sheer insanity of what had just happened and the fact that we were now lying half naked in my kitchen, covered in water, listening to a song that encouraged us to, in fact, \"get it on,\" and laughing our asses off.\n\nI finally straightened up, wiping tears from my eyes. He sat up next to me still holding his stomach.\n\n\"This is like a bad episode of Three's Company.\" He chuckled.\n\n\"No kidding. I hope someone called Mr. Furley.\" I giggled, drawing my towel tighter around me.\n\n\"Shall we get this cleaned up?\" he asked, standing.\n\nI noticed that his boxers, and anything that might be contained inside, were now at eye level. Settle, Caroline.\n\n\"Yes, I suppose we should.\" I laughed again as he held out his hand to help me up. I couldn't gain any traction, so I hung on to his hands, my feet slipping all over the floor.\n\n\"This is never going to work,\" he muttered and swooped me up. He carried me into the living room and set me down. \"Watch it there. Snoopy is drooping a little,\" he noted, gesturing to the part covering the girls.\n\n\"You'd love that, wouldn't you?\" I sassed, pulling things tighter.\n\n\"I'm going to get changed, and I'll bring you back some dry towels. Try to stay out of trouble.\" He winked and headed back to his place. I laughed again and went to the bedroom where Clive was now just a bump under the covers.\n\nI looked in the mirror over my dresser as I dug for something to put on. I was positively glowing. Huh. Must have been all that cold water.\n\nAn hour later things were back under control. We'd cleaned up the water, alerted the people downstairs in case there was leakage below, and placed a call to the maintenance guy.\n\nWe began to move toward my front door, mopping up the last little bit of water with the towels Simon had generously provided.\n\n\"What a disaster!\" I cried, pulling myself up off the floor and sinking down on the couch.\n\n\"Could have been worse. You could have had to deal with this after only three hours' sleep, and being woken up by some woman screaming at the top of her lungs,\" he said, coming to sit on the arm of the couch.\n\nI arched one eyebrow, and he recanted.\n\n\"Okay, bad example since that scenario is something you're familiar with. What are you going to do now?\"\n\n\"I dunno. I need to stay here and wait for the guy to fix this mess. In the meantime, I'm without water, which means no coffee, no shower, no nothing. Sucks,\" I muttered, crossing my arms across my chest.\n\n\"Well, I guess I'll be across the hall, drinking coffee and thinking about my shower, if you need anything,\" he said, starting for the door.\n\n\"Ass, you are totally making me coffee.\"\n\n\"Are you taking me up on the shower, too?\"\n\n\"You won't be in there with me, you know.\"\n\n\"I guess you can take one anyway. Come on, you little cockblocker,\" he huffed, pulling me up off the couch and leading me across the hall. Clive tossed one last angry cry at me from the bedroom, and I shushed him.\n\n\"Oops, wait. Let me grab breakfast.\" I snatched a foil-wrapped package from the table.\n\n\"What's that?\" he asked.\n\n\"Your zucchini bread.\"\n\nI swear he almost bit through his bottom lip. He must really like zucchini bread.\n\nThirty minutes later, I sat at Simon's kitchen table, legs curled underneath me, drinking French-pressed coffee and towel-drying my hair. He seemed really relaxed and happy, and he'd devoured the entire loaf of zucchini bread. I barely managed half a slice before he took it away from me, the entire chunk disappearing in his mouth.\n\nHe pushed away from the table and groaned, patting his full belly.\n\n\"You want another loaf? I baked plenty, you little piggy.\" I wrinkled my nose at him.\n\n\"I will take anything you want to give me, Nightie Girl. You have no idea how much I love homemade bread. No one's made anything like this for me in years.\" He winked and let out a tiny burp.\n\n\"Now that's sexy.\" I frowned and took my coffee cup into the living room, glancing out into the hallway to see if the maintenance guy had shown up yet.\n\nSimon followed me in and sat down on his big, comfy couch. I wandered around, looking at all his pictures. He had a series of black and whites on one wall, several prints of the same woman on a beach. Hands, feet, tummy, shoulders, back, legs, toes, and finally one of just her face. She was gorgeous.\n\n\"This is beautiful. One of your harem?\" I asked, looking back at him.\n\nHe sighed and ran his hand through his hair. \"Not every woman has made a trip to my bed, you know.\"\n\n\"Sorry. I'm kidding. Where were these taken?\" I asked, sitting down next to him.\n\n\"On a beach in Bora Bora. I was working on a travel photography series\u2014the most beautiful beaches of the South Pacific, very retro styled. She was on the beach one day, local girl, and the light was perfect, so I asked if I could take some shots of her. They came out great.\"\n\n\"She's gorgeous,\" I said, sipping my coffee.\n\n\"Yes,\" he agreed with a sweet smile.\n\nWe sipped silently, being okay with being quiet.\n\n\"So what were you planning to do today?\" he asked.\n\n\"You mean before my pipes revolted?\"\n\n\"Yes, before the attack.\" He smiled over the rim of his mug, blue eyes twinkling.\n\n\"I didn't have a lot planned, actually, and that was a good thing. I was gonna go for a run, maybe sit outside and read this afternoon.\" I sighed, feeling warm and comfortable and cozy. \"What about you?\"\n\n\"I was planning on sleeping the entire day before tackling a mountain of laundry.\"\n\n\"You can go sleep, you know. I can wait in my own apartment.\" I started to get up. Poor guy, he'd gotten in late, and I was keeping him from sleep.\n\nBut he waved me off and pointed to the couch. \"I know better, though. If I sleep I'll have jet lag all week. I need to get back on Pacific time as soon as I can, so it's probably a good thing your pipes attacked.\"\n\n\"Hmm, I guess. So how was Ireland? Good times?\" I asked, settling back.\n\n\"I always have a good time when I'm traveling.\"\n\n\"God, what an amazing job. I'd love to travel like that, living out of a suitcase, seeing the world, amazing...\" I trailed off, looking around again at all the pictures. I spotted a slender shelf on the far wall with tiny bottles on it. \"What's that?\" I asked, heading for the curious little shelf. They each contained what looked like sand. Some were white, some gray, some pink, and one was almost pitch black. They each had a label. As I looked I felt, rather than saw, him move behind me. His breath was warm in my ear.\n\n\"Every time I visit a new beach, I bring back a little sand\u2014like a reminder of where I was, when I was there,\" he answered, his voice low and wistful.\n\nI looked more closely at the bottles and marveled over the names I saw: Harbour Island\u2013Bahamas, Prince William Sound\u2013Alaska, Punaluu\u2013Hawaii, Vik\u2013Iceland, Sanur\u2013Fiji, Patura\u2013Turkey, Galicia\u2013Spain.\n\n\"And you've been all these places?\"\n\n\"Mmm-hmm.\"\n\n\"And why bring back sand? Why not postcards, or better yet, the pictures you take? Isn't that enough of a souvenir?\" I turned to look at him.\n\n\"I take pictures because I love it, and it happens to be my job. But this? This is tangible, it's tactile, it's real. I can feel this, this is sand I was actually standing on, from every continent on the planet. It brings me back there, instantly,\" he said, his eyes going all dreamy.\n\nFrom any other guy, in any other setting, it would have been pure cheese. But from Simon? The guy had to be deep. Dammit.\n\nMy fingers continued to trail over all the bottles\u2014almost more than I could count. My fingertips lingered on the few from Spain, and he noticed.\n\n\"Spain, huh?\" he asked.\n\nI turned to look at him. \"Yep, Spain. Always wanted to go. I will someday.\" I sighed and crossed back to the couch.\n\n\"Do you travel much?\" he asked, sinking down next to me again.\n\n\"I try to go somewhere each year\u2014not as fancy as you, or as frequent, but I try to take myself somewhere every year.\"\n\n\"You and the girls?\" He smiled.\n\n\"Sometimes, but the last few years I've enjoyed traveling by myself. There's something nice about being able to set your own pace, go where you want, and not have to run it by a committee every time you want to go out for dinner, you know?\"\n\n\"I get it. I'm just surprised,\" he said, frowning slightly.\n\n\"Surprised that I'd want to travel alone? Are you kidding? It's the best!\" I cried.\n\n\"Hell, you'll get no argument from me. I'm just surprised. Most people don't like to travel alone\u2014too overwhelming, too intimidating. And they think they'll get lonely.\"\n\n\"Do you ever get lonely?\" I asked.\n\n\"I told you, I am never lonely,\" he said, shaking his head.\n\n\"Yes, yes, I know, Simon says, but I have to say I find that a little hard to believe.\" I twisted a lock of almost-dry hair around my finger.\n\n\"Do you get lonely?\" he asked.\n\n\"When I'm traveling? No, I'm great company,\" I answered promptly.\n\n\"I hate to admit it, but I'd agree with that,\" he said, raising his mug in my direction.\n\nI smiled and blushed slightly, hating myself as I did it. \"Wow, are we becoming friends?\" I asked.\n\n\"Hmm, friends...\" He appeared to think carefully, examining me and my current state of blush. \"Yes, I think we are.\"\n\n\"Interesting. From cockblocker to friend. Not bad.\" I giggled and clinked his mug with my own.\n\n\"Oh, it remains to be seen whether you're lifted from cockblocker status yet,\" he said.\n\n\"Well, just give me a heads up before Spanx comes over next time, okay, friend?\" I laughed at his confused expression.\n\n\"Spanx?\"\n\n\"Ah, yes, well, you know her as Katie.\" I laughed.\n\nHe finally had the decency to blush and smile sheepishly. \"Well, as it happens, Ms. Katie is no longer part of what you so kindly refer to as my harem.\"\n\n\"Oh no! I liked her! Did you paddle her too hard?\" I teased again, my giggling beginning to get out of control.\n\nHe ran his hands through his hair frantically. \"I have to tell you, this is frankly the strangest conversation I've ever had with a woman.\"\n\n\"I doubt that, but seriously, where did Katie go?\"\n\nHe smiled quietly. \"She met someone else and seems really happy. So we ended our physical relationship, of course, but she's still a good friend.\"\n\n\"Well, that's good.\" I nodded and was quiet a moment. \"How does that work, actually?\"\n\n\"How does what work?\"\n\n\"Well, you have to admit, your relationships are unconventional at best. How do you do it? Keep everyone happy?\" I prodded.\n\nHe laughed. \"You're not seriously asking how I satisfy these women, are you?\" He grinned.\n\n\"Hell, no. I've heard how you do that! There doesn't seem to be any question about that. I mean, how does no one get hurt?\"\n\nHe thought for a moment. \"I guess because we were honest going into this. It isn't like anyone sets out to create this little world, it just happens. Katie and I had always gotten along great, especially in that way, so we just fell into that relationship.\"\n\n\"I like Spanx\u2014I mean Katie. So was she the first? In the harem?\"\n\n\"Enough with the harem\u2014you make it sound so sordid. Katie and I went to college together, tried dating for real, didn't work out. She's great though, she's...wait, are you sure you want to hear all this?\"\n\n\"Oh, I am all ears. I've been waiting to peel this onion since you first knocked that picture off my wall and clocked me on the head.\" I smiled, settling back on the couch and curling my knees underneath me.\n\n\"I knocked a picture off your wall?\" he asked, looking amused and proud at the same time. What a guy.\n\n\"Focus up, Simon. Gimme the skinny on your ladies in waiting. And spare no details\u2014this shit is better than HBO.\"\n\nHe laughed and put on his storyteller face. \"Well, okay, I guess it started with Katie. We didn't work out as a couple, but when we ran into each other after college a few years ago, coffee turned into lunch, lunch turned into drinks, and drinks turned into...well, bed. Neither of us was seeing anyone, so we started getting together whenever I was in town. She's great. She's just...I don't know how to explain it. She's...soft.\"\n\n\"Soft?\"\n\n\"Yeah, she's all rounded edges and warm and sweet. She's just...soft. She's the best.\"\n\n\"And Purina?\"\n\n\"Nadia. Her name is Nadia.\"\n\n\"I have a cat that says otherwise.\"\n\n\"Nadia I met in Prague. I was doing a shoot one winter. I usually never do fashion photography, but I got asked to shoot for Vogue\u2014very artsy, very conceptual. She had a house outside the city. We spent a naked weekend together, and when she moved to the States she looked me up. She's getting her masters now in international relations. It's crazy to me that at twenty-five she's on the tail end of her career, in modeling, that is. So she's working hard to do something else. She's very smart. She's traveled the entire world, and she speaks five languages! She went to the Sorbonne. Did you know that?\"\n\n\"How would I know that?\"\n\n\"Easy to make snap judgments when you don't know someone, isn't it?\" he asked, eyeing me.\n\n\"Touch\u00e9,\" I nodded, nudging him with my foot to go on.\n\n\"And then Lizzie. Oh boy, that woman is insane! I met her in London, piss drunk in a pub. She walked up to me, grabbed my collar, kissed me stupid, and dragged me home with her. That girl knows exactly what she wants and isn't afraid to ask for it.\"\n\nI remembered some of her louder moments in great detail. She really was rather specific about what she wanted, provided you could get past the giggling.\n\n\"She's a solicitor\u2014attorney\u2014and one of her main clients lives here in San Francisco. Her business is based in London, but when we're both in the same city, we make sure to see each other. And that's it. That's all she wrote.\"\n\n\"That's it? Three women, and that's it. How do they not get jealous? How are they all okay with this? And don't you want more? Don't they want more?\"\n\n\"For now, no. Everyone is getting exactly what they want, so it's all good. And yes, they all know about each other, and since no one's in love here, no one has any real expectations beyond friendship\u2014with the best possible benefits. I mean, don't get me wrong, I adore each of them, and love them in their own way. I'm a lucky guy. These women are amazing. But I'm too busy to date anyone for real, and most women don't want to put up with a boyfriend who's across the globe more often than home.\"\n\n\"Yes, but not all women want the same thing. We don't all want the picket fence.\"\n\n\"Every woman I've ever dated has said she doesn't, but then she does. And that's cool\u2014I get it\u2014but with my schedule being so crazy, it got to be very difficult for me to be involved with anyone who needed me to be something I'm not.\"\n\n\"So you've never been in love?\"\n\n\"I didn't say that, did I?\"\n\n\"So you have been in a relationship before, with just one woman?\"\n\n\"Of course, but as I said, once my life became what it is today\u2014the constant traveling\u2014it's hard to stay in love with that kind of guy. At least that's what my ex told me when she started dating some accountant. You know, wears a suit, carries a briefcase, home every night by six\u2014it's what women seem to want.\" He sighed, setting his coffee down and relaxing further into the couch. His words said he was okay with all this, but the wistful look on his face said otherwise.\n\n\"It's not what all women want,\" I countered.\n\n\"Correction, it's what the women I have dated all wanted. At least until now. That's why what I have works great for me. These women I spend my time with when I'm home? They're great. They're happy, I'm happy\u2014why would I rock the boat?\"\n\n\"Well, you're already down to two now, and I think you'd feel differently if the right woman came along. The right woman for you wouldn't want you to change anything about your life. She wouldn't rock your boat, she'd jump right in and sail it with you.\"\n\n\"You're a romantic, aren't you?\" He leaned in, bumping my shoulder.\n\n\"I'm a practical romantic. I can actually see some appeal in having a guy who travels a lot, because, frankly? I like my space. I also take up the entire bed, so it's difficult for me to sleep with anyone.\" I shook my head ruefully, remembering how quickly I used to kick my one-nighters to the curb. Some of my past wasn't all that different from Simon's. He just had his sexcapades tied up in a much neater package.\n\n\"A practical romantic. Interesting. So what about you? Dating anyone?\" he asked.\n\n\"Nope, and I'm okay with that.\"\n\n\"Really?\"\n\n\"Is it so hard to believe a hot, sexy woman with a great career doesn't need a man to be happy?\"\n\n\"First of all, bully for you for calling yourself hot and sexy\u2014because it's true. It's nice to see a woman give herself a compliment instead of fishing for one. And second, I'm not talking about getting married here, I'm talking about dating. You know, hanging out? Casually?\"\n\n\"Are you asking me if I'm fucking anyone right now?\" I shot at him, and he spluttered into his coffee.\n\n\"Definitely the strangest conversation I've ever had with a woman,\" he muttered.\n\n\"A hot and sexy woman,\" I reminded him.\n\n\"That's for damn sure. So, how about you? Ever been in love?\"\n\n\"This feels like an ABC mini-series, with all the coffee and the love talk,\" I said. I might have been stalling.\n\n\"Come on, let's celebrate this moment in our lives.\" He snorted, gesturing with his coffee mug.\n\n\"Have I ever been in love? Yes. Yes, I have.\"\n\n\"And?\"\n\n\"And nothing. It didn't end in a very good way, but what ending is ever good? He changed, I changed, so I got out. That's all.\"\n\n\"You got out, like...\"\n\n\"Nothing dramatic. He just wasn't who I thought he was going to be,\" I explained, setting my coffee down and playing with my hair.\n\n\"So what happened?\"\n\n\"Oh, you know how it goes. We were together when I was a senior at Berkley, and he was finishing up law school. It started out great, and then it wasn't, and so I left. He did teach me how to rock climb, so I'm grateful for that.\"\n\n\"A lawyer, huh?\"\n\n\"Yep, and he wanted a little lawyer wife. I should have caught on when he referred to my future career plans as a 'little decorating business.' He really just wanted someone who looked good and picked up his shirts from the cleaners on time. Not for me.\"\n\n\"I don't know you that well yet, but I can't really see you in the suburbs somewhere.\"\n\n\"Ugh, me either. Nothing wrong with the 'burbs, just not for me.\"\n\n\"You can't move to the 'burbs. Who would bake for me?\"\n\n\"Pfft, you just want to see me in my apron.\"\n\n\"You have no idea,\" he said, winking.\n\n\"It's hard to get everything you need from one person. You know what I mean? Wait, of course you do. What was I thinking?\" I laughed, gesturing to him.\n\nWe both jumped at the knocking on my door across the hall. The maintenance guy had finally arrived.\n\n\"Thanks for the coffee, and the shower, and the pipe rescue,\" I said, stretching as I walked toward the door. I nodded at the guy in the hallway and held up one finger to let him know I'd be right there.\n\n\"No problem. It wasn't the nicest way to wake up, but I suppose I deserved that one.\"\n\n\"Indeed. But thank you anyway.\"\n\n\"You're welcome, and thanks for the bread. It was great. And if another loaf happens to make its way over here, that would be okay.\"\n\n\"I'll see what I can do. And hey, where's my sweater?\"\n\n\"Do you know how expensive those are?\"\n\n\"Pffft, I want my sweater!\" I cried, slapping him in the chest.\n\n\"Well, as it happens, I did bring you something\u2014a sort of thanks-for-kicking-my-door present.\"\n\n\"I knew it. You can drop it off later.\" I walked across the hall to let the guy in. I directed him toward the kitchen and turned back to Simon. \"Friends, huh?\"\n\n\"Looks that way.\"\n\n\"I can live with that.\" I smiled and closed the door.\n\nAs the maintenance guy went about fixing the problem, I wandered to my bedroom to check on Clive. Just as I entered, my phone buzzed. A text from Simon already? I grinned and flopped down on the bed, snuggling a still-freaked-out kitty to my side. He began to purr instantly.\n\nYou never answered my question...\n\nI felt my skin heat up as I realized what he was referring to. I was suddenly warm and a little tingly, like when your foot falls asleep, but all over. And in a good way. Damn, he gave great text.\n\nAbout whether I'm fucking anyone?\n\nJesus, you're crass. But yes, friends can ask that, can't they?\n\nYes they can.\n\nSo?\n\nYou're kind of a pain in the ass. You know this, right?\n\nTell me. Don't get shy on me now.\n\nAs it happens, no. I'm not.\n\nI heard a thud from next door, and then a slight but constant banging on the wall.\n\nWhat the hell are you doing? Is that your head?\n\nYou're killing me, Nightie Girl.\n\nAs soon as I finished reading, the banging resumed. I laughed out loud as he thumped his head against the wall. I placed my hand on the wall over my bed where the thumping was concentrated and chuckled again. What a strange morning... \nChapter Ten\n\nI SAT IN MY OFFICE, gazing out the window. I had a list of things to do in front of me\u2014and it wasn't a small list either. I needed to run by the Nicholson house. The renovation was almost complete. The bedroom and bathroom were finished, and just a few details remained. I needed to get some new sample books from the design center. I had a meeting with a new client Mimi had referred to me, and on top of all that, I had a folder full of invoices to go through.\n\nBut still, I gazed out the window. I might have had Simon on the brain. And for good reason. Between the pipe explosions, the head banging, and the constant texting all day Sunday asking for more zucchini bread, my brain simply could not expunge him. And then last night, he brought out the big guns: he Glenn Miller-ed me. He even knocked on the wall to make sure I was listening.\n\nI put my head down on the desk and banged it a few times to see if it helped. It had seemed to help Simon...\n\nThat night I went straight to yoga after work and was climbing the stairs to my apartment when I heard a door open from above.\n\n\"Caroline?\" he called down to me.\n\nI grinned and continued up the stairs. \"Yes, Simon?\" I called up.\n\n\"You're home late.\"\n\n\"What, are you watching my door now?\" I laughed, rounding the last landing and staring up at him. He was hanging over the railing, hair in his face.\n\n\"Yep. I'm here for the bread. Zucchini me, woman!\"\n\n\"You're insane. You know this, right?\" I climbed the last stair and stood in front of him.\n\n\"I've been told. You smell nice,\" he said, leaning in.\n\n\"Did you just sniff me?\" I asked incredulously as I opened the door.\n\n\"Mmm-hmm, very nice. Just get back from a workout?\" he asked, walking in behind me and closing the door.\n\n\"Yoga, why?\"\n\n\"You smell great when you're all worked up,\" he said, waggling his eyebrows at me like the devil.\n\n\"Seriously, you pick women up with lines like that?\" I turned away from him to take off my jacket and squeeze my thighs together maniacally.\n\n\"It's not a line. You do smell great,\" I heard him say, and I closed my eyes to block out the Simon Voodoo currently making Lower Caroline curl in on herself.\n\nClive came bounding out of the bedroom when he heard my voice and stopped short when he saw Simon. Unfortunately, he had little traction on the hardwood floor and skidded rather ungracefully under the dining room table. Trying to regain his dignity, he executed a difficult four-foot leap from a standing position onto the bookshelf and waved me over with his paw. He wanted me to come to him\u2014typical male.\n\nI dropped my gym bag and sauntered over. \"Hi, sweet boy. How was your day? Hmm? Did you play? Did you get a good nap? Hmm?\" I scratched behind his ear, and he purred loudly. He gave me his dreamy cat eyes and then turned his gaze to Simon. I swear he cat-smirked at him.\n\n\"Zucchini bread, huh? You want some more, I take it?\" I asked, throwing my jacket on the back of a chair.\n\n\"I know you have more. Simon says gimme it,\" he deadpanned, making his finger into a gun.\n\n\"You're oddly into your baked goods, aren't you? Support group for that?\" I asked, walking into the kitchen to locate the last loaf. I might have been saving it for him.\n\n\"Yes, I'm in BA. Bakers Anonymous. We meet over at the bakery on Pine,\" he replied, sitting down on the stool at the kitchen counter.\n\n\"Good group?\"\n\n\"Pretty good. There's a better one over on Market, but I can't go to that one anymore,\" he said sadly, shaking his head.\n\n\"Get kicked out?\" I asked, leaning on the counter in front of him.\n\n\"I did, actually,\" he said, and then curled his finger to get me to lean in closer.\n\n\"I got in trouble for fondling buns,\" he whispered.\n\nI giggled and gave his cheek a light pinch. \"Fondling buns,\" I snorted as he pushed my hand away.\n\n\"Just fork over the bread, see, and no one gets hurt,\" he warned.\n\nI waved my hands in surrender and grabbed a wine glass from the cupboard over his head. I raised my eyebrow at him, and he nodded.\n\nI handed him a bottle of Merlot and the opener, then grabbed a bunch of grapes from the colander in the fridge. He poured, we clinked, and without another word, I started making us dinner.\n\nThe rest of the evening happened naturally, without me even realizing it. One minute we were discussing the new wine glasses I'd purchased from Williams Sonoma, and thirty minutes later we were sitting at the dining room table with pasta in front of us. I was still wearing my workout clothes, and Simon was in jeans and a T-shirt and his stocking feet. He'd taken off his Stanford sweatshirt before draining the pasta, something I didn't even have to ask him to do. He'd simply wandered into the kitchen behind me, and had it drained and back in the pot just as I finished the sauce.\n\nWe'd talked about the city, his work, my work, and the upcoming trip to Tahoe, and now we headed over to the couch with coffee.\n\nI leaned back against the pillows with my legs curled underneath me. Simon was telling me about a trip he'd taken to Vietnam a few years before.\n\n\"It's like nothing you've ever seen\u2014the mountain villages, the gorgeous beaches, the food! Oh, Caroline, the food.\" He sighed, stretching his arm along the back of the couch. I smiled and tried not to notice the butterflies when he said my name that way: with the word Oh right in front of it...Oh me, oh my.\n\n\"Sounds wonderful, but I hate Vietnamese food. Can't stand it. Can I bring peanut butter?\"\n\n\"I know this guy\u2014makes the best noodles ever, right on a houseboat in the middle of Ha Long Bay. One slurp and you'll throw your peanut butter right over the side.\"\n\n\"God, I wish I could travel like you do. Do you ever get sick of it?\" I asked.\n\n\"Hmmm, yes and no. It's always great to come home. I love San Francisco. But if I'm home too long I get the itch to get back out on the road. And no comments about the itch\u2014I'm starting to get to know your mind there, Nightie Girl.\" He patted my arm affectionately.\n\nI tried to feign offense, but the truth was I had been about to make a joke. I noticed he still had his hand on my arm, absentmindedly tracing tiny circles with his fingertips. Had it really been so long since I'd let a man touch me that fingertip circles sent me into a mental tizzy? Or was it that this man was doing it? Oh, God, the fingertips. Either way, it was doing things to me. If I closed my eyes, I could almost imagine O waving at me\u2014still far away, but not as far as she'd been before.\n\nI glanced at Simon and saw that he was watching his hand, as if curious about his fingers on my skin. I breathed in quickly, and my intake of breath drew his eyes to mine. We watched each other. Lower Caroline was, of course, responding, but now Heart began to beat a little wildly as well.\n\nThen Clive jumped up on the back of the couch, put his bum right in Simon's face, and killed that real quick. We both laughed, and Simon moved away from me as I explained to Clive that it was not polite to do that to company. Clive seemed oddly pleased with himself, though, so I knew he was up to something.\n\n\"Wow, it's almost ten! I've taken up your entire evening. I hope you didn't have plans,\" Simon said, standing and stretching. As he stretched, his T-shirt came up, and I bit down hard on my tongue to stop myself from licking the bit of skin showing above his jeans.\n\n\"Well, I did have a rather exciting night of watching Food Network planned, so damn you, Simon!\" I shook my fist in his face as I stood up next to him.\n\n\"And you even made me dinner, which was great, by the way,\" he said, searching for his sweatshirt.\n\n\"No problem. It was nice to cook for someone other than myself. It's what I do for any guy who shows up demanding bread.\" I finally handed him the loaf I'd left out for him.\n\nHe grinned as he grabbed his sweatshirt off the floor next to the couch. \"Well, next time, let me cook for you. I make a fantastic\u2014huh, that's weird,\" he interrupted himself, grimacing.\n\n\"What's weird?\" I asked, watching as he unfolded his sweatshirt.\n\n\"This feels damp. Actually, it's more than damp, it's...wet?\" he asked, looking at me, confused. I looked from the sweatshirt to Clive, who sat innocently on the back of the couch.\n\n\"Oh no,\" I whispered, the blood draining from my face. \"Clive, you little shit!\" I glared at him.\n\nHe jumped off the couch and darted quickly between my legs, headed for the bedroom. He'd learned I couldn't reach him behind the dresser, and that's where he hid when he'd done a bad, bad thing. He hadn't done this in a long time.\n\n\"Simon, you might want to leave that here. I'll wash it, dry clean it\u2014whatever. I am so, so sorry,\" I apologized, horrifically embarrassed.\n\n\"Oh, did he? Oh man, he did, didn't he?\" His face wrinkled as I took the sweatshirt from him.\n\n\"Yes, yes, he did. I'm so sorry, Simon. He has this thing about marking his territory. When any guy leaves clothes on the floor\u2014oh, God\u2014he eventually pees on them. I'm so sorry. I am so, so sorry. I'm so\u2014\"\n\n\"Caroline, it's okay. I mean, it's gross, but it's okay. I've had worse things happen to me. It's all good, I promise.\" He started to put his hand on my shoulder, but seemed to think better of it, probably when he realized the last thing he'd touched.\n\n\"I'm so sorry, I\u2014\" I began again as he started for the door.\n\n\"Stop it. If you say sorry one more time I'm gonna go find something of yours and pee on it, I swear.\"\n\n\"Okay, that's just gross.\" I finally laughed. \"But we had such a nice night, and it ended in pee!\" I wailed, opening the door for him.\n\n\"It was a nice night, even with the pee. There'll be others. Don't worry, Nightie Girl.\" He winked and crossed the hall.\n\n\"Play me something good tonight, huh?\" I asked, watching him go.\n\n\"You got it. Sleep tight,\" he said, and we closed the doors at the same time.\n\nI leaned back against the door, hugging the sweatshirt in my arms. I'm sure I had the goofiest grin on my face, as I remembered the feeling of his fingertips. And then I remembered I was hugging a pee-stained sweatshirt.\n\n\"Clive, you asshole!\" I yelled and ran for my bedroom.\n\nFingers, hands, warm skin pressed against mine in an effort to get closer. I felt his warm breath, his voice like wet sex in my ear. \"Mmm, Caroline, how can you feel this good?\"\n\nI moaned and rolled over, twisting legs with legs and arms with arms, pushing my tongue into his waiting mouth. I sucked on his bottom lip, tasting mint and heat and the promise of what was to come when he pushed into my body for the first time. I moaned as he groaned, and in a flash I was pinned beneath him.\n\nLips moved from my mouth to my neck, licking and sucking and finding the spot\u2014that spot underneath my jaw that made my insides explode and my eyes cross. A dark laugh against my collarbone, and I knew I was done for.\n\nI rolled on top of him, feeling the loss of his weight but the gain of my legs on either side of him, feeling him twitch and throb exactly where I needed him to be. He pushed my hair from my face, gazing up at me with those eyes\u2014the eyes that could make me forget my name but scream his own.\n\n\"Simon!\" I cried, feeling his hands grab my hips and push me against him.\n\nI sat straight up in bed, my heart racing as the last dreamy images left my brain. I thought I heard a low chuckle from other side of the wall, where the strains of Miles Davis came through.\n\nI lay back down, skin tingling as I tried to find a cool spot on my pillow. I thought about what was on the other side of that wall, inches away. I was in trouble.\n\nLater that morning I sat at my desk getting ready to meet a new client\u2014one who'd specifically requested to work with me. Still a new designer, much of my work came from referrals, and whoever had referred this guy to me I owed big time. All new interiors for some fancy apartment\u2014it was practically a gut remodel, a dream project. Whenever I prepped for a new client I pulled pictures from other projects I'd designed and had sketchbooks ready, but today I did it with particular intensity. If I let my mind wander for a second, Brain immediately returned to the dream I'd had last night. I blushed every time I thought of what I'd let Dream Simon do to me, and what Dream Caroline had done to him as well...\n\nDream Caroline and Dream Simon were some naughty kids.\n\n\"Ahem,\" I heard from behind me. I turned to find Ashley in the doorway. \"Caroline, Mr. Brown is here.\"\n\n\"Excellent, I'll be right out.\" I nodded, standing and smoothing my skirt. My hands pressed my cheeks, hoping they were not too red.\n\n\"And he is cute, cute, cute!\" she murmured as she walked beside me down the hall.\n\n\"Oh, really? Must be my lucky day.\" I laughed, rounding the corner to greet him.\n\nHe certainly was cute, and I would know. He was my ex-boyfriend.\n\n\"Oh, my God! What are the chances?\" Jillian exclaimed at lunch, two hours later.\n\n\"Well, considering my entire life now seems ruled by odd coincidences, I figure it's right on track.\" I broke off a piece of flatbread and chewed determinedly.\n\n\"But I mean, come on! What are the chances, really?\" she wondered again, pouring us another glass of Pellegrino.\n\n\"Oh, there's nothing chance about this. This guy doesn't leave things to chance. He knew exactly what he was doing when he approached you at that benefit last month.\"\n\n\"No,\" she breathed.\n\n\"Yep. He told me. He saw me, and when he found out I worked for you? Bam! He needs an interior designer.\" I smiled, thinking of how he'd always arranged things exactly the way he wanted them. Well, almost everything.\n\n\"Don't worry, Caroline. I'll move him over to another designer, or I'll even take him myself. You don't have to work with him,\" she said, patting my hand.\n\n\"Oh, hell no! I already told him yes. I'm totally doing this.\" I crossed my arms over my chest.\n\n\"Are you sure?\"\n\n\"Yep. No problem. It wasn't that we had a bad breakup. In fact, as far as breakups go, it was mild. He didn't want to accept the fact that I was leaving him, but eventually he came around. He didn't think I had the balls to do it, and boy, was he surprised.\" I played with my napkin.\n\nI'd dated James most of my senior year at Berkeley. He was already in law school, steadily moving through it on his way to a future of perfection. My goodness, he was beautiful\u2014strong and handsome, and very charming. We met at the library one night, had coffee a few times, and it grew into a solid relationship.\n\nThe sex? Unreal.\n\nHe was my first serious boyfriend, and I knew he wanted to marry me at some point. He had very specific ideas about what he wanted from his life, and that definitely included me as his wife. And he was everything I'd ever thought I wanted in a husband. Engagement was inevitable. But then I began to notice things, small at first, but over time they revealed the big picture. We went where he wanted to for dinner. I never got to pick. I overheard him telling someone that he figured my \"decorating\" phase wouldn't last long, but it'd be nice to have a wife who could make a pretty home. The sex was still great, but I was irritated with him more and more, and I stopped going along to get along.\n\nWhen I began to realize he was no longer what I wanted for my future, things got a little strained. We fought constantly, and when I decided to end the relationship, he tried to convince me I was making the wrong choice. I knew better, and he finally accepted that I was really done\u2014and not just pitching a \"feminine fit,\" as he liked to call them. We didn't keep in contact, but he'd been a major part of my life for a long time, and I cherished the memories we had together. I cherished what he'd taught me about myself.\n\nJust because we didn't work out as a couple didn't mean we couldn't work together, right?\n\n\"You sure about this? You really want to work with him?\" Jillian asked one more time, but I could tell she was ready to let it go.\n\nI thought about it again, replaying the flash of memory I'd had when I saw him standing in the lobby. Sandy blond hair, piercing eyes, charming smile: I'd been hit with a wave of nostalgia and grinned as he crossed to me.\n\n\"Hey there, stranger,\" he'd said, offering me his hand.\n\n\"James!\" I gasped, but recovered quickly. \"You look great!\" We hugged\u2014to gawking Ashley's surprise.\n\n\"Yes, I'm sure,\" I told Jillian. \"It'll be good for me. Call it a growth experience. Plus, I don't want to give up the commission. We'll see what happens tonight.\"\n\nAt that she looked up from her menu. \"Tonight?\"\n\n\"Oh, I didn't tell you? We're going for drinks to get caught up.\"\n\nI stood in front of the mirror, fluffing my hair and checking my teeth for wayward lipstick. The rest of the workday had gone quickly, and I now found myself at home getting ready for tonight. We'd agreed to just drinks, very casual, although I was leaving the option open for dinner. But skinny jeans, black turtleneck, and cropped gray leather jacket was as fancy as I was gonna get.\n\nThe time I'd spent this morning with James at the office was pleasant, and when he'd asked me to go for drinks to catch up, I agreed instantly. I was anxious to learn what he'd been up to, as well as make sure we'd be able to work together. He'd been a huge part of my life at one time, and the idea of being able to work with someone I'd once been so close to felt good to me. It felt mature. Closure? Not sure what to call it, but it seemed like the natural thing to do.\n\nHe was picking me up at seven, and I planned to meet him outside. Parking on my street was ridiculous. A glance at the clock told me it was time to get going, so I gave a quick kiss goodbye to Clive, who'd been on his best behavior since the pee incident, and let myself into the hallway.\n\nAnd straight into Simon, who was in front of my door.\n\n\"Okay, you are officially my stalker! There's no more zucchini bread, mister. I hope you made that loaf last because there is no more for you,\" I warned, pressing him back from my front door with my pointer finger.\n\n\"I know, I know. I'm actually here on official business.\" He laughed, throwing up his arms in defeat.\n\n\"Walk with me?\" I asked, nodding toward the stairs.\n\n\"I'm headed out as well. Going to rent a movie,\" he explained as we started down.\n\n\"People still rent movies?\" I joked, rounding the corner.\n\n\"Yes, people still rent movies. Just for that you're gonna have to watch whatever I pick out,\" he replied, raising an eyebrow.\n\n\"Tonight?\"\n\n\"Sure, why not. I was coming over to see if you wanted to hang out. I owe you for dinner from the other night, and I got an urge to watch something spooky...\" He launched into The Twilight Zone theme.\n\nI couldn't help but laugh at his claw hands and crossed eyes. \"Last time someone asked me to rent a movie it was code for 'let's make out on the couch.' Am I safe with you?\"\n\n\"Please! We've got that truce, remember? I am all about the truce. So, tonight?\"\n\n\"I wish I could, but I have plans tonight. Tomorrow night?\" We rounded the last stair and entered the entryway.\n\n\"Tomorrow I can do. Come on over after work. But I get to pick the movie, and I'm making you dinner. Least I can do for my little cockblocker.\" He smirked, and I punched him in the arm.\n\n\"Please stop calling me that. Otherwise I won't bring dessert,\" I said, lowering my voice and batting my eyelashes like a fool.\n\n\"Dessert?\" he asked, holding the door open as I walked out into the night.\n\n\"Mmm-hmm. I picked up some apples yesterday while I was out, and I've been craving pie all week. How does that sound?\" I asked, scanning the street for James.\n\n\"Apple pie? Homemade apple pie? Christ, woman, are you trying to kill me? Mmm...\" He smacked his lips and looked at me hungrily.\n\n\"Why, sir, you look like you've seen something you'd like to eat,\" I offered in my best Scarlett.\n\n\"You show up with apple pie tomorrow night, and I may not let you leave,\" he breathed, his cheeks rosy and his messy hair blowing in the cool air.\n\n\"That would be terrible,\" I whispered. Wow. \"Okay, so, go get your movie,\" I said, playfully shoving the six feet of hot in front of me. Remember the harem! I shouted inside my head.\n\n\"Caroline?\" a concerned voice came from behind me, and I turned to see James walking toward us.\n\n\"Hey, James,\" I called out, stepping away from Simon with a giggle.\n\n\"You ready to go?\" he asked, looking at Simon carefully. Simon straightened to his full height and looked back, just as carefully.\n\n\"Yep, ready to go. Simon, this is James. James, Simon.\" They leaned in to shake hands, and I could see that they both exerted a little extra force, neither seeming to want to be the one to let go first. I rolled my eyes. Yes, boys. You can both write your names in the snow. The question is, who would make bigger letters?\n\n\"Nice to meet you, James. It was James, right? I'm Simon. Simon Parker.\"\n\n\"That's correct. James. James Brown.\"\n\nI saw the beginnings of a laugh on Simon's face.\n\n\"Okay, James, we should get going. Simon, I'll talk to you later,\" I interrupted, ending the handshake of the century.\n\nJames turned toward where his car was double-parked, and Simon looked at me.\n\n\"Brown? James Brown?\" he mouthed, and I squelched my own laugh.\n\n\"Shush,\" I mouthed back, smiling at James when he turned back to me.\n\n\"Nice to meet you, Simon. See you around,\" James called, steering me to the car with his hand on the small of my back. I didn't think twice about it, as that's how we always used to walk together, but Simon's eyes widened a little at the sight.\n\nHmm...\n\nJames opened the door for me, then headed around to his side. Simon was still standing in front of our building when we drove away. I rubbed my hands together in front of the heater and grinned at James as he steered through the traffic.\n\n\"So, where are we headed?\"\n\nWe made ourselves comfortable in the swanky bar he'd selected. It seemed very James: chic and sophisticated, and laced with hidden sexuality. The deep red leather banquettes, thinly cushioned and cool, ensconced us as we settled in and began the process of getting to know each other after so many years apart.\n\nAs we waited for a server to come by, I studied his face. He still looked the same: closely cropped sandy blond hair, intense eyes, and a lean frame folded in on itself like a cat's. Age had only improved his good looks, and his carefully torn jeans and black cashmere sweater clung to a body I could see was in great shape. James had been a rock climber, relentless in his pursuit of the sport. He viewed each boulder, each mountain as an obstacle to overcome, something to be conquered.\n\nI'd gone climbing with him a few times toward the end of our relationship, even though I grew up skittish about heights. But watching him climb, seeing the sinewy muscles stretch and manipulate his body into positions that seemed unnatural, was a heady experience, and I'd pounced on him those evenings in the tent like a woman possessed.\n\n\"What are you thinking about?\" he asked, interrupting my musings.\n\n\"I was thinking about how much you used to climb. Is that something you still do?\"\n\n\"It is, but I don't get as much free time as I used to. They keep me pretty busy at the firm. I try and get out to Big Basin as often as I can,\" he added, smiling as our waitress approached.\n\n\"What can I get you two?\" she asked, placing napkins in front of us. \"She'll have a dry vodka martini, three olives, and for me bring three fingers of Macallan,\" he answered. The waitress nodded and left to fill our order.\n\nI studied him as he sat back, then turned his gaze to me.\n\n\"Oh, Caroline, I'm sorry. Is that still your drink?\"\n\nI narrowed my eyes at him. \"As it happens, yes. But what if I didn't want that tonight?\" I answered primly.\n\n\"My mistake. Of course, what did you want to drink?\" He waved the waitress back over.\n\n\"I'll have a dry vodka martini with three olives, please,\" I told her with a wink.\n\nShe looked confused.\n\nJames laughed loudly, and she walked away, shaking her head.\n\n\"Touch\u00e9, Caroline. Touch\u00e9,\" he said, studying me again.\n\n\"So, tell me what you've been up to the last few years.\" I put my elbows on the table and chin in hands.\n\n\"Hmm, how to encapsulate years in a few sentences? Finished law school, signed on with the firm here in the city, and worked like a dog for two years. I've been able to ease up a bit, only around sixty-five hours a week now, and it's nice seeing daylight again, I admit.\" He grinned, and I couldn't help but smile back. \"And of course working as much as I do leaves me very little time for a social life, so it was just blind luck that I saw you at the benefit last month,\" he finished, leaning forward on his elbows as well. Jillian attended many social events around town, and I accompanied her on occasion. Good for business. I should've known I'd eventually run into James at one of those shindigs.\n\n\"So you saw me, but you didn't come talk to me. And now here you are, weeks later, asking me to work on your condo. Why is that, exactly?\" I accepted my drink as it arrived and took a long pull.\n\n\"I wanted to talk to you, believe me. But I couldn't. So much time had passed. Then I realized you worked for Jillian, who a friend had recommended to me, and I thought, 'how perfect.'\" He inclined his glass toward mine for a clink.\n\nI paused for a moment, then clinked him. \"So you're serious about working with me? This isn't some kind of ploy to get me into bed, is it?\"\n\nHe looked at me evenly. \"Still direct as ever, I see. But no, this is professional. I didn't like the way we left things, admittedly, but I accepted your decision. And now here we are. I needed a decorator. You are a decorator. Works out well, don't you think?\"\n\n\"Designer,\" I said quietly.\n\n\"What's that?\"\n\n\"Designer,\" I said, louder this time. \"I'm an interior designer, not a decorator. There's a difference, Mr. Attorney Man.\" I took another sip.\n\n\"Of course, of course,\" he replied, signaling for the waitress.\n\nSurprised, I looked down to find my glass empty.\n\n\"Care for another?\" he asked, and I nodded.\n\nAs we small talked for the next hour, we also began to discuss what he needed in his new home. Jillian had been right. He was pretty much asking me to design his entire place, from area rugs to lighting fixtures and everything in between. It would be a huge commission, and he'd even agreed to let me photograph it for a local design magazine Jillian had been wanting me to submit to. James came from a wealthy family\u2014the Browns of Philadelphia, don't you know\u2014and I knew they must be footing the bill for most of this. Young lawyers didn't make enough to afford the kind of place he had, let alone in one of the most expensive cities in America. But trust funds live on, and he had a large one. One of the perks of dating him in college had been that we could actually afford real dates, not just cheap takeout all the time. I'd enjoyed that aspect of being with him. Not gonna lie.\n\nAnd I would enjoy that aspect of this project. A basically unlimited budget? I couldn't wait to get started.\n\nIn the end, it was a nice evening. As with all old flames, there was a feeling of knowing, a nostalgia you can only share with someone who has known you intimately\u2014especially at that age when you're still forming. It was great to see him again. James has a very strong personality, intense and confident, and I was reminded why I'd been attracted to him in the first place. We laughed and told stories about things we'd done as a couple, and I was relieved to find that his charm remained. We could get along quite well in a social setting. There was none of the awkwardness that could have accompanied this.\n\nAs the evening wound down and he drove me home, he got around to the question I knew he'd been dying to ask. He pulled the car to a stop in front of my building and turned to me.\n\n\"So, are you seeing anyone?\" he asked quietly.\n\n\"No, I'm not. And that's hardly a question a client would ask me,\" I teased and looked toward my building. I could see Clive sitting in the front window in his usual post, and I smiled. It was nice to have someone waiting for me. I couldn't stop myself from glancing next door to see if there was a light on in Simon's apartment, and I also couldn't stop my tummy from doing a little flippity-flop when I saw his shadow on the wall and the blue light of his television.\n\n\"Well, as your client, I'll refrain from asking those kinds of questions in the future, Ms. Reynolds,\" He chuckled.\n\nI turned back to face him. \"It's okay, James. We passed designer\/client relationship a long time ago.\" I felt triumphant as I saw a blush carve a chink in his careful fa\u00e7ade.\n\n\"I think this is gonna be fun.\" He winked, and it was my turn to laugh.\n\n\"Okay, you can call me tomorrow at the office, and we'll get started. I'm gonna fleece you blind, buddy, Get ready to work that credit card,\" I taunted as I stepped out of the car.\n\n\"Oh hell, I'm counting on it.\" He winked and waved goodbye.\n\nHe waited until I was inside, so I tossed another wave his way as the door closed. I was glad to see I could handle myself with him. Upstairs, as I turned the key in my lock I thought I heard something. I looked over my shoulder, and there was nothing there. Clive called to me from inside, so I smiled and stepped in, scooping him up and whispering softly in his ear as he gave me a tiny cat hug with his big paws around my neck.\n\nThe next evening I was rolling out the pie crust when the text came in from Simon.\n\nCome on over whenever. I'll start dinner once you're here.\n\nI'm still working on the pie, but I'll be over soon.\n\nNeed any help?\n\nHow are you with peeling apples?\n\nThe next thing I heard was a knock on the door. I walked over, hands covered in flour, and elbowed the door open. \"Well, hello there,\" I said, holding the door open with my foot.\n\n\"Looks like the end of Scarface in here,\" he observed, reaching out to touch my nose and show me the flour on the end.\n\n\"I tend to lose control when there's pie crust involved,\" I said as he shut the door.\n\n\"Duly noted. That's good information for me to have,\" he responded, swatting at my hand as I tried to slap him.\n\nHe took a good long look at me then, blue eyes dropping from my face and traveling across my body. \"Hmm, you weren't kidding about the apron, I don't know how long I'll be able to hang in here without trying a little grab-ass.\"\n\n\"Get in there and grab an apple, buddy,\" I said and walked toward the kitchen, adding a little extra swish to my hips. I heard him sigh heavily. I glanced down at my outfit, noting my tank top, old jeans, bare feet, and chef's apron that said, You should see my scones...\n\n\"Now when you said 'grab an apple,' what exactly were you referring to?\" he asked from the kitchen where he'd started taking off his sweater.\n\nI shook my head at the sight of Simon in a black T-shirt and weathered jeans. He was in his stocking feet once again, and I marveled at how at ease he seemed in my kitchen.\n\nI walked around the kitchen counter and picked up my rolling pin. \"You know, I won't think twice about whacking you over the head with this if you continue this borderline sexual harassment,\" I warned, running my hand up and down the rolling pin suggestively.\n\n\"I'm gonna have to ask you not to do that if you're serious about me peeling apples here,\" he said, eyes widening.\n\n\"I never joke about pie, Simon.\" I sprinkled a little more flour on the marble.\n\nHe was silent while he watched me pat out the pie crust, breathing through his mouth. \"So, what are you gonna do with that?\" he asked, his voice low.\n\n\"With this?\" I asked, leaning over the board, and perhaps arching my back a little as I did.\n\n\"Mmm-hmm,\" he replied.\n\n\"I'm gonna roll this crust out. See, like this?\" I teased again, thrusting the pin back and forth over the dough, making sure I arched my back each time and the forward action pushed my girls together.\n\n\"Oh my,\" he whispered, and I grinned naughtily at him.\n\n\"You gonna be okay over there, big guy? This is just the top crust, I still need to work on my bottom,\" I said over my shoulder.\n\nHis hands clutched at the edge of the counter. \"Apples. Apples. Gonna peel me some apples,\" he told himself and turned away toward the colander filled with apples in the sink.\n\n\"Let me just get you the peeler,\" I said, coming up behind him and pressing myself against him as I curled around his side to grab the vegetable peeler from the other sink. This was fun.\n\n\"Peeling apples, just peeling apples. Didn't feel your boobs. No, no, not me,\" he chanted as I openly laughed at him.\n\n\"Here, peel this,\" I said, taking pity on him and removing myself from his cooking space. I might have sniffed his T-shirt.\n\n\"Did you just sniff me?\" he asked, keeping himself turned away.\n\n\"I might have,\" I admitted, going back to my rolling pin, which I squeezed mightily.\n\n\"I thought so.\"\n\n\"Hey, if you can sniff, I can sniff,\" I shot back, taking out my sexual frustration on a defenseless P\u00e2te Bris\u00e9e.\n\n\"Only fair. So how do I rate?\"\n\n\"Good. Very good, actually. Downy?\"\n\n\"Bounce. I lost my Downy ball,\" he confessed.\n\nI laughed, and we continued to roll and peel. Within fifteen minutes, we had a bowlful of peeled and sliced apples, a perfectly rolled-out pie crust, and we'd both consumed our first glass of wine.\n\n\"Okay, what's next?\" he asked, wiping up flour and generally tidying.\n\n\"Now we spice things up and add a little citrus,\" I answered, lining up cinnamon and nutmeg, my sugar bowl, and a lemon.\n\n\"Okay, where do you want me?\" he asked, taking care to show me his hands, now covered in flour.\n\nVisions ran through my head, and I had to bite back an invitation to show him exactly where I wanted him. \"First dust yourself off, and then we'll get started. You can be my assistant.\"\n\nHe looked around for a dishtowel, and I turned to look for the one I knew I'd left out. I'd already started for it on the counter when I felt two very strong and very specifically placed hands on my ass.\n\n\"Um, hi?\" I said, freezing in place.\n\n\"Hi,\" he answered cheerfully, not releasing his hands.\n\n\"Explain yourself, please,\" I ordered, trying not to notice how my heart was trying to leave my body by way of my mouth.\n\n\"You told me to find something to clean my hands with,\" he stuttered, trying hard not to laugh as he gave each cheek a little squeeze.\n\n\"And you took that to mean my ass?\" I laughed back and turned to face him, removing his hands with my own.\n\n\"What can I say? I take liberties with my neighbors,\" he replied, his eyes darting back and forth now between my lips and my eyes.\n\n\"We have a pie to make, mister. I'll thank you to remember your manners. No one touches my ass without an invitation.\" I giggled, still holding his hands. I felt his thumb trace little circles on the inside of my palm, and my head got swimmy. This guy was going to be the death of me. \"Get over there, handsy, and behave,\" I instructed.\n\nHe smirked and turned away, which gave me the opportunity to mutter, \"Oh my Jesus Lord,\" to no one in particular before meeting him back at the apple bowl.\n\n\"Okay, you do what I tell you, got it?\" I said, sprinkling sugar into the bowl.\n\n\"Got it.\"\n\nI started tossing the apples with my hands and Simon followed my instructions to the letter. When I asked for more sugar, he sugared. When I asked for more cinnamon, he complied. When I asked him to squeeze the lemon, he lemoned so well I had trouble keeping my tongue in my mouth and off his throat.\n\nI tossed and tasted, and when they were finally right, I lifted a wedge to his mouth. \"Open up,\" I said, and he leaned in.\n\nI placed an apple on his tongue, and he snapped his mouth shut before I had to chance to remove my fingers. He let his lips close around two, and I slowly withdrew them, feeling his tongue wrap around them delicately and deliberately.\n\n\"Delicious,\" he said softly.\n\n\"Gah,\" I answered, eyes crossing a little at the sex on two legs displayed in front of me.\n\nHe chewed. \"Sweet. Sweet, Caroline.\"\n\n\"Gah,\" I managed again. Brain knew this was bad, Heart was beating out of our chest.\n\n\"Good for you?\" he asked, that knowing smile treading dangerously close to smirk territory.\n\n\"Good for me,\" I answered, on fire after the fingerlatio. Truce schmuce, harem schmarem. Who cared if there was no actual O? I needed to be in contact with this man in the very worst way.\n\nMy sexual wall had been hit, and as I prepared to rip the clothes from his body, throw him to the ground, and ride him amid a pile of apples and cinnamon with only a rolling pin to guide us, my phone rang.\n\nThank you, Jesus.\n\nI looked at the blue-eyed devil and launched myself across the room, away from the brain-scrambling voodoo. I saw his face as I ran, and he looked a little disappointed.\n\n\"Girl, what are you up to tonight?\" Mimi screeched into the phone. I held it away from my ear before the bleeding started. Mimi had three sound levels: Normal Loud, Excited Loud, and Drunky Loud. She was leaving Excited and on her way to Drunky.\n\n\"I'm getting ready to have dinner. Where are you?\" I asked, nodding at Simon who had started pouring the apples into the pie dish.\n\n\"I'm out for drinks with Sophia. What are you doing?\" she screamed.\n\n\"I just told you, getting ready to have dinner!\" I laughed.\n\nSimon came out into the living room with the pie in his hands. \"Should I put this in the oven?\" he asked.\n\n\"Hang on, Mimi. Not yet, I still need to brush it with a little cream,\" I told him, and he ducked back into the kitchen.\n\n\"Caroline Reynolds, that was a man! Who was that? Who are you having dinner with? And what are you brushing with cream?\" she fired at me, her voice growing even louder.\n\n\"Settle down. My goodness, you're loud! I'm having dinner with Simon, and we're making an apple pie,\" I explained, which she immediately screamed out to Sophia.\n\n\"Shit,\" I muttered as I heard the phone yanked away from Mimi.\n\n\"Reynolds, what are you doing? Are you baking pies with your neighbor? Are you naked?\" Sophia yelled, taking her turn to grill me.\n\n\"Okay, no, and you all need to seriously settle down. Hanging up now,\" I yelled over her yelling at me. I could hear Mimi squealing nasty things about pies and cream. Sophia was in the middle of threatening me not to hang up on her, when I did just that.\n\nI sighed and went to find Simon, with his hands full of pie. I snorted in spite of myself.\n\n\"Oh, my God, that's so good,\" I whimpered, closing my eyes and losing myself to the sensations.\n\n\"I knew you'd like it, but I had no idea you'd enjoy it this much,\" he whispered, staring at me with rapt attention.\n\n\"Stop talking, you're going to ruin it for me,\" I moaned, stretching and feeling myself respond to everything he was giving me.\n\n\"Did you want another one?\" he offered, raising up on his elbows.\n\n\"If I have another, I won't be able to walk tomorrow.\"\n\n\"Go ahead, be a bad girl\u2014you deserve it. I know you want it, Caroline,\" he teased, leaning closer.\n\n\"Okay,\" I managed, opening up to him once again. I closed my eyes and heard him fumbling about before putting it in. Sighing as I felt it, I closed my lips around what he offered.\n\n\"I've never seen a woman who could take so much in one sitting,\" he marveled, watching me come undone once more.\n\n\"Yes, well, you've never met a woman who likes meatballs as much as me,\" I moaned around another mouthful, feeling stuffed beyond belief but not wanting the meal to end.\n\nSimon had just cooked me quite possibly the most perfect meal ever, hitting every single taste bud that needed to be hit. He'd learned how to make the most amazing meatballs from a woman in Naples, and he'd sworn they'd be the best I'd ever had. After no less than seven jokes about balls and mouths, I had to agree they were the best balls I had ever had in my mouth.\n\nGod, he gave great meatball.\n\nI then proceeded to eat almost a pound of pasta myself, as well as all of my meatballs, plus half of his. I insisted he eat the last one, but he refused and brought the perfection that was his meatball to my willing mouth.\n\nSimon was a great host, insisting that I sit, drink wine, and watch rather than help. He entertained me with stories about his travels as he got everything ready, and while the food was simple, it was good. \"Nonni made me promise if she showed me how to make her polpette I would only serve them with her special sauce. If I dared serve these with a jar of Prego, she would cross the ocean to break her wooden spoon against my backside.\"\n\n\"She made you call her Nonni?\" I laughed, leaning back in my chair and unbuttoning the top button on my jeans. I had no shame. I'd eaten an obscene amount.\n\n\"You know what Nonni means?\" he asked, surprised.\n\n\"I had an Italian great-grandmother. She insisted everyone call her Nonni.\" I laughed again when his eyes went to my hands massaging my stomach.\n\n\"You gonna be okay there?\" He raised his eyebrows as he got up to clear.\n\n\"Yep, just need to breathe a little.\" I groaned, pulling myself up from the table.\n\n\"No, no, you don't have to help,\" he said, rushing to my side and grabbing my plate.\n\n\"Oh, no, I wasn't. I was gonna drop this off and pass out on that couch right there,\" I said, nodding toward the living room.\n\n\"You go relax. Anyone who just had that many balls in their mouth deserves a rest,\" he teased, and I flicked his ear.\n\n\"I said no more ball jokes! You've had your fun, now let me go die in peace.\" I shuffled into the living room. I really had made quite a little piggy of myself, but it was seriously good. I reclined and popped open another button on my jeans, relaxing into the cushions and replaying some of the finer points of the evening.\n\nWatching Simon cook was, in a word, hot. He was really at home in a kitchen, his earlier fussing about with the pie aside. Even his salad\u2014simple greens dressed lightly with lemon and olive oil, salt, pepper, and good Parmesan\u2014was easy and perfect.\n\n\"Pink Himalayan salt, thank you very much,\" he'd said proudly, producing a bag from his pantry. He'd brought it back from one of his many trips and had me taste a little before sprinkling it on the salad. Could have been pretentious, but it fit Simon. The many facets of this guy were astounding. My earliest assumptions about him were proving to be completely wrong. As assumptions tend to be...\n\nI could hear him tending to the dishes, and as much as I probably should have gone to help him, I simply couldn't remove myself from the couch. I snuggled on my side and looked around his living room again, my eyes drawn back to the tiny bottles of sand from all over the world. I marveled at how traveled he was, and how he seemed to enjoy it still. I gazed at the pictures of the woman in Bora Bora\u2014her dark, beautiful skin and the smooth planes of her body\u2014and thought about how different the three of the women in his harem were. Oops, make that two now that Katie\/Spanx was with her new man.\n\nSuddenly I could smell the apple pie and heard the oven door clank shut. I'd put it in his oven as soon as we came over so it would be ready after dinner.\n\n\"Don't you dare try to serve me pie now. I am stuffed, I tell you, stuffed!\" I yelled.\n\n\"Quiet, it's just cooling,\" he scolded, coming around the corner from the kitchen. \"You're gonna have to scooch over, sister. It's movie time,\" he instructed, pushing me with his big toe as I struggled to sit up straight.\n\n\"What is it that we're watching?\"\n\n\"The Exorcist,\" he whispered, turning off the light on the end table and leaving the room quite dark.\n\n\"Are you freaking kidding me?\" I screeched, leaning over him to turn it back on.\n\n\"Don't be a wuss. You're watching it,\" he hissed, turning it back off.\n\n\"I'm not a wuss, but there is stupid and not stupid, and stupid is watching a movie like The Exorcist with the lights off! That's just asking for trouble!\" I hissed back, turning it back on.\n\nIt was starting to look like a disco in here...\n\n\"Okay, I'll make a deal with you. Lights off, but\u2014\" he shushed me with is finger as he saw me begin to interrupt \"\u2014if you get too scared, lights go back on. Deal?\"\n\nI was still leaning across him on my way to turn the light back on again when I noticed how close I was to his face. And how I was angled across him like a girl waiting to get a spanking. And I knew he was capable of delivering one...\n\n\"Fine,\" I huffed as the opening credits came on. I returned to a normal, seated position.\n\nHe smiled triumphantly and gave me a thumbs up.\n\n\"If you show me that thumb one more time I'll bite it off,\" I growled, pulling an afghan off the back of the couch and curling it protectively around me. One minute into the movie, and I was already spooked.\n\nI was tense from that moment on, and any idea I might have had about girls being ridiculous around guys when they watched scary movies went by the wayside when Regan peed herself at the dinner party.\n\nBy the time the priest came for a little visit, I was practically sitting on Simon's lap, my right hand had a death grip on his thigh, and I was viewing the movie through the holes in the afghan, which I had draped entirely over my head.\n\n\"I actually, literally, hate you for making me watch this movie,\" I whispered in his ear, which was right in my face as I refused to leave any space between us. I'd even accompanied him to the bathroom earlier when we took a break. He insisted I stay out in the hallway, but I stood just outside the door, eyes glancing around furtively, still with the afghan over my head.\n\n\"Do you want me to stop? I don't want you to have nightmares,\" he whispered back, his eyes on the screen.\n\n\"Just no banging on the walls for a few nights, please. I won't be able to take it,\" I said, looking at him through one of my eyeholes.\n\n\"Have you heard any banging lately?\" he asked, rolling his eyes as he did every time he looked at me with the ridiculous afghan on my head.\n\n\"No, I haven't actually. Why is that?\" I asked.\n\nHe took a breath. \"Well, I\u2014\" he started, and then the most maniacal scary noises started coming from the TV, and we both jumped.\n\n\"Okay, maybe this movie is a little scary. You wanna sit closer?\" he asked, pressing pause on the remote.\n\n\"I thought you'd never ask,\" I cried, launching myself fully into his lap and settling between his thighs. \"Do you want some afghan?\" I offered, and he laughed.\n\n\"No, I can take it like a man. You stay under there, though,\" he teased.\n\nI narrowed my eyes at him through the eyeholes and poked one finger through the weave. \"Guess which finger this is,\" I said, waving it at him.\n\n\"Shhh, movie,\" he answered, wrapping his arms around me and pulling me back against his chest.\n\nHe was warm and strong and powerful, but absolutely no match for terror that was The Exorcist. What had we been talking about? Now I couldn't think about any walls banging except the one Regan was currently banging the shit out of and spraying down with pea soup. We watched the rest of that damn movie wound around each other like pretzels, and he finally succumbed to the false security that an afghan eyehole can provide.\n\nClick. Click. Click.\n\nWhat the hell was that?\n\nClick. Click. Click.\n\nOh no.\n\nI lay paralyzed in my bed, every light in my entire apartment blazing.\n\nClick. Click. Click.\n\nI pulled the covers up higher, covering my face up to my eyes, which kept a constant vigil around the bedroom. Brain knew we were safe and secure, but also kept replaying scenes from that terrible, terrible movie, making it impossible to shut off for the night and go to sleep. Nerves had everything on lockdown, blazing a trail of fiery adrenaline throughout my body. I hated Simon with every fiber of my being in that moment. I also wished he was here.\n\nClick. Click. Click.\n\nWhat was that?\n\nClick. Click.\n\nNothing.\n\nThen Clive leaped on the bed, and I screamed bloody murder. Clive puffed out his tail and hissed at me, wondering why the hell Mommy was screaming at him, I'm sure. The click-click-click was his goddamned kitty hangnail.\n\nMy phone vibrated an instant later, shaking the entire nightstand and eliciting another scream from me. It was Simon.\n\n\"What the hell is wrong? Why are you screaming? Are you okay?\" he yelled when I answered, and I could hear him through the phone and through the wall.\n\n\"Get your ass over here right now, you motherfucking scary movie pusher,\" I seethed and hung up. I pounded on the wall and ran out to unlock the door. In much the same way I'd run up the last few steps of the basement stairs when I was a kid, I hightailed it back into my room, jumping the last few feet and landing in the center of my bed. I wrapped the covers around me and peered out, waiting. He knocked, and I heard the door push open.\n\n\"Caroline?\" he called.\n\n\"Back here,\" I yelled. Sad that I'd been reduced to this, but I was glad to see him.\n\n\"I brought the pie,\" he said with an embarrassed grin. \"And this,\" he added, producing the afghan from behind his back.\n\n\"Thanks.\" I smiled at him from behind my pillow shield.\n\nA few minutes later we were settled on my bed, each balancing a plate and a glass of milk. We'd been too full, then too terrified to eat pie earlier. Clive and his phantom hangnail retired to the other room after rolling his eyes at Simon and swishing his tail.\n\n\"How old are you?\" I asked, cutting into my pie.\n\n\"Twenty-eight. How old are you?\"\n\n\"Twenty-six. We are twenty-eight and twenty-six years old and terrified of a movie,\" I mused, poking in a bite. The pie was good.\n\n\"I wouldn't say I'm terrified,\" he countered. \"Spooked? Yes. But I only came over to stop you from screaming.\"\n\n\"And to taste my pie,\" I added, winking.\n\n\"Shut it, you,\" he warned, and then he went ahead and tasted my pie.\n\n\"Jesus, that's good,\" he breathed, eyes closed as he chewed.\n\n\"I know. What is it about apples and homemade pie crust? Is there anything better?\"\n\n\"If we were eating this naked, then it would be better,\" He grinned, opening one eye.\n\n\"No one is getting naked here, buddy. Just eat your pie.\" I pointed at his plate with my fork.\n\nWe chewed.\n\n\"I feel better,\" I added a few minutes later, drinking my milk.\n\n\"Me too. Not too spooked anymore.\"\n\nHe smiled as I took his plate and set it on the nightstand. I sighed contentedly and lay back against my pillows, sated and less scared.\n\n\"So, I gotta ask...James Brown? I mean, James Brown?\" He laughed, and I kicked him as he lay down next to me. We turned on our sides to face each other, arms curling under the pillows.\n\n\"I know, I know. I can't believe you held it in as long as you did! I know you've been dying to make jokes since last night.\"\n\n\"Seriously, who is this guy?\" he asked.\n\n\"He's a new client.\"\n\n\"Ah, got it,\" he said, looking pleased.\n\n\"And an old boyfriend,\" I added, watching for his reaction.\n\n\"I see. New client but old boyfriend\u2014wait, the lawyer?\" he asked, trying to keep his expression neutral, but failing.\n\n\"Yep. Haven't seen him in a few years.\"\n\n\"How's that gonna work?\"\n\n\"Don't know yet. We'll see.\"\n\nI really didn't know how things were going to go with James. I was glad to see him, but it was going to be tough to keep things professional if he wanted more. And every instinct I had told me he wanted more. In the past he'd had more control over me than I was comfortable relinquishing. I'd found myself sucked into the gravitational pull that was James Brown\u2014lawyer, not Godfather of Soul.\n\n\"Anyway, we're just going to be working together. It'll be a great job for me. He wants his entire place redone.\" I sighed, already planning the palette. I rolled onto my back and stretched. I'd really abused my stomach tonight and was starting to get sleepy.\n\n\"I don't like him,\" Simon said suddenly, after a long pause.\n\nI turned and saw him scowling.\n\n\"You don't even know him! How could you possibly not like him?\" I laughed.\n\n\"I just don't,\" he said, now turning his gaze to mine and unleashing the power of the baby blues.\n\n\"Oh, please, you're just a stinky boy.\" I laughed, ruffling his hair. Wrong move. It sure was soft...\n\n\"I don't stink. You said yourself I was April fresh,\" he protested, lifting his arm and sniffing.\n\n\"Yes, Simon, you smell delicious,\" I deadpanned, sniffing the air around me.\n\nHe left his arm up higher on the pillow, and I knew if I rolled just a little I could slide right on into the nook. He looked at me, raising his eyebrows ever so slightly. Was he thinking what I was thinking?\n\nDid he want to nook me?\n\nDid I want to nook him?\n\nOh the hell with it...\n\n\"I'm coming into the nook,\" I announced and went full snuggle: head nestled in, left arm over chest, right arm tucked under his pillow. Legs I kept to myself\u2014I wasn't a total fool.\n\n\"Well, hello there,\" he said, sounding surprised. Then he curled himself around me immediately. I sighed again, wrapped in boy and voodoo.\n\n\"What brought this on, friend?\" he whispered into my hair, and I shivered.\n\n\"Delayed reaction to Linda Blair. I need some nook time. Friends can nook, can't they?\"\n\n\"Sure, but are we friends who can nook?\" he asked, tracing circles on my back. Him and his demon finger circles...\n\n\"I can handle it. You?\" I held my breath.\n\n\"I can handle just about anything, but...\" he started, and then stopped.\n\n\"What? What were you going to say?\" I asked, leaning up to look at him. One piece of hair uncurled from my ponytail and fell down between us. Slowly, and with great care, he pushed it back behind my ear.\n\n\"Let's just say that if you were wearing that pink nightie? You'd be in a heap of trouble.\"\n\n\"Well, it's a good thing we're just friends then, right?\" I forced myself to say.\n\n\"Friends, yes.\"\n\nHe stared into my eyes.\n\nI breathed in, he breathed out. We traded actual air.\n\n\"Just nook me, Simon,\" I said quietly, and he grinned.\n\n\"Come on back down here,\" he said and coaxed me back to his chest. I slid down, resting where I could hear his heart beat. He folded the afghan over us, and I noticed again how soft it was. It had served me well tonight, this afghan.\n\n\"I love this afghan, but I have to say it doesn't really fit your apartment\u2014the cool-dude motif you have going on,\" I mused. It was orange and pea green and very retro. He was silent, and I thought maybe he had fallen asleep.\n\n\"It was my mom's,\" he said quietly, and his grip on me became infinitesimally tighter.\n\nThere was nothing to say after that.\n\nSimon and I slept together that night, with every light in the entire place on.\n\nClive and his hangnail stayed away.\nChapter Eleven\n\nI WOKE UP A FEW HOURS LATER, startled by the warmth of the body next to me, which was decidedly bigger than the cat usually nestled against my side. I rolled carefully onto my back and away from Simon so I could see him. I could see him just fine as the lamps, along with all my other lights, continued to blaze away into the night, fighting back the evils of that awful movie.\n\nI rubbed my eyes and inspected my bedmate. He lay on his back, arms curled as though I was still in them, and I thought of how good it felt to nook with Simon.\n\nBut I shouldn't be nooking with Simon. Brain knew better. Nerves were in agreement. That was definitely a very, very slippery slope. And though the images of climbing a slippery Simon that immediately came to mind were far from innocent, I pushed them aside. I looked away and noticed the terribly wonderful afghan tangled between his legs\u2014and mine, for that matter.\n\nIt had been his mom's. Heart broke each time I thought of his sweet, timid voice sharing that little nugget with me. He didn't know I'd talked to Jillian about his past, that I knew his parents were no longer alive. The idea that he still clung to his mother's afghan was inexorably sweet, and once again my heart broke open.\n\nI was close with my parents. They still lived in the same house where I'd grown up, in a small town in southern California. They were great parents, and I saw them as often as I could, which is to say holidays and an occasional weekend. A typical twenty-something, I enjoyed my independence. But my parents were there when I needed them, always there. The idea that I would someday have to walk this earth without their anchor and misguided guidance made me wince, to say nothing of losing both of them at only eighteen.\n\nI was glad Simon seemed to have good friends and such a powerful advocate as Benjamin watching out for him. But as close as friends and lovers could be, there was something about belonging to someone completely that gave you roots\u2014roots you sometimes needed when the world battled against you.\n\nSimon stirred slightly in his sleep, and I watched him again. He murmured something that I couldn't quite pick out, but it sounded a little like \"meatballs.\" I smiled and allowed my fingers to slip into his hair, feeling the soft silk tousled on my pillow.\n\nGod, he gave good meatball.\n\nAs I stroked his hair, my mind wandered to a place where meatballs flowed endlessly and there was pie for days. I giggled to myself as sleepiness began to return, and I nestled back down into the nook. As I felt the comfort that only warm boy arms could provide, a little alarm went off in my head, warning me not to get too close. I had to be careful.\n\nClearly we were both divinely attracted to each other, and in another space and time, the sex would have been ringing out across the land and around the clock. But he had his harem, and I had my hiatus, not to mention that I did not have my O. So friends we would remain.\n\nFriends who meatball. Friends who nook. Friends who were headed to Tahoe very soon.\n\nI pictured Simon soaking in a hot tub with Lake Tahoe spread out in all its glory behind him. Which sight was actually more glorious remained to be seen. I settled back to sleep, rousing only slightly when Simon snuggled me a little closer.\n\nAnd even though it was barely above a whisper, I heard it. He sighed my name.\n\nI smiled as I slipped back to sleep.\n\nThe next morning I felt a persistent poking at my left shoulder. I brushed it away, but it continued.\n\n\"Clive, stop it, you asshole,\" I moaned, hiding my head under the covers. I knew he wouldn't stop until I fed him. Ruled by his stomach, that one. Then I heard a distinctly human laugh\u2014quiet and definitely not Clive.\n\nMy eyes sprang open, and the night before came back to me in a rush: the horror, the pie, the nook. I reached backward with my right foot, sliding it along the bed until I felt it stop against something warm and hairy. Although I was now more sure than ever it wasn't Clive, I poked with my toe, inching my way higher until I heard another chuckle.\n\n\"Wallbanger?\" I whispered, not wanting to flip over. True to form, I was spread-eagled diagonally across the entire bed, head on one side, feet practically on the other.\n\n\"The one and only,\" a delicious voice whispered in my ear.\n\nMy toes and Lower Caroline curled. \"Shit.\" I rolled onto my back to take in the damage. He was huddled in the one corner my body had allowed him. My bed-sharing habits had not improved at all.\n\n\"You sure can fill a bed,\" he noted, smiling at me from under the little bit of afghan I'd left him. \"If we're going to do this again there'll have to be some ground rules.\"\n\n\"This won't be happening again. This was in response to a terrible movie you inflicted on both of us. No more nooking,\" I stated firmly, wondering how dreadful my morning breath was. I cupped my hand in front of my face, breathed, and gave a quick sniff.\n\n\"Roses?\" he asked.\n\n\"Obviously.\" I smirked.\n\nI looked at him, exquisitely rumpled and in my bed. He smiled that smile, and I sighed. I allowed myself a moment to indulge in a fantasy where I was then quickly flipped and ravaged to within an inch of my life, but I wisely got control of my inner whore.\n\n\"What if you get scared tonight?\" he asked as I sat up and stretched.\n\n\"I won't,\" I threw back over my shoulder.\n\n\"What if I get scared?\"\n\n\"Grow up, pretty boy. Let's make coffee, and then I have to get to work.\" I whacked him with my pillow.\n\nHe slid out from under the afghan, taking care to fold it, and carried it with him into the kitchen where he set it gently on the table. I smiled, thinking of him saying my name in the night. What I wouldn't give to know what was running through his mind.\n\nWe moved about the kitchen with quiet economy, grinding beans, measuring coffee, pouring water. I put the sugar and cream on the counter while he peeled and sliced a banana. I poured granola, he milked and banana-ed the bowls for us. Within a few minutes we were seated next to each other on barstools, eating breakfast as though we'd been doing so for years. Our simple ease intrigued me. And worried me.\n\n\"Plans for the day?\" I asked, digging into my bowl.\n\n\"I need to stop by the Chronicle office.\"\n\n\"Are you working on something for the paper?\" I asked, surprised at the level of interest even I could hear in my voice. Would he be in town for a while? Why did I care? Oh boy.\n\n\"I'm spending a few days on a piece about quick getaways in the Bay Area\u2014weekend drives kind of thing,\" he answered through a mouthful of banana.\n\n\"When are you going to do that?\" I asked, examining the raisins in my bowl and trying not to look too interested in his answer.\n\n\"Next week. I leave on Tuesday,\" he replied and my stomach was instantly queasy. Next week we were supposed to go to Tahoe. Why the hell did my stomach care so much that he wouldn't be going?\n\n\"I see,\" I added, again fascinated by the raisins.\n\n\"But I'll be back before Tahoe. I was planning on just driving straight there when I finish my shoot,\" he said, looking at me over the rim of his coffee mug.\n\n\"Oh, well, that's good,\" I answered quietly, my stomach now bouncing all around.\n\n\"When are you headed up, anyway?\" he asked, seeming to now be studying his own bowl.\n\n\"The girls are driving up with Neil and Ryan on Thursday, but I have to stay in the city to work until at least noon on Friday. I'm gonna rent a car and drive up that afternoon.\"\n\n\"Don't rent a car. I'll swing through to pick you up,\" he offered, and I nodded without a word.\n\nThat settled, we finished our breakfast and watched Clive chase a stray piece of fluff around the table over and over again. We didn't talk much, but whenever we met each other's eyes, we both grinned.\n\nText between Mimi and Sophia:\n\nDid you know Caroline is working with James?\n\nJames who?\n\nJames Brown, obviously. Who else?\n\nNO! What the hell?\n\nRemember she mentioned she had a new client? She neglected to mention who he was.\n\nI'm gonna kick her ass when I see her next. She better not cancel on Tahoe. Did Ryan tell you he was bringing his guitar?\n\nYep, he told me you wanted to have some kind of fucked-up singalong.\n\nHe did? Haha. I just thought it would be fun.\n\nText between Neil and Mimi:\n\nHey, Tiny, are we still bowling with Sophia and Ryan tonight?\n\nYep, and you better bring your A game. Sophia and I are pretty severe.\n\nSophia knows how to bowl? Wow.\n\nWhy is that wow?\n\nI just wouldn't have expected her to bowl is all. See you tonight.\n\nText between Neil and Simon:\n\nYou still planning on heading up with us this weekend?\n\nYep, but I'm coming a little late, have a shoot\n\nWhen are you coming up?\n\nFri night sometime, stopping thru the city on my way\n\nWhy the hell are you going back into the city? You're doing that shoot in Carmel, right?\n\nI just need to pick up some shit for the weekend.\n\nDude, pack your shit and get your ass to Tahoe.\n\nI will, but I'm picking up Caroline.\n\nI see.\n\nYou see nothing.\n\nI see everything.\n\nYou sure about that, Big Boy? What about Sophia?\n\nSophia? Why is everyone asking me about Sophia?\n\nSee you in Tahoe.\n\nText between Mimi and Caroline:\n\nYou have some splainin' to do, Lucy...\n\nOh no, I hate it when you go Ricardo on me.  \nWhat the hell did I do?\n\nExplain to me why you didn't tell me about your new client.\n\nCaroline, don't ignore my text! CAROLINE!!\n\nOh, settle down. This is exactly why I did NOT tell you.\n\nCaroline Reynolds, this is news that obviously  \nI should have known about!\n\nLook, I can handle it okay?? He's my client, nothing more. He's going to spend an obscene amount of money  \non this project.\n\nI frankly don't care how much he's spending.  \nI don't want you working with him.\n\nListen to yourself! I will take on whatever new client  \nI damn well please! I have this under control.\n\nWe'll see...Did I hear a rumor that you're driving  \nup to Tahoe with Wallbanger?\n\nWow, subject change. Yes, I am.\n\nGood. Take the long way.\n\nWhat the hell is that supposed to mean?\n\nMimi?? You there??\n\nDamn you, Mimi...HELLO??\n\nText between Caroline and Simon:\n\nWallbanger...come in Wallbanger\n\nWallbanger isn't here, only the exorcist\n\nNot even a little bit funny.\n\nWhat's up?\n\nWhat time are you picking me up tom?\n\nI should be back in the city by noon.  \nIf you can knock off work we can beat rush hour.\n\nAlready told Jillian I'm taking a half day.  \nWhere are you right now?\n\nIn Carmel, on a cliff overlooking the ocean\n\nBoy, are you a closet romantic...\n\nI'm a photographer. We go where the money shot is.\n\nOh man, we're not discussing money shots.\n\nBesides, I thought you were the romantic one\n\nI told you, I'm a practical romantic.\n\nWell then practically speaking, even you would appreciate this sight\u2014waves crashing, sun setting, it's nice\n\nAre you alone?\n\nYep\n\nBet you wish you weren't.\n\nYou have no idea\n\nPfft...you old softie\n\nThere's nothing soft about me, Caroline.\n\nAnd we're back...\n\nCaroline?\n\nYep\n\nSee you tomorrow\n\nYep\n\nText between Caroline and Sophia:\n\nCan you give me the address again to the house  \nso I can plug it into the GPS tom?\n\nNo\n\nNo?\n\nNot until you tell me WHY YOU'RE HIDING JAMES BROWN.\n\nJesus, it's like having 2 more mothers...\n\nThis isn't about sitting up straight or eating more vegetables, but we do need to have a conversation about your posture.\n\nUnbelievable.\n\nSeriously, Caroline, we just worry.\n\nSeriously, Sophia, I know. Address please?\n\nLet me think about it.\n\nNot gonna ask you again...\n\nYes you will. You want to see Simon in that hot tub. Don't lie.\n\nI hate you...\n\nText between Simon and Caroline:\n\nYou done with work?\n\nYep, at home waiting for you.\n\nNow that's a nice visual...\n\nPrepare yourself, I'm taking bread out of the oven.\n\nDon't tease me, woman...zucchini?\n\nCranberry orange. Mmmm...\n\nNo woman has ever done breakfast bread foreplay  \nthe way you do.\n\nHa! When are you coming?\n\nCan't. Drive. Straight.\n\nCan we have one conversation where you're not twelve?\n\nSorry, I'll be there in 30\n\nPerfect, that will give me time to frost my buns.\n\nPardon me?\n\nOh, I didn't tell you? I also made cinnamon rolls.\n\nBe there in 25.\n\n\"I'm not listening to this.\"\n\n\"Like hell. It's my car. Driver picks music.\"\n\n\"Actually, you're wrong about that. The passenger always picks music. It's what you get when you give up driving privileges.\"\n\n\"Caroline, you don't even own a car, so how could you ever have driving privileges?\"\n\n\"Exactly, so we listen to what I pick,\" I chided, sitting back after changing the radio station for the hundredth time. I hit the iPod and scrolled until I found something that I thought would please us both.\n\n\"Good song,\" he admitted, and we hummed along.\n\nThe trip had been great so far. When I first met him\u2014heard him\u2014I never would have predicted it, but Simon was quickly turning into one of my favorite people. I'd been wrong about him.\n\nI glanced at him: humming along to the song, drumming his thumbs along the steering wheel. As he was concentrating on the road, I took the opportunity to catalogue some of his more swoon-worthy features.\n\nJaw? Strong.\n\nHair? Dark and messy.\n\nStubble? About two days' worth and nice.\n\nLips? Lickable, but lonely looking. Maybe I could check them out, do my own little tongue inspection...\n\nI sat on my hands to stop myself from launching over the console. He continued to hum and drum.\n\n\"What's going on over there, Nightie Girl? You look a little flushed. Need some more air?\" He started for the air conditioner.\n\n\"Nope, I'm good,\" I answered, my voice sounding ridiculous.\n\nHe looked at me strangely, but resumed his hum drum. \"I think it's time we broke out that cranberry-orange bread. Hit me,\" he said a moment later as I was indulging in a fantasy about how exactly I could maneuver myself into his lap while still maintaining a good highway speed.\n\n\"I'm on it!\" I hollered, diving into the backseat and surprising us both. I had my legs in the air and my bottom on display as I clasped my upside-down face in my hands behind the seat. I could feel how red my cheeks were, and I gave myself a little slap to snap me back into this world.\n\n\"That is one sweet ass, my friend.\" He sighed, leaning his head on it as though it were a pillow.\n\n\"Hey. Ass Man. Pay attention to the road and not my heiney, or no bread for you.\" I gave his head a bump with my bum and sent myself flailing as he took a turn.\n\n\"Caroline, you need to control yourself back there, or I'm pulling over.\"\n\n\"Oh, zip it. Here's your damn bread,\" I snapped, crawling back into my chair in a graceless way and throwing the bread at him.\n\n\"What the hell? Don't throw this. What if you'd bruised it?\" he cried, gently stroking the foil-wrapped loaf.\n\n\"I worry about you, Simon. I really do.\" I laughed, watching him struggle to open the end of the wrapper. \"You want me to cut you a piece\u2014okay, or you could just do that.\" I frowned as he took a giant bite out of the end.\n\n\"Thif if mine, righ?\" he asked, spraying crumbs.\n\n\"How do you function in normal society?\" I asked, shaking my head as he took another monster bite. He just smiled and continued, eating the entire loaf in less than five minutes.\n\n\"You're gonna be so sick tonight. That's meant to be eaten piece by piece, not ingested whole,\" I said. His only response was to burp loudly and pat his tummy.\n\nI couldn't help but laugh. \"You're one twisted man, Simon.\" I chuckled.\n\n\"You're still intrigued though, aren't you?\" He grinned, turning the blue eyes loose on me.\n\nMy panties actually disintegrated. \"Oddly, yes,\" I admitted, feeling my face flame again.\n\n\"I know.\" He smirked, and we drove on.\n\n\"Okay, the turn should be coming up just around this corner\u2014I remember that house!\" I cried, bouncing in my seat. It had been a while since I was up here, and I'd forgotten how beautiful it was. I loved Tahoe in the summertime\u2014all the water sports and everything\u2014but in autumn? Autumn was beautiful.\n\n\"Thank God. I need to pee,\" Simon groaned, as he'd been doing for the past twenty or so miles.\n\n\"That's your own fault for drinking that Big Gulp,\" I admonished, still bouncing away.\n\n\"Wow, is that it?\" he asked as we turned into the drive. Lanterns lit the way to a sprawling, two-story cedar house with a giant stone fireplace up the left side. Cars were already in the driveway, and I could hear the music spilling out from the back deck.\n\n\"Sounds like our friends have already got their party on,\" Simon observed. Squealing and laughter joined the music coming from the back side of the house.\n\n\"Oh, I don't doubt it. My guess is they've been drinking since dinner and are half-naked in the hot tub by now.\" I walked around back to grab my bag.\n\n\"We'll just have to catch up, now won't we?\"He winked, pulling a bottle of Galliano from his bag. \"I thought we could make some Wallbangers.\"\n\n\"Now isn't that interesting. I was thinking the same thing,\" I countered, pulling an identical bottle from my duffle.\n\n\"I knew you were dying to get me inside you, Caroline.\" He chuckled and grabbed my bag as we headed to the door.\n\n\"Please, you would make up a drink and call it a Pink Nightie just to have me in your mouth\u2014and don't even try to lie,\" I taunted, nudging him with my shoulder.\n\nHe stopped midway up the walk and looked at me fiercely. \"Is that an invitation? Cuz I'm a hell of a bartender,\" he stated, the eyes glowing in the darkness.\n\n\"I've no doubt,\" I breathed, the space between us now crackling with tension that was becoming ridiculously hard to ignore. I took a deep breath, and noticed he did as well.\n\n\"Come on, let's get sauced and start this weekend.\" He chuckled, nudging me with his shoulder and breaking the spell.\n\n\"Sauce away,\" I muttered, walking up the path behind him.\n\nFinding the front door open, Simon stashed our bags, and we made our way through the house to the back deck. There the lake spread out before us, just barely lit by the tiki torches dotting the dock and pathways that led to the shore. The entire back of the house was flanked with brick patios and decks, and that's where we found our friends.\n\n\"Caroline!\" Mimi screeched from the hot tub, where she and Ryan were splashing each other. Ah, we'd made it to Drunky Loud already.\n\n\"Mimi!\" I squealed back, looking around for Sophia. She and Neil were perched on the stone bench by the firepit, roasting marshmallows. They both waved merrily, and Neil gestured obscenely with his stick.\n\n\"Making them see the error of their ways might be easier than we thought, fellow matchmaker,\" I whispered to Simon, who was already mixing a cocktail at the patio bar.\n\n\"You think its gonna be that easy?\" he whispered back, giving his friends the international guy head nod that said, \"What's up, man?\"\n\n\"Hell, yes. They're almost already there without our help. All we have to do is show them what's right in front of them.\"\n\nHe handed me a cocktail. \"So, how am I?\" he asked, winking.\n\n\"Is this a Wallbanger?\"\n\n\"It is.\"\n\nI took a sip, swirling the taste around my mouth and over my tongue.\n\n\"You're as good as I knew you'd be,\" I whispered, taking a dangerously large swallow.\n\n\"To things staring you right in the face,\" he added, clinking my glass and taking his own large gulp.\n\n\"To things staring you right in the face,\" I echoed, our eyes locking over the rims.\n\nDamn Banger Voodoo.\nChapter Twelve\n\n\"WHOSE FOOT IS THAT?\"\n\n\"It's mine, Neil. Quit rubbing it.\"\n\n\"Dude! Quit trying to play footsie with me, Ryan!\"\n\n\"You're the one still holding my foot.\"\n\nRyan and Neil tried to look nonchalant as they disengaged from their footsie session under the bubbling water. I laughed as I caught Simon's eye across the hot tub, and he grinned back at me.\n\n\"Want another?\" he mouthed, nodding to my empty glass.\n\n\"I have had enough for tonight, don't you think?\" I mouthed back, as our friends cackled all around us.\n\n\"I thought you were girl who always wanted more,\" he mouthed. The characteristic smirk returned.\n\nI looked at him, the image of Hot Tub Simon that had been in my head for the last few weeks actually paling in comparison to the real thing. Strong arms stretched across the back of the hot tub, hair wet and artfully swept back. If I thought seeing him wet and half-naked on my kitchen floor was enticing, it was nothing like having him backlit by tiki torches and seen through a strong buzz.\n\nHe was now the most singularly handsome man I'd ever seen, and if I wasn't mistaken, he was trying to get me drunk. Brain was getting a bit fuzzy. Heart was beginning to sing Etta James songs.\n\n\"Are you trying to get me drunk?\" I asked, giggling as I pushed my empty glass away, resolving myself to no more alcohol.\n\n\"Nope. A sloppy Pink Nightie Girl gets me nowhere.\"\n\nHe grinned as I splashed water toward his side. Our friends had all quieted and were watching us with undisguised interest.\n\nAfter Simon and I arrived, we got our drinks, and then I showed him around the rest of the house. I left my bags at the front door, not knowing how the sleeping arrangements had been laid out. We returned to the patio to find that Sophia and Neil had joined Ryan and Drunky Mimi in the hot tub. A quick trip to the pool house left me in nothing but a dark green bikini and a smile as I approached the others. Simon had already jumped in, and I watched him watch me. As I slid under the warm water, I sipped my cocktail and drank in the sight of my neighbor, wet and in board shorts, before me. Sophia actually had to nudge me to stop the staring.\n\nNow we were smack dab in the middle of a sexual soup, bubbling away with two pairs of mismatched lovers and more pheromones than we knew what to do with.\n\nSo did I want another cocktail? Didn't matter. I couldn't afford it.\n\nI had to shake my head a little to clear it as I looked around at the rest of the group. Mimi had gotten too hot and was perched on the side, kicking Neil as she swung her feet back and forth. He indulged her in much the same way a big brother indulges his little sister. Sophia and Ryan were huddled on the other side, Sophia scratching Ryan's back as she and Neil talked back and forth about the 49ers' starting lineup or defensive line or something football-ish and, frankly, boring.\n\n\"So, what are we doing this weekend?\" I asked, focusing my attention on the group at large and not the blue eyes staring at me. Damn those eyes! They would be the death of me.\n\n\"We were thinking about going for a hike tomorrow. Who's in?\" Ryan asked.\n\nSophia shook her head. \"Count me out. No way am I hiking.\"\n\n\"Why not?\" Neil asked.\n\nSimon and I exchanged a quick glance at his sudden interest.\n\n\"Can't. Last time I hiked I took quite a spill and sprained my wrist. Can't take the chance during the season,\" she said, waving and reminding us she made her living with her hands. As a cello player, she could get out of quite a bit. Once she dodged hand jobs all winter. Investment banker Bob was not a happy camper.\n\n\"How 'bout you, Tiny?\" Neil pulled on Mimi's foot.\n\n\"Um, no, Mimi doesn't hike,\" she replied, adjusting her barely there black bikini. Her actual boy toy didn't notice, but I saw Ryan's eyes grow to the size of pies from across the hot tub as her breasts were nearly revealed.\n\n\"You gonna take a pass as well?\" Simon nodded to me.\n\n\"Hell, no. I'm hiking with the boys tomorrow!\" I laughed as Sophia and Mimi rolled their eyes. They never understood why I loved \"mountain man activities,\" as they called them.\n\n\"Nice,\" Simon purred, and for a second I calculated the distance between my mouth and his. Then we were all quiet, all six of us lost in our thoughts. I remembered the plan to out the four of them, and I jumped right in.\n\n\"So, Ryan, did you know Mimi here gives to your charity every year?\" I asked, surprising them both.\n\n\"You do?\"\n\n\"Yep, every year,\" she said. \"I've seen what having access to computers can do, especially for kids who wouldn't otherwise have the opportunity.\" She looked shyly at him, and they began a conversation about the process he used to determine which schools will receive the scholarships each year.\n\nSimon and I grinned at each other. Looking sideways at Sophia, Simon launched the second wave of the attack. \"Hey, Neil, how many seats did you get for the symphony this year?\" he asked.\n\nNeil blushed.\n\n\"You bought tickets?\" Sophia asked.\n\n\"Season tickets,\" Simon added, as Neil nodded. Sophia and Neil then launched into a discussion of where his seats were, and Simon raised his foot above the surface of the water.\n\n\"Come on, don't leave me hangin'.\"\n\n\"What?\"\n\n\"Gimme a little high five. I can't reach your hand,\" he insisted, waving his foot back and forth. I giggled and slid lower on my seat, stretching my foot out and patting his lightly.\n\n\"Ugh, pruney.\" He laughed.\n\n\"I'll give you pruney,\" I warned, dipping my foot and splashing him lightly.\n\n\"I could not be more comfortable. Seriously, I literally could not feel more cozy right now if I were actually inside a marshmallow,\" I mumbled through a thick tongue coated in Bailey's and coffee. I had curled up on top of about fifty pillows next to fireplace\u2014a fireplace with a hearth almost ten feet wide and a chimney almost three stories high. Made out of stone quarried nearby, it was massive. It was the focal point of the entire house, with rooms radiating out from its center. And it gave off massive heat.\n\nWe were chilled to the bone when we finally made it back inside. One by one, we all got too warm in the hot tub, so we hoisted ourselves out to cool off a little. By the time we realized how cold the night had gotten, we were shivering and puffing, and wanting nothing more than to curl up next to the fire. As we had yet to pick rooms, I soon learned, the girls snuck into the master bedroom to change into our pjs and rejoin the boys, who were now all decked out in T-shirts and pajama pants. We made a quick pot of coffee, and I sliced up some of the additional cranberry-orange bread I'd wisely hidden from Simon. A couple shots of Bailey's in the coffee cups, and we were all relaxing by the fire like an ad for Currier and Ives.\n\nSimon had reclined regally by the fireplace and patted the stack of pillows next to him. I dove in and a few stray puffs of feathers swirled around our heads. We'd discovered that each boy had a different method of starting a fire\u2014kindling, newspapers, kindling and newspapers\u2014when finally Sophia stuck her head up there and declared that the flue was still closed. Brought back down a few pegs, the guys at that point deferred to Ryan, if for no other reason than that he was the one holding the matches. But within minutes, they had a fire blazing, and we were now all seated around the fireplace, sleepy and content.\n\nI breathed deeply. There was nothing like the smell of an actual fire\u2014not a gas fireplace, not a bunch of candles, but an honest to goodness fireplace with snaps and crackles and funny little whizzing screeches when the steam came out of a crack in the wood.\n\n\"So, Caroline, have you asked Simon to teach you how to windsurf yet?\" Mimi asked suddenly from her perch on the arm of the couch. We'd been quiet for a while, drowsy and almost dreaming, and I started a little when she spoke.\n\n\"What? I mean, what?\" I asked, startled out of my pillows and back to the present.\n\n\"Well, these boys here all windsurf. You want to learn to windsurf, and I bet Simon here would show you, wouldn't you, Simon?\" She giggled, polishing off the last of her coffee and sliding off the arm of the chair into Ryan's conveniently placed lap. They smiled at each other for a moment before they realized what they were doing and Ryan jokingly launched her off his own lap and into Neil's. He'd not been awakened by her earlier question, but he now seemed wide awake with a lapful of Scheming Mimi.\n\n\"You want to learn to windsurf?\" Simon asked, turning toward my pillow pile.\n\n\"Actually, yes. I've always wanted to try it.\"\n\n\"It's tough\u2014not gonna lie. But totally worth it.\" He smiled, and Ryan nodded from across the room.\n\n\"Sure, Simon'll show you. He'd love to,\" Ryan chimed in, earning a wink from Mimi and an eye roll from me.\n\n\"We can plan something for when we get back to the city,\" I suggested.\n\n\"No more talk tonight. This girl has had it,\" Sophia said. \"I'm pooped. Where are we all sleeping?\" She poked her head over the back of the armchair where she'd been curled up.\n\n\"Well, how many rooms we talking about?\" Simon asked as I sat up and yawned.\n\n\"There are four bedrooms, so take your pick,\" Sophia answered, then wisely drained an entire bottle of water.\n\n\"Are we doing boy-girl, boy-girl?\" I asked, laughing when I saw Simon's surprised face.\n\n\"We can, sure,\" Mimi answered, looking a little nervously at Neil.\n\nI stifled a giggle when I saw Sophia and Ryan trade a similar spooked look. Simon caught it as well.\n\n\"Yeah, sure! Don't let Caroline and me stand in the way of the lovebirds! Mimi, you and Neil pick a room, Sophia and Ryan can pick a room, and Caroline and I will take the rooms that are left over. Perfect. Right, Caroline?\"\n\n\"Sounds perfect to me. I'll just rinse out these mugs. Now, off to bed with you all. Scoot! Scoot!\" I cried. Simon and I scurried about cleaning up while sneaking peeks over our shoulders at the four of them. They looked like they'd just begun a death march.\n\n\"Oh, man, I hope this works out...for my sake.\" I stood behind Simon as we watched the four become two pairs as they parted ways by the bedroom doors.\n\n\"Why for your sake?\" he whispered, turning his face just a little to be inches from mine.\n\n\"Because right now, behind those doors? Sophia and Mimi are trying to figure out the best way to hurt me. Physically hurt me,\" I sighed, backing away to rinse the last of the coffee cups and place them in the dishwasher.\n\nSimon added the soap and switched it on. As we walked around, turning lights off for the night, we talked about the hike we'd be taking tomorrow.\n\n\"You're not gonna slow me down, are you?\" he teased.\n\nI shoved him into the wall. \"Please, you will be eating my trail dust tomorrow, bucko,\" I warned, grabbing my bag and heading for the bedrooms.\n\n\"We'll see, Nightie Girl. Speaking of, got any nighties in there for me?\" He poked his hand into my bag as he followed me down the hall.\n\n\"Stay outta there. Nothing for you in there, or anywhere for that matter.\" I stopped at the room I was taking.\n\nHe went past me to the room next door. \"Look at that, sharing a bedroom wall once again.\" He smirked.\n\n\"Well, I know you're in there alone, so I'd better not hear any banging,\" I warned, leaning in the doorway.\n\n\"No, no banging. 'Night, Caroline,\" he said softly, leaning in his own doorway.\n\n\"'Night, Simon,\" I answered, giving him a little waggle of my fingertips as I closed my door. I placed my bag on my bed and smiled.\n\n\"Come on, guys, not that much farther,\" I yelled behind me as I surged up the final leg of the trail. We'd been hiking for about two hours now, and while everyone stayed together for a while, in the last thirty minutes or so, Ryan had slowed considerably, and Neil hung back with him. Simon and I kept the pace together, and were about to reach the crest of the trail.\n\nI'd managed to avoid being alone with Sophia or Mimi, although the puffy eyes and tired faces on all four of them proved no one had gotten a good night's sleep\u2014except Simon and me.\n\nAfter breakfast, I dodged the firing squad by changing quickly and waiting outside for the boys before the hike. I knew once I returned to the house I'd be in for it, although I admit I was curious to see how they were planning to rage without acknowledging that sleeping with the guys they'd been seeing for weeks now was not, in fact, what they wanted to do.\n\nBut as Simon had said, \"Here's to things staring you in the face.\" Tonight should be interesting.\n\nI pushed up and over the last little ridge and made it to the top. Simon was only a few yards behind me, and I could hear him on his way. I breathed deep, the clear air prickling at my lungs. It was chilly, but I was warm with exertion. It had been a while since I'd gotten out of the city, and my body had missed hiking like this. My legs were burning, my nose was running, I was sweating like a pig, and I couldn't remember when I'd felt better. I laughed out loud as I looked down at the lake below, spying a few hawks gliding on a downdraft. The steely blue of the lake, the deep green of the forest, the clean whites and creams of the rocks: it was beautiful.\n\nAnd then there was my new favorite blue. Simon appeared at my side, breathing as deeply as I was. He stretched his arms wide and took in the valley below. He'd peeled off layers as we climbed and was now wearing a white T-shirt with a flannel knotted at his waist. Khaki shorts, hiking boots, and a wide grin completed the wet dream I now stared at, instead of looking at the natural wonders all around us. And those blue eyes\u2014I could see them framing each shot as he looked around.\n\n\"Beautiful,\" I breathed, and he turned to me. I got caught staring. \"I mean, isn't it beautiful?\" I stuttered, gesturing widely with my arm.\n\nHe appeared to know exactly what I'd been doing, and I felt the blush come up in my cheeks. Luckily, I was still a bit winded from the climb, and I hoped I was already sufficiently red.\n\n\"Yes, it is beautiful actually. Very beautiful.\" He smiled, and we stared at each other. He took a few steps closer, and I felt the air shift and change. I bit my lip. He ran his hand through his hair. We smiled. There were no words, but even the woodland animals could tell there was something about to happen and wisely stayed in their hidey holes.\n\n\"Hi,\" he said quietly.\n\n\"Hi,\" I answered.\n\n\"Hi,\" he said again, taking one last step toward me and stepping inside my little circle. One more step and he'd be practically on top of me. And how.\n\n\"Hi,\" I said once more, tilting my head to the side and letting him know he could take that last step.\n\nSimon leaned toward me, just barely, but almost as if he were going to...\n\n\"Parker!\" thundered from below, and we both sprang back. \"Parker!\" It came again, and I recognized Ryan's voice underneath the jungle-man yell.\n\n\"Ryan,\" we both said and smiled.\n\nNow that the voodoo wasn't so concentrated, I could see things clearly again, and I repeated the word harem over and over again in my head.\n\n\"Up here!\" Simon yelled, and Ryan appeared from around a bend.\n\n\"Hey there! Neil is done, kaput, thrown in the towel, so to speak. You guys about ready to head back down?\" he called, jumping from rock to path to rock again with the ease of a mountain goat. He didn't even appear to be winded. Hmmmm...\n\n\"Yep, we were just about to come looking for you guys,\" I said, kicking my leg up behind me for a quick stretch.\n\n\"Is he really pussing out so close to the top?\" Simon asked, heading back down the trail.\n\n\"He's lying straight across the trail like he owns the place, refusing to go any higher.\" Ryan laughed, bounding ahead and calling down to Neil to let him know we were on our way.\n\n\"You sure you didn't want to stay up here a little longer? I mean, we worked so hard to get all the way here,\" Simon asked, reaching out to stop me running down the mountain after Ryan.\n\nI felt the warmth of his hand on my shoulder and willed my hormones to flee to the other side of my body. \"I'm sure. We should get back. Looks like a storm is coming.\" I nodded toward the horizon where a group of dark clouds had begun to build. His eyes followed mine, and he frowned.\n\n\"You're probably right. We don't want to get caught out here all alone,\" he muttered.\n\n\"Besides, if we don't hurry, we can't tease Neil about getting beat up the mountain by a girl.\" I grinned, and he laughed loudly.\n\n\"Hell, we don't want to miss that. Let's go.\"\n\nAnd down the trail we went.\n\n\"So how was your gangbang, Caroline?\" Sophia sang sweetly as she found us all in the kitchen drinking water after our hike. The three guys each did different versions of a spit take, but I calmly continued sipping like a lady.\n\n\"Fantastic, thanks. Neil especially. We practically had to carry him back down the mountain after I finished with him,\" I replied just as sweetly.\n\nThe boys recovered their game faces, but Neil could barely stop staring at Sophia's tight tank top. Her actual suitor? Playing Spot The Mimi, his head rotating so fast I could have sworn he was an owl. I shook my head and put him out of his misery.\n\n\"Where's Mimi?\" I asked.\n\n\"Shower, which you four clearly need. It's freezing outside. How could you have gotten so sweaty?\" she asked, wrinkling her nose.\n\n\"We worked hard making it up that mountain. Hiking is harder than you think,\" Neil puffed, and the rest of us wisely kept silent about the heart attack he almost had fifty feet from the summit.\n\nI grabbed an apple and headed in the direction of my room with Sophia hot on my tail, as expected. I smirked a little and contemplated going easy on her\u2014just asking her about it, giving her an out.\n\n\"Those shorts look terrible on you, Caroline,\" she remarked as she followed me into my room.\n\nNope. Not going to happen. No easy out. \"Thank you, dear. Should I have packed a little cat food for you when I packed Clive's travel bag?\" I sneered.\n\nShe collapsed on my bed, curling her body around one of the giant pillows. \"Where is he anyway? Who's watching him this weekend?\"\n\n\"He's staying with Uncle Euan and Uncle Antonio. That cat is lounging on a silk bed being handfed tuna rolls right about now. He's living the life.\"\n\n\"He has the life, that's for sure,\" she said, her face clouding briefly as she got comfortable.\n\nI peeled off my sweaty clothes and wrapped up in a terrycloth robe hanging on the back of the door. She complimented my choice of sports bra and laughed when she saw that I'd paired it with leopard panties, but then she went back to her previous wistful expression.\n\n\"What's up, Sophia?\" I asked, lying on the bed next to her and wrapping myself around a pillow as well.\n\n\"Nothing, why?\" she asked.\n\n\"You look like a sad sack.\"\n\n\"Eh, I just didn't sleep well, I guess.\"\n\n\"Oh really? Mr. Ryan keeping you up late at night, hmm? He didn't have a lot of energy on the mountain today...\" I nudged her with my elbow.\n\n\"No, no, nothing like that. I just...I dunno. I just couldn't get settled last night. Normally I sleep really well up here, but it was so quiet last night, I just...\" She beat her pillow a little with her fist, forcing it into a new shape.\n\n\"I see. Well, I slept great!\" I laughed, and she starting trying to force my head into a new shape with her fist.\n\n\"You wanna get drunk tonight?\" she asked when we finally settled down.\n\n\"Hell, yes. You?\"\n\n\"Yes, ma'am.\"\n\nThere was a knock at the door, and Mimi poked her towel-covered head in. \"Is this a private session, or can a non-lesbian get into this bed?\" she called.\n\nWe waved her in, and she vaulted from the floor to the bed and landed on top of both of us.\n\n\"What are we doing in here, ladies? Foreplay or just going for it?\" she asked.\n\n\"Please say foreplay,\" a male voice said from the now-open door. We rolled over to see the men in the doorway, different versions of the same oh-my-goodness-girls-in-bed-together look on their faces.\n\n\"Oh, get over yourselves. Like we'd ever need a guy to tell us whether we needed foreplay or not.\" Sophia giggled, kicking a foot in the air and waving at them from over my shoulder. They shifted their weight from one foot to the other and cleared their throats. So predictable.\n\n\"We're planning on getting drunk tonight. You boys up for it?\" Mimi yelled. Even though no alcohol was currently present in her system, Drunky Mimi's volume level was already making an encore appearance.\n\n\"Done and done,\" Ryan answered, giving us a weird little salute that made us giggle even harder.\n\n\"Now run away, boys, and let us have our girl time,\" Sophia tossed over her shoulder, lifting my robe a little and giving my ass a quick smack. I squealed and tried to cover myself, but it was too late.\n\n\"Dude. Leopard print,\" Neil whispered to Simon in the kind of whisper that's actually louder than just speaking.\n\n\"I know, I know,\" Simon countered, then drew his hand down his face as though he were trying to physically remove the image from his brain.\n\nSimon liked animal prints. Duly noted.\n\n\"Come on, guys. The ladies have requested a little alone time, so let's leave them to it.\" Ryan tugged them out into the hallway and closed the door behind them with a wink that made Mimi's entire neck turn bright red. Sophia examined her fingernails.\n\nI was really going to have fun with these two tonight.\n\n\"Where the hell did you learn to cook like this? Jesus, this is good!\" Neil exclaimed, taking his third helping of paella from the giant pan in the center of the table.\n\n\"Thank you, Neil.\" I laughed as he dug into another pile of rice.\n\nSimon nodded toward my wine glass, and I nodded back.\n\nI'd thought about making a quick version of paella when I saw all the wonderful seafood on sale at the local market, and when I saw their special on Spanish Ros\u00e9 and Cava, my plans came together. We'd started in on the Cava while prepping in the kitchen. The sparkling Spanish wine went perfectly with the wedge of Manchego I'd picked up, as well as the little salty olives. Once again, Simon was my helper, and we moved together in the kitchen. The other four settled on bar stools across from us while we cooked, someone popped an old Otis Redding record on the ancient turntable, and we were in business.\n\nThe wine flowed as freely as the conversation, and I could tell this had the potential to become a tight-knit group. Similar interests, similar senses of humor, but everything just different enough to keep it lively.\n\nSpeaking of lively, as the alcohol was inhaled, the walls came down. Mimi and Sophia were barely hiding their misplaced interests anymore. Not that the boys were minding. In fact, they were encouraging it. Ryan currently examined Mimi's foot for what she insisted was a spider bite. The fact that he'd been inspecting it for several minutes, and that said inspection included a calf massage did not escape my attention, or Simon's.\n\nHe grinned and motioned for me to move closer. I slid across the bench and inclined my head to his. He put his mouth next to my ear, and I inhaled. Wine, heat, and actual sex ran straight up my nostrils and invaded my brain, turning everything a bit fuzzy.\n\n\"How long before they kiss?\" he whispered, his mouth so close I swear I felt lips brush my ear.\n\n\"What?\" I asked, beginning to giggle the way I did when I'd had a little too much to drink and a little too much sexy dangled in front of me.\n\n\"How long? You know, before they kiss the wrong person?\" he asked as I turned to look into his eyes.\n\nThose eyes, oh, those eyes were now calling to me.\n\n\"You mean the right person?\" I whispered.\n\n\"Yeah, the right person,\" he answered, scooting a little closer on the bench.\n\n\"I don't know, but if the kiss doesn't come soon, I'm gonna burst,\" I admitted, knowing full well I was no longer talking about our friends. And knowing full well that he knew full well I was no longer talking about our friends.\n\n\"Hmm, I wouldn't want you to burst.\" He was now mere inches from my face.\n\nHarem. Harem. Harem. I repeated this mantra over and over.\n\n\"I wanna go in the hot tub.\"\n\nThe whining pulled me away from the voodoo and back to the kitchen. Where there were people present.\n\n\"I wanna go in the hot tub,\" I heard again and turned to address Mimi. Imagine my surprise when I saw that Sophia was actually the whiner, and she was now hanging on Neil like a backpack.\n\n\"Okay, so go in the hot tub. No one's stopping you,\" I insisted, sliding away from Simon and back in front of my plate where I began separating my peas from my lobster. I was full, but I would never leave lobster on the plate. I had standards, after all.\n\n\"You have to come too,\" Sophia whined again as I began to comprehend. Sophia was drunk. Sophia got clingy when she got drunk. Oh boy.\n\n\"Go ahead. I'll clean up the kitchen a little and then meet you guys out there,\" Simon said, taking my plate and starting to stand.\n\n\"Hey, hey, hey! Lobster bite, hello,\" I protested as I grabbed my fork.\n\n\"Here, I would never get between a woman and her lobster.\" He smiled, offering me my fork back. I accepted the bite with a smile and stood up. I was a little more drunk than I thought, and this fact made itself known as gravity began tease me.\n\n\"Whoa there, you okay?\" he asked, steadying me as Sophia started off for the bedroom.\n\n\"Yeah, I'm fine, I'm fine,\" I answered, planting my feet and winning the battle.\n\n\"Maybe you outta slow down?\" he asked, taking my wine glass.\n\n\"Oh, lighten up, it's a party,\" I cried, beginning to giggle. Suddenly everything was funny.\n\n\"Okay, party on.\" He smiled as I headed to the bedroom to change into my suit. Which proved harder than I thought. String bikinis are difficult to tie when you're more than a little buzzed.\n\n\"Okay, Caroline's next. Truth or dare,\" Mimi yelled, once again proving that Drunky Mimi only had one volume level.\n\n\"Truth,\" I yelled back, splashing Sophia in the face accidentally as I reached behind me for my glass of wine. We'd brought out the last bottle of Cava and were steadily working our way through it. And it was steadily working its way through us, our game becoming more and more dangerous. The sky crackled a bit with far-away lightning, and low rumbles of thunder were just beginning to be heard over the giggling and splashing.\n\nOnce we came outside and got settled in the hot tub, it was only minutes before Neil suggested a game of Truth or Dare, and only seconds after that before Sophia agreed to it. I laughed it off at first, saying there was no way I'd play such a childish game. But when Simon implied that I was chicken, the alcohol reared its ugly head and shouted something to the effect of, \"I will play Truth or Dare, you sucker, until you can't tell your truth from your dare!\"\n\nThis statement made perfect sense in my head, and must have seemed logical to Mimi and Sophia as well, as they immediately began offering me high fives and you-go-girls. I'm pretty sure I saw Simon shake his head, but he was smiling, so I let it go. And poured another glass of sparkly.\n\n\"Where's the one place you want to travel, and haven't been yet,\" she asked, humming along to the tunes coming through the French doors.\n\nSophia had found all of her grandfather's old records, and Simon almost had a fit when he saw the collection. He'd selected a Tommy Dorsey album, and the big band accentuated the night perfectly.\n\n\"Boring, make her take a dare!\" Simon sang, and I stuck my tongue out at him.\n\n\"It's not boring, and she chose truth so she gets truth. Caroline, where is the one place on earth you want to go?\" she asked again.\n\nI leaned my head back against the edge of the tub. I looked up at the stars and an image immediately came to mind: soft wind blowing, warm sun on my face, the ocean spread out in front of me dotted with craggy rocks. I smiled just thinking about it.\n\n\"Spain,\" I sighed quietly, the smile lingering as I imagined myself on a beach in Spain.\n\n\"Spain?\" Simon asked.\n\nI turned my face toward his. He was smiling back at me. \"Spain. That's where I want to go. But it's so expensive, it's going to have to wait a while,\" I smiled again, my head still wrapped around the image.\n\n\"Hey, wait, Simon, aren't you going to Spain next month?\" Ryan asked, and my eyes widened.\n\n\"Um, yeah. Yeah, I am actually,\" he answered.\n\n\"Great! Caroline, you can go with him,\" Mimi decided, clapping her hands and turning to Ryan.\n\n\"Ryan, you're next.\"\n\n\"No, no, wait a minute. First of all, I can't just go with Simon to Spain. And second of all, it's my turn,\" I protested, as Simon sat up.\n\n\"Actually, you could 'just go with Simon to Spain,'\" he said, turning to me fully. The other side of the hot tub got very quiet.\n\n\"Um, no, I can't. You're working. I can't afford a trip like that, and besides, I don't know that I can take time off next month.\" I felt my heart swell as I processed what he'd just said.\n\n\"Actually, I heard Jillian telling you the other day that next month would be a good time to take your vacation before the holiday season,\" Mimi piped up. She sank back into the shadows as I glared at her.\n\n\"Be that as it may, I also can't afford it, so discussion ended. Now then, I believe it's my turn. Let's see, who should I pick?\" I looked around at everyone.\n\n\"It wouldn't be that expensive. I'm renting a house, so that would be paid for. Airfare and spending money\u2014that's all you'd have to cover,\" Simon added, not letting this go.\n\n\"Hey, that's a pretty good deal there, Caroline,\" Mimi spouted, her energy making little ripples across the tub.\n\n\"Okay, Mimi, truth or dare?\" I asked, gritting my teeth and pushing ahead with the game.\n\n\"Hey, we're discussing something here. Don't change the subject,\" she objected.\n\n\"Well, I'm done discussing it. Truth or dare, you little shit,\" I said again, letting her know I meant business.\n\n\"Fine. Dare,\" she pouted.\n\n\"Great. I dare you to kiss Neil,\" I shot back, not missing a beat.\n\n\"What?\" she shouted, as the entire hot tub erupted in gasps.\n\n\"Hey, we're just playing a game, right? And Mimi, really, it's not that shocking that I would dare you to kiss the guy you've been seeing for weeks now, is it?\"\n\n\"Well, no, I just, I don't like public displays,\" she sputtered, almost going under. This from the girl who was almost arrested for public nudity when she was found under the bleachers at a football game freshman year at Berkeley.\n\n\"Oh, come on, what's the big deal?\" Simon chimed in, and I looked at him gratefully.\n\n\"Nothing, it's just\u2014\" she said again, and Neil interrupted.\n\n\"Oh, come here, Tiny,\" he exclaimed and pulled her over. They stared at each other for a second, and then Neil swept her hair out of her face. He smiled, and she leaned in. I heard Sophia inhale at the same time Ryan did, and we all watched as Mimi kissed Neil.\n\nAnd it was weird.\n\nThey broke away, and Mimi swam back over to her side. Next to Ryan. All was quiet for a moment. Simon and I looked at each other, not sure what to do next. We'd been outsmarted. And I got pissed when I got outsmarted. I began to burn. The fact that I was drunk had nothing at all to do with my overreaction.\n\n\"Okay, I guess it's my turn. Hmmm...Ryan, truth or dare?\" Neil started, and I stood up, splashing everyone around me as I did.\n\n\"No, no, no! That's not what was supposed to happen!\" I yelled, stamping my foot, losing my balance and going under in the process. Simon's strong hands brought me back to the surface, and I continued my alcohol-induced tirade. Flashes of lightning, now much closer, blazed across the sky.\n\n\"You were not supposed to let her kiss him!\" I sputtered, spitting out water and pointing at Ryan and then at Mimi. I whirled on Sophia. \"And you were supposed to get mad at her!\"\n\n\"Why would I get mad at Mimi? For kissing her boyfriend?\" Sophia mumbled, taking a sudden interest in her fingernails.\n\n\"Argh!\" I screamed and turned back to Mimi.\n\n\"Mimi, are you even remotely interested in Neil?\" I challenged, hands on my hips as I steamed into the night air.\n\n\"Neil is exactly what I've always wanted in a man. He is my type to a T,\" she countered robotically, flinching when Ryan looked at her with hurt in his eyes.\n\n\"Blah, blah, blah, have you fucked Neil yet?\" I screeched, pointing wildly as I tend to do when I drink.\n\n\"Okay, Caroline, you've made your point,\" Simon soothed, trying to get me to sit back down.\n\n\"What point? What are you two talking about?\" Sophia asked, leaning forward.\n\n\"Oh, please, the four of you are ridiculous! I don't care what you all think you want on paper. In reality, you're doing it all wrong!\" I answered, smacking the top of the water for emphasis. Why weren't they getting it? I don't know when I'd gotten so riled up, but in the last sixty seconds or so, I'd become blazing mad.\n\n\"Are you kidding?\" Mimi cried, jumping to her feet in the hot tub, which kept the water at about the same level.\n\n\"Mimi, come on! Anyone with eyes can see the way you and Ryan feel about each other! Why the hell are you wasting time on anyone else?\" I pushed.\n\nSimon pulled me back onto his lap and attempted to quiet me.\n\n\"Okay, this has gone far enough,\" Neil said, starting to get out of the tub.\n\n\"No, no! Neil, look at Sophia. Can't you see she is totally into you? Why the hell are you all so thick? Seriously? Are Simon and I the only ones that can see clearly here?\" I yelled once more, bringing Simon into the conversation whether he wanted it or not.\n\nNeil looked at Ryan, and then at Simon.\n\n\"Dude!\" Neil exclaimed.\n\n\"Dude,\" Simon answered, gesturing toward Sophia, who stood up like she was going to say something. Neil put his hand on her shoulder, and she stopped and sat back down. Neil nodded at Ryan.\n\n\"Dude?\" he asked, and Ryan nodded back. Neil took a deep breath and looked at Sophia.\n\n\"Sophia, truth or dare?\" Neil asked.\n\n\"We are not playing any more\u2014\" I tried to yell, but Simon took that moment to place his hand over my mouth.\n\n\"All clear over here,\" Simon announced as he pinned me to his lap more securely with his other hand on my waist. Thunder rolled in, blanketing the scene with an ominous air.\n\n\"Sophia?\" Neil asked again. She was quiet, and not looking in the direction of Mimi and Ryan.\n\n\"Dare,\" she whispered and closed her eyes.\n\nAlcohol made everything much more dramatic.\n\n\"I dare you to kiss me,\" Neil said, and all you could hear was the occasional loon over the lake. The loons in the tub were finally quiet. We all watched as Sophia turned to Neil and placed a hand on the back of his head, pulling him toward her. She kissed him, slowly but surely, and it went on for days. I smiled into Simon's hand, and he patted my stomach, which made me giddy.\n\nWhen they finally broke apart, Sophia was laughing into Neil's mouth, and he answered with his giant, goofy man-giggle.\n\n\"Well, it's about freaking time,\" Simon said, releasing my mouth.\n\n\"Mimi, I\u2014\" Sophia began, turning toward Mimi and finding an empty hot tub.\n\nMimi and Ryan were gone. I caught just the edge of Ryan's towel headed into the pool house\u2014with a slippery wet companion on his arm.\n\n\"Well, then, I guess we'll call it a night.\" Sophia sighed, grabbing Neil by the hand.\n\n\"'Night.\" I giggled as she walked into the house with Neil in tow. They cuddled close, already a picture in the making. I looked at the pool house, and noticed no lights had come on yet. They probably would not be coming on in the near future.\n\n\"Well, that was a fine bit of matchmaking, although your bull-in-a-china-shop delivery left a lot to be desired.\" Simon chuckled, letting his head rest against my back. I was still perched on his lap. His hand had left my mouth, and it was now drifting south, while his other hand remained tightly on my waist.\n\n\"Yes, I usually leave a lot to be desired,\" I observed wryly, not wanting to leave this exquisite spot, but knowing I needed to\u2014and soon. Simon was quiet behind me, and I started to move off his lap.\n\n\"You leave everything to be desired, Caroline,\" he said softly, and I froze. It was quiet for another moment, both of us not moving, but still moving toward each other.\n\nWithout looking back, I let out a tiny laugh. \"You know, I never really got that phrase. Does that mean I am desirable or\u2014\"\n\nHis fingers began to trace tiny circles on my skin. \"You know exactly what it means,\" he breathed into my ear. The air crackled around us, the tension as well as the actual weather. More tiny circles. In the end, it was the tiny circles that finally broke me.\n\nI lost all control. I turned quickly, catching him off guard as I wrapped my legs around his waist and threw caution, and my harem mantra, to the wind. I sunk my hands into his hair, luxuriating in the feel of wet silk around my fingertips as I pulled him toward me.\n\n\"Why did you kiss me that night at the party?\" I asked, my mouth mere inches from his. Once he realized I was driving this bus, he responded by pressing his hips into mine, bringing us closer together than we'd ever been.\n\n\"Why did you kiss me?\" he asked, running his hands up and down my back, settling into the space where his hands spanned my waist exactly\u2014thumbs in front, fingers in back\u2014and pressed me into him further.\n\n\"Because I had to,\" I answered honestly, remembering how I'd reacted instinctively, kissing him when I'd wanted to do anything but. \"Why did you kiss me?\" I asked again.\n\n\"Because I had to,\" he said, the smirk returning. Luckily I didn't see the smirk for too long. Because I'd finally discovered the secret of the ages.\n\nHow do you make Wallbanger stop smirking? You kiss him.\nChapter Thirteen\n\nTHE SKY OPENED UP, pelting us with chilly rain, which mixed with the heat around us, and between us. I looked at Simon underneath me, warm and wet, and there was nothing in the world I wanted more than his lips against mine. So even though every single cowbell in my head was ringing out the alarm, I centered myself, wrapped my legs tighter around his waist, and gazed directly into his eyes.\n\n\"Mmm, Caroline, what are you up to?\" He smiled, his hands strong on my waist as his fingers dug into my skin. His skin slipped against mine in a way that made me not right in the head, and I could feel\u2014I could actually feel\u2014his abs against my tummy. He was so strong, so powerfully delicious that Brain began to burn, and other organs began to make all my decisions.\n\nI think O even popped her head up for a moment, like a groundhog. She took a quick glance around and pronounced it much closer to spring than she'd been in months.\n\nI licked my lips, and he mirrored my actions. I could barely see him through the haze of steam from the hot tub and the lust now brewing in this little cauldron of chlorinated chemistry.\n\n\"I'm up to no good, that's for sure,\" I breathed, rising up just a little. The feeling of my breasts crushing against his skin was unimaginable. As I settled on his lap again, I felt his reaction in a very tangible way, and we both groaned at the contact.\n\n\"You're up to no good, huh?\" he said, his voice gruff and thick and maple syrup pouring over me.\n\n\"No good,\" I whispered in his ear as he pressed his mouth against my neck. \"Wanna be bad with me?\"\n\n\"You sure about that?\" he groaned, hands clutching at my back with delightful abandon.\n\n\"Come on, Simon, let's bang some walls,\" I answered, allowing my tongue to dart out from between my lips and against the skin just underneath his jaw. The scruff scratched my taste buds and gave me a sense of what that very scruff would feel like against other soft places on my body.\n\nO poked her head out just a little more at that point and went straight to Brain, which in turn spoke directly to my hands.\n\nI grasped him firmly at the base of his neck, and positioned him directly in front of me, his eyes flaring wide and turning into tiny little hypnotizers.\n\nHis grin was hard, and so was he.\n\nI leaned in and sucked his bottom lip between my teeth, nibbling lightly before biting down and pulling him closer. He came willingly, ceding control as my fingers pulled and pushed at his hair, and my tongue pressed into his mouth as he groaned into mine. Everything in my world now narrowed to just the feeling of this man, this wonderful man in my arms and threaded between my legs, and I kissed him like the world was about to end.\n\nIt wasn't sweet and tentative, it was pure carnal frustration spiked with incomprehensible lust and rolled into a giant ball of please-God-let-me-live-in-this-man's-mouth-for-the-forseeable-future. My mouth led his in a dance as old as the mountains that watched over us approvingly, our tongues and teeth and lips smacking and cracking and giving in to the sweet tension that had been building since I showed up at his door wearing the inspiration for my nickname.\n\nI shook as I felt his hands reach lower to grasp my bottom and pull me closer still, my legs scrambling as I panted like a whore in church. The Church Of Simon...where I was dying to kneel before him.\n\nMy eyes were closed, my legs were open, and I was now moaning into his mouth like some kind of rabid dog. The idea that a kiss, just a kiss, had turned me into this giant lusting bag of CarolineNeedThat was undeniable, and I knew that if he continued to make me feel this way I was going to invite him straight into my Tahoe. Great idea.\n\n\"Come into my Tahoe, Simon,\" I mumbled incoherently into his mouth.\n\nHe paused. \"Caroline, come into your what? Oh, God,\" he managed, as I pushed us off the side of the hot tub and vaulted us across the water, emptying half of its contents onto the deck and the other half sloshing us around like it was high tide. He slammed me into the opposite wall, pushing me up against the bench and rewrapping my legs around his waist, as I gamely pushed my mouth back onto his, unwilling to let go of him. At one point, I kissed him so hard, he had to push me off so he could catch a breath.\n\n\"Breathe, Simon, breathe.\" I giggled, stroking his face as he struggled before me.\n\n\"You...are...a mad woman,\" he panted, his hands looping underneath my arms and curling around the tops of my shoulders, keeping me firmly against the side while I dug my heels into his backside, nudging him to exactly where I needed him. He closed his eyes and bit down on his lower lip, an animalistic growl sounding low in his throat as I launched my second wave of the Lower Caroline-commanded attack.\n\n\"You feel uncommonly good,\" I moaned as I began to kiss him again, raining them down across his mouth, his cheeks, his jaw, slipping underneath to suck and bite at his neck as he dropped his head back to allow my assault. His hands were rough on me, dipping down low on my back and catching on my bikini strings, loosening the sides. The thought of my naked breasts against his skin drove me crazy with lust, and I removed my hands from his poor hair to snake back behind my neck and pull on the knot. As I maneuvered, I knocked into one of the empty bottles of Cava, starting a domino effect of bottles crashing to the ground. I giggled as he pulled back, startled at the sound.\n\nHis eyes were smoky blue, crowded with lust, but as they focused on me, they began to crystallize. I finally managed to get to the knot untied and could feel the water swirl across my naked skin. I started to drop the strings, when Simon grasped them tightly in his hands. He shook his head as though to clear it, then closed his eyes firmly, cutting off our connection.\n\n\"Hey, hey, hey!\" I prodded, forcing his eyes open and making him look at me. \"Where did you go just now?\" I whispered.\n\nHe wrapped his hands, still holding my strings, back around my neck. He slowly began to tie my suit back together, and I felt my face flush bright red, all the blood in my body betraying me in that instant.\n\n\"Caroline,\" he began, breathing heavily, but looking at me carefully.\n\n\"What's wrong?\" I interrupted.\n\nHis hands came to rest on my shoulders, and he seemed to be keeping a specific distance between us.\n\n\"Caroline, you're amazing, but I...I can't\u2014\" he started.\n\nNow I was the one to close my eyes. Emotions whirled behind my eyelids, shame being chief among them. Heart plummeted. I could feel his eyes on me, willing me to open my own.\n\n\"You can't,\" I stated, opening my eyes and looking anywhere but at him.\n\n\"No, I mean, I...\" he stammered, clearly uneasy as he moved away from me.\n\nI began to shake. \"You...can't?\" I asked, suddenly feeling icy cold, even in the water. I unlocked my legs from around him, allowing him the room he needed to move away.\n\n\"No, Caroline, not you. Not like\u2014\"\n\n\"Well, don't I feel like a fucking idiot?\" I managed, laughing shortly and pulling myself up and out of the water to the side of the hot tub.\n\n\"What? No, you don't understand, I just can't\u2014\" he started toward me, and I kicked out a leg, pressing my foot square in the center of his chest to keep him away.\n\n\"Hey, Simon, I get it. You can't. It's cool. Wow, what a crazy night, huh?\" I laughed again, swinging over the side and starting for the house, wanting to get away before he could see the tears I knew were on their way. Of course, as I tried to navigate the steps, I slipped in a wet spot and fell with a big thud. I could feel the back of my eyeballs begin to burn as I scrambled up as quickly as I could, panicked that I was going to cry before I could get inside. Now that I was moving, I could feel the effects of all the alcohol I'd consumed, and the beginnings of a very strong headache.\n\n\"Caroline! Are you okay?\" Simon cried, starting to get out of the hot tub.\n\n\"I'm fine. I'm fine. Just...\" I got out, my throat beginning to close as I choked back a sob. I held my hand out behind me, willing him to understand that I did not need his help. \"I'm fine, Simon.\"\n\nI couldn't turn around and see him. I just continued walking away. The cursed big band music still played on the turntable, but I still heard him say my name once more. Ignoring it, I made my way to the door, feeling foolish now in my teeny bikini that was clearly not as enticing as I thought it was.\n\nI didn't even bother to grab a towel. Instead I threw open the glass door and heard it slam shut behind me as I all but ran for my room. I left little puddles along the slate floor down the hallway, trying to ignore the giggles coming from Sophia's room. As the tears finally coursed down my cheeks, I locked the door and stripped off my bathing suit. I stumbled into the bathroom, flicked on the light, and there I stood, reflected back to me. Naked, wet hair streaming down my back, a bruise already beginning to form on my thigh from my drunken spill...and puffy, kiss-swollen lips.\n\nI wrapped my hair in a towel, and then leaned on the countertop, bringing my face within inches of the mirror.\n\n\"Caroline, my dear, you just got turned down by a man who once made a woman meow for thirty minutes straight. How do you feel?\" the naked woman in the mirror asked me, turning my thumb into a little microphone. She gestured toward me, holding out her thumb.\n\n\"Well, I drank enough wine to sustain a small Spanish village, I haven't had an orgasm in a thousand years, and I will probably die old and alone in a beautifully designed apartment with all of Clive's illegitimate children swarming around me...How do you think I feel?\" I asked back, offering Mirror Caroline her thumb.\n\n\"Silly Caroline, you had Clive neutered,\" Mirror Caroline answered, shaking her head at me.\n\n\"Go fuck yourself, Mirror Caroline, since I can't even do that,\" I finished, ending the interview and taking my naked ass back into the bedroom. Throwing on a T-shirt, I fell into bed, my drunk self exhausted from the hike and the dinner and the wine and the music and the best make-out session I'd ever engaged in. The thought of it brought my tears to the surface again, and I rolled over to grab some tissues, only to find an empty box, which made me cry even harder.\n\nStupid Wallbanger voodoo.\n\nCould this night get any worse?\n\nThen my phone rang.\n\n\"Pancakes, sweetie?\"\n\n\"Love some. Thanks, babe.\"\n\nJesus.\n\n\"Is there still cream for the coffee?\"\n\n\"I got your cream right here, honeybunch.\"\n\nSweet Jesus.\n\nListening to a new couple, much less two new couples was sometimes vomit-worthy. Add that to a hangover, and this was going to be a long morning.\n\nAfter talking to James on the phone last night, I'd fallen into a deep sleep, aided, no doubt, by all the wine I'd consumed. I woke with a thick tongue, a splitting headache, and a queasy stomach\u2014made even more queasy by the knowledge that I'd have to see Simon this morning and have that weird we-totally-made-out-last-night conversation.\n\nJames had made me feel better, though. He'd made me laugh, and I remembered how well he took care of me back in the day. It was a nice memory, and an even nicer feeling. He'd called under the pretense of checking with me about a paint color, which I quickly called as a bluff. Then he'd admitted he just wanted to talk to me, and fresh off the Great Hot Tub Rejection, I was happy to talk to someone I knew wanted my attention. Damn you, Simon. When James asked me to dinner next weekend, I agreed immediately. We'd have a great time...and since my O was back in her hidey-hole, I might as well enjoy a night on the town.\n\nNow, I was seated at the breakfast table, surrounded by two new couples who were filling the kitchen with enough sexual satisfaction to make me scream. I didn't though. I kept it to myself as Mimi perched happily on Ryan's lap, and Neil fed Sophia melon balls as though he was put on the earth for this reason and this reason alone.\n\n\"How was the rest of your evening, Ms. Caroline?\" Mimi chirped, raising a knowing eyebrow. I pressed the tines of my fork into her hand and told her to zip it.\n\n\"Wow, grumpy. Someone must have spent the night alone,\" Sophia murmured to Neil.\n\nI looked up at her in surprise. The casualness with which they were treating this was really starting to bother me.\n\n\"Well, of course I spent the night alone. Who the hell do you think I spent the night with? Huh?\" I asked, slamming back from the table and knocking my orange juice glass over. \"Ah, fuck it all to hell,\" I muttered, stomping off toward the patio, tears threatening for the second time in less than twelve hours.\n\nI sat in one of the Adirondack chairs, looking out over the lake. The cool of the morning soothed my heated face, and I wiped clumsily at my tears as I heard the girls footsteps follow me outside.\n\n\"I don't want to talk about it, okay?\" I instructed, as they took the seats opposite me.\n\n\"Okay...but you gotta give us something. I mean, I thought for sure when we left last night, I mean...you and Simon are just\u2014\" Mimi started, and I stopped her.\n\n\"Me and Simon nothing. There is no me and Simon. What, you thought we'd pair off just because you four finally figured your shit out? You're welcome for that, by the way,\" I snapped, pulling my ball cap down lower on my face, hiding my continuing tears from my best friends.\n\n\"Caroline, we just thought\u2014\" Sophia began, and I cut her off as well.\n\n\"You thought since we were the ones left over we'd just magically become a couple? How storybook\u2014three sets of perfectly matched couples, right? Like that ever happens. This isn't some romance novel.\"\n\n\"Oh, come on, you two are perfect for each other. You called us blind last night? Hi, pot. It's me, kettle,\" Sophia snapped back.\n\n\"Hi, kettle, you have about thirty seconds before this pot kicks your ass. Nothing happened. Nothing is going to happen. In case you forgot, he has a harem, ladies. A harem! And I'm not about to become his third chippie. So you can forget it, okay?\" I yelled, pushing out of the chair, turning for the house, and running right into a quiet Simon.\n\n\"Great! You're here too! And I see you two peeking through the blinds, idiots!\" I cried as Neil and Ryan backed away from the window.\n\n\"Caroline, can we talk, please?\" Simon asked, grasping me by the arms and spinning me toward him.\n\n\"Sure, why not? Let's make the embarrassment complete. Since I know you're all dying to know, I threw myself at this guy last night, and he turned me down. Okay, secret's out. Now can we please drop it?\" I wiggled out of his grip and walked toward the trail to the lake. I heard nothing behind me and turned to see all five of them, wide eyed and evidently unsure what to do next.\n\n\"Hey! Come on, Simon. Let's go,\" I snapped my fingers, and he started after me, looking a little afraid.\n\nI stomped down the trail and tried to slow my breathing. My heart was pounding, and I didn't want to talk when I was this riled up. No good could come of it. As I breathed in and out, I took in the beautiful morning all around and tried to let that lighten my heart a bit. Did I need to make this more awkward that it already was? No. I had the control here, last night notwithstanding. I could make it so last night never happened, or I could certainly try.\n\nI breathed again, feeling a bit of the tension leave my body. Despite everything that happened, I enjoyed Simon's company and had to come to think of him as my friend. I still stomped along the path, but eventually eased back into a moderately pissy stroll.\n\nI left the trees behind and didn't stop until I reached the end of the dock. The sun peeked out after last night's storm, casting a silver light on the water.\n\nI heard him approach and stop just behind me. I took one more deep breath. He was silent.\n\n\"You're not going to push me in, are you? That would be a bad move, Simon.\" He exhaled a laugh, and I smiled a little, not wanting to, but not able to help it.\n\n\"Caroline, can I explain about last night? I need to you know that\u2014\"\n\n\"Just don't, okay? Can't we just chalk it up to the wine?\" I asked, whirling about to face him and trying to beat him to the punch.\n\nHe stared down at me with the strangest look on his face. He looked like he'd gotten dressed in a hurry: white thermal, well worn jeans, and hiking boots that weren't even laced up, the strings now damp and muddy from the trek through the woods. Still, he was stunning, the early morning sun illuminating the strong planes of his face and that scruff that was so delicious.\n\n\"I wish I could, Caroline, but\u2014\" he started again.\n\nI shook my head. \"Seriously, Simon, just\u2014\" I began, but stopped when he pressed his fingers against my mouth.\n\n\"You have to shut up, okay? You keep interrupting me, and watch how fast you get tossed in that very lake,\" he warned with the twinkle in his eye I'd become so used to.\n\nI nodded, and he removed his hand. I tried to ignore the flames that licked at my lips, brought to the surface by just that little touch.\n\n\"So, last night we came really close to making a very big mistake,\" he said, and when he saw my mouth begin to open, he wagged his finger at me.\n\nI zipped my lip, miming throwing the key into the water. He smiled sadly and continued.\n\n\"Obviously I'm attracted to you. How could I not be? You're amazing. But you were drunk, I was drunk, and as great as it would have been, it would have\u2014ah, it would have changed things, you know? And I just can't, Caroline. I can't allow myself to...I just...\" He struggled, running his hands through his hair in a gesture I'd come to understand as frustration. He stared at me, willing me to make this okay, to tell him we were okay.\n\nDid I want to lose a friend over this? No way.\n\n\"Hey, like I said, it's cool\u2014too much wine. Besides, I know you have your arrangement, and I can't...Things just got away from me last night,\" I explained, trying to sell it to him.\n\nHe opened his mouth to comment, but after a moment he nodded and sighed a great sigh. \"We still friends? I don't want this to get weird for us. I really like you, Caroline,\" he asked, looking as though his world was about to come to an end.\n\n\"Of course friends. What else would we be?\" I swallowed hard and forced a smile. He smiled too, and we began to walk back up the trail. Okay, that wasn't too bad. Maybe this could work. He stopped to pick up a handful of sand from the beach and put it in a little plastic baggie.\n\n\"Bottles?\"\n\n\"Bottles.\" He nodded, and we started up the path.\n\n\"So it looks like our little plan worked,\" I began, searching for conversation.\n\n\"With those guys? Oh yeah, I think it worked well. They seem to have found what they needed.\"\n\n\"That's all anyone's trying to do, right?\" I laughed as we crossed the patio to the kitchen. Four heads disappeared from the window and began to assume positions of nonchalance around the table. I chuckled.\n\n\"Always good when what you need and what you want are the same things,\" Simon said, holding the door open for me.\n\n\"Boy, did you say a mouthful.\" A pang of sadness hit me again, but I didn't have to force the smile once I saw how happy my friends were.\n\n\"You want some breakfast? There are still some cinnamon buns, I think,\" Simon offered, walking over to the counter.\n\n\"Um, no. I think I'm gonna go pack, get my stuff together,\" I said, noticing a flash of disappointment cross his face before he smiled bravely.\n\nOkay, so this wasn't great. Well, that's what happens when two friends kiss. Things are never the same. I nodded at my girls and headed for my room.\n\nSpurred by my insistence about getting back to the city, within two hours we were all packed and deciding who was going to ride with whom. I didn't want to ride alone with Simon, so I pulled Mimi aside and instructed her to bring Ryan along with us. Now we were all outside arranging bags. As Simon piled everything into the Range Rover, I shivered a little, realizing too late that I'd packed my fleece jacket into my bag, which was now buried. As he turned back toward me, he noticed.\n\n\"You cold?\"\n\n\"A little, but it's fine. My bag's at the bottom, and I don't want you to have to rearrange everything,\" I answered, stamping my foot to keep warm.\n\n\"Oh! That reminds me, I have something for you,\" he exclaimed, rummaging in his bag, which was on top. He handed me a lumpy package, wrapped in brown paper.\n\n\"What's this?\" I asked, as he blushed deeply. Simon does blush? I rarely saw that...\n\n\"You didn't think I forgot this, did you?\" he replied, his hair falling into his eyes a little as he smiled a boyish smile. \"I was going to give it to you last night, but then\u2014\"\n\n\"Hey, Parker! Could use a little help over here!\" Neil called as he struggled to load all of Sophia's luggage. Yesterday, this would have been Ryan's job. Now it was Neil's. Yesterday. How the world had changed in one day.\n\nHe backed away from me as Mimi and Ryan got themselves settled in the backseat.\n\nI opened the package to find a very thick, very soft Irish sweater. I lifted it out of the paper, feeling the weight and the nubbly texture of the weave. I pressed it against my nose, inhaling the scent of wool and unmistakable Simon that clung to it. I grinned into the sweater, then quickly slipped it over my T-shirt, admiring the way it hung loose and low, yet still wrapped me in a comforting way. I turned to see Simon watching me from over at Neil's truck. He smiled as I twirled for him.\n\n\"Thank you,\" I mouthed.\n\n\"You're welcome,\" he mouthed back.\n\nI gave my sweater a long, deep sniff, hoping no one noticed.\nChapter Fourteen\n\nINSIDE A BLACK RANGE ROVER on the way back to San Francisco...\n\nCaroline: Okay, I can do this...It's only a few hours back into the city. I can be the bigger person here. I can act like he didn't pull an all stop at the thought of seeing my tatas last night\u2014and what the hell? What man says no to tatas? I mean, they're nice tatas. They were pushed up nice and tight, and they were wet, for Christ's sake...Why didn't he want my tatas? Caroline, just settle down...Just smile at him and act like everything is fine. Wait, he's looking over here. Smile! Okay, he smiled back...Stupid tata turner-downer...I mean, what's up with that? And he was hard!\n\n...\n\nSimon: She's smiling at me...I can smile back at her, right? I mean, we're acting natural, right? Okay, done. I hope that looked more natural than it felt. Jesus, who knew a giant sweater would look so good on a girl...But everything looks pretty good on Caroline\u2014especially that green bikini. Did I really turn her down last night? God, it would have been so easy to just...But then I couldn't. Why couldn't I??? Jesus, Simon. Well, we were drunk...Correction, she was drunk. Would she have regretted it? She might have. Couldn't risk it? Might have been a bit of a disaster...Or was it the girls? I shouldn't do that to the girls either. But it's not even really working so well with the girls these days, now is it? Huh, I didn't think about them once this weekend...because I couldn't stop thinking about Caroline. She's looking at me again...What the hell are we going to talk about the whole way back to the city? Ryan isn't even paying attention. Bastard. I told him he needed to help me out...He's helping himself to a handful of Mimi. I'm almost sorry Caroline and I worked so hard to push them together. Hmm...Caroline and I...Caroline and me in a hot tub where bikinis are outlawed...Jesus, wait a minute\u2014yep, now I've got a semi...\n\n...\n\nCaroline: Why is he twitching like that? Jesus, does he have to pee? Maybe I have to pee. Maybe this would be a good time to suggest a pee break...Then I can grab Mimi and make sure she knows the reason they're riding with us is not so they can suck face the whole way, but to run interference for me with Scared of Tatas over there. Okay, just ask him to pull over at the next gas station. Wow, he really does have to pee, I guess. I hope this gas station has Gardetto's.\n\n...\n\nSimon: Thank God she wanted to stop. Now I can adjust without looking like a pervert...oh, who am I kidding? I am a pervert. I'm riding in a car with a woman who was straddling me last night and just the thought of it makes me hard. Pervert, pervert, pervert. I hope this gas station has Gardetto's.\n\n...\n\nMimi: Ooh! We're stopping! I hope this gas station has bubble gum!\n\n...\n\nRyan: Oh, man, we're stopping already? We're not going to make it back to the city before dark. Mimi wants me to see her place, and I'm really hoping that means walk around naked and let me watch...I hope this gas station has condoms.\n\n...\n\nCaroline: Okay, you could have handled that a little better. Mimi suggesting you and Simon split the big bag of Gardetto's was not that big of a deal. Am I a little sensitive today? Yes, I suppose I am...But I know for a fact that Simon was checking out my ass as I walked away from the car. Why the hell is he checking out my ass now? Last night he didn't even want to peek under my bikini. Is he really that complicated? Why the hell is he looking at me? He's reaching his hand out. Stay still, Caroline, stay still...Oh, sesame seed on my chin. Well, if you weren't looking at my mouth, Mr. Mixed Messages, you wouldn't even have noticed it. You will never get this sesame seed now, buddy. Damn! Why does this sweater have to smell so good? I hope he hasn't noticed me sniffing this sweater the whole way.\n\n...\n\nSimon: She's really sniffly today. I hope she isn't catching a cold. We spent so much time outside this weekend...I would hate for her to come down with something. She just sniffled again. Should I offer her a Kleenex?\n\n...\n\nMimi: Busted, Caroline. I totally knew you were sniffing that sweater.\n\n...\n\nRyan: I wonder if Mimi has any more of that bubble gum? I hope she didn't notice me buying those condoms. I mean, I don't want to be presumptuous. But I definitely want to be under her again sometime very, very soon. Who knew someone so tiny could be so loud...and now I'm hard.\n\n...\n\nMimi: Ryan Hall...Mimi Reyes Hall...Mimi Hall...Mimi Reyes-Hall...\n\n...\n\nCaroline: Okay, Caroline, time to have that difficult conversation\u2014with yourself. Why exactly did you throw yourself at Simon last night? Was it the wine? Was it the music? The voodoo? Was it the combination of all those things? Okay, okay, no more bullshit. I did it because...because...Fuck, I need some more Gardetto's.\n\n...\n\nSimon: She's so pretty. I mean, there's pretty and then there's pretty...What a pussy I am. Fuck pretty\u2014she's beautiful...pussy...And she smells good...pussy...Why do some girls just smell better? Some girls smell like flowery, fruity bullshit. I mean, why would some girls want to smell like a mango? Why should a girl smell like a mango? Maybe if I think the word mango enough I won't think about pussy anymore. Caroline...mango...Caroline...pussy...God! And now I'm hard...\n\n...\n\nCaroline: He looks like he needs to pee again...He's drinking too much coffee. He's had like six cups already from that thermos. That's funny...He never has a second cup at home. Why the hell do I know how many cups of coffee he drinks? Face it, Caroline, you know so much about him because...because...\n\n...\n\nRyan: Dude, we're stopping again? We are never gonna make it home. My boy is having some serious issues today...I should probably see if he wants to get a beer or something when we get back\u2014in case he wants to come clean about what really happened last night. Should I offer? Wow, Mimi looks fantastic in those pants...I wonder if she's buying more bubble gum.\n\n...\n\nMimi: Stop sniffing your sweater, Caroline! Seriously, girl. If I could just get her alone...Okay, Simon seems to be hobbling toward the men's room. I can get her alone by the beef jerky.\n\n...\n\nCaroline: Ugh...I can't believe Mimi knew I was sniffing the sweater. I wonder if Simon noticed.\n\n...\n\nSimon: She seems better...She's not sniffling any more.\n\n...\n\nMimi: I need to text Sophia. She needs to know the Simon\/Caroline situation is not getting any better. What the hell are we gonna do with these two? I mean, seriously...sometimes people just can't see what's right in front of them. Aawww...Ryan wants me to scratch his back. I adore him...And damn, are his fingers long...\n\n...\n\nRyan: Mmmm...back...scratch...back...scratch...Mmmm...\n\n...\n\nCaroline: Okay, no more avoiding it in your own head, Reynolds. And now I'm serious because I'm using my last name. Now listen up, Reynolds...Heeheehee...I sound like such a badass!\n\n...\n\nSimon: So...she's giggling? Inside joke, she says. So maybe she is okay with how this is going\u2014oops, grabbed the wrong bag of Gardetto's. Did she just growl at me?\n\n...\n\nCaroline: Turn my tatas down and then try to steal my Gardetto's? I don't think so, buddy. Okay, Reynolds, no more giggling. You can't avoid this forever, even in your own mind. Here are the questions on deck: 1. Why did you throw yourself at Simon last night? And you're not allowed to blame alcohol or music or vacation vibes or Nerves or Heart or anything. 2. Why did he turn you down? If he didn't want to go there, why has he been flirting with you for weeks, and not just in the neighborly way? He's got a harem, for God's sake. He's not a Puritan. Agh!! 3. Does being rejected by Simon have anything to do with the date you agreed to with James? 4. How the hell do Simon and I go back to being just friends when we know what the inside of each other's mouths taste like? And his tastes very, very, very good. Okay, yes. You can sniff the sweater one more time\u2014just don't let anyone see you.\n\n...\n\nSimon: I have to figure this shit out with Caroline. She's so great, and I mean so great...Has there ever been a woman who's possessed every single quality I've been looking for? Except for Natalie Portman, of course. But Caroline? I have to stop watching so much Lifetime\u2014I mean what guy in his own mind even thinks in sentences like: \"Has there ever been a woman who's possessed every single quality I've been looking for?\" Wait, have I been looking for that woman? No, I haven't. I don't have time for that, space for that\u2014and my girls don't want the picket fence. They keep away the picket fencers. Caroline says she isn't a picket fencer...Katie found her picket fence, and I'm happy for her. When's the last time I even talked to Nadia or Lizzie? Maybe they're not right for me anymore. I don't want them the way I might want...could want Caroline. You're such a pussy, Parker...Jesus, Caroline\u2014she's a fucking keeper...Wait a minute. What the hell? Are you really entertaining the idea of a...gulp...relationship? And why the fuck did I actually think the word \"gulp\"? That was a little dramatic, Parker. Come on, think about this...If I recall correctly, you invited her to Spain! Don't run away from it. Dude, did she just sniff her sweater?\n\n...\n\nRyan: Mmmm...my girl likes beef jerky\u2014could I be any luckier? She scratches my back and eats beef jerky. I have died and gone somewhere like heaven.\n\n...\n\nMimi: I can't believe he ate all my beef jerky. What a jerky. Heehee.\n\n...\n\nCaroline: Question 1 is too hard. I can't start with that one. I'll answer them in reverse order. 4. I don't know if we can be friends, but I really want to be\u2014and not in the fake way. I really like Simon, and even though what happened last night sucked major balls, I think we can figure this out...And I would like to have some of whatever I'm smoking. 3. OF COURSE I AGREED TO GO OUT WITH JAMES BECAUSE OF WHAT HAPPENED WITH SIMON! It's funny how that shows up in all caps even in my head. 2. If I knew why he turned me down, I'd be a fucking genius. Bad breath? No. Because I was drunk? Possibly...But if it's because we were drunk that's the worst timing for chivalry in the history of the universe. He did keep saying \"I can't\" and that it was a \"mistake.\" Now, mistake perhaps. But might have been worth it...Maybe he was just being faithful to his harem? Which in an odd way is quite sweet. I know he really does care about them. Dammit, he's even great when it comes to them! But I know \"I can't\" wasn't accurate. \"Can't\" implies some kind of erectile dysfunction. And I felt that junk on my thigh. Sigh. Sigh for thigh. This sweater is doing things to my head. Sniff...\n\n...\n\nSimon: She just sniffed it again\u2014why does she keep doing that? When I wore it I didn't notice it smelling like anything other than wool. Girls are weird...weirdly wonderful...Pussy...Caroline's pussy...Aaand I'm hard. Why the hell am I even pretending I'm not totally and completely over the moon for this girl? And it has nothing to do with her pussy...and now I'm harder.\n\n...\n\nCaroline: Stop trying to get out of answering this question. Face it head on! Why did you throw yourself at Simon, forgetting about the friendship and the harem and the O drought and all of the very good reasons you had for staying away from him and his banger voodoo??? Come on, Caroline. Suck it up and say it. What was it he said when you asked him why he kissed you that night you met? \"Because I had to.\" Jesus, even in my head he sounds amazing saying that...There's your answer, Caroline: because you had to. And now you have to figure this shit out. I kissed him, and he kissed me because we had to. And the choices we made were ours and ours alone...And the fact that he stopped it and said he couldn't? Even after all the ridiculous weeks of flirting? After he invited me to Spain? Motherfucking Spain! And I want to go to motherfucking Spa\u2014wait, do I want to go to Spain with him? Argh! Spain Schmain. Anyway, he better have a damn good reason because I am a fucking catch\u2014O or no O\u2014I am a fucking catch. Yeah, you are, Reynolds. Weird how you flip back and forth between first and third person during your inner monologues, though...Thank Christ, the Bay Bridge! Enough introspection...\n\n...\n\nSimon: Shit, the Bay Bridge. We're almost home, and I have no idea how this is going to go with Caroline. We've barely said anything the entire way\u2014although I'm glad to be almost home. I smell like beef jerky, and I need to jerk off like you wouldn't believe...\n\n...\n\nMimi: Yay! The Bay Bridge! I wonder if Ryan would mind spending the night at my place!\n\n...\n\nRyan: Thank fuck, the Bay Bridge. We're almost home. I wonder if Mimi knows I'm spending the night at her place\u2014and planning on making her call in sick tomorrow? Little girl, the things I plan to do to you...But I'm never eating that much beef jerky again. This has been the quietest road trip ever.\n\nWe dropped off the new couple at Mimi's\u2014not that they particularly noticed\u2014they were in their own bubble gum world\u2014and continued on to our apartments. Though we'd mostly just been lost in our thoughts, the tension had grown during the drive, and it was even more noticeable now that we were alone in the car. Simon and I had always had things to talk about, but now that we had so much to discuss, we were silent. I didn't want things to be weird, and I knew I'd have to be the one to make sure he knew I was okay now. He'd already done his part to have a mature conversation, and once again my bull-in-a-china-shop delivery seemed to have taken care of that.\n\nA vision of me announcing on the deck, at full volume, that I'd made a pass at Simon flashed across my mind, and while my cheeks certainly heated in embarrassment, I also had a mental chuckle at how odd I must have looked, arms flailing, mouth set as though I could spit nails. And then barking at frightened Simon to follow me to the beach. He must have wondered if I was going to thrash him and dump his body in the lake.\n\nLooking at his hands on the steering wheel, the very hands that were on me in a very pronounced way the night before, I marveled at his ability to stop himself, because I know for a fact he had been in to it. Or his body had been, at least, if not his head.\n\nThe thing is, though, I did think his head was in it, at least until he thought about it too much. I glanced over at him once more, noticing we were pulling down our street. As we stopped at the curb, he looked over at me, biting down on the same lower lip that less than twenty-four hours ago I'd had the good fortune to be biting on.\n\nHe sprang from the car and ran around to my side before I even had my seatbelt unbuckled.\n\n\"Um, I'm just gonna...get the bags,\" he stammered, and I studied him closely. He ran his left hand through his hair while his right drummed against the side of the car. Was he nervous?\n\n\"So, yeah,\" he stammered again, disappearing around the back.\n\nYep, he was nervous, just as nervous as I was. He worried my bag out of the car, and we slogged up the three flights of stairs to our apartments. We were still not talking, so the only sound was our keys jangling in the locks. I couldn't leave it like this. I had to square with him. I took a deep breath, and turned. \"Simon, I\u2014\"\n\n\"Look, Caroline\u2014\"\n\nWe both laughed a little.\n\n\"You go.\"\n\n\"No, you go,\" he said.\n\n\"Nope. What were you gonna say?\"\n\n\"What were you gonna say?\"\n\n\"Hey, spit it out, bucko. I got a pussy to rescue from two queens downstairs,\" I instructed, hearing Clive call to me from the apartment below.\n\nSimon snorted and leaned against his door. \"I guess I just wanted to say I had a really great time this weekend.\"\n\n\"Until last night, right?\" I leaned against my own door, watching him flinch as I addressed the elephant in the hot tub.\n\n\"Caroline,\" he breathed, closing his eyes and letting his head fall back.\n\nHe looked like he was in actual pain as his face twisted. I took pity. I shouldn't have, but I did.\n\n\"Hey, can we just forget it happened?\" I said. \"I mean, I know we can't, but can we pretend to forget it? I know people say things won't get weird all the time, but then it always does. How can we make sure things don't get weird?\"\n\nHe opened his eyes and looked hard at me. \"I guess we just don't let it. We make sure it doesn't get weird. Okay?\"\n\n\"Okay.\" I nodded and was rewarded with the first real smile I'd seen since I unwrapped my sweater back in Tahoe. He gathered up his bag.\n\n\"Play me something good tonight, 'kay?\" I asked as I headed inside.\n\n\"You got it,\" he answered, and we shut our doors.\n\nBut he didn't play me big band that night.\n\nAnd we didn't speak again that week.\n\n\"Who peed in your chili?\"\n\nI looked up from my desk to see Jillian, composed as always with her casually elegant chignon, black pencil trousers, white silk blouse, and raspberry cashmere sweater wrap. How did I know it was cashmere from across the room? Because it was Jillian.\n\nI selected one of the five pencils currently stuck in my twisted hair bun and returned my attention to the mess that was my desk. It was Wednesday, and this week was both flying by and dragging simultaneously. No word from Simon. No texts from Simon. No songs from Simon.\n\nBut I hadn't reached out to him either.\n\nI was consumed with finishing the last few details on the Nicholson house, ordering expensive knickknacks for James's condo, and starting the sketches for a commercial design project I had lined up for next month. It looked like chaos, but sometimes it was the only way I could get work done. There were days that I needed neat and orderly, and days when I needed the mess on my desk to reflect the mess in my head. This was that day.\n\n\"What's up, Jillian?\" I barked, knocking over my cup of colored pencils as I grabbed for my coffee.\n\n\"How much coffee have you had today, Miss Caroline?\" She laughed, taking the seat opposite me and handing me the pencils that had spilled on the floor.\n\n\"Hard to say...how many cups are in a pot and a half?\" I answered, restacking some papers to clear a space for her teacup. The woman walked around drinking tea out of a bone china cup, but it worked for her.\n\n\"Wow, I take it you aren't seeing any clients today?\" she asked, leaning over the desk and casually removing my coffee cup. I hissed at her, and she wisely put it back.\n\n\"Nope, no clients,\" I answered, shoving the new sketches into color-coordinated folders and stuffing them into their appropriate drawers.\n\n\"Okay, sister, what's up?\"\n\n\"What do you mean? I'm working\u2014what you pay me to do, remember?\" I snapped, grabbing for a ring of fabric swatches and knocking my flower vase over. I'd picked out dark purple, almost black tulips for this week, and they were now all over the floor. I sighed heavily and forced myself to slow down. My hands shook from the caffeine arguing through my system, and as I sat and surveyed the state of affairs in my office I felt two fat tears forming in my eyes.\n\n\"Damn,\" I muttered and covered my face with my hands. I sat for a minute, listening to the tick of the retro clock on the wall, and waited for Jillian to say something. When she didn't, I peeked through my hands at her. She was standing by the door with my jacket and purse in her hands.\n\n\"Are you throwing me out?\" I whispered as the tears launched themselves down my face. She waved her arm and beckoned me toward the door. Grudgingly I stood, and she draped my sweater around my shoulders and handed me my purse.\n\n\"Come on, dearie. You're buying me lunch.\" She winked and pulled me down the hallway.\n\nTwenty minutes later she had me ensconced in an ornate red booth hidden partially behind two gold curtains. She'd brought me to her favorite restaurant in Chinatown, ordered me chamomile tea, and waited in silence for me to explain my semi breakdown. Actually, it was not entirely silent; we'd ordered the sizzling rice soup.\n\n\"So, you must've had a helluva weekend in Tahoe, huh?\" she finally asked.\n\nI laughed into my sizzle. \"You could say that.\"\n\n\"What happened?\"\n\n\"Well, Sophia and Neil finally got together and\u2014\"\n\n\"Wait a minute, Sophia and Neil? I thought Sophia was with Ryan?\"\n\n\"She was, she was, but truthfully she was always meant to be with Neil, so it all worked out in the end.\"\n\n\"Poor Mimi and Ryan. That must've been weird for them.\"\n\n\"Ha! Oh yes, poor Mimi and Ryan. They got it on in the pool house, for God's sake.\" I snorted.\n\nJillian's eyes grew wide. \"In the pool house...wow,\" she breathed, and I nodded.\n\nWe sizzled.\n\n\"So, Simon went to Tahoe, right?\" she asked a few minutes later, looking everywhere but at me. I cracked a small smile at her imagined stealth. Jillian was many, many things, but subtle was not one.\n\n\"Yep, Simon was there.\"\n\n\"And how was that?\"\n\n\"It was great, and then it wasn't, and now it's weird,\" I admitted, setting aside my soup to drink my tea. It was soothing and non-caffeinated, which Jillian had insisted on.\n\n\"So, no pool house for you two?\" she asked, still glancing around the restaurant as though she weren't asking me anything of importance.\n\n\"No, Jillian, no pool house. We hot tubbed, but we did not pool house,\" I said emphatically, and then I spilled my guts and told her the entire ridiculous story.\n\nShe listened, she hmm'd and groaned in the right places, and she got indignant in the right places too. By the time I was finished, I was in tears again, which was really pissing me off.\n\n\"And the stink of it all, I shouldn't have been doing it, but he is the one who stopped it, and I don't really think he wanted to!\" I huffed, angrily wiping tears away with my napkin.\n\n\"So why do you think he did?\"\n\n\"He's gay?\" I offered, and she smiled. I took a deep breath and got control.\n\nJillian looked at me thoughtfully and then finally leaned in. \"You realize we are two smart women who are not acting very smart right now,\" she said.\n\n\"Huh?\"\n\n\"We know better than to try to figure out what a man is up to. This'll get worked out when it's supposed to. And your tears? These are tension tears, frustration tears\u2014nothing more. I'll tell you one thing, though.\"\n\n\"What's that?\"\n\n\"As long as I've known Simon, I've never heard of him inviting someone on a shoot with him, ever. I mean, inviting you to Spain? That's very unlike Simon.\"\n\n\"Well, who knows if I'm even invited anymore.\" I sighed dramatically.\n\n\"You're still friends, right?\" she asked, raising an eyebrow at me. \"Why don't you just ask him?\" When I didn't respond she added, \"Stick that in your pipe and suck it.\"\n\n\"I think it's smoke it, Jillian. Stick that in your pipe and smoke it.\"\n\n\"Ah, smoke it, suck it, whatever. Eat your fortune cookie, sweetie.\" She smiled, nudging the cookie across the table. I cracked it open and removed the fortune.\n\n\"What does yours say?\" I asked.\n\n\"Fire all employees who have more than one pencil in their hair,\" she stated seriously. We laughed together, and I could feel some of the tension finally leaving my body.\n\n\"What does yours say?\" she asked.\n\nI opened it up, read the words, and rolled my eyes to the ceiling. \"Stupid fortune cookie,\" I sighed, and handed it to her.\n\nShe read it and her eyes went wide again. \"Oh, man, are you in for it! Come in, let's go back to work.\"\n\nShe laughed, tugging my hand and leading me from the restaurant. She gave the fortune back to me, and I started to throw it away, but then slipped it into my purse:\n\nBe aware of the walls you build  \nand what could be on the other side\n\nConfucius, you kill me.\n\nText from James to Caroline:\n\nHey there.\n\nHey to you.\n\nWe still on for Friday night?\n\nYep, I'm in. Where are we going for dinner?\n\nThere's a great new Vietnamese restaurant I  \n've been wanting to try.\n\nHave you forgotten I'm not really big on Vietnamese food?\n\nCome on, you know it's my favorite. You can get soup!\n\nFine, Vietnamese it is. I'll find something.  \nBTW, the last of your furniture should be delivered Monday. I'll be there to receive and place.\n\nHow much longer until the project is finished?\n\nExcept for a few pieces in the bedroom,  \nshould be all done by next weekend.  \nAhead of deadline I might add...\n\nVery good. Will you also be there  \nto finish things in the bedroom?\n\nStop it, Jamie.\n\nI hate when you call me Jamie.\n\nI know, Jamie. See you Friday night\n\nThe day had exhausted me. I literally had nothing left. I had plans to go to yoga, really I did, but as the evening approached all I wanted to do was go home. I wanted Clive, and I could no longer pretend that I didn't also want Simon. Maybe he would be home? As I walked up the stairs I could hear Simon's TV through the door. I was already turning my key in my lock when I thought about my fortune cookie. I could knock on the door, right? I could just say hi, right? As I debated, I heard his phone ring, followed by his voice through the door.\n\n\"Nadia? Hey, how are you?\" he said, and that made up my mind for me. He had his harem, and I couldn't possibly enter in to something like that. If I wanted Simon, I wanted all of Simon. I'd promised myself no more messing around. As I felt tears prick at my eyes for the thousandth time that day, I walked in to find Clive waiting for me, and I smiled through my tears. I picked him up, cuddling him to me as he told me all about his day in cat speak. I interpreted for him, and it would seem that Clive's day consisted of a light snack, a nap, about thirty minutes of grooming, another snack, another nap, and then he watched the neighborhood for the rest of the afternoon and evening. Leftover takeout with Ina and Jeffrey on the couch, a quick shower, and I packed it in early. I simply could not allow this day to go on any longer.\n\nWith Clive curled between my legs, I went to sleep, again with no music from the other side of the wall.\n\nThe following night I stood in front of my mirror, trying on different shoes for my date\/not a date\/of course it's a date with James. I'd almost called him twice today to back out, but in the end, I pushed through it and got dressed. Sometimes a girl just needs to get dressed up, and tonight I was dressed to kill: thin, fitted black blouse, tight red pencil skirt, teetery tall heels.\n\nI'd been conflicted about this event, whatever it was, all week long. But I wanted to go. Was I using James a little? Perhaps. But I did have a good time with him, and maybe it wouldn't be the worst thing in the world for us to start back up again.\n\n\"Caroline Reynolds, you heartbreaker,\" I whispered to myself in the mirror. I actually cracked myself up. Clive was embarrassed for both of us and hid his nose behind his paw. I was still laughing when I heard the knock at the door. I slipped into my heels and went for the door, Clive close behind.\n\nI took a deep breath, and opened it. \"Hey, James.\"\n\n\"Caroline, you look great,\" he murmured, stepping inside and catching me into a hug.\n\nAs his arms went around me, I knew immediately. This was a date.\n\nHe smelled spicy. I don't know why girls always say boys smell spicy, but some do. And it's a good thing, warm and spicy. But not like potpourri...\n\nI hugged him back, enjoying the way my body still fit with his. We always were good at the hugging.\n\n\"You ready to go?\"\n\n\"Yep, let me grab my bag.\" I knelt to give Clive a quick kiss. He tossed his tail angrily in James's direction and wouldn't let me kiss him.\n\n\"What's your problem?\" I asked Clive, who turned and showed me his rear end.\n\n\"You know, that's starting to become a very rude habit, Mr. Clive,\" I warned him as I picked up my purse from the table. I stuck my tongue out at Clive, grabbed James, and locked the door behind us.\n\n\"Okay, so dinner?\" I asked as we stood outside my door.\n\n\"Yep, dinner,\" he replied, standing very close to me. We stared at each other\u2014for only seconds really, but it felt much longer. He stepped a little closer, and my breath caught. Of course, just then Simon decided to open his door.\n\n\"Hey, Caroline! I was just\u2014Oh, hi. James, right?\" His smile faded slightly when he saw my dinner date. Date, date, date.\n\n\"Sheldon, right?\" James said, offering his hand.\n\n\"Simon, actually.\" He raised his trash-bag-filled hands and declined the shake. \"After you.\" He nodded to the stairs, and the three of us began to troop down together.\n\n\"So, where are you two crazy kids off to tonight?\" Simon asked as we walked ahead of him.\n\nI could feel his eyes on the back of my neck, and as I hit the landing I looked back. He had a fake smile plastered across his face, and his voice was colder than I'd ever heard it before.\n\n\"Caroline and I are headed out for dinner,\" James answered.\n\nI smiled back over my shoulder. \"Yes, some lovely little Vietnamese restaurant,\" I cooed, pretending to be thrilled.\n\n\"You don't like Vietnamese food,\" he said, frowning.\n\nThis made me smile. \"I'm going to try the soup,\" I answered.\n\nJames locked eyes with Simon as he held the door for me. He let it swing right as Simon came through with his hands full of trash bags, but I caught it just in time.\n\n\"Well, have a good night,\" I said as James walked me toward his car with his hand on the small of my back.\n\n\"'Night,\" Simon answered, lips tight. I could tell he was irritated.\n\nGood.\n\nJames bundled me into the car, and we were off.\n\nThe dinner was fine. I ordered fried rice off of the fusion side of the menu, and when it arrived, for a moment all I could think about was eating noodles on a houseboat in the middle of Ha Long Bay with Simon.\n\nBut as I said, dinner was fine, the conversation fine, the man I was with, fine. He was a fine-looking man with a great future ahead, his own adventures to be had, mountains to conquer. And tonight, I was the mountain. I kind of wanted to let him climb.\n\nHe walked me upstairs to my door, even though I could have stopped him from coming all the way up. As I dug for my keys, I could hear Simon's phone ringing, and he answered.\n\n\"Nadia? Hi. Yep, ready when you are.\" He laughed.\n\nMy heart clenched. Fine. I turned to say goodnight to James, devastatingly handsome and right there. Right there in front of me. O had been gone a long time, and she and James had once been close. Could he? Would he? I was going to find out. I invited him in.\n\nAs I pulled a bottle of wine from the fridge, I watched him scan the room, taking stock of everything: the Bose sound system, the Eames chair by the desk. He even checked out my crystal as I handed him his glass. He thanked me, his eyes burning into mine as our fingers slipped past each other.\n\nNature took over. Hands knew, skin recognized, lips teased and became reacquainted. It was new and old at the same time, and I'd be lying if I said it didn't feel good. His shirt came off. My skirt dropped, I kicked off my heels, and our arms wrapped and tucked in. Eventually and inevitably, we headed to the bedroom.\n\nI bounced lightly on the bed, watching through hazy eyes as he knelt before me on the floor.\n\n\"I missed you.\"\n\n\"I know.\" I pulled him on top of me. Everything was fine, everything was as it should be, and as I mechanically wrapped my legs around his waist, his belt buckle digging cold into my thigh, he looked deeply into my eyes and smiled.\n\n\"I'm so glad I needed a decorator.\"\n\nAnd just like that, fine was not enough.\n\n\"No, James.\" I sighed, pushing at his shoulders.\n\n\"What, baby?\"\n\nI hated when he called me baby.\n\n\"No, no, just no. Get up.\" I sighed again as he continued to kiss my neck. Tears sprang to my eyes as I realized what used to make me feel something now made me feel nothing at all.\n\n\"You're kidding, right?\" He moaned in my ear, and I pushed him again.\n\n\"I said get up, James,\" I said, a little louder this time.\n\nHe got the message. Doesn't mean he was happy to hear it. He stood up as I smoothed my shirt, which was thankfully still mostly buttoned.\n\n\"You gotta go,\" I managed, tears beginning to track down my cheeks.\n\n\"Caroline, what the\u2014\"\n\n\"Just go, okay? Just go!\" I yelled. It wasn't fair to him, but I had to be fair to myself. I couldn't go backward, not now.\n\nI clasped my hands to my face and heard him sigh, then stomp off, slamming the door. I couldn't blame him. He must've been in blue-ball hell. I was sad and mad and a little bit tipsy, and I hated my O. My eyes landed on one of my Come Fuck Me shoes on the floor, and I threw it as hard as I could into the living room.\n\n\"Ooof!\" I heard a deep voice utter, and it was not James Brown's. It was the man I did want in my bed, and the one I was most mad at right now. Holding the shoe like some kind of late-night Prince Charming to my slutty O-less Cinderella, Simon appeared in my doorway, barefoot and in his pajama bottoms. The sight of his perfect speedbump abs crossed me over from pissed off to M. A. D.\n\n\"What the hell are you doing here?\" I asked, angrily wiping my tears from my face. He was going to see me cry.\n\n\"Um, I heard you and James...Well, I heard you, and then I heard you yelling, and I wanted to make sure you were okay,\" he stammered.\n\n\"You're not here to rescue me, are you?\" I bit back, air-quoting the rescue.\n\nHe backed away as I crawled off the bed, seeming scared of my impending explosion. Even I knew this was going to be ugly.\n\n\"Why do all men seem to think they need to rescue a woman? Are we not capable of rescuing our damn selves? Why do I need to be rescued? I don't need a man to rescue me, and I certainly don't need no wallbanging, Purina-fucking, listening-at-my-wall-like-a-goddamn-psycho coming over here to rescue me! You got that, mister?\"\n\nI was pointing and waving my arms around like someone was going to take them away from me. He had every right to look scared.\n\n\"I mean, what the hell is with you men? I've got one who wants me back, and one who doesn't want anything to do with me! One who wants to be my boyfriend, but can't even remember that I'm an interior designer. Designer! Not a fucking decorator!\"\n\nI was on a roll. At this point I was just ranting, plain and simple. I stalked in a circle around Simon, pacing and shouting while he tried to follow me, finally just standing still and watching me with huge eyes.\n\n\"I mean, you shouldn't force someone to eat Vietnamese food if they don't like Vietnamese food, should you? I shouldn't have to eat it, should I, Simon?\"\n\n\"No, Caroline, I don't think you should\u2014\" he started.\n\n\"No, of course I shouldn't, so I got the fried rice! Fried rice, Simon! I'm not gonna eat Vietnamese food ever again\u2014not for James, not for you, not for anyone! You got that?\"\n\n\"Well, Caroline, I think\u2014\"\n\n\"And for your information,\" I continued, \"I did not need a rescue tonight! I took care of it myself. He's gone. And I know you think James is some kind of psycho, but he isn't,\" I said, beginning to lose momentum. My lower lip quivered again, and I fought it, but finally let go. \"He isn't a bad guy. He just...he just...he just isn't the right guy for me.\" I sighed, sinking down to the floor in front of my bed and holding my head in my hands.\n\nI cried for a moment, while Simon remained frozen above me. I finally looked up at him. \"Hello? Girl crying down here!\" I sputtered.\n\nHe swallowed a smile and sat down in front of me. He pulled me off the floor and gathered me into his arms. And I totally let him. He settled me onto his lap and held me close as I cried into his chest. He was warm and gentle, and even though I knew better\u2014oh, how I knew better\u2014I tucked into the nook and let him comfort me. His hands ran up and down my back as I sobbed, his fingertips making the tiniest of circles on my shoulder blades as I breathed him in. It had been so long since I'd been held, just held, by a man that between the tiny circles and the scent of his fabric softener I was losing my senses.\n\nFinally my sobs began to quiet as he held me close, cross-legged on my floor. \"Why didn't you play me music this week?\" I sniffled.\n\n\"My needle was broken. I have to get it fixed.\"\n\n\"Oh, I thought maybe...well, I missed it is all,\" I said shyly.\n\nHe smoothed back my hair and brought his hand under my chin, forcing me to look up at him. \"I missed you.\" He smiled gently.\n\n\"Me too,\" I breathed, and his sapphires began to spin. Oh no. No voodoo. \"How was Purina? Good? Bet she missed you too,\" I whispered and watched his face change.\n\n\"Why do you keep bringing up Nadia?\"\n\n\"I heard you on the phone with her earlier. Sounded like you were making plans.\"\n\n\"Yes, I met her for drinks.\"\n\n\"Please. You expect me to believe she didn't come over?\" I asked, noticing I was still on his lap.\n\n\"Ask your cat. Did he go crazy tonight?\" Simon pointed at Clive, who had returned and was now watching us from the back of the couch.\n\n\"No, he didn't, actually.\"\n\n\"That's because she didn't come over. We met for drinks to say goodbye.\" Simon looked at me carefully.\n\nMy heart began to beat so loud there was no way he couldn't hear it. Why did Heart have to be so in to this? \"Goodbye?\"\n\n\"Yep, she's going back to Moscow to finish her degree there.\"\n\nHeart settled down a bit. \"Oh, so you said goodbye because she was leaving, not for any other reason. Silly me.\" I lifted myself off his lap as he held me closer. I struggled.\n\n\"She's leaving, yes, but that's not why we said goodbye. I\u2014\"\n\nI continued to wiggle. \"Wow, only the Giggler left! And then there was one. I guess technically one does not make a harem, so will she be shouldering the load for the others or will you need to be interviewing for some more women? How does that work exactly?\" I snapped.\n\n\"Actually, I'm going to be having a conversation with Lizzie very soon as well. I think we're going to be just friends from now on,\" he said, watching me closely. \"What used to work for me just doesn't work anymore.\"\n\nAll stop. What? \"It doesn't work for you anymore?\" I breathed, not daring to believe it.\n\n\"Mm-hmm,\" he answered, his nose dipping down to the skin just below my ear and breathing deep.\n\nWould he notice if I licked his shoulder? Just the tiniest taste?\n\n\"Caroline?\"\n\n\"Yes, Simon?\"\n\n\"I'm sorry I didn't play music for you this week. I'm sorry that I...well, let's just say I'm sorry for a lot of things.\"\n\n\"Okay,\" I breathed.\n\n\"Can I ask you something?\"\n\n\"No, I don't have any zucchini bread,\" I whispered, and his laugh echoed through the room. I laughed along, in spite of myself. I'd missed laughing with Simon.\n\n\"Come to Spain with me,\" he whispered.\n\n\"Wait, what?\" I asked again, my voice wavering. What, what, what? \"Are you serious?\"\n\n\"I'm very serious.\"\n\nI had to remind myself to breathe. Already heady from the voodoo and fabric softener, I shook my head to clear it. He was going Spain on me?\n\nI was glad he seemed focused on the space behind my ear, because I doubted he'd be as interested if he could see how my eyes were now crossed. I needed a moment. I pulled myself away, finally standing up.\n\n\"I'm gonna go wash my face. Don't go anywhere,\" I instructed.\n\n\"Sweet Caroline, I'm not going anywhere,\" he said, his sexy smile returning.\n\nI made myself walk away. Every step I took, every thunk of my heels on the hardwood was like a chant in my head: Spain. Spain. Spain. Once in the bathroom, I splashed some water on my face, most of it going into my mouth because I couldn't stop smiling. New harem head count: two down, one to go? There were times to be cautious, and then there were times when you just needed to go balls-out and take a risk. I needed some backbone. I thought about what Jillian had said earlier today, and I went with my impulse. I steeled myself, took out my figurative balls, and headed back out.\n\n\"Okay, it's late, Simon. Time for you to go.\" I took him by the hand, pulled him off the floor, and led him toward the front door.\n\n\"Um, really? You want me to go? Don't you want to, I don't know...talk a little more?\" he asked. \"I wanted to tell you how\u2014\"\n\nI continued to pull him. \"Nope. No more talking tonight. I'm tired.\" I opened my door and ushered him out to the landing. He started to say something else, and I held up two fingers. \"I need to say two things, okay? Two things.\"\n\nHe nodded.\n\n\"First, you hurt my feelings in Tahoe,\" I began, and he tried to interrupt me. \"Shut it, Simon. I don't want a rehash. But just know you hurt me. Don't do it again,\" I finished. I couldn't stop my smile when I saw his reaction.\n\nHis eyes hit the floor, his entire body contrite. \"Caroline, I'm really sorry about all that. You have to know that I just wanted to\u2014\"\n\n\"Apology accepted.\" I smiled again and began to close my door.\n\nHis head popped up immediately. \"Wait, wait. What was the second thing?\" he called, leaning into my doorway. I stepped closer to him, bringing my body within inches of his. I could feel the heat of his skin across the tiny space between us, and I closed my eyes against the onslaught of emotions. I breathed deep and opened my eyes to look in to the sexy sapphires gazing down at me.\n\n\"I'm coming with you to Spain,\" I said. And with a wink, I closed the door in his astonished face. \nChapter Fifteen\n\n\"EGGS SUNNY-SIDE UP, bacon, wheat toast with raspberry jelly.\"\n\n\"Oatmeal with raisins, currants, cinnamon, and brown sugar, side of sausage links.\"\n\n\"Belgian waffles, fruit cup, bacon and sausage,\" Sophia said, completing our order and earning a raised eyebrow from both Mimi and me.\n\n\"What? I'm hungry.\"\n\n\"Nice to see you getting a real breakfast for a change. Must have been working up an appetite with Mr. Mitchell last night, hmmm?\" I teased, winking at Mimi over my orange juice.\n\nThe three of us were together for breakfast on a Sunday, something we hadn't done since Tahoe. They'd been busily settling into the life of new coupledom with their recently switched boyfriends, which left me out most of the time. When they were dating the wrong guys they were always more than happy to have me along\u2014the more the merrier they'd say. It helped when there was no real chemistry. But now? Mimi and Sophia were definitely with the right guys and enjoying every second of it.\n\nInitially I'd been a little worried that the Parent Trap shenanigans would make things uncomfortable, but the ladies had made me proud. They took it in stride, and since each wound up with her new better half, all my worries went by the wayside.\n\nWe giggled as we got caught up on friendly gossip, waiting until the food arrived for any big news, as was protocol.\n\n\"Okay, who's going first? Who has news?\" Mimi began, and we settled into our ritual. Sophia paused from shoveling in the waffles, indicating that she would serve the first volley.\n\n\"Neil has to go to LA for a sportswriters in television conference, and he asked me to go with him,\" she offered. Mimi and I nodded.\n\n\"Ryan is thinking of letting me reorganize his home office. You should see it\u2014his filing system alone made me break out in hives,\" Mimi reported, shuddering.\n\n\"Natalie Nicholson referred two new clients to me\u2014Nob Hill, very posh, thank you very much,\" I added, pouring myself more coffee from the carafe as they congratulated me.\n\nWe chewed.\n\n\"Neil talks in his sleep. It's the cutest thing. He calls out football scores.\"\n\n\"Ryan let me paint his toenails last night.\"\n\n\"I told Simon I'd go to Spain with him.\"\n\nHere's the thing about a spit take. In the movies, they're hysterical. In real life, they're just messy.\n\n\"Wait a minute, wait a goddamn minute...what?\" Sophia sputtered, juice still dribbling down her chin.\n\n\"Caroline, you told him what?\" Mimi managed, still choking as she waved the waiter over for more napkins.\n\n\"I told him I'd go to Spain with him. No big deal.\" I grinned. It was a big deal indeed.\n\n\"I can't believe you had the nerve to sit here and talk about random shit all morning and not tell us this. When did this happen?\" Sophia asked, leaning forward on her elbows.\n\n\"The night I went on a date with James.\" I smiled.\n\n\"Okay, that's it. No more dicking around\u2014spill it.\" Mimi rounded on me with a butter knife and a frown.\n\n\"What the hell, Caroline? I can't believe you kept all this from us. When did you go on a date with James? And don't you dare leave anything out. Tell us everything now, or I'll let Mimi loose on you!\" Sophia warned. Mimi gestured again in a menacing way with her knife\u2014in a very West Side Story menacing way, mind you. I imagined an actual fight with Mimi would involve hitch kicks and barrel turns...\n\nNevertheless, I took a deep breath and spilled. All of it. Why I went out with James, the feelings that had been percolating with Simon, how James called me a decorator, how I kicked him out. They listened intently, only interjecting occasionally when they needed clarification.\n\n\"I'm so proud of you,\" Sophia said when I'd finished. Mimi nodded in agreement.\n\n\"For what?\"\n\n\"Caroline, there was a time when if James told you to jump, you'd fucking jump. I guess we worried him showing back up in your life would take you back to being that girl again,\" Sophia explained.\n\n\"I know you were worried. You're both sweet, and no one will ever take care of me as well as you, even though you worry like old chickens in a henhouse.\" I smiled at my fierce ladies.\n\n\"So you sent James Brown packing, and then what happened?\" Sophia asked, and I finished the last of the story: Simon's entry, his apology, the disappearing Purina, his invitation...\n\n\"So you just, had this epiphany in the bathroom, just like that? Go to Spain with Simon?\" Mimi finally asked.\n\n\"Yep. I didn't really overthink it. I just, I can't explain it...I just know I should go on this trip. I mean, I've always wanted to go to Spain, and I know he'll be a good tour guide, and come on, how much fun will it be? We'll have a blast together!\"\n\n\"Bullshit,\" Sophia stated simply.\n\n\"Come again?\"\n\n\"I call bullshit, Caroline. You're going because you want something to happen there with him. Don't deny it.\" She eyed me severely.\n\n\"I deny nothing,\" I quipped, signaling the waiter for our check.\n\n\"No more harem, huh?\" Mimi asked.\n\n\"So it would seem. I'm not a fool. I know a man like him doesn't change overnight, but if the Giggler is out of the way before Spain? Well, then, that's a Simon of a different color, now isn't it?\" I grinned cheekily, wiggling my eyebrows at my girls.\n\n\"Why, Caroline Reynolds, I do believe you plan on seducing this man,\" Sophia said, and Mimi clapped her hands with glee.\n\n\"Simon's going to bring back the O!\" Mimi cheered, attracting more than a little attention.\n\n\"Oh, hush. We'll see. If, and this is a big fat if, ladies. If I ever allow anything to happen between Simon and me, it's gonna be on my terms. Which would include no harem, no drinking, and no hot tubbing.\"\n\n\"I don't know, Caroline. No drinking? I think it'd be criminal to be in Spain and not be indulging in a little sangria,\" Mimi piped up.\n\n\"Well, I do enjoy me some sangria,\" I mused. Visions of Simon and me, sipping sangria while watching the Spanish sunset. Hmmm...\n\nText between Simon and Caroline:\n\nSo are you the type of girl who wears  \na big floppy hat on the beach?\n\nPardon me?\n\nYou know, those crazy giant beach hats? Do you have one?\n\nAs it happens, yes. Is this a concern of yours?\n\nConcern, no. Just trying to get a visual of you  \non the beach in Spain...\n\nHow's that working out for you?\n\nPretty spiffy.\n\nSpiffy? Did you just say spiffy?\n\nI typed it actually. You got something against spiffy?\n\nThis explains the old records...\n\nHEY!\n\nI enjoy the old records. You know this...\n\nI do know this...\n\nAre we really going to Spain together?\n\nYep.\n\nAre you home? I didn't see the Rover this morning.\n\nChecking up on me?\n\nPerhaps...where are you, Simon?\n\nHave a shoot in LA, driving back in a few days.  \nCan I see you when I get back?\n\nWe'll see...\n\nI'll play records for you.\n\nSpiffy.\n\n\"So, since things are all completed on the Nicholson project, I was thinking...since I have a jump on the commercial project I'm starting next, and you mentioned before that I could take some time off before we get busy for the holiday season, that, well, maybe I could...\"\n\n\"Spit it out, Caroline. You trying to ask me if you can go to Spain with Simon?\" Jillian demanded, not trying very hard to hide her smile.\n\n\"Maybe.\" I winced, dropping my forehead to the desk.\n\n\"You're a grown woman, capable of making her own decisions. You know I think it's a good time to take vacation, so why should I tell you whether you should go away with Simon or not?\"\n\n\"Jillian, to clarify, I'm not going away with Simon. You make it sound like some illicit affair.\"\n\n\"Right, right, it's just two young people off to enjoy a little Spanish culture. How could I forget?\" Jillian drawled, insinuation all over her face, as well as a little satisfaction. She was enjoying my squirming.\n\n\"Okay, okay, so can I go?\" I asked, knowing I would never hear the end of it, but past caring.\n\n\"Of course you can. But can I just say one thing?\" she asked, eyebrows raised.\n\n\"Like I could stop you,\" I grumbled.\n\n\"You couldn't, actually. All I ask is that you have a good time, play hard, but take care of him while you're there, okay?\" she asked, her face taking on a seriousness I rarely saw.\n\n\"Take care of him? What is he, seven?\" I laughed, stifling it immediately when I saw she was not kidding.\n\n\"Caroline, this trip will change things. You must know that. And I love you both. I don't want either of you to get hurt, no matter what transpires while you're there,\" she said softly. I started to make a joke, but I stopped. I knew what she was asking.\n\n\"Jillian, I don't know quite what's going on between Simon and me, and I've no idea what's going to happen in Spain. But I can tell you, I'm excited about this trip. And I get the sense he is too,\" I added.\n\n\"Oh, my dear, he's definitely excited. Just...Oh, never mind. You're both adults. Go crazy on each other in Spain.\"\n\n\"First you tell me to be gentle, and now you tell me to go crazy?\" I grumbled.\n\nShe reached across the desk to pat my hand affectionately. Then she took a deep breath and changed the mood in the room entirely. \"Now then, fill me in on where we stand with James Brown. What's left to be done?\"\n\nI smiled and flipped my planner open to the end of the week, when I would be finished with All Things James Brown.\n\nA few nights later I was settling into my couch comfortably with Mr. Clive and Barefoot Contessa when I heard something in the hallway. Clive and I looked at each other, and he jumped off my lap to investigate. I knew Simon wasn't due home for another day or so based on his texts\u2014and the fact that I might have been counting the days\u2014so I followed Clive to my old post: The Peephole.\n\nAs I peered out into the hallway, there was a flash of strawberry-blond hair at Simon's door. Who was visiting Simon? Was I wrong to stare? What was that package she had? The woman the hair belonged to knocked once, then twice, and then before I knew it, she whirled about and looked directly at my door, curiously staring at my peephole. Not accustomed to anyone staring at my peephole, I froze, eyes unblinking as she appraised my door. She crossed the tiny landing, and rapped soundly on my door. Surprised, I jumped back a little, bumping into my umbrella stand and letting her know there was, in fact, someone home. I turned my face to the side and shouted, \"Coming!\" Then I proceeded to walk in place as though I was headed for the door. Clive looked on with interest, tossing his head and assuring me I was not nearly as clever as I thought I was.\n\nI made a great noise of clicking the locks, and then opened the door.\n\nWe appraised each other instantly, in the way that women do. She was tall and beautiful in a cold, patrician way. She wore a black suit, severely cut and buttoned up to the collar. Her strawberry blond hair was twisted and pinned back, although one solitary piece had marched away from her sisters and now hung in her face. She pushed it back behind her ear. Her cherry red lips pursed as she finished looking me over and offered a thin smile.\n\n\"Caroline, yes?\" she asked, a solidly British accent piercing the air as clearly as her attitude. I already knew I didn't care for this woman.\n\n\"Yes, can I help you?\" I suddenly felt underdressed in my Garfield boxers and tank top. I shifted my weight from one leg to the other, feet clad in giant socks. I shifted my weight again, realizing I probably looked like I had to pee. I also realized at the same time that this woman made me nervous, and I had no idea why. I straightened up immediately, putting my game face on. This all took place in less than five seconds, a lifetime in the world of Woman Figuring Out The Other Woman.\n\n\"I need to drop this off for Simon, and he mentioned that if he wasn't at home to leave it at the flat across from his, that Caroline would take care of it for him. You're Caroline, so here you go, I suppose,\" she finished, thrusting a cardboard box at me. I took it, taking my eyes off of hers for a moment.\n\n\"What does he think I am, a mailbox?\" I muttered, setting it on the table just inside the door and turning back to the woman.\n\n\"May I tell him who dropped this off, or will he know?\" I asked. She was still looking me over as though I were a great puzzle.\n\n\"Oh, he'll know,\" she answered, her cool tone sounding musical but clipped at the same time. As an American, I'll admit I am always fascinated by a British accent, but could do without this particular side of superiority.\n\n\"Okay, well...I'll make sure he gets it.\" I nodded, leaning my hand on the door. I closed it ever so slightly, but she didn't move.\n\n\"Is there anything else?\" I asked. I could hear Ina working on her shortbread in the other room, and I didn't want to miss any KitchenAid porn.\n\n\"No, nothing else,\" she replied, still making no move.\n\n\"Okay, then, have a good night,\" I said, almost making it a question as I started to close the door. Just as I did, she stepped forward enough so I was forced to catch the door before it hit her.\n\n\"Yes?\" I asked, my irritation beginning to show through. This Limey was stopping me from seeing the completion of the pecan squares I'd been waiting for all episode.\n\n\"I just, well, I'm really glad to have met you,\" she answered, her eyes finally softening and a hint of a smile breaking through her fa\u00e7ade. \"And you really are quite lovely,\" she added. I stared back at her. Her voice sounded oddly familiar, but I couldn't quite place it.\n\n\"Um, okay, thank you?\" I answered as she started for the stairwell. Her heel caught just slightly, and she stumbled a little. As I closed the door, she began to giggle as she worked her shoe loose. That's when I realized who'd just visited.\n\nMy eyes widened, I'm sure to the size of dahlias, and I hurled the door back open. I gaped at her, and her face broke open into the widest cheeky grin. She winked as I blushed. I'd been present for some of this lady's greatest moments.\n\nShe wiggled her fingers at me and disappeared down the stairs. Clive brought me back from my stupor by nipping me on the calf, and I closed the door.\n\nI sat on my couch, pecan squares all but forgotten as my brain processed everything.\n\nThe Giggler had said I was lovely.\n\nShe basically told me Simon had told her I was lovely.\n\nSimon thought I was lovely.\n\nWas the Giggler out of the harem?\n\nWas there even a harem left?\n\nWhat did this mean?\n\nWould I only think in questions now?\n\nAnd if so, who is Eric Cartman's father?\n\nText between Simon and Caroline:\n\nWhat are you doing?\n\nWhat are YOU doing?\n\nI asked you first.\n\nYou sure did.\n\nWaiting...\n\nMe too...\n\nJesus you're stubborn. I'm driving back from LA. Happy now?\n\nYes, thank you. I'm baking pumpkin bread.\n\nIt's a good thing I'm at a gas station right now and not driving or I would have a hard time keeping the car on the road...\n\nRight, the baking gets you worked up, doesn't it?\n\nYou have no idea.\n\nSo I probably shouldn't tell you I smell like  \ncinnamon and ginger right now?\n\nCaroline.\n\nMy raisins are soaking in brandy this very minute.\n\nThat's it...\n\nI peered out the window again, scanning the street below, and still no sign of the Rover. The fog was quite thick, and although I didn't want to be a nag, I was becoming a little concerned that he wasn't home yet. Here I sat, with cooling loaves, and no Simon had shown up to inhale them. I picked up my phone to text him, but then called instead. I didn't want him texting while he was on the road. It rang a few times, and then he picked up.\n\n\"Hi there, my favorite baker,\" he purred, and my knees clanked together. He was like the best Kegel exercise ever\u2014instant clench.\n\n\"Are you close?\"\n\n\"Pardon me?\" He laughed.\n\n\"Close to home. Are you close to home?\" I asked, rolling my eyes and unclenching.\n\n\"Yes, why?\"\n\n\"There seems to be a lot of fog tonight. I mean, more than usual...Be careful, okay?\"\n\n\"That's very sweet of you to be looking out for me.\"\n\n\"Shut up, mister. I always look out for my friends,\" I scolded, beginning to get ready for bed. I was a multi-tasker from way back. I could do my taxes while getting waxed and not bat an eye. I could certainly get undressed while talking to Simon. Ahem.\n\n\"Friends? Is that what we are?\" he asked.\n\n\"What the hell else would we be?\" I shot back, pulling off my shorts and grabbing a pair of thick woolen socks. The floor was chilly tonight.\n\n\"Hmmm,\" he muttered as I took off my T-shirt and slipped into a button-down to sleep in.\n\n\"Well, while you're hmmming, I have to tell you about a visit I had earlier this week from a friend of yours.\"\n\n\"A friend of mine? This sounds intriguing.\"\n\n\"Yep, Julie Andrews accent, buttoned-up Brit? Ring any bells? She dropped off a box for you.\"\n\nHis laughter rang out immediately. \"Julie Andrews accent\u2014that's brilliant! That must have been Lizzie. You met Lizzie!\" He laughed like this was the funniest thing ever.\n\n\"Lizzie Schmizzie. She'll always be the Giggler to me.\" I smirked, sitting on the edge of my bed and applying some lotion.\n\n\"Why do you call her the Giggler?\" he asked, playing innocent, and I could tell he was on the verge of absolute hysterics.\n\n\"You really need me to tell you? Come on, even you can't be that thick\u2014never mind, walked right into that one.\" I cut him off before he could regale me with how thick he was, indeed. I'd been pressed up against that very thick in a hot tub, so I was familiar. Kegel. And, thank you, another Kegel.\n\n\"I like messing with you, Nightie Girl. It gives me a chuckle.\"\n\n\"First spiffy, now a chuckle? I worry about you, Simon.\" I returned to the living room to turn off lights and get the place ready for bed. This included freshening Clive's water bowl and hiding a few Pounce treats around the apartment. He enjoyed playing Big Game Hunter while I slept sometimes, with the Pounce, of course, playing the part of the Big Game. Some nights the pillows were unfortunately involved, as well as any hair ties, loose shoelaces, and pretty much anything else that seemed appealing around two a.m. Some mornings my place looked like Wild Kingdom had been filmed overnight.\n\n\"Well, no worries. I'll pick it up when I get back. So, did you two have a nice chat?\"\n\n\"We chatted briefly, yes. But no dirty secrets were shared. Although with the thin walls, I'm already a bit familiar. How is the lonely haremette? Missing her sisters?\" I flipped off the lights and padded through the kitchen to fetch the Big Game. I was dying to ask him if he'd actually broken up with the Giggler. Did he, did he not?\n\n\"She may be a bit lonely, yes,\" he said, in what I thought sounded like a careful way. Hmm...\n\n\"Lonely because...\" I led, pausing in my Pounce-scattering.\n\n\"Lonely because, well, let's just say, for the first time in a very long time, I am...well...I am...you see...\" he stuttered and stalled, dancing around the issue.\n\n\"Go on, out with it,\" I instructed, barely breathing.\n\n\"Without...female companionship. Or as you would say, harem free.\" His words came out in a quiet whoosh, and my legs began a little shimmy shake. This made the Pounce shimmy-shake in their container, alerting Clive that his hunt had begun early.\n\n\"Harem free, huh?\" I breathed back, visions of Sugar Simons dancing in my head. Single Sugar Simons, Single Sugar Simons in Spain...\n\n\"Yeah,\" he whispered, and we were both silent for what seemed like months, although in actuality it was only enough time for Clive to claim his first victim: the Pounce hidden in my tennis shoe by the front door. I walked over to congratulate him on his catch.\n\n\"She said something curious,\" I mentioned, breaking the spell.\n\n\"Oh yeah? What's that?\" he asked.\n\n\"She told me that I was, and I quote, 'quite lovely.'\"\n\n\"Did she now?\" He laughed, easing back into comfortable.\n\n\"Yes, and the thing of it is, she said it like she was agreeing with something someone else had already said. Now, I'm not a girl who fishes for compliments, but it would seem, Simon, that you were talking sweet about me.\" I smiled, knowing my face was breaking into a pink glow. I'd started for the bedroom when I heard a soft knocking at the door. I walked back to unlock and open the door without looking through the peephole. I had a strong feeling I knew who was on the other side.\n\nThere he stood, phone cradled to his ear, holding his duffel bag and smiling a big, toothy grin.\n\n\"I told her you were lovely, but the truth is, you're more than lovely,\" he said, bowing his head toward mine and bringing his face to within inches of my own.\n\n\"More?\" I asked, barely drawing breath. I know my grin matched his.\n\n\"You're exquisite,\" he said.\n\nAnd with that, I invited him in. While wearing only my button-down. From far away, the O cheered...\n\nAn hour later, we sat together at the kitchen table, a decimated loaf in front of us. In between his frantic pawing, I'd managed a bite or two. The rest now lived in Simon's tummy, which he proudly thumped like a melon. We'd talked and eaten, gotten caught up, watched Clive as he finished his hunt, and now relaxed as the coffee brewed. Simon's bag rested by the front door still\u2014he hadn't even gone to his apartment yet. I was still in my button-down, feet curled beneath the chair as I stared at him. We were so comfortable, and yet that low-level hum, that electricity always sparking and snarking between us, continued.\n\n\"Fantastic touch by the way\u2014the raisins? Loved them.\" He smirked at me, poking one more in his mouth.\n\n\"You're terrible.\" I shook my head, stretching up out of my chair and collecting the plates and the few crumbs that hadn't been inhaled. I could sense him watching me as I moved about the kitchen. I grabbed the pot of coffee and raised my eyebrows at him. He nodded. I stood next to his chair to fill his mug, and I caught him peeking at my legs below my shirt.\n\n\"See something you like?\" I leaned across him to the sugar bowl.\n\n\"Yep,\" he answered, leaning toward me to take it.\n\n\"Sugar?\"\n\n\"Yep.\"\n\n\"Cream?\"\n\n\"Yep.\"\n\n\"That all you can say?\"\n\n\"Nope.\"\n\n\"Gimme something, then. Anything.\" I giggled, walking back around to my side of the table. Once again he watched me as I arranged myself in the chair.\n\n\"How about this?\" he finally said, resting on his elbows, face intense. \"As I mentioned earlier, I broke it off with Lizzie.\"\n\nI stared back, barely breathing. I tried to play it cool, so cool, but I couldn't stop the grin sneaking across my face.\n\n\"I see you are not at all broken up by this,\" he scoffed, sitting back in his chair.\n\n\"Not so much, no. Want the truth?\" I asked, the grin ushering in a sudden surge of confidence.\n\n\"Truth would be good.\"\n\n\"I mean truth truth, back-and-forth truth. No witty comebacks, no snappy banter\u2014although we do give great banter.\"\n\n\"We do, but I could go for some truth,\" he said, his voice quiet as his sapphire eyes blazed away at me.\n\n\"Okay, truth. I'm glad you broke things off with Lizzie.\"\n\n\"You are, are you?\"\n\n\"Yes. Why did you? Truth now,\" I reminded him. He regarded me for a moment, sipped his coffee, ran his hands through his hair in a maniacal way, and took a deep breath.\n\n\"Okay, truth. I broke it off with Lizzie because I didn't want to be with her any more. With any other women, in fact,\" he finished, setting his cup down. \"I'm sure we'll always be friends, but the truth is, I've been finding lately that three women? It's a lot for me to handle. I'm thinking of paring things down a bit, maybe trying just one for a while.\" He smiled, the blue getting dangerous.\n\nKnowing I was a grin and a clench away from total embarrassment, I stood quickly and went to dump my coffee in the sink. I paused there for a second, only a second, thoughts whirling. He was single. He was...single. Sweet mother of pearl, Wallbanger was single.\n\nI felt him move across the kitchen and come to stand behind me. I froze, feeling his hands gently brush my hair away from my shoulders and slip down to my hips. His mouth\u2014his ever-loving mouth\u2014barely touched the shell of my ear, and he whispered.\n\n\"Truth? I can't stop thinking about you.\"\n\nStill facing away from him, my mouth dropped open and my eyes went wide, torn between fist pumping and actual kitchen sex. Before I could decide, his mouth moved more purposefully, pressing into the skin just below my ear and making my brain burn and parts below dance a jig.\n\nHis hands gripped my hips, and he turned me toward him\u2014to face that body and grin\u2014I quickly composed my face, trying desperately to keep it together.\n\n\"Truth? I've been thinking about you since the night you banged on my door,\" he whispered, bending down to kiss the hollow of my neck with breathtaking precision. His hair tickled my nose, and I fought to keep my hands to myself. He pushed me to the side a little and surprised me by lifting me onto the counter. My legs automatically opened to allow him between them, the Universal Law of Wallbanger superseding any actual thought I had in my head. Not to worry, my thighs knew what to do.\n\nOne of his hands snuck around to the small of my back, while the other gripped the back of my neck. \"Truth?\" he asked one more time, pulling my hips to the edge of the counter, which forced me to lean back as my legs once more went on auto-pilot and wrapped themselves around his waist. \"I want you in Spain,\" he breathed, then brought his mouth to mine.\n\nSomewhere, a kitty began to call...and an O finally began her journey home.\n\n\"More wine, Mr. Parker?\"\n\n\"No more for me. Caroline?\"\n\n\"I'm fine, thank you.\" I stretched out luxuriously in my seat. First class to LaGuardia, then first class all the way to Malaga, Spain. We'd be taking a car from there to Nerja, the small coastal town where Simon had rented a house. Scuba diving, spelunking, hiking, beautiful beaches, and mountains, all set in a quaint village.\n\nSimon squirmed in his seat and shot an angry look over his shoulder.\n\n\"What? What's the problem?\" I asked, looking behind and seeing nothing out of the ordinary.\n\n\"That kid keeps banging my seat,\" he grumbled through clenched teeth.\n\nI laughed for a solid twenty minutes.\nChapter Sixteen\n\n\"WE DID IT TOO SOON. We should have waited.\"\n\n\"We waited long enough\u2014are you kidding? You know I was right. It was time to do it.\"\n\n\"Time to do it, what a crock! We could have waited just a little longer, and then we wouldn't be in the mess we're in now.\"\n\n\"Well, I didn't hear you complaining at the time. You seemed pretty pleased, as I recall.\"\n\n\"I couldn't complain, my mouth was full. But I had a feeling. I just knew this was wrong, what we were doing was inherently wrong.\"\n\n\"Okay, I give up. You tell me how to fix this.\"\n\n\"Well, for starters, you're holding it upside down,\" I shot back, grabbing the map and turning it right side up. We'd been parked along the side of the road for five minutes, trying to figure out how to get to Nerja.\n\nAfter landing in Malaga, navigating customs, navigating the rental car system, and finally navigating our way successfully away from the city center, we were now lost. Simon drove, so I was in charge of the map. And by that I mean he took it away from me every ten minutes or so, looked it over, hmm-ed and hawed, and then thrust it back my way. He didn't actually listen to anything I had to say, instead relying on his innate man-map. He also refused to turn on the GPS that had been provided for us, determined to get us there the old-fashioned way.\n\nWhich is why we were now lost. Taking a train would have been too easy. Simon needed a car to get around for his photos, which was ultimately why we were here. After flying through the night, we were both exhausted, but the best way to fight jet lag, allegedly, was to get on local time as quickly as possible. We had both agreed not to nap until we could go to sleep that night.\n\nNow we argued about where we took the wrong turn. I'd been devouring some churros from a roadside stand when the wrong turn supposedly took place, and so we played \"Place the Blame.\"\n\n\"All I'm saying is that if someone hadn't been stuffing her face and was watching for the turn, we wouldn't be\u2014\"\n\n\"Stuffing my face? Seriously? You were stealing my churros. I told you to get your own when we stopped!\"\n\n\"Well, I wasn't hungry at first, but then you were smacking your lips and licking that chocolate, and well...I got distracted.\" He looked up from the map, which he'd spread out on the hood of the car, and grinned, breaking the tension.\n\n\"Distracted?\" I grinned back, leaning a little closer. As he looked at the map, I looked at him. How could someone who'd been on a plane for the last hundred years look as good as he did? But there he was, faded jeans, black T-shirt, dark blue North Face jacket. Twenty-four hours of stubble begging to be licked. Who licked stubble? Me, that's who. He braced himself on his arms as he studied the map, his lips moving silently as he tried to figure it out. I snuck underneath his arms, draping myself across the hood of the car as shamelessly as a pinup girl in a garage calendar.\n\n\"Can I make a suggestion?\"\n\n\"It is a lewd suggestion?\"\n\n\"Surprisingly no. Can we please turn on the GPS? I'd like to make it there before I have to leave in a few days,\" I moaned. Due to my last-minute booking, I had to fly back a day before Simon. But five days in Spain...I was not complaining.\n\n\"Caroline, only pussies use GPS,\" he scoffed, turning to the map again.\n\n\"Well, this pussy is dying for some dinner, and a shower, and a bed, and to get rid of this jet lag. So unless you want to see me reenact It Happened One Night, Spanish version, turn on the GPS, Simon.\" I grabbed him by the North Face and pulled him down to me. \"Did that sound harsh?\" I whispered, giving him the tiniest of kisses on the chin.\n\n\"Yes, I'm terrified of you now.\"\n\n\"Does this mean GPS?\"\n\n\"It means GPS.\" He sighed resignedly, leaning back and pulling me off the car with him. I gave a little cheer and started for the door.\n\n\"No, no, no, you were harsh, Nightie Girl. I'm gonna need some sugar,\" he instructed, eyes twinkling.\n\n\"You need some sugar?\" I asked.\n\nHe tugged on my arm, bringing me back to him. \"Yes, I require it.\"\n\n\"You're twisted, Simon.\" I leaned into him, slipping my arms around his neck.\n\n\"You have no idea.\" He licked his lips and waggled his eyebrows like an old-timey gangster.\n\n\"Come get your sugar,\" I teased as he brought his lips to mine.\n\nI would never get tired of kissing Simon. I mean, how could you? Since the night he \"truthed\" me right up on to my kitchen counter, we'd slowly been exploring this new side of our relationship. Underneath all the snark and spark, there'd been some serious sexual tension building these many months. And we were letting it all out\u2014albeit slowly. Sure, we could've raced right back to the bedroom that night and let the sex ring out across the city for days, but Simon and I, without saying a word, seemed to be on the same page for once, and were content to let this unfold.\n\nHe was wooing me. And I was letting him woo. I wanted the woo. I deserved the woo. I needed the wow that would surely follow the woo, but for now, the woo? It was whoa.\n\nAnd speaking of woo...\n\nMy hands slipped into his hair, tugging and twisting and trying to pull his entire body inside my own. He groaned into my mouth, I felt his tongue touch mine, and I fell apart at the seams. I sighed, the tiniest whimper, and it became harder and harder to kiss him due to the giant grin overtaking my face.\n\nHe pulled back a little and laughed. \"You sure look happy.\"\n\n\"Keep kissing me, please,\" I insisted, bringing his face back to mine.\n\n\"It's like kissing a jack 'o' lantern. What's with the grin?\" He smiled down at me with a grin that looked as wide as my own.\n\n\"We're in Spain, Simon. Grinning is implied.\" I sighed contentedly, messing with his hair.\n\n\"And here I thought it was all to do with my kissing,\" he answered, kissing me again, gently, sweetly.\n\n\"Okay, cowboy, ready to see where the GPS takes us?\" I asked, stepping away. I couldn't keep my hands on him for too long or we'd never leave.\n\n\"Let's see how lost we really are.\" He smiled and we were on our way.\n\n\"I think this is the turn...Yep, this is it,\" he said.\n\nI bounced in my seat. Turned out we were closer than we thought, and we'd gotten a bit antsy. As we made one last turn, we looked at each other, and I squealed. We'd seen bits of the ocean for the last few miles or so\u2014peeking out behind a stand of trees or over a cliff. Now, as we turned down a tiny cobblestone drive, the realization that Simon had rented a house not just near the beach, but on the beach washed over me, and I was silenced by the sight.\n\nSimon pulled up to the house, the tires crunching on the rounded stones. When he turned the car off, I could hear the waves crashing against the rocky coast about a hundred feet away. We sat for a moment, just taking it all in and grinning at each other, before I scrambled out of the car.\n\n\"This is where we're staying? This entire house\u2014it's yours?\" I exclaimed as he grabbed our bags and came to stand next to me.\n\n\"It's ours, yeah.\" He smiled and gestured for me to walk ahead of him.\n\nThe house was charming and magnificent all at the same time: white stucco walls, clay-tile roof, clean lines, and soft archways. Orange trees lined the walkway from the drive, and bougainvillea climbed the garden walls. The house was a classic cottage, built to weather the sea and cocoon those inside. As Simon looked under the flowerpots for the key, I inhaled the citrus scents and the distinctly salty air.\n\n\"A-ha! Got it. Ready to see the inside?\" He struggled with the door for a moment before turning to face me.\n\nI reached for his hand, threading my fingers through his, and leaned in to kiss his cheek. \"Thank you.\"\n\n\"For what?\"\n\n\"For bringing me here.\" I smiled and kissed him square on the lips.\n\n\"Mmm, more of that sugar you promised me.\" He dropped the bag and pulled me close.\n\n\"Sugar this! Let's see the house!\" I cried, wiggling free and charging past him through the door. But as soon as I made it past the entryway, I stopped cold. Close on my heels, he bumped into me as I took it all in.\n\nA sunken living room, dotted with plush white sofas and comfy-looking chairs, opened up to what I assumed was the kitchen. French doors at the back of the house opened to several large, terraced patios, which sunk down toward the rocky beach. But what had stopped me cold was the ocean. All across the back, through the giant windows, was the deep blue of the lazy Mediterranean. The coastline curved back to the town of Nerja, where the lights were just beginning to sparkle as twilight drifted over the beach, illuminating the other white houses that clung to the cliffs. Remembering how to move, I ran to push open the doors and let the soft air spill over me and into the house, blanketing everything in the evening's perfume.\n\nI walked to the wrought iron railing, which perched at the edge of an earthen tile patio flanked by olive trees. Placing my hands on the warm metal, I looked and looked and looked. I felt Simon walk up behind me and without a word place his arms around my waist. He nestled in to me, resting his head on my shoulder. I leaned back, feeling the angles and planes of his body fit against my own.\n\nYou know those moments when everything is exactly the way it was meant to be? When you find yourself and your entire universe aligning in perfect synchronization, and you know you couldn't possibly be more content? I was inside that very moment, and fully conscious of it. I giggled a little, feeling Simon's smile stretch across his face as he pressed into my neck.\n\n\"It's good, right?\" he whispered.\n\n\"It's so good,\" I answered, and we watched the sunset in spellbound silence.\n\nAfter watching the sunset until it was totally gone, we explored the rest of the house. It seemed more and more beautiful with every room, and I squealed once again at the sight of the kitchen. It was as if I'd been transported to Ina's home in East Hampton, with a Spanish flair: Sub-Zero fridge, gorgeous granite countertops, and a Viking stove. I didn't even want to know how much Simon was paying for this house. I'd decided to just enjoy. And enjoy we did, running back and forth, laughing like kids when we found the bidet in the hallway bathroom.\n\nAnd then we entered the master bedroom. I came around the corner and saw him standing at the end of the hallway, just outside the door.\n\n\"What the hell did you find that has you so qui\u2014oh my. Would you look at that?\" I stopped next to him, admiring from the doorway.\n\nIf my life had a soundtrack, the theme from 2001: A Space Odyssey would have been playing right now.\n\nThere, in the middle of a corner room, with its own terrace overlooking the most beautiful ocean in the world, was the biggest mother-loving bed I'd ever seen. Carved out of what looked to be teak, it was as big as football field. Thousands of silky soft white pillows stacked against the headboard, spilling down over a white duvet. It was folded down just so, the million or so thread count sheets shining, actually shining, as though they were lit from within. Sheer white curtains hung from rods suspended over the bed, creating a canopy, while even more curtains hung in the windows overlooking the sea below. The windows were open and all the curtains blew gently in the breeze, giving the entire room a billowy, flouncy, windblown effect.\n\nIt was the bed to end all beds. It was the bed that all the little beds aspired to be when they grew up. It was bed heaven.\n\n\"Wow,\" I managed, still in the hallway next to Simon.\n\nIt was hypnotic. It was like a bed siren, luring us in so we could crash.\n\n\"You could say that again,\" he stammered, his eyes never leaving the bed.\n\n\"Wow,\" I repeated, still staring.\n\nI couldn't stop, and I was suddenly very, impossibly, excruciatingly nervous. I had a lovely case of performance anxiety, party of one.\n\nSimon chuckled at my weak joke, and it brought me back to him.\n\n\"No pressure, huh?\" he said, eyes shy.\n\nHuh? Nerves? Party of two? I had a choice. I could go with conventional wisdom, said wisdom being that two grownups on vacation together in a gorgeous house with a bed that was sex incarnate would immediately begin nonstop sexing...or, I could let us both off the hook and just enjoy. Enjoy being together and let things happen when they happen. Yeah, I liked this idea better.\n\nI winked and took a running leap on to the bed, bouncing pillows all over the room. I peeked over the remaining mound to see him leaning in the doorway, a sight I had seen so many times before. He looked a little nervous, but still beautiful.\n\n\"So, where are you sleeping?\" I called, and his face relaxed into a smile, my smile.\n\n\"Wine?\"\n\n\"Am I breathing?\"\n\n\"Wine it is,\" he snorted, selecting a bottle of ros\u00e9 from the generously stocked wine fridge. Simon had arranged to have some basic groceries delivered to the house before our arrival, nothing fancy but enough to nosh on and make us comfortable. It was now fully dark, and any thoughts we'd had about going into town faded away as the jet lag loomed. Instead we'd stay in tonight, get a good night's sleep, and head into town in the morning. There was a roast chicken, olives, a wedge of Manchego, some gorgeous looking Serrano ham, and enough other little odds and ends to make a meal. I assembled plates while he poured the wine, and soon we were sitting on the terrace. The ocean crashed below, and the wooden walkway down to the beach was strung with tiny white lights.\n\n\"We should go down to the beach before bed, at least take a little walk.\"\n\n\"Done. What do you want to do tomorrow?\"\n\n\"Depends, when do you need to start working?\"\n\n\"Well, I know some of the places I need to go, but I need to do a little scouting still. Want to come along?\"\n\n\"Of course. Start in town in the morning and see where that leads?\" I asked, nibbling on an olive.\n\nHe raised his glass and nodded. \"To seeing where it leads,\" he toasted.\n\nI raised my glass to his. \"I'll second that.\" Our glasses clinked and our eyes locked. We both smiled, a secret smile. We were finally alone, all to ourselves, and there was no place else on the planet I wanted to be. We ate our dinner, stealing little glances at each other throughout, and sipped our wine. It made me drowsy, and a little touchy feely.\n\nAfter that we'd picked our way carefully over the rocky shoreline to the beach. We'd grasped hands to navigate but never let go. Now we stood at the edge of the earth, the strong, salty wind whipping through our hair and clothes, buffeting us back a bit.\n\n\"It's nice, being with you,\" I told him. \"I, um, well, I like holding your hand,\" I admitted, feeling brave from the wine. Witty banter had its place, but sometimes, all you need is the truth. He didn't respond, simply smiled and brought my hand to his mouth, placing a small kiss.\n\nWe watched the waves, and when he pulled me to his chest, snuggling me to him, I breathed out slowly. Had it really been so long since I'd felt\u2014Oh, what was it I was feeling?\u2014cared for?\n\n\"Jillian told me you know what happened to my parents,\" he said so softly I could barely hear him.\n\n\"Yes. She told me.\"\n\n\"They used to hold hands all the time. Not for show, though, you know?\"\n\nI nodded into his chest and breathed him in.\n\n\"I always see these couples that hold hands and make such a show of it, calling each other baby and sweetie and honey. It seems like, I don't know, false somehow. Like, would they be doing it if they weren't in front of anyone?\"\n\nI nodded again.\n\n\"My parents? I never thought much about it at the time, but when I think about it now, I realize their hands were practically sewn together, always with the hand holding. Even when no one was looking, right? I'd come home after practice and find them watching TV, at either end of the couch, but with their hands propped up on a pillow so they could still be touching...It was just...I don't know, it was nice.\"\n\nMy hand, still tucked into his own, squeezed, and I felt his strong fingers squeeze back.\n\n\"Sounds like they were still a couple, not just a mom and dad,\" I said, hearing his breath speed up a bit.\n\n\"Yes, exactly.\"\n\n\"You miss them.\"\n\n\"Of course.\"\n\n\"Might sound weird, since I never knew them, but I feel like they would be so proud of you, Simon.\"\n\n\"Yeah.\"\n\nWe were quiet another minute, feeling the night around us.\n\n\"Want to go back to the house?\" I asked.\n\n\"Yeah.\" He kissed the top of my head as we began to make our way back\u2014hands stuck together like someone had spread Krazy Glue on them.\n\nI'd left Simon to clean up the mess from dinner. I wanted a quick shower before bed. After washing away the days of airport and travel, I threw on an old T-shirt and boy shorts, too tired for the lingerie I had packed. Yes, I had packed lingerie. Come on, I was no nun.\n\nI stood in front of the mirror in my bedroom (yep, I had totally claimed the big one) after blow-drying my hair when I saw him appear in the doorway. He was on his way to his room after his own shower, wearing pajama pants and a towel wrapped around his neck. I was exhausted, but not so exhausted I didn't appreciate the form in front of me. I watched him in the mirror as he appraised me as well.\n\n\"Have a good shower?\" he asked.\n\n\"Yes, it felt amazing.\"\n\n\"Heading to bed?\"\n\n\"I can barely keep my eyes open,\" I replied, yawning hugely to punctuate.\n\n\"Can I get you anything? Water? Tea? Anything?\"\n\nI turned to face him, as he stepped inside. \"No water, no tea, but there is one thing I'd like before I go to sleep,\" I purred, taking a few steps his way.\n\n\"What's that?\"\n\n\"Goodnight kiss?\"\n\nHis eyes darkened. \"Oh, hell, is that all? That I can do.\" He closed the distance between us and slipped his arms easily around my waist.\n\n\"Kiss me, you fool,\" I teased, falling into his embrace as if in an old-time melodrama.\n\n\"One kissing fool, coming up,\" he laughed, but within seconds no one was laughing. And within minutes, no one was standing.\n\nAfter falling into Pillow Town, we scrambled about, arms and legs twisting this way and that, kisses becoming more and more frantic. My shirt bunched up around my waist, and the feeling of his hi-there against my hoohah was indescribable. He rained kisses down upon my neck, licking and sucking as I moaned like a whore in church.\n\nTo be fair, I'd never actually heard a whore moan in church, but I had a feeling it sounded a lot like the unholy sounds pouring forth from my mouth.\n\nHe flipped me about like a rag doll and settled me on top of him, my legs on either side, the way I'd wanted to be for so long. He sighed, gazing up as I impatiently pushed my hair away from my face so I could truly appreciate the magnificence I was perched on.\n\nWe slowed our movements, then stopped altogether, staring unabashedly at each other, appraising each other without shame.\n\n\"Incredible,\" he breathed, reaching to gently cup my face as I nuzzled his hand.\n\n\"That's a good word for it, yes. Incredible.\" I turned to kiss his fingertips. He stared into my eyes again, those sex sapphires doing their voodoo that made me a puddle of voodoo goo. For him to woo. See what he did to me?\n\n\"I don't want to screw this up,\" he said suddenly, his words breaking me from my Seussian rhymes.\n\n\"Wait, what?\" I asked, shaking my head to clear it.\n\n\"This. You. Us. I don't want to screw this up,\" he insisted, sitting up underneath me, my legs wrapping around to his back.\n\n\"Okay, so don't,\" I ventured, unsure where this was going.\n\n\"I mean, you need to know, I have no experience with this.\"\n\nI raised an eyebrow. \"I have a wall back home that would disagree with that...\" I laughed, and he crushed me to his chest, inexplicably hard. \"Hey, hey...what's up? What's going on?\" I soothed, my hands rubbing up and down his back.\n\n\"Caroline, I, Jesus, how do I say this without sounding like an episode of Dawson's Creek?\" He stumbled the words while talking into my neck.\n\nI couldn't help it, I chuckled a little as Pacey flashed into my head, and that brought him back. I pulled away a bit so I could see him, and he smiled ruefully.\n\n\"Okay, Dawson's be damned, I really like you, Caroline. But I haven't had a girlfriend since high school, and I have no clue how to do this. But you need to know, that what I feel for you? Shit, it's just different, okay? And, whatever your wall would say back home, I need you to know that this? What we have, or will have? It's different, okay? You know that, right?\"\n\nHe was telling me I was different, that I was no replacement for the harem. And this, I knew. He looked at me so earnestly, so seriously, and my heart opened even more. I pressed a gentle kiss to his sweet lips.\n\n\"First of all, I do know this. Second of all, you're better at this than you think.\" I smiled, pressing his eyes closed and kissing each eyelid. \"And, for the record, I loved Dawson's Creek, and you did the WB proud.\" I laughed as his eyes sprang back open and relief rushed in. I tucked him into my nook and held him there as we rocked back and forth, the rush of the earlier hormones subsiding as we found this new space, this quiet intimacy that was becoming almost as addicting.\n\n\"I like that we're taking things slow. You give good woo,\" I whispered.\n\nHe tensed underneath me. I could feel him shaking a little.\n\n\"I give good woo?\" he laughed, tears springing to his eyes as he tried to control his laughter.\n\n\"Oh, shut up,\" I cried, smacking him with a pillow. We laughed for a few more minutes, falling back into the lush bed, and as the jet lag finally overtook us, we settled in. Together. There was no question in my mind now about sleeping in separate rooms. I wanted him here. With me. Surrounded by pillows and Spain, we nooked. The last thought I had, before slipping into sleep with his strong arms wrapped around me...I might be falling in love with my Wallbanger. \nChapter Seventeen\n\nI WAS AWAKENED THIS MORNING by a great rumbling. Forgetting where I was for a split second, I automatically assumed I was home, and we were experiencing a tremor. I was halfway out of bed with one foot on the floor before I noticed that the view outside my bedroom window was decidedly more blue than it was at home, and decidedly more Mediterranean. And the rumbling? That was no tremor. It was Simon snoring. Snoring. Snoring to beat the band, and by beat the band I mean beat that band up with his nose\u2014which was emitting the most unearthly sound. I clapped my hands over my mouth to hold in the laughter and crept back into bed, the better to appraise the situation.\n\nTrue to form, I'd taken over most of the bed in the night, and he'd been relegated to the far corner, where he was now curled into a little ball with a pillow tucked between his legs. But what he lacked in square footage, he made up for in sound. The sounds pouring forth from his nasal passages registered somewhere between grizzly bear and exploding tractor trailer. I wiggled across the mile-wide bed, curling myself around his head and looking down at his face. Even while making these horrific sounds, he was adorable. I carefully placed my fingers next to his nose, and plugged. And then waited.\n\nAfter about ten seconds, he inhaled and shook his head, looking around wildly. He relaxed when he saw me perched on the pillows next to him. He smiled a sleepy smile.\n\n\"Hey, hey, what's up?\" he mumbled, rolling into me and wrapping his arms around my waist, resting his head on my tummy. I ran my hands through his hair, delighting in the casual freedom we finally had in touching each other.\n\n\"Just woke up. Someone was quite noisy on this side of the bed.\"\n\nHe closed one eye and looked up at me. \"I hardly think someone as flaily as you can complain about anything.\"\n\n\"Flaily? That's not even a word.\" I huffed, enjoying his arms around me more than I wanted to admit.\n\n\"Flaily, as in, one who flails. As in, one who, even though she is sleeping in a bed the size of Alcatraz, still needs almost the entire mattress to spread out and kick,\" he insisted, accidentally-on-purpose pushing my shirt up so he could rest his head on my naked tummy.\n\n\"Flailing is better than snoring, Mr. Snorey Pants,\" I teased again, trying not to notice the way his stubble scraped against my skin in the most delicious way.\n\n\"You flail. I snore. Whatever will we do about this?\" He smiled happily, still half asleep.\n\n\"Ear plugs and shin guards?\"\n\n\"Yep, that's sexy. We can suit up before bed each night,\" he sighed, pressing the tiniest of kisses just above my belly button.\n\nA noise that sounded sadly like a whimper escaped my lips before I could pull it back, and my ears burned as I took in what he'd said about \"each night,\" as in sleeping together each night. Oh my...\n\nWe ate a quick breakfast at the house, then headed into town. I fell in love with the village instantly: the old stone streets, the whitewashed walls glimmering in the blazing sunlight, the beauty that poured forth from every open archway. From every speck of azure that peeked through from the coast to the friendly smiles on the sweet faces of the people who called this enchanted spot home, I was hooked.\n\nIt was market day, and we wandered in and out of stalls, picking up fresh fruit to snack on later. I've seen beautiful places on this earth, but this town was heaven for me. I'd truly never experienced anything like it.\n\nNow, I had been traveling alone for years, finding my own company quite pleasant. But traveling with Simon? It was...cool. Just, cool. He was quiet, the way I am when I'm seeing something new. He never felt the need to fill a silence with chattery words. We were content to soak up the scenery. When we did speak, it was to point out something we thought the other shouldn't miss, like the puppies playing in a dooryard, or an old man and woman talking back and forth over their balconies. He was a great companion.\n\nWe strolled back to the rental car, the afternoon sun toasting through the thin cotton covering my shoulders, when my hand tangled with his in the most unassuming way. And when he took the time to open my door for me, and leaned down to kiss me in the warm Spanish sunshine, his lips and the smell of olive trees were the only things I needed in the entire world.\n\nIn the time I'd known Simon, I'd committed several images of him to memory: seeing him for the first time, clad only in a sheet and a smirk; driving back across the bridge with him the night of Jillian's housewarming, when we called a truce; warped and blurry Simon as seen from inside an afghan; backlit by tiki torches, wet, and looking devilishly handsome by hot tub; and a recent addition to my Best of Simons? The sight of him underneath me as he clutched me close, his warm skin and sweet breath all over me as we nooked in the Giant Bed of Sin.\n\nBut nothing, and I mean nothing, was hotter than watching Simon work. I mean it. I actually had to fan myself a little\u2014which he took no notice of, because when he was working he was delightfully focused.\n\nAnd now here I sat, watching Simon work. We'd driven up the coast to get some test shots at a place a local guide had told him about, and the perilously handsome Simon now concentrated completely on the task at hand. As he'd explained to me, it wasn't about the actual pictures he was taking, it was about testing the light and the colors. So as he scrambled his way from rock to rock, I sat on a blanket we'd dug out of the trunk and observed. Perched on cliffs high above the sea, we could see for miles. The rocky shoreline stretched and curled back in on itself as millions of waves poured in from the deep sea. And while the scenery was gorgeous, what had my attention was the way the tip of Simon's tongue poked out as he surveyed the scene. The way he bit down on his lower lip as he puzzled over something. The way excitement broke over his face when he saw something new through his lens.\n\nI was glad I had something to do, something to fixate on, as the beginning of a battle was starting to wage inside my body. Ever since we'd acknowledged the pressure that giant bed could have placed on us, all I could think about was that very pressure. As well as the pressure of an O long denied, waiting patiently\u2014and sometimes impatiently\u2014for her release. The pressure was so strong, so intense, that every single part of me could feel it.\n\nCurrently taking sides in this internal debate were my brain, Lower Caroline (speaking for the distant O), Backbone, and although she'd mainly kept quiet lately, letting Brain and Nerves take control, Heart was now weighing in.\n\nIt should be noted that LC (Lower Caroline wanted a hip but abbreviated name) had somehow drafted Simon's penis into the fray, and even though his penis didn't have direct access to her yet, LC felt it necessary to speak up on his behalf. While I didn't much like the term penis, internally I felt strange about calling him dick or cock, so penis it was...for now.\n\nNow, Backbone and Brain were solidly in the wait-for-sex camp, believing this essential to the foundation of this burgeoning relationship. LC, and therefore Simon's penis, were in the have-sex-with-him-as-soon-as-possible society, obviously. O, while not officially in residence, could be counted among LC's supporters. But I felt a twinge, and just a twinge, of her floating above both camps, along with Heart, who was currently singing songs about everlasting love and warm, fluffy things.\n\nTake all this into account and what do you have? One totally confused Caroline. A Caroline divided. No wonder I had sworn off dating. This shit was tough. So was I glad to have something to think about other than the pressure cooker of sex indeterminate? Yes. Could I spend a little more time trying to come up with a more clever name for Simon's penis? Probably. It deserved it. Mammoth Male Member? No. Pulsating Pillar of Passion? No. Back Door Bandit? Hell no. Wang? Sounded like the noise those doorstopper things made when you flicked 'em...\n\nI said it out loud to myself a few times, cracking myself up a little. \"Wang. Wang. Waaaang,\" I muttered.\n\n\"Hey! Nightie Girl! Get yourself over here,\" Simon called, breaking me out of my wang study. I left behind the mental battle, picking my way carefully across the craggy rocks to where he was poised.\n\n\"I need you.\"\n\n\"Here? Now?\" I snorted.\n\nHe lowered his camera just enough to raise one eyebrow. \"I need you for scale. Get over there.\" He pointed me toward the edge of the cliff.\n\n\"What? No-no. No pictures, huh-uh.\" I backed away toward my blanket.\n\n\"Yes, yes, pictures. Come on. I need something in the foreground. Get over there.\"\n\n\"But I'm a mess! I'm all windblown and sunburned, see?\" I pulled down my v-neck just a little to show him how I was beginning to pink up.\n\n\"While I always appreciate you showing me your cleavage, save it, sister. This is just for me, just to give me some perspective. And you don't look windblown. Well, only a little.\" He tapped his foot.\n\n\"You're not gonna make me pose with a rose in my teeth, are you?\" I sighed, shuffling over to the edge.\n\n\"Do you have a rose?\" he asked, looking serious except for the shit-eating grin.\n\n\"Shut it, you. Take your pictures.\"\n\n\"Okay, just be natural. No posing, just stand there\u2014facing the water would be great,\" he instructed.\n\nI complied. He moved around me, trying different angles, and I could hear him muttering about what was working. I admit, even though I was shy about having my picture taken, I could almost feel his eyes, through the lens, watching me. He moved around for only a few moments, but it felt longer. The internal war was beginning to wage again.\n\n\"You almost done?\"\n\n\"You can't rush perfection, Caroline. I need to get the job done right,\" he warned. \"But yes. Almost done. You getting hungry?\"\n\n\"I want those clementines in the basket\u2014grab me one? Or will that mess with your masterpiece?\"\n\n\"Won't mess with it. I'll call it Windblown Girl on a Cliff with a Clementine.\" He laughed and headed back over to the car.\n\n\"You're funny,\" I said wryly, catching the tiny orange he threw me and starting to peel.\n\n\"Are you sharing?\"\n\n\"I suppose so, the least I could do for the man who brought me here, right?\" I laughed, biting into a wedge and feeling the juice dribble down my chin.\n\n\"You got a hole in your lip?\" he asked, capturing the moment as I rolled my eyes at him.\n\n\"Do you actually think you're funny, or are you just assuming you might be?\" I countered, beckoning him over with the peel. He shook his head, laughing as he took a wedge. Of course, he took a bite and no dribble. He opened his eyes wide in feigned amazement, and I took the opportunity to smash another wedge in his face. His eyes remained wide open, as juice now ran freely off the tip of his nose and on to his chin.\n\n\"Messy Simon,\" I whispered as he looked at me. In a flash, he pressed his lips to mine, getting juice all over both of us as I squealed into his mouth. \"Sweet Caroline,\" he whispered through his grin. He turned us so the sea was behind us, held up the camera, and took a picture: us covered in orange mush.\n\n\"By the way, why were you saying 'wang' earlier?\" he asked.\n\nI just laughed harder.\n\n\"This is it. This is now officially the single best thing I have ever had in my mouth,\" I announced, closing my eyes and moaning.\n\n\"You've said that about everything you've eaten tonight.\"\n\n\"I know, but I seriously can't handle how good this is. Smack me, pinch me, throw me overboard, this is too good,\" I moaned again. We sat at a little table in the corner of a small restaurant in town, and I was determined to try everything. Simon, showing off his language skills, had ordered for us. I told him to go for it, that I was in his hands and I knew he wouldn't steer me wrong. And the boy did good. We feasted.\n\nWe went with traditional tapas, of course, accompanied by glasses of the house wine. Little bowls and plates showed up at the table every few minutes after that: tiny pork meatballs, slices of ham, marinated mushrooms, beautiful sausages, grilled squid with fruity local olive oil. With each bite, I was sure that I had just eaten the best thing ever, then another wave of gorgeous food would show up and convince me once again. And then these prawns arrived. Unreal. Fried crispy in olive oil with tons of garlic and parsley, smoky paprika, and just a hint of heat. I swooned. I actually swooned.\n\nSimon? He loved it. He ate it up. My reactions as much as the food, I think. He ate it up.\n\n\"Honestly, I can't handle any more,\" I protested, dragging a piece of crusty bread through the olive oil. He smiled as he watched me shamelessly enjoy another piece of bread before finally pushing back from the table with a groan.\n\n\"Best meal ever?\" he asked.\n\n\"It really might be. That was insane.\" I sighed, patting my full tummy. Ladylike, schmadylike, I'd pounded that meal down like someone was going to take it away from me. A waiter appeared with two small glasses of a local wine. Sweet and crisp, it was the perfect after-dinner drink. We sipped slowly, the breeze coming in through the windows lightly scented with the sea air.\n\n\"This was a great date, Simon. Really. Couldn't have been more perfect,\" I said, taking another sip of my wine.\n\n\"Was this a date?\" he asked.\n\nMy face froze. \"I mean, no. I suppose not. I just\u2014\"\n\n\"Relax, Caroline. I know what you meant. It's just funny to consider this a date: two people traveling together, but only now on a date.\" He smiled, and I relaxed.\n\n\"Hmm, we haven't really followed the traditional rules so far, have we? This might even be our first date, if we wanted to get technical.\"\n\n\"Well, technically speaking, what defines a date?\" he asked.\n\n\"Dinner, I suppose. Although we've had dinner before,\" I began.\n\n\"And a movie\u2014we've already had a movie,\" he reminded me.\n\nI shuddered. \"Yes, and that was definitely a ploy to get me to snuggle with you. Scary movie, so obvious,\" I scoffed.\n\n\"It worked, didn't it? In fact, I do believe I slept with you that night, Nightie Girl.\"\n\n\"Yes, I'm cheap and easy, I admit it. I suppose we really did do this whole thing backward.\" I grinned, sliding my foot across the floor under the table and kicking him lightly.\n\n\"I like it backward.\" He smirked.\n\nI narrowed my eyes. \"Not touching that one.\"\n\n\"Seriously, though. As I've mentioned, I have no experience with this stuff,\" he said. \"How does this work? What if we were doing this...not backward? What would happen next?\"\n\n\"Well, I suppose there would be another date, and another after that,\" I admitted, smiling shyly.\n\n\"And bases. I'd be expected to try to round some bases, right?\" he asked seriously.\n\nI spluttered my wine. \"Bases? Are you for real? As in, cop a feel, over the shirt, under the shirt, those bases?\" I laughed incredulously.\n\n\"Yes, exactly. What am I allowed to get away with? As a gentleman, I mean. If this were truly a first date, we wouldn't be going home together, would we? Dating now, not hooking up. Remember, apparently I give good woo,\" he said, eyes twinkling.\n\n\"Yes, yes, you do. We wouldn't be going home together, that's true. But to be honest, I don't want you sleeping in the bedroom down the hall. Is that weird?\" I could feel my ears burning as I blushed.\n\n\"It's not weird,\" he answered quietly. I slipped off my sandal and pressed my foot against his, rubbing lightly along his leg.\n\n\"Nooking is good, right?\"\n\n\"Nooking is most definitely good,\" he agreed, nudging back with his own foot.\n\n\"As far as your bases are concerned, I think you could definitely plan on a little under the shirt action, if you were so inclined,\" I answered. Internally, Brain and Backbone gave a little cheer, while LC and Wang kicked a few chairs. Tatas were just glad someone was considering them for once, instead of being just a stopover on the way to points south. Heart? Well, she was still flitting about, singing her song.\n\n\"So, we go a little traditional, but not totally traditional. Take it slow?\" he asked, his eyes burning, the sapphires beginning to do their little hypnotic dance.\n\n\"Slow, but not too slow. We are grownups, for goodness sake.\"\n\n\"To under the shirt action,\" he announced, raising his glass in toast.\n\n\"I'll drink to that.\" I laughed as we clinked.\n\nFifty-seven minutes later we were in bed, his hands warm and sure as he slipped each button through, revealing my skin. He went slowly, purposefully, and he let my shirt fall open as I lay beneath him. He gazed down at me, his fingertips lightly drawing a line from my collarbone to my navel, straight and true. We both sighed at the same time.\n\nI can't explain it, but knowing we'd set some boundaries for the evening, silly as it may be, made it so much more sensual, something to be truly savored. His lips hovered around my neck, whispering tiny kisses against my skin, below my ear, under my chin, in the dip between my neck and my shoulder, and working his way down to the swell of my breasts. His fingers swept out, lightly, reverently, ghosting across the sensitive skin as I inhaled and then held my breath.\n\nAs his fingers gently grazed my nipple, every nerve ending in my entire body reversed and began to pulse in that direction. I exhaled, feeling months of tension begin to simultaneously flow out of me and build up even more. With sweet kisses and soft touches, he began the process of getting to know my body, and it was exactly what I needed. Lips, mouth, tongue\u2014all of it on me, tasting, stroking, feeling, and loving.\n\nAs his lips closed around my breast, his hair tickled my chin in the cutest way, and I wrapped my arms around him, holding him close. The feeling of his skin against mine was perfection, and something I'd never experienced before. I felt...worshipped.\n\nAs we explored that night, what started out as funny and cute and part of our classic banter became something more. What was crassly called \"under the shirt action\" became part of a romance, and something that could have been merely physical became something emotional and pure. And when he cradled me to him, bringing me into his nook with tender kisses and breathless giggles, we fell into a contented sleep.\n\nFlaily and Mr. Snorey Pants.\n\nFor the next two days, I luxuriated. Truly, there isn't another word in the English language to articulate the experience I indulged in. Now for some, the definition of a luxurious vacation might be endless shopping, spa pampering, expensive meals, elaborate shows. But to me, luxurious meant spending two hours napping in the sun on the terrace off the kitchen. Luxurious meant eating figs dripping with honey and dotted with crumbles of local cheese while Simon poured me another glass of Cava, all before ten a.m. Luxurious meant time alone to wander through the small, family stores of Nerja, poking through bins of beautiful lace. Luxurious meant exploring the nearby caves with Simon while he photographed, losing ourselves in the colors under the earth. Luxurious meant gazing at Simon dangling from a rock face while he searched out another foothold, shirtless. Did I mention shirtless?\n\nAnd luxurious most certainly meant that I got to spend each night in that bed with Simon. Now that's a priceless luxury, not offered on every grand tour. We rounded another base or two, teasing each other with a little over-the-panties encounter. Were we being ridiculous, waiting until the last night in Spain to consummate this \"thing\"? Probably, but who the hell cared? He spent almost an hour kissing every inch of my legs one night, and I spent about the same amount of time having a conversation with his belly button. We just...enjoyed.\n\nBut with all this enjoyment came a certain amount of, well, how shall we say, nervous energy?\n\nBack in San Francisco, we'd spent months engaged in verbal foreplay. But now, here? The actual foreplay? It was not to be believed. My body was so in tune with his, I knew when he walked into the room, and I knew when he was about to touch me, seconds before he did. The air between us was sexually charged, vibes zinging back and forth with enough energy to light up the entire town. Sexual chemistry? Had it. Sexual frustration? On the rise and getting close to critical.\n\nOh, hell, I'll say it. I was H-O-R-N-Y.\n\nWhich was why after we spent the afternoon in the caves, we found ourselves in the kitchen, kissing madly. We were both a little tired from the day, and I'd been wanting to test out that beautiful Viking range. I was preparing vegetables for the grill and stirring some saffron rice when he came in after a shower. It's almost impossible for me to explain the sight of him: worn white T-shirt, faded jeans, barefoot, scrubbing at his wet hair with a towel. He grinned, and I began to see double. I literally couldn't see through the haze of lust and need I suddenly felt surge through me. I needed my hands to be on his body, and I needed it to happen immediately.\n\n\"Mmm, something smells good. Want me to get the grill started?\" he asked, walking over to where I was chopping vegetables at the counter. He stood behind me, his body only inches from mine, and something snapped. And it wasn't just the pea pod I was holding...\n\nI turned around, and my tummy actually fluttered at the sight of him. It freaking fluttered. I pressed my hand against his chest, feeling the strength there and the warmth of his skin through the cotton. Reason waved bye-bye, and this was now purely physical. An itch that needed to be scratched, scratched, and then scratched again. I slid my hand up around the back of his neck, and pulled him down to me. My lips crashed against his, my intense need for him pouring into his mouth and down to the tips of my toes. Toes that kicked off their flip-flops and started shamelessly rubbing themselves across the tops of his feet. My body needed to feel skin, any skin, and needed it now.\n\nHe responded, matching my rough kisses with his own, his mouth covering mine as I groaned at the feel of his hands on the small of my back. I quickly spun him around and pressed him up against the counter.\n\n\"Off! I need this off, now,\" I muttered between kisses, yanking at his T-shirt. In a great whoosh of fabric, his shirt was thrown across the room as I maneuvered my body against his, sighing as I felt the contact. I was alternately trying to hug him and climb him, the lust now running freely through my body like a freight train. I reached between us and palmed him through his jeans. His eyes caught mine, and they crossed a little. I was on the right track. Feeling him getting harder by the second under my fingertips, suddenly all I wanted, all I needed, all I had to have to function in life, was him. In my mouth.\n\n\"Hey, Nightie Girl, what are you\u2014oh God\u2014\"\n\nMoving instinctively, I snapped open his jeans, dropped to my knees before him, and brought him forth. My pulse raced, and I think my blood actually boiled within me as I saw him. My breath drew in with a hiss as I regarded him, faded jeans pushed down just enough to frame this luminous sight.\n\nSimon does commando. God bless America.\n\nI wanted to be gentle, I wanted to be tender and sweet, but I simply needed him too badly. I glanced up at him, his eyes clouded but frantic, as his hands came down to brush my hair back from my face. I took his hands in my own and placed them back on the counter.\n\n\"You're gonna want to hold on for this,\" I promised. He groaned a delicious groan and, doing as he was told, leaned back a little. He pushed his hips forward, but kept his eyes on mine. Always on mine.\n\nMy lips purred as I slipped his length inside my mouth. His head dropped back as my tongue caressed him, taking him in deeper. The pure pleasure of this, the absolute pleasure of feeling his reaction to me was enough to make my head split in two. I drew him back out, letting my teeth just barely graze his sensitive skin as I saw him grip the edge of counter even harder. I ran my nails up the inside of his legs, pushing his jeans farther down for more access to his warm skin. Pressing kisses across the tip of him, I let my hands come up to grasp him, stroking and massaging. He was perfect, all smooth and taut as I took him in again, and again, and again. I felt crazed, drunk on his scent and the feel of him inside me.\n\nHe moaned my name over and over again, his words drifting down like molten chocolate sexy times, pouring inside my brain and dedicating every sense I had to him, only to him. On and on I went, making him crazy, making me crazy, licking, sucking, tasting, teasing, luxuriating in the madness that was this luscious act. To have him here, in this way, was the very definition of luxury.\n\nHe stiffened further, and his hands finally came back to me, trying to make me pull back.\n\n\"Caroline, oh, Caroline, I'm...you...first...you...oh, God...you,\" he stuttered. Luckily, I was able to interpret. He wanted me to have something as well. What he didn't realize is that this total abandon he was giving me was all I needed. I released him only for a moment, to place his hands once more on the counter.\n\n\"No, Simon. You,\" I replied, taking him in deeply once more, feeling him hit the back of my throat as my hands tended to the rest of him that my mouth could not. His hips moved once, then again, and with a shudder and the most scrumptious groan I've ever heard, Simon came. Threw his head back, closed his eyes, and let go.\n\nIt was wonderful.\n\nMoments later, crumpled into me on the floor of the kitchen, he sighed contentedly. \"Good Lord, Caroline. That was...unexpected.\"\n\nI giggled, bending down to kiss his forehead. \"I couldn't control myself. You just looked way too good, and I...well...I got carried away.\"\n\n\"I'll say. Although I don't think it's fair that I'm somewhat exposed here, and you're still fully clothed. We could remedy that pretty quickly, though.\" He pulled at the drawstring on my pants.\n\nI stopped him. \"First of all, you aren't somewhat exposed, you are hanging free on the kitchen floor, and I quite like it. And this wasn't about me, although I admit I enjoyed it immensely.\"\n\n\"Silly girl, now I want to enjoy you immensely,\" he persisted, running his fingers along the edge of my pants, dancing across the skin there.\n\nNerves began to dance the flamenco, demanding more time\u2014more time! Not ready! LC kicked some things. \"No, no, not tonight. I want to make you a nice dinner. Let me take care of you a little bit. Can't I just do that?\" I removed his devil hands and kissed them.\n\nHe smiled up at me, his hair messy and a goofy grin still adorning his face. He sighed in defeat and nodded. I started to climb off the floor when he caught me around the waist, pulling me back down.\n\n\"A word, please, before you leave me\u2014what did you say? Hanging free on the kitchen floor?\"\n\n\"Yes, dear?\" I asked, earning a raised eyebrow.\n\n\"So, using the base-rounding point of reference we've applied to this week, I'd say we just skipped ahead a few dates, yes?\"\n\n\"I should say so.\" I laughed, patting him lightly on the head.\n\n\"Then I think it's only fair to warn you...Tomorrow night? Your last night in Spain?\" he said, his eyes blazing through the twilight.\n\n\"Yes?\" I whispered.\n\n\"I'm gonna try to steal home.\"\n\nI smiled. \"Silly Simon, it's not stealing if I wave you in,\" I purred, kissing him solidly on the lips.\n\nLater that night, as I lay wrapped thickly in Simon, LC began to prepare. And Brain and Backbone began to chant...O...O...O. Wang? Well, we knew where he was, pressed rather closely against Backbone.\n\nHeart continued to float above, but was circling ever closer to home. However, an additional entity began to assert herself once again, trying to influence the others. She tinted my dreams with her quiet whispering.\n\nHello, Nerves.\n\nMy sleep was most decidedly...flaily.\nChapter Eighteen\n\n\"DID YOU ALWAYS KNOW you wanted to take pictures for a living?\"\n\n\"What? Where did that come from?\" Simon laughed, sitting back in his chair and looking at me over the rim of his coffee cup.\n\nWe were enjoying a lazy breakfast on my last day in Spain. Dark coffee, tiny little lemon cakes, freshly cut berries and cream, and a side of sunny coastline. Clad in Simon's shirt and a smile, I was in heaven. Nerves seemed very far away this morning.\n\n\"I mean it,\" I insisted. \"Did you always want to do this? You seem, well, you're very intense when you're working. You seem like you really love it.\"\n\n\"I do love it. I mean, it's a job so it has its tedious moments, but yeah, I love it. It wasn't something I always planned, though. In fact, there was a different plan altogether,\" he replied, a dark look passing over his face.\n\n\"What does that mean?\"\n\n\"For a long time I planned on following my father into his business.\" He sighed, a rueful smile slipping into place.\n\nMy hand was in his before I even realized I'd offered it. He squeezed, and then took another sip of his coffee.\n\n\"Did you know Benjamin worked for my father?\" he asked. \"Dad hired him right out of school, mentored him, taught him everything. When Benjamin wanted to go out on his own, you'd think Dad would've been pissed, but he was so proud of him.\"\n\n\"He's the best.\" I grinned.\n\n\"Don't think I don't know about the crush you girls have on him. I'm aware.\" He gave me a stern look.\n\n\"I'd hope so. We're not exactly subtle in our admiration.\"\n\n\"Parker Financial Services was getting big, really big, and Dad wanted me to come onboard as soon as I was done with college. I honestly never thought I'd leave Philadelphia. It would have been a great life: working with my dad, country club, big house in the 'burbs. Who wouldn't want that?\"\n\n\"Well...\" I murmured. It was an idyllic life, for sure, but I couldn't picture Simon there.\n\n\"I worked on our high school newspaper, taking pictures. I took the class as an easy A. You know, good for my transcript? But even though I got assignments like covering the women's field hockey tryouts, I really liked it. Like, really liked it. I just figured it would always be a nice hobby. Never really thought about it as a career. My parents supported me, though, and my mom even got me a camera for Christmas that year\u2014the year that...well...\" He paused, clearing his throat a bit.\n\n\"Anyway, after everything happened with Mom and Dad, Benjamin came out to Philadelphia for the, um, for the funeral. He stayed for a while, got things in order, you know. He was the executor of my parents' will. And since he was living out on the West Coast, well, the idea of staying behind in Philadelphia didn't sound so great. So, long story short, Stanford accepted me, I started studying photojournalism, I got really lucky with some internships, and then right-place-right-time, and bam! That's how I got into this gig,\" he finished, dunking his cake and taking a bite.\n\n\"And you love it.\" I smiled.\n\n\"And I love it,\" he agreed.\n\n\"So what happened to your dad's company? Parker Financial?\" I asked, spooning up a bite of berries.\n\n\"Benjamin took over some of the clients for a while, and over time he quietly closed up shop. The assets were transferred to me, per the will, and he manages it for me.\"\n\n\"Assets?\"\n\n\"Yep. Didn't I tell you that, Caroline? I'm loaded.\" He winced, looking out to sea.\n\n\"I knew there was a reason I was hanging out with you.\" I topped off his coffee.\n\n\"Seriously. Loaded.\"\n\n\"Okay, now you're just being an ass,\" I said, trying to lift the tension that had settled over the table.\n\n\"Well, people get weird about money. You never know,\" he said.\n\n\"When we get home you're buying our building and installing a hot tub on the landing, that's all,\" I joked, which earned me a small smile.\n\nWe sat and looked at each other, deep in our own thoughts. He'd done so much alone. No wonder he always seemed a little lost to me. Living out of a suitcase, not allowing himself to be tethered to anyone, no real sense of belonging\u2014could it really be that simple? Wallbanger had haremed because he couldn't stand to lose anyone else? Paging Dr. Freud...\n\nFreudian or no, it made sense. He was attracted to me, had been attracted to me since the beginning. But what was different this time? Clearly he'd been attracted to all the other women as well. Wow, no pressure at all...With a toss of my head, I tried to change the subject.\n\n\"I can't believe I'm leaving tomorrow. I feel like we just got here.\" I leaned forward on my elbows. He smiled, likely noticing my not-so-subtle way of changing the subject. But he seemed grateful.\n\n\"So stay. Stay with me. We can spend a few more days here, and then who knows? Where else do you want to go?\"\n\n\"Pfft. You'll recall that I'm leaving before you because it's the only flight I could get. Besides, I have to be back at work, organized, and in the right time zone on Monday. You know how many jobs Jillian has lined up for me?\"\n\n\"She'll understand. She's a sucker for a good romance. Come on. Stay with me. I'll stash you in the overhead bin for the flight home.\" His eyes twinkled over his coffee mug.\n\n\"Overhead bin, my foot. And is this what this is? A romance? Shouldn't you be embracing me on the beach? And ripping my bodice?\" I placed my bare legs in his lap, and he took full advantage of this, massaging between his warm hands.\n\n\"Lucky for you, I'm a bodice-ripper from way back. I could probably even throw together a pirate costume, if that's what you're into,\" he replied, the sapphires beginning to smoke.\n\n\"It has been quite a romantic tale, hasn't it? If someone would've told me this story, I doubt I'd have believed it,\" I mused, groaning as I finished my last bite.\n\n\"Why not? It's not that strange how we met, is it?\"\n\n\"How many women do you know who would voluntarily go to Europe with a man who'd been banging the plaster right off her walls for weeks?\"\n\n\"True, but you could also spin me as the guy who played you all those great records through the wall, and the guy who gave you, and I quote, 'the best meatball ever'?\"\n\n\"I suppose you did begin to wear me down with the Glen Miller. That got me.\" I sunk into my chair as his hands did delicious things to the bottoms of my socked feet. Socks I had also appropriated from his side of the room.\n\n\"I got you, huh?\" He smirked, leaning closer.\n\n\"Oh, shut it, you.\" I pushed his face away, smiling big as I contemplated what he said. Did he have me? Yeah. He totally had me. And would have me, sometime later that night.\n\nAt that thought, a whoosh of nerves hit my tummy, and I felt my smile falter a bit. Nerves had set up shop big time, and no matter where Brain went, eventually Nerves invaded every thought, every idea I had about where the night would go. I was ready, Lord knows I was ready, but I was damn nervous. O would come back, right? I knew she would. Did I mention I was nervous?\n\n\"So, are you almost done with your work? Do you still have a lot to do tomorrow?\" I asked, changing the subject once again. As was always the case when he talked about his work, Simon's eyes lit up. He described the shots he still needed of the Roman-style aqueduct in town.\n\n\"I wish we had time to go scuba diving. I hate that we ran out of time.\" I frowned.\n\n\"Again, something that would be solved if you stayed here with me.\" He frowned back, making a big deal of mimicking my eyebrows.\n\n\"Again, some of us have nine-to-five jobs. I have to get home!\"\n\n\"Home, right. You know there's gonna be a firing squad to face when we get home. Everyone is going to want to know what happened here between us,\" he said seriously.\n\n\"I know. We'll handle it.\" I cringed at the grilling I'd receive from the girls, to say nothing of Jillian. I wonder if a kitchen blowjob was what she had in mind when she said take care of him in Spain.\n\n\"We?\"\n\n\"What? We what?\" I asked.\n\n\"I could we with you.\" He smiled.\n\n\"Aren't we already we-ing?\"\n\n\"Yeah, we're we-ing on vacation. It's quite a different thing to be we-ing back home, in the real world. I travel all the time, and that takes its toll on the we unit,\" he said, his brow knit together.\n\nIt took all my power, all of it, not to make a joke about the we(e) unit.\n\n\"Simon, chill. I know you travel. I'm well aware. Keep bringing me pretty things from faraway places, and this girl has no problem with your we, okay?\" I patted his hand.\n\n\"Pretty things I can do. Guaranteed.\"\n\n\"Speaking of, where are you off to next?\"\n\n\"I'll be home for a few weeks, and then I'm headed down south for a bit.\"\n\n\"Down south? As in LA?\"\n\n\"No, a bit more south.\"\n\n\"San Diego?\"\n\n\"Souther.\"\n\n\"Stanford educated, right? Where are you going?\"\n\n\"Promise you won't be mad?\"\n\n\"Spit it out, Simon.\"\n\n\"Peru. The Andes. More specifically, Machu Picchu.\"\n\n\"What? Oh, man, that's it. I officially hate you. I'll be in San Francisco, planning rich people's Christmas trees, and you get to go there?\"\n\n\"I'll send you a postcard?\" He looked like a kid trying to get out of trouble. \"Besides, I don't know what you're so pissy about. You love your job, Caroline. Don't even try to tell me you don't.\"\n\n\"Yeah, I love my job, but right now I wish I was headed south.\" I huffed, snatching my feet away.\n\n\"Well, if you want to head south, I can think of something\u2014\"\n\nI placed my hand in front of his mouth. \"No way, buddy. I'm not machuuing your pichu now. Huh-uh,\" I stated firmly, not wavering one bit when he began pressing open mouth kisses against my palm. Not one little bit...\n\n\"Caroline,\" he whispered against my hand.\n\n\"Yes?\"\n\n\"One day,\" he began, removing my hand and leaving tiny kisses up the inside of my arm. \"One day...\" Kiss. \"I promise...\" Kiss kiss. \"To bring you...\" Kiss. \"And my woo...\" Kiss kiss. \"To Peru,\" he finished, now kneeling in front of me and dragging his mouth across my shoulder, peeling the fabric away to linger along my collarbone, his lips making me hot and shivery.\n\n\"You wanna woo me in Peru?\" I asked, my voice high and stupid and not fooling him for a second. He knew exactly how he was affecting me.\n\n\"True.\" His fingers tangled in my hair and brought my mouth to his. I tried for a second to come up with something that rhymed with true, but I gave up and kissed him back with all I had. And so, I let him make out with me on the terrace, overlooking the ocean. Which was...blue. Ahem.\n\nAll week long, we'd been seeing signs of a festival coming together around town. It started tonight, as if celebrating my departure, and we were headed out to dinner, to somewhere considerably more fancy than the places we'd been eating all week. I'd discovered Simon and I were very similar in many of our tastes. I was all for getting dressed up from time to time, but I much preferred smaller, casual places, as did he. So tonight, getting dressed up and going out someplace a little fancy, and then maybe hitting the festival, had a special feel to it. I was definitely looking forward to this evening, in more ways than one.\n\nThey say when a soldier loses a leg in battle, sometimes, late at night, he can still feel twinges of that leg\u2014phantom pain, they call it. I lost my O in battle, the battle of Cory Weinstein\u2014that machine-gun fucker\u2014and I was still feeling the aftershocks. And by aftershocks I mean nothing at all. But there was an end in sight. I'd been feeling twinges of the phantom O all week long, and I was very much looking forward to her return later this evening. The Return of the O. Of course I would see it as a title of some kind of action film in my head\u2014but truly, if she was returning, I would capitalize anything. Any Thing.\n\nBecause tonight, sports fans, I was gonna get me some. Not to put too fine a point on it, I was ready for some serious Simon Wang.\n\nI ran my fingers through my hair once more, noticing how the strong sun had brought out the natural honey tones. I smoothed the front of my dress, white linen with a little swing to the skirt. I paired it with some turquoise jewelry I'd bought in town and little snakeskin sandals. I was the most dressed up I'd been all week, and\u2014undercurrent of nerves aside\u2014feeling pretty good. I took one last look at myself in the mirror, noticing that my cheeks were pretty pink, and I hadn't even added blush tonight.\n\nI went to the kitchen to pour myself a quick glass of wine and wait for Simon. As I poured the Cava, I saw him on the terrace, facing the ocean. I smirked when I saw he was wearing a white linen shirt. We'd be quite matchy-matchy tonight. Khakis completed his look, and he turned just as I was walking out to meet him. My heels clicked across the stone as I sipped my bubbly wine, and he leaned back on his arms across the wrought iron railing. As a photographer, he was innately aware of the kind of imagery he was creating, I felt certain. Anytime he leaned, he oozed sex. I just hoped I didn't fall in my heels...sex ooze could be slippery.\n\nI offered my wine to him, and he let me bring the glass to his lips. Slowly, he sipped, his eyes on mine. When I removed the glass, he quickly wrapped an arm around my waist and pulled me to him, kissing me deeply, the taste of wine heavy on his tongue.\n\n\"You look...good,\" he breathed, pulling away from my lips to press his mouth against the skin just below my ear, his scruff tickling me in the most fantastic way.\n\n\"Good?\" I asked, tilting my head back to encourage everything he was doing.\n\n\"Good. Good enough to eat,\" he whispered, grazing my neck with his teeth, just enough to make me aware of them.\n\n\"Wow,\" was all I could manage as I wrapped my arms around his neck and sank into his embrace.\n\nThe sun was beginning to set, casting a warm glow all around, making the terra cotta blaze red and orange, coating us in fire. My eyes were drawn to the cool blue of the sea crashing against the rocks below, the salt in the air actually present on my tongue. I clung to him, letting myself feel and experience everything. His body, hard and warm against my own, the feel of his shaggy hair against my cheek, the heat of the railing against my hip, the rush of every cell in my body curling toward this man and the pleasure he would surely bring me.\n\n\"You ready?\" he asked, his voice gruff in my ear.\n\n\"So ready,\" I moaned, my eyes rolling back in my head at the nearness of him, the feel of him.\n\nAnd then Simon took me to town.\n\nAfter Simon had driven me to the brink with his kissing on the terrace, he'd literally driven me to the brink. We were now at a restaurant overlooking the water, which was easy to do in a coastal town. But where the little hole-in-the-wall places we'd been frequenting this week had their cozy charm, this was a romantic restaurant with an emphasis on romance. Romance was served on a platter here. It was in the wine, the pictures on the walls, the floor beneath our feet, and in case you missed the romance, it was also being piped in through the air. If I squinted, I could see the word romance floating through the air on the sea breeze...I had to really squint, but it was there, I tell you.\n\nFloor-to-ceiling window panels had been rolled back to let in the briny coastal air, and hundreds of tiny tealights sparkled in hurricane glasses. Each table was dressed in white, with low tumblers spilling over with dahlia blooms in rich shades of crimson, pomegranate, and lusty fuchsia. Tiny white Christmas lights twisted into the wooden beams overhead cast a magical sepia tone over the entire scene. In this restaurant, there were no children, no tables of four or six. No, this restaurant was filled with lovers, old and new.\n\nNow we sat, pressed closely together at an epic mahogany bar, slowly sipping wine and awaiting our own tiny table. Simon's hand settled against the small of my back, claiming me quietly and succinctly.\n\nThe bartender placed a tray of oysters on the bar in front of us. Twisted and craggy, they glistened, with slices of lemon nestled here and there. Simon raised an eyebrow, and I nodded as he squeezed the lemon, his strong and elegant fingers making short, erotic work of the oysters. He pried one from its home and brought it to my mouth on a tiny fork.\n\n\"Open up, Nightie Girl,\" he instructed, and I surely did as I was told.\n\nCold, crisp, like a burst of seawater in my mouth, I moaned around the fork as he slipped the tines back out. He grasped his own oyster and tossed it back like a man, licking his lips as I watched this little bit of food pornography play out. He winked at me as I looked away, trying not to let on how desperately turned on I was. The entire day had been like one giant, controlled ball of sexual tension, a slow burn that was now igniting into a wildfire. He slurped two more in quick succession, and as I watched his tongue dart out to lick his lips, I felt the sudden urge to help him. With no shame or sense of social propriety, I closed the distance between us and kissed him, hard.\n\nHe grinned in surprise, but kissed me back with equal intensity. The sweetness and tenderness that had been marinating between us all week now quickly deteriorated into full-on touch-me-touch-me-now, and I was all for it. My entire body turned toward him, my legs nestling in between his as his fingers found my skin\u2014the skin just above the hem of my dress. We were kissing, kissing all-out Hollywood style. Slow, sloppy, wet, and wonderful. My head tilted so I could kiss him more deeply, my tongue sliding against his, leading and then letting him lead. He tasted like sweet and salt and lemons, and it was all I could do not to grab him by his pretty linen shirt and have my way with him on top of the bar\u2014but in a very ladylike way, mind you.\n\nI heard someone clearing their throat, and I opened my eyes to see my sexy sapphires, then an embarrassed host.\n\n\"Excuse me, se\u00f1or, your table is ready?\" he asked, carefully averting his eyes from our display in his very romantic, but still very public, restaurant.\n\nI might have moaned a little as Simon removed his hands from my legs and turned my chair so I could stand. Taking my hands and pulling me, he smirked as I wobbled on my feet a bit. He grinned at the bartender.\n\n\"Oysters, man, oysters.\" Simon laughed a little as we shuffled off to our table. I was ready to let out an indignant huff until I saw him discreetly adjust himself. I was not the only one feeling the slow burn...\n\nI stuffed my huff and smiled serenely, lowering my eyes just enough so he knew I knew. As we arrived at our table, Simon pulled out my chair for me. As he scooted me in, I let my hand drift back just enough to accidentally-on-purpose graze him, feeling how worked up he was. I heard him hiss, and I smiled inwardly. Just as I went in for graze number two, he grasped my hand tightly in his own, pressing himself against me. My breath caught in my throat as I felt him harden further under our hands.\n\n\"Do I need to change your name to Naughty Girl?\" he murmured, low and thick in my ear. I closed my eyes and tried to get control as he sat across from me, grinning in a devilish way. As our waiter busied himself around us, straightening the linens and presenting menus, I only had eyes for Simon, cocksure and beautiful, across the table from me. This meal was going to take forever.\n\nThe meal did take forever, but as much as I was aching to get Simon alone again, I also never wanted this night to end. We were served a beautiful paella, coastal style with chunks of prawns and spiny lobster, chorizo, and peas. Made in the traditional way, almost impossible to recreate, the simple shallow dish it had been cooked in allowed the saffron rice on the bottom to become crunchy and nutty\u2014delicious in every sense of the word. We'd finished a lovely bottle of ros\u00e9 and were now lazily sipping tiny glasses of Ponche Caballero, a Spanish brandy with hints of orange and cinnamon.\n\nThe liquor was spicy as I rolled it around in my mouth. I was pleasantly warm and more pleasantly tipsy. Not drunk, just heady enough that I was hyperaware of my surroundings and found anything and everything sensual: the way the smooth brandy slipped down my throat, the feel of Simon's leg against my own under the table, the way my body had begun to hum. The entire population, it seemed, was out and about tonight and in a celebratory mood for the festival kicking off in the center of town. The energy was raw and a little wild. I sat back in my chair, teasing Simon with my big toe, a silly smile on my face as he stared at me hard.\n\n\"I ate your paella once,\" he said suddenly.\n\n\"Pardon me?\" I sputtered, catching the drop of brandy on my lip before it rolled off onto my dress.\n\n\"In Tahoe, remember? You made us all paella.\"\n\n\"Right, right, I did. Not like we had tonight, but it was pretty good.\" I smiled, thinking of that night. \"As I recall, we polished off quite a bit of wine as well.\"\n\n\"Yes, we ate paella and drank wine, got the others together, and then you kissed me.\"\n\n\"We did, and yes, I did.\" I blushed.\n\n\"And then I acted like an ass,\" he replied, his blush present now as well.\n\n\"You did,\" I agreed with a smile.\n\n\"You know why, right? I mean, you have to know that I, well, that I wanted you. You do know that, right?\"\n\n\"It was pressed against my leg, Simon. I was aware.\" I laughed, trying to play it off, but still thinking of how I'd felt when I ran away from him in that hot tub.\n\n\"Caroline, come on now,\" he chided, his eyes serious.\n\n\"Come on now, yourself. It really was pressed against my leg.\" I laughed again, a little weaker this time.\n\n\"That night? Jesus, it would have been so easy, you know? At that moment even I wasn't totally sure why I stopped us. I think I just knew that...\"\n\n\"You knew that?\" I prompted.\n\n\"I knew with you, it would be an all or nothing kind of thing.\"\n\n\"All?\" I squeaked.\n\n\"All, Caroline. I need all of you. That night? Would have been great, but too soon.\" He leaned across the table and took my hand. \"Now, we're here,\" he said, raising my hand to his mouth. He laid kisses across the back then opened my palm and pressed a wet kiss at its center. \"Where I can take my time with you,\" he said, kissing my hand once more as I stared back at him.\n\n\"Simon?\"\n\n\"Yes?\"\n\n\"I'm really glad we waited.\"\n\n\"Me too.\"\n\n\"But I really don't think I can wait any longer.\"\n\n\"Thank God.\" He smiled and signaled the waiter.\n\nWe laughed like teenagers as we paid the bill and began our trek up the hill to the car. The festival was in full force now, and we passed through part of it on our way back. Lanterns lit up the sky overhead as a heavy drum beat pulsed, and we saw people dancing in the streets. That energy was back, that sense of abandon in the air, and the brandy and that very energy knocked Nerves back down, way down to my gut, where LC and Wang threatened to beat her within an inch of her life. LC and Wang, it sounded like a rap duo...\n\nAs we reached the car, I went to grab the door handle when I was whirled suddenly by a very intense Mr. Parker. His eyes burned into mine as he pressed me against the car, his hips strong and his hands frantic in my hair and on my skin. His hand slid down my leg, grasping my thigh and hitching it around his hip as I moaned and groaned at the strength I was about to let run wild across my body and soul.\n\nBut I slowed him down, my hands pulling at his hair, making him moan in turn. \"Take me home, Simon,\" I whispered, pressing one more kiss against his sweet lips. \"And please drive fast.\"\n\nEven Heart seemed pleased, floating around above. She was still singing, but a song that was infinitely more dirty. \nChapter Nineteen\n\nI LOOKED AT MY REFLECTION in the mirror, trying to look objectively. When I was a kid, especially in those charming early-teen years, I used to see myself very differently. I saw dishwater-blond hair and pale, uninteresting skin. I saw flat green eyes and knobby knees that bisected skinny, bird-like legs. I saw a slightly upturned nose and a bottom lip that looked like I might trip over it if I wasn't too careful.\n\nWhen I was fifteen, one afternoon my grandmother told me she thought the pink dress I was wearing looked nice against my skin. I scoffed and immediately disagreed with her. \"Thanks, Grandma, but I got about three hours of sleep last night, and the last thing I look today is nice. Tired and pale, but not nice.\"\n\nI rolled my eyes in that way teenage girls do, and she reached for my hand.\n\n\"Always take a compliment, Caroline. Always take it for the way it was intended. You girls are always so quick to twist what others say. Simply say thank you and move on.\" She smiled in that quiet and wise way she had.\n\n\"Thanks.\" I smiled back, busying myself with the spaghetti sauce and turning my face so she couldn't see my blush.\n\n\"It breaks my heart the way young girls pick themselves over, never thinking they're good enough. You make sure you always remember, you're exactly the way you're supposed to be. Exactly. And anyone who says otherwise, well, poppycock.\" She giggled, her voice lowering a bit at that last word, the closest she would ever come to swearing. Grandma had a list of bad words and really bad words, and poppycock came close to approaching the latter.\n\nThe next day at school I mentioned to a friend that I thought her hair looked great, and her answer was to run her hands through it with disgust.\n\n\"Are you kidding? I barely even had time to wash it today.\"\n\nEven though it did look fantastic.\n\nLater on after gym class, I was changing in the locker room when I observed another friend touching up her lip gloss. \"That's pretty. What's the name of that color?\" I asked as she pursed her lips in the mirror.\n\n\"Apple Tartlet, but it looks terrible on me. God, I have no tan left over from summer!\"\n\nGrandma was right. Girls really didn't take compliments well. Now, I'm not gonna lie and say after that day I magically had no more bad hair days or never picked the wrong lipstick again. But I did make a conscious effort to see the good before the bad and really look at myself in a more clear way. Objectively. Kindly. And as my body continued to change, I became more and more aware of features I could look at positively instead of negatively. I never thought of myself as lethally gorgeous, but I did clean up well.\n\nAnd so now, as I stared into the mirror in the bathroom, knowing Simon was waiting for me, I took the time to take a little inventory.\n\nThe dishwater-blond hair? Not so much dishwater. It was shiny and golden, a little wavy and curly from the saltwater it had been cooking in all week. The pale skin? Nicely browned up and, dare I say, a little glowy? I winked at myself, holding back a maniacal giggle. My mouth had that slightly pouty lower lip, just full enough to trap me some Simon and not let him go. And the legs I saw peeking from below the lace just covering my thighs? Well, not so bird-like anymore. In fact, I think they were going to look pretty spectacular wrapping around Simon's...whatever I felt like wrapping them around.\n\nAnd so, as I smoothed my hair once more and mentally ran through all my internal checklists, I was wildly excited about the night ahead. We'd raced back to the house, practically disrobed each other in the entryway, and after begging a few moments of girl time, I was now ready to go out and claim my Simon. Because who was kidding who? I wanted this man. Wanted him for my own, and did not, would not, share him with anyone else.\n\nBrain for once was finally in agreement with LC. Especially since she'd crawled up Backbone and slapped Brain right in the stem, telling her in that special way only she could that we needed this. We deserved this, and we were ready. Nerves, well, they continued to circle in my tummy, but that was to be expected, right? I mean, it had been a long, long time, and a little bit of nerves was normal, I expect. Had I been stalling all week? Maybe.\n\nKind of.\n\nA little.\n\nSimon had been more than patient, content to take things slow, at my pace, but for crying out loud, he was only human.\n\nI was adamant that Nerves not be allowed to turn another Spanish night into the land of cuddle and coo. I turned in the mirror, trying to see myself as Simon might see me. I smiled in what I thought was a seductive way, flipped off the light, took one more deep breath, and opened the door.\n\nThe bedroom had been transformed into something from a fairy tale. Candles flickered on the dresser and nightstands, bathing the room in a warm glow. The windows were open, as well as the door to the little balcony overlooking the sea, and I could hear the waves crashing, romance-novel style. And there he stood: hair tousled, body strong, eyes blazing.\n\nI watched as he took me in, dragging his gaze down my body and back, a smile spreading across his face as he appraised my outfit of choice.\n\n\"Mmm, there's my Pink Nightie Girl,\" he sighed, holding out his hand. And when I stalled for just the tiniest second, Backbone picked up my hand and gave it to him.\n\nWe stood in the darkened room, a few feet apart but connected by our woven fingers. I could feel the rough texture of his thumb as he traced circles on the inside of my hand, the same circles he'd traced weeks and weeks before when I began to fall under his spell. Our eyes full of each other, he took a deep breath.\n\n\"It's criminal how good you look in that,\" he said, drawing me toward him and giving me a little spin so he could better see the pink baby doll nightie. As he spun me, the lacey edges flipped up just a little, showing off the accompanying ruffled panties. A low noise sounded in his throat, and if I wasn't mistaken, it was a growl? Damn...\n\nHe spun me back closer, grasping my hips and pressing me against him, my breasts crushing into his chest. He placed a tiny kiss below my ear, letting me feel just the tip of his tongue.\n\n\"So there are some things I need you to understand,\" he murmured, nuzzling with his nose, his hands brushing up under my nightie to fluff my ruffles and grab a handful of backside, catching me by surprise. I gasped.\n\n\"You listening? Don't get distracted on me now,\" he whispered again, flattening out his tongue and dragging it up the side of my neck.\n\n\"It's kind of hard to focus with your distraction poking me in the thigh,\" I groaned, letting him bend me backward just enough so that my entire lower body was pressed against him, his hard places perfectly content to mold my soft places around them. He chuckled against my neck, now dotting my collarbone with his trademarked baby kisses.\n\n\"Here's what you need to know. One, you're amazing,\" he said, his hands now traveling up to the small of my back, fingers and thumbs massaging and manipulating. \"Two, you're amazingly sexy,\" he breathed.\n\nMy hands now hurriedly unbuttoned his shirt, pushing it back off his shoulders as our pace began to transition from slow and easy to fast and frantic. Now his hands were sneaking around front, his nails lightly scraping along my tummy, lifting my nightie so we were skin to skin, nothing left between us. I ran my hands up and down his back, my nails much more aggressive, digging in and anchoring him against me.\n\n\"And three, as amazingly sexy as this pink nightie is, the only thing I want to see for the rest of this night is my Sweet Caroline, and I need to see you.\" He panted in my ear as he picked me up, straight up, and my right leg went around his waist on its own.\n\nOnce again, the Universal Law of Wallbanger dictated that legs went around hips when they were offered.\n\nHe walked me backward to the bed and set me down gently. Leaning over, he pushed me backward on to my elbows. Shirt hanging down off his shoulders, he winked at me, nodding at his state of undress. I reached forward, crooked one finger behind the button on his khakis, and snapped it open. Seeing no peek of boxers, I gently nudged his zipper down just an inch or so, exposing the happy trail that led down, down, down to where all good things were found. Sweet mother of pearl. Commando.\n\n\"You got something against underpants?\" I whispered, raising one knee and forcing him between my hips. Forcing. Right.\n\n\"I'm against your underpants, and isn't it a shame they're still there?\" He smirked, pushing his hips into me, letting me feel everything.\n\nI dropped my head back, silently pushing down Nerves when she threatened to bubble up just a smidge. Piss off, Nerves. This was happening.\n\n\"No shame. I have a feeling they won't be on for long.\" I sighed, laying back to stretch my arms over my head, lengthening my body against his and encouraging his lips to further dance along the hollow at the base of my collarbone. I could feel him licking and sucking between my breasts. I arched into him, anxious to feel more. I needed more. He began peeling the straps of my nightie down, baring me and allowing him the access he needed to make me orbit the planet.\n\nFeeling his mouth on me, on my breasts, hot and wet, tickling and sloppy, was unreal. So I told him so.\n\n\"That feels unreal,\" I moaned in to the top of his head as the scruff from his light beard roughed my skin pleasantly. His lips closed around my right nipple, and my hips went off on a tangent of their own, bucking wildly beneath him, both of my legs now wrapped firmly around his waist. Lips and tongue and teeth now lavished across my cleavage, which spilled out over the edge of the nightie as he alternated between breasts, loving them equally. I was surrounded by Simon, and even his scent was turning me on, equal parts peppery spice and thick Spanish brandy.\n\nNonsensical words poured from my mouth. I was aware of a few \"Simons,\" and one or two, \"Yes, that's good,\" but mostly what I overheard from myself were things like \"Mmph,\" and \"Erghh,\" and a rather loud \"Hyyyyaeahhh,\" for which, frankly, there is not a correct spelling.\n\nSimon sighed over and over again in to my skin, his actual breath a turn on as I felt it wash over me. My hands had been left free to roam in the wonderland that was his hair, and as I swept it back from his face I was rewarded with the amazing sight of his mouth on me, his eyes closed in clear worship. He bit down lightly, closing his teeth around my sensitive skin, and my hands almost tore the hair from his head. It felt phenomenal.\n\nHis other hand was running up and down my leg, encouraging me to grasp him tighter between my thighs as his wondrous fingers began to come ever closer to the edge of the lace. It was the last boundary we had yet to cross: the lace frontier.\n\nI felt my breathing still as he went on final approach, his fingers brushing just under the edge of my panties, barely brushing. His breathing slowed as well, and as he continued to touch me gently, his face came back up to mine, and we had this moment, this quiet moment, where we just...stared. Awe\u2014it's the only way I can describe the feeling of his hand ghosting over me, delicately, reverently. Our eyes locked as he eased his hand further underneath the lace and then, with achingly perfect precision, he touched me.\n\nMy eyes fluttered shut, my entire body awash with so many sensations. My breathing started back up again, the intense pressure that had been circling all around and inside and out was now like a low-level hum, just beneath the surface of my skin. I moved with him, feeling his fingers begin to explore me, and I let out the tiniest moan. It was all I could let out. The feelings were so intense and the energy\u2014oh my goodness, the energy that surrounded us in that moment.\n\nI was sure Simon was unaware of the entirety of the emotion that flew around behind my closed eyelids. The poor man was just finally getting a little touch. But as his fingers became more deft and sure of themselves, something incredible began to happen. That teeny tiny little bundle of nerves, which had been dormant for centuries, began to spark to life. My eyes flew open as a very specific warmth began to move through me, starting at the center of my being and working its way out.\n\nSimon was most certainly enjoying this. His eyes were hazy and crowded with lust as I writhed underneath him. I knew he could feel me tense and come alive.\n\n\"God, Caroline, you're so...you're beautiful,\" he murmured, his eyes now crowding with something a bit more than lust, and I felt tiny pinpricks behind my eyeballs.\n\nI threw my arms around his neck and held him close, tearing at his shirt to get it off, get it off him so I could feel everything. He lifted himself from me for only seconds, ripping off his shirt in an exaggerated way that made me giggle but yearn for him even more.\n\nLowering himself back on to me, he slipped further down, his lips tracing a path down to my belly button. Circling it with his tongue, he laughed into my tummy.\n\n\"What are you laughing at, mister?\" I giggled, squeezing his ear. He was below the nightie now, his face hidden from me. Poking his head back out, he let loose a slow grin that made my toes point.\n\n\"If your belly button tastes this good\u2014fuck, Caroline. I can't wait to taste your pussy.\"\n\nThere are certain things a woman needs to hear at different times in her life:\n\nYou got the job.\n\nYour ass looks great in that skirt.\n\nI would love to meet your mother.\n\nAnd when used in the just the right context, in just the right setting, sometimes, a woman needs to hear the P-word.\n\nThis could be better than Clooney.\n\nThe moan that came out of my mouth when he said that word, well, let's just say it was loud enough to wake the dead. He let his tongue trace a path from my belly button down to the edge of my ruffles, and then with loving precision, he hooked his thumbs underneath the lace and dragged them down my legs.\n\nThere I was, spread out on top of PillowTown with a pink nightie bunched up around my midriff, all pertinent parts on display, and damn happy about it. He pulled my hips just to the edge of the bed and dropped to his knees. Sweet Jesus.\n\nAs he ran his hands up and down the tops of my legs, I lifted up on my elbows so I could watch, needing to see this wonderful man tending to me, taking care of me. Kneeling between my thighs, with his khakis unbuckled and halfway unzipped, hair at atomic heights, he was stunning. And on the move.\n\nOnce again letting his tongue lead, he planted open-mouth kisses along the insides of my thighs, one side and then the other, with each pass getting closer and closer to where I needed him most. Carefully lifting my left leg, he hitched it over his shoulder as I arched my back, my entire body now aching to feel him.\n\nHe gazed at me for a moment longer, maybe even just a few seconds, but it felt like a lifetime. \"Beautiful,\" he breathed one more time, and then he pressed his mouth to me.\n\nNo quick licks, no tiny kisses, just incredible pressure as he surrounded me with his lips. It was enough to make me drop back on the bed, unable to support myself any longer. The feel, the exquisite feel of him was all-consuming, and I could barely breathe. He worked me slow and low, bringing one hand up to open me further to him, letting his mouth and fingers and perfect tongue gently and methodically coax me into the stratosphere, rising up, filling me with the sense of awe and amazement I had been missing for so long.\n\nI allowed one hand to drift down to him and tangle in his hair, running my fingers through it with as much feeling as I could. The other hand? Useless. It was fisting the sheets into some kind of ball.\n\nHe lifted his head from me once, just once, to press another kiss against my thigh. \"Perfect. Jesus, just perfect,\" he whispered, so quietly I could barely hear him over my own sighs and whimpers. He returned to me almost immediately, an urgency now to his movements, his lips and tongue twisting and pressing as he groaned into me, the vibration riding straight through.\n\nI opened my eyes for a second, just a second, and the room was glowing, almost incandescent. All of my senses came alive, and I could hear the crashing of the surf, see the candlelight flickering on our bodies. I could feel my skin break into gooseflesh, the very air caressing me and announcing what I had been missing for months, years even.\n\nThis man could very possibly love me. And he was about to bring back the O.\n\nSnapping my eyes closed again, I could almost see myself, standing at the edge of a cliff, staring down into the raging ocean below. Pressure, enormous pressure was building behind me, nudging me toward the edge where I could fall, fall freely into what was waiting for me. I took one step, then another, closer and closer as I could feel Simon grasping my hips. But wait. If the O was coming for me, I wanted Simon inside. I needed him inside me.\n\nTugging on his shoulders, I pulled him up my body, feet kicking at his khakis until they lay defenseless on the floor.\n\n\"Simon, I need, please, inside, now,\" I panted, almost incoherent with lust. Simon, schooled in Caroline shorthand, understood this completely and was poised between my legs, hips nuzzled up into mine within seconds. He leaned down, kissing me wantonly, the taste of me all over him. And I loved it.\n\n\"Inside, inside, inside,\" I kept chanting, my back and hips alternately arching, desperately trying to find what I needed, what I had to have, to push me off that cliff. He left me for only seconds to fumble in his khakis, which I had kicked halfway across the room. The telltale crinkle let me know that I was safe, that we were safe.\n\nFinally I felt him, exactly where he was meant to be. He barely nudged inside, but just the feeling of him entering me was monumental. My own needs quieted for the moment, and I watched as he began to push into me for the first time. His eyes bore into mine as I cradled his face in my hands. He looked as though he wanted to say something. What words would we speak, what wonderfully loving things would we say to commemorate this moment?\n\n\"Hi,\" he whispered, smiling as though his life depended on it.\n\nI couldn't help but smile back. \"Hi,\" I answered, loving the feel of him, the weight of him, above me.\n\nHe slipped gently into me, and at first my body resisted. It had been a long time, but the little pain I felt was welcome. It was that good kind of pain, a pain that let you know something more was coming. I relaxed a bit, allowing my legs to wrap around his waist, and as he pressed farther into me, his smile became infinitely more sexy. He bit down on his lower lip and tiny frown lines appeared on his forehead. I breathed in, inhaling his scent as I watched him pull back just the smallest bit, only to thrust once more. Now fully inside, I welcomed him the only way I could. I gave him that little internal hug, which made his eyes flash open and peer down at me.\n\n\"There's my girl,\" he murmured, raising one rakish eyebrow and thrusting into me again, with more conviction this time. My breath caught in my throat and I gasped, unwittingly rocking my hips into his with a motion as old as the waves crashing down below.\n\nSlowly he began to move within me, sliding against me with a fantastic pressure, each new angle and sensation giving way to more of that warm tingly feeling working its way out to the tips of each finger and toe. The feeling of having Simon inside me, inside my body, was more than I can articulate. I groaned, and he grunted. He moaned, and I mewed. Together. His hips pushed me higher on to the bed, up toward the headboard. Our bodies were slick with sweat, crashing and smashing into each other. I threaded my hands deeply into his hair, tugging and writhing beneath him.\n\n\"Caroline, so beautiful,\" he sighed between kisses across my forehead and nose.\n\nI closed my eyes and could see myself, once again, on the edge of that cliff, ready to jump, needing to jump. Again, that pressure began to build, that crackle of energy spinning itself wild and frantic, pulsing with every thrust, every slip and dip of his hips into mine, driving him, unrelentingly, in and out of my body.\n\nI took one final step, one foot now dangling off the edge of the cliff, and then! I saw her...O. She was in the water down below, her hair like fire dancing along the waves. She waved and I waved and just like that, Simon brought one hand down between our bodies, just above where we were joined, and he began to trace his little circles.\n\nLittle circles from a perfect hand, and I jumped. I jumped free and clear and loud and proud, announcing my approval with a lusty \"Yes!\" as I rushed toward that certain high.\n\nAnd I fell.\n\nAnd fell.\n\nAnd fell.\n\nAnd crashed. Crashed and smacked into the unforgiving surface of the water, and I didn't come up. I fell for what seemed like an age, but instead of O meeting me at the bottom with open arms, I floundered, alone and wet. Every muscle in my body, every cell was concentrated on the return of the O, as if I could will her back. I strained, body tight and taut as I caught sight of her, just the very tips of her hair, like fire under the water, slipping away from me. She was so close, so very close, but no. No.\n\nI scrambled after her, trying with sheer will to make her reappear, but nothing. She was gone, and I was left underwater. With the most beautiful man in the world inside me.\n\nI opened my eyes and saw Simon above me, saw his beautiful face as he made love to me, and that is what this was. This wasn't sex. This was love, and I still couldn't offer him all that I had. I saw his eyes heavy and thick and half closed in passion. I saw a bead of sweat running down his nose and watched as it splashed lazily on to my breasts. I saw as he bit down hard on his lower lip, the strain on his face as he delayed his own well-deserved climax.\n\nHe was everything I had hoped he would be. He was a generous lover, and I could feel my heart beat to within bursting out of my chest to be nearer to him, to love him. He was everything.\n\nI lifted his hand from between us and kissed his fingertips, then wrapped my legs tighter around his waist and anchored my hands on his back. He was waiting for me. Of course he was. I adored him. I closed my eyes once more, steeling myself for all I was able to give him.\n\n\"Simon, it's so good,\" I panted, and I meant every word of it. I bucked my hips. I clenched in all the right places, and I called his name, over and over again.\n\n\"Caroline, look at me, please,\" he begged, his voice rife with pleasure. I allowed my eyes to open again, feeling one tear spill down my cheek. A strange look stole over his features for only a second as his eyes searched mine, and then? He came. No thunder, no lightning, no fanfare. But it was stunning.\n\nHe collapsed onto me, and I took his weight. I took it all as I cradled him to my chest and kissed him over and over again, my hands soothing his back, my legs hugging him as tightly as I could. I whispered his name as he nuzzled into the space between my neck and my breast, simple touches and caresses.\n\nHeart sat to the side and quietly sighed. Nerves? You motherfucker. Don't even think about showing your face here.\n\nWe lay for a while, listening to the ocean in our own little haven, this romantic fairy tale that could have, should have been enough. When his breathing returned to normal, he lifted his head and kissed me very softly.\n\n\"Sweet Caroline,\" he smiled, and I smiled back, my heart full.\n\nSex could be amazing, even without the O.\n\n\"I'll be right back,\" he said disentangling from me and walking to the bathroom, naked backside a sight to behold. I watched him retreat, and then sat up quickly, pulling the straps of my nightie back up around my shoulders. I rolled on to my side, away from the bathroom, and curled around my pillow. This had been the single best sexual experience of my life. Every i had been dotted, every t had been crossed. And yet, I was still no-go for O. What the hell was wrong with me?\n\nI will not cry.\n\nI will not cry.\n\nI will not cry.\n\nEven though he'd only been gone from the bed a few minutes, when he came back, I panicked and pretended to be asleep. Childish? Yep. Totally childish.\n\nI felt the bed dip as he climbed back in, and then his warm and still very naked body was up against me, spooning. Arms wrapped around my middle, and then his mouth was at my ear, whispering.\n\n\"Mmm, Nightie Girl back in her nightie.\"\n\nI waited, not speaking, just breathing. I felt him shake me a little bit and let out a little chuckle.\n\n\"Hey, hey you, are you sleeping?\"\n\nShould I snore? Whenever people faked sleep on sitcoms, they snored. I let out a tiny one. He kissed my neck, my traitor skin pebbling in the wake of his mouth. I sighed in my \"sleep,\" snuggling closer to Simon, hoping he would let me pull this off. The fates were kind tonight, as he simply hugged me tighter to his chest and kissed me once more.\n\n\"'Night, Caroline,\" he whispered, and the night settled around us. I fake snored for a few more minutes until his actual snoring took over, and then I sighed heavily.\n\nConfused and numb, I was awake until dawn.\nChapter Twenty\n\nI HAD FAKED IT.\n\nFaked it with Simon. There must have been a rule written somewhere, maybe even chiseled into a stone tablet: Thou Shall Not Fake It With Wallbanger. So let it be written, so let it be done. I faked it, and now I was doomed to wander the planet forever, O-less.\n\nWas I being overly dramatic? Oh my, yes. But if this didn't call for a little drama, what did?\n\nThat next morning, I was up and out of bed before Simon was even awake, something I hadn't done the entire time we were on our trip together. Usually we stayed in bed until the other one was awake, and then lounged for a while, laughing and talking. And kissing.\n\nMmm, the kissing.\n\nBut this morning I ran quickly through the shower and was in the kitchen making breakfast when a sleepy Simon came in. Shuffling across the floor in his socks, with boxers low on his hips, he grinned through his sleep haze and burrowed into my side as I sliced melon and berries.\n\n\"What are you doing out here? I was a little lonesome. Big bed, no Caroline. Where'd you go?\" he asked, planting a quick kiss on my shoulder.\n\n\"I needed to get moving this morning. Remember the car is coming for me at ten? I wanted to make you some breakfast before I left.\" I smiled, turning to give him a quick kiss.\n\nHe stopped me from turning away and kissed me more thoroughly, not letting me hurry through anything. I could feel myself closing off, and I was almost unable to stop it. I needed some time to process this, to understand how I was feeling\u2014other than miserable. But I adored Simon, and he didn't deserve this. So I let myself fall into the kiss, be swept away by this man once more. I kissed him back feverishly, passionately, and then pulled away just before it could become something more than a kiss.\n\n\"Fruit?\"\n\n\"Huh?\"\n\n\"Fruit. I made fruit salad. Want some?\"\n\n\"Oh, yeah. Yeah. Sounds good. Coffee made?\"\n\n\"Water is boiling. French press is all ready to go.\" I patted him on the cheek as I waved him toward the pot. We coexisted in the kitchen, talking quietly, and Simon stole a kiss or two here and there. I tried not to show how messed up my brain was, tried to act as normal as I could. Simon seemed to sense something was up, but he took his cue from me, let me lead this morning.\n\nWe sat outside on the terrace one last time, eating our breakfast together and watching the breakers roll in.\n\n\"Are you glad you came?\" he asked.\n\nI bit down on my lip at the obvious. \"I'm so glad. This trip was amazing.\" I smiled, reaching across the table for his hand and giving it a squeeze.\n\n\"And now?\"\n\n\"And now what? Back to reality. What time does your flight get in tomorrow?\" I asked.\n\n\"Late. Really late. Should I call you or...\" He left off, seeming to ask me if he should come over.\n\n\"Call me when you get in, no matter what time, okay?\" I replied, sipping my coffee and watching the ocean. He was quiet now, and this time when I bit down on my lip it was to keep from crying.\n\nI had packed early, so when the driver got here, I was ready to go. Simon had tried to tempt me to join him in the shower, but I begged off, making an excuse about finding my passport. I was panicking and pulling away just when we'd been getting so close, but this had really thrown me for a loop.\n\nI had put all my Os in one basket, and the problem wasn't Simon. It was me. The sex had been unreal, amazing, perfection even with a condom on, and yet still, no.\n\nSimon walked my bags out to the car and placed them in the trunk. After speaking to the driver for a moment, he came back to me as I walked through the house one last time. It truly had been a fairy tale, and I had enjoyed every moment.\n\n\"Time to go?\" I asked, leaning back against him when he approached me at the terrace railing. I was glad for the feel of him against me.\n\n\"Time to go. You have everything you need?\"\n\n\"I think so. I wish I could figure out a way to get some of those prawns home, though.\" I laughed, and he snorted into my hair.\n\n\"I think we can find something at home that will be suitable. Maybe we can have the others over next weekend and recreate some of the stuff we ate here?\"\n\nI turned to face him. \"Make our debut?\" I grinned.\n\n\"Yeah, sure. I mean, if you want to,\" he added sheepishly, looking at me carefully.\n\n\"I do,\" I answered. And I did. Even without the stupid, blessed O, I wanted to be with Simon.\n\n\"Okay, debut over prawns. That sounds weird.\"\n\nI laughed as he hugged me to him. The driver honked, and we shuffled toward the car.\n\n\"I'll call you when I'm back, okay?\" he said.\n\n\"I'll be there. Get some good work done,\" I instructed.\n\nHe brushed my hair back from my face and leaned in to kiss me once more.\n\n\"Bye, Caroline.\"\n\n\"Bye, Simon.\" I got in the car. And drove away from the fairy tale.\n\nOnce I was ensconced in my first-class seat, I had nothing but hours to contemplate. Strike that. I had nothing but hours to sit and stew and grumble. I'd cried in the car on the way to the airport, trying all the while to assure the driver I was fine and not stone-cold crazy. I cried because, well, there was sure as shit a lot of tension in my body, and it had to come out some way. And so it did, through my eyeballs. I was sad, and I was frustrated. Now I was done crying.\n\nI tried to read. I'd stocked up on trashy magazines in the airport in Malaga. As I paged through them, titles of articles jumped out at me:\n\n\"How to Know If You're Having the Best Orgasm You Can Have\"\n\n\"Kegel Your Way to Multiples\"\n\n\"New Weight Loss Plan: Orgasm Your Way to a Thinner You!\"\n\nLower Caroline, Brain, Backbone, Heart were all lined up and throwing stones at Nerves, who was trying her best to hide.\n\nI slammed down all my new magazines, throwing them into the seatback in front of me. I grabbed my laptop, powered it up, and put in my earbuds. I'd loaded some movies on before the last flight. I could let my brain escape into a film. Yes, I could do that. I scrolled through some of the movies I had on file...When Harry Met Sally? Nope, not with that scene in the deli. Top Gun? Nope, that scene where they do it, and it's all lit blue with the breeze blowing through the gauzy curtains? No, too close to my fairy tale.\n\nI found a movie I could safely watch, took three Tylenol PM, and was asleep before Luke learned how to use his lightsaber.\n\nSomewhere between the connection at LaGuardia and the flight across the US, I downshifted from sad to mad. I'd caught up on my sleep, was done with the crying bullshit, and now I was good and mad. And on a plane where pacing was discouraged. I had to stay in my seat and try to rationalize what to do with this anger\u2014and how I was going to live my entire life with no hope of an O. And again, overly dramatic? Perhaps, but with no O in sight, it's easy to have tunnel vision.\n\nFinally, we touched down at SFO and as I followed the crowd to baggage claim, physically and emotionally exhausted, I looked up into the face of someone I never wanted to see again.\n\nCory Weinstein. That machine-gun fucker.\n\nPlastered across the newsstand was his stupid face in a giant ad campaign for Slice o' Love Pizza Parlors. I stood in front of his giant head, which wore the biggest shit-eating grin as he posed with a giant pepperoni slice, and my anger bubbled over. It now had a face. My anger had a face, and it was a stupid face. I wanted to punch it in the face, but it was only a picture.\n\nUnfortunately, that didn't stop me.\n\nNot a smart thing to do, have a fit in an international airport. Turns out they frown on that. So after a strongly worded warning from TSA, and a promise that I would never attack a poster again, I packed myself into a cab, stinking of airplane, and went back to my apartment. I kicked my own door this time, and as I threw my bags down, I saw the only two things that could make me smile.\n\nClive and my KitchenAid.\n\nWith a strongly worded meow, he came running to me, actually jumping into my arms and showing the affection he reserved for moments exactly like these. Somehow his little cat brain knew I needed it, and he lavished attention on me as only he could. Shaking his tail and purring incessantly, he butted his head up under my chin and wrapped his big paws around my neck, giving me a tiny kitty hug. Laughing into his fur, I held him close. It was good to be home.\n\n\"Did Uncle Euan and Uncle Antonio take good care of you? Huh? Who's my good boy?\" I cooed, dropping him to the floor and grabbing a can of tuna, his treat for behaving while I was gone. Turning now from Clive, who had focused solely on his bowl, my eyes laser-locked on my KitchenAid. I was going to shower, and then I was going to bake. I needed to bake.\n\nAn unknown amount of time later\u2014although I will say the sun had set and risen while I floured and stirred\u2014I heard knocking at my door. I'd been baking so long I felt my back creak and squeak as I lifted my head from slicing some of Ina's Outrageous Brownies. They took a few extra steps, but oh boy, were they worth the trouble. What the hell time was it? I looked around for Clive and didn't see him.\n\nI shuffled to the door, noticing there was sugar all over the floor, brown and white, and I was performing an accidental soft-shoe dance. There was another knock at the door, more insistent this time.\n\n\"Coming!\" I shouted, rolling my eyes at the irony. As I raised my hand to open the door, I noticed melted chocolate all over my knuckles. Not one to waste, I gave them a heavenly lick as I opened the door.\n\nThere stood Simon, looking exhausted.\n\n\"What are you doing here? You're not supposed to be home until\u2014\"\n\n\"Not supposed to be home until late tonight, I know. I took an earlier flight.\" He pushed past me into my apartment.\n\nAs I closed the door and turned to face him, I smoothed out my apron a bit, feeling bits of cookie dough clinging to it. \"You took an earlier flight. Why?\" I asked, soft-shoeing across the floor to him.\n\nHe looked around with an amused grin, noting the piles and piles of cookies, the assorted pies on the windowsills, the aluminum-wrapped bricks of zucchini bread, pumpkin bread, and cranberry orange bread, stacked like the foundation of a house all along the dining table. He grinned once more, then turned to me, picking a raisin off my forehead that I didn't even know was stuck there.\n\n\"Are you gonna tell me why you faked it?\"\nChapter Twenty-One\n\nDUMBSTRUCK, I STOOD with my mouth hanging open as he walked farther into the room to contemplate the baked goods. He shuffled through the sugar and paused to swipe a finger through a bowl lined with melted chocolate. I sighed heavily as I returned to the counter to face him and the music as I removed a ball of dough from another bowl where it was rising.\n\nHow did he know? How could he tell? I flipped and kneaded the dough\u2014a fluffy, clingy brioche\u2014feeling my face flame. I thought I'd played it pretty well. I chanced a look up at him as he licked the chocolate from his finger, his eyes growing more concerned as my thoughtful kneading turned into punching. I took my frustration out on the brioche dough as I pondered an O-less life. Dammit.\n\nHis finger now clean, he brushed a lock of hair behind my ear as I continued to punch\/knead and flip. I winced when he touched me, the glorious image of him perched on top of me impossible to ignore.\n\n\"We gonna talk about this?\" he asked quietly, dipping his nose to my neck. I leaned into his body for a scant second, then caught myself.\n\n\"What is there to talk about? I don't even know what you're talking about. Are you delirious from the time change?\" I said cheerily, avoiding his eyes as I wondered if I could pull this off. Could I convince him he was crazy one? Goddamnit, how did he know?\n\n\"Nightie Girl, come on. Talk to me,\" he prodded, nuzzling into my neck. \"If we're gonna do this, we need to talk to each other.\"\n\nTalk? Sure, I could talk. He should probably know what he was in for with me, doomed to wander the planet without an O for the rest of my life. I picked up the dough one more time and threw it against the wall. It dripped and rolled down, sticky like those creepy crawly things I used to play with as a kid. I whirled to face him, my face still red but beyond caring now.\n\n\"What was that going to be?\" he asked calmly, nodding to the dough.\n\n\"Brioche. It was going to be brioche,\" I answered quickly, my tone frantic.\n\n\"I bet it would have been good.\"\n\n\"It's a lot of work\u2014almost too much.\"\n\n\"We could try it again. I'd be glad to help.\"\n\n\"You don't know what you're offering. Do you have any idea how complicated it is? How many steps there are? How long it might take?\"\n\n\"Good things come to those who wait.\"\n\n\"Christ, Simon, you have no idea. I want this so badly, probably even more than you.\"\n\n\"They make croutons out of it, right?\"\n\n\"Wait, what? What the hell are you talking about?\"\n\n\"Brioche. It's like, some kind of bread, isn't it? Hey, quit banging your head against the counter.\"\n\nThe granite felt cool against my defeated, hot skin, but I banged with less force when I heard the edge of panic in his voice.\n\nHe knew, and he was still here. He was here in my kitchen in that blue North Face pullover that made his eyes smoky sapphires and his entire body look cuddly and warm and sexy and virile and kick-me-the-in-head gorgeous. And here I was, covered in honey and raisins, banging my head on the countertop after killing my brioche.\n\nKilling my brioche. What a great name for a\u2014focus, Caroline!\n\nHeart had damn near leapt out of my chest when she saw him at the door. LC was close behind, involuntarily clenching at the sight of him. Brain had shut down in shock and denial for a moment, but was now analyzing the situation and leaning toward pronouncing him a worthy candidate, noting the time and distance he'd committed to discovering the cause of concern. Backbone straightened now, knowing innately that proper posture created a better-looking rack\u2014could you blame her? Nerves...fluttered.\n\nWhy. Why. He wants to know why. I examined him between bangs...ahem...and saw he was getting concerned. As was I\u2014my head was really starting to hurt. I was tired, overwhelmed, and underorgasmed. And a touch slaphappy?\n\nAfter one last bang, I straightened up, then listed a little left. I caught my balance, drew in a breath, and let fly.\n\n\"You want to know why?\"\n\n\"I'd like to. Are you done banging?\"\n\n\"God bless it, no more banging. Okay, why. Why? Here goes...\" I paced in a tight circle, dodging the chocolate chips and pecans that had congregated close to the counter on the floor. I spied Clive in the corner, batting a few walnuts back and forth between his paws. Nuts all over the floor, nuts in my head. Fitting. \"Know anything about pizza parlors, Simon?\"\n\nTo his great credit, he listened. He listened as I went on and on, circling the kitchen island as I ranted and raged. I could barely make sense of it myself: \"Weinstein...one night...machine gun...It went away!...night off...Jordan Catalano...Not even Clooney!...hiatus...Oprah...lonely...single...Not even Clooney!...Jason Bourne...almost Clooney...Pink nightie...banging...\"\n\nAfter a while he looked as dizzy I was beginning to feel. But I was determined to get it all out. He tried to grab me on one pass around, but I dodged his hands, almost slipping in a patch of crushed pecans, which I had crushed further in my circling. I had worn a path through the clutter.\n\nI made one last pass, this time muttering, \"Spanish fairy tale with prawns,\" when I tripped over a muffin tin and fell into his arms.\n\nHe held me close, breathing me in, kissing my forehead. \"Caroline, babe, you gotta tell me what's going on. The mumbling? It's cute and all, but we're not really getting anywhere.\" He pressed his hands into the small of my back, holding me in place. I pulled away a little, resisting his embrace, and looked him straight in the eyes.\n\n\"How did you know?\" I asked.\n\n\"Come on, sometimes guys know.\"\n\n\"No, really. How did you know?\" I asked again.\n\nHe kissed my nose gently. \"Because all of a sudden, you weren't my Caroline.\"\n\n\"I faked it because I haven't had an orgasm in one thousand years,\" I stated matter of factly.\n\n\"Come again?\"\n\n\"I'm going across the hall to kick your door now,\" I sighed, pulling away and shuffling through the sugar.\n\n\"Wait, wait, wait, you what? You haven't had a what?\" He grabbed for my hand as I turned back to him, with everything out in the open now.\n\n\"An orgasm, Simon. An orgasm. The Big O, the climax, the happy ending. No orgasms. Not for this Nightie Girl. Cory Weinstein can give me a five-percent discount whenever I want one, but in return, he took my O.\" I sniffled, tears now coming to my eyes. \"So you can go back to your harem. I'll be entering the convent soon enough!\" I cried, the dam finally breaking.\n\n\"Convent? What? Come here, please. Get your dramatic ass over here.\" He pulled me unwillingly back to the kitchen and wrapped me in his arms. He rocked me back and forth as I let out ridiculous sobs and wails.\n\n\"You're so...so...great...and I can't...I can't...you're so good...in...bed...and everywhere else...and I can't...I can't...God...you're so hot...when you came...so hot...and you came home...and I killed my brioche...and I...I...I think...I love you.\"\n\nAll stop. Breathe. What did I just say?\n\n\"Caroline, hey, stop crying, you gorgeous girl. Mind running that last part by me again?\"\n\nI'd just told Simon I loved him. While my snot soaked into his North Face. I breathed in his scent, then peeled myself off of him and headed to the wall to peel off the dough stuck there. Nerves sprang to life, for once working for us. Could I cover? Could I rally?\n\n\"Which part?\" I asked the wall\u2014and Clive, who had stopped playing with his nuts to listen in.\n\n\"That last part,\" I heard him say, his voice strong and clear.\n\n\"I killed my brioche?\" I hedged.\n\n\"You really think that's the part I'm asking about?\"\n\n\"Um, no?\"\n\n\"Try again.\"\n\n\"I don't wanna.\"\n\n\"Caroline\u2014wait, what's your middle name?\"\n\n\"Elizabeth.\"\n\n\"Caroline Elizabeth,\" he warned, in a deep voice that unexpectedly made me giggle.\n\n\"Brioche is really good, when it's not flavored with wall,\" I blurted, my exhaustion mixing with my confession for an odd buzz. I actually felt a little relieved.\n\n\"Turn around, please,\" he asked, and so I did. He leaned against the counter, unzipping his snotty North Face. \"I'm a bit jetlagged, so a quick recap, if I could. One, you seem to have lost your orgasm, yes?\"\n\n\"Yes,\" I mumbled, watching as he took off his fleece, throwing it over the back of one of my chairs.\n\n\"Two, brioche is really hard to make, yes?\"\n\n\"Yes,\" I breathed, not able to take my gaze away from him. Underneath the North Face was a white button-down. Which was good enough on its own, but couple that with the way he was slowly and methodically rolling up the sleeves? It was mesmerizing.\n\n\"And three, you think you love me?\" he asked, his voice deep and thick, like molasses and honey and all things afghan\u2014blanket, not country.\n\n\"Yes,\" I whispered, knowing it was one hundred percent the truth. I loved Simon. Big, giant dur.\n\n\"You think, or you know?\"\n\n\"I know.\"\n\n\"Well, now. That's something to consider, isn't it?\" he replied, his eyes dancing as he drew near. \"You really have no idea, do you?\" He spread his hands along my collarbone, brushing his thumbs across the very tops of my breasts.\n\nMy breathing quickened, my body sparking to life in spite of myself. \"No idea about what?\" I murmured, allowing him to press me against the wall.\n\n\"How thoroughly you own me, Nightie Girl,\" he said, leaning in to whisper this part in my ear. \"And I know I love you enough to want you to have your happy ending.\"\n\nAnd then he kissed me\u2014Heart was in heaven\u2014kissed me like it was a fairy tale, even though in this fairy tale I had dough sticking to my back and a cat with a pawful of nuts. But that didn't stop me from kissing him back as though my life depended on it.\n\n\"Did you know I started falling for you the night you banged on my door?\" he asked, kissing my neck. \"And that I as soon as I started to get to know you, I wasn't with anyone else?\"\n\nI gasped. \"But I thought, I mean, I saw you with\u2014\"\n\n\"I know what you thought, but it's true. How could I be with anyone else when I was falling in love with you?\"\n\nHe loved me! But wait, what's this? He was backing away...where was he going?\n\n\"And now, I'm going to do something I never thought I would do.\" He sighed mournfully, looking at the stacks of bread on the table. With a deep breath and a grimace, in one fell swoop he knocked them all to the floor. Bread rained down in foil-covered bricks around us, and I can't be sure, but I think I heard a tiny whine escape as he watched them hit the floor. But then he turned to me, eyes dark and dangerous. He grabbed me and swung me up on the table before him, nudging my legs apart to stand between them.\n\n\"Do you have any idea how much fun we're going to have?\" he asked, slipping his hands inside my apron, warm and a little rough on my tummy.\n\n\"What are you up to?\"\n\n\"An O has been lost, and I'm a sucker for a challenge.\" He grinned, pulling me to the edge of the table and snugly in to him. With his hands behind my knees, he wrapped my legs around his waist, kissing me again, lips and tongue hot and persistent.\n\n\"It's not going to be easy. She's pretty lost,\" I protested between kisses, worrying his buttons open and exposing his Spanish suntan.\n\n\"I'm done with easy.\"\n\n\"You should print that on cards.\"\n\n\"Print this\u2014why do you still have clothes on?\"\n\nHe laid me back across the table as I grinned up at him. My foot hit the flour sifter and sent it crashing to the floor, dusting us thoroughly in the process. Simon's hair looked like a biscuit, powdery and puffy. I coughed and a plume of flour came out, making Simon laugh out loud. The laughing stopped when I reached down for him, finding him hard, yet still covered in denim. He groaned, my favorite sound in the world.\n\n\"Fuck, Caroline, I love your hands on me,\" he said through his teeth, dipping his mouth to my neck and leaving a trail of white-hot kisses across my skin. His tongue swept out across me, underneath the edge of my apron. Hands quickly found the bottom of my tank top, and it went sailing across the room, into the kitchen sink. Within seconds, a pair of shorts found themselves swimming alongside, quickly followed by a pair of jeans and a white button-down.\n\nThe apron? Well, we were having a little trouble with that one.\n\n\"Are you a sailor? Who tied this knot, Popeye?\" he seethed, struggling to get it undone. In his struggles, he managed to knock over a bowl of orange marmalade glaze, which now dripped down the table and on to the floor. My contribution was to flip over a carton of raisins while I craned my neck trying to see the knot behind me.\n\n\"Oh, screw the apron, Simon. Look here,\" I insisted, snapping the front of my bra and tossing it to the floor. I pulled down the top of the apron, arranging and propping up my cleavage. Pie eyed, he looked at my now-naked breasts and went in for the kill. I was pushed roughly back on to the table once more, his insistent mouth now dragging down my neck, attacking my skin like it had done something personal to him and he was exacting his revenge. And a lustful revenge it was.\n\nDipping a finger into the marmalade puddle, he traced a path from one breast to the other, circling and pressing the sticky into my skin. Bending his head, he tasted one, then the other, both of us groaning at the same time.\n\n\"Mmm, you taste good.\"\n\n\"I'm glad I wasn't making hot wings. This could be a different story\u2014wow, that's nice.\" I sighed as he responded to my smart-assery with an actual bite.\n\n\"These would be extra spicy.\"\n\nHe laughed as I rolled my eyes.\n\n\"Want me to get some celery to cool you down?\" I asked.\n\n\"No one's cooling down in this apartment, not anytime soon,\" he promised, grabbing the jar of honey from the nearby counter and pulling aside my apron. Without missing a beat, he got my panties all wet. And not in the way you think, although there was that...\n\nAs I watched, he poured the honey all over me, covering my panties and making me squeal. He stood back to admire. \"Look at that, those are ruined. They're going to have to come off,\" he said as he came close again. I stopped him with a marmalade foot.\n\n\"You first, Mr. Man,\" I instructed, nodding at his flour-covered boxers. He raised an eyebrow, and dropped the boxers. Standing naked in my wreck of a kitchen, he was insanely cute.\n\nIn that instant, Heart, Brain, Backbone, and LC lined up on one side of the playground. They beckoned for Nerves, waving her over like a game of Red Rover. I looked at Simon, naked and floury and perfect, and I sighed with a giant smile. Nerves finally, blessedly, scampered over, and we were finally all on the same page.\n\n\"I fucking love you, Simon.\"\n\n\"I love you too, Nightie Girl. Now lose the panties and gimme some sugar.\"\n\n\"Come and get it,\" I laughed, sitting up and sliding my panties down my honey-dripped legs. I threw them at him, and they hit his chest with a loud thwack, the honey dripping everywhere.\n\n\"We're going to need one helluva shower after all this,\" I remarked as he wrapped me in his sticky arms.\n\n\"That'll be round two.\" He smiled, picking me up and carrying me to the bedroom, my body aligned with his, only the apron between us. And that wasn't going to keep us apart for long.\n\nDid I need an O? I mean, was it necessary for life? Being near Simon, being so close to him, wrapped up in his arms and feeling him move inside me, was it enough?\n\nFor now, it was. I loved him, you see...\n\nHe dropped me on the bed, and I bounced a little, rolling sideways and making the headboard bang a bit.\n\n\"You gonna bang my walls, Simon?\" I laughed.\n\n\"You have no idea,\" he promised, and scrunched my apron out of the way as I sighed and threw my arms over my head. I lazed backward, with a giant smile on my face. His fingers walked down over my tummy, my hips, my thighs, finally reaching me. After a gentle nudge, I let my legs fall open. He licked his lips and sank to his knees.\n\nHe touched and tasted me as he had in Spain, but it was different. It still felt amazing, but I was different. I was relaxed. Twisting and turning his fingers, he found that spot, the one that made my back arch and my moans grow deep. He groaned into me, causing me to arch off the bed again, his lips and tongue finding me once more, deliberate. My hands sought my breasts, and as he watched, I teased my nipples, bringing them taut once more.\n\nAgain, I had the distinct honor of feeling his mouth, his wonderful mouth, on me. I seized up, my entire body tensing at the sizzle of energy that ran through me, and then I relaxed once more. I started to feel, really feel everything going on inside at that moment. Love. I felt love. And I felt loved...\n\nHere in the daytime, where nothing could be hidden, everything was on display\u2014and covered in messy stuff\u2014I was being loved by this man. No fairy tale, no waves crashing, no flickering candles. Real life. A real life fairy tale where I was being loved by this man. And I mean looooved by this man.\n\nTongue. Lips. Fingers. Hands. All of it dedicated to me and my pleasure. A girl could get used to this.\n\nI could feel the sweet tension begin to build, but this time my body received it differently. My body, perfectly in tune for once, was ready, and in my mind, behind closed eyes, I saw myself begin to approach that cliff. In my head, I grinned, because I knew this time I was gonna catch that bitch. And then? Really amazing things began to happen down below. Long gorgeous fingers pressed inside me, twisting, and curving, and finding that secret spot. Lips and tongue encircled that other spot, sucking and licking, pressing and pulsing. Tiny pricks of light began to dance behind my eyelids, intense and wild.\n\n\"Oh, God...Simon...that's so...good...don't...stop...don't...stop...\"\n\nI groaned loud, louder, and then louder still, unable to contain the sounds I was making. It was so good, so good, so very, very good, so close, so close...\n\nAnd then the screaming began. And it was not my own.\n\nOut of the corner of my eye, I became aware of some kind of furry missile racing across the floor.\n\nLike some kind of pussified dive bomb, Clive ran at Simon, leaped, and dug into his back, attacking him from behind.\n\nSimon ran from the bedroom into the hallway, then back in again, Clive still latched on like some kind of rabid coonskin back cap that would not shake off. He had his arms\u2014does a cat have arms?\u2014wrapped around Simon's neck in a way that under other circumstances would have seemed like an adorable cat hug. But right now, he meant business.\n\nI ran after them, naked except for my apron, trying to get Simon to slow down, but with those ten claws digging deeper in, he continued running from room to room.\n\nThe irony that Simon was literally trying to run away from pussy was not lost on me.\n\nIf I could have watched from outside, rather than being involved, I would have peed myself. As it was, I was having a hard time stifling myself listening to Simon's screams. I really must love him.\n\nFinally, I backed them both into a corner, turned Simon around, resisted the urge to squeeze buns, and pried Clive loose. I quickly headed out to the living room and deposited him on the sofa with a thunk, patting him on the head once as a thank you for the defense, unwarranted as it was. Clive responded with a prideful meow and began licking his whiskers.\n\nI went back into the kitchen to find Simon, still huddled against the wall. I appraised him, his eyes wild as he leaned against the wall, wincing at his back. My gaze was drawn lower. Unbelievable.\n\nHe\n\nWas\n\nStill\n\nHard.\n\nHe saw my eyes travel down his body, reminiscent of the first time we met face to face. He nodded sheepishly.\n\n\"You're still hard,\" I blurted, breathing heavily as I tried once more to untie my apron.\n\n\"Yeah.\"\n\n\"That's amazing.\"\n\n\"You're amazing.\"\n\n\"Ah, fuck,\" I huffed, giving up on the knot.\n\n\"Yes, please.\"\n\nI paused for a split second, then whirled the apron around to my back in one swift movement. I leaped across the room, my apron flying behind me like a low-rider cape and crashed into him, driving him up against the wall as I assaulted him. He caught me as I wrapped around him like a feisty blanket, kissing him furiously. My nails raked down his chest, and he gasped.\n\n\"Your back okay?\" I asked between kisses.\n\n\"I'll live. Your cat, however...\"\n\n\"He's protective. He thought you were hurting Mommy.\"\n\n\"Was I?\"\n\n\"Oh no, quite the opposite.\"\n\n\"Really?\"\n\n\"Hell yes,\" I cried, sliding against him, manipulating my body against his, honey and sugar slick and gritty between us.\n\nI dragged myself down his body, stopping to kiss the very tip of him. I pulled him down onto the floor with me and flipped him onto his back so quickly a puff of flour clouded the air. There, in the middle of the kitchen, naked with marmalade dotting my breasts, I straddled him. Raising up just a little bit, I caught his hands and encouraged him to grab my hips.\n\n\"You're gonna want to hold on for this,\" I whispered, and sank down onto him. We both sighed at the same time, the feeling of him inside me once more just amazing. I arched my back and flexed my hips experimentally...once...twice...a third time. It really was true what they said about riding a bike. My body remembered this just as quickly.\n\nWith my stupid apron riding bitch behind me, I began to move above Simon, feeling him move inside me, responding and rewarding, thrusting and never relenting. Driving, pushing, we moved together\u2014actually even moving across the kitchen floor a little. He sat up underneath me, moving deeper into me as I cried out. My hands were wild in his hair. It was standing straight up beneath my fingers as I took hold, anchoring myself as I closed my eyes and began.\n\nBegan that long march to the edge of the cliff.\n\nI could see the edge, high above the raging waters. As I peeked over the edge, I saw her. O. She waved at me, diving under and over the water like a sexual porpoise. Crafty little bitch.\n\nSimon was kissing my neck, licking and sucking my skin, making me insane.\n\nI stuck one foot over the edge, pointing my toes directly at her, rolling my ankle and waving little circles in the air in her direction.\n\nLittle circles.\n\nI pushed Simon back onto the floor, grabbed his hand in mine, and brought it between my legs. I rode him hard, pressing my fingers against his, my cries getting louder as we sped up our rocking, both of us, in tune and right there. Right there. Right, right, right...there...\n\n\"Caroline, Jesus, you...are...amazing...love...you...so...much...killing...me...\"\n\nAnd that's the little extra I needed.\n\nIn my head, I took one step back, then dove. Not jumped. Dove. Executed a perfect swan dive, thank you very much, straight into the water. Clean and true, I grabbed onto her and didn't let go as I slipped into the water.\n\nThe O had returned.\n\nWhite noise filled my ears as my toes and fingers got the news first. They tingled, tiny fizzles and sparks of energy spinning up and out, driving through every nerve and every cell that had been starving for this for months. These cells told other cells, communicating to their sisters that something fantastic was happening. Color exploded behind my eyelids, bursting brightly into tiny little sensory fireworks as the feeling continued to spread to every corner of my body. Pure pleasure shot through me, pulsing and slicing, filling me up as I shook and shimmied on top of Simon, who hung on through the entire thing.\n\nI don't know if he could see the choirs of dirty angels singing, but no matter. I could. And it was the definition of bliss.\n\nO came back, and she brought friends.\n\nWave after wave crashed through me as Simon and I continued to press and twist, arching into every single one of them. My head was thrown back as I continued to scream lustfully, not caring who or what could hear me in my own House of Orgasm.\n\nI opened my eyes at one point to see Simon below me, frantic and happy, smiling big as he stayed with me through it all, his strenuous effort clear across his face as the flour in his hair turned into a wonderful little paste.\n\nHe was becoming papier-mach\u00e9.\n\nStill onward I thrashed, passing through the land of multiples and into some kind of no man's land. Passing six and seven, my body became limp with ecstasy.\n\nBut O brought one more friend. She brought along G, the Holy Grail.\n\nStuttering like an idiot, I grasped hold of Simon, holding on for dear life as the biggest tidal wave of love and toe-curling heat hit me like a ton of bricks. Sensing I needed help for this one, Simon sat up, which positioned him even more uniquely. He found a spot deep inside, hidden to most, and he leaned into me, driving himself over and over again as I held my breath and hung on tight.\n\nI finally opened my eyes again, seeing light spark around the room as oxygen rushed back into my system. I babbled incomprehensibly into his chest as he rocked into me again and again, finally finding his own kind of amazing somewhere deep inside me.\n\nI held onto him, feeling the waves finally retreat, both of us shaking now. As we panted, the pleasure left and the love simply rushed in, filling me back up again. My mouth was too tired to move. He had taken my breath away. So I did the best I could, I placed his hand over my heart and kissed his sweet face. He seemed to understand, and kissed me back. I hummed with happiness. Humming didn't take as much effort.\n\nUtterly spent and exhausted, punch drunk and covered in sticky sweat, I lay back against his legs, not caring a bit how contorted and ridiculous I looked as tension tears ran down the sides of my face and into my ears. Sensing this was not the most comfortable position for me, Simon moved out from under me and helped to unbend my pretzel legs before cradling me in his arms on the kitchen floor.\n\nWe lay quietly, not speaking for a while. I noticed Clive sitting just inside the doorway to the bedroom licking his paws quietly.\n\nAll was good.\n\nWhen movement seemed possible, I tried to sit up, the room spinning a little. Simon kept one arm around me as we appraised the situation, the overturned bowls and bottles, the scattered bread, the chaos that was my kitchen. I laughed quietly and turned to him. He watched me with happy eyes.\n\n\"Should we clean this up?\" he asked.\n\n\"No, let's shower.\"\n\n\"'Kay,\" he answered, helping me up.\n\nI cracked my back like an old lady, wincing at the good hurt my body felt. I started for the bathroom, then changed direction, heading for the fridge. I grabbed a bottle of Gatorade and tossed it to him. \"You're gonna need it.\" I winked, flouncing my apron on my way to the shower. Now that the O was back, I planned to waste no time in summoning her again.\n\nAs Simon followed me to the bathroom, taking a swig of Gatorade, Clive suddenly flopped onto the floor, rolling over on his back. He seemed to be waving Simon over with his paws. Simon looked at me, and I shrugged. We both looked at Clive, who wiggled on his back, continuing to wave him over. Simon knelt right next to him, cautiously extending one hand. Winking at me\u2014I swear to Christ he did\u2014Clive wiggled a little closer. Knowing this could still be a trap, Simon cautiously reached down and rumpled the fur on his belly. Clive let him. I even heard a tentative purr.\n\nI left the two boys alone for a moment and went to turn on the shower so it could heat up. I finally got the apron knot undone and was able to abandon it on the floor. Stepping under the spray, I moaned at the feeling of the warm water hitting my still-sensitive skin.\n\n\"You coming? 'Cause I sure did,\" I called over the rush of the shower, laughing at my own joke. A moment later Simon poked back the corner of the shower curtain to watch me naked and covered in bubbles. He smiled like the devil as he climbed in. I drew in a breath at the sight of ten tiny punctures in his back, but he laughed it away.\n\n\"We're good. I think we just made friends,\" he assured, pulling me against him and joining me under the water.\n\nI sighed, relaxing. \"This is nice,\" I murmured.\n\n\"Yeah.\"\n\nThe water beat down around us. I was in the arms of my Simon, and it couldn't get any better.\n\nHe pulled back a little, a question on his face. \"Caroline?\"\n\n\"Hmm?\"\n\n\"Is any of that bread I threw on the floor...well...\"\n\n\"Yes?\"\n\n\"Is any of it zucchini?\"\n\n\"Yes, Simon, there's zucchini bread.\"\n\nSilence once again, but for the water.\n\n\"Caroline?\"\n\n\"Hmm?\"\n\n\"I didn't think I could love you more, but I really kind of do.\"\n\n\"I'm glad, Simon. Now gimme some sugar.\"\nChapter Twenty-Two\n\n4:37 p.m., that same day\n\n\"IS THAT THE SOAP? Don't slip on the soap.\"\n\n\"I won't slip on the soap.\"\n\n\"I don't want you to slip. Be careful.\"\n\n\"I won't slip on the soap. Now turn back around and be quiet.\"\n\n\"Quiet? Not possible, not when you...mmm...and then when you...ooohhh...and then when you\u2014ow, that hurt, Simon. You okay back there?\"\n\n\"I slipped on the soap.\"\n\nI started to turn around to see if he was indeed okay when he suddenly pressed me up against the shower wall, holding my hands flat against the tile. Lips tickled and water sprinkled down my skin and across my shoulders as his body flexed against mine. Thoughts of runaway soap slipped from my mind as he slipped inside me, hard and thick and delicious. My breath left me in a gasp, amplified by the tile walls, made sexy by the water falling, and quickly followed by another gasp as he proceeded to thrust into me, achingly slowly and purposefully, his hands now gripping my hips.\n\nI threw my head backward, turning my face to find the sight of Simon, naked and wet. His brow was furrowed, mouth open as he invaded completely and without apology. I spiraled fast, awareness and clear thought narrowing down to a pinpoint before exploding, wordless words falling out of my mouth and down to the water, circling the drain.\n\nNow that O was back, she didn't dally. So far, at least, she arrived promptly and without question, shattering the memory of days and weeks and months of waiting and crying, begging and pleading. She'd rewarded me with a steady, constant parade that left me scrambled and silly, boneless and ready for more.\n\nGroaning into my ear, shivering and pulsing, Simon failed to slow his roll. He knew inherently, as I knew, that his girl was good for a few more. And so, with agonizing dexterity, he planted a wet kiss on my neck, left my body, spun me quickly, and was back inside before I could say, \"Hey, where'd you go?\"\n\n\"Nowhere, Nightie Girl, not anytime soon,\" he muttered, roughly grabbing my bottom and lifting me against the wall, using his weight to crush me against the tile, holding me to him and holding me inside. His body flexed while mine flattened, our slippery skin feeling indescribable against each other. How had I stayed away from this man as long as I had? No matter. He was here, inside me, and about to deliver another O parade throughout. I pressed back against him just enough, opening the space between us just enough to gaze down, lust clouding my vision but not so much that I couldn't see him entering me, over and over again, filling me up like no man ever had.\n\nNow glancing down himself to see what had me so transfixed, he was captivated as well, and a sound rather like \"Mmph\" left his mouth. His movements sped up, chasing it down, that feeling, that tipping point that felt so close to pain and so close to perfection. Those blue eyes, now filled with lust and fire, flew back up to mine as we both threw ourselves off that cliff again together.\n\nSeizing. Freezing. Locked and unloaded. We came together with a roar and a grunt and a groan that left my throat raw and my hoohah thrilled.\n\nThrilled hoohah...what a great name for a...Mmmm...\n\n6:41 p.m.\n\nWalking around my apartment in only a towel, dodging flour piles and raisin clumps, Simon was a sight to behold. When he skidded on a patch of marmalade and bumped into the counter, I laughed so hard I had to sit down on the couch. He now stood in front of me with a slice of zucchini bread as I laughed, an amused look on his face. I continued to laugh, and my towel slipped down, revealing more than a little of my assets. At the sight of boobs, two things happened. His eyes popped, and something else popped. Popped out. I raised an eyebrow at this latest development.\n\n\"You realize you are turning me into some kind of machine?\" he noted, nodding down at his HiThere poking through the towel. Simon took the time to place his zucchini bread safely on the coffee table.\n\n\"How cute is that? It's like he's poking his head out from behind a curtain!\" I clapped my hands.\n\n\"You may not be aware, but as a general rule, no man likes the word cute in the same sentence as his junk.\"\n\n\"But he is cute\u2014uh-oh, where'd he go?\"\n\n\"He's shy now. Still not cute, but shy.\"\n\n\"Shy, my ass. He wasn't so shy in the shower a little bit ago.\"\n\n\"He needs his ego stroked.\"\n\n\"Wow.\"\n\n\"No, really. I think you'll find he is quite receptive to stroking.\"\n\n\"Now see, I was thinking maybe he just needed a good tongue lashing, but if you think stroking will suffice...\"\n\n\"No, no, I think a tongue lashing is quite in order. He\u2014Goddamn, Caroline!\"\n\nI leaned in, brought the shy one forth, and immediately surrounded him with my mouth. Feeling him grow harder still, I settled myself on the edge of the couch, wrapped my arms around him and dropped the towel. Pulling him closer, and therefore deeper into me, I hummed in satisfaction as I felt his hands come up into my hair and trace my face. Reverently, he placed his fingers on my eyelids, cheeks, temples, finally burying one hand in my hair and the other, well, wow. He held himself. As I concentrated all my attention on the tip of him, he stroked himself at the base, something that was quite possibly the sexiest thing I'd ever seen. Seeing his hand, wrapped around himself as he moved in and out of my mouth...oh my.\n\nSexy isn't the right word for it. It is inadequate in the face of the pure erotica playing out in front of me. And speaking of in front of me, I hummed again in appreciation, feeling myself getting worked up just at the play my mouth was getting. Lucky mouth.\n\nI fell back against the couch and pulled Simon with me. He responded by using both hands to brace against the back of the couch, thrusting in and out of my mouth with conviction. The angle allowed him to penetrate more deeply, and made it easier for me to take more of him in. I grabbed his backside, feeling the thrill of attending to him, knowing it was me, only me, who got to have him in this way.\n\nI could feel him getting close. I was already beginning to know his tells intimately. I wanted him again. I was selfish this way. Releasing him with a final strong pull, I pushed him down on to the couch and straddled him. Feeling me against him, he thrust upward as I sank down, and there was that moment\u2014you know that moment? When everything feels stretched and pulled in the most delicious way? Your body reacts: something that shouldn't be inside is now inside and for a split second, it's alien, unknown. And then your skin senses a returning champion, your muscle memory takes over, and then it's so good, that feeling of fullness, of wonder and awe.\n\nAnd then you begin to move.\n\nGrabbing his shoulders for leverage, I rolled my hips into his, noticing not for the first time that he'd been intelligently designed with my exact measurements in mind. He fit inside me perfectly, two halves of a whole, some kind of sexual Lego. He sensed it too, I could tell.\n\nHe placed his hand flat against my chest, directly on top of my heart. \"Stunning,\" he whispered as I rode him, sweet and hot. He kept my heart in his hand as I rocked into him, his other hand on my hip, guiding me, positioning me, feeling me attend to us both. He struggled to stay with me, to keep his eyes open as his release rushed in. I took his hand from my heart and placed it further down, where he began to trace those damnable perfect circles.\n\n\"Jesus, Simon...oh, God...so...soooo good...I...mmm...\"\n\n\"I love watching you fall apart,\" he groaned, and I did. And he did. And we did.\n\nI collapsed into him, watching until the room stopped spinning and the feeling returned to my fingers and toes, warmth snaking through my body as he held me to him.\n\n\"Tongue lashing. What an idea.\" He snorted, and I giggled.\n\n8:17 p.m.\n\n\"Ever think about changing the paint color in here?\"\n\n\"Are you serious?\"\n\n\"What? Maybe a lighter shade of green? Or even a blue? Blue might be nice. I'd love to see you surrounded by blue.\"\n\n\"Do I tell you how to take pictures?\"\n\n\"Well, no...\"\n\n\"Then don't tell me how to pick paint colors. And as it happens, I'm planning to change the palette in here, but it's going darker. Deeper, you might say.\"\n\n\"Deeper, you say? How's this?\"\n\n\"That's pretty good. Mmm, that's really good. Anyhow, as I was saying, I'm thinking of maybe a deep slate gray, with a new creamy sugar marble countertop, deepening the cupboards to a rich, dark mahogany. Holy shit, that feels good.\"\n\n\"Noted. Deeper is good, and very deep is even better. Can you put your foot on my shoulder?\"\n\n\"Like that?\"\n\n\"Christ, Caroline, yes, like that. So...new countertop, you say? Marble might be a little cold, don't you think?\"\n\n\"Yes, yes, yes! What? I mean, what? Cold? Well, since I'm not usually laid out like a jelly roll on the counter, the cold won't bother me. Besides, marble countertops are the best for rolling out dough.\"\n\n\"Don't,\" he warned, turning his face to kiss the inside of my ankle.\n\n\"Don't what, Simon?\" I purred, my breath hitching as I felt his pace begin to quicken slightly, unnoticeable to anyone but me, the one he was currently inside of.\n\n\"Don't try to distract me with dough talk. It won't work,\" he instructed, letting go of the countertop with his left hand and running it lightly over my breasts, back and forth, teasing my nipples into hard peaks with his fingertips.\n\nA frantic energy began to settle low, low in my hips and in my thighs, the pit of my stomach and points in between. \"No dough talk? No dirty dough talk for Simon? Mmm, but don't you think a little distraction is good from time to time? I mean, can't you just imagine me, bent over the countertop, working so hard for you...\" I trailed off, running my fingers through his hair, bending him to me to kiss him with a wet mouth, tongue and lips and teeth intent on bringing him deeper into me.\n\nI was perched on the edge of my kitchen island, very much naked, as was our fair Mr. Parker, buried inside and determined to make this last as long as possible. We wanted to see how long we could carry on a conversation while...well...doing it. So far seventeen of the most intense, sensual, fantastic minutes of my life, and that wasn't counting the foreplay. O was dancing in the periphery, wondering why she wasn't being granted immediate access. But now I had control of the bitch, and this sweet torture was incredible. Worth enduring.\n\nThat is, until Simon asked me to place my foot on his shoulder. Holy hell, he was wrecking me. One leg on his shoulder, the other leg he held open to one side, his hips rotating in maddeningly tiny circles, increasing in the smallest of increments. He was the one who insisted on the conversation, and I'd been able to keep up, until the foot on shoulder. Suddenly, parts that hadn't really been a part of it before were now being stimulated, and it was getting harder and harder to keep my wits about me. But really, who needed wits? I could be witless. As long as I could be under Simon, I was okay being witless.\n\nBut I could still play this game right now, while a few lingering wits remained.\n\n\"Don't test me, Naughty Girl. I will dirty talk you right off this island.\"\n\n\"Mmm, Simon, can't you just see me? Bent over, little apron with nothing underneath, rolling pin in hand, and a bowl full of apples?\"\n\n\"Apples? Oh boy, I love apples,\" he groaned, picking up my other foot and placing it on the opposite shoulder, his hands roughly pulling me even farther toward the edge, his pace picking up again just a bit.\n\n\"I know you do, with cinnamon? I could bake you a pie, Simon. Your very own apple pie, even a homemade crust...all for you, big guy. You know all you have to do is ask me...\" I smirked, trying to keep my eyes from crossing as he sped up again, the sound of skin slapping not even funny at all. There went another wit.\n\n\"How does that feel, Caroline. Good?\" he asked, surprising me.\n\n\"Good? It feels amazing.\"\n\n\"Amazing? Really?\" He pulled out almost all the way before sliding back into me all at once, making me feel every single inch.\n\nAnd the wit stands alone. \"You know, it does, but back to the apples. Would you like your pie served hot with vanilla ice cream? Warm and melty with\u2014oh my God...\"\n\n\"You really want to talk about this right now? Because if you keep this up, I'm going to be forced to get really dirty myself.\"\n\n\"Dirtier than apple pie talk?\" I asked, stretching and pointing my toes toward the ceiling, creating a new sensation.\n\n\"How about this, if you don't stop all this apple pie talk,\" he started, leaning down to place his mouth against my ear, making me shiver. One hand grasped my breast, roughly turning and tweaking my nipple. The other snuck down, feeling against me until he found the spot that made me tense and cry out. \"If you don't stop, I'm going to stop fucking you, and believe me when I say I haven't even begun to ravage you in all the ways I've dreamed about.\"\n\nHe stood back up and thrust. Hard.\n\nLast wit? Bye-bye. I ain't too proud to beg. \"God, Simon, I give. Just fuck me.\"\n\n\"Apple pie for me?\"\n\n\"Yes, yes! Apple pie for you! Oh, God...\"\n\n\"That's right, apple pie for me, apple pie for\u2014God, you're tight this way.\" He groaned, switching both of my legs to one side, holding them up as he pounded into me, again and again, never retreating, only advancing, looking down at me, watching as my back arched and my skin flushed, heat creeping as my climax broke over me, stunning me silent in its intensity as I was shaken to the very core of my being.\n\n\"I love you, Caroline, I love you, I love you, I love you,\" he chanted, thrusting erratically now as he sped toward his own release, sweat breaking over his brow as he clutched at my hips as I clutched him from the inside, holding him as long as I could, feeling his solid weight on me as he laid his head on my breast. How could his warm weight feel so good? It should have made it hard to breathe, constriction of the lungs and all that, but it didn't. Holding him, cradling his face as I swept his hair back, it felt the opposite of heavy.\n\n\"You're going to kill me, sure as I'm lying here,\" he moaned, kissing everywhere he could.\n\n\"I love you too,\" I sighed, gazing at my kitchen ceiling. I could feel a smile as big as the bay across my face. The O was going to be around for a very long time.\n\nNo way am I painting my kitchen blue.\n\n9:32 p.m.\n\n\"I can't believe this is the second time we're cleaning flour and sugar off each other. What's wrong with us?\"\n\n\"The sugar is good for exfoliation,\" I explained. \"Not sure what good the flour is doing us, though.\"\n\n\"Exfoliation?\"\n\n\"Yeah, I figure every time we sex it up out there, all that sugar helps us remove dead skin cells.\"\n\n\"Really, Caroline? Dead skin cells? That's hardly sexy.\"\n\n\"You weren't complaining earlier.\"\n\n\"Well no, how could I? You promised to bake me an apple pie. Don't forget that part.\"\n\n\"I won't forget, but I was somewhat under duress.\"\n\n\"You were under me, not under duress, under me.\"\n\n\"Yes, Simon, I was under you.\"\n\n\"Wash your back?\"\n\n\"Yes, please.\"\n\nWe lay on opposite sides of the tub, relaxing and soaking off yet another round of kitchen goo. At some point, I was going to have to clean all that mess up, but right now the only thing I could concentrate on was this man in front of me. This man, up to almost his neck in fragrant bubbles, strong arms snaking out now to bring me closer. I spun in the tub like a buoy, bobbing back and forth and arranging myself in front of him. He used a washcloth to gently remove the last of the sticky that covered me. Then he pulled me to his chest, leaning back against the edge of the tub. Arms encircled me, tucking me in, surrounding me with warm water and warmer Simon. I closed my eyes, relishing the feel of it all. The safety, the sweetness, the sexiness. I shifted, trying to get impossibly closer, and then I felt him against my bum. Growing.\n\n\"Why, hello there, friend,\" I murmured, sneaking my hand through the bubbles to find him, wanting and wanton.\n\n\"Caroline...\" he warned, laying his head back on the edge of the tub.\n\n\"What?\" I asked innocently, trailing my fingers along the sides of him, feeling him react.\n\n\"I'm not seventeen, you know.\" He chuckled, his voice growing husky and needy in spite of his words.\n\n\"Thank goodness, or I would have to answer for my actions\u2014corrupting a minor and all that,\" I whispered, slowly turning over to rub myself along the length of him, soap and bubbles and water making me slippery.\n\nHe hissed slightly and smiled. \"You're going to break me, you know this, right? I swear on all that's holy, I'm not a machine\u2014Christ, don't stop doing that.\" He groaned, thrusting into my hand without thought.\n\n\"Ah, break schmake. I just want to fuck you until you can't see straight,\" I purred, tightening my fist as he splashed water over the side a bit.\n\n\"I can barely see as it is. There seem to be three of you.\" He moaned, pulling my legs apart and positioning me above him.\n\n\"Aim for the one in the middle, Simon,\" I instructed and slid down.\n\nYeah, we had some water to clean up.\n\n11:09 p.m.\n\n\"I'm just going to get the food. I need sustenance, woman.\"\n\n\"Get it, then hurry back to me. I need you, Simon. Why are you crawling on the floor?\"\n\n\"I don't think I can actually stand at this point. The machine needs a break. The machine may very well need repairs. The machine, wait, what're you doin' there, Caroline?\"\n\n\"What, this?\"\n\n\"Yeah, yeah, it looks like you're\u2014wow, do you touch yourself like that a lot?\"\n\n\"I haven't lately, why? Looks good to you, yes?\"\n\n\"Yes, that's...wow...um...that's the door...the guy with the Thai is here. I...and I...Thai...I...\"\n\n\"Are you really rhyming right now, Simon? Mmm, that feels nice...\"\n\n\"Hello! Hello, anyone there? Someone called in an order for\u2014dude, how am I supposed to give you your change?\"\n\n\"Keep the change.\"\n\n\"Dude, you shoved a fifty under the door. You know that's like a thirty-dollar tip, right?\"\n\n\"Keep the change. Leave the Thai. Caroline, get on that bed.\"\n\n\"Mmm, so close, Simon. Sure you don't...want...me...to...mmm...finish...oooh. I love when you do that.\"\n\n\"Mmph, mumph, hah, hooo...\"\n\n\"Don't talk with your mouth full, Simon, Simon, Simon, Simon, Siiimmooooon...\"\n\n\"Okay, dude. I'm totally setting your food out here. Um, thanks for the tip.\"\n\n1:14 a.m.\n\nWe lay in bed, limp and a little stupid. My poor Simon, I'd ridden him to the brink of extinction. He wasn't a teenager, but even he was surprised by his...hmm...stamina. After the last round of crazytown, he crawled back to the hallway, retrieved the food, and we ate Thai sitting in the middle of the bed. I'd quickly stripped the sheets because raisins and flour clouds lingered from earlier. The amount of work I was going to be faced with in the kitchen tomorrow was daunting, but it was worth it. All of it. All of it was worth it.\n\nNow we lounged, settled but not settling. Still wrapped around each other but now clad in a pink nightie and a pair of sweatpants. To be clear, I wore the pink nightie. We lay side by side, facing each other, legs tangled and hands held.\n\n\"When do you have to go back to work?\"\n\n\"I told Jillian I'd be back Monday, although that is the last thing I can think about right now.\"\n\n\"What are you thinking about?\"\n\n\"Spain.\"\n\n\"Yeah?\"\n\n\"Yeah, it was amazing. Thank you so much for taking me, and then taking me.\" I nudged him with my elbow.\n\n\"It was my pleasure, on both counts. I'm glad you could...come,\" he snorted.\n\nNow that the O had returned, we could joke about it. We were quiet for a moment, just enjoying the music. Simon had hobbled next door a little while ago to put on a record. Even hobbled, he was sexy.\n\n\"When are you leaving for Peru? Ass, I still hate you a little for getting to go, but when are you leaving?\"\n\n\"About two weeks. And no hating on the photographer. I have to go, but I'll always come back.\"\n\n\"Oh, to be clear, I don't hate you for leaving. I hate you because I want to go too. But I digress. I love you more than I hate you, so we're good.\"\n\n\"We're good?\"\n\n\"Yes, of course. You have to travel for your job. It's not like I didn't know this.\"\n\n\"Well, knowing about it and then being the one left behind are two different things,\" he said, eyes getting a little cloudy. I smoothed my hand across his cheek, feeling his scruff and skin and watching him lean into my touch. His eyes closed, and he hummed a contented hum.\n\n\"You're not leaving me behind. We live busy lives and will continue to do so. Just because you get to stick your dick in me now, that isn't going to change us,\" I replied.\n\nA slow grin spread across his face. Eyes still closed, but grinning. \"Sometimes dicks change people,\" he said through the grin.\n\n\"Sometimes dicks change what needs to be changed. Sometimes dicks make it better.\"\n\n\"Sometimes dicks make it better\u2014what an odd thing to say.\"\n\n\"Stick around, who knows what I'm gonna say next.\"\n\n\"Sticking.\"\n\n\"Stuck.\"\n\n\"Going to kiss you now.\"\n\n\"Thank Christ.\" I giggled as he wrapped his strong arms around me. We kissed quietly, thoughtfully. I settled down into his nook, perfectly shaped and smelling like heaven.\n\n\"I adore this nook.\"\n\n\"Good.\"\n\n\"No one else gets this nook.\"\n\n\"It's yours.\"\n\n\"Yes, yes it is. Make sure you tell that to all those gorgeous Peruvian women who will try to seduce the hot American.\"\n\n\"I'll make sure to tell them my nook is spoken for.\"\n\nI smiled and yawned hugely. It had been an exhausting few days. I was jet lagged and had been rocked to within an inch of my life. Tended to make a girl tired. Simon leaned across me to shut off the light and tucked me back into the nook.\n\n1:23 a.m.\n\n\"Simon?\"\n\n\"Mmm?\"\n\n\"Are you asleep?\"\n\n\"Mm-hmm...\"\n\n\"I just wanted to say, well, I'm really glad you came home early.\"\n\n\"Mm-hmm, me too.\"\n\n\"And I'm pretty smitten with you.\"\n\n\"Mm-hmm, me too.\"\n\n\"Smitten like a kitten.\"\n\n\"Mm-hmm, me too.\"\n\n\"Who's lost her mittens.\"\n\n\"Mittens, mm-hmm...\"\n\n\"Simon?\"\n\n\"Mm-hmm?\"\n\n\"Are you asleep?\"\n\n\"Mm-hmm...\"\n\n\"I love you.\"\n\n\"I love you too.\"\n\n...\n\n...\n\n...\n\n\"Caroline?\"\n\n\"Mm-hmm...\"\n\n\"I'm really glad I came home early too.\"\n\n\"Mm-hmm...\"\n\n\"And I'm really glad you came.\"\n\n\"Enough.\"\n\n\"'Night, Caroline.\"\n\n\"'Night, Simon.\"\n\nAnd as Count Basie and his orchestra played us off into dreamland, we curled around each other and slept.\n\nText between Simon and Caroline the following Tuesday:\n\nTalked to a buddy of mine. I think I figured out how to do those prawns you went so crazy over in Spain.\n\nPerfect, they'll fit in with the Spanish feast I am planning for Saturday. Everybody's coming, even Jillian and Benjamin.\n\nSure you don't want to have it at my place?\n\nNo, it'll be easier at mine. I have the island, which is better for prepping, but I'm commandeering your oven.\n\nCan I commandeer you on the island?\n\nThat's not the correct use of the word commandeer.\n\nPlease, you know what I meant.\n\nI did, and you may.\n\nSweet. Have you seen my running shoes?\n\nYep, they're in my bathroom where you left them.  \nI tripped over them this morning.\n\nIs that the thump I heard?\n\nYou heard that?\n\nYep, woke me up.\n\nAnd yet you didn't come see if I was okay?\n\nDidn't want to disturb Clive.\n\nI can't believe he's been sleeping on your side. Traitor cat.\n\nWe're friends now...well, almost friends.  \nHe peed on my sweatshirt again.\n\nHA! I have to get back to work, cat stealer.  \nWe still watching a movie tonight?\n\nIf that's what you want to call it.\n\nMakes it seem like we actually have plans.\n\nI have plans. Oh man, do I have plans.\n\nAs do I...\n\nI'm sitting here eating your apple pie...think about that.\n\nThat's all I can think about now...hating you.\n\nYou don't hate me.\n\nThat's true. Now go eat my pie.\n\n...choking...\n\nText between Mimi and Caroline on Thursday:\n\nYou sure I can't bring anything Saturday?\n\nNah, Sophia is bringing drinks,  \nand we're taking care of the rest.\n\nSo good to hear you in a we again.\n\nYes, I'm enjoying the we.\n\nAnd the we-we?\n\nWhat are we, 7? Yes, the we-we is good.\n\nGood to hear it. So have you slept in the bed of sin yet?\n\nNo, we seem to be staying at my place.  \nI think I'd feel weird in that bed.\n\nMany walls were banged from that bed...\n\nExactly. That's my point, feels strange.\n\nMaybe it would be nice to make your mark on his bed, so to speak. New era, new girlfriend, new banger?\n\nI don't know, we'll see...I know at some point  \nI'll sleep there, just not yet. Besides,  \nhe's having too much fun bonding with Clive.\n\nWHAT? Clive hates guys! Except gay guys.\n\nThey've come to some kind of weird kitty\/man understanding. I'm not questioning it.\n\nIt's like a new world order.\n\nI know.\n\nWant me to come over early Saturday and help?\n\nYou just want to get into my drawers again.\n\nThey need to be reorganized...\n\nCome over early.\n\nWAHOO!\n\nGet some help...\n\nThursday evening all was quiet. Simon and I sat on my couch, working. I was sketching a holiday concept for someone's ballroom. Yep, ballroom. This was the world I visited. Just visited, not lived in. I was still in my yoga clothes. Simon cooked, using my kitchen, in which he was becoming very much at home. He said it would be easier since we'd just end up at my place anyway, but I caught him lifting Clive up onto the counter so he could \"watch.\" I put that in quotes because the actual word was spoken by Simon to Clive. The entire sentence, I believe, was \"Here ya go, buddy. This way you can watch! You can't see too well from down on the floor, I bet, right? Right?\"\n\nAnd Clive answered. I know it was technically impossible, but the meow he uttered sounded like, \"Thanks.\"\n\nMy boys were bonding. It was nice.\n\nSo here we sat, me sketching and Simon making his travel plans for Peru online. He had something like seventy billion frequent flyer miles, and he loved to flaunt them in my face.\n\nSo quiet it was, save the scratching of my colored pencils on the page and his clickety-clack on the keyboard. And the clicking from Clive. Most stubborn kitty hangnail in the free world.\n\nSimon finished and closed up his laptop, stretching his arms over his head and exposing his happy trail. I may have drawn outside the lines a bit. He laid his head against the back of the couch, eyes closed. Within a few moments, the tiniest of snores began, and I grinned silently. I continued my sketching.\n\nTen minutes later I felt his hand reach out across the pillows, and grasp my hand.\n\nI only needed one hand for sketching after all.\n\n\"Holy shit, Caroline, these prawns are sick!\" Mimi moaned in a way that made Ryan readjust the way he sat.\n\nIt was Saturday night, and we were all gathered around my dining room table, full of Spanish food and Spanish wine. I'd had a blast trying to recreate all the wonderful food Simon and I had eaten. Not as good certainly, but pretty close. And of course we were without the coastal ambiance, but instead had the coziness only an autumn evening in foggy San Francisco can provide. The city lights twinkled through the windows, a fire crackled in the fireplace, courtesy of Benjamin, and laughter filled the apartment.\n\nI sat in my chair, tucked in to Simon's side as we laughed with our friends. I'd been a little nervous that we'd be subjected to some kind of hazing, since our inevitable getting-together had been the topic of conversation for so long. But it was good, everyone settled into the evening with only minimal teasing. Simon and I had stuck pretty close together most of the evening, but I could already tell we would morph into one of those couples that didn't need that.\n\nI never wanted to be that couple, the one that was entirely codependent and in constant need of reassurance. I loved Simon, that much was clear. One of us traveled, for goodness sake, so we needed to roll with it. And I thought we would. I felt him next to me, and I moved just a little closer. He slipped an arm around my waist, his hand patting my arm, squeezing and just making me more aware of him. I was aware. His fingertips traced little circles around my elbow, and I sighed as he pressed a quick kiss to my forehead.\n\nI would never need the Honey and the Baby. I just needed him and his little circles. Just needed to feel him at my side, whenever he was here. Jillian caught my eye from across the table and winked.\n\n\"What was that for?\" I asked, sipping my second glass of brandy. Simon was going to have no trouble getting me into bed later that night, not that he ever did.\n\n\"Things worked out well, didn't they?\" she asked, looking back and forth between Simon and me.\n\n\"Couldn't have worked out better. Subletting your apartment to me was the best decision you ever made.\" I smiled, leaning into Simon as he rubbed my shoulder.\n\n\"Jillian giving me your number so I could text you from Ireland, now that's the best decision she ever made,\" he added, winking at Benjamin from across the table.\n\n\"Oh, I don't know. Pretending I didn't know your mysterious neighbor was a damn good decision too,\" she said, a mischievous grin lighting up her face as Simon coughed into his brandy.\n\n\"Wait, what? You knew all along I was the one living next door?\" he asked, sputtering as I handed him a napkin. \"But you've never even been to my place!\"\n\n\"She hasn't, but I have,\" Benjamin spoke up, clinking his glass with his fianc\u00e9e's.\n\nSimon and I sat pie eyed as we watched them laugh and congratulate themselves.\n\nWell played...\n\n\"Okay, that's the last of it. No more dishes,\" Simon announced, closing the dishwasher. After everyone finally left, we decided to clean up the rest of the mess instead of leaving it for the morning after.\n\n\"Thank goodness. I'm beat.\"\n\n\"And I have dishpan hands.\" He winked, showing me how red they were.\n\n\"That's the mark of a good housewife.\" I just barely sidestepped his grabby hands.\n\n\"Just call me Madge and bring that fantastic ass back over here,\" he fired back, snapping a dishtowel in my direction.\n\n\"This ass? This ass right here?\" I asked, propping myself against the island just so, leaning forward on my elbows.\n\n\"You want to play now, is that it? Thought you were beat,\" he murmured, catching my bottom in his dishpan hands and giving me a light smack.\n\n\"Maybe I'm catching my second wind.\" I giggled as he promptly swept me up over his shoulder in a fireman's hold and headed for the bedroom. Upside down, I beat my fists against his bottom and kicked, though not so much as to actually get away. His feet stopped at the bedroom door.\n\n\"Forget something today?\" he asked, turning so I could see inside: stripped bed, no sheets.\n\n\"Damn, I forgot to put the sheets in the dryer. They'll still be soaked!\" I grumbled.\n\n\"Problem solved. Slumber party at Simon's,\" he announced, pulling open my lingerie drawer. \"Pick a nightie, any nightie.\"\n\n\"You want to stay at your place tonight?\"\n\n\"Yeah, why not? We've been sleeping here since we got back from Spain. My bed's lonely.\" He ruffled through piles of lace and peekaboo.\n\nHmm, his bed was probably lonelier that it had ever been before.\n\n\"So, pick one.\" He gave my ass another slap.\n\n\"Eh, you pick out something you like. I'll model it for you.\" I grinned, talking myself into this. Come on, I could certainly spend the night in his bed. Could be fun. I saw a familiar something pink and lacy make its way under his arm, and then we were off across the hallway. I managed to kick his door on the way in, something pretty hard to do while upside down.\n\nOnce more, I found myself in a bathroom, putting on lingerie for Simon. He really liked everything I wore. Whether it was actual lingerie or one of his old shirts, he didn't seem to care. And it was rarely on for very long.\n\nWithout meaning to, I thought of all the women who'd come before me, all the women he'd enjoyed and had enjoyed him. But I was here now, and I was who he wanted. I smoothed the silk over my body with a deep breath, my skin already beginning to tingle in anticipation of his hands.\n\nI heard him messing about with his record player\u2014the telltale crackle and pop of needle on vinyl such a comforting sound.\n\nGlenn Miller. \"Moonlight Serenade.\" Sigh.\n\nI opened the door, and there he was. Standing by the giant Wallbanger bed of sin. His slow grin overtook me, and he looked me up and down.\n\n\"You look good,\" he murmured as I walked in.\n\n\"You too.\"\n\n\"I'm wearing the same clothes I was wearing earlier, Caroline.\"\n\nHe smirked as I encircled his neck with my arms. His fingertips dragged up and down my arms, tickling the inside of my elbow.\n\n\"I know,\" I replied, placing a wet kiss under his ear. \"You looked good then, and you look good now.\"\n\n\"Lemme get a better look at you,\" he whispered, responding with his own wet kiss at the base of my throat. I shivered. The room wasn't at all cold.\n\nHe spun me out, as if on a dance floor, and held me at arm's length for just a moment. The pink nightie, his favorite. He'd neglected to bring the matching panties, and I neglected to notice. He spun me back into him, and I immediately began to work the buttons on his shirt.\n\n\"Quite a night tonight,\" he remarked.\n\nTwo buttons down.\n\n\"You're telling me. I can't believe those two were matchmaking from the very beginning! Although I don't think they can take credit for the other two couples. That was all us.\"\n\n\"Who knew love was in the air when you banged on my door?\"\n\nAnother button down.\n\n\"Luckily, you were so taken by my charms, it was inevitable.\"\n\n\"It was the nightie, Caroline. It was the nightie that did me in. The charms were a bonus. I had no idea I'd be getting a girlfriend out of the deal.\"\n\nShirt untucked and on its way off.\n\n\"Really? And here I thought we were just messing around!\" I giggled, scrambling to get his belt buckle poked through.\n\n\"Well, then, here's to messing around with my girlfriend!\" Belt buckle undone, jean buttons popped. Thank goodness for the old-fashioned button fly. He picked me up, by my naked bottom I might add, and walked me to the bed as I pushed his shirt off. It hung from him by the sleeves.\n\n\"I like the sound of that,\" I whispered in his ear as he laid me down on the bed.\n\nHovering over me, placing kisses across my chest, he kept saying the word over and over again. Girlfriend, then kiss. Girlfriend, girlfriend, then kiss.\n\n\"Did you know Mimi and Neil are thinking about moving in together? Isn't that a little soon? I hope they know what they're getting into,\" I reported, arching up to meet his kisses.\n\n\"I know what I'm getting in to.\"\n\n\"What's that?\"\n\n\"You, silly,\" he said, and I heard the blessed sound of his belt buckle hitting the floor. \"I'm only concerned with our happy ending. Or two, or three even. Drank that ginseng tea you left me this morning\u2014watch out.\" He chuckled, lifting one of my legs on to his shoulder and kissing a path down the inside of my calf.\n\n\"Happy ending, huh?\"\n\n\"Don't you think we've earned it?\" he asked, kneeling now, lips trailing along the top of my thigh as I panted.\n\n\"Oh, hell, yes,\" I laughed, throwing my arms over my head and arching up to meet him. Hello, O! Nice to see you again. With his lips, he brought me one. With his tongue, he brought me another. And when he slid into me and pushed me high up on to the bed, I almost had another on contact.\n\nClothes now discarded, skin on sweaty skin, my legs wrapped solidly around his hips, which pushed against mine. His eyes burned as I felt every inch of him. Inside. Outside. All around the town.\n\n\"Oh, God,\" I moaned. And then I heard it.\n\nThump.\n\n\"Oh, God,\" I moaned again.\n\nThump thump.\n\nI giggled at the sound. We were banging.\n\nHe looked down at me, raising one eyebrow. \"Something funny?\" he asked, pausing his movements. He pushed back into me slowly, very, very slowly.\n\n\"We're banging the walls.\" I giggled again, watching his eyes change as he registered my giggling.\n\n\"We sure are,\" he admitted, chuckling a little as well. \"You okay?\"\n\nI wrapped my legs even tighter around his waist, making sure I was as close to him as I could be. \"Bring it on home, Wallbanger.\" I winked, and he complied.\n\nI was being driven up the bed with the strength of his thrusts. He drove into me with unflinching force, giving me exactly what I could take, then pushing me just past that edge. He stared down at me, hard, flashing that knowing smirk. I closed my eyes, letting myself feel how deeply I was being affected. And by deep, I mean deep...\n\nHe grasped my hands and brought them above my head to the headboard.\n\n\"You're gonna wanna hold on for this,\" he whispered and threw one of my legs up over his shoulder as he altered his hips.\n\n\"Simon!\" I shrieked, feeling my body begin to spasm. His eyes, those damnable blue eyes, bore into mine as I shook around him.\n\nHe called out my name, and no one else's.\n\nA little while later, almost asleep, I felt the mattress dip as Simon left the bed. Hearing him flip over the record, I snuggled deeper into the pillow. My body was deliciously tired, having been worked to within an inch of total exhaustion. We banged that wall, yes indeed. I owned both sides of that wall now.\n\nI heard him bumble down the hall and half wondered what he was up to. Thinking in that tired, half-awake way that he must be getting some water, I slipped back down to sleep.\n\nA few moments later I was awoken by his arms sliding around me, pulling me against his warm body. He kissed me on my neck, then cheek, then forehead as he got settled. Then I heard...purring?\n\n\"What's that?\" I asked, looking around.\n\n\"I thought he might be lonely,\" Simon admitted sheepishly. Looking over my shoulder, I saw Simon, and then Clive. Simon had gone over to get him. Clive was purring very loudly, quite pleased with all the attention he'd been receiving lately. He poked his nose in to me and settled into the nook between us.\n\n\"Unbelievable,\" I muttered, rolling my eyes at the two of them.\n\n\"Are you that surprised? You know much I love pussy,\" Simon deadpanned. Then his silent laughter shook the bed.\n\n\"You're very lucky I love you,\" I added, letting his arms hold me tight.\n\n\"I'll say.\"\n\nAnd then, as the laughter faded and sleep took hold, I pondered what the future might hold for me and my Wallbanger.\n\nI knew it wouldn't always be this easy. But it sure as hell would be a good time.\n\nAll was quiet as I set out on patrol, making sure the perimeter was secure. I padded through my new territory, taking notice of any loose Q-Tips. They would need to be dealt with if unruly. If allowed to run unchecked, they would multiply. I'd seen it happen.\n\nI came upon a curious shelf with nothing but glass bottles on it. I batted at one, watching as it fell to the floor. I would have to come back to this location, but for now I had rounds.\n\nChecking the view from the front window, I saw that I could retain control of my neighborhood from this vantage point. I scouted a possible napping station in another window with southern exposure, then stopped for a stare-off with an owl outside. Neither of us gave in willingly, and it was another fifteen minutes before I continued on to check on my people. They had finally quieted down after several rounds of caterwauling. Honestly.\n\nThe Feeder was, predictably, taking up most of the sleeping quarters. The Tall One, aptly named because he was taller than The Feeder, was making that noise again\u2014the noise I simply could not tolerate. The Feeder was beginning to toss and turn. She was not sleeping soundly. Without enough sleep, she would be unlikely to play with me the following evening, so this situation would have to be remedied. She did seem to enjoy our games, so I would once more take matters into my own paws.\n\nJumping from the floor to the bed with a natural grace\u2014a grace that was not fully appreciated by my people, I felt\u2014I navigated my way through knees and legs, arms and elbows, until I reached the pinnacle and came to rest just beneath his chin. Stretching out one paw, I placed it over his breathing holes, stopping the noise momentarily. The Tall One brushed away my effort, although once he rolled onto his side, the noise stopped. He curled in to himself, in the one corner The Feeder had allowed him. As he had done so, I remained standing, doing my best log-rolling impression and maintaining perfect balance. Again, my people just didn't get it.\n\nSettling into the nook between them, I rested. Our home was secure, and I now watched over The Feeder and The Tall One, so I allowed myself to dream. Of her. The one that got away... \nAcknowledgments\n\nThere are so many people I have to thank for helping me bring this story back out there. To Lauren, who edited this from the very beginning and always told me when I was getting it right. To Sarah M Glover for her San Francisco insight and her insistence that I do have a voice and I should be encouraged to use it. To Elizabeth for allowing me to be crazy. To Brittany and Angie for recognizing that I was one of them and allowing me to play with the curvy girls. To Deb for being the best dirty cheerleader on the planet. To my real life mentors, Staci and Janet, upon whom the character of Jillian is entirely based. To the fantastic Banger Nation, those wonderful ladies who were there from the very first chapter and enjoyed the ridiculous with me. To the Filets for their support in the wee hours and their constant gut checks. To all of the wonderful readers and friends on Twitter who make it a pleasure to communicate in 140 characters. To authors like Laura Kaye, Ruthie Knox, Jennifer Probst, Michelle Leighton, Tiffany Reisz, Karen Marie Moning, and Jennifer Crusie for writing some of my favorite stories of all time. I have always been a reader first and a writer second, and nothing makes me happier than telling a friend about a great book I just finished and can't stop thinking about.\n\nTo the online writing community that allowed me the grace and space to create something I could truly be proud of.\n\nTo Keili and Ashley for making me funny again and starting something as silly as Not Your Mother's Podcast with me.\n\nSpecial thanks to my editor, Jessica, who is the perfect blend of smart and sassy. You are a perfectionist, you are a sounding board in a padded room, you are the colon to my semi.\n\nVery special thanks to Enn for bringing me back into the fold, listening to my rants, and putting up with my commas. For working your ass off. For always having my back. There is a taco in heaven with your name on it.\n\nAnd of course big fat thanks to Peter for always taking such good care of me. I adore your giant thumbs.\n\nThank you to all the readers, to all the Nuts Girls, to all the Bangers, to all the chickens. Thank you.\n\nAlice\n\nxoxo\nAlready missing Simon, Caroline, and Clive?\n\nCheck out how they're dealing with being in a *gulp* relationship...\n\nRUSTY NAILED\n\n## chapter one\n\n\"Oh, God.\"\n\nThump\n\n\"Oh, God.\"\n\nThump thump\n\n\"Caroline, don't say those things to me when I'm so far away.\" Simon chuckled, his voice low. And still as thrilling as it ever was.\n\n\"Silly Simon, I'm simply reacting to the banging on the other side of the wall.\"\n\n\"Who's on the other side of the wall?\"\n\n\"The guy with the hammer. You should see it. It's huge.\"\n\n\"I'm going to have to ask you not to talk about some other guy's hammer.\"\n\n\"Then get home and wow me with yours.\" I laughed, closing the door to my office to reduce the noise. It wouldn't be my office much longer, though. I was moving up in the world\u2014or at least down the hall. That was the cause of the banging: renovating my new space. Bigger office, corner office, thank you very much, right next to Jillian's, my boss and owner of. Better view of the bay and almost twice the size of my old office, with a small anteroom for a possible future intern.\n\nI might one day have an intern. How was this my life?\n\n\"I'll be home tomorrow. Think you can keep your thoughts on my hammer until then?\" he asked. I glanced at the calendar on my desk, Simon's arrival home circled.\n\n\"I'm gonna do my best, babe, but you should see how thick that tool belt is. No promises.\" Simon groaned and I laughed harder. I loved torturing him across multiple time zones. \"And don't forget my present.\"\n\n\"Do I ever?\"\n\n\"No, you're a thoughtful one, aren't you?\"\n\n\"Don't forget my present either,\" he said, his voice going low again.\n\n\"Pink nightie is ready to go; I'll be in it when you get home.\"\n\n\"And then I'll be in it, on it, under it, I'll\u2014oops, gotta go, taxi's here.\"\n\n\"We'll continue the nightie talk in person. Love you,\" I said.\n\n\"Love you too, babe,\" he said, and hung up.\n\nI stared at the phone for a moment, imagining him halfway across the world in Tokyo. This year alone he'd logged more frequent-flier miles than most people accrued in a lifetime, and he was booked solid for the rest of the year.\n\nI was still smiling at the phone when Jillian knocked and breezed in, then sat on the corner of my desk.\n\n\"Something on your mind, Jillian?\" I asked, pulling a browned petal from the vase of coral tinged roses next to where she was resting her cashmere-clad bum.\n\n\"I can see something is on your mind. Was that Simon on the phone?\" she asked as I grinned. \"Only he can make your face light up like that.\"\n\n\"I say again\u2014something on your mind, Jillian?\" I repeated, poking her ever so slightly with my pencil.\n\n\"I have something on my mind that might make your face light up even brighter\u2014although it is an interesting tomato-soup color right now,\" she teased.\n\n\"Does your fianc\u00e9 find you as annoying as everyone who works for you does?\"\n\n\"Way more, way way more. You ready to hear the big news, or did you want to keep sassing me?\"\n\n\"Hit me,\" I said with a sigh.\n\nI love my boss, but she does have a flair for the dramatic. Like when she played matchmaker last year for Simon and me, playing dumb the entire time. But her heart was in the right place. It also belonged 100 percent totally and completely to Benjamin, a venture capitalist. They'd been together for years and were finally tying the knot in a few weeks, in a wedding that all of San Francisco was talking about. Benjamin was a certifiable dreamboat who made my best friends and me giddy and word-trippy whenever he was around. Jillian knew we all had a not-so-secret crush on her man, and teasingly used it against us as often as possible. Now she was finally marrying our dream man, and heading off for a dream honeymoon all over Europe.\n\n\"So remember the job we did last spring for Max Camden? The waterfront Victorian we did, before his daughter got married?\"\n\n\"Yeah, he gave it to her as a wedding present. Who does that?\"\n\n\"Max Camden, that's who. Anyway, he owns the old Clare-mont Hotel in Sausalito, and he's looking for a new design firm to update it and give it a modern twist.\"\n\n\"Fantastic! Did you do your proposal already?\" I asked, picturing the property. Right off the main drag in Sausalito, the Clare-mont had been there since the turn of the last century, one of the few to survive the Big Quake.\n\n\"No, because you're doing the proposal. You'll be the lead designer on this project if you get it,\" she clarified. \"You think I can take something like this on? Right before my wedding? I'm not giving up my honeymoon for work\u2014I've given up too many vacations over the years as it is.\"\n\n\"Me? No no no, I'm not ready for that, you're not ready for that, what are you thinking?\" I stammered, my heart leaping into my throat. This was big-time, baby.\n\n\"Please, you got this.\" She kicked me gently. \"Feel that? That's my foot, kicking you out of the nest.\"\n\n\"Um, yeah, I've been out of the nest awhile now, but this is different,\" I protested, chewing on my pencil.\n\nWhich she plucked out of my mouth. \"You really think I'd give this to you if you weren't ready? And tell me the truth, aren't you even the slightest bit intrigued?\"\n\nShe had me there. I'd always wanted to do a project this big. But to actually be the lead designer on an entire hotel redesign?\n\n\"I realize I'm asking a lot\u2014you're already going to be running the show around here while I'm on my honeymoon. Do you truly think this is too much to bite off at one time?\"\n\n\"Wow\u2014I just\u2014wow,\" I answered, taking a deep breath. When she'd initially asked me if I'd keep things running while she was on her honeymoon, it was things like making sure the alarm was set each night and that Ashley made sure to order coffee creamer. The list had steadily grown larger as projects stacked up, but still very much manageable. Now this?\n\nI let the idea sit for a moment. Could I do this? Jillian seemed to think so.\n\n\"Hmm . . .\"\n\nI pictured the hotel: great light, great location, but needed a major overhaul. I was already thinking about potential palettes when she tapped me on the head with her pencil.\n\n\"Come in, Caroline. Hello,\" she said, waving her hand in front of my face.\n\nI grinned at her. \"I'm in, let's go for it,\" I said, my head already full of ideas.\n\nShe grinned back and offered me a fist bump. \"I'll let the team know you'll be presenting.\"\n\n\"Presenting my vomit, most likely,\" I said, only half kidding.\n\n\"Just make sure it matches the drapes and we're in the clear. Now, let's celebrate by choosing a song to walk down the aisle to.\" She pulled her iPod out of her pocket and started scrolling through.\n\n\"Is that in my job description?\"\n\n\"That you indulge me? Yes, check your contract. So when I walk down the aisle, which song should I . . .\"\n\nThere was no stopping her once she'd put on her Wedding Hat, so I relaxed a bit, even though my mind was spinning. This was big-time baby, but I had this.\n\nRight?\n\nI spent the afternoon framing out the beginnings of a pitch to Max Camden. As I pulled archival photos of the hotel and the surrounding area, ideas were beginning to present themselves. Not fully formed yet, but hinting at what might be an approach interesting enough to take a chance on a young designer. I knew that the strength of my ideas would be bolstered by Jillian's reputation; anyone who was good enough to work for her was usually granted wider berth. However, it still came down to whose ideas were best\u2014and I wanted this concept to be epic.\n\nStill musing over the project as I turned my key in my front door, I heard a distinct thump, followed by a click click click padding toward me.\n\nClive.\n\nPushing through the door, I was greeted by my wonder cat, my own little piece of feline heaven. In a burst of gray fur, my ankles were surrounded by purrs and insistent nudges.\n\n\"Hi there, sweet boy, were you a good boy today?\" I asked, leaning down to scratch his silky fur.\n\nArching up into my hand, he assured me that yes, he was in fact a sweet boy, and also a good boy. Berating me for leaving him alone for a thousand years, he cooed and chirped, herding me toward the kitchen.\n\nWe talked as I readied his dinner for him, which of course I'd been put on earth expressly to do, and our conversation covered the normal subjects. What birds he'd seen from the window today, whether any dust bunnies had emerged from under the bed, and whether I'd find any toys buried in the toe of my slippers. He was noncommittal on this last question.\n\nOnce his kibble was in his bowl he ignored me completely, and I headed back to the bedroom to put on some comfy clothes. Untucking my turtleneck, I went to the mirrored dresser to grab some yoga pants. While pulling my arms out of my shirt, my heart leapt into my throat when I saw the reflection of someone sitting on my bed. Instinct kicked in and I whirled, fists clenched, a scream ready to let loose.\n\nMy brain only processed that it was Simon after my fist was flung.\n\n\"Whoa, whoa, whoa! What the hell, Caroline!\" he yelled as he grabbed his jaw.\n\n\"What the hell, Caroline? What the hell, Simon! What the hell are you doing here?\" I yelled back. Good to know if I was ever actually attacked, I wouldn't freeze.\n\n\"I came home early to surprise you,\" he managed, rubbing his jaw and grimacing.\n\nMy heart was still racing in my chest, and as I tried to calm down, I noticed the suitcase in the corner. The one I'd missed when I'd come into the room. I looked down and saw the turtle-neck still hanging around my neck like a scarf.\n\n\"I could just kill you!\" I yelled again, charging him and pushing him back onto the bed. \"You scared me to death, you idiot!\"\n\n\"I was planning on calling out to let you know I was here, but then I would've missed that entire conversation with Clive. I didn't want to interrupt.\" He grinned underneath me, threading his hands around my waist and in and out of my belt loops.\n\nI blushed. \"Traitor!\" I yelled down the hallway. \"You could have let me know someone was here\u2014you're a terrible watch-cat!\"\n\nA disinterested meow floated back.\n\n\"I'm hardly just someone. I think I rate a little higher than that,\" he told the side of my neck, which he was now feathering with the tiniest of kisses. \"So, are you going to say hi to your boyfriend who flew all the way across the globe just to show you his hammer, or are you going to punch me again?\"\n\n\"Not sure yet; I'm still a little freaked out. My heart is literally racing, can you feel that?\" I asked, pressing his hand over the left side of my chest.\n\nOnly so he could feel my heart. Yep. That's the only reason. Heart was in fact delighted to have Simon home early; she loved a good romantic reunion. Other areas were delighted as well.\n\n\"See now, I thought it was racing because of me,\" he said with a low chuckle, dipping his nose along my collarbone as he \"felt my heart.\"\n\n\"Dream on, Wallbanger,\" I said, feigning indifference. The truth? My heart was now in Simon mode, and it was pounding for him. And speaking of pounding.\n\n\"So you came home early just to see little ol' me?\" I breathed into his ear, sneaking a wet kiss just underneath it. His hands dug a little deeper into my hips as he shifted on the bed.\n\n\"I did.\"\n\n\"Think you can help me with this turtleneck?\"\n\n\"I do.\"\n\n\"And then after that, you wanna show me your hammer?\" I asked the front of his T-shirt, nuzzling at him, positioning my legs on either side of him. In answer, he thrust up and let me feel that very hammer. I chuckled. \"Mmm, am I gonna get nailed?\"\n\nHe lifted my turtleneck off, then unsnapped my bra and my breasts tumbled out, causing his eyes to flare, then focus with precision. \"No more questions,\" he directed, sitting up underneath me as he pulled me closer.\n\nI mimed zipping my lips just before he flipped me over onto my back. God, I loved this man.\n\nHis lips danced along my collarbone, nipping occasionally with his teeth in a way he always knew got me warm, fast. I got it; I'd missed him too. Arching my back, I pressed my breasts against him, twisting and turning to bring me into contact with him as much as I could be, my skin needing to feel his. After a year, he could still bring me to my knees in seconds with one touch, one kiss, one look.\n\nI pushed back against him, flipping us once more and pulling at his jeans. \"Off, now,\" I instructed.\n\nWhen his belt was gone, his buttons unbuttoned, I pulled apart his jeans to find that once more my man had gone commando.\n\nIt's like he was put on earth just to make me come out of my skin.\n\nI snuck one hand inside, grasping him firmly, feeling how warm he was; ready to take me on my own trip around the world.\n\n\"Fuck, I missed you,\" he breathed, his body lean and taut. I slid down the bed, kissing and licking at his skin hungrily. His hands came up to my face, fingers fluttering along my cheekbones, sweeping my hair back. So he could watch.\n\nI took him into my mouth, entirely. His hands clutched at my hair, freezing me in place, holding me exactly how he wanted me. \"Mmm, Caroline,\" he moaned, thrusting ever so slightly. Slightly, my ass\u2014that wasn't how this show was going down.\n\nI pulled back then took him in again, hard. Using my hands I caressed him, alternating my touch so he never knew quite where I was coming from, using my tongue and mouth to tease and tempt him, coaxing the sweetest dirty words out of that sent-from-heaven-mouth of his. That mouth that I knew would exact the sweetest dirty revenge all over my body.\n\nI loved him this way, loved that I could make him this insane. But just before he got too far gone, he pulled me up his body and took my panties off before I could say, hey, those are my panties.\n\nThen he pushed up my skirt, nudging my knees apart with his own. Gazing down at me with those piercing sapphire eyes, he ran his fingers over me, through me, making me groan and moan and shake and shimmy. \"So gorgeous like this,\" he breathed as I cried out.\n\n\"Need you, Simon\u2014need you, please!\" I was ready to tear my hair off my head and throw it at him, if I thought that would get him inside any faster.\n\nAny further thoughts vanished as he slid home. Thick, hard, and ten kinds of fantastic were all I knew the second Simon pressed inside me. \"God, that's amazing,\" I moaned, the feeling of him filling me overwhelming me.\n\nAnd when he rolled us so I was on top, and he thrust up hard inside me, it was perfection.\n\nUntil afterward, when we lay in a heap of sweaty limbs, and he asked me how I liked his hammer.\n\nThen it was beyond perfection.\nCaroline, Simon and Clive are back in USA Today bestselling author Alice Clayton's first book in her new Cocktail series!\n\nOrder now!\n\nRusty Nailed\n\n* * *\n\nORDER YOUR COPY TODAY!\n\nAbout the Author\n\nNovelist Alice Clayton makes her home in St. Louis where she enjoys gardening but not weeding, baking but not cleaning up after, and is trying desperately to get her long-time boyfriend to make her an honest woman\u2014and please buy her a Bernese Mountain dog.\n\nAfter working for years in the cosmetics industry as a makeup artist, esthetician, and educator, Alice picked up a pen (read laptop) for the first time at 33 to begin a new career: author. Having never written a thing, she soon found writing to be the creative outlet she'd been missing since walking away from the theater 10 years before.\n\nShe has a great time combining her love of storytelling with a sense of silly, and she was shocked and awed to be nominated for a Goodreads Author award in 2010 for her debut novels, the first two installments of The Redhead Series\u2014The Unidentified Redhead and The Redhead Revealed.\n\nAdditionally, Alice loves spending time with her besties on Not Your Mother's Podcast (check them out on iTunes). She also enjoys pickles, Bloody Marys, and eight hours of sleep.\n\nAlso from Alice Clayton\n\nThe Unidentified Redhead\n\nWhen Grace Sheridan returns to Los Angeles to become a working actor, it's a second shot at a life-long dream. With some help from her best friend and agent, that dream could become a reality, but at thirty-three, has Grace missed her chance at the big time? Will an unexpected sizzling romance with the entertainment industry's newest \"it\" boy shine an uncomfortable spotlight on her life?\n\nAlice Clayton spins a playful and erotic May to December romance in this, her debut novel. Clayton's funny, borderline neurotic heroine is perfect in her imperfections, and the off the charts sexual chemistry between her and her charming, yet blissfully unaware leading man are the true heart of The Unidentified Redhead. With laugh out loud dialogue and a super steamy romance that will get your heart racing, sneaking around in L.A. and dodging the Paparazzi has never been so fun.\n\nThe Redhead Revealed\n\nAs their careers catch fire, Grace and Jack \u2014 everyone's favorite foul-mouthed, funny, and feisty couple \u2014 find themselves on opposite coasts. Grace has landed in New York City, where she loves being onstage again, particularly because she's playing a fabulous character in a musical written by her old college flame, Michael. Their rekindled friendship makes exploring the city that much more fun.\n\nWait, it's just friendship, right?\n\nMeanwhile, in L.A., Jack can scarcely keep up with the swirling throngs of women who track him everywhere he goes, the endless press appearances, and the ridiculous rules his manager, Holly, keeps concocting for him \u2014 all part of the buildup to the release of Time, his steamy new film.\n\nThank goodness for phone sex.\n\nBut even when their schedules allow them to connect, Grace and Jack must keep their relationship off the radar and away from paparazzi cameras. Sure, the sex is sensational, but can this duo survive swirling rumors, the demands of their chosen professions, Grace's raging internal battles, and a whopping nine-year age difference?\n\nTick-tock, the clock is ticking. Isn't it?\n\nAlice Clayton brings the second installment of a tale told with her magical mix of humor and heat, so cuddle up under the sheets and flip on the Golden Girls. Grace and Jack are at it again.\n\nWe hope you enjoyed reading this Gallery Books eBook.\n\n* * *\n\nJoin our mailing list and get updates on new releases, deals, bonus content and other great books from Gallery Books and Simon & Schuster.\n\nCLICK HERE TO SIGN UP\n\nor visit us online to sign up at  \neBookNews.SimonandSchuster.com\n\nGallery Books\n\nA Division of Simon & Schuster, Inc.\n\n1230 Avenue of the Americas\n\nNew York, NY 10020\n\nThis book is a work of fiction. Any references to historical events, real people, or real places are used fictitiously. Other names, characters, places, and events are products of the author's imagination, and any resemblance to actual events or places or persons, living or dead, is entirely coincidental.\n\nCopyright \u00a9 2012 by Alice Clayton\n\nOriginally published in 2012 by Omnific Publishing.\n\nAll rights reserved, including the right to reproduce this book or portions thereof in any form whatsoever. For information address Gallery Books Subsidiary Rights Department, 1230 Avenue of the Americas, New York, NY 10020\n\nFirst Gallery Books ebook edition February 2013\n\nGALLERY BOOKS and colophon are registered trademarks of Simon & Schuster, Inc.\n\nFor information about special discounts for bulk purchases, please contact Simon & Schuster Special Sales at 1-866-506-1949 or business@simonandschuster.com\n\nThe Simon & Schuster Speakers Bureau can bring authors to your live event. For more information or to book an event contact the Simon & Schuster Speakers Bureau at 1-866-248-3049 or visit our website at www.simonspeakers.com.\n\nISBN 978-1-4767-4126-0\nContents\n\nChapter One\n\nChapter Two\n\nChapter Three\n\nChapter Four\n\nChapter Five\n\nChapter Six\n\nChapter Seven\n\nChapter Eight\n\nChapter Nine\n\nChapter Ten\n\nChapter Eleven\n\nChapter Twelve\n\nChapter Thirteen\n\nChapter Fourteen\n\nChapter Fifteen\n\nChapter Sixteen\n\nChapter Seventeen\n\nChapter Eighteen\n\nChapter Nineteen\n\nChapter Twenty\n\nChapter Twenty-One\n\nChapter Twenty-Two\n\nAcknowledgments\n\nRusty Nailed Excerpt\n\nAbout the Author\n\nAlso from Alice Clayton\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\nKINGDOM OF SHADOWS\n\nGreg F. Gifune\n\nFirst Digital Edition\n\nOctober 2009\n\nPublished by:\n\nDelirium Books\n\nP.O. Box 338\n\nNorth Webster, IN 46555\n\nsales@deliriumbooks.com\n\nwww.deliriumbooks.com\n\nKingdom Of Shadows \u00a9 2009 by Greg F. Gifune\n\nCover Artwork \u00a9 2009 by Zach McCain\n\nAll Rights Reserved.\n\nThis book is a work of fiction. Names, characters, places and incidents are either a product of the author's imagination or are used fictitiously. Any resemblance to actual events, locales or persons, living or dead, is entirely coincidental.\n\n\"Shape without form, shade without colour,  \nParalysed force, gesture without motion;  \nThose who have crossed  \nWith direct eyes, to death's other Kingdom  \nRemember us \u2014 if at all \u2014 not as lost  \nViolent souls, but only  \nAs the hollow men  \nThe stuffed men.\"  \n\u2014T.S. Eliot, \"The Hollow Men\"\n\n## -1-\n\nThe van rockets through darkness, swaying and bouncing along the bumpy road. The breakneck speed no longer seems necessary, but everyone is preoccupied and still racing on adrenaline and fear. For several minutes there is relative silence, but Carbone resumes his screaming and writhing about, knees pulled in close to his chest as his bloody hands clutch desperately at the mangled flesh that was once his stomach.\n\n\"Hang on, bro.\" Snow pokes his head up between the front bucket seats and looks to Rooster. \"We need to get him to a hospital!\"\n\n\"He's already dead,\" Rooster tells him.\n\nBetween screams, Carbone wheezes and literally cries for his mother.\n\n\"Christ,\" Nauls groans, \"his intestines, I\u2014I can see his fucking intestines!\"\n\nLandon, a white-knuckled grip on the steering wheel, glances quickly at the rearview then increases speed despite the rough terrain.\n\n\"Slow down,\" Starker says, his deep voice booming from his position at the rear of the van. \"This ride dies, you die with it.\"\n\n\"Whatever,\" Landon says indignantly.\n\n\"We can't just let him bleed out,\" Snow says.\n\nRooster watches the darkness through the windshield wash over them like renegade waves. He's always liked Carbone, and knows he and Snow are best friends, but they're miles from any hospital. Game over.\n\n\"Goddamn it!\" Snow leans closer. \"You hear me?\"\n\nRooster looks back at Snow. \"Stay with him, all right?\" he says evenly. \"Don't let him die alone.\"\n\nAfter several seconds, reality sinks in, and with a defeated nod, Snow disappears into the back.\n\n\"Where the hell are we?\" Rooster asks Landon.\n\n\"No clue, been following these country roads for miles now.\" He nervously paws perspiration from his face with the back of his free hand. \"You wanted the middle of nowhere. You got it.\"\n\nRooster is about to tell him to slow down when the van comes to a sudden stop. Everyone lurches forward and Carbone screams again.\n\nBefore them, fog rolls across a field of weeds and overgrown grass. In the distance, an old farmhouse sits in the darkness and mist. The moon is full but obscured by clouds, scarcely illuminating a series of hideous scarecrows nailed to rotting wooden crosses scattered throughout the property.\n\n\"What is this place?\" Nauls asks.\n\nLandon squints. \"Looks abandoned.\"\n\nWith a final gagging cough, Carbone vomits blood and bile and dies on the floor of the van in a pool of his own excrement and urine.\n\nWhat they don't understand is that his death is far more merciful than anything they'll ever know.\n\n* * * *\n\nDistant screams echoed in his mind like the sudden screech of tires. He had no idea where he was, but his first conscious thought was that something was chasing him. The sheer curtains billowed, danced before him like smoke. It seemed as if he'd been watching them for hours, though he couldn't be sure. He'd been asleep, hadn't he? Below, city streets were awakening, coming alive as the sun slowly rose over a horizon of brick and steel. It was far too cold outside for the windows to be open, but he assumed Gaby had opened them at some point during the night.\n\nRooster sat up in bed and swung his feet onto the chilly floor. He leaned forward, face in his hands. It felt like an eternity since his old life had ended. Yet there was a disconnect between the here-and-now and the past, as if one or the other wasn't quite true, falling closer to waking nightmares than reality. Even the night he and his old crew pulled their last caper was such a blur he often had difficulty piecing his scant memories into anything coherent. But for Carbone, he and the others got away. He knew that much. He remembered the final score and leaving that way of life behind him. For a long while the past had stayed buried, forgotten, perhaps even consciously ignored, but over the last few weeks, flashes of memories had returned, mostly in tiny bits and pieces. He couldn't be sure, but Rooster suspected that's what had started the awful headaches he'd been suffering from of late, his continued attempts to remember in more detail.\n\nThe farmhouse...he remembered that dark farmhouse they'd ended up at to split the take. He remembered the moon that night...and scarecrows...horrible scarecrows. He remembered them too.\n\nAnd then, like a reel of film that had run its course, the memories stopped, returning his mind to an equally unsettling darkness.\n\nThough tall, thin and wiry, with angular features and a receding swath of buzzed-down brown hair, the nickname he'd had since high school still fit, but his body was slowing with age, and for the first time he'd begun to notice it, to really feel it. He moved his hand up behind his neck and squeezed, rubbing down to his trapezius muscles. He sported the remains of a tan, his skin a deep bronze, the veins and muscles in his arms and legs defined and strong. He patted his stomach. Not quite the six-pack it had once been, but flat and tight, nothing to be ashamed of.\n\nA breeze blew through the windows, disturbing the curtains once again.\n\nThis time they were shredded and filthy with dirt and blood, dangling there like sheets of slashed flesh.\n\nRooster looked away, clenched shut his eyes.\n\nWhen he opened them the curtains were back to normal, but everything felt askew now, as if something or someone had entered the bedroom without his consent. He stood up, glanced around, eyes panning the room.\n\nNothing...no one...\n\nA chill licked his spine.\n\n\"Are you all right?\"\n\nHe turned to the doorway to find Gaby standing there bundled in a robe, dark hair mussed and the look on her face a mixture of horror and concern. \"You were screaming.\"\n\nRooster grabbed his Marlboros and a lighter from the nightstand, lit a cigarette and blew a stream of smoke at the windows. \"I wasn't even asleep.\"\n\nHer expression softened, and she leaned against the doorframe with a sigh. \"Yeah, I know. You never sleep anymore, not really.\"\n\n\"The windows...\"\n\n\"I opened them.\" She hugged herself. \"Fresh air's good for you.\"\n\nRooster drew another drag, coughed it out. \"It's freezing.\"\n\nHer brown eyes\u2014so dark they were nearly black\u2014sparkled in place of a smile. \"It's good for the soul.\"\n\n\"Nothing can live long in the cold,\" he mumbled.\n\nGaby nodded but said nothing.\n\nShadows lay across the room like fallen spirits. Rooster stepped through them, approaching the windows with caution. A cold and dreary day stared back, the sky gray and overcast, the streets beyond the housing project courtyard dirty and cracked, cold and still mostly abandoned; the buildings in the neighborhood old and rundown, many of them condemned and long forgotten. Such a bleak city, he thought. Even where bustling life should've thrived, there was only emptiness, decay and an overwhelming sense of hopelessness. He looked back over his shoulder at Gaby. It seemed to him her name should've been Grace, since she was like a savior, the only consistently good and beautiful presence in what had become an otherwise murky existence. When it was just the two of them and they held each other close in the night, the fear in him subsided. In those moments he felt alive again, perhaps even happy, but like all else he'd once believed in, Gaby would eventually leave him and he'd be alone again in the darkness with his nightmares and the awful echo of faraway screams...\n\n\"What are you thinking about?\" she asked.\n\n\"Your name.\"\n\n\"My name?\"\n\n\"It's pretty. Gabrielle.\"\n\n\"I didn't realize it interested you.\" She smiled as if letting him in on a secret. \"You've never mentioned it before.\"\n\n\"Never really thought about it until now, I guess.\"\n\n\"Its origin is Hebrew,\" she explained. \"Most people don't know that.\"\n\n\"What does it mean?\"\n\n\"God is my might.\"\n\nTheir eyes met, and for a moment he was lost in them, their depth and beauty. He knew her so well, and yet in many ways she seemed unfamiliar. How could that be? He focused on the writing table beneath the windows, the bills scattered about it.\n\n\"I better get in the shower,\" Gaby said. \"Don't want to be late for work.\"\n\nRight, he thought. Someone's got to earn some money around here.\n\n\"I'll find something,\" he promised. \"There's been some talk that big warehouse facility over on Dover Street's hiring.\"\n\n\"That place gives me the creeps,\" she said. \"What do they warehouse there anyway?\"\n\n\"I don't know. All I heard is they need some extra hands to unload trucks. It's temporary but steady work for a week or more. Word is they're only hiring a few people, so I want to get there early.\"\n\n\"What about the phone calls?\"\n\nFear rose from deep inside him. \"What about them?\"\n\nGaby came closer, padding across the chilly pockmarked floor in her bare feet, nails painted blood red and a dainty gold ankle bracelet adorned with tiny bells jingling as she moved. \"It's obviously important. He's called at least half a dozen times, and at all hours, too.\" She slid up behind him and wrapped her hands around his waist. She smelled vaguely of freshly cut flowers, and her breath caressed the back of his neck in slow, sensual intervals. \"He says he knows you.\"\n\n\"That was a long time ago.\"\n\n\"Are you going to call him back?\" When he didn't answer she leaned into him and brushed her lips against his ear. \"He sounds so frightened, the man on the phone.\"\n\nWith a sad smile, Rooster flicked his cigarette out the window. \"He is.\"\n\n## -2-\n\nBeneath an oddly gray sky, Rooster walked toward the hulking shadows cast by the enormous warehouse facility at the end of Dover Street. He strode past one alley strewn with garbage, human and otherwise, and then another, the last hope for escape from the dead-end street and the monolithic structures awaiting him. His breath spilled from his nostrils like columns of smoke, partially concealing his face as he pressed on through the cold, hands buried deep in the pockets of his battered leather jacket and chin tucked to chest in an attempt to ward off the occasional bursts of winter wind blowing in off the nearby ocean. Everything was deathly still, and though the constant din of city noises could still be heard, rather than a block or two away, they seemed impossibly far off, as if they were memories of a different city altogether, a deafeningly chaotic city recalled while passing through the mysterious solitude of another.\n\nWhen he reached the tall chain-link fence surrounding the facility, he noticed the gate was open, a thick padlock and chain dangling free as if left there mistakenly. He hesitated. A nearby security hut beyond was empty, the glass cracked and aged and looking as if it hadn't been cleaned in years. On the far side of the hut, scarred with cracks and occasional tufts of weeds, an enormous parking lot led to a series of loading docks, and amidst the larger warehouses, a smaller building marked OFFICE. Forklifts and other pieces of equipment were scattered about the property as if abandoned long before, and though most of the bays were closed, the few left open revealed enormous but empty storage areas. It looked like some time ago everyone had simply picked up and left.\n\nNo one came or went from the office building, and the lone wire-meshed window facing the street was grimy and dark. Had the place gone out of business? He could've sworn he'd passed by here a few days before and it was alive with workers and trucks coming and going, loading and unloading. He tried to remember where he'd heard about the job opportunities here. Had someone told him? Had he seen something at the Unemployment Office? Rooster watched the area a while with the experienced and trained eye of a thief. In time he looked back at the street. It was empty but for bits of trash and debris blowing about in the wind. He checked his watch then gazed at the sky. It normally wasn't so dreary this time of afternoon, but the drab winter sky conspired to cast everything in a dull pall reminiscent of dusk.\n\nAfter another quick look around, Rooster stepped through the open gate, crossed the parking lot and slipped into the office building.\n\nHe found himself in a long, dimly lit corridor that reeked of bleach. With the dull industrial tile floors, low plaster ceilings, steel-encased light fixtures and unimaginative but practical architecture, the building more closely resembled an archaic hospital or dated mental institution than office space.\n\nRooster pulled off the knit hat he was wearing and held it in his hands. Though the heavy steel entrance door had closed silently behind him he could still see his breath in the hallway. Surely they had heat here, why wouldn't it be on? A small sign protruding from the nearest doorframe read: RECEPTION.\n\nHe looked past it to the far end of the corridor, which was draped in darkness. Had something moved just then? Startled, Rooster took a step back. He was certain he'd caught a glimpse of someone shuffling into the cover of darkness, and the sudden sound of labored breath seeping down the hallway in its wake seemed to confirm it. The noise echoed along the walls, transforming into strange, indecipherable whispers.\n\nWhispers that did not sound human.\n\nRooster stuffed his hat into his back pocket, took a deep breath then ran a hand over his face, eyes trained on the shadows at the end of the hall. Calm down, he thought. It's just the nightmares again.\n\nAn unusual ticking sound drew his attention to the reception office. A lone woman well into her sixties sat behind an inordinately large desk, banging away on an old Olympia typewriter and seemingly oblivious to his presence. A series of metal file cabinets filled out the remaining space behind her. Clad in a dowdy dress and a cardigan sweater thrown over her shoulders for good measure, the receptionist's silver hair was pulled up into a bun, and a pair of half-glasses attached to a chain strung about her neck sat along the bridge of her bulbous nose.\n\nRooster stepped through the doorway. \"Are you still hiring?\"\n\nWithout looking up from her typewriter the woman retrieved a sheet of paper from a metal bin, slapped it down and slid it over to the edge of the desk. \"Fill out this application, front and back. Turn it in to me when you're finished.\"\n\nRooster took the form. \"Is it always so cold in here?\"\n\n\"Comes as a shock to most but that's the way it is.\"\n\nHe nodded like he'd understood her answer. \"Are you open today?\"\n\n\"We're always open.\"\n\n\"Then where is everybody?\"\n\nThe woman's head snapped up, her eyes glaring at him with demonic fury. \"Where are you?\"\n\nRooster watched the paper fall from his hand as the familiar torment of agonizing screams came to him again. But these were not nightmares or daydreams, he could hear them bellowing from deep within the building, as if people were being tortured in the bowels of the facility. Heart crashing his chest, he backed out into the hallway, terrified. The receptionist's mouth hung open as she panted with anger, spittle dripping from her pale, cracked lips. A quiet growl emanated from her, like the low rumbling snarl of a dog just before it attacks.\n\nHe turned and bolted for the front door, slamming into it with his shoulder and stumbling out into the parking lot as it gave way. Staggering forward, he nearly pitched face-first onto the pavement but regained his balance at the last moment and in one frantic, uninterrupted motion, broke into a full run.\n\nHe did not look back.\n\n* * * *\n\nThe payphone on the corner was occupied by a rotund woman carrying a brown paper bag filled with groceries. Across the street, Rooster waited, watching from the burned out doorway of an abandoned building only a few blocks from his apartment in the housing projects. Though he couldn't hear what the woman was saying, she was clearly upset and quite animated. He remained huddled in his hiding place until she finally slammed the phone down and stormed from the booth, a look of desperation and confusion creasing her face as she toddled toward the top of the street.\n\nHe checked the boulevard in both directions. It was empty. Not even a car or city bus to be found. Moving quickly, he crossed the street, ducked into the phone booth and dug a shred of paper from his jacket pocket. Jotted across it was the information Gaby had written down the last time a call came in. Rooster dropped a dime and punched the numbers.\n\nThe connection crackled and hissed but eventually went through and began to ring.\n\n\"Hello.\"\n\nEven after all this time Rooster knew that voice. \"Snow.\"\n\nAn exhale of relief and then: \"Rooster-man.\"\n\nHe gripped the phone tight and spun around so he could watch the street. \"You've been calling me.\"\n\n\"I can't believe it's really you. Didn't know if I'd be able to track you down after all this time.\"\n\n\"Are you here, in the city?\"\n\n\"Where else would I be?\"\n\n\"What do you want?\"\n\n\"We gotta talk.\"\n\n\"I'm not in the life anymore.\"\n\n\"You got no idea what life you're in.\"\n\nA sharp pain stabbed Rooster's temple. He flinched. \"What's that mean?\"\n\n\"What the hell you think it means? Means I need to talk to you, bro.\"\n\n\"Whatever you're into these days I'm not interested.\"\n\n\"This is serious shit.\"\n\n\"Snow, what do you want?\"\n\n\"I need to see you.\"\n\nThe receptionist's demonic eyes tore through Rooster's memory in strobe-like flashes. \"Just leave me alone, man. I got enough problems.\"\n\n\"Motherfucker, I'm trying to help you!\"\n\nThe visions faded. The fear remained. \"Stop calling me.\"\n\n\"You don't hear nothing else I say you better hear this.\" A crackling hiss bled through the line again. \"You need to know what I know.\"\n\nA burst of wind forced the phone booth door open. He pinned the phone to his shoulder with his ear and sparked a cigarette, making sure to cup the flame until he got it going. \"What do you know?\"\n\n\"I know what you're going through. The headaches, the nightmares. Hearing things, seeing things. Bad things. Evil things.\"\n\nRooster's eyes watered. He told himself the cold was to blame as a black Crown Vic with a tinted windshield and windows turned at the head of the street and slowly rolled by. Cop car, he thought, feeling the muscles in his stomach clench. He hadn't been a criminal in years, but old habits, old fears, died hard. He watched the car until it was out of sight.\n\n\"There ain't a lot of time,\" Snow pressed. \"I need to see you.\"\n\nRooster breathed heavily into the phone in quick nervous bursts. \"When?\"\n\n\"Today.\"\n\n## -3-\n\nBut for their labored breathing, the area is deathly silent. Fog rolls over the open field, cutting across the desolate country road and floating through a thick expanse of forest on the other side. The full moon, still masked by cloud cover, reveals a mist-shrouded landscape of crucified scarecrows, demonic sentries guarding a farmhouse no one would want.\n\nSnow stays in the back of the van with Carbone's body but the rest pile out of the vehicle and wander about the street amidst confusion and high emotions, attempting to gain their bearings while figuring out what to do next.\n\n\"What's with all the scarecrows?\" Landon asks. \"Nothing's grown there for years but weeds, why would they need scarecrows?\"\n\nAs he surveys the area, Starker still clutches the AK47 he used on the job, his hulking presence and enormous shaved dome daunting even in limited light. He moves to the side of the road. \"Maybe it's not crows they're looking to scare off.\"\n\n\"Well if they're meant for me they're working,\" Nauls says. \"Fucking things are creeping me out.\"\n\n\"Yeah Nauls,\" Landon quips, \"they're meant for you. Jesus, what an idiot.\"\n\n\"I'm an idiot? You're the one who stopped here.\"\n\n\"Yeah, because shit-for-brains bit it.\" Landon jerks a thumb at the van. \"And if it's OK with you I've had my fill of smelling dead ass tonight.\"\n\nSnow emerges from the rear of the van and wipes his bloody hands on his jeans. \"What did you say?\"\n\nLandon faces him. \"You heard me.\"\n\n\"Say it again, motherfucker.\"\n\n\"Hey, I'm sorry Carbone stepped off, but it's nobody's fault but his and you know it. He blew the back doors too early. Total amateur-hour horseshit, he knew better.\"\n\n\"A good man's dead.\" Snow stepped closer. \"Show some respect.\"\n\n\"He fucked up and now we've all got blood on our hands.\"\n\n\"What the hell would you know about it, wheelman?\"\n\n\"Enough to know the stupid bastard could've gotten us all killed. And I didn't hear you making any driver jokes when I was carting your sorry ass the fuck outta Dodge.\"\n\n\"You're working my last nerve.\"\n\nLandon squares his stance. \"Blow it out your ass.\"\n\nRooster steps between them. \"Both of you cool it.\" He knows he must get the crew focused, split the take and make arrangements to wrap things up one way or another. But it can't be done out in the open, even in a desolate place such as this. One local police car or nosy townsperson passing by is all it'll take to escalate things, and there's been enough escalation tonight. No one was supposed to get hurt. The job had been meticulously planned, rehearsed and timed to the millisecond. Yet there were still mistakes, and what began as a robbery ended in a homicide, one guard dead, two badly injured. And now they've lost one of their own. They have to move and move fast. \"We're still on the clock, which means I still call the shots, so get your heads out of your asses and get back in the fucking game. Now.\"\n\nSnow points at Landon. \"We ain't done.\"\n\n\"Any time, douche.\"\n\nRooster stands his ground until both men drift away in opposite directions. \"All right, let's get inside and finish our business.\"\n\nNauls, holding two large canvas duffel bags stuffed with cash, shuffles about like he needs a bathroom. \"Can't we find someplace else?\"\n\n\"I don't like this shit bin any better than you do,\" Rooster admits, \"but it's out of the way and nobody should bother us here. Nauls, you stay with me. Landon, get the van off the street and under cover. Snow, you and Starker check the place out. It looks deserted but let's be sure.\"\n\n\"OK how come the two brothers got to check the farmhouse out?\" Snow cracks. \"We more expendable, that it? We ain't special like you white folks.\"\n\n\"Just get it done.\"\n\nSnow pulls two .45s from the back of his belt and turns to Starker. \"All right, big man, let's go.\"\n\nApparently mesmerized by the field of rotting scarecrows, Starker does not respond. He stares off into the darkness as if in a trance.\n\n\"Come on biggins, time for some recon.\"\n\nStarker continues to stare at the horrible faces peering across the field through the darkness and fog. Rooster approaches him and cautiously places a hand on his shoulder. \"Starker.\"\n\nHe says nothing.\n\n\"Stay with me now,\" Rooster tells him softly. \"We need you.\"\n\nStarker remains locked on the field, one enormous finger resting on the trigger of the AK-47, the other hand sliding almost lovingly back and forth across the top of the weapon in a slow and steady motion. \"Something's not right.\"\n\n\"You see something?\"\n\n\"I feel it. So do you.\"\n\nHe's right, but Rooster can't figure out how Starker knows this. Perhaps he hasn't hidden his anxiety and uneasiness as well as he thought he has. \"Maybe we should all go,\" Rooster suggests. \"Check the place out together.\"\n\n\"It doesn't much matter.\" Starker blinks slowly, his eyes eerily reflecting moonlight. \"We're all gonna die tonight.\"\n\n* * * *\n\nMemories of Starker's bald head covered in blood flashed before Rooster's eyes, the huge man spitting and slobbering between horrific screams, choking on his own blood and bodily fluids while begging like a child for mercies he would never be granted.\n\nThe horrible sounds of that night were the last to leave him, fading gradually like the slowly dying things they were. And like the dead, a gruesome residue remained in their wake. A reminder of their power, perhaps, evidence that such figments of torturous nightmares had, in fact, existed.\n\nOut in the open air the winter wind cut like a razor. Rooster held his ground at the mouth of an alley between a seedy bar and a blown-out storefront, his jacket collar flipped up to protect the back of his neck. A red neon sign advertising the strip joint two doors down blinked with a steady rhythm, painting his face in a strange and frightening haze. His headache had weakened, but a dull pain still lingered behind his eyes. He rubbed his temple and studied the passersby. Everyone on the street seemed suspect, every car a potential menace. He swore he'd seen the same black Crown Vic twice more since he'd walked the eight blocks from the payphone to the agreed upon meeting place, but of course there was no way to know for sure if it was the same vehicle. Even if it was, what would the cops want with him? He'd been doing straight time for years.\n\nHe returned his focus to the neighborhood. It was filthy and far from the safest in the city, but Rooster had a good vantage point, as from his position he could clearly see people approaching from either direction. Though like the rest of the city many of the buildings sat vacant and rotting, this was predominantly a commercial area that still crackled with intensity and life. Heavy traffic clogged both lanes, filling the air with a glut of sickening exhaust fumes, and numerous souls of varied descriptions hurried along the sidewalks, several scowling at him as if he'd done something to personally offend them but most with their heads bowed and eyes averted. At the end of the block an old homeless man collapsed on the sidewalk and lay still. After watching him a moment Rooster realized the man's breath was no longer forming clouds in the cold air. Perhaps he'd died. No one seemed to care.\n\nIt was then that Rooster noticed a bald man of perhaps sixty standing across the street watching him, features unremarkable but for a pair of piercing ice-blue eyes. Dressed entirely in black\u2014suit, shoes and overcoat\u2014it wasn't until the man glided a bit further down the block that Rooster saw the white collar and realized he was looking at a priest. The closer the man got the more disheveled he became, his clothes wrinkled and soiled and his face creased with age and looking as if it needed a good scrubbing.\n\nIgnoring the traffic, the priest recklessly crossed the street, eyes locked on Rooster even after several drivers hit their horns and one car nearly struck him. While still several feet away, the priest raised a hand and pointed at him. \"You, I\u2014I know you!\" he called. \"I know you!\"\n\nRooster shook his head and waved the man off, though oddly enough, the closer the priest got the more familiar he became. He couldn't quite place him but was convinced he knew him from somewhere.\n\nJust as the priest made it to the sidewalk, another man appeared out of the crowd and cut him off, blocking his path.\n\nThe afro gave him away. Snow, looking like he always had, dressed in jeans, sneakers and an old army jacket thrown over a sweatshirt, extended a hand, holding it up between himself and the priest as their eyes met. Neither moved; two statues in a sea of humanity.\n\nRooster stepped out of the alley, approached them.\n\nThe priest looked over Snow's shoulder, enraged. \"I know you!\"\n\n\"Keep moving, padre,\" Snow said evenly. \"I ain't playing with you. Move.\"\n\nDefeated, the priest slipped away, looking back every few seconds until he'd been completely absorbed by the crowd, carried off down the street with the rest.\n\nRooster started after him but Snow grabbed his arm, firmly enough to stop him but with enough restraint to let him know the move wasn't a challenge.\n\n\"Let him go, man.\"\n\n\"He's right, I\u2014we know each other, I\u2014\"\n\n\"Just let him go.\" When Rooster relaxed Snow released him. \"You don't look no different.\"\n\nThey shook hands. Snow's palm was cold, rough and covered in calluses. \"Neither do you,\" Rooster sighed. \"But we are different, aren't we?\"\n\nNearby, overhead trains rumbled along rusted tracks. The noise seemed to distract Snow for a moment. \"Let's get off the street.\" He motioned to the bar behind them. \"Catch some heat.\"\n\n* * * *\n\nThe bar was dark, with scarred linoleum floors, low ceilings and only two small windows on the front wall. A scattering of tables and chairs filled the area, while a row of dark booths lined one wall and a bar filled the back. A jukebox kitty-cornered nearby sat quietly. The bartender, an overweight guy with a shock of unruly salt-and-pepper hair, chatted quietly with what was probably a regular, both staring at a small television suspended in the corner showing an old black-and-white horror movie. Otherwise the place was empty.\n\nRooster and Snow ordered a couple beers then took them over to the booth farthest from the bar and sat down.\n\n\"It's good to see you, man.\" Snow slowly caressed his beer bottle, focusing on it rather than Rooster. \"Just sucks it has to be like this.\"\n\nAfter a long swallow of beer Rooster slid a black plastic ashtray from the corner of the table into the center and lit a cigarette. \"What's going on, Snow?\"\n\nHe was about to answer when a bloodcurdling scream exploded through the bar.\n\nRooster reached to his belt for a gun that wasn't there, a gun that hadn't been there in years. Snow cocked his head in the direction of the television, where a ghoul was staggering through a cemetery shrouded in mist, closing in on a buxom young maiden with the ability to scream at octaves capable of shattering glass.\n\n\"Jesus H.\" He rubbed his temples. \"Could've lived without that.\"\n\n\"Never seen you so jumpy, Rooster-man. You were always cold as ice.\"\n\n\"The priest, who was he?\"\n\n\"I don't know.\"\n\n\"He knew me. And I knew him. I just can't remember how.\"\n\n\"You'll figure it out.\"\n\n\"Can't figure out much of anything lately. The strangest shit's happening. I can't make sense of any of it.\" He took a deep drag on his cigarette, blew out a cloud of smoke and watched it climb toward the ceiling. \"Look, I\u2014\"\n\n\"Feels like you went to sleep and woke up in the middle of your life,\" Snow interrupted, voice unusually quiet, \"and now you can't remember how the hell you got here.\"\n\nRooster stabbed the cigarette between his lips and left it there so he could put his hands flat on the table between them and better conceal the fact that they were shaking. He nodded. \"What's happening to us?\"\n\nUp close Snow's eyes were bloodshot and heavy, like he'd been crying recently, hadn't slept in a while, or both. He smelled vaguely of cheap aftershave. \"What do you know about demons?\"\n\n\"Demons? You mean like\u2014\"\n\n\"Like all kinds of crazy shit runs through your head, then you start hearing things. Screams mostly, or whispers that don't make no sense. And just when you think it can't get no worse, you start seeing shit. Not people, not...not exactly. But they look like people...least until they don't.\"\n\nThe receptionist, Rooster thought, shrugging off a chill. \"I don't believe in demons.\"\n\n\"Yeah neither do I but they don't seem to give a shit.\" Snow downed some beer then let out a quiet belch under his breath and looked to the door as if expecting someone to burst through it at any moment. \"Not too long ago I got some information.\" He leaned closer, across the table. \"And ever since then these other motherfuckers have been following me. Never up close, always a ways back, watching from their cars, Crown Vics\u2014big black bastards\u2014that's what they drive.\"\n\n\"Cops?\"\n\n\"These ain't cops.\"\n\n\"Who are they?\"\n\n\"They been following me for weeks. After today they'll be following you.\"\n\n\"Why?\" With manic repetition Rooster puffed his cigarette. \"What do they want?\"\n\n\"You remember the night Carbone died?\"\n\nRooster began to perspire as flashes of farmhouse, blood and scarecrows filled his memory. \"Some.\"\n\nNo longer able to contain his nervousness, Snow abruptly stood up and made a beeline for the jukebox. He dropped a coin in, made a selection then gave the bartender and his friend a long look that said: This is going to make hearing the television more difficult but let's not make a big deal about it or you'll force me to do some really unpleasant shit to you. Both men looked away without comment and Snow slowly strode back to the booth as The Police's Spirits in the Material World kicked in.\n\n\"You said I needed to know what you know.\" Rooster crushed his cigarette in the ashtray. \"So tell me.\"\n\n\"What do you remember about the night Carbone died?\"\n\n\"Come on, man, what the hell's going on?\"\n\n\"Do it.\"\n\n\"The armored car robbery, the last job we pulled as a crew,\" he said. \"Everything went according to plan until Carbone fucked up and blew the back doors too early. The third guard was waiting on him. Carbone took a shotgun blast dead in the gut. Starker wasted the guard, shot him in the face, killed him instantly.\" He remembered the young man's head as it exploded, a crimson mist of blood, brains and skull spraying everything, and all of them. \"You got Carbone back to the van while Nauls and I handled the other two guards and took care of the swag. Landon was the wheelman. We got out ahead of the cops, ended up in the middle of nowhere at some deserted old farmhouse. Carbone died in the van.\"\n\nSnow nodded. \"Then what?\"\n\n\"You were there.\"\n\n\"Pretend I wasn't.\"\n\nRooster fidgeted in his seat. It felt like thousands of insects were scurrying over every inch of his body. He scratched at his head and suddenly found himself checking the door every few seconds as well. \"I don't...\"\n\n\"You don't know.\"\n\nShadows along the ceiling shifted, elongated.\n\n\"We split the take,\" he finally said. \"Then we took off.\"\n\n\"That how you remember it?\"\n\n\"I think so but I can't...\" Rooster took another swig of beer. \"I can't remember exactly, it...the whole thing seems like a dream.\"\n\n\"I couldn't remember nothing either.\"\n\nThe man at the bar, a middle-aged guy wearing some sort of workman's uniform, hopped down from his stool and slipped through a nearby door marked RESTROOMS.\n\n\"The more I thought about it,\" Snow continued, \"the worse it got. I couldn't remember the rest of that night no matter how hard I tried. It was like it was just...gone. All I knew was whatever happened scared the shit out of me, made me scared like I never even knew I could be. I'm talking about the kind of fear you feel right down to your nuts, man. The kind that makes you shit in your pants like a baby sliding out lunch. You know what I mean.\"\n\nRooster did know. He swallowed so hard he gagged.\n\n\"Like you, I thought I was losing my goddamn mind.\" Snow sat back with an air of defiance. \"It's like there was something right on my ass, something evil. I couldn't take no more. I stopped sleeping, stopped eating, just locked myself up in my apartment and hid out. I wanted to kill myself but I was afraid of the other side. Ain't exactly lived the life of a saint, right?\"\n\nThe bartender was staring at them intently. When he realized Rooster had caught him he quickly looked away and busied himself.\n\n\"That's when that woman started hanging outside my apartment wanting to talk to me all the time.\" His face twisted. \"I didn't know who she was, didn't know what I'd done. I don't even remember it. I was on H when it went down and was hurting so bad for a fix I was out of my mind. I never meant to hurt her.\"\n\n\"I never knew you did heroin.\"\n\nSnow sighed helplessly. \"Neither did I.\"\n\n\"You're not making any sense. What the hell are you talking about?\"\n\n\"I tried to do straight time, man, for real. I tried.\"\n\n\"I believe you.\"\n\n\"I got a janitor gig at this office building a few blocks from my crib. I was going crazy but I showed up on time every night, did my thing and minded my own business, played by the rules, closed my eyes to the demons and the screams and that woman always staring at me. I'd walk there, work the overnight shift then walk home in the morning. I'm there about a week when I notice this old dude following me one night. Skinny little white cat with glasses. Real Poindexter-looking motherfucker. At first I think maybe he's a cop, but he don't look like no cop I ever seen, looks more like a professor or some shit. He shows up every night, tails me from my apartment to work, and then he's gone. So one night I get a lead on him, take a corner and duck into a doorway. He comes by and I grab his narrow ass.\" Snow ran a hand over his face. He too had begun to perspire. \"I'm about to rack me some old white man when he starts talking about that night at the farmhouse, all the shit I'm going through and how he can help me. Motherfucker knew more about me than I did, man. Said he had answers, said he knew what happened to us that night. He said it was time we knew the truth. And that's exactly what he laid on me. Only now sometimes I wish he didn't. Sometimes not knowing was better.\" He bowed his head in an attempt to mask the tears filling his eyes. \"Ain't that a bitch? We never had a goddamn chance, man, none of us.\"\n\n\"Who was this guy?\"\n\n\"What they done to us wasn't right, Rooster, it wasn't right. We did some bad shit but we're human beings, man, we fucking human beings.\"\n\n\"What who did to us? What are you saying?\"\n\nSnow reached into his jacket, put something on the table and slid it over to him. When he pulled his hand away a small key was revealed. \"Opens a locker at the bus station downtown,\" he said. \"Take it. Use it.\"\n\nRooster nonchalantly covered it with his palm. \"What am I gonna find?\"\n\n\"Everything I know. Everything you need to know. All the proof I got from Poindexter.\" He checked the door once, then again just a second or two later. \"They're after me, man, and they know I'm trying to pass the information to you. Once they know you got it, they're gonna come after you, too.\"\n\n\"Why are you giving this to me, why not one of the others?\"\n\nSnow shrugged. \"Carbone's dead. Nauls is a retard. Landon's an asshole, and Starker\u2014shit\u2014that boy's stone psycho. Whatever Hell them motherfuckers are burning in they deserve.\"\n\n\"I'm not so sure anybody deserves to burn in Hell.\"\n\n\"Makes sense if you're the one burning.\"\n\nRooster slid the key to the edge of the table then pocketed it. \"You know where any of them are?\"\n\n\"Last I heard Nauls and Landon were still in the city and still in the life. Starker supposedly caught his old lady banging some guy. Put a .38 in her pussy and pulled the trigger, then he beat the dude into a coma, ripped his junk off and stuffed it so far up his ass they had to do surgery to get the shit out. Couple days later they both died. Starker got away. Word was he headed down to Mexico or some shit.\"\n\n\"They ever catch him?\"\n\n\"Don't know.\" He wiped the tears from his eyes then killed his beer. \"Don't care.\"\n\n\"What happened to us that night, Snow?\"\n\n\"Go open that locker.\"\n\n\"Why don't you just tell me?\"\n\nSnow smiled, but it was the smile of the damned. \"You wouldn't believe me even if I did.\"\n\n\"Try me.\"\n\n\"You got to see for yourself.\"\n\nAcross the barroom, the restroom door opened with a scraping sound and Rooster saw the same man exit, wander back to the bar and return to the stool he'd vacated moments before. As the door slowly swung shut, he saw a moist and filthy tile floor littered with scraps of toilet paper and trash, and something else moving along the wet tiles toward the toilet stalls on the back wall of the bathroom. Like the severed appendage from some scale-covered creature, it slithered about in a snakelike motion, revealing a pale tentacle several inches thick and at least three feet long. Rooster sat up straighter, squinting through the shadows in an attempt to bring the thing into focus, but the door had closed. He glanced at Snow, who hadn't seen it but looked as if he had. Rooster turned away, hopeful he might be able to obliterate what he'd just seen and knew to be impossible, but when he returned his gaze to the bar he saw the man grinning at him with malicious glee. Both he and the bartender began to laugh.\n\nRooster shuddered. \"We need to get outta here.\"\n\n\"Don't matter for me no more.\"\n\nRooster reached across the table, took hold of Snow's wrist. It was cold as ice. He let go. \"I'm not leaving you here, man.\"\n\n\"I'm already dead. Been dead and buried for years.\" Snow's eyes suddenly looked empty, even more hopeless than before. \"And so have you.\"\n\n## -4-\n\nHe'd stood in the bus terminal for more than an hour. There was no sign of the Crown Vic or anyone following him on foot, but Rooster couldn't shake the feeling he'd been tailed. So he stayed put, watched and waited.\n\nPeople came and went, maintenance workers and ticket agents busied themselves with various duties, an occasional policeman drifted through, and a handful of homeless people sat in corners or, like many of the waiting passengers, occupied one of the numerous plastic chairs bolted to the floor in clusters and rows throughout the station.\n\nThe entire place smelled like a combination of filthy socks, urine and body odor, all of it made more oppressive by smothering bursts of forced hot air from an archaic heating system set far too high.\n\nDirectly across from the wall Rooster was leaned against stood a bank of lockers. He'd been fingering the key in his pocket since he arrived, and though he'd yet to approach it, he'd already zeroed in on the appropriate locker. He still couldn't be certain he wanted to know what was waiting for him behind that little metal door. His life was complicated and confused enough. Did he really need to up the ante? Then again, could he afford not to? Snow had assured him the answers to his torment could be found within and he had no reason to doubt him. Even if it was a Pandora's Box (and Rooster was certain it would be), how could he not open it?\n\nFuck it.\n\nPushing away from the wall, Rooster walked toward the lockers, casually sliding his hand from his pocket and holding the key down by his thigh.\n\nThey're after me, man.\n\nNo one seemed to notice as he closed on the locker, pushed the key into the slot then pulled the latch.\n\nAnd they know I'm trying to pass the information to you.\n\nRooster swung open the door, saw a black leather briefcase inside.\n\nOnce they know you got it, they're gonna come after you, too.\n\nHeart racing, he reached inside, yanked it free and walked away, leaving the locker open and the key still in the lock.\n\nMoving through the sliding front doors and into the cold but fresh air, Rooster hurried down the block and slipped into the first alley he came to, using it to cut through to the next busy street, where he disappeared into the flow of the crowd on the nearest sidewalk.\n\nNight fell across the city as darkness swept through him, awakening demons eager to tear at a soul already in ruin.\n\nThe fires of Hell burned on.\n\n* * * *\n\nHe'd always felt relatively safe at the apartment. Now he wasn't so sure. As he'd crossed town he noticed no tails, but knew he was being watched. Even when he'd hurried across the courtyard and into the projects, the area cold and empty but for one lone child sitting on a stoop a few buildings down, he still felt an overwhelming sense that someone was following him. Once inside he bolted the apartment door, pulled the shades on the windows then set the briefcase on the kitchen table. He remained still and quiet a moment, listening. Some distant sounds from neighboring units bled through the thin walls and the building settled and creaked against the increasing wind, but he could discern nothing out of the ordinary. Next he returned to the windows facing the street and courtyard, spending a few seconds at each one, pulling back the shades enough to peek out and inspect the area for intruders, strange cars or individuals. Nothing.\n\nRooster checked his watch. Gaby wouldn't get home from work for about another hour. He'd have the place to himself for a while. With only a small hanging light in the kitchen illuminating the area, he grabbed a bottle of whiskey from the cupboard and poured a shot. As the booze burned then warmed him, he pulled up a chair and sat at the table, eyeing the briefcase as if he expected it to do something other than sit there like the inanimate object it was. A basic black leather model, it had only one main zippered compartment and no markings or personalized indications of any kind. He looked at his hands. Still shaking. For Christ's sake, he thought. Get a grip. Back in the day he'd been known for his remarkable cool in the face of danger. Hadn't he? Like so much else it was lost in a dark sea of partial memories, fractured dreams and uncertain yesterdays.\n\nHe pounded down another mouthful of whiskey then held the empty shot glass out before him until he'd willed the trembling to stop. Hand finally steady, or at least reasonably close, he put the glass aside, unzipped the briefcase and reached inside. His hand returned holding a large manila folder held shut by two thick rubber bands. The only other item in the briefcase was a hardcover book. Rooster placed both on the table before him, quickly inspected the briefcase to make certain he'd gotten everything then put it on the floor by his feet.\n\nThere were no markings on the exterior of the manila folder itself but it was stuffed with various documents. The book was black, had no dust-jacket and was badly worn, the back cover blank. Rooster flipped it over.\n\nA bright red inverted pentagram filled the front cover, the title in matching color above it: DEMONOLOGY: Incantations.\n\nHe vaulted back and away from the table as if hit with an electrical charge, eyes transfixed on the pentagram as his chair tipped over backwards and fell to the floor.\n\nWhat do you know about demons?\n\nFear crashed him like a wave, surging up through his legs, guts, and into his chest, chills firing through his shoulders and neck, his eyes burning as the uncontrollable shivers returned, this time violently throttling his entire body.\n\nI'm so cold...\n\nVoices in his head...familiar voices...\n\nI'm so...so...cold...\n\nFlashes of a face stricken with horror, mouth ripped open into a bloody and devilish grin, the skin on the cheeks and forehead moving and tenting impossibly, like something was trapped beneath and trying to get out, something barbed and small slithering for purchase...\n\nHelp me...God in Heaven, help me!\n\nClutching his temples, Rooster staggered back, muttering prayers he hadn't recited since childhood.\n\nShadowy visions of a man standing over a body, the stomach cavity split open, his hands grasping a tangle of viscera\u2014ropes of blood and guts squished between his fingers\u2014laughing and squatting closer to the carnage, his face spattered with blood and colorless jellylike fluids, shards of human flesh dangling from the corners of his mouth...and all the while, horrible screams of agony bellowed amidst vicious laughter...\n\nIt wasn't until Rooster felt the far kitchen wall against his back and slid down to the floor in a heap, sobbing and moaning like a traumatized child, that the visions and voices finally retreated.\n\nBut not before he realized that the face of the man he'd seen\u2014the man in the shadows disemboweling another human being\u2014was his own. \n\n## -5-\n\nThey move across the field in a staggered line, weapons drawn, the overgrown grass and weeds nearly to their waists. The fog moves with them as they negotiate the uneven terrain, slowly, cautiously, the darkness deepening with each step they take. The scarecrows watch from their wooden crosses, some nailed, some tied with rotting lengths of rope, manlike ghouls in old and torn denim overalls and decayed work shirts, hands of straw protruding from the sleeves like talons, legs dangling, vanishing into the tall grass. With badly worn, stuffed and filthy burlap sacks for heads, their mouths are stiff grim lines of worn leather thread sewn into the fabric in a disturbing crisscross pattern, their eyes sunken black holes, as if the sockets have been long-since picked clean.\n\nStarker is in the lead. He stops and the others follow suit. His eyes pan the area, take in each scarecrow. No one speaks for several seconds. The night is unnaturally quiet.\n\n\"Come on, what is this bullshit?\" Landon moans. \"We're in the middle of nowhere. Nobody's been here in years. Why bother with the house at all? Let's split the take now. What difference does it make?\"\n\n\"We are pretty far from the road.\" Nauls looks back. \"Haven't seen any cars pass by the whole time we've been here.\"\n\nIgnoring them, Rooster looks to Starker. \"What's wrong?\"\n\n\"Notice anything about those scarecrows?\"\n\n\"I'm trying not to notice them at all,\" Nauls says.\n\nLandon rolls his eyes. \"What are you, five-fucking-years-old? There's nobody here but us, let's get on with it.\"\n\n\"Starker,\" Rooster presses, \"what is it?\"\n\n\"There are six of them,\" he says, \"six scarecrows.\"\n\nSnow shrugs. \"So?\"\n\n\"There are six of us.\"\n\n* * * *\n\nRooster studied the shadows cast throughout the kitchen, opaque swathes of darkness slashing the light. Still on the floor and covered in a thin sheen of perspiration, his flesh was clammy and hot but his breathing and heartbeat had finally returned to normal. He wasn't sure why the pentagram specifically had triggered such terror, he only knew it had. His fear had weakened, though it was still close by, and a steady throb above his eyes signaled another headache was on its way. Luckily the pain hadn't kicked in yet.\n\nWith a willful grunt he forced himself to his feet, and on shaky legs, returned to the table. Once he'd righted the chair he dropped back into it then cautiously reached for the book. The cover was old and shabby, rough in his hands. Without looking at the pentagram, he quickly flipped open the cover.\n\nIn rather ornate script, printed on the first page:\n\n\"The other shape,\n\nIf shape it might be call'd, that shape had none,\n\nDistinguishable in member, joint, or limb;\n\nOr substance might be call'd that shadow seem'd;\n\nFor each seem'd either; black it stood as night,\n\nFierce as ten furies, terrible as Hell,\n\nAnd shook a dreadful dart; what seem'd his head\n\nThe likeness of a kingly crown had on.\n\nSatan was now at hand; and from his seat\n\nThe monster, moving onward, came as fast\n\nWith horrid strides; Hell trembled as he strode.\"\n\n\u2014John Milton\n\nThe pages of the book looked even older than the cover. Made of stiff thick parchment, faded and badly furrowed, they mostly sported what appeared to be very old drawings of demons. Hideous winged creatures with leering eyes, many with horns and cloven hooves, huddled in darkness. Others perched over the beds of unsuspecting sleeping victims or sat on blasphemous thrones of human bone. Others still were illustrated engulfed in flames or in mid-flight amidst the clouds, tangled in battle with angels. But for the cover and Milton quote, the text was written in Latin, in a calligraphy-like style, as if scribed by some mad medieval monk in the bowels of a candlelit monastery. Just touching the book made Rooster uncomfortable, so he quickly flipped through the remaining pages of lurid illustrations and indecipherable text then slammed it shut. Placing it facedown, he took another shot of whiskey.\n\nWhen his nerves had settled a bit, he turned his attention to the manila folder. Six files were individually bound and stacked within, the front of each marked with a name: Paul Carbone, Terrell Snow, Anthony Starker, Perry Nauls, Thomas Landon, and the sixth and final file, his own, Michael Cantrell.\n\nRather than immediately delve into his own file, he decided to begin with someone else's. Carbone's dead, he reasoned, I'll start there. He opened the file to find a mug-shot staring back at him. He hadn't seen Carbone in anything but nightmares for years, and looking into the man's eyes now shook him to the core. He remembered Carbone as a short and stocky man of few words, with a dry but cutting sense of humor and a laid-back personality. But mostly he remembered him screaming in agony and begging for his mother as he bled to death.\n\nRooster moved to the next page. All of Carbone's stats were there: his full legal name, date of birth and social security number. Lower on the page it listed no living next-of-kin, the fact that he'd never graduated high school and had no formal education beyond the tenth grade, and that he was unmarried and had no children. The next page revealed a bullet list regarding his criminal record, which went back to his late teens and covered everything from petty theft to numerous sexual assaults and indecent exposures, to child pornography charges to assault and battery. The final entry, highlighted in yellow, documented his final arrest and conviction, the rape and stabbing death of a seven-year-old girl. He'd received two Life Sentences with no chance of parole, and according to the entry, had been serving them at the time this information had been originally compiled.\n\n\"That's bullshit,\" Rooster muttered. He hadn't known Carbone that well, but Snow had, and he'd have never aligned himself with that kind of scum. Carbone was a criminal like the rest of them for sure, but he wasn't a sexual deviant or a child killer. They were thieves, they didn't rape and butcher children. And besides, even if Carbone had been guilty of such things and given those sentences, why hadn't he been inside serving them? Had they let him out? Had he escaped? None of it made any sense.\n\nHe went back to the photograph. It wasn't an actual mug-shot, as he'd originally thought, it only looked like one. Instead it was simply a headshot of Carbone from the neck up, a black background behind him and his name stenciled along the bottom white border.\n\nThe last page of Carbone's file contained a single word: DECEASED.\n\nThe file seemed thrown together and incomplete, as if someone had hastily transcribed a few important basic points, added a photograph then bound and stuffed the information into a folder. Rooster put it aside and moved to the next one.\n\nStarker's file contained a similar photograph and described him as a former Army Ranger that had received a dishonorable discharge and had served four years in a military prison for assaulting an officer. His personal stats were listed as well, including that he was single and had no children. His civilian criminal record began after his stint in the service, and consisted mostly of assaults and illegal weapons charges. It also listed him as a member of a radical political and paramilitary group the government had labeled as a terrorist organization responsible for the numerous bombings of several government buildings. His final conviction described him as one of a three-man team that had firebombed the campaign office of a political candidate their organization opposed. Four people had been killed in the bombing, including two women, one of them eight-months pregnant. Starker, along with his accomplices, had received Death.\n\nThis information was more believable\u2014Starker had always been the most violent of the crew and the most unpredictable\u2014but again, much of it made no sense. Starker wasn't single, he was married\u2014or at least had been, according to Snow he'd since murdered his wife\u2014and although Rooster did know about Starker's prior military service, he knew nothing about this radical political organization he'd supposedly been a member of, and certainly nothing of the firebombing of a campaign office. And again, if that were true, and he'd received a death sentence and had already begun to serve time on Death Row as the information suggested, how had he been with them the night of the armored car job?\n\n\"He couldn't be.\"\n\nThis time the final page contained the word TERMINATED.\n\nRooster reached for the bottle, poured another shot of whiskey.\n\nTerminated? But Starker wasn't dead. Unless they'd killed him...whoever the hell they were.\n\nNext came Nauls. The face in the photograph showed that same narrow face with the beady eyes he remembered. A closely-cropped beard and wild nest of curly hair coupled with his thin build gave him the look of a stoner or wannabe rock musician, and in reality, he'd been a little of both. In fact it was strange to see his eyes at all, as Nauls had almost always worn a pair of dark sunglasses, the lenses small, round and tight to his face. His file described a man who had been in and out of jail from the time he'd been a teenager, and who began serving prison time at only twenty. Predominantly a thief, he'd been arrested countless times for B&Es, purse-snatching, shoplifting, and drug possession. By all accounts Nauls had been a petty thief but not the least bit violent. In his mid-twenties he'd graduated to bank robbery and done time for it in federal prison. Like so many others, Nauls had come out of prison far worse than he'd gone in, as according to the paperwork, two months after his release he was arrested for another bank robbery, one that ended particularly violently.\n\nThe report claimed Nauls, cornered in the bank, had taken several tellers and the bank manager hostage. After a fourteen-hour standoff, Nauls had been refused the helicopter he'd demanded for his escape, and as a result had executed a female teller and then the bank manager. He was shot by a SWAT sniper moments later. Hit in the upper right chest, Nauls survived.\n\nIronically, he was sentenced to Death.\n\nRooster knew Nauls to be the most harmless member of the crew, and also the least violent. He spent most of his time smoking pot, chasing women, strumming an old guitar he loved and watching cartoons. He was a thief\u2014and a good one\u2014but not that bright and generally clueless. He was damaged, the kind of guy who had done hard time and wasn't really cut out for it. Far as he knew, Nauls had an extensive criminal past but he wasn't a killer, and the idea that he could've executed two people in cold blood seemed beyond belief.\n\nThe last page was the same as Starker's. TERMINATED.\n\nLandon too looked exactly how Rooster remembered him, as a man of average build with short dark hair receded to the middle of his scalp, hazel eyes, a permanent five o'clock shadow, an aquiline nose with flared nostrils and a mouth that seemed perpetually set in a wiseass smirk. His file depicted a man with a long criminal record, the majority of his arrests involving car theft or driving violations. Landon had always been a car nut, and was one of the best drivers Rooster had ever seen\u2014certainly the best he'd ever worked with\u2014and though he had a temper, complained endlessly and never backed down from a physical confrontation, he'd never been a particularly violent individual. He had the ability to be violent, and Rooster remembered more than one occasion when Landon had handled himself competently in physical skirmishes, but for the most part it was his mouth one had to look out for. Landon could cut someone to shreds verbally without even trying. He'd begun his criminal career stealing cars as a teenager, and by the time he was in his twenties he'd done time for auto theft and for two counts of aggravated assault. In and out of prison for most of his twenties, he was later arrested as the wheelman on a jewelry store heist. He and his accomplices had escaped but not before police were on them, and in the resulting high-speed chase Landon plowed directly through a police barricade, killing two police officers. After losing control of the car he struck a group of pedestrians, killing two\u2014including an elderly man and a woman who had been holding her four-year-old child at the time\u2014and seriously injuring several others. Landon drove on. Several blocks later his tires were shot out and he crashed into a telephone pole. One of his accomplices died in the crash. The other survived but was gunned down as he attempted to flee the scene. Landon suffered several minor injuries but survived. He was given Life without parole.\n\nSame final page: TERMINATED.\n\nRooster shook his head in disbelief and turned to Snow's file.\n\nTerrell Snow looked the same in the photo as he had at the bar earlier. His record was long and varied, consisting of everything from theft to assault to attempted murder to drug charges. A lifelong criminal and former gang member, Snow had, according to the paperwork at least, struggled with heroin addiction at one point earlier in his life. Something even Snow himself had been unaware of.\n\nWhich means it's crap, Rooster thought.\n\nAfter a long criminal career, the result of which was Snow spending the majority of his adult life in prison, he'd been convicted of beating a young woman to death in her apartment during a botched robbery.\n\nI didn't know who she was, didn't know what I'd done.\n\nThe crime was listed as 'particularly vicious' in that the woman had apparently not resisted her assailant but had been beaten so mercilessly that police were initially unable to determine if the victim was male or female.\n\nI don't even remember it. I was on H when it went down and was hurting so bad for a fix I was out of my mind.\n\nSnow received Life without parole.\n\nI never meant to hurt her.\n\nLast page: TERMINATED.\n\nI'm already dead. Been dead and buried for years.\n\nOf course he'd meant it figuratively, but Rooster couldn't help but wonder if there wasn't more truth to Snow's statement than he'd originally been willing to give it. He put the file aside and eyed the final one, his own. He downed another shot, felt his head swim a bit.\n\nThere he was looking back at himself in a photograph Rooster had no memory of ever posing for. His basic stats were all correct, as were the entries concerning his criminal record. He'd served several jail sentences over the years, having been arrested numerous times for theft and assault (once with a deadly weapon), but he'd only gone to prison twice. Once for his involvement in an armed bank robbery for which he served six years of a ten-year sentence, and the other, his final conviction for which he received Death.\n\nThis is ridiculous, he thought. How could I have served time on Death Row without having any memory of it? And what am I doing out even if I did?\n\nHe continued reading. He'd been given Death for the torture and murder of a man named Roland McKay.\n\nA Roman Catholic priest.\n\nRooster's breath caught at the base of his throat, and he brought a hand to his mouth for fear a literal gasp might escape his lips. His mind replayed the memory of the priest accosting him on the street. How could this be? He had no memory of ever murdering anyone, much less a priest. He was a thief like the rest of the crew, not some sadistic psychopath. And if he'd killed this man, how could he be stalking the city streets pointing an accusatory finger at anyone?\n\nThe files were all there in front of him in black-and-white. But not one of them made any goddamn sense. The information couldn't be true.\n\nHesitantly, Rooster turned to the final page of his file.\n\nTERMINATED. \n\n## -6-\n\nHe gathered up the files and threw them back into the briefcase on the floor. As he reached for the book he saw a business card lying on the table he hadn't noticed previously. An address had been written on one side, a phone number on the other. Both had been written in ballpoint pen, and though legible, appeared hastily scribbled by a less than steady hand. Mind still reeling, Rooster considered the card a moment then grabbed the wall phone and dialed.\n\n\"We're sorry,\" a recorded female voice replied, \"the number you dialed is not in service. Please check the number and try again.\"\n\nHe hung up and tried again. Perhaps the five shots of Jack Daniels had caused him to misdial. This time he concentrated on each number to make sure he got it right, but the same recording answered. He slammed the phone down, fear and uncertainty giving way to anger. It was short-lived. Within seconds of hanging up, the phone began to ring. Startled, he slowly reached for the receiver and brought it to his ear. He could hear breathing. \"Yes?\"\n\n\"Hello, Mr. Cantrell.\" The voice was raspy and weak, like it belonged to a very tired old man. \"You dialed the number. Obviously you've seen the files.\"\n\n\"Who are you?\"\n\n\"Look on the other side of the card,\" the voice instructed. \"Do you see an address there?\"\n\n\"Yes.\"\n\n\"Be there tomorrow morning. Ten o'clock.\"\n\n\"No,\" Rooster said, \"let's do this tonight. I want this over with.\"\n\n\"Tomorrow morning. Ten o'clock.\"\n\n\"How will I know you?\"\n\n\"I'll know you. Come alone.\"\n\nThe line clicked, died and was replaced with a dial tone. Rooster grabbed the card, read the address again. It meant nothing to him, just an address. His mind on overload, he tried to consider the information in the files again but couldn't make sense of it. He knew those men. None of them were guilty of such things. And why in God's name would he have tortured and murdered anyone? Why would someone invent pasts and former crimes for him and the others? Why would they compile files with false information about things that never happened? What could possibly be the point?\n\nRooster snatched the phone up again and this time dialed the number Snow had given him. He'd promised the information would answer his questions and tell him everything he needed to know. It hadn't. The line rang several times without reply, and he was just about to hang up when he heard a soft click. The ringing ceased. \"Hello?\" he said a moment later.\n\n\"Who is this?\" The voice was strange. Though male, it had a synthetic quality to it, like the person was speaking through a machine of some sort.\n\n\"Where's Snow?\"\n\n\"Who is this?\"\n\n\"I need to speak to Snow, put him on the phone.\"\n\n\"Who is this?\"\n\n\"Who the hell is this?\"\n\nThe voice answered in what began as English but quickly morphed into an indecipherable tongue, eventually becoming a deafening screech somewhere between a scream and a rage-filled, animal-like howl. Rooster pulled the phone from his ear, holding it several inches away, but the horrible wailing continued. He knew those sounds. He'd heard them before, somewhere in a distant and blurred past. Wracked with another wave of terror, he hung the phone up and backed away, stumbling into the kitchen table as he went.\n\nA loud clap behind him sent a shiver through his body as he spun in the direction of the noise.\n\nHe'd knocked the book to the floor.\n\nHe retrieved it, tossed it on the table then grabbed the whiskey and poured another shot.\n\nThe violent tremor in his hands had returned.\n\n* * * *\n\nThe jangle of Gaby's keys in the lock startled him. Huddled at the kitchen table, Rooster had become so enthralled while further studying the book on Demonology that he hadn't heard Gaby ascending the stairs to their apartment. He'd stopped at a depiction of a particularly gruesome-looking demon with blackened wings and a hideous, half-goat, half-human face. Squatting atop a mountain of mangled and dismembered human bodies, in one of its clawed hands it held the severed head of a woman, and in the other what appeared to be a male member. Rooster rubbed his eyes, looked over at Gaby.\n\n\"Hey,\" she said, closing the door behind her. In her arms she held a brown paper bag from the neighborhood grocer. Beneath her heavy winter coat she wore a plain dress and a pair of black heels. Her hair was up and held in place with a clip but had become mussed, probably from the wind. She looked tired. \"How'd the job hunt go?\"\n\n\"Lock the door.\"\n\nShe did, then put the bag on the counter, removed her coat and walked over to the table. They kissed. \"You OK? Why is it so dark in here?\" She headed for a lamp in the den.\n\n\"Don't.\"\n\nGaby stopped, looked at him quizzically.\n\n\"Just don't. OK?\"\n\nAs if not entirely sure what to make of him, she moved back toward the table. \"What's that?\" she asked, referring to the book. Before he could answer she saw the illustration. \"What are you doing with that?\"\n\nRooster closed the book so she could see the cover.\n\n\"Demonology? I don't want that in the house.\"\n\n\"Neither do I,\" he sighed.\n\n\"Then get rid of it.\" She picked up the whiskey bottle and took it with her to the counter, where she dropped it off then began emptying the grocery bag. \"Sorry babe, I had a long day, just didn't feel like cooking.\" She held up two TV dinners. \"Got you that Salisbury steak one you like, OK?\"\n\nHe followed her to the counter, grabbed his cigarettes and lit one. \"Do you believe in them?\"\n\n\"Demons?\" she asked, busying herself with the oven. \"Do you?\"\n\n\"The book supposedly shows what they look like, and it has incantations written in Latin. Is that how people summon them?\"\n\n\"Why would anyone want to summon demons?\" Gaby unwrapped both dinners and left them on top of the stove. \"It'll just take a minute to preheat and I'll get these in.\"\n\n\"I called Snow,\" he said. \"We met this afternoon.\"\n\n\"Is that where you got the book?\"\n\n\"That and the briefcase,\" he said, motioning to it.\n\n\"Why would he give you a book like that? And what's in the briefcase?\"\n\nRooster took a couple drags before answering. \"It's better if you don't know.\"\n\n\"Is that why he kept calling? So he could tell you secrets?\"\n\n\"Gaby,\" he said, clearing his throat. \"I need to ask you something.\"\n\nShe stopped futzing about the kitchen and focused on him, dark eyes narrowed as if trying to see him more clearly. \"OK.\"\n\n\"How long have we known each other?\"\n\n\"Seems like forever, doesn't it?\"\n\n\"How did we meet?\" he asked.\n\nShe smiled uncomfortably. \"Are you serious?\"\n\n\"Yes.\"\n\n\"You don't remember?\"\n\nTears filled his eyes. He shook his head no, brought the cigarette to his lips and drew on it, hard. \"I can't...I don't know what's happening to me but\u2014something's wrong, Gaby\u2014I think I'm losing my mind or...worse.\"\n\nShe put a hand on his forehead. \"You're warm. Feels like you're running a bit of a temp. Let me get you some aspirin.\"\n\nHe gently pulled her hand away but held on tight, watching her blur through his tears. \"I know I know you but...Gaby...I don't know who you are. I'm not even sure who I am.\"\n\n\"You haven't slept, you're drinking, and now you've got a bad influence from your past giving you scary books and making things worse.\" She moved by him, grabbed the book from the table and tucked it into the briefcase. \"No wonder you're not feeling well and can't think clearly. Get this out of here or I'll take it out to the Dumpster myself. I'm serious.\"\n\n\"I need you to tell me, Gaby, please, I\u2014\"\n\n\"You need something to eat, a nice hot shower and some sleep. I'll\u2014\"\n\n\"Stop it!\" He smashed a fist on the counter. The entire room shook. \"Fucking answer me!\"\n\nGaby remained where she was, hugging herself. In a tiny voice she said, \"You're frightening me.\"\n\n\"I'm sorry.\" Rooster threw the remains of his cigarette into the kitchen sink then began pacing like a caged animal. \"I'm sorry, I\u2014Jesus Christ, what's happening to me?\"\n\nShe cautiously stepped closer. \"It's going to be all right.\"\n\nNo longer able to control it, he wept openly.\n\nClosing the gap between them, Gaby cupped his face in her hands. \"Look at me.\" He did. \"It's going to be all right.\"\n\n\"Am I crazy?\"\n\nShe pulled him into her, held his head tight to her breasts and kissed the top of his head. \"No, baby, you're not crazy. You're just trying to find your way.\"\n\n\"I think they're after me, Gaby, I think the demons are\u2014something's happened to me, I can't remember things and\u2014\"\n\n\"Nothing can hurt you while you're with me.\" She gave him a quick wink. \"My love's way too powerful for any demon, real or imagined. They mess with my man I'll kick the slithery-tailed little pukes back to Hell where they belong.\"\n\nRooster wanted to smile, but the terror remained.\n\n\"Come on. Rest while I get some chores done and dinner together.\" She led him into the den, helped him onto the couch then switched on the console television in the corner. \"Watch some TV.\"\n\nAs the set came on, Gaby retreated to the kitchen, leaving him alone. He wiped his eyes and nose and sunk deeper into the couch, hiding in the shadows.\n\nA news anchor with bad skin and an even worse comb-over sat at a stylish desk, an ACTION NEWS 8 banner on the wall behind him. Decked out in a yellow polyester blazer and ridiculously wide tie, he shuffled a stack of papers and continued relaying a story he'd begun a moment or two earlier. \"According to eyewitnesses, the black male exited the bar on Cafferty Boulevard and darted directly into traffic. He was struck by what has been described as a large black sedan, possibly a Ford, which fled the scene. Paramedics are working on the man now and we hope to have a live report from the scene very shortly.\"\n\nRooster sat up. The bar he'd met Snow at earlier was on Cafferty Boulevard.\n\n\"One eyewitness told Action News 8 the man appeared disoriented and was running as if being chased, though that did not seem to be the case. It's not yet known if the man was intoxicated or under the influence of narcotics, but\u2014one moment...\" The anchor put a hand to his ear, listened to the voice in his earpiece then paused for dramatic effect and frowned as if personally devastated. \"This just in: the victim, identified as Terrell B. Snow, has been pronounced dead on the scene. As further details become available on this horrific hit-and-run tragedy, we will\u2014\"\n\nRooster turned the television off. The apartment was quiet. He looked to the kitchen. The TV dinners were still on top of the stove but Gaby was nowhere to be found. He hurried through the apartment to the bedroom.\n\nLight filled the room as he flipped the switch. Half-expecting to see the horrible winged and long-tailed creatures in the book flying about, he was relieved to find only shadows, an aged bedroom set and the usual open window. He went directly to the closet and pulled an old shoebox down from the shelf. Inside, a 9mm, a full clip and two boxes of ammunition were wrapped in a cloth. He couldn't remember the last time he'd even touched the gun, much less fired it, but he scooped it up, deftly slapped the clip into place, chambered a round and released the safety. Something about holding the gun steadied his hand.\n\nA cold breeze blew through the room, disturbing the curtains. He moved to close the window but froze. Beneath a streetlight just beyond the courtyard, a lone man was watching the building.\n\nA priest.\n\n\"What's wrong?\"\n\nRooster glanced behind him. Gaby stood in the doorway, a laundry basket of freshly folded clothes in her arms. \"How did you know that would be on the news?\" he asked.\n\n\"How did I know what would be on the news?\" She noticed his weapon and her face went pale. \"Michael, why do you have a gun?\"\n\n\"Turn off the light,\" he instructed. \"Do it now.\"\n\nShe did. They fell into darkness.\n\nRooster looked back out the window. The priest was gone. \n\n## -7-\n\nSilence fills the night again.\n\n\"Starker's right,\" Nauls says, \"six scarecrows...six of us.\"\n\n\"Not anymore.\" Landon makes sure he smiles at Snow before he takes the lead, moves by the first scarecrow and heads for the rotting remnants of the old farmhouse. \"Scratch one Carbone. Dead guys don't count.\"\n\n\"Before this night's over,\" Snow mutters, \"I'm gonna end that fuck.\"\n\nThe others move on, following Landon now, who has gotten several yards ahead of them and is barely perceptible in the darkness and fog. When they catch up to him, they find themselves standing before a ramshackle two-story structure with a dilapidated porch. To the side of the house and further back on the property is a barn in even worse shape. From the face of the farmhouse, a series of blown-out windows stare down at them, opaque eyes gaping in judgment, perhaps in warning.\n\nA rusted metal sign has been staked a few feet from the front porch steps.\n\n~ KEEP OUT\u2014THIS PROPERTY IS CONDEMNED~\n\n\"Yeah,\" Landon says with a smirk, \"didn't see that coming at all.\"\n\nRooster immediately feels something so unsettling it leaves him breathless. He squints through the darkness at the looming structure. \"I know this place,\" he hears himself say.\n\nSnow nods, eyes fixed on the house, his mouth hanging open. \"So do I.\"\n\n\"Me too,\" Nauls says, voice shaking.\n\n\"Like we've been here before,\" Starker says.\n\n\"Don't worry about it.\" Landon tests the first step, and once satisfied it will hold his weight, climbs up onto the porch. \"These old farmhouses upstate all look alike. You're spooked, that's all. Come on.\" He ambles across the porch to the front door, which lies on its side next to the doorframe.\n\n\"Got to love Landon.\" Nauls chuckles nervously and climbs the stairs, hoisting the duffel bags of cash along with him. \"He ain't afraid of anything.\"\n\nSnow climbs the steps next. \"Too busy being an asshole.\"\n\n\"Let's get this done.\" Battling uncertainty, confusion and a growing sense of dread, Rooster forces himself up the steps. \"I don't want to be here.\"\n\nAlready fearful they will never leave this awful place, Starker, who began in the lead, is the last to enter the house.\n\nHe joins the others in a large filthy room just inside the entrance. A few broken pieces of what was once furniture are scattered about the otherwise empty area. The floor is rotted in several spots, littered with jagged holes.\n\nLandon sticks the revolver he's been carrying into his belt and pulls free a flashlight. He switches it on, punching a hole in the darkness. Countless dust motes float about in the beam. He sweeps it around. Thick spider webs dangle from the ceiling and fill every corner. A moth flits into the light then spirals off. \"Check it out, Nauls. Looks like your apartment, only nicer.\"\n\nTheir movements disturb something in the air, stirring up a pungent odor.\n\n\"What the hell is that smell?\" Nauls asks, dropping the duffels to the floor and crouching down next to them.\n\nLandon points the flashlight at Snow. \"Dude. Seriously. Put your shoes back on.\"\n\n\"You don't get that off me it's going up your ass sideways.\"\n\nDrifting deeper into the room, Starker watches the ceiling as if expecting something to attack from above. His considerable size causes the floor to creak and shift. He sniffs the air. \"It's sulfur.\"\n\nNauls opens the first duffel, stares at it dumbly a moment then scrambles to the second one and begins rifling through it. \"Landon, put the light here!\"\n\nHe illuminates the duffels. Both are stuffed with neatly banded pieces of blank paper designed to resemble money.\n\nSnow leans in for a closer look. \"Where's the cash?\"\n\n\"It was here,\" Nauls says, \"I\u2014\"\n\n\"Unbelievable!\" Landon spits. \"You assholes stole scrap paper!\"\n\nRooster steps back for a better angle on the others.\n\nNauls struggles to his feet. \"Me and Rooster loaded the cash into the bags. I saw it. It was all there. The bags were full of it.\"\n\nLandon draws his revolver. \"Yeah they're full of it all right.\" He points it at Rooster. \"Where the fuck's my money, crew chief?\"\n\nRooster, Snow and Nauls simultaneously pull their weapons and point them at each other. Preoccupied, and unconcerned with the others, Starker wanders to the back of the room, where a large unusable staircase resides. Littered with broken wood and debris, he gazes up into the shadows of the second-floor. Something dead\u2014probably an animal of some sort, though he cannot be sure\u2014lies in a mangled heap at the very edge of the landing. The walls and upper portion of the banister are streaked with what might be blood.\n\n\"Everybody calm down,\" Rooster says. \"We'll figure this out, we\u2014\"\n\n\"Fuck that,\" Landon snaps. \"Somebody switched out those bags or the money or something and one of you pricks is gonna tell me what's going on or I swear to God I'll shoot every last fucking one of you.\"\n\n\"How could we switch the bags out?\" Nauls frantically moves his gun from one person to the next then back again. \"They went straight from the armored car to the van, and we were all in the van until we got here. Nobody could switch anything out! We were together the whole time!\"\n\nSnow, who has been holding one of his .45s on Landon and the other on Nauls, lowers them both. \"He's right.\"\n\n\"I don't give a shit,\" Landon says. \"That money didn't just disappear, so where is it? Rooster, you and Nauls were the ones who loaded it, and since Nauls is a fucking mongoloid, you better start talking.\"\n\n\"Mongoloid?\" Nauls cocks an eyebrow. \"What the hell is a mongoloid?\"\n\n\"It's them little elf-looking motherfuckers,\" Snow explains, \"the ones with the pointy heads and shit.\"\n\n\"No, those are cretins,\" Landon says. \"Mongoloids are the redheads.\"\n\nNauls tucks his gun into the back of his pants. \"I don't have red hair.\"\n\nLandon sighs but keeps his attention on Rooster, who lowers his weapon as a peace offering. \"Get your piece off me,\" he says, \"and we'll figure this shit out.\"\n\n\"Nah, asshole, first you're gonna tell me where the\u2014\"\n\nAn enormous muscle-bound arm shoots out of the darkness behind him and wraps around Landon's throat, strangling him with such force that his feet leave the ground. He drops his revolver and the flashlight and clutches at the arm with both hands in a futile attempt to dislodge it. The flashlight rolls across the floor, tumbling through the room and painting the farmhouse with sweeping arcs of twisting light that eerily illuminates then plunges each man back into darkness. \"Listen to me and listen to me good,\" Starker says, holding the smaller man effortlessly, his voice just above a whisper in Landon's ear. \"We got a lot more to worry about here than that money. Now you cut the shit, keep your mouth shut and do what Rooster tells you to do or I'll snap your neck. You feel me, boy?\" Landon manages a gurgling response and Starker releases him. He crashes to the floor with a thud and one of his feet breaks through the boards.\n\nLandon lays there a moment, clutching his throat, then pulls free, retrieves his revolver and slowly returns to his feet without further comment.\n\nNauls scurries to the corner and retrieves the flashlight. As he brings it round, he stops on something beneath the old staircase. \"Hey, there's a\u2014\"\n\n\"Door under the stairs,\" Rooster interrupts. He knows he's right but has no idea how he's come to possess such information.\n\nStarker finds Rooster's face in the dark. \"It leads to another staircase.\"\n\n\"Then a hallway,\" Snow says quietly.\n\n\"And there's doors on both sides of the hallway,\" Nauls adds.\n\nEveryone looks to Landon. He rubs at his throat. \"Oh I'm allowed to talk now?\" He glares at Starker. \"Just wanna make sure it's OK with fucking Albert DeSalvo over here before I say anything.\" Nauls aims the light at him, leaving no doubt that despite his bravado, even Landon is terrified by what's happening. He finally nods reluctantly, fidgeting about tensely. \"Yeah, I\u2014I don't know how I know it either, but behind the doors there's a bunch of rooms.\"\n\n\"Even if we're right, end of the day it's just an abandoned old farmhouse with scarecrows out front and some rooms where a cellar ought to be,\" Snow says. \"Why we all so scared?\"\n\n\"There's only one way to find out for sure.\"\n\n\"Aw, fuck me running.\" The beam of light begins to tremble as Nauls heads for the porch. \"I want out right now, man, this is bullshit.\"\n\nStarker lifts the AK-47 higher on his hip, and with one short sidestep, blocks the doorway. \"We've all been here before. We need to know why.\"\n\n\"But what happened to the money?\" Snow asks, his face a mask of barely contained terror.\n\n\"Maybe there never was any money,\" Starker says. \"Maybe there wasn't even an armored car.\"\n\n\"Tell that to Carbone,\" Landon counters. \"Fuckhead died robbing it.\"\n\n\"Maybe that's not how he died. Maybe that's just what we remember. Maybe this is all some kind of sick game.\"\n\nNauls looks at the floor. \"Well I don't wanna play no more.\"\n\n\"Think about what he's saying,\" Rooster says. \"Does anybody really remember anything before the job today?\"\n\n\"Of course we know what happened today,\" Landon says.\n\n\"Do we?\" Rooster watches him, doing his best to keep his face void of emotion. \"Do any of you remember anything before the van? Because I'm not sure I do. I mean, I think I do, it feels like I do but...\"\n\n\"It's in your head,\" Starker says, \"but you don't actually remember it.\"\n\n\"Yeah,\" Snow agrees. \"What he said.\"\n\nRooster nods.\n\n\"So I'm the only one who wants to leave then?\" Nauls paces about wildly. \"Really? Are you guys fucking high?\" The light drifts back and forth across the dark room, cutting shadows and revealing quick glimpses of a long-dead house.\n\nIn that moment, eyes following the beam, fear wells in Rooster the likes of which he's never known. He's sure he sees something more, something there yet not quite there, waiting in the darkness, slipping from sight like scuttling insects just as the light passes over them. He grips his weapon tighter but it does little to calm his rising terror. \"We need to search this place.\"\n\n\"No we don't.\" Nauls shakes his head. \"We can just leave.\"\n\n\"We need to know what's happening here.\"\n\n\"We can't get upstairs,\" Starker tells them. \"Staircase is blocked with shit and it's all rotted out. But there's something dead up there and whoever killed it did some finger-painting with its blood.\"\n\n\"There's something wrong with this place, man, it's\u2014you guys all feel it too, I know you do. Shit Starker you and Rooster felt it outside, and...I don't...\" Nauls suddenly becomes strangely calm, his voice quiet and childlike. \"I don't want to die out here.\"\n\n\"Easy, Nauls,\" Landon says. \"Don't wanna trip and fall on your vagina.\"\n\n\"Bring the light around to the door under the stairs,\" Rooster tells him, his gaze moving between the horrified faces before him. \"We're going down there.\"\n\n* * * *\n\nAs daylight splintered night, it brought with it an icy rain that descended upon the city in violent torrents. Shaking off the residue of nightmares, waking and otherwise, Rooster adjusted his position in the chair. He'd placed it in front of the window and watched the street all night. Every muscle in his body hurt, his neck was stiff and sore and his temples pulsed with a dull ache. Ice ticked against the window, mixing with the sluicing rain to blur the glass and world beyond. Numerous lost souls had come and gone throughout the night, hurrying through the darkness, but the priest had not returned.\n\nThough he couldn't be certain, Rooster thought he'd briefly nodded off a few times during the night. After asking him countless times to put the gun away and come to bed, Gaby finally gave up a little after midnight and drifted off to sleep. She lay sprawled out across the bed, her breathing slow and deep. He watched her a while. She was the most beautiful woman he'd ever seen. It didn't seem right, Rooster thought, for someone so intelligent, so caring and just, so uncorrupted and faithful to be associated in any way with such madness and horror. Yet somehow it made perfect sense, a pure and tranquil soul like Gaby existing amidst the mayhem, calm beauty at the eye of an otherwise violent storm. His storm.\n\nHe sat on the bed next to her and gently caressed her face. She stirred and moaned quietly but remained asleep. Who are you? He wondered. Why are you here with me?\n\nThe pain in his temples drifted behind his eyes, lingering there as he gently kissed Gaby on the cheek.\n\nWith the 9mm tucked into the back of his pants, he threw on his jacket, swallowed a handful of aspirin and slipped into a cold and unforgiving rain. \n\n## -8-\n\nRooster found himself standing in the same rain some minutes later, having traced the address on the card to an old restaurant in a long-dead neighborhood. A small dark hole-in-the-wall, it sat alone between a series of boarded-up storefronts and a huge lot of bricks and debris that had once been a building. The street was filthy, cold and lifeless. No cars out in front of the restaurant, but the sign in an otherwise dark window blinked: Dante's. There was no one else around, and the second floor above the restaurant appeared deserted, most of the windows blown out or boarded up. Rooster looked to the end of the block, checking the corners in both directions. If he was being watched or tailed, they were the best he'd ever encountered.\n\nHe moved through the door, which alerted those inside to his arrival with the jingle of a little bell. His eyes slowly adjusted to the dim lighting as he was met with a blanket of thick, oppressive heat. A series of tables with red-and-white checkered tablecloths and small candles encased in glass orbs at their centers lined the walls to his left and right. The open area between them provided a path through the narrow restaurant to, he assumed, the kitchen in back, but it was so dark he couldn't make out much beyond the first few tables. The smell of burned food hung in the air, and although there was a podium for a ma\u00eetre de the restaurant appeared empty, perhaps closed.\n\n\"Here,\" a voice said from the rear of the room.\n\nRooster casually slid a hand to the gun in his belt and moved down the center aisle toward the direction of the voice. As the shadows parted, the candlelight danced along the floor and walls, flickering about, alive in the dark. As he cautiously approached the only occupied table in the place, the silhouette of a man's head and shoulders emerged.\n\n\"Mr. Cantrell.\" Not a question. Said with what almost sounded like adoration. \"Nasty rain out there this morning.\"\n\n\"Who are you?\"\n\n\"My name's not important,\" he said. \"Call me whatever you'd like.\"\n\nSame aged and drained voice as on the phone, Rooster was sure of it.\n\n\"Mr. Snow seemed fond of Poindexter.\" The man motioned to the chair across from him with a spindly arm, his hand brushing through the circle of candlelight cast across the table. Skeletal and liver-spotted, his pale flesh was laced with bulbous blue veins, the fingers gnarled with arthritis. \"Not terribly original, but we can go with that if you'd like.\"\n\n\"Snow's dead.\"\n\n\"Yes.\"\n\nRooster looked behind him. He could see the front door and the light beyond, though it seemed farther away than was possible.\n\n\"It's all right, Mr. Cantrell, you're safe here. Please. Sit.\"\n\nHe pulled the chair out, slid it to the side so he could still see the door then took a seat. He'd never cared for sitting with his back to doors. \"Who are you?\" Rooster pulled his gun and laid it flat on the table, barrel pointed at the man. \"I'm not asking again.\"\n\nUntil then the man's face had remained in shadow. He sat forward enough to allow the candlelight to reveal a glimpse of a loose-skinned face ravaged by age, his features sharp and birdlike. A pair of eyeglasses with black frames sat high on his needle nose, the flickering flame from the candle reflected in lenses so thick they might have been comical under different circumstances. \"Don't be an ass,\" he said wearily, \"put that away. Our time together is limited.\"\n\nRooster reluctantly returned the gun to his lap.\n\n\"Are the headaches getting worse?\"\n\nHe nodded.\n\n\"It happens as the mind recovers and remembers more and more. Truth always comes with some measure of pain.\" He folded his damaged hands before him on the table and sat back, his face again engulfed in darkness. \"Does The Kingdom Project mean anything to you?\"\n\nFaraway screams tore at him. \"No.\"\n\n\"Named for the famous Eliot poem 'The Hollow Men' which speaks of 'death's other kingdom' compiled with numerous books on demonology and the occult that consistently referred to the darkness on the other side as a 'kingdom of shadows,' The Kingdom Project was a top secret program begun in the late 1970s and continued until the mid-80s. The occult has always been of interest to the powers that be. Hitler spent a fortune on its study and possibilities. Many of the same scientists that worked for the Third Reich ended up here, in the United States, after World War II. They weren't all rocket scientists, Mr. Cantrell. Many were those who worked on the Reich's most classified occult projects. Their work not only continued here in the states, it expanded and went farther than even Hitler could've imagined.\"\n\nOutside, the muted sounds of a siren rose then fell away to silence.\n\n\"The early programs of the 50s and 60s met with failure,\" he continued. \"For much of the 70s nothing changed, and the majority of programs were scrapped. Many concentrated on psychic phenomenon or the like, but The Kingdom Project had different, more sophisticated ideas. Our goal was to discover a connection\u2014a bridge, if you like\u2014between our reality and the underworld. We weren't concerned with an afterlife that could only be entered through death, but rather alternate existences existing simultaneously with ours.\"\n\nA waiter materialized from the shadows holding a plate of spaghetti and meatballs and a goblet of red wine. He placed them before Poindexter without comment then slipped away.\n\n\"I understand you're not a man of science, so I won't bore you with the technical details, but suffice to say it all boils down to physics and mathematics. Our existence, our entire universe, this entire dimension, is based upon them. They all are. It's simply a matter of finding the correct equation then executing it via the proper tools. What we as well as the others before us failed to realize was that in a psychological sense, the physical world is essentially an illusion. The path to the other side, to the power we were searching for\u2014the darkness, that place of pure primal terror and evil\u2014isn't something one can find in the depths of the Earth or on a saucer ride through space or any of that nonsense. It exists in the limitless caverns of our minds. Our minds provide the gateway to the other side...the underworld...the darkness. It wasn't Heaven Hitler was searching for, Mr. Cantrell, and neither were we. In the end, these programs all have military\u2014or similar\u2014applications. The Kingdom Project was no different. We focused specifically on the dark side of the occult, the concept that things like demons, devils, demonic entities\u2014whatever you'd like to call them\u2014literally existed on some level, if not on a physical plane then perhaps a purely spiritual one. Think about it, beings of pure, unadulterated, unapologetic evil. Beings of pure rage, pure violence, pure hatred. Imagine if that level of evil truly existed in a conscious, intelligent form. Imagine the possibilities of literally summoning such creatures. Imagine harnessing their power, the very essence...of Hell.\"\n\n\"You're out of your fucking mind.\"\n\n\"No, but unfortunately you are. And I'm largely responsible for it.\" He took up a fork, poked at the food on his plate. \"Our push, specifically with The Kingdom Project, was largely chemical-based. We believed that once the bridge was found, if it truly existed outside theory and mathematical probability, could only be crossed in a spiritual way. In-other-words, psychologically, as the real-world applications of physics and mathematics had to be merged with spiritual, non-physical, synthetic components.\"\n\n\"Synthetic,\" Rooster asked, \"as in drugs?\"\n\n\"Yes, and it was only if and when these two areas were in perfect synchronization that our goals could be achieved. The mind itself had to be altered in order to access the other side. There was no question about that. You could get there from here, as it were, and the key was right before our eyes. Many ancient cultures, from Native Americans to countless tribes of people worldwide\u2014people we considered largely inferior savages\u2014already possessed the process we'd been searching for. These peoples used it to commune with paradise, to find nirvana, God, peace and transcendence. And they all used mind-altering substances to achieve it\u2014roots, leaves, plants, things of the Earth\u2014ingested before these journeys were taken. It's precisely that angle I studied and brought to the project. There were numerous formulas over several years that used pieces of these various concoctions from different cultures. And of course, as a chemist, I implemented my own mixtures, including LSD derivatives and other mind-altering substances. Many did nothing more than standard hits of LSD. The initial versions were far too strong and brought on brain damage, permanent insanity, even death in a few cases. Eventually we were able to isolate the aspects we required and produced what I believed was the perfect elixir for The Kingdom Project. Once the right formula was found the challenge became finding proper test subjects. No one sane would knowingly volunteer for such a thing, so we were forced to utilize subjects that hadn't volunteered.\"\n\nRooster tightened his grip on the gun but left it in his lap. \"You forced people to take a mind-altering drug you cooked up in a test tube?\"\n\n\"We did. And the results were interesting. Not what we'd hoped for, mind you, but very interesting.\" He twirled the fork around strands of spaghetti, brought it to his mouth and chewed. \"Many subjects experienced something,\" he said, \"but it wasn't the darkness we were searching for. Many believed it was nonsense, false near-death and other psychotic episodes brought on chemically. But I knew this was different. We were so close. The problem, you see, was not with the drug, but the subjects. I began to more closely study the nature of evil, the various interpretations of it in different cultures and varied religions, and though they were often vastly different, I uncovered one consistent thread throughout. According to every doctrine, evil was partly voluntary. One had to embrace it in a sense, allow it. The Devil, if you will, could not simply snatch you up in the dead of night and carry you off to Hell to do with you what he liked. Nor could his minions\u2014demons\u2014attack without provocation, their powers were limited as well. One had to let them 'in' so to speak. Simply put, if the road to Hell truly existed, one could not be dragged there. One had to voluntarily walk that path\u2014through either conscious decision or even outright deception\u2014but one had to allow it. Without that consent, evil could control no man, and no man could find or tap into pure evil. What we needed were not subjects forced into service but rather test subjects that had already embraced the darkness. We tried various subjects that practiced black magic and evil\u2014Satanists and the like\u2014but again met with failure. Evil, it seems, does not want those who so enthusiastically want it. So we began searching prisons. And that is where we found you, Mr. Cantrell. It's where we found all of you. You and your crew were chosen from thousands of potential candidates. You were all condemned, all paying for the horrible crimes you'd committed, all hopeless. If damnation was real, you were all headed straight for it. Murderers, thieves, rapists, terrorists, destroyers of innocents, you were perfect pieces to a larger puzzle of absolute darkness and depravity the likes of which even this hideous world could not begin to comprehend. You were the best of them, granted, the best of the worst, but the best just the same. As it turned out, you were also, however, a rather large fly in the ointment.\"\n\nHeart smashing his chest, Rooster attempted a deep breath. \"You're telling me everything I read in those files is true?\"\n\nPoindexter scooped up a forkful of meatball and slid it between his lips. \"That is precisely what I'm telling you.\"\n\n\"Why can't I remember?\"\n\n\"We didn't want you to remember.\" He wiped a smear of marinara from his chin with a cloth napkin. \"So your memories\u2014all your memories\u2014were wiped clean and replaced with memories we wanted you to have.\"\n\n\"Then there was no armored car job?\"\n\n\"There was not.\"\n\n\"But Carbone, he\u2014he was shot.\"\n\n\"He was killed, yes, but not from a gunshot.\"\n\nThe tremors returned. He struggled to control them. \"What then?\"\n\nPoindexter rolled more pasta onto his fork, the sauce dripping in thick globs back to his plate. \"You remember the farmhouse,\" he said, the fork shaking in his arthritic hand. \"It's coming back to you.\"\n\n\"Yes. Slowly.\"\n\n\"As I mentioned, you were the best of the worst.\" He stuffed the spaghetti into his mouth. \"You tortured and murdered a priest, claiming at your trial that you'd been repeatedly sexually molested by the man when you were a child and that's what had led to your life of crime and eventually his murder. He'd ruined you, and in turn, years later, you had ruined him.\"\n\nA spike of pain dug deep into his temple and ran down along the right side of his jaw. Rooster fought it back. \"I don't...\"\n\n\"Remember. Yes, I know. For that you should thank me.\"\n\n\"For wiping my memories away and leaving me with lies?\"\n\nIgnoring the question, he took up the goblet, sipped some wine. \"Of course the pedophilia scandal that shook the Catholic Church had not hit yet.\"\n\nRooster had no idea what scandal he was referring to. How much of his mind had these bastards destroyed?\n\n\"The idea that a respected, admired and loved parish priest would've ever done such hideous things to a little boy was unthinkable. No one, including us, believed you.\" Poindexter savored the wine a moment before continuing. \"Turns out you were telling the truth, who knew? The fact remained, however, that you tortured and murdered a priest in cold blood. Well done.\"\n\n\"What the hell did you people do to us?\"\n\n\"We sent you where no human being had ever been before...and returned from.\" He stabbed another meatball. \"You were all given the mixture. It took you to depths none of us could've imagined in our wildest dreams. You went to the core, the heart of evil, to its very soul. I must confess that until that night I hadn't counted on it actually working. But it did. As we'd hoped, you weren't alone in that boundless darkness, there was something else there with you. Something...alive.\"\n\n\"Where did we go?\"\n\nHe grinned like the demons in Rooster's nightmares. \"You touched the face of Lucifer, Mr. Cantrell. And he showed you evil in its purest, most savagely beautiful form, unbridled violence beyond comprehension.\"\n\n\"The farmhouse,\" Rooster muttered, \"the scarecrows, the rooms beneath the house...\"\n\n\"Props,\" he said, waving at the air as if to knock the words away. \"Familiar images that would elicit fear and discomfort were necessary so the mind would have something to reference. Interesting thing about the human mind, it fills in what is not there, often pulling images from a bank of previous experiences to fill the gaps. We simply helped you all with that, giving you something to experience in a pseudo-physical sense. Something terrifying that you could all relate to and understand.\"\n\n\"This is bullshit.\" Rooster stood up.\n\nPoindexter continued eating. Candlelight flickered across the plate. The spaghetti was not spaghetti at all, and it was not drenched in tomato sauce. Blood...bile...excrement... worms...human eyeballs cooked to a crisp, burned nearly beyond recognition. \"Technically the experiment was a success,\" he said. \"We did achieve what we'd set out to do, at least initially. But then it all went horribly wrong.\"\n\n\"This isn't happening.\" He pressed his palms to his temples, his head pounding now and his legs weak. \"This isn't...this isn't...\"\n\n\"Once we realized what we'd truly tapped into, that it was the equivalent of accessing the literal power of existence, and the dark side of existence at that, we knew we'd overestimated our abilities. It was actually quite beautiful in its purity, but you were all torn to shreds by its profane glory. It became an orgy of violence and blood, an orgy of death.\"\n\n\"You're lying, you sonofabitch.\" Rooster pointed the 9mm at him.\n\n\"Do you really think we could let any of you come back at that point? Or that there'd be anything left to bring back?\"\n\n\"Then where am I? I'm standing right here!\"\n\n\"The longer you struggle against truth, the longer the forces of darkness will bind you, Mr. Cantrell. There are some things human beings can never control. We're not meant to, regardless of how badly we may desire it. Evil\u2014true evil\u2014is one of those things. I understand it's hard for you to accept, but you were all thoroughly expendable, Mr. Cantrell, a bunch of hooligans and lowlifes, losers and drains on society no one cared about then or now.\"\n\n\"It wasn't enough that you used us as guinea pigs for your demented projects, crippled our minds and broke us to pieces. You had to wipe out our memories and send us back into the world haunted by nightmares you put there and with no knowledge of who we are or how we got here? You destroyed us\u2014you admit it\u2014and yet you still try to cover it up with bullshit stories about demons and Hell and\u2014\"\n\n\"Do you really believe telling yourself that long and hard enough will keep the terror at bay?\" Poindexter placed the fork next to the plate and wiped the blood from his mouth with the napkin. \"You all disappeared from the face of the Earth and not a single person noticed, much less cared.\"\n\n\"Then why come to us after all this time?\"\n\n\"Penance,\" he said softly, the air of arrogance fading. \"It's what's required of me now. Eventually, we all serve one master or another, Mr. Cantrell, whether we like it or believe in it or not. And I've come to learn that it rarely turns out to be the one we were counting on.\"\n\n\"Who are the men that killed Snow, the men in the Crown Vic?\"\n\nHe smiled blandly. \"They're not men.\"\n\n\"What do I do?\" Rooster leaned across the table so that the gun was only a few inches from the man's face. \"How do I kill these things in my head?\"\n\nHe leaned further into the light, pulled his glasses from his pale and sickly face and pushed forward until his forehead met the barrel of the gun. \"Deliver me from my sins,\" he whispered. \"Deliver us from evil.\"\n\nRooster's finger remained remarkably steady as it curled to the trigger.\n\nThe old man's eyes rolled to white.\n\nEverything else turned crimson. \n\n## -9-\n\nThe flashlight beam slides along the dirty floor to the door under the stairs. An inverted pentagram has been painted across it in blood. Above it and to the left, also in blood, are the numbers 666 and a series of words Rooster cannot decipher.\n\n\"Oh hell no, that's Devil shit right there.\" Snow backs away.\n\nRooster studies the words scrawled on the door. \"What language is that?\"\n\n\"Latin.\"\n\nThey all look to Starker. The giant shrugs. \"I took it in high school you ignorant motherfuckers.\"\n\n\"What's it mean?\"\n\n\"Abandon Hope All Ye Who Enter Here.\" Starker finds Rooster in the darkness behind him. \"Supposedly that's what it says at the gates of Hell.\"\n\n\"Why would somebody put that there?\" Nauls asks in a panic.\n\n\"Probably a bunch of drugged-out, loser, never been laid, douche bag, Devil-worshipping-wannabes.\" Landon pushes past the others. \"Who gives a shit? If we're doing this let's get it over with.\"\n\nWith that, Starker steadies his stance then kicks in the door. It implodes and tears from its hinges with a loud cracking, splintering sound, tumbling away into darkness down another set of stairs. They hear it land seconds later as an enormous cloud of dust and dirt kicks up in response, wafting out the open doorway and bursting into the room. A stale mildew odor is followed by a pungent smell similar to rotting garbage and raw sewage. They cough, block their nostrils then huddle together in the limited light until the stench weakens and the farmhouse is returned to eerie silence.\n\nNo one speaks, but before anyone can motion Nauls to lead the way with the flashlight, he hands it to Rooster. With a sigh, Rooster takes the lead, the light in one hand and his 9mm in the other. He steps through, aims the light and sees a small set of wooden stairs. Beyond them is a cement landing and what appears to be a corridor he and the others were somehow already aware of.\n\nHe begins his descent. Starker is behind him, his weight shaking the staircase with each step. Next is Landon. Snow and Nauls pull up the rear.\n\nThey reach the corridor without incident. Rooster pans the light along the walls. Several doors line either side. The far end of the hallway is draped in a darkness that the flashlight is unable to penetrate from this distance. The fear and danger is palpable now, a spiritual entity unmistakably alive and horrific, real, it drifts and moves around them like liquid, invisible to the naked eye but without question, present. Rooster sweeps the light along one wall and then the next, as together, the crew slowly moves deeper into the corridor. All the doors are closed.\n\nExcept one. He places the light on it. This door is ajar.\n\nRooster uses hand motions to let the others know what needs to be done. He sends Starker to the left side of the doorway, Snow to the right. Rooster then crouches, facing the door head-on while Landon covers his back and Nauls watches the section of hallway and stairs behind them.\n\nStarker holds the AK-47 in one hand and raises the other into the light so everyone can see. Slowly, he counts off, raising one finger, then another and finally a third. A quick nod, and the crew springs into action, rushing into the room with weapons at the ready and the flashlight leading the way.\n\nSilence returns. A mocking silence...\n\nThe light trembles in Rooster's hand. But they see. They all see.\n\nA series of metal slabs like something out of a coroner's workshop, bodies atop them in hospital johnnies, IVs attached to their arms pumping some clear fluid into their veins, oxygen tubes implanted in their nostrils, wires running from their heads and chests and limbs to machines and computers along the far wall, all of it organized and functioning in the dark bowels of an abandoned farmhouse in the middle of nowhere. Six metal tables. Six men.\n\n\"God in Heaven,\" someone says in a desperate whisper. \"It's us.\"\n\n* * * *\n\nIt might've been hours, might've been days.\n\nHe could no longer tell the difference.\n\nThe rain had stopped and the air was still, but it had gotten much colder. Bundled in a heavy coat and knit hat, the briefcase in his free hand, Rooster stood arm-in-arm with Gaby before a fresh grave. Dressed in a black dress and heels, her face partially covered with a lace veil, moments before she had placed flowers where a headstone should've been. Her lips moved in silent prayer behind the veil, dark eyes lowered. No one else was there. A life, Snow's life, had ended. Here, at this unmarked grave. And no one cared. It was like he'd never really been there at all.\n\nGaby finished her prayers, and together, they turned to leave.\n\nIt was then that Rooster saw them. Across the sea of headstones, crypts and monuments to the dead, two men watched them, their breath converted to spiraling clouds rising from their bodies like fleeing souls.\n\nGaby saw them, too. \"Do what you have to do.\" She lifted the veil, rose up on her toes so she could reach, and kissed his cheek. \"I love you.\"\n\nAs she moved away toward the gates of the cemetery, the men started toward him. Rooster lit a cigarette and smoked it until they reached him.\n\nThey looked the same.\n\nLandon stared at him, said nothing.\n\n\"Hey, Rooster,\" Nauls offered, scratching at his beard and smiling nervously, eyes concealed behind the usual sunglasses. \"Good to see you, bro.\"\n\n\"Good to see you too, Nauls.\"\n\n\"That is so precious\u2014seriously\u2014I think I just tinkled a little. How about we save the group hug for later and you two can finish jerking each other's gherkins then, OK?\" Landon stepped closer. \"Paper said the hit-and-run was probably an accident, driver just panicked. I say kiss my celluloid-dimpled ass, whoever hit Snow did it on purpose. Can't blame them\u2014I would've run the prick over if he stepped in front of my car too\u2014but sounds like somebody took him out to me.\"\n\nRooster took a final drag on his cigarette then dropped it and crushed it out with his boot. \"They did.\"\n\n\"Do you know who they are?\"\n\n\"I'm not sure yet, but\u2014\"\n\n\"You heard about Starker?\"\n\n\"Snow said he killed his old lady and ran to Mexico.\"\n\n\"He never got that far. Big bastard was hiding out in a fleabag motel right here in the city. They found him a few weeks back, in the bathtub, wrists slashed clear to the bone. Sorry, I'm not buying that one either. Same fuckers probably did him too.\"\n\n\"I keep having these dreams,\" Nauls blurted out. \"Nightmares, I\u2014\"\n\nLandon held a hand up like a crossing guard. \"Let the grownups talk.\"\n\n\"Fuck you, man! You're having them too. Tell him. Tell him.\"\n\nLandon defiantly bit his lip and looked away.\n\n\"Rooster,\" Nauls said, barely able to contain his tears, \"I've been having these dreams. There's all this screaming and yelling and blood and horrible shit. Then it gets dark and I can't see. I can't move, I can't even breathe and it feels like I'm being smothered. I try to open my mouth to scream only I can't. My mouth, it's\u2014somebody's sewn it shut. Who would\u2014the bad dreams won't stop, they\u2014I'm even starting to have them when I'm awake, I\u2014\"\n\n\"We all are,\" Rooster said evenly.\n\nNauls ran a hand through his tangle of hair. \"Every time I leave the house I see this chick and this older dude, they're dressed like they work in an office or a bank or something and they follow me and want to talk to me, but there's something not right about them. They look so familiar only I don't know who they are. And Landon, he\u2014he don't drive no more. Landon don't drive. He can't. Every time he gets behind the wheel of a car he sees this lady holding a baby.\"\n\n\"She's on every fucking corner just staring at me.\" Landon became visibly shaken as his resistance fell away. \"I know her from somewhere but...I'm pretty sure the baby's dead.\"\n\n\"What's happening?\" Nauls asked. \"What happened to us that night at the farmhouse? We can't remember nothing but bits and pieces.\"\n\n\"I've got something to show you,\" Rooster said softly, as if the dead might otherwise hear. He held up the briefcase.\n\n\"What's that?\"\n\n\"The truth.\"\n\n* * * *\n\n\"This isn't possible,\" Landon mumbles.\n\nAs if in a trance, Snow approaches the last table, the only one covered with a white sheet which has apparently been thrown there to conceal Carbone's body. \"Carbone's dead,\" he says. \"He's dead, and he's back in that van.\"\n\nMesmerized, the others gawk at their likenesses on the tables before them, confusion and fear igniting as one and slashing at them like razorblades. Rooster cocks his head, studying his own face just feet away, eyes closed and face void of expression as if in the throes of a deep, drug-induced sleep.\n\nInches from the covered metal table, Snow pokes at the sheet with one of his .45s. The sheet begins to shake in response, as whatever lies beneath convulses. Horrified, Snow yanks back the sheet.\n\nSans johnny, wires and tubes, Carbone's nude body lies quivering violently on the table. His lower abdomen and sexual organs are ripped to shreds, and the remainder of his body has sustained thousands of small but horribly deep serrated cuts, as if it's been wrapped in barbed wire then torn free. The lacerations, many blackened and scabbed over, others fresh and still bleeding, form a crisscross pattern on his savaged skin that is as strangely alluring in its symmetry as it is appalling in its brutality.\n\nAs Snow backs away, both .45s locked on the body, Carbone suddenly sits up, vaulting forward. His eyes open but they are empty raw sockets. He continues to spasm uncontrollably in seizure. \"He's coming.\" His voice is no longer exclusively his own, but many, and sounds as if it is stacked atop countless others, giving it an unsettling echo-like, inhuman tone. \"He's coming...\"\n\nHands to his ears, Nauls stumbles back into the hallway like a terrified child.\n\n\"Shoot it!\" Landon screams.\n\nSnow is frozen in place.\n\n\"He knows who we really are,\" it says. \"He knows the things we've done. Our secrets, he knows them all. He's coming...\"\n\n\"God help us,\" Rooster mutters.\n\n\"God?\" Carbone turns what remains of his butchered face in the direction of Rooster's voice. His split lips curl into a hideous, bloody-toothed grin.\n\nStarker levels the AK-47 and unloads.\n\nThe discharge is deafening in such an enclosed space, and sends the body tumbling from the metal slab. It crashes to the floor as if boneless, flesh slapping cement floor as the impact empties the remains of its internal organs from the body cavity.\n\nFrom the corridor behind them, Nauls begins to scream.\n\n* * * *\n\nAt the outskirts of the city, on a lonely dirt road, Rooster leaned against Nauls' car and smoked a cigarette. He'd waited as Nauls and Landon poured over the material in the briefcase, then he answered their questions as best he could. Both men exchanged uncertain glances throughout, and now stood watching Rooster as if expecting him to tell them what to do next.\n\n\"They used us like lab animals,\" Rooster finally said. \"They wiped our minds clean, and now that we're starting to remember they're taking us out one by one. They figure they can toss us aside like garbage.\"\n\n\"We are garbage,\" Nauls replied quietly.\n\n\"Maybe so, but we never even got the chance to make things right, to\u2014\"\n\n\"What?\" Landon interjected. \"Repent? Save our souls? Deliver ourselves from evil like this Poindexter dude told you?\"\n\nRooster stared at him.\n\n\"Maybe that's exactly what we're doing right now,\" Landon said.\n\nA breeze blew past, causing nearby trees to whisper and sway.\n\n\"We have to go back,\" Rooster said.\n\n\"To the farmhouse, are you serious?\" Landon gave a wry smile. \"You want to go back there?\"\n\nRooster nodded, smoke curling around his head like creeping vines. \"You think you could find it again after all this time?\"\n\n\"Yeah.\" Landon looked to Nauls but he had his back to him. \"I can find it.\"\n\nHe hadn't expected Landon to be so adamant. But then he hadn't expected his and Nauls' nearly blas\u00e9 reaction to the things he'd told them either. Something had changed since they'd driven out here. The moment he'd agreed to go with them they no longer seemed quite as upset as they'd been initially. He dropped his cigarette and pushed away from the car. \"You're sure?\"\n\n\"Rooster, I...we've...been there since.\"\n\n\"You've been back there since that night?\"\n\n\"You don't understand,\" Landon said. \"We never left.\"\n\n* * * *\n\nRunning...screams...confusion...\n\nPanic explodes through the darkness...\n\nThe flashlight bounces, throwing strobe-like splashes of light along the corridor, floors, walls and ceiling before finally settling on Nauls. His face protrudes from the darkness, eyes closed but with a look of horrific pain. Blood slowly trickles from his nostrils into his beard.\n\nThe others scramble about trying to cover the corridor. Landon frantically knocks Snow out of the way and climbs the stairs back to the house.\n\nNauls opens his eyes. \"He's here,\" he says in a loud whisper.\n\nHis body begins to shake. Slowly at first but gradually building in intensity, he begins to buck, wracked with increasingly violent spasms. His thin frame twists as he flails about, and his weapon falls to the floor. He brings his shaking hands to his face, screams and stabs his fingers directly into his eyes.\n\nRooster reaches out in an effort to stop him, but it's too late.\n\nNauls tears his eyes from their sockets with a spray of blood and fluid, his screams replaced with laughter as his spasms grow worse and he begins to spin like a top.\n\n\"Jesus God!\" Snow shrieks, falling away in horror.\n\n\"Go!\" Starker grabs Snow and throws him toward the stairs. \"Go!\"\n\nRooster stands paralyzed, holding the flashlight on Nauls, who comes to rest, laughing through the blood and pain, holding an eyeball in each hand as if in offering, hideous moist strings dangling from them and dripping blood. \"We're going where there are no eyes,\" he says, his voice little more than a garbled growl now. \"Where everyone is blind...yet everyone sees.\"\n\nBlood suddenly spews from his mouth, eye sockets, nose and ears. Like something has exploded deep inside him, the blood sprays free as his screams return, this time as raspy, animal-like squeals. \"He's here,\" he gurgles, choking on the blood as it pours out over his bottom lip. \"He's\u2014\"\n\nNauls flies backwards, crashes into the far wall like he's been thrown by something savage and powerful. His body slides to the floor, swallowed by the shadows there.\n\nRooster feels Starker's enormous hand clamp onto his arm and yank him back just before he fires a burst from the AK-47 into the darkness. Together, they run for the stairs. \"Don't look back!\" Starker yells out.\n\nBut it no longer matters.\n\nThe darkness, and all that dwells within it, follows.\n\nIn the room upstairs, Snow lurches about, lost in the dark, his guns at his side and his mouth open, soundlessly forming words\u2014perhaps prayers\u2014while something speaks to him from the surrounding shadows only he can hear. The voice of a woman, a young woman asking him why, her voice oddly hollow as she shuffles about nearby, hidden in darkness, her breath cold and rapid on the back of his neck. But when Snow turns there is only night, moonlight and fog beyond the blown-out windows. The scarecrows watch a field of weeds, a dead forest and a path to nowhere, an empty road no one will ever cross again.\n\nThe voice, different now\u2014neither male nor female and no longer entirely human\u2014whispers his name.\n\nSnow wants to run for the door but can't move. He knows, understands for the first time, what is coming, and still cannot move. He trembles and begins to urinate. As the .45s drop from his hands the fire appears from nowhere, sweeping over the ceiling then down the wall and across him, engulfing his body in seconds. Oddly, Snow feels no burning sensation, no pain, only sorrow and hopelessness the depths of which he never believed possible. He stumbles, flaming arms and hands held out in front of him as if to embrace some invisible presence. He sinks to his knees. Eyes wide, he stares at something through the growing inferno and laughs maniacally.\n\nThe last thing Snow sees is Starker and Rooster rushing up the stairs.\n\nOutside, Landon runs with all his might, the tall grass and overgrown weeds slowing him as he wades toward the road. The van, he thinks, just have to make it to the van and I'm free. He ignores the scarecrows' dead stares and does not look back, even when he's certain there is something right behind him, closing in with impossible speed and ready to swoop down and pluck him from the field like a hawk closing in on a mouse. He bolts through the last bit of field and jumps the final embankment down to the road. Pitching forward on landing, he catches himself, and now on pavement, takes a quick look back. No one coming, nothing behind him. He pulls the revolver from his belt just in case, sees the farmhouse in the distance. It's on fire, the flames creeping up through the roof, lapping night. He turns and runs for the van but pulls up short after only a few strides. It's gone. He looks around frantically. This isn't possible. He parked it there himself, out of the way, just as Rooster instructed.\n\n\"Yeah, I need this shit.\" He heads off down the road, running right down the center lane through the darkness; the fog-shrouded moon his only guide. Every now and then he looks back. The farmhouse, the scarecrows and the fire grow fainter and fainter until the night swallows them whole and he is alone in the darkness.\n\nHe slows his pace a few minutes later, finally opting for a fast walk. His chest heaves as he tries to catch his breath and a sharp pain digs at his side. Landon keeps moving, knowing eventually he's bound to run into something\u2014a car, a house\u2014anything. He notices a slight incline to the road. He pushes on, trying to forget the things he saw back there. All he needs is a car. He can hotwire anything and be long gone from this place for good. He kicks it back up a notch, jogging up and over the sloped portion of road. In the distance, he sees an outline of a building. Set back quite a distance from the road, it is merely a silhouette, but a hulking one. Must be a house, he reasons, then increases speed and veers off pavement onto grass.\n\nRunning across the field, he watches it become more and more defined the closer he gets. Within minutes Landon realizes it's a barn.\n\nBeyond it is a farmhouse.\n\nA farmhouse guarded by scarecrows...a farmhouse in flames.\n\n\"No fucking way.\" He comes to a stop between the barn and the house. He's gone in a circle, but how is that possible? He ran straight and in the opposite direction the entire time.\n\nShadows drift through the weeds before him. Landon steps back and raises the revolver. He can hear screams and smells a suspicious burning odor. Beyond that of burned wood, it is sickeningly sweet and similar to the stench of charred meat.\n\nA baby cries somewhere nearby. Landon whirls in the direction to find only darkness. Blind with terror, he runs but trips over something and pitches forward into the grass and dirt. He scrambles to his feet, sees what he fell over. A wooden stake...a cross of wood...\n\nThe scarecrow, he thinks, his mind shattering. It's gone.\n\nFrom behind him, shuffling movement.\n\nA strange shape comes toward him through the tall grass, hobbling like a crippled man.\n\nOnly this is not a man.\n\nLandon fires the revolver. Keeps firing even when the revolver is empty and makes only clicking sounds.\n\nAnd then something coarse covers his head, cold dead hands wrap around his throat and he hears another scream shred the night, unaware that this one is his own.\n\nIn the farmhouse, Starker and Rooster run through the burning front room, trying to find a way out in all the madness and confusion. The darkness is alive, shifting and thick with the shrieking cries of countless dead, nameless lost souls all wailing in the night with violent fury. Rooster sees a pillar of fire and realizes it is Snow kneeling before them, his body wrapped in blankets of flame.\n\nLike a cold winter wind, something follows them up the stairs, gusts into the room and cuts through them. It feeds the flames and Snow's body becomes a firestorm. Yet he doesn't topple. Instead he struggles slowly to his feet.\n\nRooster shoots him, emptying his gun.\n\nSnow finally topples over and the fire spreads, racing up the walls and along the floor in search of more victims.\n\nThe strange wind passes, surging out to the field beyond the doorway, and Rooster feels some part of himself go with it. He stumbles after it, dazed and fighting the gripping cold suddenly rising from the depths of his body. He finds Starker standing next to Snow's body, staring at it with a strange look of...satisfaction? He throws the AK-47 aside, drops down, and eyes ablaze with passion claws at the burned heap that had once been Snow, ripping charred meat in stringy handfuls he hungrily devours.\n\nAnd as the fire spreads, Rooster understands. He feels it too. Lust not for sex but violence, death, mayhem, destruction and pain...as if these things have been his destiny all along. Rather than reload the 9mm, he drops it and reaches for a combat knife tucked in his boot. He slides it free, already salivating as he closes on Starker.\n\nBehind him, Nauls slowly ascends the stairs, his hollow eyes piercing the smoke and darkness, his mouth twisted into a hideous demonic smile.\n\nRooster slams the blade deep into Starker's lower back, pulls it free and stabs him again. He seems not to notice at first, but then collapses from his knees to his side and lies there laughing, his large teeth bright in the darkness and caked with blood and human flesh.\n\nAs Rooster sets to work on him, gutting Starker from throat to pelvis, Nauls moves past, through the fire and out the doorway to the field.\n\nHis feet do not touch the floor.\n\nRooster focuses on Starker's laughter. No\u2014not laughter\u2014not anymore, cries now, screams. Beautiful screams...his face and bald head covered in blood as he spits and slobbers, each scream more horrific than the last. As Rooster tears at the enormous incision then plunges his hands inside the body, Starker chokes on the bodily fluids bubbling up into his throat and begs for mercy.\n\nBut all Rooster hears are the shrieks of souls trapped in the darkness and flames surrounding him.\n\nCords of intestines clutched in one hand and the knife in the other, he leaves Starker's now silent but convulsing body and slowly approaches the doorway. Darkness waits...a field of tall grass and weeds...six wooden crosses...three with fresh scarecrows nailed to them...three still waiting...\n\nRooster begins to laugh, bringing the intestines to his lips and eating as he steps out of the flames and into the night.\n\nSomewhere within the hurricane of violence and howling souls, a frantic, familiar and decidedly human voice screams for salvation.\n\n* * * *\n\nVisions of demonic creatures\u2014some human, some not, and others still stranded at various horrific points between the two\u2014flashed through his mind. Held in rusty metal cages, pinned, strapped or chained to medieval devices of torture and imprisonment, the creatures gawked at him in horror, several deathly still, others violently struggling to free themselves, all of them moist with blood, urine and excrement, their bodies grotesquely deformed and savaged.\n\nThe terrifying chambers of blood and death dissolved; became a roadside.\n\nLandon had already gone quite a ways up the incline on the side of the road and looked back as if he expected Rooster to follow. But Rooster knew now what lay on the other side of the tall grass blowing in the wind behind him.\n\nWith a shrug, Landon held his arms out like the victim of crucifixion and backed away over the ridge, vanishing from sight.\n\nNauls turned to him, removed his sunglasses.\n\nWe're going where there are no eyes...\n\nHis eyes were gone, just empty sockets.\n\nWhere everyone is blind...yet everyone sees.\n\nWithout warning his body shook with impossible velocity, transforming him into little more than a blur before he again fell still. \"Come with us,\" Nauls said. \"We'll all figure this out together.\"\n\nRooster shook his head no.\n\nNauls slid his sunglasses back on, slowly walked up the embankment after Landon then hesitated and looked back. \"You really think you have a choice?\"\n\n\"That's all any of us have.\"\n\nNauls reached into his jacket pocket, pulled free the car keys and tossed them to Rooster. \"We'll be waiting,\" he said sadly. \"Forever.\"\n\n## -10-\n\nHe made the car tailing him even before he'd reached his apartment. Rooster pulled over a block from the housing projects and continued on foot. As he crossed the courtyard, hurrying through the cold, the black Crown Vic crept slowly past, the windows and windshield impenetrably tinted. It continued a bit further down the street then pulled over and parked. Rooster kept checking back over his shoulder, but no one emerged from the vehicle.\n\nWhen he'd reached his floor, Rooster stopped at the incinerator shoot and dropped the briefcase in, listening to it slide away down the shaft to the fires below.\n\nBurn, he thought. Burn in Hell.\n\nHe slipped into the apartment and was met by a welcome burst of heat. Moving silently, he went to the bedroom and stopped just inside the doorway. Gaby was standing next to the bed, a blanket in her arms and a laundry basket at her feet. She'd already stripped the comforter, blanket and top-sheet from the bed but the bottom sheet remained. She seemed surprised to find him there, but smiled anyway. It was perhaps the most reassuring and comforting thing he'd ever seen.\n\nUntil he took a closer look at the bed. Rich dark soil was scattered across the sheet, blood and straw along the pillows. He narrowed his eyes and grimaced as fear clawed at what few defenses he had left.\n\n\"It's all right,\" Gaby said, quickly tossing the blanket over the bed. \"Don't look. It doesn't mean anything. They're just trying to frighten you.\"\n\nThe night sky rolled above, moving, the fog turning and twisting as the rough ground tore at his back and shoulders.\n\n\"Gaby,\" he said softly, voice breaking. \"Gabrielle...help me.\"\n\nThey were dragging him by his legs...pulling him across the field, the grass and weeds tangling and scratching him as he went, the night sky overhead, vast and ominous, the smell of death and burning flesh filling the air.\n\n\"Hell does more than burn the wicked,\" she said. \"It cleanses the lost clawing for the light. Remember what I told you. Let me help you tear them apart like they've torn at you.\"\n\nHideous hands of straw, of charred flesh and exposed bone held him down against the fallen cross of wood while shadows moved about, laughing horribly even as they drove nails through his palms, destroying flesh and shattering bone, even as they hoisted the cross up and into position, even as Rooster screamed and begged for God to save him, even as unseen filthy hands held his mouth closed while others pierced his lips with an old rusted needle, running the leather string through the holes and pulling it taut until his screams were muffled groans and his mouth could no longer open.\n\n\"Remember what I told you,\" she said again.\n\nThose in the shadows pulled the burlap sack over his head, two holes cut out in the fabric to accommodate his eyes. Eyes that could still see...inhuman eyes now, the eyes of a soulless scarecrow...impossible eyes opening, seeing, watching, frozen in time, crucified to damnation and endless suffering.\n\n\"Rooster,\" Gaby said forcefully, \"remember what I told you about my name and what it means. Do you remember?\"\n\n\"God is my might.\"\n\nAnd his eyes see the Hell he is trapped in...a Hell not of demons with pitchforks and cloven-hooves or boundless oceans of fire...but one in a small bedroom not so different than the one Rooster stood in now. A quiet and dark room where a little boy sat on a bed with crisp white sheets, crucifixes on the walls and a devil he'd believed a god sitting next to him whispering assurances that the things happening were just and right and moral and clean. Father McKay staring down at him with those striking blue eyes and telling him everything would be all right if he simply obeyed God's will.\n\nTears stain Rooster's cheeks. Rage, sorrow, fear\u2014he cannot decide. All of them, goddamn you, all of them in a tempest of blood and tears and evil.\n\n\"They're dying. You're killing them one by one.\" Gaby motioned to him with a slight turn of her head, her beauty shifting to something decidedly more sinister. \"Burn them. Burn the fuckers away like the leeches they are.\"\n\nHe smelled death...dirt...an open grave and its rotting remains...\n\nTerror strangled him, its grip desperate.\n\nThe priest stood behind him, filthy and discarded now, like the souls he'd torn from countless children years before. \"I know you,\" he said.\n\n\"I watched you die.\" Visions of Starker came to him. No. Not Starker. Father McKay, his head drenched in blood, choking on his own body fluids and gasping for forgiveness. \"I killed you. Slowly.\"\n\nBlood so dark it was nearly black trickled from the corners of his eyes. \"Did you think that would save your soul?\" the priest asked.\n\n\"I only knew it would end you.\"\n\nThe priest moved deeper into the room, stepping between him and Gaby, smiling wide like a demonic Cheshire Cat. \"But that's what you hoped for, wasn't it. Just like now, you hope it will save you from me, from this place, from those waiting for you outside, from yourself. It won't. Do you know why?\" A fat brown spider scurried across his bald head, disappeared into his ear. He didn't seem to notice. \"Because the illusion of hope is Hell's greatest joy.\"\n\n\"And Heaven's greatest weapon,\" Gaby said from behind him, her eyes rolling to black as she grabbed hold of him, sunk her teeth into his neck and pulled him to the floor with shocking strength and violence, straddling him and tearing at his throat the way a wild dog might.\n\nLight and dark merged as blood sprayed the walls.\n\nRooster backed away until he'd vanished into the safety of nearby shadows, the meager scraps of sanity he still possessed fracturing as night fell over the city of the damned.\n\n* * * *\n\nLost in time, through bloodshot eyes Rooster watched the sun rise on a new day, broken dreams collected at his feet, tarnished trophies stolen rather than won. The beautiful innocence of a little boy nailed to a cross of wood in burning fields called to him across the years, tears from a forgotten and wasted life and the sins of ghosts from a past he couldn't quite remember and perhaps never would. Perhaps he wasn't meant to.\n\nRepent? Save our souls? Deliver ourselves from evil?\n\nBut rather than destroy, the flames in those burning fields were what would eventually free him.\n\nMaybe that's what we're doing right now.\n\nRooster rubbed his hands together, they'd gone so cold. He lit a Marlboro and checked the corner. The Crown Vic was gone. From behind him, he heard heels clacking pavement. Bundled in a winter coat and hat, Gaby walked across the courtyard with her typical brisk stride.\n\nAcross the street, Nauls' car waited.\n\nGaby smiled, no longer wolf, but lamb.\n\n\"Where are we going?\" Rooster asked.\n\n\"Away from here,\" she said, offering him her hand.\n\n\"Home?\" he asked.\n\n\"Home,\" she said. \"But get rid of the cigarette. Those things'll kill ya.\"\n\nHe slipped his hand into hers, and for the first time in a long while, felt himself smile.\n\nFires burned. They always would. But Rooster's flames no longer trapped him in a Hell of his or anyone else's creation. Instead, they destroyed those things shackling him to the Devil's playground, and all the nightmares and lies that had tried so desperately to keep him there.\n\nThe longer you struggle against truth...\n\nIn a dark and distant field, a hideous scarecrow closed its sightless eyes.\n\nThe longer the forces of darkness will bind you...\n\nRooster's soul quieted as the demons fell back into the lightless abyss from which they'd come.\n\nHand-in-hand with Gaby, Rooster walked to the car. Somewhere beyond the horizon, death's other kingdom waited.\n\nA kingdom not of shadows and darkness, but of peace. \n\n## ABOUT THE AUTHOR\n\nCalled \"One of the best writers of his generation\" by both the Roswell Literary Review and author Brian Keene, Greg F. Gifune is the author of numerous short stories, several novels and two short story collections (Heretics and Down To Sleep). His work has been published all over the world, consistently praised by readers and critics alike, received starred reviews in Publishers Weekly, Library Journal and The Midwest Book Review (among others) and has recently garnered interest from Hollywood. His novels include Children Of Chaos, Dominion, The Bleeding Season, Deep Night, Blood In Electric Blue, Saying Uncle, A View From The Lake, Night Work, Drago Descending, Catching Hell, Judas Goat, and Long After Dark. In addition to working as a full-time author, he also serves as Associate Editor at Delirium Books. Greg resides in Massachusetts with his wife Carol and a bevy of cats. Greg can be reached online at: gfgauthor@verizon.net or through his official web site at: www.gregfgifune.com.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\n\nNote to the Reader\n\nCopper Canyon Press encourages you to calibrate your settings by using the line of characters below, which optimizes the line length and character size:\n\nLorem ipsum dolor sit amet, consectetur adipiscing elit. Pellentesque\n\nPlease take the time to adjust the size of the text on your viewer so that the line of characters above appears on one line, if possible.\n\nWhen this text appears on one line on your device, the resulting settings will most accurately reproduce the layout of the text on the page and the line length intended by the author. Viewing the title at a higher than optimal text size or on a device too small to accommodate the lines in the text will cause the reading experience to be altered considerably; single lines of some poems will be displayed as multiple lines of text. If this occurs, the turn of the line will be marked with a shallow indent.\n\nThank you. We hope you enjoy these poems.\n\n_This e-book edition was created through a special grant provided by the Paul G. Allen Family Foundation. Copper Canyon Press would like to thank Constellation Digital Services for their partnership in making this e-book possible._\n_for Burton Watson_\n\n# Contents\n\n  * Title Page\n  * Note to Reader\n\n  * Translator's Preface\n\nPART ONE\n\n  1. Spring Dawn\n\nMENG HAO-JAN\n\n  2. Calling on Censor Yuan without Success\n\nMENG HAO-JAN\n\n  3. Seeing Off Supply Director Kuo\n\nWANG CH'ANG-LING\n\n  4. The Roads of Loyang\n\nCH'U KUANG-HSI\n\n  5. Sitting Alone on Chingting Mountain\n\nLI PAI\n\n  6. Climbing White Stork Tower\n\nWANG CHIH-HUAN\n\n  7. On Seeing Princess Yung-lo Leave for Manchuria\n\nSUN T'I\n\n  8. Spring Complaint\n\nCHIN CH'ANG-HSU\n\n  9. A Pear Blossom in the East Wing\n\nCH'IU WEI\n\n  10. On Thinking of My Lord's Grace\n\nLING-HU CH'U\n\n  11. On Mister Yuan's Country Retreat\n\nHO CHIH-CHANG\n\n  12. Seeing Off Chao Tsung at Night\n\nYANG CHIUNG\n\n  13. Bamboo Retreat\n\nWANG WEI\n\n  14. Seeing Off Chu Ta Leaving for Ch'in\n\nMENG HAO-JAN\n\n  15. Ballad of Changkan\n\nTS'UI HAO\n\n  16. Ode to the Past\n\nKAO SHIH\n\n  17. Resigning as Minister\n\nLI SHIH-CHIH\n\n  18. On Meeting a Chivalrous Man\n\nCH'IEN CH'I\n\n  19. Traveling on the Yangtze\n\nCH'IEN HSU\n\n  20. In Reply to Li Huan\n\nWEI YING-WU\n\n  21. Ode to the Autumn Wind\n\nLIU YU-HSI\n\n  22. To Secretary Ch'iu on an Autumn Night\n\nWEI YING-WU\n\n  23. Autumn Day\n\nKENG WEI\n\n  24. Autumn Day on the Lake\n\nHSUEH YING\n\n  25. Written in the Palace\n\nLI ANG\n\n  26. Looking for a Recluse without Success\n\nCHIA TAO\n\n  27. Surprised by Autumn on the Fen\n\nSU T'ING\n\n  28. Delayed on the Szechuan Road\n\nCHANG YUEH\n\n  29. Thoughts on a Quiet Night\n\nLI PAI\n\n  30. Chiupu River Song\n\nLI PAI\n\n  31. To Vice Censor Ch'iao\n\nCH'EN TZU-ANG\n\n  32. In Reply to the Prefect of Wuling\n\nWANG CH'ANG-LING\n\n  33. Thinking of My Home in Ch'ang-an While Traveling with the Army on the Ninth\n\nTS'EN SHEN\n\n  34. A Concubine's Lament\n\nHUANG-FU JAN\n\n  35. Written at Chulin Temple\n\nCHU FANG\n\n  36. Passing the Shrine to the Master of the Three Gates\n\nTAI SHU-LUN\n\n  37. Saying Goodbye on the Yi River\n\nLO PIN-WANG\n\n  38. Saying Goodbye to Lu Ch'in-ch'ing\n\nSSU-K'UNG SHU\n\n  39. In Reply\n\nTHE ANCIENT RECLUSE\n\nPART TWO\n\n  1. Reaching Sword Gate Pass After Touring the Land of Shu\n\nHSUAN-TSUNG\n\n  2. Replying to a Poem by Prime Minister Lu of Chinling\n\nTU SHEN-YEN\n\n  3. At an Imperial Banquet in Penglai Hall Offering Praise for the Chungnan Mountains\n\nTU SHEN-YEN\n\n  4. Parting from a Friend on a Night in Spring\n\nCH'EN TZU-ANG\n\n  5. Attending a Banquet at Princess Ch'ang-ning's Eastern Estate\n\nLI CHIAO\n\n  6. Given the Word \"Forest\" by His Majesty at the Licheng Palace Library Banquet\n\nCHANG YUEH\n\n  7. Seeing Off a Friend\n\nLI PAI\n\n  8. Seeing Off a Friend Leaving for Shu\n\nLI PAI\n\n  9. Stopping at Peiku Mountain\n\nWANG WAN\n\n  10. Mister Su's Country Retreat\n\nTSU YUNG\n\n  11. Staying Overnight at the Chancellery in Spring\n\nTU FU\n\n  12. Painted on the Wall of Master Hsuan-wu's Room\n\nTU FU\n\n  13. The Chungnan Mountains\n\nWANG WEI\n\n  14. To Censor Tu at the Chancellery\n\nTS'EN SHEN\n\n  15. On Climbing Tsungchih Pagoda\n\nTS'EN SHEN\n\n  16. Climbing Yenchou Tower\n\nTU FU\n\n  17. On Deputy Prefect Tu Taking Up a Post in Shuchou\n\nWANG PO\n\n  18. Seeing Off Ts'ui Jung\n\nTU SHEN-YEN\n\n  19. The Imperial Entourage on the Road from Tengfeng\n\nSUNG CHIH-WEN\n\n  20. Written on the Wall of Master Yi's Meditation Hut\n\nMENG HAO-JAN\n\n  21. To Chang Hsu after Drinking\n\nKAO SHIH\n\n  22. Jade Terrace Temple\n\nTU FU\n\n  23. On Seeing a Landscape Painted by My Commissioner Cousin for Li Ku\n\nTU FU\n\n  24. Recording My Thoughts While Traveling at Night\n\nTU FU\n\n  25. Climbing Yuehyang Tower\n\nTU FU\n\n  26. Traveling South of the Yangtze\n\nTSU YUNG\n\n  27. Spending the Night at Lunghsing Temple\n\nCH'I-WU CH'IEN\n\n  28. The Meditation Hall behind Poshan Temple\n\nCH'ANG CHIEN\n\n  29. Written at the Sungting Relay Station\n\nCHANG HU\n\n  30. Shengkuo Temple\n\nCH'U-MO\n\n  31. Gazing across the Countryside\n\nWANG CHI\n\n  32. Seeing Off Editor Ts'ui Marching East\n\nCH'EN TZU-ANG\n\n  33. Encountering Rain at Changpa Reservoir One Evening While Enjoying a Cool Breeze with Rich Young Men and Their Singsong Girls\u2014I\n\nTU FU\n\n  34. Encountering Rain at Changpa Reservoir One Evening While Enjoying a Cool Breeze with Rich Young Men and Their Singsong Girls\u2014II\n\nTU FU\n\n  35. Spending the Night at Yunmen Temple Pavilion\n\nSUN T'I\n\n  36. Climbing Hsieh T'iao's North Tower in Hsuancheng in Autumn\n\nLI PAI\n\n  37. Overlooking Tungting Lake\u2014For Prime Minister Chang\n\nMENG HAO-JAN\n\n  38. Passing Hsiangchi Temple\n\nWANG WEI\n\n  39. For Censor Cheng on Being Banished to Fukien\n\nKAO SHIH\n\n  40. Miscellaneous Poem at Chinchou\n\nTU FU\n\n  41. Shrine to Yu the Great\n\nTU FU\n\n  42. Gazing at the Valley of Ch'in\n\nLI CH'I\n\n  43. Commiserating with Gentleman-in-Waiting Wang on Tungting Lake\n\nCHANG WEI\n\n  44. Crossing the Yangtze\n\nTING HSIEN-CHIH\n\n  45. Drinking at Night in Yuchou\n\nCHANG YUEH\n\nPART THREE\n\n  1. Casual Poem on a Spring Day\n\nCH'ENG HAO\n\n  2. Spring Day\n\nCHU HSI\n\n  3. Spring Night\n\nSU SHIH\n\n  4. Early Spring East of Town\n\nYANG CHU-YUAN\n\n  5. Spring Night\n\nWANG AN-SHIH\n\n  6. Light Rain in Early Spring\n\nHAN YU\n\n  7. New Year's Day\n\nWANG AN-SHIH\n\n  8. Attending a Banquet on the Great Beginning\n\nSU SHIH\n\n  9. Occasional Poem on the Arrival of Spring\n\nCHANG SHIH\n\n  10. On a Painting of Playing Football\n\nCH'AO YUEH-CHIH\n\n  11. Palace Ode\u2014I\n\nLIN HUNG\n\n  12. Palace Ode\u2014II\n\nLIN HUNG\n\n  13. In Praise of Huaching Palace\n\nTU CH'ANG\n\n  14. Chingping Ode\n\nLI PAI\n\n  15. Written on the Wall of an Inn\n\nCHENG HUI\n\n  16. Quatrain\n\nTU FU\n\n  17. Begonia\n\nSU SHIH\n\n  18. Grave Sweeping Day\n\nTU MU\n\n  19. Chingming\n\nWANG YU-CH'ENG\n\n  20. Festival Day\n\nWANG CHIA\n\n  21. Cold Food\n\nHAN HUNG\n\n  22. Chiangnan Spring\n\nTU MU\n\n  23. For Gentleman-in-Attendance Kao\n\nKAO CH'AN\n\n  24. Quatrain\n\nCHIH-NAN\n\n  25. Visiting a Private Garden without Success\n\nYEH SHAO-WENG\n\n  26. Traveling Away from Home\n\nLI PAI\n\n  27. Written on a Screen\n\nLIU CHI-SUN\n\n  28. Inspired\n\nTU FU\n\n  29. The Peach Blossoms of Chingchuan Hermitage\n\nHSIEH FANG-TE\n\n  30. The Peach Blossoms of Hsuantu Temple\n\nLIU YU-HSI\n\n  31. Visiting Hsuantu Temple Again\n\nLIU YU-HSI\n\n  32. Chuchou's West Stream\n\nWEI YING-WU\n\n  33. Flower Shadows\n\nSU SHIH\n\n  34. North Mountain\n\nWANG AN-SHIH\n\n  35. On the Lake\n\nHSU YUAN-CHIEH\n\n  36. Inspired\n\nTU FU\n\n  37. Spring Clearing\n\nWANG CHIA\n\n  38. Late Spring\n\nTS'AO PIN\n\n  39. Falling Flowers\n\nCHU SHU-CHEN\n\n  40. Visiting a Private Garden in Late Spring\n\nWANG CH'I\n\n  41. The Oriole Shuttle\n\nLIU K'O-CHUANG\n\n  42. Events of Late Spring\n\nYEH TS'AI\n\n  43. Climbing a Mountain\n\nLI SHE\n\n  44. Lament of the Silkmaid\n\nHSIEH FANG-TE\n\n  45. Late Spring\n\nHAN YU\n\n  46. Spring Lament\n\nYANG WAN-LI\n\n  47. Seeing Off Spring\n\nWANG FENG-YUAN\n\n  48. Seeing Off Spring on the Last Day of April\n\nCHIA TAO\n\n  49. Away from Home at the Beginning of Summer\n\nSSU-MA KUANG\n\n  50. Waiting for a Friend\n\nCHAO SHIH-HSIU\n\n  51. Waking Up in Early Summer\n\nYANG WAN-LI\n\n  52. On the Sanchu Road\n\nTSENG CHI\n\n  53. Impressions\n\nCHU SHU-CHEN\n\n  54. Visiting Chang's Garden at the Beginning of Summer\n\nTAI MIN\n\n  55. Written at the South Tower of Ochou\n\nHUANG T'ING-CHIEN\n\n  56. A Mountain Pavilion on a Summer Day\n\nKAO P'IEN\n\n  57. A Farm Family\n\nFAN CH'ENG-TA\n\n  58. Village Events\n\nFAN CH'ENG-TA\n\n  59. On the Pomegranate Flower\n\nCHU HSI\n\n  60. Village Dusk\n\nLEI CHEN\n\n  61. Written on Mister Lakeshade's Wall\n\nWANG AN-SHIH\n\n  62. Black Robe Lane\n\nLIU YU-HSI\n\n  63. Seeing Off Yuan Er on a Mission to Anhsi\n\nWANG WEI\n\n  64. On Yellow Crane Tower Hearing a Flute\n\nLI PAI\n\n  65. Written at Huainan Temple\n\nCH'ENG HAO\n\n  66. The Seventh Night\n\nYANG P'U\n\n  67. Autumn Moon\n\nCHU HSI\n\n  68. Autumn at the Gate\n\nLIU HAN\n\n  69. Autumn Evening\n\nTU MU\n\n  70. Mid-Autumn Moon\n\nSU SHIH\n\n  71. Reflections at a River Tower\n\nCHAO KU\n\n  72. Written at an Inn in Linan\n\nLIN SHENG\n\n  73. At Chingtzu Temple Seeing Off Lin Tzu-fang at Dawn\n\nYANG WAN-LI\n\n  74. Drinking on the Lake As It Clears Then Rains\n\nSU SHIH\n\n  75. Retiring After Attending a Tea at Hsuante Palace\n\nCHOU PI-TA\n\n  76. Climbing to Canopy Pavilion on a Summer Day\n\nTS'AI CH'UEH\n\n  77. Written While Serving at Jade Hall\n\nHUNG TZU-K'UEI\n\n  78. Written at Secretary Wang's Bamboo Tower\n\nLI CHIA-YU\n\n  79. Serving in the Secretariat\n\nPAI CHU-YI\n\n  80. Reflections While Reading\u2014I\n\nCHU HSI\n\n  81. Reflections While Reading\u2014II\n\nCHU HSI\n\n  82. Cold Spring Pavilion\n\nLIN CHEN\n\n  83. For Liu Ching-wen\n\nSU SHIH\n\n  84. Anchored Overnight at Maple Bridge\n\nCHANG CHI\n\n  85. Winter Night\n\nTU LEI\n\n  86. The Frost and the Moon\n\nLI SHANG-YIN\n\n  87. The Plum\n\nWANG CH'I\n\n  88. Early Spring\n\nPAI YU-CH'AN\n\n  89. The Snow and the Plum\u2014I\n\nLU MEI-P'O\n\n  90. The Snow and the Plum\u2014II\n\nLU MEI-P'O\n\n  91. Herdboy\n\nLU TUNG-PIN\n\n  92. Anchored Overnight on the Chinhuai\n\nTU MU\n\n  93. On Geese Turning Back\n\nCH'IEN CH'I\n\n  94. Written on a Wall\n\nAUTHOR UNKNOWN\n\nPART FOUR\n\n  1. Morning Court at Taming Palace\n\nCHIA CHIH\n\n  2. Responding to Secretary Chia Chih's \"Morning Court at Taming Palace\"\n\nTU FU\n\n  3. Responding to Secretary Chia Chih's \"Morning Court at Taming Palace\"\n\nWANG WEI\n\n  4. Responding to Secretary Chia Chih's \"Morning Court at Taming Palace\"\n\nTS'EN SHEN\n\n  5. On Lantern Festival at Imperial Request\n\nTS'AI HSIANG\n\n  6. On Lantern Festival at Imperial Request\n\nWANG KUEI\n\n  7. Attending a Banquet at the New Residence of Princess An-lo\n\nSHEN CH'UAN-CH'I\n\n  8. In Reply to Ting Yuan-chen\n\nOU-YANG HSIU\n\n  9. Flower Garland Song\n\nSHAO YUNG\n\n  10. Private Thoughts\n\nYEN SHU\n\n  11. Cold Food\n\nCHAO TING\n\n  12. Chingming\n\nHUANG T'ING-CHIEN\n\n  13. Drinking Wine on Grave Sweeping Day\n\nKAO CHU\n\n  14. Strolling outside Town\n\nCH'ENG HAO\n\n  15. The Swing\n\nHUI-HUNG\n\n  16. While Drinking at the Chuchiang Waterway\u2014I\n\nTU FU\n\n  17. While Drinking at the Chuchiang Waterway\u2014II\n\nTU FU\n\n  18. Yellow Crane Tower\n\nTS'UI HAO\n\n  19. A Traveler's Thoughts on a Spring Evening\n\nTS'UI T'U\n\n  20. In Reply to Li Tan\n\nWEI YING-WU\n\n  21. River Village\n\nTU FU\n\n  22. Summer Day\n\nCHANG FENG\n\n  23. Written at My Wang River Retreat after a Steady Rain\n\nWANG WEI\n\n  24. New Bamboo at East Lake\n\nLU YU\n\n  25. Spending a Summer Night with My Cousin Talking about the Past\n\nTOU SHU-HSIANG\n\n  26. Occasional Poem on an Autumn Day\n\nCH'ENG HAO\n\n  27. Visiting Crescent Pond\n\nCH'ENG HAO\n\n  28. Autumn Inspiration\u2014I\n\nTU FU\n\n  29. Autumn Inspiration\u2014II\n\nTU FU\n\n  30. Autumn Inspiration\u2014III\n\nTU FU\n\n  31. Autumn Inspiration\u2014IV\n\nTU FU\n\n  32. Aboard a Boat on a Moonlit Night\n\nTAI FU-KU\n\n  33. Autumn Longing in Ch'ang-an\n\nCHAO KU\n\n  34. Early Autumn\n\nTU FU\n\n  35. Mid-Autumn\n\nLI P'U\n\n  36. At Mister Ts'ui's Villa in Lantien on the Ninth\n\nTU FU\n\n  37. Autumn Thoughts\n\nLU YU\n\n  38. For Hermit Chu, My Neighbor to the South\n\nTU FU\n\n  39. Hearing a Flute\n\nCHAO KU\n\n  40. Winter Scene\n\nLIU K'O-CHUANG\n\n  41. Winter Solstice Eve\n\nTU FU\n\n  42. How Plum Flowers Embarrass a Garden\n\nLIN PU\n\n  43. For My Nephew, Hsiang, on My Demotion and Arrival at Lankuan Pass\n\nHAN YU\n\n  44. Spears and Shields\n\nWANG CHUNG\n\n  45. Returning to My Retreat\n\nCH'EN T'UAN\n\n  46. A Mountain Widow\n\nTU HSUN-HO\n\n  * Postscript\n  * Timeline\n  * About the Translator\n  * Index of Authors\n  * Index of Titles\n\n  * Books by Red Pine\n  * Copyright\n  * Special Thanks\n\n_Translator's Preface_\n\nPoetry is China's greatest art. No matter how wonderful its painting and calligraphy, its pottery and bronze casting, its sculpture and architecture, its music and dance, or its other literary arts, none of these has ever gained the universal appreciation and participation that poetry has enjoyed. This was especially true of the Middle Kingdom's T'ang (618\u2013906) and Sung (960\u20131278) dynasties, which Chinese ever since have called their Golden Age of Poetry. During the T'ang and Sung, poetry became the defining measure of human relationships and human understanding. Nothing was significant without a poem: no social or ritual occasion, no political or personal event was considered complete without a few well-chosen words in rhyme that summarized the subtleties of the Chinese vision of reality and that linked this vision with the beat of their hearts. Although this universal art form had been developing in China for millennia, its greatest flowering was in the T'ang and Sung, when suddenly it was everywhere: in the palace, in the street, in every household, every inn, every monastery, in every village square.\n\nI don't pretend to be able to explain the complex social changes that helped bring about this efflorescence in the T'ang and Sung, but by examining the word itself we can at least see why poetry held such an important place in Chinese culture. The Chinese word for poetry is _shih_ ( ) and is composed of two parts. The part on the left ( ) means \"language,\" and the part on the right ( ) originally meant \"administrative court\" and later came to mean \"Buddhist temple.\" But this was not the original form of the part on the right, and neither \"court\" nor \"temple\" has anything to do with the meaning of _shih._ This is because the second part was originally written _chih_ ( ), meaning \"from the heart,\" and the later form was simply the result of calligraphic shorthand and subsequent convention. Hence, the word for poetry does not mean the \"language of the court\/temple\" but the \"language of the heart.\" The Great Preface to the _Book of Odes_ says, \"When it's in the heart, it's _chih._ When it's expressed in language, it's _shih_.\" Thus, since the dawn of Chinese civilization 5,000 years ago, it has been the function of poetry to express this innermost square-inch of the Chinese heart.\n\nChina's earliest known book of this \"language of the heart\" was the _Shihching,_ or _Book of Odes._ This is a collection of ritual hymns, ballads, and folk songs that date back to the beginning of the Chou dynasty around 1100 B.C. Some are even said to have been passed down from the time of Yu the Great, founder of the Hsia dynasty in 2205 B.C. The collection itself is attributed to Confucius (551\u2013479 B.C.), who reportedly sifted through some 3,000 poems from various parts of China and finally selected 305 as suitable for use in instructing his disciples.\n\nAnother collection of equal, if not greater, importance was the _Chutzu,_ or _Songs of Ch'u._ This was a collection of the poems of Ch'u Yuan (340\u2013278 B.C.) and other shaman-literati of the state of Ch'u compiled by an anonymous editor around the beginning of the Christian Era. Unlike the _Book of Odes,_ the _Chutzu_ contained the work of known individuals. And unlike the more aloof and stylized _Odes,_ the poems in the _Chutzu_ poured forth far more personal and heartfelt joys and sorrows. As a result, this book had a much greater impact on poets. Wang Wei (701\u2013761) once said he carried two books with him wherever he traveled: the _Vimalakirti Sutra_ and the _Chutzu._\n\nAs time went on, other collections of poetry appeared. But aside from the _Book of Odes_ and the _Songs of Ch'u,_ no other anthology gained more than a limited audience\u2014until the appearance of the _Chienchiashih,_ or _Poems of the Masters._ This collection was first compiled at the end of the Sung dynasty by one of its most prolific writers, Liu K'o-chuang (1187\u20131269). Liu was also a leading literary critic, and at some point he decided to present his views on poetry through a collection of examples. Although no copy of Liu's original anthology still exists, we know that he chose about 200 poems and arranged his selections under fourteen headings: flowers, bamboo, the sky, the earth, palace life, weather, tools, music, animals, insects, seasons, festivals, daily events, and human character. It was a great success. And as it enjoyed the good fortune of being published at the beginning of China's printing revolution, it soon appeared in village schools and private academies across the country, where it established itself as eminently useful in teaching students the rhythms of language and also the heart, as well as the names of all sorts of things an educated person should know.\n\nOver the centuries, Liu's original edition was rearranged by different editors, and the number of poems increased to the present 224. In some editions, several poems dating from the Ming dynasty (1368\u20131643) were also added to this total, but I have followed most editors in deleting them as extraneous.\n\nAt the beginning of the seventeenth century, Wang Hsiang, who is better known for his works for and about women, rearranged the poems into four parts according to poetic form: 39 four-line poems with five characters to a line, 45 eight-line poems with five characters to a line, 94 four-line poems with seven characters to a line, and 46 eight-line poems with seven characters to a line. Readers interested in Chinese poetics should note that the four-line poems all follow the rhyme schemes and tonal patterns laid down for _chueh-chu_ (detached quatrains), and all eight-line poems follow the more complex rules for _lu-shih_ (regulated verse) that required a great deal of parallelism as well. Wang also arranged the poems in each section according to the seasons (more or less), and he composed the first commentary. Wang's version of Liu's anthology became so popular that the previous editions disappeared altogether, and I have generally followed Wang's arrangement.\n\nThe Chinese title of this book is _Ch'ien-chia-shih,_ which literally means _Poems of a Thousand Masters._ There are, however, only a hundred or so poets represented; hence, I have dropped the numeral. Despite the exaggeration, _Poems of the Masters_ includes the most-quoted poems in the Chinese language by the most famous poets of China's Golden Age of Poetry. And for the past eight centuries, it has been the most-memorized collection of verse in China and part of every student's education.\n\nWhen Sun Chu (1711\u20131778) published _Three Hundred Poems of the T'ang_ in 1763\u2014and _Poems of the Masters_ finally had to share the poetry stage\u2014Sun said in his preface that he based his own collection on the earlier anthology, and he admitted it was superior to his in terms of its shorter, easier-to-memorize poems. Although Sun's collection eventually proved more popular with older readers, _Poems of the Masters_ retained its preeminent place in the education of every schoolchild, as its emphasis on four-line poems made it much more useful in teaching poetry to students at an earlier age. Although China's recent detour into communism saw the _Chienchiashih_ replaced in schools with proletarian propaganda, it was still part of the national curriculum until the middle of the last century. And I suspect it will be again someday. Its success, like that of _Three Hundred Poems of the T'ang,_ has been in large part due to its inclusion of poems that could be understood and appreciated by all sectors of society. And its poems are as meaningful today as they were when they were written.\n\nIn addition to translating the poems in this collection, I have included the original text for those able to read Chinese. I have also added a certain amount of background information to help readers locate these poems in time and space and in the lives of the poets. Scholars, however, are not always in agreement concerning events of a thousand or more years ago, and my choice among different accounts simply reflects what has seemed to me the most likely. But even without such information, and even if they did not fully understand the context in which these poems were written, the Chinese have always appreciated their \"language of the heart\" and continue to quote their phrases and lines in their speech today. Hence, I can think of no better way to reveal that hidden square-inch every Chinese holds dear and depends upon for inspiration than through this modest collection of poems.\n\nRed Pine\n\nCold Dew, Year of the Horse\n\nPort Townsend, Washington\n\nTRANSLATOR'S ACKNOWLEDGMENTS\n\nMy thanks for the continuing support of the U.S. Department of Agriculture's Food Stamp Program, the Port Townsend Food Bank, the Olympic Community Action Energy-Assistance Program, the Earned Income Tax Credit Program administered by the Internal Revenue Service, and the American Optometric Association's Vision USA Program.\n\n## PART ONE\n## 1\n\nMeng Hao-jan (689\u2013740) was one of the foremost poets of the T'ang and one of the few who did not pursue an official career. When Meng was forty, the retired prime minister, Chang Yueh, invited him to the western capital in Ch'ang-an to take the civil service exam. But Meng not only failed, he annoyed Emperor Hsuan-tsung as well, and he soon returned to his family estate outside Hsiangyang. When time permitted, and it usually did, Meng visited friends at their posts along the Yangtze or enjoyed the peace of the hermitage he built on Lumenshan, twenty kilometers southeast of his home. Five centuries earlier, the recluse P'ang Te-kung also built his hut on the same mountain. Like P'ang, Meng had little interest in worldly goals and preferred to sleep late\u2014while his friends in the capital were at court before dawn. Even as winter becomes spring, Meng is still sleeping late and isn't wakened by the sun but by the sound of birds beginning their spring courtship. Still, this too fails to drive him from his bed, and he is content to wonder about the scene outside without feeling the need to do anything about it.\n\n## _Spring Dawn_\n\nMENG HAO-JAN\n\nSleeping in spring oblivious of dawn\n\neverywhere I hear birds\n\nafter the wind and rain last night\n\nI wonder how many petals fell\n\n## 2\n\nLoyang was the T'ang dynasty's eastern capital and was located at the edge of the Yellow River floodplain, 300 kilometers north of Meng's home. Although he remained an outsider, Meng's reputation as a poet made him welcome in the highest levels of society. But on this trip he failed to meet an official to whom he had written earlier. The man had been demoted for taking the responsibilities of his post too literally\u2014censors were charged with reporting improper conduct and harmful policies to the emperor\u2014and he had been banished to Chiangling, the mountainous area of the Tayu Range that separates the Kan River watershed from the semitropical South. Meng sent this poem to express both his disappointment and his sympathy. In the temperate North, the plum flowers during the lunar New Year and marks the start of spring, which is a time when families make every effort to be together.\n\n## _Calling on Censor Yuan without Success_\n\nMENG HAO-JAN\n\nIn Loyang I tried to visit you sir\n\nto Chiangling you were banished\n\nthe plum I hear flowers earlier there\n\nbut how could spring compare\n\n## 3\n\nWang Ch'ang-ling (698\u2013756) was from the Ch'ang-an area but spent most of his career in low-level posts in outlying provinces. Not long after the An Lu-shan Rebellion broke out in 755, he resigned his post in Hunan but was murdered on his way back north. He wrote this poem c. 739 while serving in Nanching after a brief banishment to Kuangtung province. The Huai is one of China's major rivers and flows parallel to the Yangtze one hundred kilometers north of Nanching. But Wang is referring to a much smaller river of the same name (usually called the Chinhuai) that emptied into the Yangtze from the south at Nanching. Although Wang did not enjoy success as an official, he was well known for his short poems, especially those on the subject of farewell. Here, he tries to detain a friend whose official duties included supervision of government granaries that supplied farmers with seed and grain at a set price. With the advent of spring planting, his friend heads west up the Yangtze, followed by the moon, while Wang takes solace in the thought that the Yangtze will not only continue to link their hearts together but also will deepen with spring rains.\n\n## _Seeing Off Supply Director Kuo_\n\nWANG CH'ANG-LING\n\nHuai River green brightens my door\n\nthis host would keep a traveler longer\n\nthe moon leaves too with an honest official\n\nbut every night spring waters grow deeper\n\n## 4\n\nCh'u Kuang-hsi (707\u2013760) was born just west of Chufu in Shantung province and passed the country's highest civil service exam at the unusually young age of nineteen. He rose to become an imperial inspector, but when the armies of An Lu-shan swept through Loyang in 755, he was forced to serve in the rebel administration. After the rebellion was crushed, he was sentenced to death but was sent into exile instead and died soon after reaching his post. This was one of five poems he wrote for his fellow poet Lu Ssu-lang. When Loyang was rebuilt during the Sui and early T'ang, it was laid out west of the old city with straight, wide roadways. During the T'ang, it served as the dynasty's eastern capital, while Ch'ang-an served as the western capital. The Wuling Hills were northwest of Ch'ang-an and named for a group of five imperial grave mounds. This was also where many well-to-do families maintained estates, and the name eventually became associated with the western capital's elite, whether or not they had estates at Wuling. Here, the sons of the elite are in the eastern capital, which was famous for its gardens during the T'ang, racing one another on horses whose bridles are decorated with cowry shells from the South Seas.\n\n## _The Roads of Loyang_\n\nCH'U KUANG-HSI\n\nThe roads are as straight as strands of hair\n\nand full of the glories of spring\n\nthe noble young lords from the Wuling Hills\n\nride by in pairs with their bridles ringing\n\n## 5\n\nLi Pai (701\u2013762) aka Li Po, and Tu Fu are ranked as China's two greatest poets. Li was born in what is now Kyrgyzstan and grew up in Szechuan north of Chengtu. For a brief period he was a favorite of the court, but he soon fell into disfavor and spent most of his life wandering along the Yangtze, a guest of those who appreciated his talent and unrestrained spirit. He wrote this poem in 753, during one of several stays in Hsuancheng, a town south of Nanching still famous for its rice paper. Chingting Mountain was five kilometers northwest of the old city wall. Although its peak was only 300 meters high, it was known for its crags and cliffs and was also where the poet Hsieh T'iao (464\u2013499) wrote many of his most famous landscape poems. On this visit, Li observes the transience of the world around him and his separation from it. But he also becomes aware of his oneness with the mountain. According to a tradition at least as old as Confucius, water represents wisdom (teaching us an awareness of impermanence) and mountains represent compassion (teaching us an awareness of the unchanging nature we share with all things). The images here also extend beyond the natural world: the birds represent officials, the lone clouds represent solitary spiritual seekers, and the mountain is the Tao.\n\n## _Sitting Alone on Chingting Mountain_\n\nLI PAI\n\nFlocks of birds disappear in the distance\n\nlone clouds wander away\n\nwho never tires of my company\n\nonly Chingting Mountain\n\n## 6\n\nWang Chih-huan (688\u2013742) was born in Taiyuan, the provincial capital of Shansi, and later moved to Hsinfeng near the southern end of the same province. When a career in the civil service failed to yield anything but minor appointments, he returned home and spent his time visiting friends and traveling to scenic spots in the area. Wang wrote this poem on a visit to Yungchi, one hundred kilometers south of Hsinfeng. The tower he climbed rose three stories above the southwest corner of the city wall and looked out on the Chungtiao Mountains to the south (which are so high\u2014and the sun's trajectory sufficiently low\u2014the noon sun disappears briefly behind their highest peaks). To the west was the Yellow River, which turned east just after it flowed past Yungchi and headed for the sea one thousand kilometers away. The tower was named for the oriental white storks ( _Ciconia boyciana_ ) that roosted on its roof, and many poets commemorated the view. Although Wang was famous for his short poems, and many became standard fare for female entertainers during his lifetime, only six survive.\n\n## _Climbing White Stork Tower_\n\nWANG CHIH-HUAN\n\nThe midday sun slips behind mountains\n\nthe Yellow River turns for the sea\n\ntrying to see for a thousand miles\n\nI climb one more story\n\n## 7\n\nSun T'i (fl. 720) was from Poshan in Shantung province and served in several prominent posts, including the Secretariat. During every dynasty, the Chinese court resorted to marriage as part of its foreign policy to foster peaceful relations with the nomadic tribes along its borders. In order to elevate their status with their prospective husbands, daughters of the nobility were honored with the title of \"princess\" prior to their departure. This particular \"princess\" was a daughter of the prominent Yang clan and was married to the king of the Khitans, Li Shih-huo, in the last month of 717. Apparently, another spring was too much for the Khitan chieftain, and he died the following year. Dragon Pass (Lungsai) was on China's northeast border with Manchuria, between what are now Blue Dragon Manchurian Autonomous County (Chinglung Mantsu Tzuchihhsien) and Black Dragon County (Lulunghsien). In the _Chuantangshih (Complete Poems of the T'ang),_ the full title is \"With Li Shao-fu of Loyang Seeing Princess Yung-lo Leave for Manchuria.\"\n\n## _On Seeing Princess Yung-lo Leave for Manchuria_\n\nSUN T'I\n\nOrioles and flowers on the border are rare\n\neven at New Year nothing looks new\n\nbut when a beauty falls from Heaven\n\nDragon Pass finally sees spring\n\n## 8\n\nChin Ch'ang-hsu (fl. eighth century) was a native of Hangchou but has left no other traces than this solitary poem. In some editions, it is entitled \"Yichouko\" (\"Song of Yichou\") and attributed to Ko Chia-yun (T'ang dynasty). While the authorship is a moot point, this alternative title is clearly a mistake, as Yichou was just west of the present oasis of Hami on China's northwest border, while Liaohsi was just inside the Great Wall along China's northeast border with Manchuria. The oriole, like the robin, is a harbinger of spring, and the twittering of mating birds only reminds this wife of her separation from her husband.\n\n## _Spring Complaint_\n\nCHIN CH'ANG-HSU\n\nChase the orioles away\n\ndon't let them sing in the tree\n\ntheir singing disturbs a wife's dreams\n\nand keeps her from reaching Liaohsi\n\n## 9\n\nCh'iu Wei (694\u2013789) was from Chiahsing in Chekiang province and served in the central government before retiring from his final post as tutor to the crown prince. The Chinese have always admired skin not darkened by the sun or dried by the wind\u2014the signs of manual labor and old age. Although this poem ostensibly refers to the imperial harem, and the imperial pear orchard was near the women's apartments, there were pear trees elsewhere in the Forbidden City. Also the title makes it clear that Ch'iu is using the pear blossom's purity to represent the virtue of such court officials as himself, who seek to be of service to the Son of Heaven despite the transience of such recognition. The \"jade\" steps of white marble at the entrance of the palace in Ch'ang-an represent the imperial presence, and the East Wing was the location of the Chancellery, where Ch'iu was serving at the time and from which he hoped to rise to the emperor's attention\u2014which he did, but as the result of his filial piety rather than his service at court.\n\n## _A Pear Blossom in the East Wing_\n\nCH'IU WEI\n\nIts pristine beauty could fool the snow\n\nits lingering scent soaks through clothes\n\ntell the spring wind not to stop\n\nblow it toward the steps of jade\n\n## 10\n\nLing-hu Ch'u (766\u2013837) was born in the Silk Road oasis of Tunhuang and reached the rank of prime minister under Emperor Hsien-tsung (r. 805\u2013820). As a court official, he was known for his elegant essays and poems, and he was once asked by the emperor to compile an anthology of poetry, which has since been lost. Here, he takes on the persona of a concubine among other concubines (the butterflies) who have fallen out of favor with the emperor (the oriole). During the Han dynasty, Emperor Wu (r. 140\u201387 B.C.) tired of his empress and had her moved to Towering Gate Palace (Changmingkung), at the rear of the Forbidden City. But she became so distraught she paid the poet Ssu-ma Hsiang-ju one hundred kilos of gold to write an ode on her behalf. The emperor was moved, and he brought her back to the main palace. Ironically, Ling-hu himself once paid a bribe of two million strings of cash for a governorship, but here he offers a poem. In Chinese poetry, birds and butterflies invariably appear together, and the absence of one accentuates the isolation of the other. Yet, though this woman (or official) no longer enjoys the emperor's attention, her (or his) sentiments, like those of the Han-dynasty empress, have not changed. Whenever the emperor visited someone, he arrived in a carriage festooned with pennants that included the iridescent blue-green feathers of the kingfisher.\n\n## _On Thinking of My Lord's Grace_\n\nLING-HU CH'U\n\nThe oriole's song is gone from my garden\n\nbutterflies dance by the towering gate\n\nI've watched another spring pass\n\nwithout seeing your kingfisher carriage\n\n## 11\n\nHo Chih-chang (659\u2013744) was a native of Shaohsing and a member of a group of poets from Chekiang province who formed a literary coterie in Ch'ang-an. Although he served in a series of prestigious posts, including vice-minister of the Ministry of Rites and director of the Palace Library, he was better known for his clever conversation and love of drinking. Tu Fu included him among the Eight Immortals of Wine, and his friends called him the Wild Man of Ssuming (a mountain range east of his hometown). With his fellow intellectuals, he also shared a love of seclusion. During the T'ang it was quite common to maintain a retreat in the Chungnan Mountains south of Ch'ang-an, and it was also during the T'ang that the expression \"Chungnan Shortcut\" came into use, whereby many would-be officials tried to gain the notice of the court through their \"seclusion.\" Shortly before he died, Ho became a Taoist priest himself and retired to the countryside near Shaohsing.\n\n## _On Mister Yuan's Country Retreat_\n\nHO CHIH-CHANG\n\nWe've never met good sir\n\nI stopped because of the woods and stream\n\ndon't worry about buying wine\n\nI have some coins in my purse\n\n## 12\n\nYang Chiung (650\u2013694) was born east of Ch'ang-an at the foot of the sacred peak of Huashan. He was recognized as a prodigy as a youth and served in the staff of the heir apparent and later in several midlevel capacities. But in 685, when a distant relation staged a minor revolt, he was banished to Szechuan. A few years later he was recalled to Empress Wu's court in Loyang. But in 693 he was banished again, this time to Chekiang, where he died the following year. This poem may have contributed to his final banishment, as it implies that the emperor (or empress, in this case) has broken his (or her) word. This connection comes from an account in Ssu-ma Ch'ien's _Shihchi (Records of the Historian):_ The King of Chao once possessed a rare jade disk for which the state of Ch'in offered to exchange fifteen cities along their common border. The King of Chao gave the disk to his chief minister to convey to the King of Ch'in. But after discovering that the King of Ch'in had no intention of honoring his pledge, the minister refused to give up the disk and returned it to his own king. Here, Yang compares his friend to the jade disk, while its celestial counterpart lights the scene of their parting on the banks of the Yellow River just north of Loyang. His friend is on his way home to Chaochou, the ancient capital of the state of Chao, whose name his friend shares. Yang was considered one of the greatest poets of the early T'ang, though only thirty or so poems survive.\n\n## _Seeing Off Chao Tsung at Night_\n\nYANG CHIUNG\n\nScion of Chao jade disk worth cities\n\nlong has your name been known\n\nseeing you off on your way home\n\nmoonlight fills the waiting river\n\n## 13\n\nWang Wei (701\u2013761) was from Taiyuan, the capital of Shansi province, and moved to Ch'ang-an as a young man. After passing the civil service exam he rose through the ranks and, despite the occasional banishment, eventually reached the post of vice prime minister. However, his interest in Buddhism blunted any political ambitions, and whenever he had time he preferred to wander in the Chungnan Mountains south of the capital. Wang was not only one of the greatest poets of the T'ang, he was also a skilled musician and one of the dynasty's greatest landscape artists. This was one of twenty poems he wrote toward the end of his life at his Wang (a different Wang) River estate, which was located in a hidden valley some sixty kilometers southeast of Ch'ang-an. Bamboo Retreat was the name he gave to one of several structures he built there. The first line is from Ch'u Yuan's \"Shankuei\" (\"Mountain Spirit\"): \"Sitting alone amid dense bamboo \/ I can't see the sky\"; and which ends, \"the wind whistles \/ trees sigh \/ thinking of you \/ how useless such grief.\" The last line of Wang Wei's poem also recalls the first of three poems titled \"Drinking Alone beneath the Moon,\" written by Li Pai the night before he left the capital for good in 744. The Chinese are fond of saying the same moon shines on those who are apart, thus joining them together.\n\n## _Bamboo Retreat_\n\nWANG WEI\n\nSitting alone amid dense bamboo\n\nstrumming my lute and whistling\n\ndeep in the forest no one else knows\n\nuntil the bright moon looks down\n\n## 14\n\nMeng Hao-jan (689\u2013740) wrote this poem for a friend who was leaving to seek his fortune (and, no doubt, an official post) in the capital of Ch'ang-an, which was situated in the middle of what was once the ancient state of Ch'in and which retained that state's name. Wuling was the name of a group of imperial grave mounds northwest of Ch'ang-an. As royalty and the rich maintained estates there, the name came to stand for the capital's elite as well as for the capital itself. No member of the gentry would consider himself properly attired at court without a sword, but the sword Meng gives his friend is of a different sort and more valuable, because it can cut through duplicity. In the background here are references to Wu Tzu-hsu (fifth century B.C.), who gave his sword to a fisherman for saving his life and said, \"This sword is engraved with the seven stars of the Big Dipper and worth a ton of gold\" ( _Yulan_ : 74), and to Chi Cha, who left on a mission and stopped to see his friend Hsu Chun. Chi was aware that Hsu secretly admired his sword, but he needed it for his mission. When he returned and learned that Hsu had died, he removed his sword and hung it on a tree beside Hsu's grave. ( _Shihchi_ )\n\n## _Seeing Off Chu Ta Leaving for Ch'in_\n\nMENG HAO-JAN\n\nFor an unemployed gentleman bound for Wuling\n\na first-rate sword is worth a ton of gold\n\nI remove this in parting and give it to you\n\na simple piece of my heart\n\n## 15\n\nTs'ui Hao (704\u2013754) was from Kaifeng and established his reputation as a poet in Ch'ang-an while still in his teens. During this early period his verse followed the carefully regulated usages of court poetics, but he soon broke free into nonconformist technique as well as unrestrained behavior. He was greatly admired by Li Pai. And like Li, he annoyed as many people as he impressed and did not enjoy a successful career. Hengtang (Cross Dike) was the embankment along the Chinhuai River, from Nanching's southwest gate to the Yangtze ten kilometers to the west; the area where the two rivers met was called Changkan (Longpole). Here, the poet speaks through a young girl who is a member of the boat people who still make up separate cultural and linguistic groups in South China. This was the first of four poems. In the second, Ts'ui presents the boy's answer: \"My home overlooks the Great River \/ I travel along the Great River's banks \/ I'm from Longpole too \/ but I've never seen you before.\" Longpole ballads, characterized by exchanges between a boy and girl from Changkan, were also written by other T'ang poets.\n\n## _Ballad of Changkan_\n\nTS'UI HAO\n\nWhere are you from good sir\n\nthis maid is from Hengtang\n\nI ask while our boats are moored\n\nperhaps we're from the same place\n\n## 16\n\nKao Shih (702\u2013765) grew up in Honan province in a family sufficiently impoverished that he had to beg for food. But through tenacity he gained an education and eventually an appointment as a secretary in the military. He was one of the few great poets of the T'ang who enjoyed a successful career. Here, he refers to the biography of Fan Shu (aka Fan Sui) in the _Shihchi._ Fan accompanied Hsu Chia as an aide on a mission from Wei to the state of Ch'i, whose king ignored Hsu but presented Fan with delicacies and gold. Hsu suspected Fan had divulged state secrets, and when they returned to Wei he reported this to the prime minister, who had Fan beaten and left for dead. Fan, however, managed to escape, changed his name, and rose to the rank of prime minister in the state of Ch'in. When Fan heard that the state of Wei was sending Hsu on a mission to Ch'in, he disguised himself as a pauper and met Hsu at his inn. When Hsu saw his former aide in such straits, he took off his silk robe and gave it to the man he had once maligned. Fan then revealed his new identity, and Hsu expected to be executed. But Fan pardoned Hsu because of his compassion for an old friend. Kao, however, condemns Hsu for his blindness and offers this oblique critique of those in power who fail to discern the true qualities of others.\n\n## _Ode to the Past_\n\nKAO SHIH\n\nHe made a gift of his robe\n\nconcerned Fan Shu was cold\n\nbut unaware of his noble rank\n\nhe still saw a common man\n\n## 17\n\nLi Shih-chih (d. 747) was a royal clansman and rose to the rank of vice prime minister. At court he was known for his poetry and love of wine, and he was ranked alongside Li Pai and Ho Chih-chang in his passion for both. As early as the third century, the Chinese referred to unfiltered rice wine as the _hsien_ (able) and filtered rice wine as the _sheng_ (sage). Thus, the poet's seemingly polite remark in the first line implies he has been replaced by someone less worthy, while he consoles himself in the knowledge of his own virtue. Despite his apparent status and prestige, Li Shih-chih ended up on the wrong side of the infamous prime minister, Li Lin-fu, and was forced to resign in 746. The following year he was sent to a provincial post and ordered to commit suicide. This poem was written following his \"resignation\" and prior to his departure. During this brief period, he served as a counselor to the heir apparent. But those who once sought out his hospitality are now avoiding him, lest they be associated with a man whose days are numbered.\n\n## _Resigning as Minister_\n\nLI SHIH-CHIH\n\nYielding my post to the able\n\nenjoying the sage in a cup\n\nI asked someone who came in the past\n\nhow many will visit today\n\n## 18\n\nCh'ien Ch'i (722\u2013780) was from Chekiang province and enjoyed a modest career as an official in both Ch'ang-an and Loyang. Though he never served in anything but minor posts, he was highly regarded as a poet. A generation younger than Wang Wei, he was considered Wang's poetic successor by many, and he was the best known of the Ten Talents of the Tali Period (766\u2013779). Although several hundred poems survive, they have suffered the neglect of later critics, and Ch'ien is not well known today. Yen and Chao were ancient states of North China. In the preface to \"Seeing Off Tung Shao-nan,\" Han Yu (768\u2013824) notes that both states were known for their troubadours. Here, the man to whom Ch'ien writes this poem takes on the persona of the troubadour, with its tragic overtones. Chi Meng lived in Loyang during the Han dynasty and was famous for his chivalrous deeds. In the third line, the feelings that cannot be expressed are perhaps best not expressed. The final line suggests this man is being banished to some distant post, perhaps on the Silk Road (where the sun sets), as punishment for his \"chivalry.\" Perhaps he was involved in the An Lu-shan Rebellion (755\u2013762) or made the mistake of praising some aspect of it. The states of Yen and Chao were An Lu-shan's base, and Loyang was the capital of his short-lived Greater Yen dynasty.\n\n## _On Meeting a Chivalrous Man_\n\nCH'IEN CH'I\n\nSinger of elegies of Yen and Chao\n\nwe meet where Chi Meng lived\n\nwords can't exhaust this inch of our hearts\n\nthe sun is setting on the road ahead\n\n## 19\n\nCh'ien Hsu (fl. tenth century) was a great-grandson of Ch'ien Ch'i, to whom this poem is incorrectly attributed in some editions. Hsu began a promising career in the capital after passing the civil service exam in 898, but he was soon implicated in a political scandal and was sent to serve as the prefect of Nanchang, the capital of Kiangsi province, just south of Lushan. During the Six Dynasties (222\u2013589), Lushan became known for the recluses who made it their home during what was one of the longest periods of instability in Chinese history. The mountain was first named Kuanglushan after one such recluse, K'uang Heng, who moved there during the reign of Emperor Ch'eng (r. 32\u20137 B.C.). In fact, _k'uang-lu_ means \"K'uang's hut.\" Later, K'uang's name was dropped, and it was simply called Lushan, or Hut Mountain. Ch'ien had hoped to climb its slopes on his way to his new post. But Lushan is famous for its mist, and Ch'ien had to content himself with viewing its half-shrouded peaks from his boat and wondering if any sages had survived the last period of disunion. As this poem was written, the T'ang dynasty (618\u2013906) was coming to end. This was sixty-ninth in a series of one hundred poems Ch'ien wrote while on this journey.\n\n## _Traveling on the Yangtze_\n\nCH'IEN HSU\n\nSo close but plagued by wind and rain\n\nI can't climb Kuanglushan\n\nI wonder if in those mist-hidden caves\n\nany Six Dynasty monks still dwell\n\n## 20\n\nWei Ying-wu (737\u2013792) was born in Ch'ang-an into a well-connected family and was allowed to serve in a number of minor posts without having to pass the civil service exam. After the usual ups and downs, he ended his career as the prefect of a series of towns along the lower reaches of the Yangtze, where he wrote this poem. Wei's poetry is often compared to that of the country poet T'ao Yuan-ming (365\u2013427), who lived in the middle reaches of the Yangtze in what was once the state of Ch'u, and who was the model for many who yearned for a life of simplicity. But Ch'u was also the home of the great shaman-poet Ch'u Yuan (340\u2013278 B.C.), who chose moral purity over adapting to the ever-changing Tao. As his friend returns to his post in Ch'u, Wei wonders which of these paths his friend will take. In the first two lines, Wei answers his friend's question concerning the direction of his own life. Wei says he has stopped reading the _Yiching (Book of Changes_ ) and cultivating a life of solitude. Now, instead of trying to predict change, he simply accepts whatever comes, like the gulls. According to a story in _Liehtzu:_ 2.11, gulls only relax around those who don't harbor thoughts of catching them. This was the third of three poems.\n\n## _In Reply to Li Huan_\n\nWEI YING-WU\n\nI left the _Yiching_ in the woods\n\nnow I drift with the gulls by the stream\n\namong the singers of the ways of Ch'u\n\nto whom do you most often turn\n\n## 21\n\nLiu Yu-hsi (772\u2013842) was born in the Grand Canal town of Hsuchou to a family of Confucian scholars. Although he held several important posts, such as advisor to the heir apparent and vice-minister in the Ministry of Rites, he was twice demoted because of poems he wrote satirizing those in power. The autumn wind reminds Liu of the end of the summer of his life and of the loss of imperial favor. Like the geese, he is headed south, away from the capital from which he has been banished. But unlike the geese, he is traveling alone. The courtyard is that of a travelers' inn.\n\n## _Ode to the Autumn Wind_\n\nLIU YU-HSI\n\nWhere does the autumn wind come from\n\nrising it sees off the geese\n\nentering courtyard trees at dawn\n\nit wakes a lone traveler first\n\n## 22\n\nWei Ying-wu (737\u2013792) wrote this poem to Ch'iu Tan, the younger brother of Ch'iu Wei. Wei Ying-wu was serving as prefect of Suchou at the time, and Ch'iu Tan had served briefly as his secretary. However, Ch'iu decided he would rather cultivate the Tao, and he quit his post, crossed the Yangtze, and retired to a hermitage on Pingshan, just outside the northwest gate of Yangchou. Pingshan was also the location of Taming Temple. Several decades earlier, the Buddhist monk Chien-chen left this temple and traveled to Japan, where he is credited with introducing Buddhism, Chinese medicine, and the Chinese language to the Japanese. Thus, the slopes of Pingshan, which was not a large mountain, were not exactly deserted. Still, this was the eighth full moon, or the Mid-Autumn Festival, when relatives and friends normally spend the night together, and Wei tries to imagine how lonely his friend must feel. Pine nuts remain an important source of food for recluses in China. But here, Wei uses them to emphasize Ch'iu Tan's isolation, where the only sound is that of falling pinecones. Note, too, that the summers are sweltering in this part of China, and any sign of autumn is welcome\u2014especially the first breath of cool air.\n\n## _To Secretary Ch'iu on an Autumn Night_\n\nWEI YING-WU\n\nOut walking and singing of cooler days\n\nI think of you on an autumn night\n\npinecones falling on deserted slopes\n\nthe recluse I suspect not yet asleep\n\n## 23\n\nKeng Wei (fl. 770) was a native of Yungchi in Shansi province and served for many years as an imperial advisor. At the peak of his career he was ranked among the court poets known as the Ten Talents of the Tali Period (766\u2013779), among whom he was the least ornate. Although little is known about his later years, this poem suggests he fell out of favor and was forced into simpler lodgings, where he is struck by the strong but fading light that returns to his lane after its journey across the southern sky (for differences of opinion over the meaning of _fan-chao_ [returning light], see Eliot Weinberger's _Nineteen Ways of Looking at Wang Wei)._ In the background is \"Swaying Millet\" from the _Shihching (Book of Odes),_ which describes the area southwest of Ch'ang-an, where the Chou-dynasty capital of Hao once stood, and where Keng is now apparently living. Its author paces the roadway, as if looking for something, and repeats the refrain: \"Those who know me \/ know cares fill my heart \/ those who don't \/ wonder what I have lost.\" Hao was destroyed by nomads as the result of the inattention of King Yu (r. 781\u2013771 B.C.). Keng is concerned that the lessons of history may not have been learned by those now in power\u2014the year is late. Thus, the \"ancient road\" refers to the Tao as well as to the road leading to a capital that no longer exists.\n\n## _Autumn Day_\n\nKENG WEI\n\nThe day's late light fills a village lane\n\nwith whom can I share my cares\n\nnobody takes the ancient road\n\nmillet sways in autumn wind\n\n## 24\n\nHsueh Ying (fl. 850) has left no information about himself other than what can be gathered from a few surviving poems. The lake on which he is sailing is Taihu (Great Lake), which was also called Wuhu (Five Lakes) because it was said to be made up of five bodies of water, like the petals of a plum blossom. It is one of the largest freshwater lakes in China and is located just south of the Yangtze, not far from where the river meets the sea. The Chinese began migrating to this region at the beginning of the first millennium B.C., and the area around the lake soon became the country's major rice- and silk-producing area. It also became the scene of some of China's most protracted wars. When this poem was written, the region around the lake was so heavily taxed, it was the source of several rebellions, and eventually the court had no choice but to recognize the area's bandit commanders as _de facto_ officials. Due to the uplifting of the Tibetan Plateau, rivers in China generally flow east, and this has become a standard metaphor for the inexorable passage of time. Here, Hsueh wonders why sorrow is its inevitable consequence.\n\n## _Autumn Day on the Lake_\n\nHSUEH YING\n\nSailing on the Great Lake at sunset\n\nmist and waves and everywhere sorrow\n\nrising and falling events of the past\n\nwho can tell me why they flow east\n\n## 25\n\nLi Ang (809\u2013840) was the name of Emperor Wen-tsung (r. 827\u2013840). Following his failed attempt to resolve the bitter factional strife at court in the autumn of 835, the eunuchs made him a virtual prisoner in the palace, and he died of a chronic illness a few years later. He was, however, a man who loved simplicity and enjoyed poetry and wine in equal measures. Here, he looks out from the balcony above the palace wall in the spring of 836 and considers his plight. Even though he is the emperor, he has lost the prerogatives of his position and cannot even visit the royal orchard, which covered thousands of acres west of the palace and which was famous for its pear blossoms. The weed-covered path reminds the emperor that he has not been able to leave the palace since his attempt to assassinate the eunuchs failed the preceding fall. As the only males besides the emperor in the inner palace, eunuchs often wielded more power than government ministers during the reigns of weak emperors\u2014or weak ministers. Implicit here is the emperor's sense that only the Son of Heaven understands the duty of caring for the realm, which even his confinement cannot diminish, while the eunuchs, and those officials who carry out their will, are only concerned with their own welfare.\n\n## _Written in the Palace_\n\nLI ANG\n\nThe Shanglin Woods are in bloom\n\nbut autumn weeds hide the royal path\n\nmy ministers no longer know\n\nhow vast is the view from on high\n\n## 26\n\nChia Tao (779\u2013843) was born near Beijing and became a Buddhist monk in his youth. He later changed his mind and decided to return to lay life so that he could devote himself to poetry. He spent all but his final years in Ch'ang-an living in impoverished, if refined, circumstances among those who shared his love. The mountain he visits here is in the Chungnan Range south of Ch'ang-an, where tens of thousands of recluses have lived over the past three millennia. It is still customary for eminent monks and nuns to be accompanied by a young attendant, even when they retire\u2014as many still do\u2014to the cloud-shrouded slopes that rise beyond China's cities and towns. Such recluses collected herbs not only for their own use in elixirs and medicinal decoctions but also to exchange for necessities such as salt and cooking oil, lamp oil, flour (in North China) or rice (in South China), and the occasional new blanket or robe.\n\n## _Looking for a Recluse without Success_\n\nCHIA TAO\n\nBelow the pines I ask the boy\n\nhe says his master has gone to find herbs\n\nhe's somewhere on this mountain\n\nbut the clouds are too thick to know where\n\n## 27\n\nSu T'ing (670\u2013727) grew up just west of Ch'ang-an and was greatly admired for his prose as well as his poetry. His father, Su Kuei, served as prime minister during the reigns of Chung-tsung (r. 705\u2013710) and Jui-tsung (710\u2013712), and Su T'ing held the same post during the early years of Hsuan-tsung's reign (712\u2013756). In 720, however, he fell out of favor and was demoted to military governor of Szechuan. He eventually returned to the capital in 724 and was able to live his final years in relative peace. But before he was allowed to do so, he was sent to inspect frontier defenses along the Great Wall in the northern part of Shansi province, which is drained by the Fen River. After coming down from Mongolia, autumn clouds move south across the river while Su, after an equally long journey from Szechuan, crosses it on his way north. The first two lines paraphrase a famous poem by Emperor Wu (r. 140\u201387 B.C.) of the Han dynasty entitled \"Autumn Wind\": \"Autumn wind stirs and white clouds fly... aboard a rocking boat I cross the Fen.\" In the last two lines Su is not only surprised by autumn but also by the first signs of his own old age\u2014not decrepitude (he is only fifty-four), but the end of his political career and his imminent forced retirement at the end of this, his last mission.\n\n## _Surprised by Autumn on the Fen_\n\nSU T'ING\n\nThe North Wind blows white clouds\n\na thousand miles and across the Fen\n\nthe hopes of my heart shudder and fall\n\nthe sounds of autumn are hard to bear\n\n## 28\n\nChang Yueh (667\u2013731) was a native of Loyang and was considered by many to be the greatest poet of the early eighth century. He was also an advisor of several emperors who held him in the highest regard. Upon Chang's death, Emperor Hsuan-tsung (r. 712\u2013756) declared two days of national mourning. Chang, however, also suffered numerous rustications to outlying provinces. He wrote this poem after he returned home from \"assignment\" to Szechuan, via the road that led north from Chengtu to Ch'ang-an and then east to Loyang. Following an earlier banishment to Hunan, Chang's contemporaries noted the change in his poetry from external description to more meditative, introspective verse. Before Chang left Chengtu, he agreed to meet a friend traveling by a different route on the Mid-Autumn Festival in Loyang. Chang arrived home later than expected. But is it the lateness of his arrival that surprises him, or his own old age, or the death of someone dear to him?\n\n## _Delayed on the Szechuan Road_\n\nCHANG YUEH\n\nA traveler races the sun and moon\n\ncoming and going according to plan\n\nbut autumn wind doesn't wait\n\nit reaches Loyang before me\n\n## 29\n\nLi Pai (701\u2013762), aka Li Po, was born in what is now Kyrgyzstan and grew up in Szechuan, north of Chengtu. Ranked with Tu Fu as one of China's two greatest poets, he left behind over one thousand poems. This is one of his most famous, and critics sigh at the effortlessness of his technique and his ability to transport the reader not only to a place but also to a state of mind. When the Chinese see the moon, it reminds them that friends and family members living elsewhere are looking at the same moon.\n\n## _Thoughts on a Quiet Night_\n\nLI PAI\n\nBefore my bed the light is so bright\n\nit looks like a layer of frost\n\nlifting my head I gaze at the moon\n\nlying back down I think of home\n\n## 30\n\nThis was number fifteen in a series of seventeen poems Li Pai wrote in 754, while visiting the town of Kueichih in Anhui province. Kueichih still serves as a minor port for Yangtze River traffic, but it is better known nowadays as the gateway for pilgrims visiting the Buddhist mountain of Chiuhuashan. In Li Pai's day, Chiuhuashan (Nine Flower Mountain) was just another mountain\u2014its name comes from one of his poems\u2014and he preferred to wander along the Chiupu River southwest of town. This poem is often cited as an example of Li Pai's romantic style. But our version of reality is forever at the mercy of our emotions, and this is about as realistic a poem as Li Pai could have written at this stage of his life. When he later died downriver from Kueichih, the story that circulated was that he drowned while trying to embrace the moon and that, while his body was buried nearby, his spirit flew up to Heaven on the back of a whale. Or maybe it wasn't a whale: between Kueichih and the place where he reportedly drowned is a nature sanctuary for the sightless Yangtze dolphin. Chinese mirrors were made of convex palm-sized pieces of polished metal that were held in the hand and covered up when not in use.\n\n## _Chiupu River Song_\n\nLI PAI\n\nMy white hair extends three miles\n\nthe sorrow of parting made it this long\n\nlooking in a mirror who would guess\n\nwhere autumn frost comes from\n\n## 31\n\nCh'en Tzu-ang (661\u2013702) was a native of Szechuan province and served in a series of midlevel posts, including that of censor. He was regarded as one of the most innovative poets of the early T'ang and was also admired for his concern with social issues, a concern that led to his death at the hands of his political enemies. Here he offers his sympathy to a friend, Ch'iao Chih-chih, whose talents (and, no doubt, Ch'en's own) he feels could be put to better use. Both men at one time served together on the Silk Road. Instead of criticizing the court directly, Ch'en aims his remarks at the Han court of eight hundred years earlier for allowing civil servants and sycophants to oversee military affairs along the border\u2014while those truly capable of such service wasted their talents standing in attendance in the capital. During the Han, the walls of the palace's Cloud Pavilion (the combined structures of Cloud Terrace and Unicorn Pavilion) were covered with paintings of those who performed meritorious service but excluded exemplars of military and civil valor, such as generals and censors. The title, \"master of dapple-gray steeds\" ( _tsung-ma-shih_ ), Ch'en bestows on his friend (and on himself), was originally an epithet of the Han-dynasty censor Huan Tien, whose carriage was pulled by a team of gray horses.\n\n## _To Vice Censor Ch'iao_\n\nCH'EN TZU-ANG\n\nThe Han court glorified clever officials\n\nCloud Pavilion disdained border service\n\npity the master of dapple-gray steeds\n\nwhat good is an old man's valor\n\n## 32\n\nWang Ch'ang-ling (698\u2013756) was from the Ch'ang-an area but spent most of his career out of favor and in low-level posts in outlying provinces. Here, he stops in Wuling on his way back from banishment to Kueichou province. This Wuling was not the Wuling Hills of Ch'ang-an but the area west of Tungting Lake whose administrative center was located in the city of Changte on the Yuan River. This was the first Chinese city of any size Wang would have encountered after leaving the frontier outpost of Liping near the headwaters of the Yuan, 700 kilometers to the southwest. Thus, he expresses his relief in having survived the dangers of the road as well as his appreciation for not having been forgotten. In ancient times, the Lord of Hsinling, Wei Wu-chi, was known for housing and caring for over 3,000 guests at a time at his estate in Taliang (Kaifeng), the capital of the state of Wei. While Wang's comparison might seem an exaggeration, it should be remembered that he is returning from a year of monitoring hilltribes in a region that must have seemed inhospitable, if not outright dangerous, to a Chinese intellectual.\n\n## _In Reply to the Prefect of Wuling_\n\nWANG CH'ANG-LING\n\nStill clutching my sword after a long journey\n\nthis servant dares but a word\n\na guest of the Lord of Hsinling\n\nnever forgets such kindness\n\n## 33\n\nTs'en Shen (715\u2013770) was born in Nanyang and grew up in Loyang, but once he began his official career he maintained an estate at the foot of the Chungnan Mountains south of Ch'ang-an. In addition to several provincial assignments, he twice served on the Silk Road. Following the fall of Ch'ang-an to the rebel army of An Lu-shan in 755, forces loyal to the throne first retreated to Yinchuan and then returned as far as Fenghsiang in 756. Ts'en joined them en route from his post on the Silk Road. T'ang forces retook the capital the following fall. Chrysanthemums bloom in fall, and a wine infused with their petals is traditionally drunk on the ninth day of the ninth moon. As nine is the perfect _yang_ number, this is the ultimate _yang_ holiday, when men celebrate their virility and congratulate one another on having lived so long. In the first line, Ts'en refers to the Chinese custom of climbing a hill or tower from which to enjoy the view on this holiday, a custom traced back to Huan Ching of the Han dynasty (206 B.C.\u2013A.D. 220). In the second line, he recalls an occasion when T'ao Yuan-ming (365\u2013427) found himself without wine on Double Ninth and sat disconsolate in his garden until the local prefect heard of his plight and sent him a jug.\n\n## _Thinking of My Home in Ch'ang-an While Traveling with the Army on the Ninth_\n\nTS'EN SHEN\n\nIf only I could climb somewhere\n\nbut no one sends me wine\n\nmy poor distant garden of mums\n\nblooms by a battlefield now\n\n## 34\n\nHuang-fu Jan (716\u2013769) was born in Kansu province but grew up near Nanching and was known for his poetry and prose even as a boy. Although he served in a number of posts, he never achieved the success he expected and here wonders why. The woman whose persona he appropriates was Pan Chieh-yu, the great aunt of the historian Pan Ku (A.D. 32\u201392) and a concubine of Emperor Ch'eng of the Han dynasty (r. 32\u20137 B.C.). Although the emperor admired her virtue and learning, he later became infatuated with Chao Fei-yen and sent Pan to live by herself in Changhsing Palace. There, she wrote poems complaining of neglect, and later writers often used her voice to point to their own overlooked abilities. In ancient China, eyebrows were considered one of a woman's greatest assets, much like breasts are today in the West. Thus, Pan compares hers to those of the other women at court. During the Han dynasty, Chienchang Palace was built by Emperor Wu and was said to have one thousand doors and ten thousand windows. Emperor Ch'eng built Chaoyang Hall directly behind it as a residence for Chao Fei-yen.\n\n## _A Concubine's Lament_\n\nHUANG-FU JAN\n\nA flowering branch grows from Chienchang Palace\n\nfrom Chaoyang Hall I hear the royal flutes\n\nI wish I could ask those favored few\n\nexactly how long are their eyebrows\n\n## 35\n\nChu Fang (fl. 750\u2013790) was from Hsiangyang in Hupei province. After a brief career in the civil service, he retired to a hermitage on the Shan River in Chekiang province and lived there as a recluse. Some years later, he accepted a sinecure as an advisor to the military commissioner of Kiangsi and moved to the provincial capital in Nanchang. Here he visits Lushan, two days' journey to the north. Chulin (Bamboo Grove) Temple was also called Holin (Crane Forest) Temple and was among the most famous sights on Lushan's mist-shrouded slopes. The temple was located along the mountain's rocky northwest ridge but has long since disappeared. Although Chu is clearly aware of the temple's separation from his life of civilized ease, he appears to have lost his ability to free himself, as he did earlier, from such mundane existence, and we can only guess why.\n\n## _Written at Chulin Temple_\n\nCHU FANG\n\nThe months and years compel our lives\n\nhere the mist and clouds abound\n\nhow many times will I again know\n\nthe welcome of Chulin Temple\n\n## 36\n\nTai Shu-lun (732\u2013789) was a native of Chintan in Kiangsu province and served for many years as a private secretary in Hunan province. At one point, he was appointed prefect of Nanchang, but during his later years he turned his back on a worldly career and became a Taoist priest. The Yuan and Hsiang are the largest tributaries of Tungting Lake, which itself is often regarded as the final section of the Hsiang before it empties into the Yangtze. It was here that Ch'u Yuan (340\u2013278 B.C.) was exiled for advice that would have saved his ruler, and it was here that he drowned himself rather than live in an unjust world. The Three Gates (San Lu) refer to Ch'u Yuan's inherited position as overseer of the three royal lineages, and also to his ancestral fief near Tzukuei in the middle of the Yangtze's Three Gorges. Some commentators place this shrine in Milo, not far from the place where Ch'u Yuan drowned himself in a tributary of the Hsiang. Others say it was on the Yuan River near the modern town of Huaihua, where Tai was serving as a private secretary at the time. The last two lines rise from the final lines of Ch'u Yuan's \"Chaohun\" (\"Summoning the Soul\"), where the red maple leaves represent the blood of his heart.\n\n## _Passing the Shrine to the Master of the Three Gates_\n\nTAI SHU-LUN\n\nThe waters of the Yuan and Hsiang never cease\n\nCh'u Yuan's grief is so deep\n\nthe autumn wind rises at sunset\n\nand blows through a grove of maples\n\n## 37\n\nLo Pin-wang (640\u2013684) was born in Chekiang, grew up in Shantung, and was ranked as one of the four great poets of the early T'ang. When he was a young man he became a Taoist priest. But later he decided to seek a position in Ch'ang-an, where he was in and out of favor. At one point he was banished to the Silk Road oasis of Turfan, and on another to a distant outpost in Southwest China. Finally, for opposing Empress Wu's usurpation, he was jailed and banished to the Chekiang coast. Not long afterward he joined the rebel Hsu Ching-yeh in Yangchou. But when Hsu's forces were defeated, Lo was among those killed in reprisal\u2014although some say he became a Buddhist monk. Here, he recalls the parting of Prince Yen Tan and Chin K'o, whom Yen was sending to assassinate the tyrant who would soon become China's First Emperor (r. 221\u2013210 B.C.). As the two said goodbye outside the capital of Yen on the banks of the Yi River, not far southwest of modern Beijing, Chin K'o clasped his sword and sang: \"The wind blows hard and the Yi is cold \/ once a hero leaves he never returns.\" Chin K'o was so emotionally aroused, his hair stood on end and lifted his hat from his head. Despite such resolve, he was killed before he could complete his mission. As Lo recalls this scene he fears that he too will die before he is able to rid the country of another tyrant, namely Empress Wu. But he takes solace in the survival of a courageous man's name.\n\n## _Saying Goodbye on the Yi River_\n\nLO PIN-WANG\n\nHere where Yen Tan said goodbye\n\na hero raised his hat with his courage\n\nthe men of the past are gone\n\nbut the water is still cold today\n\n## 38\n\nSsu-k'ung Shu (740\u2013790) was from Yungnien in Hopei province and served as director of the Bureau of Waterways and Irrigation, and later of the Bureau of Forestry and Crafts. He was ranked among the Ten Talents of the Tali Period (766\u2013779) and was known for the playfulness of his poems, which he wrote on topics such as noodle soup. For such men, spending time together discussing and writing poetry was a major part of their lives. Hence, Ssu-k'ung tries to detain a friend with wine that he compares to a great wind. This wind was named after a woman and her merchant husband. When the husband failed to return from a journey, his wife died of a broken heart. But on her deathbed she vowed, \"After I die, my spirit will become a great wind that will prevent all men going on distant journeys from leaving their wives.\" Thereafter, her name and that of her husband were combined to describe such a departure-preventing wind.\n\n## _Saying Goodbye to Lu Ch'in-ch'ing_\n\nSSU-K'UNG SHU\n\nI know we plan to meet again\n\nbut how can we part tonight\n\ndon't think an old friend's wine\n\nis weaker than a Shihyu Wind\n\n## 39\n\nNothing is known of the author of this poem, other than that he lived in the Chungnan Mountains south of Ch'ang-an and called himself T'ai-shang ying-che (The Ancient Recluse). Here, he replies to someone who has asked him why and how long he had been living there. He dismisses the first question with _ou-lai_ (somehow\/by chance) and the second question with _wu-li-jih_ (no calendar) and then laughs at the idea of time-constrained concerns. The phrase _kao-mien_ (sleeping in comfort) comes from the precept that prohibits Buddhist monks and nuns from using excessive bedding. Its use here is meant to emphasize the poet's detachment from the usual standards of comfort as well as from precepts. The last line also recalls the story of the refugees who escaped to the hidden world of Peach Blossom Spring and, upon being discovered, wondered what dynasty it was.\n\n## _In Reply_\n\nTHE ANCIENT RECLUSE\n\nSomehow I ended up beneath pines\n\nsleeping in comfort on boulders\n\nthere aren't any calendars in the mountains\n\nwinter ends but who counts the years\n\n## PART TWO\n## 40\n\nHsuan-tsung (685\u2013762) was one of China's greatest emperors (r. 712\u2013756) but was also greatly flawed, as he took little interest in government affairs and devoted himself instead to the arts, religion, and his concubine Yang Kuei-fei. After one of the longest reigns in China's long history, he fled to Western Szechuan (the ancient state of Shu) in the summer of 756, as the rebel armies of An Lu-shan approached Ch'ang-an from the east. His son, meanwhile, established a temporary capital in Ninghsia province, northwest of Ch'ang-an, and had himself enthroned as Emperor Su-tsung. When the An Lu-shan Rebellion was more or less crushed the following autumn, the new emperor invited his father back, and here Hsuan-tsung euphemistically refers to his absence as having been \"on tour.\" He also refers to the five legendary strongmen who were traveling through these mountains when they saw a huge snake disappear into a cave and tried to pull it out by its tail. Their efforts caused a landslide that crushed them to death but also resulted in the opening of a more accessible route. As the former emperor reached Sword Gate Pass (Chienmenkuan \u2014 250 kilometers northeast of Chengtu) in the winter of 757, he turned to his attendants and said, \"Although the natural barrier of Sword Gate is immense, has its ability to prevent communication since ancient times not depended on its virtue rather than its height?\" His comment and the poem he wrote on this occasion were inspired by an inscription left at the pass in A.D. 281 by Chang Tsai. Chang recalled an earlier conversation between Duke Wu of the state of Wei (fl. fourth century B.C.) and Wu Ch'i, in which the duke called mountains and rivers the treasures of the state, and Wu replied that their virtue and not their strategic significance was their true value ( _Shih-chi:_ 44). Thus, it would seem that Hsuan-tsung finally realizes that his neglect of state affairs has led to his own undoing.\n\n## _Reaching Sword Gate Pass After Touring the Land of Shu_\n\nHSUAN-TSUNG\n\nOur tour complete our carriage returns\n\nto Sword Gate's cloud-barred peaks\n\nits mile-high screen of folded jade\n\nits cinnabar walls breached by heroes\n\nour pennants weave through a tapestry of trees\n\nethereal clouds brush past our horses\n\nrising to the times depends upon virtue\n\nhow true is this inscription\n\n## 41\n\nTu Shen-yen (646\u2013708) was a native of Hsiangyang in Hupei province and moved with his father to Kunghsien, east of Loyang, which was where his grandson, Tu Fu, was born. He held important posts in the court of Empress Wu Tse-t'ien (r. 684\u2013705) but was better known for his poetry and calligraphy and is considered one of the founders of T'ang-style verse. Following Empress Wu's death he fell out of favor, and in 705 he was exiled to what is now North Vietnam. He was soon recalled and given a sinecure in the capital, but the experience had a major effect on the poetry of his final years. Here, apparently during his banishment to the South, he responds to a poem by his friend Lu Yuan-fang, who had served as prime minister during the reign of Empress Wu and who had retired to his home in Chinling (Changchou) on the Grand Canal. Lu's poem (now lost) noted how much earlier spring appeared south of the Yangtze than in the capital. Tu begins by referring to himself as a _huan-yu_ (impotent official), which was a term officials often used out of humility but also when they were sent off to the provinces. By itself, _huan_ refers to a palace eunuch. Tu's use of the term underlines his political status and loss of faith in the world's power of renewal, as well as the government's inability to break free of the chains of tradition. Thus, he is surprised by spring. The orioles remind him of his wife back in the capital, but the rootless duckweed reminds him of his life on the road. And the familiar tune is the poem to which he replies.\n\n## _Replying to a Poem by Prime Minister Lu of Chinling_\n\nTU SHEN-YEN\n\nOnly an impotent official\n\nis truly surprised when things become new\n\nred clouds giving birth to the ocean dawn\n\nplums and willows ferrying spring across the river\n\nclear skies inciting yellow birds\n\nsunshine turning duckweed green\n\nsuddenly hearing a familiar tune\n\nI think of home and dry my eyes\n\n## 42\n\nIn this poem, Tu Shen-yen is attending the emperor's birthday celebration at one of the three main halls of Taming Palace in Ch'ang-an. This hall was named for the island of immortals that still appears off the coast of Shantung from time to time and to which Taoist adepts hope to be transported after shedding this mortal form. The mountains to which Tu compares the emperor are the Chungnan Mountains, which rose thirty kilometers south of Ch'ang-an, and which were also called the Nanshan, or Southern Mountains. Their name was a metaphor for immortality, and Tu uses them here to transport the emperor's earthly realm to the celestial plane. A poem in the _Shihching (Book of Odes_ ) goes: \"The smile of the moon \/ the glory of the sun \/ the age of Nanshan \/ by change untouched.\" The Northern Dipper is a symbol not only of old age but also of leadership, as all the stars revolve around it. And the golden gates and jade halls are characteristic of the celestial realm but are also used as euphemisms for the imperial residence. Yao was a legendary emperor who ruled China during the third millennium B.C. Confucius once sighed, \"Great was Yao's rule! How magnificent he was! It is only the Celestial Realm that is great, and only Yao who modeled himself on it\" ( _Analects:_ 8:19). Yao's ancestral fief was the kingdom of T'ang, and his reign was called the \"Reign of T'ang.\" Hence, this last reference suggests this poem was written following Tu's return from exile in 706. In that year the capital also returned from Loyang to Ch'ang-an. But after the interregnum of Empress Wu, it was still uncertain if the re-establishment of the T'ang dynasty and re-enthronement of Emperor Chung-tsung (r. 705\u2013710) would last. Thus, while offering birthday congratulations, Tu also reminds the emperor of his ancestral responsibility to uphold the Mandate of Heaven for the benefit of all those under his care.\n\n## _At an Imperial Banquet in Penglai Hall Offering Praise for the Chungnan Mountains_\n\nTU SHEN-YEN\n\nThe Northern Dipper hangs beside the wall\n\nthe Southern Mountains lean before the palace\n\ngolden gates appear in distant clouds\n\njade halls float above the trees\n\na noble air spreads across the slopes\n\na propitious mist circles the central peak\n\nyour servant offers wishes for long life\n\nlong may you keep the ways of Yao alive\n\n## 43\n\nCh'en Tzu-ang (661\u2013702) was from Szechuan province and wrote this poem in 688 as he prepared to leave Chengtu for Loyang. As the candles smolder after a night of goodbyes, Ch'en and his friend hear \"High Mountains\" and \"Flowing Waters\" in each other's hearts, just as the wood gatherer, Chung Tzu-ch'i, once did in the zither playing of Yu Po-ya. And as they toast each other they recall the Weaving Maid and Herdboy stars, who were banished by the Sun King to either side of the Milky Way (River of Stars) and are only allowed to meet one night a year. Ch'en fears a longer separation. Following his arrival in Loyang he enjoyed a promising career, but he also made enemies during Empress Wu's usurpation of power. When he finally returned to Chengtu to mourn his father, he was killed by the local prefect at the direction of Wu San-ssu, the empress's nephew.\n\n## _Parting from a Friend on a Night in Spring_\n\nCH'EN TZU-ANG\n\nAs black smoke coils from silver candles\n\nwe raise gold cups across silk mats\n\nour thoughts are like zithers in this hall of parting\n\nfollowing a path over mountains and streams\n\nthe bright moon sinks below tall trees\n\nthe River of Stars vanishes at dawn\n\nthe road to Loyang leads so far away\n\nwhat year will it lead back again\n\n## 44\n\nLi Chiao (644\u2013713) served as prime minister during three reigns but was cashiered for his arrogance when Hsuan-tsung ascended the throne in 712. Although his poems on everyday objects were highly esteemed, those written at imperial request, such as this one, were rather superficial and probably included by the editor as examples of the genre. Here, Li records a visit by Chung-tsung (r. 705\u2013710) to his daughter's estate east of Ch'ang-an. Blue is the color of the east, and purple of the royal realm. Mandarins and egrets represent higher and lower officials, and the phoenixes are the princess and her husband. The Chungnan Mountains are to the south, and the Wei River is to the north.\n\n## _Attending a Banquet at Princess Ch'ang-ning's Eastern Estate_\n\nLI CHIAO\n\nHer country estate overlooks a blue domain\n\ncarriages come ringing out of a purple sky\n\nmandarins and egrets flock to an endless feast\n\nphoenixes play celestial flute serenades\n\ntree-covered mountains rise to the south\n\nmist-shrouded marshes stretch to the north\n\neveryone is drunk from drinking in such favor\n\ntheir philanthropic majesties postpone their departure\n\n## 45\n\nChang Yueh (667\u2013731) was a native of Loyang and rose to the rank of prime minister during the reign of Hsuan-tsung (r. 712\u2013756). In 725, the emperor built a new library to house state documents and also put Chang in charge of writing a dynastic history. At the banquet to honor the occasion, the emperor asked Chang to compose a poem using the word _lin_ (forest) to begin the rhyme. The library was in the East Wing, and the Hanlin Academy of scholars was in the West Garden. The _Book of Odes_ and _Book of Changes_ were the two most studied and quoted Confucian classics. The last line's use of the expression _chih-yin_ (to know another's tune or voice) refers to the emperor but originated with the wood gatherer Chung Tzu-ch'i, who was the only person who could understand what was in Yu Po-ya's heart whenever Po-ya played the zither (see also the note to verse 43).\n\n## _Given the Word \"Forest\" by His Majesty at the Licheng Palace Library Banquet_\n\nCHANG YUEH\n\nIn the East Wing's halls of maps and texts\n\nin the West Garden's forest of brushes and ink\n\nwe chant the _Odes_ and hear the royal directives\n\nwe discuss the _Changes_ and observe the moods of Heaven\n\nour posts though humble our roles are vital\n\nwell favored and supplied with wine\n\nwe compose songs on the coming of spring\n\nexhaust our hearts for the one who knows them\n\n## 46\n\nLi Pai (701\u2013762) enjoyed a brief period of fame at court but was expelled in 744 and spent the rest of his life wandering, mostly along the Yangtze, looking for patrons and hoping to be recalled to the capital. He wrote this poem in 754 while saying goodbye to a friend in Hsuancheng. Chinese cities were usually protected by two sets of walls: an inner wall of brick or stone and an outer rampart of rammed earth. Between them was enough farmland to supply the town with food during a siege. It was customary for friends to say goodbye at the outer rampart, and there were usually inns capable of supplying farewell fare for every budget. The Shuiyang River still circles what remains of the city's east wall.\n\n## _Seeing Off a Friend_\n\nLI PAI\n\nDark hills stretch beyond the north rampart\n\nclear water circles the city's east wall\n\nfrom this place where farewell begins\n\na tumbleweed leaves on a thousand-mile journey\n\ndrifting clouds in a traveler's thoughts\n\nthe setting sun in an old friend's heart\n\nas we wave and say goodbye\n\nour parting horses neigh\n\n## 47\n\nLi Pai was quite familiar with this road, as it ended near his hometown of Chinglien Village, north of Chengtu in Western Szechuan. Western Szechuan was the ancient state of Shu, and it was also called Tsantsung, after Ts'an Ts'ung, a descendent of the Yellow Emperor, who moved there in the third millennium B.C. and established the hegemony of the Han Chinese. The Tsangtsung Road was built in the third century B.C. by the state of Ch'in, in order to invade Shu. The Chinling Mountains that separated the two states were so rugged that the road often gave way to planks supported by thick pegs driven into the rock cliffs. The \"walls of Shu\" refer to those of Chengtu, and the \"currents of spring\" to those of the Min River that formed the city's moat. Yen Tsun was a famous Taoist fortune-teller who lived in Chengtu in the first century B.C. In the second half of this poem, Li tries to console his friend by suggesting he look for the silver lining in his rustication and accept his fate, however unjust; even Chengtu's greatest fortune-teller couldn't fathom the whims of those responsible for his friend's banishment.\n\n## _Seeing Off a Friend Leaving for Shu_\n\nLI PAI\n\nYou've heard of the Tsantsung Road\n\nhow rugged it is and hard to travel\n\nmountains rise before your face\n\nand clouds appear beside your horse\n\nbut the planks of Ch'in are shrouded by fragrant trees\n\nand the walls of Shu are circled by the currents of spring\n\nups and downs are surely fixed\n\nyou don't need to ask Yen Tsun\n\n## 48\n\nWang Wan (c. 690\u2013740) was from Loyang and never held high office but was well known for his poetry even as a youth. Here, he stops for the night at a small peninsula that juts into the lower reaches of the Yangtze, where the Grand Canal intersects the river and where boats often moored before continuing north or south on the canal, east and out to sea, or west and up the Yangtze. Wang's home was 800 kilometers to the north, and spring arrives much earlier here, reminding him how far away he is. During the Han dynasty, an emperor once shot a goose and found, tied to its foot, a letter written by an official being held against his will by the Huns. Wild geese have been used ever since as the mail carriers of the hopelessly separated. The fifth and sixth lines became so famous that Chang Yueh (667\u2013731) wrote them on the wall of his office as examples of inspired verse.\n\n## _Stopping at Peiku Mountain_\n\nWANG WAN\n\nMy route goes past blue peaks\n\nwhere riverboats ply green water\n\nthe current is smooth the shores are wide\n\nthe wind is perfect for hoisting a sail\n\nthe ocean sun rises in the traces of night\n\nthe river spring starts inside the old year\n\nhow shall I send a letter home\n\ntied to a goose bound for Loyang\n\n## 49\n\nTsu Yung (699\u2013746) was born in Loyang and passed the civil service exam in 725. Through the good offices of such friends as Wang Wei, he was able to serve in several minor posts in Ch'ang-an and here visits a friend's retreat southwest of the capital on the Feng River. The second line was not an idle thought. Not long after he wrote this, Tsu resigned his post and retired to a hermitage on the Jushui River, south of his family home in Loyang. The presence of mountains and water are important in choosing a residence, and bamboos represent the indomitable will of a person of humble virtue. However, some editions have _wu-fu_ (roofs are covered) in place of _chu-fu_ (bamboos bend). The darkened courtyard suggests proximity to mountains as well as the absence of worldly affairs.\n\n## _Mister Su's Country Retreat_\n\nTSU YUNG\n\nYour country retreat is such a quiet place\n\nI think of retiring each time I come\n\nthe Chungnan Mountains fill your doors and windows\n\nthe Feng River brightens your trees and garden\n\nyour bamboos bend with winter-long snow\n\nyour courtyard is dark before dusk\n\nbeyond the sound and reach of man\n\nI sit and listen to the birds of spring\n\n## 50\n\nTu Fu (712\u2013770) wrote this poem in 758 when he was serving as a censor in Ch'ang-an. This position required him to spend the night at the Chancellery next to the palace (\"the height of Heaven\") and to attend the predawn audience with the emperor. Thus, he is one of the flowers, while the lesser officials, who can go home for the night, are the birds. The palace was said to have one thousand doors and ten thousand windows. The \"jade\" bridles of the royalty were decorated with cowry shells that looked and sounded like jade (see verse 3). The An Lu-shan Rebellion was not yet over, and Tu Fu wonders about the possible bearing of the sounds he hears in the imperial night. Not long after he wrote this, he was demoted and began a period of wandering that did not end until he died.\n\n## _Staying Overnight at the Chancellery in Spring_\n\nTU FU\n\nFlowers by the palace retire at dusk\n\nnestbound birds call as they pass\n\nstars appear and a thousand doors open\n\nthe moon shines brighter near the height of Heaven\n\nthe sound of gold locks keeps me awake\n\nwere those jade bridles I heard in the wind\n\nmorning court brings more sealed dispatches\n\nand questions about last night\n\n## 51\n\nTu Fu saw this mural in 759 at a Buddhist temple near Santai, on his way to Chengtu. Ku K'ai-chih (345\u2013406) was known for his temple art, but Tu Fu simply uses his name to praise the actual artist, who includes scenes of the Taoist island paradise of Tsangchou as well as a few Buddhist monks. Emperor Wu (r. 502\u2013549) once told a Buddhist and a Taoist that whichever of them could place an object first on Chienshan could use the mountain for his hermitage. The Taoist sent his cranes, and the Buddhist sent his magic staff. The mural also depicts the story of a monk who arrived from India in the fifth century in nothing more than a wooden tub. Lieh-tzu once noted that gulls are only frightened by those who think of catching them (2.11). The monk Hui-yuan (334\u2013416) lived on Lushan and began Pure Land practice in China by vowing to be reborn in Amida Buddha's Paradise.\n\n## _Painted on the Wall of Master Hsuan-wu's Room_\n\nTU FU\n\nWhat year did Tigerhead Ku\n\ncover this wall with scenes of Tsangchou\n\nthe red sun and forest of rocks\n\nthe blue sky and ocean of waves\n\na staff forever flying next to a crane\n\na seaborne tub beside unfrightened gulls\n\nthis is like finding the road to Lushan\n\nfollowing the actual steps of Hui-yuan\n\n## 52\n\nWang Wei (701\u2013761) served in a number of prominent posts in Ch'ang-an, including that of deputy prime minister. His political ambition, however, was never equal to his love for mountains, and he spent as much time at his retreat in the Chungnan Mountains as he did at court. _Taiyi_ (The Great One) was another name for the Tao. It was also another name for the Chungnan Mountains and its highest peak. The Heart of Heaven refers to the Taoist paradise as well as to the Son of Heaven's residence. White clouds also represent a life of detachment, and blue vapors worldly aspiration. The Chinese at one time laid out the empire into twenty-eight realms, corresponding to the constellations of the Chinese Zodiac, all radiating from these mountains. Upon meeting a woodcutter, an herb gatherer, or a hermit in the mountains of China, even a stranger soon feels at home.\n\n## _The Chungnan Mountains_\n\nWANG WEI\n\nTaiyi isn't far from the Heart of Heaven\n\nits ridges extend to the edge of the sea\n\nwhite clouds form before your eyes\n\nblue vapors vanish in plain sight\n\naround its peaks the whole realm turns\n\nin every valley the light looks different\n\nin need of a place to spend the night\n\nI yell to a woodcutter across the stream\n\n## 53\n\nTs'en Shen (715\u2013770) wrote this poem in 758, one year after returning to Ch'ang-an in Emperor Su-tsung's (r. 756\u2013762) retinue, as the An Lu-shan Rebellion was winding down. He describes the morning assembly at court, where he and Tu Fu were serving at the time. Ts'en was rectifier-of-omissions in the Secretariat and was in charge of criticizing imperial documents and the emperor's own conduct, while Tu Fu was a censor in the Chancellery and was in charge of criticizing officials. The Secretariat was located in the west wing of the courtyard just outside the imperial palace, and the Chancellery was in the east wing. Both were separated by rows of purple myrtle. Not long after he wrote this poem, Ts'en Shen (the rising official) and Tu Fu (the old man) were dismissed from their posts.\n\n## _To Censor Tu at the Chancellery_\n\nTS'EN SHEN\n\nWe hurry in pairs up cinnabar steps\n\nour ranks divided by purple myrtle\n\nat dawn we follow the guards inside\n\nat dusk we return trailing perfume\n\nan old man grieves to see petals fall\n\na rising official envies wild birds\n\nthe court of a sage contains no omissions\n\nthe work of a censor I imagine is rare\n\n## 54\n\nTs'en Shen visited this Buddhist pagoda soon after returning to Ch'ang-an with Emperor Su-tsung's army in 757. The pagoda was on Nanwutai Peak, among the northernmost ridges of the Chinling Range (the Chungnan Mountains). Ch'ang-an was twenty-five kilometers to the north and the Wei River thirty-five. The Chinese count nine heavens. The smoke came from fires set to the estates in the Wuling Hills northwest of the capital by the departing rebel armies of An Lu-shan. Ts'en had spent his life in the entourage of the powerful and now sees their insignificance and relative value and wonders if it would not have been better to possess nothing at all, to live the pure life of a monk rather than that of an official. The \"golden immortal\" refers to the Buddha.\n\n## _On Climbing Tsungchih Pagoda_\n\nTS'EN SHEN\n\nI climbed a pagoda that touched the highest heaven\n\nI stood at the edge of the sun\n\nI could see ten thousand courtyards below\n\nand the dismal smoke of Wuling\n\nbeyond the railing the Chinling looked low\n\nthe Wei seemed small in the window\n\nif I had known such detachment before\n\nI would have served the golden immortal\n\n## 55\n\nTu Fu (712\u2013770) wrote this poem in 737 after failing the civil service exam and going to visit his father, who was serving as military commander of Yenchou, or East District, where the Yellow River floodplain meets the Shantung peninsula. Looking out from the city's South Tower, Tu Fu is reminded of the inexorable nature of time. Taishan, resting place of spirits, was to the northeast, and the Pohai Sea, home of immortal isles, lay beyond. The rich farmland of Hsuchou and Chingchou was to the east, but it had been impoverished by an imperial tour in 725. Similar results followed a visit in 219 B.C. by the Ch'in dynasty's First Emperor, who left a record of his visit on a stele outside nearby Chufu. Chufu was the capital of the ancient state of Lu and the site of Lingkuang Palace, built in the Han dynasty by Lu Kung. Tu Fu's father was in ill health and died the following year.\n\n## _Climbing Yenchou Tower_\n\nTU FU\n\nVisiting my father in East District\n\nI finally looked out from South Tower\n\nclouds stretched beyond Taishan to the sea\n\nbarren land spread through Hsu and Chingchou\n\nthe outline of the stele of Ch'in was still there\n\nthe walls of Lu Palace were rubble\n\nI've always been drawn to the past\n\nbut this time my heart trembled\n\n## 56\n\nWang Po (647\u2013676) was a child prodigy and later was hailed as one of the four literary lights of the early T'ang. However, he was convicted of murdering a servant and sentenced to death, and his father was banished to Hanoi. Wang was subsequently pardoned, but on his way to join his father he drowned. Here, he parts from a friend heading west from Ch'ang-an and implies that he, too, is leaving for the provinces. Ch'ang-an was in the middle of three administrative districts carved out of the ancient state of Ch'in. Shuchou, his friend's destination, was across the Min River from Chengtu in Western Szechuan (the ancient state of Shu) and was reached by taking one of the five fords along the river.\n\n## _On Deputy Prefect Tu Taking Up a Post in Shuchou_\n\nWANG PO\n\nFrom walls that guard the realms of Ch'in\n\nthrough windblown mist to the fords of Shu\n\nwe gaze and consider our separation\n\nboth of us nomad officials\n\nbut still true friends this side of the sea\n\nneighbors on either shore of the sky\n\nwhere the road forks let us not join\n\nour children drying their tears\n\n## 57\n\nTu Shen-yen (646\u2013708) wrote this poem while serving in Loyang at the court of Empress Wu. As court poet, he was required to attend such grand occasions as the departure of the army. Here, he consoles his friend Ts'ui Jung (653\u2013706), who had been assigned as court secretary to Wu San-ssu, the empress's devious nephew, who was leading an army to the northern frontier. This was a post usually filled by men of promise, but Tu's friend still needs encouraging, which Tu supplies by reminding him that he will be watching and not participating in the fighting. This poem was written in the summer of 696, shortly after the nomadic Khitans invaded Hopei province. T'ang forces were crushed during their initial encounters with the enemy and were not successful in repulsing the invasion until the following summer. Kupei was the name of a strategic pass along the Great Wall, one hundred kilometers northeast of Beijing.\n\n## _Seeing Off Ts'ui Jung_\n\nTU SHEN-YEN\n\nOur sovereign sends forth her general\n\nher secretary joins the distant campaign\n\nfarewell tents line palace gates\n\ncolumns of soldiers shake city walls\n\nwith flags and pennants to mark the dawns\n\nand flutes and drums to fill border nights\n\nsmoke and dust will rise while you watch\n\nthe autumn wind quell Kupei\n\n## 58\n\nSung Chih-wen (d. 712) was from Shansi province and served in important posts in Ch'ang-an and later in Loyang during the reign of Empress Wu. Along with Shen Ch'uan-ch'i (verse 185), he was considered one of the greatest poets of the period following the early T'ang. But when the empress died in 705 the coups and countercoups that followed left him in a vulnerable position. When Jui-tsung re-ascended the throne in 710 (his earlier reign [684\u2013690] ended when Empress Wu established her own reign [690\u2013705]), Sung was exiled to the South and forced to commit suicide. Here, he accompanies Empress Wu at the end of 696 on her way back from performing a sacrificial ceremony at Sungshan\u2014a ceremony normally reserved for emperors at the beginning of a new dynasty and normally performed at the eastern sacred peak of Taishan. Sungshan's summit was the central member of China's five sacred peaks. In ancient times, ten thousand chariots were said to constitute the power of a king. Tengfeng, seventy kilometers southeast of Loyang, was the county seat at the mountain's base.\n\n## _The Imperial Entourage on the Road from Tengfeng_\n\nSUNG CHIH-WEN\n\nA palace of brocade adorns the great peak\n\nthe ascent of an immortal is glorious indeed\n\nat dawn curtains weave through the clouds\n\nat dusk lanterns flicker among the stars\n\ncountless banners wind out of a dark ravine\n\ndown the quaking mountain ten thousand chariots roll\n\nthe royal entourage is worthy of an ode\n\nalas I lack the skill to dazzle Heaven\n\n## 59\n\nMeng Hao-jan (689\u2013740) visits a monk friend in his mountain hermitage. The lotus ( _Nelumbium_ ) blooms in early summer and was first cultivated in Egypt, from whence it spread to Assyria and India. It arrived in China, along with Buddhism, via the Silk Road, and has been used ever since as a symbol for transcending the world while remaining rooted in its mud.\n\n## _Written on the Wall of Master Yi's Meditation Hut_\n\nMENG HAO-JAN\n\nAccustomed my friend to the stillness of Zen\n\nyou built a refuge in deserted woods\n\noutside the gate a lone peak soars\n\nbeyond the steps wind countless ravines\n\nsunset follows a daylong rain\n\ntall trees cover your yard with shade\n\nnow that I've seen the lotus in bloom\n\nI know the mind impervious to stain\n\n## 60\n\nChang Hsu (fl. 750) was one of China's greatest calligraphers and was famous for his cursive script, which became more inspired as he drank. When I was first living in Taiwan, whenever I had to go to Hong Kong to renew my visa I asked my calligraphy teacher (Chuang Yen, curator of the Palace Museum's Calligraphy and Painting Collection) if I could bring him back brushes or ink, as people in Taiwan were still forbidden to travel to China. But all he ever asked for was Tachu White Lightning. He said he did his best work before dawn, after a cup or two. Chang also loved to drink and was ranked among the Eight Immortals of Wine. It's said he kept a jug beside his bed so he could drink as soon as he woke up. Blue clouds represent high position and refer to his new post as court calligrapher, which required earlier hours than he was used to. Kao Shih (716\u2013765) uses two of Chang's nicknames here: _ts'ao-sheng_ (god of shorthand) and _tien_ (crazy).\n\n## _To Chang Hsu after Drinking_\n\nKAO SHIH\n\nThe world is full of fickle people\n\nyou old friend aren't one\n\ninspired you write like a god\n\ndrunk you're crazier still\n\nenjoying white hair and idle days\n\nblue clouds now rise before you\n\nhow many times will you still sleep\n\nwith a jug of wine by your bed\n\n## 61\n\nTu Fu (712\u2013770) wrote this poem in 763 while attending the funeral of a fellow official in Langchung, 200 kilometers northeast of Chengtu on the Chialing River. This Taoist temple, on Jade Terrace Mountain north of town, was built by King T'eng, the son of Emperor Kao-tsu (r. 618\u2013626), in 679, while the son was serving as prefect of Langchung. Hsiao Shih was a flute player of the sixth century B.C. who became an immortal on Huashan. The inscriptions ostensibly record the construction of Lingkuang Palace by Lu Kung, the son of Emperor Ching (r. 156\u2013141 B.C.), in Confucius's hometown. Taoist immortals reach the islands of the blessed on the backs of cranes. Wang Tzu-ch'iao, the son of King Ling (r. 571\u2013545 B.C.), was often seen riding in this manner playing his flute.\n\n## _Jade Terrace Temple_\n\nTU FU\n\nThe palatial steps were built by a king\n\nthe terrace resembles those of the past\n\ncolored clouds welcome Hsiao Shih\n\ninscriptions record the words of Lu Kung\n\nthe shrine halls lead to the realm of the gods\n\nthe whole place recalls an immortal isle\n\npeople report hearing flutes and cranes\n\noften passing the peak to the north\n\n## 62\n\nTu Fu wrote this poem in 764 in Chengtu for his friend Li Ku. It was one of three he wrote on this landscape painted by his cousin, who also contributed the funds for Tu Fu's cottage in Chengtu. Fangchang was one of the islands where immortals lived, and the Tientai Mountains were near the Chekiang coast. The Taoist alchemist, Ko Hung (283\u2013343), said that they were the perfect place to cultivate the Tao. Fan Li was a minister of the Warring States Period (433\u2013221 B.C.) who escaped power and fame to spend his days as a recluse near Lake Taihu. And Wang Tzu-ch'iao (aka Wang Ch'iao), son of the Chou dynasty's King Ling (r. 571\u2013545 B.C.), became an immortal and flew off on a crane. Tu Fu wishes Fan's boat could hold another, or that Wang had a second crane. The dust is that of his fellow officials, who crowd the road ahead, but also refers to what Buddhists call \"the 'red dust' of sensation.\"\n\n## _On Seeing a Landscape Painted by My Commissioner Cousin for Li Ku_\n\nTU FU\n\nFangchang surrounded by water\n\nTientai all dazzling clouds\n\nI often see their pictures on walls\n\nbut I hate such tales in old age\n\nFan Li's boat is too small\n\nand Wang Ch'iao's crane flies alone\n\nI followed everyone else this life\n\nwhere can I go to escape the dust\n\n## 63\n\nTu Fu wrote this poem in the summer of 765. The death of his patron and protector, Yen Wu, prompted his departure from Chengtu in May. Looking for another place to settle with his family, he headed east. Here, traveling on one of the tall-masted boats that carried passengers on the Yangtze, he wishes he had achieved fame for something other than writing\u2014poets earned no royalties\u2014and for something more worthy of his talents. When he quit his last post he was only forty-six. But age and illness are standard reasons given to avoid criticizing the court for its lack of appreciation. Since the moon is rising after the stars are out, it must be waning\u2014an apt metaphor for Tu Fu's career. Seagulls are rarely alone.\n\n## _Recording My Thoughts While Traveling at Night_\n\nTU FU\n\nA shore of thin reeds in light wind\n\na tall boat alone at night\n\nstars hang over the barren land\n\nthe moon rises out of the Yangtze\n\nhow could writing ever lead to fame\n\nI quit my post due to illness and age\n\ndrifting along what am I like\n\na solitary gull between Heaven and Earth\n\n## 64\n\nTu Fu wrote this poem at the end of 768, after sailing through the Yangtze Gorges. He had hoped to continue north to his home near Loyang, but invading Turk armies made such a journey too dangerous. As far back as Ch'u Yuan (340\u2013278 B.C.), Tungting Lake was synonymous with exile, and the area was a common assignment for those out of favor. Yuehyang Tower was at the northeast corner of the lake\u2014not far from where it merges with the Yangtze\u2014and also where the ancient kingdoms of Wu and Ch'u met. The tower was first built during the Three Kingdoms Period (A.D. 221\u2013265) as an observation platform for naval maneuvers and was rebuilt several times. This version was built in 716 by Chang Yueh during his exile to Yuehyang. Tu Fu lived on this boat with his family until his death a year or so later. He was buried ninety kilometers southeast of Yuehyang.\n\n## _Climbing Yuehyang Tower_\n\nTU FU\n\nI heard long ago about Tungting Lake\n\nhere I am climbing Yuehyang Tower\n\nwhere Wu and Ch'u divide South from East\n\nwhere Heaven and Earth and day and night drift\n\nof family and friends I have no news\n\nold and sick I live on a boat\n\nwarhorses block the northern passes\n\nmy tears fall on the railing\n\n## 65\n\nTsu Yung (699\u2013746) was born in Loyang and served briefly in the government, but he retired at an early age to a hermitage south of Loyang. Here, he finds himself at the end of winter on an unwelcome mission to Changsha, the capital of Hunan. The Mountains of Ch'u extend from southern Hunan to northern Kiangsi and are named for the ancient state that once ruled this region. The sword represents Tsu's accoutrement of office, while his letters, flying the opposite direction of migrating geese (which are themselves the mail carriers of the hopelessly separated), convey the interests of his heart. The Southern Dipper is only visible south of the Yangtze. Changsha was known for the oranges that grew on an island in the Hsiang River. They were a welcome present at New Year, as their color represented abundance and happiness.\n\n## _Traveling South of the Yangtze_\n\nTSU YUNG\n\nThe Mountains of Ch'u never end\n\nthe only road home is bleak\n\nrain appears from a clear blue sky\n\nfloodwaters roar all night\n\nmy sword remains by the Southern Dipper\n\nmy letters buffet the distant North Wind\n\nI would send gifts of Changsha oranges\n\nbut who will take them to Loyang Bridge\n\n## 66\n\nCh'i-wu Ch'ien (692\u2013749) was from the Yangtze city of Chiangling and served in Ch'ang-an as a censor and later as editorial director of the Palace Library. He was known for his well-crafted poems but better known for those written to him by others. Some commentators say this temple was 100 kilometers southwest of Chiangling near Lingling; others say it was 300 kilometers northeast near Fanghsien. Trees are planted outside a temple's main gate as guardians of the serenity within. The lamp of wisdom lights the abbot's chamber, where he gives personal instruction. Strands of beads are used for keeping track of repetitions of the Buddha's name, but a single bead was sometimes used to fasten a monk's robe. In any case, their simplicity contrasts with the attire of officials. The lotus represents the transcendent teaching of the Buddha that remains rooted in the mud of this world. In the _Vimalakirti Sutra:_ 7, a celestial deity showers flowers on those who cultivate the Dharma. But the petals don't cling to those who are detached from their practice.\n\n## _Spending the Night at Lunghsing Temple_\n\nCH'I-WU CH'IEN\n\nI failed to leave the temple by dark\n\npine trees cooled the shrine-hall gate\n\na lamp illumined the abbot's chamber\n\nbeads held monastic robes together\n\nthe white sun taught the purity of mind\n\na blue lotus showed the subtlety of truth\n\nheavenly petals fell without cease\n\neverywhere birds carried them off\n\n## 67\n\nCh'ang Chien (708\u2013765) grew up in Ch'ang-an but failed to gain an appointment in the capital and was assigned to provincial posts south of the Yangtze. Losing interest in the civil service, he took up wandering. Here, he visits a temple on Yushan, just outside the Yangtze Delta city of Chang-shu. The temple was later renamed Hsingfu Temple, and it was one of the country's major training monasteries during the T'ang and Sung dynasties. The temple is still there, and the path behind it still winds through a mixed forest of fruit trees and conifers. The \"ten thousand noises\" refer to those of the nearby urban world. The bell is used to mark the beginning and end of meditation periods as well as the time for meals and sleep. Talking during meditation retreats is restricted to instruction in the Dharma.\n\n## _The Meditation Hall behind Poshan Temple_\n\nCH'ANG CHIEN\n\nI entered an ancient temple at dawn\n\nthe rising sun lit the tall trees\n\na trail led off to a secluded place\n\nto a meditation hall in a flowering wood\n\nwhere mountain light pleased the hearts of birds\n\nand pond reflections stilled men's minds\n\nthe ten thousand noises were hushed\n\nall I heard was a bell\n\n## 68\n\nChang Hu (792\u2013852) was born in Hopei province but grew up in Suchou, next to Lake Taihu. When he later moved to Ch'ang-an to prepare for the civil service exam, he aroused the enmity of the powerful Hanlin academician Yuan Chen and was forced to leave town. He took this as a sign that an official career was not for him, and he built a cottage near Tanyang, where he spent the rest of his life. The Sungting relay station was apparently in the vicinity of the Yangtze and Lake Taihu, which was said to comprise five bodies of water and was also called Five Lakes. This region was famous as a place where a person could avoid the notice of the authorities, and it was the home of many bandits and martial arts heroes. The friends to whom Chang refers are those, like himself, who were forced into early retirement.\n\n## _Written at the Sungting Relay Station_\n\nCHANG HU\n\nHill shapes merge with the far-off sky\n\neast of the mist-covered marshlands\n\nthe ocean glows with the day's first light\n\nthe river turns white in the distant wind\n\nsteep trails lead to a high plateau\n\nsmall paths link columns of smoke\n\nwhy are all my retired friends\n\nnot here among the Five Lakes\n\n## 69\n\nCh'u-mo (fl. 850\u2013900) was born in Shaohsing and became a monk when he was still a young man. Along with his hometown friend and fellow monk, Kuan-hsiu, he moved to Hangchou, and after several years both men headed up the Yangtze together. Ch'u-mo settled on Lushan, while Kuan-hsiu continued on to Chengtu. Prior to his departure, Ch'u-mo wrote this poem about the view from Shengkuo Temple at the summit of Fenghuangshan, outside Hangchou's South Gate. To the southeast, the Chientang River marked the border between the ancient states of Wu and Yueh (Ch'u-mo's homeland) as it flowed into Hangchou Bay. To the northwest was the city's West Lake, which was already a popular spot with revelers in the late T'ang dynasty (618\u2013906).\n\n## _Shengkuo Temple_\n\nCH'U-MO\n\nLeading up from lesser peaks\n\nthe trail winds above vine-covered slopes\n\nthe land of Wu ends at the shore\n\nthe mountains of Yueh begin across the river\n\nancient trees merge with blue mist\n\nthe distant sky joins the white waves\n\nthe city wall is so close below\n\nsingers and flutes muffle the bell\n\n## 70\n\nWang Chi (585\u2013644) was from Shansi province and retired twice, the last time to Tungkao near his family home in Hochin. Tungkao means \"Eastern Paddy\" and appears as a place-name in a poem by T'ao Yuan-ming, on whose life of simplicity Wang modeled his own. Here, the cowherds remind him of the leaders of the new T'ang dynasty, and the hunters call to mind its generals. But watching them go by, he sees no friends and sings instead about the blameless pursuit of gathering ferns, just as the recluses Po-yi and Shu-ch'i did when they refused to serve King Wu (r. 1122\u20131116 B.C.), the founder of the Chou dynasty. Retiring to a mountain just south of Tungkao, they sang: \"Climb that west slope \/ pick those ferns \/ trading evil for evil \/ don't they know is wrong \/ the sage kings are gone \/ where can we go \/ oh we're both finished \/ life doesn't last.\"\n\n## _Gazing across the Countryside_\n\nWANG CHI\n\nWatching dusk fade at Tungkao\n\nI look for something to lean on\n\nevery tree is the color of fall\n\npeak after peak loses its light\n\ncowherds lead their calves home\n\nhunters ride by with their prey\n\nI look but don't know their faces\n\nI sing about gathering ferns\n\n## 71\n\nCh'en Tzu-ang (661\u2013702) was a censor at the court of Empress Wu, and Ts'ui Jung was editor of the court diary and calendar. In 696 Ts'ui was also assigned as court secretary to the army sent to pacify the Khitans. Dark skies of windblown loess are common in fall in North China, and \"White Dew\" is the name of a solar period in early September. A poem in the _Book of Odes_ envisions the army as a tool for pacification rather than for war, and Lao-tzu says, \"Fine swords aren't auspicious tools\" (31). The fifth and sixth lines refer to areas of Hopei province, including the Pohai seacoast, occupied by the Khitans. Lulung Pass was just inside the Great Wall east of Beijing, and the walls of Unicorn Hall were covered with portraits of heroes by Emperor Hsuan (r. 73\u201349 B.C.).\n\n## _Seeing Off Editor Ts'ui Marching East_\n\nCH'EN TZU-ANG\n\nGolden skies have turned forbidding\n\nwhite dew marks the start of your mission\n\nthe empress's armies don't enjoy war\n\nher generals shun fine swords\n\nbut ocean storms trouble the South\n\nand border winds sweep the North\n\ndon't sell out at Lulung Pass\n\nor seek your fame at Unicorn Hall\n\n## 72\n\nChangpa Reservoir was located halfway between the southwest corner of Ch'ang-an and the nearby Chungnan Mountains, and supplied water to the capital via the Tungchi Canal. The scene is midsummer, when lotuses are in full bloom and a little relief from the heat is welcome. Ice was stored underground in straw-filled boxes during winter and sold to those who could afford it during the summer. Lotus roots are about the size of a yam. Once their thick skin is removed they are sliced and either stir-fried or, in this case, eaten raw and washed down with chilled rice wine.\n\n## _Encountering Rain at Changpa Reservoir One Evening While Enjoying a Cool Breeze with Rich Young Men and Their Singsong Girls\u2014I_\n\nTU FU\n\nSunset is just right for boating\n\na light breeze stirs a few waves\n\nboaters pause by the dense bamboo\n\nand breathe in the cool lotus-flower air\n\nthe young men add ice to their drinks\n\nthe women scrape lotus roots clean\n\na cloud overhead turns dark\n\nsurely the rain will bring poems\n\n## 73\n\nTu Fu's friends have hired a boat to take them out on the water but have now returned to the shore, soaked by the sudden downpour. The sitting mats were made of woven grass, and the awning was also woven, but probably from thin strips of bamboo or rattan. The women of the ancient states of Yueh in Southeast China and Yen in the Northeast were known for their beauty and also for their voices, and they were sought after as entertainers. The reservoir was about ten kilometers south of Ch'ang-an but only a few kilometers west of Tu Fu's home on the Tungling Plateau. At least he didn't have to travel as far as his rich young friends and their companions.\n\n## _Encountering Rain at Changpa Reservoir One Evening While Enjoying a Cool Breeze with Rich Young Men and Their Singsong Girls\u2014II_\n\nTU FU\n\nRain drenches the mats\n\nwind beats against the prow\n\ngirls from Yueh wring out their red skirts\n\ngirls from Yen lament their mascara\n\neven with the boat tied to a willow\n\nthe awning still lifts in the spray\n\nthe road home looks desolate now\n\non a fall day in May at the lake\n\n## 74\n\nSun T'i (fl. 720) was a native of Shantung and served in several prominent posts in the capital, including the Secretariat. Here, while waiting to be recalled from an assignment in Chekiang province, he visits a Taoist temple at the foot of East Peak, fifteen kilometers south of Shaohsing. The Taoist immortal Wang Tzu-ch'iao was said to have lived here during the Han dynasty, and brightly colored clouds were often reported in its vicinity. Hence its name: Yunmen (Cloud Gate). As Sun looks south into the myriad peaks of the Kuaichi Range he feels the first cool air of autumn rising from the lake region to the north. Seeing the images of wild geese on the wall and, outside, two constellations that were associated with this region to which officials out of favor were banished, he dreams of returning to the capital, but apparently as an immortal.\n\n## _Spending the Night at Yunmen Temple Pavilion_\n\nSUN T'I\n\nAt Incense Pavilion below East Peak\n\nthe flowers in the mist were from another world\n\nI held up a lantern on a deep mountain night\n\nand pulled back the curtain on a lakeland fall\n\nthe swans stayed behind on the walls\n\nthe Dipper and the Ox spent the night in the window\n\nthe road to Heaven seemed so close again\n\nI dreamed I was traveling with clouds\n\n## 75\n\nHsieh T'iao (464\u2013499) was known for his landscape poems and built this tower while serving as prefect of Hsuancheng, to which he had been sent at the urging of jealous officials at the Southern Ch'i (479\u2013502) court in Nanching. Li Pai (701\u2013762) also sees himself at the mercy of the T'ang court and combines his own sense of injustice with a landscape poem in Hsieh's honor. The tower was south of town on the slopes of Lingyang-shan and faced north toward two bridges that spanned one of the two rivers that circled the town. In the second line, some editions have _hsiao_ (dawn) in place of _wan_ (dusk), but Li Pai was never known for getting up early. The paulownia is a large-leafed shade tree that loses its leaves in early autumn.\n\n## _Climbing Hsieh T'iao's North Tower in Hsuancheng in Autumn_\n\nLI PAI\n\nThis river town looks like a painting\n\nmountains at dusk against a clear sky\n\ntwo rivers frame an unblemished mirror\n\ntwin bridges form a rainbow\n\nkitchen smoke and winter oranges\n\nfall colors and leafless paulownias\n\nwho thinks of climbing North Tower\n\nof facing the wind and remembering Hsieh T'iao\n\n## 76\n\nMeng Hao-jan (689\u2013740) cultivated the life of a recluse but clearly harbored hopes of public service. He gave this poem to Chang Chiu-ling during a visit to Ch'ang-an in 733, shortly after Chang became prime minister. Once the high water of summer recedes, much of Tungting Lake becomes a marsh. Cloud Dream Marsh included two such sections of the western shore. Yuehyang is on the eastern shore and is still a major port for river traffic. Most commentators interpret the waves here as referring to the lake's water. But the lake is flat, and the only waves are those of the vapors that rise from the lake's broken hearts: Ch'u Yuan (340\u2013278 B.C.) drowned himself in the Milo River just south of Yuehyang, and the two wives of Emperor Shun (c. 2250 B.C.) also drowned themselves near here and became the lake's resident spirits. In the last couplet, Meng recalls a story in _Huainantzu_ : \"Rather than covet the fish in the deeps, better to go home and make a good net\" (17).\n\n## _Overlooking Tungting Lake\u2014For Prime Minister Chang_\n\nMENG HAO-JAN\n\nIn September the lake is so flat\n\nthe great hollow joins the celestial void\n\nvapors rise from Cloud Dream Marsh\n\nthe waves rock Yuehyang walls\n\nbut there's no boat to take me across\n\nand retirement is shameful in a golden age\n\nsitting watching other people fish\n\nto covet their catch is vain\n\n## 77\n\nWang Wei (701\u2013761) spent much of his time wandering in the mountains. Hsiangchi Temple was fifteen kilometers southwest of Ch'ang-an, on top of one of the plateaus formed by the loess that blew down from Mongolia in winter and settled at the foot of the Chungnan Mountains. The clouds on this occasion are below the plateau and envelop the capital. The temple was built around A.D. 700 at the confluence of the Hao and Chueh Rivers by the disciples of Shan-tao (d. 681), one of the patriarchs of Pure Land Buddhism. The serpent in the pool represents the delusions of the mind. For a comparison of Pure Land with Zen practice by the temple's recently-deceased abbot, see _Road to Heaven,_ pg. 95.\n\n## _Passing Hsiangchi Temple_\n\nWANG WEI\n\nUnaware of Hsiangchi Temple\n\nI walked for miles past mountains of clouds\n\nancient trees an empty path\n\nsomewhere in the hills a bell\n\nstreamsound murmuring boulders\n\nthe sun through cold green pines\n\na silent pool in fading light\n\nwhere Zen subdued the serpent\n\n## 78\n\nKao Shih (702\u2013765) tries to mollify a friend being sent to Fukien as punishment for performing too well his job of criticizing breaches of policy and law. Geese are a reminder of exile, but they can fly north in spring. Kao would have us believe they don't fly as far south as Fukien. Still, one cannot avoid hearing the eerie howl of a gibbon and being reminded of one's desolation. The Coast Road skirts the East China Sea in Chekiang and Fukien. What Kao means by \"the South\" is \"in this part of the South.\" Elsewhere in the South the climate was equivalent to a death sentence for Northerners. At the end of this poem Kao encourages his friend by trying to assure him that he will enjoy imperial favor (rain and dew), but only if he can avoid stirring up more trouble (wind and waves).\n\n## _For Censor Cheng on Being Banished to Fukien_\n\nKAO SHIH\n\nGo into exile but bear no grudge\n\nI was once in Fukien myself\n\ngeese for the most part are rare in fall\n\nthough gibbons are common at night\n\nthe Coast Road merges with cloud-high peaks\n\nbut miasmas and plagues in the South are mild\n\nyou'll know the rain and dew again\n\ngo but watch out for the wind and waves\n\n## 79\n\nTu Fu (712\u2013770) wrote this poem in 759 after resigning his post near Huashan and moving briefly to Tienshui (Chinchou), before continuing on to Chengtu to escape the chaos that followed the An Lu-shan Rebellion. Fenglin and Yuhai were border posts west of Tienshui, under attack by the Turks. The Northern Court (Peiting) was the name of a major Chinese garrison that had already fallen on the Silk Road near Urumuchi. Signal towers burned straw at night and wolf dung, which emitted an intense black smoke, during the day. Li Kuang was a hero of China's earlier conquest of this region and was known as the Flying General. Tu Fu suggests placating his maligned spirit might help the country overcome its current crisis. The reference here is an oblique one to Kuo Tzu-yi, who defeated the armies of An Lu-shan but suffered the slander of jealous officials at court.\n\n## _Miscellaneous Poem at Chinchou_\n\nTU FU\n\nThe fighting goes on in Fenglin\n\nthe road to Yuhai is still blocked\n\nsignal fires fill the sky with smoke\n\ntroops in the field find only dry wells\n\nthe wind shakes even the Western stars\n\nthe moon turns cold above the Northern Court\n\nan old man recalls the Flying General\n\nwhen will they build him an altar\n\n## 80\n\nAfter his patron died in 765, Tu Fu left Chengtu and headed down the Yangtze. Here, he stops outside Chunghsien at a shrine to Yu the Great. Yu founded the Hsia dynasty (c. 2200 B.C.) and was credited with taming China's rivers, not by building dikes but by dredging them. He is also credited with introducing the orange to Szechuan. But here his shrine has been neglected, and the fruit hangs unpicked, much like Tu Fu's own talents. It was said that Yu adapted himself to every situation: he rode chariots on land, boats on water, sledges on mudflats, and palanquins in the mountains. But Tu Fu has achieved no such mastery. The Land of Pa included the eastern half of Szechuan. It was the Pa (dragon) tribe that first settled this region.\n\n## _Shrine to Yu the Great_\n\nTU FU\n\nA shrine to Yu on a desolate slope\n\nin autumn wind and the sun's last rays\n\na tree full of oranges in an overgrown courtyard\n\na dragon mural in an ancient hall\n\nvapor rising from a rocky cliff\n\nroar of the river scouring the sand\n\nonce he mastered the art of riding\n\nhe opened up this Land of Pa\n\n## 81\n\nLi Ch'i (690\u2013751) served mostly in provincial posts but enjoyed a close association with a number of major poets in the capital. Here, he finally resigns and leaves for his home in Szechuan. Standing atop Ch'ang-an's north wall he sees the sun rising behind Lishan (a spur of the Chungnan Mountains to the east), the Wei River to the north (its watershed was also called the Valley of Ch'in, as the state of Ch'in once controlled this region), and the imperial grave mounds of Wuling to the northwest. He also hears the soughing of bamboo in the myriad palace gardens below. In the seventh line he recalls Confucius's regret in not finding anyone to employ his teachings and his anxiety that his disciples would not be able to pass them on ( _Analects:_ 5.21). The last line paraphrases the _Lichi_ ( _Book of Rites_ ): \"When a gentleman walks on frost, sadness fills his heart\" (24).\n\n## _Gazing at the Valley of Ch'in_\n\nLI CH'I\n\nGazing at the Valley of Ch'in at dawn\n\nthe rising sun behind the east peak\n\nmountains and rivers near and far so clear\n\nundulating wall upon palace wall\n\nthe sound of autumn from a world of bamboo\n\nthe look of winter among Wuling pines\n\n_let's go home_ a traveler sighs\n\nthe frost and dew are so thick\n\n## 82\n\nChang Wei (720\u2013770) was a native of the Loyang area and served as a minor official in various posts in the central government. He wrote this poem in 747 during his rustication to Changsha, the capital of Hunan province, south of Tungting Lake. The title _cheng-chun_ (gentleman-in-waiting) was an honorary epithet for someone who had been recommended to office, but it was not a formal position. Wang was thus exempt from the civil service exam (\"no need to open a book\") but was clearly still \"waiting.\" Tungting Lake is fed by several rivers, but the largest is the Hsiang, which is joined near its source by the Hsiao. Chang watches their waters flow north through the lake, into the Yangtze, and toward Loyang, while he and his friend wait to be called back to the capital.\n\n## _Commiserating with Gentleman-in-Waiting Wang on Tungting Lake_\n\nCHANG WEI\n\nThrough Tungting Lake in the middle of fall\n\nthe waters of the Hsiao and Hsiang flow north\n\nbut home is a thousand-mile dream away\n\nand a guest greets dawn with sorrow\n\nthere's no need to open a book\n\nfar better to visit an inn\n\nCh'ang-an and Loyang are full of old friends\n\nbut when will we join them again\n\n## 83\n\nTing Hsien-chih (fl. 713\u2013741) was a native of this area where the Grand Canal crosses the Yangtze. He was known for his love of wine and travel and once served as military commander of Hangchou. Here, he takes the ferry south of Yangchou\u2014it was this stretch between Yangchou and the sea that was called the Yangtze. On the south shore are Junchou (Chenchiang) and its temple-covered hills, and beyond Junchou is Tanyang, Ting's hometown. He has been away so long he can hear its leaves falling even in the middle of the river. Oars of _Cinnamomum cassia_ represent virtue rebuffed. Ch'u Yuan (340\u2013278 B.C.) used such oars in his failed rendezvous in \"The Princess of the Hsiang.\" Ting's use of this metaphor suggests that he, too, considers his lord fickle.\n\n## _Crossing the Yangtze_\n\nTING HSIEN-CHIH\n\nMy oars of cassia I gaze from midstream\n\nthe sky and waves and both shores are clear\n\nthe treeline parts at the Yangtze ferry\n\nhills rise up from the Junchou walls\n\nthe edge of the sea is dark and silent\n\na chill wind comes from the river's cold\n\nagain I hear maple leaves falling\n\nthe brittle sounds of another autumn\n\n## 84\n\nChang Yueh (667\u2013731) was sent, around 720, to serve as commander-in-chief at Yochou (an administrative area equivalent to that of modern Beijing, with its headquarters in Tahsing in the suburbs south of the present city); forces were stationed there to defend the empire against incursions by the nomadic Khitans. Chang had previously served as chief advisor to the emperor but had subsequently been demoted to a series of provincial posts and was uncertain whether this new post was a promotion or another demotion. Some commentators read the last couplet as an attempt to see it as a promotion. But such an assignment only serves to remind Chang how much he misses his family. Thus, while \"grace\" ostensibly refers to that of the emperor, it also refers to that of his wife and children. Ironically, Chang's ancestors were from Yuchou, and he would have found a warm welcome there. Still, the sixth line suggests homesickness rules the night. While the sword dance impresses with its flashy display, the sound of a flute travels farther and carries those who hear it back to those they miss\u2014in Chang's case, to his home in Loyang.\n\n## _Drinking at Night in Yuchou_\n\nCHANG YUEH\n\nA cold wind fills the night with rain\n\nand whistles through a leafless wood\n\nattending a sumptuous banquet\n\nwe can forget old age\n\nsoldiers are fond of a sword dance\n\nbut favor the flute on the border\n\nif I never served as a frontier general\n\nhow could I know the extent of grace\n\n## PART THREE\n## 85\n\nCh'eng Hao (1032\u20131085) was born near Hankou in Hupei province and grew up in Loyang. He and his younger brother, Ch'eng Yi, were major advocates of the neo-Confucian revival that dominated the intellectual life of the Sung dynasty. During the ten years he taught in Loyang, his students numbered in the thousands. He also served at the Sung capital in Kaifeng as a companion to the crown prince and as an investigating censor. But due to his opposition to the policies of Wang An-shih, he was demoted and sent to Chenning in Kuangsi province, where he served as an administrative assistant. He was later recalled but died on his way back. The third line recalls a story in _Chuangtzu_ in which Chuang-tzu is out walking with Hui-tzu and pauses to comment on the joy of the fishes playing under a bridge. Hui-tzu says, \"You're not a fish. How do you know if the fishes are happy?\" The Taoist sage replied, \"You're not me. How do you know I don't know the fishes are happy?\" (17.13).\n\n## _Casual Poem on a Spring Day_\n\nCH'ENG HAO\n\nThe clouds are thin the wind is light the sun is nearly overhead\n\npast the flowers through the willows down along the stream\n\npeople don't see the joy in my heart\n\nthey think I'm wasting time or acting like a child\n\n## 86\n\nChu Hsi (1130\u20131200) was born and died south of the Wuyi Mountains in Fukien province, although his family's ancestral home was in Kiangsi. He served in a number of posts, including senior compiler in the imperial archives and governor of several provinces, but he is better known for carrying on the work of the Ch'eng brothers in promoting the Sung dynasty's neo-Confucian revival. He also made lasting contributions in the areas of literature and history. Here, he visits a river just north of Confucius's hometown of Chufu in Shantung province. One day Confucius asked his disciples what they most desired. After several related their political ambitions, Tseng Hsi said, \"In the last month of spring, when the new clothes are ready, I would like to go down to the river with a couple of students and perform the rites of spring.\" The Master sighed and said, \"I'm with you\" ( _Analects:_ 11.7). According to the conceptions that underlie the _Book of Changes,_ spring begins in the East.\n\n## _Spring Day_\n\nCHU HSI\n\nAlong the Ssu River it's a fine day for blossoms\n\nthe landscape is endless and suddenly new\n\nI recognize the East Wind's familiar face\n\na thousand pinks and purples and everywhere spring\n\n## 87\n\nSu Shih (1037\u20131101), aka Su Tung-p'o, was a native of Meishan in Szechuan province and one of the greatest poets, essayists, calligraphers, and personalities of the Sung dynasty. He also served in a number of high-level posts, although his opposition to the policies of Wang An-shih resulted in a brief imprisonment and several banishments, the last of which was to the southern borders of the empire. Not long after the death of his wife he was allowed to return, but he died within months of retiring to a piece of land he had bought earlier outside Changchou in Chekiang province. The swing is said to have arrived in China during the first millennium B.C. via one of the nomadic groups on the country's northern border. It still plays an important part in the harvest celebrations of such hilltribes as the Hani (Aini), along China's southwest border, where its ropes are often thirty or forty feet in length and it literally takes one's breath away.\n\n## _Spring Night_\n\nSU SHIH\n\nA spring night hour is worth a ton of gold\n\nthe pure scent of flowers the moon's pale light\n\nmusic from the terrace finer than silk\n\nswinging in the courtyard far into the night\n\n## 88\n\nYang Chu-yuan (755\u2013820) was from Yungchi in Shansi. He served as vice-governor of the southern half of the same province and also as director of studies at the imperial university. The Chinese love to go looking for the first signs of spring, especially the blossoms of such fruit trees as the plum and the apricot. The Royal Woods (Shanglin) consisted of a large tract of pear and chestnut trees planted adjacent to the palace in Ch'ang-an during the Ch'in dynasty. They were expanded during the Han dynasty to well beyond the western suburbs. The Pa River, at the eastern edge of Ch'ang-an, was famous for its willow-lined banks. Yang suggests that the true connoisseur of spring is able to appreciate the willow's emerging golden catkins without feeling the need to wait for the more ostentatious show to come. In his commentary, Wang Hsiang says this also refers to recognizing men of talent before they become known.\n\n## _Early Spring East of Town_\n\nYANG CHU-YUAN\n\nThe best time for a poet is when spring is new\n\nwhen willows turn gold but not completely\n\nif you wait until the Royal Woods look like brocade\n\nthe whole town will be out gawking at flowers\n\n## 89\n\nWang An-shih (1021\u20131086) was from Fuchou in Kiangsi province and became one of the most famous prime ministers in Chinese history. His financial policies, however, resulted in a profound split among government officials and also in Wang's retirement to the hills east of Nanching. He wrote this poem in the spring of 1069, the year before he became prime minister. Spring marks the beginning of the agricultural calendar, and the reforms he was hoping to implement were primarily aimed at helping farmers. Hence, Wang has stayed up all night anticipating the new year. But the only flowers he sees are the shadow flowers on the railing outside his window. Sticks of incense and devices that dripped water were used to keep time during the night. The last three lines paraphrase lines in poems by Han Wo (844\u2013923), Lo Yin (833\u2013909), and Yao Ho (fl. 831).\n\n## _Spring Night_\n\nWANG AN-SHIH\n\nThe burner is out of incense the dripping has almost stopped\n\nthe wind comes in gusts the cold in waves\n\nspringtime disturbs me and keeps me from sleep\n\nthe moon casts shadow flowers on the balustrade\n\n## 90\n\nHan Yu (768\u2013824) was a native of Menghsien in Honan province. Although he spent a good part of his life serving in the upper echelons of government, he was banished on several occasions, most famously for his suggestion that the emperor's veneration of the Buddha's finger-bone would shorten his life. He was also a distinguished literary figure of the T'ang and its most ardent supporter of Confucian ideals. This was written in 823 shortly before his death and was the first of two poems he wrote on the appointment of his friend Chang Chi, entitled \"For Chang Shih-pa [the eighteenth son], Vice-Director of the Bureau of Waterways and Irrigation, in Early Spring.\" The \"streets of Heaven\" and the \"royal city\" refer to Ch'ang-an, the residence of the Son of Heaven. Here, the angle of view condenses the faint colors on the horizon, whereby green appears in the distance but not nearby, and the myriad new willow catkins look like so much mist.\n\n## _Light Rain in Early Spring_\n\nHAN YU\n\nThe streets of Heaven glisten from light rain\n\ngrass appears far off but not nearby\n\nthis is truly the best time of spring\n\nwhen the sight of misty willows fills the royal city\n\n## 91\n\nWang An-shih (1021\u20131086) wrote this poem soon after being appointed prime minister in 1070 and beginning implementation of his controversial agricultural reforms. Hence, while he records customs associated with the lunar New Year, he also has his new policies in mind. The Chinese viewed the year as a spirit named _Nien_ (Year), which needed to be encouraged to make way for its replacement. In ancient times the Chinese heated sections of bamboo that chased the Year away when they exploded. During the Sung these were replaced by paper tubes filled with a new concoction\u2014gunpowder. In preparation for the New Year the Chinese still steep a combination of herbs in grain alcohol during the winter. This combination (called _Tusu_ ) includes cinnamon, pepper, _Atractylis lancea, Siler divaricatum,_ and balloon flower root. According to the T'ang-dynasty herbalist Sun Ssu-miao, drinking Tusu wine on New Year's Day drives off infections. Because _t'ao_ (peach) was a homophone of _t'ao_ (chase away), the wood of the peach tree was thought to have the power of exorcising evil spirits and was used to make the block prints of guardians and sages that were pasted outside doorways on New Year's Day. These prints originated just south of Kaifeng in Chuhsienchen, which became one of the world's first centers of printing in the tenth and eleventh centuries.\n\n## _New Year's Day_\n\nWANG AN-SHIH\n\nFirecracker sounds chase the Year away\n\nspring wind infuses herbal wine with warmth\n\noutside countless doorways in the rising sun\n\nnew peach prints are pasted over last year's charms\n\n## 92\n\nThis was one of three poems Su Shih (1037\u20131101) wrote at imperial request while attending a dawn banquet for court officials on the occasion of Shangyuan (the Great Beginning), which celebrates the first full moon of the year, when the moon sinks in the west at dawn prior to its return at sunset for Yuanhsiao (Lantern Festival). But the sinking moon and fading stars also represent Wang An-shih and his supporters at court, whose policies Su opposed and whose light pales in the glory of the imperial presence. Chienchang Court was built during the Han dynasty as part of the palace in Ch'ang-an, but here it refers to the assembled officials in the Sung capital of Kaifeng. Tungming Palace is where Yu Huang-ti (the Jade Emperor\/Sage on High) resides in the highest of the Taoist heavens. But here the palace is in Kaifeng, rather than the heavens, and the sage is the Sung emperor. In honor of the occasion, officials wore red robes, the color of life. Standing in the courtyard perfectly still and _en masse,_ they resembled the horizon at dawn.\n\n## _Attending a Banquet on the Great Beginning_\n\nSU SHIH\n\nPale moon and scattered stars encircle Chienchang Court\n\ncelestial winds waft incense from the royal censer\n\ninside Tungming Palace officials stand like storks\n\nan offering of red clouds for the Sage on High\n\n## 93\n\nChang Shih (1133\u20131180) was born in Mienchu in Szechuan province and later moved to Hengyang in Hunan. The son of the famous general and prime minister, Chang Chun, he served in the Southern Sung capital of Hangchou and eventually reached the post of senior compiler of the dynastic history. He was also known for his essays advocating Confucian values. Chang, Chu Hsi, and Lu Tsu-ch'ien were ranked as the three illuminati of their age. The Arrival of Spring ( _Lichun_ ) is a solar period that begins shortly after the lunar New Year, usually in the first week of February. The division of the year into phases of light and dark, or _yin_ and _yang,_ goes back to the Yellow Emperor. In the ancient system of trigrams devised by Fu Hsi around 2800 B.C. and reworked by King Wen around 1100 B.C., east is the direction of spring and west is the direction of autumn. In the different but complementary systems of these two men, east is also seen as the direction of either fire or thunder, while west is the location of water or lakes.\n\n## _Occasional Poem on the Arrival of Spring_\n\nCHANG SHIH\n\nThe light is back the year is past ice and frost are rare\n\nplants and trees all know spring is in the world\n\nwe see the force of life spread before our eyes\n\nthe East Wind blowing water ripples green\n\n## 94\n\nCh'ao Yueh-chih (1059\u20131129) was from Chuyeh in Shantung province and served as a member of the Hanlin Academy in Kaifeng, where he was assigned the task of recording imperial pronouncements during the emperor's morning meeting with his officials, which apparently inspired this poem. Ch'ao was also known for his skill as a painter, and here he portrays Emperor Hsuan-tsung (685\u2013762), the third son of Emperor Jui-tsung (662\u2013716), indulging in a game similar to soccer that employed a leather ball stuffed with feathers\u2014while the country went to ruin. At the beginning of Hsuan-tsung's reign his chief ministers, Chang Chiu-ling and Han Hsiu, submitted memorials criticizing his lack of attention to governmental affairs. But both were eventually forced to resign or retire and thereafter the emperor seldom received frank advice, much less criticism, during his dawn audience. Ch'ao could only have written this poem about an emperor of a previous dynasty, but his reference is clearly to Emperor Hui-tsung (r. 1101\u20131126), whose passion for the arts and whose inattention to governmental affairs are blamed for the loss of North China to the Khitans and for the flight of the court from Kaifeng to Hangchou, where his successors established the Southern Sung dynasty (1127\u20131279).\n\n## _On a Painting of Playing Football_\n\nCH'AO YUEH-CHIH\n\nA thousand doors and windows open in the palace\n\nthe Third Son is drunk and gives the ball a kick\n\nChiu-ling is too old and Han Hsiu is dead\n\nno longer are memorials submitted in the morning\n\n## 95\n\nLin Hung (fl. 1250) was a native of Fukien province, but little else is known about him. One source says he was a seventh-generation descendant of the Hangchou recluse Lin Pu (967\u20131028). This and the following poem are from a series of one hundred verses written by the poet on life in the palace in Hangchou. On New Year's Day the Chinese pay respects to their ancestors. And since the emperor is the Son of Heaven he greets his progenitors in semimythical fashion, reminiscent of the ascent of his shaman predecessors during their annual communion with ancestral spirits. At the palace in Hangchou there were two pillars, on top of which stood two immortals offering up jade lotuses to the gods. The flowers also served the function of collecting dew unsullied by earthly dust. The dew was then used in elixirs. A number of commentators point to this and to the following poem as examples of the lower standards used by the compiler in including a handful of poems by friends, and they are deleted in some editions. Among those editions that include them, several incorrectly attribute them to Wang Chien (751\u2013830), who also wrote a series of one hundred poems on palace life in the T'ang. In ancient times, horses over seven feet tall were called \"dragon steeds.\"\n\n## _Palace Ode\u2014I_\n\nLIN HUNG\n\nAbove the gilded hall's purple-tiered pavilion\n\nimmortals offer up lotuses of jade\n\nthe Emperor of Peace greets the New Year dawn\n\nhis chariot of colored clouds drawn by dragon steeds\n\n## 96\n\nThe poet describes the examination of candidates seeking appointment as officials, all of whom are required to write three-thousand-word essays according to forms established in the canons of ritual and prosody. Meanwhile, the emperor looks on, wearing a robe embroidered with images and symbols of the sun and moon. The Chinese make their ink by grinding sticks of soot and pine resin on flat stones, onto which small amounts of water are added. When the proper consistency is reached the ink is drained into a depression at one end of the stone. Here, the flapping banners hanging from the rafters are reflected in these pools of ink. Finally done, the candidates stand before the court's cinnabar-colored steps and present their work.\n\n## _Palace Ode\u2014II_\n\nLIN HUNG\n\nAt court the royal robes reflect the sun and moon\n\nbanner shadows move like dragons on the inkstones\n\nthe length and breadth of rites and music in three thousand words\n\nline the cinnabar steps before the sun goes down\n\n## 97\n\nTu Ch'ang (fl. 1080) was a descendant of Empress Chao-hsien (fl. 950) and held a series of midlevel posts in and around Kaifeng. Here, he visits the ruins of Huaching Palace (destroyed in 755 by An Lu-shan) east of Ch'ang-an, and he likens his journey to that of the relay riders who brought lychees from South China for Hsuan-tsung's favorite concubine, Yang Kuei-fei. The palace at Lishan was where the two lovers spent much of their time, while the T'ang dynasty went to ruin. Overlooking this testament to indulgence was Chaoyuan (Sunrise) Pavilion at the summit of Lishan. Normally, the West Wind indicates the coming of autumn in China. But west of Hanku Pass it means rain. The tall willows ( _ch'ang-yang_ ) are interpreted by some as referring to the former Han-dynasty palace of that name, west of Ch'ang-an. But Huaching was also known for these trees, and it seems unlikely that the poet would have confused the two places, which were fifty kilometers apart. This was apparently written while Tu was traveling west to Tienshui to take up his post as senior judge for the province, and he is concerned that his in-laws back at the Sung capital might not have learned the lesson that Hsuan-tsung learned too late. With some trepidation, I have amended the second line, reading _hsiao-yun_ (dawn clouds) for _hsiao-feng_ (dawn wind) to avoid redundancy in the third line.\n\n## _In Praise of Huaching Palace_\n\nTU CH'ANG\n\nA journey of countless stages from the Southland ends\n\nwith the pale moon and dawn clouds of Huaching\n\nand the West Wind rushing past Sunrise Pavilion\n\nentering the tall willows it sounds like rain\n\n## 98\n\nLi Pai (701\u2013762) was born in Tokmak in what is now Kyrgyzstan and moved to Szechuan when he was five. After failing to secure a post through normal channels, he was given a sinecure at the Hanlin Academy in 742 based on his reputation as a poet. A year or so later, when he was drinking one day with friends, he was summoned to the palace, where Hsuan-tsung and Yang Kuei-fei were enjoying the peony blossoms. Asked to compose a few verses to the tune of \"Chingping,\" he complied with this ode to erotic beauty and two other poems. However, he was so drunk he aroused the enmity of the eunuch, Kao Li-shih, who had to help him take off his boots and to grind his ink. Yang Kuei-fei also became incensed after Kao interpreted allusions in the other two poems as referring to women who brought the country to ruin. Li Pai was soon sent packing and never returned to the capital. The jade peaks are in the land of immortals, and Alabaster Terrace is at the court of Hsi-wang-mu (Queen Mother of the West), dispenser of the elixir of immortality.\n\n## _Chingping Ode_\n\nLI PAI\n\nHer cloudlike clothing and flowerlike face\n\nspring wind at the threshold caresses her dewy luster\n\nif this isn't a scene from the land of jade peaks\n\nit must be Alabaster Terrace in the moonlight\n\n## 99\n\nCheng Hui (fl. 1210) was a minor poet of the Sung about whom we know next to nothing. He wrote this poem in his wife's voice while traveling from Hangchou to Kiangsi province, probably at an inn in Changshan. Changshan was a major stop on the post road that led southwest from the Southern Sung capital. Cheng's home was in Kueihsi, still another 150 kilometers away. The variety of climbing raspberry he imagines in his wife's dream, _Rubus rosifolius_ var. _commersonii,_ blooms in this region at the end of spring. Swallows represent conjugal bliss but they, too, are waiting for warmer weather. The green is that of early spring foliage. Red candles are used for weddings, New Year's, and any other happy occasion. Women often used their hairpins as scissors to trim candle wicks, but here they lie broken from overuse. In some editions this poem is attributed to Cheng Ku (fl. 890\u2013930), a prominent poet of the independent state of Nanping. Nanping controlled the area of the Yangtze east of the Three Gorges during the Five Dynasties Period (907\u2013960) between the T'ang and Sung.\n\n## _Written on the Wall of an Inn_\n\nCHENG HUI\n\nA raspberry-scented dream dispels the chill of spring\n\nthe swallows are resting behind doors shrouded green\n\nmy jade hairpins lie broken the red candles are cold\n\naccording to my count he should be in Changshan\n\n## 100\n\nTu Fu (712\u2013770) wrote this poem in the early summer of 764 while he and his family were living in Szechuan in a thatched cottage on a minor side-stream of the Min River, just outside the west wall of the provincial capital of Chengtu. Orioles are the harbingers of spring, as egrets are of summer. To the west of Chengtu are the Min Mountains, which formed the country's border with Tibet and which included peaks in excess of 6,000 meters. The Min River flowed out of the mountains and past Chengtu and connected Western Szechuan, via the Yangtze, with the rest of China. Tu Fu may have been living in relative security far from the chaos of North China, but he is already thinking of leaving. A few kilometers southeast of his house was Ten Thousand Mile Bridge, where Chu-ko Liang (181\u2013234) bid goodbye to an emissary heading for Chekiang province with the words, \"a journey of ten thousand miles begins right here.\" When Tu Fu's patron died the following year, the poet and his family began their own journey down the Min and Yangtze Rivers in search of another sanctuary. This was the third in a series of four poems.\n\n## _Quatrain_\n\nTU FU\n\nA pair of golden orioles sings in green willows\n\na column of snowy egrets flies off in blue sky\n\nmy window contains peaks with a thousand years of ice\n\nmy gate harbors boats from ten thousand miles downriver\n\n## 101\n\nSu Shih (1037\u20131101) was from Meishan in Szechuan province. Along with his father and younger brother, he was ranked among the eight greatest essayists of the T'ang and Sung dynasties. He was equally famous for his calligraphy and poetry and also served at the highest levels of government. But his opposition to the policies of Wang An-shih led to repeated banishments and even a brief imprisonment. Here, the East Wind represents the arrival of spring, and this harbinger of the season reminds Su of Yang Kuei-fei, concubine of Emperor Hsuan-tsung, who called her his \"begonia.\" On one occasion she got so drunk that she slept for two nights. Hence, Su hopes to see her garden incarnation before she retires. Su wrote this poem at the beginning of his banishment to Huangchou, down the Yangtze from Wuhan, at a garden known for its begonias.\n\n## _Begonia_\n\nSU SHIH\n\nThe East Wind gently spreads her celestial glow\n\nthe moon slips behind her veil of perfumed mist\n\nafraid this flower won't stay up much longer\n\nI light a tall candle to view her crimson face\n\n## 102\n\nTu Mu (803\u2013852) was born in Ch'ang-an to a once-prominent family that had lost its wealth and power by the time he was a boy. Hence, he grew up in poverty and began his studies relatively late. Still, he excelled in the literary arts and served in a number of posts, though of ever-decreasing importance. When he wrote this poem he was serving south of Nanching as prefect of Kueichih, and found himself far from home on a day in early April when family members make every effort to return to honor their deceased parents and ancestors. Since ancient times, Grave Sweeping Day (Chingming) has been the day when the Chinese pay their respects at their ancestral graves and cut back the weeds that have grown up since their last visit. The weather during this time of year is usually fine in North China, but not along the Yangtze. Still, this traveler is in luck. Apricot Blossom Village (Hsinghuatsun) was just west of Kueichih and was famous throughout China for its rice wine. The well that supplied the water is still there.\n\n## _Grave Sweeping Day_\n\nTU MU\n\nOn Grave Sweeping Day the rain pours down\n\na traveler on the road feels his heart sink\n\nwhere he asks can he buy some wine\n\na herdboy points off to Apricot Blossom Village\n\n## 103\n\nWang Yu-ch'eng (954\u20131001) was from Chuyeh in Shantung province. Although he was quite poor as a youth, he was recommended to the court by a local official and served in a series of posts that alternated between the highest and lowest echelons. While serving as a censor he was known for his forthright criticism of government policies and was eventually banished to the South. According to an ancient tradition, people were forbidden to eat hot food for the two days before Chingming, when families paid their respects at ancestral graves and made offerings of wine and flowers. After two days of cold hearths, people made a new fire by drilling willow wood. Here, the poet is so poor he has to borrow a light from his neighbor. However, he borrows it so that he can study before dawn in preparation for the civil service exams. In many ways the life of a would-be official was not unlike that of a poor monk who begs for his living. And yet, despite his poverty, the poet honors his ancestors with his diligence and willingness to suffer on behalf of the family name.\n\n## _Chingming_\n\nWANG YU-CH'ENG\n\nCelebrating Chingming without flowers or wine\n\nbereft of excitement like a monk in the wilds\n\nlast night I begged a neighbor for new fire\n\nfor my lamp by the window to read before dawn\n\n## 104\n\nWang Chia (851\u2013890) was from near Yungchi in Shansi province and reached the post of vice-director of one of the bureaus in the Ministry of Rites in Ch'ang-an. Here, he is in Kiangsi province serving as prefect of Yenshan. Gooselake Mountain (Ohushan) was a few kilometers southeast of town. During the following century it was the location of a major Confucian academy. In ancient times the Chinese held a neighborhood festival in honor of the Earth God; it included twenty-five families and was held twenty-five days after the beginning of spring and fall. In the first line I have read _tao-t'ien fei_ (fields are fertilized\u2014therefore, ready) in place of _tao-liang-fei_ (grain is fat). This emendation seems inescapable, as grain could not possibly be _fei_ (fat) a month after New Year. On the other hand, _fei_ was commonly used to describe fields. The closed sties and coops suggest that sows have just given birth and chicks have just hatched, and that they are being protected from predators in a family's absence. Mulberry trees supply the leaves silkworms feed on. They are pruned in winter and put forth new leaves about this time, at least enough to cast shadows in the setting sun. Taxes were usually paid in silk; hence, this was a major concern of every family. Some editions attribute this poem to Chang Yen (fl. 880).\n\n## _Festival Day_\n\nWANG CHIA\n\nThe fields below Gooselake Mountain are ready\n\nthe pigsties and chicken coops are all shut tight\n\nmulberry shadows mean springfest is over\n\nfamilies all help their drunken men home\n\n## 105\n\nHan Hung (fl. 750\u2013780) was a native of Nanyang in Honan province. Although he served in midlevel posts in the capital, he was far better known for his poetry and was ranked as one of the Ten Talents of the Tali Period (766\u2013779). One day when the chief minister informed Emperor Te-tsung that there was an opening in the Secretariat, the emperor said, \"Give the post to Han Hung.\" The minister replied, \"But there are two Han Hungs. Which one does his majesty mean?\" Te-tsung said, \"The one who wrote 'The city in spring isn't safe from flying petals.'\" Although nowadays this poem is read as a critique of the privileged class, the emperor read it as a praise of imperial largesse. As noted in the commentary to verse 103, during the two days before Grave Sweeping Day no hot food was eaten. This observance was said to have begun 2,600 years ago with Duke Wen of the state of Chin, to commemorate the death of Chieh Chih-t'ui, who refused to come out of seclusion to serve at the Duke's court. Furious at Chieh's refusal, the Duke ordered the mountain on which Chieh lived burned. Chieh still refused to come down and died in the fire. Here, too, the poet notes the imperial custom, begun by Han-dynasty emperors, of sending lighted candles to the rich and powerful, who were thus allowed to ignore the ban on fire.\n\n## _Cold Food_\n\nHAN HUNG\n\nThe city in spring isn't safe from flying petals\n\non Cold Food Day before the East Wind royal willows lean\n\ncandles are sent from palace halls at sunset\n\nsmoke curls faintly inside the great estates\n\n## 106\n\nTu Mu (803\u2013852) summarizes the sights of a journey through a region the Chinese call Chiangnan (South of the Yangtze)\u2014to which he was rusticated from 838 to 842. Not only does this region south of the river's lower reaches enjoy a milder climate and more rainfall than North China, it was also more hospitable to mercantile activities and Buddhism during a series of southern dynasties (420\u2013589). The Yangtze remains China's greatest highway, and from its river towns and hillside hamlets wineshops and inns advertised their presence with pennants suspended from bamboo poles. Emperor Wu of the Liang dynasty (502\u2013556) was said to have built nearly 500 temples in this region, and Buddhist architecture was as much a part of the landscape as bamboo and lotuses. Thus, as he describes the sights and sounds, Tu Mu also suggests the reasons for the decline in dynastic fortunes: indulgence in worldly passions and the emptying of imperial coffers to support the otherworldly way of life of Buddhist monasteries. Pagodas usually contained the ashes and relics of eminent monks.\n\n## _Chiangnan Spring_\n\nTU MU\n\nA thousand miles of oriole songs and red among the green\n\nof wine flags flapping along the shore and in the hills\n\nfour hundred and eighty temples built by the Southern Court\n\nand how many pagodas in the land of mist and rain\n\n## 107\n\nKao Ch'an (fl. 850\u2013890) grew up in Tsangchou near the mouth of the Yellow River and moved to Ch'ang-an as a young man. Although he became well known for his poetry, he had to take the civil service exam several times before finally succeeding in 876. Still, he eventually reached the post of deputy chief censor. Here, he compares those who enjoy imperial favor (dew and clouds) to varieties of fruit trees known for their blossoms, and he likens himself and his friend, Kao Pin, to the lotus that will have its day\u2014but in late summer or early fall, not in spring when the East Wind blows. Kao Ch'an wrote this poem following one of his earlier failures to pass the civil service exam, when he learned his friend had been appointed military commissioner of the Huainan region, which was known for its lotuses. Hence, this poem is about the poet's own disappointment rather than about Kao Pin's. The second line was later quoted as part of a couplet in the Ming-dynasty novel _Dream of the Red Chamber._\n\n## _For Gentleman-in-Attendance Kao_\n\nKAO CH'AN\n\nThe green peach of heaven thrives in the dew\n\nthe red apricot of the sun flourishes by the clouds\n\nthe lotus that rises from a river in fall\n\ndoesn't blame the East Wind for not blooming sooner\n\n## 108\n\nChih-nan (fl. 1200) was a Buddhist monk who lived near the foot of the Tientai Mountains at Kuoching Temple. A friend of many literati of his day, he was also responsible for publishing the first printed edition of the poems of Han-shan (Cold Mountain) in 1184. In this poem he describes a scene reminiscent of the fisherman who tied up his boat and traced peach blossoms floating down the stream back through a crevice in the rocks to the lost world immortalized by T'ao Yuan-ming in \"Peach Blossom Spring.\" Here, apricot blossoms replace those of the peach. The fisherman, too, is replaced by an old monk who needs the help of a walking staff of _Chenopodium album_ but whose spirit is invigorated by the spring weather. East is where the sun rises and is the direction associated with spring. The famous front gate of Kuoching Temple also faces east. Just past the gate is Fengkan Bridge, and beyond the bridge is the road (or trail in Chih-nan's day) that leads up the mountain.\n\n## _Quatrain_\n\nCHIH-NAN\n\nI tied up my sampan in the shade of an ancient tree\n\nand headed east across the bridge with the help of my pigweed staff\n\nmy clothes were soaked by the apricot-blossom rain\n\nbut my face wasn't numbed by the willow-catkin wind\n\n## 109\n\nYeh Shao-weng (fl. 1200\u20131250) was from Lishui near the southern border of Chekiang province. Little else is known about him, other than that he served as an academician in the imperial archives in Hangchou and wrote in the unadorned style advocated by the Rivers and Lakes school of poetry. The Chinese often use moss as a ground cover in their gardens, and the clogs, which the Japanese still wear at home and at public baths, had two high wooden ridges on the bottom, one in front and one in back. The front ridge helped when going uphill, the back ridge when going downhill. And together they kept one's feet above the mud. The red blossoms are those of the apricot. The last line is quite famous but was originally part of an earlier poem by Lu Yu (1125\u20131210) entitled \"Written on Horseback.\" There is also a nearly identical poem by Chang Liang-ch'en, a fellow member of the Rivers and Lakes school of poetry, entitled \"Occasional Poem,\" but it's impossible to tell who is quoting whom. A variant of the second line reads: \"I knocked ten times, nine times without success.\" But that would suggest the owner finally opened his gate, which would seem at odds with the title as well as with the last couplet. Another variant has \"I knocked lightly, but his gate stayed closed.\" But why lightly? Finally, some editions attribute this poem to Yeh Shih (1150\u20131223).\n\n## _Visiting a Private Garden without Success_\n\nYEH SHAO-WENG\n\nIt must be because he hates clogs on his moss\n\nI knocked ten times still his gate stayed closed\n\nbut spring can't be kept locked in a garden\n\na branch of red blossoms reached past the wall\n\n## 110\n\nLi Pai (701\u2013762) wrote this poem around 740 while visiting the sites associated with the founders of the Confucian tradition in Shantung province. In addition to paying his respects at the graves of Confucius and Mencius in Chufu and Tsouhsien, he also traveled to Lanling near the southeastern border of Shantung. This was the location of the grave of Hsun-tzu (third century B.C.), who is credited with establishing the teachings of Confucius at court. Lanling was also known for a wine infused with turmeric, or _Curcuma longa,_ which was fragrant and reddish yellow in color and which was mixed with the wine drunk on ceremonial occasions during Confucius's day. To establish their reputations and connections, poets did quite a bit of traveling. Thus, much of Chinese poetry is dominated by the sentiments associated with being away from home.\n\n## _Traveling Away from Home_\n\nLI PAI\n\nThe fine wine of Lanling with its turmeric scent\n\nfills jade cups with its amber light\n\nif only a host can keep his guests drunk\n\nthey'll soon forget about their hometowns\n\n## 111\n\nLiu Chi-sun (fl. 1070) was born in the Sung capital of Kaifeng, but he spent most of his career as a minor official in the provinces. He wrote this poem in Kiangsi province while serving as inspector of the government winery in the town of Poyang on the eastern shore of Poyang Lake. Officials seldom took their families with them to their provincial posts, and the swallows, which represent marital harmony, remind Liu of his wife. Finding no sympathy at work, Liu decided to take the day off and left this memento for his fellow officials. Mushroom Mountain (Chihshan) was just north of town and represents the call of reverie, in which he finds solace from his sense of separation. One day not long after Liu wrote this, Wang An-shih (1021\u20131086) was serving as judicial commissioner for the same region and saw this poem on a screen in the office while checking for irregularities at the winery. He was so taken by the poem, he left without making an inspection. As a result, Liu enjoyed a brief moment of fame in literary circles, and though he was too honest to capitalize on it, he became deputy director of the Crafts Institute and later prefect of Hsihsien in Shansi province. When he died, the only property he left was a collection of thirty thousand books and several hundred paintings.\n\n## _Written on a Screen_\n\nLIU CHI-SUN\n\nTwittering swallows gossip in the rafters\n\nwhy do they disturb my daydream reverie\n\nthe clerk beside me shrugs when I ask\n\nI grab my staff and jug and head for Mushroom Mountain\n\n## 112\n\nThis was the fifth in a series of nine poems Tu Fu (712\u2013770) wrote in the late spring of 761. He was living on the bank of a minor branch of the Min River, just beyond the west gate of Chengtu, in a thatched cottage that sheltered him and his family during one of the few idyllic periods of his life. The second line recalls the end of Ch'u Yuan's \"Lady of the Hsiang\" in the _Chutzu_ : \"Sweet pollia I've picked on the flowering shore \/ to give to the maiden who lies below [one of the wives of Emperor Shun, who drowned herself in the Hsiang River] \/ what is gone cannot be found again \/ let us then wander and enjoy what is left.\" The petals recall T'ao Yuan-ming's story of a fisherman who sees peach blossoms floating down a stream and traces them upstream through a cleft in the rocks to an idyllic paradise untouched by the chaotic world of his time. Tu Fu feels that the hope of reaching such a paradise, illusory or not, has once again faded with the end of another spring\u2014and how many springs does he have left?\n\n## _Inspired_\n\nTU FU\n\nThe heartbreaking flood of spring is nearly over\n\nI poke along with my staff and stand on a flowering shore\n\nwillow catkins dance wildly in the wind\n\npeach petals float diaphanous in the current\n\n## 113\n\nHsieh Fang-te (1226\u20131289), aka Hsieh Ping-te, was from Yiyang in Kiangsi province and served in Hangchou as an examiner of civil service candidates. His selection of topics, however, led to his dismissal and subsequent demotion to the provinces, and eventually to the prefecture next to his hometown. When the Mongols finally brought the Southern Sung dynasty to an end in 1279, Hsieh formed an army of resistance in Yiyang, but he was defeated and fled to Fukien. He was later captured and taken to the new capital in Beijing, but he starved himself in prison rather than serve the rulers of the new Yuan dynasty. Here, he visits a hermitage in the hills just south of Yiyang. T'ao Yuan-ming's story, which supplies the background for this poem, recounts how a fisherman followed peach petals that were drifting down a stream until he reached their source in a cleft in the rocks. Squeezing through the crevice he came out into an idyllic valley, where he met people whose ancestors had come there several hundred years earlier to escape the brutal rule of the Ch'in dynasty (221\u2013207 B.C.). After returning to tell others of his discovery, the fisherman was unable to relocate the valley, as the refugees had obscured the trail and the crevice. Hsieh uses the Ch'in here to represent the Yuan (1280\u20131368), from whose encroaching dominion he hoped to find refuge.\n\n## _The Peach Blossoms of Chingchuan Hermitage_\n\nHSIEH FANG-TE\n\nIn Peach Blossom Valley they escaped the Ch'in\n\npeach blossom red means spring is here again\n\ndon't let flying petals fall into the stream\n\nsome fisherman I fear might try to find their source\n\n## 114\n\nLiu Yu-hsi (772\u2013842) was a native of Loyang and a prominent, though often banished, poet. He wrote this poem in 815 upon returning to Ch'ang-an after a ten-year banishment to Hunan for siding with the reform faction of Wang Shu-wen. The combination of allusions in this poem\u2014the purple (i.e., royal) paths of the imperial city, the red dust (of illusion and sensory indulgence), the name of the temple ( _hsuan-tu_ \u2014celestial capital), the thousands of officials promoted since his banishment, and the thinly veiled reference to himself\u2014together resulted in a second banishment. When he returned he wrote the next poem (verse 115) after visiting the same temple, and was banished for a third time. The Taoist temple to which he refers was just outside Ch'ang-an, and the people he meets are on their way back from going to see its blossoms. The peach trees were planted by the abbot after Liu's earlier banishment. While \"Mister Liu\" is clearly a mask for the author, it ostensibly refers to Liu Ch'en, an herb gatherer who became lost in the Tientai Mountains and survived on peaches until he finally returned two hundred years later.\n\n## _The Peach Blossoms of Hsuantu Temple_\n\nLIU YU-HSI\n\nRed dust from purple paths swirls before their faces\n\neveryone says they've been to see the flowers\n\nthousands of peach trees at Hsuantu Temple\n\nand all of them planted since Mister Liu departed\n\n## 115\n\nLiu wrote this poem in April of 828 and included this preface: \"In the year 805, when I was serving as vice-director of the State Farm Bureau, there were no flowers at this temple. That was the year I was demoted to serve as prefect of Lienchou [in Kuangtung province] and later as vice-prefect of Changte [in Hunan province]. After ten years I was called back to the capital, and everyone told me there was a Taoist priest who had planted heavenly peach trees that filled his temple with clouds of red blossoms. So I wrote the previous poem to record such an unusual sight. Again I was sent away, and now after fourteen years I have returned once more, this time to serve as director of the Bureau of Receptions. Once more I visited Hsuantu Temple, but not a single tree was standing\u2014nothing but useless weeds swayed in the spring breeze. Therefore, I am writing four more lines in anticipation of my next outing.\" The temple courtyard is a reference to that of the palace, and the Taoist priest to the former prime minister, Wang Shu-wen. Again Liu went too far in his sarcastic references. Not long after writing this poem he was sent off on another \"outing.\"\n\n## _Visiting Hsuantu Temple Again_\n\n_LIU YU-HSI_\n\nThe temple's vast courtyard is now home to moss\n\nvegetables flower where peach trees once bloomed\n\nwhere is the priest who planted the trees\n\nOld Mister Liu is back here again\n\n## 116\n\nWei Ying-wu (737\u2013792) grew up in Ch'ang-an, where he was recognized as a prodigy and given his first appointment at fifteen as a member of the heir apparent's retinue. After serving in a number of posts in the two capitals, he resigned in 765 and devoted himself to poetry, his family, and his friends. He returned to government service in 781 but was soon rusticated to a series of prefectures near the Yangtze. He wrote this poem in the spring of 785 while serving as prefect of Chuchou, one hundred kilometers west of Nanching. After spending the day wandering through the countryside across the Shangma River (Chuchou's West Stream) west of the city, he returned so late he had to pull himself across on the cable boat kept there for ferrying people back and forth. Commentators are divided over whether this is a simple descriptive poem or whether political implications lie beneath its surface, in which case the neglected plants would refer to virtuous officials and the orioles to those whose voices are heard but ignored. The last couplet recalls similar lines in the _Book of Odes_ that refer to spurned genius. It also suggests that political conditions are worsening, and that there is nothing left to do but save oneself. Then again, maybe this is just a poem about having the countryside to oneself on a rainy day.\n\n## _Chuchou's West Stream_\n\nWEI YING-WU\n\nI love unnoticed plants that grow beside a stream\n\norioles singing overhead somewhere in a tree\n\nat dusk the current quickens fed by springtime rain\n\nI pull myself across an unmanned country ferry\n\n## 117\n\nMany commentators interpret this poem by Su Shih (1037\u20131101) as a political critique: the alabaster terrace (which appears in verse 98 as part of the courtyard of the Queen Mother of the West) represents the Sung court, and the flowers represent the reforms of Wang An-shih (1021\u20131086). In this light, their shadows signify their perceived benefits, the boy stands for the poet or those who listened to his advice, and the sun stands for Emperor Shen-tsung (r. 1068\u20131085), who supported Wang's reforms. As soon as the emperor died, the reforms were overturned by Empress Dowager Hsuan-jen, who took over as Regent until 1093, when her son was old enough to rule. Unfortunately, Emperor Che-tsung (r. 1086\u20131100), represented here by the moon, reintroduced the reforms, and Su Shih was once more banished. While such symbolism might well lie beneath the surface, the poem does fine without such baggage.\n\n## _Flower Shadows_\n\nSU SHIH\n\nLayer upon layer on the alabaster terrace\n\nI tell the boy to sweep them up in vain\n\njust as the sun takes them all away\n\nthe full moon brings them back again\n\n## 118\n\nWang An-shih (1021\u20131086) grew up in Hangchou and spent most of his career in the Sung capital of Kaifeng, where he rose to the rank of prime minister during the reign of Shen-tsung (r. 1068\u20131085). In an attempt to relieve the economic plight of farmers, Wang introduced a series of financial reforms that divided the court and that actually increased the hardships of the rural poor due to their irregular implementation. But regardless of what his enemies thought of his politics, everyone agreed Wang was one of the country's greatest poets and essayists. He wrote this poem toward the end of his life, after he had retired to his country estate on North Mountain, just outside Nanching's Unicorn Gate. Nowadays it's called Burnished Gold Mountain (Tzuchinshan) and is better known for the mausoleum where Sun Yat-sen's remains are kept. Looking south, Wang's view included a small crescent-shaped lake next to Pipa Village at the foot of the mountain, the long, narrow moat along the city's east wall, and the embankment of the Chinhuai River at the southern edge of the city. The expression \"sweet-smelling plants\" ( _fang-ts'ao_ ) also referred to men of virtue. The phrase was made famous by Ch'u Yuan (340\u2013278 B.C.), whose insistence on the fragrance of righteousness, as opposed to the stench of self-interest, resulted in his banishment.\n\n## _North Mountain_\n\nWANG AN-SHIH\n\nNorth Mountain sends down green flooding the embankment\n\nthe city moat and crescent lake shimmer in the light\n\ncounting every falling petal I forget the time\n\nsearching for sweet-smelling plants I return home late\n\n## 119\n\nHsu Yuan-chieh (1196\u20131245) was a native of Shangjao in Kiangsi province. At the Southern Sung capital in Hangchou he was known for his outspoken views, but he still managed to serve as vice-minister in the Court of Imperial Sacrifices and also in the Ministry of Works. Here, he describes Hangchou's West Lake on a fine day in late spring. The lake was created during the previous four centuries by dredging its mud to create several small islands and a series of dikes. When the Sung dynasty transferred its capital to Hangchou, the city's new lake soon became the talk of the country. And despite its relatively small size\u2014four square miles\u2014it has remained the most famous lake in China.\n\n## _On the Lake_\n\nHSU YUAN-CHIEH\n\nOrioles chatter madly in trees of red blossoms\n\negrets converge on a lake of tall grass\n\neveryone loves a clear mild day\n\nboats return at dusk on waves of flutes and drums\n\n## 120\n\nThis was the seventh of a series of nine poems Tu Fu (712\u2013770) wrote in 761 when he was living just outside the provincial capital of Chengtu. The fifth poem in this series is verse 112. Here, he describes the simple yet remarkable scene around his cottage as spring gives way to summer.\n\n## _Inspired_\n\nTU FU\n\nWillow fuzz lines the path with bolts of white felt\n\nlily pads dot the stream with stacks of green coins\n\nthe offspring of bamboo haven't yet appeared\n\nducklings on the shore sleep beside their mother\n\n## 121\n\nWang Chia (b. 851) has left little information about himself other than that he was from Yungchi at the southwest corner of Shansi province and that he served as prefect of several outlying districts as well as vice-director of one of the bureaus in the Ministry of Rites. Although this poem ostensibly describes the poet's garden, many critics read it as a commentary on the fickleness of human sentiment: the center of attention one day, without friends the next. Even when they enjoyed the emperor's support, the only thing of which officials were certain is that it wouldn't last. But the only alternatives for someone educated in the Confucian classics and literary models was to teach in the village school, to serve as someone's secretary, or to farm a poor patch of ground\u2014hence the popularity of the hermit tradition in China. Two hundred years after this poem was written, Wang An-shih (1021\u20131086) took such a liking to it he rewrote it and included it in his own collection.\n\n## _Spring Clearing_\n\nWANG CHIA\n\nBefore the rain there were buds among the flowers\n\nwhen it cleared even those below the leaves were gone\n\nall the bees and butterflies flew across the wall\n\napparently spring has moved to the neighbor's\n\n## 122\n\nTs'ao Pin (1170\u20131249) was from Wenchou in Chekiang and served in a number of posts, including judicial commissioner of his home province, rectifier of omissions, and member of the Hanlin Academy. Apparently he, too, suffered the loss of imperial favor. While the poet sighs at the transience of spring, he also voices a complaint about the replacement of the melodious orioles at court with the croaking of lesser talents.\n\n## _Late Spring_\n\nTS'AO PIN\n\nNo one comes out to look at fallen flowers\n\na canopy of green slowly veils the sky\n\nthe orioles in the trees have finally stopped singing\n\nI only hear frogs in the grass-filled pond\n\n## 123\n\nChu Shu-chen (d. 1233) grew up in Hangchou in a well-off family and became adept at poetry and painting as a young girl. However, her education far outstripped the needs of her social position, and she spent most of her life in Suchou as the wife of a minor official who was often away on extended assignments and with whom she had little in common. Instead of sewing, she wrote poems and became known throughout the country for her plaintive verses on a woman's unequal place in Chinese society. In fact, during the Southern Sung dynasty her poetry was better known than that of her fellow woman poet Li Ch'ing-chiao (1084\u20131155). When she died, Chu's parents burned most of what she had written as an offering to her departed spirit. Still, many of her poems were so well known they survived, in an edition published several decades after her death. The intertwined branches represent the union of lovers, while the King of Green (Ch'ing Ti) is the god of spring.\n\n## _Falling Flowers_\n\nCHU SHU-CHEN\n\nWhenever intertwined branches bloom\n\nthe jealous wind and rain strip away their flowers\n\nif only the King of Green could perpetuate his rule\n\nthey wouldn't end up scattered across the moss\n\n## 124\n\nWang Ch'i (Sung dynasty) has left no information about himself other than what can be derived from a few poems. Here, he provides us with the sequence of attention in this small garden, which is probably somewhere south of the Yangtze, perhaps in the Southern Sung capital of Hangchou. It was during this period that the plum came to symbolize Chinese resistance to Khitan aggression and the loss of North China, and it was planted extensively in the Hangchou area. See, for example, Sung Po-jen's _Guide to Capturing a Plum Blossom._ As plum blossoms fall in early spring they are replaced in this garden by the ruby-tipped flowers of the begonia, which are joined by the white, grandiflora blooms of the Chinese raspberry, or _Rubus rosifolius_ var. _commersonii,_ both of which then give way to summer greenery. The name _t'ien-chi_ (some editions have _yao_ for _t'ien,_ apparently a copyist error) sometimes referred to the Chinese asparagus vine, but here it refers to the willow.\n\n## _Visiting a Private Garden in Late Spring_\n\nWANG CH'I\n\nOnce the plum casts off its faded charms\n\nfresh rouge graces the begonia\n\nwhich lasts until raspberry petals are gone\n\nand willow catkins hang across the mossy wall\n\n## 125\n\nLiu K'o-chuang (1187\u20131269) was from Putien in Fukien province and was one of the leading literary critics of his time. He was also its most prolific writer and was considered the most important poet of the Rivers and Lakes school of poetry. Although he served in important posts in the Ministry of Works and the Central Secretariat, he became disgusted by court politics and spent his final decade living in simple circumstances in the countryside. Liu was reportedly the original compiler of this anthology\u2014though his arrangement was topical rather than based on formal and seasonal categories, and this poem must have been added by a later compiler in honor of Liu's earlier efforts. The images of the first line are indebted to the poem \"Cutting Wood\" in the Hsiaoya section of the _Book of Odes_ : \"They come from dark valleys \/ they settle among treetops \/ twittering they call \/ in search of a friend.\"\n\n## _The Oriole Shuttle_\n\nLIU K'O-CHUANG\n\nThrough the willows to the treetops so full of feeling\n\ntwittering on the wing they sound like a loom\n\nLoyang in April is a tapestry of flowers\n\nbut it takes so much effort to weave\n\n## 126\n\nYeh Ts'ai (c. 1190\u20131240) was a native of Shaowu in Fukien province. Although he was a famous Confucian scholar of his day and director of the Palace Library in Hangchou, he makes fun here of his own immersion in the philosophy of change and his failure to pay attention to the changes taking place in the world around him. The _Yiching (Book of Changes_ ) was the primary text used by neo-Confucians during the Sung dynasty for developing something comparable to Taoist cosmology and Buddhist psychology. The book dates back to the beginning of the Chou dynasty, or 1100 B.C., and divides the dynamic forces in the world into sixty-four modes, each represented by a series of six lines and explained by a written commentary. But the sight of swallows busy building their nests, and willow fuzz in his inkwell, suddenly brings Yeh back to the mundane world. During the Sung the Chinese still sat on floor mats, and scholars used a small table or bookrest about one foot high for their texts and writing materials. The Chinese version of the inkwell consisted in a depression at one end of a slate slab, where ink accumulated once it was ground with drops of water on the slab's surface. The small window suggests poverty. Often the mouth of a broken pottery container was used for the frame. It also suggests the narrow-minded focus on studies characteristic of those preparing for the civil service exams.\n\n## _Events of Late Spring_\n\nYEH TS'AI\n\nHouse swallows swoop in pairs above my desk\n\nwillow fuzz floats in my inkwell\n\nbelow my small window reading the _Book of Changes_\n\nI wonder how long spring has been over\n\n## 127\n\nLi She (fl. 830) was the son of a prominent Loyang family, but he preferred a simpler, unfettered life. When he was still a young man he and his younger brother, Li Po, lived as recluses on Lushan, overlooking the Yangtze. Later, he was recommended to the court and served briefly in Ch'ang-an as an advisor to the crown prince. But he did not get along with those in power and was banished, called back, and then banished again. When he wrote this poem he was serving not far downriver from Nanching in Chenchiang. In the _Chuantangshih_ this poem is entitled \"Written at the Monks' Quarters of Holin Temple.\" This temple was located on Yellow Crane Mountain outside Chenchiang's South Gate, beyond the estates of the town's well-to-do. Two hundred years later, one of China's greatest calligraphers, Mi Fei, became so fond of this temple that he asked to be buried by its front gate. The temple is gone but Mi Fei's grave is still there.\n\n## _Climbing a Mountain_\n\nLI SHE\n\nAll day I feel lost as if drunk or in a dream\n\nthen I hear spring is over and force myself to climb\n\npassing a bamboo courtyard I meet a monk and talk\n\nand spend another afternoon beyond this floating life\n\n## 128\n\nHsieh Fang-te (1226\u20131289) was from Yiyang in Kiangsi province. Prior to his imprisonment in Beijing he served in several posts in the Southern Sung capital of Hangchou, which is still famous for its silk brocade. During this period, taxes were paid in lengths of silk, and each household had a quota to meet and someone assigned to fill it. To the Chinese, the cuckoo's cry sounds like the phrase _pu-ju kuei-ch'u_ (\"better go home\"), and during spring it often cries all night. In an effort to guard against thieves and fires, Chinese in urban areas divided the night into five two-hour watches. Here, the servant in charge of the household silk quota has been kept awake by the cuckoo's cry and gets up after midnight to feed the silkworms, which were kept on wicker trays in a warm part of the house and fed a diet of mulberry leaves until they spun themselves into their cocoons. But not only does this silkmaid have a hard life, so does her mistress, who has to stay up all night entertaining others. This poem, as with verse 113, was not part of Liu K'o-chuang's original anthology and was probably added to include something from one of the major writers of the Sung dynasty's final years.\n\n## _Lament of the Silkmaid_\n\nHSIEH FANG-TE\n\nUntil the fourth watch the cuckoo cries\n\nshe gets up to see if the silkworms have leaves\n\nsurprised at the moon between rooftops and willows\n\nand her mistress not back from the party\n\n## 129\n\nHan Yu (768\u2013824) was born into a family of scholars in Menghsien in Honan province. Orphaned at the age of two, he grew up in the family of his elder brother, under whose guidance he received his early education while moving from post to post in the provinces. After finally passing the civil service exam he enjoyed success in the central government\u2014where he eventually rose to the rank of chief minister of the Secretariat\u2014and also in the literary realm, where he was admired as one of the dynasty's greatest poets and essayists. Han was an ardent follower of Confucian tradition but was adept at adapting it to the times. In his poetry he strove to be clear and concise, and he often used the language of ordinary speech to this end. Here, he portrays a scene in late spring, which many commentators read as a comparison of ostentatious sycophants to simpler officials who have no other means of gaining the court's attention than through their pure-hearted virtue.\n\n## _Late Spring_\n\nHAN YU\n\nEvery plant and tree knows spring will soon be gone\n\na hundred pinks and purples compete with their bouquets\n\nwillow fuzz and elm pods lack such clever means\n\nthey only know how to fill the sky with snow\n\n## 130\n\nYang Wan-li (1124\u20131206) was from Chishui in Kiangsi province. He was so devoted to literature, it was said, that he favored a good poem over a beautiful woman. Despite such devotion, or perhaps because of it, he managed to become a member of the Institute of Academicians and, eventually, director of the Palace Library. However, his frequent disagreements with official policy resulted in a career spent mostly in the provinces. In 1178, while serving on the Grand Canal as prefect of Changchou, he came to a new understanding of poetic technique. Breaking with the more formal style favored by members of the ruling elite, Yang created poems that were known for their clarity and humor and their use of everyday speech. He retired to his hometown in 1192, and by the time of his death he was considered one of the greatest poets of the Southern Sung. In the _Sungshihchiao_ this poem is entitled \"Hiking Up Wanhua Valley at Dawn to See Begonias.\" The East Wind rises with the sun as it returns from its annual southerly migration in spring.\n\n## _Spring Lament_\n\nYANG WAN-LI\n\nI was hoping joy would overflow this spring\n\nas usual I've wasted another East Wind\n\nit's been years since I could look at flowers\n\nif not because of cares then illness\n\n## 131\n\nWang Feng-yuan (1032\u20131059), aka Wang Ling, was a native of Yangchou, where the Grand Canal meets the Yangtze. Although he was quite poor he was such a brilliant student that he came to the attention of Wang An-shih, before he became prime minister. Wang An-shih admired his talents and virtue so much he considered him an equal and arranged for him to marry his wife's younger sister. Unfortunately, Wang Feng-yuan did not live long enough to realize his potential: he died of an illness before he was thirty. The swallow is the bird most commonly associated with summer, while the oriole and cuckoo are the birds of early and late spring. The East Wind is the wind of spring, just as the West Wind is the wind of autumn.\n\n## _Seeing Off Spring_\n\nWANG FENG-YUAN\n\nIn April flowers fade and fall and more appear\n\nswallows fly below the eaves back and forth all day\n\nthe cuckoo cries at midnight as if its voice would break\n\nconvinced it can still call the East Wind back\n\n## 132\n\nChia Tao (779\u2013843) was a native of the Beijing area but moved to Loyang and later Ch'ang-an as a young man. At first he chose the life of a monk but then changed his mind and decided he would rather devote his life to poetry. Although he failed to attain even minor distinction as an official, he became well known for his plaintive, well-crafted verses, some of which took years to write, and his poetry greatly influenced later generations of poets. Han Yu (768\u2013824), who was a major influence on him, once wrote, \"When Meng Chiao [751\u2013814] died and was buried at Peimang [the graveyard outside Loyang] \/ the sun moon and stars suddenly felt bored \/ worried that literature had come to an end \/ Heaven gave the world Chia Tao.\" The dawn bell is that of a monastery, which will soon rouse Chia Tao's former colleagues in their quest to transcend their own attachments. In some editions this poem is entitled \"For Judge Liu on the Last Day of April.\" In the Chinese calendar April (the third lunar month) is the last month of spring.\n\n## _Seeing Off Spring on the Last Day of April_\n\nCHIA TAO\n\nWhen April reaches its thirtieth day\n\nyour wind and light forsake a poor poet\n\nI don't want to sleep with you tonight\n\nuntil the dawn bell you're still spring\n\n## 133\n\nSsu-ma Kuang (1019\u20131086) was born in Hsiahsien in Shansi province and grew up in Loyang. In the course of a distinguished career at the Sung capital in Kaifeng, he led the political faction that opposed the financial reforms of Wang An-shih. Not long after Shen-tsung (r. 1068\u20131085) ascended the throne, Ssu-ma Kuang was demoted and sent to what was once Ch'ang-an, where he wrote this poem. He was allowed to return to his home in Loyang the following year (1071) on the condition that he avoid politics, which he did until the young Che-tsung ascended the throne in 1086 and made him prime minister. Ssu-ma Kuang died later that same year. Here, he looks out his door during the summer he spent in Ch'ang-an and remarks on the seasonal changes that remind him of life at court. These mountains south of Ch'ang-an were a common metaphor for imperial and dynastic long life, while the willow fuzz of late spring represents officials who float on any passing breeze, and the sunflowers of early summer stand for loyal officials such as the poet himself. According to this analogy the summer sun would be the newly crowned emperor, Shen-tsung, for whom Ssu-ma Kuang has not yet given up hope.\n\n## _Away from Home at the Beginning of Summer_\n\nSSU-MA KUANG\n\nIn May when it's mild and the rain finally stops\n\nthe Chungnan Mountains appear in my door\n\nwillow fuzz no longer swirls in the wind\n\neverywhere sunflowers lean toward the sun\n\n## 134\n\nChao Shih-hsiu (d. 1219) was a native of Yungchia in southern Chekiang province and was one of the founders of what became known as the Rivers and Lakes school of poetry, which championed the works of such poets as Chia Tao and the subjects of everyday life, and which opposed the use of refined language and courtly themes favored by the Kiangsi school. The Chinese still call this period (when early summer rains coincide with the ripening of plums) _mei-yu_ (plum rains), which keeps everyone but farmers indoors and keeps friends away. The game of _wei-ch'i_ (Japanese: _go_ ) originated in China more than three thousand years ago: it involves the placement of small black and white stones on a chessboard of intersecting lines, with the object being the occupation of territory and removal of the opponent's pieces.\n\n## _Waiting for a Friend_\n\nCHAO SHIH-HSIU\n\nDuring plum season it rains on every roof\n\naround the grassy pond frogs are everywhere\n\nwaiting after midnight for a friend who doesn't come\n\nI play a game of chess until the lamp goes out\n\n## 135\n\nWhen Yang Wan-li (1124\u20131206) was serving as prefect of Changchou, he experienced a kind of enlightenment he equated with that of Zen and began writing in a simple, spontaneous style. Eventually he produced over 20,000 poems, some 4,000 of which survive. Here, he has been taking a midday nap and quenches his thirst with some plums. Gazing outside his windows, screened with silk the density of cheesecloth to keep out insects in summer, he watches his children chase willow fuzz, which is itself a sign that spring is over. Such are the pleasures of early summer for those who don't chase fame or fortune.\n\n## _Waking Up in Early Summer_\n\nYANG WAN-LI\n\nThe sour trace of plums squirts between my teeth\n\nthe light green of bananas fills my window screen\n\nwaking up at noon without a thought or care\n\nI sit and watch my children chasing willow fuzz\n\n## 136\n\nTseng Chi (1084\u20131166) was born in Kanchou near the southern border of Kiangsi province and grew up in Loyang. He is better known to history as the teacher of the Sung-dynasty lyric poet Lu Yu (1125\u20131210), but during his career he served as judicial commissioner of Kiangsi and Chekiang and eventually headed the Ministry of Rites and the Secretariat. At one point he was forced into temporary retirement when Ch'in Kuei was prime minister, but he returned upon Ch'in's death. On this occasion he is leaving the Southern Sung capital of Hangchou to take up his new post as prefect of Chuchou at the southwest corner of Chekiang province, which most officials would have viewed as a demotion. Normally, as noted in verse 134, this season is marked by constant light rains. But this is a poem about finding good fortune where one doesn't expect it\u2014the silver lining in rustication to Chuchou. Taking advantage of good weather, Tseng travels from Hangchou by boat up the Fuchun and Chuchiang Rivers as far as he can go and then continues on foot over Sanchu Mountain for the final part of the journey. As he nears his destination he hears the welcome songs of like-minded poets.\n\n## _On the Sanchu Road_\n\nTSENG CHI\n\nPlums are yellow and days are sunny\n\nwhere the river turns shallow I take the trail\n\nthe canopy of green doesn't get less shady\n\nthe occasional sound of orioles though is new\n\n## 137\n\nChu Shu-chen (fl. 1200) grew up in Hangchou in a family that produced a number of officials. Her uncle was the great neo-Confucian philosopher Chu Hsi (1130\u201312o0). She was somewhat of a prodigy herself as a young girl, and as a result of her family background she received an unusually good education. After she married she moved to Suchou. But her marriage was not a happy one and her husband was often away. She was also a skilled musician and calligrapher, but her fame rested on her poems, especially those employing the voice of the abandoned woman. The houses of the rich included a series of courtyards, and the place where the wife lived would have been in the back, hidden from the view of visitors. The pairs of birds remind Chu of her own isolation, and the disappearance of the signs of spring of her own fading beauty. This part of China, near the Yangtze Delta, is known for its hot, humid summers.\n\n## _Impressions_\n\nCHU SHU-CHEN\n\nSwaying bamboo shadows shade my secluded window\n\nsummer birds chatter in pairs in the sunset\n\nbegonias have faded and willow fuzz has flown\n\nthe enervating days are starting to get longer\n\n## 138\n\nTai Min (fl. 1170) was a native of Huangyen in Chekiang province and the father of Tai Fu-ku, who was a prominent poet of the Rivers and Lakes school of poetry. Neither the father nor the son (to whom some editions attribute this poem) served in the government. Here, the poet spends a day following the sun across a friend's garden\u2014from the eastern half that greets the sun to the western half that sees it off. The loquat looks similar to a small apricot and ripens in early summer, about the same time as the plum. Both are often enjoyed with rice wine or stronger spirits. It was during the Sung dynasty that garden landscape became a major art form, especially in such cities as Suchou and Hangchou. One mainland Chinese commentator complains that there is no social value in a poem such as this. Indeed, one cannot put a price on it.\n\n## _Visiting Chang's Garden at the Beginning of Summer_\n\nTAI MIN\n\nDucklings in a pond of deep and shallow water\n\nplum-season weather partly clear and cloudy\n\ntaking wine to the East Garden we get drunk in the West\n\nbeneath a loquat tree picked clean of gold\n\n## 139\n\nHuang T'ing-chien (1045\u20131105) was from Hsiushui in Kiangsi province and was ranked with his mentor, Su Shih (1037\u20131101), as one of the great calligraphers and poets of the Sung. He was also the founder of the Kiangsi school of poetry and was known for his layered allusions and irregular rhythms. Although some critics consider him too careful in his poetic constructions, and others judge his poetry superficial, he responds here to a poem by Su Shih with a masterful treatment of the simplest of scenes. During his banishment in 1080 to Huangchou, just downriver from Wuchang (Ochou), Su wrote his famous \"Red Cliff Ode,\" which includes the lines: \"the cool breeze on the river \/ the bright moon above the mountains \/ the ear turns things into sounds \/ the eye turns things into sights \/ the endless treasury of these creators \/ you and I enjoy together.\" Huang also suffered several banishments. On this occasion he was being rusticated farther up the Yangtze to serve as prefect of Chiangling. Stopping in Wuchang, he wrote this poem while looking out onto the lily-covered expanse of the city's South Lake. Political overtones rise to the surface at the end, where the poet turns his back on the human politics of the northern capital. The South Wind became a common metaphor for benevolent government after Emperor Shun (r. 2300 B.C.) composed the lines: \"The touch of the South Wind can relieve my people's pains.\" Also, while the sun's ecliptic moves north during the summer, the path of the full moon moves south. Some editions wrongly attribute this poem to Wang An-shih.\n\n## _Written at the South Tower of Ochou_\n\nHUANG T'ING-CHIEN\n\nMountain light meets water light everywhere I look\n\nfrom the railing I can smell miles of water lilies\n\nthe soft wind and bright moon aren't controlled by man\n\ntogether from the south they bring something cool\n\n## 140\n\nKao P'ien (821\u2013887) was from Beijing. After serving in senior military posts along China's northern, western, and southern frontiers, he was appointed military commissioner of the Huainan section of the Yangtze in 879 and successfully defended the key city of Yangchou against the rebel forces of Huang Ch'ao. However, he also intentionally avoided attacking the rebels in favor of protecting his own position, thus ensuring the end of the T'ang dynasty. Kao later died at the hands of Pi Shih-to, another bandit leader. Here, he anticipates the end of the dynasty, which is represented by the upside-down pavilion, and the beginning of a new dynasty, which is suggested by the moving curtain and rising breeze. The prickly rose, or _Rosa acicularis,_ is known for its perfume. Here, its scented flowers refer to men of greater virtue than those who filled the court when this poem was written. No doubt Kao had in mind his friends and associates in the Yangchou area.\n\n## _A Mountain Pavilion on a Summer Day_\n\nKAO P'IEN\n\nThe shade from trees is dense and summer days are long\n\nan upside-down pavilion is mirrored in the pond\n\ncrystal curtains shimmer in the faintest wind\n\na trellis of roses fills the courtyard with perfume\n\n## 141\n\nFan Ch'eng-ta (1126\u20131193) was from Suchou in Kiangsu province. He served in a number of posts but is chiefly remembered for his staunch refusal, while serving as ambassador to the Chin Tartars in North China, to betray national interests even when threatened with death. He was also ranked as one of the four greatest poets of the Southern Sung dynasty and was known for his objective, realistic style and his focus on the everyday events of farm life. Eventually, court politics became too much for him, and in 1183 he retired to the countryside just outside Suchou. This is one in a series of sixty poems he wrote not long afterwards entitled \"Occasional Poems on the Four Seasons of Farm Life,\" a series that served as the model for many later poets. The mulberry trees produced the leaves that fed the silkworms that made the silk that paid the taxes that supported the imperial lifestyle.\n\n## _A Farm Family_\n\nFAN CH'ENG-TA\n\nWeeding fields at sunup twisting hemp at night\n\nvillage boys and girls all have their chores\n\neven little children too small to plow or weave\n\nlearn to plant melons in the shade of mulberry trees\n\n## 142\n\nSome editions attribute this poem to Weng Chuan (fl. 1250), one of the Four Poet Laureates of Yungchia. The white water here is the result of the \"plum rains\" that fall in May and June (the fourth lunar month) in Central and Southern China. During this time of year the cuckoo's cry, which normally sounds like _pu-ju kuei-ch'u_ (\"better go home\"), sounds like _k'uai k'uai ch'a-ho_ (\"hurry up and plant rice\"). About the same time that silkworms wake from their winter slumber, farmers transplant rice sprouts from seedlings started a month or more earlier.\n\n## _Village Events_\n\nFAN CH'ENG-TA\n\nGreen covers hills and plains and white fills the rivers\n\nin the cuckoo's cry there's a misty rain\n\nidle folks are rare the Fourth Month in a village\n\nonce silkworms are fed it's time to transplant rice\n\n## 143\n\nChu Hsi (1130\u20131200) was born in Fukien province but made his home in Anhui. He enjoyed a relatively successful career in the government, but he was far better known as one of the leading proponents of neo-Confucianism, which dominated intellectual life during the Sung dynasty. In fact, ever since the Sung, every student's education in the Confucian classics has begun with Chu Hsi's commentaries. The pomegranate arrived in China from Central Asia via the Silk Road more than a thousand years before this poem was written. But within a few hundred years of its arrival it became so popular that people started calling the fifth lunar month Pomegranate Month. This is when its red fruit begins to form, while the flower is still in bloom. The fruit itself often appears in Chinese art as a symbol for a womb that contains countless offspring. The poet is also reminded that, like the pomegranate, his own neo-Confucian ideas are full of promise, and yet the country's elite (on horseback or in horse-drawn carts) do not pay them the attention they deserve. Some editions attribute this poem to Han Yu (768\u2013824).\n\n## _On the Pomegranate Flower_\n\nCHU HSI\n\nPomegranate flowers brighten eyes in June\n\nas soon as they appear their fruit begins to form\n\nbut neither carts nor horses visit this poor place\n\nwhere ruby blossoms lie upon the emerald moss\n\n## 144\n\nLei Chen ( _lei_ means \"thunder\" and _chen_ means \"shake\") lived during the Sung dynasty, but nothing else is known about him\u2014which is surprising, as commentators marvel at the combination of effortlessness and craftsmanship in this poem. Perhaps he was a friend of the compiler and preferred anonymity. Among the duties assigned children in farming communities was the tending of oxen (in North China) or water buffaloes (in the South). The back of the water buffalo, though, is much broader and provides a better seat. The transverse flute arrived in China in prehistoric times and apparently originated with one of the nomadic groups along the country's northwest border, where eagle bones were used before the Chinese replaced them with bamboo.\n\n## _Village Dusk_\n\nLEI CHEN\n\nGrass lines the pond and water lines the bank\n\nthe sun sinks in the mountains and in the icy ripples\n\na herdboy returns on the back of his ox\n\naimlessly blowing a flute to no tune\n\n## 145\n\nWang An-shih (1021\u20131086) was the most controversial political figure of the Sung dynasty, as well as one of its greatest calligraphers, writers, and poets. He wrote this near the end of his life, while living in retirement outside Nanching at the foot of Tzuchinshan. Political undertones seem to have finally disappeared from his verse as he writes a poem to his neighbor, Yang Te-feng, whose pen name was Mister Lakeshade (Hu-yin Hsien-sheng). Sweeping the path before one's house was normally done when expecting a visitor\u2014in this case, the poet. This was the first of two poems.\n\n## _Written on Mister Lakeshade's Wall_\n\nWANG AN-SHIH\n\nBelow your thatched eaves you've swept away the moss\n\nyou yourself planted the path of flowering trees\n\na stream guards your fields encircling it with green\n\ntwo peaks swing open and welcome the blue\n\n## 146\n\nLiu Yu-hsi (772\u2013842) was from the Grand Canal town of Hsuchou in Kiangsu province. He rose to the position of censor at court but was banished for siding with a group of reformers opposed to the power of the eunuchs, the military, and the aristocracy. This poem is directed against the latter group, though it is conveniently set in the past to avoid political repercussions. In 826 Liu was sent south to serve as prefect of Hohsien. This was one of five poems he wrote while visiting Nanching, not far downriver from his post. During the Six Dynasties (222\u2013589), Red Bird Bridge spanned the Chinhuai River just outside Nanching's South Gate, and Black Robe Lane was just inside the gate. During the Wu dynasty (222\u2013277) this was the location of Black Robe Palace, named for the black-robed officials who hovered around the court. During the subsequent Chin dynasty (278\u2013419) this was where the powerful Wang and Hsieh clans lived. It was also the location of Laiyentang (Arriving Swallow Hall), where they welcomed visiting relatives. Both of these clans were wiped out at the beginning of the T'ang (618\u2013906). Here, black-robed swallows replace the officials.\n\n## _Black Robe Lane_\n\nLIU YU-HSI\n\nWildflowers bloom by Red Bird Bridge\n\nthe fading light slants down Black Robe Lane\n\nwhere swallows once lived among Hsiehs and Wangs\n\nthey now frequent homes of ordinary people\n\n## 147\n\nWang Wei (701\u2013761) was born in Taiyuan in northern Shansi but grew up in Yungchi at the southern end of the same province. Although he rose to the post of deputy prime minister, he also suffered several banishments. Here, he sees off a friend who was banished to Anhsi, the westernmost part of the Silk Road under Chinese control. As the Silk Road left the Tunhuang oasis it divided into a northern and a southern route, both of which skirted the Taklamakan Desert. Jade Pass marked the beginning of the northern route through Hami and Turfan to Kuche, where the main garrison of the Anhsi Protectorate was located. Yang Pass was twenty kilometers to the south and marked the beginning of the southern route to Khotan and Kashgar. Weicheng (Wei City) was just west of Ch'ang-an on the north shore of the Wei River, near the former Ch'in dynasty (221\u2013207 B.C.) capital of Hsienyang. This was where people held going-away parties for friends heading west\u2014hence the dust. The Chinese word for willow is a homophone of another word meaning \"stay behind.\" Thus, the Chinese broke off willow catkins to give to friends as a parting memento. The custom apparently began in Ch'ang-an, where the willow's new catkins appeared about the same time that snow melted in the passes. This poem was later set to music and called \"The Song of Weicheng.\" According to Su Shih, the first line was sung once and the last three lines three times.\n\n## _Seeing Off Yuan Er on a Mission to Anhsi_\n\nWANG WEI\n\nMorning rain dampens the dust in Weicheng\n\nnew willow catkins turn an inn green\n\ndrink one more cup of wine my friend\n\nwest of Yang Pass there's no one you know\n\n## 148\n\nLi Pai (701\u2013762) wrote this poem in 758 when he stopped to visit Wuchang's Yellow Crane Tower after being released from prison for his involvement in the revolt of the emperor's brother. In some editions this poem is titled \"Written on the North Gazebo Stele.\" On one of the tower's stories were four gazebos from which to view the four directions. The north gazebo faced the place where the Han River joins the Yangtze from the west. Li Pai is on his way south to Changsha, the provincial capital of Hunan, and looks back up the Han toward Ch'ang-an. This tower was built in A.D. 223, shortly after the end of the Han dynasty, in honor of the Taoist immortal Fei Wen-wei, who rode off from here on a yellow crane and who sometimes returned to play his jade flute. \"Plum Blossoms Fall\" was the name of a tune for the flute. The plum is said to transcend winter's adversity and normally blooms during New Year. In the background of this poem is the Han-dynasty poet Chia Yi (200\u2013168 B.C.), who was unjustly banished to Changsha for his frank and unwelcome advice and with whom Li Pai clearly identifies.\n\n## _On Yellow Crane Tower Hearing a Flute_\n\nLI PAI\n\nSuddenly an exile on the way to Changsha\n\nlooking back toward Ch'ang-an I don't see a soul\n\nfrom Yellow Crane Tower I hear a jade flute\n\nplum blossoms fall in this city in June\n\n## 149\n\nCh'eng Hao (1032\u20131085) was from Loyang. He and his younger brother, Ch'eng Yi (1033\u20131107), were among the earliest and most eloquent spokesmen for the Sung dynasty's neo-Confucian philosophy, which incorporated elements of Taoism and Buddhism. Here, the older brother demonstrates the detachment of such understanding, as he stops at a Taoist temple just outside Yangchou where the Grand Canal meets the Yangtze. Arriving too late to cross the river, he watches the sun setting on the hills along both shores and notices that the floating duckweed, whose white blooms line the Yangtze in early autumn, has already finished flowering. Duckweed also represents the poet's own rootless existence as an official, subject to imperial whim or displeasure. The region along both shores of the Yangtze was controlled at one time or another by the ancient state of Ch'u, one of its famous poets being Sung Yu (290\u2013223 B.C.), who began his \"Nine Arguments\" with the line: \"How mournful the season of autumn.\"\n\n## _Written at Huainan Temple_\n\nCH'ENG HAO\n\nTraveling north heading south stopping where they can\n\nduckweed blooms depart Ch'u waterways by fall\n\na man of the Way doesn't mourn autumn\n\nlet the evening hills share each other's sorrow\n\n## 150\n\nYang P'u (921\u20131003) was from Chengchou in Honan province and spent a good deal of his life in the countryside near the Yellow River, where he was known for riding everywhere on his ox. Trying to avoid the conflicts of the Five Dynasties Period (907\u2013960), he lived as a recluse for several years on Sungshan. But he was finally called out of seclusion at the beginning of the Sung dynasty, and he served briefly as an official in the capital of Kaifeng. According to an ancient Chinese folktale, the Emperor of Heaven had a daughter who wove heavenly cloth with her golden loom. Realizing she was lonely, he arranged for her to marry a herdboy. But after they were married she forgot all about weaving. This made the emperor so angry he banished the lovers to different shores of the Milky Way and only allowed them to meet one night a year, on the seventh night of the seventh lunar month. But instead of making love this night, the Herdboy asks his wife to show the world how to weave celestial cloth. Until modern times, it was customary for women to make offerings to the Weaving Maid on this night in exchange for her blessings on their own handiwork. In contrasting the simple generosity of the celestial couple with the wily ways of the world, the poet reflects his own view of the times in which he lived.\n\n## _The Seventh Night_\n\nYANG P'U\n\nI've never understood the Herdboy's thoughts\n\nwhy he asks the Weaving Maid to work her golden loom\n\nevery year offering her skill to the world\n\nunaware the world has enough skills already\n\n## 151\n\nChu Hsi (1130\u20131200) wrote this poem around 1180 while traveling through the mountains to Kao-an, southwest of the capital of Kiangsi province. Chu Hsi was serving as governor at the time. The neo-Confucian point of view is as clear as the moonlight in this poem. Everything is part of everything else, and yet everything has its own nature. Red dust ( _hung-ch'en_ ) was a term used by Chinese Buddhists when referring to the world of sensation. Some editions attribute this poem to Ch'eng Hao.\n\n## _Autumn Moon_\n\nCHU HSI\n\nA pure stream flows past a jade green peak\n\nthe water and the sky have the clear look of fall\n\ndozens of miles from the world of red dust\n\nwhite clouds and yellow leaves appear without end\n\n## 152\n\nLiu Han (fl. 1190), aka Liu Wu-tzu, was from Changsha in Hunan province, but little else is known about him. In addition to their calendar of twelve lunar months, the Chinese also divide the year into twenty-four solar periods. Autumn at the Gate ( _Li-ch'iu_ ) occurs at the end of the first week of August. Most commentators are convinced the jade screen here refers to an elegant, jade-colored silk or paper screen\u2014as it does in the next poem. But why would anyone put up a screen to block the wind at this time of year, when the first cool breeze of autumn is so welcome? It makes more sense to read it as a metaphor for a cloudless (and duckless) sky. Paulownias are planted for their shade. Their huge leaves are a foot or two across, and when they fall at the beginning of autumn, they all fall at once.\n\n## _Autumn at the Gate_\n\nLIU HAN\n\nThe quacking is gone the jade screen is empty\n\nmy pillow feels cool fanned by the wind\n\nautumn sounds wake me but where are their traces\n\npaulownia leaves cover the steps in the moonlight\n\n## 153\n\nTu Mu (803\u2013852) was born in Ch'ang-an to a family that had lost its wealth and power by the time he was a boy. Although he began his studies late, he excelled in the literary arts and served in a number of posts, though of ever-decreasing importance. Here, he expresses his disappointment through the persona of one of the emperor's concubines. With the passing of summer, her fan has lost its function, and her ability to affect her own life is reduced to trying to swat fireflies. The \"streets of Heaven\" refers to those in the forbidden precincts where only the Son of Heaven and his concubines and eunuch attendants lived. During the hours after sunset on the seventh night of the seventh lunar month, the Weaving Maid and Herdboy stars (Vega and Altair) are directly overhead on either side of the Milky Way. And though they only meet one night a year on a bridge formed by the wings of magpies, this is still more frequent than this woman's rendezvous with the emperor. Likewise, Tu Mu, who was known for his concern with social injustice, feels bereft of access to the royal presence and the means to effect change. The silver lantern ( _chu,_ \"candle,\" is a euphemism here for the celestial lanterns) is the moon, and the chill is that of loneliness, not the weather\u2014which is still hot this time of year.\n\n## _Autumn Evening_\n\nTU MU\n\nSilver lantern autumn light chills her painted screen\n\nshe swats at passing fireflies with her small silk fan\n\nat night the streets of Heaven look as cool as water\n\nlying down she gazes at the Weaving Maid and Herdboy stars\n\n## 154\n\nSu Shih (1037\u20131101) wrote this poem in 1078 while serving as prefect of Hsuchou on the Grand Canal. He was joined on this occasion by his younger brother, Su Ch'e (1039\u20131112), with whom he had an unusually close relationship. Both brothers held numerous posts during their careers, and they spent much of their lives out of favor and far from the capital and their families. Still, as the last line suggests, both brothers made every effort to be together on this night, which was the eighth full moon and second in importance only to the lunar New Year as a time of family celebration. The wheel of jade that moves through the sky is the moon, and the Silver River, also known as the Long River or River of Heaven, is the Milky Way. Su composed this poem to be sung to the tune of \"Yang Pass.\"\n\n## _Mid-Autumn Moon_\n\nSU SHIH\n\nAs evening clouds withdraw a clear cool air floods in\n\nthe jade wheel passes silently across the Silver River\n\nthis life this night has rarely been kind\n\nwhere will we see this moon next year\n\n## 155\n\nChao Ku (815\u2013856) was born in Huai-an in Kiangsu province. After passing the civil service exam in 842, he served in a number of posts and finally as commandant of Weinan, to which he eventually retired. Weinan is located on the south shore of the Wei River just east of Ch'ang-an and served as the capital's entrep\u00f4t for goods coming up the Yellow River. Towers were built along rivers in China to monitor boat traffic. But they were also used for enjoying the view, especially on moonlit nights. Tu Mu (803\u2013852) was so taken with the line in another Chao Ku poem (\"I lean from a tower listening to a flute\"), he gave Chao the nickname \"Tower Leaning Chao.\"\n\n## _Reflections at a River Tower_\n\nCHAO KU\n\nAlone on a river tower my thoughts full of sorrow\n\nthe moonlight like the water the water like the sky\n\nwhere is the person with whom I shared the moon\n\nthe view isn't quite the same as last year\n\n## 156\n\nLin Sheng (fl. 1180) was a native of Putien in Fukien province. Lin witnessed the transfer of the Sung capital from Pienchou (Kaifeng) in North China to Linan (Hangchou) south of the Yangtze, and here he describes the carefree decadence of the new Sung capital and laments the lack of concern among the \u00e9migr\u00e9 officials and nobles for the loss of the old capital to the Khitan Tartars. The Khitans destroyed Pienchou in 1127 and the Sung court fled south, first to Nanching and then, when Nanching proved too difficult to defend, to Hangchou, which was renamed Linan and made the new capital in 1129. It is now more than fifty years later, and who thinks of returning to Kaifeng? But the last line is not only a lamentation; it is also a warning. Hangchou's West Lake was surrounded on three sides by hills, and its shores were lined with one pavilion after another. Its most famous lakeshore restaurant is still called Lou-Wai-Lou (Pavilion Beyond Pavilions).\n\n## _Written at an Inn in Linan_\n\nLIN SHENG\n\nHills upon hills pavilions beyond pavilions\n\nthe singing and dancing never stop at West Lake\n\nrevelers are so drunk on the warmth\n\nthey mistake Hangchou for Pienchou\n\n## 157\n\nYang Wan-li (1124\u20131206) was from Kiangsi province and served in a number of minor posts in the provinces and in the capital of Hangchou. While serving as prefect of the Grand Canal town of Changchou, he came to a new understanding of poetic technique that he equated with the enlightenment of Zen. He threw away his earlier work and began writing a dozen or more new poems every day. When he died he left behind over 20,000 poems, of which some 4,000 are still extant. Chingtzu Temple was one of the largest Buddhist monasteries in Hangchou and was located on the south shore of West Lake. Yang's friend Lin Tzu-fang had been serving in the capital's Palace Library but is leaving on some unspecified journey. Lotus leaves rise four or five feet above the water on long stems, and their flowers, which are various shades of pink, rise to about the same height before opening.\n\n## _At Chingtzu Temple Seeing Off Lin Tzu-fang at Dawn_\n\nYANG WAN-LI\n\nFinally West Lake in the month of July\n\nthe scene isn't like any other time of year\n\nsky-high lotus leaves are an endless blue-green\n\nand their sunlit flowers are a different kind of red\n\n## 158\n\nSu Shih (1037\u20131101) wrote this poem in 1073 when he was serving as controller-general of Hangchou, which would not become the capital of the Southern Sung for another fifty years. Hsi-tzu (Western Beauty) was another name for Hsi Shih, a famous beauty of the state of Yueh. Following a disastrous defeat by the state of Wu in 494 B.C., the King of Yueh (r. 496\u2013465 B.C.) gave Hsi Shih to the King of Wu in hopes that she would distract him from state affairs, an assignment in which she succeeded famously. As a result of this poem, some people started calling West Lake \"Western Beauty Lake.\" According to an account in _Lengchi yehhua,_ \"When Tung-p'o [Su Shih's pen name] was serving in Hangchou, there was not a day he did not visit West Lake.\"\n\n## _Drinking on the Lake As It Clears Then Rains_\n\nSU SHIH\n\nThe shimmering waves are translucent when it clears\n\nthe mist-veiled hills are transcendent when it rains\n\nI think of West Lake as the Beauty of the West\n\nequally lovely in powder or paint\n\n## 159\n\nChou Pi-ta (1126\u20131204) was from Chi-an in Kiangsi province. As an official, he was known for his unflinching honesty and willingness to criticize policies he thought harmful. On this occasion, which would have been one of the great events in any official's life, he was summoned to a private audience with Emperor Hsiao-tsung (r. 1163\u20131189). At the time, Chou was serving in the Hanlin Academy, which was in charge of preparing official documents. It was also called Jade Hall, and during the T'ang, when it was in Ch'ang-an, it acquired the appellation Purple Myrtle Court, after the flowers planted around its courtyard and along the walkways outside. After returning from his audience Chou sees in the crescent moon his own prospects waxing as well. In 1189 he was appointed deputy prime minister; hence, this poem would have been written sometime earlier. Pagoda trees ( _Sophora japonica_ ) were known for their shade and also for their pagoda-like panicles of fragrant yellow blossoms that bloomed during summer. Thus, they were often planted in front of temples and palaces whenever a deciduous tree was favored over an evergreen. The crows also refer to black-robed officials returning from the palace.\n\n## _Retiring After Attending a Tea at Hsuante Palace_\n\nCHOU PI-TA\n\nPagoda trees crowd the path crows flock at dusk\n\na messenger brings a summons to a tea\n\nreturning to Jade Hall how can I sleep\n\nwith a new crescent moon above the myrtle\n\n## 160\n\nTs'ai Ch'ueh (1037\u20131093) was from Chuanchou in Fukien province. He rose to the rank of minister during the reign of Emperor Shen-tsung (r. 1068\u20131085) and at first supported Wang An-shih's financial reforms. But when the emperor changed sides, so did Ts'ai. In addition to his opportunism, he was also known for his arrogance, and he delighted in harassing others. He was later appointed prefect of Anlu in Hupei province and wrote a series of ten poems while visiting this pavilion outside Anlu that overlooked a tributary of the Han River. When the military governor of Hanyang heard these poems, he took advantage of a few double entendres to balance accounts and accused Ts'ai of disrespect to the throne. Ts'ai was arrested and died in detention. Until modern times, Chinese preferred porcelain or stone pillows during summer, as they \"cool the brain.\" Mats of woven bamboo or rattan are also used in the summer to reduce the body's heat. And a square bed is meant to be shared. The fisherman here recalls the figure in the poem titled \"The Fisherman\" in the _Chutzu,_ in which a fisherman criticizes Ch'u Yuan (340\u2013278 B.C.) for his rigid adherence to principle and rows away singing: \"When the waves are clear I wash my hat [serve in office] \/ when the waves are muddy I wash my feet [retire].\"\n\n## _Climbing to Canopy Pavilion on a Summer Day_\n\nTS'AI CH'UEH\n\nA paper screen a stone pillow a square bamboo bed\n\na book falls from my hand during a midday dream\n\nI wake up pleased and smile to myself\n\nat the sound of a fisherman's flute on the waves\n\n## 161\n\nHung Tzu-k'uei (1176\u20131235) was a native of the Hangchou area. In addition to serving as vice-minister of justice, he also served as a scholar in the Hanlin Academy, which was also known as Jade Hall. The Academy was in charge of drafting edicts for the emperor's approval, in this case, the edict appointing the chief ministers empowered to oversee government affairs. On such an occasion, members of the Hanlin Academy would have stayed up all night in order to have the appointments ready for the emperor's vermilion seal at the predawn audience. The poet pauses to savor the hour before the sun rises, as the moon sets on the purple myrtle lining the steps and courtyard outside the Academy.\n\n## _Written While Serving at Jade Hall_\n\nHUNG TZU-K'UEI\n\nThe forbidden gates are locked and silent\n\nthick ink soaks ministerial appointments\n\nthe crier announces the last watch before dawn\n\nmoonlight floods the myrtle-lined steps\n\n## 162\n\nLi Chia-yu (719\u2013781) was a native of Chaohsien in Hopei province and the uncle of Li Tuan, who was one of the Ten Talents of the Tali Period. He served as a proofreader in the Palace Library, but due to some unspecified offense he was banished to the South. Although he was called back to the capital, he didn't stay long and was sent back to the provinces, where he served as magistrate of a series of prefectures south of the Yangtze. The phrase \"upright official\" ( _ao-li_ ) comes from the story about Chuangtzu in Ssu-ma Ch'ien's biography of him, in which the Taoist sage refuses the post of prime minister in favor of a minor post near his hometown north of Minchuan on the Honan\u2013Shantung border. Here, the phrase refers to Secretary Wang, whose job would have been to draft documents in some government office. The tower Wang built was intended as a sleeping platform during the hot and humid Yangtze summer. The South Wind is the wind of compassionate government, but it also represents the less severe cultural milieu south of the Yangtze. An ancient poem goes: \"The touch of the South Wind can relieve my people's pains.\" Li also recalls a story in _Liehtzu_ (2.11) in which gulls are said to perceive the thoughts of those who might want to harm or catch them. Clearly, this official has no such thoughts. Also, although he is \"idle\" and thus not currently employed, he still wears his hat, the emblem of office, suggesting his readiness to serve should the occasion arise.\n\n## _Written at Secretary Wang's Bamboo Tower_\n\nLI CHIA-YU\n\nAn idle upright official laughs at the high and mighty\n\nwest of the river he built a tower of bamboo\n\nwho needs a palm leaf when the South Wind blows\n\nwearing his silk hat he naps beside gulls\n\n## 163\n\nPai Chu-yi (772\u2013846), aka Pai Le-t'ien, was born across the Yellow River from Loyang but grew up east of Ch'ang-an near Weinan. A man of high principle, he fell out of favor early in his career and was banished to Kiangsi, where he served as that province's military commissioner. When he was called back to Ch'ang-an he refused to take sides in court politics and asked instead to be sent back to the provinces to serve as prefect of Hangchou and then Suchou. This poem was written in 821 after he returned to Ch'ang-an. Gossamer Hall (Ssulunko) was another name for the building where imperial decrees were drafted and where Pai was serving as Minister of Justice. However, at this time the real power at court was in the hands of the eunuchs\u2014hence, the lack of decrees to draft. Pai identified with the sufferings of the common people. But having failed to achieve any reforms that might have helped their lot, he is simply waiting for another day at the ministry to end. He eventually retired from government service and spent his remaining years in the company of the monks at Hsiangshan Temple, across from the Lungmen Caves south of Loyang. His grave is still there, facing the big stone buddha carved to resemble Empress Wu Tse-t'ien. Periods of time at court were calculated using a water clock and were announced by a huge bell and an even bigger drum in the twin towers just inside the main gate of the imperial courtyard. As noted previously, purple myrtle was planted extensively around the courtyard of the Secretariat and bloomed from mid-June until the end of September.\n\n## _Serving in the Secretariat_\n\nPAI CHU-YI\n\nAt Gossamer Hall the writing has stopped\n\nthe hours drip slowly in the bell and drum towers\n\nsitting here at dusk who are my companions\n\npurple myrtle flowers face a purple myrtle man\n\n## 164\n\nChu Hsi (1130\u20131200) was born in Fukien province, but he spent most of his life elsewhere. After a precocious childhood he served in a series of provincial posts and academic positions in the central government in Hangchou. But he was better known as one of the great neo-Confucian writers and philosophers of the Sung dynasty. On this occasion he incorporates his philosophy of the mind in a poem about a pond. The Chinese called the mind the \"square-inch\" ( _fang-ts'uh)._ Here, like a Chinese bronze mirror kept covered until needed, the mind opens to reflect the world of light and dark, sunshine and clouds. And like a pond, it is constantly refreshed by the inexhaustible water of the Tao.\n\n## _Reflections While Reading\u2014I_\n\nCHU HSI\n\nA small square pond an uncovered mirror\n\nwhere sunlight and clouds linger and leave\n\nI asked how it stays so clear\n\nit said spring water keeps flowing in\n\n## 165\n\nIn writing this poem, Chu Hsi probably had the first chapter of _Chuangtzu_ in mind, in which the Taoist sage says, \"Pouring a cupful of water into a depression, a mustard seed can serve as a boat. But if you place the cup in the water, it gets stuck. The water is too shallow, and the boat too large.\" Like Chuang-tzu, Chu Hsi preferred to express his philosophy through metaphor rather than abstraction. Here, his point is that an understanding of the Tao does not depend so much on an expenditure of effort as it does on an awareness of the nature of change. These particular warships were covered in leather in order to withstand projectiles and were used during battles to ram other ships. Thus, they were quite heavy.\n\n## _Reflections While Reading\u2014II_\n\nCHU HSI\n\nLast night spring waters rose along the river\n\neven great warships seemed light as a feather\n\ntrying to row earlier would have been useless\n\ntoday in midstream they travel with ease\n\n## 166\n\nLin Chen (fl. 1180) was from Suchou in Kiangsu province but has left us no other information about himself than what can be gathered from this poem. Some commentators attribute this poem to another Lin Chen (fl. 1050) from Fukien province. The last couplet, however, suggests it was written after the Sung court moved to Hangchou, which would support the attribution to the later man. Still others assign the poem to a man named Lin Hung, about whom nothing else is known. In any case, Cold Spring Pavilion was at the foot of a rock formation in Hangchou known as Feilai Peak, opposite the front gate of Lingyen Temple. From here, the spring flows along a rocky bed for several kilometers and then empties into the northwest corner of West Lake. Following the relocation of the court to Hangchou in 1129, West Lake became the scene of constant parties by an expatriate government that lacked the will or the power to retake North China from the Khitans. Lin, like most Chinese, accepted Mencius's view ( _Mencius_ : 6b.8) that our nature is originally pure but becomes impure as the result of improper involvement in our social and cultural environment, and also because of poor nourishment. When Zen masters try to explain enlightenment to their disciples, they often compare it to drinking water: \"Whether it's warm or cold is something you have taste for yourself.\"\n\n## _Cold Spring Pavilion_\n\nLIN CHEN\n\nA stream of pure water can soothe a poet's soul\n\nit alone knows how warm or cold the years have been\n\nflowing into West Lake it carries entertainers\n\nlooking back it's changed since the mountains\n\n## 167\n\nSu Shih (1037\u20131101) wrote this poem in 1090 when he was out of favor and had been sent to Hangchou as the city's prefect. Earlier, Su had become close friends with Liu (1033\u20131092) when Su was rusticated to the same area in 1080. Liu was from Kaifeng and was serving as commander-in-chief of military forces in the province of which Hangchou was the capital. The two men became such close friends that Su compared Liu to K'ung Jung (153\u2013208), a Han-dynasty Confucian noted for his erudition and courage. Among the many poems the two men exchanged in later years was this one, in which Su reminds his friend of their time together in Hangchou and of the region's citrus trees, which produced their fruit long after the flowers of summer and fall had faded and after all other fruits had been harvested. Such fruit is also a metaphor for the pleasures of old age.\n\n## _For Liu Ching-wen_\n\nSU SHIH\n\nLotuses are gone and their rainproof umbrellas\n\nchrysanthemums have faded but not their hardy leaves\n\nthe year's best scene though surely you'll recall\n\nis when oranges are yellow and tangerines are green\n\n## 168\n\nChang Chi (fl. 760) was born in Hsiangfanyang in Hupei province. During the Tali Period (766\u2013779), he served in Ch'ang-an as vice-director of the Bureau of Sacrifices and later in Nanchang as assistant commissioner of the Salt and Iron Monopoly Bureau, where he is said to have died. Here, he is traveling along the Grand Canal and has moored for the night in the western suburbs of Suchou. The bridge where he dropped anchor was named Feng Ch'iao (Enfeoffment Bridge), but Chang chose the more auspicious homophone _feng_ (maple). As a result of the popularity of this poem\u2014and it is perhaps the most famous poem in the Chinese language\u2014the name of the bridge was changed. It's still there, a few hundred meters north of Hanshan Temple. Although most accounts claim the temple was named for the reclusive poet Han-shan (Cold Mountain), Han-shan was Chang's contemporary and his poetry did not become well known until the end of the next century. The temple was actually named for a nearby hill. Temple bells were seldom rung so late, but it was customary in Suchou and elsewhere during the T'ang to ring the \"Bell of Impermanence\" at midnight. The river maples recall the final lines of Ch'u Yuan's (340\u2013278 B.C.) \"Calling Back the Soul\": \"On and on the river flows \/ maple trees line the shore \/ I wish I could see for a thousand miles \/ but my spring heart breaks in sorrow \/ O soul come back \/ I grieve for the lands of the South.\" Fishermen used lanterns and torches at night to attract fish, and cormorants to catch them.\n\n## _Anchored Overnight at Maple Bridge_\n\nCHANG CHI\n\nCrows caw the moon sets frost fills the sky\n\nriver maples fishing fires care-plagued sleep\n\ncoming from Cold Mountain Temple outside the Suchou wall\n\nthe sound of the midnight bell reaches a traveler's boat\n\n## 169\n\nTu Lei (d. 1240) was from Linchuan in Kiangsi province. Toward the end of his life he served as private secretary to Hsu Kuo, the minister of the imperial treasury. However, on a mission to Shantung he was killed along with his superior by General Li Ch'uan. When the Chinese first began drinking tea in the fifth and sixth centuries, they boiled it along with ingredients as diverse as cloves and scallions and drank it mostly as a medicinal broth. It wasn't until the ninth and tenth centuries that they began drinking tea by itself\u2014though as a powder they whisked into boiling water rather than in the form of leaves. This small, portable tea stove was made of clay and included a woven bamboo exterior that allowed it to be moved about while the coals inside it were still hot. Only a close friend would appreciate such simple pleasures. And only such a friend, who no doubt also came to appreciate a friend's plum blossoms, would visit on such a night.\n\n## _Winter Night_\n\nTU LEI\n\nFor a winter-night guest tea serves as wine\n\nboiling on a wicker stove as the coals turn red\n\noutside the window is the same old moon\n\nbut with plum blossoms now it's different\n\n## 170\n\nLi Shang-yin (813\u2013858) was born in Chinyang in Honan province and grew up in Chengchou and Loyang. He was considered one of the best poets of his day, and he is still ranked among the great poets of the T'ang. He was related through marriage and friendship to the two competing factions at court led by Li Te-yu and Niu Seng-ju; unable to resolve the contradictions in his relationships, he eventually incurred the enmity of his friend Ling-hu T'ao (802\u2013879), who held the position of prime minister and who made it impossible for Li to serve in anything but minor posts. Li died without office in Chengchou. Here, he cloaks these factions in the guise of two celestial maidens. After ingesting a magic elixir, Ch'ang O floated up to the moon and has been its resident goddess ever since. Ch'ing Nu, whose name means \"Girl from the Blue,\" is the Goddess of Frost. According to the _Lichi (Book of Rites),_ \"In the first month of fall, the cold cicadas sing. In the second month of fall, the geese fly past. In the third month of fall, the frost begins to fall\" (6). Viewed from afar, the moonlit water of Ch'ang O and the frost-filled sky of Ch'ing Nu don't appear different, their claims notwithstanding. So, too, are the two factions at court equally deserving of praise but also of criticism for their narrow-minded views of one another.\n\n## _The Frost and the Moon_\n\nLI SHANG-YIN\n\nBy the time I hear geese the cicadas are gone\n\nfrom a hundred-foot tower the water is like the sky\n\nCh'ang O and Ch'ing Nu don't mind the cold\n\nin the frost and moonlight they contest each other's charms\n\n## 171\n\nWang Ch'i (c. 1150) has left no information about himself other than what can be gathered from a few surviving poems. Ever since Lin Pu (967\u20131028), aka Lin Ho-ching, began singing the praises of the plum blossom, the flower came to represent a life of simplicity and beauty in the face of adversity. Especially famous were his lines: \"Its scattered shadows fall lightly on clear water \/ its subtle scent pervades the moonlit dusk.\" But Wang is concerned that as a result of Lin's infatuation with the plum, others now exhaust themselves trying to excel him, and that more often than not they fail to understand or to cultivate the plum's true spirit of purity and simplicity. Lin lived as a recluse on a small island at the edge of Hangchou's West Lake when the city was still a provincial backwater, long before it became the capital of the Southern Sung dynasty. He never sought public office, never married, and amused himself by teaching cranes to dance. People say the plum was his wife, and the cranes were his children.\n\n## _The Plum_\n\nWANG CH'I\n\nImmune from the slightest contaminating dust\n\ncontent beside a thatched hut or bamboo fence\n\nthen you met Lin Pu by mistake\n\npoets ever since haven't stopped talking\n\n## 172\n\nPai Yu-ch'an (fl. 1195\u20131230), aka Ko Ch'ang-keng, was born on Hainan Island in South China and became a Taoist priest at an early age. After studying with various masters, he finally settled in the Wuyi Mountains of Fukien province. He is ranked as the Fifth Patriarch of the Southern School of Taoism and composed a number of well-known works on inner alchemy. He was also known for his calligraphy and painting. In this poem he uses the southernmost tip of a plum branch to pinpoint the end of winter and the first sign of spring. In the second line he likens the plum blossom to a woman wearing perfume and makeup. And in the last two lines he views the blossoms under different conditions and with different backdrops. The wording of these last two lines is indebted to Tu Mu's \"Anchored Overnight on the Chinhuai\" (verse 176). In the second line, the word _fen_ can mean \"powder\" as well as \"pollen,\" and most commentators interpret it as referring to the blossom's powdery appearance. But as the great plum connoisseur Sung Po-jen notes in the introduction to his _Guide to Capturing a Plum Blossom,_ \"I breathe on their petals, inhale their fragrance, and taste their pollen.\"\n\n## _Early Spring_\n\nPAI YU-CH'AN\n\nAs soon as a southern branch unveils its first buds\n\nI savor the perfume and pollen in the snow\n\npaler in the mist whiter in the moonlight\n\ndarker on the water lighter on the sand\n\n## 173\n\nLu Mei-p'o (Sung dynasty) has left nothing behind but these two poems. The name by which he was known means Plum Tree Slope and, no doubt, was a pen name. Some editions attribute these poems to Fang Yueh (1198\u20131262), who was a native of Chihsien in Anhui province. Either way, they are good examples of the Chinese poet's view of his art. In this first poem the poet demonstrates that there are no absolutes. Everything has its own incomparable quality. But this leaves us with a conundrum: in a world where everything is unique, what can we use as a standard, and how can we speak of beauty?\n\n## _The Snow and the Plum\u2014I_\n\nLU MEI-P'O\n\nThe plum and the snow both claim the spring\n\na poet gives up trying to decide\n\nthe plum must admit the snow is three times whiter\n\nbut the snow can't match a wisp of plum perfume\n\n## 174\n\nIn this second verse the poet resolves his conundrum by introducing the poem\u2014and thus the poet\u2014as the catalyst that transforms the separate worlds of the plum and the snow into one world of beauty. Thus, the poet justifies his existence as well as his passion by his ability to perceive the spirit and the appearance of things without dividing one from the other, which is what a poem aims to do.\n\n## _The Snow and the Plum\u2014II_\n\nLU MEI-P'O\n\nThe plum without the snow isn't very special\n\nand snow without a poem is simply commonplace\n\nat sunset when the poem is done then it snows again\n\ntogether with the plum they complete the spring\n\n## 175\n\nLu Tung-pin (fl. 870) was from Yunglo in Shansi province. After failing to pass the civil service exam, he met Chung-li Ch'uan (aka Han Chung-li) at a wineshop in Ch'ang-an, got drunk, realized the impermanence of human existence, and joined Chung-li in leading a group of Taoists who became known as the Eight Immortals. In some texts, such as Hu Tzu's _Tiaohsi yuyin tsunghua_ (pub. 1148), this poem is entitled \"In Reply to Chung Juo-weng,\" and the author is listed as \"a herdboy.\" Chung (fl. 1105) was an otherwise unremarkable scholar from Kiangsi province who received an appointment to the Palace Library in Kaifeng as the result of a recommendation and was subsequently dismissed for issuing false reports of military successes. Although Chung was clearly in need of the advice offered by this poem, I have, instead, gone along with the _Chuantangshih,_ which attributes the poem to Lu Tung-pin and entitles it \"Herdboy.\" In either case, it presents the Taoist vision of the well-lived life of a person at peace with the world and in harmony with its changes. In South China, farmers wove their raincoats from the bark of the coir palm ( _Trachycarpus excelsa)._ In North China, where Lu lived, they used a coat of woven reeds in summer and one of animal skins in winter.\n\n## _Herdboy_\n\nLU TUNG-PIN\n\nAcross the countryside grass spreads for miles\n\nI blow a few notes on the evening wind\n\nback home I eat after the sun sets\n\nand lie in the moonlight still wearing my raincoat\n\n## 176\n\nTu Mu (803\u2013852) grew up in Ch'ang-an in a prestigious family that had fallen on hard times, and his only posts of significance were those he held during his rustications. Here, he drops anchor on the Chinhuai River outside Nanching's South Gate. The river also acted as the city's moat before it joined the Yangtze a dozen kilometers to the west, and this was where many of those visiting the city disembarked. Along the moat's south shore were the \"willow-lined lanes\" of brothels and inns that catered to travelers and merchants. And behind the city wall on the north shore was the site of the Ch'en-dynasty (557\u2013588) palace, which had since become a residential area for the rich and powerful. From one such home, Tu Mu hears the song \"Rear Palace Flowers on a Tree of Jade\" ( _Yushu houtingkunghua),_ which was composed by Ch'en Hou-chu (r. 583\u2013588), who drank away his throne and brought the Ch'en dynasty to an end. Hearing it still sung, and, no doubt, the lines, \"Rear palace flowers on a tree of jade \/ don't bloom again for a long long time,\" Tu Mu can only sigh that its lessons have fallen on deaf ears. The eunuchs were now the real power behind the throne, and most of Northern China had, in effect, become a number of separate kingdoms.\n\n## _Anchored Overnight on the Chinhuai_\n\nTU MU\n\nVapor shrouds the icy water moonlight shrouds the sand\n\nI anchor near a brothel on the Chinhuai for the night\n\noblivious of the sorrow from a dynasty's demise\n\nacross the moat girls still sing \"Rear Palace Flowers\"\n\n## 177\n\nCh'ien Ch'i (722\u2013780) was from Huchou in Chekiang province but he spent most of his adult life in the Ch'ang-an area, where he served as a bureau director in the Ministry of Personnel and as an academician in the Hanlin Academy. Included among the Ten Talents of the Tali Period (766\u2013779), he was considered a poetic successor to Wang Wei and Meng Hao-jan. Here, he asks wild geese why they turn around when they reach the Hsiao and Hsiang Rivers in southern Hunan province, where the environment seems so welcoming. The answer is the spiritual malaise of the region. When Emperor Fu Hsi (c. 2850 B.C.) heard the fifty-string zither, the sound so saddened him that he ordered the instrument limited to twenty-five strings, which are being played here by the two sisters both of whom married Emperor Shun (c. 2250 B.C.). When their husband died in battle south of Hunan, they drowned themselves where the Hsiang flows through Tungting Lake and became the river's and the lake's resident spirits. Two thousand years later they were joined in the same river by Ch'u Yuan (340\u2013278 B.C.), whose suicide in the Milo River, which also empties into Tungting Lake, is still commemorated by the Dragon Boat Festival on the fifth day of the fifth month. The Hsiao flows into the Hsiang, and the phrase _Hsiao-Hsiang_ simply indicates their combined watershed. I've read _yeh-yueh_ (night moon) as descriptive. But it could also be taken as the name of a song for the zither. Or it could even refer to the pavilion of that name outside Yiyang, where the Hsiang joins Tungting Lake. In any case, while such melancholy drives off the geese, exiled officials must wait to be recalled.\n\n## _On Geese Turning Back_\n\nCH'IEN CH'I\n\nWhy do they turn back when they reach the Hsiao and Hsiang\n\nthe water is green the sand is bright and both shores are mossy\n\ntwenty-five strings echo beneath the moon at night\n\nunable to bear such melancholy they all fly away\n\n## 178\n\nIn his _Kueierchi_ , Chang Jui-yi quotes this poem and says it was written on the wall of a Buddhist temple on the sacred peak of Sungshan east of Loyang. According to Hsu Yen-chou's _Choushihhua,_ when Ssu-ma Kuang (1019\u20131086) saw this poem there, he wrote \"Don't erase this poem,\" then added: \"When you're climbing a mountain, if you walk slowly, you won't become exhausted. And if you keep your feet on solid ground, you won't get hurt.\" Although this poem was ostensibly written to remind pilgrims and passersby of the true focus of spiritual practice, Ssu-ma Kuang saw in it a commentary on the complex, radical economic reforms of Wang An-shih, and many commentators have agreed with him. Like many similar poems in this collection, it probably would not have been included if it had not been open to such double entendre.\n\n## _Written on a Wall_\n\nAUTHOR UNKNOWN\n\nA pile of dry rushes in total disarray\n\nsuddenly lights the sky and suddenly is gone\n\nno match for a stove full of old stump wood\n\nslowly steadily giving off heat\n\n## PART FOUR\n## 179\n\nChia Chih (718\u2013772) was born in Loyang and enjoyed a distinguished career under several emperors, all of whom admired his skill in polishing their edicts, a skill he inherited from his father, who served in the same capacity. The son wrote this in late spring of 758 while serving in the Secretariat under Emperor Su-tsung. Although written as a court poem, it was such a good example of the genre that several colleagues responded in the same vein (three such poems follow). Taming Palace housed the royal family and highest offices of the central government in Ch'ang-an. Built on the site of the Han dynasty's Chienchang Court, it also retained the earlier name. In the first line, courtiers light the sky with their lanterns as they proceed by carriage to the palace. Phoenix Pond was just outside the Secretariat.\n\n## _Morning Court at Taming Palace_\n\nCHIA CHIH\n\nSilver lanterns light the sky along imperial streets\n\nin spring forbidden walls turn bright green at dawn\n\ncountless hanging catkins veil the painted gates\n\na hundred twittering orioles encircle Chienchang Court\n\nthe sounds of swords and pendants echo up jade steps\n\nevery robe and hat is lined with incense soot\n\nand bathed in waves of grace at Phoenix Pond\n\nand daily stained with ink in the service of our lord\n\n## 180\n\nNot long after Emperor Su-tsung's return to Ch'ang-an in 757, Tu Fu (712\u2013770) was reappointed censor but was demoted the next year. The water clock's arrow indicated when it was time for court, and the sleeves of Chinese robes were like wings. \"Gossamer\" refers to the emperor's silken and never-ending edicts. Tu Fu likens Chia Chih to Hsieh Chao-tsung (whose name meant \"phoenix\"), who also followed his father in composing imperial edicts, and of whom Emperor Wu (r. 502\u2013549) of the Liang dynasty said, \"You are so capable, you must have phoenix feathers.\" The feather here is Chia Chih's poem.\n\n## _Responding to Secretary Chia Chih's \"Morning Court at Taming Palace\"_\n\nTU FU\n\nThe sound of nightlong dripping speeds the shaft of dawn\n\nspring within the inner gates intoxicates the peach trees\n\ndragons writhe on tapestries in the warming sun\n\ntiny birds soar above on the faintest breeze\n\nincense spills from our sleeves after court\n\npearls pour from your brush into a poem\n\nhow do generations learn such gossamer art\n\nby the pond today I found a phoenix feather\n\n## 181\n\nWang Wei (701\u2013761) was reappointed to the Secretariat in 758 and became its deputy director the following year. The dawn crier wore a red cap to mimic a rooster and announced dawn by beating bamboo on a wooden board. The steward was in charge of dressing the emperor, and nine is the number of levels of Heaven. At morning court, two huge fans were held behind the emperor to block the rising sun. Also, the bottom of his robe was decorated with nine dragons playing with the sun and moon. After court, Chia Chih and the other members of the Secretariat returned to their offices to prepare documents generated by the morning audience, which they wrote on multicolored paper. As they went about their duties, the pendants attached to their sashes chimed.\n\n## _Responding to Secretary Chia Chih's \"Morning Court at Taming Palace\"_\n\nWANG WEI\n\nWhen the crimson-capped crier strikes the dawn clapper\n\nthe royal steward enters with azure cloud robes\n\nthe ninefold gates of the palace swing open\n\ncourtiers from every land bow before the throne\n\nthen the sun rises and celestial fans wave\n\nincense smoke swirls beside dragon brocade\n\nafter court flowery decrees must be drafted\n\nthe sound of jade pendants returns to Phoenix Pond\n\n## 182\n\nTs'en Shen (715\u2013770) was from Chiangling in the middle reaches of the Yangtze. After beginning his career as a military aide on the Silk Road, he returned to Ch'ang-an as part of Emperor Su-tsung's entourage in 757 and was serving as rectifier of omissions when he wrote this response. Swords and pendants were worn at the waist, and the banners accompanied different administrative units. The scene in the second half of the poem is that of officials returning to their offices after morning court. Phoenix Pond was just outside the Secretariat where Chia Chih and those who responded to his poem worked. \"Yangchun\" (\"Sunny Spring\") was the name of a tune that was too difficult for most people to sing.\n\n## _Responding to Secretary Chia Chih's \"Morning Court at Taming Palace\"_\n\nTS'EN SHEN\n\nRoosters announce dawn in the capital is cold\n\norioles proclaim spring in the royal realm is over\n\nthe golden gate's morning bell wakes ten thousand households\n\nattendants line jade steps and crowd a thousand officials\n\npendants and swords reflect the fading stars\n\nbanners brush dew from the willows\n\nhow rare is the man at Phoenix Pond\n\nand how hard to match his \"Sunny Spring\"\n\n## 183\n\nTs'ai Hsiang (1012\u20131067) was from Hsienyu in Fukien province and rose to the rank of minister in the Ministry of Rites and the Department of State Affairs. He was also one of the greatest calligraphers of the Sung dynasty. This poem describes the first night of the first full moon of the year's first season, when Sung emperors traveled in a great procession down the main street of Kaifeng. Huafeng, east of Ch'ang-an, was where emperors performed sacrifices to the sacred mountain of Huashan to ensure the birth of sons, their own longevity, and the prosperity of their reign. Emperor Jen-tsung (r. 1023\u20131063) reigned for forty-one years, and this turned out to be his last lantern festival.\n\n## _On Lantern Festival at Imperial Request_\n\nTS'AI HSIANG\n\nA forest of jeweled candles forms a thousand peaks\n\nthe royal gate is graced by his iridescent presence\n\nthe procession isn't for the year's first moon\n\nbut for sharing the joy of the people\n\nthe sky's clear light lasts all night\n\nthe world's mild air foreshadows the spring\n\nthe reason all wish him the blessings of Huafeng\n\nfor more than forty years his benevolence has grown\n\n## 184\n\nWang Kuei (1019\u20131085) was from the Chengtu area of Szechuan. During the reigns of Jen-tsung (1023\u20131063) and Shen-tsung (1068\u20131085), he directed the Hall of Scholarly Worthies and also served as prime minister. Here, he describes a royal banquet celebrating the first full moon and the images meant to remind the audience of their host's immortal heritage. The two phoenixes are yoked to the celestial carriage of Hsi-wang-mu, Queen Mother of the West, dispenser of the elixir of immortality. And the three pairs of tortoises support three islands in the East Sea where immortals live. Wang also recalls similar feasts held during the Chou dynasty (1122\u2013255 B.C.) at its capital of Hao and during the Han dynasty (206 B.C.-A.D. 220), when Emperor Wu (r. 140\u201387 B.C.) banqueted on the Fen River and wrote poems about the autumn wind. It was the royal custom to eat and drink more in good times and less in bad times. Hence, Wang hopes for another cup of wine.\n\n## _On Lantern Festival at Imperial Request_\n\nWANG KUEI\n\nIt isn't snow but moonlight on the terrace of immortals\n\ncandles line the balconies and jeweled fans part\n\nfrom the clouds two phoenixes lead a carriage down\n\nfrom the sea six tortoises carry mountains in\n\nspring wine in old Hao drenched the feasts of Chou\n\nautumn wind on the Fen shamed the bards of Han\n\neveryone rejoices in a song of lasting peace\n\nmay our lord lift up his rosy cup once more\n\n## 185\n\nShen Ch'uan-ch'i (656\u2013715) was from Neihuang in Honan province and served at the court of Empress Wu. Following her death in 705, he was accused of corruption and exiled to Vietnam but was soon recalled and served as a court poet under Chung-tsung (r. 705\u2013710) and Jui-tsung (r. 710\u2013712). Here, he records Chung-tsung's visit in 709 to the villa of his favorite daughter. The princess was as ambitious as Empress Wu and is said to have exceeded her in depravity and licentiousness. However, she was not as long-lived and was killed in 710 during a coup that led to the enthronement of Hsuan-tsung (r. 712\u2013756). The River of Heaven was another name for the Milky Way. Singing Phoenix Ridge was in Shensi province, and Drinking Dragon Stream refers to a stretch of the Wei River just west of Ch'ang-an. But here both are meant to summon images of a Taoist paradise. \"Chuntien Kuanglo\" (\"Joy Pervades Heaven\") was an ancient court tune.\n\n## _Attending a Banquet at the New Residence of Princess An-lo_\n\nSHEN CH'UAN-CH'I\n\nHer Royal Highness loves immortals and gods\n\nshe built her estate near the River of Heaven\n\nwith hills like those of Singing Phoenix Ridge\n\nand ponds that rival Drinking Dragon Stream\n\nher green-curtained balconies invite the spring to stay\n\nher gold-decked pavilions imitate the sun\n\nas our imperial majesty's entourage arrives\n\nshe toasts his long life with \"Joy Pervades Heaven\"\n\n## 186\n\nOu-yang Hsiu (1007\u20131072) was from Chi-an in Kiangsi province and was one of the great literary figures of his day as well as one of the leaders of government reform and, during his final years, opposition to the financial policies of Wang An-shih. Thus, he was banished on several occasions and wrote this poem in 1036 during his rustication as magistrate of Yichang at the mouth of the Yangtze Gorges. The man to whom this poem replies was serving in Yichang as district judge. Ou-yang Hsiu once served in Loyang, which was famous for its private gardens, and the \"illness\" to which he refers was probably the cause of his exile.\n\n## _In Reply to Ting Yuan-chen_\n\nOU-YANG HSIU\n\nSpring wind I guess doesn't reach this side of Heaven\n\nthis mountain town in March is still devoid of flowers\n\noranges still hang from snow-laden branches\n\ndreaming bamboo shoots are startled by cold thunder\n\nhonking geese at night make me think of home\n\nnursing last year's illness I feel the season's pulse\n\nformerly a guest in the gardens of Loyang\n\nwhy should I care if country plants bloom late\n\n## 187\n\nShao Yung (1011\u20131077) grew up in Yaohsien in Honan province. Instead of pursuing a career as an official, he devoted himself to neo-Confucian studies and lived as a farmer-recluse on Mount Sumen outside his home town. He later moved to Loyang and was repeatedly offered positions at court in Kaifeng, all of which he refused, preferring to spend his time in the company of friends such as Ssu-ma Kuang. During the Sung, men wore their hair long and in a bun, and officials were required to wear hats. Hence, only those who were without a position or retired could afford to wear a few flowers for fun.\n\n## _Flower Garland Song_\n\nSHAO YUNG\n\nThe flowers on my head shine in my cup\n\nmy cup contains beautiful flowers\n\nI've seen two generations of peaceful days\n\nand witnessed four reigns of prosperous times\n\nthen too my body is more or less sound\n\nalso the season is at the height of its bloom\n\nmy cup glows red with flower reflections\n\nhow can I face them and not go home drunk\n\n## 188\n\nYen Shu (991\u20131055) was from Fuchou in Kiangsi province. Despite growing up poor he managed to receive a good education, and his home in Kaifeng became a meeting place for the poets of his day. He eventually reached the post of prime minister. But only the ruthless hold power for long, and he spent his last decade in provincial posts. Here, he recalls an extramarital affair in late spring. The walls of a woman's carriage consisted of waterproof material, and the gorge clouds refer to the Sorceress of the Yangtze Gorges, whose name and cloud-wreathed image were associated with romantic liaisons, especially those that were impossible to maintain. Fires, even those for heating wine, were forbidden during the two days in April preceding Chingming, or Grave Sweeping Day.\n\n## _Private Thoughts_\n\nYEN SHU\n\nHer lacquered carriage no longer arrives\n\nwhere do gorge clouds go when they vanish\n\na pear-blossom courtyard in waves of moonlight\n\na willow-lined pond in the lightest of winds\n\nso many days of loneliness and drinking\n\nand now desolation and no stove fire\n\nI'd send a letter in a fish if I could\n\nbut everywhere rivers and mountains are endless\n\n## 189\n\nChao Ting (1085\u20131147) was from Wenhsi in Shansi province and twice served as prime minister, but he incurred the wrath of the eunuch Ch'in Kuai and was banished to Chaochou on the southeast coast. Here, he notes the extent to which Chinese customs honoring ancestors were followed in this area where other ethnic groups such as the Yueh predominated. Willow wood was known for its regenerative power and was used to start the new fire after the two-day cooking ban. P'ang Te-kung (fl. 220) lived with his family on a mountain south of Hsiangyang, and his name was later linked with the custom of clearing weeds from one's ancestral grave. Chao, too, retreats to the countryside and tries to ignore the flute that reminds him of home. He died here soon afterwards as the result of a hunger strike.\n\n## _Cold Food_\n\nCHAO TING\n\nOver the most remote poorest village gate\n\npeople stick willow wood to mark the end of spring\n\nin the Land of Yueh the fire ban's unknown\n\nbut like old P'ang they visit ancestral graves\n\nhere the royal tombs see no sign of grain\n\nbut mountain streams and paths are lined with fallen blossoms\n\nafter a jug of wine I lie down on the moss\n\nand try to ignore the evening flute I hear on the city wall\n\n## 190\n\nHuang T'ing-chien (1045\u20131105) was from Hsiushui in Kiangsi province. Despite the support of Su Shih, he did not enjoy a successful career and died in exile. Although he was considered Su's poetic heir, his style was unconventional and full of obscure allusions. Here, however, he recalls two well-known Chingming stories. The first is about a man who often came home drunk and told his wife he had been out with rich friends. His wife followed him the next day and watched him go from gravesite to gravesite begging for leftovers ( _Mencius:_ 4b\/33.1). The second is about Chieh Chih-t'ui, who refused to serve Duke Wen (r. 635\u2013628 B.C.). Even when the Duke set fire to the mountain he lived on, Chieh still refused and was burned to death. The Duke was so contrite, he began the practice of banning fire on the two days prior to Chingming.\n\n## _Chingming_\n\nHUANG T'ING-CHIEN\n\nPeach and plum trees smile on Chingming Day\n\nweed-choked fields and graves can only sigh\n\nthunder wakes the serpents in the earth and sky\n\nrain fills the countryside with tender new plants\n\none begged for funeral scraps and tried to fool his wife\n\nanother died in flames rather than be enfeoffed\n\nafter a thousand years was the fool or wise man right\n\nboth share the same bramble-covered hills\n\n## 191\n\nKao Chu (fl. 1200) was from Chekiang province and chose a life in the countryside outside his hometown of Yuyao over a career in Hangchou. A follower of the Rivers and Lakes school of poetry, he was called the Poetry King of Yuyao and his verses were often quoted in the capital. Here, he describes the scene when weeds are cleared from graves and paper money is burned to provide funds for departed spirits in the underworld. Because azaleas bloom during Chingming, they are planted around graves. Foxes often make their homes in coffins and are thought to be the transformed spirits of the dead. Just as the heavens were thought to have nine levels, the underworld inhabited by the dead also had nine levels, the first of which was marked by sulfur springs at the earth's surface.\n\n## _Drinking Wine on Grave Sweeping Day_\n\nKAO CHU\n\nHillsides north and south are overrun with graves\n\nsweeping rites on Chingming are nothing but a mess\n\npaper ashes fly like snow-white butterflies\n\ntears from broken hearts stain azaleas red\n\nfoxes sleep in tombs once the sun goes down\n\nchildren play in lamplight on the way back home\n\nwho has wine this life should drink until they're drunk\n\nno drop has ever reached the ninefold springs below\n\n## 192\n\nCh'eng Hao (1032\u20131085) was from Loyang and served in Kaifeng as a companion to the crown prince and as an investigating censor. However, due to his opposition to the policies of Wang An-shih, he was demoted and sent to the southernmost part of the empire. He was later recalled but died on his way back. He and his younger brother, Ch'eng Yi, were major spokesmen for the neo-Confucian revival that dominated intellectual life in the Sung. Ch'eng Hao thought that separating the world into internal and external realities was the root of all ill, and that cultivating impartiality and spontaneity was the path all men should follow. In a letter to his fellow neo-Confucian Chang Tsai, he once wrote, \"Why should a man think it wrong to follow things that are external, and right to seek what is internal?\" He was a man for all seasons, not only spring.\n\n## _Strolling outside Town_\n\nCH'ENG HAO\n\nIn the sweet green countryside I walk where I want\n\nspring is in the distant hills jade on all four sides\n\ninspired I chase every red down every willow lane\n\ntired I sit on mossy rocks beside a rushing stream\n\ndon't refuse wine unless you're already drunk\n\nfear only that flowers will leave with the wind\n\nespecially on Chingming when the weather is fine\n\nwhy not go wandering just remember to return\n\n## 193\n\nHui-hung (1071\u20131128) was from Yifeng in Kiangsi province. Not much else is known about him other than that he was a Buddhist monk who lived at Chingliang Temple in Nanching. Clearly, he did not spend all his time in the meditation hall. The swing was introduced to the Chinese by northern nomads sometime during the first millennium B.C. For reasons I have yet to discover, it became part of the pastimes associated with Chingming, especially along the Yangtze in the land of Ch'u, where people usually stood rather than sat on the board. Toad Palace is another name for the moon, which is inhabited by a three-legged toad that is the transformation of Ch'ang O, Goddess of the Moon.\n\n## _The Swing_\n\nHUI-HUNG\n\nA pair of blue ropes swing from a painted frame\n\na beauty enjoys spring by a small pavilion\n\nher crimson skirt flutters as it scrapes the ground\n\nher beguiling jade face rises to the sky\n\nthe carved board glistens with apricot-blossom rain\n\ncolored ribbons trail in the green-willow haze\n\ndown she steps in silence and stands nonchalant\n\na banished immortal from Toad Palace it seems\n\n## 194\n\nTu Fu (712\u2013770) returned to Ch'ang-an at the end of 757 and was reappointed as a censor. But the post was merely a sinecure and he felt powerless to do anything to help the country recover from the An Lu-shan Rebellion. He wrote this and the next poem the following April while visiting the serpentine lake formed by an artificial canal that brought water from the Chungnan Mountains to the southeast corner of the capital. After passing the civil service exam in 752, Tu Fu and his fellow graduates were entertained by Emperor Hsuan-tsung at Purple Cloud Pavilion at the southwest corner of this waterway. But the pavilion and surrounding structures had since become home to kingfishers, which never nest near places frequented by humans, and the stone statues in front of the nearby grave mound of the Ch'in dynasty's Second Emperor had been knocked down.\n\n## _While Drinking at the Chuchiang Waterway\u2014I_\n\nTU FU\n\nEach flying petal diminishes the spring\n\nten thousand on the wind break a person's heart\n\nwatching the final flowers fall before my eyes\n\nhow could too much wine pass between my lips\n\nkingfishers nest below the waterway pavilions\n\nand the unicorns have fallen by the garden tomb\n\nto ponder such things is to turn to pleasure\n\nwhat use is mere fame if it weighs a person down\n\n## 195\n\nIn ancient times the Chinese made new clothes that they reserved for the end of spring. Nowadays they do the same, but they put them on as soon as the new year begins. At court this meant multiple sets of garments and various kinds of regalia. Tu Fu is so poor he pawns each set as soon as he's used it and buys wine to ensure his obliviousness to the sad state of affairs in the country and at court. Even his reappointment to the post of censor is no solace but, rather, a burden. Several months after he wrote this he was demoted and transferred to Huachou. But instead of taking up his new post a hundred or so kilometers east of Ch'ang-an, he resigned, left for Chengtu, and never returned to the capital, though he never stopped hoping to be recalled.\n\n## _While Drinking at the Chuchiang Waterway\u2014II_\n\nTU FU\n\nEvery day after court I pawn my spring clothes\n\nevery day from the waterway I come home drunk\n\nwherever I go I owe money for wine\n\nbut living until seventy has always been rare\n\nbutterflies float half-seen among the flowers\n\ndragonflies flit here and there across the water\n\nI urge you to flow with the wind and light\n\nenjoy your time together and don't fight\n\n## 196\n\nTs'ui Hao (704\u2013754) was from Kaifeng and moved to Ch'ang-an as a young man to take the civil service exam. Although he soon became a member of the salon of the Prince of Ch'i and a close friend of Wang Wei, his unconventional behavior resulted in a series of rustications, first to the frontier, then to various posts along the Yangtze. Here, he stands on the river's south shore, across from Hanyang, on a tower that commemorated the Taoist immortal Fei Wen-wei, who flew off from here on the back of a yellow crane, leaving the clouds behind. The locust and gingko trees on Hanyang's Phoenix Hill were so large their reflections reached the middle of the Yangtze. Parrot Isle, a sandbar in midstream, was a popular place to hold parties, until it disappeared in a flood. Li Pai liked this poem so much he quoted its ending in his own \"Climbing Chinling's Phoenix Terrace.\" The mist reminds the poet of his loss of direction and of his separation from loved ones.\n\n## _Yellow Crane Tower_\n\nTS'UI HAO\n\nA man rode off on a crane long ago\n\nYellow Crane Tower is all that remains\n\nonce the crane left it never returned\n\nfor a thousand years clouds have wandered in vain\n\nthe trees of Hanyang shine in midstream\n\nthe sweet plants of spring overrun Parrot Isle\n\nat sunset I wonder which way is home\n\nmist on the river only means sorrow\n\n## 197\n\nTs'ui T'u (fl. 900) was born along the Fuchiang River south of Hangchou but spent most of his life far from home. He witnessed the end of the T'ang dynasty and the rise of a series of independent states along the Yangtze, whose middle reaches were once controlled by the state of Ch'u, and where he now finds himself. In _Chuangtzu:_ 2 a man wakes up from a butterfly dream to wonder if he isn't a butterfly dreaming he's a man. Here, the poet hears the cuckoo's cry, _pu-ju kuei-ch'u_ (\"better go home\"), in the middle of the night. But why go home to a state where men compete over the meaningless mists of Lake Wuhu (aka Taihu) west of Suchou?\n\n## _A Traveler's Thoughts on a Spring Evening_\n\nTS'UI T'U\n\nFlowing water and falling petals have no pity\n\nI see the East Wind off past the walls of Ch'u\n\na butterfly dreams ten thousand miles from home\n\na cuckoo perches beneath the midnight moon\n\nno letters from home for more than a year\n\nthe gray at my temples all due to spring\n\nI could go home if I wanted but don't\n\nwhy fight over the mists of Wuhu\n\n## 198\n\nWei Ying-wu (737\u2013792) was from Ch'ang-an and wrote this in 785 when he was serving as magistrate of Suchou. He had been rusticated to the South in 781, and this was the last of a series of posts he held along the lower reaches of the Yangtze. Some commentators say this poem was written in 784 at his previous post in Chuchou, which also had a West Tower\u2014but not one that faced the road from the capital to the northwest, as did Su-chou's. The refugees were most likely the result of the wars that took place (781\u2013786) between the T'ang court and the independent governors farther north along the Yellow River. But severe floods remain common in this region. Li Tan was a friend of the poet who was serving as a palace historian and with whom Wei exchanged poems.\n\n## _In Reply to Li Tan_\n\nWEI YING-WU\n\nWe parted last year among flowers\n\nthis year they're blooming again\n\nthe haze of the world is hard to penetrate\n\ntroubled by the cares of spring I fall asleep alone\n\nmy body is ill my thoughts are in the fields\n\nI'm ashamed of my salary with refugees in town\n\nI heard you were planning to come for a visit\n\nhow many moons have I watched from West Tower\n\n## 199\n\nAfter quitting his post in Huachou in 758, Tu Fu (712\u2013770) stayed for a while in Tienshui then moved to Szechuan province in 759. With the help of a relative he managed to buy a small piece of land and build a cottage outside Chengtu's West Gate on the banks of the Huanhua River, a minor tributary of the nearby Min. He wrote this poem in 760 during his first summer there. His cottage, or at least a reconstructed version, is still on the same spot, and authorities have even begun excavating Tu Fu's old trash heap. Another variant of the seventh line reads: \"all I need is medicine for my countless ills.\" Officials were paid in units of grain they could use as food or convert into currency.\n\n## _River Village_\n\nTU FU\n\nA clear river winds around the village\n\nall summer long village life is peaceful\n\nswallows in the rafters come and go at will\n\nseagulls on the water visit friends and kin\n\nmy wife draws a chessboard on a piece of paper\n\nmy children make fishhooks out of sewing needles\n\nthankfully an old friend shares his office rice\n\nwhat else does this poor body need\n\n## 200\n\nChang Feng (1052\u20131112) was from Chingchiang in Kiangsu province and served in the Sung capital of Kaifeng, where he rose to the rank of vice-minister in charge of ritual observances. He was one of the four most famous members of Su Shih's literary coterie and was known for the realism of his poetry as well as for his honesty and love of simplicity. Not surprisingly, many of his surviving poems are about life in the countryside. This was one in a series of three he wrote soon after his retirement.\n\n## _Summer Day_\n\nCHANG FENG\n\nLate summer on the river the sun and wind are mild\n\nthe little birds below the eaves are grown\n\nsun-drenched butterflies dance among the flowers\n\nnewly spun spiderwebs brighten every room\n\nthreadbare curtains invite the moon's reflection\n\na pillow made of clay echoes with the current\n\nmy long-graying temples recall the frost and snow\n\nlet me pass this life chopping wood and fishing\n\n## 201\n\nThe retreat Wang Wei (701\u2013761) bought on the Wang River was sixty kilometers southeast of Ch'ang-an and once belonged to the poet Sung Chihwen. Although Wang rose to the post of deputy prime minister, he was a lifelong Buddhist and a vegetarian, and toward the end of his life he spent more time meditating at his retreat and hiking around the mountains than working in the capital. Hibiscus flowers only last a day, or two at most, and mallow leaves are best picked after the dew dries. Hence, Wang Wei is too hungry to wait. In _Chuangtzu:_ 27 the arrogant Yang-tzu Yu returns to his inn after receiving instruction from Lao-tzu. But where he was once waited on hand and foot, he now has to fight for a place for his mat. \"Seats\" here also refer to positions of authority, concerning which the poet no longer has any interest. In _Liehtzu:_ 2.11 the author recounts how seagulls flock around a man, until he conceives of a plan to catch them.\n\n## _Written at My Wang River Retreat after a Steady Rain_\n\nWANG WEI\n\nSteady rain deserted woods and finally kitchen smoke\n\nsteamed greens and millet for those in the eastside fields\n\nsnowy egrets fly above a sea of flooded fields\n\ngolden orioles sing in the shadows of summer trees\n\nsitting in the mountains I regard the day's hibiscus\n\nand cut dewy mallow leaves for a meal below the pines\n\nliving in the country I've stopped fighting over seats\n\nwhy then do the seagulls still suspect me\n\n## 202\n\nLu Yu (1125\u20131210) was an orphan who was raised by monks at a monastery in Tienmen in Hupei province. He was one of the most romantic figures of his day and was known for his support of efforts to recover North China from the Khitans, as well as for the travel diary he wrote about his journey to Szechuan and for his book on tea. But he was also the most prolific lyric poet of the Sung and left behind nearly 10,000 poems. He spent the last twenty years of his life in Chekiang province in Shaohsing and built a retreat at the city's East Lake, which was once a quarry and was known for its precipitous rock formations, shaded bays, and bamboo-lined shores. Some editions attribute this poem to Huang T'ing-chien.\n\n## _New Bamboo at East Lake_\n\nLU YU\n\nI planted thorns and built a fence to keep them safe\n\ntheir growing emerald canes shimmer in the ripples\n\nautumn arrives first where a breeze cools the earth\n\nthe summer sun overhead is far away at noon\n\nthey make a rustling sound as they discard their wrappers\n\nand cast spindly shadows as they put forth new branches\n\nI plan to visit often as soon as I stop working\n\nand take my mat and pillow when I go\n\n## 203\n\nTou Shu-hsiang (fl. 780) was from Fenghsiang west of Ch'ang-an and was a close friend of Ch'ang Kun. When Ch'ang became prime minister, Tou was appointed censor. But when Ch'ang was ousted in 779, Tou was banished to Lishui, southeast of Nanching, where he wrote this poem. When officials were sent to such distant posts, family members usually remained behind. Despite his long absences, Tou's five children all became known for their poetry. The flowers here are identified by some as _Magnolia pumila_ and by others as _Pergularis ordoratissia._ The blue flags are those of wineshops on the Kan River, a tributary of the Chinhuai that flows northwest into the Yangtze at Nanching.\n\n## _Spending a Summer Night with My Cousin Talking about the Past_\n\nTOU SHU-HSIANG\n\nMagnolia perfume inundates the courtyard\n\nthe wine doesn't last through a late night drizzle\n\nhow can I reply to the distant words of loved ones\n\nor bear to hear about the dismal past\n\nour children all are grown\n\nour friends are mostly gone\n\nanother boat leaves tomorrow\n\nI hate those blue flags by the bridge\n\n## 204\n\nCh'eng Hao (1032\u20131085) taught thousands of students at his home in Loyang and served briefly in the nearby capital of Kaifeng. But due to his opposition to the policies of Wang An-shih, he was banished to South China and died before he could return. Still, with his younger brother, Ch'eng Yi, he was among the leading lights of neo-Confucianism. This poem reflects his philosophy whereby all things are seen as part of the Tao and also as part of the mind. Thus, man shares the same nature as Heaven and Earth and all creation. The harmony of this nature is easily upset by desire and ignorance, but it can be restored through the cultivation of such virtues as kindness and equanimity. The last two lines paraphrase _Mencius_ : 3b\/2.3.\n\n## _Occasional Poem on an Autumn Day_\n\nCH'ENG HAO\n\nWhen I'm at peace I let everything go\n\nI wake by the east window long after sunrise\n\nviewed without passion everything is fine\n\nseasonal glories hold true for man\n\nthe Tao fills the world the formed and the formless\n\nour thoughts are in the ever-changing wind and clouds\n\nnot troubled by wealth content in poverty\n\nthe person who reaches this is truly noble\n\n## 205\n\nCrescent Pond was located inside the palace grounds in Kaifeng where Ch'eng was serving at the time. The tower he refers to was Iron Pagoda, which was part of Kaipao Temple northeast of the palace. The tower was originally built of wood c. 975. After a fire, it was rebuilt of bricks in 1049 and covered with glazed tiles that made it look as though it were made of iron. The rebuilt version, which Ch'eng sees in the distance and which still stands, is 56 meters high, whereas the original was 120 meters.\n\n## _Visiting Crecent Pond_\n\nCH'ENG HAO\n\nWe circle the shore of Crescent Pond\n\nto the north is a tower that touches the sky\n\nthe world has changed in the autumn air\n\nwe pour a cup for the evening chill\n\nthe image of a cloud pauses on the water\n\nthe sound of a stream lingers beneath the trees\n\nour tasks are endless there's no need to count\n\nlet's meet again our next day off\n\n## 206\n\nAfter leaving Chengtu in 765, Tu Fu (712\u2013770) traveled down the Yangtze with his family and stopped in Fengchieh. He stayed there for two years as the guest of the prefect and during his second autumn wrote a series of eight poems, four of which appear here. The place where he lived was a few kilometers east of Fengchieh in the town of Paiti, overlooking the western entrance to the first of the Three Gorges. Wuhsia, or Sorceress Gorge, is the second of the three and was named for a mountain that served as a fair-weather beacon for river travelers. Paiti was once the capital of an independent state and also where Liu Pei died while trying to reunite China in the third century. Jade Dew is another name for the solar period known as White Dew that occurs in early September\u2014when the leaves turn red and chrysanthemums bloom. Women used mallets of wood or stone to beat clothes while washing them or to toughen new material.\n\n## _Autumn Inspiration\u2014I_\n\nTU FU\n\nJade dew wounds a forest of maples\n\nSorceress Mountain looks bleak in the gorge\n\nwaves in the river crash against the sky\n\nclouds in the passes darken the earth\n\nchrysanthemums once more have seen my tears about the past\n\nmy boat is still tied to thoughts of my garden\n\neverywhere people are making winter clothes\n\nin Paiti at dusk the mallets beat faster\n\n## 207\n\nTu Fu describes his daily visit to a pavilion in Paiti that overlooked the Yangtze. Fishermen often spent several nights on the river before returning. But the two nights also refer to Tu Fu's two-year stay in Paiti. The swallows of summer are also staying longer than expected. Liu Hsiang (79\u20136 B.C.) and K'uang Heng took part in reforming religious observances of the Han court and its interpretation of the Confucian canon, often through the submission of memorials. Even though Tu Fu grew up near Loyang, he spent much of his youth and early career in the village of Shaoling in the southern suburbs of Ch'ang-an. Wuling was the location of five Han-dynasty imperial tombs northwest of Ch'ang-an, and its name was later used for the capital and its elite. The robes are light because they're made of silk.\n\n## _Autumn Inspiration\u2014II_\n\nTU FU\n\nA hillside town of a thousand homes rests in morning light\n\nevery day from the river tower I look out on the landscape\n\nfishermen are still drifting after two nights away\n\nand swallows are still flying in the cool fall air\n\nK'uang Heng was heard but my voice is faint\n\nLiu Hsiang could teach but I am ignored\n\nand yet few friends of my youth are poor\n\nin Wuling their robes are light and their horses well-fed\n\n## 208\n\nTu Fu imagines himself back at court. Penglai Hall was located to the rear of Taming Palace in Ch'ang-an. The name came from a mythical island off the Shantung coast where immortals lived. The Chungnan Mountains were south of Ch'ang-an. As they seemed impervious to change, they were often used as a metaphor for immortality. The bronze pillars were erected by Emperor Wu of the Han dynasty, for collecting pure dew to use in elixirs. The Queen Mother of the West (Hsi-wang-mu) dispensed an elixir that ensured long life and lived at Jade Lake somewhere in Central Asia. Hanku Pass in westernmost Honan province was reportedly where Lao-tzu wrote the _Taoteching_ before he disappeared into the mountains. Lao-tzu was especially honored during the T'ang, as his last name, Li, was the imperial family's surname. Pheasant tails were used for fans. The dark waters refer to those of Ch'u Yuan's \"The Fisherman,\" which recalls the dilemma of the exiled poet who would still save his lord from disaster. And the last line concerns the poet's dreams.\n\n## _Autumn Inspiration\u2014III_\n\nTU FU\n\nPenglai Hall looks south toward the mountains\n\nits pillars of bronze collect dew from the stars\n\nto the west the Queen Mother appears at Jade Lake\n\nto the east purple mist fills Hanku Pass\n\na cloud of pheasant fans parts before the throne\n\nsunlit dragon scales reflect our sage's face\n\nliving by dark waters I fear the year is late\n\nI often return to court below the painted gates\n\n## 209\n\nEmperor Wu of the Han dynasty (r. 140\u201387 B.C.) constructed an artificial lake southwest of Ch'ang-an to conduct naval maneuvers and named it after the kingdom in Southwest China he planned to attack. On the east and west shores were statues that resembled the Weaving Maid and Herdboy\u2014the two lovers forever separated by the Milky Way\u2014and in the middle was a stone whale covered with jade scales. During the T'ang the lake was better known for its Manchurian wild rice, _Zizania latifolia,_ and lotuses, both of which were harvested in the fall. Here, though, they have been neglected, as the country is in chaos. Tu Fu feels cut off from the capital in the North and adrift in the watery Yangtze watershed synonymous with recluses, bandits, and exiled officials.\n\n## _Autumn Inspiration\u2014IV_\n\nTU FU\n\nKunming Lake was the jewel of the Han\n\nEmperor Wu's banners appear before my eyes\n\nthe Weaving Maid weaving in vain below the moon\n\nthe stone whale shimmering in the autumn wind\n\nscudding black clouds of windswept wild rice\n\ndrifting pink powder from dry lotus pods\n\nbut sky-high passes are only for birds\n\nrivers and marshes are a fisherman's world\n\n## 210\n\nTai Fu-ku (1167\u20131248) was from Huangyen in Chekiang province. He spent many years roaming the rivers and lakes of the lower reaches of the Yangtze and was the most celebrated member of the Rivers and Lakes school of poetry. Especially famous are his poems criticizing the life of ease of the Sung court and its refusal, or inability, to retake North China from the barbarians. Tai wrote this poem at Hangchou's West Lake. Chinese boats often use a single scull at the stern for propulsion, rather than a pair of oars. Water pepper is the common name for _Polygonum hydropiper,_ a variety of knotweed or smartweed. Broken Arch Bridge connected the Southern Sung capital to the small island of Kushan, which was once the home of the reclusive poet Lin Pu (967\u20131028).\n\n## _Aboard a Boat on a Moonlit Night_\n\nTAI FU-KU\n\nMoonlight fills the boat and floods an empty sky\n\nnight air pours across the green and glassy water\n\nideas for a poem sway in the shadows of the mast\n\nmy dream spirit rows to the sound of the scull\n\nstars are scattered on the jade-colored lake\n\nwild geese cry out in the water-pepper wind\n\na few fishing lanterns mark the ancient shore\n\ndew drips from paulownias on Broken Arch Bridge\n\n## 211\n\nChao Ku (815\u2013856) was from Huai-an in Kiangsu province, which was once part of the ancient state of Ch'u. Although he wrote this while serving as commandant of Weinan just east of Ch'ang-an, his sentiments must have changed, as this was where he later retired. During the Western Chin dynasty (265\u2013316) Chang Han was serving one autumn in Loyang, when he thought about sea perch stew on the Grand Canal. He became so homesick he quit his post and went home. Chao recalls Chang's example but is unable to emulate it and likens himself, instead, to Chung Yi. Chung was the prime minister of Ch'u, captured during a war with the state of Cheng and given to the state of Chin. When the Duke of Chin saw him, the duke asked, \"Who is this in chains wearing a strange hat?\" An attendant answered, \"A prisoner from Ch'u\" ( _Tsochuan:_ King Ch'eng, Ninth Year).\n\n## _Autumn Longing in Ch'ang-an_\n\nCHAO KU\n\nDesolate cloud shapes brush past the dawn\n\nthe full force of autumn descends on the palace\n\na few fading stars and migrating geese\n\nleaning from a tower I hear a flute\n\nchrysanthemums are silently starting to turn purple\n\nwithout their pink robes lotuses look sad\n\nsea perch are in season but I can't go home\n\na prisoner from Ch'u I wear a strange hat\n\n## 212\n\nTu Fu (712\u2013770) wrote this poem in 761 in Chengtu. Some editions, however, attribute it to Sun Chin (960\u20131017). As autumn brings relief from summer heat, Tu Fu is more concerned that the time has nearly passed for an appointment equal to his aspirations. Hence, he considers submitting a poem to the throne, as he had done previously without success\u2014and as the poet Ssu-ma Hsiang-ju (179\u2013117 B.C.) once did during the Han dynasty at Chinma (Bronze Horse) Gate outside the Secretariat. The Chinese pounded their clothes on flat rocks when washing them, but they also pounded new material in autumn to toughen the fabric for winter. The last line quotes a poem in the _Book of Odes_ that expresses disappointment in growing old without achieving one's ambition.\n\n## _Early Autumn_\n\nTU FU\n\nBefore the fiery clouds have quenched their fearsome peaks\n\nresting on a pillow I'm surprised by a breeze\n\nsomewhere garden trees are sighing\n\nand someone pounds a solitary mallet\n\nthrobbing cicadas mourn the setting moon\n\nzigzagging fireflies light the evening sky\n\nI planned to take a poem to Chinma Gate again\n\nbut late at night I scratch my head and sigh at my thin hair\n\n## 213\n\nLi P'u (1063\u20131127) was from Kanchou in Kiangsi province and was known as an honest, forthright official. Here, he describes the eighth full moon, the celebration of which is second only to the lunar New Year in the Chinese calendar. The autumn equinox also takes place around this lunar holiday. The hare that lives on the moon produces the elixir of immortality. When the moon is full the hare is upside down and looks as if it is falling off. The three-legged toad, which consumes the hare's elixir, is visible earlier in the month but hard is to make out when the moon is full. It's said that the sea and sky were once joined and that immortals went back and forth on a magic raft. The Silver River is the Milky Way, which is only navigable when the sky is clear.\n\n## _Mid-Autumn_\n\nLI P'U\n\nA bright spirit in the sky a jeweled mirror rising\n\nsilences the music of immortals in the clouds\n\na perfect wheel at the height of autumn\n\nshines for a thousand miles on a never-ending path\n\nthe clever hare falls off its rim\n\nthe ugly toad stays out of sight\n\nlet's go together on a magic raft\n\nthe next time the Silver River is clear\n\n## 214\n\nTu Fu (712\u2013770) wrote this poem in 758 while attending a banquet in Lantien, southeast of Ch'ang-an at the foot of the Chungnan Mountains. Lantien was named for a nearby mountain known for its jade and was also called Yushan, or Jade Mountain. The banquet was held at a friend's villa to celebrate the ninth day of the ninth month, the ultimate _yang_ holiday. According to ancient custom, men consumed a decoction of prickly ash ( _Zanthoxylum ailanthoides_ ) and drank chrysanthemum-infused wine on this day to expel evil influences and to lengthen their lives. A hat was a gentleman's badge of honor, and to be without one was considered a breach of etiquette. Once, when Chia Meng was at a banquet his hat blew off. Instead of apologizing or being embarrassed he acted as if nothing had happened. Tu Fu shows similar nonchalance by doing the opposite.\n\n## _At Mister Ts'ui's Villa in Lantien on the Ninth_\n\nTU FU\n\nAn old man mourning fall I try to console myself\n\nhappy to have shared this day with friends\n\nthinking my hair too short or my headgear insecure\n\nI laughed and asked someone to fix my hat\n\nblue water falls from a thousand distant streams\n\nJade Mountain is high and its two peaks are cold\n\nthis time next year who will still be healthy\n\nlet's find some prickly ash after we get drunk\n\n## 215\n\nLu Yu (1125\u20131210) was one of the most beloved poets of the Sung dynasty but was too much of an idealist to ever achieve high office. Here, he turns to the countryside for solace. In 279 B.C. the state of Ch'i outfitted a herd of oxen with knives on their horns and oil-soaked rushes on their tails, then set fire to the rushes and drove the oxen into the army of the state of Yen and routed it. The Chinese used wooden or stone mallets to launder clothes, but they also used them for toughening material for winter. Paulownia leaves are bigger than dinner plates. Once, when Hsu Fan was a guest of Ch'en Yuan-lung, Yuan-lung made him sleep on the lower bunk. When Hsu later complained to Liu Pei, Liu reprimanded him, saying, \"It's a small-minded person who wants to sleep in a hundred-foot tower.\" Here, Lu turns this around to his own amusement.\n\n## _Autumn Thoughts_\n\nLU YU\n\nLike oxen on fire we're driven by desire\n\nor we drift like seagulls among rivers and lakes\n\ndays can last years when a person finds peace\n\ngreat concerns vanish for those who stay drunk\n\nmallet sounds fade below a country-lane moon\n\npaulownia leaves blanket my garden in fall\n\nI need somewhere high to let my eyes roam\n\nwhere can I find Yuan-lung's old tower\n\n## 216\n\nTu Fu (712\u2013770) wrote this poem in 762 when he was living in his cottage on the Huanhua River just outside the West Gate of Chengtu, the capital of Szechuan province. A kilometer or two to the south, his reclusive friend Chu Hsi-chen lived on the Chin River. Wearing some sort of hat or head covering was so universal among men in ancient China that even recluses wore something, usually a simple bandanna that they knotted on either side of the head and that looked like a pair of horns. The chestnuts here are water chestnuts. Tu Fu likens his friend to one of the Four Worthies of the Chungnan Mountains who was known as Mister Chiao Village.\n\n## _For Hermit Chu, My Neighbor to the South_\n\nTU FU\n\nMister Chin Village with his blackhorned bandanna\n\ngrows taro and chestnuts and can't be called poor\n\nhis children are glad to see a guest arrive\n\nand birds aren't afraid to eat food on his steps\n\nthe river in autumn isn't five feet deep\n\nand the ferry holds maybe three people\n\nin the sand and bamboo and the river-village dusk\n\nhe sees me to his gate when the moon is new\n\n## 217\n\nChao Ku (815\u2013856) was born in Huai-an in Kiangsu province. After passing the civil service exam in 842, he served in a number of posts and finally as commandant of Weinan just east of Ch'ang-an, where he eventually retired. Huan Yi (d. 392) was known for his military prowess and his adherence to decorum. He was also the most famous flautist of his time. Once, while traveling down a river Wang Hui-chih was informed of his presence in a nearby boat and sent a messenger to request an example of his skill. Huan replied by playing three tunes but made no attempt to meet the son of the famous calligrapher Wang Hsi-chih. Ma Jung (79\u2013166) remains famous today for his commentaries on the Confucian classics, but he also wrote odes, one of them on the flute.\n\n## _Hearing a Flute_\n\nCHAO KU\n\nWhose flute is that in the painted tower\n\nblowing and pausing in harmony with the wind\n\nits sound stops the clouds traveling across the sky\n\nits notes reach my curtain with the winter moon\n\ninspired like the tunes of Huan Yi\n\nreminiscent of Ma Jung's old ode\n\nbut where is the person when the song is done\n\nand the notes continue to float in the air\n\n## 218\n\nLiu K'o-chuang (1187\u20131269) was from Putien in Fukien province and was a leading literary figure of the late Sung dynasty. He was also the compiler of the original edition of _Poems of the Masters._ Hence, this poem must have been added sometime later. The sound of fall is that of the north wind. The \"leaves\" in the fifth line refers to tea leaves. The lower reaches of the Yangtze remain famous for freshwater crabs, which reach their peak of maturity in the ninth and tenth lunar months. Both their meat and their roe are still sought-after delicacies in Shanghai and other Delta cities this time of year. The rose mallow, or _Hibiscus mutabilis,_ produces large rose-colored blooms in the eighth and ninth lunar months in China.\n\n## _Winter Scene_\n\nLIU K'O-CHUANG\n\nI love to wake to morning light beneath a frosted window\n\nthe sound of fall growing louder in the bamboo grove outside\n\nI tell a serving maid to heat my new pavilion\n\nand call a young attendant to iron last winter's clothes\n\nthe leaves are green and tender the wine is freshly made\n\nthe oranges sweet and golden the crabs are truly fat\n\nand mallows and chrysanthemums lovely in the garden\n\nmay I never part from such delights again\n\n## 219\n\nCommenting on the hexagram _Fu_ (The Return), the _Yiching_ : 24 says, \"When a single _yang_ line appears below, this is the time of the winter solstice.\" Many customs developed in China in association with this solar date when days finally begin to grow longer again. On the solstice, women began embroideries to be completed by New Year, and men stuffed reed ashes into flutes and blew. The ashes were supposed to float upward if Heaven's and man's seasons were in harmony. Prior to the solstice they floated downward. The color and shape of clouds were also noted as forecasting prospects for the coming year. Tu Fu wrote this poem in 766 while staying in Fengchieh in the Yangtze Gorges. It was still unsafe to return to his home, as the An Lu-shan Rebellion had left North China at the mercy of local warlords as well as the Turks. Hence, Tu Fu has no enthusiasm for wine.\n\n## _Winter Solstice Eve_\n\nTU FU\n\nHeaven's times and man's affairs hurry us along\n\non winter solstice _yang_ appears and spring returns\n\nto silk embroideries a thread is added\n\nout of long flutes reed ashes fly\n\nthe shores wait for New Year to set willows free\n\nthe hills battle cold to liberate plum trees\n\nthe shapes of clouds are the same as back home\n\nI tell my son to finish my wine\n\n## 220\n\nLin Pu (967\u20131028) spent most of his life near Hangchou and steadfastly refused to become an official, preferring to live as a recluse just outside the city's West Gate. During one stretch he didn't enter the city for twenty years. He also never married and doted instead on his pet cranes and plum trees. In the fifth and sixth lines, migrating birds confuse the shimmering white petals with snow, and if butterflies could only survive the cold, they would surely mistake the plum's blossoms for white-winged colleagues from a higher realm. Poets often tapped on a sounding board to keep time while singing poetry.\n\n## _How Plum Flowers Embarrass a Garden_\n\nLIN PU\n\nWhen everything has faded they alone shine forth\n\nencroaching on the charms of smaller gardens\n\ntheir scattered shadows fall lightly on clear water\n\ntheir subtle scent pervades the moonlit dusk\n\nsnowbirds look again before they land\n\nbutterflies would faint if they but knew\n\nthankfully I can flirt in whispered verse\n\nI don't need a sounding board or winecup\n\n## 221\n\nHan Yu (768\u2013824) was one of the most famous literati of the T'ang and an ardent Confucian. When Emperor Hsien-tsung (r. 805\u2013820) welcomed the Buddha's finger-bone to the palace in 819, the ceremonies were too much for Han, and he submitted a memorial criticizing such worship, suggesting it would shorten the emperor's life span. For his temerity Han was banished to the southeast coast, from which many officials never returned. Here, he finds himself southeast of the capital, stuck at Lankuan Pass in the Chinling\/Chungnan Mountains. While he was waiting for the snow to melt, his nephew, Han Hsiang, heard of his plight and paid him a visit. Han Hsiang later took up residence in a cave not far to the west of Lankuan Pass and was eventually ranked among Taoism's Eight Immortals.\n\n## _For My Nephew, Hsiang, on My Demotion and Arrival at Lankuan Pass_\n\nHAN YU\n\nI submitted a memorial to the palace at dawn\n\nby dusk I was bound for Chaoyang two thousand miles away\n\nI hoped to rid the court of evil ways\n\nbut dared in my senility to begrudge a few more years\n\nChinling clouds now bar me from my home\n\nLankuan snows still block the path ahead\n\nthere must be a reason you've traveled this far\n\nno doubt to collect my bones from some infested river\n\n## 222\n\nWang Chung (fl. thirteenth century) lived in North China when the Mongols began a series of invasions that brought the Sung dynasty to an end. Wang Ts'an (177\u2013217) wrote odes that described the suffering of people during war, and many of Tu Fu's (712\u2013770) best-known poems described a country in turmoil and his own separation from home. In the _Book of Odes_ the wagtail represents the closeness of brothers who sing out when one of them is in trouble. The magpie is wary of traps and builds its nest at the tops of trees where several branches join. But here it is forced to make do with a single branch. Ts'ao Ts'ao's \"Short Ballad\" is also in the backgound: \"The moon is bright and stars are few \/ a magpie flies south \/ circling a tree three times \/ on which limb can it rest?\" According to Chang Hua's _Powuchih,_ a Taoist immortal named Chung-shan brewed a wine that could keep a person drunk for three years.\n\n## _Spears and Shields_\n\nWANG CHUNG\n\nWhere can I go when spears and shields are clashing\n\nmy one success has been to turn my temples white\n\nmy steps more or less have followed a Wang Ts'an ode\n\nand my heart feels like a Tu Fu poem\n\na wagtail can't be heard a thousand miles away\n\na magpie spends the winter on a single moonlit branch\n\nwhere can I get some of Chung-shan's Three Year Wine\n\nand stay completely drunk until peaceful times begin\n\n## 223\n\nCh'en T'uan (871\u2013989) was from Liuyi in Honan province, Lao-tzu's old hometown. Although he hoped to become an official during the Later T'ang dynasty (923\u2013935), he was so disappointed by the chaos of the times that he returned to the hermitage he had built earlier at the foot of Hua-shan and lived there as a recluse for the rest of his long life. Ch'en cultivated Taoist meditation and often remained in a sleeplike trance for months. His _Wuchitu (Diagram of the Limitless_ ) was a major influence on early proponents of neo-Confucianism. Red dust refers to the world of the senses, purple cords were attached to the seals of high officials, and red paint was reserved for the households of the elite. The \"spring\" of the last line refers to the season.\n\n## _Returning to My Retreat_\n\nCH'EN T'UAN\n\nThrough the red dust I tramped for ten years\n\ngreen mountains though were often in my dreams\n\na purple cord brings fame but can't compare to sleep\n\ncrimson gates are grand but having less is better\n\nhow sad to hear swords guarding a feeble lord\n\nhow depressing the songs of noisy drunks\n\nI'm taking my old books back to my retreat\n\nto wildflowers and birdsongs and the same old spring\n\n## 224\n\nTu Hsun-ho (846\u2013907) was from Shihtai, just south of the Yangtze port of Kueichih and the sacred mountain of Chiuhuashan, in whose foothills he spent most of his life studying for the civil service exams. When he finally passed at the age of forty-five, he was appointed to the Hanlin Academy under the patronage of the former warlord Chu Wen (852\u2013912), whose political machinations also brought the T'ang dynasty to an end. Tu's poetry, nearly all of which dates from before his success in the exams, is noteworthy for its focus on the distress of common people and the disorder of the times. It was common for the bereaved, after the death of a parent or spouse, to live in a thatched hut for up to three years and to wear rough hemp. Mulberry trees were used to feed silkworms. Taxes were imposed on land regardless of crop yields, and \"seedling money\" was often assessed by local officials and landlords when revenues fell short. Also, corv\u00e9e\u2014forced labor on government work projects\u2014was calculated according to the number of males in a family.\n\n## _A Mountain Widow_\n\nTU HSUN-HO\n\nMy husband died in battle my home is now a hut\n\nmy clothes are made of hemp my hair is like a broom\n\nthey keep collecting taxes on mulberry stumps\n\nand still expect seedlings from overgrown fields\n\nevery day I forage for roots and plants to eat\n\nor chop green wood and gather leaves to burn\n\neven in the deepest depths of the mountains\n\nthere's nowhere to go to escape corv\u00e9e\n_Postscript_\n\nThe system of romanization used in this book to represent the sounds of modern Chinese is a modified version of the one that first appeared in R.H. Mathews's _Chinese-English Dictionary_ (1931), which itself was based on the earlier publication of Thomas Wade's _Syllabary_ (1867) and Herbert Giles's _Chinese-English Dictionary_ (1912). All romanization systems have their shortcomings, but I have found the Wade-Giles the least bizarre. According to this system, aspirated consonants are unvoiced ( _t_ ' as in \"toy\", _ch_ ' as in \"church\") and unaspirated consonants are voiced (t as in \"dog\" and _ch_ like _j_ in \"jam\")\u2014reflecting the system's European origin. The only unfortunate romanization is the letter j, which is pronounced r, as in \"row.\" (Compare this to the x, the z, and the _q_ of the Pinyin system devised by the Cultural Revolution's masters of torture and to which most scholars now adhere for fear of being out of step.) Vowels are generally longer than in English, but the final _e_ or _u_ is given little or no emphasis. Thus, the _tzu_ in _Lao-tzu_ is pronounced the same as the final sound of the word \"adze.\" In the interest of simplicity I have omitted the umlauts of the Wade-Giles system. And for the names of places and books I have usually omitted apostrophes and hyphens and run words together (e.g. Loyang, instead of Lo-yang, but Ch'ang-an, as it has become customary and avoids ambiguity of where to break the syllables).\n\nAs far as I know, this edition is the first in a Western language to include translations of all 224 poems in the _Chienchiashih (Poems of the Masters)._ Admiral Ts'ai T'ing-kan (1861\u20131935) translated all the four-line poems and published the resulting collection in 1932 with the University of Chicago Press, under the title _Chinese Poems in English Rhyme._\n\nDozens of Chinese commentaries to the _Chienchiashih_ have been published over the years, especially over the past few decades. Although most are too superficial to deserve mention, I have found the following five useful:\n\nChang Che-yung: _Ch'ien-chia-shih p'ing-chu_ (Shanghai: Huatung Normal University Press, 1982);\n\nT'ang Lin and Yao Feng: _Ch'ien-chia-shih chu-hsi_ (Lanchou: Kansu People's Press, 1982);\n\nYang Hung-ju: _Ch'ien-chia-shihp'ing-yi_ (Beijing: Huawen Publishing Company, 1999);\n\nWang Ch'i-hsing: _Ch'ien-chia-shih hsin-chu_ (Wuhan: Hupei People's Press, 1981);\n\nWen Chieh: _Hsiang-chieh ch'ien-chia-shih_ (Hong Kong: Wenkuang Publishing Company, 1987).\n_Timeline_\n\nHsia Dynastyc. 2200 B.C.\n\nChou Dynasty1122\u2013255 B.C.\n\nWarring States Period733\u2013221 B.C.\n\nCh'in Dynasty221\u2013207 B.C.\n\nHan Dynasty206\u2013220\n\nThree Kingdoms Period221\u2013265\n\nWu Dynasty222\u2013277\n\nSix Dynasties Period222\u2013589\n\nWestern Chin Dynasty265\u2013316\n\nChin Dynasty278\u2013419\n\nLiang Dynasty502\u2013557\n\nCh'en Dynasty557\u2013588\n\nSui Dynasty589\u2013618\n\nT'ang Dynasty618\u2013906\n\nAn Lu-shan Rebellion755\u2013762\n\nGreater Yen Dynasty755\u2013762\n\nTali Period766\u2013779\n\nFive Dynasties Period907\u2013960\n\nLater T'ang Dynasty923\u20131643\n\nSung Dynasty960\u20131278\n\nSouthern Sung Dynasty1127\u20131279\n\nYuan Dynasty1280\u20131368\n_About the Translator_\n\nBill Porter assumes the pen name Red Pine for his translations. He was born in Los Angeles in 1943 and grew up in Northern Idaho. He served a tour of duty in the U.S. Army, graduated from the University of California, Santa Barbara, and attended graduate school at Columbia University. Uninspired by the prospect of an academic career, he dropped out of Columbia in 1972 and moved to a Buddhist monastery in Taiwan. Following four years with the monks and nuns, he struck out on his own, eventually working at English-language radio stations in Taiwan and Hong Kong, which afforded him the opportunity to travel extensively throughout China. He has visited the homesteads and graves of all the major poets in this collection, as well as the places where they wrote many of these poems. A Chinese translation of his book _Road to Heaven_ has re-inspired Chinese interest in their own hermit tradition.\n\nIn recent years, Bill Porter has devoted himself to the translation of Buddhist texts. Currently, he is is working on the Heart and Lankavatara sutras.\n_Index of Authors_\n\nAncient Recluse, The\n\nAuthor Unknown\n\nChang Chi\n\nChang Feng\n\nChang Hu\n\nChang Shih\n\nChang Wei\n\nChang Yueh, Delayed..., Drinking..., Given the Word...\n\nChao Ku, Autumn Longing..., Hearing a Flute, Reflections...\n\nChao Shih-hsiu\n\nChao Ting\n\nCheng Hui\n\nChia Chih\n\nChia Tao, Looking..., Seeing Off...\n\nChih-nan\n\nChin Ch'ang-hsu\n\nChou Pi-ta\n\nChu Fang\n\nChu Hsi, Autumn Moon, On the Pomegranate Flower, Reflections I, Reflections II, Spring Day\n\nChu Shu-chen, Falling Flowers, Impressions\n\nCh'ang Chien\n\nCh'ao Yueh-chih\n\nCh'en Tzu-ang, Parting..., Seeing Off..., To Vice Censor Ch'iao\n\nCh'en T'uan\n\nCh'eng Hao, Casual Poem..., Occasional Poem..., Strolling..., Visiting..., Written...\n\nCh'i-wu Ch'ien\n\nCh'ien Ch'i, On Geese..., On Meeting...\n\nCh'ien Hsu\n\nCh'iu Wei\n\nCh'u Kuang-hsi\n\nCh'u-mo\n\nFan Ch'eng-ta, A Farm Family, Village Events\n\nHan Hung\n\nHan Yu, For My Nephew..., Late Spring, Light Rain...\n\nHo Chih-chang\n\nHsieh Fang-te, Lament..., Peach Blossoms...\n\nHsu Yuan-chieh\n\nHsuan-tsung\n\nHsueh Ying\n\nHuang T'ing-chien, Chingming, Written...\n\nHuang-fu Jan\n\nHui-hung\n\nHung Tzu-k'uei\n\nKao Chu\n\nKao Ch'an\n\nKao P'ien\n\nKao Shih, For Censor Cheng..., Ode..., To Chang Hsu...\n\nKeng Wei\n\nLei Chen\n\nLi Ang\n\nLi Chia-yu\n\nLi Chiao\n\nLi Ch'i\n\nLi Pai, Chingping Ode, Chiupu River Song, Climbing..., On Yellow Crane Tower..., Seeing Off a Friend, Seeing Off a Friend..., Sitting Alone..., Thoughts..., Traveling...\n\nLi P'u\n\nLi Shang-yin\n\nLi She\n\nLi Shih-chih\n\nLin Chen\n\nLin Hung, Palace Ode--I, Palace Ode--II\n\nLin Pu\n\nLin Sheng\n\nLing-hu Ch'u\n\nLiu Chi-sun\n\nLiu Han\n\nLiu K'o-chuang, Oriole Shuttle, Winter Scene\n\nLiu Yu-Hsi, Black Robe Lane, Ode..., Peach Blossoms..., Visiting...\n\nLo Pin-wang\n\nLu Mei-p'o, Snow and Plum--I, Snow and Plum--II\n\nLu Tung-pin\n\nLu Yu, Autumn Thoughts, New Bamboo...\n\nMeng Hao-jan, Calling..., Overlooking..., Seeing Off..., Spring Dawn, Written...\n\nOu-yang Hsiu\n\nPai Chu-yi\n\nPai Yu-ch'an\n\nShao Yung\n\nShen Ch'uan-ch'i\n\nSsu-k'ung Shu\n\nSsu-ma Kuang\n\nSu Shih, Attending..., Begonia, Drinking..., Flower Shadows, For Liu Ching-wen, Mid-Autumn Moon, Spring Night\n\nSu T'ing\n\nSun T'i, On Seeing..., Spending...\n\nSung Chih-wen\n\nTai Fu-ku\n\nTai Min\n\nTai Shu-lun\n\nTing Hsien-chih\n\nTou Shu-hsiang\n\nTseng Chi\n\nTsu Yung, Mister Su's Country Retreat, Traveling...\n\nTs'ai Ch'ueh\n\nTs'ai Hsiang\n\nTs'ao Pin\n\nTs'en Shen, On Climbing..., Responding..., Thinking..., To Censor Tu...\n\nTs'ui Hao, Ballad..., Yellow Crane Tower\n\nTs'ui T'u\n\nTu Ch'ang\n\nTu Fu, At Mister Ts'ui's Villa..., Autumn Inspiration--I, Autumn Inspiration--II, Autumn Inspiration--III, Autumn Inspiration--IV, Climbing Yenchou Tower, Climbing Yuehyang Tower, Early Autumn, Encountering Rain... I, Encountering Rain... II, For Hermit Chu..., Inspired, Inspired, Jade Terrace Temple, Miscellaneous Poem..., On Seeing..., Painted..., Quatrain, Recording..., Responding..., River Village, Shrine..., Staying..., While Drinking... I, While Drinking... II, Winter Solstice Eve\n\nTu Hsun-ho\n\nTu Lei\n\nTu Mu, Anchored..., Autumn Evening, Chiangnan Spring, Grave Sweeping Day\n\nTu Shen-yen, At an Imperial Banquet..., Replying..., Seeing Off...\n\nWang An-shih, New Year's Day, North Mountain, Spring Night, Written...\n\nWang Chi\n\nWang Chia, Festival Day, Spring Clearing\n\nWang Chih-huan\n\nWang Chung\n\nWang Ch'ang-ling, In Reply..., Seeing Off...\n\nWang Ch'i, Plum, Visiting...\n\nWang Feng-yuan\n\nWang Kuei\n\nWang Po\n\nWang Wan\n\nWang Wei, Bamboo Retreat, Passing..., Responding..., Seeing Off..., Chungnan Mountains, Written...\n\nWang Yu-ch'eng\n\nWei Ying-wu, Chuchou's West Stream, In Reply to Li Huan, In Reply to Li Tan, To Secretary Ch'iu...\n\nYang Chiung\n\nYang Chu-yuan\n\nYang P'u\n\nYang Wan-li, At Chingtzu Temple..., Spring Lament, Waking Up...\n\nYeh Shao-weng\n\nYeh Ts'ai\n\nYen Shu\n_Index of Titles_\n\nAboard a Boat on a Moonlit Night\n\nAnchored Overnight at Maple Bridge\n\nAnchored Overnight on the Chinhuai\n\nAt an Imperial Banquet in Penglai Hall Offering Praise for the Chungnan Mountains\n\nAt Chingtzu Temple Seeing Off Lin Tzu-fang at Dawn\n\nAt Mister Ts'ui's Villa in Lantien on the Ninth\n\nAttending a Banquet at Princess Ch'ang-ning's Eastern Estate\n\nAttending a Banquet at the New Residence of Princess An-lo\n\nAttending a Banquet on the Great Beginning\n\nAutumn at the Gate\n\nAutumn Day\n\nAutumn Day on the Lake\n\nAutumn Evening\n\nAutumn Inspiration\u2014I\n\nAutumn Inspiration\u2014II\n\nAutumn Inspiration\u2014III\n\nAutumn Inspiration\u2014IV\n\nAutumn Longing in Ch'ang-an\n\nAutumn Moon\n\nAutumn Thoughts\n\nAway from Home at the Beginning of Summer\n\nBallad of Changkan\n\nBamboo Retreat\n\nBegonia\n\nBlack Robe Lane\n\nCalling on Censor Yuan without Success\n\nCasual Poem on a Spring Day\n\nChiangnan Spring\n\nChingming (Huang T'ing-chien)\n\nChingming (Wang Yu-ch'eng)\n\nChingping Ode\n\nChiupu River Song\n\nChuchou's West Stream\n\nChungnan Mountains, The\n\nClimbing a Mountain\n\nClimbing Hsieh T'iao's North Tower in Hsuancheng in Autumn\n\nClimbing to Canopy Pavilion on a Summer Day\n\nClimbing White Stork Tower\n\nClimbing Yenchou Tower\n\nClimbing Yuehyang Tower\n\nCold Food (Chao Ting)\n\nCold Food (Han Hung)\n\nCold Spring Pavilion\n\nCommiserating with Gentleman-in-Waiting Wang on Tungting Lake\n\nConcubine's Lament, A\n\nCrossing the Yangtze\n\nDelayed on the Szechuan Road\n\nDrinking at Night in Yuchou\n\nDrinking on the Lake As It Clears Then Rains\n\nDrinking Wine on Grave Sweeping Day\n\nEarly Autumn\n\nEarly Spring\n\nEarly Spring East of Town\n\nEncountering Rain at Changpa Reservoir One Evening While Enjoying a Cool Breeze with Rich Young Men and Their Singsong Girls\u2014I\n\nEncountering Rain at Changpa Reservoir One Evening While Enjoying a Cool Breeze with Rich Young Men and Their Singsong Girls\u2014II\n\nEvents of Late Spring\n\nFalling Flowers\n\nFarm Family, A\n\nFestival Day\n\nFlower Garland Song\n\nFlower Shadows\n\nFor Censor Cheng on Being Banished to Fukien\n\nFor Gentleman-in-Attendance Kao\n\nFor Hermit Chu, My Neighbor to the South\n\nFor Liu Ching-wen\n\nFor My Nephew, Hsiang, on My Demotion and Arrival at Lankuan Pass\n\nFrost and the Moon, The\n\nGazing across the Countryside\n\nGazing at the Valley of Ch'in\n\nGiven the Word \"Forest\" by His Majesty at the Licheng Palace Library Banquet\n\nGrave Sweeping Day\n\nHearing a Flute\n\nHerdboy\n\nHow Plum Flowers Embarrass a Garden\n\nImperial Entourage on the Road from Tengfeng, The\n\nImpressions\n\nIn Praise of Huaching Palace\n\nIn Reply\n\nIn Reply to Li Huan\n\nIn Reply to Li Tan\n\nIn Reply to the Prefect of Wuling\n\nIn Reply to Ting Yuan-chen\n\nInspired (\"The heartbreaking flood\")\n\nInspired (\"Willow fuzz\")\n\nJade Terrace Temple\n\nLament of the Silkmaid\n\nLate Spring (Han Yu)\n\nLate Spring (Ts'ao Pin)\n\nLight Rain in Early Spring\n\nLooking for a Recluse without Success\n\nMeditation Hall behind Poshan Temple, The\n\nMid-Autumn\n\nMid-Autumn Moon\n\nMiscellaneous Poem at Chinchou\n\nMister Su's Country Retreat\n\nMorning Court at Taming Palace\n\nMountain Pavilion on a Summer Day, A\n\nMountain Widow, A\n\nNew Bamboo at East Lake\n\nNew Year's Day\n\nNorth Mountain\n\nOccasional Poem on an Autumn Day\n\nOccasional Poem on the Arrival of Spring\n\nOde to the Autumn Wind\n\nOde to the Past\n\nOn a Painting of Playing Football\n\nOn Climbing Tsungchih Pagoda\n\nOn Deputy Prefect Tu Taking Up a Post in Shuchou\n\nOn Geese Turning Back\n\nOn Lantern Festival at Imperial Request (Ts'ai Hsiang)\n\nOn Lantern Festival at Imperial Request (Wang Kuei)\n\nOn Meeting a Chivalrous Man\n\nOn Mister Yuan's Country Retreat\n\nOn Seeing a Landscape Painted by My Commissioner Cousin for Li Ku\n\nOn Seeing Princess Yung-lo Leave for Manchuria\n\nOn the Lake\n\nOn the Pomegranate Flower\n\nOn the Sanchu Road\n\nOn Thinking of My Lord's Grace\n\nOn Yellow Crane Tower Hearing a Flute\n\nOriole Shuttle, The\n\nOverlooking Tungting Lake\u2014For Prime Minister Chang\n\nPainted on the Wall of Master Hsuan-wu's Room\n\nPalace Ode\u2014I\n\nPalace Ode\u2014II\n\nParting from a Friend on a Night in Spring\n\nPassing Hsiangchi Temple\n\nPassing the Shrine to the Master of the Three Gates\n\nPeach Blossoms of Chingchuan Hermitage, The\n\nPeach Blossoms of Hsuantu Temple, The\n\nPear Blossom in the East Wing, A\n\nPlum, The\n\nPrivate Thoughts\n\nQuatrain (Chin-nan)\n\nQuatrain (Tu Fu)\n\nReaching Sword Gate Pass After Touring the Land of Shu\n\nRecording My Thoughts While Traveling at Night\n\nReflections at a River Tower\n\nReflections While Reading\u2014I\n\nReflections While Reading\u2014II\n\nReplying to a Poem by Prime Minister Lu of Chinling\n\nResigning as Minister\n\nResponding to Secretary Chia Chih's \"Morning Court at Taming Palace\" (Ts'en Shen)\n\nResponding to Secretary Chia Chih's \"Morning Court at Taming Palace\" (Tu Fu)\n\nResponding to Secretary Chia Chih's \"Morning Court at Taming Palace\" (Wang Wei)\n\nRetiring After Attending a Tea at Hsuante Palace\n\nReturning to My Retreat\n\nRiver Village\n\nRoads of Loyang, The\n\nSaying Goodbye on the Yi River\n\nSaying Goodbye to Lu Ch'in-ch'ing\n\nSeeing Off a Friend\n\nSeeing Off a Friend Leaving for Shu\n\nSeeing Off Chao Tsung at Night\n\nSeeing Off Chu Ta Leaving for Ch'in\n\nSeeing Off Editor Ts'ui Marching East\n\nSeeing Off Spring\n\nSeeing Off Spring on the Last Day of April\n\nSeeing Off Supply Director Kuo\n\nSeeing Off Ts'ui Jung\n\nSeeing Off Yuan Er on a Mission to Anhsi\n\nServing in the Secretariat\n\nSeventh Night, The\n\nShengkuo Temple\n\nShrine to Yu the Great\n\nSitting Alone on Chingting Mountain\n\nSnow and the Plum\u2014I, The\n\nSnow and the Plum\u2014II, The\n\nSpears and Shields\n\nSpending a Summer Night with My Cousin Talking about the Past\n\nSpending the Night at Lunghsing Temple\n\nSpending the Night at Yunmen Temple Pavilion\n\nSpring Clearing\n\nSpring Complaint\n\nSpring Dawn\n\nSpring Day\n\nSpring Lament\n\nSpring Night (Su Shih)\n\nSpring Night(Wang An-shih)\n\nStaying Overnight at the Chancellery in Spring\n\nStopping at Peiku Mountain\n\nStrolling outside Town\n\nSummer Day\n\nSurprised by Autumn on the Fen\n\nSwing, The\n\nThinking of My Home in Ch'ang-an While Traveling with the Army on the Ninth\n\nThoughts on a Quiet Night\n\nTo Censor Tu at the Chancellery\n\nTo Chang Hsu after Drinking\n\nTo Secretary Ch'iu on an Autumn Night\n\nTo Vice Censor Ch'iao\n\nTraveler's Thoughts on a Spring Evening, A\n\nTraveling Away from Home\n\nTraveling on the Yangtze\n\nTraveling South of the Yangtze\n\nVillage Dusk\n\nVillage Events\n\nVisiting a Private Garden in Late Spring\n\nVisiting a Private Garden without Success\n\nVisiting Chang's Garden at the Beginning of Summer\n\nVisiting Crescent Pond\n\nVisiting Hsuantu Temple Again\n\nWaiting for a Friend\n\nWaking Up in Early Summer\n\nWhile Drinking at the Chuchiang Waterway\u2014I\n\nWhile Drinking at the Chuchiang Waterway\u2014II\n\nWinter Night\n\nWinter Scene\n\nWinter Solstice Eve\n\nWritten at an Inn in Linan\n\nWritten at Chulin Temple\n\nWritten at Huainan Temple\n\nWritten at My Wang River Retreat after a Steady Rain\n\nWritten at Secretary Wang's Bamboo Tower\n\nWritten at the South Tower of Ochou\n\nWritten at the Sungting Relay Station\n\nWritten in the Palace\n\nWritten on a Screen\n\nWritten on a Wall\n\nWritten on the Wall of Master Yi's Meditation Hut\n\nWritten on Mister Lakeshade's Wall\n\nWritten on the Wall of an Inn\n\nWritten While Serving at Jade Hall\n\nYellow Crane Tower\n\nBooks by Red Pine\n\n_The Diamond Sutra: The Perfection of Wisdom_ (Counterpoint Press, 2001)\n\n_The Collected Songs of Cold Mountain_ (Copper Canyon Press, 2000)\n\n_The Zen Works of Stonehouse: Poems and Talks of a Fourteenth-Century Chinese Hermit_ (Mercury House, 1999)\n\n_The Clouds Should Know Me by Now: Buddhist Poet Monks of China_ (co-edited with Mike O'Connor, Wisdom Publications, 1998)\n\n_Lao-tzu's Taoteching_ (Mercury House, 1996)\n\n_Guide to Capturing a Plum Blossom by Sung Po-jen_ (Mercury House, 1995)\n\n_Road to Heaven: Encounters with Chinese Hermits_ (Mercury House, 1993)\n\n_The Zen Teachings of Bodhidharma_ (North Point Press, 1989)\n\n_Pu Ming's Oxherding Pictures and Verses_ (Empty Bowl, 1983)\nCopyright 2003 by Red Pine\n\nAll rights reserved\n\nCover art: _Scholar,_ Anonymous, Sung Dynasty.\n\nCourtesy of the National Palace Museum, Taipei, Taiwan, Republic of China.\n\nISBN: 978-1-55659-159-0\n\neISBN: 978-1-61932-088-8\n\nSupport Copper Canyon Press:\n\nIf you have enjoyed this title, please consider supporting Copper Canyon Press and our dedication to bringing the work of emerging, established, and world-renowned poets to an expanding audience through e-books:\n\nwww.coppercanyonpress.org\/pages\/donation.asp\n\nContact Copper Canyon Press:\n\nTo contact us with feedback about this title send an e-mail to:\n\nebooks@coppercanyonpress.org\n\nThe Chinese character for poetry is made up of two parts: \"word\" and \"temple.\" It also serves as pressmark for Copper Canyon Press.\n\nPoetry is vital to language and living. Since 1972, Copper Canyon Press has published extraordinary poetry from around the world to engage the imaginations and intellects of readers, writers, booksellers, librarians, teachers, students, and donors.\nWe are grateful for the major support provided by:\n\nAnonymous\n\nArcadia Fund\n\nJohn Branch\n\nDiana and Jay Broze\n\nBeroz Ferrell & The Point, llc\n\nMimi Gardner Gates\n\nGull Industries, Inc.\n\non behalf of William and Ruth True\n\nMark Hamilton and Suzie Rapp\n\nCarolyn and Robert Hedin\n\nSteven Myron Holl\n\nRhoady and Jeanne Marie Lee\n\nMaureen Lee and Mark Busto\n\nNew Mexico Community Foundation\n\nH. Stewart Parker\n\nPenny and Jerry Peabody\n\nJoseph C. Roberts\n\nCynthia Lovelace Sears and Frank Buxton\n\nThe Seattle Foundation\n\nCharles and Barbara Wright\n\nThe dedicated interns and faithful volunteers of Copper Canyon Press\n\nTo learn more about underwriting Copper Canyon Press titles, please call 360-385-4925 ext. 103\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nCopyright \u00a9 2005\n\nby Paul Elder\n\nAll rights reserved, including the right to reproduce this work in any form whatsoever, without permission in writing from the publisher, except for brief passages in connection with a review.\n\nCover design by Marjoram Productions\n\nCover digital imagery: eye \u00a9 2004 PictureQuest\/Image Ideas; angel \u00a9 2004 Jonathan Friedman. All rights reserved\n\nHampton Roads Publishing Company, Inc.\n\nCharlottesville, VA 22902\n\nwww.hrpub.com\n\nLibrary of Congress Catalog Card Number: 2004028314\n\nISBN 978-1-57174-429-6\n\nPrinted in the United States of America\n\nSB\n\n10 9 8 7 6 5\n\nwww.redwheelweiser.com\n\nwww.redwheelweiser.com\/newsletter\n\n# Contents\n\nAcknowledgments\n\nIntroduction: Eyes of an Angel\u2014\"Windows to Your Soul\"\n\n1: The Awakening\n\n2: Where Was God?\n\n3: Search for Truth\n\n4: Expanding Horizons\n\n5: To Die Again\n\n6: Guidance\n\n7: We Are Not Alone\n\n8: The Ultimate Mission\n\n9: Spiritual Connections\n\n10: Exploration\n\n11: Our Journey Together\n\n12: Removing Walls\n\n13: Healing Hearts\n\n14: The Reading\n\n15: Affirmation\n\nPoem: Eyes of an Angel\n\nAbout the Author\n\n# Acknowledgments\n\nThere are many people who have contributed to the completion of this book to whom I am eternally grateful. To Robert Friedman, Frank DeMarco, Ginna Colburn, and the entire staff at Hampton Roads Publishing Company, I offer my sincerest appreciation. Sarah, Richard, Grace and all the rest\u2014thank you for your dedication to excellence.\n\nTo an extraordinary agent and friend, Bill Gladstone of Waterside Productions in California, thank you for your enthusiasm and faith.\n\nTo my friend and colleague, Skip Atwater, author, former Operations Officer for U.S. Army Intelligence \"Project Star Gate,\" and current Research Director at The Monroe Institute, I offer my heartfelt gratitude and admiration.\n\nTo Dale Graff, author and former civilian director of \"Project Star Gate,\" thank you for your friendship and tutoring.\n\nTo my friend, Will Murray, prolific author and extraordinary writer, thank you so much for your assistance, advice and tutoring. I am forever grateful for your guidance and friendship.\n\nTo Laurie Monroe, Darlene Miller, and all my fellow facilitators and friends at The Monroe Institute, in Virginia, thank you for your years of friendship and support.\n\nTo Paul Bernstein, editor of the International Association for Near Death Studies magazine, _Vital Signs_ , thank you for your encouragement. To the people of Swift Current, Saskatchewan, Canada: thank you for your trust and faith over these many years.\n\nTo my friends, Steve Garrison, Ron Reinhart, and my sister, Josie McGuire. Thank you for spending so much of your time reviewing my manuscripts, providing valuable feedback, and keeping me on track. And to the rest of my brothers and sisters, Dale, Joe, Marg, Evalina, Pauline, Eva, Anna, Elizabeth, and Lena, thank you for sharing your love.\n\nI would also like to offer special thanks to numerous other dear friends for their reviews and encouragement: Margaret Whelan, Ariane de Bonvoisin, Duff Marshal, Carol Hansen, Donalee Campbell, David Hughes, Tom Coates, Winston Fuller, Ed Thrall, Tamera Knobel, Celeen Vizer, and Professor Buddy Wynn. Thank you all for your wonderful contributions.\n\nThank you, Patricia, for returning once again to this world to touch my heart and soul, reminding me of our greater purpose and ultimate goal.\n\nAnd to a most wonderful, gentle soul, thank you, Lisa, for all your love and support.\n\nAnd finally, I want to thank three very special people who have shared with me a lifetime of love, laughter and tears. Candace, Stacey, and David, you will always be my special Angels, and I will love you forever.\n\n# Introduction\n\n# Eyes of an Angel\u2014\"Windows to Your Soul\"\n\nThis world we live in is truly a remarkable place: so many mysteries to solve, incredible wonders to marvel at, and so much to learn. But, all things considered, what's even more astonishing than the power of the human mind is the fact that none of this is new. We already know the answers to life's mysteries, we've only forgotten. It's all there, just below the surface, tugging at us, begging us to take the time to look. And the best place to start is in your own eyes, and the eyes of your loved ones.\n\nHere's a common experience. You're meeting someone for the first time. Reaching out to shake his hand, you glance briefly into his eyes. And then a strange thing happens\u2014you get the oddest feeling you've met this person before. Another place perhaps, another time? Even though you're sure it's not the case, you're left with an inner turmoil, a fluttering in your heart that you can't explain. Perplexed, you shrug it off as some strange quirk of the mind. But, wait a minute! What if it isn't just some weird anomaly? What if there _is_ more to it than meets the eye?\n\nIt's often said that our eyes are the windows to our souls. What if this were true? What if we could somehow learn to recognize and identify these subtle, innocuous feelings and intuitions? Would it not completely alter our perception of the world? Could it possibly lead to a better, more understanding, loving universe?\n\n_Eyes of an Angel_ addresses some of the most revealing and astonishing aspects of the relationships binding each and every soul in our universe. It's the story of the evolution of consciousness of an avowed skeptic to that of an ardent spiritual believer. It's the kind of story I wouldn't have believed had it not happened to me.\n\nWithin these pages you'll find the details of some remarkable conversations and messages from the spirit realms\u2014concepts, thoughts, and lessons from souls who have been through it all before, but choose, out of love, to help the rest of us remember who we really are and where we're heading. These conversations and events really did take place. Since there wasn't, obviously, a way to record the actual dialogues, their reproduction was dependent on my reconstruction of the material. To the best of my ability, I've tried to accurately present these experiences and messages just as they happened. It is my sincerest hope that from within these timeless messages of the soul, you too might recognize your own truths and pathways.\n\nTrue in every respect, _Eyes of an Angel_ offers a detailed, first-hand account of a remarkable spiritual adventure. It's a story about soul travel, angels, spirit guides, and soul mates. But most of all, it's a story about everlasting love.\n\n# 1\n\n# The Awakening\n\n_Through shaded eyes we view the world_\n\n_Present, past, and future swirl_\n\n_Only fleeting glimpse there revealed_\n\n_The mysteries of life concealed_\n\nWhat would you do if you suddenly awoke to find yourself floating near the ceiling of your bedroom? It happened to me, and it changed my life.\n\nI first heard the term \"out-of-body experience\" in the summer of 1979. A local television station had broadcast an interview with a man named Robert Monroe. He had written a popular book, _Journeys Out of the Body_. Intrigued by the incredible story he told, I borrowed a copy from the library and read it from cover to cover.\n\nIn the book, Monroe claimed that in his early forties he began to spontaneously leave his body in spirit form while remaining fully conscious. He gave fantastic detailed descriptions of traveling in this spirit body to places on Earth as well as into the spirit world itself.\n\nWhen the experiences first began, Monroe was afraid that he had become ill, or was actually losing his mind. Submitting himself to intense medical and psychiatric examinations, he was relieved when all tests found him to be healthy and normal. Eventually he just quit worrying about it. Meticulously documenting his experiences, he decided to take advantage of the continuing phenomenon, pursuing every opportunity to explore and learn.\n\n\"Fascinating story,\" I thought, \"but the guy's obviously a brick short of a load.\"\n\nHaving been raised in a poor farm family with ten brothers and sisters, my grasp on reality was far too entrenched in the physical world to give this kind of fantasy much sway. Returning the book to the library, I soon forgot all about it.\n\nTo my amazement, eleven years later, my perception of reality would be changed forever.\n\nThis life-changing event took place at the end of a long and hectic week. It was late Friday night and, looking forward to a quiet weekend, I sat at my desk thinking about the recent events that were complicating my life. It had been more than five years since I was first elected to city council, and everything in my life was running smoothly. Then, one day, some important government people showed up, asking me to make a big commitment. They wanted me to run as a candidate in an upcoming election to the Legislature. With more than a few reservations, I had agreed to let my name stand. But it wasn't long before I was second-guessing my decision.\n\nWith only days remaining before the nomination meeting, I waged an internal battle. Was this what I wanted to do? Would it be good for me? With all the time I'd have to spend away, would our family life suffer? Would it be worth it? Finally, around 1:30 in the morning I gave up and sleepily crawled into bed, vowing to sleep until noon. The second my head hit the pillow, it was lights out.\n\nThe next thing I knew, I was wide-awake. But something was terribly wrong. In dazed confusion, I scrambled to get my bearings. What I was seeing was impossible! Instead of being in my bed, I was floating like a balloon near the ceiling, bumping lightly against its stippled surface.\n\nDisoriented, a landslide of possibilities flooded my mind. There had to be a rational explanation. Below me in bed were two sleeping bodies that seemed so familiar I couldn't divert my attention from them. As I looked down, a stunning realization hit me: the woman in the bed was my wife, and the guy sleeping beside her was me.\n\nThis couldn't be happening. Here I was, floating around my bedroom ceiling, without a clue as to how I'd gotten there. Closing my eyes, I shook my head, praying that when I reopened them, everything would be back to normal again. But nothing changed. I was still floating in midair.\n\nMy thoughts reeling, I knew I wasn't dreaming. But there was another possibility. I began to wonder if I had somehow died. It didn't seem likely, though; I felt completely normal, my mind had never been clearer.\n\nThe memory of Robert Monroe's book flashed into my awareness. Maybe, I thought, Monroe wasn't crazy after all. This was absolutely real! Although it seemed beyond belief, somehow the experience felt strangely familiar. It had the feeling of routine, like I had done this many times before. I knew I was just now returning from some sort of journey, but I had no idea where I might have been.\n\nAs this insight filtered into my mind, a strange calming sensation flooded through me. Most of the fear and apprehension I had been entertaining slowly evaporated into the surrounding darkness. Far from feeling out-of-sorts, I could not remember when I had ever felt better. How could I have felt better? I was doing the impossible\u2014I was flying.\n\nAlthough it seemed such a familiar feeling, it was, nevertheless, unsettling. After a few moments I wondered if I shouldn't be returning to my body. However, I wasn't sure how to make that happen or, for that matter, if it was possible. I didn't have a lot of time to worry about it, though, because with just that simple thought, I began to slowly descend towards the bed.\n\nTo my dismay, when I finally came to a stop, I wasn't in my body. Instead, I hovered beside the bed, about two feet above the floor. At first, I worried about having somehow missed the target, but within moments my concerns subsided as I quickly became engrossed in studying this quiet, peaceful form beside me.\n\nThere didn't seem to be anything unusual or out of place. My body appeared to be in a normal sleeping state. Thinking how odd this was, I chuckled in nervous amazement. Here I was, staring at my own face from an outside perspective, examining it as if it belonged to someone else. It occurred to me that other than looking into a mirror, we never actually see our own faces, so this felt more than a little strange.\n\nIn the darkness, my face seemed a lot older than the one I usually saw in the mirror. It looked weary and surprisingly gray and colorless. The more I thought about it, the more I began to feel compassion for my seemingly abandoned body, and I had to admit that I should probably be taking better care of it.\n\nEmotions building inside me, I thought I had better try again to get back into my body. To my surprise, at the very thought of it, I simply floated across the two foot span, and just like that, I was back inside. For the briefest moment, I felt faint. A spinning dizziness overcame me, everything faded to black, and then, with a startling jolt, I opened my eyes.\n\nBack in my body, I could feel my thoughts slowing down. Like a genie being sucked back into a bottle, I felt squeezed and confined. And I didn't like it at all.\n\nAs I lay in bed lamenting my increasing heaviness, I became aware of a strange vibration gently rocking me. Every molecule in my body seemed about to burst with this bizarre, bubbling electrical energy. As I focused on the vibrations, they began to surge in waves back and forth from my head to feet. It was the most energized, exciting physical feeling I had ever had. My body felt like it was expanding, contracting, and rolling with the sensations. Waves of energy flooded my awareness. I could not conceive of anything more strange or delightful.\n\nFor several minutes I lay analyzing the intense energy until the feeling in my arms and legs returned. I tried to move, but the lower part of my body felt heavy, almost paralyzed. Finally managing to turn my head, I noticed the time on my nightstand clock. It was 3:02 A.M. I had been asleep for only 90 minutes.\n\nClosing my eyes, I lay in bed struggling with my emotions, my mind desperately searching for a rational explanation. When I began to wonder why it happened, a feeling of melancholy came over me. \"My God,\" I begged. \"What on earth is happening to me?\"\n\nThe word \"God\" stuck in my mind. It had been a long time since I'd paid much attention to God, but for some reason I felt the most compelling urge to be thankful for His blessings. As I contemplated the gift of this experience, love and gratitude slowly overwhelmed me. Tears began trailing down my cheeks.\n\nIn this euphoria, my mind began to wander. Dreamlike images of events in my life took shape and then quickly dissolved. And there were faces, so many faces. Some I knew, but most I couldn't place even though they seemed familiar. I was becoming lost in my thoughts when a sudden snap of my body brought me back to the present.\n\nOpening my eyes, my attention was drawn to the ceiling. My eyes had to be playing tricks on me. Not only could I see the stipple on the ceiling but I could also see right through to the clouds and stars in the night sky. Hoping to correct this obvious anomaly, I closed my eyelids tightly and then quickly opened them again. But nothing changed.\n\nMy senses seemed to be reaching overload. My heart was pounding with excitement, while powerful vibrations continued flowing through my body. I couldn't imagine a more incredible sensation, but, at the same time, I worried where it was leading.\n\nA sudden sharp surge in the vibrations gripped my attention. The electrical tingling became so intense I began to wonder if it would be possible to leave my body again. The more I thought about it, the more convinced I became that I could.\n\nI had just begun to consider my next step, when another wave of energy swept through me. My awareness seemed to expand outward until I thought I could feel the air pressure increasing in the room. And then it hit me! A chill ran through my body. I had the creepiest feeling that I wasn't alone. Someone was watching me. I sensed the presence of other energies or beings around me. Appearing as pinpoints of light, they seemed to be everywhere. I started to panic. The thought of having unknown spirits in my room terrified me. I wanted so desperately to float out of my body again but there was no way I could force myself to do so with these \"things\" around.\n\nSummoning up every ounce of courage, I projected a thought into the room asking the intruders to please leave. I waited, but nothing happened. The feeling persisted. Again I asked, this time more forcefully. Still nothing. With mounting anxiety, I mentally yelled into the room, demanding that everyone leave. There was no change. Finally, in desperation, I found myself returning to the Catholic teachings of my youth, and I prayed to God and Jesus to help and protect me. Incredibly, the moment the words became articulated in my mind, I felt a sudden change around me. In an instant every single unwanted presence in the room blinked out and disappeared, leaving me alone in the darkness.\n\nElated, I thought of floating up into the air, and that's all it took. A surge of energy rushed through my body while a high-pitched sound began to resonate in the center of my head, spreading through me. With the sound of air hissing in my ears, I felt a slight wobbling sensation, but then firmly and steadily rose into the air. The rush of vibrations and sounds, so consuming just a moment before, quickly faded as I separated from my physical body and floated silently to the ceiling.\n\nAs I hovered in midair, trying desperately to make sense of it all, a tidal wave of understanding swept through me. Throughout most of my life, I had doubted the existence of God. The God that we were taught about as children seemed an all-too-human, punishing, and vengeful being. I could not fathom nor accept a God like that, and decided I wanted no part of it. To me, the concept was nothing more than a fabrication by organized religion to keep control of the masses.\n\nSince childhood, the image of God that stuck in my mind was that of a stern-faced, bearded old man\u2014someone to be feared. But, as these thoughts flooded through me, I suddenly understood that my long-held perception of God was wrong.\n\nAlthough I had no idea where these mental images originated, insights continued to fill my mind. The truth became clear. I knew about God. I had merely forgotten. God is the collective consciousness, the universal power of love and creation that gives life and meaning to everything in the universe. I found this simple revelation emotionally humbling.\n\nHow, I wondered, could I have veered so far off course? Surrounded by the beauty of creation, how could I have lost this truth? Floating in my body of energy, I also knew that this is who we really are. This pure, cosmic energy is our natural state, and the loving, powerful vibration within all of us is the vibration of God.\n\nReturning my awareness to my floating body, I discovered that I didn't have to actually turn my head in order to see in any direction. My visual field was a full 360 degrees and was attained instantly with just a thought. As I marveled at this, the urge to test the boundaries of my new reality grew, and I began to wonder if it would be possible to float around to other parts of the house. I slowly rotated my body in midair until I faced the doorway. Setting my intention to coast out of the bedroom, and impelled by just that thought, I began moving effortlessly toward the door.\n\nApproaching the doorway, I realized that I was too high to make it under the doorframe. But it was too late. My head and upper body were on a collision course with the wall above the doorway. Closing my eyes, I braced for the impact. And then an amazing thing happened: there was no impact. I floated through the wall like it wasn't there. The reality of this seemingly impossible act left me stunned. I had just gone through a solid wall without the slightest resistance. In disbelief, I spun around, surveying the wall as I descended to the floor in the hallway outside the bedroom, about ten feet from the door.\n\nComing to a gentle landing, I could feel my feet sinking into the rug. I felt the fibers of the carpet, the underlay, and finally the smoothness of the plywood below.\n\nMotionless, I stared in wonder at the wall I had just floated through. It seemed inconceivable. If I hadn't seen and felt it, I wouldn't have believed it. But I couldn't deny the experience. It had really happened.\n\nStanding in the hallway, I could see through the doorway into the bedroom. There, in bed, some thirty feet away, lay my physical body. I tried to rationalize how this could be possible. How could I be standing in the hallway, completely whole, thinking and feeling, while my body including my brain\u2014the organ supposedly required for thinking\u2014was in a different location? Pondering this dilemma, I was surprised by the sound of a soft, whispered voice.\n\n\"You are not your brain. You are not your body. You are continuous conscious energy.\"\n\nStartled, I spun around. No one was there. I had no idea where the voice had come from. It felt different than my own inner voice, yet natural and familiar. The more I thought about it, the more convinced I became that it had to be only my own inner thoughts. \"Just my imagination,\" I told myself. \"What else could it be?\"\n\nPutting the question out of my mind, I decided to continue the tour. I was standing in the upper hall balcony looking down a flight of stairs into the open living room below. Reaching out with my right hand, I gripped the rounded top of the railing post leading to the stairway. The hard, smooth wooden ball felt completely normal. But when I put some weight on it to adjust my stance, my hand suddenly pushed right through the post. The feeling was incredible! Astonished, I had to try it again and again. Each time, my hand swished through the \"solid\" column like it was made of water.\n\nIntent on continuing my excursion, I finally pulled myself away from this mesmerizing diversion. Looking down the stairway, I paused, wondering whether I would float down the stairs or end up falling flat on my face. Bracing myself, I pushed off. A surge of energy swept through me, and ever so gently I lifted off the floor. I gawked around unsteadily as I slowly floated higher into the air. Traveling a couple of feet above the floor, I cautiously made my way down the stairway, heading for the landing below. I prepared to touch down, and then just as my toes brushed the carpet, I felt a jolt, and in an instant, I was back in my body.\n\nThe vibrations had disappeared and the paralyzing numbness in my legs was gone. Wide-awake, I sat up in bed. The clock on my nightstand showed 3:05 A.M. Less than three minutes had passed since I decided to leave my body. It didn't seem possible. So much had occurred, I thought it could have been a half hour.\n\nUnable to contain my excitement, I shook my wife awake, blurting out every detail. I didn't care if she thought I was crazy. This was the most fantastic thing that had ever happened to me, and I had to tell someone. To my surprise, Candace believed every word. In fact she was excited about the possibilities. \"That is so amazing,\" she gushed. \"I'm really jealous. Why couldn't it happen to me?\" Candace had always been more receptive to this type of thing than I, and I was thankful for her open mind. Had our positions been reversed I doubted I would have believed the story.\n\nHaving had so little sleep, I was surprised that I was no longer tired. Pumped with energy, I jumped out of bed and got dressed. I put on a pot of coffee, and listlessly read through a couple of magazines, returning occasionally to my living room and bedroom to retrace the experience.\n\nAlthough my rational mind wanted to dismiss the event as some sort of aberration or hallucination, I knew that would not be possible. This had been more real than anything I had ever experienced. My senses had been working at hyper-speed and sensitivity. I had been able to see and feel more clearly than I had ever imagined possible.\n\nThe memory of the snapping, jerking feeling I experienced upon reentering my body kept churning in my mind. There was something familiar about it, but I couldn't place it. Then it suddenly dawned on me. I had felt that jolt thousands of times. Often when I was falling asleep, especially if I had been really tired, a jolt would bring me back to consciousness. How many times had I watched my wife and children make that same jerking reaction as they were falling asleep? I wondered if it was our spirits returning to our bodies that caused the jolt. Could it be that we all leave our bodies every time we fall asleep? I had to find answers to these questions.\n\nWhen morning arrived, my decision was made. I telephoned the party president to inform him that I would be withdrawing my name from the candidacy. This was not the path I should be following. There was new meaning and direction in my life, and it had nothing to do with politics.\n\nThe library was the first stop on my morning agenda. Candace and I were sitting in the parking lot waiting when the doors finally opened. We borrowed every book we could find on paranormal and spiritual subjects. Fortunately, they had a copy of Robert Monroe's book _Journeys Out of the Body_ , which we snapped up. I wanted to learn everything possible about the experience. I needed to know why it happened, but more importantly, I needed to know if I could make it happen again.\n\nMonroe's book was even more intriguing the second time around. This time, it didn't seem so far-fetched. My own experience had opened my mind to the possibilities and power of the human spirit. This was something I could learn to do.\n\nAlthough I hadn't yet realized the impact of my out-of-body experience, my world was changed. I was no longer sure about who I was or who I should be, but I knew one thing: I was on a mission, and I couldn't wait to see where it would lead me.\n\n# 2\n\n# _Where Was God?_\n\n_Though evidence in abundance shouts_\n\n_Thoughts of creation are lined with doubt_\n\n_Life's grand design in darkness robed_\n\n_Hints of spirit left unprobed_\n\nI was born and raised on a farm in the west-central plains of Canada. My parents, along with everyone we knew in a forty-mile radius, were German Catholics. They worked hard all week and went to church on Sundays. That was the only day that we kids got a break from the never-ending cycle of farm chores.\n\nOur mother, Anne, was a creature of habit. She liked the security of a routine, but mastering the rhythm method of birth control was apparently not her forte. She had eight baby girls in a row and then, when she figured out what was causing it, promptly gave birth to three baby boys. I was the tenth of 11 children.\n\nGrowing up in such a large family wasn't easy. These were the '50s. The world was recovering from the Great Depression and war. It was a time when large families were common. For many, there wasn't a lot of money to go around, but with 13 people in our house we seemed just a bit poorer than everyone else.\n\nIn the years of hand-me-down shoes, clothes, and schoolbooks, we often took life on the chin, sometimes bearing the brunt of cruel, hurtful remarks by our schoolmates. Nevertheless, probably the one thing that seemed to keep us going was our humor. When you don't have much, it helps if you can find a way to laugh at everything.\n\nI'll never forget a schoolyard incident that happened when I was ten or eleven. A small convent of Catholic nuns operated our rural school. Classes were over for the day and, as my little brother, Dale, and I were walking across the playground, one of the nuns made a derogatory comment about how we looked like ragamuffins. Although embarrassed, I'd be damned if I was going to let it show.\n\n\"That's because we're poor, Sister,\" I sarcastically quipped. \"Hell, it's a good thing we're boys or we'd have nothing to play with.\"\n\nI couldn't believe what had just come out of my mouth. The silence was deafening. My ears and face burned as blood rushed to my head. The words echoed in my mind.\n\nFor a moment the good sister just stood there. I didn't think she had grasped what I had said, but my little brother couldn't contain himself. His hand shot up over his mouth to squelch an eruption of laughter. But it was too late.\n\nThe old nun didn't budge an inch. She just stood there in the schoolyard glaring at me for what seemed an eternity. Nervously, I glanced at my brother. Frozen in place, his hand was glued to his mouth. Tears were now running down his cheeks. It was all too surreal and funny. I couldn't believe the stunned look on the nun's face or the pathetic, muffled whimpering coming out of my brother. It was just too much. Dropping to my knees, I rolled, howling with laughter.\n\nAs expected, the rest of the afternoon did not go well for me. Unfortunately, the uptight nun didn't see the humor in my little stand-up routine. That wee bit of insolence earned me nine whacks with a leather belt across my backside\u2014one for each remaining day of lent, wouldn't you know. Why couldn't I have waited with my smart mouth until it was a lot closer to Easter weekend?\n\nWe grew up, I suspect, like many other kids during the fifties and sixties, without any closeness to our parents. Our father was not the kind of person who could show a lot of love, or even much kindness for that matter. He probably would have considered that to be some kind of weakness.\n\nFor the most part, we grew up as a bunch of tough little realists. If we couldn't see it, hear it, or feel it, we couldn't believe it. So, although religion played a big part in our lives, and we were forced to go to church every Sunday, I wasn't really convinced there was a God. The lyrics of that old Blood, Sweat, and Tears song, \"I swear there ain't no heaven, but I pray there ain't no hell,\" probably described my beliefs all too well.\n\nMy younger brother and I were forced by our parents to serve as altar boys until we were almost 16. That's when we finally rebelled, refusing to buy into the mind control of \"hell fire and damnation.\" Although we would never dare talk about these things with our parents, we began to fervently resist the pressure of going to church to be preached at.\n\nIn those early years, the Catholic Church, and I suppose just about every other Christian denomination, preached \"fear of the Lord\" as a main plank in their religious dogma. If you did this or that, a merciless and vengeful God could fry your butt in Hell for the rest of eternity. My brother and I always had trouble with this concept. We were not about to believe that God\u2014if there actually was a God\u2014could possibly be so mean.\n\nWe had great difficulty believing what the nuns and the priests of the day were trying to teach us. They professed that there were basically two kinds of people in the world. There were the Catholics who would be going to Heaven, and then there were all those other people who weren't! These other people would undoubtedly be spending eternity in Hell or Purgatory, or maybe even a place called Limbo.\n\nAs young, impressionable boys, this was more than a bit unsettling to us. We had two young Presbyterian cousins who happened to be from the only non-Catholic family in the community. They were our good friends, and it made us very sad to think they were going to have to spend eternity in Limbo just because their parents weren't Catholic. We couldn't understand or accept the apparent injustices of our religion. At the very least, we desperately hoped that these things weren't true.\n\nDuring our later teenage years my brother and I would occasionally go to church to please our mother, but after leaving home we rarely attended. We had seen far too much hypocritical behavior by the holier-than-thou \"pillars of our community\" to believe that going to church made you a good person. It seemed to me that most, if not all, of these rules were man-made. They couldn't be God's rules because every now and then the church would actually change them.\n\nFor example, back in the early 1960s, it was a \"big-time sin\" to eat meat on Fridays. If you did and were unlucky enough to die before you made it to confession, yours would be a most horrible fate. God would surely send you to Hell for eternity. What was equally amazing, though, was that many years later, they changed the rules and it was no longer a sin to eat meat on Fridays. One of my favorite comedians, George Carlin, had similar thoughts about this: \"I always wondered,\" George would say, \"what ever happened to all of those dudes doing time in Hell on a meat rap?\"\n\nBeyond one significant event, I think most of my childhood wasn't much different than that of a lot of other kids growing up in this era. This unusual incident took place on a warm Saturday in July of 1963, when I was 12 years old. It was an exciting day for my little brother and me. Our two cousins from the city had arrived for a weekend visit, and we had a lot of plans. My cousin Brian and I were best friends. We were the same age, we liked the same things, and we shared the same fantasies. Unfortunately we weren't able to spend a lot of time together, so whenever we had a chance to visit we tried to make the best of it. It was bound to be a fun weekend.\n\nAbout a half mile from our farmhouse was a water reservoir the department of highways had dug beside a road. The purpose of these little ponds was twofold: they provided the contractors with the dirt they needed to build the road, and the remaining hole provided valuable storage for runoff water from the summer rains and melting spring snow.\n\nDale and I had been warned to stay away from the dugouts. We couldn't swim a stroke and had no one to teach us. Our cousins, however, had taken lots of swimming lessons in big swimming pools, and as a result were very good swimmers.\n\nWe hadn't planned on going near the overflowing dugout; we were just crossing a pasture looking for prairie dogs when one of the cousins noticed the pool of water. Tied to a stake at one end of the dugout was a small wooden raft our neighbor's kids had obviously abandoned. Well, it was just too tempting to pass up. Before long we were all aboard the raft, bobbing precariously, paddling around in the middle of the twelve-foot-deep pond.\n\nWe were having a great time until the older of the cousins discovered that my brother and I couldn't swim. Realizing we were afraid of water, he thought it would be good fun to violently rock the raft while the rest of us hung on for dear life.\n\nOn the wet, slippery planks, I lost my balance, and the next thing I knew I was toppling backwards into the water. Terrified, I didn't have the presence of mind to even try to swim. With numbing quickness, shock overcame me; my head slipped below the surface and, before I knew it, I was on my way to the bottom. In my panicked state, it wasn't long before the last bubbles of air had escaped me and my lungs filled with water.\n\nI had always imagined that drowning would be a horrible way to die\u2014the mental terror while one's lungs desperately screamed for air\u2014but it wasn't like that at all. In fact, as soon as my lungs filled with water, the struggle ended. There was no more suffocating or fighting for air. Instead, an absolute peace came over me.\n\nWith my eyes wide open, I continued a slow descent. The water grew darker and darker, and soon I was up to my ankles in mud. After pausing there for a few seconds, I'm not sure why, but ever so slowly I began to float upwards. Within a few moments I was nearing the surface. I could see and feel the warm sunlight radiating-into the water. My head briefly broke through, and then in a dreamlike state, without fear or panic, I began to sink again.\n\nMy senses numbing, I felt no particular discomfort, just the greatest urge to fall asleep. Soon I could feel my feet sinking again into the mud, and then everything seemed to grind to a halt. Time stood still as I hung suspended in the water, my surroundings fading. Too tired and sleepy to be concerned, I simply let go and drifted into the blackness.\n\nThe next thing I knew, I snapped back into consciousness, opening my eyes to an astonishing sight. I was being bathed in a shimmering kaleidoscope of soft, warm colors. Swirling and gyrating, they seemed to pass through my body into the core of my being. I knew I was still surrounded by water, but somehow I had gotten into a beautiful, comfortable bubble. It felt like I was inside a rainbow. Mysteriously, the colors seemed to be causing a wonderful vibrating sensation throughout my body, each shade carrying its own distinct frequency. Fanning my hands in front of me, I watched, in childish delight, the multicolored energy swirling through my fingers and hands, electrifying my senses.\n\nI had never felt more alive and energized. In awe of the warmth and beauty of my surroundings, I sensed something important was happening\u2014that I had somehow changed. It occurred to me that perhaps I had died. And for some unknown reason, I recalled the story of Tom Sawyer\u2014the part when he was believed to have drowned in the Mississippi, but arrived home just in time to witness his own funeral.\n\nJust as these thoughts settled in my mind I suddenly found myself shooting out of the water and into the air. A moment later I was hovering over the choir loft in the back of our church.\n\nIt felt so strange. There I was, floating around in our church, anxiously awaiting my own funeral. Then I noticed there wasn't a soul in the building. If this was my funeral, I thought, where were all the people? I would have expected to see my mom and dad, my brothers and sisters, and all of the other people who normally went to church. But no one was there, and I hadn't a clue what to make of it.\n\nFascinated, I floated over the empty pews marveling at the impossibility of the situation. I was flying\u2014actually flying! And there didn't seem to be a darn thing holding me up. How could this be? I hadn't read anything about Tom Sawyer flying. Puzzled, I floated for several more moments, deep in thought.\n\nSuddenly, I felt a violent convulsion. Spasms erupted through my stomach. The next thing I knew, I was back on the shore of the dugout spewing water from my lungs, gasping for air.\n\nMy face bouncing in the mud, cousin Brian had his arms locked around my waist, dangling me upside down, draining the water from my body. Sick to my stomach, I vomited for a couple of agonizing minutes.\n\nWhen I finally began to collect my wits, I looked up to see the horrified faces of my brother and cousins. We were all, to some degree, in a state of shock. To make matters worse, we knew we were going to be in a heap of trouble when we got home.\n\nWe sat quietly beside the dugout, contemplating what had just happened. After a few moments, as if on cue, everyone started talking at once. Chances were we would get a good whipping if anyone found out about our little incident, so we made a pact to never tell a soul. Then we got up, brushed ourselves off, and walking as slowly as possible to dry our clothes, headed for home.\n\nPumped with fear and adrenaline, Brian wasn't feeling too well. He told me he had thought for sure that I was dead because I'd been under for such a long time. He had spent several terrifying minutes diving and groping around in the murky water. When he finally found me in the mud at the bottom, it took every ounce of energy he could muster to get me to the surface.\n\nAs Brian told me this, I put my arms around my little brother and hugged him. Dale had been deeply affected. Unable to speak, tears flowed down his freckled cheeks.\n\nAs we walked back to the farm, Brian and I lagged behind the other two. I told him about the strange things that happened while I was in the water, about my memories of Tom Sawyer's funeral, about floating midair in the church, and my surprise that there were no people in the building. Brian made a simple observation. Of course there wouldn't be any people in the church. It was four o'clock in the afternoon and Saturday to boot.\n\nI told him about all the bright colors that surrounded me in the water. But what I was recounting seemed strange even to me. I remember saying \"Jeez, I didn't know you could feel colors.\" Brian, however, didn't say much. Maybe he thought I had water on the brain or something because I'm sure this couldn't have sounded very rational.\n\nI told him again about what it was like to fly. He listened quietly, and then as if he hadn't heard my comments, he questioned, \"Tom Sawyer? Why would you be thinking of Tom Sawyer?\" I admitted that I hadn't the slightest idea, but assured him that everything had seemed so real.\n\n\"Jesus, Paul,\" he scolded. \"You scared the hell out of us. I couldn't find you. I was praying like crazy. It's a good thing you ain't dead, 'cause they'd sure be killing the rest of us when we got home.\"\n\nAlthough exceedingly nervous, when we finally reached the farm, we tried to be as nonchalant as possible. But as twelve-year-olds, we were probably too nonchalant. It must have been obvious we were hiding something because it didn't take more than ten minutes before we were running for cover.\n\nI can't remember who caved in, but the result was dramatic. After a lot of yelling, cussing, and recriminations, two kids were given a sound strapping on that day. Oddly enough, I was one of them.\n\nIt was, to say the least, a very strange experience. My little brother and cousins had been horrified, and our parents extremely upset. I, on the other hand, hadn't experienced it as a horrible thing at all. And I couldn't understand why everyone was so uptight. I was still alive and feeling just fine.\n\nOn thinking back to the event, it occurs to me that at no time in my young life did I ever attach to the experience any kind of spiritual or religious significance. Even though I had ended up in a church, it didn't seem to carry any consequence or lasting effect\u2014at least that's what I thought at the time. Although I could never forget the incident, I didn't spend a lot of time thinking about it in the following years. I could not, however, have even begun to imagine the importance of this event until much later in my life.\n\nBack then, I had been grateful to Brian for saving my life, and I told him so repeatedly. But not once did I think of thanking God. In retrospect, I'm glad that God doesn't hold grudges. Well, at least I hope He doesn't.\n\nOne of the great things about childhood on a farm is that life moves quickly. There are so many exciting things to do and lots of ways to get into trouble, so you don't spend a lot of time worrying or contemplating life. You just go out to the shed, build yourself a new slingshot, and try to improve your prowess as a marksman by shooting the glass insulators off the utility poles.\n\nAs far back as I can remember I always had a knack for getting my brother and myself into trouble. Often, the word \"trouble\" didn't even come close to it. Our mother didn't raise fools, but my brother and I seemed to excel at doing foolish things. By any account, it's remarkable we lived past our sixteenth birthdays.\n\nBoredom, and sibling rivalry can lead kids to do dumb things, and on a warm, sunny day in the summer of 1966 we must have been as bored as anybody could get. We had been picking and hauling rocks from the fields, when our father announced that he had to drive 30 miles to a nearby town to pick up parts for the cultivator. And just like that, he drove off, leaving us stranded more than three miles from home. Fortunately, Dale and I had snuck along our .22-caliber rifle, which, for a while, gave us something to pass the time. But out in the middle of nowhere, we soon ran out of targets to shoot at, or things to amuse us.\n\nBored, we eventually ended up in the shade of an old abandoned farm building. Cradling the rifle like a marine on a mission, I made a slow pivot looking for something, anything, to shoot at. My brother suggested I looked like a dork and couldn't hit a damn thing anyway. As usual, that was more than enough to get things started. He was on his knees in the grass about 30 feet away, fiddling with a small piece of cedar roof shingle. I told him to hold the shingle up in the air and I'd show him what I could hit. He obviously didn't believe for a moment that I would act on such a crazy notion. He held up the shingle, a half-inch wide and a foot long. I shot it out of his hand. Letting out a couple of surprised expletives, he quickly struck back. \"Alright, you bastard\u2014my turn!\"\n\nThis was becoming a bit serious. In Dale's mind, he was the Second Coming of Davy Crockett, but in reality he was a notoriously bad shot. Backing down in front of him was unthinkable, and there would be no purpose in arguing he was a terrible shot, because he would just call that a \"chicken shit excuse.\"\n\nExpecting to lose at least a finger, I finally held up the shingle. The gun sounded, and the little piece of wood snapped in my hand, and the game of \"chicken\" began. We passed the rifle back and forth. With each shot the shingle was whittled down until all that remained was a stub. We knew we'd have to find another target or one of us was going to be sorry.\n\nLooking around, the only other thing we could find was an empty salmon tin. It was a standard tin, about three inches in diameter and an inch and a half high. However, in the tall grass, we couldn't find a decent place to set the tin to get a good shot at it. That's when I suggested I could easily shoot it off his head from where I was (about 30 feet away), but we might as well forget it, because he would be too chicken. \"Bullshit,\" he said, placing the tin on his head.\n\nMy stomach muscles knotted. But with every impulse in my brain screaming, \"Don't be crazy,\" I still found myself thinking, \"Hell, there is no way I could miss from this distance.\" I slowly lowered the rifle to sight-in the tin. Immediately, Dale began to complain that it looked like the rifle barrel was pointed right between his eyes. I assured him that it was not, but sarcastically suggested that if it bothered him, he could just close his eyes. I should well have known the effect that that statement would have.\n\nEyes wide and unblinking, he steadied his stance. \"Oh yeah,\" he yelled, \"go ahead. Do it!\"\n\nMy finger flexed. The shell exploded, and in an instant the can spun off his head.\n\nLetting out a huge gasp, my brother then recited the five most horrifying words in the English language: \"Okay, now it's my turn.\"\n\nI couldn't believe the stupidity of the game we had gotten into. But it was too late. I was going to have to stand there and let my lousy-shot brother try to knock this little tin can off my head. I cussed myself for not having at least worn a hat, preferably one made of titanium.\n\nThere was no turning back. I was just about to place the can on my head when I had an idea. Reaching up, I fluffed up my long hair as high as I could. My brother, however, couldn't let this go unchallenged. \"Chicken shit!\" he jeered. My heart sank.\n\nIt was useless. There was no way I could get out of it. I put the tin back on my head and he lowered the rifle into position.\n\nIt looked like he was aiming too low; he had to be aiming too low. I could have sworn the barrel was centered right between my eyes. \"Jesus,\" I hissed, \"don't be pissin' around. You're aiming at least four inches too low.\"\n\n\"So, how does it feel?\" he whispered, one eye peering down the sights.\n\nThe delight in his voice was more than I could bear. But there was no way I could back down.\n\n\"Quit your damn whining,\" he snarled. \"I'm aiming right at the can.\"\n\nSweat now flooding from every pore, I wanted to throw up. I began to think how, in a strange and twisted way, I had gained a new respect for my brother. I was practically ready to faint. How in hell had he been able to stand still and let me shoot?\n\nThe abrupt sound of the exploding shell shattered the silence. I flinched as the empty tin twirled off my head.\n\nMy brother was now grinning like a horse eating thistles. I, however, thought I was about to have a heart attack. My knees almost gave out. \"Okay,\" I wheezed. \"It's a tie. That's enough. I have to take a leak.\"\n\nWe both relieved ourselves, and then, as we were zipping up, for some reason\u2014nervousness, adrenaline, we didn't know why\u2014we began to giggle, then laugh. Soon we were howling, joking about how ridiculous we looked at the moment of truth. Within a couple of minutes, though, our laughter and horseplay faded into an uneasy silence. We were both visibly shaking. With eyes wide as saucers, the insanity of the act we had just committed began to sink in. Without saying a word, we hugged each other and set off on our long walk home.\n\nFollowing that day, we never again engaged in that degree of craziness and we never again intentionally put each other in harm's way. Well, almost never.\n\nRaised in virtual poverty, it seemed necessary for me to develop a practical attitude about most things in life. I quickly learned that nothing comes without a price, and the only way you're going to make it in this world is through hard work and sacrifice.\n\nAt an early age, I learned from my mother that if anything was worth doing, it was worth doing to the best of my ability. I learned the lessons of hard work almost to the point of obsession. Driven by two factors, my motivation for success was simple: I needed to succeed so when I grew up, I would have an easier life than my parents, and if I ever had kids, they would never have to know the agony and humiliation of being dirt poor.\n\nBy the time my brother and I reached the ages of 15 and 16, we were the only two children left at home. We had been born late in our parent's lives, so it felt like we were living with our grandparents. They were already in their late fifties and the generation gap grew increasingly wide.\n\nOur mother, whom we loved dearly, hadn't had an easy life. Aside from the fact that she had been pregnant with one child after another for eight and a half years while coping with a brood of 11 kids, she had a husband who didn't spend a lot of time being caring or helpful. She would just bite her lip and do her best for us kids while going without so many things for herself. She never enjoyed anything close to a luxury. We had no running water, television, or a phone, but we never heard her complain. She was truly a saint.\n\nOne of my greatest disappointments in life was that I didn't have a father who would ever show an emotion other than anger. My brother and I would work our butts off trying to please him, but never once would he ever say, \"Thanks\" or \"You did a good job.\" It seemed the only response we could elicit from him was criticism. At an early age we learned that we could never please our father. He always seemed to be angry, so we just stayed out of his way.\n\nAs a child, I did what I could to protect myself from hurt, and it didn't take long before a strong wall was built around my heart. I had learned early in life that the old nursery rhyme was wrong. It should have been changed to \"Sticks and stones may break my bones, but _words can simply destroy me.\"_\n\nSometimes when things became particularly repressive, as if to distance myself from the hurt, I would say to my brother and myself, \"Quit feeling sorry for yourself and get over it.\" I would not understand the degree of hurt this would inflict upon me until much later in my life.\n\nNow, don't get me wrong: If I had to do it over again, there's probably not a lot I would change. I am what I am precisely because of the hardships and lessons of my upbringing. I believe there's a lot of truth to the statement \"Whatever doesn't kill you, makes you stronger.\" In that respect, our father may have been a good and valuable teacher.\n\nAt the age of 17, I was to have another close encounter with death. It was a chilly November evening when two of my teenage friends and I were excitedly cruising down a country road. With Gerald behind the wheel, and my cousin Mervin riding shotgun, I was sandwiched in the middle of the front seat of a 1963 Ford. We were on a testosterone mission to the Taylor farm, where awaiting our arrival were two of Mr. Taylor's finest daughters. They were fabulous and we couldn't wait to get there. Our youthful anticipation, however, could only be outdone by our lack of experience in the operation of a motor vehicle.\n\nAs we rocketed down the gravel road at 85 miles per hour, the sound of the right front tire blowing out at first registered only as an annoying hiccup in our mission. Within seconds, however, the car began to swerve wildly before spinning out of control and sliding into the ditch sideways. The wheels abruptly dug into the frozen dirt and the car catapulted into the air. Time and consciousness stood still.\n\nWith a violent explosion, the car slammed back onto the ground. Metal twisted and the windshield disintegrated into thousands of pieces. In a surreal numbness I felt the shards of glass raining in on me, then everything faded to black.\n\nThe next thing I knew, I was awakening from what seemed like a nightmare. My head was pounding. Fighting the heaviness of sleep, I tried to sit up, but fell dizzily back to the ground. Baffled, I lay for a few moments trying to get a grip on what had happened. Here I was, lying on the frozen earth in the middle of nowhere without a clue as to how I'd gotten there. After falling a couple more times, I finally made it to my feet. It was freezing cold, and I shivered uncontrollably.\n\nLooking around, I spotted a car about 20 feet away. The door was wide open so I crawled into the driver's seat to warm up. I reached for the ignition to start the engine, but there wasn't a sound. I tried again, but still nothing. I tried to close the door, but it wouldn't budge. Something was terribly wrong!\n\nDazed and confused, I began to crawl back out of the car, when a stabbing pain ripped through my left side. It practically brought me to my knees. I couldn't imagine what was happening. Glancing back at the car, my breath caught in my throat. The car was horribly mangled, almost completely crushed. My eyes had to be playing tricks on me. I stood there staring, disbelieving, and then it hit me: we had crashed the car. Where were Gerald and Mervin?\n\nPanicking, I began frantically searching for my friends. There was no one inside the car, so I crawled out of the wreckage, heading into the dark field. As I rounded the front corner of the car, I tripped and crashed heavily to the ground. Screaming in pain, I pulled myself onto my hands and knees. There beside me lay a crumpled body. In the moonlight I recognized Gerald's ashen, dirt-covered face.\n\nA chill ran up my spine. I was frozen in place. Although my horrified mind demanded I get up and run, I couldn't move a muscle. I just knelt there, staring at his lifeless face, unable to move, unable to look away. A low growling noise began to build in my throat as I struggled against the terror that held me in its grip. Then I heard my own screams shattering the night air.\n\nMy paralysis broken, I got to my feet. My mind clicked into gear. I had to find Mervin. Through burning pain I spun around, scanning the area. About 25 feet away lay another body. Staggering to where Mervin lay, I jammed my fingers against the artery in his neck but could feel no pulse. The horrifying thought that he was dead crept through me like a slow suffocation. I became aware of great gasping sobs coming from my throat.\n\nFrantically, I stumbled back to Gerald. He didn't seem to have a pulse either, and it looked like he wasn't breathing. I put an ear to his chest and held my breath. Nothing: no sound, no movement. In my numbing mind I heard my own pleading voice, \"God, please don't let this happen.\"\n\nCalling Gerald's name, I grabbed his arms to pull him up. There was no response. His lifeless body slumped back to the ground. Finally, in despair, sobbing uncontrollably, I surrendered to the cold ground. Ever so slowly, the spinning darkness overcame me.\n\nThrough the fog of drowsiness I became aware of a distant siren, and my consciousness again fought for control. In the distance, headlights and flashing red lights were approaching. Soon a police car roared into the field beside me. Two Mounties sprinted from the car, their flashlights blazing into my eyes.\n\nI motioned in the direction of Mervin's body. The policemen checked his and Gerald's body, assuring me an ambulance was on its way and everything would be okay. Helping me to my feet, they walked me to the warm police car. It was then that I realized I had lost my shoes. My nearly frozen feet felt like blunt stubs.\n\nI sat by myself in the back of the police car trying to warm up. Shaking, my body was numb with hypothermia and I couldn't keep my eyes open. One of the policemen opened the back door of the car and slipped in beside me. He kept telling me to try and stay awake, but I faded in and out of consciousness.\n\nWhen the ambulance finally arrived, the medics loaded my friends and sped off toward the hospital. I rode in the back of the police car and drifted into sleep.\n\nThe abrupt sound of a siren rising to life startled me awake as we neared the emergency entrance of the hospital. A doctor and a nurse waiting with a stretcher wheeled me into the emergency room. Then, after taking numerous X-rays, they began to work on me. They seemed to be concentrating on the right side of my head, which up until then had been feeling numb. I could feel the bite of a needle as the anesthetic was injected. Soon, I could no longer keep my eyes open and the beckoning darkness crept over me. It would be two days before I regained consciousness.\n\nWhen I finally opened my eyes, I found myself in a bright room, utterly confused. My head felt like it was about to explode. Reaching up to scratch my brow, I discovered that with the exception of my nose, mouth, and left eye, my head was wrapped in bandages.\n\nI tried to sit up, but such a pain shot through the left side of my body that the room began to spin and I collapsed back onto the bed. Every inch of me seemed to hurt. I was becoming more panicked by the moment; I had no idea where I was or what had happened to me.\n\nSoon a nurse entered. Noticing I was conscious, she sprinted off to fetch a doctor. The physician told me about the car accident and assured me that my two friends were okay. Gerald had suffered a smashed pelvis and a concussion, while Mervin's skull had been cracked. He told me that I was the only one conscious when they brought us in. Apparently, after the accident, I had crawled to the road and flagged down a passing car. I remembered nothing.\n\nThe doctor filled me in on my injuries. I had a ruptured left kidney, a cracked skull, two cracked ribs, a bruised lung, and my right ear had been nearly severed.\n\nOver the next few days, memories of the accident slowly began to return, but my recollection of the two days leading up to the accident was gone. For some unknown reason it would take several more months before I had any recall of that lost time.\n\nRecovery was a long and painful process. Physical activity was a struggle, and much of my life had to be put on hold.\n\nThis close call with fate only seemed to reinforce some of the philosophical doubts I had been harboring since my early teens when I began to question the existence of God. I had now been involved in a serious accident. I had been unconscious for a long time, but not once did I have an experience of any kind that might lead me to believe there was a God. One minute I was alive and well, and the next minute I was gone. It was like I had ceased to exist for that period of time. There was no conscious awareness of anything. In my mind, the concept of death became an act of surrender to nonexistence....\n\nEventually I healed and returned to normal teenage life.\n\nMy brother and I could hardly wait to finish school and leave the farm, but strangely, when that day finally arrived, we did not leave without some degree of sadness.\n\nA short time later I began dating a girl I had known in high school. She was everything I dreamed of and I fell madly in love. There was nothing I wanted more than to spend the rest of my life with her. But life, I had found, sometimes deals out harsh realities. One day she told me she needed to \"find herself\" and walked out the door.\n\nDevastated, I couldn't imagine how life could continue, but somehow it did. I spent a lot of time feeling sorry for myself, and then at some point I vowed, at least subconsciously, that I would never let anyone hurt me that way again. And the walls around my heart grew ever thicker.\n\nFor a couple of years after leaving home, I drifted around, much the same as many kids in the 1970s. I worked at a number of jobs until I woke up one morning from a vivid dream and knew that I would become a radio announcer. This was an era of hippies, war protests, and most of all, rock and roll music. I couldn't imagine a better way to make a living.\n\nThe top rock station around was CKOM. I dropped in to see the manager and told him I wanted to be a radio announcer. After quickly determining that I had neither experience nor training, he said they had no openings and sent me on my way.\n\nTalk about feeling dejected. I couldn't believe he had turned me down. It had been so clear in my dream.\n\nA few weeks later, I stopped by to see the manager again. This time my pleading seemed more like begging. I heard myself offering to work in the station for free. I would be available day or night, I said. All they had to do was teach me about radio.\n\nI must have been sufficiently pathetic in my pleading, because he thought about it for a moment, and was in the middle of saying, \"I'm sorry,\" when he unexpectedly stopped, stuck out his hand and said, \"Okay, I like your guts. You've got yourself a deal.\"\n\nI loved radio from the start. Spending almost every available moment at the station, I filled my end of the bargain. Filing record albums, producing commercials, newscasts, and generally anything they wanted me to do, I learned all I could about broadcasting. Everything went so well that within a few months I was offered a position as a disc jockey. For me, it was a job made in Heaven. I couldn't believe they were _paying_ me to do something so enjoyable.\n\nA couple of years later, the love of a woman would again enter my life. Four years my junior, she was just a year out of high school, but carried herself with the grace, charm, and elegance of a mature woman. Candace was pretty, intelligent, witty, she made me laugh, and most of all she loved me. Our romance intensified, and a year later we were married.\n\nFor the next eight years I gave everything I could to my career. In order to advance I moved on to larger radio and television markets. Candace, a banker, happily followed me from city to city. Eventually we grew tired of the hectic pace and decided to make some changes in our lives. I said goodbye to broadcasting, bought a small insurance brokerage, and we settled down to raise a family.\n\nNormally, what we understand about ourselves becomes less of a puzzle as we age. But little did I know that up to that point in my life, things were never entirely within my control. So much of my life had been guided and assisted, preparing me for the life-changing experiences still to come, experiences that would rock my world and challenge even my most basic concepts of reality.\n\nThis brings me again to the conscious beginning of my life's intended journey that began when I found myself floating near the ceiling of my bedroom. Virtually overnight my understanding of the world had been obliterated. It was, however, only the beginning of a long search for the ultimate truth and meaning of life.\n\n# 3\n\n# Search for Truth\n\n_In days of science and truth unclear_\n\n_Man and religion fraught with fear_\n\n_Purpose and destiny remain obscured_\n\n_Life without meaning long endured_\n\nIn the aftermath of my first out-of-body experience, I became obsessed with finding even the slightest bit of information that would enlighten me on the cause of the phenomenon. It had been the most fantastic thing that had ever happened to me and I desperately wanted to have the experience again. To the exclusion of almost everything else, I pursued this goal with a religious fervor.\n\nThe folks at the public library must have wondered about the volume of metaphysical books that I began borrowing. I read voraciously, and spent the rest of my free time trying to go out-of-body. Several books offered methods to achieve the necessary altered state. I would spend hours lying in bed, trying to reproduce the vibrations that seemed necessary for me to escape my body.\n\nThe process, however, became frustrating. The more I learned, the more I yearned to have another out-of-body experience. I prayed and I begged, but to no avail; it seemed like it was not meant to be. More than two months had passed since that first experience and I didn't seem to be any closer to my goal. It occurred to me that maybe I was trying too hard, so for several days I tried to not even think about it. But there was no escaping my obsession with spiritual ecstasy, and it wasn't long before I was back at it.\n\nThen one night, out of the blue, it happened. I had been trying to meditate my way into an altered state. Dozing off for what seemed like just seconds, I abruptly awoke to powerful vibrations coursing through my body. My lower torso and legs were numb and heavy.\n\nExcited by this development, I knew I had to be careful. I didn't want to risk losing the vibrational state and ruining the experience. With effort I tried to calm my mind and maintain control. The shuddering intensity of the vibrations surprised me. I didn't recall them being as choppy and strong in my previous experience. The sensations became so jarring and uncomfortable I concentrated on mentally speeding them up as a possible solution. It worked! Within moments the vibrations smoothed into a delightfully energizing wave. In my head, a high-pitched sound began to slowly intensify into a sharp oscillating tone. There was something hauntingly familiar about it; it consumed my thoughts. For a while it was as if nothing else existed.\n\nMy mind drifting, I had to remind myself of my goal. I sorted through all the possible things I could do after separation from my body. Opening my eyes, I glanced about the bedroom and found that my vision swept through the ceiling and into the night sky as it had during my first experience.\n\nAn ominous feeling swept through me. It was the unmistakable sensation that I was not alone. A chill crept up my spine. Sparks of energy invaded my senses. They were all around my room! Things. Beings. Whatever they were. I felt them everywhere. Frantically I fought to overcome the fear building inside me, but it was useless. My panic turned to terror. I wasn't ready for this. As much as I wanted to leave my body again, there was no way I could force myself to deal with this unknown horror.\n\nStraining against the suffocating paralysis, I forced my legs to move and then my head. In an instant the spell was broken. The vibrations, along with the things or beings, disappeared.\n\nAngry with myself, I sat up in bed. This was the opportunity I had been praying for. This was what I wanted to experience more than anything else in the world, and I had let it slip by. How could I have allowed my fear to get the best of me? I vowed never to let it happen again.\n\nInsatiably, I read anything to do with spirituality. From the biblical teachings of Christianity to Hinduism and Buddhism, I consumed it all. Robert Monroe's books were always a comfort. He had been through it all, overcame his fear, and went on to explore the spirit worlds. I noted his and other writers' assurances that there was no reason for fear. They claimed that in our astral or soul bodies, we are pure, indestructible energy. Nothing in the ethereal realms could harm us.\n\nSome authors recommended offering a prayer for protection before attempting astral travel. Robert Monroe had developed a prayer-like affirmation for students attempting to learn out-of-body travel. The affirmation apparently worked well in allaying the debilitating fear many students faced. Much like a security blanket, I found the affirmation a great help. I memorized it, and used it religiously before every subsequent attempt.\n\nI am more than my physical body. Because I am more than physical matter, I can perceive that which is greater than the physical world. Therefore, I deeply desire to expand, to experience, to know, to understand, to control, and to use such greater energies and energy systems as may be beneficial and constructive to me and to those who follow me. Also, I deeply desire the help and cooperation, the assistance, the understanding of those individuals whose wisdom, development, and experience is equal to or greater than my own. I ask their guidance and protection from any influence or any source that might provide me with less than my stated desires. (Reprinted by permission of The Monroe Institute.)\n\nArmed with these words and assurances that nothing could harm me, I knew that the real test of my resolve would come only if I could recreate the necessary conditions to achieve an out-of-body state.\n\nOver the following month, I spent at least an hour each night meditating and experimenting with some of Monroe's methods, but nothing seemed to work. The process was frustrating. Night after night, I managed to reach a certain degree of energy and vibration, but failed to carry it to the required intensity. The best I had been able to achieve was a deep relaxation which, unfortunately, usually resulted in my falling asleep. I began to worry that I might never be able to reproduce the experience. Then, one night, without warning, it happened again.\n\nI had gone to bed early, quickly falling into a deep sleep. After about an hour and a half, I jerked awake as intense vibrations coursed through my body. Every molecule of my being seemed to be buzzing with electricity. A high-pitched squeal resonated in my ears.\n\nI was ecstatic. This was what I had been working for! This time fear would not stop me. I was in such a hurry to make it happen that I even forgot to offer my prayer for protection. I simply thought \"Up,\" and in an instant, I was floating near the ceiling of my bedroom.\n\nIt was almost too fast. A momentary feeling of disappointment came over me. I hadn't experienced the increase in vibration or the elation of pulling away from my body. I had just transported instantly to the ceiling.\n\nBelow me in the bed lay my body. Mesmerized, I studied the sleeping form. It seemed so odd to be doing this. I felt compassion and gratitude towards this body that had served me well for so many years. The blankets had slipped down to just above my waist, and it looked like I might be chilly. My wife was asleep on her right side, with her left arm draped across my chest.\n\nI thought about the many years Candace and I had been together and our constant disagreement over what the temperature in the house should be. She preferred the bedroom cool at night, while I liked it a bit warmer. It had been warm enough when I got into bed, but as usual, it appeared that sometime during the night she had turned down the thermostat and opened a window. I turned to my right. Sure enough, a cool breeze rustled the curtains in front of the open window.\n\nI remembered why I liked to be warm at night. There were far too many vivid childhood memories of winter nights when the fire in the woodstove, the only source of heat in our house, had gone out. Often, after nights like that, my brother and I would wake up, clinging to each other like puppies trying to keep warm. These were things I didn't want to experience again, but they were far removed from the life my wife enjoyed while growing up in the city.\n\nI had to pull myself away from my musings. It might not be long before I would have to return to my body. With increasing excitement, I was ready for a new adventure. Having decided to first try a tour through the house, I pivoted around to face the doorway, but the door was closed.\n\n\"Damn,\" I cussed. \"Why does she always have to close the door?\" But I remembered I had floated through the wall during my first excursion. Wondering if it would work again, I braced myself and headed slowly towards the solid door. Half expecting to crash headlong into the panel, I raised my hands in front of me to absorb the impact. Instead, I sailed through it with ease. I could feel the material of the door as I passed through. It felt a bit like moving through warm water, but with no resistance.\n\nOn the other side of the door I stopped and floated to the floor. It was too tempting to resist, so just for the thrill of it, I reached back and pushed my arm elbow-deep into the door. Talk about a weird sensation. I could see only half my arm. The rest of it was undoubtedly waving about on the other side of the door. Chuckling, I envisioned what this might look like back in the bedroom. I knew that if I were on the other side seeing something like this, I'd be under the covers with the phone, dialing \"Ghostbusters.\"\n\nWith my arm still embedded in the wood, I slowly put my forehead against the smoothness of the door and pushed. Instantly my head passed through, and I found myself looking back into the bedroom. Everything was as I had left it, my wife and my body seemingly sound asleep. Stepping fully through the door into the bedroom, I began talking to myself. \"Un-frickin-believable!\" Standing still for a moment, I looked around, and then confident everything was okay, I turned and stepped back through the door into the hallway.\n\nA world of possibilities opened before me. Was there any kind of structure I couldn't go through? \"Maybe steel would be a problem,\" I thought. \"I'll have to check it out sometime.\"\n\nIt seemed nothing could hold me back. Leaping into the air, I floated down the stairway into the living room. Slowly, I flew up to the two-story open ceiling just to see what it would be like to be that high off the floor. It was exhilarating.\n\nI was flying! I was actually flying, and I could hardly contain myself. Like a kid whose wildest fantasy had come true I swooped down from the ceiling, passed through the wall into the dining room, and continued on into the kitchen. The feeling of flying through solid objects was irresistible. Before long I was zooming back and forth through the walls like Casper, the friendly ghost. As I zipped through the wall of the family room into an adjacent bedroom, I came to a midair stop. It was my eight-year-old son's bedroom. Floating silently near the ceiling, I gazed down as David slept.\n\nMy heart flooded with compassion for this gentle little soul. He was one of the happiest people I had ever known. Always a smile on his face, nothing seemed to get him down for long. Even at this young age, he had a wonderful sense of humor. With a warm and generous heart, he had empathy for anyone less fortunate. David's was the bright and cheery attitude I wished I had. From my perspective, he was beautiful.\n\nAs I floated near the ceiling, my daughter Stacey came to mind. Her bedroom was directly above David's. Traveling straight up through the floor, I was soon hovering over her bed.\n\nWe were so proud of our little girl. Pretty and petite, Stacey had the dedication, intelligence, and perseverance to accomplish anything that she wanted in life. She possessed such great inner strength and, at the same time, a wonderful sensitivity to others. At eleven, she was entering that awful hormone stage of life. We were often amazed and amused at the transformation taking place before us. One moment, she would be our sweet little girl singing her dollies to sleep. The next she would become a mature young lady trying to convince her mother that she was definitely old enough to stay at Alicia's party until 10 P.M.\n\nI was about to move closer so I could see more of that dimpled little face, when a huge spasm suddenly jolted through me and, in an instant, I snapped back into my body.\n\nWhat could possibly have happened? Opening my eyes, I sat up. Candace was now facing in the other direction. She must have turned over in her sleep, bumping my body as she withdrew the arm that had been lying across my chest. Disappointed at being forced back so soon, I still rejoiced at having been able to go out-of-body again.\n\nFor several minutes, I lay in bed processing the experience. It had been a great outing. Finally I got up and, retracing my journey through the house, I ended up in our children's bedrooms. I thanked God for sharing these wonderful little souls. I knew that when the time was right, I would have to tell them about the astonishing events taking place in my life.\n\nEncouraged by this small success, I renewed my efforts to find a reliable way to achieve an out-of-body state. Every night, before going to sleep, I experimented with various methods. Some I had read about, others I thought up. But nothing seemed to work. As the weeks dragged by, my obsession grew. It was, however, becoming more like work than pleasure.\n\nAs summer approached, my wife began preparations for a trip to Quebec where she had enrolled in a French language-immersion program. Crawling into bed a few nights before her departure, Candace asked me if I had seen her umbrella. She had searched the house without any luck. I couldn't recall having seen it for months. Speculating that we had probably left it somewhere she sighed, \"Well, I guess I'd better go out and buy a new one.\" With those last thoughts, I drifted off to sleep.\n\nAs if awakening from some obscure dream, I suddenly realized that I was staring at the umbrella. It was lying on a closet shelf in front of me. My awareness bolted to attention.\n\nThere was no question: this was her umbrella. I reached out to grab it. Swish\u2014my hand swept right through the handle. \"Oops!\" It hadn't registered on my sleepy mind that I was no longer in my body. Momentarily puzzled, I wondered how this could have happened without my being aware of it. I decided it didn't matter. The important thing was I had found the umbrella. I couldn't wait to tell Candace.\n\nIn a flash, I snapped back into my body. \"Honey,\" I gushed. \"I found the umbrella!\" Then it hit me. I hadn't the slightest clue as to where I'd been. All I knew for certain was that someone else now owned our umbrella, and it would probably stay that way. I continued with dismay to tell Candace the story, admitting that in my enthusiasm I had foolishly bolted out of the closet without knowing where I was. After considerable giggling, we finally fell asleep. I was going to have to learn how to contain my excitement.\n\nAgain, I had to grapple with something new. Without warning, I had been instantaneously transported to an unknown location without any awareness that I was not in my body. How had this happened?\n\nAs summer matured, I grew more practiced at producing out-of-body experiences, but I was still far from being able to make it happen at will. Occasionally, I would feel the choppy, fluttering vibrations starting in the center of my chest, before moving slowly into the rest of my body. I would try to gain control, attempting to step up the frequency to a higher pitch. But it always seemed so difficult to maintain. More often than not, the vibrations would eventually fade, and frustrated, I would fall asleep.\n\nFortunately, every now and then, I would find myself briefly dozing off, then abruptly snapping back to the awareness of intense vibrations already coursing through my body. I would hear a familiar, high-pitched tone and simply wish myself into the air. Then, after checking on my body, I would be off on another adventure. I didn't know why, but I always looked back at my body. It wasn't necessary, but I felt compelled to do it.\n\nOn one of those occasions, I happened to be cruising through the kitchen looking for something to do. When I spotted the fridge, I had an idea. Poking my head through the door, I checked for any leftover dessert I could eat when I finished my excursion. For some reason, the darkness inside the fridge caught me by surprise. Reminded of a silly old joke, I had to chuckle at my expectation that the light would automatically switch on. After a few amusing moments, I simply stated my intension to see better and, in an instant, I could identify every object in the fridge. Unfortunately, there was no dessert. But I soon returned in my physical body to gobble up the leftover ham and scalloped potatoes.\n\nIn the early stages, I felt safer staying within the familiar walls of my home, but soon that became boring. One night near the end of summer, I awoke from a deep sleep to the pressure of rapidly escalating vibrations. Looking up, I found I could see right through the ceiling into the night sky. A beautiful full moon beckoned, and I decided that it was about time I sucked up my courage and ventured outdoors.\n\nFloating effortlessly to the ceiling, I wondered what it would feel like to go through that yucky, itchy fiberglass insulation in the attic. With some nervousness, I moved slowly through the ceiling into the attic space. The insulation was hardly noticeable. Relieved, I continued up through the top of the rafters. A moment later, I was standing on the roof.\n\nThe freedom was exhilarating. Delighted with the ease of the maneuver, I thought how wonderful it would be to float up into the sky. Impelled by this simple thought, I lifted off of the roof and ascended. Rising to about a hundred feet, I stopped and hovered. I could see the entire neighborhood. It was after midnight, so very few houses were lit. I mused about how freaked out my neighbors would be if they only knew what was happening in the sky above them.\n\nSlowly, I moved higher, stopping at around five hundred feet. It was beautiful. Lights were visible from miles around. I thought it might be fun to take an aerial tour of the city, but just as I began to move down the block, a strange tugging sensation began to draw me back to my body. I didn't want to end the experience, but the tugging was persistent. When I could no longer resist, I gave in, surrendering to the pull. In an instant, I slammed back into my body with such force that it shook the bed. It was no big deal, as it turned out, just my bladder begging to be relieved.\n\nThe next opportunity came sooner than expected. Just three days later, too tired to think about spiritual journeys, I flopped into bed and immediately fell asleep. Less than an hour later, I awoke to a shrill frequency resonating in my ears. Sleepily, I closed my eyes and mentally recited my little prayer for assistance and protection. The sound in my ears seemed to intensify as I tried to gain control. Perhaps, I thought, I could use the sound as a trigger to evoke the necessary increase in vibrations. Experimenting, I found I was able to control its intensity, but could detect little change in my inner body vibration. After a while, when I could no longer maintain my concentration, I gave up, and simply fell asleep. Unexpectedly, just a few moments later, I again jerked back into full consciousness. The sound was now a high-pitched squeal. A strong vibration flowed through my body. I was ready for another journey.\n\nDeciding to take it slower this time, I floated leisurely to the ceiling. Skimming horizontally, I was heading for the bedroom door when I decided to try something different. Changing directions, I plunged through the north wall of my bedroom. Expecting to come out the other side near the high open ceiling of the living room, I found myself, instead, in a small, darkened space, about four feet long and two feet wide. There was a large metal pipe in the middle of it.\n\nIt took me a few seconds to figure out where I was. I had forgotten the large fireplace chimney on the other side of the wall. As I drifted inside the brick enclosure, I realized that my arm and part of my right shoulder were imbedded in the chimney pipe. I pulled back in disgust. Imagining how black and dirty my hand and arm were going to be, I pushed through the stone mantel into the living room. My arm, of course, was perfectly clean. I began to laugh at how bizarre and silly the whole thing was. When every touchstone of our reality is based in the physical world, it's difficult to leave old concepts behind.\n\nSoot or no soot, I was going to enjoy this outing. Rather than going up through the roof, I decided this time to exit through the front wall of the house into the street. I slid easily through the inner wall and the outer brick facade. But as I was coming out the other side, something struck me as odd. I was sure that I had automatically closed my eyes as I passed through the wall. I wondered if I could ever get used to the differences between the astral and material worlds.\n\nEverything was _strange_. To begin with, I wasn't sure how I was able to see. Obviously, it wasn't through my physical eyes. I decided to pay more attention to these pesky little physical quirks.\n\nPropelled by a small burst of energy, I floated out into the quiet street and up into the night sky. Stunningly beautiful, the sky appeared to have a special glow. As several clouds floated nearby, I noticed what appeared to be an energy halo around them. It reminded me of the aurora borealis. Shades of luminous blue, pink, and violet began to swirl, merge, and dance together in a brilliant display of choreography.\n\nCaptivated, I watched in awe, my attention drawn to the iridescence of one cloud. It seemed to be increasing in brightness and intensity, becoming more and more animated. Suddenly, a terrific flash lit up the sky. A dazzling arc of blue-white light erupted from the bottom of the cloud and slammed into the ground. Startled, I realized it was nothing more than lightning. But in my out-of-body state, it was a sight to behold. I hadn't noticed the approaching thunderstorm, but the developing view was sensational. I could actually see electrical energy building up in the cloud before it exploded into the ground.\n\nI studied the sky as the same cloud began to sparkle and dance, building in intensity once more. Soon it was a boiling vortex of power. I watched intently as the shimmering energy, swirling like a miniature hurricane, slowly edged its way to the top of the cloud. The bubbling formation arched across the top towards the center, and then, like a whirlpool draining through a funnel, collapsed in on itself, firing off a bolt of electricity into the ground below. Strangely, a jagged arc of orange light seemed to erupt simultaneously from the ground, leaping into the air to greet the incoming charge.\n\nIt was _beautiful_. I would never again look at a prairie thunderstorm in the same way, and I thanked God for this wonderful gift. If this was what it was like to be an angel, if this was what angels were able to see, then I wanted to be an angel.\n\nAs I continued to soak up the spiritual beauty of nature, my physical self-preservation senses began to kick in. I wondered whether I should be out in this kind of weather. As I think back on it now, I find it amusing. But at that point in my journey, I had no idea what boundaries existed. I knew that you don't mess around in a lightning storm. I may have wanted to be an angel, but not right at that moment.\n\nReluctantly, I headed back home through the large kitchen window. Passing through the double-paned glass was easy, almost imperceptible. Back inside I floated through the house and into my bedroom. I didn't want to end the excursion just yet. I wanted to play and thought it would be nice to have a playmate to share the experience.\n\nFloating down closer to my sleeping wife, I hovered a couple of feet above her. I wondered if it would be possible to get her to leave her body. The problem was that I hadn't the slightest clue as to how this might be accomplished. First, I needed to get her attention without actually waking her physical body. Even that thought seemed a bizarre contradiction.\n\nI recalled that when I was about to leave my body, the vibrations often seemed most intense in my feet and legs. Perhaps this was the key? Turning slowly in the air, I floated down towards her feet. Thinking I might be able to speed up her vibrations by sending energy into her body, I placed my hands on the blankets covering her feet and began to concentrate. After a few moments, however, it became apparent that nothing was happening. There has to be a way, I thought, and concentrated harder. All of a sudden, my right hand swished right through her foot. In surprise, I jerked my hand back. It was tingling with electrical energy.\n\nThe idea of passing through another living being hadn't occurred to me, but it was kind of fun. I began to wonder if it might be considered improper behavior. But, heck, it was way too cool. I couldn't resist. Placing my index finger gently against Candace's lower calf, I began to slowly increase the force. The tip of my finger easily passed through her skin and slowly sank to the knuckle in her leg muscle. Minute jolts of electricity tingled through my finger and up into my hand. After a few moments, I gently withdrew my finger and then poked it back in. \"Boy,\" I thought, \"this is way more fun than going through walls.\"\n\nOther than an exhilarating diversion, this didn't seem to accomplish much, so I floated towards Candace's head. Hoping for better results, I focused instead on sending a telepathic signal into her brain. For nearly a minute, I continued to project the message, \"Candi, come out and play,\" but nothing happened.\n\nI had another idea. Placing the middle finger of my right hand on her forehead, I concentrated on the same message. Several moments passed. I could feel the energy flowing easily from my hand into her brow. Suddenly, she began to move. \"Okay,\" I thought. \"Now we're getting somewhere.\" Unfortunately, to my dismay, she just kept right on moving\u2014all the way over onto her side. As she rolled over, I watched helplessly as her sleeping arm hung in the air for a couple of seconds, and then dropped onto the chest of my sleeping body. That's all it took: with a sudden jolt, I was instantly back inside. The vibrations were gone, and I was wide-awake.\n\nWhile she was still partially asleep, I asked Candace if she had felt me trying to communicate with her. Sleepily, she said she didn't know, but thought that something must have awakened her.\n\nWhen I casually mentioned passing my astral hand through her body, that certainly got her full attention. For some reason, she didn't think it would be so cool. She let me know, in no uncertain terms, that it was a rude thing to do, and I would be well advised to keep my damn hands to myself when she's sleeping. I tried to explain that it was purely for scientific purposes, but she wasn't buying it. The next morning, when fully awake, she did admit that it would have been great if I had been able to coax her out of her body, but she still didn't think she wanted anything passing through her.\n\nThroughout the summer and into the fall, I continued to study anything even remotely connected to the paranormal. One author explained that one of the best times to attempt astral travel was during the pre-sleep hypnagogic state. It's the term that alludes to that narrow balance of consciousness between awake and asleep that we all experience while we're in the process of drifting off each night.\n\nCalled theta, it is the relaxed state of mind in which dream images begin to flash through our consciousness. The condition is often accompanied by small involuntary muscular jolts. In my experimentation, I had found that theta was, indeed, an ideal mental state for exploration. The problem was in trying to maintain the condition.\n\nTry as I may, I couldn't seem to hold that level of consciousness for long. With continuous practice, I found that I could manage to extend my time hovering in the borderland, but it usually ended up with me falling asleep and waking up frustrated the next morning.\n\nThat same author also suggested that one way to help prolong the hypnagogic condition was to lie on your back with your hand and forearm held in the air. If you started to fall asleep, your arm would automatically begin to fall, which in turn would trigger you into raising your level of consciousness. You could then pull your arm back into position and in this manner stay on the edge of consciousness, extending the hypnagogic state indefinitely. I tried the method many times. After I got used to the discomfort, I found that it worked quite well, but it wasn't as successful as I had hoped. It did, however, provide me with a couple of exceptional experiences.\n\nLate one night, while experimenting, I lay in bed for more than 20 minutes, my body practically asleep while my mind remained awake. At some point, I could feel my arm slowly collapsing onto the bed. Reacting, I opened my eyes, but to my surprise my arm was still upright. It hadn't moved.\n\nDoubting my senses, I shook my head and shifted my awareness to my physical arm. Immediately its feeling and position were restored. Closing my eyes, I soon drifted back into my altered state. Not more than a minute later, I again felt my arm slowly dropping to the bed. Opening my eyes yet again, I was alarmed to find my arm still in an upright position.\n\nThe whole thing was intriguing. Vowing to maintain a stronger awareness of what was happening, I returned to my meditation. Moments later, I felt it begin to topple again. This time I snapped open my eyes to an unbelievable sight. As before, my physical right arm remained locked in an upright position, but now a lighter, whitish shadow of it slowly sank to the bed beside me. I realized that I was observing my physical arm and my astral arm at the same time.\n\nThe reality of the situation jarred me to full awareness. It had come upon me so gradually; I had no idea that I'd moved into an out-of-body condition. I couldn't believe how subtly it had taken place. Had I simply been so deep that I hadn't noticed the change? Or was it just becoming easier to accomplish?\n\nI had only begun to realize the scope of the changes that were taking place in my life. My explorations were becoming more and more fascinating, and I couldn't wait to see where this new freedom would take me. Even in my wildest fantasies, I could not have imagined what lay in store for me.\n\n# 4\n\n# Expanding Horizons\n\n_Promises of paradise, and wonders to behold_\n\n_Man-made dogma, imposed upon the fold_\n\n_But in frightened hearts there flickers still_\n\n_Desire for truth that guilt can't kill_\n\nAs time passed my obsession to learn and experience continued unabated. Reading anything paranormal that I could get my hands on, I came upon Robert Monroe's second book, _Far Journeys_. True to the author's style, it contained some fascinating revelations. Practically everything in the book resonated a feeling of recognition and truth for me.\n\nMost importantly, the book included an address for The Monroe Institute. It was a nonprofit organization that Monroe had founded to research the out-of-body phenomenon. I phoned the Institute. They promised to send me an information package, and for ten days I waited anxiously for the material to arrive.\n\nThe Monroe Institute sounded like the answer to my prayers. They had developed successful programs to assist people achieve altered states of consciousness. Thousands of people had attended their seminars. Due to great variances in the abilities and sensitivities of each individual, the Institute would not guarantee that participants would have an out-of-body experience, but they were confident that one would not leave the program without being convinced that awareness _can_ extend beyond the physical world.\n\nThe establishment of the Institute near Charlottesville, Virginia, came about as a direct result of public reaction to Monroe's books, his curiosity about the phenomenon, and his abilities as a researcher. Following the success of _Journeys Out of the Body_ , Monroe received a large volume of correspondence from all over the world. Many claimed to have had experiences similar to his, but were afraid to talk about them for fear of ridicule or concern that something might be wrong with them. Monroe's books helped people understand that out-of-body travel was something every human was capable of experiencing. It was a gift, something to cherish, not fear.\n\nDue to escalating interest, Monroe, along with a number of volunteers, set up a private lab to investigate the phenomenon. Out of the many letters and calls he received were a number of people who, like him, had the ability to travel out-of-body at will. With some of these people as volunteer test subjects, he and his technical team built a small lab containing a specially designed isolation chamber where participants' levels of consciousness could be monitored. Test subjects were connected to various biofeedback devices to study and map their states of relaxation and brainwave activity.\n\nAfter hundreds of hours of monitoring and testing, they found that subjects who attained out-of-body states while in the lab exhibited very similar levels of consciousness. Monroe and his assistants postulated that if they could somehow re-create these brainwave frequencies, they might be able to help others achieve an out-of-body experience.\n\nAn accomplished audio engineer, Monroe concentrated on sound as the medium to influence the brainwaves of test subjects. He and his team spent thousands of hours experimenting with various audio tones and electronic pulses, trying to produce an effective combination of frequencies that could be fed to a subject through stereo headphones.\n\nTheir hard work and perseverance finally paid off. Through the manipulation of a small electronic pulse known as a \"binaural beat,\" they were able to perfect an innovation in audio technology. The pulsing electronic signal was found to create a frequency following response in the human brain, which enabled automatic synchronization of the left and right hemispheres. Monroe was granted a patent for the process he called Hemispheric Synchronization, or simply Hemi-Sync.\n\nThe new technology made it possible to easily move test subjects into a state of deep relaxation in which their bodies slept while their minds remained awake and alert. In this state, subjects were able to experience expanded levels of consciousness, opening the doorway to enhanced sensory perception and the ability to access an apparent sixth sense, including the out-of-body state.\n\nThe Hemi-Sync process proved extremely successful. With the assistance of trained helpers, Monroe began to offer weeklong seminars to the public.\n\nThe more I learned about it, the more intrigued I became. There was no doubt in my mind. This was something I had to try. Calling the Institute, I signed up for \"Gateway,\" their weeklong beginners' course, and the time until it started couldn't pass quickly enough.\n\nWhen the day arrived, I could hardly contain myself. At the end of a long flight, as we approached the airport in Charlottesville, I looked out over the peaceful beauty of the Blue Ridge Mountains, and something inside me clicked. It was a moment of deja vu. I had the feeling I'd been there before.\n\nAfter a 45-minute car ride, we finally drove into the parking lot at The Monroe Institute. Like hallowed ground, the understated elegance of the rustic buildings and quiet surroundings commanded reverence. A virtual Mecca for body, mind, and spirit, it was making history here. The energy was palpable. As I walked through the front entrance, I knew that my world would never be the same.\n\nAfter dropping off luggage in my room, I headed to the dining room where the other participants and instructors were gathering. The excitement was contagious. With mounting anticipation, we were all eager to get started. Once everybody had arrived, we assembled in a comfortable meeting room where the instructors began to lay out the agenda for the upcoming week. An intriguingly diverse group, we were from all over the world, including the U.S., Canada, Australia, Germany, Switzerland, and England.\n\nEach of the twenty participants was assigned to a specific room, which would serve as our sleeping quarters as well as our own experimental lab. The instructors referred to these rooms as CHEC units (an acronym for Controlled Holistic Environmental Chamber). With a constant supply of fresh air feeding into a climate-controlled environment, the rooms were constructed to be as comfortable and quiet as possible. When the heavy black curtain was pulled across the opening to the enclosed bed, the unit passed into total darkness. It made no difference if our eyes were open or closed, everything would be black. Each encapsulated unit, however, was also equipped with special dimmer-controlled lights for those who preferred a bit of illumination. Complete with warm blankets, tape recorders, built-in wall speakers, and stereo headphones, the rooms contained everything we would need.\n\nFinishing the orientation early, we headed off to bed. Our training would begin in the morning. With the speakers emitting the soft relaxing sounds of ocean surf, I quickly drifted into a deep, peaceful sleep anticipating my first encounter with Hemi-Sync technology.\n\nMorning came too quickly. Still tired from the long day in crowded airports and cramped jetliners, I bolted out of bed to prepare for the exciting day to come. Following breakfast and an orientation gathering, we all retired to our CHEC units for the inaugural session that would introduce us to the deep relaxation of what Robert Monroe referred to as Focus 10. This was a level of consciousness he described as \"Mind Awake, Body Asleep.\" After each session, we would meet in the conference room to debrief, compare notes, and share our experiences.\n\nClosing the black curtain, I crawled into the bed and pulled on the headphones. The sound of ocean surf soon lulled me into a quiet meditative state. Within a few minutes, the surf sound faded away and a new mixture of soft pure tones began to flow through the headphones. It was a fascinating progression. In a slow controlled vibrato, the sound seemed to move back and forth, from ear to ear. Heaviness collected in my eyes. Within moments it spread down through my body and into my legs and feet. Slowly but surely, the sounds carried me to the edge of sleep.\n\nIt was a wonderful feeling. Calm and serene, I basked in the euphoria of relaxation for what seemed like 15 or 20 minutes. Much like the hypnagogic state one encounters when falling asleep, a constant stream of dreamlike images flowed through my perception. Not focusing on anything in particular, I allowed my mind to roam freely. The sensation was exquisite, and I soon lost all sense of time or connection with my body.\n\nAll of a sudden, I found myself jerking awake to the sound of Robert Monroe's voice calling us back to normal consciousness. Almost an hour had passed. To my great disappointment, I had fallen asleep. What had I missed? The trainers had warned us that this might happen. But even if we did fall asleep, they said, our subconscious would still receive the benefits of Hemi-Sync. Although they assured us it was nothing to worry about, I wasn't convinced. The long plane trip to Virginia was taking its toll.\n\nMost of my sessions during the first couple of days were similar. I often fell asleep and had little to report. But the energy of the place was starting to have its effect on me. Each day we were introduced to progressively more powerful Hemi-Sync signals. We moved from the \"Mind Awake, Body Asleep\" level of consciousness on to Focus 12, the state of \"Expanded Awareness,\" and finally to Focus 15, a level of consciousness beyond our concept of linear time.\n\nBy the third day, I was nearly bouncing off the walls. The energy flowing through my body was so strong that a tingling vibration rippled from my thighs all the way up to the back of my neck. The high-pitched tone, normally only present while I was in an altered state, now seemed to be with me all the time. By late afternoon, as we headed to our rooms for another session, the sound in my ears had become so loud, I worried it might be distracting.\n\nAfter settling in to the relaxing Hemi-Sync tones, I could feel the changes slowly spreading through my body. At first, as my thoughts drifted, I began to lose all awareness of anything physical. But when I finally refocused my attention, I couldn't believe the amount of electrical activity playing through my muscles. I could feel my lower legs tense up and then slowly relax. The stiffness and tightness moved to my thighs, which also contracted and then relaxed. Soon, the lower half of my body began to feel heavy and immoveable.\n\nFloating ever closer to the borderland of sleep, as had happened so many times before, my consciousness began to click in and out. One moment I would be awake and alert, and the next I'd be snapping back to awareness, knowing only that I had somehow fallen asleep. It was from one of these unconscious voids that I suddenly clicked back into awareness, just as I was lifting off the bed, floating toward the ceiling. My excitement was almost uncontainable.\n\nAs I approached the ceiling, I turned as usual to look down at my body. But this time there was something different\u2014I didn't seem to be in control of what was happening. I kept on rising through the ceiling of my CHEC unit. Before I knew it, I was sailing right through the lower body of a participant in the unit above me. Startled, but only somewhat concerned, I floated up through the roof of the Institute.\n\nWith every molecule buzzing, I began rising into the sky. One hundred feet, five hundred feet, a thousand feet, I quickly picked up speed, moving ever higher. Soon, watching the landscape shrink below me, I felt like a rocket blasting into space. This, I thought, must be what the Shuttle crews see as they leave the Earth.\n\nThe Virginia landscape became a blur of green as I moved higher. To my right, the vast sweep of ocean blue came into view, and still I continued climbing. Within moments, the entire planet became visible as I hurtled into space. The sun took on a distinct brilliance I had never seen before. Moving ever faster, I continued on until the Earth itself appeared as a tiny blue ball. Finally, many thousands of miles into space, I slowed and came to a stop.\n\nThe panorama was stunning. In every direction, I could see and feel the beauty of creation. A blanket of stars lay in every direction. It occurred to me how small and insignificant I was in the midst of this beautiful sea of creation, yet I knew I was a part of it all. Everything was intimately connected, and perhaps even the tiniest particle was no less important than the largest.\n\nFor an indeterminate period of time, I floated quietly in space, absorbed in the beauty around me. Monroe was right. This level of consciousness, Focus 15, was unaffected by time.\n\nIn my contented solitude, I became aware of a strange, distracting sound. It seemed to surround me, while at the same time resonate within me. Although I couldn't pinpoint the source, it appeared to be a deep, low frequency like the roaring of the wind or pounding ocean surf. Continuing to grow louder, I soon found it impossible to ignore. Concerned, I thought perhaps I should get back to my body.\n\nIn one swift movement, only a flicker in time, I snapped back into my body in the CHEC unit. To my astonishment, I could still hear the sound, and it was even louder. Then the realization hit me. _The sound was coming from me!_ It was my body, snoring!\n\nMy mind completely awake, I lay in the darkness, amused and astonished at the power of this anomaly. I was listening to my own body snore as if it were someone else. There was no doubt about it. Our minds and bodies could certainly operate independently of each other.\n\nMoments later, Robert Monroe's voice was calling us back to end the session. Briefly, I entertained a strange notion. I wondered how I was going to awaken my body. With a mere thought, though, I coaxed my fingers to move, then my arms. Soon I was back in control.\n\nTaking off the headphones, I stretched out. The debriefing session would have to go on without me. I needed some additional time to process the significance of this latest experience.\n\nMy flight into space had been profound. Almost too powerful, it had the effect of creating within me a high degree of performance anxiety. I became so determined to have another out-of-body experience that I'm sure I ruined the next few sessions by trying too hard. Rather than just relaxing and letting it happen, I kept trying to _make_ it happen. I was defeating myself.\n\nBy the last day of the seminar, I was so energized by the Hemi-Sync process, I was practically buzzing. We had been warned about this possibility and had been advised to do a grounding exercise. Running, swimming, hugging a tree\u2014any of these activities would help to ground the energy and bring us down to Earth. Most of the participants paid serious attention to these suggestions. I, however, didn't want to. I loved being high on psychic energy. It wasn't until later, when I was finally back trying to manage in the business world, that I began to understand why grounding was a good thing.\n\nDuring the week, we had progressively made our way into deeper levels of consciousness. The final and deepest level of the \"Gateway\" seminar was Focus 21. Bob Monroe called it \"The Bridge to the Other Side.\" It was from this level of consciousness that one could access the spirit world, visit with angels and spirit guides, or even deceased relatives and loved ones. Although intrigued by the possibility, I didn't think I was ready for it. I wasn't sure I wanted anything to do with anyone who had died.\n\nAs one of our last sessions got underway, I began to contemplate how I might integrate some of the new things I had learned into another meaningful experience. Offering my usual prayer for protection, I also asked to be given an experience that would most benefit me at this stage in my development. I sent the prayer out into the universe and trusted that all would go as intended. Then I relaxed and waited.\n\nAs the Hemi-Sync frequencies carried me into deeper and deeper levels of consciousness, time seemed to drag. I was beginning to wonder if I would get any response, when a sudden, powerful surge in vibration swept through me. In the next instant, I almost leapt out of my skin. There at the foot of my bed, bathed in light, stood a blond-haired boy. It startled me so much I had to fight the urge to jump out of bed and turn on the lights. I saw him clearly. He appeared to be about 12 years old. His freckled face and sparkling blue eyes beamed with delight. I didn't know what to do. Who was he? Should I try to talk to him? Say, \"Hi\"?\n\nAs I struggled to control my fear, an expression of understanding and compassion spread across his face. I was sure he knew my thoughts. As if in apology for frightening me, he smiled ever so gently, lowered his eyes, and in an instant he was gone.\n\nA heavy sadness flooded through me. I couldn't get over the feeling that this young spirit had wanted to communicate with me, and I had let him down. Why hadn't I been able to deal with my fears? Perhaps he could have helped me. I wondered if I would ever see him again or find out who he was. Through the remainder of the session I repeatedly invited him back, but it was not to be.\n\nWhen the session ended, I sat quietly during the debriefing. I had been deeply affected. The image of the boy would not leave my mind. There was something hauntingly familiar about him and I couldn't shake the feeling I'd known him before. It felt like he could have been a part of me, a distant memory. I wanted to call him back, to tell him I was sorry.\n\nWhen the final session got underway, my consciousness dropped quickly as a persistent numbing sensation enveloped my legs. Ready for another adventure, I knew that I would shortly start to feel electrical spasms in my body. For a moment I seemed to fall asleep, but snapped back with the usual jolt. I had been startled yet again by the sound of my own snoring. This was amazing, as ordinarily I seldom snore. On rare occasions when it did happen, I would usually be immersed in a deep sleep.\n\nI began to think about the autonomic processes of the body and how complex and intricate our physiology is. How ridiculous the official accepted view of our reality had become. Our scientific community was still insisting that we, and all that we have become, were the result of a freak accident in which precise quantities and types of amino acids came together in primeval pond scum and evolution took care of the rest.\n\nHow ignorant that was. At one time, I may have bought into such concepts because I wanted to deny the existence of a Creator. But even though I had preferred this left-brain linear view of the universe, deep inside, it didn't sit comfortably. I knew intuitively that there had to be something more. Now I was living and learning about this \"something more.\" Now I was experiencing firsthand the beauty and complexity of creation. I was beginning to understand the order in which everything existed.\n\nThe more I thought about our existence, the more intrigued I became with the concept of consciousness. Where exactly was the seat of consciousness within a human? If it is within the brain, as science dictates, then what was it that allowed me to travel thousands of miles away from my brain and remain a thinking, self-aware individual? It was becoming obvious to me that consciousness is not at all dependent upon the brain. The brain is only a secondary factor in control of our physical existence. It is merely an electronic relay system\u2014something that translates the mind's thoughts into actions and reactions within our physical bodies.\n\nI began to wonder if it would be possible to focus the thinking process at another location within the body other than the brain. Would it be possible, for example, to move my consciousness into my foot? I wanted to try, but couldn't imagine how this might be accomplished.\n\nI decided to just go for it. Almost immediately, I felt a change in internal vibration, and my perception became clouded and confused. I hadn't a clue where I was, but I could feel the flow of liquid surging past me in a steady rhythm. Too disoriented to make any sense of what was happening, I decided to withdraw to an external point of view. After a brief feeling of movement, I found myself floating several inches away from my right foot.\n\nI had somehow become a mere microscopic version of myself. With no perception of having any type of body, it felt like I had been reduced to the tiniest spark of conscious energy. Reorienting myself, I settled on re-entering my leg just above the ankle. In an instant I found myself back where I had started. It felt like I was inside a main artery, but with my vision extremely blurred, I couldn't be sure. If this was going to be of any value at all, I needed to be able to see. I recalled the incident within the darkness of my refrigerator and thought if it worked once, it should work again. I asked for clarity. Almost as quickly as the thought entered my mind, everything around me came clearly into focus.\n\nThe sensation was overwhelming, almost too much to handle. Surrounded in a flood of red and pink, a beam of light seemed to come from within me, lighting the way. It reminded me of TV medical shows where tiny cameras had been inserted through small holes into a patient's body. I could feel the flow of the liquid surging past me, and I could hear the sound of blood swishing through the artery. It seemed as though I was moving steadily along, against the flow of blood, pulled in the direction of my upper body.\n\nIn just a few moments, the sound of swooshing liquid swelled and I knew I had arrived at the center of my body. Surprised at the loudness of my pumping heart, I felt jostled about, the motion and flow of blood, agitated and powerful. But why, I wondered, was my heart beating so fast? It seemed a lot faster than usual. Considering the possibility that my own mental excitement might be contributing to the increased pounding, I decided I should find another place to hang out. This was new territory for me, and I wasn't sure how this might be affecting my body.\n\nAs I thought of other places to go, I felt a tug at my consciousness, distracting me, drawing me out of my altered state. I tried to resist, but to no avail. A moment later, I was back to normal awareness, wondering what had happened.\n\nAt some point during the session, I had rolled over onto my side. The headphones digging into the side of my head had become extremely uncomfortable\u2014and that's all it took.\n\nThrilled with my short excursion, I lay in bed contemplating the potential inner bodywork such as this might have for medicine. If trained medical people could be taught to make similar journeys into patients' bodies, the diagnostic possibilities could be enormous. Maybe this was how Edgar Cayce, the famous medical psychic, had been able to give accurate readings on the medical conditions of his clients. If only this inner-body exploration could be duplicated at will, I thought, the ramifications for the future could be exciting.\n\nOur last session over, everyone was sad that we would soon be leaving such a wonderful place. There were hugs and tears all around. It had been a great week. We had learned so much about consciousness, the reach of the mind, and the power of the spirit. But most of all, we had all come to a greater understanding of our own existence.\n\nBack home, the energy I had generated during my time at the Institute stayed with me for weeks. My recall of nightly dreams increased dramatically, the images and emotions often remaining throughout the day.\n\nI thought about how much simpler my life had been before these experiences began. I knew the answers to many questions that previously I hadn't even contemplated. It was becoming increasingly obvious to me that we know so little about the nature of our own reality. We just go about our lives, working, eating, and sleeping, trying to survive and maintain our existence in this material world. Forgotten are the times in our childhood when almost anything was possible, a time when we truly wondered about who we were and how we got here. I felt like I was inching closer to some answers, but there was so much more to learn.\n\nOccasionally, while dreaming, I encountered the boy who had visited me in spirit form at the Institute. Although it was never entirely clear, I had the impression we were brothers, spending hours together in boyhood fantasies. Intrigued by the familiarity and friendship his presence evoked, I was nonetheless shocked by my next discovery.\n\nMy sister, Josie, and I had been close for years and I often shared with her my experiences and the things I was learning. It was during one of these long conversations I mentioned my encounter with the boy and my subsequent dreams. When I suggested that he felt like a brother, I was taken aback by her reply. \"Well, don't be surprised if he really is your brother,\" she stated, excitement in her voice.\n\n\"What do you mean?\" I asked. \"How could he be my brother?\"\n\nJosie then went on to tell me a story I had never heard, or at least didn't recall. She recounted that when I was about two years old, a year after my brother Dale was born, our mother became pregnant with her twelfth child. Several months into her pregnancy, she had gone outside to fetch a pail of water from a well in the yard. A thunderstorm had just passed through the area and the pump-handle was still wet with rain. While she was working the handle, a bolt of lightning struck the pump. A huge jolt of electricity exploded down the wet handle sending our mother flying. Although she wasn't seriously injured, the baby didn't make it. It was a boy.\n\nI couldn't believe my ears. How could I not have known this? Why hadn't I been told? Josie explained that we were very young at the time, and it was something our mother never felt comfortable talking about. There was no reason to tell us.\n\nAlthough angry at being left out of the loop on this bit of family history, I was grateful for the validation. Strangely enough, since that day, other than one fleeting glimpse of his smiling face, I have neither seen nor dreamt about him. Perhaps, the contact served its purpose and there is no longer a need to maintain the connection. But I know in my heart that someday we'll be together again.\n\nDuring the summer of 1992, my sister Evalina was rushed to the hospital with a ruptured pancreas. It's a very serious ailment and, within a short time, it advanced to the point where there was little medically that could be done for her. Within a couple of days, she lapsed into a coma and was put on life support with little chance of recovery.\n\nOur family was in shock. Other than our brother-in-law, Albert, and our mother's earlier miscarriage, we hadn't lost any members of our immediate family. Suddenly we found ourselves facing our own mortality. Even though the doctors had expected her to live only a few days more, we took up a vigil at the hospital, praying that they were wrong and that Evalina would somehow make a miraculous recovery. To everyone's surprise, the days stretched into weeks, the weeks into months. Some days she would show small signs of improvement, only to dash our hopes in the following days, sliding ever closer to death.\n\nWhile at The Monroe Institute I had learned methods of remote healing. We had been taught how to send healing energies to others, even at great distances. Scientific studies had shown compelling evidence that the healing process could be dramatically assisted through focused prayer. Sending healing energies while out-of-body could be even more effective.\n\nAlthough I felt the need to contribute to the healing efforts of my family's prayers, I struggled with the appropriateness of doing so without my sister's permission. But how do you get permission from someone in a coma?\n\nA statement I had heard at The Monroe Institute contributed to my mental struggle. \"Every disease known to man can be cured, but not every person can be healed.\" They said that some things were just meant to be. According to some, everyone has a plan as to how their life will play out, the challenges they'll face, and the lessons they hope to learn. I felt the truth of this statement. As much as I loved my sister, I wasn't sure I should mess with her soul's karmic choices.\n\nEventually, I decided I couldn't just sit back and do nothing. I didn't know if there were consequences for someone who interferes with another's spiritual plans, but on several occasions, when I was able to move into an altered state, I attempted to contact my sister's spirit to ask permission to send her healing energy. I could never seem to get through. Perhaps it was because I was too involved emotionally. Or maybe I hadn't yet learned enough about the spirit world to make the connection. Each time, after trying unsuccessfully to connect with her spirit, I asked for forgiveness if I was intruding and then with great focus, mentally bathed her body in healing energy. Each time, I hoped and prayed I was doing the right thing, but if her healing was not meant to be, I knew that we might only be delaying the inevitable.\n\nStaying in contact with the family, we checked on Evalina's condition almost daily. I did find on a few occasions that her condition seemed to improve shortly after I had sent the healing energy. However, in my still skeptical mind, I could never be sure that it wasn't just coincidence.\n\nFor months, our roller coaster ride continued. Day to day, our hopes were raised and then dashed. Beyond the doctor's amazement that she continued to defy the odds, it seemed that we had little to be hopeful for. It was rare for anyone to survive as long as she had, and this in itself gave us hope that perhaps another miracle was possible.\n\nOne night in early November, after going to bed late, I attempted another spirit journey to help Evalina. After trying for an hour without headway, I gave up and dropped off to sleep.\n\nThe next thing I knew, a rush of vibration jarred me back into awareness. There followed a swift movement and I found myself floating over the nurses' station in the hospital's intensive care unit. It took me a moment to figure out where I was, but I was able to look directly into my sister's room. Although I couldn't actually see them, I was aware of beings on either side of me. They had brought me to this place, and I was not afraid.\n\nFocusing my attention on Evalina's room, I saw her lying on the bed. To the right side of the bed sat her daughter, Jackie; on the other side, a nurse was busily checking monitoring equipment. Realizing that it had to be nearly 3:00 A.M. I wondered why Jackie would be there at that time of the night.\n\nIt soon became apparent that something was wrong. While the nurse scurried about, Jackie appeared agitated. As I watched intently, a wispy, shimmering energy-form began to emerge from my sister's body. Floating into a vertical position a few feet above the bed, she paused for a moment to look in Jackie's direction, and then turned toward me.\n\nAlthough appearing a bit confused, Evalina looked radiant. A warm smile of recognition spread across her face. I tried to smile back, but the excitement was too much. In a flash, I was back in my body.\n\nImmediately awakening my wife, I told her I thought Evalina had either just died or had suffered a major trauma. Candace suggested that I phone the hospital, but, in the wee hours of the morning, it might have seemed weird, so I resisted. I reasoned that if something had happened, we would undoubtedly receive a call in the morning. I didn't want to be spooking anybody.\n\nMorning arrived and then midafternoon, but no one called. I'm not sure if Candace was beginning to doubt me, but I knew what I had seen. I also knew one thing more: Evalina was ready to make her transition. I somehow knew that she was now looking forward to the release. It was meant to be. I would not be attempting any more healing sessions.\n\nTwo days later, Candace and I made the three-hour trip to the hospital. While I sat with Evalina, my wife visited with other family members. After a while, I headed back to the waiting room where Candace and my niece were in conversation. Jackie addressed me immediately, \"Uncle Paul, how did you know about mom?\"\n\nA bit surprised, I asked, \"How did I know what?\"\n\n\"Aunt Candi was just telling me that on Wednesday night you knew that Mom's heart quit, and you knew that I was at the hospital with her. That's so amazing, it's scary.\"\n\nI sat down and told her what I had seen. Wiping away tears, she confirmed that indeed, early on Wednesday morning, at about ten minutes to three, her mother's heart had temporarily stopped. The nursing staff had previously been instructed to not attempt resuscitation. Believing Evalina might be dying, they had called Jackie in from the waiting room.\n\nJackie told us that it was just coincidence that she was in the hospital at the time. She normally would have been home with her family, but her husband was spending a few days away with their sons. Rather than stay at home by herself she had decided to spend the night at the hospital. She was sleeping in the waiting room when the nurse came to get her. Evalina's heart had stopped for more than two minutes before starting again. Even though I had been certain of what I had seen, the validation sent chills up my spine.\n\nI still wondered who had brought me to the hospital at that precise moment. Somebody had been guiding me. I knew it would be just a matter of time before I found out who.\n\n# 5\n\n# _To Die Again_\n\n_So is it still by chance or sad intent_\n\n_Destruction looms with our consent_\n\n_While thoughts, the thief within our heads_\n\n_Leave us fearful in our beds?_\n\nThe autumn of 1992 proved to be very mild across the Midwest. While our sister continued to defy all odds by clinging to life, distracted as the family was, our lives went on.\n\nI had been an athlete and hockey player all my life. But, by the age of 41, I was already too slow and old to be playing a young man's game, so I contented myself with a position on a local old-timers' team. It was a lot of fun, and we got a bit of exercise at the same time. We were, however, your typical \"weekend warriors,\" going from relative inactivity during the week to all out battles on weekends. It was during one of these robust outings that providence dealt me another unexpected hand.\n\nAlthough my leg muscles felt stiff, everything else seemed normal as I made my way to the arena for our Sunday morning scrimmage. Looking forward to a good workout, I laced up my skates and headed out onto the ice with my team. It wasn't long before we were racing up and down the ice, firing frozen pucks at our aging goaltenders.\n\nAfter only a few minutes of play, I was becoming surprisingly tired, when midstride, it felt like I had been hit in the chest with a sledgehammer. My legs nearly buckled beneath me. I had no idea what was happening. Utterly exhausted, I stumbled off the playing surface.\n\nI couldn't believe how winded I had become. Even after several minutes on the bench, I still struggled to catch my breath, and, to top it all off, a sharp, deep pain began to develop in my left elbow. I couldn't remember hurting myself and had no idea why it should be so sore. The pain, however, seemed to grow more severe with each passing moment. Nauseous and sweating profusely, I was beginning to think that I might have come down with food poisoning or flu.\n\nAs the minutes dragged, the pain in my elbow became unbearable, my breathing difficult and labored. It was obvious that I was too sick to continue playing, so I decided to go home. Heading to the dressing room, I changed into my street clothes, and in less than five minutes I was ready to leave.\n\nMy equipment bag in tow, I pushed the door open and, struggling to keep my balance, staggered into the hallway. I was beginning to think that my problems might be more than just food poisoning. Instead of heading home I should be going to the hospital. Fortunately, an alert rink attendant took one look at me, and rushed off to call an ambulance.\n\nWith each passing minute my condition seemed to worsen. Before long, I was leaning against a wall, trying to maintain my balance. Just the simple act of breathing was painful. No longer able to stand, I dropped to my knees and slumped to the floor just as a couple of my teammates popped around the corner to see how I was doing. Alarmed, they propped my head up on a towel as the rink attendant returned to announce an ambulance was on its way.\n\nSoon the rest of the team gathered around me in the hallway. Their hushed tones and concerned looks suggested that this probably wasn't one of my better days. Closing my eyes, I tried to ignore the numbing pain and slowly I drifted towards sleep.\n\nThe sound of running feet and clattering wheels jarred me back to awareness. I opened my eyes as two paramedics, aided by a couple of my teammates, lifted me onto a stretcher. In a moment, we were whizzing through the halls of the arena and out the front doors to the ambulance.\n\nThe female attendant quickly strapped an oxygen mask to my face, and the siren wailed as we roared out of the parking lot and onto the street. She asked my name and address, but as I tried to respond, it sounded like I was speaking in an empty barrel, my voice strangely hollow. I fell silent. Aware that the young lady was still talking to me, I tried hard to concentrate on her words, but it proved impossible. Nothing she was saying made any sense.\n\nI became aware of a buzzing sensation building in my ears. The sound grew louder, and for the first time I experienced fear. Panicking, I fought to maintain control, but it was a losing battle. Soon I was struggling to keep my eyes focused. Everything turned hazy and then, with the sound of whooshing air, the world around me began to close in. Within moments I could no longer see nor hear and an eerie gray mist flooded my perception. Nervously, I waited for whatever was next.\n\nTime seemed to slow. I drifted at the edge of consciousness, lost in my fading thoughts. Finally, too tired to even think, I let go, surrendering to the inevitable. Darkness engulfed me. How long I remained unconscious I could not be sure, but the next thing I knew, from deep within, came the abrupt sound of a loud, pinging bell. Crashing into my awareness, it jolted me to the core. Then, to my surprise, I started to move.\n\nSlowly, like a feather wafting on a breeze, I lifted off the stretcher and into the air. It was a peculiar feeling. Although the movement felt similar to my previous out-of-body experiences, I knew that this time I was definitely not in control. Strangely, the pain, so excruciating only moments earlier, simply evaporated.\n\nFrom a deep darkness, my vision began to open around me as I felt myself bumping up against the roof of the ambulance. I watched the paramedic below as she adjusted the oxygen mask on my face and checked the time on her wristwatch.\n\nHaving experienced a touch of fear moments earlier, I now wondered why I had been so worried. Although realizing this might be the last time I would be leaving my body, I remained calm and collected. There was no doubt in my mind that I was dying, and I began to consider the consequences.\n\n\"Jesus,\" I thought, \"Candace is going to be pissed off.\" I remembered that just a couple of weeks earlier she had been giving me hell for thinking that I could just run off and play hockey without first getting into proper shape. In all the years that we'd been married, she had never brought up that concern. But she had her reasons. A few months earlier, a friend of ours had died of a heart attack after an evening of floor hockey. Several weeks later, another friend barely survived a heart attack at the age of 39. \"You're not 20 anymore,\" she warned. \"You're 41. Do you want to kill yourself?\"\n\nI had, of course, just laughed her off. \"What are you talking about?\" I argued. \"I've never been in better shape.\" Now I wondered if this would be my contribution to the collection of \"famous last words.\"\n\nThe thought of leaving her and the kids behind weighed heavily on my mind, and an intense sadness came over me. Then suddenly I found myself floating in the living room of my home. Candace and David scurried about, getting ready for church service. I remembered it was Sunday morning.\n\nI loved my family intensely, but I knew that even though my death would be tough on them for a while, they would survive. They understood that we don't really die; we just move on to a different realm. I was confident that they would be fine and someday we'd be together again. Whispering a sad goodbye, I felt another shift in perception, and a moment later, I was back near my body.\n\nAs I floated again inside the ambulance, I began to feel a tug, something pulling at me. Then, slowly, as if being drawn by an unknown force, I started to move upwards. Within moments I was pushing through the roof of the ambulance. I felt the various layers of materials, the fabric, the insulation, and finally the metal, as I slowly emerged out the top into the crisp morning air.\n\nRising to about 20 feet, I flew alongside the ambulance as it roared through the streets. After a few blocks, the tugging became stronger. Somehow I knew it was time to go. So without concern or apprehension, I simply gave in, accepting my fate. In a moment the streets, the houses, and finally, the ambulance faded into a gray mist. Euphoria overcame me. My body may have been dying, but I had never felt better.\n\nThe mist around me grew steadily thicker, turning to black. I was floating in a dark void. Remarkably, I wasn't afraid\u2014instead I became increasingly energized with delight and anticipation. Powerful vibrations coursed through me, and then, like a jet accelerating down a runway, I started to move through the darkness. Before long it felt like I was traveling at a tremendous rate. I was drawn toward a pinpoint of light in the distance ahead and could barely contain my excitement. The urgency and yearning to reach the light became consuming. It seemed I had waited so long for this moment. I was finally going home, and there was nothing I wanted more.\n\nOblivious to everything else, I became absorbed in my goal when, suddenly, a huge spasm virtually exploded through my awareness, jarring every particle of my being. The next instant, I was back inside my body.\n\nI couldn't believe the assault on my senses as I opened my eyes in the emergency room at the hospital. The pain had returned, flooding through my body in waves. In shock, I knew only one thing for sure: I didn't want to be back.\n\nA nurse struggled to insert an IV needle into a vein in my left wrist while a doctor barked his instructions. Everything around me seemed too bright, too harsh. My jacket and shirt were pulled back, exposing my chest, and I found myself screaming, \"No, please, this is a mistake. Let me go. Please let me go!\" The words, however, seemed caught up in my mind; there was no sound.\n\nIn a surreal haze I could see and hear everything around me, but I had lost control of my body. I felt like a rag doll as a doctor pulled off my shirt and flopped me back down onto the table. While one of the physicians injected me with drugs, another put his face in front of mine. Slowly, but firmly, he spoke to get my attention. \"Paul,\" he said, \"you're having a heart attack. You've got to relax.\"\n\nHis words sounded bizarre and incomprehensible. Me, having a heart attack? Ridiculous! And if it was true, how in hell did he expect me to relax?\n\nThe pain was more than I could handle. It felt like there was a huge weight on my chest squeezing the life out of me, but it was nothing compared to the pain in my left elbow. If I could have spoken, I would have begged them to cut it off. I would rather go through the remainder of my life missing an arm than have to endure another minute of that pain.\n\nIV bottles flying, they rushed me out of the emergency room and into the intensive care unit. For more than an hour, the doctors worked frantically to clear the blockage from my heart. The pain, somewhat lessened by morphine, continued to be excruciating. In my horrified state, time dragged agonizingly slowly.\n\nThe most mind-bending part of the whole experience came about while _watching_ the drama unfold around me. It was intriguing. If I hadn't been feeling so lousy, I'm sure I would have enjoyed watching all the action even more.\n\nEspecially spellbinding was the minidrama of the EGG machine as it displayed a continuous graph of my heart rhythm. It would show a steady rhythm of heartbeats, and then it would go crazy. There might be as many as five rapid beats in a row as my heart fibrillated, and then it would stop and miss several beats. Continuing this way for almost an hour, the erratic display was more than a little scary and surreal. Often when my heart stopped beating or began missing beats, I wondered if this would be the time when it wouldn't start up again.\n\nAs time wore on, the doctors became increasingly concerned I would not survive. They decided they had better allow my wife in to see me. Candace appeared shaken, but not as badly as I thought she might be. To her credit, not once did she say, \"I told you so,\" although she later confided that she had certainly thought about it.\n\nThe situation was becoming critical in more ways than one. If I were to survive, the longer the blood-flow was blocked, the more damage there would be to my heart muscle. Too much permanent damage wouldn't leave me with great prospects for a future.\n\nFinally, after an eternity and an arsenal of drugs, the blockage dissolved. I became nauseated for about 30 seconds, and then, like someone had thrown a switch, the pain disappeared and I felt instantly better. My heart rhythm returned to normal and so did my thought processes. When it was all over, exhausted, I fell into a deep, deep sleep.\n\nLater, when I awoke, the episode seemed like a bad dream. I couldn't believe it had happened to me. Several days would pass before the totality of the experience settled in. My recollection of the pain was all too real, but the memory of the near-death experience was wonderful.\n\nSuch a paradox: one of the most wonderful experiences in my entire life had been dying. For the first time in years, I recalled my previous encounter with death at the bottom of the murky dugout when I was 12. All the details flooded back into my mind, releasing with it countless long-forgotten childhood memories and emotions. It was too much to bear. Tears welled up in my eyes, and I cried.\n\nDeath, I realized, would be a glorious event. Rather than something to fear, it was something to look forward to. The physical pain of a heart attack I wouldn't want to repeat, but the dying part was beautiful. Strangely, although thankful for the spiritual experience, I was somewhat disappointed. I had read numerous accounts of other people's near-death experiences. Some of them had had fabulous encounters with light-beings who guided them through life reviews. This hadn't happened to me. I hadn't received the full deal, and I felt gypped.\n\nIn the weeks following the heart attack, I went through a range of emotions. In the beginning, I didn't want to believe that it had happened to me, but when it was confirmed that 25 percent of my heart muscle had been permanently damaged, self-pity often overwhelmed me. I had been physically active through most of my life. When it became apparent that I could hardly walk up a flight of stairs without becoming winded, it shook me to the core.\n\nMy family seemed to handle everything in stride, but they were at times perplexed by my emotional swings. One moment I would be fine and the next practically reduced to tears. The doctors explained that this gamut of emotions was normal for most heart attack victims, but to me it didn't matter. For the first time, I didn't seem to be in control of my own body. The possibility that I could, at any moment, drop dead in front of my children was greatly distressing. Over the next year, there was hardly a day when I didn't consider how vulnerable I'd become.\n\nFollowing the heart attack, I came to realize the degree to which my children had been affected by my previous out-of-body experiences. Prior to the incident, I had shared many of the stories with them, and they had been accepting of the spiritual changes taking place in my life. Candace told me that after receiving a call from the emergency room that Sunday morning, she had bundled the kids into the car, apprising them of the seriousness of my situation. During their long stay in the waiting-room, our daughter Stacey became teary-eyed and said, \"But Mom, Daddy likes it on the Other Side so much, what if he doesn't come back?\" To this day, I cannot think about that without a lump in my throat.\n\nA month after I was released from hospital, my sister Evalina finally made the transition to the Other Side. Along with the rest of our family, I was relieved she would no longer suffer. Her six-month coma had been hard on the family. Everyone was emotionally and physically exhausted. The funeral seemed anticlimactic as most of us had all but completed our grieving. I, however, had a different perspective on her death from the others. I knew that she was in good hands and that I would be seeing her again.\n\nIn the first few months following my second near-death experience, I found out-of-body travel virtually impossible. Try as I might, I couldn't free my mind enough to relax. Whenever I tried, the vivid images of my dying would play over and over in my mind. I was haunted by strong emotions. The experience itself had been extraordinary, and even though I had been initially upset at being brought back to this painful physical world, I wasn't ready to leave. I still had a lot to learn, and there were so many things I needed to do. Time, however, heals all things, and although it took more than a year, my life, along with my spiritual quest, slowly returned to normal.\n\nReading in bed one night, I became so droopy-eyed I set the book aside and turned off the lights. After tossing and turning for several minutes, it was becoming apparent I wasn't ready to sleep, so I decided to try to move into an altered state. Within a short time, a high-pitched tone began in the center of my head and I thought it might be possible to do some exploring.\n\nJust as the familiar vibrations started, I felt an annoying itch in the calf muscle of my right leg. I didn't want to ruin the altered state by scratching, but it was driving me crazy. So, while lying on my back, I thought that perhaps if I moved my arm very slowly I could reach the itch and still maintain my level of consciousness. Slowly I inched my fingers forward, but it soon became obvious that I might have to either twist my body to the side or lift my legs in order to reach the spot. Concentrating, I continued to stretch, reaching farther and farther until I realized that my fingertips were already well below my knee. This was impossible.\n\nThere was no way my physical arm could stretch that far. My body, however, felt completely normal, and I began to wonder how far I might be able to reach.\n\nAgain I stretched with ease and I soon felt the tips of my fingers moving down my calf to the bottom of my leg. What a bizarre feeling! An image of a monkey popped into my mind and I almost laughed out loud. Regaining control, I pushed on. In no time, my hand was reaching past my ankle and making its way through the blanket at the bottom of the bed. The absurdity of it was too much to believe, and I had to giggle at the strangeness of it all.\n\nCommitted, I was having too much fun to stop. As I continued to reach out into the darkness, I thought, \"Jeez, I sure hope Candi doesn't wake up. She'd be in for a hell of a surprise.\"\n\nMy arm, or at least my perception of it, continued to stretch until the tips of my fingers bumped into a solid object. Feeling around, I discerned a smooth, raised wooden surface. It had to be the dresser, but that was at least 12 feet away from the foot of the bed. For this to be possible, my arm would have to be at least 18 feet long. The whole thing struck me as terribly funny, and I struggled to maintain control.\n\nSlowly but firmly I pushed against the smooth surface of the dresser. After a slight resistance, my fingers slid easily through the cabinet door and into the sweaters stacked on a small inner shelf. I kept going. Soon I was through the thin panel at the back of the dresser and into the wall behind it. As I continued on through the drywall and into the wall cavity, my hand brushed lightly along the side of a two-by-four-inch stud where I felt something sharp. Sticking out of the edge of the stud was a drywall nail. Poorly aimed, it had almost missed the stud entirely. I'm not sure why this stood out in my mind, but it occurred to me that perhaps some day I could cut into the wall to check it out, if for no other reason than to verify for myself that what I felt inside the wall was really there.\n\nI had just begun to consider what to do next, when something pulled at my awareness. A cough was building in my throat. I tried to resist for a while, but the urge finally overcame me. In an instant my consciousness was back to normal, and so was my arm.\n\nOver the next several months I occasionally thought about the bent nail I had found in the wall. Although my curiosity had been piqued, the prospect of all of the work involved in cutting open the wall and repairing it kept me from checking it out. Almost a year later, however, I ended up doing some renovations in an adjoining bathroom and my curiosity got the best of me. The work involved some drywall patching on another part of the room, so I thought if I have to patch and paint anyway, why shouldn't I just cut a hole in the other wall too?\n\nIn no time I had finished cutting a sizeable hole through the wall where I calculated the bent nail might be. With my heart pounding, I slowly removed the piece of drywall, looked inside, and there it was, exactly as I believed it to be. Running my fingers along the stud, I grasped the nail. You can't imagine how delighted I was with the confirmation. Astral and physical realities had blended perfectly.\n\nEven though my out-of-body successes had been few and far between, I continued my crusade. One night, while Candace was away for the weekend, I tuned our bedroom television to an easy-listening music channel and got into bed with a book.\n\nAt some point, I must have drifted off while reading. Later, through the fog of sleep, I slowly became aware of my surroundings. For some reason, the music playing on the TV began to cause a similar vibration in my body. The sensations became more and more jarring, until I felt like a huge tuning fork. Finally I could stand it no longer. I had to get up and turn off the television.\n\nSitting up, I swung my feet over the edge of the bed and stood up. An odd feeling came over me. Something didn't feel right. A chill ran up my spine. Out of the corner of my eye I caught a glimpse of someone in the bed behind me. In a rush of adrenaline I spun around to face the intruder and stopped dead in my tracks. There was no one in the bed but me. It was my own body that I had just left behind.\n\nRelieved, I turned away from the bed, intending to take advantage of my out-of-body state. I had only taken a couple of steps when I felt a pull near the back of my neck. Something was holding me back. Again I looked behind but saw nothing, felt nothing. It occurred to me that it might be the so-called silver cord. (I had read accounts of people who reported noticing, while out-of-body, a thin silvery cord connecting their energy bodies to their physical bodies. Even the Bible contains references to a \"silver cord.\")\n\nWhatever it was, there was no way I was going to let it stop me. Straining like a horse in harness, I pulled with all my might. Suddenly, as if someone had let go of the other end of a rope, the resistance released and I cartwheeled across the room. Half expecting a burst of laughter from some hidden audience, I spun around. There was nobody there but my sleeping body. A little chagrined but more determined than ever, I headed across the room towards the TV.\n\nThe small television sat on the left side of a long dresser. At the center of the dresser was a large mirror. As I passed in front of it, it occurred to me that this would be an opportunity to examine myself in a mirror while out-of-body. But to my surprise, there was nothing there, only the reflection of the room behind me. It was so weird to be looking directly into a mirror from less than two feet away and not see myself.\n\nIn the mirror, I could clearly see my sleeping body on the bed behind me. Looking for some rational explanation, it occurred to me that maybe my eyes weren't open. Concentrating, I made an effort to open my vision and instantly found myself looking through my physical eyes. All I could see was my hand draped across the open book splayed across the bed covers. Well, that didn't seem to help, so I let my eyes fall shut and immediately I was back in front of the mirror, which again reflected only the room behind me.\n\nAgain I opened my eyes, and again I was instantly transported back to my body, seeing only the fallen book in the light of the reading lamp. Closing my eyes once more, I found myself back in front of the mirror, staring at the room behind me, marveling at the peculiar physics of the out-of-body state.\n\nThe annoying music, however, was still playing so I decided to try another experiment. I wondered if it would be possible to affect physical matter and shut off the TV while in my energy body. Reaching the knob on the front panel, I twisted it counterclockwise. There was a distinct clicking sound and, to my amazement, both picture and sound went dead. But almost immediately it came back on. I tried again. There was a click, the TV seemed to shut down, but then came back on.\n\nThis was beginning to unnerve me. The irritating music continued, and I was no further ahead. I had to find out what was going on. In frustration, I turned back towards the bed and dove into my body. Sitting up, I then leapt out of bed and headed over to the TV. As I got to within a couple of feet of it, I stopped in disbelief. To shut off this particular set, you had to push the button in, not turn it. I reached out with my physical hand, pushed the button to \"Off,\" and returned to bed.\n\nSleep, however, was not possible. After tossing and turning, I got up and retraced my steps through the room. Why had I so clearly felt the button turn and click off? Previously, I had read accounts of others who reported similar anomalies while out of body. On astral journeys, they had noticed oddities such as furniture or other objects out of place. Since that night, I've never again encountered such an anomaly. Over the years, I have often contemplated the details of that particular experience, and the TV quirk still remains a mystery to me.\n\nFor some time, I also wondered why I hadn't been able to see my astral body in the mirror. I have, however, come up with a possible answer to that question. A mirror in our physical world can only reflect that which is visible in the physical world. Our astral or energy bodies operate at much higher rates of vibration, beyond the spectrum of physical light. So even though I could look down and see my energy body, I would not be able to see its physical manifestation in a mirror, because it is simply not physical.\n\n# 6\n\n# _Guidance_\n\n_Then steadfast courage and honor bequest_\n\n_The hand of God we'll squarely test_\n\n_Until lust for battle be lost in the fray_\n\n_And restless spirit lights the way_\n\nThroughout our lives, most of us encounter moments of deja vu, those elusive memory jolts that give us the impression that we may have had the same experience before. It might be a conversation, an incident, a feeling of familiarity with a place we've never been, or a knowing that we've had the identical experience at a previous time, but just can't remember it clearly. Other times, we receive flashes of intuition or receive answers to difficult problems seemingly out of nowhere. Occasionally I had such experiences and wondered what it was all about.\n\nTo my delight, I had been able to arrange another trip to The Monroe Institute. They referred to this second weeklong seminar as \"Guidelines.\" As its name suggested, the program was designed to help participants search within to identify and experience the subtle communication from so-called spirit guides.\n\nBased on the experiences of Robert Monroe and numerous other out-of-body explorers, it was postulated that every person has one or more spirit guides assisting them throughout life. This wasn't a new concept to me. I had been taught as a child that we all had guardian angels looking after us. My problem was that I had never seen evidence of anything like an angel and had difficulty buying into the idea. However, considering some of the seemingly impossible experiences I'd been through, it was certainly worth checking out.\n\nThe \"Guidelines\" protocol was the same as before. We were each assigned to our own specially designed rooms, where all sensory distractions could be blocked out, enabling us to move easily into controlled altered states.\n\nUnfortunately, the first couple of days were a rerun of my first visit to the Institute. Tired after the long trip, I could hardly make it through a single session without falling asleep. I decided not to waste time worrying about it, and by the afternoon of the second day, an energizing vibration began building within me.\n\nAs the morning session began, it wasn't long before I felt myself settling into a deep hypnagogic trance with the now familiar flashes of hyper-consciousness. Slowly my thoughts and attention drifted away from my body, leaving me in a warm, comfortable void.\n\nI loved this part. The clarity of mind was such that I could drift for hours in the euphoria of expanded consciousness. I always felt a strong connection with what seemed like a never-ending universal awareness of all things. Images, thoughts, concepts, and living scenes played out in my mind. Sometimes it felt as though I was watching my own dreams, but without the confusion and scattered emotion that normally accompanies them. Absorbed in my thoughts, I lost track of time; I could spend an hour or a moment, it didn't matter. It was so utterly enjoyable.\n\nThe abrupt sound of Monroe's voice shunted my attention back to the present. He was asking us to return to normal waking consciousness. I couldn't believe that 45 minutes had passed.\n\nI didn't want the session to end. A powerful energy had just begun rippling through me. It was an opportunity I couldn't pass up; they would have to do the debriefing without me. Concentrating on the vibration, I thought about separating from my body. The next thing I knew, I was floating towards the ceiling.\n\nIt would be just a matter of minutes before the waking signal would pull me back to my body, so I had to make a decision. Where to go? I thought about my wife back home and wondered if it would be possible to make the long trip to see her. In the next instant, I found myself hovering in bright sunlight over the front entrance of a building with a strange roof design. I wasn't sure where I was, but the street seemed familiar. A few moments later, as I glanced around for clues, a vehicle pulled up to the sidewalk and stopped. It was a car I knew well.\n\nI realized I was floating over the front entrance of the primary school where my wife worked as a secretary. Delighted, I watched intently as Candace hurried up the long sidewalk, and passing right below me, opened the door and rushed into the building.\n\nI took a mental note of what she was wearing: a matching outfit with a blouse and walking shorts, white with light blue flowers throughout. These details would help verify the fact that I had made the three-thousand-mile trip home in a fraction of a second. I would be able to describe the weather as well as my wife's clothing.\n\nSomething, however, seemed amiss. It occurred to me that it had to be nearly ten o'clock. Why was Candace only now arriving for work? She had to be two hours late.\n\nI was just about to drop down through the roof to follow her into the office, when I felt an insistent tug. It was the beta signal on the headphones calling me back to Virginia. The session had obviously ended. Surrendering to the pull, I slammed back into my body.\n\nPuzzled, I remained still for a couple of minutes, contemplating my journey. Why would Candace be going to work at almost ten o'clock? Then it occurred to me. I had forgotten about the two-hour time difference between Virginia and home, where it was only 8 A.M.\n\nDuring the debriefing and chat sessions on the previous day, our instructors talked about the various experiences that some participants commonly reported during Guidelines seminars. Some reported encounters with spirit guides, loved ones who had passed on, and some even talked about previous lifetimes.\n\nI had serious reservations about things such as past lives. As a Catholic, reincarnation wasn't part of our belief system. To us, it was just some silly illusion of those Eastern religions. In retrospect, I can't believe what a skeptic I was. For almost two years I had been floating through walls and flying into outer space. I had seen the spirit of my dying sister. Yet I still resisted the possibility that we could live more than one life.\n\nThe concept of guardian angels had been somewhat easier to swallow as these beings were actually alluded to, albeit in vague terms, in the Bible. I reasoned that the term \"spirit guides,\" might well refer to the same beings. It could have been just a different name for angels. I filed this stuff under the \"I'll believe it when I see it\" category, and got up to join the group.\n\nThe afternoon session began later than usual. But it didn't take me long to settle into a relaxed state. While my mind wandered in a dark, comfortable void, I soon forgot about my sleeping body.\n\nThe purpose of the session was to introduce us to a new type of frequency. Designed to create a shift in our vibrations, it would give us the opportunity to experience a new dimension. As Monroe's calm voice reverberated over the headphones, I was having difficulty concentrating on his directions. The energy within me was strong, but I kept drifting off into my own thoughts, barely aware of Monroe's instructions. To make matters worse, my mind also seemed to be falling asleep. I kept clicking in and out of consciousness.\n\nIt was from one of these lapses that I became aware of a spinning vibration in the pit of my stomach. Different than anything I had experienced before, it was nearly nauseating. I was trying to get a grip on it, when I suddenly found myself floating in strange surroundings, about 15 feet above the ground. Wherever I was, it was nighttime.\n\nIn the dim glow of a small yard light, my attention was drawn to a gleaming object lying in the thick, dew-covered grass below. It appeared to be a polished, nickel-plated handgun like some of the semiautomatic pistols I had seen on television. I was puzzled. I hadn't the foggiest idea where I was or why.\n\nAs I floated in place, trying to figure out what was happening, I felt another surge of energy and, a moment later, I was looking down on a pair of hands over a washbasin. Beside the basin, an old, hand-operated water pump was bolted to the counter top. Below me, someone reached out, pumped water into the basin, and began to wash his hands. To my surprise, the wash water quickly turned to red. It looked like blood.\n\nMy thoughts were all questions. Whose hands were these? My hands? It didn't feel like they were. Whose blood? Had I killed someone? Had somebody killed me?\n\nThe scene faded and I next found myself in a long frame building looking down at a dirty wooden floor. Feeling suddenly fatigued, my mind reeled at the realization that I was now inside a physical body wracked with pain. At the front of the building, a uniformed army officer sat at a wide official-looking desk. Pacing in front of the desk was another soldier in combat gear. He was screaming at me in a Slavic tongue. I couldn't understand a word he was saying, but somehow I knew that this man was going to kill me. Almost faint from prolonged, intense pain, I couldn't bear much more. I just wanted him to get it over with.\n\nA moment later, I was again floating in the night air, looking down on tall green grass. I had a startling realization. This was indeed another place and time. It was a scene from a previous life. As if in answer to an unstated question, I heard the words \"nineteen-forty-four\" in my mind. I understood. This was the year of my previous death.\n\nThe next thing I knew, I was back in my body in the present world. Pulling off my headphones, I drew a deep breath and stretched out on the bed. There was a lot to think about. No longer did I doubt the possibility of past lives. Reincarnation was a reality.\n\nUpset that the session ended so quickly, I had many unanswered questions. Who was I in that lifetime? How did it end? As I finally got up and prepared for the debriefing room, I could only hope that someday these questions would be answered.\n\nThe rest of the day was a blur. I spent most of the time wrapped up in my own thoughts. I couldn't shake the images. Each scene played over and over in my mind. The questions and possibilities were endless. As I finally drifted off to sleep for the night, it was clear that there was so much more I needed to learn.\n\nThe next morning I awoke early and trudged off to the washroom. Upon checking my wristwatch, I noted that it was only 5:30. Since the wakeup call on the room speakers came at 7:00 A.M., I headed back to bed, hoping to get at least another hour of sleep.\n\nWith the soothing sound of ocean surf playing in the background, I settled in under the covers, reflecting on the experiences of the past week. There hadn't been anything earthshattering, but I had learned a few new things. Most significantly, I had actually experienced a piece of a past life. That meant another big shift in my perception of reality.\n\nAs I pondered the recent changes to my worldview, I began to notice a subtle vibration exerting pressure against the inside of my forehead. A new sensation, it seemed to be slowly but steadily increasing in intensity. With the exception of the vibration, it felt a lot like the beginning of a sinus headache.\n\nDeciding to let the process unfold, I relaxed and waited for what would follow. The pressure continued to build, growing increasingly uncomfortable. When I began to wonder if I could stand much more, it abruptly released. It was as if a jet of air had puffed through the center of my forehead, creating an aperture that I could see through. At first only a small speck of light appeared, but the vibration inside my head began to surge and, ever so slowly, the opening grew wider. Soon, it seemed like I was peering through a long dark tunnel. I wondered if this vision was what some referred to as the third eye, or was it instead the type of tunnel people described in near-death experiences. Then I began to worry. Why was this happening? Could I be dying? No sooner had this thought crossed my mind, than a feeling of peace and well-being began to flow through me.\n\nThe exquisite vibration, which earlier seemed to be confined to my head, now surged through my entire body, intensifying as it spread. It wasn't long before my every molecule felt as if it were being gently massaged from within. As I drew nearer to what appeared to be the end of the tunnel, my anticipation became almost unbearable. I was shaking like a leaf in a breeze.\n\nSoftly, like sunlight filtering through the morning mist, a wonderful golden light began pouring into the opening, enveloping me, bathing me. Overwhelmed by powerful emotions, I found myself irresistibly drawn into the light. There was nothing I had ever wanted more.\n\nFor several moments I floated in place, soothed by the vibrating glow. Whatever this light was, it filled me with the warmth of complete love and acceptance. I would have been willing to stay forever, but as the minutes passed I felt a compulsion to move on. I thought I heard someone calling my name, and I could not resist. Someone was waiting for me to move through the brightness to the other side.\n\nI had to know what the next step would bring. With increasing excitement, I moved through the opening. The golden light faded, and before me appeared the most beautiful vista I had ever seen. Billowing white clouds floated through a pristine sky. In the distance, my eyes were drawn to a horizon of beautiful snow-covered mountain peaks. Spilling out of the mountains, a forest of magnificent blue green trees cascaded down the slopes to the valley below. Alive with every imaginable color and variety of flower, dazzling grass-covered hills swept up from the valley and across a small clearing to a stand of vibrant fruit trees beside me. The smell of the blossoms, flowers, and grass was intoxicating. \"If this isn't Heaven,\" I mused, \"then it sure ought to be.\"\n\nWhile soaking in the natural beauty of my surroundings, I began to have the feeling that someone was watching me. Cautiously turning to my left, I was surprised by the presence of two men a few feet away. Dressed in tan, monklike robes, warm smiles spread across their faces as they bowed their hooded heads in greeting. My mind raced with excitement. Who were they? They looked so familiar. I was certain I had seen them before.\n\nAs if in answer to my thoughts, the man on the left reached up and pulled back his hood. Deeply tanned, with short black wavy hair and a full beard, his beautiful dark eyes burned their way into my soul. Appearing to be around 35, he had a trim, but stocky physique. With eyes twinkling, he greeted me. \"Hi Paul. My name is Meldor.\" The other man, a bit younger and slimmer, was cleanshaven with medium brown hair and blue eyes. Smiling warmly, he just said, \"Hi.\"\n\nI was stunned. They had actually spoken to me. It was the same as speaking with any person in the physical world. I clearly and distinctly heard their voices. But how could they know my name? For some reason I thought communicating with a spirit would be different from this. I'm sure that my thoughts must have been as plain as the confusion written all over my face.\n\nMeldor's expression took on a look you might expect from an amused, loving parent answering a child's question. \"We have known you through eternity, in many, many lifetimes. It's taken a while in this incarnation, but we are delighted that we have finally been able to make you aware of our presence.\"\n\nWith that, he smiled, and a beautiful golden aura began to form around the two beings. Growing in brightness, it enveloped both of them, slowly radiating towards me. Cautiously, I pulled back, bracing myself as the wave of energy hit. Within seconds, my body began to vibrate with the purest feeling of love and acceptance I'd ever experienced. At first, it practically took my breath away. But when I felt the love and trust it engendered, I closed my eyes, allowing the wonderful energy to envelop me.\n\nHow long I basked in their aura I didn't know, but at some point I began to feel the embrace of Meldor's mind within my own. I could feel his vibration and hear his thoughts: \"We are pleased that we have been able to spend this short time with you, but it is now time for us to withdraw. In a moment your consciousness will be drawn back to your body and you will not be able to see us, but please know that we are always available to you. You need only to search within. Call us and we will be with you.\"\n\nThe love and appreciation that I felt for these two beings seemed unlimited. My heart ached. I wanted them to know this, but I was speechless. I managed to express only two simple words. \"Thank you.\"\n\nA moment later, an electrical spasm shot through my body. I was being called back. I didn't want the experience to end, but the pull to return to the physical was powerful and unrelenting. The morning wakeup music had begun to play over the room speakers, along with Monroe's voice bidding us to \"rise and shine.\" For a moment it seemed that my consciousness was simultaneously in two places. Although I was back in my bed, I could still see through the aperture. Finally, when I could no longer maintain this level of consciousness, the vision of the two beings faded and the aperture closed. Completely back in my body, I remained alert, but emotionally shaken.\n\nTurning down the speaker volume, I lay back to process the experience. I couldn't believe the time that had actually passed. It felt like I had been gone for only a few minutes and yet nearly two hours had evaporated.\n\nPreviously, I had read about spiritual helpers and guides. I had never given the idea much consideration, but now I somehow knew that these two beings were my spirit guides. I sounded his name over and over in my mind, \"Meldor. What a strange name.\" My thoughts then turned to the other spirit, and it dawned on me that he hadn't told me his name.\n\nThere had been such a powerful connection between us. I had the feeling that nothing about me was or could ever be hidden from them. Meldor had said that all I had to do was call, and that was exactly what I planned to do, at the first opportunity.\n\nThe remainder of the day was a washout. I couldn't chase the experience out of my mind. For some reason an uncomfortable pressure formed in the center of my chest. I wondered if it was my heart reminding me that it still required more time to heal. Perhaps the opening of my third eye and meeting my spirit guides had been a bit too much emotional stress.\n\nDuring each of the remaining sessions, I had difficulty remaining focused. When I was able to relax into an altered state, I felt my heart pounding so loudly it seemed to drown out the Hemi-Sync tones. It destroyed any chance of success through the last day of the program.\n\nI would be heading home the next morning, and even though I had lost a day of sessions, I certainly wasn't disappointed. It had been an incredible week. I had been blessed with experiences that would keep my mind busy for days to come.\n\nBack at home, the power of the visions and experiences continued to occupy my mind. Floating invisibly through walls or flying home to watch my wife arrive for work paled in comparison to experiencing a past life and meeting my own spirit guides.\n\nAs usual, I eagerly shared all my experiences within minutes of walking through the front door. Always an enthusiastic listener, Candace couldn't wait to hear about my new adventures. When I told her I had been floating above the school just as she arrived for work, she confirmed that on that day she had, indeed, been wearing the outfit I had seen. A small, relatively unimportant confirmation, but to me, it was greatly reassuring.\n\nThe new energy that had built up in my body during my stay at The Monroe Institute continued unabated. Still buzzing from the revelations I had experienced, I wanted so much to reconnect with the two spirit entities. The memory of their faces, their vibration, and the love I felt in their presence was continuously with me.\n\nA couple of weeks after arriving home, the end of a long business day found me sitting in my office chair daydreaming about the encounter with my guides. Wondering if I would ever see them again, I lolled deep in thought when the abrupt sound of a distinct male voice jolted me to attention.\n\n\"We are here!\"\n\nIt startled me so much I bolted upright. At first I thought there had to be someone else in the room. Where else could the voice have come from? I glanced around, but I knew that there was no one else in the building. The more I thought about it, the more I felt that the voice had to be inside my head. It sounded like the same voice I had heard a few weeks earlier at The Monroe Institute. It had to be Meldor.\n\nSitting back in my chair, I calmed my mind and moved into a meditative state. Within a few minutes I was hovering at the edge of sleep, when a sudden chill went up my spine. The hair on the back of my neck stood on end. Fighting every impulse to jump up and run, I refused to move. Moments later a warming vibration began to move down the back of my skull to the bottom of my spine, and then back up to the top of my head. With it, a wave of heat spread through my upper body. Everything began to spin. I felt like I was about to faint, and then I heard the voice again.\n\n\"We're sorry we startled you.\" Meldor's words were quiet and soothing. \"We have been working with your internal energies to help you become more receptive to our communication. We wanted you to know that it's not necessary for you to be separated from your body in order to be aware of our presence.\"\n\nBreathing in deeply, my body shuddered as I exhaled. I had a vague sense of slumping into the chair, my mind riveted to the new impressions inside my head. It felt different than anything I had ever experienced. I could clearly hear words and thoughts which were not my own.\n\nI wondered if the spirits were able to hear my thoughts. Meldor's clear and measured response came quickly.\n\n\"Yes, we can hear your thoughts and your spoken words, but only if you have directed them to our attention. If you call us, we hear you.\"\n\nI tried to remain calm. There were so many things I wanted to know. Although unsure of my ability to communicate, I began, \"I think I've seen you in my dreams. You seem so familiar, but I need to know who you are. Are you my guides or guardian angels?\"\n\nI heard a soft chuckle. \"No, we are not angels in the sense that you have come to understand. We are helpers. We have been with you through many lifetimes in this physical world. We have agreed to assist you in the challenges and goals that you've set for yourself. The circumstances of your early years were tougher than expected, but you have overcome many barriers and are now beginning to remember and reconnect with your true essence.\"\n\n\"But if you've always been with me,\" I questioned, \"why haven't I been aware of you before?\"\n\n\"When you first begin an incarnation as an infant,\" Meldor replied, \"you are still closely bound to the realms of the spirit. As your physical form grows and you become more and more involved in the physical aspects of survival, your connection to the world of spirit begins to fade. Incarnated spirits, for the most part, slowly forget their origin and true identity. The consciousness of the soul, operating through the human brain, can be very slow and dense. Therefore contact between the helper and the soul is easier when the body is asleep or in an expanded state of awareness. We are able to communicate with you in your dreams and meditative states, sometimes to simply put a thought into your mind.\"\n\nThis was all so fantastic. I didn't doubt his words, but being the type of person I am, I needed proof. Meldor was way ahead of me. With a small titter, he admonished me, \"Okay, Missouri.\"\n\nInstantly I was transported back in time driving my car through British Columbia. It was a long haul through the mountains and I was having difficulty staying awake. At some point I had fallen asleep while approaching a long curve in the road. All of a sudden, I became aware of a voice screaming in my ears, \"Paul, wake up!\"\n\nJerking back to consciousness, I found to my horror that I was in the wrong lane, heading straight for a truck coming from the opposite direction. Jamming on the brakes, I cranked the steering wheel. With tires screeching, the car lurched across the centerline, missing the truck by inches.\n\nThe mental imagery came to an end, and I recalled the entire incident. I remembered having had to pull over to the side of the road while I waited for my heart to crawl back into my chest.\n\n\"That was you?\" I asked. The question needed no answer.\n\nBefore I could refocus my thoughts, another memory started playing out in my mind. It was the terrible car accident I had been in years before. I was back in the car as it careened down the ditch, crashed, and spun wildly into the air. Everything went black, just as it had during the crash. The next scene, however, was not part of that memory. It was something new. As I lay unconscious in the frozen field, I became aware of someone calling my name, urging me awake, telling me to get up. I had to get help or I would die.\n\nOnly vaguely aware of my body, I drifted in and out of consciousness. Then, out of the darkness, a luminous mist began to appear beside me. It continued to expand and grow in intensity until it surrounded me. At its center appeared a radiant being. As he loomed over me, I was enveloped by an overwhelming feeling of love and compassion. The glowing form bent down and, putting his arms around me, drew me close. I could feel the warmth flowing from his body into mine. I thought that I must be dreaming or hallucinating, but it didn't matter. Completely at peace, I slipped again into the blackness of sleep. The next thing I knew, I heard my name called again, this time more forcefully.\n\n\" _Now_ , Paul, you must get up _now_.\"\n\nJerking awake, I sat up in the field. The being was gone. I was alone. In the distance I could see headlights approaching. Although it had to be more than two hundred feet to the road, I had to get someone's attention. Desperately, I began crawling. But it soon became apparent that I wouldn't get there in time. Struggling to my feet, I ran for all I was worth. Just as the car was approaching, I lurched to the edge of the road, frantically waving my arms. The car careened by, but then with a spray of gravel, it came to a skidding stop. The driver put the car into reverse and backed to where I stood. The passenger side window rolled down, and someone exclaimed, \"Jesus! Take it easy, man. We'll go and get help.\"\n\nWith that, the engine roared, gravel flew from the tires, and the car sped off into the night.\n\nThe scene faded, and a moment later I was back in my office chair.\n\nI was stunned. This episode had never been part of my conscious memory. All of these years I had no idea that a spirit had come to help me in that frozen field. As feelings of love and appreciation flooded through me, Meldor continued his narrative.\n\n\"We would never interfere with your intentions or free will, but there have been times when we have interceded to prevent a transition before your intended time. There have been many instances in your life when we have been in direct contact with you, in times of trauma or at critical points in your development. The two examples you have been shown are the extreme. Normally, we merely try to provide a subtle guidance to nudge you along, to help you achieve your goals.\"\n\nI was deep in thought. What an opportunity! A new question occurred to me.\n\n\"Meldor, you said that you have been with me through many lives. A few days ago at The Monroe Institute, as I'm sure you're aware, I had an experience that seemed to be from another lifetime. It left me with a lot of confusion and a number of questions. Was that a past life? Did I kill someone, or was it me that was killed?\"\n\n\"Yes, it's unfortunate that the experience was cut short,\" Meldor replied. \"As you now understand, you have had many incarnations. The one you experienced was the most recent life in your linear concept of time. We know that having the experience interrupted left you in confusion. If you wish, we can assist you with a continued look at that short life.\"\n\nAlthough a bit apprehensive, I certainly wanted to unravel the mystery. \"Yes, please,\" I responded, tentatively. \"I'd like to know.\"\n\nImmediately, I began to feel a slight pressure at the center of my forehead. There followed a surge in vibration and a blur of colors. The next thing I knew, I was gasping for air, wracked with pain.\n\nThe assault on my senses was unbearable. A moan escaped from my throat. My head hung down, my chin resting on my chest. Opening my eyes, I realized it was nighttime. I was outdoors in the cool air. Looking down at my body, a wave of shock spread through me. I was naked and filthy, my chest blackened and covered with blood. On my inner left thigh, a large gaping wound was crusted over with blood trailing down my leg to a pool of scarlet beneath my feet. The entire left side of my body felt like it had been crushed. An incessant pain throbbed in the left side of my chest and I was having difficulty breathing. The salty taste of blood filled my mouth. I knew there must have been a hole in my lung; I could hear bubbles as I drew each painful breath.\n\nMy hands had been tied behind my back and bound tightly to a pole. Overhead a dim yard light shone from the top of the pole, and about 15 feet in front of me stood a gable-ended building covered with tin. Exhausted and in agony, I couldn't take much more. I knew they were going to kill me, and I just wanted them to get it over with.\n\nMoments later, two soldiers stepped out of the building, slamming the door behind them. One of them, a man I had seen in my first vision of this life review, strode up to me, raised his pistol to my left temple, and pulled the trigger. In an instant I found myself floating above my body, which now hung awkwardly from the pole. With a dizzying spin, everything faded to black, and I was back in the quiet of my office.\n\nEmotionally drained, I needed time to recover from the intensity of the experience. Meldor's voice focused my mind once again. \"We are pleased that we had this time to communicate with you. But we must now withdraw our vibration so that your energy may be restored. We are with you always.\"\n\nWith that, the vibrations disappeared and the ringing telephone brought me back to the physical world. It was Candace, wondering when I'd be home for dinner. It was almost 6:30. The encounter had gone on for more than an hour.\n\n# 7\n\n# _We Are Not Alone_\n\n_Though the stock of angels be divine_\n\n_Must we yet refuse to see the sign?_\n\n_Why is it only when troubles abound_\n\n_He finds us kneeling on the ground?_\n\nLife in the aftermath of meeting my spirit guides grew ever more compelling. My days were divided between operating my business and performing my duties at City Hall, but at night I would shut out the rest of the world to investigate the nature of human existence and who I really was. My experiences with Meldor led me to realize that my understanding of life had been rather limited. There was so much more that I needed to learn about the universe and my place within it.\n\nSo many profound experiences had occurred during my last trip to The Monroe Institute that I had almost overlooked a significant event\u2014the opening of my third eye. Although it had been a fairly recent discovery for me, it was a powerful and intriguing portal to the spirit world. In those early days, however, this new psychic vision would, more often than not, leave me confused.\n\nI found that the best time for successful out-of-body exploration was just after awakening from a night's sleep. One morning, I woke up but stayed in bed watching dreamlike images in my mind.\n\nAs I lolled in comfort, the high-pitched sound I had become accustomed to while entering an altered state gripped my attention. My eyes closed, I began to see a tiny ray of light in the center of my forehead. Anticipating another trip to the Other Side to see Meldor, I was surprised when the opening merely widened into an oval-shaped viewing portal. Through this aperture\u2014about an inch in diameter\u2014a clear and vivid scene materialized.\n\nIt appeared that I was in a lecture hall with people seated around me listening intently to a presentation. I could hear a male voice droning in the background. Everyone looked like ordinary, everyday people, but the vividness and color of their faces was astounding. My own flawed physical eyesight had never revealed such depth and clarity. Where was this place? Who were these people? Opening my physical eyes, I shook off the altered state. The vision disappeared and the aperture closed.\n\nI couldn't imagine where I had been. Had I been watching an event in the spirit realm or had I somehow remotely viewed another physical location on Earth? It was unfortunate I hadn't maintained the vision for a while longer. If I had listened to the lecture it may have given me some idea as to where I had been. Whatever was happening to me, I was grateful and excited by the possibilities. There seemed to be no end to the changes taking place in my life. As if meeting my spirit guides wasn't enough, a new means of travel and perception had opened up for me.\n\nOver the next few weeks, it became clear that my sensitivity to many different forms of psychic phenomena had increased dramatically. I noticed subtle changes occurring when I hovered in the borderland before falling asleep. I began to see faces\u2014all kinds of faces. Usually, a face would momentarily appear in my third eye, then, after a few seconds it would fade out, and another would replace it. Where they were coming from, or who they were, I didn't know. Sometimes as many as a dozen vivid beings passed before me. I couldn't tell if they were checking me out or I them. Some looked vaguely familiar, but I could never place where I might have seen them.\n\nOne of the strange things I noticed about these visual montages was the difference in the quality of those occurring in the morning and those I saw at night. The morning apparitions were usually clear and in color, while the nighttime visions were mostly black and white and not nearly as vivid. The only explanation that made sense was that in the morning, after a good sleep, I was more relaxed than at night. Perhaps the degree of relaxation had a direct effect on my ability to see the Other Side. The frequent visions became so interesting that most nights I could hardly wait to get to bed to see who or what would show up.\n\nOver a period of three months, \"the parade of faces,\" as I began to call them, materialized on numerous occasions. Mostly, it was like looking at a series of still photos. Their features didn't seem animated, and there was no indication that the spirits were either willing or able to communicate with me. Although I had become comfortable with it, I wasn't sure why the phenomenon continued to work that way. That, however, was about to change.\n\nDuring a weekend convention in another city, I sat in a lecture hall with comfortable seats and a boring speaker (not a good combination at any time). Late to bed the previous night, I grew drowsy as the speaker droned on. The lights were dimmed, so I thought nobody would notice if I were to doze off. Closing my eyes, I quickly relaxed into a hypnagogic state.\n\nWithin minutes, a procession of faces, male and female, began parading before me. After several had passed, the face of an attractive young woman appeared. Struck by her beauty, I thought to myself, \"Hmm, pretty.\" To my surprise, the moment the thought formed in my mind, a smile spread across her face. Batting her long eyelashes, she beamed with delight. Completely surprised, I exclaimed, \"Jesus, she heard me!\" In an instant, the vision was gone, and I opened my eyes.\n\nI didn't realize I had blurted these words out loud until a guy sitting in front of me turned around and said, \"What?\"\n\nEmbarrassed, I mumbled a feeble apology.\n\nFor some reason, I had always assumed that the beings I saw couldn't hear or communicate with me. It was a surprise to discover they could do both. Another bridge had been crossed. The journey was becoming even more interesting.\n\nAs I continued to read and learn, I occasionally came across new spiritual concepts I had difficulty believing or accepting. One of these notions was the suggestion that animals had souls and that they, too, could be found in many of the spirit realms. During a previous seminar at the Institute, Robert Monroe spoke of being visited by a couple of his house cats that had passed on. I had also read books on shamanism that often referred to \"power animals.\" These were the spirits of animals who, in much the same manner as spirit guides, dedicated themselves to the guidance and protection of humans. Refusing to allow for even the possibility of such a concept, I had always considered this some sort of uneducated primitive myth.\n\nMost nights, I read for an hour or two before falling sleep. One night, I had become so engrossed in a book that I kept reading into the wee hours of the morning. As I read, a peculiar feeling came over me. It felt like someone was in the room watching me. Setting the book aside, I glanced around, but noticed nothing out of place. I returned to my reading, but couldn't shake the persistent sensation. It was becoming more powerful and distracting. Putting the book down, I again scanned the room. This time, my attention caught something moving just outside the door.\n\nFloating in the doorway, two feet above the floor was a round, swirling mass of energy the size of a basketball. The iridescent globe seemed to be made up of millions of small particles, orange and black in color. I had the impression that I was seeing something at a molecular level.\n\nI watched as the shimmering sphere expanded, spreading outward from the center until it took the shape of a huge spotted leopard. Standing in profile 20 feet away from me in the doorway, the cat was spectacular. It was entirely real and alive. I could hear and see it breathing, its belly moving with each breath. Slowly it turned its head in my direction. When our eyes met, an instant flash of familiarity swept through me. A warm vibration swirled in my chest.\n\nThe leopard's beautiful eyes seemed to stare directly into my soul. I felt I had known this spirit for an eternity. Captivated by its gaze, I became immersed in the memories of its love and protection. Finally turning its head away, the majestic beast promptly dissolved into millions of multicolored molecules, and then abruptly disappeared.\n\nMy throat felt dry. I was having trouble breathing. I was wide-awake and yet, right in front of me a ball of molecules had appeared, turned into a leopard, and then disappeared. It was real in every sense. I had heard it breathing. I could even smell it.\n\nShakily, I got out of bed and headed cautiously to the doorway. I wanted to be sure of my own senses. This had not been an out-of-body experience. I had watched all of this with my own physical eyes, yet I was still having trouble believing it. I stood in the doorway where the leopard had materialized. The unmistakable, musky odor of the big cat lingered, but all other remnants of its visit had disappeared. Something magical had just happened and I felt truly blessed. Why had this beautiful animal visited me? What was the connection that I felt? Was the shamanic belief true? Could this be my _power animal?_ I lay awake for hours, pondering it all.\n\nFor weeks following the visitation, I often found myself reliving it. It was difficult to pass through my bedroom doorway without thinking about the leopard, re-experiencing its mesmerizing eyes, or feeling that vibration in my chest.\n\nAbout a month after the incident, I had another strange experience. One night, I snapped awake with a falling sensation in the pit of my stomach. It felt as though I had just returned from an excursion, but I had no idea from where. Vibrations still coursed through my body so I prepared myself for another adventure.\n\nTrying to decide on a destination, I floated to the ceiling. Curious as to where I had been, I concluded that although my conscious mind didn't know, my subconscious could probably return me to the same place. I decided to go with the flow, and almost instantly I felt a shift and found myself floating in an unfamiliar room in what looked like the lower level of a two-story home. There was nobody around, but I could hear a commotion on the upper floor. Deciding to check it out, I slid up through the floor into the next level. Just as my head and shoulders began to pull through the carpet, I was startled by the hiss of a cat and the sensation of something slashing through my face. If I had been wearing pants, I'm sure they would have been soiled. I had come up into the hallway right beside where a cat was sitting. As soon as it saw me, it tried to claw a chunk out of me.\n\nGathering my courage, I floated up to the ceiling high above the hissing ball of fur. In terror, it finally tore out of the hallway and down the stairs like it had seen a ghost. I didn't know whether to feel good about it or bad. This was the first living animal I had encountered while out-of-body. I couldn't believe the poor little creature had seen me. I had scared the crap out of it, and it returned the favor.\n\nWhen I finally got my wits about me, I noticed an open door at the end of a hallway. The light was on, and I could hear the sound of giggling kids. Floating through the doorway, I rose up to a corner of the ceiling. Below, four young boys, about ten years old, were having a great time jumping on the bed, batting each other with pillows. This seemed to be an adult's bedroom. Their parents obviously away, the boys were taking advantage of it.\n\nAs I watched their amusing high jinks, one of the boys stopped dead in his tracks, looking up in my direction. His mouth fell open. He stood there, frozen. I was astounded that he could see me! When a couple of the other boys stopped to look, I decided to make my exit. I thought of returning to my body and, in an instant, I was back in my bed.\n\nI didn't think I would be trying that again soon. Whether or not my subconscious had taken me back to where I was previously, I didn't know, but it didn't seem likely. Somehow, I had gone off course. I wondered why the boy had been able to see me. I hoped I hadn't been responsible for scaring the poor kid into therapy. I couldn't help but laugh, thinking of the wild story the boys would be telling when their parents got home. Some guy in blue jockey shorts floating around the ceiling isn't the type of report most parents want to hear.\n\nAs time wore on, I became more familiar with the feeling and vibration of my spirit guides. Although my life had become extremely busy, I tried to spend at least a few minutes each day communicating or attempting to make contact. Depending on my level of stress and ability to relax, sometimes I felt their presence and sometimes I didn't.\n\nOne night I awoke at about 3 A.M. with the familiar high-pitched tone in my ears. I knew that if I could keep from falling asleep, it would be a good opportunity to leave my body. Within just a few minutes my mind calmed and I dropped through the threshold into superconsciousness. Before long a wave of energy flooded the top of my head and began to radiate throughout my body. Concentrating on the frequency of the vibration, I pushed the energy higher. When every molecule in my body seemed to be buzzing with electricity, I was ready to leave. With just the thought of floating, there came an abrupt surge and I lifted off the bed towards the ceiling.\n\nElation momentarily overwhelmed my senses. When I regained control, I had difficulty deciding what to do. Should I fly around the house? Head out into the night sky? Perhaps I could find my way back to the beautiful park where I had met my spirit guides. The thought had hardly registered when I felt a shift, a feeling of movement, and I was standing in a parklike setting. Sensing a familiar presence nearby, I turned in the direction of the vibration and was greeted by two smiling faces.\n\n\"Welcome back. We're glad you could join us.\" Meldor's voice was clear and strong. \"We know you have many questions. Reopening awareness to your own spiritual reality can be very exciting. We are pleased to assist you in any way possible. You need only to ask.\"\n\nI wasn't sure what I should be asking. My mind was racing. Finally, I thought of a good beginning. \"Thank you for helping me,\" I said. And referring to the second guide I added, \"But you know, I don't know your name.\"\n\n\"In our plane of energy,\" he replied, \"names are not necessary. Every being has an identifiable and unique vibration. You will know us by our vibration and feeling, but if you feel more comfortable with a name, you may call me Raylon.\"\n\n\"Sometimes,\" I continued, \"when I try to connect with you, I'm only able to feel Meldor's vibrations. I can't seem to feel you.\"\n\n\"Raylon,\" Meldor injected, \"is my assistant. He is what you might call an apprentice, learning to work as a teacher. He has other assignments and is therefore not always able to accompany me.\"\n\nI thought of another question. \"Sometimes, when I'm trying to communicate with you, the volume of information is overwhelming, and yet now I can clearly hear and understand what you're saying. Why does this happen?\"\n\nMeldor nodded his head and smiled. \"Communication in the spiritual realm is more encompassing and proficient than in the physical world. Human language concepts are extremely limiting. When the consciousness of your mind is operating in this realm, it is easy for you to assimilate the complexity of the communication. However, when you are back in your body, trying to process the information through your physical brain, you may have difficulty remembering or interpreting aspects of the information. The human brain, in its present development, simply cannot handle the assimilation of so many levels of data.\n\n\"You may have noticed this phenomenon in your dream states. While you are engaged in the dream, your mind is operating at its spiritual level. Although everything may seem entirely understandable during dream consciousness, as soon as you awaken and begin to process the data through your physical brain, the details may seem completely nonsensical.\n\n\"To answer your question, we are communicating in this simple manner to enable you to fully process and remember the experience when you are back in your physical body. As you become more adapted to our form of communication, you will find that you are able to assimilate information at more complex levels.\"\n\nA thought that had been nagging at me ever since my near-death several months earlier came to mind. I had heard about near-death experiences where people had gone through a review of their life. I hadn't made it to that point before being brought back from the Other Side. I had always wondered what the experience would be like.\n\nReading my thoughts, Meldor was way ahead of me. \"As you are beginning to realize,\" he said, \"when each incarnated soul returns to its origin, there is no judgment by any other entity. Each soul making the transition is assisted by loving guides in a review of their immediate past life. Very often, perfected higher beings such as the Christ Consciousness, Krishna, Mohammad or other deities and masters\u2014depending on the individual's beliefs\u2014will be present to offer support and guidance to the returning soul. They do not judge, nor does anyone mete out punishment of any kind. The concept of a judgmental, punishing God is purely dogmatic, a manmade construction. God does not punish. God only loves.\n\n\"We know,\" he continued, \"that you have wondered about the experience of a life review, and to further your understanding, we will assist you in this aspect of the higher vibration. Nothing of energy is ever lost. All thoughts, actions, and deeds are retained to provide the basis for assessment and reflection. In a moment,\" he warned, \"you may feel a shift in vibration, which is necessary to engage the higher energies that hold the imprint of the life vibration.\"\n\nI felt a burst of energy, brief motion, and then a montage of images began to play through my mind. Like fanning through the pictures in a photograph album, my life flashed before me. Everything was there: incidents, events, thoughts long forgotten, good experiences, traumatic and uncomfortable experiences\u2014all whirled by in chronological succession. If at any time I wanted to take a closer look at a particular event, the process would stop and, instantly, like a videotape playing in my mind, I would be immersed in reliving the occasion. I was on an emotional roller coaster\u2014the highs, the lows, the sadness and grief, along with jubilation and happiness, forever imprinted on my mind.\n\nI could not be sure of the amount of time the processing took. It could have been a minute or hours, I didn't know. I saw why there was no need for someone else to pass judgment on my life. Nothing was lost. I was the only judge necessary, and I would be the harshest critic. I knew when I had failed and when I had succeeded, but there was an unexpected twist.\n\nNot only could I relive how I had felt during any particular incident, I could also feel the pain or happiness of others whom I had affected through my words and actions. I could feel again the anger I felt as I lashed out at a playmate in the schoolyard, but I could also feel the pain and injury he felt at the humiliation. The good and the bad all passed before me. I watched the effect that even a small act of kindness could have on others. Something as small as my cheery greeting had lifted someone's spirits, and they, in turn, felt better about themselves and treated others more kindly. I was astonished at how the lives of so many people could be inadvertently affected by my actions, whether positive or negative.\n\nInevitably the chronology of my life brought me back to the time of my heart attack and near-death experience. I needed to see no more. The images stopped. I felt the love and support in Meldor's embrace. I was emotionally drained, yet at the same time rejuvenated. I had a better understanding of who I was and what I needed to do.\n\nMeldor's soft voice interrupted my reverie. \"It's time now to return to the physical vibration. There is much processing that you will need to do, and your energy needs to be restored. We are pleased to have been able to assist you.\"\n\nIn a snap, I was back in my body. Wide-awake, I rolled over in bed and glanced at my nightstand clock. It was just after 5 A.M. More than two hours had passed.\n\nFor another hour I lay in bed, reviewing and contemplating the experience. The images of my past life were engrained in my consciousness. I would never forget them. The message was clear: it didn't matter one bit in life who won or who lost, or how rich or successful we became. When we finally pass from this world, the only thing that will matter, the only thing of lasting importance, is how we treated others along the way.\n\n# 8\n\n# _The Ultimate Mission_\n\n_When horrors plague our daily bread_\n\n_And grieving thoughts we cannot shed_\n\n_Is it then we pray our Lord to see?_\n\n_Shallow promises made in desperate plea_\n\nJust when I thought that I'd seen and heard more than my share of strange things, I learned of a bizarre yet intriguing concept from Robert Monroe and his researchers at The Monroe Institute.\n\nIn the late '70s Monroe had spent a great deal of time monitoring lab sessions with a gifted out-of-body explorer, Rosalind McKnight. Following numerous sessions, he became familiar with McKnight's spirit guides who often communicated through her vocal chords as she lay in an altered state. McKnight's guides began to reveal the reality of so-called \"lost souls.\" They explained that due to the circumstances of some people's deaths, instead of moving on to the spirit realm, occasionally some souls become trapped in a moment of their own reality. Unaware that they have died, these poor beings are incapable of escaping the confines of their own trauma or belief systems.\n\nNormally, when the physical body dies, the spirit passes easily into the higher vibration of the spirit world. It will be met by spirit guides and loved ones who have previously passed on. From there, the soul is lovingly assisted and escorted into the realms of spirit, often referred to as Heaven. But in some cases, due perhaps to the extreme violence or trauma of their deaths, some souls become trapped in a reality of their own construction for varying periods of Earth time. As they don't realize they have died, they're unable to perceive the nonphysical world they now inhabit. Although their guides and loved ones may be trying desperately to get their attention to assist them home, they are unable to make the transition. Locked into the slower Earth vibration, they cannot easily escape until they realize the truth of their existence and consciously seek the Light.\n\nWhat McKnight's guides further imparted to Monroe seemed to be even more astonishing. They claimed that living human beings in an out-of-body state could be of invaluable assistance to these lost spirits. As they are still of the Earth vibration, astral travelers can be easily seen and heard by the trapped souls. Due to their lower physical vibration, living humans can gain their attention and make them aware that they have died. When these souls realize the truth of their predicament, they are then able to see and communicate with their own guides and loved ones who are waiting to help them complete the transition.\n\nAfter hundreds of hours of research with McKnight and other out-of-body explorers, Monroe mapped out the specific level of consciousness in which these lost souls could be found. Thanks to improving technologies they were able to duplicate the required brainwave frequencies through their Hemi-Sync process, allowing them to guide an out-of-body explorer into this realm. Monroe and his staff also committed themselves to developing a new program to teach what they considered the ultimate service to humanity: \"Lifeline.\"\n\nAlthough a bit skeptical, I was intrigued by the possibilities, and knew I had to give it a try. Calling the Institute, I registered for the first available program.\n\nThrough several previous months, my work and other commitments had kept me extremely busy and I hadn't had much contact with my guides. I was sure that another week at the Institute would help to bridge the gap.\n\nTime passed quickly. Soon I was back in the beautiful blue hills of Virginia meeting a new group of explorers. Although the relaxing surroundings of The Monroe Institute were just what I needed, it was not without some anxiety that I approached the week to come. The idea of trying to help people who had already died was more than a little unnerving.\n\nThe first days of the program reacquainted us with territory we had explored in previous programs. As usual, I ended up falling asleep during most of the early sessions with very little to report. The full day in noisy jets and airports always seemed to take its toll. I came to the conclusion that arriving a day earlier, to catch up on sleep, would be a good idea for future seminars.\n\nOn the second day, I awoke rested and eager to get into the program. Although I couldn't remember much, I knew that I had been dreaming about my spirit guides. I looked forward to reconnecting with them.\n\nThe morning session began and, in the quiet darkness of my CHEC unit, the relaxing Hemi-Sync tones carried me quickly into a deep, altered state. Within moments I began clicking in and out of consciousness, hovering tentatively at the edge of sleep. After three or four clicking-out episodes, I found myself in the grip of a strong vibration. My body shook from head to toe. Fighting for control, I willed the slow choppy cycles into higher, smoother frequencies. In a short time, the pulsing tremor rose into a high-pitched whine. Soon, a series of small energy rushes coursed through my body. Although I couldn't seem to exert any control over them, I lay back and welcomed the experience. It wasn't long before I realized that someone else had taken charge. My body rose into the air as if being drawn by a huge magnet. A light briefly flashed before my eyes, and the next thing I knew I was speeding towards a familiar energy.\n\nThe iridescent beauty of what I now called the Park greeted me with the warmth of a long lost friend. Meldor's powerful presence was pervasive. I could feel his distinct vibration flowing through me. Within moments he was standing beside me, softly touching my shoulder. \"Hello again,\" he greeted me. \"Welcome home.\"\n\n\"Thank you,\" I eagerly responded. \"Meldor, this is so incredible. Why can't I come here everyday?\"\n\n\"Actually, you do, quite often. Most nights while your physical body sleeps, at some point you travel here in your spirit body to join with those who love you. You come to learn and to rejuvenate your energy within the vibration of _The All.\"_\n\n_\"The All?\"_\n\n\"Yes, _The All That Is_. Many refer to it by various names, but the energy you feel here permeates all things. It's the universal energy of Love. Nothing can avoid or escape its power. It is essentially the energy vibration of all souls. Considering the religious concepts that you've been taught, you may think of it as the energy of the Holy Spirit or the essence of God. It's all the same, and it's the source of _All That Is.\"_\n\nDeep within, I felt the truth of Meldor's revelations, but I was still puzzled. \"It's a wonderful feeling, but if I come here every night, why don't I remember it? In fact why doesn't everyone remember it? Wouldn't the world be a much better place if everyone knew and remembered that this is our real home? If everyone could experience this and remember it, there would be no wars, there would be no crime, and no one could hurt another being.\"\n\n\"That is true,\" Meldor acknowledged. \"But if everyone knew and remembered, there would be little reason to incarnate. There would be little challenge to the human experience. The challenge and the quest to improve, to overcome and to remember, is what gives real value to the experience. The value and power of human existence is in the experience itself. We all benefit from the lessons and experiences of each soul. We are, in reality, all one.\"\n\nHis answer made sense to me, but I struggled with the idea of being all one. I needed more of an explanation. Meldor sensed my confusion. \"To truly experience something,\" he continued, \"is the best tool for understanding. Knowing is better than believing. The human mind experiences itself as distinct and separate, but this is an illusion. To fully understand the reality of 'all as one,' and the connection of all things to each other, it is helpful to experience the polarity of separation and connection.\"\n\nWith that, Meldor placed his index and middle fingers on my forehead. A surge shot through me and I felt as though I was about to lose consciousness. Where just moments before I felt clear and rational, now I couldn't seem to hold a coherent thought in my mind. I was dizzy. Nothing made sense, and very quickly my entire existence seemed to be closing in on me. As I fought to maintain consciousness, the image of a television screen loomed before me. It was as if my entire view of the universe was from inside a television tube from where I looked out.\n\nThere was an electronic snap as if someone had shut off the power to the television. The screen to my perception of the world went blank. All that seemed to remain of me was a tiny dot of energy in the center of the screen. A virtual sea of blackness surrounded me. The feeling of isolation and darkness creeping into my spirit was almost too much to bear. I was alone in a universe of emptiness. A profound sadness overcame me. I wanted to cry out in desperation, but even my thoughts had begun to fade. My consciousness, my awareness of self was dying, and I knew that in another moment I would cease to exist. Then everything went black, and I was no more.\n\nAs if from an eternity of sleep, I slowly began to reemerge from the nothingness. Vestiges of dream images and fleeting thoughts tugged at me like the rescuing hands of a loving parent. In a few moments I was again awake, my consciousness fully restored. I was still me, but I felt somehow different\u2014changed in some unfathomable way.\n\nI'm not sure how to best describe this, but I was no longer aware of being in any particular form. I felt instead like I was a floating sphere of microscopic particles. I had no idea where I was, but it felt as if I was hovering in outer space.\n\nWith each passing moment I felt the power of spiritual energy growing within my form. Even though I was composed of hardly more than a mass of seemingly nonconnected particles, I felt more alive and together than I had ever been. My awareness of self was acute. Somehow, every individual particle of my being seemed to contain the totality of my consciousness.\n\nAs I floated in space, relishing the immensity of the experience, I became aware of a concentrated power approaching me. It was nothing that I could see, but the energy of this presence, whatever it was, was awesome. Every fragment of my awareness began to shake and then to resonate in harmony with the vibrations of this enormous force. Suddenly, I found myself exploding into the universe.\n\nIt felt as if the very essence of my spirit was being blown like dust into the far reaches of the cosmos. Expanding and growing in perception, it seemed that I had somehow joined with the energy of every living thing. I felt a baby's cry and the anguish of its mother. I experienced the joy of a puppy as it bounded happily after a child. I felt the power of a whale swimming in the ocean. From the mindless destruction of war to the purposefulness of a hummingbird in flight, I experienced and became a part of everything that I touched. It was at the same time an explosion of emotion and a mystical encounter with the love of God. I suddenly understood the perfection of creation. The entire experience was so overwhelming, so illuminating, that I became lost in the beauty and complexity of it all.\n\nIt began slowly at first\u2014small shifts of consciousness tugging at me, reminding me that I was still attached to a physical body. Tears of gratitude and joy started welling up in my eyes until I could no longer resist the pull of the physical. In an instant the experience was over and I was back in my CHEC unit.\n\nMy heart pounding, I remained in the silence of my room, contemplating the emotions and images newly imbedded in my mind. The montage of revelations had been almost overpowering. So much had happened simultaneously that I couldn't even hope to remember, much less describe the totality of the experience. Sitting up, I tried to make notes of the session, but the effort seemed futile. After scratching out a few inadequate phrases, I summarized it with one word: \"Wow.\"\n\nInitiations complete, we were ready to get down to business. With the help of Hemi-Sync, the trainers slowly led us into new territories of exploration. The basic plan of action called for us to be guided by the frequencies to the Park. This was a place Monroe often came across in his journeys to the Other Side. Others had described it as the \"meeting place,\" the \"receiving station,\" or, more elegantly, as Heaven. This was where the souls of the newly deceased arrived when they first made the transition. Here, they were greeted by spirit guides and deceased loved ones, who then accompanied them to their place for orientation to their spiritual surroundings. In the program's Hemi-Sync terminology, the level of consciousness where the Park could be found is referred to as Focus 27.\n\nThis part excited me. I felt sure that I had already been to the Park. Focus 27 must have been where I first met my guides. We were told that this level was where we could unite with our guides or other helping spirits who would then escort us to other levels of consciousness where the lost souls were trapped.\n\nAlthough nervous, I couldn't wait to get going as we began our first guided tour of the various levels. With the powerful, brainwave-altering frequencies humming in the background, Monroe led us through the steps we would repeat many times in the coming days. After several palpable changes in vibration, to my delight, I suddenly found myself standing in the Park welcomed by Meldor and Raylon.\n\n\"This is a wonderful service you've made yourself available for,\" Meldor began. \"And we are very happy to assist you.\"\n\nNervous, I remained silent as Meldor continued. \"There are many beings that require assistance and we hope to help as many as we can. We do want to make you aware that you may be faced with some very powerful emotions and disturbing scenes, but we are certain you will be able to handle it. We will first accompany you on a tour of this level to help you understand and acclimate yourself to the coarse and sometimes uncomfortable vibrations that you may encounter.\"\n\nWith those words, Meldor and Raylon positioned themselves on either side of me. \"When we begin to move through the various levels,\" Meldor continued, \"we may not be visible to you, but you will be able to feel our energy and know that we are with you.\"\n\nAt that point, they placed their hands on my shoulders. I felt a dizzying shift in vibration and my senses seemed to turn inward. Within moments I had the sensation of moving at high speed through a dark, mistlike atmosphere. On and on we flew, until without warning, it felt as if I had burst through the other end of a dark tunnel in the sky. The landscape dropped away below me, and I found myself shooting through the air a couple of hundred feet above the ground. A scream caught in my throat. I felt like Wile E. Coyote shooting off the edge of a cliff, desperately clawing the air when he discovers there's no ground beneath his feet. Shocked, I virtually hit the brakes in midair, finally coming to something of a skidding stop.\n\n\"Holy shiiiit!\" The words blew out of my mind before I could stifle them. Floating motionless in the air, I needed a moment to calm my thoughts and regain my bearings. \"Jesus! Meldor,\" I muttered, \"take it easy.\"\n\nFrom somewhere beside me, I heard Meldor chuckling. \"Oops, sorry about that.\"\n\nThe whole thing suddenly struck me as terribly funny, and I burst out laughing. It occurred to me that Meldor had probably done it on purpose. I had been nervous, perhaps a bit too uptight, and needed to relax a tad before taking on whatever lay ahead.\n\nMy wits somewhat recaptured, I began to check out my surroundings. It seemed like a real Earth environment. Below me was a large body of water, like an inland lake. A 12-foot-high cliff wound its way along the perimeter overlooking a thin strip of beach at the water's edge. I noticed a couple of people swimming in the lake, perhaps a hundred feet or so from the shore. I floated down for a closer look.\n\nAs I drew closer to the surface of the water it dawned on me that these people weren't swimming. They weren't even moving. The bodies of a man and a woman appeared to be floating just below the surface. They were dead, obviously drowned. It was then that I first noticed a man sitting on the beach, agitated and distraught. Just as I was about to move closer to him, I heard Meldor's voice in my mind. \"Don't worry about him; we'll come back for him later. He'll be okay. But for now, we should continue on. There are other souls we want you to see as well.\"\n\nWith Meldor's reassurance, we resumed our travels. The scene below began to fade and I was soon floating over a small country home in a flat agricultural area. I noticed the absence of roads in the prairie below. There was only a small, worn, rutted path leading to the property. A short wooden fence surrounded a large, well-treed back yard containing a huge vegetable garden. Dropping lower I was drawn to the back of the house and a pathway into the garden. It was then that I understood why I was there. A rather plump elderly woman in apparent difficulty lay on her side at the edge of the path. I knew we'd be returning to try to help this poor soul, and at Meldor's bidding, we continued on.\n\nThrough scene after scene our tour continued: from visages of armored soldiers fighting bloody sword battles in medieval times to more modern forms of warfare, we journeyed on. The brutal reality was numbing. I wasn't sure that I was prepared for the depth of the emotional trauma I was witnessing. Still we continued, mostly just flying over the sites, but occasionally stopping to get a closer look at some of the souls we would return to help later.\n\nTwo of the more traumatic scenes stunned my senses, leaving a heaviness in my heart. While flying over a barren, hot and dry grassy plain of what I took to be Africa, my attention was drawn to a small, one-room, stone building. The roof was gone. It looked like the structure had been destroyed by fire. Meldor's voice surfaced in my mind, telling me that two African tribes, the Hutus and the Tutsis, had been at war. In one of the skirmishes, a number of the local people had been rounded up and locked into this building, which was then torched. There was a young soul there who needed assistance. We would return for him.\n\nIn the last encounter, we floated down into a jungle in South America, perhaps Columbia or Peru. Vegetation was thick and green. As I neared a small clearing I spied the body of a young man draped over the bottom rail of a ranch-type fence. It looked as if he had been crawling through the fence when he died. A large wound cut through the back of his neck, nearly severing it. I wondered about the circumstances of this gruesome scene, but knew we'd be back.\n\nWith that, Meldor brought our tour to an end. \"It's time for us to depart,\" he said. \"Your emotional energy needs to be restored. You will have much to process before we return for this work.\"\n\nA second later I was back in my body. The session hadn't ended yet, but I pulled off my headphones and stretched out on the bed. Meldor was right; I would need some time to think about all I had witnessed. As shocking as these scenes were, I was surprised at my calmness and detachment. In my physical body, the horror of the experience would have been difficult to deal with, but in my spirit body I understood that there was no such thing as death. There were no accidents. Each and every event had its purpose.\n\nStark images still playing in my mind, I got up and headed out into the warm Virginia sunshine. Enjoying the peaceful sounds of nature, I wandered aimlessly through the lush grounds of The Monroe Institute. In the shade of a large old tree I sat down, resting my back against its trunk, and while the birds and frogs sang a lazy lullaby, I fell asleep.\n\nSometime later, I awoke to the chirping of an angry squirrel in the branches just above my head. He was, no doubt, scolding me for invading his territory. With the fading images of a dream retreating into my subconscious, I crawled to my feet, apologized to the squirrel, and headed off to dinner. At the dining room I was surprised to find that most of my friends had already finished their meals. I had slept for more than an hour.\n\nThe following morning, rested and alert, I couldn't wait to get going on my first soul rescue mission. Following breakfast and an early briefing session, I anxiously headed to my CHEC unit. By the time the vibrations of the Hemi-Sync tones surfaced in my headphones, I was already deep in an altered state.\n\nAs Bob Monroe's voice guided us through the various levels of consciousness, I was having trouble focusing on his words. I kept clicking in and out of awareness. The stereo frequencies were designed to lead us through the various levels in ascending order until we finally reached the Park at Focus 27. Halfway there, I began to feel a familiar vibration that could only be coming from Meldor. In anticipation, I relaxed and waited for the sound of his voice. Instead, I was unexpectedly jolted by the sound of a squirrel. It was so startling, I almost sat up in bed. Then I heard his distinct laugh.\n\n\"That was _you_ waking me up under the tree yesterday?\" I asked.\n\n\"What, you didn't recognize me?\" Meldor chuckled. \"You'd probably still be out there if I hadn't awakened you.\"\n\n\"Very funny, Meldor,\" I replied. \"You do that type of thing often?\"\n\n\"Only when I need a good laugh,\" he chortled. \"You'll never be able to look at a squirrel in the same way again, will you?\"\n\n\"Not likely,\" I mused. \"It's a neat trick, though. You'll have to teach it to me some day.\"\n\n\"Oh, I can assure you, that was no trick. What makes you think I can't be a squirrel if I want to?\"\n\n\"I don't know. But trick or not, I'd still like to learn how to do it.\"\n\n\"You already know how to do it, you've only forgotten. Someday you'll remember.\"\n\nMeldor changed the subject. \"So, are you ready to get down to business? We have a busy day ahead, you know. Places to go, people to meet.\"\n\n\"To tell you the truth, Meldor, I'm still a bit nervous about it. How will I know what to say? What if I screw up or can't get through to them?\n\nMeldor's voice softened. \"Don't worry. You'll be just fine. You were becoming a bit too emotionally involved yesterday. You've got to lighten up a little or you may end up blocking your own efforts. That's why I've been trying to cheer you up. Just remember, even if you can't see me, I'll be right beside you.\"\n\nWith that, there came a surge in vibration, and a moment later I found myself hovering in the air above the lake we had visited the previous day. Below me the same two bodies floated beneath the surface, one male, the other female. About a hundred yards from shore I noticed a small boat lying on the bottom in relatively shallow water. I somehow knew that it had been swamped in a sudden storm. The man and the woman had drowned.\n\nMy attention was then drawn to the lone male sitting at the edge of the water. Rocking back and forth in apparent grief, he appeared extremely agitated. Tears streamed down his face.\n\nAt that point I became aware of another being floating in the air beside me. I felt his vibration but I couldn't get a close look\u2014his shimmering image faded in and out of view. Not alarmed in the least, I wondered who he was and why he was there beside me. Meldor interjected, telling me telepathically that this entity was the spirit guide of the man on the beach. His was the soul we had come for, and his spirit guide was along to assist in leading him to the Light. It was my job to get this poor man's attention and bring him to the realization that he had died.\n\nAs I descended to the shore, I wasn't sure what to expect. From there I walked slowly in the sand until I drew to within just a few feet of the man. He appeared to be in his mid-thirties, a bit stocky, but of average height. With a mop of dark matted hair, and a week's growth of stubble, he looked like he'd seen better days. Undoubtedly supplied by his spirit guide, the name \"Randy\" resounded in my mind.\n\nI was about to step closer, when a wave of energy almost toppled me. Inundated with a jumble of frantic, overwhelming emotions, I realized I was actually hearing and feeling the man's thoughts. Imprisoned in his own reality of unthinkable horror, he was paralyzed with shock. The sheer intensity of his mental anguish was staggering. I found myself backing away, begging, \"Oh God, I can't do this. How am I supposed to help him?\" Instantly, Meldor's reassuring voice calmed my fear.\n\n\"It's okay, Paul. You'll be just fine. Nothing can harm you, and this soul can use your help. Use the power of your mind to help him relax. Send him loving energy. You'll see.\"\n\nI decided to give it another try. Hanging back, I tried to project soothing, calming thoughts that everything would be all right. To my surprise, after two or three minutes, his thoughts began to slow, his terror subsiding. When he had calmed down a bit, I tried probing his mind to see if I could make sense of his dilemma.\n\nSlowly the details began to emerge. This poor soul had somehow made it to shore after their boat capsized and sank, but his wife remained in the lake. Unable to swim, he was terrified of water. Frantically crawling on his hands and knees at the edge of the lake, he had screamed in horror and disbelief. He wanted desperately to dive back in to save her, but found himself frozen with dread. His fear was simply too great. Not realizing that he had actually drowned without making it to shore in his physical body, he now sat trapped in the horror of his own reality. He was already in his spirit body, but because of his dilemma he could not escape the confines of his own illusion.\n\nHow could I get him to realize that he was dead? I wondered if I could somehow get him to float into the air with me. Perhaps if he saw his own body he would realize the truth. Not sure of anything, I decided to play it by ear. Summoning up my courage, I took a couple of steps forward and called his name. \"Randy.\" There was no response. I repeated my call, this time a bit louder. \"Randy, can you hear me?\" Suddenly, jerking his head in my direction, he looked up at me with wild eyes, screaming.\n\n\"My wife! She's in the water. Please help her. I can't swim!\"\n\nShaken by his outburst, I tried to remain calm. \"She's okay, Randy,\" I stuttered. \"Marlene is . . . she's just fine.\" My own words surprised me. Where had her name come from? How did I know it?\n\n\"What do you mean, she's okay?\" He stammered, wide-eyed, disbelieving.\n\n\"She's okay, Randy. She made it out of the water and went to get help.\"\n\n\"She's okay? How did she do that? I didn't see her.\"\n\n\"She's just fine, Randy. You might have been unconscious for a while and didn't see her. Marlene is just fine, but right now we're more concerned about you.\"\n\n\"Oh God, thank you God,\" he wailed. \"I thought I'd lost her. Thank you. Thank you.\" Staggering to his feet, he hugged me. I gently patted his back, thinking to myself, Well, that wasn't too bad. Now how the hell am I going to tell him he's dead?\n\n\"Where's Marlene?\" Randy asked as he stepped back from the embrace. \"God, I can't believe she's okay. She's not hurt or anything, is she?\"\n\n\"No,\" I said, wondering how I was going to get out of the hole I was digging. \"She's okay, but now you're the one we need to help.\"\n\n\"Me? I'm okay. There's nothing wrong with me.\"\n\nNow what would I tell him? I began to grope for the right words. Would mere words be enough to convince him that he was now in the spirit world? Not likely.\n\n\"Show him his body.\" The sound of Meldor's voice filtered into my mind carrying with it images of looking down at the two bodies from above the water. I wasn't quite sure how to pull it off, but decided to give it a try.\n\n\"When our bodies die, Randy, we may not even notice the difference for a while. It might seem like we're just our normal selves, but there is a difference; we can do things we couldn't possibly do if we were still in our bodies.\"\n\nRandy's eyes widened and a sort of disbelieving \"Say what?\" expression came over his face.\n\nHaven't you noticed anything different since you got out of the water?\" I asked. \"Don't you feel a bit strange and tingly, a little lighter?\"\n\n\"Aaah, no, not really.\" His incredulous look said it all. Knitting his brows, he looked me in the eyes and in an agitated voice retorted, \"What the hell! Of course I feel a bit tingly. I just about drowned, for Christ's sake.\"\n\nTrying to soften my demeanor and wishing I had used a different tack, I continued. \"What would you think, Randy, if I told you that you _did_ drown?\"\n\n\"What... then, how the hell could I be standing here talking to you?\"\n\nI needed to somehow get the message across without further upsetting him. I had an idea. \"I'm sorry, Randy, I didn't mean to upset you,\" I said. And in a conciliatory gesture I reached out to shake his hand. His features softened as he tentatively accepted my offer. In that moment I begged Meldor to help me out. Projecting every ounce of energy I could muster into the connection between our hands, my hope was to try to boost his vibrations beyond the physical field that limited his perception. The flow was powerful and instantaneous.\n\nIt hit him with such force that he appeared stunned. His eyes widened. For a moment he hesitated, trying to pull away, but I maintained my grip. Then all hell broke loose! To my surprise, I began to rise slowly into the air. Disbelief spread across Randy's face as he too started lifting off the sand. Stretching his toes, he tried desperately to reach back down to the beach, but we continued our slow ascent. As we approached a height of about eight feet, he desperately threw his left arm around my neck, hanging on for dear life. I had no idea what to expect next, but I knew one thing for sure: Meldor had taken charge of this operation.\n\nLeveling off at around 12 feet, I tried to calm my racing thoughts. Still clinging to me tightly, Randy seemed to be taking my cue, and I could feel his stress begin to subside. As we began to slowly float out over the water, he looked around in amazement. When we were finally in position overlooking the two bodies, our movement stopped. Releasing my grip on Randy's right hand, I directed his attention towards his own limp remains. He looked down, and upon seeing the body he flinched, tightening his grip around my neck. With intense curiosity, he continued to stare at his corpse until a look of sheer amazement spread across his face. Finally recognizing it as his own, a tremor rolled through him. And then, ever so slowly, his vibrations seemed to change. Without diverting his gaze he began to speak in low subdued tones.\n\n\"I didn't know. This is so strange. I feel perfectly fine, but that's my body. I guess I don't really need it.\" He then turned his head, looked up at me and asked, \"Who _are_ you?\"\n\nA feeling of relief and compassion came over me. \"Just a friend,\" I whispered. \"I came to help you to the Other Side.\"\n\nPerhaps without even realizing it, Randy released his grip, floating freely in the air beside me.\n\nI knew the next step might be a bit more painful for him, but I wanted him to understand that his wife would be waiting for him on the Other Side. I pointed again to the water, but this time towards the woman's body floating about 50 feet away.\n\nHis eyes followed my direction. \"Oh no! Marlene. It's Marlene,\" he sobbed. \"But you said she was okay.\"\n\n\"But she _is_ okay, Randy. She's never been better. She just went on ahead of you. If you want, just take my hand and I'll try to take you to her.\" Without hesitation he reached out, gripping my left hand with his right.\n\nThe moment of truth was approaching. Nervously, I closed my eyes to send a quiet thought-message. \"OK, Meldor, it's up to you. Please take us where we need to be.\"\n\nA rush of energy swept through me. My consciousness plummeted, and everything began to turn black. Seconds later we emerged through what seemed like a dense mist into the warmth and brightness of the Park.\n\nThe power of the place was stupefying. For a moment Randy tried to shield his eyes from the tremendous light, but quickly realized that it wasn't necessary. Dropping his hand, he let out a whistled exclamation as he drank in the beauty of the surroundings. Soon, our attention was drawn to the sound of someone approaching. On a pathway just to the left of us appeared a dark-haired woman beaming with light and happiness.\n\n\"Mar?\" Randy could hardly believe his eyes. \"Oh Jesus, Mar. You're okay!\" He bolted down the path to embrace his happy wife.\n\nThis was one of the most wonderful and rewarding encounters I had ever experienced. I felt blessed and honored to be able to help another soul in this way. Breathing a quiet thank you to God, I felt the warmth of Meldor's voice in my mind.\n\n\"A wonderful feeling, isn't it? You've done well. We're very grateful for your assistance with this soul, but it's now time to return to your body. You'll need to become a bit more grounded and rested before our next journey. As always, we are with you.\"\n\nSuddenly dizzy, I seemed to lose consciousness. A moment later I was back in my body.\n\nThe rest of the day flew by. Although we followed up with two more sessions, I just couldn't get my mind into it. When I did manage to stay awake, my thoughts kept returning to the previous experiences. Meldor was right. I needed grounding, some physical activity to take the psychic edge off my vitality. Heading outdoors, I ran the half-mile trail to a small lake. There, the Institute kept a couple of canoes for participants' use. I crawled into one and paddled hard around the pond before returning. The sweating and hard work did the trick. My body and my mind were once again attached to the Earth.\n\nAfter a good night's sleep, I was ready for another adventure in the spirit realm. Soon into the first session, I made contact with Meldor. Now that I had a bit of experience helping lost souls, he indicated that he would leave me more on my own. He felt confident I'd be able to handle almost any situation, but assured me he would still be there if I needed assistance. All I had to do was ask.\n\nThe Hemi-Sync frequencies carried me quickly to the Park. Once there, I was immediately approached by a wonderful, loving spirit who'd been trying to break through to a soul trapped in her country garden. It was the elderly woman I had seen a couple of days earlier.\n\nThis time the transition to the necessary level of consciousness was quick and direct. After only a moment I found myself floating in the air directly above the elderly woman. She lay on her side on a well-worn garden path. Beets and tomatoes she had been carrying remained strewn across the path. Although I knew there was no connection, she reminded me of my grandmother.\n\nThere didn't seem to be a lot of trauma surrounding the woman's death, and I wondered why she'd become trapped. Her guide told me her name was Maria. He explained that throughout her life, she had maintained an inordinate sense of duty to her husband. In addition, she had never believed in God or any kind of afterlife. Her rigid belief, or lack thereof, dictated that when someone died, that was it. There was nothing more. So, refusing to accept existence beyond the body, when Maria died, she was unable move on. Thus for her, time stood still.\n\nAs her guide telepathically communicated these messages to me, I wondered how long she had been caught in her own time warp. The crude construction of the cabin, along with the apparent absence of any utilities or even roads, suggested that her death must have occurred many years earlier. I was wondering what year it might have been, when the voice of her guide popped into my mind: \"1853.\"\n\nUnbelievable! More than 140 years had passed in the physical world since this poor soul had become locked in her illusory reality. Her guide, however, assured me that it was not nearly as bad as it seemed. Our perception of time didn't exist on the Other Side. In her mind, very little time had passed.\n\nI began to consider the possibility that this particular soul might be tougher to get through to than I had at first imagined. Convincing her that she was dead might not be so easy. Asking Meldor for assistance, I immediately felt him contributing to my thought process. With his guidance, I decided to try something that might be in opposition to her beliefs.\n\nI floated down near the woman to a position about three feet off the ground. Seeking to determine her state of mind, I focused my attention on her thoughts. To my surprise, I found she was almost frozen with fear. Terrified of her husband, Maria knew he would be very angry if his lunch wasn't ready by the time he got home. She had long ago convinced herself that Herbert was a good, hardworking man, and it was her duty to see that he was happy. If he wasn't, it would always somehow be her fault.\n\nEven more significant to the present moment was her fear of dying. She did not want to cease to exist. For several weeks before she died she had felt increasingly strong pains in her chest that began whenever she worked too hard or became upset. She hadn't said anything about it to Herbert. Now, even though the severe pains had passed, she was perplexed and upset that she couldn't get up. It became clear to me that she had died of a heart attack. Her fear of both living and dying had locked her into her illusion.\n\nFrom my position, hovering a few feet above the ground, I softly called her name. \"Maria.\"\n\nShe lifted her head in my direction, but said nothing. Through squinting eyes, she appeared to be trying to make out what she was seeing. After a few seconds she finally responded, \"What do you want?\" Her voice revealed a strong Polish or Ukrainian accent.\n\n\"I came to help you, Maria. Everything will be all right.\" I said.\n\n\"But I just can't get up,\" she replied.\n\n\"Sure you can, Maria. You can get up. All you have to do is take my hand, and I'll help you.\"\n\n\"But you're not even touching the ground. How can you do that?\" She asked suspiciously. \"Who are you? Are you an angel?\"\n\n\"No, Maria. I'm not an angel.\" I said softly, \"But I can help to take you to Heaven.\"\n\n\"Heaven, there's no such thing,\" she argued.\n\n\"It doesn't matter if you believe in Heaven or not, Maria. It still exists, and it _is_ real. You didn't believe what the Bible said, but it's true. You remember the pains in your chest? You knew there was something wrong with your heart, didn't you? Well, now your body has died. You can leave it behind and go to Heaven. All you have to do is take my hand.\"\n\nI floated down to the ground beside her. As if testing the reality of what she was seeing, Maria closed her eyes and shook her head. When she reopened them and saw that I was still there, she let out a sigh of resignation and slowly reached for my hand. Grasping her arm, I gently guided her to her feet. Although she didn't realize it, in her energy body, she was virtually weightless.\n\nA look of relief and amazement spread across her face. She was finally ready to leave. Holding her hand, I focused my thoughts on her spirit guide, asking for assistance. Almost instantly, a vibration surged through my body like a high-voltage transformer. I felt a shift in consciousness, and in less than a heartbeat, we were standing in the beautiful green grass of the Park.\n\nMaria's eyes widened. Overwhelmed by the beauty and power of the place, she mumbled in disbelief. There was hardly time for a quick look around as her attention was drawn to a group of people approaching on a small pathway beside us. Tears of joy rolled from her eyes as her two children, her mother, and her husband welcomed her home.\n\nAn emotional moment for me as well, I felt a lump building in my throat, and decided it was time to leave. Sensing my thoughts, Maria's guide expressed his gratitude with a wonderful parting gift. Placing the palm of his right hand flat on my chest, he closed his eyes in concentration. A moment later, like someone had opened a tap, my body was flooded with a radiating energy. I closed my eyes to more fully enjoy the exhilarating vibrations. The force began to swirl and expand throughout my chest. It felt like my heart was about to burst. Wonderfully dizzy, my consciousness began to slip. Soon everything turned to black, and the next thing I knew, I was back in my body.\n\nElated, I got up and headed out into the morning sun. The energy infusion had been so powerful, I felt like I could take on the world. Grateful for the tremendous opportunity for spiritual growth, I offered a quiet prayer of thanks.\n\nThe last day of the program was eventful and emotional. I had been having the best results during the morning sessions, and this morning would not be a disappointment.\n\nWithin minutes of the session's start, I floated quickly through the levels of consciousness. Meldor's familiar vibration accompanied me as we began to move into the spirit planes. By the time we reached the area of the lost souls, we were joined by a bright and intensely focused being. There was something new and different about this guide. Though I caught only glimpses of it, it didn't seem to have a physical form at all. Instead, it appeared only as a bright, vibrating oval of light. Even without a physical shape, this entity seemed to project a powerful feminine energy.\n\nWith a determined but gracious manner, she led us quickly to the soul who had been entrusted to her. Without any feeling of movement or direction, we were suddenly standing before a mudstone building on a wide, open plain. It was the burnt-out building I had been shown a few days earlier on my tour with Meldor. The building had been burned black. The roof was gone, and what had once been a window and door, were now just bleak sooty openings. I stepped closer, peering in through the window.\n\nThe sight of more than a dozen burnt and charred human bodies almost staggered me. Repulsed, I wanted to get away from the place. Even though it was difficult to discern the features of most of the bodies, it appeared that nearly all of them were either quite young or very old. They were male and female, strewn across the floor. At least five of them were small children. The suffering that must have occurred seemed too horrible to comprehend. This had been done deliberately! What kind of fear and hatred would cause someone to do such a thing?\n\nIt was then that I began to sense the presence of a soul trapped in deliberation. Near the far corner of the room, a young black African boy, perhaps 10 or 11 years old, lay face up in a crumpled heap along the wall. His thin hands were clasped to the sides of his head covering his ears. The horribly burnt body of an older woman lay across his legs, pinning him down.\n\nThis was simply too much. I didn't want to see what this young man had experienced. Closing off my perception, I begged Meldor for help.\n\nAlmost immediately the warmth of his energy began flowing through me. Like a powerful sedative, the effect was calming. For several moments I stood, languishing in the intense vibrations. Soon Meldor's reassuring words filled my mind. \"The more horrific the prison,\" he said, \"the more wonderful the release.\"\n\nThe words pierced powerfully through my being. He was right. This unfortunate soul certainly needed our help. I began to imagine how wonderful it would be to see the young man's transformation from desperation to joy. There was no question; this was something I had to do.\n\nReopening my perception, I began to focus on the boy's thoughts. Bracing myself for the mental shock that would surely follow, I prepared for the worst. But it didn't happen. Instead, I heard only a small voice quietly singing a simple tune. To my surprise, I found his mind engrossed in thoughts of being safely hidden beneath thick, comforting blankets. It was like he was absorbed in a daydream. Probing further, I understood what was happening.\n\nAs a young boy, he had endured a terrible existence. All too often, he had witnessed atrocities of mind-numbing consequence. His mother had been frightfully abused and beaten so many times that he had learned to protect himself by crawling into the safety of his bed. Pulling the blankets over his head, he would put his fingers into his ears to block out the sound. Now, at the horrible ending of his young life, he had again protected himself by mentally crawling into his bed and blocking out the terrible reality. This time, however, he was locked into his self-induced daydream.\n\nAlthough saddened by what this young boy had endured, I was immensely relieved I didn't have to experience his terror. This would be a lot easier than I expected.\n\nStepping through the doorway, I crossed the floor until I stood over him. He looked like such a lost little creature it tore at my heart. Asking for God's help, I reached down and placed my hand on top of his head. Startled, his eyes flashed open. He jerked his hands away from his ears, staring up at me in complete surprise. I couldn't help but smile. Feeling a deep empathy and tenderness for this child, I began softly. \"Hi,\" I said. \"What's your name?\"\n\n\"Lelanee,\" he whispered back.\n\n\"Lelanee,\" I repeated. \"That's a nice name.\" His eyes widened. I slowly continued. \"You're okay now, Lelanee. You can get up if you like. How would you like to see your mother?\"\n\nHe nodded his head.\n\nI reached down, took his hand, and began to help him to his feet. All of a sudden, the building, the horrific scene, and everything around us dissolved before my eyes and we were instantly transported to the Park. Lelanee's eyes were wide with amazement as he scanned the surroundings. Before I could say a word, he bounded past me yelling, \"Mama, Mama!\" Elated, I turned to watch as his mother swept him into her arms, and hugging him close, spun in circles, laughing happily.\n\nGratified, I just stood there, grinning. When tears began building in my eyes, I wanted to be back in my body, and a moment later I was.\n\nThere was so much to think about. I lay in bed contemplating the power and meaning of all that I had witnessed. At the beginning of the week, I had been more than a little apprehensive and skeptical about the program, but it had turned out to be a miraculous learning opportunity.\n\nEven though I had experienced many remarkable encounters in the spirit world, the final session of the week proved to be the most profound. After moving quickly through the various levels, I was joined by two spirits who wanted to take me to their lost charge. One of the entities was a spirit guide, while the other had been the grandfather of the soul we were to retrieve. Although I wasn't given the name of the guide, I was told that the grandfather's name was Alejandro. The name of the young man we were searching for was Marcos.\n\nFollowing a short burst of movement, we emerged on the ground in a heavily vegetated, jungle-like area. It seemed to be about midafternoon. The sun shone brightly in the warm, humid air. This was the same place Meldor had taken me on our first tour. My impression was of a rural area somewhere in South America. I could feel the vibrations of both spirits that accompanied me, but could only make out the faint form of the grandfather.\n\nAs I stood in the green grass of a small clearing, the memory of the disturbing scene of my earlier visit returned. We would be looking for a body with a vicious neck wound, draped over a fence. Our immediate surrounding was familiar. The old weathered three-board ranch fence was right in front of us, but there was no one to be seen.\n\nI stood for a few moments peering about when, to my left, the sounds of someone running caught my attention. A soldier in green camouflage came charging out of the dense underbrush and stopped short in the clearing. Obviously looking for someone, he crouched, anxiously scanning the fence line. The view of the other side of the railing, however, was obscured by dense brush and foliage. From a distance I could hear the sounds of machine-gun fire and barking dogs moving in our direction. The thought entered my mind that this was a drug raid.\n\nThere came a flurry of motion, and I spotted a dark-haired young man crashing through the foliage on the other side of the fence. Diving to his hands and knees, he began desperately crawling through the horizontal planks towards the clearing. The soldier, now less than a hundred feet away, quickly raised his rifle and fired twice. The bullets ripped through the back of the young man's neck just as he was part way through the fence. His head dropped, and his body collapsed lifelessly over the bottom rail. Then it was as if time stood still.\n\nLong moments passed as I stood riveted to my spot. Stunned by what I had witnessed, I was also puzzled. This was different than the other retrievals. All of the earlier cases had been presented after death had occurred, not _while_ it was happening. Previously I had either been given the details by my guides or had probed the thoughts of the affected soul. This time, however, I was right there, as if it were just happening. Could it be that we had actually traveled back in time to the exact moment of the event? Not knowing what to make of it, I returned my mind to the task at hand.\n\nMarcos's thoughts were frantic. In that instant he knew his neck was broken and that he would soon be dead. In shock and disbelief, he began to pray.\n\nMoving to within just a few feet of him, I called his name. \"Marcos,\" I said softly. Continuing his Spanish prayer, he obviously hadn't heard me.\n\n\"Marcos,\" I repeated more loudly and firmly. This time he stopped praying and fell into a dead silence. Then as if deciding that he must have been hearing things, he returned to his prayers with renewed vigor.\n\nAt this point, I wasn't sure what to do, but I had to get his attention. Offering a quick prayer of my own, I asked for help and guidance. A moment later I felt a strange new energy swirling through my body; it intensified until it began to literally radiate from my fingertips. My hands felt scorching hot.\n\nKnowing there must be a good reason for the energy flow, I bent down and, placing my hand on the back of his head, I again called out his name.\n\nBreakthrough! In the middle of a word, he stopped praying. Then with gushing sobs he began to cry.\n\n\"Marcos,\" I repeated softly, trying to sooth him. \"Everything is okay. Your neck is all right. You can get up now. Look, your grandfather, Alejandro, is here to see you.\"\n\nAs the words were coming out of my mouth, a stunning realization hit me. I couldn't believe what I was hearing. My thoughts were in English but my words were not. I clearly heard myself speaking Spanish, yet I hadn't a clue how to speak Spanish! Momentarily dumbfounded, I didn't have time to give it much thought.\n\nMarcos began to stir. Slowly raising his head, he glanced at me, and then turned in the direction of his deceased grandfather. With a loud cry, he let go a burst of unintelligible expletives. I didn't know if he was cursing or praying, but suspected a bit of both.\n\nSeemingly forgetting his injuries, Marcos jumped to his feet, his energy-body passing through the upper fence boards like they weren't there. A big smile spread across his face. Fixated, he gingerly stepped around me and rushed to his grandfather's embrace. As I turned to follow him, in the next instant, Marcos, his grandfather, and everything around us, abruptly vanished. With a resounding jolt, I slammed back into my body. The session was over.\n\nAs I lay in the darkness reliving the fascinating images, numerous questions plagued my mind. Had I actually gone back in time to view an event _while_ it was happening? A fluent Spanish dialogue had flowed from my mind. Where did it come from? Had I spoken Spanish before? Perhaps in another lifetime?\n\nI knew I might never know the answers to these questions, and perhaps it didn't matter. The questions themselves could be more intriguing than the answers.\n\nOur final debriefing session ended with the expected rounds of teary-eyed hugs and goodbyes. We would soon be heading back to our busy lives in every corner of the globe. It had been a most remarkable week for me. I hadn't shared many of my own experiences with the group, but I had certainly been enthralled by some of the emotional stories I heard. Just a week earlier, I had had serious reservations, but now there was no doubt in my mind. We _had_ made an important difference in the lives of some people.\n\n# 9\n\n# _Spiritual Connections_\n\n_Then comes an angel, eyes ablaze_\n\n_Heart secure and love unfazed_\n\n_A glimpse of what was meant to be_\n\n_Pure light upon reality_\n\nLife, I had always believed, was for the most part a crapshoot. Fraught with unexpected twists and turns, it could lead us down some peculiar paths. I was beginning to discover, however, that regardless of the roads traveled, eventually we all arrive at the same place\u2014home. Although often an endlessly confusing struggle, the path we end up following is likely the very same one we ourselves designed before we came here.\n\nI was gradually learning that there weren't many coincidences or accidents in life. Instead, there seemed to be a definite plan including some built-in roadblocks and obstacles that each of us envisioned before incarnating.\n\nFollowing my return home from the Lifeline Program, I spent a lot of time in contemplation and meditation. My recent experiences had been profound, and I needed to know how it all fit into the path I was taking.\n\nIt felt like my life was becoming something of a giant jigsaw puzzle without the benefit of a guiding photo. Every now and then I was given a new piece to put into place, but I couldn't yet grasp the entire picture. There were still more pieces missing than found. On top of that, I had no idea how big the completed puzzle was supposed to be or if I could ever hope to put all the pieces together.\n\nSome things I knew to be true without a doubt. I knew there is definitely a God but I wasn't as sure about what or who God is. From Meldor's revelations, it seemed that we are all part of God, and even more incredibly, as a collective unit, we _are_ God.\n\nI accepted the premise that we are eternal and that the great majority of us do not come into the world just once, but incarnate many, many times. Working through our goals and trials, we hope to eventually achieve perfection and ultimately the realization of being one with God.\n\nI also understood that we\u2014and apparently every other living thing in the universe\u2014are connected at a spiritual or pure energy level. Being all part of the experience, we could also experience all of the parts. These things felt right and true in my heart, but there were still so many things that I didn't know and couldn't comprehend. In terms of spiritual knowledge, I was still in kindergarten. My education had only just begun....\n\nHaving lived a difficult and tormented life, my mother, wracked with cancer and Parkinson's disease, finally gave up her body on her 81st birthday. Although grief-stricken at her passing, our family was relieved that she would no longer have to endure the suffering of her final years. Most of us were with her in the hospital when she made her transition. By that time in my life, I had a completely different view of death than my devoutly Catholic siblings.\n\nA few months earlier, as it became apparent her condition was terminal, she had been afraid of dying, uncertain if she was good enough to be allowed into Heaven. Undoubtedly, throughout her life, she had heard far too many sermons of hellfire and damnation. Although we all thought she could be sainted, she worried she might be somehow lacking.\n\nIn those final moments, while holding her hand, I spoke to her with my thoughts, encouraging her to let go, to go to the Light. I promised her that she would be going to a wonderful place where she would be happy and free of her crippling diseases. I assured her that Grandma and Grandpa would be there to greet her, and I told her that she would be able to fly.\n\nAlthough my sisters and brothers wept with grief, and a large lump formed in my throat, I tried to maintain positive thoughts. This, I told myself, was a wonderful occasion, and we should be happy for our mother's return home. I would not and could not cry.\n\nNear the end, as her life force began to withdraw, a noticeable change came over the energy in her body. I knew she was ready to leave and closed my eyes, hoping to be able to witness her transition. But it was not to be. With all the anxious thoughts flooding my brain, I wasn't able to step up my vibrations or see beyond the insides of my eyelids. Although I couldn't see her, I was able to feel her subtle vibration as she floated to the ceiling in the upper right hand corner of the room. I sensed her elation and wonder. Moments later, while I strained to open my inner vision, her vibration faded and she was gone. My eyes still closed, I projected a final message. \"Bye, Mom. I'll see you soon.\"\n\nThis had all happened about a year before my Lifeline seminar. At various times during the ensuing year I attempted to make contact with my mother while out-of-body, but to no avail. Out of frustration I stopped trying. I knew that if it were meant to be, someday I would meet her again on the Other Side. I had learned through trial and error that if I became obsessed with trying to achieve a certain experience, it would usually elude me until I least expected it.\n\nAbout a month after my return from The Monroe Institute, it was my mother who came to visit me.\n\nI had gone to bed late. As I floated at the edge of sleep, a fluttering vibration began to slowly surface in my awareness. Tired as I was, I couldn't resist the opportunity for an adventure. Taking mental control of the vibrations, I moved quickly through the threshold of consciousness, attempting to home in on the frequency that led to the Park. But nothing happened. For several more minutes I lay in bed, trying to focus my thoughts while fighting off sleep. It appeared that I was going to lose the struggle and slowly I drifted into oblivion.\n\nA loud thud quickly brought me back to awareness. To my surprise, I found myself floating near my bedroom ceiling, the sound of a woman's soft voice calling my name. Startled, I turned in the direction of the voice. To my total bewilderment, a large portion of the bedroom wall began to swirl into a dim mist before opening into a tunnel-like vortex about six feet in diameter. Near the center of the opening moving in my direction was an oval of shimmering light.\n\nProjecting my senses, I probed the light to identify the approaching entity. Almost immediately a vibration of familiar love permeated my entire being. I had not the slightest doubt. It was my mother!\n\nThe anticipation was exhilarating. I watched as the light moved slowly in my direction. Was this the form that my mother now inhabited? Was this the true form of all spirits?\n\nWithin moments the glowing oval floated through the wall and into my bedroom. Once inside the room it stopped, and with a gentle mesmerizing swirl, it began to transform into a familiar human shape. My mother did not appear at all like the withered, tired, 81 -year-old I had last seen. She looked to be in the prime of her life, between 35 and 40 years old. She was simply radiant.\n\nEven though I wasn't in my body, I could feel the pounding of my heart. I had not anticipated anything like this. My mother exuded such a wonderful peacefulness about her that I ached with happiness. As she stretched out her arms, I moved to embrace her. The instant we touched, I felt a rush of energy; the bedroom faded, and the next thing I knew, we were in the Park.\n\nSurprised, I had no idea how this movement had taken place. Pulling back from her embrace, I looked into her eyes. They were beaming with happiness and reassurance. In my mind I heard her voice. \"It's easier to talk to you here.\"\n\nFor a moment I felt Meldor's vibration nearby, then it faded. I had the impression that he had been assisting in this reunion and then withdrew to give us privacy.\n\n\"Mom, you look wonderful,\" I gushed. \"It's so great to see you.\"\n\n\"It's beautiful over here,\" she said. \"All the pain is gone, and see [she held up her hands], I don't shake anymore.\"\n\nI couldn't take my eyes off hers. Beaming brilliantly back at me, they carried the knowledge and memories of so much of my life. How many times had I looked into those wonderful eyes to see the pain and tears of a hard and unrewarding life? How many times had I fought back my own tears for the deep disappointment I saw in those eyes? Disappointment and frustration at the things that she wanted to give but couldn't. How many times, as a boy, had I been comforted by the warm, unconditional love that shone in those eyes when the rest of the world seemed cruel and unforgiving?\n\nAll of these thoughts and feelings flowed though my mind. For the all-too-few times that I would see happiness in those eyes, I would also see years of sadness and yearning. Now it was different. No longer was there sadness or disappointment. I now saw only happiness and the kind of contentment that could perhaps only be realized by someone who had gone through great tribulation and triumphed.\n\nBursting with joy, I wouldn't have traded that moment for all the gold in the world.\n\n\"It's okay,\" she said, knowing my thoughts. \"It's all part of the learning experience that we chose. It might have seemed pretty hard at the time; sometimes I wanted to quit, but it was worth it.\"\n\nI was puzzled. \"You mean you wanted a life like that?\"\n\n\"Yes, of course. It wasn't actually so bad, you know. We've been through tougher lifetimes than that. In order for us to get what we need out of it, it's important that we don't remember that we ourselves chose the kind of life we wanted to experience. You and the rest of your brothers and sisters are part of the plan we all put together. That's how it works.\"\n\n\"We all put together?\" I was confused. \"You're saying we all wanted a life like that? You mean we came here knowing what was going to happen and how it was going to turn out?\"\n\n\"Well, sort of.\" She smiled coyly, a gleam in her eyes. \"We don't always know how things will turn out, but we put ourselves into circumstances where there can be a number of possibilities and outcomes. It all depends on the choices we make and how we deal with the situations we put ourselves into. Before I was born, I already knew how I was going to die.\"\n\n\"Jeez, Mom, that's kind of spooky,\" I said in a serious tone. Almost immediately, the silliness of my statement became glaringly apparent. We both burst into laughter. Here I was, talking to the spirit of my dead mother, proclaiming that what she was telling me was spooky. We giggled about this for some time.\n\nFor several moments we stood with arms entwined, enjoying each other's vibrations and our own memories. This meeting was so completely different than I had ever imagined it might be. I marveled at the remarkable change in my mother. Yes, she was still the same wonderful person I had always known, but it was our relationship that was different. We seemed to be relating to each other like old friends meeting again after a brief separation. It wasn't the same type of mother\/son relationship we had experienced in our physical bodies. It was much deeper and older than that. It was like we had known each other for a long, long time.\n\nFinally, I broke the silence. \"You know, it's been amazing. Sometimes I can hardly believe everything that's been happening. My life has changed so much in the last few years.\"\n\n\"Yes, I know,\" she smiled. \"I've been keeping track of you.\"\n\n\"You have . . . ? Thank you,\" I responded. \"I think about you a lot. Sometimes I dream about you.\"\n\n\"I know,\" she said, her eyes hinting of sadness. \"You've been punishing yourself because you thought you could have spent more time with me in those last few years. But you didn't need to worry. I knew how you felt. I knew how it hurt you to see me wither away, and I hurt for you. I still felt your love. You don't need to feel that way anymore.\"\n\nI was now beginning to fight off tears; I didn't want to be crying. I didn't want to be sad. I changed the subject. \"So, how does it feel?\" I asked.\n\n\"Oh, you mean how does it feel to die? You mean you don't remember? Oh, of course you don't, but you've come close a couple of times, haven't you?\n\n\"It was wonderful! The dying part, that is. The sickness and pain beforehand wasn't so easy, but the actual dying was great. Oh, and I want to thank you for your help when I was finally able to leave. And thank the rest of the kids for their prayers too. It did help a lot. I was really scared. It was kind of strange at the time, because I could hear your voice in my mind. I thought I was dreaming, but I could hear you and some of the other kids. It made me feel better, though, and then Grandpa came and told me I could come with him.\"\n\nPausing for a moment, she had an idea. \"Here's a better way to show you what it was like.\"\n\nA visible stream of mental energy began flowing from her to me, and I became immediately immersed in her reality. I felt her emotions. I saw what she saw. I experienced what she experienced.\n\nInstantly transported back to her old hospital room, I felt myself and her float to the ceiling. Below me I could see myself sitting beside her bed. I could see her frail withered body, and my sisters and brothers sitting quietly in the hospital room, tears in their eyes. I felt the sympathy and love that she felt. I felt the tug at her spirit as my grandfather's face appeared in her mind. Amazement transformed to elation as I felt myself drawn through a tunnel-like darkness, moving towards a powerful vortex of light.\n\nAt first only a small glow in the distance, the light became brighter as I sped along. Anticipation and yearning consumed me, pulling me along. Then, as waves of energy swept through me, the journey ended and I felt myself bathed in a beautiful golden light. The warmth and love of all consciousness overcame me like a narcotic. Soon, everything began to fade, and in a dizzying, spinning euphoria, I lapsed into a blissful sleep.\n\nI wasn't sure how long we slept, but my next awareness was of awakening from a peaceful, dream-filled sleep. Through my mother's eyes I awoke to a softly lit, immaculately clean, white hospital room. Around her were a number of loved ones who had passed on before. I recognized the beaming faces of my grandmother and grandfather, my brother-in-law, Albert, and a number of aunts and uncles, all there to welcome my mother home.\n\nWith that final scene, the thought-transference ended, the vibrations subsided, and I was left with a wonderful sense of peace and happiness.\n\nIt took me a few moments to recover. This method of communication\u2014telepathy, thought-transference, or whatever it was\u2014was incredible. Complete in every respect, there was no room for error or interpretation. I experienced what she had experienced. It did, however, leave me puzzled. Why had my mother awakened on the Other Side in a hospital room? I thought everyone was whole and healthy in the spirit realm. There should be no need for hospitals.\n\nPicking up on my thoughts, Mom explained. \"They tell me that if a person has had a long, painful death in an Earth hospital, their spirit energy can also become weak and run-down. Disease in the physical body can also affect the energy body and vice versa. People in that condition can also become very accustomed to their surroundings. It helps make the transition to spirit so much easier if they wake up in a familiar room where they're being cared for. It can also take time for a person to restore their energy after they arrive. A hospital is a comfortable and natural way to do it. There's nothing but love and caring here.\"\n\nWhile she was explaining these things to me, a remote tug on my awareness began urging me back to my body. I tried to resist the pull. I needed more time. There were still so many questions to ask. I wanted to find out about other lives that we might have lived together, how my brothers and sisters were connected, and how she now spent her time. I was about to ask if she planned on reincarnating when it became apparent that I could no longer fight off the pull to return. Finally, I told her that I had to leave. My body was calling me back. With an understanding smile she reached out to draw me near. I had time for only one last parting thought. \"I love you, Mom.\"\n\nA moment later, I was back in my body. My right arm was throbbing. I had rolled over onto it, cutting off the circulation. As I sat up and began rubbing my arm, my eyes were drawn to the wall in front of me. I would never forget the sight of my mother in her brilliant energy form, floating into my life once again.\n\nSome of the concepts my mother had told me about intrigued me. Do we really lay out a plan for our lives before we take on a physical body? I couldn't imagine that she would have chosen the type of life she had endured for 80 years. For that matter, if I had any choice in the matter, why would I have chosen the circumstances of my childhood? Certainly there must have been any number of easier options. Why wouldn't I have chosen to be born into a rich family? And did this mean that we choose our parents?\n\nWhat about the starving, emaciated children we often see on television in some far-off country? Why would they have chosen such terrible existences where they could die of starvation or disease before reaching the age of five?\n\nThe questions were endless, the possibilities mind-boggling. Fortunately, I knew just where to go for the answers. At my first opportunity I would pay Meldor a visit.\n\nMy schedule during those next few weeks was hectic and opportunities for a spiritual adventure were few. The rare times when I did make an attempt to connect usually resulted in my falling asleep within minutes.\n\nThen, one Saturday morning, it was Meldor who decided to take the initiative. I remember awakening in a dreamy relaxed state. But as I lay in bed, I began to feel a gentle tingling in my forehead. This usually meant that the aperture of my third eye was about to open. Anticipating the usual parade of faces, I was surprised to find myself peering out into the peaceful setting of the Park. On a large boulder sat Meldor, looking for all the world like a canary-swallowing cat. With a twinkle in his eyes, he greeted me.\n\n\"Good morning, sleepy head. Are you ready for today's lesson?\"\n\nHe looked so funny I couldn't help but laugh. \"Aahh, it's the squirrel guide,\" I chorded. \"Where have you been?\"\n\nAt this, Meldor burst into such an uproarious laugh that he almost rolled off the boulder. \"Very funny,\" he howled. \"Squirrel guide, girl guide. And you do this while you're still half asleep.\n\n\"Hey, Mister,\" he roared, \"you wanna' buy some cookies?\"\n\nMeldor continued laughing so hard; he appeared to be hyperventilating. Tears streamed down his face.\n\nAs infectious as his mirth may have been, I tried to remain unaffected, looking on with feigned bemusement. \"A regular bloody comedian,\" I quipped. This, however, only sent him into more spasms. No longer able to hang on to his perch, he rolled off the boulder and fell howling, to his hands and knees where he continued to carry on.\n\nWhen he finally began to catch his breath and settle down, Meldor let out a big sigh. \"Boy, it's times like these I sure miss having a body. There's nothing quite like laughing until your sides hurt. Thank you for the memory.\"\n\nAfter a few moments of chuckling, he got down to business. \"So, you have some questions, and think I have the answers.\"\n\n\"I'm not so sure that there are answers to my questions,\" I began, \"but I thought if anyone would know, you would.\"\n\n\"Well, I don't know if I'll have all the answers, but I'll certainly give it a try.\"\n\n\"I want to learn more about how things work in the spirit world,\" I continued. \"Do we actually decide, before we come here, the type of life we'll have and even how we'll die? Do we choose our own parents? Do we set up our own challenges to overcome? How does it all work?\"\n\n\"Aahh, the big question,\" Meldor replied. \"Yes, I think it's something we can communicate to you.\" He sat back down on the rock, pondering where to begin.\n\n\"First, as I have alluded to before, you should know that you and most every other incarnated being already knows most of the answers to these questions. It's not so much a matter of learning about these things as it is a matter of remembering what you already know.\n\n\"Most of us have made the transition from spirit to physical life and back, many, many times. Each time we return, the memory of our origin slowly fades from awareness. By the time a child has reached the age of six, most of the memory of its true spiritual nature has been filtered out by physical consciousness, leaving the imprint only in the subconscious, and the soul. The denseness of the physical body puts a tremendous restriction on the sensory perceptions of the spirit. As a child grows in size and physical awareness, the genetics of the species, with its powerful built-in survival instinct, becomes more and more dominant, leaving a decreasing memory of its true spiritual reality.\n\n\"Our teachers have told us that we all began as part of the one great consciousness. In order to experience diversity and to honor ever-continuing creation, we, as parts of the Creator, agreed to become individuated, and split off from the source. By doing so, _The All_ , which you refer to as God, could experience itself in endless evolving form and reality. Like individual drops of water in the ocean, we are still one, and flow together as part of the whole ocean of consciousness. Therefore, what we experience as individuated beings, seemingly in isolation, is actually also experienced by _The All_ at various levels of understanding and consciousness.\n\n\"God, as you understand it, continues to seek perfection. It is never at rest, but continuously moving, creating, and experiencing through all aspects of its colossal energy. As individuated parts of the Creator's energy, we seek the same perfection that is sought by God; that is to become fully aware, flowing and vibrating together as one unified consciousness. We seek, as a part of God, to bring every individuated aspect, every soul, to the perfect love of God realization.\"\n\n\"Meldor,\" I interrupted, \"you made reference a few moments ago to what your teachers have told you. Are you saying you have teachers?\"\n\n\"Oh, most definitely. We all have teachers. There are many, many levels of spiritual vibration and realization. The physical world that you now occupy is among the lowest levels of vibration and consciousness. Because the Earth is of such a dense physical nature, with harsh realities, it is one of the most powerful levels to gain experience and meaningful development. There is tremendous opportunity for growth and advancement in the Earth plane.\n\n\"Each soul or aspect of _The All_ continues always to work toward higher and higher levels of vibration and understanding. As we are in reality all one, it is a natural desire for all souls to want the very best for every other soul. The evolution and advancement of each individuated soul is extremely important to souls in all planes of consciousness. Regardless of the levels of vibration and realization that we have attained individually, we can never be completely happy or content until every aspect of _The All_ has attained the God realization and perfection that we seek.\n\n\"We are our brother's keepers,\" Meldor added, \"not because we have to be, or because we are told to be, but because we want to be. For this reason, out of pure unconditional love, many advanced souls volunteer to assist others in progressing through levels that they, themselves, have already assimilated. I have enjoyed many teachers and I will have many more.\"\n\n\"But, if we were already in the spirit world before we came here,\" I asked, \"why would we bother to incarnate into a slow and dense physical body?\"\n\n\"The greatest opportunity that the physical world offers is the experience of human emotions. It is for these powerful learning tools that most souls choose to incarnate, and return again and again to Earth.\"\n\n\"You said _most_ souls choose to incarnate,\" I interrupted. \"Do you mean there are souls that don't?\"\n\n\"Oh, Heavens, yes,\" Meldor explained. \"Billions upon billions of souls have never incarnated into a physical being as you understand it. Many souls choose to experience and contribute to _The All_ entirely from the spirit form. Every soul has complete freedom to contribute to the well-being of the collective consciousness as it sees fit.\"\n\n\"But why would a soul be attracted to experiencing human emotions?\"\n\n\"Let me put it this way,\" Meldor explained. \"In this higher vibration, a spirit, other than vicariously, cannot really experience or understand the many powerful emotions that an incarnate spirit can. Being part of _The All_ , a spirit on this side, who has never been exposed to human or similar incarnations, is not normally able to experience feelings other than the emotion of _love_. A spirit could not, for example, experience or understand emotions such as pain, hate, fear, jealousy, sexuality, or lust.\n\n\"Knowledge gained from experience is the key. Some of my compatriots argue that to really know and appreciate a complete understanding of love, it may be necessary to experience the opposite of love through emotions like fear and hatred. The challenge and experience of all these powerful emotions can be like a very strong magnet, an almost irresistible drug, to those in spirit. These are very potent and compelling learning tools.\n\n\"But let me get back to your original questions. Through various incarnations, souls develop special friendships and bonds with certain other souls. They quickly learn that with each other's help they can achieve better results, and a more profound learning experience. So, together they agree to return again and again to the Earth environment, learning, evolving, and experiencing. Often they will switch roles in each other's lives. In one life, a soul might play the role of a parent, only to return in the next life as the other's child. Numerous scenarios can be played out time and time again, all for experience and accumulated knowledge. Experience and knowledge are, after all, the only things you can take with you when you leave the physical world.\n\n\"When souls become intricately involved in the numerous rotations of incarnations, the learning can be great. A group or family of souls may become involved in planning out the challenges and general circumstances they hope to experience or overcome in the next lifetime. The lessons and roles they agree to play can be very harsh in the physical realities of Earth. But the soul knows and understands that the experience itself will be like a fleeting dream, the batting of an eyelash in the cosmic reality of nonlinear time.\n\n\"There is also great sacrifice by individual souls to help fulfill another's chosen lessons. Like your most recent Earth mother, they may agree to put themselves into a difficult lifetime to help facilitate the learning and experience of others. For those souls who wish to accelerate their growth and spiritual advancement, the quest for experience and knowledge is tremendous.\n\n\"The human experience is a very powerful lesson. With few exceptions, those who choose to incarnate make their way through hundreds, even thousands of lives. They will experience both male and female bodies. Most, at some point, will also become murderers, thieves, and vagabonds. They'll experience lives of happiness and productivity as well as lives filled with grief, despair, and human failure. Always seeking to improve, a soul might make significant gains in one life, only to fall and digress in the next. The paths are as many and varied as the number of souls participating.\n\n\"As you have heard and read, there are many who refer to life experiences as karma. They may refer to karma as a spiritual system of justice and balance. They believe that if a soul performed badly, perhaps causing injury or even death to others, that soul would be forced to endure the consequences of its actions and make amends. This is not necessarily true. No one is forced to do anything.\n\n\"From the physical point of view, karma can be considered with both positive and negative connotations. But in the larger spiritual sense it is neither good nor bad. It is merely experience. For example, human societies may view the actions of some souls as incurring negative or bad karma where an apparent wrong has been done to another. What if that same soul had actually agreed to perform the seemingly cruel act to assist another soul in its learning experience and development? In reality it may be that an act judged by human society to be cruel and hurtful was actually a great sacrifice by the soul carrying out the act. The soul may have agreed to provide the lesson or experience for others, all the while knowing that he would incur the wrath and punishment imposed by human laws.\n\n\"But, please do not misunderstand my message here. These types of laws that hold people accountable for their actions in Earth society are very necessary for the order and well-being of the physical world. In the spiritual world, retributions and punishments are not necessary. The responsibility for actions is readily accepted by souls in the course of their development.\n\n\"For example: a soul, having lived a life as a cruel and violent womanizer, may very well choose to return in a subsequent life as a woman who may become the victim of abuse. But, I should also make it very clear: a soul caught up in cycles as an abuser or abused should not accept this as their fate, but rather strive to conquer and triumph over these powerful lessons. The soul learns and experiences from both polarities, and the slow wheel of cause and effect in human endeavor continues to turn.\"\n\nMy mind was spinning. Meldor's words were profound and engaging. I only hoped that I would be able to remember and integrate the message in my physical world.\n\nI thought of another question. \"I can understand, Meldor, that it may be possible for two or more souls to plan on a relationship by being born into the world as another soul's child or as brothers and sisters. But, how could two souls plan to meet and become involved with each other as husband and wife? Given the freedom of choice by their parents and the many variables involved, how could they ever hope to overcome the tremendous odds against finding each other?\"\n\n\"Well,\" Meldor smiled, \"that's where we come in. Each and every soul that chooses to incarnate into the physical world has the help of one or more beings to guide and assist them towards achieving their goals. Guides and other helpers also work together in this realm to assist the individual souls with the lessons they hope to accomplish. This is by no means foolproof, and there is certainly no guarantee that the intended outcome will be achieved. Even as incarnated souls, you always have the freedom of choice, which could negate the possibilities, but the power of the guidance can be quite strong.\n\n\"Usually, guidance will appear in small, almost undetectable, subliminal ways. A simple thought or idea could be put into your mind, causing you to turn onto a certain street or to go into a certain store where you may encounter another specific soul or circumstance designed for your development. It could also involve working with totally unrelated people, causing them to take a certain action that would assist in bringing the two souls together. The possibilities and circumstances are endless. It's very interesting, enjoyable work.\"\n\n\"Jeez,\" I interjected. \"That's amazing. I never thought of it that way. That would certainly improve the possibilities for almost anything if you guys on the Other Side decided you wanted something to happen.\"\n\n\"Well, not exactly.\" Meldor grinned. \"We would never do anything to interfere with a soul's higher purpose or wishes. But you should understand that our assistance might also help to achieve what may be considered by human society as bad or destructive. If a soul, for example, wanted to experience a tragedy resulting in his body or brain being handicapped, we would also assist in providing the circumstances for the accident to happen.\n\n\"We realize that from a human's physical perspective, it might seem unthinkable or cruel that anyone would assist in what may seem like an unfortunate or needless tragedy. But, from a spiritual perspective, it is only the temporary physical body that is negatively affected. The soul is enriched and empowered by the experience.\n\n\"You may also be surprised to know that many, many souls voluntarily choose to incarnate into bodies that will be born deformed or handicapped, or will become handicapped due to accident or disease at some point during life. There is literally a waiting list of souls offering their love and energy to the experience itself, or to helping others in the experience. In view of the challenge, the opportunity for spiritual growth and development can be significant, not only for those handicapped but also for the parents, siblings, and caregivers.\"\n\nMy mind was bursting with new questions and ideas. I was about to ask yet another question on the relationship between guides and their loved ones, when my alarm-radio clicked on so abruptly it jolted me into a sitting position in my bed. Frustrated, I got up to start my day.\n\nI was astonished at some of the information and concepts Meldor had shared with me. Even though I had read a few books that tried to explain how things worked on the Other Side, I had never given them much credence. Now it was different. Meldor had planted a seed in my mind. I began to realize how little I knew about myself and everyone else around me.\n\n# 10\n\n# Exploration\n\n_And oh what joys will they evoke_\n\n_When God and angels first uncloak_\n\n_Life's greatest treasures gleaned at last_\n\n_Our hearts and souls in light are cast_\n\nAutumn of 1994 brought even more dramatic change to my life when I was elected mayor of our city. At the time, I had no idea how much this new duty would affect my life. While still operating my own business, the additional commitment left little time for leisure or family. My days were long and tiring. More often than not I would arrive home late and exhausted.\n\nAs the hectic pace of my life sped up, contact with Meldor and the spirit world waned. There weren't enough hours in the day. It became a rare occasion when I could slow my mind and body sufficiently to achieve a working altered state.\n\nInexorably, life's continuing demands relegated my quest for spiritual connection to the realm of dreams. On occasion, Meldor would visit me in my sleep, but dreams being what they are, most left me with only fleeting impressions. After more than a year of this maddening pace, I desperately needed a holiday to relax, and I couldn't think of a better place than Virginia.\n\nAlmost on cue, a letter arrived from The Monroe Institute inviting me to a newly developed program they called \"Exploration 27.\" Designed to carry participants to new levels of consciousness, up to and beyond the Park, it was just what the doctor ordered. As its name suggested, the program's goal was to explore new levels of consciousness to find and identify what they contained. It sounded too good to be true. I signed up for the first available seminar.\n\nAfter a chilly midwest winter, the unseasonable warmth of the Virginia air was a welcome relief. The beautiful, serene surroundings and familiar rustic buildings of The Monroe Institute welcomed me like an old friend. It was good to be back.\n\nOur explorer group consisted of seasoned veterans of the Institute's programs and research. There were new faces as well as old friends. Flocking around the trainers, we coyly probed for previews of what awaited us. However, in traditional Institute solidarity, they deflected our probing with a standard Bob Monroe response. \"You'll just have to wait and see for yourself.\"\n\nThe first day of the program was a familiar reconditioning to the numerous levels of Hemi-Sync. The process never ceased to amaze me. I loved the sensations the frequencies generated and quickly settled into the routine. Lying in bed, I took note of the changes slowly creeping through my body. It always seemed to start in my legs with an intense vibration tickling the soles of my feet. From there it would slowly move up through my ankles and into my calves. The electrical sensation riding the front of this wave always seemed to trigger small jolts and spasms in my larger muscles as it moved along. In its wake, a peculiar numbness and rigidity took hold. I would feel my muscles tightening, becoming immobile, and within a few moments all physical feeling would cease. The process usually took about ten minutes to work its way through to the top of my head. My body finally asleep, my mind would remain awake and alert.\n\nAs it turned out, that first morning was not exceptional. I slid smoothly into a relaxed altered state, but nothing more happened. Through several sessions I just lay in the womblike darkness trying to guide myself into some meaningful experience, but to no avail. I called on Meldor and anyone else who might be around to lend a hand, but still nothing happened.\n\nPerhaps I was trying too hard. This had happened many times before, and it was frustrating. I took note of the incessant chatter going on in my mind. It was that little voice we all have, the one that originates within the left hemisphere of the brain, trying to analyze the hell out of everything. I would have to figure out how to shut off all the mental noise.\n\nBy the last session of the day I was becoming perturbed. Finally, I decided to let go of the wanting. I quit trying to control the process and allowed my mind to drift to wherever it would take me. Fifteen minutes into the session, I started clicking in and out of consciousness, my mind finally free.\n\nDrifting closer to the edge of sleep, I felt a sudden falling sensation in the pit of my stomach. After a few moments the motion stopped and I found myself floating in a deep void. Enveloped in darkness, I hadn't the slightest inkling of where I was. Well, if we're supposed to be here for exploration, I thought, I'd better check this place out.\n\nPushing forward through the blackness, I strained my senses trying to perceive whatever lay ahead. Minute after minute I continued on, but there seemed to be nothing, no sights and no sounds, only darkness.\n\nSuddenly, from out of nowhere, the wrinkled face of a gray, longhaired old man popped up in front of me. \"BOO!\" he screamed. And then waiting just long enough to see my startled reaction, he disappeared, howling with laughter.\n\nI'm sure my heart quit.\n\nThe old guy, however, was practically splitting his gut. It sounded like he was rolling around on the floor.\n\n\"That'll sure git yer heart a-goin',\" he yelped, sounding every bit like Gabby Hayes in an old Western.\n\nI was so shaken it took me a few moments to get my wits back. Meantime, the old boy continued to howl like it was the funniest thing he had ever seen. Listening to his infectious giggling, I couldn't help but laugh myself. When he finally began to move away, I could still hear him cackling as he headed off into the distance.\n\n\"You crazy old bastard!\" I yelled after him. \"If I catch you, I'll kick your wrinkled old ass!\" This only brought new howls of laughter as he continued his invisible retreat. Eventually his cackling faded out, and I was left chuckling to myself.\n\nI couldn't believe what had happened. Here I was, hovering in the middle of God-knows-where, it was blacker than the ace of spades, and some old guy's spirit had just earned a great laugh at my expense. I looked around warily, but couldn't perceive a thing. After a couple more minutes, I decided that it would probably be wise to get my butt out of there before somebody else showed up for a laugh. Focusing on my body, I felt a flurry of movement and a moment later I was back in my bed.\n\nBefore I could get the headphones off I was laughing. The crazy old bugger had practically scared the snot out of me. I wished I knew where I had been. Who was that crazy old devil? And why was he the only person I had encountered? I began to wonder if Meldor had anything to do with it. I wouldn't have put it past him.\n\nThe following day saw a big improvement in my perceptual abilities, and I could feel the energy building inside me. We were being introduced to new frequencies that would take us beyond the Park into a special meeting place. The routine was the same for each excursion. We would all try to home in on a certain area of the Park, gathering there, before heading off individually to explore other areas and levels of consciousness.\n\nDuring the first few sessions I was having difficulty connecting with the other members of the group. I wasn't, however, the only one having a problem, so I didn't worry about it. We would be naturally drawn to wherever it was that our spirits needed to go. Sometimes some of us didn't seem to go anywhere at all.\n\nAlthough I occasionally sensed his energy, I hadn't yet met up with Meldor. I wasn't particularly concerned because I knew that he would be there whenever I really needed him.\n\nDuring one of the morning sessions, I was having a great deal of difficulty staying focused on our destination. Believe me, during an altered state, this can be a significant problem. The spirit-mind has the ability to move so quickly, but it can also become bored when nothing is happening. You can sometimes find yourself heading off in unwanted directions, triggered by a mere thought.\n\nStretched out in bed, I tried concentrating on the meeting place, but with little success. My mind kept wandering. I thought about home and some silly little joke my son, Dave, had told me. Wondering how he was doing with his school project, I unexpectedly found myself hovering over the street in front of his school, some three thousand miles away.\n\nA group of kids were heading in through the front doors, but my attention was drawn to one blond-haired boy. Even though he faced away from me, I knew it was Dave. But he was wearing a jacket I hadn't seen before. Striped in blue, green, and purple, it had the strangest combination of colors. I thought, Well, maybe it's not Dave. He wouldn't wear such an ugly jacket. A moment later, I was back in my body in Virginia.\n\nI hadn't the time, though, to give much more than a passing thought to what I'd just witnessed. Steady vibrations rippled through my body, and I was set to head off on another adventure.\n\nThere was an abrupt spinning motion, and seconds later I found myself standing in front of an enormous building. It looked like something out of ancient Rome or Greece. Built of marble or granite, it must have been at least fifty feet high. Huge pillars spanned the front of the imposing structure and an immense stairway led up to a large arched entry. Throngs of people moved in and out of the building while a boisterous group of children played on the steps. I noticed that only two or three of the children were dressed in modern clothing. The rest wore plain, loose-fitting toga-style robes. A sudden pang of paranoia gripped me, and for a moment I panicked. Quickly glancing down to make sure I hadn't shown up in my birthday suit, I let out a sigh of relief. Thankfully I was clothed in the same tan-colored robe I usually traveled in.\n\nFeeling more secure, I clambered up the wide steps to the massive landing. Everyone seemed happy. Small clusters of adults cheerfully greeted me as I made my way through the arched entrance and into the foyer.\n\nThe beauty of the interior was breathtaking. Ornate pillars drew my eyes upwards to astonishing domed ceilings. Flawlessly painted in flowing murals, the intricate images were stunning. Perhaps this was where Michelangelo perfected his art. The paintings seemed to move and flow with a life of their own. I had never seen anything so exquisite. But, somehow, it all seemed oddly familiar, like I had been there before.\n\nI was so entranced, I hardly noticed a woman clearing her throat in front of me. She had been patiently trying to get my attention. Lowering my eyes from the ceiling I realized that my mouth was hanging open.\n\n\"Hi, Paul. My name is Larissa,\" she warmly greeted me. \"We've been waiting for you.\"\n\nShe was a tall, elegant woman in her mid-thirties. Her long dark hair was swept back from her face, revealing a soft creamy complexion and the most extraordinary blue green eyes. A white braided sash tied around her waist accented a loose fitting, light blue, floor-length gown. She emanated a wonderful air of happiness and love.\n\n\"You've been waiting for me?\" I stuttered.\n\n\"Oh, most certainly,\" she smiled. \"We knew you'd be arriving and you're right on time. Come, I'll show you around.\"\n\n\"What is this place?\" I asked.\n\n\"This is our library and Hall of Records,\" she responded as we began walking along a huge corridor. On either side were row after row of shelves filled with books and odd cylinders. It occurred to me that the cylinders might contain scrolls.\n\n\"Yes, you're right.\" She read my thoughts. \"We have every mode of communication stored here. There are other beautiful libraries on this level, but this one contains the life records of all incarnated souls. Energy imprints of all sentient beings are constantly gathered and recorded here. Nothing is ever missed. You could, I suppose, call this a living library. It's constantly growing, changing, and evolving.\"\n\nAs Larissa spoke, a strong feeling of deja vu came over me. Somehow, this felt uncannily familiar. \"Have I been here before?\" I asked.\n\n\"Oh, yes. Many times,\" she laughed. \"You come here often between lives to study the records, to review what you've learned, and to plan for your next incarnation. Everybody does. This is an important step in your continuing development.\"\n\nAs we continued wandering through the aisles I noticed for the first time that my charming guide wasn't wearing shoes. I'm not sure why this struck me as odd, but looking around I realized that nobody else wore shoes, and neither did I.\n\nThe building was massive and appeared much larger inside than out. As we passed through the maze of shelves, aisles, and adjacent reading rooms, I caught sight of two other members of our group. My friend Bruce was leaning over a desk in a tiny alcove chatting with a guide, while Robyn, a most wonderful, gentle soul, scanned the titles of several books at the end of the row. Noticing me, she smiled in recognition. I was thrilled to see them. It would be great to compare notes later to verify that we had shared the same experience.\n\n\"Would you like to see your book?\" Larissa's soft voice interrupted my contemplation.\n\n\"My book . . . ?\" I asked. \"Yes, I guess so.\" For some reason I felt apprehensive. What would the book reveal? What would it say about me? Who had written it? I wasn't so sure I wanted to know.\n\n\"Oh, please don't worry,\" she whispered with a reassuring smile. \"You wrote every bit of it yourself. You didn't sit down and write actual words for it, but you provided all the details through the energy of your existence. Everything that you do, say, feel, or even think is an exchange of energy. These life-force imprints are recorded and stored at this level so you can track them and learn from your experience. Every nuance of energy exchange in the astral and lower vibrations is stored here. There are also many higher levels of vibration, but they have their own repositories to record and store their energy impressions.\n\n\"Aahh, here we are.\" Larissa stopped near the end of an aisle, reached up and withdrew a book from the shelf. The manuscript itself seemed fairly thick as books go, but I couldn't imagine how it could possibly hold all the details of my life.\n\n\"Oh, it doesn't just contain the information from your present life,\" she said. \"It holds the story of every incarnation you've experienced on this level, and there are many hundreds of them. It also details your time between lives. It's all here.\"\n\nWith that, Larissa led me to a nearby alcove. Inviting me to take a chair, she placed the book on a pedestal in the center of a small round table, and randomly opened it near the middle. Soon a shimmering, tubular beam of light appeared above the pages. Although transparent, the light\u2014or energy field-\u2014had a bluish tinge to it. It reminded me of a miniature version of the transporter beam from _Star Trek_. The field was about 16 inches in diameter and about three feet high.\n\nAmazed, I watched closely as a lifelike, three-dimensional moving image took shape inside the field. A dark-skinned, dark-haired woman, clothed in hardly more than animal skins, sat cross-legged in front of a small shelter nursing a newborn baby. Somehow, I knew I was that baby.\n\nI felt the power of that moment, the peaceful, intimate connection, the soothing warmth of my mother. Emotions and feelings swirled through my mind like a long forgotten memory.\n\nI would have continued the experience indefinitely, but the light touch of Larissa's hand on my shoulder brought me back to the present. Looking up at her smiling face, I saw in those beautiful eyes the reflection of my own delight and amazement in the timeless splendor of our existence. Flashing a warm, understanding smile, she flipped the pages to another section.\n\nI returned my attention to the energy field. The image of a dirty, white-skinned young man with a filthy mop of long brown hair began to take shape. He lay shivering under a small dirty blanket and heaps of rubbish in what appeared to be a dark, dank corner of some old abandoned building. As I looked more closely it appeared that he was quite young, possibly only 13 or 14. Coughing violently, he seemed to be burning with fever. Slowly, sad recognition came over me. I knew he was dying, and I knew this boy was me, in another lifetime, another place.\n\nI didn't want to become involved in the scene I was witnessing. I didn't want to feel the resignation and grief already weighing on me. Turning away from the images, I wished Larissa would flip to a new page. Again I felt her hand on my shoulder, but this time it was to comfort and soothe. With her other hand, she closed the book and withdrew it from the pedestal. The energy field immediately dissolved, leaving me sitting in contemplation. After several moments, Larissa gently interrupted my reverie. \"It's time to go now,\" she whispered. \"You can come back any time you like.\"\n\nSmiling sweetly, she led me from the table. Hand-in-hand, we walked back to the shelf where she returned the book to its place. As we strolled along, I began to feel a tug on my awareness. The familiar pull of the session-ending beta tones became more pronounced. Larissa clasped my hand between both of hers. \"It was so nice to see you again,\" she offered cheerily. \"I look forward to the next time.\" I felt a small jolt to my consciousness and her image began to fade. Although I fought to stay focused, it was a losing battle. The next thing I knew, I was back in my body.\n\nSitting up, I removed the headphones. The room seemed intensely hot, and I was sweating profusely. This had never happened before. If anything, I usually felt a bit chilly after a session. Worried that I might be coming down with something, I headed to the bathroom to wash my face. By the time I returned, my body heat had normalized, and the temperature in the room had subsided. I struggled for an explanation. Could this have been a side effect of watching myself burning with fever as a boy in a past existence?\n\nReplaying the vivid scenes in my mind, I sat on the edge of my bed making notes. The skeptical, analytical part of me still wanted to put everything in order. I tried to determine the culture and time period of the image of myself as a baby. My mother's skin color had been dark. The large leafed branches covering the shelter seemed tropical. I settled on Polynesian or Philippine culture, but hadn't a clue about the era.\n\nThe images of myself as a boy slowly succumbing to death, also returned to my mind. The boy's fair skin, along with the musty, old, decaying stone walls, reminded me of Europe. England, I thought. Perhaps Scotland. Maybe Ireland. Here again, I had no idea what time period this might have been. Finishing up my notes, I headed downstairs to join the group for debriefing.\n\nSome of my fellow participants had also enjoyed vivid, powerful experiences in the same library-type building. A few had visited other places of note, while some hadn't had any luck at all. Most important for me, however, was the unsolicited confirmation by Robyn. She had indeed seen me in the library. As I walked into the debriefing room, she was already seated. With a mischievous smile, she winked at me. \"Hey Paul, who was the good-looking woman with you in the library?\" Bruce, on the other hand, couldn't recall having seen me, but he and Robyn had seen each other. The validation was both interesting and reassuring.\n\nSuccess during the remaining sessions continued to be a hit and miss proposition. Sometimes I would fall asleep or end up spending my time staring into the darkness. Then, in the next one, I'd be off on a new adventure. There didn't seem to be any rhyme or reason to it.\n\nI was beginning to wonder about Meldor. Three days had gone by and I hadn't seen him. Hoping that he wasn't off on vacation, I vowed that at the next opportunity I would try to find him.\n\nWhen the late afternoon session began, I plunged rapidly through the threshold into an altered state. As the energy in my body began to oscillate, I deliberated whether to send out a call for Meldor to join me or to attempt to travel to him. Deciding that it might be fun to track him down for a surprise visit, I concentrated on his familiar vibration. Almost instantly a wave of energy plowed through my body. In a heartbeat I was standing in front of him.\n\nI couldn't believe my eyes. I was standing on the peak of a high, desolate mountain. Meldor sat before me in an oversized chair, like the kind we sometimes saw in hotel lobbies years ago. It was so large, his feet hardly reached the front of the cushion. He looked every bit like a three-year-old in an adult chair. A childlike scowl knitted his bearded face, and with hands clasped across his lap he vigorously twirled his thumbs.\n\n\"What took you so long?\" he snapped.\n\nI didn't know what to think. He looked annoyed, but at the same time hilariously silly in the gigantic chair. I had no idea whether he was serious or not. Nervously I began to search for words. \"Well, I aahh ... I thought that... aahh ...\"\n\n\"Ha, ha,\" he interrupted. And in the wee small voice of a three-year-old, he taunted, \"Bet you can't sneak up on me.\"\n\nEyes twinkling, an impish smile spread across Meldor's face. He swung his legs over the edge of the chair and slid to the ground.\n\n\"Jeez, Meldor,\" I stammered. \"You are some piece of work.\"\n\n\"Hey, my boy, you gotta lighten up. This is funny stuff,\" he chortled, reaching out to hug me. \"You should've seen the look on your face.\"\n\n\"It's good to see you,\" I grinned, returning his embrace.\n\nWhat a comedian, I thought, looking around. The wind-blown rocky terrain, wisps of cloud floating by in the thin air, and this huge chair on a mountain peak in the middle of nowhere. The guy had style!\n\n\"What on earth are you doing out here on a mountain?\" I asked.\n\n\"Oh, just waiting for you,\" he grinned. Then, pointing to the picturesque surrounding valley, his tone softened, becoming reflective, appreciative. \"Isn't it a great view? A perfect place for contemplation, don't you think?\"\n\nHe was right. The ambiance of the landscape was exhilarating, yet peaceful. The only thing out of place was the chair.\n\n\"What's with the big old chair?\" I asked.\n\n\"What chair?\" Meldor's eyes were beaming.\n\nI turned to look, but there was nothing behind us but the barren mountaintop.\n\n\"The chair was just a creation for your amusement,\" Meldor explained. \"It was a manifestation of my thoughts. I put it there to show you that in the realms of pure energy, a thought can be real and easily manifested.\"\n\n\"I've been wondering about that,\" I admitted. \"I've seen so many things in different realms. Sometimes it's hard to tell what's real and what isn't.\"\n\n\"In a sense it's all real,\" Meldor responded. \"In the spirit realm, a person can create their own reality, including what you would perceive to be actual physical matter. It is the density of the matter and the rate of vibration that is relative to one's perception. On the Earth plane, where matter is extremely dense and the rate of vibration is slow, physical beings have a comparative perception of reality. A rock is real. It's hard, heavy, and relatively unchanging in terms of Earth time. So from a human perspective it's easy to form concepts that material things are permanent and immutable. But in the spirit realms, the concept of permanence is much different.\"\n\n\"I'm not sure I understand what you mean,\" I said, looking for a place to sit. I chose a bare outcropping of rock. Meldor found a place beside me and continued his explanation.\n\n\"Take, for example, the chair I was enjoying when you arrived,\" he said. \"It was completely real to your perception, was it not? You could have crawled onto it and sat as I did. It was very real, but at the same time it was only a temporary construction of my mind; whereas the library building you visited yesterday was much more permanent and substantial. It too, however, was created by thought. But, it is reinforced by the conscious energy of billions of souls. Each and every visitor to the building maintains and perpetually reinforces the reality of the structure.\"\n\n\"You knew I was at the Records Hall?\" I asked.\n\n\"Yes, of course. I was there with you.\"\n\n\"But I didn't see you. Why didn't you let me know you were there?\"\n\n\"It was better for you to have the experience by yourself, without my influence or interpretation. Your evolution is more complete through different perspectives.\"\n\n\"The building was phenomenal,\" I recalled. \"It reminded me of pictures I've seen of some of the ancient buildings in Rome and Greece.\"\n\n\"A good comparison,\" Meldor agreed. \"It's one of the places from which the builders and architects of your ancient civilizations developed their skills. They built these beautiful structures in the spirit realm before they returned to build them on Earth. It is often in the spirit realm that creation and construction concepts are developed and honed to perfection. But it is a much greater challenge to create and build on the dense Earth plane.\n\n\"As you know, in the lower vibrations, thoughts, in and of themselves, are not so easily manifested without the necessary physical actions needed to bring ideas into reality. On Earth, for example, before a builder of stone is able to bring his creative concepts into reality, he must first create the tools with which to shape the stone as well as bring together the proper mixture of available raw materials to create the mortar.\"\n\nAs Meldor spoke, I closed my eyes to absorb his message. In this manner his communication was more comprehensive and thorough. He spoke directly from his mind to mine.\n\n\"There are many millions of talented spirits with such an intense love of design and innovation, they constantly work at improving their creations both in the spirit realm and in the physical worlds. While in the spirit plane, they continue their endeavors, developing new ideas, solutions and inventions. It's much easier for them to work from this level where thoughts can be manifested instantly. The more difficult part is in transferring their ideas and knowledge to the physical planes. Many accomplish this through the slower process of reincarnation.\n\n\"But there are countless souls who choose instead to transmit their ideas and inventions directly to the minds of receptive physical beings. There are great numbers of creative people on Earth who make these connections, receiving inspiration during sleep or in altered states. They may not remember, or even be aware of their journeys, but the impressions stay with them. This is one reason why people in different parts of the world often seem to come up with the same ideas or patents at virtually the same time. They have been guided.\"\n\nMeldor's revelations made a great deal of sense to me. In my travels to the Other Side I had witnessed many amazing things and was slowly beginning to understand the larger picture. There was no question. We are all connected.\n\nI sat quietly for a few moments before opening my eyes. With the exception of the valleys far below, the mountain on which we sat was relatively barren. I wondered where we were.\n\nMeldor broke into my thoughts. \"I believe you would call it the Himalayas.\"\n\n\"You mean we're on the Earth plane?\" I asked. \"How did you make the chair appear in the physical world? I didn't think that was possible.\"\n\n\"Oh, it's certainly possible,\" Meldor replied. \"But don't forget, at this moment you are not in your physical body. You're in an elevated level of consciousness where these manifestations are visible and real.\"\n\nFor a few moments Meldor sat quietly, reflecting. \"I still come here occasionally,\" he mused. \"This used to be a favorite place of mine on a couple of previous incarnations. I always found the energy here very relaxing.\"\n\nI was on the verge of asking Meldor to tell me about his lifetimes on Earth when, without warning, I slammed back into my body. I couldn't imagine what had happened.\n\nA sharp, heavy object was pressing into my left temple. Somewhat disoriented, it took me a few moments to figure it out. Before the session began, I had been resting in bed reading a book. When we were ready to begin, I had simply set the book aside on an extra pillow beside my head. Somehow, it had slipped off the pillow, and banging into my face, jolted me out of my altered state.\n\nAlthough the session was still underway, I didn't think I'd be able to reconnect with my guide, so I removed my headphones and stretched out. I thought about Meldor's explanations of the creation of buildings and structures on the Other Side, and their relationship to our physical world. The structures that I had seen, such as the library, seemed absolutely real. The stone steps leading to the front entry were as solid as anything I had experienced. Inside the building, I had felt the smooth hardness of the beautiful marble floor beneath my feet. I had touched the solid ornate pillars. I had sat upon the bench with my elbows resting on a firm table as I watched the past life images in the energy field. These things were real, perhaps more real than the structures I was used to on Earth.\n\nAs I lay in my bunk pondering the creative possibilities in the spirit realms, a pressure began to build in the center of my forehead. I was again falling into an altered state.\n\nFocusing my attention on the spot between my eyes, I concentrated on opening the portal. Soon a small pinpoint of light appeared. But, before I could even react, the aperture flashed open and a moment later I found myself peering down the hallway of what seemed to be a very long building. Numerous doorways led off to both sides of a wide corridor. The walls themselves emanated a soft, warm green light. White tile floors reflected spotlessly. It had the feel of a hospital.\n\nTo my left, a wide-open doorway displayed the number 17 above it. I wondered if there was any significance attached to the number, but more importantly, I questioned why I had been drawn to this particular place. It occurred to me that perhaps this was the kind of hospital room my mother had arrived in after making her way to the Other Side. Moving anxiously through the doorway, I half expected to see my mother sitting up in bed waiting for me. To my disappointment, there was no one there. Feeling a small twinge of loneliness, I slowly surveyed the room. It was indeed a hospital.\n\nThe whole room was bathed in a soft, warm light. Against an adjacent wall stood a standard hospital-type bed, made up in soft white linens. A couple of white chairs had been placed on either side of the bed. What struck me as odd, though, was the absence of any medical equipment. But when I thought about it, it seemed reasonable. There was no need for medical equipment here. There were no physical bodies to heal. Hospitals in this realm served only to provide a comfortable transition for those souls who may have died in an earthly hospital environment. This was a natural, peaceful place for newly arrived souls to restore their energy until they were reoriented to the Other Side.\n\nOn the right side of the bed, a vase filled with beautiful flowers adorned a small round table. Exotic and vibrant, the flowers filled the room with a wonderful fragrance. Enthralled by their form and texture, I carefully examined several of them. They were like nothing I had seen before.\n\nA warm breeze drew my attention to a large, partially opened window. Moving across the floor, I peered out. Immediately outside the room was a beautiful garden overflowing with flowers and plants so vibrantly colorful they defied description. The plot extended for what seemed like hundreds of yards, finally giving way to the rich green grass of a large valley. A small blue stream meandered through the gorge into a background of bright green forest crowned with snow-covered mountain peaks. I knew this had to be an area of the Park.\n\nWith a sigh of appreciation, I finally withdrew from the window and turned back into the room. I could certainly understand how wonderful and peaceful it would be to wake up in this environment.\n\nI wondered about the rest of the building, and whether there might be other buildings in the area. Heading for the doorway, I felt a swirl of energy as I passed through the opening. A moment later I was standing in a beautiful, high-walled, cathedral-style building.\n\nIt was massive. At first impression, it seemed like a cathedral, but it was different from any church I remembered. \"Remembered\" was the key word. As I stood, absorbing the images, a distinct feeling of recognition came over me. I had definitely been there before. Intuitively I knew that this structure was for revitalizing and tuning soul energy.\n\nAlthough enclosed by high, sturdy, perimeter walls, there didn't appear to be a ceiling over the main structure. Throughout the long, open, marble corridors there were an inestimable number of pillared structures looking a lot like tall gazebos. Approximately 12 feet in diameter and 20 feet high, they consisted of six marble pillars placed in a circle, supporting a beautiful domed ceiling.\n\nThe center of each dome contained a massive multifaceted crystal at its peak. Having the appearance of a perfectly cut diamond, each one was at least four feet in diameter. In the shape of a child's toy top, they had been set upside-down into the roof structure. The crystal peaks protruded through the tops of the domes, while their convex bottoms formed the center of the inner ceiling. The crystals, or whatever they were, were dazzling. Every one of the hundreds of facets seemed to reflect a different color and vibration. On the floor directly below each crystal stood a large round bed, about two feet high and eight feet in diameter.\n\nFrom my position about 20 feet from one of these structures, I noticed three spirits approaching. A middle-aged man and woman seemed to be giving instructions to a younger man of about 20. I watched intentiy as the youth, dressed in blue jeans and t-shirt, crawled to the center of the bed and lay down. I could sense his uncertainty as he cautiously prayed to God for healing energy and then nervously prepared himself to receive. Within moments, a dazzling field of shimmering, multicolored energy cascaded from the crystal, flooding the area below, permeating his body.\n\nWith a look of surprise and relief the young man appeared nearly overwhelmed in the bliss of the experience. So intense was the energy infusion, that after a minute or so in ecstasy he floated off the bed. His body began to shimmer and fade from view until only an energy outline remained. Within the form of his now translucent body I began to notice a row of bright, glowing spheres of light. About the size of tennis balls, they seemed to radiate a swirling power. Could these be the chakras, I wondered? Were these the energy centers that are supposedly contained in every living body? Ever so slowly, the luminous orbs continued to intensify and expand until they finally enveloped each other, becoming a single, glowing oval of energy.\n\nThe entire spectacle was mesmerizing. The beam from the crystal had begun a process of shirting colors and intensity. Every thirty seconds or so it would blend from one color to another until it had gone through every hue of the rainbow. Finally, having made its way through the color spectrum, the beam subsided, softening to a warm golden hue. The young man's body, now a radiant sphere of energy, slowly descended to the bed below.\n\nI had been so engrossed in the scene that I hadn't noticed that the middle-aged couple had left. I glanced around but couldn't see them anywhere.\n\nWith the process apparently now complete, the crystal energy began to fade and then disappear. The rejuvenated soul lifted off the bed and floated gently to the marble floor. As he landed, he was joined by two other light-beings who appeared out of nowhere. A moment later, all three vanished.\n\nFascinated, I wanted badly to give the crystal a try. However, not being so sure of the effects it might have on someone still occupying a physical body, I decided to let it go for the time being. Perhaps, I thought, Meldor might be able to advise me on it later.\n\nI had no idea how much time had passed back in the physical world. I remembered that I had removed my headphones, so there was no way to know if the session was over. I decided I'd better get back into my body, and a moment later I was.\n\nThe speakers were silent. The session had ended. I got up, made a few notes in my journal and headed downstairs. The debriefing session had long since broken up. Most of my fellow explorers were already gathering for dinner.\n\nAs the last day of the program wound down, a group of us had been discussing the various forms of communication employed on the Other Side. I had already experienced several of these methods. When I first met my guides, it seemed that our conversation was completely normal in the physical sense. Their lips moved, and I heard their voices just as I would in everyday conversation. I have since learned, however, that the sound of their voices was perceived only in my mind.\n\nI had also experienced the speedier process of communicating by thought transference alone. I simply projected my questions or comments and received a telepathic response, hearing and feeling the other person's thoughts in my mind.\n\nThe most powerful means of communication, though, was the method my mother had shown me. I heard, felt, and saw everything she had experienced. It was a process of thought transference containing a great deal of information all at once. Robert Monroe referred to this type of telepathic communication as a \"rote\" or a \"thought ball.\"\n\nIt seemed to me, however, that I still had some major unanswered questions regarding communication in the spirit realm. Knowing that people of all cultures and languages were able to communicate on the Other Side, I wondered about the process that allows for this apparent translation of languages. I was aware that whenever I thought about something, my thoughts were in English. But, there had to be thousands of different languages and dialects on our planet alone. How, for example, could a Spanish-speaking spirit understand what I was talking about? I resolved that the next time I connected with Meldor, I would ask him.\n\nAs we settled in for our final session of the week, I hoped to connect with my guide. Within just a few minutes, my racing mind had dropped through the threshold of normal consciousness into a peaceful altered state. Projecting a mental message to Meldor, I asked him to join me, and then waited. Several minutes passed. I repeated the message a number of times, but still there was nothing but darkness. Quickly becoming bored, I found myself drifting towards sleep. All of a sudden, a loud commanding voice startled me into awareness. \"Go deeper!\" It was Meldor.\n\nI concentrated again on following the Hemi-Sync signals into a lower level of consciousness. But it felt like I was already in synch with the frequency, and it was at its maximum. How could I go deeper? The only solution, I thought, was to leave it in Meldor's hands. Carefully, I turned off the headphones and relaxed into the silence.\n\nMinutes passed, and ever so gently I started to feel an unfamiliar energy building within me. Soon, I began to experience a duality of sorts. Still acutely aware of being in my body, I felt as if I was experiencing myself in an abstract manner from another place or dimension. Brief flashes of images and colors began to pierce my awareness, too brief, however, for me to make any sense of them. I could feel my energy changing, evolving to something I had never before experienced. I knew I was entering a new field of consciousness.\n\n\"In order to provide meaningful answers to your questions,\" Meldor's voice jolted me to attention, \"my assistants and I have decided to bring your perception to a higher level than you are accustomed to. It is from this perspective, while your consciousness is still within your physical body, that we hope to be able to explain and show you the sometimes difficult-to-understand concepts of all-knowing awareness.\n\n\"As we have relayed to you previously, the energy of all consciousness is connected. Separateness is merely an illusion of the human experience. In the higher realms, there are no barriers to the communication of sentient consciousness. From infinite collections of like-minded souls, through numerous levels of expanded awareness, to the ultimate level of God-Realized beings, the energy of thought is the same for all.\n\n\"At its purest level, thought is merely directed conceptual emotion. Interpretation or translation is not necessary because in the spirit realms it is universal and automatic. In as much as the principals of mathematics are universal and applicable to all matter and energy, so too is the relationship of consciousness between sentient beings. Pure communication is, in a sense, an expression of the mathematics of conscious awareness.\"\n\nMeldor's words burnt into my mind, but I wasn't at all sure that I understood their meaning. In my physical world, I was far from academically astute, and when it came to mathematics\u2014I had spent my entire life severely handicapped. Engrossed in Meldor's dissertation, I lacked the presence of mind to ask a question or to even respond. I knew this was important information, but I could only hope to remember half of it when I returned to physical awareness.\n\n\"The communication of consciousness,\" Meldor continued, \"in its more universal application, is the creative expression of the energy of light and the vibration of sound. It is pure energy, and therefore embodies the elements of wave and frequency. Through these constant principles, beings of all planets and universes are able to communicate at higher levels without confusion. We are all connected and related to each other.\"\n\nThe realization of what Meldor was saying gripped my attention like a vice. Had I heard right? Had he actually said there were intelligent beings on other planets and universes? I didn't have to phrase the question; Meldor had picked up my thoughts.\n\n\"Most certainly,\" he responded. \"There are infinitely more sentient beings throughout the cosmos than your world can even imagine. But, as we have discussed before, we are all part of the same collective consciousness.\"\n\nMy mind raced with the possibilities. I'm not sure why this was having such a profound effect on me. I had always believed there had to be other intelligent beings in our universe. But now Meldor's confirmation made it all the more plausible and real. Trying to put this all together in my mind, it occurred to me that if all beings throughout the cosmos were part of the collective consciousness, then I should be changing my perception of God to an infinitely more awesome power. Again, Meldor knew my thoughts and responded.\n\n\"The power of _The All_ is beyond ordinary comprehension or imagination. Only God-Realized beings can truly know the magnitude of the massive collective energy that is _The All.\"_\n\nGetting back to his original message, Meldor continued. \"Although, in your present state you may not fully grasp the potential or complexity of this higher form of communication, we feel that you may benefit from a new experience.\"\n\nUntil this point, I had been in darkness, earnestly listening to Meldor's words. But now, a small, single point of blue light appeared in the upper left hand corner of the dark space in front of me. As I watched, the speck of light began to oscillate and started moving across my field of vision towards the right. Like a fine laser, it flowed in a slow wave motion, tracing its path with a continuous trail of light. Soon, a purple beam appeared just below the first, beginning its flowing journey across the darkness. Almost immediately, the two lasers were joined by one after another, each a different color, until the space before me was filled with wave upon wave of beautiful, oscillating colors.\n\nAs I watched in fascination, the whole thing started over again at the top, working its way down. But this time each wave of color came accompanied by a sound, a distinct pure tone. Every color vibrated its own tone. In increasing succession, wave after wave of light and sound were flawlessly added to the mixture, every tone in harmony with the previous one.\n\nTo my increasing delight, each tone seemed to elicit a different feeling and vibration in my body. Within moments, the area around me was filled with a gyrating spectacle of light, sound, and vibration. As I grew more absorbed, it increased in complexity and richness. Before long it had built into a living symphony for the senses. For some reason I couldn't help thinking about the spaceship communication scene from _Close Encounters of the Third Kind_.\n\nI had no idea what it all meant or why it was affecting me so dramatically. I felt like I was riding on an emotional roller coaster. Depending on the color, sound, and speed at which it moved I could be carried from absolute joy to deep emotional sadness. As I delighted in flowing with the vibrations, colors, and emotions, time passed unnoticed, until the montage came to an abrupt stop and I found myself in the silent darkness of my CHEC unit. Meldor was gone. The vibrations were gone. The session had ended.\n\nI lay in my bunk for at least another twenty minutes, going over the things I had seen and heard. It had been another remarkable week. I had learned a great deal more about who we are and where we came from. Silently, I offered a prayer of thanks for the wonderful gifts I had been given. And I thanked Meldor for his love and guidance.\n\nThe trip home was long and uneventful, but it gave me time to contemplate the new knowledge I'd absorbed. More importantly, it gave me time to wind down my energy before I plunged back into the demanding world of business and politics.\n\nWhen I finally arrived home, Candace and I sat at our kitchen table as I filled her in on the great experiences I'd had. Listening intently, she eagerly asked questions while I recalled the significant events. I almost forgot to mention, though, that I had zipped home one day while out-of-body and watched a boy I thought was Dave heading into the school. I allowed however, that it might not have been Dave because this kid was wearing a weird-colored jacket I hadn't seen before.\n\nWith an incredulous look on her face, Candace asked what the jacket looked like. I described it as best I could. Without saying a word, she abruptly got up from the table and left the room. A few moments later she reappeared from the back closet carrying a jacket. \"Is this it?\" she asked. I didn't have to say a word; the look on my face said it all. Without a doubt it was the same ugly jacket. Shaking her head in amazement she added, \"We bought it while you were gone.\"\n\n# 11\n\n# _Our journey Together_\n\n_The message comes from Heaven clear_\n\n_His wrath was never lent to fear_\n\n_You see, God has only love to give_\n\n_It's man distorts the way we live_\n\nSome of the strangest mysteries in life are oftentimes those fleeting, innocuous synchronicities that seem to be just beyond our grasp and understanding. For example, how is it that we may be meeting someone for the first time, yet feel as though we had known him or her before? Isn't it odd how, on occasion, we can be instantly attracted to some people and inexplicably repulsed by others? Could it be that we have, in fact, known them before, in another time and place?\n\nLike most everyone else, I occasionally noticed these anomalies, but usually didn't give them more than a passing thought. Given what I have learned about the reality of reincarnation, these intuitive recognitions of other people now make a lot more sense. If most of us have lived hundreds of lives, chances are we'll run into people we've known previously, some who have affected us positively and others negatively. Although I was gaining a clearer understanding of how things really worked, my next contact with Meldor would open my mind to the intriguing realities of the tremendous support group we all have on the Other Side.\n\nSeveral weeks following my return from Virginia, I had gone to bed earlier than usual to get a good rest before an intense series of meetings the following day. As I approached the borderland of sleep, a distinct popping sound captured my attention. I could feel the presence of someone else in the room. Expecting to see my wife sleepily trudging to the bathroom, when I opened my eyes, I was surprised to see Meldor floating quietly beside the bed. Sitting with his legs crossed lotus style, he was hovering about three feet off the floor, looking like the Dalai Lama in meditation. Intrigued, I watched in silence as he continued to hover without opening his eyes or saying a word. After a couple of long minutes, I could no longer stand the suspense. I had to find out what he was doing.\n\n\"Pssst, Meldor,\" I whispered. \"What are you doing?\"\n\nOpening one eye, he looked at me briefly and then closed it again. Another few moments passed. Then, as if resigned to the interruption, he sighed and opened his eyes. A smile spread across his face.\n\n\"Just floated in for a visit,\" he whispered. \"I haven't heard from you in a while, and there are some things I want to share with you.\"\n\n\"That's great,\" I whispered back. \"I'm really happy to see you, but I can't visit for too long. I have to get some sleep. Tomorrow's a busy day.\"\n\n\"Don't worry,\" Meldor laughed. \"You'll be just fine.\" And then with a curious look on his face, he asked, \"By the way, why are we whispering?\"\n\n\"I don't want to wake up Candi,\" I replied.\n\n\"But she can't hear us!\" Meldor looked mildly perplexed. \"We aren't using physical voices.\"\n\nFor a moment I didn't grasp what he was saying. I could have sworn I was still grounded in my body and very much physically awake. There was no doubt in my mind that I was speaking with my normal physical voice.\n\nMeldor just smiled knowingly. Slowly unfolding his legs, he straightened his body until he was standing upright on the bedroom floor. \"Come,\" he said, reaching for my wrist, \"take a look for yourself.\"\n\nGrasping his arm, I pulled myself to a sitting position, dropped my feet over the edge of the bed and stood up. Turning back towards the bed, I was in for a bit of a shock. There, still under the covers, sleeping peacefully was my body. The transition out of my physical form had been so smooth and effortless I couldn't believe I had simply sat up and walked away from it. Meldor was right. We didn't have to whisper.\n\n\"So, where are we going tonight?\" I asked.\n\nMeldor's eyes twinkled. \"How about to the heart of your soul?\"\n\n\"The heart of my soul?\" I questioned. \"Am I not the heart of my soul? Where else could it be?\"\n\n\"The reality of your soul and who you really are is much more complex than you realize, or rather I should say, remember. There are many aspects to the reality of every soul. Your reality is a combination of what you experience personally as well as the experiences you share with others.\"\n\nWith that Meldor sat on the foot of the bed, motioning for me to join him. Being careful to avoid sitting on my own body, I selected a spot beside him and sat down. Reaching out, he placed his hand on top of mine and quickly got into the heavy stuff.\n\n\"In order for you to better understand the realities of your existence, it is important for you to know the context of your creation and origin. My teachers tell me that when the Creator first manifested the individuated aspects of _The All_ , they were created by pure thought in the form of energy, light, and vibration. Each unit of energy, being part of the Creator, carried with it an aspect of the Deity's consciousness.\n\n\"As we've discussed before, these individual aspects of God formed part of the Creator's wish to experience itself. The design and intent of the intricate light-beings thus created was and is a manifestation of perfect love. It is in this manner that we have all been created from the energy and likeness of God.\n\n\"In order to instill an innate yearning for reunion and completion, as well as diversity of experience, each of these energy manifestations was further split into two individuated beings. In much the same manner as the nucleus of a fertilized egg begins its life experience by first growing into two individual cells, so too do these identical aspects of God begin their conscious existence. Although identical in source and vibration, they develop their own realities through differing experience.\n\n\"The two souls thus created are never far apart in their thoughts and vibration. Attracting each other like magnets, they interact continuously in the spirit realms, and if they have chosen to incarnate, they may be together in the physical world as well. Their bond is strong and eternal.\"\n\n\"So the talk about soul mates is actually true?\" I breathed. But it was more a statement than a question. The revelation astounded me. I had often heard people using the term, but to me it always seemed nothing more than a wishful romantic notion. Now I had to consider that there might be more to it than I'd previously been willing to believe.\n\n\"The term 'soul mates,' is actually an appropriate description.\" Meldor had obviously read my thoughts. \"Although, I think it has become something of an overused term in the reality of the world today.\"\n\nA question arose in my mind. \"Meldor, you said that they _may_ be together in the physical world. Wouldn't soul mates _always_ want to be together?\"\n\n\"Actually,\" Meldor replied, \"it's not as common as people would like to believe. Most incarnates spend occasional physical lives with their soul mates in various relationships, but most of their time together is spent in the spirit realms. You should understand that if it were common practice, it could seriously hinder the development of a soul who most often seeks to learn and evolve by challenge and diversity. Repeating life after life with the same soul would not provide the diversity of experience conducive to spiritual development.\"\n\nAnother question popped into my mind and I couldn't resist asking. \"Is my soul mate in the physical world right now? Have I met her?\"\n\n\"To begin with,\" Meldor chuckled, \"it might be unwise to assume that your soul mate is someone of the opposite sex. Secondly, although I understand your curiosity, it would not be appropriate for me to provide an answer to your questions. It is your life to live as you have chosen. We may help and guide you toward your intended direction, but we would never short-circuit your desired path. Suffice it to say that everything will eventually be revealed to you.\n\n\"You should also know,\" Meldor added, \"that there are many other levels of soul interaction that are just as important to our development, if not even more so. In our quest to grow and develop from experience, we also find that there are other souls whose vibrations and energy resonate in almost perfect harmony with ours. The attraction amongst these harmonious like-minded souls is formidably strong and nurturing, resulting in a very special bond and relationship. In seeking ultimate perfection and unity, these groups of souls may bond together as powerful units of energy, essentially becoming a single larger and more powerful entity. They work together so harmoniously that their combined energies vibrate as one.\"\n\nI had also heard various theories about \"soul groups\" or \"soul clusters,\" but the descriptions given were mostly vague and speculative. A number of questions and doubts came to mind.\n\n\"Meldor,\" I asked, \"when you talk about these groups of souls, are you talking in terms of a family? For example, would members of my family here on Earth be from my soul group?\"\n\n\"Usually that is not the case,\" Meldor replied. \"Although members of a group may incarnate into the same family, it usually doesn't involve more than a couple of members in one family at the same time. The reason for this is that members of the same soul group would not have as much to gain by continuously incarnating with each other. Their bond and love for each other may override the challenge and intensity of the lessons that were intended. For this reason, among others, members of the same soul group do not normally incarnate together.\"\n\n\"So, then,\" I asked, \"what is the purpose of souls spending time in groups?\"\n\n\"As you know,\" Meldor explained, \"the main goal of a soul's journey is to attain the perfection of unlimited love through the accumulation of knowledge by experience. Most soul groups accomplish this through a cycle of incarnations, challenging themselves, and working to overcome the obstacles. Every entity in the cluster adds to the experience, knowledge, and energy of the group. Although each soul has its own plan and life blueprint, it is loved, encouraged, and supported unconditionally by the entire group.\n\n\"The life experience of every member of the group contributes immeasurably to the strength of the whole. This is a very important aspect of the soul group. As the entire cluster of souls vibrates as one, the individual life experience of any single member becomes an experience shared by the rest of the group. This is of tremendous benefit to the advancement of the entire group. With several members of the group incarnated at the same time, the process of learning and accumulating experience is significantly improved. As the combined experience of the cluster grows, so does the power of its overall spiritual vibration. This acceleration of the developmental process allows the group to move more quickly to higher levels of understanding and vibration.\"\n\nMeldor's statements intrigued me. \"So, are you saying that other members of my soul group are able to share the experience of the life I'm now living?\"\n\n\"Yes, precisely,\" he replied. \"If you could have the experience of several lives simultaneously, would it not accelerate your understanding and accumulation of knowledge?\"\n\nI began to wonder about possible connections with my own group, if in fact I was part of one. How many souls were in my group? Was there a set number, or could it vary? Why hadn't I met any of them in my spiritual journeys? Or had I, and just didn't realize it?\n\nMeldor read my thoughts and responded. \"Yes, most certainly. You are a valuable part of your group's activities and experiences. There are 14 individual aspects in your present group, which seems to be a bit smaller than average. There are groups that have as few as three or four, but there are also others that contain many more souls. While you may not be consciously aware of other members of your group, you do often join them during sleep. Although you had no way of knowing it, some of the faces you have seen through your mind's eye are theirs.\"\n\n\"Are any of my group living on Earth right now?\" I asked.\n\n\"Yes,\" Meldor responded. \"Besides you, there are currently four other aspects of your group involved in a life experience.\"\n\n\"Do I know them?\" I asked. \"Have I met any of them?\"\n\n\"No, not in the physical world, but you have interacted with one of them briefly during an altered state.\"\n\nThis was a complete surprise to me. I couldn't recall anything that would lead me to believe I had been in contact with a member of my soul group.\n\nI felt a surge in vibration and knew that Meldor was retrieving an energy imprint that would help me remember. Within a few short moments the forgotten experience began to play in my mind. I recognized it immediately.\n\nAbout four months earlier I had awakened one morning and was drifting in and out of sleep, when a subtle vibrational change swept through me. The next thing I knew I was riding in the back of a pickup truck in some unfamiliar place. The truck was traveling down a paved road past a cluster of large, high-roofed buildings resembling warehouses. With me in the back of the truck were a number of orange road pylons and what appeared to be a collection of ropes and pulleys.\n\nPuzzled, at first I couldn't imagine where I was. But as we rode along, the strangest thoughts began surfacing in my mind. I somehow knew that we were in the midst of a large property containing numerous film studios and soundstages. Fatigued, I found myself looking at a wrist-watch I didn't recognize, thinking, \"Shit, it's only 7:00 A.M. and I'm already wiped.\"\n\nAs the truck turned a corner and pulled through a large overhead door into one of the warehouses, I was thinking about the upcoming week when I would actually get to do a couple of car crashes. All of a sudden, my surroundings began to fade, and a moment later I was back in my own body.\n\nThe whole incident was so curious and confusing I had simply discarded it as a weird anomaly or vivid dream. Now Meldor was telling me that this wasn't just a dream, it had actually happened.\n\nThe concept was entirely new to me. I remembered the confusion I experienced at the time. Although it felt like I was in my own body, the mental process was perplexing. I could identify my own thoughts as I sat in the back of the truck trying to make sense of what was happening, yet I seemed to be simultaneously experiencing someone else's thoughts. I wondered if this other person had been aware of my intrusion. Was he as confused as I was? Perhaps, if I had gotten out of bed immediately following the experience, I could have examined it more closely and come to some conclusion. Instead, I had simply rolled over and gone back to sleep. When I finally awoke, my memory of the episode was hazy and dreamlike.\n\n\"The vibration of your two energies is so similar,\" Meldor broke into my thoughts, \"it is understandable that you may not have realized that you were interacting with the physical body of another soul.\"\n\nMy mind spun. There were so many things I wanted to know.\n\nWhat was this person like? Was he like me? Who was he? Where did he live?\n\n\"Your curiosity is certainly understandable,\" Meldor smiled in sympathy, \"but the information you seek cannot be shared with you at this time. It could only serve to alter the individual paths that both of you are following. You will, at the appropriate time, know and remember.\"\n\n\"Yeah, I suppose you're right,\" I acknowledged. \"But it would sure be neat to meet him while we're still on Earth. I am curious, though; can you tell me why or how this happened? Could it happen again?\"\n\n\"Yes, I can do that,\" Meldor confided. \"It is actually a fairly common experience for incarnated souls. Because of the extremely close ties among members of a group, various personalities may spend a great deal of time with each other individually or within the overall group consciousness. As I have mentioned, incarnated souls often rejoin their soul groups for short periods of time while their bodies sleep. As this is a very powerful relationship, many souls do in fact find themselves connecting in the same manner that you did, but most often it is remembered only as a dream.\"\n\nWe sat in silence for a few moments as I contemplated these astonishing revelations. I couldn't come up with any new questions, and I wasn't so sure I even wanted to. I knew that everything would work out exactly as it was supposed to. And then I began to wonder, as I had so often in the past\u2014why me? Why were these things happening to me? And what about this latest experience? Why was Meldor bringing it to my attention? Where was this supposed to lead?\n\n\"This is your wish,\" Meldor's gentle voice broke into my thoughts. \"This is the challenge and goal of your experience. There is a definite plan and a specific purpose to all things. This is the path that you have chosen. _Remember, it's the journey that's important, not the destination_. You are beginning to understand that it is impossible to not reach your destination. It will always be the same for everyone.\"\n\nWith a warm smile, Meldor then placed his hand over my heart. \"Now it's time for me to leave,\" he whispered. \"Until next time, know that you are loved.\" For just a brief moment I seemed to lose consciousness. A second later, I was back in my body.\n\nIn a sleepy state, I lay thinking about the possibilities that seemed to inhabit our lives. Again, I had learned something new, yet it seemed I was barely scratching the surface.\n\nIn my busy world, time passed quickly, and several weeks went by without any significant spiritual activity. Then out of the blue I was presented with another intriguing experience that would eventually have a remarkable impact on my perception of life.\n\nI awoke early one morning from a long and restful sleep. Not ready to rise, I decided to snooze for a while, and settled into a meditative state. A montage of dreamlike images had just begun to waft through my mind, when I seemed to lose consciousness. A moment later I found myself walking along the concrete floor of a long corridor in what appeared to be a large stadium. \"Hmm,\" I thought. \"This is interesting. I wonder where I am this time?\" Its concrete walls painted a light shade of green, the hallway was about ten feet wide with a well-worn path down the middle of the floor.\n\nAs I approached a corner, I found myself thinking that it had been a good concert. There had been a couple of screw-ups, but overall everything had gone well. The adrenalin was still pumping, and I was feeling pretty good.\n\nSuddenly it dawned on me that these could not be my own thoughts. They had to be someone else's. Although comfortable and completely at ease with the circumstance, I was, nonetheless, more than a little confused.\n\nTurning right, I continued down an adjacent hallway before reaching the doorway to a dressing room that I knew was mine. Opening the door, I crossed the floor to a chair and makeup table in front of a large mirror. As I lowered myself into the chair and looked up into the mirror, I was in for a shock.\n\nIt wasn't me! The face staring back from the mirror wasn't mine. The surprise jolted me so badly, I instantly slammed back into my body.\n\nI couldn't believe it. For a second time I had somehow ended up in someone else's body. This time I had actually seen his face. He appeared to be much younger than me, perhaps in his early twenties.\n\nA good-looking young man, he had long jet-black hair and stunning blue eyes. Mesmerized by the intensity of his eyes, the image burned its way into my mind and soul.\n\nFrom the brief moments when I shared his thoughts, I knew that he was a musician or a singer. Probably, I decided, just one of millions of young musicians in the world, struggling to make his way. Could he, I wondered, be another member of my soul group?\n\nThe powerful images of this experience stayed with me for a long time. I often found myself returning to them in my quiet moments. On a couple of occasions I even dreamed about the young man, but in the morning couldn't recall any more than vague impressions. Eventually, as with all things, time and life took their toll, and the incident faded into the recesses of my mind.\n\nI love sleeping in on weekends. To me, there are no finer moments than lounging in bed after a long sleep, dozing and dreaming. A couple of months had gone by since my last contact with Meldor, when one of those glorious, lazy Saturday morning opportunities arrived. Relaxing in bed, I enjoyed the luxury of a do-nothing-day, drifting in and out of consciousness, when the abrupt sound of a male voice jolted me to attention. \"Knock, knock.\"\n\nIt was Meldor.\n\nWith unreserved delight, I immediately locked in on his vibration, laughing to myself, \"Oh no, not a dumb knock-knock joke.\" I was about to play the straight man with the inevitable \"Who's there?\" when a wave of vibration swept through me. A moment later I was floating in a void.\n\n\"Oh, just me.\" Meldor's amused voice came from behind me. \"Why\u2014you were expecting someone else?\"\n\nAnticipating his usual robed appearance, when I turned around, I was surprised to behold a shimmering sphere of energy. I had never seen Meldor without a physical form. For just a moment, I felt a twinge of anxiety. Recognizing my uncertainty, he began to radiate a subtle shade of violet light. Warmth and love permeated my being. \"If you prefer,\" he said, \"I can revert to the human projection of my body.\"\n\n\"No, it's quite all right,\" I stammered. \"It just caught me by surprise. It seems strange though. I don't know how to describe it, but it's like I can feel or sense your body in the same way as before. Even though you don't look the same, you still feel the same.\"\n\n\"The feeling that you're experiencing,\" Meldor explained, \"is due in part to your recognition of my vibration, as well as the imprint that my physical appearance has left in your mind. As we have discussed, the vibration of every soul is unique. It is our true and indisputable identity. You would know me even if I was here with only a small part of my energy.\"\n\n\"What do you mean?\" I asked. \"How could you be here with anything less than your total self? Are you saying that you can be in more than one place at a time?\"\n\n\"Aren't you?\" I could sense Meldor grinning. \"Aren't you here talking to me while your physical body is somewhere else?\"\n\n\"Jeez, I guess so. Good point,\" I sheepishly conceded. \"You would think I should have learned by now to take you at your word.\"\n\nMy curiosity had been piqued. I just had to ask. \"So, right now, are you here with all your energy, or is part of you somewhere else too?\"\n\n\"Well, right at this moment, as you perceive time, the complete focus of my attention is with you. But there are often times when I am with you, and part of me is elsewhere.\"\n\n\"But, where would you go?\" I asked. \"You don't have a physical body to maintain in another place.\"\n\n\"That is true,\" Meldor acknowledged. \"But I do have responsibilities and commitments in other places. Often while I am in this plane, part of my energy is in another level, learning and studying. Besides my commitment to you, I help to train other souls to become guides, and I am also working with a wonderful group learning to control and transmute emotional energy. The work we do is very important in helping returning souls shed or transform some of the intense emotional baggage they often bring with them.\"\n\n\"That sounds like a lot of work,\" I interjected. \"I had no idea. Does everyone keep so busy in your level?\"\n\n\"No. But a great many are even more involved than I am. You must understand that there is no requirement or pressure on anyone to become involved. But everyone, at some point, happily donates their efforts and skills for the betterment of all.\"\n\nAs we chatted, I looked around, but there was absolutely nothing to see. I perceived only an endless, dark void. There was something foreboding about it. It seemed to exude an atmosphere of depression and negativity, and it was beginning to bother me. Increasingly curious, I finally asked, \"Where on earth are we?\"\n\n\"Well, we're definitely not on Earth,\" Meldor sighed. \"We are in the void between the physical world and the astral realms. More correctly, this is actually part of the astral realm. It is the lowest vibration of the first spirit levels.\"\n\n\"But there's nothing here,\" I observed, wondering why he had brought me to this place.\n\n\"On the contrary,\" Meldor corrected me. \"There are, unfortunately, a great number of souls in this realm. They are cloistered here by their own actions and intentions. The souls in this territory have been severely affected by the negativity and despair they brought on themselves during their last incarnations. This is the result of guilt, along with self-imposed punishment and confinement. This is the level of consciousness closest to the human concept of Hell.\"\n\n\"So that's why I've been feeling so anxious,\" I mused. \"I don't like the vibrations. It's really quite uncomfortable, like something's eating away at me.\"\n\n\"Yes, I understand. Due to the tremendous emotional energy given off by the souls isolated here, the atmosphere in this realm can be very menacing.\n\n\"As you know,\" Meldor continued, \"none of these souls are forced to be here. No one is preventing them from leaving. They have not been judged or punished by God or anyone else. The All manifests only love and compassion; it does not judge or punish.\n\n\"You will remember from your recent experience of a life review, how you felt upon reliving the memory of some of the uncomfortable moments. All of your actions and thoughts were clear and transparent. Nothing was hidden or shielded. No one had to tell you what was negative or positive. You knew and judged yourself accordingly. Suppose that during your life you had committed seemingly vile, unforgivable acts; you would certainly feel the intensity of the impact you had on others. Depending on your mental state and level of consciousness, you may become so despondent and distressed that you might put yourself into a deep state of tormented contemplation and self-review. This is the state of the souls in this level. They have closed themselves off from everything while they deal with the internalized misery they have inflicted on themselves and others.\"\n\n\"But they do eventually get out, don't they?\" I asked.\n\n\"Oh yes,\" Meldor replied. \"With the assistance of loving spirits, eventually all of them are brought along through the reorientation process.\n\n\"The cycle of numerous incarnations can be very difficult, and most souls, through one life or another, will spend some time in this state. As we have discussed before, the development of each and every soul is of great importance to the rest, and we all want to help. All are loved and nurtured through the many levels, until they reach the goal of God-Realization.\"\n\nPerhaps it was due to concerns with my own ego, and I wasn't so sure if I really wanted to know, but I just had to ask, \"Have I spent any time here?\"\n\n\"More than once,\" Meldor laughed. \"So you see, there is certainly hope for everyone.\"\n\nI tried to laugh along with him, but I couldn't hide the disappointment in myself. My all-too-human ego was a bit bruised. I began to wonder what kind of despair and confusion it would take to close oneself off so completely. I wondered what I could have done to put myself into a self-imposed Hell.\n\nMeldor broke into my thoughts. \"But there is another reason I brought you here. There is something special I want you to experience and remember. There are some very powerful light-beings that have attained the ultimate goal of God-Realization. Although immersed in the massive consciousness of _The All_ , they continue to dedicate their existence to serving humanity. Their knowledge and awareness is virtually incomprehensible to any entity that is not itself God-Realized. Truly they embody the greatest love of all, giving freely of their love and compassion. They are perfected souls, the greatest manifestations of God.\n\n\"These special light-beings are able to manifest on any level at any time, or indeed on all levels at the same time if that is their desire.\n\nHowever, it is to the souls in this lower vibration that they dedicate much of their energy. The radiation of their love is so immense that often it is the only power that can penetrate the cloistered consciousness of these tormented souls. It is here that these marvelous deities perform their greatest service. And it is this power that you are about to experience.\"\n\nNot knowing what to expect, I was becoming more anxious with each passing moment. Meldor had used the word \"deities,\" and called them \"the greatest manifestations of God.\" What would their energy be like? Who was I to be experiencing a manifestation of God?\n\nSensing my growing anxiety, Meldor reassured me, \"There's no reason to be concerned. I'm sure you will find the experience very enlightening.\"\n\nSilently we waited in the darkness. After two or three minutes passed with agonizing apprehension, I began to feel a warm tingling sensation in the center of my chest. Ever so slowly the sensation grew stronger and stronger until every particle of my being was vibrating with overwhelming emotion. And then it appeared\u2014a blinding white light, radiating energy from its center, growing in intensity as it approached. I could not even begin to grasp the size or magnitude of this light-being as it moved slowly by. My sense of relative space seemed inadequate and delusional. My perception reaching overload, I closed my eyes.\n\nTo this day I struggle to describe the feelings and emotions that permeated my being. My words seem frustratingly inadequate.\n\nEyes closed, my consciousness was flooded with a wonderful feeling of tenderness and love. All tension and uncertainty drifted away, leaving me with a peaceful, undisturbed clarity of mind. It seemed as if I had been reduced to a state of childlike trust and wonder. It was simply blissful. Loved and accepted, I wanted nothing more than to become one with this vibration.\n\nAs I floated in euphoria, a sudden surge of vibration forced me to open my eyes. To my astonishment, the likeness of Jesus Christ materialized before me. Obviously choosing to appear in a form I would recognize, he looked every bit like the painting of Christ my mother kept on a kitchen wall when I was a child. Manifesting as a most wonderful, kind looking man in his thirties, he had a light brown beard and shoulder-length hair. His eyes beaming the very essence of love, he quietly smiled in acknowledgment, and then faded into the darkness. As quickly as that, it was over.\n\n\"A powerful presence, don't you think?\" Meldor's voice cut the silence.\n\n\"Wow!\" I exclaimed. \"That was incredible!\" I could think of nothing else to say. It would take a few minutes to center my thoughts.\n\n\"As you can see,\" Meldor continued, \"the energy of a perfected light-being is awesome to behold. Because of their pervasive power, these deities are able to break through to the despondent consciousness of the souls in this realm when they are finally ready for healing.\"\n\nAs Meldor spoke, I wondered if other beings of the stature of Christ visited this plane.\n\n\"Yes, most certainly,\" Meldor answered. \"Having been raised in the Christian faith, you are naturally more familiar with the Christ Consciousness. There are, however, many other perfected light-beings that assist the souls in this level. Due to their perceived affiliation with major Earth religions, the better-known deities and masters such as Muhammad, the Buddha, Krishna, and Christ undoubtedly spend more time in this realm.\n\n\"These perfected souls continue to be guiding lights as they were during their Earthly incarnations. Considering the awesome power and awareness they possess, it was with great sacrifice and love that they voluntarily reincarnated into the gross vibration of the physical form, entirely for the benefit of humanity. Even though man has misconstrued their messages into differing religious dogmas, these Masters are one and the same in _The All_. As I have said before, they are the highest manifestation of God.\"\n\n\"But some of these other weird religions can't be right?\" I reasoned. It was an old thought, a relic from my past, but ever so fleetingly, it surfaced in my consciousness. Meldor picked up on it immediately, and although I felt a bit foolish for even thinking such a thing, later I was glad that I had.\n\n\"The answer to your question is a paradox,\" he said. \"They are all correct and yet none are correct. All of the major religions are correct in that they teach about the love of God in all things, but their individual dogmas are the construct of human minds and have little significance to God.\n\n\"God is not Catholic or Muslim or Hindu. Nor is God associated with any other man-made belief system. Within _The All_ there is only one universal precept, and that is _love_. God commands nothing, but only gives and asks for love. If this one universal principle were adhered to, all the \"shaft nots\" of man-made religions would be rendered redundant. Any soul seeking the perfection of God-Realization needs only to perfect its love through every expression and manifestation. As you know, from the perception of an incarnate, this may sound simple in theory, but in practice it is not.\"\n\n\"No kidding!\" I was wrapped up in Meldor's observations. Everything was beginning to make a lot more sense. Even for those souls trapped in their own self-imposed Hell, there was hope. I knew that there was no place anywhere where God did not exist.\n\nMeldor interrupted my thoughts. \"The time has come to return to your body. The negative vibrations of this realm can have a disturbing effect on one's vitality. You may need to restore the balance of your energy. As always, it is a pleasure to assist you on your journey. Until we meet again, remember, my love for you is eternal.\"\n\nWith those last words echoing in my mind, I felt a sudden swirling motion, my consciousness faded, and a moment later I was restored to my body.\n\nAfter working a small kink out of my neck, I propped up a couple of pillows and settled into a light meditation. I certainly had a lot to think about. Contemplating how enlightening the experience had been, I chuckled to myself. The irony of it all. An enlightening experience\u2014I had literally been to Hell and back. And apparently it wasn't the first time.\n\nConsidering the task it would be to reach the point of spiritual perfection, I thought about my continuing struggle to overcome my own intolerance and impatience with others. Could I ever hope to love without condition and expectation? Could I learn to love and accept others for who and what they were? Perhaps more importantly, could I learn to love and accept myself for who and what I was? It sure seemed like an awfully long road ahead.\n\nThis particular Saturday morning experience has always remained a special memory for me. I am unable to recall it without strong emotion and nostalgia. Not only because of the lessons learned or the experience with the Christ Consciousness, but more significantly, because it turned out to be the last conscious meeting I would have with my cherished guide, Meldor. Although it marked the climax of an important phase in my journey of self-discovery, it also established a solid base to prepare me for what was still to come.\n\n# 12\n\n# Removing Walls\n\n_But in angel eyes love grows complete_\n\n_While strong hands tremble in defeat_\n\n_The truth of God is soon revealed_\n\n_It's through our hearts that we are healed_\n\nSometimes there doesn't seem to be enough time to stop and smell the roses. There's always another deadline, another commitment, another fire to put out. Regardless of all good intentions, for me, life's busy wheels kept on turning.\n\nMaintaining two jobs, raising a family, and trying to stay ahead of the game, my life became increasingly convoluted and demanding. Working 12- and 14-hour days, the weeks and months rolled by in numbing succession. Each day, I was drawn deeper into the dense vibrations of the physical world and further away from the spiritual yearnings of my heart.\n\nWhile at one time, hardly a day would go by without some contact with my guides, my connections to the spirit world had all but vanished. When I did find time to attempt out-of-body travel or contact with Meldor, I often fell asleep. Or, after hours of staring into the darkness, I would give up in frustration. Eventually, I quit trying.\n\nAfter nearly two years with precious little spiritual activity, I was beginning to feel a great void in my life. I missed the wonderful excitement and connectedness I had once cherished. I'd read somewhere that our guides can change at different stages of our lives. I wondered if Meldor had perhaps moved on and been replaced by new guides whose vibration I couldn't perceive. In my more despondent moments, I wondered if I had been left entirely on my own. Although my physical life had never been more active, I had never felt more lost and abandoned.\n\nI could not shake the loneliness. More often than not, before going to bed at night, I would end up in the quiet of my backyard, absorbed in the beauty of the star-studded prairie skies. Often, tears would fill my eyes as I struggled to stem the flood of emotions that threatened to overcome me. My heart was breaking, and I didn't know why.\n\nI needed to get away from the rat race to recharge my energy and restore my soul. The best place for this was The Monroe Institute. It was the only place that could provide the peace and relaxation I needed to reconnect. Hoping to get into the first available \"Guidelines Program,\" I called the Institute in the spring of 1999. To my disappointment, I was told that there were no seminars available until mid-July. Although longer than I thought I could possibly wait, I eagerly grabbed one of the last openings.\n\nWaiting was difficult. The stress of my spiritually barren existence was becoming unbearable, but the day finally came. For me, there couldn't have been a more welcome feeling in the world than when I arrived at the front doors of the Institute. Stepping through the threshold, I could feel a great weight lifting from my soul. The anticipation was thrilling. But even in my wildest dreams, I could not have imagined what lay in store for me during the coming week.\n\nAfter stowing my luggage, I headed off to the dining hall where most of the participants were gathering. Scanning for familiar faces, I didn't see anyone I knew. I was, however, struck by the presence of an attractive, blond-haired woman sitting at one of the tables. From across the room, I watched her for a few moments. I didn't know what it was, but there was something about her that seemed hauntingly familiar. Deciding that she probably just reminded me of someone else, I shrugged off the notion, and moved on to meet some of the others.\n\nLater that evening we all met in the lecture hall for formal introductions and the usual preliminary briefing. To facilitate breaking the ice, the trainers decided to partner each of us with another participant. Each person would interview the other, learning all they could about their background before introducing their new friend to the group.\n\nI thought this novel idea might be fun. When we were told to select a partner, I was about to turn in my chair to check out the people milling around, when from behind me came a cheery female voice. \"Hi! Can I be your partner?\"\n\nTurning, I found myself face-to-face with the pretty lady I had noticed in the dining room. \"Hi. I'm Patricia. Can I be your partner?\" she softly repeated, extending her hand to greet me.\n\n\"Hi, I'm Paul.\" I stood quickly, returning her smile as I reached out to shake her hand. \"It would be my pleasure.\"\n\nOur hands touched and, as I looked into her eyes, I'm sure my heart skipped a beat. In an instant, I was rocked by an electrical vibration that swept from her hand into my body like a wave of warm water. As our eyes met, it was as if some powerful energy had reached deep into my chest and squeezed my heart. It took my breath away. My knees suddenly weak, I dizzily returned to my chair.\n\nI had no idea what had just happened to me. I was trying not to let it show, but I'm certain the bewilderment must have been clearly visible on my face. I was having trouble thinking, much less speaking.\n\nWhen I finally recovered enough to continue, we began, as instructed, to interview each other for our later introductions. Patricia, I learned, had been born and raised in the mountains of Oregon. Throughout her life, even as a child, she had been able to see and hear things that other people couldn't. She had always experienced intuitive flashes, but until she was a teenager, she thought that everyone had the same experiences. She described herself as someone searching for truth. She loved The Monroe Institute, and thought it was a great place to further her search.\n\nAlthough virtually mesmerized, I managed to stumble through a quick review of my life and background. Patricia smiled in understanding as she made her notes.\n\nThe nine teams finally began their introductions. It was a fascinating process. At the end of the introduction, each partner was supposed to provide a teaser, an intriguing little insight about their new friend that would impel the rest of the group to want to get to know them too.\n\nWhen it came to our turn, Patricia began. She eloquently recounted my background and keen interest in the spirit world. Then she made an astonishing announcement. In a quiet, almost blushing fashion, she said she just knew I was a kind and gentle person with a warm heart, but the most intriguing thing about me, she said, were my eyes.\n\nWith a perplexed look on her face, she said, \"He has the most amazing eyes. They're like a painting. You can lose yourself in them. It's as if you're drawn right into his soul.\"\n\nA huge lump quickly formed in my throat; my eyes began to sting.\n\nIt was my turn to speak next. I swallowed hard. Glassy-eyed, I glanced at my notes, trying to pull my thoughts together, hoping that nobody, including Patricia, would notice the emotional turmoil I was in.\n\nAfter clearing my throat, I shakily began. I hardly heard my own words or knew what I was saying. To hide my nervousness, I tried to lighten the situation by making a couple of funny remarks, but failed miserably. Soon I was at the end of the introduction and couldn't think of anything more to say.\n\nI was relieved when the group's focus finally moved on to the next pair of introductions. Glancing at Patricia, I found her staring at me with wide glistening eyes, a bewildered smile on her face. Finally, lowering her eyes, she shyly looked away.\n\nIntroductions concluded, we spent the rest of the evening chatting with other members of the group, talking at length about other programs and experiences. Throughout the many conversations, my thoughts often returned to the rush I had felt upon first touching Patricia's hand and looking into her eyes. I had never experienced anything like it before, and didn't know what to make of it. She _must_ have felt something too, I thought. The look on her face, the way she introduced me, the things she said.\n\nAs the evening wore on, Patricia and I occasionally found ourselves in idle, if not uncomfortable, conversation. I was becoming certain that she had also felt something unusual, but I was too afraid to even talk about it. Finally, along with everyone else, we said our good nights and headed off to bed.\n\nThe next morning, I awoke eager to get into the program. Of the various seminars I had attended at the Institute, \"Guidelines\" had been my favorite. It was during Guidelines that I had first met Meldor, and I was desperate to reestablish our connection.\n\nIt came as no surprise to me when the first sessions yielded little in the way of experience or insight. I had come to expect it. Whether it was jetlag or just my need to unwind and relax, it proved a pretty normal start to the week. I spent most of my time either sleeping or staring into the darkness of my CHEC unit. It didn't matter to me. I was thrilled just to be back in peaceful surroundings.\n\nAlthough I tried to put the introduction experience with Patricia out of my mind, it was easier said than done. Whenever I saw her between sessions, I would feel a strange vibration in my chest. Occasionally during the group discussions, I could sense her energy and when I looked up I would find her staring at me. Without realizing it, I often found myself doing the same thing. Something about her was so compelling it attracted me like a magnet.\n\nLater that evening, I was relaxing at a picnic table in the yard, when I spotted her timidly approaching. Following an exchange of pleasantries, I invited her to sit and chat. It turned out to be a lengthy conversation that would have a profound effect on my life.\n\nPatricia acknowledged that she too felt a strange sensation when she first saw me. It became even more intense when she touched my hand or looked into my eyes. She felt as if she had somehow known me before. I suggested that it was perhaps from another lifetime. She countered that she really wasn't sold on reincarnation, but had no explanation for the familiarity we both felt.\n\nAlthough Patricia had grown up in the mountains and I on the prairies, our experiences and values were strikingly similar. She too had endured a difficult upbringing. Her mother had died while she was still very young, and she and her little brother had been raised by their father. Actually, if the truth were known, the raising of her brother had been Patricia's responsibility. No time to play or to just be a kid, there was only hard work and responsibility. In addition to looking after the house and her brother, she could not remember a time when she didn't have to work in her father's small lumber mill.\n\nAs Patricia talked about her childhood, so deeply did I identify with her feelings and emotions that I often felt a lump in my throat and a burning sensation in my eyes. Like me, Patricia had grown up with a constant yearning for something better. Other than attending school, she was rarely able to get away from home and, as a result, had very few childhood friends. At an early age she developed a great love for reading. But beyond schoolbooks, the only other reading material in her house was a set of 1937 encyclopedias and a Bible. She had read them all from cover to cover, voraciously consuming the knowledge they offered. When she turned 18, Patricia left the mill and the mountain, and never looked back.\n\nAs I listened to her story, my heart went out to her. It brought back a flood of my own childhood memories. I thought that I had had it tough, but I couldn't even imagine how hard it would be to grow up without a mother. It was apparent that, like me, Patricia had also learned to place a strong wall around her heart to keep from getting hurt.\n\nPatricia confided that in the past couple of years, she had also been feeling listless and disconnected. There was something missing in her life. She loved to spend time at night looking into the starlit sky and she too felt a yearning and loneliness that plagued her.\n\nAs we continued to chat about everything from world affairs to the stock market, I became more amazed at her insight and intuition. She was without question one of the most intelligent and knowledgeable people I had ever met, Extremely well read, she seemed to have developed a near photographic memory.\n\nEngrossed as we were in our conversation, the hours flew by. Finally, after midnight, we bid each other good night and headed off to bed. Sleep, however, did not come easily. I had become utterly intrigued by the connection I felt with this sweet, quiet soul. In all my life I had never felt such an extraordinary attraction, and I was unsure of how to deal with it. Finally, after what seemed like hours of tossing and turning, mercifully I fell asleep.\n\nThe next day, although suffering from lack of rest, I sensed a huge improvement in my psychic energy. The vibration within my body was steadily increasing as I became more relaxed. By midmorning, as we began the second session of the day, I felt confident that I was moving closer to a meaningful experience. I looked forward to finding answers to my questions about Meldor, but I could not have anticipated the strange encounter I was about to have.\n\nAs the Hemi-Sync signals lulled me into a deep altered state, I settled into my breathing rhythm, prayed for help, and mentally projected my questions into the ether. I asked if my feeling that Meldor was no longer with me was correct. I asked if new spirit guides had taken his place. And finally, I asked that if I did have new guides, they present themselves to me as soon as possible. I desperately needed to know.\n\nWithin seconds of sending out these thoughts, I felt a shift in vibration, and almost immediately the form of a male spirit appeared. He seemed to be about 40 feet away, and although he looked familiar, at that distance I couldn't get a clear picture of his face. Thanking the spirit for answering my call, I asked him to come closer. In one quick movement, he drew to within four feet of me and stopped.\n\nI couldn't believe my eyes! The face staring back at me was my own! The spirit was me, or at least a good duplicate of me. For several moments, I stared at his features. It was as if I was looking into a mirror. The problem was that the person in the mirror was moving and acting independently of me. I began to wonder if this was just some mischievous spirit playing tricks on me. Picking up my thoughts, the other me looked directly into my eyes, smiled, and slowly shook his head in reply. I was about to ask who he was and what this was all about, when he abruptly vanished.\n\nTalk about being confused. This was hardly an answer to my questions. Try as I may, I could not get the spirit to return or provide any other insights or answers. I became so perplexed I ended up breaking out of my trance. Moments later I was back to normal consciousness.\n\nI didn't know what to make of this strange occurrence. Was this a message showing me that, I, myself, was the only one responsible for my guidance? Could this entity have been some aspect of my higher self, or did it mean that the answers I sought were actually within myself? For the life of me, I couldn't figure out what this was all supposed to mean.\n\nOn the fourth day of the program I could feel my vibrations increasing steadily throughout the morning sessions, but I was still falling asleep. There always seemed to be a lot of activity going on around me. I could feel subtle changes in vibration, but I just couldn't break through the veil.\n\nSoon after the start of the first afternoon session I had fallen asleep yet again, when unexpectedly, I found myself jerking back into full awareness. To my surprise, I was locked in the grip of a wildly fluctuating vibration. My body tingled from head to toe as wave after wave of energy coursed through me. I began to sense a powerful presence, the unmistakable feeling that I was not alone.\n\nI opened my eyes to a dazzling display of multicolored lights. Shimmering waves of blue and violet energy swirled around my body. I didn't know what was happening, but I didn't want it to stop; the vibration and feeling it created in me were exquisite. Every molecule of my body felt electrically supercharged.\n\nThen I noticed them. At first, only a fleeting glimpse of a hand and arm moving in a circular motion above me became visible. But when I focused my attention, through the swirling haze I could make out the form of a woman. Although it was difficult to get a good look at her through the energy field, she appeared to be a rather petite lady in her early thirties. Her light blonde hair was swept back over her shoulders, seemingly blending into the material of her loose-fitting ivory-colored robe.\n\nAs the energy waves and swirling light continued, I became aware of another pair of hands, and then another. At the foot of the bed was a male form, and on my left, another female. I tried to get a better look at them, but the energy field was so thick, they were difficult to see. From what I could tell, the male and the second female were also wearing light-colored robes, but both had hoods drawn over their heads, partially concealing their features.\n\nAll three were busy with some sort of energy work. Their hands slowly circling at various distances above me, they seemed to be smoothing and shaping my aura while at the same time supercharging the energy in my body. Like strong electromagnetic fields sweeping in different directions, I could feel their power as each pair of hands moved over me. Every movement seemed to set off a new wave of energy crashing into the next, sending strong electrical currents spinning throughout my physical form. No words were spoken, and no messages were received. I simply lay quietly, relishing the wonderful feelings of warmth and love that permeated me.\n\nEver so slowly the urge to sleep began to overtake me. For some reason, even with the energy gyrations going on around me, I was having difficulty staying awake. In the presence of these three loving souls, completely safe and relaxed, I finally stopped resisting, allowing the energy to carry me into a deep sleep.\n\nWhen I awoke, it felt like a huge weight had been lifted from me. But, I also felt a strange and haunting sadness. I was experiencing a sensation that could only be described as homesickness.\n\nWhen I walked into the debriefing room, Patricia immediately noticed something different about me. Letting out a breath, she exclaimed, \"Wow! What on earth happened to you? You should see your eyes.\"\n\n\"I'm not really sure,\" I smiled weakly, shrugging. \"I think I might have just met some new guides.\"\n\nWithout taking her eyes off mine, Patricia reached out to give my hand a reassuring squeeze. As our fingers touched, I felt again that surprising flood of energy that jolted me when we first met.\n\nWithout saying a word, I placed my other hand over hers, and closing my eyes, I stood still, quietly absorbing her energy. A feeling of peace and tranquility came over me. I could practically feel my brain producing a flood of endorphins, soothing and relaxing. Mesmerized, I didn't want to let go, but I began to sense other people moving around us. With a sigh, I opened my eyes. Patricia stood in front of me, her eyes closed, eyelids twitching rapidly as if in a dream state.\n\nOther people had begun filtering into the room for the debriefing. I cleared my throat to get Patricia's attention. Startled, she opened her eyes and stepped back, a puzzled look on her face. \"Thank you!\" She whispered and, as if slightly embarrassed, moved away, taking her place on the soft carpet.\n\nWhile the other participants shared their experiences, I was practically oblivious to my surroundings. Absorbed in my own thoughts, I shifted back and forth between the experiences of the last session and the strange energy I felt whenever I was near Patricia. There was a lot to think about.\n\nIt wasn't long into the next session before I found myself clicking in and out of awareness. At some point, I felt a slight movement and knew I was about to separate from the physical. Prepared for the inevitable surge of vibration that usually precipitated my launch into the spirit world, I was surprised when nothing happened. Instead, my body began gently rocking from side to side. Within moments, I again sensed the presence of other beings in my room. Opening my eyes, I was greeted by the same three spirits from the last session. They were already at work sending energy through my body.\n\nVisually, all three spirits appeared unchanged. But the energies they projected seemed more pronounced. As their hands passed rhythmically over me, their energy and vibration became more intense. Shimmering blue, green, and violet waves rolled through me, growing in power and magnitude, electrifying my body. Soon the energy flow expanded to such intensity that it enveloped all three spirits in one continuous field. Their combined radiance flooded the room and permeated every part of me. Vibrating into my body and soul, it filled me with incredible feelings of love.\n\nDetermined not to fall asleep again, I reached out with my senses to draw in any impressions I could get from the trio. Soon, myriad thoughts and images percolated into my mind. Scenes of gut-wrenching emotional pain began to surface; scenes of longing, rejection, and unrequited love. There was something so familiar about them: they felt more like memories. And then it dawned on me. They were indeed memories, memories of my own past lives, previous heartaches, past hurts. The message became startlingly clear. My fear of rejection, and the compulsion to put walls around my heart had a much longer history than this present life. I had spent previous lives literally afraid to fall in love.\n\nTo protect myself from hurt, I had long ago closed off my heart to deep and total love. I would not allow myself to risk again the possible pain and helplessness that love could bring. But now I felt the error of living in a shell. I understood that unless we open up our hearts to love, we will never know what it's like to truly live. To know the pain of a broken heart, however intense, is far better than never having loved at all. Having our hearts broken time and time again is still better than putting up walls to keep our love in, and others out. This, I somehow knew, was one of the main reasons we were here, to know and experience love.\n\nThe spirits' thoughts continued to flow through my mind. Nothing in our physical dimension is of any consequence or lasting reality. The only enduring reality is love. To live without opening our hearts denies the most beautiful and enabling aspect of the human experience.\n\nFeeling the awesome power and compassion of these three wonderful beings, I began to sense a history of our companionship. It seemed like I had known them for a very long time, but couldn't place the context of our relationship. Were they angels? Spirit guides? Was there a difference between the two? I didn't know.\n\nRaw with emotion, I thought about how much I missed Meldor. Why hadn't I been able to see or feel his presence for such a long time? Had I been too engrossed in the physical world? Or had he completed his task and moved on? Were these radiant beings my new guardians? If they were, why hadn't I seen them before? Recalling the pain of loss and abandonment and praying for an answer, I projected these thoughts into the room.\n\nAlmost instantly, it seemed as if the air pressure in my CHEC unit plummeted. I could feel my eardrums reacting like I was in a descending airplane. The shimmering waves of energy slowed and began to change, coalescing into a beautiful golden light, enveloping me with love and warmth. Radiating a powerful feminine aura, the sound of a woman's soft whispered voice filled my mind.\n\n\"We have always been with you. Although part of our energy has been changed to meet your needs, we would never leave you. Our beloved Meldor has completed his work with you on this level and has returned to his higher vibration. You must know in your heart that you will again be reunited with him.\n\n\"We have been with you throughout, and will continue to be. Even when your conscious mind becomes too dense and physical to perceive us, we are with you. We come to you in your sleep and when your body is resting or meditating. Your well-being, your evolvement, and the very essence of your soul are intricately bound to our energy. You are a wonderful part of our existence and we love you beyond measure.\"\n\nAs she spoke, I closed my eyes. Powerful images flooded my mind. Drawn from memory, I saw an old picture of footprints in the sand and I recalled the story of a man who was carried by the Lord during his darkest hours when he thought that he had been abandoned. The vision carried too much pain. Emotion overwhelmed me. Clinging to her every word, tears welled up in my eyes, trailing down my cheeks.\n\n\"In this physical world,\" she continued, \"the greatest aspect of one's spiritual awareness has to do, not with the intellect, but with the heart. Although you have grown much in this respect, there is much more that you must learn. In order to fully assimilate the beauty of love, you must seek a greater balance to the masculine and feminine energies of your ego. We have been working with your energies to help dissolve the protective walls built around your heart. But in order to fully experience this reality, you must also open your heart to embrace the feminine aspects of your being.\"\n\nWaves of energy rolled through me. From deep within I could feel emotion building, lodging in my throat, choking me. I couldn't speak; I could hardly even think. With eyes closed, I felt the warm loving arms of my angels drawing me close, embracing and comforting me. At first it was just a trickle. And then it felt as if a great dam had broken inside me. The walls finally gave way.\n\nAs emotion overcame me, I began to cry. Through deep wracking sobs, I released the pain of this lifetime and perhaps countless lifetimes before. I didn't know if it was for sadness or joy, but from the depths of my soul I wept. I cried for all the love I had been given, and for all the love I had lost by shielding myself from it. I cried until my mind was numb and I could cry no more. Finally I drifted into a deep healing sleep.\n\nWhen I awoke, the session was over; the angels had disappeared, and I was left alone with the fading remnants of their energy, a light unburdened heart, and a clarity of consciousness I had never known.\n\nAs I sat up in bed, I could hardly contain my happiness and relief. It was as if I had been given a new lease on life. I hadn't a clue as to how this might affect me, but I knew it would be significant.\n\nThe debriefing session flew by. I couldn't seem to get the grin off my face. I'm sure I looked a bit daft, just sitting there, smiling at everyone. I couldn't even begin to explain what had happened to me, much less share my experience with the group. For the moment, at least, it was far too personal.\n\nFollowing the debriefing and a late dinner, Patricia and I headed outdoors to enjoy the warm evening. We hadn't been able to talk since before the last session, and I was curious as to what she had experienced when we had clasped our hands earlier in the debriefing room. After a bit of pleasant chitchat, I broached the subject. I talked about the vibration and the peaceful feeling that came over me every time I touched her hand, and I asked if she had been feeling anything similar.\n\nReaching across the picnic table, she placed her hand in mine, covered it with her other hand, and closed her eyes. Quiet seconds passed, and then finally she began to speak. \"When I'm holding your hand,\" she started slowly, searching for the right words, \"I feel warmth and tenderness. I feel completely safe and protected. Nothing can hurt me.\"\n\nSeveral moments passed as she sat motionless, and then without opening her eyes, she continued, \"This is so strange. I don't know what to say. I've never felt anything like it before. It's like I've known you for a long time. It's wonderful, but I don't know what to think. Weird things are happening. This afternoon, before the other people came into the room, I had the strangest vision. When we were holding hands, I closed my eyes and all of a sudden I saw a beautiful silver locket, like the kind you see people wearing around their necks. It looked really old, like an heirloom. The two compartments opened up and in each side was a beautiful butterfly. They spread their wings and flew out into the air. They were so happy to see each other. They came together in the air, touched each other, and then they just disappeared.\"\n\nPatricia opened her eyes. \"It was really quite beautiful. What do you think it means?\"\n\nMy mind was reeling. I struggled for words. \"I don't know,\" I stammered. \"You really saw that? That's incredible.\"\n\n\"I see a lot of things,\" she replied, \"but usually it's when I'm meditating or really relaxed. I don't know why I saw butterflies, but it must have something to do with you and me. Maybe it's a symbolic thing.\"\n\nI felt a swirling queasiness in my stomach. \"Hey, maybe we did know each other in a past life,\" I quipped, trying to make light of the situation.\n\n\"Well, maybe,\" she smiled, \"if you believe in that sort of thing.\"\n\n\"Yeah, if you believe in that sort of thing,\" I laughed, squeezing her hand.\n\nShe was such a wonderful paradox. How could she not believe in reincarnation? She was one of the most spiritually adept and knowledge able people I had ever met, yet she questioned the possibility of other lifetimes.\n\nAs if reading my mind, she responded to my thoughts. \"I'm not saying I don't believe in reincarnation. For all I know, it could be absolutely true. I just haven't experienced anything that would convince me. I guess I'll just have to keep my mind open to the possibility.\"\n\n\"Okay,\" I beamed back, \"we'll just have to see if I can convince you.\"\n\nAt that point, several other group members gathered around the picnic table and we spent the rest of the evening visiting. It was after midnight before we finally bid each other goodnight and headed off to bed.\n\nThe following morning I awoke energized. Although I looked forward to what the day would bring, I was sad that we would soon be saying goodbye and heading home.\n\nThe two morning sessions flew by without any significant happenings. Although I tried to reconnect with my angels, it was not to be. Other than a few innocuous images, my time was spent staring into the darkness or dozing. I was having difficulty focusing, and it was becoming more and more frustrating.\n\nThe first session of the afternoon wasn't much better. Even though my energy had been building, I still found myself drifting off. To make matters worse, all of the pent-up energy was causing a peculiar reaction in my body. I was becoming aroused. This certainly wasn't making things any easier. Finally I tried to focus on suspending all feelings in my body. It worked too well\u2014I fell asleep, and didn't wake up until the end of the session.\n\nThe next exercise started off a bit more promising. As I drew ever closer to the threshold of expanded awareness, an intense vibration permeated my body. That was usually a sign of good things to follow. As I focused on taking myself deeper, at some point I fell asleep. When I clicked back into awareness, to my frustration, I discovered that I had again become aroused. Determined not to let this distraction ruin another session, I began a mental struggle against the condition. I wasn't making much headway, though. Finally out of frustration I decided to allow myself to fall asleep, hoping that when I awoke the problem would be gone and forgotten. But just as I was drifting off, I was suddenly jolted back into awareness with the distinct impression of someone kissing my lips.\n\nThe feeling was unmistakable, an intense physical sensation. There had to be someone in my CHEC unit. I opened my eyes. Instead of the usual darkness, the room was bathed in a warm golden light. To my amazement, floating beside me, just a few inches above the bed was a very pretty, scantily clad woman.\n\nI couldn't believe what I was seeing. It was clear that this was not a physical person, but a full-bodied spirit. In instant panic reaction, I tried to pull away, but couldn't move a muscle. Immediately, as if in response to my fear, from out of nowhere, a feeling of warmth and love enveloped me, dissolving and washing away my anxiety.\n\nAs my shock subsided I tried to get my wits about me, but could do little more than lie there staring at her. There was something so familiar about her, it made my head spin.\n\nLying on her side, she was an angelic vision. Propped up on her left elbow, her long dark curls flowed past her rounded shoulders to the bed below. Dark eyes beaming brightly, she gazed intently into my eyes. It was as if she were looking for a sign, a recollection, perhaps a fond memory. As I stared back, I began to feel little sparks of recognition. Then all of a sudden it hit me. I knew where I had seen her before. She had been in my dreams.\n\nAs I lay on my back in disbelief, I drank in her image. She took my breath away. Invitingly open from her neck to her feet, a full-length creamy white gown hung loosely, caressing her body, accenting and revealing her sensuous form. She virtually radiated love and desire. Permeating my mind and body, her intense vibration filled the room with a raw sexual energy. Although I had seen her in my dreams, I could recall only fleeting glimpses. Something about her seemed hauntingly familiar, but I couldn't place her.\n\nI lay speechless. Helpless in the aura of her intoxicating feminine energy, my desire increased by the moment. The sparkle in her eyes grew stronger, and a loving, understanding smile spread across her face. Ever so gently she reached out with her right hand to lightly touch my face, then slowly, sensuously, traced her fingers down my chest and across my stomach. As her hand moved lightly across my body, it created within me a surging vibration that rippled through my entire being. The power was so electrifying. I felt like I was lifting off the bed in response to her touch.\n\nIt was as if I was being zapped by an electrical transformer. But the sensation was pleasurable beyond belief.\n\nTingling and shaking with the most intense desire I had ever experienced, I lay with the vibrations coursing through me, my mind practically numb from the rush of adrenaline. Moving her right leg across me, my celestial lover slowly positioned herself above me, straddling my aching body. Intense vibrations began to rocket through me. My heart was pounding, my brain begging for oxygen.\n\nThe feeling was so exquisite I felt like I was about to faint. Her eyes glistened like diamonds as she began to slowly lower her body to mine. Our lips met! For a brief moment I was aware of her breasts touching my chest. Then to my astonishment, her entire body seemed to dissolve, sinking into mine.\n\nIn panic, I found myself suffocating, fighting for air. It was, at the same time, one of the most terrifying yet exhilarating moments of my life. In just seconds, however, my thoughts carried me from fear and anxiety to surrender, and then to ecstatic passion. As our bodies flowed together, our energy and molecules seemingly occupying the same space, a growing heat began to vibrate in my heart. Quickly spreading to my stomach and genitals, it soon enveloped my whole body. With each moment the intensity increased, flowing, bubbling, and expanding until I thought I could take no more. Every particle of my being on fire with desire, I couldn't tell whether it was a physical sensation or a spiritual one.\n\nAs my breathing turned to short hyper bursts, my heart pumped wildly. With building intensity, a powerful contraction began to spread through my stomach, gripping and paralyzing me. My mind and body screamed for release.\n\nBeginning at first as a low rumbling vibration, it hit me like a freight train. From deep within, a wave of tremendous orgasmic energy exploded through every cell in my body. Overwhelmed, I nearly lost consciousness. Almost immediately, my muscles contracted yet again and, from the center of my being, a second volcanic wave carried me to virtual oblivion. In a frenzy of ecstasy, I hung on for dear life.\n\nWave after wave of electrifying energy swept through my body, locking me in the throes of uncontrollable passion. Then, as if seeking relief from the intensity of the experience, my mind began to slip. Flashes of light and vibrant energy flowed into my awareness, and then everything turned quiet and peaceful.\n\nI found myself looking out over a beautiful valley of green grass caressed by a warm morning sun. Sitting on a small grassy knoll, my lover's slim arms were wrapped around my waist. Against my back, I felt the warmth and comfort of her gentle embrace. Overflowing with unreserved love and passion, I could wish for nothing more. My life was complete. Love was all that mattered. Suddenly, my senses began to swirl and fade. Consciousness escaped me. A moment later I was back in my body, startled again by the light touch of tender lips on mine.\n\nOpening my eyes, I watched in awe as the female spirit rose above me. In radiant elegance, she floated to a standing position at the foot of my bed. Slowly and sensuously she traced her left hand across her breast until it came to rest over her heart. Eyes glowing with warmth and affection, a gentle smile spread across her face. Then, without diverting her gaze, she quickly faded from view.\n\nFor several moments I stared into the unrevealing darkness. My heart only now beginning to return to a normal rhythm, a flood of questions arose in my mind. Who was this spirit? Why had she come to me? What had happened? Had I just experienced a sample of the spirit world's version of sex? Could I even _call_ it sex? There was certainly nothing in the physical world to compare with it.\n\nThe last session was over, but I was in no hurry to join the others in the debriefing room. This had been a strange and amazing week. It occurred to me that, although I would normally have been traveling all over the place, during this seminar I had not once ventured out of my CHEC unit. I had, however, had some of the most incredible experiences of my life.\n\nWhen I finally headed downstairs, Patricia was waiting for me. She'd had some extraordinary experiences of her own and wanted to share them with me. Knowing that this might be the last time we would ever see each other, we talked through the evening and into early morning. We were both amazed and intrigued by the familiarity we found in each other. Somehow, somewhere, we had known each other before, and now we promised to stay in touch.\n\nWhen it finally came time to say goodnight, I drew her into a long heartfelt hug and, as her soothing vibrations flowed into the deepest recesses of my soul, sadness welled up in me. I knew from the glistening in her eyes that she was feeling the same. We were both happily married. Would it be possible to stay in touch? Would our spouses understand our friendship and connection? With a lump in my throat, I headed off to bed.\n\nMorning, as usual, came too soon. I was always sad to leave the Institute, but this time my reluctance was even stronger. After saying our goodbyes to everyone else, the inevitable moment arrived. With tears in our eyes and sorrow in our hearts, Patricia and I quickly hugged each other and left for the airport.\n\nAs we made our way into the Virginia sky, I sat back, reflecting on the week's events. Every nuance of the trip had been intense and mystifying. From meeting Patricia to encountering my angel guides, it had been an emotional affair. The angels had said they were helping to tear down the walls I'd placed around my heart so I could learn to love again. I didn't know if it was the things they had done or the emotional circumstances of the week, but I was feeling completely raw and exposed. My heart ached and I didn't know why. But in some strange way, it felt good.\n\nThe first leg of my trip home took me to the Washington, D.C., airport for a two-hour layover. Plopping myself down into the most comfortable chair I could find, I pulled out a book to read. After just a few minutes, having difficulty concentrating on the words, I gave up. Setting the book aside, I closed my eyes to rest. Almost immediately I found myself sinking into an altered state. Shortly, I became aware of a subtle pressure building in my forehead. It had been some time since I had had a third eye vision, so I quickly focused my attention on the spot. The air around me began to change. The portal burst open, and a moment later I found myself looking at the front cover of a book.\n\nWhat on earth is this? I wondered. This was not at all what I expected. The cover was fascinating: simple and striking, it bore the image of a beautiful, blue eye, containing the reflection of an angel's wings. It's tide was _Eyes of an Angel_.\n\nI had no idea what this was supposed to mean. Was this a book the angels wanted me to read? Scanning the cover for the author's name, I couldn't believe my eyes. At the top of the book was my own name.\n\n\"No way!\" I instantly protested. \"I'm not writing a book. Besides, what do I know about angels?\"\n\nThis had been a long-standing argument of mine. A number of friends had previously suggested that I write a book about my spiritual experiences. And I had always resisted the notion. There were lots of good metaphysical books on the market. Most of the things I'd experienced had already been written about. I didn't feel I had anything new to offer.\n\nIn the previous two years, three different psychics had predicted that I would write a book. In fact, just two months earlier, I had become upset with a psychic who kept insisting that I was going to write a book. It was all she could talk about. She went on and on about a book, until I thought, \"Look, lady, if you're not getting any messages, just admit it and let me out of here. Forget it! I'm not writing a book!\"\n\nNow, it appeared that even the spirit world was on my case.\n\nI continued my mental protest. \"No, I can't do it. I'm not a good writer. Where will I find the time or energy?\"\n\nI had hardly finished enunciating these thoughts when the book faded, and in its place appeared a blank sheet of paper. As I looked on, words began to form on the page, quickly flowing from line to line until the entire page was filled. While this was taking place, I heard a woman's whispered voice. \"Don't worry. We'll help you.\"\n\nWith that, the vision disappeared, and I bolted to an upright position.\n\nI couldn't believe what I'd been told. How could I write a book? I didn't have the time or, for that matter, the desire. And I certainly didn't feel I had the skills necessary for such an undertaking.\n\nAs I boarded the plane for the last leg of my flight home, I couldn't chase the images out of my mind. Me, write a book? It just didn't seem possible.\n\nWe hadn't been long in the air before I fell into a deep, dream-filled sleep. I recall fleeting images of butterflies and Patricia. I dreamed about grass-covered valleys, the ocean, and a dark-haired woman I seemed to have known so long ago. As the plane finally touched down, I awoke with a feeling that these two wonderful beings were somehow connected. Their energies felt so similar, so familiar.\n\n# 13\n\n# Healing Hearts\n\n_And now in slumber's sweet delight_\n\n_Dream angels carry us in flight_\n\n_When there are no more souls to mend_\n\n_Who is the angel God will send?_\n\nBack from Virginia, I felt like I had returned from a lifetime away. Nothing seemed to be the same. I was having a great deal of difficulty picking up where I left off, and determining how I fit into the scheme of things. Everything around me seemed too harsh, too crowded, and too noisy. My biggest problem, though, was figuring out how I should be feeling. Completely open, emotional, and vulnerable, I wasn't so sure I liked it. Things that I previously held as important now seemed insignificant and petty. Each day I spent hours lost in thought, reviewing and reliving the experiences that had turned my world inside out.\n\nThe integration of feminine energy into my life was a lot tougher than I would have believed. When the angels first told me about it, I couldn't even imagine what it would be like, much less deal with the powerful effect it held over me. Through their ministrations and energy treatments, they had facilitated a change in my vibration. But most significantly, they had opened my heart to accept and embrace the feminine energies within my being.\n\nAlthough worried about her reaction, I shared all of the week's experiences with Candace. From the meeting with Patricia, to the emotional reunion with my angels and the sensuous encounter with the female spirit, I told her everything. Her reaction was amazing. Although she admitted to a bit of jealousy, Candace found the account fascinating and intriguing. She thought it would be wonderful to meet someone with whom you felt a soul connection. She also thought that I should stay in touch with Patricia. I was sincerely grateful for her thoughtfulness and understanding.\n\nAs the days went by, I became obsessed with thoughts of Patricia. Wondering about the possible spiritual connections we shared, I yearned for answers. Why had I been so affected by her? Were there any special reasons for our meeting? The questions were endless and bewildering.\n\nOn top of problems focusing on work, at night I had difficulty sleeping. I would lie awake for hours, fighting the barrage of thoughts that often kept me awake until dawn.\n\nSeveral nights after my return home, I settled into bed with the usual scattering of images cluttering my mind. As I began to drift into the twilight of sleep, thoughts of Patricia surfaced in my awareness. In frustration, more to myself than anyone else, I pleaded, \"Why can't I get her out of my mind?\" From out of nowhere came a distinct, whispered reply.\n\n\"Because she is a part of you.\"\n\nThe presence of my angel guides swept through me in a wave of energy. Although I heard their words clearly, I struggled to grasp the meaning. \"She's a part of me? What's that supposed to mean?\"\n\n\"You and she are of the same energy.\" The soft female voice was lucid and precise.\n\nPuzzled, I asked, \"The same energy?\"\n\n\"You and she share the same soul energy. You are what some refer to as soul mates.\"\n\n\"Soul mates?\" The words reverberated in my mind. I felt certain that Patricia and I had known each other in a previous existence, but I sure wasn't prepared for a soul mate connection. Even though Meldor had once described the concept to me, I gave it little consideration. Now I was stunned by the revelation.\n\n\"You and Patricia,\" the angel continued, \"are part of the same individuated energy. The intense emotional pull that you feel is the memory of another part of you that is in harmony with the total vibration of your being. She is the balance of your energy. When two aspects of the same energy source meet in the physical world, the reaction can be powerful and unsettling. That is why you feel the way you do. Your reunion in this lifetime is not by accident or chance, but rather due to specific purpose and need. Through your heart, you will soon come to realize that need and purpose.\"\n\nAs the angel spoke, her voice began to fade, and I found myself struggling to stay awake. The more I fought to maintain consciousness, the further I sank into the heaviness of sleep. The next thing I knew, it was morning, and I was awakening from a dream of a hauntingly familiar classroom and a group of wonderful, loving, light-beings.\n\nThroughout the day, I carried with me an odd feeling of homesickness. It left me perplexed. Which home was I missing?\n\nAlthough the angelic message had brought me some relief as to the nature of my connection with Patricia, it created a multitude of new questions. The concept staggered me. Had I truly found a soul mate? But there was an even bigger question: \"Now what?\"\n\nWe had promised to keep in touch, and thanks to e-mail, we did. As the weeks sailed by, Patricia and I learned more about each other and we quickly became the best of friends.\n\nCandace remained supportive and understanding throughout. Firmly believing that our meeting was not just a coincidence, she encouraged us to explore our spiritual connection.\n\nWhile all of this unfolded, a strange mixture of feelings haunted me. It seemed that just about everything affected me in an overboard, emotional way. Where in the past I could suppress or ignore my emotional responses, I now found it impossible. I began to feel and internalize even the slightest bit of emotional pain in the people around me, especially my family and friends. The way I reacted or thought about things was changing dramatically, and there didn't seem to be a thing I could do about it.\n\nThrough the years of raising children, Candace and I never seemed to have had much time to just sit and talk. But following my return from Virginia, we would spend hours each day chatting about our goals, relationships, and spirituality.\n\nI found that the connection with my new guides, or angels, was much more subtle than my previous associations with Meldor. Reaching their vibrational state seemed more difficult. I knew they were always with me, but their communication was mostly soft and gentle. I very seldom saw them, yet in times of need I would often hear their thoughts in my mind, always reassuring, always loving, and always feminine. Sometimes I had the impression that if something was in my best interests, they simply made it happen.\n\nAll of these things added to the raw emotion I was already experiencing. There was no question about it; the angels had certainly managed to open my heart, and I felt every ounce of the love and pain that came with it.\n\nIn the months that followed, I grew more and more homesick. Hoping it might provide some clues to my nostalgia, I reread my old notes, especially Meldor's discussion of soul mates and soul groups. The answers, however, were not apparent. With the persistent sadness continuing to weigh heavily, it became obvious I was going to have to work through it myself.\n\nOne night as I lay in meditation, I suddenly found myself floating in a bright, vibrant space. Before me, a massive, rotating sphere of energy began to materialize.\n\nI had no idea what, or who, it was, but something about it was so familiar, it started a reaction within my body. Before I knew it, my own energy had begun to change. Within seconds, I was vibrating in harmony with what appeared to be a single colossal spirit.\n\nAlthough it didn't seem rational, I had the urge to allow myself to become absorbed into its energy. Like a giant magnet, drawing me ever closer, it exerted a compelling power over me. But just as I was about to give in to the pull of this giant being, it began to soften and change in appearance. Soon I could discern a number of individual, swirling bright spots within its aura. As I drew closer, some of these oscillating energy clusters began to break away from the larger body. Within moments I was surrounded by a group of bright, shimmering forms. Then, one by one, they transformed into distinct physical beings. Faces appeared, male and female, smiling reassuringly, their loving radiance melting into my heart.\n\nSlowly, feelings of recognition and longing came over me. Then suddenly I understood. This was the source of my homesickness. These were the members of my soul group\u2014my classmates, my spiritual family. They knew me intimately, perhaps better than I knew myself, and they loved me unconditionally.\n\nOverwhelmed with the emotional energy of their combined presence, I remained motionless, absorbed in the power of their vibrations, wishing I could stay. In the euphoria of their blissful energy, my awareness began slipping away, and momentarily I succumbed to sleep. An instant later, I was back in my body.\n\nI didn't want the experience to be over. Fighting to maintain the connection, I struggled to hold onto their vibration. For a moment my third eye reopened to bright smiling faces. Then the vision faded and I was alone in the darkness of my bedroom. Tears flowed from my eyes. Whether from sadness or happiness, it was impossible to contain my emotions. Hugging my pillow, I cried until sleep again overcame me.\n\nIn the morning I awoke with vigor and renewed direction in my life. Reflecting on the significance of my soul group encounter, I understood they had come to support and encourage me. Awed by the reality of their existence, buoyed by their unconditional love, I felt immensely better. A burden had been lifted from my heart.\n\nPatricia and I continued our correspondence. The bond between us was too powerful to ignore. Anxious to see each other again, we came upon a solution\u2014a spiritual conference in Utah. With my wife's encouragement, I agreed to go.\n\nCandace never failed to amaze me. Although I'm sure she struggled with the possible risks this could present to our marriage, she didn't believe in coincidences. Convinced there must be an important reason for our meeting again in this lifetime, she felt we'd be ignoring our own spiritual callings if we didn't try to find some answers.\n\nThe beauty of southern Utah was stunning. Surrounded by gigantic, naturally formed rock statues and canyons, I had never been to a place more spiritual. In awe of its immense, desolate beauty, Patricia and I found the sanctity of the place intoxicating.\n\nWe knew our time together would be short. Outside of the workshop, we spent every available minute deep in conversation, learning more about each other's lives and dreams. Our connection deeply spiritual, even though Patricia was still skeptical, I became increasingly intrigued at the possibility of exploring any past lives we might have shared.\n\nFor several years I had been investigating numerous forms of consciousness exploration. An important aspect of my research dealt with various modes and applications for hypnotism. Experimenting with a number of possibilities, including past-life regression, I had occasionally enlisted some of my own family and friends to develop a few of my own techniques. At the end of the conference schedule, Patricia and I decided to experiment to see what we could uncover.\n\nAs Patricia relaxed on a couch, I guided her through a process of ever-deepening relaxation, leading her into an altered state. The plan was to move back in time, through various stages of her life, until she was able to re-experience herself in the womb. Upon reaching that point, I would then suggest she move even further back, to a time prior to this present life, where she could conceivably recall previous lifetimes. At least that was the plan, but it didn't quite turn out that way.\n\nThe session seemed to get off to a good start. Through suggestion, I encouraged Patricia to return in her mind to a happy time when she was 16. Although I hadn't asked her to describe what she was experiencing, from the contented, animated expression on her face, I could tell that she was indeed reliving a happy time. Allowing her to enjoy her memories, I then gave the suggestion that she return to an earlier age, when she was perhaps nine or ten. At this point a change began to take place.\n\nAs Patricia settled into the suggestion, her eyelids started to twitch and the expression on her face shifted to sadness. Slowly moving her head from side to side, she looked as if she was trying to block what she was experiencing. I asked her to tell me what was happening.\n\n\"It's my puppy,\" she said in a slow, faltering voice.\n\n\"Your puppy?\" I repeated.\n\n\"Yes, he's just sleeping in the yard.\"\n\n\"How old are you?\" I asked.\n\n\"I'm nine and a half,\" she replied. Her voice seemed distant, almost childlike.\n\n\"Tell me about your puppy,\" I said, trying to sound upbeat.\n\n\"He's so cute,\" she responded quickly, her voice a bit brighter. \"He's just little, only four months old.\"\n\n\"He sounds like a very nice puppy,\" I offered. \"Was he a good puppy?\"\n\n\"Yes, he was good,\" she responded adamantly, almost defensively. And then she became very quiet, seemingly deep in thought. After a few moments she continued, \"Well, he was mostly good.\" Her voice trailed off.... She seemed to be struggling with her emotions. Her lips began to quiver. She was trying not to cry. \"He didn't mean to be bad. He was just sleeping.\"\n\n\"Did something happen to your puppy?\" I asked softly.\n\n\"Yes! Something bad,\" she sobbed. \"Something very bad.\"\n\nWith her eyes scrunched up and lips quivering, she now seemed to be fighting for control. Taken aback by the raw power of her emotions, I was unsure how to proceed. I knew that I needed to calm her down.\n\n\"It's okay, Patricia,\" I said, touching her hand to comfort her. \"There's nothing to worry about. This is something that happened a long time ago. You don't have to experience it again if you don't want to. Take a deep breath now, and just allow yourself to relax. Let all of your unhappy feelings go.\"\n\nWithin a few minutes, Patricia appeared to be resting comfortably. I wondered if it would be helpful and healing for her to revisit the trauma she had obviously experienced with her puppy. Gently, I broached the subject again.\n\n\"There, now that you're relaxed and comfortable again, if you want, we can talk about your puppy. You don't have to be a little girl again. You can just watch what happens as if you're seeing it on television. Any time you want, you can just turn the TV off, or change the channel. Would you like to tell me more about your puppy?\"\n\nAlmost immediately her body stiffened, her breathing became shallow. \"No, please,\" she whispered pleadingly, \"I don't want to see it. It's awful.\"\n\nConsidering the emotional state the mere mention of the puppy elicited, I thought it best to move away from the subject. If it was important for her to relive the experience, she could explore it again another time. Slowly and carefully, I counted her out of her trance. After a few moments of quiet, she stretched her arms and neck, and finally opened her eyes.\n\nI felt bad that she had had a negative experience, and wanted to console her. I told her I was sorry about how the experiment had gone, and hoped that she wouldn't let it deter her from trying again in the future. With a puzzled look on her face, Patricia, however, said she was glad for the experience. She said it must have been previously blocked from her mind, but somehow she could now remember everything. It had been a very sad moment in her life, but she felt the need to share it with me.\n\nTaking a deep breath, Patricia launched into the most compelling and heart-wrenching story I have ever heard.\n\n\"It happened the summer that I was about nine years old,\" she began. \"My dad and my little brother and I were living on a small parcel of land on the side of a mountain in Oregon. We had a stallion that we kept in a corral close to the house. Sometimes the horse would get out of the corral and run away. Our closest neighbor was a bit further down the mountain, about a half-mile away. He kept a couple of mares in his pasture. Whenever our horse got away, that's where he would go. It had already happened a number of times and our neighbor was getting really mad about it, and that bothered my dad.\n\n\"One Saturday afternoon dad went into town to get some things, and left me to look after my brother. The puppy was just a few months old then. It was a nice warm day, and he just curled up and went to sleep in the middle of the yard. The horse wasn't in the corral. My dad had tethered him to a stake outside the corral so he could eat some grass in the yard. The horse was grazing along when he noticed the puppy. He bent his head down and sniffed at him. It must have startled the puppy, because he jumped up barking. That frightened the horse. He bucked back, yanked the stake out of the ground, and started to run away. I tried to stop him, but I couldn't, and he ran all the way down to our neighbor's property.\n\n\"When my dad got home, I told him what happened. He became really angry, and drove down to the neighbor's yard to get the horse.\n\nWhen he got back he was even madder. After he put the horse in the corral, he came across the yard, grabbed the puppy, and tied his leash to a clothesline post. I didn't know what he was going to do. I was sitting on the front step of the house. I told him it wasn't the puppy's fault, but I didn't say anything else because I'd never seen him so angry.\n\n\"He walked around the side of the house, and I couldn't see him for a while. I thought he was going somewhere. But a few seconds later he came back around carrying a steel pipe.\"\n\nAt this point Patricia's lips began to quiver. She was straggling to control her emotions. \"And, and then,\" she stuttered, \"he started hitting the puppy with the pipe. I screamed, but I couldn't do anything. The puppy was crying, but he just kept hitting him. I didn't know what to do. He was so mad. I ran into the house and hid in my room. But I could still hear the noises. So I just put my hands over my ears and cried and cried.\"\n\nTears now flowing down her face, Patricia began to shake. I couldn't believe what I was hearing. I saw only a terrified little girl, traumatized by what she had witnessed. Choking on a burning lump in my throat, I clutched her hand tightly.\n\n\"Oh God, please!\" I found myself silently begging for the strength to remain in control. At that moment, I couldn't even imagine how awful this would be. But Patricia's horror wasn't over, far from it.\n\nHyperventilating, with tears streaming down her face, she continued, \"And then my dad started yelling at me. He said, 'Patricia, you better get out here right now, and bury this goddamn dog.' I was so scared. I'd never seen him like this. But I had to go outtside. I picked up the puppy; he was covered with blood. I didn't want to even look. I took him to an old flower garden behind the house, and I got the shovel. Then I dug a hole and put him in.\"\n\nHer speech now punctuated with deep uncontrollable sobs, Patricia forced herself to continue. \"I started to cover him up. But when I threw the dirt on him, he started to move.\" Her voice turning to pure horrified anguish, she cried, \"Oh God, I'm so sorry! I couldn't stop. I couldn't look at him. I just kept on covering him up. And then I just ran and ran, and I didn't want to ever come back.\"\n\nFinally, with heart-wrenching sobs, Patricia turned away, buried her head in her arms and wept.\n\nSeconds turned into minutes, and the minutes into eternity.\n\nMy heart was breaking. Gasping for breath, I couldn't think what to do. My own tears were now streaming down my face, falling like rain onto my hands. I drew her close, cradling her in my arms. I wanted to pray to God to take away her pain, but instead I found myself silently cursing a humanity that could inflict such devastating pain.\n\nWith my own grief and anger slowly boiling into a rage, I was suddenly jolted by an internal burst of light. Brief flashes of imagery began to strobe through my mind, piercing my awareness. For a moment I wondered if I was having a stroke. Then in the next instant, my inner vision cleared, and I was rocked by a memory. It was the uncovering of my own horrific trauma. Somehow, I had blocked this memory from my mind.\n\nI was suddenly back in my five-year-old body. It was early on a Saturday morning, and my older sister Josie was shaking me awake. \"Come on, Paul, the puppies have been born. Don't you want to see the new puppies?\" Half awake, but excited, I scrambled to my feet. Josie pulled a sweater over my head, hurried me through the house, and out the front door.\n\nMy excitement mounting with each step, we ran across the farmyard to an old shed behind the workshop. We had two adult German Shepherds on our farm. My sisters had told me that Sheena was pregnant and would be having her babies very soon. I couldn't wait to see them.\n\nBefore we even rounded the corner of the shed I could hear little whimpers from the newborn pups. Spotting them immediately, I counted four little balls of fur. In amazement, I sat down beside them as they bumped and bobbed, nosing through their mother's fur. Josie reached into the churning brood, gently withdrew a whimpering, shivering ball of fluff, and placed it in my lap. He was beautiful. I giggled with delight at his closed eyes and clumsy movements. It was a perfect moment.\n\nWe had been sitting for several minutes holding and playing with each of the puppies when around the corner of the shed strode our father. \"What the hell are you doing back here?\" he barked. In instant fear, Josie jumped to her feet. Catching sight of the puppies, he let out a stream of German cuss words. \"Jesus Christ, I thought so,\" he yelled. \"Josephine, you're supposed to be helping Mom in the garden.\"\n\nIn a rage, he grabbed the puppy out of my lap. \"I told you I don't want any more dogs around here,\" he screamed. And swiftly raising the puppy over his head, he smashed it to death on the ground in front of us. One by one, he grabbed the rest of the puppies, hurling them to their deaths.\n\nFrozen with shock and fear, I couldn't speak, I couldn't cry. I just sat there shaking. The next thing I knew I was in Josie's arms and she was running as fast as she could towards the house. She put me down on the porch, and through terrified sobs, ordered me back to bed. I couldn't get there fast enough. Racing through the house, I dived under the covers. Mercifully I fell asleep, putting the entire horrible memory forever out of my mind.\n\nCutting through my thoughts, I felt a hand touching my face, wiping away the tears. I opened my eyes. Patricia was looking up at me, blinking through her own tears, a deeply saddened, concerned expression on her face. \"Are you okay?\" she whispered.\n\n\"I think so,\" I managed to croak out. \"God, Patricia, I'm so sorry you had to go through something like that.\"\n\n\"Shhhh,\" she hushed me, putting her fingers to my Hps. \"It's okay now.\"\n\nTrying to get my wits together, I decided not to tell her about my experience until later. I knew we would have to help each other in our healing.\n\n\"What happened when you ran away,\" I finally asked. \"Where did you go?\"\n\nPatricia sighed and took a deep breath. \"I didn't go very far. I ran up into the mountain where there was a stream about a half-mile away. I used to go there sometimes to just be by myself. But, this time I wanted to die\u2014I wanted to just disappear.\"\n\nHer lips began to quiver again as she fought to remain in control. The pained expression on her face ripped into my heart. She seemed so small and vulnerable. It felt like I was looking into the eyes of a little girl. I gently squeezed her hand and, with tears welling up in her eyes, she continued her story.\n\n\"At first I was just going to wash the blood off my clothes, but I thought how easy it would be to die. So I walked right into the water. I wanted to drown myself. But I just couldn't make myself do it. Finally, I got out of the water and just sat on the bank for a long time. The sun went down, but I stayed there. I didn't want to ever go home again. It was a few hours later, maybe two in the morning; it got so cold I was shaking. I walked down the mountain to a logging-truck road about a half-mile from our house. I wanted to go all the way down to the highway and run away, but I didn't know where to go. I just couldn't get it out of my mind. It was so horrible. I wanted to die.\"\n\nPatricia put a hand over her mouth to muffle a sob. After a few moments, with tears freely flowing, she continued. \"There was a curve in the road as it came up the hill, so I lay down in the middle of the road, hoping a truck would kill me. But no one came. It was really dark and cold. I couldn't cry anymore and I didn't know what else to do, so after about an hour, I got up and walked home. And that was all.\" With that, Patricia put her head back, closed her eyes, and lay quietly with her thoughts and tears.\n\nAfter a few minutes, when she appeared to have settled down somewhat, I broke the silence. \"What do you mean, that was all?\" I asked. \"Wasn't your dad worried and looking for you? Didn't you get into trouble?\"\n\n\"No, I thought I would. But nothing happened. There were no lights on in the house. Everyone was sleeping. So I just crawled into bed and went to sleep. And that was all. The next morning, life went on like nothing happened. Nobody ever spoke a word about it. You're the only one I've ever told.\"\n\nI sat quietly, holding her hand. I thought about the unbelievable horror that she had endured. What disturbed me most was that she had spent the greater part of her life carrying a guilt that she had been responsible for her puppy's death when she covered it with dirt.\n\nIt occurred to me that just sharing her story was probably a big step in a healing process, and I was grateful for having played a part. I felt fortunate my own similar childhood horror had also been blocked from my memory. It had been strange, recalling an experience that had been erased from my conscious mind.\n\nFeeling that we were both cried-out, I told Patricia about uncovering my own blocked experience with the puppies. By the time I finished the story, we both sat with tears in our eyes and lumps in our throats. We reflected on the similar nature of our experiences, wondering what effect these events had in shaping our lives. We considered that perhaps this was one of the reasons we had been brought together again, to initiate a healing process.\n\nEven though we felt better, it was apparent that we both still carried a deep psychological hurt. The trauma of our experiences had forever changed the way we related to the world. It was as if an innocent, sacred part of our souls had been damaged or lost. The discussion brought to mind the training I had received at The Monroe Institute during the soul retrieval sessions. Some of the participants had talked about retrieving parts of their own souls. They felt that during particularly traumatic events, it was actually possible to lose a part of one's own spiritual energy. They believed that these shadow bits of our soul could be locked into the emotional time-vibration of the event itself. But they also claimed that it was indeed possible for an aware person to find and reintegrate these lost aspects of energy into his or her soul. Doing so often resulted in tremendous healing.\n\nPatricia and I talked about the possibility of trying something of this nature. Although she was dubious, we agreed to experiment. Unsure where it would lead us, we decided on another hypnosis session.\n\nWith only a loose idea of the direction we were heading, I guided Patricia through a process of deepening relaxation, and then counted her into a stable hypnotic trance. Allowing her to rest, I closed my eyes and prayed for guidance. Almost immediately, I detected the subtle vibration of my guides.\n\nThoughts and ideas forming in my mind, I began with the suggestion that whenever she was ready, she could return safely to any time in her life. She could review and re-experience any event or incident, but if she chose to do so, she could also remove herself from the scene and become only an observer. With these final suggestions, we were ready to explore.\n\nSlowly, gently, I guided her back to the awful day her puppy died. I reinforced the concept of retrieving any lost aspects of her energy that may have left her during times of extreme stress or trauma. Allowing that it could very well be that she had no missing aspects, I suggested that if there were, she should locate these lost portions and restore them to her soul.\n\nIn anticipation of what was to come, Patricia took a deep breath. I then took her back to the time immediately after the puppy was put into the ground.\n\n\"Patricia,\" I began, \"you told me that after the puppy was buried, you ran up into the mountain to a stream. Do you remember how to get there?\"\n\n\"Yes, I know where it is,\" she replied, her voice registering a touch of anxiety.\n\n\"You'll be just fine, Patricia. Nothing can hurt you.\" I tried to sound as soothing as I could. \"Now, in your mind, return to that place near the stream where you went years ago. I'll just be quiet so you have time to make your way there.\"\n\n\"I'm there.\" Her immediate response surprised me.\n\n\"Very good.\" I tried to remain calm. \"Now, take a few moments to look around and tell me what you see.\"\n\nPatricia's eyelids began to twitch ever so slightly. There was a pause in her breathing. She seemed to be concentrating on something. \"I see a little girl sitting on some dead-fall beside the water.\" Her voice was slow and absorbed. \"There's a nice lady sitting beside her, but I don't think the girl can see her.\" For a moment, she paused, captivated by what she was seeing. \"Her pants and shirt are all wet,\" she said, coughing slightly, a sound of concern in her voice. \"She's shivering, and she's crying.\" Patricia lay quietly for a couple of moments, and then with quivering lips, she added, \"I think she's me.\"\n\nI could hardly contain my excitement. This was just what we had hoped to accomplish. I searched for the right words to continue the process. \"You're doing very well, Patricia. How do you feel now?\"\n\n\"I feel really sorry for her.\" Her voice was filled with emotion. \"I just want to hug her.\"\n\n\"I think that's a good idea,\" I encouraged. \"I'm sure she could use a hug.\"\n\nFor several moments everything was quiet. The tension in Patricia's face began to soften. Then in a mystified voice, she broke the silence. \"She's gone... I was hugging her and then she just disappeared.\"\n\n\"That's wonderful, Patricia,\" I whispered, struggling to control my emotions. \"How do you feel now? Are you okay?\"\n\n\"Yes, I feel a lot better. I'm really glad I found her.\"\n\nTears filled my eyes as I contemplated the power of what I had just witnessed. I knew that regardless of the validity of the experience, whether it was real or imaginary, it must have been healing.\n\nUncertain what to do next, I suggested that Patricia relax and reflect on all that she had experienced. While she rested, I decided on the next step. I would attempt to guide her back to the dark road where she had lain in the gravel waiting for a truck and then finally to the place where she had buried the puppy.\n\nThe process seemed to go well. In the reality of her altered state, Patricia returned easily to the road and then to the puppy's grave. Although she couldn't perceive any further lost aspects of her soul at either site, I encouraged her to go through the process of welcoming any possible stranded energy to reunite with her. When she arrived at the site of the puppy's grave, Patricia began to re-experience some strong emotions, but she was able to maintain control, making it through the difficult final processing.\n\nWith the main purpose completed, I suggested that she could instantly travel to any peaceful place she chose. This would give her a few minutes to relax before I brought her out of her trance. Patricia drew in a deep breath, and I watched all signs of tension slowly leave her body. Although curious about where she had gone, I remained quiet. Within minutes her breathing had slowed to a steady relaxed pace. Appearing at rest, her eyelids began to lightly flutter and an expression of peaceful contentment spread across her face. I waited a couple of minutes, and then began the process of returning her to normal consciousness.\n\nStretching her arms, Patricia opened her eyes. A delighted smile spread across her face. \"Wow, that was incredible,\" she exclaimed. \"I don't know where I was, but the most amazing thing happened.\"\n\nConcerned that she may not have remembered all of the aspects of the session, I asked, \"You do remember everything that happened, don't you? You remember finding a lost part of yourself?\"\n\n\"Oh, yes,\" she smiled, \"that was incredible too. But at the end when you told me to just rest for a few minutes, that's what I'm talking about.\" Her eyes beamed with excitement. \"I don't know where I was, but all of a sudden I felt somebody holding me, and I felt really different. It was like I was a baby again. There were people standing in a circle. For some reason, I couldn't see much of their faces, but they were so happy, just hugging me, and passing me around. It was wonderful. I could feel all of their love, and it was just beautiful. Then they started to fade away and I fell asleep, until you woke me up.\"\n\nPatricia sat up, a puzzled look spreading across her face. \"They seemed so familiar, like I knew them, but just forgot. I think they were from my soul group. What do you think?\"\n\nThis was completely unexpected. Not knowing what to say, I just sat there, grinning. Frankly, we couldn't have hoped for a better outcome. As I reached out to give her a hug, I silently thanked God and our angels for their help.\n\nThe episode had been healing for both of us. Although it was Patricia who had gone through the exercise of uncovering a childhood trauma, her experience was the trigger that allowed me to do the same. Her ordeal had helped in my healing by putting me in touch with my own forgotten past. We had shared a powerfully liberating experience. And we were in awe of the spiritual bond that had brought us together at a time when we needed it most.\n\nPatricia's experience with the rest of her soul group had been particularly gratifying for her. Even though I had shared with her Meldor's insights regarding soul groups, she had remained unconvinced. Now, it seemed, we had crossed another bridge.\n\nOur time together in Utah seemed all too short. Before we knew it, we were at the airport saying goodbye again. With heavy hearts and a few more tears, we hugged each other and headed off on our separate journeys home.\n\n# 14\n\n# The Reading\n\n_Though late, we find that in the end_\n\n_Matters not if we break or bend_\n\n_It wasn't God we were meant to see_\n\n_For only love can set us free_\n\nThroughout this intense period of change, I continued to share everything with my wife. I'm sure there must have been times when Candace simply humored me and only pretended to listen. She seemed happy that I had found such a wonderful friend, and occasionally she and Patricia spoke on the phone. The depth to which Candace supported our friendship was remarkable, and I was grateful for her understanding. Within a few weeks of my return from Utah, however, she had a dream so vivid, it drew her irretrievably into a relationship that may have spanned centuries.\n\nWhen Candace first told me about the dream, I hadn't offered her more than an amused snicker. It did, however, cause me to worry that beneath the surface, she was more than a little uneasy about my friendship with Patricia. At least, that's what I believed at the time.\n\nCandace recorded the dream in her journal:\n\nI had a dream last night, and woke up at 2:00 A.M. in a cold sweat. I was feeling really distressed. In my dream, Patricia had come to our home; I believe it was to help Paul with some project he was working on. I had no idea prior to her arrival that she was coming, and was not introduced when she arrived. We ended up being seated side-by-side at the dinner table, but it was like there was a very dark shadow between her face and mine, which didn't allow us to look at each other. I interpreted this as Paul's reluctance for us to meet.\n\nAfter the meal, I decided it was time to cut the foolishness, so I shook her hand and introduced myself. The next thing I recall, Paul and I were alone, and I was asking him how long she would be staying. He ignored my question and instead took me to task for being rude to a couple of men who were doing some work around the house. At that point, I held his face in my hands, and said to him over and over again, \"When are you going to admit that you don't love me anymore?\" He denied it several times, and then finally said, \"All right, I don't love you anymore.\" I felt instantly relieved. I said \"Good, now I can get on with my own life.\" That's when I awoke.\n\nAs I woke up, my mind was still in turmoil and I was quite upset. But I had the overwhelming sense that everything was okay. And then I heard these distinct words in my mind, \"Don't worry, this was another lifetime.\"\n\nCandace told me it was the clearest, most powerful dream she had ever had, but admitted she didn't know what to make of it. Never before had she experienced what she felt was direct contact with the spirit world. She joked about the possibility that all three of us could have known each other in previous lives. For my part, I felt strangely uncomfortable with the idea. I swept the notion aside. \"It was just a dream,\" I said. \"I wouldn't read too much into it.\" Dreams, I argued, were mostly symbolic. There could be any number of interpretations for what the subconscious mind is experiencing.\n\nCandace, however, wasn't so quick to write it off. She explained that the dream hadn't taken place in any of the homes we had ever lived in. This one was a very large two or three story home similar to those she had seen in movies of the old South. She sensed that we had employed a number of servants in a plantation-type setting. Although she wasn't sure how to interpret it, she felt the dream was too vivid to ignore.\n\nA couple of weeks later, Candace surprised me by announcing that she wanted to attend a program at The Monroe Institute. We had talked about it before. Although she always felt it would be a good experience, she was waiting for the right time. Now she was ready to give it a try.\n\nDelighted with her decision, I knew that she would have a wonderful time. Hopefully, she could experience firsthand some of the things I had shared with her. When the day finally arrived, it was hard to tell who was more excited, Candace or me.\n\nFor me, the week seemed to drag on forever, but when she finally got home, Candace's eyes were beaming with wonder and excitement. We spent hours talking about the life-changing insights and revelations she had experienced. But there was something that she particularly wanted to tell me. She had shared the story of the soul mate connection between Patricia and me with a couple of the other participants. One of them had done a lot of past-life work, and seemed to possess a remarkable psychic ability. She told him about her unsettling dream, and asked if it was at all possible that she and Patricia had met in a previous lifetime.\n\nClosing his eyes to contemplate, he replied, \"Yes, definitely!\" He went on to suggest that the three of us had actually been involved with each other over several lifetimes. His guidance told him that in their most recent connection, Candace and Patricia had in fact been brothers. A serious rift had developed between the two of them, and Candace had supposedly treated Patricia very badly-during their last time together.\n\nI'm not sure if I was more surprised or amused by the revelation, but I confess to not giving it any serious consideration. It seemed all too contrived, and I brushed it off as absurd. Although I had visited a few psychics over the years, none of them had impressed me. I surmised that this guy was just blowing smoke. The whole notion was a bit too much of a stretch. \"Well, you can think what you want,\" Candace smiled, \"but I think there might be something to it.\"\n\nMy life continued as always: busy, busy, busy. By now, I was becoming accustomed to the strange anomalies and revelations being delivered to me on a regular basis. Sometimes it felt like I was being guided through a specific curriculum. Lessons were laid out for me, but only when I was ready to accept them.\n\nOther than my obsession with reading, the only other pastime I enjoyed was watching movies. I love the big screen and the surround-sound stereo. On one occasion, Candace and I were engrossed in Kevin Costner's movie _Message in a Bottle_. Near the end, there is a particularly dramatic scene in which Costner leaps into the raging sea to save a drowning woman. At the time, I had no idea why it was having such a powerful effect on me, but well before the fateful moment, I found myself becoming emotional. My heart began to pound, and I grew anxious.\n\nTrying to get a grip on my emotions, I closed my eyes. I felt a shift in my perception, and in an instant I was in a small boat caught in the midst of a tremendous storm. In horror I watched as a huge wave slammed over the bow, snatching a young woman off her feet, sweeping her overboard into the dark churning waters. Terrified, I screamed her name, \"Yoshi!\" and dove into the water after her. The next thing I knew, I was beneath the waves fighting for air. Then as quickly as it had begun, I was transported back to my theatre seat, tears streaming down my face.\n\nThe rest of the movie was a blur. Sitting in shock, I knew without doubt that I had been shown yet another scene from a previous lifetime. All of it seemed so real. But what was most surprising was my intuition that the young woman I had tried to save was in this lifetime my daughter, Stacey.\n\nReplaying the scene in my mind, I was puzzled by the questions it raised. Especially unclear was the nature of our relationship in that lifetime. The fact that Stacey is now my daughter only served to confuse the possible scenarios. She could have been my daughter, or maybe even my wife; I didn't have a clear sense of it. I knew only that she had been precious to me. The one detail that was clear, however, was the distinct feeling that we were Asian.\n\nOver the next few days, the past-life episode was often on my mind. Although I tried several times to connect with my spirit guides for clarification, all of my attempts were unsuccessful. Three days later, I awoke from a fleeting dream that involved both Stacey and Patricia. Although I recalled very little detail, it awakened my mind to a new concept. Since Patricia and I were connected through our souls' energy, and Stacey and I had shared previous lives, was it possible that Stacey and Patricia had also shared past lives? I got out of bed with the conviction that this was a very real possibility.\n\nBy the time winter had tightened its icy grip on the prairies, Candace and I were looking for a break. We had friends in Calgary, Alberta, and we needed a change of scenery. Our friends had told us about an excellent Tarot reader named Janine, encouraging us to see her. Although we knew nothing about Tarot, just for fun, we decided to give it a try.\n\nJanine worked out of a small room in the back of a downtown bookstore. She blew away every doubt or suspicion I harbored about professional psychics. It was the most accurate reading I had ever witnessed. Neither vague nor generalized, she effortlessly nailed down every aspect of my physical and spiritual life.\n\nTo give her a good run for my money, I decided to remain tight-lipped throughout the session. As we sat down to begin, I supplied only my first name and date of birth. Janine shuffled the oversized cards, asked me to cut the deck, and then laid them out in an organized spread across the table. A tape recorder captured every word:\n\n**Janine** : Okay, in Chinese, that's the Year of the Rabbit. So, we can combine that, Gemini, and the rabbit. Those are very similar. It's like being a double Gemini. Do you use a lot of verbalization in your work?\n\n**Paul** : Yes, I suppose so.\n\n**J** : And what about writing? Because, I feel that you definitely came here to communicate ideas through writing. It feels like you're full of ideas, and you need to be writing.\n\n**P** : That's possible.\n\n**J** : There's something going through a big birthing here, well, like yourself, your own energy. It's very emotional. And it's a spiritual thing ... can you feel it coming? Like you're really changing?\n\n**P** : Yes, I think so.\n\n**J** : Well, it's going to continue. Okay, and there's a huge healing going on around the ego self. There is a sense of healing. Are you headed in the direction of doing some healing yourself, or in healing others?\n\n**P** : Sometimes healing others, yes, and myself.\n\n**J** : Are you working in a community right now? Is it a spiritual community or are we talking of a physical world community? There's definitely a sense of bringing people together, being a liaison even. Like you're a politician or diplomat in the work you do. You are a Gemini trying to integrate your male and female parts in this lifetime, and really work with those energies in a combined effort.\n\nI was floored. Could this Tarot reader be intuiting the emotional changes that had been taking place in my life? She had even correctly identified that I was a politician. Could she be picking up on the energy work my angels had initiated to bring me into balance with female energies?\n\n**P** : That's certainly true. I've been dealt some things recentiy.\n\n**J** : It seems like you're really integrating, though, and you end up putting into your work in the world that sense of integration, wherever it works for you.\n\nOkay, I keep getting that this is a real healer card here. It's like the healer in you, mentally, coming up. There's a gift here that you're maybe just beginning to use; are you doing anything like hands-on healing or Reiki or anything?\n\n**P** : Yes. I'm trained in Reiki. [Several years earlier, I had taken classes in Reiki energy healing.]\n\n**J** : I get that you have quite a gift and it's necessary to use it more often. This is a healing gift that wants to come out, and it's the more feminine part of you, most certainly. It's like goddess energy, and it's mental and verbal too.\n\nOkay, now I'm getting a victory around a youth. This is a big, positive, trophy card. This could be about childhood issues. Did you just work through a really big childhood issue?\n\nI thought of the repressed childhood memory of the puppies. I could hardly believe what I was hearing.\n\n**J** : Because I get that you've come to a personal victory with some part of yourself as a young person. There is also a big unconditional love card here, and I'm picking up that you've really come to a place of loving yourself. It's not only about loving yourself, but it's going to help you to also love others. This is really nice.\n\nNow, I get a sense of patience, and deep roots in a relationship that, down the road, will grow great harvest. And the more you do your own personal healing work, the more your relationship with another very specific person will grow because of the work you're doing. I get more independence emotionally, but I'm not getting the ending of a relationship; I'm getting a rebirthing or something. Is there a new relationship, or could this be a rekindling of an old one, because this is so much like roots and birthing.\n\n**P** : I don't know. Perhaps both.\n\n**J** : There's beautiful stuff going on there, and I get a fruitful relationship, and it's all because of your personal work, the integration with yourself. Relationships seem to be different, but they're becoming deeper or better.\n\nOkay, here I get a really big restructuring through communication. Have you had a big talk with somebody that first leveled them, and now there's a restructuring going on? Is that in one of your relationships? Almost like something shook up your world and caused you to restructure from the ground level and it's like you're in the process of doing that.\n\n**P** : Yes, that's very accurate.\n\n**J** : Only a Gemini could work that fast. You're working very fast at the restructuring. Some people take years and years. It might be a bit intense, though, a shock to the people who know you. Okay, now did you have a really incredible dream or vision recently?\n\n**P** : Yes, I've had many.\n\n**J** : Yeah, because it's like your dreams are really challenging you to move through fear-based ideas and really get in touch with what your fears are, so you can release the fear and get on with whatever it is you're doing. I get a lot of dreams or visions here; like you're a dream catcher right now and you're receiving a lot of messages through your dreams. It's karmic that this happens this way right now. It's all predestined. Not everything is; we create a lot of our own destiny, but for some reason there's an overriding energy right now of destiny around you.\n\nAnd there is also a big sense of vulnerability around you. It's almost in an intimate sense, somehow. There is a sense of you really being open. Are you feeling like you're open in an intimate way around people? It feels like you're opening up like you haven't in a long time. And you're really worried about it.\n\n**P** : Uh, yes. That's true.\n\n**J** : It's very healing, but it feels like you are very vulnerable at the same time. Your ego is worried that it's going to completely disappear in this process. But, I'm getting that you really heal through this and that your ego sense actually gets stronger in a more positive way. You will learn how to use it properly.\n\nYou need to write. How much do you write now? Ever?\n\n**P** : No, not a lot. But I know that I should.\n\n**J** : I get that this could go back lifetimes. I get you bringing this from a place that you developed quite intensely in a past lifetime or something, and you could tap into that. And it could be public speaking, or working with the public in some way. I get like a lot of charisma and you talking to large groups. I don't know where your spiritual path is going, but this is very New Age, very healing, and [involves] working with groups.\n\nYou will have, or have already had, a big revelation through dreams or some spiritual process. Now it is time to write or speak about it. It will come from your inner voice. You have quite a lot in store, but you've done this before. This is lifetimes of work. It is like the power of many others; it's like you'll be channeling stuff. It's like there's a whole stored up thing going on here that you can tap into.\n\nYour writing could really heal and shift a lot of that energy. You need more downtime to get that process happening. You will definitely publish something. It might take a whole year to accomplish this, but you will.\n\nYour emotions are sometimes high, and sometimes low, so take some cave time. You will get your shift and messages will come. There's some pure channeling going on here. It's already coming through, and it's going to really move you.\n\nActually, I get communication on a worldwide scale. It's like an Aquarian card with a Pleiadian knowledge and influence; you are really tapping into it, and will end up being a channel for a very large body of energy that is universal. This is needed in the world, and it's very healing. And you seem to do it through written word, and verbally, or both.\n\nAt this point Janine picked up the cards and reshuffled the deck. She had been extremely accurate in identifying so many of the changes taking place in my life. From the emotions of balancing male and female energies to identifying the spirit world's assistance in writing a future book, she was bang on.\n\nShe finished shuffling the cards, placed the stack in front of me, and asked if I had any questions. At first I was still too shaken to think of anything, but then I thought of Patricia.\n\n**P** : This past summer I met a unique person at a seminar, and we seemed to have a very strong connection. Do you have a sense of what this meeting or relationship is all about?\n\n**J** : Okay, shuffle the cards and separate them into three piles and indicate with your left hand which pile to use.\n\nI pointed to one of the piles. Janine picked up the cards and spread them out.\n\n**J** : Wow, this is a very powerful feminine card here. I'm assuming it's a woman. She has a powerful energy. There is past life stuff going on here, and it's not bad stuff. She is someone who developed some big esoteric skills in a past life. She could just use them a lot more than she does. This is a very empowered person, with a very sensual energy.\n\nI get that you and she are in a place of healing. There is a sense of finding balance and patience. You are learning to meld and integrate your energies, and you'll be very successful at this healing. This person gives a very strong sense of Scorpio energy. She is very intense and emotional. Has she lived through some hard stuff?\n\n**P** : Uh, yes. Definitely.\n\n**J** : Everything she knows she learned the hard way. This is a survivor who has come into herself through a lot of hard work. But there is a sense of sadness or failure.\n\nThis is a soul intention; one of you is a messenger to the other, or both to each other. Something is coming out that you both need to hear right now. There is a strong karmic reason why you're pulled together, which is part of your work here.\n\nThis is all about integration. It's a very positive influence. This is definitely a soul mate. There is a sense of wholeness when you're together. The challenge, though, is in how to bring it into the world you've created so far.\n\nAnd this is a surrender card. Surrender to what you're feeling, and also let go of your doubting. But expect the unexpected. What is really going on, and what you both think is going on, may be really different here. I get that it's going to be really surprising and unexpected how this manifests and fits into your life. But it's very good.\n\nI get that this person and you were meant to meet again to help each other quantum leap to where you're supposed to be. So there's a bit of bumping you forward, speeding up your process. I think it has a lot to do with physically manifesting a book. I don't know why, but this is about speaking and putting something down in written form. It's like she speeds up your process. You have definitely done lifetimes together with this person and were one and the same energy. This is definitely soul mate stuff here.\n\nOkay, we have time for another question.\n\nJanine again picked up the cards, shuffled them expertly and placed them in front of me. My mind was reeling. I could hardly think, but finally settled on asking her about the possibility of a previous relationship between Candace and Patricia. I still felt skeptical about a past life involving the two of them and I fully expected that Janine would put the whole thing to rest.\n\n**P** : Is there a possibility of a relationship in a previous lifetime between the person I recently met and my wife?\n\nWith a surprised expression on her face, Janine looked up at me from her cards. Her words almost bowled me over.\n\n**J** : What about your daughter and this person?\n\nI could hardly believe what I had just heard. How could she know that I even had a daughter, much less my suspicions about a past-life connection between Stacey and Patricia? I finally managed an almost inaudible reply.\n\n**P** : Uh, quite possibly. I, uh, uh, suspected that there might be.\n\n**J** : Well, I get that this person certainly has been involved with your daughter. Now usually it's all intertwined, but let's have a look. Okay, now think about your wife and this woman. Cut the cards into three piles and indicate with your left hand, which pile to use.\n\nI quickly made my selection. Janine laid out the cards and began interpreting.\n\n**J** : Yes. Definitely. Most definitely, and somebody already got this message intuitively. Somebody has had either a dream or an awareness of the other person and what is actually going on. So somebody actually has all the information ... one of these two people. Have you talked to either one of them about it?\n\nI was shaking. How could she have possibly picked up on Candace's dream? This was too unbelievable. I had to clear my throat before replying.\n\n**P** : Well, we only joked about it. I wouldn't have believed it.\n\n**J** : Well, there have definitely been lifetimes between everybody involved here. It's been quite intense, actually. I'm not getting a very pleasant time on some of these lives. There's some sense of a rift somewhere between the two females, but somebody already knows this. It's like one of them has had a dream scenario that relates to that. Somebody has the full information. It feels like your wife might be the one.\n\nThere's an awareness of what exactly went on, and they may be in some denial about it. And there's some ego struggle here too. But yeah, there is an awareness and it will heal a lot of things between these two people.\n\nThere were some tough times, at least in one lifetime between these two, and it wasn't always pleasant. This is interesting, though; I get that they were both males.\n\nI get that a lot of healing goes on between these two energies that's meant to happen. This is fascinating. But you have to have patience. Over the next six months there seems to be a complete death and rebirth in this whole situation that seems to be quite different than anyone thinks. I get a lot of letting go of guilt and fear. A sense of humor is necessary in this situation, and diplomacy on all sides. And you're right in the middle of it all, _again_.\n\nWith those final words, Janine ended the reading. In a dazed state, I thanked her for the session. It was, without doubt, the most remarkable display of psychic intuition I had ever witnessed.\n\nCandace was waiting for me as I opened the door and made my way back into the bookstore. \"Are you okay?\" she asked. \"You look like you've just seen a ghost. How was your reading?\"\n\nI managed only a bewildered shrug. \"You won't believe it,\" I whispered. \"Wait until you hear the tape.\"\n\nDuring our long drive home, we listened to the recording at least three times. Amazed by Janine's accuracy, we talked for hours about the changes taking place in our lives. There was no question about it: we both shared a renewed reverence and respect for the mysteries of life.\n\n# 15\n\n# Affirmation\n\n_If it's proof you seek, there's no denying_\n\n_The truth is only found in dying_\n\n_But in search of God, and where to begin_\n\n_You need look no further than within_\n\nIf there is one thing that I've come to realize over time, it is the more I learn, the more convinced I am how precious little I actually know.\n\nIf someone had told me just ten years ago that I would become immersed in a life of spirituality, I would have suggested they go for a drug test. But now I couldn't imagine anything better.\n\nFriends often ask if I have any idea why all of these astonishing things have happened to me. The simple answer is, \"I haven't got a clue.\" It would seem that I must have planned it this way. Or, it could be that\u2014as two different psychics suggested years ago\u2014I had set out a special goal to accomplish in this lifetime, but I was procrastinating, allowing self-doubt and insecurity to defeat my purpose. At the time I had no idea what they were talking about. Perhaps because of my procrastination, my guides and helpers decided to put Plan B into action\u2014the one where they virtually had to hit me over the head to get my attention; the one where I had to be dragged kicking and screaming into territory and beliefs I would not have otherwise accepted. I'm still not sure, but one thing is for certain: I'm forever grateful that they did.\n\nSo many inconceivable occurrences have taken place around me that I've come to expect the unexpected. Candace and I often marveled at the steady progression of events and experiences that had been presented to us. There were many times when I couldn't wait for a new day to begin, just to see what would happen next. We were on a remarkable journey, but shortly after we returned from the visit to the Tarot reader, another unexpected piece of the puzzle fell into place.\n\nAs winter finally gave way to spring, we looked forward to warmer weather and green grass. While surfing on the Internet we accidentally discovered a place known as the Omega Institute. Located in a beautiful rural area of upstate New York, it was a former boys' camp renovated and updated to provide a large multi-use facility. Advertised as a center for holistic studies, Omega offers a wide variety of spiritual workshops and seminars hosted by many famous authors and teachers.\n\nWhile Candace and I perused their web site, one particular class caught our attention. Billed as a workshop on soul survival, it was a weekend seminar slated for mid-May. To our delight, one of the main presenters was a favorite author of ours. A prominent psychiatrist, Dr. Brian Weiss had written several best-selling books on the subject of reincarnation and past-life regression. The list included _Many Lives Many Masters, Through Time into Healing, Only Love is Real_ , and _Messages from the Masters_. Thinking it would be a wonderful opportunity for a fun weekend, we registered for the workshop and booked our airline tickets.\n\nOmega was everything it promised to be and more. We had a great time listening to the various presentations, but it was Brian Weiss that we had come to see. With his quiet, modest demeanor, Dr. Weiss was every bit the gentle soul we expected him to be. Speaking softly about the importance of love, he radiated genuine caring and trustworthiness. He seemed to practice what he preached.\n\nAs part of his presentation, Dr. Weiss announced that he would guide the audience in a group hypnosis session to explore any important past lives that might surface. Even though I had developed a few of my own methods to help others through hypnosis, I personally had very little success in being hypnotized. I'm not sure if I blocked the experience for fear of losing control, but it never seemed to work for me.\n\nThe chairs we sat in were comfortable, but I always needed absolute silence to achieve an altered state. With over three hundred people in the room, I expected to be continually distracted by people coughing, clearing their throats, and shuffling about. I didn't give it much chance for success.\n\nBeginning the session with his standard induction into deepening relaxation, Dr. Weiss's soft hypnotic voice lulled the crowd into an eerie expectant silence. I found myself becoming deeply relaxed. Within a short time, I began to notice that his voice was becoming more and more distant, drifting out of my perception. I tried to focus on his fading instructions, but to no avail. Soon there was only silence. My mind awake and alert, I slowly lost all awareness of my body.\n\nTime seemed to stand still. I was lost in my thoughts when I became aware of small jolts of electricity stabbing through my body. My muscles reacted in jerky spasms. Lasting for only seconds, as quickly as the jolts began, they faded away. I started to feel a mounting pressure closing in on me. It was as if my own consciousness was somehow compressing into a small ball. Moments later, I found myself looking down into the top of a bright, golden chalice.\n\nMomentarily bewildered, it took me a few seconds to steady my mind. It seemed as if I was the one holding the chalice. Swirling in the bottom was a small quantity of red wine. It was then that I heard the distinct Latin dialogue of a Catholic Mass. The words echoed around me. I could feel them in my mind and in my mouth\u2014they were my words. J was saying them. I seemed to be inside the body of a Catholic priest in the middle of a communion service.\n\nHolding the chalice in my left hand, I watched as my right hand came to rest above the goblet holding a thin, round, white wafer between my index finger and thumb. Who and where was I? I wanted to lift my head to look around the room, but was unable to do so. I saw only what this priest saw as he ceremonially lifted the chalice into the air holding the communion wafer above it. His eyes followed the wafer as he mouthed the memorized words. But his heart and mind wandered, drifting far away\u2014preoccupied.\n\nFocusing more closely on his thoughts and feelings, I became increasingly gripped by an overwhelming crush of hopelessness. I could hardly breathe. It felt like my heart was breaking\u2014and I wanted to know why.\n\nIn response to my query, myriad memories instantly downloaded into my conscious mind. For several intense moments I was besieged by an onslaught of emotions. The emotional weight this priest carried was almost unbearable. His heart and, indeed, his very life, lay in the clutches of anguish and despair.\n\nAlthough this understanding was almost instantly perceived, I'm not sure any words could adequately describe the intensity of the moment. As nearly as I can relate it, this is the story that unfolded in my mind.\n\nProcessing his thoughts and memories, I understood the reasons for his, or should I say _my_ , desperation. When he\/I was a young man, I had fallen helplessly in love with a young woman from a nearby village. We had planned to be married. Our families, however, did not approve of the relationship. I was Catholic and she was Protestant. Finally, due to family pressure, she had broken off the relationship, leaving me in heartbroken desolation. I couldn't imagine how I could possibly live without her. In my grief, I had allowed myself to be talked into leaving the village to join the priesthood.\n\nAs these thoughts unraveled, I would catch short intriguing glimpses of a young woman. There seemed to be something familiar about her. Focusing on her energy, a complete image began to form in my mind. The recognition stunned me. It was she! It was the woman I had seen in my dreams, the same woman who had come to me in spirit form while I lay in my bunk at The Monroe Institute.\n\nI followed the chain of events unfolding before me. Several years had passed, and by chance, our paths crossed again. From our first encounter in the streets, our obsession was rekindled. Even though she was now married and I was a priest, we could not resist the temptation of our hearts. We would steal away in secrecy, allowing our passions to carry us to forbidden ecstasy, swearing our undying love.\n\nAll of these memories and thoughts flooded into my mind in hardly more than a heartbeat. It was both an instant revelation, and an inner knowing.\n\nShifting my awareness back to the priest, I felt a numbing despair slowly choking away my life. Continuing mechanically through the ceremony of the Holy Sacrament, I fought for control against the tears of self-pity that burned my eyes, scorching my soul. The guilt of my sin was too much to bear. I had doomed myself to the fires of Hell. As a priest, I found my sin unforgivable. I was trapped between the aching of my heart and the guilt I felt at betraying my church and God. It was all so unfair. Why had I let myself be talked into becoming a priest? Why had she come back into my life?\n\nMy dilemma and torment were inescapable. I would be condemned by the church and most certainly by God. In the agony of my guilt, I knew that it would be impossible for us to be together. But I also knew I could no longer live without her.\n\nSuddenly, with a resounding thump, I found myself snapping upright in my chair. The abrupt explosion of another participant coughing directly behind me was all it took to catapult me out of my hypnotic state. Just like that, the vision disappeared.\n\nFrustrated but undeterred I quickly concentrated on reconnecting with this previous life's energy. Within moments I was again floating in a void. Frantically, I tried to home in on the priest's vibration. Minutes ticked by. Then, without warning, I felt my consciousness plummet, and a dizzying burst of energy overtook me. Finally, the darkness evaporated, and I found myself back in the priest's body. Again I felt his crushing desolation and despair.\n\nAs my vision opened, I was surprised to find that I was no longer in the middle of the church service. I was outdoors, staring off into distant rolling hills of blue-green grass. My heart was breaking\u2014I seemed to be saying goodbye.\n\nFinally, with tears in my eyes, I spun around. The next thing I knew I was plunging over the edge of a cliff to the rocks and sea below. The shock and sensation of falling was so powerful and frightening, it snapped me out of my trance. In an instant I found myself jerking upright in my chair, my hands clawing the air in front of me.\n\nOpening my eyes, I was thankful the room lay in semidarkness. The session was still underway. No one had seen my nailing arms.\n\nLost in thought, I sat quietly in my seat waiting for the session to end. The room had been cool, but now I felt sweat trickling down my neck. The powerful scenes played over and over in my mind. I had been a priest who committed suicide. But more importantly, I now knew who the woman was, why I had seen her in my dreams, and why she had come to me in my room at The Monroe Institute.\n\nMy sense was that this experience had taken place somewhere in Ireland at least a hundred years ago. Yet, as unbelievable as it seemed, it appeared that my long-lost love was still on the Other Side, apparently still in love, and awaiting my return.\n\nThe disturbing episode left me with a jumble of emotions. Torn between the sorrow of this tragic affair and the elation of my discovery, an uneasy melancholy settled over me. There were still too many unresolved issues, too many questions left unanswered. I realized that it might be some time before I could come to terms with the effects of this revelation.\n\nWhen the session ended, Candace and I shared our experiences. We marveled at the power of the workshop. Dr. Weiss had truly developed a remarkable technique for uncovering the past to heal the present.\n\nFollowing our return home from New York, I frequently reflected on the extraordinary events that had taken place, and how they might affect my life. I thought about the young woman I had left behind in another lifetime, and wondered what became of her after my self-inflicted death. One night, about a month later, I had my answer.\n\nAs I lay in bed waiting for sleep, fragmentary thoughts about that distant past life in Ireland haunted me. When the hypnagogic images of presleep began to flutter through my mind, I offered a little prayer and again asked my angels to show me what had become of the woman. I had posed the question a number of times before without receiving any kind of response, but this time I seemed more connected.\n\nWithin moments of sending out the question, I felt a shift in vibration and in an instant I found myself floating near the ceiling of a rather spacious older home. Wherever it was, the owners must have been fairly well-to-do. Although not overly large, the living room was ornately decorated, containing a large stone fireplace and expensive furnishings. Feeling drawn to the next room, I floated through the wall into an adjacent bedroom. As I came through the other side, my attention was drawn to the body of a woman laid out on top of a bed. I knew instantly that it was the woman I had loved during my life as a priest. A feeling of sadness came over me. I somehow knew that she, too, had taken her own life.\n\nHer long dark hair had been done up in ringlets, meticulously placed on the pillow beneath her to accentuate her ashen but serene face. I contemplated how different she now looked without her normal color, vitality, and beaming eyes.\n\nLaid out on the bed in a beautiful long gown, she still exhibited grace and elegance. Crossed upon her chest, her hands cradled a small golden cross. Her rich, royal blue gown was accented by white and gold lace around the neck and sleeves. Obviously designed for a funeral wake, its long skirt flared out at the bottom, extending past her feet, over the foot of the bed, almost to the floor.\n\nWith mixed emotions, I hovered in place, absorbed in the scene, when I became aware of the strangest feelings. Thoughts of Patricia began to permeate my mind. There was no question that the woman on the bed was the one I had loved in that previous lifetime, but I was also receiving strong impressions of Patricia. Their vibrations seemed so much alike. As my confusion grew, I felt a shift in consciousness and I was back in my bed.\n\nOpening my eyes, I lay back, staring at the ceiling, contemplating all that had taken place. Although the journey had given me some answers, it had created just as many questions. Why had I been receiving impressions of Patricia? Was there some sort of connection between the two of them? Could they actually be from the same soul energy? It didn't seem possible. If Patricia was the reincarnated energy of the woman from Ireland, how could she still be on the Other Side in her previous form? The questions hung in the air, but the answers weren't evident. Eventually I fell asleep.\n\nWhen morning arrived, I awoke with the answers still beyond my grasp. \"Someday,\" I thought, \"I'll get to the bottom of it, even if I have to die trying.\" The irony of this sleepy statement made me chuckle. It occurred to me that it might indeed be the only way I'd uncover the rest of the story.\n\nWhile the lessons of my life slowly unfolded, time itself was quickly passing. I had a job to do. After months, perhaps years, of procrastination, I could put it off no longer. I sat down to begin the long task of writing this book.\n\nI wish I could say that writing has been an easy job, that my angels came through as they said they would in the Washington airport. But that might be stretching it a wee bit. To begin with, I wasn't even sure where to begin.\n\nAs I started going through my journals containing years of notes, it became not so much a question of whether I had enough material for the book, but rather what would I leave out. The easiest part was the title. I already knew that. But there were so many revelations, incidents, and insights. How could I pick and choose?\n\nAfter mapping out a chapter outline, I did the next best thing. I lay in bed, opened my consciousness to the Other Side, and asked for help. Within moments, I felt a wave of energy\u2014and the download began. It was like a floodgate had opened. Practically overwhelmed by the flow of information, I realized that if I was already overloading while in an altered state, I had little hope of processing everything during normal consciousness. The angels, however, were probably more aware of what I could handle than I was. As I came out of my meditation, their message was clear. Before each writing session, I was to open myself to guidance, taking it one step at a time.\n\nAll of this, however, was easier said than done. In my busy life, it wasn't always easy to find the time to write, and often the circumstances were not ideal. Sometimes the words and information would flow. Other times I might sit and stare at the same paragraph for two hours.\n\nEventually, I developed a bit of a routine. I found that my most productive sessions came when I first took the time to meditate and offer a prayer for help before I sat down to write. Strangely, I also discovered that my best effort was when I was physically tired or sleepy. I seemed to be able to make a better connection with my more creative side in the quiet of early morning. Consequently, I frequently found myself working through the night until dawn.\n\nThe times when I did find myself connected to guidance were wonderful. Concepts and words would form in my mind faster than I could type. I regularly dropped into a deep altered state while sitting at my computer. It was in this profound, focused level of consciousness that my memories returned to life. When this happened, the detail of past events was startlingly clear. Sometimes I became so absorbed that time would simply disappear. Frequently moved to tears, I worked my way through the powerful emotions that accompanied my memories.\n\nOf particular significance, however, were the typing sessions recalling my many visits with Meldor. Often, in a trance, typing while practically unconscious, I would peck away until four or five in the morning before falling exhausted into bed. On my next return to the computer, I was repeatedly amazed at the words that appeared before me. Although I knew the messages well, the diction and words weren't mine; they were Meldor's.\n\nOverall, the experience of writing this book has been an adventure in itself\u2014mystical, gratifying, and very healing.\n\nNow, as I near the end of my account, I am having difficulty finding the right words to convey how thankful I am for the wonderful lessons and blessings I've been given. There have been some tough times, and some great times, but I probably wouldn't change a minute of it. It's been a remarkable journey, an incredible education. But I have still so much to learn, so much more to experience.\n\nAs never before, I have learned to love life and everything in it. Hardly a day passes in which I am not reminded of the power of the human spirit, and our relationship to God and to each other. We are not merely the bodies we occupy. We are pure indestructible energy. More powerful than we can even imagine, we\u2014and all the things we experience\u2014live forever. As so many others have discovered, there is but one thing, one essential element that binds everyone and everything together. And that is the power of _love_.\n\nThroughout my life, there has been one main revelation\u2014the single most important message I have ever received\u2014that puts it all into perspective. At the end of our lives, when our bodies finally release us back to the spirit world, it won't matter one little bit who won or who lost, or how rich or famous we became. The only thing that will truly matter is how we treated others along the way. That's the importance of love.\n\nPerhaps the toughest lesson of the human experience, _the reality of love_ can sometimes be the most difficult to fully embrace. When we are caught up in the struggle for survival, when seemingly awful things happen to us or others around us, when we are hurt or annoyed by the things other people do, we must strive to see past our biggest fear\u2014the human illusion of separation. There is no real separation. _We are all one!_\n\nWe need only to try to remember, to look deeply within ourselves and within others to see the beauty of the unified soul that we all share with God.\n\nIf there were any advice that I could offer, it would be this: Don't hold back! Love for all you're worth! That's what hearts are for. Having your heart broken a thousand times is still far better than not having loved at all. That would be the greatest tragedy.\n\nLife is a wonderful gift, so live it to the fullest. Take every single opportunity to hug someone. Take the time to hug your kids, your parents, your brothers and sisters, friends and even strangers. And take the time to look closely into their eyes. Perhaps, when you least expect it, you just might feel a fluttering sensation in your heart and a deep stirring within your soul. You might find yourself looking into the _eyes of an angel_.\n\nAs for me, my journey continues with new experiences and revelations almost daily. Where it will take me, and where it will end, I have no idea. I do, however, know one thing for sure. Life is an addictive evolution of steps, lessons, and wonderful mysteries to solve. I can't even imagine how much more I need to learn. But I can hardly wait to find out.\n\n# Channeling\n\nWhen I had completed the body of my book\u2014even after several rewrites\u2014I still had the feeling that something was missing. It needed something more. It occurred to me that I could, perhaps, insert a poem or some quotations at the beginning of each chapter like I'd seen done in other books. After a lengthy search, however, I was unable to find anything to my liking and was resigned to dropping the idea.\n\nThen, early one morning as I lay in bed drifting in and out of sleep, the following whispered words jolted me to attention: _Through shaded eyes we view the world\u2014Present, past, and future swirl_. A chill ran up my spine. It was the unmistakable voice of one of my spirit guides giving me the first two lines of a poem.\n\n\"Hey, that sounds wonderful,\" I thought, repeating the words in my mind. \"So what's the rest of it?\"\n\n\"The rest,\" she whispered, \"is up to you.\" And her vibrations faded away.\n\nImmediately, a battle began to rage in my mind. What did I know about poetry? Throughout my life, I'd never tried to write poetry. It just wasn't me. This I knew would be an impossible task.\n\nMonths passed and I continued my procrastination. Often I would roll those first two lines over in my mind, trying to assemble at least one complete stanza, but nothing ever seemed to fit. It was almost a year later when I awoke one morning from a dream in which my spirit guides had again chided me for not finishing the poem. Having the concept refreshed in my mind, the following night around ten o'clock, I finally sat down at my computer to give it a shot.\n\nWhat an exercise in frustration! Hour after hour went by as I sat staring at those first two lines, typing concoction after silly concoction onto the screen, then deleting it all and starting again. By 3 A.M. I had accomplished virtually nothing and was considering giving up and going to bed. But as I sat at my keyboard struggling to stay awake, I found myself dropping deeper and deeper into an altered state. Then, when it seemed that my brain had all but retired for the night, a new energy took control of my fingers\u2014and a flood of words began. A short time later I crawled, zombie-like, into bed.\n\nWhen I finally awoke nine hours later, I couldn't believe the words stored in my computer. They weren't my words; they were theirs. But I am eternally grateful for having been honored as the messenger. From the hearts of Angels to your heart, this is their message.\n\n# Eyes of an Angel\n\n_Through shaded eyes we view the world_\n\n_Present, past, and future swirl_\n\n_Only fleeting glimpse there revealed_\n\n_The mysteries of life concealed_\n\n_Though evidence in abundance shouts_\n\n_Thoughts of creation are lined with doubt_\n\n_Life's grand design in darkness robed_\n\n_Hints of spirit left unprobed_\n\n_In days of science and truth unclear_\n\n_Man and religion fraught with fear_\n\n_Purpose and destiny remain obscured_\n\n_Life without meaning long endured_\n\n_Promises of paradise, and wonders to behold_\n\n_Man-made dogma, imposed upon the fold_\n\n_But in frightened hearts there flickers still_\n\n_Desire for truth that guilt can't kill_\n\n_So is it still by chance or sad intent_\n\n_Destruction looms with our consent_\n\n_While thoughts, the thief within our heads_\n\n_Leave us fearful in our beds?_\n\n_Then steadfast courage and honor bequest_\n\n_The hand of God we'll squarely test_\n\n_Until lust for battle be lost in the fray_\n\n_And restless spirit lights the way_\n\n_Though the stock of angels be divine_\n\n_Must we yet refuse to see the sign?_\n\n_Why is it only when troubles abound_\n\n_He finds us kneeling on the ground?_\n\n_When horrors plague our daily bread_\n\n_And grieving thoughts we cannot shed_\n\n_Is it then we pray our Lord to see?_\n\n_Shallow promises made in desperate plea_\n\n_Then comes an angel, eyes ablaze_\n\n_Heart secure and love unfazed_\n\n_A glimpse of what was meant to be_\n\n_Pure light upon reality_\n\n_And oh what joys will they evoke_\n\n_When God and angels first uncloak_\n\n_Life's greatest treasures gleaned at last_\n\n_Our hearts and souls in light are cast_\n\n_The message comes from Heaven clear_\n\n_His wrath was never lent to fear_\n\n_You see, God has only love to give_\n\n_It's man distorts the way we live_\n\n_But in angel eyes love grows complete_\n\n_While strong hands tremble in defeat_\n\n_The truth of God is soon revealed_\n\n_It's through our hearts that we are healed_\n\n_And now in slumber's sweet delight_\n\n_Dream angels carry us inflight_\n\n_When there are no more souls to mend_\n\n_Who is the angel God will send?_\n\n_Though late, we find that in the end_\n\n_Matters not if we break or bend_\n\n_It wasn't God we were meant to see_\n\n_For only love can set us free_\n\n_If it's proof you seek, there's no denying_\n\n_The truth is only found in dying_\n\n_But in search of God, and whereto begin_\n\n_You need look no further than within_\n\n\u2014Paul Elder\n\n# About the Author\n\nA former long-term mayor of a small city, Paul Elder has spent a lifetime studying human nature from a unique perspective. He is a survivor of three near-death experiences: a drowning at the age of 12, a car accident at 17, and a heart attack while playing hockey at the age of 41. Along with the trauma and subsequent beauty of death came a series of spontaneous spiritual events\u2014events that would rock his world, turning his belief system upside down, leaving him with some profound insights as to the true nature and ultimate purpose of human life.\n\nAfter spending nearly a decade as a Radio and Television news reporter, he ventured into a career of politics and business spanning more than 20 years. Forever grateful for his near-death experiences, his spiritual insights molded and deeply affected his approach to politics, business, and life.\n\nNow retired, and living on Vancouver Island, British Columbia, Paul spends his time writing, teaching, and speaking throughout the United States and Canada on the phenomenon of near-death and out-of-body experiences.\n\nTo contact the author, visit his web site at www.paul-elder.com.\n\nThank you for reading _Eyes of an Angel_. Hampton Roads is proud to publish an extensive array of books on the topics discussed in _Eyes of an Angel_ , topics such as out-of-body experiences, spiritual guidance, near-death experiences, the exploration of other dimensions, past lives, and more. Please take a look at the following selection or visit us anytime on the web: www.hrpub.com.\n\n**_Soul Journeys_**\n\n_My Guided Tours through the Afterlife_\n\nRosalind A. McKnight\n\nIn _Soul Journeys_ , McKnight relates her explorations of the afterlife with a being she calls Radiant Lady. With Radiant Lady as her spiritual guide, McKnight explores and reports on the non-physical energies in the afterlife. _Soul Journeys_ reveals the inner workings of the other dimensions including the afterlife and emphasizes that all of us have spirit guides we can work with at any time.\n\n**Paperback \u2022 272 pages \u2022 ISBN 978-1-57174-413-5 \u2022 $15.95**\n\n**_Cosmic Journeys_**\n\n_My Out-of-Body Explorations with Robert A. Monroe_\n\nRosalind A. McKnight\n\nForeword by Laurie Monroe\n\nMcKnight was one of the first and most successful out-of-body research explorers to work with Robert Monroe. In _Cosmic Journeys_ she recounts the early days of The Monroe Institute and her sessions in the famed \"black box.\"\n\n**Paperback \u2022 304 pages \u2022 ISBN 978-1-57174-123-3 \u2022 $16.95**\n\n**_Out-of-Body Exploring_**\n\n_A Beginners Approach_ Preston Dennett\n\nNovice, intermediate, or expert\u2014anyone interested in the spiritual art of astral travel will benefit from Dennett's 20 years of out-of-body traveling. Including his initial forays into expanded consciousness, he shares the many techniques he experimented with, and discusses the people and sights he encountered. Plus, he offers tips on how to go reliably out-of-body, and how to control and maintain this enhanced awareness for extended periods.\n\n**Paperback \u2022 200 pages \u2022 ISBN 978-1-57174-409-8 \u2022 $13.95**\n\n**_The Unanswered Question_**\n\n_Death, Near-Death, and the Afterlife_ Kurt Leland\n\nSeasoned astral traveler Leland explores the deepest reaches of consciousness in an attempt to answer the great question: Is a near-death experience the same as real death? What he finds will fascinate and enlighten you, and more importantly, convince you that everything you think you know about Near-Death may be dead wrong.\n\n**Paperback \u2022 496 pages \u2022 ISBN 978-1-57174-299-5 \u2022 $16.95**\n\n**_Otherwhere_**\n\n_A Field Guide to Nonphysical Reality for the Out-of-Body Traveler_ Kurt Leland\n\nA luminous travelogue of the ethereal world we travel to in dreams, OBEs, and NDEs. Noted psychic Leland guides you to various points of interest in this other-world including alternate realities, the living archives, and offers translation tables that categorize nonphysical energies by function rather than form.\n\n**Paperback \u2022 288 pages \u2022 ISBN 978-1-57174-241-4 \u2022 $18.95**\n\n**_Ordered to Return_**\n\n_My Life After Dying_ George G. Ritchie, Jr., M.D.\n\nIn 1943, Army private George Ritchie died of pneumonia. Nine minutes later, he came back profoundly changed. Here, Ritchie recounts one of the most fascinating NDEs ever recorded\u2014and then tells what happened later, including the real miracles that he has seen in his years as a doctor.\n\n**Paperback \u2022 184 pages \u2022 ISBN 978-1-57174-096-0 \u2022 $16.95**\n\n**_Return of the Revolutionaries_**\n\n_The Case for Reincarnation and Soul Groups Reunited_ Walter Semkiw, M.D.\n\nPresenting evidence that facial structure, personalities, and peer groups stay the same lifetime after lifetime, Semkiw shows how America's revolutionary heroes are reemerging today as the force behind America's spiritual rebirth. Semkiw even includes hundreds of startling photos and clues to tracking your own past lives.\n\n**Paperback \u2022 464 pages \u2022 ISBN 978-1-57174-342-8 \u2022 $22.95**\n\n**_Soul Agreements_**\n\nDick Sutphen with Tara Sutphen\n\nYour karmic road map was programmed at a soul level before you were born. Soul agreements established how your life's relationships, career directions, and major triumphs and challenges would unfold. Explore your past lives and your destiny and learn how you can create your own reality.\n\n**Paperback \u2022 192 pages \u2022 ISBN 978-1-57174-442-5 \u2022 $14.95**\n\n**_Fast Lane to Heaven_**\n\n_A Life-after-Death Journey_ Ned Dougherty\n\nIn the midst of a heated argument, Ned Dougherty died\u2014only to be reborn from his self-destructive lifestyle into the divine light of salvation. That unforgettable near-death experience included meeting both the Supreme Being and the Archangel Michael, as well as with an enigmatic Lady of Light, who set Ned on a 15-year mission to communicate her urgent message to the world.\n\n**Voted \"NDE Book of the Year\" bynear-death.com**\n\n**Paperback \u2022 304 pages \u2022 ISBN 978-1-57174-336-7 \u2022 $18.95**\n\n# Hampton Roads Publishing Company\n\n_...for the evolving human spirit_\n\nHAMPTON ROADS PUBLISHING COMPANY   \npublishes books on a variety of subjects,   \nincluding spirituality, health, and other   \nrelated topics.\n\n**_For a copy of our latest trade catalog_** ,   \ncall 978-465-0504 or   \nvisit our website at www.hrpub.com\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n_Mrs. Lincoln's Dressmaker_\n\n_M rs. Lincoln's  \nDressmaker_\n\n_a novel_\n\nJENNIFER CHIAVERINI\n\nALSO BY JENNIFER CHIAVERINI\n\n_The Giving Quilt_\n\n_Sonoma Rose_\n\n_The Wedding Quilt_\n\n_The Union Quilters_\n\n_The Aloha Quilt_\n\n_A Quilter's Holiday_\n\n_The Lost Quilter_\n\n_The Quilter's Kitchen_\n\n_The Winding Ways Quilt_\n\n_The New Year's Quilt_\n\n_The Quilter's Homecoming_\n\n_Circle of Quilters_\n\n_The Christmas Quilt_\n\n_The Sugar Camp Quilt_\n\n_The Master Quilter_\n\n_The Quilter's Legacy_\n\n_The Runaway Quilt_\n\n_The Cross-Country Quilters_\n\n_Round Robin_\n\n_The Quilter's Apprentice_\n\n\u2022 _Elm Creek Quilts_ \u2022\n\n_Quilt Projects Inspired by the Elm Creek Quilts Novels_\n\n\u2022 _Return to Elm Creek_ \u2022\n\n_More Quilt Projects Inspired by the Elm Creek Quilts Novels_\n\n\u2022 _More Elm Creek Quilts_ \u2022\n\n_Inspired by the Elm Creek Quilts Novels_\n\n\u2022 _Sylvia's Bridal Sampler from Elm Creek Quilts_ \u2022\n\n_The True Story Behind the Quilt_\n\n\u2022 _Traditions from Elm Creek Quilts_ \u2022\nPLUME\n\nPublished by the Penguin Group\n\nPenguin Group (USA) LLC\n\n375 Hudson Street\n\nNew York, New York 10014\n\nUSA | Canada | UK | Ireland | Australia | New Zealand | India | South Africa | China\n\npenguin.com\n\nA Penguin Random House Company\n\nFirst published in the United States of America by Dutton, a member of Penguin Group (USA) Inc., 2013\n\nFirst Plume Printing 2013\n\nCopyright \u00a9 2013 by Jennifer Chiaverini\n\nPenguin supports copyright. Copyright fuels creativity, encourages diverse voices, promotes free speech, and creates a vibrant culture. Thank you for buying an authorized edition of this book and for complying with copyright laws by not reproducing, scanning, or distributing any part of it in any form without permission. You are supporting writers and allowing Penguin to continue to publish books for every reader.\n\n  REGISTERED TRADEMARK\u2014MARCA REGISTRADA\n\nCIP data is available.\n\nISBN: 978-1-101-60964-4 (eBook)\n\nThis is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, businesses, companies, events, or locales is entirely coincidental.\n\nVersion_3\n_To Marty, Nicholas, and Michael,  \nwith love and thanks_\n\n# Table of Contents\n\nChapter One\n\nChapter Two\n\nChapter Three\n\nChapter Four\n\nChapter Five\n\nChapter Six\n\nChapter Seven\n\nChapter Eight\n\nChapter Nine\n\nChapter Ten\n\nChapter Eleven\n\nChapter Twelve\n\nChapter Thirteen\n\nChapter Fourteen\n\nChapter Fifteen\n\nChapter Sixteen\n\nChapter Seventeen\n\nChapter Eighteen\n\nAuthor's Note\n\nAcknowledgments\n\nAbout the Author\n\nExcerpt from _Mrs. Lincoln's Rival_\n\nPrologue\n\n# Chapter One\n\n## NOVEMBER 1860\u2013JANUARY 1861\n\nOn Election Day, Elizabeth Keckley hurried home from a midafternoon dress fitting to the redbrick boardinghouse on Twelfth Street where she rented two small rooms in the back. Although she never failed to carry her license attesting to her status as a freedwoman whenever she ventured out, on that day of the presidential election of 1860 she was eager to be safely indoors well before curfew. The city hummed with breathless excitement even though the white citizens of Washington City, District of Columbia, were not enfranchised to vote. In this, the capital's colored residents, both slave and free, were their equals, although Elizabeth prudently refrained from remarking upon this similarity to the wives of the city's social and political elite for whom she sewed the beautiful gowns they wore to balls and levees and receptions. Her patrons were united in their suspicion of and disdain for the Republican candidate, a lawyer from Illinois they disparaged as an unpolished rube from the West and a radical abolitionist. They disagreed, however, on which of his three rivals ought to succeed President Buchanan, who, if ineffectual, had at least done their home states and the South's \"peculiar institution\" no enduring harm.\n\nIf a spontaneous parade sprang up and turned into a riot, as happened far too often those days, Elizabeth wanted to be well away from the furor. Already the streets were filling with men hurrying from tavern to hotel for news of the election, gathering on corners with like-minded fellows and glaring across the way at their rivals, crowding anxiously around the doors of the telegraph office on Fourteenth Street although the returns wouldn't be in for hours yet. Many folks had obviously been enjoying the free whiskey dispensed by the various political clubs that dotted the blocks near the White House, and from time to time their bursts of raucous laughter drowned out even the unceasing clip-clop of horses' hooves and the more distant whistles of passing trains. As she made her way home, Elizabeth tightened her grip on her sewing basket and kept her bearing serene and composed, flinching only once, when a young man wearing a campaign button boasting a tintype of Mr. Breckinridge jostled her in his haste to reach the bulletin board outside the National Hotel.\n\nShe breathed a sigh of relief when she reached her own quiet neighborhood, a haven in what had been an unfamiliar city only months before. She had come to Washington City that spring after a few failed weeks in Baltimore, where her struggle to find work convinced her to seek her fortune elsewhere. Not long before that, she had lived in St. Louis, where, after years of toiling and saving, she had purchased her freedom and her son's. Now George was a student at Wilberforce University in Ohio, and she was a successful mantua maker, a businesswoman with an admirable reputation, independent and free. She could more easily bear the miles separating her from her only child knowing that he was acquiring the education she herself had always longed for and had been denied, and that no man could claim him as property ever again.\n\nVirginia Lewis, her landlady and dear friend, must have seen her approach through the front window, for she met her at the door. \"What's the news?\" she asked, breathless, studying her expression as if to read the answer there. \"I know how your ladies talk. If anyone knows what's happening, they would.\"\n\n\"I'm afraid they don't know any more than we do.\" Elizabeth set down her sewing basket and unwrapped herself from her shawl. \"Nothing's changed from what we heard this morning. Mr. Lincoln's favored to win, but we won't know for a fact until they count the ballots.\"\n\n\"I suppose your ladies wouldn't care much for a President Lincoln.\"\n\n\"Not one bit. Most of their husbands like Mr. Breckinridge, and so they do likewise. A few of them like Mr. Bell too.\" Elizabeth lowered her voice conspiratorially. \"As for Mr. Lincoln, they fear he wants to free all the slaves.\"\n\nThey shared a laugh. \"Well, then, God bless him,\" declared Virginia. \"If that's true, I pray he wins.\"\n\nElizabeth nodded, but private doubts troubled her. No matter what the Southern partisans inhabiting the city said about Abraham Lincoln, as far as she knew, he had never promised to end slavery, only to keep it from spreading. But even if he had placed his hand on a Bible and declared himself a staunch abolitionist, her few months in Washington had taught her that candidates often made promises that they found impossible to keep after taking office. Whatever Mr. Lincoln's intentions, and whether they boded ill or good for Elizabeth and the Lewises and other colored folk, when Mr. Lincoln's people swept into the city to take over, most of Mr. Buchanan's would be swept out, among them the husbands of several of her best patrons. She could only hope that enough of them would remain behind to keep her business thriving.\n\nIt was perhaps too much to hope that the new First Lady would have a taste for finery, and that Elizabeth might somehow attract her notice.\n\nIt was well after midnight when she was startled awake by the sounds of shouting on the streets outside, of jubilant songs and speeches, and soon thereafter, of the firing of pistols and shattering of windows. She sat up in bed and strained her ears in the darkness, sifting sense from the jumble of noise and fury until she understood that the votes had been counted, the result announced.\n\nAbraham Lincoln would become the sixteenth president of the United States.\n\nWhen Elizabeth had first arrived in Washington, she had no money, no friends, no place to call home, but she had soon found work as an assistant seamstress for two and a half dollars a day and took a room in a boardinghouse. Before long she decided to strike out on her own, and she had a sign and business cards made. She advertised herself as a skilled mantua maker, capable of sewing the complicated, snug-fitting bodice of the style that well-dressed ladies most desired. Slowly but surely, she acquired a few patrons, who recommended her to their friends and acquaintances. One generous lady, a friend of the mayor, persuaded him to waive the fee for the license that, like all free Negro females over the age of fourteen, Elizabeth was required to obtain within thirty days of her arrival if she wished to remain in the city. And she had already decided that she did wish to stay, even though the daily sight of slaves in chains being led through the muddy streets from shipyard to auction house and the restrictions upon freedmen like the license and curfews sometimes made her feel as if she were not truly free.\n\nIn October, a few weeks before the election that threw Washington City into uproar, Elizabeth sewed a gown for the wife of the cavalry officer Colonel Robert E. Lee. \"Spare no expense,\" the colonel had urged Elizabeth as he entrusted her with a hundred dollars to purchase lace and buttons and ribbons for his wife's attire. Elizabeth gladly did as he bade her, and when Mrs. Lee wore the gown to a private dinner party at the White House in honor of the Prince of Wales, the other ladies in attendance were so impressed by the gown's beauty and elegance and excellent fit that they sought out Elizabeth's services. Her reputation grew as one delighted patron after another recommended her to their friends, and soon she had almost more work than she could handle.\n\nNot long after the election, one of Elizabeth's patrons recommended her to a neighbor, Varina Davis, the wife of Senator Jefferson Davis of Mississippi. Informed that the Davises were late risers, Elizabeth offered to come to their I Street residence in the afternoons rather than first thing in the morning. That way she would have steady work sewing for the senator's family in the afternoons, and her mornings would be free to devote to other patrons. Mrs. Davis agreed to the arrangement, and so the autumn days passed pleasantly and productively, despite the growing tension in the city that the election seemed to have augmented rather than diffused. The Davis residence was a hub of Southern activity, with politicians and statesmen from the South racing in at all hours and disappearing into Senator Davis's study for clandestine meetings in hushed voices that occasionally broke out into angry shouts. Mrs. Davis's mouth would tighten a trifle at each outburst, but usually she would barely glance at the door and carry on as if nothing had happened. If she were in a particularly cheerful mood, she might make a small joke about men and their tempers, but Elizabeth suspected that she was as concerned as they were about the widening divide between North and South, if not quite so loud about it.\n\nElizabeth liked the Davises. The senator struck her as a considerate, dignified gentleman, and Mrs. Davis\u2014dark-haired, with large dark eyes and a complexion less fair than was fashionable\u2014was well educated and witty, and she seemed to have friends in every corner, Republicans and Democrats alike. Mrs. Davis looked to be in her early thirties, perhaps a decade younger than Elizabeth, perhaps a little less. She loved to read and to quote poetry, and whenever Elizabeth called, she usually found her in the midst of writing a letter, reading a book, or enjoying a visit from a friend.\n\nOne gray December afternoon, Elizabeth arrived at the Davis residence just in time to see one of Mrs. Davis's friends, the pretty and vivacious Mrs. Samuel Phillips Lee, leaving in a huff. \"She doesn't look well pleased,\" Elizabeth murmured to Jim, the colored doorman, a slave who had come north from Mississippi with the Davises. \"I'm almost afraid to come inside and find out why. Did she and Mrs. Davis argue?\"\n\nJim nodded. \"Not long, and not too loud, but enough. I heard Mrs. Davis say she's not gonna associate with Republicans anymore.\"\n\n\"Why, that's just silly.\" Elizabeth shifted her satchel to her other hand and gestured to Mrs. Davis's departing guest, who was picking her way carefully down the street. The winter chill had frozen the mud into hard, deep ruts, which made walking treacherous. It was difficult to say which was worse, the thick, sodden mud that soiled skirts and stole shoes from one's feet or the frozen variety that tripped one up and turned one's ankles. \"Mrs. Davis has lots of Republican friends.\"\n\n\"Not many of them show up around here anymore.\" Jim glanced over his shoulder and leaned toward her confidentially. \"Guess how many angry Democrats can squeeze inside Mr. Davis's study.\"\n\n\"Ten?\" Elizabeth replied, curious. \"A dozen?\"\n\n\"Twice that, I saw it for myself. They came in yesterday at dusk, senators and secretaries and the like, one by one, all quiet, and they stayed till near three o'clock this morning, talking secession.\"\n\nElizabeth's heart thumped. The first time she had heard the word secession, she had been obliged to ask her landlord Walker Lewis what it meant, and the sound of it still had the power to unnerve her. \"Talking war too?\"\n\n\"Talk of one usually leads to talk of the other, don't it?\"\n\nElizabeth nodded. It did, and ever more so. \"Maybe it won't come to that. Maybe it's all just a lot of political bluster.\" She knew that Senator Davis was working day and night to come up with a compromise to keep the slave states in the Union. She wanted to believe that something might yet come of his efforts, but the two sides' differences were so sharp and strongly felt that an agreement seemed impossible.\n\n\"Maybe so,\" Jim said as he showed her in, but she knew he was merely being polite. Perhaps he wanted war, if the war would bring him his freedom. She could not fault him for that.\n\nShe found Mrs. Davis in the parlor, a book on her lap, her gaze faraway. \"Good morning, ma'am,\" she greeted her. \"I've finished basting the lining for the day dress if you'd care to try it on.\"\n\nMrs. Davis looked up with a start. \"Certainly, Elizabeth.\" She rose and beckoned Elizabeth to follow her upstairs to her boudoir.\n\nThe senator's wife said little as Elizabeth helped her into the muslin bodice and skirt. Elizabeth firmly believed that investing the time and effort into the cut and fit of the lining was the secret to a well-fashioned dress, and she could only shake her head at seamstresses who hastened through that most essential part of the process. First Elizabeth would make a pattern for the lining by pinning inexpensive muslin or paper upon her patron's figure. Next, she would remove the pieces and baste them together. Her patron would try on this rough garment, which Elizabeth would fit and adjust and offer to her patron to try on again, over and over, for more fittings and adjustments until Elizabeth was satisfied. Only then would she risk putting shears to the more expensive satins and silks and moir\u00e9s that would comprise the outer garment. After that, the carefully constructed pattern was the only model Elizabeth needed to sew the gown, a step she would complete in the privacy of her own rooms. The process was laborious and time-consuming, but Elizabeth insisted that it was the only way to ensure a perfect fit. She wondered what her patrons would think if she told them she had learned the technique of cutting on the figure by sewing clothing for her fellow slaves back in Virginia\u2014not for the sake of fashion, of course, but so as not to waste any of the rough osnaburg cloth the mistress provided.\n\nAs Elizabeth took in the vertical pleats in the back of the bodice and pinned them carefully, Mrs. Davis sighed and said, \"I wonder...I wonder if I should have chosen a fabric more suitable for a Southern summer.\"\n\n\"The sateen complements your complexion, and it won't be too warm except on the very hottest days.\" Studying the bodice, Elizabeth plucked two more pins from her pincushion and slipped them in place. \"Why should you wonder? Are you going home to Mississippi for the summer?\"\n\n\"Home, indeed.\" Mrs. Davis laughed shortly. \"Mississippi is the state of my birth, but Washington is the home of my heart. I should hate to leave it, even if the place is overrun by Republicans.\"\n\n\"I should hate to see you go.\" Elizabeth thought of Mrs. Samuel Phillips Lee, her pretty features drawn with irritation as she stormed away from the Davis residence. \"As I'm sure your friends would too.\"\n\n\"I doubt my dearest friends will linger. Some of them have already declared themselves eager to depart for friendlier states. None of us has much choice in the matter.\" Sighing, Mrs. Davis obligingly allowed Elizabeth to move her arms apart so she could better fit the sleeves. \"It's our duty as wives to accompany our husbands. If they say we stay in the Union, and thus in Washington City, then we stay. If not\u2014\" Mrs. Davis forgot herself and waved a hand in helpless dismissal, tugging the muslin sleeve from Elizabeth's grasp. \"Well, we go where they go, willingly or not.\"\n\n\"If you must go, and it seems to me there's still a good chance you won't,\" said Elizabeth, \"you'll be the best-dressed woman in Mississippi\u2014if you hold still and let me finish.\"\n\nIn spite of herself, Mrs. Davis laughed. She patted Elizabeth's hand where it rested on her waist, but after that, she stood perfectly motionless as Elizabeth bade her until the task was done.\n\nAs December passed, the subject of war came up often in the Davis household\u2014between the senator and his wife, Mrs. Davis and her friends, the senator and his numerous visitors. The senator worked increasingly long and erratic hours, and when Mrs. Davis wasn't tending to her young children, she distracted herself with writing letters, reading books, and entertaining friends she probably thought she must soon part with and might never see again. A deep, hushed anxiety permeated the city, the sense that time was running out, and Elizabeth felt it most intensely within the Davis household.\n\nAs the holidays approached, Mrs. Davis kept Elizabeth busy sewing everyday clothes for her and the children, and one special garment for her husband\u2014a fine silk dressing gown meant for his Christmas present. As Mrs. Davis wanted it to be a surprise, she asked Elizabeth to keep it out of sight whenever the senator was home. Once, Mrs. Davis dashed into the sitting room, where Elizabeth was busy sewing, and flung a quilt over her\u2014the pieces of silk, her sewing basket, and Elizabeth herself from shoulders to shoe tops\u2014only moments before Mr. Davis walked in, home earlier than expected from the Capitol. There was no time to adjust the quilt so that the draping was less ridiculous, so Elizabeth merely sat and nodded a greeting while Mrs. Davis whirled about to kiss her husband. The senator accepted the welcome with distracted affection, but then he looked past his wife to peer curiously at Elizabeth. \"Are you cold, Lizzie?\"\n\n\"Not anymore, Mr. Davis,\" she replied. Mrs. Davis flashed her a quick, secret smile and led her husband from the room, asking about his day. Muffling laughter, Elizabeth threw off the quilt, folded up the silk, and hid the pieces in her sewing basket before the senator returned.\n\nIt was a rare lighthearted moment in a season of increasing worry and dread. A few days before Christmas, all of Washington was shaken by the news from the South\u2014South Carolina had voted to secede from the Union. Within the Davis household, the debates Elizabeth overheard immediately shifted from whether it was legal and right to dissolve the Union to which states would follow South Carolina and how soon. As the calls for secession increased, Mrs. Davis's ambivalence persisted. One day, she would champion the cause of states' rights and praise the slave system, while on the next, she would confide to a friend that she could not bear the thought of leaving Washington City, where she had resided so long and forged such enduring ties. \"I would rather remain in Washington and be kicked about than go south and be Mrs. President,\" she said. And when her friend exclaimed in surprise, Mrs. Davis assured her that it was the true sentiment of her heart.\n\nThat more Southern states would follow South Carolina out of the Union seemed all but certain. In all the disquiet, Elizabeth had found very little time to work on Mr. Davis's dressing gown. After weeks of swift, clandestine, off-and-on sewing, on Christmas Eve, the dressing gown remained unfinished.\n\nAlthough Mrs. Davis did not rebuke her, Elizabeth knew she was anxious to have the dressing gown completed, lovingly wrapped and ready to give to her husband Christmas morning. \"I'll stay and finish it, if you like,\" Elizabeth offered as the afternoon waned, though she was already weary after a long day of sewing, her fingers tired, her head and back aching.\n\nMrs. Davis gladly, gratefully accepted, so Elizabeth kept on, sewing with neat, quick, flawless stitches in the sitting room by gaslight while her patron decorated a Christmas tree in the parlor for the children. Her thoughts wandered to her son, who would be spending the holidays with a school friend in Cleveland. It would be their first Christmas apart. Elizabeth had sent George a gift\u2014candies, a book, and two fine shirts\u2014but she had not heard from him. Watching Mrs. Davis bustle about preparing for her family's celebration made her miss George more acutely than ever. When she was still a young woman, her beauty and natural grace had drawn the attention of a powerful white man, Alexander Kirkland, who could not be fought off and would not be denied. Her master had done nothing to protect her, and so George had come to be\u2014the child she had not wanted but had come to love with all her heart.\n\nThe closing of a door elsewhere in the house snatched her from her reverie, and when her gaze fell upon the clock on a nearby table, she saw that it was a quarter to midnight, well past the ten o'clock curfew for colored people. She could not walk home now, even if she dared brave the streets at so late an hour, what with crime on the upswing even in fine white neighborhoods like the Davises' and respectable middle-class colored neighborhoods like her own. She decided that when she finished the dressing gown, she would present it to Mrs. Davis and request a bed for the night in the attic with the servants. With any luck she could snatch a few hours' sleep and be well rested enough to enjoy the Christmas service at Union Bethel Church in the morning. Virginia and Walker Lewis had kindly invited her to join them and their young daughters for a Christmas feast later that day, and Elizabeth had gratefully accepted. She was accustomed to a good deal of time alone\u2014after more than thirty-seven years as a slave denied all privacy, a bit of solitude still felt luxurious\u2014but on the day of the Lord's birth, she wanted to celebrate in the company of friends.\n\nShe was adjusting the tie cords when she heard Senator Davis in the hallway between the sitting room and the parlor. \"How festive you've made everything, my dear,\" she heard him praise his wife, though his voice was devoid of merriment. \"The children's eyes will light up with joy when they behold that tree.\" He leaned against the doorway, and suddenly he glanced over his shoulder into the sitting room. \"That you, Lizzie?\" he exclaimed, turning her way. \"Why are you here so late? Still at work? I hope Mrs. Davis isn't too exacting.\"\n\n\"No, sir,\" she answered, startled by his gray and careworn look. Elizabeth knew he was ten years older than herself and that he often suffered from poor health, but clearly the events of the past few weeks had aged him drastically. Taken aback, she had missed her brief opportunity to hide the dressing gown. She had no choice but to explain herself, although she tried to conceal as much of the surprise as possible, without lying outright. \"Mrs. Davis was very anxious to have this gown finished tonight,\" she told him carefully, glancing toward the parlor to be sure Mrs. Davis would not overhear, \"and so I volunteered to remain and complete it.\"\n\n\"Well, well, the case must be urgent.\" The senator crossed the room and took the hem of his Christmas surprise in his hand. \"What is the color of this silk? This gaslight is too deceptive to my aging eyes.\"\n\n\"It's a drab changeable silk, sir.\" She almost added that it was a rich, handsome color and would suit him well, but she decided to let him discover that for himself when he opened his gift the next morning and could admire it in daylight.\n\nHe nodded, smiled curiously, let go of the hem, and left the room without another word. Elizabeth knew then that he had easily guessed that the dressing gown was his wife's Christmas gift to him, and he did not want her to know that the surprise had been spoiled. She found herself touched by his thoughtfulness. She would never concur with his views on slavery and secession, but she couldn't help admiring him for the small kindness he showed his wife.\n\nShe finished the dressing gown just as the clock struck midnight. Her thoughts drifted unexpectedly to her own husband, how he had deceived and disappointed her, until she had been obliged to tell him that they must live forever apart. James Keckley, who had pretended to be a freedman among other things when she agreed to marry him, was likely still a runaway slave hiding out in St. Louis if his drinking had not killed him yet.\n\nCarefully she folded the dressing gown and pushed thoughts of her estranged husband aside. It was Christmas, and she would not wish anyone ill on that sacred day, especially not the man she had once loved.\n\nJust before midnight on December 31, the bells at Trinity Church at Third and C streets rang out \"Hail, Columbia\" and \"Yankee Doodle\" to welcome the New Year, but any lingering chance for reconciliation between North and South had been left behind in the old. New Year's Day was usually a festive, merry holiday in the city, or so its longtime residents told Elizabeth, but on the first day of 1861, no crowds celebrated in the inns and taverns, and few neighbors ventured out to pay calls. Those who did bore anxious and troubled expressions, as if urgent business rather than a wish to spread New Year's greetings had compelled them from the comfort of their homes on that bright, frigid morning. A neighbor, a former slave employed as a butler at the White House, later told the Lewises that only a small fraction of the usual number of guests had attended the traditional New Year's Day reception. The guests were as courteous to one another as gentility required, but some boldly wore Union or secession cockades in their hats so that no one would mistake their loyalties. Not even President Buchanan was immune from barbs and bitter words as he shook hands in the receiving line, but as the host as well as the leader of the land, if only for a few more weeks, he bore it with diplomacy and tact.\n\nOn January 9, Mississippi voted to secede, and Jefferson Davis told his wife that he was resolved to follow his beloved state out of the Union. Within days, the senator fell ill and took to his bed, which was where the unceasing stream of friends and politicians and statesmen conferred with him. Elizabeth did not doubt that the strain upon him as the leader of the Democrats' Southern faction contributed in no small part to his poor health. Mrs. Davis nursed him tenderly, and when Elizabeth remarked upon her sure hand in the sickroom, Mrs. Davis replied matter-of-factly, \"I've had a great deal of practice.\"\n\nOne evening, as Elizabeth was dressing Mrs. Davis for one of the few social gatherings the couple attended that bleak winter, Mrs. Davis suddenly said, \"Elizabeth, you are so very handy that I should like to take you with me.\"\n\nSomething in her voice told Elizabeth that this was no idle compliment. \"When do you go South, Mrs. Davis?\" she asked.\n\n\"Oh, I cannot tell just now, but it will be soon. You know there is going to be war, Elizabeth?\"\n\nShe had long suspected, but Mrs. Davis spoke as if she had certain knowledge of it\u2014which, as Senator Davis's wife, she very well could have. \"War? You cannot mean it.\"\n\n\"I tell you, yes. The Southern people will not submit to the humiliating demands of the abolition party. They will fight first.\"\n\nElizabeth fought to keep her voice even as she fastened the last of Mrs. Davis's buttons. \"And who do you think will win?\"\n\n\"The South, of course,\" replied Mrs. Davis. \"The South is impulsive, and the Southern soldiers will fight to conquer. The North will yield when it sees the South is in earnest, rather than engage in a long and bloody war.\"\n\nImpulsive and earnest, perhaps, but Elizabeth didn't think that was enough to win a war. She also thought, although she would not be impudent enough to say so aloud, that Mrs. Davis underestimated the determination of Northerners. Those of Elizabeth's acquaintance did not seem particularly yielding, or any more afraid of a fight than Southern folks. \"Mrs. Davis,\" she said instead, in her most reasonable tone, \"are you certain that there will be war?\"\n\n\"I know it.\" Suddenly she turned in her chair and clasped Elizabeth's hand. \"You had better go South with me. I will take good care of you.\"\n\nElizabeth was so startled that without thinking she snatched her hand away.\n\nMrs. Davis seemed not to notice her rudeness. \"When the war breaks out, the colored people will suffer in the North. The Northern people will look upon them as the cause of the war, and I fear, in their exasperation, will be inclined to treat you harshly.\"\n\nReluctantly, Elizabeth acknowledged the truth of her patron's words with a nod. Secession would cause the war, and the state delegates who had voted to leave the Union would have to bear the responsibility. Still, Elizabeth had no reason to doubt that somehow the blame would shift to the people of her race, as it so often, so unfairly, did in other matters.\n\n\"I may come back to Washington in a few months, and live in the White House,\" Mrs. Davis mused, turning around to examine the drape of her dress in the mirror. \"The Southern people talk of choosing Mr. Davis for their president. In fact, it may be considered settled that he _will_ be their president. As soon as we go South and secede from the other states, we will raise an army and march on Washington, and then I shall live in the White House.\"\n\n\"Mrs. Davis,\" Elizabeth managed to say, \"I'm very pleased that you've placed so much confidence in me. However, I\u2014\" She had to stop, to take a breath, to find a moment to think. \"I have my business to consider. I have my church and my friends.\"\n\n\"We do have churches in the South, you may recall, as well as many colored women who would surely like to count you among their friends.\" A faint, amused smile turned up the corners of Mrs. Davis's mouth. \"As for your business, it will surely thrive. I'll have plenty of work for you within my own household, but if that doesn't suffice, with my recommendation you'll have no trouble finding many eager new customers in Montgomery.\"\n\nSo the Davises were planning to remove to Alabama, not Mississippi. Elizabeth wondered why her patron had not mentioned this before. \"I\u2014I don't know what to say. Forgive me my uncertainty. I'm very grateful you think so highly of my work.\"\n\n\"Not only of your work, but also of you.\" Mrs. Davis caught her gaze in the mirror and held it. \"Promise me you'll consider my proposal\u2014although time is of the essence. I'll need your answer soon.\"\n\nThat much Elizabeth could do. \"I promise.\"\n\nElizabeth kept her word, pondering Mrs. Davis's proposal, praying over it. She was tempted to accept. She liked the Davis family, and Mrs. Davis's reasoning seemed plausible. But to go so far south, so deep into the land of slavery\u2014even as a freedwoman, life there would be difficult for her, far more difficult than in the slaveholding District of Columbia. But as much as she liked Mrs. Davis, she liked the Lewises more, and she would miss her friends in the congregation of Union Bethel Church. And though the Northerners might, as Mrs. Davis predicted, blame the colored race for the inevitable war and turn upon them in anger, weren't Southerners as likely to do the same?\n\nAfter pondering the question alone, and with the deadline for her decision approaching, Elizabeth turned to her friends for guidance. One and all, freeborn and former slave, urged her to remain in Washington. They were astonished that Mrs. Davis would even presume to ask such a thing upon such a short acquaintance. Elizabeth had been in the family's employ less than three months, and Mrs. Davis expected her to leave her home and place herself in unimaginable risk in a land she herself expected soon to be torn by war? They did not believe, as Elizabeth did, that Mrs. Davis's offer was generous, that it was a sign of respect. \"Don't go,\" Virginia implored after one late-night talk in her parlor. \"If you change your mind, you may not be able to come home.\"\n\nElizabeth knew her friends were right. She also knew that the North was far stronger than Mrs. Davis seemed to believe\u2014in spirit as well as might. Mr. Lincoln's people were powerful and eager for victory, and Elizabeth could not believe that they would let the Southern states go without a fight or that they would give up as soon as the Southerners resisted. In the end, after all her questioning and pondering and prayer, her decision came down to one irrefutable fact: She was a colored woman, and she would be far wiser to cast her lot with the people of the North, many of whom supported abolition, than those of the South, most of whom believed she belonged in chains.\n\nElizabeth had not yet told Mrs. Davis of her decision when she arrived at the Davis residence a few days later to find that her patron had gone out earlier that morning to purchase several yards of floral chintzes, pretty but less fine than the fabrics she usually favored. \"I'd like you to make me two wrappers,\" Mrs. Davis said, draping the fabrics upon the sofa.\n\n\"From chintz?\"\n\n\"Yes, Elizabeth, from chintz.\" Mrs. Davis's smile twisted as if she were fighting back tears. \"I must give up expensive dressing for a while. Now that war is imminent, I\u2014and I daresay all Southern people\u2014must learn to practice lessons of economy.\"\n\n\"Of course.\" Elizabeth gathered up the fabrics. \"I'll get started right away.\"\n\n\"Thank you.\" After a moment, she added, \"Elizabeth?\"\n\n\"Yes, Mrs. Davis?\"\n\n\"I think...\" Her voice trailed off, and she inhaled deeply. \"To be prudent, it would be best to finish the wrappers sooner rather than later.\"\n\nElizabeth understood.\n\nThe Davises' course was fixed, irrevocable. It was only a matter of time.\n\nElizabeth finished the wrappers a few days before the Davis family left Washington. When she presented the finished garments to Mrs. Davis, she admired them, set them aside, and handed Elizabeth a bundle of fine needlework, a difficult work-in-progress of her own that she wanted Elizabeth to finish. \"You can send it to me by post when it is done,\" she instructed, and then she paused to offer Elizabeth a hopeful smile. \"Or perhaps you can call on me in the parlor of my new home and hand it to me yourself. Perhaps you won't have to walk very far. I'm sure that wherever we settle, we can arrange a room in our residence for you.\"\n\nElizabeth could delay no longer. \"I'm very sorry, Mrs. Davis. I'm happy to finish your sewing, but I'll have to send it to you. I've decided to stay in Washington.\"\n\nMrs. Davis pressed her lips together and nodded as if she had expected Elizabeth to refuse, and yet she could not quite give up her cause for lost. \"Aren't you tempted, even a little, by the prospect of being a First Lady's personal modiste?\"\n\nElizabeth laughed shakily. \"I am indeed, but not enough to leave my home. I promise, ma'am, if you return to Washington, I'd be pleased to sew for you again. More than pleased\u2014I would be delighted.\"\n\n\"Oh, Elizabeth.\" Mrs. Davis regarded her with sad affection. \"You betray yourself. You said _if_ I return, not when.\"\n\nOn the twenty-first of January, Jefferson Davis and several other Southern senators resigned their seats and left Washington, casting their lots with their home states. Later Elizabeth would read in the papers that Mr. Davis had expressed love for the Union and a desire for peace, but he had also asserted his right to own slaves and the right of states to secede. \"I am sure I feel no hostility to you, senators from the North,\" he had told the assembly. \"I am sure there is not one of you, whatever sharp discussion there may have been between us, to whom I cannot now say, in the presence of my God, I wish you well.\"\n\nSoon thereafter, Mrs. Davis left Washington with her husband and children and slaves. The Southern states elected Jefferson Davis as their president, and Varina Davis became First Lady of the Confederacy.\n\n# Chapter Two\n\n## FEBRUARY\u2013MARCH 1861\n\nAll of Washington City was abuzz with anticipation\u2014and in certain quarters, apprehension\u2014for the arrival of President-Elect Lincoln. He and his family were approaching the capital by a circuitous train route, both to greet as many supporters along the way as he could and to thwart anyone who might attempt to do him harm. \"He hasn't even taken office yet and those secessionists are already threatening his life,\" said Walker Lewis one morning, offering Elizabeth his newspaper, which disgusted him too much for him to continue reading. \"They won't bother to wait and see what he might do in office. They hate him on principle.\"\n\nThe ongoing, escalating conflict over Fort Sumter in Charleston Harbor also dominated conversation. Ever since December 26, when United States major Robert Anderson had moved his command from the vulnerable Fort Moultrie on Sullivan's Island to the stronger, more defensible fortifications controlling the harbor, the handful of federal troops there had been essentially under siege. In early January, on the same day Mississippi seceded from the Union, South Carolinian forces had fired upon the _Star of the West,_ an unarmed merchant ship President Buchanan had sent to resupply and reinforce Major Anderson and his men. The ship had been forced to turn back, and reports from Fort Sumter had become increasingly dire as the men ran low on food, arms, and supplies. Although many Republicans called for an immediate military response, President Buchanan seemed inclined to wait out the last few remaining weeks of his presidency and let Abraham Lincoln worry about it when he took over.\n\nA few officers' wives had been living on Sullivan's Island with their husbands, but when Major Anderson moved his troops from Fort Moultrie to Fort Sumter, the ladies had been sent over to Charleston for their safety. There they found every door closed to them. Not a single boardinghouse would offer them lodgings, and one landlady bluntly declared that if she offered the officers' wives a safe haven, she would lose all her other boarders. Discouraged and angry, the women had been obliged to leave their husbands to their defense of Fort Sumter and seek refuge in the North. When they arrived in Washington, bitter and defiant, they found themselves warmly welcomed by the Republicans and celebrated as the first martyrs of the war.\n\n\"I cannot imagine such a state of feeling,\" one of Elizabeth's patrons declared as Elizabeth dressed her for a levee at the White House one evening. Margaret Sumner McLean was the daughter of the Massachusetts-born Major General Edwin Vose Sumner and the wife of Captain Eugene McLean, a Maryland native with unabashed Southern sympathies. Her father's cousin was the abolitionist senator Charles Sumner, who had been savagely caned and nearly murdered on the Senate floor almost five years earlier by a colleague from South Carolina who had taken great offense to one of his antislavery speeches, which was not surprising considering that it had been full of personal insults. To say that Mrs. McLean's loyalties were probably divided in those troubled times was, in Elizabeth's opinion, a grave understatement. \"To turn away helpless women, to leave them homeless and unprotected! I am quite indignant with so-called Southern chivalry.\"\n\n\"I hear a few Southern gentlemen offered the ladies rooms in their own homes,\" said Elizabeth, adjusting the fall of lace around Mrs. McLean's shoulders. \"The ladies declined.\"\n\nMrs. McLean laughed in brusque dismissal. \"They had no choice, and the gentlemen of Charleston must have known that. How could officers' wives accept the hospitality of men who had openly avowed enmity toward their husbands? I can imagine the turn the negotiations would have taken: 'Ho, there, Major Anderson! Still determined to hold Fort Sumter, are you? Did I mention that we have your men's wives?'\"\n\n\"You think those Charleston men would have harmed the ladies?\"\n\n\"Oh, probably not. They were gentlemen, not common ruffians. But even if their safety had been guaranteed and notarized, the ladies couldn't put their brave husbands under any obligation to their enemies. Nor should it have been necessary. Where were the kindly widows of Charleston, the dutiful landlords? How would they feel if their women were treated so contemptuously in Washington or New York?\"\n\n\"I imagine they would not think well of it.\"\n\n\"No, indeed. They would consider it an act of war.\" Her gown and hair in order, Mrs. McLean dismissed Elizabeth with her thanks and a promise to send along Elizabeth's good wishes in her next letter to Varina Davis, a dear friend whom she missed terribly. Washington society wasn't the same without her, and the excitement of new acquaintances was a poor substitute for the company of longtime, loyal friends.\n\nElizabeth didn't see Mrs. McLean again until nearly two weeks later, when she unexpectedly called at the Lewis boardinghouse. Elizabeth was hard at work sewing in her rooms when she heard footsteps and the rustle of hoop skirts in the hallway. Glancing up from her work, she was astonished to discover Mrs. McLean standing there, regarding her imperiously over a muslin-wrapped bundle tied with twine. Elizabeth's heart sank, but she kept her face pleasantly expressionless. She did not like it when her patrons came to her rooms, especially without any notice whatsoever. It was more appropriate to their status\u2014and more protective of her privacy\u2014if she went to them.\n\n\"Elizabeth, I am invited to dine at Willard's next Sunday,\" Mrs. McLean declared by way of a greeting. \"I positively have _not_ a dress fit to wear on the occasion. I have just purchased material, and you must commence work on it right away.\"\n\nElizabeth did not allow even the smallest sigh of exasperation to escape her lips. \"I have more work promised now than I can do,\" she replied evenly. \"It is impossible for me to make a dress for you to wear on Sunday next.\"\n\n\"Pshaw! Nothing is impossible.\" Mrs. McLean swept the room with her gaze as if to confirm that no other lady stood waiting to assert her claim upon Elizabeth's time. \"I must have the dress made by Sunday.\"\n\n\"I am sorry\u2014\"\n\n\"Now, don't say no again. I tell you that you must make the dress.\" Mrs. McLean raised a hand to forestall Elizabeth's objections. \"I have often heard you say that you would like to work for the ladies of the White House. Well, I have it in my power to obtain you this privilege. I know Mrs. Lincoln well, and you shall make a dress for her provided you finish mine in time to wear at dinner on Sunday.\"\n\nFor a moment, Elizabeth wondered how a Massachusetts-born wife of a Southern-sympathizing Marylander could have any acquaintance at all with the new First Lady, late of Illinois and not yet arrived in Washington. Then she remembered that Mrs. McLean's father was one of the cavalry officers who had volunteered to accompany the Lincolns from Springfield to the capital city. Perhaps the families were old friends from Mr. Lincoln's term in Congress years before. Perhaps Mrs. McLean could indeed bring about what she promised.\n\nElizabeth needed no further inducement. \"Very well,\" she said, setting aside the dress she had been embroidering for another client, a kind, sweet-faced, grandmotherly woman who would never dream of complaining if Elizabeth finished her gown a day or two later than expected. Elizabeth felt a pang of guilt over the injustice, but she could not let such a rare and promising opportunity slip from her grasp. She resolved to make it up to her other patron somehow\u2014but not before next Sunday.\n\nElizabeth persuaded Mrs. McLean to let her begin fitting the lining that very moment by assuring her that it was the only way she had any chance at all of finishing the dress in time. They met the next day at Mrs. McLean's residence across the street from Willard's Hotel, and the next, and then Elizabeth had enough completed so that she could work on her own. But she was not entirely on her own; in order to meet the deadline, she hired two young ladies to assist her, skilled if inexperienced seamstresses she had hired for small tasks before. Even with their help, Elizabeth was obliged to work all day long and late into the night, barely pausing to eat and sleep and work the knots out of her neck, back, and fingers.\n\nIt took a small miracle, but somehow, after much worry and trouble, Elizabeth completed the dress by midmorning on the day of the dinner. She hurried to the McLean residence as soon as the last stitch was put into the hem; had her patron try it on; made a few last-minute, minuscule adjustments that likely no one else but another skilled mantua maker would notice; and then\u2014at last and just in time\u2014the dress was finished.\n\n\"Elizabeth, you are a marvel,\" Mrs. McLean declared, admiring herself in the full-length mirror. She had a lovely figure and the bodice fit as tightly as wallpaper, and the ashes of rose silk lent a warm elegance to her alabaster skin. \"One glimpse of this dress is all the recommendation to Mrs. Lincoln you will need.\"\n\n\"All the same,\" said Elizabeth, \"I would be grateful if you would put in a good word for me too.\"\n\nMrs. McLean turned and peered over her shoulder at her own reflection, smiling with satisfaction at the snug lines of the bodice and the graceful drape of the flounce. \"I'm tempted to keep you my little secret.\" She glimpsed Elizabeth's expression in the mirror and quickly added, \"I'm only teasing. Of course I'll mention your name to the First Lady. You kept your part of our bargain and I'll keep mine.\"\n\nElizabeth murmured her thanks and hid her disappointment. Apparently she and Mrs. McLean remembered their arrangement quite differently. She had not toiled so long and so hard and with so little sleep for the mere mention of her name to the First Lady. \"You shall make a dress for her\" had been her patron's exact words. She had made it sound so certain. Now Elizabeth wondered if it had ever been within Mrs. McLean's power to grant her fondest wish.\n\nElizabeth cherished each precious letter George sent from Wilberforce University. She savored every word of his descriptions of his studies, his new friends, their merry antics, and his long hours bent over books in the library. He asked her as many questions about Washington as he answered about himself, eager for news of the election, the transition of power, and the famous personages she encountered. \"Your observations make me the envy of my classmates,\" he told her. \"They have to learn what little they can from the papers, while I receive your firsthand, eloquent dispatches at regular and frequent intervals. The fellows come to me with their questions about what is really going on in the capital (don't worry, Mother, I never pass on gossip about your patron ladies) and I am happy to oblige. I will never share the fine shirts you sew for me, Mother, but I am less selfish when it comes to the news.\"\n\nGeorge's gentle teasing made her smile, as did his generous praise. Her lack of formal schooling had always troubled her, although she knew she was fortunate to be literate at all. It was illegal for slaves to read and write, but none of Elizabeth's former owners had forbidden their slaves to learn if they were bright enough to pick it up on their own or if another slave taught them on their own time. Elizabeth did not think she was a very good writer, so she could not help but be flattered and charmed when her darling son called her eloquent.\n\nShe was laughing over George's comic description of a recent snowball fight on the college's main quadrangle when a knock sounded on her door. It was an unexpectedly mild Sunday, a welcome glimpse of spring in early March, and she and Virginia had made plans to go walking later to watch the preparations for the next day's inauguration. Elizabeth hastened to the door, expecting her friend, but instead a young messenger stood outside in the hallway, a colored boy of about fourteen years, breathless from haste and clutching a letter. \"I'm supposed to wait for a reply, ma'am,\" he said, panting.\n\nCurious, Elizabeth unfolded the letter, skimmed the first line\u2014and uttered a sound that fell somewhere between a groan and a laugh. \"My dear Elizabeth,\" Mrs. McLean had written. \"Do be so kind as to call on me at my home today at four o'clock P.M. With my best regards, etc., Mrs. Eugene McLean.\"\n\nShaking her head, Elizabeth read the brief letter again, as if she could make more words appear by sheer force of will. \"Did she say anything else?\" she asked the young messenger. \" _Why_ she wants me to come, perhaps?\" And on such short notice, she almost added, but thought better of it.\n\n\"No, ma'am,\" he said, shaking his head. \"Just that I should bring this to Mrs. Elizabeth Keckley at the Lewis boardinghouse on Twelfth Street.\"\n\n\"Well, you've done as you were told.\" She folded the letter, slipped it into her pocket, and offered the messenger a coin. \"Please return to Mrs. McLean and tell her that I thank her for her note, and I will call on her tomorrow morning at my earliest convenience.\"\n\nHe nodded and hurried off, and Elizabeth closed the door behind him. Sunday was supposed to be her day off, as her patrons ought to have known. She was a free woman, free to ignore unreasonable demands. Monday morning would be soon enough for whatever Mrs. McLean wanted, and if not, she could hire another modiste\u2014although Elizabeth did not think it vain to doubt that her patron would find anyone as skilled and accommodating as herself.\n\nThe morning of Abraham Lincoln's inauguration dawned raw and overcast, the busy streets clouded with dust stirred up by a thin, intermittent wind. It was not yet nine o'clock when Elizabeth left her boardinghouse and joined the throngs of people already out and about, buzzing with anticipation for the day's events. She saw not one familiar face among the thousands as she walked the few blocks to Mrs. McLean's house. Along the way she spotted evidence of out-of-town visitors who had been unable to find room in the city's packed hotels and had instead spent the night on the streets\u2014a makeshift bed on a pile of lumber, a line of people waiting to wash up at a public pump. An excited crowd was milling about Willard's Hotel when Elizabeth arrived, and she struggled to work her way through it to the McLeans' residence on the opposite side of the street. Breathless, she knocked loudly upon the front door, uncertain whether she would be heard at all, but before long the doorman answered. When she gave her name and explained her errand, she was told that Mrs. McLean was not at home. Just as she was wondering whether she should wait there or try to return home, one of Colonel McLean's aides appeared and told her she was wanted at Willard's.\n\nPerplexed, Elizabeth again braved the throng, crossed the street, and somehow managed to push her way into the hotel through the crowded entrance. \"There you are,\" someone exclaimed, and a hand seized her shoulder and whirled her about. It was Mrs. McLean, her gaze sharp and incredulous. \"Why did you not come yesterday, as I requested?\"\n\n\"Your note didn't say that it was urgent,\" Elizabeth reminded her.\n\n\"On the eve of the inauguration, how could it be otherwise?\" Mrs. McLean's mouth thinned in disapproval. \"Mrs. Lincoln wanted to see you, but I fear that now you are too late.\"\n\n\"Mrs. Lincoln wanted to see _me_?\"\n\nMrs. McLean nodded impatiently. \"A week ago, someone spilled coffee on the gown Mrs. Lincoln intended to wear today. She needed a dressmaker, so I recommended you. Lo and behold, she recognized your name. Apparently you've worked for some of her lady friends in St. Louis, not that it matters now.\"\n\n\"I'm sorry, Mrs. McLean,\" said Elizabeth, her heart sinking. If Mrs. Lincoln had requested a dressmaker, why on earth had Mrs. McLean waited a week to summon her? \"You did not say what you wanted with me yesterday, so I judged that this morning would do as well.\"\n\n\"You should have come yesterday,\" Mrs. McLean scolded, but then she relented, if only a trifle. \"Go on up to Mrs. Lincoln's room. She may find use for you yet.\"\n\nAs soon as Mrs. McLean gave her the number of the suite, Elizabeth hurried off to find parlor number six. When she knocked upon the door, a cheerful voice invited her to enter, and when she stepped into the room, she found herself face-to-face with a dark-haired woman just over forty, inclined to stoutness but with a lovely complexion and clear blue eyes that boasted a quick, keen gaze. All about her were well-dressed ladies helping her prepare for the inauguration.\n\nThe dark-haired woman did not introduce herself, nor did she need to. \"You are Elizabeth Keckley, I believe.\"\n\nElizabeth bowed her assent.\n\n\"The dressmaker that Mrs. McLean recommended?\"\n\n\"Yes, madam.\"\n\n\"Very well.\" Mrs. Lincoln returned to her dressing table and examined her face in the mirror, touching the delicate skin beneath her eyes, frowning at what might have been newly discovered or newly imagined lines. \"I have not time to talk to you now, but would like to have you call at the White House, at eight o'clock tomorrow morning.\" Turning in her seat, she caught Elizabeth's gaze and held it. \"Where I shall then be.\"\n\nThe brief meeting was over. Elizabeth bowed herself out of the room and returned home, insensible to the ever-increasing crowds, the gathering of horses and men for the grand parade, the distant strains of martial music. Only a few years before, she had been a slave in St. Louis, working herself into a state of near collapse and wondering if she would ever earn enough money to purchase her freedom and her son's. Now she had an invitation to meet with the First Lady at the White House\u2014and an extraordinary opportunity to win her as a patron.\n\nElizabeth wished George could be there to walk into the White House at her side. She would remember every detail and describe everything to him\u2014every sight, every word of conversation, filling pages and pages if necessary so that it would be as if he had experienced it with her.\n\nShe spent the rest of the day alone in her rooms, sewing when she could keep her mind on her work, but more often letting her thoughts drift to her upcoming interview with Mary Lincoln.\n\nLater her friends would tell her about the thrilling inauguration ceremony, about how the sky had finally cleared and how the proud cavalry had surrounded the president-elect's carriage as it rumbled at a stately pace up the cobblestones of Pennsylvania Avenue to the Capitol. They would tell her about Mrs. Lincoln, how she had glowed with pride as she observed the ceremony with her family from her seat on the platform erected on the Capitol steps. Elizabeth would smile when they described how the tall, gaunt president-elect had stepped forward, removed his hat, and then suddenly halted, realizing only then that he had no place to put it while he took his oath. His former opponent, the Illinois Democratic senator Stephen A. Douglas, had gallantly taken the hat and had held it for the new president until the last words were spoken. Elizabeth would be moved, later, when she read a transcription of President Lincoln's speech and learned how he had said, of the disagreeing citizens of North and South, \"We are not enemies, but friends. We must not be enemies. Though passion may have strained, it must not break our bonds of affection. The mystic chords of memory, stretching from every battlefield and patriot grave to every living heart and hearth-stone all over this broad land, will yet swell the chorus of the Union when again touched, as surely they will be, by the better angels of our nature.\"\n\nKnowledge of all this would come later. From the solitude of her rooms on Twelfth Street, Elizabeth heard only the low booming of cannon fire that marked the moment Abraham Lincoln became the sixteenth president of the United States. More than thirty thousand people had packed the fenced grounds of the Capitol to witness the historic occasion, but Elizabeth, lost in her own thoughts and eagerly anticipating what could turn out to be the most important day of her life, was not among them.\n\n# Chapter Three\n\n## MARCH\u2013APRIL 1861\n\nA few minutes before eight o'clock on the morning after the inauguration, Elizabeth walked one-third of a mile from her home to the White House, crossing Lafayette Square, passing the bronze statue of Thomas Jefferson in the semicircular drive. When she ascended the portico leading to the front entrance of the Executive Mansion, the short, burly, elderly Irish doorman admitted her into the vestibule, where moments later she spotted a familiar figure approaching from down a wide hall. \"Good morning, Mrs. Keckley,\" the butler greeted her. \"Welcome to the White House.\"\n\n\"Thank you, Mr. Brown,\" she replied, pleased and unexpectedly comforted to see a friendly face. Like her, Peter Brown was a former slave. He and his family lived only three blocks from Elizabeth, and they had become acquainted through the Lewises. \"I've come to meet with Mrs. Lincoln to see if she would care to hire me as her dressmaker. Any advice to smooth my way?\"\n\nPeter Brown chuckled and escorted her up a busy central staircase. Men of all types hurried by, both coming and going, almost certainly job seekers like herself, eager to secure a post within the new administration. \"I haven't known the new First Lady long enough to be a good judge of her likes and dislikes,\" Peter confided in an undertone. He paused for a moment on the landing, allowing others to pass. \"I'll say this much: Whatever you see, don't let it trouble you.\"\n\nMystified, Elizabeth wondered what he could possibly mean, but when he led her into a waiting room full of worn mahogany furniture, and three other well-dressed women looked up at their approach, she understood at once. She should have realized that the First Lady would have asked several of her acquaintances to recommend their favorite dressmakers, and naturally each lady would have been eager to curry favor by putting forth her favorite. As the women exchanged nods of greeting, Elizabeth felt their eyes upon her taking in every detail of her attire, just as she was assessing theirs. No one trusted a dressmaker who was not herself garbed in the most becoming costume she could afford. She was thankful she had worn her newest, most stylish dress, sewn from the finest blue, red, and tan plaid wool and perfectly tailored to her figure, with the colored lines so painstakingly matched that the pattern seemed unbroken from bodice to skirt.\n\n\"Mrs. Lincoln is still at breakfast,\" Peter told her, showing her to a chair. \"She'll summon you presently. Chin up.\"\n\nElizabeth thanked him with a smile, but as she sank into her chair, her hopes plummeted. She had not expected to face any rivals that morning, and she doubted she would be chosen over these women, who were almost certainly better established in Washington City than she was. To make matters worse, each of them had the distinct advantage of being white. But she could do nothing to change those plain facts, so she sat straight in her chair, serene and patient, until at last one of the ladies Elizabeth had seen with Mrs. Lincoln at Willard's appeared and summoned the first mantua maker into an adjoining room.\n\nElizabeth was the last to be called.\n\nShe was taken into a family sitting room, oval shaped with a high ceiling and tall windows that looked out upon the White House lawn, which sloped downhill to a tall iron fence and a marsh leading to the Potomac just beyond. Mrs. Lincoln stood beside one of the windows chatting animatedly with a companion Elizabeth did not recognize, but she glanced up as Elizabeth entered and came to greet her. \"You have come at last,\" she said warmly.\n\n\"Thank you for seeing me, madam,\" said Elizabeth. She reached into her bag and withdrew several papers, which she had protected from creases and tears between two stiff pieces of cardboard. \"I bring you several letters of recommendation from my customers in St. Louis.\"\n\n\"Yes, I know your reputation well.\" Mrs. Lincoln scanned the letters, nodded with satisfaction, and returned them. \"For whom have you worked in Washington City?\"\n\n\"Among others, Mrs. Senator Davis has been one of my best patrons.\"\n\nMrs. Lincoln's eyebrows rose. \"Mrs. Davis! So you have worked for her, have you? Of course you gave great satisfaction?\"\n\n\"Certainly, madam.\"\n\n\"So far, so good. Can you do my work?\"\n\n\"Yes, Mrs. Lincoln,\" Elizabeth assured her emphatically. \"Will you have much work for me to do?\"\n\nMrs. Lincoln cupped her chin in her hand as if considering the question. \"That, Mrs. Keckley, will depend altogether upon your prices. I trust that your terms are reasonable. I can't afford to be extravagant.\" She threw a rueful glance to her companion, who shook her head in commiseration. \"We are just from the West, and are poor. If you do not charge too much, I shall be able to give you all my work.\"\n\n\"I don't think there will be any difficulty about charges, Mrs. Lincoln.\" In Elizabeth's experience, white ladies of Mrs. Lincoln's status defined \"poor\" rather differently than she herself would. \"My terms are reasonable.\"\n\n\"Well, if you will work cheap, you shall have plenty to do.\" A slight frown belied the breezy lightness of her tone. \"I can't afford to pay big prices, and so I frankly tell you so in the beginning.\"\n\nThey quickly arranged satisfactory terms, and only then did Mrs. Lincoln mention that she wanted Elizabeth to begin immediately. She beckoned to her friend, who promptly produced a bright, rose-colored moir\u00e9-antique dress that Mrs. Lincoln wished to wear to the first levee of her husband's presidency, a grand public reception to be held on Friday evening. Mrs. Lincoln described the alterations she wanted, which were not at all daunting, so although Elizabeth would have only three days to work on it, she assured Mrs. Lincoln that the dress would be finished on time. After taking the First Lady's measurements, she carried the dress home, where she worked on it until late into the night. She knew that this garment, though not of her own design and inelegantly begun, was an audition of sorts. If she did well, Mrs. Lincoln would trust her with an original gown, and perhaps many more after that. If her alterations were judged unsatisfactory, she should expect no second chances.\n\nThe next day Elizabeth returned to the White House to fit the gown and found Mrs. Lincoln in excellent spirits, the center of a lively group of ladies, relations from Illinois, Kentucky, and elsewhere, all come to Washington City to enjoy the festivities. Mrs. Lincoln was dressed in a cashmere wrapper with fine quilting worked down the front and a simple headdress, while her companions wore morning robes. They chatted and teased one another cheerfully as Elizabeth helped Mrs. Lincoln into the dress and marked a few necessary adjustments. All the while Mrs. Lincoln was so merry and gracious that Elizabeth wondered what could have inspired the disparaging rumors about her temperament. Elizabeth had overheard some of her patrons\u2014all Southerners, now that she thought about it\u2014refer to the new First Lady as an ill-mannered, ignorant, and vulgar country bumpkin, but Mrs. Lincoln was certainly none of those things. Clearly Elizabeth would do well to trust the evidence of her own observations and dismiss such unkind remarks as the product of jealousy and politics.\n\nLater, back in her own rooms, Elizabeth worked diligently upon the gown, adorning it with pearls and a point-lace cape with little time to spare. On Thursday afternoon, she was surprised to receive another summons to the White House, but having learned from the fiasco with Mrs. McLean, she quickly packed up the dress and her sewing basket and hurried off without a moment's delay. Upstairs in the oval sitting room once again, Mrs. Lincoln informed her that the levee was postponed until Tuesday, but before Elizabeth could breathe a sigh of relief, Mrs. Lincoln added that she had conceived of a few more improvements for the gown. Elizabeth hid her dismay as Mrs. Lincoln described what she wanted and it became evident that these were no simple adjustments but rather a significant alteration of the style. Still, she thought she could accomplish them in time and said so, and when Mrs. Lincoln added a waist of blue watered silk for her cousin Mrs. Grimsley to the order, she agreed.\n\n\"All will be ready in plenty of time for you to dress for the levee on Tuesday,\" Elizabeth promised, taking her leave as quickly as she could, knowing she had not a moment to spare.\n\nFor days she sewed and sewed, rising early and working late, barely pausing for meals and church and a newly arrived letter from George. She completed the last stitches early on Tuesday evening, just in time to fold it carefully with Mrs. Grimsley's completed blue silk waist and hurry the one-third of a mile to the White House. Peter Brown greeted her in the vestibule, but she knew the way well enough by then that he let her hasten upstairs on her own.\n\nThere she found Mrs. Lincoln in her dressing gown, in a terrible state of excitement despite her companions' attempts to soothe her. \"I cannot go down,\" Mrs. Lincoln exclaimed, tearing herself from a cousin's embrace. \"How could I possibly? I have absolutely nothing to wear. Imagine what people will say. Think of how they'll mock me.\"\n\n\"No one will mock you,\" said Mrs. Edwards, Mrs. Lincoln's patient elder sister, clad in a subdued gown of brown brocade.\n\n\"Wear the gray velvet,\" said Mrs. Grimsley, who wore a blue watered silk gown with a long train adorned with turquoises and pearls, and a headdress of white roses. \"It's lovely and no one here has seen it yet.\"\n\nJust then, the youngest of the ladies spotted Elizabeth lingering uncertainly in the doorway. \"She's here,\" she cried out, gesturing, and everyone turned Elizabeth's way.\n\n\"There, now,\" said Mrs. Edwards, visibly relieved. \"Didn't we tell you she would come?\"\n\nHer lips pressed together tightly, Mrs. Lincoln strode two paces toward Elizabeth before stopping short. \"Mrs. Keckley, you have disappointed me\u2014deceived me.\" Her face was pale with outrage, her eyes red rimmed. \"Why do you bring my dress at this late hour?\"\n\nElizabeth took a deep breath and carefully unfolded the gown, Mrs. Grimsley's smaller garment still draped over her arm. \"Because I have just finished it, and I thought I should be in time.\"\n\n\"But you are not in time, Mrs. Keckley.\" Mrs. Lincoln shook her head, wrung her hands, and began pacing. \"You have bitterly disappointed me. I have no time now to dress, and, what is more, I will _not_ dress, and go downstairs.\"\n\nHer ladies protested and reached out to her, but she brushed them off.\n\n\"I am sorry if I have disappointed you, Mrs. Lincoln, for I intended to be in time.\" Humiliated, Elizabeth took care to keep her voice calm, even, and firm. She saw no reason for such distress. She had plenty of time to get Mrs. Lincoln ready for the levee if the good woman would only consent to it. \"Will you let me dress you? I can have you ready in a few minutes.\"\n\n\"No, I won't be dressed.\" Mrs. Lincoln halted, threw one glance at the window, another at the door, and flung her hands into the air, helpless. \"I will stay in my room. Mr. Lincoln can go down with the other ladies.\"\n\n\"But there is plenty of time for you to dress, Mary,\" said Mrs. Grimsley.\n\n\"Let Mrs. Keckley assist you,\" implored Mrs. Edwards, \"and she will soon have you ready.\"\n\nUncertain, Mary looked from her sister to her cousin and back. \"Oh, very well,\" she said, subdued. \"She may try.\"\n\nQuickly Elizabeth helped Mrs. Lincoln out of her dressing gown and into the moir\u00e9 antique before she could change her mind. She arranged Mrs. Lincoln's dark hair with red roses that complemented the hue of her gown. The dress was very becoming, and when Mrs. Lincoln examined herself in the mirror it was as if a freshening breeze blew away the storm clouds of only moments before, for she was all sunshine and smiles, very pleased with her appearance. She did indeed look very elegant in the rose moir\u00e9 antique, accented beautifully by her pearl necklace, earrings, and bracelets.\n\nA knock sounded on the door, and President Lincoln himself came in, with young Tad and Willie trailing after him. He greeted the ladies with smiles and compliments, and then threw himself on the sofa without any apparent fear of wrinkling his evening attire. His sons were immediately upon him, wrestling and laughing, and he made a great show of bravely fending them off, joking all the while. Amused, Elizabeth suppressed a smile. She never could have imagined the leader of the land roughhousing with his boys like any fond father.\n\nBefore long, Mrs. Lincoln said pointedly, \"Perhaps the boys have had enough of that for now.\"\n\n\"No, Mother,\" said Willie. \"We're not at all tired.\"\n\nMr. Lincoln laughed. \"Maybe not, but your father is.\" He swung his long legs around and rose from the sofa as the boys darted off to play. Pulling on his gloves, he quoted a few lines from a poem about a barefoot boy's carefree play, and then another about a blacksmith whose hard work earned him a night's repose.\n\nThe other ladies were charmed, but his wife seemed barely able to contain her impatience. \"You seem to be in a poetical mood tonight.\"\n\n\"Yes, Mother, these are poetical times.\" His smile deepened as he looked her over. \"I declare, you look charming in that dress. Mrs. Keckley has met with great success.\" As he offered compliments to the other ladies\u2014Mrs. Grimsley in the blue watered silk, Mrs. Edwards in an understated gown of brown and black, Miss Edwards in crimson, Mrs. Baker in ashes of rose, and Mrs. Kellogg in lemon-colored silk\u2014Elizabeth flushed with pleasure and pride. The president himself had praised her handiwork and referred to her by name, although until that moment she had not realized that he even knew it.\n\n\"Shall we go downstairs, Mother?\" the president asked his wife.\n\n\"In a moment,\" she said, frowning as she searched her dressing table. \"I cannot find my lace handkerchief.\"\n\nEveryone swore that they had seen it on the table only moments before, but no one had any idea what had become of it. Elizabeth joined in the search with the others, but after a few minutes, Mr. Lincoln's laughter boomed again, and he raked a long-fingered hand through his hair and sent an aide after Tad and Willie. When the boys were brought back to face their parents\u2014Willie, somber and good; Tad, bursting with mischievous smiles\u2014it was easy enough to deduce what had become of the handkerchief. Eventually the playful thief was persuaded to relinquish his prize and the handkerchief was restored to its rightful owner. Only then did Mrs. Lincoln smile, take her husband's arm, and lead the other ladies downstairs to the levee, as elegant and regal as any queen.\n\nFor a long moment, Elizabeth stood watching after them, a bit confounded, marveling at all that she had seen. \"George will never believe this,\" she murmured as she drew on her shawl and collected her satchel. Yet she had observed the scene with her own eyes.\n\nThat night, Elizabeth would later hear, a gracious and cordial President Lincoln had shaken hands with hundreds of well-wishers for more than two hours, welcoming strangers as warmly as friends and greeting many by name. Some of Elizabeth's longtime patrons had attended the event, but most Southern ladies of the established Washington social elite had stayed away, a calculated snub that could not have escaped the First Lady's notice. Those who did attend mingled freely, Northerners and Southerners alike, but their disagreements were surely never far from their thoughts, what with militias forming in their home states and Major Anderson's men languishing at Fort Sumter. While all eyes were fixed on the Lincolns\u2014measuring, appraising\u2014mischief had broken out in the unwatched cloakroom; when the guests departed, they discovered that their coats and wraps had been haphazardly mixed up and some had been stolen, inspiring one wag to remark that only one in ten guests left the gala clad in the same outer garments he had worn upon his arrival.\n\nMrs. Lincoln herself was well pleased with her first levee, having perhaps forgotten her distress and anger in the hours preceding it. She was less sanguine on the morning after the president's first state dinner held for members of his cabinet a few weeks later. Elizabeth had dressed her in a striking blue silk gown, beautifully embroidered, and had watched her take her husband's arm happily and descend the grand staircase as if she expected to have a perfectly lovely time. The next day, however, when Elizabeth returned to the White House to sew and to present various ribbons for Mrs. Lincoln to consider for a new bonnet, she found her patron in a state of bewildered distress. The previous night, as the dinner guests were leaving, the secretary of the treasury's daughter, the twenty-year-old, auburn-haired beauty Miss Kate Chase, had slighted Mrs. Lincoln in front of all their guests.\n\n\"Including men from the papers,\" lamented Mrs. Lincoln. \"Soon everyone will hear of it.\"\n\nElizabeth decided it would be kinder not to warn her that gossip would spread the tale swiftly enough without help from the press, if something deliciously shocking had indeed happened. Miss Chase was one of the most popular young ladies in Washington society, praised for her charm and wit as well as her beauty, and if she had affronted the First Lady, it would be quite a story. \"What did she do?\"\n\n\"Well...\" Mrs. Lincoln hesitated. \"It sounds silly when I describe it.\"\n\n\"Then perhaps it's really nothing after all.\"\n\n\"Oh, no, no, it's something.\" Mrs. Lincoln picked up a spool of ribbon, turned it over in her hands, and set it aside without really seeming to examine it. \"I was bidding my guests good-bye, and when it came to be Miss Chase's turn, I said, 'I shall be glad to see you anytime, Miss Chase.' She replied, in a thoroughly lofty tone, 'Mrs. Lincoln, I shall be glad to have _you_ call on _me_ at anytime.' See how she puts herself above me? I will have to go to her. She does not intend to visit me.\"\n\nElizabeth frowned. Miss Chase's remark did possess a certain air of disrespect, but her reputation was that of a perfectly lovely and admirable, if ambitious, young woman. \"That was not kind of her.\"\n\n\"Not kind? It was more than that. It was impertinent and unbecoming a young lady.\" Mrs. Lincoln rose and went to the window, looking out over the Potomac to the green hills of Virginia on the other side. \"I suppose I should expect nothing better from the daughter of Secretary Chase. You know she and her father expected the Republican Party to nominate him instead of my husband.\"\n\n\"I did not know.\"\n\nMrs. Lincoln turned away from the window, nodding. \"It's true, and since her mother is dead, if her father had been elected president, Miss Chase would have been his hostess. She believes that their rightful place is here in the White House, and that she should be First Lady now, not I.\"\n\n\"Well,\" said Elizabeth mildly, \"he wasn't, and she isn't.\"\n\nAfter a moment, Mrs. Lincoln laughed. \"Yes, that's right. Still, I'm certain it remains her greatest ambition.\"\n\nFrom that day forward, Mrs. Lincoln and Miss Chase were social rivals, each considering herself the highest lady in Washington society and resenting the other's attempts to demonstrate her superior rank. Miss Chase had the advantage of beauty, popularity, charm, and long-standing ties to the established elites, but Mrs. Lincoln had the president, the White House, and the title of First Lady. And, Elizabeth liked to flatter herself, Mrs. Lincoln had the advantage of a particularly skilled dressmaker who would make sure she always went out in public\u2014or into battle, as it sometimes seemed\u2014perfectly turned out.\n\nFor Elizabeth had become Mrs. Lincoln's regular modiste, and throughout the spring of 1861, she would sew more than fifteen gowns for the First Lady. She also often dressed Mrs. Lincoln in her finery and arranged her hair for balls, dinners, and levees. One evening, as the president observed how skillfully Elizabeth tended to his wife, he asked her if she were brave enough to attempt to subdue his own unruly locks.\n\n\"If you didn't make such a habit of running your hands through your hair, it wouldn't be such a tangle,\" Mrs. Lincoln admonished him.\n\nThe president merely smiled and sat down in his easy chair. \"Well, Madam Elizabeth,\" he asked, \"will you brush my bristles down tonight?\"\n\n\"Yes, Mr. President,\" she replied, taking his comb and brush in hand. When she finished, he examined himself in the mirror and declared that his hair looked as if it had been taught a lesson. He was so pleased that it became his custom to ask Elizabeth to attend to his hair after she finished dressing his wife, and Elizabeth did what she could with it.\n\nAs the weeks went by, Elizabeth took on other duties within the White House, such as running errands for Mrs. Lincoln and tending to Willie and Tad when they fell ill with the usual mild childhood ailments, but modiste to the First Lady remained her most prominent role. Over time she would learn that Mrs. Lincoln preferred to wear white but that she was also fond of pink, crimson, bright yellow, deep purple, and royal blue. She loved to wear pearls against her skin and flowers in her hair, and she favored low, open necklines with short sleeves to show off her well-formed neck and shoulders, ignoring whispered criticism that such styles were more appropriate for younger women.\n\nElizabeth soon learned that scathing criticism of her newest and most important patron would be unrelenting, coming from all corners in copious amounts, much to the dismay and consternation of its unhappy subject, who could do nothing to staunch the flood.\n\nOn the morning of April 12, Washington was jolted awake by shocking and often contradictory reports from Charleston. Before sunrise that day, Confederate cannon had fired upon Fort Sumter. No, indeed\u2014both parties were still engaged in serious negotiations. No, that was but wishful thinking\u2014shots had been fired. No one knew what to believe. A furious battle was ongoing, or Major Anderson had surrendered. The fort was destroyed utterly and its defenders slaughtered, or the starving, exhausted Union troops had marched out under a flag of truce and were being held prisoner. The citizens of Washington crowded telegraph offices and hotels, demanding news and spreading rumors, but no one knew precisely what was happening, what might have already happened. Secessionists who had kept their opinions to themselves since Mr. Lincoln's administration took over the capital now cheered the start of war. Southern sympathizers openly sought recruits for the Confederate army, while loyal Union men rushed to join militias. In the streets, arguments turned into fistfights, and then the most alarming rumor of all swept through the city: Rebels were marching on Washington with an army twenty thousand strong.\n\nEventually the truth of Fort Sumter reached the capital: After his troops had exchanged fire with Confederate guns for thirty-four hours, Major Robert Anderson had been forced to surrender. On April 14, five additional Washington militia companies were called into active duty, for a total of about twenty-five hundred local soldiers serving throughout the district. Mounted soldiers were posted at all approaches to Washington City. Twenty cavalrymen guarded the White House, with hundreds more stationed in the immediate surroundings and at the Capitol, the Treasury, and the post office. The following day, President Lincoln issued a nationwide call for seventy-five thousand recruits, assigning a quota to each state. These troops, who would enlist to serve for ninety days, would surely be sufficient to put down the rebellion.\n\nAs Washington awaited reinforcements from the North and fears of an imminent Confederate invasion grew, the young colored men of Elizabeth's comfortable middle-class neighborhood as well as the less fortunate who lived along the alleyways were as eager as their white counterparts to take up arms in defense of their city. Even Peter Brown's eleven-year-old son, who worked as a shoeshine boy on the grounds of the Treasury Building, proudly told Elizabeth of his plans to enlist as a drummer boy as soon as he turned twelve. But every young man of color who tried to enlist, regardless of his age, strength, or status, was turned away.\n\n\"Either the need for soldiers is very small or the foolishness of Mr. Lincoln's recruiters is very great,\" Virginia Lewis remarked to Elizabeth one Sunday afternoon as they went on their customary stroll and discovered a few militia soldiers drilling on the grounds of the Capitol. The city had taken on the appearance of an armed camp, and everywhere, apprehensions were on the rise. Although it was the capital of the Union, Washington was essentially a Southern city, surrounded by the Union slaveholding state of Maryland on one side and Virginia, which had seceded after Mr. Lincoln issued the state quotas for recruits, on the other side, with only the Potomac separating them.\n\nA few days earlier, the husband of one of Elizabeth's favorite patrons, Colonel Robert E. Lee, a Virginian, had been offered the command of the entire United States Army, but after careful consideration, he had declined and had gone home to his plantation in Arlington on the other side of the river. Elizabeth knew from conversations overheard in the White House that President Lincoln had fought valiantly to keep Virginia in the Union, and she knew from remarks Mrs. Lee had made in her presence that her husband had not wanted Virginia to secede. For every firebrand eager for war, there seemed to be two or more who had been drawn into the conflict reluctantly but were nonetheless resolved to do their duty with all their might. And with the conflict so obviously pivoting upon the point of slavery, was it any wonder that Negro men too wanted to do their part to help the Union triumph?\n\n\"The recruiters must think they'll have more than enough white volunteers to fill their quotas,\" Elizabeth said. \"If the fighting goes on longer than they expect, maybe they'll let colored men enlist later.\"\n\nAlthough the day was balmy, Virginia shivered. \"I'd rather have the fighting over before that day could come. How terrible this rebellion would be if seventy-five thousand men weren't enough to finish it. Can you imagine the bloodshed?\"\n\nElizabeth inhaled shakily, the acrid odors of camp refuse and coal smoke so heavy in the air that her eyes stung and watered. \"I can imagine it all too well.\"\n\nShe tucked her arm through Virginia's and they turned toward home. Although she thought it was an outrage that men of color were forbidden to enlist, she was secretly relieved that her son would not be required to lay down his life for his country.\n\nElizabeth wondered if her husband, James, would try to enlist. As a young man he had been full of fight, and after John Brown's failed raid on Harpers Ferry, he had declared that if he had been there, he would have taken up arms and stirred up the slave revolt John Brown had intended. Suddenly James's visage appeared so clearly to her mind's eye that it was as if he stood before her, not rambling and drunk as he had been in their last years together but smiling, bold, and handsome as he had been when they first met.\n\nElizabeth gasped and stopped short, shaken by the vision. She had not thought of her husband in weeks, perhaps months. Why would he come unbidden to her thoughts now?\n\n\"Elizabeth?\" Virginia had been brought to an abrupt halt when Elizabeth stopped. \"What's the matter? You look as if you've seen a ghost.\"\n\nElizabeth managed a shaky laugh. \"I haven't, not as far as I know. At least not recently.\"\n\nVirginia smiled tentatively at her joke, but she looked uncertain. \"What's troubling you, then?\"\n\nElizabeth hesitated. Virginia knew she was married and that she and her husband were estranged, but Elizabeth disliked speaking ill of James, and as a consequence, she had told Virginia little about him. She wasn't sure why, but she was reluctant to admit that she had been thinking about him then.\n\nSo instead she nodded to the scenes of preparation for the defense of the city\u2014all around them, and all inadequate. \"Aren't there reasons enough for all of us to be troubled these days?\"\n\nVirginia nodded. They watched a few moments more before continuing on home.\n\nSoon, Elizabeth would wonder whether James's restless, wistful spirit had indeed visited her in that moment she had imagined him so vividly.\n\nNot two days after her stroll with Virginia, a letter came from Missouri, written in a deliberate yet shaky hand, full of misspellings and apologies.\n\n_Dear Mrs Keckley_\n\n_It greves me to writ and tell you that your husband James past on from this life in Feb of an alement of the liver. He did not suffer long and he was not alone at the end. Being as there was no money for a funerl he was layd to rest in the slaves field I hope this suits you. It was a gud Christian service there were prayers and hyms._\n\n_I know you and he livd apart but I thot you should be told becauseyou are his wife and only kin. He was a gud man in his way as you know and his frens will miss him._\n\n_I am sorry I culd not tell you sooner but I did not know were you are. But your old landlady gave me this adres and I hope this letter will find you there and well._\n\n_Most Truly Yours I remain_\n\n_Ephraim Johnson_\n\nElizabeth held the letter for a long moment before folding it deliberately and returning it to the envelope. She was relieved to hear that James had not suffered long, but it pained her to think he had suffered at all.\n\nShe wondered who Ephraim Johnson was. She knew no one by that name.\n\nNo tears came to her eyes, and she wondered what that said about her, that she did not weep for her husband, a man she had once loved so dearly. The news of his death saddened her, but she did not feel grief stricken, perhaps because she found no small measure of relief in knowing that at last he was at peace. His earthly torment had ended.\n\n\"So,\" she said softly to the empty room, \"I am now a widow.\"\n\nLet James's faults be buried with him. She had no desire to think ill of him now that he was gone.\n\nThe residents of Washington City waited apprehensively to see which would arrive first, trained militia companies from the north or invaders from the south. Union troops traveling by train to the capital from Northern states would have to pass through Baltimore, about forty miles to the northeast. This should have been no concern; although Maryland was a slave state, it had remained in the Union. But rumors abounded that thousands of Marylanders with Southern sympathies were plotting to block the passage of Northern troops through the city, and since Baltimore had a history of street-mob violence, the rumors could not be ignored. Complicating matters was a quirk of Baltimore's railway system that meant Washington-bound trains would arrive at President Street Station, but would then have to be towed by teams of horses several blocks west through the city streets to Camden Station, from which they could resume their journey by rail. The system, merely inconvenient in peacetime, was potentially disastrous in war.\n\nOn the morning of April 19, the Sixth Massachusetts left Philadelphia on a train bound for Washington, arriving in Baltimore with weapons loaded. The wary men hoped for unimpeded passage through the city, but they had been warned that in the interim between stations they would likely receive insults, abuse, and possibly assault, all of which they had been ordered to ignore. Even if they were fired upon, they were not to fire back unless their officers gave the command.\n\nThe train carrying the Sixth Massachusetts arrived in Baltimore unannounced, and cars carrying seven of its companies were towed through the city unhindered. But word of the soldiers' presence spread quickly, and soon a crowd massed in the streets, shouting insults and threats. The mob tore up the train tracks and blocked the way with heavy anchors hauled over from the Pratt Street piers, forcing the last four companies of the Sixth to abandon their railcars and march through the city. Almost immediately, several thousand men and boys swarmed them, hurling bricks and paving stones, and dishes and bottles rained down upon them from upstairs windows. As the mob's rage grew, some few among them broke into a gun shop, and from somewhere, the soldiers heard pistol shots. The companies pushed onward at quick time, but when the furious mob blocked the streets ahead, the soldiers opened fire. The crowd dropped back and the soldiers managed to fight their way to the Camden Street Station, and after repairing other tracks sabotaged in the melee, the train sped off to Washington.\n\nWhen the Sixth Massachusetts finally arrived, battered and bloodied, their appearance brought more alarm than relief to the citizens who had awaited them so anxiously. Four soldiers and at least nine civilians had been killed and scores more injured on the streets of Baltimore, and as reports came in of other railway lines destroyed, bridges burned, and telegraph lines severed, panic ignited as the people of Washington City realized they had been cut off from the North. As she walked to the White House, Elizabeth was shaken to observe citizens frantically piling their belongings onto wagons and into coaches and fleeing the city.\n\nWithin the Executive Mansion, Mrs. Lincoln worked valiantly to maintain a sense of calm, of normalcy. She fulfilled her role as hostess at official events the ladies of the entrenched Washington elite disdained, she enrolled Willie and Tad in the Fourth Presbyterian Sunday School, and she cajoled her husband out of his melancholy, which deepened as the crisis worsened. \"I begin to believe there is no North,\" Elizabeth once heard the president say, and indeed, with no reinforcements arriving, no telegraph reports, no mail, she too felt the strange gloom of isolation, of being alone and surrounded by hostile, unseen enemies. It did not help that Southern newspapers managed to make their way into Washington with an ease that mocked their defenses. Time and again the _Richmond Examiner_ proclaimed that Washington would make an excellent capital for the Confederacy, noting that most of the city's residents were from Virginia or Maryland anyway and would likely welcome the Confederate army as liberators, with cheers and flowers, rejoicing to be restored to the South.\n\nWashington waited and prepared, until finally, at noon on April 25, the Seventh New York regiment arrived at the B & O Station. Relieved citizens cheered them as they marched to the White House to report to President Lincoln. Fears of an imminent Confederate invasion diminished as more troops arrived from Massachusetts, Rhode Island, and elsewhere, settling into the House chamber and the Capitol rotunda, while later arrivals quartered in the White House, the patent office, and the seminary at Georgetown, or pitched tents on the south lawn of the White House. Massachusetts masons built twenty brick ovens in the cellar of the Capitol to bake enough bread to feed the soldiers, and the noise of drums and bugles and musket-fire practice was so constant that Elizabeth could almost forget what Washington had sounded like before the soldiers came.\n\nWhile the city transformed around her, Elizabeth sewed for Mary Lincoln, usually at the White House\u2014which she preferred\u2014but sometimes in her own rooms, where the First Lady enjoyed coming to have dresses fitted. One by one, most of Mrs. Lincoln's friends and family had returned to their own homes in Illinois and elsewhere, until only her loyal and sensible cousin Mrs. Grimsley remained, although she often dropped hints that she too missed her own home. As the Washington elite continued to snub Mrs. Lincoln, she found herself increasingly lonely and alone. Elizabeth, sympathetic and kind, became her confidante, and she soon discovered how unsettled Mrs. Lincoln felt in her new surroundings and in the elevated role she had so desired. Never before had she lived among strangers who were thoroughly unimpressed with her family name, which had always carried great influence back in Lexington, thanks to the prominent businessmen and politicians among her relations. Her husband surrounded himself with male colleagues who regarded her notes about policies and appointments as annoying and meddlesome, so that she had to struggle against his aides even for control over the very White House functions for which she played hostess. Excluded from her husband's inner circle, missing her departed sisters and cousins, disdained by the popular ladies of Washington, Mrs. Lincoln often told Elizabeth\u2014sometimes sadly, sometimes in defiance\u2014that Elizabeth was her only true friend within a hundred miles.\n\nElizabeth\u2014who had been readily received into the elite of colored Washington society by virtue of her natural grace and dignity, her status as a White House intimate, and, ironically, the impressive bloodline she had inherited from Colonel Armistead Burwell, her father and former master\u2014gently tried to steer Mrs. Lincoln down paths that might lead to her greater acceptance. While sewing for other clients, she had detected hopeful signs that the hearts of some of the ladies were softening toward the First Lady. \"Nowadays the womenkind of Washington are united only in giving the cold shoulder to Mrs. Lincoln,\" Elizabeth Blair Lee, who had remained Varina Davis's friend despite their political differences, remarked to Elizabeth during a fitting. \"We Republicans at least ought to rally around her.\"\n\nBut not even the friendship of the popular, imperious, and generous Mrs. Samuel Phillips Lee could redeem Mrs. Lincoln in the eyes of those who disparaged her for behaving as if the nation were not at war. When she discovered that Congress allotted twenty thousand dollars to each administration to refurbish the White House, Mrs. Lincoln promptly set about spending the allowance with unrestrained delight. Elizabeth would be among the first to agree that the White House had been sorely neglected; even upon her first visit, when the importance of her interview preoccupied her thoughts, she had not failed to notice the shabby state of the mansion's threadbare rugs, broken furniture, torn wallpaper, and ruined draperies, from which souvenir collectors had snipped pieces until they hung in tatters. But as necessary as the purchases were, it did not look well for the First Lady to be spending so much on carpets and china when some poor, brave soldiers went without tents and blankets.\n\nMrs. Lincoln's loyalty to the Union was also questioned\u2014entirely without justification, as Elizabeth well knew. \"Why should I sympathize with the rebels?\" Mrs. Lincoln had once declared, angrily tossing aside a _Harper's_ magazine that insinuated she might. \"Are they not against me? They would hang my husband tomorrow if it was in their power, and perhaps gibbet me with him. How then can I sympathize with people at war with me and mine?\"\n\nWhile it was true that Mrs. Lincoln's brother, three half brothers, and three of her brothers-in-law were serving in the Confederate army, Mrs. Lincoln herself was a staunch Unionist, and despite growing up in a slaveholding family, she had stronger abolitionist leanings than her husband. This did not prevent Northern newspapers from printing entirely false tales that Mrs. Lincoln's youngest half sister, Emilie, whose husband was a general in the Confederate army, had used a presidential pass to smuggle supplies across Union lines to the rebels. Only Southerners correctly surmised where Mrs. Lincoln's loyalties truly resided, and they condemned her for it. They considered her a traitor whose repudiation of her Southern heritage disgraced her family's good name. Union and Confederate alike, each side believed her loyal to the other, and thus neither would claim her.\n\nAlthough the immediate threat of invasion seemed to have passed for the moment, the Confederate army never felt very far away. Virginian militia companies drilled in Alexandria just across the river, and for every Union picket guarding the Washington side of bridges leading into the capital, there were Virginia militiamen posted on the other end. A Confederate flag flew boldly above an Alexandria hotel, easily visible from Washington City to anyone with a good vantage point and sharp eyes or a telescope. Elizabeth sometimes observed President Lincoln standing at a window in the White House, studying the flag in silence. Sometimes he sat back in his armchair, rested his feet on the windowsill, and watched ships on the Potomac through a telescope, occasionally letting his gaze drift upward to the rebel flag whipping in the wind. If it seemed an especial taunt, he never said so in Elizabeth's presence.\n\nThe war was very close, and it would come closer still.\n\nTelegraph and mail service resumed, and in early May, Elizabeth received a letter that had been delayed in the crisis.\n\n_April 24, 1861_\n\n_Dearest Mother,_\n\n_I hope this letter finds you safe and well. The news from Washington has been so alarming of late that I can only pray that the papers are following their usual custom of exaggeration._\n\n_I write in haste, and yet I write reluctantly, for I fear that you will not welcome what I must tell you. Mother, I am a soldier. You may well ask how this can be, since colored men are not welcome in the ranks of the Union Army. This I discovered firsthand when I and some of my fellows went to enlist at Columbus and were rejected, withthe jeers and insults of the white recruits burning in our ears. My friends had no choice but to return to school, but I was too restless to remain, so instead I went home to St. Louis. There, alone, without my duskier-skinned friends, no one had cause to think me other than a white man, and so I signed my name._\n\n_Mother, I know you will not be pleased that I have left school without finishing the term, and I am sorry for grieving you, but I am confident I will be allowed to resume my studies when my service is done. It is only a three-month enlistment, but everyone says it will be over well before then, and I could not miss my chance to strike a blow for the Union. I am ever mindful that if not for you I would still be a slave, and thus I have a special obligation to help deliver others from bondage into freedom. I am willing to give this noble cause the three months they ask, and my life if necessary._\n\n_Please write to me often, and pray for me always. I am and will always be_\n\n_Your Devoted Son,_\n\n_George W. D. Kirkland_\n\n_Private_\n\n_First Missouri Volunteers, Company D_\n\nElizabeth was in tears well before she reached his closing lines. He thought she would be upset because he left college before the end of the term? _That_ was what he believed would grieve her the most\u2014not his deception, not his youthful, impetuous enlistment, not the possibility of his death on the battlefield?\n\nBlinded by tears, dizzy with anguish, she groped for a chair and sank into it, crumpling the letter in her hand. She pressed her lips together to still her weeping, before Virginia or Walker or one of the other tenants heard her distress and came running to see what was wrong. She trembled in silence until she calmed herself, desperately chasing from her mind's eye visions of her precious only child lying wounded or dying in some distant Southern meadow.\n\nBut even in the midst of her fear, she was proud of her son\u2014deeply, profoundly proud. He was right. God had blessed them with the means to free themselves from the misery of enslavement, and therefore they both were obliged to help the many others of their race still held in bondage\u2014George in his way, and she in hers.\n\nShe prayed the Lord would reward his noble willingness to sacrifice his life by sparing it.\n\n# Chapter Four\n\n## MAY\u2013AUGUST 1861\n\nDuring the winter of secession, Elizabeth's dressmaking business had declined as many of her best patrons departed for the South, but as spring flourished, so did her fortunes. By that time Mrs. Lincoln had proudly displayed Elizabeth's handiwork at many a White House gathering, and suddenly she found herself the best-known and most coveted mantua maker among the loyal Union women of Washington. With more work than she could handle on her own, she rented a workroom across the street from her boardinghouse and hired assistants. Her patrons understood that Elizabeth must always give the First Lady preference, so whenever Mrs. Lincoln traveled, other ladies rushed to place their orders.\n\nMrs. Lincoln could scarcely stroll down Pennsylvania Avenue without reporters setting telegraph lines abuzz with the news, so when she and her cousin Mrs. Grimsley traveled to New York on a shopping expedition in early May, newspapermen hounded their every step. Stories of their attending the theater, inspecting carriages at a manufacturer, dining, enjoying soirees, and visiting local luminaries filled newspaper columns and invited spiteful commentary. When Elizabeth read of Mrs. Lincoln's expenditures upon carpets, china, mantel ornaments, and other furnishings for the White House, she winced in sympathy, wishing it were possible for her patron to be more discreet. She also worried at the amount Mrs. Lincoln seemed to be spending, not only because the papers depicted her as wasteful, but also because Elizabeth could not imagine how the congressional allowance could stretch far enough to cover it all.\n\nBut it was impossible not to be caught up in her patron's delight when she returned to Washington, flush with excitement, eagerly anticipating the delivery of her goods. \"I am determined to transform the White House into a showplace worthy of our nation,\" she told Elizabeth as she led her and Mrs. Grimsley room to room, describing her purchases and where she intended to arrange them. Repairs and restoration would have to be completed first, of course, but Mrs. Lincoln planned to escape the noise and dust, as well as the heat and disease of summertime Washington, by taking Tad and Willie north as frequently as she could.\n\n\"I'd like to take Mr. Lincoln as well,\" she confided, \"if only I could pry him away from his cabinet.\"\n\nNo one in Washington could forget the Confederate threat looming ever nearer. For weeks the Confederate flag had waved and snapped in the breeze above Alexandria, taunting them with the threat of invasion, which set their nerves on edge and sent prices for flour, coffee, and other groceries soaring. One morning in late May, Elizabeth woke to the tolling of firehouse bells and went down to breakfast to learn from the Lewises that shortly after dawn, ten Union regiments had quietly crossed the Potomac and had captured Alexandria\u2014but the leader of the New York Zouaves, Colonel Elmer Ellsworth, had been shot and killed, the only casualty of the mission.\n\nElizabeth's heart sank. \"Are you certain?\" She had seen Colonel Ellsworth often in the White House and knew him to be a particular favorite of the president. She knew that sometime before the election, Mr. Lincoln had met the young man in Chicago and had urged him to move to Springfield to study the law. He had been part of the honor guard that had accompanied the president-elect on the train to Washington, and after secession, Mr. Lincoln had used his influence to obtain him a good position in the military. He was only a few years older than Mr. and Mrs. Lincoln's eldest son, Robert.\n\n\"There's no mistake,\" Walker told her, shaking his head grimly. \"His is the first blood shed on rebel soil.\"\n\n\"The first drops of many,\" said Virginia quietly.\n\nElizabeth's thoughts flew to George\u2014so near in age to the colonel, so eager to join in the fight\u2014and then to the Lincolns. Mrs. Lincoln had liked Colonel Ellsworth nearly as much as her husband did. She would be heartbroken.\n\nAlthough she was not expected that day, Elizabeth quickly made her way through the muddy streets to the White House, determined to offer Mrs. Lincoln whatever comfort she could. At the door, a somber Mr. McManus told her that the household had plunged into mourning. \"Mr. Lincoln was in his library with visitors when word came of the colonel's death,\" the elderly doorman confided. \"Peter Brown says he was so overcome by emotion that he could not speak.\"\n\n\"I'd best see to the First Lady,\" Elizabeth said, hurrying off.\n\nShe found the First Lady in her boudoir, clad in a cheerful floral day dress and sitting at her dressing table, staring straight ahead at nothing. She looked up when Elizabeth entered, a faint light of surprise cutting through her grief. \"Ah, Elizabeth,\" she said. \"How did you know I needed you?\"\n\n\"I heard the bells tolling for the colonel and thought you might.\" Elizabeth went straight to the wardrobe and began sorting through the First Lady's dresses for a more suitable garment. Suddenly she was seized by the cold realization that she would likely be asked to sew many black dresses in the months ahead as her patrons lost husbands, sons, and brothers, and donned the somber colors of mourning that custom demanded. Shaking off the thought, she spied a black silk dress Mrs. Lincoln must have brought with her from Illinois and held it up. \"What do you think of this?\"\n\n\"It will be fine, I'm sure.\" Mrs. Lincoln barely glanced at the dress. \"We're going out this afternoon, Mr. Lincoln and I, to the navy yard to view the body and pay our respects. After that I believe\u2014I believe the colonel will be removed here, to lie in state in the East Room.\"\n\nElizabeth nodded and began unbuttoning Mrs. Lincoln's bodice.\n\n\"That dreadful flag provoked him so,\" said Mrs. Lincoln, her voice distant. \"He promised my husband he would tear it down. That's exactly what he did, and he was killed for it.\"\n\n\"Killed for a flag?\" asked Elizabeth, without thinking. It seemed like such a waste.\n\n\"It should never have happened.\" Mrs. Lincoln knotted her handkerchief in her lap. \"Before a single shot was fired, a lieutenant was sent into Alexandria under a flag of truce, to warn the Confederate commander that they faced an overwhelming force, and he had until nine o'clock to evacuate or surrender.\"\n\nThe commander must have chosen one or the other rather than fight; surely Elizabeth would have heard some distant sounds of battle in Washington if the rebels had resisted. \"Which did they choose?\"\n\n\"They chose to retreat. The lieutenant reported to Colonel Ellsworth that the rebels said they would not resist because the town was full of women and children. Most of the rebel troops boarded a train and left Alexandria well before the deadline, but a few stayed behind. I don't know why. They were captured and imprisoned in a slaver's pen.\" Mrs. Lincoln sighed as Elizabeth helped her out of her day dress. \"I should tell you, Elizabeth, this is only what I've gathered here and there, not an official report of any kind.\" A trace of anger made her voice tremble. \"My husband confides in me less and less.\"\n\n\"Tell me anyway,\" Elizabeth prompted gently as she straightened Mrs. Lincoln's chemise. She wanted to know what had become of the young officer, and the effort of telling the story seemed to keep Mrs. Lincoln calm.\n\n\"Well, from what I've heard, Colonel Ellsworth set off with some of his men to capture the telegraph office, but then he happened to pass the hotel where that flag was flying as bold as brass. He knew how it vexed my husband, what an eyesore it had become.\"\n\n\"For everyone in Washington,\" Elizabeth agreed, assisting Mrs. Lincoln into the black silk dress. Mrs. Lincoln moved as Elizabeth willed, as compliant as a doll.\n\n\"Perhaps he thought the president was watching the hotel that very moment, and perhaps he wanted to signal that the town had been captured. We'll never know. But in any case, Colonel Ellsworth marched into the hotel and up to the roof, and he took hold of that flag and tore it down. He returned downstairs to his men\u2014\" Mrs. Lincoln pressed her handkerchief to her lips, steeled herself, and continued. \"As he carried the captured banner downstairs, the owner of the hotel stepped out of nowhere and shot him in the chest, from just a few feet away.\"\n\nElizabeth's hands froze in the midst of buttoning Mrs. Lincoln's bodice. \"Lord have mercy.\"\n\n\"One of Colonel Ellsworth's soldiers promptly avenged him\u2014he killed the man with a musket round to the head.\" Mrs. Lincoln looked over her shoulder at Elizabeth, her expression full of pain. \"But he was too late, you know. Too late to save him.\"\n\nElizabeth pressed her lips together, shaking her head. \"I am truly very sorry, Mrs. Lincoln. I know you and the president were very fond of him.\"\n\n\"We were indeed.\" Mrs. Lincoln fell silent as Elizabeth finished dressing her. \"I believe he was like another son to Mr. Lincoln. And to both of us, he was a bit of home, do you understand? With all the promise of youth, all the vigor and courage\u2014\" A sharp intake of breath. \"I shall have to write to his mother. Mr. Lincoln will, of course, but I should too. Although I can't imagine what I will say.\" Her voice broke, and she sank back down into her chair.\n\n\"Mrs. Lincoln,\" said Elizabeth steadily, \"is there anything I can do for you, anything at all, for you or for Mr. Lincoln?\"\n\n\"You've done what I needed most, Elizabeth, as you always do.\" Mrs. Lincoln offered her a sad, tight smile. \"Sometimes I don't know how I'd endure living in this\u2014\" She waved a hand as if to indicate all of Washington City. \"Well. You know how it is, and how I rely upon you.\"\n\nA small glow of pride warmed Elizabeth's heart. She did indeed know. \"It is my great pleasure, Mrs. Lincoln.\"\n\nA few days later, when Mrs. Lincoln next summoned her to the White House, Elizabeth learned that thousands of mourners had come to pay their respects to the young colonel as he lay in state in the East Room. Among them was Robert Lincoln, who had traveled all the way from Harvard to mourn and to comfort his parents. They seemed shaken, not only by their personal loss, but also by the sense that this early, violent, and sudden death heralded the sacrifice of many young, valiant men who would perish in the months ahead.\n\nAfterward, the flag that Colonel Ellsworth had given his life for was presented to the First Lady. Mrs. Lincoln was honored and deeply moved, and kept it always.\n\nThe death of this soldier she had barely known made Elizabeth ever more fearful for the one she loved most of all.\n\nGeorge wrote often, at least twice a week, entertaining her with tales of camp life and his humorous early mishaps as a novice soldier. Elizabeth had seen enough of soldiering in Washington to realize that he gave her only the most optimistic, uncomplaining version of his new life, and that it was certainly more difficult than he let on. As far as he could tell, he was the only man of color in the regiment, which was comprised nearly entirely of German immigrants, with some Irish and native-born Americans mixed among them. \"I am whiter than most of my comrades,\" he said in an early letter, \"so you need not worry that I will be found out.\" Many civilian Missourians disliked the Germans, but George admired their industry and stoicism, and he shared their abolitionist views.\n\nThe slave state of Missouri was a curious case, having voted in March to remain in the Union but not to supply men or weapons to either side. That decision had not prevented Unionists and secessionists alike from forming militia units on their own and jostling for control over the various federal armories located throughout the state. Within weeks of George's enlistment, the First Missouri under the command of Captain Nathaniel Lyon marched on Camp Jackson, where secessionist Missouri Volunteer Militia troops were holding captured Union heavy artillery and munitions, which had been given to them by the Confederates so they could attack the St. Louis Arsenal. Nearly seven hundred militiamen were forced to surrender, but when they refused to take an oath of allegiance to the United States, Captain Lyon decided to arrest them and march them through the streets of St. Louis to the arsenal, where he intended to parole the humiliated men and order them to disperse.\n\nGeorge's account of how Captain Lyon's plans went terribly wrong, though rendered in a son's carefully selective prose, reminded Elizabeth alarmingly of the altercation in Baltimore. As the First Missouri marched the captured secessionists through the city, indignant residents shouted insults and hurled rocks and paving stones upon the soldiers. \"The refrain 'Damn the Dutch!' was shouted at our German comrades so often and with such anger that I thought, at last there is someone hated more than the Negro,\" George wrote. Next, some accounts claimed, a drunken civilian stumbled into the path of the marching troops, fired a pistol at them, and fatally wounded a captain in the Third Missouri. The soldiers responded by opening fire, first above the heads of the civilians and then into the crowd. Twenty-eight people were killed, including women and children, and some fifty more were injured. Several days of rioting broke out, during which anti-German hatred ran rampant throughout St. Louis, civilians shot at soldiers from the windows of their businesses and homes, and troops once again fired upon crowds in the streets. Although George described the events lightly, with a young man's bravado, Elizabeth could tell he was shaken.\n\nIn the middle of June, George told her in a later letter, the First Missouri and other federal regiments marched on Jefferson City to find that the secessionist governor had abandoned the state capital. The Union forces easily captured the city and pursued the governor and his Missouri State Guard to the town of Boonville, about fifty miles to the northwest. \"It was a minor skirmish, short and sweet,\" George wrote, \"but we whipped the State Guard, drove the secesh away, and secured the Missouri River Valley for the Union. Not a bad day's work if I do say so myself.\"\n\nThey were encamped at the time he wrote his letter, and he did not know when they would march again or to where. He urged her not to worry, and asked her to pray for him, and promised he would write again soon. \"I am ever mindful of my special obligation,\" he wrote in closing. \"Nothing I suffer on the battlefield is of any consequence if it brings freedom to all our race. I only wish my friends from college would be allowed to take up arms as I have done. Sometimes when I hear my comrades talk about the cowardice and inferiority of the Negro I am tempted to do more than speak up in 'their' defense. I want to leap to my feet and shout, 'Haven't I fought as bravely as you? Haven't I marched as far and endured as much? If my blood is shed, will it not be as red as yours? Well, I too am a Negro, and I defy you to explain how I am not as good a soldier as you!' But of course I cannot say it\u2014not yet. When the war is over the truth will be known, and then let no man call me his inferior.\"\n\nElizabeth was too proud of him to burden him with her worries and misgivings. George had enlisted to prove himself, to preserve the Union, to deliver others of their race from bondage. She, who had lived nearly forty years a slave, knew all too well that his purpose was noble and necessary.\n\nTo think that when she first discovered she was carrying him, more than twenty-four years ago, she had been despondent. She had not wanted to have relations with Alexander Kirkland, and she had dreaded bearing a child into slavery. But after George was born, her greatest fear had been that he would be taken away from her.\n\nBy midsummer, the newspapers were filled with stories of battles here, skirmishes there, advances and retreats upon towns Elizabeth had never heard of. Prisoners were taken on both sides, but Mr. Lincoln and his advisers did not know what to do with the men in their custody. As rebels, they had committed treason and should by law have been hanged, but that would be abhorrent on such a large scale\u2014and would almost certainly result in retaliation against captured Union soldiers. An exchange of prisoners, a well-established practice between nations at war, would be construed as recognizing the Confederacy as a legitimate, sovereign government, which the president was disinclined to do. So captured rebels were held indefinitely, sometimes moved from slave pens to navy brigs to civilian Washington prisons, while Mr. Lincoln and his advisers debated what to do with them\u2014and with rebel property the Union Army had seized on the other side of the Potomac. Robert E. Lee's captured Arlington plantation was perhaps the most valuable estate in Virginia, and some of Mr. Lincoln's advisers urged him to sell it to help fund the war and to act as a warning to other planters. Elizabeth understood that the gentleman who had once smiled at her as he pressed one hundred dollars into her hand and told her to spare no expense on his wife's gown was now a rebel, not only a rebel but their general, but she could not help feeling sorry for his wife, her former patron. The plantation had come to the Lees from her side of the family, descendants of Martha Washington. It pained Elizabeth to think that Mrs. Lee might never see her ancestral home again.\n\nElizabeth had not forgotten other former patrons who had fled Washington for their seceded states. In June she completed the fine needlework Varina Davis had left in her care, but she was confounded by the puzzle of how to get it to her. The postal service had stopped delivery to the Confederacy on the first of June, and although Elizabeth heard tales of smugglers who carried goods freely between North and South, she had no idea how to make such arrangements nor would she feel comfortable trusting anyone who engaged in the practice. In the end she entrusted the embroidery to Mrs. Davis's friend Matilda Emory, the wife of Major William Emory. A Marylander serving alternately in Indian Territory and the capital, he had quit the Union Army in May against his wife's wishes. Not content merely to protest, Mrs. Emory had had her husband reinstated by retrieving his letter of resignation herself. Despite Mrs. Emory's strong Union sentiments, she had remained close to Mrs. Davis, and, without explaining how she would accomplish the task, she assured Elizabeth that she would get the embroidery safely to her friend.\n\nSince leaving Washington City barely six months before, Mr. Davis had been elected president of the Confederacy and Mrs. Davis had become their First Lady. Sometimes Elizabeth thought back to her months working in the Davis household and marveled at the changes time had wrought. Mrs. Davis and her husband had moved from Alabama to Virginia, or so the papers claimed, and they now resided in another White House in Richmond, the new Confederate capital. Elizabeth wondered if her former patron still entertained hopes that the Confederates would capture Washington City and that she would take her place in the first grand residence to bear that title. Observing the new Union defenses all around the district, Elizabeth suspected that conquering the city would not be as easy as Mrs. Davis had once believed.\n\nThe First Lady presently occupying the White House would certainly have something to say about Mrs. Davis's confident declarations, if Elizabeth were foolhardy enough to stir up her temper by telling her about them. Mrs. Lincoln was, as she had ever been, a patriotic Unionist, and although the president's cabinet repeatedly thwarted her efforts to influence her husband on policy matters, she used her position to support the Union cause in other ways. She hosted dinners for dignitaries and, hoping to raise people's spirits, she arranged for the Marine Band to perform concerts every Wednesday and Saturday when the White House grounds were open to the public. She toured regimental encampments and visited soldiers in the hospital, often distributing delicacies from the White House kitchen and gardens with her own hands. She obtained weapons from the War Department and had them sent to a Union colonel in her home state of Kentucky along with a sincere, heartfelt letter professing her admiration for him as well as her loyalty and love for the nation. She reviewed troops with her husband and was charmed when an army colonel broke a bottle of champagne over a carriage to christen the field where his men were bivouacked as \"Camp Mary\" in her honor.\n\nElizabeth wished Mrs. Lincoln's sincere and helpful efforts received more than an occasional complimentary aside in the press. Unfortunately, reporters and gossips alike were far more fascinated by stories of her secessionist relatives and her lavish expenditures on fringed satin bed curtains and a purple-and-gold dinner service emblazoned with the seal of the United States on each piece\u2014as well as a second set adorned with her own initials. The press hounded her so that she could not escape them, not even on her trips away from Washington to escape the heat, the disease, and the swarms of flies and mosquitoes that plagued the city throughout those oppressive summer days.\n\nMrs. Lincoln was at home at the end of July when word began to spread that troops were gathering near Manassas Junction, Virginia. Thousands of citizens eager for diversion packed picnic hampers and hired carriages to take them out to watch the spectacle. Politicians determined to witness history, reporters chasing the story, curious workmen, ladies with parasols thrilled by the prospect of danger and heroism\u2014all wanted to watch Brigadier General Irvin McDowell and his mighty Army of Northeastern Virginia soundly defeat the rebels before marching on to take Richmond and bring a quick and decisive end to the conflict.\n\nSeveral of Elizabeth's young assistant seamstresses had been invited to accompany a few young men in their wagon. \"Come with us, Mrs. Keckley,\" urged Emma Stevens, a former slave from Maryland who rented a small attic room in the Lewises' boardinghouse. As a very young child, she and her mother had been granted their freedom when their old mistress passed away, but the woman's heirs had contested the will. Emma and her mother had been kept in slavery for ten long years more while the lawsuit Emma's mother brought against the heirs dragged out in court. Upon miraculously winning their case, and their freedom, Emma and her mother had adopted the last name of the lawyer who had courageously represented them in a hostile courtroom.\n\nElizabeth smiled and shook her head. \"Thank you, no.\"\n\n\"Oh, do join us,\" Emma said persistently. \"There will be plenty of room in the wagon. And it's a Sunday. You shouldn't work on a Sunday.\"\n\n\"I must,\" Elizabeth said. \"Mrs. Lincoln needs the white day dress finished before her trip to the seaside. You young people go and have a good time, but do take care.\"\n\nShe breathed a sigh of relief when Emma gave her a small disappointed pout and set off on her own, immediately brightening at the thought of her friends and the charming escorts awaiting them. Elizabeth didn't want to admit to the younger woman, her favorite of her assistants, that she could not bear to watch the scenes of battle and to be painfully, vividly reminded of what George might be facing at that very moment somewhere in Missouri.\n\nShe sewed all day in her rooms with the windows and doors open to stir the stifling air, breaking only for lunch and for a walk with Virginia along the river, where the stench diminished the pleasure of the cooling breeze. They heard the rumble of gunfire to the west, and they wondered how the battle was faring. \"We may be sorry later that we missed all the excitement,\" Virginia remarked.\n\n\"I'm certain I won't,\" Elizabeth replied.\n\nAs they walked back to Twelfth Street, they passed a telegraph office, where men and boys lingered, waiting for news of the battle. The Confederates were on the run, from the sound of it, and they overheard two men say that the Union Army was expected to be in Richmond within a week. Gladdened by the remarkably good report, they returned home to tell Walker what they had learned. Back in her rooms again, Elizabeth took up Mrs. Lincoln's dress, threaded a needle, and allowed herself to hope that the war could be over by the autumn, in time for George to muster out of the army and resume his classes at the start of the term.\n\nHer hopes were dashed before morning.\n\nSometime after nightfall, noises outside roused her from sleep\u2014carriage wheels, horses' hooves, voices raised in alarm. She hurried to the windows, but they looked out over the garden and she could see nothing but the backs of other houses and the glow of a few lighted lamps visible through the windows. As the commotion continued, more lights joined them. Elizabeth dressed quickly and went outside to the front sidewalk, where she found Virginia, Walker, and a few of their neighbors watching in stunned amazement as carriages and wagons and men on horseback came streaming through the city as fast as their tired horses could carry them, their expressions stricken and terrified.\n\n\"What happened?\" Elizabeth asked Virginia, who only shook her head, pressed her lips together, and clutched her arm. They held on to one another as the ominous parade went by, realization dawning that these were the spectators who had so enthusiastically set out for Centreville to observe the battlefield earlier that day.\n\nFinally Walker waved down a colored man on a dray, who slowed his team but would not stop. \"It was a rout,\" he called down from the driver's seat. \"Hundreds dead. Whole companies lost. McDowell's army's retreating back to the city if they haven't been caught or killed yet, and the rebels are right behind them.\"\n\nVirginia gasped, and Elizabeth felt a chill. Emma and two of her other young prot\u00e9g\u00e9es were in the midst of all that chaos. \"Were any civilians hurt?\" she called out to the driver, but he had already hastened away.\n\nAll through the night the civilians made their way back to the capital, shaken and exhausted, with wild tales of their narrow escape from certain death. They had been at some distance from the fighting, closer to the river, so they were the first to return. Elizabeth had almost given up hope when Emma appeared, escorted by a young man who solicitously entrusted her to the care of Walker, Elizabeth, and Virginia. \"It was a nightmare,\" Emma told them, wide-eyed and trembling. \"I've never seen anything like it. If we hadn't been near the back of the crowd, close to the wagon\u2014\" She shook her head and fell silent as Virginia led her into the house.\n\nNot until dawn did the first soldiers reach the city, their expressions stunned and haggard, their uniforms torn and disheveled, their ranks diminished. Gone were the waving of regimental colors and the stirring martial music of fife and drum. Famished and exhausted, some of the troops dropped their kits in doorways, on sidewalks, on empty lots, and lay down to sleep where they were. Some observers cast off their shock and hurried back into their homes, quickly returning with bread, cheese, apples, and other food to distribute to the passing soldiers. Elizabeth and Virginia joined in, offering water from pitchers and pails as quickly as one man could pass the dipper to the next.\n\nThe wounded came too, brought into the city by the wagonload. There were not enough beds for all the soldiers who needed them, nor enough bandages, nurses, food\u2014nor hospitals for that matter. A few gray-clad rebel prisoners were marched in on foot and under guard. Some citizens shouted curses and threw stones at the captives, but a few bold Confederate sympathizers among them called out encouragement as they were led off to the Old Capitol on First and A streets Northeast, which had been turned into a federal military prison.\n\nFor hours the people of Washington waited in terror for the Confederate army to press their advantage and take the city, but the invasion never came.\n\nIn the days that followed, the people of the North wanted revenge and President Lincoln wanted answers. Even as volunteers filled recruiting offices, the president closeted himself with his advisers, determined to ascertain how the battle had gone so terribly wrong and how they could ensure it would never happen again. Already the press had taken to calling the Union's disorderly retreat from the battlefield \"the Great Skedaddle,\" shaming the Union soldiers and heartening their enemies.\n\n\"They were green troops,\" Mrs. Lincoln told Elizabeth, staunchly defending the soldiers\u2014and by extension her husband, who had sent them into battle. \"They aren't green anymore. They will never again suffer such an embarrassing rout.\"\n\nElizabeth hoped she was right, and her thoughts flew to her son. In his last letter, George had written that the First Missouri was preparing to march on Springfield, Missouri, but he did not know when they would set out. She wished she knew whether they had departed yet, whether they had arrived, whether the battle had been fought, and who had emerged victorious. George was still green himself, she thought, despite his regiment's involvement in the conflict in the streets of St. Louis and the minor skirmish at Boonville. He had yet to face the kind of warfare McDowell's men had faced at the creek called Bull Run, and she felt sick at heart when she imagined him facing rebel guns and cannon.\n\nThe days between George's letters elapsed in a ceaseless misery of waiting and worry. The arrival of each letter brought only momentary joy and relief, because although she savored each word, she knew that the letters offered no assurances of his safety any longer, only proof that he had been alive and unhurt at the time he had written them.\n\nBut that was something, at least. She kept his letters safely tucked away in a rosewood box her former mistress had given her as a farewell gift when she departed St. Louis. Every Sunday evening she read the letters in the order he had written them, a ritual that grew lengthier every week. They were talismans that closed the distance between them and, she hoped, would bring him back to her someday.\n\nThe first days of August were oppressively hot and humid, with no relief on the horizon. After hosting a state dinner for Prince Napol\u00e9on III, Mrs. Lincoln took Willie and Tad and her cousin Mrs. Grimsley on a vacation to Long Branch, Manhattan, and upstate New York, but her absence left Elizabeth busier than ever as other ladies rushed in with their dressmaking demands. She sewed for Mary Jane Welles, the wife of the secretary of the navy; Margaret Cameron, the wife of the secretary of war; and Adele Douglas, the young widow of the late Senator Stephen A. Douglas from Illinois. Recently widowed, the lovely Mrs. Douglas dressed in deep mourning but with excellent taste, so that even in her sorrow other ladies were jealous of her beauty and grace.\n\nIn the middle of the month, with summer storms worsening the humidity and bringing no relief from the heat, Elizabeth received a letter sent from Missouri but addressed in an unfamiliar hand. It could not be from George, she thought, quickly opening the envelope, or from her former mistress Anne Garland, whose writing she knew well. Heart pounding with sudden apprehension, she unfolded the letter and read it slowly, word by word, afraid to reach the end.\n\n_Near Springfield, Missouri_\n\n_August 11, 1861_\n\n_To Mrs. Elizabeth Keckley, Washington, DC:_\n\n_Dear Madam,_\n\n_It is with great pain that I write to inform you of the death of your son, George Kirkland. By his good conduct and bravery while with me, he had won the respect of myself and his fellow soldiers, and should he have lived I would have promoted him soon. He was shot through the body at the Battle of Wilson's Creek, where his last words were of our noble struggle and how we must see it through. He is buried here beside his brothers in arms who also fell that day. His effects I shall send home at the earliest opportunity._\n\n_Yours truly,_\n\n_Charles W. Anderson_\n\n_Captain, Commanding Company D_\n\n_First Missouri Volunteers_\n\nElizabeth felt the room shift and turn around her before all went dark.\n\n# Chapter Five\n\n## AUGUST 1861\u2013MARCH 1862\n\nWhen George's personal effects arrived a week later, Elizabeth kept them on her bureau for two days before mustering up the courage to examine them. They included all of her letters, neatly bound with a ribbon; a book of psalms; a sewing kit that could be rolled up and tied, commonly called a housewife; a rosewood ink bottle; a tin cup; a small tobacco box, also of tin; and a pair of bone dice. She touched the items one by one, imagining them in his hands. She was glad that he had found solace and inspiration in Holy Scripture while at war, but she did not like to think that he had taken up smoking and gambling.\n\nIt no longer mattered.\n\nIn the midst of her bereavement, Elizabeth found comfort in a heartfelt, compassionate letter from Mrs. Lincoln and from the gentle kindness of Virginia, Walker, and Emma, whose eyes brimmed with tears when she told Elizabeth she wished she would have known her fine, heroic soldier. For a brief moment, Elizabeth allowed herself to imagine George and Emma meeting, falling in love, marrying, raising children\u2014but then she banished such thoughts forever.\n\nHe had died a free man. That much, at least, she had done for her son.\n\nIn autumn Mrs. Lincoln returned to Washington, and duty again called Elizabeth to the White House. The battlefront had moved off from the outskirts of Washington, but every day brought new reports of intense fighting and grisly scenes of death and destruction. War raged in several states, and the Union Army endured one demoralizing defeat after another. In Washington, a stretch of rainy weather flooded the Potomac, washing corpses of Union soldiers killed weeks before at the Battle of Ball's Bluff downstream until they were fished out between the bridges at Fourteenth Street. Near the river and canals, sewers overflowed into the streets, spreading a miasma of stench and sickness for blocks. Soldiers camped in the surrounding hills, bunked close together in soggy tents where smallpox and typhoid fever struck them down in great numbers.\n\nUnion soldiers fought Confederate, doctors and nurses struggled against disease, police and volunteers alike battled the fires that broke out with alarming regularity, and Mr. Lincoln struggled with his cabinet and with General McClellan, who seemed incapable of pressing his advantage on the battlefield or even recognizing when he had it. All the while, Mrs. Lincoln engaged in a very different sort of battle. Her renovations to the White House had garnered her much criticism in the press and had run well over budget, and to Elizabeth's dismay, the gardener, James Watt, had taught her how to pad bills and hide expenses in his account. Ignoring the commissioner of public buildings' warnings that she had no money left to spend, Mrs. Lincoln continued running up debts until it became impossible to conceal them from her husband anymore. They argued furiously on her forty-third birthday, and afterward Mrs. Lincoln begged the commissioner to intercede with the president on her behalf. Reluctantly he did so, and although Elizabeth did not witness Mr. Lincoln's explosive reply\u2014\"I swear I will never approve the bills for flub dubs for this damned old house!\"\u2014everyone heard of it later.\n\nMrs. Lincoln's reputation suffered further damage thanks to the questionable characters who populated her evening salons\u2014a coterie of favorites, almost exclusively men, who flattered her vanity and may have betrayed her confidences. It was said that one of her regular callers provided a copy of the president's annual message to Congress to the _New York Herald,_ which published the speech before he could deliver it. After that, Mr. Lincoln warned his wife against idle talk, banished the man blamed for the leak from the White House, fired the gardener, and to Mary's sorrow, ceased confiding in her altogether where the work of his government was concerned.\n\nElizabeth soon learned firsthand what Mrs. Lincoln's troubles had already demonstrated: Unscrupulous people were eager and determined to wriggle their way into the White House for their own wicked purposes. As soon as Elizabeth became well-known throughout Washington as Mrs. Lincoln's modiste, strangers crowded around her affecting friendship, hoping that she could use her influence to obtain them a job, secure them a favor, or provide gossip they could use to their own benefit.\n\nOne day, a woman Elizabeth had never met called on her at her rooms, placed an order for a dress, and insisted upon paying her partly in advance. For quite some time she came daily to see Elizabeth for fittings and adjustments, never failing to be gracious and kind. On the day the dress was complete and she came to pick it up, she hesitated before saying, \"Mrs. Keckley, you know Mrs. Lincoln?\"\n\n\"Yes,\" said Elizabeth, folding the dress carefully for her new patron to take home.\n\n\"You are her modiste, are you not?\"\n\n\"Yes.\"\n\n\"You know her very well, do you not?\"\n\nElizabeth kept her expression smooth and pleasant, wondering where the questions would lead. \"I am with her every day or two.\"\n\n\"Don't you think you would have some influence with her?\"\n\nSo at last, there it was. \"I cannot say. Mrs. Lincoln, I presume, would listen to anything I should suggest, but whether she would be influenced by a suggestion of mine is another question.\"\n\n\"I am sure that you could influence her.\" The woman offered her an ingratiating smile. \"Now, Mrs. Keckley, I have a proposition to make. I have a great desire to become an inmate of the White House. I have heard so much of Mr. Lincoln's goodness that I should like to be near him, and if I can enter the White House no other way, I am willing to go as a menial.\" At this Elizabeth tried to speak, but the woman hurried on as if she hadn't noticed. \"My dear Mrs. Keckley, will you not recommend me to Mrs. Lincoln as a friend of yours out of employment, and ask her to take me as a chambermaid? If you will do this you shall be well rewarded. It may be worth several thousand dollars to you in time.\"\n\nElizabeth regarded her with amazement. \"Madam, you are mistaken in regard to my character. Sooner than betray the trust of a friend, I would throw myself into the Potomac. I am not so base as that. Pardon me, but there is the door, and I trust that you will never enter my room again.\"\n\nThe woman sprang to her feet, her expression utterly astonished and outraged. \"Very well,\" she snapped as she strode from the room. \"You will live to regret your action today.\"\n\n\"Never, never,\" Elizabeth exclaimed, slamming the door shut behind her. A moment later, she heard a rap upon it and tore it open only to find Emma standing in the hallway.\n\nEmma inclined her head down the hallway. \"Why is she leaving in such a huff?\"\n\n\"Oh, it's nothing,\" said Elizabeth shortly, beckoning Emma inside. \"Just another opportunity hunter. She claims it's her heart's desire to work as Mrs. Lincoln's chambermaid.\"\n\n\"She doesn't look like a chambermaid,\" observed Emma. \"But she does look somewhat familiar.\"\n\n\"I'd never seen her before she ordered that dress, and I hope never to see her again.\"\n\nShe wanted nothing more than to forget the incident, but Emma's curiosity had been piqued, and she promptly launched her own investigation. Before long she discovered that the woman was an actress, who had confided to several friends her determination to enter the White House as a servant, learn the secrets of its inhabitants, and publish a scandalous account to the world. \"She underestimated your principles, and your loyalty,\" said Emma, clearly relishing her triumph at discovering the scheme.\n\n\"Let that be a lesson to us both.\" Elizabeth never would have accepted a bribe, but she knew others would have had no qualms about doing so. When she thought of how long and how hard she had worked to raise the twelve hundred dollars to purchase her freedom and George's, she could well understand how someone else in her place would have given in to temptation. But Mrs. Lincoln was no longer only a patron, or even just the First Lady. She had become a friend, and Elizabeth would rather die than betray her.\n\nFor Mrs. Lincoln, the autumn was marked by changes in the White House staff, some she desired and others that were forced upon her. After all the schools in the district were shut down due to the wartime emergency, the First Lady decided to open a classroom in the White House rather than send Willie and Tad off to boarding school. She hired a tutor, arranged desks and a chalkboard, and brought in her sons' best friends, Bud and Halsey Taft, as classmates. Until then, the youngest Lincoln boys' education had been sorely neglected by their indulgent parents. Willie had a scholarly bent and often read and composed poetry on his own, but Tad was barely literate, something Mrs. Lincoln resolved to remedy. Both Mr. and Mrs. Lincoln valued education and hoped their younger sons would come to love learning as much as they did, and perhaps in time follow their eldest brother, Robert, on to Harvard. Privately, Elizabeth heartily approved of the new plan, which she hoped would give the mischievous boys some much-needed discipline along with the usual lessons. She had long regretted that she had been given no formal education, and she disliked seeing any opportunity for learning squandered.\n\nSome new members of the White House staff should have been reassuring, but instead their presence made Elizabeth uncomfortably aware of the dangers confronting the president in those perilous times. New doormen\u2014some of them officers from the new Metropolitan Police, dressed in civilian attire and carrying concealed weapons\u2014could be found in the public rooms, and uniformed sentries were posted on the grounds. In the year since his election, the president had received so many threatening letters that it was impossible to keep track of them all, although their frequency and virulence had increased since Bull Run. Mrs. Lincoln worried more about her husband's safety than he seemed to, and she urged him to travel unannounced and accompanied by guards whenever he moved about the city. He considered such precautions unnecessary and consequently ignored his wife's requests, which made her fret all the more.\n\nMrs. Lincoln was also concerned about threats from within. She and the president often discussed his cabinet members in Elizabeth's presence, and Elizabeth had observed that the First Lady was a shrewd judge of character, and her intuition about other people's sincerity was usually more accurate than her husband's. Elizabeth had learned early on that Mrs. Lincoln despised Secretary of the Treasury Salmon Chase, whom she called a selfish politician instead of a true patriot, but Elizabeth attributed some of her dislike to the fact that he was the father of her most bitter social rival, the lovely Miss Kate Chase.\n\nMrs. Lincoln thought no better of Secretary of State William Seward. One morning, Elizabeth arrived at the White House earlier than usual to find Mr. Lincoln sitting in a chair, holding the newspaper in one hand and stroking little Tad's head with the other. While Elizabeth was basting a dress, a servant entered with a letter for Mr. Lincoln that had just arrived by messenger. He broke the seal and read the letter in silence.\n\n\"Who is the letter from, Father?\" asked Mrs. Lincoln.\n\n\"Seward.\" Mr. Lincoln tucked the letter into his pocket. \"I must go over and see him today.\"\n\n\"Seward! I wish you had nothing to do with that man. He cannot be trusted.\"\n\nMr. Lincoln regarded her mildly, but Elizabeth thought she saw the corner of his mouth twitch as if he were trying not to smile. \"You say the same about Chase. If I listened to you, I should soon be without a cabinet.\"\n\nMrs. Lincoln looked as if she thought that would be a significant improvement. \"Better to be without it than to confide in some of the men that you do. Seward is worse than Chase. He has no principle.\"\n\n\"Mother, you are mistaken.\" The brief glint of humor had turned sober. \"Your prejudices are so violent that you do not stop to reason. Seward is an able man, and the country as well as myself can trust him.\"\n\n\"Father, you are too honest for this world! You should have been born a saint.\" The president scoffed at that, but Mrs. Lincoln persisted. \"You will generally find it a safe rule to distrust a disappointed, ambitious politician. It makes me mad to see you sit still and let that hypocrite Seward twine you around his finger as if you were a skein of thread.\"\n\n\"It is useless to argue the question, Mother.\" Mr. Lincoln returned his gaze to the newspaper. \"You cannot change my opinion.\"\n\nThat did not dissuade her from trying. She called Andrew Johnson a demagogue and warned her husband that if he placed him in a position of power, he would regret it one day. When the popular General McClellan was promoted, she declared that he was a humbug, because he talked so much and did so little. When Mr. Lincoln protested that he was a patriot and an able soldier, Mrs. Lincoln retorted, \"You will have to find some man to take his place\u2014that is, if you wish to conquer the South.\"\n\nGeneral Ulysses S. Grant was definitely not the officer she would have chosen to replace McClellan. \"He is a butcher,\" she often said, \"and is not fit to be at the head of an army.\" When Mr. Lincoln pointed out that he had been very successful in the field, she replied, \"Yes, he generally manages to claim a victory, but such a victory! He loses two men to the enemy's one. He has no management, no regard for life. If the war should continue four years longer, and he should remain in power, he would depopulate the North.\" She shook her head, indignant and angry. \"I could fight an army as well myself. According to his tactics, there is nothing under the heavens to do but march a new line of men up in front of the rebel breastworks to be shot down as fast as they take their position, and keep marching until the enemy grows tired of the slaughter. Grant, I repeat, is an obstinate fool and a butcher.\"\n\n\"Well, Mother, supposing that we give you command of the army,\" said Mr. Lincoln, a merry gleam in his eye and a ring of irony in his voice. \"No doubt you would do much better than any general that has been tried.\"\n\nIf the offer had been made in earnest, Elizabeth would not have been surprised if Mrs. Lincoln had accepted. Then her husband would have had to listen to her counsels.\n\nThe war cast a somber mood over the holidays, but Mrs. Lincoln was determined to celebrate. Elizabeth made her a blue velvet gown to wear to a performance at the National Theatre, as well as other dresses for her afternoon receptions and the president's formal levees. The two youngest Lincoln boys were merry, but Mr. Lincoln seemed always melancholy, his cares weighing heavily upon him. Mrs. Lincoln endeavored to cheer him out of his gloomy spells, but the reliable tactics of the past were losing their power to affect him.\n\nRobert Lincoln came home from college for the holidays, and his presence added joy to what already had turned out to be a surprisingly triumphant season for the First Lady. Her much-maligned renovations had transformed the reception areas of the White House into elegant showplaces, earning rave reviews in the papers and grudging praise from even her most persistent detractors.\n\nAs for Elizabeth, she celebrated Christmas quietly, with church services on Christmas Eve and the morning of Christmas Day. As they had the year before, the Lewises invited her to join their family for a midday feast, and this time Emma too was a welcome guest. Her friends were so gracious and easy that it wasn't until later that Elizabeth realized that they were each taking care to amuse and divert her, knowing that her first Christmas without her son was bound to be mournful. She was so touched by their kindness that she endeavored to be of good cheer. She reminded herself that thanks to the Savior whose birth they celebrated that day, her son would have eternal life. Surrounded by friends, comforted by the certainty that she and George\u2014and her father and mother too\u2014would be reunited in heaven someday, she could not grieve, if only for that day, if only on Christmas.\n\nThe social success Mrs. Lincoln enjoyed during the Christmas season extended into the New Year. Thousands visited the White House for the traditional New Year's Day reception, and although some guests continued to criticize Mrs. Lincoln's decorating expenses and others gossiped about corruption scandals within the administration, many more visitors praised the refurbished Executive Mansion as tastefully and elegantly done, befitting a glorious nation.\n\nA brilliant levee followed shortly after the New Year's Day reception, and the next morning, Mrs. Lincoln spoke of it in glowing terms while Elizabeth fitted her for a dress. \"I have an idea,\" she mused. Although she spoke as if suddenly inspired, Elizabeth had the distinct impression that she had been pondering whatever she was about to say for quite some time. \"These are war times, and we must be as economical as possible. You know the president is expected to give a series of state dinners every winter.\"\n\n\"Yes, of course.\" Elizabeth had dressed Mrs. Lincoln for almost every one of them.\n\n\"These dinners are very costly.\" Mrs. Lincoln shook her head and sighed as if no words would suffice to describe the outrageous expense. \"I thought that if I gave three large receptions, the state dinners could be scratched from the program. What do you think?\"\n\nElizabeth considered. A large reception would allow the Lincolns to entertain a far greater number of guests for the cost, but what Washington society would think of abandoning tradition was another matter. In any event, Mrs. Lincoln had surely already made up her mind, in which case it was simply best to agree with her. \"I think you're right, Mrs. Lincoln.\"\n\nMrs. Lincoln brightened. \"I am glad to hear you say so. If I can persuade Mr. Lincoln, I shall not fail to put the idea into practice.\"\n\nBefore Elizabeth finished her work for the day, Mr. Lincoln joined them, his expression clouded with frustration. \"I've just come from the sickbed of my recalcitrant general,\" he said, sighing gloomily as he settled into a chair. \"At least today he can blame typhoid fever for his reluctance to budge.\"\n\nMrs. Lincoln murmured sympathetically for a moment, and then, either to distract him from his gloom or because she simply couldn't wait, she proposed her changes to their social calendar. The president mulled it over, then frowned and said, \"Mother, I am afraid your plan will not work.\"\n\n\"But it _will_ work, if you will only determine that it _shall_ work.\"\n\n\"It is breaking in on the regular custom,\" he noted mildly. The protocol was indeed complex, with the rules of etiquette and the ranking of guests carefully noted. The president and the secretary of state alternated hosting evening receptions from the last week of January through March, and the White House also gave weekly dinners for various members of the government as well as receptions for military officers, diplomats, and Supreme Court justices. Breaking these well-established Washington traditions had the potential to offend.\n\n\"But you forget, Father, that these are war times, and old, impractical customs can and should be set aside,\" Mrs. Lincoln said. \"The idea is economical, you must admit.\"\n\n\"Yes, Mother, but we must think of something besides economy.\"\n\nElizabeth hid a smile. Matters of economy plagued both Lincolns, due in no small part to Mrs. Lincoln's notorious spending habits, but frugality had become a tool each employed when it served them and dispensed with when it did not. Mrs. Lincoln was usually the spendthrift and her husband the more cautious, and it was amusing to see them trade roles.\n\n\"I do think of something else,\" Mrs. Lincoln retorted. \"Public receptions are more democratic than stupid state dinners\u2014are more in keeping with the institutions of our country, as you would say if called upon to make a stump speech. There are a great many strangers in the city, foreigners and others, whom we can entertain at our receptions, but whom we cannot invite to our dinners.\"\n\nMr. Lincoln pondered this. \"I believe you are right, Mother,\" he finally said. \"You argue the point well. I think that we shall have to decide on the receptions.\"\n\nWhen her husband's gaze was turned elsewhere, Mrs. Lincoln shot Elizabeth a look of triumph. She had won the day.\n\nFor all her talk of economy, for the first of these receptions, Mrs. Lincoln decided to host a grand ball in the East Room. As word of her lavish plans spread, she once again invited criticism\u2014not only from the usual suspects in the press and the popular circles, but also from her husband's cabinet secretaries, who took to calling her Hellcat behind her back, though not behind Elizabeth's. She wouldn't carry hurtful tales of their bold hostility to her patron, but she knew cruel remarks usually managed to make their way from the offices, through the servant's quarters, and on to the subject of their derision.\n\nIf the men's sniping troubled Mrs. Lincoln, she feigned indifference and threw herself into her preparations. She collaborated on a new dress with Elizabeth, an off-the-shoulder white satin gown with a low neckline, flounces of black lace, black and white bows, and a long, elegant train. She planned an elaborate menu of roast turkey, foie gras, oysters, beef, duck, quail, partridge, and aspic, complemented by an assortment of fruits, cakes, and ices, and fanciful creations of spun sugar. She sent out seven hundred invitations to prominent men in government and their wives, as well as to certain favorite friends, important Washington personages, and visiting dignitaries. \"Half the city is jubilant at being invited,\" Elizabeth overheard Mr. Lincoln's personal secretary remark, \"while the other half is furious at being left out in the cold.\" Expectations soared after the _New York Herald_ predicted that the ball would be \"the most magnificent affair ever witnessed in America.\"\n\nBut such raptures in the press were not enough to entice everyone. The usual detractors, and many more besides, expressed astonishment and disgust for the vain spectacle of the ball and its hostess. A great many of the invitations were brusquely declined, and nearly one hundred were returned with indignant notes protesting such excessive frivolity when the nation was distracted, mournful, and impoverished by the war. \"I am astonished by such impertinence from a gentleman,\" Mrs. Lincoln exclaimed to Elizabeth one afternoon as she read her mail. \"Listen to what Senator Benjamin Wade writes: 'Are the President and Mrs. Lincoln aware that there is a civil war? If they are not, Mr. and Mrs. Wade are, and for that reason decline to participate in dancing and feasting.'\" Mrs. Lincoln slapped the letter down on the table. \"Are we aware there is a war? We are scarcely aware of anything else!\"\n\n\"I cannot imagine that anyone is more mindful of the war than Mr. Lincoln,\" said Elizabeth.\n\nOutraged, Mrs. Lincoln bolted from her seat and stalked to the window. \"If canceling the ball would bring a swift end to the war, or even a single hour of respite to a weary soldier, then I would be the first to propose it.\"\n\nElizabeth murmured soothing words until Mrs. Lincoln calmed herself. If only the war could be ended so easily. At the moment, Elizabeth would have settled for an end to the tempests that sprang up whenever someone affronted Mrs. Lincoln. Her critics judged her actions without understanding the motives behind them, which meant they would never be fair and only rarely accurate. As they were unlikely to change, it would serve Mrs. Lincoln well to cultivate a sense of calm and learn to ignore them. A dignified silence was often the best response to spiteful gossip\u2014but that was not Mrs. Lincoln's way.\n\nSoon thereafter, Mrs. Lincoln was distracted from her frenzy of planning when Willie caught a severe cold while riding his pony in foul weather. A few days before the ball, his condition worsened into a fever. Elizabeth had nursed both children through measles and a host of other illnesses, and this time too Mrs. Lincoln summoned her to his bedside. With tender efficiency, Elizabeth cared for Willie\u2014a kind, gentle, thoughtful boy, everyone's favorite\u2014and tried to comfort his mother, who fretted and worried incessantly, as she always did when any of her children fell ill. She had lost her second-born, Eddie, to chronic consumption when he was not quite four years old, and she lived in terror of losing another son.\n\nSoon it was evident that Willie was becoming steadily weaker. One afternoon, Elizabeth was at his bedside pressing a cooling cloth to his brow while Mrs. Lincoln hovered nearby, telling Willie cheerful stories of his pets, although he was too drowsy to pay much attention. Mr. Lincoln entered to check on his son, as he often did throughout the day. \"How is my boy?\" he asked.\n\n\"He seems to be unchanged since this morning,\" Elizabeth replied. \"No worse, but no better either.\"\n\n\"Perhaps he'll turn the corner soon,\" said Mr. Lincoln, but his eyes did not brighten as they would have had he felt true hope.\n\nElizabeth managed what she hoped was an encouraging smile. \"I've seen sick little boys recover from much worse.\"\n\n\"I should cancel the ball,\" Mrs. Lincoln fretted. \"It's ridiculous to think of hosting such an event with our boy in his sickbed. I can recall the invitations, postpone it until after he recovers.\"\n\n\"No, the reception should go on,\" said Mr. Lincoln. \"You've already gone to too much trouble and expense to cancel now.\"\n\n\"I don't mind the trouble.\" Mrs. Lincoln's voice carried an edge. \"Surely our guests will understand, and if I send word to the caterers immediately, they shouldn't have any difficulty accommodating a delay of a week or two.\"\n\nThe president fell into a thoughtful silence for a moment and then said, \"Why don't we consult Dr. Stone first?\"\n\nHis wife agreed, so the Lincoln family physician was sent for, and in due time he arrived to examine his patient. \"Your son is much improved,\" Dr. Stone announced after checking the boy's pulse, listening to his breathing, and questioning him about his symptoms. \"There is every reason to expect a full recovery soon.\"\n\n\"Oh, thank heavens,\" said Mrs. Lincoln fervently. \"And\u2014what do you think about the reception?\"\n\n\"I see no reason why it shouldn't go on as planned,\" the doctor replied, packing his instruments into his bag. \"I assure you, Mr. President, madam, your son is in no immediate danger.\"\n\nMrs. Lincoln clasped her hands together as if in prayer and thanked Dr. Stone profusely, but after he left, her momentary brightness faded. \"As glad as I am with this good news, I don't feel like dancing with Willie suffering so.\"\n\n\"If you don't feel like dancing, we won't have dancing,\" Mr. Lincoln replied. \"Come now, Mother. We won't be far away. We can come upstairs and look in on him as much as you like, and I'm sure we can rely upon Madam Keckley to nurse him in our absence, can we not?\"\n\n\"Certainly,\" said Elizabeth. \"I'll stay the night too, if you like.\"\n\nThus reassured, Mrs. Lincoln agreed to the arrangements, and so plans for the reception continued. Even after Tad too fell ill, the Lincolns reminded themselves of Dr. Stone's diagnosis and carried on, hopeful that their boys would be up and around soon.\n\nOn the evening of the reception, Elizabeth arrived early to dress Mrs. Lincoln and found her sitting at Willie's bedside, drawn and pensive, holding his small hand in hers. \"His fever worsened overnight,\" she told Elizabeth softly. \"The doctor insists he is in no danger, but I believe he's taken a turn for the worse.\"\n\nThe boy's sweet face did appear more flushed than Elizabeth had yet seen it, and his breathing seemed labored. \"How is Tad?\"\n\n\"Not well, but he's faring better than his brother.\" Mrs. Lincoln sighed and rose, pressing the back of her hand to her forehead. \"I'd better dress for the ball. To think, at one time I had been so looking forward to it.\"\n\nMrs. Lincoln summoned a servant to sit with Willie until Elizabeth could return after helping her prepare for the evening. In the boudoir, Elizabeth helped the First Lady don the white satin gown with its long train and black and white embellishments, and then arranged her hair in a headdress of black and white flowers. Mrs. Lincoln wore more flowers on her bosom, a half-mourning bouquet of crepe myrtle in honor of the recently widowed Queen Victoria.\n\nMr. Lincoln came in to escort his wife downstairs to the reception before she was quite ready, and while he waited for them to finish, he stood with his back to the fireplace, his hands clasped behind him. His expression was solemn, and although his gaze was fixed on the carpet, his thoughts seemed very far away.\n\nWhen Mrs. Lincoln was dressed, she admired herself in the mirror, her long train sweeping the floor. The rustle of satin roused Mr. Lincoln from his reverie, and he regarded his wife for a moment before the barest of smiles touched his lips. \"Whew! Our cat has a long tail tonight.\"\n\nMrs. Lincoln raised her eyebrows at him but otherwise made no reply.\n\n\"Mother,\" he said, taking in her bare arms and neck, \"it is my opinion that, if some of that tail was nearer the head, it would be in better style.\"\n\nElizabeth did not often disagree with the president, but in this case, she did not share his opinion in the slightest. Mrs. Lincoln's beautifully formed shoulders and neck were her best features, and the gown's low neckline set them off to great advantage. Mrs. Lincoln turned away from her husband with a look of offended dignity, but before long she consented to take his arm, and together they went downstairs to welcome their guests.\n\nElizabeth returned to the sickroom, where Willie languished in a fitful, sweaty doze. Before long she heard the Marine Band begin to play in the reception halls below, their music drifting through the ceiling and down the halls like the low, muted whispers of grieving spirits.\n\nThroughout the evening, Mrs. Lincoln frequently left the party to come upstairs and look in on her darling boy, sometimes with her husband by her side. Anxiously she would ask Elizabeth if there had been any change, but each time Elizabeth could only shake her head. Shortly before the ball ended, Willie seemed to struggle for breath, but after a worrisome hour, his breathing eased. The moment her guests departed, Mrs. Lincoln joined Elizabeth at Willie's bedside, and together they kept vigil until the first light of dawn crept above the horizon.\n\n\"You should rest, Elizabeth,\" Mrs. Lincoln told her in a voice raspy from exhaustion and worry.\n\n\"I'm fine, Mrs. Lincoln,\" she replied gently. \"Why don't you go to bed instead?\"\n\n\"It would be no use. I couldn't sleep. No, Elizabeth; you've been keeping watch over Willie longer than I. I must insist that you rest.\"\n\nElizabeth knew that it would do her no good to argue. \"Very well, Mrs. Lincoln.\" She rose wearily, arching her back to relieve the stiffness. \"I'll sleep for a time, but I'll be back down as soon as I wake so you can take a turn.\"\n\nMrs. Lincoln nodded distractedly, her gaze fixed on her suffering child.\n\nElizabeth found the bed that had been made up for her in the servants' quarters, undressed to her chemise, and lay down with the quilt pulled up to her chin. She slept restlessly and woke shortly before noon with vague memories of foreboding dreams. Someone had kindly left a full pitcher beside the washbasin, and after making a quick toilet she hurried downstairs to relieve Mrs. Lincoln, only to discover that Willie had not rallied during her absence.\n\nOver the next few days, Willie declined, steadily and inexorably, while his parents watched and waited and prayed. The president canceled a cabinet meeting and Mrs. Lincoln a levee rather than stray too far from their son's sickbed. Willie's best friend, Bud Taft, visited, and fell asleep on the floor, so determined was he not to leave his favorite companion's side. Elizabeth attended them all, not neglecting little Tad, who was not as seriously ill as his brother and had been confined to a separate bedroom. She stayed overnight at the White House when she was needed, and hurried away to her own home for rest and a change of clothes when she could be spared. All the while, Mrs. Lincoln kept vigil at Willie's bedside, surely haunted by memories of the long, sorrowful days more than a dozen years earlier when she had watched her son Eddie slip away.\n\nOn February 20, a mild, sunny day, Willie finally breathed his last. For all the long hours Elizabeth had spent at his sickbed, when he passed away she happened to be at home, where a messenger immediately summoned her. She hurried to the White House, and once there she asked to see Mrs. Lincoln, but the First Lady had been taken to her bed, inconsolable and keening. Elizabeth paused to check in on Tad\u2014the poor child was feverish, grief stricken, and terrified\u2014before she went to assist in washing and dressing Willie, laying him out upon the bed in the Green Room, and covering his face gently with a white cloth.\n\nElizabeth was keeping vigil alone when Mr. Lincoln entered the room, his face ashen and haggard, his eyes red and tormented. Elizabeth was too shocked and sorrowful to address him, so she nodded respectfully and stepped aside to the foot of the bed as he approached. She had never seen a man so bowed with grief, and she had seen many, far more than she could ever bring herself to count.\n\nHe lifted the sheet from his son's face and gazed at him long and tenderly. \"My poor boy, he was too good for this earth,\" he murmured. \"God has called him home. I know that he is much better off in heaven, but then we loved him so. It is hard, hard to have him die!\"\n\nGreat, heaving sobs choked off his words. Tears filled Elizabeth's eyes as she watched the anguished father bury his head in his hands, his angular frame shaking with grief. She wanted more than anything to offer words of comfort, but she knew none would suffice, and she found herself unable to speak.\n\nLater Elizabeth sat with Mrs. Lincoln in her rooms, where the curtains were drawn and the mirrors covered. She came again the next day and found Mrs. Lincoln wild with lamentations and sorrow. She was inconsolable, shattered by loss, unable to rise from her bed even to care for Tad, who was still struggling to recover from his illness. As the White House was draped in the black crepe of mourning, she collapsed in paroxysms of grief, shrieking and wailing, frightening Tad and worrying Elizabeth, who could do nothing to help her. Dr. Stone prescribed laudanum, but whenever the stupor of a dose lifted, Mrs. Lincoln seemed worse off than before. Elizabeth had nursed many people through illnesses, but she had never faced anything like Mrs. Lincoln's hysteria and felt powerless to help her. To her relief, Mr. Lincoln quickly understood that she was out of her depth and arranged for Rebecca Pomroy, a nurse who had been working in a military hospital, to be reassigned to the White House so she could look after Mrs. Lincoln and Tad.\n\nUnder Nurse Pomroy's care, Mrs. Lincoln regained the presence of mind to write to Mrs. Taft, begging her to keep her boys at home during the funeral because the sight of the children with whom her son had passed so many happy hours would devastate her. Elizabeth never told Mrs. Lincoln that her husband had secretly allowed Bud to come to the White House to see Willie one last time before he was placed in his casket. Shortly thereafter, Mrs. Lincoln managed to leave her bed for a final, private farewell with her husband and two surviving sons in the East Room, where the public services would be held later that day. Mrs. Lincoln was too distressed to endure the rituals of grief, so she retreated to her rooms while members of Congress, cabinet secretaries, diplomats, generals, and dignitaries filed in to pay their respects and offer condolences.\n\nOn the one-week anniversary of Willie's death, the president locked himself in the Green Room for a time\u2014Elizabeth did not know but could well imagine\u2014to be alone with his thoughts, to remember his beloved son, to pray. He observed the private mourning ritual every Thursday for several weeks thereafter, and it seemed to offer him solace. Tad improved day by day under the watchful eye of Nurse Pomroy, but as he would take his medicine from no one but his father, the president was frequently called out of meetings to administer the dose. Robert grieved the loss of his young brother, but he endeavored to maintain a brave, manly front, and the effort seemed to sustain him. Of the survivors, Mrs. Lincoln alone found no lessening of her grief. She was alternately paralyzed by sorrow or frantic with despair, and her sudden bouts of keening frightened Tad and alarmed the entire household.\n\nElizabeth was present the day Mr. Lincoln inadvertently revealed that he feared his wife's nerves would never recover. Once, when Mrs. Lincoln was caught up in one of her spells of unmitigated anguish, Mr. Lincoln took her gently by the arm, led her to the window, and solemnly pointed to St. Elizabeths Hospital in the distance.\n\n\"Mother,\" he said, \"do you see that large white building on the hill yonder?\"\n\nMrs. Lincoln nodded, her eyes widening. Everyone in Washington recognized the lunatic asylum, an imposing landmark on the city skyline.\n\n\"Try and control your grief,\" the president continued steadily, \"or it will drive you mad, and we may have to send you there.\"\n\nElizabeth muffled a gasp of horror. That this might be Mrs. Lincoln's fate was an idea too terrible to contemplate, and she was shocked that Mr. Lincoln would speak so to his grieving wife. And yet, in the days that followed, it seemed that the warning alone had compelled her to try to regain control of her nerves. Even so, Robert was so concerned by his mother's slow progress that he implored for his aunt Elizabeth Edwards to return to the White House, hoping that she would be able to comfort his mother even though the sisters had had a falling out the previous autumn. When she arrived, she discovered her younger sister bedridden with grief and her nephew Tad sobbing that he would never see his brother again. But Elizabeth observed that Mrs. Edward's presence brought an immediate sense of relief to the family. It was she who was able to persuade Mrs. Lincoln to leave her bed and dress, and later, to venture out to attend church services.\n\nBut despite these incremental improvements, Mrs. Lincoln was often overcome and overwhelmed by sorrow and spent most of the next month in seclusion within the White House. Elizabeth was often with her, sewing her mourning wardrobe, offering whatever comfort she could. Once, when Mrs. Edwards was not with them, Mrs. Lincoln suddenly said, \"If Willie had lived, he would have been the hope and stay of my old age.\" Her voice was quiet, her thoughts faraway. \"But Providence did not spare him.\"\n\nElizabeth understood, perhaps as only another grieving mother could. She had hoped that George would go far in life and that she would be able to rely upon him in her later years, but Providence had not spared her son either.\n\nIn the same distant voice, Mrs. Lincoln added, \"I believe Willie's death is God's punishment for my sins, for my vanity and self-indulgence.\"\n\n\"What?\" exclaimed Elizabeth. \"You cannot mean that.\"\n\n\"His decline began on the night of my lavish reception,\" she said bitterly. \"What clearer sign do I need of God's judgment than that?\"\n\n\"Oh, Mrs. Lincoln, no.\" Elizabeth shook her head. \"No. God is not so cruel.\"\n\n\"Isn't He?\" Mrs. Lincoln's gaze drifted to the window, but Elizabeth could not tell what it rested upon. The church steeple? The hospitals filled with wounded soldiers? St. Elizabeths across the Anacostia River? \"If His punishment is just, who are we to call it cruelty?\"\n\n# Chapter Six\n\n## MARCH\u2013APRIL 1862\n\nIn the weeks following Willie's death, Mrs. Lincoln struggled to get through each day. She avoided public appearances and clung to her sister Mrs. Edwards, who often stood in for her when she did not feel up to receiving visitors. She shunned the Green Room, where her dear son had been laid out, and she never again invited the little Taft boys to the White House. She saved the casket flowers from Willie's funeral and a poignant eulogy written by the poet Nathaniel Parker Willis, but she gave away all his toys and games. She could not bear to be reminded of him, and yet he was always in her thoughts.\n\nMr. Lincoln mourned also, but he did not have the luxury of secluding himself away from the world. Squabbles in Congress continued, the profound question of what to do about the slaves persisted, and the war went on.\n\nThe Confederates had salvaged the steam frigate _Merrimack_ from the Norfolk navy yard, cut off its upper hull, armored it with iron, and rechristened it the _Virginia_ before sending it out upon the James River to decimate the Union fleet. The next day the Union navy retaliated with its own ironclad ship, the _Monitor,_ whose unconventional deck design and revolving gun turret astonished all who witnessed the battle from other Union and Confederate ships on the water and the riverbanks on both sides. Neither warship was able to inflict much damage upon the other, but the North was able to maintain its blockade of Norfolk and Richmond, and thus claim victory.\n\nA few weeks later in early April, the Union defeated the Confederates at Shiloh in Tennessee, but it was the bloodiest battle of the war so far, with more than thirteen thousand killed, wounded, or missing on the Union side and nearly eleven thousand for the South. Among those killed was Mrs. Lincoln's half brother Samuel B. Todd, an officer serving with the Twenty-fourth Louisiana. His death reminded Mrs. Lincoln's critics of her family ties to secessionists and stirred up the old, groundless questions about her loyalties\u2014a monstrous thing to do at such a time, Elizabeth thought, showing utter disregard for Mrs. Lincoln's grief. After the costly victory, General Ulysses S. Grant was vilified in the press and many called for his removal, but President Lincoln replied, \"I can't spare this man; he fights.\"\n\nPresident Lincoln too fought, on other fronts. Recent antislavery lectures at the Smithsonian Institution had inspired a rising chorus of voices among abolitionists and radical Republican factions in Congress calling for emancipation. Some of his military officers had taken the lead in emancipating slaves under certain circumstances, but not always with the president's sanction. For more than a year, abolitionists had been arguing that the Union could not hope to win the war unless it deprived the Confederates of their labor force by emancipating their slaves. Frederick Douglass, a former slave and tireless advocate for the colored race\u2014and someone Elizabeth greatly admired\u2014concurred, writing in his newspaper, \"Arrest that hoe in the hands of the Negro, and you smite the rebellion in the very seat of its life.\" In July 1861, General Benjamin Butler had put three escaped slaves to work at Fort Monroe, declaring that they were \"contraband of war\" and that he was not obliged to return them to their masters under the Fugitive Slave Law because they had come from a state that had left the Union. Congress upheld the legality of General Butler's policy, and thereafter, other \"contraband\" were offered work to help the war effort, and their numbers rapidly grew. At the end of August, and without any authorization from the president whatsoever, Major General John C. Fr\u00e9mont, commander of the army's Department of the West, issued a proclamation freeing the slaves of Confederates within the state of Missouri. The president was furious when he learned what Fr\u00e9mont had done and commanded him to rescind the proclamation; when Fr\u00e9mont refused, Lincoln revoked it himself, angering Northern abolitionists and bringing a storm of criticism down upon himself.\n\nPresident Lincoln's ostensible reluctance to free the slaves puzzled and disappointed Elizabeth, but the longer she worked in the White House and the better she knew him, the more she understood that he indeed wished to abolish slavery, but on his own terms and on his own timetable. She knew he worried that immediate and total emancipation would drive the slaveholding border states out of the Union and into the Confederacy, and she did not envy him his burdens. But by early spring of 1862, attitudes had shifted and opportunities had manifested. Perhaps encouraged by the increasing demands from the public, and perhaps inspired by reports from his field commanders who employed \"contraband\" as laborers, cooks, and nurses, President Lincoln must have decided the time was right to nudge the idea of abolition forward. One morning in early March, he called Massachusetts senator Charles Sumner to the White House to read him the draft of a bill that would grant individual states the right to establish emancipation within their borders. Slave owners would be compensated with federal funds for the loss of their property, and freed slaves would be given the opportunity to immigrate to proposed colonies in Africa or Central America. In the past, the abolitionist senator had decried such incremental, compensatory measures, but recognizing the opportunity for great and lasting change the bill offered, he resolved to support it. Within hours, he and his fellow Massachusetts senator Henry Wilson began to push through Congress a bill calling for emancipation in the District of Columbia.\n\nOn the Senate floor, Henry Wilson spoke eloquently of the contributions the free colored population had made to the district\u2014how they strove for self-improvement and had created schools, churches, businesses, and benevolent societies. All their accomplishments had come about despite the unfair \"black codes\" that constricted their lives and freedoms. They paid taxes for schools that their own children were not permitted to attend, a grievous injustice. It was time for them to become truly, fully free, and no honest citizen had any reason to doubt that emancipated slaves would follow their fine example and become industrious members of society upon gaining their liberty.\n\nNot everyone agreed with Senator Wilson, of course, or with another important supporter of the bill, Thaddeus Stevens, the leader of the radical Republican faction and the lawyer who had represented Emma and her mother in the legal suit that had granted them their long-deferred freedom. Strong opposition to the bill was led by Peace Democrat Clement Vallandigham of Ohio, and the debate played out over several days. Walker Lewis and several other respected colored men of Washington City came to the Senate gallery day after day to observe the speeches, evoking displeasure and alarm from some senators, especially Garret Davis of Kentucky, who interrupted a bitter speech against the bill to snarl, \"I suppose in a few months they will be crowding white ladies out of the galleries!\" Undeterred, Walker and his companions continued to attend the debates and report on the progress of the bill to the colored community, who awaited the vote for passage with great anticipation and hope.\n\nNewspaper reporters covered the story in a more official capacity; the _National Republican_ praised Senator Wilson, but the Washington _Evening Star_ warned that the aim of the Emancipation Bill was to enforce Negro equality upon white men. In New York, Horace Greeley, who for years had lambasted President Lincoln in the pages of his _Tribune,_ was so elated by the proposal that he thanked the Lord Almighty that Lincoln had become so wise a ruler. Mr. Lincoln showed no reaction to Mr. Greeley's endorsement in Elizabeth's presence, but Mrs. Lincoln made no effort to hide that she was unimpressed. \"Today he showers my husband in praise,\" she said, waving a hand dismissively. \"Tomorrow it may well be criticism again.\"\n\nIt seemed to Elizabeth that most white Washingtonians did not share Mr. Greeley's opinion of the president. In letters and petitions to congressmen, editors, and anyone else with any degree of influence, they demanded that the bill be voted down. Those who owned slaves were against the bill for reasons they hardly needed to explain, but some citizens who had never owned a slave denounced the measure also. Emancipation would open the floodgates to a tide of free Negros who would compete with white citizens for jobs, they warned. Runaway slaves would flee to Washington and become a burden to the public. With the war on, the government couldn't afford to compensate district slave owners up to three hundred dollars per slave as the bill called for; it would be more prudent, more frugal, and all over better to let them keep their slaves for now and be responsible for their upkeep than to have the government shoulder that burden.\n\nThe bill's opponents could complain and they could protest, but the federal government did not need their approval to bring the measure to a vote. On April 3, the Senate passed the Emancipation Bill twenty-nine to fourteen, and a few days later, the House voted in favor by a margin of ninety-three to twenty-nine. After that, President Lincoln waited four days, a time in which he listened respectfully to impassioned arguments on both sides of the issue, and then, on April 16, a beautiful spring day, he signed the bill into law.\n\nWith a few strokes of his pen, he had abolished slavery forever in the capital of the United States.\n\nThe colored residents of Washington responded with unrestrained jubilation. Voices rose in joyful cheers and reverent hymns throughout colored neighborhoods, where community leaders, well aware that their every action was being watched and judged, urged them to respond with quiet dignity. Such self-possession came naturally to Elizabeth, but even she laughed and cheered and danced a little jig with Virginia in the parlor when they heard the news. \"If I had known this was coming, I would have saved my money,\" remarked another boarder, an elderly shoemaker who had recently bought his freedom. He said it so comically that all who heard him burst out laughing, the sound of it ringing out with exultation and joy, echoing other happy celebrations throughout the city. Their faith in the president had been renewed, as was their resolve to see slavery abolished everywhere, for everyone, for all time.\n\nOn Sunday, Elizabeth joined her fellow believers at Union Bethel Church, where freeborn, freedmen, and the newly, suddenly free rejoiced and gave thanks. She gloried in the minister's sermon, heartfelt words of rejoicing and thankfulness and praise to God, who had delivered them from bondage. And yet there was a somber undercurrent to their worship, for they were ever mindful of the great many people of their race who had not benefitted from the new law, and they understood that freedom brought with it great responsibility. \"We must resolve,\" the minister declared, as surely other ministers were also doing in the sixteen other colored churches throughout Washington, \"that by our industry, character, and moral righteousness we will prove ourselves worthy of the glorious privilege that has been conferred upon us. We have always been an orderly and law-abiding people, and so in the future we must strive to live so morally and industriously that anyone who is today disappointed with the passage of this law will forget why they ever opposed it.\"\n\nElizabeth's hopes for those outraged white citizens of Washington were less ambitious. She wouldn't ask for them to forget their objections altogether, but she would gladly settle for their acceptance of the fact that emancipation was the new law of the land\u2014and for their greater tolerance for the colored folks who had also made the city their home.\n\nThe day after Mr. Lincoln signed the bill into law, Elizabeth was sewing in her rooms, her windows wide open to the faintly pungent spring breeze and the chirping of songbirds, when a knock sounded upon her door. In the hallway stood a dark-haired white woman of about thirty years, sensibly attired in a brown wool dress. \"Good afternoon,\" Elizabeth greeted her. \"May I help you?\"\n\nThe woman's brown eyes lit up with expectation. \"You are Elizabeth Keckley, the mantua maker?\"\n\n\"Yes, I am.\"\n\n\"How do you do?\" The younger woman extended a hand, and Elizabeth shook it. \"My name is Mary Ames. I'm a Washington correspondent.\"\n\n\"Oh, dear,\" said Elizabeth without thinking, surprise driving away her usual composure. \"That is\u2014I beg your pardon, Mrs. Ames. How do you do?\"\n\n\"I'm quite fine, thank you.\" She glanced past Elizabeth's shoulder into her rooms, and Elizabeth was profoundly grateful that Mrs. Lincoln had not chosen that day of all days to surprise her with a visit to discuss a new dress design. \"I wondered if we might chat for a bit?\"\n\n\"I'm terribly sorry, but if you're seeking information about the president and Mrs. Lincoln\u2014\"\n\n\"Oh, no, no, that's not my purpose at all,\" Mrs. Ames interrupted. \"I am pursuing a story, but you are my intended subject, not the Lincolns.\"\n\n\"Why would you want to write about me?\" Curiosity compelled Elizabeth to open the door wider. \"For that matter, why would anyone want to read about me?\"\n\n\"Because of the new emancipation law, of course. Everyone wonders what the consequences will be, for Washington itself as well as all the new freedmen. I would like to write a piece describing the accomplishments of several former slaves who have done well for themselves since coming to the city.\" She smiled winningly. \"And who has accomplished more than the celebrated dressmaker Elizabeth Keckley? From slave cabin to White House\u2014quite a remarkable rise.\"\n\n\"That's very kind.\" Elizabeth managed a small, uncertain smile. \"I've always tried to be industrious, and I've been blessed by generous patrons who have graciously recommended me to their acquaintances.\"\n\n\"Oh, you're too modest,\" protested Mrs. Ames. \"You've succeeded thanks to your talent and your unmatched reputation. Surely you know how highly the ladies of Washington regard you. When I asked around for recommendations for successful late slaves to interview, your name came up more than any other.\"\n\nElizabeth firmly warned herself not to succumb to flattery, but she felt herself wavering. \"If I agree to an interview, it is with the understanding that I never gossip about my patrons.\"\n\n\"Of course.\" A shadow of disappointment darkened Mrs. Ames's amiable expression for a moment, but it quickly lifted. \"Discretion is essential between a dressmaker and her customer, whether she lives in the White House or the house around the corner.\"\n\n\"I'm glad you understand.\"\n\n\"I don't intend to publish a scandal, Mrs. Keckley.\" Mrs. Ames paused for a moment, considering. \"No matter how much my editor might prefer one. After all, scandalous tales sell papers. But so do newsworthy and inspirational pieces, which is what I intend this to be. Your story will inspire these newly emancipated slaves to practice economy, industry, and morality so that they might achieve as you have done.\" Almost as an aside, she added, \"It might also serve to reassure the white citizens of Washington that they have nothing to fear. Sudden change can be frightening, you know, and emancipation has brought dramatic change.\"\n\nElizabeth hesitated. \"I suppose it would do no harm if I spoke to you. The story might even do some good.\"\n\n\"There's no question that it shall,\" Mrs. Ames declared as Elizabeth beckoned her inside.\n\nElizabeth prepared tea, and they chatted in her front room. At first Elizabeth spoke cautiously, with great reserve, mindful of how her patrons\u2014Mrs. Lincoln in particular\u2014would react to her words. Then, halfway through her second cup of tea, she lost herself in reminiscing about her slavery days, about George, about her astonishing new life in Washington, and she revealed more about her past sorrows than she had intended. It was only after the teapot had cooled and Mrs. Ames had filled up several pages with notes that Elizabeth, too late, wondered if she had said too much.\n\n\"When will your story be printed?\" Elizabeth asked after the interview concluded and she had shown Mrs. Ames to the door.\n\n\"Perhaps as early as tomorrow, if I apply myself,\" Mrs. Ames replied cheerfully.\n\nElizabeth's heart thumped anxiously as they exchanged good-byes. She had expected to have more time to\u2014not to _warn_ Mrs. Lincoln, exactly, but to inform her that the article was forthcoming, and to reassure her that Elizabeth had not carried any tales from the White House. She didn't think she had said anything that could possibly offend, but with the First Lady, that was sometimes difficult to anticipate. With any luck, she would be able to read the piece before Mrs. Lincoln did\u2014and perhaps Mrs. Lincoln wouldn't see it at all.\n\nThe very next day, one of Elizabeth's neighbors in the Lewis boardinghouse rapped on her door and handed her the newest edition of the _Evening Post_. \"I know you must be the 'Lizzie' in this story,\" Miss Brown said, her dark eyes shining with excitement. \"Congratulations, my dear.\"\n\n\"Thank you,\" said Elizabeth, taking the paper. They chatted for a few minutes, but as soon as Miss Brown departed, Elizabeth closed the door, sank into a chair, and held the paper in her lap, eager and yet apprehensive to see what Mrs. Ames had written. She took a deep breath and began to read, her eyes lighting on the title, \"Lives of the Late Slaves,\" and the first subtitle, \"A Slave-girl's Story.\" Elizabeth skimmed the tale of a nine-year-old girl from the Congo and stopped when the column was broken by another subtitle, \"A Stylish Black Woman.\" Then she steeled herself and read on.\n\nLizzie\u2014\u2014\u2014 is a stately, stylish woman. Her cheek is tawny, but her features are perfectly regular, her eyes dark and winning; hair straight, black, shining. A smile half sorrowful and wholly sweet makes you love her face as soon as you look on it. It is a face strong with intellect and heart, with enough of beauty left to tell you that it was more beautiful still before wrong and grief had shadowed it.\n\nThat was not such a bad beginning, Elizabeth thought. Her name was spelled differently than she wrote it, but the editor had also removed her last name to shield her privacy, so perhaps the alternate spelling was meant to do the same. And it was good to hear that she had \"enough of beauty\" remaining at her age, she thought wryly before returning her gaze to the newspaper.\n\nLizzie's father was a gentleman of \"the chivalry,\" and in her mother's veins ran some of the best blood of the Old Dominion.\n\nI cannot tell the wrongs of her childhood and early youth; if I were to try my hand would stiffen with horror, my heart, in its strong indignation, would stifle the words I might utter.\n\nIn her girlhood she was sold to a family who took her to a great city of the Northwest. For years the \"gentleman,\" the \"lady,\" and their large family of children were supported by the labors of this young slave. Lizzie's great skill and taste made her the fashionable dressmaker of this metropolis. She earned thousands of dollars, and it all went to the support of her master's family. The young ladies went in fashionable society and enjoyed their fine costumes none the less that they were first earned and then made by the young slave, who, thrice as intelligent and quite as handsome as themselves, sat through the weeks and months in a chamber at home, spending her life for them.\n\nElizabeth felt her first stirrings of dismay. She had never called Armistead Burwell her father; that honorific she reserved for George Hobbs, her mother's devoted husband, whom Elizabeth had loved dearly. She had also never referred to Colonel Burwell as a member of \"the chivalry\"; that turn of phrase was entirely Mrs. Ames's. But that was not the worst of it. She had never claimed to be \"thrice as intelligent\" as her former young mistresses. They could not sew as well as she, but to be fair, few people could. They were as bright and clever as any young ladies Elizabeth had ever known, and Elizabeth cared for them very much, despite the regrettable tension ever present in the bonds of their affection that prevented them from being true friends. What on earth would they think of her if they believed that she had boasted of her superior intelligence? She could only pray that the fracture in the nation would prevent the newspaper from falling into their hands, or if it did, that they would know that Elizabeth never would have said such a thing and that the reporter must have misunderstood her.\n\nThe article went on to describe how Elizabeth had earned fame as a skilled dressmaker in that \"great city of the Northwest,\" how she had purchased her freedom, how her son had died on the battlefield wearing Union blue and taking with him \"the last earthly hope and consolations of his mother's heart,\" and how she had \"turned her face from the West, and came here, sorrowfully, wearily, to begin life anew.\" She winced at a few factual and chronological errors, but overall, Mrs. Ames had described her experiences nearly as Elizabeth had shared them. A few mistakes could be forgiven.\n\nThe article went on to praise her accomplishments since arriving in Washington City:\n\nIt is Lizzie who fashioned those splendid costumes of Mrs. Lincoln, whose artistic elegance has been so praised during the past winter. It was she who \"dressed\" Mrs. Lincoln for \"the party,\" and for every grand occasion. Stately carriages stand before her door, whose haughty owners sit before Lizzie docile as lambs while she tells them what to wear. Lizzie is an artist, and has such a genius for making women look pretty, that not one thinks of disputing her decrees. Thus she forgets her sorrows, interesting herself to serve each one who comes, as if to dress _her_ was the chief business of her existence. But to the woman who stretched out her hand to her as a sister, she broke into passionate tears, saying, \"I am alone in the world. I have nothing to live for any longer. I try to interest myself in these things, but cannot.\"\n\n\"Oh, dear,\" Elizabeth murmured to the empty room. She did not mind being called an artist, or a genius, but she doubted very much that any of her patrons would be pleased to know that they had been described as haughty in one sentence and docile as lambs and subject to her commands in the next. But the conclusion of the paragraph utterly bewildered her. At first she did not know who the \"woman who stretched out her hand to her as a sister\" was meant to be\u2014Mrs. Le Bourgois, who had organized the ladies of St. Louis to help her attain her freedom? Mrs. Lincoln?\u2014until it came to her that the reporter was speaking of herself, a rather strange claim to make based upon their short acquaintance. It was also astonishing to read how she had wept during their interview, for Elizabeth had not shed a single tear. She might have said that she lived alone, but she had never lamented that she was alone in the world, nor would she ever say that she had nothing to live for any longer. She had her work, her friends, her freedom, her faith\u2014she would never succumb to despair. How embarrassing it was to think that now her friends and patrons would believe she did, because of course they would recognize her as quickly as Miss Brown had done. What had possessed Mrs. Ames to invent such sentimental phrases, and to place them in Elizabeth's mouth?\n\nThe article went on to describe where Elizabeth lived, although thankfully Mrs. Ames had not provided her address or named the Lewises as her landlords. She concluded with a dramatic final assessment of Elizabeth, saying, \"A woman of thought and refinement, a woman of deep affection and high aspiration, she stands alone in her womanhood, alone in the universe.\"\n\nWhy did Mrs. Ames emphasize Elizabeth's solitude? Elizabeth was not friendless, helpless, and forsaken. She had lost her son, but in this she was not unlike many other mothers in those perilous days of war. She had many friends. She had the patronage of Mrs. Lincoln and many other ladies of quality in Washington. Elizabeth did not understand what she had said or done to make such an impression of loneliness upon the reporter.\n\n\"If one is going to write the life of another,\" Elizabeth said aloud, rising from her chair with the newspaper still clutched in her hand, \"one should always, always write the truth. Anything else is a waste of paper and ink and words.\"\n\nShe should have thrown the paper away and forgotten it, and yet some faint pride or vanity compelled her to fold it neatly and tuck it away for safekeeping in the same trunk in which she saved fabric from Mrs. Lincoln's gowns, leftover scraps too small for dressmaking that Mrs. Lincoln generously allowed her to keep.\n\nSuddenly Elizabeth realized that the article could have been much worse. For all her mistakes and fictions, Mrs. Ames had said nothing unflattering about Mrs. Lincoln and had written several things that could be construed as praise for the First Lady's style and elegance. That was no small blessing, and as Elizabeth latched the trunk shut, she breathed a sigh of relief and resolved to avoid speaking to reporters in the future. She would not tempt fate by risking embarrassment and scandal in the press a second time.\n\n# Chapter Seven\n\n## MAY\u2013OCTOBER 1862\n\nIn May, although Mrs. Lincoln still grieved deeply, she made an effort to resume her usual routines and former duties. She forced herself to stroll the White House grounds with visiting friends and to take carriage rides. She sent bouquets from the conservatory to acquaintances and important dignitaries, and she gave Nurse Pomroy delicacies from the White House kitchen and garden to distribute at military hospital wards. Her remaining sons, Tad and Robert, the latter of whom had come home from college for the summer, provided her with the most compelling distraction from her sorrow.\n\nFor weeks she had mentioned idly in conversation that she believed fresh air and a change of scenery would do her good, so Elizabeth was not surprised when Mrs. Lincoln announced that in the middle of June, the Lincoln family would relocate to their summer residence. Anderson Cottage, she told Elizabeth, was a charming two-story dwelling of stucco and gables on the 240-acre grounds of the Soldiers' Home, an asylum established as a place for wounded veterans to convalesce. It was about two miles north of the city, a cool, wooded, secluded haven on a hilltop, far enough away from the Capitol and the White House to act as a restful retreat, but near enough for Mr. Lincoln to travel back and forth as needed.\n\nAlthough Mr. Lincoln returned to the city almost daily, Mrs. Lincoln and her sons did not, so Elizabeth saw the First Lady very little throughout the late spring and summer. Although many other ladies left Washington to escape the heat, the insects, the pervasive illnesses, and the sights, sounds, and smells of a city given over to the care of thousands of ill and wounded soldiers, enough of Elizabeth's patrons remained to keep her sewing busily during Mrs. Lincoln's absence.\n\nAt times Elizabeth wished she too could ride off to a cool summer retreat, but she made the most of the early mornings and temperate evenings to go for walks, attend lectures, or enjoy outings with the Lewises.\n\nOne evening, she and Virginia were coming home from a choir concert at the Fifteenth Street Presbyterian Church when they heard a band playing merry music a few blocks away. Curious, they both instinctively slowed their pace to listen. They exchanged a glance, and Elizabeth asked what they were both thinking: \"Shall we see what that's about?\"\n\nVirginia eagerly agreed, and so they followed the enchanting melody to the home of Mrs. Farnham. The yard was brilliantly illuminated with lanterns, and white ladies and gentlemen were strolling about, enjoying conversation in the night air, glasses of punch, and lovely music.\n\nVirginia tucked her arm through Elizabeth's and, with a nod, indicated the colored butler standing sentinel at the gate. By unspoken agreement, they crossed the street and approached him. \"Good evening,\" Elizabeth greeted him pleasantly. \"Would you kindly satisfy our curiosity? What happy occasion is Mrs. Farnham celebrating?\"\n\n\"It's a festival, ma'am,\" he said. \"A benefit for the sick and wounded soldiers.\"\n\n\"A benefit?\" echoed Virginia.\n\nThe sentinel nodded. \"Yes, ma'am. The guests pay twenty-five cents to enter, and then they can buy tickets to exchange for punch and delicacies. For a few tickets more, they can request a special tune from the band. All the money will go to help the suffering soldiers get the things they need\u2014medicine, bandages, blankets, good food\u2014everything that's in short supply.\"\n\n\"What a marvelous idea,\" said Elizabeth. \"I hope it's a complete success.\"\n\nThe sentinel thanked them politely as they moved on.\n\nAs she and Virginia made their way home, Elizabeth mulled over what they had seen. \"Virginia,\" she said thoughtfully as they turned onto Twelfth Street, \"something occurs to me.\"\n\n\"And what is that?\"\n\n\"If white people can give festivals to raise funds for the relief of suffering soldiers, why shouldn't the well-to-do colored people work for the benefit of the suffering of our race?\"\n\n\"I don't see any reason why we should not,\" said Virginia. \"Heaven knows we have suffering colored folks all around us in abundance, and more coming every day.\"\n\nIt was a sobering truth. Just as the opponents of the president's bill had warned, Washington after emancipation had become the refuge of contraband, runaways, and freedmen emigrating from slave states. Before the law abolished slavery in the district, the contraband population had numbered only a few hundred. Newcomers found places to live in the town houses along Duff Green's Row on East Capitol Street and were taken up into the ebb and flow of the city with little difficulty. After emancipation became law, however, their numbers had swelled into the thousands. They arrived in the city alone or with their families in tow, footsore, hungry, exhausted, most of them field hands with no trade or training except farm labor, almost all of them illiterate. Very few contraband could find or afford rooms in colored homes or boardinghouses, and even fewer had set out for the capital with any thought of what they would do once they arrived. At first they were accommodated in Camp Barker, a complex of empty soldiers' barracks, stables, and tents, but when these places filled to overflowing, the refugees built themselves shacks of blankets, mud, and scraps of wood in camps that sprouted up near military hospitals, beside the forts on the outskirts of the city, or tucked away in alleys. Sometimes the government administrators failed to distribute rations efficiently, and entire families went hungry. Diseases like dysentery, smallpox, and typhoid flourished in the overcrowded, filthy camps, and the dead were buried in makeshift cemeteries not far away.\n\nWhite Washingtonians, even some of those who had supported emancipation, had become increasingly alarmed by the rising tide of impoverished refugees. They were not alone in their worries. Some among the colored elite and middle class, whose loftier status within the city social hierarchy seemed tenuous even at its best, were determined to distinguish themselves from their uneducated, destitute, darker-skinned brethren. In Washington's colored community, rank was determined according to one's wealth, distance from slavery, and lightness of skin tone. Thus freeborn Negros of means were at the pinnacle, with light-skinned former slaves a rung below, darker-skinned freedmen beneath them, and slaves lowest of all, with barely a toehold on the ladder. It was an order not unlike what Elizabeth had known as a slave, wherein light-skinned house slaves\u2014often the children, grandchildren, or siblings of their masters\u2014held themselves above the darker field hands. The unspoken rules were strictly observed even within the White House, and even the president was powerless to enforce equality. Mrs. Lincoln had told Elizabeth regretfully of William Johnson, Mr. Lincoln's dark-skinned personal valet, who had come with them to Washington from Springfield but had left soon thereafter, before Elizabeth came into her service. The lighter-skinned White House servants had snubbed William and subjected him to such adamant scorn that after enduring two days of it, he resigned and asked the president to help him find employment elsewhere.\n\nThankfully, most middle-class colored Washingtonians overcame these class prejudices to assist the contraband. They acted out of compassion, and they acted out of self-interest, understanding well that many white people saw no difference between the prosperous colored merchant whose wife and daughters dressed in silk and the shoeless, whip-scarred contraband who had been a slave only weeks before. As the lowest among them fared, so everyone of their race would be perceived, and thus for their own sakes it was essential to forgo snobbery and raise up all colored people. Elizabeth wished no one needed to be reminded that it was also simply the right thing to do, to be charitable to those in need.\n\nThe following Sunday, with encouragement from Virginia, Walker, and Emma, Elizabeth received permission from her minister to address the congregation after his sermon. She was not accustomed to speaking before so great a crowd, but she summoned up her courage, stepped up to the pulpit, and reminded herself that the need was too great to allow nerves to get the better of her.\n\n\"We have all observed the great migration of newly freed slaves to Washington,\" she began. \"They come with great hope in their hearts and all their worldly goods on their backs. Fresh from the bonds of slavery, from the benighted regions of the plantation, they come to the capital looking for liberty, but many of them don't know it when they find it. Many good friends have reached out to them in the spirit of charity and compassion, but for each kind word spoken, two harsh ones have been uttered.\"\n\nA few women nodded as they fanned themselves, their hats bobbing emphatically.\n\n\"The newly liberated have not been welcomed, but repelled,\" said Elizabeth, \"and their bright, joyous dreams of freedom are fading in the presence of that stern, practical mistress, reality.\"\n\nMurmurs of agreement went up from the pews.\n\n\"Instead of flowery paths, days of perpetual sunshine, and bowers hanging with golden fruit, the road has been rugged and the garden full of thorns,\" Elizabeth continued. \"Appeals for help too often are answered by cold neglect. Poor children of slavery, men and women of our race\u2014the transition from slavery to freedom has been too sudden for them! They are not prepared for the new life that has opened before them, and now the great masses of the North look upon their helplessness with indifference. Our white neighbors, observing those who have come to us as refugees, have learned to speak of our entire race as idle and dependent.\"\n\nA more emphatic response followed these words\u2014grim nods of assent, a scattering of amens, and encouraging calls for her to speak on.\n\n\"It is our sacred duty to help our own,\" said Elizabeth, \"and these poor unfortunates are indeed our own. Our Lord Jesus Christ instructed us to love our neighbor. He told us that whatever we do unto the least of His brethren, that we have also done unto Him. Brothers and sisters, I ask you to look toward the contraband camps and see your neighbors, and to join me in creating a society of colored people to work for the benefit of the suffering freedmen.\"\n\nA loud crash of applause greeted her proposal, and as Elizabeth nodded her thanks and returned to her pew, the minister took the pulpit and invited anyone who would be interested in founding a relief association to stay after services for its inaugural meeting.\n\nTwo weeks later, the Contraband Relief Association officially commenced, with forty women members and Elizabeth as its elected president. They planned benefit concerts and festivals, and they organized volunteers to work in the contraband camps. Their most prominent members solicited donations from well-to-do Northerners, white and colored alike. With the funds they raised, they purchased essential supplies for the camps\u2014shoes, clothing, bedding, food, tools\u2014and helped the contraband settle into permanent homes.\n\nElizabeth often visited the camps to teach sewing, reading, and simple housekeeping to the colored women struggling to care for their families in the squalid conditions. Because of her dignified, well-spoken manner and status as the president of a relief organization, officials often asked her to escort visiting dignitaries who came to inspect the camps. On one such occasion, she guided a Miss Harriet Jacobs, a former runaway slave from North Carolina a few years older than herself and a prominent, influential abolitionist. Miss Jacobs's knowing, compassionate gaze took in every detail as Elizabeth showed her around and described the contrabands' plight. \"They're willing and ready to work, but they've never worked for wages,\" Elizabeth noted as they passed a group of young men listening intently to a well-dressed colored man in spectacles who addressed them from atop an overturned crate. Suddenly one of the young men folded his arms over his chest, lifted his chin, and frowned thoughtfully up at the speaker in a manner so reminiscent of her lost son, George, that Elizabeth had to look away, pained. \"We have to teach them how to seek work for pay, how to settle on fair wages, and how to leave an employer who treats them unfairly.\"\n\nMiss Jacobs nodded, studying the young men as she followed Elizabeth past. \"And is there work to be found?\"\n\n\"Oh, indeed, yes. The war has created ample quantities of that.\" Elizabeth guided Miss Jacobs around a tent where a soldier was distributing loaves of bread to a cluster of gaunt, silently determined women, some of whom clutched young children dressed in tatters. \"Do you know about President Lincoln's new law, the one that allows him to employ 'persons of African descent' to help put down the rebellion?\" Miss Jacobs nodded. \"Then perhaps you already know that soon after the law was passed, the government set about registering contrabands and distributing rations, clothing, and wages in exchange for their labor in support of the Union. The men of the camps cut firewood, haul water, dig ditches, police hospital grounds, construct roads, build whatever needs to be built, and repair whatever needs fixing.\"\n\n\"And the women?\"\n\n\"They work as laundresses and cooks, and they take turns watching one another's children. For wages, men receive ten dollars a month. Single women hire at four dollars a month, while a woman with one child earns two and a half, perhaps three.\" Suddenly Elizabeth halted and placed a hand on Miss Jacobs's arm, bringing her to a stop. \"Make no mistake, the most unpleasant tasks always fall to the contraband, the work no one else wants to do\u2014cleaning privies, burying dead horses and mules, removing the human refuse of the hospital wards. It's important, necessary work, but it's also hard, exhausting toil, and not what they had expected of freedom.\"\n\n\"Their dreams have failed to live up to their expectations,\" Miss Jacobs remarked as she resumed walking.\n\n\"That's true,\" said Elizabeth, hurrying to catch up with her, \"but until I spent more time among them, I did not realize how extravagant their dreams had once been\u2014and how thoroughly demoralized the dreamers have become.\"\n\nSomething, some embarrassment or uncertainty, restrained Elizabeth from confiding that whenever she visited the camps, refugees would crowd around her, lamenting in their distress and pleading for relief. For some, the bitterness of their disappointment cast their memories of slavery in a false, rosy glow, so that they pined for their old lives. Elizabeth had been dismayed the first time she had heard one old woman declare that she would prefer to return to slavery in the South, where everything was familiar and her master had provided for her, rather than suffer the miserable freedom of the North. Elizabeth told herself that the old woman and others who felt as she did were not to blame for their despair, and she did not hold their words against them. After a lifetime of dependence, they knew nothing else, and the worries and cares of poverty had given them a harsh introduction to freedom.\n\n\"Are all the camps in such a state?\" Miss Jacobs asked. \"Everything seems to be in short supply except for disease, misery, hunger\u2014and the will to triumph over them, which is heartening to see.\"\n\n\"The need for the simple necessities of life and comfort is great everywhere,\" Elizabeth said, \"but I have heard that conditions are somewhat better across the river.\"\n\n\"Across the river, in Virginia?\"\n\n\"Oh, yes. Part of General Lee's estate at Arlington has been transformed into a settlement called Freedman's Village. I haven't visited it myself, but you certainly should, to make your report complete. I've heard that all of the men and most of the women are gainfully employed. They also have the benefit of plenty of exercise in the open air, something that is denied the people here.\"\n\nMiss Jacobs nodded thoughtfully. \"I'd certainly like to see that for myself.\"\n\nWhen the tour was over, Elizabeth accompanied Miss Jacobs to the street where her carriage waited. She waved farewell as the driver assisted Miss Jacobs inside, but she paused, bewildered, when the driver addressed her as Miss Brent, and she answered to the name.\n\n\"You mean you didn't know?\" one of her fellow volunteers exclaimed later, when Elizabeth told her about the curious exchange. \"Miss Harriet Ann Jacobs is also known as Miss Linda Brent.\"\n\n\"You don't mean the author?\"\n\n\"Is there any other Miss Linda Brent? When she writes, she uses a nom de plume for her own protection, but I assure you they are one and the same.\"\n\nElizabeth was so astonished she had to laugh. She had read _Incidents in the Life of a Slave Girl_ soon after its publication, and she had been greatly moved by the author's courageous escape to freedom in the North and her tireless efforts to see her two children freed as well. So many anecdotes from Miss Brent's\u2014Miss Jacobs's\u2014life were painfully familiar to Elizabeth, and the story had lingered in her thoughts long after she finished the book. She had even been inspired to compose a few memory sketches of her own, describing her childhood as a slave and the harrowing years of her young womanhood, when she had been preyed upon by Alexander Kirkland and a cruel mistress to whom she had been \"lent\" had tried to break her spirit with regular beatings. She had thought one day to share her writings with her son, but after his death, her interest in the project had waned.\n\nPerhaps she should take pen in hand again someday. Even if she never published a memoir, as Miss Jacobs had done, she might share her story with her closest friends.\n\nIn the weeks that followed, she heard that Miss Jacobs had indeed visited Freedman's Village in Alexandria, and that she had been so moved by their plight and impressed by their progress that she had decided to stay and work. For her part, Elizabeth continued her efforts in the camps in the capital, teaching necessary skills and offering guidance. It was heartening to see that while some disillusioned freedmen and women yearned for a past that had never really existed, others set themselves to the work of rising above their humble beginnings. They built sturdy cabins for their families and cultivated gardens. They saved their earnings and bought chickens and pigs. They joined churches and sent their children to the camp schools, where their teachers marveled at their swift and steady advancement. Clear-eyed, proud, and determined, they planned carefully for the future and strove forward to meet it. As the months passed, the more hesitant and anxious among them started to follow the others' example, and Elizabeth and her fellow volunteers rejoiced as their labors began to bear fruit.\n\nThe work of the Contraband Relief Association was worthwhile, never-ending\u2014and expensive. Elizabeth contributed all that she could afford from her own earnings, but she knew the organization needed to find other sources of funds if it were to thrive.\n\nAlthough Elizabeth had gone to the White House only rarely that summer since Mrs. Lincoln spent most of her time at the Soldiers' Home or traveling with Robert and Tad, she had seen enough of the inner workings of President Lincoln's administration to know that he was surely embattled and struggling. The war was going badly. General Stonewall Jackson's outnumbered Confederate forces had stymied the Union Army in the Shenandoah Valley, forcing it to draw back to the Potomac. In the Seven Days' Battles in Virginia, General McClellan had again retreated despite defeating the Confederates on three consecutive days. A Union attack on Vicksburg failed, thwarting their attempt to gain control of the Mississippi River. In Richmond, Kentucky, Confederate troops soundly defeated a smaller band of Union soldiers, taking most of them prisoner. And at a second battle at the stream called Bull Run, General John Pope suffered a disastrous loss, with more than fifteen thousand troops reported killed, wounded, missing, or captured and the rest of his army driven back across the river toward Washington by General Robert E. Lee.\n\nElizabeth's heart went out to the president, knowing full well that each defeat would weigh heavily upon him, but she was less sympathetic and utterly bewildered by another battle he fought. In July, President Lincoln had tried valiantly to persuade congressmen from the border states to enact a plan for gradual, compensated emancipation, and although the effort failed, it had assured Elizabeth that his heart was in the right place. Within weeks, however, he made statements in the press that forced her to question whether she truly understood anything about his position on abolition. On August 14, the president received a delegation of Negro leaders at the White House and made his case for the colonization of freed slaves in Africa or Central America. The presence of the colored race on the American continent had caused the war, the president asserted, and enmity between races was certain to persist after the conflict was resolved. Even when they ceased to be slaves, the colored race would never be equal to the white, and thus it would be better for both if they were separated.\n\nThe delegation left the White House unconvinced and angry, and they immediately shared the disappointing outcome of their historic meeting with sympathetic members of the press. A few days later, in a _New York Tribune_ editorial titled \"The Prayer of Twenty Millions,\" Horace Greeley furiously censured the president for failing to execute the laws of the land by not demanding that his generals immediately obey the emancipating provisions of the new Confiscation Act. He accused the president of \"being unduly influenced by the counsels, the representations, the menaces, of certain fossil politicians hailing from the Border Slave States,\" appeasing them to the detriment of the nation. No champion of the Union cause on earth believed that the rebellion could be put down unless slavery, its cause, was ended as well, Mr. Greeley insisted, warning that \"every hour of deference to Slavery is an hour of added and deepened peril to the Union.\"\n\nThree days later, President Lincoln responded to the editorial with a terse letter, which Mr. Greeley published on August 25 along with his own lengthy rebuttal. Excusing Mr. Greeley's \"impatient and dictatorial tone\" out of deference to their friendship, the president emphatically stated that his goal was to save the Union, and nothing else. \"My paramount object in this struggle is to save the Union,\" he wrote, \"and is not either to save or to destroy slavery. If I could save the Union without freeing any slave I would do it, and if I could save it by freeing all the slaves I would do it; and if I could save it by freeing some and leaving others alone I would also do that. What I do about slavery, and the colored race, I do because I believe it helps to save the Union; and what I forbear, I forbear because I do not believe it would help to save the Union.\"\n\nEveryone in Washington either read the heated exchange in the papers or heard about it from friends and neighbors. Unionists ambivalent about emancipation were satisfied by the president's stated priorities, but Elizabeth knew of no one else who was. Outraged abolitionists insisted that the president seemed incapable of understanding that the surest and swiftest way to win the war and save the Union was to emancipate all slaves everywhere and to allow them to don Union blue and take up arms in service to the nation. As for Elizabeth, she certainly wanted slavery abolished everywhere. She wanted colored men to be allowed to enlist as her son, George, had done. But she also wanted the contraband to be healthy, well fed, educated, employed, and prosperous, and she knew that no amount of wishing could make it so\u2014only hard work and careful planning. She wanted to believe that the president too had to work hard and plan carefully to bring about what he desired, and that there were good reasons for his delay, and that that was why he did not yet free the suffering slaves, although he could with a few strokes of a pen.\n\nShe wanted to have faith in him, but she hoped he would not delay much longer.\n\nIn mid-September, in a costly battle along Antietam Creek in Maryland, General McClellan managed to repulse General Lee's advance into the North. Although the president was displeased that General McClellan had allowed the battered Confederate army to withdraw to Virginia without pursuit, the stalemate was victory enough to hearten him.\n\nLess than a week later, newspapers across the North published a proclamation in which the president declared that \"on the first day of January in the year of our Lord, one thousand eight hundred and sixty-three, all persons held as slaves within any State, or designated part of a State, the people whereof shall then be in rebellion against the United States shall be then, thenceforward, and forever free.\"\n\nWhen Elizabeth read the entire preliminary Emancipation Proclamation in the _National Republican_ on September 23, her spirits soared, and she felt her faith in Mr. Lincoln renewed. Now, at last, details that had been kept secret leaked out, and the president's tactical delays made sense. Mr. Lincoln had wanted to free the slaves all along, his supporters insisted, despite his earlier statements that the war was being fought only to preserve the Union. He had written his Emancipation Proclamation weeks or perhaps months earlier and had presented it to his cabinet, but he had been obliged to wait until after a decisive Union victory before he could announce the proclamation to the American people or it would appear an act of desperation. Mr. Lincoln had also wanted to determine whether freeing the slaves was constitutional, and he had come to believe that it was indeed legal for him, by virtue of his war powers as commander in chief, to free the slaves in areas under rebellion.\n\nThe colored community and abolitionists of all races rejoiced, but in the days that followed, as the president's words were discussed and debated, their celebration was tempered by concerns that it did not go far enough. The proclamation called for the abolition of slavery only in states that were in rebellion as of January 1, 1863, so in theory, if a state agreed to return to the Union before that date, slavery would be permitted to continue there. The proclamation did nothing to free the enslaved people living within the loyal Union border states of Delaware, Kentucky, Maryland, and Missouri, as well as Tennessee and parts of Louisiana, Confederate territory that had come under Union control. What practical good did it do to declare slaves free in regions where the people did not respect Mr. Lincoln's authority and therefore were highly unlikely to obey his laws? It seemed to Elizabeth that Mr. Lincoln had emancipated slaves where the Union could not free them and kept them enslaved in places where the Union did enjoy the power to give them liberty.\n\nAnd yet, despite its weaknesses, the proclamation was worth celebrating as proof that the nation was moving forward with single-minded determination toward greater freedom for all. The old Union was gone forever. When the nation was restored, it would be a new United States.\n\nThe day after the proclamation was published, a large crowd complete with a band gathered outside the White House to serenade the president and offer speeches of praise. The president stepped outside to thank them, saying, \"I have not been distinctly informed why it is this occasion you appear to do me this honor, though I suppose it is because of the proclamation.\" The crowd applauded and shouted back that he indeed had it right. The president went on to say, with his characteristic humility, \"What I did, I did after very full deliberation, and under a very heavy and solemn sense of responsibility. I can only trust in God I have made no mistake.\" The crowd roared back their assurances that he had not.\n\nElizabeth too believed he had made no mistake, and that greater freedoms would be forthcoming.\n\nShe also believed that very soon, her Contraband Relief Association would be more necessary than ever.\n\nEarlier in September, the First Lady had left Washington to visit New York with Tad. In October, plagued by severe headaches, buffeted by storms of grief, and desiring a sympathetic companion, she asked Elizabeth to join her at the Metropolitan Hotel in Manhattan. Elizabeth eagerly agreed, not only because she had never visited New York and desired very much to see it, but also because the trip would enable her to promote her cause to a new audience. Armed with credentials and letters of recommendation, Elizabeth took the train to Manhattan, settled into the accommodations Mrs. Lincoln had arranged, and, the following morning, told the First Lady about her work.\n\n\"This is truly a noble effort,\" Mrs. Lincoln declared. She was swathed in mourning black from head to toe and had suffered unpleasant symptoms throughout her travels, but for a moment she seemed restored to her former self, energetic and bustling with plans. \"I insist upon joining your list of subscribers. Will a contribution of two hundred dollars suffice?\"\n\n\"Why, yes, Mrs. Lincoln,\" said Elizabeth, delighted by her generosity. \"Thank you very much.\"\n\n\"It is the very least I can do for all you've done for me,\" Mrs. Lincoln said, and set about writing to her husband to secure the funds.\n\nDuring her time in New York, Elizabeth circulated among the colored community, soliciting and acquiring donations to purchase much-needed supplies for the contraband camps. She was introduced to the Reverend Henry Highland Garnet, who led a meeting on behalf of her association at his Shiloh Presbyterian Church. After Elizabeth told the steward of the Metropolitan Hotel about her mission, he promptly raised an impressive sum from among the colored dining room waiters.\n\nWhen Mrs. Lincoln decided to take a side trip to visit Robert at Harvard, she asked Elizabeth to accompany her, and again she seized the opportunity. In Boston she was introduced to Mr. Wendell Phillips, a lawyer, orator, and abolitionist of such devotion that for years he had refused to taste cane sugar or wear cotton because both were produced by slave labor. He and other Boston philanthropists contributed generously and pledged her their support for her cause. She also met Reverend Leonard A. Grimes, who presided over a mass meeting at the Twelfth Street Baptist Church, where his wife established a Boston branch of the Contraband Relief Association. Reverend Grimes happened to be a friend of Mr. Frederick Douglass, whom Elizabeth had long admired, and he offered to write to the renowned abolitionist and orator on her behalf. From his lecture tour in England, Mr. Douglass not only made a sizable contribution to the association, but he also raised money for them from several English antislavery societies.\n\nThroughout her autumn travels, and with the encouragement and support of Mrs. Lincoln, Elizabeth worked tirelessly to gather donations for contraband relief, ever mindful of her son George's belief that as freed slaves, they had a special obligation to help deliver others of their race from bondage. Elizabeth agreed wholeheartedly, but she also believed that their obligation to help those in need endured even after their chains and shackles were removed. They who had already crossed the river into the land of freedom were obliged to turn and offer a hand to those who were taking their first tentative steps upon the shore, and Elizabeth resolved to do exactly that.\n\n# Chapter Eight\n\n## DECEMBER 1862\u2013MAY 1863\n\nOn the evening of December 31, Elizabeth and Emma went to Union Bethel Church to attend a freedom vigil, just like thousands of other colored folks in hundreds of colored churches all across the North. By sundown every pew was full, and while they waited for the minister to begin, the worshippers prayed, sang, and testified about their experiences as enslaved people. At ten o'clock, the minister stood before the congregation, opened his well-worn Bible, and led them in prayer before preaching a sermon about God, Satan, Mr. Lincoln, and the coming day of eternal freedom. He asked the Lord to bless the New Year's morning only a few hours away, and to bless the hand of President Lincoln when he held the pen to sign the Emancipation Proclamation. As he celebrated the imminent glorious event, he also spoke of what freedom meant for the people of their race and reminded them of the new responsibilities they must now willingly and cheerfully shoulder. Then, almost as if he believed their white neighbors and leaders of government were among them, he began to address them directly. \"Your destiny as white men and ours as black men are one and the same,\" he declared. \"We are all marching toward the same goal. Give us therefore the same guarantee of life, liberty, and the pursuit of happiness that you have enjoyed since the founding of our great nation, and make the very same demands of us to support the government as you make of yourselves.\"\n\nA chorus of amens went up from the congregation.\n\n\"Give us the vote and give us arms to fight,\" the minister said, his voice rising to a shout. \"Let us don Union blue and shoulder our rifles. Do not put vulgar prejudice before necessity and national preservation. Do not refuse to receive the very men who have a deeper interest in the defeat of the rebels than any others. Cannot black men wield a sword, fire a gun, march and countermarch, and obey orders like any man?\"\n\nThe congregation called out that yes, they could, of course they could.\n\n\"A man who wants to win a fight doesn't approach his enemy with one hand tied behind his back,\" the minister said. \"If your house is on fire, and a black man offers you a bucket of water, do you refuse it? If you are drowning, and a black man reaches out to haul you to shore, do you tread water and hope a white man happens by before you go under, or do you seize that dusky limb and live?\"\n\nA roar of approval greeted his words. Elizabeth applauded until her palms stung, the minister's stirring words lifting her to her feet and sending righteous determination pumping through her veins, as warm and strong as life.\n\nJust before midnight, the minister's tone changed again. \"At this time I want no one to pray standing up with bowed head,\" he intoned. \"No sister sitting down, with bended neck praying, and no brother kneeling on one knee, because his pants are too tight for him. I want all of us to get down on both knees to thank Almighty God for our freedom and for President Lincoln too.\"\n\nSide by side, Elizabeth and Emma knelt. From the corner of her eye, Elizabeth saw Emma's lips moving in silent prayer. Silence fell over the church, broken from time to time by a worshipper calling out to the Lord for guidance when freedom dawned, just as He had guided them through the dark night of slavery. People called out for the Lord to guide President Lincoln as well.\n\nIn reverent silence, Elizabeth echoed those prayers with all her heart.\n\nDespite her late night, Elizabeth rose at her usual hour the next morning and went to the White House to dress Mrs. Lincoln for the annual New Year's Day reception at eleven o'clock. The First Lady, pale and drawn, had already chosen a black dress and bonnet and was awaiting Elizabeth in her boudoir. \"I cannot help but think of all we have passed through since we last welcomed a New Year,\" she murmured as Elizabeth buttoned her dress up the back. \"Every day from now until spring will mark an unhappy anniversary.\"\n\nElizabeth knew she was thinking in particular of Willie's death. \"When the first anniversary of my son's passing came, I made sure to keep busy.\" In truth, the day that she suffered the worst was not the anniversary of the date he had fallen on the field but rather the day she had received the letter announcing his death. \"There's always so much to do at the contraband camps that I had no trouble finding work enough to occupy my thoughts so I couldn't dwell upon my grief.\"\n\n\"Perhaps I should do something of that sort,\" said Mrs. Lincoln. \"The soldiers always need care and attention, especially those languishing in the military hospitals. They do so seem to like it when I write letters home for them when they cannot write themselves.\" She managed a tremulous smile, and she seemed relieved to have a plan. \"I shall do as you suggest, Elizabeth, and distract myself with the needs of others. Even if it does not ease my pain, it will at least accomplish some good for the soldiers.\"\n\n\"I hope it does both,\" said Elizabeth.\n\nWhen Mrs. Lincoln finished dressing, she asked a servant if the president was ready to go downstairs, and she was told that he was in his office, writing. \"Still?\" she asked, and to Elizabeth she added, \"He's been working on the final draft of that proclamation since last night.\"\n\n\"The Emancipation Proclamation?\" Elizabeth asked, suddenly uneasy. What else could Mrs. Lincoln mean? From the moment the newspapers had published the preliminary decree, President Lincoln had been bombarded by criticism. Northern Peace Democrats declared that they were not going to fight a war to free slaves, and in the midterm elections, opponents of emancipation made their displeasure known by electing Democratic governors and congressmen. The Democrats worried that despite this apparent rejection of Mr. Lincoln's policies by the voting populace, he would go ahead regardless and sign the proclamation into law, while abolitionists and radical Republicans worried that he would not. Rumor had it that the president's cabinet had been urging him to make changes to the document up to the last minute, and that rumor appeared to be true.\n\nElizabeth could only pray that the changes the president was making\u2014even at that very moment, not very far away\u2014would not cut the heart out of the new law.\n\nElizabeth left the White House before Mr. Lincoln appeared, passing guests arriving for the reception on her way out. She wished she could have seen the president\u2014not to question him about the proclamation, because he had enough burdens without bearing her inquiries too, but to see if she could discern from his expression whether his revisions meant good or ill for her race.\n\nInstead of turning toward home, she walked to Union Bethel Church, where the vigil continued. Along the way she observed a crowd gathered outside the telegraph office, where the moment a messenger arrived from the White House to confirm that the deed was done, the news would be sent with electric speed to newspaper offices throughout the North. Other men, white and colored alike, clustered near the doorway of a printer's, where clerks waited to typeset the proclamation as soon as the official phrasing was known. Elizabeth suspected they would be waiting quite a long time. The New Year's Day reception was by custom a three-hour affair. Mr. Lincoln would stand in the East Room shaking hands and welcoming visitors\u2014foreign diplomats first, then ranking officials, and lastly the public, anyone who wished to come. After shaking all those thousands of hands, it would be a wonder if his own hand was not too worn out to hold a pen.\n\nWhen Elizabeth arrived at Union Bethel Church, the minister and several of the nearly two dozen members of the congregation keeping vigil broke off their prayers and hurried over to question her. She told them what she knew, which amounted to little more than the expectation that they were in for a lengthy wait. They nodded resignedly and resumed their prayers or hushed conversations, listening to the bells toll the passing hours, glancing up quickly at the sound of the door, settling back down to wait again.\n\nAnd then, in the late afternoon, a deacon burst into the chapel, breathless, sweat on his brow, a paper clutched in his right hand, the smell of ink still fresh upon it. \"It's done!\" he shouted. He was too winded from his run to read the proclamation aloud, so he handed it to the minister, who stepped up to the pulpit and read it slowly and emphatically, each word ringing through the church. Elizabeth felt relief and joy wash over her as with every line it became more evident that the president had made only a few changes to the preliminary proclamation she had read so often that she had learned it by heart. The list of territories under Union control had been revised due to the advances the army had made in the interim\u2014but far more significant were two new paragraphs that had not appeared in the preliminary version released the previous autumn. In the first of these, President Lincoln enjoined the people newly freed by the proclamation \"to abstain from all violence, unless in necessary self-defense.\" A murmur arose from the congregation as the meaning of the words sank in. Never before had they been permitted to defend themselves physically, to fight off a vicious beating by a cruel white master or mistress. Now they could stand their ground and fight to preserve their lives, if confronted by such a choice. It was a revelation.\n\nThe second addition was more astonishing yet. \"And I further declare and make known that such persons\"\u2014the newly emancipated\u2014\"of suitable condition, will be received into the armed service of the United States to garrison forts, positions, stations, and other places, and to man vessels of all sorts in said service.\"\n\nCheers rang out, and shouts of astonishment. In the rebellious territories, slaves were free and men of color would be allowed to fight for the Union.\n\nThe day many among them thought would never come had arrived at last.\n\nThe congregation broke into prayers of thanksgiving and songs of rejoicing, and together they paraded to the White House, where hundreds of other jubilant citizens both white and black sang and shouted praise to the president, who briefly appeared at a window to acknowledge them with a humble, solemn bow.\n\nLater Elizabeth went to the contraband camps, where the struggling freedmen forgot their cares for a while and celebrated the dawn of a new day of liberty, the first day of their new lives when they would not fear the whip, live in terror of the auction block, or dread having their children and husbands and wives sold away from them, never to be seen again.\n\nIn the evening, a hush fell over the throng as the Reverend Danforth B. Nichols, superintendent of freedmen, read the Emancipation Proclamation aloud, intoning each word with careful clarity. When he finished, and after the applause and cheers faded, he raised his hands for their attention and cautioned them that the law did not apply everywhere\u2014not in neighboring Maryland, for one. But not even that sobering counsel could diminish their joy.\n\nAt long last, freedom was at hand.\n\nAs the anniversary of Willie's death approached, Mrs. Lincoln became almost feverishly restless, and she poured her excess energies into her social obligations as First Lady. She resumed receiving callers on Saturday afternoons, and although she still dressed in deep mourning, she and the president began to venture out in the evenings again, attending recitations or going to the theater, which was Mr. Lincoln's favorite entertainment.\n\nMrs. Lincoln continued to find her greatest comfort in her two surviving sons. She brightened considerably every few months when Robert visited, and although Elizabeth privately found the young man severe and prickly, she was always happy when he came home from school for the joy his company brought his parents. His visits were not entirely cheerful, however, for he was very eager to quit Harvard and join the army, and he would always use his few days at home to press his case.\n\nMrs. Lincoln adamantly opposed his enlistment, as she often told her husband when they were alone\u2014alone except for Elizabeth, sewing nearby and quietly listening. \"We have lost two sons, and their loss is as much as I can bear, without being called upon to make another sacrifice,\" Mrs. Lincoln declared one day in mid-February when the subject resurfaced yet again.\n\n\"But many a poor mother has given up all her sons,\" Mr. Lincoln replied mildly. At that, Elizabeth half expected them both to glance sympathetically her way, but they did not. It stung to think that they might have forgotten her loss, or forgotten that she was there. \"Our son is not more dear to us than the sons of other people are to their mothers.\"\n\n\"That may be, but I cannot bear to have Robert exposed to danger. His services are not required in the field, and the sacrifice would be a needless one.\"\n\n\"The services of every man who loves his country are required in this war,\" Mr. Lincoln replied. \"You should take a liberal instead of a selfish view of the question, Mother.\"\n\nMrs. Lincoln was firmly resolved that Robert should stay in school, and since neither could persuade the other they let the matter drop for a while, perhaps wearily certain that it would not be long before Robert again implored them to permit him to enlist.\n\nAlthough the argument had ended for the moment, Mr. Lincoln's words lingered in Elizabeth's thoughts. The services of every man were required, he had said. Perhaps she was still hurt that it had not occurred to Mr. and Mrs. Lincoln that one of those mothers who had given up all her sons to the Union cause sat among them, because she felt a strange, unfamiliar desire to rebuke the president. Every man who loved his country, he had said. She was tempted to inquire whether he included men of color in that calculation.\n\nWhy did he _not_ include men of color\u2014 _every_ man of color, not only the newly emancipated slaves living in captured Southern territories\u2014if every man who loved his country was needed so desperately?\n\nElizabeth's prediction that the Contraband Relief Association would be more essential than ever after emancipation proved true. The needs of the men, women, and children in the camps were so great that even the generous contributions of concerned friends were insufficient to supply everyone with the basic necessities of health and comfort. Throughout the winter, Elizabeth gave increasingly more of herself to the association\u2014more of her time, and more, too much more, of her money. For the first time since she had established herself comfortably in Washington, she found herself struggling to make ends meet, and she could not endure the worry and uncertainty. For most of her life, she had served masters and mistresses who badly managed their finances, incurring debt and moving from one town to another in a futile pursuit of a better living. While a slave, she had promised herself that if she ever gained her freedom, she would never be so careless with money. Now it seemed that she was following her old masters down the same road of precarious economy. It made no difference that her debts sprang from altruism rather than reckless pursuit of a loftier style of living than she could afford; she would end up at the same miserable destination.\n\nShe would have confessed her troubles to her friends, but she didn't want to worry Emma needlessly about the state of the dressmaking business, and she was reluctant to upset Virginia, who was with child. Instead she brooded silently and secretly, or so she believed, until one day at the White House. She was sorting trims for Mrs. Lincoln to consider for a new dress and mulling over her worries when Mrs. Lincoln suddenly said, \"Good heavens, Lizzie, why are you scowling at those ribbons? If you dislike them so much, we'll send you to the shops for something else, although I don't know what else they'll have in black that we haven't seen already.\"\n\n\"Forgive me,\" said Elizabeth, embarrassed. \"The ribbons are fine. I confess my thoughts were elsewhere.\"\n\n\"I knew something was wrong.\" Mrs. Lincoln sat beside her and patted her hand. \"Tell me. You've been sighing and frowning for days. Are you unhappy? Indisposed?\"\n\n\"Merely worried.\" Elizabeth hesitated, but she felt cornered, and she had to offer some explanation. \"I find that I am living beyond my means.\"\n\n\"You?\" said Mrs. Lincoln, astonished. \"I don't believe it. You've never been a spendthrift. You are always reliably sensible.\"\n\nElizabeth managed a small, wan smile. \"I don't live lavishly, if that is what you mean, but lately I've been spending much of my earnings\u2014whatever does not go to food and rent\u2014on the contraband camps.\"\n\n\"Well, then...\" Mrs. Lincoln thought for a moment as if tallying the ways Elizabeth might economize. \"You must simply stop doing that.\"\n\n\"But the need is so great.\" Elizabeth steeled herself and said aloud what she had been dreading to acknowledge, even to herself. \"I must earn more money.\"\n\nMrs. Lincoln looked taken aback. \"Lizzie, dear, as much as I would love to assist you, I cannot possibly pay more for my dresses than I already do. Mr. Lincoln already believes I spend too much\u2014not on dresses, mind you, but on other things.\"\n\nElizabeth tended to agree with the president on that count. \"Oh, no, Mrs. Lincoln. I didn't mean to suggest that I intend to raise my prices. I wouldn't dream of asking any of my patrons to pay more than is fair, especially not you.\" She hesitated. \"However, I do think I need to find some way to increase my income.\"\n\n\"Could you take on new customers, sew more dresses?\"\n\nElizabeth thought back to those hard, long years in St. Louis when she had worked herself almost to collapse from exhaustion and injury trying to earn the money to buy her freedom and George's. She nearly shuddered from the memory of pain, of constant headaches and backaches and eye strain and sore fingers. \"I don't think I could possibly sew more than I already do.\"\n\n\"Well, there must be another way you could earn a little extra,\" said Mrs. Lincoln confidently. \"And there are surely other sources of funding for your association other than your wages. We simply haven't given it enough thought. We'll think of something.\"\n\nElizabeth thanked her, her spirits lifting even though she wasn't truly any closer to a solution. It was good to see Mrs. Lincoln more like her old self again, briskly efficacious and intent on a new scheme.\n\nA few days later, Mrs. Lincoln rose to meet her when she arrived and clasped Elizabeth's hands in her own. \"Elizabeth, I think I have a solution.\"\n\n\"It would be very welcome news if you do,\" Elizabeth replied, as surprised as she was relieved.\n\nMrs. Lincoln explained that the Treasury Department needed employees for the cutting room, workers who trimmed the printed sheets of currency into bills. Seamstresses were highly desirable for these posts due to their proficiency with scissors. \"The work is not arduous, and the pay is reasonable,\" said Mrs. Lincoln, clearly delighted with herself. \"Best of all, it should not take away time from your dressmaking business.\"\n\n\"I\u2014I hardly know what to say,\" stammered Elizabeth, overwhelmed. \"It sounds ideal.\"\n\n\"Good! You'll need to apply, of course, but I'm confident that with my recommendation you'll have no trouble.\" Mrs. Lincoln hesitated for a moment and seemed to steel herself. \"There's one small hurdle that you'll have to overcome first\u2014an ugly, unpleasant, dishonest hurdle.\"\n\nThe choice of adjectives was so bewildering that for a moment Elizabeth could only look at her. \"What do you mean?\"\n\n\"I think it would ease your way if you were introduced to Mr. Chase before you apply.\"\n\nShe meant Salmon P. Chase, the secretary of the treasury and the father of Mrs. Lincoln's biggest social rival, the lovely and charming Miss Kate Chase. \"But you despise Secretary Chase.\"\n\nMrs. Lincoln lifted her chin and inhaled deeply. \"I shall put aside my revulsion long enough to arrange the interview.\"\n\nAnd so she did. Later that afternoon, when Mr. Chase came to the White House to confer with the president, Mrs. Lincoln managed to be outside her husband's office as he departed. Expressing delight at the chance encounter, Mrs. Lincoln told him there was someone she would very much like him to meet, and she led him to the Lincolns' private sitting room, where Elizabeth waited.\n\nElizabeth had seen Mr. Chase in passing often, but they had never been formally introduced. She set aside her sewing and rose when Mrs. Lincoln escorted him in, keeping her features carefully composed even though the obvious annoyance and impatience in his expression disconcerted her. To his credit, he did his best to be gracious when Mrs. Lincoln introduced them. \"Mrs. Keckley, of course,\" he said, shaking her hand. \"The renowned dressmaker. My daughter speaks very highly of you.\"\n\n\"Thank you, sir,\" she replied. \"I think very highly of her as well.\" She ignored Mrs. Lincoln's raised eyebrows, which, fortunately, Mr. Chase did not observe.\n\nMrs. Lincoln got right to the point, and when she finished explaining the purpose of the meeting, Mr. Chase gave Elizabeth an appraising look and said, \"So, Mrs. Keckley, you'd like to give up sewing for a job in the cutting room? From what I've heard, that would distress a great many ladies throughout Washington.\"\n\n\"I would not give up my sewing, sir,\" she said. \"I simply wish to supplement the income from my business.\"\n\n\"There is no one in the district more qualified to wield a pair of shears than Elizabeth,\" Mrs. Lincoln broke in. \"The Treasury Department can certainly entrust the task of cutting paper in straight lines to her. She does far more complicated work than that every day.\"\n\n\"I have no doubt she does.\" Mr. Chase studied Elizabeth for a moment. \"Very well, Mrs. Keckley. If this is what you desire, you should apply to the Treasury Department. I shall tell the assistant secretary that we spoke, and after that, I'm sure your qualifications will speak for themselves.\"\n\n\"Thank you very much, Mr. Secretary,\" Elizabeth said. He gave her a nod and strode from the room, with Mrs. Lincoln hurrying after as if to thank him with the courtesy of an escort. When Mrs. Lincoln returned, she told Elizabeth that she would write to Mr. George Harrington too, to add her own recommendation to Mr. Chase's.\n\nMrs. Lincoln immediately sat down with pen and paper, and before long she was blotting the ink on her composition. \"I've assured him that you are industrious and will perform your duties faithfully,\" she said, well satisfied with her work. \"I also informed him that you will not be able to begin working for him until the middle of April, after the busy social season, when I will need you less frequently. I hope that suits you.\"\n\n\"The timing would be perfect,\" said Elizabeth. \"Thank you very much, Mrs. Lincoln.\"\n\nBeaming, Mrs. Lincoln summoned a servant to take the letter. \"Now we await their reply.\"\n\nIt would only be a matter of time, Mrs. Lincoln assured her the next day, and again the day after that. Perhaps Mr. Chase had not yet found a moment to broach the subject with Mr. Harrington. Every department was managing the best it could with the war on, but sometimes even important matters were delayed.\n\nA week went by, and another. Elizabeth's hopes diminished, but she did not complain. Mrs. Lincoln had shown her a great kindness in recommending her to Mr. Chase and Mr. Harrington, and Elizabeth did not wish to appear ungrateful. Mrs. Lincoln, however, had no reservations about complaining when she thought complaints were warranted. Indignant that the department had shown neither Elizabeth nor herself the courtesy of a reply, she declared that she would discover the reason why even if she had to go to the Treasury Department herself.\n\nElizabeth knew she had found her answer when she arrived at the White House one morning to find Mrs. Lincoln waiting pensively by the window, her expression vexed and faintly embarrassed.\n\n\"You've heard from the Treasury Department,\" Elizabeth said flatly. \"I didn't get the job.\"\n\n\"I'm very sorry, Elizabeth,\" said Mrs. Lincoln. \"All agreed that your qualifications are exemplary, but I'm told the cutting-room supervisor had some...reservations. He had his current employees' feelings to consider.\"\n\nImmediately Elizabeth understood, and she felt humiliated for ever entertaining the slightest expectation of being offered the position. \"His other employees don't wish to work side-by-side with a woman of color.\"\n\nMrs. Lincoln crossed the room and placed her hands on Elizabeth's shoulders. \"It's wrong, I know it, and I'm so very sorry. If it were up to me, you should have the post and anyone who does not care for your company would be free to resign.\"\n\n\"Regrettably, it is not up to you.\"\n\n\"No.\" Mrs. Lincoln frowned. \"It is not\u2014although I believe it _should_ be up to me, as I am the First Lady.\"\n\nNodding, Elizabeth turned away and sank wearily into a chair. She had not felt so hurt and discouraged and outraged all at once since her slavery days.\n\n\"We'll think of something else,\" Mrs. Lincoln said, offering her an encouraging smile. \"Don't give up, Elizabeth.\"\n\n\"I am not in the habit of giving up,\" she replied, but she had no idea what to do next.\n\nTo celebrate Tad's tenth birthday, Mrs. Lincoln proposed a family outing to visit the Army of the Potomac encampment near Falmouth, Virginia, on the north bank of the Rappahannock. At sunset on Saturday, April 4, in the midst of an unexpected spring snowstorm, the president, Mrs. Lincoln, Tad, and a small entourage of friends, administration officials, and trusted members of the press boarded the steamer _Carrie Martin_. They set off south down the river, passing Alexandria and Mount Vernon, where the ship's bells tolled a salute to George Washington in keeping with the time-honored custom of the river. Blinding, swirling snow forced the captain to put in to a sheltered cove for the night, and although the storm worsened overnight, the next morning they continued on to the crowded supply port of Aquia Creek, arriving on Easter Sunday. The party transferred to a flag-adorned train and chugged through fierce winds past huge snowdrifts to Falmouth Station, where General Hooker welcomed them.\n\nThe weather delayed their official plans for a day, but in the week that followed, the president and First Lady reviewed the troops, enjoyed a grand parade of cavalry, visited patients and staff in field hospitals, and ventured down to the shore of the Rappahannock, close enough to see Confederate troops waving to them from the other side. The family returned to Washington on April 11, their spirits much improved by the time away from the city. When Elizabeth came to the White House the next day, she observed that Mr. Lincoln seemed heartened by the readiness, strength, and spirit of the troops he had met, while Mrs. Lincoln appeared much more relaxed and contented\u2014despite one irritating incident in a receiving line in which a vivacious, flame-haired equestrienne who had married a Prussian prince and acquired the title Princess Salm-Salm had ardently kissed the president rather than shaking his hand. Delighted, several other ladies had been emboldened to kiss him too, until he had been all but undone by an assault of kisses. Princess Salm-Salm had laughingly explained that General Sickles had put the ladies up to it, in hopes that their kisses would cheer the gloomy president. Mrs. Lincoln had been greatly affronted, but she refused to let those silly women ruin the excursion for her. What she always wanted above all else was time in the company of her husband and children, and the brief trip with Mr. Lincoln and Tad had clearly done her a world of good.\n\nMrs. Lincoln had returned to Washington with a new plan to help Elizabeth out of her troubling circumstances. In conversation with the staff at a field hospital, she had been informed about a new law that entitled Elizabeth, as a widow whose only son had been killed in action, to a government pension. \"All you need to do is apply at the United States Pension Office,\" Mrs. Lincoln assured her. \"In my opinion, it is the very least the nation can do to express our gratitude for your sacrifice.\"\n\nElizabeth thanked her, and as soon as she could, she went to the appropriate office and collected the necessary forms. The clerk looked askance at her as he explained the various signatures and affidavits she would need to collect, but she was used to such looks from white men who found her in places where they did not expect to see a woman of color, and so she ignored him. It was not until later, when she spoke with several of the men whom she asked to testify on her behalf, that she realized she was again suspended in that strange middle ground between the white world and the black, and that circumstances not of her making would prevent her from obtaining the pension she so desperately needed.\n\nEven though a few scattered contraband regiments existed and it seemed likely that someday soon colored men would be welcomed into their own army regiments, George had enlisted nearly two years before by passing himself off as white. He had been able to do that, but one glance at Elizabeth would reveal that she was not a white woman. It would be difficult enough to explain how a black woman had given birth to a white man, but it was not even the most painful problem: Her son was, in the eyes of the law, illegitimate. George was the product of rape, the offspring of a liaison she had never desired, and yet, due to the curious morality surrounding race and marriage and the \"peculiar institution\" of slavery, Elizabeth would suffer for what would be perceived as her sexual indiscretion.\n\nDiscouraged, Elizabeth feared that would be the end of it, but just when her prospects were at their bleakest, Miss Mary Welch, a longtime patron whom she had first met in St. Louis, drew the story out of her. \"I believe I know someone who can help you,\" the kind woman said, indignant that Elizabeth should be caught in such a tangle of bureaucratic unfairness. \"Leave it to me.\"\n\nA few days later, Miss Welch introduced Elizabeth to her friend Illinois congressman Owen Lovejoy, a staunch abolitionist with a great deal of experience unsnarling the knots of racially biased laws. He gladly agreed to help her prepare her application, and when he was obliged to return to Illinois before the task was done, his younger brother, Joseph, took over. A third brother, Elijah, the eldest, had been shot and killed about twenty-five years earlier while trying to defend the printing press of his abolitionist newspaper from an angry pro-slavery mob.\n\nBoth surviving brothers were present when Owen Lovejoy summoned her to his office to discuss his solution to her dilemma. \"The crux of the matter is that the Union Army believes your son is a white man,\" said Owen Lovejoy. He had strong, intelligent features, a keen gaze, and dark hair receding from a bold brow. \"We need to show that George was, in fact, the legitimate mulatto son of you and Mr. Alexander Kirkland.\"\n\nElizabeth felt a sudden stir of apprehension. \"How will we do that?\"\n\nThe brothers exchanged a look, and then Joseph Lovejoy spoke. \"You will testify that you and Mr. Kirkland were married.\"\n\n\"But...we never were,\" she said faintly. The very thought of it made her sick to her stomach. \"Even if the idea were not wholly, utterly repugnant to me, I could never place my hand on the Holy Bible and swear to something that was not true.\"\n\nThe brothers exchanged another glance, and Elizabeth knew at once that they had anticipated her reaction. \"Mrs. Keckley,\" said Owen Lovejoy gently, \"we have studied the matter backward and forward, and it is our professional opinion that this is the only way you will be able to obtain the pension.\"\n\n\"You _are_ the widowed mother of a soldier killed in the line of duty, are you not?\" asked Joseph Lovejoy.\n\n\"Of course,\" Elizabeth replied, taken aback.\n\n\"And does not the law declare that widowed mothers who have suffered this loss are entitled to the pension?\"\n\n\"Yes.\"\n\n\"Then I hope you agree that it would be an injustice to deny you that pension simply because other, lesser laws weigh against you.\"\n\n\"Your son did not claim to be a white man,\" said Owen Lovejoy. \"He stood in line to enlist, and when he put his signature to paper, the recruiters assumed he was a white man. They not only accepted him, they were glad and eager to have him.\"\n\n\"Your son did not lie to enlist,\" his brother added, as if to reassure her. Indeed, it did comfort her to have her son's honesty confirmed.\n\n\"As for the other complication...\" Owen Lovejoy hesitated. \"Forgive me for addressing a subject that I can only assume must bring you great pain. The circumstances of your son's conception are not your fault. It was not your choice to bear Mr. Kirkland a child out of wedlock.\"\n\nIt was not her choice to bear him a child at all, Elizabeth almost retorted, but she could not utter the words that would sound, to anyone who did not know her, like a rejection of her son, like an admission that she had not loved him. The greatest truth of her life was that she _had_ loved George, fiercely and proudly, loved him as strongly and passionately as she had hated his father.\n\n\"It would be a grave injustice, therefore,\" Owen Lovejoy continued, \"to hold the letter of the law above that which is truly right\u2014and it is most certainly right for you to receive the pension that your son's patriotic sacrifice earned for you.\"\n\n\"And if you _are_ denied the pension,\" his brother added, with a defiant edge to his voice, \"it will only help our political cause.\"\n\nElizabeth wasn't quite sure what cause he meant\u2014abolition? Equality? Republicanism?\u2014but what she needed at the moment was something that would do her immediate, practical good, not intangible political victories that might benefit her in years to come. \"You both speak like lawyers,\" she said shakily. \"You don't need to coax me along with sweet words that poorly conceal what it is you want me to do. I know you want to help me, so please, let's speak plainly with one another. Your recommendation is that I should lie in order to secure my pension, isn't that right?\"\n\n\"Well...\" Joseph Lovejoy hesitated. \"Yes. It is wrong to lie, but in this case, a small lie would serve the greater truth of justice.\"\n\n\"And you believe that only by lying will my application succeed?\"\n\n\"I know it,\" said Owen Lovejoy in a voice that allowed no room for doubt.\n\n\"Then that is what I shall do.\"\n\nThe Lovejoy brothers nodded, satisfied that they had persuaded her. She wondered if they understood precisely which of their carefully measured phrases had ultimately convinced her. Owen Lovejoy had said\u2014and the law maintained\u2014that her son had earned that pension for his mother by giving his life for the Union. That was what did it. She would not cast aside George's last gift to her out of deference to foolish laws governing race and marriage that had not been designed with colored people in mind and that had never helped a single slave, not one, not ever. Why should those laws apply to her now when they had never applied before, during the long, difficult years when they might have eased her suffering?\n\nShe trusted that the Lord would forgive her for the lies she intended to tell, freely and without reservation.\n\nIn the middle of April, Elizabeth stood before Justice of the Peace William S. Clary and two respected witnesses to offer her sworn testimony. She was the widowed mother of George W. D. Kirkland, private, First Missouri Volunteers, Company D, killed on August 10, 1861, in the Battle of Wilson's Creek. George's father was her first husband, Mr. Alexander Kirkland, a white man, who died when George was only eighteen months old. The Garland family later took mother and son to St. Louis, where Elizabeth married James Keckley, but for the past three years they had lived apart and Elizabeth received no support from him. She had purchased her freedom and her son's for twelve hundred dollars and infinite toil and labor, and she had relied upon her son's contributions from his wages to help pay back the debts she had incurred by doing so. She still owed two hundred dollars upon that debt, as well as one hundred more yet unpaid for George's education. George had left no widow or minor child under the age of sixteen. Elizabeth had not, in any way, been engaged in, or aided or abetted, the rebellion in the United States, and she had willingly laid her son's life upon the altar of her country.\n\nAny lingering guilt she might have felt about lying was dispelled when two respected ministers, Daniel A. Payne, bishop of the African Methodist Episcopal Church, and John M. Brown, clergyman, separately swore that they had known her for ten years and that every word of her testimony was true. Elizabeth wondered how anyone could possibly be fooled by that, when she had never denied\u2014and it could be easily proven\u2014that she had not come to Washington City until 1860.\n\nBut Elizabeth did so solemnly swear, and the documents were signed and the seals affixed, and within a few weeks her application was approved. From that day forward, and for the rest of her life, she received a pension from the federal government. As compensation for her loss it was hopelessly inadequate\u2014eight dollars a month, later raised to twelve, for the life of her son\u2014but it often made the difference between contentment and worry, so she was grateful.\n\nThe fair weather of spring helped to lift spirits a trifle throughout the beleaguered city, but to Elizabeth it seemed as if Mr. Lincoln was becoming gaunt and aged beyond his years by the cares of war. The Union Army was in such great need of soldiers that Congress had been obliged to pass a draft law applying to married white men between the ages of twenty and thirty-five, and single white men up to the age of forty-five. Black men were exempt because they were not considered citizens, and anyone who could afford to hire a substitute could avoid service. The unfairness of a system that sent poor men to war and let rich men buy their way out created great anger and resentment throughout the North.\n\nAt the end of April and into May, Union and Confederate forces met at Chancellorsville, a crossroads town west of Fredericksburg in Spotsylvania County, Virginia. Although the Confederates lost their celebrated general Stonewall Jackson to friendly fire, General Lee won a decisive victory in what became the bloodiest battle the war had yet manifested, with over seventeen thousand casualties for the North and nearly thirteen thousand for the South. Devastating news and thousands of wounded flooded Washington City, turning it again into one expansive, sprawling field hospital, with the stench of death and decay pervading the air so that Elizabeth kept her windows closed day and night in a futile attempt to block out the smell and the images of battlefield horror it evoked.\n\nA few weeks later, in mid-May, General Grant led his troops across the Big Black River in Mississippi and on to Vicksburg, but two direct assaults on the city were repulsed, and the Union forces settled back to the dispiriting, tedious business of the siege.\n\nAlthough Mrs. Lincoln did not like her husband to walk about Washington City without a military escort because of the frequent letters he received threatening assassination, the president often went alone back and forth between the White House and telegraph office in the War Department to collect the most recent bulletins from the field. Once, when Elizabeth was with Mrs. Lincoln in the family's sitting room fitting a dress, the president returned from one of those solitary walks, his step slow and heavy, his expression careworn and unhappy. Like an exhausted boy he flung himself upon a sofa and shaded his eyes with his long, bony hands, a complete picture of dejection.\n\nMrs. Lincoln exchanged a quick, worried look with Elizabeth and asked, \"Where have you been, Father?\"\n\n\"To the War Department,\" he replied shortly.\n\nMrs. Lincoln did not let his tone discourage her. \"Any news?\"\n\n\"Yes, plenty of news, but no good news. It is dark, dark everywhere.\"\n\nMrs. Lincoln sighed softly, and Elizabeth's heart went out to the president. When she tried to imagine what it must be like to bear his burdens, she did not know how one man could carry so much upon his shoulders.\n\nShe continued adjusting the fit of Mrs. Lincoln's bodice, watching from the corner of her eye as Mr. Lincoln stretched out one of his long arms, took a small Bible from a stand near the head of the sofa, opened it, and leafed through the pages almost idly until a passage caught his eye. Soon he was engrossed in reading, and after about fifteen minutes had passed in near silence, Elizabeth glanced his way and observed that his expression seemed greatly changed, almost cheerful. The dejected frown had vanished, and a new resolution and hope lit up his countenance. Curious, Elizabeth wondered what he had read to have discovered so much comfort so quickly. Murmuring an excuse about a misplaced pincushion, Elizabeth quietly walked around the sofa pretending to search for it, but really stealing a glimpse of the open Bible, and she discovered that the president was reading from the Book of Job. Greatly moved, Elizabeth could almost imagine hearing the Lord speaking to him in a thunderous voice from the whirlwind of battle: \"Gird up thy loins now like a man: I will demand of thee, and declare thou unto me.\" The sublimity of witnessing the ruler of a mighty nation turning to Holy Scripture for comfort and courage, and finding both in his darkest hour, brought tears to her eyes, and she was obliged to quickly compose herself before returning to Mrs. Lincoln's side.\n\nMrs. Lincoln too looked to the Bible for solace and guidance, but after Willie's death, she delved more and more often into the mysteries of spiritualism. She consulted mediums and invited them to hold s\u00e9ances in the White House, hoping to communicate with the spirits of her dead children. Not only did the most gifted spiritualists help her to see and speak with Willie and Eddie, Mrs. Lincoln affirmed, but they also warned her about daring new offensives General Lee was planning and treachery within her husband's cabinet. \"I have been warned that there is not one man among them except the postmaster general who would not stab my husband in the back if the opportunity arose,\" Mrs. Lincoln told Elizabeth one morning after a s\u00e9ance. While Elizabeth believed that some of the spiritualists certainly did possess unexplained powers, she knew that long before any message from the beyond told her so, Mrs. Lincoln already had been convinced that the cabinet secretaries were motivated by avarice and the lust for power rather than love of country and loyalty to her husband.\n\nIn matters of spiritualism, Elizabeth also encouraged the First Lady to develop a little healthy skepticism. Unscrupulous frauds were known to prey upon widows and bereaved parents, interpreting the ringing of an unseen bell or mysterious tappings as messages from their departed loved ones, coded messages of undying love and reassurance, when in fact the mediums themselves made the noises with devices concealed beneath the table. The previous summer, Mrs. Lincoln had been taken in by just such a scoundrel, a stout, debonair British gentleman who called himself Lord Colchester and claimed to be the illegitimate son of an English duke. He earned quite a reputation in Washington for his prowess, so Mrs. Lincoln invited him to the White House for a demonstration. Sometime after that, gossip about the alleged nobleman raised Mr. Lincoln's suspicions, so he asked Dr. Joseph Henry, head of the Smithsonian Institution, to investigate. Dr. Henry in turn enlisted the aid of Noah Brooks, a correspondent for _The_ _Sacramento Union_ and President Lincoln's closest friend within the press. Mr. Brooks attended one of Lord Colchester's s\u00e9ances in a darkened room at the residence of a true believer, but just when the unseen spirits were at their most loquacious, Mr. Brooks reached beneath the table, seized a hand that was rapping upon a drum with a bell, and shouted, \"Strike a match!\" Before any of the astonished participants could comply, the drum struck Mr. Brooks a strong blow across the head. When someone finally lit a lamp, it illuminated a startling scene\u2014Mr. Brooks, blood trickling down his face, grasping the arm of Lord Colchester.\n\nIn the uproar that followed, the Englishman slipped away, but a few days later, Mrs. Lincoln summoned Mr. Brooks to the White House. She had received a note from Lord Colchester demanding that she arrange a War Department pass for him and hinting at blackmail if she did not obey. Together Mrs. Lincoln and Mr. Brooks arranged for the false clairvoyant to come to the White House, ostensibly to collect his pass, and when he arrived Mr. Brooks confronted him as a swindler and a humbug, and ordered him to leave Washington immediately or be thrown into the Old Capitol Prison. Lord Colchester promptly fled the city and did not trouble Mrs. Lincoln again. When Mrs. Lincoln later told Elizabeth the entire story, she laughed so merrily that one would almost think she had been amused by a farce rather than deceived by a charlatan. But even though Lord Colchester had been discredited, Mrs. Lincoln retained her faith in other spiritualists and clung to her belief that her dear sons Willie and Eddie did indeed appear to her, not only in the midst of a s\u00e9ance, but beside her bed at night, waking her from sleep to assure her they were together and happy in heaven.\n\nNot all of Mrs. Lincoln's messages from the spirit realm offered comfort. Sometimes when Elizabeth arrived at the White House, she found Mrs. Lincoln shaken and pale, with dark shadows beneath her eyes. Those were the mornings Mrs. Lincoln had been visited the night before by terrifying premonitions that her husband would be murdered. They were only nightmares, Elizabeth tried in vain to assure her, terrible to endure but still nothing more than unpleasant dreams. Mrs. Lincoln was not convinced, and even Elizabeth had to admit she had good reason. It was wartime. President Lincoln moved freely about in a city full of strangers and secessionists. He regularly received anonymous letters full of wild threats and promises that he did not have long to live. Sometimes his friends became so worried on his behalf that they took turns sleeping in the White House to watch over him. For his part, although he too sometimes endured sleepless nights and suffered foreboding dreams, Mr. Lincoln ignored the dangers, or laughed off the threats, and only reluctantly conceded to the guards assigned to escort him when he traveled longer distances from the mansion.\n\nElizabeth was bent over her sewing in the Lincolns' sitting room one afternoon when Mrs. Lincoln broke off their conversation at the sight of Mr. Lincoln putting on his overshoes and shawl. \"Where are you going now, Father?\" she asked.\n\n\"I am going over to the War Department, Mother,\" he said resignedly as he stood and adjusted his collar, \"to try and learn some news.\"\n\n\"But, Father, you should not go out alone. You know you are surrounded with danger.\"\n\n\"All imagination,\" he said. \"What does anyone want to harm me for? Don't worry about me as if I were a little child, for no one is going to molest me.\" Utterly unconcerned, or perhaps simply fatalistic, he departed for his solitary walk, closing the door behind him.\n\nMrs. Lincoln could not emulate his studious confidence. She seemed to read impending danger in every rustling leaf, in every whisper of the wind, and Elizabeth could not honestly say that she was wrong to do so.\n\n# Chapter Nine\n\n## JUNE\u2013DECEMBER 1863\n\nFor months, President Lincoln had rejected appeals to allow freed slaves and freeborn black men to enlist as combat soldiers, but over time, Elizabeth observed, his resistance appeared to be weakening. Perhaps the rising pressure of demands from radical Republicans, appeals from respected black leaders like Frederick Douglass, and, most of all, the army's overwhelming need for more troops had forced him to take a more pragmatic\u2014and less prejudiced\u2014view of the matter. One of the White House doormen once told Elizabeth that he had witnessed an occasion when Vice President Hamlin introduced the president to his son and several other white officers who had volunteered to command black soldiers in combat. \"The president seemed deeply moved,\" the doorman confided, \"and then he said, 'I suppose the time has come.'\"\n\n\"If you ask me, 'the time' arrived months ago and has been sitting idle outside the president's office ever since,\" said Elizabeth dryly. It was unlike her to say anything critical of the president, whom she admired and respected greatly, but on this subject, like emancipation, she found herself made impatient by his inexplicable, unhelpful delays. But as always, she told herself he likely had good reason and that he would take action when the time seemed right.\n\nThen, in late May, the War Department issued General Order Number 143, which established the Bureau of Colored Troops to recruit and train black soldiers. By the middle of June, the First Regiment United States Colored Troops was organized in Washington and began training on Analostan Island in the Potomac River near the Virginia shore, a ferry ride away from Georgetown. Their presence brought a great swell of pride to Washington's Negro community, and to the refugees still living in the contraband camps. Whenever Elizabeth went there to teach and to distribute necessities, she heard men and women referring to them as \" _our_ soldiers.\" They had become every colored family's sons and brothers.\n\nAs the June weather grew sultry, Mrs. Lincoln traveled north, as she had in summers past, to escape the heat and the other afflictions of summer. She and Tad spent a week visiting her friend Sally Orne in Philadelphia before returning to Washington to prepare for the family's annual retreat to the cool, quiet haven of the Soldiers' Home. While Mrs. Lincoln was away, Virginia gave birth to a beautiful baby girl, whom she and Walker named Alberta Elizabeth. The delivery was blessedly easy, and both mother and child were in good health and spirits. Elizabeth was proud and honored when the Lewises asked her to be Alberta's godmother, a role she gladly accepted.\n\nIt was an unexpected stroke of good fortune that the birth coincided with Mrs. Lincoln's absence, freeing Elizabeth to care for Virginia as she recovered and to help Walker look after the older children. Elizabeth did not expect to see the First Lady again until later summer or perhaps early autumn, but on July 2, as Washington stirred anxiously with rumors that General Lee was on the move toward the Potomac, Elizabeth was shocked by news that Mrs. Lincoln had been badly injured in a carriage accident.\n\nElizabeth immediately raced to the White House, where she found Mrs. Lincoln in bed with her head bandaged, the able Nurse Pomroy nearby. \"Mrs. Lincoln,\" she exclaimed, hastening to her side. \"What happened? Are you all right?\"\n\n\"Oh, Elizabeth, you're so good to come to me. I'm somewhat bruised but I'm really quite fine.\" Even so, Mrs. Lincoln looked pale and anxious. She waited for Nurse Pomroy to step out of the room before adding, \"Mr. Lincoln and I were at the Soldiers' Home. He was going to take the carriage into the city, but at the last minute he decided to depart earlier on horseback instead, so I came along in the carriage later, alone.\"\n\n\"And Tad?\" prompted Elizabeth, worried. \"Was he riding with you?\"\n\n\"No, thank heavens. He remained behind with friends.\" Mrs. Lincoln took a deep, shaky breath. \"I was riding along as pleasantly as always when suddenly, without warning, the horses bolted. The carriage absolutely broke apart. I had to jump to safety or I would have\u2014well, I hardly dare to think of what could have happened.\"\n\n\"Did you injure yourself in the carriage or in the jump?\"\n\n\"Neither,\" said Mrs. Lincoln with a wry twist to her mouth. \"In my clumsy landing. I stumbled and fell, and I hit my head on a rock. It bled a frightening amount and I confess I was a bit dazed afterward, but thankfully, I happened to fall almost directly in front of an army hospital, of all things, and some dutiful soldiers serving there ran to my aid.\"\n\n\"Thank heavens they were nearby.\"\n\n\"Thank heavens, indeed.\" Mrs. Lincoln reached for her hand and pulled her to a seat on the edge of the bed. \"Elizabeth, I'm frightened. It was no accident.\"\n\nElizabeth felt a chill of dread. \"What do you mean?\"\n\nMrs. Lincoln's voice trembled, and her hand tightened around Elizabeth's. \"After the horses were brought under control, it was discovered that someone had unscrewed the bolts holding down the driver's seat of the carriage. When the seat came loose, it frightened the horses. That's why they jumped and ran.\"\n\n\"How terrible,\" Elizabeth exclaimed. \"Someone deliberately tried to injure you?\"\n\n\"Not me\u2014Mr. Lincoln. He was supposed to be riding in the carriage, remember?\"\n\n\"Oh, my goodness.\" Elizabeth felt faint with alarm. \"Do you have any idea who's responsible?\"\n\nMrs. Lincoln shook her head. \"My husband has entire states full of enemies. It could have been one of the authors of any of those dreadful letters he's received. We should have been keeping a list of their names. It could have been a Southern spy chosen by Jefferson Davis himself.\"\n\nElizabeth could not imagine Mr. Davis arranging for Mr. Lincoln's murder, but she had known him in peacetime, not at war. He could have changed.\n\n\"After this, Mr. Lincoln simply must agree to increase his guards.\" Mrs. Lincoln settled back weakly against her pillow. \"I simply must insist upon it and he must agree. You can imagine how terrible and guilty he feels knowing that I was injured in his stead.\"\n\nElizabeth managed a helpless laugh. \"Then I suppose now is the time to press your advantage.\"\n\nMrs. Lincoln joined in, her laughter weaker than Elizabeth's and bearing a slight note of hysteria. \"I hadn't considered that. Perhaps today I should also confess my debts.\"\n\n\"Oh, dear me, no. The shock would injure him far worse than any carriage accident.\"\n\nMrs. Lincoln smiled, too fatigued to laugh anymore. She closed her eyes, and Elizabeth held her hand until she drifted off to sleep.\n\nBy the next morning, Mrs. Lincoln felt much improved, and so she continued to prepare for a glorious Independence Day celebration. She had agreed to a grander, more elaborate event than she had allowed since Willie's death, with the Marine Band scheduled to perform on the White House grounds, a large viewing stand erected on the Mall, firecrackers, and a parade of regiments to march alongside twelve councils of the Masons, Odd Fellows, and Union League. The spectacle was meant to inspire patriotism and boost morale, but with increasing ire over Congress's new draft measures and worries about the Confederate army's bold advances, some administration officials wanted to cancel the celebration. All of Washington was in a state of nervous agitation as General Lee led his army across the Potomac and into Pennsylvania, and as military engagements in that state seemed increasingly likely, fears that Washington sat helpless and undefended soared. Nevertheless, when the president's private secretary shared these worries with Mrs. Lincoln, she told him stoutly that her husband was confident that the Union forces would halt Lee's advance and that they should not let their resolve crumble or it would dishearten the entire city.\n\nSo the celebration went on as planned\u2014but Mrs. Lincoln did not join in. On the morning of July 4, while her husband anxiously awaited reports from Vicksburg, where Grant was attempting a new assault on the city, and from a small farm town in Pennsylvania called Gettysburg, where Union and Confederate forces had finally met, Mrs. Lincoln's injury took a sudden turn for the worse. A messenger came for Elizabeth while she was out with Emma enjoying the festive day, so it was not until midafternoon that she realized she was needed and she hurried off to the White House.\n\nShe found Mrs. Lincoln in bed, fretful and feverish. Earlier that day, Nurse Pomroy had discovered that her wound was badly infected, and she had been obliged to reopen it to drain the laudable pus. The injury became inflamed, and Mrs. Lincoln was in significant distress.\n\nElizabeth sat with her that day and the next, and at times her fever became so dangerously high that Mr. Lincoln sent a telegram to Robert to urge him to return home at once. The president paced from his wife's sickroom to the telegraph office and back, but the welcome news of the Union victory at Gettysburg and promising but unconfirmed reports of Grant's success at Vicksburg distracted him only momentarily from his wife's decline.\n\nElizabeth assisted Nurse Pomroy as best she could, but she could do frustratingly little more than bathe Mrs. Lincoln's brow, read to her, assure her that all would be well, and pray. Within days General Grant's success in Vicksburg was confirmed and it seemed that the momentum of the war had finally shifted in favor of the Union, but Mrs. Lincoln was no better and Robert had still not arrived, nor had he sent word of when he might be expected. Worried anew when no one could confirm his eldest son's whereabouts, Mr. Lincoln telegraphed Robert again and urged him to make haste to Washington. Then draft riots broke out in New York, where Robert was last known to have been; over the course of several terrifying days, mobs of white men, most of them Irish immigrants, attacked draft offices, looted shops, destroyed black-owned businesses, burned to the ground an orphanage for colored children, and violently attacked people of color in the streets, brutally murdering more than one hundred. Finally the New York State Militia and other troops were sent in to restore order, but the death and destruction sent waves of shock throughout the North and sparked fears that the draft would evoke similar violence in other cities.\n\nTwo weeks after Mrs. Lincoln's accident, Mr. Lincoln seemed so overwhelmed by worry and strain that Elizabeth began to fear for his health as much as for his wife's. While most of the North celebrated the recent military victories, Mr. Lincoln was preoccupied with calamities\u2014New York City was in a shambles, General Meade had failed to pursue General Lee and had allowed him to escape, Robert Lincoln was unaccounted for, and Mrs. Lincoln was fading away, struck down by a blow meant for himself.\n\nFinally, days later, Robert arrived home, greatly easing his father's worries. If the young man offered an explanation for his lengthy absence and his silence, Elizabeth never learned what it was. Under Nurse Pomroy's attentive care, Mrs. Lincoln gradually improved. Her fever relented; her wound closed. To Elizabeth's relief, every day she seemed a little stronger, but the midsummer heat and humidity impeded her recovery. When malaria began to spread through the city, it was decided that Mrs. Lincoln could not endure the climate of Washington in her condition and should relocate to the north until she regained her health. Mr. Lincoln quickly made the necessary arrangements: As soon as she was strong enough, Robert would escort his mother and brother to a resort in Manchester, Vermont, nestled in the sublimely beautiful Green Mountains and famed for the healing properties of its natural mineral waters. It sounded to Elizabeth like a cool, restful haven, the ideal place for Mrs. Lincoln to convalesce.\n\nElizabeth packed Mrs. Lincoln's trunks for her, and on the day of her departure, Elizabeth came to the White House to see her off. \"You should come with me,\" Mrs. Lincoln urged again, as she had several times before. \"I feel so much better in your company.\"\n\nElizabeth was tempted, but she had accepted too much work from her other clients to pack a bag and quit the city on such short notice. \"I wish I could,\" she said, \"but my commitments oblige me to stay behind.\"\n\nMrs. Lincoln sighed her acceptance, and soon the carriage departed with her and her sons to meet the train that would speed them northward.\n\nAs Elizabeth walked along the curved path in front of the White House on her way home, she heard a man behind her remark to a companion, \"The Hellcat gave the Tycoon quite a scare, didn't she?\"\n\nStung, Elizabeth halted, drew herself up, and turned to face them. \"If you please,\" she said crisply, looking up at the men, quietly furious, \"do not ever again use that loathsome word to describe Mrs. Lincoln in my presence.\"\n\nShe recognized the stupefied men gaping back at her as two very junior secretaries within two obscure departments. She did not even know their names, but she would bet they knew hers.\n\n\"I beg your pardon, ma'am,\" the younger of the two managed to reply. The men quickly tipped their hats and shamefacedly hurried past her down the sidewalk.\n\nHer anger spent, Elizabeth watched them scurry off, and the ridiculousness of the scene obliged her to smother a laugh. Perhaps no woman of color had ever addressed them like that before. They had better learn to mind their manners or they would be hearing from her again.\n\nSuddenly she realized that she had objected to \"Hellcat\" but not the equally disrespectful, if less spiteful, \"Tycoon,\" and then she truly did laugh.\n\nMrs. Lincoln stayed away from Washington for nearly two months. In all that time, Elizabeth sewed for other ladies and dedicated the rest of her hours to the Contraband Relief Association, raising funds from abolitionists and well-to-do people of color from throughout the North and teaching sewing and other practical skills to the women and girls residing in the camps. She took on a few promising young freedwomen as apprentice seamstresses, and with their earnings they were able to afford rooms in pleasant boardinghouses and escape the camps, which seemed perpetually squalid despite the bedding and other small comforts Elizabeth and the other volunteers provided for them. Her young assistants bloomed in their new lives, mentored by Elizabeth and Emma, their eager guides to all things a young woman of color needed to know in order to successfully navigate Washington City.\n\nDuring Mrs. Lincoln's absence, the president was preoccupied with the hard business of war. The Union victories at Gettysburg and Vicksburg seemed, to him, to mark a turning point, but the weeks following those costly triumphs were not without setbacks. The valiant but ultimately bloody and unsuccessful attack of the Fifty-fourth Massachusetts upon the Confederate stronghold Fort Wagner near Charleston was especially heart wrenching for the Negro community, for the Fifty-fourth Massachusetts was one of the first official colored regiments and a great source of pride for their race. Their casualties in the fierce assault were shockingly high, so reports of their courage and heroism were bittersweet. Elizabeth was proud that the men of her race had acquitted themselves so nobly and had inspired more colored men to enlist, but she mourned for them, and prayed for their wives and mothers.\n\nShortly before Mrs. Lincoln returned to the capital, Elizabeth heard other sad news from the battlefield\u2014sad for the Lincolns personally, at any rate, although not for the Union. Mrs. Lincoln's brother-in-law, the Confederate general Ben Helm, husband of her beloved half sister Emilie, had been killed at the Battle of Chickamauga in Georgia. Years before, on the advent of war, Mr. Lincoln had offered his favorite sister-in-law's husband a generous commission as paymaster of the Union Army, but Mr. Helm had declined and had enlisted with the Confederates instead. After Shiloh, he had been promoted to brigadier general and had gone on to lead the famed \"Orphan Brigade,\" Kentucky's most celebrated infantry unit. Now he was gone, and their darling \"Little Sister\" was in mourning, but the Lincolns could not publicly grieve for a rebel.\n\nSoon thereafter, only a day or two after Mrs. Lincoln returned home, word spread that her youngest half brother, Captain Alexander Todd, had been killed the day before his brother-in-law General Helm while serving as his aide-de-camp at the Battle of Baton Rouge. Aleck had been only an infant when Mrs. Lincoln had left home, but the red-haired, cheerful boy had been everyone's favorite, and Elizabeth knew Mrs. Lincoln was very fond of him.\n\nWhen Elizabeth next received a summons to the White House, she departed at once, eager to see her best patron again after so many weeks apart. Mrs. Lincoln greeted her warmly, clasping her hands and smiling, and declared theirs the happiest of reunions. Elizabeth was relieved to see that Mrs. Lincoln's health seemed almost entirely restored to her, but an air of sadness lingered about her that Elizabeth could only assume sprang from her recent losses.\n\nThey had been discussing new dresses for the upcoming social season for only a few minutes when Mrs. Lincoln suddenly said, \"Elizabeth, I have just heard that one of my brothers has been killed in the war.\"\n\nElizabeth was taken aback by the lack of emotion in her words. \"I also heard the same, but I hesitated to speak of it, for fear the subject would be a painful one to you.\"\n\n\"You need not hesitate.\" Mrs. Lincoln attempted a matter-of-fact smile, but her lips trembled. \"Of course, it is but natural that I should feel for one so nearly related to me, but not to the extent that you suppose.\"\n\nElizabeth did not know what to say. \"Indeed?\"\n\nMrs. Lincoln clasped her hands in her lap and studied them. \"Aleck made his choice long ago. He decided against my husband, and through him against me. He has been fighting against us, and since he chose to be our deadly enemy, I see no special reason why I should bitterly mourn his death.\"\n\n\"I suppose not,\" Elizabeth replied slowly, torn between relief and regret. She was thankful that Mrs. Lincoln did not appear possessed by the same madness of despair that had seized her after Willie's death, but she knew Mrs. Lincoln felt deep sorrow over the loss of her kin, rebel or otherwise, all the same. To feign indifference was an act for her critics, for the spiteful masses who eagerly snatched up any crumb of proof, however tenuous, of the First Lady's disloyalty. It was not a pretense she needed to maintain in front of Elizabeth, and Elizabeth was very sorry that her patron and friend could not be perfectly honest with her so that she could offer her the comfort her broken heart surely needed.\n\nMonths earlier, in springtime, Mrs. Lincoln's longtime social rival Miss Kate Chase had become engaged to the wealthy former Rhode Island governor and current United States senator William Sprague IV, and their November wedding in the parlor of the Chase mansion was anticipated to be the social event of the season. Fifty guests, including President and Mrs. Lincoln, the cabinet secretaries and their wives, and certain senators, congressmen, and generals had been invited to the ceremony, and five hundred more would join them at the reception. When Elizabeth mentioned to Mrs. Lincoln that she would be sure to keep that afternoon open so she would be available to dress her for the occasion, Mrs. Lincoln told her that would not be necessary. \"I believe I will have a terrible headache that day,\" she said lightly.\n\n\"But, Mrs. Lincoln,\" protested Elizabeth, \"aren't you worried about how it will look if you don't attend? Everyone of quality in Washington will be there.\"\n\n\"I rather worry what it would say if I did attend,\" Mrs. Lincoln retorted. \"You know those Chases, father and daughter alike, have always believed that they belong in the White House rather than us. Mr. Chase has spent the last three years in the high office my husband bestowed upon him making friends and setting himself up as an alternative candidate, and he would be all too delighted to snatch the Republican nomination away from Mr. Lincoln. I absolutely refuse to promote the daughter through any political favor to the father, nor will I promote the father by offering social preference to the daughter. They are my husband's rivals, Elizabeth, and therefore also mine.\"\n\nElizabeth understood Mrs. Lincoln's point of view, and yet it filled her with misgivings. Although Mrs. Lincoln was frequently indisposed, surely no one would believe her excuse of illness if she failed to appear at the wedding, and her critics would concoct their own wild theories that would invariably cast her in an unflattering light. Mr. Lincoln must have concurred, for when his entreaties for her to accompany him failed, he attended the wedding alone and lingered at the reception for two hours as if to compensate for his wife's absence.\n\nElizabeth wished she could have attended the wedding reception, if only to observe the guests in their finery. Several of the ladies had been attired in gowns of her creation\u2014Mrs. Mary Jane Welles's lovely rose moir\u00e9 antique and Mrs. Elizabeth Blair Lee's stunning off-the-shoulder crimson silk were especially fine examples of her handiwork, she thought\u2014and she would have taken both pride and pleasure in comparing her handiwork to that of other mantua makers. Like many other Washingtonians who had not been invited, Elizabeth eagerly read descriptions of the wedding in the papers the next day. The new Mrs. Sprague was said to have been resplendent in a bridal gown of white velvet with a needlepoint lace veil, a diamond solitaire ring worth four thousand dollars sparkling on her graceful hand. As she had entered the room, the Marine Band had played \"The Kate Chase March,\" composed by Mr. Thomas Mark Clark especially for the occasion. Most of the papers listed the most prominent guests, among whom President Lincoln ranked highest. A few reporters pointedly noted that Mrs. Lincoln was not present, but only one cattily mentioned her \"sudden and curious\" illness and added a wholly insincere wish for a swift recovery from her \"uncannily timely affliction.\"\n\nBut it was Tad Lincoln who truly did fall ill only a few days after the wedding. His symptoms were frighteningly reminiscent of the illness he had suffered in the winter of 1862, the same illness that had claimed his brother's life. His parents became increasingly worried as he began a familiar decline. Elizabeth tended him as she had his brother, and Nurse Pomroy was always on call. Mr. Lincoln had agreed to offer a few appropriate remarks at the dedication of a new national cemetery in Gettysburg, but the day before he was scheduled to depart, Mrs. Lincoln begged him not to go. \"Mother, it is my duty,\" Elizabeth heard him tell his wife, and when she burst into tears and declared that he was a better bureaucrat than a father, the wounded, dispirited look in his eyes pained Elizabeth so much that she had to look away. She had never known another man with such nobility of soul and greatness of heart, and she wondered why Mrs. Lincoln sometimes seemed blind to her husband's exemplary qualities. Over time, Elizabeth had come to believe that Mr. Lincoln was unselfish in every respect and that he loved his children and their mother very tenderly. He asked for nothing but affection from his wife, but he did not always receive it. When one of her wayward, impulsive moods seized her, she often said and did things that wounded him deeply. If he had not loved her so much, she would have been powerless to hurt him, but he did care about her and about her opinion of him. She often hurt him in unguarded moments, but afterward, in times of calm reflection, she would not fail to regret her cruel words.\n\nThis, Elizabeth thought, would surely be one of those occasions.\n\nThe next day, Elizabeth arrived at the White House to find Tad bedridden, Mrs. Lincoln hysterical, and President Lincoln deeply melancholy as he prepared to leave for the train station. \"Tad was too ill to eat his breakfast this morning,\" Mrs. Lincoln told her, wringing her hands as she turned back to her husband. \"Please, Father, don't go. Mr. Everett will be offering the oration. You said yourself that your remarks are secondary. You would not be missed.\"\n\n\"I flatter myself that I would be,\" he said, wearily putting on his coat and hat. Mrs. Lincoln stopped pacing long enough to submit to his kiss good-bye, but as soon as he departed, she burst into tears. Elizabeth hurried to offer her a handkerchief, led her to the sofa, and tried to calm her with soothing words, but Mrs. Lincoln had worked herself into a frenzy and would not be comforted. Eventually Elizabeth persuaded her to sit quietly, and she sent for a soothing cup of tea, and by noon, the hour Mr. Lincoln's presidential train was scheduled to depart, she had managed to compose herself. Reminding Mrs. Lincoln that she must remain calm or she would frighten her son, Elizabeth accompanied her to Tad's sickroom, where he promptly sat up in bed and asked for something to eat.\n\nRelief illuminated Mrs. Lincoln so vividly that it was as if a shaft of sunlight had broken through storm clouds. She immediately sent word to the kitchen to prepare her son's favorites, and when they were brought up to him on a tray, he ate slowly, but with a steady appetite. When he lay down again, Mrs. Lincoln and Elizabeth left the room to let him rest under Nurse Pomroy's watchful eye. \"Mr. Lincoln was not feeling well himself this morning,\" Mrs. Lincoln remarked in an undertone as they returned to the sitting room. \"I do hope he won't exhaust himself on this trip.\"\n\nElizabeth nodded, carefully keeping her expression mild. Just as she had predicted, now that the crisis had passed, Mrs. Lincoln regretted the harsh words she had hurled at her husband before his departure. \"Mr. Hay and Mr. Nicolay will look after the president,\" she reminded the First Lady. \"They are traveling with him, aren't they?\"\n\n\"Oh, yes. They've made up quite a party.\" There was a slight edge to Mrs. Lincoln's voice. \"Although it's smaller than it could have been.\"\n\n\"Tad was too ill to travel,\" Elizabeth protested, \"and you were needed here to look after him.\"\n\n\"Oh, dear me, I wasn't speaking of us, Elizabeth. Certain others, who were invited to come, declined the president's invitation.\"\n\n\"Who?\" asked Elizabeth, surprised. \"And why? I should think it would be quite an honor to travel with the president for such an important occasion.\"\n\n\"One _would_ think so, unless one were Secretary Chase\u2014not that his refusal surprises me\u2014or Secretary Stanton, or Senator Stevens, who used to be a reliable friend.\" Mrs. Lincoln lowered her voice. \"They do not believe my husband will win reelection, and they wish to put some distance between themselves and him so that none of his so-called failures will reflect upon them.\"\n\n\"Of course he'll win,\" said Elizabeth, indignant. \"As for those men\u2014I'm astonished by their disloyalty after all he's done for them.\"\n\n\"With few exceptions\u2014very few\u2014his cabinet secretaries are loyal to no one but themselves,\" Mrs. Lincoln replied darkly, and then she felt silent for a moment, thinking. \"I believe I'll send my husband a telegram to let him know that our Tad is feeling better. That will ease his mind, I have no doubt.\"\n\nElizabeth agreed, pleased that Mrs. Lincoln was thinking of her husband's feelings and the difficulties he struggled with as a matter of course and not merely her own.\n\nTad continued to improve, but two days later, Mr. Lincoln returned from Gettysburg with a fever, which soon turned into varioloid, a mild form of smallpox. He was quarantined in the White House for three weeks, but Mrs. Lincoln confided to Elizabeth that he was rather cheerful about it. Since he had been president, he had noted, people had been crowding about him, always asking for something. \"Now let the office-seekers come,\" he joked weakly from his sickbed, \"for at last I have something I can give all of them.\"\n\nIn the first week of October, not long after Mrs. Lincoln's return to Washington City, President Lincoln had issued the Proclamation of Thanksgiving. Despite the destruction of the war, he noted, the year 1863 had been bountiful, with fruitful fields, steady industry, widening national borders, increasing population, and plenteous mines. Even in the midst of a civil war of unprecedented magnitude and severity, peace had been preserved with foreign nations, order had been maintained everywhere except the theater of war, and laws had been made, respected, and obeyed. \"It has seemed to me fit and proper,\" the president declared, that these gifts of a merciful Lord \"should be solemnly, reverently and gratefully acknowledged as with one heart and one voice by the whole American People. I do therefore invite my fellow citizens in every part of the United States, and also those who are at sea and those who are sojourning in foreign lands, to set apart and observe the last Thursday of November next, as a day of Thanksgiving and Praise to our beneficent Father who dwelleth in the Heavens.\"\n\nAfter the president's proclamation was released, Mrs. Lincoln and Elizabeth agreed that it was a wonderful idea, but when the day Mr. Lincoln had selected for the National Day of Thanksgiving finally arrived, it found the president still confined to his sickbed. Elizabeth imagined that he gave thanks nonetheless\u2014for his son Tad's improving health, for recent military victories that offered a glimmer of hope that the Union would ultimately triumph, and for loyal friends and wise counselors, and that most precious rarity of all, loyal friends who offered wise counsel.\n\nOn that Thursday of Thanksgiving, Elizabeth attended a special service at her church. Among the many blessings for which she gave thanks, she included President Abraham Lincoln. She could not imagine a better man to lead the nation through those dark years than he.\n\nOne blustery December day soon thereafter, Elizabeth was making her way to the White House sitting room where she usually sewed when a servant took her aside and cautioned her that Mrs. Lincoln was with a visitor, and sounds of intermittent weeping had been heard through the door all morning. Thanking him for the warning, Elizabeth prepared herself before opening the door.\n\nInside, Elizabeth found Mrs. Lincoln seated on the sofa, her hands clasped in those of a small figure swathed in black crepe\u2014a dark-haired, sweet-faced woman who looked to be nearly two decades younger than Mrs. Lincoln, with pale cheeks and large, tragic, red-rimmed eyes. The ladies were engaged in impassioned conversation, but at the sound of the door, they fell abruptly silent and two tear-streaked faces turned her way. \"Ah, Elizabeth,\" said Mrs. Lincoln, rising, her hand still in the younger woman's. \"Allow me to present my dear sister Mrs. Emilie Helm.\"\n\n\"How do you do, Mrs. Helm,\" said Elizabeth cordially, with a polite bow of the head, but Mrs. Helm was too distraught to do more than press her lips together in a pained semblance of a smile and nod.\n\nMrs. Lincoln crossed the room, placed a hand on Elizabeth's elbow, and guided her back toward the door. \"My sister and her daughter arrived only yesterday, and we have so much to discuss. Would you come back tomorrow\u2014no, the day after? And would you please\u2014\" Her voice dropped to a murmur. \"What I mean is, we would not like it whispered about that Little Sister is staying with us.\"\n\nElizabeth agreed, somewhat bemused, and made her way home. She had worked in the presence of Mrs. Lincoln's guests many times before, but then again, Mrs. Helm was a most unusual visitor, being the widow of a Confederate general. Elizabeth wondered what the press would make of her finding sanctuary within the White House.\n\nTwo days later, Elizabeth resumed sewing for Mrs. Lincoln, and before long she was able to piece together the story of Mrs. Helm's arrival at the White House. The young widow and her daughter had been traveling to Mrs. Lincoln's stepmother's home in Kentucky when they were detained at the border because Mrs. Helm refused to take the oath of allegiance to the United States. To do so would dishonor the memory of her beloved husband, Mrs. Helm insisted, and since the stymied border guards did not know what else to do with her, they held her at Fort Monroe, hoping she would change her mind and swear the oath so they could send her on her way. When her resolve did not falter, they telegraphed President Lincoln to ask him what they should do. He promptly telegraphed back, \"Send her to me.\"\n\nMr. and Mrs. Lincoln adored the young widow they both called Little Sister, and both found comfort in her presence. Each was anxious to hear her opinion regarding the health of the other; to Mary, Mrs. Helm said that she thought the president looked well, only thinner, but when Mr. Lincoln confided that he thought Mrs. Lincoln's nerves had gone to pieces, she replied that Mrs. Lincoln seemed very nervous and excitable, and her rapturous insistence that she saw Willie's and Eddie's spirits at the foot of her bed at night was frightening, abnormal.\n\nAs the days passed, Elizabeth overheard Mr. Lincoln imploring Mrs. Helm to extend her visit, because Mrs. Lincoln seemed better in her sympathetic company. Elizabeth too had noticed a marked improvement in her manner with Little Sister there, but she observed that for all the good she did, Mrs. Helm also introduced an uncomfortable tension into the household. Word had spread throughout Washington that the Lincolns were harboring an unrepentant rebel beneath their roof, stirring up displeasure and contempt and more aspersions about Mrs. Lincoln's suspect loyalties. One morning, Tad and his cousin got into a shouting match over who was the real president, Tad's father or Jefferson Davis. Worst of all was a confrontation that took place in the Blue Room one evening, which a distressed Mrs. Lincoln described to Elizabeth the next day. Senator Ira Harris, father of a colonel in the Army Ordnance Department and stepfather of an officer in the regular army Twelfth Infantry Regiment, and General Daniel Sickles, who had lost a leg at Gettysburg, had come to the White House to meet Mrs. Helm, ostensibly to inquire about mutual acquaintances. The conversation took a disconcerting turn when Senator Harris, perhaps influenced by an excess of alcohol, began taunting the young widow with praise for the Union's recent victories in the West, where her husband had been killed.\n\n\"And, madam,\" he had said, fixing his glare upon her, \"if I had twenty sons they should all be fighting rebels.\"\n\n\"And if I had twenty sons, Senator Harris,\" she had tearfully retorted, \"they should all be opposing yours.\"\n\n\"And then,\" Mrs. Lincoln told Elizabeth, indignant and outraged, \"Senator Harris turned his bleary eye upon me and demanded to know why Robert wasn't in the army!\"\n\n\"Oh, dear,\" said Elizabeth. \"What did you reply?\"\n\n\"I told him that Robert was not a shirker and that he is preparing even now to enter the army. I declared that if fault there be it is mine, as I have insisted that he should stay in college a little longer since I believe an educated man can serve his country with more intelligent purpose than an ignoramus. The senator merely harrumphed at me. I wish I had thought of something more clever, but I was too upset. Robert knows he is criticized for not enlisting, and he would do so this very day if I permitted it.\" Mrs. Lincoln frowned and wrung her hands. \"I fear that someday soon, my husband will overrule me and let him do it.\"\n\n\"Perhaps Mr. Lincoln could use his influence to find Robert a safe post.\"\n\n\"I don't believe there is any such thing. Oh\u2014but then, after Senator Harris flung that question in my face, General Sickles hobbled out of the room to go harass my husband on his sickbed.\"\n\n\"No!\"\n\n\"Yes! Can you believe the nerve? Honestly, the general asked my husband, lying ill in his private chamber, how he could have a rebel in the house. My husband gave him all the courtesy due him as a wounded veteran, and then he said very solemnly, 'Excuse me, General Sickles, my wife and I are in the habit of choosing our own guests.'\"\n\n\"Mrs. Helm is family,\" said Elizabeth. \"Union or Confederate, family must come first.\"\n\n\"Oh, Elizabeth\"\u2014Mrs. Lincoln shook her head ruefully\u2014\"if everyone felt as you do, we might not have had any war at all. I could fill pages and pages if I listed all the families I know that have been divided by this war. I would start with my own and go on and on until it broke my heart.\"\n\nBefore the month was out, the strain in the household indeed became too much, and Mrs. Helm returned to Kentucky. Before she departed, she consented to take the loyalty oath, and so the president granted her amnesty.\n\nAfter Little Sister was gone, Mrs. Lincoln sighed wistfully to Elizabeth that it would have been lovely if she could have remained at the White House throughout the holidays, to help them reclaim some joy from the season, but it was not to be.\n\nBut as Christmas approached, and as Mr. Lincoln recovered from his illness, and after the ordeal of Willie's birthday with its painful memories of all they had lost was behind them for another year, something of the spirit of the season must have shone a little brighter for Mrs. Lincoln. One morning, she announced that she would put off her mourning weeds on the first day of January and begin the New Year afresh. Delighted, Elizabeth immediately began working on a new gown for Mrs. Lincoln to wear to the annual New Year's Day reception. What a pleasure it would be to attire her most visible patron in something other than black silk and crepe after such a long time.\n\nThe gown would be magnificent, rich purple velvet adorned with Valenciennes lace and white satin fluting, with a sweeping train, finished with a headdress boasting a large white plume. \"It will be good luck to begin the New Year finely attired in new clothes,\" Elizabeth told Mrs. Lincoln as she cheerfully fit the muslin lining to her form.\n\nMrs. Lincoln sighed. \"We will need more than good luck if my husband is to win the war and keep his high office in the year to come.\"\n\n\"That's certainly true,\" Elizabeth replied, pinching two folds of fabric with her thumb and forefinger as she slid a pin into place. The Lincolns would also need perseverance, hard work, faith, and courage. Without those, all the luck in the world wouldn't make any difference.\n\n# Chapter Ten\n\n## JANUARY\u2013NOVEMBER 1864\n\nGuests who met the president and his wife at the New Year's Day reception that year took note of Mrs. Lincoln's emergence from official mourning, which some believed was long overdue. Elizabeth knew that although Mrs. Lincoln still felt the pain of Willie's death acutely, she realized that the demands of the upcoming election obliged her to put aside the solace of ritual for the sake of her husband's political future\u2014and her own.\n\nMr. Lincoln's nomination and reelection was by no means certain. History and custom were against him; no sitting president had been reelected in more than three decades, when Andrew Jackson had won his second term in 1832. The people's approval of President Lincoln reflected the outcomes on the battlefield, soaring in the aftermath of victory and plummeting after defeat. With an eventual Union victory seeming ever more possible, political discourse had begun to address the question of how to bring the South back into the Union after the war was over. Plans were under way for the reconstruction of Tennessee, Louisiana, and Arkansas, which were already largely under federal military control, but not everyone agreed with the president's approach. As the incumbent, he could not avoid receiving the blame for every wartime failure, and for every partial success that wasn't succeeding quickly enough. Most Democrats declared his entire presidency a massive failure\u2014an inaccurate but hardly unexpected assessment, given the source\u2014but even his own party was divided in its opinions about his performance in office. Radical Republicans complained that he was too lenient with vanquished Confederates and that his plans for postwar reconstruction were insufficiently harsh. To secure the nomination, Mr. Lincoln would somehow have to convince the Republicans to unite around him, but this would be no easy task.\n\nMrs. Lincoln was determined to do all she could to help. In the cold, gray days of early January, she mustered up her will and resumed her Sunday receptions, taking care to invite anyone who would be well placed to support her husband. The first presidential levee of the year came on January 9, followed by the round of balls and dinners that ushered in the winter social season. Mrs. Lincoln accepted as many invitations as was proper, and entertained her own guests in fine style, and afterward wrote gracious letters and sent gifts of bouquets from the White House greenhouses to prominent figures in government and business.\n\nIt was inevitable, perhaps, that Mrs. Lincoln's schemes would set her at odds with members of her husband's cabinet, who had their own ideas for the president's social calendar. The worst conflict, and the most upsetting for Mrs. Lincoln, was with John George Nicolay, the president's personal secretary, who was charged with the responsibility of arranging state dinners. Single-minded in her resolve to do nothing to promote the ambitious Secretary Chase, Mrs. Lincoln told Mr. Nicolay not to invite him, his daughter, or his son-in-law to a cabinet dinner scheduled for the end of January. Mr. Nicolay balked and told the First Lady he could not possibly exclude one of the secretaries, not only because it went against custom, but also because it would make the president appear spiteful and overly wary of a potential rival. Mrs. Lincoln insisted, and Elizabeth had the dubious privilege of witnessing more than one argument between her and the adamant Mr. Nicolay, who\u2014never within the First Lady's presence but often within earshot of the White House staff\u2014began referring to Mrs. Lincoln as Her Satanic Majesty. When Mr. Lincoln became aware of the conflict, he put an abrupt end to it by ordering Mr. Nicolay to invite Mr. Chase and the Spragues, and by telling his wife to let the matter drop.\n\n\"You see how little power I have in this house of men,\" Mrs. Lincoln lamented to Elizabeth, bitter and humiliated in defeat. \"Everything I do is for the benefit of my husband, but if I am not overruled, I am ignored entirely.\"\n\nElizabeth consoled her as best she could\u2014although privately she thought Mrs. Lincoln had exercised poor judgment in attempting to exclude Mr. Chase and his family\u2014but Mrs. Lincoln worried so much that she hardly slept for two days following the incident. To her credit, on the afternoon of the dinner, she sent Mr. Nicolay a contrite note of apology through the White House doorman. If Mr. Nicolay responded in kind, or at all, Mrs. Lincoln never mentioned it in Elizabeth's presence.\n\nAs much as Mrs. Lincoln regretted the dispute with Mr. Nicolay, she remained as full of antipathy and mistrust for Mr. Chase as ever, and she urged her husband to investigate his loyalties. \"If he thought he could make anything by it, he would betray you tomorrow,\" she insisted. \"I am not the only one who has warned you.\"\n\nThat was true enough, for there were a great many witnesses to Mr. Chase's political ambitions. For months he had been traveling throughout the North, making speeches and shoring up support, expressing his personal hopes that Mr. Lincoln would be reelected while coyly avoiding going on the record about his own potential candidacy. Mr. Chase was a particular favorite among the abolitionists and radical Republicans who believed that the Proclamation of Amnesty and Reconstruction that President Lincoln had issued the previous December was too mild and cautious.\n\nEven to Elizabeth it was evident that Mr. Lincoln disapproved of Mr. Chase's forays into campaigning, but it seemed that he merely observed the secretary's activities with mild, watchful tolerance, waiting to see if they would get out of hand. Mrs. Lincoln believed they already had. \"Why do you not ask for his resignation?\" she demanded in one especially heated moment. It upset her greatly that Mr. Chase showed her husband such disrespect, and she could not comprehend why Mr. Lincoln allowed a rival to remain in a lucrative, influential post where he acquired more power day by day.\n\n\"Fire him, and openly acknowledge that he is my opponent for the nomination?\" replied Mr. Lincoln. \"I am not ready to declare that I am so doubtful that I retain the confidence of the people.\"\n\nThe heated subject came up again one day while Elizabeth was dressing the First Lady for a levee. \"Even Dr. Henry believes that the secretary's behavior has been so disgraceful that he ought to be dismissed from the cabinet,\" the First Lady told her husband heatedly as Elizabeth tied her satin sash into an attractive bow. Dr. Anson Henry was a longtime family friend from Illinois and Mr. Lincoln's trusted personal physician. \"He himself discovered that Mr. Chase's people have been spreading unkind rumors about me, so you can't say I'm merely making that up.\"\n\n\"I would never say that to you, Mother,\" said Mr. Lincoln mildly.\n\n\"You might not say it to me, Father, but you might say it to others, and you might think it,\" she retorted. \"You must promise me to get rid of Secretary Chase once and for all.\"\n\n\"I cannot and will not do that, Mother.\"\n\nNone of her appeals would persuade him, so finally she threw her hands in the air in frustration and strode to the other side of the room, fuming. She yanked the chair back from the desk, sat down stiffly, and began scratching out a furious letter to someone\u2014but with her in such a temper, Elizabeth dared not peek over her shoulder to see whom.\n\nAfter a moment, the president sighed and settled down wearily in his easy chair. \"Well, Madam Elizabeth,\" he asked, his voice hoarse and tired, \"will you brush my bristles down today?\"\n\n\"Certainly, Mr. President.\"\n\nTaking up his comb and brush, Elizabeth dressed his hair, arranging it as neatly as she could. Mr. Lincoln sat in glum, brooding silence while she worked, until suddenly his mouth curved in a faint smile and he said, \"Madam Elizabeth, you have lived on a farm, have you not?\"\n\n\"Yes, Mr. President,\" she said. \"When I was a child in Virginia.\"\n\n\"Then you know what a chin-fly is.\"\n\nElizabeth smiled. \"Sadly, sir, I am acquainted with the pesky species.\"\n\n\"When I was a boy, my brother and I were once plowing corn, I driving the horse and he holding the plow,\" Mr. Lincoln reminisced. \"The horse was lazy, but on one occasion, he rushed across the field so that I could scarcely keep pace with him running as fast as I could. On reaching the end of the furrow, I found an enormous chin-fly fastened upon him, and so I swatted it off. My brother asked me why I did it. I told him I didn't want the poor old horse bitten in that way. 'But Abe,' said my brother, 'that's all that made him go!'\"\n\nElizabeth laughed.\n\n\"Now, Madam Elizabeth,\" said Mr. Lincoln, more cheerful than she had yet seen him that day, \"if Mr. Chase has a presidential chin-fly biting him, I'm not going to swat it off, just in case that is what makes the Department of the Treasury go.\"\n\nElizabeth considered. \"Sometimes a chin-fly can become such a nuisance that the horse flicks its tail and cranes its neck trying to bite it and is so busy trying to get at it that he forgets to plow.\"\n\nMr. Lincoln frowned thoughtfully. \"I suppose in that case, it would be right to swat that chin-fly without delay. The trick is knowing when the chin-fly goes from being a help to a hindrance.\"\n\n\"It's good that you know how to properly deal with chin-flies,\" said Elizabeth pertly, and she was rewarded with the president's laugh. On the other side of the room, Mrs. Lincoln gave a loud, drawn-out, exasperated sigh.\n\nAs the dreary, muddy winter dragged on, it began to appear that \"presidential chin-flies\" were not the only pests driving the Department of the Treasury. For months, Washington gossip hinted at irregularities in business and immorality among the Treasury staff. One outraged citizen wrote to President Lincoln accusing Secretary Chase of speculation in stocks, gold, and cotton. Women employees were reportedly hired for their personal attractions rather than their skills, and several young ladies claimed that they were refused employment until they yielded to the passionate embraces of the superintendent of the Bureau of Engraving and Printing. (When Elizabeth heard that particular rumor, she was greatly relieved that she had not been hired to work in the cutting room the year before.) More shocking yet, dozens of the department's young, unmarried, female employees were said to be with child.\n\nAlarmed, Mr. Chase brought in a detective from the War Department to investigate the allegations, and when the detective found outrage and scandal everywhere he looked, a special congressional committee began a formal inquiry. After hearing the testimony of a series of witnesses, including two young clerks who swore that they had been coerced into intimate encounters with their employer, the committee could not unanimously conclude whether the charges were true or false. The public preferred to believe the most scandalous, salacious version of events, and so even if the Department of the Treasury was not the \"most extensive Whorehouse in the nation,\" as one critic claimed, its reputation was tarnished\u2014as was Mr. Chase's, at a time he most wanted to appear to be a responsible, trustworthy leader.\n\nA more cautious man might have reined in his presidential ambitions at that point, at least until someone else's more shocking story distracted the public, but Mr. Chase did not\u2014nor did his most ardent supporters wish him to. His friends in Congress organized a committee to promote his candidacy, and Mr. Chase encouraged his supporters even if he did not directly participate in their activities, which they sometimes undertook without his knowledge. In February, his campaign supporters drafted and circulated two documents: one that criticized President Lincoln's first term in office without mentioning Mr. Chase, and another, which became known as \"the Pomeroy Circular\" after its author, Mr. Chase's campaign chairman, Kansas senator Samuel C. Pomeroy. The Pomeroy Circular denounced President Lincoln and put forth Mr. Chase as \"a statesman of rare ability and an administrator of the highest order\" who possessed \"more of the qualities needed in a President, during the next four years, than are combined in any other available candidate.\" It asserted that reelecting Mr. Lincoln would be next to impossible, and that to avoid the disaster of a Peace Democrat victory in November, all loyal Republicans had to rally to Mr. Chase to ensure that he won the nomination.\n\nIf Mr. Chase's supporters had hoped to keep the widely distributed documents confidential, they were sadly disappointed. The Pomeroy Circular was leaked to the press and was soon reprinted in the _National Intelligencer_ and other papers. Outraged Union loyalists who received copies of the memo by mail in envelopes marked with the congressional frank of Mr. Chase's supporters forwarded their copies to Mr. Lincoln at the White House, often including personal notes expressing their disgust with Mr. Chase and their steadfast allegiance to the president.\n\nElizabeth heard whispers that the intrigue greatly embarrassed Mr. Chase, and she learned from snatches of conversation she had overheard in the Lincoln family sitting room that Mr. Chase had written to Mr. Lincoln disavowing any knowledge of the Pomeroy Circular until it was published in the press. She also knew that upon receiving Chase's chagrined letter, the president had replied with a brief note acknowledging that he had received it and promising a longer reply when he had time to compose one. That longer letter went out almost a week later, Elizabeth believed, but what it said she did not know, and of course she was in no position to ask. Almost every secret she knew about the inner workings of the White House she had learned by accident, by happening to be present at moments when important words were spoken.\n\nWhatever it was that Mr. Lincoln had written to Mr. Chase, the outcome of the embarrassing scandal was less than Mrs. Lincoln had hoped, for he remained secretary of the treasury. \"I don't know how my husband can bear being in the same room with such a traitor,\" Mrs. Lincoln muttered irritably to Elizabeth one day at the end of February when she realized he wasn't going anywhere. Nevertheless, Mr. Chase appeared greatly subdued, and on March 5, he publicly announced that he was not a candidate for the Republican nomination. \"I suppose that's something,\" said Mrs. Lincoln, remarkably more cheerful.\n\nThe incident had inspired Mr. Lincoln's staunchest supporters to rally around him and go politicking within their home states on his behalf, but even with Mr. Chase out of contention, the way was by no means clear. A small group of Republicans urged Vice President Hannibal Hamlin to declare his candidacy, but he quickly put a stop to that, declaring that he stood by the president. A few radical Republicans championed General Benjamin Butler, the Union military commander of New Orleans, who they believed could unite prowar Democrats and the radical and moderate wings of the Republican Party. The general had won admirers for his strong opposition to slavery and his uncompromising occupation of Baltimore and New Orleans, but when he failed to garner widespread popularity, he eventually dismissed the faction's overtures.\n\nIn February, the _New York Herald_ had called for the nomination of General Ulysses S. Grant, but the very idea seemed to horrify the Union commander, who publicly and adamantly insisted that Mr. Lincoln's reelection was essential to the Union cause. That suited Mrs. Lincoln very well; she often disparaged General Grant as a butcher who did not care how many troops were slaughtered so long as he could claim victory in the end. Elizabeth had often heard Mrs. Lincoln declare that if General Grant should ever be elected president of the United States, she would desire to leave the country and remain absent throughout his term of office.\n\nJohn C. Fr\u00e9mont, who had been the fledgling Republican Party's first presidential nominee in 1856, was strongly popular among abolitionists like the German-American population of St. Louis. Unlike General Grant, General Fr\u00e9mont was willing and eager to supplant his commander in chief. For years he had been harboring resentment against the president, who had revoked his 1861 proclamation freeing the slaves of Confederates within the state of Missouri and had subsequently fired him as commander of the Union Army's Department of the West. General Fr\u00e9mont alone seemed unlikely to follow the course of President Lincoln's other potential challengers and remove himself from consideration.\n\nMrs. Lincoln refused to sit idly by and hope everything worked out in her husband's favor. She campaigned for him in her own way, cultivating friendships with politicians and businessmen of dubious character but with ample wealth, influence, or, ideally, both. Gossip swirled about her yet again, scathingly impugning her for surrounding herself with a certain class of men. When Elizabeth, ever attuned to appearances, delicately ventured that Mrs. Lincoln's new friends and correspondents did not seem to be the sort of gentlemen Mr. Lincoln would choose as friends, Mrs. Lincoln replied, \"I have an object in view, Elizabeth. In a political canvass it is policy to cultivate every element of strength.\"\n\nDubious, Elizabeth asked, \"And you consider these gentlemen from New York to be a strength, despite the unpleasant chatter your friendship with them inspires?\"\n\n\"These men have influence, and we require influence to reelect Mr. Lincoln,\" the First Lady explained. \"I will be clever to them until after the election, and then, if we remain at the White House, I will drop every one of them, and let them know very plainly that I only made tools of them. They are an unprincipled set, and I don't mind a little double-dealing with them.\"\n\nElizabeth was shocked that Mrs. Lincoln could so blithely plan to use unscrupulous, possibly dangerous men to her own advantage, dismiss them when they had served their purpose, and then mock them by announcing what she had done. \"Does Mr. Lincoln know what your purpose is?\"\n\n\"God, no! He would never sanction such a proceeding.\"\n\n\"Perhaps that should be enough to recommend against it.\"\n\n\"Oh, Elizabeth.\" Mrs. Lincoln shook her head as if she should have known that Elizabeth would understand no better than her husband. \"I keep him in the dark, and will tell him when all is over. He is too honest to take the proper care of his own interests, so I feel it to be my duty to electioneer for him.\"\n\nAs always, Mrs. Lincoln professed the best of intentions and truly seemed to believe she was doing what was good and necessary\u2014and once again, Elizabeth found herself dismayed. She wished Mrs. Lincoln would be more prudent. The First Lady was quite a schemer, but she was surely out of her depth among such shrewd, experienced New York politicians, for whom double-dealing and electioneering were a way of life. Elizabeth feared that it was far more likely that the crafty, flattering gentlemen were using Mrs. Lincoln than that she was using them. She did not see any way the First Lady's alliances with such men would end well.\n\nElizabeth had another reason for disliking the First Lady's trips to New York. Whenever Mrs. Lincoln was worried or upset, she found comfort in shopping. Whenever she traveled to the North, the newspapers were full of snide reports about how she had ransacked the treasures of Broadway stores, filling her carriage with shawls, boas, capes, handkerchiefs, parasols, fans, bonnets, boots, and gloves. Elizabeth fought to conceal her astonishment if she happened to be present when Mrs. Lincoln's purchases were delivered, but one eighty-dollar handkerchief made her jaw drop, and a two-thousand-dollar shawl caused her to gasp aloud, hand to her heart. Estimating the bills, Elizabeth could not see how Mrs. Lincoln could possibly afford all that she acquired, not when her rough calculations strongly suggested that Mrs. Lincoln's personal expenses that spring surpassed her entire budget for the White House refurbishing during the first year of her husband's presidency.\n\nA member of the White House staff was charged with the thankless task of approving all of Mrs. Lincoln's official expenditures, but no one monitored her personal spending. Perhaps everyone assumed it would be constrained by her husband's salary, but that, of course, was not so. If the First Lady ordered luxurious goods from a fine shop on Broadway or Pennsylvania Avenue, every shopkeeper therein would be all too delighted to extend her credit\u2014but eventually, the bills would come due, and payment in full would be expected, and later demanded.\n\nAs soon as the spring sunshine dried the muddy roads of Virginia enough to make them passable, the armies would be on the move, and for the first time, General Grant would face General Lee. Everyone in the North realized that defeating General Lee was crucial to ending the rebellion, not only because General Lee was a brilliant strategist, but also because his army protected the Confederate capital of Richmond.\n\nOn April 25, at midday, Elizabeth, Emma, Virginia, and the Lewis girls joined the crowds lining Fourteenth Street to watch General Burnside's thirty thousand troops march out to reinforce the Army of the Potomac. In the early months of the war, every parade of soldiers had drawn throngs of cheering onlookers, but since then the sight of passing regiments had become so commonplace that they attracted little notice. But this was no ordinary procession. This time the column included seven regiments of United States Colored Troops, three of them recruited in nearby Maryland, and it seemed that every person of color in Washington and many more besides had come to see them set out to confront General Lee.\n\nPride surged through Elizabeth's veins as she waited, listening as the stirring sounds of fife and drums heralded the column's approach. Down New York Avenue they came, smart and polished, turning south onto Fourteenth Street past cheering crowds. Virginia lifted toddler Alberta up to her shoulder while the older Lewis girls, Jane and Lucy, rose up high on tiptoe, craning their necks to see over the heads of the crowd. \"There they are,\" Virginia cried, gesturing for her children to look. \"Do you see how well they march? Do you see? Those are _our_ soldiers, our brave colored soldiers.\"\n\nAs the girls, eyes shining, assured their mother that they did indeed see, Elizabeth gazed at the dark, proud, eager faces of the colored soldiers and felt her throat constricting with emotion. Their splendid uniforms, the rousing music, the bold and steady marching, the cheering crowd\u2014in that glorious moment it seemed to Elizabeth that there might be no limit to what the people of her race could accomplish in the years to come, unhindered by slavery, when peace reigned over a nation united once again. It was the most sublime spectacle she had ever witnessed, and she prayed that the men would acquit themselves bravely. Everyone would be watching them, she knew, and many would maliciously hope for them to fail. They must succeed, and they would succeed, and in so doing they would disprove every false, slanderous word uttered about the folly of allowing men of color to enlist.\n\nThe marching corps approached Willard's Hotel, where President Lincoln and General Burnside stood on the eastern portico to review the parade. When the colored troops passed the president, they waved their hats in the air and cheered for the Great Emancipator, the man who had set their people free. Mr. Lincoln stood with his hat off, bowing and nodding, showing them the same respect and courtesy he had shown every white soldier.\n\nThe column needed more than four hours to cross Pennsylvania Avenue. After the soldiers came ambulances, then thousands of cattle to feed the troops, all heading across the river to Virginia. A renewed sense of purpose and determination filled the city, from the marching soldiers to the people lining the streets shouting blessings and good wishes upon them. And then they were gone, leaving hope and fear and anticipation and apprehension in their wake.\n\nThe crowds dispersed, and the people went home. Now, all knew, they had to brace themselves for the inevitable onslaught of casualties.\n\nThey did not have long to wait.\n\nWhile the Union and Confederate armies clashed in the Wilderness, the dead and wounded came flooding into Washington from field hospitals, just as they had after the battles at Bull Run, the Peninsula, Antietam, and Gettysburg. The wounded arrived in ambulances, one train a day, but the trains were miles long and had jolted and jerked their suffering passengers over great distances, without food and comforts, filthy and fainting, limbs gone, wounds untended. Injured, sick, and dying soldiers, corpses, prisoners, officials\u2014the choked docks and stations and roadways could not be cleared swiftly enough to make room for new transports. The noxious odor of bodies in the summer heat hung sickly sweet over every street and alley, and the remains of the dead piled up faster than the embalmers could attend to them. One Washington undertaker fell so far behind as he raced to work through his backlog of deceased that he was briefly arrested and cited for causing a public nuisance. All of Washington seemed to be one great, terrible hospital, and no corner of it was spared the miasma of death.\n\nFrom the Wilderness the fighting moved on to the Spotsylvania Court House, and from there to the North Anna River, and then on to Cold Harbor. Casualties were massive on both sides, disproportionately so for the Union, but the outcomes of the battles were tactically inconclusive. More revealing was what General Grant did each time he failed to destroy General Lee's army: In circumstances where his predecessors had always chosen to retreat, General Grant regrouped and moved his army forward, again and again, keeping General Lee on the defensive and inching ever closer to Richmond. The people of the North realized then that General Grant possessed a very different military mind than they had witnessed thus far in the war.\n\nIn the last major battle of the campaign, General Grant surprised General Lee by directing his engineers to construct a pontoon bridge 2,100 feet across the James, stealthily crossing the river, and threatening Petersburg, the most important supply base and railway depot for the entire region, including the Confederate capital of Richmond. If General Grant could capture Petersburg, Richmond would inevitably fall. The Union troops settled in for the siege.\n\nThe split within the Republican Party widened throughout the spring. At the end of May, the radical Republican faction convened in Cleveland to select their own candidate for the general election, determined that he should win enough delegates to make Mr. Lincoln's nomination irrelevant. Styling themselves as the Radical Democracy Party, they were expected to choose General Fr\u00e9mont as their nominee, although rumors floated about that some among them held out hope for General Grant, who could not have made his disinterest in the office more clear. He had enough to do fighting the war, Elizabeth thought, without taking on all the additional fighting that went on in Washington.\n\nSoon after the convention closed, Elizabeth was dressing Mrs. Lincoln for an evening at the opera with Postmaster General Blair and his daughter when Mr. Lincoln entered carrying a newspaper. \"Nicolay brought me the _Herald,_ so I can examine the news from Cleveland at my leisure,\" he told his wife, his expression nonchalant but his eyes shining with suppressed amusement.\n\nMrs. Lincoln's brows drew together. \"Didn't you read the report at the telegraph office yesterday?\"\n\n\"I did,\" he said, settling down on the sofa and stretching out his long legs in front of him, \"but Nicolay must have believed I wanted it.\"\n\n\"Has the news changed?\" inquired Mrs. Lincoln, in an ironical tone that implied she knew it had not.\n\n\"No, every word is the same. Fr\u00e9mont is their man\u2014the man, anyway, of the four hundred people who showed up for the convention.\"\n\n\"Was that all?\" exclaimed Elizabeth, forgetting herself. \"Only four hundred?\"\n\nMr. Lincoln's mouth quirked in a smile. \"That's right, Madam Elizabeth. A mere four hundred.\" Suddenly inspired, he sat upright and reached for the Bible on the wooden stand nearby. \"That reminds me,\" he said, turning pages. \"First Samuel, chapter twenty-two, verse two.\" He found the passage, cleared his throat, and began to read the scripture. \"'And every one that was in distress, and every one that was in debt, and every one that was discontented, gathered themselves unto him; and he became a captain over them; and there were with him about four hundred men.'\" He replaced the open Bible on the stand. \"Four hundred again. An interesting number, it seems to me.\"\n\nIn this way, Elizabeth discovered that Mr. Lincoln was not terribly concerned about Mr. Fr\u00e9mont being put up against him.\n\nA few days later, at their convention in Baltimore, Republicans loyal to Abraham Lincoln renamed themselves the National Union Party to distinguish themselves from the gentlemen who had met in Cleveland. They also hoped the new name would appeal to War Democrats, with whom they wanted to forge a coalition. Like themselves, the War Democrats were in favor of the war, and they wanted to break away from the antiwar Peace Democrats to support a candidate who reflected their views\u2014but they could not bring themselves to vote for a Republican. A candidate from the new National Union Party, on the other hand, might be tolerable.\n\nIn the proceedings, Republicans and War Democrats united to nominate Mr. Lincoln as their candidate. The nomination would have been unopposed but for a delegation of twenty-two radical Republicans from Missouri, who first nominated General Grant before changing their votes so Mr. Lincoln's nomination would be unanimous. The delegates also established their party platform, which praised the president for his management of the war and called for, among other critical issues, the pursuit of the war until the Confederacy surrendered unconditionally, a constitutional amendment to abolish slavery, assistance for disabled Union veterans, and the construction of a transcontinental railroad.\n\nNext the agenda turned to the selection of a vice president. Previously Mr. Lincoln had expressed his desire not to interfere and to let the convention decide, and once the debate began, he stuck to his resolution. Vice President Hannibal Hamlin wanted to be renominated, but much had changed since the election of 1860, and this time his candidacy stirred up little enthusiasm. Many delegates believed that they should select a War Democrat from a border state to broaden the appeal of the ticket. After some wrangling, they eventually chose Andrew Johnson, the Union military governor of Tennessee, a War Democrat and Southern Unionist who was nominated overwhelmingly on the first ballot.\n\nLater that same month, the assistant secretary of the treasury resigned. Mr. Lincoln made his criteria for selecting a successor quite clear, but Secretary Chase disregarded the president's wishes. In the ensuing disagreement, Secretary Chase loftily submitted his resignation as a matter of principle, as he had many times before\u2014but this time, the president astounded him by promptly accepting.\n\nElizabeth expected Mrs. Lincoln to be overjoyed. At last, two things she had greatly desired had come to pass: Mr. Lincoln would be on the ballot in the November presidential election, and Mr. Chase was out of the cabinet. Mrs. Lincoln's worries should have eased, at least a little, but instead she seemed more agitated and anxious than ever. \"What do you think about the election?\" she asked Elizabeth one sultry morning at the end of June as she gazed out the open window upon the Potomac.\n\nElizabeth looked up from her work, fabric on her lap, needle in hand. \"I think that Mr. Lincoln will remain in the White House four years longer.\"\n\nMrs. Lincoln turned away from the window, her face a curious mixture of hope and apprehension. \"What makes you think so? Somehow I have learned to fear that he will be defeated.\"\n\n\"Because he has been tried, and has proved faithful to the best interests of the country,\" replied Elizabeth. \"The people of the North recognize in him an honest man, and they are willing to confide in him, at least until the war has been brought to a close. The Southern people made his election a pretext for rebellion, and now to replace him by someone else, after years of sanguinary war, would look too much like a surrender of the North.\"\n\nMrs. Lincoln looked as if she wanted very much to believe her. \"So you believe Mr. Lincoln is likely to be reelected?\"\n\n\"Mr. Lincoln is certain to be reelected,\" said Elizabeth emphatically. \"He represents a principle, and to maintain this principle the loyal people of the loyal states will vote for him, even if he had no merits to commend him.\"\n\nMrs. Lincoln pondered this for a long moment in silence. \"Your view is a plausible one, Elizabeth, and your confidence gives me new hope.\" Her expression suddenly clouded over with worry again, belying her words. \"If he should be defeated, I do not know what would become of us all. To me, to him, there is more at stake in this election than he dreams of.\"\n\nElizabeth studied her, wary and wondering. \"What do you mean, Mrs. Lincoln?\"\n\nMrs. Lincoln hesitated, took a deep breath, and said, all in a rush, \"I have contracted large debts, of which he knows nothing, and which he will be unable to pay if he is defeated.\"\n\nElizabeth's heart sank. She had suspected as much for ages. Steeling herself, she asked, as perhaps she should have asked months before, \"What are your debts?\"\n\nMrs. Lincoln began to pace in front of the open window, wringing her hands. \"They consist chiefly of store bills. I owe altogether about twenty-seven thousand dollars, the principal portion at Stewart's, in New York.\"\n\nElizabeth dropped her threaded needle and fell back against her chair. It was a shockingly enormous sum, more than Mr. Lincoln's entire annual salary.\n\n\"You understand, Elizabeth, that Mr. Lincoln has but little idea of the expense of a woman's wardrobe. He glances at my rich dresses, and is happy in the belief that the few hundred dollars that I obtain from him supply all my wants.\" She stopped pacing and threw Elizabeth a beseeching look. \"I must dress in costly materials. The people scrutinize every article that I wear with critical curiosity. The very fact of having grown up in the West subjects me to more searching observation. To keep up appearances, I must have money\u2014more than Mr. Lincoln can spare for me. He is too honest to make a penny outside of his salary; consequently I had, and still have, no alternative but to run in debt.\"\n\nShaking her head in disbelief, Elizabeth tried to sort out Mrs. Lincoln's rationalizations. \"And Mr. Lincoln does not even suspect how much you owe?\"\n\n\"God, no!\" she exclaimed. \"And I would not have him suspect. If he knew that his wife was involved to the extent that she is, the knowledge would drive him mad. He is so sincere and straightforward himself that he is shocked by the duplicity of others. He does not know a thing about any debts\u2014and I value his happiness, not to speak of my own, too much to allow him to know anything. This is what troubles me so much. If he is reelected, I can keep him in ignorance of my affairs, but if he is defeated, then the bills will be sent in, and he will know all.\"\n\nA hysterical sob escaped her then, and Elizabeth was compelled to assure her that Mr. Lincoln would be reelected, of course he would. She was also tempted to warn her that Mr. Lincoln's reelection would only delay the inevitable, but the First Lady was already in such a dreadful state that Elizabeth couldn't bear to make matters worse with more harsh truths. Instead she murmured words of comfort and platitudes about frugality that she resignedly expected Mrs. Lincoln to ignore.\n\nThe subject resurfaced from time to time throughout those oppressive summer days, for anxiety seized Mrs. Lincoln with every bill that arrived in the mail or political setback that caused President Lincoln's popularity to sag. Sometimes Mrs. Lincoln feared that her husband's enemies would discover the particulars about her debts and use them against him in the campaign. Whenever this thought occurred to her, she became almost wild with agitation and fear.\n\nSometimes too she seized upon a way out of her troubles that seemed, to Elizabeth, a vain hope. \"The Republican politicians must pay my debts,\" she would declare. \"Hundreds of them are getting immensely rich off the patronage of my husband, and it is but fair that they should help me out of my embarrassment. I will make a demand of them, and when I tell them the facts they cannot refuse to advance whatever money I require.\"\n\nElizabeth thought that they certainly _could_ refuse, and very likely _would,_ and that all Mrs. Lincoln would accomplish with this scheme would be to place herself in the way of malicious gossip yet again. She gently discouraged her from confessing her debts to anyone but her husband and her trusted sisters, but she could not compel Mrs. Lincoln to heed her counsel\u2014on this, or on practicing frugality, or on any other matter. What frustrated Elizabeth most was that all the while the First Lady fretted about her debts, she continued to spend, buying shawls and gloves and expensive trinkets she did not really need. The pleasure of buying pretty things seemed to help her forget her misery for a brief moment, but all the while, she was really only making matters worse.\n\nNever in her life had Elizabeth known a more peculiarly constituted woman.\n\nIn the first week of July, General Sherman was making little headway in his offensive maneuvers upon Atlanta, and in the Shenandoah Valley, Confederate lieutenant general Jubal Early halted the Union major general David Hunter's thrust south and then turned his rebel army north toward the Potomac. Most Northerners assumed this was yet another summer raid and paid it little attention, but General Early's Army of the Valley kept advancing, skirting Harpers Ferry to cross the Potomac at Shepherdstown and moving on into Maryland. State officials in New York and Pennsylvania were worried enough to call out more than twenty-four thousand militia to provide defense, but most people of Washington City had such confidence in General Grant that they could not believe the rumors that their city might soon be in danger.\n\nGeneral Early captured Hagerstown and then Frederick, demanding cash, clothing, food, and other necessary supplies from the citizens. Rebel soldiers fanned out into the countryside, claiming cattle and horses and harvesting at will from local orchards. This seemed to prove that they were indeed on a simple plundering expedition, especially since the War Department released no information to the contrary.\n\nThen worrisome rumors began to circulate that General Early had crossed the Potomac with nearly twenty thousand troops and was advancing upon Washington. The newspapers printed the alarming stories, retracted them, and printed them again: General Lee had sent General Early north to menace Washington and Baltimore in the hopes of forcing General Grant to divert troops from Richmond to defend them. General Early intended to kidnap President Lincoln and hold him hostage, forcing the Union to capitulate. General Early planned to invade the Union capital in order to convince foreign nations that they must recognize the Confederates as a legitimate government. Mrs. Lincoln was with her family at the Soldiers' Home, so Elizabeth could not hope to overhear anything that might dispel or confirm the talk on the streets. Like everyone else, she could only gather whatever useful news she could, and wait.\n\nOn July 9, the governor of Maryland and the mayor of Baltimore declared an emergency and called their citizens to arms. \"It may without exaggeration be said today that we are having something of an excitement,\" the Baltimore correspondent of _The_ _New York Times_ breathlessly reported. Union major general Lew Wallace, commander of the Middle Department and Eighth Army Corps, moved his meager forces\u2014about 6,300 troops, mostly Hundred Day Men\u2014in place to resist General Early's advance, not knowing whether he intended to move toward Baltimore or Washington. The two armies met on the banks of the Monocacy about forty miles northwest of the capital, but General Wallace's army was badly outnumbered.\n\nAlmost immediately, refugees from the countryside of Frederick and upper Montgomery counties began streaming into Washington on wagons piled high with their household goods, babes in arms, livestock trailing along behind. They told harrowing stories of advancing armies and narrow escapes, and it soon became evident that General Early's forces had pushed forward to Rockville, a mere twenty miles away. The War Department had withheld information from the public to avoid causing a panic, but now there could be no more pretense. General Early was coming, and the city was not prepared to withstand him.\n\nAlthough miles of trenches and earthworks surrounded Washington, the most experienced soldiers had been gathered into the Army of the Potomac for General Grant's march on Richmond, and the soldiers left behind, mostly National Guardsmen from Ohio, had not been trained to use their forts' heavy artillery. Every able-bodied citizen was called to defend the city. The Pennsylvania Bucktails who guarded the White House left their customary posts for the front lines. Quartermaster clerks took up arms and formed ranks. Eighteen hundred men from the convalescent camps and thirty-two hundred more from the Invalid Corps were put on active duty. Nearly one thousand marines and mechanics from the navy yard traded their tools for rifles. Civilians were quickly recruited into a Loyal League militia, and dozens of colored men were compelled to serve as teamsters.\n\nPresident Lincoln was at Fort Stevens when the defenders began firing upon the Confederate advance; later it was said that he had stood on a parapet, a perfect target amid the flying bullets, until a nearby soldier roughly ordered him to get down unless he wanted his head knocked off. In a potential disaster worthy of Bull Run three years before, thousands of eager civilians\u2014men, women, and children\u2014rushed to the fort to watch the spectacle, and when soldiers forced them away, they scaled fences, trees, and hills to get a better look at the fight.\n\nBut even as General Early's forces reached the breastworks at Fort Stevens and he began gathering his troops for a full-scale attack, Union reinforcements from the Sixth and the Nineteenth Corps under Major General Horatio G. Wright began arriving in southwest Washington by steamer. The numbers of these desperately needed veteran fighters were few, but General Early must not have known that, because after two days of skirmishing\u2014during which time additional Union troops arrived, further strengthening their defenses\u2014he withdrew before dawn on the morning of July 13, so stealthily and unexpectedly that the Union defenders did not realize the Confederates had gone until it grew light enough to look out from the fort and see that they had disappeared.\n\nExultant, the civilian defenders left the battlements and resumed their ordinary lives, but President Lincoln was greatly displeased that yet another Confederate army had gotten away. Elizabeth was simply grateful, and enormously relieved, that they had gone.\n\nAugust waned, oppressively hot and sultry, with little good news from the battlefield and even less from the presidential campaign. At the end of the month, the Democratic National Convention met in Chicago and nominated President Lincoln's disgruntled former general in chief of the Union Army, George B. McClellan, on a peace platform that called for a cease-fire and negotiated settlement with the Confederacy. With General Grant unable to advance upon Richmond and General Sherman stalled near Atlanta, the war seemed to have ground to a dispiriting halt.\n\nMrs. Lincoln, too, was noticeably discouraged. She took to visiting Elizabeth in her rooms, ostensibly to discuss necklines and trims and trains, but the election loomed ever larger in her thoughts. \"Elizabeth,\" Mrs. Lincoln said forlornly, \"where do you think I will be this time next summer?\"\n\n\"Why, in the White House, of course.\"\n\n\"I can't believe so.\" Tears appeared in her eyes. \"I have no hope of the reelection of Mr. Lincoln. The canvass is a heated one, the people begin to murmur at the war, and every vile charge is brought against my husband.\"\n\n\"No matter,\" Elizabeth replied firmly. \"Mr. Lincoln will be reelected. I am so confident of it, that I am tempted to ask a favor of you.\"\n\nMrs. Lincoln's eyebrows rose. \"A favor. Well, if we remain in the White House I shall be able to do you many favors. What is this special favor?\"\n\n\"I should like for you to make me a present of the right-hand glove that the president wears at the first public reception after his second inaugural.\"\n\nMrs. Lincoln laughed, astonished. \"It will be so filthy when he pulls it off that I shall be tempted to take the tongs and put it in the fire. I cannot imagine what you want with such a glove.\"\n\nSmiling, Elizabeth lifted her chin, refusing to be dissuaded. \"I shall cherish it as a precious memento of the second inauguration of the man who has done so much for my race. He has been a Jehovah to my people\u2014he has lifted them out of bondage, and directed their footsteps from darkness into light. I shall keep the glove, and hand it down to posterity.\"\n\nMrs. Lincoln shook her head. \"You have some strange ideas, Elizabeth,\" she remarked. \"Never mind, you shall have the glove; that is, if Mr. Lincoln continues as president after the fourth of March next.\"\n\n\"It is a certainty,\" said Elizabeth, as she had many times before throughout that long, hot summer. Mrs. Lincoln only smiled, wistful and worried.\n\nA few days later, General Sherman captured Atlanta.\n\nThe news came to Mr. Lincoln by telegram on September 2. Exultant, the president commanded every arsenal and navy yard to fire a one-hundred-gun salute in General Sherman's honor, and at Petersburg, General Grant saluted his brother in arms by ordering all of his batteries to fire live rounds at the enemy, which they did within the hour, with much rejoicing.\n\nThe people of the North were jubilant. After a dismal summer full of stalemate, discouragement, and defeat, the Union Army suddenly surged toward victory\u2014and so too did Mr. Lincoln. Overnight he had become a victorious commander in chief, and in the transformed political environment, the radical Republican effort to replace him seemed dangerously unwise. The possibility that the Republican electorate would divide their votes between Mr. Lincoln and General Fr\u00e9mont, and thereby allow the Democrats to seize the presidency, began to worry members of the new National Union and Radical Democracy parties. In September, President Lincoln asked for the resignation of Postmaster General Blair, who among all the members of his cabinet particularly infuriated the radical Republican faction. In the meantime, though no one would claim or could prove a connection, General Fr\u00e9mont withdrew from the race.\n\nThe political distinction between the two remaining candidates could not have been more clear\u2014President Lincoln, leader of a victorious army and savior of the Union, and General McClellan, a once-popular and perpetually hesitant military leader whose party insisted upon a peace platform he himself did not advocate.\n\nIn the weeks leading up to Election Day, Secretary of War Edwin Stanton made sure that soldiers were given absentee ballots, if the laws of their states permitted, or furloughs so they could travel home to vote. President Lincoln himself wrote to several of his generals asking them to grant leave to soldiers from states where the election would likely be a close call\u2014Indiana, Pennsylvania, Missouri, and New York\u2014assuming that the Union soldiers would overwhelmingly support the Republican ticket as they had in the off-year elections. Even the former secretary Salmon Chase, still stinging from his abrupt dismissal from the Department of the Treasury, began praising President Lincoln in public and then campaigning for him in crucial Midwestern states. His support was no less beneficial for all that he was obviously angling for the newly vacated position of chief justice of the Supreme Court.\n\nAlthough the outcome of the election had never looked brighter for Mr. Lincoln, Mrs. Lincoln had her own campaigns to wage to help ensure his victory. She had come to trust Elizabeth's taste and judgment, so she had asked Elizabeth to accompany her to New York for one last autumn shopping trip. When Mrs. Lincoln needed to return to Washington in time for the election, Elizabeth remained in New York to attend to her business there\u2014making purchases on her behalf, obtaining estimates for fabrics, placing or canceling orders, paying or deferring bills, and carrying on the First Lady's business affairs as directed, sending telegrams back and forth, often several times a day.\n\nElizabeth remained in New York for all of November, which meant that she spent Election Day not at home in the nation's capital as she had four years before, but in the fraught, anxious city where memories of violent draft riots still evoked pain and anger, and where Mr. Lincoln had never been popular.\n\nIt was there too that Elizabeth heard the glorious news that Mr. Lincoln had been reelected\u2014not only elected but decisively so, receiving fifty-five percent of the popular vote and an enormous margin in the Electoral College, two hundred and twelve votes to General McClellan's twenty-one.\n\nAlone in her rented room, suddenly overwhelmingly homesick, Elizabeth nevertheless rejoiced.\n\nAll of her predictions had proven true. The good people of the loyal states had wisely decided that Mr. Lincoln should continue at the nation's helm.\n\n# Chapter Eleven\n\n## DECEMBER 1864\u2013APRIL 1865\n\nThe war continues,\" Mr. Lincoln began his annual presidential message to Congress in December, but after that somber opening, it took a more optimistic turn, certainly more so than his previous three State of the Union addresses. The Union Army was steadily advancing, and the results of the November elections proved that the people of the North were resolved to see the war through to victory. Despite their significant losses, the North still had more men and resources than the South. As if he expected his address to be read in the Confederate capital\u2014and indeed it likely would be printed in the Richmond papers within days\u2014Mr. Lincoln noted that the overmatched South could have peace the moment they decided to lay down their arms and submit to federal authority. His administration would not, however, compromise in any way on the matter of slavery; in fact, the president called on the House to approve the constitutional amendment abolishing slavery that the Senate had already passed. The end of slavery throughout the nation was only a matter of time, he asserted, and \"the sooner the better.\"\n\nA few days later, while Salmon P. Chase was being sworn in as the new chief justice of the Supreme Court, word arrived in the capital that General Sherman had reached the Atlantic, the terminus of his march across Georgia. On Sunday, December 25, the general sent the president a telegram bearing most unusual but nonetheless heartening holiday greetings: \"I beg to present to you, as a Christmas gift, the city of Savannah, with 150 heavy guns and plenty of ammunition, and also about 25,000 bales of cotton.\"\n\nThat year Washingtonians celebrated Christmas with patriotic jubilance, and a week later they welcomed the New Year with reinvigorated hopes. Nearly four thousand citizens attended the annual New Year's reception at the White House, filling the public rooms to overflowing so that when they were ready to depart, some of the more agile guests had to exit through a first-floor window in the East Room and clamber down a long ramp of wooden planks to the ground.\n\nFor Elizabeth, the last day of January was a greater, more joyful harbinger of new beginnings than the first, for it was on January 31 that the House voted to approve the Thirteenth Amendment, abolishing slavery throughout the United States. For the first time, people of color were allowed into the congressional galleries, where they watched the final speeches and heard the vote taken in breathless quiet, breaking into cheers and joyful weeping when the measure passed. Although three-fourths of the states still had to ratify the amendment before it would become the law of the land, the colored residents of Washington rejoiced, certain that slavery had been dealt a fatal blow.\n\nDespite the ongoing war, there was much to be thankful for in the first months of 1865, but Mrs. Lincoln found herself caught up in a frenzy of discontent. After Mr. Lincoln was reelected and various resignations and appointments were made, Mrs. Lincoln made some staffing changes of her own, beginning with firing the longtime White House doorman Edward McManus. Elizabeth wasn't sure why; apparently he had failed to deliver some documents on time or he had divulged a secret Mrs. Lincoln had entrusted to him\u2014the other servants were not sure and Mrs. Lincoln would not say. When Mr. Lincoln heard about the dismissal, he and Mrs. Lincoln had a terrible quarrel in front of a member of her New York coterie, embarrassing her greatly. Worse yet, a disgruntled Mr. McManus carried tales of her overspending, duplicities, and ill tempers to Thurlow Weed, a New York newspaper editor and political organizer whom Mrs. Lincoln particularly despised. Scandalous stories began to make the rounds, compelling Mrs. Lincoln to frantically write a flurry of letters in her own defense to the New York elite in an attempt to salvage her reputation.\n\nWhile Mrs. Lincoln waged war for her social status, her husband and eldest son embarked on a secret campaign of their own. After graduating from Harvard College in July 1864, Robert had again sought his parents' blessing to enlist in the army, and Mrs. Lincoln had again adamantly refused. Angered, Robert declared that if he could not live as he wanted he would at least escape the \"glass house\" of Washington, so he returned to Cambridge and enrolled in Harvard Law School. Six months later, with the Union Army sweeping relentlessly through the South, Mr. Lincoln decided to grant his son's longtime wish. Without consulting his wife, he wrote to General Grant and asked him to find a post for Robert that would neither place him on the front lines nor bestow upon him a coveted position that ought to go to a more deserving veteran soldier. The general wrote back that he would gladly welcome Robert into his \"military family,\" and so it was that on February 17, Robert Todd Lincoln entered the army as a captain and began serving as an assistant adjutant general on General Grant's own staff. Mrs. Lincoln kept up a brave, tremulous front, assuring the nation that she was very proud of her son, but she confided to Elizabeth that she was nervous and afraid, even though Robert was so well placed that he would likely never see a single battle.\n\nOn the night before Mr. Lincoln's second inauguration, a terrible storm struck Washington City. Elizabeth was jolted awake by the crash of thunder and the scour of hail upon the roof, and she sat up in bed, her quilt gathered around her, heart pounding, until it subsided enough for her to lie back down and try to fall asleep. It was not an ill omen, she told herself firmly. It was only a storm, perhaps more severe than others they had seen in that damp early spring, but still, only a storm.\n\nThe next morning, she woke to a gray day. She had planned to attend the parade with Emma and two of her other assistant seamstresses, and later to join Virginia and Walker on the grounds of the Capitol to witness Mr. Lincoln take the oath of office and give his second inaugural address, but the torrential rain had turned the streets of Washington to canals of mud ten inches thick, and she had no desire to wade through them, fighting the crowds for the rare dry patches of earth. She would have to go out later, first to the White House to dress Mrs. Lincoln for the White House reception, and then to an inaugural party with the Lewises and other friends, but with any luck the streets would be drier by then.\n\n\"Oh, do come,\" Emma protested when she knocked upon Elizabeth's door and she apologetically explained her change of heart. \"How can you think of staying away the first time colored folks are allowed on the Capitol grounds for an inauguration? The streets aren't so bad everywhere, and the little boys are out in full force.\"\n\nEnterprising youngsters relished muddy days, because they would carry wooden planks around the streets and charge ladies a few pennies to have an improvised boardwalk placed before them to walk upon. \"You go,\" said Elizabeth. \"You go and tell me all about it later.\"\n\n\"I won't tell you a thing,\" Emma retorted. \"It's your own choice to stay home, and you'll have to suffer the consequences.\"\n\n\"If you tell me about the president's address,\" Elizabeth said, \"I promise to describe for you every detail of the gown Mrs. Lincoln will wear to the inaugural ball\u2014and I'll show you how I fashioned the train.\"\n\nEmma hesitated, tempted, but when she turned to go she merely lifted her chin airily and said, \"I'll think about it.\"\n\nEmma's desire to learn a new dressmaking skill won out, and so when she returned to the boardinghouse in the early afternoon, she came to Elizabeth's rooms straightaway, eyes shining, features bright and animated. The parade had been splendid despite the mud, with horses and men marching proudly and bands playing as merrily as music was ever heard. A team pulled a model ironclad gunboat, complete with a revolving turret that startled and delighted onlookers by firing blanks as it made its way down Pennsylvania Avenue. Smartly attired representatives from fire departments of Washington and Philadelphia, civic organizations, and fraternal lodges from across the North had marched proudly, carrying banners and flags. A local printers' society had mounted a hand press on a wagon, and as they processed along, they cheerfully printed broadsides and distributed them to spectators they passed along the way. Most gratifying of all, for the first time people of color had marched in the parade too, a battalion of colored soldiers as well as distinguished leaders of several Negro civic associations. People of their race were at last included in the inauguration, fully part of the celebration and ceremony, not merely onlookers in the crowd or the unseen workers who cooked the food and cleaned up afterward.\n\nLater, on the muddy Capitol grounds, Emma and her friends had stood in the crowd beneath overcast skies threatening rain, waiting with thousands of others for President Lincoln to emerge from within and take his place on the East Portico with the newly completed Capitol dome above. \"And then he stepped out, a sheet of paper in his hand,\" Emma said, glowing with remembered awe. \"As soon as the people recognized him, they let out a great roar of welcome and gladness, and just then\u2014oh, you should have seen it, Elizabeth!\u2014at the moment he took his place, the clouds parted and the sun broke through, and a bright shaft of sunlight shone down upon him like a blessing from heaven.\"\n\nElizabeth leaned forward eagerly, captivated. \"And what did he say?\"\n\n\"I don't recall,\" Emma said lightly, with an indifferent shrug. \"You can read about it in the papers tomorrow.\"\n\n\"Emma!\"\n\nShe laughed. \"I'm only teasing, but you deserved it. Oh, it was a marvelous speech. Brief, but all the better for it in my opinion.\"\n\nElizabeth nodded. \"Brevity is his custom in daily speech too, unless he's telling a story or reading aloud from one of his favorite authors.\" As soon as she spoke, she could have kicked herself for how puffed up and pompous she sounded, boasting about her familiarity with the president.\n\nIf Emma thought she was showing off, she gave no sign of it. \"It was a lovely address, clear and sad and warm, full of forgiveness and reconciliation,\" she said. \"He talked about the war, and how slavery was the cause of it, and how four years ago everyone, North and South alike, had wanted to avoid war, but one side would make war rather than let the nation survive, and the other would accept war rather than let it perish. He talked about the Lord too, and how strange it is that each side prays to the same God and invokes His aid against the other.\"\n\nElizabeth nodded. It was something she had often thought in the months since George was killed. She had prayed for her son every day and every night he was at war, and somewhere, the mother of the young man who had killed George had been praying for her son too.\n\nSuddenly Elizabeth was struck by the realization that after so many years of war, the rebel who had killed George could very well have fallen as well, cut down by a Mini\u00e9 ball or disease or dreadful accident. Another woman might have felt a surge of righteous satisfaction at the thought, but Elizabeth felt only sorrow.\n\n\"Mr. Lincoln suggested that the Lord sent us this terrible war as punishment for the offense of slavery,\" Emma went on, \"and that the war may be a mighty scourge to rid us of it.\"\n\n\"Perhaps it is,\" said Elizabeth softly.\n\n\"He ended with words so profound that I wrote them down as soon as he finished.\" Emma withdrew a scrap of paper from her pocket and unfolded it. \"These were the last lines as best I could remember them: 'With malice toward none, with charity for all, with firmness in the right as God gives us to see the right, let us strive on to finish the work we are in, to bind up the nation's wounds, to care for him who shall have borne the battle and for his widow and his orphan, to do all which may achieve and cherish a just and lasting peace among ourselves and with all nations.'\" Emma smiled self-consciously, folded the paper, and tucked it into her pocket with a shrug. \"I probably didn't do it justice. The whole speech will likely be in the paper tomorrow, so you'll be able to read it properly then.\"\n\n\"I'm sure you captured the spirit of it, if not every word.\" Elizabeth's throat tightened with emotion, and she blinked away tears. \"I'm glad I didn't have to wait for the morning to hear those words.\" So compassionate, so true, just like the man who had spoken them.\n\n\"You should have been there,\" Emma scolded her fondly, shaking her head.\n\n\"I wish I had been, despite the ankle-deep mud and the crowds. And now, Emma,\" said Elizabeth, lowering her voice conspiratorially, \"how would you like to be the only person in Washington\u2014the entire United States, rather\u2014besides Mrs. Lincoln and myself who knows what she will be wearing to the inaugural ball?\"\n\nEmma's pretty face lit up with joy as Elizabeth described the rich, pure white satin, the needlepoint lace shawl, the elegant draping and exquisite trims. Emma hung on every word as intently as if she thought she might be required to re-create the gown from memory later. She already knew how to sew the narrow vertical pleats in the back of a mantua bodice so it would fit the body like wallpaper; Elizabeth herself had taught her to make the tiny, interlocking stitches that could withstand the strain across the figure and avoid unattractive gaping. She had also shown Emma how to use longer, looser stitches for the skirts so the seams would not pucker and ruin the lines of the gown, and after much practice Emma could do so flawlessly.\n\nEmma was becoming quite a skilled seamstress, but Elizabeth had so much more to teach her. With her natural talent, bright mind, and deft fingers, Emma would surely master every technique, and in time she could become as accomplished as Elizabeth herself\u2014perhaps even more so.\n\nElizabeth believed that a student who surpassed her would indeed be a fine legacy\u2014more precious and enduring and gratifying than all the beautiful gowns she had made for the ladies of Washington City, even those she had made for the First Lady of the land.\n\nLater that evening she picked her way through the muddy streets to the White House, where she found Mrs. Lincoln in a state of nervous excitement. \"I'm sure I'm not the first to offer you congratulations on this momentous day,\" Elizabeth told her warmly, \"but I offer them all the same, and I hope you'll find them among the most sincere.\"\n\n\"Thank you, Elizabeth,\" Mrs. Lincoln said, sighing, \"but now that we have won the position, I almost wish it were otherwise. Poor Mr. Lincoln is looking so brokenhearted, so completely worn out. I fear he will not get through the next four years.\"\n\n\"Of course he will,\" said Elizabeth stoutly. \"The campaign taxed him, but that's over now, and spring is here, and news from the front has never been more cheering.\"\n\n\"Well, that's certainly true,\" admitted Mrs. Lincoln, without looking at all heartened. \"If only this terrible war would be over! I confess that I live in dread of the inauguration ending, because when it does, Robert must return to the war.\"\n\n\"I thought his post was safe.\"\n\n\"He isn't marching out with the infantry if that's what you mean, but he goes wherever General Grant goes, and the general is at the front.\"\n\nHer voice broke with fear and worry, and Elizabeth's heart went out to her. The war must end soon, she almost said, but people had been saying that for so long that the words had ceased to have any meaning. But surely now, even after so many false hopes and disappointments, the phrase finally rang with truth. On every side the Confederates were losing ground and the lines of Union blue advanced in triumph. Almost every day, Elizabeth could look out her window and see artillery going past on the way to fire a salute in honor of some new victory. Even so, she understood why Mrs. Lincoln would worry incessantly until the war was over and Robert was entirely out of danger.\n\nWhen it seemed as if Mrs. Lincoln might sink into a brood, Elizabeth endeavored to draw her out by asking for her impressions of the inauguration. \"Mr. Lincoln spoke brilliantly,\" Mrs. Lincoln replied, brightening a trifle. \"Did you know he missed the entire procession?\"\n\n\"I didn't.\"\n\n\"It's true! He had so much business to attend to that he went to the Capitol ahead of time in his barouche, and he was there signing bills until the last minute.\" She uttered a small laugh, an encouraging sign. \"So all the people lining the streets later and cheering him as his carriage passed at the head of the parade\u2014why, they were cheering only me. I doubt they would have carried on so had they known.\"\n\nElizabeth smiled. \"I'm sure many would have.\"\n\nMrs. Lincoln laughed again, scoffing and yet pleased. \"Oh, Elizabeth, that small deception was not even the greatest scandal of the hour.\" She paused dramatically. \"When the vice-president\u2013elect arrived to take his oath of office, he was drunk.\"\n\n\"No!\"\n\n\"Indeed he was, and he downed nearly two tumblers of brandy right there in the Senate chamber.\" Mrs. Lincoln had become more animated as she warmed to her subject. \"Mr. Hamlin made some perfectly lovely, gracious remarks to introduce his successor, and then Mr. Johnson came up and began spouting the most astonishing, rambling harangue I've ever had the misfortune to witness.\"\n\n\"How shocking,\" exclaimed Elizabeth. \"What on earth was he thinking?\"\n\n\"I don't know that he was thinking at all. He was red-faced and barely coherent, and when the secretary of the Senate tried to bring the appalling performance to a close, Mr. Johnson persisted as if he were quite deranged.\" Mrs. Lincoln shook her head. \"My poor husband entered in the middle of this disaster and stood with his head bowed, enduring the embarrassment in dignified silence and waiting patiently until Mr. Johnson finished and took his oath of office.\"\n\n\"It was not the president's embarrassment, but Mr. Johnson's,\" said Elizabeth.\n\n\"Why, certainly, but it spoiled the occasion all the same.\" Mrs. Lincoln pursed her lips and shook her head. \"What a dreadful debut. I doubt he will ever live this down. None of us who were there will ever forget it.\"\n\nElizabeth could tell by her sharp frown of disapproval that Mrs. Lincoln would never forgive him either.\n\nElizabeth was arranging Mrs. Lincoln's hair when the president entered, so Elizabeth went to him, extended her hand, and offered her sincere congratulations. \"Thank you,\" he said, grasping her outstretched hand warmly and holding it. \"Well, Madam Elizabeth, I don't know whether I should feel thankful or not. The position brings with it many trials. We do not know what we are destined to pass through. But God will be with us all. I put my trust in God.\" He released her hand and crossed the room to sit down upon the sofa, his expression solemn.\n\nElizabeth felt painful sympathy for both wife and husband, First Lady and president. From every region of the fractured nation came glorious news of the Union Army's successes, and yet, alone in their private chambers, the Lincolns looked careworn, sad, and anxious on a day that should have been their triumph. In her quiet way, she tried to cheer them with pleasant conversation, and by the time she finished dressing Mrs. Lincoln, it did seem that their spirits had risen at least a little. Mrs. Lincoln took the president's arm, and as he led her off downstairs where thousands of citizens waited to meet them, Mrs. Lincoln called over her shoulder, \"I'll have that glove for you Monday night, Elizabeth, when you come to dress me for the Inaugural Ball.\"\n\nElizabeth smiled, pleased that she had remembered.\n\nWhile the Lincolns welcomed the public and graciously accepted their congratulations, hiding their weariness and worry rather than spoil the occasion for their many well-wishers, Elizabeth attended a smaller but not less joyful gathering of Washington's colored elite. With an effort, Elizabeth put aside her concerns for the dispirited Lincolns for the moment and joined her friends and acquaintances in celebrating Mr. Lincoln's victory, certain that he would accomplish great things for the nation and its people of color in his second term. Mr. Frederick Douglass was in attendance, and as a longtime admirer Elizabeth was very pleased to have the chance to speak with him. He captivated all within earshot by relating an incident that had occurred at the White House scarcely two hours before. Many people of color had come to Washington for the inauguration, and dozens of them had desired to attend the levee, but they had not been permitted to enter. Mr. Douglass had stood on the edge of the crowd, already mentally composing a righteously indignant letter of protest, when a member of Congress spied him, remarked about the press of the immense crowd, and asked, \"You are going in, of course?\" When Mr. Douglass told him, regretfully, that he would not, the congressman exclaimed, \"Not going in to shake the president by the hand! Why, pray?\"\n\n\"The best reason in the world,\" Mr. Douglass had said, his tone dignified but ironic. \"Strict orders have been issued not to admit people of color.\"\n\nElizabeth could not imagine Mr. Lincoln issuing such a command on such a day, and she wondered which of the secretaries had been responsible for it, or if some misunderstanding had occurred between the crowd and the doormen. Just as she was about to suggest that Mr. Douglass return to the White House and try again, Mr. Douglass continued his tale, explaining that the congressman had been quite perturbed that he had been \"placed under ban.\" He had taken the famed orator inside, led him through the crowd to the president, and asked permission to introduce them. Mr. Lincoln readily agreed, and soon Mr. Douglass stood face-to-face with the president, who shook his hand and said, \"Mr. Douglass, I am glad to meet you. I have long admired your course, and I value your opinions highly.\"\n\nMr. Douglass was obviously proud of the manner in which the president had welcomed him, and all who heard him tell his story were proud too, and pleased that the president had treated one of their leaders with such respect and interest. Elizabeth was not at all surprised, not only because she had observed the president receiving guests often enough to know that he never failed to be courteous, but also because she herself had granted a friend's request to arrange a meeting between the president and the former slave and abolitionist Sojourner Truth the previous October. She had not witnessed their conversation, but afterward she learned that Sojourner Truth had spoken well of the president and had been honored and pleased by his attention. Mr. Lincoln did not have a perfect record of dealing with the colored race, as Elizabeth would be among the first to admit, but he was learning, and she was confident that his abundant compassion for his fellow man would guide him to an even greater understanding of their unique concerns and hopes for the future.\n\nAlthough she would never exaggerate her position by claiming the title of adviser, Elizabeth liked to think that she too had played some small part in helping President Lincoln know the desires and worries of colored people better. She hoped she had used, and would always use, her acquaintance with the president and her time in the White House for the good of her race.\n\nThe crowds who had come to the capital to mark President Lincoln's second inauguration departed soon after the ceremonies, but the city remained full of strangers, with more arriving every day. Confederate soldiers were abandoning General Lee's army in droves, and while most simply went home, others crossed into Union lines and surrendered. Some straggled into Washington on foot, the tatters of their gray or butternut uniforms hanging from their emaciated frames, but most arrived around four o'clock every afternoon on the \"deserters' transport,\" disembarking on railway platforms one or two hundred at a time. Once in the Union capital, they took the oath of allegiance and were assigned to work on farms, in factories, or upon the western frontier. Until they shipped out to their new posts, they were permitted to wander the city as they pleased, striking up conversations and sharing tobacco with Union soldiers who had been their mortal enemies not long before, their Confederate uniforms drawing curious, suspicious glances until they became commonplace. Indeed, the sight of the thin, ragged former rebels became so typical throughout Washington that at least one newspaper reporter worried that they might in fact represent a devious invasion by the enemy, massing their numbers and awaiting the order to strike at the heart of the Union from within. But the truth seemed to be far simpler. The Confederate soldiers were starving, and they had realized that an army that could not feed its soldiers could not withstand its opponents much longer. They were hungry and tired and sick of war, and many resented futilely persisting in what they called a \"rich man's war but a poor man's fight.\" So they had withdrawn, trusting that they would not be captured, and that if they were, they would not be shot for their crime. Indeed, it seemed to Elizabeth that there could not be enough soldiers left in the rebel army to shoot all the deserters.\n\nLater, Elizabeth would wonder if the Confederate government's new measure to increase their army had compelled some of their soldiers to desert. Soon after the inauguration, Northern newspapers announced that the Confederate congress had voted to allow slaves to enlist in the rebel army, and thereby earn their freedom. Elizabeth and every other person of color she knew, from her friends in the boardinghouse to her fellow churchgoers at Union Bethel to the former slaves she assisted in the freedmen's camps, wondered how anyone of their race could agree to fight to preserve the institution that had kept them and their families in bondage and degradation. They were shocked by reports that, a mere nine days after Jefferson Davis signed the measure into law, three companies of Confederate Negro soldiers were drilling in Richmond's Capitol Square. Elizabeth felt the sharp sting of betrayal whenever she thought of colored men in rebel gray, but she felt profoundly sorry for them too. They surely did not understand what was happening in the wider country around them, or they never would have made such a bewildering choice or, as Virginia grimly called it, a deal with the devil.\n\nThe inauguration brought newcomers not only to Washington but also into the most intimate circles of the White House, for between the November election and the commencement of President Lincoln's second term, several members of his cabinet resigned, and others were named to replace them. Iowa senator James F. Harlan assumed the post of secretary of the interior, a staffing change that seemed to please Robert Lincoln, who was\u2014not as secretly as he seemed to believe\u2014courting Mr. Harlan's daughter, Mary. Elizabeth too was glad to see Mr. Harlan elevated to such an important post, because his wife, a kind, gracious woman, was one of her favorite patrons, and Elizabeth was very happy for her.\n\nOn April 3, a Monday, Mrs. Harlan had come to Elizabeth's reception room at the boardinghouse with material for a new dress, a lovely green-and-white striped silk. \"I'm not certain about the color,\" Mrs. Harlan mused as Elizabeth examined the fine fabric. \"I fear it will make me appear sallow.\"\n\n\"Oh, I don't think it shall.\" Elizabeth beckoned Mrs. Harlan to step closer to the window, where she draped the fabric over Mrs. Harlan's shoulder and bosom and stepped back to study the effect. \"I think it's very becoming to your complexion and it suits your hair and eyes well.\"\n\n\"Mr. Harlan likes me in green.\" Mrs. Harlan had to raise her voice to be heard over a sudden clamor in the streets, a cacophony of passing artillery.\n\n\"All the more reason to wear it often,\" Elizabeth replied, nearly shouting as whistles and cheers joined the din.\n\n\"What is going on out there?\" Mrs. Harlan wondered aloud, peering out the window.\n\n\"They must be on their way to fire off a salute. We've become quite accustomed to the show around here.\" Then, frowning thoughtfully, she added, \"I admit this does seem more exuberant than usual.\"\n\nMrs. Harlan's eyebrows rose. \"This must mean good news, then.\"\n\n\"It surely must.\"\n\nA look of understanding passed between them, and together they hurried outside. \"Excuse me, sir,\" Elizabeth called to a man whistling cheerfully as he strode after the artillery. \"What's happened? What's the news?\"\n\n\"What's the news, you ask?\" The man whooped and threw his hat into the air. \"Richmond's fallen, that's the news!\"\n\nAs the man hurried off to retrieve his hat and rejoin the impromptu parade, Elizabeth gasped, Mrs. Harlan cried out, and then they joined hands and twirled about in a circle, laughing and cheering. \"I must tell the girls,\" Elizabeth exclaimed, and dropping Mrs. Harlan's hands, she ran across the street to her workrooms. \"Emma, girls,\" she called out as she burst into the room where her assistant seamstresses sewed. \"Richmond has fallen!\"\n\n\"We heard! We know!\" Elated, Emma threw her arms around her, and only then did Elizabeth notice her young assistants were laughing and crying and embracing all around her. \"Did you hear the best part? It was colored soldiers that took the city. _Our_ soldiers!\"\n\nElizabeth's spirits soared. Speechless with happiness, she clasped her hands to her heart and laughed aloud.\n\n\"That's not the best part,\" another seamstress called out joyfully. \"The best part is that you promised us a day off when Richmond fell.\"\n\nEveryone burst out laughing, and Elizabeth joined in, helpless to do otherwise, until, catching her breath, she shook her head and waved them to silence. \"I can't send you all home,\" she protested. \"Mrs. Harlan is waiting across the way with silk for a new dress.\"\n\nA chorus of dismay greeted her words. \"Mrs. Harlan can't want to stay and fit the lining for a dress now,\" Emma protested. \"Surely she wants to celebrate too.\"\n\nWhen the other seamstresses chimed in, urging her to at least go and ask, Elizabeth wavered. \"Don't leave yet,\" she said, but her irrepressible smile undercut whatever sternness she might otherwise have mustered. She dashed back across the street, where she found Mrs. Harlan in her reception room, gathering up the silk they had let fall to the floor in their excitement.\n\n\"Mrs. Harlan,\" Elizabeth exclaimed, hurrying to help. \"I'm terribly sorry. I had to share the good news with my assistants, and as it happened they already knew, and they reminded me of a promise I had made them months ago, that when Richmond fell I would give them the day off, although I realize it's terribly inconvenient\u2014\"\n\n\"Not at all,\" Mrs. Harlan assured her, smiling. \"For such good tidings, I would gladly wait another day or two for my dress. You must keep your promise. Give your girls a holiday and a treat, by all means.\"\n\nElizabeth thanked her profusely and promised to begin fitting her lining the next day. Mrs. Harlan agreed, and as she departed for her home, Elizabeth put the green-and-white striped silk in a safe place, snatched up her bonnet, and hurried back to her workrooms. \"More good news, girls,\" she called out, but she didn't need to say any more, because they had guessed that their long-promised holiday had come at last. They implored Elizabeth to join them, and this time she agreed. Arm in arm, they joined the celebration already spilling over into the streets, their hearts overflowing with joy, their happiness reflected in the faces of the people they passed, clerks and shopkeepers and housemaids and waiters whose businesses had also declared a holiday. Residents draped patriotic banners and bunting from their windows, and bands quickly formed up on street corners and in parks to play spirited marches and merry jigs. Crowds gathered outside the homes and offices of various dignitaries and called for them to come out and address them, but of the many who complied, only the few loudest could be heard over the din. An eight-hundred-gun salute shook the city, three hundred booms for the fall of Petersburg, five hundred for Richmond. As the afternoon passed, Elizabeth observed many young men\u2014and many more without the excuse of the foolishness of youth\u2014celebrating by indulging in too much liquor, and she was alternately scandalized and amused to observe neighbors she knew to be sober, responsible folk tottering down the streets, singing and proclaiming the glory of President Lincoln, General Grant, and the Union Army in loud, slurring voices. Tomorrow they would regret their overindulgence, but for the moment, nothing could diminish their rejoicing, or Elizabeth's.\n\nPresident Lincoln had been in Virginia since the last week in March, and he was meeting with General Grant in Petersburg when Richmond fell. He decided to tour the captured Southern capital the next day, and when Mrs. Lincoln heard of his plans, she proposed to meet him at City Point on the James River and accompany him into the conquered city.\n\nThis would be Mrs. Lincoln's second trip to General Grant's City Point headquarters within two weeks. She and Tad had traveled there with the president and a small party the week before, but Mrs. Lincoln had returned early, and alone, leaving Tad behind with his father. Mrs. Lincoln had not explained why she had cut her visit short, and although she had been abrupt with Elizabeth when asked about her travels, she had revealed enough for Elizabeth to conclude that the tour had been something of a disaster. When the party had gone out to review the troops, the president had ridden ahead on horseback accompanied by General Grant and two officers' wives, but Mrs. Lincoln and Mrs. Grant had been obliged to trail after them in an ambulance slowed to a crawl by shin-deep mud. Mrs. Lincoln finally arrived to find that the review had already begun and that the beautiful wife of Major General Ord was riding alongside Mr. Lincoln in a place of honor that properly belonged to herself. Seized by one of her jealous fits, she gave the young Mrs. Ord a terrible tongue-lashing, hurled abuse upon the astonished Mrs. Grant, and ferociously attacked her husband in front of everyone, demanding that he fire Major General Ord immediately, a command her husband quite reasonably ignored. Elizabeth did not know whether Mrs. Lincoln had fled back to Washington in shame or if Mr. Lincoln had ordered her away, but apparently, time to reflect and knowing that the incident had cost her a chance to see her son Robert had chastened her. When she mentioned that she was determined to try again to see Richmond, Elizabeth asked if she might accompany her. Years before, Petersburg had been her home, and she was curious to see it again, to stroll along its familiar streets as a free woman.\n\nOn April 5, at eleven o'clock in the morning, Mrs. Lincoln, Tad, Elizabeth, and the rest of their party, which included Senator Charles Sumner; Secretary of the Interior James Harlan; Mrs. Harlan, whose green-and-white striped silk dress Elizabeth had belatedly begun; Miss Mary Harlan, the young lady Robert Lincoln was courting; and several other gentlemen departed Washington on the steamer _Monohasset_. When they arrived at Fort Monroe the following morning, Mrs. Lincoln learned that President Lincoln had entered Richmond two days before.\n\n\"Are you sure you meant to say Richmond?\" the First Lady queried the aide-de-camp, crestfallen. Elizabeth knew she had strongly hinted to her husband that she wanted him to wait for her so they might tour the fallen capital together. \"In his telegram he said he went to Petersburg.\"\n\nHe had done that first, Mrs. Lincoln was promptly assured. It was the following morning that he had entered Richmond, scarcely a day after the Confederate army had evacuated, while flames still licked at the ruins. The aide-de-camp, apparently oblivious to Mrs. Lincoln's deepening frown, went on to tell them that a group of colored workmen had recognized the president from a distance and, to his embarrassment, had shouted, \"Glory, hallelujah!\" at his approach and had fallen to their knees to kiss his feet. \"Please don't kneel to me,\" President Lincoln had told them. \"You must kneel only to God and thank Him for your freedom.\" Escorted by the German-born general Godfrey Weitzel, Mr. Lincoln had toured the Confederate White House and had sat at Jefferson Davis's desk. Later he and his escort had passed the infamous Libby Prison, where thousands of captured Union soldiers had suffered starvation, disease, and unimaginably cruel treatment\u2014and where Mr. Lincoln's own brother-in-law, Confederate captain David Humphreys Todd, had served as a warden.\n\n\"I see,\" said Mrs. Lincoln flatly when the aide-de-camp finished his report. Immediately she stormed to the telegraph office and sent her husband several urgent messages imploring him to wait and to allow her party to join him on his boat, since theirs was dreadfully uncomfortable and she much desired to see Richmond by his side. After some back-and-forth, the arrangements were made, and soon the First Lady's party joined the president's aboard the _River Queen,_ steaming up the James River, traveling with swift ease along a route that not long before had been impassible even to Union gunboats. Elizabeth spent hours on deck, holding on to the railing and lifting her face to the sun, savoring the breeze and the pure, balmy air. The river flowed along majestically, and its banks were lovely and fragrant with the first sweet blossoms of spring. Beyond them stretched fair fields, the very image of peacetime bounty\u2014but all too often, the illusion of prosperity was broken by glimpses of deserted army camps and ruined forts, the ugly detritus of war.\n\nElizabeth had not glimpsed the fertile fields and green hills of her birthplace for years, and she had long yearned to see them again. Virginia would forever remind her of her mother, and her father, her aunts and uncles and cousins, her son, George, as a babe in her arms, her mother's warm kisses, the hours spent in the company of those she loved best. Her childhood had been difficult, full of pain and fear and loss, and yet she treasured a few precious, golden memories of those years. When she had embarked for City Point, a part of her so deep she was unaware of it had expected the journey to reunite her somehow with all she had lost. As the _River Queen_ carried her along, the sad, unsettling truth of her own expectations dawned upon her, and she felt the first stirrings of regret.\n\nPerhaps she should not have come.\n\nAt long last the _River Queen_ arrived in Richmond, and, brimming over with curiosity, Mrs. Lincoln's party entered the smoldering city. Elizabeth would not admit it aloud, but she also felt the smallest stirring of apprehension. The streets were unnaturally subdued, save for a few citizens who averted their eyes as they hurried past on business of their own, and occasional patrols of Union soldiers. The Virginia statehouse where the Confederate congress had met was in a state of disarray that spoke of fear and haste\u2014desks broken, papers scattered, chairs overturned as if their last occupants had fled in alarm. While her companions walked ahead, marveling at the desolation the Confederate congress had left in the wake of their flight, Elizabeth gathered her skirts in one hand and bent to pick up a handful of documents. She skimmed the first page, and her abrupt, ironic laugh when she realized what she held caused Mrs. Lincoln and Mrs. Harlan to glance over their shoulders at her, curious. \"It's a resolution,\" Elizabeth said, indicating the papers. \"It prohibits all free colored people from entering the State of Virginia.\"\n\nSenator Sumner shook his head and snorted, but another gentleman in the party, the Marquis de Chambrun, smiled and said, \"My dear madam, I think you need not fear that anyone will come to arrest you on that account.\"\n\n\"Thank you, sir,\" Elizabeth replied. \"I confess that had not even occurred to me.\"\n\n\"Are you going to save those papers as a keepsake?\" Mrs. Lincoln inquired. She often teased Elizabeth about her attachment to objects that evoked fond memories.\n\n\"I think not,\" said Elizabeth shortly, and let the papers fall to the floor.\n\nThey moved on to the senate chamber, where Elizabeth sat in President Jefferson Davis's chair and gazed around the room, imagining it filled with rebel senators and contemplating the many decisions he must have made from that very place, and what they had cost him, what they had cost them all. When she moved on to the vice president's chair to give the others a chance to try the president's, her thoughts flew to Mrs. Davis and her children. She imagined them in flight or in hiding, hastening south to the imperfect sanctuary of their Mississippi home or cowering beneath the sheltering roof of a brave and loyal friend. Wherever they were, Elizabeth hoped Mrs. Davis and the children were safe. They were blameless, and she prayed no harm would come to them. As for Mr. Davis, before the war she had liked him and had considered him a gentleman, but he had made his choices and would have to live with the consequences. She would never wish suffering upon him, but she thought it unlikely that he would survive the war unscathed.\n\nLater the party visited the Confederate executive mansion on K Street, which had also been called the White House, although it was gray stucco. The Richmond ladies in charge of its safekeeping glared at them darkly as they toured the elegant three-story residence, but they were in no position to bar any of them from exploring wherever they pleased. Elizabeth could not help but search each room for signs of her former patron\u2014a forgotten quilt in the second-floor nursery, a dress that could not be stuffed into an overfull satchel\u2014but she found nothing, and the absence made her feel oddly despondent.\n\nWith a will, she endeavored to cease brooding and to adopt her companions' cheerful curiosity. Eventually their lively spirits elevated hers, and by the time they returned to the _River Queen,_ she too felt a flush of triumph and renewed hope that the end of the war could not be long in coming.\n\nIt was a merry party that gathered around the dining table in the ship's cabin that evening. For Elizabeth it was a special pleasure to dress Mrs. Lincoln for an event to which she herself was invited. Introductions were made all around, and they were chatting pleasantly about their impression of Richmond when one guest, a young captain attached to the Sanitary Commission, turned to the First Lady and said, \"Mrs. Lincoln, you should have seen the president the other day, on his triumphal entry into Richmond. He was the cynosure of all eyes. The ladies kissed their hands to him, and greeted him with the waving of handkerchiefs. He is quite a hero when surrounded by pretty young ladies.\"\n\n\"Is he, indeed,\" said Mrs. Lincoln, her voice brittle with frost.\n\nThe young officer's pleasant smile faltered. \"Why, yes, Mrs. Lincoln. Quite a hero.\"\n\nElizabeth's heart sank, but before she could think of a way to gracefully change the subject, Mrs. Lincoln fixed him with a steely glare and said, \"And do you find him that way often? Surrounded by pretty young ladies, I mean?\"\n\nThe captain flushed. \"Why, no, Mrs. Lincoln, that is to say\u2014\" He glanced frantically down the length of the table, but his friends were too startled to come to his rescue.\n\n\"That is to say what, precisely?\" Mrs. Lincoln prompted him sharply.\n\nElizabeth muffled a sigh and resisted the urge to slouch in her chair and stare at her plate as Mrs. Lincoln needled the poor young man, who would surely never forget his wretched evening aboard the president's steamer. Mrs. Lincoln made quite a scene before she wore herself out, and Elizabeth felt both indignant and embarrassed as she observed the other guests exchanging knowing looks and regarding the president with sympathy. They did not understand what Mrs. Lincoln had been through those past few days, how overtired she was from her travels, or they would have shown her more compassion and understanding.\n\nEven so, Elizabeth found herself wishing that Mrs. Lincoln could learn to rein in her tempers, so it would not always fall to others to accommodate them.\n\nThe next morning, the entire party decided to visit Petersburg, and Elizabeth was only too eager to accompany them. As the president's special train sped them ever closer to the city where Elizabeth had been a slave twenty years before, her heart pounded with excitement and dread. She did not know what she would find there, what, if anything, would remain of the life she had once known\u2014what, if anything, she wanted to remain.\n\nWhen they disembarked at the station, Elizabeth parted company with the others; while they headed out to inspect forts and hospital camps and to confer with generals, she wandered off on her own in search of people she had known in days gone by. The city was at once both unsettlingly familiar and greatly changed. Many houses she had once admired had suffered neglect or damage from artillery; some shops that she had often visited on errands for her mistresses had shuttered their windows or displayed different names in their windows or upon their doors. She glimpsed familiar faces among the people she passed, but they walked by her without a glimmer of recognition in their eyes. Almost unwillingly, she made her way down to the riverfront where she and James had once strolled, sharing conversation and hopes and dreams\u2014and in James's case, lies. Her heart sank deeper into sorrow with every step, and she was just about to turn and hasten back to the train station when she heard a woman behind her call, \"Elizabeth? Lizzie Hobbs?\"\n\nWith a start, Elizabeth whirled around and spotted a face she knew well\u2014twenty years older, much thinner, with silver in her black hair and lines around her mouth, but still as dear to her as ever. \"Martha? Can it truly be you?\"\n\nThe woman nodded, tears filling her eyes. \"It is.\"\n\n\"Oh, my dear friend,\" Elizabeth gasped as she hurried to close the distance between them and embrace her. \"I never thought to see you again.\"\n\n\"I never thought to see you.\" Martha held her out at arm's length, and when she smiled, Elizabeth saw that her bottom front teeth were missing. \"What on earth are you doing here? I thought you'd bought your freedom and disappeared up north.\"\n\n\"I didn't disappear,\" Elizabeth replied, laughing tearfully, hiding her shock. Martha was thin, so very thin, and her dress was worn and patched. Martha was a talented seamstress, a freeborn colored woman who had worked at Mrs. Miller's dry goods store. Though she was five years older than Elizabeth, they had become fast friends thanks to Elizabeth's frequent visits to purchase fabric, thread, needles, and other notions for her mistresses. \"I've been living in Washington City since before the war. I've set myself up as a dressmaker there.\"\n\n\"How wonderful!\" Martha shook her head in admiration. \"You always did work marvels with fabric.\"\n\n\"How is your husband? Did he ever buy that land north of the river he liked so well?\"\n\n\"No, no, he never did.\" Martha's smile faded. \"He passed on ten years ago from smallpox, but my girls are doing well. My eldest is married now and has two little ones. How is your George?\"\n\nElizabeth dug her fingernails into the palm of her hand to hold back the grief that suddenly welled up within her. \"He's gone. He was killed in the war, in the Battle of Wilson's Creek in Missouri.\"\n\n\"Oh, my goodness. He was a soldier, you mean. I'm so sorry he's gone.\"\n\n\"I'm sorry for your loss too.\"\n\n\"But you must be so proud.\"\n\n\"Yes,\" Elizabeth forced herself to say. \"Yes, very proud.\"\n\nThey walked along together for a while, and Martha reunited her with several mutual friends who remained in the city. Throughout the war years, every one of them had endured hardships and deprivations Elizabeth could not have imagined while she dwelled comfortably in her boardinghouse on Washington's pleasant Twelfth Street, but none of them complained or said more than was necessary about their suffering. They all admired her dress, which was only her second best but was well made and whole, and expressed joy without even the smallest taint of envy that she had done well for herself. Her heart went out to them, and she wished with all her strength that she could help them. She had a few Union dollars in her pocket, which they first refused and then gratefully accepted when she reminded them of debts she owed them from her Petersburg years, debts they did not recall because she had invented them on the spot. When she bade her old friends good-bye, she urged them to write to her if ever she could do anything for them.\n\nShe said good-bye to Martha last of all, and remembering her skill with needle and thread and cloth, she urged her to make her way to Washington City if she could. \"I have too many dresses to make and not enough hands,\" she said. \"You could earn a good living, working with me.\"\n\nMartha looked as if she wished with all her heart she could accept, but she shook her head. \"My girls need me,\" she said, \"and Richmond is my home. I can't leave.\"\n\n\"But it will be so hard for colored folks in the South when the war is done.\"\n\nMartha managed a smile. \"It's always been hard, and we've always gotten by.\"\n\nBefore they parted ways, Elizabeth made Martha promise that she would at least keep the offer in mind, and then, heavyhearted, she made her way back to the place where the presidential party had agreed to meet. The sad sights of the day mingled with distant memories so painfully that she was eager to board the train and depart, but the president did not wish to set out right away. On an earlier visit, he had seen a large, peculiarly shaped oak tree that he was quite keen to show them, so they willingly accompanied him to the outskirts of the city, where it grew in stately solitude. Only after admiring it\u2014and even Elizabeth, in her melancholy, had to admit that it was indeed a magnificent specimen\u2014did they return to the train station and leave Petersburg.\n\nElizabeth was not sorry to put the city behind her, but as if to jeer at her eagerness to go, the train moved along at a crawl back to City Point, for what reason, Elizabeth could not fathom. Their pace was so slow that President Lincoln was able to observe a terrapin basking in the warm sunshine on the wayside. He called for the conductor to stop the train and had one of the brakemen bring the creature to him, and he and Tad amused themselves with it all the way to the James River, where their steamer waited. The president's obvious delight in the terrapin's lethargic, ungainly movements\u2014and the ridiculous similarity between the train's pace and the reptile's\u2014lifted Elizabeth out of her gloom, and so by the time they boarded the _River Queen,_ she was feeling somewhat restored, although her wistfulness lingered.\n\nFor a week the _River Queen_ remained on the James River, usually anchored at City Point, offering a pleasant, comfortable respite to all on board. General Grant and his wife visited the steamer several times, as did other officers and dignitaries, and whenever the president was not off on an excursion, he lounged about with the rest of the party, as comfortably at his ease as if they were all old friends.\n\nOn the day before they were to return to Washington, Mr. Lincoln went out for one last review of the troops, and in the evening he returned to the steamer thoroughly exhausted. \"Mother,\" he told his wife wearily, \"I have shaken so many hands today that my arms ache tonight. I almost wish that I could go to bed now.\"\n\nMrs. Lincoln murmured sympathetically and urged him to excuse himself and retire, but the president hated to disappoint his companions, so as the twilight shadows lengthened and the lamps were lit, he remained among them. When all was brilliantly illuminated, Elizabeth found herself utterly entranced by the lovely vision of the ship shining on the waters like a floating palace. A military band was on board, and as the night deepened, an air of enchantment enveloped the scene. Several officers came aboard to bid the president farewell, and around ten o'clock, they called upon him to make a speech. With some effort, the president rose and said, \"You must excuse me, ladies and gentlemen. I am too tired to speak tonight. On next Tuesday night I will make a speech in Washington, at which time you will learn all I have to say. And now, by way of parting from the brave soldiers of our gallant army, I call upon the band to play 'Dixie.' It has always been a favorite of mine, and since we have captured it, we have a perfect right to enjoy it.\"\n\nThe moment he sat down, the band struck up the tune, and when the last sweet notes faded away, the listeners applauded, and to Elizabeth they seemed almost reverential, as if each understood that the great, solemn duty that had occupied them for so long was almost accomplished, but the great task of rebuilding the Union yet awaited them.\n\nAt eleven o'clock, the last farewells were spoken, those who were to remain behind disembarked, the festive lights were taken down, and the _River Queen_ set out for Washington.\n\nAll the next day the steamer slowly traveled up the bay and the Potomac. To avoid debating reconstruction with Senator Sumner, the president read aloud from _Macbeth,_ dwelling somberly on the Scottish king's torment. When the _River Queen_ passed Mount Vernon, the marquis declared that in years to come, Mr. Lincoln's home in Illinois would be honored as reverently as President Washington's in Virginia. \"Springfield,\" Mr. Lincoln said, and to Elizabeth his voice seemed full of quiet longing. \"How happy I shall be four years hence to return there in peace and tranquility!\"\n\nThe steamer arrived at the capital at six o'clock on Sunday evening, where the travelers disembarked one last time and went their separate ways. It was not quite sunset, so Elizabeth walked home alone, enjoying her solitude after so much time in the company of others. The trip had been wonderful, enlightening, but she was looking forward to crossing the threshold of her own rooms and sleeping in her own bed. As she strolled along, she observed that the streets were rather full of people for a Sunday evening, especially a Palm Sunday evening, and that bonfires were burning as if to illuminate her way home. \"What's happening?\" Elizabeth asked a neighbor as she turned onto Twelfth Street and spotted him taking in the scene from his front steps. \"Is all this to welcome the president and First Lady home?\"\n\n\"It's a celebration,\" he called back, \"and a vigil. We're all just waiting. It won't be long now!\"\n\nElizabeth felt a thrill of excitement and gladness, and she almost thought she could guess his reply when she asked, \"Won't be long until what?\"\n\n\"Until it's over,\" her neighbor shouted gleefully. \"General Grant cut off Lee's retreat at Appomattox Court House. The rebels are surrounded!\"\n\n# Chapter Twelve\n\n## APRIL 1865\n\nAt daybreak, Elizabeth woke to a five-hundred-gun salute that shook her bed and rattled the windows. Through the walls of the boardinghouse, she heard other residents respond with sleepy cheers, and she knew at once that General Lee must have surrendered.\n\nQuickly she dressed and went downstairs to the foyer, where Virginia, Walker, Miss Brown, Emma, and a few other neighbors were gathering to share what little news they had. \"Does this mean the war is over?\" Elizabeth asked, but no one could say for certain. No one knew precisely what was happening in North Carolina between General Sherman and his Confederate counterpart, General Johnston. Walker proposed heading to the telegraph office for official word, but thunderclaps had picked up where the cannons had left off, and a heavy rain was falling. Elizabeth and Virginia had no desire to venture out in the downpour, but nothing would keep Walker and Emma from satisfying their curiosity, and so they set out together.\n\nWatching from the windows, Elizabeth estimated that thousands of others had headed out into the streets despite the storm, laughing and embracing and cheering. Elizabeth felt so lighthearted that she thought she might burst out in laughter or song, so as soon as the rain tapered off, she and Virginia gathered the Lewis girls and joined the celebration. Impromptu parades formed as civilians linked arms with soldiers and sang \"Rally Round the Flag\" and cheered as they followed bands marching through the muddy streets. Steam fire engines adorned with flags and bunting shrilled their whistles. Soldiers and mechanics towed a battery of six howitzers in from the navy yard and fired off salutes at whim. Elizabeth felt herself carried along with the crowd as it made its way to the White House, where she added her voice to the exultant, thankful chorus singing \"The Star-Spangled Banner.\" Shouts rang out for the president to come and make a speech. A cheer went up when Tad poked his head out of a window, and a louder cheer followed when he waved a captured rebel flag for the crowd's amusement. Before long Mr. Lincoln himself appeared, and a great roar went up, and hundreds of hats were flung into the air. When the din subsided enough for the president to be heard, he said, \"I am very greatly rejoiced to find that an occasion has occurred so pleasurable that the people cannot restrain themselves.\"\n\nElizabeth laughed and applauded along with the crowd.\n\n\"I suppose that arrangements are being made for some sort of a formal demonstration,\" Mr. Lincoln mused aloud, \"this, or perhaps tomorrow night.\"\n\n\"We can't wait,\" someone called out.\n\n\"We want it now,\" another cried, and soon hundreds of other voices chimed in with their agreement.\n\n\"If there should be such a demonstration, I, of course, will be called upon to respond,\" Mr. Lincoln protested, \"and I shall have nothing to say if you dribble it all out of me before.\"\n\nHis eager listeners responded with laughter and applause.\n\nMr. Lincoln looked out upon the crowd. \"I see you have a band of music with you.\"\n\n\"We have two or three,\" a man called back.\n\n\"I propose closing up this interview by the band performing a particular tune which I will name,\" Mr. Lincoln said. \"Before this is done, however, I wish to mention one or two little circumstances connected with it. I have always thought 'Dixie' one of the best tunes I have ever heard. Our adversaries over the way attempted to appropriate it, but I insisted yesterday that we fairly captured it.\" A great shout of assent met his words. \"I presented the question to the attorney general, and he gave it as his legal opinion that it is our lawful prize.\" As laughter and applause again rang out, the president raised his hand to the musicians. \"I now request the band to favor me with its performance.\"\n\nNever had Elizabeth heard the tune played more merrily. Afterward the band immediately struck up \"Yankee Doodle,\" and the crowd clapped along.\n\nAs the last notes faded, Mr. Lincoln said, \"Now give three good hearty cheers for General Grant and all under his command.\" The crowd did so eagerly, and Elizabeth joined in as loudly as anyone there. \"Three more cheers for our gallant navy.\" This command too was promptly obeyed.\n\nThen, with a modest bow, the president disappeared from the window, to the applause and whoops and shouts of the people below. Shortly thereafter, word spread through the crowd that they ought to march to the Department of War next and call for Secretary Stanton to address them too, but Elizabeth, Virginia, and Emma decided to leave the throng and take the children home.\n\nThat evening, Elizabeth reflected upon Mr. Lincoln's impromptu speech at the window, and she considered that for all the years she had known him, and for all the many times they had chatted while she worked at the White House, she had never heard him make a public speech. On their last night aboard the _River Queen,_ and again that day at the White House window, he had promised a formal speech for the following day. Perhaps because she regretted missing his inaugural address, she became very anxious to hear his speech, the first after General Lee's surrender.\n\nThe next morning, Mrs. Lincoln called on Elizabeth at the boardinghouse to discuss her ideas for a new gown. As she was leaving, Elizabeth asked her if she might come to the White House that night and attend the president's speech.\n\n\"Certainly, Lizzie,\" said Mrs. Lincoln. \"If you take any interest in political speeches, come and listen in welcome.\"\n\n\"Thank you, Mrs. Lincoln.\" After a moment's hesitation, she added, \"May I trespass further on your kindness by asking permission to bring a friend with me?\"\n\n\"Yes, bring your friend also,\" Mrs. Lincoln said graciously, and then, as if she had almost forgotten, \"and do come in time to dress me before the speaking commences.\"\n\n\"I will,\" Elizabeth promised. \"You may rely upon that.\"\n\nMrs. Lincoln nodded and swept from the room. Moments later, through the window, Elizabeth watched her step from the house to the street, where she ascended into her carriage and drove away.\n\nElizabeth thought of Virginia, and she thought of Emma, and she wished that she had asked Mrs. Lincoln if two friends could accompany her. But since it was too late to amend her request, she decided to invite Emma. Her young assistant attended every speech of Mr. Lincoln's that she could, including the second inaugural address, which she had described so well for Elizabeth. Since she admired the president so much, it seemed only right to offer her the opportunity to hear him speak from a place of honor within the White House.\n\nThrilled, Emma immediately accepted the invitation, and so at seven o'clock, they entered the White House through the front door, as Elizabeth had done alone countless times before. On the way upstairs to Mrs. Lincoln's chamber, Elizabeth touched Emma's arm, a silent signal that they should tread softly. As they passed Mr. Lincoln's room, they slowed their pace and glanced through the half-open door. The president was seated at his desk, looking over his notes and muttering to himself, his expression thoughtful, his manner abstracted. Elizabeth paused for a moment to watch him, knowing that he was rehearsing and refining the words he would soon speak not only to the crowd gathering outside, but to the entire nation and beyond, for everyone would read his remarks in the papers in the days to come. When the president spoke, his words traveled around the world, so each one had to be selected with care.\n\nWhen they reached Mrs. Lincoln's rooms, Emma waited outside while Elizabeth swiftly dressed Mrs. Lincoln in yellow silk and arranged her hair with early spring blossoms. Mr. Lincoln appeared just as Elizabeth was finishing, and they both graciously agreed to allow Elizabeth to introduce Emma to them. Elizabeth hid a smile as the young woman shook their hands and managed to chat politely with them for a brief moment, even though the unexpected honor left her quite tongue-tied.\n\nGreat crowds had gathered in front of the White House, and over the music of the Marine Band, loud, eager calls were made for the president to appear. When he finally advanced to the center window above the door, a thrilling roar went up from the throng assembled in the darkness below. Looking out from another second-story window nearby, where she and Emma stood as equals among many distinguished ladies and gentlemen, Elizabeth could scarcely breathe from amazement. She had never seen such a mass of people before, like a black, gently swelling sea in the night, the motion of the crowd like the ebb and flow of the tide upon the stranded shore of the ocean. The faces near the front were clearly discernible, but they faded into ghostly outlines farther away. Lending the scene a weird, spectral beauty was the indistinct hum of voices that rose above them all, reminiscent of the subdued, sullen roar of an ocean storm or the wind sighing through a dark, lonely forest. It was a grand, imposing scene, and when the president regarded it all with a piercing, soulful gaze as he waited for the cheers to subside, he seemed to Elizabeth more like a demigod than a mortal man.\n\nEvery window of the White House was illuminated by hundreds of tiny candles arranged in tiers on slender strips of wood, but the flickering candlelight must have seemed insufficient, for at once a cry went up for someone to fetch a lamp. When it was brought, Elizabeth heard little Tad cry, \"Let me hold the light, Papa! Let me hold the light!\"\n\nMrs. Lincoln gestured and said something that Elizabeth could not make out, but she must have asked for her son's wish to be granted, for the lamp was passed to him.\n\n\"We meet this evening, not in sorrow, but in gladness of heart,\" the president began, and a hush fell over his listeners. \"The evacuation of Petersburg and Richmond, and the surrender of the principal insurgent army, give hope of a righteous and speedy peace whose joyous expression cannot be restrained.\"\n\nYes, Elizabeth thought, yes, and the compassion and thankfulness in his voice filled her heart until it seemed to lift her upward. Emma drew in a breath and touched Elizabeth on the arm, and Elizabeth knew that her young friend was as moved by the striking tableau as she was\u2014father and son standing together in the presence of thousands of free citizens, the elder pronouncing eloquent ideas for the fate of a nation, the younger looking up at him with proud admiration.\n\nElizabeth stood not far from the president, and after he praised the military and turned to the fraught subject of reconstruction, the light from Tad's lamp fell fully upon him, so that he stood out boldly against the night. A sudden, chilling thought struck Elizabeth then, and drawing closer to Emma, she murmured, \"What an easy matter it would be to kill the president, as he stands there! He could be shot down from the crowd, and no one would be able to tell who fired upon him.\"\n\nBleakly, Emma nodded. Elizabeth could hardly pay attention to the rest of the speech, so afraid was she that one of the many vicious men who had sent him violent, threatening letters lurked within the shadowed throng below. Only a few days before, Elizabeth had overheard Mrs. Lincoln requesting additional protection for her husband, and soon thereafter officers from the Metropolitan Police had been posted at the White House to handle new threats of arson, kidnapping, and other terrors. But since then, General Lee's surrender seemed to have eased Mrs. Lincoln's perpetual worries, at least a little. Even so, Elizabeth found herself seized by an intense, foreboding dread that the president's enemies had not abandoned their hatred when General Lee conceded defeat at Appomattox.\n\nSuddenly Emma clutched her arm, and a smile lit up her face. Pulled from her dark reverie, Elizabeth quickly picked up the threads of the president's speech. \"It is also unsatisfactory to some that the elective franchise is not given to the colored man,\" Mr. Lincoln said, referring to criticism of the new Louisiana state constitution. \"I would myself prefer that it were now conferred on the very intelligent, and on those who serve our cause as soldiers.\"\n\nElizabeth managed just in time to muffle a gasp. Unless her ears were deceiving her, the president had just told the world that he approved of enfranchisement for black Union soldiers and certain other men of color.\n\n\"Can he mean it?\" Emma asked in a whisper. \"Will our men be permitted to vote?\"\n\n\"I think they will be,\" Elizabeth whispered in reply, a thrill of excitement putting a tremble in her voice. Perhaps that would be only the beginning. Perhaps the lady suffragists would finally have their way too. Elizabeth could imagine the obstacles tumbling over one after the other like books on a too-crowded shelf: First colored soldiers would be allowed to vote, and then prominent black men, and then all black men, and then white women, and last of all but finally, women of color like her and Emma and Virginia. It could happen. Some people had said that slavery would be with them forever, and yet it had been abolished. Good people with strong convictions could overturn any injustice if they simply refused to quit. But even as she took heart, Elizabeth could not forget that the march to justice had ever been arduous and long, and the changes she yearned for might not come within her lifetime.\n\nBut she had already witnessed so many remarkable events since coming to Washington City. Why might not universal suffrage be another?\n\nElizabeth's thoughts were still full of the night's splendors when she went to the White House the following day, and yet, after thanking Mrs. Lincoln for allowing her and Emma to attend, something compelled her to mention the sudden apprehension she had felt when Mr. Lincoln stood dangerously illuminated and vulnerable before the crowd.\n\n\"Yes, yes, Mr. Lincoln's life is always exposed,\" Mrs. Lincoln agreed, sighing. \"Ah, no one knows what it is to live in constant dread of some fearful tragedy. The president has been warned so often that I tremble for him on every public occasion. I have a presentiment that he will meet with a sudden and violent end.\"\n\n\"I suppose it is only natural that you should worry,\" said Elizabeth, thinking again of the dreadful letters Mr. Lincoln received nearly every day. Mrs. Lincoln received her fair share too, though not as many as her husband.\n\nMrs. Lincoln shook her head. \"I said _presentiment_. I'm not speaking of ordinary worries and fears that might plague anyone in my circumstances, any wife whose husband has made enemies. The sensation is much too powerful for that.\"\n\nMrs. Lincoln seemed so certain, so despondent, that Elizabeth wished she had never encouraged her when she had consulted spiritualists after Willie's death. Surely one of them had planted this sepulchral notion in her head when she was tormented with grief.\n\nShe was quiet too long, for Mrs. Lincoln frowned and said, \"I know that look. You think I'm being foolish, but you're too polite to say so. Well, what would you think if I told you Mr. Lincoln shares my beliefs?\"\n\n\"I\u2014would not know what to think,\" said Elizabeth, taken aback.\n\n\"He has had several premonitions himself.\" Mrs. Lincoln pressed her lips together and inhaled deeply, and a furrow appeared between her brows. \"The first came to him a few days after the election in 1860, when we were still in Springfield. The weight of his new responsibilities was settling upon him, and he was having trouble sleeping. He was in his office chamber reclining on a lounge, when his gaze fell upon the mirror and he saw his image reflected with two faces, one much paler than the other. He was very unsettled by the sight, and little wonder. I believe the vision meant that he would be elected twice, but not live out his second term.\"\n\nShaken, Elizabeth nonetheless mustered up enough skepticism to ask, \"Mr. Lincoln himself believes this was a vision, not merely a trick of the light upon a poor glass and weary eyes?\"\n\n\"Well, if he did not say so, it was evident from his manner. But there have been others. He has a strange recurring dream preceding events of great significance. He describes himself as aboard a ship\u2014he cannot describe it but he knows it always to be the same vessel\u2014moving swiftly toward a dark and indefinite shore. He had this dream before Antietam, Murfreesboro, Gettysburg, and Vicksburg.\"\n\n\"Goodness.\" Elizabeth shivered as if Mrs. Lincoln's very words carried a chill. \"But if this dream is somehow prophetic, it seems to presage victory, not death.\"\n\n\"I suppose that's true.\" Suddenly tears sprang into Mrs. Lincoln's eyes. \"But I have not yet told you the worst of his dreams. I truly wish he had never told me.\"\n\n\"Why, Mrs. Lincoln.\" Alarmed, Elizabeth took her by the elbow and guided her to a seat on the sofa. \"Perhaps we've dwelt too long on this subject\u2014\"\n\n\"No. No. I've begun and I must tell you the rest.\" Mrs. Lincoln took out her handkerchief, dabbed at her eyes, and distractedly twisted the fine white cloth into a rope on her lap. \"Only a few days ago, Mr. Lincoln and I were chatting with his guard Mr. Lamon and a few others when the conversation turned to the abundance of dreams in the Bible. 'If we believe the Bible,' my husband said, 'we must accept the fact that in the old days God and His angels came to men in their sleep and made themselves known in dreams.' When I asked what had prompted the remark, he first asserted that he did not believe in dreams, and then he went on to describe a dream that he'd had a few nights before, and which has strangely annoyed him ever since.\"\n\n\"What was this dream?\" asked Elizabeth uneasily, though she almost didn't want to know.\n\n\"He told us that about ten days before, he had retired very late because he had been up waiting for important dispatches from the front. He was not long in bed when he fell into a weary slumber and began to dream. He said there seemed to be a deathlike stillness about him, and then he heard subdued sobs, as if a number of people were weeping. In the dream he left his bed and wandered downstairs, where the silence was broken by the same pitiful sobbing, but the mourners were invisible. Though he searched from room to room, and saw no one, the same mournful sounds of distress met him everywhere he went. The rooms were lit, and every object was familiar, but where were all the people who were grieving as if their hearts would break? He became puzzled and alarmed, wondering what could be the meaning of all this. Determined to find the cause of circumstances so mysterious and so shocking, he kept on until he arrived at the East Room\u2014where he met with a sickening surprise. Before him was a catafalque, on which rested a corpse wrapped in funeral vestments. Soldiers were standing guard all around while throngs of people gazed mournfully upon the deceased, whose face was covered, and some wept pitifully. 'Who is dead in the White House?' my husband demanded of one of the soldiers. 'The president,' came the answer. 'He was killed by an assassin.' Then a loud, terrible wail of grief went up from the mourners and woke him.\"\n\n\"How dreadful,\" exclaimed Elizabeth. No wonder the president had grown so gaunt and weary, if such terrible imaginings plagued him at night. \"But, Mrs. Lincoln, you must not fear that this nightmare will come to pass.\"\n\nMrs. Lincoln regarded her flatly. \"You said yourself that it would have been an easy matter for an assassin to kill him as he stood at that window last night.\"\n\nAnd how Elizabeth regretted saying so. \"What I mean is that you should not believe that these are anything more than troubled dreams. They are not glimpses into the future. It would be astonishing if Mr. Lincoln did _not_ have nightmares prompted by the threats made against him\u2014in fact, it is a testament to his strength that he does not have more of them.\"\n\nMrs. Lincoln looked as if she wished she could believe her. \"I've ordered the guards increased, but I don't know what else I could\u2014\"\n\nShe broke off abruptly as the door opened and Mr. Lincoln entered. Mrs. Lincoln quickly composed herself, but something in Mr. Lincoln's expression told Elizabeth that he had detected her mood in that brief interval before she concealed it. He greeted them, peered curiously at them for a moment, and went to the window, where he gazed out upon the yard, smiling. \"Madam Elizabeth,\" he said suddenly, turning to her. \"You are fond of pets, are you not?\"\n\n\"Oh, yes, sir,\" she answered.\n\n\"Well, come here and look at my two goats.\"\n\nElizabeth caught Mrs. Lincoln's eye; she discreetly nodded and gestured for Elizabeth to go ahead. When she stood beside him at the window, he nodded to the goats frolicking on the grass below. \"I believe they are the kindest and best goats in the world,\" he declared, his eyes twinkling with good humor. \"See how they sniff the clear air, and skip and play in the sunshine. Whew! What a jump,\" he exclaimed as one of the goats bounded high over a stony patch. \"Madam Elizabeth, did you ever before see such an active goat?\"\n\nShe smiled. \"Not that I can recall, sir.\"\n\n\"He feeds on my bounty, and jumps for joy,\" Mr. Lincoln mused. \"Do you think we should call him a bounty-jumper?\"\n\nElizabeth laughed at his play on words. \"If you'd like to, Mr. President, I don't see why not.\"\n\n\"But I flatter the bounty-jumper,\" he said. \"My goat is far above him. I would rather wear his horns and hairy coat through life, than demean myself to the level of the man who plunders the National Treasury in the name of patriotism.\" A sudden shadow fell over his face. \"The man who enlists into the service for a consideration, and deserts the moment he receives his money but to repeat the play, is bad enough, but the men who manipulate the grand machine and who simply make the bounty-jumper their agent in an outrageous fraud are far worse. They are beneath the worms that crawl in the dark hidden places of earth.\"\n\nBefore Elizabeth could reply that with the national draft ended, the bounty hunters would no longer be able to profit as they had, both goats looked up at the window and shook their heads in a friendly sort of way. \"See, Madam Elizabeth?\" said the president, brightening. \"My pets recognize me. How earnestly they look. There they go again; what jolly fun!\" He laughed out loud as the goats bounded swiftly to the other side of the yard.\n\n\"Come, Lizzie,\" Mrs. Lincoln called out sharply. \"If I want to get ready to go down this evening I must finish dressing myself, or you must stop staring at those silly goats.\"\n\nElizabeth exchanged a quick look of understanding with the president before hurrying back to Mrs. Lincoln. The First Lady was not overly fond of pets, and she could not understand why Mr. Lincoln took such delight in his goats. Elizabeth would never say so, but she thought Mrs. Lincoln ought to be glad for any pleasant distraction Mr. Lincoln found from his cares. His goats, his favorite authors, conversations with intelligent friends, and the occasional night at the theater were rare respites from the unrelenting pressures of his high office.\n\nElizabeth finished dressing Mrs. Lincoln quickly, unconcerned with her patron's frown of annoyance. It would fade soon enough, and in the meantime, Elizabeth would rather have Mrs. Lincoln irritable than anxious and brooding over dark, imagined omens, the cruel offspring of a mind weary from strain and sorrow and too much toil.\n\nTwo days later, early on the morning of Good Friday, Elizabeth walked to the White House with the bodice, skirt, and sleeves of a new spring frock to fit for Mrs. Lincoln, an embroidered French muslin with cap sleeves and a delicate lace trim around the neckline. Already the First Lady was looking forward to moving to the Soldiers' Home for the summer, where she hoped her husband would rest and regain his vigorous good health. The war had demanded much of him, and reconstruction would likely demand more, but as General Sherman advanced in North Carolina, it seemed that the war would soon end, at long last. The worst was certainly behind them.\n\nElizabeth turned onto Fifteenth Street just as colored soldiers marched past flanking columns of captured rebel troops. Onlookers did not mock them, but instead offered them sympathetic glances or pretended not to notice them rather than add to their disgrace. They could have hurled insults\u2014or bricks, if they were truly vengeful\u2014but most Washingtonians seemed to feel as Elizabeth did, that these unfortunate captives surely wanted only to go home, just as Northern fathers, sons, brothers, and sweethearts wanted more than anything else to return to their families. With the war so near its conclusion, and the outcome certain, everyone wanted nothing more than for it to end before any more blood was shed.\n\nThe sight of colored soldiers guarding white Confederate prisoners was no longer uncommon, and yet Elizabeth halted and watched the columns and guards pass, marveling. How the city had changed during the past four years\u2014and even just in the past week. The exuberant rejoicing of the first days after General Lee's surrender had settled into a calm sense of hope, gratitude, and peace, despite the ever-present concerns about what yet lay ahead. In all but the most radical, punitive hearts, Mr. Lincoln's recent speech had inspired a mood of forgiveness and clemency. Elizabeth thought of the old friends she had seen in Petersburg the previous week, and all those she had known in Virginia and Missouri and even North Carolina, where she had been subjected to such torment, and she hoped the president's plans for reconstruction would treat them gently.\n\nAt the White House, she was pleased to find Mrs. Lincoln in a cheerful mood. Captain Robert Lincoln and General Grant had arrived that morning from Virginia, and over breakfast Robert had given his parents his firsthand account of General Lee's surrender at Appomattox. \"What a momentous occasion it was,\" Mrs. Lincoln remarked as Elizabeth helped her into the skirt and bodice, \"and how fitting that the son of the president was there to witness it.\"\n\nMrs. Lincoln was in especially good spirits, not only because Robert was home and safe, and not only because the previous night Mr. Lincoln had again dreamed of the ship carrying him swiftly toward a distant shore, a dream that he believed foretold General Sherman's imminent victory against General Johnston in North Carolina. \"My husband sent me a note this morning,\" she confided, \"inviting me to go for a drive this afternoon.\"\n\nElizabeth had to smile. \"He sent you a note? Wouldn't it have been faster simply to ask you as you went in to breakfast together?\"\n\n\"Faster, but not half so charming,\" rejoined Mrs. Lincoln. \"It reminds me of the days of our courtship back in Springfield. We've endured so much since then that it's often difficult to recall what it felt like to be young sweethearts.\"\n\nSmiling, Elizabeth adjusted the neckline of the dress, slipping one pin and then another in place. \"Perhaps a pleasant day together will help you remember.\"\n\n\"We both need a pleasant, restful day for other reasons besides, but I fear that my husband is not likely to see one today.\"\n\n\"Why not? Isn't he entitled to a little holiday on Good Friday?\"\n\n\"If you could persuade him of that, you would have my undying gratitude. Since breakfast he has been sitting through conferences with legislators and petitioners, and after that his cabinet will be meeting. I may not see him at all until this afternoon\u2014at which point I will insist that he have a proper luncheon before our ride instead of merely munching an apple at his desk. He grows gaunt and gray and people blame me for not feeding him enough.\"\n\n\"They should blame the White House cook before you,\" remarked Elizabeth, \"but even that is unfair.\"\n\nMrs. Lincoln heartily agreed, but at that moment her disappointment that she might not see her husband until later vanished, because Mr. Lincoln strode into the room. \"I've been to the War Department,\" he told his wife after greeting them both. \"I saw General Eckert there, and told him of our plans to attend the theater tonight, and I invited him to come along.\"\n\nMrs. Lincoln's eyebrows rose. \"Did you? Well, I suppose there's room in the box for him and us and General and Mrs. Grant too.\"\n\nMr. Lincoln flung himself onto the sofa. \"There is, Mother, but we won't need it, because General Eckert cannot join us.\"\n\nMrs. Lincoln's laughter carried the slightest hint of amused exasperation. \"Then why mention it at all?\"\n\n\"I thought you liked me to tell you every detail of my day.\"\n\nMr. Lincoln looked as if he might tease her more, but just then Robert entered carrying a small portrait. \"Here it is, Father,\" Robert said, handing him the object. \"General Robert E. Lee, as promised.\"\n\n\"Yes, thank you, son.\" Mr. Lincoln set the picture on the table before him and studied the general's visage thoughtfully. \"It is a good face; it is the face of a noble, brave man.\"\n\n\"I think it's a fine likeness,\" said Robert, clearly pleased with himself.\n\nMr. Lincoln nodded appreciatively and then looked up at him from beneath raised brows. \"Well, my son, you have returned safely from the front. The war is now closed, and we will soon live in peace with the brave men that have been fighting against us. I trust that the era of good feeling has returned with the end of the war, and that henceforth we shall live in peace.\"\n\n\"Amen,\" Elizabeth murmured, so softly that not even Mrs. Lincoln heard her.\n\n\"Now listen to me, Robert,\" the president continued. \"You must lay aside your uniform and return to college. I wish you to read law for three years, and at the end of that time I hope that we will be able to tell whether you will make a lawyer or not.\"\n\n\"Yes, sir,\" said Robert solemnly, and his father rose and shook his hand, looking more cheerful than Elizabeth had seen him in a long time.\n\nElizabeth was happy to see the family enjoying a moment of domestic harmony after so much strife and worry. It _was_ scarcely a moment, though; Mr. Lincoln hurried off soon thereafter to attend the meeting of his cabinet, and Robert left too, perhaps to rejoin General Grant. Alone again, Elizabeth and Mrs. Lincoln resumed fitting the dress, and they were not quite finished when a servant came in with a message from General and Mrs. Grant, explaining that they had decided to visit their children in New Jersey and would be unable to accompany the Lincolns to the theater. \"How disappointing that they would cancel on such short notice,\" said Mrs. Lincoln, flinging the letter on the table. \"Lately Mrs. Grant seems to seize upon any excuse to avoid spending time in my company.\"\n\nElizabeth could hardly blame the general's wife for that, considering how Mrs. Lincoln had unfairly castigated her in the carriage at City Point. \"Could you invite another couple instead?\"\n\n\"It would be a waste of paper and ink to try. Who would be free at this late hour?\"\n\n\"Who would not eagerly abandon other plans for the honor of sharing the presidential box with Mr. and Mrs. Lincoln?\"\n\n\"General and Mrs. Grant, evidently,\" Mrs. Lincoln retorted, but as Elizabeth continued to adjust and pin her dress, she reconsidered. \"I suppose we could find other guests, and if not, I wouldn't mind keeping my husband all to myself for a change.\"\n\n\"To yourself, and hundreds of other theatergoers,\" Elizabeth reminded her, and was pleased to see Mrs. Lincoln smile.\n\nLater, as Elizabeth packed up her things, she asked, \"Do you want me to come back later to dress you for the theater?\"\n\n\"Well\u2014\" Mrs. Lincoln hesitated. \"I think not. I'm not altogether certain we shall go after all. I feel a headache coming on, and Mr. Lincoln has been worn out from unrelenting toil. I suppose we shall decide after our ride, but I don't want to oblige you in the meantime.\"\n\n\"If you decide you need me,\" Elizabeth reminded her, \"you know you can always send for me.\"\n\nMrs. Lincoln smiled, grateful. \"Yes, Elizabeth. I know.\"\n\nElizabeth woke abruptly in the dark of night to an incessant pounding on her door. \"Mrs. Keckley,\" someone called. \"Mrs. Keckley, please get up!\"\n\nDisoriented, Elizabeth sat up, groped for her dressing gown, and pulled it on. The pounding and calling persisted as she climbed from bed and hastened to the door. \"Miss Brown?\" she said, confused, as she discovered her boardinghouse neighbor in the hall. \"What's the matter? Is there a fire?\"\n\n\"No, no, not a fire, but terrible news, terrible news!\" Miss Brown wrung her hands, tears on her cheeks. \"Mr. Lincoln has been shot, and the entire cabinet has been assassinated!\"\n\nElizabeth's heart lurched. \"Assassinated?\" Images of the men she had seen through the years at the White House\u2014smiling, scowling, whispering urgently in corners, striding intently about the president's business, laughing at his stories\u2014flew through her mind. Her blood felt as if it had been frozen in her veins, and her lungs as if they would collapse for want of air. \"Dear God, all of them dead? And Mr. Lincoln? Was he badly hurt?\"\n\nMiss Brown shook her head. \"No, praise heaven. He was not mortally wounded.\"\n\n\"Oh, thank God. And Mrs. Lincoln?\"\n\n\"Unharmed\u2014at least, I have heard nothing about any injury done to her.\"\n\nHeart pounding, Elizabeth thanked her neighbor and hurried back into her room to dress. It was after eleven o'clock, but sleep had fled, and she could not bear to sit patiently awaiting news when it felt as if the house could not contain her. She dashed outside, where the streets were full of wondering, fear-struck people. Rumors flew thick and fast, some confirming Miss Brown's account, others contradicting it, still more offering even wilder, more terrible reports. Elizabeth grew ever more frustrated and frightened until worry compelled her back indoors, where she woke Walker and Virginia and told them the president had been shot. \"I must go to the White House,\" she said, her voice breaking. \"I cannot remain in this state of uncertainty.\"\n\n\"Elizabeth, dear,\" said Virginia, her face stricken beneath her muslin nightcap. \"You must calm yourself. Mr. Lincoln must be recovering just fine, or the bells would be tolling all over the city.\"\n\n\"Virginia's right,\" said Walker, but he too looked apprehensive. \"You've had a shock. Go back to sleep. You can call on Mrs. Lincoln first thing in the morning.\"\n\n\"No, no.\" Elizabeth shook her head, her heart jumping and fluttering like a trapped bird in her chest. \"I must go to her now. Even if the president was only slightly injured, Mrs. Lincoln will be upset, and she will want me.\"\n\nThe Lewises exchanged a silent look, and then Walker agreed that Elizabeth should go, but she should not go alone. Elizabeth paced in the foyer while the couple dressed and told Jane to look after her younger sisters. It seemed an eternity until Virginia and Walker joined her and they headed out into the night. They made their way swiftly through the milling throng, scores of confused and alarmed people questioning one another and repeating rumors and sometimes breaking into loud, gasping sobs. On the way through Lafayette Square, they passed the residence of Secretary Seward and were shocked to find it surrounded by armed soldiers, bayonets drawn to keep back all intruders. Alarm quickening their pace, they hurried on until they reached the White House and found it too surrounded by soldiers. Every entrance was strongly guarded, and no one was permitted to pass.\n\nElizabeth drew herself up, summoned all her strength of will, and addressed a guard at the gate firmly. \"I am Mrs. Elizabeth Keckley, the First Lady's personal modiste and friend. If she is in distress, she will want me.\"\n\nThe soldier ignored her, his gaze moving steadily over the crowd gathering in front of the White House, his grip tightening around his rifle. Elizabeth felt Virginia nudge her discreetly, and so she quickly took another breath and said, \"Ask the doorman who I am. I am here at the White House several times a week. He will know me.\"\n\nThe soldier again ignored her, but another standing a few feet away took pity on her. \"I'm sorry, ma'am, but no one is allowed to enter tonight.\"\n\nQuickly, desperately, Elizabeth whirled upon him. \"Would you please have the doorman send word to Mrs. Lincoln that I am here? I know she will tell you to let me in.\"\n\nThe guard, a red-haired, freckled lad little older than George would have been, shook his head. \"She isn't here, and the president hasn't been brought home either.\"\n\nThe first guard sucked in his breath through his teeth and glared at the red-haired soldier, who promptly tore his gaze away from Elizabeth and snapped to attention. Neither of them would say anything more, so reluctantly, and with an increasing sense of sickening dread, Elizabeth, Virginia, and Walker left the White House. As they made their way through the milling throng toward home, Elizabeth felt weak from grief and anxiety. A few blocks along, she glimpsed a gray-haired elderly man passing, and something of the kindness and sorrow in his expression compelled her to reach out, touch his arm gently, and imploringly ask, \"Will you please, sir, tell me whether Mr. Lincoln is dead?\"\n\n\"Not dead,\" he said, \"but dying. God help us!\" With a heavy step he continued on his mournful way.\n\n\"Not dead but dying,\" said Virginia, her voice trembling. \"Then indeed, God help us all!\"\n\nEventually, in bits and fragments they collected on their way to Twelfth Street, they assembled the shocking story. The president lay mortally wounded, some said in a residence across the street from Ford's Theatre, but no one knew for certain. The popular actor John Wilkes Booth had stolen into the president's private box while he watched the performance with Mrs. Lincoln and their guests, and had shot him in the back of the head. Mr. Lincoln was not expected to live until morning.\n\nHearts crushed and broken, Elizabeth, Virginia, and Walker returned home.\n\nOnce there, Elizabeth could not sleep. She imagined Mrs. Lincoln wild with grief and wanted to go to her, but she did not know where she might be, and she had no choice but to wait until morning. Never had the nighttime hours dragged so slowly. Every minute seemed an eternity, and Elizabeth could do nothing but pace and wait and watch the eastern sky through her window for the coming of dawn, hugging her arms to her chest as if warding off a bitter wind.\n\nMorning came at last, gray and somber. At half past seven, a distant church bell began to toll, and then another joined it, and another, until all the bells in Washington resounded with the terrible news. The president was dead, Elizabeth thought numbly, and then a sob escaped from her throat, and she flung herself onto her bed, anguished and weeping. She covered her ears with clenched fists, but nothing would block out the mournful sound.\n\nShe lay there still, exhausted and drained, at eleven o'clock, when a carriage drove up to the boardinghouse and a messenger rapped on her door. \"I come from Mrs. Lincoln,\" he said. \"If you are Mrs. Keckley, come with me immediately to the White House.\"\n\nElizabeth hastily threw on her shawl and bonnet, and within moments she was seated in the carriage and speeding toward the White House through a cold and dismal rain. The gas streetlamps were usually extinguished by that time of day, but that morning they had been left burning. The streets were unnaturally hushed for a late Saturday morning. Flags hung in cheerless folds at half-staff, and here and there, shops and houses were draped in black crepe. A lone sentry paced back and forth in front of a house, and in front of another farther along, and after a moment of bewilderment Elizabeth recognized the residences as those belonging to cabinet members. Miss Brown had been mistaken when she had told Elizabeth that they had all been assassinated, but at nearly the same time Mr. Lincoln was shot, Secretary Seward had been attacked in his own home, in bed where he lay recovering from a carriage accident. If not for the neck brace he wore due to his injuries, he likely would have been stabbed to death. As it was, he had been terribly wounded and might yet perish, and no one could say what plots against the other secretaries had been thwarted, had been abandoned, or had yet to be enacted.\n\nSoon the carriage reached the White House, where Elizabeth saw hundreds of colored folks, mostly women and children, milling about on the lawn, weeping and lamenting their loss. Unlike the previous night, this time she easily passed through the gate and by the guards and up to the very front door, but once she crossed the threshold into the silent, somber house, she felt as if a heavy shroud of grief had been draped over her shoulders. Steeling herself, she removed her bonnet and hurried upstairs, but Mrs. Lincoln was not in her bedroom, nor in the family sitting room. Before long, a servant with red-rimmed eyes guided her to a small spare bedroom that had been made up for Mr. Lincoln to use during the summer, when the family resided mostly at the Soldiers' Home and Mr. Lincoln spent only occasional nights alone at the mansion. \"The First Lady refused to be taken into any of the family bedrooms,\" the servant confided. Elizabeth thought of how Mrs. Lincoln still avoided the room where her son Willie had been laid out for his funeral, and she understood why Mrs. Lincoln could not have gone to the chambers she had shared with her husband.\n\nElizabeth hesitated with her hand on the door, taking deep breaths to push back the sorrow that threatened to rise up and choke her. When she had composed herself, she entered the room and plunged into darkness. Her eyes adjusted, and she made out the dim shapes of a woman tossing and turning restlessly upon the bed, and another sitting in a chair by the window, where the curtains had been drawn so barely a beam of pale sunshine leaked in. After a moment Elizabeth recognized Mrs. Mary Jane Welles, the wife of the secretary of the navy and one of her patrons. Nodding to her, Elizabeth hurried to Mrs. Lincoln's bedside and murmured her name.\n\n\"Elizabeth?\" Mrs. Lincoln's voice was the barest of whispers. \"You have come at last?\"\n\nElizabeth took her hand. \"Yes, Mrs. Lincoln. I'm here.\"\n\nMrs. Lincoln slowly rolled over and looked up at her bleakly, her eyes red and swollen, her cheeks pale, her expression shocked and haggard. \"Why did you not come to me last night, Elizabeth? I sent for you.\"\n\nElizabeth blinked back tears. \"I did try to come to you, but\u2014\" She fought to keep her voice steady as she lay her palm upon Mrs. Lincoln's brow, which felt strangely feverish to the touch. \"I could not find you.\"\n\n\"I sent three messengers.\"\n\n\"I'm sorry.\" Elizabeth sat down on the edge of the bed, clutching Mrs. Lincoln's hand in both of hers. \"I did come. I did come, but you weren't here.\"\n\n\"They did not bring us here.\" Mrs. Lincoln's voice was distant, disbelieving. \"They took my husband across the street from the theater to Mr. Peterson's boardinghouse. My husband was too tall for the bed. They had to lay him upon it diagonally.\" She choked on a sob and began to keen.\n\n\"Hush now,\" Elizabeth said, stroking her brow. \"Hush.\"\n\nWhile Elizabeth soothed her, Mrs. Welles, who did not look at all well and had left her own sickbed to answer Mrs. Lincoln's summons, quietly excused herself and departed for home, entrusting Mrs. Lincoln to Elizabeth. She was nearly exhausted from grief, but Elizabeth endeavored to calm her. After she had quieted, Elizabeth gently asked if she might go pay her respects. When Mrs. Lincoln nodded, Elizabeth slipped her hand free and quietly left the darkened chamber and went alone to the guest room, where the president lay in state. As she crossed the threshold, heart pounding and legs trembling, she suddenly recalled the sight of young Willie Lincoln lying in his coffin in the same place where his father's body now lay in repose. In her mind's eye she watched again as the president wept over the pale, beloved face of his boy, and she thought of how kindly Mr. Lincoln had spoken to her the last time she saw him alive, how generously and respectfully he had described the vanquished General Lee. The Moses of her people had fallen in the hour of his triumph. His tragic death was all the more heartbreaking because he had not lived to enjoy the peace he had toiled so long to achieve.\n\nSeveral members of the president's cabinet, a few army officers, and other dignitaries were grouped around the body of their fallen chief, some weeping openly, but they made room for her as she approached. Reverently she lifted the white cloth from the pale face of the man that she had so greatly admired\u2014not merely admired, but looked upon as someone as divine as he was human. Her pounding heart grew calm as she looked upon his face; notwithstanding the violence done to him, she discovered something beautiful as well as grandly solemn in his expression\u2014the sweetness and gentleness of childhood, the stately grandeur of inspired intellect. She gazed long upon him, until she was obliged to turn away, her vision blurred by tears, her throat constricted in grief.\n\nFighting to regain her composure, she returned to Mrs. Lincoln's room to find her in a new paroxysm of grief\u2014wailing, keening, convulsing terribly. Robert bent over her, murmuring, his expression pained and tender, while young Tad crouched at the foot of his mother's bed, his face a world of agony. Elizabeth quickly went to her and bathed her brow with cool water, struggling to soothe the violent tempest as best she could. Tad's grief was no less than his mother's, but her frightening outbursts had shocked him into silence. Suddenly he threw his arms around her. \"Don't cry, Mama,\" he begged, his voice muffled as he pressed his face against her neck. \"Don't cry, or you will make me cry too! You will break my heart.\"\n\nMrs. Lincoln could not bear to hear her youngest son cry, and with a great effort, she clasped him to her heart and struggled to calm herself, but she could not restrain her grief for long, and soon it burst forth again, alarming and heart wrenching to behold.\n\nIn the days following the assassination, every room in the White House was darkened, every voice low and subdued, every footstep heavy and muffled. Mrs. Lincoln sequestered herself in the small guest room with the curtains drawn, sometimes silent, sometimes weeping or shrieking. She could not bear to view her husband's body, but the construction of a tall platform in the East Room to display her husband's coffin intruded upon her seclusion. \"When will they be finished?\" she asked through clenched teeth, sitting up in bed, rocking back in forth in misery. \"Every nail they drive is like a pistol shot.\"\n\nOn April 17, the president's casket was set upon the finished catafalque beneath a black canopy, mirrors and chandeliers draped with black crepe, seats on risers covered in black cloth erected for the mourners. The mood was hushed and somber as an honor guard of two generals and ten other officers kept watch day and night. Nearly twenty-five thousand people passed through the East Room to pay their respects the following day, and on April 19, invited guests returned for the funeral service. Mrs. Lincoln and Tad could not bear to attend, but Robert did, the sole representative of the family. Elizabeth wanted to see the president one last time before he was laid to rest, but Mrs. Lincoln needed her more, so she remained at her side.\n\nAfter the service, a funeral procession carried the president's remains to the Capitol, where he lay in state in the rotunda so that thousands more could file past and pay their respects. On Friday, April 21, nearly a week after the president's death, a nine-car funeral train bedecked with bunting, crepe, and a portrait of Mr. Lincoln on the cowcatcher left Washington on a seventeen-hundred-mile journey westward to Springfield, carrying three hundred passengers and the remains of the president and his young son Willie. The Lincoln Special traveled at only five to twenty miles per hour out of respect for the thousands of mourners who had assembled along the rail lines, lighting the way with bonfires at night. The train made scheduled stops in twelve cities, where tens of thousands gathered to mourn and to bid farewell to their fallen leader.\n\nFor six weeks Elizabeth remained at the White House with the distraught widow, sleeping on a lounge in her chamber at night, comforting and soothing her as best she could throughout the long, sorrowful days, rarely leaving her side. Mrs. Lincoln's closest friends in Washington, Mrs. Mary Jane Welles and Mrs. Elizabeth Blair Lee, attended her sometimes too, but she denied admittance to nearly everyone else. The new president, Mr. Andrew Johnson, did not try to come to see her, nor did he send even so much as a brief note to express his sympathies. It was said that on that terrible night he had tried to enter the Peterson residence to see the president on his deathbed but had been turned away lest the sight of him upset the First Lady. However much that might have offended him, in Elizabeth's opinion, the only right, respectful, and dignified thing to do was to put aside his hurt feelings for the sake of the grieving widow. At first Elizabeth attributed his neglect to the demands of his sudden elevation to head of state, but as time went on and he still sent no condolences, she became appalled and indignant. She only hoped that Mrs. Lincoln in her sorrow would not take notice of his inexplicable affront.\n\nMrs. Lincoln did not often speak of Mr. Booth, her husband's murderer, but Elizabeth knew she brooded over who else might have been complicit in the assassination. A new messenger had accompanied the Lincolns to the theater that terrible night, and it had been his duty to stand at the closed door of the box throughout the performance, guarding the president and his party from intrusion. It soon came to light that this new messenger had become engrossed in the play, and had neglected his duty and gone off to watch it, allowing Mr. Booth easy admission to the box. Mrs. Lincoln had convinced herself that this man was implicated in the plot against her husband.\n\nOne evening, Elizabeth was lying on her lounge near Mrs. Lincoln's bed when a servant entered the room and Mrs. Lincoln asked her who was on watch that night.\n\n\"Mr. Parker, madam,\" she replied. \"The new messenger.\"\n\n\"What?\" Mrs. Lincoln exclaimed. \"The man who attended us to the theater on the night my dear, good husband was murdered! He, I believe, is one of the murderers. Tell him to come in to me.\"\n\nFrom Mr. Parker's wide-eyed and wary expression upon entering, Elizabeth knew he had overheard Mrs. Lincoln's words through the door, which had been left ajar.\n\n\"So you are on guard tonight,\" Mrs. Lincoln fiercely addressed him. \"On guard in the White House after helping to murder my husband!\"\n\n\"Pardon me, but I did not help to murder the president,\" Mr. Parker protested, trembling. \"I could never stoop to murder\u2014much less to the murder of so good and great a man as the president.\"\n\n\"But it appears that you _did_ stoop to murder.\"\n\n\"No, no! Don't say that,\" he pleaded. \"God knows that I am innocent.\"\n\n\"I don't believe you. Why were you not at the door to keep the assassin out when he rushed into the box?\"\n\n\"I\u2014I did wrong, I admit,\" he stammered, \"and I have bitterly repented it, but I did not help to kill the president. I did not believe that anyone would try to kill so good a man in such a public place, and the belief made me careless. I was attracted by the play, and did not see the assassin enter the box.\"\n\n\"But you should have seen him,\" Mrs. Lincoln snapped. \"You had no business to be careless. I shall always believe that you are guilty. Hush!\" she exclaimed as he attempted to reply. \"I shan't hear another word. Go now and keep your watch.\" She dismissed him with an imperious wave of her hand. White-faced and stiff, Mr. Parker turned and left the room. As soon as the door closed behind him, Mrs. Lincoln fell back upon her pillow, covered her face with her hands, and burst into sobs.\n\nRobert Lincoln was tender and solicitous of his mother in her grief, but his haggard looks revealed that he suffered greatly and shouldered his new responsibilities as head of the household heavily, though he never failed to put up a brave, strong front in his mother's presence. Robert endured his mother's fits of anguish better than his poor younger brother, who had lost one parent to death and feared losing another to grief. Often at night, when the bereft Tad was awakened by his mother's sobs, he would pad down the hall from his room to hers and climb into her bed. \"Don't cry, Mama,\" he would say, hugging her tightly. \"I cannot sleep if you cry! Papa was good, and he has gone to heaven. He is happy there. He is with God and brother Willie. Don't cry, Mama, or I will cry too.\" And Mrs. Lincoln would again endeavor to stop crying for his sake.\n\nElizabeth's heart went out to the boy. He and his father had adored each other, but as much as Mr. Lincoln had doted on him, especially after losing Willie, Tad had never grown spoiled. Despite his youth, he had seemed to understand that he was the son of a president, and that this had bestowed upon him both privileges and responsibilities. Early one morning, Elizabeth passed by his room when his nurse was dressing him. \"Pa is dead,\" she heard him say to his nurse soberly. \"I can hardly believe that I shall never see him again. I must learn to take care of myself now.\"\n\nHer heart aching for him, Elizabeth drew closer to the door to listen.\n\n\"Yes, Pa is dead,\" he went on, mournful and matter-of-fact, \"and I am only Tad Lincoln now, little Tad, like other little boys. I am not a president's son now. I won't have many presents anymore.\" He sighed, a soft and forlorn sound. \"Well, I will try and be a good boy, and will hope to go someday to Pa and brother Willie, in heaven.\"\n\nElizabeth pressed her lips together and hurried on before she broke down. She had to be brave. Mr. Lincoln would have wanted her to be strong for his wife and family. It was the last service she could do for him, and she could not bear to fail.\n\nMr. Lincoln's sons understood the practical matter of their new circumstances sooner than their mother, whose grief had consumed her every thought, waking and sleeping, since her husband's death. President Johnson, in a singular act of generosity, had allowed the grieving family to remain in the White House while he lived under guard at a residence on Fifteenth and H streets and worked out of a small office in the Treasury Building. Eventually, however, Mrs. Lincoln realized that his patience would not last forever, and she had to prepare to leave.\n\n\"God, Elizabeth,\" she once exclaimed, after the bleak awareness dawned, \"what a change! Did ever a woman have to suffer so much and experience so great a change? I had an ambition to be Mrs. President; that ambition has been gratified, and now I must step down from the pedestal. My poor husband! Had he never been president, he might be living today. Alas! All is over with me!\"\n\nElizabeth tried to comfort her, but her words rang hollow, and she knew it.\n\nMrs. Lincoln folded her arms and rocked herself back and forth. \"My God, Elizabeth, I can never go back to Springfield! No, never, until I go in my shroud to be laid by my dear husband's side, and may heaven speed that day! I should like to live for my sons, but life is so full of misery that I would rather die.\"\n\nKeening, she pressed her fists against her eyes and wept hysterically, and although Elizabeth desperately wished to give her solace, she knew nothing she could do would suffice, nothing she could say would bring her peace. Time would do what she could not, Elizabeth hoped grimly, as she watched and waited by Mrs. Lincoln's side.\n\n# Chapter Thirteen\n\n## MAY\u2013JUNE 1865\n\nMrs. Lincoln could not bear to go to Springfield, but she had to go somewhere.\n\nSome of Mr. Lincoln's closest friends urged her to return to Springfield, to the home she still owned there, at least until her husband's estate was settled. He had died without a will, so although his property would eventually go to his wife and children, they would not receive their shares of his estate until the legal knots were untangled. Mrs. Lincoln adamantly rejected the idea of returning to Springfield. She had burned too many bridges in her former hometown and had become the subject of gossip. She was estranged, at least in part, from her sisters and half sisters, even Little Sister Emilie and the faithful Elizabeth Edwards. Most of all, as she confided to Elizabeth, she could not bear to set foot in her once-happy home on Eighth and Jackson, where she knew she would be tormented by memories of the early years of her marriage and the husband and children she had lost. But Elizabeth knew, as the well-meaning gentlemen did not, that Mrs. Lincoln also had to take care to find a residence she could afford, since her debts to her favorite stores had climbed to at least seventy thousand dollars. Elizabeth thought it a blessing that Mr. Lincoln had not learned of his wife's debts before he died, and had been spared that anger and embarrassment.\n\nIn the meantime, letters of condolence continued to arrive from all corners of the nation and from foreign heads of state. Mrs. Lincoln read them all, quoted aloud to Elizabeth from the most tender and sympathetic, and replied to as many as she could, tears streaming down her face as she wrote. One letter in particular she cherished above all the others.\n\n_Osborne, April 29, 1865_\n\n_Dear Madam,_\n\n_Though a stranger to you I cannot remain silent when so terrible a calamity has fallen upon you & your country, & must personally express my_ deep & heartfelt _sympathy with you under the shocking circumstances of your present dreadful misfortune._\n\n_No one can better appreciate than I can who am myself_ utterly _brokenhearted by the loss of my own beloved husband, who was the light of my life\u2014my stay\u2014my all\u2014what your sufferings must be; and I earnestly pray that you may be supported by Him to whom alone the sorely stricken can look for comfort in this hour of heavy affliction._\n\n_With the renewed expression of true sympathy, I remain, dear madam,_\n\n_Your sincere friend, Victoria_\n\n\"The queen indeed understands my suffering,\" Mrs. Lincoln said the first time she read the letter, and said again, with greater feeling, each time she reread it. Once, as she returned the letter to its envelope, she added that the queen was truly blessed in that no one had driven her from her home the moment she became a widow. \"That, Elizabeth,\" she noted with a sad, tearful smile, \"is the difference between being a widowed First Lady of America, and a widowed Queen of England.\"\n\nElizabeth was glad Mrs. Lincoln found comfort in Queen Victoria's words, and, wanting to be kind, she did not point out that no one had driven Mrs. Lincoln from her home the moment she had become a widow either. Elizabeth would allow that President Johnson had been more than patient, permitting her to stay on for weeks and weeks, forgoing the use of the White House residence, offices, and reception rooms that were rightfully his. But that scant, unspoken acknowledgment was all that Elizabeth would grant him. Mrs. Lincoln had become aware that Mr. Johnson had never sent any written condolences, nor had he called upon her to express his sympathies. Robert, who had taken responsibility for communicating with the new president's staff during the transition, complained indignantly about Mr. Johnson's neglect of proper form, but his mother was even more upset and angry. If the Queen of England could find time to write a thoughtful letter, surely a new president elevated by his predecessor's death could too. It was shameful, Mrs. Lincoln declared, absolutely shameful, that a letter could cross the Atlantic Ocean in less time than Mr. Johnson required to cross the street.\n\nBut the White House _was_ his, after all, and though Mrs. Lincoln desperately clung to the tatters of her former life as long as she could, eventually, reluctantly, she decided to settle in Chicago. Mr. Lincoln had intended to retire there after his second term, she said, and it was a city that had always been good to him. It was in Chicago that he had received his first nomination as the Republican candidate for president, a place reminiscent of triumph, not despair. The city was also reasonably close to Mr. Lincoln's tomb in Springfield, where Mrs. Lincoln imagined she might seek solace in the years to come.\n\nAlmost as soon as the matter was settled, Mrs. Lincoln asked Elizabeth to come with her.\n\nStartled, Elizabeth could not speak for a moment, so intensely did she not wish to go. \"I cannot go west with you, Mrs. Lincoln.\"\n\n\"But you must go to Chicago with me, Elizabeth,\" Mrs. Lincoln implored. \"I cannot do without you.\"\n\n\"You forget my business, Mrs. Lincoln.\" Though she knew Mrs. Lincoln's moods by that time, Elizabeth was still astonished that she would make such a request. \"I cannot leave it. Just now I have the spring trousseau to make for Mrs. Douglas, and I have promised to have it done in less than a week.\"\n\nMrs. Lincoln dismissed her objections with the wave of a hand. \"Never mind that. Mrs. Douglas can get someone else to make her trousseau.\" Seeing that Elizabeth was unpersuaded, she tried a more practical line of argument. \"You may find it to your interest to go. I am very poor now, but if Congress makes an appropriation for my benefit, you shall be well rewarded.\"\n\n\"It is not the reward, but\u2014\"\n\n\"Now don't say another word about it, if you do not wish to distress me.\" Already Mrs. Lincoln's mouth was tightening, her eyes becoming tearful and beseeching. \"I have determined that you shall go to Chicago with me, and you must go.\"\n\nElizabeth had been with Mrs. Lincoln so long, and Mrs. Lincoln had become so dependent upon her, that she felt as if she could not refuse. She clung to one thread of hope: that Mrs. Douglas, the lovely young widow of the late senator Stephen A. Douglas from Illinois and one of Elizabeth's favorite patrons, would insist that she remain in Washington to complete her trousseau as agreed. But when the gracious Mrs. Douglas learned of Mrs. Lincoln's request, she told Elizabeth, \"Never mind me. Do all you can for Mrs. Lincoln. My heart's sympathy is with her.\"\n\nNever before had Elizabeth wished for a patron to be unkind and selfish, just once.\n\nRealizing that no excuse would suffice, Elizabeth prepared to go to Chicago with Mrs. Lincoln and her sons, packing a satchel, explaining her absence to Virginia and Walker and paying a few months' rent in advance, and distributing her sewing among her assistants.\n\nAlthough Emma had admired Mr. Lincoln very much and thought his widow deserved the nation's sympathy and consideration, she thought Elizabeth was making a terrible mistake. \"When will you return?\"\n\n\"I don't know.\" Elizabeth looked around the workroom and let her gaze rest on the young women in her employ, her heart sinking with a distinct sensation of dread. She intended to leave her most important, difficult sewing in Emma's capable hands, but while her favorite assistant was honored and had assured Elizabeth she would satisfy their clients' every request, Elizabeth was nonetheless worried. In her absence, would her loyal patrons place their orders with Emma, trusting that her assistant could attend to the easier tasks until Elizabeth returned to finish the more difficult, or would they choose another dressmaker, one of her competitors? \"I'll come back as soon as Mrs. Lincoln doesn't need me anymore.\"\n\nA corner of Emma's mouth turned down in a wry grimace. \"In other words, you're never coming back?\"\n\n\"Now, Emma\u2014\"\n\n\"What shall I tell your patrons? Some of them insist that you sew every stitch yourself.\"\n\n\"Show them some of your handiwork. I'm certain you'll win their confidence.\"\n\n\"I must tell them something,\" Emma insisted. \"Will you return in a week? A month? Two?\"\n\n\"Tell my patrons\u2014\" Elizabeth hesitated. \"Tell them I will return as soon as I am able.\"\n\nEmma nodded, relenting, but Elizabeth knew she was not pleased\u2014and that she was nearly as worried about the future of the business as Elizabeth herself.\n\nOnce Mrs. Lincoln resigned herself to leaving Washington, she threw herself into the tedious work of packing\u2014but first she gave away nearly everything intimately connected with her late husband, just as she had done with Willie's belongings after his death. She could not bear to be reminded of the past, and so, with Elizabeth acting as her agent, she gave away articles to those whom she regarded as the warmest and most sincere of Mr. Lincoln's admirers. Mr. Lincoln's faithful messenger, William Slade, received one of Mr. Lincoln's many canes and his heavy gray shawl, while his wife received the black-and-white striped silk dress Mrs. Lincoln had worn to the theater on the night of the assassination. Mrs. Lincoln sent other canes from his collection to colored abolitionists Frederick Douglass and the Reverend Henry Highland Garnet, the minister of the exclusive Fifteenth Street Presbyterian Church, where Virginia and Walker worshipped and Elizabeth hoped to someday, if her application and interview met with approval. Another cane went to Senator Sumner, with a note explaining that the gift of \"this simple relic\" paid tribute to his \"unwavering kindness to my idolized Husband, and the great regard _he_ entertained for you.\" She gave the suit Mr. Lincoln was wearing when he was shot to a favorite White House guard, and the last hat he wore to Reverend Dr. Gurley, who had officiated at both Mr. Lincoln's and Willie's funerals. The lively goats whose antics had given the president so much enjoyment went to Mrs. Elizabeth Blair Lee, one of the few friends who had looked after Mrs. Lincoln in the days of her most intense, anguished grieving. To Elizabeth Mrs. Lincoln presented the bonnet and cloak she had worn that terrible night, stained with the president's blood, as well as Mr. Lincoln's overshoes and the comb and brush that Elizabeth had often used to dress his hair. They were precious mementoes of the great man, and Elizabeth accepted them with deepest gratitude. Her vow to cherish them always brought a rare smile to Mrs. Lincoln's face.\n\nWhat were not distributed as relics and mementoes were loosely packed into fifty or sixty boxes and a score of trunks. Privately Elizabeth believed that many of the things Mrs. Lincoln insisted upon taking with her were not worth carrying away, but she went along with it when she observed that the work of sorting and folding and packing occupied Mrs. Lincoln so completely that she had far less time for lamentation.\n\nInto the boxes and trunks went all the bonnets Mrs. Lincoln had brought with her four years before from Springfield, along with every one she had purchased since coming to Washington. \"I may find use for the material someday,\" she replied when Elizabeth carefully asked if she meant to take even those that were no longer in fashion and had not been worn in years. \"It is prudent to look to the future.\"\n\nElizabeth, still unhappy with her reluctant decision to accompany Mrs. Lincoln, pressed her lips together rather than declare that she wished Mrs. Lincoln's foresight with regard to the future had not been confined to the present moment, and to worn-out clothing. Patience, she counseled herself. She knew she was tired and disgruntled, and grieving too in her own way. Mrs. Lincoln needed her to be steadfast and sensible. First Ladies\u2014and queens too, she supposed\u2014could fall apart from grief, but women like Elizabeth could not.\n\nDuring their time in the White House, Mrs. Lincoln and her children had received many gifts from admirers and dignitaries, and those too were packed up for Chicago. Mrs. Lincoln took no furniture with her save a dressing stand her husband had particularly liked and that the commissioner had given her permission to keep for Tad. Mrs. Lincoln replaced it with another, equally fine piece, but Elizabeth observed that other furnishings had disappeared from the executive mansion, carried off by servants and visitors after the steward was dismissed and no one was superintending affairs. Elizabeth was dismayed to see so much of Mrs. Lincoln's lovely and expensive refurbishment of the White House being stealthily undone, day by day, but Mrs. Lincoln seemed too distracted to notice.\n\nRobert was often in the room where his mother and Elizabeth were packing boxes, and he argued in vain that she should set fire to her vast stores of old goods, or at the very least leave them behind. \"What are you going to do with that old dress, Mother?\" he asked, scowling and nudging a box with the toe of his boot as she folded yet another garment and tucked it away.\n\n\"Never mind, Robert,\" she replied. \"I will find use for it. You do not understand this business.\"\n\n\"And what is more, I hope I never may understand it,\" retorted Robert, gesturing impatiently to the piles. \"I wish to heaven the car in which you place these boxes for transportation to Chicago would take fire, and burn all of your old plunder up.\" He turned on his heel and strode from the room.\n\nElizabeth had watched his tirade from the corner of her eye, saying nothing. She agreed with him that Mrs. Lincoln would probably never put any of the old clothes to good use, but she disapproved utterly of his arrogant, disrespectful tone.\n\n\"Robert is so impetuous,\" Mrs. Lincoln said, making excuse for her son, as if she could read Elizabeth's thoughts. \"He never thinks of the future. Well, I hope that he will get over his boyish notions in time.\"\n\n\"I'm sure he would not speak to you this way if he were not grieving.\"\n\n\"He has spoken to me this way for years,\" Mrs. Lincoln reminded her, and then she sighed. \"Elizabeth, I may see the day when I shall be obliged to sell a portion of my wardrobe.\"\n\n\"What do you mean?\"\n\n\"If Congress does not do something for me, then my dresses someday may have to go to bring food into my mouth, and the mouths of my children.\"\n\n\"Surely it will never come to that,\" Elizabeth hastened to assure her. She did not like to imagine the dresses she had so painstakingly fashioned being haggled over like apples in a market, sold off for as much as they could fetch before they spoiled.\n\nLater, upon reflection, Elizabeth realized that perhaps Robert thought quite a bit about the future. His had certainly undergone enormous change in the weeks since his father's murder. On April 14, he was a proud Union officer, courting the lovely young Miss Mary Harlan and intending to study the law. Now he was the head of a household, planning to leave Washington for Chicago\u2014and like herself, he knew not for how long. In the midst of his own grief, he surely also felt terribly disappointed for himself, responsible for his only surviving brother, and worried about his unstable mother. That did not excuse his impertinence, but it did make his behavior more understandable, and more forgivable.\n\nAt last everything, worthless and invaluable alike, was packed, and the day of their departure arrived. As she accompanied Mrs. Lincoln from the White House, Elizabeth was stunned almost breathless by the stark contrast with Mr. Lincoln's final leave-taking, when his casket was carried from the hall in a grand and solemn state. Thousands had gathered to bow their heads reverently as the plumed hearse bore him off to the Capitol rotunda surrounded by the mournful pomp of military display\u2014battalions with reversed arms, the riderless horse with boots turned about in the stirrups, the flags at half-staff, the melancholy strains of funeral dirges. Mrs. Lincoln left to complete indifference, the only music the chirping of birds, with scarcely anyone to bid her farewell. The silence was almost painful.\n\nOn the threshold, Mrs. Lincoln paused for a moment, drew a deep, shaky breath, and took Tad's hand in hers. \"Come along,\" she said, eyes fixed straight ahead. She left the White House without looking back, boarded her carriage, settled herself as the rest of her party climbed in after her, and said nothing more as they drove to the depot to board the private green railcar that had so often carried Mrs. Lincoln to and from the capital and New York City.\n\nBefore long the train puffed and chugged away from the station. Until they left the city limits, every exhalation of steam from the engine seemed to Elizabeth a great sigh of relief\u2014Washington City, glad to see the last of Mrs. Lincoln, who had never been good enough for their great, martyred president and now could be forgotten.\n\nThey were a small party\u2014Mrs. Lincoln, Robert, and Tad; Elizabeth; Dr. Anson Henry, a longtime friend of the family and Mr. Lincoln's former personal physician; and Thomas Cross and William Crook, two White House guards who had been assigned to escort the Lincoln family back to Illinois. Not long after the train headed westward, Mrs. Lincoln began to complain of one of her terrible, head-splitting migraines, so Dr. Henry dosed her with laudanum, and Elizabeth bathed her temples with cool water. \"Lizzie, you are my best and kindest friend,\" Mrs. Lincoln told her drowsily, reclining with her eyes shut, her face pale. \"I love you as my best friend. I wish it were in my power to make you comfortable for the balance of your days.\"\n\n\"That's very kind of you, Mrs. Lincoln,\" said Elizabeth, touched. \"For now, let's think about how to make _you_ comfortable.\"\n\nMrs. Lincoln grasped her arm. \"If Congress provides for me, depend upon it, I will provide for you.\"\n\nGently Elizabeth patted her hand, thanked her, and urged her to rest quietly.\n\nMrs. Lincoln slept soundly that night, and the next morning she felt well enough to sit up and gaze out the window. For hours she seemed distracted, dazed, while Elizabeth sat sewing quietly nearby, keeping an eye on her.\n\n\"What's that you're making?\" Mrs. Lincoln suddenly asked. \"A quilt?\"\n\n\"That's right.\" Elizabeth set her needle and fabric pieces on her lap and handed her one of the completed sections, seven small pieces joined together, a light center hexagon encircled by six dark. \"I've just begun. You know I'm not accustomed to idleness, and dressmaking is too difficult to do well with the motion of the train.\"\n\n\"Such small pieces, and such fine stitches.\" Mrs. Lincoln peered closer. \"These fabrics look familiar.\"\n\n\"They're scraps left over from making your dresses,\" said Elizabeth. \"I'm glad you recognize them. Each one seems to me like an old friend, and when my gaze falls upon one, I remember the gown cut from the same cloth, and the grand occasion for which it was made.\"\n\n\"What a lovely idea.\" Mrs. Lincoln returned the quilt pieces, and her gaze went back to the window. \"A memory album made of fabric. It is just the thing.\"\n\nShe paused so long, Elizabeth thought she was finished, but then she added in a barely audible whisper, \"Unless it is too painful to remember.\"\n\nFifty-four hours after they set out from Washington, their train arrived in Chicago. No one met them at the station; Elizabeth did not know who might have met them, if indeed anyone knew they were coming. Mrs. Lincoln had arranged for rooms at the Tremont House, a luxurious hotel on the southeast corner of Lake Street and Dearborn. \"My husband began his Senate campaign from that balcony,\" Mrs. Lincoln said as they descended from the carriage, indicating the place above by lifting her chin. \"Back in 1860, it served as the headquarters for the Illinois Republican Party during the Republican National Convention. My, how they rallied around my husband here in those days!\"\n\nRobert glanced at the impressive edifice, tugging on his ear thoughtfully. \"Isn't this also where Senator Douglas died?\"\n\nMrs. Lincoln's lips thinned. \"Yes,\" she said sourly. \"It is that too.\" She swept inside, and without another word, they all trailed after her.\n\nElizabeth was astounded by her sumptuous room. Never had she stayed anyplace so fine, and her heart sank with dismay when she estimated the likely expenses. Mrs. Lincoln had become accustomed to the grandeur of the White House, and she apparently expected to keep herself in that style. Elizabeth realized that as her friend and companion, it fell to her to warn Mrs. Lincoln that she would soon bankrupt herself if she persisted in that way. Just as she gathered her courage, Mrs. Lincoln spared her the onerous duty by reaching the same conclusion herself. \"Everything here is so very fine, and so very dear,\" she confessed to Elizabeth, sighing heavily. \"We cannot stay. I cannot fall into further financial embarrassment.\"\n\nShe dispatched Robert to find them less expensive accommodations, and within a week Robert proposed they move to the Hyde Park Hotel, a quiet retreat seven miles from the city center on the shore of Lake Michigan at Fifty-third Street. The village of Hyde Park, population five hundred, was a cool, lovely spot that had become a popular summer escape for well-to-do Chicagoans. The hotel's owner, Mr. Paul Cornell, a Chicago lawyer and developer for whom Mr. Lincoln had once done some legal work, told Robert that he would consider it a great honor if Mrs. Lincoln decided to reside there.\n\nAnd so she did.\n\nThey traveled by train, arriving in Hyde Park at about three o'clock on a Saturday afternoon. Elizabeth was struck by the newness of the hotel, which had opened only the previous summer and still smelled faintly of pine boards. The accommodations were markedly different from Tremont House, the rooms comfortable but small and plainly furnished. Most of Mrs. Lincoln's boxes and trunks had been stored in a warehouse upon their arrival, but what remained they unpacked. Elizabeth helped Mrs. Lincoln put away her clothes, and then she assisted Robert as he unpacked his books and arranged them on shelves in the corner of his room. They chatted pleasantly all the while, and when they were finished, Robert folded his arms, stood by the mantel, and gazed into space as if the weight of his change of fortune, the dramatic contrast between the past and the present, had just become real to him. \"Well, Mrs. Keckley,\" he eventually said, \"how do you like our new quarters?\"\n\n\"This is a delightful place,\" said Elizabeth, \"and I think you will pass your time pleasantly.\"\n\nHe studied her quizzically for a moment, as if he had expected a different answer. \"You call it a delightful place. Well, perhaps it is.\" He looked around the small but neat room, and she realized then that he saw it quite differently than she\u2014cramped and spartan. \"Since you do not have to stay here, you can safely say as much about the charming situation as you please. I presume that I must put up with it, as my mother's pleasure must be consulted before my own. But candidly, I would almost as soon be dead as be compelled to remain three months in this dreary house.\"\n\nIf he had not said \"almost,\" Elizabeth thought wryly, she might have accused him of exaggeration. She watched as he went to the window and gazed out upon the lovely scenery with a moody, querulous expression. Muffling a sigh, she excused herself, left him to his brood, and went to check on Mrs. Lincoln, who had retired to her room to rest before they sent down to the kitchen for their supper. Elizabeth had listened to Mrs. Lincoln's sobbing for eight weeks, so she was not surprised to find her lying on the bed, weeping as if her heart was broken. Elizabeth backed quietly into the hallway and slowly began to pull the door shut, but Mrs. Lincoln had heard her enter and rolled over to see who had disturbed her.\n\n\"What a dreary place, Elizabeth,\" she lamented, propping herself up on her elbow and wiping tears from the corner of her eyes with her other wrist. \"And to think that I should be compelled to live here, because I have not the means to live elsewhere. Ah! What a sad change has come to us all.\"\n\n\"It is not so bad,\" Elizabeth said, seating herself in the spindle chair beside the bed. \"The views are lovely, and the breeze from the lake is refreshing.\" She reached out her hand. \"Come and see it with me.\"\n\nMrs. Lincoln shook her head and fell back against the pillow. \"I couldn't bear it. How could I enjoy any small beauty when my husband lies in his grave and I know not what will become of me and my poor sons?\"\n\nElizabeth sighed softly. \"Very well. As you wish.\" She excused herself, found Tad playing alone in his room, and took him outside instead. He held her hand and chattered excitedly as they walked to the lakeshore, and once there he tore himself free and ran whooping and hollering down to the water. Together they walked along the beach, picking up smooth, round stones, stacking the prettiest and most interesting in a pile by the grass to take back to their rooms, flinging the rest into the lake one by one so they might enjoy the satisfying splashes.\n\nSunday dawned quiet and peaceful. From her window, Elizabeth looked out upon the beautiful lake, only one of many enchanting views from the hotel she had discovered. The wind rippled the broad, blue expanse of the water, and sunbeams made the waves sparkle like scattered jewels. Here and there a sailboat silently glided by or disappeared below the faint blue line of the horizon. Her thoughts turned toward the heavenly realm to which she aspired, the sunbeams on the water suggesting crowns studded with the jewels of eternal life. Elizabeth could not fathom how anyone could consider Hyde Park a dreary place, when it shone so brilliantly with light and life. She would be happy to rest there. She had seen so much trouble in her life, and she was weary, and knowing that she had to shepherd Mrs. Lincoln through her misery made her wearier still, and reluctant to leave her room and face the day. She would almost prefer to fold her arms and sink into an eternal slumber, so that the great longing of her soul for peaceful rest would at last be gratified.\n\nRobert spent the day in his room with his books, while Elizabeth remained in Mrs. Lincoln's, describing the many charming features of their new accommodations Mrs. Lincoln had perhaps not noticed in her grief, speaking plainly but gently about how Mrs. Lincoln's present circumstances were different from what she had come to expect as her due and encouraging her to plan for the future. Mrs. Lincoln refused to think beyond the summer, insisting that she wanted to live in seclusion all that while. \"Old faces will only bring back memories of scenes I wish to forget,\" she said, \"and new faces could not possibly sympathize with my distress, or add to the comforts of my situation.\"\n\nElizabeth disagreed but could not persuade her otherwise. Overnight, however, Mrs. Lincoln evidently relented enough to allow herself to ponder Tad's future, for on Monday morning, after Robert went into Chicago on business, she told Tad that he was going to have a lesson every morning, beginning that very day.\n\nTad protested that he did not want a lesson, to which Mrs. Lincoln replied that in that case she supposed he wanted to grow up to be a great dunce. \"You must do as Mother tells you, Tad,\" she said firmly. \"You are getting to be a big boy now, and must start school next fall. You would not like to go to school without knowing how to read.\"\n\nTad considered her words, perhaps imagining his humiliation if he were the only boy in his class who could not read, and then bounded to his feet, declaring that he did want a lesson after all and that he must have his book and start right away. Elizabeth looked on, amused, as Mrs. Lincoln seated herself in the easy chair and Tad pulled his own smaller chair up alongside, his book on his lap. The scene would have pleased Mr. Lincoln very much, Elizabeth thought. Tad had been humored and pampered by his parents, especially by his father. Tad suffered from a lisp and had never been sent to school, and so he did not know his book at all well. Elizabeth had never understood how two parents who valued knowledge and learning as much as the Lincolns did could have neglected Tad's education, so she was pleased to see that Mrs. Lincoln meant to make up for lost time. It was not only for Tad's sake that Elizabeth approved of this new plan. As much as Tad needed to learn his letters, Mrs. Lincoln needed some worthy endeavor to occupy her time and thoughts.\n\nTad opened his book and slowly spelled the first word. \"A, P, E.\"\n\n\"Well,\" his mother prompted, \"what does A, P, E spell?\"\n\nTad glanced at the small woodcut illustration above the word. \"Monkey.\"\n\n\"Nonsense,\" Mrs. Lincoln exclaimed. \"A, P, E does not spell monkey.\"\n\n\"Does too spell monkey!\" Tad pointed triumphantly at the picture. \"Isn't that a monkey?\"\n\n\"No, it is not a monkey.\"\n\nTad's mouth fell open in disbelief. \"Not a monkey! What is it, then?\"\n\n\"An ape.\"\n\n\"An ape! That's not an ape. Don't I know a monkey when I see it?\"\n\n\"Not if you claim that is a monkey.\"\n\n\"I do know a monkey,\" he insisted. \"I've seen lots of them in the street with the organ grinders. I know a monkey better than you do, 'cause I always go out to see them when they come by and you don't.\"\n\n\"Tad, listen to me. A monkey is a species of ape. It looks much like a monkey, but it is not a monkey.\"\n\n\"It shouldn't _look_ like a monkey then. Here, Yib\u2014\" This was the name he had adopted years before for Elizabeth, when his youth and speech impediment had conspired to render her name impossible to pronounce. \"Isn't this a monkey, and don't A, P, E spell monkey? Ma don't know anything about it.\" And with that he thrust the book at Elizabeth, earnest and excited.\n\nElizabeth could not help it; she burst out laughing. Tad drew back, very much offended. \"I beg your pardon, Master Tad,\" she gasped, fighting to chase away her mirth. \"I hope that you will excuse my want of politeness.\"\n\nHe bowed his head, forgiving her as graciously as a little lord, but then he persisted, saying, \"Isn't this a monkey? Don't A, P, E spell monkey?\"\n\n\"No, Tad,\" she said kindly. \"Your mother is right. A, P, E spells ape.\"\n\n\"You don't know any more than Ma.\" Indignant, Tad slammed the book shut. \"Both of you don't know anything.\"\n\nAt that moment, Robert entered, home from Chicago, and Tad immediately posed the question to him. It took some doing, but eventually Robert convinced his younger brother that Mrs. Lincoln and Elizabeth did indeed know what they were talking about, and that A, P, E did not and never would spell monkey. Once Tad accepted this irrefutable truth, the rest of the lesson proceeded with far less difficulty.\n\nElizabeth watched from the corner of her eye while she pieced her quilt, biting her lips together until the urge to laugh had passed. Then it occurred to her that if Tad had been a colored boy rather than the son of a president, and a teacher had found him so difficult to instruct, he would have been ridiculed as a dunce and held up as evidence of the inferiority of the entire race. Tad was bright; Elizabeth knew that well, and she was sure that with proper instruction and hard work, a glimmer of his father's genius would show in him too. But Elizabeth knew many black boys Tad's age who could read and write beautifully, and yet the myth of inferiority persisted. The unfairness of the assumptions stung. If a white child appeared dull, he and he alone was thought to suffer from a lack of intelligence or a deficient education, but if a colored boy appeared dull, the entire race was deemed unintelligent. It seemed to Elizabeth that if one race should not be judged by a single example, then neither should any other.\n\nThomas Cross and William Crook returned to Washington to resume their duties at the White House, taking with them the last vestiges of Mrs. Lincoln's former rank. As the days passed, having lost her husband and then her status, Mrs. Lincoln became consumed with fears of her incipient poverty. It had been demoralizing enough to leave the elegant Tremont House because she could not afford to live in such fine style, but to have fallen so far, so suddenly, felt like shameful exile. At first, acquaintances attempted to call on her at the Hyde Park Hotel, but Mrs. Lincoln could not bear for anyone to see her in such reduced circumstances, so she turned them all away. \"I had hoped for a much different reception when I decided to move here,\" she confided to Elizabeth. She had expected the Chicago elite to embrace her upon her arrival in their city; she had anticipated an outpouring of sympathy for her as the widow of the slain president. Elizabeth suspected that she had also hoped that some wealthy Republican benefactor would settle her in a residence more befitting her status and insist upon paying her expenses. But no guardian angel appeared, no offers were forthcoming. Time after time Mrs. Lincoln grasped thin straws of hope that one or another of her husband's wealthiest supporters or a friend who had acquired great wealth due to his patronage would come to her aid, and time and time again, her hopes were dashed to pieces.\n\nHumiliated, Mrs. Lincoln withdrew further into her seclusion, seeing no one, expressing her grief and worry to confidantes in lengthy letters. The weight of her debts threatened to crush her, but she was determined to pay back every cent. She penned elaborate pleas for government compensation in recognition of her husband's sacrifice for the nation, asking for all of the salary he had expected to earn in his second term as president. When Congress did not promptly make a provision for her, she decided to write directly to some of the newly wealthy, influential men who had earned fortunes from the appointments Mr. Lincoln had granted them, hoping that by reminding them what they owed the man, they would be moved to help the widow. She might have succeeded had the press not reported that Mr. Lincoln had left behind an estate worth seventy-five thousand dollars. Of course, it was bound up in a snarl of legalities, and it would be split three ways between Mrs. Lincoln and her sons, and even when her inheritance did come to her, she would not be given the principal to live on, but only the annual interest. But since the newspapers neglected to mention those details, Mrs. Lincoln's claims of poverty rang hollow. No one, not even her husband's closest friends and greatest admirers, was inclined to offer her sympathy or charity when her plaintive appeals seemed to paint a false portrait of her circumstances.\n\nWorsening matters were reports that when the Johnson family moved into the White House, they found the place ransacked, the public rooms nearly emptied of furniture. Although unscrupulous visitors and unfaithful servants had spirited away the goods, gossip lay the blame upon Mrs. Lincoln's shoulders\u2014or rather, within the scores of boxes and trunks she was known to have taken with her to Chicago. Thus while Mrs. Lincoln passed her lonely days feeling neglected and impoverished in remote Hyde Park, the public was gleefully relishing the new campaign against her, devouring each new sordid tale, no matter how improbable it would seem to a less prejudiced reader.\n\nThe public's scornful perception of her drove Mrs. Lincoln deeper into self-pity and despair. She rejected the ongoing, insistent assertions of her husband's executor\u2014Justice David Davis, Mr. Lincoln's campaign manager in 1860 and his nominee to the Supreme Court in 1862\u2014that she could live comfortably on her existing income if only she would move back to her home in Springfield. She begged friends to write to congressmen on her behalf, to use their influence to secure her the government pension she urgently needed. As First Widow, she believed that her sacrifice should be acknowledged by the government as well as the public, and lifting her above indigence was the very least they should do. Her tireless, frenzied letter writing took on the air of a crusade\u2014to redeem her reputation, to regain the status she had lost, to receive the honors she felt herself entitled to, and to preserve her husband's legacy. She pored over newspapers to stay informed about what was being said about Mr. Lincoln and by whom, and she filled her letters with astute observations about politics and current affairs. Any honors or provisions bestowed upon others greatly affronted her, for she believed they did not give enough credit to the slain president to whom they owed everything.\n\nMemorializing President Abraham Lincoln became her great cause\u2014in addition, of course, to raising Tad and seeing Robert independent, established, and content. She often confided to Elizabeth that if not for her sons, she would be quite relieved to take her own life.\n\nWhile Mrs. Lincoln wrote letter after letter, Elizabeth too kept up her own correspondence, sending courteous letters to favorite patrons eager to know when she might return, and writing more intimate missives to Virginia and Emma, to whom she poured out her concerns about Mrs. Lincoln and her business. Virginia offered her sympathy and urged her to come home at the earliest possibility, while Emma told her, simply and frankly, that her business was faltering in her absence. Emma and her other assistants had completed the dresses Elizabeth had begun before her departure, but new orders had become few and far between once word spread that the renowned Madam Keckley would not wield the needle herself. Out of necessity, Emma had let go half of the assistants, for there simply was not enough work for them all. \"All the ladies ask when you will return,\" Emma wrote. \"If you come back soon I believe we can entice them back in time to sew dresses for the next social season etc., but I fear if you stay away too long they will find someone else to attend them. They will not look half as pretty but they must wear something I guess.\"\n\nElizabeth's worst fears seemed to be coming to pass. She had invested everything into her dressmaking business\u2014her money, her time, her toil\u2014and now she felt as if she were watching it crumble to pieces from across a great chasm. The Lincolns were settled in their new home, though less happily than Elizabeth wished, and she had begun to wonder how much longer Mrs. Lincoln would require her to stay. The government had given Mrs. Lincoln a small sum to hire Elizabeth as her paid companion\u2014thirty-five dollars a week for her services, one hundred dollars for travel expenses and lodging, and fifty dollars for mourning attire\u2014but the income she lost neglecting her business in Washington vastly exceeded what she earned in Chicago. Also, although she would never distress Mrs. Lincoln by inquiring about the fund, she was certain that it would be depleted soon.\n\nElizabeth kept up with the news too, though not as fervently as Mrs. Lincoln, who read several newspapers from New York, Washington, and Chicago each day, and Elizabeth's heart skipped a beat whenever her gaze fell upon a story about Mr. John Wilkes Booth and his conspirators. All along, Mrs. Lincoln had seemed strangely indifferent to news about the death of her husband's assassin, the rounding up of his conspirators, and their upcoming trial, but Elizabeth hung on every word. She also found herself searching the columns of print for any mention of Mr. and Mrs. Jefferson Davis, who had fled deep into the South until they and their small party were finally captured early one May morning near Irwinville, Georgia. Newspapers reported that when the fugitives heard Union soldiers approaching their camp, Mr. Davis threw on one of his wife's dresses in an attempt to disguise himself and began to walk off into the surrounding forest. But a sharp-eyed corporal noticed that the very tall woman was wearing a man's boots and ordered him to halt, while another soldier raised his rifle. Still in her nightclothes, Mrs. Davis ran to her husband and flung her arms around him, and the soldier, unwilling to shoot a woman in the back, lowered his weapon. Elizabeth knew that Varina Davis had probably saved her husband's life, since the officers had orders to take the Confederate president dead or alive.\n\nElizabeth wished she could send a kind word to Mrs. Davis, whom she had always liked despite their disparate views on the very significant matters of slavery and secession, but she did not know where to send a letter. According to newspaper accounts, Mr. Davis was in prison at Fort Monroe, but Elizabeth did not know what had become of his wife and children.\n\nIn the last week of May, Elizabeth read that Chicago would soon host its second Great Northwestern Sanitary Fair. Throughout the war, the United States Sanitary Commission and its chapters throughout the North had raised essential funds in support of the war effort, its volunteers hosting countless fairs to earn money to purchase food, blankets, bandages, hospitals, uniforms, bedding, and nearly everything else the soldiers desperately needed. Even though President Johnson had declared the fighting officially over, the work of the Sanitary Commission went on, for the soldiers' needs, though different in nature, were as great as ever. Proceeds from the Chicago fair would benefit impoverished veterans, wounded soldiers who could not work, and the widows and orphans of soldiers killed in battle.\n\nFor the first Great Northwestern Sanitary Fair in the autumn of 1863, President Lincoln had donated the original draft of the Emancipation Proclamation, and it had sold for three thousand dollars. President Johnson or one of his advisers must have decided that a similar gesture was expected of him, for Elizabeth learned that the catafalque upon which President Lincoln's casket had rested had been sent from Washington for the event. It was hoped that ticket sales to view the exhibit, which would include other artifacts from the war, would raise an impressive sum for the worthy cause.\n\nIntrigued, Elizabeth wanted very much to attend, but she could not persuade Mrs. Lincoln to accompany her. \"An outing would do you good,\" Elizabeth told her, \"and I know the cause of the Union soldiers is very dear to your heart.\"\n\nMrs. Lincoln shuddered. \"The soldiers are indeed very dear to me, but not even for them do I wish to look upon something so intimately associated with my husband's death.\"\n\n\"The pavilion is so vast, that even if you do not go near it, you will still have many other things to see.\"\n\n\"And what would the people say if I refuse to go near my husband's catafalque?\" Mrs. Lincoln countered. \"They would say I do not show him proper respect. They would say I do not mourn him and that I am indifferent to his memory. Such lies would fill the papers tomorrow. No, Elizabeth, I do not wish to be seen there, and even if I wore a heavy veil, I would be known.\"\n\nElizabeth was disappointed, but she understood, and so she decided to go alone. She took the train north, and after a short ride into the heart of the city, she arrived at the station and walked the short distance to the fair's Trophy Hall on Michigan Avenue. She paid her admission fee and began touring the exhibits, reading the placards and studying the artifacts. She deliberately chose her route to save President Lincoln's display for the end, and her heart felt heavier with every step toward it.\n\nThe catafalque looked nearly as it had in the East Room, but strange and small and out of place in the unfamiliar, noisy, bustling setting of the pavilion. The simple bier was about seven feet long, two and a half feet wide, and perhaps two feet tall, draped in black silk trimmed in white, with heavy black tassels hanging at the four corners, deep swags of silk along the sides, and long fringe adorning the edges all around. A glass dome covered it, studded with stars, and at its base lay souvenirs of slavery taken from Southern plantations during the war\u2014an enormous ball and chain, a pair of heavy shackles, an assortment of whips, and other items so bleakly, painfully reminiscent of her youth that Elizabeth had to turn away. She remembered how distraught Mrs. Lincoln had been when the catafalque was being constructed in the East Room, because every strike of a hammer against a nail sounded to her like a pistol shot. How thoughtless she had been to invite Mrs. Lincoln to attend the fair with her! Why on earth would any grieving widow subject herself to the sight of the bier upon which her husband's casket had rested?\n\nShaking her head and silently chiding herself, Elizabeth quickly walked away from the catafalque. She had seen nearly all she had wanted to see and was inclined to leave, but then her gaze fell upon a crowd gathered around another display on the opposite side of the hall. Curious, she drew nearer and spied an iron bell with a sign proclaiming that it had been taken from the Mississippi plantation of Jefferson Davis\u2014but that object, though it drew many a curious eye, did not account for the crowd. The fairgoers were waiting in line to pay twenty-five cents to see a wax figure depicting Jefferson Davis at the moment of his capture in Georgia.\n\nElizabeth could not resist. She joined the queue, paid her fee, and when her turn came, stepped beyond the curtain with a handful of other spectators to view the display. A wax figure bearing a fairly good resemblance to Mr. Davis stood on a platform, clad in a man's suit with a woman's floral garment over it, a strangely familiar garment\u2014\n\n\"Oh, my goodness,\" Elizabeth exclaimed. \"It's Mrs. Davis's chintz wrapper!\"\n\nThe other visitors in her group started at her outburst. \"It's what now?\" one gentleman queried as all eyes turned her way.\n\nHer gaze fixed on the garment, Elizabeth gestured, utterly disbelieving. \"It's\u2014it's one of the two chintz wrappers I made for Mrs. Davis back in January of 1861, right before she and her husband left Washington.\" From the corner of her eye, she saw two ladies regarding her with astonishment as they whispered to each other. \"I am a seamstress,\" she explained. \"I'm visiting from Washington City, where I have a dressmaking business. Mrs. Davis was one of my customers, and Mrs. Lincoln was too, and still is.\"\n\nShe was not sure why she added the last, perhaps because the doubt in their eyes unsettled her. She became aware then of other fairgoers peering around the curtain, drawn by her exclamations or wanting to see what was holding up the line.\n\nThe first gentleman to speak to her told them, \"This colored woman claims she made this wrapper for Mrs. Davis.\"\n\nThe people murmured excitedly, and some pushed their way to the front while still more peered around the curtain. Elizabeth instinctively took a step backward.\n\n\"Is that so?\" a woman asked excitedly. \"Can you prove it?\"\n\nOthers joined in the call for her to prove her claims, and Elizabeth looked from one to another to the next in dismay, wishing she could flee. \"How?\" she managed to say. \"How shall I prove it?\"\n\n\"Tell us how you made it,\" an elderly woman prompted.\n\nThat Elizabeth could do well enough. Steeling herself with a deep breath, she stepped closer to the wax figure and told them how Mrs. Davis had purchased the fabric herself, how the specter of war had compelled her to economize, how she had needed Elizabeth to complete the garments quickly because she anticipated\u2014correctly, as it turned out\u2014that her husband would soon decide to leave Washington. These first details came out in a rush, but then Elizabeth began to talk about how she had made the garment, how a particular technique was a signature of her style, and that part came easily.\n\nWhile she spoke, the curtain was shoved out of the way so a large crowd could gather around her. It grew by the minute as Elizabeth proved, as best she could, that she knew firsthand that the garment had belonged to Mrs. Davis because she was the seamstress who had made it. She knew that words were not proof\u2014for all her listeners knew, she could be a very good actress or a particularly skilled liar\u2014but her self-confident manner and knowledgeable descriptions of the wrapper's finer details seemed to convince them of her veracity.\n\n\"This colored woman says she made the dress old Jeff Davis wore when he was captured,\" a man shouted over his shoulder, beckoning others to gather around.\n\n\"I make no such claim,\" Elizabeth protested. \"That is to say, I did indeed sew this garment for Mrs. Davis, but I have no idea whether Mr. Davis ever wore it as a disguise. I would think that he did not, that he could not have done, for his shoulders are far too broad and the bodice too narrow\u2014\"\n\nThey interrupted her explanation with jeers and protests that she must not defend him, so she quickly steered the subject back to the making of the wrapper. In the meantime, someone had run to fetch the chairwoman of the fair, who stood among the crowd, listening as eagerly as the others. The spectators had expected to enjoy amazing sights at the fair, but nothing quite like this, and they had not even had to pay the extra twenty-five cents for the privilege.\n\n\"Will you swear to all this?\" someone called out when she finished. The crowd was pushing forward and she was becoming a little uneasy.\n\nElizabeth agreed, because she could not imagine what the mob's disappointment would compel them to do if she refused. The fair's chairwoman, Mrs. Bradwell, dashed off to find a notary public, and while Elizabeth waited, she answered questions and demurely avoided others about the Davis family and the Lincolns. At last Mrs. Bradwell returned with a short, dark-haired, portly gentleman who quickly produced paper, pen, and the seals of his office. He and Mrs. Bradwell advised her what to say, but she changed the wording slightly as she wrote her oath:\n\nI hereby certify that I, Elizabeth Keckley, was originally the dressmaker for Mrs. Jefferson Davis, that I have recently been dressmaker for Mrs. President Lincoln and have attended her from Washington to Chicago; that I have seen the figure of Jefferson Davis now on exhibition at Trophy Hall, and recognize the dress upon said figure as made by me for Mrs. Jefferson Davis, and worn by her.\n\nElizabeth Keckley\n\nChicago, June 6, 1865\n\nThree witnesses, including Mrs. Bradwell, signed the document after she did, and the notary stamped and sealed it and held it up for all to see. They burst into cheers, and in the confusion Elizabeth managed to slip away, flustered and breathless, but not before she glimpsed Mrs. Bradwell affixing the signed and notarized document to the display.\n\nThe next day, Elizabeth was surprised and chagrined to read about her little adventure in the Chicago _Evening Journal_. Her oath had been printed in its entirety, and it was noted that after she had verified the authenticity of the wrapper, ten thousand fairgoers had spent a total of twenty-five hundred dollars on lottery tickets for a chance to win the garment.\n\n\"Twenty-five hundred dollars,\" Elizabeth mused aloud, impressed. She decided that her small embarrassment had been well worth it, since it had helped to raise a substantial amount of money for a very worthy cause.\n\nIt was almost mid-June when the fund to pay for Elizabeth's services ran out and Mrs. Lincoln could no longer afford her room and board. Mrs. Lincoln begged her to remain a little while longer regardless, but Elizabeth, missing her own home and worried about her dressmaking business, was secretly not altogether disappointed that she could not.\n\nBy that time she had persuaded Mrs. Lincoln to leave her rooms occasionally and take the air, and as the day of Elizabeth's departure approached, they often strolled together in a nearby park that Mrs. Lincoln had become quite fond of, or along the lakeshore, where they enjoyed the refreshing breezes.\n\nOn their last day together, Mrs. Lincoln was melancholy and tearful, and since Elizabeth realized that nothing she could say would ease the pain of her departure, she spoke very little but instead listened attentively while Mrs. Lincoln lamented. \"It almost appears to me that I am on the seashore,\" said Mrs. Lincoln sadly, gazing out at the lake. \"How wide is it, do you suppose?\"\n\n\"I believe it is some seventy-five miles in breadth.\"\n\n\"Well, then, it is little wonder I cannot discern the opposite shore.\" Mrs. Lincoln sighed heavily, dejected. \"My friends thought it would be quieter here during the summer months than in the city, and they were right, but it will be far too quiet without you, Lizzie.\"\n\n\"If I could stay, I would.\" Elizabeth spoke truly. She needed not hurt Mrs. Lincoln's feelings by adding that she was relieved that she could not stay.\n\n\"Tell me, how can I live without my husband any longer?\" Mrs. Lincoln suddenly cried. \"This is my first awakening thought each morning, and as I watch the waves of the turbulent lake under our windows I sometimes feel I should like to go under them.\"\n\nElizabeth felt a chill every time Mrs. Lincoln spoke thus, but she said firmly, \"You thought you couldn't live after Willie died, and yet you did. You have Tad and Robert to live for if you don't care about living for yourself. You must think of your sons, and rally.\"\n\nMrs. Lincoln inhaled deeply and shuddered. \"You are the only person who talks to me this way. Tad becomes frightened when I sink into despair, and Robert becomes impatient. You are the only good, kind friend I have anymore, and I don't know how I shall get along without you.\"\n\n\"You would perhaps have more friends if you would allow them to see you instead of sending them all away.\"\n\nMrs. Lincoln was silent for a long moment. \"I suppose there is something to what you say.\" She threw Elizabeth a beseeching look. \"Promise me, that if Congress makes an appropriation for my benefit and I can bear your expenses, you will come with me to visit my husband's tomb on the first anniversary of his death.\"\n\n\"I have already promised you that,\" Elizabeth reminded her, smiling fondly, \"but if it will comfort you to hear me say it again, then I promise, again, that I will.\"\n\n\"And promise me that you will write.\"\n\n\"I will, but\u2014\" Elizabeth hesitated. \"I confess that I am not a good writer.\"\n\n\"You don't think you are, but you are,\" Mrs. Lincoln declared with a little of her old fire, \"and I wouldn't care if you were not. Write to me as often as you can.\"\n\nSo Elizabeth promised that she would.\n\n# Chapter Fourteen\n\n## JUNE 1865\u2013SEPTEMBER 1866\n\nIn the middle of June, Elizabeth returned to Washington with Mrs. Lincoln's best wishes for her continued success in business. She also assured Elizabeth that she still considered her to be her personal modiste, though hundreds of miles would soon separate them. \"You know my form as well as I do myself,\" Mrs. Lincoln remarked. \"I will still ask you to make new dresses for me, when I have the occasion and the means to have a new dress made. I must insist that you sew every stitch yourself, as you have always done; I am no longer First Lady, and I'm sure your assistants are quite good, but they cannot be your equal.\"\n\n\"I will always consider it a great privilege to sew your dresses with my own hands,\" Elizabeth assured her.\n\nThe train ride east was long and tiresome, but each mile brought her closer to home. The front door of her boardinghouse on Twelfth Street was a welcome sight, and as she approached, Virginia and Walker hurried out to greet her, taking her satchel and embracing her and declaring how good it was to see her again. After she settled herself in her room, which Virginia had kindly aired and dusted in anticipation of her arrival, she accepted the couple's invitation to join the family for dinner.\n\nAfter they finished eating and the children left the table, little Alberta in Jane's arms and Lucy trailing along behind, Virginia and Walker queried Elizabeth about her time in Chicago and what she planned to do next. \"First I intend to salvage my dressmaking business,\" she said. \"I have some duties to attend to at the White House on behalf of Mrs. Lincoln, and of course I want to resume my work with the Freedmen and Soldiers' Relief Association.\" The charitable organization that she had founded to assist the contraband seeking refuge in Washington City had changed its name in 1864 to reflect the changing times and emancipation.\n\n\"Have you sent business cards to President Johnson's family?\" asked Walker. \"Or maybe you plan to leave one at the White House when you take care of Mrs. Lincoln's business?\"\n\n\"I don't intend to do so at all,\" Elizabeth said. \"I have no desire to work for the new president's family.\"\n\n\"Why not?\" asked Virginia. \"Mrs. Lincoln was your best customer. Why wouldn't you want to sew for the new First Lady?\"\n\n\"Or in this case, all three of them,\" said Walker. \"Mrs. Johnson is said to be so often indisposed that her daughters have taken responsibility for most of the duties of White House hostess.\"\n\n\"Is that the problem?\" said Virginia, smiling as if she couldn't believe it. \"Too many First Ladies to please and too much work?\"\n\n\"That's not it at all.\" How lovely it was to be among her friends again, enjoying their teasing. \"Mr. Johnson was never a friend to Mr. Lincoln, and he failed to treat Mrs. Lincoln with common courtesy in the hour of her greatest sorrow.\"\n\n\"You don't have to sew for him,\" Walker observed. \"His wife and daughters surely did nothing wrong.\"\n\nElizabeth hesitated. He made a fair point, and yet she could not overcome her reluctance. She would feel disloyal to Mrs. Lincoln if she sewed for her successor. Also, if she sewed at the White House again, she was likely to encounter Mr. Johnson as frequently as she had seen Mr. Lincoln, and she had no desire to spend any time in his company.\n\nShe took a day to rest and recover from her journey, and then she called at the White House to transact Mrs. Lincoln's business. Her heart filled with dread as she passed through the front doors, for every familiar sight and sound and smell bitterly reminded her of the past. Inside, she discovered that its new occupants had already altered the mansion significantly, and more refurbishments were in progress. The walls had been painted here and new wallpaper hung there. Stained and damaged upholstery had been concealed within linen slipcovers. The wood floors, doors, and trim had been refinished and repainted to a glossy sheen. Taking in the scene, Elizabeth was reminded painfully of Mrs. Lincoln's extensive beautification efforts, and how all her hard work had been undone by unscrupulous staff and greedy treasure seekers.\n\nWhen Elizabeth completed her errand and departed, which she did as quickly as she could, she fervently hoped that she had crossed the threshold for the last time.\n\nElizabeth had promised Mrs. Douglas that she would create her long-overdue spring trousseau as soon as she returned from Chicago, so after setting Emma to the task of delivering notes to her favorite patrons to inform them she was back in business, she called on Mrs. Douglas to meet their engagement. Mrs. Douglas looked very pleased to see her, but also quite surprised. \"Why, Mrs. Keckley,\" she exclaimed, \"can it really be you? I did not know you were coming back so soon. It was reported that you would remain with Mrs. Lincoln all summer.\"\n\nElizabeth acknowledged that she had expected to stay longer too. \"Mrs. Lincoln would have been glad to have kept me with her had she been able.\"\n\n\"Able?\" Mrs. Douglas echoed. \"What do you mean by that?\"\n\n\"Only that she is already laboring under pecuniary embarrassment, and was only able to pay my expenses, and allow nothing for my time.\"\n\n\"You surprise me. I thought she was left in good circumstances.\"\n\n\"So many think, it appears,\" said Elizabeth ruefully. \"I assure you, Mrs. Lincoln is now practicing the closest economy.\" She went on to tell her of Mrs. Lincoln's fruitless efforts to obtain a widow's pension from the government and the withholding of her inheritance due to the delays sorting out her husband's estate.\n\nIn the days and weeks to come, Elizabeth would share the tale of Mrs. Lincoln's woes to mutual acquaintances and sympathetic patrons\u2014Mrs. Lee, Mrs. Welles, anyone kind enough to listen and perhaps to advocate her cause to their influential husbands and friends. If the truth about Mrs. Lincoln's circumstances came to light, perhaps Congress would be compelled to make a provision for her.\n\nWith Mrs. Douglas's spring trousseau at last under way, Elizabeth gathered her assistants, polished her sign, and, to her great relief, soon had her business going along at a steady pace. As word of her return to Washington spread, orders soon began to come in faster than Elizabeth could fill them. One day in late June, the girl attending the door found Elizabeth in the cutting room, where she was hard at work on a lovely rose silk gown. \"Mrs. Keckley,\" the youngest of her assistants said, \"there is a lady below who wants to see you.\"\n\nCaught in the middle of a difficult section, Elizabeth finished cutting before she answered. \"Who is she?\"\n\n\"I don't know. I didn't learn her name.\"\n\nElizabeth did not want to interrupt her work at that moment, but she also didn't wish to offend an important patron. \"Is her face familiar? Does she look like a regular customer?\"\n\nThe girl shook her head. \"No, she is a stranger. I don't think she was ever here before. She came in an open carriage, with a colored woman for an attendant.\"\n\n\"It might be the wife of one of Johnson's new secretaries,\" Emma mused.\n\n\"Do go down, Mrs. Keckley,\" urged another assistant.\n\nTheir curiosity had fanned the flames of her own, so she set down the shears, brushed loose threads from her skirt, and went below. When she entered the parlor, a tall, brown-haired, plainly dressed woman rose and asked, \"Is this the dressmaker, Mrs. Keckley?\"\n\n\"Yes,\" Elizabeth replied. \"I am she.\"\n\n\"Mrs. Lincoln's former dressmaker, were you not?\"\n\n\"Yes, I worked for Mrs. Lincoln.\"\n\nThe woman smiled. \"And are you very busy now?\"\n\nElizabeth spread her hands and laughed, indicating the workroom just beyond, where the sounds of her industrious assistants hard at work could not be mistaken. \"Very, indeed.\"\n\n\"Can you do anything for me?\"\n\n\"That depends what is to be done, and when it is to be done.\"\n\nThe woman tapped her chin with her forefinger, thinking. \"Well, say one dress now, and several others a few weeks later.\"\n\nElizabeth quickly ran through her mental list of work she had already accepted. \"I can make one dress for you now, but no more,\" she said with a hint of polite regret. \"I cannot finish the one for you in less than three weeks.\"\n\n\"That will answer,\" the woman said, her manner cheerfully decisive. \"I am Mrs. Patterson, the daughter of President Johnson. I expect my sister, Mrs. Stover, here in three weeks, and the dress is for her. We are both the same size, and you can fit the dress to me.\"\n\nFor a brief, disquieting moment, Elizabeth wished she had asked the woman for her name before agreeing to sew for her, but they soon arranged satisfactory terms. After Elizabeth measured Mrs. Patterson, she bade her good morning, entered her carriage, and drove away.\n\nWhen Elizabeth returned to the workroom, her assistants were naturally eager to learn who her visitor had been. \"It was Mrs. Patterson,\" she replied. \"The daughter of President Johnson.\"\n\n\"What?\" exclaimed one of the girls. \"The daughter of our good Moses. Are you going to work for her?\"\n\nWhen Elizabeth spoke, it felt like an admission of guilt. \"I have taken her order.\"\n\n\"I fear that Johnson will prove a poor Moses,\" said Emma, frowning, \"and I would not work for any of the family.\"\n\nSeveral of the young women murmured agreement. It was not until that moment that Elizabeth realized how little they liked Mr. Lincoln's successor. That Mrs. Lincoln disliked him, Elizabeth knew, and that he had made a bad impression when he had arrived drunk for his own inauguration, all of Washington was well aware. But since Mr. Johnson had taken office, Elizabeth had been either sequestered with the grieving First Widow in her White House chambers or hundreds of miles away in Hyde Park. She knew very little about any policies he might have enacted or speeches he had made in the past few weeks, but clearly, he had not won over the women in that room.\n\nElizabeth wondered if Mr. Johnson would turn out to be as poor a leader as Emma predicted, or if her assistants were merely biased against him because he was not Mr. Lincoln, the Great Emancipator they had all admired and respected. That, Elizabeth thought, with the first pang of empathy she had felt for him, was not his fault, and he should not be condemned for it.\n\nBefore long she finished the first dress for Mrs. Patterson\u2014or rather, her sister\u2014and was pleased when it was received with great satisfaction. She agreed to make additional dresses for the sisters, and as the summer passed, she discovered that both Mrs. Patterson and Mrs. Stover were kind, plain, unassuming women, making no pretensions to elegance. One day when she called at the White House, she found Mrs. Patterson busily at work with a sewing machine. The novelty of the sight struck her, because although Mrs. Lincoln knew how to sew and had owned a lovely sewing machine in a solid redwood full case, silver plated and adorned with inlaid pearl and enamel, Elizabeth had never seen her use it, nor could she recall having ever seen Mrs. Lincoln with a needle in her hand.\n\nBut as pleasant as the sisters were, and as kindly as they treated her, Elizabeth was never entirely happy in their employ. She rarely glimpsed Mr. Johnson, so he was not the problem, although she could not forget how he had slighted Mrs. Lincoln by neglecting to offer his condolences after Mr. Lincoln's death. She could not cross the threshold of the White House without remembering the pleasant hours she had once spent there or the kind familiarity Mr. Lincoln had always shown her. She missed collaborating with Mrs. Lincoln on a stunning new gown, as the sisters took little interest in fashion and preferred simple, long-sleeved garments with high necklines and collars and scant ornamentation. She missed hearing Mr. Lincoln address her as \"Madam Elizabeth\" and \"combing down his bristles\" before he escorted his wife to a levee. She even missed his silly, rambunctious goats. The White House held so many vivid associations for her that every step she took, every direction she turned, evoked a memory from a past more satisfying than the present. It pained her to be in the White House when Mr. and Mrs. Lincoln no longer could be.\n\nIn August, Elizabeth had recently finished two light summer linen dresses, one for each sister, when Mrs. Patterson sent her a note requesting her to come to the White House to cut and fit another, warmer dress for her in anticipation of autumn. Some strange mood possessed Elizabeth in that moment, and she curtly wrote back that she never cut and fitted work outside of her workrooms. This brought her business relations with the president's daughters to an abrupt end.\n\nEmma regarded Elizabeth curiously when she explained why they would not be sewing anything else for the White House hostesses. \"You told them you never cut and fitted dresses except within your workrooms, but you used to do so for Mrs. Lincoln.\"\n\n\"Yes, on occasion, but Mrs. Lincoln never objected to coming here. In fact, I think she enjoyed it\u2014perhaps even preferred it.\"\n\n\"I remember.\" Emma regarded her from beneath raised brows. \"I also recall that you used to say that you never approved of ladies attached to the presidential household coming to your rooms. You said it was more consistent with their dignity to send for you, and have you go to them.\"\n\n\"Well\u2014\" Elizabeth paused, thinking, but it was no use. \"You're quite right. I have said that, and I did feel that way. I _do_ feel that way. I cannot explain why I responded to Mrs. Patterson as I did.\"\n\n\" _I_ can explain,\" said Emma, as if it were obvious. \"You don't want to work for them, but you don't know how to refuse them.\"\n\n\"I must be mad to do so,\" said Elizabeth, pressing a hand to her forehead. \"They've been perfectly agreeable patrons, and who turns down work from the president's daughters?\"\n\n\"The most popular modiste in Washington City, that's who.\" Emma swept her arm toward the busy workroom, where all of her assistants were industriously sewing, sitting up straight as she had taught them to avoid backaches and neck strain. \"You're not mad at all, nor were you impertinent. You didn't refuse to work for Mrs. Patterson; you agreed to work for her, so long as the cutting and fitting took place here, and she declined.\"\n\nElizabeth sighed. \"I suppose that is one way of looking at it.\"\n\n\"It's the only way to look at it.\" Emma smiled fondly and shook her head. \"You can choose your customers, Elizabeth, and you can afford to be particular.\"\n\nLooking around the workroom, Elizabeth realized that indeed, perhaps she could.\n\nBy the end of summer, Elizabeth's business had prospered so much that she opened a second shop in the market, but Mrs. Lincoln's prolific correspondence revealed that she was faring far less well. She said virtually nothing about the June trial and July execution of the four conspirators condemned to die on the gallows for their part in her husband's assassination, but she poured out her sorrow when she informed Elizabeth of the death of Dr. Anson Henry, one of the few friends from her Springfield days who had not abandoned her. \"I will never forget how tenderly and solicitously he cared for me in the weeks after my beloved husband's death,\" she lamented. \"To think of him lost at sea is almost more than I can bear.\"\n\nMrs. Lincoln also felt control of her husband's legacy slipping from her grasp. She had won an important early skirmish when she overruled the Illinois dignitaries in choosing the location of her husband's memorial and tomb, but another battle was brewing against an opponent she had not seen coming. Her husband's former Springfield law partner, William H. Herndon, hoped to write a revelatory book about the president's \"inner life\" and had taken to poking about asking Mr. Lincoln's friends and acquaintances to confide their memories to him. Mrs. Lincoln was troubled by Mr. Herndon's actions, but Robert was incensed. It was one matter to study a politician, he declared, because having his private life exposed to the public was part of the price he paid for his office. It was another thing altogether to subject his wife and children to such uninvited scrutiny, to compel them to live inside a \"glass house.\"\n\nAll the while, the beauties of Hyde Park continued to elude her. \"I am miserable,\" Mrs. Lincoln wrote later in July. \"I remain sequestered in my rooms except to take an occasional walk in the park, and of course I see no one, becalmed as I am on the _far off_ shores of Lake Michigan.\" In August, she quit the Hyde Park Hotel and moved herself and her sons to the Clifton House, a respected residential hotel at Wabash Avenue and Monroe Street in central Chicago. Tad enrolled in a local school and was determined to catch up to his peers, and Robert apprenticed at Scammon, McCagg & Fuller, a prominent Chicago law firm. But any satisfaction she might have found in her new residence and her sons' accomplishments was dimmed by her ongoing torment. Grief stricken and feeling abandoned, Mrs. Lincoln remained terrified of poverty and debt, and it seemed she could think of little else. Her creditors had hesitated to pester her about her overdue bills when she was First Lady, but recently they had begun threatening to sue her and to publish lists of her debts in the papers. She wrote to friends, former White House staff, and members of the House of Representatives pleading her case and asking them to use their influence to assist her, and Elizabeth could well imagine the alternately desperate, hectoring, flattering, and relentless tone of her letters. Although her husband's frequent critic Horace Greeley, editor of the _New York Tribune_ , astonished Mrs. Lincoln by taking up a subscription to raise money for \"the late president's grieving widow and her fatherless sons,\" her own efforts seemed all in vain.\n\nElizabeth offered Mrs. Lincoln what encouragement and comfort she could through the mail, but it was never enough. It was a national shame, Elizabeth thought indignantly, that President Lincoln's widow could not be better provided for. He had given his life for his country the same as any soldier, the same as her own dear George, and the least the government could do was to care for his widow and his children, the same as any soldier's.\n\nIn autumn, a woman\u2014a stranger, Elizabeth believed\u2014called on her at her boardinghouse. \"You are surprised to see me, I know,\" the woman greeted her happily. She was not a customer, nor anyone Elizabeth recognized from Washington, and yet her face was familiar. \"I have just come from Lynchburg, and when I left cousin Anne, I promised to call on you if I came to Washington.\" The woman beamed, spreading her hands. \"I am here, you see, according to promise.\"\n\n\"Cousin Anne?\" said Elizabeth, bewildered. \"Pardon me, but\u2014\"\n\n\"Oh, I see you do not recognize me,\" the woman exclaimed. \"I am Mrs. General Longstreet, but when I was a girl, you knew me as Bettie Garland.\"\n\n\"Bettie Garland,\" Elizabeth gasped. Bettie Garland was the cousin of her former master, Hugh Garland, and had often visited the family at their former residence near Dinwiddie Court House in Virginia. \"Is this indeed you?\"\n\nThe woman nodded, beaming, and they clasped hands, exclaiming with the delight that only an unexpected reunion could bring.\n\n\"I am so glad to see you,\" said Elizabeth, offering her guest a chair and seating herself. After buying her freedom, she had kept in touch with her former mistress and her children\u2014especially the daughters whom she had raised and loved\u2014but their ties had been severed by the outbreak of war. Elizabeth had often wondered what had become of them, although whenever she mentioned their names and expressed concern for their welfare, her Northern friends would roll their eyes and ask how she could possibly spare a kind thought for those who had kept her in bondage. Try though she might, Elizabeth could not make them understand that despite the grave injustice done to her, and without condoning any part of it, she still felt a deep and abiding affection for a few particular members of the families that had owned her\u2014though certainly not all of them. \"Where does Miss Anne live now?\"\n\n\"Ah! I thought you could not forget old friends,\" said Mrs. Longstreet. \"Cousin Anne is living in Lynchburg. All the family are in Virginia. They moved there during the war.\" Then her jubilance dimmed. \"Fannie is dead. Nannie has grown into a woman and is married to General Meem. Hugh Junior was killed in the war, and now only Spotswood, Maggie, and Nannie are left.\"\n\n\"Fannie, dead!\" Fannie was the Garlands' third-eldest daughter and had been especially fond of Elizabeth's mother, who had served as her nurse. \"And poor Hugh! You bring sad news as well as pleasant.\" Her thoughts flew to Nannie, who had been her special charge. She had shared Elizabeth's bed, and Elizabeth had watched over her as if she had been her own child. Indeed, Elizabeth could not have loved her more if she had been. \"I can hardly believe it. She was only a child when I saw her last.\"\n\n\"Yes, Nannie is married to a noble man. General Meem belongs to one of the best families in Virginia. They are now living at Rude's Hill, up beyond Winchester, in the Shenandoah Valley. All of them want to see you very badly.\"\n\n\"I should be delighted to go to them,\" Elizabeth declared. \"Miss Bettie, I can hardly realize that you are the wife of General Longstreet, and just think, you are now sitting in the very chair in the very room where Mrs. Lincoln has often sat!\"\n\n\"The change is a great one, Lizzie,\" she said, laughing ruefully. \"We little dream today what tomorrow will bring forth. After fighting so long against the Yankees, my husband is now in Washington, suing for pardon, and we propose to live in peace with the United States.\"\n\nElizabeth was very pleased to hear it.\n\nShe had many questions about old friends, and the time passed swiftly in conversation, but all too soon Mrs. Longstreet's visit ended. Before she left, she gave Elizabeth the Garlands' address, and the next day Elizabeth wrote to them, telling them of her life in Washington and expressing hope that she would be able to see them before long.\n\nWhen she told Virginia and Emma about Mrs. Longstreet's visit and the letters she had sent to Miss Anne and her daughters, Emma shook her head in wonder, frowning. \"I don't know why you miss them so. I never, never wish to see any of my masters or mistresses again.\"\n\n\"I don't wish to see _all_ of my old masters,\" Elizabeth pointed out. \"There are some I can never forgive. But I understand why you feel the way you do, Emma. Your break with your last master and mistress was particularly unpleasant. They should never have refused to abide by your mistress's will, so they have only themselves to blame for the lawsuit. I bought my freedom. Perhaps that makes the difference.\"\n\n\"I hope you won't be disappointed, Elizabeth,\" said Virginia, her brow furrowing with concern. \"I suspect your old mistress and her daughters have forgotten you. Surely they're like all of their kind, too selfish to give a single thought to you now that you're no longer their slave.\"\n\n\"Perhaps so,\" said Elizabeth, \"but I cannot believe it. Did they not ask Miss Bettie to call on me? You don't know the Southern people as I do. Though master and slave, we had a warm attachment.\"\n\nVirginia and Emma exchanged a dubious look, and Elizabeth suspected that they could debate the matter forever and her friends still would not understand her point of view. \"You have some strange notions, Elizabeth,\" Emma remarked, shaking her head.\n\nOf all her acquaintances, only Mrs. Lincoln seemed to understand her enduring affection. \"Certainly the Garlands will not have forgotten you,\" she responded after Elizabeth wrote to her about Mrs. Longstreet's visit. \"I have never forgotten my beloved Sally, and how tenderly she cared for me throughout my early years. I do not mean to speak ill of my dear mother when I say that Sally raised me. After my mother died, I do not know what I should have done without Sally, as my father's new wife considered all of us stepchildren a burden, and was much preoccupied with her own children. No, Elizabeth, they cannot have forgotten you.\"\n\nHeartened, Elizabeth waited anxiously for a reply to her letters, and she did not have long to wait. Her heart soared when the first of many long missives came from various members of the family, warm affection filling every line. For months they exchanged letters, and in the winter, Miss Nannie\u2014now Mrs. General Meem\u2014wrote that she and her husband would be very glad to have Elizabeth visit them in the summer. \"You must come to me, dear Elizabeth,\" Miss Nannie entreated. \"I am dying to see you. We are now living at Rude's Hill. Ma, Maggie, Spot, and Minnie, sister Mary's child, are with me, and only you are needed to make the circle complete. Come\u2014I will not take no for an answer.\"\n\nElizabeth was delighted to accept, and after consulting with Emma and considering the likely state of dressmaking orders at one time and another, she wrote back telling Miss Nannie to expect her in August.\n\nMrs. Lincoln was delighted for her but sorry for herself, that someone else would have the pleasure of Elizabeth's company when she could not. Throughout the autumn, she had tried to sell some of her jewels and other luxury items, and had tried to return others to the stores where she had purchased them, but the effort proved futile. Shortly before Christmas, Congress informed her that they would not give her Mr. Lincoln's salary for his entire second term, as she had asked, but would only part with one year's pay, which after deductions amounted to little more than twenty-two thousand dollars, equal to only a small fraction of her debts. Later that winter, Congress granted President Johnson seventy-five thousand dollars to refurbish the White House, and humiliating criticism appeared in the New York _World_ and other papers saying that Mrs. Lincoln had left behind a ransacked mansion and mountains of overdue bills that continued to plague merchants throughout the city. The persistent Mr. Herndon had succeeded in collecting letters, interviews, and statements from people who had known Mr. Lincoln\u2014some quite well, some barely at all\u2014and was delivering lectures drawn from the material, which he hoped to publish as a book. Still eager to interview Mrs. Lincoln, he had sent his request to Robert, but both he and his mother were unsettled by his phrasing: \"I wish to do her justice fully\u2014so that the world will understand things better. You understand me.\" Mrs. Lincoln did not clearly understand him at all, she fretted to Elizabeth. \"What precisely does he mean by _do me justice fully_?\" she had written to Elizabeth. \"He puts on a foreboding tone that I do not like. And yet I feel it may be necessary to speak with him. If I do not give him _my truth,_ he may invent his own.\"\n\nWorse yet had been the anniversaries of days that had brought Mrs. Lincoln anguishing memories of loss\u2014November 4, her first wedding anniversary without her husband; December 13, her forty-seventh birthday; December 21, Willie's birthday; New Year's Day, marking the start of another year without the loved ones she mourned; February 1, the date of her son Eddie's death; and February 12, Mr. Lincoln's birthday. All of these melancholy dates built up to the worst, most unbearable anniversary, April 15, the date of her husband's assassination. She suffered too on March 30, Good Friday, for it was on the night of Good Friday that he had been shot. \"I am desperately unhappy and do not think I will be able to get through the day without you by my side,\" she wrote to Elizabeth as the end of March loomed nearer. She reminded Elizabeth of the promise she had made upon her departure from Chicago, that if Congress had granted her a widow's pension, Elizabeth would return and accompany her to Springfield to visit her husband's tomb on the anniversary of his death. The appropriation was not made, and so Elizabeth could not go. Mrs. Lincoln traveled with Tad instead, arranging her travel plans at odd times and along circuitous routes to avoid encountering any old friends.\n\nElizabeth could not go to Illinois that spring, but she did go to Virginia in summer. On August 10, the fourth anniversary of her son George's death, Elizabeth boarded the train for Harpers Ferry, eagerly anticipating her reunion with the Garlands. The journey was not without mishaps. The train arrived at Harpers Ferry at night, but Elizabeth slept through the stop and was carried to the next station, where she was obliged to wait for another train to take her back. Once there, she intended to change cars for Winchester, but she had missed the train and was detained another day. Arriving at last in Winchester, she learned that the only way to reach Rude's Hill was by a series of stagecoaches. The drive commenced in the evening and would last through the night, but Elizabeth was so exhausted she could scarcely keep her eyes open. A young gentleman riding in the stage told Elizabeth that he knew General Meem well and that he would tell her when they had reached the proper place for her to disembark. Thus reassured, Elizabeth drifted off to sleep.\n\n\"Aunty.\" Someone was shaking her. \"Aunty, didn't you want to get out at Rude's Hill?\"\n\n\"Yes, I did.\" Elizabeth straightened in her seat, rubbing her eyes. \"Are we there?\"\n\nAs she spoke, her gaze fell upon the young man who had promised to wake her, and she discovered him softly snoring.\n\n\"More than there,\" the man who had woken her said. \"We have passed it.\"\n\n\"Passed it?\"\n\n\"Yes. It is six miles back. You should not sleep so soundly, Aunty.\"\n\n\"Why did you not tell me sooner?\" Elizabeth cried. \"I am so anxious to be there.\"\n\n\"Fact is, I forgot it,\" he said with a shrug. \"Never mind. Get out at this village, and you can find conveyance back.\"\n\nElizabeth had little choice but to do exactly that. The town, New Market, was in a sad, dilapidated condition that spoke plainly of the heartless destruction of war. Climbing down from the stage and collecting her satchel, she found her way into a hotel, really little more than a house, where she was able to buy a cup of coffee and gather her wits. When she inquired about a ride back to Rude's Hill, the landlord told her that the stage would return that evening.\n\n\"This evening?\" Elizabeth's spirits plummeted. It was only just dawn. \"I want to go as soon as possible. I should die if I had to stay all day in this lonely place.\"\n\nShe didn't mean to insult the residents of the tiny hamlet, but fortunately few of them were present to hear. The landlord shrugged and said there was nothing he could do, so she settled down for a long wait, utterly dejected.\n\nShe had not sat there long, sipping her cooling coffee, when the colored man behind the bar came over to her table. \"I'm sorry for your troubles, ma'am.\"\n\nShe managed a tremulous smile. \"That is very kind of you.\"\n\n\"I know Gen'ral Meem's place. I can drive you over in 'bout an hour.\"\n\nElizabeth seized her chance. \"Oh, could you? That would be the most joyful news I've heard in days.\"\n\nHe assured her it would be his pleasure, and in turn she thanked him and urged him to set out as soon as possible.\n\nShe finished her coffee and waited outside the door of the hotel for her courteous driver to bring his wagon around. While she stood there, fighting to conceal her impatience, a fat old lady spied her from across the street and waddled over to greet her. \"Ain't you Elizabeth?\"\n\n\"Yes, I am,\" she replied, startled that the stranger should know her name.\n\n\"I thought so.\" The woman smiled, revealing several missing teeth. \"They been expecting you at Rude's Hill every day for two weeks, and they do little but talk about you. Mrs. Meem was in town yesterday, and she said that she expected you this week certain. They will be mighty glad to see you.\"\n\nAt this news, Elizabeth's spirits rose slightly. \"I will be even gladder to see them.\"\n\n\"Well, as to that I couldn't say,\" the woman said, chuckling hoarsely. \"They've kept a light burning in the front window every night for ten nights, in order that you might not go by the place should you arrive in the night.\"\n\n\"Thank you,\" Elizabeth said fervently, feeling much restored. \"It's pleasant to know that I'm expected. I fell asleep in the stage, and failed to see the light, so here I am instead of at Rude's Hill, where I meant to be.\"\n\nAs the woman clucked sympathetically, the man from the hotel pulled up in his wagon. Elizabeth climbed aboard, and soon they were on the road to General Meem's country seat. \"That's Rude's Hill,\" her driver remarked, nodding to a picturesque green rise encircled with tall trees as they approached. Elizabeth shaded her eyes with her hands as they climbed the hill, eager for her first glimpse of her old acquaintances. She spied a young man standing in the front yard, quickly tallied the years and subtracted from his apparent age, and deduced that he must be Spotswood\u2014or Spot, as everyone called him, shortening the old family name. She had not seen him in eight years, but when she beckoned to him, he cried out with joy and came running. His happy shout drew the attention of the rest of the family, who had been waiting by the windows or on the veranda but now came hurrying to the wagon. \"It is Elizabeth! It is Elizabeth,\" she heard them cry happily, and eager to be among them, she stepped from the wagon to the top of the stile\u2014but when she attempted to leap down, her hoopskirt caught on one of the posts, and she fell sprawling to the ground.\n\n\"Elizabeth,\" someone exclaimed. Her palms stung and her right knee throbbed from the impact, and she lay there, dazed, the smell of fresh Virginia soil with a pungent undercurrent of manure filling her senses.\n\nSpot reached her first and hauled her to her feet, and a moment later she was in the arms of Miss Nannie, Miss Maggie, and Mrs. Garland. She scarcely had time to properly thank her kind driver before they whisked her into the house, tended to her soiled hands and skirts, and settled her into an easy chair by the hearth. All the while, the servants looked on in amazement.\n\nBeaming with joy, Miss Nannie clung to her hand as if she would never let go. \"Elizabeth, you are not changed a bit,\" she declared, kissing Elizabeth again on the cheek, her eyes filling with happy tears. \"You look as young as when you left us in St. Louis, years ago.\"\n\n\"I cannot say the same,\" Elizabeth replied, laughing. \"You have quite grown up!\"\n\nMiss Nannie smiled proudly and beckoned to a tall, graceful young woman. \"Here, Elizabeth, this is Minnie, Minnie Pappan, sister Mary's child. Hasn't she grown?\"\n\n\"Minnie,\" Elizabeth exclaimed, extending her hand to the young woman, who smiled, took it, and sat down on the footstool. \"I can hardly believe it. You were only a baby when I saw you last. It makes me feel quite old to see how tall you have grown. Miss Minnie, you are bigger than your mother was\u2014your dear mother whom I held in my arms when she died.\" Elizabeth had to pause to compose herself, wiping a tear from each eye.\n\n\"Have you had your breakfast, Elizabeth?\" asked Mrs. Garland.\n\nWhen Elizabeth shook her head, the children let out a chorus of vows that they would take care of that right away. \"It is not necessary that all should go,\" Mrs. Garland said, laughing, as Nannie, Maggie, and Minnie headed for the kitchen. \"The cook is there. She will get breakfast ready.\"\n\nBut the three young ladies did not heed her. They rushed to the kitchen, and soon they brought Elizabeth a delicious hot breakfast on a tray. While Elizabeth ate, the cook observed the commotion from the doorway, astonished. \"I declare, I never did see people carry on so,\" she said, shaking her head. \"Wonder if I should go off and stay two or three years, if all of you would hug and kiss me so when I come back?\" The Garlands laughed and teased her, saying that they could not spare her so she must not test the theory.\n\nSoon after Elizabeth finished her breakfast, Miss Nannie's husband arrived. \"Elizabeth, I am very glad to see you,\" he greeted her. \"I feel that you are an old acquaintance, I have heard so much of you through my wife.\"\n\n\"And me,\" Miss Maggie interjected.\n\nGeneral Meem smiled. \"Yes, and you, and your mother. Welcome to Rude's Hill, Elizabeth.\"\n\nIn the days that followed, Elizabeth learned that during the war, General Stonewall Jackson had used Rude's Hill as his headquarters, and he had slept in the very room Elizabeth was given as a sitting room. General Jackson was the Southern ideal of a soldier, and admirers from far and near still came to Rude's Hill to pay tribute to their fallen hero, to walk in his footsteps. Elizabeth observed that nearly every visitor would tear a splinter from the walls or windows of her sitting room, which they would carry away and treasure as a priceless relic. The Garlands' plantation was beautiful, but the scars of war were visible everywhere upon the house and the landscape. General Meem had taken up planting, and he employed many laborers to tend the fields and servants to care for the home.\n\nElizabeth soon discovered that she evoked great curiosity in the neighborhood. Her association with Mr. Lincoln, and her attachment to the Garlands, her former owners, had garbed her in the disguise of a tragic heroine from a sentimental romance. Elizabeth thought it was nonsense, but she did not complain. She was comfortably quartered at Rude's Hill, and the Garlands showed her every attention. They passed the days sewing together or talking of old times, and every day they either drove about the countryside or rode on horseback.\n\nElizabeth and Mrs. Garland\u2014Miss Anne, as Elizabeth would always think of her\u2014had many long talks alone. For the first time Elizabeth searched her former mistress's face and discovered features that resembled her own. She wondered if the long, quiet looks Miss Anne often gave her meant she was doing the same. Miss Anne was only eight years older than she, they were children of the same father, and yet their lives could not have been more different.\n\nOnce, while they were out walking through the new flower garden Miss Anne had only recently begun to cultivate, Elizabeth asked her what had become of her aunt Charlotte, her mother's only sister. She had been maid to the elder Mrs. Burwell, Miss Anne's mother.\n\n\"She is dead, Lizzie,\" Miss Anne said gently. \"She has been dead for some years.\" She sighed, and her eyes seemed to gaze back through the years. \"A maid in the old time meant something different from what we understand by a maid at the present time. Your aunt used to scrub the floor and milk a cow now and then, as well as attend to the orders of my mother.\"\n\n\"I remember,\" said Elizabeth, clasping her hands behind her back as they walked. She supposed she probably knew better than her former mistress what her aunt's chores had included.\n\n\"My mother was severe with her slaves in some respects, but then her heart would be full of kindness.\"\n\nElizabeth mulled that over. \"I suppose that's fair to say.\"\n\n\"She had your aunt punished one day\u2014\"\n\n\"For what offense?\"\n\n\"I don't recall what she did, or didn't do,\" Miss Anne admitted, \"but my mother punished her. Not liking her sorrowful looks, my mother made your aunt two extravagant promises on the condition that she would look cheerful, and be good and friendly with her again.\"\n\nSilently, Elizabeth cheered for her stubborn aunt. \"What did she promise?\"\n\n\"First, that Charlotte might go to church the following Sunday, and second, that my mother would give her a silk dress to wear on the occasion.\"\n\n\"Extravagant indeed,\" remarked Elizabeth. \"I assume that my aunt accepted?\"\n\n\"Oh, certainly. Now, my mother had but one silk dress in the world, silk being not so plentiful in those days as it is now, and yet she gave this dress to her maid so they would be friends again.\" Miss Anne laughed merrily.\n\n\"Did your mother's plan work?\" Elizabeth asked. \"Were they friendly again?\"\n\n\"Oh, her plan worked all right, and it was fortunate for Mother that it did. Two weeks afterward, she was invited to spend the day at a neighbor's house, but when she inspected her wardrobe, she discovered that she had nothing fit to wear in company.\"\n\nElizabeth felt a slow smile grow on her lips. \"Is that so?\"\n\nMiss Anne nodded, amused. \"She had but one alternative, and that was to appeal to the generosity of your aunt Charlotte. So, she was summoned, and the problem was explained to her, and the maid offered to loan her silk dress to the mistress for the occasion, and the mistress was only too glad to accept. She made her appearance arrayed in the silk that her maid had worn to church on the preceding Sunday.\"\n\nThey laughed together over the incident, although perhaps not for the same reasons.\n\n\"Elizabeth\u2014\" Miss Anne broke off, and they walked in silence for a long moment before she spoke again. \"During the entire war I used to think of you every day, and have longed to see you so much. When we heard you were with Mrs. Lincoln, the people used to tell me that I was foolish to think of ever seeing you again\u2014that your head must be completely turned.\"\n\n\"And of course you believed their wise counsel,\" said Elizabeth lightly, \"because people who have never met me are always the best judge of my character.\"\n\n\"Of course not,\" Miss Anne protested. \"I knew your heart, and I could not believe that you would forget us. I always insisted that you would come to see us someday.\"\n\n\"How could I forget the people I grew up with from the time I was a baby?\" asked Elizabeth. \"My Northern friends used to tell me that you would forget me, but I told them I knew better, and I didn't lose hope.\"\n\n\"Love is too strong to be blown away like gossamer threads,\" Miss Anne said. \"The chain is strong enough to bind life even to the world beyond the grave.\" Abruptly she halted and placed a hand on Elizabeth's forearm to bring her to a stop too. \"Elizabeth,\" she said, suddenly anxious. \"Do you always feel kindly toward me?\"\n\nElizabeth chose her words carefully. \"To tell you candidly, Miss Anne, I have but one unkind thought, and that is that you didn't give me the advantages of a good education, which was my heart's desire. All I have learned has been the study of later years.\"\n\nMiss Anne pressed her lips together and nodded. \"You're right,\" she said unhappily. \"I didn't look on things then as I do now. I've always regretted that you weren't educated when you were a girl.\" She paused and managed a wan smile. \"But you haven't suffered much on this score, since you get along in the world better than we who enjoyed every educational advantage.\"\n\nElizabeth did not contradict her. Miss Anne was a widow dependent upon the generosity of her brother-in-law, whereas while Elizabeth was also a widow, she was independent and self-sufficient, a prosperous businesswoman. She had been an intimate of the Lincoln White House, and except for the death of her son, she had not suffered from the war as the Garlands, Meems, and Pappans had.\n\nShe would not trade places with her former mistress, Elizabeth realized, despite Miss Anne's privileges and advantages. She was proud of all she had done for herself without them, and she was proud of the woman she had become.\n\nElizabeth remained at Rude's Hill for five weeks, and parted from Miss Anne and her children with heartfelt wishes on all sides that they would meet again someday.\n\n# Chapter Fifteen\n\n## OCTOBER 1866\u2013FEBRUARY 1868\n\nWhen Elizabeth returned to Washington, she had much business to attend to and mail to sort. Most of the letters were from Mrs. Lincoln, who struggled on as lonely and miserable as ever. In the summer she had become so weary of boarding that she spent nearly all of her husband's remaining 1865 salary granted to her by Congress on a fine stone home on West Washington Street in Chicago. She and Tad had settled there, in a popular neighborhood near Union Park, while Robert had moved into a bachelor apartment, where he was no doubt much happier.\n\nAfter much consideration, Mrs. Lincoln had also warily consented to speak with Mr. Herndon, but when they met in Springfield in September, Mrs. Lincoln had tried to flatter and charm her husband's would-be biographer into leaving her out of his book altogether. \"I told him that it was not unusual to mention the existence of the wife, in the biography of her husband, with nothing more than to note that the two were married on this particular date in such and such a place,\" she wrote to Elizabeth. \"I _wish_ he would say nothing at all of me, but I _hope_ he will say no more of me than that. I do not know if my pleas will convince him. He has disliked me since we met at a dance at the home of Colonel Robert Allen, shortly after I first came to Springfield. Mr. Herndon engaged me for a waltz, and afterward, he told me that I had glided through the dance with the grace of a serpent. A serpent, Elizabeth! Hotly I replied, 'Mr. Herndon, comparison to a serpent is rather severe irony, especially to a newcomer,' and I promptly left him. I believe he has harbored a grudge against me ever since, but as he was my husband's law partner, we were often thrown together, and our relations were civil, if not friendly. I can only trust that his fond memory of my husband will prevent him from taking out any lingering resentment upon me. I do believe he means to exalt my husband in his book, in which case it is my duty as his widow to share my memories of him, which are more intimate than anyone else's.\"\n\nElizabeth received Mrs. Lincoln's letter too late to write back and advise her against the interview, but she consoled herself with the realization that it might not have mattered in any case. Mrs. Lincoln could have disregarded Elizabeth's misgivings, or Mr. Herndon might have written whatever he pleased even without speaking with Mrs. Lincoln. Elizabeth hoped that whatever Mr. Herndon chose to do, it would be of so little consequence that no one would ever hear of it, but in this matter she was disappointed. In November, Mr. Herndon delivered another lecture, which he arranged to distribute widely on broadsides in advance and which was reprinted in newspapers throughout the nation. In his lecture, the perfidious Mr. Herndon claimed that Mr. Lincoln had never loved his wife, but instead pined for Miss Ann Rutledge, to whom he had been engaged until her untimely death in 1835. After that, he had never addressed another woman with love and affection. He had even signed his letters to Miss Mary Todd \"Your Friend Abraham Lincoln\" rather than \"Yours affectionately,\" and he had eventually married her only out of obligation to honor. Thus Mary Lincoln was not to blame for the well-known difficulties in the marriage, because Mr. Lincoln had never loved her.\n\nWhen Elizabeth read the shocking assertions, her heart went out to Mrs. Lincoln. If Mr. Herndon's intention had been to wound the grieving widow, he could not have chosen a more devastating tactic. On the first anniversary of her husband's death, Mrs. Lincoln had written to Elizabeth, \"It was always music in my ears, both before and after our marriage, when my husband told me that I was the only one he had ever thought of, or cared for. That will solace me to my grave.\" Now Mr. Herndon had stolen that comfort from her. And on what shaky grounds? Was Mr. Herndon constantly at Mr. Lincoln's side, day and night, year after year, so that he could with all certainty confirm that Mr. Lincoln had never addressed any other woman but Miss Ann Rutledge with love and affection? Had he read every letter Mr. Lincoln had written to his wife, heard every word uttered? Elizabeth could not count how many kind, affectionate phrases she had heard Mr. Lincoln speak to his wife through the years. Admittedly, they had quarreled from time to time, but so did every husband and wife of Elizabeth's acquaintance, and however hotly their tempers flared, Mr. and Mrs. Lincoln had always been anxious to make up soon afterward.\n\nIt comforted Elizabeth very little\u2014and as she would learn in letters yet to come, Mrs. Lincoln not at all\u2014that Mr. Herndon was roundly disparaged for his lectures. The people were shocked and appalled that he would violate all standards of decorum by addressing such intimate details of the martyred president's life, and many were sure that Miss Rutledge was a figment of his imagination. A furious Robert Lincoln took measures to discredit and silence the aspiring biographer, warning that the subject of the Lincoln family was strictly off-limits for his manuscript in progress. The Lincolns' pastor from their Springfield days, Dr. James Smith, wrote a scathing rebuke, addressed to Mr. Herndon but widely published in the newspapers, including the _Chicago Tribune,_ which was where Mrs. Lincoln discovered it. The pastor had read Mr. Herndon's lecture \"with feelings of mingled indignation and Sorrow, because coming as it did from his intimate friend and law partner, it was calculated to do the character of that great and good man an incalculable injury, deeply to wound the feelings of his heart broken widow and her orphan boys, and to place that whole family both the dead and living, in their family relations, in a most unenviable light before the public.\" He emphatically asserted that no man was better placed to know Mr. Lincoln's heart than his pastor, esteemed and respected by the family, entrusted with their spiritual care, intimate with all the joys and sorrows of their lives, relied upon for his advice and counsel. \"During the seven years when he and myself were at home,\" he wrote, \"scarcely two weeks ever passed during which I did not spend a pleasant evening in the midst of that family Circle.\" Dr. Smith's intimacy with the Lincoln family had convinced him that Mr. Lincoln was \"utterly incapable of withholding from the Bride he led to the Altar that which was her due, by giving her a heart dead and buried in the grave of Another; but that in the deep and honest sincerity of his Soul, he gave her a heart overflowing with love and affection.\" He was certain that Mr. Lincoln \"was to the Wife of his bosom a most faithful, loving and Affectionate husband.\"\n\nMrs. Lincoln must have felt a certain sense of vindication to have her husband's own pastor publicly confirm that Mr. Lincoln had loved her most tenderly, but even so, Mr. Herndon's claims had dealt her a terrible blow. \"There are hours of each day, that I cannot bring myself to believe, that it has not _all_ been some hideous dream,\" she lamented to Elizabeth. \"In my bewildered state, I sometimes feel that my darling husband, _must and will_ return to his sorrowing loved ones. This I know shall never be, in this World, and if not for Tad I would all too willingly join him in the Next.\"\n\nIn the first few weeks of 1867, Mrs. Lincoln's letters often alluded to her sinking fear that she would not be able to afford her home on West Washington Street much longer, and that she would be obliged to take cheap rooms for herself and Tad elsewhere, rent out the house, and live off the income. In March she wrote again to confess that her fear had become a certainty. She had struggled long enough to keep up appearances, but that mask at last had to be thrown aside, for she simply could not live on her meager allowance. \"As I have many costly things which I shall never wear,\" she wrote, \"I might as well turn them into money, and thus add to my income, and make my circumstances easier. It is humiliating to be placed in such a position, but, as I am in the position, I must extricate myself as best I can. Now, Elizabeth,\" she continued, \"I want to ask a favor of you. It is imperative that I should do something for my relief, and I want you to meet me in New York, between the 30th of August and the 5th of September next, to assist me in disposing of a portion of my wardrobe.\"\n\nElizabeth knew that Mrs. Lincoln's income was modest, only seventeen hundred dollars a year, and that her collection of elegant gowns, packed away in boxes of trunks since her move from Washington, was of no tangible value to her any longer, since she would almost certainly never wear the dresses again. Elizabeth decided that since Mrs. Lincoln's need was urgent, it would be prudent to dispose of the gowns quietly, and that New York would be the best place to transact such delicate business.\n\n\"Why do you take this on?\" Emma asked when Elizabeth explained why she might have to leave the dressmaking business in her care or shut it down altogether while she traveled on behalf of Mrs. Lincoln. \"You have already done so much for her, and she is never any better for it.\"\n\n\"I think she is better for knowing she can rely on me,\" Elizabeth countered. \"Everyone else has betrayed or abandoned her, except her sons. The question should not be why I help her so much but why other people help her so little.\" Mrs. Lincoln was the wife of the Great Emancipator, the martyred president who had done so much good for their race. How could Elizabeth refuse to do anything that would be to her benefit?\n\nOn September 15 Elizabeth received a letter from Mrs. Lincoln announcing that she would arrive in New York City on the night of the seventeenth. She instructed Elizabeth to come beforehand and secure rooms for them at the St. Denis Hotel under the name Mrs. Clarke, an alias she had sometimes employed while traveling as First Lady.\n\nStartled, Elizabeth read the letter again to be sure she had not misunderstood it. She had never heard of the St. Denis Hotel, which suggested that it was not a first-class establishment and was unlikely to be up to Mrs. Lincoln's standards. She also was perplexed by Mrs. Lincoln's decision to travel without protection under an assumed name, and forgo the trust and deference due to her as the First Widow. Most dismaying of all, she knew it would be difficult if not impossible for her as a colored woman to engage rooms at a strange hotel for a person about whom the proprietors knew nothing.\n\n\"What is she thinking?\" Elizabeth murmured, shaking her head as she scanned the letter. If only she could ask! Mrs. Lincoln would already be en route to Washington before a letter could reach her in Chicago, and a telegram was out of the question, because Elizabeth could not expose the delicate business to every curious operator along the line. Caught in an impossible predicament, Elizabeth's only hope was that at the last moment, Mrs. Lincoln would send word that she had changed her mind. So Elizabeth remained in Washington, waiting for a letter or a telegram, her anxiety increasing each day. When Mrs. Lincoln sent no word by September 18, the morning after she had said she would arrive in New York, Elizabeth immediately telegraphed \"Mrs. Clarke\" at the St. Denis Hotel and told her she would join her there soon.\n\nShe took the next train to New York, and after an anxious ride by rail and then by stage, she arrived at the hotel, a six-story building at Broadway and East Eleventh Street. Pulling the bell at the ladies' entrance, she inquired with the boy who answered whether a Mrs. Clarke was staying there. He did not know, but went off to check with the manager, and soon returned to reply that Mrs. Clarke was indeed their guest. \"Do you want to see her?\" he asked.\n\n\"Yes.\"\n\n\"Well, just walk right there.\" He gestured in an indefinite direction. \"She's down here now.\"\n\nHesitating, thinking that perhaps Mrs. Lincoln was in the parlor with company, Elizabeth gave him one of her business cards. \"Take this to her, if you please,\" she said, but at that moment, Mrs. Lincoln came into the hall, drawn by the sound of her familiar voice.\n\n\"My dear Elizabeth, I am so glad to see you,\" she exclaimed, crossing the room and giving Elizabeth her hand. \"When I arrived last night and you were not here, I was simply frantic.\"\n\nElizabeth had not seen Mrs. Lincoln in more than two years, and her pale, haggard appearance momentarily rendered her speechless. \"I sent a telegram,\" she managed to say.\n\n\"Yes, but I have only just received it. It has been sitting here all day but it was never delivered until this evening. Come, and let us find out about your room.\"\n\nShe led Elizabeth into the office, where the clerk, like all modern hotel clerks, was exquisitely arrayed, highly perfumed, and too self-important to be courteous. He eyed Elizabeth with disdain as Mrs. Lincoln approached. \"This is the woman I told you about,\" she said. \"I want a good room for her.\"\n\nThe clerk's eyebrows rose. \"We have no room for her, madam.\"\n\n\"But she must have a room. She is a friend of mine, and I want a room for her adjoining mine.\"\n\n\"We have no room for her on your floor\" was his pointed reply.\n\nElizabeth understood his meaning perfectly, and she regarded him with steady, dignified silence. She would like nothing more than to quit his establishment and find a room for herself at a hotel run by a courteous colored proprietor, but night had fallen and she dared not venture out again, nor could she leave Mrs. Lincoln alone.\n\nMrs. Lincoln drew herself up, frowning. \"That is strange, sir. I tell you that she is a friend of mine, and I am sure you could not give a room to a more worthy person.\"\n\n\"Friend of yours or not, I tell you we have no room for her on your floor.\" He paused and reluctantly added, \"I can find a place for her on the fifth floor.\"\n\n\"That, sir, I presume, will be a vast improvement on my room,\" declared Mrs. Lincoln imperiously. \"Well, if she goes to the fifth floor, I shall go too. What is good enough for her is good enough for me.\"\n\n\"Very well, madam.\" The clerk heaved a sigh and checked his register and room keys. \"Shall I give you adjoining rooms, and send your baggage up?\"\n\n\"Yes, and have it done in a hurry. Let the boy show us up. Come, Elizabeth.\" Mrs. Lincoln turned away from the clerk with a haughty parting look. The boy who had met Elizabeth at the door led them to the stairs, which they climbed and climbed until Elizabeth began to suspect that they would never reach the top. When they did, and the boy opened the doors to their rooms, Elizabeth could not have said which of them was more appalled. They had been given a pair of cramped, dingy, scantily furnished, three-cornered rooms in the servants' garret that smelled of dust and damp and sweat. Never in her life would Elizabeth have imagined a president's widow in such humble accommodations.\n\n\"How provoking,\" Mrs. Lincoln exclaimed, sinking heavily into a chair, panting from the effort of scaling the tower of stairs. \"I declare, I never saw such unaccommodating people. Just to think of them sticking us away up here in the attic. I will give them a regular going over in the morning.\"\n\n\"You forget they do not know you,\" Elizabeth reminded her. \"Mrs. Lincoln would be treated differently from Mrs. Clarke.\"\n\n\"True, I do forget. Well, I suppose I shall have to put up with the annoyances.\" Mrs. Lincoln's expression turned woebegone. \"Why did you not come to me yesterday, Elizabeth? I was almost crazy when I reached here last night, and found you had not arrived. I sat down and wrote you a note\u2014I felt so badly\u2014imploring you to come to me immediately.\"\n\n\"I thought perhaps you would change your mind,\" Elizabeth admitted. \"I also knew I would have great difficulty securing rooms for 'Mrs. Clarke.'\"\n\n\"Well, based upon what we have seen so far I cannot fault you for suspecting as much.\" Then she gave a little start. \"You have not had your dinner, Elizabeth, and you must be hungry. I nearly forgot about it in the joy of seeing you. You must go down to the table right away.\"\n\nElizabeth was famished, but the thought of a good meal partly revived her. Mrs. Lincoln pulled the bell rope, and when a servant appeared, she ordered him to give Elizabeth her dinner. Elizabeth followed him downstairs, where he led her into the dining room and seated her at a corner table. She was giving him her order when the steward approached. \"You are in the wrong room,\" he said gruffly.\n\nElizabeth regarded him mildly. \"I was brought here by the waiter.\"\n\n\"It makes no difference. I will find you another place where you can eat your dinner.\"\n\nElizabeth's stomach rumbled as she got up from the table and followed him from the dining room. \"It is very strange,\" Elizabeth said tightly when they reached the hall, \"that you should permit me to be seated at the table in the dining room only for the sake of ordering me to leave it the next moment.\"\n\nThe steward halted and regarded her over his shoulder. \"Are you not Mrs. Clarke's servant?\"\n\n\"I am with Mrs. Clarke,\" Elizabeth replied, emphasizing the distinction.\n\n\"It is all the same.\" He turned back around and continued down the hall. \"Servants are not allowed to eat in the large dining room. Here, this way. You must take your dinner in the servants' hall.\"\n\nHumiliated and hungry, Elizabeth followed the steward through the rear corridors of the hotel, knowing that it was the only way she was likely to get a bite to eat. On reaching the servants' hall, the steward tugged on the knob only to find the door locked. He left Elizabeth standing in the passage while he went to inform the clerk. A few minutes later, the obsequious clerk came blustering down the hall, the scent of his perfume preceding him. \"Did you come out of the street, or from Mrs. Clarke's room?\"\n\n\"From Mrs. Clarke's room,\" she replied politely, refusing to mirror his ill temper.\n\n\"It is after the regular hour for dinner. The room is locked up, and Annie has gone out with the key.\"\n\nFor a moment Elizabeth hoped that he might allow her to return to the dining room, but when he said nothing more, her pride would not allow her to stand waiting in the hall any longer. \"Very well,\" she said, turning toward the staircase. \"I will tell Mrs. Clarke that I cannot get any dinner.\"\n\nHe scowled as she began to climb the stairs. \"You need not put on airs,\" he called after her. \"I understand the whole thing.\"\n\n\"I don't think you do,\" Elizabeth muttered under her breath as she reached the first of far too many landings. \"If you understood the whole thing,\" she huffed as she climbed, \"it is strange that you should put the widow of President Abraham Lincoln in a three-cornered room in the attic of this miserable hotel.\"\n\nMurmuring to herself what she could not say to his face only made her feel worse. When she finally reached Mrs. Lincoln's room, tears of humiliation and frustration blurred her vision.\n\nAt the sight of her downcast expression, Mrs. Lincoln's brow furrowed in concern. \"What is the matter, Elizabeth?\"\n\n\"I cannot get any dinner.\"\n\n\"Cannot get any dinner? What do you mean?\"\n\nElizabeth sank into a chair and told her all that had happened since the servant had led her downstairs. \"Those insolent, overbearing people,\" Mrs. Lincoln exclaimed, furious. She seized the armrests of her easy chair and pulled herself fiercely to her feet. \"Never mind, Elizabeth. You shall have your dinner. Put on your bonnet and shawl.\"\n\n\"What for?\"\n\n\"What for?\" Mrs. Lincoln put on her bonnet and stood at the mirror, tying the strings. \"Why, we will go out of the hotel, and get you something to eat where they know how to behave decently.\"\n\nWarily, Elizabeth said, \"Surely, Mrs. Lincoln, you do not intend to go out on the street tonight?\"\n\n\"Yes I do. Do you suppose I am going to have you starve, when we can find something to eat on every corner?\"\n\n\"But you forget. You are here as Mrs. Clarke and not as Mrs. Lincoln. You arrived alone, and the people here already suspect that everything is not quite as you say. If you go outside the hotel at night, they will accept that as evidence against you.\"\n\n\"Nonsense. What do you suppose I care for what these low-bred people think? Put on your things.\"\n\n\"No, Mrs. Lincoln,\" Elizabeth said firmly, though her stomach rumbled a protest. \"I shall not go outside of the hotel tonight, for I understand your situation, even if you do not. Mrs. Lincoln has no reason to care what these people may say about her, but Mrs. Clarke wishes to remain incognito, so she must be more prudent.\"\n\nWith some difficulty, Elizabeth finally persuaded Mrs. Lincoln to act with caution. She was so frank and impulsive that she never gave enough thought to how her words and deeds could be misconstrued. Elizabeth bade her good night and went off to her own room, but not until she was settled in bed and had turned down the lamp did it occur to her that Mrs. Lincoln could have ordered dinner to be served to Elizabeth in her room, so that she would not have had to retire hungry.\n\nThe next morning, Mrs. Lincoln knocked on Elizabeth's door before six o'clock. \"Come, Elizabeth, get up,\" she called. \"I know you must be hungry. Dress yourself quickly and we will go out and get some breakfast. I was unable to sleep last night for thinking of you being forced to go to bed without anything to eat.\"\n\nElizabeth too had slept poorly in her uncomfortable, lumpy bed, kept from restful slumber by her growling stomach. Swiftly she dressed, and before long she and Mrs. Lincoln were taking their breakfast at a restaurant on Broadway about a block away from the St. Denis Hotel. Afterward, they strolled up Broadway and entered Union Square Park, where they seated themselves on a bench beneath a canopy of trees bright with autumn colors, watched the children at play, and discussed Mrs. Lincoln's plan to sell her wardrobe. Mrs. Lincoln soon revealed that the previous day, while Elizabeth was en route from Washington, she had called on a diamond broker after seeing an ad in the _Herald_ at the breakfast table. \"I tried to sell them a lot of jewelry,\" Mrs. Lincoln said. \"I gave my name as Mrs. Clarke. The first gentleman I spoke with was pleasant, but we were unable to agree on a price. He stepped back into the office to confer with another gentleman, and just as I concluded they were plotting to hurry me out the door, a third gentleman entered the store. He looked over my jewelry\u2014he was, as I later learned, Mr. Keyes, a silent partner in the firm\u2014and discovered my name engraved inside of one of my rings.\"\n\n\"Oh, dear,\" said Elizabeth.\n\n\"I had forgotten about the engraving,\" Mrs. Lincoln confessed. \"When I saw him looking at the ring so earnestly, I snatched it from him and put it in my pocket.\"\n\nElizabeth smothered a laugh. \"I'm sure that didn't provoke his curiosity at all.\"\n\n\"I did not want to wait around to find out. I hastily gathered up my jewelry and started to leave, but they had become much more interested in my wares, as you can imagine. I left my card, Mrs. Clarke at the St. Denis Hotel. They are to call to see me this forenoon, when I shall enter into negotiations with them.\"\n\n\"Is that wise?\" Elizabeth asked carefully. \"Surely they've figured out that you are Mrs. President Lincoln.\"\n\n\"Or that I am her thieving maid.\"\n\n\"Not likely, I think.\" Earnestly, Elizabeth said, \"With such delicate business, should you trust a firm that you have never heard of before, one that you chose only because you spied their ad in the paper? Wouldn't it be more prudent to seek a recommendation from an acquaintance, or work through a jeweler you've done business with before?\"\n\n\"More prudent but impossible,\" Mrs. Lincoln replied. \"I could not ask friends for introductions or go to my favorite jewelers without exposing myself. Even if Mr. Keyes knows who I am, the rest of the city must not suspect. I could not bear the humiliation.\"\n\nElizabeth understood, and after enjoying the sunshine and cool breezes a little while longer, they returned to the hotel. Shortly after their arrival, Mr. Keyes called, and after they withdrew to a parlor to chat, Mrs. Lincoln confirmed her true identity. He was, as Elizabeth suspected he would be, elated, and he willingly admired the shawls, dresses, and fine laces Mrs. Lincoln displayed for him. When Mrs. Lincoln explained why she was compelled to sell her wardrobe, Mr. Keyes was much affected by her story and severely denounced the government for its ingratitude. He was disgusted by the tale of their ill treatment at the St. Denis too, and he urged Mrs. Lincoln to move to another hotel without delay.\n\nMrs. Lincoln agreed, and as they rode to the Union Place Hotel, Elizabeth said, \"Perhaps when we arrive, you can confide in the proprietor, and give him your true name without registering, to ensure the proper respect.\" Hesitantly, Mrs. Lincoln agreed, but by the time they reached the hotel, she had changed her mind again, and registered as Mrs. Clarke. Even so, they were treated with far more courtesy\u2014perhaps because Mrs. Lincoln's real name was discernible on some of her trunks, if one looked carefully, and the staff knew who she was but pretended otherwise.\n\nAfter they had settled into their new accommodations, Mr. Keyes and Mr. Brady called on Mrs. Lincoln often over the next few days, for they had much to discuss. Mrs. Lincoln was anxious to dispose of her garments and return to Chicago as quickly and discreetly as possible, but to Elizabeth's consternation, the gentlemen would hear nothing of this. \"Put your affairs in our hands,\" Mr. Brady declared, \"and we will raise you at least one hundred thousand dollars in a few weeks. The people will not permit the widow of Abraham Lincoln to suffer. They will come to her rescue when they know she is in want.\"\n\nThis was precisely what Mrs. Lincoln wanted to hear, in both financial and emotional terms, and so she agreed to work with W. H. Brady & Co. They advanced her six hundred dollars for her expenses while she remained in New York, and they assured her that their plan, once they devised it, would succeed.\n\nThe Union Place Hotel was comfortable, and the brokers' confidence was heartening, but even so, Mrs. Lincoln and Elizabeth remained wary of discovery. On Sunday they took a carriage ride through Central Park, but they took no pleasure in it, because despite the heavy veil Mrs. Lincoln wore to conceal her identity, they could not throw open the window for fear of being recognized. They also narrowly escaped being run into by another carriage, and they were terribly alarmed not only for the harm that might have come to them, but because an accident would have drawn attention and their masquerade would have been exposed.\n\nElizabeth disliked the subterfuge more every day, but when Mr. Brady and Mr. Keyes came to Mrs. Lincoln with their plan for disposing of her goods, she began to dread that the scheme was a disaster in the making. Mr. Brady believed that the prominent Republican men who owed their fortunes to Mr. Lincoln would be willing to advance her money rather than let it be made known that his widow was so impoverished that she was compelled to sell her wardrobe. To that end, he urged Mrs. Lincoln to write letters describing her unhappy circumstances, addressed to him but dated as if she had written them weeks earlier while still in Chicago. The brokers wanted her to imply that she had saved letters compromising various politicians and businessmen who had profited from wartime contracts. Mr. Brady and Mr. Keyes would show the letters to the gentlemen in question in hopes of embarrassing them into providing for her. If that failed\u2014if her plight, their loyalty to the Republican Party, and their own sense of self-preservation did not move them\u2014Mr. Brady would threaten to publish the letters.\n\nTo Elizabeth the plan smacked of blackmail, and the moment Mr. Brady and Mr. Keyes left, she urged Mrs. Lincoln not to go along with it. \"I fail to see how any good could come of dishonesty and threats,\" she said. \"You said you wanted to dispose of your wardrobe quickly and quietly. I fear these letters will only draw the matter out and bring you unwanted attention.\"\n\n\"I have been ignored long enough.\" Mrs. Lincoln sat down at her table and took out pen, ink, and paper. \"Mr. Brady and Mr. Keyes believe that I must do something to gain the attention of the gentlemen best placed to assist me.\"\n\n\"You will gain their attention, certainly, and also the attention of the wider world.\" Elizabeth despaired of convincing her, and yet she could not be silent. \"The worst of what the papers have written about you in the past will be nothing compared to the disparagement and scorn they will heap upon you over this.\"\n\n\"'Dear Sir,'\" Mrs. Lincoln said aloud as she began her first letter, making a show of ignoring her that Elizabeth thought unkind. \"'A notice that you sold articles of value on commission, prompts me to write to you.' See, Elizabeth, there is an honest beginning. You cannot fault me for following Mr. Brady's advice if I say nothing that is not true.\"\n\n\"The date, as well as your location.\" Elizabeth gestured to the page, barely keeping her frustration in check. \"Chicago, September 1867. That is already untrue.\"\n\n\"Those are small details, and of no consequence.\"\n\nElizabeth muffled a sigh and resisted the urge to fling her hands in the air in exasperation. She peered over Mrs. Lincoln's shoulder as she wrote, ostensibly to Mr. Brady, of how urgent necessity compelled her to part with valuable gifts from dear friends. Mrs. Lincoln finished that letter and began another, dated Chicago, September 14. \"My dear sir,\" she addressed Mr. Brady. \"Please call and see Hon. Abram Wakeman. He was largely indebted to me for obtaining the lucrative office which he has held for several years, and from which he has amassed a very large fortune. He will assist me in my painful and humiliating situation, scarcely removed from want. He would not hesitate to return, in a small manner, the many favors my husband and myself always showered upon him. Mr. Wakeman many times excited my sympathies in his urgent appeals for office, as well for himself as others. Therefore he will be only too happy to relieve me by purchasing one or more of the articles you will please place before him.\"\n\nElizabeth's dismay deepened with every dip of Mrs. Lincoln's pen into the inkwell. Mr. Wakeman was a former New York congressman and postmaster of New York City, whom President Lincoln had appointed surveyor of the Port of New York, second only to the Custom House as the most coveted patronage post the administration could bestow. He was also a sharp, shrewd politician, and Elizabeth could not imagine his fearing embarrassment so much that he would allow himself to be bullied by a diamond broker with questionable ethics.\n\n\"Please, Mrs. Lincoln, take care,\" Elizabeth urged her. \"If you insist upon writing these letters, at least use the mildest language possible.\"\n\n\"Never mind, Elizabeth,\" said Mrs. Lincoln briskly, signing her name. \"Anything to raise the wind. One might as well be killed for a sheep as a lamb.\"\n\nSo Elizabeth sighed and said no more, not even when Mrs. Lincoln began a new letter with \"You write me that reporters are after you concerning my goods deposited with you...and also that there is a fear that these newsmen will seize upon the painful circumstances of your having these articles placed in your hands to injure the Republican Party politically.\" Not for the world would Mrs. Lincoln do anything to injure their cause, she wrote, \"notwithstanding the very men for whom my noble husband did so much unhesitatingly, deprived me of all means of support and left me in a pitiless condition.\" The phrases were designed to put the fear of exposure in the press into the gentlemen's hearts, and considerable guilt regarding Mrs. Lincoln's circumstances besides, but Elizabeth could not imagine powerful men meekly throwing money Mrs. Lincoln's way to make the potential scandal disappear. They were far more likely to throw Mr. Brady from their offices in utter contempt.\n\nBut Mrs. Lincoln would not hear her.\n\nThe letters were written and delivered to Mr. Brady and Mr. Keyes, who immediately commenced showing them to the gentlemen Mrs. Lincoln had named. Soon thereafter, it occurred to her that the brokers' scheme was designed to bring her income without parting with any of her clothes, so she decided to dispose of her wardrobe another way. She instructed Elizabeth to make appointments for several dealers in secondhand clothing to call on \"Mrs. Clarke\" at the Union Place Hotel, but although they duly came, and seemed interested in her goods, they could not agree on a price. Thwarted, but not daunted, a few days later Mrs. Lincoln and Elizabeth packed a bundle of dresses and shawls into a carriage and drove along Seventh Avenue, stopping at first one store and then another, seeking buyers. They soon discovered that the dealers wanted the goods for little or nothing, and although Mrs. Lincoln met them squarely, her tact and shrewdness accomplished nothing. Discouraged, they returned to the hotel disgusted with the whole affair, but they could not abandon it.\n\nIn the meantime, their strange behavior had drawn the attention of the hotel staff and other guests, who cast curious, suspicious looks upon them wherever they went. The large trunks in which Mrs. Lincoln's wardrobe was stored had been kept in the main hall rather than carried upstairs, and they had become the objects of scrutiny and speculation. First one reporter, then another, noticed that the faint outlines of Mrs. Lincoln's name appeared on the lid of one trunk, even though the letters had been rubbed out. Mrs. Lincoln and Elizabeth became ever more wary and uncomfortable with the speculative glances and whispers, until Mrs. Lincoln finally decided that they must escape. They packed their smaller trunks, had the larger ones delivered to the offices of W. H. Brady & Co. at 609 Broadway, paid their hotel bills, and quickly departed for the country, where they remained for three days to throw the reporters off the scent. When they returned to the city, Elizabeth suggested that Mrs. Lincoln go to the Metropolitan Hotel, where she had stayed on previous occasions, and confide in the proprietor, who had always been courteous and respectful to her. Mrs. Lincoln refused. Instead they took rooms at the Brandreth House, where Mrs. Lincoln registered as \"Mrs. Morris.\"\n\nElizabeth had grown weary of the aliases and the clandestine skulking, but she endured it, wanting to believe that eventually it would benefit Mrs. Lincoln. As the weeks passed, she lost hope that any good would come of their efforts, but she felt no vindication when Mr. Keyes and Mr. Brady were finally forced to admit that their scheme had failed. They had shown the letters to numerous prominent Republicans, but not one had responded favorably. With the exception of a few dresses sold at low prices to secondhand dealers, Mrs. Lincoln's wardrobe was still packed away in her trunks. The six hundred dollars the brokers had advanced was nearly spent, and Mrs. Lincoln had nothing to show for her time in New York\u2014in fact, with gossip circulating about the mysterious, heavily veiled widow peddling her wardrobe, it would be fair to say she was worse off than when she had left Chicago.\n\nIn the first week of October, her funds and patience spent, Mrs. Lincoln reluctantly agreed to allow Mr. Brady to exhibit her wardrobe in their showrooms for sale, and to widely publicize that it was hers. \"People who were not interested in Mrs. Clarke's clothing will be very eager to acquire Mrs. Lincoln's,\" Mr. Brady declared, but Elizabeth was not reassured. Mrs. Lincoln also agreed to allow the brokers to publish her letters in the New York _World,_ a Democratic newspaper more than willing to expose Republicans to ridicule. After resigning herself to the new plan, discouraged, anxious, and missing young Tad dreadfully, Mrs. Lincoln left New York the same morning the letters appeared in the paper, leaving Elizabeth behind in charge of her affairs.\n\nElizabeth wanted very badly to return home too, and to reopen her business. Mrs. Lincoln had promised to pay her a commission from the sale of her wardrobe, but no sales meant no pay. She could not afford a hotel, so she found lodgings in a private home, economized, and hoped that putting Mrs. Lincoln's wardrobe on display would stir up interest that would lead to profitable sales.\n\nIn the end, she received only part of what she hoped for.\n\nThe first sign of impending disaster came not from 609 Broadway or the press, but from a letter Mrs. Lincoln sent to Elizabeth within hours of returning to Chicago. On the train west, she found herself in the peculiar situation of sitting behind two gentlemen who had just read her letters in the _World_ and were discussing her pecuniary embarrassment. Later, she went to the dining room and was shown to a table where sat none other than her friend from Washington Senator Charles Sumner, who recognized her immediately despite the black veil doubled over her face. His pity was so tangible that she knew he too was thinking of the letters and the public display of her wardrobe, and she was so discomfited that she invented an ill friend who needed her and fled from the table. She was embarrassed to tears, later, when the compassionate senator brought a cup of tea to her car. In a second letter, written only hours after the first, Mrs. Lincoln lamented, \"I am writing this morning with a broken heart after a sleepless night of great mental suffering. R came up last night like a maniac, and almost threatening his life, and looking like death because the letters of the _World_ were published in yesterday's paper. I could not refrain from weeping when I saw him so miserable.\"\n\nElizabeth needed a moment to parse out her meaning, and when she deduced that Mrs. Lincoln meant the falsified letters had been printed in the Chicago papers, her heart plummeted so quickly it left her lightheaded. She should have expected the scandal to be too big and too interesting to remain confined to New York.\n\n\"I weep whilst I am writing,\" Mrs. Lincoln wrote on. \"I pray for death this morning. Only darling Taddie prevents my taking my life.\" She was nearly losing her reason, she concluded, and she instructed Elizabeth to tell Mr. Brady and Mr. Keyes not to put a single line more of hers into print.\n\nIn the weeks that followed, more and more letters came from Chicago, and although Mrs. Lincoln begged Elizabeth to write to her every day, it was all she could do to take care of the business already entrusted to her. The curious and the contemptuous visited the offices of W. H. Brady & Co. to examine dresses piled up on a long table, shawls hanging over the backs of chairs, furs and laces and jewels in a glass case. The publication of the letters had indeed stirred up interest, but while many people browsed, no one bought. As one reporter sniggered in the _Evening Express,_ some of the dresses \"if not worn long, have been worn much; they are jagged under the arms and at the bottom of the skirt, stains are on the lining, and other objections present themselves.\" The extravagantly high prices on the labels dangling from the articles, the reporter said, had no doubt been conceived by the dressmakers, overproud of their labors. \"The peculiarity of the dresses,\" he added, \"is that the most of them are cut low-necked\u2014a taste which some ladies attribute to Mrs. Lincoln's appreciation of her own bust.\"\n\nThat report, and other similar pieces that followed in other papers, only worsened from there. Reporters and gossips alike began referring to the situation as \"Mrs. Lincoln's Old Clothes Scandal.\" Democrats sensed an opportunity to injure their opponents by pouncing upon the _World_ letters as evidence that Republicans had bought favors from Mr. Lincoln's White House. In response, Republican newspapers denounced Mrs. Lincoln's rash, improper, and unwomanly actions and fell over one another in their haste to repudiate her. \"It appears as if the fiends have let loose, for the Republican papers are tearing me to pieces in this border ruffian West,\" Mrs. Lincoln wrote Elizabeth on October 9. \"If I had committed murder in every city in this _blessed_ _Union,_ I could not be more traduced. And you know how innocent I have been of the intention of doing wrong.\"\n\nBut Elizabeth could not consider her entirely innocent, for she had, after all, agreed to compose the false letters. Always Elizabeth wanted to believe that Mrs. Lincoln's intentions were good, but she could not quite convince herself this time.\n\nWhile Mrs. Lincoln pored worriedly over the papers and tried to make amends to Robert and groused about her maltreatment in letters to Elizabeth and other sympathetic friends, Elizabeth remained in New York, assisting the brokers as she could, trying to prevent the damage to Mrs. Lincoln's reputation from worsening. Eventually, as the scandal dragged on, so many false reports were circulated that Elizabeth, at a suggestion from Mrs. Lincoln, granted interviews to the sympathetic _Herald_ and the _Evening News_ to set the record straight. None of it seemed to make any difference. The vitriol increased, the wardrobe sat pawed over and collecting dust at 609 Broadway, and Elizabeth worked herself close to exhaustion. She began to take in sewing to make ends meet, but such intermittent piecework brought in a meager living compared to what she had earned running her own dressmaking business. She moved to a boardinghouse on Broome Street, whose proprietress, Mrs. Bell, was a cousin of William Slade, Mr. Lincoln's former messenger. But those measures were not enough.\n\nElizabeth remained in New York throughout the fall and winter, struggling to serve her former patron's best interests while neglecting her own. At her lowest moments, she wondered why she persisted when she had lost almost all hope of success. By November she was wretchedly low-spirited and wanted desperately to return to Washington, but Mrs. Lincoln urged her to remain in New York a little while longer. Perhaps later, she suggested, after the whole sorry business was resolved, Elizabeth could join Mrs. Lincoln in Chicago. \"Had you not better go with me and share my fortunes, for a year or more?\" she implored, apparently oblivious to the pleasant, productive, satisfying life Elizabeth had left behind in the capital, and how she longed to return to it.\n\nNevertheless, duty compelled her to persist in the cause of Mrs. Lincoln's relief. Drawing upon the connections she had made on behalf of the Contraband Relief Association, Elizabeth wrote to leaders of the colored community proposing that collections for Mrs. Lincoln should be taken up in colored churches. The idea met with strong approval, because freeborn and freed slave alike recognized Abraham Lincoln as their great friend, and they were anxious to show their kind interest in the welfare of his family in some way more earnest and substantial than simple words. \"You judge me rightly,\" Mr. Frederick Douglass wrote in his reply. \"I am willing to do what I can to place the widow of our martyr President in the affluent position which her relation to that good man and to the country entitles her to.\" To that end, he proposed arranging a series of lectures by the best speakers in the country\u2014and he himself would be honored to participate. The fees this venture would raise, combined with the generous donations from the colored community, would at last relieve Mrs. Lincoln's financial distress and allow her a great measure of comfort.\n\nGratified, Elizabeth wrote to Mrs. Lincoln about the plan, elated to have good news to share at last. But to her consternation, Mrs. Lincoln resisted. As desperate as she was for relief, she did not want to accept help from Negros. \"I want neither Mr. Douglass nor Garnet to lecture on my behalf,\" she decreed abruptly on the second day of November. Bewildered, and not a little disappointed, Elizabeth had no choice but to inform Mr. Douglass of Mrs. Lincoln's wishes, and the project was immediately abandoned. Less than two weeks later, Mrs. Lincoln changed her mind, but her initial rebuff had surprised and disappointed colored leaders, and their enthusiasm for helping her had cooled.\n\n\"Write to me, dear friend, your candid opinion about everything,\" Mrs. Lincoln begged as the scandal swirled about her. \"I pray God there will be some success,\" she wrote on another occasion, \"although, dear Elizabeth, entirely between ourselves, I fear I am in villainous hands.\" \"Alas! Alas!\" she lamented the very next day. \"What a mistake it has all been!\" Mrs. Lincoln asked Mr. Brady and Mr. Keyes to return her wardrobe, and ordered Elizabeth to collect it, only to have the brokers defer her requests again and again. With Mrs. Lincoln losing heart, Elizabeth found it next to impossible to keep her own spirits aloft.\n\nIn early January, in one last, desperate attempt to raise money from the \"Old Clothes,\" Mr. Brady and Mr. Keyes took a portion of the wardrobe to Providence, intending to launch a traveling expedition at Remington's Hall, charging the \"moderate consideration\" of twenty-five cents each for admission. \"The exhibition will bring in money,\" Mr. Keyes told Elizabeth, \"and as money must be raised, this is the last resort.\" He assumed Mrs. Lincoln would approve, and Elizabeth assumed that he had consulted her beforehand, but both were mistaken. \"Why did _you_ not urge them _not_ to take my goods to Providence?\" Mrs. Lincoln demanded by return mail when informed of the plan, already under way. \"For heaven's sake see K & B when you receive this, and have my wardrobe immediately returned to me, _with their bill_.\" Although advertisements ran in _The_ _Providence Journal_ announcing the exhibit, it never opened, for the Providence board of aldermen refused to grant the brokers a license. Nevertheless, Mrs. Lincoln was not mollified.\n\nElizabeth soon inadvertently upset Mrs. Lincoln again. The needs of her own race, especially their education, were always in her thoughts, and it pained her that Wilberforce University, the school her son had attended, had burned to the ground on the day of President Lincoln's assassination. Although she was preoccupied with discharging Mrs. Lincoln's business, Elizabeth had agreed to lead a fund-raising drive to rebuild the college. She arranged with Reverend Daniel Payne, one of the college's founders and a witness to her pension statement, to loan some of her Lincoln relics\u2014Mr. Lincoln's hat, cloak, gloves, comb, and brush, as well as Mrs. Lincoln's bonnet and blood-spattered cloak\u2014for an exhibition that would tour Europe. At the news of yet another exhibit linked to her name, Mrs. Lincoln became frantic. \"Your letter announcing that my clothes were to be paraded in Europe\u2014those I gave you\u2014has almost turned me wild,\" she immediately responded. \"Robert would go _raving distracted_ if such a thing was done. If you have the _least_ _regard_ for _our reason,_ pray write to the bishop that it _must_ not be done. How little did I suppose you would do _such a thing;_ you cannot imagine how much my overwhelming sorrows would be increased. May kind heaven turn your heart...For the sake of _humanity,_ if not _me_ and my children, _do not_ have those black clothes displayed in Europe. The thought has almost whitened every hair of my head.\"\n\nThe withering rebuke was almost more than Elizabeth could bear, and she immediately did as Mrs. Lincoln demanded.\n\nWith dwindling hopes of ever receiving a commission from the sale of Mrs. Lincoln's wardrobe, and with increasing worries about the damage to her reputation for her very visible role in the \"Old Clothes Scandal,\" Elizabeth realized that she must do something for herself, and rely no longer upon ephemeral expectations.\n\nShe had accumulated several memory sketches since the summer day she had learned that the woman she had escorted around Washington City's contraband camps was Miss Harriet Jacobs, the runaway slave from North Carolina who had bravely published her autobiography in 1861, using a pseudonym to protect herself from recapture and return to her horrifically abusive owner. Many other former slaves had also written their own life stories, including Mr. Frederick Douglass, whom she greatly admired. Though illiterate, Sojourner Truth had dictated her memoirs to a friend, and they had been published to great interest and acclaim. Why could Elizabeth not do the same? Mrs. Lincoln and Mr. Douglass, as well as the Garlands, had complimented her writing. Why should she not tell her own story?\n\nAnd so she took her memory sketches and began to fashion them into a memoir. She envisioned her book as tracing her life from her birth into slavery, through the sorrowful years of her youth, and on to her triumphant emergence into freedom. She would describe her years as Mrs. Lincoln's personal modiste and White House intimate, and share her observations of the great, martyred president. She hoped to place Mrs. Lincoln in a better light before the world by revealing the innocent motives that had spurred her often misunderstood actions. Lastly, she would offer a detailed account of the so-called \"Old Clothes Scandal,\" to redeem her own good name as well as Mrs. Lincoln's.\n\n\"Do all you can, dear Mrs. Keckley,\" Mr. Douglass had exhorted her as they planned the thwarted lecture series for Mrs. Lincoln's benefit. \"Nobody can do more than you in removing the mountains of prejudice toward that good lady.\"\n\nElizabeth believed he was right. No one knew Mrs. Lincoln the way she did, and no one could explain and justify her actions better than she, a friend and confidante who understood her good intentions.\n\nElizabeth could do this much for Mrs. Lincoln, and since no one else would, she had to. She owed that much to the grieving widow\u2014and even more so, to the memory of the noble, martyred president who had done so much for her race. For both of them, she resolved to try.\n\n# Chapter Sixteen\n\n## MARCH\u2013JUNE 1868\n\nAt first, Elizabeth was too embarrassed to confide in any of her New York acquaintances that she was writing her memoirs, even though many friends and patrons throughout the years had told her she ought to, since she had lived such an extraordinary life. But as the pages accumulated, she mentioned her manuscript to one friend, and then another, and eventually the tale found its way to Mr. James Redpath, a friend of Mr. Frederick Douglass and an editor with the publishing house G. W. Carleton & Company. Elizabeth had already met Mr. Redpath, but only in passing, and it was not quite accurate to say they knew each other. When she was living in St. Louis, some of her friends had spoken with the staunchly abolitionist Mr. Redpath when he came to the city to interview slaves for a book he was writing. He had also visited the White House often to advise President Lincoln on matters concerning Haiti, but although she had seen him there, they had never spoken. Having heard of her fledging memoir, however, he was eager to make her acquaintance, and mutual friends introduced them.\n\nThe red-haired, fiery Scotsman called on her at her Broome Street boardinghouse, where she sat anxiously in the chair opposite his while he read the opening chapter of her memoir, the story of her birth and early childhood. She held her breath, waiting for his verdict, perching on the edge of her seat until he set her handwritten pages on his lap and said, \"This is a promising start.\"\n\nShe exhaled deeply. \"Thank you, sir.\"\n\n\"The story of Little Joe sold off by the pound and his poor mother running after the wagon as it sped away from the plantation...\" Mr. Redpath shook his head, grim. \"Tragic. Absolutely heartbreaking. I don't suppose you know if they ever reunited?\"\n\n\"I wish I knew,\" said Elizabeth. \"Perhaps, after the war, they were able to find each other, but Little Joe was no more than four years old when he was sold to the Petersburg slaver. I imagine it would have been very difficult for him to retrace the path of dim childhood memories.\"\n\nMr. Redpath grunted and nodded. \"Tragic indeed.\" He held up the pages of her manuscript. \"Where have you taken the story after this, and where do you intend to go?\"\n\nShe explained her plan for the book, and his eyes lit up with eagerness when she mentioned that she hoped to devote a significant portion to her White House years and \"Mrs. Lincoln's Old Clothes Scandal.\" She intended to share the profits of her book with Mrs. Lincoln, and to reveal the truth, the good and the bad, so that the world would better understand the former First Lady, who had been so unfairly maligned.\n\n\"I admire your loyalty,\" Mr. Redpath remarked. \"Rest assured, Carleton and Company wants the truth presented to the world as much as you do\u2014the good and the bad.\"\n\nWithin days Mr. Redpath brokered an agreement between Elizabeth and his publisher, and told her that they would like to publish her memoir as soon as possible, by spring at the latest. \"But I have only just finished writing about how I purchased my freedom,\" Elizabeth replied, her joy and alarm creating a dizzying muddle of her thoughts.\n\nHe assured her that he would help, and so they agreed to meet several times a week to collaborate on the rest of the manuscript\u2014daily, if necessary, as the deadline approached.\n\nFrom then on, by day Elizabeth attended to Mrs. Lincoln's business and took in sewing to earn her keep, and by night she wrote in the quiet seclusion of her garret room. In the evenings, Mr. Redpath would call on her at her boardinghouse, and they would meet in the public parlor to review her previous night's work. Sometimes, necessity and haste required her to merely jot down her recollections rather than writing lengthy paragraphs of elegant prose. On such occasions she would read aloud from her notes to Mr. Redpath, elaborating where needed, while her editor took notes of his own, which he took back to his office, combined with hers, and revised.\n\nElizabeth passed her fiftieth birthday working with Mr. Redpath on the twelfth chapter of her memoir, an account of Mrs. Lincoln's departure from the White House after her husband's assassination. The next day, she received a letter from Mrs. Lincoln, who was despondent because she had forgotten her pocketbook, which held her entire allowance for the month, on the streetcar. \"The loss I deserve for being so careless,\" she lamented, \"but it comes very hard on poor me. Troubles and misfortunes are fast overwhelming me; may _the end_ soon come.\"\n\nMrs. Lincoln had so often said that only Tad kept her from taking her own life that Elizabeth did not know whether by \"the end\" she meant the end of her troubles or the end of everything. After so much repetition, laments of that sort should no longer have bothered Elizabeth, but they unsettled her anew every time.\n\nEvery other scheme having failed, Mr. Brady and Mr. Keyes decided to put what remained of Mrs. Lincoln's wardrobe\u2014which was nearly all of it\u2014up for public auction. Unwilling to dispose of her goods in such a manner, Mrs. Lincoln adamantly insisted that the brokers return everything to Elizabeth and settle her account. They took their time about it, but eventually they allowed Elizabeth to collect the unsold goods\u2014at which time they presented her with a bill for $820, the charge for their services. They also kept several hundred dollars more that they had received for selling Mrs. Lincoln's diamond ring and a few other items\u2014for their expenses, they explained. Elizabeth promptly packed up Mrs. Lincoln's wardrobe and shipped it back to her in Chicago, and when Mrs. Lincoln sent her a check for the bill, Elizabeth delivered it to Mr. Keyes personally. When she left his office on March 4, receipt in hand, she felt a wave of relief, resignation, and disgust wash over her. The whole ugly, ill-conceived business was over at last. Not only had it failed to raise any money for Mrs. Lincoln's support, it had also cost her hundreds of dollars in brokers' fees and incalculable damage to her reputation. But finally it was done.\n\nAlthough Elizabeth missed her home and friends in Washington City and urgently needed to reopen her dressmaking business, she decided to remain in New York until her manuscript could be completed. Since she no longer needed to reside within a convenient distance of W. H. Brady & Co., she moved to a less costly, fourth-floor garret room at No. 14 Carroll Place, the home of Mrs. Amelia Lancaster, a successful hairdresser to New York City's elite.\n\nAs she drew closer to the final chapters of her book, a certain sense of unease compelled Elizabeth to write to Mrs. Lincoln and tell her about it. After Mr. Herndon's shocking lectures, Elizabeth expected Mrs. Lincoln to have misgivings, so she emphasized in the warmest terms that everything she wrote was designed to place Mrs. Lincoln before the world in the best possible light. Mrs. Lincoln acknowledged Elizabeth's confession with a single line in her next letter: \"It would appear that every former inmate of my beloved husband's White House _must_ inflict a memoir upon the public\u2014I can only trust that my _dearest friend_ would not write anything _unkind,_ or betray any confidences.\" Elizabeth had hoped for her to more explicitly bestow her blessing upon the project, but she accepted Mrs. Lincoln's remarks as tacit permission to proceed. She could not have abandoned the book then in any case; she had invested too much of her time and industry, as well as Mr. Redpath's.\n\nA few days later, Elizabeth finished writing her account of the disastrous attempt to sell Mrs. Lincoln's wardrobe, the final chapter of her book. After Mr. Redpath read it, he praised her for her unflinching frankness and honesty in describing what surely must have been as embarrassing an episode for herself as it was for Mrs. Lincoln.\n\n\"Unflinching?\" echoed Elizabeth. \"On the contrary, I did flinch, quite often, as I wrote it. I confess I'm very concerned about divulging Mrs. Lincoln's private conversations and personal thoughts to the world.\"\n\nMr. Redpath's brow furrowed in sympathy. \"I understand completely. I promise you, nothing published in the book will hurt Mrs. Lincoln.\"\n\nElizabeth was relieved to hear it. \"That pleases me very much.\"\n\n\"You're nearly finished,\" Mr. Redpath remarked, gathering the day's pages into a neat stack. \"You must be looking forward to completing the manuscript and returning to Washington.\"\n\n\"I am,\" said Elizabeth tentatively. She had thought the manuscript _was_ finished. \"What else would you like me to add?\"\n\n\"Well, there's the preface, which should describe your purpose and qualifications for writing a book of this nature.\"\n\n\"Oh, of course. I'll begin it right away.\"\n\n\"I had another thought as well. The letters you've included from Mr. Douglass and the Garland ladies are most illuminating. Do you have any letters from Mrs. Lincoln? I'm sure your readers would find them fascinating.\"\n\nElizabeth felt a thrill of excitement and fear when he spoke of her readers, as if they already existed in eager multitudes. \"I have many, but I simply could not include them in the book, fascinating or not. Mrs. Lincoln would not approve.\"\n\nMr. Redpath shrugged, thoughtful. \"She allowed several letters to be printed in the _World_.\"\n\n\"Yes, and she deeply regretted it afterward.\"\n\n\"Ah.\" Mr. Redpath nodded, thinking, and then ventured, \"I don't suppose you would let me read them?\"\n\n\"Are you merely curious?\"\n\n\"I am _very_ curious, but not _merely_ curious. Reading Mrs. Lincoln's letters would give me a sense of her manner and intonation, which will be of great help to me as I edit your manuscript. The facts her letters include, too, will help me verify dates and details you have provided. And there is the matter of your credibility. Some foolish people will argue that a former slave never could have written such a fine book, nor gained admission to the most intimate circles of the White House in the first place. I cannot emphasize enough the importance of personal correspondence in establishing the authenticity of a biography.\"\n\n\"I understand,\" said Elizabeth. She had heard of other former slaves' memoirs that had been unfairly dismissed as works of fiction, and she knew many people who, after knowing her but a little while, insisted that she was far too dignified and talented to have ever been a slave. She could not bear to have anyone question her integrity, but such accusations were probably inevitable. If she could forestall them or at least limit their number by sharing Mrs. Lincoln's letters with her editor, she would be wise to do so.\n\nShe went to her garret room and returned with a bundle of Mrs. Lincoln's letters, neatly bound with a black ribbon left over from trimming one of her many bonnets. She gave them to Mr. Redpath with the understanding that they were precious mementoes and must be returned to her undamaged as soon as he was finished with them, and that they were only meant to assist him in editing her manuscript, not to become a part of it.\n\nMr. Redpath assured her that he would publish nothing from Mrs. Lincoln's letters that would embarrass her, and with that, he left Elizabeth to write her preface.\n\nAlone in her room, she held her pen hovering above the page, unable to begin. A twinge of doubt struck\u2014who was she, to think she could write a book? Immediately she chased the thought away, impatient with her timidity. Who was she, indeed. She _had_ written a book, so the question of whether she _could_ had already been answered. Now all that remained was to explain why she had, and to what purpose.\n\n\"I have often been asked to write my life, as those who know me know that it has been an eventful one,\" Elizabeth wrote. \"At last I have acceded to the importunities of my friends, and have hastily sketched some of the striking incidents that go to make up my history. My life, so full of romance, may sound like a dream to the matter-of-fact reader, nevertheless everything I have written is strictly true; much has been omitted, but nothing has been exaggerated.\"\n\nElizabeth paused. A sudden vision filled her mind's eye\u2014Mrs. Lincoln picking up her book, examining the cover, and turning the pages, her frown deepening with every paragraph. Elizabeth drew in a deep breath, exhaled slowly, and refreshed her pen in the ink. \"In writing as I have done, I am well aware that I have invited criticism.\" Elizabeth figured she could probably compose an accurate list of those who would be first and loudest to complain. \"But before the critic judges harshly, let my explanation be carefully read and weighed.\"\n\nThe critic might refuse, but it wouldn't hurt to ask.\n\nElizabeth next ruminated briefly on slavery, a topic she would return to in a more personal, revelatory way early in her memoir. Then her thoughts again turned to Mrs. Lincoln. \"It may be charged that I have written too freely on some questions, especially in regard to Mrs. Lincoln,\" she acknowledged. \"I do not think so; at least, I have been prompted by the purest motive. Mrs. Lincoln, by her own acts, forced herself into notoriety. She stepped beyond the formal lines which hedge about a private life, and invited public criticism.\"\n\nBut was that fair? The nagging question brought Elizabeth's pen to a halt. Mrs. Lincoln had not run for office and had not been elected First Lady; the role had been bestowed upon her by virtue of her husband's choices and actions. It was, admittedly, a role she had relished, a title she had desired since childhood. She had reveled in the attention, so long as it was favorable, and she had certainly enjoyed the benefits and privileges of being, as she had sometimes called herself, Mrs. President. So yes, Elizabeth decided, it was fair to say that Mrs. Lincoln had chosen a public life.\n\nEven so, the people had judged her too harshly. \"The people knew nothing of the secret history of her transactions, therefore they judged her by what was thrown to the surface.\" Indignantly, Elizabeth rebuked a few prominent personages who had been especially unkind and certain newspapers who had taken excessive glee in exposing Mrs. Lincoln's faults, but then she thought better of it and scratched out the lines. \"Mrs. Lincoln may have been imprudent, but since her intentions were good, she should be judged more kindly than she has been.\"\n\nIf she could persuade her hypothetical readers of nothing else, Elizabeth hoped she would convince them of this. Suddenly she wondered if Mr. Herndon had entertained similar thoughts as he composed his lectures.\n\nHer spirits dipped, but she firmly reminded herself her book and Mr. Herndon's scribblings had nothing in common except for purporting to be about Mr. and Mrs. Lincoln. Their motives\u2014and their degree of truthfulness\u2014could not be more dissimilar.\n\nAnd yet worry nagged at her. Would Mrs. Lincoln agree?\n\n\"If I have betrayed confidence in anything I have published, it has been to place Mrs. Lincoln in a better light before the world,\" Elizabeth wrote, her hand firm and steady around the pen. \"A breach of trust\u2014if breach it can be called\u2014of this kind is always excusable.\" She took a deep breath and steeled herself; the next admission would not be easy. \"My own character, as well as the character of Mrs. Lincoln, is at stake, since I have been intimately associated with that lady in the most eventful periods of her life. I have been her confidante, and if evil charges are laid at her door, they also must be laid at mine, since I have been a party to all her movements.\"\n\nTo defend herself, she had to defend the lady she served, and as Elizabeth wrote the words, she realized this truth had compelled her throughout the entire writing of her memoir. She had long been admired for her integrity and dignity, but the unfortunate business at 609 Broadway had tainted her sterling reputation with scandal. She had to redeem herself, and she could not do that without redeeming Mrs. Lincoln.\n\nElizabeth finished the preface, hoping she had written everything Mr. Redpath wanted her to write, that she had said everything she needed to say. Readers might disregard her lengthy explanation and decide for themselves that her motives were not pure, but she knew the truth. She also knew that nothing within the pages of her book could cause Mrs. Lincoln to be regarded in a worse light than that in which she presently stood, so the secrets Elizabeth revealed could do Mrs. Lincoln no harm.\n\n\"I am not the special champion of the widow of our lamented President,\" Elizabeth emphasized. Mistrustful readers would say that she could not be honest about the former First Lady's faults, since they were friends. She must make them believe otherwise. \"I wish the world to judge her as she is, free from the exaggerations of praise or scandal. The reader of the pages which follow will discover that I have written with the utmost frankness in regard to her\u2014have exposed her faults as well as given her credit for honest motives.\"\n\nHad she inadvertently weighted the scales too much in one direction or the other? The question had plagued her with every word she had written. She had tried her best to be fair, and Mr. Redpath assured her she had been. She hoped the world would agree.\n\nShe especially hoped Mrs. Lincoln would agree.\n\nOn April 1, Mr. Redpath brought her a copy of the _American Literary Gazette and Publishers' Circular,_ opened it to a page he had marked, and twice tapped an advertisement at the top of the left column. \"Mr. Carleton and I and everyone at our publishing house are tremendously proud to have you as one of our authors,\" he said. \"We have great expectations for the success of your book.\"\n\nElizabeth glanced away from the page to thank him, but she immediately turned back to it and began to read:\n\n**G. W. C ARLETON & CO.**\n\n_Will publish early in April_\n\nA REMARKABLE BOOK ENTITLED\n\n**BEHIND THE SCENES**\n\nBy Mrs. Elizabeth Keckley, for thirty years a household slave in the best Southern families, and since she purchased her freedom, and during the plotting of the rebellion, a confidential servant of Mrs. Jefferson Davis, where, \"Behind the Scenes,\" she heard the first breathing of that monster, SECESSION. Since the commencement of the rebellion, and up to date, she has been Mrs. Abraham Lincoln'smodiste (dressmaker), confidante, and business woman generally: a great portion of her time having been spent in the White House in the president's own family. Being thus intimate with Mrs. Lincoln and her whole family, as well as with many of the distinguished members of Washington society, she has much to say of an interesting nature in regard to men and things in the White House, Congress, Washington, and New York. She discloses the whole history of Mrs. Lincoln's unfortunate attempt to dispose of her wardrobe, etc., which when read will remove many erroneous impressions in the public mind, and place Mrs. L. in a more favorable light.\n\nThe book is crowded with incidents of a most romantic as well as tragic interest, covering a period of forty years. It is powerfully and truthfully written, and cannot fail to create a wide-world interest, not alone in the book, but in its gifted and conscientious author. It is perfectly authentic. One vol. 12 mo. 400 pp. Cloth. Illustrated with portrait of the author. Price $2.\n\n\"What do you say to that?\" said Mr. Redpath, smiling.\n\n\"I say it is quite wonderful,\" said Elizabeth, with a warm but tremulous laugh. Although she had been working on her memoir for months, somehow the advertisement made it seem real, immediate, tangible. Hesitantly, she asked, \"It says that I was thirty years a slave, but I was actually a slave for thirty-seven years, almost thirty-eight.\"\n\n\"An insignificant difference,\" Mr. Redpath assured her. \"Thirty is a nice round number. Although you are right to say that it is not entirely accurate, it sounds better in the advertisement.\"\n\n\"Of course,\" Elizabeth said, regretting her criticism. \"Thank you for your kind words about me and my book. You make it sound so intriguing that if I had not written it, I would be first in line to buy it.\"\n\nMr. Redpath smiled. \"You deserve abundant praise. The book is remarkable, and its author even more so.\" To Elizabeth's surprise, he took her hand and raised it to his lips. \"It has been an honor working with you, Mrs. Keckley. The privilege would not have been greater if I had worked with the First Lady herself.\"\n\n\"Don't ever tell her that,\" Elizabeth quickly warned him, without thinking. Laughing, Mr. Redpath assured her he would not.\n\nA few days later, Mr. Redpath called on her again and placed her finished book in her hands. For a long moment she held it, unmoving, disbelieving. \"Learn your book,\" her father had urged her in his letters. How proud he would be to know that she had become an author.\n\nElizabeth's hands trembled, and as Mr. Redpath looked on smiling, she opened the red clothbound cover and turned the first few blank pages. She paused at the engraved portrait of the author\u2014she didn't think it was a very flattering likeness, but never mind. Next she came to the title page:\n\nBEHIND THE SCENES.\n\n_by_\n\n_Elizabeth Keckley_ ,\n\nFORMERLY A SLAVE, BUT MORE RECENTLY MODISTE,\n\nAND FRIEND TO MRS. ABRAHAM LINCOLN.\n\n**OR,**\n\nTHIRTY YEARS A SLAVE, AND FOUR YEARS IN THE WHITE HOUSE.\n\nSo Carleton & Company had not corrected the number of years she had been a slave. Well, she supposed, what harm would it do to have people think she was a few years younger?\n\n\"How does it feel to see your words in print for the first time?\" asked Mr. Redpath.\n\n\"It feels wonderful,\" Elizabeth said. Her heart was pounding so quickly that she had to take a deep breath to quiet it. She turned to her preface and read again the familiar words that until then she had seen only in her own imperfect handwriting. Then an unfamiliar turn of phrase caught her eye, and another. Her brief musings on slavery had been embellished greatly. The sense of her original words was still evident, but it had been cloaked in more ornate prose than she liked. Feeling Mr. Redpath's eyes upon her, she was careful not to allow her expression to betray her surprise and disappointment. Everyone knew that editors changed an author's words, she chided herself. An author of a first book should expect even more assistance, especially from such an experienced editor.\n\nShe paged through the book carefully, her joy soon returning when she discovered that the rest of the memoir had not been altered as drastically. Then she reached the end of her tale, and discovered an unexpected appendix.\n\nJoy turned to shock when she discovered the letters from Mrs. Lincoln she had lent to Mr. Redpath to assist him in editing her manuscript. They had been reproduced almost exactly as Elizabeth remembered, with only a few inconsequential phrases omitted. Ironically, Mrs. Lincoln's warning that her confidences were meant to be \"BETWEEN OURSELVES\" had been preserved, including the emphatic capitalization.\n\nShe felt as if all the air had been squeezed from her lungs. \"Mr. Redpath,\" she managed to say in a strangled voice. \"Mrs. Lincoln's private letters\u2014\"\n\n\"Yes. We thought they contributed to the authenticity of your work.\"\n\n\"But I let you borrow them only to assist you in your editing,\" protested Elizabeth, distressed. \"You agreed not to publish them.\"\n\nMr. Redpath's brow furrowed. \"No, Mrs. Keckley, no,\" he said, shaking his head. \"We agreed that I would not publish anything from the letters that would embarrass Mrs. Lincoln.\"\n\nShe held out the book to him and quickly paged through the appendix. \"I assure you, this will embarrass her!\"\n\n\"I disagree,\" he replied. \"The letters reveal her thinking, the motives behind her actions. You always said that if people understood her good intentions, they would be more forgiving of her...outbursts and mishaps, as it were. I fail to see how that should embarrass her.\"\n\nHe failed to see because he did not want to see. Sick at heart, Elizabeth sank into a chair, the book on her lap. Mrs. Lincoln would view the publication of her private correspondence as the worst sort of betrayal. She would never forgive her.\n\n\"Mrs. Keckley, please do not distress yourself.\" Mr. Redpath either truly was utterly bewildered by her reaction or he was a shrewd actor. \"These letters\u2014in fact, your entire memoir\u2014will serve their purpose. They will inspire Mrs. Lincoln's critics to abandon their misconceptions and rally to her side. Mark my words.\"\n\nShe pressed her lips together and nodded, hoping he was right. There was no point in arguing with him. What was done was done. She could not go from bookstore to newsstand tearing the appendix from every copy of _Behind the Scenes_.\n\nAnd, as it happened, Mr. Redpath's prediction proved true to some extent, but not as he had expected.\n\nOn April 12, a woman for whom she did some sewing, a native of Boston, greeted her with the cryptic remark \"I'm glad to see you doing so well. You should not give a care for what they say in Springfield.\"\n\nElizabeth's heart thudded, and her thoughts flew to Mrs. Lincoln. \"What do they say in Springfield?\"\n\nHer patron's quick, guarded expression revealed that she assumed Elizabeth already knew. She didn't want to explain, but after some coaxing, she reluctantly told Elizabeth that a brief item about her forthcoming book had appeared in the _Springfield Daily Republican_. Although Elizabeth was relieved to learn that her patron referred to Springfield, Massachusetts, rather than the former hometown of the Lincoln family, she still needed to steel herself before asking her patron to bring her the clipping.\n\nThe headline read \"Kitchen and Bed-Chamber Literature,\" and the review\u2014if it was right to call it a review, before the book was published\u2014was worse than Elizabeth could have imagined.\n\nA disreputable New York publishing house announced a book to be entitled _Behind the Scenes_ , by Mrs. Elizabeth Keckley, who professes to have been Mrs. Jeff Davis's confidential servant while the rebellion was being cooked, and Mrs. Abraham Lincoln's dressmaker while the rebellion was going on. Such a book may be written in good taste, be interesting, be instructive, be decorous; but the chances are as the Bank of England to a filbert, that it is a sensational enterprise of the worst sort, after the order of Gen. Baker's \"Secret Service,\" unprincipled, false, scandalous, indecent. The very idea of domestic servants being persuaded to write books about the secrets of their employers, being crammed by literary adventurers with what they ought to say, and their lumbering and halting narration being helped at every stage by perhaps the very class of men who edit the flash papers of our cities, must be repulsive to every person of an ordinary degree of refinement. We hope it will prove to be a good book. We greatly fear it will be an exceedingly bad one.\n\nAt first Elizabeth was too flabbergasted to speak. \"Should they not first _read_ my book before they condemn it?\" she eventually managed to say. \"Indecent? My memoir? I was told what to write by 'literary adventurers'? How could they possibly believe such terrible things?\"\n\n\"I'm so very sorry,\" her patron said, thoroughly miserable. \"I wish I had never mentioned it.\"\n\n\"You're not to blame.\" Elizabeth blinked back angry tears and fought to maintain her composure. \"You are only the bearer of bad news. You didn't write these cruel things.\"\n\n\"Those newspapermen will eat their words after they read your book,\" her patron said comfortingly. \"I for one am no less interested in reading your memoir now than I was before they attempted to discourage me.\"\n\nElizabeth managed a wry smile. \"Perhaps, but you know me.\"\n\n\"Think of it, though,\" her patron mused. \"The _Republican_ might have done you a favor.\"\n\n\"In what way? I don't see how.\"\n\n\"They have made the people curious about your story. Everyone will want to buy your book now. They will read it and make up their own minds. You shall see.\"\n\nElizabeth hoped she was right, but an ominous cloud seemed to blot out the sun, casting shadows upon what she had expected to be a bright and happy day.\n\nUndeterred by the newspaper's grim predictions, her New York friends were immensely proud of her and excited about her book, and her landlady threw a party for her on the first day of its release. Emma, the Lewises, and other friends sent congratulatory telegrams from Washington City. Their enthusiasm lifted Elizabeth's spirits somewhat, but Mrs. Lincoln had not responded to her letter confessing the matter of their private correspondence made public, and silence from such a prolific letter writer made Elizabeth uneasy.\n\nOn April 15, a new advertisement appeared in the _American Literary Gazette and Publishers' Circular,_ but with a dramatic new headline that described her memoir as \"A Literary Thunderbolt.\" Most of the book description remained the same, although what Elizabeth had to say \"in regard to men and things in the White House\" was no longer described as merely \"interesting,\" but now also \"startling.\"\n\n\"Oh, no, no, no,\" Elizabeth murmured, setting the paper aside. Her insightful, poignant memoir had been transformed into a spectacle.\n\n_Behind the Scenes_ outraged the press, who were swift, scathing, and merciless in their response.\n\nThe New York _Citizen:_ \"Has the American public no word of protest against the assumption that its literary taste is of so low grade as to tolerate the back-stairs gossip of Negro servant girls?\"\n\nThe Washington _National Intelligencer:_ \"Where will it end? What family that has a servant may not, in fact, have its peace and happiness destroyed by such treacherous creatures as the Keckley woman?\"\n\n_The_ _New York Times,_ after offering three columns of excerpts and noting that Mrs. Lincoln was in financial distress: \"Mrs. Keckley, we also learn, is likewise in trouble. Mrs. Lincoln is unable to pay her, and she supports herself by taking in sewing\u2014and by writing a book. She would much better have stuck to her needle. We cannot but look upon many of the disclosures made in this volume as gross violations of confidence. Mrs. Lincoln evidently reposed implicit trust in her, and this trust, under unwise advice no doubt, she has betrayed. But only in a restricted sense can the book be called her own. It is easy to trace, all through its pages, the hand of a practiced writer\u2014of one who has prejudices to gratify and grudges to repay. As mere gossip, the book is mainly a failure. Mrs. Keckley really knew very little about life in the White House, and she ekes out her scant stock of story and anecdote with extracts from newspapers, moral reflections and other expedients of like character. The public will be disappointed when they come to read her book. They will find it less piquant, less scandalous, than was expected, considering its source, while as a literary work it can lay claim to very little merit indeed.\"\n\nAnd the _Springfield Daily Republican,_ which seemed all too delighted to discover that its dire predictions about the quality of her book had been fulfilled: \"One would suppose the public had been treated to Mrs. Lincoln and her affairs _ad nauseam,_ \" the reporter sneered, \"but scandal is always a marketable commodity, and this book contains plenty of it.\"\n\nThe theories about the diffusion of knowledge and the education of the masses, are all very fine, and within certain limits work very well. It is not pleasant, to be sure, to have a cook so literarily inclined as to be continually removing all your pet books from the library to the kitchen, and who insists on the first reading of the morning paper while she is getting breakfast; or a housemaid who prefers reading your letters to attending to her own proper duties. But all these can be patiently endured in consideration of the many benefits that are supposed to accrue to Bridget and Dinah on account of a smattering of knowledge. But when Bridget or Dinah takes to writing books instead of reading them, and selects for themes the conversations and events that occur in the privacy of the family circle, we respectfully submit that it is carrying the thing a little too far. The line must be drawn somewhere, and we protest that it had better be traced before all the servant girls are educated up to the point of writing up the private history of the families in which they may be engaged.\n\nThe vitriol stunned and sickened Elizabeth, and yet she was compelled to read on, nor could she ignore the other withering criticisms and condemnations that threatened to bury her in an avalanche of newsprint and ink. The reviews were overripe with ridicule and condemnation, and often contradictory. _Behind the Scenes_ was both badly written, and written so well that it could not have come from the pen of a \"treacherous Negro servant.\" Her memoir was worthless trash, and yet deserving of multiple newspaper columns for excerpts. Except for the _Chicago Tribune_ , which emphasized the presence of notable Illinoisans in her memoir, praised certain sections as \"interesting\" and \"affecting,\" and concluded that she was a woman \"of more than ordinary intelligence,\" the reviews were unanimous in their disgust and outrage, and they seemed less concerned with commenting on her book than with denouncing Elizabeth for writing at all. To her astonishment, the same papers that had spent the past eight years gleefully pillorying Mrs. Lincoln now became her staunch defenders against the villainous \"White House Eavesdropper\" she had unwisely trusted.\n\nIn that sense, Mr. Redpath's prediction that _Behind the Scenes_ would compel the public to rally around Mrs. Lincoln came true.\n\nMr. Redpath no longer called on her at No. 14 Carroll Place, most likely because they had parted on uncomfortable terms over the printing of Mrs. Lincoln's letters, but when the attacks upon her and her book became unbearable, Elizabeth visited him at his office and asked him what she should do about the terrible, unfair reviews. \"Ignore them,\" he advised her simply.\n\n\"I cannot,\" Elizabeth insisted. \"Nor should I. They are in the wrong. These reporters cannot offer any specific examples of poor writing in my book. They can only claim that it is worthless in a general sense while they impugn my character. And all the while they insist upon referring to me as Mrs. Lincoln's Negro servant as if I were a\u2014a housemaid or a kitchen girl rather than an accomplished seamstress. They say I desecrated the memory of President Lincoln, a man I admired and respected with all my heart. I cannot let those accusations stand unchallenged.\"\n\nSeeing that she was resolved, Mr. Redpath reluctantly suggested that she write a rebuttal, which he would forward to the editor of the New York _Citizen,_ a gentleman who happened to be one of Carleton & Co.'s bestselling authors. Immediately Elizabeth took up her pen in her own defense, challenging her critics to read her book alongside the pages and pages that respected newspapers from across the nation had committed to the \"bitter crusade\" against Mrs. Lincoln. Which of them, she asked, had truly betrayed and scandalized the former First Lady? \"Is it because my skin is dark and that I was once a slave that I am being denounced?\" Elizabeth demanded. \"As I was born to servitude, it was not fault of mine that I was a slave; and, as I honestly purchased my freedom, may I not be permitted to express, now and then, an opinion becoming a free woman?\"\n\nThe resoundingly unanimous response of the public was that she could not.\n\nNot long after her refutation appeared in the _Citizen,_ someone left a small, flat package for her on the doorstep of her boardinghouse, wrapped in brown paper with \"Mrs. Kickley\" printed in block letters upon it. Curious, Elizabeth unwrapped it\u2014and suddenly discovered that the misspelling of her name had been no accident. The gift of her anonymous enemy was a booklet entitled _Behind the Seams; by a Nigger Woman who took work in from Mrs. Lincoln and Mrs. Davis_. The ostensible author, \"Betsey Kickley (nigger),\" had signed the book with an X, indicating that she was illiterate.\n\nHer heart thudded as she opened the booklet and allowed her gaze to skim over the lines of text, enough to determine that it was a cruel, vicious parody of her memoir. Sickened, she threw it away, but she could not erase the taunting image from her thoughts\u2014stacks and stacks of the booklet flying off newsstands and bookstore shelves, people throughout the country savoring its ugliness with malicious glee.\n\nHer own book, she soon discovered, had become much harder to find. The controversy and the accusation that it was an \"indecent\" book had discouraged many booksellers from stocking it, and as time passed, Elizabeth heard convincing rumors that a furious Robert Lincoln had ordered her publisher to recall the book, and when that failed, he and his friends bought up all the copies they could, and had them burned.\n\nElizabeth felt as if she had plunged into a nightmare. She wrote impassioned, apologetic letters to Mrs. Lincoln but never received so much as a single word in reply. When she discovered that Robert Lincoln was visiting the city, she sent him a note imploring him to let her call on him so she could apologize in person, but he flatly refused to see her. White people closed ranks against her, the colored woman who had dared scrutinize someone so high above her station, and even those who disliked Mrs. Lincoln denounced Elizabeth for betraying her and for dishonoring Mr. Lincoln's memory by revealing intimate scenes from his family life. Her own people were upset with her because they worried that white employers would be reluctant to hire colored servants, for fear that their own secrets would be written up in scandalous books. Despondent, smoldering with indignation and disappointment and embarrassment, Elizabeth nonetheless held her head high, even when several of the seamstresses for whom she had sewed suddenly discovered that they no longer had any piecework to assign her.\n\nWith her income sharply diminished, sales of her book became more important than ever. When she called on Mr. Redpath to inquire about receipts, he said that the book had not yet sold enough to pay for the costs of printing and distribution. To help increase sales, he proposed arranging public lectures, one in New York and one in Boston, and if those went well, they would consider expanding the series to other cities. \"Chicago, perhaps?\" prompted Elizabeth, eager for an opportunity to call on Mrs. Lincoln and offer her in person the apologies and explanations she deserved.\n\n\"We'll see,\" said Mr. Redpath affably.\n\nIn the third week of June, Elizabeth read aloud from her book to an audience of no more than three dozen ladies and gentlemen at a Manhattan bookstore, a third of them her own good friends who had come to show their support. As for the rest, Elizabeth soon realized from their queries and comments that they could be divided into three groups: the curious, drawn by the press's claims of indecency and scandal and hoping to witness some of each; people who despised Mrs. Lincoln and were glad to see her brought low; and unrepentant secessionists and other political enemies seeking evidence to confirm their suspicions of treachery, indecency, and corruption in the Lincoln administration. At the end of the evening, only Elizabeth's friends went home satisfied.\n\n\"Mrs. Elisabeth Keckley, the eavesdropper of the White House, seeks to increase the sale of her dirty book by reading selections from it,\" the _Springfield Daily Republican_ sneered two days later. \"She attempted at New York, Tuesday evening, but failed utterly and deservedly, and to-night will make an effort at Boston. She reads more poorly than she writes, and that is as bad as can be.\"\n\nNotwithstanding the power of the newspaper's warning to beckon curiosity seekers, the turnout at the Boston event at Lee & Shepard was slighter than that of her first reading but composed of much of the same sort of people. Thus Elizabeth was relieved rather than disappointed when Mr. Redpath informed her that Carleton & Co. did not intend to arrange any more lectures for her, although they certainly encouraged her to do so on her own, if she wished.\n\nElizabeth emphatically did not.\n\nShe had too much dignity to rail against the unfairness and injustice of her circumstances. She understood that she was mired in scandal and that every movement she made to climb out of it only dragged her in deeper.\n\nShe could not count how many times through the years had she observed Mrs. Lincoln similarly entangled. Mrs. Lincoln's instinct had been to fight ferociously to salvage her reputation, sending scores of letters to garner support, evoking the power of her husband's high office to demand the deference due to her as his wife, using patronage to forge alliances and appease enemies, and, when all else failed, fleeing the scene and leaving faithful friends to clean up the mess she left behind.\n\nMrs. Lincoln's tactics had never succeeded, not that Elizabeth could remember.\n\nIf Mrs. Lincoln's example had taught her nothing else, it had shown her that the only way to redeem oneself from scandal was to live an exemplary life every day thereafter.\n\nAnd that was precisely what Elizabeth intended to do.\n\n# Chapter Seventeen\n\n## 1868\u20131893\n\nWhen the furor subsided, Elizabeth quietly moved back to Washington, where Virginia, Walker, and Emma welcomed her warmly\u2014and kindly said nothing of her book or the scandal. She reopened her dressmaking business, summoned her former assistants, and sent out letters to her patrons announcing her return, but during her long absence, many of her assistants had found other employment or had gone into business for themselves. Some of her favorite patrons soon placed orders with her for garments for the winter social season, but many more demurred. Although they never said so, Elizabeth was certain they stayed away because of her book.\n\nShe wrote again to Mrs. Lincoln seeking forgiveness, to let her know she had returned to Washington if Mrs. Lincoln wanted to reply. The next time Mrs. Lincoln visited Washington, Elizabeth vowed, she would apologize sincerely and profusely in person. And Elizabeth would present to her a gift of her heart and hands three years in the making, and as yet unfinished\u2014the quilt she had begun on the train to Chicago, when she had accompanied the grieving widow on her departure from the White House.\n\nPieced of silks left over from Mrs. Lincoln's gowns, the quilt struck the eye as predominantly red, blue, gold, and white, although other colors appeared too, tans and lavenders and ashes of rose. In the center Elizabeth had appliqu\u00e9d and padded a bold eagle clasping a flag in his talons, wings outspread, the word \"Liberty\" embroidered beneath him in gold. Around the central emblem she had added three concentric borders, the first of gold silk embroidered with flowers along the sides and black silk cornerstones. The second was simpler, tan pieces along the top and bottom and a black-and-white stripe along the sides. Next she had attached borders of dark blue and blue-gray, and cornerstones of light tan, intricately embroidered with flowers, sprigs, and other ornaments. She planned to frame the central medallion with borders of Grandmother's Flower Garden hexagons joined in clusters of seven and separated by contrasting rhombuses, giving the traditional design the effect of a radiating star, but those segments were still in progress. After that, she did not think the quilt would be quite finished; it would need another concentric border or two, and perhaps another embellishment, such as a scalloped edge or fringe, something she could not yet envision. For a while she worked swiftly\u2014fewer patrons meant fewer dresses, which meant more idle hours to fill\u2014hoping to complete the quilt in time to present it to Mrs. Lincoln when she next returned to Washington.\n\nBut although Mrs. Lincoln returned to the capital in September for her son Robert's marriage to Miss Mary Harlan, she seemed to abhor the city, for to the best of Elizabeth's knowledge, she made no other visits. Elizabeth was sure she would have heard if she had, for the newspapers continued to follow her comings and goings, her struggles with her finances, and her attempts to persuade the government to raise her pension.\n\nIt was from the newspapers that Elizabeth learned Mrs. Lincoln and her son Tad had left for Europe, where they were expected to embark upon a lengthy tour. Mrs. Lincoln reportedly hoped to recover her health, which had worsened of late, and to seek respite from the disappointments and humiliations at home. Some wags, recalling the notorious memoir of Mrs. Lincoln's dressmaker, noted that she had once declared that she would prefer to leave the country and remain absent throughout his term of office should General Grant be elected president of the United States. Perhaps, they impertinently suggested, that was why she planned to stay abroad so long. Elizabeth did not believe for a moment that Mrs. Lincoln had taken Tad abroad to avoid the Grant administration. Surely Mrs. Lincoln was fleeing grief and unhappy memories\u2014one of which, Elizabeth could only assume, was the ruin of their friendship.\n\nWith a pang of regret and sorrow, Elizabeth carefully packed the unfinished quilt in the trunk where she kept the leftover fabric from Mrs. Lincoln's gowns. She hadn't the heart to work on it anymore, not knowing when, or if, she would ever see her estranged friend again.\n\nEven practicing the closest economy, Elizabeth found it difficult to make ends meet. The book that had cost her so much had paid her nothing, so quietly she sued Carleton & Company for half of the profits from her memoir. She did not prevail in court, and therefore she did not receive a single cent, nor were Mrs. Lincoln's letters ever returned to her.\n\nFrom time to time, Elizabeth would spot Mrs. Lincoln's name in the newspaper and read about her travels in Germany, Scotland, England, France, and Italy. In 1870, while Elizabeth was struggling to rebuild her dressmaking business, she learned that Senator Charles Sumner had pushed through Congress a bill granting Mrs. Lincoln an annual pension of three thousand dollars, which President Ulysses S. Grant promptly signed. Delighted, Elizabeth immediately wrote Mrs. Lincoln a congratulatory letter, expressing her heartfelt joy at the well-deserved benefit and long-overdue justice. She said nothing of their estrangement, hoping that perhaps Mrs. Lincoln's long months away had given her time to reflect on the great many occasions Elizabeth had demonstrated her loyalty and friendship, and that perhaps she might extend a hand in forgiveness. Elizabeth sent her letter to the residence in Frankfurt am Main last reported in the papers, but no reply came. Not long thereafter, in May 1871, she read that Mrs. Lincoln and Tad had left Germany for England a few months after Elizabeth had written to her and from there had returned to Chicago. Wistfully, she wondered if her letter had arrived too late, and if even at that moment it sat unopened on a table in Mrs. Lincoln's unoccupied rented rooms in Germany. How tragic it would be, she thought, if that letter would have provoked the tender, forgiving reply she had longed for, but the opportunity for reconciliation had been lost, all for the mischance of one letter gone astray.\n\nYet knowing that Mrs. Lincoln had returned to the United States rekindled Elizabeth's hopes, spurring her to unearth the abandoned quilt and resume working on it in earnest. Though Mrs. Lincoln was hundreds of miles away, she had not been so close in years, and if she returned to the capital\u2014and was it not likely that eventually some official business would summon her there?\u2014Elizabeth wanted the quilt to be ready.\n\nBut to her shock, the next news Elizabeth read of Mrs. Lincoln was not of an impending visit to Washington, but of the death of her son Tad.\n\nTears ran down Elizabeth's face as she read the grim report. As mother and son had sailed to America from England, where eighteen-year-old Tad had been enrolled in boarding school, the young man's weak lungs had suffered in the storms and damp. Upon his arrival in Manhattan, he was diagnosed with a serious chest ailment and put on bed rest at a hotel until he regained enough strength to continue on to Chicago by train. Eventually Tad's doctors pronounced him fit enough to travel, and after a long train ride that Elizabeth could well imagine from her own experience of it, they reached Chicago and settled into Robert Lincoln's new home, which he shared with his wife and daughter. Mrs. Lincoln and Tad soon moved into Clifton House, where Mrs. Lincoln could better nurse him, but her tender efforts were all in vain. On July 15, Tad Lincoln died from a dropsy of the chest.\n\nElizabeth sent condolences but neither expected nor received a reply. Elizabeth prayed for Mrs. Lincoln and for Robert, and worried about them both, but especially Mrs. Lincoln. She could not count how many times in the years since President Lincoln's assassination the despondent widow had declared that if not for Tad, she would gladly join her husband in the grave. By her own admission, only Tad and her responsibility for him had kept her from taking her own life.\n\nWhat would Mrs. Lincoln live for now?\n\nElizabeth kept her beloved, familiar rooms in the Walker Lewis boardinghouse for a while longer, but eventually her financial troubles obliged her to move elsewhere. For the next few years she boarded with one family and then another, always within the capital. Later, after her dear friend Virginia passed on, she moved back into the Lewis boardinghouse to help Walker care for their youngest daughters, still at home\u2014pretty Alberta, her goddaughter, and sweet Elizabeth, born after the war. Sewing work still came in sporadically, but fashions had changed with the times, and other dressmakers, including Emma, now married with a son and daughter, were more eagerly sought after by the Washington elite. In her success Emma had never forgotten who had trained her and launched her career, and she frequently sent work Elizabeth's way. Elizabeth was grateful for the assistance and very proud of her most accomplished apprentice.\n\nFrom the little she gleaned from the press, her own troubles were nothing compared to Mrs. Lincoln's, whose erratic behavior had only worsened in the wake of Tad's death. In May 1875, after a shocking trial, every sordid detail of which was breathlessly recounted in the papers, Robert Lincoln had his mother declared insane and committed to an asylum in Batavia, Illinois, about forty miles west of Chicago. Powerless to help or to comfort her former friend, Elizabeth followed the heartbreaking story in the newspapers, stunned by reports that Mrs. Lincoln had tried to commit suicide by taking an overdose of laudanum the day before the verdict was delivered. The attempt had been thwarted by a wary druggist, who had recognized her and had substituted a solution of burnt sugar and water for the medicine she demanded.\n\nFor weeks afterward, Elizabeth was haunted by visions of Mrs. Lincoln languishing in a cold, ominous institution, devoid of all warmth and comfort. She imagined her keening endlessly as she had when Willie died, weeping and shrieking as she had after Mr. Lincoln had been killed. Elizabeth had been her most faithful companion in those dark days, but she had been unable to offer Mrs. Lincoln any solace when Tad passed away, nor could she do anything for her now.\n\nThen she remembered the quilt, and she worked upon it feverishly. She finished the Grandmother's Flower Garden borders, and then added one final border, ivory silk with four more proud eagles, one on each side echoing the dark eagle bearing a flag in the center, but made of gold silk, stuffed and embroidered, with more intricate floral embroidering all along the sides. As it neared completion, she wrote to Dr. Richard J. Patterson at Bellevue Place and asked if she might be permitted to visit Mrs. Lincoln. \"I have made her a precious gift, which I hope she will accept as a token of my enduring sympathy and friendship,\" she wrote. \"It is my great hope that this quilt, with its pieces reminiscent of happier days of years gone by, will offer her diversion, comfort, beauty, and solace in her confinement.\"\n\nShe had grown so accustomed to sending letters to Illinois only to receive no reply that she was almost startled when Dr. Patterson responded within a fortnight. He offered her his compliments and said that he was \"most gratified to know that Mrs. Abraham Lincoln was not forgotten by her Washington friends,\" but that he must discourage Elizabeth from attempting to call on Mrs. Lincoln, especially since she would be obliged to travel hundreds of miles to do so. Often Mrs. Lincoln was unfit to receive callers, he explained, and when she was, she almost always refused to see anyone.\n\nRemembering how Mrs. Lincoln had turned away many kindhearted visitors during the early days of her widowhood in Hyde Park, Elizabeth was disappointed but not surprised.\n\nAs for her gift, Dr. Patterson wrote, \"It is with great admiration for your generosity that I must respectfully urge you not to send this quilt\u2014which sounds truly extraordinary, a masterpiece of the form\u2014until Mrs. Lincoln is better able to appreciate it. At this time she abhors all tokens of the past. Objects, places, and anniversaries that evoke memories a sane person would find pleasantly diverting, are an anathema to her. I regret that as finely made as your quilt surely is, Mrs. Lincoln would find no comfort in it. I encourage you to keep the quilt safe, and to deliver it to her after she has been restored to reason, at which time I am sure she will thank you for it.\"\n\nShe wouldn't, Elizabeth realized. She never would. It was not a symptom of Mrs. Lincoln's insanity that she could not bear souvenirs of the past. She had always shunned them. She had quickly, almost frantically, given away Willie's toys and books after his death. She had readily parted with Mr. Lincoln's relics in the weeks following his assassination, not only to express gratitude and respect for his dearest friends, but also to get them away from her, far away, where she would never have to look upon them again.\n\nBlinking back tears, Elizabeth packed up the quilt, certain that this time would be the last. What a fool she had been to think Mrs. Lincoln would ever want to look upon scraps of the dresses she had worn at the height of her power as First Lady, dresses she had tried to part with in what turned out to be the most humiliating scandal in a life that had known too many.\n\nWhat a fool Elizabeth had been to assume that Mrs. Lincoln would want anything from her.\n\nFrom time to time, brief news reports would address Mrs. Lincoln's incarceration. One journalist from the _Chicago Post and Mail_ who had come to view the sanatorium's grounds was inexplicably granted an interview with the ailing widow, and when Elizabeth read that he found her clad in a shabby dress with her hair gone completely white, her heart sank. When the reporter noted that she rambled in conversation, and that if left alone in her room she spoke to imaginary companions, Elizabeth felt so heartsick for her that she could not finish reading the article. Expecting Mrs. Lincoln's decline to continue inexorably, Elizabeth was delighted to read little more than a month later that she had been pronounced well enough to leave the asylum and visit her sister, Mrs. Elizabeth Edwards, in Springfield. \"It is not likely that she will return to Bellevue Asylum,\" the paper's Chicago correspondent reported. \"She is decidedly better, sleeps and eats well, and shows no tendency to any mania; but whether the cure is permanent or not the test of active life and time will prove.\"\n\nElizabeth rejoiced at this good news, and again when, in June 1876, headlines declared that Mrs. Lincoln had officially been declared restored to reason. She resided with her sister Elizabeth in Springfield, their old estrangement apparently forgotten. Elizabeth's faint hope that Mrs. Lincoln might find it within her heart to forgive her too diminished yet again when she learned that Mrs. Lincoln had sailed for France on October 1 and would likely spend many years abroad.\n\nMrs. Lincoln did not return to the United States until the autumn of 1880, and only then because, at sixty-two years of age, she was too ill to live alone. She returned to her sister Elizabeth's home and care, and Elizabeth knew then that Mrs. Lincoln would never again visit Washington.\n\nOn July 2, 1881, Robert Lincoln was walking with the newly elected president James Garfield through a Washington railway station when a disgruntled office seeker shot the president twice in the chest. Mr. Garfield survived the attack, but as the summer waned, infection set in and he declined, suffering eighty days until death finally claimed him on September 19. As shocked and horrified as the rest of the nation, Elizabeth found herself unable to sleep, worrying about Mrs. Lincoln, who was surely in anguish as the new tragedy forced her to relive her own husband's assassination.\n\nElizabeth was tempted to write to her again\u2014to offer sympathies, to assure Mrs. Lincoln she had at least one friend thinking about her in those troubled times\u2014but she sat at her desk staring off into the distance, waiting for words that would not come. Eventually she abandoned the letter as a futile effort, having never once touched pen to paper.\n\nElizabeth read about Mrs. Lincoln twice more in the national papers. The first occasion came in November of that year, shortly after Congress granted the newly widowed Mrs. Garfield an annual pension of five thousand dollars, two thousand dollars greater than Mrs. Lincoln's. Her campaign to get Congress to increase her own pension to match put Mrs. Lincoln in the headlines once again. In January of 1882, Congress agreed, and with unexpected generosity also voted to grant her back pay as well as a fifteen-thousand-dollar bonus. Elizabeth privately cheered the decision, glad to know that Mrs. Lincoln would surely at last be released from the financial worries that had plagued her for years.\n\nBut Mrs. Lincoln did not have long to enjoy her triumph.\n\nThe next time Elizabeth discovered her name in the Washington papers was the day after she died in Springfield on July 16, 1882.\n\nWith all hopes of reconciliation forever lost, Elizabeth no longer had any reason to finish the medallion quilt, but while the nation mourned and remembered and eulogized the former First Lady, she found herself compelled to take the unfinished top from the trunk. At first she considered adding a wide band of black silk all around the outer edges to signify mourning, but just as she was about to put shears to fabric, she decided against it. Mrs. Lincoln had spent nearly two decades in mourning, and although Elizabeth knew it was a strange notion, she could not bear to consign the quilt to the same unhappy end. Instead she trimmed the quilt in long red fringe, which reminded her of the glamour and patriotism of Mrs. Lincoln's best days in the White House. Then, in recognition of Mrs. Lincoln's passing, Elizabeth added four red tassels, one in each corner, because they reminded her of the tassels that had hung from the black silk drape on Mr. Lincoln's catafalque. That was the single note of mourning Elizabeth allowed in the quilt, although she knew it was unlikely anyone else gazing upon the quilt would recognize it for what it was.\n\nIt's over, Elizabeth thought when the quilt was finished.\n\nShe had been working doggedly until then, keeping herself so busy that she had no time for contemplation, but the realization that it was truly over cracked the fa\u00e7ade of her serene resignation, and all the grief pushed out through the fissures, and she wept.\n\nIn 1890, eight years after Mrs. Lincoln was beyond caring what Elizabeth did with the precious relics entrusted to her, Elizabeth found herself in such difficult financial circumstances that she was obliged to sell her mementoes of Mr. Lincoln, which she had cherished and protected for thirty-five years. Overcoming her wariness of brokers, she enlisted the services of W. H. Lowdermilk & Co. and sold her treasures to Mr. Charles F. Gunther of Chicago, a candy manufacturer and collector of curiosities. To Mr. Gunther went the bloodstained cloak Mrs. Lincoln had worn to the theater on the night of her husband's assassination, the right-hand glove Mr. Lincoln had worn at the first public reception after his second inauguration, and everything else. The only mementoes Elizabeth kept for herself were a pair of Mrs. Lincoln's earrings, the pieces of fabric left over from sewing Mrs. Lincoln's dresses, and the medallion quilt, which to Elizabeth seemed a relic of that time, although she had finished it later.\n\nWord of the sale found its way to Wilberforce University near Xenia, Ohio, where the news must have evoked Bishop Daniel Payne's curiosity and concern. He had never forgotten how Elizabeth had once offered to donate her Lincoln relics to the university, so they might be exhibited in Europe to raise money to rebuild the school's main building, which had burned down on the day President Lincoln was assassinated. Mrs. Lincoln had objected so vehemently that Elizabeth had apologetically withdrawn her offer, but Bishop Payne had understood her dilemma and had respected her decision. Mrs. Keckley must have fallen upon hard times indeed to sell her cherished mementoes now.\n\nElizabeth was pleasantly surprised to receive Bishop Payne's letter inquiring about her health, and she enjoyed his pleasant reminiscences and his description of how the university had grown and prospered since the days her son, George, had studied there. Then he proposed a change to the faculty that astonished her so much she had to ease herself into a chair: He offered her a position as head of Wilberforce University's Department of Sewing and Domestic Science.\n\nShe immediately wrote back to thank him profusely, but to decline. She was no professor, she reminded him. It was true that she had learned her letters in childhood, even though it had been illegal for her to read and write, and she had endeavored all her life thereafter to obey her father's wish that she \"learn her book.\" She had always loved to read, especially the Bible, but she had enjoyed nothing in the way of a formal education. Would not university students expect a more learned instructor?\n\nNot at all, Bishop Payne promptly responded. Students in the Department of Sewing and Domestic Science expected instructors who knew their craft inside and out and could skillfully pass on their knowledge to others. Elizabeth fit that description perfectly, and her experience as a mantua maker in the highest of Washington's social circles would be invaluable. That she had accomplished so much without the benefit of formal schooling attested to her strength of character\u2014her determination, perseverance, and love of learning, values she could impart to her students.\n\nAlthough Elizabeth had taught sewing to freedwomen in contraband camps and ambitious young ladies in her workrooms, she had never taught in a classroom. How very different it would be, she reflected, imagining herself standing before a class of eager pupils, patiently demonstrating how to make flawless darts and lecturing them on the benefits of proper posture in avoiding neck and back strain.\n\nShe agreed to visit the campus, to meet the faculty and students and to learn more about the position. She traveled by train and was entertained so graciously by everyone she met there that after two days at Wilberforce University, she gladly accepted Bishop Payne's generous offer. And so it was that after thirty years in Washington, at the age of seventy-four, she packed up her few belongings, bade farewell to Walker Lewis and his daughters and Emma and her many friends from the Fifteenth Street Presbyterian Church, and moved out west to Ohio.\n\nNever once in her blessedly peaceful and rewarding years there did she regret her decision.\n\nElizabeth thrived in the classroom and on peaceful strolls around the campus chatting with her cheerful, inexhaustibly curious, impossibly young students, all of them born after the war, never having known a single day when slavery was the law of the land. Dressmaking had changed since she had honed her skills in the art, but she adapted, and she soon discovered that experience trumped even the most newfangled tricks and contraptions. Among themselves, her students whispered stories about her and would come to her later, wide-eyed and awestruck, to ask what President Lincoln had really been like and whether it was true that his wife had been insane. Elizabeth would never fail to respond with serene praise for president and First Lady alike, and she would never allow anyone to disparage Mrs. Lincoln in her presence.\n\nSometimes Elizabeth would invite a group of favorite, promising students to her home and show them the silks and satins and other fine fabrics she had saved from making Mrs. Lincoln's garments. On occasion, she would reward an accomplishment with a gift of a small scrap of the precious fabric, which the lucky student could make into a pincushion, a small, useful White House treasure of her own. Her students and a few faculty members alike marveled at her quilt, and they sympathized when she expressed her regret that she had been unable to give it to Mrs. Lincoln as she had intended.\n\nElizabeth found it strange that not one of her students seemed aware of the scandal that had led to her estrangement from Mrs. Lincoln. Some of them had heard that she had written a memoir, although none of them had ever read it, which surprised Elizabeth not at all since Robert Lincoln had done his utmost to reduce the inventory.\n\nEventually it dawned on her that perhaps she had triumphed over the scandal after all.\n\nIn 1893, Wilberforce University participated in the World's Columbian Exposition in Chicago, a grand and glorious event in commemoration of the four hundredth anniversary of Columbus's discovery of the New World. When the event was in the planning stages, exhibit proposals from colored Americans were summarily rejected, but after strong protests from the black community led by Frederick Douglass and Ida B. Wells, the exposition organizers relented and allowed very limited participation, including displays of needlework and drawings in the Women's Building. Knowing that many skilled artisans and scientists of her race had been unfairly excluded diminished Elizabeth's pleasure in the honor bestowed upon Wilberforce University, and if it had been up to her, she would have withdrawn from the exhibition in solidarity. But it was not her decision, and, as she told herself, her hardworking students deserved to have their excellent work displayed and their achievements noted. Only by taking their rightful place among whites and presenting their handiwork before the eyes of a skeptical public could they hope to influence prejudiced minds.\n\nElizabeth felt herself drawn along by the strong pull of memory as she and several of her students traveled by train to Chicago. When she closed her eyes, she could see her former patron and lost friend as she had been in those grief-stricken weeks in Hyde Park, lamenting alone in her darkened room, staring unhappily across the vast dark blue of Lake Michigan. She had never ceased mourning, Elizabeth knew. She had never gathered up the shards of her broken life and reassembled them into something new, something endurable, if not happy. Even though Mrs. Lincoln had passed beyond all human suffering, and had been reunited at long last with her youngest sons and her beloved husband, Elizabeth was moved to pity for her anew.\n\nThe exposition was a wonder to behold. Elizabeth had assumed that it would be much like the Great Northwestern Sanitary Fair she had attended in the city in June of 1865, but the World's Columbian Exposition far surpassed it. Fifty nations participated, and when she wasn't tending her department's exhibit, Elizabeth and her students toured the various buildings and pavilions, marveling at clever new inventions, tasting strange and often delicious foods, and listening to music from different countries far across the sea. A wistful ache filled Elizabeth as she waited safely on solid ground while her bold, young students braved a ride on a dangerous-looking contraption called a Ferris wheel. The exposition's vast grounds were so close to Mrs. Lincoln's former Hyde Park residence that Elizabeth could have walked there, if she wished\u2014but she knew she would find no comfort or satisfaction in revisiting that unhappy chapter of her past.\n\nShe would rather continue to explore the exposition, which seemed to represent the future with all its newness and wonder and innovation, and to pass pleasant hours at the Wilberforce University exhibit, which was firmly rooted in the present. Their exhibit resided within the Manufacturers and Liberal Arts Building, and in their modest portion of the vast space, wooden figures clad in garments designed and made by Elizabeth and her students showed off the department's skill and artistry, and a revolving showcase displayed photographs and biographies of their graduates.\n\nObserving passersby as they admired the garments, and watching her students as they answered questions with grace, intelligence, and poise, Elizabeth felt her wistfulness recede as the warm glow of joy and pride filled her heart almost to bursting. This was her legacy, she realized, not the beautiful wardrobe she had sewn for Mrs. Lincoln or even the book she had written with such good intentions. These young women, and the apprentices she had instructed and advised back in Washington, and the runaways and freedwomen she had provided with the fundamental skills to care for themselves and their families\u2014they were her legacy. Their success and independence and confidence were her true gifts to the world.\n\nHer greatest legacy could not be measured in garments or in words, but in the wisdom she had imparted, in the lives made better because she had touched them.\n\n# Chapter Eighteen\n\n## 1901\n\nElizabeth gazed out upon the streets of Washington through the carriage window, thinking of how much the capital had changed since she had first settled there as a younger, more ambitious woman in 1860. In her mind's eye she could still envision it as it had been then\u2014the unfinished Capitol dome, the muddy streets, the slaves in chains being led from harbor to holding pen.\n\nHow the world had changed since then. How she herself had changed.\n\nHer weekly carriage rides were prescribed by her doctor, who insisted that she get out and about to increase the vigor of her constitution. Ever since she had moved back to Washington after a mild stroke had obliged her to resign from Wilberforce University, she had resided in a basement room at the Home for Destitute Colored Women and Children on Fifteenth Street. It amused her to refer to it as _her_ home, not only because she lived there, but because it had been endowed in part by donations from the Contraband Relief Association she had founded so long ago. She liked her solitary room, small and neat, the little table with a pitcher and bowl in one corner, a straight chair in the other, a rocking chair near the bed, and the old trunk containing her clothes and what remained of her cherished mementoes. Over the dresser hung a picture of Mrs. Lincoln, and through the window Elizabeth could watch young colored men and women on their way to and from classes at Howard University. No one but her pastor and a few friends knew that she resided there, who she was and who she had been. And that was the way she liked it.\n\nWhen the carriage pulled to a stop in front of the home, Elizabeth slowly and gingerly stepped down, accepting the driver's assistance with thanks. Straight and tall, unbowed by age or infirmity, she crossed the threshold, exchanging polite greetings with the residents and staff she passed. \"You have a visitor in the parlor,\" one woman told her, and so Elizabeth headed down the hall, curious, because she was not expecting anyone, or she would have gone driving earlier.\n\nIn the parlor she discovered a most welcome guest, her goddaughter Alberta, or Mrs. Alberta Elizabeth Lewis-Savoy as she was now called, whom she had known since the moment of her birth. \"My dear girl,\" Elizabeth greeted her happily, embracing her and kissing her cheek. Alberta would always be a girl in Elizabeth's eyes, though she was thirty-seven with children of her own. \"To what do I owe this unexpected pleasure? I didn't expect to see you until Sunday.\"\n\nAlberta smiled warmly, but her manner was somehow hesitant, a guarded look in her eye. \"I stumbled upon an interesting article in the newspaper this morning, and I thought you might want to see it.\"\n\nShe took a clipping from her handbag and gave it to Elizabeth, who held the page without unfolding it. \"It was so fascinating you could not wait a few days?\"\n\nAlberta looked as if she might speak, but then she shook her head, smiled weakly, almost apologetically, and guided Elizabeth to a chair. Elizabeth muffled a sigh as she seated herself, wondering what on earth it could be this time. Peering closely at the paper, she read the headline aloud: \"'Negro Authors.' Hmm.\" Suddenly wary, she glanced up at Alberta, who gestured for her to read on.\n\n# NEGRO AUTHORS.\n\nTHREE HUNDRED BOOKS BY THEM  \nON EXHIBITION IN BUFFALO\n\nBUFFALO, NY\u2014Some three hundred books written by American Negroes form a part of the Negro exhibit at the Pan-American Exposition. The collection contains the best work of the race in the field of authorship, and is unique. Examination of these books furnishes new data by which to rate the civilization of the Negro.\n\nAt this, Elizabeth sighed and shook her head.\n\nWe may as well be entirely frank in the appraisal. Much of it is rubbish. None of it is very great.\n\nThis judgment evoked an ironic laugh.\n\n\"Keep reading,\" Alberta urged her, and Elizabeth reluctantly complied.\n\nThere has been no Negro Homer, Shakespeare, or Dumas\u2014no American counterpart of the great French mulatto. But a great deal of this work has better qualities than the world has reason to expect, when it remembers the condition of its origin. Its chief value, the one thing that makes it worthy of attention, has no concern with the graces of literary form, but lies in the fact that here is the world's best record of the evolution of the Negro recorded by the Negro himself.\n\n\"Do you suppose,\" Elizabeth mused aloud, \"that when our white friends write and speak in this manner, they have any idea how insulting they are?\"\n\n\"I believe most of them don't,\" Alberta replied. \"And of those that do, many wouldn't care.\"\n\nElizabeth sighed and read on, through admiring reviews of the poems and letters of Phillis Wheatley, a pamphlet by Reverend Daniel Crocker, and several scholarly works. Praise for the exceptional works of Mr. Frederick Douglass and Wilberforce University professor W. E. B. Du Bois followed, and then Elizabeth held her breath as she glimpsed her own name one paragraph below.\n\nAs may be supposed the collection is rich in what may be called literary curios. One of them, _Behind the Scenes_ , by Elizabeth Keckley, \"formerly a slave, but more recently modiste and friend to Mrs. Abraham Lincoln,\" had considerable vogue in its day (1868) because of its singular revelations.\n\n\"'Considerable vogue'?\" Elizabeth echoed, unsure whether she should be amused or alarmed. The reporter apparently had not done his research, or he would have chosen a much different descriptor.\n\n\"I thought you should be aware that your book is on display, and apparently attracting notice,\" said Alberta, looking profoundly sorry. She knew well the story of how Elizabeth's slim red volume of remembrances had been received upon its publication. Considerable vogue, indeed.\n\n\"Thank you, my dear,\" Elizabeth said, reaching for her hand and smiling fondly. \"I am safely forewarned. Let us hope nothing else comes of this. It is nice, I suppose, to have been included in the exposition, but the world has moved on, and I suspect my 'literary curio' will soon be forgotten again.\"\n\nShe hoped so, but she was much mistaken.\n\nLess than a week later, she received a letter from a journalist, Mr. Smith D. Fry. He had read about the Negro exhibit, and, intrigued by the brief description of _Behind the Scenes,_ he wished to interview the authoress. \"I was only a lad when your memoir was published,\" he recalled, \"but my mother owned a first edition. It had a red cover, I recall, and I remember her saying that it was so vividly written that she felt as if she had been in the room with you and Mrs. Lincoln during some of the most significant events of the Civil War years, some moments triumphant and others tragic.\"\n\nElizabeth realized then that she was apparently as susceptible to flattery as the next woman, because although she had been quite determined to decline his request, that paragraph changed her mind. The detail about the red cover convinced her that he described a true memory and not a fiction meant to win her over. The newspaper had not mentioned the color of her book's cover, nor was it likely that Mr. Fry had glimpsed a copy more recently. She was also both charmed and amused by his proud boast that his mother had owned a first edition, as if there had been a second.\n\nStill, she gave herself a day to reflect before replying that she would grant his request.\n\nOn the morning of the appointed day, she dressed herself with her usual care in a black silk dress with a white fichu tied about her shoulders. Mr. Fry was waiting for her in the parlor when she came down, and as they exchanged greetings she silently gave him credit for punctuality. He looked to be about fifty years old, stout and a bit jowly, and mostly bald, but he was courteous and pleasant, and in his eye she detected keen intelligence and a strong desire to learn all he could about his subject\u2014dissecting her, if necessary. She would have to be on her guard.\n\nShe settled herself into a high-backed, upholstered chair, fixed a demure expression in place, and awaited his first question.\n\n\"Mrs. Keckley,\" he began, pencil and paper at the ready, \"I regret that I must begin with a delicate question.\"\n\nShe prepared herself for the worst. \"Very well.\"\n\n\"My readers will want to know, but I dread asking.\" He hesitated. \"May I say that you are eighty years of age?\"\n\n\"You may,\" said Elizabeth, though she was eighty-three.\n\nHe nodded and scribbled something on his pad. \"It is said that you were Mrs. Lincoln's dressmaker, but you were really much more than that, weren't you?\"\n\n\"I was her modiste,\" Elizabeth replied, as if the title were explanation enough.\n\n\"And that means you sewed for her, and dressed her for important occasions, is that not so?\"\n\nIt was so much more. \"I dressed Mrs. Lincoln for every levee,\" she said. \"I made every stitch of clothing that she wore. I dressed her hair. I put on her skirts and dresses. I fixed her bouquets, saw that her gloves were all right, and remained with her each evening until Mr. Lincoln came for her. My hands were the last to touch her before she took the arm of Mr. Lincoln and went forth to meet the ladies and gentlemen on those great occasions.\"\n\n\"I imagine you must have seen the president quite often then.\"\n\n\"Yes.\"\n\n\"They considered you part of the family?\"\n\nElizabeth reflected briefly upon how Mrs. Lincoln had behaved toward the women of her family. \"I would not make such a claim. I would say that I was a dear and trusted friend.\"\n\n\"To Mrs. Lincoln?\"\n\n\"To them both, although I was certainly much closer to Mrs. Lincoln.\" For a moment she was lost in thought, but then she fixed her gaze upon Mr. Fry and said, \"Mr. Lincoln was a good friend of mine, but he never knew what a good friend of his I was, and have ever been.\" When Mr. Fry watched her expectantly, waiting, she added, \"I was Mr. Lincoln's friend, am his friend now, and will always protect his memory by keeping my mouth closed concerning the many things which he unhappily suspected or imagined were going on around him officially and unofficially.\"\n\n\"I assume you feel very loyal to him,\" Mr. Fry remarked. \"So many Negroes do, or so I am told, because he was the Great Emancipator.\"\n\nElizabeth wondered at his line of questioning. She thought he had come there to discuss her book, but he seemed to be more interested in learning about Mr. Lincoln. That suited her fine. She would much prefer to talk about the great president, but if Mr. Fry entertained any hopes that she would giddily rattle on with malicious gossip, he would be sorely disappointed. \"I was born a slave, but bought my freedom, and so was under no obligations to Mr. Lincoln for emancipation,\" she said. \"But I loved him for his kind manner toward me and for his great act of giving freedom to my race. I know what liberty is, because I remember what slavery was.\" She smiled, and for a moment she almost felt sorry for the reporter, who was likely a father and perhaps a grandfather but to her seemed very young. \"You who have never suffered cannot understand the full meaning of liberty.\"\n\nMr. Fry had been writing swiftly the entire time she spoke, but at that, he paused. \"No,\" he said thoughtfully. \"I don't suppose we do. I don't suppose we can.\"\n\nSuddenly she warmed to him, although she could not bring herself to trust any journalist completely. \"I have almost worshipped Abraham Lincoln,\" she admitted. \"He was as kind and considerate in his treatment of me as he was of any of the white people about the White House. In that he manifested the consistency of his belief that all human beings are created equal in the sight of God.\"\n\nUnlike his successor, Elizabeth thought, but prudently did not say aloud. Mrs. Lincoln had judged Mr. Johnson's character rightly from the moment he stumbled drunkenly onto the national stage. He had undone many of the reforms Mr. Lincoln had put into place, and his policies had flung obstacles into the path of people of color as they marched toward equality. Many of the gains they had made during the war had been thwarted under President Johnson, but progress moved ever forward, and Elizabeth had not lost her hope for a better future for her race.\n\n\"And did you feel the same about Mrs. Lincoln?\"\n\nElizabeth considered his question. \"She was a very different person than her husband. He was a remarkable man, the greatest man I have ever known. If I were to use him as a measuring stick for Mrs. Lincoln or anyone else, including myself, we should all fail to measure up.\"\n\n\"We should indeed.\" Mr. Fry's gaze was on his pad as his pencil rapidly scratched upon the page. \"Do you ever feel that you failed to measure up to the trust the Lincolns placed in you?\"\n\n\"I never betrayed a secret in the days when secrets were worth gold, and gold was scarce,\" Elizabeth said, a trifle sharply.\n\n\"And yet, there is the matter of your book.\"\n\nSo he _was_ aware of the controversy. \"Yes,\" she said evenly. \"I wrote a book about my life, which included my years in the White House. I did it with the best of intentions, but as you are no doubt already aware, it did not turn out at all as I had expected.\"\n\nHe studied her for a moment, brow furrowing, and then he asked, \"Were you troubled at all, afterward, to know that you profited from exposing Mrs. Lincoln's foibles to the world?\"\n\n\"Sadly, her foibles were well-known before I ever wrote a word about them.\" Suddenly Elizabeth felt very tired. \"As for profit, I never received a dollar from the publication of that book. They kept it all. However, I need not regret that, because they printed many things which ought not to have been printed, many things which caused heartaches, because they were untrue.\" She sighed and glanced away, feeling tears threatening. \"The book was printed, and my name was on the title page as the author of everything contained between its lids.\"\n\n\"And you were not?\" Mr. Fry asked, prompting her, when she had been silent for a while.\n\n\"No, indeed.\" She could tell from his expression that he wanted an example. Oh, how the press rejoiced when you gave them enough rope for them to hang you by! But perhaps she was being unfair. This man had never written an unkind word about her. It was unfair to hold him accountable for the actions of others of his profession. \"They told the story of how Mrs. Lincoln tried vainly to sell her expensive and costly wardrobe in New York, but they did not tell it right. I was with Mrs. Lincoln at that time and did my best to help her dispose of her valuable things\u2014\" Suddenly, prudence prevailed and she thought better of telling him how Mr. Redpath had strayed from her version of events. \"But it would do no good now to say anything about the details of that venture. I merely mention it because they made it a part of the book which made money for them, but the publication of which brought me no money, while it made some enemies for me who should have always been my friends.\"\n\nShe felt herself sinking into a mournful reverie, but after a time, Mr. Fry quietly asked, \"You made dresses for other well-known ladies too, didn't you?\"\n\nElizabeth nodded. \"I made dresses for Mrs. Lincoln, the ladies of President Andrew Johnson's family, and also for the ladies of the Grant administration. And Mrs. Jefferson Davis too, before secession.\" She recalled that Mrs. Grant and Mrs. Davis had become quite good friends after the war. At the time she had thought that if the wives of Union General Grant and Confederate President Davis could become friends, then surely she and Mrs. Lincoln could reconcile. But it was not to be.\n\n\"You must have been quite famous in those days.\"\n\nElizabeth regarded him skeptically. \"I was acquainted with famous people. That does not mean that I garnered any fame for myself.\"\n\n\"Mrs. Keckley, I believe you're being too modest,\" he protested, smiling. \"You should be proud of your fame.\"\n\nHe _was_ young, not to understand how foolhardy it was to take pride in something so fickle, so fleeting, as fame. \"I was famous, in my line, for many years,\" she acknowledged. \"I was proud; yes, very proud. But fame and pride do not last, as I have found to my sorrow. They don't bring food to an old woman when she is forgotten or when her friends have passed away\u2014\"\n\nAbruptly she fell silent. She had not meant to broach that unhappy subject, nor would she care to return to it, no matter how long Mr. Fry sat there regarding her with plaintive, hopeful, puppy-dog eyes beneath a furrowed brow. \"I understand you have fallen upon hard times,\" he eventually said, all frank sympathy.\n\nSlowly drawing in a deep breath, Elizabeth nodded. She did not want his pity. \"When I am in most distress,\" she said with an effort, keeping her voice calm and even, \"I think of what I often heard Mr. Lincoln say to his wife: 'Don't worry, Mother, because all things will come out right. God rules our destinies.'\"\n\nMr. Fry's pencil hovered above the paper. \"He truly said so? He believed that?\"\n\nElizabeth nodded. \"Many and many a time have I heard him say those words, when disasters surrounded the Union armies, and also when domestic troubles came.\" His words had comforted Mrs. Lincoln, and Elizabeth too, though he had not intended them for her. \"Yes, that good man's memory helps me, for I recollect very many of his sayings, showing faith in God and His goodness.\"\n\n\"It is good to find comfort in memories,\" Mr. Fry said gently.\n\n\"Yes.\" Elizabeth smiled wistfully, her gaze faraway. \"I have always thought so.\"\n\nThey chatted awhile longer, and Elizabeth tried not to lament too much about her poverty, or her loneliness, as she had outlived so many of her friends. When she grew tired, Mr. Fry noticed and promptly stood, shook her hand, thanked her profusely, and showed himself out.\n\nShe wondered, as she watched him leave, what he had seen that she had not intended to reveal to him. She wondered if he would invent dialogue and sighs and long, sad looks to fill in the gaps in the story he had surely already composed in his head before she had spoken a word, gaps she had neglected to fill because she did not have the pieces his story required. Why should one write the story to fit the facts, she thought wryly, when nothing could be easier than to invent one's own facts to suit a more provocative story?\n\nShe sighed, pulled herself slowly to her feet, and made her way alone to her solitary room. She would learn soon enough what Mr. Fry thought of her, and after his story ran in the papers, a brief flash of notoriety might again illuminate her quiet life before it faded into embers. And when it was gone, it would be gone. Such was the nature of fame. Soon she would be forgotten again, except for the few dear friends yet living who mattered to her most.\n\nHer life was smaller than it had been at the height of her success and fame as Mrs. Lincoln's dressmaker, but she was older now, and wiser, and did not require so much as she once had.\n\nShe was a free woman in a nation united and at peace. She had lived a full and fascinating life. She had known the most remarkable people of the age, and she had never refused to help the humble and downtrodden. Despite its disappointments and losses and heartbreaks, she would not have wished her life a single day shorter\u2014nor, when the time came for her to join the many friends and loved ones who had gone on before her, would she demand an hour more.\n\nShe was ready to lay down her burdens and rest, and perhaps to find in heaven the reunion and reconciliation that had eluded her on earth.\nAUTHOR'S NOTE\n\nElizabeth Hobbs Keckley lived out her remaining years in Washington, DC, at the Home for Destitute Colored Women and Children. There, after a brief illness, she died in her sleep of a paralytic stroke on May 26, 1907, at the age of eighty-nine. According to John E. Washington, author of _They Knew Lincoln_ (Dutton, 1942), she was laid to rest in the Columbian Harmony Cemetery \"on a beautiful knoll, facing the east beneath a spreading elm tree\" among \"most of the illustrious colored people who loved and served Lincoln while he was in Washington.\"\n\nThe controversy surrounding Elizabeth Keckley's memoir lingered on after her death. In 1935, a Civil War historian, journalist, and self-proclaimed \"unreconstructed Southerner\" named David Rankin Barbee contended that not only had Elizabeth Keckley not written _Behind the Scenes,_ but no such woman had ever existed. The memoir was a fraud, he insisted, a fiction created by the outspoken \"abolitionist sob sister\" and Washington correspondent Jane Swisshelm. Barbee apparently had not considered how many friends and acquaintances of Elizabeth Keckley were yet living, and one can well imagine his feelings when they promptly came forward to refute his claims and to testify that Elizabeth Keckley had indeed existed. In the face of such adamant opposition, Barbee partially retracted his statements, explaining that what he had meant to say was that \"no such person as Elizabeth Keckley\" could have written _Behind the Scenes_. \"Scholars, with whom I cannot class myself, have long been bothered over the authorship of this book,\" he stated in Washington's _Evening Star_ on November 26, 1935. But despite the attempts of Barbee and others to deny Elizabeth Keckley's authorship and to erase her from memory, her memoir, so denounced in its day, is now widely regarded as a significant historical artifact offering invaluable insight into the Lincoln White House and the private lives of Abraham and Mary Lincoln.\n\nThe provenance of the exceptional quilt attributed to Elizabeth Keckley is even less certain than that of her memoir. At the time of this writing, the earliest documented appearance of the Mary Todd Lincoln quilt was in 1954, when its owner, the quilter and author Ruth Ebright Finley, mentioned it in a lecture. Finley reportedly claimed that Elizabeth Keckley pieced the quilt from scraps of fabric left over from sewing Mary Lincoln's dresses, and that she had presented the quilt as a personal gift to the First Lady, who used it as a counterpane on her bed in the White House and took it with her when she departed after President Lincoln's assassination. When Finley died in 1955, her quilt collection passed to her nephew, Bill Dague, who kept the quilt until 1967, when he and his wife sold it at a yard sale at their family farm in Sharon, Ohio. The quilt was purchased by Ross Trump, an antiques collector and dealer and family friend of the Finleys and Dagues. In 1994, Trump donated the quilt to the Kent State University Museum, where it remains to this day. However, other accounts question whether Elizabeth Keckley ever gave the quilt to Mary Lincoln, or indeed whether she was the quiltmaker who made it. I was unable to establish the quilt's provenance before Ruth Ebright Finley's ownership, and so, in the absence of historical evidence, I created a history for the quilt that best suited my story.\n\nIn the 1950s, after years of neglect, the Columbian Harmony Cemetery was sold to a land developer, and the remains of approximately thirty-seven thousand people\u2014including Elizabeth Keckley\u2014were moved to the new National Harmony Memorial Park Cemetery in Landover, Maryland. Whether due to neglect or mischance, the headstones were not relocated, and thus Elizabeth Keckley's remains were reinterred in an unmarked grave. For decades, historians believed that her gravesite would remain forever unknown, but in 2009, a self-described amateur historian named Richard Smyth was researching historical graves within the cemetery's ledgers and records when he discovered the plot and section where Elizabeth Keckley had been buried. For two years, Smyth worked with National Harmony Memorial Park and several other organizations, including the Surratt Society, Black Women United for Action, the Lincoln Forum, and the Ford's Theatre Society, to raise funds to purchase a memorial honoring Elizabeth Keckley's role in presidential history. On May 26, 2010, the 103rd anniversary of her death, the new bronze and granite grave marker was erected. The inscription reads:\n\n**E LIZABETH KECKLEY**\n\n1818\u20131907\n\nENSLAVED MODISTE CONFIDANTE\n\nBORN INTO SLAVERY, ELIZABETH KECKLEY PURCHASED HER FREEDOM USING HER EXCEPTIONAL SKILLS AS A SEAMSTRESS. AFTER ESTABLISHING HER OWN BUSINESS, SHE WAS EMPLOYED AS A MODISTE (DRESSMAKER) BY MARY LINCOLN, BECOMING HER TRUSTED FRIEND AND CONFIDANTE. MRS. KECKLEY'S AUTOBIOGRAPHY \"BEHIND THE SCENES\" PROVIDES INTIMATE DETAILS ABOUT LIFE INSIDE THE LINCOLN WHITE HOUSE.\nACKNOWLEDGMENTS\n\nI offer my heartfelt thanks to Denise Roy, Maria Massie, Liza Cassity, Christine Ball, Brian Tart, Kate Napolitano, and the outstanding sales teams at Dutton and Plume for their support of my work and their contributions to _Mrs. Lincoln's Dressmaker_.\n\nI owe a debt of gratitude to the people who graciously assisted me during the research and writing of this novel. Geraldine Neidenbach, Heather Neidenbach, Marty Chiaverini, and Brian Grover were my first readers, and their comments and questions proved invaluable throughout the writing of this book. Nic Neidenbach willingly responded to my frantic questions when technology failed me, proving once again how wonderful it is to have a computer guru in the family. I always appreciate the support and encouragement of Marlene and Len Chiaverini, and Marlene's enthusiasm for this story in particular was especially heartening. Karen Roy and Alyssa Samways went on library research expeditions and sent me copies of crucial historical documents that I could not have obtained on my own; and Sara Hume of the Kent State University Museum not only provided me with a personal tour and viewing of the Mary Todd Lincoln quilt but also offered intriguing and informative replies to the many questions that occurred to me after my visit. Many thanks to you all.\n\nI am fortunate indeed to live so close to the Wisconsin Historical Society, whose librarians, staff, and excellent archives I have come to rely upon in my work. Of the many resources I consulted, the following proved especially valuable and instructive: Jean H. Baker, _Mary Todd Lincoln: A Biography_ (New York: Norton, 1987); Joan E. Cashin, _First Lady of the Confederacy:Varina Davis's Civil War_ (Cambridge, MA: Belknap Press of Harvard University Press, 2006); Catherine Clinton, _Mrs. Lincoln: A Life_ (New York: HarperCollins, 2009); Daniel Mark Epstein, _The Lincolns: Portrait of a Marriage_ (New York: Ballantine Books, 2008); Jennifer Fleischner, _Mrs. Lincoln and Mrs. Keckly: The Remarkable Story of the Friendship Between a First Lady and a Former Slave_ (New York: Broadway Books, 2003); Ernest B. Furgurson, _Freedom Rising: Washington in the Civil War_ (New York: Knopf, 2004); Becky Rutberg, _Mary Lincoln's Dressmaker: Elizabeth Keckley's Remarkable Rise from Slave to White House Confidante_ (New York: Walker and Company, 1995); Justin G. Turner and Linda Levitt Turner, _Mary Todd Lincoln: Her Life and Letters_ (New York: Knopf, 1972); and John E. Washington, _They Knew Lincoln_ (New York: Dutton, 1942).\n\nOf course, no work was more important than Elizabeth Keckley's own memoir, _Behind the Scenes_ (New York, G. W. Carleton & Company, 1868). Though I regret the unhappiness its publication brought her, I am deeply grateful that Elizabeth Keckley left behind such a rich and evocative account of her life.\n\nAs always and most of all, I thank my husband, Marty, and my sons, Nicholas and Michael, for their continuous love, support, and encouragement. I tell you every day how much I love you, but there are times when words cannot suffice, and this is one of them.\nABOUT THE AUTHOR\n\nJENNIFER CHIAVERINI is the author of the _New York Times_ bestselling Elm Creek Quilts series, as well as five collections of quilt projects inspired by the novels. _Mrs. Lincoln's Dressmaker_ is her twenty-first novel and first stand-alone work of historical fiction. A graduate of the University of Notre Dame and the University of Chicago, she lives with her husband and sons in Madison, Wisconsin.\n\n## Prologue\n\n* * *\n\nMARCH 28, 1861\n\nO  \nn the occasion of President Lincoln's first state dinner, carriages, carts, and hundreds of men on foot crowded the circular drive in front of the White House nearly all the way to Lafayette Square. Kate Chase studied the scene through the window of her father's carriage, forgetting, for the moment, her misgivings that she was attending the event as a guest rather than the hostess. The crush of people forced their horses to slow to a walk long before they reached the bronze statue of Thomas Jefferson in the center of the driveway, and the tall white columns of the front portico suddenly seemed to be an interminable distance away.\n\n\"Father,\" Kate said, touching his hand where it rested on the black leather seat between them. \"Didn't you say that the president and his wife had invited only cabinet members, a few dignitaries, and their wives to tonight's dinner?\"\n\n\"Wives or daughters, as the case may be.\"\n\n\"Or daughters,\" amended Kate, smiling. \"I ask because if the size of this crowd is any indication, it would seem that either you were misinformed, or the vast population of Washington City was.\"\n\n\"I trust I didn't misunderstand the president's invitation,\" her father replied. \"No, my dear Katie, what you see before you is the capital's most persistent plague\u2014patronage seekers. They know they won't be allowed past the doorman tonight, but that won't stop them from clutching at the sleeves and pleading in the ears of any unfortunate official whom they can accost on the way to the door.\"\n\nKate lowered her voice to a conspiratorial whisper. \"Then we shall have to be quick and clever to avoid them.\"\n\n\"Yes, and we'll keep the distance from carriage to threshold as short as possible.\" Salmon P. Chase frowned out the window. Impeded by the throng, the carriage had slowed to a crawl until it finally halted several yards away from the portico. \"Although that might prove difficult.\"\n\n\"I'm sorry, Mr. Secretary,\" the driver called down. \"I can't get any closer until those other drivers clear away.\"\n\nKate had no desire to spend any more of that lovely spring evening gazing longingly at the brightly lit White House through a carriage window and imagining all she was missing within. \"Let's brave the gauntlet,\" she proposed. \"The president and his wife are expecting us.\"\n\nHer father nodded, more impatient than she to join the gathering inside. He deplored tardiness and did not tolerate it in himself or anyone else. \"This will do, George,\" he called to the driver. \"We'll walk from here.\"\n\nHer father helped her down from the carriage, and as she took his arm, she felt a glow of warmth and happiness fill her heart despite the small, sad ache of disappointment that had nagged her ever since the Republican Party had selected Mr. Lincoln rather than her father as its candidate. If not for the perfidy of the delegates from his own state of Ohio, her father surely would have been sworn in as president earlier that month rather than as secretary of the treasury, and Kate, as the widower's eldest daughter, would have become First Lady.\n\nSalmon P. Chase was, Kate knew to the very core of her being, the better man\u2014better educated than Mr. Lincoln, more experienced, more committed to the noble cause of abolition, and vastly more qualified. Father had been a senator and governor, while Mr. Lincoln\u2014well, Kate liked him, but if she set sentiment aside and forced herself to be strictly objective, she had no choice but to admit that a kindly country lawyer from the West with but one term in Congress to his credit was ill prepared to steer the ship of state, especially through the rough waters the nation faced. Any observer could see that, and in fact, many had, and had said so in the streets and in the press. No one expected much of the new president, and they expected even less of his overeager, overanxious wife, a matron in her midforties who had thus far failed to make a favorable impression on Washington's social elite.\n\nFortunately, Mr. Lincoln would have Salmon P. Chase to advise him. As for Mrs. Lincoln, for the good of the nation and the Republican Party, some kind lady ought to befriend her, become her confidante, and help her navigate the thorny maze of Washington society.\n\nKate resigned herself to the likelihood that no one was better suited for the role than herself, though she was more than twenty years younger than Mrs. Lincoln.\n\n\"Mr. Secretary,\" a voice rang out. Glancing to her left, Kate glimpsed a freckled young man in a brown suit two sizes too large for his bony frame, waving a handful of papers and grinning hopefully. \"Mr. Secretary, a moment of your time, if you please.\"\n\n\"He's no one,\" Kate murmured, not unkindly. Her father was terribly nearsighted, but he hated to wear his spectacles in public and often relied upon Kate to identify people at a distance. He had been known to pass good friends and acquaintances on the sidewalks or halls of Congress without recognizing them, an unfortunate habit that contributed to his reputation as being aloof and uncongenial.\n\nAs other eager, avaricious faces turned their way, Father offered Kate his arm. \"Let's make haste.\"\n\nQuickly Kate slipped her hand into the crook of his elbow and hurried off beside him. \"Perhaps you should throw a few minor treasury appointments after us to distract them,\" she teased, breathless. \"Rather like Aphrodite's golden apples, but in reverse.\"\n\n\"I'll have no Hippomenes catch my Atalanta,\" her father declared, quickening his pace as he guided her through the crowd. Laughing, Kate did her best to keep up with him, though she was shorter than her father by nearly half a foot and encumbered by her corset and hoopskirt.\n\nAt last they reached the portico, where the burly, white-haired doorman greeted them in an Irish brogue and admitted them into the vestibule. They passed through the main hall into the Blue Room, a graceful ellipse with tall windows overlooking the south lawn and the Potomac River. It seemed to Kate to be in better repair than the other public rooms of the Executive Mansion, which were shabbily furnished with threadbare and tobacco-stained rugs, broken furniture, torn wallpaper, and ruined draperies, from which souvenir collectors had snipped pieces until they hung in tatters. Here, however, all was in elegant order. The chairs and settees were upholstered in rich blue and silver damask, the woodwork brilliantly gilded. Ornate mirrors on the marble mantel reflected the light from the chandeliers hanging high above from a frescoed ceiling of cerulean blue, beneath which men of influence clad in evening black and their ladies in elegant gowns of every hue mingled and chatted, the soft blue-and-white carpet muffling their footsteps.\n\nBefore passing from the brightly lit hall into the Blue Room, the Chases paused in the doorway, long enough for almost thirty pairs of eyes to turn their way. Conversations paused as the guests took in the newcomers, and as Kate smiled warmly and nodded gracefully to one acquaintance after another, she drew herself up proudly, knowing how she and her father looked to them. Salmon P. Chase\u2014tall, broad-shouldered, and powerfully built, his features strong and regular beneath a high, clear brow projecting intelligence, courage, and dignity\u2014was the very image of a statesman. And Kate herself\u2014auburn-haired, hazel-eyed, young, slender, vivacious, becomingly attired in a gown of pale-yellow silk, her hair arranged in a simple, elegant twist and adorned with white flowers\u2014was the very ideal of the accomplished, dutiful daughter. \"You look like the king and queen of Washington,\" her younger sister, Nettie, had sighed wistfully upon their departure from home. Father had taken a moment to lecture his youngest child on the superiority of American democracy to European monarchy, which she accepted with a good-natured shrug, with none of the shame and remorse Father's admonishments evoked in Kate.\n\nAs John Nicolay, the president's private secretary, made the customary introductions, Father escorted Kate across the room, where Mr. and Mrs. Lincoln waited to receive their guests. The president's gaunt features became warmer, his eyes brighter, when he smiled, as he did when he shook Father's hand and then Kate's. \"How is little Nettie?\" Mr. Lincoln asked, his thick, dark brows rising quizzically, an amusing complement to the high, thin quality of his voice.\n\n\"Very well, thank you,\" Father replied formally, bowing to Mrs. Lincoln.\n\nSeeing her father and the president together, Kate almost laughed aloud, taken anew by the remarkable distinction between them. If Father looked every inch a statesman, Mr. Lincoln resembled a frontier schoolmaster with no wife to remind him to brush his hair and straighten his coat. \"Nettie is indeed well\"\u2014she amended her father's reply\u2014\"but I confess she was disappointed to be left at home.\"\n\n\"Why didn't you bring her along?\" Mr. Lincoln asked, genuinely perplexed.\n\n\"Oh, Mr. Lincoln,\" his wife chided him, laughing shortly. Her eyes were sharply blue, her complexion white and smooth, her neck and arms elegantly molded, but she was otherwise plain and tended toward stoutness, which her short stature and her husband's great height unfortunately exaggerated. \"Children, at a state dinner for the cabinet? Not even Tad and Willie will be in attendance tonight.\"\n\nMr. Lincoln smiled benignly down upon his wife. \"I'm fortunate to have you here to remind me of such things.\" He turned back to the Chases with a self-deprecating shrug. \"When I consider the reams of paper piled upon my desk, which are almost certainly accumulating greater heights even as we speak, I can only hope that an instructive book on presidential manners lies at the bottom.\"\n\n\"At the _very_ bottom, do you mean?\" teased Kate, detecting an ironic note in his tone. \"So deeply buried that no one can expect you to unearth and to read it?\"\n\nMr. Lincoln's laugh rang out, rich and full. \"Why, yes, Miss Chase. You understand me perfectly.\"\n\nFather beamed proudly, but as she shook Mrs. Lincoln's hand, Kate detected a flicker of annoyance in the First Lady's eyes above her gracious smile, and she was not sorry to move on. There were wives and daughters to befriend, gentlemen to charm, and people of influence to impress, but as she and her father made their way around the room, Kate often thought she still felt Mrs. Lincoln's blue eyes upon her, taking her measure with scrupulous, unforgiving precision.\n\nKate soon forgot Mrs. Lincoln's displeasure, swept up in the pleasure of making new acquaintances and engaging in lively conversation that occasionally, and rather delightfully, leaned toward debate. United States attorney general Edward Bates, black-haired and white-bearded, spoke longingly and endearingly of the Missouri home he had only recently departed, while General Simon Cameron, with his keen, deep-set eyes and thin mouth, struck her as insistent and shrewd. One by one Kate addressed them all, and the ladies who had accompanied them, complimenting their attire and inquiring about their children. Kate especially liked the wife of the secretary of state, Frances Seward, whose dark, intelligent eyes belied the frailty of her form. Unfortunately, Mrs. Seward spent little time in Washington City, preferring her gracious family home and more temperate climate of Auburn, New York\u2014but her frequent absence, and that of the vice-president's wife, did leave Kate the second-highest-ranking woman of the executive branch according to protocol and tradition, behind Mrs. Lincoln, who never let her forget it.\n\nA moment came when Kate found herself unaccompanied, but just as it occurred to her that they should have been called in to dinner by then, Mrs. Lincoln appeared at her side. \"Dinner will be delayed somewhat longer,\" the First Lady explained, drawing her apart from the others. \"General Scott has not yet arrived.\"\n\n\"Oh, of course we must wait for him,\" Kate replied. \"I hope he wasn't swallowed up in the crush outside.\"\n\n\"I'm sure he wasn't. A crowd of patronage seekers is no match for a gentleman who commands entire armies.\"\n\nKate smiled. \"I'm sure you're right.\"\n\n\"Be careful what you say in this city, even in jest,\" Mrs. Lincoln cautioned, linking her arm through Kate's and strolling away from the other guests so that Kate was compelled to come along. \"Rumors fly so swiftly in Washington that if I were to cross this room right now, I would not be surprised if the people on the other side greeted me with the dreadful news that General Scott met his demise at the hands of a mob on our front doorstep.\"\n\nKate laughed easily. \"Thank you for the warning. I'll take heed, I assure you.\"\n\n\"Speaking of rumors.\" Mrs. Lincoln halted, slipped her arm free from Kate's, and fixed her with an inscrutable look. \"You do know the rumors circulating about us, don't you?\"\n\n\"Why, no,\" said Kate. \"Why should there be any rumors about you and me?\"\n\n\"Because people enjoy gossip even more than they relish believing that accomplished women cannot get along.\" Mrs. Lincoln smiled, but she glanced past Kate's shoulder as if wary that they would be overheard. \"They say that you and I are embroiled in a terrible feud.\"\n\nKate was so astonished she laughed. \"And what reason do they give for it?\"\n\n\"My overwhelming jealousy at the attentions my husband showed you at the military ball given in his honor when we stopped in Columbus in February on our way to Washington.\" Mrs. Lincoln's smile tightened as she nodded to a cabinet official passing nearby. \"They say he danced with you more often than was seemly and showered you in compliments, while I looked on, weeping by some accounts and seething according to others.\"\n\n\"That's nonsense,\" Kate exclaimed, quickly lowering her voice. \"I wasn't even in Columbus when your train passed through.\"\n\n\"Why, no, you weren't, were you?\" mused Mrs. Lincoln as if she had only just remembered\u2014although an edge to her voice immediately told Kate that the perceived slight had been in the forefront of her thoughts all evening. \"You knew the president-elect and his wife and sons would be passing through your own city, and yet some urgent business of far greater importance compelled you away.\"\n\n\"Not of greater importance, but essential nonetheless.\" Kate drew herself up to her full height and regarded the First Lady steadily. \"Governor Dennison had appointed my father a state delegate to the Peace Convention. My family had already come to Washington City by the time you reached Columbus, or I would have been there to meet you. I trust Governor Dennison's wife welcomed you as well as I would have done.\"\n\nMrs. Lincoln's smile deepened and hardened. \"Or better, perhaps. Almost certainly better.\"\n\nWith that, she moved off in a swirl of silk skirts, leaving Kate watching after her, utterly astonished.\n\nSoon thereafter, word came that General Scott was ill and would not be able to join them after all. Mr. Nicolay signaled for the Marine Band to strike up a spirited march, and as the brisk, merry tune played, Kate quickly composed herself, found her father, and let him lead her into the state dining room. Mr. Lincoln's place was in the middle of one side of the long table rather than at the head, while Mrs. Lincoln sat opposite him, and the others were seated all around according to rank. Kate found herself across the table from a Mr. William Howard Russell, a correspondent for the _London Times_ , who seemed as charmed by Kate's conversation as she was by his wit and accent.\n\nThe food was excellent, the talk around the table bright and lively and quick, but whenever Mr. Lincoln spoke, all other voices hushed and all faces turned to him expectantly as he spun an amusing tale or made a point with a clever witticism. From the first course to the sweets, Mrs. Lincoln was so merry and chatty and smiling that Kate found herself wondering if she had imagined the strange confrontation in the Blue Room.\n\nAfter the meal, the gentlemen withdrew to the Red Room, but just before the ladies were led off to an adjoining drawing room, Kate observed Mr. Nicolay whisper in her father's ear, and then in Mr. Seward's, and on to each member of the cabinet in turn. Just before a servant closed the door between the rooms, Kate glimpsed Mr. Lincoln and his cabinet quietly disappearing into another chamber while the other gentlemen lit cigars and poured brandy, apparently oblivious to their quiet departure.\n\nKate frowned at the closed door. Her curiosity would have to remain unsatisfied until the drive home, when her father would surely tell her everything.\n\nShe chatted easily with the other ladies, even Mrs. Lincoln, but her thoughts were with her father, wondering what intriguing subjects of national importance the men were discussing. Had the president finally decided whether to send provisions to Major Anderson's men holding Fort Sumter in Charleston Harbor? Had Virginia at last declared its intention to remain in the Union or to secede with the South? How she wished she could put an ear to the wall and listen. Her father was open and frank when he confided to her the substance of such clandestine conferences, but sometimes he missed the subtleties of tone and implication and expression, and thereby a significant amount of any conversation. He relied upon her for those observations, and for a great deal more besides.\n\nShe managed to stifle a sigh of relief when at last the gentlemen rejoined the ladies. She sensed a new tension in the air as she studied the cabinet secretaries and tried to read their expressions, but like her father, they were careful to maintain a facade of the former joviality of the party. She noticed that Mrs. Lincoln's keen gaze was often upon the president's face, and she knew that Mrs. Lincoln was as eager to hear his account of the secret meeting as Kate was to hear her father's.\n\nWhen the evening at last drew to a close, Kate slipped on her shawl, took her father's arm, and bade Mr. and Mrs. Lincoln good night. \"I do hope we will prove all the rumors false and become good friends,\" Kate told the First Lady quietly while her father and the president were otherwise distracted. \"There is already too much division in our country for us to contribute to it. If malicious gossips are eagerly anticipating a fight between us, let's conspire to disappoint them.\"\n\n\"Why, I hope we will become good friends too,\" Mrs. Lincoln replied grandly, too loudly and too brightly, not for Kate but for everyone else. For Kate she reserved a private, haughty glare that announced she meant not a word of it. \"I shall be glad to see you at any time, Miss Chase.\"\n\nKate's temper flared. She had spoken with utter frankness and sincerity, but Mrs. Lincoln was determined to be disagreeable. \"Mrs. Lincoln,\" she said, smiling graciously, \"I shall be glad to have _you_ call on _me_ at any time.\"\n\nMrs. Lincoln's eyes widened with shock at her impudence, but Kate's smile only deepened as she turned and left the White House on her father's arm.\n\nThose who had overheard the exchange might conclude that Kate had innocently misspoken, that because of her youth and inexperience she was unaware of the custom that decreed that the First Lady did not call on others. Mrs. Lincoln was first in Washington society by virtue of her husband's exalted position, and so, as an inviolable rule, others came to her. But it had been no girlish mistake. Kate understood precedent perfectly well, and she knew that by assuming that Mrs. Lincoln would call upon her like any other lady of Washington society might, she was claiming a higher rank than the First Lady.\n\nKate knew it, and Mrs. Lincoln knew it too.\n\nKate had tried to befriend her, but she had been coldly and unreasonably rebuffed. She would not try again.\n\nIf Mrs. Lincoln was determined to have a rival, Kate would be happy to oblige.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Robert Cicconetti, Fred Salzer and the Online\nDistributed Proofreading Team at http:\/\/www.pgdp.net (This\nfile was produced from images generously made available\nby The Internet Archive\/American Libraries.)\n\n\n\n\n\n\n\n\n\n  THE SEARCH PARTY\n\n  BY\n\n  G. A. BIRMINGHAM\n\n  AUTHOR OF \"SPANISH GOLD,\" \"LALAGE'S LOVERS,\"\n  \"THE SIMPKINS PLOT,\" ETC.\n\n  [Illustration]\n\n  HODDER & STOUGHTON\n\n  NEW YORK\n\n  GEORGE H. DORAN COMPANY\n\n\n\n\nTHE SEARCH PARTY\n\n\n\n\nTHE SEARCH PARTY\n\n\n\n\nCHAPTER I\n\n\nDr. O'Grady, Dr. Lucius O'Grady, was the medical officer of the Poor\nLaw Union of Clonmore, which is in Western Connacht. The office is\nnot like that of resident magistrate or bank manager. It does not\nnecessarily confer on its holder the right of entry to the highest\nsociety. Therefore, Dr. O'Grady was not invited to dinner, luncheon,\nor even afternoon tea by Lord Manton at that season of the year when\nClonmore Castle was full of visitors. Lady Flavia Canning, Lord\nManton's daughter, who was married to a London barrister of some\ndistinction, and moved in smart society, did not appreciate Dr.\nO'Grady. Nor did those nephews and nieces of the deceased Lady Manton\nwho found it convenient to spend a part of each summer at Clonmore\nCastle. They were not the sort of people who would associate with a\ndispensary doctor, unless, indeed, he had possessed a motor car. And\nDr. O'Grady, for reasons which became obvious later on, did not keep a\nmotor car.\n\nOn the other hand, he was a frequent guest at the Castle during those\nearly summer months when Lord Manton was alone. In April and May, for\ninstance, and in June, Dr. O'Grady dined once, twice, or even three\ntimes a week at Clonmore Castle. The old earl liked him because he\nfound him amusing; and Dr. O'Grady had a feeling for his host as\nnearly approaching respect as it was in his nature to entertain for\nany man. This respect was not of the kind which every elderly earl\nwould have appreciated. The doctor was constitutionally incapable of\nunderstanding the innate majesty of a peerage, and had not the smallest\nveneration for grey hairs in man or woman. Nor was he inclined to bow\nbefore any moral superiority in Lord Manton. In fact, Lord Manton,\nthough grown too old for the lavish wildness of his earlier years,\nmade no pretence at morality or dignity of any kind. What Dr. O'Grady\nrespected and liked in him was a certain cynical frankness, a hinted\ncontempt for all ordinary standards of respectability. This suited\nwell enough the doctor's own volatile indifference to anything which\nthreatened to bore him.\n\nWhen Lord Manton returned to Clonmore in May, 1905, after his usual\nvisit to his daughter in Grosvenor Street, he at once asked Dr.\nO'Grady to dinner. There was on this occasion a special reason for the\ninvitation, though doubtless it would have been given and accepted\nwithout any reason. Lord Manton wanted to know all that could be known\nabout a new tenant who had taken Rosivera for six months. Rosivera,\nlong used as a dower house by Lord Manton's ancestors, was not an easy\nplace to let. It stood eight miles from the village of Clonmore, on\nthe shore of a small land-locked bay. It was a singularly unattractive\nbuilding, rectangular, grey, four storeys high, and lit by small\nineffective windows. There was no shooting connected with it nor any\nfishing of the kind appreciated by a sportsman. There were, it was\nbelieved, small flat fish to be caught in the bay, but no one thought\nit worth while to pursue these creatures earnestly. Occasionally an\nadventurous Englishman, cherishing some romantic idea of the west of\nIreland, rented the house for August and September. Occasionally a\nwealthy Dublin doctor brought his family there for six weeks. None of\nthese tenants ever came a second time. The place was too solitary for\nthe social, too ugly for the amateurs of the picturesque, utterly dull\nfor the sportsman, and had not even the saving grace of an appeal to\nthe romantic. The mother and grandmother of Lord Manton had died there,\nbut in the odour of moderate sanctity. Their ghosts wandered down no\ncorridors. Indeed, no ghosts could have haunted, no tradition attached\nitself to a house with the shape and appearance of Rosivera.\n\nThere was, therefore, something interesting and curious in the fact\nthat a tenant had taken the place for six months and had settled down\nthere early in March, a time of year at which even a hermit, vowed to a\nlife entirely devoid of incident, might have hesitated to fix his cell\nat Rosivera.\n\n\"The first thing that struck me as queer about the man,\" said Lord\nManton, after dinner, \"was his name. Did you ever hear of anybody\ncalled Red? Scarlett, of course, is comparatively common.\"\n\n\"So is Black,\" said Dr. O'Grady, \"and Brown, and Grey, and White. I've\nheard of Pink, and I once met a man called Blue, but he spelt it 'ew.'\"\n\n\"Guy Theodore Red is this man's name. Guy and Theodore are all right,\nof course, but Red----!\"\n\n\"Is he safe for the rent, do you think?\"\n\n\"He has paid the whole six months in advance,\" said Lord Manton, \"and\nhe never asked a question about the drains. He's the only tenant I ever\nheard of who didn't make himself ridiculous about drains.\"\n\n\"He hasn't got typhoid yet,\" said Dr. O'Grady. \"If he's the kind of man\nwho pays six months' rent in advance and asks no questions, I hope he\nsoon will.\"\n\n\"Unfortunately for you he seems to have neither wife nor children.\"\n\n\"No, nor as much as a maid-servant,\" said Dr. O'Grady. \"And from the\nlook of him, I'd say he was a tough old cock himself, the sort of man a\nmicrobe would hesitate about attacking.\"\n\n\"You've seen him, then?\"\n\n\"I happened to be standing at Jimmy O'Loughlin's door the day he drove\nthrough in his motor car.\"\n\n\"You would be, of course.\"\n\n\"But I've never seen him since. Nobody has. He has a servant, an\nEnglishman, I'm told, who comes into the village every second day in\nthe motor, and buys what's wanted for the house at Jimmy O'Loughlin's.\"\n\n\"Jimmy makes a good thing out of that, I expect,\" said Lord Manton.\n\n\"Believe you me, he does. Jimmy's the boy who knows how to charge, and\nthese people don't seem to care what they pay.\"\n\n\"I hear he has two friends with him.\"\n\n\"He has, foreigners, both of them. Jimmy O'Loughlin says they can't\neither of them speak English. It was Jimmy who carted their things down\nto Rosivera from the station, so of course he'd know.\"\n\n\"Byrne told me that,\" said Lord Manton, chuckling as he spoke. \"There\nseems to have been some queer things to be carted.\"\n\nThe conversation turned on Mr. Red's belongings, the personal luggage\nwhich the English servant had brought in the train, the packing-cases\nwhich had followed the next day and on many subsequent days. Byrne,\nit appeared, had also met Mr. Red and his party on their arrival; but,\nthen, Byrne had a legitimate excuse wherewith to cover his curiosity.\nHe was Lord Manton's steward, and it was his business to put the new\ntenant in possession of Rosivera. He had given a full report of Mr.\nRed, the foreign friends and the English servant, to Lord Manton. He\nhad described the packing-cases which, day after day, were carted from\nthe railway station by Jimmy O'Loughlin. They were, according to Byrne,\nof unusual size and great weight. There were altogether twenty-five\nof them. It was Byrne's opinion that they contained pianos. The\nstation-master, who had to drag them out of the train, agreed with him.\nJimmy O'Loughlin and his man, who had ample opportunities of examining\nthem on the way to Rosivera, thought they were full of machinery,\npossibly steam engines, or as they expressed it, \"the makings of some\nof them motor cars.\"\n\n\"No man,\" said Lord Manton, commenting on this information, \"even if\nhis name happens to be Red, can possibly want twenty-five grand pianos\nin Rosivera.\"\n\n\"Unless he came down here with the intention of composing an opera,\"\nsaid Dr. O'Grady.\n\n\"Even then--three, four, anything up to six I could understand, but\ntwenty-five! No opera could require that. As for those cases containing\nsteam engines or bits of motor cars, what on earth could a manufacturer\nof such things be doing at Rosivera?\"\n\n\"My own belief,\" said Dr. O'Grady, \"is that the man is an artist--a\nsculptor, engaged in the production of a statue of unusual size.\"\n\n\"With blocks of marble in the packing-cases?\"\n\n\"Yes, and the two foreigners for models. They look like models. One\nof them had a long black beard, and the other was a big man, well\nover six foot, blond, seemed to be a Norwegian; not that I ever saw a\nNorwegian to my knowledge, but this fellow looked like the kind of man\na Norwegian ought to be.\"\n\n\"It will be a pretty big statue,\" said Lord Manton, \"if it absorbs\ntwenty-five blocks of marble, each the size of a grand piano.\"\n\n\"He looked like an artist,\" said Dr. O'Grady; \"he had a pointed beard,\nand a wild expression in his eye.\"\n\n\"A genius escaped from somewhere, perhaps.\"\n\n\"He very well might be. Indeed, I'd say from the glimpse I had of him\nthat he's worse than a genius. He had the eye of a mad gander. But,\nof course, I only saw him the once, sitting in his motor, the day\nhe arrived. He hasn't stirred out of Rosivera since, and, as I said\nbefore, I haven't been sent for to attend him for anything.\"\n\n\"The queerest thing about him was the message he sent me,\" said Lord\nManton. \"By way of doing the civil thing, I told Byrne to say that I\nshould make a point of calling on him as soon as I got home.\"\n\n\"And he sent you word that he'd be thankful if you'd stay away and not\nbother him. I heard all about that. Byrne was furious. That is just one\nof the things which makes me feel sure he's a genius. Nobody except a\ngenius or a socialist would have sent a message of that kind to you;\nand he clearly isn't a socialist. If he was, he couldn't afford to pay\nsix months' rent in advance for Rosivera.\"\n\nDr. O'Grady spoke confidently. He was not personally acquainted with\nany of the numerous men of genius in Ireland, but he had read about\nthem in newspapers and was aware that they differed in many respects\nfrom other men. No ordinary man, that is to say, no one who is\nperfectly sane, would refuse to receive a visit from an earl. Mr. Red\nhad refused, and so, since he was not a socialist, he must be a genius.\nThe reasoning was perfectly convincing.\n\n\"I expect,\" said Lord Manton, \"that his statue, in spite of its\nimmense size, will be a melancholy object to look at. Rosivera is the\nmost depressing place I know. It was built to serve as a dower house\nby my grandfather, and he evidently chose the site and the style of\narchitecture with a view to making his widow feel really sorry he was\ndead. If I had a wife whom I disliked intensely I should try to die\nat once so that she should have as long a time as possible to live at\nRosivera.\"\n\n\"I wouldn't care to spend a winter alone there,\" said Dr. O'Grady, \"and\nI'm a man of fairly cheerful disposition.\"\n\n\"I suppose there's a lot of talk about Red in the village?\"\n\n\"There was at first; but the people are getting a bit sick of him now.\nIt's a long time since he's done anything the least exciting. About a\nfortnight after he came he sent a telegram which had the whole place\nfizzing for awhile.\"\n\nTelegrams in the west of Ireland, are, of course, public property. So\nare postcards and the contents of the parcels carried by his Majesty's\nmails. Lord Manton, whose taste for the details of local gossip was\nstrongly developed, asked what Mr. Red's telegram was about.\n\n\"That's what nobody could tell,\" said Dr. O'Grady. \"It began with\nfour letters, A.M.B.A., and then came a lot of figures. Father Moroney\nworked at it for the best part of two hours, with the help of a Latin\ndictionary, but he could make no more out of it than I could myself.\"\n\n\"Cipher,\" said Lord Manton; \"probably quite a simple cipher if you'd\nknown how to go about reading it.\"\n\n\"At the end of the week, another packing-case arrived, carriage paid\nfrom London. It was as big as any of the first lot. Byrne and I went\nup to the station to see it before Jimmy O'Loughlin carted it down to\nRosivera. He seemed to think that it was another piano. Since then\nnothing of any sort has happened, and the people have pretty well given\nover talking about the man.\"\n\nLord Manton yawned. Like the other inhabitants of Clonmore he was\nbeginning to get tired of Mr. Red and his affairs. A stranger is only\ninteresting when there are things about him which can be found out. If\nhis affairs are public property he becomes commonplace and dull. If, on\nthe other hand, it is manifestly impossible to discover anything about\nhim, if he sends his telegrams in cipher, employs a remarkably taciturn\nservant to do his marketing, and never appears in public himself, he\nbecomes in time quite as tiresome as the man who has no secrets at all.\n\n\"Any other news about the place?\" asked Lord Manton. \"You needn't\nmention Jimmy O'Loughlin's wife's baby. Byrne told me about it.\"\n\n\"It's the tenth,\" said Dr. O'Grady, \"the tenth boy.\"\n\n\"So I believe.\"\n\n\"Well, there's nothing else, except the election of the inspector of\nsheep dipping. I needn't tell you that there's been plenty of talk\nabout that.\"\n\n\"So I gathered,\" said Lord Manton, \"from the number of candidates for\nthe post who wrote to me asking me to back them up. I think there were\neleven of them.\"\n\n\"I hear that you supported Patsy Devlin, the smith. He's a drunken\nblackguard.\"\n\n\"That's why I wrote him the letter of recommendation. There's a lot of\nstupid talk nowadays about the landlords having lost all their power\nin the country. It's not a bit true. They have plenty of power, more\nthan they ever had, if they only knew how to use it. All I have to do\nif I want a particular man not to be appointed to anything is to write\na strong letter in his favour to the Board of Guardians or the County\nCouncil, or whatever body is doing the particular job that happens to\nbe on hand at the time. The League comes down on my man at once and he\nhasn't the ghost of a chance. That's the beauty of being thoroughly\nunpopular. Three years ago you were made dispensary doctor here chiefly\nbecause I used all my influence on behalf of the other two candidates.\nThey were both men with bad records. It was just the same in this\nsheep-dipping business. I didn't care who was appointed so long as it\nwasn't Patsy Devlin. I managed the labourers' cottages on the same\nprinciple. There were two different pieces of land where I particularly\nobjected to their building cottages. I offered them those two without\nwaiting to be asked. Of course, they wouldn't have them, insisted in\nfact on getting another bit of land altogether, thinking they were\nannoying me. I was delighted. That's the way to manage things nowadays.\"\n\n\"Do you suppose,\" said Dr. O'Grady, \"that if I wrote to Mr. Red saying\nI sincerely hoped he wouldn't get typhoid for a fortnight, because I\nwanted to go away for a holiday--do you suppose he'd get it to spite\nme?\"\n\n\"That's the worst of men in your profession. You're always wanting\neverybody to be ill. It's most unchristian.\"\n\n\"I want Red to get typhoid,\" said Dr. O'Grady, \"because he's the only\nman in the neighbourhood except yourself who would pay me for curing\nhim.\"\n\n\n\n\nCHAPTER II\n\n\nDr. O'Grady spoke the simple truth when he said that the people of\nClonmore had ceased to take any interest in Mr. Red and his household.\nThe election of an inspector of sheep dipping, a man from their own\nmidst to a post with a salary attached to it, was a far more exciting\nthing than the eccentricity of a chance stranger. When the election was\nover a new and more thrilling matter engaged their attention. Mr. Red\nwas entirely forgotten. The monotonous regularity of the visits of the\nsilent English servant to Jimmy O'Loughlin's shop no longer attracted\nattention. The equally monotonous regularity of his cash payments for\nthe goods he took away with him was extremely satisfactory to Jimmy\nO'Loughlin, but gave absolutely no occasion for gossip. The man who\nmakes debts and does not pay them is vastly more interesting to his\nneighbours than the morbidly honest individual who will not owe a penny.\n\nDr. O'Grady owed a good deal, and just at the time of Lord Manton's\nreturn to Clonmore, his money difficulties reached the point at which\nthey began to attract public attention. Like most good-humoured and\neasy-going men, Dr. O'Grady lived beyond his income. There was a good\ndeal of excuse for him. He enjoyed, as dispensary doctor, a salary of\nL120 a year. He received from Lord Manton an additional L30 for looking\nafter the health of the gardeners, grooms, indoor servants and others\nemployed about Clonmore Castle. He would have been paid extra guineas\nfor attending Lord Manton himself if the old gentleman had ever been\nill. He could count with tolerable certainty on two pounds a year for\nushering into the world young O'Loughlins. Nobody else in his district\never paid him anything.\n\nIt is unquestionably possible to live on L152 a year. Many men,\ncurates for instance, live on less; face the world in tolerably\nclean collars and succeed in looking as if they generally had enough\nto eat. But Dr. O'Grady was not the kind of man who enjoys small\neconomies, and he had certain expensive tastes. He liked to have a\ngood horse between the shafts of a smart trap when he went his rounds.\nHe liked to see the animal's coat glossy and the harness shining. He\npreferred good whisky to bad, and smoked tobacco at 10_s._ 6_d._ a\npound. He was particular about the cut of his clothes and had a fine\ntaste in striped and spotted waistcoats. He also--quite privately,\nfor in the west of Ireland no one would admit that he threw away his\nmoney wantonly--bought a few books every year. The consequence was\ninevitable. Dr. O'Grady got into debt. At first, indeed for more than\ntwo years, his debts, though they increased rapidly, did not cause any\nuneasiness to his creditors. Then a suspicious tailor began to press\nrudely for the payment of a long account. Other tradesmen, all of\nthem strangers who did not know Dr. O'Grady personally, followed the\ntailor's example. A Dublin gentleman of large fortune and philanthropic\ntastes, a Mr. Lorraine Vavasour, having somehow heard of these\nembarrassments, offered to lend Dr. O'Grady any sum from L10 to L1000\nprivately, without security, and on the understanding that repayment\nshould be made quite at the borrower's convenience.\n\nThere was an agreeable settlement with the tailor who lost Dr.\nO'Grady's custom for ever, and with several others. Life for a time\nwas pleasant and untroubled. Then Mr. Lorraine Vavasour began to act\nunreasonably. His ideas of the payment of instalments turned out to\nbe anything but suitable to Dr. O'Grady's convenience. The good horse\nwas sold at a loss. The competent groom was replaced for an inferior\nand cheaper man. Mr. Lorraine Vavasour showed no signs of being\npropitiated by these sacrifices. He continued to harass his victim with\na persistency which would have made most men miserable and driven some\nmen to excessive drinking. Dr. O'Grady remained perfectly cheerful.\nHe had the temperament of an unconquerable optimist. He used even to\nshow Mr. Vavasour's worst letters to Jimmy O'Loughlin, and make jokes\nabout them. This, as it turned out afterwards, was an unwise thing to\ndo. Jimmy himself had a long account against the doctor standing in his\nbooks.\n\nAfter awhile the miserable screw which succeeded the good horse in\nDr. O'Grady's stable was sold. The smart trap and harness were sold.\nThe incompetent substitute for the groom was dismissed. Dr. O'Grady\nendeavoured to do his work with no better means of getting about than\na dilapidated bicycle. It was generally known that his affairs had\nreached a crisis. His housekeeper left him and engaged a solicitor\nto write letters in the hope of obtaining the wages due to her. It\nseemed very unlikely that she would get them. Mr. Lorraine Vavasour\nwas before her with a claim which the furniture of Dr. O'Grady's house\nwould certainly not satisfy. Jimmy O'Loughlin was before her too.\nHe would have been willing enough to wait for years, and if left to\nhimself would not have driven a friend to extremities for the sake of\na few pounds. But when he saw that Mr. Vavasour meant to use all the\nresources of the law against Dr. O'Grady he thought it a pity to let\na complete stranger get the little there was to get. He apologized to\nDr. O'Grady and summoned him before the County Court judge. The usual\nthings happened. The end appeared to be at hand, and the Board of\nGuardians began to discuss the appointment of a new dispensary doctor.\n\nIt is very much to the credit of Dr. O'Grady that, under these\ncircumstances, he slept soundly at night in his solitary house; rose\ncheerful in the morning and met his fellow-men with a smile on his\nface. He continued to dine frequently at Clonmore Castle, and Lord\nManton noticed that his appetite improved instead of failing as his\ntroubles increased. In fact, Dr. O'Grady frequently went hungry at\nthis time, and Lord Manton's dinners were almost the only solid meals\nhe got. Then, just before the bailiffs took possession of his house\na curious way of escape opened. It was at the beginning of August.\nDr. O'Grady spent the evening reading a new book about germ plasm,\npan-genesis, determinates, and other interesting things connected with\nthe study of heredity. He was obliged to go to bed early because his\nlamp went out at ten o'clock and he had no oil with which to refill\nit. Once in bed he went comfortably to sleep. At two o'clock in the\nmorning he was roused by a ponderous, measured knocking at his door. He\nused the sort of language commonly employed by doctors who are roused\nat unseemly hours. The knocking continued, a series of heavy detached\nblows, struck slowly at regular intervals. Dr. O'Grady got up, put his\nhead out of the window, and made the usual inquiry--\n\n\"Who the devil's that? And what do you want?\"\n\n\"It is I. Guy Theodore Red.\"\n\nEven then, freshly roused from sleep, Dr. O'Grady was struck by the\nanswer he received. Very few men, in search of a doctor at two o'clock\nin the morning, are so particular about grammar as to say, \"It is\nI!\" And the words were spoken in a solemn tone which seemed quite\ncongruous with the measured and stately manner in which the door had\nbeen hammered. Dr. O'Grady put on a pair of trousers and a shirt, ran\ndownstairs and opened the door. Mr. Red stood rigid like a soldier at\nattention on the doorstep. In the middle of the road was the motor car\nin which the English servant used to drive into Clonmore to do his\nmarketing.\n\n\"Is it typhoid?\" said Dr. O'Grady; \"for if it is I ought to have been\nsent for sooner.\"\n\n\"No.\"\n\n\"It can't be a midwifery case in your house?\"\n\n\"No.\"\n\n\"You're very uncommunicative,\" said Dr. O'Grady. \"What is it?\"\n\n\"A gun accident.\"\n\n\"Very well. Why couldn't you have said so before? Wait a minute.\"\n\nDr. O'Grady hurried into his surgery, collected a few instruments\nlikely to be useful, some lint, iodoform, and other things. He stuffed\nthese into a bag, slipped on a few more clothes and an overcoat.\nThen he left the house. He found Mr. Red sitting bolt upright in the\nmotor car with his hands on the steering wheel. Dr. O'Grady got in\nbeside him. During the drive Mr. Red did not speak a single word. He\ndid not even answer questions. Dr. O'Grady was left entirely to his\nown thoughts. The fresh air had thoroughly awakened him, and, being\nnaturally a man of active mind, he thought a good deal. It occurred to\nhim at once that though gun accidents are common enough in the daytime\nthey very rarely occur in the middle of the night. Good men go to bed\nbefore twelve o'clock, and no men, either good or bad, habitually clean\nguns or go out shooting between midnight and two A.M. Dr. O'Grady began\nto wonder how the accident had happened. It also struck him that Mr.\nRed's manners were peculiar. The man showed no sign of excitement. He\nwas not exactly rude. He was not, so far as Dr. O'Grady could judge, in\na bad temper. He was simply pompous, more pompous than any one whom Dr.\nO'Grady had ever met before. He seemed to be obsessed with an idea of\nhis own enormous importance.\n\nThe impression was not removed when the car drew up at Rosivera.\nMr. Red blew three slow blasts on the horn, stepped out of the car,\nstalked up to the door, and then stood, as he had stood in front of Dr.\nO'Grady's house, upright, rigid, his arms stretched stiffly along his\nsides. The door was opened by the foreigner with the long black beard.\nNo word was spoken. Mr. Red raised his left hand and made some passes\nin the air. His bearded friend raised his left hand and imitated the\npasses with perfect solemnity. Mr. Red crossed the threshold, turned,\nand solemnly beckoned to Dr. O'Grady to follow him.\n\n\"I see,\" said the doctor, in a cheerful, conversational tone, \"that you\nare all Freemasons here. It's an interesting profession. Or should I\ncall it a religion? I'm not one myself. I always heard it involved a\nman in a lot of subscriptions to charities.\"\n\nMr. Red made no reply. He crossed the hall, flung open a door with\na magnificent gesture, and motioned Dr. O'Grady to enter the\ndining-room. The doctor hesitated for a moment. He was not a nervous\nman, but he was startled by what he saw. The room was brightly lit with\nfour large lamps. The walls were hung with crimson cloth on which were\nembroidered curious beasts, something like crocodiles, but with much\nlonger legs than crocodiles have, and with forked tongues. They were\nall bright yellow, and stood out vividly from their crimson background.\n\n\"Enter,\" said Mr. Red.\n\nDr. O'Grady faced the crocodiles. In the course of his medical\nexperience he had often met men who had seen such beasts in unlikely\nplaces and been haunted by them unpleasantly; but his own conscience\nwas clear. He was strictly temperate, and he knew that the pictures on\nthe walls in front of him could not be a symptom of delirium. Mr. Red\nfollowed him into the room and shut the door. It was painted crimson on\nthe inside, and a large yellow crocodile crawled across it.\n\n\"I suppose,\" said Dr. O'Grady, \"that you got leave from Lord Manton to\npaper and paint the house. I dare say this sort of thing\"--he waved his\nhand towards the crocodile on the door, which was surrounded with a\nlitter of repulsive young ones--\"is the latest thing in art; but you'll\nexcuse my saying that it's not precisely comfortable or soothing.\nI hope you don't intend to include one of those beasts in your new\nstatue.\"\n\nMr. Red made no reply. He crossed the room, opened a cupboard, and\ntook out of it a bottle and some glasses. He set them on the table and\npoured out some wine. Dr. O'Grady, watching his movements, was inclined\nto revise the opinion that he had formed during the drive. Mr. Red was\nnot merely pompous. He was majestic.\n\n\"Drink,\" said Mr. Red.\n\nDr. O'Grady looked at the wine dubiously. It was bright green. He was\naccustomed to purple, yellow, and even white beverages. He did not like\nthe look of the stuff in the glass in front of him.\n\n\"If,\" he said, \"that is the liqueur which the French drink, absinthe,\nor whatever they call it, I think I won't venture on a whole claret\nglass of it. I'm a temperate man, and I must keep my hand steady if\nI'm to spend the rest of the night picking grains of shot out of your\nfriend.\"\n\n\"Drink,\" said Mr. Red again.\n\nDr. O'Grady felt that it was time to assert himself. He was a friendly\nand good-tempered man, but he did not like being ordered about in\nmonosyllables.\n\n\"Look here,\" he said, \"I'm not a Freemason, or a Rosicrucian, or\nan Esoteric Buddhist, or the Grand Llama of Thibet, or anything of\nthat kind. I don't deny that your manner may be all right with other\nsculptors, or with those who are initiated into your secrets, and I\ndare say you have to live up to this thing in order to produce really\nfirst-rate statues. But I'm only an ordinary doctor and I'm not\naccustomed to it. If you have whisky or any other civilized drink, I\ndon't mind taking a drop before I see the patient; but I'm not going\nto run the risk of intoxicating myself with some strange spirit. And\nwhat's more, I'm not going to be talked to as if you were a newly\ninvented kind of automatic machine that can only utter one word at a\ntime and won't say that unless a penny has been dropped into the slot.\"\n\n\"Your fee,\" said Mr. Red, laying an envelope on the table.\n\nDr. O'Grady took it up and opened it. It contained a ten pound Bank of\nEngland note. His slight irritation passed away at once. Never before\nin the course of his career as a doctor had he received so large a fee.\nThen a sharp suspicion crossed his mind. A fee of such extravagant\namount must be meant to purchase something else besides his medical\nskill. Men, even if they are as rich as Mr. Red appeared to be, even\nif they have the eyes of a mad gander and an eccentric taste in house\ndecoration, do not pay ten pounds to a country doctor for dressing a\nwound. Dr. O'Grady began to wonder whether he might not be called upon\nto deal with the victim of some kind of foul play, whether he were\nbeing paid to keep his mouth shut.\n\n\"Follow me,\" said Mr. Red.\n\nDr. O'Grady followed him out of the dining-room and up two flights of\nstairs. He made up his mind that his silence, supposing silence to be\npossible, was worth more than ten pounds. He determined to keep Mr.\nRed's secret if it did not turn out to be a very gruesome one, but\nto make Mr. Red pay handsomely. One hundred pounds was the amount he\nfixed on. That sum, divided between Mr. Lorraine Vavasour and Jimmy\nO'Loughlin, would pacify them both for a time.\n\nMr. Red stopped outside a bedroom door, and Dr. O'Grady saw on it four\nlarge white letters, A.M.B.A. Mr. Red opened the door. On a bed at the\nfar end of the room lay the servant who used to drive into Clonmore and\nbuy things at Jimmy O'Loughlin's shop. He was lying face downwards and\ngroaning.\n\n\"Exert your skill as a physician,\" said Mr. Red, waving his hand in the\ndirection of the bed.\n\n\"Don't you be a damneder ass than you can help,\" said Dr. O'Grady\ncheerfully.\n\nHe crossed the room and examined the man on the bed.\n\n\"Look here,\" he said, turning to Mr. Red, \"you told me that this man\nwas suffering from the result of an accident he had had with a gun.\nWell, he isn't. I defy any man to scorch the skin off the backs of both\nhis own legs with a gun. The thing simply couldn't be done.\"\n\n\"Exert your skill as a physician, and be silent,\" said Mr. Red.\n\n\"You may fancy yourself to be the Cham of Tartary,\" said Dr. O'Grady,\n\"or Augustus Caesar, or Napoleon Bonaparte, or a Field Marshal in the\narmy of the Emperor of Abyssinia, but you've got to give some account\nof how that man flayed the backs of his legs or else I'll have the\npolice in here to-morrow.\"\n\nMr. Red smiled, waved his hand loftily, and left the room.\n\nDr. O'Grady, his professional instinct aroused, proceeded to dress the\nman's wounds. They were not dangerous, but they were extremely painful,\nand at first the doctor asked no questions. At length his curiosity\nbecame too strong for him.\n\n\"How did you get yourself into such a devil of a state?\" he asked.\n\nThe man groaned.\n\n\"It looks to me,\" said Dr. O'Grady, \"as if you'd sat down in a bath of\nparaffin oil and then struck a match on the seat of your breeches. Was\nthat how it happened?\"\n\nThe man groaned again.\n\n\"If it wasn't that,\" said Dr. O'Grady, \"you must have tied a string\nround your ankles, stuffed the legs of your trousers with boxes of\nmatches, and then rubbed yourself against something until they went\noff. I can't imagine anything else that could have got you into the\nstate that you're in.\"\n\n\"I was smoking,\" said the man at last, \"in the Chamber of Research.\"\n\n\"In the what?\"\n\n\"It's what 'e calls it,\" said the man. \"I don't know no other name for\nit.\"\n\n\"Perhaps the floor of the Chamber of Research was covered with\ngunpowder behind where you were standing, and you dropped a lighted\nmatch into it.\"\n\n\"'Ow was I to know the stuff would go off?\"\n\n\"If you knew it was gunpowder,\" said Dr. O'Grady, \"you might have\nguessed it would go off if you dropped a match into it.\"\n\n\"It weren't gunpowder, not likely. It were some bloomin' stuff 'e made.\n'E's always messing about making stuff, and none of it ever went off\nbefore.\"\n\n\"If you mean Mr. Red,\" said Dr. O'Grady, \"I can quite imagine that the\nstuff he made wouldn't go off. Unless, of course, it was intended not\nto. From what I've seen of him so far, I should say that his notion\nof manufacturing dynamite would be to take a hundredweight or so of\ntoothpowder, and say to it, 'Powder, explode.' Still, you ought to have\nbeen more careful.\"\n\n\"'E's a damned ass,\" said the man.\n\n\"He is,\" said Dr. O'Grady. \"Still, even an ass, if he goes on\nexperimenting for four months in a chamber specially set apart for\nresearch, is sure to hit upon something that will explode by the end\nof the time. By the way, do you happen to know where he got that\ndining-room wall-paper with the crocodiles on it?\"\n\nThe door opened and Mr. Red stalked into the room.\n\n\"Follow me,\" he said to Dr. O'Grady.\n\n\"All right. I've finished with this fellow's legs for the present. I'll\ncall again to-morrow afternoon, or rather, this afternoon. He'll get\nalong all right. There's nothing to be frightened about. You may give\nhim a little beef-tea and---- Damn it all! Augustus Caesar has gone!\nGood-bye, my man. I'll see you again soon. I must hurry off now. It\nwon't do to keep the Field Marshal waiting. The crocodiles might get on\nhis nerves if he was left too long in the room alone with them.\"\n\n\n\n\nCHAPTER III\n\n\nDr. O'Grady left the room and closed the door behind him. His spirits,\nowing to the ten-pound note which lay in his breast pocket, were\ncheerful. He whistled \"The Minstrel Boy\" as he walked along the\npassage. Just as he reached that part of the tune which goes with the\ndiscovery of the boy in the ranks of death he stopped abruptly and\nswore. He was seized from behind by two men, flung to the ground with\nsome violence, and held there flat on his back. A few useless struggles\nconvinced him that he could not make good his escape. He lay still and\nlooked at his captors. The foreign gentleman with the long black beard\nwas one of them. The other was the man whom Dr. O'Grady had declared\nto be a Norwegian. He was a powerful man, adorned with a mass of fair\nhair which fell down over his forehead and gave him a look of unkempt\nferocity. Behind these two who knelt beside and on Dr. O'Grady stood\nMr. Red.\n\n\"Hullo, Emperor!\" said the Doctor, \"what's the game now? If you want\na gladiatorial show, with me and these two swashbucklers as chief\nperformers, you ought to have given me fair notice. You can't expect a\nman to put up much of a fight when he's caught from behind just as he's\nin the middle of whistling a tune.\"\n\n\"You have learned too much,\" said Mr. Red, with fierce intensity.\n\"It is necessary in the interests of the Brotherhood to secure your\nsilence.\"\n\n\"Right,\" said Dr. O'Grady. \"You shall secure it. One hundred and fifty\npounds down and the secrets of the Brotherhood are safe. Or if prompt\ncash inconveniences you in any way, I'll be quite content with your\nname on the back of a bill. Jimmy O'Loughlin would cash it.\"\n\n\"I have passed judgment on you,\" said Mr. Red, \"and the scales are\ndepressed on the side of mercy. Your life is spared. You remain a\ncaptive until the plans of the Brotherhood are matured and discovery\ncan be set at defiance. Then you will be released.\"\n\n\"If that's all,\" said Dr. O'Grady, \"you needn't have knocked me down\nand set these two brigands to kneel on my chest and legs. I haven't\nthe slightest objection to remaining a captive. I shall enjoy it. Of\ncourse, I shall expect to be paid a reasonable fee for my time. I'm a\nprofessional man.\"\n\n\"Number 2 and Number 3,\" said Mr. Red, \"will bind you and convey you to\nthe place of confinement.\"\n\nHe spoke a few words to his assistants in a language which Dr. O'Grady\ndid not understand. Two ropes were produced.\n\n\"If you choose to tie me up,\" said Dr. O'Grady, \"you can do it of\ncourse. But you'll simply be wasting time and energy. I've told you\nalready that I don't in the least mind being a captive. Just you tell\nme the place you want me to go to, and if it isn't an insanitary,\nunderground dungeon, I shall step into it with the greatest pleasure,\nand stay there without making the least attempt at escape as long as\nyou choose to go on paying me my fees.\"\n\n\"Give your parole,\" said Mr. Red.\n\n\"Parole? Oh, yes, of course; I know the thing you mean now. I'll give\nit, certainly--swear it if you like. And now, like a good man, tell\nyour fair-haired pirate to get off my legs. He's hurting my left ankle\nabominably.\"\n\nMr. Red gave an order, and Dr. O'Grady was allowed to stand up.\n\n\"Now for the cell,\" he said. \"I know this house pretty well, and I\nshould suggest that you give me the two rooms on the top floor which\nopen into each other. And look here, Emperor, I'm a first-class\npolitical prisoner, of course. I'm not going to do any hard labour, or\nget out of bed before I want to in the morning. I must be decently fed,\nand supplied with tobacco. You agree to all that I suppose?\"\n\n\"Lead the prisoner upstairs,\" said Mr. Red.\n\n\"One minute,\" said Dr. O'Grady. \"We haven't settled yet about my fee.\nLet me see, what would you say--my time is valuable, you know. I have\na very extensive practice, including the nobility and gentry of the\nneighbourhood; Lord Manton, for instance, and Jimmy O'Loughlin's wife.\nWhat would you say to----? Good Lord! Emperor, put that thing down, it\nmight go off!\"\n\nMr. Red had taken a revolver from his pocket, and pointed it at Dr.\nO'Grady's head.\n\n\"Lead the prisoner upstairs,\" he said.\n\n\"I'm going all right,\" said Dr. O'Grady. \"But, like a good man, put\ndown that pistol. I dare say it's not loaded, and I'm sure you don't\nmean to pull the trigger; but it makes me feel nervous. If you injure\nme you will be in a frightful fix. There isn't another doctor nearer\nthan Ballymoy, and he's no good of a surgeon. Do be careful.\"\n\nMr. Red took no notice of this remonstrance. He held the revolver at\narm's length, pointed straight at Dr. O'Grady's head. The doctor turned\nquickly and walked upstairs. He was ushered into a large empty room,\nand bidden to stand in a corner of it. Still covered by the threatening\nrevolver he watched various preparations made, first for his security,\nthen for his comfort. There were two windows in the room. The\nblack-bearded foreigner nailed barbed wire across them in such a way as\nto make an entanglement through which it was impossible to thrust even\na hand.\n\n\"That's quite unnecessary,\" said Dr. O'Grady. \"I'm familiar with this\nhouse, have been over it half a dozen times with Lord Manton, and I\nknow that there's a sheer drop of thirty feet out of those windows on\nto the paved yard at the back of the house. I shouldn't dream of trying\nto jump out.\"\n\nMr. Red stood with the revolver in his hand glaring at Dr. O'Grady. His\ntwo assistants left the room.\n\n\"I do wish,\" said the doctor plaintively, \"that you'd put that gun\ndown.\"\n\nBy way of reply, Mr. Red settled himself in an heroic attitude,\nsomething like that usually adopted by the hero on the cover of a\nsixpenny novel when he is defending his lady from desperate villains.\nHe kept the revolver levelled at Dr. O'Grady's head. The bearded man,\nNumber 2, returned, dragging a small iron bedstead after him. Number 3\nfollowed him with a mattress, pillows, and some blankets.\n\n\"For me?\" said Dr. O'Grady. \"Thanks. Now fetch a washhand-stand, a jug\nand basin, a table, a couple of chairs, some food, tobacco, and a few\nbooks. Then I'll be able to manage along all right.\"\n\nOne thing after another was added to the furniture of the room until\nit began to look fairly comfortable. Dr. O'Grady observed with\nsatisfaction that a substantial meal was spread on the table, and a box\nof cigars laid on the washhand-stand.\n\n\"Would it be any harm my asking,\" he said, \"how long you intend to keep\nme here? I have some rather pressing engagements just at present, and\nI should like to have an idea when I'll get home. Of course, I don't\npress the question if it inconveniences the Brotherhood to answer it\nbefore the plans are matured.\"\n\n\"You shall be paid at the rate of L4 a day during the time that you are\ndetained,\" said Mr. Red.\n\n\"Make it L5,\" said Dr. O'Grady, \"and I'll stay a year with you and\nsettle my own washing bills.\"\n\n\"In four weeks,\" said Mr. Red, \"the plans of the Brotherhood will be\nmatured, and you can be released.\"\n\n\"I'm sorry it's no longer,\" said Dr. O'Grady. \"The arrangement is\nperfectly satisfactory to me. But look here, Emperor, have you taken\ninto consideration that I shall be missed? Before four weeks are out\nthey'll be certain to start out looking for me. Search parties will go\nout with lanterns and bloodhounds. You know the kind of thing I mean.\nThey won't come straight here, of course; nobody has any reason to\nsuppose that I'm in this house; but sooner or later they certainly will\ncome. I don't mind telling you that there are a couple of men--Jimmy\nO'Loughlin for one, and Lorraine Vavasour for another--who will be\nparticularly keen on finding me. What will you do when they turn up?\"\n\n\"The waters of the bay are deep,\" said Mr. Red grimly. \"Your body will\nnot be found.\"\n\n\"I catch your meaning all right,\" said Dr. O'Grady, \"but I think\nyou'll make a mistake if you push things to extremes in that way.\nYou've got the usual idea into your head that Ireland is a country\nin which every one kills any one they don't like, and no questions\nare ever asked. I don't in the least blame you for thinking so. Any\nintelligent man, reading the newspapers, would be forced to that\nconclusion; but, as a matter of fact, Ireland isn't that sort of\ncountry at all. We have our little differences with each other, of\ncourse; all high-spirited people quarrel now and then, but we really\nhardly ever drown anybody. We don't want to; but even if we were ever\nso keen we couldn't without great risk. The country is overrun with\npolice, and---- I beg your pardon, did you speak?\"\n\nMr. Red had not actually spoken. He had snarled in a curious and\nvicious way.\n\n\"The police----\" said Dr. O'Grady.\n\nMr. Red snarled again.\n\n\"If you object to my mentioning them by name,\" said Dr. O'Grady,\n\"I won't do it. All I wanted to say was that in Ireland they live\nextremely dull lives, and any little excitement--a cattle drive, or\nan escaped lunatic--is a positive godsend to them. A murder--perhaps\nI ought to say an informal execution, such as you contemplate--would\nbring them down to this neighbourhood in thousands. There'd be so many\nof them that they simply wouldn't be able to help tripping over my body\nwherever you hid it. Don't imagine that I'm saying all this with a view\nto preventing your cutting my throat. What I'm really thinking about,\nwhat you ought to be thinking about, is the Brotherhood. How will its\nplans ever be matured if you get yourself hanged? And they will hang\nyou, you know.\"\n\n\"I am prepared to die,\" said Mr. Red majestically, \"in the cause of the\nAnti-Militarist Brotherhood of Anarchists.\"\n\n\"Of course you are. Anybody who knows anything about military\nanarchists knows that. My point is that your life is too valuable\nto be thrown away. How would poor Long Beard get on? And the other\nfair-haired highwayman? Neither of them knows a word of English.\"\n\n\"If the accursed minions of an effete tyranny seize me----!\"\n\n\"Quite so. I see your point. Death before dishonour, and all that kind\nof thing. But why let it come to that? I am perfectly willing to stay\nhere as long as you like at the liberal salary you offer, cash down\nevery evening. I'm quite as anxious as you are to keep the accursed\nminions of the what-do-you-call-it away from Rosivera. I don't mind\ntelling you in confidence that I have reasons of my own for avoiding\nany contact with the law at present. In my particular case it isn't\nnearly so effete as you appear to think it ought to be. But I needn't\ngo into all that. It wouldn't interest you, and it's no pleasure to\nme to talk about that beast Lorraine Vavasour. What I want to suggest\nis a simple and practicable way of avoiding all fuss, and keeping the\naccursed minions quiet in their barracks.\"\n\n\"Speak,\" said Mr. Red.\n\n\"I am speaking. For a man who hasn't had any breakfast this morning, I\nflatter myself I'm speaking pretty fluently. Don't be captious, Field\nMarshal. I don't mind your manner a bit, now that I'm getting used to\nit. I know that it's quite the right kind of manner for a military\nanarchist, but there's no use over-doing it.\"\n\n\"Your plan?\" said Mr. Red, fingering the revolver.\n\n\"I wish you'd lay that weapon down, Emperor. I've told you half a dozen\ntimes that I haven't the least intention of trying to escape, and it\nwill be a horrid nuisance if the thing goes off and injures me. My\nsuggestion is simply this. I'll write a letter blotted all over with\ntears, saying that driven to desperation by Lorraine Vavasour and Jimmy\nO'Loughlin I've committed suicide, and that all search for my body will\nbe vain. Owing to circumstances which I need not explain, circumstances\nnot unconnected with Lorraine Vavasour, the story will be believed in\nClonmore and no further steps will be taken in the matter. All you will\nhave to do is to drop the letter into the pillar-box which is only half\na mile from your gate. I happen to know that that box is cleared at\neight P.M., so any time to-day will do. I'll address it to the police\nsergeant.\"\n\nMr. Red gave an order to one of the two foreigners. The man left the\nroom and returned in a few minutes with a supply of note-paper, a pen,\nand a bottle of ink. He laid them beside the food on the table in the\nmiddle of the room.\n\n\"Write,\" said Mr. Red.\n\n\"I forgot to mention,\" said Dr. O'Grady, \"that I'm engaged to be\nmarried to a young lady in Leeds. Miss Blow is her name--Adeline Maud\nBlow. I dare say you've heard of her father in connection with cigars.\nHe's a tobacconist and advertises a good deal. 'Blow's beauties,\ntwopence each.' You must have heard of them. They're beastly things as\na matter of fact, and I don't recommend them to friends, but they're\namazingly popular.\"\n\n\"Write,\" said Mr. Red.\n\n\"I am going to write. Don't hustle me, like a good man. What I want to\nsay to you is this, that I must send a line to Adeline Maud as well as\nto the police sergeant. I want to tell her that I'm not really dead,\nonly bluffing.\"\n\n\"That,\" said Mr. Red, \"is impossible.\"\n\n\"Nonsense. There's nothing impossible about it. It's just as easy to\nwrite two letters as one. I shan't mention the Brotherhood to her, and\nif I did she would have more sense than to talk about it. If you don't\nbelieve me you can read the letter yourself.\"\n\n\"I trust no woman.\"\n\n\"That,\" said Dr. O'Grady, \"is a most illiberal sentiment, and I'm\nsurprised to hear you utter it. If you'd been an old-fashioned Tory\nnow, or an Irish landlord, or a Liberal Cabinet Minister, I could have\nunderstood your position; but in a military----\"\n\n\"Anti-militarist,\" said Mr. Red.\n\n\"That's what I meant. In an anti-militarist, that sort of prejudice\nagainst women is most inconsistent. Who was it that hammered a nail\ninto Sisera's head? A woman, and an anti-military woman. Who was it\nthat stuck a knife into that horrid beast Marat, when he was sitting\nin his bath? A woman again. Who was it that shot that Russian governor\nthe other day? I've forgotten her name for the minute, but you know who\nI mean. It was a woman. She did for him on a railway platform. And yet\nyou stand up there calling yourself an advanced kind of anarchist, and\nsay that you can't trust a woman. Emperor, you ought to be ashamed of\nyourself. Just think the matter out and you'll see that when it comes\nto thorough-going, out-and-out revolutions women are quite the most\ntrustworthy kind of people there are.\"\n\nMr. Red gave another brief order in his foreign language. The\nfair-haired anarchist stepped forward and took away the note-paper, pen\nand ink.\n\n\"What are you at now?\" said Dr. O'Grady. \"Surely to goodness you're not\ngoing back on the suicide plan? Oh, very well. I can't help it. But\nyou'll be sorry afterwards when the police come here looking for me.\"\n\n\"I have spoken,\" said Mr. Red.\n\n\"You have not. You've growled occasionally, but nobody could call your\nremarks speaking.\"\n\n\"I leave you,\" said Mr. Red. \"Remember.\"\n\n\"Remember what? Oh, you're going, are you? Just wait one instant. You\nrefuse to let me write to Adeline Maud. Very well. You don't know\nAdeline Maud, but I do. Even supposing the police can't find me, or my\nbody after you've cut my throat, and supposing that Jimmy O'Loughlin\nand Lorraine Vavasour give up the pursuit--from what I know of Lorraine\nI think it most unlikely that he will--you'll still have to reckon with\nAdeline Maud. She's a most determined young woman. All the perseverance\nwhich has gone to making 'Blow's beauties' the popular smokes they are\nat twopence each has descended from her father to her. When she finds\nout that I've disappeared she'll go on searching till she finds me.\nThe ordinary sleuth-hound is absolutely nothing to her for persistence\nin the chase. It will be far wiser for you--in the interests of the\nBrotherhood I mean--to let me head her off, by telling her that I'll\nturn up again all right.\"\n\n\"Farewell,\" said Mr. Red.\n\n\"I ought to mention before you go,\" said Dr. O'Grady, \"that Adeline\nMaud may be in Clonmore to-morrow. I'm expecting a visit---- Damn it!\nThe fool is gone and shut the door behind him.\"\n\nMr. Red had, in fact, entirely ignored the announcement of Miss\nBlow's impending arrival. He and his two friends left the room, and,\ndistrustful of the parole which had been given, locked the door behind\nthem. Dr. O'Grady turned to the breakfast provided for him with an\nexcellent appetite.\n\n\"Silly old ass that Emperor is,\" he murmured. \"I suppose now he'll go\nand sit down beside his yellow crocodiles in the dining-room and try to\ninvent some new kind of dynamite. He ought, as a matter of fact, to be\nin an asylum. Some day he will be if he doesn't blow himself up first.\nAnyhow this is a jolly good business for me. If he chooses to pay me\nfour pounds a day for sitting here twiddling my thumbs, I'm quite\ncontent. I only hope it will be some time before they think of looking\nfor me at Rosivera. I must try and hit on a plan for putting the police\noff the scent. I wonder how long it will last. The Field Marshal\nsuggested four weeks. Seven times four--even putting it as low as four\npounds a day--and I'll try to screw him up a bit--seven times four are\ntwenty-eight. And four times twenty-eight is a hundred and twelve. That\nwith the tenner I've got will make L122. I'll make a clear L100 out of\nit anyway, and they won't have time to elect another dispensary doctor\nbefore I get out.\"\n\n\n\n\nCHAPTER IV\n\n\nPatsy Devlin strolled into the Imperial Hotel at noon. He found Jimmy\nO'Loughlin, the proprietor, behind the bar, and was served at once with\na pint of porter.\n\n\"It's fine weather for the hay, thanks be to God,\" he observed.\n\nIn Connacht the hay harvest is gathered during the month of August, and\nPatsy's comment on the weather was seasonable.\n\n\"I've seen worse,\" said Jimmy O'Loughlin. \"But what's on you at all,\nPatsy, that you haven't been next or nigh the place this two months or\nmore?\"\n\n\"Be damn! but after the way you behaved over the election of the\ninspector of sheep dipping, the wonder is that I'd ever enter your door\nagain. What would hinder you giving me the job as soon as another?\"\n\nJimmy O'Loughlin did not wish to discuss the subject. He was, as a\ntrader ought to be, a peaceful individual, anxious to live on good\nterms with all possible customers. He realized that the election was a\nsubject on which Patsy was likely to feel bitterly. He filled another\nglass with porter from the tap and handed it silently across the\ncounter. Patsy tendered a coin in payment.\n\n\"I'll not take it from you,\" said Jimmy O'Loughlin heartily. \"It would\nbe a queer thing if I wouldn't give you a sup at my own expense now\nthat you are here after all this length of time. How's herself?\"\n\nPatsy Devlin took a pull at the second pint of porter.\n\n\"She's only middling. She was complaining these two days of an\nimpression on the chest, and a sort of rumbling within in herself that\nwouldn't let her rest easy in her bed.\"\n\n\"Do you tell me that? And did you fetch the doctor to her?\"\n\n\"I did not then.\"\n\n\"And why not?\"\n\nPatsy Devlin finished the porter and winked across the bar at Jimmy\nO'Loughlin. Jimmy failed to catch the meaning of the wink.\n\n\"If it was a red ticket you wanted,\" he said, \"you know very well that\nyou've nothing to do but ask me for it. But Dr. O'Grady, the poor man,\nwould go to you without that.\"\n\n\"If I did be wanting a red ticket,\" said Patsy, \"it wouldn't be you I'd\nask for it. There's them would give it to me and maybe something along\nwith it, and what's more, did give it to me no later than this morning.\"\n\n\"Well,\" said Jimmy, who guessed at the identity of the unnamed\nbenefactor, \"and if so be that his lordship is after giving you the\nticket, why didn't you go and fetch the doctor to herself?\"\n\n\"I went for him right enough.\"\n\n\"And do you mean to tell me that he refused to attend the call and you\nwith the red ticket in your hand? For if he did----\"\n\n\"He wasn't within when I went for him.\"\n\nThe explanation was perfectly simple and natural; but Jimmy O'Loughlin,\nnoting the manner in which it was given, realized there was something\nbehind it.\n\n\"What do you mean?\" he asked.\n\nPatsy Devlin winked again. Jimmy, vaguely anxious, but not knowing what\nto fear, handed his visitor a third pint of porter.\n\n\"I'm thinking,\" said Patsy, \"that it's about time for us to be making a\nmove in the matter of collecting funds for the horse races and athletic\nsports. The season's going on and if we don't have them before the end\nof the month the days will be getting short on us. I suppose now I may\nput you down for a pound the same as last year?\"\n\n\"You may,\" said Jimmy. \"But what was it you were after telling me about\nDr. O'Grady?\"\n\n\"Does he owe you any money?\"\n\n\"He does, a power.\"\n\n\"Then you'll not see it. Devil the penny of it ever you'll handle, no\nmatter how you try.\"\n\nPatsy chuckled. He had nourished a grudge against Jimmy O'Loughlin ever\nsince the election of the inspector of sheep dipping.\n\n\"And why will I not?\"\n\n\"Because the doctor's gone, that's why. He's off to America, and every\nstick of furniture he owns is gone along with him.\"\n\n\"He is not,\" said Jimmy. \"He couldn't. It was only last night he passed\nthis door, looking the same as usual. I was speaking to him myself, and\nhim on his way home after being out beyond in the bog.\"\n\n\"It was last night he went.\"\n\n\"He couldn't. Sure you know as well as I do there's no train.\"\n\n\"There's the goods that passes at one o'clock. What would hinder him\ngetting into one of them cattle trucks? Who'd see him? Anyway, whether\nit was that way he went or another, he's gone. And you may be looking\nfor your money from New York. Be damn! but I'd take half a crown for\nall of it you'll ever get.\"\n\n\"I don't believe you,\" said Jimmy; but it was evident that he was\nfighting desperately against conviction.\n\n\"Go round to the house then yourself and you'll see. He's not in it,\nand what's more he didn't leave it this morning, for old Biddy Halloran\nwas watching out for him along the road the way he'd renew the bottle\nhe gave her for the rheumatism, and if he'd gone out she'd have seen\nhim.\"\n\n\"It might be,\" said Jimmy, \"that he was called out in the night and\ndidn't get back yet.\"\n\n\"It might; but it wasn't. I'm after spending the morning since ten\no'clock making inquiries here and there, and devil the one there is in\nthe parish sick enough to be fetching the doctor out in the middle of\nthe night. If there was I'd have heard of it. And Father Moroney would\nhave heard of it, but he didn't, for I asked him this minute.\"\n\n\"And what did he say?\"\n\n\"He said very much what I'm saying myself, that the doctor's gone.\n'And small blame to him,' says Father Moroney, 'with the way that old\nreprobate of a Jimmy O'Loughlin, who's no better than a gombeen man,\nhas him persecuted for the trifle of money he owes.' It's the truth I'm\ntelling you.\"\n\nIt was not the truth, but it could scarcely be called a lie, for the\nessence of a lie is the desire to deceive, and Patsy Devlin invented\nthe speech he put into the priest's mouth without the least hope of\nits being believed. The best he expected was to exasperate Jimmy\nO'Loughlin. Even in this he failed.\n\n\"If so be he's gone,\" said Jimmy, \"and I wouldn't say but he might,\nI'd as soon he got clear off out of this as not. I'll lose upwards\nof thirty pounds by him, but I'd sooner lose it than see the doctor\ntormented by that bloodsucker of a fellow from Dublin that has a bill\nof his. I've a great liking for the doctor, and always had. He was an\ninnocent poor man that wouldn't harm a child, besides being pleasant\nand agreeable as e'er a one you'd meet.\"\n\nPatsy Devlin felt aggrieved. He had sprung his mine on Jimmy\nO'Loughlin, and the wretched thing had somehow failed to explode. He\nhad looked forward to enjoying a torrent of oaths and bitter speeches\ndirected against the absconding debtor. He had hoped to see Jimmy\nwrithe in impotent rage at the loss of his money.\n\n\"Be damn!\" he said helplessly.\n\n\"And anyway,\" said Jimmy O'Loughlin, \"there's the furniture of the\nhouse left, and it'll be a queer thing if I don't get a hold of the\nbest of it before ever the Dublin man--Vavasour or some such they call\nhim--hears that the doctor's gone.\"\n\nThere was no more spirit left in Patsy Devlin. He did not even repeat\nthe obviously incredible statement that the doctor had secretly carried\nhis furniture away with him in the middle of the night.\n\n\"The train's in,\" he said, changing the subject abruptly, \"for I hear\nthe cars coming down from the station. If so be now that there should\nbe a traveller in belonging to one of them drapery firms, or Campbell's\ntraveller with the flour, you might give me the word so as I can get a\nsubscription out of him for the sports. The most use those fellows are\nis to subscribe to one thing or another where subscriptions is needed.\"\n\nJimmy O'Loughlin nodded. He realized the importance of the commercial\ntraveller as a contributor to local funds of every kind. He left the\ndoor and reached the bar of the hotel just as his bus, a ramshackle,\ndilapidated vehicle drawn by a sickly horse, drew up. It contained a\nlady. Jimmy O'Loughlin appraised her at a glance as she stepped out of\nthe bus. She was dressed in a grey tweed coat and skirt of good cut and\nexpensive appearance. She wore gloves which looked almost new, and she\nhad an umbrella with a silver handle. She was tall and carried herself\nwith the air of one who was accustomed to command service from those\naround her. Her way of walking reminded Jimmy O'Loughlin of Lady Flavia\nCanning, Lord Manton's daughter; but this lady was a great deal younger\nand better looking than Lady Flavia. Jimmy O'Loughlin allowed his eyes\nto leave her for an instant and seek the roof of the bus. On it was a\nlarge travelling trunk, a handsome bag, and a bundle of rugs and golf\nclubs. Jimmy's decision was made in an instant. He addressed his guest\nas \"my lady.\"\n\nShe made no protest against the title.\n\n\"Can I get a room in this hotel?\" she asked.\n\n\"Certainly, my lady. Why not? Thomas, will you bring the lady's luggage\nin at once and take it up to number two, that's the front room on the\nfirst floor. Your ladyship will be wanting a private sitting-room?\"\n\n\"If I do,\" she said, \"I shall ask for it.\"\n\nJimmy O'Loughlin was snubbed, but he bore no malice. A lady of title\nhas a right to snub hotel-keepers. He stole a glance at the label\non her luggage as Thomas, the driver of the bus, passed him with the\ntrunk on his shoulders. He discovered that she was not a lady of title.\n\"Miss A. M. Blow,\" he read. \"Passenger to Clonmore.\" The name struck\nhim as being familiar, but for a moment he could not recollect where he\nhad heard it. Then he remembered. Miss Blow passed upstairs guided by\nBridgy, the maid. Patsy Devlin emerged from the bar.\n\n\"It's the doctor's young lady,\" whispered Jimmy O'Loughlin.\n\n\"Is it, be damn? How do you know that?\"\n\n\"Didn't he often tell me,\" said Jimmy, \"that he was to be married to\na young lady out of Leeds or one of them towns beyond in England, and\nthat her name was Miss Blow? And didn't I see it on her trunk, 'Miss A.\nM. Blow'? Would there be two in the world of the name?\"\n\n\"And what would bring her down to Clonmore?\"\n\n\"How would I know? Unless maybe she's heard of the misfortunes that has\nthe doctor pretty near bet. She might have seen his name in the paper\nwhen the judgment was gave against him.\"\n\n\"She might; but if she did wouldn't she keep away from him as far as\never she could?\"\n\n\"She would not,\" said Jimmy O'Loughlin. \"That's not the sort she is. I\nseen her and you didn't.\"\n\n\"I did.\"\n\n\"Well, and if you did you might have known that she'd be the sort that\nwould come down after him the minute she got word of the trouble that\nwas on him. Believe you me, Patsy Devlin, that's a fine girl.\"\n\n\"She's a good-looking one anyway,\" said Patsy, \"but mighty proud, I'd\nsay.\"\n\n\"You may say that. I'd sooner she married the doctor than me, and\nthat's the truth.\"\n\n\"What'll she do now,\" said Patsy, \"when she finds that the doctor's\ngone and left her?\"\n\n\"It'll be best,\" said Jimmy, \"if we keep it from her.\"\n\n\"How can you keep it from her when the man's gone? Won't she be asking\nto see him?\"\n\n\"There's ways of doing things. What would you say now if I was to tell\nher that the doctor had gone off on a holiday for six weeks with the\npermission of the Board of Guardians and that there'd have to be a\nsubstitute appointed in his place? Would she be contented with that, do\nyou think?\"\n\n\"She might,\" said Patsy, \"but she might not. She'd be wanting his\naddress anyway.\"\n\n\"If she wanted it, it would be mighty hard to keep it from the like of\nthat one.\"\n\n\"You haven't got it to give, and so you can't give it,\" said Patsy.\n\nMiss Blow came downstairs as he spoke and walked up to Jimmy O'Loughlin.\n\n\"Will you kindly have some luncheon ready for me,\" she said, \"at two\no'clock?\"\n\n\"Certainly, miss, why not? Is there any particular thing that your\nladyship would fancy, such as a chop or the like?\"\n\nHe reverted to the \"ladyship\" again, although he knew her name and\ndegree. The girl's manner seemed to force him to. She deserved\nsomething better than a mere \"miss.\"\n\n\"In the meanwhile will you be so good as to tell me where Dr. O'Grady\nlives?\"\n\n\"Is it Dr. O'Grady? Well, now, never a nicer gentleman there is about\nthe place, nor one that's more thought of, or better liked than Dr.\nO'Grady. It's him that does be taking his dinner up at the Castle with\nthe old lord and attending to his duties to the poor the same as if he\nwas one of themselves. Many's the time I've said to him: 'Dr O'Grady,'\nsays I, 'if anything was to take you away out of Clonmore, and I don't\ndeny but what you ought to be in a less backward place, but if ever----\n'\"\n\n\"Will you be so good as to tell me where he lives?\" said Miss Blow.\n\nPatsy Devlin interposed at this point of the conversation with an air\nof contempt for Jimmy O'Loughlin.\n\n\"Can't you stop your talking,\" he said, \"and tell the lady where the\ndoctor lives?\"\n\nJimmy cast a venomous glance at him.\n\n\"I will tell your ladyship to be sure. Why not? But it will be of no\nuse for you to go to call on him to-day. Patsy Devlin here is after\ntelling me this minute that he's not at home.\"\n\nMiss Blow turned to Patsy.\n\n\"Do you know,\" she asked, \"when he's likely to be back?\"\n\n\"I do not, my lady. But I'd say it wouldn't be for a couple of days\nanyway.\"\n\n\"A couple of days! Where has he gone to?\"\n\n\"It's what Mr. O'Loughlin there was just after telling me, your\nladyship, and he's the Chairman of the Board of Guardians, that the\ndoctor did ask for leave to go on a holiday. But I wouldn't say that\nhe'd be away for very long.\"\n\n\"When did he ask for a holiday?\" said Miss Blow to Jimmy O'Loughlin.\n\n\"It was Patsy Devlin told me,\" said Jimmy; \"and six weeks was the time\nthat he mentioned.\"\n\nMiss Blow turned again to Patsy Devlin; but he had vanished. Having\ncommitted Jimmy O'Loughlin, as Chairman of the Board of Guardians, to\nthe fact of the doctor's holiday, he slipped quietly into the bar.\n\n\"I don't believe,\" said Miss Blow, \"that you're telling me the truth.\"\n\n\"He was not,\" said Jimmy, sacrificing his friend with the utmost\npromptitude. \"It's seldom he does that same. Devil the bigger liar,\nbegging your ladyship's pardon for the word, devil the bigger liar\nthere is in Connacht than that same Patsy Devlin, and it's what every\none that knows him would tell you.\"\n\n\"I don't,\" said Miss Blow severely, \"see very much to choose between\nyou and him.\"\n\nIn England people have a great regard for the truth so long as it\ndoes not interfere with business. Miss Blow expressed her scorn for\nthe two men who had tried to deceive her quite plainly both by her\nwords and the tone in which she spoke them. In Connacht truth is less\nrespected. Good manners and consideration for other people's feelings\nare looked upon as virtues superior to blunt accuracy of statement.\nJimmy O'Loughlin lied feebly, but he lied with the best intentions. He\nwanted to spare Miss Blow the knowledge that her lover had deserted\nher. In return she insulted him; but even under the sting of her words\nhe recollected that courtesy is due to every lady, especially to one as\ngood-looking as Miss Blow. It was not until she had turned her back on\nhim and left the hotel that he murmured under his breath--\n\n\"May the Lord help the poor doctor if it ever comes to his being\nmarried by the like of her!\"\n\n\n\n\nCHAPTER V\n\n\nMiss Blow came back for her luncheon, and then, asking no more help\nor advice from Jimmy O'Loughlin, went out and made her way to Dr.\nO'Grady's house. It stood a few hundred yards from the village in the\nmiddle of a small field. Miss Blow knocked and rang at the door, though\nshe had no real expectation of its being opened to her. She walked\nround the house and peered in at the windows. The rooms on the ground\nfloor showed every sign of having been recently occupied by a person\nof untidy habits. She reached the yard, surveyed the coach house and\nthe stable which had once sheltered a good horse. She tried the kitchen\ndoor and found it bolted against her. The kitchen had a disused and\nneglected appearance which puzzled her. She returned to the front of\nthe house and sat down on a stone to think out the position in which\nshe found herself.\n\nPatsy Devlin, who had followed her from the hotel, watched her\nproceedings from a distance with great interest. He afterwards made\na report to Jimmy O'Loughlin, a masterly report which interpreted\nher actions, and added a picturesque touch at the end. Patsy Devlin\nwould have written good histories if fortune had made him a university\nprofessor instead of a blacksmith.\n\n\"You'd have been sorry for the creature,\" he said, \"if you'd seen her\nsitting there on a lump of a stone with the tears running down the two\ncheeks of her the same as if you were after beating her with a stick.\"\n\n\"I am sorry for her,\" said Jimmy. \"It's herself has her own share of\ntrouble before her when she finds out that the doctor's off to America\nwithout so much as leaving word for her to go after him.\"\n\nIt did not seem likely that Miss Blow would easily arrive at a\nknowledge of the full extent of her misery. Biddy Halloran, the\nrheumatic old lady who had waited long on the roadside for the doctor\nin the morning, was still lurking near the house when Miss Blow reached\nit. She, like Patsy Devlin, watched the examination of the premises\nwith deep interest. When Miss Blow sat down on the stone, Biddy\nHalloran hobbled up to her.\n\n\"Is it the doctor you're looking for?\" she said. \"For if it is, it's\nhardly ever you'll see him again.\"\n\nMiss Blow was startled, and demanded an explanation of the words.\nBiddy, who was slightly deaf, pretended to be very deaf indeed. Miss\nBlow's clear voice and determination of manner subdued her in the end.\nShe professed to be the only person in Clonmore who really knew what\nhad happened to the doctor.\n\n\"Holidays, is it?\" she said, recollecting what Patsy Devlin had told\nher, \"no, nor work either. It's to Dublin he's gone, and it's little\npleasure he'll find there. Och, but he was a fine man and it's a pity\nof him!\"\n\n\"Tell me at once,\" said Miss Blow, \"what he went to Dublin to do.\"\n\n\"There was a lump in the inside of him,\" said Biddy, \"a gathering like;\nand many's the time he told me of that same. It was the size of a young\npullet's egg, and you'd feel it lepping when you put your hand on it,\nthe same as it might be a trout. 'Biddy, agra,' he says, speaking to\nme, as it might be to yourself or to some other young lady that would\nbe in it, instead of an old woman like myself, 'medicine's no good,'\nsays he, 'but the knife is what's wanted.' 'Would you not be afeared,'\nI said, 'to be trusting yourself to them murdering doctors up in\nDublin, and maybe a young lady somewhere that would be crying her eyes\nout after you, and you dead?' 'I would not be afeared,' says he--och,\nbut he was a fine man!--'only I wouldn't like the girl that's to be\nmarried to me to know,' says he; 'I'd be obliged to you if you'd keep\nit from her,' says he; 'and what's more, I'll go to-morrow.'\"\n\nMiss Blow did not believe a word of it, but old Biddy Halloran reaped\nher reward. Jimmy O'Loughlin, when the conversation was reported to\nhim, sent her a present of a bottle of patent medicine which had been\na long time in the shop and appeared to be unsaleable. It professed to\ncure indigestion, and to free the system from uric acid if taken in\nteaspoonfuls after meals. Biddy Halloran rubbed it into her knees and\nfelt her rheumatic pains greatly relieved.\n\nMiss Blow sought and, after many inquiries, found the woman who had\nacted as Dr. O'Grady's house-keeper, and had basely deserted him in\nthe hour of his extremest need. She had taken refuge, as a temporary\nlodger, with Patsy Devlin's wife. It was understood that she would pay\nfor her board and lodging when her solicitor succeeded in recovering\nthe wages due to her. The news of the doctor's flight had depressed\nher. She felt that she was greatly wronged; but even when smarting from\nher loss, she was not so heartless as to revenge herself by telling the\nterrible truth to an innocent and beautiful creature like Miss Blow.\nShe gave it as her opinion that the doctor, driven to desperation,\nperhaps almost starved, had poisoned himself. He had, she asserted,\nbottles enough in his surgery to poison the whole country. His body,\nshe believed, was lying in the house behind the locked doors.\n\n\"If so be,\" she added, \"that the rats haven't him ate; for the like of\nthat house with rats, I never seen. Many's the time, when the doctor\nwould be out, I've sat the whole evening on the kitchen table, with my\nlegs tucked up under me, and them running across the floor the same as\nhens would come to you when you'd be calling them. You couldn't put\ndown a dish out of your hand, but they'd whip the bit off of it before\nyour eyes, without you'd have some sort of a cover to put over it.\"\n\nNo one who was even slightly acquainted with Dr. O'Grady could suppose\nhim capable of suicide under any conceivable circumstances. Miss Blow,\nwho of course knew him well, was quite unimpressed by the housekeeper's\nhorrible suggestion. But she realized that the truth, whatever it was,\nwas not to be reached by inquiries. Jimmy O'Loughlin and Patsy Devlin\nlied to her. So did Biddy Halloran. So did the house-keeper. There was\nevidently an organized conspiracy among the people of Clonmore for\nthe concealment of the truth. Miss Blow had a logical mind. It seemed\nplain to her that if everybody agreed to tell lies the truth must be\nsomething of a dangerous or uncomfortable kind. She had some knowledge\nof Ireland, gleaned from the leading articles of English newspapers.\nShe knew, for instance, that it was a country of secret societies, of\nmidnight murders, of defeated justice, of lawlessness which scorned\nthe cloak of hypocrisy. She had heard of reigns of terror, emphasized\nby the epithet \"veritable.\" She was firmly convinced that the lives\nof respectable people were not safe on the west side of the Shannon.\nHer father, Mr. Blow of the cigars, was an earnest politician, and at\nelection times his house was full of literature about Ireland which his\ndaughter read. Her experience of the people of Clonmore went far beyond\nher worst expectations. She made up her mind that Dr. O'Grady had been\nmurdered; that everybody in the place knew the fact; and that, either\nthrough fear or an innate fondness for crime, no one would help to\nbring the murderers to justice.\n\nIt is very much to her credit that she did not take the next train\nhome; for she must have thought that her own life was in great danger.\nBut she was a young woman of determination and courage. She made up her\nmind to discover and bring to the scaffold the men who had done away\nwith Dr. O'Grady. Her suspicions fastened, in the first instance, on\nJimmy O'Loughlin and Patsy Devlin.\n\n\"Mr. O'Loughlin,\" she said, when she returned to the hotel after her\ninterview with the housekeeper, \"kindly tell me who is the nearest\nmagistrate.\"\n\n\"You haven't far to go to look for a magistrate, miss, if that's all\nyou want. I'm one myself.\"\n\n\"I don't believe you,\" said Miss Blow, rudely.\n\n\"Maybe not,\" said Jimmy; \"but I'm telling you the truth for all that.\nLet you go into the Petty Sessions Court to-morrow, and see if I'm not\nsitting there on the bench; with the police and Mr. Goddard himself,\nthat's the officer, if he happens to be over from Ballymoy, doing what\nI bid them, be that same agreeable to them or not; and oftener it's\nnot, for them police think a lot of themselves. When you see me there\nadministering the law you'll be sorry for what you're after saying.\nIt's the Chairman of the Urban District Council I am, and an ex-officio\nmagistrate, thanks be to God.\"\n\n\"Is there any other magistrate in the neighbourhood?\"\n\n\"There is not; for the R. M. lives away off at Ballymoy, and that's\nbetter than twelve miles from this. There's ne'er another, only myself\nand Lord Manton up at the Castle, and he never sits on the bench from\none year's end to another, unless maybe there's a job on that he'd like\nto have his finger in.\"\n\nThe title produced its effect on Miss Blow. Earls are much less common\nin the industrial districts of England than they are in Ireland.\nThe statistics have never been exactly worked out, but there can be\nlittle doubt that there are far more earls to every thousand common\nmen in Ireland than in any other part of the three kingdoms. This is\nnot because governments are more generous to the Irish in the matter\nof titles. The explanation is to be found in the fact that untitled\npeople in Ireland tend to disappear, thinned out by famine, emigration,\nand various diseases, while the earls survive. In England it is the\nnoblemen who die away, being, as every reader of popular English\nnovels knows, a degraded set of men, addicted to frightful vices,\nwhereas the working men and the great middle class increase rapidly,\ntheir morality being of a very superior kind. Curiously enough, the\nEnglish, though perfectly aware of the facts, respect their debauched\nearls greatly, on account, it may be supposed, of their rarity. The\nIrish, on the other hand, think very little of an earl, regard him as\nin many respects similar to an ordinary man; earls being, as has been\nsaid, comparatively common in Ireland. Miss Blow, who had never to her\nknowledge seen an earl, brightened up at the mention of Lord Manton.\n\n\"I'll go up to the Castle,\" she said, \"and see him to-morrow morning.\"\n\nJimmy O'Loughlin sent a message to Patsy Devlin, asking him to call\nat the hotel that evening. The fact that he had not been elected\ninspector of sheep dipping still rankled in Patsy's mind. He blamed\nJimmy O'Loughlin more than any one else for his rejection. He made up\nhis mind to obey the summons, but not to be seduced from the path of\nrighteous wrath by porter or whisky. He would refuse contemptuously to\noblige Jimmy in any way.\n\nHe was received in Jimmy O'Loughlin's private office, a small room\nat the back of the hotel, which looked out on the yard. The walls\nwere adorned with two pictures, enlarged photographs of eminent\necclesiastics with small eyes and puffy cheeks. The table was mahogany\nand was covered with circular stains of various sizes. There was a\nsideboard with a very dilapidated cruet-stand and two teapots on it.\nThe chairs were all rickety. A writing-desk, which stood under the\nwindow, was littered with a number of exceedingly dirty papers. On the\ntable in the middle of the room, by way of preparation for Patsy's\nvisit, were arranged a jug of porter, a bottle of whisky, a water\ncroft, and several tumblers.\n\n\"Fill your glass,\" said Jimmy hospitably, \"and light your pipe. You can\nstart on the porter, and finish up with the spirits.\"\n\nPatsy poured out the porter suspiciously, and drank a tumbler full\nwithout any sign of appreciation.\n\n\"There isn't one about the place,\" said Jimmy, \"that's better\nacquainted with the old earl up at the Castle than yourself. He thinks\na deal of you, and well he may.\"\n\n\"He gave me a letter,\" said Patsy, \"at the time of the election. But\nit's little heed you or the rest of them paid to it.\"\n\nJimmy was anxious to avoid the subject of the election.\n\n\"I'm told,\" he said, \"that whatever you might ask of him, he'd do.\"\n\nPatsy was susceptible to flattery of this kind.\n\n\"He always thought a deal of me and my father before me,\" he said.\n\"You could tell the opinion he had of me by the letter he wrote. And\nwhy wouldn't he when either my father or myself put the shoes on every\nhorse that's come and gone from the Castle this fifty years.\"\n\n\"I could tell what he thought of you,\" said Jimmy. \"Sure anybody could.\"\n\n\"You could tell it, if so be you read the letter.\"\n\n\"The doctor's young lady,\" said Jimmy, \"is going up to see the earl\nto-morrow. The Lord save her! but she's half distracted with grief this\nminute.\"\n\n\"And what good will going to the Castle do her? Sure he doesn't know\nwhere the doctor is no more than another.\"\n\n\"He might tell her the truth,\" said Jimmy.\n\n\"Be damn! but he might, not knowing.\"\n\n\"And if he did, the girl's heart would be broke.\"\n\n\"It would surely.\"\n\n\"We've kept it from her,\" said Jimmy, \"and may the Lord forgive us for\nthe lies we're after telling, fresh ones every hour of the day. And if\nso be that now, at the latter end, she hears how the doctor has gone\nand left her it'll go through her terrible, worse than the influenza.\"\n\n\"And what would you consider would be best to be done?\" asked Patsy.\n\n\"I was thinking that maybe, if you was to see him to-morrow, early,\nbefore ever she gets at him with her questions, and if you were to give\nhim the word, that it might be, coming from a man like yourself that he\nhas a respect for, that he'd hold off from telling her.\"\n\n\"He might.\"\n\n\"And will you do it, Patsy Devlin? Will you do it for the sake of the\nfine young girl that's upstairs, this minute, heart scalded with the\nsorrow that's on her?\"\n\n\"It's little you deserve the like from me,\" said Patsy, \"you nor the\nrest of the Guardians. But I'll do it for the sake of the girl.\"\n\n\"I knew you would,\" said Jimmy. \"It's a good heart you have in you,\nPatsy Devlin, and a feeling for them that's in distress. But the\nporter's finished. Will I draw you another jug of the same, or will you\ntry the whisky for a change?\"\n\nPatsy indicated the whisky bottle with his thumb. He remained lost in\ndeep thought while the cork was drawn and a considerable quantity of\nthe spirit poured into the tumbler before him. Indeed, so complete was\nhis abstraction that the glass might have been absolutely filled with\nundiluted whisky if Jimmy had not, of his own accord, stayed the flow\nof it.\n\n\"I'm collecting the town and the neighbourhood,\" said Patsy, \"for the\nsports, and there's no reason that I can see why I shouldn't call on\nhis lordship to-morrow and ask for a subscription.\"\n\n\"You might.\"\n\n\"And in the course of conversation I could draw down about the doctor\nand the young lady and give him the word.\"\n\n\"Take care now would she be beforehand with you, if so be you were a\nbit late in going.\"\n\n\"Let you see to that,\" said Patsy.\n\n\"I might try,\" said Jimmy; \"but she's that headstrong and determined\nit's hard to stop her once she takes the notion into her head.\"\n\n\"Be damn!\" said Patsy, \"but however you manage you'll have to stop her.\nThe old earl doesn't have his breakfast took till near ten o'clock, and\nif I was to try to see him before half-past ten, he'd eat the face off\nme.\"\n\n\"I'll do the best I can,\" said Jimmy. \"I'll tell Bridgy to have the\nbreakfast late on her. She does be wanting it at half-past eight.\"\n\n\"Let her want. If she gets it by half-past nine itself oughtn't she to\nbe content? There's many a house where she wouldn't get it then.\"\n\n\"Content or not,\" said Jimmy heroically, \"it's at half-past nine she'll\nhave it to-morrow anyway.\"\n\n\"And after that,\" said Patsy, \"it could be that the horse might be lame\nthe way she'd have to walk.\"\n\n\"It could.\"\n\n\"And if you sent her round by the big gate,\" said Patsy, \"it would put\na couple of miles on her beyond what she'd have to walk if she was to\ngo up through the deer park.\"\n\n\"It would,\" said Jimmy; \"but the talk she'll give after will be\nterrible to listen to.\"\n\n\"Don't tell me. A young lady like her wouldn't know how to curse.\"\n\n\"It's not cursing,\" said Jimmy, \"but it's a way she has of speaking\nthat would make you feel as if the rats beyond in the haggard was\nChristians compared to you.\"\n\n\"Let her talk.\"\n\n\"And she looks at you straight in the face,\" said Jimmy, \"the same as\nif she was trying to see what would be in the inside of your head, and\nfeeling middling sure all the time that there wasn't much in it, beyond\nthe sweepings of the street.\"\n\n\"It's for her own good you're doing it,\" said Patsy.\n\nThere was some consolation in the thought. But Patsy, even while making\nthe suggestion, felt that a good conscience is not always a sufficient\nsupport in well-doing.\n\n\"You might,\" he added, \"be out about the place and let herself talk to\nher till the worst of it was over.\"\n\nThis plan, which perhaps would not have suited Mrs. O'Loughlin,\ncommended itself to Jimmy; but it did not make him altogether\ncomfortable about the future.\n\n\"I might,\" he said, \"and I will, but she'll get me for sure at the\nlatter end.\"\n\nIf he had done as his conscience suggested, Patsy Devlin would have\ngone home at once after settling Miss Blow's business for her. But the\nwhisky bottle was still more than half full, and it seemed to him a\npity to break up a pleasant party at an early hour. He started a fresh\nsubject of conversation, one that he hoped would be interesting to his\nhost.\n\n\"Tell me this now,\" he said. \"Do you think that fellow down at\nRosivera, the same that brought the pianos along with him, would give a\nsubscription to the sports?\"\n\n\"I don't know,\" said Jimmy. \"He's queer. I never set eyes upon him\nmyself since I finished carting the packing-cases down to Rosivera.\"\n\n\"They tell me that he does be calling at your shop for his bread and\nthe like, and leaving a power of money with you.\"\n\n\"I wouldn't say he left so much at all,\" said Jimmy cautiously. \"And\nanyway it's a servant that did be coming every day till to-day, and\nthen it was some sort of a foreigner with a written order, him not\nbeing able to speak English.\"\n\n\"Would you see your way to asking him for a subscription?\"\n\n\"How would I do it, when he can't know a word I say to him, nor him to\nme? Why won't you talk sense?\"\n\n\"And where's the man himself, and the fellow that did speak English?\"\n\n\"How would I know? If it's a subscription you want from him, you'd\nbetter go over to Rosivera and ask for it.\"\n\n\"They say,\" said Patsy thoughtfully, \"that he has plenty to give. A\nman like that with a motor car running on the road every day, and two\nforeign gentlemen, let alone an Englishman, to wait on him, must have a\npower of money. I wouldn't wonder now, if I took him the right way, but\nhe'd give five pounds. I might drop him a hint that five pounds is the\nleast that any of the gentry would give to the sports.\"\n\n\"Let you see what you can get out of him,\" said Jimmy, \"and the more\nthe better.\"\n\nJimmy had got all he wanted out of Patsy Devlin. He did not care very\nmuch whether Mr. Red subscribed to the sports or not. He took the\nwhisky bottle and drove the cork home into its neck with a blow of his\nfist. Patsy looked regretfully at it, but he was a man of self-respect.\nHe would neither ask for, nor hint that he wanted, a drink which his\nhost did not seem inclined to offer him freely. He realized what the\ndecisive blow given to the cork meant. There would be no more whisky\nfor him unless he chose to go out to the bar and pay for it there in\nthe usual way. This he was unwilling to do. Later on in the month,\nwhen the collection for the sports was complete, he might be in a\nposition to spend lavishly; but for the present he felt it necessary to\neconomize.\n\n\"It'll be off with me,\" he said. \"It'll be best, seeing I have to be\nup at the Castle early to-morrow. Later on in the day I might be going\nover to Rosivera to see what's to be got out of the man that's there.\nIf he's as rich as you're after telling me he'll never miss a pound, or\nfor the matter of that five pound. I'll have a try at him anyway.\"\n\n\n\n\nCHAPTER VI\n\n\n\"Is there any news of the doctor?\" asked Lord Manton.\n\nHe was standing on the steps outside the door of Clonmore Castle.\nHe had just given Patsy Devlin a sovereign for the Horse Races and\nAthletic Sports, and was endeavouring to cut short the thanks with\nwhich the subscription was received.\n\n\"There is not, your lordship, devil the word; and why would there? It\ncould be that he's on the sea by this time, and, anyway, why would he\nbe wanting to tell us where he is? Isn't it enough of their persecuting\nhe had without going out of his way to ask for more?\"\n\nLord Manton, like everybody else, regarded Dr. O'Grady's flight to\nAmerica as the natural result of his financial embarrassment. He was\nsorry; but he recognized that the doctor had taken the wisest course.\n\n\"Might I be speaking a word to your lordship about the doctor?\"\n\n\"Certainly, Patsy.\"\n\n\"It's what Jimmy O'Loughlin was saying to me that there'd be no need,\nif your lordship was agreeable to the same, to be telling the young\nlady the way the doctor is gone off and left her without a word. She\nhas trouble enough, the creature, without that.\"\n\n\"What young lady?\"\n\n\"Be damn!\" said Patsy hurriedly, \"if there isn't herself coming up the\navenue. It wouldn't do for her to see me talking to your lordship. I'd\nbetter be going before she's on top of us.\"\n\nPatsy Devlin slipped round the corner of the Castle, and dodging\nthrough a plantation of laurels, made his way to the stable-yard.\nLord Manton was left to watch the approach of Miss Blow, without any\nvery clear idea of what she was likely to want of him; or how Jimmy\nO'Loughlin and Patsy Devlin expected to keep the doctor's flight a\nsecret from her. He observed with pleasure that she was more than\ncommonly good-looking, that she carried herself well, and wore clothes\nwhich set off a fine figure. He had heard from Dr. O'Grady of the\ndaughter of the Leeds tobacconist, and had formed a mental picture of\nher which in no way corresponded to the young lady who approached him.\nHe reflected that she was probably in deep distress, and he looked\nforward with some pleasure to an interview in the course of which she\nwas almost certain to cry. He had no objection to playing the part of\ncomforter to a charming girl. His face expressed fatherly benignity\nwhen Miss Blow reached him.\n\n\"Am I addressing Lord Manton?\" she asked.\n\n\"Certainly. Is there anything I can do for you?\"\n\nHe almost added the words \"my dear,\" but there was a look in Miss\nBlow's fine eyes which checked him. He decided that paternal affection\nwould come in more appropriately after she began to cry.\n\n\"I am Miss Blow,\" she said.\n\n\"Come in,\" said Lord Manton, \"come in. You must be tired after your\nwalk. Let me lead the way into the library. I have often heard of you\nfrom my friend Dr. O'Grady, and if there is anything I can do to help\nyou I shall be most happy to do it.\"\n\nHe set Miss Blow in a deep chair near the window, pulled over another\nchair for himself, and sat down beside her.\n\n\"I am entirely at your service,\" he said. \"It will be a pleasure to me\nto give any help in my power to a charming young lady. I----\"\n\nMiss Blow's eyes warned him again. There was a hard glitter in them\nvery little suggestive of tears. He stopped abruptly.\n\n\"I understand that you are a magistrate,\" she said.\n\nLord Manton bowed. Then he sat up straight in his chair and tried to\nexpress in his attitude a proper judicial solemnity.\n\n\"I want,\" said Miss Blow, \"to have Dr. O'Grady found at once.\"\n\n\"A very natural and a very proper wish,\" said Lord Manton. \"I am in\nentire sympathy with you. I should like very much to find Dr. O'Grady.\nBut----\"\n\n\"Dead or alive,\" said Miss Blow firmly.\n\n\"My dear Miss Blow!\" The \"my dear\" came quite naturally to his lips\nthis time. The words expressed sheer astonishment. There was no\nsuggestion of affection, paternal or other, in the way he uttered them.\n\n\"Dead or alive,\" said Miss Blow again.\n\n\"Don't make such horrible suggestions, Miss Blow. I assure you there's\nnot the slightest reason for supposing that Dr. O'Grady is anything but\nalive and well.\"\n\n\"Then where is he?\" Miss Blow spoke sharply, incisively. Lord Manton\nbegan to think that she must be some new kind of girl, quite outside of\nhis experience, one who felt more indignation than sorrow at the loss\nof her lover.\n\n\"I understand,\" he said, \"that he is absent from home, temporarily\nabsent. I have no doubt----\"\n\nMiss Blow rose from her chair and took up her umbrella.\n\n\"You're like all the rest,\" she said. \"You are as bad as the\nhotel-keeper and his friend. You are simply trying to put me off with\nlies. Good morning.\"\n\n\"Wait a moment. Please do not hurry away. I am not like all the rest,\nreally. I assure you I'm,--compared to Patsy Devlin, for instance,--I'm\na miserably inefficient liar. Please sit down again.\"\n\nMiss Blow allowed herself to be persuaded.\n\n\"Tell me the truth,\" she said; \"and then find his body.\"\n\n\"The truth,\" said Lord Manton, \"is painful--very painful. But it's not\nso bad as that. Dr. O'Grady has been for some time past in a position\nof considerable pecuniary embarrassment.\"\n\n\"He was up to his neck in debt,\" said Miss Blow bluntly. \"I know that.\nThat's what brought me here. And now I find he's gone.\"\n\nThere was just a hint of a break in her voice as she spoke the last\nwords. Lord Manton thought that tears were at last imminent. He felt\nmore at his ease, and ventured to take her hand in his and to stroke it\ngently. She snatched it from him.\n\n\"You're worse than the others,\" she said. \"How dare you?\"\n\nFor a moment Lord Manton thought that she was going to box his ears. He\ndrew away from her hurriedly and attempted an apology.\n\n\"I am sincerely sorry,\" he said. \"For the moment I forgot that you\nwere not my daughter. She always came to me with her troubles ever\nsince she was quite a child. I got into the way of taking her hand----\"\n\n\"Never mind about my hand. Tell me the truth about Dr. O'Grady.\"\n\nLord Manton saw that she was mollified. To be mistaken for the daughter\nof an earl is a soothing thing under any circumstances. He thought\nfor an instant of trying to repossess himself of her hand; but Miss\nBlow's eyes, though no longer passionate, were steely. He felt himself\naggrieved, and spoke with brutal directness.\n\n\"To put the matter plainly,\" he said, \"Dr. O'Grady has run away from\nhis creditors.\"\n\n\"I don't believe a word of it,\" said Miss Blow.\n\n\"I have no doubt that he intended to let you know where he was going.\nI expect he wants you to go after him and join him there--make a fresh\nstart, you know, in the New World, and build up a happy home, where the\nmiserable past may be forgotten. That's what he means. I'm convinced\nof it. Only he had to leave this rather hurriedly in the middle of the\nnight. But don't be despondent, Miss Blow; you'll get a letter from him\nsoon.\"\n\n\"That's all nonsense,\" said Miss Blow. \"He's done nothing of the sort.\"\n\n\"But, my dear young lady, how can you possibly speak so confidently?\nHe's not the first man who has run away under such circumstances.\nPlenty of people do it, I assure you. It's not even considered\ndisgraceful.\"\n\n\"I know he didn't.\"\n\n\"But how do you know?\"\n\n\"Because I wrote to him a week ago, when I first heard he was in\ntrouble, and told him I was coming over here to see him. I said that\nfather would help him out of his difficulties, whatever they were. Do\nyou think that, after that, he'd run away and not so much as tell me he\nwas going?\"\n\nLord Manton did not know what to think. Dr. O'Grady had disappeared.\nThere was no getting out of that. It was a patent fact. On the other\nhand, if Mr. Blow had really offered to pay the doctor's debts, there\nseemed to be no reason why he should disappear. No doubt the wealthy\nproprietor of the well-known twopenny Beauties could afford to pay Mr.\nLorraine Vavasour's bill twenty times over if necessary. Still, Dr.\nO'Grady had disappeared.\n\n\"You are all,\" said Miss Blow passionately, \"a lot of slanderous\nbusy-bodies, telling lies and meddling with everybody's business\nbecause you have no business of your own to attend to. My poor Lucius\nhas been murdered among you, and now not one of you will help me to get\nat the truth; but I'll do it in spite of you.\"\n\nLord Manton looked at her. She was undeniably handsome; handsomer than\never now that she was in a rage. It occurred to him suddenly that Dr.\nO'Grady might have a reason for disappearing, quite unconnected with\nthe money he owed. He was engaged to be married to Miss Blow. It was\npossible that the idea of home life with this masterful and passionate\nyoung woman for a partner might be rather terrifying. Besides, the\nwife who pays her husband's debts for him has a hold over him ever\nafterwards; and Miss Blow seemed exactly the kind of lady who would\ntake advantage of such a position. She would certainly make him aware\nof the fact. Lord Manton thought he understood at last why Dr. O'Grady\nhad run away. Miss Blow's face was buried in her handkerchief. She\nwas not crying, but she was flushed after her outburst, and preferred\nto keep her face covered. Lord Manton ventured on a smile and a gentle\nchuckle.\n\n\"I assure you,\" he said soothingly, \"that he hasn't been murdered.\nWho would murder him? Everybody in the neighbourhood was fond of him.\nI don't think there was a man, woman, or child but loved him. I did\nmyself.\"\n\n\"If you loved him,\" said Miss Blow, \"show it now.\"\n\n\"I will, with pleasure; but how?\"\n\n\"Give me a search warrant.\"\n\n\"A search warrant! But----\"\n\n\"Yes, a search warrant; and I shall insist upon the police executing\nit.\"\n\n\"I haven't the least doubt you will; but--but what do you mean to\nsearch?\"\n\n\"Every house in the neighbourhood. Every house until I find him.\"\n\n\"But he isn't in a house. Do try to be reasonable, Miss Blow. Even\nif he's murdered--and I'm quite sure he's not--he wouldn't be in a\nhouse. His body would be hidden in a wood or a bog-hole or a river, or\nwherever it is that murderers usually do hide bodies.\"\n\n\"You admit then that he has been murdered.\"\n\n\"No, I don't. You mustn't catch up my words like that. All I said was\nthat, if he had been murdered, he wouldn't be living in a house, and\nso a search warrant wouldn't be any use to you. You don't really want\na warrant at all. You don't even want the police. All you have to do\nis to go prowling round the country, poking into any shadowy-looking\nhole you see with the point of your umbrella until you come across his\nbody.\"\n\nThe interview was beginning to tire Lord Manton. He was not accustomed\nto being bullied by handsome girls, and he did not like it.\n\n\"Perhaps you'd like to start at once,\" he said politely.\n\n\"It's impossible,\" she said, \"for me to search the country by myself.\"\n\n\"Not at all. Nothing is impossible for a young lady of your energy.\nStart with the wood behind this house; it's very thick in parts, quite\na likely spot for a corpse; and come in here for lunch when you've\nfinished.\"\n\n\"Give me a written order to the police,\" said Miss Blow, \"commanding\nthem to aid me in my search.\"\n\n\"It wouldn't be the least use to you, I assure you. You've no idea what\nindependent people the police are. An order from me would simply put\ntheir backs up and make them determined not to help you.\"\n\n\"Give me the order, and I'll see that they execute it.\"\n\n\"My dear Miss Blow, I can't, I really can't. Try the Chief Secretary.\nYou'll find him in his office at Dublin Castle. He's a most agreeable\nman. You needn't be the least bit afraid of him. Not that it's likely\nyou would be. He's much more likely to be afraid of you. It won't take\nyou long. You can run up by the night mail and----\"\n\n\"Give me the order.\"\n\nLord Manton surrendered. He crossed the room, sat down at his desk, and\nwrote--\n\n\"Sergeant Farrelly, R.I.C. Kindly give all the assistance in your power\nto Miss Blow, the bearer of this note, who wishes to search the country\nfor a dead body.--MANTON.\"\n\n\"If that is any use to you,\" he said, \"you're welcome to it. Let me\nknow how you get on. Any time you happen to find yourself near this\nhouse, drop in for luncheon or tea. Good-bye.\"\n\nMiss Blow rose, bowed, and left the room. Lord Manton rang the bell.\n\n\"Wilkins,\" he said to his butler.\n\n\"Yes, my lord.\"\n\n\"You saw that young lady who left the house just now? Very well, if\nshe calls again and I happen to be out, you are to give her breakfast,\nluncheon, tea or dinner according to the hour of the day. If I happen\nto be at home you are to stay in the room during the time she is with\nme, as a chaperone. You understand, Wilkins?\"\n\n\"Yes, my lord.\"\n\n\"She doesn't require a chaperone, but I do. I don't feel safe when\nI'm alone with her. And Wilkins, if she brings a corpse along with\nher, either Dr. O'Grady's corpse or any other, you will provide proper\naccommodation for it. Put it on the table in the servants' hall with a\nsheet over it, and send out to the garden for flowers--white flowers.\"\n\n\"Yes, my lord.\"\n\n\"One thing more, Wilkins; if Sergeant Farrelly or any policeman comes\nup here from the barracks either to-day or to-morrow and asks to see\nme, tell him I'm out, and that it won't be the least use his waiting\nbecause I won't be in before midnight and probably not then.\"\n\n\"Yes, my lord.\"\n\nWilkins left the room, and Lord Manton, taking the chair in which Miss\nBlow had been sitting, lit a cigarette. There was a stealthy step on\nthe gravel outside. He looked up and saw Patsy Devlin's face pressed\nagainst the window. He rose, opened the window, and asked Patsy what\nhe wanted.\n\n\"Might I be so bold as to put a question to your lordship?\"\n\n\"Is it about Miss Blow or Dr. O'Grady?\"\n\n\"It is,\" said Patsy. \"It's about the both of them.\"\n\n\"Out with it, then. That young woman and her relations with the doctor\nform quite the most interesting subject I know at present.\"\n\n\"Did your lordship keep the truth from her?\"\n\n\"I did my best not to, Patsy; but I think I may safely say that I did.\"\n\nPatsy pondered this saying. The meaning was not immediately obvious to\nhim.\n\n\"It's what Jimmy O'Loughlin was saying to me last night,\" he said,\n\"that if so be she heard that the doctor had left her, the creature's\nheart would be broke, and her as handsome a young girl as any you'd\nsee.\"\n\n\"At the present moment,\" said Lord Manton, \"she believes that you and\nJimmy O'Loughlin have murdered the doctor and concealed his body in a\nbog hole.\"\n\n\"The Lord save us and deliver us! Was there ne'er another story you\ncould tell her only that? Sure the police will be out after us.\"\n\n\"She went straight from this house to the barracks,\" said Lord Manton,\n\"and I shouldn't wonder if you were arrested before night.\"\n\n\"Be damn!\" said Patsy, \"saving your lordship's presence; but they\ncouldn't take me for the like of that. There isn't one in the country\nbut knows that I wouldn't lay a hand on the doctor, drunk or sober, not\nif it was to save my soul.\"\n\n\"Don't you be too sure,\" said Lord Manton. \"My own belief is that if\nMiss Blow doesn't come across the doctor in the course of the next\ntwenty-four hours, she'll have you hanged for murdering him.\"\n\n\"It's joking you are. She couldn't do it.\"\n\n\"I am not joking. I defy any man to joke after spending half an hour\nwith Miss Blow. She is the most determined young woman I ever met. She\ncould do anything, absolutely anything. There isn't a judge or a jury\ncould stand out against her for an hour. If I were you, Patsy, I'd make\na bolt for it and join the doctor in America. Your life's not safe in\nthis country. I expect by this time you have as much collected for the\nsports as would pay your passage out.\"\n\n\"I have,\" said Patsy; \"but I'll not go. I'll stand my trial, if it\ncomes to that, sooner than have people saying it of me that I ever laid\na hand on the doctor.\"\n\n\"It's all very fine talking that way, but you won't feel at all so\nconfident when you see the judge putting on the black cap; and a little\nlater on, when the rope's round your neck, you'll be sorry you didn't\ntake my advice. The thing for you to do is to skip while you've time.\nIt'll take Miss Blow a good while to persuade the police to arrest you,\nbut she'll do it in the end.\"\n\n\n\n\nCHAPTER VII\n\n\nA poet, writing, as some of them will, a parody on the work of another\npoet, has these words--\n\n  \"To every Irishman on earth\n   Arrest comes soon or late.\"\n\nPatsy Devlin did not read much poetry, and had never come across the\nlines. If he had met them, he would have recognized at once that they\nexpress a great truth. His experience of life convinced him that\nthe law in Ireland, though erratic in its methods, may be relied on\nin the end to get the upper hand of either daring or innocence. The\nproceedings of the police, depending as they do on the view which\nsome complete stranger has promised his constituents in England to\ntake of Irish affairs, are quite incalculable. Patsy himself had been\npraised by political orators, had been favourably mentioned by eminent\nstatesmen in the House of Commons itself, for actions which he would\nhave kept concealed if he could. What seemed to him to be serious\nmisdemeanours, if not actual crimes, had more than once turned out to\nbe virtuous deeds. It seemed likely, therefore, that the law might\noccasionally regard entire innocence as highly culpable. He would\nhave been indignant, but he would not have been greatly surprised,\nif the Government, that remote deity from which there is no appeal,\nhad decreed his arrest, trial, and execution for the murder of Dr.\nO'Grady. Patsy reasoned the matter out with himself. If, he thought,\na man is not punished for the crimes he does commit, it is probable\nthat he will be punished for those he has avoided committing. This\nconsideration, coming on top of Lord Manton's friendly warning, made\nhim uneasy. He determined to keep clear of the police barrack when he\nleft Clonmore Castle.\n\nRosivera is a remote and lonely spot. It was extremely improbable that\nthere would be any police lurking near it. It was not the sort of\nplace to which a police sergeant would think of going if he were bent\non the arrest of a murderous blacksmith. Patsy felt that he might,\nwithout running any undue risk, venture on a visit to Mr. Red. He was\nnot willing to forego the chance of getting an additional subscription\nto the sports fund. He had in his pocket money enough to take him to\nAmerica, but another pound, two pounds, perhaps even five pounds, would\nbe very welcome to him. He was a good man, with a tender heart and a\nstrong sense of his duty to those dependent on him. He wanted to be\nin a position to make some provision for the support of his wife and\nfamily when he left them. Mr. Red's subscription, if Mr. Red turned out\nto be a generous man, would enable him to leave Mrs. Devlin and the\nchildren comfortably provided for.\n\nHe went to Rosivera by a circuitous route, avoiding the public roads as\nmuch as possible, walking over fields and finally along the sea-shore.\nWhen he came in sight of the house, he reconnoitred it carefully, and\napproached it very much as a skilful scout might advance on an enemy's\ncamp, availing himself of all the cover which the country afforded.\nSatisfied at last that there was no police patrol in the neighbourhood,\nhe made a circuit of the house, and finally reached the yard gate\nby way of the kitchen garden. He entered the yard and made sure that\nthere was no one in it. He peeped into the stable and the cow-house and\nfound that they were both empty. He opened the door of the coach-house\nand took a long look at the motor car which stood there. It was well\ncleaned; its lamps and other metalwork shone brilliantly; it was a very\nhandsome vehicle. Patsy felt reassured. Mr. Red might be eccentric,\nas Jimmy O'Loughlin hinted, might even be vicious, but he was\nunquestionably opulent. No one but a rich man could keep such a motor\ncar. Patsy closed the coach-house door quietly and took a long look at\nthe back windows of the house. They were all shut and veiled with drawn\nblinds; all of them, except one small window in the top storey. It was\nwide open. Patsy stared at it.\n\nSuddenly, something flew from the window and dropped at Patsy's feet.\nIt was a pellet of paper. Patsy looked round him cautiously and then\nstooped down and picked it up. He unwrapped the paper and discovered\nthat it contained a small piece of china, apparently a fragment of a\nbroken saucer. He put this into his waistcoat pocket and examined the\npaper in which it had been wrapped. He discovered a few words on it\nscribbled in pencil--\n\n\"Come back into the yard in twenty minutes. Don't wait\nnow.--O'GRADY.\"\n\n\"Be damn,\" said Patsy softly, \"it's the doctor himself!\"\n\nBeing a man of high intelligence, with a natural taste for conspiracy,\nhe acted in the wisest possible way. Without the smallest display of\nemotion, or a single glance at the window from which the communication\nhad come, he turned and slouched carelessly towards the yard gate.\nIt was flung open before he reached it, and Mr. Red, a revolver in\nhis hand, strode forward. Patsy displayed great presence of mind and\nresource.\n\n\"I'm just after knocking at the back door, your honour,\" he said,\n\"thinking that it might be more agreeable if I didn't go round to the\nfront, where maybe you'd be entertaining company. It was that I was\ncollecting a trifle from the gentry round about for the grand annual\nhorse races and athletic sports that does be held every year up beyond\nin Jimmy O'Loughlin's big field. And the committee would feel pleased,\nyour honour, if you'd act as a vice-president or a starter, or the\nlike, along with Lord Manton from the Castle.\"\n\nMr. Red raised the revolver and pointed it at Patsy's head.\n\n\"Hand me that note,\" he said.\n\n\"Sure your honour's joking. What would a poor man like me be doing with\na note? If you're the gentleman they say you are, it's yourself will be\ngiving a note to me, and maybe a five-pound note, for the sports.\"\n\nMr. Red took four steps forward, and stood so that Patsy had every\nopportunity of looking into the barrel of the revolver.\n\n\"Right about turn,\" he said; \"march!\"\n\n\"I was in the militia one time,\" said Patsy, \"and I know well what\nyou're saying. If it's into the house you want me to go through the\nback door, I'm willing. But there's no need for you to be looking at me\nthat way or to be reaching out at me with your pistol. If you think I'm\nhere trying to steal a motor car on you, you're making a big mistake.\nAnybody can tell you that I wouldn't do the like. If I wanted to itself\nI wouldn't be able. I couldn't drive one of them things no more than\nfly.\"\n\n\"March!\" said Mr. Red.\n\nHe held the revolver within a couple of inches of Patsy's head.\n\n\"A gentleman like yourself,\" said Patsy, \"likes his bit of a joke. I\nknow well it's only funning you are and that it's not loaded; but I'd\nbe obliged to you if you'd point it the other way. Them things goes off\nsometimes when you're not expecting them.\"\n\nBy way of demonstrating that it was loaded, and that he was not\n\"funning,\" Mr. Red fired a shot. The bullet went quite close to Patsy's\nhead and buried itself in the kitchen door. Patsy, convinced that he\nhad to do with a dangerous lunatic, turned quickly and walked into the\nkitchen. From the kitchen he was forced, at the point of the revolver,\nup several flights of stairs. He was bidden to halt at last opposite a\ndoor. Mr. Red produced the key from his pocket, and, still keeping the\nrevolver levelled at Patsy, opened the door.\n\n\"Enter,\" he said.\n\n\"Be damn! but I will, and I'll be all the better pleased if you'll stay\noutside yourself.\"\n\nThis was exactly what Mr. Red did. The door was locked again, and Patsy\nfound himself face to face with Dr. O'Grady.\n\n\"I'm sorry,\" said the doctor; \"I'm infernally sorry. I was an ass to\nthrow you out that note. I might have known that the Field Marshal\nwould be spying round somewhere. It's just the kind of absolutely\nidiotic thing he does rather well.\"\n\n\"You needn't be sorry at all. Now that I know I'm not shot, I'd as soon\nbe here as anywhere else.\"\n\n\"Would you? I'm glad you're satisfied. All the same I wish you were out\nof it. Now that there are two of us here, the police are bound to come\nafter us and find us.\"\n\n\"They're out after me, anyway,\" said Patsy. \"That's why I say I'd as\nsoon be here as anywhere else.\"\n\n\"And what do they want you for? Is it any of your League work?\"\n\n\"It is not. It's nothing to do with the League, good or bad. It's for\nmurdering you and concealing your body after.\"\n\n\"Can't you talk sense, Patsy Devlin?\"\n\n\"It's the truth I'm telling you, and I couldn't say different if I was\nput on my oath this minute.\"\n\n\"But, damn it all, I'm not murdered; I'm alive.\"\n\n\"That may be,\" said Patsy. \"All I know is that Lord Manton's after\ntelling the young lady that you are murdered; and what's more, he said\nit was Jimmy O'Loughlin and myself that done it.\"\n\n\"Tell me the truth now, Patsy. Is Miss Blow in Clonmore?\"\n\n\"She is.\"\n\n\"You're sure of that?\"\n\n\"I am sure. She came the day after you went to America. Why wouldn't I\nbe sure when she has the whole of us riz ever since with the questions\nshe did be asking about you; and not one in the place but told her\nlies, be the same more or less, for fear the creature would break her\nheart if she heard what you were after doing. And at the latter end\nhis lordship told her we had murdered you, to quiet her like, for fear\nshe might hear that you had gone to America, leaving her behind you,\nwithout ever a word to her, good nor bad.\"\n\n\"Good God, man! But I haven't gone to America.\"\n\n\"I see that well enough now. But tell me this, doctor, why didn't you\nsend us word, so as we'd know what to say to her?\"\n\n\"I couldn't. The first chance I got was when I dropped that note out of\nthe window to you. If you'd come back to the yard the way I told you,\nI'd have had a letter written to Jimmy O'Loughlin that you could have\ntaken back with you. I'd have explained the whole situation.\"\n\n\"I was meaning to come back just as you bid me. Wasn't I walking out\nof the yard quiet and easy, so as I'd be able to come back at the end\nof half an hour or thereabouts, when that murdering villain came at me\nwith a pistol and went very near shooting me dead?\"\n\n\"I'll tell you what it is, Patsy. The sooner you're out of this the\nbetter. There's not the least difficulty about escaping. The Field\nMarshal thinks he's a tremendous swell at conspiracies of all sorts;\nbut as a matter of fact he's a perfect fool, and I have the lock\nloosened on the door this minute. You can walk out any time you like;\nand the best time in my opinion will be to-night.\"\n\n\"And why wouldn't you go yourself, doctor, if it's as easy as all that?\"\n\n\"I don't want to go,\" said the doctor. \"I'm very well contented where I\nam. It's much better for you to go.\"\n\n\"How would it do if the both of us went?\"\n\n\"It wouldn't do at all. I tell you I want to stay. I don't want to\nescape. But you must. I don't want the police here searching for me.\"\n\n\"Be damn, then, but I won't go either! As sure as ever I went they'd\nhave me hanged for murdering you and that wouldn't suit me at all.\"\n\n\"Don't be a fool, Patsy. How can they hang you for murdering me when\nI'm alive?\"\n\n\"But--but--without I'd bring them here to see you they'd never believe\nthat you weren't dead. What with the young lady going round the country\ncursing like mad at them that killed you, and the old lord telling the\npolice it was me and Jimmy O'Loughlin done it, what chance would a poor\nman like myself have against them? They'd have me hanged, I'm telling\nyou.\"\n\n\"I'll give you two notes,\" said Dr. O'Grady; \"one to Lord Manton, and\none to Jimmy O'Loughlin, and between the three of you it'll be a queer\nthing if you can't keep the police quiet and stop all this silly rot\nabout my being murdered.\"\n\n\"Will you give me a writing to the young lady herself?\"\n\n\"I will not. I know very well what she'd do if she heard I was here.\nShe'd come straight down after me.\"\n\n\"I'm not saying but what she might. From what I seen of her I'd say\nshe's just the sort of a young lady that would.\"\n\n\"Well, then, can't you understand that the last thing in the world I\nwant is for her to know where I am? If I could have got a note to her\nat the first go off, before she came to Clonmore, it would have been\nall right. I'd have told her that I was detained here in attendance\non an important case, and that she was to stay at home and not come\nnear Clonmore till I sent her word. But that silly old ass of a Field\nMarshal thought he knew better than I did how to deal with a girl, and\nhe wouldn't let me send the note. Then you and Jimmy O'Loughlin and\nLord Manton, and I suppose every soul about the place, go stuffing her\nup with a pack of lies until----\"\n\n\"Be damn, doctor! but that's a hard word. What we did was for the best.\nYou wouldn't have us tell the creature the truth, and her thinking all\nthe time that you'd rather die than desert her?\"\n\n\"Tell her the truth! What truth?\"\n\n\"That you were off to America so as you wouldn't have to pay Jimmy\nO'Loughlin what you owed him.\"\n\n\"But--oh, damn it all, Patsy, what a fool you are! That's not the\ntruth. Can't you see that it's not the truth? Will I be obliged to\nleave a mark on your ugly face with my fist to prove to you that I'm\nnot in America?\"\n\n\"It was all the truth we had for her, anyway. But we wouldn't tell her.\nAnd why not? Because she was a fine girl, and we didn't want to see her\ngoing off into a decline before our eyes and maybe dying on us. And\nbecause we had a respect for your memory; and that's more than you had\nfor yourself, hiding away here from a girl that any man might be proud\nto own. And it's more than you have for us, putting the hard word on\nus, and we doing the best we could from the start.\"\n\nDr. O'Grady was a reasonable man. His anger cooled. He came to see\nthat his friends had acted with the best intentions. He apologized\nhandsomely to Patsy Devlin.\n\n\"All the same,\" he added, \"you will have to go. I tell you what it is,\nif the police do come here, the Field Marshal will shoot the two of us.\nHe told me himself that that's what he'd do. And, whatever else he may\nbe, he's a man of his word.\"\n\n\"He dursn't, not with the police in the house. He'd be hanged.\"\n\n\"He doesn't care a pin whether he's hanged or not. As a matter of fact,\nI expect he'd rather like to be hanged. He's an anti-militarist.\"\n\n\"I was just thinking,\" said Patsy, \"when he gave the word of command to\nme there in the yard, that he'd been in the militia himself some time.\"\n\n\"Well, he hasn't, so you're out there. So far as I can make out, one\nof his main objects in life is to blow up the militia, and the regular\narmy along with it. He's an anarchist of the most advanced kind.\"\n\n\"Be damn, and is he that?\"\n\n\"He is. And I can tell you an anarchist isn't what you'd call a\nplayboy. Anarchism isn't a bit like your futile old League. It doesn't\ngo about the country making speeches and pretending it's going to\nboycott people that it hasn't the least notion of doing any harm to. A\ngenuine anarchist, a man like the Field Marshal, for instance, doesn't\nsay a word to anybody, but just goes quietly and blows up a town.\"\n\n\"I'll not have you speaking against the League, doctor. I've been a\nmember of it since ever there was a branch started in Clonmore. I'd\nbe a member of it still, if it wasn't that they went against me the\ntime of the election of the inspector of sheep dipping. I can tell you\nthere's them in it would think very little of making the country hot\nfor the man that went against the will of the people in the matter of\ngrazing ranches or the like.\"\n\n\"I don't want to argue about the League either on one side or another.\nWhat I'm trying to get you to understand is simply this. You've got to\ngo, and to go to-night, as soon as ever the Field Marshal is tucked up\nin his little bed and the house quiet. Listen to me now, and I'll make\nthe position plain to you. As long as I was here by myself I was more\nor less safe. The disappearance of one man doesn't make much difference\nin a neighbourhood like ours; but when it comes to two men vanishing in\nthe inside of one week there's bound to be a fuss. The police will take\nthe matter up to a certainty. If they come here--and they will come\nhere when they've tried everywhere else--you and I will be shot by the\nField Marshal.\"\n\nHe looked at Patsy as he spoke, and noticed with regret that he was\nproducing little or no impression.\n\n\"And what's more, I'll lose well over a hundred pounds; a hundred\npounds that I want badly.\"\n\n\"Why didn't you tell me that before?\" said Patsy. \"Is it likely now\nthat I'd want to stand between you and a lump of money like that? I\nwouldn't do it to any man, much less one like yourself, that I have a\nrespect for. Give me the writings now that we were speaking of, and\nI'll start at once.\"\n\n\"You can't start till night; but I'll write the notes at once if you\nlike.\"\n\n\"And the one to the young lady along with the other two.\"\n\n\"I told you before,\" said Dr. O'Grady, \"that I won't give you a note to\nher.\"\n\n\"Then I'll stay where I am. It's more than I dare to go back without a\nline of some sort to quiet her. Don't I tell you she'd have me hanged?\nAnd when that's done she'll be down here after you with the police, and\nyou'll be as badly off as you were before.\"\n\n\"She'll not be able to do that. Lord Manton would stop the police.\"\n\n\"She'll come without them, then. That sort of a young lady would do\nanything.\"\n\n\"If she does,\" said Dr. O'Grady, \"the Field Marshal will shut her up\ntoo. He won't do her any harm, and I'll be delighted to have her here.\"\n\n\"You want her here along with you?\"\n\n\"I'd be glad to have her, of course. Don't you know that she and I are\nengaged to be married?\"\n\n\"Well, aren't you the queer man? Anybody'd think you were trying to\nhide yourself from her for fear she'd marry you against your will.\"\n\n\"Is that one of the lies you and Jimmy O'Loughlin have been telling\nMiss Blow?\"\n\n\"It is not, of course.\"\n\n\"Well, don't let me catch you saying anything of the sort, or it'll be\nthe worse for you. Now, leave me in peace till I write the notes.\"\n\n\"I'm not going with them,\" said Patsy; \"so you needn't trouble yourself\nto be writing.\"\n\n\"All right. If you prefer to stay here and be shot, you can. You'll be\nsorry afterwards, that's all. I tell you the Emperor is not a man to be\ntrifled with. There's a fellow downstairs here that's sick, and I go\ntwice every day to attend him. I give you my word, Patsy, all the time\nI'm dressing his wounds I have the muzzle of that revolver stuck up\nagainst the back of my neck. I'd be uncommonly nervous if I didn't know\nthat the poor old Emperor is a good sort and reliable, in spite of his\nfondness for yellow crocodiles.\"\n\n\"Is there crocodiles in this house?\"\n\n\"There are; large yellow ones. The dining-room is crawling with them.\"\n\n\"That settles it, then,\" said Patsy. \"If I was ever so keen to get out\nof this, I wouldn't do it after that. I'd be in dread of them beasts;\nand I dursn't face them. They'd be out after me, and me going down the\nstairs in the dark; and I wouldn't know how to speak to them the way\nthey wouldn't bite.\"\n\n\"Don't be an ass, Patsy. They're not real crocodiles. They're dead.\"\n\n\"Alive or dead, I'll not face them; so it's no use your talking.\nThey're not what I'm used to, and I'd rather stay here along with you\nwhere I'd be safe.\"\n\n\n\n\nCHAPTER VIII\n\n\nMiss Blow went straight from Clonmore Castle to the police barrack.\nShe was received at the door by Constable Moriarty, who happened to be\non duty at the time. He was a young man who had only recently joined\nthe force. Miss Blow, after a glance at his smooth boyish face, asked\nto see the sergeant. She was shown into a small room, known as the\noffice, and kept waiting while Sergeant Farrelly, who was digging\npotatoes in the garden, \"cleaned himself.\" Her manner, when he joined\nher, was peremptory. She demanded that a search party should start at\nonce and scour the country for Dr. O'Grady's body. Sergeant Farrelly\nwas puzzled, and scratched his head. Miss Blow handed him Lord Manton's\nnote. He read it, was very perplexed, and scratched his head again.\nMiss Blow pressed her demand.\n\n\"It will be better,\" said the sergeant at last, \"if I go up to the\nCastle, and speak to his lordship myself. If you'll have the kindness,\nmiss, to leave your name and address, I'll communicate with you.\"\n\n\"I'll wait here,\" said Miss Blow, \"till you return.\"\n\n\"Is it in the barrack, miss?\"\n\n\"Here,\" said Miss Blow firmly.\n\nSergeant Farrelly looked at her helplessly. He did not want a handsome\nyoung lady in the barrack; he thought his office an unsuitable place\nfor Miss Blow; but he saw no way of altering her determination. He left\nher and summoned Constable Moriarty.\n\n\"The young lady within,\" he said, \"will wait in the office until such\ntime as I come back from the Castle, where I'll be speaking on business\nto his lordship. I leave it to you, Constable Moriarty, to see that\nshe's treated with proper respect.\"\n\n\"Is it me?\" said Moriarty.\n\n\"It is you. You can take her in yesterday's paper, and if it happens\nthat she's read it already, you can talk to her, making yourself as\npleasant and agreeable as you know how.\"\n\nWilkins, Lord Manton's butler, was a good servant. He opened the door\nto Sergeant Farrelly at about twelve o'clock, and blandly gave the\nmessage with which he had been charged.\n\n\"His lordship is out, and it is uncertain at what hour he will return.\"\n\nSergeant Farrelly was baffled. He went back to the barrack. He\nfound that Miss Blow had moved from the office, which was small and\nincommodious, and had settled herself in the men's day room. Constable\nMoriarty, embarrassed and very pink about the face, stood in front\nof her. Constable Cole, who was off duty, was grinning in a corner.\nMiss Blow at once reiterated her demand that a party should go out to\nlook for Dr. O'Grady's body. Sergeant Farrelly reasoned with her. When\nreason failed, he tried to impress her with the majesty of the law. He\nwas a portly man and over six feet in height. On patrol duty, on guard\nover a public-house on Sunday, or giving evidence in court as to the\namount of drink taken by a prisoner, he was an impressive man. He did\nnot impress Miss Blow. Being an English woman, she held the curious\ntheory that the police exist for the protection of the public, and\nthat they ought to engage willingly in the investigation of crime.\nSergeant Farrelly knew, of course, that this was not true; but he was\nunable to explain his position to Miss Blow, because she would not\nlisten to what he said.\n\nAt two o'clock, Miss Blow being still immovable in the day room of the\nbarrack, Sergeant Farrelly started again for Clonmore Castle. This\ntime he was accompanied by Constable Moriarty, who, reckless of the\nconsequences of not obeying orders, refused to be left to entertain\nMiss Blow. Constable Cole slipped quietly out into the garden and took\na turn at the potatoes which the sergeant had been obliged to leave\nundug.\n\nWilkins said politely what he had said before. Sergeant Farrelly and\nMoriarty sat down in the hall to wait. They waited till four o'clock.\nThen they returned to the barrack, hoping that Miss Blow would have\ngone home. They found that she had not gone and showed no signs of\ngoing. She was sitting in the men's room, eating biscuits out of a\npaper bag. It appeared afterwards that Constable Cole had gone out and\nbought the biscuits for her, fearing that she might be hungry. Sergeant\nFarrelly reprimanded him sharply for this.\n\nMiss Blow gave the sergeant and his men her opinion of the Royal Irish\nConstabulary. She used plain and forcible language, repeating such\nwords as incapacity, inefficiency, and cowardice at frequent intervals.\nShe spoke for nearly half an hour, and then demanded again that the\nwhole force should set out at once and search for Dr. O'Grady's body.\nConstable Cole grinned, and was caught in the act. The sergeant snubbed\nhim promptly. Miss Blow took off her hat and jacket, and said she\nintended to stay where she was until a search party went out.\n\nSergeant Farrelly and his men withdrew and held a counsel in the\nkitchen. Constable Moriarty suggested that Miss Blow should be arrested\non a charge of drunkenness, and locked up for the night.\n\n\"If she isn't drunk,\" he argued, \"she wouldn't be behaving the way she\nis.\"\n\nHis advice was not taken. In the first place, she was a well-dressed\nand good-looking young woman; and Sergeant Farrelly, being unmarried,\nwas a courteous man. In the second place, she had come to the barrack\nbearing a note from Lord Manton, and however unintelligible the note\nmight be, it had unquestionably been written by a peer of the realm. In\nthe third place, as Constable Cole pointed out, their object was not to\nkeep Miss Blow in the barrack, but to get her out. Pressed by Moriarty\nand the sergeant for an alternative scheme, Cole suggested vaguely that\nthey should resort to what he called a \"stratagem.\" This sounded well;\nbut it turned out in the end that he had nothing better to suggest than\nthat Jimmy O'Loughlin should be induced to send a message to her to\nthe effect that her dinner was ready and would be spoiled if she kept\nit waiting. The plan received no support. Sergeant Farrelly pointed\nout that it would be most unwise to confess their difficulty to Jimmy\nO'Loughlin.\n\n\"That fellow,\" he said, \"would take a delight in turning the police\ninto ridicule, and setting the whole country laughing at us. And\nbesides,\" he said, with a look of withering contempt at Constable Cole,\n\"it's not likely she'd be caring about her dinner after you giving her\nsixpennyworth of biscuits and more. Believe you me, she wouldn't mind\nthis minute if she never saw dinner again.\"\n\nHalf an hour later Sergeant Farrelly himself offered Miss Blow a cup\nof tea. He was a kindly man, as most police sergeants are, and it\ngrieved him to think that the young lady who had established herself\nin his barrack was spending a whole day with nothing to eat except dry\nbiscuits. She took the tea without thanks, and again demanded that\nthe search for Dr. O'Grady should begin. Sergeant Farrelly became\ndesperate. He set out at once again for Clonmore Castle. This time he\nwas accompanied by both constables, and Miss Blow was left in sole\npossession of the barrack. He learned that Lord Manton was still out.\nAfter a short consultation, he and the two constables sat down in the\nhall to wait. They waited till ten o'clock, and would have waited\nlonger still had not Wilkins informed them that it was his duty at that\nhour to lock up the house for the night. The police returned to their\nbarrack, disputing hotly about the best way of disposing of Miss Blow\nfor the night. To their immense relief they found that she had gone.\n\nAt nine o'clock next morning she walked into the barrack again and took\nher seat in the men's day room. This time she had with her a brown\npaper parcel. Constable Cole gave it as his opinion that it contained\nprovisions for the day.\n\n\"I shall stay here,\" she said firmly, \"until you choose to do your\nduty.\"\n\nSergeant Farrelly, who was refreshed and invigorated by his night's\nsleep, began to argue with her. The two constables stood near the\ndoor of the room and admired him. Miss Blow, adopting a particularly\nirritating kind of tactics, refused to pay any attention to his\nremarks. Whenever he paused to give her an opportunity of stating her\ncase, she said--\n\n\"I shall sit here until you choose to do your duty.\"\n\nShe had just repeated her formula for the ninth time when a groom rode\nup to the door of the barrack. He brought a note from Lord Manton.\n\n\"Sergeant Farrelly, R.I.C. Lord Manton is seriously annoyed to hear\nthat the police spent the greater part of yesterday afternoon and\nevening in the hall of Clonmore Castle. Lord Manton has not asked\nfor police protection, and knows no reason why it should be forced\non him. Lord Manton will not be at home to-day, and he requests that\nany communication by way of apology or explanation be made to him in\nwriting.\"\n\nThe note was passed round and read with dismay. Sergeant Farrelly and\nthe two constables, moved by a common impulse, withdrew together to the\nkitchen. For a time they stood silent and dejected. Then the sergeant,\nassuming an air of confident authority, gave his order.\n\n\"Constable Moriarty,\" he said, \"you will take that note over to\nBallymoy and hand it to the District Inspector. You will kindly explain\nat the same time the way we find ourselves fixed here.\"\n\n\"Maybe,\" said Moriarty, \"it would be as well if I was to take the other\nnote along with it--the one his lordship was after sending with the\nyoung lady about the corpse of Dr. O'Grady.\"\n\n\"It would be as well,\" said the sergeant; \"I'd be glad he'd see that\nnote too. But it's the young lady has it and not me. Did you happen to\nthink now of e'er a way that it could be got from her?\"\n\n\"Would you ask her for it?\" said Constable Moriarty.\n\n\"I might ask her for it, and I might ask the King if he'd lend me the\nloan of his crown to go courting in. I'd be as likely to get the one as\nthe other by asking. If you can think of no better way of getting it\nthan that, Constable Moriarty, you may go and ask for it yourself; and\nyou can come back here and let us have a look at you, when she smacks\nyour face.\"\n\n\"We might try a stratagem with her,\" said Constable Cole, who had made\na similar suggestion the day before. \"I was reading a book one time\nabout a man that was great on stratagems. There wasn't a thing would\nhappen but he'd---- I'm sorry now I haven't the book by me.\"\n\n\"Stratagems be damned,\" said Sergeant Farrelly. \"What stratagem\nwould you be proposing to try? Maybe you'd like me to send for Jimmy\nO'Loughlin and him to tell the young lady that his wife was wanting the\nnote the way she could use it for lighting the kitchen fire to boil the\nkettle for tea. Is that your stratagem? Tell me now.\"\n\n\"It is not,\" said Constable Cole, with dignity, \"nor it isn't like\nit. If I was the sergeant here, I'd go to the young lady and I'd tell\nher, speaking civil and pleasant, that the District Inspector beyond\nin Ballymoy had sent a man over for the note, so as he could set the\npolice all over the country looking for Dr. O'Grady, and that he\nwouldn't be able to do that same without he got the note, on account of\nthe way the law does be at the present time.\"\n\n\"Is that what you call a stratagem?\" said the sergeant. \"It's a lie I'd\ncall it myself, a whole pack of lies, and it's just what they might\ntake the stripes off me for saying, if so be I was fool enough to say\nit. Is it looking to be sergeant yourself in the place of me you are,\nthat you'd suggest the like of that?\"\n\n\"All stratagems is lies,\" said Constable Cole soothingly. \"The one I'm\nsuggesting is no worse than another.\"\n\n\"Go and try her with it yourself, then,\" said the sergeant, \"and see\nwhat you'll get out of it.\"\n\nConstable Cole, pursued by the sniggering laugh of Moriarty, left the\nkitchen and went into the day room. Miss Blow had made herself quite\nat home. On the iron-legged table with which the police barrack was\nprovided lay her hat, her jacket, and her gloves. She was knitting a\nsilk tie, meant perhaps as a present for her father, perhaps as an\nadornment for the corpse of Dr. O'Grady when she found it. Constable\nCole drew himself up stiffly to attention and addressed her--\n\n\"I beg your pardon, miss, but Sergeant Farrelly will be obliged to you\nif you'll lend him the loan of the note that Lord Manton gave you. He's\nthinking of sending a man over to Ballymoy to the D. I.\"\n\n\"What's a D. I.?\" asked Miss Blow.\n\n\"He's an officer, miss, a gentleman by the name of Mr. Goddard.\"\n\n\"Is he your superior officer?\"\n\n\"He is, miss.\"\n\n\"Then I'll go and see him myself, and take the note with me.\"\n\nThe reply was quite unexpected. Constable Cole hesitated.\n\n\"It's better than twelve miles of a drive,\" he said, \"and the road's\nnone too good. And it could be, miss, that the D. I. might be off\nsomewhere, shooting or the like, when you got there, and then you\nwouldn't find him.\"\n\n\"If he is out, I shall wait for him.\"\n\n\"I wouldn't doubt you, miss; but it could be----\"\n\n\"No, it couldn't,\" said Miss Blow. \"At all events, it won't. Kindly go\nand get the car at once.\"\n\nConstable Cole returned to the kitchen grinning broadly.\n\n\"It's yourself that's in luck this day, Moriarty,\" he said. \"It's not\nevery man that gets the chance of driving round the country on a car\nwith the like of that one. Be careful now what you're saying to her, or\nyou'll have the doctor out after you with a stick if he has to come all\nthe way back from America for the purpose.\"\n\n\"What do you mean?\" said Moriarty.\n\n\"She said she'd go with you to Ballymoy, as soon as ever she heard\nit was you that was going. But if I hear tell of any impropriety of\nconduct, I'll send word to the red-haired girl that you used to be\nwalking out with on Sundays when you were up in the depot learning your\ndrill. I heard of you.\"\n\nMoriarty was young, very young. He blushed hotly.\n\n\"I'd be ashamed of my life to be seen with her,\" he said. \"I'd never\nhear the last of it.\"\n\n\"Off with you at once and get the car,\" said the sergeant. \"In the name\nof God, if the girl's willing to go out of this, will you take her\nalong with you before she changes her mind? Haven't we had enough of\nher this two days?\"\n\nIn less than half an hour the car--Jimmy O'Loughlin's car--was at the\ndoor of the barrack. Constable Moriarty, in full panoply, his grey cape\nrolled round his chest, his carbine between his knees, sat on one\nside. Miss Blow, looking very handsome, got up on the other. Sergeant\nFarrelly wrapped a rug round her knees and tucked the end of it under\nher feet. Then he presented her with a sixpenny box of chocolates. He\nhad gone round to Jimmy O'Loughlin's shop and bought this offering\nwhile the horse was being harnessed.\n\n\"It's a long drive you have before you,\" he said, \"and I was thinking\nmaybe you'd like something to comfort you on the way. It's no more\nthan a trifle, and not what you'd be accustomed to, but Clonmore is a\nbackward place, and it's the most thing of the kind there was in Jimmy\nO'Loughlin's shop.\"\n\nConstable Cole rushed from the barrack bareheaded, just as the car was\nstarting. He had Miss Blow's brown paper parcel in his hand.\n\n\"You've forgotten your lunch, miss,\" he shouted. \"You'll be wanting it\nbefore you're back.\"\n\nHe stowed the parcel in the well of the car, and was able as he did so\nto still further embarrass the unfortunate Constable Moriarty.\n\n\"By rights,\" he whispered, \"you ought to be sitting on the same side\nwith her. It's what she'd expect of you; and if you don't do it when\nyou get off to walk up Ballyglunin Hill, she'll be in a mighty bad\ntemper against you have her safe with the D. I. If you're half a man,\nMoriarty, you'll do it.\"\n\nThere was a good deal of excitement in the town when Miss Blow drove\noff under the escort of Constable Moriarty. The news that Jimmy\nO'Loughlin's car had been ordered for her and the constable spread so\nrapidly that by the time the start was actually made a small crowd\nhad gathered in the street to see it. Afterwards, for more than an\nhour, men stopped casually at the barrack door, chatted on indifferent\nsubjects with Sergeant Farrelly or Constable Cole, and then asked one\nor two leading questions about Miss Blow and her business. The police\nwere very reticent. Sergeant Farrelly was an impressive man with a\ngreat deal of personal dignity. He knew that contemptible witticisms\nwould be levelled at him and the force of which he was a member, if\nit came to be known that Miss Blow, a solitary young woman, had held\nup the police of Clonmore for the whole of one day and a part of the\nnext. He dreaded the remarks of irreverent small boys if they heard how\nnearly he and his men had been forced to go in search of Dr. O'Grady's\nbody. He was haunted by a terrible fear that the story might get into\nthe newspapers.\n\n\"There's them,\" he said to Constable Cole, \"who'd be only too glad to\nget a handle against the police--fellows up in Dublin writing for low\npapers. Believe you me, it'll be mighty unpleasant for us if you aren't\nable to keep your mouth shut.\"\n\nCole was no more willing than the sergeant to give information. The\ninquirers, baffled at the barrack, moved on to the bar of the hotel,\nand asked their questions there. Jimmy O'Loughlin had no information to\ngive them. He did not know, any better than his customers, the reason\nfor Miss Blow's expedition; but he liked to pose as being well up in\nthe whole business. He shook his head gravely, made cryptic remarks,\nparried questions with other questions, and at last, without in the\nleast meaning to, conveyed the impression that Miss Blow, by some\nmysterious process of law, had been arrested for Dr. O'Grady's debts.\nThe opinion gained ground in the town as one after another of the\ninquirers emerged from the bar. Strong sympathy was felt with Miss\nBlow, and there was some talk of summoning a special meeting of the\nLeague to consider her case. It was generally agreed that a unanimous\nresolution would be appropriate, and that a series of questions might\nvery well be asked in the House of Commons by one of the members for\nthe county.\n\n\n\n\nCHAPTER IX\n\n\nBefore any definite action was taken, the public interest was diverted\nfrom Miss Blow and her affairs by a new sensation. At about half-past\none o'clock Mrs. Patsy Devlin was seen advancing along the street\ntowards the barrack with a crowd of women and children after her. Her\nappearance suggested that she was suffering from an extremity of grief.\nHer hair hung loose over her shoulders in picturesque grey wisps. Her\nbodice had only one fastening, a white pin, driven through it near the\nneck. Below the pin the garment gaped, down to the point at which,\nstill gaping, it was tucked into a crimson petticoat. Her boots, a pair\nso large that they might have been, and probably were, her husband's,\nwere unlaced, and clattered on the ground every time she lifted her\nfeet.\n\n\"Himself is gone from me,\" she wailed, when she reached the door of the\nbarrack, \"gone and left me, and me sick in my bed with an impression\non my chest and a rumbling within in the inside of me as it might be\na cart, or two carts, and they with turf in them going along on the\nstreet.\"\n\n\"My good woman,\" said Sergeant Farrelly, \"go home out of this, and\ndon't be making a disturbance on the public street.\"\n\n\"Home, is it?\" wailed Mrs. Devlin; \"and where will I find a home when\nhimself is gone from me?\"\n\n\"Go to your bed,\" said the sergeant; \"and if so be that you're sick\nthe way you're after telling us, get the doctor to attend you.\"\n\nThen he recollected that there was no doctor in Clonmore, and suggested\nas an alternative that she should send one of the children up to Jimmy\nO'Loughlin's shop to buy \"some sort of a bottle that would do her good.\"\n\n\"And what good would a bottle be to me, if I had the money to pay for\nit itself, and where would I get the money, with himself gone from me?\nIt was a bad head he was to me, and many's the time I've been sorry\nthat ever I married him, but sure it's worse I'll do without him now\nhe's gone.\"\n\n\"And that's true for her, the creature,\" said old Biddy Halloran from\nthe outskirts of the crowd.\n\n\"Where is he gone to?\" asked the sergeant.\n\n\"Is it where has he gone? If I knew that, would I come to you to find\nhim for me? Where has he gone? Och! but I'd be the thankful woman this\nday if I could lay my eyes on him.\"\n\nMrs. Devlin wept wildly.\n\n\"Stop your crying, woman dear,\" said a sympathizer; \"sure the police\nwill have him found for you in two twos. Isn't that what they're here\nfor?\"\n\n\"When did you miss him?\" asked the sergeant.\n\n\"Miss him! Amn't I missing him every minute since he went from me, and\nthe children along with me crying for their da, and them with no da\nleft them?\"\n\n\"When did you see him last?\" said the sergeant, varying the form of his\nquestion.\n\n\"I seen him,\" said Mrs. Devlin, \"e'er last night. He came home, and him\nafter spending the evening at Jimmy O'Loughlin's.\"\n\n\"Had he drink taken?\"\n\nMrs. Devlin hesitated. One of her supporters in the crowd encouraged\nher to speak out boldly.\n\n\"Don't be afeard, Mrs. Devlin, ma'am. Speak up to the sergeant when he\nasks you the question. It's for your own good that he's trying to get\nthe truth out of you.\"\n\n\"I won't,\" said Mrs. Devlin at last, \"be telling lies to you. He had\nnot. There wasn't a sign of it on him, though I won't deny that he\nmight have had a glass of porter or the like.\"\n\n\"And did he spent the night with you in the house?\"\n\n\"He did--he did,\" wailed Mrs. Devlin, overcome afresh by the\nrecollection. \"And in the morning, as it might be yesterday morning,\nwhen he had his breakfast took, 'I'm off,' says he, 'up to the Castle,'\nsays he, 'to see the old lord that always was a good friend to me and\nmy father before me.' And I never seen him since.\"\n\n\"Did he go to the Castle?\" asked the sergeant.\n\n\"How would I know whether he did or not? Amn't I after telling you that\nI was sick in my bed?\"\n\n\"He did go,\" said a man from the crowd, \"for I seen him getting over\nthe wall into the deer park, and him looking most determined, the same\nas if he had business to do.\"\n\n\"And why didn't you come up and tell me before?\" said the sergeant.\n\"He's gone for twenty-four hours and more, and it's only now you find\nit out.\"\n\n\"I was expecting him to step into the house every minute,\" said Mrs.\nDevlin, \"and I wasn't willing to lay my mind down to it that he was\ngone, till there was no help for it.\"\n\n\"The Lord save us!\" said old Biddy Halloran. \"It's an afflicted\ncreature you are this day, Mary Devlin.\"\n\nSergeant Farrelly buttoned his tunic and took his cap. He summoned\nConstable Cole and they marched together down the street towards Jimmy\nO'Loughlin's hotel. The crowd, Mrs. Devlin at the head of it, followed\nthem. Constable Cole turned.\n\n\"Go home out of that the lot of you,\" he said, \"and take Mrs. Devlin\nalong with you. The matter is in the hands of the police now, and that\nought to content you.\"\n\nJimmy O'Loughlin's customers deserted him as soon as the noise of\nMrs. Devlin's wailing was heard in the street. He stood alone behind\nhis bar when the police entered the hotel. He greeted the sergeant\nheartily, for he was a man of good conscience and unaware of any reason\nwhy he should dread a visit from the police. He was struck by the\nsolemn severity with which Sergeant Farrelly replied to his greeting,\nand became vaguely uneasy. The business of a publican is beset with\nlegal snares which only the most fortunate men succeed in avoiding\naltogether. Jimmy examined himself rapidly, but failed to discover\nanything in his immediate past which could bring him under the lash of\nthe law. He was quite sure that batch of convivial friends who absorbed\na few dozen bottles of porter on the previous Sunday had entered his\nhouse unseen. They had certainly entered it cautiously--by the back\ndoor, after climbing over a wall into his yard. His mental attitude was\nthat of the people of the town of Bethlehem when the prophet Samuel\ncame unexpectedly among them. They were not conscious of deserving any\nkind of denunciation, but they were anxious, and said to the great\nman: \"Comest thou peaceably?\" Jimmy O'Loughlin might have repeated\ntheir exact words, but he had never heard the story, and was therefore\nunable to quote from it. His face wore an expression of anxious\ninterrogation.\n\n\"Did you hear,\" said Sergeant Farrelly, \"what they're after telling me\nabout Patsy Devlin?\"\n\n\"What about him?\" said Jimmy cautiously.\n\n\"He's left the town, and deserted his wife and family.\"\n\n\"Do you say that now?\"\n\n\"It's not me that says it,\" said the sergeant; \"but it's being said.\"\n\n\"I wouldn't wonder,\" said Jimmy, \"but there might be some truth in it.\"\n\nNeither he nor his public-house could be held in any way responsible\nfor the disappearance of Patsy Devlin. He felt free to discuss\nthe event in a friendly way with the police and to give them any\ninformation he could so long as he said nothing likely to lead to the\ncapture of Patsy.\n\n\"I'm told,\" said the sergeant, \"that the last night he was in Clonmore\nhe spent the most of it along with yourself.\"\n\n\"He might,\" said Jimmy.\n\nHe did not quite see the point of the sergeant's remark, and felt that\nhe must be cautious.\n\n\"Could you give me any information about what he intended doing with\nhimself the next day?\"\n\n\"He told me,\" said Jimmy, after thinking the matter over, \"and it\ncould be that he was telling me the truth--he told me he was going up\nto the Castle to try if he could get a pound, or maybe two pounds,\nout of Lord Manton for the sports. He was collecting the town and the\ndistrict; and he said to me himself that he'd done well. The money was\ncoming in better than ever it did.\"\n\n\"Ah!\" said the sergeant with deep meaning.\n\n\"Just so,\" said Jimmy.\n\n\"And was that what you meant this minute when you said that you\nwouldn't wonder if there might be some truth in what they're saying\nabout him being gone?\"\n\n\"He's not the first,\" said Jimmy, \"nor he won't be the last. There was\nCooney that was treasurer of the League, and nobody ever heard of him\nafter. It was upwards of twenty pounds he had. There was----\"\n\n\"It's larceny,\" said Constable Cole.\n\n\"You're wrong there,\" said the sergeant. \"It's misappropriation of\npublic funds under trust, besides the charge that might be brought by\nthe subscribers of obtaining money under false pretences.\"\n\n\"You'll never get him,\" said Jimmy.\n\n\"It isn't him I'm thinking of,\" said the sergeant, \"but his wife that's\nleft with a long family dependent on her and not knowing where to look\nfor the bit to put into their mouths.\"\n\n\"It's a pity of the creature,\" said Constable Cole. \"It's badly she'll\nbe able to do without him.\"\n\n\"If so be,\" said Jimmy, \"that Patsy's gone the way you say, he'll have\nleft a trifle behind him for the widow, be the same more or less. He\nalways had a good heart, and it wouldn't be like a thing he'd do to\nleave his children to starve.\"\n\n\"Devil the penny she owned up to anyway,\" said the sergeant.\n\n\"Then he'll send it,\" said Jimmy; \"he'll send it from America.\"\n\n\"He might,\" said the sergeant.\n\n\"The world,\" said Constable Cole, \"is full of trouble, any way you look\nat it.\"\n\n\"Does Father Moroney know he's gone?\" asked Jimmy.\n\n\"I'm thinking he must,\" said the sergeant; \"he could hardly miss\nhearing the way the creature was going on in the street, crying all\nsorts.\"\n\n\"He'll be apt to be raising a subscription for her,\" said Jimmy, \"to\nput her over until such time as Patsy sends home the trifle he has for\nher.\"\n\n\"I'll give something towards it myself,\" said the sergeant, \"and I'll\nsee that the men in the barrack contributes.\"\n\nJimmy O'Loughlin was not to be outdone in generosity by the members of\nthe constabulary.\n\n\"I have the pound by me,\" he said, \"that I'd promised Patsy Devlin, the\npoor boy, for the sports. I hadn't it paid over to him, thanks be to\nGod. I'd be thankful to you, sergeant, if you'd take it and hand it on\nto Father Moroney. It's no more than due to the woman, seeing that her\nhusband could have had it if he'd thought of taking it, and I'll add\nanother five shillings to it from myself.\"\n\nHe handed the whole sum over to Sergeant Farrelly, who put it in his\npocket.\n\n\"He was always a bit of a lad, that Patsy Devlin,\" said Constable Cole.\n\n\"He might be a bit foolish at times,\" said Jimmy; \"but there was no\nharm in him.\"\n\n\"It's as good for you,\" said the sergeant, \"that you didn't make him\nthe inspector of sheep dipping that time.\"\n\n\"It was that preyed on his mind,\" said Jimmy. \"He never rightly got\nover it. I don't say he'd have been elected--there was better men up\nfor the job than him--but he destroyed himself altogether when he went\ngetting a testimonial to his character from Lord Manton. The League\nwouldn't stand the like of that, and small blame to them. It couldn't\nbe expected that they would.\"\n\n\"We'll go up to the Castle,\" said the sergeant to Constable Cole, \"and\nfind out whether Patsy went there before he left, if so be that his\nlordship has come home.\"\n\n\"I didn't hear any talk of his being away,\" said Jimmy.\n\n\"Well, he was away. Yesterday and to-day.\"\n\n\"That's queer now,\" said Jimmy, \"for it was only this morning I was\ntalking to Byrne the steward, and he told me that his lordship was\nwalking round yesterday afternoon looking into the new drain he's\nthinking of making across the top of the deer park, and that he went in\nfor his tea the same as usual. What's more, he was speaking to Byrne\nthis morning about the disgraceful way the roads is kept by the County\nCouncil, and the rates being so high and such-like.\"\n\n\"It couldn't be,\" said the sergeant, \"for I was up there three times\nyesterday, and I wasn't able to see him.\"\n\n\"Take care but he didn't want to see you.\"\n\n\"And I had a letter from him this morning telling me that he'd be away\nfrom home all the day.\"\n\n\"Take care,\" said Jimmy again, \"but he mightn't have wanted to see\nyou.\"\n\n\"And why wouldn't he?\"\n\n\"I'm not saying it is, mind you,\" said Jimmy; \"but it might be that he\nknows more about Patsy Devlin than he'd care to tell. Him and Patsy was\nmighty thick.\"\n\n\"Talk sense, can't you?\" said the sergeant. \"Is it likely now that a\nman like his lordship would be conniving at the escape of a criminal\nfrom justice?\"\n\n\"I said no such thing,\" said Jimmy; \"and I'll thank you, Sergeant\nFarrelly, not to be putting it out that I did. What I said was that\nhe might know more than he'd care to tell. Would you think now that a\ngentleman like him--and I'll say this for him, that he always was a\ngentleman--do you think now that, if so be he did know where Patsy was\ngone, he'd be wanting to tell you and maybe get a poor man into trouble\nthat he had a liking for? Didn't you tell me this minute that he had\nhimself hid away from you when you were up at the Castle looking for\nhim? Why would he do the like? Tell me that now. Why would he do it?\"\n\n\"Come along out of this,\" said the sergeant to Constable Cole. \"We've\nno business standing here listening to such talk. I'm going up to the\nCastle now, Mr. O'Loughlin, and if I hear so much as another word of\nthat nonsense out of your head I'll tell his lordship what you're after\nsaying.\"\n\n\"You may tell him,\" said Jimmy, \"when you get a hold of him to tell;\nbut it's my belief that if he hid on you yesterday, he'll hide on you\nagain to-day.\"\n\nIt turned out that Jimmy O'Loughlin was perfectly right. Constable Cole\nsaid that he was prepared to swear, if necessary, that he saw Lord\nManton looking out of one of the windows of the Castle. But Wilkins\nwas as impenetrably suave as he had been the day before.\n\n\"His lordship left word,\" he said, \"in case you called, that he was\naway from home and couldn't say precisely when he might return.\"\n\n\n\n\nCHAPTER X\n\n\nMr. Goddard, the District Inspector of Police, was a young man and\nstood on the lowest rung of his professional ladder. It was recognized\nby his superiors, it was even feared by the man himself, that he was\nnever likely to rise very high in the service. He was, in fact, an\ninefficient officer. He had a natural sense of humour, which was a\ngreat misfortune, because it led him to see situations like those of\ncomic operas in the course of the duties which he was called upon to\nperform. He was wise enough not to laugh loudly at the things which\nhappened; but the fact that a great deal of what he had to do struck\nhim as ludicrous prevented his doing his duty heartily and thoroughly.\nAn Irish police officer ought to have a good opinion of himself and\nhis position. He ought to recognize his official kinship with the\npotentates who draw large salaries for administering the affairs of the\nIndians, the Egyptians, and other barbarous peoples. He is humbler, of\ncourse, and is paid less; but he is engaged in the same kind of work.\nHe is securing to a conquered people the blessings of law and order.\nUnfortunately, Mr. Goddard could never see himself steadily in this\nlight. Another cause of his inefficiency was indolence. The duties\nwhich did not strike him as comic bored him intolerably; but like many\nlazy men, he was subject to spasms of vigour when irritated. It was his\ncustom to avoid doing anything for as long as possible, and at last,\nif goaded beyond endurance, to act with a violence which surprised his\nsubordinates. He had a taste for literature, even when it took the form\nof poetry, and used to read a good deal in the evenings.\n\nHe was reading a novel when Miss Blow and Constable Moriarty drove up\nto his house. He received them in his dining-room, and was chiefly\nanxious at first to get rid of them as soon as possible. He did not\nrealize for some time that Miss Blow was the kind of woman who ought to\nbe offered a chair and invited to sit down on it. She came, apparently,\nin the custody of a policeman, and Mr. Goddard did not look closely at\nher. She saved him from any embarrassment he might afterwards have felt\nby walking over to an armchair and settling herself comfortably in it.\n\nConstable Moriarty produced Lord Manton's note of complaint and handed\nit over to his officer. He explained that Sergeant Farrelly had called\nthree times at Clonmore Castle and had waited in the hall for some\nhours in the hope of seeing Lord Manton. Mr. Goddard read the note\nand asked a number of questions. He succeeded in greatly embarrassing\nConstable Moriarty. That young man had the feelings of a gentleman. He\nwould not say anything likely to cause pain or discomfort to a lady.\nHe hesitated in his account of Miss Blow's invasion of the barrack. He\ncontradicted himself three times in trying to explain the views of the\npolice about the disappearance of Dr. O'Grady.\n\nMiss Blow cut into the conversation sharply.\n\n\"I demand,\" she said, \"that the police should investigate a case of\nmurder, and they refuse to do so.\"\n\nMr. Goddard, for the first time, took a good look at her. He realized\nat once that she was an extremely handsome girl. Even so it seemed\nimprobable that any one would murder a dispensary doctor well known to\nbe very popular.\n\n\"Murder,\" he said tentatively, \"is perhaps too strong a word.\"\n\n\"Lord Manton agrees with me,\" said Miss Blow, \"that he has been\nmurdered.\"\n\nShe took Lord Manton's note from her pocket and handed it to Mr.\nGoddard. He read it, reread it, and then turned inquiringly to the\nconstable. Moriarty reluctantly admitted that Lord Manton's words might\nbear the interpretation which Miss Blow put on them.\n\n\"Then why on earth did you not investigate the matter?\" said Mr.\nGoddard.\n\nConstable Moriarty became very confused. With Miss Blow's fine eyes\nfixed on him he could not bring himself to blurt out the naked truth.\nHe was as unwilling as everybody else had been to break the heart of a\nbeautiful girl by saying that her lover had basely deserted her.\n\n\"It could be,\" he said feebly, \"that the doctor's alive and well yet.\"\n\nThis was very much Mr. Goddard's own opinion. He read Lord Manton's\nnote again, and then turned to Miss Blow.\n\n\"I am very sorry,\" he said, \"that you should have been put to so much\ntrouble and inconvenience. I hope I am not tiring you too much, but\nwould you mind telling me what reasons you have for supposing that Dr.\nO'Grady has been murdered?\"\n\nMiss Blow's eyes suddenly filled with tears. She caught at her skirt\nwith both hands, clenching the folds of it tightly.\n\n\"We were to have been married this year,\" she said, \"and oh----\"\n\nThen, fumbling hurriedly for her pocket-handkerchief, she burst into\na flood of tears. There was every excuse for her. She had been driven\nto the belief that her lover was murdered. She had gone through three\ntrying days, the last two of them very trying. She was a stranger among\npeople who seemed heartless and cruel to an extraordinary degree. She\nhad every right to an outbreak of hysterical weeping. Yet it should\nbe noted to her credit that she chose as the witness of her breakdown\nthe man, of all those whom she had met, most likely to be influenced\nby tears. Lord Manton, if she had wept in his study, would have\ncomforted her; but he would also have enjoyed his task and would have\nappreciated the appearance of her slobbered cheeks. Sergeant Farrelly\nwould have sympathized with her if she had wept in the police barrack,\nbut he would not have gone out to search for Dr. O'Grady's body merely\nbecause she made a sponge of her pocket-handkerchief. Mr. Goddard was\ndifferent. He was young, and though he had a sense of humour, the sight\nof a beautiful girl shaken with sobs embarrassed him.\n\nHe did not exactly know what he ought to do. He looked round, hoping\nfor some suggestion from the constable; but Moriarty had slipped\nquietly from the room. Mr. Goddard made a hasty and impassioned vow\nthat he would give Constable Moriarty a severe lesson in the respect\ndue to his superior officer. Then he looked at Miss Blow again. She was\nsobbing convulsively. He watched her helplessly for several minutes,\nand then asked her if she would like a cup of tea. He had to repeat\nthe question twice, because, owing to nervousness, he was inaudible\nthe first time he asked it. Miss Blow, when she heard what he said,\nshook her head vigorously. Mr. Goddard felt that there must have been\nsomething insulting about the suggestion, and was sorry he had made it.\nHe wished very much that he knew how to behave under the circumstances.\nBy way of relieving the tension of the situation he got up and stood\nbehind Miss Blow. Then he walked round her chair and stood in front\nof her. Neither position availed to check her weeping. Her head was\nbowed almost to her lap and her face was covered with her hands. It\noccurred to him that it might soothe her if he patted her back and\nshoulders gently. He stretched out his hand. Then he paused. He was a\nman of chivalrous feeling towards women, the result perhaps of reading\nTennyson's poetry, and it struck him that it would be unfair to pat a\ngirl who was obviously incapable of resisting. He stood irresolute, his\nhand still stretched out. His attitude was not unlike that of a priest\nwho bestows a benediction upon a deeply contrite penitent.\n\n\"Miss Blow,\" he said at last, \"please stop crying.\"\n\nCuriously enough this appeal produced its effect upon her.\n\n\"How can I help crying,\" she said, though her utterance was broken with\nsobs, \"when he's dead, and no one will help me even to find his body?\"\n\nMr. Goddard's resolution was taken in an instant. He did not believe\nthat Dr. O'Grady was dead. He knew that he was laying up trouble for\nhimself in the future; but it was absolutely necessary to stop Miss\nBlow crying and, if possible, to get her out of the house.\n\n\"I'll help you,\" he said. \"I'll do all that can be done to find him. I\nshall put all the men in my district to work. I shall leave no place\nunsearched until I find him, alive or dead.\"\n\nMiss Blow looked up at him, and smiled through her tears. Even an\nordinary girl, with no particular pretensions to beauty, looks very\ncharming when she succeeds in smiling and crying at the same time.\nMiss Blow seemed radiantly lovely. Mr. Goddard felt that he was losing\ncommand of himself. He felt strongly inclined to quote some poetry. He\nknew that there must be poetry suitable to the situation, but for the\nmoment he could think of nothing except four lines out of _Maud_.\n\n  \"Oh, that 'twere possible\n    After long grief and pain,\n  To feel the arms of my true love\n    Around me once again.\"\n\nThere was a certain appropriateness about the verses, and yet he\nhesitated to quote them. He was not sure that Miss Blow would care to\nadmit in plain words to a total stranger that she wanted Dr. O'Grady's\narms round her. Miss Blow saved him from his uncertainty. She gave her\neyes two rapid dabs with her wet pocket-handkerchief, and said--\n\n\"When shall we start?\"\n\n\"Some day next week,\" said Mr. Goddard. \"Suppose we say----\"\n\nMiss Blow collapsed again, and showed every sign of more tears.\n\n\"Sooner, if you like,\" said Mr. Goddard hurriedly.\n\n\"At once, then,\" said Miss Blow, rallying; \"at once; this very moment.\"\n\nMr. Goddard looked at his watch. It was three o'clock. It would take\nhim at least two hours to drive to Clonmore. He could do very little\nthere that evening.\n\n\"At once,\" repeated Miss Blow; \"this very instant.\"\n\nMr. Goddard gave in. He believed that he was going to make a fool of\nhimself, but saw no way of escape. Miss Blow's vigorous manner of\ncrying convinced him that she was quite capable of sitting in his\ndining-room and continuing to cry until he set out in search of Dr.\nO'Grady's body.\n\n\"Will you excuse me,\" he said, \"while I go upstairs and put on my\nuniform?\"\n\nMiss Blow nodded and smiled again. The thought of a uniform comforted\nher. A man can hardly fail in his duty when he puts on a uniform for\nthe express purpose of performing it. Mr. Goddard took courage from her\nsmile.\n\n\"I shall tell my housekeeper,\" he said, \"to bring you a cup of tea. I\nam sure you must need it. You can drink it while you are waiting. I\nshall have my horse harnessed and drive you back with me to Clonmore.\"\n\nThis time Miss Blow did not refuse the tea. She drank two cups of it\nwhen it was brought to her, and finished a plate of bread-and-butter.\nIt took Mr. Goddard some time to array himself, because he paused\nfrequently to try and hit upon some way of getting rid of Miss Blow\nwithout driving her into Clonmore. He tried in vain. Miss Blow, after\ndrinking her tea, was able to devote a few minutes to her hair and the\nposition of her hat. There was a mirror over the chimney-piece in Mr.\nGoddard's dining-room.\n\nMr. Goddard's horse was a good one, and could undoubtedly have done the\njourney to Clonmore in less than two hours, but Mr. Goddard did not\npress him. He found the drive agreeable. Miss Blow had quite stopped\ncrying before she got into the trap, and she talked with the utmost\nfrankness about her affairs. There is something very delightful in the\nconfidences of a beautiful girl. Miss Blow had a way of looking up with\nher eyes wide open, which sent a curious thrill of pleasure through\nMr. Goddard. There was only one thing which marred the enjoyment of\nthe drive. Her confidences were of the most puzzling and embarrassing\nkind. She told how everybody in Clonmore had lied. And so far, Mr.\nGoddard sympathized with her. But he could not help asking himself why\nthey lied. Even Jimmy O'Loughlin, as Mr. Goddard understood, would\nnot lie persistently without a motive of some kind. Of course, Miss\nBlow had an explanation ready at hand. Dr. O'Grady was murdered, and\neverybody was interested in concealing the fact. But Mr. Goddard could\nnot believe this. He did not attempt to argue with her, for he had no\nwish to reduce her to tears again; but he knew very well that nobody in\nthe neighbourhood would murder Dr. O'Grady. He was inclined at first\nto think, as every one else did, that the doctor had run away from his\ncreditors; but Miss Blow demonstrated the absurdity of this theory.\nShe had come to Clonmore with a blank cheque signed by her father, and\nfull authority to pay every penny the doctor owed. She had told him\nbeforehand in a letter that he might expect some relief. Mr. Goddard\nwas forced to admit that under the circumstances it was very unlikely\nthat Dr. O'Grady had allowed himself to be chased away to America by\nhis creditors.\n\nAt the same time, the doctor had undoubtedly disappeared. That seemed\nthe one solid fact there was. Miss Blow referred frequently to Lord\nManton's note. It puzzled Mr. Goddard quite as much as anything else\ndid. He knew Lord Manton, and he could not understand how he came to\nbelieve, as apparently he did believe, in the murder of Dr. O'Grady.\nMr. Goddard made up his mind that he would go up to Clonmore Castle\nand talk the whole matter over with Lord Manton. This was the solitary\ndecision he was able to come to; though he was obliged to pledge\nhimself over and over again to institute a strict search for the\ndoctor's body. He arrived at last in Clonmore and drove straight up to\nthe hotel.\n\n\"You must be very tired,\" he said to Miss Blow, \"quite worn out. Now\nthat the matter is in my hands, you need have no further anxiety. I\nhope you will stay quietly in the hotel and rest till I see you again.\"\n\nHe spoke quite sincerely. He did hope that Miss Blow would go into the\nhotel and stay there. He feared that she might feel it necessary to\nfollow him about and watch what he did.\n\n\"I shall inform you at once,\" he went on, \"of everything which\ntranspires. I must spend this evening in making some preliminary\ninquiries; but there is no necessity for you to fatigue yourself\nfurther.\"\n\nMiss Blow looked at him long and searchingly. Mr. Goddard felt that she\nwas judging of his strength and determination, was deciding whether she\ncould fully trust him or not. He endeavoured to assume the expression\nof face which he believed to be common to those strong, silent\nEnglishmen, whom the heroines of novels learn, after other people have\nturned out to be frauds, to trust to the uttermost. He was, apparently,\nquite successful in his effort. He deserved to be; for he really felt\nfor the moment, with Miss Blow's eyes on him, that he was exactly the\nkind of man he wished to appear. Miss Blow, without a word, stretched\nout her hand to him. He took it and ventured to press it slightly. He\nwould have carried it to his lips and pressed a reverent kiss upon it,\nif Jimmy O'Loughlin had not been standing at the door of the hotel.\nAfterwards he was sorry that he had not defied Jimmy O'Loughlin. A\nreverent kiss, even if it had to soak through a glove, would have been\nmost appropriate to the occasion.\n\nMiss Blow got out of the trap and went into the hotel. Mr. Goddard\ndrove on to the police barrack.\n\nHe found Sergeant Farrelly and Constable Cole drawn up before the door.\nThey had seen their inspector drive into the town and were ready for\nhim. Mr. Goddard at once ordered the sergeant to follow him into the\nbarrack. Constable Cole mounted guard over the horse and trap.\n\n\"Now, sergeant,\" said Mr. Goddard, \"what have you to tell me about the\ndisappearance of Dr. O'Grady?\"\n\n\"There's another man gone since then, sir,\" said the sergeant gloomily.\n\n\"What do you mean? Who's gone?\"\n\n\"I thought it right to let you know, sir--I was within writing a\nreport on the matter when I seen you drive into the town--that Patsy\nDevlin the smith is gone. His wife was up at the barrack shortly after\nConstable Moriarty left with the young lady, and she says he's missing.\"\n\n\"It's a queer thing,\" said Mr. Goddard, \"that two men should disappear\nin this sort of way within a couple of days of each other. It looks\nbad. Let's take them one at a time and see what we can make out of\nthem. What have you to say about the doctor?\"\n\n\"Everybody knows the reason he's made off,\" said the sergeant, \"only\nnobody'd like to be telling the young lady, and that's what has us all\nput about the way we are.\"\n\n\"Do you mean debt?\"\n\n\"I do, sir. It's common talk. Jimmy O'Loughlin told me himself----\"\n\n\"It's not true anyhow, whether Jimmy O'Loughlin said it or not.\nWhatever it was made the doctor bolt, it wasn't that.\"\n\n\"Then I don't know what it would be.\"\n\n\"No more do I; but I'm going to see Lord Manton and talk to him about\nit.\"\n\n\"It could be,\" said Sergeant Farrelly, \"that he'd know. Did you take\nnotice of the note that he gave to the young lady?\"\n\n\"Yes; I saw it.\"\n\n\"Well now, his lordship couldn't be believing that the doctor's\nmurdered, and whatever made him write that note it's my opinion that\nthere was something behind it. And what's more, Jimmy O'Loughlin\nsays----\"\n\n\"Damn Jimmy O'Loughlin!\"\n\n\"Jimmy O'Loughlin says,\" went on the sergeant, \"that his lordship knows\nsomething, be the same more or less, about Patsy Devlin. I wouldn't\nwonder now if he'd be able to tell you where the both of the two of\nthem is gone and why.\"\n\n\"What about Patsy Devlin?\" asked Mr. Goddard. \"What sort of a man is\nhe?\"\n\n\"He's no great things any way you take him. He's a bit foolish at\ntimes, and takes more than is good for him. I hear them say that he\nfretted a deal when they didn't make him the inspector of sheep\ndipping. It might be that the disappointment preyed on him, that and\nthe drink, so as he wouldn't be rightly responsible for what he did.\"\n\n\"Was he mixed up with the League?\"\n\n\"He was one time, but there was a falling out between him and them over\nthe sheep dipping. Patsy wasn't what you'd call great with the League\nsince then. I'm told he had a deal of money collected for the sports.\nJimmy O'Loughlin let out to me that----\"\n\n\"It seems to me,\" said Mr. Goddard, \"that Jimmy O'Loughlin knows more\nabout these disappearances than anybody else about the place.\"\n\n\"Unless it would be Lord Manton,\" said the sergeant.\n\n\"I'll see Lord Manton anyhow,\" said Mr. Goddard. \"You can tell the\nconstable to take my horse and trap round to the hotel. I'll walk up to\nthe Castle.\"\n\n\"It'll be well,\" said the sergeant, \"if you get seeing his lordship.\"\n\nHe spoke meaningly. Mr. Goddard, who was half way to the barrack door,\nturned back.\n\n\"What's that you say?\"\n\n\"It'll be well,\" said the sergeant, \"if you're not told that his\nlordship's away from home.\"\n\n\"What do you mean by that?\"\n\n\"I was up there yesterday,\" said the sergeant; \"off and on I was there\nfor the most of the day; and I was up there again to-day, and all I got\nby it was word that his lordship was away from home. Jimmy O'Loughlin\nwas saying----\"\n\n\"Go on,\" said Mr. Goddard; \"Jimmy O'Loughlin's remarks are always\nvaluable.\"\n\n\"He was saying that his lordship was within all the time. It was his\nopinion--I'm not saying was he right or wrong--but it was his opinion\nthat his lordship didn't want the police next or nigh him.\"\n\n\"Jimmy O'Loughlin,\" said Mr. Goddard, \"appears to be as big a fool as\nanybody about Clonmore.\"\n\n\n\n\nCHAPTER XI\n\n\nWilkins was a little puzzled when he opened the door to Mr. Goddard.\nHis orders were definite. Lord Manton was not at home when the police\ncalled. Mr. Goddard in his uniform, complete to the sword, the whistle\nand the spurs, was undeniably a policeman. But Wilkins was a good\nservant, a very good servant. He was accustomed to interpreting his\norders as well as obeying them. He knew that Mr. Goddard was a superior\nkind of policeman. He dined occasionally at Clonmore Castle, and\nWilkins waited on him. After a moment's hesitation, Wilkins offered to\ngo and find out whether Lord Manton was at home or not. Mr. Goddard\nwas shown into a large, desolate drawing-room, and left there. Wilkins\nwas glad afterwards that he had appreciated the difference in standing\nbetween a district inspector and a sergeant. It appeared that Lord\nManton was quite willing to see this visitor. Mr. Goddard was shown\ninto the library.\n\n\"Sit down,\" said Lord Manton. \"I'm very glad to see you. You'll stay\nand dine, won't you? Since poor O'Grady left us I haven't had a soul to\nspeak to at meals.\"\n\n\"It was about Dr. O'Grady's disappearance that I called to see you,\"\nsaid Mr. Goddard.\n\n\"There's no use coming to me about that. I'm a magistrate, I know; but\nI very seldom act. Why not go to Jimmy O'Loughlin? He loves signing\npapers.\"\n\n\"I'm rather puzzled over the case. The fact is----\"\n\n\"My advice to you is to leave it alone. Don't do anything. Masterly\ninactivity is plainly the policy for you.\"\n\n\"That's all well enough. I'd be very glad to leave it alone. There's\nnothing I'd like better. But the fact is, I can't. I'm more or less\npledged to---- My hand has been, so to speak, forced.\"\n\n\"Had a visit from Miss Blow?\"\n\n\"Yes.\"\n\n\"She's a wonderful woman. She was here yesterday; spent half an hour\nwith me in this very room.\"\n\n\"She's a very good-looking girl,\" said Mr. Goddard.\n\n\"She is. I admit that. Her eyes, for instance. Grey, I thought them;\nbut they looked quite blue in certain lights; and a very good figure,\na remarkable figure. All the same I couldn't have her settling herself\ndown for good and all in my house. I had to get her out of it somehow.\nI gave her a note for Sergeant Farrelly.\"\n\n\"Oh! That's the meaning of your note. It rather puzzled me.\"\n\n\"I'm afraid it must have rather puzzled Sergeant Farrelly too. I felt\nsorry for him; but what could I do? She evidently meant to stay here\ntill I gave her a note of the sort she wanted. I thought it better to\nshift the responsibility of dealing with her on to the sergeant. After\nall, he's paid for looking into things of the kind. I'm not.\"\n\n\"The sergeant sent her on to me. It was extremely awkward. She cried\nlike anything.\"\n\n\"I thought she'd have cried here,\" said Lord Manton; \"but she didn't.\nDid you comfort her?\"\n\n\"No, I didn't. At least I suppose I did in the end. I didn't know what\nto do. I'm not a married man, and I'm not accustomed----\"\n\n\"How did you comfort her in the end? It would interest me very much to\nhear, if the details are of the sort that will bear repeating without\ngiving Miss Blow away. Did you hold a pocket-handkerchief to her eyes?\"\n\n\"No, nothing of that kind. I----\" Mr. Goddard hesitated.\n\n\"Go on,\" said Lord Manton. \"I'll treat whatever you say as strictly\nconfidential.\"\n\n\"Well, I promised to do what she asked.\"\n\n\"Do you mean to say that you've pledged yourself to go searching the\ncountry for Dr. O'Grady's body?\"\n\n\"I couldn't help it. What on earth else could I do?\"\n\n\"And when do you start?\"\n\n\"I don't mean to start at all.\"\n\nLord Manton pretended not to hear this remark.\n\n\"Do the thing in style if you do it at all,\" he said. \"Get bloodhounds.\nI'll give you the address of a man in England who breeds them. Fish out\nan old sock of the doctor's; let the bloodhound get the scent, and then\nwe'll be off across country.\"\n\n\"I don't mean to do any such fool thing.\"\n\n\"We'll have glorious paragraphs in all the papers,\" said Lord Manton.\n\"MYSTERIOUS DISAPPEARANCE OF A DOCTOR. VIGOROUS ACTION OF THE ROYAL\nIRISH CONSTABULARY. DISTRICT INSPECTOR GODDARD THINKS HE HAS A CLUE.\nBLOODHOUNDS USED. FIANCEE IN TEARS. Your portrait will appear\nalong with Miss Blow's.\"\n\n\"It's all very fine to laugh,\" said Mr. Goddard; \"but of course I'm not\ngoing----\"\n\n\"You'll have to. You've promised. You can't go back on a promise made\nto a lady. Her portrait will be published in the papers and everybody\nwill see how charming she is. You'll be an object of universal hatred\nand contempt if you go back on your word.\"\n\n\"If ever it gets into the papers at all, you'll look quite as great a\nfool as I shall. Your note will be published. And, after all, you know,\nthe girl has something to say for herself. The doctor's gone. Now, why\nthe devil did he go, and where has he gone to?\"\n\n\"Can't you give a guess at his motives?\"\n\n\"No, I can't. It wasn't debt. Miss Blow told me herself that she was\nready to pay every penny that he owed.\"\n\n\"No; it's not debt. I thought it was at first, but it appears I was\nwrong.\"\n\n\"He didn't drink,\" said Mr. Goddard, \"that ever I heard of. Besides,\neven if he did, that would be no reason for bolting. Whisky's as plenty\nhere as anywhere in the world.\"\n\n\"No; it's not drink. Try the other thing. There's only one more. 'Love,\nor debt, or whisky'--you know the old saying. If it isn't either of the\nlast two, it must be the first.\"\n\n\"Love!\" said Mr. Goddard.\n\n\"Precisely.\"\n\n\"But, hang it all! Miss Blow's quite ready to marry him.\"\n\n\"Too ready,\" said Lord Manton; \"that's my point.\"\n\nMr. Goddard thought hard for a couple of minutes.\n\n\"Do you mean to suggest,\" he said at last, \"that he's run away from\nher?\"\n\n\"In the absence of any other conceivable reason for his bolting,\" said\nLord Manton, \"I am unwillingly driven to the conclusion that he wants\nto escape from Miss Blow.\"\n\n\"But, hang it all! why should he? The girl's uncommonly good-looking.\"\n\n\"Good looks aren't everything,\" said Lord Manton; \"when you come to\nmy time of life, you'll understand that. Just put yourself in the\ndoctor's place for a minute. You've had some little experience of Miss\nBlow. So have I. But the poor doctor knew her a great deal better than\nwe do. Just think of what his feelings must have been when he heard\nthat she was coming over here to pay his debts. He'd be bound to marry\nher straight off after that. And then--just think of sitting down to\nbreakfast every morning opposite a young woman of her character. I\nadmit her good looks, but she's masterful. She'd bully a prize-fighter.\nThe poor doctor wouldn't have had the ghost of a chance, especially as\nmarried life would begin by her paying his debts. That would give her\nthe whip hand of him at once; and she's just the sort of girl who would\nmake the most of her opportunities.\"\n\n\"I don't know--it's possible, of course.\"\n\n\"It's certain, man. Be reasonable. Here's a fact, a perfectly\nundeniable fact. The doctor's gone. Unless you're going to adopt Miss\nBlow's hypothesis----\"\n\n\"Oh, he's not murdered, of course. I know that.\"\n\n\"Very well, then, my explanation of his disappearance is the only one\nthat's left. And it's quite a probable one in itself. Nine men out of\nten in the doctor's position would do exactly what he's done.\"\n\n\"Then what the devil am I to do?\"\n\n\"You're in a deuced awkward position. I don't know what the end of it\nwill be. The authorities certainly won't stand your taking the police\naway from their ordinary duties, and setting them to scour the country\nfor the body of a man who isn't dead. There'd be questions asked in\nParliament about it, and all sorts of fuss. Besides, you'd look such an\nass.\"\n\n\"I know I should.\"\n\n\"All the same, you're in for it now. Unless you choose to go and tell\nMiss Blow the truth. She might believe you, though I very much doubt\nit.\"\n\n\"I couldn't possibly do that. No man could tell a girl that her----\nBut, look here, Lord Manton, your theory may be all very well so far as\nO'Grady is concerned; but there's another man gone now.\"\n\n\"Patsy Devlin,\" said Lord Manton. \"I heard about that.\"\n\n\"Do you think he has run away through fear of his wife?\"\n\n\"No, I don't. There are plenty of other ways of accounting for his\ndisappearance. Besides, in the case of Mrs. Devlin--you know her,\nperhaps?\"\n\n\"No; I never set eyes on the woman in my life.\"\n\n\"Well, she's not equal to Miss Blow in personal appearance; but she has\na certain charm of her own. You wouldn't meet a quieter, less obtrusive\nsort of woman anywhere. Nobody would run away from her unless he was\nforced to. You take my word for it, Patsy will send for her to follow\nhim wherever he's gone to. I knew both Patsy and his wife well, and\nthey always got on splendidly together. The poor fellow was something\nof a _protege_ of mine. I think he regarded me as a friend, and was\ninclined to confide in me. I gave him a letter of recommendation to the\nBoard of Guardians, at the time of the election of the inspector of\nsheep dipping.\"\n\n\"I understood from Sergeant Farrelly,\" said Mr. Goddard, \"that the man\nwas rather a blackguard.\"\n\n\"A horrid blackguard,\" said Lord Manton. \"That's why I didn't want them\nto elect him.\"\n\n\"But I thought you said----\"\n\n\"So I did; but there's no use discussing that now. It's all over and\ndone with. Poor Patsy will never inspect the dipping of a single sheep\nnow. Besides, it's almost dinner-time. You'll stay, of course. Never\nmind about dressing.\"\n\nMr. Goddard was tempted. Lord Manton would give him a good dinner;\nJimmy O'Loughlin--and the choice lay between the two--would almost\ncertainly give him a bad one. Inclination struggled with conscience. In\nthe end conscience prevailed.\n\n\"I can't,\" he said. \"I must see Miss Blow this evening.\"\n\n\"Oh, of course, if you have an appointment with Miss Blow--I suppose\nthe poor doctor won't mind now.\"\n\n\"You're quite wrong. I haven't that sort of an appointment at all. The\nsimple fact is that I'm afraid of her. If I don't see her and manage to\nkeep her quiet somehow, she'll be over at the barrack again making a\nnuisance of herself. You couldn't tell what she'd do.\"\n\n\"She might take it into her head that you were murdered, and set\neverybody searching for your body.\"\n\n\"She might do anything. That's the reason I won't stay to dine with\nyou, though I'd like to.\"\n\n\"Good night,\" said Lord Manton. \"Let me know how things go on; and if\nyou are driven to bloodhounds, remember that I can put you on to the\nbest in England.\"\n\nWhen Mr. Goddard got back to the hotel, he found that his\nself-sacrifice was wasted. Miss Blow had retired to her room for the\nnight.\n\n\"It could be,\" said Jimmy O'Loughlin, \"that the young lady was tired.\n'Bridgy,' says she, when she came in, 'I'm off up to my bed; and I'd be\nthankful to you if you'd bring me up a cup of tea when it's convenient\nto Mrs. O'Loughlin to wet it.'\"\n\nThis did not sound like a thing Miss Blow would have said. Mr. Goddard\nfelt that Jimmy O'Loughlin was adding a varnish of politeness to the\noriginal remark. The next words reassured him. There was at least a\nfoundation of fact beneath the version of the story which he had heard.\n\n\"'And take care,' says she, 'that the kettle's boiling, for the last\ncup you made for me was poison, and smoked at that.'\"\n\nThe words were not yet the words of Miss Blow, but the meaning might\nvery well have been hers.\n\n\"She's mighty particular about her tea,\" said Jimmy. \"She has the life\nfair plagued out of Bridgy; not but what Bridgy deserves it. And what\nwould you be wishing for yourself, Mr. Goddard?\"\n\nMr. Goddard wished that he had accepted Lord Manton's invitation. Since\nit appeared that Miss Blow was safely in her room, perhaps actually in\nbed, he might just as well have dined in comfort at Clonmore Castle.\nBut he did not make this reply to Jimmy O'Loughlin. He said that what\nhe was wishing for himself was a chop and a bottle of porter. He might\nhave said whisky instead of porter, but he knew that there was no hope\nof getting anything else in the way of food except the chop. Jimmy\nO'Loughlin accepted the order and ushered his guest into the commercial\nroom, which happened to be empty.\n\nIn due time Bridgy entered with the chop. It was served on a plate\nwith a round tin cover over it; a cover which meant well, but failed\nto keep the chop warm. Mr. Goddard contemplated the frozen grease\nwhich clung round the edges of the plate, and then, lest it too should\nfreeze, plunged his knife and fork into the chop. Bridgy uncorked his\nbottle of porter, and set it on the table beside a dish of potatoes.\nYellow froth oozed rapidly from the mouth of the bottle and ran down\non to the tablecloth. It added one more brown stain to those which\nthe last four commercial travellers, eating in haste, had made with\nWorcester sauce, mustard, or gravy. Mr. Goddard, who had a fastidious\ndislike of dirty table linen, seized the bottle, and then discovered\nthat he had no tumbler. He set the bottle in the fender, and rang the\nbell furiously. Bridgy half opened the door, and put her head into the\nroom. The rest of her body remained outside. This was her ordinary way\nof presenting herself to people who rang bells. She looked as if she\nexpected to have plates thrown at her, and meant to be ready for a\nswift retreat.\n\n\"A tumbler,\" said Mr. Goddard.\n\nBridgy smiled pleasantly. \"It's hardly ever,\" she said, \"I lay the\ntable but I do be forgetting something. It might be the salt, or--have\nyou the salt? Glory be to God! you have, and the spoon along with it.\"\n\nThere was a pool of considerable size in the fender round the porter\nbottle, and the froth was still oozing out persistently.\n\n\"A tumbler,\" said Mr. Goddard again.\n\nHis tone startled Bridgy. She disappeared, closing the door behind her.\nIn a few minutes she was back again without the tumbler.\n\n\"Where is it?\" said Mr. Goddard.\n\n\"The mistress is giving it a bit of a rinse the way it'll be clean for\nyou; and I came back to tell you that Mr. Moriarty from the barrack\nbelow is at the door, and he says he wants to see you. I'm thinking\nit's a telegram he has in his hand.\"\n\nMr. Goddard rose. \"Get a cloth,\" he said, \"if there is a cloth in the\nhouse----\"\n\n\"Sure there is,\" murmured Bridgy, \"there's dozens.\"\n\n\"And mop up that abominable mess. Take the chop down to the kitchen and\nheat it up again. Get another bottle of porter; and for heaven's sake\nlet me have a meal I can eat when I come back.\"\n\nHe went out and discovered Constable Moriarty, who had, as Bridgy\nobserved, a telegram in his hand.\n\n\"It came, sir,\" said the constable, \"and you just after leaving\nBallymoy with the young lady. The sergeant beyond said I'd better bring\nit with me the way you'd get it at once, in case it might be important.\"\n\nMr. Goddard opened it. \"From Inspector-General, Dublin Castle,\" he\nread. \"Party of Members of Parliament arrive Clonmore to-morrow, noon,\nfrom Dublin. On tour. Provide vehicles to meet train. Show every\nattention. Five in party.\"\n\n\"Damn it!\" said Mr. Goddard.\n\nEvery one in Ireland had heard of the tour of the Members of\nParliament. It was well advertised by means of paragraphs in all\nthe daily papers, so well advertised that an exaggerated opinion\nwas formed in Ireland of the importance of the party. It was\ngenerally believed--and the language of the newspapers fostered the\ndelusion--that at least two Cabinet Ministers were coming, ten or\ntwelve influential politicians, and that more than the usual number of\njournalists would be in attendance. It was felt that the tour offered\na unique opportunity for producing a lasting effect on English public\nopinion. There was, consequently, a severe struggle in Dublin among\nthe numerous people who wanted to conduct the strangers round Ireland.\nNot only all the heads of all the Boards and Departments, but all the\nPresidents and Secretaries of the Leagues and Associations unconnected\nwith the Government were anxious to secure the honour. In the end it\nfell to a high official, who, in order to obtain it, made what was felt\nto be an unfair use of the influence he possessed in England. When he\nfound at the last moment that the party consisted after all of only two\nMembers of Parliament, and they men of inferior calibre and no real\nstanding, he was disgusted and withdrew his offer of a motor car. When\nhe found, further, that there were to be no journalists, and that the\nMembers of Parliament were to be accompanied only by three women, two\nwives and one aunt, he lost his temper and offered the party to any\nLeague which liked to apply for it. In the end he made over his whole\nresponsibility to the police. The party went from place to place in\nthe usual way. It was met at railway stations by polite inspectors of\npolice, allowed to ask questions of the people who could be relied upon\nto supply the proper answers, and given every opportunity of seeing\nwith its own eyes the things that inquiring Englishmen ought to see.\nThe Members enjoyed themselves immensely, and insisted on prolonging\ntheir tour after all the places originally marked out for them had been\nvisited. Then the Inspector-General of Police, who was getting tired\nof making arrangements for them, sent them to Clonmore. It was a very\ndistant place, the terminus of the line of railway on which it stood,\nand it was supposed that no harm could possibly come of their visiting\nit. They were told to drive from Clonmore round a particularly\ndesolate coast, to stop at hotels which were quite abominable, and to\npick up another railway fifty miles off. The Inspector-General reckoned\nthat the trip would take at least four days, and that at the end of it\nthe party would have had enough of Ireland. In any case he would not\nbe bothered with them again until they reached the fifty miles distant\nrailway station.\n\nWhen Mr. Goddard read the telegram he was greatly irritated. He did not\nwant to conduct a party of Members of Parliament round Clonmore, and\ntheir coming would certainly not help him to deal with Miss Blow. He\nforesaw frightful complications. It was possible that the Members of\nParliament, rejoicing in the unexpected discovery of a side of Irish\nlife hitherto unknown to them, might insist on joining in the search\nfor the body of Dr. O'Grady. He determined, if he possibly could, to\nprevent their arrival. He went back into the hotel and wrote a telegram.\n\n\"Inspector-General of Police, Dublin. This district quite unsuited\nto Members of Parliament. Am investigating cases of mysterious\ndisappearance. Inhabitants greatly excited. Disaffection feared. Send\nparty elsewhere.\"\n\n\"Take this,\" he said to Constable Moriarty, \"and dispatch it at once.\"\n\n\"The office shuts at eight, sir,\" said the constable; \"but I'll see\nthat it's sent off first thing in the morning.\"\n\nMr. Goddard looked at his watch. It was twenty minutes to nine o'clock.\nAt eight the next morning the party would leave Dublin. He swore again\nand tore up his telegram.\n\n\n\n\nCHAPTER XII\n\n\nMr. Goddard met Miss Blow at breakfast next morning, and nerved himself\nto the task of telling her that the search for Dr. O'Grady's body must\nbe put off for twenty-four hours on account of the visit of the Members\nof Parliament to the district. He did not know exactly how she would\ntake the news. He half hoped she might get angry and say something\nwhich would give him an excuse for washing his hands of her affairs.\nHe feared that she might cry again as she had cried in his dining-room\nthe day before. He did not want to comfort her under Jimmy O'Loughlin's\nroof. Either Bridgy or Mrs. O'Loughlin might enter the room at any\nmoment. Even Jimmy himself, under some pretext, might interrupt the\naffecting scene. Mr. Goddard was conscious that an account of his\ndealing with a tearful Miss Blow given by Jimmy O'Loughlin would add\nto the gaiety of the neighbourhood. His hope, as it turned out, was\nquite vain and his fear unfounded. Miss Blow took the news in a most\nunexpected way.\n\n\"I'm glad to hear it,\" she said.\n\nMr. Goddard was surprised. Miss Blow explained herself, and he came to\nsee that she had not given up her plan of a search party. It appeared\nthat she had a very high regard for Members of Parliament. Next to the\nHabeas Corpus Act, which did not apply in Dr. O'Grady's case because no\none knew who had the body, and trial by jury, a notoriously uncertain\nthing in Ireland, she looked to Parliament as the great safeguard of\nindividual rights and liberties.\n\nIt became obvious to Mr. Goddard that Miss Blow expected the touring\nMembers to take up her case at once and vigorously, perhaps to make\na special law about it on the spot. He tried to explain that these\nparticular Members were travelling unofficially. Miss Blow did not\nseem to think that fact made any difference. A Member of Parliament,\naccording to her view, is a Member of Parliament, whether he is\nactually delivering a vote at Westminster or not. Mr. Goddard then gave\nit as his opinion that there were too few Members of Parliament in the\nparty to do anything effective.\n\n\"It isn't,\" he said, \"as if we had enough of them to constitute a\nmajority of the House.\"\n\nMiss Blow, by way of reply, stated her intention of meeting the party\nat the railway station. She said she was sure that as soon as they\nheard her story they would bestir themselves just as Lord Manton\nand Mr. Goddard had, but rather more vigorously. She even suggested\nthat a telegram might be sent to the Prime Minister. Mr. Goddard\ndiscouraged her. He felt that he was being rapidly edged into a very\nawkward position. It was utterly impossible to calculate the effect\nthat a story like Miss Blow's might produce on an enthusiastic Member\nof Parliament. Almost anything might happen. Publicity, newspaper\nparagraphs, and questions in the House of Commons would be certainties.\nAnd he had been warned to show the approaching party every civility. It\nwould certainly not be civil to plunge them into a vortex of mysterious\ncrime. The Inspector-General would naturally be vexed if such a\nthing happened, and Mr. Goddard's prospects of promotion, never very\nbrilliant, would be injured.\n\n\"I don't think it would do,\" he said, \"for you to meet them at the\nrailway station.\"\n\n\"Why not?\"\n\nThe Miss Blow at the other side of the breakfast table with a teapot in\nfront of her and a decisive way of handling it seemed quite different\nfrom the Miss Blow who had wept so pleasingly in his dining-room the\nday before.\n\n\"I rather think,\" he said feebly, \"that they have ladies with them.\"\n\n\"All the better,\" said Miss Blow.\n\n\"And their tour is quite unofficial. We're not supposed to know that\nthey are Members of Parliament.\"\n\n\"But we do know,\" said Miss Blow.\n\nThe argument ended by Mr. Goddard promising to lay the case before the\nMembers. It was only by making the promise that he was able to induce\nMiss Blow to refrain from going to the railway station. Having made\nit, he slipped out of the room. It was after ten o'clock, and he still\nhad to make arrangements for the comfort of the party. He found Jimmy\nO'Loughlin in the yard behind the hotel.\n\n\"There's a party coming in by the train to-day,\" he said, \"and I\nwant a brake and four horses to meet them. They'll be driving to\nPool-a-donagh.\"\n\n\"Is it the Lord Lieutenant?\" said Jimmy O'Loughlin.\n\n\"No, it's not. It's a private party.\"\n\n\"I asked the question,\" said Jimmy, \"because, if it was the Lord\nLieutenant itself, he couldn't get the brake.\"\n\n\"Oh, well, that's all right. It's not the Lord Lieutenant.\"\n\n\"And if the Lord Lieutenant wouldn't get it,\" said Jimmy, \"you may take\nyour oath that another won't.\"\n\n\"And why not?\"\n\n\"Because the two front wheels is off the only brake there is in the\ntown, and Patsy Devlin has them up at the forge fixing them, and he's\ngone from us. That's why.\"\n\n\"And there isn't another brake?\"\n\n\"There is not.\"\n\n\"There are five in the party,\" said Mr. Goddard. \"We'll have to get two\ncars.\"\n\n\"It's a good eight miles to Rosivera,\" said Jimmy, \"and better than ten\non from that to Pool-a-donagh. I wouldn't say that there was a horse in\nthe town fit to do the journey with four on the car, and there'll have\nto be four on one of the cars, if there's five in the party, that'll be\ncounting the drivers. My own mare was over at Ballymoy yesterday with\nConstable Moriarty and the doctor's young lady. It's a day's rest she\nought to have by rights, and not to be going off on the road again.\"\n\n\"It can't be helped,\" said Mr. Goddard. \"I must have the horses.\"\n\n\"There isn't another gentleman about the country,\" said Jimmy, \"that\nI'd do it for only yourself; but seeing that the party is friends of\nyour own, I'll let my mare go, and I'll get Patsy Devlin's grey pony\nthat was promised to Mr. Byrne for the day to be carting home the turf,\nthe same pony that the priest was thinking of buying. It's little use\nPatsy's widow will have for a pony now her husband's gone from her, the\ncreature. I don't know another that you could put under a side car for\na gentleman to sit behind, and it's badly able for the road the grey\npony will be this minute.\"\n\n\"Give her a feed of oats between this and the time the train comes in,\"\nsaid Mr. Goddard.\n\n\"I will; and I'll see if I can't get the loan of the priest's cushions\nfor the old car. The ones that are on it are terrible bad.\"\n\nJimmy O'Loughlin was certainly not guilty of raising any false hopes\nabout the quality of his cattle or his vehicles; but Mr. Goddard had\na feeling of cold disgust when he saw the two cars standing together\noutside the railway station. Jimmy O'Loughlin's mare was the better of\nthe two animals, and she looked extraordinary feeble and depressed. The\ngrey pony, which should have been drawing home Mr. Byrne's turf, was\nclearly unfit for the journey before him. He may have been fretting\nfor the loss of Patsy Devlin. He may have been insufficiently fed by\nMrs. Devlin. He looked as if his spirit was completely broken either by\nstarvation or great grief. The cars were dilapidated, and the harness\nevidently untrustworthy. On the other hand the drivers were full of\nlife and vigour. They looked forward to receiving handsome tips from\nthe strangers at the end of their day's work, and were quite prepared\nto earn them by saying all the things which tourists expect from Irish\ncar drivers. They were primed with stories of the most popular kind\nabout every point of interest along the road. They had a store of bulls\nand humorous repartees ready to their lips. They touched their hats\njauntily to Mr. Goddard as he passed them. He was a benefactor, and\nthey owed him respect and thanks.\n\nThe train drew up at the platform. The door of a first-class\ncompartment was flung open, and a gentleman bounded from it. His face\nexpressed a feeling of irresponsible holiday happiness. His movements\nand the pose of his body suggested abundant vitality and energy. He\nturned and assisted three ladies to alight. One, the youngest of them,\nhe lifted out in his arms and deposited three or four paces from the\ncarriage door. He laughed merrily as he did so. She also laughed. The\nother two ladies, who had not been embraced, laughed. The holiday\nspirit was strong in all of them. Then, still laughing, he turned to\nthe carriage again, and received from some one inside a number of\nbags, boxes, coats and parcels. He took them two by two and laid them\nin a long row at the feet of the ladies. Then a second gentleman got\nout of the train, a long, lean man, of sallow complexion and serious\nexpression.\n\nMr. Goddard felt that it was time to introduce himself. He approached\nthe party cautiously, skirting an outlying hold-all. The first\ngentleman, who was also a good deal the younger of the two, spied him\ncoming and bounded forward to meet him.\n\n\"You are Mr. Goddard,\" he said joyously. \"I'm sure you must be. I'm\ndelighted to meet you. My name is Dick, surname, you know. Christian\nname similar--Richard Dick, M.P., not yet in the Cabinet.\"\n\nHe had to the highest possible degree that air of breezy joviality\nassumed by many Englishmen when they cross St. George's Channel. It\nwas as if, having at last reached the land of frolicsome recklessness,\nhe was determined to show himself capable of wild excess and a very\nextremity of Celtic fervour.\n\nMr. Goddard bowed, and murmured that he felt it a pleasure and an\nhonour to make Mr. Dick's acquaintance.\n\n\"This,\" said Mr. Dick, \"is Mr. Sanders. Sorrowful Sanders we call him,\non account of the expression of his face. He's not in the Cabinet\neither; but he soon will be.\" He caught Mr. Goddard by the arm and\nwhispered a clearly audible aside. \"He's a Scot, and that's nearly the\nsame thing, you know.\"\n\n\"Shut up, Dick!\" said Mr. Sanders.\n\n\"How can you talk such nonsense?\" said the eldest of the three ladies,\nwhose appearance suggested some strength of character, and suddenly\nbrought back Miss Blow to Mr. Goddard's mind.\n\n\"This,\" said Mr. Dick, indicating the lady, \"is Miss Farquharson,\nSanders' aunt. This\"--he pointed to another lady--\"is Mrs. Sanders,\nhis wife. And this\"--he drew forward by the hand the lady whom he had\nlifted out of the carriage--\"is Mrs. Dick, the 'woman that owns me.'\nIsn't that the way you express it in this country?\"\n\nMr. Goddard bowed three times, and then glanced doubtfully at the bags\nwhich lay on the platform.\n\n\"I've only been able to get two cars for you,\" he said. \"Have you much\nmore luggage in the van?\"\n\n\"Oh, but an Irish car can carry any quantity of luggage,\" said Miss\nFarquharson. \"I've seen them absolutely packed, and still there was\nroom for more.\"\n\nMr. Goddard admitted that this was true; but he thought of Patsy\nDevlin's grey pony and the long miles between Clonmore and\nPool-a-donagh.\n\n\"And if there isn't room,\" said Mrs. Dick, \"I'd like to sit in the\nmiddle--on the well--isn't that what they call it?--with my back\nagainst the driver. I saw a boy sitting like that the other day, and it\nlooked lovely.\"\n\n\"It's all right,\" said Mr. Dick. \"We have two bikes with us. Sanders\nand I will ride. You shall have my machine, Sanders, and I'll take my\nwife's. Come along. The spin will do you all the good in the world.\nWe'll pedal along and let the ladies have the cars.\"\n\nMr. Sanders protested strongly against this plan. He had, he said, a\nweak heart, and cycling did not agree with him. He was overborne by a\ncommand from his aunt, and towed down the platform towards the luggage\nvan by Mr. Dick. Miss Farquharson confided to Mr. Goddard that her\nnephew's heart was not nearly so bad as he thought it was, and that\nthe exercise would do him good. She was, unquestionably, a lady of\ncommanding character.\n\n\"He's so full of energy,\" said Mrs. Dick, watching her husband's\nprogress admiringly. \"I say the air of Ireland has got into his head.\"\n\n\"Is he not so energetic at home?\" asked Mr. Goddard.\n\n\"Indeed he isn't. Just fancy if he was!\"\n\nShe giggled convulsively at the thought. It was evident that Mr.\nDick, though not a Scot, belonged to some part of the country where\ndecorum of demeanour and a certain gravity are expected from Members of\nParliament.\n\n\"'Fare thee well,'\" shouted Mr. Dick from the end of the platform,\n\"'and if for ever, still for ever fare thee well.'\"\n\nHe was wheeling the two bicycles out of the station, and was followed\nby the obviously reluctant Mr. Sanders.\n\n\"Richard, Richard!\" said his wife.\n\nMr. Dick paused and looked round.\n\n\"Have you got your sandwiches? Don't go without your sandwiches. I'm\nsure they're in my hold-all.\"\n\nBut Mr. Dick patted his coat pocket triumphantly. He had the\nsandwiches. He was not the kind of man, so his attitude suggested,\nthe feeble and inefficient kind of man, who goes off without his\nsandwiches. A moment later, having compelled Mr. Sanders to mount, he\nwas cycling gaily down the road towards Clonmore.\n\n\"Richard, Richard!\" shouted his wife again. But it was too late. Her\nvoice did not reach him.\n\n\"I'm sure,\" she said, turning to Mr. Goddard, \"that he doesn't know the\nway. He has never been here before. He'll get lost. Whatever are we to\ndo?\"\n\nMr. Goddard consoled her. He pointed out that Mr. Dick had started in\nthe right direction; that it was generally possible to make inquiries\nwhen in doubt; that, as a matter of fact, once clear of Clonmore, there\nwas only one road on which anybody could ride a bicycle, and that it\nled straight to Pool-a-donagh. Miss Farquharson helped to reassure her.\n\n\"Where there's a will there's a way,\" she said sententiously.\n\n\"Do stop him,\" said Mrs. Dick. \"You must stop him. He's got my\npocket-handkerchief in his pocket. I gave it to him to keep for me in\nthe train.\"\n\n\"I can't,\" said Mr. Goddard. \"He's gone. He's out of sight. Even if\nhe wasn't, I don't think that I could stop him. But I can lend you a\npocket-handkerchief. I have two. This one is quite clean.\"\n\nThen came the business of packing the ladies and their belongings on\nthe cars. Mr. Goddard, after consultation with the station master and a\nporter, gave all the luggage to the driver of Patsy Devlin's grey pony.\nJimmy O'Loughlin's mare was the more likely of the two animals to reach\nPool-a-donagh, and the station master pointed out that if there were\nto be a break down it would be better for the ladies to arrive without\ntheir luggage than for the luggage to arrive while the ladies were left\non the side of the road. Mrs. Dick, recovering her spirits, insisted\non carrying out her plan of sitting on the well of the car. There was\na small crowd outside the railway station, which watched with reverent\nwonder her climbing and wriggling. She waved both hands to Mr. Goddard\nas soon as she had settled herself comfortably, and was very nearly\nthrown off the car when the mare started with a jerk. Afterwards she\nclung to Miss Farquharson and Mrs. Sanders, who sat one on each side of\nher.\n\nIt was not until the two cars were well on their way down the road that\nMr. Goddard recollected the promise he had made to Miss Blow. He had\nreally intended to fulfil it. He had it in his mind to say something\nof a light and jocular kind about the disappearance of the doctor,\nsomething which would redeem the letter of his promise without exciting\nthe Members of Parliament. It was not, he reflected, in any way his\nfault that he had failed. He had no opportunity of speaking. Mr. Dick's\nimpetuous energy had made it quite impossible to approach the subject\nof Dr. O'Grady. But, while his own conscience absolved him, he was\nquite sure that he would not be able to explain himself satisfactorily\nto Miss Blow. She would not believe that Members of Parliament could\npossibly behave as Mr. Dick had behaved. She would not understand\nthe effect of the Irish air upon naturally staid men. There was some\ncomfort for him in the thought that the cars, with Mrs. Dick's legs\nswinging off the foremost one, must have passed the hotel, and that\nMiss Blow might have seen for herself that the party was in no mood for\ninvestigating murders. The bicyclists, unless they deliberately turned\naside before reaching the town, must also have passed the hotel. Mr.\nDick very probably sang some song of the open road as he sped through\nClonmore. Miss Blow might have heard it, might have seen for herself\nthe sort of people these tourists were. If she did, Mr. Goddard's\nreputation as a man of honour would be safe. She could not possibly\nexpect him to redeem his promise.\n\nThen a fresh and most depressing thought attacked him. The Members of\nParliament had come and were gone; but there was another promise of\nhis unfulfilled. Miss Blow would certainly expect him to start at once\nand search for Dr. O'Grady. He knew that he could not postpone the\nmatter any longer. She would pin him to his word, insist upon immediate\naction, refuse to rest satisfied with excuses. He walked very slowly\ndown the hill from the station.\n\nA cowardly way of escape presented itself to him at the last moment.\nHis horse and trap were in the hotel yard. If he could get them\nwithout being seen by Miss Blow he might drive back to Ballymoy. Miss\nBlow, since the Members of Parliament had got the only available\ncars, could not follow him. Forgetful of honour and chivalry, of Miss\nBlow's tear-stained face, of the pressure of her hand, of the kiss\nwhich he had nearly given to her glove, he made up his mind to fly. He\napproached the hotel very cautiously.\n\nLike a thirsty man on a Sunday who has not been able to travel the\nnumber of miles which make drinking legal, he climbed over a back wall\ninto the yard. He glanced nervously at the windows, hoping that Miss\nBlow's room looked out on the front and that she would be expecting him\nto reach the hotel along the road. He caught sight of Bridgy staring\nout of the scullery window. She had watched him climb the wall and was\nmost anxious to discover what he intended to do next. It seemed to her\nunnatural that an officer of police should enter an hotel in such a\nway. Mr. Goddard, taking shelter in the stable, beckoned to her through\nthe door. Filled with curiosity, she crossed the yard and joined him in\nthe stable.\n\n\"Bridgy,\" he said, \"here's a shilling for you. Is Mr. O'Loughlin\ninside?\"\n\n\"He is, sir,\" said Bridgy.\n\n\"Then tell him to come out here. I want to speak to him.\"\n\n\"Is it out to the stable?\"\n\nMr. Goddard had sacrificed his own self-respect when he yielded to\ntemptation and made up his mind to escape. He now flung away all hope\nof ever being respected by Bridgy.\n\n\"Yes; here in the stable. And if you meet Miss Blow, don't tell her\nwhere I am.\"\n\n\"I will not. Why would I? But sure----\"\n\n\"Go on now, like a good girl, and don't waste your time talking to me.\"\n\nJimmy O'Loughlin was a man of tact and good manners. He greeted Mr.\nGoddard cheerfully as if then were nothing surprising in the choice of\na stable for the scene of an important interview. He had been warned\nby Bridgy that Mr. Goddard, for some reason, stood in terror of Miss\nBlow; but he made no allusion to her. He opened the conversation with a\nremark on a safe, indifferent topic.\n\n\"Them was queer people,\" he said, alluding to the Members of Parliament\nand their party.\n\n\"I want my horse and trap,\" said Mr. Goddard, \"and I want to pay my\nbill. I am going back to Ballymoy at once.\"\n\n\"I wouldn't say but you're right,\" said Jimmy, \"if them ones is likely\nto be back here in the course of the day.\"\n\n\"It's not that. I don't mind about them. It's business that's taking me\nhome--important business.\"\n\nThis was too much for Jimmy O'Loughlin. His tact and manners were good,\nbut he was not going to allow Mr. Goddard to escape without an allusion\nto Miss Blow.\n\n\"The doctor's young lady is within, waiting for you,\" he said.\n\n\"I know that; but I haven't time to talk to her now. In fact, it is\nmost important that I should get away without her seeing me--on account\nof my business.\"\n\n\"I wouldn't say but you might be right there too,\" said Jimmy.\n\nThey set to work together and harnessed Mr. Goddard's horse. They\nled him into the yard and put him between the shafts of the trap as\nsilently as possible.\n\n\"I'm thinking,\" said Jimmy, \"that maybe it would be better for you not\nto be paying me the trifle that's due for your bed and your dinner\nuntil the next time you're over.\"\n\n\"Very well. Then I'll be able to start at once.\"\n\n\"You will. And, what's more, when the young lady asks me what's\nhappened to you, I'll be able to say that I don't know because you went\noff without paying your bill.\"\n\nJimmy O'Loughlin had a sensitive conscience. He could lie without\nhesitation when circumstances required it of him, but he preferred,\nwhere possible, to deceive without departing from the literal truth.\n\n\n\n\nCHAPTER XIII\n\n\nMr. Dick, who was a man of energy, rode fast. Mr. Sanders toiled behind\nhim, but was able to keep him in sight because Mr. Dick dismounted\nand waited for him at the top of every hill. The day was hot, and\nthere was very little breeze. Neither of the men was in good training.\nBoth of them became thirsty. Five miles outside of Clonmore the road\ncrosses a small river. Mr. Dick stopped on the bridge, and, when Mr.\nSanders overtook him, proposed that they should take a drink. They\nmade their way down to the stream, lay on their stomachs, and sucked\nup large quantities of luke-warm water. Then they rode on again, and,\nas might have been expected, became much hotter and thirstier than\nthey were before. Mr. Dick stopped again, this time at a pool. Mr.\nSanders, though very thirsty, expressed doubts about the wholesomeness\nof the water. Mr. Dick explained briefly that, as there was no human\nhabitation in sight, the pool could not possibly be polluted by drains.\nThen he lay down and drank as eagerly as a camel at an oasis. The pool\nwas shallow, and the violence of his sucking stirred up a good deal\nof mud. Mr. Sanders, realizing that every moment's delay increased\nthe chance of his getting typhoid fever from his draught, chose the\ncleanest corner of the pool and drank a great deal more than was good\nfor him. The water of the pool was even warmer than that of the river.\n\nThen it was discovered that one of the tyres of the bicycle ridden by\nMr. Dick had burst. He had pumped it too vehemently, and the heat of\nthe sun had swelled the air inside until the strain was too great for\nthe cover. Both tool-bags were examined, but no materials for repair\nwere found in them. Even under these circumstances Mr. Dick remained\ncheerful.\n\n\"We'll walk on,\" he said, \"until we come to a police barrack. We're\nbound to come to one soon.\"\n\n\"What good will that be?\"\n\nMr. Sanders did not want to walk on if he could help it.\n\n\"In Ireland,\" said Mr. Dick, \"the police are experts at repairing\nbicycles. One of the people we met last week told me that. He was an\ninspector of something, and was always going about the country, so he'd\nbe sure to know.\"\n\n\"In any case,\" said Mr. Sanders, \"there isn't likely to be a police\nbarrack about here. There are no houses. We haven't passed a single\nhuman habitation for the last twenty minutes.\"\n\nMr. Dick overruled this objection at once. He had been studying the\nIrish question for a whole fortnight, and he thoroughly understood the\ncountry.\n\n\"In Ireland,\" he said, \"the most likely place to find a police\nbarrack is where there are no houses. The reason for that is that the\nuninhabited districts of the country are those from which the people\nhave been evicted. They naturally want to get back again, and so police\nbarracks are built to prevent them doing so. You will always find a\nbarrack where there are no people. A man who was greatly interested in\nthe land question told me that the day before yesterday.\"\n\n\"I think,\" said Mr. Sanders, \"that I'll sit down here and wait till\nthe cars overtake me. You can take your own bicycle and go on if you\nlike.\"\n\n\"You'd better not, because the cars may never overtake you. We may be\non the wrong road altogether. I didn't ask the way. I simply steered a\ncourse by the sun like an explorer in central Africa.\"\n\n\"What an ass you are, Dick!\"\n\n\"Not an ass, Sanders, not an ass, an adventurer. I love risk for its\nown sake. The blood of the ancient Bersekers is in my veins. I feel\nlike the man in the song 'Fiddle and I,' that is to say, in this case,\n\n  'Biky and I\n   Wandering by\n   Over the world together.'\n\nIf you don't come on with me, Sanders, you will get lost like a babe in\nthe wood, and then\n\n  'When you are dead\n   The robins so red\n   Will take strawberry leaves and over you spread.'\"\n\nMr. Sanders shrank from such a fate. Also he had sat down during the\ndiscussion and felt rested. He agreed to go on. Mr. Dick, the Berseker\nspirit strong in him, wheeled both bicycles. They climbed a long hill\nand found walking even hotter work than bicycling. But they had their\nreward. From the top of the hill the entrance gate and the trees of\nRosivera were visible.\n\n\"A sail, a sail!\" cried Mr. Dick. \"We are saved!\n\n  'We shall hear the sweet music of speech,\n    Nor finish our journey alone.'\"\n\n\"It looks like a gentleman's place,\" said Mr. Sanders, \"a dilapidated\ngentleman's place. I mean to say, of course, the dilapidated place of a\ngentleman.\"\n\nMr. Dick was not a stickler for the nice arrangement of adjectives.\n\n\"Of course,\" he said, \"it's a dilapidated gentleman's place. All\ngentlemen's places in Ireland are dilapidated. That is one of the\nconsequences of the recent land legislation.\"\n\n\"Rents,\" said Mr. Sanders, who took a special delight in all kinds of\nfigures, \"were reduced twenty per cent. all round on an average at the\nfirst fixing. Then they were reduced again by----\"\n\n\"Come on,\" said Mr. Dick. \"The proprietor, whoever he is, must have\nenough left to buy a repair outfit. Let's go and borrow it.\"\n\nHe had heard quite as much as he wanted to hear about rent fixing and\nland purchase during the fortnight he had spent in Ireland. He did not\nwant to go into the matter again. It required intricate calculations,\nand Mr. Dick had no taste for arithmetic. Mr. Sanders sighed and\nfollowed his friend down the hill.\n\nThey passed through the gate of Rosivera and went down the avenue, Mr.\nDick leading the way with the two bicycles. Turning a corner, they came\nsuddenly upon a view of the house. Beyond it, at the bottom of the\nlawn, lay the bay.\n\n\"Dilapidated!\" said Mr. Sanders, in disgust. \"I knew it would be\ndilapidated, but I didn't expect it to be as bad as this. It looks to\nme as if it were uninhabited.\"\n\nMr. Dick paid no attention to the appearance of the house. The sight of\nthe sea seemed to intoxicate him. He was very hot and very dusty. The\nidea of bathing in cold water was delightful.\n\n\"We'll have a swim,\" he said. \"First we'll knock up the proprietor,\nborrow the repair outfit, and mend the tyre. Then we'll go round the\ncorner, out of sight of the house, and wallow in the briny wave.\"\n\nBut Mr. Sanders was cautious, more cautious than he had been about\ndrinking the water out of the pool.\n\n\"You've no towel,\" he said.\n\n\"What do we care for towels? We are primitive men, savages on our\nnative wild, cave dwellers of the paleolithic age. I should spurn a\ntowel if it was offered me.\"\n\n\"That's all very well for you, Dick; but I have a weak heart, and I\nhave to be careful. In the heated state in which I am at present, I\ndaren't risk bathing, especially without a towel.\"\n\n\"Then I'll bathe by myself. You can get the bicycle repaired. By the\ntime you have it settled I shall be ready to start again.\"\n\nThey reached the house. Mr. Dick commented, laughing, on the fact that\nthere was no electric bell. Mr. Sanders sighed at this fresh evidence\nof dilapidation. There was no knocker either. Mr. Dick hammered on the\ndoor with his fist until he thought he had made noise enough to attract\nattention. Then he walked on towards the shore, leaving his friend\nalone to enjoy the hospitality or face the wrath of the inmates of the\nhouse. Mr. Sanders waited for some time and then knocked again. After\nanother pause he tried kicking the door. Then he rang both the bicycle\nbells. At last he began to despair of attracting attention. Some\nmen, under the circumstances, would have gone away. Mr. Sanders was\npersevering and little troubled with delicacy of feeling. He pushed\nopen the door and walked in.\n\nMr. Dick, yearning for a swim, soon found a spot which seemed\nsufficiently secluded. It was not particularly attractive as a\nbathing-place. The beach was covered with small rough stones; and, the\ntide being out, there was a considerable stretch of beach. The water\nnear the shore was full of jellyfish and brown seaweed. But Mr. Dick\nwas too hot and too eager to care much about these inconveniences. His\ndesire was to get as quickly as possible into the sea. He undressed\nbeside a large stone which lay just above high-water mark. Then his\ntroubles began. He had never before walked on such trying stones.\nThe pain which they gave his feet caused him to stumble and fall\nsuddenly forward on his hands. Part of the journey he accomplished on\nall-fours. The seaweed was a relief when he reached it. The jellyfish\nwere deliciously soft under his feet. He floundered out through them\nand over them until the water reached his knees. Then he flung himself\nforward and struck out. The weed brushed his limbs and body. The\njellyfish, incredible numbers of them, slipped past him.\n\n\"This,\" he said, \"is delicious.\"\n\nHe got past the belt of weed and jellyfish into deep water. He shouted\naloud in his joy--a wild inarticulate whoop which went sounding across\nthe waters of the bay. He lay on his back. He kicked with his legs,\nraising what seemed to him splendid fountains of water. He shouted\nagain. Then he swam further out, using a side stroke which he had\nlearned in a swimming-bath, burying his head each time his arm left\nthe water, and then turning his face up and snorting like a porpoise.\nAfter awhile he lay on his back again and began to sing--\n\n  \"Rule, Britannia! Britannia rules the waves.\"\n\nThe ditty was appropriate enough. Mr. Dick M.P., represented in\nRosivera Bay the world's greatest maritime power, and he had, plainly,\nso far got the better of the sea that it was obliged to bear him on its\nbreast and minister to his delight. He ruled it. There were, indeed, no\nwaves; but that was not Mr. Dick's fault. If there had been waves, he\nwould have ruled them.\n\n  \"Rule, Britannia!\" he sang again, \"Britannia rules the waves.\n   Britons never, never, never shall be slaves.\"\n\nMr. Dick had a poor ear for music, and his rendering of the tune\nwas far from correct; but he had a fine voice and it rang out\nsatisfactorily. He felt more than ever like an ancient Berseker. His\nsong was a kind of triumphant challenge to man and nature alike.\n\n\"This,\" he said breathlessly, when he had finished the song for the\nthird time, \"is better than mending bicycles. I wonder how poor Sanders\nis getting on?\"\n\nHe wallowed round and faced the shore. He saw a man approaching from\nthe direction in which the house lay.\n\n\"Hullo--lo--o!\" he shouted. \"Coo--ee, Sanders!\"\n\nThen he saw that it was not Mr. Sanders. He came to the conclusion that\nit must be one of the inhabitants of the house. He was perfectly right.\nIt was Mr. Red.\n\n\"Funny-looking old cock he is,\" said Mr. Dick.\n\nMr. Red stalked majestically along the shore. Mr. Dick swam to meet\nhim. His heart was light. He broke into a song of greeting--\n\n  \"Come o'er the sea,\n   Stranger to me,\n   Mine through sunshine, storm and cloud!\"\n\nMr. Red walked straight to the place where Mr. Dick's clothes lay. It\nseemed possible that he was bringing down a towel. Mr. Dick swam on\ntowards the shore, intending to express his gratitude for the civility.\nMr. Red reached the clothes, picked them up one by one and walked\naway with them. Mr. Dick shouted after him, but without effect. He\nswam for the shore as quickly as he could. If the proprietor of the\nplace objected to people bathing on the shore, Mr. Dick was prepared\nto apologize. He would apologize humbly, get back his clothes, and\nthen point out that a notice ought to have been erected to warn the\npublic not to bathe. Mr. Red deposited the clothes on the grass at\nsome distance from the beach, turned round, and walked towards the sea\nagain. Mr. Dick felt bottom with his feet, and plunged forward until he\nstood in water which only reached his knees. Mr. Red, standing on the\nvery brink of the sea, took a revolver out of his pocket and pointed it\nat Mr. Dick.\n\n\"Come on shore,\" he said.\n\nBy way of reply Mr. Dick sat down suddenly, rolled over on his side,\nand lay with no part of him above water except his head. He did not\nlike standing naked in front of a revolver which might be loaded.\n\n\"Come!\" said Mr. Red.\n\nMr. Dick put his head under water, and kept it there as long as he\ncould. When he came up, gasping, he saw the revolver still levelled at\nhim.\n\n\"Come at once,\" said Mr. Red, \"or I shall fire!\"\n\nMr. Dick struggled to his feet and stumbled forward a few steps. Then\nhis self-respect, the self-respect which is the birthright of every\nfree-born Briton, asserted itself within him.\n\n\"I presume,\" he said, speaking with a certain cold dignity, \"that this\nbeach is private property, and that I have trespassed by bathing here.\nIf so, I offer my apologies; but I cannot refrain from saying at the\nsame time that the aggressive violence of your conduct----\"\n\nMr. Red pulled the trigger of the revolver. A bullet struck the water\nso close to Mr. Dick that his leg was splashed. He stopped short in his\nprotest and waded as rapidly as possible towards the shore. He pushed\nhis way through the seaweed and jellyfish and stood at last, a pitiful,\ndripping figure, on dry land.\n\n\"Go on,\" said Mr. Red, fingering the revolver.\n\nThe fear of instant death impelled Mr. Dick over the sharp stones. He\ntravelled rapidly for five or six yards, and then collapsed suddenly on\nto all-fours.\n\n\"Go on quickly,\" said Mr. Red.\n\nMr. Dick went on as quickly as he could on his hands and feet. He\nlooked like a large hairless ape of some bleached kind. Then, overcome\nby the pain of both hands and feet, he sat down and faced Mr. Red.\n\n\"This is an outrage,\" he said. \"I protest against it in the strongest\npossible manner. You have absolutely no right----\"\n\n\"Get up,\" said Mr. Red, \"and precede me to the house.\"\n\nMr. Dick got up with a groan. He was really suffering considerably.\nThe Berseker spirit, active in him during an earlier part of the day,\nwas almost dead. He staggered on a few steps. Then a new notion seized\nhim, a fear which was actually stronger than the fear of death. He\nunderstood suddenly that he was not, as he had boasted, a primitive\nman; that he was, on the contrary, highly civilized. He turned on Mr.\nRed and faced the revolver without a tremor.\n\n\"I won't go to the house without my clothes,\" he said. \"There may be\nladies. There may be servants.\"\n\nThere was a determination in the way he spoke and in the expression of\nhis face which was quite unmistakable. Mr. Red tried to bully him, but\nfailed. No threat was of any avail.\n\n\"I won't go,\" said Mr. Dick, \"unless I can have my shirt at least.\"\n\n\"You can have your shirt,\" said Mr. Red. \"Go and get it.\"\n\nMr. Dick made his way slowly across the beach. He looked over his\nshoulder from time to time, and saw that Mr. Red was following closely\nwith the revolver. He reached the grass, felt it soft under his feet,\nand moved more easily. It occurred to him that he might make a dash for\nliberty. Mr. Red would shoot, no doubt; but he might miss. He might\nhit, and still only wound--wound in some trifling way which would not\nprevent further flight. There was a grove of trees not fifty yards\ndistant. They were small, scrubby trees, but they would afford some\nshelter. He made up his mind to risk it. Then the awful prospect before\nhim made him pause. How could he--even supposing that he was not shot\ndead at once--wander stark naked through a strange country?\n\nHe reached the pile of clothes, stooped down and picked up his shirt.\nThen he took up his trousers.\n\n\"Drop that,\" said Mr. Red.\n\nDean Swift noted the fact that an unarmed man in his shirt is likely\nto get the worst of a struggle with eight armed men who are fully\ndressed. The chances are more equal, if there is only one armed man, if\nthe unarmed man gets his trousers on. Mr. Red knew this, and strictly\nlimited Mr. Dick's clothing. He was not prepared to run any unnecessary\nrisk with his prisoner. The warmth of the shirt, which was flannel,\nrestored Mr. Dick's self-respect and courage. He turned on Mr. Red once\nmore.\n\n\"If this is the way that Irish landlords habitually behave,\" he said,\n\"I don't wonder that there has been an agrarian revolution in the\ncountry. I always had a certain sympathy with your class before, but\nnow that I know what you really are I shall vote for the next Land Bill\nthat is brought in, whatever it is.\"\n\n\"Precede me to the house,\" said Mr. Red.\n\nMr. Dick walked on. He crossed the grass rapidly and came to the gravel\nsweep in front of the house. Here his feet began to give him great pain\nagain. The gravel was in some ways worse than the stones on the beach\nhad been. Mr. Dick felt it severely.\n\n\"I've heard stories,\" said Mr. Dick, \"of the way you Irish landlords\ntreated your tenants in the past. I know now that I didn't hear half\nthe truth. Anything more abominable, more outrageous, more utterly\nillegal----\"\n\nHe reached the door of the house as he spoke, and stepped with a sigh\nof relief on to the smooth step.\n\n\"Look here,\" he said, \"where's Sanders? Have you murdered Sanders?\"\n\nHe went forward again, passed through the door, and stood in the hall.\n\nThere, in front of him, bound hand and foot, guarded by the\nlong-bearded anarchist, lay Mr. Sanders.\n\n\n\n\nCHAPTER XIV\n\n\nMr. Red treated Dr. O'Grady and Patsy Devlin very well. They could\nnot have fared better if they had been political prisoners awaiting a\ntrial for inciting people to boycott each other. They had abundance\nof excellent food, three meals a day, brought up to them by one of\nthe foreign anarchists. They had a sufficiency of whisky and tobacco.\nA second bed was supplied when the doctor objected to sleeping with\nPatsy Devlin. Twice every day Dr. O'Grady was taken downstairs and\nallowed to attend the injured man, who was recovering rapidly. Every\nevening Mr. Red, adhering honourably to his bargain, handed over five\nsovereigns to the doctor. A large number of books was supplied to the\nprisoners. They were chiefly treatises on the theory and practice of\nanarchism, accounts of the revolution in Russia, and kindred matters.\nDr. O'Grady was perfectly content with them. The subject was new to\nhim, and he read with excited curiosity. It was a favourite boast of\nhis that no book of any kind bored him, provided he understood the\nlanguage in which it was written. Unfortunately, Patsy Devlin did not\nlike reading. He worked slowly through the accounts of the murders of\na few Grand Dukes, and displayed some slight interest in the tortures\ninflicted on a female anarchist. Then he became bored and refused to\nread any more. He used to walk about the room whistling loudly. There\nwere only two tunes which he cared to whistle--\"The wearing of the\nGreen\" and \"God save Ireland.\" The constant repetition of them began to\nirritate Dr. O'Grady at the end of the second day of Patsy's captivity.\nHe expostulated, and Patsy agreed to stop whistling. He was a man of\nkindly heart and had a real affection for the doctor.\n\n\"I wouldn't,\" he said, \"be doing what might annoy you; and if so be\nthat the tunes is disagreeable, there won't be another note of them\nheard from this out.\"\n\nHe meant what he said, but he promised more than he was able to\nperform. It soon appeared that he could not help whistling. He did not,\nin fact, know when he was whistling. The tunes came bubbling from his\npursed lips against his will.\n\nDr. O'Grady recognized that Patsy could not accomplish the impossible.\nHe gave up reading about anarchists, and asked Mr. Red for a pack of\ncards. The request, a reasonable one, was refused, and Dr. O'Grady\nsnubbed for making it. Mr. Red disapproved of card-playing on\nprinciple. He said that games of chance were demoralizing to the human\nrace. As a consistent anarchist he could not and would not allow them\nto be played in his house. Then Dr. O'Grady invented a game which could\nbe played with coins on a table. A penny was placed at a short distance\nfrom the edge of the table and driven along by the impact of another\npenny flipped at it from the edge. The object was to hit the first\npenny as frequently as possible before it was driven over the opposite\nedge of the table. Patsy displayed a great aptitude for the game. After\nan hour's play he was fairly expert. Before the end of the afternoon\nhe had won ninepence from Dr. O'Grady. He practised assiduously next\nmorning while the doctor read anarchist books. During the afternoon\nand evening he won, to his great delight, sums which amounted\naltogether to one-and-sevenpence. He did not whistle while he practised\nor played. In order to concentrate his whole energy on the game he was\nobliged to keep his mouth wide open.\n\nA game was in full swing on the third afternoon of Patsy's captivity\nwhen the key turned in the lock and the door of the room was flung open.\n\n\"Now what,\" said Dr. O'Grady, \"can the Emperor possibly want with us at\nthis hour of the day? It's not tea-time.\"\n\nMr. Dick, clad only in his grey flannel shirt, walked in.\n\n\"Hallo!\" said Dr. O'Grady, \"are you an anti-military anarchist getting\ninto training for the simple life? Or are you a new recruit undergoing\nthe ceremony of initiation into the brotherhood? Or is it nothing but\nthe heat of the day?\"\n\nMr. Red and the bearded anarchist dragged Mr. Sanders into the room and\ndeposited him, still bound, on the floor.\n\n\"Oh,\" said the doctor, \"another captive! Good. But don't overdo the\nthing, Emperor. You can't you know, go on storing up men in this way\nwithout attracting public attention. Patsy Devlin and I are all right,\nof course. We're not important people. Nobody misses us in the least.\nBut that man on the floor looks to me like a commercial traveller, and\nif he happens to belong to any English firm, a search will sooner or\nlater be made for him.\"\n\n\"We are Members of Parliament,\" said Mr. Dick.\n\n\"You don't look as if you were,\" said the doctor. \"Does he, Patsy?\"\n\n\"Be damn, but he does not! I've seen some queer fellows made members of\nParliament in my day. There was one time I was very near going in for\nit myself; but I never heard tell of e'er a one yet but owned a pair of\nbreeches--to start with, anyway.\"\n\n\"You hear what Patsy Devlin says,\" said the doctor. \"He quite agrees\nwith me that you don't look like a Member of Parliament.\"\n\nMr. Red, with his assistant anarchist, left the room and locked the\ndoor behind him. Dr. O'Grady shouted after him.\n\n\"Hullo! Emperor! I say, are you there? Wait a minute before you go\naway. Are you listening to me?\"\n\n\"I hear.\"\n\nMr. Red's voice, coming through the shut door, sounded more solemn than\nusual.\n\n\"I'm glad you do,\" said Dr. O'Grady. \"Don't go away now till I've\nfinished speaking. I want you to understand that we can't possibly have\nthese two fellows billeted here on us. It's all very fine for you to go\nabout picking up all sorts of people off the public roads and dumping\nthem down here; but Patsy Devlin and I don't like it. It's not fair. I\ntold you before that we don't in the least object to being imprisoned;\nbut we bar having an escaped lunatic with nothing on but a shirt, and a\nwretched commercial traveller shoved in here into a room which we have\ncome to regard as our private apartment.\"\n\n\"We are Members of Parliament,\" said Mr. Dick. \"We demand to be taken\nat once before the British Consul.\"\n\nHe appeared to think, not at all unnaturally, that he had strayed\nbeyond the bounds of His Majesty's dominions.\n\n\"Do shut up,\" said Dr. O'Grady, in a whisper. \"We don't believe you're\nMembers of Parliament. Nor does the Emperor. As a matter of fact,\nyou're not up to the level of the average county councillor in the way\nof respectability. Saying absurd things like that will simply enrage\nthe Emperor. He's frightfully touchy, and the moment he loses his\ntemper he shoots off his pistol. I say,\" he went on in a loud tone,\n\"are you there still? Hang it, I believe he's gone. That's your fault.\"\nHe addressed Mr. Dick. \"Why didn't you keep your mouth shut and let me\ntalk to him? Now the Emperor won't be back till tea-time, and we'll\nhave to put up with you till then. Perhaps, as you are here, you'll\nkindly explain what you mean by bursting in on us unannounced in this\nway. You might at least have knocked at the door.\"\n\n\"I was bathing----\" began Mr. Dick.\n\n\"Where?\"\n\n\"I don't know. Just outside the house. He came and took away my\nclothes.\"\n\n\"Serves you jolly well right,\" said the doctor. \"What made you come\nhere, of all places in the world, to bathe? Surely to goodness the\nAtlantic Ocean is big enough to bathe in without your picking out the\nexact spot in which the Anti-Military Anarchists are maturing their\nplans. Why did you do it?\"\n\n\"Anarchists?\" said Mr. Dick.\n\n\"Yes, Anti-Military Anarchists--the very worst sort there is.\"\n\n\"I thought he was a landlord.\"\n\n\"Well, he isn't. He's as nearly as possible the exact reverse.\"\n\n\"Are you an anarchist too?\"\n\n\"No, I'm not,\" said Dr. O'Grady, \"nor is Patsy Devlin. So far we\nhaven't been asked to join the organization. We're simply prisoners.\nWe've been captured by the brotherhood. But we had sense enough to wear\nour ordinary clothes. You may think it's the proper thing to go about\nin nothing but your shirt because you happen to be in the house of an\nanarchist, but I can tell you----\"\n\nMr. Sanders, who still lay on the floor, groaned dismally.\n\n\"Loose that fellow, Patsy,\" said the doctor. \"He appears to be in pain\nof some sort.\"\n\n\"Anarchists!\" said Mr. Dick. \"Good heavens! How frightful! My wife! My\npoor wife!\"\n\n\"She'll be all right,\" said the doctor. \"The Emperor won't do her any\nharm. He's a thorough gentleman in every way, a chivalrous gentleman,\nand I'm perfectly certain he wouldn't ill-treat a woman. You may\nrely on it that she'll be made quite comfortable. She was with you,\nI suppose. If so, I don't in the least wonder that you were run in.\nThe Emperor has frightfully strict, old-fashioned ideas about lots\nof things. He objects to cards, for instance, as demoralizing. I'm\nsure that mixed bathing would simply horrify him. There's no greater\nmistake than to think that just because a man's an anarchist, you can\ndo what you like without shocking him. You can't. The Emperor has his\nprejudices just like the rest of us.\"\n\nPatsy Devlin set Mr. Sanders on his feet and rubbed him down carefully.\nThe poor man seemed dazed and bewildered.\n\n\"Do you belong to the bathing party too?\" asked Dr. O'Grady. \"Or are\nyou a separate and distinct capture, unconnected with the gentleman in\nthe flannel shirt?\"\n\n\"I don't bathe,\" said Mr. Sanders, \"because I have a weak heart and it\ndoesn't agree with me.\"\n\n\"Then what were you doing? You must have been doing something which\nannoyed the Emperor. He may be a bit touched in the head. Most of us\nare, more or less; but he's not so mad as to saddle himself with the\nexpense of boarding and lodging a fellow like you unless you have been\ndoing something he dislikes. What were you at?\"\n\n\"I was mending a bicycle, Mrs. Dick's bicycle; at least----\"\n\n\"Who is Mrs. Dick?\" asked Dr. O'Grady. \"The lady who was bathing?\"\n\n\"She wasn't bathing,\" said Mr. Dick.\n\n\"All right. Don't get angry. I thought your name might be Dick and that\nthe lady might be your wife.\"\n\n\"My name is Dick, and she is my wife.\"\n\n\"I don't understand you in the least,\" said Dr. O'Grady. \"Have you got\ntwo wives? Has that anything to do with the way you're going about with\nnothing on you but your shirt?\"\n\n\"No, it hasn't; and I've not got two wives.\"\n\n\"You must have two wives,\" said Dr. O'Grady. \"You told me this instant\nthat your wife was bathing with you on the shore. And now you say Mrs.\nDick wasn't bathing. Those two statements can't possibly both be true\nabout the same woman. But I won't go into that yet. Later on you shall\nhave an opportunity of clearing yourself if you can. First of all, I\nwant to get to the bottom of this bicycle business, which seems to be\nless complicated.\"\n\nHe turned to Mr. Sanders. \"You say that you were engaged in mending\nMrs. Dick's bicycle when the Emperor came on you.\"\n\n\"Yes,\" said Mr. Sanders. \"At least I wanted to mend it.\"\n\n\"That's not exactly the same thing. I wish you'd try to be accurate.\nThere's no use my attempting to unravel this tangle and get at the\ntruth for you if you won't be careful what you say. Go on.\"\n\n\"I knocked for some time at the door, but nobody came to me. Then I----\"\n\n\"I expect,\" said the doctor, \"that the Emperor was in the Chamber of\nResearch at the time, concocting some new kind of dynamite. That's his\nfavourite occupation.\"\n\n\"Good God!\" said Mr. Sanders.\n\n\"It's all right. Don't be afraid. The mixtures he makes hardly ever\ngo off. And in any case he won't want to blow you up unless you are\na soldier. You're not in the army, are you? You don't look as if you\nwere.\"\n\n\"No.\"\n\n\"Or in the militia? The Emperor has a perfect horror of militiamen.\nHasn't he, Patsy?\"\n\n\"He damn nearly shot me,\" said Patsy, \"when I told him I'd been in the\nmilitia.\"\n\n\"Let's get back to the bicycle,\" said Dr. O'Grady. \"Mrs. Dick's\nbicycle, as I understand, which you wanted to mend. When you found\nthat nobody took any notice of your knocking, what did you do?\"\n\n\"I pushed the door open--it wasn't locked or bolted--and stepped in.\"\n\n\"That strikes me,\" said Dr. O'Grady, \"as pretty fair cheek on your\npart, considering that you're not an Anti-Military Anarchist. You might\nhave guessed that it would irritate the Emperor, especially as he was\nmaking dynamite at the time. What happened next?\"\n\n\"I looked round and saw no one. I rang the dinner gong, which was\nstanding in a corner of the hall. When that didn't attract anybody's\nattention I tried the door on the right, and found it locked.\"\n\n\"It's a pity you didn't try the one on the left. If you had, you'd have\nseen the yellow crocodiles, and they'd have frightened you out of the\nhouse.\"\n\n\"Then, just as I was turning to go away, two men sprang on me and bound\nme. After that I knew no more until----\"\n\n\"I wish,\" said Mr. Dick, interrupting his friend's story, \"that\nsomebody would lend me a pair of trousers.\"\n\n\"I haven't got any spare trousers,\" said Dr. O'Grady, \"and if I had,\nI'm not at all sure that I'd lend them to you. You say you're a Member\nof Parliament; but I've no proof of that. And even if you are it\ndoesn't seem to me to follow that you'd return the trousers.\"\n\n\"What am I to do?\" said Mr. Dick. \"I can't go about in this state all\nday.\"\n\n\"You won't go about much in any case,\" said the doctor. \"But for the\nimmediate present I think you'd better get into Patsy Devlin's bed.\nIt's no pleasure to us to see you standing about in your shirt. When\nyou're in bed I shall ask you a few questions, and if it turns out that\nthe Emperor really has got your clothes, I'll do my best to persuade\nhim to give them back to you when he comes up here at tea-time. I\nsuppose you don't mind his getting into your bed, Patsy, just for the\npresent?\"\n\n\"I do not,\" said Patsy, \"so long as he's out of it before I'm wanting\nit myself.\"\n\nMr. Dick crept in between the blankets.\n\n\"Now,\" said the doctor, \"we'll take up your story. You have, as I\nunderstand, two wives, one of whom bathes, and the other owns a\nbicycle.\"\n\nMr. Dick sat up and protested strongly. He appealed to Mr. Sanders to\nclear him of the charge of bigamy.\n\n\"All right,\" said Dr. O'Grady; \"I'll accept the statement that you've\nonly one. Was she, or was she not, bathing with you when the Emperor\ncame on you? Be careful how you answer.\"\n\n\"Certainly not. She's--I trust she's miles away, and safe.\"\n\n\"Then why did you express anxiety about the way the Emperor was likely\nto treat her?\"\n\n\"I didn't.\"\n\n\"You did. You kept saying, 'My poor wife.' What did you mean by that,\nif you didn't mean that the Emperor had captured her?\"\n\n\"I meant that she'd be anxious about me.\"\n\n\"She can't be as fond of you as all that,\" said the doctor. \"Nobody\ncould.\"\n\n\"She's very fond of me. We're only quite lately married.\"\n\n\"Have you a wife too?\" said the doctor to Mr. Sanders.\n\n\"Yes. I have a wife and an aunt. They're both with me in Ireland.\"\n\n\"Do you suppose that your wife is as fond of you as Mrs. Dick is of her\nhusband?\"\n\n\"I don't know. How could I know that?\"\n\n\"The reason I ask the question,\" said Dr. O'Grady, \"is this. There's\na girl I'm engaged to be married to, a Miss Adeline Maud Blow, who\nis probably scouring the country in pursuit of me this minute. Patsy\nDevlin has a wife who may be out looking for him.\"\n\n\"The minute she'd find out I was gone,\" said Patsy, \"she'd be up at the\nbarrack telling the police, the way she'd have me brought back to her.\"\n\n\"And now,\" said the doctor, \"it turns out that Mr. Dick has a wife, or\nperhaps two, of an unusually affectionate kind. And you have a wife and\nan aunt who have some regard for you. That makes five women altogether,\nall of them more or less energetic, and all of them bent on finding us.\nThe question is, how long will it be before they think of coming to\nRosivera?\"\n\n\"Not long,\" said Mr. Sanders, \"not long, I hope.\"\n\n\"For the sake of my poor wife,\" said Mr. Dick, who had covered himself\nwith the bed-clothes again, \"I trust it will not be long.\"\n\n\"Be damn,\" said Patsy Devlin, \"but to listen to the way you're talking,\na man would think nobody in the world but yourself ever had a wife. I\nhave one myself, as the doctor was saying this minute, and I wouldn't\nwonder but she might be a better one than yours. But you don't hear me\nlamenting over her every time I open my mouth.\"\n\n\"Tell me this,\" said the doctor: \"did you ever escape from your wife\nbefore?\"\n\n\"Escape from her!\"\n\n\"I mean, did you ever temporarily desert her, either through being\ntaken prisoner or otherwise?\"\n\n\"Never,\" said Mr. Dick. \"We've only been a year married, and we've\nnever been parted, even for a single day.\"\n\n\"That's all right,\" said the doctor. \"Then she won't have had any\npractice in looking for you. Patsy Devlin's wife, as you heard, is\nlikely to go straight to the police barrack when she misses him. But\nthen she's more or less accustomed to his not turning up regularly.\"\n\n\"I wouldn't say,\" said Patsy, \"that the police would be paying too much\nattention to what she might tell them.\"\n\n\"What about your wife, and your aunt?\" said the doctor to Mr. Sanders.\n\"Are they accustomed to having to hunt you up?\"\n\n\"Do I gather from what you say,\" said Mr. Sanders, \"that you don't want\nto be rescued?\"\n\n\"We certainly do not,\" said the doctor. \"Patsy and I are perfectly\ncomfortable where we are. We know when we are well off.\"\n\n\"Good God!\" said Mr. Sanders. \"In the hands of a bloodthirsty\nanarchist!\"\n\n\"Don't abuse the Emperor,\" said the doctor, \"for I won't stand it. He\nmay be an anarchist, but he's a thoroughly good sort.\"\n\n\"I'm thankful to say,\" said Mr. Sanders, \"that my aunt is a woman of\ngreat vigour and determination. She will do everything that can be done\nto discover where we are and to rescue us. I place implicit confidence\nin her. Nothing will daunt her.\"\n\n\"If, as I gather from your description of her, she is any kind of\nSuffragette,\" said the doctor, \"I hope, for her own sake, that she'll\nkeep clear of the Emperor. He has the strongest possible prejudice\nagainst advanced women. I happened one day to mention the name of Jael\nto him in the course of conversation. You know the woman I mean, the\none who hammered the nail into the man's head. I naturally thought he'd\nadmire her immensely, but he didn't in the least. He flew into the most\nfrightful rage. Didn't he, Patsy?\"\n\n\"I heard you saying so,\" said Patsy, \"and of course I believed you. But\nI wasn't here myself at the time. You know that, doctor.\"\n\nMr. Sanders brightened up suddenly.\n\n\"Are you a doctor?\" he said.\n\n\"I am, or I was, the dispensary doctor of Clonmore Poor Law Union. By\nthe way, Patsy, was there any talk of their electing a new man when\nthey found out that I'd gone?\"\n\n\"I wouldn't be telling you a lie. There was.\"\n\n\"If you're a doctor,\" said Mr. Sanders eagerly, \"you'll be able to\ncertify that I have a weak heart, and that confinement will seriously\ninjure my health. Then he'll be bound to let me go.\"\n\n\"I haven't a stethoscope with me,\" said the doctor, \"so I can't. But\neven if I wrote you out quite a long certificate I don't suppose that\nit would influence the Emperor in the least. He doesn't care about your\nhealth. Why should he?\"\n\n\"But I've always understood---- Why, the Home Secretary only the other\nday----\"\n\n\"And I may as well tell you straight,\" said the doctor, \"that if I had\na stethoscope, and if the Emperor would let you out on my certificate,\nI wouldn't give it. In fact, if I certified at all, I'd certify that\nyou are a particularly strong and enduring kind of man, and that a\nlittle imprisonment would do you good.\"\n\n\"But why----? Why should----?\"\n\n\"Because the first thing you'd do when you got out would be to bring\nthe police down on us here, and that's exactly what we don't want.\"\n\n\n\n\nCHAPTER XV\n\n\nMiss Blow stood at the window of her bedroom in Jimmy O'Loughlin's\nhotel, and saw Mr. Dick and Mr. Sanders ride by on their bicycles. She\ndid not for a moment suppose that they were the Members of Parliament\nwho had arrived by the train. They looked like hilarious but shabby\ntourists. It is not with such faces or in such clothes that candidates\nfor election woo their constituents in Merrie England. She afterwards\nsaw the three ladies drive by, followed by the car which carried their\nluggage. She did not see Mr. Goddard. She supposed, very naturally,\nthat he and the Members of Parliament were consulting together\nsomewhere, perhaps in the waiting-room at the railway station, devising\nplans for the rescue of what still might remain of Dr. O'Grady's body.\nShe waited patiently, sustained by the hope of heroic measures to be\ntaken in the near future.\n\nAt half-past one Bridgy knocked at the door, and told her that her\ndinner was ready for her.\n\n\"Will Mr. Goddard be in for dinner?\" asked Miss Blow.\n\nBridgy did not feel that it was her duty to disclose the fact that Mr.\nGoddard had driven away in the direction of Ballymoy. She answered\ncautiously.\n\n\"He might,\" she said.\n\nThis was unquestionably true. It was possible that Mr. Goddard would\nrepent as he went on his way, and come back to dine with Miss Blow. It\nwas not, in Bridgy's opinion, at all likely that he would; but she did\nnot say this to Miss Blow.\n\n\"Will the other gentlemen be with him?\"\n\n\"Well, I don't know, miss.\"\n\nShe did know, but she held charity to be a higher virtue than truth. If\nMiss Blow hoped for the company of other gentlemen, Bridgy would not\nsubject her to a disappointment sooner than was absolutely necessary.\n\nMiss Blow dined alone. After dinner she waited another half-hour. Then\nshe rang the bell for Bridgy.\n\n\"Has Mr. Goddard come back yet?\"\n\n\"Well, I don't know, miss.\"\n\n\"Are the other gentlemen in the hotel?\"\n\n\"Well, I don't know.\"\n\n\"Go and find out, then,\" said Miss Blow.\n\nBridgy went downstairs and summoned Jimmy O'Loughlin from behind the\nbar.\n\n\"The young lady above,\" she said, \"is asking me this two hours if Mr.\nGoddard is come back.\"\n\n\"You know as well as I do,\" said Jimmy, \"that he hasn't, and, what's\nmore, won't.\"\n\n\"Sure, I do know that. Didn't I see him making off out of the yard by\nthe back gate the way she wouldn't see him go?\"\n\n\"Well, then, go and tell her so.\"\n\n\"Is it me? I dursn't.\"\n\n\"Do it at once,\" said Jimmy. \"It was you she asked, and it's you that\nought to answer her. Off with you now, and not another word out of your\nhead.\"\n\nBridgy, who saw no reason why she should face the wrath of Miss Blow,\nretired to a small room which opened off the scullery, and spent an\nhour cleaning her own boots. Miss Blow waited, and after a while got\ntired of waiting. She rang the bell again. Bridgy heard it, and, for\ngreater security, went into the yard and concealed herself in the cow\nbyre. Miss Blow came downstairs and found Jimmy O'Loughlin. He had not\nthought of hiding, and was caught unawares.\n\n\"Is Mr. Goddard in the hotel?\" she asked.\n\n\"He is not,\" said Jimmy. \"It's himself was sorry to go; but there\ncame a messenger over from Ballymoy, a police sergeant, to say he was\nwanting, and wanting badly. It might be cattle driving that's in it, or\nit might be a meeting about the land or maybe an eviction. They're a\nqueer lot down in them parts. Anyway, he was wanting, and he went.\"\n\n\"It's a curious thing,\" said Miss Blow, \"that he went off without\nsaying a word to me.\"\n\n\"Well, now,\" said Jimmy, \"if I amn't the fool this day; but I thought\nBridgy would be sure to tell you.\"\n\n\"Tell me what?\"\n\n\"The last words he said to me as he was getting up into the trap was\nthese; 'Jimmy,' says he, 'it goes to my heart to be leaving Miss Blow\nbehind me, and her in trouble. She'll never forgive me.' 'Believe you\nme,' says I, striving to encourage him, for I could see he felt it,\n'she will.' 'She will not,' says he. 'Don't I know she won't?' 'She\nwill,' says I. 'There isn't a young lady ever I met with a better\nheart.' 'Will you tell her,' says he, 'that it was forced on me or I\nwouldn't do it, not if you was to give me the full of my hat of golden\nsovereigns?' 'I'll tell her,' says I; 'I'll break it to her quiet and\neasy the way she won't be feeling it.' And, glory be to God, I have.\"\n\n\"You might have done it a little sooner,\" said Miss Blow.\n\n\"I might, and that's a fact,\" said Jimmy apologetically.\n\n\"Where are the other gentlemen?\" said Miss Blow.\n\n\"The other gentlemen! Is it them ones that come in on the train? Did\nyou not see them going by on their bicycles? They're the most of the\nway to Pool-a-donagh by this time.\"\n\nIt occurred to Miss Blow that she had been deceived, tricked,\nabominably insulted by Mr. Goddard. She distinctly remembered the\nbicyclists. They had passed down the street almost immediately after\nthe arrival of the train. It was quite impossible for Mr. Goddard to\nhave had any serious conversation with them. She hesitated, on the\nverge of tears. Then the utter futility of weeping in the presence of\nJimmy O'Loughlin struck her. She pulled herself together and, with\nprompt decision, settled her course of action.\n\n\"Get the car,\" she said, \"at once. I shall drive over to Ballymoy and\nsee Mr. Goddard.\"\n\n\"The car's not in it,\" said Jimmy.\n\nMiss Blow stamped her foot.\n\n\"Get any car,\" she said. \"I don't care whether it's your own or\nanother.\"\n\n\"Divil the car there is in Clonmore this minute,\" said Jimmy. \"The\nladies that came in by the train has the both of them took, and they're\naway off with them to Pool-a-donagh, if so be that the horses can do\nthe journey.\"\n\n\"Do you mean to say that there isn't a horse and car to be got in the\nplace?\"\n\n\"There is not. It's the truth I'm telling you; and if you don't\nbelieve me, go up and ask Sergeant Farrelly at the barrack.\"\n\n\"Then,\" said Miss Blow, \"I'll walk.\"\n\n\"Is it to walk to Ballymoy? You'd never over it, and the weather as hot\nas it is. It's better than twelve miles.\"\n\n\"I'll do it,\" said Miss Blow.\n\nShe did. A little footsore, very tired, but still blazing with\nindignation, she arrived in Ballymoy at a quarter to six o'clock.\n\nJust outside Ballymoy, close to the Clonmore road, is the ground\nbelonging to the tennis club. There are three courts and a galvanized\niron hut, known as the pavilion. On club days, when some lady gives\ntea to the players, there are usually twenty or thirty people on\nthe ground. On other days only a few enthusiasts go to the club. It\nhappened that when Miss Blow came in sight on the road there was no\none at the club except Mr. Goddard and Captain Fielding, the Resident\nMagistrate. They were playing a vigorous game, having a final practice\nin preparation for the tournament which was to be held next day. Mr.\nGoddard was serving in the fifth game of the set when he caught sight\nof Miss Blow. He paused, stared at her, dropped the two balls which he\nheld in his hand, and fled without a word into the pavilion.\n\nCaptain Fielding gaped with astonishment, looked up and down the road,\nsaw nothing very alarming, and followed Mr. Goddard into the pavilion.\n\n\"What the devil----\" he began.\n\nMr. Goddard took no notice of the remark. He was staring out of a small\nwindow which commanded a view of the road.\n\n\"What the----\" said Captain Fielding again.\n\nMr. Goddard turned suddenly.\n\n\"Fielding,\" he said, \"jump on your bicycle and ride like hell to my\nhouse. Don't lose a moment. Get a hold of my housekeeper, and tell her\nthat if any lady calls she's to say I've gone away on leave and won't\nbe back for six weeks. Don't stop to ask questions. I'll explain it all\nto you when you get back.\"\n\nHe pushed Captain Fielding out of the pavilion.\n\n\"You'll be in time,\" he said, \"if only you'll go. Thank goodness my\nhouse is at the far end of the town.\"\n\nCaptain Fielding did as he was told. He rode fast, because he was most\nanxious to hear Mr. Goddard's explanation as soon as possible. He\ndelivered his message to the housekeeper, and then rode back.\n\n\"Did you pass a girl walking along the road?\" said Mr. Goddard.\n\n\"I did; a pretty girl in a grey dress, a stranger.\"\n\n\"Well, she's mad--stark mad. She's been bothering me out of my life\nthese two days. I'm sick and tired of talking to her. If she catches me\nnow I'm done.\"\n\n\"If she's mad,\" said the R. M., \"why don't you have her locked up? I'll\nsign the papers for you if you get a medical certificate.\"\n\n\"She's not mad in that sort of way,\" said Goddard, \"although she is\nmad.\"\n\n\"I see,\" said the R. M.; \"temper. It's not a breach of promise case, is\nit, Goddard?\"\n\n\"Nothing of the sort. I never set eyes on her in my life till\nyesterday. In fact, she's engaged to quite another man.\"\n\n\"I see,\" said the R. M. doubtfully.\n\n\"The fact is,\" said Goddard, \"she has a theory that there has been a\nmurder over at Clonmore. There hasn't, of course; but she thinks there\nhas--and--well, the fact is, I promised to investigate it. She cried,\nyou know, and that sort of thing. The whole business is utterly absurd\nfrom start to finish.\"\n\n\"And what do you propose to do now?\"\n\n\"I don't know. Perhaps she'll go away when she gets the message about\nmy being on leave. I shall stay here till after dark, anyhow, and then\nsneak down to the house and find out what's happened.\"\n\n\"I think,\" said Captain Fielding, \"that I'll stay with you. I'd rather\nlike to see this business through. Besides, when she finds she can't\nget you she'll probably go up to my house for me.\"\n\nMiss Blow received the message from Mr. Goddard's housekeeper with open\nincredulity.\n\n\"I shall wait here,\" she said, \"until Mr. Goddard comes home.\"\n\nThe servant was in a difficult position. She recognized Miss Blow as\nthe lady to whom she had given tea the day before, who had driven\noff, apparently on the most friendly terms, with Mr. Goddard. It was\nplainly impossible to slam the door as doors are slammed in the faces\nof drunken tramps. She was a woman of kindly disposition. She saw\nthat Miss Blow was dusty, tired, in need of rest and refreshment. Her\nnatural impulse was to be hospitable. At the same time she had no doubt\nas to what Mr. Goddard's wishes were with regard to this particular\nvisitor. She was a woman of intelligence, and she realized that the\nmessage delivered by Captain Fielding was urgent and important. She\nhesitated, standing in the middle of the doorway. Miss Blow put an end\nto all uncertainty by walking past her into the house, opening the\ndining-room door, entering the room, and sitting down.\n\nMr. Goddard's housekeeper retired to the kitchen and meditated on the\nsituation. It was plainly impossible to remove Miss Blow without the\nuse of actual force. At seven o'clock pity triumphed over her sense of\nduty to her master. She made some tea and took it in to Miss Blow on a\ntray. At eight o'clock a police constable came to the door and asked\nto see Mr. Goddard. He refused to believe that the District Inspector\nhad gone on leave. The housekeeper, despairing of anything except the\nactual truth, confessed that there was a young lady in the house, and\ngave it as her opinion that Mr. Goddard was afraid to meet her. The\nconstable disappeared, grinning. The housekeeper was uneasily conscious\nthat he was putting a wrong meaning into the fact of Miss Blow's\npresence in the house.\n\nAt nine o'clock Mr. Goddard, still in tennis flannels, and accompanied\nby Captain Fielding, climbed over the wall of his own back garden, and\nslipped through the yard into the kitchen.\n\n\"Mary,\" he said to the housekeeper, \"is that young lady gone?\"\n\n\"She is not; and, what's more, it's my opinion that she won't go.\"\n\n\"There,\" said Mr. Goddard to Captain Fielding, \"what did I tell you?\nShe's an amazingly persistent woman.\"\n\n\"Go and turn her out,\" said Captain Fielding. \"Tell her it's highly\nimproper for her to be sitting here in your house in the middle of the\nnight.\"\n\nMr. Goddard sighed. The advice was well meant, but it was useless. He\nknew Miss Blow, and Captain Fielding did not.\n\n\"Mary,\" he said, \"slip upstairs as quietly as you can, and get my\npyjamas and tooth brush. I'll have to go to the barrack for the night.\nThey'll give me some sort of a shake-down.\"\n\n\"And will the young lady be sleeping here?\" asked Mary.\n\n\"I expect so,\" said Mr. Goddard. \"If she does, she'll tell you so\nbeforehand quite plainly. Make her as comfortable as you can.\"\n\nMary went upstairs on tiptoe. Unfortunately, owing to nervousness\ninduced by excess of caution, she upset a hot-water can, which fell\nwith a hideous clatter against Mr. Goddard's bath. She hastened from\nthe bedroom, and was met on the stairs by Miss Blow.\n\n\"What have you got there?\" said Miss Blow, eyeing the pyjamas\nsuspiciously.\n\n\"It's clothes from the wash,\" said Mary. \"Now that the master's away, I\nthought I might as well be doing that as nothing.\"\n\n\"You can tell him,\" said Miss Blow, \"that I'm going to the hotel now,\nand that I'll be round to see him the first thing in the morning.\"\n\nShe left the house, slamming the door behind her. Mary carried the\npyjamas and the tooth brush upstairs again. Then she went into the\nkitchen and broke the happy news to her master and Captain Fielding.\n\n\"Thank God!\" said Mr. Goddard. \"I'm safe for to-night, anyway.\"\n\n\"She'll be back with you in the morning,\" said Captain Fielding,\ngrinning. \"I don't see that you've much to congratulate yourself\nabout.\"\n\n\"Something may happen before then,\" said Mr. Goddard.\n\nIt is very seldom that things happen just when they are wanted to. The\nprisoner on the verge of execution hopes, but as a rule hopes in vain,\nfor an earthquake at early dawn. The debtor whose bill is due the next\nday prays for, but scarcely expects, a fire in the bank premises during\nthe night, an effective fire destructive of iron safes. Mr. Goddard's\nfeeling that something might happen before Miss Blow caught him again\nwas hardly a hope. Nothing short of some wholly unprecedented event\nwould be any use to him. But Mr. Goddard was exceptionally fortunate.\nSomething did happen. He was roused at six o'clock in the morning by\na violent knocking at his door. He looked out of his window and saw\nConstable Moriarty standing in the street with a bicycle.\n\n\"What on earth do you want at this hour?\" said Mr. Goddard.\n\nHe had slept badly during the early part of the night, and was greatly\nannoyed at being roused from a doze at six o'clock.\n\n\"If it's Miss Blow you're after,\" he went on, \"I haven't got her here.\nShe's at the hotel. Go there and get her. Take her with you, if you\nwant her. She's no use to me.\"\n\nConstable Moriarty grinned. He did not want Miss Blow any more than his\nofficer did. Then, in the very middle of his grin, he grew grave again.\n\n\"It's what Sergeant Farrelly is after sending me over to tell you, sir,\nthat the two English gentlemen that went off to Pool-a-donagh on their\nbicycles is after getting lost.\"\n\n\"Lost! What do you mean by lost?\"\n\n\"According to the word that came to the barrack at Clonmore, sir, the\nladies arrived at Pool-a-donagh at four o'clock or thereabouts and\ndidn't find the gentlemen. It was thought that maybe they'd taken a\nwrong turn, though it's hard to know how they could, seeing there isn't\na wrong turn to take, and that they might have come back to Clonmore.\nSo the sergeant at Pool-a-donagh sent a wire to Sergeant Farrelly; but\nhe didn't know where they were, and no more did Jimmy O'Loughlin, for\nwe asked him. We sent a few more wires to the barracks round about, but\nwe got no tidings of them good nor bad. There was men out searching\nall night from Pool-a-donagh, and Constable Cole and the sergeant took\na look round when they were on patrol. They do say the ladies was in\na terrible state. There was a mounted man came into Clonmore between\nthree and four this morning, and Sergeant Farrelly sent me over to you\non a bicycle the way you'd be able to tell him what he'd better do.\"\n\n\"I never heard of such a thing in my life,\" said Mr. Goddard. \"How\ncan they be lost? The thing is impossible. Hang it all! they must be\nsomewhere.\"\n\n\"So you'd say, sir. But there isn't a police station anywhere round but\nit's been wired to, and not one has seen the gentlemen, dead or alive.\"\n\n\"Don't talk about their being dead,\" said Mr. Goddard. \"Good heavens,\nman! They're Members of Parliament. If they've gone and committed\nsuicide there'll be a most frightful row, and everybody will say it's\nthe fault of the police. They must be found at once.\"\n\nMr. Goddard was in his pyjamas and dressing-gown. He told Constable\nMoriarty to go into the yard and harness the horse while he shaved\nand dressed. He was inclined at first to be angry with the Members of\nParliament. It is a stupid thing to get lost, and men have no right to\nbe stupid. He did not want to spend the day searching bohireens and\nbog-holes. He wanted to play in the Ballymoy tennis tournament. He had\nengaged himself as partner to Mrs. Fielding, the Resident Magistrate's\nwife, and it vexed him to have to disappoint the lady. By degrees the\nmatter began to present itself to him in a brighter light. He was now\nobliged, absolutely forced, to leave Ballymoy, and by leaving early\nhe would escape Miss Blow, escape her for a time at all events. He\nreflected that if she sought him out with her usual relentlessness he\nwould, in any case, have been unable to play in the tournament. A man\ncannot with any decency appear on a tennis court while a beautiful\nand angry girl hurls reproaches at him from the side lines. Even if\nhe succeeded in evading her, his nerve would be shattered by the\nconsciousness that she might appear beside him at any moment. It seemed\nto him that the loss of the Members of Parliament was, after all, a\nblessing in disguise; that, in fact, his wish had been fulfilled beyond\nhis hope--something had happened.\n\nHe sat down and scribbled a note of apology for his absence.\n\n\"Dear Fielding,\" he wrote, \"will you ask your wife to be so kind as to\nexcuse me. I am exceedingly sorry to disappoint her, and I assure you I\nwould not do it if I could possibly help it. Two Members of Parliament\nhave disappeared during the night, and I have to go out and look for\nthem. Perhaps the rector would play with Mrs. Fielding if you asked\nhim. He's no earthly good, but he's on the handicapping committee, and\nwould let himself in soft. In any case she would get a game of some\nsort. If you happen to come across that good-looking girl in the grey\ndress, you might tell her that I promised to play to-day with Mrs.\nFielding, and that you expect me up at the club between twelve o'clock\nand four. That will keep her in Ballymoy for most of the day, anyhow.\nThe Lord knows I don't want her on my track. I shall have worry enough\nwithout that.\"\n\nHe stuffed some biscuits into his pocket and went out into the yard.\nConstable Moriarty promised to deliver the note to Captain Fielding.\nMr. Goddard got into his trap and drove off. He had to pass the hotel\non his way, and was half afraid that Miss Blow might sally out and\nseize the horse's head. She did not. It was not yet seven o'clock, and\nthe blinds of all the upstairs rooms were drawn down. Miss Blow was,\napparently, sleeping off the fatigue of the day before.\n\nMr. Goddard reached Clonmore at half-past eight. He found Sergeant\nFarrelly and Constable Cole at breakfast. They had been up all night,\nand were looking fagged, nervous, and harassed.\n\n\"There's two more wires, sir, that's come in since eight this morning.\nThe one of them says that there's no further news of the missing\ngentlemen in Pool-a-donagh, and the other says that the ladies started\nback to Clonmore at seven this morning, and is in hopes that you'll be\nhere to meet them when they arrive.\n\n\"If I'm not,\" said Mr. Goddard--\"and it's quite possible that I\nwon't--if I'm not, you'll have to see them, sergeant. They're very nice\nladies,\" he added, noticing a gloom on Sergeant Farrelly's face, \"not\nthe least like Miss Blow.\"\n\n\"It's a queer thing,\" said Sergeant Farrelly, \"that a man could get\nlost between this and Pool-a-donagh. It's a straight road, and they got\nstarted right on it. Jimmy O'Loughlin was telling me last night that he\nseen them going through the town, and them heading straight on the same\nas if they knew the place all their lives. Without they took a boat and\nwent out to sea in it--and I don't know where they'd get a boat along\nthat road--it's a queer thing.\"\n\n\"It is a queer thing,\" said Mr. Goddard. \"But what's the use of\nspending the morning saying so? Show me all the telegrams that came\nlast night.\"\n\nHe worked his way through a sheaf of pink forms. The messages were,\nfor the most part, very monotonous. In language which hardly varied\nand with a strict attention to the number of words which can be sent\nfor sixpence, the sergeants of all the police barracks within a wide\ncircle disclaimed any knowledge of the missing gentlemen. No one had\nseen them. The most careful inquiries brought no information. Here and\nthere, among these cold, official statements, Mr. Goddard came upon a\nmessage which breathed anxiety, heartbreak, and despair, a despatch\nfrom a wife or an aunt. These were verbose, reckless of expense, and\nalways ended by urging the necessity of immediate action on the part of\nthe police.\n\n\"I can't do anything,\" said Mr. Goddard; \"at all events, I won't do\nanything till I've had some breakfast. It's after nine now. I'll go\ndown to the hotel and see what I can get out of Jimmy O'Loughlin.\"\n\nHe found, when he reached the hotel, that the table in the commercial\nroom was laid for breakfast. A pleasant scent of frying bacon reached\nhim from the kitchen. He became aware that he was extremely hungry.\n\n\"Jimmy!\" he called. \"Jimmy O'Loughlin!\"\n\n\"Is that yourself, Mr. Goddard?\" said Jimmy, speaking from the top of\nthe stairs. \"They're just getting the breakfast ready for the doctor's\nyoung lady. I'll tell them to fry a couple more eggs for you. She'll be\nglad of your company. It's lonely for the creature taking her meals by\nherself, and her in trouble about the loss of the doctor and all.\"\n\nJimmy O'Loughlin in his shirt-sleeves and without a collar leaned over\nthe banisters as he spoke. The grin on his face was malicious.\n\n\"Is Miss Blow here?\" said Mr. Goddard. \"I don't believe she is. How\ncould she?\"\n\n\"She is not here,\" said Jimmy, \"but I'm just after getting a wire from\nher to say that she'll be in for her breakfast at half-past nine. It\nmust be that now, if it isn't past it. Bridgy! Are you there, Bridgy?\nI say, will you fry a couple of eggs and some rashers for Mr. Goddard?\nHe'll be taking his breakfast along with the doctor's young lady as\nsoon as she comes.\"\n\n\"Look here,\" said Mr. Goddard, \"there's no use your trying to pull my\nleg in this way. Miss Blow can't be here. She was sound asleep when I\nleft Ballymoy at seven, and she couldn't possibly do the drive in the\ntime.\"\n\n\"Not if she drove,\" said Jimmy; \"but she'd do it on a bicycle.\"\n\n\"How do you know she has a bicycle?\"\n\n\"I don't know. How would I? All I say is that, being the sort of a\nyoung lady she is, it's likely she'd get a bicycle, or for the matter\nof that a motor car, if so be she wanted one. Anyway, she'll be here\nat half-past nine, for that's what she said.\"\n\n\"Then,\" said Mr. Goddard, \"I'll not stay for breakfast. I'm going up\nto the Castle. I have business with Lord Manton. And look here, Jimmy,\nwhen Miss Blow arrives, if she does arrive, there'll be no need for you\nto say anything about my being here. It would only upset her.\"\n\n\"I will not. Why would I? Hasn't she trouble enough without that? But\ntell me now, Mr. Goddard, is it true what I hear them saying about them\ngentlemen that was through the town yesterday on their bicycles?\"\n\n\"Is what true?\"\n\n\"That they're off.\"\n\n\"They're lost,\" said Mr. Goddard, \"if that's what you mean.\"\n\nJimmy O'Loughlin winked. \"The same as the poor doctor,\" he said, \"and\nPatsy Devlin.\"\n\n\"Not in the least the same. We know what happened to them well enough.\nBut these fellows are quite different. But I can't stop here talking\nto you all morning. I've something else to do. Don't say anything to\nanybody about my being up at the Castle.\"\n\n\"I will not. If I open my mouth about you at all, I'll say you're\nsearching high and low for the men that's gone.\"\n\n\n\n\nCHAPTER XVI\n\n\n\"His lordship is still at breakfast, sir,\" said Wilkins.\n\n\"Will you ask him if I can speak to him for a few minutes as soon as he\nhas finished?\" said Mr. Goddard.\n\nHe was shown into the library, where Lord Manton's letters and\nnewspapers were arranged on a table near the window. He was an old\ngentleman who declined to do business of any kind, even open a letter,\nbefore breakfast. He attributed his good health to his habit of facing\nthe kind of worries which the post brings only when he was fortified by\na solid meal. Mr. Goddard glanced at the columns of a Dublin evening\npaper of the day before, half fearful of a scare headline announcing\nthe loss of two Members of Parliament in Connemara. He was delighted to\nfind that the editor had been able to discover nothing more exciting\nthan a crisis in the Balkans and a speech by the Prime Minister.\nWilkins entered the room.\n\n\"His lordship's compliments,\" he said, \"and he will be pleased if you\nwill join him at breakfast.\"\n\nMr. Goddard accepted the invitation gladly. The smell of Jimmy\nO'Loughlin's bacon and eggs had whetted his appetite. He was conducted\nto the dining-room by Wilkins, and found Lord Manton seated by himself\nat the end of a large table. The dining-room in Clonmore Castle is\na spacious and well-proportioned apartment. A tradition exists that\nin the time of Lord Manton's father, on the occasion of a county\nelection, fifty gentlemen once sat down in it together at dinner. In\nthose days, no doubt, people did not object to being crowded. But even\nto-day thirty people could very comfortably have dined in the room,\nand there would still have been space for the servants to make their\nway about. It was the largest room in the Castle, and its present\nproprietor looked singularly small, even insignificant, seated alone in\nthe middle of it. Lord Manton greeted his guest cheerfully.\n\n\"Good morning, Mr. Goddard. This is an early visit. Have you by any\nchance come across the body of Dr. O'Grady? Won't you help yourself to\nsomething to eat? You'll find whatever there is on the side table.\"\n\nMr. Goddard secured a kidney and some bacon.\n\n\"No,\" he said; \"there's no news of Dr. O'Grady.\"\n\n\"Have you got engaged to be married to Miss Blow?\" said Lord Manton.\n\"That's the way this business will end, I expect. That young woman came\nover here for a husband, and now that the doctor has bolted there is\nonly you or me for her to choose. So far as I can see at present, she\nseems to prefer you. I'm very glad she does. In fact, I'm keeping out\nof her way as much as I can.\"\n\n\"So am I; but it's rather difficult.\"\n\n\"I suppose it is. I hear that she went the whole way over to Ballymoy\nafter you last night; walked every step of it. I'm surprised that you\nescaped her after that. How did you manage it? I should have thought\nyou could hardly have refused to marry her without being actually rude.\nBut perhaps that story is only gossip.\"\n\n\"It's quite true,\" said Mr. Goddard. \"But, Lord Manton, I came up here\nthis morning to speak to you about--the fact is, I am in a serious\ndifficulty.\"\n\n\"You are. I can scarcely imagine anything more serious.\"\n\n\"I wanted,\" said Mr. Goddard, \"to ask your advice about----\"\n\n\"You shall have it. Give in and marry her at once with a good grace.\nThat's my advice. After all, you might do a great deal worse. She's a\nvery good-looking girl, and by her own account she'll have some money.\nIt's far better for you to pretend you like it. It's no earthly use\nyour trying to escape if her mind is made up. Your plan of dodging\nup and down between Clonmore and Ballymoy can't be kept up for ever.\nSooner or later she'll overtake you in one place or the other; or else\nshe'll meet you on the road between the two, where you'll be quite at\nher mercy.\"\n\n\"I dare say you're right,\" said Mr. Goddard. \"But it's not about Miss\nBlow that I want to consult you this morning. Did you hear about the\nMembers of Parliament?\"\n\n\"I heard that two Members of Parliament went through the town on\nbicycles yesterday,\" said Lord Manton, \"with a lot of women after\nthem on cars; Suffragettes, I suppose, pursuing them for votes. It's\nastonishing how they track those poor fellows to the remotest ends of\nthe earth.\"\n\n\"They've disappeared,\" said Mr. Goddard.\n\n\"Sensible men. That was by far the wisest thing they could do. But I\nwonder how they managed it? You haven't been able to disappear from\nMiss Blow. We must find out about it. I may have to disappear myself\nwhen that question comes up to the House of Lords.\"\n\n\"You've not got it right,\" said Mr. Goddard. \"The women on the cars\nwere their wives. Or rather two of them were. The other was their aunt.\"\n\n\"Dear me! Is that legal? That sort of group marriage with a common aunt\ndoesn't seem to me quite the thing for Members of Parliament.\"\n\n\"They had one wife each, of course,\" said Mr. Goddard, \"and the aunt\nonly belonged to one of them. They have the police roused all over the\ncountry looking for them. That's what brings me here this morning. I've\ngot to do something to find them.\"\n\n\"Do you mean to tell me that these two men have got lost in such a way\nthat they can't be found?\"\n\n\"They have. It's really a most extraordinary thing. They started\non bicycles yesterday from Clonmore to ride to Pool-a-donagh. They\nnever arrived. That's really all we know about the matter. The ladies\nfollowed them on a car and didn't overtake them. We've telegraphed to\nevery police barrack in the neighbourhood, and they haven't turned up\nanywhere.\"\n\n\"I'll tell you what it is, Goddard. This is an uncommonly awkward\nbusiness for you. Here you are, responsible for the safety of the\ninhabitants of this district, and no less than four men disappear\ncompletely in the inside of a single week. First there was Dr. O'Grady.\nThen poor Patsy Devlin vanished, leaving a wife and family behind him.\"\n\n\"We know all about them,\" said Mr. Goddard.\n\n\"And now no less than two Members of Parliament. It doesn't look at\nall well. I'm greatly afraid there'll be a fuss when it gets into the\nnewspapers. The public will take the keenest interest in it. We shall\nhave columns and columns every day. Do you happen to know whether\nthese men were Liberals or Conservatives?\"\n\n\"No, I don't. I don't see that it makes any difference. I've got to\nfind them whichever they are.\"\n\n\"It makes the greatest difference,\" said Lord Manton. \"If they're\nsupporters of the Government, there'll be a much bigger fuss than if\nthey're on the Opposition side. The Prime Minister isn't likely to\nsit down quietly under the loss of two votes, especially with the way\nbye-elections are going at present. You'll simply have to find them.\"\n\n\"I wish I could,\" said Mr. Goddard. \"I'd do it quick enough, whatever\ntheir politics are, if I knew how to go about it.\"\n\n\"Of course, if they belonged to the Opposition,\" said Lord Manton, \"the\nmatter won't be pressed by the Government. But even so you'll have the\nnewspapers to reckon with, and the curiosity of the general public. Why\ndon't you go and look for them?\"\n\n\"I would go at once, if I had the remotest idea where to look. But I\nhaven't. That's what I want your advice about.\"\n\n\"You say,\" said Lord Manton, \"that there are three ladies belonging\nto them. If I were you, I should begin by introducing them to Miss\nBlow. Then get a hold of Mrs. Patsy Devlin and let her talk to the\nwhole party. You could put them all together into a room in Jimmy\nO'Loughlin's hotel. They'll interest each other. They have so much in\ncommon.\"\n\n\"I might do that, of course. But I don't see what good it would be. I\nshouldn't be any nearer finding the men.\"\n\n\"No, you wouldn't; but you'd have time to turn round and think things\nout a bit. They'd be sure to talk for a good while once you get them\nstarted, and it's quite possible that by the time they'd finished, the\ntwo men might have turned up somewhere or other.\"\n\n\"I don't see what can possibly have happened to them,\" said Mr.\nGoddard. \"After you pass Rosivera the road runs the whole way between\nthe bog and the sea. They couldn't get off it if they tried. And they\nmust have got as far as Rosivera when they started right, for there\nisn't a cross road between this and there. Even if they were fools\nenough to go up some bohireen or other, they'd only have to turn around\nand come down again.\"\n\n\"That seems to me,\" said Lord Manton, \"to knock the bottom out of the\ntheory that they've got lost. As you say, they can't be lost. They\nmight sit down on the side of the road and cry; but if they did the\nwomen on the cars would have seen them.\"\n\n\"I've thought all that out. I can't see how they've got lost; but the\nfact is they are lost.\"\n\n\"I don't believe it,\" said Lord Manton.\n\n\"Surely you don't suppose that they're playing off a practical joke on\nus.\"\n\n\"Come into the library and have a smoke,\" said Lord Manton.\n\n\"I'm not sure that I ought to. With this business on my hands I\nscarcely feel justified--I think I should----\"\n\n\"Oh, nonsense. You can't do anything until you've thought. Premature\nand imperfectly considered action in a case like this is always a\nmistake, and you can't possibly think properly without tobacco.\"\n\nLord Manton kept excellent cigars for his guests. Mr. Goddard, who had\nsmoked them before, lit one with pleasure, and stretched himself in a\ndeep chair. He had breakfasted heartily, in spite of anxiety and worry.\nHe felt that a short rest was due to him. He was prepared to believe\nthat a period of calm reflection was due to the case of the two errant\nMembers of Parliament. Lord Manton did not smoke his own cigars, and\nrarely lay back in his own very comfortable chairs. He preferred an\nupright kind of seat, and he smoked cigarettes, lighting them one after\nanother in rapid succession, while he talked, wrote or thought. On this\noccasion he did not remain seated for more than a couple of minutes.\nBefore his first cigarette was half smoked he stood up and talked down\nto Mr. Goddard from a commanding position beside the chimney-piece.\n\n\"I don't believe,\" he said, \"I don't for one moment believe, that those\nMembers of Parliament are lost.\"\n\n\"Lost or disappeared or stolen,\" said Mr. Goddard, \"it's all the same\nso far as I am concerned. I've got to find them.\"\n\n\"It's not in the least the same thing. If they were simply lost in\nthe way a child or a dog or a collar stud gets lost, then you'd know\nwhere to look for them. Draw a circle of, say, ten miles in diameter,\nwith Rosivera for its centre, and they must be inside of it somewhere.\nYou've only got to put men enough on the job and you're bound to find\nthem. But if they have disappeared of their own accord, vanished\nintentionally and deliberately, then you will have to proceed in quite\na different way.\"\n\n\"It's all very well to talk of Dr. O'Grady and Patsy Devlin\ndisappearing like that; but it's different with these men. Why the\ndevil should they disappear?\"\n\n\"They can't have got lost,\" said Lord Manton. \"You satisfied me, if\nyou did not satisfy yourself, about that; and if they're lost they must\nhave bolted. How else can you account for their not turning up?\"\n\n\"But why should they? What had they to run away from?\"\n\n\"That, of course, I can't say for certain, for I don't know the men.\nBut, taking into consideration the little I do know about them, I can\nmake a guess. Can't you imagine it? They had with them three women, two\nwives and an aunt. The one who hadn't his aunt with him very probably\nhas one at home, and it's quite possible that they both have sisters.\"\n\n\"I dare say they have. I didn't ask.\"\n\n\"Well now, think. They were English Members of Parliament, and\ntherefore presumably pretty well to do. They probably had businesses\nsomewhere, and residences in a suburban district. What does all that\nmean? Respectability. Imagine to yourself the appalling weight of\nrespectability involved in the possession of a wife, an aunt, a sister,\na business, investments, and a seat in Parliament. Can't you fancy\nthe poor fellows coming to find it all perfectly intolerable; saying\nto each other: 'Hang it, let's get off and run loose for a while'? I\ndon't say I'm right. I can't be sure, because I don't know the men, and\nI didn't see the women; but what I'm offering you is an intelligible\nexplanation. What sort of women were they? What were they like?\"\n\n\"I don't know. I didn't notice anything particular about them. They\nwere just ordinary women.\"\n\n\"There you are,\" said Lord Manton. \"It's your ordinary, well-behaved\nwoman who drives a man perfectly frantic if he has any spirit in him.\nI couldn't live with an ordinary woman for three months at a stretch.\nLook at my daughter, for instance. How long do you suppose I could\nstand her? Certainly not three months, not two months. I'm at the end\nof my tether after a six weeks' visit to her. And I'm not married to\nher. I haven't got to be very close to her. I haven't got to see her\nin a red dressing-gown combing her hair. I haven't got to listen to\nthe noise she makes when she's washing her teeth. You don't understand\nthese things because you're not married. Very likely you never will\nunderstand them, because Miss Blow's not an ordinary woman----\"\n\n\"She is not,\" said Mr. Goddard.\n\n\"But these two poor fellows,\" said Lord Manton, \"had got to do with\nordinary women. They had to give their opinions on new hats, had to\nlisten to stories about the things which the servants did or didn't do.\nOf course they bolted.\"\n\n\"I wish to goodness they'd have bolted out of some other district,\nthen, if they had to bolt. Why should they come here? You'd think it\nwould be much more convenient to run away from a place like London than\nfrom Clonmore. But I don't believe they have bolted.\"\n\n\"They can't have got lost. The thing's absurd on the face of it. Take\na single point in addition to all those that you've made already.\nTake their bicycles. A man might accidentally lose himself, but his\nbicycle couldn't possibly get lost. He must leave it on the side of\nthe road somewhere, and wherever he leaves it you're sure to find it.\nYou can get rid of a bicycle on purpose, though even that's not so\neasy. Sinking it in the sea is about the only way I can imagine of\neffectually disposing of it. But you can't lose it and yourself. Nobody\ncould lose both himself and a bicycle unless he did it on purpose.\"\n\n\"But----\" said Mr. Goddard.\n\nHe had a whole series of criticisms utterly destructive of Lord\nManton's hypothesis. He had seen the Members of Parliament. He felt\nthat he knew them. He found it impossible to imagine their breaking out\ninto wild Bohemianism, sickening of respectability, rebelling against\nthe monotony of any woman, however ordinary. There was Mr. Dick, who\nhad bounded out of the railway carriage, volatile, debonnair, gay\nwith familiar quotations. There was Mr. Sanders, rigid, mathematical,\na little morose on account of his weak heart. Lord Manton might\nconceivably find an ordinary woman maddening; but Mr. Dick, once he\nbegan, would go on kissing her contentedly for years and years. He\nwould appreciate the way in which his dinner was cooked for him. Mr.\nSanders would appreciate that too. He might be a little exacting,\nwould, no doubt, expect his weak heart to be taken seriously, would\nput a woman in her proper place in the settled order of things, and\nkeep her there. But both Mr. Dick and Mr. Sanders would recognize that\na woman must be allowed to wash her teeth. They would not resent the\nnoise of the water gurgling in her mouth. They would both know that\nwomen must comb their hair, and that a red dressing-gown is a suitable\nand convenient garment under certain circumstances. They would not be\nangry when they were told about the misdeeds of maid-servants, a trying\nand exhausting class to deal with. They were men with well-balanced\nminds, men of practical common sense, men who would not fly blindly in\nthe face of facts. They were not the men to run away from their wives,\nor even from their aunts.\n\n\"But----\" he began again.\n\nLord Manton lit a fresh cigarette, his fourth, and waited. Mr. Goddard\nsought for words in which to express his feelings. He might have found\nthem in time, but time was not given him. Wilkins entered the room.\n\n\"There is a lady to see you, sir,\" he said to Mr. Goddard. \"I showed\nher into the small drawing-room. She said she wished to see you\nparticular.\"\n\n\"Good God!\" said Mr. Goddard. \"It's Miss Blow!\"\n\n\"Is it Miss Blow?\" said Lord Manton to Wilkins. \"Don't attempt to break\nthe news to him if it is. Tell him straight out. It's kinder in the\nend.\"\n\n\"It's the same young lady,\" said Wilkins, \"that called on your lordship\ntwo days ago.\"\n\n\"The young lady that was talking about bringing a corpse here? I told\nyou, I remember, to get white flowers from the gardener. Is it that\nyoung lady?\"\n\n\"Yes, my lord.\"\n\n\"And has she brought it this time?\"\n\n\"Not that I saw, my lord.\"\n\n\"If you didn't see it, Wilkins, I think we may take it for granted that\nit isn't there.\"\n\n\"Damn that girl!\" said Mr. Goddard. \"You'd think I had enough on my\nhands without her.\"\n\n\"From the way you're speaking of her,\" said Lord Manton, \"I suppose\nyou don't feel inclined to go into the drawing-room and offer to marry\nher. That, as I said before, is the proper thing for you to do. If you\nwon't, the only other course I see open to you is to get away out of\nthis as fast as you can.\"\n\n\"What's the good? She'd be after me again at once.\"\n\n\"Not at once,\" said Lord Manton. \"I think I can arrange to give you a\ngood long start. Wilkins will go back and ask her to be good enough\nto remain where she is for a few minutes. You could tell her, Wilkins,\nthat Mr. Goddard is engaged with me at present, discussing very\nimportant business, but that he'll be with her in less than a quarter\nof an hour. You could do that, Wilkins, couldn't you?\"\n\n\"Yes, my lord.\"\n\n\"Then Mr. Goddard could get out of the house and walk across the deer\npark. You could run if you like, Goddard, keeping carefully out of\nsight of the drawing-room windows. After a time, say at the end of half\nan hour, Miss Blow will begin to get impatient. What do you suppose\nshe'll do when she gets impatient, Wilkins?\"\n\n\"I'd say, my lord, that's she's likely to ring the bell.\"\n\n\"Exactly. That's what I thought myself. I suppose, Wilkins, that you\ncould arrange for the under housemaid to go to her when she rings.\"\n\n\"Yes, my lord.\"\n\n\"And the under housemaid would, of course, know nothing about Mr.\nGoddard. What would the under housemaid do, Wilkins, when she was asked\nabout Mr. Goddard, and knew nothing about him?\"\n\n\"I'd say, my lord, that she'd tell the young lady she'd go to make\ninquiries.\"\n\n\"Quite so. She'd be sure to do that. But she needn't go back again to\nMiss Blow. She could resume her interrupted duties. She could start\noff under-housemaiding again at the place she left off when Miss Blow\nrang the bell. I presume that, after a decent interval, say a quarter\nof an hour, Miss Blow would ring the bell again; this time the upper\nhousemaid could go to her. That could be arranged, I suppose, Wilkins?\"\n\n\"Certainly, my lord.\"\n\n\"She, of course, would know nothing about Mr. Goddard; but she would\npromise to go and make inquiries. She would then get back to her\nupper-housemaiding and completely forget about Miss Blow. After another\ninterval, this time probably a shorter one, say ten minutes, Miss Blow\nwould ring the bell again. Then the cook could go to her, and, of\ncourse----\"\n\n\"Beg pardon, my lord, but the cook wouldn't go.\"\n\n\"Couldn't you arrange it, Wilkins?\"\n\n\"No, my lord; it's not the cook's place to answer bells.\"\n\n\"I forgot that,\" said Lord Manton. \"It was stupid of me. I should have\nremembered. I'm afraid, Wilkins, that you'd have to go yourself the\nthird time. You would tell her that Mr. Goddard had left the house an\nhour before. It would be about an hour, wouldn't it, Wilkins?\"\n\n\"As near as I can go to it, it would be about that, my lord.\"\n\n\"After that I should recommend you to leave the room at once, Wilkins.\nYou can stay if you like and see what happens, but I rather recommend\nyou to leave at once.\"\n\n\"Yes, my lord.\"\n\n\"Very well. That gives you a clear hour's start, Goddard. You'd better\nbe off at once.\"\n\n\"Where am I to go to?\"\n\n\"I really don't know. I'm not advising you to run away. I think it\nwould be wiser for you to stay here and marry Miss Blow without any\nfurther resistance. If you act contrary to my advice, you mustn't\nexpect me to make your plans for you.\"\n\nMr. Goddard and Wilkins left the room together. Lord Manton lit a\nfresh cigarette and tried to make up his mind whether he would like\nto see Miss Blow before she left the house. He felt sure that the\ninterview, if he ventured on it, would be an interesting one, and that\nMiss Blow would say very amusing things. On the other hand, he had a\nvivid recollection of the masterful way in which she compelled him to\nwrite a note to Sergeant Farrelly. It seemed likely that she would\nattempt something of the same sort again. He was most unwilling to get\nfurther entangled in the mystery of Dr. O'Grady's disappearance. He\nwas still turning the question over in his mind when Mr. Goddard burst\nnoisily into the room.\n\n\n\n\nCHAPTER XVII\n\n\n\"Hullo!\" said Lord Manton. \"Back again. What have you forgotten? It's\nrather rash of you to venture, I think. If it's only your cigar-case or\nsomething of that sort that you've left behind, I should have had it\nposted after you as soon as you sent me your address. I can't keep Miss\nBlow here permanently, you know. In dealing with a lady like her you\nought not to take these risks.\"\n\n\"The others are coming,\" said Mr. Goddard breathlessly.\n\n\"Oh, indeed! What others?\"\n\n\"I met them in the deer park. They are coming up here. They had Jimmy\nO'Loughlin's boy with them to show them the way.\"\n\n\"What others?\"\n\n\"I turned at once and ran back,\" said Mr. Goddard. \"I don't think they\nsaw me; in fact, I'm sure they didn't. But what are we to do now?\"\n\n\"I'd stand a better chance of answering that question,\" said Lord\nManton, \"if I knew who you were talking about.\"\n\n\"The other women, Mrs. Dick and----\"\n\n\"Oh, the wives of the lost Members of Parliament.\"\n\n\"Yes. What shall we do?\"\n\n\"Don't say, what shall _we_ do,\" said Lord Manton. \"I'm not going to\ndo anything except sit here and watch the progress of events. I think\nthat's as much as can be expected of me. Many men wouldn't do even\nthat. I know lots of people who'd object to your way of filling up\ntheir houses with strange women; but I'm giving you every latitude. If\nyou choose to interview your lady friends in my drawing-room, I make no\nobjection. But I won't be dragged into any complications myself. That's\nthe reason I don't like your saying 'we.' The question is not what\nshall we do, but what will you do.\"\n\n\"What shall I do, then?\" said Mr. Goddard.\n\n\"If you'd taken my advice,\" said Lord Manton, \"and married Miss Blow\nhalf an hour ago, you wouldn't be in this difficulty now. Your course\nwould have been perfectly plain. You'd simply have referred the other\nladies to Miss Blow. She would have dealt with them, and not allowed\nthem to do you any harm. However, I don't want to rub in your past\nmistakes. The only course open to you now is to introduce these three\nto Miss Blow and let them talk the matter over quietly together while\nyou get Mrs. Patsy Devlin to join them.\"\n\nWilkins entered the room while Lord Manton was speaking.\n\n\"There are some ladies to see your lordship,\" he said.\n\n\"To see me? To see Mr. Goddard, you mean.\"\n\n\"It was your lordship they asked for.\"\n\n\"There must be some mistake,\" said Lord Manton. \"Go to them again,\nWilkins, and say that Mr. Goddard is here and will be delighted to see\nthem.\"\n\n\"I don't see what good that will be,\" said Mr. Goddard. \"Wait a minute.\"\n\nBut Wilkins was gone. Mr. Goddard made a protest.\n\n\"Why did you give me away like that?\" he asked. \"If you hadn't told\nthem I was here, they would never have thought of asking for me.\"\n\n\"I gave you away to save myself,\" said Lord Manton. \"I always give\nother people away when there's any kind of unpleasantness going on.\nThat has been a fixed rule of mine through life, and I've always found\nit work well.\"\n\n\"Beg pardon, my lord,\" said Wilkins, who returned, \"but the ladies say\nit's your lordship they want to see; but they'll be very glad to meet\nMr. Goddard too.\"\n\n\"Did you ask their business, Wilkins?\"\n\n\"I did, my lord. They said they heard that you were a magistrate,\nand----\"\n\n\"That,\" said Lord Manton, \"sounds about as bad as anything could be.\nHow many ladies were there, Wilkins?\"\n\n\"Three, my lord.\"\n\n\"The whole three,\" said Mr. Goddard. \"I told you so.\"\n\n\"Where have you put them, Wilkins?\"\n\n\"I showed them into the big drawing-room, my lord. The other lady was\nin----\"\n\n\"Well, go and change them out of that into the small drawing-room. Or,\nwait a minute. It may be easier to move Miss Blow. Go and put her into\nthe big drawing-room along with the other three. Then shut the door and\nleave them. Do you think you can manage that, Wilkins?\"\n\n\"I could try, my lord.\"\n\n\"Very well. Go and try. And if you succeed, don't go to them again\nuntil they've rung the bell at least a dozen times.\"\n\n\"Certainly not, my lord.\"\n\n\"It's rather hard on a man of my age,\" said Lord Manton, \"to be hunted\nout of my own house in this way by a lot of strange women. I'm not\nblaming you, Goddard. All the same, if you had been reasonable about\nMiss Blow we wouldn't be in the position we are. Now, of course,\nthere's nothing for it but to fly. It's very undignified for me, being\na peer and that sort of thing. It will also, I'm afraid, be most\nuncomfortable. I mind that much more than the humiliation. But there's\nnothing else for it. If I stay here they'll catch me sooner or later.\"\n\n\"What will you do?\"\n\n\"I shall get out a horse and trap if I can without being noticed. I\nshall drive down to the station, and lie hid in the ticket office until\nthe next train is due. Then I shall go to London. You'd better come\nwith me.\"\n\nLord Manton's idea was to reach the stable-yard by way of the servants'\nquarters, so as to avoid passing the doors of either the big or the\nsmall drawing-room. To do this from the library it was necessary to\ngo along a corridor and then cross an angle of the central hall off\nwhich both the drawing-rooms opened. The corridor was safe. The passage\nacross the hall was dangerous. Unless Wilkins actually locked the\nladies in, a door might be opened at any moment.\n\nLord Manton and Mr. Goddard went on tiptoe along the corridor. A voice\nreached them from the hall. It was Wilkins' voice.\n\n\"This way, miss, if you please.\"\n\n\"One of them has escaped,\" said Mr. Goddard.\n\n\"No,\" whispered Lord Manton; \"it's only Wilkins moving Miss Blow into\nthe big drawing-room. Wait a minute.\"\n\nThey heard Wilkins' voice again. This time it was nearer than it had\nbeen.\n\n\"I beg your pardon, miss, but if you'll allow me to show you the way.\nYou're going in quite the wrong direction, miss.\"\n\nMr. Goddard clutched Lord Manton's arm. \"She's coming down here,\" he\nsaid. \"We'd better----\"\n\n\"It's all right,\" whispered Lord Manton. \"Wilkins will head her off.\"\n\n\"I'm going in the direction in which I mean to go.\"\n\nThis voice was unmistakably Miss Blow's. It was clear, resonant,\ndetermined, and sounded very near at hand.\n\n\"Good heavens!\" said Mr. Goddard.\n\n\"She's quite right,\" said Lord Manton. \"She is going in the direction\nin which she means to go. I rather respect her for it.\"\n\nWilkins, walking sideways, and expostulating vehemently, appeared\nat the end of the corridor. Behind him, mistress of herself and the\nsituation, strode Miss Blow. She caught sight of Lord Manton and Mr.\nGoddard at once. She pointed to them with a finger and fixed her eyes\nupon them with a terrible glare. They stood still, fascinated. The idea\nof escape by running and a leap through the library window occurred\nto Lord Manton. But with Miss Blow's eyes on him he was incapable of\nthe effort. Wilkins, faithful to the last, walked backwards along the\ncorridor, in front of Miss Blow. He looked as if he meant to sacrifice\nhimself in order to stand between his master and Amazonian violence. He\nwas a good servant.\n\n\"So there you are,\" said Miss Blow. \"You----\" Her finger, pointed at\nMr. Goddard, trembled with indignant scorn. \"And you, my lord.\"\n\nThere was a fine note of contempt, bitter and furious contempt, in her\nvoice, as she uttered the words \"my lord.\" There was something terrible\nin the association of a title reckoned honourable by the world with the\nbaseness which she evidently attributed to Lord Manton.\n\n\"Yes,\" said Lord Manton; \"we're both here. But why aren't you in the\nbig drawing-room with the other ladies? You oughtn't to come down here,\nyou know, especially without a chaperone.\"\n\n\"You call yourselves gentlemen,\" said Miss Blow.\n\n\"No,\" said Lord Manton; \"we don't. I certainly don't. And I should\nbe surprised to hear that Mr. Goddard did. The word is quite out of\nfashion, I assure you. Nobody uses it nowadays. Don't bring unjust\ncharges against us, Miss Blow. There are lots of things we may be\naccused of with truth. We're not the men we ought to be, especially\nMr. Goddard. Charge us with the things we've done, and we'll confess\nat once and apologize. But don't be unjust. We never called ourselves\ngentlemen.\"\n\n\"You ran away from me yesterday,\" said Miss Blow, addressing Mr.\nGoddard, \"after promising faithfully that you'd help me. You ran away\nagain to-day. You would be running away now if I hadn't caught you in\nthe act.\"\n\n\"He did and he would,\" said Lord Manton. \"I've just been speaking to\nhim about it. I told him his conduct was disgraceful. I'm glad now that\nhe'll hear what you think of it from your own lips. It'll do him good.\"\n\nMr. Goddard frowned and shuffled uneasily. Even though he had been\nfairly warned of the principle on which Lord Manton treated his friends\nin emergencies, he did not expect to be sacrificed so completely and\nremorselessly.\n\n\"You're as bad yourself,\" said Miss Blow.\n\n\"No,\" said Lord Manton; \"I'm not. Try to be just, Miss Blow. I didn't\nrun away from you.\"\n\n\"You're just as bad as he is,\" said Miss Blow.\n\nHer voice was clear and loud. Lord Manton glanced anxiously towards the\nhall. It was quite possible that the noise of the denunciation might\nreach the drawing-room.\n\n\"Wilkins,\" he said, \"did you shut the door of the big drawing-room?\"\n\n\"No, my lord; I understood your lordship to say that this lady----\"\n\n\"Then go and do it at once. I can't have the other three---- Miss Blow\nis speaking to us in a confidential manner and doesn't want----\"\n\n\"Who are you shutting up?\" said Miss Blow. \"Who is your lackey going to\nimprison?\"\n\n\"Some ladies,\" said Lord Manton. \"There are, I believe, three of them.\nBut I'm not imprisoning them. I'm only trying to keep them where they\nare for a few minutes. I'm doing it entirely on your account. They\nwouldn't be at all cheering company for you. They have, unfortunately,\njust lost their husbands under the most mysterious and trying\ncircumstances. Members of Parliament, you know. Excellent fellows every\none of them. The whole thing is unspeakably sad.\"\n\n\"Are those the ladies----?\" said Miss Blow. \"But of course they are.\nI heard about that. Do you mean to say that you're going to sit here\nand do nothing, nothing whatever, while men are being murdered in\nthis wholesale manner every day? Will you make no effort to bring the\ncriminals to justice and prevent the loss of more human life? You,\nsir,\" she addressed Mr. Goddard, \"you wear his Majesty's uniform; you\nare an officer in what is supposed to be a police force----\"\n\n\"It is a police force,\" said Mr. Goddard feebly. \"It really is,\nalthough I am an officer in it.\"\n\n\"And you,\" she went on, turning to Lord Manton, \"you are a magistrate\nbesides being a peer.\"\n\n\"Miss Blow,\" said Lord Manton, \"won't you come into the library and sit\ndown? We could talk so much more comfortably if we were sitting down.\nBesides, this is rather a public place for the discussion of private\naffairs.\"\n\nHe looked past Miss Blow towards the end of the corridor. Something\nin his expression made Miss Blow turn her head. She saw, gathered in\na knot in the hall, the cook, the kitchen-maid, the upper and under\nhouse-maid, and Wilkins. All of them, except Wilkins, were grinning.\nThey had forgotten all decency and the respect due to their master.\nThey were eagerly listening to every word Miss Blow said. She allowed\nherself to be led into the library.\n\n\"Now sit down,\" said Lord Manton. \"You must be thoroughly tired out\nafter your long walk yesterday and all this excitement to-day. Will you\nallow me to offer you a glass of wine and a biscuit? Goddard, ring the\nbell, like a good man.\"\n\n\"No,\" said Miss Blow.\n\n\"A cup of tea, then? No? Or an egg flip? The cook would have it\nready in a moment. I often have an egg flip myself when I'm feeling\nover-done. It's an excellent thing, I assure you.\"\n\n\"No,\" said Mrs. Blow; \"I'll take nothing--nothing from you. I----\"\n\n\"Well, just allow me to say one word,\" said Lord Manton, \"before you\nbegin again.\"\n\n\"If you've any excuse to make for your behaviour,\" said Miss Blow,\n\"make it. I shall listen to you.\"\n\n\"I haven't,\" said Lord Manton. \"Not a shred. Nor has Mr. Goddard. Don't\ninterrupt me, Goddard. You haven't any real excuse, and you know it.\nBut you mustn't be too hard upon us, Miss Blow. Try to put yourself in\nour position, in Mr. Goddard's position, for I really haven't anything\nto do with the business one way or other. It wasn't his fault about\nthose Members of Parliament. He's just as sorry about it as anybody\nelse. If he'd known that they intended to run away from their wives\nhe'd have stopped them; but how could he know?\"\n\n\"Oh!\" said Miss Blow. \"That is the latest theory, is it? Their husbands\nran away from them? Do you expect to get any one to believe that? I\nsuppose the husband of that poor woman down in the village ran away\nfrom her. I suppose you mean to try and prove that she ill-treated\nhim, that she, a half-starved, delicate woman, bullied a great hulking\nblacksmith. I suppose you'll say that Dr. O'Grady ran away from me.\nLast time I was here you said he ran away from his creditors. When I\nproved that to be a lie, you have the assurance to say that he ran away\nfrom me.\"\n\n\"I hadn't mentioned you or Dr. O'Grady,\" said Lord Manton. \"But come\nnow, Miss Blow, be reasonable. If he has run away from you, he wouldn't\nbe the only man that has. You can't deny that Mr. Goddard ran away\nfrom you. He did it twice. You said so yourself. In fact, you more than\nhinted that he was in the act of a third flight when you caught him.\nThere's nothing inherently absurd in supposing that one man would do\nwhat another man has done several times. I needn't say I wouldn't do\nit myself. But that's another matter. It's far better for you to look\nfacts straight in the face, however unpleasant they are.\"\n\nWhether Miss Blow looked at the facts or not, the facts as Lord Manton\nrepresented them, she certainly looked at Mr. Goddard. It seemed for\na moment as if she was about to re-open the question of his flight\nto Ballymoy and his subsequent flight back again to Clonmore. He\nfelt greatly annoyed with Lord Manton for calling fresh attention to\nthese performances. There ought, he was convinced, to be some limit\nto the extent to which a man may give away his friends. But Miss Blow\nrecognized that these hurried flittings of his and the causes of them\nwere side issues. She got back, with an evident effort, to the main\npoint immediately under discussion.\n\n\"And why should you suppose that the husbands of the ladies you have\nshut up in your drawing-room have run away from them?\"\n\nIt was of Lord Manton that she asked the question; but Mr. Goddard\nanswered her. He saw his opportunity and seized it. Having been\nsacrificed more than once as a burnt-offering to Miss Blow's wrath, he\nwas perfectly ready, now he got the chance, to show up Lord Manton, as\na man who also deserved strong denunciation.\n\n\"Lord Manton says,\" said Mr. Goddard, \"that their husbands couldn't\nbear to live with them any more, because they were ordinary women\nand----\"\n\n\"As well as I recollect,\" said Lord Manton, \"it was you who used the\nword 'ordinary.' I hadn't seen the ladies at the time. For that matter,\nI haven't seen them yet.\"\n\n\"And,\" said Mr. Goddard, speaking slowly and with emphasis, \"because\nthey wear red dressing-gowns and wash their teeth.\"\n\nHe glanced at Lord Manton with an expression of triumph on his face.\nMiss Blow stared first at one of the men before her and then at the\nother. She was amazed. In spite of the white heat of her virtuous\nindignation she was reduced for the moment to a silence of sheer\nastonishment. The nature of the charge brought against the wives of the\nMembers of Parliament took her aback. It was totally unexpected.\n\n\"Those,\" said Mr. Goddard, striking home after his victory, \"are the\nexact expressions Lord Manton used.\"\n\n\"Of course,\" said Lord Manton, in an explanatory and half-apologetic\ntone, \"I didn't mean to suggest that those were Patsy Devlin's reasons\nfor absconding. I don't suppose that his wife has a dressing-gown of\nany colour, and as for her teeth----\"\n\nMiss Blow began to recover a little from her first shock.\n\n\"Do you mean to say,\" she said, \"that you consider a man is justified\nin deserting his wife because she wears a red dressing-gown and washes\nher teeth?\"\n\n\"Certainly,\" said Lord Manton. \"If you consider the matter fairly\nand impartially, without bias in favour of either sex, you will see\nthat there can't be two opinions about it. The mere act of wearing\na dressing-gown, red or blue, is of course nothing in itself. But\nconsidered as an expression of a certain spirit, of what I may perhaps\ncall the spirit of persistent, puritanical domesticity; regarded as an\nevidence of an oppressive kind of civilized respectability, taken in\nconjunction with a whole series of trifling, or apparently trifling,\nmental and physical habits,--ordering dinner, for instance, engaging\nservants, doing needlework, paying weekly bills, keeping a visiting\nlist, taking a holiday every year at the seaside--you will, I am sure,\nunderstand the sort of things I mean--taken in conjunction with these\nand regarded as an expression of the kind of spirit which takes a\ndelight in doing these things and doing them continuously year after\nyear--considered in this way, the wearing of a red dressing-gown does\njustify a man, a certain sort of man, in deserting his wife. You catch\nmy meaning, I am sure, Miss Blow.\"\n\nOnce more Miss Blow was silent from sheer astonishment. Then, after a\npause, she spoke, and Mr. Goddard, like the governor Felix before the\nApostle Paul, trembled. Lord Manton, although it was to him that her\nremark was specially addressed, bore himself more bravely.\n\n\"You think it very fine,\" she said, \"to bully and badger a helpless\ngirl, and to allow innocent men to be murdered under your very eyes.\nBut you'll have to answer for it. You'll be held responsible, both of\nyou. It's--it's intolerable.\"\n\n\"My dear Miss Blow,\" said Lord Manton.\n\n\"Don't dare to say 'my dear' to me.\"\n\n\"I didn't mean it,\" said Lord Manton, quite truthfully. \"Nothing was\nfurther from my mind than any idea of expressing affection, although\nof course I have a great regard and esteem for you. But do try to be\nreasonable. We're quite ready, both of us--I'm sure I may speak for Mr.\nGoddard as well as myself--to do anything in our power. But what can we\ndo? What do you suggest our doing? What do you want us to do?\"\n\n\"Arrest the murderers,\" said Miss Blow.\n\n\"Certainly,\" said Lord Manton. \"Goddard, go at once and arrest the\nmurderers. You're a policeman. It's your business to arrest murderers.\nDon't waste time. Do it.\"\n\n\"Who am I to arrest?\" said Mr. Goddard. \"I don't know any murderers. I\ndon't so much as know the name of a single murderer. If I did, I'd be\noff after him at once.\"\n\n\"Who is he to arrest, Miss Blow?\" said Lord Manton. \"You'll have to\ngive him the name and address of your murderer. I suppose you know who\nhe is and where he lives.\"\n\n\"Yes, I do,\" said Miss Blow. \"I didn't at first, but now I do.\"\n\nWilkins entered the room as she spoke. \"I beg your pardon, my lord,\" he\nsaid.\n\n\"Yes, Wilkins, what is it? Be as quick as you can, Wilkins. We are at a\nmost interesting point of our conversation. Miss Blow is just going to\nreveal to us the name of a murderer, and then Mr. Goddard is going out\nto arrest him.\"\n\n\"I beg pardon, my lord, but the ladies in the big drawing-room have\nbeen ringing the bell.\"\n\n\"Much, Wilkins? I mean to say, have they been ringing it an excessive\nnumber of times, or very hard? There is no harm in their ringing\nmoderately. I should wish them to ring when they want anything.\"\n\n\"They've been ringing a good deal, my lord.\"\n\n\"Are they still ringing?\"\n\n\"Yes, my lord. That's the reason I mentioned it. I thought there might\nbe something you would wish me to say to them.\"\n\n\"Quite right,\" said Lord Manton. \"I'm very glad you told me. My idea\nwould be for you to offer them a cup of tea or something of that\nsort to keep them quiet. Or what would you say, Miss Blow, if we had\nthem down here? They might take to smashing up the furniture and the\nornaments if I keep them there much longer. There's some china in the\nroom that I'd be sorry to lose. Not very valuable, you know, but still\nthings that it would be difficult to replace. Don't tell us the name\nof your murderer for a minute or two. Mr. Goddard and I will restrain\nour impatience. I'd like the other ladies to be here when you make the\nrevelation. I'm sure they'd enjoy hearing all about it. Wilkins, will\nyou kindly go to the big drawing-room and ask the three ladies to be\ngood enough to come down here?\"\n\n\n\n\nCHAPTER XVIII\n\n\nAfter a delay of about five minutes Wilkins opened the door of the\nlibrary again.\n\n\"Miss Farquharson, Mrs. Sanders, and Mrs. Dick,\" he announced, giving\nthe names in the order in which he had received them from Miss\nFarquharson, who had taken command of the party.\n\nShe entered the room first. Her face was pale with anxiety. Her manner\nand expression were those of a woman who was very much in earnest. Her\nhat, a severe garment of grey felt adorned with a single bow of black\nribbon, was pushed to one side of her head. Her hair, hurriedly pinned\ninto place early in the morning, was untidy. She looked as if she had\nbeen working so hard as to have had scant leisure for attention to the\ndetails of her toilet. Mrs. Dick followed her. She was tremulous and\nshowed signs of having wept frequently and bitterly during the earlier\npart of the day. Mrs. Sanders, a sallow, lean woman of about five and\nthirty, seemed frightened and bewildered.\n\n\"I hope,\" said Miss Farquharson, \"that we are not intruding on you. Our\nbusiness is really very important. That must be our excuse.\"\n\n\"Not in the least,\" said Lord Manton. \"We are delighted to see you.\nAs a matter of fact, you couldn't have come at a more appropriate\nmoment. Miss Blow is just--but won't you sit down? You'll be much more\ncomfortable sitting down.\"\n\nHe pulled a chair forward towards Mrs. Dick, who collapsed into it\nand took out her pocket-handkerchief. Mrs. Sanders perched herself\nuncomfortably on a corner of the sofa. Miss Farquharson sat upright in\nthe writing-chair which stood in front of Lord Manton's desk.\n\n\"Miss Blow,\" said Lord Manton, \"is just going to tell us the name of\nthe man who has murdered your husbands.\"\n\nMrs. Dick gave a queer cry, half wail, half whoop, a loud cry, ending\nin a gasp.\n\n\"Murdered;\" she cried. \"Murdered! Oh, no, no, not murdered!\"\n\n\"Certainly not murdered,\" said Lord Manton. \"That's only what Miss Blow\nsays. I don't believe they're murdered. Neither does Mr. Goddard, and\nif anybody knows the ins and outs of this unfortunate business it's Mr.\nGoddard. He's a policeman, the chief policeman of this district. It\nstands to reason that he must know. Don't be the least uneasy, ladies.\nThere's no necessity for tears. Nobody is murdered.\"\n\nMrs. Dick allowed herself to be a little comforted by this strong\nassurance. She stopped making any very loud noise and let her crying\nsubside into a subdued and inoffensive whimper. On the other hand, Miss\nBlow was evidently enraged by Lord Manton's scepticism. Her voice, when\nshe spoke, sounded defiant and extremely angry.\n\n\"I denounce the inhabitants of the house called Rosivera,\" she said,\n\"especially the man Red. I denounce them as the murderers of Dr.\nO'Grady, Patrick Devlin, and of the husbands of these ladies.\"\n\n\"Go,\" said Lord Manton to Mr. Goddard, \"and arrest the man Red at\nonce.\"\n\nHe liked the phrase \"the man Red.\" It sounded as if it came out of a\nnewspaper report of a criminal trial. It was evident that Miss Blow had\na feeling for appropriate expression.\n\n\"Arrest the man Red,\" repeated Lord Manton, seeing that Mr. Goddard had\nnot moved.\n\nHe was perfectly willing that Mr. Red should be arrested, tried,\nimprisoned, hanged; or arrested, imprisoned, and hanged without a\ntrial, on a charge of murder or any other charge. The really important\nthing was not to obtain justice for Mr. Red or anybody else, but to get\nMiss Blow out of his house.\n\n\"Do go and arrest the man Red, Goddard,\" he said again.\n\n\"I can't,\" said Mr. Goddard. \"How can I possibly go and arrest a man\nwithout a single scrap of evidence against him?\"\n\n\"You hear what he says,\" said Lord Manton to Miss Blow. \"He won't\nact without evidence. Why don't you produce your evidence? You have\nevidence, of course.\"\n\n\"You shall hear the evidence,\" said Miss Blow. \"I have evidence that\nfour, if not five abominable murders have been committed by this man\nand his confederates.\"\n\n\"There now, Goddard,\" said Lord Manton, \"what more can you want? Good\ngracious, what's that?\"\n\nIt was Mrs. Dick. Miss Blow's last words had been too much for her.\nShe was uttering a series of wild shrieks. Mrs. Sanders was sobbing\nconvulsively on the sofa.\n\n\"We'll have to go into another room,\" said Lord Manton, \"to hear the\nevidence.\"\n\n\"No,\" said Miss Farquharson. \"Now we are here we'll stay here. I am\nanxious to hear what this lady has to say.\"\n\n\"But,\" said Lord Manton, \"we can't possibly hear anything while----\"\n\nMiss Farquharson approached Mrs. Dick, grasped her two hands, and spoke\nsternly to her. She repeated the treatment with her niece. It was\nmost efficacious. Both the younger ladies seemed to be afraid of Miss\nFarquharson. They had a cowed, terrified look when she left them; but\nthey had stopped making a noise.\n\n\"Thank you,\" said Lord Manton. \"Now, Miss Blow.\"\n\n\"I begin,\" said Miss Blow, \"with the two latest cases. These gentlemen,\nMembers of Parliament, as I understand, set out on their bicycles, to\nride to Pool-a-donagh. They did not arrive there. They were seen in\nsafety three miles out of Clonmore by a man who was carting turf.\"\n\n\"How do you know that?\" said Mr. Goddard.\n\n\"The hotel-keeper told me,\" said Miss Blow.\n\n\"Oh, Jimmy O'Loughlin! I see. I wouldn't take every word he says for\ngospel if I were you, Miss Blow.\"\n\n\"I don't see,\" said Miss Blow, \"that there's much difference between\nhim and the rest of you. I haven't heard six consecutive words of truth\nsince I came to Clonmore.\"\n\n\"There you are now, Goddard,\" said Lord Manton. \"That's what you get by\ninterrupting. Don't mind him, Miss Blow. Please go on. What you say is\nmost interesting.\"\n\n\"There is just one house between the spot at which they were seen and\nPool-a-donagh,\" said Miss Blow. \"The hotel-keeper told me that too.\"\n\n\"I think that's true,\" said Miss Farquharson. \"I noticed that there\nwere very few houses while we drove along yesterday.\"\n\n\"Therefore,\" said Miss Blow, \"they were murdered in that house.\"\n\nMr. Goddard started violently. The sequence of Miss Blow's reasoning\nhad the effect of a strong electric shock on him. He would have\nprotested if Mrs. Dick had not begun to wail again. When she was\npacified by Miss Farquharson's scowls, Lord Manton began to speak.\n\n\"Perhaps----\" he said.\n\n\"I know what you're going to say,\" said Miss Blow. \"You are going to\nsuggest that they are not murdered, but that they have deserted their\nwives.\" She glanced at Miss Farquharson as she spoke.\n\n\"I wish you wouldn't talk of us as if we were all their wives,\" said\nMiss Farquharson. \"I am not a married woman. I am Mr. Sanders' aunt.\nThat is his wife.\"\n\nShe pointed to the pale Mrs. Sanders, leaving Miss Blow to infer that\nthe lady who shrieked was the wife of the other Member of Parliament.\n\n\"That they have deserted these ladies,\" said Miss Blow, accepting the\ncorrection, \"because they wear red dressing-gowns, and----\"\n\n\"But I don't,\" wailed Mrs. Dick; \"I've never had a red dressing-gown.\nRichard always liked me in blue. He couldn't bear red. He used to\nsay---- Oh, poor Richard!\"\n\n\"There!\" said Lord Manton, with an air of triumph; \"what did I tell\nyou, Miss Blow? You see for yourself now that the man had the strongest\npossible objection to a red dressing-gown.\"\n\n\"And,\" said Miss Blow firmly, ignoring the interruptions, \"because they\nwashed their teeth.\"\n\n\"I never in my life,\" said Miss Farquharson, \"heard such a pack\nof nonsense. Are you all mad, or am I? What on earth have red\ndressing-gowns or that unfortunate little Mrs. Dick's teeth got to do\nwith the disappearance of my nephew and Mr. Dick?\"\n\n\"Shall I ring for some tea?\" said Lord Manton. \"I think we'd all be the\nbetter for a cup of tea. Then we could go on. We'd be much better able\nto understand each other afterwards.\"\n\n\"I pass on to the next case,\" said Miss Blow calmly; \"that of Patrick\nDevlin. He is, I am informed----\"\n\n\"Jimmy O'Loughlin again,\" said Mr. Goddard.\n\n\"By others as well as the hotel-keeper,\" said Miss Blow, showing that\nshe placed no implicit trust in Jimmy O'Loughlin's statements. \"I am\ninformed that he is a blacksmith. He was, it appears, collecting money\nfor some local sports.\"\n\n\"Grand Annual Regatta and Horse Races,\" said Lord Manton. \"I'm a\nmember of the committee, and so I know. Please excuse me interrupting\nyou, Miss Blow, but in a case like this it's as well to be perfectly\naccurate.\"\n\n\"He informed the hotel-keeper in Clonmore----\" said Miss Blow.\n\n\"I propose,\" said Mr. Goddard, \"that we send down for Jimmy O'Loughlin,\nand let him give his evidence himself.\"\n\n\"That he intended to call on Lord Manton for a subscription and then go\non to Rosivera to see what he could get from the man Red. Did he call\nhere?\"\n\n\"Certainly,\" said Lord Manton. \"I gave him a sovereign.\"\n\n\"After that,\" said Miss Blow, \"he was seen no more. It seems to me\nperfectly obvious that he went on to Rosivera and was there murdered.\"\n\nPoor Mrs. Dick wailed again, and was again suppressed, this time very\nrapidly, by Miss Farquharson.\n\n\"There remains,\" said Miss Blow, \"the case of Dr. O'Grady; but before I\ngo into that I have to inform you that there is another man missing.\"\n\n\"My goodness!\" said Lord Manton. \"The thing is becoming a perfect\nepidemic. Who is it now?\"\n\n\"Jimmy O'Loughlin, I hope and trust,\" said Mr. Goddard. \"He's been\ndragged into this business by everybody that has said anything. It's\nalways Jimmy O'Loughlin told me this or Jimmy O'Loughlin told me that.\nIf he's gone off himself now it serves him jolly well right.\"\n\n\"It's not Mr. O'Loughlin,\" said Miss Blow, \"but the man Red's own\nservant, an Englishman. This man used to drive the motor car into\nClonmore to do the marketing for the party at Rosivera.\"\n\n\"At Jimmy O'Loughlin's shop, of course,\" said Mr. Goddard.\n\n\"He did so for the last time the day before Dr. O'Grady's\ndisappearance. The inference is perfectly plain. The man was mortally\nwounded. Dr. O'Grady was decoyed to Rosivera because his services as\na medical man were required. Then he too, to secure his silence, was\nfoully murdered.\"\n\nMr. Goddard gasped. For the second time Miss Blow's logic took away\nhis breath. He tried to speak, but failed. Three times he got as far as\nuttering the word \"but\" and then stuck fast. Lord Manton, who remained\ncomparatively calm, offered a mild criticism.\n\n\"There is just one point,\" he said, \"in the course of your extremely\nable and lucid statement, on which I should like to have a word of\nexplanation. I have no doubt that you have thought the matter out\ncarefully, and will be able to meet my difficulty at once. You say--and\nof course I don't contradict you--that Mr. Red, I mean the man Red, of\ncourse, first tried to murder his own servant and then sent for Dr.\nO'Grady to cure him. Now, why should a man get a doctor for the person\nhe's trying to kill? Wouldn't it have been simpler--I mean to say, do\nmurderers generally summon medical assistance for their victims?\"\n\n\"I am not concerned with his reasons for acting as he did,\" said Miss\nBlow. \"I am dealing simply with what has occurred, with plain facts.\nNow, perhaps, you will do your duty, Mr. Goddard, and arrest the\nmurderers.\"\n\n\"But,\" said Mr. Goddard, \"you haven't given us any evidence at all.\nYou've spun out a lot of wild hypotheses, supported by information\ngiven you by Jimmy O'Loughlin, who is the biggest liar in Connacht.\nIt's perfectly absurd to suppose----\"\n\n\"I join with this lady,\" said Miss Farquharson, \"with this lady whose\nname, as I understand, is Miss Blow, in demanding the arrest of the\nsuspected persons.\"\n\n\"Oh, no, no,\" wailed Mrs. Dick. \"Don't arrest them. Let us forgive\nthem; but bring Richard back to me. It's cruel, cruel. I can't\nlive--oh, I can't----\"\n\nHer voice died away to a whisper. Miss Farquharson was gazing at her\nwith a very stern expression.\n\n\"I can't,\" said Mr. Goddard, \"and won't allow myself to be hustled into\na perfectly illegal act. The thing's----\"\n\n\"You'll have to do it sooner or later,\" said Lord Manton. \"Why not do\nit at once? After all, what does it matter about the man Red? It won't\ndo him any harm to be arrested. If he doesn't deserve it for murdering\nDr. O'Grady and the rest of them, he's sure to deserve it for something\nelse that we know nothing about.\"\n\n\"I'm not thinking about Mr. Red's feelings in the matter. I don't care\nif he's hanged, drawn, and quartered, so long as Miss Blow does it\nherself. But I'm not going to be dragged into a----\"\n\n\"Look here,\" said Lord Manton in a whisper, \"you must do it, Goddard.\nIf you don't, I shall never get these ladies out of the house.\"\n\n\"Very well,\" said Mr. Goddard, \"I'll do it; but I'll do it on your\nauthority. Make out a warrant in proper form and sign it. Then I'll go\nand arrest Red.\"\n\n\"Certainly,\" said Lord Manton. \"I haven't got the--ah--the necessary\npapers here, you know. I don't keep them in the house. Go down to the\nClerk of Petty Sessions, Miss Blow. You'll find him in his office. Get\nhim to fill the form in the proper way. He'll understand. Then send\nit up to me by one of the police. I'll sign it and hand it on to Mr.\nGoddard. I think that will be the most satisfactory arrangement we can\nmake. Don't you?\"\n\nHe appealed to Miss Farquharson, carefully avoiding Miss Blow's eye.\n\n\"Certainly,\" said Miss Farquharson. \"Everything ought to be done in a\nstrictly legal manner.\"\n\n\"Then I suppose,\" said Lord Manton, \"that our interview is at an end.\nShall I order the carriage, Miss Blow, to take you down to Clonmore? It\nwill save time.\"\n\n\"It would be very kind of you to do so,\" said Miss Farquharson. \"I am\nquite able and willing to walk, but my niece and Mrs. Dick are upset\nand greatly tired.\"\n\n\"I shall stay here,\" said Miss Blow; \"I shall not let Lord Manton out\nof my sight till the warrant is signed.\"\n\nMiss Farquharson started. This expression of want of faith in the\ngood will of a peer shocked her. She expostulated with Miss Blow, but\nwithout effect.\n\n\"I shall then go with Mr. Goddard,\" said Miss Blow, \"and see that he\nexecutes the warrant.\"\n\n\"Really,\" said Miss Farquharson, \"I don't understand--I altogether\nrefuse to associate myself with this discourteous language.\"\n\n\"I know these gentlemen,\" said Miss Blow, \"and you don't. I've had some\nexperience of the way they keep their promises.\"\n\n\"I dissociate myself entirely----\" said Miss Farquharson.\n\n\"I appreciate Miss Blow's feeling,\" said Lord Manton. \"I quite\nunderstand it. I even sympathize with it. There has been a good deal\nin Mr. Goddard's conduct during the last few days which justifies her\nsuspicions. I----\"\n\n\"And in your own conduct,\" said Miss Blow.\n\n\"And in my own conduct, of course,\" said Lord Manton. \"Didn't I say\nthat? I meant to. We have acted for the best. You, at least, will\nbelieve that, Miss Farquharson. If Miss Blow has not fully realized our\ndifficulties, that is not her fault. I don't in the least blame her for\nthe attitude she has taken up. Nor does Mr. Goddard.\"\n\nMr. Goddard looked as if he did blame her, but he said nothing. The\nswift glances of appeal which Lord Manton shot at him were sufficient\nto keep him silent.\n\n\"What I propose now,\" said Lord Manton, \"is that Mr. Goddard and I\nshould accompany you to the village, so that Miss Blow shall have the\nopportunity of seeing with her own eyes that her wish with regard to\nthe man Red is carried out.\"\n\n\"That will be giving you far too much trouble,\" said Miss Farquharson.\n\n\"Not at all,\" said Lord Manton. \"I'm delighted to do it. Mr. Goddard,\nthe bell is just beside you; will you be so good as to ring it? I shall\norder the waggonette to take us down. And I think you must allow me\nto offer you some tea. You can drink it while the horses are being\nharnessed, and so waste no time.\"\n\n\"It's very kind of you,\" said Miss Farquharson, \"most kind; we shall be\nvery glad----\"\n\n\"I shall neither eat nor drink in this house,\" said Miss Blow.\n\n\"I quite understand your feeling,\" said Lord Manton. \"There was a\nprophet once who said the same thing. As well as I recollect, a lion\nate him afterwards; but of course that won't happen in your case, Miss\nBlow. There aren't any lions in Connacht.\"\n\n\"_I_,\" said Miss Farquharson, with strong emphasis on the pronoun,\n\"shall be very pleased to accept Lord Manton's hospitality in the\nspirit in which it is offered.\"\n\n\"Thank you,\" said Lord Manton. \"And I'm sure Miss Blow won't have the\nleast objection to your doing so. She holds her own opinions, as we all\ndo; but from what I know of her I'm convinced that she doesn't want to\nforce them upon other people.\"\n\n\"It is understood,\" said Miss Blow, \"that I accompany Mr. Goddard when\nhe goes to arrest the murderers.\"\n\n\"He may not go himself,\" said Lord Manton. \"You will understand, Miss\nBlow, that it is not usual for a man in Mr. Goddard's position, for an\nofficer, to make arrests in person. It is probable that he will send\nSergeant Farrelly and perhaps one of the constables.\"\n\n\"I shall accompany the sergeant, then,\" said Miss Blow.\n\n\n\n\nCHAPTER XIX\n\n\nLord Manton's waggonette was a roomy vehicle, used chiefly for picnic\nparties in the summer, when Lady Flavia and her children were at\nClonmore Castle. The four ladies, Lord Manton, and Mr. Goddard packed\nthemselves into it quite comfortably. Lord Manton gave his order to the\ncoachman.\n\n\"Down to the village, Thomas, and stop at the hotel.\"\n\nMiss Blow was alert and suspicious. \"Why the hotel?\" she asked.\n\n\"I'll go to the police barrack if you like,\" said Lord Manton. \"I only\nsuggested the hotel because it is a convenient central sort of place\nwith a room in it large enough to hold the whole party.\"\n\n\"The hotel is decidedly the most suitable place,\" said Miss Farquharson.\n\nShe was beginning to dislike Miss Blow, whose manner struck her as\naggressive to a degree quite unsuitable in a young woman. On the other\nhand, she highly approved of Lord Manton, who was most courteous and\nhad given her tea, which she wanted badly.\n\n\"Besides,\" said Mr. Goddard, \"we shall be sure of seeing Jimmy\nO'Loughlin at the hotel.\"\n\n\"I don't see why Mr. O'Loughlin should be mixed up in our business,\"\nsaid Miss Blow.\n\n\"Nor do I,\" said Mr. Goddard. \"I've always protested against the way\nhe's dragged in. But everybody does it. Sergeant Farrelly can't say a\nsimple sentence without quoting Jimmy as his authority; and you did the\nsame thing yourself repeatedly when you were elaborating your theory of\nmurder.\"\n\n\"Besides,\" said Lord Manton, \"Jimmy O'Loughlin is a magistrate. He's\nreally far more of a magistrate than I am.\"\n\n\"If he's a magistrate,\" said Miss Farquharson, \"we ought to consult\nhim.\"\n\n\"He's a liar,\" said Miss Blow definitely.\n\n\"Really!\" said Miss Farquharson.\n\nThen she turned her back on Miss Blow, who sat next her, and looked\nwith great interest at the horses. Mr. Goddard leaned across the\nwaggonette and whispered to Lord Manton.\n\n\"What do you mean to do when we get to the hotel?\"\n\n\"Hush!\" said Lord Manton.\n\nMiss Blow's eyes were fixed on him, and he felt that confidences were\ndangerous.\n\nJimmy O'Loughlin greeted the party at the door of his hotel. He was\nsurprised to see them. He was still more surprised when Lord Manton\ndemanded the use of a private sitting-room.\n\n\"There's the commercial room,\" he said, \"and there's the drawing-room.\nYou can have the two of them if I you like, for there isn't a soul\nstopping in the house this minute barring the doctor's young lady.\nIt'll be better, maybe, for you to take the commercial room by reason\nof there being a key to the door, so as you'll be able to lock it if\nyou heard Bridgy coming along the passage.\"\n\n\"What are you going to do now?\" said Mr. Goddard, catching Lord Manton\nby the arm, as the ladies entered the commercial room.\n\n\"I'm going to persuade Jimmy O'Loughlin to sign the warrant, if I\npossibly can,\" said Lord Manton.\n\n\"He'll not do it. Jimmy O'Loughlin's not a born fool.\"\n\n\"Then I'm afraid,\" said Lord Manton, \"that you'll have to arrest the\nman Red without a warrant.\"\n\n\"I won't,\" said Mr. Goddard.\n\n\"Send the sergeant, then.\"\n\n\"I won't.\"\n\n\"Couldn't you send the sergeant and a constable and tell them to\ninquire civilly of Mr. Red whether he'd seen anything of the Members\nof Parliament? It's quite a natural thing to ask. They passed his gate\nyesterday, and he might have seen them. We could give the sergeant some\nsort of a blue paper in the presence of Miss Blow and pretend that he\nwas going to make the arrest.\"\n\n\"That's no good,\" said Mr. Goddard. \"She said she'd go and see the\narrest made herself, and she'll do it.\"\n\n\"Send the men on bicycles,\" said Lord Manton, \"then she won't be able\nto keep up with them.\"\n\n\"The objection to that is that she has a bicycle herself. I don't know\nhow she got it, but she rode over on it from Ballymoy this morning.\nMoriarty might get ahead of her, but she'd knock spots out of the\nsergeant in a race; he's fat.\"\n\n\"Jimmy O'Loughlin,\" said Lord Manton, \"come here.\"\n\nJimmy was leaning in a careless attitude against the doorpost of his\nhotel. He appeared to be entirely uninterested in what was going on,\nand was surprised when Lord Manton called him.\n\n\"Jimmy O'Loughlin,\" said Lord Manton, \"Mr. Goddard is sending a couple\nof police over to Rosivera to see if there's any news there of the\ngentlemen that have got lost, and he doesn't want Miss Blow to go with\nthem. Can you think of any way of stopping her?\"\n\n\"Her mind's made up to go,\" said Mr. Goddard.\n\nJimmy meditated on the problem.\n\n\"If so be,\" he said at last, \"that I could get the boots off of her,\nand had them hid in the haggard where she wouldn't find them easy, she\ncouldn't go.\"\n\n\"She could not,\" said Mr. Goddard. \"But how do you propose to get them\noff her?\"\n\n\"I was thinking,\" said Jimmy, \"that if his lordship here would draw\ndown the subject of boots, and was to say that all boots was the better\nof being cleaned, and then if I was to say that Bridgy was doing\nnothing particular and would be glad to give a rub to any lady's boots\nthat liked, that maybe she'd take them off herself.\"\n\n\"If that's the best you can do in the way of a suggestion,\" said Mr.\nGoddard, \"you might as well have kept it to yourself. Is it likely\nshe'd take off her boots at this hour of the day to please you?\"\n\n\"It wouldn't be to please me,\" said Jimmy; \"it would be his lordship\nthat would ask her at the latter end.\"\n\n\"I'm afraid,\" said Lord Manton, \"she wouldn't do it for me. She doesn't\nlike me. You'd think she would, but as a matter of fact she doesn't.\"\n\n\"How would it be,\" said Jimmy, \"if I was to have a telegram for her?\nThe young lady that minds the post-office is a niece of my own. It\nmight be in it that it was from the doctor himself and came from New\nYork. That would turn her mind away from the police.\"\n\n\"She wouldn't believe it,\" said Mr. Goddard. \"Not if you were to go in\nand swear to it on a Bible. She says you're an awful liar.\"\n\n\"Begad, then, I don't know what it would be best to do.\"\n\nSergeant Farrelly, Constable Cole, and Constable Moriarty arrived at\nthe door. They had been summoned by Lord Manton's coachman after he had\ndeposited the party at the hotel. The situation was explained to them\nby Mr. Goddard. Sergeant Farrelly expressed perfect readiness to go to\nRosivera and make any inquiries that were considered necessary. When\nasked whether he could escape without Miss Blow, he looked blank. As a\nmatter of fact, Miss Blow had opened the door of the commercial room\nand was watching the party in the hall with suspicious eyes.\n\n\"Unless,\" said Constable Cole, \"we could hit on some kind of a\nstratagem.\"\n\n\"You and your stratagems,\" said the sergeant; \"we've heard enough of\nthem.\"\n\n\"If you have a stratagem in your mind,\" said Mr. Goddard, \"trot it out.\nBut there's no use your suggesting taking her boots, or sending her a\nbogus telegram. We've discussed those two plans already.\"\n\n\"Don't you give heed to him,\" said the sergeant. \"Stratagems is never\nout of his mouth, and there's no sense at all in what he says.\"\n\n\"How would it be,\" said Constable Cole, \"if the sergeant and myself was\nto go off to Rosivera on our bikes?\"\n\n\"That's been suggested before,\" said Mr. Goddard, \"and it's no good.\nShe has a bicycle herself, and she'd go with you.\"\n\n\"Sure he knows she has a bicycle,\" said Jimmy O'Loughlin. \"Didn't he\nsee her riding in on it this morning, the time you were off up at the\nCastle? And didn't he remark on its being mighty like the machine that\nthe sergeant's wife beyond in Ballymoy is after buying?\"\n\n\"I did,\" said Constable Cole.\n\n\"If you knew all that,\" said Mr. Goddard, \"what on earth was the good\nof your suggesting that you and the sergeant should go on bicycles?\nDon't you know she'd be after you?\"\n\n\"Stratagems!\" said Sergeant Farrelly scornfully. \"Do you call them\nstratagems?\"\n\n\"How would it be,\" said Constable Cole, \"if you was to go in to her\nand tell her that it was Constable Moriarty that was to go with the\nsergeant to Rosivera?\"\n\n\"I dislike telling gratuitous and entirely useless lies,\" said Mr.\nGoddard. \"She simply wouldn't believe me.\"\n\n\"She'd believe Moriarty,\" said Cole. \"She has a great wish for Moriarty\nsince the day he took her for a drive on Jimmy O'Loughlin's car. If\nMoriarty was to go in as soon as ever you were done telling her, and\nwas to say he'd be glad if she'd go along with him----\"\n\n\"I couldn't say the like to a young lady,\" said Moriarty. \"I'd be\nashamed.\"\n\n\"Be quiet, Moriarty,\" said Mr. Goddard. \"And let's hear the rest of the\nstratagem.\"\n\n\"And if he was to say at the same time----\" said Cole.\n\n\"If who was to say?\" asked Lord Manton. \"I'm getting mixed.\"\n\n\"Constable Moriarty,\" said Cole; \"if he was to say that it would be a\npleasure to him to go round to the yard and get her bicycle for her the\nway she'd be ready to go with him and the sergeant----\"\n\n\"I see,\" said Lord Manton.\n\n\"She'd go with him to the yard,\" said Mr. Goddard. \"Don't forget that.\"\n\n\"She would,\" said Cole, \"and on the way there she'd see Moriarty's\nbicycle and another, as it might be the sergeant's.\"\n\n\"But of course it wouldn't be the sergeant's,\" said Lord Manton.\n\n\"It would not. The sergeant and myself might be off by that time a good\nbit along the road to Rosivera. But the bicycle she'd see would have a\nlook about it as if it might be the sergeant's. It could have his cape\nstrapped on it and maybe his name somewhere about so as she'd notice\nit.\"\n\n\"It's just possible,\" said Mr. Goddard, \"that if she has the sort of\nconfidence you say she has in Moriarty she'll be taken in for five\nminutes; but at the end of that time she'll be off after you.\"\n\n\"I wouldn't wonder,\" said Cole, \"but the back tyre of her own bicycle\nmight be flat.\"\n\nHe looked at Jimmy O'Loughlin as he spoke. His face was entirely devoid\nof any expression, but Jimmy was a man of quick wit.\n\n\"Bridgy?\" he said.\n\n\"Bridgy'd do,\" he said.\n\n\"Do what?\" said Lord Manton. \"I'm getting mixed again.\"\n\n\"With the blade of a knife?\" said Jimmy O'Loughlin.\n\n\"That, or a three-pronged fork,\" said Cole, \"unbeknown to any but\nherself.\"\n\n\"It could be done,\" said Jimmy O'Loughlin, \"and Bridgy's the girl for\nthe job.\"\n\n\"Constable Moriarty,\" said Cole, \"would offer to mend it for her,\nwhile they'd be waiting for the sergeant, who'd be down at the barrack\ngetting the handcuffs.\"\n\n\"Oiling them,\" said Lord Manton.\n\n\"Oiling or such,\" said Cole.\n\n\"Where's the use of oiling handcuffs?\" said the sergeant.\n\n\"Shut up, sergeant,\" said Mr. Goddard. \"That's part of the stratagem.\"\n\n\"Constable Moriarty,\" said Cole, \"would be talking to her pleasant and\nagreeable the way he'd distract her mind and him mending the tyre. It's\nhimself knows how to talk to a young lady.\"\n\n\"I do not,\" said Moriarty. \"She'd be laughing at me.\"\n\n\"She'll not laugh,\" said Mr. Goddard. \"That's one thing you may feel\nquite sure about. Whatever else she does, she'll not laugh.\"\n\n\"I'm not fit to talk to her,\" said Moriarty.\n\n\"You're not fit to mend a tyre either,\" said Cole; \"mind that now. When\nyou have the hole there is in it with a patch on it, and you're putting\nback the cover on the wheel you'd nip the inner tube so as there be a\nbit took out of it.\"\n\n\"Do you take me for a fool?\" said Moriarty. \"Haven't I mended----\"\n\n\"I'll take you for a fool,\" said Mr. Goddard, \"if you don't do exactly\nwhat you're told.\"\n\n\"After that,\" said Cole, \"you'll have to mend the tyre again; and I'd\nsay, if you're any kind of good at all, that by the time you've done\nwith it it'll be beyond the help of man in the way of holding the air.\"\n\n\"It's a great stratagem,\" said Lord Manton; \"I never heard a better.\"\n\n\"It's what I was reading in a book one time,\" said Cole. \"You know the\nbook, sergeant----\"\n\n\"I've heard you speaking about it,\" said the sergeant, \"many a time.\"\n\n\"Well,\" said Cole, \"that stratagem was in it. It was a young fellow\nthat was off with a girl that he was wishing to marry and her father\nwas after them. It was bicycles they had. And the young fellow gave\nhalf a crown to the man in the hotel to do the like to the old chap's\nbicycle the way he'd get off with the girl.\"\n\n\"Whether the idea is absolutely original or not,\" said Lord Manton,\n\"you deserve the greatest credit for applying it to this particular\ncase.\"\n\n\"It was in my mind,\" said Cole, \"and I just said to myself that maybe,\nif I didn't forget it, it might come in handy some day.\"\n\n\"At the latter end,\" said Moriarty, \"she'll be asking where the\nsergeant is.\"\n\n\"At the latter end,\" said Mr. Goddard, \"when the bicycle's quite past\nmending, you can tell her that he was obliged to start to Rosivera\nwithout her and take Cole along with him.\"\n\n\"She'll have the face ate off me when I do,\" said Moriarty.\n\n\"If she does,\" said Mr. Goddard, \"it won't make much matter to you or\nany one else. The new one you'll get can't be worse to look at than the\none you have.\"\n\n\"I don't know,\" said Sergeant Farrelly, \"how will the sergeant's wife\nat Ballymoy like having them tricks played with her bicycle. She's a\ncousin of my own.\"\n\nHe had never had a high opinion of Cole's stratagems, and it pained him\nto have to listen to the praise bestowed on this one.\n\n\"She can get a new tyre,\" said Mr. Goddard; \"and serve her jolly well\nright for lending her bicycle to Miss Blow. She ought to have had more\nsense.\"\n\nA few minutes later Lord Manton and Mr. Goddard entered the commercial\nroom of the hotel. They had with them a warrant for the arrest of\nTheodore Guy Red of Rosivera on a charge of wilful murder. They spread\nthis out on the table and invited Miss Blow to inspect it. She did so,\nscanning every line carefully.\n\n\"It's not signed,\" she said.\n\n\"No,\" said Lord Manton. \"We thought you'd prefer to have it signed in\nyour presence. Kindly ring the bell, Mr. Goddard.\"\n\n\"What for?\" said Miss Blow.\n\n\"I want a pen and ink, for one thing,\" said Lord Manton. \"And I want\nJimmy O'Loughlin. He's going to sign it too.\"\n\nMiss Blow sniffed, but she made no objection to the second signature.\n\n\"Now,\" said Lord Manton, when he and Jimmy O'Loughlin had signed their\nnames, \"call in the sergeant and the constable.\"\n\nMr. Goddard opened the door and summoned the police. They marched into\nthe room and stood upright, rigid and impressive, near the door. They\nmade a great impression on Miss Farquharson. Sergeant Farrelly, in\nparticular, struck her as a kind of embodiment of the spirit of law\nand order. Mr. Goddard held the warrant in his hand and addressed the\nmen.\n\n\"Sergeant Farrelly,\" he said, \"will take this warrant, proceed at once\nto Rosivera, and effect the arrest of Theodore Guy Red, the person\nnamed in it.\"\n\nHe looked round as he finished his sentence, and noticed with pleasure\nthat Miss Blow was listening intently to what he said.\n\n\"Constable Moriarty,\" he went on, \"will accompany the sergeant, and\nwill be prepared to act vigorously in the event of the use of force\nbeing necessary to effect the capture of the prisoner.\"\n\nHe looked round again at Miss Blow. Her face was beginning to assume\nquite an amiable expression.\n\n\"With a view to saving time,\" said Mr. Goddard, \"the police will\nproceed to Rosivera on bicycles, starting as soon as possible. Sergeant\nFarrelly, is your bicycle ready?\"\n\n\"It is, sir,\" said the sergeant. \"It's at the door of the hotel this\nminute, and my cape is strapped on to the handle bars.\"\n\n\"Is Constable Moriarty's bicycle ready?\" asked Mr. Goddard.\n\n\"It is, sir,\" said the sergeant.\n\n\"The police,\" said Mr. Goddard, \"will be accompanied on this expedition\nby Miss Blow. Every effort, consistent with the effecting of the\narrest, will be made by the police to protect Miss Blow in the event of\nriot.\"\n\n\"It will, sir.\"\n\n\"By the way, Miss Blow,\" said Mr. Goddard, \"have you got a bicycle?\"\n\n\"Yes,\" said Miss Blow. \"I borrowed one this morning in Ballymoy. It's\nin the hotel yard now.\"\n\n\"It's in the stable,\" said Jimmy O'Loughlin. \"I'm after telling Bridgy\nto give it a bit of a rub over with a soft cloth the way it'll be\ndecent like, when the young lady wants it.\"\n\n\"Sergeant Farrelly,\" said Mr. Goddard, \"will now proceed to the barrack\nand provide himself with handcuffs.\"\n\n\"Carefully oiled,\" said Lord Manton.\n\n\"He will be accompanied by Constable Cole, who will remain on guard\nat the barrack. Having obtained the handcuffs, Sergeant Farrelly will\nreturn to the hotel and join the rest of the party. Constable Moriarty\nwill proceed to the back yard, take Miss Blow's bicycle and wheel it\nround to the front door, so that everything will be in readiness for an\nimmediate start when Sergeant Farrelly returns with the handcuffs.\"\n\n\"I can get my bicycle for myself,\" said Miss Blow.\n\n\"It would be too much trouble for you, miss,\" said Jimmy O'Loughlin.\n\"But of course, if the young lady's doubtful about the way Moriarty\nmight handle it, she's right to go. It's a good bicycle,\" he added,\n\"though I'd say that maybe the back tyre of it was a bit worn.\"\n\nMiss Blow, accompanied by Moriarty, who looked extremely uncomfortable,\nleft the room. Sergeant Farrelly and Constable Cole marched rapidly\ndown the street towards the barrack.\n\n\"I think, ladies,\" said Lord Manton, \"that Mr. Goddard and I will leave\nyou for the present. We shall see you this evening again, so we need\nnot say good-bye.\"\n\n\"That,\" said Lord Manton to Mr. Goddard as they walked together across\nthe deer park, \"was a good stratagem. I don't altogether envy Moriarty\nwhen it comes to its climax; but Cole certainly deserves promotion.\"\n\n\"I don't see that it's much use in reality,\" said Mr. Goddard. \"It's\nonly putting off the evil day, you know. When the sergeant comes back\nfrom Rosivera we'll have the whole thing to do over again.\"\n\n\"You told him not to hurry, I hope,\" said Lord Manton.\n\n\"I told him to spend as long over the job as he possibly could. I told\nhim not to be back before six this evening at the very earliest. But\nwhat's the good of that? He's bound to come back some time.\"\n\n\"Still, it's always so much time gained. We may hear something of the\nmissing men before then. You'll telegraph all over the country, of\ncourse.\"\n\n\"I'll go back at once,\" said Mr. Goddard, \"and take entire possession\nof the telegraph office.\"\n\n\"Come up and have some luncheon first. The afternoon will be time\nenough for the telegrams.\"\n\n\n\n\nCHAPTER XX\n\n\nThere are, as Patsy Devlin reminded Jimmy O'Loughlin on the occasion\nof Miss Blow's first visit to Lord Manton, two ways of getting to the\nCastle from the village of Clonmore. There is the longer way by the\ngreat avenue which leads through the demesne and is remarkable for its\nfine rows of beech trees. By it all visitors who drive must go. They\nleave the public road a mile to the east of the village, having passed,\nsupposing them to start from Jimmy O'Loughlin's hotel, both the police\nbarrack and the railway station. There is also the shorter way through\nthe deer park, available only for foot passengers, because it is\nnecessary in the first place to climb a boundary wall. The visitor who\ngoes by this route, supposing once more that he starts from the hotel,\nleaves the village, and walks in a westerly direction until he comes to\nthe spot where the wall is partially broken down and therefore easy to\nclimb. He passes no building of any importance on his way, because the\nhotel is almost the last house at the west end of the village street.\n\nWhen luncheon was over Mr. Goddard spoke of going down to Clonmore to\nsend off his telegrams. Lord Manton offered to order the dog-cart and\nhave him driven down. Mr. Goddard refused.\n\n\"You'd much better let me,\" said Lord Manton. \"It will save you a lot\nof time and do the cob good. He hasn't been out for two days. I've\nbeen afraid to put my nose outside the place for fear of meeting the\npolice.\"\n\n\"I'd like to drive well enough,\" said Mr. Goddard, \"but I daren't. The\nfact is, I want to get into the telegraph office without being seen,\nif possible. Miss Blow is sure to be at the barrack, and I'm a little\nnervous about passing the door.\"\n\nThe post-office in Clonmore is a sort of bye-product of Jimmy\nO'Loughlin's commercial activity. The business is carried on in a\ncorner of his shop, and the shop itself is an adjunct of the hotel.\nApproaching the village from the west you come upon the shop door\nfirst, then that of the hotel.\n\n\"I dare say you're right,\" said Lord Manton. \"Unless Moriarty is a\nyoung man with quite remarkable powers of persuasion, Miss Blow's\ntemper is likely to be very bad indeed.\"\n\n\"She'll find out, of course,\" said Mr. Goddard, \"that the sergeant\nreally has gone off in the direction of Rosivera, and taken Cole with\nhim. That ought to pacify her to some extent. Still, I think I'll avoid\nan interview as long as I can.\"\n\nBy walking through the deer park and approaching the village\ncautiously, Mr. Goddard succeeded in getting into the post-office\nunseen. After a short search he discovered Jimmy O'Loughlin's\nniece, a red-haired girl, who sold stamps and sent off and received\ntelegrams. She was indulging in what looked like a flirtation with the\nstation-master, in the millinery department of the shop. Mr. Goddard\ncalled her away from her companion.\n\n\"Susy Lizzie,\" he said, \"come here. I want you.\"\n\n\"Is it stamps?\" she asked, \"or is it a postal order?\"\n\n\"It's neither the one nor the other,\" said Mr. Goddard; \"it's a\ntelegram. In fact, it's as many telegrams as will keep you busy for the\nrest of the afternoon, so there won't be any use the station-master\nwaiting for you till you've done.\"\n\nSusy Lizzie tossed her head and walked defiantly down the shop to her\nproper counter. She established herself behind a sort of wire screen\nand pushed a sheaf of telegraph forms towards Mr. Goddard.\n\n\"Take those things away,\" he said. \"I'm coming inside there to watch\nyou send off my wires and to wait for the answers.\"\n\nSusy Lizzie by way of reply drew Mr. Goddard's attention to a printed\nnotice which forbade members of the general public entering the inner\nprecincts of the office.\n\n\"If you go on with any more of that nonsense,\" said Mr. Goddard,\n\"I'll tell your uncle the way I found you this minute with the\nstation-master.\"\n\nIt was one thing to be bullied by Miss Blow, quite another thing to\nendure the official insolence of Susy Lizzie. Mr. Goddard felt that he\nwas man enough to make a stand against that.\n\n\"There'd be nothing to tell if you did,\" said Susy Lizzie.\n\n\"Wouldn't there? I suppose you'd try and make out he was talking to you\nabout buying a new hat for his mother, and you were showing him the way\nit ought to be trimmed.\"\n\nSusy Lizzie grinned. Then she removed the official notice of no\nadmission, lifted up a slab of her counter, and invited Mr. Goddard\nto enter. The first telegram he handed her completely restored her\ngood-humour. It was a message of a delightfully exciting kind.\n\n\"To the Inspector-General of Police, Dublin Castle. Your Members of\nParliament are unfortunately lost. Am making inquiries. Goddard,\nDistrict Inspector.\"\n\nAfter this came eight much longer messages. They were addressed to\ndetective sergeants in Derry, Larne, Belfast, Greenore, Dublin,\nKingstown, Rosslare, and Queenstown. They contained inadequate\ndescriptions of the appearance of Mr. Dick and Mr. Sanders; and an\nearnest request that all steamers leaving for America, Scottish, or\nEnglish ports might be watched. While Susy Lizzie was tapping her\nway through these messages, Mr. Goddard unrolled an ordnance survey\nmap, and made a list of all the railway stations within fifty miles\nof Clonmore. He then wrote out a series of messages to the police\nsergeants of these places, directing them to make inquiries at the\nrailway stations about all strangers who had left by train that day. He\nappended, for the further guidance of the police, his descriptions of\nthe Members of Parliament, and a note to the effect that one of them\nwas riding a lady's bicycle.\n\nSusy Lizzie handed him a reply to his first message.\n\n\"From Under Secretary, Dublin Castle. Inspector-General of police in\nBelfast quelling riot. Wire forwarded. Lord Lieutenant anxious to know\nwhether ladies of party safe.\"\n\nMr. Goddard replied at once, \"Ladies perfectly safe, but anxious and\ntearful.\"\n\nSusy Lizzie, by this time in a state of extreme excitement and perfect\ngood temper, set to work again at the seaport messages. She had got\nas far as Greenore when she had to stop to receive another incoming\nmessage.\n\n\"From Chief Secretary, Dublin Castle. Please explain first wire.\nUnintelligible here.\"\n\nThis was followed immediately by one from Belfast.\n\n\"Inspector-General supposes mistake in transmission of wire. Kindly\nrepeat.\"\n\nMr. Goddard wrote out two replies.\n\n\"To Chief Secretary, Dublin Castle. Members of Parliament completely\nlost during night. Deeply regret occurrence. Goddard, District\nInspector.\"\n\n\"To Inspector-General of Police, Belfast. First message probably\ncorrectly transmitted. Members of Parliament cannot be found. Goddard,\nDistrict Inspector.\"\n\nSusy Lizzie, working at high speed, got these messages despatched and\nwent on with those to Dublin, Kingstown, Rosslare, and Queenstown. Then\nMr. Goddard handed her a bundle of forms which contained his appeals\nto the police at the railway stations. He felt that he had done all\nthat could be done to discover the escaped gentlemen. It is impossible,\nas Mr. Goddard knew, to get out of Ireland without going either to\nGreat Britain or America. Derry and Queenstown were the ports of exit\nwestwards. If, as was far more likely, the fugitives had made a rush\nfor England or Scotland, he would get news of them at one of the other\nplaces. It was possible, of course, that they were still loitering\nabout Ireland. In that case he would hear of them from one of his\nrailway stations. Even the most energetic Member of Parliament would\nnot be likely to do more than fifty miles on his bicycle over west of\nIreland roads, and Mr. Sanders was afflicted with a weak heart.\n\n\"Isn't there some way of getting from the hotel to the shop,\" said Mr.\nGoddard, \"without going out into the street? I want to speak to your\nuncle.\"\n\n\"There is surely,\" said Susy Lizzie. \"If you step across to the grocery\ncounter, the young gentleman that's there will show you the door.\"\n\nThe door, as Mr. Goddard found when the young gentleman opened it for\nhim, led directly to the hotel bar. Jimmy O'Loughlin was serving out\nbottles of porter to about a dozen customers. There was a babble of\ntalk, which ceased abruptly as Mr. Goddard entered.\n\n\"Jimmy,\" he said, \"I want to speak to you for a minute.\"\n\n\"Affy Ginnetty,\" said Jimmy, \"come here and attend the bar.\"\n\nThe young gentleman who had opened the door for Mr. Goddard left the\ncare of the bacon, flour, and tobacco which strewed his counter, and\ntook his place behind the bar. Jimmy led the way to the ironmongery\ncorner of his shop.\n\n\"We'll have this place to ourselves,\" he said. \"There's nobody comes\nto buy them things\"--he indicated an assortment of lamps, pots, and\nrat-traps--\"unless it would be of a fair day.\"\n\n\"Jimmy,\" said Mr. Goddard, \"where are the ladies?\"\n\n\"There's two of them,\" said Jimmy, \"that's in their beds.\"\n\n\"In their beds?\"\n\n\"I suppose it's in them they are. Anyway, they said they were going to\nlie down, and Bridgy brought up a can of hot water apiece for them, and\nI didn't see them since. There was talk,\" he added, \"of their being up\nand dressed again to be down ready at the barrack at four o'clock,\nthat being the hour at which they were expecting Sergeant Farrelly to\nbe back.\"\n\n\"Those,\" said Mr. Goddard, \"are probably Mrs. Dick and Mrs. Sanders.\"\n\n\"They might be.\"\n\n\"And what about the other two?\"\n\n\"There's one of them that's writing letters above in the drawing-room.\nShe sent Bridgy for twopenny-worth of paper and envelopes, and I gave\nher the loan of a bottle of ink and a writing pen. I heard her say\nshe'd be down along with the other two at the barrack at four o'clock\nto see the sergeant.\"\n\n\"That's most likely to be Miss Farquharson.\"\n\n\"I wouldn't say,\" said Jimmy, \"whether the sergeant would be back at\nfour.\"\n\n\"He will not, nor yet at five.\"\n\n\"I was thinking as much.\"\n\n\"Where's Miss Blow?\"\n\n\"The doctor's young lady,\" said Jimmy, \"is down at the barrack along\nwith Moriarty.\"\n\n\"I'm glad to hear that.\"\n\n\"She wasn't as much put out as you'd expect,\" said Jimmy, \"when she\nfound she couldn't go on the bicycle. I was thinking she might be in a\nbad way and that maybe she'd be too much for Moriarty, so I sent Bridgy\ninto the yard to her, not caring to go myself----\"\n\n\"You were right there,\" said Mr. Goddard.\n\n\"Bridgy told me after, that she never saw in all her born days anything\nto equal the state that Moriarty had the tyre in. 'You could have run\nthe wisps of it,' she said, 'through the teeth of a fine comb.'\"\n\n\"Get Bridgy,\" said Mr. Goddard. \"I'd like to hear the story from\nherself.\"\n\n\"I'm not sure could I get her, for she's busy with the pig's food and\nwould be wanting to clean herself before she'd come. But there's no\nneed anyway, for I can tell you the way she quietened the doctor's\nyoung lady as well as she'd tell it herself, and maybe better, for she\nmight be backward in speaking out before a gentleman. 'The sergeant bid\nme say,' says Bridgy, 'that he's off this quarter of an hour, and has\ntook Constable Cole along with him.' Only for Moriarty being there and\nlistening to her Bridgy says the young lady would have cursed awful\nat hearing that. 'Wild horses,' says Bridgy, 'wouldn't have held the\nsergeant back from going, he was that set on catching the blackguards\nthat ill-treated the poor doctor.' Bridgy says the young lady seemed\nmore pacified at them words. 'The sergeant,' says Bridgy, 'is a\nterrible man when his temper is riz, and riz it is this day if ever\nit was. I'd be sorry for the man that faces him.' And so she went on\ntelling stories about the sergeant and Constable Cole that would make\nyou think they were the blood-thirstiest villains in Ireland, and that\nnothing delighted them only getting a hold of murderers and the like.\"\n\n\"And did Miss Blow believe all that?\"\n\n\"I'm not sure did she, but it quietened her. She didn't say a word to\nMoriarty, good nor bad, but she went out into the street and she asked\nthe first three men she met where was Sergeant Farrelly and Constable\nCole.\"\n\n\"Well, and what did they tell her?\"\n\n\"They told her the truth, of course. They told her they'd seen the\nsergeant and Cole going off on their bicycles back west in the\ndirection of Rosivera. I'm thinking that after that she began to be\nof opinion that there was, maybe, some truth in what Bridgy had been\nsaying. Anyway, she went off down to the barrack, and Moriarty after\nher, and she's there yet.\"\n\nSusy Lizzie emerged from behind the wire screen of the post-office\ncounter while Jimmy O'Loughlin was speaking. She held in her hand a\nbundle of telegraph forms. Her uncle caught sight of her.\n\n\"What is it you're wanting, Susy Lizzie?\" he said. \"Why wouldn't you\nstay where you're paid to stay and attend to the wants of the public?\"\n\n\"It's a dozen wires and more,\" said Susy Lizzie, \"that's after coming\nfor Mr. Goddard, and I thought maybe he'd like to get them at once.\"\n\n\"You thought right,\" said Mr. Goddard. \"Hand them over.\"\n\nThe first he looked at was from Derry.\n\n\"Two men, answering the description given, but without bicycles, left\nthis last night at eight P.M., on steamer _Rose_ for Glasgow.\nHave wired Chief of Police there.\"\n\n\"That'll hardly be them,\" said Jimmy O'Loughlin, who was looking over\nMr. Goddard's shoulder.\n\n\"It can't possibly be them,\" said Mr. Goddard. \"I defy them to get\nfrom this to Derry between twelve o'clock yesterday and eight in the\nevening. I hope to goodness those silly asses won't go and arrest two\ntotal strangers and dump them down on us here. We have worry enough\nwithout that.\"\n\n\"I don't know,\" said Jimmy, \"if so be the men they got were decent sort\nof men, it might be----\"\n\n\"What's that you're saying?\" said Mr. Goddard.\n\nHe had glanced at five more telegrams which came from the police who\nlived in towns where trains stopped. They all denied any knowledge of\nthe Members of Parliament.\n\n\"I was saying,\" said Jimmy, \"that maybe since the gentlemen that's\nwanted can't be found, the ones they'd be sending over to us from\nGlasgow might be some comfort to the ladies. Susy Lizzie, what are you\ndoing standing there with a grin on your face that a man could post a\nletter through? Aren't you ugly enough the way God made you without\ntwisting the mouth that's on you into worse than it is? Get back with\nyou and mind the telegraph machine. I hear it ticking away there as if\nthe devil was in it. Be off with you now, and don't let me be obliged\nto speak to you twice. I was saying, Mr. Goddard----\"\n\n\"I heard what you were saying,\" said Mr. Goddard, \"and I never heard\ngreater nonsense in my life.\"\n\nHe was scattering more pink papers on the floor as he spoke. Every\nrailway station about which he inquired had been drawn blank.\n\n\"Do you suppose,\" he went on, \"that a lady like Mrs. Dick, who has been\ncrying the whole day because she's lost her husband, would take up\nstraight off with any strange man the police happened to send her over\nfrom Glasgow? Have some sense, Jimmy.\"\n\n\"It's them ones that cries the most,\" said Jimmy, \"that is the quickest\nto get married again if so be there's anybody willing to take them.\"\n\n\"I don't deny that,\" said Mr. Goddard; \"but, hang it all! you must give\nher time to make sure that the first one's really dead. I don't believe\nhe is myself. Damn it all! look at this.\"\n\nHe held out a telegram, the last of his batch, to Jimmy O'Loughlin.\n\n\"From Inspector-General of Police. Matter of disappearance of Members\nof Parliament serious. Keep news out of papers if possible. Am leaving\nBelfast to-night. Shall reach Clonmore to-morrow noon. Meet train and\nreport.\"\n\n\"I don't know,\" said Jimmy, \"will he be expecting to stay the night at\nthe hotel, for if he is there's no place for him to sleep. There's the\ndoctor's young lady has the big front room, and yourself and the other\nthree ladies has all the rest of the bedrooms there is in it, and the\nlike of a high-up man such as the Inspector-General will be looking for\nsomething better than a sofa in the drawing-room. I don't know either\nis Bridgy fit to cook the sort of dinner he'd be accustomed to.\"\n\n\"I don't care a straw where he sleeps. What I'm thinking about is the\nabominable fuss there'll be when he arrives. One comfort is he won't be\nable to find these wretched Members of Parliament any more than I can\nmyself.\"\n\n\"He might, then,\" said Jimmy hopefully.\n\n\"He will not; but I tell you what he'll do. He'll find out about the\ndoctor being gone and Patsy Devlin. Then he'll come to the conclusion\nthat there's some sort of a conspiracy on foot in the country. He'll\ndraft in a lot of extra police, and he'll have the life worried out of\nme. Look here, Jimmy, I can't stand much more of this sort of thing\nto-day. You'll have to keep those women off me somehow. I'll count on\nyou to do it. I shall go and shut myself up in the telegraph office\nalong with Susy Lizzie, and if you let one of them in on me I'll never\nforgive you. Let them fight it out with Sergeant Farrelly when he gets\nback. Let Cole try them with another stratagem if he likes. All I ask\nis to have them kept off me.\"\n\n\"I'll do the best I can,\" said Jimmy; \"and if you send Susy Lizzie into\nthe hotel any time, Bridgy'll give her a cup of tea for you. You'll be\nwanting it.\"\n\n\n\n\nCHAPTER XXI\n\n\nAt four o'clock Miss Farquharson, Mrs. Dick and Mrs. Sanders went down\nto the police barrack. They found Miss Blow seated by herself in the\nmen's day room. Constable Moriarty was digging potatoes in the garden\nat the back of the house. He had been questioned and cross-questioned\nby Miss Blow for more than an hour after he had completed the\ndestruction of the bicycle tyre. He felt jaded and nervous. He stood on\nthe brink of a frightful exposure. A trifling accident, an incautious\nword, might at any moment betray the part he had played in Constable\nCole's stratagem. Some men under the circumstances would have steadied\nthemselves with whisky, but Moriarty was a strict teetotaller. Others\nwould have smoked pipe after pipe of strong tobacco. Moriarty, much\nwiser, went out and dug potatoes. There is nothing more soothing\nto racked nerves than digging in the ground, and there is a mild\nexcitement about driving a spade into potato ridges which distracts the\nmind from painful thoughts and terrifying anticipations. The turning up\nof the roots of any particular plant may display an amazing wealth of\ntubers, may expose to the gaze of the delighted gardener some potato of\nhuge size or very unusual shape. You cannot tell beforehand what will\nbe unearthed. Expectations and hope run high. There is also present a\ncertain fear. It is always possible, unless you are very skilful at the\nwork, that a spade may slice a potato, leaving you face to face with\ntwo reproachful, earth-soiled, flat surfaces, useless for the pot, a\nmanifest disgrace to your digging. Moriarty was not a highly skilled or\nvery experienced potato digger. He enjoyed to the full the pleasures of\nanticipation. He suffered from the bitterness which follows mistakes.\nHe almost forgot Miss Blow and the torment of her questioning.\n\nThen, at a quarter past four, Miss Blow's voice brought him back from\nhis security. She called him by name from the back door of the barrack.\nMoriarty scraped the clay off the sides of his boots, shuffled on his\ncoat, and gave his hands a rub on the front part of the legs of his\ntrousers. Then he joined Miss Blow and the other ladies in the day room.\n\n\"Why isn't Sergeant Farrelly back?\" asked Miss Blow.\n\n\"I don't know, miss.\"\n\n\"It's after four,\" said Miss Blow.\n\n\"Well, now,\" said Moriarty, \"that's a queer thing, so it is.\"\n\n\"He started, I am told, at about half-past twelve.\"\n\n\"It might have been that,\" said Moriarty, \"or it might have been more.\nI didn't take notice on account of being engaged in mending your tyre\nat the time.\"\n\n\"It's eight miles to Rosivera. Allow three-quarters of an hour to get\nthere----\"\n\n\"It could be,\" said Moriarty, \"that he was punctured on the way. It's a\nbad road.\"\n\n\"Allowing for all possible accidents, even supposing he walked the\nwhole way there and back, they should be here by this time.\"\n\n\"They should. You're right, there, miss. They should.\"\n\n\"Will you,\" said Miss Blow, \"kindly go and fetch Mr. Goddard, your\nofficer? He ought to know that his men have not returned.\"\n\n\"I don't know will I be able to find him.\"\n\n\"Go and try.\"\n\n\"I will,\" said Moriarty. \"I'll go and ask Jimmy O'Loughlin. If anybody\nknows where the officer is, it'll be Jimmy O'Loughlin.\"\n\nJimmy O'Loughlin told Moriarty to go back to the ladies and keep them\namused for a while by showing them over the barrack. Moriarty refused\nto do this. He said that the situation was urgent and critical, and\nthat he would not take the responsibility of dealing with Miss Blow\nwithout help. Jimmy O'Loughlin reluctantly went down to the barrack.\nMiss Blow attacked him at once.\n\n\"Where is Mr. Goddard?\" she said.\n\n\"He's beyond,\" said Jimmy, \"in the post-office, and he left word that\nhe wasn't to be disturbed on account of his being terrible busy.\"\n\n\"He ought to be told that his men have not returned from Rosivera.\"\n\n\"I'd be in dread to go near him. It's telegrams he's sending, telegrams\nto the Lord Lieutenant and more of them high-up gentlemen, and that's\nthe kind of work that would be preying upon a man.\"\n\n\"Sergeant Farrelly left this at a quarter past twelve,\" said Miss Blow;\n\"and he's not back yet.\"\n\n\"It's a long road,\" said Jimmy, \"longer maybe than you'd think.\"\n\n\"It's eight miles.\"\n\n\"And eight back along with that.\"\n\n\"That only makes sixteen,\" said Miss Farquharson, who shared her\nnephew's fondness for intricate calculations.\n\n\"So they ought to be here by this time,\" said Miss Blow.\n\n\"It could be,\" said Jimmy, \"that they might be a long time looking\nfor the gentleman they're after before they'd find him. A fellow like\nthat would be as cunning as an old fox, hiding himself when he saw the\npolice after him.\"\n\n\"Oh,\" said Mrs. Dick, \"I do hope nothing has happened to them. It would\nbe too terrible.\"\n\n\"The sergeant,\" said Jimmy, \"is a heavy man. He wouldn't be as quick as\nanother at doing a run on a bicycle.\"\n\n\"Still,\" said Miss Blow, \"he's had four hours and it's only sixteen\nmiles.\"\n\n\"He'll have to walk back,\" said Jimmy, \"and if so be the prisoner\nwasn't willing to come with him it might be a long time before he got\nhim along the road, for he wouldn't like to be beating him. He has a\nkind heart, the sergeant; it's hardly ever you'd see him as much as\nlaying a stick across a child.\"\n\nMiss Blow seemed more or less satisfied. The idea of Mr. Red driven by\nthreats of violent batoning along a dusty road comforted her. Jimmy\nO'Loughlin made his escape from the barrack.\n\nHe found Mr. Goddard scribbling a fresh telegram in the post-office.\n\n\"Look at that,\" he said, handing a form to Jimmy.\n\n\"From Chief Secretary, Dublin Castle. Why have you not reported\nrecovery of Members of Parliament?\"\n\n\"That's not what I'd call a civil message,\" said Jimmy. \"What did you\nsay to him?\"\n\n\"Susy Lizzie,\" said Mr. Goddard, \"give your uncle the answer I sent to\nthe last telegram but two.\"\n\nSusy Lizzie fumbled among a pile of papers and finally handed one over\nto Jimmy.\n\n\"To Chief Secretary, Dublin Castle. Impossible to report recovery of\nMembers of Parliament with any truth because still at large. Goddard,\nDistrict Inspector.\"\n\n\"Be damn,\" said Jimmy, \"but you had him there, as neat as ever I seen.\"\n\n\"He's not satisfied, though. Look at this.\"\n\nHe handed another form to Jimmy.\n\n\"From Chief Secretary, Dublin Castle. Lord Lieutenant requests\nexplanation of disappearance of Members of Parliament.\"\n\n\"Here's the answer to that,\" said Mr. Goddard. \"Read it out to your\nuncle, Susy Lizzie, before you send it off.\"\n\nSusy Lizzie grinned broadly.\n\n\"Go on,\" said Jimmy, \"and do what the gentleman bids you. You've no\nright to be laughing at the business of the Government.\"\n\n\"Theory current locally,\" read Susy Lizzie, \"that Members of Parliament\nhave deserted their wives and aunt.\"\n\n\"That'll give him his 'nough of telegraphing for this day anyhow,\" said\nJimmy. \"I'd like to see the way he'll be rampaging up and down the\nstairs of the Castle when he gets that. It'll show them fellows that\nyou think mighty little of them anyway, Mr. Goddard.\"\n\n\"How are the ladies getting on?\"\n\n\"I've quietened them for a bit. They were annoyed on account of\nSergeant Farrelly not coming back; but I told them it would take him\na long time to be dragging Mr. Red along the road by the hair of his\nhead, and I didn't think he'd come without. They'll be all right for\nanother hour, anyway. Will I be going into the hotel and telling Bridgy\nto fetch you over a cup of tea? It's after five o'clock.\"\n\nIt was nearly six o'clock before Bridgy came to the shop with a tray\nin her hands. She was followed by Affy, the young gentleman from the\ngrocery department, who carried a loaf of bread and a pot of jam. Mr.\nGoddard and Susy Lizzie began a comfortable meal together. They were\ninterrupted twice by telegrams from Dublin, but they did not allow\nthese to trouble them much. A very much more serious interruption was\ncaused by a breathless whisper from Affy to the effect that four ladies\nwere entering the shop. Mr. Goddard, carrying his teacup with him,\nconcealed himself behind a screen of muffed glass, originally erected\nby the post-office authorities in order that letters might be sorted\nout of sight of the public.\n\n\"Susy Lizzie,\" he whispered, \"run like a good girl and get your uncle.\nTell him that the ladies are here, and that he'll have to come at once\nand pacify them.\"\n\nSusy Lizzie ran. She was enjoying her afternoon immensely. The monotony\nof her life was seldom broken in any way half so agreeable.\n\n\"If the thing is to be done, I'll do it,\" said Jimmy, when he received\nthe message.\n\nHe left his bar and went into the shop. He faced Miss Farquharson, who\nwas asking Affy where Mr. Goddard was likely to be found.\n\n\"I was thinking, my lady,\" said Jimmy, \"that maybe you'd be wanting\na bit of supper. What would you say now to a chop, or a couple of\nrashers and some eggs? I could have them got for you in a minute.\"\n\n\"I want Mr. Goddard, the police officer,\" said Miss Blow, who stood\nbeside the post-office counter.\n\n\"Wouldn't it be better for you now,\" said Jimmy, turning to her, \"to\nbe eating your supper quietly instead of rampaging about the town\nfrightening the wits out of a poor man that's doing his best for you?\nCome now, sure I'm old enough to be your father, and I know what's good\nfor you. It's moidered you are with the trouble that's on you, and\nthere isn't one in the place but is sorry for you and for all the rest\nof the ladies this night. But what's the good of making yourselves sick\nover it, and tormenting the officer? If they're gone, they're gone, and\nall the talking in the world won't bring them back to you.\"\n\n\"Man----\" said Miss Blow.\n\n\"Look at poor Susan Devlin,\" said Jimmy, \"she's lost a husband as well\nas the rest of you, and barring that she might be crying an odd time\nwhen she'd be thinking of him, she's as quiet as a lamb. Why can't you\nbehave yourselves like her? I'm not setting up to teach ladies like\nyourselves what ought to be and what ought not, but I'd say myself that\nthe men that would run away from yez, from the like of yez\"--he spoke\nwith a smile that was meant to flatter--\"isn't worth looking after.\"\n\n\"Do you dare to suggest----\" said Miss Farquharson.\n\n\"It's not the first time that suggestion has been made,\" said Miss\nBlow. \"It's part of the scandalous conspiracy in which every man, woman\nand child in this place is involved.\"\n\n\"If you won't take your supper when it's offered you,\" said Jimmy,\n\"maybe you'll sit inside in the hotel. You're interrupting the business\nof the shop standing where you are, let alone preventing the public\nfrom buying the stamps they have a right to buy to put on their\nletters.\"\n\nThe appeal produced its effect on Miss Blow. She had the blood of a\nbusiness man in her veins. She understood that her father would have\nresented any interference with the sale of his twopenny Beauties, and\nwould not have admitted either grief or anger as legitimate reasons\nfor damming the flow of trade. Also she and her companions belonged\nto a law-abiding race. They had a natural respect for any department\nof the State. They felt it wrong to stop the sale of postage stamps.\nThey trooped into the hotel and sat down in the commercial room. Once\nthere Miss Farquharson's strong common sense asserted itself. She\nsuggested that it might be wise after all to order bacon and eggs,\nquoting the advice given by St. Paul to the sailors who were threatened\nwith shipwreck. Mrs. Dick and Mrs. Sanders declared with tears that\nit was impossible for them to eat in their heart-broken condition.\nThis confirmed Miss Farquharson in her resolve to have a regular meal.\nShe gave the necessary order to Jimmy O'Loughlin. Miss Blow agreed\nto eat on condition that the meal was served in the commercial room,\nthe windows of which looked out on the street. She knew that Sergeant\nFarrelly, Constable Cole, and the prisoner must pass the hotel on their\nway to the barrack.\n\n\n\n\nCHAPTER XXII\n\n\nAt eight o'clock Mr. Goddard, who had enjoyed some fried bacon and a\nbottle of porter in the telegraph office, began to feel surprise at\nthe prolonged absence of Sergeant Farrelly. It was all very well to go\nslowly to Rosivera and to return without undue hurry, but it was hard\nto imagine how eight hours could possibly be occupied in travelling\nsixteen miles. He sent Susy Lizzie, who remained in attendance on him,\nto call her uncle.\n\n\"I'm doing the best I can,\" said Jimmy, \"to keep them quiet; but\nthey'll be out after you in spite of me soon. I left herself and Bridgy\ntalking to them; but what use are they against the four? And the\ndoctor's young lady is the worst of them.\"\n\n\"It's not the ladies I'm bothered about now,\" said Mr. Goddard. \"Why\nthe devil isn't Sergeant Farrelly back? What's keeping him?\"\n\n\"Faith, I don't know, unless maybe he'd be in dread!\"\n\n\"Nonsense. I told him not to hurry, but I didn't tell him to stay out\nall night.\"\n\n\"It's queer, so it is,\" said Jimmy. \"What would you think now of\nsending Constable Moriarty out a bit along the road to look and see if\nthere's e'er a sign of them coming?\"\n\n\"I'll go myself,\" said Mr. Goddard.\n\n\"I wouldn't say,\" said Jimmy, \"but what it might be just as well if you\ndid. The ladies is sure to be here in a couple of minutes now. I can't\nkeep them.\"\n\nMr. Goddard walked a mile along the road towards Rosivera, and then\nsat down on a ditch. There was no sign of Sergeant Farrelly. At nine\no'clock he got up, and walked another mile and sat down again. Still\nthere was no sign of the missing policemen. He walked a third mile and\nonce more waited. He was puzzled and began to be uneasy. He turned and\nwalked back towards Clonmore. Half a mile outside the town he climbed\nthe demesne wall and crossed an angle of the deer park to the Castle.\nIt was a quarter past ten when he reached the door.\n\nHe found Lord Manton in the library with a book in his hand and a glass\nof whisky and soda water on the arm of his chair. There was a pile of\ncigarette ends on the tray which stood within reach.\n\n\"Well, Goddard,\" he said, \"I suppose you've tucked all your ladies up\nin bed and come up here for a quiet smoke. You deserve it.\"\n\n\"No,\" said Mr. Goddard; \"I've come to consult you. I don't\nunderstand----\"\n\n\"Surely nobody else has bolted?\"\n\n\"Sergeant Farrelly and the constable have not returned from Rosivera.\"\n\n\"Dear me! I always heard that these things were infectious, like\nmeasles. One suicide, half a dozen suicides. We appear to be in for\nan epidemic of bolting. But I'd never have suspected the sergeant. He\nseemed such a solid sort of man, not the least hysterical; but you can\nnever tell. I hope you won't vanish to-morrow, Goddard. If you feel it\ncoming on you, you'd better put yourself under arrest at once. Was the\nsergeant a married man?\"\n\n\"No, he wasn't.\"\n\n\"Then there'll be no widow to make lamentation in his case. That's a\ngood job for you, Goddard. I don't see how you could have got on with\nanother woman running round after you. What about the constable?\"\n\n\"He's not married either. He's not long enough in the service.\"\n\n\"Poor fellow!\" said Lord Manton. \"I suppose now that this will ruin his\nprospects, even if he comes back.\"\n\n\"Lord Manton,\" said Mr. Goddard, \"what do you know about that tenant of\nyours at Rosivera?\"\n\n\"Surely you're not coming round to Miss Blow's murder theory, are you?\"\n\n\"I don't know. It's a very queer business. Is Red--that's the man's\nname, isn't it--respectable?\"\n\n\"I don't know. I never saw the man in my life. I know nothing about him\nexcept that he paid his rent and came here in a motor-car. That looks\nas if he had money, doesn't it?\"\n\n\"I don't understand it,\" said Mr. Goddard. \"The sergeant and Cole\ncertainly went to Rosivera to-day. I don't see what was to stop them\ngetting there. They weren't likely to lose their way. They certainly\nhaven't come back.\"\n\n\"What does Jimmy O'Loughlin say? I suppose you've consulted him.\"\n\n\"Jimmy thinks,\" said Mr. Goddard, \"that the sergeant's afraid to come\nback on account of Miss Blow. But that's all rot, of course.\"\n\n\"I'm not at all sure that Jimmy's not right. He's a very shrewd man,\nJimmy O'Loughlin. I shouldn't wonder a bit----\"\n\n\"But what the deuce am I to do?\"\n\n\"I'll tell you what it is,\" said Lord Manton. \"We'll make a descent\nin force on Rosivera to-morrow. You shall collect all the police you\npossibly can, fifty of them, if there are fifty available. We'll\ntake all the ladies interested in the matter in my waggonette. Jimmy\nO'Loughlin and I will accompany the army in the capacity of civil\nmagistrates, each of us armed with a Riot Act and a writ of Habeas\nCorpus. We'll make that fellow Red sit up if he's been at any games.\"\n\n\"I really think we'll have to. It seems very absurd, but what else are\nwe to do? I've been harried with telegrams from everybody in Dublin\nCastle all day. The Inspector-General is coming down here to-morrow,\nthough I don't see what on earth he thinks he'll be able to do.\"\n\n\"We'll forestall him,\" said Lord Manton. \"He can't possibly get here\nbefore noon. We'll start at eleven A.M. sharp. We'll have the mystery,\nwhatever it is, probed to its inmost recesses before he gets at it. The\nwhole credit will be yours, Goddard.\"\n\n\"I'll make any one that likes a present of the credit.\"\n\n\"In the meanwhile,\" said Lord Manton, \"you'd better sleep here. I\nexpect Jimmy O'Loughlin's hotel is pretty well full up.\"\n\n\"Thanks,\" said Mr. Goddard. \"The fact is, I don't particularly care\nabout going back to the village to-night.\"\n\n\"Will they be waiting up for you?\"\n\n\"They will.\"\n\n\"Miss Blow,\" said Lord Manton, \"is a wonderful woman.\"\n\n\"She's not bad looking,\" said Mr. Goddard magnanimously, \"but she's\nrather----\"\n\n\"I know what you're going to say--vehement, wasn't that it? A good deal\nof life force about her? I quite agree with you. Now, what do you\nthink? Supposing it turns out that the man Red has really been up to\nany kind of tricks; supposing he's engaged in a business of kidnapping\ndispensary doctors, blacksmiths, Members of Parliament and policemen,\nfor the purpose of shipping them off as slaves to the Sultan of\nZanzibar. I don't say for certain that that's exactly what he's doing.\nI don't know yet. But if he's at anything of the sort it would serve\nhim jolly well right if we made him marry Miss Blow.\"\n\n\"He wouldn't do it.\"\n\n\"He might. He seems to be a man of adventurous disposition. If he\ndoesn't like her, we could offer him Miss Farquharson.\"\n\n\"I should think he'd refuse them both. He'd see Miss Farquharson, and\nhe'd be absolutely certain to hear Miss Blow.\"\n\n\"I think we could put pressure on him,\" said Lord Manton. \"In fact, it\nwould be a choice for him between marrying and a criminal prosecution.\nA man can't kidnap people in this wholesale sort of way without\nsuffering for it. He'll understand that when we put it to him. But\nyou've had an exhausting day, Goddard, and I dare say you'd like to\ntoddle up to bed. Wilkins will leave you whatever you want. We'll have\na great time to-morrow, raiding Rosivera.\"\n\n\n\n\nCHAPTER XXIII\n\n\nMiss Blow was very angry when she discovered that Mr. Goddard was not\nin the telegraph office. Nothing Jimmy O'Loughlin said soothed her in\nthe least. He pointed out that the officer's absence was caused by his\nexcessive zeal for the cause they all had at heart; that he had in\nfact gone in person to investigate the mystery of Rosivera. Miss Blow\nrefused to believe him, and expressed her contempt for habitual liars\nin plain language.\n\n\"If you don't believe me, miss, ask Constable Moriarty, and he'll tell\nyou the same.\"\n\n\"I shouldn't believe him either,\" said Miss Blow; \"and in any case\nConstable Moriarty is a fool.\"\n\n\"He was standing by,\" said Jimmy, pursuing the subject without\nregarding the interruption, \"the same as it might be yourself, and he\nheard every word that passed between us. 'Mr. Goddard,' I says, 'them\nladies is in a terrible state, and getting worse. It's hardly ever\nthey were able to take the cup of tea I had wetted for them.' 'I know\nit, Jimmy,' says he; 'I know it well; and if cutting off my right hand\nwould be any ease to them I'd do it this minute.' 'You would,' I says,\nfor I knew well the way it was going through him; 'but what good would\nthat be? Wouldn't it be better now if you was----' 'If I was to what?'\nsays he, catching me up like. 'If you was to send Moriarty for a bit of\na stroll along the road,' I says, 'to see could he hear any news of\nthe sergeant.' 'I'll not do it,' says he, looking terrible determined.\n'I'll not do the like. What's Constable Moriarty but a boy, Jimmy? A\nboy with maybe a mother breaking her heart after him somewhere. If\nthere's murder going,' says he, 'it's not Constable Moriarty it'll\nlight on, but myself.' 'Do you mean that?' says I. 'I do,' he says; 'I\nmean it; I'll go myself. What murder's done to-night will be done on\nme, for I'll not take it on my conscience to be the cause of Moriarty's\ndeath.' 'Mr. Goddard,' says I, 'you're a fine man. I'll give it in\nto you that you're as brave as any one ever I met; but you'll be\ntaking your sword with you, promise me that now.' 'I will, Jimmy,'\nsays he; 'I'll take my sword, and I'm thankful to you for making the\nsuggestion.' And with that, miss, he was up and off out of the door,\nand that was the last I seen of him.\"\n\n\"If you hadn't been telling me lies ever since I've been in Clonmore,\"\nsaid Miss Blow, \"there'd be some chance of my believing you now.\"\n\n\"There's a man going past the door of the hotel this minute,\" said\nJimmy, \"with an ass load of turf that he's after fetching in off the\nbog. Will you go out now and ask him did he meet Mr. Goddard on the\nroad? Maybe you'll believe him when he tells you.\"\n\nMiss Blow accepted the challenge. She waylaid the man with the donkey,\nwho proved to be very deaf. She raised her voice and shouted at him. He\nreplied in a low tone. She shouted again, and the man made what seemed\na long answer. Miss Blow returned to Jimmy O'Loughlin.\n\n\"You've told me the truth for once,\" she said ungraciously. \"That man\nmet the officer a mile out of town on the road to Rosivera.\"\n\nJimmy was generous. He did not attempt to humiliate Miss Blow. He\npursued his policy of trying to soothe her. An hour passed. Two hours\npassed. Even Miss Blow's anger began to give way to anxiety. What if\nMr. Goddard was himself a victim to the mysterious gang which had\nalready made away with seven men? He might have gone the whole way to\nRosivera. He might have fallen into some craftily arranged ambush on\nthe road. Fear laid hold on her heart. Jimmy O'Loughlin, who was a\nlittle puzzled but not particularly anxious, seized his opportunity.\n\n\"It'll be better for you,\" he said, \"to go to your bed, you and the\nrest of the ladies, where you'll be safe till the morning. You'll\nhardly be expecting any other man to be going out into the darkness of\nthe night, risking his life maybe, to satisfy you. Not but what there's\nmany a one would do it. I'd do it myself if I saw any good would come\nout of it.\"\n\nMiss Blow and Miss Farquharson consulted together anxiously. This\nfresh disaster had gone a long way towards cowing them. They were not\nprepared to insist on the sacrifice of more human life. Mrs. Dick wept\nnoisily and unrebuked. Mrs. Sanders became very white and her hands\ntrembled.\n\n\"You'll be safe in your beds anyway,\" said Jimmy. \"You can turn the\nkey in the door of every room in this house barring the one I sleep in\nmyself, for the lock of it is gone wrong on me, and since poor Patsy\nDevlin went from us there isn't a man about the place fit to settle it.\"\n\n\"Let us go,\" said Mrs. Dick, sobbing. \"I want to be somewhere at peace.\nI don't care---- Oh, poor Richard!\"\n\n\"You're right, ma'am,\" said Jimmy; \"it'll be better for you. I have a\ndog in the yard that'll bark fit to wake the dead if e'er a one comes\nnear the house during the night; and I'll leave word with Bridgy that\nshe's to waken you in good time if so be anything was to happen; but\nwith the help of God there'll be no need for that.\"\n\n\"Come,\" said Miss Blow at last; \"we can do no more to-night. Let us\nget some sleep, if any of us are able to sleep.\"\n\n\n\n\nCHAPTER XXIV\n\n\nIt was three o'clock in the afternoon of the day which followed their\ncapture, and the two Members of Parliament showed no signs of becoming\nreconciled to their situation. Mr. Sanders grumbled and occasionally\nswore. Mr. Dick passed from bursts of violent rage to fits of\nlamentation over the desolate condition of Mrs. Dick. Dr. O'Grady and\nPatsy Devlin bore with them patiently for a long time. But there are\nlimits to human endurance. After a consultation with Patsy, the doctor\nundertook to speak seriously to the unreasonably afflicted men. The\nbearded anarchist who usually attended to the wants of the prisoners,\ncarried off the dinner things. Dr. O'Grady pulled Mr. Dick's bed out to\nthe middle of the floor.\n\n\"Now,\" he said, \"sit down on that, the two of you in a row, till I try\nif I can't talk sense into you.\"\n\n\"Why,\" said Mr. Sanders sulkily, \"why should we sit there and be talked\nto by you?\"\n\n\"There are two reasons why you should,\" said Dr. O'Grady. \"The first is\nbecause I want to talk to you, and I can do that much more conveniently\nif you're seated in a row in front of me than if you're scattered about\nall over the room. Does that satisfy you, or must I give you the second\nreason?\"\n\n\"I won't be talked to by you,\" said Mr. Dick. \"You're in league with\nthe infernal scoundrels who have locked us up here.\"\n\n\"My good man----\" said Mr. Sanders.\n\nThis pacific form of address produced no more effect on Dr. O'Grady\nthan Mr. Dick's blunt denunciation did. Mr. Sanders was given no time\nto finish his remark.\n\n\"The second reason why you should is because it will be the worse for\nyou if you don't.\"\n\n\"Do you mean to threaten us with violence?\" said Mr. Dick.\n\n\"Patsy,\" said Dr. O'Grady, \"take off your coat, roll up your sleeve\nand show your arm. I may mention, gentlemen, that Patsy Devlin was a\nblacksmith by trade before he took to being a captive. He's used to\nhammering.\"\n\nMr. Dick and Mr. Sanders watched Patsy Devlin bare his arm, but they\nmade no move towards the bed.\n\n\"Patsy,\" said Dr. O'Grady, \"roll up your other sleeve. If you want to\nfight, gentlemen, I'd recommend you to take off your coats.\"\n\n\"I can't fight,\" said Mr. Sanders, \"on account of my heart. It's weak,\nand the doctor expressly forbade any form of excitement or violent\nexercise. If it wasn't for that----\"\n\nHe sat down on a corner of the bed.\n\n\"I haven't fought for years,\" said Mr. Dick with spirit; \"but I'm not\ngoing to be bullied by a damned Irish doctor. Come on.\"\n\nThe speech was worthy of a man who had once at least felt the blood of\nthe ancient Bersekers coursing through his veins. It was followed by\nprompt action. He took off his coat. Patsy Devlin spat on his hands and\nthen rubbed them together.\n\n\"Mr. Dick,\" said Dr. O'Grady, \"I don't want to have you smashed up,\npartly because from what you've told us about your wife I expect she'd\nbe sorry, and partly because it'd be a nuisance to me to have to fit\nyou together again. If you don't sit down on that bed, I'll ring the\nbell for the Emperor and get him to take away your clothes. You didn't\nlike going about in nothing but your shirt yesterday evening. Just\nrecollect that, and be careful.\"\n\nMr. Dick thought better of his resolution. It may have been the\nbusiness-like way in which Patsy spat on his hands which daunted him,\nor it may have been Dr. O'Grady's second and very horrible threat. He\nsat down sulkily beside Mr. Sanders.\n\n\"Now,\" said Dr. O'Grady, \"I'm going to speak plainly to you for your\nown good. You've behaved uncommonly badly since you came here. You've\nsulked and you've whimpered, and you've raved in such a way as to make\nPatsy Devlin and me quite uncomfortable. We made every excuse for you\nyesterday afternoon. We recognized that Mr. Dick couldn't be expected\nto be cheerful when he hadn't got any trousers. We knew that what had\nhappened would upset Mr. Sanders' heart and bring on palpitations. We\ndid our best to make things easy for you. When the assistant anarchists\nbrought up your beds, we made them for you, and allowed you to get into\nthem, although it was barely five o'clock, and the habit of going to\nbed at that hour is most unsociable. Patsy brought your tea over to you\nlater on before he drank a drop himself, to save you the trouble of\ngetting out of bed. That was pure kindness of heart on Patsy's part,\nand you didn't so much as say 'thank you.'\"\n\n\"They did not,\" said Patsy, who stood behind the bed with sleeves\nstill rolled up; \"and it will be long enough before I do the like\nagain.\"\n\n\"After tea,\" said Dr. O'Grady, \"I sent for the Emperor and persuaded\nhim to let Mr. Dick have his clothes back. I needn't have done that. It\ndidn't matter to me if he had to go about stark naked for the rest of\nhis natural life. Did you show any gratitude this morning? Not a bit.\nYou sulked and whimpered again in the most unbearable manner. We put up\nwith it. We tried to cheer you. There was an egg short at breakfast.\nWho did without? Patsy again; although he deserved an egg a great deal\nmore than either of you. When breakfast was over, I suggested that we\nshould all join in the game of flipping pennies across the table. I\ndidn't do that because I wanted to play. As a matter of fact, I don't\nusually play in the morning; I read. Patsy will bear me out in that. I\nwas prepared to sacrifice my own time and inclination to amuse you. How\ndid you receive the proposal? You sulked again.\"\n\n\"I was brought up strictly,\" said Mr. Sanders; \"I belong to the Free\nKirk, and my conscience will not allow me to gamble.\"\n\n\"That,\" said Dr. O'Grady, \"is a paltry excuse, which you ought to be\nashamed to make. No man could be brought up to think it wrong to flip\npennies. Besides, what is it that's wrong about gambling? It's the\nexcitement created by the element of risk associated with all gambling.\nNow, in this case, as you know perfectly well, there would have been\nno excitement, because there was no element of risk. The thing was a\ndead certainty. Patsy would have given you ten points in every game and\nbeaten the head off you. If you had to make an excuse, why didn't you\ntrot out your weak heart again? That would have been more reasonable.\nWhen you wouldn't play that, I proposed another game, called Moggy, at\nwhich you wouldn't have had a chance of winning either. You refused it\ntoo. Then I said that if you liked we'd have a debate on Home Rule or\nTariff Reform. I said that you two could choose your own side and that\nPatsy and I would take the other, whatever it was. I thought that would\ninterest you. It would have bored both Patsy and me frightfully; but we\nwere prepared to put up with that for your sakes.\"\n\n\"How could you expect us to take an interest in Tariff Reform,\" said\nMr. Dick, \"when our minds were full of----\"\n\n\"Patsy,\" said Dr. O'Grady, \"if that man mentions his wife again, hit\nhim with the flat of your hand on the side of the head. Now go on, Mr.\nDick.\"\n\n\"When--when we were imprisoned,\" said Mr. Dick.\n\n\"That ought not to have stopped you,\" said Dr. O'Grady. \"The great\nproblems of your country's welfare ought to come before considerations\nof your personal convenience. Think of the ancient Romans. Remember\nHoratius, and Coriolanus, and the fellow that jumped into the hole in\nthe Forum. That's the way you ought to be behaving instead of grumbling\nand growling. Now I'm going to give you another chance. If you choose\nto behave like reasonable human beings, well and good. Patsy and I will\ndo everything in our power to make your stay here pleasant for you. If\nyou won't, you shan't have a bite or sup until you do. I'll give you\nthree minutes by my watch to make up your minds. Will you or will you\nnot be sociable and pleasant? Will you join us in a game of Hunt the\nSlipper? Come over to the window, Patsy, and leave them to make up\ntheir minds together.\"\n\nPatsy and Dr. O'Grady stood looking out at the yard; the doctor held\nhis watch in his hand; there was dead silence in the room.\n\n\"One minute gone,\" said Dr. O'Grady.\n\nThere was a sound of whispering which ceased abruptly.\n\n\"Two minutes gone.\"\n\nThere was more whispering. Then Mr. Sanders spoke.\n\n\"Don't choose such a silly game,\" he said. \"We can't play Hunt the\nSlipper. We really can't. Dick says he'd rather starve.\"\n\n\"I'm not particular about what game you choose,\" said Dr. O'Grady,\n\"so long as it's a possible game. There's no use your saying golf or\ncricket or lawn tennis, because we've no way of playing them here. You\ncan have Drop the Handkerchief, if you like, or Oranges and Lemons.\"\n\nThis time there was a good deal of whispering, a kind of debate\nconducted with great earnestness.\n\n\"I'll play Hop Scotch, if you like,\" said Mr. Sanders; \"but Dick won't.\"\n\n\"Never mind about Mr. Dick,\" said the doctor. \"He'll join in when he\nsees how pleasant it is. You can play Hop Scotch, I suppose, Patsy?\"\n\n\"I cannot,\" said Patsy. \"I never heard tell of it.\"\n\n\"I can't either,\" said the doctor. \"But it'll be all right: Mr. Sanders\nwill teach us.\"\n\n\"I shall want a piece of chalk,\" said Mr. Sanders.\n\n\"There's no chalk here,\" said Dr. O'Grady. \"Will nothing else do you?\nWhat's the chalk for?\"\n\n\"I have to mark out a figure on the boards,\" said Mr. Sanders.\n\n\"I've a small bottle of Condy's Fluid in my bag,\" said the doctor,\n\"that I carry about with me for disinfecting my hands. You could manage\nwith it, I dare say.\"\n\n\"You can stain the floor with Condy's Fluid,\" said Mr. Dick, who was\nreally a sociable man, and was beginning to be interested in the\nproceedings. \"When we were first married and went into our new house,\nmy wife----\"\n\n\"Now, don't start talking about your wife,\" said Dr. O'Grady, \"just as\nyou're beginning to cheer up. You'll upset yourself again.\"\n\nMr. Sanders went down on his hands and knees. He made a little pool of\nCondy's Fluid on the floor and drew lines from it with his forefinger.\nThe rest of the party watched with great interest. Suddenly he stopped\nand knelt bolt upright.\n\n\"What's that noise?\" he said.\n\n\"I didn't hear any noise,\" said Dr. O'Grady. \"There wasn't any noise.\nGo on Hop Scotching.\"\n\n\"There was a noise. I heard it. A noise like a fall. I have very sharp\nhearing.\"\n\n\"That always goes with a weak heart,\" said Dr. O'Grady. \"But----\"\n\nHe stopped abruptly. This time there was an unmistakable noise, a shout\nuttered somewhere in the lower part of the house, which reached even\nthe remote room where the captives were. They drew together and waited,\nbreathless. Mr. Sanders grew very white.\n\n\"They're fighting downstairs,\" he said.\n\n\"Perhaps the police have come,\" said Mr. Dick. \"Perhaps we shall be\nrescued. I knew that my wife would do everything to find me. I knew she\nwould find me.\"\n\n\"You didn't seem to think so yesterday,\" said Dr. O'Grady. \"And I\nwouldn't make too sure now, if I were you. If they are fighting\ndownstairs, and I can't hear plainly enough to be certain----\"\n\n\"I can,\" said Mr. Sanders.\n\n\"I expect,\" said Dr. O'Grady, \"that the Emperor will get the best of\nit. He'll probably be in on us here in a minute or two as proud as\nPunch at having bagged another chance traveller. If the Emperor has a\nfault at all, it's his extraordinary fondness for kidnapping people. I\ncan't think why he does it.\"\n\n\"They're coming upstairs,\" said Mr. Sanders. \"I hear them distinctly.\"\n\n\"They are,\" said Dr. O'Grady; \"a whole lot of people. I wonder who he\nhas got this time.\"\n\nThe door of the room was unlocked. There was a short scuffle outside,\nand then Sergeant Farrelly and Constable Cole were thrust in. They both\nlooked as if they had been roughly handled. The sergeant's tunic was\ntorn, his right eye was beginning to swell, and there was blood on his\nlower lip. Mr. Red, looking very grim and determined, stalked into the\nroom behind them.\n\n\"Emperor,\" said Dr. O'Grady, \"this is too much. I complained to you\nyesterday about the habit you have got into of thrusting strange people\nin on top of me and Patsy. I put up with the last two, but this is more\nthan I am going to stand.\"\n\n\"They remain here,\" said Mr. Red, \"as captives.\"\n\n\"Not at all,\" said Dr. O'Grady. \"Think it over, Emperor, and\nyou'll come to see that you can't possibly leave them here. You\nare an anarchist, an anti-military anarchist. You've often told\nme so yourself. Now, an anarchist, as I understand his position,\nis absolutely pledged to every kind of social reform. Whatever\nanybody else may do, an anarchist can't consistently go in for the\nover-crowding of tenement houses, or tolerate insanitary prisons. You\nsee that, don't you? If ever it got out that you'd shut up six men in\none room and kept them there, your reputation would be gone. There\nwouldn't be a decent anarchist anywhere in the world who'd recognize\nyou as belonging to his party or so much as speak to you. You'd be a\nsort of Suraja Dowlah with a horrible Black Hole of Calcutta story\ncropping up against you at every turn. You simply must give these two\nmen---- Oh, you're going, are you? Very well. But think over what I've\nsaid. You'll realize that I'm right.\"\n\nMr. Red shut the door and locked it. Sergeant Farrelly turned fiercely\non Constable Cole.\n\n\"You born fool,\" he said, \"why didn't you strike him when I had him\ndown?\"\n\n\"I'd have----\"\n\n\"I suppose it was planning a stratagem you were,\" said the sergeant,\n\"instead of striking when you got the chance.\"\n\n\"I'd have broke his head,\" said Constable Cole, \"if so be I'd struck\nhim and me in the rage I was in at the time. How would I know but that\nit might be murder?\"\n\n\"Serve you right if he broke yours,\" said the sergeant.\n\n\"Come now,\" said Dr. O'Grady, \"there's no use making a fuss. You put up\na middling good fight to judge by the look of you, and you ought to be\ncontent. None of the rest of us did as much.\"\n\n\"Is that yourself, doctor?\" said the sergeant.\n\n\"It is.\"\n\n\"And, by the holy,\" said Constable Cole, \"he has Patsy Devlin along\nwith him!\"\n\n\"And us thinking that the two of yez was off to America,\" said the\nsergeant.\n\n\"I heard that you thought I'd gone,\" said Dr. O'Grady; \"but what made\nyou suspect poor Patsy?\"\n\n\"Hadn't he the funds collected for the sports?\" said the sergeant.\n\n\"Be damn!\" said Patsy, \"but that wasn't the cause of my going off, and\nwell you know it, sergeant. It's little call you have to be saying\nthat about me, and you out after me at the time for the murder of the\ndoctor. Wasn't that enough for you to be saying, without the other?\"\n\n\"My wife?\" said Mr. Dick. \"Did you see her? Was she well?\"\n\n\"She's distressed,\" said the sergeant. \"The whole of them's distressed,\nand small blame to them. I couldn't rightly tell this minute which of\nthe ladies was your wife; but they were all round at the barrack this\nmorning, and it would have gone to your heart to see the way they were.\"\n\n\"As we're on the subject of ladies,\" said Dr. O'Grady, \"I suppose\nMiss Blow is out after me. I heard from Patsy that she'd arrived in\nClonmore.\"\n\n\"You may say she is,\" said the sergeant. \"She has the officer's heart\nfair broke, pursuing him here and there, and not letting him rest in\nhis bed at night the way he'd have us all scouring the country for your\ndead body.\"\n\n\"And was it looking for me that brought you here, sergeant?\" said Dr.\nO'Grady.\n\n\"It was after they left the barrack,\" said the sergeant, \"that the\nladies went up to the Castle--Jimmy O'Loughlin's boy showing them the\nway--thinking that maybe, being a magistrate, Lord Manton would help\nthem to find yez. What happened there I couldn't say; but it wasn't\nlong after us eating our dinner when the officer came down and Lord\nManton along with him and Miss Blow and the other ladies. Such language\nyou never heard. There was one lady, tall she was, and dark, with a\nkind of a grey dress on her, and a big umbrella under her arm with a\nstone knob on the end of the handle of it----\"\n\n\"My aunt, I think,\" said Mr. Sanders.\n\n\"She might be. Anyway, barring the doctor's young lady, she was the\nworst of them for wanting to have Mr. Red hanged for murdering the lot\nof yez.\"\n\n\"I wouldn't wonder if he was hanged,\" said Constable Cole, \"the way\nhe's conducting himself. It's scandalous.\"\n\n\"The orders I got,\" went on Sergeant Farrelly, \"was to proceed\nto Rosivera, and ask Mr. Red whether he'd noticed two Members of\nParliament going by his gate on bicycles the day before.\"\n\n\"And not to be in too great a hurry over the job,\" said Constable Cole,\n\"because you was only doing it to satisfy the ladies.\"\n\n\"Whisht!\" said the sergeant; \"what call have you to be repeating\nconfidential orders? So we came along----\"\n\n\"Wait a minute,\" said Dr. O'Grady. \"How did you get off without taking\nMiss Blow with you? I'd have expected her to insist on going too.\"\n\n\"She did,\" said the sergeant; \"and it was only by means of a stratagem\nwe got clear of her. It was Constable Cole invented the stratagem.\nHe's great at that.\"\n\n\"How did he manage?\"\n\n\"Never mind about the stratagem,\" said Mr. Sanders; \"go on with your\nnarrative.\"\n\n\"I'll tell you about it after,\" said the sergeant to Dr. O'Grady in a\nwhisper. Then he went on delivering himself of his tale very much as if\nhe were giving evidence in a court of law.\n\n\"We reached Rosivera at half-past three P.M. We then knocked at the\ndoor and asked for Mr. Red. We stepped into the hall of the house,\nConstable Cole being about two paces behind me. A tall man with a black\nbeard, whom I can identify and swear to if necessary, came out of a\ndoor on the left, and without warning struck at my head with a stick. I\naimed a blow at him with my fist, not having time to draw my baton, and\nknocked him down. I was then engaged by two other men. One of them was\nMr. Red, to whom I am prepared to swear. I shouted to Constable Cole\nto strike the man on the ground with his baton, in order to keep him\nquiet. Constable Cole did not obey my order.\"\n\n\"I did not,\" said Constable Cole; \"sure I might have killed him.\"\n\n\"We were then overpowered,\" said the sergeant, \"the man whom I knocked\ndown coming to the assistance of his comrades. That's all, gentlemen.\"\n\n\"I suppose,\" said Mr. Dick, \"that your officer will immediately\ntelegraph for the military?\"\n\n\"He might,\" said the sergeant; \"but he won't do it sooner than\nto-morrow, anyway. He won't be expecting us home till late to-night.\"\n\n\"I must warn the Emperor,\" said Dr. O'Grady. \"He had better clear out\nat once. I shouldn't like any harm to happen to the Emperor, and I\nknow he hates soldiers. His whole life is given up to the destruction\nof standing armies. He'd break his heart if the military captured him\nin the end.\"\n\n\"Do you mean to say that you intend----\" said Mr. Sanders.\n\n\"Of course I do,\" said Dr. O'Grady. \"The Emperor is a thoroughly decent\nsort, and has always treated me well. I'm not going to allow him to be\nbullied simply to gratify your passion for revenge. You pretend you\nthink it wrong to gamble, but you're not above entertaining a spite\nagainst the poor Emperor and wanting to stick a knife into him. I\ncan tell you, Mr. Sanders, that sort of spirit is a jolly sight more\nunchristian than losing a few pence to Patsy Devlin over an innocent\ngame.\"\n\n\"The police,\" said Mr. Sanders, confidently, \"will see that you do not\nassist this criminal to escape from justice.\"\n\nHe looked at Sergeant Farrelly as he spoke. The sergeant scratched his\nhead.\n\n\"Tell me this now,\" he said to Dr. O'Grady; \"is this any kind of League\nwork?\"\n\n\"It is not,\" said the doctor. \"I don't suppose the Emperor ever heard\nof the League till I mentioned to him the other day that Patsy was a\nmember of it.\"\n\n\"If it's not the League,\" said the sergeant, \"and if the doctor will\nanswer for it that the man's a respectable man----\"\n\n\"He is,\" said the doctor. \"Why, my goodness, sergeant, he owns a\nmotor-car. You've seen it yourself, a remarkably fine motor-car; one\nthat couldn't have cost less than eight hundred pounds.\"\n\n\"I don't know,\" said the sergeant, \"that there's any call for me to\ninterfere with the doctor in the matter. It would be a queer thing now,\nand a thing that wouldn't suit me at all, if I was to be preventing the\ndoctor from speaking to a friend of his anyway that pleases him.\"\n\n\"He attacked you,\" said Mr. Sanders; \"he knocked you down. He has\nimprisoned you, and yet you say----\"\n\n\"If he's brought up before the Petty Sessions,\" said the sergeant,\n\"I'll tell what he's done; but till he is I don't see what right I\nhave to put the handcuffs on the doctor on account of what might be no\nmore than some kind of a joke that's passing between him and another\ngentleman....\"\n\n\n\n\nCHAPTER XXV\n\n\nAt ten o'clock next morning Jimmy O'Loughlin entered the commercial\nroom of his hotel. Miss Blow was breakfasting by herself. Miss\nFarquharson, who had finished her breakfast an hour earlier, was\nwriting letters at one end of the long table, having folded back the\nwhite cloth. Mrs. Dick and Mrs. Sanders were still in their bedrooms,\nmourning for their husbands.\n\n\"There's five police after coming into the town from Ballymoy,\" said\nJimmy excitedly; \"and it's what I'm after hearing from Moriarty that\nthere's more expected.\"\n\nMiss Blow looked up from her breakfast. Her face expressed irritation\nand incredulity.\n\n\"It's the truth I'm telling you,\" said Jimmy. \"Begad, but Mr. Goddard's\nthe fine man.\"\n\n\"You told us last night,\" said Miss Farquharson, \"that Mr. Goddard had\nvanished like every one else. Has he appeared again?\"\n\n\"He never was lost, thanks be to God! He was up with Lord Manton\nbeyond at the Castle, devising plans and concocting stratagems for the\nsettling of the matter that's been troubling you; and settled it'll be\nnow the one way or the other.\"\n\n\"I know the sort of plans he and Lord Manton would be likely to\ndevise,\" said Miss Blow, scornfully. \"I've had some experience of them.\"\n\nShe poured out a cup of tea as she spoke and devoted herself to\nher breakfast. Bridgy burst into the room. She appeared to be in a\ncondition of violent excitement.\n\n\"There's four police on a car,\" she said, \"driving up to the barrack,\nand two more along with them on bicycles.\"\n\n\"Is that four more on top of the first five?\" asked Jimmy O'Loughlin.\n\n\"It is, it is,\" said Bridgy. \"The Lord save us and help us! There'd\nhardly be more of them if there was to be a Member of Parliament making\nspeeches about the land!\"\n\n\"There now,\" said Jimmy O'Loughlin, reproachfully, to Miss Blow. \"What\ndid I tell you? Sure, Mr. Goddard's as fine a man as e'er a one that's\nin it. It's himself will do a job in fine style when once he takes it\nin hand at all. That's eleven men, and Constable Moriarty makes twelve,\nand there was a sergeant with the first lot that I seen myself. Was\nthere e'er a sergeant on the other car, Bridgy?\"\n\n\"There was,\" said Bridgy. \"I took notice of him passing, and one of\nthem two that came on bicycles had two stripes on his arm.\"\n\n\"That makes two sergeants and an acting sergeant,\" said Jimmy\nO'Loughlin. \"What more would you expect? What more would anybody want,\nunless it would be a gunboat sent round from Cork? And that's what you\ncould hardly expect, unless it might be for an eviction on one of the\nislands.\"\n\n\"I'll see what all these men are going to do,\" said Miss Blow, \"before\nI give an opinion about them. I've been----\"\n\n\"Here's Mr. Goddard himself,\" said Jimmy O'Loughlin, who was standing\nnear the window, \"and his lordship along with him. And they have the\nbig waggonette from the Castle and the dog-cart with the yellow cob in\nit. Be damn, but it's great!\"\n\nMiss Farquharson stood up and looked out of the window. Miss Blow,\nobstinately sceptical, continued to eat her breakfast. Mr. Goddard and\nLord Manton entered the room.\n\n\"Ladies,\" said Mr. Goddard. \"In a quarter of an hour we start for\nRosivera.\"\n\n\"At the head of a small army,\" said Lord Manton. \"Twelve men armed with\ncarbines, not counting Mr. Goddard, who wears a sword.\"\n\n\"Fifteen men,\" said Mr. Goddard. \"I'm expecting three more. They may\narrive at any minute.\"\n\n\"I suppose,\" said Miss Blow, \"that this is some new kind of trick.\"\n\n\"Come and see,\" said Mr. Goddard. \"Lord Manton has placed his\nwaggonette at your disposal. We invite your presence. We insist upon\nit.\"\n\n\"My niece and Mrs. Dick,\" said Miss Farquharson, \"are still in their\nbedrooms.\"\n\n\"Get them out,\" said Lord Manton, \"as quickly as possible. There is no\ntime to be lost. Military expeditions of this sort cannot possibly be\ndelayed.\"\n\n\"If so be,\" said Jimmy O'Loughlin, \"that you'll be wanting me, as a\nmagistrate, to be taking depositions or the like----\"\n\n\"We don't absolutely require you,\" said Lord Manton; \"but we'll be glad\nto have you with us.\"\n\n\"It'll be better, then,\" said Jimmy, \"if I go upstairs and put on a\ndecent coat and shave myself. Bridgy, will you run like a good girl and\nget me a cup of hot water?\"\n\n\"You may change your coat,\" said Mr. Goddard, \"but you can't shave.\nThere isn't time.\"\n\nMiss Blow and Miss Farquharson left the room together.\n\n\"Would you have any objection to telling me,\" said Jimmy O'Loughlin,\n\"what is it that you're thinking of doing?\"\n\n\"It's Lord Manton's plan,\" said Mr. Goddard, \"not mine. The fact is,\nwe're going to Rosivera to marry Mr. Red either to Miss Blow or Miss\nFarquharson.\"\n\n\"And is that the reason you have the police gathered from the four\ncorners of the county?\"\n\n\"It is,\" said Mr. Goddard.\n\n\"I wouldn't wonder but you're right. It's ten to one he won't care for\nthe notion; not but what Miss Blow is a fine-looking young lady, and\nthat's what I've always said since the first time ever I set eyes on\nher.\"\n\nThe street of Clonmore, the single street which runs from end to end\nof the village, presented a most unusual appearance when Miss Blow,\nfollowed by the other three ladies, emerged from the hotel. A long\nline of vehicles stretched from the door of the post-office to the\nbarrack. There were two cars, each holding four policemen. They had\ntheir carbines between their knees and presented a most warlike and\ndetermined appearance. Next came a third car, Jimmy O'Loughlin's, with\ntwo policemen on one side of it, the side left vacant being intended\nfor Mr. Goddard. Behind it was Lord Manton's dogcart. Jimmy O'Loughlin\nwas on the back seat, Lord Manton sat beside the groom in front. Next\ncame the waggonette with Mrs. Patsy Devlin in it. The acting sergeant\nand the constable who had arrived on bicycles stood beside their\nmachines immediately behind the waggonette. They formed a kind of\nrear-guard, and could be counted on to frustrate any attempt which Mr.\nRed might make to attack the party from behind.\n\nAn eager crowd thronged the footpaths and broke into a cheer when the\nladies appeared.\n\n\"Miss Blow, Miss Farquharson and ladies----\" said Mr. Goddard, when the\nnoise of the cheering had subsided.\n\n\"We thought you were murdered too,\" said Mrs. Dick.\n\n\"Kindly do not interrupt me,\" said Mr. Goddard. He felt that he was at\nlast in a position to assert himself even in the face of Miss Blow. A\nman acquires self-confidence when he is in command of an armed force.\n\n\"We start,\" he said, \"for Rosivera, to discover whether Mr. Red, the\nman Red, as he has been well described, the tenant of that house, has\nanything to do with the extraordinary series of disappearances which\nhave disturbed the peace of this neighbourhood.\"\n\nLord Manton said, \"Hear, hear!\" from his seat in the dogcart, and the\ncrowd cheered again.\n\n\"We invite your presence, ladies, and place the waggonette which you\nsee at your disposal. When you are seated in it we start at once. God\nsave the King!\"\n\n\"Hear, hear!\" said Lord Manton again.\n\nThe four ladies, a little bewildered by this oration, took their seats\nin the waggonette. Mr. Goddard got up on his car and gave the order to\nmarch. The expedition started.\n\nAn hour's steady driving brought the party to the top of the hill from\nwhich the gate of Rosivera is visible. Mr. Goddard gave the order to\nhalt. It was passed forward to the leading car, and the expedition came\nto a standstill on the summit of the hill. Mr. Goddard got down from\nhis car and walked up to Lord Manton.\n\n\"It's a damned awkward thing,\" he said, \"to march up to a man's house\nat the head of a body of men like this.\"\n\n\"Don't say you're thinking of going back,\" said Lord Manton. \"It would\nbe a shame to disappoint Miss Blow.\"\n\n\"I'm not going back; but all the same it's awkward. What excuse shall I\nmake?\"\n\n\"If he's been kidnapping people,\" said Lord Manton, \"he won't expect\nyou to make any excuse.\"\n\n\"Oh, of course, if he really has. But has he?\"\n\n\"The only way of finding out for certain is to go and see. Miss Blow\nwon't be satisfied with anything less.\"\n\n\"Damn Miss Blow. Anyhow, we needn't drive up to the house as if we were\na funeral.\"\n\nHe gave an order to the police, who dismounted. Mr. Goddard marched\nat the head of them down the hill. Lord Manton and Jimmy O'Loughlin\nfollowed. The ladies, led by Miss Blow, also followed. At the gate of\nRosivera, Mr. Goddard halted his party. He ordered the police to remain\noutside the gate. He invited Lord Manton to accompany him to the house.\nThen with a glance at the ladies he told one of the sergeants not to\nallow any one else to pass the gate.\n\nMiss Blow reached the bottom of the hill and prepared to follow Mr.\nGoddard along the avenue. She was stopped by the sergeant. Suspecting\nsome trick to be played on her at this last and critical moment, she\nsuggested to Miss Farquharson that they should force their way through\nthe cordon of police, by making, all four of them, a simultaneous rush.\nMiss Farquharson refused to do anything of the sort, and gave it as\nher opinion that Mrs. Sanders and Mrs. Dick would be quite useless in\na hand-to-hand conflict. Miss Blow left her, walked a little way along\nthe road, crossed a ditch, and began to climb the wall which enclosed\nthe Rosivera grounds. The police eyed her doubtfully. They did not want\nto lay violent hands on Miss Blow. They excused themselves to their own\nconsciences. Their orders were to prevent her passing through the gate.\nMr. Goddard had said nothing about what was to be done if she climbed\nthe wall.\n\nMr. Goddard and Lord Manton surveyed the house. It looked peaceful, too\npeaceful. There was no sign of its being inhabited.\n\n\"Are you sure,\" said Mr. Goddard, \"that there's anybody here?\"\n\n\"I'm not sure,\" said Lord Manton. \"I only know that I let the house to\nthe man Red. He may or may not be living in it. I never saw him.\"\n\nMr. Goddard walked up to the door and knocked. There was no answer.\nHe knocked again. Still there was no answer. He turned the handle and\npushed. The door resisted his push. It was locked.\n\n\"What shall we do now?\" he said to Lord Manton.\n\nThen he caught sight of Miss Blow, who was crossing the gravel sweep.\n\n\"Go back,\" he said; \"go back at once. This is no place for ladies.\"\n\n\"I won't go back,\" said Miss Blow. \"If I did you'd not enter the house\nat all. You would come back in ten minutes and say you had searched it\nand that there was nobody inside.\"\n\n\"My dear Miss Blow,\" said Lord Manton, \"Mr. Goddard may perhaps\ndeserve that, but surely I don't. Be just. Give me credit for common\nhonesty.\"\n\n\"I'm sorry I can't do that,\" said Miss Blow.\n\n\"You might,\" said Lord Manton. \"I gave you tea yesterday.\"\n\n\"You can come with us if you like,\" said Mr. Goddard; \"but how do you\npropose to get in? The door is locked.\"\n\n\"The top of that is open,\" said Miss Blow, pointing with her finger to\nthe window at the left side of the door, \"so I suppose the bottom can\nbe pushed up.\"\n\nMr. Goddard felt like a burglar, which is an unpleasant sensation for a\npolice officer, but one which he was getting gradually accustomed to.\nHe had experienced it when he hid in the stable of Jimmy O'Loughlin's\nhotel and when he entered his own house by way of the back garden.\nHe had experienced it when he drove out of Ballymoy in the early\nmorning and when he tried to escape from Lord Manton's library. He\nhad experienced it again when he concealed himself in the Clonmore\npost-office. He opened the window and climbed in. Miss Blow followed\nhim. Lord Manton, moving rather stiffly, for he was not used to\nclimbing, followed her. They stood together in the dining-room.\n\n\"Dear me,\" said Lord Manton, \"what a very remarkable taste Mr. Red has\nin wall decoration! Yellow dragons on a crimson ground! Did you ever\nsee anything like that before, Miss Blow?\"\n\nMiss Blow made no answer. She was opening the doors of the small\ncupboards in the sideboard, in the hope, perhaps, of discovering the\nmutilated remains of some of Mr. Red's victims. She found nothing but\nempty bottles and some wine-glasses.\n\n\"Even the door,\" said Lord Manton, \"has been painted, and there is a\nlarge yellow dragon on it, a mother dragon with several young ones.\nYou'd better follow her, Goddard.\"\n\nHe added this hurriedly, and obviously did not refer to the female\ndragon. Miss Blow had opened the door and passed through it into the\nhall.\n\n\"The door opposite to you,\" said Lord Manton, \"leads into the\ndrawing-room.\"\n\nIt did not look like a drawing-room when they entered it. A heavy deal\ntable, like a carpenter's bench, stood in the middle of the floor, and\non it were a number of curiously shaped metal flasks. There was a pile\nof long brass tubes in one corner of the room, which looked like empty\ncartridge cases, intended to contain ammunition for some very large\ngun. There were wooden shelves all round the walls stocked with thick\nglass bottles, such as are seen in chemists' shops, bottles with glass\nstoppers. In one corner the floor was charred, as if a fire had been\nlighted on it.\n\n\"Nobody here,\" said Mr. Goddard, looking round.\n\nLord Manton was staring curiously at the things about him. He picked up\none of the brass tubes.\n\n\"A.M.B.A.,\" he read. \"What do you suppose that means, Miss Blow? Hullo!\nShe's gone again. After her, Goddard! We can't allow her to escape.\nThere may be an explosion at any moment. This place looks uncommonly\nexplosive, and if she is shattered into little bits her father will\nhold us responsible.\"\n\nMiss Blow stood at the bottom of the stairs, listening intently.\n\n\"I hear a noise in the upper story,\" she said. \"The house isn't empty.\"\n\nMr. Goddard and Lord Manton listened.\n\n\"There is a noise,\" said Lord Manton. \"Goddard, unsheathe your sword\nand proceed cautiously up the first flight of stairs. We will protect\nyour rear.\"\n\nThey reached the first floor of the house and the noise became much\nmore plainly audible.\n\n\"It sounds to me,\" said Lord Manton, \"as if Mr. Red--the man Red, I\nmean--and his friends were having a hurdle race in the attics.\"\n\n\"Go on,\" said Miss Blow.\n\n\"Let us pause for a moment,\" said Lord Manton, \"and consider the\nsituation before we rush headlong into some unknown danger. I am\ninclined to think that several men are jumping hurdles, and from the\noccasional violence of the bumps I should say that one of them is a\nheavy man, much heavier than the others. Can you infer anything else\nfrom the noise we hear, Miss Blow? Or you, Goddard? Very well, if\nneither of you can, we may as well go on, bearing in mind that there\nare several men and that one of them is large.\"\n\nThey climbed two more flights of stairs and reached the top story of\nthe house. The noise sounded very loud. Guided by it they reached the\ndoor of a room at the end of the passage. Mr. Goddard knocked. There\nwas no reply, but this did not surprise him, for the noise inside was\nso great as to drown the knock.\n\n\"Go in,\" said Miss Blow.\n\nMr. Goddard turned the handle. \"The door is locked,\" he said.\n\n\"The key is on the outside,\" said Miss Blow. \"Turn it.\"\n\n\"As landlord of the house,\" said Lord Manton, \"allow me.\"\n\nHe turned the key and flung open the door. Inside were Dr. O'Grady, Mr.\nDick, Patsy Devlin, Sergeant Farrelly and Constable Cole. They were\nplaying leapfrog. Mr. Sanders sat on a bed in the corner and watched\nthem. Dr. O'Grady, executing a splendid bound over Sergeant Farrelly's\nbroad back, landed on his feet opposite the door just as it opened.\n\n\"Adeline Maud,\" he said, \"I'm delighted to see you. Have you been here\nlong? I hope you weren't kept waiting at the door.\"\n\n\n\n\nCHAPTER XXVI\n\n\n\"Dear me!\" said Lord Manton; \"all our lost sheep found again. One, two,\nthree, four, five, six. Yes, the whole six of you.\"\n\n\"You're not dead,\" said Miss Blow, clutching at Dr. O'Grady's arm;\n\"you're not dead then, after all.\"\n\n\"My wife,\" said Mr. Dick, \"my poor wife. Can any of you tell me----\"\n\n\"Sergeant Farrelly,\" said Mr. Goddard. \"Will you kindly explain how it\nis that I find you here playing leapfrog with----\"\n\n\"I thought it was a hurdle race,\" said Lord Manton. \"It sounded exactly\nlike a hurdle race.\"\n\nSergeant Farrelly drew himself up to attention.\n\n\"In obedience to your orders, sir,\" he said, \"Constable Cole and I\nproceeded to Rosivera on our bicycles, leaving Clonmore at a few\nminutes after twelve o'clock. Acting on instructions received, we rode\nas slowly as possible----\"\n\n\"You can leave out that part,\" said Mr. Goddard, with a glance at Miss\nBlow, \"and go on.\"\n\n\"We have been kidnapped and imprisoned here,\" said Mr. Sanders; \"and I\ndemand that the police shall instantly pursue----\"\n\n\"Lucius,\" said Miss Blow, \"have you been imprisoned?\"\n\n\"Certainly not,\" said Dr. O'Grady; \"we've been the guests of Mr. Red;\nhaven't we, Patsy?\"\n\n\"We have, be damn!\" said Patsy Devlin. \"What was there to hinder us\ngoing off any time we wanted?\"\n\n\"I don't understand,\" said Miss Blow.\n\n\"I demand----\" said Mr. Sanders again.\n\n\"We were prisoners,\" said Mr. Dick; \"and I join with my friend Sanders\nin insisting that the men who captured us shall be brought to justice.\nI insist upon it for the sake of my poor wife----\"\n\n\"You shut up,\" said Dr. O'Grady. \"If either you or that ass Sanders\nsays another word, I'll tell how you came to be here.\"\n\n\"I demand----\" said Mr. Sanders.\n\n\"Very well,\" said Dr. O'Grady. \"Excuse me, Adeline Maud, but would you\nmind leaving the room for a moment? The story I have to tell is not\nexactly one for a lady to listen to.\"\n\n\"If it's very improper,\" said Lord Manton, \"perhaps I'd better go too.\nI was very carefully brought up when I was young.\"\n\n\"It's simply this,\" said Dr. O'Grady, \"that Mr. Dick was found by the\nEmperor bathing on the shore in company with----\"\n\n\"I wasn't,\" said Mr. Dick.\n\n\"You may try to wriggle out of it now,\" said Dr. O'Grady; \"but you\ntold me at the time that you were, and the other fellow, who says he's\na Member of Parliament, but looks like a commercial traveller, was\nmending a bicycle for a lady, who----\"\n\n\"If these stories,\" said Lord Manton, \"are the sort which are likely to\nbreak up the happiness of two homes, I hope you won't tell them. Mrs.\nDick and Mrs. Sanders are outside, as well as Miss Farquharson, who is\nan aunt.\"\n\n\"I don't want to tell them,\" said Dr. O'Grady. \"I'd much rather keep\nthem to myself. But I won't have the Emperor pursued.\"\n\n\"Sergeant,\" said Mr. Goddard, \"proceed with your evidence, leaving out\nall the part about my orders.\"\n\n\"Why?\" said Miss Blow. \"Are you ashamed of your orders? What were they?\"\n\n\"If you've anything to be ashamed of in the orders you gave, Goddard,\"\nsaid Dr. O'Grady, \"you'd better not have that story told either. I warn\nyou fairly that if any attempt is made to molest the Emperor, I shall\nhave those orders of yours, whatever they are, produced in court.\"\n\n\"We might get on a little,\" said Lord Manton, \"if some one would tell\nus who the Emperor is.\"\n\n\"He's an anarchist,\" said Mr. Sanders.\n\n\"An anti-military anarchist, a most dangerous man,\" added Mr. Dick.\n\n\"I've warned you once already,\" said Dr. O'Grady, \"what will happen if\nyou persist in talking that way. Even supposing the poor old Emperor is\nall you say, isn't it a great deal better to blow up a few armies, than\nto go about the country deceiving innocent women when each of you has a\nwife at home? Patsy Devlin will bear me out in saying that the Emperor\nis a most respectable man, large-hearted, generous to a fault; a little\neccentric, perhaps, but a thoroughly good sort.\"\n\n\"Do explain,\" said Lord Manton, \"who the Emperor is, and how you and\nPatsy Devlin and the Members of Parliament and the police all come to\nbe here, playing leapfrog in an attic.\"\n\n\"The Emperor,\" said Dr. O'Grady, \"is Mr. Red. He sent for me to\nattend a servant of his who had unfortunately scorched the back of his\nlegs while assisting in some scientific experiments in the Chamber of\nResearch.\"\n\n\"The drawing-room?\" asked Lord Manton.\n\n\"The room that used to be the drawing-room,\" said Dr. O'Grady. \"I have\nbeen in attendance on the man ever since, earning a fee of five pounds\na day. That doesn't look as if I was badly treated, does it; or as if I\nwas kept here against my will? There's no other reason, so far, why the\npoor Emperor should be ruthlessly pursued, as these gentlemen suggest.\"\n\n\"No,\" said Lord Manton. \"So far his record is clear. Go on.\"\n\n\"Patsy Devlin,\" said Dr. O'Grady, \"came here to take refuge from the\npolice. He hadn't done anything particularly wrong, nothing worse than\nusual; but some one had put out the absurd story that he had murdered\nme, and advised him to fly to America.\"\n\n\"I'm afraid I was responsible for that,\" said Lord Manton. \"I\nwanted----\"\n\n\"Very well,\" said Dr. O'Grady. \"If the Emperor is pursued and caught,\nthat story will come out too. You will have to explain in open court\nwhy you treated Patsy Devlin in such a way.\"\n\n\"And why you treated me as you did,\" said Miss Blow.\n\n\"And why you treated Adeline Maud as you did,\" said Dr. O'Grady. \"I\ndon't exactly know how you did treat her, but if you pursue the Emperor\nI'll insist on finding out.\"\n\n\"I won't pursue him,\" said Lord Manton. \"I promise not to. As a matter\nof fact, I don't want to pursue him in the least. He paid his rent in\nadvance.\"\n\n\"Patsy Devlin,\" said Dr. O'Grady, \"was sheltered, lodged, and fed by\nthe Emperor, and has no complaint whatever to make. Have you, Patsy?\"\n\n\"I have not. Only for him, they'd have had me hanged for murdering you,\ndoctor; which is what I wouldn't do, and never thought of.\"\n\n\"And you don't want to have the Emperor pursued?\"\n\n\"I do not,\" said Patsy.\n\n\"We pass on,\" said Dr. O'Grady, \"to the case of the two Members of\nParliament--if they are Members of Parliament. I don't want to make\nmyself unnecessarily unpleasant, especially as I understand that their\nwives, their real wives, are waiting for them----\"\n\n\"I protest----\" said Mr. Dick.\n\n\"I always told you, Goddard,\" said Lord Manton, \"that there was\nsomething of this sort, something uncommonly fishy behind the\ndisappearance of these two gentlemen. You'll recollect that. But I must\nsay I didn't expect it to be as bad as this.\"\n\n\"I protest----\" said Mr. Dick and Mr. Sanders together.\n\n\"I need say no more about them,\" said Dr. O'Grady. \"The Emperor, out of\nsheer kindness of heart, saved them from what might have been a very\nugly scandal. I don't want to drag the whole thing into the light of\nday; but if they insist upon the pursuit of the Emperor, I shall tell\nthe truth so far as I know it, and the Emperor, when you catch him,\nwill fill in the details.\"\n\n\"There's nothing,\" said Mr. Dick, \"absolutely nothing----\"\n\n\"There may be nothing,\" said Lord Manton; \"but from the little I've\nheard I should say that Mrs. Dick will have a distinct grievance;\nand as for your aunt, Mr. Sanders--you know her better than I do, of\ncourse, but she doesn't strike me as the kind of lady who will treat\nthe doctor's story as a mere trifle.\"\n\n\"As to the police,\" said Dr. O'Grady, \"I don't profess to explain\nexactly how they came here. Goddard seems to have given them some\nvery peculiar orders, orders that won't bear repeating. I don't want\nto probe into the secrets of the force. I have a respect for Sergeant\nFarrelly; I used to have a respect for Goddard----\"\n\n\"You won't have any respect for him when you hear how he has treated\nme,\" said Miss Blow.\n\n\"You hear that, Goddard?\" said Dr. O'Grady. \"Adeline Maud says you've\nbeen ill-treating her. That's a thing I can't and won't stand from any\nman living, and if you make the smallest attempt to annoy the Emperor\nin any way, I'll publish her story in the newspapers, and what's more,\nI'll hire the best barrister in Dublin to cross-examine you about the\norders you gave to Sergeant Farrelly and Constable Cole.\"\n\n\"What you suggest, then,\" said Lord Manton, \"is to leave the whole\nmatter wrapped in a decent obscurity--to let the dead past bury its\ndead. I quite agree. None of us want our share in the proceedings of\nthe last few days made public. But will Miss Blow consent to allow the\nman Red--it's her phrase, doctor, not mine; so don't be angry with\nit--to allow the man Red to escape scot free? After all, it was she who\nurged us on to have him hanged.\"\n\n\"Adeline Maud,\" said Dr. O'Grady, \"has more sense than to quarrel with\na man who has been paying me five pounds a day. He suggested four\npounds at first; but he sprang it to a fiver the moment I made the\nsuggestion.\"\n\n\"Of course, if Miss Blow is satisfied----\" said Lord Manton.\n\n\"She is,\" said Dr. O'Grady. \"Aren't you, Adeline Maud?\"\n\n\"Now that I know you're safe----\" said Miss Blow.\n\n\"In any case,\" said Dr. O'Grady, \"your pursuit would be quite useless.\nThe Emperor started at ten o'clock last night in his motor-car. It must\nbe after twelve now, so he has fourteen hours' start of you. By the\ntime you get back to Clonmore and send off telegrams----\"\n\n\"I shall insist,\" said Mr. Dick; \"I shall never consent----\"\n\n\"You go out at once to your wife,\" said Dr. O'Grady. \"Didn't you hear\nLord Manton say that she is outside waiting for you? You've been\nswaggering about the way you loved her ever since I first met you\nwalking about with nothing on you but your shirt. I don't believe you\ncare a pin about her. If you did, you'd be with her now, relieving her\nanxiety, instead of standing about here talking like a born fool. As I\nwas saying, Goddard, by the time you've sent off telegrams----\"\n\n\"I don't want to send off any more telegrams,\" said Mr. Goddard; \"I had\nenough of that yesterday.\"\n\n\"In any case,\" said Dr. O'Grady, \"I don't think you'd catch him, when\nhe'd have fifteen or sixteen hours' start of your telegrams. It was a\ngood car, and I don't believe you so much as know the number of it.\"\n\n\"The only thing that troubles me,\" said Mr. Goddard, \"is the\nInspector-General. He'll be in Clonmore by this time.\"\n\n\"Is he mixed up in it?\"\n\n\"Yes,\" said Lord Manton. \"He and the Lord Lieutenant, and the Chief\nSecretary, and the Prime Minister; though I'm not quite sure about the\nPrime Minister. It's a State affair. The whole Empire is on the tip-toe\nof excited expectation to find out what has happened.\"\n\n\"All you can do,\" said Dr. O'Grady, \"is to tell them the truth.\"\n\n\"The truth?\" said Lord Manton.\n\n\"Yes; the simple truth, just as I've told it to you; doing your best,\nof course, to spare Mr. Sanders and Mr. Dick, especially Mr. Dick, on\naccount of his poor wife.\"\n\n\"I'm not sure,\" said Mr. Goddard, \"that they'd believe--I mean to say,\nI'm not sure that I could venture to tell them the truth--that exact\nkind of truth, I mean.\"\n\n\"If you don't care to tell it yourself,\" said Dr. O'Grady, \"get Jimmy\nO'Loughlin to tell it for you. He'd do it; wouldn't he, Patsy?\"\n\n\"Be damn, but he would,\" said Patsy Devlin. \"He'd tell it without as\nmuch as turning a hair, so soon as ever he knew what it was you wanted\nhim to tell.\"\n\n\nTHE END\n\n\n\n\nTRANSCRIBER'S NOTES\n\nThis text has been preserved as in the original, including archaic and\ninconsistent spelling, punctuation and grammar, except as noted below.\n\nObvious printer's errors have been silently corrected.\n\nIllustration notation has been moved to below any enclosing paragraph.\n\nText in italics in the original work is represented herein as _text_.\n\nSmall capitals in the original work are represented herein as all\ncapitals.\n\n\n\n\n\nEnd of the Project Gutenberg EBook of The Search Party, by G. A. Birmingham\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nThank you for downloading this Scribner eBook.\n\n* * *\n\nJoin our mailing list and get updates on new releases, deals, bonus content and other great books from Scribner and Simon & Schuster.\n\nCLICK HERE TO SIGN UP\n\nor visit us online to sign up at  \neBookNews.SimonandSchuster.com\n\n## Contents\n\n1: Nose of a Dog\n\n2: Smeller\n\n3: Sniffing the Wind\n\n4: Walking While Smelling\n\n5: Plain as the Nose on Your Face\n\n6: My Dog Made Me Smell It\n\n7: Nose to Grindstone\n\n8: Nose-wise\n\n9: Stink-waves\n\n10: Civet Cats and Wet Dogs\n\n11: Nosed Out\n\n12: Smelloftheworld\n\nAbout the Author\n\nNotes and Sources\n\nThanks, an Index of\n\nIndex\nfor my Dad\n\n## 1: Nose of a Dog\n\nFinnegan's is ebony-black, moist, and dappled, two cavernous bass clefs at its front. Upton's is cleft with a visible valley, the whole thing guarded by short whiskers standing at attention.\n\nThese are my dogs, and these are my dogs' noses.\n\nBefore I became a research scientist studying dog cognition, I didn't think much about the dog nose. It may have been frowned at when poking impolitely at the privates of visitors to my home, or dabbed with peanut butter to encourage the swallowing of a pill. Even then, though, little regard was given to the nose itself\u2014to its form, to its movement, to the impossibly convoluted and complex vault it opens into.\n\nThis oversight isn't restricted to dog noses. We rarely study the nose on each other's face. At the nose prominently displayed\u2014indeed, jutting out, leading the way for the rest of the body. Without looking, try to describe your partner's nose, or your mother's. If it's not beakish or buttony, well, it's just a nose. Two nostrils dangling from a squashed, fleshy tetrahedron.\n\nI gaze at my son's nose, but mostly at its surface\u2014where the freckles have begun to assemble on his fair skin. But the snout of the dog gets my full attention. Now I look at dogs nose-first. For I am besotted with dogs, and to know a dog is to be interested in what it's like to be a dog. And that all begins with the nose.\n\nWhat the dog sees and knows comes through his nose, and the information that every dog\u2014the tracking dog, of course, but also the dog lying next to you, snoring, on the couch\u2014has about the world based on smell is unthinkably rich. It is rich in a way we humans once knew about, once acted on, but have since neglected.\n\nBy smelling, tapping into this sensory resource that we have but that we largely ignore, the dog has become an informant. Working dogs, trained to tell us what they naturally know, detect the presence of illegal substances and unwanted pests. But the dog also knows about the upcoming weather, the way afternoon smells, and whether you are sick or upset. Every inhaled gulp of air is full of information. It holds the odors of people who have recently passed by, leaving olfactory trails in their wake. It catches pollens and plant notes carried on the breeze. Each noseful captures the traces of animals who have walked, run, cowered, eaten, or died nearby. It traps the electric charge and round humid molecules from distant rainstorms.\n\nThis book is an exploration of what the dog's nose knows as has never been done before. What does your dog smell on you, on the ground, or with his nose deep in the fur of another dog? What does he know about you\u2014that you might not know yourself? What is it like to smell the world with that amazing nose leading the dog through his days?\n\nTo find out, I tracked the tracking dogs. Over the last years I have watched detection dogs grow up, be trained, and find their quarry, be it drugs, food, or people. With my Dog Cognition Lab at Barnard College, I have researched the pet dog's experience of himself, other dogs, and the smells of the human world in which he lives. I have spoken to scientists who study and model the dog nose, and to trainers and handlers who follow it. It is an examination of all aspects of the dog's olfactory world, and the magnificent organ that leads to it.\n\nBut it is also an exploration of the noses on our faces. We have untrained ourselves, we humans\u2014unlearned how to smell, over millennia. We are out of the habit of smelling. Why, you may not even have smelled this book yet, though it is but inches from your face. I find people who are smelling, and practice their methods.\n\nHaving myself committed a lifetime of not smelling, I let my dogs' behavior counsel me: I have ventured to be a little more like a dog myself. In my book Inside of a Dog I began an imaginative leap into what it might be like to be a dog\u2014and here I leap with four feet. I try to take my nose to the places the dog nose goes. And I sniff.\n\nI begin this process by learning more about our own sense of smell. And then I train my nose to better conjure what it might be like to have the mind and nose of a dog.\n\n\u2022 \u2022 \u2022\n\nMy inspiration and guides are our family's own dogs, Finnegan and Upton. Both are highly charismatic mutts. My husband and I met Finnegan's nose through the bars of a crate at a shelter that imports unwanted dogs from the South. He was four months old, had had ringworm and parvovirus, and, though recovering, was skinny and a bit sickly. I should say that I don't go to shelters much, because when I do, I inevitably walk out with a new animal. When I first saw him, in that crate at eye level, he wagged mightily, accepted a finger poked through the bars for licking, poked his nose through reciprocally, and then, as we moved on, sat down patiently. I looked back at him often: he was sitting and . . . waiting. We took him out of the crate to meet him, and he moved between my husband and me, looking at each of us in the face. Weary, he leaned against me. That was that. We took him home.\n\nFinn is now eight years older. He still has the look of that puppy who leaned. While his coat is a glossy black, as though we polish him daily, it is his way of looking that most captures him. One gets the feeling that he is always aware of what is happening. His eyes penetrate us. They follow us, they check with us when another animal is misbehaving, they look to us dolefully when we head out the door. Ears back, eyes full, he is hard to leave. He does not look simply with his eyes: his nose examines us. When we return home, he sniffs us, as close as we will let him, in an exploration of where we've gone, what we've eaten, whom we've touched or pet. Never have I returned home, having met a dog on the street, and escaped Finn's examination.\n\nI tend to think of Finn as a \"professional dog.\" He is eminently civilized: without our explicitly teaching him, he fell into line with what we expect from a dog in the house. He swallowed the little culture of our family whole. Upton, by contrast, whom we met three years into his life, is feral by comparison. He had been relinquished to the shelter from which he was initially adopted three years prior. We've seen his first photos: a small-bodied hound, ears too large for his head. Nose a blur. Well, his head and body grew up: he's a large, brindle, hound mix, with full eyes and a corkscrew tail. Whiskers punctuate his snout; his jowls droop. He is a dog's dog, incurably friendly with any dog, and has a gangly, goofy run. There is no photo of Upton in which he looks streamlined, athletic, or svelte. In running, his jowls flap, he tilts, his ears go akimbo. He is generously silly. Alas, he was also not a city dog before we knew him, and he is easily startled by every and any sound\u2014car door, garbage truck, garage door opening, street sign shaking in the wind, jackhammer, a bag flapping in a tree, a person suddenly appearing from around the corner . . . you name it. For this reason, while I have followed both of their noses to whatever sundry and sordid spots they're smelling, it is only Finn who I bring out to do new investigative smelling work. He could half write this book himself.\n\nTake a breath (through the nose, please). We are going on a journey through scents and smelling, with a tour through the improbable science about the olfactory abilities of the dog\u2014and the abilities of our own noses, waiting for us to discover them. By following the dog's lead, we can learn from him about what we are missing\u2014some of which is beyond our ability to sense, and some of which we simply need a guide to see. The world abounds with aromas, but we are spectacle-less. The dog can serve as our spectacles.\n\nIn so doing, we may also see how to return to that perhaps more primal, so-called animal state of knowledge about ourselves and the world that we have forgotten in a culture wrought of technology and lab tests. To follow animals is to become more attuned to our own existence. To follow dogs is to begin to apprehend the experience of our silent, loyal partners through our days.\n\n## 2: Smeller\n\nThe smell of that buttered toast simply talked to Toad, and with no uncertain voice; talked of warm kitchens, of breakfasts on bright frosty mornings, of cosy parlour firesides on winter evenings, when one's ramble was over and slippered feet were propped on the fender; of the purring of contented cats, and the twitter of sleepy canaries.\n\n\u2014Kenneth Grahame, The Wind in the Willows\n\nIn my home, the library is rife with reminders of how little we might understand what an animal smells. Not my library\u2014my six-year-old son's. Some of the most fabulous writers for children were highly attuned to smell. Roald Dahl's child-smelling monster and sweet-smelling chocolate palaces are matched by the animal protagonists from William Steig's imagination. His character Dominic, a nomadic, adventurous, and canine soul who leaves his barnyard-animal friends to explore the world, gives them an apt good-bye: \"I embrace you all and sniff you with love.\" Dominic's \"all-knowing\" nose directs him in his travels. He sniffs out an evil fox\u2014whom he \"would have recognized [by] smell merely by sniffing something he had touched only lightly a year earlier.\" He smells out tea, sugar, and milk for a snack. His nose discovers a sick pig, a witch-alligator, the residents of an unknown town. \"Dominic always noticed smells first,\" Steig wrote. All of the animal residents of Kenneth Grahame's Willows notice the \"warm, rich, and varied smells\u2014that twined and twisted and wreathed themselves at last into one complete, voluptuous, perfect smell that seemed like the very soul of Nature taking form and appearing to her children.\"\n\nEven that is not the half of it.\n\nI live with two quadrupedal reminders of the keenness of the dog nose in my house. We are agog when they find minute particles of food that long ago skittered off a young child's plate. But clearly this is not the extent of their olfactory acuity; it is only the extent of my daily awareness of it.\n\nScientific measures of the sensitivity of the dog nose are limited more by the capacity of measuring devices\u2014and dogs' interest in submitting to them\u2014than by their noses. Both pet dogs and tracking dogs have been put through their paces in a variety of threshold detection tasks that ask how diluted an odorant can be before a dog stops noticing that it's there. Pick out a canister with the smell of banana, amyl acetate, from a series of non-banana-smelling canisters, for instance. Dogs keep finding the banana until it is diluted to 1\u20132 parts per trillion: a couple drops of amyl acetate, one trillion drops of water. Early research with one very cooperative fox terrier concluded that she could essentially detect one milligram of butyric acid\u2014think smelly socks\u2014among 100 million cubic meters of air. You'll notice your spouse's smelly socks the moment after they are removed in the bedroom: that's around forty cubic meters of very socky-smelling air. The dog knows if someone's removed his socks in a room bigger than the gargantuan vehicle assembly building at NASA's Kennedy Space Center in Florida, made to put the space shuttles together. Any dog in the nearly four million cubic meters of Space Center would be alert to sweaty astronauts.\n\nExplosives-detection dogs smell as little as a picogram\u2014a trillionth of a gram\u2014of TNT or other explosive. What might it be like to notice a picogram of an odor? Since explosives dogs come to have very fond associations with their search odors, let's think of an aroma pleasing to our noses: cinnamon rolls cooking in a home kitchen. The average cinnamon roll has about a gram of cinnamon in it. Sure, the human nose is on it, from the moment we open the door of the house. Now imagine the smell of one trillion cinnamon rolls. That's what the dog coming in with us smells when we enter.\n\nThe sensitivity of the dog's nose can also be gauged by simply looking at his behavior. Hunting and tracking dogs naturally follow scent trails of prey or people who have passed before\u2014in some cases, several days before, over difficult terrain. Watch video of a Discovery channel show's host trying to outwit a bloodhound by crossing a river, spraying himself with deodorant, laying down sausages as a diversion, then doubling back and going a different way. Then watch the dog follow the path he had run, crossing the river, noticing (but forsaking) the sausages, and then doubling back and easily catching himself a TV personality.\n\nEncountering a trail in its middle, a tracking dog needs to sniff only five footprints\u2014laid in less than two seconds\u2014to tell which way the person who made those prints was going. Each footprint holds a certain amount of that person's odor, and as the intensity increases between step one and five, the dog has his answer. Run other people over the same path, intersect with other trails; the dog can still find his man.\n\nDogs are so good at finding their man that in the Netherlands, Germany, Poland, and a handful of other countries, courts have sanctioned evidence from dogs presented with scent-identification lineups. The human-scent lineup is not what it sounds like: a dog sniffing his way down a line of suspects and innocent people (called fillers), stopping at each, sniffing, and assessing.I Instead, the dog sniffs his way down an array of metal bars that have been handled by the suspect and the fillers. Dogs pick out the smell that matches an odor from a crime scene\u2014thus picking out the criminal.\n\n\u2022 \u2022 \u2022\n\nYour own dog, beside you right now, performs surprising and sometimes alarming feats of olfactory perception every day. Many of these behaviors are familiar to us; what is not familiar is the smell behind it.\n\nAs a first step toward discovering the acuity of the dog's nose, then, let us look at what a dog can and does smell in the course of an ordinary day. Our dogs exist in parallel with us, at our feet and by our sides, living in lockstep. Though we look them in the eyes and see how they look\u2014gazing at us, peering off toward a distant barker\u2014most of their behavior is about their noses, smelling the world.\n\n### NICE TO SNIFF YOU\n\nLeft to their own devices, most dogs will not pursue the dissipating trail of a stranger, or sniff metal bars one after the other. Dogs like sniffing other dogs. People like looking at people\u2014and left in a room by themselves will look at images, static or moving, of people. Dog pinups are not popular among the dogs I know, but were I able to bottle the odor of the lean black-and-white pup down the street, it might make for a good diversion when our dogs are alone and bored.\n\nEvery owner has surely observed this. But do not think that the mutual sniff is meaningless, a mere sneeze. When one dog sneezes, his companion does not. But when two dogs meet, they sniff\u2014and let themselves be sniffed\u2014as a real communication. There is likely a pleasure in sniffing other dogs, but what we cannot see is the information gleaned. There is a measured flow to their smelling: both sniff each other at once or take turns politely, burrowing noses into fur. The fur bears odors from skin glands located in sniffable locations. And these odors are the key: they hold news of the dog from which they effuse.\n\nWatching male and female dogs sniff one another, researchers found that males like to go for what they call \"the tail area\" (read: rump) first. Skin glands circle the anus, secreting odor. At either side of the anus\u2014at \"four and eight o'clock,\" one writer helpfully guides the clock-readers among us\u2014are anal sacs producing a strong odor of dog. More specifically, they seem to convey stressed dog. When a dog is anxious, the sacs produce a skunky smell. Secretions from the sacs also serve as a topping to every poo. For this reason, some scientists consider this odor likely to be the dog's \"signature\" odor\u2014his identification tag, written for smellers. Forty years ago, Dr. George Preti of the Monell Chemical Senses Center and his colleagues squeezed out the contents of some (semi) cooperative beagles' anal sacs. \"I was a pioneer!\" he told me. \"With no followers.\" What they found after examining the constituents was that, though the smell might seem similar\u2014and noxious\u2014to most human noses, the samples varied greatly: enough to be a marker of each individual. Because dog researchers will do just about anything (apparently) in search of knowledge about their subjects, we now know that, even for humans, there are noticeable differences between individuals' secretions\u2014ranging from a \"somewhat neutral or slightly pleasant, doglike odor\" on the one hand to a \"sharp acrid\" odor on the other. Thank you, scientists, for sniffing that.\n\nCanids have distinct tail glands as well, at the base of the tail. You can locate them by either watching a male sniff a new dog or by looking for an oily patch of hair where the tail meets the back\u2014so oiled from glandular secretions. In the fox, this supracaudal gland produces odors notable even to our noses: a hint of violet in the red fox, muskiness in the gray fox. Insofar as these glands reflect differences in the levels of sex hormone, part of what those male dogs are keen on finding out about a new dog is who she is, sure, but also whether she is ready to mate.\n\nFemale dogs more often sniff faces first. Wolves, too, are especially keen on studying the smell of each other's head and muzzle. You may have a dog who noses into your ears or pokes around your nose and eyes. He is treating you like a proper dog: there are plenty of apocrine and sebaceous glands and secretions in dogs' ears, and eccrine glands on their noses. Perhaps this examination is less about mating and more about determining health and diet. Saliva smells wherever saliva gets, which in dogs is plentifully spread around the face and snout.\n\nIf you get close enough, you will smell the unique odor of your dog. What your dog smells like to you is probably due to the secretions of the apocrine glands, which are over his entire body. There is also an individual smell on the bottom of his foot pads. If you have not ever sniffed your dog's feet, for goodness' sake, now's that time. (You can be sure your dog knows the smell of your feet.) On dogs, glands grace the pads themselves and hide between the toes. They emit an odor distinctive enough\u2014to other dogs, at any rate\u2014that it might explain one of the all-time most puzzling dog behaviors: scratching the ground after peeing or pooing. I've known dogs who in moments of great excitement of any sort\u2014a fine-smelling dog passes by; they have just finished a rousing, frenetic run\u2014furiously scratch long, deep rivulets in the ground, as though rendering an exclamation point on the scene. If every scratch releases drops of odor, then this behavior probably serves as a signpost to lead other dogs to where they might find the mother lode of smells: the excreta or urine the scratcher left behind.\n\nA sniff of another dog reveals who he is, but also information that only the more undignified gaze would catch as it falls. Dogs do not glance under the belly to see if the new dog is male or female. They smell male or female. Moreover, they may smell ready to mate, of being recently ill, or of having recently eaten. They smell their age: age is but a metabolic process, chemistry. And chemistry smells. Dogs smell of having had a bath or not having had one; they smell of having recently peed or very much having to pee. They smell of themselves, their status, and, likely, whether they are scared or happy or anxious.\n\n### MESSAGES LEFT BEHIND\n\nEven without an actual dog to nose on her walk, your dog is not bereft. Happily, dogs have left behind an embarrassment of scented calling cards. Every footstep on a porous surface leaves odor; the clump of fur an owner has brushed off has oils and secretions from hair follicles. We carry home with us the smells of dogs whose owners we have visited; of melancholy dogs leashed outside stores who have let us tickle their ears; of indiscriminately friendly puppies met on the sidewalk who tumbled, slobbered, and dribbled all over us.\n\nAnd then there is the pee. Anyone who has spent hours on hours on grassy knolls or lawns popular with dogs has witnessed the tragic Person Mark. A dog owner, letting down her guard, perhaps feeling a little weary, sits on the grass as dogs gambol about. Suddenly, without warning, a single dog tears away from the games, approaches the person's side or back, raises a leg and . . . urinates.\n\nThe person has been \"marked.\" Should she stay seated, it would not be long before she would be counter-marked by another dog. She does not stay seated. She rises and people commiserate and scold the dogs between laughs. But the dogs do not see this as bad behavior, of course. This behavior has long standing. Dogs do it, bees do it, even hippopotami do it.\n\nScent-marking is the leaving of urine or other bodily secretions on a stone, stump, bush, or other protruding feature of the landscape: fire hydrants on city streets, tractor wheels in farming climes. The marked item becomes a scent post: an olfactory flag holding information about the marker, ready for the sniffing.\n\nClassically, scent-marking is thought to be territorial. Classically, it has been\u2014in most animals. And the means and placement are wonderfully more complicated than human flag-leaving. Muskrats leave an oily scent on a blade of grass; beavers leave a distinctive yellowish oil, castoreum, right on top of a pile of pond muck that they've gathered on the shore. Otters do them one better, making a whole scent-region by the water's edge by rolling downhill and defecating to top it off. Snowshoe hares even each-other mark: when courting, one leaps over the other, balletically, while spraying his partner. Dik-diks paw a communal dung heap, then scent-mark their path with their footprints; spotted hyenas leave anal-gland scents and scents from between their toes and use communal latrines at the edge of their hyena town. A cat \"bunts\" a post with his face, spreading odor from glands on his chin and cheeks. Both the hippopotamus and rhinoceros deliver a forceful urine mark; the rhino projectile-pees into a bush he's just torn to pieces with his horn, for good measure. The badger \"bum-presses\" a scent on the ground; the mongoose and female bush dog have been known to do handstands to spread their urine or anal-gland scents.\n\nWhen scent posts are at the edge of social animals' territories\u2014areas often patrolled and defended\u2014it seems apt to call them territorial. By fencing your property, you are indicating that anyone who steps over the fence is trespassing. No other signage needed. But many posts are not left on an animal's territory at all: they are left as one walks a new area, or in a shared social space. Marks on what are called runways, or shared paths or trails, are not about claiming the path. In these cases the mark is probably conveying social information about who left it and the kind of animal he is.\n\nAfter that first marking, some species counter-mark: peeing or rubbing over the mark another animal has left. While eau de spotted hyena might carry well on the breeze, most animals will bother to go right up to the fence-mark and examine it closely first. Any counter-marks may be seen as a challenge to the territory-holder, or a kind of response to the social-information leaver: Here I am, too. We know that counter-marking is not simply territorial, because it is not regularly followed by territorial challenges or relinquishments. It is also socially competitive: among house mice, the top-marking counter-marker is often the most popular house mouse.\n\nSo what about with dogs? Both wild and domestic dogs conspicuously mark and counter-mark objects. This is typically achieved through a sometimes gymnastic \"raised leg display\" (what scientists, in a fit of unneeded acronymizing, call the RLD), balancing on three legs while raising the fourth high. Done by both males and females, the RLD allows for a directional flow of urine, aimed to land (whether it does or not) on a vertical or semivertical object. Note that marking is not just peeing willy-nilly: urine-marking is the leaving of just a spray here or a mist there.II\n\nHere's a surprise: in contrast to these other marking animals, domestic dogs do not mark territorially. Yes, you read that right. Dogs are not \"marking their territory.\" How do we know this? Simply by looking at where dogs do\u2014and do not\u2014pee. Owned dogs do not mark the periphery of their homes. The apartment-living dog does not pee along the walls and threshold. (Does yours? That's another topic . . . ) When living in fenced suburban yards, dogs do not assiduously line the border of the property with pee. Research in India on the massive free-ranging dog population there\u2014stray dogs who actually might have home territories at risk of being wandered into by others\u2014found that they, too, rarely mark on the boundary of a territory. Dogs walked along shared paths and parks could not verily consider these areas their \"territories,\" given the occasionalness of their occupying them\u2014and indeed they show no accompanying behaviors that would indicate that dogs feel that a path is \"theirs.\"\n\nInstead, dogs are great runway markers. Consider where your dog pees: on the lamppost along a shared path, the one tiny bush along the country road, the garbage pail at a driveway's end that wasn't there yesterday. Notably, they spend a lot of time sniffing all the possible marking sites, but they do not counter-mark all of them. A sniff may be followed by a quick look around, a scratching at the ground, or even a chattering of the teeth\u2014part of smelling the hormones in the urine.\n\nSo what are dogs saying to each other as they load a fireplug with layers of pee? Most likely, it is social information being left. Calling it pee-mail is not far off. They are telling each other who they are, and, intentionally or not, revealing a host of other information: their sex, whether they are a female in estrus, what they ate, how they're feeling, their health. The few studies asking how and when dogs mark found that non-neutered male dogs marked more, counter-marked more, and teeth-chattered more than their neutered brothers or than females. But all mark and counter-mark (although some \"adjacent\" mark, wildly missing the target, intentionally or not). Simply the amount of time that dogs, left to their own devices, spend sniffing indicates that the mark has reams of information.\n\nBut the dog scent-mark is the perfect graffiti: you must have a secret nose key to unlock its specific message. Thus far, human researchers have failed to decode the mark. Part of the reason for this is that we are not asking the dogs: while few animals answer questions about a behavior in easy-to-parse sentences, often what he does after the behavior serves as an answer. If a firefly flashes his light three times and a dozen female fireflies come hurtling toward him to mate, we get a pretty good idea what \"three flashes\" roughly means.\n\nThis is the reason that I applied to the NYC Parks Department to do research in their parks. While it would cost nothing, disturb no habitat or organism, and be unobtrusive, the proposal was, well, unusual. I proposed to put a \"pee post\" in one of their parks, Riverside Park, and watch what happened. How many dogs would sniff a post that's been peed on? How often do they over-mark? How high and accurate are they in their aim? Do they come back to see their own handiwork? And what do they do after sniffing and peeing?\n\nSix weeks later I learned that my proposal had been accepted. On a London plane tree I mounted and secured a camouflaged motion-activated camera. I aimed it toward a short post\u2014about the size of a standard metal fencepost\u2014which was conspicuously placed by a popular dog-walking path.\n\nFor a week at a time the post brought curious noses, and the camera witnessed it all. What it captured was dogs taking in the information in marks\u2014and leaving marks for others\u2014but rarely following through. Dogs who sniffed marks along the path often then looked back and forth on the trail for the mark-leaver. When the other dog was still nearby, they showed signs of wanting to pursue him. But, as a species whose walking route is usually determined by a person, not by themselves, they were usually prevented from pursuing the good-smelling dog by a leash. Counter-marking was surprisingly rare: sniffings outpaced markings. When a dog marked, it seemed to take the place of actual interaction: If I can't sniff you, I'll leave my card for you here. But even off-leash dogs never returned to see if their messages had been scribbled over. As for what the dogs read in the bulletin board of marks that quickly dotted the post, that remains a mystery. Without a territory to guard, they still post tiny, odorous flags. But they never check who is saluting.\n\n### ROLLING IN IT\n\nAs a scent-mark gets the animal's scent on the ground, scent-rolling or -rubbing gets the ground's scent on the animal. Or, rather, the scent of whatever-it-is that they are rolling in. Dogs are unrepentant scent-rollers\u2014and the scents they roll in are often extremely stinky. In this they share their tastes with other mammals: a list of common desirables includes \"meat (fresh or decaying), vomit, intestinal contents, cheese, engine oil, perfume, insecticide, and the feces of other species.\" Other researchers have added \"raisins, beetles . . . cigarette butts, hard candy, human bed pillows, and many spots with nothing detectable to a human nose or eye\" to the list of rollables.\n\nCommonly, a dog sniffs at the scent, getting right down into it\u2014although in the case of decay or discharge this seems a bit unnecessary\u2014and then hurtling his head or shoulder into the source, followed by his neck and back, often squirming supine with apparent glee. I am reminded of nothing more than the cat's enrapture with catnip. In both cases it seems to spasm the neural circuitry responsible for play, sex, and eating all at once. Thus both species may roll ecstatically, bite and paw at the scent, and rub their faces in it.\n\nWhy the dog does this is the source of uncertainty by scientists and owners alike. Theories come in a few flavors. One is the \"camouflage\" theory: by matching their own bodies to the environmental odors, they are more likely to be seen as territory holders. African wild dog females roll in the urine of the males whose packs they then attempt to join: they may be more likely to be accepted if they smell like home. Another is the \"popularity\" theory: it might enhance their social standing to be covered with something as clearly desirable as a very, very stinky stink. Spotted hyenas who were scented with carrion on their shoulders got more grooming attention by their pack than those who were dotted with camphor. Finally, the hedonic theory: it is simply pleasurable. Perhaps by festooning themselves with the perfume of a decaying animal, the fragrance can be enjoyed later. New sources of smell are particularly interesting: should you want your dog to smell like your perfume, try pouring a little on the grass. Next to the days-old cat poo.\n\n\u2022 \u2022 \u2022\n\nThese are but the most ordinary examples of dogs' use of smell. But then there are the seemingly extraordinary dogs, the detection dogs, weaned off pee-marking and now smelling professionally. They smell things we not only cannot see, but often things we cannot imagine.\n\nDetection dogs have been trained to find just about anything, from the lock to the stock and barrel. Dogs find, we know, explosives, accelerants, and land mines. They find missing people, still alive\u2014and cadavers, on land or underwater. They can smell out drugs and counterfeit goods. But they can also detect illicit cell phones in prison and imported sharks' fins in suitcases; termites, fire ants, and the red palm weevil, which kills palm trees used both ornamentally and for their dates; screwworm, nematodes, and bedbugs; invasive knapweed in Montana and invasive brown tree snakes in Guam; the hard-to-spot Northern right whale in the sea and the Amur tiger on land; the feces of black bears, fishers, bobcats, maned wolves, bush dogs, and turtles; birds killed on wind farms and dairy cows who have come into estrus. Provided that it has an odor, it can be smelled out by a dog. There are now dogs put into service to find other, lost dogs.\n\nAs we will see, these are extraordinary acts, but these are not necessarily extraordinary dogs. Any dog can perform surprising feats of discovery and identification. But only some dogs have learned to bother to tell us if they notice where a missing person has got to, or if a traveler is bringing a single guava across the border. Those are the ones trained in persistence and communication with their handlers, but they all have the same quality nose. Dogs seek out smells, they roll in smells, and if we ask them to, they act on smells.\n\nThis has made me realize what else in my dogs' lives has an existence through smell. Things that smell before they appear, things that do not appear to have a smell. Things that we might think don't smell (us) and things that don't smell as we think (them). Things that start to draw a different picture of what it might be like to see the world nose-first.\n\n### SMELL OF US\n\nOne of the cheeriest genres of YouTube video compilations is that of the short, gleeful scenes of dogs greeting soldiers returning home. Whether the owner's deployment was long or short, the dogs erupt into that singular, irrepressible doggy celebration: bounding, tails wagging maniacally, whimpering, rolling and wiggling on their backs, grinning, spastically weaving under the soldiers' legs or between their arms\u2014sometimes all at once. There is little doubt that the dogs remembered, loved, and missed their people.\n\nIn some of the videos, though, this recognition is at first in doubt: as the (often uniformed) person arrives or enters the home, the dog barks, approaching guardedly, tail down and ears back. He does not know this person. But then comes a magical moment of transformation. Look closely at the videos . . . pause on the frames when the dog's allegiance is in doubt. Watch his nose. Each dog lifts his nose in the air, catching a whiff on the breeze. Or sniffs first, one offered hand, then the other. In an instant, the stranger is transformed\u2014into the person the dog knows and longed to see.\n\nTo our dogs, each of us is encircled by a cloud of scent that is as familiar to them as our image in the mirror is to us. We are our scent\u2014and it is not the smell of our shampoo. Your dog would have no trouble picking you out from a line of people, even if you didn't dissolve into cooing, bend down, and tickle his head. Instead, he could pick out the particular human bouquet of you that is a mix of oleic, palmitic, and stearic acids; trained dogs notice if any of the ingredients in the mix is a few micrograms more or less present.\n\nYou may see a glimpse of evidence of this in your dog's anticipation or seeming foreknowledge of when you are coming home. Because such a perception is foreign to us, some label psychic anything the dog foresees\u2014the hour of our return home, the earthquake that we notice only with the shakes. But with our smells and sounds, we announce ourselves with olfactory cowbells and the noisiness of a skunk's spray. To the dog, we arrive before we do and stay after we leave.\n\nMany people have pointed to dogs who seem to \"know\" when their owners are returning as a special kind of skill. I think, rather, it is the special skill of smell. A few years ago I concocted a thought experiment to test just how much smell mattered to a dog in sensing when his owner was returning home. Rather than assume that the dog was smelling or hearing the owner through the door, I suggested that there was a potent combination of two forces leading to these dogs' abilities. The first is the distinctness of our smell to our dogs. The second is the ease with which dogs learn our habits: we leave and return reliably\u2014if not always at precisely the same time. So, how can your pup know when you are returning from work, given that the sun sets at a different time each day? Well, it might be that the odors that we leave around the house when we leave lessen in a consistent amount each day. Over the hours that we are gone, our home begins to smell less of us. We could test this, I suggested, by bringing in a \"fresh\" smell of the owner. If the dog then assumes that the owner has just left, he will be surprised if the owner then returns.\n\nThat is just what happened. Working with a couple whose dog seemed to have magical powers of prediction as to when his owner returned, producers of a science program arranged to have a smelly t-shirt from one of them snuck into the house many hours after he left. Thus to the dog the house then smelled more strongly of the owner\u2014as if he had left not long ago.\n\nSure enough, the dog was not waiting by the door when his owner came in, as he usually did. He was snoring on the couch: surely it would be hours before his owner returned, with that strong smell in the air . . .\n\n### SMELL OF THEMSELVES\n\nIt seems reasonable to suggest that dogs know their own smell. And, moreover, that they like it: any owner who has watched her dog roll in the dirt after getting a hose-down can attest to their interest in not smelling like shampoo. On the other hand, does this indicate that they know it is their smell\u2014or is it just that they don't like the green tea shampoo that you chose?\n\nIn other words, do they have a sense of themselves\u2014of who they are? In animal cognition, this is the question of \"self-recognition,\" which is considered a fairly complex cognitive ability\u2014and which requires more than an observation of post-bath rolling to prove. The only test of self-recognition that has been reliably used with animals is the famous \"mirror mark\" test. In essence, if your face or body has been subtly marked or changed, will you move to examine the mark when you spy your reflection in a mirror? While we would expect that every normal adult would\u2014and does, after eating spinach or poppy seed bagels\u2014we are not born with self-recognition. But by the time a child is eighteen months old, he will reach up to remove a sticker an adult has surreptitiously placed on his head when he sees it in a mirror, thus passing the test. Chimpanzees pass (after being inked on their foreheads), an elephant named Happy passed (when an X of tape was placed above her eye), and captive dolphins pass (by doing bodily convolutions in order to examine the ink marks in reflective glass).\n\nDogs do not. Imagine showing your dog the mirror when his face is covered with stickers. He will, no doubt, express indifference. What looks foolish to us is not of moment to him. But this is not sufficient evidence to say that dogs fail the test and thus have no sense of themselves. For one thing, dogs do not groom themselves (like primates) and show little concern for maintenance of appearance. So they are simply unlikely to want to correct an errant mark on their faces. Neither are they visually oriented as primates are. While the mirror test is appropriate for some species, this paradigm offers challenges for dogs, who show little interest in a mirror.\n\nSome research hints that dogs might nonetheless be able to pass such a test, if a kind of olfactory mirror were designed: something that smelled like them, but was a small bit different. While walking his dog in the winter in the foothills of Colorado, researcher (and my colleague) Dr. Marc Bekoff wondered if every \"yellow spot\" in the snow was equally interesting to his dog Jethro. Bekoff began carefully noting where his dog peed and where his dog sniffed. He even ported some yellowed snow to new locations to see what happened. He found that Jethro avoided smelling his own urine but smelled others': a kind of recognition of himself, written in the snow. Jethro looked as if he were recognizing his smell, but I decided to put this hypothesis to a formal test. I wanted to see if any dog, smelling a reflection of himself, thinks, That smells like me.\n\nTo do this, my research lab and I set out to create a kind of olfactory mirror. Instead of a reflective surface, we used a canister exuding odor. When you look in the mirror, you see yourself. When you smell the canister, you smell . . . yourself. I used a dog's own scent as well as a revised \"scent image\": the scent, altered (or \"marked\"). We were asking whether dogs would tell the difference\u2014and whether their marked self would be more interesting to smell.\n\nThis is how it happened that we started collecting pee.\n\nOne might not ordinarily think about handling, examining, or, perhaps especially, presenting urine to dogs. But pee is oddly central to our lives with dogs. Not only is it the great communicative medium between dogs; it is also a major part of the dog-human relationship. Perhaps you live with a dog in an apartment. Undoubtedly you must take your dog for a series of walks\u2014or, at minimum, outings\u2014for her to urinate outside. Should you be unable to get home during the day while working elsewhere, you might have a dog walker come and \"out\" your dog. If you live in a house, the dog needs to be let out in a timely fashion, or you need to design a way that he can let himself out.\n\nMy own social life as a young adult was to a great extent arranged around the fact that I needed to get home to take my dog, Pumpernickel, out to pee. Of course, I also wanted to give her the social companionship and exercise she needed and deserved as my loyal and bed-warming companion. But surely some of those outings were just about urine.\n\nThen we leave it, of course. Dogs urinate, and we literally leave it.\n\nUnless you begin to duck under their bellies as they squat\u2014or reach around a raised leg\u2014and stick a little cup in the way of the yellow stream. Then you gather it.\n\nAnd that's just what we began to do. For if urine is so important for dogs, it is important for dog scientists.\n\nGathering urine is trivial work for a dog owner accustomed to handling a warm, soft pile of her pup's excrement with only the thinnest of plastic bags between skin and poo. Still, it takes some getting used to, for both persons and dogs. I established a methodology with my lab manager, Julie, who grew to have a proclivity for the task. Finnegan was our test subject. We went out for a walk together and both eyed him closely. He did not seem worried that Julie had latex gloves on and carried a sterile, orange-capped plastic cup. As Finn's owner, having accompanied him for thousands of walks, I could predict to a fare-thee-well when he was about to pee. Every owner, I suspect, has a strange body of knowledge about the set of behaviors that precede her dog's eliminations. So when I saw Finn take aim I raised my eyebrows at Julie, who ducked a cup midstream. Caught.\n\nAs we proceeded, there were considerations we had never imagined. First off, we needed to strategize the best way to scooch under a dog with a plastic cup without, well, the dog getting startled by a person reaching under him with a plastic cup. Some dogs balked at this\u2014though they had given their urine away freely for years, they suddenly had proprietary inclinations. Or, more likely, that person's arm was just awkwardly close to their privates.\n\nThen we had to determine just how much pee was enough to be smelled. We wildly overestimated on our first go: a small cotton ball dampened with pee. I called Finnegan over, showing him something in my gloved hand. He trotted right up to me. But on arrival, his reaction was quick and decisive. Have you seen the \"disgust look\" of a dog? A wonderfully clear judgment had been levied.\n\nEventually we got the proportions right (down to minuscule amounts of urine), and we also recruited owners to do the \"materials\" gathering for us. When they arrived at our experimental site, we made the first sample: a tiny bit of their dog's pee in a closed container with airholes on the top. A second sample had an added smidgen of diseased tissue from a deceased dog (specially acquired from a veterinary school postmortem). Other containers held the urine of an unknown dog, or the pee of a friend. The containers were paired and laid out in a room with no other visual cues that anything or anyone had been by. Would the dogs find these smells equally smellable?\n\nThirty-six wagging, highly cooperative dogs and owners participated in this very odd experiment for us. We have videos from various angles showing each nostril twitch, each eyebrow raise of surprise or alarm. We timed where they sniffed, how long they sniffed, and how many times they came back for another look. And the result: dogs passed the test. Our subjects spent way more time sniffing their olfactory image when it had been marked, as though looking at the mirror most when there was something funny between their teeth. They looked at (sniffed) themselves, to be sure, but not as much as when something was different about that smell. The smells of other dogs were also highly sniffed: as when another person looks over your shoulder in the mirror. You look back.\n\nAs in any experiment, some subjects offered additional behaviors, not falling under the strict rubric of the experimental design. Chimpanzees in the mirror mark task use the mirror to check out parts of their bodies they can't usually see: the insides of their mouths, their own rear ends, their nostrils. They make great faces at themselves. Our dogs did the equivalent. They scratched at and licked the containers. They turned back to their owners with looks of despair or excitement, sharing the news of this peculiar smell before them. Sometimes, I'll admit, they counter-marked the container. Any good dog cognition lab has lots of paper towels and disinfectant around. But they peed only on other dogs' containers, not their own. They saw themselves.\n\n### SMELL OF THE NEW DAY\n\nApart from the concrete\u2014recognizing themselves or us\u2014the dog's nose notices a world of abstraction. Why, to a dog, the new day has a smell.\n\nYour morning dog walk may follow a route around the block, to the park, down the driveway and back. We take our dogs on these walks precisely because of their familiarity, their reliability. We humans are creatures of habit, and we assume that dogs are as habitual\u2014and are experiencing the same habit, at that.\n\nThey almost certainly are not. Each departure from the house brings a new scene, one never visited. Each day, each hour, wears a new smellscape. For as people pass, shedding their scents and foods; as cars warm the street and line it with particulate dust; as the clouds coat the scene with rain pulled from alpine lakes; as breezes carry the scent from downtown uptown and the seeds from forest to plain; as the universe of animals, from bug to bird to dog, passes and leaves footsteps, excreta, and skin\u2014the world outside your door is transformed. There is no such thing as \"fresh air\" to a dog. Air is rich: an olfactory tangle that the dog's nose will diligently unknot.\n\n### SMELL OF TIME\n\nAs each day wears a new smell, its hours mark changes in odors that your dog can notice. Dogs smell time. The past is underfoot; the odors of yesterday have come to rest on the ground. Carried by the morning's first breath or sloughed off the backs of nighttime animals, the message lies on the doorstep with the folded newspaper. The odor of the future is carried from around the corner, reaching the dog's nostrils before reaching our eyes. Smell rubber-bands time for dogs, pulling some of the past and future into the now.\n\nMuch of this information is on the breeze. Heads out the car window, ears sailing on the updraft of air, dogs enter into the breeze.III We are distracted by what we see of the wind: its effect on our skirts and hair, the flag's crisp responsiveness to the vagaries of the breeze. The dog experiences the wind from within, the messages it carries and the stories it brings from faraway climes. It brings word of a storm's arrival to the dog's nose before to ours. When low pressure arrives, and the air above the ground feels extra roomy, the earth loosens its grip on the odors it holds in reserve and begins to exude aroma into that space. Dogs, inadvertently trained by us to care about associations\u2014to notice that \"picking up the leash\" precedes \"going for a walk,\" that \"scraping my plate\" means \"tidbits from dinner\"\u2014can easily learn that a bouquet from the soil means \"a storm's afoot.\"\n\n### SMELL OF IT ALL\n\nFaced with bedbug or thunderstorm, the dog notices the details. You would not recognize the smell of your house from your dog's point of nose. Yes, your home itself has a smell. With few exceptions, every item of furniture, every book and plate, every throw pillow thrown and table lamp tabled has a smell. Each exists in the visual world that we see, with a straight edge demarcating the space it occupies and the space it neighbors. But in the world of smell, the edge is fuzzy. It is clouded, and its cloud morphs from lamp to the shape of a bunny to train as it is touched or moved, as it is warmed by its light bulb quivering or cooled by its sleep.\n\nIt is not that dogs can smell each molecule of their environment. To appreciate their abilities one need not hyperbolize. But the \"roseness\" of a rose's odor profile may be less important, relevant, or perceptible to the dog than some of the other notes: citronellol and rose oxide compounds, say\u2014a smell that sometimes is likened to metal or urine. Imagine being able to distinguish the parts from the whole, the past from the present, only with your nose.\n\nHow do they do it? The story of how those odors get into his mind, form the substance of his thoughts, and send his head upward in search of the telltale odor that his owner is approaching the door, begins with his konker, his snuffbox, his snout. Ladies and gentlemen, let me introduce you to the inside of the dog's glorious nose.\n\n* * *\n\nI. Although, how terrifying that might be for the suspect! And how potentially effective: surely a suspect would release those odorant molecules from each and every sweat gland available. Unfortunately, so might any simply anxious fillers waiting for their turn as sniffee and wondering what secrets this seemingly miraculous dog might have.\n\nII. Physics researchers have found consistency among bladder-emptying peeing, which they dubbed the Law of Urination: on average, all animals, from dog to vole to elephant, take twenty-one seconds to pee everything out. Marking, by contrast, takes one second or three.\n\nIII. \"The blind person enters into the windiness of the day at first hand,\" John Hull wrote about his experience of being blind.\n\n## 3: Sniffing the Wind\n\nUpton is a scent-on-the-air dog. He stands and sniffs in the breeze for many minutes, his head proudly up, eyes with a faraway look, cheeks puffing slightly with every exhale. I almost feel I can see his moist nose, lightly spritzed with mucus, catching odorous words from distant callers.\n\nTo understand how the dog can do what he does, you've got to follow his nose. What a dog is experiencing is formed of what he is smelling: as we see images in our minds, he glimpses scents; as we speak in words, he communicates in fumes.\n\nHave you toured the dog nose? Ridden on a corkscrew of air into the dark vault, bumped along its curves, caught a breeze up to the chamber where a molecule will settle into the wetlands and begin to tickle the nerves to the brain?\n\nI have\u2014at least, near enough for my liking. I've been down the nose in a simulation of an odor molecule's point of view as it is sniffed in. This unlikely video was created using airflow models generated by Dr. Brent Craven, who does work in computational fluid dynamics. Craven doesn't have a stake in dog olfaction, as such. His research into how fluids and air move is primarily basic science, geared to understanding animal systems. Its application (and grant money) is often to reverse-engineer particularly good noses, by using airflow models for the creation of artificial noses for military purposes.\n\nThe nasal airway Craven and his team modeled is actually that of an eastern gray squirrel\u2014a perfectly good smeller, but \"much simpler\" to model, he said. Imaginatively superimposed with MRI images of the innards of the more complicated canid nasal cavity, though, the video provides a sense of what a bumpy, hurly-burly, complex journey it is from the point of view of the sniffed odorant.\n\nYou're riding on an odorant molecule. A miniature soap bubble comfortably bouncing and sailing on the briefest of wind currents, small and light and volatile. You're in the vicinity of a dog nose, then suddenly, abruptly closer. The nostrils gape wider and approach. The odor bubble is sucked inside. Its speed is immense. Terrifying roller coasters could be modeled on the beginning of a long-nosed animal's nasal cavity. For after a quick ascent, slowing as the path steepens, the odor's arrival at the peak brings a heart-stopping view: only air ahead. A rush downward, curving this way and that, slowing for a second, then resuming. New projections pop from the walls as the odor races toward them, forcing you to bounce to your side, hit your head on the ceiling, and drop with a stomach-lifting thump. There are convoluted curves, perilous edges, and, all along, a gravity-defying force pushing you ever deeper. A tenth of a second later (slowed down two hundred times in the simulation), you are summarily deposited in a marsh well stuffed with moist grasses standing soldierlike, awaiting your arrival.\n\nAnd this is all before you get to the brain.\n\n### NOSTRILY\n\nFinn snortles up close to me to wake me. I reluctantly peek out with one eye. Its field of vision is entirely taken up with Finn's right nostril, twisting this way and that, as we might do to make a silly expression. The result is that I twist my face into a smile and rise to greet him.\n\nWhat has happened? At its essence, olfaction begins with the detection of as little as a few molecules of an odorant. The detector is the nose, the Hoover vacuuming in that trace amount. So let's begin there.\n\nIf you have a dog near (and I hope you always do) bend down and take a good look at the dog's nose. Get close, really close\u2014one hopes, with a friendly dog who, at worst, will greet your examination with a peremptory lick of your own neb.\n\nThe hairy muzzle of the dog is what we usually look at\u2014the snout, reaching back to the eyes and sloping on either side into jowls. Its length matters: while it isn't the part of the nose responsible for the detecting, per se, nor the sniffing, it is the well-designed hallway down which the sniff will tumble.\n\nOr we look at the moist, hairless tip of the snout, the so-called planum nasale or rhinarium. This is where the action begins. Dog nose tips are mesmerizing: as different as thumbprints, and as full of individual details. What most wet nose tips share is the pebbled topography of their surface, polygonal cells made visible in their opal darkness. This surface is wet so that more odors can be picked up and absorbed into the nose. Moreover, they have temperature gauges that can direct them toward a cooling breeze that might carry odors.\n\nAnd their nostrilness! The nose tip carries two gaping nostrils\u2014or nares\u2014which will lead to the real nose an inch or six north, where special tissue lines the nasal canals. For those who find it hard to love a nostril, I say: You haven't spent long enough looking at your dog's. While to the uninformed they are just \"holes in the front of the nose,\" to those who study how dogs sniff, they are \"articulating aerodynamic inlets,\" their bulbous shape facilitating high airflow. Ringing the nostrils are the highly developed muscles in the alar folds. It is this anatomy that allows the nostrils to be an active part of the sniff, and that gives nostrils such a terrific range in shape from dog to dog. Many look like commas curled tight, or rounded into disks. Others are mere blunted openings, as if drawn hastily with a large-tipped pen. In some countries a noseprint is used as an identification: the inexpensive microchip. Ink the thin outer layers of the nose tip and press it against paper. A dog's noseprint is the newborn's footprint all parents will stare at for some clue as to their progeny's future.\n\nDogs can use their nostrils separately and differentially: when sniffing something new and \"nonaversive\"\u2014neutral or likeable odors\u2014they begin with the right nostril, then switch to the left. By videotaping down the length of a cotton swab dabbed with an odorant while dogs sniffed, researchers found that lemon, food, and the secretions of a female dog elicited this right-then-left sniffing. When faced with the smells of adrenaline or the sweat of a kennel veterinarian (who swabbed his armpit for the sake of science), though, dogs sniffed only with the right nostril. It is thought that this nostril preference is due to the side of the brain involved. The right nostril connects to the right hemisphere of the brain (ipsilaterally, in contrast to the other senses, which cross sides)\u2014which is more often associated with fear or aggressive behavior than the left, which analyzes familiar stimuli. If a dog is sniffing you right nostril only, he may be feeling suspicious.\n\nBack to those muscles around the nares. Outside in a mild breeze, I watch as my dog Upton turns his nose\u2014just his nose tip\u2014to the right and left, bending in a way you don't think a nose should or will bend. His nose gymnastics get the nostrils facing the breeze, picking up turbulent air, and snagging an odor. Then they open the aperture of the nostrils wide to maximize the amount of air that is pulled inside. An odor looks inside the business end of the telescope that is the nostril and closes its eyes for the ride.\n\n### SNIFFER\n\nWhen I return to the house after an absence, I bend to Finnegan to let him sniff me, and to let me hear him sniff me. His sniff-snuffle is almost communicative\u2014nosed phonemes in strings of merry sentences.\n\nWhat we know about what happens when the odor faces its future up the dog nose comes largely from the work of mechanical engineer Dr. Gary Settles, one of Craven's former professors. Settles, now emeritus at Penn State, has brought fluid dynamics\u2014a field associated more closely with designing smooth-flying airplanes\u2014to the study of noses. To my great delight, what Settles studies is the sniff. He and his team ask dogs to wear a specially designed muzzle, give them good things to smell, and measure the fluid dynamics of their sniffing.\n\nSettles speaks of the dog's nose as a classic \"variable-geometry aerodynamic sampler.\" The \"sampler\" (the nose) approaches a vapor cloud (air with smelly stuff in it) and transfers that cloud into its internal sensor chamber (the back of the nose, where the neurochemical magic happens).\n\nThe transfer happens through sniffing. As an invisible means of pulling in what are invisible odors, the sniff has long been downplayed. Further, while vision happens through light \"hitting\" the eye, in olfaction, odors do not just hit the nose; they sail right in. For that reason, smell feels intrusive. It is common for people to suggest that this is so because if one is alive and breathing, one can't help but also be smelling: a gulp of air enters the nose with each breath, carrying, presumably, various odors. While it's true that a nose-sized swig of air is bound to have smellable stuff in it, it's not true that smelling happens by being alive and having a nose. It turns out that to smell, one needs to sniff. Not just breathe, not just sit with nostrils open. In the mid-nineteenth century, the start of the heyday of scientific discovery based on self-experimentation,I the physician Ernst Heinrich Weber, a founder of modern experimental psychology, first demonstrated the importance of the sniff. Weber lay down, had a solution of water and cologne poured into his nostrils, and waited, motionless. This is just the kind of experiment for which it is hard to recruit a lot of volunteers. What Weber found was that the normally ripe cologne was undetectable: he could not smell it when it was passively introduced to his nose.\n\nWeber subjected himself to a noseful of cologne in the interest of discovering whether smells need to be delivered in vaporous form or can be detected in liquid. His was not the final word. A hundred years later, researchers renewed experimentation on the means of smelling: some repeated Weber's supine nose-filling experiment; others presented strong smells before a person blowing through his lips like a trumpeter; and still others tried injecting odors intravenously to people during sleep apnea, between breaths. These studies had one common design element: in no case could the subjects sniff. The results? None of the bemused subjects perceived any odor. During typical breathing, only a very small amount of the air you inhale makes it up to the part of the nose holding the cells that will receive and manage the smell. One simply must energetically sniff air in.\n\nSo sniff, already! Sniffing is the inhalation part of a breath (you sniff in, but not out\u2014that part's more rightly a snort), often audible, requiring a modicum of effort. The variety of sniffs in the animal kingdom is certainly underplayed. Elephants sniff strongly, with trunk hovered over a scent or raised in the air for a \"periscope\" sniff. Gerbils sniff rapidly, noses twitching; tortoises, by contrast, have a slow-motion exploratory sniff in which they extend their necks, orient their heads down, and open their nostrils. A small puff of dust may appear on exhale. New Guinea singing dogs even \"huff-sniff\" when pursuing prey such as voles in a hole or under vegetation, blowing vigorously out the nose before sniffing vigorously in.\n\nWhen canids first appeared, some tens of millions of years ago, their sniffing was enabled by the bellows that were the lungs, probably through a fairly straightforward nose. For fish, frogs, and reptiles, the snout is just a cavity allowing water and air to wash directly over the olfactory cells. This is still more complicated than the current invertebrates, many of which smell a lot, but not by way of sniffing smells in. Invertebrates have olfactory detectors outside their bodies, such as on antennae, so they must actually run into the odor source. Some\u2014like lobster\u2014send their sensory organ forward to sniff; others sniff simply by flying or wiggling their bodies and letting the air or water hit them.\n\nNot for dogs, that crude method. Dogs not only have multiple sniffs, at different rates and for different purposes, but they have an ingenious method of exhaling that aids their smelling.\n\nSettles discovered how dogs sniff by watching what he quaintly calls \"canine nares airflows.\" He recruited a handful of pets as well as detection dogs to come into the lab to be observed and recorded. There were golden retrievers, Airedales, Labs, German shepherds: all manner of noses, trained and untrained. The researchers put a variety of objects\u2014highly desirable food, some inedible and novel scents, even a tiny bit of TNT and the sweet smell of marijuana\u2014either right in front of the dogs or a distance away. Without being asked twice, the willing subjects sniffed. Detection dogs and pets sniffed alike, whether they were presented with pot or pet food. Both groups had two kinds of sniffs. When the smelly stuff is far away, unreachable, dogs do a \"long sniff\"\u2014a highly directional, air-pulling sniff that lasts two seconds. In the long sniff, nostrils dilate and the alar fold opens; the mouth may be slightly open. Imagine the regal-looking large dog on a hill, his chest forward, his nose in the breeze. That's the long sniffer. This sniff is often the culmination of a bout of sniffing, many weaker sniffs followed by a whopping one. In fact, one English pointer skilled at hunting birds was found to be able to maintain an extra long sniff. \"Sir Satan\"\u2014the only of the tested dogs who was willing to suffer a sensor fastened into his nostril\u2014maintained a continuous inhalation through his nose for forty seconds while running into the wind after a scent.\n\nOn the other hand, when the smell is nearby and on the ground, dogs sniff in bursts. First, they scan the surface. Have you ever noticed your dog nosing for a toy in a grassy area and seemingly not finding it, though she's right on top of it? Her nose isn't deficient: she is simply doing a survey of the area, as we scan a scene with our eyes. Typically a dog will walk her nose right up to the toy, actually pause over it, snort a little exhale, then continue on. To the vision-obsessed bipedal creatures nearby, it seems she's missed it\u2014rather stupidly\u2014but she has not. She returns to the toy. She is simply assessing the concentration of all the smells in the vicinity to find the strongest source\u2014something like a visual inspection of all the brunch buffet selections before you zoom in on the Belgian waffle (that you probably knew you wanted all along). Along the way, the dog does rapid \"short sniffs,\" from five to twelve every second, enough to make me hyperventilate just imagining it. This sniffing occurs at about the same rate as a dog's panting\u20145.3 tongue flaps a second, on average\u2014which is energy-efficient, and so might be their sniffing.\n\nA hemisphere of air from four inches away is pulled up and in\u2014this is called the nose \"reach.\" Given the chance, as every horrified dog owner knows, dogs will work to decrease that reach to a centimeter or full-on contact with the odor. Owners sometimes ask me, If their sense of smell is so good, can't they smell it from here\u2014i.e., at a polite, safe distance? But we mistake their noses for ours. They are not trying to detect it at all; they are trying to discern its contours, to take in all of its features, to make a measure of the smell.\n\nIndeed, it is at a centimeter out that the smelly air is drawn in fastest. At this distance, the dog can get different odor samples from each nostril, which wind up giving the dog a bilateral vision that is a kind of \"stereo\" olfaction. Just as the images from our two eyes are constructed into a three-dimensional image of the world, the differences in the strength of the smell image in each nostril help the dog locate the source of the smell in space, whether it's to his left or right, fore or aft.\n\nGiven this, asking why a dog is sticking his nose right in that dog's rump is the equivalent of asking why you like to experience van Gogh's The Starry Night from close enough to see each brushstroke\u2014rather than peeking in the doorway from the next room.II\n\n### CANINE AIR JETS\n\nBut the most profound difference between the dog sniff and our own sniffing comes when dogs exhale. Our exhalations go straight out the entrance, out through the in door, pushily shoving any new air out of the way and preventing it from getting in. This can be a terrific relief when you want to purge a horrible odor from your nose, but it also sends lovely odors out almost as soon as they have arrived. When a dog exhales, he creates what Settles calls, charmingly, \"expired turbulent canine nostril air jets.\" Through high-speed videography of nostril and air motion, Settles found that dogs create tiny wind currents by exhaling not straight out, but out the side slits of their noses. This strategy minimizes the odor displaced\u2014what Settles calls the \"sample blow-off\"\u2014by the puff of air. The wings of the nostrils flare, a nose-plane ready for liftoff, and expired air leaves through a sneaky side exit. Not only does it not push the odor out of the way: the exhale creates a puff that lifts more smelly particles off the surface and a suction that hurries the next noseful of smell inside the snout. The exhaled nostril jets are little rotating funnel clouds that pull Dorothy, her house, and her little dog, too, right up into the nose.\n\nRemember your dog's short, thoughtful pause over the toy she is seeking? This is a pause of moment. She is sending her \"expired air jets\" right onto the source. More clouds of odorous particles come up from the toy and the ground. These jets are essentially increasing the reach of the nose, blowing and vacuuming in sync.\n\nOne scientist I spoke with likened this kind of sniffing behavior to the circular breathing a player of a wood or brass instrument might learn to do. It is the sniff without punctuation, allowing dogs to get a continuous read of the world\u2014just in the way that we see the world without pauses while we blink.\n\nSettles saw this jet-enabled sniffing by using specialized video called schlieren photography, which uses mirrors and slow-motion cameras to capture a picture of airflow. The photos make warmed air visible as blurred clouds emanating from the nose and mouth. In slow-motion schlieren videos, dogs' snouts turn nearly prehensile, reaching and retreating to move the air; the muzzle seems to undulate, like a jellyfish pressing himself through the depths. But you can see some of this with the naked eye, too: simply watch your dog sniffing over a dusty plot of earth. After a particularly enthusiastic bout of sniffing some invisible thing, you can easily spot the puff of dirt and dust and odor pushed into the air and into the dog's face with his own snorty exhale.\n\nCould a sniff really be responsible for dogs' keen sense of smell? Well, it is one of a number of key components. We know this because of evidence from when dogs lose their nose: when they are panting. Hot dogs can't smell much. Dogs do not have sweat glands allowing them to release heat through pores across their skin. Dogs have only their waggy, pulsing tongues. They must pant\u2014and panting, schlieren photos show, forces out such a wash of air that no smelly air can get to the nose. The panting dog must close his mouth to sniff well.\n\nMy Finnegan's sniffing is audible, his own particular combination of huffing and grunting and spluttering. Outside, he is a nose-on-the-ground dog, trailing invisible stories in the grass. Give him a tree trunk that is popular with the local dogs, and he will serenade it with his rapid huff-splutter and his turbulent-jet exhales. At the other end of the leash, he will bring me to a stop with his nose set into an odor, his whole body cemented to the ground to allow the nose free smelling. Our Upton, the second dog, learned sniffing from Finn\u2014learned that it was fine in this house to stand fast and observe the scents before moving on.\n\n### SNOUT\n\nIf one didn't know which sense was most dominant in humans, it would take only a few minutes of observation to discover. Everything we want to perceive or examine, we parade before our eyes. Something to the side? We turn our heads (moving our ears away) so our eyes are nakedly upon it. Hear something overhead or underfoot? We don't try to listen up or smell down: we turn our eyes. Our faces, while giving equal billing to the nose and mouth, have an array of protective devices around the eyes\u2014eyebrows, top and lower eyelid lashes\u2014and many of us use our noses simply as a perch for large spectacles to help us see better. Do you see? we ask, as a stand-in for understanding. Not Do you smell? Do you taste? Seeing is understanding, for us. When we meet each other, we greet each other with our eyes: to not look is considered rude, even abnormal. When walking, we start to turn our heads\u2014and eyes\u2014toward a corner seconds before we direct our legs around it.\n\nSo, too, with the dog's nose. There is a fair bit about its simple position on the dog's body that enables finely tuned sniffing. The snout is plainly protuberant. Not by coincidence, it is at the end of a head which, via a highly flexible neck, can reach the ground\u2014where most odors lie. Dogs don't spend a lot of time worrying about sniffing treetops: they sniff objects emanating from and landed upon the earth. As well as parts of other dogs' anatomy that are, well, about nose-height.\n\nDog snouts are long for a good reason: evolution commits large amounts of anatomical real estate only to very useful causes. The cavelike, moist interior vaults under skin and fur are filled with air filters, humidifiers, and warming devices. When you're sniffing dog rumps and decaying squirrel carcasses for a goodly portion of your life, you'd better have an excellent air filter. Inhaled air is cleaned and conditioned before it gets to the back of the nose. It must be dressed and ready for courtly presentation, as in the back it will meet kingly neurons that travel directly into the brain.\n\n\"So, you've been caught by a nostril,\" Craven says. He's walking me, riding that intrepid odor, through the interior snout. \"The flow is pretty high in the vestibule,\" the antechamber when the odor gets in the nostril. The dog is pulling in with his lungs through the nasal pharynx; the air is turbulent and chaotic\u2014and then there is a fork in the nose. Inhaled air can go one of two ways: the breathing route or the sniffing route. If you go the respiratory route, you are heated up and humidified and proceed to the lungs. But if you're sniffed to be smelled by a dog, you take a whole 'nother route, a high-speed ride toward the olfactory region. A current of air zips along a labyrinthine, tortuous path past a series of thin, curvy bones called turbinals; in cross section they resemble a big brain, convoluted and branched, folded into a small space. The turbinals, too, are part of the cleaning system\u2014and, farther back, some are also covered with tissue useful for smelling. The turbinals create the roller-coastery pathways along which the odor rides on its sniffed journey.\n\nSince air flows only one way across the long olfactory road up the nose (odors then are exhaled via the breathing route or are broken down by enzymes), dogs may be able to do something extra fancy with their noses: sort smells into groups as they fly in. Some odors are absorbed more readily than others, which means they will be grabbed by the sensory cells earlier in their route through the nose. For instance, researchers found that one component of the explosive TNT\u2014DNT\u2014is snagged more readily than other odors; this may partially account for sniffer dogs' seeming ease at identifying it. It is more soluble, so dissolves earlier in the nose, than a molecule like amyl acetate (smells of bananas), which is itself more soluble than limonene (smells of lemon), which, like many other kinds of odors, will make it back to the olfactory recess before being absorbed into the mucus and finding a receptor to bind to. While we rarely ask dogs to find bananas for us, we could.III Given the layout of different sensory receptors throughout the nose, a dog can start to identify and discriminate smells in his nose before the brain even gets involved.\n\nToward the back of the channel the odor suddenly slows. The turbinal bones here are lined with olfactory epithelium\u2014brown tissue that holds the sensory cells, which will snag odors out of the air and is where the magic conversion from \"odor\" to \"odor you smell\" begins to happen. The cells are coated in mucus.\n\nThen, \"At some point, you will be deposited or absorbed by the mucus that's lining these airways, and you'll slowly diffuse,\" Craven warns. It's not as grim as it sounds. A thin lining, about ten microns thick, forms the transition from the outside air to the internal neurons. The odor you are riding will migrate through the mucus in a tenth of a second, about the time it takes for a sniff to travel from the nostril to the back of the nose. But for now you can relax.\n\n### RECESS FOR ODORS\n\nYou've arrived at the cul-de-sac of the snout, the olfactory recess. This is the part farthest back in the nose, at a point a half inch into the skull, just between the eyes. In the olfactory recess odors can linger, trying to find a sensory cell to nuzzle up with for many rounds of inhaling and exhaling. Dogs get a chance to really ruminate on the smells they've sniffed in before the air is scooted out.\n\nThe recess\u2014as well as some of the bones along the way\u2014is lined with the aforementioned epithelial tissue. When newspaper articles say that a dog's sense of smell is ten thousand times better than a human's, or one million times better, or whatever-exponent-of-ten times better, one of the most cited bits of anatomical proof is the amount of olfactory epithelium\u2014the expanse of nose with cells specialized for smelling smells. Though the comparative numbers are suspect (and quite variable for different odors), dogs are massively better endowed than humans in olfactory sensory cells. If his olfactory epithelium were spread out along the outer surface of the dog's body, it would completely cover it. In humans, ours would about cover a mole on our left shoulder.\n\nThe epithelium here is covered with a dense mat of cilia, little hairlike branches that extend from the sensory neurons. A few dozen cilia sprout from each nerve, and each is coated with dozens of proteins called olfactory receptor cells. Receptors do what they sound like: they receive odors. To do so, they stick, untroubled, right out there in the mucosal environment of the nose, perfectly designed to snag odorant molecules coming by on a sniff.\n\nDogs cram more smellability into their snouts at every level: they have many cilia on each neuron and more receptors on each cilia than humans do. Indeed, every dog has hundreds of millions more cells devoted to detecting smelly stuff than humans do. Dogs have from two hundred million to one billion receptor cells, depending on the breed, compared to the six million in our noses. In the dog's case, more nose mass also enables more kinds of receptors\u2014over 800\u2014that themselves can encode more information about the odors.\n\nThis number\u2014eight hundred and change\u2014gives researchers pause. The eye, conveyer of the brilliance of a sunset dancing off the clouds after a thunderstorm, uses just three receptors to draw this colorful scene in our heads. With eight hundred more receptors, the possibilities for the odor landscape are staggering. The number of odors the dog could detect could theoretically be \"billions,\" Dr. Stuart Firestein, a neuroscientist at Columbia University who studies olfaction, has written. But \"in fact, the question is probably not relevant, just as it makes little sense to ask how many colours or hues we can see.\"\n\nEven when odors land in their receptors, they are still undercover. The nose does not know what they are. There is no \"cheese\" receptor, activated by the Stiltons and cheddars my dogs nose out on our kitchen counter. Despite the dog's alacrity in locating the corpse of an unlucky squirrel in the park, there is no \"dead squirrel\" receptor. Simply, each odor activates many receptors; there is not one receptor for each odor.\n\nAlthough the means of reception has not yet been conclusively determined, the most popular theory of how it happens uses a lock and key metaphor. In this model, receptors are the locks, of various shapes and lengths, and the various molecules that make up odors are the keys. A related theory suggests that the reception of odors is less specific than a key in a lock; instead, it's more like a key in a pocket, in which many differently shaped keys can bind to a receptor and prompt it to fire. Apt for dogs, smeller of pockets.\n\nWhen biologists Drs. Linda Buck and Richard Axel won the Nobel Prize for their work in olfaction, it was for discovering the genes that code for these receptors. Amazingly, olfactory genes are wildly overrepresented in the mammalian genome. Dogs have around eleven hundred olfactory receptor genes, some eight hundred of which are operative.IV Keep in mind that your dog's genome\u2014the blueprint for making his entire body, from his charismatic curled tail to dark expressive eyes\u2014is just over 19,000 genes. Nearly 5 percent of the genome is committed to making smell receptors alone\u2014plenty of kinds of locks with which to smell the world's keys (or pockets).\n\nBreeds vary in their smelling abilities, and it may be because they have varying numbers of operative olfactory genes: boxers (short-nosed, compressing that turbinate-filled snout) have slightly fewer functional genes than poodles (long-nosed and respectable smellers). Though research has just begun on the topic, there is some evidence that specific genes might even be linked to detecting specific odors. In one study, a tiny substitution on a particular gene was seen in dogs who did less well on explosives-detection work.\n\nIf genetic differences lead to differences in detection ability, one may well ask, does this mean that certain breeds are genetically superior sniffers? Insofar as some undetermined subset of a dog's genes may lead him to notice a smell that another dog does not, yes. Whether the genetic difference in smell abilities is part of the genetic difference in breeds is another question\u2014an unanswered one, for now.\n\n### THE SMELL HITS THE FAN\n\nAll that work to get to the receptor cell\u2014but the odor is not done yet. The smell of dead squirrel\u2014or any other substance\u2014is discovered only when components of its odor, having snuggled into receptor sites, cause the neurons to fire\u2014changing voltage\u2014and sending an action potential down its length, leaving the nose and entering the nose brain: the olfactory bulb. Tens of millions of neurons converge into a few thousand bundles and sneak through small openings in the bone into the brain. Long ago, it was thought that brains did the smelling, and the nose was just the conduit. Even in the twentieth century, with the rise of brain\/computer analogies, it was not uncommon to hear the nose be called simply the fan of the massive supercomputer that is the brain. Now we know it's a bit more electrical. Santiago Ram\u00f3n y Cajal, an early and influential anatomist, mapped the route from the nose to the brain at the end of the nineteenth century, observing that nerves (carrying word of odors smelled) enter the brain, not the odors themselves.\n\nThe olfactory bulb sits right behind the back of the nose, jammed in under the frontal lobe. The nose is the quickest route to the brain: one neuron goes from the warmed, dinner-smelling, light-spritzing-of-dog-hair-wafting air in your living room to, on the other side, the highly conditioned environment of the brain. Before the olfactory bulb, all a neuron \"knows\" is that it's firing; in the bulb, axons from thousands of neurons from the same kind of receptors all converge on a single target site and flood it with activity. What this seems to enable is the cobbling together of a smell sensation. Just as an odor molecule is grabbed and deconstructed by the receptors, causing many to fire, it is reconstructed in the topographical layers of the olfactory bulbs. Little traces of evidence from many cells are translated into the fetid, cadaveric smell sensation that is your dog's discovery.\n\nOne might expect the dog's olfactory bulb to be humongous. It's not. But it is 2 percent of his entire brain, two pennies to the silver dollar of the brain. (In humans it is vanishingly small: less than a thirtieth of a penny.) And this matters: the olfactory bulb translates those receptor cells firing into something like an experience of an odor.\n\nFrom the olfactory bulb the odor information is zipped through the dog's brain on a massive and near-instantaneous search for recognition, how to feel about the smell, what memory it might awaken, and what behavior it might prompt. The bulb connects straight to the olfactory cortex for some of these decisions, as well as directly with the subcortical limbic system, which adds the emotional tone\u2014fear, excitement\u2014to the odor.\n\nResearch on the brain's response to odors has always had one common element: present the subject with something truly stinky. Measure the result. An early researcher studying the brain's response was physiologist Edgar Adrian, who exposed the humble hedgehog to various stinks, including the odor of decayed worms. \"Water in which an earthworm had been allowed to rot\" prompted a big reaction in the hedgehog.\n\nMore contemporary researchers have presented rabbits with store-bought sharp cheddar, lured tsetse flies with a combination of oxen and buffalo urine, exposed terrified rats to anal gland secretions from weasels and red foxes (rat predators), and, most recently, stuck the smell of their owners' armpits under the noses of dogs.\n\nTo back up. Neuroscientists can be fond of trying to answer the big questions about the mind with machines. MRI machines, when you are in one or looking at images generated from one, can seem superficially like mind reading. In \"functional\" MRI, a subject lies on a platform surrounded by walls housing a very powerful magnet. Through disturbance of the magnetic field, images of the blood flow in the brain (indicating neural activity) can be captured. Lie down, think of your grandmother, and the areas of the brain where your memory of her crooked smile and glasses, the smell of talcum powder, and the melancholy memory of the toys she made for you as a child reside will light up on the computer screen.\n\nfMRI will never answer the questions of what it is like to have those memories or to revisit her smell. Images of brain location do not explain how the odor of talcum prompts me to remember my grandma sitting in her special chair in her dark, stuffed living room. Instead, the device lets us view the activity from afar: watching the Perseid meteor shower without answering the questions of the universe. And so a few research programs have begun looking at dog brains in MRI machines. This is itself a feat of training and patience\u2014though quite doable for both species\u2014as the dogs need to be very much awake in the machine, and yet stay perfectly still. One early study looked at what areas of the dogs' brains were triggered by their owners' smells.\n\nBy \"owner's smell\" I mean \"smell of the person's armpit,\" caught on a gauze pad and waved in front of the dog prostrate in the MRI magnet. They found an area called the caudate nucleus dancing with excitement at the armpit gauze. Researchers had looked there because it's easy to visualize in the MRI, and it is associated with reward.\n\nI suggest that another approach to discovering where the olfactory information goes in the brain is to watch your own dog. If you live with a dog, you know what happens after dead squirrel is identified by the brain. No dog waits and considers the wisdom of his next move before doing it.\n\nThe dog rolls in it.\n\nThis jibes with a theory Stuart Firestein advocates. \"My craziest idea, you want to hear it?\" I do want to hear it. \"I think the actual percept of an olfactory object\"\u2014the sense of experiencing a smell\u2014\"may not occur until you hit something very close to or actually in the motor cortex,\" Firestein tells me. \"Because most of what we do in olfaction is very tied to decision-making: do you put this in your mouth, do you flee from it, do you fuck it, do you whatever-you-want-to-do-with it?\"\n\nIndeed, some signals from the olfactory bulb do go straight to the motor cortex, a part of the brain that controls movement. It directs the muscles in the dog's legs that cause them to buckle gracefully, the precise angle of the head to squish perfectly into the smelly stuff, the intensity of supine squirm necessary to distribute the scent down his back. Note that when the dog gets up from throwing his body on and thrashing about in that squirrel, the first thing he wants to do again is get another whiff.\n\n### VNO\n\nHe licks. He licks my knee, my face, my ear. He licks the air, tongue reaching for me, as he approaches. It feels like affection, and I smile. But he also licks the corner of the building, the very smelly-looking patch of grass, and\u2014sigh\u2014the cat's bottom.\n\nSo much nose! But in dogs, and in many mammals, the olfactory system is bigger than just the nose. Dogs have a kind of \"second nose,\" right under the bone separating the nostrils and above the roof of the mouth. Two scrolls of cartilage house what's called the vomeronasal organ\u2014snappily abbreviated to VNO by researchers wanting to save themselves four syllables\u2014which is as much a part of smelling as the nose is. Despite its evocative name, vomer describes the shape of the organ: ploughshare-like, the part of the plow that does the cutting. Secreted under the nose, the organ cannot be reached by simply sniffing; instead, odorants must dissolve into the tissue and be sucked inside. The pumping mechanism is prompted either by touching the molecule directly or through making a horrifyingly silly face called flehmen. Should you see a horse curl back his upper lip, seeming to scowl and shuddering a bit, you are witnessing classic flehmen: it pulls the odors back to the nasal tissue to be absorbed. To flehmen, the pig opens his mouth wide; the cat holds her mouth a small amount open, giving her a disconcertingly muddled appearance. A snake's forked tongue flicks to pick up odors and send them to each side of its VNO.\n\nMost dogs do not do a classic flehmen lip-curl, but they have their own methods. At times, after sniffing, a dog may wrinkle his nose in an apparent grimace and chatter his teeth\u2014that's a canine version of flehmen. Even better, they lick. The dog's extravagant, long tongue is good for reaching the bits at the bottom of the peanut-butter jar and cleaning your legs of post-exercise sweat, yes, but it is also a perfect mechanism for bringing odors to the VNO for investigation. Lick ground, lick nose, smell.\n\nWhat the VNO enables dogs to detect is a kind of molecule that the ordinary olfactory route often cannot, such as pheromones. A pheromone was originally defined as a signal sent between two members of the same species\u2014which causes the receiver of that signal to behave or develop in a very specific way. Androstenone, produced by a boar, causes a sow to assume, rather robotically, a mating posture; bombykol, given off by the female silk moth, wafts onto the antennae of the male silk moth and causes him to seek her out. Pheromones are used by an incredible diversity of organisms, from lobsters and rabbits to ants and bacteria.\n\nWhat makes these pheromones detectable by the VNO is that they are typically water-soluble chemicals, nonvolatile, low-molecular weight molecules. So are lots of other molecules, such as hormones and \"signature mixes\" that may hold information about an animal's identity or the particular family or pack it belongs to. The VNO receptors are tuned to be highly specific and sensitive, unlike the more broadly receptive olfactory receptors in the nose. The rump-to-face-to-rump sniffing dance between your dog and another dog is a chemical communication of their sex, readiness to mate, their health\u2014and also who each of them is. Each animal's urine and saliva may hold the same.\n\n### SNEEZER\n\nApart from the snout proper, the featured player on the face, the dog's sense of smell commandeers other dog features and behaviors for its olfactory ambitions. Not for nothing are bloodhounds so well ear-endowed. James Thurber's famous sketches of the breed have as much ear as head. And ears help make members of the breed such outstanding smellers. With his nose pointed down, the bloodhound's long ears\u2014over thirteen inches, in extraordinary cases\u2014sweep up odors from the ground for better smelling. They are dual fans attached to the face to direct possible smells to the smeller. Even the drool that hangs from a bloodhound's mouth might help to bring odors upward to be absorbed by the vomeronasal organ for consideration.\n\nI discovered two other ordinary behaviors involved in smelling through simply watching dogs. Many of my research programs with the Dog Cognition Lab are rooted in observations of dog behavior in natural environments\u2014for instance, in public parks, around people and other dogs. I have spent untold hours videotaping dogs playing together, then coding their behavior by reviewing the tapes one-thirtieth of a second at a time. I code the video in this super-slow-motion way because it enables me to see things we miss in real time. But there is another benefit: watching video at that pace defamiliarizes the dog. While it is terrific fun to watch dogs playing (believe me, it was hard to convince a dissertation committee that this might be actual work), our natural human tendency is to see the play but not see what is really going on. To see, but not see.\n\nWhen we see dogs, we are so immediately and confidently sure of what they are doing\u2014Oh, they're friends; look, he wants to join the play; she is shy\u2014that we preempt actual examination of what they are doing.\n\nIt was while out watching dogs in this way that I made a surprising observation: dogs wag their tails a lot while meeting someone new or greeting a dog they know.\n\nWait, you say: that is not new! Of course. We know what tail-wagging is for. A loose, jangly, high tail-wag is a sign of friendliness. A fast, low tail-wag indicates anxiety. This is true\u2014but simply knowing this makes it harder to see what else tail-wagging is doing. It is spreading scent. Whether intentionally or not, when a dog wags her tail, all the very fascinating odors (to a dog) from her anal sacs spread in a bloom around her body. She is not only telling other dogs how she's feeling; she's scenting who she is.\n\nThere is precedent for the use of the tail-wag as scent-spreader in other animals. The hippo madly fans his tiny tail while urinating and defecating, better to atomize that scent. Some rodents and squirrels wave their tails when a possible mate is nearby. Woodland caribou, whose tail is covered with scent glands producing a musky odor, use it as a \"scent brush\" to disseminate an alarm odor. So though a dog wagging his tail looks simply like a dog wagging his tail happily, he is also saying Come smell the wafty odor that is me.\n\nSimilarly, walking with my dog Finnegan one day, I noticed another usual but unusual behavior. As we passed a tall, prancing black poodle being walked on a tight leash across the street, our dogs looked at (smelled at) each other. Suddenly, the poodle did a full body shake\u2014while still walking. I could very nearly see the plumes of scent from her curly hair hurtling toward Finn's nose in a fume cloud, and he was fixated. Could a shake be deliberate? I wondered. It might be the canine equivalent of flirtingly running my fingers through my hair as I talk with a suitor, perfuming the air around me with the scent of some slightly overpriced shampoo I use.\n\nTrying to see through the familiar dog behavior also allowed me to learn about the sneeze. Sneezes are sneezes, to be sure: reflexive ways to clear the nose of something ticklish or alien. Dogs sneeze only through their noses, as opposed to the human mouth-nose sneeze. The nose-only route marks an interesting secondary use of sneezes: to clear the nose of an undesired smell. In this case, I believe dogs use sneezes intentionally to end a session of smelling one odor and to ready themselves for a different one. Watch your dog sniff a strong, recently laid smell at a corner, then see if she doesn't sneeze before moving on down the street. Relatedly, trainers of tracking dogs have noted that some dogs clear their noses by lifting their heads up from the ground, taking in a bit of less scented air. It is not a pause or a miscue; it is an essential part of his tracking.\n\n\u2022 \u2022 \u2022\n\nIn the end, it is not the size of the olfactory bulb itself that makes the dog's nose so keen. It is not the numbers of receptors per se, or the way that the dog sniffs. It is not just the length of his nose. It is all of it. It is sniffing in the way he does, through the nose he has, into the many receptors he houses, to the brain that has evolved. The result is astonishing.\n\nYou have surely seen the sleeping dog who, eyelids fluttering, toes dancing, barking muffled through a closed mouth, seems to be dreaming. Dogs do have REM sleep\u2014the time when we most often dream\u2014so it is likely that they are also dreaming: chasing, with those dancing toes; announcing, with that muffled bark. Next time you see your dreaming pup, look for his active nostrils. For a creature of the nose is almost certainly smelling in sleep\u2014following the odor of his friend down the street or of dinner newly plopped in a bowl, or investigating a curious smell arriving under the door.\n\nGo and look at your dog's nose again. I love to look at my dogs in their eyes\u2014a shared glance is full of understanding, an agreement that we belong to each other. That dogs look at us is what bootstrapped an ancient proto-dog, lurking around the edges of early villages, into the dog who sits on your lap and wags at the sight of you.\n\nBut now I look at my dog's nose, too\u2014the whole length of it, but especially its wet tip\u2014and my heart also skips. What moves me is the depth of what that nose knows about the world.\n\n\u2022 \u2022 \u2022\n\nTalk to human psychophysicists and neuroscientists, and they'll say, You know, it's a myth that dogs smell so much better than humans do. And you can look at them and ask them what they smelled today. The answers vary from \"nothing\" to the poetic: \"grassy meadows, prairie grass,\" smell scientist and author Dr. Avery Gilbert reports to me, of his new home in Colorado. Pretty good. Now consider asking a dog. If a dog could talk, his answer would be an epic poem recited over several hours. In fact, humans have fine noses. But most of us simply don't bother to smell. I mulled this over as I sniffed Finnegan's fur one night (smell: fresh running river, water burbling over river rocks). Hmm. Maybe I'd bother to: to try to make my nose a worthy companion to my dogs'.\n\n* * *\n\nI. Practitioners included Freud, who famously experimented widely with cocaine; plenty of others who ate or drank questionable (including radioactive) materials; and researchers who self-catheterized or gave themselves viruses or unfinished vaccines.\n\nII. Yes, I am equating the dog rump with The Starry Night. Dogs have perfectly artistic rumps.\n\nIII. And, in fact, there are now many detection dogs trained to find bananas\u2014as well as any other agricultural contraband brought across the border illegally.\n\nIV. The rest are \"pseudogenes\": genes that have a mutation such that they no longer result in the development of a receptor\u2014that is, they no longer make the thing that they are supposed to. For dogs, 20 to 25 percent of their olfactory receptor genes are pseudogenes. For humans, it's more than 50 percent.\n\n## 4: Walking While Smelling\n\nMost dogs are not employed to find bedbugs or spot cancers. Most live in owner-controlled smell environments, asked to do little smelling beyond locating and cleaning up any food the resident child has dropped from his high chair. Moreover, in dogs' peculiar indoor world, the smells of the day that accumulate on hardwood and carpet are periodically vacuumed up (which period depends on the dirt tolerance of the co-inhabitants) and scoured with substances rude to the nose. In this world, crotch-sniffing is frowned upon, and even a surreptitious lick of an arm or mouth is only sometimes tolerated. The evocative and emphatic smells that people saturate their clothes with during the day\u2014piles of smelly reminders of their presence\u2014are mysteriously spun and heated away in large, noise-making machines, emerging as replacement \"fresh\" odors that call to mind nothing at all. And on very, very bad days the dogs are led to the fecal-urinous room, piled into a slippery-bottomed vat, wetted, and made to smell like taxi air freshener.\n\nBut oh! when it's time for The Walk: the olfactory scene changes. The great outdoors provides stimulation on the ground and on the breeze, on objects passed and passing objects. Each time the front door opens a new scene arrives, a scene of what has recently happened, what is now happening, and even a bit of what might be happening down the street. Not for lack of seeing, but clearly my dogs' journeys down the block are not made of visual landmarks. We launch outside a few steps above the sidewalk, and before we turn onto it the dogs' noses are in the slipstream of sidewalk currents, three feet above the ground, occasionally raised up high for a tall sniff. As we proceed, we pass a loamy tree pit, whose iron rail is messy with the passing dogs of the day; the sulphurous emissions from a work crew on the street, wafting from between parked cars; a frightening garage door that sometimes belches open with activity and rushes of air; the busy passage of a community of birds from apartment house to low tree branches, the berries dropped by the birds on the pavement, from mouth or ass; the edge of the corner building, where winds race uphill from points south; the slick-slippery feel of the marble stairs down, requiring more deliberate steps; the bench where a man sleeps at night, littered with his leavings and the stale, fetid odor of his clothes.\n\nWhen I came to realize how central smelling can be to the dog, I began doing concerted \"smell walks\" with both of my dogs. On these walks we are not trying to make good time. Never do I pull them away from that spot they've been at for impossibly long; I celebrate rather than worry over their abiding interest in other dogs' rumps. We do not hurry to get around the block, to get home, to get anywhere but wherever their noses lead us. There is no place we need to get to, or time by which we need to get back, to satisfy smell-walkness. Instead, the walk is defined precisely by how long and how much my dogs can sniff in. Sometimes the walk involves a lot of not-walking: standing, nose buried into the earth; nostrils pivoting their heads around to find the dog who left that message; even lying down with nose high in the air.\n\nOur dogs' smell-walk routes are almost never the routes\u2014or the paces\u2014of the walks that we humans concoct for them. As with many owners, when I head out on a walk, there is often a rectangularity to the episode: down, over, up, back. A walk in the human book has a time limit: \"until I need to leave for work,\" \"until he does his business,\" \"until one of us is tired out.\" A walk in the dog's book is much more determined by circumstance: the course is irregular, doubling back and turning suddenly. On many days, there is no end in sight\u2014until one suddenly arrives at it.\n\nIt occurred to me that, should I truly want to grasp the dog's sensory experience, it might make sense to begin with the dog's walk. I'm already along for the ride. As dog owners, we are implicit\u2014if rather uninformed\u2014accomplices as our dogs sniff their way down sidewalks, along road edges or park paths. How much more might it take to convincingly experience what they smell? After all, everyone who walks a dog surely sees, after a few goes around the block, the dog's heightened interest in fireplugs, a tree branch fallen overnight, or a building scaffold newly erected. Over many, many walks I began to get better at predicting what would be of interest to my various dogs, following their noses. That fencepost looks recently moistened, I found myself thinking. Or, Oh, that leaf pile looks pretty appealing. This is, of course, but a tiny minority of the dogs' smelling experience. Everything they (seem to) put their eyes to, they are nosing: the passersby, the wash of air from a car door opening, the ground around the park bench, the plastic bag being swelled up by a breeze, or the helicoptering maple seed being sent down by the trees. I see it. But do I dare to smell it?\n\n\"Do you ever smell what the dog is smelling?\" Avery Gilbert first says when I ask him for a suggestion of how to become better at using my nose. \"You want to get as close to the ground as you can get to smell things,\" Stuart Firestein tells me, \"because that's where the molecules are.\" While I know this logically, I had not followed it through, to, as Gilbert tells me, \"get down there and sniff at it.\"\n\nIndeed, we don't generally sniff it, whatever it is. And so I determined to sniff at it. To smell my neighborhood as my dogs did.\n\nI found myself surprisingly unworried about the obvious: that the reason we don't sniff what dogs are sniffing is, essentially, simple self-preservation and survival. There are funky, foul, downright fetid smells on the sidewalk. Our species may be the robust, successful species it is because we do not ingest the noningestibles down there. Instead, I considered the logistics: how could I get at the smells? To start, I figured, I must be willing to lose some of my restricting bipedal configuration. If indeed being upright, nose too far from the ground, is what keeps us from smelling, why, I'd go on all fours. When my dog lingers with nose planted at the base of a tree, so shall I plant my nose. If he gets intimate with a patch of invisible interest in the grass, so shall I pursue that interest. If he makes inquiries about the pant cuff of a passerby, so shall I. Passerby willing.\n\nI began strongly. On a cool summer morning Finnegan, Upton, and I launched out off the stairs in front of our apartment building. Right away Finn angled for a tree guard: the impotent short iron fences that ring street trees in New York City. He examined the edge, which looked recently moistened, with the precision of a watchmaker diagnosing the innards of an ill watch. When he came up for air, I took over. I had to kneel awkwardly, one hand ringed with leashes and the other on the rail. I bent very, very close\u2014too close. Emboldened by Finn's thorough vetting of the spot, I sniffed with abandon. A strong, bright smell hit me. I sniffed again. It was not urine, I thought with appreciation. It was, simply, the cool astringency of paint on metal. As I pulled back my head, my dogs stood to the side, watching me. A couple walking up the hill took a wide berth. I got up, suddenly self-conscious, brushed off my knees, and let Finn pull me away and down the sidewalk.\n\n\"No one questions a dog when they smell the environment,\" says Kate McLean, a self-described \"multisensory\" artist from the UK who has herself been pursuing urban odors. It was only too apparent to me and everyone else that I did not have the canine bona fides to get away with sniffing tree guards. After that brief foray into sidewalk sniffing, I lost my courage. To build some smelling bravado, I followed the brave\u2014in this case, McLean, who bravely smells her way through cities with an interest in depicting the particular smell clouds to be found in each.\n\nMcLean was visiting New York and invited me along on a \"smell-mapping project\" in Williamsburg, Brooklyn. I found her, a bit flushed and with a black stingy-brim fedora atop her head, waiting on the sidewalk on a warm September evening. She is lean and fine-featured, with an easy smile. Her shoes were well worn, presumably having walked her nose along many miles. Umbrella in hand, a cheeky nod to the UK style of \"tour leader,\" she addressed a cluster of two dozen smell-interested folk gathered around her. They included artists who worked in multimedia, photography, and with olfaction; memoirists and science writers; a couple of interested hangers-on; and two professional children. We were, McLean instructed, to take a tour of six or eight blocks and simply smell, recording what we noticed.\n\nWell, it was not so simple.\n\n\"Be aware that there are many different kinds of smells,\" she cautioned. \"There are episodic smells, which drift past in a moment\u2014a person, a draft from smoking, a truck\u2014brought on the breeze. Especially at street corners. So at corners, stand and turn around.\"\n\nApart from the wafting smells, there are also static smells that have been absorbed into materials. \"So sniff walls, touch plants, go into stores,\" she instructed, watching smiles creep onto every face. A few people exchanged raised-eyebrow glances.\n\nAnd furthermore, \"smell voids\" are just as important, she said. Given that we get acclimated to smells, we should be alert to when we are not smelling anything. McLean recommended using one's own smell as the natural resuscitator of one's sense of smell: by burrowing your nose in your own skin, the receptor cells have a moment's pause to recharge and resume their attention to the environmental odors.\n\nMcLean handed out handmade accordion-folded maps of our route, with \"nose points\" along the way to remind us to pause; to smell high, low, close, and deeply; to breathe in passing smells and constant smells. At each stop we were to record five scents. Five! I inhaled through my nose as we listened to her. I smelled nothing, but by the walk's end my handwritten notes would be overflowing the page, turning around corners and ducking under other notes much like the routes of the smells we were chasing.\n\nMcLean has traveled the globe smell-mapping, from Amsterdam to Pamplona; Glasgow to Newport, Rhode Island; Milan to Edinburgh; and Paris to Singapore. After each walk, she has translated the walkers' records into beautiful maps with a colored, topographic styling marking the sources and spread of odors. Washes of colored dots mark migrating smells. Each city, she suggests, has a background smell that specifies the place. In Amsterdam in the spring, it is the \"sugary, powdery sweetness of waffles\" and the water of the canals. And each city has distinguishing particulars: Edinburgh's map, for instance, includes fish and chips, malt spewing from breweries, and the scent of \"boys' toilets in primary schools.\" Heaven knows how she knew that one.\n\nOn an earlier visit to New York, McLean mapped what she described as the city's \"smelliest [square] blocks\"\u2014on the Lower East Side between Allen and Eldridge, south of Delancey. While this part of the city has a varied history\u2014partly manufacturing, partly sordid\u2014its recent sprouting of multimillion-dollar condominiums challenges her claim. Still, the final smell map included landmarks of sawdust, trash, car oil, and cabbage, as well as long swaths of dried fish and cheap perfume.\n\nGeographer J. Douglas Porteous called olfactory landscapes like those McLean maps \"smellscapes.\" Cities, it has been claimed, are identifiable by their scent. As a freshly baked baguette invokes contemporary Paris, the characteristic odor experience of a city may come from food or spices sold on the street, the marine air that fills the city's avenues, or the detritus of the teeming populace. Certainly there have been smellscapes, celebrated or not, for thousands of years: in ancient times, temple builders mixed milk and saffron into the plaster; mosques were built with musk and rose water worked into their mortar. On being rained on or warmed by the sun, the buildings effused fragrance. For many years there were regular weekly smells: the warm wet smells of washing, the hot iron on linen; the scent of \"baking day.\"\n\nThe idea of smellscapes has caught on in the field of urban design, whose proponents have an eye to celebrating and improving the sensory experience of city residents. Some of the programs cities have enacted are pleasingly quirky. In the Netherlands, pedestrian plazas have been designed to include plants that may have relaxing, therapeutic effects. Since 2001, the Japanese Ministry of the Environment has supported the maintenance of \"One Hundred Sites of Good Fragrance\" countrywide: national landmarks as significant to the history and culture and life in the country as visual landmarks. They include \"the smell of deer seen at Jinhua mountain\"; a house smelling of the animal glue used to paint papier-m\u00e2ch\u00e9 dolls; \"one hundred thousand peach blossoms at a glance\"; and, in the biggest city in Japan, \"Kanda's streets of used bookstores.\"I\n\nThis turn into designing and celebrating smells follows a long history of complaining and worrying about smells in cities. And most of this cogitation was because cities smelled utterly, horrifyingly disgusting.\n\nThe streets of Manhattan were designed as a grid not only for ease of way-finding but also for ease of smell-letting. What this implies is that smells needed to be let out. Indeed, the curvy, narrow streets of old European cities\u2014Paris and London, for instance\u2014were widely seen as attracting and providing breeding ground for stinks. \"Paris may be smelt five miles before you arrive,\" it was claimed; the atmosphere around the cities of Italy was saturated with garlic. The smells of nineteenth-century Paris were described as \"intolerable\"; the \"monstrous\" city of London was \"strewn with excrement, mud, decomposing animals, meat, vegetables, and blood\"; odiferous industries like tanneries and breweries were cheek by jowl with residential sections. Both London and Paris had episodes known as the Great Stink\u2014both caused by a failure of the prevailing sewage treatment approach.II\n\nThe idea with the Manhattan grid, reaching from river to river, was that smells would swirl down the streets and out to sea. The New York commissioners reported that this would \"promote the health of the city\" and used words like free and circulation to vaunt their approach.\n\nThe grid did not save the city from its smells. And \"smells\" in this context were mostly \"noxious, horrible smells.\" Consider the state of New York City in the mid-nineteenth century, when horses\u2014then the city's main transport devices\u2014shat freely and plentifully in the street and were often left where they fell and died; the contents of chamber pots were summarily dumped out of windows; and fear ran rampant that miasmas (foul gases emerging from the ground) were leading to a cholera epidemic. Conditions were sufficiently bad that a governmental \"smelling committee\" was officially called together, tasked with seeking out sources of ill smells by nose(s).\n\nThe notion of contamination by smell continues today, in small doses, with fear and retreat from people who smell bad, as though disease were conveyed by odor. Though long discredited, this anxiety had its root hundreds of years ago, when \"deodorization\" projects began. In particular, streets and sidewalks were paved to tamp down the miasmic emanations; plaster covered walls that might seep with foul-smelling vapors; industries, reeking of paint solvent (vaguely banana) or cleanser (soapy), were moved away from residential areas.\n\nToday's city has done away with the horse and chamber-pot leavings, but it still smells of the city's occupants and wares. Concentrate as you walk down the street and you'll notice the wash of odors out of open store doors, warmed inside and released in bursts by customers' entrances and exits. People are perfumed with bottled fragrances\u2014or the bodily odors that bespeak a lack of fragrances. But one can walk many miles in an American city without being hit by any specific smells. One of the responses to the overwhelming array of noxious odors in cities has been to try to do away with odors altogether (as \"deodorization\" implies) or to cover them. Ubiquitous chain stores homogenize cities, and there is a booming business of branded scented environments, in which retailers diffuse a fragrance intended to invoke \"car showroom\" or \"fancy hotel.\" The result may be the waning of the characteristic odor of a city. Should Paris stop smelling like bakeries and Gauloises, Vancouver like the spray of sea salt, or midtown New York of hot garbage and food carts' emissions, some part of the place will disappear.\n\nThe British urban planner and designer Victoria Henshaw started thinking about characterizing\u2014and perhaps commemorating\u2014the remaining smellscapes. Building off the idea of city \"soundwalks,\" in which urban explorers actively listen instead of just listening in the background, Henshaw thought to bring the notion to collecting smells, too: smelling actively, searchingly, and intentionally\u2014instead of passively and accidentally. McLean has used the noses of others, in addition to her own, to take off from Henshaw's early, labor-intensive work.\n\nOn the street in Williamsburg, our group was slow to begin smelling. To rally us, McLean brought our attention to a handout she had prepared. Stay hydrated, it instructed. Find hidden corners. Be not embarrassed. Finally, \"Sniffing in public is completely legal,\" she thought to add.\n\nApparently, the activity is sufficiently nonstandard that she has had to consider this.\n\nWe set out at a meandering pace. On the sidewalk, among the generally steadfast and quick-footed New York City pedestrians, our group had a vaguely disoriented look. I came to learn that this is characteristic of those who are sniffing the air: a faraway, unfocused look to the eyes, head cocked or raised, expression somewhere between \"Did I leave the oven on?\" and \"I just remembered a dream I had last night. I was in a car, with no pants . . .\" Passersby step around you.\n\nAt our first \"nose point\" we formed a scrum of confused loiterers, nosing around for something to zoom in on. Stepping to the curb I got my first waft: a nose-pinching, clean-but-not-clean smell. A warm, scrubbed sidewalk: chlorine at battle with filth. Across the street, a food truck gave off an unmistakable taco-shell\/chips smell: fried corn and used oil. The evening was sufficiently warm, we realized, to be full of smells: just as hot foods have more odor than cold, summer days have more odor than winter ones. Warmth makes many substances airborne, volatile, wafting up to meet any sniffing noses.\n\nOur group began to self-organize, alternating poses by curb and by building, sniffing, then scribbling down a report. By the next street we were all but choreographed in our movements, flocking in pairs to local landmarks\u2014a tree, an outdoor bench, an exhaust fan\u2014and pointing our noses in unison into the air. People themselves become clues as to where to sniff. A photographer who had traveled with McLean from the UK, Sam Vale, bent under a bench backed against a food shop. I followed his nose. Wheatgrass (source: juice shop) prevailed on the sidewalk. But at the altitude of the bench, the pith of a leaf mixed with a definite spring-onion smell coming out of the exhaust fan. \"This smells amazing!\" he said and smiled.\n\nWe were into it. I sniffed the base of a tree: urinous. People walking by brought a chaos of fragrant odors: hair products, lotions, perfumes. Fried food followed a person with a takeout bag. \"Use your other senses to guide you,\" McLean reminded us. I touched and crushed a leaf (pleasingly fresh). I followed my ears to a dripping air-conditioning unit (dank basement) and to a clean towel spanking the air as it was shaken out (dryer sheets). Each new or different feature spotted on the sidewalk aroused interest. A construction site's temporary fencing yielded a peephole into the site (dust, caulk, and warm brick) and, on its surface, a place for posters (fresh paper, paste). Even sights I would ordinarily veer around I began veering toward. With but a moment's hesitation I dipped my head into the olfactory space above an open trash can. It smelled sweet, almost tangibly so. The remains of recently chewed mint gum cast upward to my nose. What was usually stomach-turning had turned into simply news of the street.\n\nVirginia Woolf once tried her hand at smell-walking, in a manner of speaking, through her biography of Flush, Elizabeth Barrett Browning's cocker spaniel. He wanders off into Florence \"to enjoy the rapture of smell\" on the streets\u2014\"the rough, the smooth, the dark, the golden\"\u2014smelling brass being battered, bread being baked, hair combed, cloth beaten, and men spitting.\n\nI thought of Flush locating the \"swooning smells\" of the gutter as we hovered around a sewer drain and tried to catch a spiral of air. Far from swooning, it walloped me. Was it salsa? It bit us with its sharpness. \"Chinese food\" was offered. The photographer, Vale, poised over the drain. His head cast around, neck extended, the very image of a newborn animal lurching for its mother. After a moment, he said, simply, \"Garlic.\" Our group erupted in murmurs of appreciation. The question of whether or not we wanted to be smelling garlic from the drain was trumped by the great satisfaction of identifying it.\n\nNight began to fall. We had been walking for hours. We were shooed away from a restaurant where, even at a distance, our interest in the outdoor diners' food was a little too conspicuous. At the walk's final odor-promising corner, I faced a plain bricked building, the spent neon sign in a high window signifying a particular kind of dark, stale-beer-smelling bar within it. Instead of stale beer, I caught a very pleasant savory smell. It was suspended in just one small invisible cloud of air, and I had to stand on tiptoes, head raised and nose up, in order to catch it. When I stepped off the curb, it was replaced by a cloud of something darker, gray and waxy. I took a quick look around for likely sources\u2014perhaps someone was walking by with a plate of food\u2014and found none. Come over here; there are smells here waiting to be caught I indicated to one of the very good smell-catchers. He beelined over, stood facing me on the curb's edge, and we sniffed. Gas? Tar? Not quite. Then he stepped off the curb in front of a vehicle parked on the street and inclined his head toward the grille\u2014just about where you'd put your head if you wanted to be squarely concussed by a car. I went in for a sniff. Warm air greeted my face: the car, an SUV, had been recently parked. A smoky, waxy note curled out from the engine. It smelled delicious.\n\nHow did it come to smell delicious? When one begins actively searching for smells, one finds . . . not a lot. Just opening your mind (and nose) to the possibility of smells isn't enough to actually smell anything. And actively sniffing is an oddly tiring exercise. Try sniffing now, and keep it up for just thirty seconds. Done? Chances are, you quit about halfway there (or wanted to). And furthermore, you probably noticed not a single smell.\n\nSo catching a whiff at all can be exhilarating. But here our undercooked relationship with odors kicks in. In English, most words for smells are words for their sources. While the sommeliers and perfumers among us may have a vocabulary to describe that whiff, most of us need something more. To name it\u2014to know it\u2014we want to know where the smell comes from. If the name and the source are not the same, our work is to resolve them satisfyingly. McLean remembers someone worrying to her that Paris seemed to have a background note of honey. Why should the city, not a center of beekeeping, smell of it? She tracked it to its source: not to a hive nor a covetous Pooh Bear, but to the wax polish popular in the city rife with parquet floors.\n\nIf a smell wanders off before being pinned, a beetle to an entomologist's board, the frustration is stark. If the smell is traced to its source and its name, it feels truly caught, captured, collected. The car grille smelled delicious because it was, very clearly, hot oil on hot metal. I recognized the smell, but my certainty was buttressed by my recognition that I was, after all, sniffing into a recently used engine.\n\nBuoyed, I stepped into the street, momentarily free of traffic, in search of the savory smell still at large. Catercorner to the bar across the broad intersection was a shop radiating light into the dusk. Its glass doors, trimmed in red and thrown open, faced the intersection. Aha! This was the source: a bakery. A terrifically obvious smell, now that I saw its source. The bakery was outputting a caramely, buttery smell, some of which was surviving being tossed this way and that by passing cars and weaving its way across the street to my nose as I stood on tiptoes. \"I think I need to go there,\" said one of the other walkers, catching the drift and darting toward the light. I did not. I had my source, and the moment was complete.\n\nAt the walk's end, McLean changed senses on us. Pulling out a flip book of paint colors, she asked for the color we would use to describe one of our smells. I chose Pantone 1245C, an unlovely yellow-brown with a hint of green, for the first smell I pinned, the chlorine of the washed sidewalk. \"Everyone always picks a yellow-green,\" she said. But another walker found the caramel smell to be maroon, and our shared wheatgrass was remembered in a light mint. Garlic and the tarmac were an ashen purple. Gilbert and others have found correspondences between color names and odors, even with smells that don't have clearly colored odor sources like lime or banana do. In one study, odor of civet\u2014an artificial version of the odor from the anal sacs of civets, used in perfumery\u2014was roundly considered \"brown,\" and another perfume favorite, bergamot oil, was most often seen as \"yellow.\"\n\nI headed down into the nearby Bedford Avenue subway station to catch a train home. To my great surprise, even as I approached the entrance, I could smell it. Now, I'm sure that the subway always smelled\u2014but never before had the smell reached out and poked my consciousness. I slowed on my descent down the stairs to the station. The smell of youth (wet, shampooed hair mixed with teenage body odors) and of decay (the moldering of the walls under accumulated dirt and water). I smiled.\n\nWhen you notice the smell of decay and smile, either something is very wrong with you, or your relationship with smell has changed. The smell walk had begun to change us. Smells were to be noticed, collected, considered, not just avoided or spurned. The deep funk of the subway in the summer is plainly awful. But sometimes, awful smells are awful because of their incongruity: a classroom with an eggy smell, a restaurant with the smell of air freshener. Like the garlic and car grille, the stink of the subway suddenly felt honest: it was what one would expect the subway to smell like. What is truly off-putting are disingenuous smells: ones inconsistent with how a place ought to be. We have a sense of familiarity born of experience more than consciousness. We know the sounds of our home\u2014and hence might notice if it is \"too quiet\"\u2014and we know what a baseball game sounds like\u2014and would be disturbed if it sounded like golf. We expect vision and taste to align, so we want our orange juice to be orange, not purple, and any food that is flavored deceptively (bubble gum chocolate) is foul.\n\nSo, too, with smell. In 2005, all of New York City was up in arms because of a sudden odor of maple syrup oozing over the city. While maple syrup might rank among the most preferred smells of many residents, the overarching response was terror\u2014as the smell made no sense in our urban jungle. Later, once the weather pattern (a cool night in winter and a \"lid\" of warm air trapping odors caught on the wind near the ground) and source (probably a flavor manufacturer across the river in New Jersey) were identified, the smell could be enjoyed for itself.\n\nAfter a little time walking and thinking, Dominic decided to smell out Grandville, his usual practice in towns he was visiting for the first time. He raced up and down avenues and alleys, rubbed himself against various poles, lampposts, cornerstones, and trees, inquired about the population and the town's history\u2014how many members of each species it contained, the birth rate, when the town was founded and by whom, and why\u2014looked up the oldest landmarks, smelled them carefully, asked about the climate at different times of the year, learned what the salary of schoolteachers was, and the price of tangerines . . .\n\n\u2014William Steig, Dominic\n\nOn the five-minute walk between the subway stop and where I rest my head, I caught curls of air from basement dryers, grilled-meat trucks, circular-sawed wood; urinous wafts of mysterious source; a single smoker exhaling as he passed; curry from an open window; and menthol from a jogger's sweaty legs. This, I realized, is more or less the smell of my own block. One psychological study that asked undergrads to find their way, blindfolded, in a room divided into grids of different scents, found that they could navigate using smell alone. Could I find my way home by smell?\n\nMy longtime and well-loved dog Pumpernickel once strayed from our home in a coastal town in California. I returned home late at night to find the front door wide open, light streaming out, and a house that was definitely too quiet. Pump had, apparently, opened the door (these were days before dead bolts were de rigueur) and walked out. I raced up and down the street calling for her. Stoppering my growing panic, I began to devise a plan of action, retracing all of our walks together. I phoned some friends who ran a dog-food shop in town to ask them to come by and stay at the house while I searched by car. Imagining where she would go, if entirely left to her own devices, was trying: I was bereft.\n\nWhen my friends pulled up to the house twenty minutes later, Pump jumped out from the back of their pickup. Between jumpings of joy, I asked how they found her. They had driven by their store on the way over, they told me. Pump was sitting out front, awaiting opening hour.\n\nNow, we had many times gone to the dog-food store on our walks. It was less than a mile away. But we had approached it from so many different angles: how, I wondered, had Pump gotten there? Had she turned right at the eucalyptus, headed seaward, then taken a left at the bagel shop? Did she follow the crow-flies path, cutting across backyards and going through back alleys? Did she smell her way?\n\nWalking through my own block's smells, I considered her olfactory navigation. Though the smells appear vague to us most of the time, they are very particular\u2014enough to be landmarks. Sailors use smell in navigation: \"(A)n old salt is said to be able to smell fog, rain, wind and snow. In calm, fluky weather, especially near shore, the knowledgeable shellback can often sniff out a breeze to keep his vessel going. He does this in large measure by sorting out what he smells\u2014dampness from off the sea tells of a sea breeze or a fog while the aroma of new-mown hay, clam flats, or a pigsty warns him before the first ripple is seen that the breeze will be off the land.\" The homing pigeon's gift to return home, over hundreds of miles, appears to be a result of overlapping sensory awareness\u2014including (but not limited to) smell. So, too, for dogs. In the First World War dogs were used by Britain as messengers or liaisons between the front trenches and the home camp, finding their way through some combination of navigational techniques including, presumably, the general smell of the area and the smell of (what was standing in as) home.\n\nI turn left at the smell of curry, hop up the stairs, and am home. It's only my first trip to the trenches and back.\n\n* * *\n\nI. A recent topic of research for scientists who publish in the Journal of Pulp and Paper Science or who study \"material degradomics,\" the \"smell of old books\" has been deconstructed and seen to overlap with vanilla, mushroom, nylon, and \"grassy notes with a tang of acids.\" Since the smell is literally the degradation of paper, ink, binding, and glues, it is not obvious that Tokyo's bookstore blocks will smell the same in another century.\n\nII. At the time (1850s) in London, cesspits under residential buildings filled with the wastewater and excreta of their inhabitants were transferred to the Thames and summarily dumped in it. Apparently this worked well enough, with the exception of a particularly dry June, when the river ran low and the wastewater ran high. The river \"ripened\" to \"a peak of pungency\" and lasted for two weeks, until the heat abated.\n\n## 5: Plain as the Nose on Your Face\n\nYou can't miss the Monell Chemical Senses Center building in West Philadelphia. It's the one with an enormous golden nose jutting out of the building by the front door. A daring move, even for a center of taste and smell research: the triangle of skin and cartilage that fronts our faces is not widely admired. I have come to Monell to speak with an olfactory scientist, but I pause at the nose for a while and just look at it. How strange the nose is. While the human face looks bereft without a nose, it looks fairly silly with one. With our species' affection for gazing into each other's eyes and our focus on kissing or stuffing food into our mouths, we nearly overlook the organ in between. Well, not entirely. We break our noses; we surgically fix our noses; we pick our noses; we powder or sunscreen them. The nose leads our face as we walk and fills our bedrooms with snores as we sleep.\n\nBut it is often unlovely and largely unloved. It is reduced to an upside-down 7 by a child's hand, and may never evolve beyond this rendering. It is the stepchild of the face\u2014the overlooked space between more esteemed anatomical parts.\n\nFor a facial feature that is so conspicuous, so apparent that it prompts its own aphorism, the nose is surprisingly unfamiliar to us. We might gaze at a cute turned-up nose or a prominent bulbous nose. We might suffer a sinusitis that arrives with the pollens of spring. Surely we savor the rush of dinner smells that hits us when we arrive home, just as we bristle at the wash of sewage odor emanating from the so-called sanitation department's hub.\n\nWe have many fanciful\u2014if moderately obscure\u2014words for nose in English. The beak, nozzle, snoot, beezer, bill, gnomon, nib, snot-gall; occasionally rendered as a bowsprit, a vessel, a boko, a snooter. The snitch, trunk, index, horn, the spectacles-seat. What we do not have is much knowledge about what happens after we sniff with that scent-box. Most of us know little about the smells of our world, little about how the nose takes in odor, and even littler about the smell system of the human brain. Scientists, surprisingly, feel the same way. George Preti, the researcher at Monell whose group studies primarily \"human odors,\" found the territory relatively untrammeled: \"They were probably the last thing that chemists tackled for some reason.\" Scientific knowledge is only nose-deep, another olfactory researcher, Dr. Leslie Vosshall, told me: \"The basics are easy,\" she said, \"and then the hard stuff [about how smell works], we have no idea.\"\n\n\"Nobody knows about olfaction,\" Stuart Firestein consoles me, as I admit that I haven't thought much about smelling in all my forty-some years. His statement extends not just to the layperson, but to biologists, and even olfactory neuroscientists like himself. Human olfaction was one of the last sensory systems to receive the scrutiny of science, upstaged by vision\u2014how we love and admire our eyes!\u2014and even the bench team, hearing and taste. But it deserves a look. It is not a coincidence nor an accident that we, dogs, sharks, and voles have noses. Noses house special cells that allow us to smell\u2014and smelling is an evolved strategy to find out more about the world.\n\nSmelling dates back to the ancient and single-celled prokaryotes, enabling them to avoid things that are toxic and head toward things that are beneficial. Today almost all living creatures in air, sea, or on land can smell in some way.I Smelling is chemosensation, the detection of chemicals, and we live in a world of chemicals.II The nose, then, is for seeing them, and figuring out which to pursue and which to avoid. \"The biological nose,\" Firestein has written, \"is the best chemical detector on the face of the planet.\" At the same time, he admits the olfactory system \"is probably, like everything else in evolution, a bit of a kludge job\"\u2014a few systems all plonked in or around the nose, doing more or less the same thing, though in slightly different ways.\n\nHumans have an odor problem. It involves our use of smell (scanty when compared to many other animals). Primates, including the human primate, are considered microsmatic\u2014what is (probably unfairly) called feeble scented\u2014as contrasted with macrosmatic dogs.III It involves our cultural sensitivities about smell (given that artificially scented \"air fresheners\" are not considered ironic or aberrant). Smell is reliably the sense that people suggest they would be most willing to lose. But also, most fundamentally, our aversion comes from a deep misunderstanding of smell. We distrust smells. Invisible somethings find their way into our noses: it can feel horrifying at worst, peculiar at best. Though we will put all manner of steaming, dripping, oddly colored foodstuffs into our mouths with alacrity, we can feel embarrassed, alarmed, or disgusted by an odor showing up in our noses.\n\nThe invisibility of odors accounts for some of this reaction. We rarely search them out; we more often experience them happening to us, catching us unawares. And there is uncertainty about what smells actually are. Here's what they are: they are molecules, flowing about in the air. Since the world is molecular, they can come from just about anything\u2014gases, liquids, or solids (which continuously release a haze of molecules into the air).IV In particular, biologists specify that odors are \"small, low-molecular weight organic molecules.\" These molecules also must be a certain amount volatile\u2014able to evaporate into the air, be caught by a nose, and cause the sensory cells to hum and purr. When we smell something, we are really ingesting it, after a fashion: the molecule is being absorbed by the mucus layer of the nose. In this way smell is different than our other, less alarming senses: what we see (through light reflecting into our eyes) stays out there; what we hear (through vibrations drumming our ear canals) ends at the ear; what we feel does not burrow into our skin, but glances off it. But in smell a bit of the source itself comes into our bodies.\n\nMost adults see smells as incidental, and as binary\u2014very good or very bad. \"The principal axis of human odor perception,\" write neurobiologists struggling with our detachment, \"remains odor pleasantness\"\u2014whether we like a smell or not. Not only the unwashed masses but also \"the greatest poets in the world,\" wrote Virginia Woolf, \"have smelt nothing but roses on the one hand, and dung on the other. The infinite gradations that lie between are unrecorded.\" Sigmund Freud came out on the \"very bad\" side in general, equating the diminishment of our olfactory powers with the rise of our rational minds. \"The organic sublimation of the sense of smell,\" he claimed, \"is a factor of civilization.\" Sights are information; smells are judged. Smelly never means anything but \"stinking.\" We like those who smell like our own social group and distrust those who are \"smelly.\" \"Being odorous,\" writes Jim Drobnick in The Smell Culture Reader, \"is tantamount to being odious.\"\n\nNone of us began life smelling so monochromatically. We are born smellers. In early development, nasal chemosensors emerge before any other sensory system. And we have been in contact with odors from the get-go: as fetuses, we are in a wash of fluid carrying the odors from our mother's food. By birth the olfactory nerve is intact: better to search for the smell of mother's nipple and the milk it promises. Tiny glands around the nipple send beacons to the baby's nose.\n\nIn our first hours and days, we are tiny but macrosmatic animals, exploring by nose more than by sight. A newborn recognizes his parents by the cloud of fragrance coming off them; his vision is still too blurry to see them clearly. A child's security blanket, or any favored raggedy, one-eyed teddy bear, is so loved due to its smell. If washed, it is changed\u2014and sometimes rejected. Children are ambivalent about what adults find to be clearly horrible smells: they must learn to hate the smell of spoiled milk and flatus. They don't know that \"skunk\" is a bad smell; that \"flower\" is terrific.\n\nBut now, reader, take a moment to consider: What have you smelled today? Chances are, nothing. And if anything, it was involuntary\u2014the baked bread as you entered the house, the thoroughly-sodden-dog smell that you'd sealed up in your car after that trip to the lake yesterday. When I ask people what they have smelled today, I often get a lot of searching looks. By adulthood, we have mostly forgotten that smell developed as a means of discovery. Most animals\u2014our ancestors among them\u2014use smell purposefully: to sniff out potential mates, to find delicious and nutritious food, to notice predators before being noticed. We have not entirely neglected these important discoveries, but instead of smelling a person, we smell his shampoo; the food is not found in the wild, but cinnamon buns and pizza joints can be navigated to by nose; and while we might not smell danger, we know the acrid smell of smoke and the added smell (the rotten-eggy compound mercaptan) of natural gas.\n\nTo restore smell to its rightful place we must undertake a simple three-step process: first, notice the smell at all. We must sniff it in, let it settle into the warm lining of the nose and snuggle into a receptor cell. Second, we have to be able to distinguish the smell from other smells\u2014to simply note and remember their differences. Finally, we want to name what it is, or locate its source.\n\nTo begin, we'd better be clear about our noses.\n\n### SCHNOZZOLA\n\nNosed animals come in terrific varieties: mollusks smell with their tentacles; male silk moths with their feathery antennae; the simple nematode worm detects chemicals via an opening near its front tip. The elephant's periscopic sniff is enabled by its trunk\u2014which is also used to examine objects and to caress other elephants. The domestic pig's nose has expanded into a perfectly lovely implement for rooting around. Star-nosed moles have a spectacular, fleshy nose with twenty-two radiating appendages that function as tactile sense organs, doing no smelling at all. Semiaquatic animals like the water shrew blow air bubbles to trap scent and then re-inhale the bubbles to smell them. Leslie Vosshall and her colleagues at Rockefeller University discovered that the mosquito repellant DEET works on the insects' \"nose\"\u2014receptors on its antennae. The repellant is a \"molecular confusant,\" they write, that scrambles the message to the mosquito about a warm-blooded target nearby.\n\nThe world of nosed animals can be divided into those with hidden noses and those with flagrant noses. We are in the latter group, naturally. Among primates there are also two nose types: the curved noses (strepsirrhines)\u2014think of the cute-faced lemur, with its surprised eyes and tufted ears\u2014and the simple noses (haplorhines) of most primates, including humans. The curved noses have the wet, naked rhinariums of dogs and cats. Among the simple noses, we can be further divided into downward-nosed (hominoids and old world primates) or flat-nosed (new world primates), based on the direction of the nostrils. So we're a flagrant, simple, downward-nosed creature.\n\nThe human nose is, anatomically, a soft organ, layers of skin and muscle, braced only by cartilage and fat on the inside. The outside is rife with sebaceous glands, the inside lined with mucus. A squishy, unwieldy vessel, moist and oily.\n\nAnd it is just a vessel, most of it. \"In man,\" Isaac Asimov wrote, the nose is \"primarily an air vent and has no exotic uses.\" (That might depend on what, exactly, one chooses to smell.) Notably, the exposed \"projection in the midface region,\" as it is romantically referred to, is not actually the smelling nose. Just as with dogs, while odors are hurried into its dark depths, most of the visible nose is just a cavern and humidifying chamber en route to the treasure of olfactory tissue deep at the back.\n\nThe smelling part of the nose\u2014the olfactory epithelium where odors are received and translated into neural signals that make the brain say cake! or kimchi!\u2014is the very end of the cavern. In the depths of the human nose, at about the point where the outer nose flattens into forehead\u2014the midpoint between the eyes\u2014is a postage stamp\u2013sized plot of epithelial tissue. \"You can't reach it with your finger,\" Stuart Firestein cautions, as though I might try. Sitting across from me, he wears a look marrying perpetual amusement and skepticism. His whitish hair leans toward unruly, but never delivers on the threat. My fingers stay in my lap, but I squirm reflexively. I make a note to refrain from telling my six-year-old this bit of news.\n\nLet's not forget the importance of that pyramidal, protuberant vent, though. If it's congested and the sinuses swollen, smelling is temporarily disabled: we often lose our sense of smell\u2014and thus of taste\u2014when we have a head cold. Food designed for astronauts must be highly spiced and flavorful, for they are permanently congested: the fluid in their heads does not drop nicely toward their toes with gravity. As it clogs them, it deprives them of the enjoyment of what were beloved foods back on terra firma.\n\nThe appearance of the outside of the human nose does not reflect its internal architecture. The large-nosed among us have no more tissue dedicated to smelling, relatively, than the tiny-nosed. In both, our olfactory postage stamps are a very small amount of the entire nose. Hence the reason that humans are generally considered poor-smelling: we have less space for olfactory cells, and thus less sensitivity to odors, than macrosmatic animals like dogs. \"It's this little teeny-weeny nose,\" Firestein says. \"And it's very, very tight up in there.\"\n\nA nasal septum divides the nose up the middle, making two vestibules: the most perfunctory of waiting rooms en route to the back. Each vestibule is lined with special glands that produce up to two liters of mucus a day. This soda-bottleful of mucus helps moisten the air (good for both breathing and smelling) and helps protect against any large or irritating molecules flying up into the nasal tissue. For when you give a good sniff, air flows through the nose at twenty-seven liters per minute\u2014\"gale-force speeds.\"\n\n### LIKE DOG, LIKE HUMAN?\n\nThe human olfactory system is more developed than we usually admit. But is it near dog levels? Anatomically, the comparison is stark: our nose is smaller and our sniff is less complex. The human sniff is not unlike that of the ancient canid: a bellowsful, a crude, imprecise pull of air in and out. Unlike the modern dog, our sniff is long and slow: we take a second and a half to sniff even once, pulling in as much air as might fill a regulation-sized softball. We have half as many genes coding for olfactory cells, and more of ours are not functional. We have less space for smelling\u2014only one to two square centimeters of epithelial tissue; our noses house hundreds of millions fewer olfactory receptors and half as many kinds of receptors. If there's less territory to make sense of an odorant, then even if it manages to land in the human nose, our sensation of it goes . . . nowhere. We may notice an odor but not be able to identify, locate, puzzle out, or even react to it, before it dissipates and we move on.\n\nArchitecturally, our noses are children's block towers next to dogs' modern architecture: made of similar stuff but in a much simpler, more brutalist formulation. While the human nose does hold turbinate bones like the dog's, there are only three small bones, and they don't bear nearly as much olfactory tissue. Turbinates in a human nose are like minimalist modern art: simple Mir\u00f3 figures compared to the dog nose's healthy branching tree. And alas, my friend, you lack the \"olfactory recess,\" the deepest part of the dog's nose, partially segregated from the rest. This matters: there is no place in the human nose for air to pull up a chair, sit, and be repeatedly smelled. Some scientists have suggested that with the movement of human eyes forward to the front of the head, we lost the space for a nose alcove. As a result, we exhale any inhaled smell right back out, scrubbing it from the gentle embrace of the receptors. This accounts for the sometime success of our frantic effort to purge a foul smell by blowing our nose.\n\nWe entirely lack the vomeronasal organ that forms the dog's second smelling route: in humans, it is vestigial and disappears before we are born. All our VNO genes are pseudogenes, no longer functional, so we produce no cells, no receptors, and make no connection to the brain. We do not seem to detect pheromones at all. \"Sadly,\" writes Tristram Wyatt, an expert in all things pheromonal, \"there is no good evidence for a human pheromone to make the wearer irresistible to potential partners.\"V\n\nPsychologically, we are also different than dogs. We will trust our eyes over our noses. If there's disagreement between the senses, vision wins. Cherry juice made to look green tastes to us like lime; color a white wine red and even enology students taste it as red. Not only are we largely ignoring the input through our snouts, we sniff so infrequently that only an unusually strong odor reaches our consciousness. Any attention we bring to odors as infants is quickly learned away: \"Baby smells an odor, mother says nothing,\" Mary Roach quotes a researcher at Monell as telling her. And so baby ignores the next smell coming her way. Our brains develop around non-smell things; the dog's brain is developed around smells. What the dog ignores, by contrast, is all else apart from the smell he has his nose in. As anyone who has tried to pull his dog from a long investigative bout knows, the dog practices impressive attention. Fitting that the Latin root of attention means \"to stretch,\" to direct the faculties toward. The dog's nose stretches in all ways.\n\nNor do we humans celebrate the intimacy of smell. Should someone be close enough that we can smell him, we find that too close. \"Most people in Western cultures don't smell at social distances,\" George Preti tells me\u2014referring to both people's bodies and their smelling habits. We seem to bathe at levels matching our sense of normal personal space; in the United States, it's an eighteen-inch buffer radiating from us. Notably, vision and hearing allow for interaction with someone in social space without any violations of personal propriety\u2014we can both see and hear someone comfortably. But to smell them would be intrusive at best. When I ask Preti if he smells people often, he laughs and says: \"I don't want to get smacked!\"\n\nFundamentally, though, our noses work in the same way as the dog's. As with the receptors in the dog nose, there is no one-to-one odor-and-receptor correspondence. There is no single \"vanilla\" receptor nor \"smoldering cigar\" receptor, despite the instant familiarity of each. We specialize in our olfactory sensitivities: some odors we can smell in impressively small amounts, such as the banana-y odor of amyl acetate in .01 parts per million; others need to be thousands of times more intense for us to notice them. We are great at detecting and recognizing coffee, which, like most food and drink, has hundreds of constituent parts, but we are utterly anosmic to plenty of other molecules. Some other animals smell carbon dioxide; we do not. For an especially puzzling case, look at carvone, in the class of naturally occurring chemicals called terpenoids. It comes in two identical forms that are mirror images of each other. One form smells like caraway seeds, the rye bread of an old Jewish deli; the other smells like spearmint gum. Our brains read the same molecule as entirely different smells. Any model of receptor processing tries to bring a rhyme or reason to why a molecule smells the way it does to us.\n\nOur olfactory neurons function just as dog neurons do: they serve to transmit the message of an odor's arrival to the brain, which then scrambles around trying to figure out what in the world it is. That's when you \"smell\" it: when the brain registers there's something there. In some sense receptor cells \"know\" what the smell is\u2014insofar as each will allow only specific molecular shapes to bind to it\u2014but they don't really know. It is the brain that knows (or doesn't), and that swoons with the rush of a memory of hot chocolate after a long winter's day playing outside, or balks at a urine smell in the subway, source unseen.\n\nOlfactory neurons themselves are pretty special. In all animals, these cells regenerate about every thirty days. You trade out old summer neurons, which may have conveyed the loveliness of the lavender garden and the rank odor of warmed manure to your brain, for new fall ones, ready for apples fermenting and coats being unmothballed. This fact is extraordinary. Aging usually means deterioration: all our senses dim through damage and cell loss. Our hearing diminishes over time as we damage our auditory cells by the fact of merely living (and listening to loud music through headphones, waiting for subway trains, and standing too close to the fireworks). In the course of a perfectly normal life we age into glasses, then reading glasses, then bifocals. But there is not an odor correlate to staring at the sun, turning the headphones volume to 11, or touching a scorching cast-iron pan. Unlike the neurons that allow you to see, hear, or touch, whose damage can lead to permanent loss, the nose keeps growing shiny new cells.\n\n### EVERY MAN TO HIS SMELL\n\nDespite the qualitative and quantitative differences between dog and human noses, throughout my investigation of our snoots, I heard a surprising comment from various psychophysicists and neuroscientists. The human \"schnozzola,\" as Stuart Firestein sometimes calls it, is \"quite good,\" he says. Dr. Noam Sobel, neurobiologist at the Weizmann Institute in Israel, holds nothing back: in his papers he writes, variously, that humans have a \"superb\" or even \"astonishingly good\" sense of smell.\n\nAt first, these claims are puzzling. Every walk outdoors with my dogs seems evidence to the contrary. To see them suddenly stop, turn on a dime, and hightail it back five steps to nose something invisible on the curb is to see that my nose is an inferior model (perhaps, in this case, given what might be emanating from the curb, to my great satisfaction). If my nose were \"astonishingly good,\" then I should, in theory, be able to experience some dogness by just remembering how to turn the thing on.\n\nNot quite.\n\nThe neuroscientific research does indicate that, in essence, our olfactory equipment is reasonably good. What it overlooks, though, is what you and I know intuitively: how we use our noses. I knew that I was not smelling the way my dog was: she loved to loiter on invisible odor markings on every surface, in every breeze; I rarely bothered to sniff.\n\nOn the other hand, there is one convincing demonstration of the range of our noses: breakfast. How was breakfast? Did you taste it? If so, you have just confirmed your sublime sense of smell: taste is 80 percent smell. When we chew food, we are basically loosening odorant molecules from their tethers, warming them, and sending the odor-laden air backward in the mouth, where it is but a quick journey up the chimney of the throat to the nose. If in your childhood you ever experienced or witnessed the classic milk-out-the-nose result of getting the giggles in the cafeteria, you have experienced the short connection between the mouth and the nose. Smelled food needn't go all the way out; indeed, it only has to rise to the olfactory epithelium. As we exhale while eating, air from the lungs pushes past the back of the mouth, grabs some warmed food odors, and sends them up into the nose's backdoor. At least, if you're being polite and chewing with your mouth closed.\n\nThis top-secret route is called retronasal olfaction. Humans are terrific back-of-the-mouth smellers. The retronasal route is largely responsible for the fact that we experience food as having any flavor at all. While the taste buds impart experience\u2014of sweet, sour, bitter, salty, or umami\u2014these experiences will never add up to what we mean when we think breakfast. \"The sense of flavor produced is a mirage,\" Gordon Shepherd, another neuroscientist, has written. \"It appears to come from the mouth.\" Breakfast's deliciousness comes largely from the experience of its odor, as you can quickly confirm by plugging your nose while you take a bite. The feeling of the food\u2014the crunch of the toast collapsing into a softness\u2014is still all there. Indeed, the feeling of the food may be more present than you'd like: repeated chewing and toast suddenly feels gummy on the tongue, not an experience most are hoping for with their morning toast. Unplug your nose and the taste comes back in waves: a yeastiness, perhaps a caraway seed, the rich note of butter. Your nose did that!\n\nYou might experiment with an orange. Pick out a firm, well-colored specimen. A delicious, bright-smelling odor will greet you as your thumbnail punctures the rind: pith, zest, the effervescence of orange. Peel off a segment, slice it in two, and pop a half in your mouth. Let it rest on your tongue, but do not bite. You might feel its juiciness, sense that it might be sweet\u2014but, notice, you cannot taste it. Now chew! The orange has returned to your senses. Pinch your nose and it disappears. Unpinch it and it's back to citrus central.\n\nThe reason it tastes so pleasing may be what the neuroscientists mean when they say we have great noses.\n\nSo it is that our orthonasal (through the nostrils, breathing in) olfaction is surpassed by our retronasal (through the back, breathing out) olfaction. If you've ever watched a dog eat, you have seen that the conditions seem to be reversed for the dog. Though your dog may delight in rolling in dead squirrel and may lick a passing dog rump with seeming pleasure, if you put something unappetizing in his bowl, he sniffs it and turns up his nose at it. Dogs use orthonasal for examination. If, on the other hand, the food passes muster, it is usually gulped down. Chances are that there is not much or indeed any retronasal olfactory experience: airflow in the nose hinders smells from rising up the very long route from the mouth. Nor is the food even in the mouth long enough to be smelled\u2014let alone savored.\n\n### SMELL, MEMORY\n\nSince the olfactory neurons reach into nose on one end and into brain on the other, a single synapse\u2014the connection between neurons\u2014separates the world of subway odors and over-perfumed teenagers from our fragile central processing unit. In two synapses word of odor has traveled all the way to the cortex. \"One of the reasons I work on olfaction is that it's a very shallow circuit,\" Firestein told me. \"You can get from the outside world to cortical tissue in the brain in two synapses\u2014two synapses! In the visual system you'd still be in the outer retina.\" The nose reaches the cortex lickety-split.\n\nOnce in, olfactory information cascades through the brain, knocking us into sensation and remembrance. This process\u2014the creating of the olfactory experience\u2014is not deeply understood, even in humans. Every smell researcher I talked to, though, held out hope for the answer to how the actual experience of smelling is formed. Firestein, who, like many olfactory researchers I spoke with, came to his research topic inadvertently, told me: \"The promise in olfaction is that it is one of the systems in the brain where we really could get from an initial stimulus interaction\"\u2014from the odorant\u2014\"to some kind of a percept.\" Such a thing is within reach of the science of vision. When we spot a parent's face in a crowd, we now know that there are specific cells in the visual cortex that identify the vertical and horizontal lines on the face and others that recognize the fact of it being a person's face and not a balloon face\u2014even before it is linked to memory and we can smile and say, \"Dad!\" The scientific knowledge of what happens past the olfactory bulb, by contrast, is a little scantier.\n\nWhat happens upstream? It is not uncommon for an academic paper on some component of olfaction to say that an interesting-looking part of the system is \"unknown.\" How the brain translates a pattern of neural firing into recognition of a scent is still a mystery. Even the mechanism of the very first step, the receptors, is still partly unconfirmed. \"On all these different fronts we have no idea,\" Vosshall tells me. \"We don't know what the odor space is, we don't know how the receptors capture that odor space, and we have no idea how the brain takes all of this information and makes a picture.\" Avery Gilbert puts it more bluntly: once you get past the bulb, \"All bets are off. Nobody has a fricking clue.\"\n\nWhat we do know is that once we are two synapses in, many regions of the brain then hear about the smell: this includes various other parts of the cortex, the amygdala, the hippocampus, and the cerebellum. These landmarks are clues to unpacking some of our experience of smells. First, noticing and responding to a smell can feel automatic, unmediated by reflection. There is a reason for this: olfactory information goes straight to the forebrain, missing the stopover point\u2014the thalamus\u2014where all other sensory systems arrive when entering the brain. Without air traffic control, the scent is flying in under the radar. Our reaction to a smell is often just as fast as the noticing of the smell. Second, olfaction is the quickest route into the amygdala, considered the emotional center of the brain. \"The memories you get from olfaction are always emotional memories,\" Firestein confirms. \"You don't smell something and remember an equation, or a page of text. It's always Grandmother's house, somebody's closet, first day of school, an ex-lover.\"\n\nThird, the hippocampus is involved: the seahorse\u2013shaped part of the brain involved in the making of memories. Sitting in a too-large overstuffed chair in the dark of your grandmother's living room; coming upon a decaying animal in the woods; a new boy at school scooching in next to you on the bus after gym. As we begin to process these memories, a rush of odor sneaks in on the memory's coattails. Later, the odor itself sparks the whole scene.\n\nIndeed, if smell has a good reputation with us, it is for its role in igniting memories long hidden from view. Scents cause a scene to shine suddenly\u2014a sun blazing out from behind clouds, coloring the space before it. Often the memory is accessible only through the trigger of a smell: as a molecule wafts into your nose, you are transported into a childhood far, far away, and into the head of the child who inhabited that time and space. The limits of brain science on olfaction seem apt, somehow, given the limited access the conscious mind has to the memories preserved by unknown odors. These memories are not Proustian, as the term has begun to be used: they are not held in crystalline suspension, undistorted by subsequent experiences and learning. But they are evocative, calling forth, as often as not, a summary of an experience.\n\nAsk anyone for their oldest smell memories and you get these characteristic, emotive responses.\n\nMy father:\n\nGene, who worked in Dad's [furniture] shop, he was the guy who varnished and worked with the woods and he always smelled of the stuff. That was the smell of Dad, a bear of a guy. And Mom's attic smelled like\u2014was it mothballs? It had a strident smell . . . she was so tough on Dad.\n\nMy mother:\n\nMy grandmother would share a room with me when she visited, each of us sleeping in a twin bed. She would put this talcum powder on herself in the morning\u2014poof poof poof\u2014and it went everywhere. I did not like that.\n\nThe inside of Daddy's hat. Holding it before handing it to him, hanging it up . . .\n\nA rooftop tar warmed by summer, the smell of your grandparents' attic, the brewery or river or tree grove on a childhood walking route, blackboard chalk and rubber cement, a struck pipe or a rolled cigarette, Play-Doh and suntan lotion, wet wool. A smell caught by an inhalation can be friction to matchstick, igniting the dormant memory. Let loose from the downy dandelion head of memories is a seemingly single soft moment with a thousand threads that spin away. Interestingly, these memories are rarely brought on by the foul and noxious odors that vex us in our ordinary days; instead, they are colored the nostalgic hue of childhood. I catch a sharp, bright smell like that of old tapes and am brought to the round, woody, tobacco smell of my father's desk in his study, the wide drawers opening to reveal packs of cigarettes, untold sundry desk supplies, and long pads of paper decorated in his scribbled hand\u2014and there he is, sitting before me, large and smiling, ready to pause to greet a daughter poking her head in. It is not one moment but all such moments that I see: a childhood stitched together by drafts of smells.\n\n### BECOMING SMELLY\n\nDo we all smell the same? The answer, you surely know if you have a functioning nose and have not been living in a cave, is clearly not. And this is true of both meanings of the question: How do we smell\u2014is it the same? and What do we smell (or taste) of the world\u2014is it selfsame? For the former, a trip on the subway at 8:30 on a weekday morning will serve to disabuse us of the notion that it is applied fragrances\u2014perfumes, colognes, scented shampoos\u2014that make us smell different from one another. If anything, fragrances unify us in smells, whereas our individual odor is unique.\n\nAnd just as the philosopher might puzzle over whether we all see the same color that we call red, it is an open question whether we all smell the same odor that we call by a name\u2014the strawberry or spice that accompanies the red. We all smell different odor scenes: exactly what you smell of the world is different from what the person standing right next to you smells. This is part biology and part autobiography. On the one hand, there is good evidence that every person's olfactory genome is slightly different, resulting in individual variations in what odors each of us can notice and attend to. Selective anosmia\u2014smell-blindness\u2014to particular scents can be inherited. Some people with a particular genetic construction cannot smell isovaleric acid, a component of body odor, at all. People differ in the threshold at which they can detect the presence of various scents by several orders of magnitude.\n\nOn the other hand, we each learn odor preferences and aversions, and even cultivate degrees of attention or inattention. While we both may say we see red and smell strawberry, your red may be brighter than mine. And my strawberry may smell sweeter\u2014and may bring to mind the intertwined smells of tomato-plant stems and warmed, pocked strawberries on the vine on a Sunday afternoon in the garden by the house. (Boy, those were delicious nuggets of berries.)\n\nThere are better or worse smellers, of course. \"Many people claim they don't have a good sense of smell\u2014\" Firestein begins, then implicates himself: \"like me. I don't have a very good sense of smell. It's almost invariably not a neurological issue.\" Instead, \"it's sinusitis, it's inflammation, it's allergies . . .\" The conduit to the epithelium must be clear. \"You can then say to them, 'Well, how is food for you? Do you like food?' That, they taste fine.\" Flavor is intact, so retronasal smell is intact. They have a sense of smell and do not know it.\n\nBut the majority of the experienced difference between individuals in their sense of smell is within their control. Every perfumer is made, not born. Even each detection dog is trained for years before being sent out to find explosives or bedbugs or pine martens or illegally imported guava fruit.\n\nHow much are you willing to push out your nose and sniff at the world?\n\n* * *\n\nI. Notably, cetaceans are an exception: dolphins, for instance, do not have an olfactory bulb, though they have a \"nose.\"\n\nII. It is worth noting, in this age of distinction between natural and man-made or artificial products, that the term chemicals includes it all: every product of nature is wrought of chemistry.\n\nIII. Even so, some nonhuman primates can detect smaller amounts of fruit odors called aliphatic acetic esters than dogs can. This makes sense for primate frugivores\u2014or at least fruit lovers\u2014as contrasted with the historically carnivorous ancestors of the dog. (Lest any primates feel too proud, know that Asian elephants are also very good with their aliphatic odorants.)\n\nIV. With exceptions: metal does not have a smell, for instance. Just as the \"smell of sun\" is the smell of things warmed by the sun, not the smell of the sun, the \"smell of metal\"\u2014an iron banister after touching it, a handful of pennies\u2014is actually the smell of the metal interacting with our own sweat, not a smell of the metal per se. Indeed, the researchers who discovered this describe metal's smell as a \"type of human body odor.\"\n\nV. This is not to say that we might not implicitly detect each other's biology in some other way: researchers are looking at \"trace amine associated receptors\" as perhaps involved in detecting the presence of bacteria. If they do so, then Firestein says, \"The types of things that people would have said the pheromones are doing . . . judging the relative health of a potential mate or rival,\" could be done by these bacteria receptors.\n\n## 6: My Dog Made Me Smell It\n\nIt is humbling to read the words of Helen Keller on the topic of smell. Left without two of her senses, it is perhaps not terribly surprising that as a child she became exquisitely sensitive to her remaining senses, especially smell and touch. Still, her descriptions of experiencing what she calls the \"elusive person-odor\" sound as if she could be channeling a dog outright. From someone's simple exhale, she could read \"the work they are engaged in [for] the odors of wood, iron, paint, and drugs cling to the garments of those that work in them. Thus I can distinguish the carpenter from the iron-worker, the artist from the mason or the chemist. When a person passes quickly from one place to another I get a scent impression of where he has been\u2014the kitchen, the garden, or the sick-room.\"\n\nMight anyone be able to notice the scent impressions evident to Keller\u2014and any attentive canine?\n\nThere may be people who are born noses. But most of us are just born with noses. Plenty good ones, as we have seen. Apart from retronasal olfaction, the other bit of evidence of the considerableness of the human nose is more abstract than comes with breakfast orders. The recent coding of the human genome alerted us to the astounding fact that around one percent of our entire genome is devoted to coding for the olfactory receptors of the nose.\n\nOne percent! At first blush, that might not seem like a lot, but our time spent smelling is much less than one percent of our lives, between what we are remembering, scheming, seeing, wondering, daydreaming, feeling, swallowing, uttering, digesting, respiring, suggesting, moving, or thinking from moment to moment. And yet the genetic blueprint for you reserved a one-percent slice for smelling, spit-spot and ready for use.\n\nPsychological research has provided plenty of examples of the inborn skill of the human nose. They are perfectly normal abilities, and yet they are striking. Mothers can pick out the smell of their newborn baby's shirt from among other newborns' shirts within two days of birth; and newborns, in turn, can distinguish their mom's smell from that of other moms\u2014as well as recognize the amniotic fluid they swam in for nine months. These abilities are, evidently, innate. Nor do we lose them after infancy. Children can pick out the smell of their siblings from that of other kids of the same age\u2014even after not seeing each other for two years; what is more, there is evidence they can identify their friends by smell. We recognize our own smells\u2014our own body perfume, captured in our clothes, or our preferred applied perfumes, woven between strands of our hair. An experiment asking college students not to bathe or use soaps or fragrances for twenty-four hours while they wear a plain t-shirt found three-quarters then able to pick out their own t-shirt from among nine other similarly unwashed tees. We have the same ease with our partner's smelly t-shirt. When asked, we find it trivially easy to tell genders apart by smell alone.\n\nRecognizing one's infant or partner makes a certain amount of biological sense, even if you've never inhaled the odor of a t-shirt in the hamper and thought, Ah, great, that's my brother! Indeed, given how common it is in the animal kingdom to recognize family members by their odors\u2014everything from the paper wasp to the Belding's ground squirrel to the spotted hyena uses odor to recognize packmates and kin\u2014it would be more surprising if we did not.\n\nHappily, if oddly, researchers have provided sundry evidence that our olfactory prowess ranges a bit farther afield than family and friends alone. In one study, nearly 90 percent of owners asked to sniff a pair of blankets, each rolled on, slept in, and generally dribbled on by a dog (one their own), could tell which of two blankets had their own dog's odor. (Notably, the owners did not always find their dog's blanket the more pleasant-smelling of the two.) Just as the dog recognizes his person by smell, we can recognize our dog.\n\nIn another case, blindfolded subjects were able to reliably distinguish closely related strains of laboratory mice. They could tell them apart by the smell of their bodies, fecal pellets, or urine. When I questioned the experimenter, Avery Gilbert, then at Monell, about the motivation for this experimental design, he laughed. For many years researchers had been trying to fractionate mouse urine, he told me, \"to find the active molecule that was the differentiator between the two strains. It's a huge project chemically; you have to have lab techs squeezing mouse bladders into test tubes . . . I was just\u2014Can we smell it?\"\n\nThey could.\n\nThe physicist Dr. Richard Feynman described feeling prompted by reading about the skills of bloodhounds to explore his own latent olfactory powers. Since dogs can identify items that had been briefly touched by a person, this is what Feynman tried: he asked his wife to handle a single book from their bookshelf while he was out of the room, then return it. Feynman picked it out. Later this became a party trick of his, matching three books to the three people who had touched them. \"[N]othing to it!\" he wrote. \"It was easy. You just smell the books.\"\n\nYou just smell the books. \"People's hands smell very different,\" he suggested. \"All hands have a sort of moist smell\"\u2014and, of course, the smoker and the well-perfumed and the person who habitually jingles the coins in his pocket all wear their habits on their hands.\n\nYou just smell the books. That very evening, I set out to test some of these abilities with my unpracticed nose. Having no lab mice at hand, but having good availability of both dogs and books, I cobbled together a few home trials. My dogs both rest on a short blue sofa during the day. Lying rump to rump, they bisect the sofa but seem to choose sides randomly. So I asked my husband to monitor their resting poses and let me know when they awoke, stretched, and wandered out of the room in search of other diversions. Then with no evidence left of their presence but their scent, I entered to sniff my sofa.\n\nBelated apologies to anyone who has sat on that sofa. It does smell very much like the fur of dogs\u2014not well known as a popular human perfume. I stuck my nose right into the cushions, smushing nose and cushion alike. The sofa faces north; its west end smelled clearly of Finnegan, musty and close and tangy. The east, Upton\u2014a peculiar, somewhat unwholesome smell, recognizable also by it being not-Finnegan. My husband confirmed my choice. The dogs had returned to the room and were gazing at me dolefully. Upton took the opportunity of my kneeling posture to hump my leg.\n\nOf course, I had a decent shot, one in two, of getting this in one go. So I have done it again and again. I'm now beating the house: as I smell, I improve and am more reliably right.\n\nKnowing my son to have the moistest hands in the house, I then pulled out a handful of well-read books of similar size and backing, and I set them on the sofa. \"Handle one, touching it as much as you want, then put it back where it was and don't tell me a thing about which one you touched,\" I instructed my son, then six. This is the perfect game for a six-year-old, incidentally, should you have any around. Delighted, he did my bidding, an adult witness elbowed the books back in line, and I came in to Feynman the scene. I followed his very specific instructions\u2014\"You put each book up to your nose and sniff a few times\"\u2014and I was at a loss. Newish book covers are much more secretive about their identities than the pages they shelter. But one book did smell more . . . warm-blooded, maybe, than the others. I guessed.\n\nSpot-on.\n\nSo clearly it's not that we can't smell; it's that we largely don't. And it's not only that we choose not to\u2014although we do choose not to. One theory of the demotion of smell from our sensory lives is that our lost urge is traceable to our newfound bipedalism. When, in the course of human evolutionary events, our distant ancestors were equipped to stand upright, we not only began \"flashing our genitals\" to each other, as Freud worried; we also distanced our noses from the ground. The ground is a terrific source of smells\u2014not only do things that emerge from the ground have an odor, most smells wind up on the ground, settling there after a trip on gossamer wings (or a current of air). Another, not-incompatible theory suggests that our sense of smell lost importance as our vision gained it, and thus began the shift on our faces and in our brains to visual primacy. As three-color vision appeared in primates' history, there was a decrease in the number of functional olfactory genes. Relatedly, a shorter snout allowed our eyes to develop closer to each other on the face, allowing for more overlap of our visual field and better binocular (three-dimensional) vision, which we cherish.\n\nAs I glance in the mirror, I'll admit, my nose doesn't look as up to the task as my dog's nose. Instead of projecting proudly forth like a dog nose, with nostrils open for all smelly comers, we house the nostrils at the bottom of the pyramid perched above our mouths. To be sure, it is the most prominent part of our face; indeed, the bones of our face are split in the middle to make room for it. Still, it hardly reaches out into the environment with bold and curious gestures. Instead, we tend to move our faces\u2014noses with them\u2014away from any approaching surface that might have a strong smell. One of the oddest things to do with your face is to push it right up into something: a person (often strongly frowned upon), the ground (only when we've been tackled), our food (we bring food delicately to the mouth\u2014not vice versa\u2014held on long silver implements to avoid even our hands touching it).\n\nBased on the biological facts alone, the major determining factor in whether I smell an odor or not seems to be just this: whether I bother to try and smell it. Breathing normally about twelve to sixteen times a minute, that gives me twenty thousand opportunities each day, plus or minus, to smell something. We are, most of us, perfectly equipped. Sure, there is variation among people from the get-go, based on their distinctive genomes. But the major difference between me, on the one hand, and the sommelier and the perfumer, on the other, is one thing: time spent sniffing. Committed sniffing. Practice. Time spent putting our noses in things: under the horizon of a wide wineglass, over a small brown bottle containing a chemical formulation. Is there something in your hand now? Draw it to your nose and smell it. Likely, there's an odor there, but to trace that line from hand to nose\u2014or even, heavens, to bend to smell the roses\u2014is a bit out of fashion.\n\nPractice is more familiarly applied to piano-playing, rifle-shooting, or tightrope-walking, but the upshot is the same for smelling. All these skills, in fact, actually depend on sensory practice: adjusting one's fingers according to what one hears; coordinating pressure on the trigger with sight of the target; vision and proprioception\u2014the sense of the position of one's body in space\u2014as one toes the tightrope. If you are a baker, you come to know when the apple cobbler is finished cooking. This ability comes from baking many apple cobblers, to be sure\u2014and also from noting the intensity of smell that accompanies a cooked one. William James tells of a woman who could sort the cleaned linen of the residents at the Hartford Asylum by smell, from simple repetition of the daily task, and a man who came to be able to taste the difference between the bottom and top halves of a bottle of Madeira. That was a man who had tasted a lot of Madeira.\n\nWhat Helen Keller had was obligatory practice. Deprived of hearing and vision, she was obliged to learn about the world with her remaining senses, and smell came to the foreground. But with some voluntary practice, could we learn to tell the iron-worker from the chemist? Or learn to smell when our children are home, a friend's illness, if someone else has been kissing our spouse, or how long ago a dog had been in a now-empty house?\n\nExperiments have tested the added value of time spent practicing smelling. One study found people capable of learning the difference between two pairs of nearly identical molecules\u2014one pair \"green\"ish-smelling, the other \"oily-vinous.\" They merely sniffed the bottles repeatedly. Oh, and they were given an electric shock with one of each of the pairs.\n\nThat might have done it. Motivation might be required, but it needn't be aversive. Have we learned nothing from the years of miserable results of aversive dog training? Punishment does not lead to efficient learning; reward does.\n\nAnd so another experiment was founded on chocolate, not electric shocks. First, a ten-meter line of twine was buried in a grass field in Berkeley, California, forming a simple trail with two straight segments connected by an angle. The trail smelled of chocolate: the twine had been soaked in diluted chocolate essential oil. Untrained volunteer subjects donned opaque goggles, earmuffs, work gloves, elbow pads, and knee pads to reduce their sensory experience to the olfactory. They were set on their hands and knees on the field three meters from the trail; each wore an airflow monitor below their nostrils to record their sniffing. It is not clear whether the experimenter said, Go find it! as one might to a working dog. But twenty-one subjects\u2014two-thirds of the group\u2014did successfully find the trail and follow it to its end within the ten minutes they were given.\n\nAerial video of one of the trials, played at four times actual speed, shows a definite sniffing creature, if not quite a doglike creature. Hands planted wide as she crawls along, the well-padded and sensorially restricted subject finds the trail without too much trouble, turns efficiently onto it, and zigzags her way down the rest in less than a minute. Her head veers back and forth between her hands, as one might vacuum a hallway rug absentmindedly.\n\nSome of the subjects then agreed to practice the task: they sniffed along similar versions of the trail three times a day for three days. All of them found and followed the track each time\u2014and they got faster with practice. By the end they were twice as fast as when they'd begun, speedily sniffing to the trail's end. They deviated less from side to side along the way. And they sniffed faster: going from one sniff every three seconds to two sniffs in that same time. Through various manipulations that must have required subjects with particularly good humor, the researchers also discovered that sniffing with both nostrils was better than with just one, and that it was the fact that each nostril gulps a slightly different square centimeter of air that made the difference.\n\nNow, these were no dogs. All the humans were generally pretty laggardly, taking minutes to crawl ten meters\u2014approximately the pace of an infant who has not quite learned crawling. The odor source was still lying there on the line in the grass\u2014quite unlike a trail left by a source who then flees. And the trail was a simple, continuous path, which varied little in the practice rounds.\n\nLet's not forget that these subjects were asked to smell for chocolate, not bottom-of-criminal's-shoe. No tracking dog is at any risk of losing his employment to these subjects. But the simple but significant result was: they got better at it. Practice made reliable chocolate-trackers of them.\n\n### \"THE LOST MUSCLES OF THE NOSE\"\n\nSo I resolved to practice. To simply sniff\u2014and sniff again with the other nostril. To, first, bother to smell the objects that I touch and the air that I stride through. As I go through my days, to remember to inhale through my nose with the same intent with which I gaze with my eyes.\n\nAnd also to sniff more assiduously, to practice in the manner of the chocolate trackers or the wine and perfume experts\u2014expertise wrought of training, not genetics.\n\nKnowing about the role of the humble sniff in dog olfaction proved practicable: if I was going to try to learn how to smell well, I'd better get my muscles in shape. My nose muscles.\n\nYou would be forgiven for not attending to your nasal musculature; indeed, one might not even realize that one has muscles in the nose. Editors of the redoubtable Gray's Anatomy apparently paid so little attention to the nose that of the seven \"intrinsic,\" or interior, muscles listed in their 1901 edition, three mysteriously vanished by the 1989 edition.I This excising didn't have to do with a rapid devolution of the human nose, nor with exciting developments in our understanding of facial anatomy; instead, it may simply have represented neglect. Noting the muscles' absence from medical school anatomy textbooks, some physicians concerned with what happens to nose muscles after rhinoplasty (plastic surgery on the nose) confirmed in 1996 that those muscles still exist in our schnozzes. Not only that, these small, neglected muscles are critical for the nose to do its nosely duty.\n\nA dozen muscles are dedicated to movement of the human nose. Each has an important expressive or expansive task. There are muscles responsible for helping you wrinkle your nose with disdain or disgust; for the contracting and dilating that leads to, in the physiologist's terms, \"a marked appearance of the countenance\" and a look of contempt (that's the Levator labii superioris, should you need working on your contempt face); for changing the size of the nostril opening for breathing, working hard against atmospheric pressure; for enabling an expression of anger (think cartoon fire-snorting dragons and bone-tired racehorses); and, finally, for sniffing in odors.\n\nSome of the sniffing work is done by \"alar flaring\": the energetic widening of the nostril\u2014pulling out the wings\u2014with the intrinsic muscles. In some people (sometimes those who go in for the above mentioned rhinoplasty) the nose wings are always flared, giving an appearance of moderate excitement or being perpetually mid-sniff. One study found that 40 percent of subjects, including a majority over the age of seventy, could not flare when asked to. Try to compensate with other muscles of the face, and the result is a combination grimace\/snarl. (Which, temperamentally, might be expressive of the experience of not being able to use the nose properly.)\n\nAs it turned out, I'm in the 60 percent, and no workouts were needed, merely the knowledge of what I can do with a sniff. Just an awareness of the crosshatched muscles running along my nose made focusing on using them easier. Even the most inexperienced sniffers can voluntarily change how much air is taken in, how fast they take it in, how long the sniff lasts, and how many sniffs they take. Your brain, meanwhile, is not unimportant in this moment. It is the brain that determines what the smell is, not the nose. It takes into account the strength of the sniff: if you strain a nostril muscle sniffing hard, then the perceived intensity of the odor is less\u2014you needed to work to get it upnose. In fact, just sniffing at all is sometimes enough to trick your brain into thinking you are smelling something: if you sniff and there is only pure, clean, odor-free air (inasmuch as this is possible), the primary olfactory cortex will probably register an odor. Both the flow of air and the activity of the muscles contribute to the brain's miscue.\n\n### PRACTICE\n\nIt didn't start well. My first training at smelling led me to bouts of nausea, repugnance, and fear.\n\nThe solicitation was blunt. \"The study involves smelling odors and evaluating them. The first visit will take about two hours.\" An olfaction lab at Rockefeller University run by Leslie Vosshall was looking for subjects. The call for participants was not in and of itself tempting. And yet there was an implied evaluation of the smeller herself that appealed to my competitive spirit: \"There is the possibility that you may be invited back for up to ten visits in a relatively short period of time to evaluate more odors.\" This seemed the perfect opportunity to practice sniffing: regular bouts of compulsory smelling, in a school environment, no less.\n\nI neglected to consider what ten visits at two hours a pop might be like.\n\nThe eligibility requirements are:\n\nAge: over 18 under 50. Nonsmoker.\n\nNo allergies to odors or fragrances. No asthma.\n\nNo history of nasal illness or surgery that would affect your sense of smell.\n\nNo currently active seasonal allergies or upper respiratory infection.\n\nNot taking medication that would affect your sense of smell.\n\nI was eligible. I wrote to the study coordinator.\n\nJanuary 22. It's snowing in the morning as I make my way across town to Rockefeller's campus, perched on the far eastern side of Manhattan, behind a set of gates and with no crossroads inviting passers-through. Traffic slows and the city is partially silenced. The piling snow busily buries any city smells. With the wind, sensation is reduced to the tactile\u2014cold prick points of snow pummeling my face.\n\nAt the university, I work to pull open the doors, extra heavy and nearly vacuum sealed. Inside, they've bottled the waiting room's perfume: part hand sanitizer, part wet newspaper and magazine. I am given an informed-consent form to sign that explains, gently, that 20 percent of subjects will have a temporary diminishment in their sense of smell after participating. Taking in the distinctive, acrid smell of Purell, I wonder if that would be all bad.\n\nIn the experimental room, three women sit stoically in front of computer screens, each with a large bin of small bottles at her side, and tap occasionally on keyboards. They do not remark on my entry. One woman rummages among the bottles, unscrews the cap of one, brings it to her nose, and unemotionally keys in some response. One might take them for professional sniffers. But when I joined them it became clear that they were just hundreds of sniffs further into the game than I.\n\nPeggy Hempstead, wearing a sweater and warm smile, greets me. She is the study coordinator and walks all the subjects through what kind of smelling adventure they are about to embark on. She brings a wide tray of tiny bottles over to me. I feel a shiver of excitement. One hundred white-capped brown glass bottles jostle and tinkle against one another as she sets the tray down. I will be asked to evaluate the odor of the contents of each bottle: first, its strength, pleasantness, and familiarity, but also its characteristics, its allusions, its basic \"notes.\" Does it smell like fruit? A bit fishy? Or is it grassy? Urinous? Edible? They provide seventeen options. I hardly feel prepared.\n\nHempstead puts on a grave expression. \"Some of the smells are very strong,\" she warns. She recommends taking a slow, serpentine route from uncapping to nose.\n\nI listen but begin recklessly. Peering inside the first bottle, I see a few drops of liquid resting at its bottom. The merest hint of a liquid. In some bottles there will be enough to pool at the bottom, a tiny puddle I will swirl around like a wine snob to try to get more of a particularly weak scent. In others the liquid has crystallized and begun to climb the side of the bottle, lunging toward escape.\n\nI open my nostrils wide for the first bottle and sniff until I'm full of air. I have no idea what I am perceiving. Clearly, this smells, I think, as I raise this, then the second, and the next bottles to meet my nose. But what is the smell? I rate each on \"fishiness\" and \"acidity,\" but my assignments seem haphazard, random; I recognize nothing.\n\nA dozen bottles in, I am reassured: a familiar scent. Coconut. The full, sweet roundness of it wallops me as the bottle is on its way northward. My childhood birthday cake; the macaroons my father coveted (and grandmother indulged); the pleasant, sweet nuttiness alongside a warm curry. Swimming in memories, I am momentarily victorious.\n\nI am also about eighty-eight scents from finishing that day.\n\nBy bottle thirty I am disoriented, dizzied by the odors strobe-lighting my olfactory bulbs. There are innumerable odd smells that take me down long hallways with no clues where I am going: a deep sour note turns sharp\u2014Or is it spicy? Is it the smell of a person?\u2014and then disappears behind an unseen door. My head swirls with echoes of smells, but nearly all are utterly unplaceable\u2014like a tip-of-the-tongue phenomenon, with no chance to retrieve the lost word. A tip-of-the-nose phenomenon. In this verbal purgatory one can judge whether an odor is familiar, and even name similar odors, but can come up with nothing about the name of the source. Some odors ring a bell, but more often than not it is a distant, muffled bell covered with blankets and smothered by traffic noise.\n\nThere are a very few glimmers of light. In the middle of a long streak of utterly unidentifiable smells, a particularly weak smell penetrates my brain and comes out with arms up in celebration: Hello! I'm pencil shavings! I sniff it again, redundantly, out of pure satisfaction.\n\nAlong the way, I also meet and identify bubble gum and almond extract. Another arrives at my nose with a whomp of recognition: it is my local health food store. I can almost hear the bells that announce my opening its front door\u2014which simultaneously brings this distinctive odor. The next week I search out the owner in his shop. \"Did you ever notice that there's a distinct smell . . .\" I begin with some hesitation. People might not want to hear about their \"distinct smells.\"\n\n\"Wheatgrass!\" he cuts me off. \"It's wheatgrass. Everyone says I smell like it.\"\n\nOne scent is unthinkably horrible. Worse, even as I suppress my own reflexive retching and jerk the bottle away, I know I'll have to smell it again\u2014for the sake of Science. There is little that prepares one for mustering the courage to voluntarily bring a sweaty, vomitous wash of air close to one's face.\n\nAfter some coughing, furious nose-blowing (as though I could force the odor molecules out of my cells), and a sip of blissfully odorless water, I get a reprieve with the next scent: Smarties. I loiter on it, letting the sweet, chalky pleasantness of it blunt my recent foul memory.\n\nAs I cap the final bottle, another scent suddenly fills the room: a sharp, sweet grassiness. I look around. Clearly, someone has spilled a bottle and released its smell into the room. But not only can I not see the errant bottle, no one else is looking around for the source of the odor. Hempstead, the coordinator, is walking calmly to a back room\u2014not at all in the hurried way one might to escape an olfactory catastrophe. I think to call after her, Hey! There's a smell! and catch myself. This is clearly an odor coming from my own head. An olfactory hallucination\u2014or maybe an aftereffect. I wait until Hempstead returns from the next room. \"Do people experience, um, imaginary smells . . . ?\" I ask. \"Oh, yes!\" She sounds delighted. \"All the time! Smell echoes, even.\"\n\nShe predicts my future. It takes me two hundred minutes to get through those one hundred bottles\u2014and that includes ones holding odors that I cannot detect, even with repeated and strenuous sniffing. I stumble out of the room. The entire episode has been stultifying. After I have put the final bottle back in its place and walked outside into the snowstorm, I search for the smell of the city. The only note that comes through is the exhaust from the idling cars, trucks, and buses stuck in snowstorm traffic. But an hour or two later, back inside, the scent echoes begin: mostly sweet, chemical-y smells coming from out of nowhere to suddenly punch my nose. Foods still have a smell\u2014I am not one of the 20 percent who experience temporary anosmia\u2014but they often have an additional odor: blueberries have an overlay of something rotting. Plain crackers are nauseating. Not only did my nose seem truly mediocre, unable to translate the odors into smells, it is now delivering smell when there is no odor. The afternoon is dotted with these smell apparitions until, finally, sleep blunts them all.\n\nThe next day I receive mixed news. \"I am delighted to let you know that you successfully completed the screening visit for the smell study,\" the email from Peggy Hempstead reads. The good news: despite my impression, I can actually smell worth a damn. The bad news: there are nine hundred more smells to go.\n\n\u2022 \u2022 \u2022\n\nOver the following months, I begin to catch on. I develop a method of really sniffing: inhaling as if my life depends on it. In my most ardent sniffs, I both pinch the nares in toward each other and flare them below, forming a belted skirt that rises in a breeze. Various nasal gymnastics\u2014elevate, dilate, compress, repeat\u2014become second nature.\n\nAs it turns out, there are many ways to sniff. And as with the dogs, the method of transporting odorant molecules into the nose turns out to be nontrivial. Without sniffing, only 5 to 10 percent of the air we inhale through our nose ever makes it to the epithelium. Unless we sniff, we do not smell. Olfaction is an \"active process,\" says one academic paper on sniffing. Without any air flowing through the nasal cavity, the \"olfactory scene\" in front of us vanishes.\n\nIndeed, by making careful recordings of the turbulent air movement around the upper lip, researchers found that we begin with a moderate sniff. Within fifty milliseconds, though, we adjust our so-called sniff vigor\u2014a gauge of the force of our nostril pull and how much air we vacuum in\u2014based on what we have detected. Given the speed of this adjustment, it appears that we are processing the odor subcortically, without awareness, and looping back to the nostrils to tell them to sniff harder or let up. Should the odor be very weak, or very lovely, we change the length of our sniff from sniff to sniiiiiiiffffff, increasing the force of the sniff and keeping at it. If it's a very strong odor, or very unpleasant, we sniff less vigorously and less long.II\n\nWe usually sniff many times, although one good sniff would be sufficient to get the odor in the nose. One \"good\" sniff would be a second or so long, taking in two cupfuls of air. And when looking to deconstruct an odorant mixture into its components, two sniffs are much better than one.\n\nOnce in a while I pause and take in the style of the experiment's other subjects. Nearly everyone in the study seems to sniff one nostril at a time, matter-of-factly delivering the bottle first to one, then the other. As it turns out, our nostrils do different work for us: we alternate being left- or right-nostriled. One of your nostrils is a low-flow nostril. It is not worse; it just works harder or longer. So each nostril gets a slightly different view of the odor. Dr. Rachel Herz, a psychologist who specializes in olfaction, has shown that right-nostril sniffing seems to be strongly linked with \"hedonic\" responses. We perceive a neutral smell as more pleasant when right-nostril sniffed. People also are better able to discriminate unknown, new odors when using the right nostril than when using the left. The left nostril, leading ipsilaterally back to the more verbal left hemisphere, is what enables us to name the smell, once it becomes familiar.\n\nMost of the time I feel smell-bereft. Many bottles, though housing something viscous or liquid, have no odor to my nose at all. I begin wondering about a prosthetic super-nose, able not just to leap buildings in a single bound, but to smell each resident inside as it did. The closest our society has come to inventing such a contraption came in the late-nineteenth century, when one Hendrik Zwaardemaker invented the olfactometer\u2014a device that has never taken off. It consisted of two porcelain tubes, one of which could telescope out within the other. The user was to stick out from his nose, playing elephant. A nose-shaped fitting made the business end, and a handle underneath allowed it to be wielded like a radar gun. Rather than being a magnifier, it was intended to measure how much of an odorant was necessary to be detectable: the outer tube was saturated with the odor. But one could imagine it consolidating the smell of the thing it is pointed at, a reverse megaphone enabling odors to shout at us.III\n\nIdentifying a smell was the highlight of any visit. Identification, though, was hard: the bane of my experience. I was missing words for smells, and the smells themselves were slippery creatures, eluding my attempts to pin them down with name or memory. Where was my language for smells?\n\nOdors can be smelled, certainly, without the odor having a name. But our species\u2014verbal and language-centered\u2014has a hard time fully perceiving without words. The more words for smells we have, the better we will be able to see them.\n\nBut as it happens, most humans are lacking in two ways. First, most languages have very few words for smells. English has a countable number\u2014countable on two hands. (Being generous, one might include alliaceous, barny, farinaceous, fusty, hircine, mephitic . . . and I've got four fingers left.) No \"pink,\" \"loud,\" or \"rough\" equivalents exist, no basic words for fundamental smells\u2014or for lovely or dreadful smells. Most of the time we are describing the source of the odor: oh, that smells like coffee; it smells of cow dung. Or the source might be even more abstract: it might smell like \"early summer,\" but what the thing that smells that way in the summer is\u2014the honeysuckle, the cut grass, the damp earth prodded by tiny, forceful plant shoots trying to come through\u2014is unknown. Instead, our words are more evaluative\u2014the lovely or disgusting\u2014or abstract\u2014\"green,\" \"fetid.\"\n\nNotably, not all languages are so smell-word-poor. Researchers describe the olfactory richness of the language of two Malay hunter-gatherer communities, the Maniq and the Jahai. Among the Maniq, whose culture values olfactory knowledge\u2014about dangers, for use in foraging, in medicine\u2014everyday language has dedicated words for scent. Odor words represent classes of perceptions, much in the way we use color words: as our \"blue\" covers the sky, the sea, and my son's eyes. For the Maniq, a word\u2014such as \"mi? huh\u0169\u03a6\"\u2014might be used in reference to a whole category of similarly smelling things and experiences: snakes, soil, mushrooms, sweat, or walking in the forest. The Jahai name odors as easily as we name colors. There are a dozen basic odor terms, including one encapsulating \"bat droppings, smoke, gingerroot, and petroleum.\"\n\nBy contrast, most of us (not living on the Malay Peninsula) are simply not trained, schooled, or accustomed to naming or identifying odors. Noam Sobel, one of the parties responsible for the chocolate-trained Berkeley subjects, describes putting a jar of peanut butter under a family member's nose while she closed her eyes. This person, he notes, eats peanut butter \"every day.\" The result? Though \"neurologically intact,\" she was unable to identify the smell. And this is a person who lives with a smell scientist. Research has found that when they encounter odors used or applied every day out of context, people are unable to name about half of them.\n\nTo be fair to humans, it has not seemed to matter to most of us whether we could wax eloquent about what we smelled, evolutionarily. A smell might lead us to flee, mate, or eat (or avoid those things). Not to speak. We still seem to have that thirty-million-year-old tendency to lead with an emotional reaction first, and search for the word to plug in second.\n\n\u2022 \u2022 \u2022\n\nEven when I cannot put a single name to the odor in one of those bottles, the satisfaction of recognition matches the great sensory pleasures of life: the thonk of an old refrigerator door closing; the smoothness of a ball bearing in a hand; goggling at the fitted intricacy and rhythm of a watch's gears. Over many visits I identify crayon; violet; something peppery that makes me sneeze for three minutes; flowers that have been left in the vase too long; nail polish remover; VapoRub; Red Hots; rum; apple lollipop; cooking meat; icing; Band-Aid; a biology classroom; Play-Doh; springtime; grass on a sweaty day; unspecified \"candy\"; charcoal; new vinyl.\n\nI make anatomical discoveries. My sneezing turns out to be controlled by an accessory part of the olfactory system. The nerve that senses touch to the face and the top of the head, the trigeminal nerve, notes when irritants of various sorts are inside the nose, mouth, or eyes. We experience all manner of sensation\u2014\"cooling, numbness, tingling, itching, burning, and stinging\"\u2014as something we are smelling, though we are not. Spicy foods can be felt as hot and stinging; the trigeminal nerve detects pollutants in the air, prompting the nose to tingle and the eyes to water.\n\nIn one protracted sniffing bout, I discover the magic of adaptation. When one keeps smelling the same odor, the olfactory receptors, best at noticing something new, get weary and the neurons stop firing. This is a physiological phenomenon with daily application: enter your local coffee shop and, if you are a coffee lover, the smell of beans recently pulverized by expensive burr grinders hits you at once. A few minutes later you may still be in line, but the smell is mysteriously gone. The coffee shop has not changed; the receptors that noticed its odor are just momentarily quiet. You can awaken them by leaving the olfactory space and reentering it later. So when I have a particularly tough odor to identify, the phenomenon of adaptation enforces a rest: simply sniffing more isn't going to bring it to mind.\n\nPractice brings new methods of searching for the odor's name. One extremely familiar smell (many were \"familiar\") causes me to put down the bottle, close my eyes, and look for the memory attached to it. It is in a house. One of the houses I've lived in. Where? The bathroom? No. Is there another person here? No. I look through a dozen empty rooms in my memory. This, I'd guess, is what it might be like not to have any thoughts\u2014living \"in the moment\" (the wide, vacuous moment). I put the bottle to my nose again and spend five more minutes in the void. Lysol? Sunscreen?\n\nMagic Marker. Out it pops.\n\nEven the most ordinary, obvious smells sometimes require circuitous routes to deliver themselves, smelled\u2014as they are\u2014out of context. I smell one and try to imagine what kind of place that smell might come out of. It's a spice. Cardamom? (Do I even know what that smells like?) It's a spice my mother would use, but only every once in a while. . . . It's an occasional spice. I picture the spice cabinet in my mother's kitchen, the shelves made just the width of the spice bottles, each with a rounded safety bar in front of it to hold in the spice kids on their wild cabinet-opening rides. Then I see it. Mom used it to cook some kind of loaf of meat. There: black, pointy\u2014ah. Cloves.\n\nDespite the grin-making pleasures of discovery and recognition, over the visits I begin having a physical reaction in anticipation of my arrival at Rockefeller. Just a very few of the hair-raising smells have left their mark. As I settle in with a new tray, my whole body braces for each approach of a bottle to my nose. My stomach curdles; my throat constricts. It is a major leap of faith to bring a small, condensed odor right up to my nostrils: to deliberately raise the bottle to my nose when I know the odor Pandora that emerges, every once in a while, out of it.\n\nAfter a time, Peggy Hempstead emails me to schedule another visit. I put her off for weeks on end, until my mind begins to forget the close, cloying smells that I might find in her bottles.\n\n\u2022 \u2022 \u2022\n\nOn the day I arrive back on Rockefeller's campus, months after my last visit, it is spring perfection. Magnolias celebrate their two weeks of bloom, cherry trees are aflame with blossoms, the air is mild, and the sky is blankly blue. It is a sharp contrast from the last time I left this campus, with winter lingering and the odors of hundreds of smells ricocheting in my nose.\n\nI find Leslie Vosshall on the third floor of a stark glass building, the Collaborative Research Center, wedged between two other, more austere, older buildings. The building hums with the activity of students preparing tables with displays for a \"Science Saturday\" event to commence in an hour. Everyone is in tie-dyed shirts, less for fashion's sake than to display \"a reaction where we show chemical bonding, basically,\" one of them says of the end product of fiber + dye. Booths on soil bacteria, the microbiome of the subway system, the freezing point of ice cream, and the DNA of strawberries are spaced across each floor. \"That's what actually gives dirt its smell, is the microbes,\" a young researcher effuses to the visitors to his table peering at the petri dish he is presenting them to sniff.\n\nI enter a glass-walled room in which Vosshall, also tie-dyed, her hair in low pigtails, is preparing a display of familiar-looking bottles across five tables for the day's demonstrations. The week prior, she was elected to the prestigious and storied National Academy of Sciences, an acknowledgment of the terrific breadth and skill of her work, from mapping the fruit fly Drosophila melanogaster's olfactory circuitry to discovering the odorant gene that makes humans appealing mosquito meals; today she is numbering large index cards to be used by visitors to guess the odors they will smell. An industrial fan is on in the corner, because, her assistant says ominously, scents \"tend to gather.\"\n\nStraightaway, she thanks me for participating in what was apparently Study 0780. \"It's brutal,\" Vosshall confesses. Apparently the other subjects and I faced 480 odors presented at two concentrations, with twenty control odors as a warm-up\u2014partly to ensure that everyone could smell\u2014\"because five percent of the people who come in are blind to smells.\"\n\nThe study had been, in essence, basic science: investigating how one goes from a molecule to an experience of odor. In Study 0780, most of the bottles held single molecules, not necessarily known smells. \"We were just looking at the chemical structure,\" Vosshall says, \"to say, What does it smell like? There is no rational explanation for why things smell the way they do.\" Light comes in wavelengths, across a spectrum, and we have visual receptors tuned to different components of that wavelength, determining what color we see. For smell, there is no sense of what kind of odor \"spectrum\" or space exists.\n\n\"If we didn't know what the rules for vision would be, we would bring in one hundred people and have them look at cards and tell us, This feels blue to us. This feels purply,\" and that's what Vosshall and her team were doing with odors. The evaluative words we were given was a way to give us a preliminary vocabulary to talk about smells. \"The words are very imperfect,\" she admits. Indeed, I had often wished for more descriptive options than the seventeen they included.IV Some odors were cloying, savory, or gymnasium-like. There were definitely soft smells and sharp smells. Pit-of-the-stomach smells. There were pointy smells and fresh smells, and I began to feel as though many odors indeed had a color. There were moments when, in all seriousness, I jotted down, That is sweet acid tree. Only later did that description seem like the product of a well-addled nose.\n\nI ask her about those very, very bad smells that stuck in my nose for days. \"What's probably happening there is one of two things,\" she replies, looking down at the cards she is numbering 4. \"The odorant\u2014the particular receptor or receptors it's binding to\u2014causes persistent neural activity. Basically it's bound to the receptor and it won't let go. And the receptor is still saying gross gross gross gross gross. The other possibility is that, there are all sorts of proteins dissolved in your mucus in your nose, whose job it is to clear odorants\u2014make the 5s so they're clear 5s,\" she interrupts herself to instruct an assistant. \"They're overwhelmed, because there's too much.\"\n\nAvery Gilbert calls this a noseworm\u2014the olfactory equivalent of an insipid tune that gets stuck in your head on constant replay. He speculates that it could also be a kind of \"precognitive sensory rehearsal\" of the smell\u2014as one might \"rehearse,\" or ruminate on, new flavors or sounds, especially odd or unpleasant smells. In which case, had I continued for a few thousand more bottles, maybe I would have gotten over it.\n\nBefore I leave, I stop to smell the first display they have prepared for the day's visitors: vitamin jar\u2013sized containers with cotton balls stuffed inside, saturated with classic smells. Synthetic vanilla\u2014the first synthesized odor, in the nineteenth century\u2014is terrifically recognizable: I name it on the first nostril. I find cis-3-Hexen-1-ol awfully sharp, definitely grassy; the accompanying label identifies it as an insect attractant given off by cut grass. I make the discovery that I am in the 30 or so percent of the population who does not strongly smell androstadienone (from which androstenone is synthesized), a component of men's sweat and a potent mating pheromone in pigs. The nearly identical R- and S- carvone molecules turn out to smell similarly to me, not distinctively spearminty or caraway-like. I head outside, looking for a newsstand at which to buy some spearmints, to try to imagine them with a caraway overtone.\n\n\u2022 \u2022 \u2022\n\nThere seemed no way I was going to smell as well as my dogs do. And I could lay the blame broadly. My bipedalism might've shot my chances. Also the very anatomy and physiology of my snoot. Worst of all, I had paid no attention to smells in my life, and they were in need of attention. But whether it served to improve my nose or just to get me sticking my nose close to things, the result of this self-experimenting was that I began spontaneously sniffing the world.\n\nI stick my nose in wineglasses before swilling; idly waiting in line at the grocery store, I nose various aisle-lining packages; when the season turns I open drawers to new clothes and take in their hibernating scents. I begin keeping a journal of daily smells. Early entries are pithy: July 24: peppermint. July 31: horse dung. August 10: sudsed sidewalk. They expand with time: March 21: sugar house\u2014can smell it before step in, wood smoke and steam of sugared water evaporating; sweet, dense, hazy. On airplanes, I notice, smells are contained and bounce over to me more frequently: almonds next to me, coconut lip balm, airplane food heated up, my hands of the bathroom soap.\n\nOne morning I smell a fire on leaving my building. We google fire, Upper West Side and get nothing. Later we learn there was a building collapse at 116th Street and Park Avenue, two miles away, minutes earlier. Plausibly, the human sense of smell was originally useful as predictive of toxins and danger. Could I be tuning in to that primitive state?\n\nMany detections are unintentional and, indeed, unwanted. I walk into my classroom before teaching a morning seminar and immediately know that someone has eaten garlic the night before. When I tell them this, they all blanch: even if it wasn't them, could I know something about them by walking by and catching a drift of air? I decide not to report how obvious cigarette smokers are when they enter the room, long after putting out their smoke. I smell a friend's bad tooth before he tells me about it.\n\nIn summer, I begin entering our budding garden with nose up, searching for something fresh, resinous, honeylike, sultry, or melon; for a hint of spicy, piney, spring, coarse. My nose catches Band-Aid, tomato stem, and turpentine, with a base of fetidness: inside of a closed cottage, mud, still pond.\n\nAfter a hiatus to clear my nose of any echoes and to still my stomach, I would begin to concertedly attend to smells again. First, though, I determined to watch the experts.\n\n* * *\n\nI. The lost muscles have since been found and are back in the current Gray's Anatomy.\n\nII. Even in sleep, our brain registers smell\u2014and knows whether it is a pleasant odor or not (sniffing more deeply when it is pleasant). The odor tends not to wake us, though, unless it tickles what is known as the trigeminal nerve.\n\nIII. A contemporary olfactometer, the Nasal Ranger is a kind of de-magnifier, allowing precision dilutions of an odor until it passes below the threshold for detection (thus permitting determinations of just how toxic a very stinky industrial odor is, for instance). Electronic noses, promising to do all the work of the dog's nose without the dog, are regularly developed but have not achieved near-dogness.\n\nIV. edible \/ bakery \/ sweet \/ fruit \/ sour \/ acid \/ chemical \/ sweaty \/ musty \/ urinous or ammonia \/ decayed \/ wood \/ grass \/ flower \/ spices \/ fishy \/ garlic\n\n## 7: Nose to Grindstone\n\nI'm crouched under a desk in a back room in an old DuPont chemical building. The building is giant: three stories of impossibly long hallways and once-bustling laboratories. It is now abandoned. The room I am in had been an office, perhaps, for one of the researchers who worked in the adjoining lab. Stripped wires hang from the ceiling of the lab; chemical hoods and faucets collect filmy dust. Everything about the room is decrepit: stained carpet underfoot; stained and broken ceiling tiles overhead; stained and broken Levolor blinds half-mast over casement windows on the walls, unopenable from the inside. The entire building is basement-cold; the halls bare. Though lights flicker on here and there, every room down the long hallway is unkempt, as if the building had not only been cleaned out, it had been cleaned out in a hurry. Bookends are scattered near shelves; drawers are left half opened and half empty. I look on the floor around me. A scattering of small washers peers up at me, yelling oooooo silently.\n\nWhen the door closed behind me, it did not feel good. The room is an insulated chamber. I feel a bit as though I will need to be rescued. I hear a trainer's muffled yell\u2014\"Dog on site!\"\u2014downstairs, and then I wait, hands gloved and clinging on to a leather tug toy. \"Gus on site!\" the trainer shouts. Gus is a yellow Labrador retriever with a taut body and velvet ears. I sit in silence, three stories and a long hallway from where he would have entered the building. Then, suddenly, I hear him charging through the halls: I can almost see the muscular body of a dog through his galloping vibrations in the floors. I think of suspects being pursued by a police dog; I get a glimmer of nervousness. When the dog closes in, when that strong, determined animal approaches, I experience a little of what it might be like to be the destined prey of a skilled predator.\n\nHe thunders into the next room. I can hear his breathy panting and sniffing. My predator-cum-savior barks twice, then stops. Then two more barks. Rummaging sounds come from his side of the door. I place myself fully by the door, ready to open it, tug toy in hand. More barks\u2014three, four. At eight barks I open the door and we encounter each other. At the sight of me his eyes seem to widen; he shifts suddenly back. I thrust the toy into Gus's mouth, as I've been directed to do. He takes it, half play-bows, half barks, and tugs with force and pleasure. He's found me.\n\n\u2022 \u2022 \u2022\n\nWorking dogs are the real smelling experts. While there are working dogs who do not use their noses for a living\u2014therapy dogs, Seeing Eye dogs\u2014many are \"detection\" dogs. What they are detecting is some particular thing that has a smell: a lost person, an illegal drug, an invasive plant, a cancerous cell, a destructive insect. Vision and hearing might aid the search, but olfaction is the driver.\n\nBut no dog begins life with interest in sniffing out palm weevils. Dogs begin life as wiggly puddles of puppies, eyes and ear canals closed, unable to walk in a coordinated way, mostly seeking out the warmth and soft touch that represents both mother and food. To discover how a wiggly puddle of puppy becomes a highly motivated, serious detection dog, I had to follow their noses from the very beginning. I went to the Working Dog Center.\n\nIn late summer, the walk from the train station to the University of Pennsylvania Veterinary School's two-year-old Working Dog Center is a journey in shedding comforts. From the cool, spacious vault of Philadelphia's 30th Street Station one walks through redly-bricked Drexel and Penn, bejeweled in leafy trees, before veering south through a more industrial, impersonal medical building corridor and finally out onto a broad auto viaduct where pedestrians are scarce. A 1929 drawbridge, apparently left untouched since 1929, looks slightly ajar, and offers a view of the Schuylkill River and train tracks beneath it. I am its only crosser. Across the river, an unpromising intersection appears: two gas stations flank a highway entrance. But this, I've been told by the Center's founder and heart, Dr. Cindy Otto, is the penultimate landmark: one is nearly arrived.\n\nBy the time a well-warmed and -moistened visitor is waved through the gates of 3401 Grays Ferry Avenue, the accoutrements of organized, polished campus life have been thoroughly left behind. For here stand the remains of the twenty-three-acre DuPont research and paint-testing plant, which closed in 2009 and was bought by Penn in 2010, but has barely changed since the chemists split the scene. The footprint of the site was originally the Harrison Brothers Chemical Company, makers of sulfuric acid and paint. Today, broad parking lots are punctuated by lonely trees giving little shade. Warehouses loom over empty footpaths. A three-story industrial building, long and broad, smells of another decade. The large chemical buildings seem to have spawned a smattering of single-story buildings, windowless or without windows anyone would look into or out of.\n\nAround a corner there is movement. A yellow Labrador puppy is prancing along a dusty stretch of landscaping. She mouths at her leash, at a wafting leaf, at the hand attached to the leash that pulls the leaf from her mouth. This puppy is a sign. Now peer through the weeds and past the squat abandoned building: you might see a black Lab charging across a parking lot, landing and stopping on a cooler, then pivoting, leaping down, and charging to another. Over there, behind that wire fence, espy a broad spread of detritus and abandoned objects\u2014concrete bits and broken wood pallets, a totaled car with its trunk ajar. But look closely: there is a German shepherd dog there, racing up a hill of rubble and disappearing into its bowels. The abandoned site teems with dogs.\n\n### FIRST DAY: \"SEEK!\"\n\nThe Working Dog Center (WDC) is run out of an entirely unprepossessing low building shoved between parking lots full of dormant transportation vans. An identifying sign can be found\u2014by squinting. Before finding any sign, the glimpses of dogs and the din of barking orient a visitor to the door.\n\nInside there are double gates\u2014to prevent unwitting escapees\u2014and audio cacophony. Kennel cough has hit the Center, and the afflicted, which is now the greatest majority of the dogs and all of the puppies, are kenneled in the front room, their barks resonating well against bare floors and walls. Cindy Otto, professor of veterinary medicine at Penn, appears at once to greet me. This is her center, her brainchild: a training center for dogs who will become detection dogs and a rehab center for dogs already on the job. Otto seems undisturbed by the noise. After all, these are not disobedient dogs; they are just about the most obedient dogs one could meet. As I will soon see, these are super-dogs, even in their youth\u2014already able, or becoming able, to find a person hiding in a three-story building, to locate someone trapped in a broad pipe under rubble, to pick out malignant plasma from benign.\n\nOtto wears a shirt with the Center's motto: \"The art and science of working dogs.\" While it has engaged in classic scientific research programs since it opened its doors in 2012, the Center's staff has always included a number of highly accomplished trainers and dog handlers whose skill comes from their practice, not from reading up on original research. \"We want to capture it,\" Otto says of their knowledge. \"Steal it. Reproduce it.\" To take the practical wisdom of the handlers and reify it, make it policy. \"Eventually will that t-shirt say 'The science and science of working dogs'?\" I ask. \"The artful science,\" Otto corrects me and smiles a broad, confident smile.\n\nWait a minute. Let's pause. There are, today, thousands of working dogs all over the country. Dogs working in airports, at the borders, in police departments. Bomb-sniffing dogs, drug dogs. There are ever-increasing numbers of dogs working in conservation settings or in pest control. There are dogs who are able to find cancer cells at rates as good as or better than medical methods. Isn't there already a fully proven science to creating a detection dog? Isn't there already a standard method to determine who is ready to work and who is not? While there are federal and state certification standards, today, Otto says, \"There is inadequate scientific research to validate optimal training or testing methodology for detection dogs.\"\n\nThere is no standard procedure for training a detection dog. Nor any one way of testing whether they're good at it. What Otto's Center is doing is working to change that. Much of the work here is preparatory, encouraging certain behaviors and fitness to increase each dog's chance of success once he is assigned a job.\n\nThe social center of the facility is a small kitchen with doors on either end. Someone has left a birthday cake (for a dog) on the counter; trainers walk through slowly and coo in admiration. A double-wide sink holds well-chewed Kong dog toys and various bowls and coffee cups. Bags of marshmallows sit next to bags of dog food on a nearby table; a jar of peanut butter sports a handwritten label: PEOPLE PEANUT BUTTER. PLEASE DO NOT USE PLASTIC BAGS AROUND PACKER, a pinned announcement reads, in urgent capital letters. HE EATS THEM.\n\nAround the corner a hallway serves as a makeshift operations room; a whiteboard on the wall lists the dogs' names and various projects and notes\u2014Hydration; Puppy run; Foster visit. Pat Kaynaroglu mans the board; trainers are in movement, in and out, gathering leashes and fanny packs to fill with handfuls of kibble. \"I'll take Osa, Rookie\u2014the medium dogs,\" Kaynaroglu says, moving through the day's assignments. No one acknowledges her out loud, but everyone has heard her. In three minutes the hallway is empty and each person is en route to her dog.\n\nKaynaroglu seems to be in perpetual motion. Each day she is arrived before I come and remains after I leave. As the Center's training manager, she takes a decidedly hands-on approach: she is rarely without dog in hand, hard hat on head, or walkie-talkie crackling. She wears her shoulder-length perfectly-blond hair pulled back in a ponytail, sunglasses propped atop her head. She is a search-and-rescue handler, but also, since almost the inception of the WDC, she has been among a few vital heartbeats of the place.\n\nI follow her outside to where the puppies are playing. It is little but a fenced, dirt enclosure with a plastic pool under a tarp. No matter. Five yellow Labs, eleven weeks old, are delightedly gallivanting. Various volunteers and trainers circle them, grinning at the puppies' antics and pulling things out of their mouths without skipping a beat. These puppies are the WDC's first litter. While breeding working dogs has long standing\u2014and many facilities will get their dogs only from specified breeders or breed lines\u2014the advantage of raising their own puppies is that the dogs can be observed, molded, and shaped from the very beginning. Dogs destined to be working dogs are not trained before they are twelve or eighteen months old. So while they are growing up, it seems reasonable to try to prepare them for their future life, as well as to fend off any potential problems or direct them toward the kind of job they may be best suited for.\n\nIn front of us are the comically named Pak, Packer, Patterson, Parsons, and Pinto\u2014the P litter\u2014at this point discriminable only by the color of their collars. Even as we watch, though, we begin to see small differences in behavior: Parsons (pink collar, female) is often the leader of the troop; Packer (dark blue collar, male) is trying to put his mouth around everything, dogs included. In addition to these five, Punches, Philip, Pierce, and Pearl are living with foster families. They will visit the Center once a week, but, by living apart, serve as a kind of control group to this uncontrolled experiment of raising the puppies. The Center group will have early training and assessment and daily social rumbles. The fosters will be living, well, as pets. Whether that will make them less suitable to future employment remains to be seen.\n\nFor now, the puppies are playing. Later in the week they will have their first assessment of their developing personalities. Kaynaroglu's walkie-talkie squawks and she heads to a broad lawn where a dog has splayed himself on the ground, his tongue hanging long out of his mouth. Two men take measurements of him; he does not protest. Kaynaroglu approaches: \"How're you doing, buddy?\" She smiles at him.\n\nOhlin, a shiny-eyed chocolate Lab with a glossy coat, is taking part in a study. In essence, handlers play fetch with the dogs until they are worn out; then their temperature and blood work are taken, to see the efficacy of different kinds of hydration. \"One hundred forty-four,\" a technician reports his heart rate when Kaynaroglu appears. A well-slobbered ball lies next to Ohlin. He has stopped retrieving, a straightforward behavioral marker of hot. Kaynaroglu takes a look at him and says, \"One-oh-five?\" The technician smiles: \"one hundred four point nine.\" She has guessed his internal temperature. \"I look at their respiration, their tongue\u2014is it cupped, how far out is it?\u2014are their eyes squinty?\" she tells me. \"Easy things\u2014how long they have been working . . .\" The outside of the dog tells her about the inside.\n\nAnother trainer walks by, noting a graffiti of urine on the sidewalk. \"Was Jesse here?\" she asks. She could recognize the dog even in his absence.\n\nAll of the staff have this kind of second sense about the dogs. They know where they've been, sense their temperature, and talk about dogs' dimensions as though they are part of each other. And, indeed, for the time that they are at the Center, the dogs' lives are supervised and overseen by these acute lookers. Everyone is always watching the dogs' behavior, assessing them, and the chatter is all about their latest suspicions and working hypotheses about the personality, abilities, and inner life of each dog.\n\nWe head over to the hulking old DuPont laboratory building. They call it \"227,\" for its former address. It is here that I will hide, to be rescued or apprehended by Gus. At its entrance, Annemarie DeAngelo and Bob Dougherty, two experienced handlers, hand out assignments: a person to take videos; another to keep track of the various tug toys; a hider, the \"target.\" DeAngelo, the WDC training director, is retired from the New Jersey Police Department; Dougherty is a volunteer here, still an officer at a local police department. They radio over to the Center to have someone bring the first dog by. As the designated hider, Dougherty slips inside the building before one of the \"pointy-ears,\" as everyone refers to the young Dutch and German shepherd dogs, is brought over. Felony, seven months old, is small for her breed and has a dark, lovely face. A handler holds her close. When Dougherty's in place, he radios over the all-set; Felony's handler says \"Seek!\" and seek goes Felony. Though new to this game, she has already connected her handler's exclamation-point enthusiasm with getting to run through the building, and so she does, handler in tow.\n\nOn the first floor there is a draft down the hallway out the open door. Felony pulls strongly and pokes her nose in a few rooms, but then heads to the stairs. Everyone trundles up after her. She is straining at her leash. The stairwell is greatly musty; the air is close. Smell has been deeply carved into the walls. On the second floor she pauses, and her handler encourages her to look in one of the first rooms. Dougherty pops up from behind a door and gives her his arm, with a bite guard on it. At this delicious offer, Felony lunges.\n\nAfter watching a few dogs seek\u2014and being sought-after myself\u2014it dawns on me that there is no \"smell training,\" as such, happening at 227. Though no dog is born knowing how to search for a person, what the trainers are doing is not teaching them to smell a person out. The training assumes that the dog already knows how to do that. That comes built in.\n\nInstead, what needs to get developed is the drive to search for that smell at any time, at a person's behest. To search when there are dogs nearby wanting to play; to search past piles of toothsome smells and cooing people; to search when one seems not to be getting any closer to the target. To search simply so that at the end one gets to tug on a leather toy or the moving arm of a target.\n\nSearch training begins with small steps, bringing the dogs to the building's front door and having the \"target\" appear from behind it, then naturally evolving through small increases in duration to looking in rooms, encouraged by the handler holding them on leash. The dogs catch on quickly. Over the course of an hour I watch one pup go from playing with an octopus-shaped tug toy outside, to following it down the hall, to finding the toy (held by a trainer) behind a closed door. At each step the dog is rewarded for his \"discovery,\" however large or small, by getting to play with the tug or wrap his mouth around a slobbered tennis ball. Later the leashes will be dropped, the searches made steadily more difficult. The dogs will be asked to \"alert\"\u2014to tell their handlers that they have found the hidden person, usually through barking.\n\nBarking is not natural for all dogs. The trainers keep calling Rookie, another pointy-ears, a possible \"dad\" dog, which I take to mean one that will make some father a good family pet. Later I learn this is shorthand for diabetic alert dog (DAD) candidates: dogs with good noses, but not as much drive to pursue a hidden scent like gangbusters. And in Rookie's case, no particular inclination to bark with excitement when she does find its source. We watch as she finds hidden Dougherty (upstairs closet, behind three doors and around a corner) with ease, but then just sets her rump on the floor and waits. She has a classic three-month-old-puppy shape: large head and ears, small gangly body with a hint of muscling developing underneath. Patiently, she stares at the door that conceals him. Eventually, she looks back at us, the observers half hidden outside the room, quietly hoping for her. Her expressive eyebrows mark concern, endearingly. Perfect pet dog behavior. But Rookie is not being trained to be a perfect pet dog. We wait. She waits. Finally, she lets out a single, shrill note\u2014a request, a plea. A bark. Everyone cheers, and Dougherty appears, sticks a tug toy in her mouth, and lets her play.\n\nThis is the unstated mantra of the design of the working dog: create an anti-pet. We want our pets to be motivated to sit still, not tear up the house looking for a scent; to look back at us in request, not bark endlessly; to keep their mouths closed in enthusiasm, not to bite. At the Center, the trainers build up those bad habits one by one, assiduously rewarding them. Some dogs need training to bark to alert. The pointy-ears are working on \"bite\"\u2014which is just as it sounds: bite and hold the person (for now, the covered arm of a person) they find. Each of those natural behaviors\u2014search, bark, bite\u2014is highly useful in the context of work. Each had a function for the dog in the species' evolution\u2014functions that are often moot for pet dogs. In training dogs to work, we are giving them reason to use those behaviors they evolved.\n\nInside the Center you can see the result of this anti-pet development. The rooms ring with successfully trained barkers. And while in most contexts people are the focus of the dog's gaze, here the dog seems to be looking beyond the person to something . . . some smell . . . else. The dogs are not vying to be pet or tickled or to curl up next to you on the floor. Nor does anyone but the occasional visitor even try to pet or tickle them.\n\nNeither are they, after puppyhood, especially interested in dogs. On their regular walks around the outside of the Center to pee, they are hustled along, encouraged to ignore other dogs' scents, distracted by obedience games and the fanny pack full of kibble that brings their attention to the handler. Dogs are walked separately, run on searches separately, and kenneled separately and often with visual barriers to even seeing dog passersby. As we wait in front of 227 for the next dog, a good-natured yellow Lab and his handler approach\u2014the handler attuned to his every pause and suggestion of loitering: \"We're not gonna mark here; we're not marking.\" She suddenly sees us and, like an owner with a difficult dog, turns on her heel and walks away.\n\nWhile the dogs certainly know of each other's presence\u2014the sound of dogs is ubiquitous; the smell of dogs hangs in the air\u2014they are learning that other dogs are not the most important thing in their lives. As they must not be later, in their working lives. To the uninitiated, the Center looks like it is running an elaborate game of dog tag, with dogs being moved around but rarely catching up to each other.\n\nInstead of encouraging smelling or sociality, these dogs are encouraged in their drive. Drive to pursue the scent, the toy that arrives at the finding of the scent\u2014and to be frustrated when they cannot get at it. In many ways the dogs are being kept (when working) in constant vacillation between immense frustration and satiation. There is something they hugely want\u2014usually a game with a toy\u2014and they have to wait and wait and wait until they can go after the scent that lets them get it. Many times I have seen the dogs shaking with excitement to get started.\n\n### SECOND DAY: \"FIND DOPE!\"\n\nI follow Bob Dougherty into another, smaller DuPont building. Dougherty is an active K9 handler and comes to the Center on his days off. He wears serious work shoes and an easy manner. His comfort around dogs is apparent.\n\nPApa Bear, a full, strong chocolate Lab, is by his side.I Today he will be initiated into the world of narcotics: PApa Bear is being trained to sniff out drugs. He has been a generalist thus far, and Dougherty is tasked with trying to get him to be more of a specialist, so that he can be usefully placed with a permanent handler. Up a set of worn stairs, through a dizzying set of doors and small anterooms, we arrive in a big double-sized room with a chem-lab kitchenette along one side: heavy on the sinks. The room has a huge multicolored sunburst on the floor, some late-night \"paint test\" accident.\n\nDougherty takes out a large plastic bag and pulls out a towel. It has been cavorting overnight with a smaller plastic bag containing a small amount of the scent source. Quantity matters: a dog trained on a subtle scent may not know to alert if he has discovered a large amount of the scent: it's almost a different stimulus. On the other hand, starting with a large amount of the scent in the room is too much. \"If I put two pounds of marijuana in here,\" Dougherty explains, \"the odor will be all over the place.\" A dog just learning won't be able to localize it.\n\nEven so, I can easily smell the presence of groggy, giggling teenagers in the park as the bag is opened. We can all smell it, surely. The task being presented to PApa Bear, though, is to learn that this sweet smell\u2014more or less, stronger or weaker\u2014is the smell he needs to find in order to get a partner for a vigorous game of tug.\n\nWithout being able to explain this to PApa Bear, Dougherty simply works backward from the desired result: he begins with a vigorous game of tug with the towel, rolled on itself and secured with rubber bands. As PApa Bear mouths the towel intensely, his strong tail slicing the air, Dougherty tells me, \"Dogs can eat this without getting ill. Not so with other drugs\u2014we use pseudo-narcotics for training there. Plus, drugs like heroin are sometimes cut with rat poison\"\u2014nothing you'd want your dog keen to put in his mouth. Then Dougherty tosses the towel; PApa Bear readily pursues it, bringing it back to him for another tug. After this, Dougherty begins to feign-toss the towel, making PApa Bear work to find the source. The dog is thrilled, breathing heavily with excitement and effort. But when he begins to search he closes his mouth until it is barely ajar, his head cocked in contemplation. Already, something has happened: the dog is \"going to get\" the odor and \"bringing it\" to his handler. The start of narcotics search.\n\nLike all the handlers at the Center, Dougherty is continually improvising new variations and new steps, based on knowing the dog, how he's performing relative to other dogs in training, and what the conditions are like\u2014is the dog hot, overworked, distracted, ill? PApa Bear looks to be getting the game, and Dougherty starts ramping up the difficulty level. Within minutes he hides the towel in plain sight, then out of sight, then hides the bag alone. \"Find dope!\" he instructs, introducing new vocabulary to match the new game. PApa Bear finds the target at each step without a hiccup. He's now finding dope. It's been about forty-five minutes since he walked up the building stairs to this room, never having smelled the stuff before.\n\nOnce the target is set, Dougherty guides PApa Bear around the room, working on \"detailing\"\u2014helping him learn that after he's barreled toward an area that has the odor, he should explore the source in detail. For dogs, who are not furniture users (except when they sneak up on your sofa), human-world objects that have surfaces and undersides, insides and partitions are not obvious. By being guided to the top, middle, and low reaches of a counter, for instance, the dog can learn that a general \"counter odor\" may be different at different altitudes, or inside or behind it. Then when he hits the source\u2014BAM!\u2014it's a big, different smell.\n\nWe head back to 227 to \"try PApa Bear out\" in a new space. Dougherty will hide the baggie in a half dozen new spaces before calling it a day and letting the dog rest. I stand in the middle of the first room\u2014one of the old laboratories\u2014with a video camera, hoping both to keep up with the dog and stay out of his way. A pipe with a faucet fitting and a sleeve reading NITROGEN hangs from the ceiling, entangled with insulated and cut wires. PApa Bear enters the room running. He cruises all the way to the back wall, nose up, then turns heel and starts careering back. He takes barely a peek at me. The way he examines the counters and each drawer beneath feels habitual, as though he's just checking where he left that file folder or test tube. Only his assessment is rapid-fire. He rears on his hind legs and appears to look at a cloud of air about six feet off the ground. His nose guides him to the northerly wall. Then he pivots, finds a cabinet, goes behind, then returns to the front and barks to alert, staring at the top drawer, his jaws violently snapping shut with each bark as if grabbing at the scent. And yes, PApa Bear, the baggie is inside the top, closed, drawer.\n\nAs the team moves on to their next search, I go back into the room to pick up the sample left behind: when it was found, Dougherty tossed PApa Bear the tug and the sample was immediately forgotten. Since I know where he alerted, I stop and sniff the spot myself. The drawer is closed, and I smell nothing. I pry open the drawer a few inches; it is bound by tape\u2014prepped for moving day and then never moved. I still can't smell the sample; nor do I see it. I move to visual search, looking deeper into this drawer, opening the other three drawers to check them. Then I return to the first and look again. The baggie is right at the front; I simply missed it. Drawer ajar, with my eyes on it, in the place where I knew it was, I missed it.\n\n\u2022 \u2022 \u2022\n\nAs PApa Bear keeps searching, I wander back past the Center to a field with multiple apparatuses for balance, strength, and skill work scattered around. It smells of recently cut grass. Under a high sun, Kaynaroglu and DeAngelo are standing on either side of a ladder. It is propped at a low angle and leads nowhere. Kaynaroglu holds a ball near its end, though, and Gus wants that ball. To get it, he is on the rungs, tentatively inching forward. He does not look happy to be there.\n\nTeaching the dogs \"body awareness\" is one of Otto's real missions. Not just to have proprioception, but to be fit, agile, physically confident. Any working dog out in the world is going to encounter uneven surfaces and tricky terrain, to need to negotiate an uncomfortable path. Otto wants them to be ready. Her interest began as a veterinarian, with seeing working dogs with substantial injuries from being unfit or unable to use their bodies strongly.\n\nOver my few days at the Center, I saw the ladder brought out many times. It seemed to stand as the epitome of dog fitness: while a dog may never need to climb a ladder in his working life (although many will), when he can, it is only because he has mastered using his body skillfully.\n\nOr, as the trainers say, \"Dogs have to understand they have back legs.\" Gus might be able to get to the ball because he is muscular and can brute-force his way up the ladder. But they will not let him move forward unless he moves his feet one at a time, settling firmly on a rung and knowing that he has before he deigns to move the next foot. \"He has to make a connection between here\"\u2014Kaynaroglu points to Gus's rear foot\u2014\"and between the ears.\" A trainer uses a clicker to reward him each time he moves his rear foot at all. I leave the group after a grueling several-minute exercise in which Gus makes it about two rungs. Before the month is out, he'll be walking that ladder like a champ.\n\n### THIRD DAY: \"PUPPY PUP PUP PUP PUP PUP!\"\n\nToday the dogs are in the \"Field,\" a westerly part of this old campus that Center staff say, sighing, used to be much better when it had great piles of dirt on it. Now it has been leveled and is one and a half football fields' worth of cracked concrete, with weeds considering taking it over, and various large containers and broken vehicles scattered about. Large PVC barrels are rolled hither and thither into the Field, blue beacons in the desolation. On the other side of a fence and some marginal wilderness is a sanitation transfer center: one can hear, then see a small bulldozer scuttling along its perimeter. Kaynaroglu looks for the wind direction: there is little in the Field that sways in a breeze, but at once we are hit with the smell of the trash pile. The wind is from the southwest. The smell starts out sweet and becomes nauseating: there is no \"adaptation\" relief for our human noses, because the wind keeps shifting and bringing new wafts.\n\nThe dogs are brought out one at a time from their crates. None seems to notice any of this sensory scene, except the barrels. For today the barrels, or any large item, may hold a person\u2014Kaynaroglu or another trainer, volunteer, or visitor willing to stuff themselves in a hot, close space, pull a lid over the opening, and wait to be searched for and found. Some of the dogs have been here before; for others, this is their first go. But all know that the game will be to discover the person\u2014any person\u2014who is not in sight.\n\nSirius is out first. He is one of the older dogs here, at two years; his panting face gives him a smiling expression. \"Go find!\" He races past observers, uninterested in the already-found, as he is released into the Field. Kaynaroglu has been hiding in a barrel on the other side of the Field since before he came out. He finds her in forty-two seconds and barks. She tosses a ball on a rope in his mouth and tugs it as a reward. \"You're just a model guy,\" she gushes.\n\nKaynaroglu moves to a new hiding place while the next dog is brought out. Gus. He starts quickly, tours the grounds a bit more, and suddenly settles down to poop. While doing so, he spots a scent and beelines toward her. \"Hi! Nice to see you! Yay!\" Kaynaroglu beams. One minute thirty-two seconds. Slowing down seemed to allow him to see.\n\nNext, Jake bolts out, a yellow Lab like Sirius, with asymmetrical ears and a deeply dark nose against his blond fur. As he passes we can see his eyebrows working and hear him whining with excitement. He touches his nose to each barrel, then lifts his nose into the air and licks it. Fishing for molecules on the wind, surely. He runs circles around Kaynaroglu's barrel, barking, thirty-four seconds after his release. \"Someone else!\" she says. He pivots and finds a second trainer squirreled away in a large pipe, tugging rope in her hand.\n\nThe dogs keep coming out, each with his own style, and each a model of efficiency. Quest (bright-eyed, tall-eared young German shepherd): twenty-seven seconds, in a great unhurried lope.\n\nLogan (also a young shepherd male, with soft eyes and a dark face): one minute forty-nine seconds. His sniffing is impossible to see when he is running, but when he turns abruptly, it is the nose that is leading him. It turns, and pokes to the ground, or into the air.\n\nFelony (a dusky dark Dutch shepherd, the baby of the group): thirty-seven seconds.\n\nThese dogs, finding their way along the Field, are in essence drawing a picture for those of us standing and watching their paths. They are drawing a picture of how air moves around the Field. When Logan begins, he first heads toward a Dumpster and a pile of detritus, far off from Kaynaroglu's hiding place. But this area is downwind of where all the searchees have been, and a distant scent of them must remain. Finding no one, he pokes his nose into the wind. He moves farther out into the Field, and again his nose is up. Finally, he is downwind of Kaynaroglu, in a triangle of space behind a mound of concrete chunks\u2014where a large pool of Kaynaroglu-scented air would have been collecting. Seconds later, he's nosed her out.\n\nThey are drawing a picture of the way dogs process smells. By looking at the barrels first, they are using visual landmarks to aid their olfactory search. By retreating from old, deteriorating smells\u2014representing a person who was in a hiding place but no longer there\u2014they are using quantity and age of odors to tell them something about the present moment. By passing by the observers with nary a look, they are using their budding knowledge of what a search is to ignore the very ripe smells of people right in front of them, who don't need to be searched for.\n\nThey are drawing a picture of the mind of a dog. Being olfactory is living in an impermanent space, where seeming \"objects\"\u2014fixed for us, with our visual approach\u2014are fixed only as long as their smell remains.\n\n\u2022 \u2022 \u2022\n\nThe handlers walk Quest around the corner, staying on the outside edge of the campus. Unseen, behind a fence, the Schuylkill River wends its way through Philadelphia. On its opposite bank, two of the Center's previous dogs\u2014Socks and ZZisa\u2014now work with the Penn campus police.\n\nBehind a locked fence is a treasure. The rubble pile: mounds of rebar and concrete and slabs; wooden pallets; PVC pipes; miscellaneous junk; a smashed and gutted car, its trunk creepily poised open, as though someone has just casually retrieved his belongings after a disastrous accident. A few corrugated rubber pipes are jammed downward into the mess, like portals to the innards of the earth. The rubble pile is not an unsightly mistake; it is carefully constructed to provide a place for the dogs to practice searching while also becoming fearless and physically flexible. There are dozens of places where a person might be trapped or hidden: the pile is riddled with awkward gaps and crevices, mimicking the disorderliness of a sudden building collapse. If you assent when the trainer asks you, Do you want to hide?, you leave the comfort of human company and breezy airflow and may scoot into the slippery inside of a broad pipe. It is too wide and smooth inside to get traction and you settle in, deeper than you'd like, pulling a makeshift lid over the opening. It fits awkwardly, a scratched window showing a piece of azure sky. The bottom of the lid is suddenly the only sign that you are not truly lost in a pile of catastrophe-site rubble. From this sepulchral vantage, you will grow sweaty and hear your own heart beating in your ears. Then this sound is replaced by the galloping vibrations of a dog close by. You might see the beautiful moist nose of a dog like Quest appear in the slit between lid and barrel, almost bending to reach in and confirm inside\u2014then EFF-fuh-fuh-fuh, quick strong sniffs\u2014then hear an alert bark, twenty alert barks. Good dog.\n\n\u2022 \u2022 \u2022\n\nOver where Gus climbed the ladder yesterday, six puppies tumble over each other, mauling ears, grabbing tails, bumping bodies, and squirming as they wait in a small fenced corral on the grass. Five are the twelve-week-old yellow Labradors; the sixth is one of the pointy-ears, Dargo, a week their senior, whose bearing is less \"cuddly puppy\" than \"detection preprofessional.\" They are all about to meet their first obstacle course. Suddenly one of the corral's walls has an opening and the nearest puppy\u2014Parsons (pink collar)\u2014hops through it. She has begun the \"puppy run,\" as the trainers call it: an improvised, narrow runway along which she will confront obstacles that require hopping over (a tire on its side) or walking carefully through (a ladder lying flat) or bravely negotiating (a wobbly narrow board, suspended a few inches from the ground) or barreling into (a dark tunnel). The other puppies continue their antics, until one notices the opening, too, and follows Parsons's tail through the hole. Then another puppy, then they all charge through.\n\nA half dozen volunteers and trainers watch the scene. Annemarie DeAngelo stands a little apart. She is another one of the Center's heartbeats. A former law enforcement officer, she developed the New Jersey State Police's canine program. Even out of uniform\u2014\"I'm a civilian now,\" she says\u2014she has a no-nonsense manner. She does not suffer fools gladly. But when one of the golden-haired puppies looks momentarily disoriented, DeAngelo approaches the fence and melts into a high-pitched \"Puppy pup pup pup pup pup! Good girl, good girl!\" to encourage her along.\n\nThe puppies are not being explicitly trained at this age. But most of their day's activities are cleverly designed to sneakily prepare them to be more trainable in whatever working position they wind up. The obstacle course is designed to expose them to new environments\u2014shadows, uneven or wobbly surfaces, gates\u2014in a safe, non-pressured way, and at an early enough age that they learn not to be fearful of new or unknown things.\n\nRunning the lot of puppies through the course also takes advantage of learning that happens between dogs. When one dog charges into a dark tunnel, any puppy watching is more likely to follow him: this is called social facilitation by cognition researchers. Here it is called puppy see, puppy do.\n\n\"C'mon, munchkins,\" one of the trainers calls out to the puppies considering the business end of the tunnel. Though the dogs vary in their willingness to move through the course, by their fourth round, they are all barreling onto the wobbliness, over the obstacles, into the tunnel. Simply getting the chance to be exposed to these new things, and watch others do it, has turned a riotous play session into a context for real development.\n\nAs tongues hang low, a few of the puppies are taken aside and brought to a quieter area, a sidewalk leading to 227 where a trainer is waiting for them, clipboard in one hand and a basket of Very Strange Objects in the other. This is the puppies' first assessment test. They are not being assessed for their skill at smelling. While the volunteer who whelped the puppies and housed them for their first weeks did expose them early in their lives to odors\u2014a cut-up tennis ball, leather, other novel scents\u2014to see how they reacted, it is not yet clear that subtle differences in smelling as a puppy lead to differences as an adult. In the meantime, these trainers are using an early test of \"drive and courage\" designed and used by the U.S. Border Patrol. In its first stage, a puppy is simply presented with various loud or surprising objects, and the trainer makes a qualitative impression\u2014from one to five\u2014of his reaction.\n\nOne of the litter wags merrily as the trainer shakes a little bottle with coins inside, tosses a rattle under an upturned Tupperware container, drops small metal pipes on the ground, drags keys along a metal ramp. An umbrella is suddenly opened and dropped on the ground. The puppy races after the pipes, paws the Tupperware, follows the keys up the ramp until she catches and mouths them, and is unfazed by the umbrella. For now, fives all around.\n\n### FOURTH DAY: \"ROTTING MEAT AND PROM CORSAGES\"\n\nAs I walk to the Center on the last day of my visit, the wind carries an odor of sulfur along the river. This will not be the worst smell I smell today.\n\nToday we go back inside 227. \"Do you want to run Gus on HRD?\" Kaynaroglu asks Dougherty, who already has the leash and is heading out. I jog after him.\n\nOne day, this part of campus will be renovated into housing or new facilities for the university, but for now, this old industrial building is a great training surface for the Center's dogs. Not only is it expansive, with hundreds of rooms inside of rooms; the rooms all present different olfactory challenges. \"You work here awhile, you start learning about all sorts of air currents\" in the building, DeAngelo says. While she can't articulate what she knows about the air currents, all the longtime trainers have learned that certain rooms and times are more difficult for the dogs. The way the air flows determines whether one can get a sniff at the doorway of someone inside, or whether their smell loiters and collects in a corner of the room, discoverable only by close investigation.\n\nOne room in particular\u2014a lab on the second floor, old chemical hoods in its center and closed doors on the perimeter\u2014is especially difficult for the dogs. Over an hour, I watch three dogs pass the door concealing Dougherty multiple times before clearly paying a different kind of attention to it: what might be called examining, instead of browsing. When they do, concertedly sniffing, they all find their man straightaway.\n\n\"We sometimes spray baby powder to follow airflow,\" Dougherty says: in essence, using a powder gun to see what the dog sees. In other words, to get a sense of how air moves in the room\u2014where the smells might be lingering, where they are rushed upward or straight out a window\u2014one can follow a dog, or one can put something into the air. In the Field, the dogs were our source. Here, a powder gun or the like makes visible the invisible (to us).\n\nIf you reflect on it, we have all watched airflow. The mesmer of a lit match comes from its phosphorous flame, to be sure, but also from the curl of smoke that slips up and away from it. We are seeing airflow when we watch the \"fog\" used in theater ooze across the stage and increase the moodiness of a scene. Smoke and other particulates, though, dissipate when air is turbulent\u2014which is much of the time. Other methods use a \"neutrally buoyant\" agent, like helium bubbles, which slide into the slipstream of air. With a specialized light, this method can be used to trace air as it moves naturally around a room. In an ordinary empty room, air warmed by the day creeps up the walls onto the ceiling, where some is bumped downward again and crashes, wavelike, into air coming from the other side. The two streams tumble together, paints swirling on the canvas of a hallucinating van Gogh.\n\nIn essence, that is the path of odors, too: the path that these dogs see in every room they walk into.\n\n\u2022 \u2022 \u2022\n\nDougherty holds a small glass jar. We are on the second floor of 227, hiding the \"HRD.\" This turns out to be one of those initialisms that is lovelier in brief form than long form: human remains detection. In the jar are a few pieces of human remains: bits of knee from a cadaver. The supply catalog for these folks must be fantastic.\n\n\"Do you want to smell it?\" he asks, unscrewing the top. I do not, but I do. I lean over the jar, instinctively pulling my hair back from my face. At first I detect nothing. Then: a sweet and vomitous waft, a carnation rotted in a bin. Dougherty smiles at me. This is one of the only times one knows just what is in someone else's head.\n\nUpton is in the distance, over-interested in something on the ground. I hurry over. His nose points at a recently deceased squirrel being visited by a small number of flies. His tail wags and he weaves his head around, mimicking rolling in it. I ogle, draw him away, but then go back. I must smell it. I bend over. It is not difficult to smell. Sweet and foul together\u2014not the kind of smell that is appropriate for a squirrel who recently was chattering and racing up trees.\n\nMost people might not be attracted to it as dogs are, but there is a fascination and a depth to the smell of death, of things rotten and gone. Hemingway described the odor's many parts as though it were a perfume: combining the kiss of a woman who has tasted blood; the morning streets mixing last night's cigarette pails and wastewater with the soap used to clean them; and decayed flowers; all experienced with the swimmy belly of a seasick ship passenger. This \"saccharine putrescence\"\u2014P. J. O'Rourke writes of eau de corpse as invoking \"rotting meat and prom corsages\"\u2014sickens us even more for our knowing the cause: death sickens us.\n\nMortuary technician Carla Valentine says, \"You have to smell what's going on in an autopsy because you can diagnose things with smell . . . no professional examiner of the dead would ever try and mask the smell.\" For a mortician, the \"natural smell\" of decomposition is not so much offensive as it is appropriate for the context. Context is, if not everything, highly determinant with smells, at least in one direction. Any enjoyed odor smelled in the wrong context\u2014lavender in our coffee cup; smells of the barbecue on our beloved\u2014can become foul.\n\nWhat the dogs are smelling, whether they are intent on a decomposing animal in the park or are HRD dogs, is death: death of cells, and the odor of the biological processes that take over as cells are broken down. That dogs find it notable makes them particularly good death-detectors\u2014and may also enable some of their abilities to notice diseased cells in their living owners.\n\nDogs' apparent interest in the smell of death does not mean that they will instinctively lead you to the dead smell in the room, though. At the Center, HRD is introduced just in the way all the new scents are introduced: in baby steps, slowly shaping the dog's behavior first to notice the scent, then to pick it out from other scents, and then to bother to tell the person about it. And at the end of every search, a tennis ball or a tug toy awaits. The dog's reliable satisfaction with his payment is written in the strong wag of his tail.\n\n\u2022 \u2022 \u2022\n\nA year later, I write to Otto to check on the dogs. She reports that PApa Bear, who had his first exposure to marijuana in a paint-splotched building with me, is now a drug and search dog for the Gloucester County Sheriff's Department in New Jersey. Gus, who found me and found his rear legs, is working as a certified search-and-rescue dog in New Mexico, as are Felony and Sirius; Jake is doing search and rescue in Pennsylvania. Quest and Logan are police dogs with SEPTA, Philadelphia's transit system; Rookie is also a police dog. The former puppies Packer and Parsons are search-and-rescue dogs; Pinto is an HRD dog. Pak and Patterson are trained in search and rescue and awaiting the right team to bring them on. A new litter of six\u2014the Q litter\u2014is now at the WDC learning the ropes.\n\nSpending the better part of a week with these dogs infects you. When I return home to my lovely, goofy, non-ladder-walking dogs, who sniff my eyes closely and look to me to solve problems of tennis balls under couches, I also see how much they are not workers. This fact, to be sure, cuts both ways: though unemployed, our dogs are engaged in heavy bouts of mutual adoration and silliness with our family. On the other hand, this seems to be their only occupation. Is it enough?\n\nWhat impresses me most about the Center's dogs is how much they are treated on their own merits, dealt with as members of a species with specific skills\u2014and not treated as furry, quadrupedal humans. A dissatisfaction I feel with my own field\u2014comparative cognition, broadly conceived\u2014is that it begins with the premise that the most interesting topic of investigation is whether nonhuman animals can perform at the level of humans at various tasks.\n\nBy contrast, the WDC dogs prompt me to turn the comparison on its head. Given what these dogs can do\u2014find a stranger hidden in a three-story building in two minutes; learn the smell of a target odor of any sort in a single session\u2014what I think is more apt is seeing if we can do what dogs do.\n\nAnd so I aimed to follow what some of these detection dogs are doing further\u2014and then ask if people, too (including me) can learn to detect what the dog does. Comparative cognition, written by the dogs.\n\n* * *\n\nI. All of the Center's dogs are named in homage to someone (dog or person) who was on the scene at the World Trade Center explosion and its long aftermath. PApa Bear is named after a working dog named Bear from Pennsylvania (\"PA\").\n\n## 8: Nose-wise\n\nAs I rise slowly up out of sleep I become aware of Finnegan, his nose millimeters from my mouth, snuffling. I open my eyes and he looks surprised to find me attached to whatever-that-smell-is. I mentally note to call the doctor.\n\n### -DOG-\n\nIn the UK, in the late eighties, a Border collie\u2013Doberman mix noticed something. Her owner's left thigh. At the same time, her owner, a forty-four-year-old woman, began noticing her dog's extreme attention to a mole that had cropped up\u2014on her left thigh. Minutes would pass while the dog examined the mole with her nose, even through her pants. When the weather warmed and she dressed in shorts, the dog began nipping at it, as though to bite it off.\n\nA few years later, a Labrador named Parker began paying particular attention to his owner's left thigh. The sixty-six-year-old man was constantly pushing the dog's nose out of his pants leg as Parker tried to get at the patch of itchy eczema that had developed there.\n\nIn the United States, a Dachshund puppy began showing unusual interest in his owner's left armpit. The forty-four-year-old woman, in good health, endured some time of this sniffing while they shared the couch in front of the TV. One day she moved the dog away and, feeling the area, discovered a lump.\n\nIn each of these cases, the object of the dogs' attention was a malignant cancer. The first case resulted in the discovery of a melanoma\u2014which removal may have saved the woman's life. The man's eczema turned out to be carcinoma. And the axillary lump was biopsied and discovered to be breast cancer. After a mastectomy, the woman still had the Dachshund's attention at her armpit. The woman had radiation and chemotherapy, but, a year later, died of her cancer.\n\nAs the authors of a paper reporting the first case wrote in the medical journal The Lancet, \"Perhaps malignant tumours such as melanoma, with their aberrant protein synthesis, emit unique odours\"\u2014a kind of signature scent\u2014\"which, though undetectable to man, are easily detected by dogs.\" Before this case in 1989, the idea of dogs detecting\u2014essentially, diagnosing\u2014cancer would have been laughable. But these dogs, all of them untrained to do anything remotely medical, were able to contribute to their owners' lives\u2014in some cases, lengthening their owners' lives\u2014simply through being themselves.\n\nThere is little better proof that the nose is how detection dogs are able to detect their quarry than when they are after that most invisible and sneaky of villains, cancer cells. Now, the authors of each of these reports were careful to describe their case studies as \"only anecdotal.\" But the stories were provocative. Moreover, the standard business of cancer detection was expensive, lengthy, and sometimes painful. The hope of replacing office visits, biopsies, and CAT scans with a dog scan, so to speak, was irresistible. Those first anecdotes birthed a small industry of research studies designed to suss out exactly what the dogs were noticing, and if they could be trained to notice it on anyone at all. A cancerous growth certainly produces compounds that are volatile, and that would be effused in blood, urine, or breath. Except for the strangeness of dogs wagging into the medical arena, there was no reason that dogs wouldn't be able to become diagnosticians.\n\n\u2022 \u2022 \u2022\n\nAt the Penn WDC, another trainer, Jonathan Ball, is \"doing cancer.\" In other words, he is running dogs in trials designed to entrain them on the smell of cancerous cells, and to train them to alert to that smell.\n\nThe door to the training room opens and Ball lets me in. He has a boyish haircut and is dressed in jeans with a treats bag at his hip. The room is dominated by a specially designed training \"wheel\"\u2014set on its side, with twelve radiating spokes. At the end of each arm sits a small glass vial, the size of a diner saltshaker. They are covered by mesh screens to conceal any possible visual cues\u2014and to prevent overenthusiastic lickers. In three, there are minuscule donated specimen samples. The target vial has fifty microliters of plasma generated from a pooled group of cancer patients with malignant ovarian cancer. Two contrasting vials hold the same volume of either benign-tumor patient plasma or normal plasma. The remaining nine jars are empty.\n\nI peer into the vials. They all appear empty: fifty microliters is not a visible quantity. I smell the pizza from the kitchen next door; I smell the astringent tang of isopropyl alcohol, with which they clean the wheel; I smell the soap on my own hands; I do not smell the plasma. A long, lean yellow Lab, Ffoster, is gracefully padding her way through the kitchen with her handler. En route from her crate through the kitchen, she noses the food bins extra carefully. She takes a quick tour of the room: the single cabinet, the two people sitting discreetly along a wall. Ball turns to her, and her attention is on him at once. He plies her with treats, saying \"Focus!\" until she is looking at his face. She actually seems to look right into his eyes, maybe three inches into his skull, her pools of brown eyes turning inward somewhat, reflectively. She ever-so-slightly tilts her head. \"Go find!\" he says, and she heads obediently for the wheel.\n\nFfoster sweeps her jowls smoothly across each mesh top, taking less than a second. Without breaking her trot, she repeats her analysis with the next one, until she pauses, just barely, at one. She looks at Ball, who is staring unwaveringly at the due center of the wheel. His eyes are narrowed, his posture fixed. He is \"blind,\" as they say, to the position of the malignant sample in the wheel\u2014only another person in the room, back turned to him, knows where it is\u2014but he is also being careful not to cue the dog in any way. Since the time of Clever Hans, the horse who was alleged to be doing arithmetic but was actually just a highly skilled reader of his trainer's inadvertent body language, anyone who trains or studies animals has been preternaturally attentive to whether they are \"cuing\" the animal. Experiments ensure they are always blind: that they do not know which cup the dog is \"supposed\" to tip over to get the hidden treat, or which person the clever dog \"should\" beg to get a reward. Similarly, in training, one wants the dog to learn that the cue the trainer devises\u2014a word or gesture, often\u2014is instruction, not some accidental sound or movement. By contrast, many pet dogs make themselves seem very smart indeed precisely by noticing owners' inadvertent cues. Hence their magical knowledge of when it is time to eat, walk, sleep, go to the vet, have a brush, or have a bath: they are reading their owners, and their owners are happy to be read.\n\nFfoster gets nothing from Ball and continues a very light pace around the wheel. She traces it twice, carefully, but does not alert. Then she sneezes and sniffs extra hard, almost speaking Hmm-huh-huhn-mm-huhn, then Phrunh and sits. Alert. The knowledgeable trainer assents, and Ball breaks out of his statue stance and rewards her for her find.\n\nBall sends Ffoster out of the room to have a quick lunch. \"We have to remember not to do cancer at lunchtime,\" he says. He considers her slight delay in response to be evidence of her distraction by the dogs' doled-out lunch helpings and four boxes of pizza she passed in the kitchen en route over.\n\n\"C'mon, girlie,\" her handler says as she is led back. Time for another tour of the wheel. Ffoster's training is an exercise in shaping: gradually encouraging a desired behavior by rewarding all the subcomponents leading up to that behavior. To teach a dog to water-ski, you don't set the dog down on the beach to watch other water-skiing dogs, or drag the dog out into the water with you unawares. Instead, you begin with the smallest of steps: getting the dog on the ski\u2014on the beach. Simply approaching the ski might earn a treat. Once the dog is approaching reliably, you delay the reward until he has stepped on the board. When he is happy stepping, ask him (by delaying the reward again until he acts) to step with two feet, or four. Soon enough a dog can be taught to stand and stay standing on the ski, without ever being explicitly asked, or, indeed, even knowing what he is doing. In later steps the \"water\" part will be slipped in. When a dog is water-skiing, the behavior has been \"shaped.\"\n\nHere, too, Ffoster and the other dogs are led through days of shaping. Instead of water-skiing, they are only being asked to care about, and eventually find and sit next to, one particular smell. Ball and the other WDC trainers begin by presenting the dogs with a sample of the diseased plasma (combined from many patients). If the dog sniffs, he is rewarded with a click and a little treat. Sniff, click. Sniff, click. Eventually the diseased plasma will be brought out next to plasma from healthy persons. In that case, the game is to sniff the diseased sample, but not the healthy sample. Sniff, reward. If he sniffs the other sample, no reward. Repeat. They do this hundreds of times, but each session is short so as not to exhaust, discourage, or bore the dogs. Once they get to finding the sample at the wheel, training continues: ten times at the wheel, twice a day.\n\nEventually, dogs like Ffoster and McBaine, a black and white springer spaniel with the yearning look characteristic of the breed, become so proficient that when they miss a sample, the trainers know it's not because of the dogs; it's because of something else: a growling stomach, a budding cold.\n\nAfter Ffoster leaves, McBaine comes in with his handler, Annemarie DeAngelo. Between dogs, the room is dust-mopped, the bins and mesh spray-cleaned with isopropyl alcohol. \"Where are we?\" DeAngelo chatters with McBaine as they enter. \"What are we doing?\" McBaine sniffs the non-wheel parts of the room: the floor leading from the kitchen door to an outside door, the chairs for visitors, a visitor (perfunctorily). \"Hi, dog,\" she says, to get his attention. \"Are you going to sniff around? Hi, goober.\"\n\nAfter he settles, DeAngelo steps behind a screen\u2014Clever DeAngelo\u2014and says \"Seek!\" McBaine sets to it. He addresses the wheel with a trotting, not overly rushed pace. He nearly rests his muzzle on each bin, making full swipes across the mesh, leaving moist streaks on its top. Long eyelashes frame his face; his feathery ears droop. One can't help wondering if he is hoping to find a partridge in one of these vials. On the first trial he sits, alerting, to one of the \"distractors\"\u2014the normal plasma. No reward. The vials are emptied, reloaded, and he goes again. On the second trial, same thing. Finally, on the third he hits his stride, easily finding the sample, and earning a delighted response\u2014and a tiny edible treat\u2014from DeAngelo as reward.\n\nLater, via Skype, I watch Tsunami\u2014Tsu, as everyone refers to this handsome, long-eared German shepherd\u2014run through her training. She is \"reactive,\" the trainers say\u2014meaning, in this case, she isn't fond of unfamiliar people or dogs around, so I watch from an adjacent room. Even on the small computer screen Tsu is formidable. She cruises around the wheel like a circus horse circling the ring, seeming barely interested. Her tongue dangles lazily. Then suddenly she turns, as though to address the wheel, and finds the sample. DeAngelo tosses her a ball on a rope and she proudly prances with it. Then she circles the wheel nine more times, finding the sample each time with barely a hiccup or an eyebrow-raise.\n\nIn the WDC protocol, the dog needs to be at 83 percent specificity to go to the next level\u2014that is, correctly ignoring the noncancerous samples ten out of twelve attempts. This level of performance is statistically highly distinguished from random guessing, which would have them correct one in twelve times. Indeed, in the next months the dogs are finding the correct sample (sensitivity) and not alerting to the incorrect samples (specificity) more than 85 percent of the time. But . . . why are they not perfect?\n\nA puzzle. The first plasma they encounter is \"pooled,\" combined from many patients, so it could be that training on pooled samples makes it difficult to generalize to an individual sample. It could be that the dogs are picking up on some, but not all, of the smell profile of the samples.\n\nOr it could simply be that the game is not sufficiently interesting to them every time. Humans rarely achieve 100 percent success at even the skills and tasks we have long since mastered\u2014hence the ubiquity of tripping while walking, hearing your own voice saying runny babbit or other such spoonerisms, or having to look up which way daylight savings time goes every year. We get distracted, have bad days, and grow sleepy; a working dog does, too. The dogs are being only human here.\n\nMany other dogs, at many other research groups, have been tested on their cancer detection skills. Though the training has been less assiduous than at Penn's Center, the results are stunning.\n\n\u2022 \u2022 \u2022\n\nResearch studies on the topic have, for the most part, shown dogs to be eminently good cancer detectors, via a range of means and substrates. Given that there is not one cancer but many cancers\u2014in lungs and other internal organs, on skin, in blood\u2014researchers generally take samples related to the source.\n\nTo few people's surprise, dogs were plenty interested in the cups with small amounts of urine that they were presented to sniff. Perhaps to more people's surprise, six mixed-breed dogs were able to distinguish bladder-cancer patient urine from healthy urine at rates better than chance. Urine, carrier of the final throes of whatever metabolic processes have been happening in the body, seems to have a different cast when there is disease in the body as well. Prostate cancer, too, might leave traces in urine; in one study, a Belgian Malinois used it to correctly spot the disease 91 percent of the time. Two German shepherds retired from explosives detection spent six months instead sniffing urine cups from hundreds of patients and healthy volunteers; at the term's end, they were nearly perfect in identifying the cancer patients' samples.\n\nThe studies go on, testing dogs on tissue samples from biopsies and even right on the person: another two dogs were terrifically reliable at finding the melanomas from among thirty bandages placed on human volunteers. Perhaps the best medium for this research, though, has been exhaled air. Linus Pauling, who won both a Nobel Prize in Chemistry and the Nobel Peace Prize, was less well known for, in 1971, discovering that there were hundreds of volatile organic compounds in an ordinary person's exhaled breath. By those who study it, \"bad breath\"\u2014or even \"breath\"\u2014is not a simple phenomenon. The constituents of breath represent a couple of things: what is in air that's being breathed in, and a gaseous impression of the metabolic process that has happened in the body before being exhaled. It now appears that everyone's breath is different, reflections of themselves and their innards. Some two dozen compounds are identical across people, and then another two hundred or so are unique to you, from among thousands of compounds researchers have caught in their sticky tape.\n\nBreath also holds information about disease in the breath-organ, the lungs. To collect breath samples, researchers ask people to exhale a few times into a test tube in which a bit of polypropylene \"wool\" has been stuffed. Just as smoke sticks on clothing and chlorine in hair, the volatile compounds in breath stick on the wool. A top is snapped on the tube and it is sealed in a zippered baggie. A few weeks of clicker-training, and one study's five young dogs\u2014Labrador retrievers and Portuguese water dogs being trained as guide dogs for the blind\u2014were snappily finding the samples from patients with lung cancer.\n\n\u2022 \u2022 \u2022\n\nAgain, though, a puzzle: what is it, exactly, that the dogs are smelling? Diseased tissue might have hundreds of volatile molecules; is there one that is the cancer \"tell\"? If so, it could be isolated and presented, without the bother of actual cell samples. And can we be sure that what dogs are detecting is the disease at all? A body fighting disease will have inflammatory and immunological responses\u2014which themselves might emit smellable stuff. Accompanying illnesses\u2014and even the depression or anxiety that might come with a serious disease\u2014might also be the noticed scent.\n\nAt the Monell Chemical Senses Center, George Preti is trying to find the silver bullet. Having moved on from his research on canid anal sacs, he is now analyzing the arguably lovelier plasma samples that the WDC dogs are training on. When I visit him, on a hot afternoon in August, West Philadelphia, where Monell is located, smells dazzlingly pungent. Vents on the street fume with the fury of smelly, imprisoned dragons in the sewers below. Monell is blissfully antiseptic by comparison. I speak to Preti in a cool room lined with bookshelves with titles on flavor, taste, and olfaction: Sugar Research 1943\u20131972 jostles for room with I Was a Food Writer for the C.I.A.: A Dietary Confession, Hunger: A Biopsychological Analysis, and Umami: Proceedings of the Second International Symposium on Umami.\n\nAs I puzzle over how he is analyzing the component parts of the smell of cancer, Preti perks up. \"Well, I'll show you.\" He pushes his chair away and starts out of the library into the hallway. I follow him downstairs into a classic-looking laboratory room: beakers, tabletop lights, piles of partially boxed equipment scattered about. He stops at what looks like office copier equipment that you hope you never have to learn how to operate. I would need to make no copies. Instead, I was about to meet the most thrilling technology you've never heard of: \"This,\" Preti says with moment, \"is a gas chromatograph.\"\n\nA large, sealed safelike box with an elaborate keypad on one side, its front piece opens, microwave-like, to reveal its innards. \"It's not on now,\" he reassures me, opening a thickly insulated door. What it does is, in the science of smells, nothing short of futuristic. Insert a sample of something smelly\u2014an orange, an old book, a violet, the captured airspace above a baby's head\u2014and the \"GC\" provides a listing of all the volatile molecules found in it.\n\n\"The guts of it is really the column,\" he says\u2014a column that holds fifty meters of coiled glass, half a millimeter in diameter, and is coated along its walls with a polymer. \"So. You inject the sample up here\"\u2014pointing to the top of the GC\u2014\"into the hot injector\"\u2014kept at between 200 and 300 degrees. Helium gas is pushed through the coil, and the sample materials and the polymer interact. \"Then you gradually increase the temperature of the column at a set rate, say four degrees a minute. As you go up in temperature you start to volatilize a lot of the things that are condensed on the column. They all start to proceed down the column\u2014but they proceed at different rates\" according to their different molecular weights. The components are thus separated, a chord pulled apart into an \"arpeggio,\" as Avery Gilbert writes of the process.\n\n\"Now this particular GC is set up as gas chromatography (with) olfactometry.\" Preti points to the output nozzle. As the solution is separated into different compounds, each compound emerges at a different time. \"If you have a complex mixture, and you want to know when certain odorants are coming out, or certain kinds of smells, you can sit here and sniff and record your olfactory impressions,\" he explains.\n\nI am a little startled. This fancy machine, helium, polymers, and temperature gradients, and at the end there is just someone sitting there . . . sniffing? \"Oh my goodness. You must be trained to do that . . .\" I suggest.\n\n\"Well, I could train you to do it in five minutes. It's easy. Just sit here and sniff. The problem is applying the right vocabulary.\"\n\nHe shows me a piece of paper by the nozzle with a list of possible smell descriptors, three columns long, including quantity characterizations (slight or nothing); familiar foods (pizza, pickles, hot dog, popcorn, coffee); and qualitative impressions (swimming pool, clean). \"Olfactometry\" is \"person sitting with nose in nose-shaped inhaler at nozzle's end, smelling.\"\n\nAlternately, \"GC\" ends in \"MS\": mass spectrometry, in which the different compounds are identified by a machine that graphically represents the compounds as they emerge. The smell is translated into an image; the compounds, a series of drawn peaks on paper. The bigger the peak, the more of that compound is in the mix. The GC-MS, then, tells you all the volatile\u2014potentially smelly\u2014components of any substance: coffee, lilac, citrus, soil. If you put urine in the GC, it will come out as many different compounds as well, including those of the foods eaten by the urinator.\n\nWhat the GC-MS does not tell you, though, is which compound is the smelliest\u2014to us or to dogs. The biggest peak is not necessarily the biggest smell; nor do all the peaks contribute equally to the smell you experience. In some cases, a tiny trace component is actually responsible for the odor. Our sense of coffee might be mostly captured by a mere handful of the six hundred or so compounds separated by the GC.\n\nAs of the day we spoke, Preti had not yet found the compound that makes the cancer patients' plasma samples distinctive to the dogs. Should he find it, that biomarker would not only make training dogs more efficient; it could transform human diagnosis of disease as well.\n\nThe inside of the dog's snout, with receptor cells grabbing passing volatile odors from the sniffed airstream at different places in the nose, is in essence acting as a biological gas chromatograph. The olfactometer at the end, analyzing the samples, is the dog's brain. Oh, to get a glimpse of its list of smell descriptors.\n\n\u2022 \u2022 \u2022\n\nGiven dogs' abilities, does it make sense to put your book down, strip, lie on the floor, and ask your dog to sniff you for melanoma tout de suite? I cannot recommend it. He may indeed detect that errant mole on your torso\u2014but he does not know to tell you about it. It may smell different to him, if he is familiar with your smell, as any dog who spends time on the laps, couches, or beds of their persons is bound to be. It may even smell \"diseased\": one dog-olfaction researcher proposed to me, tongue not entirely in cheek, that wolves' sensitivity to the weakest or most ill of their prey may be related to disease-detection dogs' reactivity to human diseases. But the dog does not have a concept of disease; we must draw that for him. If he sniffs a skin ailment, he will simply notice it and say nothing.\n\nIt is not only cancer that has captured the attention of detection-dog trainers. Work is well under way training dogs in detecting hypoglycemic and hyperglycemic episodes (in diabetics) as well as in alerting prior to an owner's epileptic seizures. One research paper describes dogs, glowingly, as a \"fully biocompatible and patient friendly alarm system\" for hypoglycemia.\n\nAt Penn, the trainer I saw working with the cancer-detection dogs, Jonathan Ball, designed a special \"diabetic suit\"\u2014coveralls with pockets sewn variously over them into which samples of the odor could be stashed (and searched for). \"What are they smelling?\" I ask him. \"Good question,\" he replies, shrugging. Nonetheless, at the time of my visit, the Center had just, to some fanfare, placed its first fully trained diabetic-alert dog, Bretagne, a sweet-tempered young golden retriever, with her new owner. Bretagne's training began by imprinting her on the smell of saliva swabs from someone whose blood sugar level was in the 50 to 70 ml range\u2014on the low side of normal. For nondiabetics, this blood sugar range is experienced as the slow, sluggish feeling that might descend about half an hour after a big meal, when the insulin released for a moment overtakes the glucose level it is regulating. In training, Bretagne is shaped to alert at a blood sugar level of about 80 ml, as it is falling. The test swab and control swabs are put in vials and pocketed, and Bretagne ticklingly sniffs the coverall-wearer all over until she reliably spots the one.\n\nBut what it is in the saliva sample\u2014what volatile odor, exactly what compound or set of compounds that has the indicative odor\u2014is not yet known. Just as with cancer, until the signature cocktails of disease are known, biological samples will need to be used.\n\nIn Philadelphia, once Bretagne went to live with her person, the training process only shifted. Weekly training is needed in detection so that the dog continues to get some reward for alerting. Unlike other working dogs, a diabetic-alert dog may have no \"successes\"\u2014positive identifications of precariously dropping blood sugar\u2014during the day if her person is stable. Although that's good for the owner, it can be frustrating for a dog: trained to do something, and do it well, she wants to get a chance to succeed. Training sessions with samples off the body keep the dog's motivation high and nose focused.\n\nIn the UK, the first empirical test of the success of diabetic-alert dogs was published in 2013. The first indication that dogs might be clueing in to their owners' health came from owners' reports of their pet dogs spontaneously responding to their hypoglycemic episodes; stories with so-called epileptic-alert dogs have similar evidence. Now, owners may remember only those instances in which it seemed the dog warned them\u2014not those when the dog did not, or when he showed these behaviors without an episode under way: a foible particularly typical of all humans. If that were the case, the levels of owner-reported dog success may have been overreported. So the researchers asked owners to take blood glucose samples of themselves throughout the day and also to note when they believed their dog alerted\u2014by nuzzling, pawing, even retrieving the blood-test kit. It turned out that the times when these dogs alerted was more likely to be out of the desirable blood sugar range than the other routine sample times. Dogs did not get all the hits, nor did they always hit correctly, but it was promising. Dogs might have been smelling a change in their owners' sweat or even breath\u2014but also, as acute observers of their people, might have been noticing some change in behavior that served as corroboration.\n\nIn the cases where these detection dogs are living with their focus of detection attention\u2014their people\u2014as opposed to other working dogs, it could very well be that some of the life improvement of diabetics and epileptics is due to the general positive effect on well-being that living with a dog can impart. A person who had never been able to be left alone could now be alone (with her dog); people restricted from travel would now be free to travel. Dogs did that.\n\n### -PERSON-\n\nDogs' capacities at cancer detection are, to my thinking, the perfect example of what seems most mysterious about the dog nose. Cancer is not only something we fear or dread; it is something opaque to the mind and often invisible to the eye\u2014certainly not smellable. To the dog, freed of our conceptions of mortality and our obsession with sight, cancer is just a smell.\n\nOn the other hand, a bit of our own, mostly neglected history is relevant here. It is this: humans have been diagnosing disease by smell for thousands of years. Only in recent times have we essentially stopped smelling.\n\nSo here I must go back in time. We have grown so antiseptic, so machine-addled, that we no longer bother to sniff (or, sometimes, look at) patients. This has not always been so. Ancient thinkers and cultures were aware of the role of smell in disease. Smelling one's patient and his diseases goes back at least to Hippocrates, who advised doctors to keep an \"open nose.\" The ancient Greeks thought about smell as symptomatic: to Plato, smell arose in transformation between two elements: \"As water changes to air, and air to water, all odours have arisen in between.\" The physician Galen characterized smells on people, from their mouths, as either \"according to nature\" or \"against nature.\"\n\nThe interest in smells led to some bizarre ancient medical treatments. The belief that a pregnant woman's uterus could travel up to her throat and suffocate her led physicians to wave horrible-smelling material near the woman's mouth (to send the womb away), and pleasant-smelling stuff around her genitals (to draw it near). Pliny noted that the \"rank\" smell of the armpit was thought to be addressable by drinking a white Falernian wine\u2014and then urinating out the smell. Whether this method replaced the smell with a smell of wine, or merely made the drinker sufficiently besotted that he did not notice his smell, is unknown. Other recipes involved chewing gum; perfumes, too, were thought to be health-giving.\n\nLinnaeus's original eighteenth-century classification of the fundamental kinds of scents was actually medically inspired: he was interested in botanical smells and the curative (or ill) effect they would impart on their consumer. While a \"fragrant\" plant (lime, lily) might be healthful, \"repulsive\" or \"nauseating\"-smelling plants\u2014such as veratrum, highly toxic\u2014were clearly to be avoided. Plants with no odor at all were no good.\n\nDespite Hippocrates' \"open nose,\" the fear that bad odors themselves caused disease loomed large. Galen warned people to avoid \"those who exhale such putrid humours that the houses in which they lie in bed become stinky.\" Given Hippocrates' continued influence in modern medicine, perhaps the contemporary worry that some still hold that foul odors breed contagious illness is not surprising. One helpful sixteenth-century instructional manual for doctors suggested readying oneself when approaching an ill-smelling patient with a fragrant branch of burning juniper and speaking to the patient \"from a certain distance away.\" If one must approach, it suggested, venture forth with back turned, and take the patient's pulse by groping behind you with your hand, never getting any closer than necessary\u2014and with an assistant there to hold the branch directly underneath your nose.\n\nNonetheless, smelling diagnostically continued. After some research on guinea pigs, breath-sniffing had become a medical tool (little is said about the breath of the guinea pigs). In the eighteenth century there even developed a medical specialty\u2014called osphresiology, forgettably\u2014that reified a list of characteristic smells of disease. Odors from the skin as well as bodily effluvia\u2014vomit, urine, sweat, fecal matter\u2014were all informative. A sweet smell of onions indicated smallpox (something that distant cheetahs are apparently drawn to, should you see a cheetah sniffing after you)I; freshly baked bread, typhoid; the smell of a butcher's shop was a sign of yellow fever. A fruity, ripe-banana or \"fake fruit\" smell indicates diabetic ketoacidosis. Dozens of conditions and diseases have henceforth been added to the list. Smell of cat? Ringworm. Tickle of freshly plucked goose feathers in the air? Measles. Stale beer, sour bread, old straw, and sweet, urinous, or putrid smells are all indicators. Even psychiatric disorders may have characteristic smells: pungent and sweaty, urinous, and vinegary odors have been smelled in schizophrenic, psychotic, and anxious patients, respectively.\n\nOften, odors can indicate the ingestion of something toxic: arsenic smells like garlic on the breath, iodine is metallic, mothballs or eucalyptus reek of camphor, and any glue-sniffing is exuded from the body as the smell of petroleum. Cyanide can smell like the bitter almonds it can come from; alcohol poisoning, the alcohol imbibed: juniper berry (gin), fermented grape (wine), hops (beer).\n\nBy the nineteenth century olfaction was not unusual in medical diagnoses. Infections, in particular\u2014sepsis, tooth or bone decay\u2014were common, and carried distinctive smells. The tide then turned. Professional interest in smells rapidly diminished as infections\u2014and their pungent odors\u2014were reined in. The decrease in olfactory diagnostic techniques coincided with the steady increase in the stigma of smells and smelling. Many cultures grew concerned with ridding the environment of smell, and, coincident with that, technologies that could do the job of \"smelling\" bodily effluvia began to replace human noses. In the twentieth century, the gas chromatograph supplanted all but the keenest of noses, with its ability to separate and identify the molecules in a complex odor\u2014be it from mouth, armpit, groin, or palms.\n\nToday, little attention is explicitly paid to smell in Western medicine. While doctors and nurses seem to regularly refer to \"patient odors\" (especially malodors), what those odors may denote\u2014apart from some unpleasantness\u2014is getting forgotten.\n\nWhen I ask physicians and nurse practitioners how they use odors in their practices, I get remarkably similar regrets: \"unfortunately . . .\"; \"alas\"; \"sad to say . . .\"; \"sorry.\" Dr. Jane Orient, editor of the masterly Sapira's Art and Science of Bedside Diagnosis, which has an entire section on the \"bouquet\" of odors one might detect from a patient's breath, tells me that she doesn't know of anyone who uses smell prominently in their medical practice. \"It is a much neglected subject,\" she says.\n\nContemporary medical texts mirror this waning of interest in smell as diagnostic. One exception is in the realm of infectious disease. Infection is, simply, the state of being affected by some non-endogenous organism, like a foreign bacteria in an otherwise healthy body. Infectious-disease textbooks write about smell as though it were natural sense to use in diagnosis. Indeed, when trying to figure out if an anaerobic infection is involved, \"Only the foul or putrid odor of a lesion or its discharge is specific,\" one states; other clues are only suggestive at best.\n\n\"The characteristic foul odor of the sputum suggests anaerobic involvement\" is a not-unusual sentence in medical literature. \"Almost all doctors have come to know that a really foul smell\"\u2014of cadaverine or putrescine, say\u2014\"coming from the lung or an abscess or a wound represents the presence of anaerobic bacteria,\" Dr. Bennett Lorber tells me. \"Foul-smelling\" is one of the major categories of medical smelling, apparently. Even the petri dishes of bacteria that pass through microbiologists' hands\u2014Lorber is a professor of microbiology and immunology at Temple University's School of Medicine\u2014might be smelled. Some evoke grapes; some bleach; and some have a \"mousy smell,\" Lorber adds. While specimens are now usually examined by machine more than by nose, if Lorber gets ahold of one of those agar plates himself, he sniffs.\n\nSo doctors are smelling, to be sure. The same practitioners who claimed to not wittingly use smell in their practice all casually mentioned some ways they in fact notice patient smells. \"Alcohol, of course; some poisons like organophosphates; uremia; hepatic failure; blood in stool if a patient has a big GI bleed . . .\" The list tumbles out of Orient. Others mention the distinctive smell of the combination of alcohol and tobacco, which might indicate that other drugs are being used. Dr. Abraham Verghese, who has written and spoken widely about the value of bedside-exam techniques also demurs when asked about the use of smell in his practice. The next second, though, he observes that he did notice odor \"in the course of (medical) rounds, incidentally\u2014(such) as the fruity odor of diabetic coma.\"\n\nWhat there currently is not in Western medicine is a practice of teaching smelling for diagnosis. It would not be impracticable: Orient recommends that a battery of vials with substances like chloroform, the carbon tetrachloride in fire extinguishers, and other toxins be kept available for sniff-training. In fact, emergency room workers at Bellevue Hospital were once trained on a \"ten test tube sniffing bar\" holding the characteristic odors of poisons. In the 1970s some medical schools used similar means to teach students about possible odors, but its use seemed to wane quickly. It is hard to imagine why, given the possible smell training field trips. What doctor would decline a visit to a brewery, to become familiar with the oaty smell of oasthouse urine disease, an inherited metabolic disease, or to reacquaint themselves with the smell of hops absent the drinking of it?\n\nSure, along the way, the observant doctor will lean close to a patient and notice what that intimacy reveals. But Western medical schools largely ignore smells. There are machines for that.\n\nEnter Eastern medicine. Or, rather, reenter: the medical traditions from the East far precede Western practices, of course. And in traditional Chinese medicine (known as TCM), smell has always mattered\u2014and still does. The corporeal body itself is the source of information about what is wrong with the body: the means are asking, looking, touching, smelling, and listening. Noting the breath, sweat, saliva, mucus, urine, feces\u2014and even the odor of a room that's had you in it\u2014are part of TCM's diagnostic method.\n\n\u2022 \u2022 \u2022\n\nTurn off the main drag in Northampton, Massachusetts, and the character of the town comes through. Bay-windowed Victorians with small, well-tended lawns; an ambling citizenry; a vegetarian restaurant that does not allow patrons to wear fragrances. On the corner of State and Center, two flights up, is the office of Leta Herman. Her waiting room is cool on a hot day. Incense burns and I hear the trickle of water. Herman practices acupressure and what she calls the Five Element strain of Chinese medicine, which differs from TCM in some of its approach.II But boy, can she smell.\n\n\"It took me eight years to get to this,\" she says happily of her olfactory acuity. Herman has a positively ebullient air: she approaches smile first, her curly brown hair framing a youthful face. \"I was not a dog going into it.\" When Chinese medicine gave her relief from a stubborn medical condition, she enrolled in a class to see how it worked. Soon she quit her job in the computer industry and now, sixteen years later, she owns her own practice. And smells a lot.\n\nIn TCM's conceptual scheme, five fundamental constituents\u2014wood, fire, earth, metal, and water\u2014are invoked as contributing to the harmonious balance or imbalance of a system (such as a body). To practitioners like Herman, the elemental types smell. That is, they are considered to have distinctive odors, similar across people despite differences in race, age, cleanliness, interest in perfumery, or recency of running in a polyester tracksuit. Odor descriptions can be blunt\u2014a person may smell burnt, rotten, or rancid\u2014and sometimes impressionistic: some people effuse the effervescence of \"ginger ale or vinegar,\" an odor that \"hit you in the nose\u2014and it's like pow, and then it disappears,\" Herman says. Or the smell of \"clothes on the line, drying\"; a honeysuckle blossom that \"follows you down the street and entangles you.\"\n\nAfter talking to Herman for a while, I have to ask: \"Do I smell like something?\"\n\nHerman offers me a table to lie down on, then softly lifts my wrist to take my pulse. She looks at me penetratingly\u2014smelling at me, I realize. Many of her patients don't know that she is smelling them: \"I don't really bring it up with people,\" she says. Given the strangeness of being sniffed\u2014as well as our own self-consciousness about our body odors\u2014that seems a sensible choice. She and her colleagues have developed tricks for getting a sense of someone's smell without actually burrowing into the person's neck. For one, they leave the room for a good amount of time\u2014twenty minutes should do it. When they return, the person's odor has filled the room, and\u2014boom!\u2014it hits them in the face. Another trick for some of what Herman describes as \"dense\" smells that tend to descend (something like \"the smell of vitamins, or gym clothes that have been in the gym bag a long time\")\u2014is to \"accidentally\" toss a pen on the floor under the treatment table. In bending over to pick it up, one can cast about for any gym-bag\/vitamin notes in the air.\n\nShe furrows her brow slightly, fixing her gaze somewhere in the middle of my forehead. \"I'm getting a little hint of water,\" she says. That means an ammonia smell\u2014putrid, in the jargon. I nod, trying not to look too aghast. Herman switches to the other side of the table and gives a big exhale: \"You haven't been in here very long.\" So at least the room isn't filled with the smell of urine. \"I could smell your neck if you want to know your [foremost] element. It's kind of cheating.\"\n\nI agree to cheating. Sitting up, I experience the unusual interaction of having a stranger intentionally lean in toward the back of my neck for a sniff.\n\n\"Hmm. Hmm. No, I don't think it's water, then. Yeah. Okay, so metal and earth are predominant.\"\n\nI ask what use it is knowing someone's smell. \"Then you can tell if their smell is off,\" she says. \"So I might be looking at their stomach and notice Oh, this is getting sour; it's fermented; they have smelly feet. Their metal smell is getting really rotten; their sweet earth smell is sickly sweet. Then I'm trying to bring them back to their good smells.\"\n\n\u2022 \u2022 \u2022\n\nSurely TCM is not the only bastion of human olfactory diagnosis, I reason. So, smelling of earth, I dig into the medical literature. A few more reports of smell diagnostics are still bouncing around in the twenty-first-century journals. They share references to some of the now classic smells\u2014cyanide, the fruitiness of diabetes, the \"sewer breath\" indicating a gastrointestinal infection or periodontal disease. They also share a kind of surprised delight at discovery. These doctors and nurses have become olfactory Sherlock Holmeses, able to sniff out something invisible and elusive in their patients. An alcoholic trying to avoid drink smells fruity, betraying his consumption of the cleaning fluid isopropyl alcohol, a severe central nervous system depressant. A corn farmer who is sweating and vomiting smells of garlic\u2014indicating that he's been affected by the pesticide he was applying to his crops. A couple of coal miners arrive at the hospital unconscious. Their odor of rotten eggs\u2014as well as the blackened coins in their pockets\u2014reveals that they inhaled deadly hydrogen sulfide underground.\n\nMedical personnel are still encouraged to follow their noses if they suspect some of the conditions that produce distinctive odors: phenylketonuria (PKU), with its musty, murine odor; maple syrup urine disease, pancakes-evocative; infections; toxic ingestions. Early suspicions that lead to further investigation can preempt development of serious problems or death.\n\nBut remaining evidence for disease smellers is few and far between. As an example of few, George Preti, at Monell, though not trained as a medical doctor, turns out to do his fair share of sniffing people. \"You'd be surprised how many people write to me and say, Oh, my head stinks, it oozes, it comes out in my hair.\" Preti tells them to come in having not washed their hair for several days. Then? \"Oh yeah, I smell 'em, yeah,\" Preti says. \"We sniff the patient. It's probably the only place on the planet where people are diagnosed this way.\" From head odor to concerns over having a very bad body odor, people find their way to Monell. \"Oftentimes people are way out of line in terms of how bad they think they smell,\" he says reassuringly. One of Monell's research interests is a metabolic condition called trimethylaminuria (TMAU), an uncommon and debilitating disease that can easily be identified by smell. Why? It is also known as fish odor syndrome.\n\nAnd far between: in 2012 an untrained Scottish woman reported that she could smell her husband's Parkinson's disease, a motor disease that often involves sensory changes. He smelled suddenly \"musty,\" she was quoted as saying. When tested on her ability to diagnose which of twelve t-shirts\u2014six from Parkinson's patients, six from healthy volunteers\u2014were from Parkinson's patients, she got an impressive eleven correct, incorrectly identifying one as having the smell when he did not have the diagnosis. The collars of the shirts, where oily secretions from sebaceous glands tend to collect, seemed to be the culprit.\n\nAs for that one incorrect t-shirt? Its wearer was diagnosed with Parkinson's eight months later.\n\nThis woman's ability forms an interesting twist, because smell is being used diagnostically in a different way. In particular, when a patient has a sharp decline in his own sense of smell it is itself indication of possible disease\u2014especially the early stages of Parkinson's. Alzheimer's, too, though a disease of memory, is often marked by olfactory dysfunction. Apart from self-reporting smell loss, patients can take what is now a standardized olfactory test. It uses \"scratch-and-sniff,\" a microtechnology more often associated with foul-smelling stickers and ill-advised novelty books. To make scratch-and-sniff patches, tiny capsules holding slurried oils are sprayed onto paper, waiting to be scratched\u2014and the capsules burst\u2014by an aging fingernail.\n\nAs I consider Finnegan's close sniffing investigation of me in the morning, I rummage around in the closet and unearth a forty-three-year-old board game named, punnily, Smell & Tell, billed as a \"scent-sational game\" that is centered around scratching-and-sniffing cards. Incredibly, odors of banana, chocolate, root beer, and garlic have waited, patiently, for decades before meeting my family's apprehensive noses. Finnegan, for his part, will not suffer the game and retreats to a distant room.\n\n(Though Finn had not, happily, sniffed out a serious illness on me, some consider pregnancy to be a disease\u2014and he had detected that, I found.)\n\n\u2022 \u2022 \u2022\n\nThe German philosopher Friedrich Nietzsche, a man fond of smelling (he once wrote, \"My genius resides in my nostrils\"), spoke in an offhand way about the \"keenness\" of one's nostrils: an ability to smell something out, rather than just smell something. This kind of wisdom of the nose used to be reified in the English language as the word nose-wise. In the twenty-first century, it has lost its precocious, keen-nosed meaning and is now just defined as \"with regard to the nose.\" Some demotion. Nose-wisdom has a place still in our society; it just needs to be given the respect that the dog's nose is getting.\n\n* * *\n\nI. \"It is stated that the odour emitted from a small-pox patient will attract cheetahs from afar\" (The Lancet, 1906): while this may seem far-out, we are certainly accustomed to the flies attracted by our smell.\n\nII. For instance, the language of TCM is more likely to refer to the \"Five Phases of change\" (wu-hsing).\n\n## 9: Stink-waves\n\nThe otherwise cheerful musical film Chitty Chitty Bang Bang, starring Dick Van Dyke as a mildly dotty Caractacus Potts, takes a dark turn about halfway through. The two Potts children find themselves holed up in a barony where children are outlawed, and whose baron employs a \"child catcher\" with a spectacularly long and bulbous-tipped nose who can smell out children. In no time at all he sniffs the guileless kids out, then lures them with candy into his candy-wagon-slash-cage. In retrospect, it is no surprise to learn that this film script was written by Roald Dahl.\n\nDahl wrote preposterous noses. Of one of the eponymous, child-hating subjects in The Witches: \"She can actually smell out a child who is standing on the other side of the street on a pitch-black night.\" When a child protests that he is reasonably clean (having recently bathed), he learns, \"It isn't the dirt that the witch is smelling. It is you. The smell that drives a witch mad actually comes right out of your own skin. It comes oozing out of your skin in waves, and these waves\u2014stink-waves, the witches call them\u2014go floating through the air and hit the witch right smack in her nostrils.\"I\n\n### \\- DOG -\n\nDetection dogs (known in various dogly circles as sniffer dogs, detector dogs, or working dogs) detect much more than disease, of course. Just how much more has not yet been discovered. Until recently, we assumed that the dog's nose was only as good as our imagination was in applying the nose to finding things. Our imagination has proven more fertile of late\u2014hence the expansion from training dogs to find drugs and land mines to orienting dogs toward mealybugs, smuggled agricultural products, and endangered ribbon snakes. Dogs have been trained to find minute amounts of environmental contaminants: derivatives of gasoline production and various toxic products at industrial disposal and waste sites. They work sniffing out sea cucumbers being illegally exported from the Gal\u00e1pagos, and the smuggled ivory and horns unfairly removed from their elephant and rhinoceros hosts.\n\nAnd dogs detect us: tracking, trailing, search-and-rescue, and scent-identification dogs pursue the missing, escaping, lost, or dead; the criminal, confused, unseen, or unlucky. Their noses' authority is acknowledged by our legal system: no less than the U.S. Supreme Court described the detector dog's sniff as \"sui generis,\" a tool unlike any other. Tracking's earliest appearance among canids was no doubt as hunting: predators like canids cannot just wait until prey shows up under their claws or leaps into their open mouths. Being able to follow the tracks of (hoped for) prey is a necessary adaptation. Since the time of their wild ancestry, domesticated dogs have modified the hunt into a hunt-minus-consumption. Pliny wrote of huntsmen carrying even their aged and infirm hunting dogs with them, so good were they at finding their quarry by \"snuffing with their muzzles at the wind.\" But in common with those hungry wolves, the dogs find us primarily through one thing: our stink-waves.\n\nThe smell of a person is so strong that dogs can follow it over time, underwater, after the person is long gone, and even after the thing the person touched has actually blown up. In one study, researchers found trained bloodhounds able to identify who had touched a pipe bomb\u2014after the pipe bomb exploded. The person's scent, laid on the pipe by touching it while setting it up, \"survived\" when little of the actual pipe did. Dogs have been trained to find drowned persons: the odor of decomposition rises to the surface of a lake or other still body of water; some dogs can even track in flowing stream or river water. Where sonar, divers, and underwater cameras fail, a dog may start pointing at the location from the dock, and then from a boat narrow the search space to just twenty feet across. Cadaver dog handler Cat Warren writes that when a dog alerts during an underwater search, it is \"like stepping from one room to another\": from fumbling in the dark to a room aglow. Avalanche-rescue dogs have found people buried under twenty-four feet of snow: the person's scent travels to the surface and the dog, after some digging to convince himself, alerts there.\n\nIn some cases of search and rescue and trailing, the dog is given an article of clothing to \"search out\"; but in most tracking, all the dog needs is to know that he is after some person. For each of us exudes a perfectly loud and boisterous smell to a dog. What smell-science researchers have shown us is that \"without a trace\" really is impossible: we are always leaving a trace. A flurry of skin flakes trails from us as we move. Even when quite still, we are wafting out odor from our skin and the things on and in it. Not only that, it is still there long after we are gone. To a dog, wherever you go, there you still are.\n\nShould this sound fanciful, consider the lowly mosquito. One could adjudge that it, too, \"tracks\" people. Leslie Vosshall has studied what makes people differently attractive to mosquitoes by bringing in hundreds of volunteers, putting them in what she calls a tropical room, and testing how many mosquitoes will fly upwind to bite an inch-wide patch of their exposed skin. (Being a volunteer at the Vosshall lab is not for the faint of heart.) Your body chemistry affects the number of bites you receive, but the number can also be reduced by simply creating a breeze. \"A very helpful side effect (of ceiling fans) is that the mosquitoes get very confused,\" she offers: the air becomes turbulent, and though mosquitoes smell you nearby, they can't find you.II\n\nThe tracking dog cannot be so easily eluded. Unlike Paul Newman's character in Cool Hand Luke, who successfully confuses the bloodhounds on his scent with chili powder, pepper, and curry, prison escapees are unlikely to so easily muddle a tracker. The tracking dog's skill is not just to find the person's scent, but, perhaps more impressively, to distinguish it from the thousands of other scents gurgling and enticing his nose. A bit of pepper may cause a sneeze, but not a system collapse.\n\nShould a track grow dim, dogs search for the \"scent cone\"\u2014the invisible blooming of smelly air that spreads out from an odor source. Smell radiates from its source, getting weaker but its reach getting wider. By zigzagging, crossing perpendicular to the odor corridor, all the while moving forward, dogs zero in on their target. Along the way they engage in a kind of on-the-fly geometry to determine when the scent has weakened sufficiently relative to their previous sniff to warrant changing direction.\n\nNot only do we effuse a strong smell, our odor lingers incredibly well on things: on gauze, on paper, on plastic, on metal. We mostly touch things with our hands, and hand odor can stick around for months on some materials. Porous objects like gloves and clothing are utterly saturated with the smell of you, but even your stainless-steel watch or gold wedding ring has you on it and in its crevices. Indeed, in training and testing handlers use \"scent samples\" made up, simply, of a cloth that someone held in her hand for fifteen minutes\u2014or, for the better funded handlers, by using a specialized vacuum that traps the air above a person and deposits it on gauze. That does it.\n\nIn researchers' parlance, our stink-wave is multilayered. We wear a primary odor all the time, part of being biological creatures. Merely sitting, doing little but turning the pages of a book, we slough off some two billion polygonal skin cells every day, along with their population of bacteria and funguses, and generate a pint of sweat (or up to five pints an hour should we up and exercise).III These include carbonyl compounds like aldehydes and ketones, alkanes, and organic fatty acids\u2014the proportion of which distinguishes us from one another. On top of that, we are fragranced with a secondary odor of where we've been and what we've eaten. Incredibly, either in an ill-conceived attempt to hide our other odor layers or in the similarly ill-conceived notion that we don't yet smell enough, we often add tertiary odors: perfumes, soaps, hand sanitizers, lotions, hair products, aftershaves.\n\nThe tracking dogs get all that. Lavender lotion does not put them off the particular funkfest that is your bacteria-topped, aldehydic, fatty-acid, alkane mix. And then, of course, you leave footprints. The clues in a footprint are numerous: shoe size, walker's weight (leading to a lighter or deeper track), shoe tread. The prints themselves churn up dirt and grass and leave small bits of shoe-smell in them. But also you-smell.\n\nDon't think your smell comes through your shoes? Here's an experiment for you: Stick a pair of shoes in your gym bag, zip it up, drive home, open the bag, take your shoes out, and closet them. Return to the bag. Sniff it. Smell like \"shoe\"?\n\nIf you're still unconvinced, you could mimic the experiment a pair of K9 trainers did to see how much liquid\u2014potentially carrying odor\u2014comes through shoes. They poured water up to the shaft of a waterproofed leather hiking boot. In half a minute, water was seeping through to the outside. When walking, the boot's exterior got more sodden. A shoe emits the liquid or gas from within it, even without the wearer sweating like mad and pressing it out. And here is how the foot odor\u2014also gaseous, even sneakier than liquid\u2014gets out. The soles of the human foot have hundreds of eccrine glands per square centimeter\u2014more than anywhere else on the body\u2014which sweat. Sebaceous glands also emit odor; the fatty acids from these glands get right through shoes and socks onto the treaded surface. Your smell, via the sweat in your feet that you are constantly, normally producing, is squeezed out in every footstep.\n\nWhile tracking is thus a simple matter of following smell, what is more startling is that dogs do not need to be trained to track by smell. Of course, there is plenty of smell-training to shape good tracking behavior. Early training encourages dogs to keep searching for a scent that has disappeared; handlers walk paths that point orthogonally to the wind to help catch a smell on a breeze; dogs learn to eliminate all the other cues present in the world and focus on just the one relevant right now. This training is just as much about the handler learning to help her dog do the task as it is about teaching the dog to track.\n\nThe one necessary attribute every kind of detection dog must have, by nature or training, is this: motivation. Sometimes overwhelming, house-wrecking motivation. Successful working dogs are not gung ho to track because they want to please their handlers, find the bad guy, or experience the most intense scent. No, they are gung ho to track because tracking a smell to its source gets them one thing\u2014the thing they want more than anything: a grubby tennis ball or the business end of a shredded tug toy. Dr. Simon Gadbois, an olfactory researcher at Dalhousie University in Halifax, calls natural working dogs the \"dopamine breeds\"\u2014breeds like Border collies, Jack Russell terriers, Belgian Malinois, and huskies. They are tenacious and motivated and can focus their attention on the task they need to do to get their ball.\n\nIt needn't be a purebred dog, though. A passion for toys, a yen for fetching and tugging: these represent characteristics of a dog who can be trained to do anything in order to sate his desire to play. If a puppy is not especially interested in playing tug, a trainer may train him to tug\u2014confident that if it takes, the rest of the scent-training will fall tidily into place.\n\nTo describe their work, the skilled handlers of working dogs speak almost in tautologies: \"The core of searching and tracking exercises,\" one training handbook reads, \"lies primarily in generating the desire to search and to track.\" Later, the authors confess, \"No dog needs to learn to search.\" The point is this: dogs are natural searchers, hunters, trackers, smellers, pursuers. You only have to create the conditions for them to do it, and encourage it\u2014ensure they keep liking the game. Such encouragement goes against how most of us deal with our dogs. It is almost as though most owners are assiduously teaching their dogs not to track. The dog who will patiently sit outside a store, waiting for you, who heels nicely and points his nose politely forward as you walk together\u2014that dog is not destined to be a tracker. After watching tracking dogs devotedly follow their noses in search of a missing person, zigzagging in front of their handlers, seeing the familiar behavior of an owner walking her dog on leash down a city sidewalk is like watching the result of careful untracking training.\n\nStill, training or no, watching a dog find his target through attention to invisible tracks is like gazing at a dark sky and knowing the universe sees you, but you do not see the universe. Our only access to what the dog is doing is to, well, just watch the dog. In Norway, researchers strapped a microphone on the nose of four German shepherds and set them to find the tracks of people who had secretly trodden through grass or over concrete sometime earlier that day. That the dogs could all do it (they could, in a handful of seconds) was not the issue. Instead, the researchers were investigating three distinct phases of tracking: an initial \"searching\" phase, ten to twenty seconds long, during which the dogs tried to find the track. Upon finding the track\u2014about two footsteps into it\u2014they slowed way down and entered what the researchers call the \"deciding\" phase. Not \"deciding how I feel about this\" but \"deciding which way the tracks go from here.\" Nose in near contact with the grass or concrete, fewer than five seconds were needed for each dog to make his decision and quickly move on to the actual \"tracking\" phase. The head microphones captured each snort, sniffle, and sneeze, and conveyed the news that the dogs sniffed six times per second throughout. Actual breathing was only about 10 percent of what they were doing with their nose and mouth. As with other studies, the dogs were comparing the strength of the smell of each footstep\u2014footsteps that had been laid a second apart, many minutes earlier.\n\n### MOTHER LODE\n\nIt was a garden for the blind: a constant offense to the eyes, a pleasure strong if somewhat crude to the nose. The Paul Neyron roses . . . had changed into things like flesh-colored cabbages, obscene and distilling a dense, almost indecent, scent which no French horticulturist would have dared hope for. The Prince put one under his nose and seemed to be sniffing the thigh of a dancer from the Opera. Bendic\u00f2 [the dog], to whom it was also proffered, drew back in disgust and hurried off in search of healthier sensations amid dead lizards and manure.\n\n\u2014Giuseppe di Lampedusa, The Leopard\n\nWhile dogs can reasonably, and with good cheer, be trained to detect bladder cancer or diabetes, to locate bedbugs, methamphetamine, or missing persons, every dog owner knows that these quarries are not their true specialty. If there is one thing that dogs seem to have a natural propensity for, it is, instead, finding feces. Every dog cohabiting with a cat has deep, visceral knowledge of the placement of the litter box; outside, should a neighborhood dog have moved his bowels unattended, your dog can locate it with alacrity. An urban dog might pinpoint for you, without your even requesting that he do so, either the wandering coyote's scat or the mess left behind a tree by a local human. What's your pleasure?\n\nHappily, dogs are now fulfilling their natural m\u00e9tier. No, sorry, Upton, not \"finding feces of the homeless person behind the tree in the park.\" Instead, detection dogs are being put into service to find the droppings of various endangered or difficult-to-locate populations and species.\n\nAt the University of Washington, Dr. Sam Wasser is quietly heading up a dedicated facility for training and placing what he calls \"professional poop chasers.\" Or, in the terse nomenclature of science, \"scat-detection dogs.\" I have come to meet Wasser on an early morning at the Seattle campus, whose broad open spaces are mostly empty of students at this hour. A dense fog hides Mount Rainier, which should be proudly rising to my southeast as I head into Johnson Hall.\n\nThough I have come to talk to Wasser about dogs, wildlife is his main concern. His detection-dog program, Conservation Canines, did not grow out of an interest in dogs. In the late '70s, Wasser worked in Mikumi National Park in Tanzania, Africa, studying yellow baboons. Like many other wildlife researchers, he was in the business of collecting information about the behavior and health of his chosen animal population\u2014in his case, the reproductive competition between female baboons. Typically this information is gathered by stealth, trying to keep out of the animals' way and, instead, setting up camera \"traps\" that catch their movement, hair snags that pull hair samples (yielding DNA) as the animals wander past, or tagging or collaring animals and releasing them. Each method has shortcomings, from sampling bias (some animals are more likely to be caught than others), to the time and expense required to catch and release animals, to the stress and sometimes fatal injury from the tagging. Lures can change the target animals' behavior; trapping can injure them; even wearing a tag can fundamentally change the population dynamic.IV\n\nA better way to find out about living populations would be to treat them archaeologically: to see what they leave behind. Like bad visitors to a hotel, animals leave evidence of their activity in their wake. They graze, abandon nests, trample ground\u2014and, most shamelessly, poop as they go. It is that poo which captured Wasser's interest.\n\nA bit about poo. Generally, we think of excreta as worthless, without value. To call something shit is, pithily if not wittily, to observe that it is devoid of use or merit. Not wildlife shit. Researchers see shit and think: gold mine. There is information in that scat: information about health, reproductive status, what the animal is eating, and how it is feeling. The DNA allows identification of individual animals\u2014who it is, the age and sex\u2014and provides a way to determine relatedness among the animals. Scat samples allow researchers to reconstruct how large a species population is and how widely it's ranging. Oddly, this nearly perfect collection program is such that animal researchers can gather considerable information about their subjects without ever seeing the animals.\n\nNow Wasser's concern is the demands that human growth puts on wildlife. The scat samples allow him to suss out how the health of animal populations is changing. But the poop business began with the baboons: \"I was on a mission to try to get DNA out of feces,\" Wasser tells me. Collecting feces could be a trial: trying to be there at the moment of production, as it were, or ferreting around in the dirt when animals had moved on. \"I realized, wow, if I could figure out a better way to collect it, that would be great.\" At a conference on bears, where the banning of hound hunting\u2014using hounds to find bears to shoot\u2014was being discussed, he met a hound hunter bemoaning what he would now do with his dog. The dog, he claimed, could go forward and backward to follow bear tracks. Wasser was transfixed. \"I was like, Oh my God, that's my scat-detection dog.\"\n\nAs it turned out, he really needed an air-scenting dog, not a ground tracker like a hunting hound, and over time Wasser met and got involved with narcotics-detection-dog programs, including at McNeil Island Corrections Center in Puget Sound, to observe their dogs and training methods. \"The dogs were un-be-LIEV-able,\" he swoons. \"The prisoners would take off their clothes and go to the laundry and the dogs would just smell everyone and go yup\"\u2014he snaps brightly\u2014\"that guy's got drugs\"\u2014snap\u2014\"that guy's got drugs\"\u2014snap\u2014\"and that guy's got drugs.\" Wasser was convinced.\n\nHe began training dogs. His lab's first study was searching for grizzly bear in Washington State, in a place called Goat Peak. They never saw the grizzly\u2014but the dog told them one was there. Then he performed a large study of grizzly bears in Alberta, Canada. Wasser regularly experienced the whiz-bang spectacle of his dog's nose: \"I'm sitting there,\" he remembers, \"and the dog sticks his nose in this hole\u2014I reach down in there and here's this grizzly bear poop.\" Another time, they encountered a raging river, \"and the dog jumps over the river and hits\"\u2014alerts on\u2014a \"tiny poop\" in it.\n\nWasser's projects grew in size. Before long, his dogs were surveying over twenty-five hundred square kilometers in Alberta, Canada. Even in the deep snow in winter, they found thousands of scat samples from caribou, moose, and wolves, which let Wasser determine that oil-exploration activity and roads were changing the behavior of the animals. The closer the herds were to the crew activity, the higher their stress levels. He was able to tell, from scat alone, what each species was eating (wolves: mostly deer) and their seasonal variations in nutrition and stress levels. Human activity turned out to have the biggest effect on caribou populations, not the wolves that so many accused of consuming them.\n\nSince then, Conservation Canines has trained dogs to find the scat of jaguars, tigers, and wolverines; Townsend's big-eared bat, the Sierra Nevada red fox, and the Pacific pocket mouse. Nor are the dogs mammal-centric, crossing the animal class borders to find sea turtle nests after the BP oil spill, determine the population of the northern spotted owl in northern California forest, and count the numbers of endangered Jemez Mountains salamanders, which live in dead logs and come out only one month a year during monsoons. Dogs may be trained on up to twenty species: once trained on one, adding another species comes easily. And yet they have little trouble distinguishing scat from different animals, even from closely related species, and will ignore the universe of nontarget scat around them. To alert the handler that they have found the scat, they simply sit. Not pick it up in the mouth or roll in it. Sit.\n\nUnlike many detection programs, Wasser's dogs are mostly young mixed-breeds from shelters\u2014because that's where dogs with excessive energy and borderline-obsessive personalities wind up. A dog with what he calls, gently, \"fixation with the ball,\" a strong play drive, and high energy is that classically motivated dog that all programs love. \"These are the kinds of dogs people think, Oh my God, you'll never be able to control this dog,\" he says. But \"they never run away\u2014because you've got their ball.\"\n\nNext to Wasser's computer is a calendar featuring a number of the Conservation Canines. May 2015's dog is Tucker, who has perhaps the most unlikely and spectacular job of them all. In the calendar he sits patiently on a boat with a yellow life preserver around his neck. What this sweet-faced black Labrador retriever mix does is detect the slimy scat of orcas\u2014killer whales\u2014who live in Puget Sound.\n\nThe resident population of orcas in the area has declined precipitously, and Wasser thought that information about their diet, hormone levels, and possible toxins could help identify why this is happening. That's in their scat\u2014but orca scat, though smelly (think: fish) and sometimes even yellow or orange (though usually brown or green), is buoyant for only a short time and then sinks. It is not easy to find in the great wide sound. Not for Tucker. \"Tucker is amazing,\" Wasser says. \"He smells a sample over a nautical mile away and is able to track it over a fast current.\" Wasser is talking quickly and then stops for that fact to sink in. Though orcas are large, their scat is not highly visible. To find it before it sinks, the team has to be speedy: in choppy water, it sinks nearly at once; even in calm seas they've got only thirty minutes to find it.\n\nThe researchers take Tucker out on the boat and head in the direction that orcas have been seen, driving downwind of the animals with the wind at the boat's side. That way, they are potentially maneuvering into the scent cone (the V of the odor dispersing from the scat), perpendicular to the wind. \"Tucker is asleep on the bow of the boat. But you can see his nostril going\"\u2014Wasser flares his own right nostril, which comes off like a wry grimace\u2014\"and then all of a sudden as soon as you hit it\"\u2014snap\u2014\"he is up.\"\n\n\"This is the most convincing of all the work that we do,\" he says. \"There's no landmarks; the current is pushing these samples . . .\" But when they enter the scent cone, Tucker stands up over the bow, his nose down, and essentially points with his nose toward the source. If the boat goes too far by it, he runs to the side of the boat. His nostrils go up and down, side to side\u2014wet-nose rudders directing his handler how to steer the boat to reach the source.\n\nWasser smiles remembering how hard it was for them to see at first what Tucker was doing. In the first year of the project they never found scat. But it turned out not to be because Tucker wasn't spotting it, \"but because we couldn't believe he had it that far [away]\"\u2014and so they would stop the boat and turn back. When they began baiting pie pans with samples, letting one boat follow the pan visually and the dog boat try to find it, they realized that even a mile away, Tucker was actually leading them to the pan.\n\nAt the scat, the researchers gently scoop up the sample with a little net at the end of a telescoping pole. Meanwhile, the handler gives Tucker his reward: his well-loved tennis ball. Eventually, Wasser gets his tennis ball: a well-founded result. Given the hormonal and nutritional levels that varied seasonally, he concluded that a reduction in the animals' primary prey, Chinook salmon, seemed to be driving the population's decline.\n\nWhile one could, in theory, put the orca scat through the gas chromatograph to figure out what the volatile elements are that Tucker might be detecting, for Wasser, the evidence that he can find it at all is sufficient. Still, he acknowledges that there is some ambiguity about what part of the smell the dogs actually believe they should detect. For instance, he discovered that dogs trained on the sound's resident killer whale population\u2014what Wasser called the \"fish eating dogs\"\u2014were not alerting to the \"transient\" orcas\u2014the ones just passing through. The transients are mammal-eaters. \"We had trained him on killer whale [smell] plus fish,\" Wasser realized. Without the fish smell, the dog didn't interpret the smell as \"orca.\"\n\nBut the dogs are ingenuous: they are just trying to get it right, based on what their handlers reward them for. \"The handler is the hard part\" of training the dogs, Wasser says. \"Way harder than the dog.\" When training a dog, there comes a point when the dog-handler team moves from working with samples in a controlled setting to beginning to search in the wild. But scat in the wild will never smell just the same as the frozen, thawed, and refrozen sample from early training: it may be newer or older; the bacteria on it is different; it is from a new animal. The dog has to see what is the same between both samples, and begin alerting to that. Sometimes, though, if the team does not find a sample quickly, \"You've got this highly motivated dog that wants its ball more than anything and an extremely anxious handler\u2014because the handler loves the work and is thinking Oh my God, I'm a failure. And now the dog stops and checks something out and goes, Will this work?\" Wasser looks at me from the side of his eyes, eyebrows up\u2014the questioning, hopeful dog. \"Is this what you're after? And then the handler goes, Oh, well, let me see, and the dog goes, Oh, I guess I'm right, and he sits down, and the handler goes, Oh, he's right and throws the ball. Next thing you know you've layered the wrong species onto the dog.\"\n\nEven with that nose, a detection dog will look to his handler to confirm he's correct. The handler has to trust the dog enough to let him finish his job. There is not just dog anatomy involved in detection; there is human psychology.\n\nTo Wasser, there is no downside to working with dogs. Sure, it might be hard to move them around sometimes\u2014they can't be stashed under the seat in front of you on the plane\u2014and they are individuals, with their own temperaments and moods. But a piece of equipment\u2014an electronic nose\u2014would never replace the dog. \"The one thing about a dog is, they improve with time,\" he says. And, I might add, if you're spending the day finding animal scat, there must be nothing like having an energetic, furred, devoted partner in crime.\n\n\u2022 \u2022 \u2022\n\nLyall Watson writes of a hound tracking a missing man by following a week-old trail through a bank and a grocery store, crossing traffic, and ending at the bus station. Indeed, at the very bench where the man was later determined to have briefly rested before boarding the bus. When I hear of stories like that, or of Tucker's orca-tracking, and remember, by contrast, my immediate and complete bafflement as to where my dog, Pumpernickel, had got to on that day she left the house, I think that dogs and humans have nothing in common. It seems a dog can find another animal, but it is very hard for a human animal to find a dog. But maybe I wasn't hanging out with the right human animals.\n\n### \\- PERSON -\n\nMembers of the Kanum-Irebe tribe in New Guinea, as described by anthropologists, have a parting ritual that may be alien to most Westerners. When two friends are saying good-bye, one reaches into the armpit of the other, \"sniffs at his hand, and rubs the smell on himself,\" therewith eliminating the disturbing possibility that \"he cannot stand his smell.\"\n\nWe do use each other's smell\u2014if usually more unwittingly. At some level, we know that our own smell reflects what we eat; that we smell our age; that we smell like our smoking, drinking, and swimming habits: and so do others. Our odor reflects our mood, our health, our occupation, our medication.\n\nRarely do we overtly attend to or seek out these smells of others, though. But, like the Kanum-Irebe, we may actually collect others' smells\u2014if subliminally. The now notorious research result that the menstrual cycles of women who live together spontaneously synchronize shows, in essence, a detection of others' odors, with a specific physiological result. And recently, psychologists covertly videotaped hundreds of subjects' behavior right before they took part in a (mock) experiment. The actual experiment happened in this pre-experimental time: greeted with a handshake from the researcher, people actually appeared to sniff their shaken hand shortly afterward. With same-sex researchers in particular, the subjects brought their hands near or to their noses after shaking hands, and sniffed. It was as though they were sampling the smell: what the researchers called \"chemo-investigation of conspecifics.\" Smelling you.\n\nCertainly most of this chemo-investigation is unconscious. Can people intentionally track other people or other animals by smell? I resolved to find out.\n\nThe porcupine is a messy animal. Compact, his body covered with tens of thousands of hairs modified into barbed quills\u2014which, famously, a feisty porcupine can release and drive into the flesh of anyone foolish enough to come near\u2014he is well armored. Before he does that, though, the porcupine will clack his teeth at a potential predator, and release a penetrating odor from glands on his back. He would rather not get into a scuffle.\n\nIt is, perhaps, the many layers of defense that the porcupine, Erethizon dorsatum,V has, that allow him to be so, so slovenly. The porcupine pees where he pleases, often while walking along, not bothering to slow or stop. The footprints of a porcupine may be peppered with bits of dirt or defecation, as he sleeps in a den piled with his own excreta. Unsurprisingly, he does not groom himself. When animals groom, they are doing so less out of a sense of the importance of cleanliness than out of a fear of predators scenting them or pests injuring them, or out of an interest in mating with other like-minded animals. The porcupine handles predators with aplomb, seems to have a hearty immune system, and has figured out the mating game even with all those quills. So he gets to be unkempt with impunity.\n\nHis grubby manner is how it came to pass that I could tell a porcupine was near us in the forest, one cold January day. Reader, I caught a waft of porcupine urine.\n\n\u2022 \u2022 \u2022\n\nThe day had begun eight hours before, under three layers of night that gradually yielded to day along my drive from New York City to western Massachusetts. I had risen early to go animal tracking. If there was anyone who could rival the scat-detection dogs, it was scat-detecting animal trackers. I was headed two highways and a string of desultory industrial towns away from where I awoke, where I planned to meet Charley Eiseman and Noah Charney, naturalists with an ecological, philosophical, and purely hedonic interest in finding the tracks and sign (read: often scat) of animals.\n\nIn the dark of the city, I kissed the top of my son's headVI and headed outside to my car. I thought to sniff the city's nighttime air, knowing that I would soon have a contrast in the bright, cold western Massachusetts climate. I had to work my nostrils a dozen times to notice anything at all. The city smelled ashy\u2014dusty, maybe. It smelled like . . . absence.\n\n\u2022 \u2022 \u2022\n\nIn the excitement over dog-tracking ability, one might forget that humans have a long and storied history as trackers themselves. Certainly, as an omnivorous species, and before being an armed species, we humans found and trapped animals for food simply by knowing their habits: where they lived, what they ate, when to find them out and about and vulnerable. These days, apart from sport hunters, people rarely track animals to find dinner; instead, the extant \"animal trackers\" are hunting animals largely to photograph them, to survey the population, or to feed their own inquisitive spirit. In this way, contemporary animal tracking is a curiosity rather than a necessity. So, too, for the kind of \"tracking\" a contemporary American pet dog might do: for the great majority of twenty-first-century dogs, following their nose is less about finding a mate or dinner or minding territories than it is simply noticing what's around them.\n\nThe phrase animal tracking gives the impression that the end result is finding an animal, but this is not so. The most likely outcome is that one will not see the animal at all. Nearly every wild animal will notice a human wandering in its midst well before the human notices the animal in his. Instead, tracking is about discovering the various indications that the animal has been by. The classic sign is an animal's tracks, or footprints. Since we were headed out into a mostly untrammeled forest, where snow had fallen a few days before, there would be clear, discernible animal footprinting for us on this day. Mud is also a fine substance for catching the footfall of the passing coyote, turkey, or moose. Most terrain, though, does not capture the forest's populations' light footsteps (such that humans can perceive it), and most weather\u2014rain, continuous snow, wind\u2014is well designed to wash any traces away. A tracker often doesn't have, and doesn't need, actual tracks to track an animal. Any forest is, down to each hillock, to every tree and shrub, wearing evidence of the animals that live there. All of our senses, and all sorts of evidence, are used to trace the passage of an animal\u2014including their smell. Deer smell. Squirrels smell. Bears, bobcats, foxes, and moose smell.VII\n\nWhat might a moose smell like, you ask?\n\n\"Oak-y,\" says Noah Charney, as, nose in the air, he cuts a corner on our path and wanders among a grove of trees. \"I thought I smelled a moose.\" He pauses. \"But it [both the moose and its smell] is gone now.\" Watching Charney nose-hunting for moose is like watching an art appraiser eyeballing the ostensive masterpiece placed before him: while one can see him looking, one has no idea what he is looking at.\n\nWhat Charney, and every animal tracker, does is learn to embody the animal he is tracking. To understand where the animal might be, one has to think like that animal\u2014to imagine its life. The skill of tracking has been compared to learning to read: only one reads not the grade school primer, but the forest. As a book's hieroglyphics morph into legible prose to the young reader, a wilderness begins to look the way it might to the animals who live in it. What bush would thoroughly conceal you, if you were a rabbit? What manner of tree presents the perfect height and visibility for a bear to announce her presence by rubbing and clawing along it? Which secret hollow appeals to a flying squirrel? What time of day might the nearsighted possum feel safe foraging with her pink grubs of pups? Once one sees the world like the animal, becoming aware of oneself in that space, just as the animal is in that space, one finds traces of the animal.\n\nOf the animal tracker's bag of tricks, it is the tracking of smells that I am after, of course: the odorous sign of an animal's passage. Heading up north in my sealed-off car, I have only a vague idea exactly what those signs might be. When I nuzzle my nose in the scruff of Finnegan's neck, I surely recognize his smell. I have learned that when guests come to our door, they note the distinct odor of \"dog\"; we, who inhabit the house, have become so accustomed to the fog of dog that we cannot smell what they smell.VIII Surely Finn wears, or emits, other odors, but even in our close companionship I have never thought to loiter on them. This is about to change.\n\n\u2022 \u2022 \u2022\n\nIt is still the early hours of dawn when I pull up to the tracking-school classroom building. It is deeply cold: eight degrees Fahrenheit. The deepness of the chill is heightened by the stillness of the morning. Cold, calm winter days are good for ground tracking, the animal-tracking manuals say, presumably because, apart from the recent animal smells, all the other volatile odors of the world\u2014from tree, plant, and earth\u2014are quietly asleep beneath the cover of snow. The warm touch of an animal is a beacon of odor in a cold scene; this warmth even volatilizes whatever it touches, creating bubbles of odor in a barren landscape.\n\nI want to properly prepare my nose. Admittedly, this preparation is pretty straightforward: I blow. Make room, remnant city and car smells: I need to smell coyote! For good measure, I've brought along a nasal steroid recently prescribed to clear some congestion plugging my ears. It can't hurt, I figure, to be a nose on steroids. \"If I wanted to improve my [scent] detection,\" Stuart Firestein had admitted to me, \"I would use steroids.\" He was not making a doctor's recommendation, just an observation based on his own past experience. Due to an oral surgery, he needed a nasal steroid temporarily. \"I smelled things I had never smelled before,\" he waxed rhapsodically.\n\nThe spray is like a sudden injection of a lily into my nose, petals, pistils, pollen, and all. As I inhale, I can feel my nasal passages opening, stretching their arms and widening their eyes to the daylight. If there's any time that my nose might be a super-nose, it is now.\n\nInside, I find Charney and Eiseman in a small, busy classroom; Charney lifts his eyes in the briefest of greetings. They both share the same unruffled manner. Both are dressed comfortably in muted layers. Two packs are laid out across the chairs, spilling supplies. Their equipment has the specificness of those who are prepared to be outside for long periods, but who do not like to be over-prepared.\n\nShortly, a half dozen tracking students stumble in, all heavily dressed for the day's expedition. \"The bulk of our time will be spent in the field visiting local habitats,\" the course description had warned, telling students to \"be prepared for long days in cold, wet, and strenuous conditions.\" Hand warmers are in evidence.\n\nCharney approaches and wordlessly hands me something. It is an eighty-page tan Steno Notes notebook, its pages plumped with wear and water. His tracking notebook: the repository of notes and ephemera from past expeditions. \"Noah's scratch-and-sniff book,\" Eiseman calls it, because it contains physical leavings and evidence in the form of fur, leaves, sticks\u2014and in the form of tissue paper sodden with urine.\n\nI handle it gently, as though it were an ancient scroll. On one page, a long, tassel-like hair is taped on the page. \"Moose\" is scribbled by it. On another, \"Fox, 12\/27\/01.\" Some pages hold a series of twigs with incisor marks, or clumps of fur that seem not to have fallen from their bearers naturally. There is sign of rabbit, caribou, bobcat, possum, bison. \"3\/03 Beaver\" holds a patch of beaver hair; under it, \"Coyote,\" and a tissue once moistened with, presumably, coyote urine. I wonder whether that coyote and that beaver met on that day; I sniff the page. Riffling the pages through to another tissue, I sniff that one, too. Apart from the surprisingly clear \"Steno notebook\" smell, I can detect that there are some kinds of smells, but they betoken nothing.\n\nJammed between two of the final pages is a sample with High Security, for the field: a crumpled tear of aluminum foil is wrapped around a ziplock bag\u2014itself securing a folded tissue. A note on the bag reads, \"Red fox, 1\/9\/03.\" Charney carefully extracts the sample, which has not seen fresh air in twelve years and four days. He barely touches it to his nose, with the gentleness of a man sniffing a woman's handkerchief. \"Oh!\" He has a physical response to the odor, pulling his head back and striding across the room away from it. I take a turn with the tissue. My nose catches something sweet, animaly. Eiseman leans toward it. \"Skunky!\" he says at once, smiling.\n\nCharney nods. The simple syntax of animal-urine odors seems to be: name of animal evoked (skunk) or food source (oak), plus punctuation to reflect the intensity of the odor. Skunky! works for both skunk and red fox. Musty, for a bear. To both Charney and Eiseman, this odor is exclamation-point loud\u2014and patently from the elderly bladder of a red fox.\n\nEyeing the tissue, I feel a bit like a color-blind person in a room of rainbows: I sense there is something to be seen, but my searching eyes can't find it. Charney hands the tissue to the students, hesitantly poised at the table in full winter garb. Each willingly brings it to her nose and sniffs it. Last week, before the class, such an offer would certainly not have been accepted by any of them\u2014or, I daresay, by many outside of this class.\n\nCharney is the exception. \"I've always smelled everything,\" he tells me, \"for as long as I remember.\" He tells a story of living temporarily in a wigwam in the woods while he took classes by day. Returning to the forest in the depths of night, he had to navigate by smell. \"One time I did wake up and I didn't know where I was,\" he admits. Usually, though, he found his way by following his nose. Remembering my lack of familiarity with the smells of my own block after smell-walking through Brooklyn, I comment that this may be atypical. He shrugs it off. \"I'm always surprised [at the things] that people can't smell.\" He laughs, a bit vexed. \"I'm like, Why not?\"\n\nThat Charney does not see his exceptionalism is obvious in his description of meeting his wife for the first time. What he remembers, he says, is that \"when we first started dating, she smelled like Coca-Cola.\" He recalls his childhood as redolent of his toys: \"The plastic tires on those little plastic toys? They smell this really strong vinyl-y kind of smell . . . And the whole reason to play with Play-Doh is the smell of Play-Doh.\" As with perfumers and wine experts, when he describes the vinyl smell of toy car tires or his wife's cola ambience, he is not just remembering the smell abstractly: he is smelling it in his head.\n\nThis ability to invoke odors, to smell-daydream, as it were, is a clear marker of those who know how to use their noses, and those who do not. It marks attention, the brain's willingness to loiter on an experience, which is later conjured up. Some people experience smells in dreams\u2014though even normal smellers seem to have their dreams affected emotionally by odors in their environment.\n\nThough odors have emotional content, for Charney, they are simply a part of the scene, and a customary note of his daily experience. He has odor preferences, as we all do; where he differs is in his approach to the vast expanse of odors that are not naturally pleasing. \"There are smells I hate. But I still like to smell them\u2014because it's like, Ooh, that's horrible.\" He grins. What kind of things, I wonder. \"You know, vomity kinds of things,\" he replies.\n\nOh.\n\n\u2022 \u2022 \u2022\n\nEveryone piles into a van and we head farther from highway and habitation. Our destination is the northern end of the Quabbin Reservoir, a water source for Boston surrounded by state forest. Charney drives and Eiseman plays bird songs on the radio. Apart from being in the outdoors, this is clearly their element, listening and smiling and nodding in recognition of whoever sings tuh-wheee!\n\nAfter an hour we park and tumble out of the van. The forest at the road's edge is a wall of undergrowth and fallen trees and branches; there is no obvious path. Charney simply pushes a branch aside and walks in. After a beat, we all follow him. The woods immediately swallow us. There are occasional animal \"runs\" that serve as makeshift paths, but we mostly bushwhack through tall, tight reeds and bushes, forging serpentine routes between low branches of close-growing hemlock and white pine.\n\nA dozen minutes in, the only signs I can detect of where we might be are our own footprints in the snow and the ascending late-morning sun from the southeast. I am engrossed in the activity of minding where my foot is going to step, and notice little else about our surroundings. But the trackers have their heads up: while I walk, they investigate.\n\nCharney stops. \"Do you see this? Have a look.\" And he walks off, leaving us to examine the tree he's gestured toward. On first examination, it is, unquestionably, a tree. Nothing remarkable about that. Then our eyes shift. It is damaged\u2014there is a tear, or some holes. Peering closely there is a single hair\u2014a hair!\u2014sticking horizontally out of the bark. It feels like a miracle to see it. Charney returns. \"Did you find the hair?\" It had popped out at him, bogglingly, while he was striding by. He then diagnoses the scene: this is a tupelo tree, in a bit of clearing, on a small hillock over what are now-frozen wetlands. Nearby are a handful of larger tupelos.\n\nThat is what the tracker sees first. They see that setting as a great fire hydrant on which a female bear might leave her mark. The trunk damage: bite marks and scratches from her broad claws\u2014both ways to leave her scent behind as a warning to competitors. The hair? From rubbing her back against the tree's trunk. Those larger tupelos could be \"nursing trees\"\u2014up which a mother bear can safely stash her cubs while she forages for the tree's coveted berries.\n\nI peek around me carefully. No bears in sight. Still, the air is charged with the scene evoked by the traces left behind.\n\nAs we carry on, we slowly but surely collect animal sign: the track of a flying squirrel, which begins suddenly (its landing site) and quickens to the safety of a tunnel at the base of a nearby tree; cottontail rabbit; deer so frequent they are unremarked upon. Sign can be claw marks or shed hairs; traveling paths forged through close undergrowth; depressions in grass or fur where an animal lay and nested. And excretions: droppings, secretions, other often mephitic exudations. \"There is a whole novel in an owl's pellet,\" a tracking guide book rhapsodizes. The book is chock-filled with full-color photographs of scat and mucuslike blobs, centered respectfully in their frames. The novels written by excreta are portraits of the excreter: his species and sex, health and diet, preoccupations and companions, and daily routine. A coyote's scat is filled with fur or hair, contrasting it with dogs', rife with the grains that make up packaged dry dog food. A pile of lake muck, topped with a yellow-orange smudge, is sign of a beaver feeling territorial.\n\nThe human tracker may constitutionally be directed by his eyes, but his nose brings more information still. An experienced tracker like Charney can air-scent, if the air is sufficiently humid and the wind is at his face. But to do as the animals do, it is also useful to get at animal-height: the \"stick your nose right in it\" method. The general instruction to \"get down on your knees, put your nose as close as possible . . .\" (as one tracking bible says) is anathema to most people. But Charney regularly drops on all fours, his face a half inch from a moss-covered stump or tree trunk. He races ahead into a clearing; when we catch up with him, he is getting up from sniffing a wide stump. Everyone takes a turn on their hands and knees, nose up to the broad vertical shelf of the trunk. \"Remember to blow some warm air on it so the smells come up,\" he advises. Whatever odor is on that trunk, it needs the warmth from our lungs to make it volatile, to evaporate into the air and be captured by a sniff. Similarly, the lightest of rains can \"breathe new life\" into otherwise quiescent odorants on the ground. I warm the trunk with a breath, close my eyes, and sniff. I smell a mustiness, reminiscent of basement. Eiseman is smiling: he already knows what animal has been here, just from the context. He confirms it with his nose. \"Cats in a basement,\" he corrects me. Here, bobcat. What I recognize as basement is actually the smell that my past cats have left there.\n\nThere are no footsteps, no other obvious sign of an animal being by. So how did Charney know to sniff just here, then? \"You have to think about whatever is prominent for the animal. I just look around for all the places that might be interesting to mark,\" if you're an animal that urine-marks\u2014a single item sticking out in an otherwise undifferentiated scene, say. And \"context-wise, this area\"\u2014with a stump alone in a small clearing\u2014\"looks good\" to attract a marking feline or canid. And then you smell.\n\nBobcats are so specific in their marking habits that animal trackers can be likewise specific in where to look for their marks. \"Urine deposits are usually eight to nineteen inches above the ground,\" one book instructs. \"The most common scent post is a short, decaying stump, usually not more than six inches in diameter and four and a half inches high. If the stump is leaning, [bobcats] usually deposit urine on the underside where it is better shielded from the elements.\" We had aimed about fourteen inches above the ground, on a decaying stump with a lean.\n\nIndeed, a good amount of smell-tracking lies in using vision, too. That's fair. Dogs, certainly, use their vision. Their being olfactory creatures does not preclude their looking: they don't find another dog's rump only by air-scenting. They may see the body first and follow it to its derriere: then they sniff. The whole point of scent-marking on a fireplug is that it is easy for the next sniffer to locate. The sight of a stump in a forest clearing presumably does for the forest-dwelling animal what the sight of a fireplug does to an urban dog. And I, at the other end of that dog's leash, see it, too, and know where he will lead me. That behatted, upright fireplug planted right along a boring flat stretch of sidewalk: that's a fine place to put a mark. Footprints\u2014another visual clue\u2014might lead one to a likely smell site. And so it is with animal tracking.\n\nOn top of the contextual clues, we could eliminate the possibility of the marker being a canid even before putting nose to stump. If it had been one of the local wild canids, the red fox, \"we'd be able to smell it from here,\" Charney says, stepping back eight paces. \"A red fox really assaults your nose.\" Trackers tend to call red fox urine \"skunklike,\" much stronger than domestic dog, coyote, or gray fox urine. Unlike the relatively modest\u2014if not exactly pleasant\u2014ammonia of the bobcat urine.\n\nWe eat lunch on the frozen wetlands, glad for the cloudless sky. I turn my face to the sun and curl my fingers into fists after quickly downing my thermos of tea and sandwiches. For a while, everything will smell to me like peanut butter.\n\nWhen we continue on, conversation has dimmed, but the sound of our footsteps obliterates any other sound. We are not quiet animal trackers. At a broad trail I hang back from the group to listen to the forest. The day has warmed under the steady sun, prompting the crystalline sound of tree ice showering onto lower branches. I take sniff-samples of the air; I sidle up to trees and surreptitiously give a sniff. I feel the cold in my nose and can almost taste the clarity of the country air.\n\nAhead of me, footprints form a track. To my naive eyes, it appears to be bisected by one or more indistinct sets of tracks running across it. I follow the track down off the path and into an area of low brush. There, near a little spring of a hemlock, is a dribbling trail of light yellow. I stop.\n\nNow, I've seen a lot of pee. I live with dogs, after all: which is to say, I spend some time every day overseeing my dogs' urination. Indeed, one of the strangest things about being an urban dog owner is that we owners have great insight and attention to the evacuations of our charges. We know their methods (leg-lift, squat, squat-and-scoot, sniff-and-circle), their preferred locations (in the curb, on the hydrant, with grass or leaf litter underfoot), and even the expected quantity of output. I'd hazard that, accidentally, we are familiar with the smells of our dogs' urine, just as parents become intimately acquainted with the sweet odor of their own newborns' blissful defecations and micturitions, simply by constant exposure.\n\nSo I barely hesitate before kneeling in the snow, crouching low, and sticking my nose ever closer to the yellowed snow. And I smell . . . porcupine.\n\nPorcupine pee smells\u2014there can be no question\u2014piney. In the winter, the animals subsist mostly on the bark, leaves, and buds of pine trees. Scattered hemlock branches on the ground, incised at 45 degrees to vertical, are evidence of the foraging of this particular porcupine. They climb trees, gripping with their impressive claws, and make their way out to the ends of branches, holding the newest of the new growth. This arboreal habit, while providing them with ample calories, is also sometimes responsible for their ends: a porcupine who falls off a tree lands on many sharp, impaling quills\u2014his own.\n\nI stand up, beaming, and glance at Eiseman. \"That might be my favorite animal-pee smell.\" He smiles. Well, if you're starting with the \"smell of urine,\" I have to agree: it isn't too bad. Woodsy and strong, but not overpowering; I've smelled much worse in NYC taxi air fresheners. We follow the footprints leading away from my find: the back feet come down slightly ahead of where the front feet lay\u2014an \"indirect register.\" The toes point out, befitting a waddling, short-legged, chubby-bodied animal. Aside the toe holes are lightly rendered lines drawn by the quills grazing the snow; within them are small chunks of dirt and forest debris. The profligate peeing, the messy trail, the distinct shuffle: porcupine. \"Porcupines and red foxes are the animals that I will often smell a hundred feet away,\" Eiseman adds. \"I only sometimes see the animal.\"\n\nDo their tracks themselves smell? \"I'm sure that to a fisher\"\u2014their most common and successful predator\u2014\"they smell.\" To a porcupine, too, perhaps. Nosing a four-toed footprint,IX I confirm that I have neither porcupine nor fisher genes.\n\nWe follow the trail to its end\u2014a den with remnants of a stone wall on three sides\u2014and peer in hopefully. This might have been an icehouse a century ago, before the state flooded the four towns in the area and created the nearby reservoir. Instead of porcupine or porcupette, there is a towering pile of pellets, many feet high, in either corner: insulation for a cold winter. As the wind turns, the fecal smell floods the air. If the quills aren't deterrent enough, this could prevent other animals from trying to steal the den.\n\n\u2022 \u2022 \u2022\n\nMidafternoon. We stop at a frozen beach, taking in some more of the sun's warmth and gazing out at the long expanse of iced lake at our toes. Twenty feet out on the ice, the trackers are looking down.\n\n\"Hey, Alexandra, there's coyote pee!\"\n\nRarely has this utterance inspired excitement, anticipation, or thrill. Or, I'd guess, a quick jog toward the informant\u2014as I set to do. As I approach, I can see that Charney's nose is inclined toward a small twig frozen into the ground, protruding only a few inches. A single dried leaf shivers from its top, and an expanse of moss is wrapped around it. At a few inches tall, it isn't much to look at\u2014but it is the tallest thing around. Looking out to the ice and around my feet on the beach, numerous trails clearly converge on this local landmark. It's perfect for a coyote, who prefers to mark highly undistinguished items. The coyote passes by the towering oak without breaking his stride; he snubs the ancient hemlock, the prehistoric rock outcropping. No, it's \"small bushes, tree trunks, rocks, blocks of ice, mounds of snow\" that shine brightly in the coyote's eyes. This twig is just right: though only noticeable to the human when she trips over it, it is nonetheless a single bump in a sea of flatness. The tracks are about a foot from the twig, indicating that this was probably a male, who could reach the twig with a spray under a lifted leg, rather than a squatting female.\n\nI crouch and hover over the stick. There is a splash of yellow on the snow, a more profligate leaving than the single droplets doled out in the forest. I sniff. Smoky, earthy. Incredibly strong.\n\n\"Doesn't have the skunkiness of a fox,\" Charney offers. \"Pure ammonia,\" Eiseman says. \"A lot milder than fox.\" Charney leans in and takes another whiff\u2014\"Whoa!\"\u2014and he keels over, flat on his back, in mock (partially real) horror. \"There might be a domestic dog involved here.\" Meaning, this smells like an animal that is eating a different sort of diet (one coming out of bags and off of kitchen counters).\n\nIndeed, though the large, loping footprints converging on the stick look like coyote, some tracks are much more disorderly than the others. The dog's gait is, in fact, considered \"sloppy.\" The dog indirect registers\u2014leaving twice the footprints. His nails make distinct nail-marks, spreading out from the pad, unlike the tucked-in pithiness of the coyote print. The dog's course is meandering. \"You just don't see wild animals making some of these tracks,\" Eiseman explains, \"because they're trying to conserve energy.\" Coyotes' paths are clean, straight vectors from A to B; they \"direct register,\" back feet stepping in the front feet's track. They cannot afford to explore every new thing, every which way. This wandering track is from someone that's getting fed.\n\nWe linger a bit at this perfect scene. Not an animal in sight, but sign of animal all around.\n\nThe sun's quickening descent tells us that it's time to find our way out of the forest. Though we have wandered for hours, over many miles, my guides seem to know just where we are and beeline for the van. \"Every forest has a smell,\" Charney suggests as we walk. \"This one smells like the leaf litter when there is no snow,\" says Eiseman, \"of the blueberry and gorseberry undergrowth.\" If they were dropped into this scene, blindfolded, with no wind on their faces or sound in their ears to go on, they could still identify where they were.\n\nEyes open, ears on alert, nose exercised, I try to take it in. Back in the city, hours later, I can distinctly smell my urban forest. In the first half of the twentieth century, the MTA\u2014the Metropolitan Transportation Authority, responsible for running the city's subway system\u2014employed James Patrick \"Smelly\" Kelly to walk the tracks and sniff out any gas or water leaks. His skill was legion. Once, presented with the complaint of a bad odor beneath the subway a block off of Times Square, he arrived at the site to find distinct odor of elephant. Indeed, it turned out that the Hippodrome, which housed many a traveling circus, had been at Sixth Avenue and Forty-third Street, right above the station. Apparently the circus elephant dung had been left behind, and its smell was reanimated by a nearby water leak.\n\nI sniff for, and do not find, elephant in the city air. Often the city's smells are noticed only when foul, but now they feel neutral, informative. I breathe them in with satisfaction. But I do mourn the lack of a certain pure, icy northern snowflake in my nose.\n\n* * *\n\nI. To the great delight, no doubt, of Dahl-reading youngsters everywhere, the tactic to avoid witches is to \"never have baths\"\u2014thus curbing the stink-wave-dispersal with layers of dirt. Alas, in Dahl's version of the fairy tale Jack and the Beanstalk, Jack only escapes being smelled out by the giant (\"FEE FI FO FUM . . .\") by bathing, emerging \"smelling like a rose.\"\n\nII. If you are, like the author, a person who not only attracts many mosquitoes, but also redirects the mosquitoes initially keen on your nearby friends and family, take these two bits of wisdom from Vosshall's work: stay downwind of the critters, and stay near fans.\n\nIII. One researcher fancifully quantified the smell \"pollution\" generated from one \"standard person\" (skin surface 1.8 meters; takes .7 baths a day), when sitting, to be 1 olf. Someone exercising may produce up to 11 olfs; a smoker, 25 olfs.\n\nIV. This was well represented by the study of reproductive choices of zebra finches: females turned out to prefer the males not with physical extravagance or prowess, but those accidentally assigned red, not black, leg bands for tracking.\n\nV. Which translates roughly as \"animal with the vexing back.\"\n\nVI. smell: sweet, sweaty hay.\n\nVII. In her wonderful volume A Nosegay, Lara Feigel quotes the story told by Edmund Snow Carpenter in Eskimo Realities (1973). He recorded this exchange between an Inuit woman (W) and an anthropologist (A): W: \"Do we smell?\" \/ A: \"Yes.\" \/ W: \"Does the odour offend you?\" \/ A: \"Yes.\" \/ W: \"You smell and it's offensive to us. We wondered if we smelled and if it offended you.\"\n\nVIII. This phenomenon is called habituation: no longer noticing an odor after repeated exposure. While it is similar to adaptation, the latter operates at the level of the receptor cells; habituation is a result of the brain no longer bothering to notice a smell.\n\nIX. Its front foot; a porcupine's back feet have five toes.\n\n## 10: Civet Cats and Wet Dogs\n\nAnd every day, as he came near to it, he would lift his small pointed nose high in the air and sniff the wonderful sweet smell of melting chocolate. Sometimes, he would stand motionless outside the gates for several minutes on end, taking deep swallowing breaths as though he were trying to eat the smell itself.\n\n\u2014Roald Dahl, Charlie and the Chocolate Factory\n\nIt is May 19, and the state of Washington is gray and evergreen. The highway out of Seattle is a course of improbably-tall and improbably-taller trees, until finally, far enough north, they tickle the undersides of the clouds.\n\nI'm traveling, and missing my own dogs. Were I home now, I'd be snapping on leashes and heading for the park, the dogs jostling each other to be the first out the door, out the elevator door, out the building door. As I drive I chuckle remembering a typical Monday morning tour through our local park, during which Finnegan locates the weekend's best picnicking spots and scours the grass for any edible contributions. Today's outing would certainly be of interest to him\u2014and to any dog who tends to arrive unbidden from across the room to snarf the tiniest morsel of cheese left after sandwich-eating.\n\nI'm going hunting for truffles.\n\n### -DOG-\n\nIt seems a shame that dogs aren't employed to work at finding smidgens of cheddar. One might charitably call the yellow Lab who steals bagfuls of bagels from unsuspecting picnickers a \"bagel-detection dog,\" but rarely is one in need of this particular kind of breaded acuity. For a short time when our new cat was still introducing herself to our home hiding-places first, we employed Finnegan as a kitten-detection dog. Now that the kitten sleeps on my face, the dog's services are no longer needed. Alas, there is no employment for hot dog\u2013detection dogs, awesome though most untrained dogs would be at this work.\n\nEnter truffles. Truffle-detection may be the perfect convergence of dogs' desires and our own. For to dogs, truffles smell fantastic; and to humans, whether or not you find truffles fantastic-smelling, they smell like gold. A golf ball\u2013sized P\u00e9rigord black truffle from Italy or France retails, at last check, for over one hundred dollars; enormous 2.5-pound truffles have sold for hundreds of thousands of dollars. Truffles are \"hypogeous fruit bodies of the ascomycetes Tuber spp.\"\u2014that is, an underground mushroom, one that has its spores in its belly. It's unusual among mushrooms in that it forms a symbiotic relationship with the trees to which it attaches itself: the tree gives the truffle a landing spot, around their tendrilly roots, and carbohydrates for growth. But the truffle, extending its own reach, seems to gather nutrients for the tree. Truffles don't grow without trees, and trees grow better when entruffled.\n\nAn industry has formed around unearthing these delicacies. They are often only inches underground\u2014even a centimeter in some cases\u2014but are not easily spotted. So how to find them? You could dig willy-nilly, but given the rarity of truffles, therein lies madness\u2014hours of digging and little to no yield. Some rake the ground, hoping a truffle will be pulled out on a prong, but this reveals more immature, untasty specimens than not, and damages the forest floor to boot.\n\nTruffles have a related problem. As a spore-driven reproductive system, they need to find their way to the light to send their spores far and wide. So the truffle has figured it out. When it matures, it lets out a hair-raising odor. The odor draws animals to pursue and dig up the truffle, consume it, and thereby spread its spores wherever the animal travels and defecates. Scientists have identified some likely culprits for the particular eau de truffle: it includes dimethyl sulfide (think boiled cabbage) and androstenone (a hormone also in our sweat).\n\nSo-called truffle flies love the smell\u2014and you could try to track a scrum of truffle flies as they tumble over each other and dive toward it\u2014but nicer might be a quadrupedal vertebrate. One who likes smelly stuff.\n\nAh, there's a lead: androstenone, which we met before, is a pig and boar pheromone, and whether or not pigs find truffles erotic-smelling, they have no trouble locating the source of the odor, even underground. As a result, pigs have been used to hunt truffles in Europe for centuries\u2014their snouts turned down, already half in the mud and dirt, unwavering in their course. This same smell has an unusual human history: it is one of the rare cases in which an olfactory receptor has been identified (named, unpoetically, OR7D4) for which variations in the gene responsible for the receptor lead to different reported experiences of the smell. Some people find the smell to be like urine or a sweaty sock; others find it pleasant, even flowery. Other people (including yours truly) appear to have a variation in the gene such that they cannot detect it at all.\n\nThose people would not be good truffle pigs. But sometimes truffle pigs are not good truffle pigs either: they find the truffles naturally, but also love to eat them. Points for efficiency: they are discoverer and consumer at once. Enter the truffle dogs. While dogs need to be trained to care about the particular smell of truffles, they can easily find them for their owners or handlers once they do. And they won't eat their find. Not all of the time, anyway.\n\nI pull my rental car off the highway into a nondescript rest stop where we have agreed to meet. I see the trainers first. Their dogs\u2014and they have brought four of them\u2014lie quietly in separate crates in the backs of their cars. Alana McGee hops out of her car and presents me with a nondisclosure form I must sign. She is dressed in a Truffle Dog Company jacket\u2014the company she cofounded\u2014with a logo of a dog digging himself into a hole; her hair is pulled back casually into a single ponytail. From a second car, cofounder Kristin Rosenbach greets me guardedly, an occasional smile transforming her face.\n\nThe nondisclosure form is typical of the mystique that surrounds truffle hunting: see the possible prices for the fungus. In this case, McGee is hunting on private land, where she has been granted special permission to sniff and dig. While the presence of truffles in North America is not widely known, in the areas such as Oregon where truffles have been found, an identified hunting site will attract opportunistic hunters, who ransack the forest using destructive methods.\n\nAfter signing and peeking in at the dogs, we all drive separately to their first location. The roads get narrower until we turn at a former game trail with no identifying signs and park. As she gets her gear together, McGee describes how she got interested in truffles. She is a certified dog trainer, but only began training truffle dogs in earnest after visiting Italy and watching the tartufaro work with i cani da tartufo. Returning to Washington, she realized that she was living in a potentially rich environment for truffles; indeed, Oregon has truffles, and the habitat is very similar one state north. With the help of another truffle dog trainer in Vancouver, Kelly Slocum, she trained her black Lab mix Duff on the scent. Now all she needed was the land to search on.\n\n\"I just started approaching timber companies,\" she remembers. \"Wow\u2014they are not used to getting approached like that\"\u2014by a young woman who wants to nose around the land with a truffle dog. With each, she simply suggested that she and Duff explore some of their land, with the notion that there may be mutual economic benefit should she find a rich vein of truffles. A number of companies were receptive. She explored, following foresters' leads as to where she might find success\u2014but also looked for characteristic features of a forest that make it a likely truffle forest. This includes a wide range of ingredients: the age of the forest, types of trees, the level of disturbance or clearing, the amount of sunlight, the soil consistency. Possible trees include pines, oaks, birches, willows: all can host the fungi that are truffles. In the Northwest, Douglas fir, Scotch pine, and noble fir are likely\u2014and common\u2014hosts. Black truffles \"really like\" ferns, as they keep the ground moist: so she would look for fern undergrowth. Given a plausible forest, one then has to get dirty. \"First look around on the forest floor,\" one truffling field guide recommends. \"Have the local experts\u2014the small animals\"\u2014like squirrels, mice, even coyote or raccoon\u2014\"been creating little digs in the earth?\" All these animals look for truffles as part of their diet. One simply needs to dig gently for a few inches; truffles are often just under the top layer of the forest floor. After some searching, and despite some harvest-killing cold snaps for a few years, McGee pinpointed a handful of good sites. \"I still have keys to several of the blocks. They would pretty much give me free rein as long as I reported back on what we were finding.\"\n\nSo grew her business. Rosenbach, who has a background in human education, found her by googling Washington state truffles after her dog Callie dug up a truffle unbidden while they were walking together in the forest. They soon joined forces.\n\nAt her name, Callie, a Border collie with a becoming half-white-half-black face, perks up. Rosenbach has opened the back of her vehicle, but it is Cash, a shelty, who is going to hunt first. He barks\u2014arr-rup!\u2014lifting himself with each exclamation. He has runway-worthy long hair and broad furry feet.\n\n\"We might find a truffle on the side of road,\" McGee says. \"We work here.\" Since they both hunt truffles and train students at this site, there is always the possibility that a truffle falls out of someone's pocket. If so, her dog will find it.\n\nLolo jumps out of McGee's car. She wears a brown-and-white fur coat, her hair in tight curls, and a fitted jacket to protect her from errant twigs, barbed wire, and tossed bottles one might encounter in the forest. Lolo is a Lagotto Romagnolo, a breed you have likely never seen around the neighborhood. Bred in Italy specifically for truffle hunting, the Lagotto has been slow to catch on in the United States and remains a distinct minority. It's hard to see why: the dog is personable, engaged, sportif. Her fur is captivatingly soft and dense, the teddy bear of imagination, good for childhood bedtime snuggling.\n\nThere is a certain resemblance between dog and owner, as so often happens. While the dog alone has the curly coat and the lolling tongue, Lolo is, like her handler McGee, compact and athletic. Lolo's head is a darker tone than her body hair; it's hard to tell if this is from the holes she winds up plunging her head in, or if it's her natural coloration. Ringlets of fur hang down over her eyes\u2014but she doesn't much need to see. She is on the scent.\n\nTraining a dog is what McGee describes as \"reward-based detective work,\" an exercise in expanding and contracting the challenge to dogs. Once dogs learn the game of searching and rewarding and are imprinted on a scent, the process can begin in earnest. An early training situation might involve setting up a simple hunt inside: a truffle oil\u2013infused cotton ball in a box. Once a dog reaches a level of success or comfort there, the scenario is complicated\u2014the target is hidden up high, or underneath other objects\u2014or the criteria for success are increased\u2014the target odor is weaker; a subtler alert is accepted. From there, one might move outside\u2014expanding the challenge, but reducing the criteria for success\u2014contracting it. All along, the challenge is to teach the dogs what the game is\u2014finding the scent, not \"finding boxes\" or \"barking\"\u2014and teach the handlers to be sensitive to their dogs.\n\nThe forest of towering fir trees smells like growth, fertility, promise. Moss grows everywhere. Pine needles cover the floor; ferns, some still in their fiddleheads, are profligately scattered. The dogs dive in, trailing long, brightly colored nylon leashes. \"Yeah,\" McGee says, speaking to the unasked question about their motivation. \"We have to use games to get them out of a truffle forest.\" She walks quickly after them. Once we have stepped inside the forest, the road is invisible behind us. It is mesmerizingly silent but for our own sounds: our steps crunch small twigs underfoot. A pouch holding a shovel and a spoon from McGee's belt clangs against her leg.\n\nMcGee starts the search at the bottom of an incline, heading generally upward, because odors tend to cascade downhill. To check airflow McGee may bring matches, watching the smoke as they are struck, or a device called a wizard stick, which gives off a visible vapor. Just as indoors at the Working Dog Center, making air visible helps us nose-poor creatures help the nose-rich.\n\nLooking up at the hill ahead and catching no whiff of truffle, I imagine the forest's dark depths as the arboreal haystack into which we will mine for a singularly elusive needle. The search seems impossible: the forest surrounds us, uniform in its ambivalence to our presence. But almost at once my analogy is proven utterly inapt. In the dog's hands, as it were, the needles are rife: we can't but turn without being pricked by one.\n\n\"Stay with me!\" says McGee to Lolo, hurrying a bit. \"Lolo, good girl. You can do it. Where is it? Here? Show me?\" She takes out her red-handled spade and jogs over to Lolo, nosing deep into the fern. Twenty yards away, Cash gives a little sneeze. He is nose-bent to the forest floor\u2014what they call duff\u2014and barks.\n\nWithin minutes, both dogs have found truffles. McGee and Rosenbach quickly dig them out, cache them, and pat the duff back into place. Lolo is already out of sight. McGee keeps up a constant one-sided conversation with her. \"This way, please! Let's go! Thank you . . . come here, love. Can you show me? Good girl.\"\n\nWhat was her alert? I wonder, having seen nothing more than a dog pausing and then taking off into a run again. \"She digs,\" McGee admits. \"I'm trying to work on nose touches\"\u2014pointing the truffle out by touching it with her nose. \"Dogs tend to offer alert behaviors organically\"\u2014behaviors like sitting, lying, even a pause\u2014\"and we tend to go with those. And there are those we teach, like nose touches.\" Rosenbach adds, \"The first alert with Cash: he looked at me\u2014but didn't bark. The look is what organically started with him. I built in the bark.\"\n\nMcGee suddenly runs off to follow her fast-moving charge. \"This way, sweetheart! Did you find something? Good girl.\" McGee reaches her and looks down. \"Oh, spit it out. So,\" she turns back to me, jogging up to them, \"if I am not on top of her, she will self-reward. And with black truffles, she will occasionally eat them. So that's what that was.\" (She has also been known to eat bugs, snails, and the large, thick banana slugs of the forest.)\n\nFor the next thirty minutes we all engage in an exercise in chasing after dogs, McGee ready with pieces of hot dog, chicken, and cheese to ply Lolo with before she self-rewards again. They gather a dozen truffles, leaving behind countless pinhead-sized ones that the dogs nosed\u2014but hardly worth keeping. Both dogs hunt naturally, without request, as if once they see the forest they know the game. The dogs alternate deep sniffing and panting, tongues very red, pulsing, and curling. In counterpart, the handlers keep up a stream of encouragement. Lolo tears through the bramble, running low to the ground, her rear legs moving together and giving a little hop to her step. \"She's what you might call a rustic Lagotto,\" McGee says, watching her. \"She is more compact, heavier, and squatter\" than other lines of the breed. It seems not to affect her ability to maneuver\u2014or her sometimes sophisticated search methods. At one site she \"brackets\" the truffle, going back and forth as though diligently examining each square on a grid until the truffle is found.\n\nEventually the handlers call their dogs off the search: \"Free dog!\" This slows them but does not stop them. \"Lolo, wait for me please! Lo? Lo? Can you come? Good for you!\" Lolo, pausing, comes, then scarfs down the contents of McGee's palm. \"Searching behavior with Lagotto in general tends to be pretty hardwired; it's rewarding itself. It's like taking a herding dog and watching cars go by: it's like chasing cars, basically.\" For the first time, Lolo seems to notice me. She suffers a tickle under her chin. As I lean over I catch a whiff of truffle breath.\n\n\u2022 \u2022 \u2022\n\nSome think the North American black truffle smells of \"pineapple and a little bit of chocolate\"; others simply call it \"absolutely disgusting,\" with a texture \"between moist parmesan cheese and ground almonds.\" They are not \"mushroom\" smelling fruits; instead, truffles are variously described as like smelly socks or hot dogs; cucumber or green apple; molasses or musk; garlic or gasoline. What at first glance looks like a rock covered with dirt reveals itself to be a black truffle, its surface marked with warts. Inside it is peppered white, generally hard but with plenty of give under a thumbnail. If it's ping-pong-sized, you may have found fifty bucks; if it's a grapefruit, you've hit the jackpot.\n\nThe truffles are the fruit, bearing spores, of a fungus: the fungal \"organism\" is actually a network of filaments called mycelium that wraps around tree roots. Given the value people place on the fruits, attempts have been made to agriculturalize the process by planting likely host trees and inoculating the roots with spores. Thus far this method is not sufficiently successful or controlled that Lolo has to worry about her job security. (And even on a tree farm, dogs are needed to root out the stock.)\n\nFor training dogs, McGee and Rosenbach make a solution from grape-seed oil\u2014relatively odorless among oils\u2014infused with the headspace of the various truffles that they find in the region. They don't steep them in the oil\u2014\"because botulism,\" McGee points out. They just let the oil and the truffles hang out in a closed area together, and the fat soaks up the odor. If they are affiliating for a short time, the oil is relatively odorless for humans, but the dogs have no trouble detecting it. On the other hand, McGee warns me against keeping a truffle unsealed in the refrigerator (essentially a closed box) for any length of time. The volatiles are picked up by any fats\u2014cheese, butter\u2014and \"it will truffle everything in there. It can be gross.\"\n\nWe head to another site to let Rosenbach's other dogs have a go. Another fir forest, this one is littered with ferns and salmonberries, proudly poking out like new raspberries. The undergrowth is dense; low dead fir branches catch my hair and backpack. The \"open area\" that we are headed to has sword ferns that climb up to my neck; in another month this will be impassable.\n\nDa Vinci, a Belgian Tervuren, and Lolo are on the hunt. Da Vinci is a calm dog, with the stately bearing of a larger dog. Though he performed a short protest howl when left in the car, his approach is measured. He is the least experienced of the dogs out today, having searched in forests for only one year. Rosenbach keeps an extra close eye on him. \"Trovarlo,\" she calls out\u2014\"find it\" to the tartufaro. Da Vinci apparently speaks Italian, for he heads straight to a tree and noses around, doing nearly a headstand as he reaches under himself. His style is less frenetic than courtly; he kicks up some dirt, snorts, and waits for Rosenbach. \"He's very literal,\" she says, coming to see his find. As I approach, the whole space smells cloyingly strong: truffle air.\n\nWe head in deeper, the two dogs zigzagging through the dappled sunlight. At one point they pass each other, pause to wag and perhaps share information, then continue their respective ways. As we watch the dogs, McGee muses, \"A lot of people have the idea that oh, the dogs just go do it. It's like, (a) they're machines, and (b) you just train them once and that's it\u2014or that I can train a dog and give it to you, and you'll be able to operate it.\" While they are called truffle dogs, the handlers are integral to the dogs' success. Both handlers are highly aware of what their dogs are doing in the forest, and work to subtly help them if they encounter an obstacle. They have to continually read the dogs' behavior, a conversation without words. If a dog who lies down as an alert isn't alerting, but is paused, it might be because there is a bed of nettles under his body; if a dog is digging but not finding the truffle, the handlers might move some dirt around themselves to unearth it. And training, as with all the detection dogs, never ends. If a long search ends with no finds, a good handler may manufacture a success, tossing a truffle nearby, to give the dog the sense of fulfillment. \"My job is to make sure that they are always satisfied\" with the outing, Rosenbach says. \"He should always thinks he's a genius, a rock star.\"\n\nI head Lolo-way. McGee suddenly breaks into a run\u2014\"Good girl, good, good for you\"\u2014and approaches Lolo at once rapidly and stealthily, careful not to startle her. She bends over Lolo with her spade and pail out. Declined downward, Lolo is only a little curly rump, tail an immobile exclamation point, madly digging into the ground. She snurfles, a half-grunt half-sneeze, and kicks mounds of moist, dark earth toward McGee, now kneeling next to her uttering assurances, \"Good girl, want help? You're doing a good job.\" Lolo digs steadily and heartily, keeping her head down, occasionally pausing to push her nose in as far as it can go, snort-inhaling the smell. The curly hairs of her dangling ears are painted with inky brown earth that has never seen sunlight. Then the slightest of pauses\u2014McGee notices, \"Did you . . . ?\"\u2014and Lolo lifts her head out, up to her eyebrows in dirt, her panting tongue curling up at its end. \"Wooooow!\" McGee says, looking into the hole she has dug. She keeps Lolo at arm's distance and calls me over. While Lolo is inhaling her cheese reward, I peer into the hole. The fertile, earthy smell of soil and green plant life is suddenly supplanted by the smell of earth from the other side: piercingly fungal, on the edge of sweet and revolting. There are roots and clusters of displaced moss, and McGee points out a dark dirt clod, about an inch and a half across. It looks terrifically undistinguished. But to Lolo's nose and McGee's eyes, it is clearly a handsome sized black truffle. \"A nice size,\" McGee says admiringly, fingering it and plopping it into her pail. \"Thank you! Good girl! Yay for you!\" Lolo is off running again.\n\nThe day is waning but Rosenbach's third dog, Callie, needs a go. Once the door to her crate is opened, she is out like a rocket. She gets a ball in her mouth, squeaks it twice, and looks at Rosenbach expectantly. They disappear into the trees. McGee slips me a small plastic container, sealed. It contains a brown, lumpen mass and a paper towel. I get into my car before daring to peep inside, then stash my precious cargo in a large ziplock bag. Heading back to the city by thousands of acres of forest, it feels that truffles might be everywhere, under each fir that proudly guards the dark forest. An hour later, the scent in the car is overpowering. I open all my windows and spread a trail down the highway for anyone with the nose to follow it.\n\n### -PERSON-\n\nThey haven't got no noses,\n\nThe fallen sons of Eve;\n\nEven the smell of roses\n\nIs not what they supposes;\n\nBut more than mind discloses\n\nAnd more than men believe.\n\n\u2014G. K. Chesterton, \"The Song of Quoodle\"\n\nEarly in June, I beeline for Central Park, head two miles south, and stop under an American linden tree, Tilia americana. I did so on recommendation of Raymond Matts. Matts is a fragrance designer, maker of a line of niche perfumes. While he concocts odor combinations\u2014scents that are never found in a natural setting\u2014he also celebrates odors that appear naturally. Like that of the blossoms of the linden tree in June.\n\nIts branches are beginning to bend under the weight of uncountable tiny, cream-colored flowers. They grow in clusters, each flower really an explosion of petals and dozens of tiny stamens. As I loiter underneath the tree, the odor is heady: honey-sweet and full. A cloud of perfume greeted me well before I saw the tree, but the smell was so widespread that I could not have found its source without knowing where to look.\n\nTo stand for ten minutes under a linden is to have one's senses undone. It is to swim in odor, to bear the blows of friendly attacks to the face. It is probably something like what it is like to be a perfumer.\n\n\u2022 \u2022 \u2022\n\nIf there is an equivalent for humans of the heady smell of truffles to dogs, it may be perfume. We have been rolling in it, so to speak, for thousands of years. Cuneiform tablets provide evidence that fragrances were added to wines four thousand years ago; ancient Greeks and Romans perfumed their clothes chests and, very specifically, their bodies: \"mint . . . for the arms; palm oil for the face and breasts; marjoram extract for the hair and eyebrows; and essence of ivy for the knees and neck.\"\n\nThe popularity of fragrances has waxed and waned since then, but given the fears, at various times, that bad smells brought disease, perfumes stepped in as a safeguard against malodor. At times perfumery was considered a perfectly reasonable substitute for a cleansing bath. On the other hand, the increase in public spaces in the late seventeenth and early eighteenth centuries\u2014which brought more attention to \"personal space\"\u2014led to a movement against perfumes. As sensory historian William Tullett has noted, a fragrance fundamentally \"extends [one's body] into the space beyond which it should\"\u2014and into the personal space of others. Through an act of Parliament, perfume was deemed false advertising in England: if a woman \"seduced or betrayed into matrimony\" any gentleman by use of scents (or, it should be noted, artificial teeth, false hair, or high heels) the marriage was rendered null and void, and the woman subject to misdemeanor punishment.\n\nLet us acknowledge, here, the strangeness of perfume. A perfume\u2014from the French, fumigate or \"to smoke thoroughly\"\u2014is an odorous chemical combination, usually suspended in alcohol. It is an olfactory intrusion, an odor without a reason, and what is more, it is a kind of \"dishonest\" signal, in biological terms. That is, animals, earth, flowers in bloom all smell\u2014like themselves. We smell the skunky presence of an animal startled or killed, and we expect that it comes from . . . a skunk. Smell indicates presence. By contrast, perfumes are concoctions, odors separated entirely from their origins and mixed together, sometimes hundreds at a time. Instead of smelling the \"rear end of an Asian cat,\" as Luca Turin describes it, when we smell Chanel No. 5, we smell the soft velvet whole, reminiscent to many people of their mothers or grandmothers who wore it habitually. But it includes the greasy rear-end secretion that the animal called civetI uses socially as a fear response or to mark a territory. The nocturnal, low-slung civet, which looks like the unlikely love child of a feral cat, mongoose, and raccoon, was formerly kept captive in small cages and bothered in order that it might produce this secretion; today, synthetic compounds have largely replaced this practice.II\n\nThose who work with fragrance\u2014as perfumers, mixing the odorants, or designers\u2014and those who work with wine\u2014as sommeliers or tasters\u2014are widely considered to be the best smellers around. Their powers of detection, distinction, identification, and simple noticing are used professionally, and inevitably change their way of experiencing the quotidian odors that they share with us novice noses. We smell a rose as a rose, but Coco Chanel claimed to smell \"the hands that picked\" the flower as well.\n\nThere are myriad routes to becoming a perfumer, but if you want to work at a big house, evaluating or making perfumes, there are internal perfumery schools, with a curriculum that involves learning to distinguish blindly between hundreds of raw materials. Ron Winnegrad, the director of the perfumery school at International Flavors & Fragrances (IFF), designed a multiyear curriculum for the five or six students he has at one time\u2014who \"show promise on olfactory and psychological [application] tests.\" Winnegrad tests the students every day on raw materials. They are also tested on \"duplications\"\u2014they must try to duplicate a fragrance that is on the market\u2014and, if Winnegrad adds or decreases a little of one ingredient, the students are tasked with identifying what has changed.\n\nNot only are most perfumers not super-smellers, many have specific anosmias: that is, they are entirely unable to smell some odors, such as musk. But \"although they can't smell it in isolation, they can tell if it's been added to a perfume,\" Leslie Vosshall tells me. \"They can paint with it.\" And they pay attention to smells, bothering to try to remember what the building blocks smell like: flash cards where one side is a puff of phenylethyl alcohol (a floral odorant), not high school vocabulary words. As with other forms of expertise, the perfumer's brain reorganizes itself with practice; indeed, one could say that it is the reorganization that makes the expert. While a part of the brain called the primary olfactory\u2014or piriform\u2014cortex is active in all people when smelling, fMRI images show that professional perfumers' brains process smell differently than others. They seem to rely less on memory areas; instead, perceiving or even imagining odors is a more automatic process for expert perfumers.\n\nAs important, expert perfumers helpfully remove evaluation of an odor from their consideration of it. It is not that they do not themselves experience smelly socks as \"bad\" and coffee as \"good\"; it is that they are more interested in whether the smell will work well in admixture with other smells to make a perfume. \"I hate the smell of coffee,\" Raymond Matts says, frowning\u2014by which he means not only that he finds it disagreeable on its own, but that it interferes with good smelling. On the other hand, many perfumes that he no doubt admires have evocations of coffee deep in their olfactory heart. To say nothing of the civet, ambergris (formed in the intestines of the sperm whale), and castoreum (beaver perineal secretion) of the fragrance world. \"Has the highly distinctive pungent odor of sperm whale feces,\" a researcher of the whales writes of fresh ambergris. The very common \"musk,\" used by male deer to mark territory, is now accepted by most people as the clean smell of \"laundry\": it is used as a maskant to cover up what Matts describes as \"the horrible base\"\u2014enzymes needed to clean clothes\u2014\"the smell of eyeballs; the worst fish you can imagine.\" (Use only \"unscented\" products? Until recently \"unscented,\" too, was a scent\u2014used to cover up the musk.) Clinique's perfume Happy, which Matts worked on, is based around lily of the valley, but there are also what he merrily describes as \"pukey\" notes\u2014\"that keep the fresh floral notes trailing in the air,\" he explains.\n\nFor a \"nose,\" as fragrance experts are sometimes called, Matts has an ordinary looking smeller: slightly broad, classically nose-shaped. It is partially hidden by the thickness of the bridge of his eyeglasses\u2014resting, it should be noted, just above where the epithelial tissue with his well-trained receptor cells lies. Matts is youthful in appearance, belying the thirty years he has worked in fragrance. His career has ranged from a job as evaluator of fragrances of products like laundry detergents and industrial soaps, to working at various houses and helping to design Liz Taylor's White Diamonds perfume and T, a fragrance of Tommy Hilfiger. T was one of his remarkable hits, though Matts says that it might be the least respected one. He describes smelling the fragrance on a man seated next to him, and asking him after it. \"The guy denied it,\" Matts says. \"But no one [but me] ever used saffron to that level.\" The non-perfumer can be forgiven for the blank expression she might wear at that comment.\n\nMatts has no facial hair. It does not become someone who deals in fragrances. While Michel de Montaigne noted that his \"mustachios, which are full\" served admirably as vehicles to keep wanted and unwanted smells resting under his nose for hours, the perfumer cannot afford this distraction.\n\nWhen Matts introduces students to smelling, he begins with \"families\" of smells, combinations of \"accords\"\u2014from three to ten odorants that combine well. At his class The Technique and Language of Perfumery at the Pratt Institute in New York, an enticing olfactorium\u2014a kind of traveling trunk that houses forty-eight bottles, each in its own soft housing\u2014sits open on the table. Each bottle holds an odorant, one of the palate of colors used to make a fragrance in the \"Oriental\" (vanilla, sandalwood, patchouli) or \"Foug\u00e8re\" (lavender, vetiver grass, oakmoss, florals, coumarin) families, for instance.III A blotter, a long strip of paper with which to gather scent, is bent and dipped into the bottle. Matts dips his in a bottle with the number 16 on it. He pauses significantly to let the paper dry\u2014otherwise \"you're only smelling the top notes,\" the scents that volatilize and waft off first. Perfumes are made to have three tiers of \"notes,\" usually called top, middle (or heart), and base (or background). The top is what you smell when spritzed as you race through the perfume section of the department store; the base is the odor that is on your clothes the next day. My mother's now-vintage \u00c9cusson eau de cologne will not smell the same as when she bought it: it oxidized every time she opened the bottle. The top notes\u2014citrus, aldehydes\u2014break down, and the heavy, base notes of oakmoss and vanilla might stick around. \"It will resemble the fragrance it was,\" Matts says.\n\n\"Mmm,\" he says, after drawing the number 16 blotter under each nostril. \"Cedarwood. From South Morocco and Virginia.\" This is one of the \"woody\" notes. Like a tea expert who can identify not just the origin of the tea, but the time of year it was harvested and whether it was planted near a plum tree, a fragrance expert detects real differences in the quality and source of what to the rest of us appears to be one category of smell. This one is \"pencil shavings\" or \"hamster cage\"\u2014less smooth than the air that rushes out on opening your grandmother's cedar chest. \"Can you smell the waxiness?\" He lifts the blotter to his nose, sniffs, pulls it away, considers, lifts it again. \"Think about a Crayola. Do you see the dryness and depth?\" Even over a few minutes, the sharpened-pencil smell resolves a little, gets smoother.\n\nWith another wood note, Matts says confidently, \"You can smell the grain of the wood\"\u2014as though it has a distinct olfactory fingerprint. Sandalwoods come creamy or raw. Some woods are oily: the synthetic Norlimbanol, dry, harsh, and strident. Dried patchouli smells of earth and roots. Another smells like nothing more than fresh-cut twig to me. But to Matts: \"Velvet. On top, it's the transformer of the electric train set when you turn it on\u2014the metallic odor.\"\n\nReflexively, one might think that \"wood\" smells, in essence, of trees. No longer: the range of woodiness suddenly appears vast. \"I'm always trying to get the smell of the construction site: the circular saw going through wet wood,\" Matts confesses. \"I haven't got it yet.\"\n\nMatts is frequently trying to capture a found smell to bottle it. When he had a young baby, he became entranced at his smell: milk coming through skin. He thought the ubiquitous baby blanket covered in spit-up \"smelled good\"\u2014the sour notes evaporated, and leaving the creamy essence. He spent some time trying to capture the smell, and gave up after his wife balked when he brought out a hypodermic needle.\n\nIndeed, there is a way to capture the kind of essence that enchants so many of us: the odor that emerges from the baby's head, from the tomato vine, from the dried and raked leaves. Perfumery could be described as initiating with that urge\u2014although it has gone far beyond simply trying to capture natural smells. Various methods have been used over the years, from enfleurage (similar to the truffler's method, absorbing scent of, say, a rose into a fat), extraction (rinsing the source with a solvent and extracting the oil), distillation (creating a vapor by heating and then chilling to separate the oil), expression (crushing citrus peel, rinsing, and cold-pressing it), to infusion (macerating a plant in alcohol).\n\nBetter for the non-maceratable odors (like the baby's head), though, is headspace technology: a glass bulb is put over the item, capturing the air that wafts naturally off of it onto an odor-absorbing material. The captured odor can then be analyzed in a lab with gas chromatography, and a perfumer can try to re-create it. The possibilities of this method are most fantastically represented by Roman Kaiser, fragrance and flavor chemist for Givaudan, for instance, who traveled the world for a decade to preserve the headspace of endangered flowers, should the source disappear. Self-described \"professional in-betweener\" Sissel Tolaas has amassed a library of thousands of odors via headspace, including the smell of her daughter. Matts describes using the technology to capture the distinctive odors of Cool Whip and Wheat Thins when designing a fragrance for Abercrombie & Fitch.\n\nStanding in front of an array of tiny bottles, Matts picks one up and spins off its black cap, bearing the number 42. It's a leather note, and begins sharp, almost medicinal. Matts smells the blotter and a faraway look appears in his eyes: a smeller's look. He is seeing something. He begins speaking, describing the process of creating the odorant: \"Originally it was made by infusing scraps of leather\u2014which itself smelled of the tanning, done with birch-smoking.\" On saying birch the woodiness of the odor rings brightly, like a child called on in class. \"Think of August in Amish country,\" Matts says. \"Tobacco leaves drying, hanging by barn slats. That juicy, sweet smell in the air.\" With his words, the other odors in the class jump to attention, one by one. Dried tobacco leaps forward, the moistness of the end of a well-chomped cigar; the aura of barn, fecal and weathered. Indeed, it has been suggested that part of the practice that transforms a budding perfumer or wine taster into an expert is the need to \"verbalize their olfactory experience\"\u2014to put a language to their experience. Both wine and perfume have their own jargons\u2014for perfume, the Orientals and Foug\u00e8res\u2014but all individuals in the field also generate their own vocabulary to remember and classify scents, serving as imaginative touchstones. These landmarks are a significant part of not only experts' discrimination and identification abilities, but also their ability to invoke a smell without the smell being present.\n\nMatts's linguistic landscape for smells is rich with evocations of familiar objects and childhood memories. Ylang-ylang, a floral note, is \"Necco wafer\"; jasmine, \"the horses of Central Park on a hot day.\" Over a few hours, he references rubber doll heads, the smell of crayons, white paste, basements, latex gloves, a Lionel train set, wet paper, carrots. His specificity is humbling: to Matts, a floral note doesn't smell like the flower, but \"like the air when you're walking by the flower.\"\n\nAs language can be used to bootstrap smell sensation into memory, the other senses are also useful. Give a roomful of people a bottle with a note of heliotrope, a purple summer flower, and rather than asking them to describe the smell, ask them to give the smell a shape and a color. Something surprising happens: almost everyone smells it as yellow-pink, or a variation thereof, and sees it as round or almond-shaped. The aldehyde C12, used to impart a sense of \"freshness,\" is most often experienced as some form of blue and square. We share an unspoken, untapped knowledge of smell associations that we are simply rarely asked to relate.\n\nAfter a few hours of smelling, the room we are in becomes a monster of odors. An overwhelming wash of something disorients my nose, like I need a psychic sneeze. I begin to get a little wobbly. Olfaction is a kind of \"change detection\" system, so if nothing changes, it turns itself off. But if new smells keep getting distributed around you\u2014even variations on an old smell\u2014the nose does not adapt, and desperately tries to keep up. With every bottle we open, we are restimulating those receptor cells trying to catch a few winks.\n\nBefore I leave, Matts makes a recommendation for how I might become a better smeller. \"This summer,\" he suggests, \"pick things up. And just squeeze them between your fingers\"\u2014a kind of on-the-fly extraction of odor\u2014\"and smell.\"\n\n\u2022 \u2022 \u2022\n\nI decide to do Matts one better. Not only do I begin bringing things to my nose, I start bringing my nose to things: I again bend down to the sidewalk; I sidle up to trees and then nose them. I follow the advice of an animal-tracking handbook to \"calibrate\" my sense of smell by sniffing at different walking paces, in different directions, and at different latitudes (nose-height and toes-height). \"Go for a very slow walk,\" the authors suggest. \"Stop every ten steps, or every time you get a new rush of scent that seems different. Do this for about ten minutes,\" alternating regular sniffing and doglike sniffing. I do.\n\nTo counter, or perhaps encourage, these habits, one day my husband delivers to me a large package.\n\nThe box is designed to appeal to the buyer of Wine with Fancy Labels: it is dense and solid. LE NEZ DU VIN reads the cover. An outer case, lined in fire-engine-red bound cloth, slips off either side to reveal another case, redoubling the mystery. The inner case splays like a book; inside are fifty-four small, rectangular bottles, each snugly secured in its own fitted slot and beguilingly filled with mysterious liquids. Some are perfectly clear; others, amber, or, worryingly, lake-dark.\n\nA box of bottled wine notes is the kind of thing that must be approached slowly. The first day I sit down with my bottles of curious smells and rub my hands together, a rare gesture of real excitement: oh, the smells that await me! But then I stop. I am already smelling, and it is a cloyingly sweet, strong smell. And it is coming from me. My shirt: it has just come out of the dryer. A major blunder; now I am steeped in a musky lavender, smelling of \"clean laundry.\" I strip, find a shirt that I'd aired out by wearing the day before, and begin anew.\n\nI open the first bottle and bring it bravely to my nose. Left nostril, sniff; right, sniff. Lemon pops up in my head at once. I check my experience against a deck of scent note cards and move to the next bottle, feeling self-congratulatory. Bottle number two is also of the citrus family; I'm going to say orange. Wrong: grapefruit. I return for a second sniff. Okay, I can smell it now, with an image of a grapefruit in my head, imagining its heft in my hand and the thickness of the skin yielding under my thumbnail.\n\nI continue through the deck, joined periodically by a member of my family or a dog, both intent on sniffing, too. The dog, Finnegan, waits patiently for a bottle and then, when I offer it to him, sets to his inspection like a professional. Without reading too much into it, I think it is fair to say that he looks genuinely puzzled. He shoots me a look, then sniffs again, ending with an exhale that makes the bottle toot like a one-note recorder.\n\nMy early success with \"lemon\" is not often matched through the run of the bottles. I mistake \"toast\" for \"vanilla\"\u2014impossible, I would think. I confuse \"peach\" with \"melon\"; every form of berry (and there are six) smells to me just like \"candy.\" \"Saffron\" and \"hawthorn\" ring exactly no bells in my sensory memory. While \"smoke\" is a cinch, I struggle with \"leather\" and \"butter.\" I have the classic tip-of-the-nose problem: I know this smell, but I cannot name it.\n\nFor professional smellers, naming the odor is not the goal, per se; it is simply a step toward more understanding. But for my novice nose, a name is necessary. Over time, I invent a strategy to try to come up with the name: an image-matching complement to the memory-search I learned when participating in the smell study. Recognizing that context matters for identifying scent\u2014it is much easier to identify the smell of \"apple\" in an orchard, rather than in a spa\u2014I decide to use my visual imagination for good. As I sniff, I close my eyes and conjure random objects to appear in my mind's eye. They begin to appear out of nowhere, hallucinations out of my conscious control\u2014a ceramic sink, a wool car coat with a sheen of frost on its shoulders, a school classroom pencil sharpener\u2014and when one looks promising I zoom in on it. A small tree, bare of leaves. I hold it tight and check if it matches the odor rollicking up my nose. Not quite. Switch it out for another tree, hold it close, sniff . . . repeat.\n\nIn this way, I make my way round to \"pine.\" An image of the prototypical pine tree shimmers into my head; the odor of note number 35 suits it. After I identify a scent, it's obvious, like a dumb joke badly done.\n\nAnd so begins my practice, a kind of smelling introspection, refined over the months. I like to start my day with coffee, but on the recommendation of Ray Matts, I have to give up this habit, or plan a session later in the day when I will be relatively coffee-free. Ideally I would rise, have a glass of water, and sit to smell. But often my stomach grumbles, or the dogs lick me to go for a walk, or my son is raring to start the day, and I have to put off the smelling practice to another hour. This is a tricky business, because, as I learn, the day is actually full of smells near my face.\n\nMy hair is redolent of the perfumed shampoos I wash it with, as the fragrance gets caught between the layers of hair and the oils grip the shafts (for this reason Matts recommends putting perfume in your hair over dabbing it on the wrist). My face has a vaguely Hawaiian suntan-oil trace from my preferred moisturizer. Hand soap smells hugely\u2014and every time I lift my hand to my nose I get a whiff of whatever it was. Soap in public restrooms begins to worry me: what impossible stink will I be painting myself with for the mere interest in cleansing my hands of bathroom germs? Anything in my mouth is off-limits. Since taste is mostly smell, any echo of the rosemary bread or cucumber or peanuts that I consumed an hour ago presents a major roadblock to smelling the wee bit of odor wafting out of a bottle.\n\nThere are plenty of days I simply cannot spend an hour with The Smelly Box. A stuffy nose is a disqualifier; relatedly, simply being chilly makes it harder for me to concentrate on any smell at all. Even when I am warm and clearheaded, the episodes are brief. Our noses tire easily and shut down with too much stimulation. I hoped that I could conquer at least the nose-fatigue with training.\n\nOnce I have made it all the way through the deck of bottles, I can no longer be surprised at the odors, so I practice distinguishing between smell groupings: animal, vegetal, toasty. I sit down with three or six bottles in an odor group, shuffle them around, and try to identify each blindly. When I get nose-weary, I take a cue from Matts: I smell my own shirtsleeve, at the crook of my arm. This olfactory interruption with, presumably, one's own odor works as blinking does to restore vision after staring at a bright light.\n\nTo resolve the ongoing orange\/grapefruit confusion, I step into my kitchen, where I have both grapefruit and oranges. I hold each to my face. To my surprise, the actual grapefruit smells \"citrusier\" than the orange, which I think of as the model citrus. I have the same difficulty with two of the stone fruits voted most likely to appear in my mouth: peach and apricot. While they are dramatically different fruits in my mouth, I am unable to reliably smell the difference.\n\nIt is at this point that I visit, profoundly and fully, my lifetime smell agnosticism. These are among the simplest of smells. They are common. I have ingested thousands of oranges and grapefruit, peaches and apricots, exactly for the reason that I relish their piquant smells and tastes. Was I not really attending even to these most loudmouthed and big-breasted of smells?\n\nNope. I have, apparently, spent my \"eating\" life simultaneously training myself not to pay attention to the subtle scent differences between foods. Each orange I encountered was, well, orange-tasting, and I did not have to check to be sure I was not eating a grapefruit\u2014and vice versa. I could tell the difference in my mouth; indeed, I enjoyed each one differently. But I had gotten lazy about noticing anything but that it was a confirmed \"citrus fruit.\" So my citruses have collapsed into each other. I untrained myself out of what children notice naturally (try to sub a grapefruit for an orange and see how a child reacts). I grew out of noticing.\n\nIn this, I have great company among English speakers. While \"lemon\" had, in my nose, preserved its own distinctive spot, in cross-cultural research asking people to identify common smells, English-speaking subjects were likely to identify an out-of-context lemon odor as \"air freshener,\" \"berry,\" \"Magic Marker,\" \"citrus,\" \"hard candy,\" \"some kind of fruit,\" and, the most depressing of all, \"lemon-fresh Pledge.\" For some people, lemon-fresh Pledge is a more vivid olfactory memory than an actual, honest-to-God lemon.\n\nThese subjects' vague characterizations of smells contrast with the whiplike precision, relatively, of the olfactorily-sensitive Jahai people of the Malay Peninsula. Though probably less fluent in the odors of lemons, turpentine, onion, cinnamon, and the other odorous stimuli the researchers presented them with than the English-speaking subjects, they were much better at naming the smells.\n\nIt takes me days to get the citruses reliably right. Now I can pull it out like a great party trick\u2014though I have no doubt plenty of people would have had the same ease on the first go. The ability to detect the presence of an odor varies widely from person to person; we are similarly individual in our ability to identify the odor. But while the former has to do with our genome, the latter has to do with our life. I am much better, from the get-go, at the smells I have registered as dislikes: almond, smoke, licorice. My attention is, it seems, well-tuned to spot the unlovely.\n\nThe Le Nez du Vin box itself, with all the bottled scents cloistered inside, is a hugely busy odor. It is the wine that you would never sip, and evidence that pleasant scents are not additive. Put together, the sum diminishes the parts. Walking away after a morning episode comparing almond, walnut, peach, apricot, cherry, and prune, I find that the whole room smells like prune. I step outside with Finnegan; a wind wrests the screen door from my grip. His nose rises to attention at the passing air. I smell . . . prune. I have been afflicted with a prune noseworm. The smell sticks with me for an hour. As I continue to challenge my nose, eventually sorting nine at a time\u2014cedar, rose, green pepper, thyme, vanilla, cut hay, acacia, caramel, banana!\u2014and finally all fifty-four, I seem to become more susceptible to the occasional noseworm or errant olfactory hallucination. I do not worry: they seem more to mark my increasing sensitivity to odor than any disorder. Chronic hallucinations, or phantosmias, though, are far less gentle. They are frequently reported by psychiatric patients or people who have experienced head trauma, and less often take the form of lovely prunelike odors than of burning, foul, spoiled, vomitous, or rotten smells.\n\nWithin a few months I have made a library of smell notes in my head, with idiosyncratic descriptions that reliably help me along: prune is vanilla, dried out; apricot is more cloyingly sweet than peach; cherry is cough syrup; walnut is dry almond; \"lees\" (the yeasty sediment left at the bottom of a wine barrel) smell like \"wine\"; violet is small hard candy, dusted with sugar, in an oval tin with a decorative top; bell pepper is \"green.\" I have become a Nez du Vin expert. What I am not, is any more sensitive to the notes of actual wine. Each glass I bring to my nose I sniff, I swirl, I nose almost to the point of touching the liquid. I do not smell apricot, peach, grapefruit, orange, jam jar, mushroom cap, or Gauloises. I smell\u2014not soda pop, not water. Definitely wine.\n\nBut that is about it. And so I fly across the country to meet John Buechsenstein.\n\n\u2022 \u2022 \u2022\n\nBuechsenstein, longtime winemaker and wine judge, stands at the head of his class at the Culinary Institute of America in St. Helena, California. Drought conditions have rendered Napa Valley preternaturally yellow: hills covered with a fuzz of browned grass roll into soldiered rows of trellised, green-leafed Vitis vinifera grapevines. Inside the so-called CIA, a huge bricked structure that was formerly a wine vault, the classroom is deeply cool: a wine refrigerator in which students now chill. The tiered lecture hall tables and chairs face a lectern and whiteboard in a familiar collegial way\u2014but each table is lined with spittoons and sinks, and eight perfectly poured glasses of wine sit at its center. This is the Sensory Analysis of Wine class, boys and girls, and you will be spitting.\n\nThe room exudes spilled wine, the intermixed aromas of the first samples, ranging from light yellow (\"pale straw,\" to Buechsenstein's eyes) to darker (\"rose,\" \"garnet,\" and the \"purple\" typical of a petite Syrah). There is plenty to see in wine: from the relevance of a brilliant or slightly hazy pour, to the faults (sometimes rendering the wine undrinkable) indicated by a cloudy glass or one with a constellation of crystals at its bottom. And of course there is plenty to taste. But \"smelling is the most glorious part of wine tasting,\" Buechsenstein says, smiling. Buechsenstein smiles a lot, furrowing his brow and nose while he does, as though working to keep his glasses up. And he smells a lot. The alacrity with which he will stick his nose into a glass is at first surprising, and then contagious. His mandate in this class, which he developed and has taught for three decades at UC Davis, seems to be to give wine enthusiasts and professionals the means and mettle to stick their noses into glasses and really sniff.\n\nFor Buechsenstein, this begins with simple attention. \"If you do nothing more than pay attention to what's in that glass, you'll be miles ahead\" of the person next to you\u2014or the person you were the moment before. To focus that attention, he gets his students\u2014of which, for two eight-hour episodes, I am one\u2014to bring a lot of glasses to their nebs, and to flirt with talking about what they notice. Initially, this is difficult for everyone. Wine smells like wine, it seems. We are so used to greeting a glassful of wine by glugging it, we rarely pause to reflect on that moment before it pours past our lips. And we haven't learned a language to describe what we smell.\n\nBut of course there is a lot of wine language. \"Low-keyed, solidly structured, with a youthful buoyancy\"\u2014what Buechsenstein calls the \"poetic school of prismatic lumination\"\u2014dominates wine reviews and prose. This way of talking about wine seems intended to exclude all but the cognoscenti. Buechsenstein's own descriptions of aromas are a bit more accessible, invoking shared experiences, memories, and conjurings:\n\n\"That tasted to me like I just bit my Bic pen open.\"\n\n\"Ooh\u2014St. Joseph chewable fake-orange aspirin! I take one of those every night.\"\n\n\"This is the charred log from your campfire, smelled the next morning.\"\n\n. . . and familiar food smells:\n\n\"chopped celery\"\n\n\"when the pot boils over and beans burn on the stove\"\n\n\"baked bananas\"\n\n. . . and just plain effusion:\n\n\"Now that's aromatic to beat the band.\"\n\n\"Whoa, fruit-and-vegetal double whammy.\"\n\n\"I put it up to my nose and it was like Dorothy-landing-in-Oz-Technicolor.\"\n\nTo enable his students to get to Oz, Buechsenstein concocts a series of \"flights\" of wine\u2014usually six or eight glasses to be compared and contrasted. Many consist of a neutral wine and variations therefrom, doctored with flavors that one might find in the wine. A red might be steeped with a sliver of bell pepper, another with a couple of peppercorns, a third with some crushed raisins. To pull out the notes of a neutral white, he spikes a glass with a bit of canned asparagus juice (just as horrifying as it sounds) or marinates it with a martini olive. On another occasion Buechsenstein cuts a white wine with varying levels of sugar or acid, and a red with acid and enough tannins to seal my teeth to the inside of my lips. Over a few days I come to pick up on\u2014if not always correctly identify\u2014\"all that funkadelia\" that Buechsenstein finds in wines.\n\nWatching him approach a glass of unknown wine is its own master class in tasting. He eyeballs the glass and waves his nose over its top with a perfunctory sniff. Picking up the tulip-shaped glass by the stem so as to not inadvertently warm the contents with his hands, he tilts it to the point of near-dumping to see its color and makeup. Then an assertive swirl of the wine, still gripping the stem. The swirl washes high on the sides of the bulb, creating petals of pink and violet and red that replace themselves at once. In the same gesture he places the glass directly under his nose\u2014or, more precisely, places his nose directly into the glass. \"Five bunny sniffs,\" he says, audibly sniffing, then hoovering a last sniff. His gaze is directed to an invisible point two feet south and in front of his head. Swirl-sniff-repeat. Then, suddenly, he brings the lip to his mouth and takes a sip. We have all sipped drinks\u2014but never with the complete confidence that we will not spill anything on our shirt that John Buechsenstein has. This wine does not go the way of all wine that enters my mouth: down the gullet. It is audibly swirled around his mouth. Though his mouth is closed, we can almost see it going over his tongue, across the hard palate, around his teeth. It is gargled, to its great surprise. He pulls his top lip down as though clearing his nose. His moustache quivers. And then he unceremoniously\u2014but actually ceremoniously\u2014turns and spits it out.\n\nThis tasting technique has been refined over the dozens of years as winemaker and as graduate student at UC Davis, which has a renowned viticulture and enology program. As a TA at Davis, the methods for learning smells were sometimes a bit ad hoc: \"When we had undergrads who couldn't smell Brett\"\u2014Brettanomyces, a yeast that leads to a classic wine fault\u2014\"we'd go to the Davis polo team and we'd say, Give me your funkiest, sweatiest, ready-to-throw-out blanket,\" Buechsenstein describes, rubbing his hands together. \"And we'd stuff it in a ziplock bag and seal it up. And say, Who was it who doesn't know that smell?\"\n\nMore formally, smellers now consult the wine aroma wheel introduced by Ann C. Noble and her colleagues in the 1980s. In the world of wines, it is the primary color group: the reds, blues, and yellows that fundamentally characterize all wine odors. There are twelve main groups\u2014fruity, floral, pungent, earthy, chemical, vegetative, nutty, caramel, woody, oxidized, microbiological, and spicy\u2014with secondary and eighty-six tertiary subcategories beneath them. If you're smelling \"chemical\" it could be plastic or tar, wet wool or garlic, rubber or mercaptan, the odor added to natural gas. \"Fruity\" covers everything from lemon to strawberry jam. In the academic paper introducing the wheel, published in American Journal of Enology and Viticulture, the authors suggest other ways to reproduce the aromas in wines for beginners. Your butterscotch Life Savers will be useful for generating the caramel aroma that might come up in a muscat. Wet wool added to a neutral white will perfectly mimic that \"wet dog\" smell of corked wine. Have any extra roofing tar? Let it sit overnight in a cup of wine; that will give you the sense of tar that a Rh\u00f4ne-style wine might exude. Only seven or eight granules of powdered tropical punch Kool-Aid get you that fruity smell in some reds. A single 5x10 mm piece of bike tire, a burnt match, a mildewed cloth? Add 150 ml wine, stir, and taste. Your pantry, supply cabinet, and wood shop are equally useful in this exercise. Elsewhere, Noble suggests putting a single Froot Loops loop or a Handi Wipe (preferably before cleaning the roofing tar off your hands) into a wineglass to get a sense of the aroma of Riesling.\n\nWinemakers employ in-house odor and taste evaluators who not only ensure that a wine is without flaws\u2014elements that some drinkers may like, but that mask the true flavors of the wine\u2014but that there is consistency between yields. They sit in positively pressurized rooms that resist the encroaching of environmental odors and use a descriptive analysis (such as the wine aroma wheel) to characterize the odors in a quantifiable way. Ideally, each panelist's graph of a wine's aromas will superimpose perfectly over each of the others'. If not, winemakers can adjust their wine's aromatics through changing the temperature of fermentation, the length of fermentation, and the time the wine sits on lees. \"Flavoring\" is never added to wine; a wine's character is simply the result of the grapes and the environment in which the grapes are crushed, then ferment and loiter while awaiting bottling.\n\nBottling itself is a tumultuous process. \"The wine is not really happy about it,\" Buechsenstein says. Plenty of faults can be introduced in bottling, too\u2014many of those make it to the store and your kitchen. All of the faults can be smelled. If you open a bottle and are greeted with a kind of moldy, musty, old-wet-newspaper, the wine is corked: a compound called TCA\u2014more formally, 2,4,6-trichloroanisole\u2014has developed from fungi or bacteria on the cork and has overwhelmed the wine.IV\n\nTasting is meditation with spitting. Occasional mutterings of pleasure or alarm interrupt the gentle chorus of glasses clinking on tabletops as they are picked up and returned. On a blind tasting I consult my wine aroma wheel in order to induce images into my head to match the waftings from the glass: Is that a really ripe banana? No. Something artichoke, green-beany? No. A field of hay, laced with honey? Ooh, maybe! I scribble a note and clink my glass with the sopranos. Looking up, I see Buechsenstein poking one side of his nose into a glass, as though examining it with a nose monocle. On his glasses hand he has drawn a carbon group in explanation of an earlier aroma; a C=H double bond flashes us on his second finger and pinky for the remainder of the day. He lifts his head: \"Did you get it? Think Orville Redenbacher popcorn.\"\n\nPop! There it is: a butter smell, diacetyl, one of the side results of malolactic fermentation, a common process with reds. And, since this wine is a straw color\u2014with the honeyed hay\u2014with chardonnay.\n\nOn another I get a trigeminal nerve jangle with one wine, the sensation of a match having been struck. My throat is coated. A match is a good stand-in for some of the wine aromas, Buechsenstein says. When it is first struck, there is sulfur; then one gets a smoky odor; finally, a charred mercaptan smell.\n\nOver two days we run through a half dozen training flights. Whites feel easier\u2014their odors are more distinct; in reds, the odors combine and hide in each other. But I begin to find my own language. I get vinyl in one glass; pencil shavings in another. Some of the familiar descriptors\u2014cat pee in a sauvignon; deeply ripe berry in a zinfandel; oaky, buttery chardonnay\u2014become comfortably familiar.\n\nAfter sixteen hours, my nose has had enough. My mouth tastes only of the tannins of an unpeeled grape. I drive away, board a plane, and look forward to the smells of home and my dogs.\n\n* * *\n\nI. Not a cat, but one of various species of the Viverridae family, which also includes other animals you have not heard of, such as the genet and fossa.\n\nII. Chanel No. 5 was also one of the first fragrances to use synthetic components\u2014aldehydes C10\u2013C12, imparting \"freshness\" and \"citrus pith\"\u2014when introduced in 1921.\n\nIII. You might know an Oriental: Johnson & Johnson's baby powder. The fact that this is more familiar to most readers, I'd hazard, than the description of the family of odors, is a testament to how our vocabulary of smells has been hijacked by those used in retail products. It would be as if all our musical knowledge were based on commercial jingles.\n\nIV. Recent research has shown that, in fact, it is not that TCA causes new, \"mold\" olfactory receptors to fire; rather, it stops other olfactory receptors\u2014that notice wine fragrance\u2014from firing: the DEET of the wine world.\n\n## 11: Nosed Out\n\nOne January day, I took only one of my dogs out. Finnegan. He is the nosey one, the snuffler, the one whose snout pokes deep in pockets and postal packages. Finn was delighted at this rare single-dog outing. He positively pranced over to the car and hopped in. As we merged into rush-hour traffic leaving the city, his mood turned somber. I had no way of telling him where we were going\u2014to what would become his Favorite Place on Earth.\n\n\u2022 \u2022 \u2022\n\nThe truffle, disease-detection, scat-detection, and other working dogs serve as evidence to the power of the nose of a trained dog. What about the nose of the one lying at your feet right now, though\u2014that of the well-loved, well-fed, well-pet companion dog? Sure, Finnegan can sniff out a fallen crumb encountered on our walk, and quickly distinguishes between people we meet who-have-ever-had-dog-treats-in-their-pockets and people who have not. But how would he do if faced with a real test of his nose, I wondered.\n\nMy lab and I set up such a test for dogs. We started with a simple one: a quantity test. Any owner knows (and research has confirmed) that dogs have no trouble distinguishing one piece of hot dog from five pieces: if given the choice between plates with each, they choose to eat five pieces. It is the rare dog who goes for the plate with one and wags away satisfied. Dogs have perfectly good vision, but certainly any creature with a nose such as theirs should be able to do just as well or better if the plates are covered, and they can use only their sense of smell to discern the contents.\n\nSixty-four hungry dogs and their people agreed to an afternoon of sniffing and eating hot dogs. Each dog showed great interest in the (covered) hot-dog plates that the experimenter presented to him to examine. Then she put the plates down and the dog was free to approach the plate of his choice. Based on smell alone, the dogs did something I did not expect: they chose randomly. Though they had sniffed the five-hot-dog plate more, they did not use their noses to find that plate again.\n\nDogs are losing their noses. Other research has supported the odd and disturbing result that companion dogs are not only not using their smelling abilities to their capacity; they are forgetting how to be sniffers. In a human-defined, visual world, it seems that it does not pay to notice all the smells around the house, to sort their way through the world by smell. Instead, the typical owned dog gets a mound of food in a bowl once or twice a day whether he sniffs it out or not. He may be discouraged from sniffing the sidewalk, the lamppost, and even other dogs' rear ends as his owner walks with him\u2014out of the person's disinterest, press for time, or horror. We talk to the dog in words and point at him with hands, but rarely give him smells to learn and live by. The sad result has been that pet dogs are letting their noses go dormant.\n\nBut this is easy to change. I look at Finn, idly waiting for me to lead him through his walk, and I decide to change it.\n\n\u2022 \u2022 \u2022\n\nThe Port Chester Obedience Training Club building is tucked nearly under an interstate highway, a collector of noise and truck debris. Of more interest to Finn and the other dogs converging on its glass door, though, are the local highlights: a well-visited pile of frozen, blackened snow, graffitied with yellow letters drawn by barely literate urinators; a trail of crumbs that might have dribbled from a dog owner's pocket.\n\nUp a set of stairs marked with helpful yellow pawprints is George Berger, our coach and the current president of the Port Chester club. He greets us cordially, the faintest of smiles wrinkling his bearded cheeks. He wears a fleecy vest, khakis, and sensible shoes. Behind him is a large, bright room, mostly empty but for a smattering of tables, chairs, and agility-course obstacles pressed against three of the walls. The fourth is lined with crates, five of them occupied by dogs. Their owners sit scattered nearby, chatting and peeking over at their charges.\n\n\"The game, of course, is nosework,\" Berger begins. \"It's about using the dog's greatest ability, scenting\": what the dog trainer and writer Turid Rugaas has called \"channel nose.\"\n\nWe have come to this bright, empty room for Finn to smell with impunity. The sport of \"nosework\" makes a game out of what to working dogs is their occupation: simply using their noses to find something hidden\u2014the \"hides.\" The dog has to improvise, to problem-solve, to correct and redirect himself. Finn has been encouraged and indulged in using his nose\u2014we have spent long minutes at tree trunks and building corners, and I have probably had every inch of my face mapped by his examining snout\u2014which is at least an advantage over dogs who have been yanked from every good-looking smell they nose out. This means Finn's nose is open, ready; but it is not yet tuned. Nosework is the tuning. He will learn that if he uses his nose, and his nose alone (not following what he sees, or what I direct him to do), he will discover hidden treasures and get great rewards (read: copious treats of salmon or cheese).\n\nBerger gently instructs the owners present in the mantras of what is called nosework but is clearly nosegame. First: the dog is the leader. While pet dogs are typically expected to look to their owners for advice on what to do and when to do it (or when to do nothing), here the owners are expected to follow the dogs. \"I want them to learn that whatever I or you are doing is BS,\" Berger says. Eyebrows silently raise a quarter-inch. But, dear eyebrows: to do nosework, the dog has to figure out where the hides are on his own.\n\nSecond, the dog must be motivated. At the beginning, this means finding the food treat that the dog loves more than anything\u2014for which he will drop whatever is in his mouth or on his mind and lunge. Later, the reward can be a favorite toy\u2014the one he always yearns to grab\u2014or a game of tug, as with detection dogs. But in all events the dog has to know that something incredibly terrific happens when he uses his nose.\n\nThird, when training, the dog has to find the hide every single time. Until the dogs are experienced at the game, the owners must help them be successful, setting up situations to enable them to find the hide on their own\u2014lest the dogs learn that once in a while, they can give up, look at their owners pleadingly, and get fed anyway.\n\nThese are exactly the principles of detection-dog training: independence, motivation, success. Indeed, nosework is modeled on detection-dog training. For the ambitious owner, nosework is even reified as a trial sport with various levels of competition. Just as a narcotics-detection dog might search containers, early-stage nosework dogs learn to find particular odors secreted among a lineup of a couple dozen closed cardboard boxes. As explosives dogs might examine traffic at national borders or airports, elite nosework teams search for odor in outdoor arenas and around cars.\n\nThe thrill of this sport, or of watching detection dogs, is that the dog is shedding some of the raiments of domestication. We are reminding the dog\u2014or reminding ourselves\u2014that the dog comes from a hunter who was intrinsically motivated and autonomous. Domestication has allowed dogs to be good sitters-on-couches, but has robbed them of the pursuit of some of their instinctual drives.\n\nWhat nosework does not train is the dog's nose to smell. The nose is perfectly good as it is. It just needs waking up.\n\nWithin our small group of six at Port Chester, the constraints that owners have unwittingly put on their dogs' abilities to follow their noses are immediately clear. One dog stands immobile if his owner is still, a perfect I have absolutely no ideas look on the dog's jowly face. He is not driven to do anything at all without his owner's direction. In some contexts, this might look like a \"polite\" dog. In this context, it is heartbreaking: the dog has no internal motivation, no striving to pursue, no ambition to be doggish. Another dog is unwilling to lead her owner: a very good, cooperative heeler on leash, she won't pull ahead. The pair stands transfixed and bemused. A third dog will take weeks to learn that it's okay, here, to jump up on the table and grab the good-smelling food that he's found. This behavior would be called counter surfing in some circles\u2014nosing for food left untended on the dining table or kitchen counter (depending on the height of the dog)\u2014and is frowned upon by polite owners and their polite dogs.\n\nIndeed, all of the well-trained and obedient dogs have something to overcome here. Obedience is \"totally unnecessary,\" for nosework, one trainer suggests. Another: \"in many cases, obedience is a hindrance\" for tracking. If the dog is concerned about what his owner wants him to do at every moment\u2014if his concerns are reduced to when to heel or when to sit\u2014then he has lost some part of his dogness.\n\nHappily, Finnegan has not been well trained. While he is the more civilized of my two dogs, highly responsive to me, after a fashion, he is allowed to be a dog with us, and to poke his nose in our bags\u2014Where have you been, and what have you brought me?\u2014in a way that turns out to be incredibly useful if you want your dog to find a bit of cheese hidden in a stranger's suitcase. Should that come up in your life.\n\nThe first stage of nosework is simply to introduce the dogs to the possibility that, when you walk into a space with your person, she will let you sniff wherever you want, and you may find Very Good Treats. Among our small group, this is a terrific and heady leap. After only a few rounds, some of the crated dogs are whimpering with excitement as they wait for their next turn.\n\nWhite, lightweight boxes sit scattered in the room, ringed by movable gates. \"They look like they could easily be destroyed,\" the owner of an extremely energetic pit mix worries. For Finn's first tour of the arena, Berger takes a few pieces of a treat from me, places them obviously in a box, jiggling the box for good measure and then dropping it on the ground. Finn goes right to the box. At this \"discovery\" I am to reward him copiously with further treats. While this seems a bit overkill now, the idea is that eventually he will be able to race ahead and find a hide on his own\u2014and I will want him to stay there (eating the treats) until I can catch up with him.\n\nThe other dogs all have their turns and Finn's comes up again. This time the hide is a bit subtler. Finn takes off when I release him with \"Go find it!\"\u2014but he doesn't find the hide straightaway. Instead, he's vacuuming the floor of the crumbs left by classes past. \"The good thing is that he keeps searching,\" Berger spins it. I don't know that I would call it \"searching,\" but I wait Finn out. After a very long floor-cleansing, Finn comes to the hide. We all applaud as though he's discovered the moon.\n\nOver the next half dozen rounds Finn continues his wet-vacuum ways, but less and less. His gait changes: no longer casually ambling along; instead, he leaps toward the boxes. By the end of the session, Finn is standing up in his crate while he waits for his turn, watching the other dogs' attempts. En route home in the car, he reclines the seat, curls into a crescent, and falls asleep.\n\nOne week later, at the appointed hour, I leash up Finnegan and we head out to the car for the long commute to our class. \"We're going to nosework!\" I propose\u2014at which, Finn begins lunging down the street, a huge enthusiasm powering him forward. Although I know this is anatomically impossible, his eyes seem to widen into saucers. His ears are laid back against his head as though to streamline him for faster-getting-there-now.\n\nThis week, there is a new twist in the hides: the search space, as defined by accordion fences creating a nosework theater in the room, is larger. Indeed, from this day on, each class brings a slight change of the game. This is Berger's way of \"expanding the world\" of the dogs. One class, he aims to teach the dogs that \"the world won't always be boxes\"\u2014and hides are concealed in baskets, egg cartons, buckets, under a sheet. Dogs learn prepositions: hides can be on, under, in.\n\nOn another day, the world is \"tricky\"; the scattered objects on the floor unexpectedly never hold hides and instead are distractors from where the hides really are: along the wall, or in plain sight. Some hides will be in corners, because, Berger says, \"Corners are typically repugnant to dogs.\" I consider whether I have ever seen a dog heading for a corner, and cannot find an image to match that query. Clearly it is not part of typical hunting drive, say, to go for the corners.\n\nOn some days, the world the dogs will encounter has \"problems\": hides are in long plastic tubes with an opening too small for a head. They are in zipped suitcases. They are near the owners waiting with their charges, stoically not responding when dogs come around looking for a friendly scratch behind the ears. (Berger admits, \"Some other instructors here wouldn't do this\"\u2014planting hides around the waiting owners\u2014\"because they think owners should be out-of-bounds. I think everything should be in bounds.\")\n\nOne day, \"the world is not just here\"\u2014Berger motions by the ground. \"It's also up here,\" he says, fluffing a pillow of air at the height of his head. Hides are set on chairs, on boxes piled onto boxes. Later, they will be in cups nailed to walls or perched on windowsills.\n\nAll of the dogs are, initially, stymied by the elevated hides. Finn waltzes by one, then jerks his head back just after we passed. It takes an extra beat before, powered by his rapid motor of a wagging tail, he finally gets the hide from atop a tower of boxes. He then has no trouble finding a hide on a chair, sticking his head through the back to reach the hide on the seat. (Well, we have trained him for that with a young child in the house, I think.) Soon, the moment when the dogs suddenly sniff an elevated hide is visible to us all. Their nostrils turn obliquely as they pass, or, with the small-nosed, their heads cock, their bodies tense. Their gait changes from a walk to a prance with intent.\n\nEventually, the hidden scent will be not a treat but a \"neutral\" odor\u2014one dogs are not predisposed to either like or dislike: the essential oils of birch, anise (the smell of licorice), or clove. Along the way, other incremental changes are introduced: searching outside, among vegetation; in hot weather, obliging smell-inhibiting panting; against the wind.\n\nWe think of bloodhounds, beagles, and their brethren as the \"nose dogs,\" but what is especially satisfying about nosework is the range of dogs who can do it. Watch a nosework class or trial and maybe a quarter will be mixed-breed dogs\u2014and of the others, plenty of Chihuahuas, schnauzers, and unlikely breeds appear. While law enforcement detection dogs swing strongly toward Labrador retrievers, German shepherds, and Belgian Malinois, none of these is in our class. Instead, the dogs in our first class range from a twenty-pound mixed-breed with long furry ears, to a mellow long-coated retriever, to a muscular pit mix who might top a hundred pounds. And yet each dog gets the game. Their working styles are as individual as their appearances. One Border collie habitually heads out in a measured but lively clockwise tour of the space. The long-eared dog is very laid-back about the whole endeavor, and alternates gentle sniffs with searching glances at his owner. The pit mix defines the antithesis, charging into the game with little owner consultation. While a few dogs will bravely stick their noses into a box or bag, this one works out a puzzle by stepping on it, grabbing it in his mouth, or pushing it out of the way. One day he actually ends up with a cone on his head, befitting his headstrong enthusiasm.\n\nIn my admittedly biased estimation, Finn is turning out to be a champion sniffer. He merrily launches out on every bout with a sprightly gait. He tours the periphery and works inward, poking his nose in crevices and nooks. He snuffles loudly and charmingly. I learn to see when Finn is telling me he thinks he has found something\u2014the subtle change in his demeanor that is his \"alert.\" His nosing lingers just a little longer; he paws the ground or air, referencing digging rather than actually digging. His searching is exhaustive, never giving up the hunt for that tiny piece of salmon, or waft of birch, that he knows is stashed somewhere.\n\nThis is miraculous to the human-nosed among us. On one go around the ring with Finn I try to sniff out the hide myself. It's birch; before Berger hides it I bring the container to my nose and smell a sharp mint note, which resolves into a eucalyptus chord. I watch where Berger hides it\u2014between some folded chairs stacked against the wall\u2014so I am, I'd think, nose-advantaged. As Finn and I approach the chairs, I direct my nostrils toward them and inhale. Finn, ahead of me, has his nose on the ground, leaving a slug trail of moisture on the mats. He's walked by the hide. I sniff again. While I am ineffectively washing my nose with dog-gym air, Finn has stopped walking and lifts his head high. He sniffs the wall where the chairs lean against it, then draws a line with his nose down, over, down . . . to right where the hide is. Before I can catch a whiff, he has wolfed down his reward.\n\nAfter many weeks of nosework, it has become clear that the instruction is not just, or even primarily, for the dogs. It's for us. Sure, dogs need to learn the game. But they get it at once, and steadily proceed through each level of difficulty. It is the owners who need training\u2014even to understand how good their dogs are. Owners do not get it. At this stage, by which point every dog in the bunch has displayed prodigious abilities to catch the scent of a minuscule salmon morsel in a closed box in the corner of the room, owners still walk out of the ring and say, Do you think he was using his nose? Or, I think she just got it accidentally. One afternoon Berger directs every owner to handle someone else's dog in a search; after a few bemused dog glances, each dog sets to his task with alacrity. And all of us get to see what our dogs actually do\u2014without our assumptions getting in the way.\n\nEvery search by a blind yellow Lab in a later class is the ultimate proof: he cannot see visual landmarks, so he really air-scents. Often he seems to sense an obstacle before hitting it: smelling it in front of him. He always finds the hides.\n\nNot only do we have trouble seeing what the dogs are doing, we have trouble playing our role. Our role, incidentally, is mostly to stay out of the way: follow the dog, create the space for him to search, and step in only to reward him for finding the hide. Somehow, this is unintuitive. Even at the course's end Berger will be encouraging us to bother to really praise our dogs\u2014by petting, with more treats, with praise\u2014when they've succeeded. \"Do them all,\" he says. \"It's a big deal\" that they've found it. \"The quality of your payment,\" Norwegian trainer Anne Lill Kvam says, is directly connected to \"the quality of your results. If you pay the dog with tiny little pieces\" of food, she says, \"you tell them okay\": that's not a bad thing they did. \"If you give them big ones, you tell him This is it!\"\n\nWorse, we cheat. Even in this no-stakes game, we unintentionally bamboozle even ourselves. If I see where Berger has stashed the treat, I may unconsciously lead my dog . . . roughly right over to it. In the class, as we watch each other with our dogs, we witness this happening dozens of times. The leading is more subtle than calculated: for instance, if a dog is idling in one area, an owner may simply angle her body toward the part of the room that holds the hide. This cue is like shouting through a bullhorn to the dog: GO OVER THERE! IT'S OVER THERE!!! Or an owner dawdles when the dog runs toward what she knows to be a barren area, resulting in tension on the leash (screaming STOP to the dog). If the owner keeps walking as the dog slows, the dog takes that as a clue to keep moving along. Even the side of the dog's body on which an owner lets the leash fall can influence where the dog goes next. An owner's tension as the dog innocently saunters right by a hidden canister is carried down the leash to the dog. Just as the detection dog's handler's job is to be responsive to the dog, not leading of him, here, too, the owners are tasked with following, not steering the dog.\n\nIt is difficult to be entirely neutral. When I put on my stone-coldest face, try to follow Finn wherever he goes and give him no clues or pushes in the \"right\" direction, I still find my mind going to the hide. I wonder how my body is following it.\n\nThe cure for our unwitting cheating is simple, happily: blind hides, in which neither the dog nor the owner knows where the treat has been hidden. Once we graduated to blind hides the dogs' performance convinced even the most skeptical of owners. As one heads into the arena, one must truly observe the dog, follow him, and trust that wherever he goes, there's a good reason for it.\n\nHere I find the one reliable place in my life where my own ignorance is a pleasure. I love not knowing: it is a great discovery to have my dog be the informant for me. On walks in both city and country I begin to view Finn's now-prodigious sniffing as telling me a story of what has happened on this path before we got here. An unearthed set of tracks shows where a flock of turkeys walked, then alighted; a small burrow turns out to be fur-lined, perhaps the feeding den of a coyote. In the city, I know if our neighbor and his dog have been outside by watching Finn's examination of the air around our stoop, which they pass en route to the park.\n\nAfter a few months, all the dogs are changed. One dog who began as a barker, reactive to other dogs, mostly ignores them in search of a quarter-inch of cubed chicken. Others have learned to look to their owners to tell the people something, not just to ask them or to get instruction; or that it's okay to put their nose on a table or gently jump up on a wall. Another dog, a collie mix, is like a cartographer. She tours the perimeter and will exhaustively examine every square meter within. When she is done with her survey, she beelines back to the smelly spot. All the dogs begin to lick their noses, better to snag the scent out of the air, in preparation to search. Finn's nose has become nearly periscopic, air-scenting up above the fray of the odor sea and sighting the hide on the horizon. I soon learn that I can't hide the tiniest dog treat on a windowsill and expect Finnegan not to find it. He slows as he walks by, checks the air, snorts, then pivots on a dime, sniffing great gulps right up to the smelly source. I feel perhaps unduly proud of him.\n\nMy own vision is changed: through the dogs, I've begun to see how the air in an ordinary room is moving. \"Scent corridors\" along walls\u2014invisible tunnels of air that hurry odors in their wake\u2014lead sniffing dogs to a hide from well across the room. Vents in the ceiling push air down and to the sides, creating small vortices along the middle of the room. Baseboard heaters push scent up the wall with the heated air. Currents flow from under doors and wash the room with the scent from outside.\n\nAt home, we begin doing nosework games in the living room, and Upton, watching the game, quickly picks it up. Their styles are different\u2014Finn is an assiduous searcher, exhaustively visiting the home landmarks (couch, rocking chair, baseboards, bookshelves); Upton simply wanders around, nose in the air, and seems to catch odors on the rise\u2014but both find their targets.\n\nI begin trying other nose-search games with Finn, such as asking him to pick up something I've dropped on our walk: first a highly valued orange ball, then a glove that smelled strongly of dog treats, then my keys. One day he finds my datebook for me, nosing it out with a satisfied expression. I stare at him in wonder\u2014not only that he could find it, but that he knew what to find. How much more do you know, my friend?\n\n## 12: Smelloftheworld\n\nThe act of smelling something, anything, is remarkably like the act of thinking itself.\n\n\u2014Lewis Thomas, Late Night Thoughts on Listening to Mahler's Ninth Symphony\n\nThe great pleasure in having spent the last years thinking about smelling is that my world has changed color. It smells. Well, it has always smelled, just as the light bouncing amidst the reds and greens of our palette are the unnoticed ultraviolets and infrareds that the honeybee and pit snake attend to. But I had not bothered to open my mind to the smells.\n\nWhat has not happened is that I have become an expert in smelling. Nor am I constantly ringing with Proustian experiences, inflating ephemeral bubbles of memories with each inhale. I lived for over forty years not attending particularly to the smelloftheworld, in e. e. cummings's phrasing; so now there are only a handful of memories to reawaken. But I am now smelling all over the place. I put my nose to things without fear and with interest. I smell the skunkiness of bitter coffee beans swirled into ice cream. The loamy smell of the park after rain.I I smell when the previous occupant of our building's elevator has opened a new glossy magazine as she rode home. I smell someone's sudden urge to clean his hands with a sanitizing wipe while on the bus.\n\nA summer trip in the car is not complete without notes of gasoline, cut grass, honeysuckle, warmed vinyl, sunscreen, overheated-dog breath, and wet sandals, stirred by wind from the open windows or wafting up from the floorboards.\n\nI smell a thunderstorm approaching on my last visit to Colorado, the home of my family during my childhood, when I come to help clean out the house after my father's death. The watery, fresh smell of sea air comes, I now know, from ozone carried down from higher altitudes on the winds of a storm.II It is also the odor of the city when I emerge after swimming, the receptors pinging chlorine! silent for a long enough moment for me to smell the world in its absence.\n\nI smell gin on the man who sits next to me in 10C.\n\nI smell the acrid, lingering piles of freshly turned, festering wood chips on the other side of the park.\n\nI see two people with a dog; then a second later smell that dog's poo, which must've recently been deposited in a trash bin.\n\nI smell the art room at kindergarten before seeing it.\n\nI smell every book I open.\n\nI smell the clove antiseptic paste that screams dental office.\n\nOn return from a trip, I smell the way our apartment smells to those who do not live in it. I see that it is not difficult to know there are dogs living in it.\n\nI smell how long ago an office door was last opened.\n\nMy nose captures vernal smells: the incomplete and jagged smells of fall, caught in winter, and finally released from their hibernation in spring.\n\nOne summer day I retronasally smell a good peach\u2014slurpilly juicy\u2014and orthonasally take in scrubbed sidewalks, a fetid sewer puddle, a vinegar sharpness.\n\nI smell blue cheese in the building; Magic Marker on the street; cucumber on the east side of the park, steeping tea on the west side.\n\nI smell my friends (sorry, friends, I have smelled you). Each person in my world has a \"smell-face,\" as my great colleague Dr. Oliver Sacks once described it, and it is neither terrific nor terrifying; it simply rings of them.\n\nI smell the wet wool on my fingers after coming in from a winter storm; coffee on my fingers after handling a takeout cup.\n\nWhen I wake up, I take a moment to smell if my son or husband\u2014or both\u2014are still in bed. When I return home, I smell who has returned home before me.\n\nWhen I awaken from the rare nap, I try to smell the time of day.\n\nI smell burnt rubber by the train, smoked mozzarella in the Bronx, warmed pine needles in the forest.\n\nPerhaps I will revisit these smells in another forty years and be brought back to these days, when I lived with a young boy, a youngish man, two great dogs, and an accidental cat, and headed into my books nose-first.\n\nI have now dreamed of smells\u2014smells of people I love; and unknown, unsmelled smells concocted by my unconscious brain. Awake, I can now call forth a smell imaginatively, in my mind's nose. If I consider the smell of a penny . . . there, I've got it, the familiar waft that is really the smell not of copper and zinc but of copper and zinc handled by persons. As I write the word lavender (lavender . . . lavender . . . ) I can conjure up the soapiness of the dried plant, sachet-ready. I look at a photograph of my father's desk, and the slightly sharp and resinous smell of its inside, mixed with pencil shavings and loose tobacco, hits me, just as if I had opened the drawer itself.\n\nSmells precede us and remain after we leave: our presence is thus extended in a place. Watching the dogs has allowed me to extend the dimensions of perception.\n\nUltimately what I have learned to do is simply to bother to attend to smells. This was enabled by making associations\u2014with words and with images\u2014to fix my mind on a smell and then to curl it into a slip of memory. The pictorial and verbal vocabulary I am collecting is what clears access not just to current perception, but to my being able to smell more next time I nose a rose. In some important way, I realize that this is the opposite of being a dog, though. I am taking advantage of something that we have that the dog does not\u2014the words that describe, and simply accompany my experience. So I must assume that I am not reproducing Finn's experience.\n\nWhile smells now appear to me more public\u2014they are out there to be detected by a nose\u2014I am evermore appreciative of the privacy of smells. Odors may blow in from across the river or waft through an open window from outside, but the greatest majority of smells need to be nearly touched to be perceived. Smell is a private sense, reserved for those people and objects we literally bring close to us. By smelling what my dog does, it brings me closer to him.\n\nToday\u2014as I will do tomorrow, and the next tomorrow\u2014I head out the door with both my dogs. I watch their idle sniffs, their exploratory sniffs, their communicative sniffs, their pushy sniffs. When my dogs poke their noses in the earth, I pause to let them. I feel a frisson of excitement\u2014both from knowing some of what is happening, and from the realization that I will never really know what is happening.\n\nI will never smell as a dog does. I accept it. It is dogs' difference I celebrate\u2014and their ways of smelling\u2014their very noses\u2014are different. Quiet distillers of a world that we have stood up from and forgotten.\n\n* * *\n\nI. geosmin: \"a metabolic by-product of bacteria or blue-green algae.\"\n\nII. Its \"freshness\" led to the earlier erroneous claim that ozone was health-bringing.\n\n## About the Author\n\n\u00a9 VEGAR ABELSNES\n\nAlexandra Horowitz is the author of the #1 New York Times bestseller Inside of a Dog: What Dogs See, Smell, and Know (2009) and On Looking: A Walker's Guide to the Art of Observation (2013). She teaches at Barnard College, where she runs the Dog Cognition Lab. She lives with her family and two large, highly sniffy dogs in New York City.\n\nMEET THE AUTHORS, WATCH VIDEOS AND MORE AT\n\nSimonandSchuster.com\n\nauthors.simonandschuster.com\/Alexandra-Horowitz\nALSO BY ALEXANDRA HOROWITZ\n\nInside of a Dog: What Dogs See, Smell, and Know\n\nOn Looking\nWe hope you enjoyed reading this Scribner eBook.\n\n* * *\n\nJoin our mailing list and get updates on new releases, deals, bonus content and other great books from Scribner and Simon & Schuster.\n\nCLICK HERE TO SIGN UP\n\nor visit us online to sign up at  \neBookNews.SimonandSchuster.com\n\n## Notes and Sources\n\nBeing a dog emerged from conversations, interviews, classes, and experiences with dogs and people who work with or think about smell. On the person side, this includes Jonathan Ball, George Berger, John Buechsenstein, Noah Charney, Brent Craven, Annemarie DeAngelo, Bob Dougherty, Charley Eiseman, Stuart Firestein, Simon Gadbois, Avery Gilbert, Leta Herman, Pat Kaynaroglu, Raymond Matts, Alana McGee, Kate Mclean, Cindy Otto, George Preti, Kristin Rosenbach, Leslie Vosshall, and Sam Wasser.\n\nMany books on smell served as reference, inspiration, and merely good riffling. I recommend them all:\n\nAckerman, D. 1990. A natural history of the senses. New York: Vintage Books.\n\nDoty, R. L., ed. 2003. Handbook of olfaction and gustation, 2nd ed. New York: Marcel Dekker, Inc.\n\nDrobnick, J. ed. 2006. The smell culture reader. Oxford: Berg.\n\nFeigel, L., ed. 2006. A Nosegay: A literary journey from the fragrant to the fetid. London: Old Street Publishing.\n\nGerritsen, R., and R. Haak. 2015. K9 Scent training: A manual for training your identification, tracking, and detection dog. Canada: Brush Education.\n\nGilbert, A. 2008. What the nose knows: The science of scent in everyday life. New York: Crown Publishers.\n\nHenshaw, V. 2013. Urban smellscapes: Understanding and designing city smell environments. London: Routledge.\n\nRezendes, P. 1999. Tracking and the art of seeing: How to read animal tracks and sign. New York: HarperCollins.\n\nRouby, C., B. Schaal, D. Dubois, R. Gervais, and A. Holley, eds. 2002. Olfaction, taste, and cognition. Cambridge: Cambridge University Press.\n\nAdditional sources for each chapter are listed below. (I also include additional notes here and there, for the extra-interested reader.)\n\n2: SMELLER\n\non dogs' sensitivity in finding banana, butyric acid, and human odors:\n\nWalker, D. B., J. C. Walker, P. J. Cavnar, J. L. Taylor, D. H. Pickel, S. B. Hall, and J. C. Suarez. 2006. Naturalistic quantification of canine olfactory sensitivity. Applied Animal Behaviour Science, 97, 241\u2013254.\n\nNeuhaus, W., and S. R. Lindsay. 2000. Handbook of applied dog behavior and training, Vol. 1: Adaptation and learning. Ames, Iowa: Blackwell Publishing.\n\nSulimov, K. T., V. I. Starovoitov, T. F. Moiseeva, I. I. Poletaeva, and E. P. Zinkevich. 1995. Dogs distinguish by scent quantitatively different mixtures of three higher fatty acids. Sensory Systems, 9, 99\u2013102.\n\non following scent trails:\n\nSommerville, B., and M. Green. 1989. The sniffing detective. New Scientist, 122, 54\u201357.\n\nDiscovery Channel show\n\nMythBusters: \"Dog Myths.\" Aired March 14, 2007.\n\nfive footsteps\n\nHepper, P. G., and D. L. Wells. 2005. How many footsteps do dogs need to determine the direction of an odour trail? Chemical Senses, 30, 291\u2013298.\n\nscent-identification lineups:\n\nSchoon, G. A. A. 1996. Scent identification lineups by dogs (Canis familiaris): Experimental design and forensic application. Applied Animal Behaviour Science, 49, 257\u2013267.\n\n\"signature odor\" of anal sac:\n\nBradshaw, J. 2011. Dog sense: How the new science of dog behavior can make you a better friend to your pet. New York: Basic Books.\n\nanal sac research:\n\nPreti, G., E. L. Muetterties, J. M. Furman, J. J. Kennelly, and B. E. Johns. 1976. Volatile constituents of dog (Canis familiaris) and coyote (Canis latrans) anal sacs. Journal of Chemical Ecology, 2, 177\u2013186.\n\nDoty, R. L., and I. Dunbar. 1974. Attraction of beagles to conspecific urine, vaginal and anal sac secretion odors. Physiology & Behavior, 12, 825\u2013833.\n\nsmell of fox:\n\nDoty 2003.\n\nfemales sniff faces first:\n\nBradshaw 2011.\n\napocrine glands on pads of dogs' feet:\n\nHepper, P., and D. Wells. 2015. Olfaction in the Order Carnivora: Family Canidae. In R. L. Doty, ed. 2015. Handbook of olfaction and gustation, 3rd ed (pp. 591\u2013603). Hoboken, NJ: Wiley-Blackwell.\n\nother animals' urine-marking:\n\nSNOWSHOE HARES:\n\nRezendes 1999; Liebenberg, L. 1990. A field guide to the animal tracks of Southern Africa. South Africa: David Philip Publishers.\n\nRHINO AND HIPPO:\n\nWatson, L. 2000. Jacobson's organ and the remarkable nature of smell. New York: W. W. Norton & Company.\n\nBUSH DOG:\n\nPorton, I. 1983. Bush dog urine-marking: Its role in pair formation and maintenance. Animal Behaviour, 31, 1061\u20131069.\n\nmice counter-marking:\n\nRich, T. J., and J. L. Hurst. 1999. The competing countermarks hypothesis: Reliable assessment of competitive ability by potential mates. Animal Behaviour, 58, 1027\u20131037.\n\nFerkin, M. H., and A. A. Pierce. 2007. Perspectives on over-marking: Is it good to be on top? Journal of Ethology, 25, 107\u2013116.\n\nthe \"Law of Urination\":\n\nYang, P. J., J. C. Pham, J. Choo, and D. L. Hu. 2014. Duration of urination does not change with body size. Proceedings of the National Academy of Sciences, 111, 11932\u201311937.\n\ndomestic dog marking behavior:\n\nBerthoud, D. 2010. Communication through scents: Environmental factors affecting the urine marking behaviour of the domestic dog, Canis familiaris, kept as a pet. PhD thesis, Anglia Ruskin University.\n\nLisberg, A. E., and C. T. Snowdon. 2011. Effects of sex, social status and gonadectomy on countermarking by domestic dogs, Canis familiaris. Animal Behaviour, 81, 757\u2013764.\n\nobjects and style of scent-rolling:\n\nGosling, L. M., and H. V. McKay. 1990. Scent-rubbing and status signalling by mammals. Chemoecology, 1, 92\u201395.\n\nKoler-Matznick, J., I. Lehr Brisbin Jr., and M. Feinstein. 2005. An ethogram for the New Guinea Singing (Wild) Dog (Canis hallstromi). The New Guinea Singing Dog Conservation Society, U.S.A.\n\ntheories of scent-rolling:\n\nDrea, C. M., S. N. Vignieri, S. B. Cunningham, and S. E. Glickman. 2002. Responses to olfactory stimuli in spotted hyenas (Crocuta crocuta): Investigation of environmental odors and the function of rolling. Journal of Comparative Psychology, 116, 331\u2013341.\n\nMcCormick, J. 1993. In praise of stinks. The Lancet, 341, 1126\u20131127.\n\nRyon, J., J. C. Fentress, F. H. Harrington, and S. Bragdon. 1986. Scent rubbing in wolves (Canis lupus): the effect of novelty. Canadian Journal of Zoology, 64, 573\u2013577.\n\nsoldiers returning home:\n\nsee e.g., <https:\/\/www.youtube.com\/watch?v=eZ6oS5dUT30>\n\nmirror mark studies:\n\nGallup Jr., G. G. 1970. Chimpanzees: Self-recognition. Science, 167, 86\u201387; Plotnik, J. M., F. B. M. de Waal, and D. Reiss. 2006. Self-recognition in an Asian elephant. Proceedings of the National Academy of Sciences, 103, 17053\u201317057; Reiss, D., and L. Marino. 2001. Mirror self-recognition in the bottlenose dolphin: A case of cognitive convergence. Proceedings of the National Academy of Sciences, 98, 5937\u20135942.\n\nstudy of dog urine in snow:\n\nBekoff, M. 2001. Observations of scent-marking and discriminating self from others by a domestic dog (Canis familiaris): Tales of displaced yellow snow. Behavioural Processes, 55, 75\u201379.\n\nsmells in wind:\n\nHull, J. M. 1990. Touching the rock: An experience of blindness. New York: Vintage Books.\n\nsmell of storm approaching:\n\nAckerman 1990. From Ackerman's observation about cows' anticipatory behavior before a storm.\n\n3: SNIFFING THE WIND\n\nright- and left-nostril sniffing:\n\nSiniscalchi, M., R. Sasso, A. M. Pepe, S. Dimatteo, G. Vallortigara, and A. Quaranta. 2011. Sniffing with the right nostril: Lateralization of response to odour stimuli by dogs. Animal Behaviour, 82, 399\u2013404.\n\nself-experimenters:\n\nFiks, A. P. 2003. Self-experimenters: Sources for study. Westport, CT: Praeger.\n\nWeber's experiments:\n\nMainland, J., and N. Sobel. 2006. The sniff is part of the olfactory percept. Chemical Senses, 31, 181\u2013196.\n\nNew Guinea singing dog sniff:\n\nKoler-Matznick et al. 2005.\n\nairflow of dog sniff:\n\nSettles, G. S., D. A. Kester, and L. J. Dodson-Dreibelbis. 2003. The external aerodynamics of canine olfaction. In F. G. Barth, J. A. C. Humphrey, and T. W. Secomb, eds. 2003. Sensors and sensing in biology and engineering (pp. 323\u2013355). New York: SpringerWein.\n\nSir Satan:\n\nSteen, J. B., I. Mohus, T. Kvesetberg, and L. Wall\u00f8e. 1996. Olfaction in bird dogs during hunting. Acta Physiologica Scandinavica, 157, 115\u2013119.\n\nsize of olfactory epithelium\n\nGerritsen and Haak 2015.\n\nnumber of odors detectable\n\nFirestein, S. 2001. How the olfactory system makes sense of scents. Nature, 413, 211\u2013218.\n\nlock and key theory of receptors:\n\nThis model is not without its detractors. Nineteenth-century models suggested that smell perception resulted from odor \"undulations\" or waves, as sound and vision perception do; in the twentieth century, Luca Turin elaborated this model and posited a vibrational theory of odor reception. (see Block et al. 2014. Implausibility of the vibrational theory of olfaction. Proceedings of the National Academy of Sciences, 112, e2766\u2013e2774.) For the most recent response to this idea, see Vosshall, L. B. 2015. Laying a controversial smell theory to rest. Proceedings of the National Academy of Sciences, 112, 6525\u20136526.\n\n\"key in pocket\" theory:\n\nShepherd, G. M. 2012. Neurogastronomy: How the brain creates flavor and why it matters. New York: Columbia University Press.\n\non dog genome, proportion committed to olfactory receptor genes:\n\nLindblad-Toh, K., C. M. Wade, and T. S. Mikkelsen, et al. 2005. Genome sequence, comparative analysis and haplotype structure of the domestic dog. Nature, 438, 803\u2013819.\n\nOstrander, E. 2007. Genetics and the shape of dogs. American Scientist, 95, 406\u2013413.\n\ngenetic differences in performance on olfactory tests:\n\nHepper and Wells 2015.\n\n\"brains did the smelling, and the nose was just the conduit\":\n\nsuch as Galen, see Totelin, L. 2015. Smell as sign and cure in ancient medicine. In M. Bradley, ed. 2015. Smell and the ancient senses (pp. 17\u201329). New York: Routledge.\n\nnose as brain's fan:\n\nas said by David Chudnovsky. In Preston, R. March 2, 1992. \"The Mountains of Pi.\" The New Yorker.\n\non Cajal's work:\n\nFigueres-O\u00f1ate, M., Y. Guti\u00e9rrez, and L. L\u00f3pez-Mascaraque. 2014. Unraveling Cajal's view of the olfactory system. Frontiers in Neuroanatomy, 8, 55.\n\nsize of olfactory lobe in dog and human:\n\nLaska, M., and L. T. Hernandez Salazar. 2015. Olfaction in nonhuman primates. In R. L. Doty, ed. 2003. Handbook of olfaction and gustation, 2nd ed. (pp. 605\u2013621).\n\ntopographical layers of olfactory bulb:\n\nBakker, J. 2013. Olfaction. In D. W. Pfaff, ed., Neuroscience in the 21st century: From basic to clinical (pp. 815\u2013837). New York: Springer-Verlag.\n\n\"odor of decayed worms\":\n\nAdrian, E. D. 1942. Olfactory reactions in the brain of the hedgehog. Journal of Physiology, 100, 459\u2013473.\n\ncheddar-based experiments:\n\nShepherd 2012.\n\nresearch exposing rats to predators' glandular secretions:\n\nZibrowski, E. M., and C. H. Vanderwolf. 1997. Oscillatory fast wave activity in the rat pyriform cortex: relations to olfaction and behavior. Brain Research, 766, 39\u201349.\n\nFor those interested in this kind of thing, the weasel odorant was 2-propylthietane; the red fox, trimethyl thiazoline.\n\nbrain response to \"owner smell\":\n\nBerns, G. S., A. M. Brooks, and M. Spivak. 2014. Scent of the familiar: An fMRI study of canine brain responses to familiar and unfamiliar human and dog odors. Behavioural Processes, 110, 37\u201346.\n\nflehmen response:\n\nSommerville, B. A., and D. M. Broom. 1998. Olfactory awareness. Applied Animal Behaviour Science, 57, 269\u2013286.\n\nGalizia, C. G., and P-M. Lledo. 2013. Olfaction. In C. G. Galizia, and P-M. Lledo, eds. Neurosciences: From molecule to behavior: A university textbook (pp. 253\u2013284). Heidelberg: Springer-Verlag.\n\npheromones and VNO:\n\nKarlson, P., and M. L\u00fcscher. 1959. \"Pheromones\": A new term for a class of biologically active substances. Nature, 183, 55\u201356.\n\nWyatt, T. D. 2014. Pheromones and animal behavior: Chemical signals and signatures. 2nd ed. Cambridge: Cambridge University Press.\n\nandrostenone:\n\nSell, C. S. 2014. Chemistry and the sense of smell. Hoboken, NJ: John Wiley & Sons, Inc.\n\nbombykol:\n\nBarnard, C. 2003. Animal behaviour: Mechanism, development, function and evolution. Canada: Pearson Education.\n\nVNO:\n\nBarrios, A. W., P. S\u00e1nchez-Quinteiro, and I. Salazar. 2014. Dog and mouse: Toward a balanced view of the mammalian olfactory system. Frontiers in Neuroanatomy, 8, 106.\n\nuse of the tail-wag as scent-spreader:\n\nHickman, G. C. 1979. The mammalian tail: A review of functions. Mammal Review, 9, 143\u2013157.\n\nLewin, V., and J. G. Stelfox. 1967. Functional anatomy of the tail and associated behaviour in woodland caribou. Canadian Field-Naturalist, 81, 63\u201366.\n\nsome air-scenting dogs clear their noses by lifting their heads:\n\nGerritsen and Haak 2015.\n\n4: WALKING WHILE SMELLING\n\n\"smelliest [square] blocks\":\n\n<http:\/\/sensorymaps.com\/portfolio\/new-yorks-smelliest-block\/>\n\nPorteous's smellscapes:\n\nPorteous, J. D. 1990. Landscapes of the mind: Worlds of sense and metaphor. Toronto: University of Toronto Press.\n\nDrobnick 2006.\n\nin ancient times, temples mixed milk and saffron into their plaster:\n\nClassen, C., D. Howes, and A. Synnott. 1994. Aroma: The cultural history of smell. London: Routledge.\n\nmosques were built with musk and rose water worked into their mortar:\n\nAckerman 1990.\n\nsmells of the week:\n\nPorteous 1990.\n\nNetherlands pedestrian plazas:\n\nfrom Kate McLean.\n\n\"One Hundred Sites of Good Fragrance\":\n\nJapanese Ministry of the Environment, <https:\/\/www.env.go.jp\/air\/kaori\/>. Retrieved July 2015.\n\nsmell of old books:\n\nStrli\u010d, M., J. Thomas, T. Trafela, L. Cs\u00e9falvayov\u00e1, I. Kralj Cigi\u010d, J. Kolar, and M. Cassar. 2009. Material degradomics: On the smell of old books. Analytical Chemistry, 81, 8617\u20138622.\n\nBuchbauer, G., L. Jirovetz, M. Wasicky, and A. Nikiforov. 1995. On the odor of old books. Journal of Pulp and Paper Science, 21, 398\u2013400.\n\nManhattan grid design:\n\nDrobnick 2006, p. 114.\n\nHenshaw 2013.\n\nsmells of Paris and London:\n\nKeate, G. 1802. Sketches from nature: Taken, and coloured, in a journey to Margate (1802). London: T. Hurst.\n\nMargolies, E. 2006. Vagueness gridlocked: A map of the smells of New York. In J. Drobnick, ed. (pp. 107\u2013117).\n\nHenshaw 2013.\n\nReinarz, J. 2014. Past scents: Historical perspectives on smell. IL: University of Illinois Press.\n\n\"deodorization\" projects:\n\nDrobnick 2006.\n\nhomogenization of city smells:\n\nDrobnick 2006.\n\nHenshaw walks to capture smellscapes:\n\nHenshaw 2013.\n\nhoney smell of Paris:\n\nStromberg, J. June 7, 2013. \"Mapping the smells of New York, Amsterdam and Paris, block by block.\" Smithsonian magazine.\n\ncorrespondences between color names and odors:\n\nGilbert, A. N., R. Martin, and S. E. Kemp. 1996. Cross-modal correspondence between vision and olfaction: The color of smells. The American Journal of Psychology, 109, 335\u2013351.\n\nmaple syrup smell:\n\nDePalma, A. October 29, 2005. \"Good smell vanishes, but it leaves air of mystery.\" New York Times.\n\nLindeman, S. June 14, 2010. \"The mystery of the maple syrup smell.\" The Atlantic.\n\nblindfolded undergrads navigating by smell:\n\nJacobs, L. F., J. Arter, A. Cook, and F. J. Sulloway. 2015. Olfactory orientation and navigation in humans. PLOS ONE, 10, e0129387.\n\nSailors use smell in navigation:\n\nBeck, H. 1973. Folklore and the sea. Middletown, CT: Wesleyan University Press.\n\nhoming pigeon:\n\nJacobs, L. F. 2012. From chemotaxis to the cognitive map: The function of olfaction. Proceedings of the National Academy of Sciences, 109, 10693\u201310700.\n\ndogs in the First World War:\n\nRichardson, E. H. 1920. British war dogs: Their training and psychology (pp. 171\u2013172). London: Skeffington & Son Ltd.\n\n5: PLAIN AS THE NOSE ON YOUR FACE\n\nalmost all living creatures smell:\n\nDoty 2003.\n\nsense people are most willing to lose:\n\nDrobnick 2006\u2014though people who come to suffer from anosmia tend to disagree.\n\nAsian elephant olfaction:\n\nRizvanovic, A., M. Amundin, and M. Laska. 2013. Olfactory discrimination ability of Asian elephants (Elephas maximus) for structurally related odorants. Chemical Senses, 38, 107\u2013118.\n\n\"smell of metal\":\n\nGlindemann, D., A. Dietrich, H-J. Staerk, and P. Kuschk. 2006. The two odors of iron when touched or pickled: (Skin) carbonyl compounds and organophosphines. Angewandte Chemie International Edition, 45, 7006\u20137009.\n\n\"principal axis of human odor perception . . . remains odor pleasantness\":\n\nYeshurun, Y., and N. Sobel. 2010. An odor is not worth a thousand words: From multidimensional odors to unidimensional odor objects. Annual Review of Psychology, 61, 219\u2013241.\n\n\"the greatest poets in the world\":\n\nWoolf, V. 1933. Flush: A biography. New York: Harcourt Brace Jovanovich.\n\nFreud: sublimation of smell:\n\nFreud, S. 1978. In Le Gu\u00e9rer, A. 2002. Olfaction and cognition: A philosophical and psychoanalytic view. In Rouby et al. 2002 (pp. 3\u201315).\n\nodorous is odious:\n\nDrobnick 2006, p. 14.\n\nsmell of the security blanket:\n\nWyatt 2014.\n\nthey don't know that \"skunk\" is a bad smell:\n\noriginally posited by Freud (1929\/1961). Civilization and its discontents, trans. J. Strachey. New York: W. W. Norton & Company; later confirmed (with alterations) experimentally.\n\nkinds of animal noses:\n\nMOLLUSKS:\n\nAche, B. W., and J. M. Young. 2005. Olfaction: Diverse species, conserved principles. Neuron, 48, 417\u2013430.\n\nNEMATODES:\n\nHart, A. C., and M. Y. Chao. 2010. From odors to behaviors in Caenorhabditis elegans. In A. Menini, ed. The Neurobiology of Olfaction. Boca Raton, FL: CRC Press\/Taylor & Francis.\n\nMOLES:\n\nCatania, K. C. 1999. A nose that looks like a hand and acts like an eye: The unusual mechanosensory system of the star-nosed mole. Journal of Comparative Physiology A, 185, 367\u2013372.\n\nWATER SHREW:\n\nCatania, K. C. 2006. Underwater 'sniffing' by semi-aquatic mammals. Nature, 444, 1024\u20131025.\n\nDEET as molecular confusant:\n\nPellegrino, M., N. Steinbach, M. C. Stensmyr, B. S. Hansson, and L. B. Vosshall. 2011. A natural polymorphism alters odour and DEET sensitivity in an insect odorant receptor. Nature, 478, 511\u2013514.\n\ncategories of noses:\n\nLaska and Hernandez Salazar 2015.\n\n\"has no exotic uses\":\n\nAsimov, I. 1963. The human body: Its structure and operation. Cambridge, MA: The Riverside Press.\n\n\"gale-force speeds\":\n\nClerico, D. M., W. C. To, and D. C. Lanza. 2003. Anatomy of the human nasal passages. In R. L. Doty, ed. 2003. Handbook of olfaction and gustation, 2nd ed (pp. 3\u201331). New York: Marcel Dekker, Inc.\n\nhuman pheromones:\n\nWyatt 2014.\n\ngreen cherry juice tastes like lime:\n\nSela, L., and N. Sobel. 2010. Human olfaction: A constant state of change-blindness. Experimental Brain Research, 205, 13\u201329.\n\n\"Baby smells an odor, mother says nothing\":\n\nRoach, M. 2013. Gulp: Adventures on the alimentary canal. New York: W. W. Norton & Company.\n\neighteen-inch buffer of personal space:\n\nHediger, H. 1950. Wild animals in captivity. London: Butterworth.\n\nability to smell banana odor:\n\nLaska, M., A. Seibt, and A. Weber. 2000. 'Microsmatic' primates revisited: Olfactory sensitivity in the squirrel monkey. Chemical Senses, 25, 47\u201353.\n\nsome other animals smell carbon dioxide:\n\nMice and rats smell CO2: Jones, W. 2013. Olfactory carbon dioxide detection by insects and other animals. Molecules and Cells, 35, 87\u201392.\n\nretronasal olfaction:\n\nShepherd 2012.\n\nretronasal olfaction in dogs:\n\nCraven, B. A., E. G. Paterson, and G. S. Settles. 2010. The fluid dynamics of canine olfaction: Unique nasal airflow patterns as an explanation of macrosmia. Journal of the Royal Society Interface, 7, 933\u2013943.\n\nselective anosmia can be inherited; genetic thresholds to detection:\n\nZhang, X., and S. Firestein. 2007. Nose thyself: Individuality in the human olfactory genome. Genome Biology, 8, 230.\n\n6: MY DOG MADE ME SMELL IT\n\ninborn skill at smelling:\n\nPorter, R. H., J. M. Cernoch, and F. J. McLaughlin. 1983. Maternal recognition of neonates through olfactory cues. Physiology & Behavior, 30, 151\u2013154.\n\nSchaal, B., L. Marlier, and R. Soussignan. 1995. Responsiveness to the odour of amniotic fluid in the human neonate. Biology of the Neonate, 67, 397\u2013406.\n\nMallet, P., and B. Schaal. 1998. Rating and recognition of peers' personal odors by 9-year-old children: An exploratory study. Journal of General Psychology, 125, 47\u201364.\n\nIn addition to citations, some of this comes from Sela and Sobel 2010. Also Gilbert 2008; Gerritsen and Haak 2015; Wyatt 2014.\n\nnonhuman animal scent recognition:\n\nPAPER WASP AND BELDING'S GROUND SQUIRREL:\n\nAlcock, J. Animal Behavior.\n\nspotted hyena:\n\nEmery, N. et al., eds. Social intelligence: From brain to culture.\n\nowner's rating of smell of dog's blanket:\n\nWells, D. L., and P. G. Hepper. 2000. The discrimination of dog odours by humans. Perception, 29, 111\u2013115.\n\nsmell of lab mice:\n\nGilbert, A. N., K. Yamazaki, G. K. Beauchamp, and L. Thomas. 1986. Olfactory discrimination of mouse strains (Mus musculus) and major histocompatibility types by humans (Homo sapiens). Journal of Comparative Psychology, 100, 262\u2013265.\n\n\"You just smell the books\":\n\nFeynman, R. P. 1985. \"Surely you're joking, Mr. Feynman!\" (pp. 105\u2013106). New York: W. W. Norton & Company. Found via Gilbert 2008.\n\nbipedalism demoting smells:\n\nE.g., Sigmund Freud (1929\/1961) and Stuart Firestein (interview), among others.\n\ndemotion of smell with promotion of vision:\n\nas laid out in Wyatt 2014 and Shepherd 2012.\n\nJames on sensory component of practice:\n\nJames, W. 1890. The Principles of Psychology, vol. 1. New York: Henry Holt & Co.\n\nexperiment on smell learning:\n\nLi, W., J. D. Howard, T. B. Parrish, and J. A. Gottfried. 2008. Aversive learning enhances perceptual and cortical discrimination of indiscriminable odor cues. Science, 319, 1842\u20131845.\n\nchocolate trail following:\n\nPorter, J., B. Craven, R. M. Khan, S-J. Chang, I. Kang, B. Judkewitz, J. Volpe, G. Settles, and N. Sobel. 2007. Mechanisms of scent-tracking in humans. Nature Neuroscience, 10, 27\u201329.\n\n\"The lost muscles of the nose\":\n\nThis title comes from a nearly selfsame paper in the journal Aesthetic Plastic Surgery.\n\nLevator labii superioris:\n\nStandring, S. 2015. Gray's anatomy: The anatomical basis of clinical practice. 41st ed. New York: Elsevier.\n\n5 to 10 percent of air makes it to olfactory epithelium:\n\nRoach 2013.\n\nOlfaction is an active process:\n\nMainland, J., and N. Sobel. 2006. The sniff is part of the olfactory percept. Chemical Senses, 31, 181\u2013196.\n\nsniff vigor, volume, flow, and value:\n\nMainland and Sobel 2006.\n\nEven in sleep, our brain registers smells:\n\nArzi, A., L. Shedlesky, M. Ben-Shaul, K. Nasser, A. Oksenberg, I. S. Hairston, and N. Sobel. 2012. Humans can learn new information during sleep. Nature Neuroscience, 15, 1460\u20131465.\n\ntwo sniffs are better than one:\n\nJoel Mainland, personal communication. May 13, 2015.\n\nright- and left-nostril differences:\n\nHerz, R. S., C. McCall, and L. Cahill. 1999. Hemispheric lateralization in the processing of odor pleasantness versus odor names. Chemical Senses, 24, 691\u2013695.\n\ndiscriminate unknown, new odors when using the right nostril:\n\nSavic, I., and H. Berglund. 2000. Right-nostril dominance in discrimination of unfamiliar, but not familiar, odours. Chemical Senses, 25, 517\u2013523.\n\nolfactometer:\n\nTotelin 2015.\n\nno basic words for fundamental smells:\n\nSperber, D. 1975. Rethinking symbolism. Cambridge: Cambridge University Press.\n\nMalay olfactory language:\n\nWnuk, E., and A. Majid. 2014. Revisiting the limits of language: The odor lexicon of Maniq. Cognition, 131, 125\u2013138.\n\nMajid, A., and N. Burenhult. 2014. Odors are expressible in language, as long as you speak the right language. Cognition, 130, 266\u2013270.\n\ndifficulty naming familiar odors:\n\nYeshurun and Sobel 2010.\n\ntrigeminal nerve:\n\nShusterman, D. 2009. Qualitative effects in nasal trigeminal chemoreception. Annals of the New York Academy of Sciences, 1170, International Symposium on Olfaction and Taste (pp. 196\u2013201).\n\n7: NOSE TO GRINDSTONE\n\nold DuPont plant, now Penn's WDC:\n\n<http:\/\/www.upenn.edu\/pennnews\/current\/2012-09-13\/features\/penn's-south-bank-23-acres-pure-potential>. Retrieved October 15, 2015. <http:\/\/www.workshopoftheworld.com\/south_phila\/dupont.html>. Retrieved October 15, 2015.\n\nHemingway on the smell of death:\n\nHemingway, E. 1940. For whom the bell tolls. New York: Charles Scribner's Sons.\n\n\"saccharine putrescence\":\n\nO'Rourke, P. J. 1988. Holidays in hell. New York: Atlantic Monthly Press.\n\nMortuary technician Carla Valentine:\n\nA Life in Scents podcast, <http:\/\/bit.ly\/1WrtaHY>. Retrieved November 1, 2015.\n\n8: NOSE-WISE\n\nearly cancer-finding dogs:\n\nWilliams, H., and A. Pembroke. 1989. Sniffer dogs in the melanoma clinic? The Lancet, 333, 734.\n\nChurch, J., and H. Williams. 2001. Another sniffer dog for the clinic? The Lancet, 358, 930.\n\nWelsh, J. S., D. Barton, and H. Ahuja. 2005. A case of breast cancer detected by a pet dog. Community Oncology, 2, 324\u2013326.\n\nvolatiles in cancer:\n\nWells, D. L. 2012. Dogs as a diagnostic tool for ill health in humans. Alternative Therapies in Health and Medicine, 18, 12\u201317.\n\nbladder cancer study:\n\nWillis, C. M., S. M. Church, C. M. Guest, W. A. Cook, N. McCarthy, A. J. Bransbury, M. R. T. Church, and J. C. T. Church. 2004. Olfactory detection of human bladder cancer by dogs: Proof of principle study. British Medical Journal, 329, 712\u2013714.\n\nsmells in urine:\n\nShirasu, M., and K. Touhara. 2011. The scent of disease: Volatile organic compounds of the human body related to disease and disorder. Journal of Biochemistry, 150, 257\u2013266.\n\nprostate cancer study:\n\nCornu, J. N., G. Cancel-Tassin, V. Ondet, C. Girardet, and O. Cussenot. 2011. Olfactory detection of prostate cancer by dogs sniffing urine: A step forward in early diagnosis. European Urology, 59, 197\u2013201.\n\nmelanoma detection study:\n\nPickel, D., G. P. Manucy, D. B. Walker, S. B. Hall, and J. C. Walker. 2004. Evidence for canine olfactory detection of melanoma. Applied Animal Behaviour Science, 89, 107\u2013116.\n\nodors of breath:\n\nPhillips, M., J. Herrera, S. Krishnan, M. Zain, J. Greenberg, and R. N. Cataneo. 1999. Variation in volatile organic compounds in the breath of normal humans. Journal of Chromatography B, 729, 75\u201388.\n\nlung cancer study:\n\nMcCulloch, M., T. Jezierski, M. Broffman, A. Hubbard, K. Turner, and T. Janecki. 2006. Diagnostic accuracy of canine scent detection in early- and late-stage lung and breast cancers. Integrative Cancer Therapies, 5, 30\u201339.\n\nGC as generating an \"arpeggio\":\n\nGilbert 2008.\n\npeaks of GC:\n\nShepherd 2012.\n\nGilbert 2008.\n\n\"fully biocompatible and patient friendly alarm system\":\n\nChen, M., M. Daly, N. Williams, S. Williams, C. Williams, and G. Williams. 2002. Non-invasive detection of hypoglycaemia using a novel, fully biocompatible and patient friendly alarm system. British Medical Journal, 321, 1565\u20131566.\n\ndiabetic-alert dogs:\n\nRooney, N. J., S. Morant, and C. Guest. 2013. Investigation into the value of trained glycaemia alert dogs to clients with type I diabetes. PLOS ONE, 8, e69921.\n\nHippocrates \"open nose\"\n\nLe Gu\u00e9rer 2002.\n\n\"As water changes to air . . .\":\n\nPlato, via Totelin 2015\n\nearly uterine theory; Pliny cure for smelly armpits:\n\nTotelin 2015, p. 27.\n\nLinnaeus:\n\nLinnaeus. 1764. Odores medicamentorum. Amoenitates Academicae, vol. 3 (pp. 183\u2013201). Stockholm: Lars Salvius.\n\nNB: Gilbert 2008 made this observation first.\n\nmedicinal effect of fragrant and repulsive plants:\n\nSchiller F. 1997. A memoir of olfaction. Journal of the History of the Neurosciences, 6, 133\u2013146.\n\nGalen:\n\nTotelin 2015.\n\nhelpful sixteenth-century instructional manual:\n\n\"Rem\u00e8des, Pr\u00e9servatifs et Curatifs de Peste,\" 1562. In Feigel 2006.\n\nsmell of breath:\n\nKwak, J., and G. Preti. 2011. Volatile disease biomarkers in breath: A critique. Current Pharmaceutical Biotechnology, 12, 1067\u20131074.\n\nsmell of diseases, psychiatric problems, and toxins:\n\nWatson 2000.\n\nOrient, J. M., ed. 2010. Sapira's art and science of bedside diagnosis, 4th ed. Philadelphia: Lippincott Williams & Wilkins.\n\nKenny, J. C. 1989. The valuing, educational preparation and diagnostic use of the olfactory sense in nursing practice. Dissertation, Adelphi University.\n\nuse of odors in Western medicine:\n\nE.g., footnote in L. Goldfrank, R. Weisman, and N. Flomenbaum. 1982. Teaching the recognition of odors. Annals of Emergency Medicine, 11, 22.\n\n\"The characteristic foul odor of the sputum suggests anaerobic involvement\":\n\nChung, G., and M. B. Goetz. 2000. Anaerobic infections of the lung. Current Infectious Disease Reports, 2, 238\u2013244.\n\nbattery of vials for sniff-training:\n\nOrient 2010.\n\na \"ten test tube sniffing bar\":\n\nKenny 1989.\n\nTCM's method:\n\nLiu, Z., and L. Liu eds. 2010. Essentials of Chinese medicine, vol. 1. London: Springer-Verlag.\n\nFive elements\/phases:\n\nUnschuld, P. U. 1985. Medicine in China: A history of ideas. Berkeley: University of California Press.\n\nalcoholic, corn farmer, coal miners:\n\nBehrman, A. D., and S. Goertemoeller. 2009. What is that smell? Journal of Emergency Nursing, 35, 263\u2013264.\n\nconditions that produce distinctive odors:\n\nDoty 2001; Wyatt 2014; Kenny 1989.\n\nwoman smelling Parkinson's:\n\n<http:\/\/www.scientificamerican.com\/article\/one-woman-s-ability-to-sniff-out-parkinson-s-offers-hope-to-sufferers\/>. Retrieved November 22, 2015.\n\nsense of smell in early Parkinson's:\n\nDoty, R. L., S. M. Bromley, and M. B. Stern. 1995. Olfactory testing as an aid in the diagnosis of Parkinson's disease: Development of optimal discrimination criteria. Neurodegeneration, 4, 93\u201397.\n\nscratch-and-sniff:\n\nDoty 2009; Wyatt 2014.\n\n\"My genius resides in my nostrils\":\n\nNietzsche, F. 1911\/2004. Ecce Homo (p. 132). A. M. Ludovici (transl). Mineola, N.Y.: Dover.\n\nnose-wise:\n\nOxford English Dictionary, OED online. Retrieved March 2015.\n\n9: STINK-WAVES\n\nRoald Dahl:\n\nThe Witches (p. 24): 1983. London: Puffin; \"Jack and the Beanstalk,\" from Revolting rhymes. 1982. New York: Alfred A. Knopf.\n\nribbon snakes:\n\nGadbois, S., and C. Reeve. 2014. Canine olfaction: Scent, sign, and situation. In A. Horowitz, ed. Domestic dog cognition and behavior: The scientific study of Canis familiaris. Heidelberg: Springer-Verlag.\n\nenvironmental contaminants:\n\nArner, L. D., G. R. Johnson, and H. S. Skovronek. 1986. Delineating toxic areas by canine olfaction. Journal of Hazardous Materials, 13, 375\u2013381.\n\nsui generis:\n\nUnited States v. Place (1983).\n\nPliny:\n\n1855. The natural history of Pliny, vol. 2. (Book VII, p. 314). Bostock, J., and H. J. Riley (transl). London: Henry G. Bohn.\n\nsmell of person on pipe bomb:\n\nCurran, A. M., P. A. Prada, and K. G. Furton. 2010. Canine human scent identifications with post-blast debris collected from improvised explosive devices. Forensic Science International, 199, 103\u2013108.\n\ndetection of cadavers in water:\n\nKillam, E. W. 1990. The detection of human remains. Springfield, IL: Charles C. Thomas.\n\nWarren, C. 2013. What the dog knows (p. 211). New York: Touchstone.\n\navalanche-rescue dogs:\n\nKillam 1990.\n\nlongevity of human smell on objects:\n\nCurran, Prada, Furton 2010.\n\nSyrotuck, W. G. 1972. Scent and the scenting dog (p. 106). Mechanicsburg, PA: Barkleigh Productions.\n\nhuman \"scent samples\":\n\nLesniak, A., M. Walczak, T. Jezierski, M. Sacharczuk, M. Gawkowski, and K. Jaszczak. 2008. Canine olfactory receptor gene polymorphism and its relation to odor detection performance by sniffer dogs. Journal of Heredity 99, 518\u2013527.\n\nCurran, A. M., S. I. Rabin, and K. G. Furton. 2005. Analysis of the uniqueness and persistence of human scent. Forensic Science Communications, 7, 2.\n\nskin cell sloughing:\n\nAllen T., and G. Cowling. 2011. The cell: A very short introduction (p. 10). Oxford: Oxford University Press.\n\nsweat generated in sitting or exercising:\n\nMedeiros, D. M., and R. E. C. Wildman. 2012. Advanced human nutrition, 2nd ed. Sudbury, MA: Jones & Bartlett Learning.\n\nWatson 2000.\n\nolf:\n\ncoined by Fanger, P. O. 1988. Perceived quality of indoor and ambient air. Proceedings of the Indoor Ambient Air Quality Conference, London, 365\u2013376. In McCormick 1993.\n\ncomponents of sweat:\n\nCurran, A. M., S. I. Rabin, P. A. Prada, and K. G. Furton. 2005. Comparison of the volatile organic compounds present in human odor using SPME-GC\/MS. Journal of Chemical Ecology, 31, 1607\u20131619.\n\nclues in a footprint:\n\nGerritsen and Haak 2015.\n\nWright, R. H. 1982. The sense of smell. Boca Raton, FL: CRC Press.\n\nwet shoe experiment:\n\nGerritsen and Haak 2015.\n\nsweat from soles of feet:\n\nSyrotuck 1972.\n\n\"dopamine breeds\":\n\nGadbois and Reeve 2014.\n\nNorway sniffing study:\n\nThesen, A., J. B., Steen, and K. B. D\u00f8ving. 1993. Behaviour of dogs during olfactory tracking. Journal of Experimental Biology, 180, 247\u2013251.\n\n\"professional poop chasers\":\n\nWasser, S. K. 2008. Lucky dogs. Natural History, 117, 48\u201353.\n\nyellow baboons:\n\nWasser, S. K., ed. 1983. Social behavior of female vertebrates. New York: Academic Press.\n\neffect of zebra finch leg bands:\n\nBurley, N. 1988. Wild zebra finches have band-colour preferences. Animal Behaviour, 36, 1235\u20131237.\n\neffect of human and wolf activity on caribou populations:\n\nWasser, S. K., J. L. Keim, M. L., Taper, and S. R., Lele. 2011. The influences of wolf predation, habitat loss, and human activity on caribou and moose in the Alberta oil sands. Frontiers in Ecology and the Environment, 9, 546\u2013551.\n\ndogs with excessive energy:\n\nWasser 2008.\n\nTucker's Orca-scat detection:\n\nAyres, K. L., R. K., Booth, J. A., Hempelmann, K. L., Koski, C. K., Emmons, R. W., Baird, K., Balcomb-Bartok, M. B. Hanson, M. J. Ford, and S. K. Wasser. 2012. Distinguishing the impacts of inadequate prey and vessel traffic on an endangered killer whale (Orcinus orca) population. PLOS ONE, 7, e36842.\n\nscent at bus stop:\n\nWatson 2000, p. 72.\n\narmpit sniffing of Kanum-Irebe:\n\nEibl-Eibesfeldt, I. 1971. Love and hate: The natural history of behavior patterns (p. 191). New York: Holt, Rinehart and Winston.\n\nSchultze-Westrum 1968, as cited in Mykytowycz, R. 1985. Olfaction\u2014A link with the past. Journal of Human Evolution, 14, 75\u201390.\n\nsynchronization of menstrual cycles:\n\nStern, K., and M. K. McClintock. 1998. Regulation of ovulation by human pheromones. Nature, 392, 177\u2013179. This result is still debated.\n\ncold, calm winter days are good for ground tracking:\n\nGerritsen and Haak 2015.\n\n\"novel in an owl's pellet\":\n\nRezendes 1999.\n\nrains can \"breathe new life\":\n\nGerritsen and Haak 2015.\n\nscent posts:\n\nRezendes 1999.\n\nsome porcupine facts:\n\n<http:\/\/www.nwf.org\/news-and-magazines\/national-wildlife\/animals\/archives\/1994\/prying-into-the-life-of-a-prickly-beast.aspx>. Retrieved November 1, 2015.\n\ndog's gait is \"sloppy\":\n\nRezendes 1999.\n\n\"Smelly\" Kelly:\n\nJones, P. 1978. Under the city streets: A history of subterranean New York. New York: Holt, Rinehart and Winston.\n\nThe MTA now uses electronic \"sniffers\" that analyze air samples, alas. (Neuman, W. October 3, 2006. \"M.T.A. to Upgrade Chemical-Detection System.\" New York Times.)\n\n10: CIVET CATS AND WET DOGS\n\ntruffle biology:\n\nRubini, A., C. Riccioni, S. Arcioni, and F. Paolocci. 2007. Troubles with truffles: Unveiling more of their biology. New Phytologist, 174, 256\u2013259.\n\nKunzig, R. 2000. The biology of . . . truffles. Expensive and delectable, truffles are one crop modern agriculture can't tame. Discover.\n\n\"Have the local experts . . . been creating little digs in the earth?\":\n\nTrappe, M., F. Evans, and J. Trappe. 2007. Field guide to North American truffles: Hunting, identifying, and enjoying the world's most prized fungi. New York: Ten Speed Press.\n\ntruffle descriptors:\n\nFrom Alana McGee and Trappe, Evans, and Trappe 2007.\n\nChesterton:\n\nChesterton, G. K. 1914. The flying inn. New York: John Lane Co.\n\nGreeks and Romans:\n\nDoty 2003.\n\n\"extends [one's body] into the space beyond which it should\":\n\nTullett, W. A Life in scents podcast, <http:\/\/bit.ly\/1XMs36f>\n\nif a woman \"seduced or betrayed into matrimony\" any gentleman by use of scents:\n\nDoty 2003.\n\n\"rear end of an Asian cat\":\n\nTurin, L. 2006. The secret of scent: Adventures in perfume and the science of smell (p. 90). New York: HarperCollins.\n\nChanel No. 5:\n\nSell, C. S., ed. 2006. The chemistry of fragrances: From perfumer to consumer, 2nd ed. Cambridge, UK: The Royal Society of Chemistry.\n\nSmelling notes from Raymond Matts.\n\nsmell \"the hands that picked\" the flower:\n\nFeigel 2006, p. 73.\n\n\"show promise on olfactory and psychological tests\":\n\nwww.ifraorg.org.\n\nperfumers' brains:\n\nPlailly, J., C. Delon-Martin, and J-P. Royet. 2012. Experience induces functional reorganization in brain regions involved in odor imagery in perfumers. Human Brain Mapping, 33, 224\u2013234.\n\nambergris:\n\nRice, D. W. 2008. Ambergris. In W. F. Perrin, B. W\u00fcrsig, and J. G. M. Thewissen, eds. Encyclopedia of marine mammals, 2nd ed (p. 28). San Diego, CA: Academic Press.\n\nMontaigne's \"mustachios, which are full\":\n\nMontaigne M. de. 1580. C. Cotton (transl). Essays of Montaigne. New York: Edwin C. Hill.\n\ntea expert who knows whether the plant grew near a plum tree:\n\nAs invoked by Huysmans, J-K. 1884. Against the grain. In Feigel 2006.\n\nperfumery methods:\n\nCinqui\u00e8me Sens, \"Introduction to the techniques and language of perfumery.\"\n\npreserve the headspace of endangered flowers:\n\nThe flowers are also memorialized in Kaiser's book, Scent of the Vanishing Flora.\n\n\"verbalize their olfactory experience\":\n\nRoyet, J-P., J. Plailly, A-L. Saive, A. Veyrac, and C. Delon-Martin. 2013. The impact of expertise in olfaction. Frontiers in Psychology, 4, 928.\n\nverbal landmarks to invoke a smell:\n\nGilbert, A. N., M. Crouch, and S. E. Kemp. 1998. Olfactory and visual mental imagery. Journal of Mental Imagery, 22, 137\u2013146.\n\n\"change detection\" system:\n\nThis is adaptation, introduced in chapter 6, My Dog Made Me Smell It. \"Change detection\" from Herz, R. 2007. The scent of desire: Discovering our enigmatic sense of smell. New York: William Morrow.\n\nadvice of an animal-tracking handbook:\n\nYoung, J., and T. Morgan. 2007. Animal tracking basics. Mechanicsburg, PA: Stockpole Books.\n\nJahai:\n\nMajid and Burenhult 2014.\n\nolfactory hallucinations:\n\nLeopold, D. 2002. Distortion of olfactory perception: Diagnosis and treatment. Chemical Senses, 27, 611\u2013615.\n\nNoble wine wheel aromas:\n\nNoble, A. C., R. A. Arnold, J. Buechsenstein, E. J. Leach, J. O. Schmidt and P. M. Stern. 1987. Modification of a standardized system of wine aroma terminology. American Journal of Enology and Viticulture, 38, 143\u2013146.\n\nNoble, A. C. 2009. Using the wine aroma wheel. Downloaded August 27, 2015, from <http:\/\/winearomawheel.com\/Websites\/aromawheel\/Images\/userguide_2010.pdf>.\n\nTCA blocks olfactory receptors:\n\nTakeuchi, H., H. Kato, and T. Kurahashi. 2013. 2,4,6-Trichloroanisole is a potent suppressor of olfactory signal transduction. Proceedings of the National Academy of Sciences, 110, 16235\u201316240.\n\n11: NOSED OUT\n\nsmelloftheworld:\n\ncummings, e. e. 1925. \"and this day it was Spring.\"\n\nquantity-smelling study:\n\nHorowitz, A., J. Hecht, and A. Dedrick. 2013. Smelling more or less: Investigating the olfactory experience of the domestic dog. Learning and Motivation, 44, 207\u2013217.\n\ndogs are not using smell to capacity:\n\nSee, e.g., Polg\u00e1r, Z., \u00c1. Mikl\u00f3si, and M. G\u00e1csi. 2015. Strategies used by pet dogs for solving olfaction-based problems at various distances. PLOS ONE, 10, e0131610.\n\nObedience is \"totally unnecessary\" . . . \"a hindrance\":\n\nTurid Rugaas; Anne Lill Kvam.\n\n\"lost keys\" game:\n\nKvam, A. L. and T. Rugaas. 2012. Nosework Search games DVD.\n\n12: SMELLOFTHEWORLD\n\ngeosmin:\n\nYuhas, D. July 18, 2012. \"Storm scents.\" Scientific American.\n\nozone:\n\nR. Matts, personal communication; Yuhas 2012. Etymology: Oxford English Dictionary.\n\nsmell of a penny:\n\nGlindemann, Dietrich, Staerk, Kuschk 2006.\n\n## Thanks, an Index of\n\nbook\n\nadvocacy of, superagent Kris Dahl, Caroline Eisenmann\n\nenactors of, Nan Graham, Colin Harrison, Roz Lippel, Susan Moldow, and Scribner\n\nmaking of\n\nart, Vegar Abelsnes, Jaya Miceli\n\neditorial, Colin Harrison (my ideal reader)\n\neditorial, assistance of, Sarah Goldberg\n\nmarketing and publicity, Katie Monaghan\n\nproduction editorial and design, Mia Crowley-Hald, Erich Hobbing\n\nspace for writing, New York Society Library\n\nwilling and useful interlocutors on, Brian Boyd, Betsy Carter, Alison Curry, Holly Fairbank, Glen Finkel, Elizabeth Hardin, Damon Horowitz, Ogden Thelonious Horowitz Shea, Jay Horowitz, Daniel Hurewitz, Elizabeth Kadetsky, Maira Kalman, Sally Koslow, Aryn Kyle, Maria Popova, Douglas Repetto, Ammon Shea, Timea Szell, Andy Tuck, Jennifer Vanderbes, Carlin Wing, Mark Woods\n\ndog nose\n\namateur, every dog in New York City, Finnegan, Upton, volunteer subjects in my studies\n\nexperts on, Brent Craven, Simon Gadbois, Gary Settles\n\ninstructor of, George Berger\n\nmodel, Finnegan\n\nprofessional, Penn Working Dog Center dogs; truffle dogs Callie, Cash, Da Vinci, Lolo; scat-detection dogs; tracking dogs\n\nstudies of, Barnard College, John Herrold and the NYC Parks Department, members of the Dog Cognition Lab, owner participants\n\ntrainers of, Jonathan Ball, Annemarie DeAngelo, Bob Dougherty, Pat Kaynaroglu, Cindy Otto, and everyone at the WDC; Alana McGee and Kristin Rosenbach; Sam Wasser\n\nhuman nose\n\ndemonstrators of abilities of, John Buechsenstein, Noah Charney, Charley Eiseman, Leta Herman, Ray Matts, Kate McLean\n\nexperts on, Stuart Firestein, Avery Gilbert, George Preti, Leslie Vosshall\n\npedagogical, John Buechsenstein, Ray Matts\n\nrespondents to questions of, Bennett Lorber, Jane Orient, Oliver Sacks, Gary Settles, Abraham Verghese, Ron Winnegrad\n\nusers of, with me, John Gillespie, Ogden Thelonious Horowitz Shea, Susan Orlean, Ammon Shea, Peter Sokolowski\n\ninspiration\n\nconstant, Ogden Thelonious Horowitz Shea, Ammon Shea\n\n## Index\n\nA note about the index: The pages referenced in this index refer to the page numbers in the print edition. Clicking on a page number will take you to the ebook location that corresponds to the beginning of that page in the print edition. For a comprehensive list of locations of any word or phrase, use your reading system's search function.\n\nadaptation, , ,\n\nAdrian, Edgar,\n\nair, olfactory abilities for smells in, \u201330,\n\nAiredales,\n\nair fresheners, , ,\n\nair movement\n\nmosquitoes confused by,\n\nnosework training and, \u201368\n\nsearches affected by, , \u201353\n\ntruffle hunting and, \u201326\n\nair-scenting, , , ,\n\nalar folds, , ,\n\nAlzheimer's disease, \u201382\n\nambergris,\n\namygdala, ,\n\nanal-gland scents, , ,\n\nandrostadienone,\n\nandrostenone, , , \u201322\n\nanimal state of knowledge,\n\nanimal tracking, \u201317. See also tracking\n\nair-scenting in,\n\nanimal smells in, \u20133, \u20135\n\nfootprints in, , \u201316\n\ngoal of,\n\nhumans' long history of, \u20133\n\njournal kept in, \u20137\n\nnose preparation for, \u20136\n\nporcupines identified on, \u201315\n\nscent-markings in, , \u201316\n\nsmells used in, \u20135, \u201311\n\nsnow cover and,\n\nstudent training for, \u20137,\n\nthinking like animals tracked during,\n\ntree marks used in, \u201310\n\nanosmia (smell-blindness), , ,\n\napocrine glands,\n\nAsimov, Isaac,\n\navalanche-rescue dogs,\n\nAxel, Richard,\n\nbabies, smell in, , ,\n\nbaboons, \u201393,\n\nBall, Jonathan, ,\n\nBarnard College,\n\nbears, , \u201395, , , ,\n\nbedbugs, , ,\n\nBekoff, Marc, \u201325\n\nBelgian Malinois, , ,\n\nBelgian Tervuren,\n\nBerger, George, , , \u201361, , , \u201366\n\nbipedalism, , \u20135, \u201327\n\nbladder cancer, \u201365\n\nblind dogs,\n\nblindness\n\nguide dogs for, ,\n\nuse of smell in, \u20138\n\nblindness to smells (anosmia), , ,\n\nbloodhounds, , , \u20134, , ,\n\nblowing, to heighten odors,\n\nbobcats, , , , \u201312\n\nbombykol,\n\nbooks, recognizing by smell, \u20134,\n\nBorder collies, , ,\n\nBorder Patrol,\n\nbrain\n\nmemory areas in, , ,\n\nnostril side preferences in sniffing and, \u201337\n\nof perfumers,\n\nin smelling, \u201354, , \u201391, \u201396, \u201312\n\nin vision-processing, ,\n\nbreast cancer,\n\nbreathing\n\nairflow in, ,\n\nmucus and,\n\nnasal muscles for,\n\northonasal olfaction and,\n\nsniffing as part of, , ,\n\nbreath-sniffing, , , , , , ,\n\nbreeding of working dogs,\n\nbreeds. See also specific breeds\n\nolfactory variations and,\n\ntracking abilities of, , \u201390, ,\n\nBretagne (working dog),\n\nBuck, Linda,\n\nBuechsenstein, John, \u201351, \u201354\n\ncadavers, detection of, ,\n\nCallie (working dog), ,\n\ncancer detection, \u201369,\n\nanecdotal reports on dogs and, \u201359\n\nbreath samples used in,\n\ncomponent parts of smell of cancer in, \u201369\n\nresearch studies on, , \u201365\n\ntraining dogs for, \u201364\n\nurine samples used in, \u201365\n\ncarcinoma, dog's detection of,\n\ncaribou, ,\n\ncartilage, nasal, , ,\n\nCash (working dog), , \u201327\n\ncastoreum, ,\n\ncats, , , \u201355\n\ncaudate nucleus,\n\ncerebellum,\n\nChanel, Coco,\n\nChanel No. 5 perfume,\n\nCharney, Noah, , , , \u20138, \u201310, , , , \u201317\n\nchemical detector, nose as,\n\nchemosensation,\n\nchildhood, memories from, \u201398,\n\nchildren\n\nsense of smell in,\n\nwitch's smelling out, \u201384\n\nchildren's books, smells in, \u20138\n\nchimpanzees, ,\n\nChinese medicine, , \u201380\n\ncilia, nasal,\n\ncities\n\ncontamination by smell in,\n\ndeodorization projects in, \u201370\n\nsmell-mapping of, \u201367, \u201375\n\nsmellscapes of scents in, \u201368\n\nstreet layout for smell-letting in, \u201369\n\ntracing smells to sources in, \u201375,\n\ncivet secretions, , ,\n\nclothing, smell on, , ,\n\ncompanion dogs, ,\n\nConservation Canines, , ,\n\ncortex, , , , , ,\n\ncounter-marking, , , , , ,\n\ncoyotes, , , , , ,\n\nCraven, Brent, \u201334, \u201346,\n\nCulinary Institute of America,\n\ncultural sensitivities about smell, , \u201375\n\ncummings, e. e.,\n\ncyanide poisoning,\n\nDahl, Roald, , , n\n\ndanger, smell as predictive of,\n\nDa Vinci (working dog),\n\ndaydreaming and smells,\n\nDeAngelo, Annemarie, , , \u201350, , \u201363\n\n\"deciding\" phase of tracking,\n\ndeer, , , ,\n\ndetection dogs, \u201356, \u201391. See also tracking dogs\n\ncancer cells and, \u201359. See also cancer detection\n\ncertification standards for,\n\ncharacteristics of,\n\ndiabetes and, , \u201371\n\nepileptic seizures and, ,\n\ngenetic differences affecting,\n\nhandlers' relationship with,\n\nin law enforcement,\n\nlegal recognition of noses' authority in,\n\nnosework modeled on training of,\n\nolfactory abilities of, \u201321,\n\nperson's scent used by, \u201388\n\nrange of items found by, , ,\n\nscent cones used by, , \u201397\n\nscientific measures of olfactory sensitivity in, \u20139\n\nsearch and rescue by, , , \u201386\n\nas smelling experts,\n\nsniffing by,\n\nsorting smells by,\n\nstink-waves and, , \u201388\n\ntracking abilities of breeds in, \u201390,\n\ntraining of, \u201321, \u201330\n\ntruffle hunting by, , , \u201328, \u201331\n\nvision and hearing as aids for, , ,\n\ndiabetic alert dogs, , \u201371\n\ndiabetic coma, ,\n\ndiseases\n\nChinese medicine and, \u201380\n\nmedical smelling for diagnosis of, \u201378, \u201382\n\nperfumes as safeguards against malodors in,\n\nsmells in detection of, , \u201376\n\nDog Cognition Lab at Barnard College, , ,\n\ndog trainers, . See also training of working dogs\n\nat Working Dog Center, , ,\n\nDougherty, Bob, , \u201339, \u201343, , , ,\n\ndreaming\n\ndogs' smelling during, \u201359\n\nsmells experienced during, ,\n\nDrobnick, Jim,\n\ndrug detection, \u201344, ,\n\nDuff (working dog),\n\nDutch shepherds, ,\n\nears\n\nas aids in smelling,\n\ngland secretions in,\n\nEastern medicine,\n\neccrine glands, ,\n\nEiseman, Charley, , , , , , , \u201317\n\nelephants, , , n, ,\n\nemotional content, in olfaction, , ,\n\nEnglish language\n\nwords for nose in, ,\n\nwords for smells in, ,\n\nEnglish pointers,\n\nepileptic-alert dogs, ,\n\nepithelium, olfactory, \u201348, , , ,\n\nexhaling, in sniffing, , \u201344\n\nexplosives-detection dogs, , , , , ,\n\nface recognition,\n\nfeet, odors of, ,\n\nFelony (working dog), , ,\n\nFeynman, Richard, \u20134,\n\nFfoster (working dog), \u201361,\n\nFinnegan (author's dog),\n\nexperience of living with, ,\n\nexperiment to recognize by smell, \u20135\n\nfirst meeting and adoption of,\n\nas kitten-detection dog,\n\nnosework games with,\n\nnosework training for, \u201358, \u201361, \u201363, ,\n\npregnancy detected by,\n\nself-recognition experiment in, \u201327\n\nsmelling by, \u201359\n\nsniffing by, ,\n\ntest of nose of, \u201356\n\nwine notes smelled by,\n\nFirestein, Stuart, \u201349, , , , , , , n, , , \u201395, , ,\n\nflavor\n\nruminating on, rehearsing,\n\nsmell related to, ,\n\nvision and,\n\nin winemaking,\n\nflehmen, , \u201355\n\nfMRI analysis, ,\n\nfood, smell and taste of, , ,\n\nfoot pads, skin glands on, \u201313\n\nfootprints, in tracking, , , , , , \u201315,\n\nforebrain,\n\nfoxes, \u20138, , , , , , ,\n\nFreud, Sigmund, n, ,\n\nGadbois, Simon,\n\nGalen, ,\n\ngas chromatography, \u201368, , ,\n\nGerman shepherds, , , , , , \u201365\n\nGilbert, Avery, , \u201364, , , , ,\n\nGrahame, Kenneth,\n\ngrooming behavior, , ,\n\nground\n\nbipedalism and distance of nose from, , \u20135, \u201327\n\nlifting heads from, for change of scented air,\n\nproximity of nose to,\n\nrain stimulating odors on,\n\nscent-rolling or -rubbing to pick up scents from, \u201320\n\nscratching behavior on,\n\nsniffing in bursts for smells on, \u201341\n\nweather changes and aromas from, ,\n\nground tracking, ,\n\nguide dogs, ,\n\nGus (working dog), , , \u201346, ,\n\nhabituation, n\n\nheadspace technology, \u201340\n\nhearing\n\naging and deterioration of,\n\nas aid for detection dogs,\n\nsocial space in interactions during, \u201390\n\nHemingway, Ernest,\n\nHempstead, Peggy, \u201314, , ,\n\nHenshaw, Victoria,\n\nHerman, Leta, \u201380\n\nheroin,\n\nHerz, Rachel,\n\nhippocampus, ,\n\nHippocrates, ,\n\nhippopotami, , ,\n\nhorses, ,\n\nhound hunting,\n\nhuman development, smell during, ,\n\nhuman remains detection (HRD), \u201355\n\nhuman-scent lineups, \u201310\n\nhunting, tracking in, \u201385\n\nhunting dogs, ,\n\nhuskies,\n\nhyenas, , ,\n\nhypoglycemic episodes, \u201370,\n\nillegal substances, detection of, , , ,\n\ninfants, smell in, , ,\n\ninfections, smell of, , \u201376\n\nInside of a Dog (Horowitz),\n\nJack Russell terriers,\n\nJake (working dog), ,\n\nJames, William,\n\njournals\n\nof daily smells,\n\nof tracking results, \u20137\n\nKaiser, Roman,\n\nKanum-Irebe tribe,\n\nKaynaroglu, Pat, \u201335, , , \u201347,\n\nKeller, Helen, , \u20138\n\nKelly, James Patrick \"Smelly,\" 217\n\nkennel cough,\n\nKvam, Anne Lill,\n\nLabrador retrievers, , , , , , , , \u201358, , \u201371, , ,\n\nLagotto Romagnolo, , ,\n\nLancet, The (medical journal),\n\nlanguage\n\nof perfume notes, \u201341\n\nof wine tasting, ,\n\nlanguages, words for smells in, \u201321\n\nlaw enforcement detection dogs,\n\nlicking behavior, , ,\n\nlimbic system,\n\nlinden tree blossoms,\n\nlineups, human-scent, \u201310\n\nLinnaeus,\n\nLogan (working dog), ,\n\nLolo (working dog), \u201325, , \u201328, , \u201331\n\nLondon, smells of, \u201369\n\nLorber, Bennett,\n\nlung cancer,\n\nmalignant tumors. See cancer detection\n\nmarijuana, , \u201342,\n\nmarking behavior. See scent-marking\n\nmass spectrometry,\n\nmating, pheromones in,\n\nMatts, Raymond, , \u201337, \u201339, \u201340, , ,\n\nMcBaine (working dog), \u201363\n\nMcGee, Alana, \u201323, , \u201327, \u201329, ,\n\nMcLean, Kate, \u201367, \u201371, , ,\n\nmedical diagnosis, use of smell in, \u201377, \u201382\n\nmelanoma, , ,\n\nmemory\n\nbrain area related to, , ,\n\nperfumers' use of,\n\nsmell and, \u201398, , , , , ,\n\nwine aromas and, \u201350\n\nmirror mark test, ,\n\nmissing persons, , , , , ,\n\nmixed-breed dogs, , , ,\n\nmolecules, odors as,\n\nMonell Chemical Senses Center, , , , , ,\n\nMontaigne, Michel de,\n\nmoose, , , ,\n\nmosquitos, ,\n\nmotor cortex, \u201354\n\nmouth, connection between nose and,\n\nMRI analysis, , \u201353\n\nmucus, , , , , , , , ,\n\nmuscles, nasal, , , \u201312\n\nnaming of smells, \u201320, \u201344\n\nnarcotics-detection dogs, \u201342, ,\n\nnares. See nostrils\n\nNasal Ranger, n\n\nnavigation by smell, \u201378, \u20138\n\nneurons, olfactory, , , , , , , , ,\n\nNew York City\n\nsmell-mapping of, \u201367, \u201375\n\nsmells of, , ,\n\nstreet layout for smell-letting in, ,\n\nsubway smells in, \u201376,\n\nNew York City Parks Department, \u201318\n\nNietzsche, Friedrich,\n\nNoble, Ann C.,\n\nnoseprints, as identification,\n\nnoses, of dogs, \u20136\n\nbreathing route using,\n\ndescription of exterior of, \u201334\n\neccrine glands on,\n\nEnglish words for,\n\nexhaling and, \u201344\n\nexternal hairiness of,\n\nhuman noses compared with, \u201391,\n\ninformation learned through,\n\ninhaled air filtering in, \u201346\n\nlength of, ,\n\nmuscles around, ,\n\nnostrils' functions in, \u201337\n\nolfactory recess at back of, , \u201348\n\nposition on head of,\n\nrichness of world based on smell and,\n\nscientific measures of sensitivity of, \u20139\n\nsmelling and. See smelling by dogs\n\nsniffing mechanism in, \u201342\n\nsorting smells in, \u201347\n\ntraining and tuning of. See nosework training\n\nwet and pebbled surface of,\n\nnoses, of humans, \u201380, \u201387\n\nbringing closer to things to be smelled, \u201342\n\nas chemical detector,\n\nconnection between mouth and,\n\ndog noses compared with, \u201391,\n\nEnglish words for,\n\nhidden or flagrant,\n\ninborn smell-recognition skill of, \u20133\n\nmucus in, ,\n\nmuscles of, \u201312\n\nneuroscientific research on quality of, \u201392\n\nodor molecules entering body through,\n\nolfactory epithelium in, ,\n\nas overlooked part of faces, \u201380\n\npersistent smells in, \u201326\n\nsize and architecture of,\n\nsmelling and. See smelling by humans\n\nas soft organ, \u201386\n\ntwo types of,\n\nunlearning of smell by,\n\nnosework training, \u201368\n\nblind hides used in, \u201367\n\ncues from owners in, \u201366\n\nexpanding dog's world using, \u201362\n\nfinding \"hides\" in,\n\nfirst stage of, \u201361\n\nground rules in,\n\novercoming aspects of domestication in, \u201360\n\nowners' need of training and, \u201365\n\nrange of dogs in, \u201364\n\nrewards for success in,\n\ntuning of nose in,\n\nnoseworms, ,\n\nnostrils\n\nmuscles for changing shape of and opening, ,\n\nright- or left-side preference while using, \u201337,\n\n\"stereo\" approach to smelling using both nostrils,\n\nnotes (smell notes)\n\nmental library of, \u201348\n\nof perfumes, , \u201338, \u201341\n\nsmell training and, , ,\n\nof wines, \u201344, , , ,\n\nodors. See also scents; smells\n\nability to invoke,\n\nair movement in path of, \u201353\n\nblowing warm air to heighten,\n\nchemical structure and experience of,\n\ndefined as molecules,\n\ndisease detection using, \u201374,\n\ndistinguishing compounds in,\n\ninformation gleaned from,\n\nlearning preferences and aversions for,\n\nlingering of, on all materials,\n\nrecognizing family members by their, ,\n\nscent cones radiating from sources of, , \u201397\n\nscent-identification lineups and, \u201310\n\nfrom skin glands on dogs, \u201313\n\nwords for naming,\n\nolfaction. See also smelling by dogs; smelling by humans\n\nactive process of sniffing in,\n\nas change detection system,\n\nemotional memories in,\n\nlack of knowledge about, \u201381,\n\nlimits of brain science on,\n\northonasal, , ,\n\nretronasal, \u201394,\n\nolfactometer,\n\nolfactory abilities of dogs, , \u201331\n\nbeing able to distinguish parts from whole in, \u201331\n\nbreed variations for,\n\nchanges to smells over time and, \u201330\n\nchildren's books describing, \u20138\n\ndetection dogs and, \u201321,\n\ndogs losing capacity for,\n\nexamination of owners returning home using,\n\nhuman-scent lineups to identify criminals and, \u201310\n\ninformation gleaned from scent trails by,\n\nknowing humans by their specific scents using, \u201323\n\nmale or female identification using,\n\nnew scents encountered each day and, \u201328\n\nduring ordinary day, \u201321\n\nscent-marking by other dogs read by, \u201318\n\nscent-rolling or -rubbing and, \u201320\n\nscientific measures of sensitivity of, \u20139\n\nself-recognition of their own smells and, \u201328\n\nsniffing other dogs as communication using, \u201313\n\nweather change detection and,\n\nolfactory bulb, \u201351, , ,\n\nolfactory cortex, ,\n\nolfactory nerve,\n\nolfactory receptor genes, \u201350, , , \u201322\n\nolfactory receptors, \u201391,\n\nandrostenone and, \u201322\n\ncancer detection and,\n\ngenetic factors and, \u201350, , , \u201322\n\nnumber of and kinds of, \u201349\n\npersistent binding of odorants to, \u201326\n\nolfactory recess, , \u201348,\n\nolfactory tests, diagnostic, \u201382\n\norcas, \u201398,\n\nOrient, Jane, ,\n\nO'Rourke, P. J.,\n\northonasal olfaction, , ,\n\nOtto, Cindy, , \u201334, ,\n\nPak (working dog), ,\n\npanting, , , ,\n\nPApa Bear (working dog), \u201344,\n\nParis, smells of, , , , ,\n\nParker (working dog), ,\n\nParkinson's disease,\n\nParsons (working dog), , ,\n\npatient odors,\n\nPatterson (working dog), ,\n\nPauling, Linus,\n\nperfumers,\n\nanosmia of,\n\nbrain function of,\n\ndaily time spent sniffing by,\n\nmixing of smells by, \u201336\n\nnatural odors used by,\n\nnose of,\n\nsmelling ability of, , , ,\n\ntraining of, \u201335,\n\nperfumes, \u201341\n\ncapturing essences of odors for, \u201340\n\ncreating, \u201340\n\ndabbing on hair rather than wrists,\n\ndescriptive language used for, \u201341\n\ningredients in, \u201334, \u201338,\n\npopularity over time,\n\nsmelling example using,\n\npersonal smells\n\nKanum-Irebe tribe's greetings with,\n\nsniffing shaken hands for,\n\nsocial space and experience of, \u201390\n\npersonal space\n\ninteractions with others and, \u201390\n\nperfume use and,\n\npesticide poisoning,\n\npests, detection of, ,\n\npet dogs,\n\ndetection tasks and,\n\nhypoglycemic episodes and,\n\nloss of smell capabilities by,\n\nnosework training for, \u201366\n\nowners' cues to, \u201361\n\nscent-marking by, ,\n\nworking dogs differentiated from, \u201340,\n\npheromones, , \u201389, ,\n\nphysical reactions to smells, \u201322,\n\npigs, ,\n\nPinto (working dog), ,\n\npipe bombs,\n\nPlato,\n\nPliny, ,\n\npoisons, detection of, , \u201377,\n\npolice departments, \u201310, , ,\n\npolice dogs,\n\nporcupines, , \u201315\n\nPort Chester Obedience Training Club,\n\nPorteous, J. Douglas, \u201368\n\nPortuguese water dogs,\n\npractice, in developing sense of smell, \u20139,\n\nPratt Institute,\n\npredictive aspect of smell,\n\npregnancy, ,\n\nPreti, George, , , \u201390, \u201367, \u201369, \u201381\n\nprimates, ,\n\nprostate cancer,\n\npsychiatric disorders,\n\nPumpernickel (author's dog), \u201326, \u201378,\n\nQuest (working dog), , \u201349,\n\nrabbits, , , , ,\n\nRam\u00f3n y Cajal, Santiago,\n\nrats, smelling by,\n\nreceptor cells, olfactory, \u201350, \u201391,\n\nred foxes, , , , ,\n\nREM sleep, \u201359\n\nretronasal olfaction, \u201394, ,\n\nRoach, Mary,\n\nRockefeller University, \u201318, \u201326\n\nRookie (working dog), , \u201339,\n\nRosenbach, Kristin, , , \u201327, , , ,\n\nRugaas, Turid,\n\nrumps (tail area)\n\nanal glands around, , ,\n\ndogs' sniffing of, , , \u201356,\n\nrunway marking, , \u201317,\n\nSacks, Oliver,\n\nsaliva\n\nin Chinese medicine,\n\ndiabetic alert using,\n\nsmells from, in dogs, ,\n\nscat-detecting animal trackers, . See also animal tracking\n\nscat-detection programs, \u201399\n\ninformation gathered in, \u201394\n\nmixed-breed dogs suitable for, \u201397\n\norcas tracked in, \u201398,\n\ntraining dogs for, \u201395\n\nwildlife research using, \u201393\n\nscent cones, , \u201397\n\nscent-identification dogs. See detection dogs\n\nscent-identification lineups, \u201310\n\nscent-marking, \u201318\n\ncounter-marking in, , , , ,\n\ndomestic dogs' behavior in, ,\n\nobservational research in park on, \u201318\n\nin other species, \u201315\n\nof people,\n\nraised leg behavior in,\n\non runways (shared paths), , \u201317,\n\nsocial information in,\n\nterritorial definition using, \u201315\n\ntracking and identifying animals by placement of, , \u201316\n\nurination differentiated from,\n\nworking dogs trained to not exhibit,\n\nscent-rolling or -rubbing, \u201320\n\nscents. See also odors; smells\n\nbody shaking to spread, \u201359\n\nmedical classification of,\n\nsmellscapes identified by, \u201368\n\ntail-wagging to spread,\n\nscent samples, in detection training,\n\nschizophrenia, smell of,\n\nscratch-and-sniff technology,\n\nin animal-tracker's book,\n\ngames,\n\nfor olfactory test,\n\nscratching behavior on ground,\n\nsearch-and-rescue dogs, , , \u201386\n\nsearches, dogs used for. See detection dogs\n\n\"searching\" phase of tracking,\n\nsearch training. See nosework training; training of working dogs\n\nsebaceous glands, , , ,\n\nSeeing Eye dogs, ,\n\nself-recognition, \u201328\n\nsenses. See also hearing; smell; taste; vision\n\naging and deterioration of,\n\nseptum, nasal,\n\nSettles, Gary, , , ,\n\nshampoos, smell of, ,\n\nshaping behavior, \u201355, \u201362\n\nShepherd, Gordon,\n\nshoes, smells of, \u201389\n\nsickness. See diseases\n\nsight. See vision\n\nSirius (working dog), ,\n\nskin cells and flakes, , \u201388\n\nskin glands, \u201313\n\nskin odors, in disease detection, ,\n\nsleep, smelling during, , \u201359, n\n\nSlocum, Kelly,\n\nsmell\n\naging and,\n\nof cancer, \u201369\n\ndisease detection using, \u201376\n\nfetal and infant experience of, ,\n\nhumans' unlearning of ability to,\n\nindividual differences in sense of,\n\nas least used human sense, \u201382\n\nas means of discovery,\n\nnavigation by, \u201378, \u20138\n\nnotion of contamination by, \u201370\n\npersistent, stuck in nose, \u201326\n\npractice and repetition to develop sense of, \u20139,\n\nas predictive of toxins and danger, , ,\n\ntaste related to, , ,\n\nthree-step process of restoring rightful place of,\n\nvision as more important sense than, ,\n\nsmell-blindness (anosmia), , ,\n\nsmell-daydreaming,\n\nsmell-faces,\n\nsmelling abilities. See olfactory abilities of dogs\n\nsmelling by dogs, \u201359. See also olfaction; olfactory abilities of dogs\n\nairflow models of, \u201334\n\nair movement and, , \u201353\n\nbody shaking to spread scents for, \u201359\n\nbrain processing in, \u201354, \u201391\n\nbreed variations for,\n\nexhaling and, \u201344\n\nhuman-dog comparisons for, \u201394\n\ninhaled air filtering in, \u201346\n\nlength of, ,\n\nlicking as part of, , ,\n\nlifting heads from ground for change of air during,\n\nmovement and motor cortex in, \u201354\n\nmuscles around nose and, ,\n\nnose's exterior and, \u201334\n\nnose's position on head and,\n\nnostrils' functions in, \u201337\n\nnumber of and kinds of receptor cells in, \u201349\n\nnumber of odors detectable during, \u201349\n\nobservations of, in natural environments, \u201357\n\nolfactory bulb in, \u201351, ,\n\nolfactory genes and,\n\nolfactory recess and, , \u201348\n\northonasal route in,\n\nowner's experiment with imitating, \u201365\n\nof pheromones, , \u201389, ,\n\nreceptors in, \u201350, \u201391\n\nscientific measures of sensitivity in, \u20139\n\nduring sleeping and dreaming, \u201359\n\nsneezing to clear smells from nose during, ,\n\nsniffing mechanism in, \u201344\n\nsorting smells in, \u201347\n\n\"stereo\" approach to, using both nostrils,\n\ntail-wagging to spread scents for,\n\nvomeronasal organ (VNO) in, \u201356,\n\nduring walking, \u201378\n\nsmelling by humans, \u201399. See also olfaction\n\nactively sniffing by, \u201374\n\nadaptation in,\n\nbipedalism and, , \u20135, \u201327\n\nbrain processing in, \u201391, \u201396, \u201312\n\nbringing closer to things to be smelled in, \u201342\n\nchildren's learning of, \u201384\n\ndaily time spent sniffing during, \u20137\n\nfailure to identify familiar odors in,\n\nfetal and infant experience of,\n\ngenes reserved to,\n\nhuman-dog comparisons for, \u201394\n\ninborn skill of human nose in, \u20133\n\nindividual differences in, \u201399\n\nlack of knowledge about, \u201381,\n\nas least used sense, \u201382\n\nmisunderstanding and distrust of smells and,\n\nnotion of contamination by smell and, \u201370\n\nodor preferences and aversions in,\n\nolfactometer for,\n\nolfactory neurons in, , ,\n\northonasal route in,\n\nowner's experiment to imitate dogs' smelling during walks, \u201365\n\nowners' recognition of their dogs' smells and,\n\npersistent smells stuck in nose and, \u201326\n\nphysical reactions to smells in, \u201322,\n\nrecognizing and naming smells out of context during, \u201323\n\nretronasal route in, \u201394, ,\n\nsmell as means of discovery in,\n\nsmell-mapping project with, \u201367, \u201375\n\nsmellscapes and, \u201368\n\nsmells as binary (good or bad) in, \u201383\n\nsmells as molecules and,\n\nsniffing during, \u201318\n\nsocial space and, \u201390\n\nstreet layout for smell-letting and, \u201369\n\ntaste and, , ,\n\ntendency to move nose away from rather than toward smells during,\n\nthree-step process of restoring importance of smell in,\n\ntraining experiment for, \u201318, \u201326\n\nsmell notes. See notes (smell notes)\n\nsmells. See also personal smells; scents\n\nadaptation to,\n\nof animals during tracking, \u20133, \u20135\n\nbeing blind to (anosmia), , ,\n\ncalibrating,\n\nchemical structure and experience of,\n\nchildren's books mentions of, \u20138\n\ncolors for describing,\n\ncultural sensitivities about, , \u201375\n\ndefined as molecules,\n\ndescriptive words for,\n\ndilution percentages of, in olfactory sensitivity, \u20139\n\ndisease detection using, \u201376\n\ndogs' processing of. See smelling by dogs\n\ndogs' self-recognition via their own, \u201328\n\nEnglish words for, ,\n\nexperiments to recognize objects by, \u20136\n\nfailure to identify familiar smells,\n\nof forests, \u201317\n\nhumans' approaches to, during walks, \u201365\n\ninformation gleaned from,\n\nkeeping journal of daily,\n\nknowing owners by their specific smells, \u201323\n\nmemories related to, \u201398, , , , , , ,\n\nmisunderstanding and distrust of,\n\nnavigation by, \u201378, \u20138\n\nneed to name, \u201320\n\nnumber detectable during smelling, \u201349\n\nphysical reactions to, \u201322,\n\nrecognizing and naming out of context, \u201323\n\nscent cones radiating from sources of, , \u201397\n\nscent-identification lineups and, \u201310\n\nscent-marking by other dogs and, \u201318\n\nscent-rolling or -rubbing for picking up, \u201320\n\nseen as binary (good or bad) by humans, \u201383\n\nsmell-mapping of, \u201367, \u201375\n\nsmellscapes to capture, \u201368\n\ntracing to their sources during walks, \u201375,\n\nof truffles, , \u201322,\n\nvague characterizations of, \u201346\n\nof wine notes, \u201344, ,\n\nin wine tasting, \u201352\n\nsmellscapes, , \u201368,\n\nsmell walks, \u201378\n\nactively sniffing during, \u201374\n\ncolors for describing smells during,\n\nnavigation by smell during, \u201378\n\nsmell-mapping project during, \u201367, \u201375\n\nsmellscapes and, \u201368\n\nsubway smells and, \u201376\n\ntracing smells to their sources in, \u201375,\n\nsmuggled items, detection of,\n\nsneezing, during smelling, ,\n\nsniffer dogs. See detection dogs\n\nsniffing by dogs\n\nbreathing (respiratory) route in nose different from,\n\ndogs losing capacity for,\n\ndrug searches using, \u201344\n\nexhaling as part of, , \u201344\n\nfluid dynamics model of, \u201338\n\ninhaled air filtering and, \u201346\n\nlength of noses and,\n\nlong sniffs for smells at a distance,\n\nnasal mechanisms in, \u201343\n\nnose reach in, ,\n\n\"nosing\" an object as part of, \u201342,\n\nin other species, \u201340\n\npanting and, , ,\n\nposition of nose on head and,\n\npulling in air through, \u201339\n\nrate of,\n\nof rumps between dogs, , , \u201356,\n\nshort sniffs in,\n\nsniffing in bursts for smells nearby and on the ground, \u201341\n\nsorting smells during, \u201347\n\nsniffing by humans\n\nactively, during walks, \u201374\n\nadaptation in,\n\nairflow rate in,\n\ndaily time spent, \u20137\n\ndifferent ways of, \u201318\n\nmuscles for alar flaring during,\n\nnostril side preferences in,\n\nphysical reactions to smells during, \u201322,\n\npractice to develop,\n\nspontaneous use of, \u201328\n\ntraining experiment for developing, \u201318, \u201326\n\nsniff vigor, \u201318\n\nsnow, tracking through, ,\n\nsoaps, smell of, , , \u201345\n\nSobel, Noam, ,\n\nsocial facilitation,\n\nsocial space, \u201390\n\nsommeliers, , ,\n\nspotted hyenas, , ,\n\nspringer spaniels,\n\nsquirrels, , , , ,\n\nSteig, William, \u20138, \u201377\n\nsteroids, nasal, \u20136\n\nstink-waves\n\nmultilayered, \u201388\n\ntracking dogs' use of,\n\nwitches' use of, \u201384\n\nstreet layout, for smell-letting, \u201369\n\nsubcortical limbic system,\n\nsubway smells, \u201376,\n\nSupreme Court,\n\nsweat, disease detection using, , ,\n\nsweat glands,\n\nsweating, of feet,\n\ntail area (rump)\n\nanal glands around, , ,\n\ndogs' sniffing of, , , \u201356,\n\ntail-wagging,\n\ntaste\n\ncolor of food and,\n\nsmell related to, , ,\n\nterritory, marking of, \u201315. See also scent-marking\n\nthalamus,\n\ntherapy dogs,\n\nThurber, James,\n\ntime-related changes to smells, \u201330\n\nTolaas, Sissel,\n\ntoxic products, detection of, n, , ,\n\ntoxins, smell of, , ,\n\ntracking. See also animal tracking\n\nfootprint clues in, , , , , , \u201315,\n\nhunting as early use of, \u201385\n\nlifting heads from ground for change of scented air during,\n\nnavigation by smell during, \u20138\n\nphases of,\n\ntraining for,\n\nwatching dogs find targets in, \u201391\n\ntracking dogs. See also detection dogs\n\nbreeds for, , , ,\n\nhandlers' learning to work with,\n\ninformation gleaned from scent trails followed by,\n\nmissing persons and, , , , , ,\n\nmotivation of, \u201390\n\nnoses of,\n\nresearch on,\n\nscientific measures of olfactory sensitivity of, \u20139\n\nstink-waves used by, , \u201388\n\nunderwater searches and, ,\n\n\"tracking\" phase of tracking,\n\ntraditional Chinese medicine (TCM), , ,\n\ntraining experiment for smelling, \u201318, \u201326\n\ntraining of working dogs, \u201321, \u201356. See also nosework training; Working Dog Center (WDC)\n\nanti-pet behavior needed in, \u201340\n\nbarking as part of, , , , , ,\n\nbody awareness and fitness as part of, \u201345\n\nbreeds suitable for,\n\ncancer cell detection and, \u201364\n\ncharacteristics of dogs suitable for,\n\ncontinuous assessment of behavior during,\n\nof diabetic alert dogs, \u201371\n\n\"drive and courage\" test in,\n\ndrive of dogs in,\n\nfinding hidden persons in, \u201349\n\nfocus of search training in, ,\n\nhuman remains detection (HRD) in, \u201355\n\nlevel of performance needed in, \u201364\n\nobstacle course in, \u201350\n\nplacement of dogs after,\n\nscat detection and, \u201395\n\nscent samples used in,\n\nseek exercise in, \u201330, \u201338\n\nshaping behavior in, \u201355, \u201362\n\n\"smell training\" not needed in, \u201338\n\nsniffing out drugs (\"find dope\") as part of, \u201344\n\nsocial facilitation in,\n\nfor truffle hunting, , , \u201327, \u201329,\n\ntree marks, during tracking, \u201310\n\ntrees, truffles' symbiotic relationship with, ,\n\ntrigeminal nerve, n, ,\n\nTruffle Dog Company,\n\ntruffle flies,\n\ntruffle hunting, \u201331\n\ndogs used for, , , \u201328, \u201331\n\nhunting sites for, ,\n\npigs used for, \u201322,\n\ntraining of dogs for, , , \u201327, \u201329,\n\ntypical hunts in, \u201328, \u201331\n\ntruffles\n\nlocation of, in ground, \u201324\n\nodor of, , \u201322,\n\ntrees' symbiotic relationship with, ,\n\nTsunami (working dog),\n\nTucker (working dog), , \u201398,\n\nTullett, William,\n\nturbinate bones, , ,\n\nTurin, Luca,\n\nU.S. Border Patrol,\n\nU.S. Supreme Court,\n\nUpton (author's dog),\n\nexperience of living with, ,\n\nexperiment to recognize by smell, \u20135\n\nfirst meeting and adoption of, \u20135\n\nnosework games with,\n\nsniffing by,\n\nurban design\n\ndeodorization projects in, \u201370\n\nsmellscapes used in, \u201368\n\nstreet layout for smell-letting in, \u201369\n\nurine\n\ncancer detection using, \u201365\n\ndisease detection using, ,\n\nscent-marking using, \u201318\n\ntracking and identifying animals by placement of,\n\nVale, Sam, , \u201373\n\nValentine, Carla,\n\nVerghese, Abraham,\n\nvestibules, nasal,\n\nvision\n\naging and deterioration of,\n\nas aid for detection dogs, , ,\n\nanimal tracking using,\n\nbrain processing in, ,\n\nas more important sense than smell, ,\n\nsocial space in interactions involving, \u201390\n\nvomeronasal organ (VNO), \u201356,\n\nVosshall, Leslie, , , , , \u201325, ,\n\nwalking\n\nactively sniffing during, \u201374\n\ncolors for describing smells during,\n\nnavigation by smell during, \u201378\n\nnew scents encountered each day on, \u201328\n\nsmelling during, \u201378\n\nsmell-mapping project during, \u201367, \u201375\n\nsmellscapes and, \u201368\n\nsubway smells and, \u201376\n\ntracing smells to their sources in, \u201375,\n\nWarren, Cat,\n\nWasser, Sam, \u201398,\n\nwater, tracking through, , , \u201397\n\nwater leaks, sniffing out,\n\nWatson, Lyall,\n\nweather patterns\n\ndog's detection of, , , ,\n\nground aromas affected by, ,\n\nsailors' knowledge of, during navigation,\n\nsmelling,\n\ntracking affected by,\n\nWeber, Ernst Heinrich, \u201339\n\nwine aroma wheel, \u201352,\n\nwine making\n\nbottling in,\n\nodor and taste evaluators in, \u201353\n\nwine notes, \u201344, , ,\n\nwine tasting, \u201354\n\napproaching unknown wine in, \u201351, \u201354\n\nlanguage used in, ,\n\nlearning smells in, \u201352\n\nnotes in, \u201349\n\npaying attention to wine in,\n\nseries of \"flights\" of wine in, , \u201354\n\nsmelling in,\n\ntraining in,\n\nWinnegrad, Ron, \u201335\n\nwolves, , , ,\n\nWoolf, Virginia, ,\n\nWorking Dog Center (WDC)\n\ndescription of, \u201332\n\nfoster families used for control group of dogs from,\n\nhydration research at,\n\nkitchen as social center of,\n\nlevel of performance at,\n\nmission of,\n\nmotto of,\n\nraising their own dogs at, ,\n\ntrainers at, , ,\n\ntraining manager at, \u201335\n\nworking dogs\n\nhandlers' relationship with,\n\ninformation learned through noses of,\n\npet dogs differentiated from, \u201340,\n\nrange of duties performed by,\n\nas smelling experts in detection,\n\ntracking abilities of,\n\ntruffle hunting by, , , \u201328, \u201331\n\nWyatt, Tristram, \u201389\n\nyellow baboons, \u201393,\n\nZwaardemaker, Hendrik,\n\nSCRIBNER\n\nAn Imprint of Simon & Schuster, Inc.\n\n1230 Avenue of the Americas\n\nNew York, NY 10020\n\nwww.SimonandSchuster.com\n\nCopyright \u00a9 2016 by Alexandra Horowitz, with illustrations by the author\n\nAll rights reserved, including the right to reproduce this book or portions thereof in any form whatsoever. For information, address Scribner Subsidiary Rights Department, 1230 Avenue of the Americas, New York, NY 10020.\n\nFirst Scribner hardcover edition October 2016\n\nSCRIBNER and design are registered trademarks of The Gale Group, Inc., used under license by Simon & Schuster, Inc., the publisher of this work.\n\nFor information about special discounts for bulk purchases, please contact Simon & Schuster Special Sales at 1-800-506-1949 or business@simonandschuster.com.\n\nThe Simon & Schuster Speakers Bureau can bring authors to your live event. For more information or to book an event, contact the Simon & Schuster Speakers Bureau at 1-866-248-3049 or visit our website at www.simonspeakers.com.\n\nDesigned by Erich Hobbing\n\nJacket design by Jaya Miceli\n\nJacket photograph \u00a9 Roberta Tancredi\/Flickr Flash\/Getty Images\n\nLibrary of Congress Cataloging-in-Publication Data is available.\n\nISBN 978-1-4767-9599-7\n\nISBN 978-1-4767-9604-8 (ebook)\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":" \n# LE POINT DE VUE DES \u00c9DITEURS\n\nEn pleine R\u00e9volution culturelle, le pouvoir chinois construit une base militaire secr\u00e8te destin\u00e9e \u00e0 abriter un programme de recherche de potentielles civilisations extraterrestres. Ye Wenjie, une jeune astrophysicienne en cours de \"r\u00e9\u00e9ducation\", parvient \u00e0 envoyer dans l'espace un message contenant des informations sur la civilisation humaine. Le signal est intercept\u00e9 par les Trisolariens, qui s'appr\u00eatent \u00e0 abandonner leur plan\u00e8te m\u00e8re, situ\u00e9e \u00e0 quatre ann\u00e9es-lumi\u00e8re de la Terre et menac\u00e9e d'un effondrement gravitationnel provoqu\u00e9 par les mouvements chaotiques des trois soleils de son syst\u00e8me. En raison de la distance, Ye Wenjie met pr\u00e8s de huit ans \u00e0 recevoir la r\u00e9ponse des Trisolariens. Elle tient d\u00e9sormais entre ses mains rien de moins que le destin de l'esp\u00e8ce humaine.\n\nPremier volume d'une trilogie culte d'une ambition folle, r\u00e9compens\u00e9 par le Hugo du meilleur roman en 2015, _Le Probl\u00e8me \u00e0 trois corps_ signe l'arriv\u00e9e d'un auteur majeur.\n\n# LIU CIXIN\n\n_N\u00e9 en 1963, Liu Cixin est une v\u00e9ritable l\u00e9gende de la SF en Chine, o\u00f9 il a remport\u00e9 le Galaxy Award (prix le plus prestigieux d\u00e9di\u00e9 au genre) \u00e0 neuf reprises._\n\nIllustration de couverture : \u00a9 YOOZOO Pictures\n\nTitre original :\n\n_San Ti_ (\u4e09\u4f53)\n\n\u00c9diteur original :\n\nChongqing Publishing House Co., Ltd., Chongqing\n\n\u00a9 Liu Cixin (\u5218\u6148\u6b23), 2006\n\nFrench translation rights authorized by China Educational\n\nPublications Import & Export Corp., Ltd.\n\n\u00a9 ACTES SUD, 2016\n\npour la traduction fran\u00e7aise\n\nISBN 978-2-330-07103-5\nLIU CIXIN\n\nLe probl\u00e8me\n\n\u00e0 trois corps\n\nroman traduit du chinois\n\npar Gwenna\u00ebl Gaffric\n\n_ACTES SUD_\n\n# I\n\n# UN PRINTEMPS SILENCIEUX\n\n## 1\n\n## LES ANN\u00c9ES DE FOLIE\n\n_Chine, 1967._\n\nL'assaut de l'Union rouge contre le quartier g\u00e9n\u00e9ral de la brigade du 28 Avril durait depuis d\u00e9j\u00e0 deux jours. Tout autour de l'\u00e9difice, les drapeaux de la brigade claquaient au vent, telles des torches attendant d'\u00eatre raviv\u00e9es.\n\nL'officier en chef de l'Union rouge \u00e9tait terriblement inquiet, non qu'il ait peur des miliciens en faction dans la tour \u2013 les deux cents et quelques hommes de la brigade du 28 Avril ne pesaient pas bien lourd face aux gardes rouges de l'Union, fond\u00e9e au d\u00e9but de l'ann\u00e9e 1966 et dont la puissance \u00e9conomique et militaire \u00e9tait sans rivale. Ce qu'il craignait, c'\u00e9taient les quelques dizaines de fours de m\u00e9tal en fusion entrepos\u00e9s \u00e0 l'int\u00e9rieur du b\u00e2timent. Ceux-ci, reli\u00e9s par des d\u00e9tonateurs \u00e9lectriques, \u00e9taient pleins \u00e0 ras bord d'explosifs. Il ne pouvait les voir, mais il pouvait sentir leur pr\u00e9sence magn\u00e9tique. Un doigt sur un bouton et tous seraient r\u00e9duits en cendres.\n\nLes jeunes gardes rouges de la brigade du 28 Avril \u00e9taient capables d'une telle folie. Par comparaison avec les v\u00e9t\u00e9rans qui avaient m\u00fbri \u00e0 mesure qu'ils affrontaient les temp\u00eates, cette g\u00e9n\u00e9ration de rebelles \u00e9tait une meute de loups cavalant sur des charbons ardents. Plus fous que la folie elle-m\u00eame.\n\nUne silhouette gracile \u00e9mergea au sommet de la tour, celle d'une belle jeune fille brandissant un grand \u00e9tendard sur lequel on lisait \"28 avril\". Son irruption fut accueillie par une vol\u00e9e de coups de feu. C'\u00e9tait un cocktail h\u00e9t\u00e9rog\u00e8ne d'armes : des vieilles carabines \u00e0 l'am\u00e9ricaine, en passant par des ZB-26 tch\u00e8ques, des Arisaka Type 38, jusqu'\u00e0 des armes plus r\u00e9centes, comme des fusils d'assaut et des pistolets-mitrailleurs \u2013 ces derniers avaient \u00e9t\u00e9 d\u00e9rob\u00e9s \u00e0 l'arm\u00e9e au lendemain de la parution de \"L'\u00e9ditorial d'ao\u00fbt1\". On trouvait m\u00eame des armes blanches comme des coutelas et des lances, qui formaient avec les armes \u00e0 feu un v\u00e9ritable condens\u00e9 de l'histoire moderne...\n\nLes miliciens de la brigade du 28 Avril connaissaient la chanson par c\u0153ur. Outre l'\u00e9tendard, le camarade perch\u00e9 sur le toit se servait parfois d'un haut-parleur pour vocif\u00e9rer des slogans ou bien il lan\u00e7ait des tracts au pied du b\u00e2timent. Chaque fois, ceux qui parvenaient \u00e0 sortir indemnes de la pluie de projectiles \u00e9taient couverts de gloire.\n\nLa fille qui venait d'appara\u00eetre sur le toit croyait sans doute qu'elle conna\u00eetrait aujourd'hui le m\u00eame destin. En secouant ce drapeau de guerre, c'\u00e9tait son adolescence passionn\u00e9e qu'elle agitait. L'ennemi serait bient\u00f4t consum\u00e9 par les flammes et demain, un monde id\u00e9al jaillirait de son sang bouillonnant... Alors qu'elle s'enivrait de ce triomphal r\u00eave \u00e9carlate, la balle d'un fusil lui transper\u00e7a le corps.\n\nLa poitrine de cette jeune fille de quinze ans \u00e9tait si fr\u00eale qu'elle ne freina m\u00eame pas la vitesse du projectile qui, une fois pass\u00e9 au travers, \u00e9mit un sifflement dans son dos. La jeune garde rouge s'\u00e9croula en m\u00eame temps que son drapeau, mais sa silhouette d\u00e9licate s'effondra moins vite que l'\u00e9tendard. Elle plana jusqu'au sol, comme un oisillon amoureux du ciel.\n\nLes gardes rouges de l'Union tir\u00e8rent des salves de joie. Certains grimp\u00e8rent jusqu'au toit de la tour, o\u00f9 ils arrach\u00e8rent l'\u00e9tendard de la brigade du 28 Avril et brandirent fi\u00e8rement la d\u00e9pouille de la milicienne comme un troph\u00e9e de guerre, avant de la lancer depuis cette hauteur jusqu'au portail en fer de la grande cour.\n\nLes barreaux aux extr\u00e9mit\u00e9s pointues des grilles m\u00e9talliques du portail avaient \u00e9t\u00e9 arrach\u00e9s pour servir de lances au d\u00e9but du combat. Il n'en restait plus que deux sur lesquels la jeune garde rouge vint par malchance s'empaler. L'espace d'un instant, la vie sembla regagner ce corps gracieux.\n\nLes gardes rouges de l'Union firent quelques pas en arri\u00e8re et mitraill\u00e8rent le corps suspendu de la fille comme une cible d'entra\u00eenement. Ce feu nourri de balles fut pour elle aussi doux que la pluie, elle ne ressentait plus rien, ses bras se soulevaient par moments telles des branches de lierre, comme si son corps \u00e9tait balay\u00e9 par des gouttes de pluie, jusqu'\u00e0 ce qu'enfin la moiti\u00e9 de sa jeune t\u00eate se d\u00e9tache et qu'il ne lui reste plus qu'un \u0153il, magnifique, en train de contempler le ciel bleu de cette ann\u00e9e 1967. Un regard o\u00f9 la souffrance \u00e9tait absente et o\u00f9 ne demeuraient plus qu'un espoir et une ardeur inflexibles. Elle avait en r\u00e9alit\u00e9 eu plus de chance que les autres : elle, au moins, avait sacrifi\u00e9 sa vie au plein c\u0153ur d'un \u00e9pisode de passion id\u00e9aliste.\n\nCe fanatisme s'empara de la ville enti\u00e8re, comme si des processeurs fonctionnaient en parall\u00e8le. La \"R\u00e9volution culturelle\" \u00e9tait devenue un seul et unique combat. Un d\u00e9luge informe et fr\u00e9n\u00e9tique s'abattait maintenant sur la cit\u00e9, s'engouffrant dans la moindre br\u00e8che, se faufilant dans le moindre recoin.\n\nAux confins de la ville, sur le terrain de sport d'une universit\u00e9, une s\u00e9ance de critique publique \u00e0 laquelle assistaient plusieurs milliers de personnes avait commenc\u00e9 depuis pr\u00e8s de deux heures. En ces temps o\u00f9 coexistaient de multiples factions, des conflits complexes \u00e9clataient partout entre diff\u00e9rentes ob\u00e9diences. Rien que sur le campus, des hostilit\u00e9s brutales opposaient gardes rouges, membres du Groupe de travail de la R\u00e9volution culturelle, miliciens de la brigade de propagande des travailleurs ou de la brigade de propagande des soldats. Chaque groupe finissait souvent par imploser et donnait naissance \u00e0 de nouvelles coalitions internes qui revendiquaient chacune des principes et des programmes uniques. Leurs antagonismes d\u00e9cha\u00eenaient des actes d'une cruaut\u00e9 plus inf\u00e2me encore. Ils partageaient pourtant un ennemi commun : l'Autorit\u00e9 acad\u00e9mique r\u00e9actionnaire, si bien que les malheureux qu'on disait appartenir \u00e0 cette tendance devaient endurer des attaques f\u00e9roces provenant de tous c\u00f4t\u00e9s.\n\nLes membres de l'Autorit\u00e9 acad\u00e9mique r\u00e9actionnaire poss\u00e9daient des particularit\u00e9s qui les distinguaient des autres adversaires malfaisants de la r\u00e9volution. Lorsqu'ils avaient \u00e9t\u00e9 attaqu\u00e9s, leur premi\u00e8re r\u00e9action avait \u00e9t\u00e9 de r\u00e9pondre avec orgueil et t\u00e9nacit\u00e9, mais c'\u00e9tait cette attitude qui leur avait valu le plus d'ex\u00e9cutions. Rien que dans la capitale, en l'espace de quarante jours seulement, plus de mille sept cents personnes furent battues \u00e0 mort au cours de s\u00e9ances de critique publique. Pour \u00e9chapper \u00e0 cette folie, beaucoup d'entre eux choisirent une voie plus rapide : Lao She, Wu Han, Jian Bozan, Fu Lei, Zhao Jiuzhang, Yi Qun, Wen Jie, Hai Mo et tant d'autres d\u00e9cid\u00e8rent de mettre eux-m\u00eames un terme \u00e0 des vies autrefois tr\u00e8s estim\u00e9es.\n\nCeux qui avaient surv\u00e9cu \u00e0 cette \u00e9tape devenaient progressivement insensibles aux coups r\u00e9p\u00e9t\u00e9s de fureur r\u00e9volutionnaire ; c'\u00e9tait comme s'ils avaient d\u00e9velopp\u00e9 une carapace spirituelle qui les prot\u00e9geait d'un effondrement final. Lors des s\u00e9ances de critique publique, ils plongeaient dans un \u00e9tat de demi-sommeil que seules les menaces vocif\u00e9r\u00e9es \u00e0 leur encontre \u00e9taient \u00e0 m\u00eame de secouer, ils r\u00e9p\u00e9taient m\u00e9caniquement un nombre incalculable de fois les m\u00eames aveux. Puis, le moment venait pour certains d'entre eux de passer \u00e0 la troisi\u00e8me \u00e9tape : apr\u00e8s une s\u00e9ance de critique ayant dur\u00e9 jusqu'au soir, la glorieuse id\u00e9ologie politique de la r\u00e9volution se d\u00e9versait comme du mercure dans leur conscience, \u00e9branlant les fondations d'un \u00e9difice intellectuel jadis fa\u00e7onn\u00e9 par le savoir et la raison. Ils se mettaient \u00e0 se croire effectivement coupables, constataient les d\u00e9g\u00e2ts qu'ils avaient v\u00e9ritablement caus\u00e9s \u00e0 la grande cause du peuple, et ils pleuraient de douleur. Les confessions des intellectuels \u00e9taient toujours beaucoup plus profondes et sinc\u00e8res que celles des autres ennemis de la r\u00e9volution. Mais pour les gardes rouges, les adversaires qui parvenaient jusqu'aux deux derni\u00e8res \u00e9tapes manquaient de piquant. Seuls les dangereux ennemis qui restaient bloqu\u00e9s \u00e0 la premi\u00e8re \u00e9tape \u00e9taient capables de susciter une ferveur dans leurs esprits depuis longtemps fanatis\u00e9s. Ces adversaires \u00e9taient des morceaux de tissu rouge agit\u00e9s par un matador, mais ils se faisaient de plus en plus rares. Il n'en restait plus qu'un seul dans cette universit\u00e9, un professeur dont la rare intransigeance lui avait valu l'honneur d'\u00eatre le dernier convoqu\u00e9 en s\u00e9ance de critique publique.\n\nYe Zhetai avait surv\u00e9cu \u00e0 la R\u00e9volution culturelle jusqu'\u00e0 ce jour et il en \u00e9tait toujours \u00e0 la premi\u00e8re \u00e9tape : non seulement il refusait de reconna\u00eetre ses crimes, mais il ne s'\u00e9tait pas donn\u00e9 la mort et n'\u00e9tait pas devenu insensible aux accusations port\u00e9es contre lui. Lorsque le professeur de physique arriva \u00e0 la barre, l'expression dans son regard disait clairement : Alourdissez encore ma croix ! Mais si le fardeau que les gardes rouges lui faisaient porter \u00e9tait en effet \u00e9crasant, ce n'\u00e9tait pas une croix. Les personnes condamn\u00e9es \u00e0 des s\u00e9ances de critique publique devaient en temps normal \u00eatre coiff\u00e9es d'un haut chapeau tress\u00e9 en bambou, mais le sien \u00e9tait en acier soud\u00e9. Il portait aussi une pancarte sur sa poitrine, pas en bois comme d'ordinaire, mais une plaque en fer qui n'\u00e9tait autre que la porte du four de son laboratoire, barr\u00e9e en diagonale d'une grande croix rouge et sur laquelle \u00e9tait \u00e9crit son nom en caract\u00e8res noirs clinquants.\n\nLes gardes rouges qui escortaient Ye Zhetai \u00e9taient deux fois plus nombreux que d'habitude : ils \u00e9taient six, deux gar\u00e7ons et quatre filles. Les deux gar\u00e7ons marchaient d'un pas s\u00fbr et soutenu, leur apparence \u00e9tait celle de jeunes bolcheviks aguerris. Tous deux \u00e9taient en quatri\u00e8me ann\u00e9e de licence de sciences physiques, mention \"physique th\u00e9orique\", et par ailleurs anciens \u00e9tudiants de Ye Zhetai. Les quatre filles, beaucoup plus jeunes, \u00e9taient \u00e9l\u00e8ves de deuxi\u00e8me ann\u00e9e du lyc\u00e9e affili\u00e9 \u00e0 l'universit\u00e9. Ces petites combattantes v\u00eatues d'un uniforme serr\u00e9 \u00e0 la taille par une ceinture militaire irradiaient de l'\u00e9nergie bouillonnante propre \u00e0 l'adolescence. On aurait dit quatre braseros verts encerclant Ye Zhetai. L'apparition de ce dernier provoqua l'embrasement de la foule qui se trouvait sous la tribune et les slogans que l'on rechignait \u00e0 clamer quelques instants plus t\u00f4t r\u00e9sonn\u00e8rent de plus belle, comme une mar\u00e9e emportant tout sur son passage.\n\nApr\u00e8s avoir patiemment attendu la fin de ce d\u00e9ferlement, les deux gar\u00e7ons debout sur l'estrade se tourn\u00e8rent vers l'accus\u00e9 :\n\n\u2014 Camarade Ye Zhetai, vous \u00eates un expert en sciences m\u00e9caniques, vous devez bien vous rendre compte que la force unie contre laquelle vous r\u00e9sistez est si puissante que si vous continuez \u00e0 vous obstiner, elle vous conduira \u00e0 la mort ! Nous allons aujourd'hui reprendre la s\u00e9ance l\u00e0 o\u00f9 nous nous \u00e9tions arr\u00eat\u00e9s la derni\u00e8re fois. Ne perdons pas de temps en palabres. Professeur Ye, r\u00e9pondez \u00e0 la question suivante : de 1962 \u00e0 1965, lors de vos cours d'introduction \u00e0 la physique, avez-vous, oui ou non, donn\u00e9 sans permission un grand nombre d'informations sur la th\u00e9orie de la relativit\u00e9 ?\n\n\u2014 La th\u00e9orie de la relativit\u00e9 est un principe de base des sciences physiques, comment pourrais-je ne pas en avoir parl\u00e9 durant mes cours de physique fondamentale ? r\u00e9pondit Ye Zhetai.\n\n\u2014 Menteur ! hurla une jeune garde rouge \u00e0 c\u00f4t\u00e9 de lui. Einstein \u00e9tait un repr\u00e9sentant de l'Autorit\u00e9 acad\u00e9mique r\u00e9actionnaire ! Et, comme les mauvais esprits se rencontrent, il est parti fabriquer la bombe atomique pour les imp\u00e9rialistes am\u00e9ricains ! Si nous voulons fonder une science v\u00e9ritablement r\u00e9volutionnaire, il nous faut d'abord mettre \u00e0 bas le pavillon noir de la th\u00e9orie de la relativit\u00e9 qui repr\u00e9sente les int\u00e9r\u00eats de la classe bourgeoise !\n\nYe Zhetai garda le silence, il encaissait la souffrance de devoir porter une coiffe de m\u00e9tal et une plaque sur la poitrine, restant muet aux questions qui ne m\u00e9ritaient aucune r\u00e9ponse. Dans son dos, ses \u00e9tudiants fron\u00e7aient les sourcils. La jeune fille qui venait de prendre la parole \u00e9tait la plus sagace des quatre jeunes gardes rouges. Elle avait manifestement pr\u00e9par\u00e9 sa tirade, car il l'avait vue un peu plus t\u00f4t r\u00e9p\u00e9ter derri\u00e8re la tribune. Cependant, pour affronter Ye Zhetai, il ne suffisait pas d'\u00e2nonner des slogans. Aujourd'hui, ils avaient choisi de faire usage d'une arme nouvelle pour vaincre le professeur. Une jeune garde rouge fit un signe de main \u00e0 quelqu'un dans l'assistance.\n\nAssise au premier rang, Shao Lin, \u00e9pouse de Ye Zhetai et coll\u00e8gue de celui-ci dans la facult\u00e9 de sciences physiques, se leva et monta sur l'estrade. Elle \u00e9tait affubl\u00e9e d'une veste vert gazon qui ne lui allait pas du tout, mais qui se voulait tout proche de la couleur des uniformes des gardes rouges. Cependant, pour tous ceux qui avaient connu Shao Lin et savaient qu'elle aimait dans le pass\u00e9 se v\u00eatir d'\u00e9l\u00e9gantes robes mandchoues, cet accoutrement paraissait saugrenu.\n\n\u2014 Ye Zhetai ! cria-t-elle \u00e0 l'adresse de son \u00e9poux.\n\nIl \u00e9tait \u00e9vident qu'elle n'avait gu\u00e8re l'habitude de ce genre de spectacle, et \u00e9lever le ton lui co\u00fbtait des efforts, au point m\u00eame que les tr\u00e9molos dans sa voix avaient redoubl\u00e9 d'intensit\u00e9.\n\n\u2014 Tu ne t'imaginais certainement pas que je me dresserais devant toi aujourd'hui pour te d\u00e9noncer et te critiquer, n'est-ce pas ? Oui, tu m'as autrefois bern\u00e9e, tu m'as aveugl\u00e9e avec ta vision r\u00e9actionnaire du monde et de la science ! Mais j'ai ouvert les yeux \u00e0 pr\u00e9sent ! Gr\u00e2ce \u00e0 l'aide des petits g\u00e9n\u00e9raux de la r\u00e9volution, je me tiens d\u00e9sormais du c\u00f4t\u00e9 de la r\u00e9volution, du c\u00f4t\u00e9 du peuple !\n\nElle se tourna vers l'assistance :\n\n\u2014 Camarades ! Petits g\u00e9n\u00e9raux de la r\u00e9volution ! Vous tous, personnels et \u00e9ducateurs r\u00e9volutionnaires ! Nous devons prendre conscience de la nature r\u00e9actionnaire de la th\u00e9orie de la relativit\u00e9 d'Einstein. Son aspect r\u00e9actionnaire est encore plus explicite pour ce qui est du principe de la relativit\u00e9 gravitationnelle ! Il propose un mod\u00e8le statique de l'univers et nie la nature dynamique de la mati\u00e8re. Cette th\u00e9orie est antidialectique ! Elle pr\u00e9tend que l'univers a des limites, ce qui n'est autre chose que de l'id\u00e9alisme r\u00e9actionnaire...\n\nEn entendant son \u00e9pouse d\u00e9verser son \u00e9loquent discours, Ye Zhetai eut un sourire amer :\n\n\u2014 Lin, je t'ai donc aveugl\u00e9e ? \u00c0 dire vrai, tu as toujours \u00e9t\u00e9 une \u00e9nigme pour moi. Un jour, j'ai lou\u00e9 ton intelligence devant ton p\u00e8re \u2013 il a d'ailleurs de la chance d'\u00eatre d\u00e9j\u00e0 parti, cela lui \u00e9pargne d'\u00eatre aujourd'hui t\u00e9moin de ce d\u00e9sastre \u2013, il a secou\u00e9 la t\u00eate et m'a affirm\u00e9 que sa fille n'apporterait jamais aucune contribution \u00e0 la science. Puis, il a prononc\u00e9 cette phrase qui s'est grav\u00e9e en moi pour le reste de ma vie : \"Lin est trop intelligente. Pour r\u00e9ussir dans le domaine des th\u00e9ories fondamentales, il faut \u00eatre stupide.\" Les ann\u00e9es ont pass\u00e9 et je n'ai cess\u00e9 de m\u00e9diter la signification de cette phrase. Lin, tu es trop intelligente. Voil\u00e0 d\u00e9j\u00e0 quelques ann\u00e9es que tu as senti le vent politique tourner dans le monde acad\u00e9mique. Tu as assur\u00e9 tes arri\u00e8res, par exemple en changeant dans tes cours les intitul\u00e9s des param\u00e8tres et des lois de la physique : la loi d'Ohm est devenue la loi de la r\u00e9sistance \u00e9lectrique, les \u00e9quations de Maxwell sont devenues les \u00e9quations \u00e9lectromagn\u00e9tiques, la constante de Planck est devenue la loi des quanta... Tu expliques \u00e0 tes \u00e9tudiants que les r\u00e9sultats de la science ne sont que la cristallisation de la sagesse des masses laborieuses, que les autorit\u00e9s acad\u00e9miques r\u00e9actionnaires ne faisaient que d\u00e9rober les fruits de cette sagesse. Mais malgr\u00e9 ton z\u00e8le, tu n'as toujours pas \u00e9t\u00e9 admise dans le courant r\u00e9volutionnaire, regarde-toi maintenant, tu ne portes m\u00eame pas sur tes manches le brassard rouge du \"personnel \u00e9ducatif de la r\u00e9volution\". Tu es venue les mains vides, sans m\u00eame avoir le droit d'emporter avec toi un ouvrage des citations des plus grands r\u00e9volutionnaires... Pourquoi le destin t'a-t-il fait na\u00eetre dans une illustre famille de la Chine ancienne ? Et dire que tes parents sont tous deux d'\u00e9minents scientifiques.\n\nTiens, en parlant d'Einstein, tu as sans doute davantage \u00e0 confesser que moi. Cet hiver de l'ann\u00e9e 1922, lorsque Einstein est venu \u00e0 Shanghai pour donner une conf\u00e9rence, ton p\u00e8re, qui parlait tr\u00e8s bien l'allemand, a \u00e9t\u00e9 d\u00e9sign\u00e9 pour faire partie de ses accompagnateurs. Tu m'as racont\u00e9 plusieurs fois que c'est avec la b\u00e9n\u00e9diction d'Einstein lui-m\u00eame que ton p\u00e8re avait suivi la voie des sciences physiques et que, si tu avais choisi de faire de la physique ton m\u00e9tier, c'\u00e9tait sous l'influence directe de ton p\u00e8re. Ce qui faisait donc du grand Albert ton mentor indirect ! Quel bonheur, quelle fiert\u00e9 tu en tirais ! Puis plus tard, j'ai appris que ton p\u00e8re t'avait racont\u00e9 un mensonge bienveillant, car il n'avait en r\u00e9alit\u00e9 \u00e9chang\u00e9 qu'un bref instant avec Einstein.\n\nC'\u00e9tait le matin du 13 novembre 1922, alors qu'il marchait dans la rue de Nankin, accompagn\u00e9, si je ne me trompe, de Yu Youren, pr\u00e9sident de l'universit\u00e9 de Shanghai, de Cao Gubing, patron du journal _Ta Kung Pao_ , et d'autres chercheurs... Tandis qu'ils passaient \u00e0 c\u00f4t\u00e9 d'un chantier de voirie, Einstein s'est arr\u00eat\u00e9 \u00e0 hauteur d'un ouvrier en train de briser des pierres, il a regard\u00e9 en silence ce gar\u00e7on aux mains et au visage sales simplement v\u00eatu d'une chemise en loques dans le vent glacial de l'hiver. Il a demand\u00e9 \u00e0 ton p\u00e8re : \"Combien gagne cet homme par jour ?\" Apr\u00e8s avoir interrog\u00e9 l'ouvrier, ton p\u00e8re a r\u00e9pondu : \"Cinq centimes.\"Ce fut le seul \u00e9change entre ton p\u00e8re et ce scientifique qui a chang\u00e9 la face du monde. Il n'a \u00e9t\u00e9 question ni de physique ni de th\u00e9orie de la relativit\u00e9, simplement de la duret\u00e9 de la r\u00e9alit\u00e9. Selon ton p\u00e8re, Einstein est rest\u00e9 debout en silence encore un moment au m\u00eame endroit en observant le labeur languissant du jeune gar\u00e7on. La pipe qu'il tenait dans sa main s'est \u00e9teinte alors m\u00eame qu'il n'en avait pas tir\u00e9 une seule bouff\u00e9e. Lorsqu'il m'a racont\u00e9 ce souvenir, ton p\u00e8re a l\u00e2ch\u00e9 un soupir en disant : \"En Chine, toutes les pens\u00e9es libres et contestataires, apr\u00e8s avoir pris leur envol, finissent toutes un jour ou l'autre par s'\u00e9craser sur le sol, car la gravit\u00e9 de la r\u00e9alit\u00e9 est trop lourde.\"\n\n\u2014 Baisse la t\u00eate ! ordonna en hurlant un garde rouge.\n\nYe Zhetai n'aurait pas su dire pourquoi, mais il avait senti dans cette injonction comme la manifestation d'un reliquat de sympathie d'un \u00e9tudiant pour son professeur. Car en lui accordant de baisser la t\u00eate, le chapeau de m\u00e9tal tomberait et tant qu'on lui ordonnerait de maintenir la t\u00eate baiss\u00e9e, il n'y aurait aucune raison pour le forcer \u00e0 porter \u00e0 nouveau le chapeau. Cependant Ye Zhetai garda la t\u00eate haute, soutenant \u00e0 la force de son maigre cou le pesant couvre-chef.\n\n\u2014 Baisse la t\u00eate ! Esp\u00e8ce d'insolent r\u00e9actionnaire !\n\nUne garde rouge qui se tenait \u00e0 c\u00f4t\u00e9 de lui se d\u00e9fit de sa ceinture de cuir et la lan\u00e7a en direction de Ye Zhetai. En heurtant son front, la boucle en laiton de la ceinture laissa une marque visible qui fut tr\u00e8s vite escamot\u00e9e par un gros h\u00e9matome violac\u00e9. Ye Zhetai chancela, mais il parvint \u00e0 retrouver son \u00e9quilibre. Un garde rouge reprit ses semonces :\n\n\u2014 Lors de vos cours sur la m\u00e9canique quantique, vous avez aussi propag\u00e9 une grande quantit\u00e9 de contrev\u00e9rit\u00e9s r\u00e9actionnaires.\n\nCela dit, il fit signe de la t\u00eate vers Shao Lin pour lui indiquer qu'elle pouvait poursuivre.\n\nShao Lin n'attendait que cette occasion, elle devait continuer \u00e0 parler sans rel\u00e2che pour pr\u00e9server son esprit d'un effritement total.\n\n\u2014 Ye Zhetai, voil\u00e0 un fait que tu ne pourras pas nier ! Combien de fois as-tu distill\u00e9 \u00e0 tes \u00e9tudiants l'hypoth\u00e8se contre-r\u00e9volutionnaire de l'\u00e9cole de Copenhague ?\n\n\u2014 Mais ce n'est ni plus ni moins que l'hypoth\u00e8se reconnue comme \u00e9tant la plus conforme \u00e0 l'interpr\u00e9tation des r\u00e9sultats exp\u00e9rimentaux, r\u00e9torqua Ye Zhetai.\n\nLe ton de sa voix, toujours aussi calme m\u00eame apr\u00e8s qu'il eut re\u00e7u un tel coup, emplit Shao Lin de stupeur et d'angoisse.\n\n\u2014 Cette interpr\u00e9tation postule qu'une observation ext\u00e9rieure conduit \u00e0 une r\u00e9duction du paquet d'ondes. Encore une autre manifestation outrageante de l'id\u00e9alisme r\u00e9actionnaire !\n\n\u2014 Est-ce la philosophie qui doit guider l'exp\u00e9rience ou l'exp\u00e9rience qui doit guider la philosophie ? l'interrogea Ye Zhetai.\n\nCette soudaine contre-attaque plongea un moment ses accusateurs dans l'embarras.\n\n\u2014 C'est bien entendu la juste philosophie marxiste qui doit diriger les exp\u00e9riences scientifiques ! lan\u00e7a un garde rouge.\n\n\u2014 Cela revient donc \u00e0 dire que cette pr\u00e9tendue \"philosophie\" juste tombe du ciel. C'est donc remettre en cause l'id\u00e9e que les connaissances v\u00e9ritables proviennent de la pratique, ce qui \u2013 soit dit en passant \u2013 va pr\u00e9cis\u00e9ment \u00e0 contre-courant des principes marxistes de la connaissance du monde naturel.\n\nNi Shao Lin ni ses deux \u00e9tudiants gardes rouges n'\u00e9taient en mesure de justifier convenablement ce paradoxe. Contrairement aux jeunes coll\u00e9giens gardes rouges, ils ne pouvaient pas riposter en faisant fi de toute logique. Mais ce n'\u00e9tait pas le cas des quatre jeunes filles, persuad\u00e9es que la r\u00e9volution ne devait souffrir aucune attaque. La fille qui venait tout juste de fouetter Ye Zhetai lui ass\u00e9na un nouveau coup de ceinture. Ce fut ensuite le tour des trois autres filles de le cingler de la m\u00eame mani\u00e8re, car d\u00e8s lors qu'un camarade faisait la r\u00e9volution, on se devait d'\u00eatre encore plus r\u00e9volutionnaire, ou \u00e0 tout le moins autant. Les deux gar\u00e7ons ne bronch\u00e8rent pas. S'ils \u00e9taient intervenus, on les aurait suspect\u00e9s \u00e0 leur tour d'\u00eatre des contre-r\u00e9volutionnaires.\n\n\u2014 Durant vos cours, vous avez \u00e9galement propag\u00e9 la th\u00e9orie du Big Bang, certainement la th\u00e9orie scientifique la plus r\u00e9actionnaire de l'univers ! dit un garde rouge en essayant de changer de sujet.\n\n\u2014 Peut-\u00eatre cette th\u00e9orie sera-t-elle retoqu\u00e9e dans le futur, mais les deux grandes d\u00e9couvertes astronomiques de notre si\u00e8cle, la loi de Hubble et le fond diffus cosmologique, font du Big Bang la th\u00e9orie la plus cr\u00e9dible de l'origine de l'univers.\n\n\u2014 Foutaises ! s'\u00e9gosilla Shao Lin, puis elle se lan\u00e7a dans un interminable r\u00e9quisitoire contre la th\u00e9orie du Big Bang, en n'oubliant naturellement pas de diss\u00e9quer en profondeur sa nature r\u00e9actionnaire.\n\nCependant le caract\u00e8re parfaitement insolite de cette th\u00e9orie intrigua la plus sagace des jeunes gardes rouges, qui ne put s'emp\u00eacher de demander :\n\n\u2014 Mais alors le temps aussi a pris naissance avec cette singularit\u00e9 initiale ? Qu'y avait-il avant la singularit\u00e9 ?\n\n\u2014 Rien du tout, dit Ye Zhetai, comme s'il r\u00e9pondait \u00e0 une question pos\u00e9e par une petite fille.\n\nIl tourna la t\u00eate et la regarda avec bienveillance, mais son chapeau en m\u00e9tal et ses blessures rendirent ce mouvement tr\u00e8s difficile.\n\n\u2014 Comment ? Rien ? Mais c'est r\u00e9actionnaire, absolument r\u00e9actionnaire ! hurla la jeune fille avec fureur.\n\nNe sachant quoi r\u00e9pondre d'autre, elle se tourna vers Shao Lin pour solliciter un soutien qu'elle obtint aussit\u00f4t.\n\n\u2014 Cette hypoth\u00e8se laisse une place pour l'existence de Dieu, sugg\u00e9ra Shao Lin \u00e0 la jeune fille en hochant la t\u00eate.\n\nLa jeune garde rouge, encore confuse, se trouva ragaillardie et mena\u00e7a Ye Zhetai avec sa ceinture :\n\n\u2014 Vous... pr\u00e9tendez que Dieu existe ?\n\n\u2014 Je n'en sais rien.\n\n\u2014 Que dites-vous ?\n\n\u2014 Je dis que je n'en sais rien. Si par Dieu, vous entendez une sorte de superconscience au-del\u00e0 de l'univers, alors je n'ai aucune id\u00e9e de son existence. Que l'on soit pour ou contre, la science ne fournit pour l'instant aucune preuve tangible.\n\nLe cauchemar qu'il \u00e9tait en train de vivre en ce moment faisait plut\u00f4t pencher Ye Zhetai vers la conviction de l'inexistence de Dieu.\n\nCes propos r\u00e9actionnaires outranciers soulev\u00e8rent une indignation g\u00e9n\u00e9rale sur le terrain de sport et, sous l'impulsion d'un garde rouge situ\u00e9 juste sous l'estrade, retentirent \u00e0 nouveau des salves de slogans.\n\n\u2014 \u00c0 bas le pouvoir scientifique r\u00e9actionnaire de Ye Zhetai !\n\n\u2014 \u00c0 bas les autorit\u00e9s scientifiques r\u00e9actionnaires !\n\n\u2014 \u00c0 bas les doctrines r\u00e9actionnaires !\n\n\u2014 Dieu n'existe pas ! Les religions sont des subterfuges cr\u00e9\u00e9s par les classes bourgeoises pour museler l'esprit des classes laborieuses ! cria la jeune fille une fois que les slogans se furent apais\u00e9s.\n\n\u2014 C'est un point de vue \u00e0 sens unique, r\u00e9pliqua froidement Ye Zhetai.\n\nBless\u00e9e dans son amour-propre, la garde rouge \u00e9mit un jugement radical et sans appel, estimant qu'aucun discours ne pourrait jamais faire changer d'avis le dangereux ennemi qui se dressait devant elle. Arm\u00e9e de sa ceinture, elle se pr\u00e9cipita sur lui, aussit\u00f4t imit\u00e9e par ses trois comparses. Mais comme Ye Zhetai \u00e9tait de grande taille, les quatre gamines de quatorze ans durent faire de grands moulinets pour atteindre la t\u00eate que Ye refusait de baisser. Apr\u00e8s quelques coups, le chapeau de m\u00e9tal qui lui servait malgr\u00e9 tout de protection tomba et d\u00e8s lors, les boucles en laiton des larges ceintures s'abattirent sur lui comme un d\u00e9luge. Ye Zhetai finit par s'\u00e9crouler sur le sol. Enhardies par cette issue, les jeunes gardes rouges se jet\u00e8rent corps et \u00e2me dans ce glorieux combat, persuad\u00e9es de lutter pour des id\u00e9es, pour des id\u00e9aux, enivr\u00e9es par la perspective de participer \u00e0 une mission historique, fi\u00e8res de leur acte h\u00e9ro\u00efque...\n\n\u2014 Le pr\u00e9sident Mao nous enseigne de combattre par le verbe et non par la violence ! se d\u00e9cid\u00e8rent enfin \u00e0 crier les deux \u00e9tudiants de Ye Zhetai, et ils se pr\u00e9cipit\u00e8rent pour \u00e9carter les quatre filles, devenues \u00e0 moiti\u00e9 folles.\n\nMais il \u00e9tait d\u00e9j\u00e0 trop tard, le physicien \u00e9tait allong\u00e9, immobile, sur le sol. Ses yeux \u00e0 moiti\u00e9 clos semblaient fixer le filet de sang qui s'\u00e9coulait de son cr\u00e2ne. L'assistance, aussit\u00f4t gagn\u00e9e par la folie, se figea en un instant dans un silence de mort. La seule chose en mouvement \u00e9tait ce filet de sang qui rampait en ondulant tel un serpent jusqu'au bord de l'estrade o\u00f9 il tombait goutte \u00e0 goutte et en rythme dans un carton vide, comme des bruits de pas, tant\u00f4t proches, tant\u00f4t \u00e9loign\u00e9s.\n\nUn rire myst\u00e9rieux, \u00e0 glacer le sang, brisa le silence. Il semblait provenir de l'esprit d\u00e9j\u00e0 disloqu\u00e9 de Shao Lin. Certains commenc\u00e8rent \u00e0 quitter les lieux et ce fut bient\u00f4t la d\u00e9bandade. Tous fuyaient la sc\u00e8ne le plus rapidement possible, jusqu'\u00e0 ce qu'il ne reste plus qu'une jeune fille sous l'estrade.\n\nC'\u00e9tait Ye Wenjie, la fille de Ye Zhetai.\n\nLorsque les quatre filles s'en \u00e9taient prises \u00e0 son p\u00e8re, elle avait voulu se ruer sur l'estrade, mais deux vieux concierges de l'\u00e9cole qui se tenaient debout \u00e0 ses c\u00f4t\u00e9s l'en avaient emp\u00each\u00e9e en lui glissant \u00e0 l'oreille qu'elle aussi risquait de perdre la vie. Son apparition n'aurait fait qu'exacerber la violence d'une sc\u00e8ne d\u00e9j\u00e0 guid\u00e9e par la folie. Elle avait pleur\u00e9 et hurl\u00e9 de toutes ses forces, mais sa voix avait \u00e9t\u00e9 noy\u00e9e par les vagues de slogans et d'encouragements de la foule. Maintenant que tout \u00e9tait redevenu calme, elle n'\u00e9tait plus capable de faire s'\u00e9chapper le moindre son de sa bouche, elle ne pouvait que fixer le corps sans vie de son p\u00e8re. Tout ce qui n'avait pas pu sortir sous forme de pleurs ou de cris s'\u00e9tait dissous dans son sang et l'accompagnerait le restant de sa vie.\n\nAlors que tout le monde \u00e9tait parti, elle \u00e9tait debout, l\u00e0, le corps et les membres ayant conserv\u00e9 la m\u00eame position immobile dans laquelle les vieux concierges l'avaient contrainte de demeurer, comme si elle s'\u00e9tait fossilis\u00e9e. Ce ne fut qu'apr\u00e8s de longues minutes qu'elle rel\u00e2cha ses bras encore tendus et qu'elle se dirigea lentement vers la tribune, s'assit \u00e0 c\u00f4t\u00e9 du cadavre et serra la main d\u00e9j\u00e0 froide de son p\u00e8re, dont les yeux vides se perdaient dans le lointain. Lorsque quelqu'un vint pour emporter le corps, Ye Wenjie sortit de la poche de sa chemise un objet qu'elle glissa dans la main de son p\u00e8re : sa pipe.\n\nQuand Wenjie quitta silencieusement les lieux, il n'y avait d\u00e9j\u00e0 plus personne au milieu du chaos du terrain de sport. Elle prit le chemin qui la ramenait chez elle. Lorsqu'elle arriva devant le dortoir des enseignants, elle entendit au premier \u00e9tage des gloussements provenir de leur maison : ce rire \u00e9manait de la bouche de la femme qu'elle appelait autrefois sa m\u00e8re. Wenjie tourna les talons sans bruit, laissant ses pieds la conduire o\u00f9 bon leur semblait.\n\nElle s'aper\u00e7ut que ceux-ci l'avaient transport\u00e9e jusque devant la porte de Ruan Wen, son professeur principal lorsqu'elle \u00e9tait en quatri\u00e8me ann\u00e9e de licence, et aussi son amie la plus fid\u00e8le. \u00c0 compter des deux ann\u00e9es de ma\u00eetrise d'astrophysique, puis ensuite, lorsque les cours furent arr\u00eat\u00e9s avec le lancement de la r\u00e9volution et jusqu'\u00e0 aujourd'hui, le Pr Ruan avait \u00e9t\u00e9, avec son p\u00e8re, la personne dont elle s'\u00e9tait sentie le plus proche. Ruan Wen \u00e9tait jadis partie \u00e9tudier \u00e0 Cambridge. Sa maison \u00e9tait pour Ye Wenjie un v\u00e9ritable cabinet de curiosit\u00e9s : elle regorgeait de livres aux belles reliures, de peintures et de disques ramen\u00e9s d'Europe, et elle poss\u00e9dait m\u00eame un piano. Elle avait \u00e9galement une collection de pipes occidentales dispos\u00e9es en ordre sur une exquise \u00e9tag\u00e8re en bois. C'\u00e9tait elle qui avait offert la pipe \u00e0 son p\u00e8re. Les pipes \u00e9taient faites de diff\u00e9rents mat\u00e9riaux : rhizomes de bruy\u00e8re m\u00e9diterran\u00e9enne, s\u00e9piolite de Turquie... C'\u00e9tait comme si chacune d'entre elles avait \u00e9t\u00e9 impr\u00e9gn\u00e9e de la sagesse de l'homme qui les avait jadis tenues dans les mains et port\u00e9es \u00e0 sa bouche. Toutefois Ruan Wen ne lui avait jamais parl\u00e9 de lui. Ce chaleureux petit monde raffin\u00e9 \u00e9tait devenu aux yeux de Wenjie un port o\u00f9 jeter l'ancre pour \u00e9chapper aux temp\u00eates de la terre. Mais cela, c'\u00e9tait avant que la maison de Ruan Wen ne soit fouill\u00e9e et ses biens, saisis. Comme son p\u00e8re, elle aussi subissait de terribles \u00e9preuves depuis le d\u00e9but de la R\u00e9volution culturelle. Durant la s\u00e9ance de critique o\u00f9 elle s'\u00e9tait retrouv\u00e9e sur le banc des accus\u00e9s, les gardes rouges lui avaient accroch\u00e9 ses chaussures \u00e0 talons hauts autour du cou et avaient barbouill\u00e9 son visage de rouge \u00e0 l\u00e8vres, pour montrer la d\u00e9cadence de sa vie bourgeoise.\n\nLorsque Ye Wenjie poussa la porte de Ruan Wen, elle d\u00e9couvrit que la maison avait \u00e9t\u00e9 remise en ordre depuis qu'elle avait \u00e9t\u00e9 d\u00e9valis\u00e9e. Les peintures \u00e0 l'huile arrach\u00e9es avaient \u00e9t\u00e9 raccroch\u00e9es au mur, le piano qui avait \u00e9t\u00e9 renvers\u00e9 \u00e9tait retourn\u00e9 \u00e0 sa place d'origine \u2013 bien que les touches soient cass\u00e9es et l'instrument injouable, il \u00e9tait parfaitement propre \u2013 tandis que les lambeaux de livres qui n'avaient pas \u00e9t\u00e9 confisqu\u00e9s \u00e9taient align\u00e9s sur l'\u00e9tag\u00e8re... Ruan Wen \u00e9tait assise sur un fauteuil pivotant devant son bureau, les yeux ferm\u00e9s. Elle semblait paisible. Ye Wenjie alla se placer \u00e0 sa hauteur, elle lui toucha le front, le visage et les mains. Glac\u00e9s. D\u00e8s son entr\u00e9e dans la pi\u00e8ce, Wenjie avait en effet remarqu\u00e9 le flacon de somnif\u00e8re vide renvers\u00e9 sur le bureau. Elle demeura un instant debout avant de se retourner. Elle se sentait plus triste que jamais. Elle \u00e9tait \u00e0 pr\u00e9sent devenue un compteur Geiger bloqu\u00e9 sur le chiffre 0 pour avoir \u00e9t\u00e9 soumis \u00e0 trop de radiations. Mais lorsqu'elle voulut sortir de la maison et qu'elle tourna la t\u00eate pour jeter un dernier regard \u00e0 Ruan Wen, elle vit que la professeur se portait comme un charme. Elle se mettait du rouge \u00e0 l\u00e8vres et enfilait ses chaussures \u00e0 talons hauts.\n\n* * *\n\n1 En ao\u00fbt 1967, la revue _Le Drapeau rouge_ publia un \u00e9ditorial intitul\u00e9 \"D\u00e9busquer les contre-r\u00e9volutionnaires au sein de l'arm\u00e9e\" qui contribua \u00e0 la flamb\u00e9e des attaques et des pillages des armureries et des poudri\u00e8res nationales, provoquant une explosion des conflits entre diff\u00e9rentes factions de gardes rouges \u00e0 l'\u00e9chelle de tout le pays. _(N.d.A.)_\n\n## 2\n\n## LE PRINTEMPS SILENCIEUX\n\n_Deux ans plus tard,_\n\n_sur la cha\u00eene de montagnes du Grand Khingan._\n\n\"Timber !\"\n\nAussit\u00f4t apr\u00e8s ce cri retentissant, un m\u00e9l\u00e8ze gros comme un pilier du Parth\u00e9non s'effondra avec fracas sur le sol, et Ye Wenjie sentit la terre trembler sous ses pieds. Elle ramassa sa hache et sa petite scie et se mit \u00e0 d\u00e9biter les branches de l'arbre. Chaque fois, elle avait cette impression d'\u00eatre en train de d\u00e9pecer un gigantesque cadavre. Il lui arrivait m\u00eame d'imaginer que ce g\u00e9ant \u00e9tait son propre p\u00e8re. Les sentiments qui avaient \u00e9t\u00e9 les siens deux ans auparavant lors de cette terrible nuit o\u00f9 elle avait d\u00fb nettoyer la d\u00e9pouille de son p\u00e8re \u00e0 la morgue refaisaient surface. Les larges \u00e9corces des branches devenaient les profondes blessures qui recouvraient le corps de son p\u00e8re.\n\nPlus de cent mille travailleurs, appartenant \u00e0 six divisions et \u00e0 quarante et un r\u00e9giments du Corps de production et de construction de la Mongolie-Int\u00e9rieure, \u00e9taient diss\u00e9min\u00e9s dans les vastes for\u00eats et prairies de la r\u00e9gion. Tout juste d\u00e9barqu\u00e9s de la ville dans ce monde inconnu, beaucoup de jeunes instruits ayant rejoint le Corps entretenaient un espoir romantique : lorsque les tanks imp\u00e9rialistes des Sovi\u00e9tiques passeraient la fronti\u00e8re de la Mongolie-Int\u00e9rieure, ils se mettraient rapidement en ordre de bataille, et leur chair et leur sang formeraient le premier rempart de la R\u00e9publique. C'\u00e9tait bien l\u00e0 l'une des appr\u00e9ciations strat\u00e9giques qui avaient pr\u00e9sid\u00e9 \u00e0 la constitution de ces unit\u00e9s. Mais la guerre que tous esp\u00e9raient tant \u00e9tait comme ces montagnes que l'on distingue au loin sans jamais parvenir \u00e0 les atteindre. Ils devaient donc se contenter de d\u00e9frichage, de p\u00e2turage et de coupe du bois. Ces jeunes adolescents enflamm\u00e9s par la fi\u00e8vre de la R\u00e9volution culturelle s'apercevaient vite que, par rapport \u00e0 ces terres sauvages, les plus grandes villes du continent n'\u00e9taient en r\u00e9alit\u00e9 que de simples bergeries. Leur ardeur fr\u00e9n\u00e9tique n'avait aucun sens au milieu de ces prairies et de ces for\u00eats froides et insondables o\u00f9 une giclure de sang chaud refroidissait plus vite qu'une bouse de vache, et o\u00f9 le purin avait davantage de valeur que le sang. Mais ils \u00e9taient de la g\u00e9n\u00e9ration des enflamm\u00e9s, leur destin \u00e9tait de se consumer dans le brasier de la r\u00e9volution. C'est pourquoi, sous les assauts r\u00e9p\u00e9t\u00e9s de leurs tron\u00e7onneuses, d'immenses for\u00eats touffues devenaient des collines chauves et arides ; sous les roues de leurs tracteurs et de leurs moissonneuses-batteuses, de vastes prairies \u00e9taient labour\u00e9es avant de devenir des champs de c\u00e9r\u00e9ales, puis des d\u00e9serts.\n\nYe Wenjie n'avait qu'un seul mot pour d\u00e9crire la d\u00e9forestation dont elle \u00e9tait t\u00e9moin : \"folie\". Les hauts et nobles m\u00e9l\u00e8zes de Dahurie, les pins sylvestres \u00e0 feuilles persistantes, les gracieux bouleaux de Mandchourie, les trembles qui touchaient les nuages, les sapins de Sib\u00e9rie, les bouleaux noirs, les ormes des montagnes, les fr\u00eanes de Mandchourie, les chosenias, les ch\u00eanes de Mongolie : ils coupaient tout ce qui se trouvait sur leur chemin. Les tron\u00e7onneuses fondaient sur les arbres comme un essaim de criquets d'acier. Partout o\u00f9 \u00e9taient pass\u00e9es les unit\u00e9s du Corps, il ne restait plus qu'un parterre de souches.\n\nUn tracteur \u00e9tait sur le point d'emporter le m\u00e9l\u00e8ze abattu. \u00c0 l'autre extr\u00e9mit\u00e9, Wenjie caressa doucement la section sci\u00e9e de l'arbre. Elle le faisait souvent. Elle se disait que c'\u00e9tait une immense cicatrice et qu'ainsi elle pouvait ressentir la douleur intense \u00e9prouv\u00e9e par l'arbre. Soudain, elle vit une autre main en train de caresser une autre souche, et les frissons qui parcouraient cette main r\u00e9sonn\u00e8rent en elle. La main semblait diaphane, mais on voyait que c'\u00e9tait celle d'un homme. Ye Wenjie leva la t\u00eate, elle d\u00e9couvrit que c'\u00e9tait Bai Mulin, un jeune homme mince et fr\u00eale, journaliste au _Journal de la grandeproduction_. Il \u00e9tait arriv\u00e9 l'avant-veille pour faire un reportage sur son r\u00e9giment. Ye Wenjie avait d\u00e9j\u00e0 lu certains de ses articles, il avait une plume \u00e9l\u00e9gante et faisait preuve d'une sensibilit\u00e9 et d'une minutie qui tranchaient avec la brutalit\u00e9 de l'environnement, ce qui lui avait fait grande impression.\n\n\u2014 Ma Gang, viens par ici, cria Bai Mulin \u00e0 un gar\u00e7on un peu plus loin.\n\nLe gar\u00e7on, aussi robuste que le m\u00e9l\u00e8ze qu'il venait tout juste d'abattre, s'approcha, et le journaliste Bai l'interrogea :\n\n\u2014 Sais-tu quel \u00e2ge a cet arbre ?\n\n\u2014 C'est facile \u00e0 compter, r\u00e9pondit Ma Gang en indiquant les cernes sur la souche.\n\n\u2014 C'est ce que j'ai fait. Cet arbre a trois si\u00e8cles. Combien de temps t'a-t-il fallu pour le scier ?\n\n\u2014 \u00c0 peine dix minutes, tu peux me croire, je suis le tron\u00e7onneur le plus rapide du r\u00e9giment. Peu importe l'\u00e9quipe o\u00f9 je suis affect\u00e9, la banni\u00e8re rouge des travailleurs mod\u00e8les me suit et ne bouge plus.\n\nL'enthousiasme de Ma Gang \u00e9tait manifeste, comme d'ailleurs celui de tous ceux auxquels le journaliste avait pr\u00eat\u00e9 attention. Avoir son nom dans les lignes d'un reportage du _Journal de la grande production_ \u00e9tait un insigne honneur.\n\n\u2014 Plus de trois si\u00e8cles, une dizaine de g\u00e9n\u00e9rations. \u00c0 l'\u00e2ge de ses premiers bourgeons, la Chine \u00e9tait encore \u00e0 l'\u00e9poque de la dynastie Ming. Pendant ces longues ann\u00e9es, combien de temp\u00eates a-t-il \u00e9prouv\u00e9es, de combien d'intrigues a-t-il \u00e9t\u00e9 le t\u00e9moin ? Mais toi, en quelques minutes, tu l'as abattu d'un coup de tron\u00e7onneuse, tu ne ressens donc rien ?\n\n\u2014 Que veux-tu que je ressente ? (Ma Gang resta un moment interdit.) Ce n'est qu'un arbre, \u00e7a ne manque pas ici, il y a un bon paquet de pins bien plus anciens.\n\n\u2014 Retourne donc travailler.\n\nBai Mulin secoua la t\u00eate, il se rassit sur la souche et poussa un l\u00e9ger soupir.\n\nMa Gang soupira lui aussi, tr\u00e8s d\u00e9\u00e7u que le journaliste ne souhaite pas l'interroger davantage.\n\n\u2014 Les intellectuels font toujours beaucoup de bruit pour pas grand-chose.\n\nEt au moment m\u00eame o\u00f9 il parlait, ses yeux se pos\u00e8rent sur Ye Wenjie, assise non loin de l\u00e0. Son jugement l'englobait aussi manifestement.\n\nLe m\u00e9l\u00e8ze fut emport\u00e9 par le tracteur. \u00c0 force de racler contre les pierres et les souches qui jonchaient le sol, son \u00e9corce s'ouvrit et d\u00e9voila sa chair. \u00c0 l'endroit o\u00f9 il \u00e9tait enracin\u00e9 \u00e0 l'origine, l'humus form\u00e9 par les \u00e9paisses couches de feuilles tomb\u00e9es depuis plusieurs ann\u00e9es sur le sol se creusa en un canal d'o\u00f9 suintait de l'eau que les feuilles mortes coloraient de rouge carmin, comme du sang.\n\n\u2014 Wenjie, viens te reposer un peu par ici.\n\nBai Mulin lui d\u00e9signa une autre souche. Elle \u00e9tait effectivement \u00e9reint\u00e9e, elle posa ses outils et s'approcha avant de s'asseoir dos \u00e0 dos avec le journaliste.\n\nApr\u00e8s un moment de silence, Bai Mulin d\u00e9clara subitement :\n\n\u2014 Je peux deviner quels sont tes sentiments. Ici, nous sommes les seuls \u00e0 les partager.\n\nWenjie ne r\u00e9pondit rien. Bai Mulin ne s'attendait pas \u00e0 une r\u00e9ponse de sa part. Elle \u00e9tait d'ordinaire d'un temp\u00e9rament taciturne et ne communiquait que tr\u00e8s peu avec les autres. Les nouvelles recrues croyaient m\u00eame qu'elle \u00e9tait muette.\n\nBai Mulin poursuivit, comme s'il se parlait \u00e0 lui-m\u00eame :\n\n\u2014 Je suis venu ici il y a un an, avant le Corps de production et de construction. Je me souviens qu'il \u00e9tait midi quand je suis arriv\u00e9. Mes h\u00f4tes m'ont dit que nous mangerions du poisson, j'ai jet\u00e9 un coup d'\u0153il dans la petite baraque en bois, mais je n'ai vu sur le feu qu'une marmite remplie d'eau. O\u00f9 \u00e9tait le poisson ? Une fois l'eau bouillie, j'ai vu la personne en charge de la pr\u00e9paration du repas sortir avec un rouleau \u00e0 p\u00e2tisserie et se rendre \u00e0 une rivi\u00e8re qui passait devant la baraque. En quelques coups de rouleau, il a attrap\u00e9 plusieurs saumons. L'endroit abondait en ressources, mais vois ce que la rivi\u00e8re est maintenant devenue, c'est un filet d'eau vaseuse vide de poissons. Je me demande vraiment si c'est un Corps de production ou de destruction.\n\n\u2014 D'o\u00f9 te viennent ces r\u00e9flexions ? demanda Ye Wenjie, en se gardant bien de dire si elle les approuvait ou si elle les rejetait.\n\nCependant, rien que pour cette phrase, Bai Mulin lui \u00e9tait reconnaissant.\n\n\u2014 Je viens de finir un livre qui m'a beaucoup marqu\u00e9... Lis-tu l'anglais ?\n\nEn voyant Wenjie approuver de la t\u00eate, Bai Mulin sortit de son sac un livre \u00e0 la couverture bleue. En le tendant \u00e0 Wenjie, il regarda autour de lui, peut-\u00eatre pour s'assurer que personne ne le voyait faire.\n\n\u2014 Ce livre est sorti en 1962, il a eu un grand retentissement en Occident.\n\nYe Wenjie se tourna pour prendre le livre, elle lut le titre : _Silent Spring_ , par Rachel Carson.\n\n\u2014 O\u00f9 te l'es-tu procur\u00e9 ? lui demanda-t-elle \u00e0 voix basse.\n\n\u2014 Le bouquin a attir\u00e9 l'attention des \u00e9lites. J'y ai eu acc\u00e8s car je suis responsable de toutes les traductions concernant les for\u00eats pour les publications internes r\u00e9serv\u00e9es aux cadres.\n\nWenjie feuilleta le livre. Elle fut tr\u00e8s vite captiv\u00e9e. Dans une br\u00e8ve introduction, l'auteur d\u00e9crivait un village devenu silencieux et sur le point de mourir \u00e0 cause des poisons pr\u00e9sents dans les pesticides. Derri\u00e8re une \u00e9criture simple et limpide battait un c\u0153ur anxieux.\n\n\u2014 Je voudrais \u00e9crire au comit\u00e9 central pour d\u00e9noncer les actes irresponsables du Corps de production et de construction, d\u00e9clara Bai Mulin.\n\nWenjie leva la t\u00eate du livre. Elle mit un certain temps \u00e0 comprendre de quoi il parlait, puis sans rien dire, elle replongea dans sa lecture.\n\n\u2014 Si tu veux le lire, je te le pr\u00eate, mais il vaut mieux que personne ne le voie, tu sais, ce genre de choses..., dit Bai Mulin en jetant un \u0153il autour de lui.\n\nPuis il se leva et partit.\n\nTrente-huit ans plus tard, \u00e0 l'heure de rendre son dernier soupir, Ye Wenjie se rappela l'impact qu'avait eu le _Printemps silencieux_ sur sa vie. Avant cette lecture, la m\u00e9chancet\u00e9 des hommes avait imprim\u00e9 une plaie incurable dans sa jeune \u00e2me, mais ce livre lui avait pour la premi\u00e8re fois permis de r\u00e9fl\u00e9chir de fa\u00e7on rationnelle. Il n'avait en soi rien d'extraordinaire et son sujet \u00e9tait assez restreint, il ne faisait apr\u00e8s tout que d\u00e9crire les pr\u00e9judices provoqu\u00e9s sur l'environnement par l'abus de pesticides, un ph\u00e9nom\u00e8ne que Wenjie consid\u00e9rait autrefois comme un acte l\u00e9gitime, normal ou du moins neutre. Cependant le livre lui avait fait prendre conscience que, du point de vue de la nature, il n'y avait pas de diff\u00e9rence entre cet acte malfaisant et la R\u00e9volution culturelle. Les dommages provoqu\u00e9s sur notre monde \u00e9taient tout aussi consid\u00e9rables. Mais alors, combien y avait-il encore d'actes humains qu'elle croyait l\u00e9gitimes ou justes mais qui se r\u00e9v\u00e9laient en fait nuisibles ? Plus elle y pensait, plus cette id\u00e9e la faisait frissonner, plus elle chavirait dans la peur. Et si le rapport entre l'homme et le mal \u00e9tait celui qui unissait l'oc\u00e9an et l'iceberg ? Deux \u00e9normes masses faites de la m\u00eame mati\u00e8re, \u00e0 cette diff\u00e9rence pr\u00e8s que l'iceberg se distingue plus facilement que l'oc\u00e9an. Mais ce n'est qu'une diff\u00e9rence de forme, car l'iceberg n'est en fin de compte qu'une infime partie de la gigantesque entit\u00e9 qu'est l'oc\u00e9an. Elle avait jadis d\u00e9masqu\u00e9 le mal profond de la R\u00e9volution culturelle que la plupart de ses contemporains trouvaient juste et glorieuse, comme Rachel Carson avait d\u00e9masqu\u00e9 celui de l'usage des pesticides que Wenjie trouvait normal et l\u00e9gitime. Il \u00e9tait bien possible que toutes les actions humaines soient mauvaises par nature et que diff\u00e9rentes personnes puissent d\u00e9masquer diff\u00e9rentes formes de mal. L'humanit\u00e9 ne saurait jamais atteindre une v\u00e9ritable conscience morale, de la m\u00eame mani\u00e8re que l'homme ne pouvait s'\u00e9lever du sol en tirant sur ses propres cheveux. Pour r\u00e9ussir, il fallait l'aide d'une force ext\u00e9rieure \u00e0 l'homme.\n\nEt c'est cette id\u00e9e qui d\u00e9terminerait le cours du destin de Ye Wenjie.\n\nQuatre jours plus tard, elle alla rendre le livre \u00e0 Bai Mulin. Il vivait dans la seule chambre d'h\u00f4te de tout le r\u00e9giment. En poussant la porte, Wenjie l'aper\u00e7ut allong\u00e9 sur son lit, harass\u00e9 de fatigue, le corps couvert de boue et de sciure de bois. En la voyant, il se leva pr\u00e9cipitamment.\n\n\u2014 Tu as travaill\u00e9 aujourd'hui ? demanda Wenjie.\n\n\u2014 Cela fait un moment que je suis ici sans rien faire d'autre que musarder. Je devais mettre la main \u00e0 la p\u00e2te, c'est \u00e7a la solidarit\u00e9 r\u00e9volutionnaire ! Nous sommes all\u00e9s travailler au pic du Radar, la for\u00eat y est luxuriante, les couches d'humus nous arrivaient jusqu'aux genoux, j'ai peur d'avoir attrap\u00e9 le chol\u00e9ra.\n\n\u2014 Le pic du Radar ? sursauta Wenjie.\n\n\u2014 Oui ! Le r\u00e9giment a re\u00e7u l'ordre urgent d'y \u00e9tablir un p\u00e9rim\u00e8tre de s\u00e9curit\u00e9.\n\nLe pic du Radar \u00e9tait un endroit myst\u00e9rieux. C'\u00e9tait une aiguille aux pentes abruptes qui ne portait aucun nom \u00e0 l'origine mais avait acquis cette appellation \u00e0 cause de l'immense antenne qui se trouvait maintenant \u00e0 son sommet. En r\u00e9alit\u00e9, tous ceux qui avaient un peu de bon sens savaient qu'il s'agissait d'une antenne radar qui, bien qu'elle change quotidiennement de direction, ne connaissait pourtant jamais de rotation r\u00e9guli\u00e8re. On pouvait entendre le vrombissement sourd de l'antenne \u00e0 des kilom\u00e8tres. Les membres du r\u00e9giment savaient que le pic abritait une base militaire. Selon les habitants de la r\u00e9gion, lorsque celle-ci avait \u00e9t\u00e9 construite trois ans plus t\u00f4t, elle avait n\u00e9cessit\u00e9 le travail d'un grand nombre d'ouvriers. Une ligne \u00e0 haute tension avait \u00e9t\u00e9 install\u00e9e et une route avait \u00e9t\u00e9 ouverte pour rallier le sommet, sur laquelle avait \u00e9t\u00e9 achemin\u00e9 beaucoup de mat\u00e9riel. Mais une fois la construction de la base achev\u00e9e, la route avait curieusement \u00e9t\u00e9 d\u00e9truite et il ne restait plus qu'un sentier exigu passant \u00e0 travers la for\u00eat, tandis que des h\u00e9licopt\u00e8res venaient se poser en haut du pic. L'antenne radar n'\u00e9tait pas sortie souvent. Quand le vent \u00e9tait trop fort, elle restait au sol, mais quand elle \u00e9tait mise en marche, il se passait des choses \u00e9tranges : les animaux de la for\u00eat s'agitaient, des nu\u00e9es d'oiseaux s'affolaient brusquement, et il arrivait que les humains souffrent de maux de t\u00eate ou de naus\u00e9es. Les gens d'ici perdaient plus facilement leurs cheveux qu'ailleurs et, si on les \u00e9coutait, ces sympt\u00f4mes \u00e9taient apparus apr\u00e8s l'installation de l'antenne. De nombreuses l\u00e9gendes couraient au sujet du pic du Radar. Les jours de grande neige, d\u00e8s que l'antenne \u00e9tait mise en route, les flocons se transformaient en gouttes de pluie. Les temp\u00e9ratures \u00e9tant tr\u00e8s basses, les gouttes d'eau gelaient au contact des arbres et des stalactites de glace apparaissaient suspendues aux branches. La for\u00eat devenait un palais de cristal, dans lequel on entendait sans cesse des bruits de branches bris\u00e9es et de stalactites tombant sur le sol. Parfois, quand l'antenne \u00e9tait en route, un ciel bleu pouvait soudain devenir violemment orageux, et on apercevait parfois dans le ciel nocturne de bien \u00e9tranges halos de lumi\u00e8re...\n\nLe pic du Radar \u00e9tait sous haute surveillance. D\u00e8s l'\u00e9tablissement du camp, le premier avertissement du commandant du Corps de production et de construction fut de rappeler \u00e0 chacun de ne pas approcher du pic, car les sentinelles de la base avaient ordre de tirer sans sommation. La semaine pr\u00e9c\u00e9dente, deux soldats du r\u00e9giment partis chasser avaient poursuivi un chevreuil jusqu'au pied du pic du Radar, et ils furent aussit\u00f4t accueillis par une salve de tirs convulsifs en provenance de la gu\u00e9rite situ\u00e9e \u00e0 mi-hauteur sur le flanc du pic. Par chance, la for\u00eat \u00e9tant \u00e9paisse, ils avaient pu rentrer sains et saufs ; l'un d'entre eux avait eu si peur qu'il en avait mouill\u00e9 son pantalon. Le lendemain, ils furent convoqu\u00e9s et re\u00e7urent chacun un bl\u00e2me. Peut-\u00eatre \u00e9tait-ce pr\u00e9cis\u00e9ment la raison pour laquelle la base du pic du Radar avait d\u00e9cid\u00e9 de cr\u00e9er un p\u00e9rim\u00e8tre de s\u00e9curit\u00e9. Le fait qu'ils puissent utiliser les soldats des r\u00e9giments \u00e0 leur guise montrait d'ailleurs bien \u00e0 quel point ceux qui l'administraient \u00e9taient haut plac\u00e9s.\n\nBai Mulin r\u00e9cup\u00e9ra son livre et le pla\u00e7a d\u00e9licatement sous son oreiller. Il en profita pour sortir du m\u00eame endroit quelques feuilles de brouillon barbouill\u00e9es de caract\u00e8res et les tendit \u00e0 Wenjie :\n\n\u2014 C'est le brouillon de la lettre, pourrais-tu la lire et me donner ton avis ?\n\n\u2014 La lettre ?\n\n\u2014 Je t'en ai parl\u00e9, celle que je vais envoyer au Comit\u00e9 central.\n\nLa lettre \u00e9tait \u00e9crite en pattes de mouche, et Wenjie eut du mal \u00e0 la lire jusqu'au bout. Toutefois le contenu \u00e9tait riche, et les arguments avanc\u00e9s solides. La lettre commen\u00e7ait par expliquer comment la d\u00e9forestation avait transform\u00e9 les montagnes Taihang, c\u00e9l\u00e8bres dans l'histoire pour leur luxuriance, en pitoyables monts chauves. Elle \u00e9voquait ensuite la quantit\u00e9 croissante de boue et de sable dans le lit du fleuve Jaune, pour conclure en indiquant que les op\u00e9rations de d\u00e9forestation men\u00e9es par le Corps de production et de construction provoqueraient des d\u00e9g\u00e2ts irr\u00e9parables. Wenjie remarqua que sa prose \u00e9tait proche de celle de Rachel Carson, pr\u00e9cise, claire, mais pas exempte de po\u00e9sie. Malgr\u00e9 sa formation scientifique, elle se laissa \u00e9mouvoir.\n\n\u2014 C'est tr\u00e8s bien \u00e9crit, lui dit-elle avec sinc\u00e9rit\u00e9.\n\nBai Mulin la remercia d'un hochement de t\u00eate.\n\n\u2014 Eh bien, je vais l'envoyer.\n\n\u00c0 ces mots, il prit une nouvelle feuille pour recopier la lettre au propre, mais sa main tremblait tant qu'il n'arrivait pas \u00e0 \u00e9crire convenablement le moindre caract\u00e8re. La premi\u00e8re fois que quelqu'un utilise une tron\u00e7onneuse, sa main tremble tant qu'il ne parvient m\u00eame pas \u00e0 tenir son bol de riz en main, alors \u00e9crire...\n\n\u2014 Je vais l'\u00e9crire pour toi, dit Ye Wenjie, puis elle prit la feuille que lui tendait Bai Mulin et se mit \u00e0 recopier le brouillon.\n\n\u2014 Tu as une tr\u00e8s belle \u00e9criture, dit Bai Mulin en regardant la premi\u00e8re ligne de caract\u00e8res trac\u00e9e par Wenjie.\n\nIl lui donna un verre d'eau, mais sa main tremblait encore, et une bonne quantit\u00e9 d'eau tomba du verre ; Wenjie s'empressa de d\u00e9caler un peu la lettre.\n\n\u2014 Tu as \u00e9tudi\u00e9 la physique ? demanda Bai Mulin.\n\n\u2014 L'astrophysique. Mais cela ne m'est d'aucune utilit\u00e9 aujourd'hui, r\u00e9pondit Wenjie, sans lever la t\u00eate.\n\n\u2014 Tu \u00e9tudies les \u00e9toiles, comment cela pourrait n'\u00eatre d'aucune utilit\u00e9 ? Les universit\u00e9s ont rouvert, mais ils peinent \u00e0 trouver des \u00e9tudiants de troisi\u00e8me cycle. Qu'une brillante chercheuse comme toi se retrouve dans un coin perdu comme celui-ci, c'est vraiment...\n\nWenjie se tut et continua \u00e0 \u00e9crire, la t\u00eate entre les \u00e9paules. Elle n'avait pas envie de r\u00e9torquer \u00e0 Bai Mulin qu'elle s'en \u00e9tait en fait tr\u00e8s bien sortie en int\u00e9grant le Corps de production et de construction. Elle ne voulait rien dire qui concerne la r\u00e9alit\u00e9. Il n'y avait rien \u00e0 dire.\n\nLa chambre recouvra sa qui\u00e9tude, on n'entendait que la pointe du stylo plume gratter sur le papier. Wenjie pouvait sentir l'odeur du bois de pin sur le corps du journaliste. Pour la premi\u00e8re fois depuis la mort tragique de son p\u00e8re, elle \u00e9prouvait une sensation de chaleur ; pour la premi\u00e8re fois, elle se sentait un peu d\u00e9tendue et elle rel\u00e2cha provisoirement sa m\u00e9fiance \u00e0 l'\u00e9gard du monde qui l'entourait.\n\nElle finit d'\u00e9crire la lettre plus d'une heure plus tard, elle recopia ensuite l'adresse et le nom de la personne que lui avait donn\u00e9s Bai Mulin, puis elle se leva et prit cong\u00e9. Arriv\u00e9e \u00e0 hauteur de la porte, elle se retourna et lui dit :\n\n\u2014 Donne-moi donc ta veste, je te la laverai.\n\nElle fut elle-m\u00eame surprise de cette sollicitude.\n\n\u2014 Non ! Comment oserais-je ? r\u00e9pondit prestement Bai Mulin en agitant la main. Vous, les femmes du Corps de production et de construction, vous faites toute la journ\u00e9e le m\u00eame travail que les hommes ! Rentre vite te reposer, tu dois partir \u00e0 6 heures pour la montagne, demain. \u00c0 ce propos, je rentrerai apr\u00e8s-demain au si\u00e8ge central, je parlerai de ta situation \u00e0 mes sup\u00e9rieurs hi\u00e9rarchiques, peut-\u00eatre pourront-ils t'aider.\n\n\u2014 Merci, mais je me sens bien ici. Je suis au calme, lui confia Wenjie en regardant l'oc\u00e9an d'arbres brumeux baign\u00e9s par les rayons de la lune sur les montagnes du Grand Khingan.\n\n\u2014 Ne serais-tu pas en train de fuir quelque chose ?\n\n\u2014 J'y vais, fit Wenjie \u00e0 voix basse, puis elle sortit.\n\nBai Mulin observa sa silhouette gracile dispara\u00eetre sous le clair de lune, puis il leva la t\u00eate vers les arbres que venait de contempler Wenjie. Au loin, sur le pic du Radar, il vit se dresser lentement la gigantesque antenne. Elle luisait d'une lumi\u00e8re froide et m\u00e9tallique.\n\nUn midi, une semaine plus tard, alors qu'elle travaillait dans le chantier forestier, Ye Wenjie fut convoqu\u00e9e d'urgence au quartier g\u00e9n\u00e9ral du r\u00e9giment. \u00c0 peine entr\u00e9e dans le bureau, elle sentit que quelque chose ne tournait pas rond : le capitaine du r\u00e9giment et l'instructeur politique \u00e9taient tous les deux dans la pi\u00e8ce, ainsi qu'un homme au regard glacial qu'elle ne connaissait pas. Une serviette noire \u00e9tait pos\u00e9e sur le bureau. Un livre et une enveloppe qui devaient probablement se trouver \u00e0 l'int\u00e9rieur un peu plus t\u00f4t avaient \u00e9t\u00e9 sortis et pos\u00e9s \u00e0 c\u00f4t\u00e9. L'enveloppe \u00e9tait ouverte, tandis que le livre en anglais \u00e9tait celui qu'elle avait lu quelques jours plus t\u00f4t : _Printemps silencieux_.\n\nPendant la R\u00e9volution culturelle, les individus avaient tous d\u00e9velopp\u00e9 une sensibilit\u00e9 concernant leur situation politique. Celle qu'\u00e9prouvait Ye Wenjie \u00e9tait plus intense encore. Elle sentit aussit\u00f4t que le monde se refermait sur elle comme une poche.\n\n\u2014 Ye Wenjie, voici le directeur Zhang, du Bureau politique central. Il est venu enqu\u00eater dans notre r\u00e9giment, lui expliqua l'instructeur politique. Nous attendons de toi que tu collabores et nous dises toute la v\u00e9rit\u00e9.\n\n\u2014 As-tu \u00e9crit cette lettre ? demanda le directeur Zhang, en sortant la lettre de l'enveloppe.\n\nWenjie tendit la main pour s'en saisir, mais le directeur la garda entre les mains, se contentant de d\u00e9tacher chaque feuille et de les lui montrer jusqu'\u00e0 la derni\u00e8re page \u2013 celle qu'elle voulait voir. Elle ne portait aucun nom \u00e0 l'endroit de la signature, il \u00e9tait simplement \u00e9crit : \"Les masses r\u00e9volutionnaires\".\n\n\u2014 Non, ce n'est pas moi qui l'ai \u00e9crite, nia Wenjie, tremblante, en secouant la t\u00eate.\n\n\u2014 C'est pourtant ton \u00e9criture.\n\n\u2014 Oui, j'ai aid\u00e9 quelqu'un \u00e0 la recopier au propre.\n\n\u2014 Qui ?\n\nEn temps ordinaire, lorsque Ye Wenjie \u00e9tait injustement pers\u00e9cut\u00e9e au sein du r\u00e9giment, elle ne prenait que rarement la peine de se d\u00e9fendre, ravalant son honneur sans rien dire, endurant silencieusement les punitions et ne compromettant personne d'autre. Mais cette fois, la situation \u00e9tait diff\u00e9rente, elle en \u00e9tait parfaitement consciente.\n\n\u2014 C'est un journaliste travaillant pour le _Journal de la grande production_ arriv\u00e9 la semaine derni\u00e8re pour faire un reportage, son nom est...\n\n\u2014 Ye Wenjie, l'interrompit le directeur Zhang en la d\u00e9visageant avec des yeux pareils \u00e0 des canons de fusil. Laisse-moi t'avertir qu'accuser \u00e0 tort un autre camarade ne fera qu'aggraver ta situation. Nous avons fait notre enqu\u00eate au sujet de Bai Mulin, il a simplement post\u00e9 cette lettre \u00e0 Hohhot sur ta demande, mais il n'en connaissait pas le contenu.\n\n\u2014 C'est... ce qu'il a dit ?\n\nYe Wenjie eut l'impression qu'un voile noir \u00e9tait tomb\u00e9 devant ses yeux.\n\nLe directeur Zhang ne lui accorda aucune r\u00e9ponse, il brandit le livre :\n\n\u2014 Pour \u00e9crire cette lettre, tu as certainement d\u00fb \u00eatre influenc\u00e9e par ceci.\n\nIl montra un instant le livre au capitaine et \u00e0 l'instructeur politique en disant :\n\n\u2014 Ce livre a pour titre _Printemps silencieux_ , il a \u00e9t\u00e9 publi\u00e9 aux \u00c9tats-Unis en 1962, et a eu un impact consid\u00e9rable dans le monde capitaliste.\n\nIl sortit alors un autre livre de la serviette, dont la couverture \u00e9tait compos\u00e9e de caract\u00e8res noirs sur un fond blanc uni :\n\n\u2014 Voici la traduction chinoise de ce livre, publi\u00e9e en interne \u00e0 titre de r\u00e9f\u00e9rence n\u00e9gative. \u00c0 l'heure qu'il est, les autorit\u00e9s ont d\u00e9j\u00e0 rendu un verdict juste concernant ce livre : c'est un ouvrage toxique de propagande r\u00e9actionnaire. Le livre s'appuie sur la th\u00e9orie de l'id\u00e9alisme historique pour faire l'apologie des eschatologies religieuses. Sous couvert de pr\u00e9occupations environnementales, l'auteur cherche \u00e0 justifier les d\u00e9g\u00e2ts ultimes du capitalisme. Son contenu est profond\u00e9ment contre-r\u00e9volutionnaire.\n\n\u2014 Mais ce livre... ce n'est pas le mien non plus, murmura Wenjie d'une voix sans force.\n\n\u2014 Le camarade Bai Mulin a \u00e9t\u00e9 d\u00e9sign\u00e9 pour \u00eatre l'un des traducteurs de cet ouvrage par les autorit\u00e9s. Il est tout \u00e0 fait l\u00e9gitime qu'il ait ce livre en sa possession. Bien entendu, il \u00e9tait toutefois de sa responsabilit\u00e9 d'en prendre soin, il n'aurait pas d\u00fb te laisser l'opportunit\u00e9 de le voler pendant son travail, car \u00e0 pr\u00e9sent, tu as trouv\u00e9 \u00e0 travers lui une arme intellectuelle qui peut \u00eatre utilis\u00e9e contre le socialisme.\n\nYe Wenjie garda le silence. Elle savait qu'elle \u00e9tait d\u00e9j\u00e0 tomb\u00e9e au fond du pi\u00e8ge qu'on lui avait tendu et que toute lutte serait d\u00e9sormais superflue.\n\nContrairement \u00e0 ce que rapport\u00e8rent plus tard les r\u00e9cits historiques rendus publics, Bai Mulin n'avait \u00e0 l'origine pas d\u00e9lib\u00e9r\u00e9ment eu l'intention de compromettre Ye Wenjie. C'\u00e9tait en toute bonne foi qu'il avait envoy\u00e9 cette lettre aux autorit\u00e9s. \u00c0 cette \u00e9poque, beaucoup de gens avaient des raisons d'envoyer des lettres au gouvernement, mais la plupart \u00e9taient des cailloux jet\u00e9s \u00e0 la mer. D'autres, plus rares, valaient \u00e0 leurs auteurs une promotion express ou une totale destruction. En ce temps-l\u00e0, le syst\u00e8me nerveux de la politique en Chine \u00e9tait extr\u00eamement complexe. En tant que journaliste, Bai Mulin avait cru conna\u00eetre les trajectoires et les sensibilit\u00e9s de ce syst\u00e8me nerveux, mais il avait \u00e9t\u00e9 trop s\u00fbr de lui, sa lettre avait atterri dans un champ de mines dont il n'avait pas connaissance. Une fois qu'il fut conscient de cette r\u00e9alit\u00e9, la peur prit le dessus et il d\u00e9cida de sacrifier Ye Wenjie pour se prot\u00e9ger.\n\nPlus d'un demi-si\u00e8cle plus tard, les historiens seraient unanimes pour consid\u00e9rer que cet \u00e9pisode de l'ann\u00e9e 1969 fut une date clef dans l'histoire de l'humanit\u00e9.\n\nMalgr\u00e9 lui, Bai Mulin devint un personnage embl\u00e9matique de l'histoire, mais il n'eut jamais l'occasion de le savoir. Les historiens rapport\u00e8rent non sans d\u00e9ception la suite sans histoire de son existence. Bai Mulin continua \u00e0 travailler pour le _Journal de la grande production_ jusqu'en 1975, ann\u00e9e durant laquelle le Corps de construction et de production de la Mongolie-Int\u00e9rieure fut d\u00e9mantel\u00e9. Il d\u00e9m\u00e9nagea dans une ville du Nord-Est o\u00f9 il travailla pour l'Association pour la science et la technologie jusqu'au d\u00e9but des ann\u00e9es 1980. Il partit ensuite vivre \u00e0 Ottawa, o\u00f9 il enseigna la langue chinoise jusqu'en 1991 et o\u00f9 il mourut d'un cancer du poumon. Durant le restant de sa vie, il ne mentionna \u00e0 personne l'histoire de Ye Wenjie. Quant \u00e0 savoir s'il \u00e9prouva du remords ou du repentir, personne ne le saura jamais.\n\n\u2014 Wenjie, nous t'avons toujours trait\u00e9e avec tous les \u00e9gards possibles dans ce r\u00e9giment ! cracha le capitaine en m\u00eame temps qu'une grosse bouff\u00e9e de tabac Mohe.\n\nPuis il poursuivit en regardant le sol :\n\n\u2014 Souviens-toi d'o\u00f9 tu viens, des ant\u00e9c\u00e9dents dans ta famille, pourtant jamais nous ne t'avons trait\u00e9e comme une \u00e9trang\u00e8re ! Lorsque tu t'\u00e9cartais du groupe, que tu ne t'investissais pas totalement dans le progr\u00e8s, combien de fois l'instructeur Wang et moi-m\u00eame t'avons tendu la main ? Jamais nous n'aurions imagin\u00e9 que tu pourrais te rendre coupable d'un tel crime !\n\n\u2014 Je savais depuis le d\u00e9but que sa r\u00e9sistance \u00e0 la R\u00e9volution culturelle \u00e9tait profond\u00e9ment enracin\u00e9e, rench\u00e9rit l'instructeur politique.\n\n\u2014 Faites venir deux hommes et, cet apr\u00e8s-midi, escortez-la, elle et les preuves de son dossier, au Bureau politique, soupira le directeur, toujours impassible.\n\nUne fois les trois autres d\u00e9tenues emmen\u00e9es l'une apr\u00e8s l'autre, Ye Wenjie se retrouva seule dans sa cellule. La pile de charbon situ\u00e9e dans un coin de la pi\u00e8ce s'\u00e9tait \u00e9puis\u00e9e et personne ne l'avait ravitaill\u00e9e, le foyer allait bient\u00f4t s'\u00e9teindre. Il faisait d\u00e9sormais si froid dans la cellule que Wenjie dut s'envelopper dans sa couverture.\n\nLa nuit tomb\u00e9e, deux personnes vinrent la voir. L'une d'entre elles \u00e9tait une cadre politique plus \u00e2g\u00e9e qu'elle. Elle \u00e9tait accompagn\u00e9e d'un soldat qui la pr\u00e9senta comme l'\u00e9missaire militaire du Tribunal populaire interm\u00e9diaire1.\n\n\u2014 Cheng Lihua, se pr\u00e9senta la cadre.\n\nElle avait la quarantaine, \u00e9tait v\u00eatue d'une veste militaire et portait sur le nez des lunettes aux verres \u00e9pais. Les traits de son visage \u00e9taient doux et il \u00e9tait \u00e9vident qu'elle avait d\u00fb \u00eatre une belle femme dans sa jeunesse. Elle parlait sans se d\u00e9partir d'un sourire qui donnait une impression d'affabilit\u00e9. Ye Wenjie savait pertinemment qu'il \u00e9tait inhabituel qu'une cadre de ce rang vienne rendre visite \u00e0 un d\u00e9tenu attendant sa sentence. Elle fit prudemment un geste de la t\u00eate vers Cheng Lihua et se leva de son lit \u00e9troit, lui offrant une place pour s'asseoir.\n\n\u2014 Il fait si froid ici. Et le foyer ?\n\nCheng Lihua jeta un regard insatisfait vers le ge\u00f4lier-chef en faction devant la porte de la cellule, puis se tourna vers Wenjie :\n\n\u2014 Mmmh, jeune, plus jeune que ce que je pensais.\n\nCela dit, elle s'assit sur le lit et se rapprocha de Wenjie. Elle plongea la t\u00eate dans sa serviette et murmura sur un ton tr\u00e8s maternel :\n\n\u2014 Wenjie, tu es une sotte ! Vois ce que tu as fait de toi. Plus tu lis de livres, plus tu deviens sotte ! Ah, vraiment...\n\nElle trouva enfin ce qu'elle cherchait et agita devant sa poitrine une liasse de documents, puis elle leva la t\u00eate vers Ye Wenjie et lui dit avec un regard plein de sollicitude :\n\n\u2014 Ah, les jeunes ! Mais qui n'a jamais fait de sottises ? Moi aussi, j'ai \u00e9t\u00e9 jeune et sotte. \u00c0 cette \u00e9poque, j'\u00e9tais membre de la troupe d'art prol\u00e9taire de la IVe arm\u00e9e, j'\u00e9tais soliste dans la chorale de chants sovi\u00e9tiques. Une fois, lors d'une r\u00e9union d'apprentissage politique, j'ai sugg\u00e9r\u00e9 que nous devrions rejoindre l'Union sovi\u00e9tique pour devenir une nouvelle R\u00e9publique au sein d'une grande alliance socialiste, afin de renforcer le communisme \u00e0 l'\u00e9chelle internationale... J'\u00e9tais si na\u00efve mais qui n'a jamais \u00e9t\u00e9 na\u00eff ? Ce qui est fait est fait, il faut simplement reconna\u00eetre qu'on a eu tort et rectifier son point de vue, puis continuer \u00e0 servir la r\u00e9volution !\n\nLes paroles de Cheng Lihua semblaient destin\u00e9es \u00e0 lui faire gagner la confiance de Wenjie, mais cette derni\u00e8re avait dans son malheur appris \u00e0 \u00eatre prudente. Elle n'osait pas prendre cette bonne intention pour argent comptant.\n\nCheng Lihua posa la pile de documents sur le lit devant Wenjie, puis elle lui tendit un stylo :\n\n\u2014 Allons, signe, puis nous pourrons parler entre adultes et tout reprendre \u00e0 z\u00e9ro en \u00e9liminant les furoncles qui infectent ton esprit.\n\nElle avait pris le ton d'une m\u00e8re qui voulait faire avaler un biberon de lait \u00e0 son enfant.\n\nYe Wenjie observa silencieusement le rapport, mais elle resta immobile et ne saisit pas le stylo qui lui \u00e9tait tendu.\n\nCheng Lihua sourit avec indulgence :\n\n\u2014 Tu peux me faire confiance, je te garantis personnellement que le contenu de ce dossier n'a rien \u00e0 voir avec ce qui t'a amen\u00e9e ici. Signe donc !\n\nL'homme qui l'accompagnait et qui \u00e9tait rest\u00e9 debout ajouta :\n\n\u2014 Ye Wenjie, la responsable Cheng est l\u00e0 pour t'aider, elle a fait beaucoup d'efforts pour toi.\n\nCheng Lihua leva la main pour le faire taire :\n\n\u2014 Il faut la comprendre, cette gamine est effray\u00e9e. De nos jours, il y a des camarades dont la conscience politique n'est pas assez \u00e9lev\u00e9e, c'est le cas au sein du Corps de construction et de production, mais aussi au Tribunal populaire. Ils sont parfois si brutaux et emploient des m\u00e9thodes si simplistes ! Bien, Wenjie, lis donc ce document, lis-le attentivement.\n\nYe Wenjie prit le dossier et le feuilleta \u00e0 la lueur blafarde de la cellule. La responsable Cheng ne lui avait pas menti, ces documents n'avaient effectivement rien \u00e0 voir avec son affaire, mais avec son p\u00e8re. Certains \u00e9taient des rapports d'information sur les relations entre son p\u00e8re et d'autres individus et sur le contenu de leurs conversations. Les informations rassembl\u00e9es dans le document avaient visiblement \u00e9t\u00e9 fournies par Ye Wenxue, la petite s\u0153ur de Ye Wenjie. Garde rouge des plus radicales, Wenxue avait d\u00e9nonc\u00e9 son p\u00e8re avec beaucoup de z\u00e8le. C'\u00e9tait elle qui avait r\u00e9dig\u00e9 de nombreux chefs d'inculpation, dont certains avaient directement contribu\u00e9 \u00e0 la mort atroce de Ye Zhetai. Mais d\u00e8s le premier coup d'\u0153il, Wenjie sut que sa s\u0153ur n'\u00e9tait pas l'auteur de ce rapport. Le style avec lequel Wenxue avait d\u00e9nonc\u00e9 son p\u00e8re \u00e9tait violent, on avait l'impression en la lisant d'une cha\u00eene de p\u00e9tards qui explose, mais celui du dossier qu'elle tenait entre les mains \u00e9tait froid et l'\u0153uvre d'un auteur aguerri. Le contenu \u00e9tait clair et pr\u00e9cis : qui avait vu qui, en quelle ann\u00e9e, quel mois, quel jour, ce dont ils avaient parl\u00e9. En dehors de ce contexte, n'importe qui y aurait vu une chronique sans int\u00e9r\u00eat, mais le funeste dessein qui se cachait derri\u00e8re ce document \u00e9tait sans rapport avec les d\u00e9lires juv\u00e9niles de Ye Wenxue. Wenjie ne comprenait pas vraiment o\u00f9 voulait en venir le rapport, mais elle devinait vaguement que cela avait un rapport avec un grand projet national de d\u00e9fense. En tant que fille de physicien, Wenjie comprit qu'il s'agissait du \"programme des deux bombes\" (H et nucl\u00e9aire) qui faisait trembler le monde depuis 1964. Durant la p\u00e9riode de la R\u00e9volution culturelle, pour \u00e9liminer une personne haut plac\u00e9e, il \u00e9tait n\u00e9cessaire d'obtenir aupr\u00e8s de tous les organismes impliqu\u00e9s les informations secr\u00e8tes susceptibles de le faire tomber. Cependant le programme des deux bombes \u00e9tait un sujet d\u00e9licat pour les conspirateurs, car le projet \u00e9tait sous haute surveillance du gouvernement, afin d'\u00e9viter que quiconque ne profite des troubles de la R\u00e9volution culturelle pour glaner des informations sur celui-ci. Son p\u00e8re ayant \u00e9t\u00e9 condamn\u00e9 pour des raisons politiques, il n'avait pas eu l'occasion de participer directement \u00e0 la conception des deux bombes, il n'avait fait que mener un travail th\u00e9orique en parall\u00e8le. Mais il \u00e9tait plus facile de se servir de lui plut\u00f4t que de ceux qui travaillaient sur le projet depuis le d\u00e9but. Wenjie ignorait si ce qui \u00e9tait \u00e9crit \u00e9tait vrai ou faux. Ce dont elle \u00e9tait s\u00fbre, en revanche, c'\u00e9tait que le moindre signe de ponctuation \u00e9tait porteur d'une mortelle force de frappe. En dehors de la cible ultime vis\u00e9e par ce rapport, d'innombrables individus avaient peut-\u00eatre \u00e9t\u00e9 entra\u00een\u00e9s dans l'ab\u00eeme \u00e0 cause de ces rapports. En toute fin du document, elle vit la signature grossi\u00e8re de sa petite s\u0153ur. Ye Wenjie devait y ajouter la sienne. Elle remarqua que trois t\u00e9moins avaient d\u00e9j\u00e0 appos\u00e9 la leur.\n\n\u2014 J'ignore si mon p\u00e8re a eu ces conversations avec ces gens-l\u00e0, dit-elle \u00e0 voix basse en reposant les documents \u00e0 leur place d'origine.\n\n\u2014 Comment peux-tu l'ignorer ? Beaucoup ont eu lieu chez vous. Si ta s\u0153ur le sait, toi aussi.\n\n\u2014 Je l'ignore vraiment.\n\n\u2014 Mais ces conversations ont r\u00e9ellement eu lieu, tu dois nous faire confiance.\n\n\u2014 Je n'ai pas dit qu'elles n'avaient pas eu lieu, simplement que je ne les ai jamais entendues. Donc, je ne peux pas signer.\n\n\u2014 Ye Wenjie..., dit l'homme en s'avan\u00e7ant, mais la responsable Cheng l'arr\u00eata aussit\u00f4t une nouvelle fois.\n\nElle s'assit encore plus pr\u00e8s de Wenjie et, lui prenant la main, elle lui dit :\n\n\u2014 Wenjie, laisse-moi t'expliquer ce qui se passe. Le sort d'une personne dans ton cas peut changer du tout au tout. Au mieux, tu seras jug\u00e9e comme une jeune instruite aveugl\u00e9e par un livre r\u00e9actionnaire, ce qui n'est pas si grave. Il n'y aura m\u00eame pas besoin d'avoir recours \u00e0 une proc\u00e9dure judiciaire. Tu devras simplement participer \u00e0 une r\u00e9union d'apprentissage et \u00e9crire quelques rapports autocritiques, et tu pourras retourner au sein du Corps de construction et de production. Mais dans le pire des cas, Wenjie, tu le sais au fond de toi, tu seras inculp\u00e9e pour trahison contre-r\u00e9volutionnaire. Et pour ce qui est de ce genre de condamnation politique, le tribunal pr\u00e9f\u00e9rera pencher vers la gauche, quitte \u00e0 para\u00eetre trop radical, que d'\u00eatre accus\u00e9 de droitisme. Toutefois, il appartiendra au responsable militaire de trancher. Bien s\u00fbr, cela reste entre nous.\n\nL'homme qui l'accompagnait ajouta :\n\n\u2014 La responsable Cheng essaie de t'aider. Tu le vois bien, il y a d\u00e9j\u00e0 trois signatures, que tu signes ou non ne changera rien. Ye Wenjie, ne sois pas stupide cette fois.\n\n\u2014 Il a raison, mon enfant ! Quand je vois une fille aussi brillante que toi dans un tel p\u00e9trin, \u00e7a me fait de la peine ! Je veux vraiment te sauver, mais tu dois coop\u00e9rer. Regarde-moi. Me crois-tu capable de te faire du mal ?\n\nYe Wenjie ne voyait plus la responsable militaire, elle ne voyait plus que le sang de son p\u00e8re.\n\n\u2014 Responsable Cheng, j'ignore si les faits rapport\u00e9s dans ces documents sont vrais, je ne signerai pas.\n\nCheng Lihua se tut, elle fixa Wenjie pendant un long moment, comme frigorifi\u00e9e par l'air glac\u00e9 de la cellule. Puis lentement, elle repla\u00e7a les documents dans sa serviette et se leva. L'expression de bienveillance n'avait pas disparu de son visage, mais celui-ci semblait fig\u00e9, comme si elle portait un masque de pl\u00e2tre. Et c'est sans se d\u00e9partir de ce rictus bienveillant qu'elle se dirigea vers le coin de la pi\u00e8ce o\u00f9 \u00e9tait pos\u00e9 le seau d'eau destin\u00e9 \u00e0 la toilette des d\u00e9tenus. Elle s'en saisit et en reversa la moiti\u00e9 sur Wenjie et l'autre sur son lit, placide et m\u00e9thodique, puis elle envoya valser le seau et sortit de la cellule en lan\u00e7ant un juron : \"B\u00e2tarde but\u00e9e !\"\n\nLe ge\u00f4lier-chef fut le dernier \u00e0 sortir, il jeta un dernier regard glacial sur une Wenjie enti\u00e8rement tremp\u00e9e, puis claqua la porte et tourna le verrou.\n\nLe froid hivernal de la Mongolie-Int\u00e9rieure traversa les habits humides de Wenjie comme une gigantesque main qui se cramponnait \u00e0 elle. Elle entendit le claquement martial de ses propres dents, puis le son disparut. En p\u00e9n\u00e9trant dans sa moelle, le froid changea le monde r\u00e9el devant ses yeux en un enfer blanc comme le lait. Elle sentit que l'univers tout entier devenait un immense bloc de glace \u00e0 l'int\u00e9rieur duquel elle \u00e9tait la seule cr\u00e9ature vivante. La petite fille qu'elle \u00e9tait et qui allait bient\u00f4t mourir de froid n'avait aucune allumette, seulement des illusions. Le bloc de glace devint peu \u00e0 peu diaphane, et une grande tour \u00e9mergea devant elle. \u00c0 son sommet, une jeune fille agitait un drapeau, la maigreur de son visage contrastait furieusement avec la majest\u00e9 du drapeau. C'\u00e9tait sa petite s\u0153ur, Wenxue. Depuis que cette derni\u00e8re avait coup\u00e9 les ponts avec sa famille de scientifiques r\u00e9actionnaires, Wenjie n'avait plus eu aucune nouvelle, jusqu'\u00e0 il y a peu, lorsqu'elle apprit que sa s\u0153ur \u00e9tait morte brutalement deux ans plus t\u00f4t lors d'une lutte entre gardes rouges. Dans le tumulte de cette sc\u00e8ne, la personne qui brandissait le drapeau se mua en Bai Mulin, dont les lunettes refl\u00e9taient les flammes qui br\u00fblaient au pied de la tour. Puis il devint \u00e0 son tour la responsable Cheng, puis sa m\u00e8re Shao Lin, puis Ruan Wen, et enfin son p\u00e8re. Alors que le porteur de l'\u00e9tendard ne cessait de se m\u00e9tamorphoser, le drapeau, lui, flottait en continu comme le pendule d'une horloge \u00e9ternelle comptant le peu d'ann\u00e9es qu'il lui restait \u00e0 vivre. Bient\u00f4t, le drapeau se brouilla, comme tout autour d'elle, et le bloc de glace qui renfermait l'univers l'emprisonna une nouvelle fois. Mais cette fois-ci, tout \u00e9tait devenu noir.\n\n* * *\n\n1 \u00c0 cette \u00e9poque de la R\u00e9volution culturelle, la plupart des organes et des institutions de s\u00e9curit\u00e9 publique \u00e9taient sous la responsabilit\u00e9 de l'arm\u00e9e. Les repr\u00e9sentants militaires avaient le dernier mot concernant toutes les d\u00e9cisions judiciaires. _(N.d.A.)_\n\n## 3\n\n## C\u00d4TE ROUGE I\n\nYe Wenjie ne savait combien de temps avait pass\u00e9 lorsqu'elle entendit un bourdonnement sourd qui semblait provenir de toutes les directions. Dans son esprit encore embu\u00e9, il lui apparaissait comme le vrombissement d'une \u00e9norme machine en train de forer ou de d\u00e9couper le bloc de glace dont elle \u00e9tait prisonni\u00e8re. Le monde \u00e9tait encore assombri par une chape obscure, mais le son lui paraissait de plus en plus r\u00e9el et, bient\u00f4t, elle put affirmer que la source du bourdonnement ne venait ni de l'enfer ni du paradis. Elle r\u00e9alisa qu'elle avait encore les yeux clos. Elle s'effor\u00e7a d'ouvrir ses paupi\u00e8res lourdes. Elle vit d'abord une lampe incrust\u00e9e profond\u00e9ment dans le plafond qui semblait prot\u00e9g\u00e9e par un treillis m\u00e9tallique et \u00e9mettait une lueur tamis\u00e9e. Le plafond lui-m\u00eame semblait \u00eatre en m\u00e9tal. Elle entendit une voix d'homme l'appeler \u00e0 voix basse.\n\n\u2014 Vous avez beaucoup de fi\u00e8vre.\n\n\u2014 O\u00f9 suis-je ? demanda Wenjie d'une voix si faible qu'elle eut l'impression que ce n'\u00e9tait pas elle qui parlait.\n\n\u2014 Dans un h\u00e9licopt\u00e8re.\n\nWenjie se sentit \u00e0 nouveau f\u00e9brile, et elle se rendormit, berc\u00e9e par le m\u00eame \u00e9trange bourdonnement. Peu de temps apr\u00e8s, elle s'\u00e9veilla une nouvelle fois. Cette fois-ci, son engourdissement avait disparu, mais la douleur \u00e9tait revenue : sa t\u00eate et ses articulations la faisaient souffrir et son haleine \u00e9tait bouillante. Elle avait mal \u00e0 la gorge. C'\u00e9tait comme si elle avalait de la braise \u00e0 chaque d\u00e9glutition. Mais le fait qu'elle puisse \u00e9prouver ces sensations indiquait en soi que son corps fonctionnait de nouveau.\n\nWenjie tourna la t\u00eate et vit \u00e0 ses c\u00f4t\u00e9s deux hommes qui portaient la m\u00eame veste d'uniforme que la responsable Cheng, \u00e0 la diff\u00e9rence pr\u00e8s qu'eux \u00e9taient coiff\u00e9s d'un bonnet de fourrure brod\u00e9 de cinq \u00e9toiles rouges. Leurs cols l\u00e9g\u00e8rement ouverts d\u00e9voilaient un insigne rouge accroch\u00e9 \u00e0 leur poitrine, et l'un d'eux portait des lunettes. Wenjie remarqua qu'on l'avait bord\u00e9e avec un grand pardessus militaire rouge qui lui tenait chaud. Elle voulut se lever et se surprit de l'aisance avec laquelle elle r\u00e9ussit. Elle regarda \u00e0 travers le hublot de l'autre c\u00f4t\u00e9 de l'appareil. Des cylindres de nuages pivotaient lentement. L'\u00e9clat du soleil l'aveugla, elle s'empressa de diriger son regard ailleurs et vit que l'int\u00e9rieur \u00e9troit de la cabine \u00e9tait jonch\u00e9 de malles militaires vertes en m\u00e9tal. Par-del\u00e0 les hublots de l'autre c\u00f4t\u00e9, elle put remarquer les ombres des rotors, et elle eut la confirmation qu'elle se trouvait bien \u00e0 bord d'un h\u00e9licopt\u00e8re.\n\n\u2014 Vous devriez vous recoucher, dit le militaire \u00e0 lunettes, puis il l'aida \u00e0 s'allonger et la recouvrit du pardessus.\n\n\u2014 Ye Wenjie, est-ce vous qui avez \u00e9crit cet article ?\n\nL'autre militaire brandissait une revue en anglais devant ses yeux. Elle lut le titre de l'article : \"De l'existence d'une tension superficielle dans la zone de radiation du soleil et ses particularit\u00e9s en termes de r\u00e9flexivit\u00e9\". La couverture de la revue indiquait qu'il s'agissait du volume 66 de _The Journal of Astrophysics_.\n\n\u2014 \u00c9videmment que c'est elle, est-ce qu'il y a encore besoin de le prouver ?\n\nLe militaire \u00e0 lunettes reprit la revue et fit les pr\u00e9sentations :\n\n\u2014 Voici Lei Zhicheng, commissaire politique \u00e0 la base de C\u00f4te Rouge, je suis Yang Weining, ing\u00e9nieur en chef de la base. Nous sommes encore \u00e0 une heure de notre lieu d'atterrissage, reposez-vous.\n\nYang Weining ? Ye Wenjie ne r\u00e9pondit rien, elle se contenta de le regarder avec surprise. Elle remarqua qu'il n'avait pas l'air serein ; manifestement, il n'avait pas l'intention de r\u00e9v\u00e9ler \u00e0 son compagnon qu'ils se connaissaient. Yang Weining avait autrefois \u00e9t\u00e9 l'\u00e9tudiant de Ye Zhetai. L'ann\u00e9e o\u00f9 il avait obtenu son dipl\u00f4me, Wenjie venait tout juste d'entrer \u00e0 l'universit\u00e9. Elle se rappelait encore aujourd'hui tr\u00e8s bien de la premi\u00e8re fois o\u00f9 Yang Weining \u00e9tait venu chez eux. Il avait tout juste r\u00e9ussi son entr\u00e9e en troisi\u00e8me cycle et \u00e9tait venu discuter avec son p\u00e8re des orientations de son sujet de recherche. Yang Weining avait exprim\u00e9 son d\u00e9sir de travailler dans le domaine des sciences exp\u00e9rimentales et appliqu\u00e9es, et voulait autant que possible se tenir \u00e0 l'\u00e9cart des th\u00e9ories fondamentales. Wenjie se souvenait de ce que son p\u00e8re avait r\u00e9pondu : \"Je ne m'y oppose pas, mais n'oublie pas que nous sommes sp\u00e9cialistes de physique th\u00e9orique. Pourquoi une telle r\u00e9solution ?\" Yang Weining avait r\u00e9pondu qu'il souhaitait apporter une contribution r\u00e9elle \u00e0 son \u00e9poque. Ye Zhetai avait r\u00e9torqu\u00e9 que la th\u00e9orie \u00e9tait le fondement de l'exp\u00e9rience : d\u00e9couvrir les lois de la nature n'\u00e9tait-il pas une contribution tout aussi importante ? Yang Weining avait h\u00e9sit\u00e9 un moment avant de l\u00e2cher que les \u00e9tudes th\u00e9oriques menaient plus facilement \u00e0 des fautes id\u00e9ologiques. Cet argument avait plong\u00e9 le p\u00e8re de Ye Wenjie dans le silence. Yang Weining \u00e9tait un gar\u00e7on talentueux, il avait un esprit vif et de solides connaissances math\u00e9matiques, mais pendant la br\u00e8ve p\u00e9riode durant laquelle il effectua ses \u00e9tudes de troisi\u00e8me cycle, il maintint avec son professeur une relation aussi distante que respectueuse. \u00c0 cette \u00e9poque, Wenjie et Yang Weining se croisaient souvent mais, peut-\u00eatre en raison de l'influence de son p\u00e8re, elle ne faisait gu\u00e8re plus attention \u00e0 lui. Quant \u00e0 savoir s'il s'int\u00e9ressait \u00e0 elle, Wenjie l'ignorait. Peu de temps apr\u00e8s l'obtention de son dipl\u00f4me, Yang Weining coupa les ponts avec son professeur.\n\nApr\u00e8s s'\u00eatre sentie une nouvelle fois faible, Wenjie ferma les yeux. Les deux militaires la quitt\u00e8rent pour aller s'accroupir derri\u00e8re une malle et chuchoter \u00e0 voix basse. Mais dans cette cabine \u00e9triqu\u00e9e, le vrombissement du moteur ne parvenait pas \u00e0 couvrir enti\u00e8rement leur conversation et elle saisit quelques bribes :\n\n\u2014 J'ai quand m\u00eame l'intuition que c'est une mauvaise id\u00e9e.\n\nC'\u00e9tait la voix de Lei Zhicheng. Yang Weining r\u00e9torqua :\n\n\u2014 Aurait-on trouv\u00e9 une personne avec ses comp\u00e9tences par la voie l\u00e9gale habituelle ?\n\n\u2014 Eh, ce n'est pas faute de m'\u00eatre d\u00e9men\u00e9 ! Je n'y suis pour rien si aucun militaire n'est sp\u00e9cialiste de ce domaine. Mais avec les civils, on n'est jamais au bout de nos peines. Tu connais le degr\u00e9 de confidentialit\u00e9 de ce projet. Pour y prendre part, il faut d'abord avoir servi dans l'arm\u00e9e. Le r\u00e8glement exige de rester isol\u00e9 pendant des ann\u00e9es. Comment font les militaires qui ont des familles ? Elles aussi, on doit les retenir \u00e0 la base, mais personne n'accepte une telle contrainte. J'ai trouv\u00e9 deux candidats qui auraient fait l'affaire, mais ils pr\u00e9f\u00e9raient encore \u00eatre envoy\u00e9s dans les \u00e9coles du 7 Mai plut\u00f4t que de venir s'enterrer ici1. On peut bien s\u00fbr leur forcer la main mais, pour ce genre de travail, il vaut mieux des personnes venues de leur plein gr\u00e9.\n\n\u2014 Nous n'avons donc pas le choix.\n\n\u2014 Mais c'est contraire au r\u00e8glement.\n\n\u2014 C'est de la nature du projet de contourner le r\u00e8glement. Si quelque chose arrive, j'en assumerai la responsabilit\u00e9\n\n\u2014 Ah, Yang, de quelle responsabilit\u00e9 parles-tu ? Tu as la t\u00eate plong\u00e9e dans les probl\u00e8mes techniques, mais C\u00f4te Rouge n'est pas un programme national de d\u00e9fense ordinaire. Sa complexit\u00e9 va bien au-del\u00e0 des questions techniques.\n\n\u2014 Sur ce point, je peux difficilement te contredire.\n\nQuand ils atterrirent, le soleil \u00e9tait d\u00e9j\u00e0 en train de se coucher. Ye Wenjie d\u00e9clina l'aide de Yang Weining et de Lei Zhicheng et descendit toute seule tant bien que mal de l'appareil. Un vent violent manqua de la faire s'envoler. En soufflant sur les h\u00e9lices encore en mouvement de l'h\u00e9licopt\u00e8re, le vent fit entendre un son strident. Ye Wenjie \u00e9tait famili\u00e8re de ce vent qui soufflait dans la for\u00eat, elle connaissait ce vent et ce vent la connaissait, c'\u00e9tait celui des montagnes du Grand Khingan. Bien vite, elle entendit un autre bruit, un bourdonnement grave, \u00e9nergique, vaste et puissant, qui semblait servir de d\u00e9cor \u00e0 ce monde, celui de l'antenne parabolique battue par le vent. C'\u00e9tait en l'approchant qu'on pouvait v\u00e9ritablement mesurer l'immensit\u00e9 de ce voilage c\u00e9leste. En un mois, la vie de Wenjie avait effectu\u00e9 une grande boucle, et elle se trouvait maintenant au pic du Radar.\n\nWenjie ne put s'emp\u00eacher de tourner la t\u00eate pour regarder dans la direction du campement du Corps de construction et de production, mais elle ne vit qu'un oc\u00e9an d'arbres brumeux berc\u00e9s par les couleurs du cr\u00e9puscule.\n\nL'h\u00e9licopt\u00e8re ne paraissait pas avoir \u00e9t\u00e9 affr\u00e9t\u00e9 sp\u00e9cialement pour elle : des soldats approch\u00e8rent pour embarquer les malles qui jonchaient le sol de la cabine. Ils pass\u00e8rent devant elle, sans m\u00eame lui jeter un regard. Lei Zhicheng, Yang Weining et elle continu\u00e8rent leur chemin tout droit. Wenjie remarqua \u00e0 quel point la superficie du pic du Radar \u00e9tait \u00e9tendue. Du pied de l'antenne, on distinguait un complexe de b\u00e2timents blancs qui, par contraste avec l'antenne, ressemblaient \u00e0 de d\u00e9licats cubes de construction. Ils se dirig\u00e8rent vers l'entr\u00e9e de la base, gard\u00e9e par deux soldats arm\u00e9s. Ils s'arr\u00eat\u00e8rent devant la porte.\n\nLei Zhicheng se tourna vers Ye Wenjie et lui dit avec gravit\u00e9 :\n\n\u2014 Ye Wenjie, vous avez \u00e9t\u00e9 convaincue d'un acte de trahison contre-r\u00e9volutionnaire. Le tribunal vous a condamn\u00e9e \u00e0 une peine \u00e0 la hauteur de votre faute, mais vous avez aujourd'hui l'opportunit\u00e9 d'expier vos crimes en travaillant ici. C'est \u00e0 prendre ou \u00e0 laisser.\n\nIl indiqua la direction de l'antenne :\n\n\u2014 Vous vous trouvez dans un complexe militaro-industriel. L'un des projets de recherche men\u00e9s ici requiert vos comp\u00e9tences. Pour des d\u00e9tails plus concrets, je laisserai l'ing\u00e9nieur en chef Yang vous expliquer. R\u00e9fl\u00e9chissez bien.\n\nCela dit, il fit un signe de la t\u00eate \u00e0 Yang Weining et suivit la file de soldats qui transportaient les caisses de mat\u00e9riel \u00e0 l'int\u00e9rieur de la base.\n\nYang Weining attendit que les hommes soient partis avant de faire signe \u00e0 Ye Wenjie de s'\u00e9loigner un peu. Il \u00e9tait \u00e9vident qu'il craignait que les soldats plac\u00e9s en sentinelle devant la base n'entendent leur conversation. Cette fois, il ne pr\u00e9tendit plus ne pas la conna\u00eetre :\n\n\u2014 Wenjie, je vais te parler franchement, ce n'est pas une opportunit\u00e9. J'ai d\u00e9fendu ta cause devant la Commission militaire et, malgr\u00e9 la lourde sentence pr\u00e9conis\u00e9e par la responsable Cheng Lihua, ta situation te vaudrait au maximum dix ans de prison qui pourraient \u00eatre ramen\u00e9s \u00e0 six ou sept ans en fonction des circonstances. Mais ce lieu \u2013 il balaya la base du regard \u2013 c'est un complexe de recherches ultra-confidentiel et, dans ton cas, cela signifie peut-\u00eatre...\n\nIl s'interrompit, comme s'il souhaitait que le bourdonnement de l'antenne battue par le vent aggrave le ton de sa voix :\n\n\u2014... que tu ne sortiras jamais d'ici.\n\n\u2014 J'accepte, murmura Ye Wenjie.\n\nYang Weining fut surpris par la rapidit\u00e9 de sa d\u00e9cision.\n\n\u2014 Rien ne presse, tu peux r\u00e9fl\u00e9chir encore un peu. L'h\u00e9licopt\u00e8re ne repart que dans trois heures. Si tu as l'intention de refuser, je peux te raccompagner l\u00e0-bas.\n\n\u2014 Je ne veux pas y retourner, entrons.\n\nLa voix de Wenjie \u00e9tait encore tr\u00e8s faible, mais son timbre \u00e9tait cette fois dur comme du m\u00e9tal. Dor\u00e9navant, \u00e0 part peut-\u00eatre le monde des morts dont elle ignorait toutefois l'existence, le seul endroit o\u00f9 elle accepterait de se rendre \u00e9tait ce pic coup\u00e9 du monde. Ici, elle \u00e9prouverait un sentiment de s\u00e9curit\u00e9 qu'elle n'avait pas ressenti depuis longtemps.\n\n\u2014 Sois tout de m\u00eame plus prudente, tu dois comprendre ce que cela signifie.\n\n\u2014 Je suis pr\u00eate \u00e0 rester ici toute ma vie.\n\nYang Weining se tut, il regarda l'horizon, comme s'il voulait forcer Ye Wenjie \u00e0 prendre le temps de la r\u00e9flexion. Wenjie aussi \u00e9tait silencieuse, emmitoufl\u00e9e dans son pardessus battu par le vent, les yeux \u00e9galement plong\u00e9s dans le lointain. Les montagnes du Grand Khingan avaient disparu dans l'\u00e9paisse obscurit\u00e9 de la nuit. On ne peut rester longtemps dehors \u00e0 r\u00e9fl\u00e9chir en plein hiver. Yang Weining d\u00e9cida de se diriger vers l'entr\u00e9e, il marchait vite, comme s'il voulait laisser Wenjie derri\u00e8re lui, mais elle s'accrochait pour suivre son allure. Ils franchirent ensemble la grande porte de la base de C\u00f4te Rouge. Les deux gardes referm\u00e8rent les deux lourds battants de m\u00e9tal derri\u00e8re eux.\n\nApr\u00e8s avoir march\u00e9 un moment, Yang Weining s'arr\u00eata. Il d\u00e9signa l'antenne \u00e0 Wenjie :\n\n\u2014 Il s'agit d'un projet de d\u00e9veloppement d'une arme de grand calibre. Si l'op\u00e9ration est un succ\u00e8s, le r\u00e9sultat sera encore plus r\u00e9volutionnaire que l'invention de la bombe nucl\u00e9aire ou de la bombe H.\n\nArriv\u00e9 devant le plus gros b\u00e2timent du complexe de la base, Yang Weining poussa directement la porte sur laquelle Ye Wenjie vit qu'il \u00e9tait inscrit : \"Console de contr\u00f4le des transmissions\". \u00c0 peine \u00e9taient-ils entr\u00e9s qu'elle re\u00e7ut au visage une bouff\u00e9e de vapeur chaude qui empestait l'huile de moteur. Elle d\u00e9couvrit que le vaste hall \u00e9tait envahi de tous types de machines et d'instruments. La pi\u00e8ce \u00e9tait \u00e9clair\u00e9e par des signaux lumineux et les vacillements des oscilloscopes. Plus d'une dizaine de techniciens y \u00e9taient assis, mais on les aurait plut\u00f4t crus accroupis au fond de profondes tranch\u00e9es face \u00e0 des rang\u00e9es d'appareils pr\u00eats \u00e0 les submerger. Les mots de code des diff\u00e9rentes op\u00e9rations en cours se succ\u00e9daient par vagues, et l'ambiance paraissait \u00e0 la fois nerveuse et d\u00e9sordonn\u00e9e.\n\n\u2014 Il fait meilleur ici. Attends-moi un moment. Je vais veiller \u00e0 ce qu'on te pr\u00e9pare une chambre, dit Yang Weining \u00e0 Wenjie, avant de lui montrer du doigt une chaise \u00e0 c\u00f4t\u00e9 d'une table.\n\nWenjie vit que quelqu'un d'autre \u00e9tait assis \u00e0 la table, un militaire arm\u00e9 d'un fusil.\n\n\u2014 Je pr\u00e9f\u00e8re attendre dehors, l\u00e2cha Wenjie en faisant demi-tour.\n\nYang Weining lui sourit aimablement :\n\n\u2014 D\u00e9sormais, tu es une employ\u00e9e de la base. Hormis quelques endroits que je t'indiquerai, tu peux te d\u00e9placer o\u00f9 tu le souhaites.\n\nD\u00e8s qu'il eut fini de parler, une expression de d\u00e9sarroi se dessina sur son visage. Il avait compris que l'autre sens de cette phrase \u00e9tait : \"Tu ne pourras plus jamais partir d'ici.\"\n\n\u2014 Je pr\u00e9f\u00e8re attendre dehors, insista Wenjie.\n\n\u2014 Comme tu voudras...\n\nYang Weining observa le militaire, qui ne les avait pas remarqu\u00e9s. Il comprit presque Wenjie, et l'accompagna hors de la console de contr\u00f4le.\n\n\u2014 Reste ici \u00e0 l'abri du vent, je reviens dans quelques minutes. Je vais simplement demander \u00e0 quelqu'un d'allumer un feu dans ta chambre. Les conditions de vie \u00e0 la base sont rudimentaires, il n'y a pas de chauffage.\n\nPuis il s'\u00e9loigna.\n\nWenjie resta debout \u00e0 c\u00f4t\u00e9 de la porte de la console de contr\u00f4le. L'immense antenne se dressait derri\u00e8re son dos, elle occupait tout juste la moiti\u00e9 du ciel. De l\u00e0 o\u00f9 elle \u00e9tait, Wenjie pouvait entendre distinctement les bruits \u00e0 l'int\u00e9rieur de la console de contr\u00f4le. Brusquement, la cacophonie des mots de code s'\u00e9teignit et un silence s'abattit sur la salle. Tout juste pouvait-on entendre les ronronnements irr\u00e9guliers des appareils. Puis la voix d'un homme rompit le silence :\n\n\u2014 2e corps d'artillerie de l'Arm\u00e9e populaire de lib\u00e9ration. Projet C\u00f4te Rouge. Transmission 147 autoris\u00e9e, 30 secondes avant lancement.\n\nCat\u00e9gorie de la cible : 3A ; Code de position : BN20197F ; Fin de la v\u00e9rification de la transmission, 25 secondes !\n\nNum\u00e9ro de fichier de la transmission : 22 ; Annexe : nulle ; Extension : nulle ; Fin de la v\u00e9rification du fichier, 20 secondes !\n\nRapport du p\u00f4le d'\u00e9nergie : OK !\n\nRapport du p\u00f4le de codage : OK !\n\nRapport du p\u00f4le d'amplificateur : OK !\n\nRapport du p\u00f4le de d\u00e9tection des perturbations : dans les limites autoris\u00e9es !\n\nEntr\u00e9e dans la phase irr\u00e9vocable du programme : 15 secondes !\n\nTout redevint silencieux, puis dix secondes plus tard, une alarme retentit et un signal rouge se mit \u00e0 clignoter \u00e0 toute vitesse sur l'antenne.\n\n\u2014 Transmission ! \u00c0 chaque p\u00f4le, \u00e0 vos postes de surveillance !\n\nWenjie sentit comme un l\u00e9ger picotement sur son visage. Elle savait qu'un puissant champ \u00e9lectrique s'\u00e9tait cr\u00e9\u00e9. Elle leva la t\u00eate pour voir la direction dans laquelle s'orientait l'antenne, elle aper\u00e7ut dans le ciel nocturne un nuage \u00e9mettant un halo bleu fonc\u00e9, si faible qu'elle crut tout d'abord \u00e0 une illusion n\u00e9e de son esprit. Mais lorsque le nuage fut souffl\u00e9 par le vent et que la lueur s'\u00e9teignit, un autre nuage vint prendre sa place dans le ciel. Dans la console de contr\u00f4le, le concert de mots de code avait \u00e0 nouveau repris, mais elle pouvait encore saisir quelques phrases :\n\n\u2014 P\u00f4le d'amplificateur endommag\u00e9, magn\u00e9tron 3 grill\u00e9.\n\nP\u00f4le d'assistance : retour \u00e0 la normale !\n\nSeuil no 1 atteint ! Continuez la transmission.\n\nDans l'obscurit\u00e9 qui r\u00e9gnait, Wenjie entendit une autre vibration. Elle vit une ombre noire de la forme d'une colonne \u00e9merger de la for\u00eat sur le flanc du pic et monter en spirale jusqu'au ciel. Elle n'avait pas imagin\u00e9 que, m\u00eame au plus rude de l'hiver, il y ait autant d'oiseaux dans la for\u00eat. Elle fut alors le t\u00e9moin d'un spectacle atroce : une nu\u00e9e d'oiseaux s'envola dans le p\u00e9rim\u00e8tre o\u00f9 pointait l'antenne et, derri\u00e8re le nuage diaphane, elle les vit distinctement retomber sur le sol, comme foudroy\u00e9s en plein vol.\n\nL'op\u00e9ration se prolongea environ quinze minutes, puis le voyant rouge de l'antenne s'\u00e9teignit et la sensation de picotement sur la peau de Wenjie disparut. Dans la console de contr\u00f4le, les murmures des techniciens reprirent tandis que la voix sonore de l'homme continuait \u00e0 communiquer des informations.\n\n\u2014 Op\u00e9ration de transmission no 147 du projet C\u00f4te Rouge termin\u00e9e. Fermeture du syst\u00e8me de transmission. C\u00f4te Rouge entre en mode veille. Veuillez laisser l'\u00e9quipe de veille prendre le contr\u00f4le du syst\u00e8me et transf\u00e9rez les donn\u00e9es sur les seuils de contr\u00f4le.\n\nMerci de remplir scrupuleusement le journal de transmission. Les responsables des diff\u00e9rents p\u00f4les sont attendus en salle de r\u00e9union pour un compte rendu g\u00e9n\u00e9ral. Termin\u00e9.\n\nTout redevint alors silencieux. Seule l'antenne continuait \u00e0 bourdonner dans le vent. Ye Wenjie observa dans le ciel nocturne les oiseaux qui regagnaient peu \u00e0 peu la for\u00eat. Elle leva une nouvelle fois les yeux vers l'antenne ; elle eut l'impression que c'\u00e9tait la gigantesque paume d'une main qui s'ouvrait vers la vo\u00fbte c\u00e9leste, une main dot\u00e9e d'un myst\u00e9rieux pouvoir surhumain. Mais alors qu'elle regardait le ciel, elle ne parvint pas \u00e0 voir la cible no BN20197F atteinte par cette main. Derri\u00e8re la volute de nuages, elle ne voyait que le ciel \u00e9toil\u00e9 d'un hiver glacial de l'an 1969.\n\n* * *\n\n1 Durant la R\u00e9volution culturelle, les \"\u00e9coles du 7 Mai\" ou \"\u00e9coles des cadres du 7 Mai\" \u00e9taient des camps de \"r\u00e9\u00e9ducation\" pour les cadres administratifs, techniques et politiques, o\u00f9 ceux-ci prenaient notamment part \u00e0 des travaux agricoles. _(N.d.T)_\n\n## 4\n\n## LES FRONTI\u00c8RES DE LA SCIENCE\n\n_Trente-huit ans plus tard._\n\nLes quatre individus venus interroger Wang Miao formaient une bien curieuse \u00e9quipe. Deux flics et deux militaires. Wang Miao aurait \u00e9t\u00e9 moins surpris si les deux militaires avaient \u00e9t\u00e9 de simples soldats, mais il se trouvait nez \u00e0 nez avec deux officiers.\n\nLes deux flics lui avaient fait mauvaise impression d\u00e8s le premier regard. Si le jeune en uniforme avait fait montre d'une certaine politesse \u00e0 son \u00e9gard, celui en civil avait l'air d\u00e9sagr\u00e9able. Le gaillard, grand et robuste, avait une mine patibulaire. Il \u00e9tait v\u00eatu d'une veste en cuir sale, empestait le tabac et, pour couronner le tout, grognait plus qu'il ne parlait. Il \u00e9tait l'incarnation m\u00eame du type imbuvable.\n\n\u2014 Wang Miao ?\n\nCette fa\u00e7on de l'interpeller directement par son nom l'irrita au plus haut point, d'autant que l'homme ne daigna pas lever la t\u00eate, trop occup\u00e9 \u00e0 allumer sa cigarette. Sans m\u00eame attendre une r\u00e9ponse, il fit un signe au plus jeune qui pr\u00e9senta \u00e0 Wang Miao sa carte de police. Sa cigarette allum\u00e9e, le policier s'engouffra aussit\u00f4t dans l'appartement.\n\n\u2014 Merci de ne pas fumer \u00e0 l'int\u00e9rieur de mon appartement, l'arr\u00eata Wang Miao.\n\n\u2014 Je vous demande pardon, monsieur le professeur. Permettez que je vous pr\u00e9sente notre commissaire de police, M. Shi Qiang, expliqua le jeune policier dans un sourire, avant de lancer un clin d'\u0153il au commissaire.\n\n\u2014 Bon. Eh bien, bavardons sur le palier, l\u00e2cha Shi Qiang.\n\nIl inspira une profonde bouff\u00e9e qui consuma la moiti\u00e9 de sa cigarette, mais \u00e0 la surprise de Wang Miao, il n'expira pas la moindre fum\u00e9e.\n\nIl fit un nouveau geste de la t\u00eate \u00e0 son jeune coll\u00e8gue :\n\n\u2014 Demande-lui.\n\n\u2014 Professeur Wang, nous souhaiterions vous interroger au sujet de la Soci\u00e9t\u00e9 scientifique des fronti\u00e8res de la science. Certains de ses membres ont r\u00e9cemment pris contact avec vous, n'est-ce pas ?\n\n\u2014 La Soci\u00e9t\u00e9 des fronti\u00e8res de la science est une association de recherche tout \u00e0 fait l\u00e9gale, tr\u00e8s influente au niveau international. Elle est constitu\u00e9e d'\u00e9minents scientifiques. En quoi serait-il ill\u00e9gal d'\u00eatre en contact avec une telle association ?\n\n\u2014 \u00c9coute-le ! beugla Shi Qiang. Vous nous avez entendus dire que c'\u00e9tait ill\u00e9gal ? Est-ce qu'il vous a \u00e9t\u00e9 signifi\u00e9 \u00e0 un seul instant de ne pas entrer en contact avec eux ? dit-il en recrachant enfin au visage de Wang Miao la fum\u00e9e qu'il avait aspir\u00e9e quelques instants plus t\u00f4t.\n\n\u2014 C'est une question de vie priv\u00e9e et je n'ai aucune obligation de r\u00e9pondre \u00e0 vos interrogations.\n\n\u2014 De vie priv\u00e9e ? Un scientifique aussi r\u00e9put\u00e9 que vous devrait savoir qu'il a des responsabilit\u00e9s en termes de s\u00fbret\u00e9 publique, vous ne croyez pas ? r\u00e9torqua Shi Qiang en jetant son m\u00e9got avant de tirer une nouvelle cigarette de son paquet \u00e9cras\u00e9.\n\n\u2014 J'ai le droit de ne pas vous r\u00e9pondre. Au revoir, messieurs, se contenta de dire Wang Miao en tournant le dos pour rentrer dans son appartement.\n\n\u2014 Minute ! mugit Shi Qiang tout en faisant un signe de la main au plus jeune des officiers. Donne-lui l'adresse et le num\u00e9ro de t\u00e9l\u00e9phone. Cet apr\u00e8s-midi, on va faire un tour.\n\n\u2014 Mais que voulez-vous \u00e0 la fin ? riposta Wang Miao, en col\u00e8re.\n\nDes voisins, alert\u00e9s par la dispute, sortirent la t\u00eate de leurs appartements pour voir ce qui se passait sur le palier.\n\n\u2014 Commissaire Shi, vous aviez dit que vous...\n\nLe jeune policier tira avec rage le commissaire par la manche. Visiblement, Wang Miao n'\u00e9tait pas le seul \u00e0 \u00eatre indispos\u00e9 par la trivialit\u00e9 du commissaire.\n\nLe commandant militaire se pr\u00e9cipita devant la porte :\n\n\u2014 Professeur Wang, ne vous m\u00e9prenez pas sur notre compte. Une r\u00e9union importante se tient cet apr\u00e8s-midi o\u00f9 seront pr\u00e9sentes plusieurs personnalit\u00e9s de la communaut\u00e9 scientifique nationale. Nous avons pour mission de vous y conduire.\n\n\u2014 J'ai d'autres choses pr\u00e9vues pour cet apr\u00e8s-midi.\n\n\u2014 Nous sommes au courant. Notre sup\u00e9rieur a d\u00e9j\u00e0 contact\u00e9 le directeur de votre Centre de recherches en nanotechnologie. Votre pr\u00e9sence est indispensable. Nous n'aurions d'autre choix que de la reporter.\n\nShi Qiang et son coll\u00e8gue rest\u00e8rent silencieux, ils tourn\u00e8rent les talons et descendirent l'escalier. En les voyant s'\u00e9loigner, les deux militaires parurent pousser un long soupir de soulagement.\n\n\u2014 C'est quoi, son probl\u00e8me ? demanda \u00e0 voix basse le commandant \u00e0 son coll\u00e8gue.\n\n\u2014 Ses bavures sont de notori\u00e9t\u00e9 publique. Il y a quelques ann\u00e9es, lors d'une prise d'otages, il a jou\u00e9 au loup solitaire, au m\u00e9pris de la s\u00e9curit\u00e9 des otages. R\u00e9sultat : trois morts de la m\u00eame la famille, abattus par les ravisseurs. On raconte aussi qu'il fricote avec la p\u00e8gre, il collaborerait avec des organisations mafieuses dans le but d'en faire tomber d'autres. Et cerise sur le g\u00e2teau, l'ann\u00e9e derni\u00e8re, il a tellement tortur\u00e9 un suspect qu'il s'est retrouv\u00e9 handicap\u00e9 moteur. Depuis, il a \u00e9t\u00e9 mis \u00e0 pied...\n\n\u2014 Qu'est-ce qu'un mec comme lui fout au Centre d'op\u00e9rations militaires ?\n\n\u2014 Ordre du g\u00e9n\u00e9ral. Il doit avoir quelque chose de plus que les autres. Mais les restrictions le concernant sont claires : mis \u00e0 part les questions de s\u00fbret\u00e9 publique, nous ne lui donnons aucune autre information.\n\nLe Centre d'op\u00e9rations militaires ? De quoi parlaient-ils ? Wang Miao observa les deux militaires avec perplexit\u00e9.\n\nLa voiture qui vint le chercher le conduisit dans un grand complexe situ\u00e9 \u00e0 la p\u00e9riph\u00e9rie de la ville. Wang Miao remarqua qu'aucune indication ne figurait sous le num\u00e9ro du b\u00e2timent. Il en d\u00e9duisit que le site appartenait \u00e0 l'arm\u00e9e et non \u00e0 la police.\n\nLa r\u00e9union avait lieu dans une grande salle de conf\u00e9rences. Wang Miao fut stup\u00e9fait du chaos qui r\u00e9gnait \u00e0 l'int\u00e9rieur. La salle \u00e9tait truff\u00e9e de mat\u00e9riel informatique. Certains ordinateurs, n'ayant pu trouver place sur les tables, \u00e9taient pos\u00e9s \u00e0 m\u00eame le sol, tandis que des fils \u00e9lectriques et des c\u00e2bles Ethernet emm\u00eal\u00e9s \u00e9taient \u00e9parpill\u00e9s un peu partout. Les commutateurs r\u00e9seau n'\u00e9taient pas rang\u00e9s dans des racks \u2013 comme l'exigeaient pourtant les normes de s\u00e9curit\u00e9 habituelles \u2013 mais directement empil\u00e9s sur les serveurs. Des \u00e9crans de vid\u00e9oprojecteur \u00e9taient install\u00e9s aux quatre coins de la salle selon des angles diff\u00e9rents. On aurait cru des tentes de gitans. La fum\u00e9e des cigarettes flottait dans l'air comme une nappe de brume. Wang Miao ignorait s'il se trouvait dans le fameux Centre d'op\u00e9rations militaires dont avaient parl\u00e9 les deux officiers, mais il \u00e9tait au moins s\u00fbr d'une chose : ce qui se tramait ici ne souffrait aucune autre pr\u00e9occupation.\n\nLa table de conf\u00e9rence de fortune \u00e9tait jonch\u00e9e de documents et d'objets divers. La plupart des participants avaient l'air \u00e9puis\u00e9s, leurs costumes \u00e9taient froiss\u00e9s et leurs cravates d\u00e9nou\u00e9es, comme s'ils venaient de passer une longue nuit blanche. Le pr\u00e9sident de s\u00e9ance \u00e9tait un g\u00e9n\u00e9ral de l'arm\u00e9e de terre nomm\u00e9 Chang Weisi, et la moiti\u00e9 des participants \u00e9taient des militaires. Apr\u00e8s un bref tour de table, Wang Miao apprit que l'assistance comptait aussi un certain nombre de membres de la police et d'autres experts scientifiques comme lui. Certains d'entre eux \u00e9taient d'ailleurs des chercheurs prestigieux travaillant dans le domaine des sciences fondamentales.\n\nCe qui l'\u00e9tonnait davantage, c'\u00e9tait la pr\u00e9sence de quatre \u00e9trangers, et plus encore, l'identit\u00e9 de ces derniers : un colonel de l'arm\u00e9e de l'air am\u00e9ricaine et un colonel de l'arm\u00e9e de terre britannique, tous deux \u00e9missaires de l'Otan. Plus surprenant encore, deux agents de la CIA, pr\u00e9sents en tant qu'observateurs.\n\nWang Miao lisait une seule et m\u00eame phrase sur leurs visages : \"Nous avons fait l'effort de venir, alors ne tra\u00eenons pas !\"\n\nWang Miao aper\u00e7ut le commissaire Shi Qiang. La rudesse du matin fit place \u00e0 un salut poli. Pour autant, le sourire nigaud du commissaire ne parvint pas \u00e0 le rendre plus sympathique \u00e0 ses yeux. Il n'avait aucune envie de prendre place \u00e0 c\u00f4t\u00e9 de lui, mais il ne restait qu'un si\u00e8ge de libre et il dut s'y r\u00e9soudre. L'odeur d\u00e9j\u00e0 pesante du tabac devint encore plus oppressante apr\u00e8s l'arriv\u00e9e de Shi Qiang.\n\nAu moment o\u00f9 furent distribu\u00e9s les documents, Shi Qiang se pencha vers Wang Miao et lui glissa :\n\n\u2014 Professeur Wang, je crois que vous faites des recherches sur euh... les nouveaux mat\u00e9riaux, c'est bien \u00e7a ?\n\n\u2014 Les nanomat\u00e9riaux, r\u00e9pondit simplement Wang Miao.\n\n\u2014 J'ai entendu dire que c'\u00e9tait du costaud. \u00c7a pourrait \u00eatre utilis\u00e9 \u00e0 des fins criminelles \u00e7a, non ?\n\nUne expression \u00e0 moiti\u00e9 goguenarde se lisait sur son visage, sans que Wang Miao parvienne \u00e0 savoir s'il \u00e9tait s\u00e9rieux ou s'il se moquait de lui.\n\n\u2014 Que voulez-vous dire ?\n\n\u2014 On m'a racont\u00e9 que des trucs pas plus \u00e9pais que des cheveux suffisaient \u00e0 soulever un camion et qu'un criminel qui arriverait \u00e0 s'en procurer pourrait forger une machette capable de d\u00e9couper une voiture en deux.\n\n\u2014 Mmh, pas besoin de s'en faire une machette. Un fil de l'\u00e9paisseur d'un centi\u00e8me de cheveu con\u00e7u avec certains nanomat\u00e9riaux et qu'on tendrait sur une route pourrait trancher un v\u00e9hicule aussi facilement que si c'\u00e9tait du fromage... Mais vous savez, n'importe quel objet peut servir \u00e0 des fins criminelles : on peut tuer quelqu'un avec un \u00e9cailleur \u00e0 poisson...\n\nShi Qiang sortit la moiti\u00e9 d'un document de la sacoche et l'y replongea aussit\u00f4t, n'y voyant visiblement aucun int\u00e9r\u00eat.\n\n\u2014 Pas faux. On peut m\u00eame tuer avec un poisson ! Je me suis occup\u00e9 d'une affaire o\u00f9 la nana avait tranch\u00e9 l'engin de son mari avec vous savez quoi ? Un tilapia sorti du cong\u00e9lateur ! L'\u00e9pine dorsale congel\u00e9e de la poiscaille \u00e9tait aussi coupante qu'un couteau.\n\n\u2014 \u00c7a ne m'int\u00e9resse pas. On ne m'a pas fait venir \u00e0 cette r\u00e9union pour \u00e7a tout de m\u00eame ?\n\n\u2014 Les poissons ? Les nanomat\u00e9riaux ? Non, non, aucun rapport.\n\nShi Qiang rapprocha sa bouche de l'oreille de Wang Miao :\n\n\u2014 N'ayez aucune tendresse pour ces types, ils nous prennent pour des abrutis. Ils nous soutirent des infos, mais ils ne nous racontent rien. Regardez-moi, \u00e7a fait un mois que je fais partie du plan, et je n'en sais pas plus que vous.\n\n\u2014 Chers camarades, la r\u00e9union va d\u00e9buter, les interrompit Chang Weisi. L'endroit o\u00f9 vous vous trouvez aujourd'hui est le point de ralliement de diff\u00e9rentes op\u00e9rations militaires men\u00e9es dans le monde entier. Je commencerai par pr\u00e9ciser aux camarades ici pr\u00e9sents la situation telle qu'elle est \u00e0 l'heure actuelle.\n\nWang Miao fut d\u00e9concert\u00e9 par l'utilisation du terme d'\"op\u00e9rations militaires\". Il remarqua cependant qu'il n'\u00e9tait pas dans l'intention du g\u00e9n\u00e9ral d'expliquer aux nouveaux arrivants comme lui les tenants et les aboutissants de l'affaire, ce qui allait dans le sens de ce que venait de dire Shi Qiang. Durant son court discours d'ouverture, le g\u00e9n\u00e9ral Chang utilisa par deux fois l'expression \"chers camarades\". Wang Miao jeta un \u0153il sur les militaires de l'Otan et les agents de la CIA et se demanda si le g\u00e9n\u00e9ral n'avait pas oubli\u00e9 de leur donner \u00e0 eux du \"chers messieurs\".\n\n\u2014 Eux aussi, ce sont des camarades. De toute fa\u00e7on, tout le monde s'interpelle comme \u00e7a ici, chuchota Shi Qiang en d\u00e9signant les quatre \u00e9trangers avec la cigarette qu'il tenait entre les doigts.\n\nWang Miao fut \u00e0 la fois d\u00e9concert\u00e9 et impressionn\u00e9 par le sens de l'observation du commissaire.\n\n\u2014 Shi Qiang, \u00e9teins ta cigarette, l'odeur de fum\u00e9e est d\u00e9j\u00e0 suffisamment forte comme \u00e7a, lan\u00e7a Chang Weisi avant de replonger la t\u00eate dans ses documents.\n\nLa cigarette qu'il venait d'allumer \u00e0 la main, Shi Qiang jeta un regard circulaire autour de lui. Ne trouvant pas trace d'un cendrier, il \u00e9crasa dans un gr\u00e9sillement son m\u00e9got dans sa tasse de th\u00e9. Profitant de l'occasion, il leva la main pour prendre la parole et, sans m\u00eame attendre la permission de Chang Weisi, il \u00e9ructa :\n\n\u2014 G\u00e9n\u00e9ral, j'ai une r\u00e9clamation que j'avais d\u00e9j\u00e0 exprim\u00e9e : nous voulons une transparence totale sur les informations !\n\nLe g\u00e9n\u00e9ral Chang leva la t\u00eate :\n\n\u2014 Il n'existe pas d'information d'ordre militaire qui puisse \u00eatre totalement transparente. Je demande aux chercheurs pr\u00e9sents aujourd'hui de bien comprendre que nous ne sommes pas en mesure de vous divulguer davantage d'informations.\n\n\u2014 Mais nous, ce n'est pas pareil, nous sommes flics, nom d'un chien ! Nous participons aux activit\u00e9s du Centre d'op\u00e9rations militaires depuis sa fondation et, encore aujourd'hui, nous n'avons pas la moindre id\u00e9e de ce \u00e0 quoi il sert ! Sans compter que vous \u00eates joliment en train de nous mettre de c\u00f4t\u00e9 : vous profitez de notre boulot et il faudrait qu'on s'\u00e9carte une fois qu'il est fini !\n\nPlusieurs officiers de police pr\u00e9sents dans la salle tent\u00e8rent \u00e0 voix basse de faire taire Shi Qiang. Wang Miao \u00e9tait quelque peu estomaqu\u00e9 que celui-ci ose s'adresser ainsi \u00e0 quelqu'un du grade du g\u00e9n\u00e9ral Chang Weisi. La contre-attaque de ce dernier fut cinglante :\n\n\u2014 Eh bien, Shi Qiang, il semblerait que tes d\u00e9plorables mani\u00e8res n'aient pas chang\u00e9. Tu penses t'exprimer au nom de toute la police ? Tu as d\u00e9j\u00e0 \u00e9t\u00e9 mis \u00e0 pied plusieurs mois pour ton comportement et tu risques de te retrouver t\u00f4t ou tard exclu du corps des forces de l'ordre. Si je t'ai demand\u00e9 de venir, c'est parce que j'ai du respect pour ta carri\u00e8re dans les forces de police urbaine. Tu devrais plut\u00f4t me remercier de te donner cette chance.\n\nShi Qiang \u00e9leva la voix :\n\n\u2014 Une chance, en reconnaissance des services rendus ? Et moi qui croyais que j'avais des m\u00e9thodes de truand...\n\n\u2014 Elles pourraient bien s'av\u00e9rer utiles, r\u00e9pondit Chang Weisi en hochant la t\u00eate, c'est la seule chose qui compte aujourd'hui. Nous sommes en guerre.\n\n\u2014 Plus rien ne compte \u00e0 pr\u00e9sent, reprit un agent de la CIA dans un mandarin parfait. Il nous faut rompre avec les conventions du pass\u00e9.\n\nLe colonel britannique, qui semblait lui aussi comprendre le chinois, approuva de la t\u00eate et l\u00e2cha :\n\n\u2014 _To be or not to be_...\n\n\u2014 Qu'est-ce qu'il a dit ? demanda Shi Qiang \u00e0 Wang Miao.\n\n\u2014 Rien, r\u00e9pondit-il m\u00e9caniquement.\n\nIls avaient tous l'air de divaguer : en guerre ? Mais o\u00f9 voyaient-ils une guerre ? Il tourna la t\u00eate vers les baies vitr\u00e9es de la salle. Au loin, derri\u00e8re la cour, on pouvait apercevoir la ville, ses art\u00e8res et ses cort\u00e8ges de voitures. Sur les pelouses, on distinguait des ma\u00eetres promenant leurs chiens et des enfants en train de jouer... Lequel des deux \u00e9tait le plus r\u00e9el, le monde de l'int\u00e9rieur ou celui de l'ext\u00e9rieur ?\n\nLe g\u00e9n\u00e9ral Chang poursuivit :\n\n\u2014 R\u00e9cemment, l'ennemi a significativement intensifi\u00e9 ses attaques. Ses cibles sont maintenant les hautes sph\u00e8res scientifiques. Veuillez \u00e0 pr\u00e9sent consulter la liste des noms distribu\u00e9e dans vos documents.\n\nWang Miao se saisit de la premi\u00e8re feuille. Les noms, retranscrits en chinois et en anglais et \u00e9crits en gros caract\u00e8res, semblaient avoir \u00e9t\u00e9 tap\u00e9s \u00e0 la h\u00e2te.\n\n\u2014 Professeur Wang, quels sentiments vous inspire cette liste ? demanda le g\u00e9n\u00e9ral.\n\n\u2014 Je connais trois d'entre eux. Ce sont tous de remarquables physiciens, de vrais pr\u00e9curseurs, r\u00e9pondit-il, l'esprit distrait.\n\nSon regard \u00e9tait en r\u00e9alit\u00e9 rest\u00e9 riv\u00e9 sur le dernier nom de la liste. Dans son esprit, les deux caract\u00e8res apparaissaient sous une couleur diff\u00e9rente de celle des caract\u00e8res figurant au-dessus. Comment son nom \u00e0 elle s'\u00e9tait-il retrouv\u00e9 ici ? Que lui \u00e9tait-il arriv\u00e9 ?\n\n\u2014 Une connaissance ? demanda Shi Qiang en pointant le nom avec un doigt bouffi et jauni par la cigarette.\n\nWang Miao fit la moue.\n\n\u2014 Ah, pas plus que \u00e7a ? Vous souhaiteriez la rencontrer ?\n\nWang Miao comprenait \u00e0 pr\u00e9sent pourquoi Chang Weisi avait eu du nez de faire appel \u00e0 Shi Qiang. Sous ce regard un peu lourdaud se cachaient en r\u00e9alit\u00e9 des yeux aussi ac\u00e9r\u00e9s que des poignards. Ce n'\u00e9tait peut-\u00eatre pas un flic int\u00e8gre, mais il \u00e9tait loin d'\u00eatre idiot.\n\nC'\u00e9tait il y a un an de cela, Wang Miao \u00e9tait responsable de la section des nanocomposants du projet d'acc\u00e9l\u00e9rateur de particules \"China II\". Un apr\u00e8s-midi, lors d'une br\u00e8ve pause sur le site de Liangxiang, quelque chose avait attir\u00e9 son regard. En tant que passionn\u00e9 de photographie de paysage, il lui arrivait souvent que des sc\u00e8nes de la r\u00e9alit\u00e9 se mat\u00e9rialisent dans ses yeux en une composition artistique. Ce jour-l\u00e0, le motif principal de la composition \u00e9tait form\u00e9 par la bobine supraconductrice que l'\u00e9quipe \u00e9tait en train d'installer. La moiti\u00e9 seulement de la bobine de trois \u00e9tages avait \u00e9t\u00e9 mont\u00e9e. En la regardant, on aurait cru voir une cr\u00e9ature monstrueuse faite d'\u00e9normes blocs de m\u00e9tal et d'un \u00e9cheveau de tuyaux r\u00e9frig\u00e9rants cryog\u00e9niques. Comme le tas d'ordures d'une grande \u00e8re industrielle, la machine r\u00e9v\u00e9lait \u00e0 elle seule la brutalit\u00e9 et la froideur m\u00e9tallique d'une technologie inhumaine. En face de ce monstre d'acier avait \u00e9merg\u00e9 la silhouette mince d'une jeune femme. La luminosit\u00e9 de la composition \u00e9tait parfaite : le monstre \u00e9tait immerg\u00e9 dans l'ombre du plafond temporaire de la salle, ce qui intensifiait sa texture froide et rugueuse, tandis qu'un halo argent\u00e9, presque cr\u00e9pusculaire, transper\u00e7ait les trous du plafond pour venir se projeter sur la silhouette. La douceur de la lumi\u00e8re \u00e9clairait ses cheveux satin\u00e9s ainsi que la blancheur de son cou qui saillait du col de sa combinaison de travail. On aurait dit une fleur d\u00e9licate qui venait de s'ouvrir sur de gigantesques ruines de m\u00e9tal apr\u00e8s un violent orage...\n\n\u2014 Qu'est-ce que tu regardes ? Au travail !\n\nWang Miao avait sursaut\u00e9, avant de remarquer que le directeur du Centre de recherches en nanotechnologie ne s'\u00e9tait pas adress\u00e9 \u00e0 lui mais \u00e0 un jeune ing\u00e9nieur dont le regard semblait tout comme le sien happ\u00e9 par la silhouette. Wang Miao \u00e9tait redescendu de l'art \u00e0 la r\u00e9alit\u00e9 et il s'\u00e9tait aper\u00e7u que la jeune femme qui venait de p\u00e9n\u00e9trer dans la pi\u00e8ce n'\u00e9tait pas un membre ordinaire du personnel, car elle \u00e9tait accompagn\u00e9e de l'ing\u00e9nieur en chef qui lui pr\u00e9sentait les lieux d'un air affable.\n\n\u2014 Qui est-ce ? avait demand\u00e9 Wang Miao au directeur.\n\n\u2014 Tu dois avoir entendu parler d'elle, avait-il r\u00e9pondu en faisant de grands gestes. C'est son mod\u00e8le th\u00e9orique des super-cordes qui sera test\u00e9 lors de la premi\u00e8re mise en marche de l'acc\u00e9l\u00e9rateur \u00e0 deux milliards, lorsque sa construction sera achev\u00e9e. La r\u00e8gle de l'anciennet\u00e9 du monde scientifique fait qu'elle n'aurait sans doute pas d\u00fb \u00eatre la premi\u00e8re \u00e0 visiter les lieux, mais aucun des vieux briscards n'ose montrer patte blanche, ils ont peur d'avoir l'air ridicule. Cons\u00e9quence : ils lui laissent une chance de se faire un nom.\n\n\u2014 Comment ? Alors, Yang Dong est... une femme ?\n\n\u2014 En effet. Nous ne l'avons appris qu'il y a quelques jours lorsque nous l'avons rencontr\u00e9e pour la premi\u00e8re fois.\n\nLe jeune ing\u00e9nieur avait alors demand\u00e9 :\n\n\u2014 Elle doit souffrir de troubles psychologiques, sinon comment expliquer qu'elle ne se montre jamais dans les m\u00e9dias ? C'est comme l'\u00e9crivain Qian Zhongshu qui jusqu'\u00e0 sa mort n'a jamais accept\u00e9 de passer \u00e0 la t\u00e9l\u00e9 ?\n\n\u2014 Oui mais, quand m\u00eame, Qian Zhongshu, on connaissait au moins son sexe, non ? Elle a d\u00fb conna\u00eetre des exp\u00e9riences traumatisantes dans son enfance pour d\u00e9velopper un tel autisme, avait poursuivi Wang Miao, un peu par amertume.\n\nYang Dong et l'ing\u00e9nieur en chef s'\u00e9taient approch\u00e9s d'eux et, au moment de les d\u00e9passer, celle-ci les avait salu\u00e9s de la t\u00eate en souriant. Elle n'avait pas dit un mot, mais la puret\u00e9 de son regard s'\u00e9tait ancr\u00e9e dans les souvenirs de Wang Miao.\n\nLe soir m\u00eame, Wang Miao s'\u00e9tait assis dans sa biblioth\u00e8que pour admirer fi\u00e8rement ses photos de paysages. Ses yeux \u00e9taient tomb\u00e9s sur une sc\u00e8ne qu'il avait captur\u00e9e au nord de la Grande Muraille : une vall\u00e9e d\u00e9serte derri\u00e8re laquelle per\u00e7ait la blancheur des montagnes enneig\u00e9es. \u00c0 l'extr\u00e9mit\u00e9 de la vall\u00e9e, des branches de bois mort qui semblaient avoir v\u00e9cu mille histoires occupaient presque un tiers de l'espace. Dans son imagination, Wang Miao avait incrust\u00e9 dans cette sc\u00e8ne la silhouette f\u00e9minine qui baignait encore dans son esprit. Il l'avait imagin\u00e9e au plus profond de la vall\u00e9e, o\u00f9 elle avait l'air minuscule. \u00c0 cet instant, Wang Miao avait eu la surprise de constater que le tableau paraissait prendre vie, comme si le monde \u00e0 l'int\u00e9rieur de la photographie avait reconnu cette silhouette, comme si tout cela n'avait jamais exist\u00e9 que pour elle. Il avait encore superpos\u00e9 dans son esprit la silhouette sur quelques autres images. Les yeux de la fille devenaient parfois m\u00eame la toile de fond de plusieurs sc\u00e8nes, et toutes celles-ci s'\u00e9veillaient, r\u00e9v\u00e9lant une beaut\u00e9 que Wang Miao n'avait jamais soup\u00e7onn\u00e9e auparavant. Wang Miao avait toujours trouv\u00e9 qu'il manquait une \u00e2me \u00e0 ses photos, il savait maintenant que ce qui manquait, c'\u00e9tait elle.\n\n\u2014 En moins de deux mois, tous les physiciens dont les noms apparaissent sur la liste se sont donn\u00e9 la mort, expliqua Chang Weisi.\n\nCe fut comme un coup de tonnerre. Wang Miao eut un grand blanc, derri\u00e8re lequel des images naquirent bient\u00f4t : les m\u00eames paysages bicolores de ses photographies, mais d'o\u00f9 la silhouette avait disparu. Le ciel avait effac\u00e9 ses yeux, le monde n'\u00e9tait plus qu'une nature morte.\n\n\u2014 C'\u00e9tait... quand ? demanda Wang Miao, ahuri.\n\n\u2014 Il y a moins de deux mois, r\u00e9p\u00e9ta le g\u00e9n\u00e9ral.\n\n\u2014 Vous voulez sans doute parler du dernier nom sur la liste, lan\u00e7a cr\u00e2nement Shi Qiang, qui \u00e9tait assis \u00e0 ses c\u00f4t\u00e9s.\n\nPuis il baissa la voix :\n\n\u2014 C'est la derni\u00e8re \u00e0 s'\u00eatre suicid\u00e9e. Avant-hier soir, overdose de somnif\u00e8res. Sa mort a \u00e9t\u00e9 rapide, elle n'a pas souffert.\n\nL'espace d'un instant, Wang Miao \u00e9prouva un soup\u00e7on de gratitude pour Shi Qiang.\n\n\u2014 Pourquoi ? demanda-t-il.\n\nComme des diapositives, les images des paysages morts tournaient en boucle dans son esprit.\n\nChang Weisi r\u00e9pondit :\n\n\u2014 Pour l'heure, nous ne sommes certains que d'une chose : c'est la m\u00eame raison qui les a tous pouss\u00e9s \u00e0 mettre fin \u00e0 leurs jours. Mais il est difficile \u2013 peut-\u00eatre m\u00eame impossible pour des non-sp\u00e9cialistes comme nous \u2013 de d\u00e9terminer pr\u00e9cis\u00e9ment cette raison. Vous trouverez dans les documents distribu\u00e9s une partie du contenu de leurs lettres d'adieu, vous pourrez les lire plus en d\u00e9tail apr\u00e8s la r\u00e9union.\n\nWang Miao feuilleta les photocopies de ces lettres, toutes tr\u00e8s longues.\n\n\u2014 Docteur Ding Yi, peut-\u00eatre pourriez-vous transmettre au Pr Wang Miao celle de Yang Dong ? C'est la plus courte, mais aussi celle qui r\u00e9sume le mieux la situation.\n\nL'homme resta silencieux et t\u00eate baiss\u00e9e pendant un long moment avant de r\u00e9agir. Il sortit une enveloppe blanche et la fit passer \u00e0 Wang Miao. Shi Qiang se pencha \u00e0 son oreille :\n\n\u2014 Le petit ami de Yang Dong.\n\nWang Miao se souvint seulement alors qu'il avait d\u00e9j\u00e0 rencontr\u00e9 le Dr Ding Yi sur le site de l'acc\u00e9l\u00e9rateur de Liangxiang. Il \u00e9tait membre du comit\u00e9 th\u00e9orique. Il s'\u00e9tait fait un nom en d\u00e9couvrant le macro-atome lors de ses recherches sur la foudre en boule. Wang Miao sortit de l'enveloppe un objet aux senteurs odorantes et \u00e0 la forme irr\u00e9guli\u00e8re. Ce n'\u00e9tait pas une feuille de papier, mais un morceau d'\u00e9corce de bouleau de Mandchourie. Il y \u00e9tait \u00e9crit avec un trait gracieux :\n\n_Tout \u00e7a pour en arriver \u00e0 cette conclusion : la physique n'a jamais exist\u00e9 et n'existera jamais. Je reconnais que c'est irresponsable, mais je n'ai pas d'autre choix._\n\nPuis elle \u00e9tait partie, sans m\u00eame une signature.\n\n\u2014 La physique... n'existe pas ?\n\n\u00c9tourdi, Wang Miao regarda autour de lui.\n\nLe g\u00e9n\u00e9ral Chang referma le dossier.\n\n\u2014 Nous poss\u00e9dons des informations concr\u00e8tes concernant les r\u00e9sultats des exp\u00e9riences obtenus \u00e0 partir des trois acc\u00e9l\u00e9rateurs de particules existant dans le monde. Celles-ci sont tr\u00e8s techniques et nous n'en discuterons pas ici. La premi\u00e8re chose sur laquelle nous devons concentrer notre enqu\u00eate est la Soci\u00e9t\u00e9 scientifique des fronti\u00e8res de la science. L'ann\u00e9e 2005 a \u00e9t\u00e9 d\u00e9clar\u00e9e \"Ann\u00e9e mondiale de la physique\" par l'Unesco, la soci\u00e9t\u00e9 est n\u00e9e au lendemain des nombreux colloques et \u00e9changes acad\u00e9miques internationaux ayant vu le jour cette ann\u00e9e-l\u00e0. Il s'agit d'une association scientifique \u00e0 dimension internationale. Docteur Ding, vous \u00eates vous-m\u00eame sp\u00e9cialiste de physique th\u00e9orique, pouvez-vous nous donner plus de d\u00e9tails sur cette organisation ?\n\nDing Yi acquies\u00e7a :\n\n\u2014 Je n'ai jamais eu de rapport direct avec la Soci\u00e9t\u00e9 des fronti\u00e8res de la science, mais c'est une organisation r\u00e9put\u00e9e dans le monde scientifique. Elle part du constat suivant : depuis la seconde moiti\u00e9 du si\u00e8cle dernier, l'abandon des concepts \u00e9l\u00e9mentaires de la physique classique a fait place \u00e0 des th\u00e9ories de plus en plus complexes, de plus en plus vagues et de plus en plus incertaines. Les v\u00e9rifications exp\u00e9rimentales, elles, deviennent de plus en plus difficiles, ce qui montre selon eux que l'exploration scientifique se retrouve aujourd'hui face \u00e0 des obstacles importants. La Soci\u00e9t\u00e9 des fronti\u00e8res de la science se consacre \u00e0 la promotion de nouvelles perspectives de pens\u00e9e. Pour faire simple, leur objectif consiste \u00e0 utiliser des m\u00e9thodes scientifiques pour sonder les limites m\u00eames de la science, afin de d\u00e9terminer si la connaissance scientifique de la nature ne se heurte pas \u2013 en surface ou en profondeur \u2013 \u00e0 une ligne de fond au-dessous de laquelle la science ne pourrait acc\u00e9der. Pour eux, le d\u00e9veloppement de la physique moderne semble plus ou moins avoir atteint cette ligne de fond.\n\n\u2014 Merci, poursuivit Chang Weisi. Selon nos informations, la plupart des scientifiques suicid\u00e9s ont \u00e9t\u00e9 \u00e0 un moment donn\u00e9 en contact avec la Soci\u00e9t\u00e9 des fronti\u00e8res de la science. Certains en \u00e9taient m\u00eame membres. Mais rien ne nous permet aujourd'hui de conclure \u00e0 une manipulation spirituelle occulte ou \u00e0 une consommation collective de drogues. Ce qui signifie en d'autres termes que m\u00eame si les Fronti\u00e8res de la science ont exerc\u00e9 une quelconque influence sur ces scientifiques, cela s'est fait au moyen d'\u00e9changes universitaires parfaitement l\u00e9gaux. Professeur Wang, ils ont r\u00e9cemment pris contact avec vous, et nous aimerions avoir des informations \u00e0 ce sujet.\n\nShi Qiang rajouta aussit\u00f4t d'un ton bourru :\n\n\u2014 Y compris le nom de vos contacts, le lieu et l'heure des rencontres, le contenu des conversations, si vous avez \u00e9chang\u00e9 des documents ou des courriels...\n\n\u2014 Shi Qiang ! le coupa Chang Weisi, avec autorit\u00e9.\n\nUn policier se pencha vers Shi Qiang et lui murmura :\n\n\u2014 Ce n'est pas parce que tu n'ouvres pas ta gueule une minute qu'on va croire que tu es muet !\n\nCe dernier souleva la tasse de th\u00e9 pos\u00e9e devant lui et, apr\u00e8s un regard sur le m\u00e9got qui baignait \u00e0 l'int\u00e9rieur, la reposa bruyamment sur la table.\n\nCette r\u00e9action fut aussi insupportable pour Wang Miao que s'il avait gob\u00e9 une mouche. L'infime sentiment de gratitude qu'il avait ressenti plus t\u00f4t disparut aussit\u00f4t. Mais il se ma\u00eetrisa et r\u00e9pondit \u00e0 la question :\n\n\u2014 Mon premier contact avec la Soci\u00e9t\u00e9 des fronti\u00e8res de la science remonte \u00e0 ma rencontre avec Shen Yufei, une physicienne japonaise d'origine chinoise, qui travaille maintenant dans une entreprise \u00e0 capital japonais. Elle habite P\u00e9kin. Elle menait autrefois des recherches dans un laboratoire de Mitsubishi Electrics, et nous nous sommes rencontr\u00e9s pour la premi\u00e8re fois lors d'un s\u00e9minaire technique au d\u00e9but de l'ann\u00e9e. Gr\u00e2ce \u00e0 elle, j'ai aussi fait la connaissance d'autres physiciens, tous membres de la Soci\u00e9t\u00e9 des fronti\u00e8res de la science. Lors de nos discussions, nous abordions des questions... comment dire... extr\u00eames, essentiellement autour de la ligne de fond de la science que vient d'\u00e9voquer le Dr Ding. Au d\u00e9but, je ne me passionnais gu\u00e8re pour ces d\u00e9bats, je prenais plut\u00f4t cela comme un divertissement. Je fais de la recherche appliqu\u00e9e et mes connaissances sur ces sujets sont limit\u00e9es. Je me contentais d'\u00e9couter leurs discussions et leurs controverses. Mais leurs r\u00e9flexions \u00e9taient profondes, et leurs perspectives originales. J'avais l'impression que mon horizon de pens\u00e9e s'\u00e9largissait \u00e0 leur contact et, peu \u00e0 peu, je me suis davantage impliqu\u00e9. Toutefois leurs sujets de discussion se limitaient \u00e0 ces d\u00e9bats passionn\u00e9s purement th\u00e9oriques. Ils m'ont invit\u00e9 une fois \u00e0 rejoindre les Fronti\u00e8res de la science, mais assister aux d\u00e9bats serait devenu un devoir. Ne m'en sentant pas la force, j'ai pr\u00e9f\u00e9r\u00e9 d\u00e9cliner.\n\n\u2014 Professeur Wang, nous aimerions que vous acceptiez cette invitation d'int\u00e9grer la Soci\u00e9t\u00e9 des fronti\u00e8res de la science. C'est aussi la raison principale pour laquelle nous vous avons fait venir aujourd'hui, annon\u00e7a le g\u00e9n\u00e9ral Chang. Nous esp\u00e9rons par votre interm\u00e9diaire obtenir des informations de l'int\u00e9rieur.\n\n\u2014 Vous voulez dire que je servirai de taupe ? demanda Wang Miao, anxieux.\n\nShi Qiang explosa de rire :\n\n\u2014 Ha ha !, de taupe !\n\nLe g\u00e9n\u00e9ral lui jeta un coup d'\u0153il r\u00e9probateur et s'adressa \u00e0 Wang Miao :\n\n\u2014 Il s'agit simplement de nous fournir quelques renseignements. Et puis, nous n'avons pas d'autre interm\u00e9diaire.\n\nWang Miao secoua la t\u00eate :\n\n\u2014 Je suis navr\u00e9, g\u00e9n\u00e9ral. Je ne peux pas faire \u00e7a.\n\n\u2014 Professeur Wang, la Soci\u00e9t\u00e9 des fronti\u00e8res de la science est une organisation qui regroupe les meilleurs sp\u00e9cialistes du monde entier. Enqu\u00eater sur ses activit\u00e9s est une chose \u00e0 la fois tr\u00e8s complexe et tr\u00e8s sensible. Avec eux, nous marchons sur des \u0153ufs. Sans appui dans le monde scientifique, nous n'avancerons pas, c'est pourquoi nous nous permettons de vous soumettre si brusquement cette requ\u00eate. J'esp\u00e8re que vous pouvez nous comprendre. Toutefois nous respecterons votre d\u00e9cision si vous d\u00e9cidez de rejeter notre sollicitation.\n\n\u2014 Je... suis tr\u00e8s occup\u00e9, je n'ai pas le temps, pr\u00e9texta Wang Miao.\n\nChang Weisi hocha la t\u00eate :\n\n\u2014 Bien, professeur Wang, nous n'abuserons plus de votre temps. Merci d'\u00eatre venu \u00e0 cette r\u00e9union.\n\nWang Miao resta distrait quelques secondes avant de comprendre qu'il \u00e9tait temps pour lui de quitter les lieux. Chang Weisi le raccompagna poliment jusqu'\u00e0 la porte de la salle de conf\u00e9rences. Derri\u00e8re eux, Shi Qiang cria \u00e0 voix haute :\n\n\u2014 Et voil\u00e0, depuis le d\u00e9but, j'\u00e9tais contre ce plan. Trop d'intellos se sont d\u00e9j\u00e0 foutus en l'air, infiltrer celui-l\u00e0, \u00e7a aurait \u00e9t\u00e9 jeter une boulette de viande \u00e0 une meute de chiens !\n\nWang Miao fit demi-tour et se dirigea vers Shi Qiang. Il lui glissa en se mordant les l\u00e8vres :\n\n\u2014 Vous n'\u00eates vraiment pas un mod\u00e8le de policier, commissaire.\n\n\u2014 Je ne l'ai jamais \u00e9t\u00e9.\n\n\u2014 La raison du suicide de ces scientifiques n'a pas encore \u00e9t\u00e9 \u00e9lucid\u00e9e, vous ne devriez pas parler d'eux sur un ton si m\u00e9prisant. Leurs travaux ont beaucoup contribu\u00e9 au d\u00e9veloppement de l'esp\u00e8ce humaine, tout le monde ne peut pas en dire autant.\n\n\u2014 Vous voulez dire qu'ils sont plus malins que moi ?\n\nShi Qiang se balan\u00e7a sur sa chaise et pencha la t\u00eate vers Wang Miao :\n\n\u2014 Au moins, je n'ai jamais tent\u00e9 de me faire sauter la cervelle apr\u00e8s avoir entendu les d\u00e9lires d'un coll\u00e8gue.\n\n\u2014 Et vous pr\u00e9tendez que ce sera mon cas ?\n\n\u2014 Vous \u00eates responsable de votre propre s\u00e9curit\u00e9, lui r\u00e9pondit-il en le d\u00e9visageant et en d\u00e9voilant le m\u00eame sourire nigaud qui devait \u00eatre sa marque de fabrique.\n\n\u2014 Dans ce cas, je suis plus en s\u00e9curit\u00e9 que vous. Vous devriez savoir que le pouvoir de discernement d'un homme est proportionnel \u00e0 son intelligence.\n\n\u2014 Je n'en suis pas si s\u00fbr, voyez-vous...\n\n\u2014 Shi Qiang, encore un mot et je te mets dehors ! le r\u00e9primanda s\u00e9v\u00e8rement Chang Weisi.\n\n\u2014 \u00c7a ne fait rien, laissez-le dire, j'ai chang\u00e9 d'avis, j'accepte votre proposition d'int\u00e9grer les Fronti\u00e8res de la science.\n\nShi Qiang hocha la t\u00eate :\n\n\u2014 Tr\u00e8s bien. Une fois int\u00e9gr\u00e9, soyez fut\u00e9, il y a des choses qui peuvent \u00eatre faites en douceur : jeter un coup d'\u0153il \u00e0 leurs ordinateurs, m\u00e9moriser une adresse mail ou un site Internet, etc.\n\n\u2014 \u00c7a suffit. D\u00e9trompez-vous, je ne suis pas une taupe, je veux seulement prouver \u00e0 tous votre ignorance et votre stupidit\u00e9.\n\n\u2014 Si dans quelque temps vous \u00eates encore vivant, alors oui, vous l'aurez prouv\u00e9... Mais j'ai bien peur que... h\u00e9 ! h\u00e9 !\n\nSon sourire niais se changea en ricanement sardonique.\n\n\u2014 Bien s\u00fbr que je serai encore en vie ! J'esp\u00e8re simplement ne plus avoir \u00e0 vous revoir !\n\nChang Weisi raccompagna Wang Miao jusqu'au rez-de-chauss\u00e9e et appela une voiture pour le raccompagner chez lui. Au moment de le quitter, il lui glissa :\n\n\u2014 Shi Qiang a mauvais caract\u00e8re, mais il a une grande exp\u00e9rience dans la police judiciaire et la lutte antiterroriste. Il y a vingt ans, il a servi dans ma compagnie.\n\nArriv\u00e9 \u00e0 hauteur de la voiture, Chang Weisi ajouta :\n\n\u2014 Professeur Wang, vous avez certainement beaucoup de questions.\n\n\u2014 En quoi tout ce qui a \u00e9t\u00e9 dit tout \u00e0 l'heure concerne-t-il l'arm\u00e9e ?\n\n\u2014 La guerre concerne \u00e9videmment l'arm\u00e9e.\n\nWang Miao regarda autour de lui. Tout scintillait sous ce soleil printanier.\n\n\u2014 O\u00f9 voyez-vous une guerre ? Il n'y a m\u00eame pas un seul point chaud dans tout le globe. Nous devons \u00eatre \u00e0 l'\u00e9poque la plus pacifique de toute l'histoire de l'humanit\u00e9.\n\nChang Weisi se fendit d'un sourire imp\u00e9n\u00e9trable :\n\n\u2014 Vous saurez bient\u00f4t, le monde entier saura bient\u00f4t. Professeur Wang, avez-vous d\u00e9j\u00e0 connu un grand chamboulement dans votre vie ? Un chamboulement tel qu'il a compl\u00e8tement chang\u00e9 la face de celle-ci, comme si, l'espace d'une nuit, votre monde n'\u00e9tait soudain plus le m\u00eame ?\n\n\u2014 Jamais.\n\n\u2014 Vous \u00eates donc chanceux. Il existe tant de facteurs impr\u00e9visibles dans ce monde, et, pourtant, votre vie n'a jamais connu de chamboulement.\n\nWang Miao r\u00e9fl\u00e9chit un long moment, mais il ne comprenait pas.\n\n\u2014 C'est le cas de la plupart des gens, non ?\n\n\u2014 Eh bien, disons que la plupart des gens sont chanceux.\n\n\u2014 Mais... des g\u00e9n\u00e9rations enti\u00e8res de gens ont une vie aussi terne que la mienne.\n\n\u2014 Tous des chanceux.\n\nWang Miao secoua la t\u00eate en souriant :\n\n\u2014 Je dois avouer que je ne comprends pas grand-chose aujourd'hui, est-ce que vous voulez dire que...\n\n\u2014 Oui, toute l'histoire humaine est une chance. Depuis l'\u00e2ge de pierre jusqu'\u00e0 nos jours, elle a eu le bonheur de ne jamais conna\u00eetre de chamboulements majeurs. Mais ce bonheur devait prendre fin t\u00f4t ou tard. Ce jour-l\u00e0 est arriv\u00e9, je vous le dis, il faut vous pr\u00e9parer psychologiquement.\n\nWang Miao aurait souhait\u00e9 encore l'interroger, mais le g\u00e9n\u00e9ral lui serra la main et le salua, l'emp\u00eachant de poursuivre ses questions.\n\nDans la voiture, le chauffeur demanda l'adresse de Wang Miao. Ce dernier r\u00e9pondit puis s'excusa aussit\u00f4t :\n\n\u2014 Oh pardon, ce n'est pas vous qui \u00eates venu me chercher \u00e0 l'aller ? C'est la m\u00eame voiture.\n\n\u2014 Ce n'\u00e9tait pas moi. Je suis all\u00e9 chercher le Dr Ding.\n\nWang Miao changea soudain d'id\u00e9e. Il demanda au chauffeur l'adresse de Ding Yi, qui la lui donna. Le soir m\u00eame, il rendit visite \u00e0 ce dernier.\n\n## 5\n\n## LE BILLARD\n\nEn poussant la porte du trois-pi\u00e8ces tout neuf de Ding Yi, Wang Miao sentit une odeur d'alcool. Il aper\u00e7ut Ding Yi allong\u00e9 sur le canap\u00e9 avec la t\u00e9l\u00e9vision allum\u00e9e, ses yeux riv\u00e9s sur le plafond. Wang Miao balaya la pi\u00e8ce du regard. Elle n'\u00e9tait pour ainsi dire pas d\u00e9cor\u00e9e et ne poss\u00e9dait presque aucun meuble ou accessoire, si bien que le vaste salon paraissait tr\u00e8s vide. Un \u00e9l\u00e9ment de la pi\u00e8ce sautait aux yeux : une table de billard dispos\u00e9e dans un coin.\n\nDing Yi ne parut pas s'offusquer de la visite impromptue de Wang Miao. Il \u00e9tait \u00e9vident qu'il cherchait quelqu'un \u00e0 qui parler.\n\n\u2014 J'ai achet\u00e9 cet appartement il y a trois mois, dit Ding Yi. Pourquoi ? Ai-je vraiment pu croire qu'elle \u00e9tait pr\u00eate \u00e0 fonder une famille ?\n\nIl secoua la t\u00eate en partant d'un rire alcoolis\u00e9.\n\n\u2014 Vous deux...\n\nWang Miao voulait apprendre le moindre d\u00e9tail de la vie de Yang Dong, mais il ne savait par o\u00f9 commencer.\n\n\u2014 C'\u00e9tait une \u00e9toile, toujours si lointaine... M\u00eame la lumi\u00e8re qu'elle projetait sur moi \u00e9tait sans cesse froide.\n\nDing Yi s'approcha de la fen\u00eatre pour regarder le ciel nocturne, comme s'il cherchait des yeux cette \u00e9toile disparue.\n\nWang Miao ne dit rien. C'\u00e9tait \u00e9trange, il avait maintenant envie d'entendre sa voix. Pourtant, lorsqu'il y a un an leurs regards s'\u00e9taient crois\u00e9s un instant, alors que le soleil se couchait \u00e0 l'ouest, ils n'avaient pas \u00e9chang\u00e9 une parole. Il n'avait jamais entendu le son de sa voix.\n\nDing Yi secoua sa main, comme s'il voulait chasser quelque chose. Il \u00e9mergea de son humeur triste.\n\n\u2014 Tu as raison. Ne fais pas alliance avec les militaires et les policiers, c'est une meute de corniauds pr\u00e9tentieux. Les suicides des physiciens n'ont aucun lien avec les activit\u00e9s des Fronti\u00e8res de la science, je le leur ai d\u00e9j\u00e0 dit, mais ils ne veulent rien comprendre.\n\n\u2014 Ils semblent pourtant avoir enqu\u00eat\u00e9...\n\n\u2014 Oui, enqu\u00eat\u00e9... Et sur un p\u00e9rim\u00e8tre international qui plus est ! Ils devraient savoir que deux personnes de la liste n'ont jamais \u00e9t\u00e9 en contact avec les Fronti\u00e8res de la science, parmi lesquelles... Yang Dong.\n\nPrononcer son nom semblait \u00eatre un supplice pour Ding Yi.\n\n\u2014 Ding Yi, comme tu le sais, je suis maintenant malgr\u00e9 moi impliqu\u00e9 dans cette affaire. Pourrais-tu me dire ce qui a pouss\u00e9 Yang Dong \u00e0... agir ainsi ? Tu dois s\u00fbrement savoir quelque chose, l\u00e2cha maladroitement Wang Miao, en essayant de dissimuler sa v\u00e9ritable intention.\n\n\u2014 Si tu le savais, tu te retrouverais encore plus gravement impliqu\u00e9. Pour l'heure, tu es concern\u00e9 par cette histoire parce qu'on t'a invit\u00e9 \u00e0 t'y int\u00e9resser, mais lorsque tu sauras ce qui se trame r\u00e9ellement, ton esprit sera pris dans le tourbillon et c'est seulement alors que commenceront les vrais ennuis.\n\n\u2014 Je travaille dans le domaine des sciences appliqu\u00e9es, je n'ai pas votre sensibilit\u00e9 de th\u00e9oricien.\n\n\u2014 Eh bien, soit. Tu as d\u00e9j\u00e0 jou\u00e9 au billard ?\n\nDing Yi se dirigea vers la table de billard.\n\n\u2014 Quelquefois, \u00e0 l'\u00e9poque o\u00f9 j'\u00e9tais \u00e9tudiant.\n\n\u2014 Elle et moi nous aimions beaucoup jouer, cela nous faisait penser aux collisions des acc\u00e9l\u00e9rateurs de particules, expliqua Ding Yi.\n\nPuis il prit deux boules, la noire et la blanche. Il pla\u00e7a la boule noire au bord d'un trou et la boule blanche \u00e0 une distance d'environ dix centim\u00e8tres de celle-ci, puis il demanda \u00e0 Wang Miao :\n\n\u2014 Tu es capable de rentrer la boule ?\n\n\u2014 N'importe qui en serait capable \u00e0 cette distance.\n\n\u2014 Montre-moi.\n\nWang Miao prit la queue et tamponna l\u00e9g\u00e8rement la boule blanche pour faire rentrer la noire.\n\n\u2014 Bien. Viens, nous allons maintenant changer la position de la table, indiqua Ding Yi \u00e0 un Wang Miao perplexe.\n\nIls d\u00e9plac\u00e8rent ensemble la lourde table sous une fen\u00eatre du salon. La table une fois stable, Ding Yi ramassa la boule noire dans le r\u00e9ceptacle et la d\u00e9posa \u00e0 nouveau \u00e0 c\u00f4t\u00e9 d'un trou, puis la boule blanche, qu'il approcha \u00e0 une distance de dix centim\u00e8tres.\n\n\u2014 Tu peux toujours la rentrer dans le trou ?\n\n\u2014 Bien s\u00fbr.\n\n\u2014 Apr\u00e8s toi.\n\nWang Miao fit encore rentrer la boule, sans aucune difficult\u00e9.\n\n\u2014 D\u00e9pla\u00e7ons encore la table.\n\nIls manipul\u00e8rent une nouvelle fois la table pour l'installer dans un troisi\u00e8me coin diff\u00e9rent de la pi\u00e8ce. Encore une fois, Ding Yi disposa les boules aux deux m\u00eames endroits que les coups pr\u00e9c\u00e9dents.\n\n\u2014 Joue.\n\n\u2014 Je pense que nous...\n\n\u2014 Joue.\n\nWang Miao sourit, impuissant. La troisi\u00e8me boule noire rentra une fois de plus dans le trou.\n\nIls d\u00e9plac\u00e8rent encore la table de billard \u00e0 deux reprises. La premi\u00e8re fois, ils l'install\u00e8rent \u00e0 c\u00f4t\u00e9 de la porte du salon, et la seconde, \u00e0 sa place originelle. Ding Yi pla\u00e7a encore deux fois la boule \u00e0 c\u00f4t\u00e9 du trou et Wang Miao parvint chaque fois \u00e0 la faire rentrer. Mais cette fois, tous deux commen\u00e7aient \u00e0 transpirer un peu.\n\n\u2014 Bien, l'exp\u00e9rience est termin\u00e9e, analysons les r\u00e9sultats, annon\u00e7a Ding Yi en allumant une cigarette. Nous avons en tout effectu\u00e9 cinq essais, dont quatre dans des espaces et \u00e0 des moments diff\u00e9rents, et deux dans la m\u00eame position, mais \u00e0 un moment diff\u00e9rent. N'es-tu pas surpris par les r\u00e9sultats ?\n\nIl ouvrit exag\u00e9r\u00e9ment les bras :\n\n\u2014 Cinq essais ! Et les r\u00e9sultats de l'exp\u00e9rience de la boule sont les m\u00eames !\n\n\u2014 Mais que veux-tu dire \u00e0 la fin ? soupira Wang Miao.\n\n\u2014 Commence par expliquer ces incroyables r\u00e9sultats. Avec le langage de la physique.\n\n\u2014 Eh bien... durant ces cinq exp\u00e9riences, les deux boules n'ont pas chang\u00e9 ; pour ce qui est de leurs positions, si le cadre spatial de r\u00e9f\u00e9rence est bien la table de billard, celles-ci n'ont pas boug\u00e9 non plus. Le vecteur de vitesse de la boule blanche heurtant la boule noire est \u00e0 peu de chose pr\u00e8s le m\u00eame. Aussi, il n'y a eu aucune variation de l'impulsion des deux boules. C'est pourquoi, \u00e0 chaque essai, la boule noire a \u00e9t\u00e9 envoy\u00e9e dans le trou.\n\nDing Yi ramassa une bouteille de cognac pos\u00e9e \u00e0 c\u00f4t\u00e9 du canap\u00e9 et en offrit un verre \u00e0 Wang Miao qui d\u00e9clina poliment.\n\n\u2014 Tu devrais c\u00e9l\u00e9brer \u00e7a, nous avons fait la d\u00e9couverte d'un grand principe : les lois de la physique restent invariables dans des cadres spatiotemporels diff\u00e9rents. Toutes les th\u00e9ories physiques de l'histoire de l'humanit\u00e9, du principe de la flottabilit\u00e9 \u00e0 la th\u00e9orie des cordes, de m\u00eame que toutes les d\u00e9couvertes scientifiques et les r\u00e9alisations de la pens\u00e9e humaine sont \u00e0 ce jour des sous-produits de cette grande loi. Compar\u00e9s \u00e0 nous, Einstein et Hawking n'\u00e9taient que de vulgaires techniciens.\n\n\u2014 Je ne comprends toujours pas o\u00f9 tu veux en venir.\n\n\u2014 Imaginons \u00e0 pr\u00e9sent une autre s\u00e9rie de r\u00e9sultats : la premi\u00e8re fois, la boule blanche pousse la boule noire dans le trou ; la deuxi\u00e8me, la boule noire d\u00e9vie sur le c\u00f4t\u00e9 ; la troisi\u00e8me fois, la boule noire grimpe au plafond ; la quatri\u00e8me, la boule noire volette dans la pi\u00e8ce comme un moineau affol\u00e9, puis vient finalement se ficher dans ta poche ; la cinqui\u00e8me, la boule noire s'envole \u00e0 la vitesse de la lumi\u00e8re, fait un trou dans le bord de la table, transperce le mur et d\u00e9passe la plan\u00e8te Terre, puis le syst\u00e8me solaire, comme dans la nouvelle d'Asimov1. Qu'en d\u00e9duirais-tu ?\n\nDing Yi fixa Wang Miao. Ce dernier garda le silence un long moment avant de r\u00e9pondre :\n\n\u2014 C'est ce qui est arriv\u00e9, n'est-ce pas ?\n\nDing Yi vida dans son gosier les deux verres d'alcool, ses yeux toujours riv\u00e9s sur la table de billard comme s'il s'agissait d'un engin d\u00e9moniaque.\n\n\u2014 Oui, c'est ce qui est arriv\u00e9. Ces derni\u00e8res ann\u00e9es, le financement des projets de recherche en sciences fondamentales est enfin progressivement arriv\u00e9 \u00e0 maturit\u00e9. Trois co\u00fbteuses tables de billard ont \u00e9t\u00e9 construites : une en Am\u00e9rique du Nord, une en Europe et une, comme tu le sais, \u00e0 Liangxiang en Chine. Votre Centre de recherches en nanotechnologie en a d'ailleurs tir\u00e9 un certain profit. Ces acc\u00e9l\u00e9rateurs de particules \u00e0 haute \u00e9nergie augmentent l'\u00e9nergie de collision \u00e0 un ordre de grandeur jusque-l\u00e0 jamais atteint par l'homme. Dans ce nouveau r\u00e9gime, en d\u00e9pit des m\u00eames particules, des m\u00eames niveaux d'\u00e9nergie, ainsi que des m\u00eames conditions d'exp\u00e9rience, les r\u00e9sultats se sont r\u00e9v\u00e9l\u00e9s tr\u00e8s diff\u00e9rents. Les variations ne sont pas seulement notables entre diff\u00e9rents acc\u00e9l\u00e9rateurs, mais aussi sur un m\u00eame acc\u00e9l\u00e9rateur \u00e0 des moments diff\u00e9rents de l'exp\u00e9rience. \u00c7a a \u00e9t\u00e9 la panique chez les physiciens. Ils ont r\u00e9it\u00e9r\u00e9 encore et encore leurs exp\u00e9riences de collisions \u00e0 haute \u00e9nergie, mais les r\u00e9sultats obtenus ont \u00e9t\u00e9 chaque fois diff\u00e9rents, ils ne semblent r\u00e9gis par aucune loi.\n\n\u2014 Qu'est-ce que cela signifie ? demanda Wang Miao.\n\nEn voyant Ding Yi l'observer sans rien dire, il rench\u00e9rit :\n\n\u2014 Oh, tu sais, je travaille sur les nanotechnologies, et il m'arrive d'avoir affaire \u00e0 des microstructures, mais c'est autrement plus superficiel que vos recherches. Peux-tu \u00e9clairer ma lanterne ?\n\n\u2014 Cela signifie que les lois de la physique varient dans le temps et dans l'espace.\n\n\u2014 Et donc ?\n\n\u2014 Et donc, ce qui suit, tu peux le d\u00e9duire par toi-m\u00eame, le g\u00e9n\u00e9ral Chang \u2013 qui est un homme intelligent \u2013 l'a bien compris.\n\nWang Miao regardait \u00e0 travers la fen\u00eatre, perdu dans ses pens\u00e9es. Dehors, la ville \u00e9tait illumin\u00e9e par des flots de lampadaires qui dissimulaient les \u00e9toiles.\n\n\u2014 Cela signifie que les lois de la physique que nous croyions applicables partout dans l'univers n'existent pas, que la physique... n'existe pas, dit Wang Miao en se retournant.\n\nDing Yi poursuivit :\n\n\u2014 \"Je reconnais que c'est irresponsable, mais je n'ai pas d'autre choix.\" C'est la derni\u00e8re phrase de la lettre qu'elle a laiss\u00e9e. Sans le vouloir, tu viens de prononcer la premi\u00e8re. Maintenant, tu peux sans doute davantage la comprendre.\n\nWang Miao prit dans la main la boule blanche qu'il avait frapp\u00e9e \u00e0 cinq reprises, il la caressa un instant avant de la reposer d\u00e9licatement.\n\n\u2014 Pour les plus pr\u00e9curseurs des chercheurs en physique th\u00e9orique, cela semble en effet \u00eatre un d\u00e9sastre.\n\n\u2014 Ceux qui r\u00e9ussissent dans ce domaine ont pour les lois de la physique une foi presque religieuse. Il est facile de sombrer dans l'abysse.\n\nAu moment de se quitter, Ding Yi transmit une adresse \u00e0 Wang Miao :\n\n\u2014 Quand tu auras le temps, va rendre visite \u00e0 la m\u00e8re de Yang Dong. Yang Dong a toujours v\u00e9cu avec elle. Sa fille, c'\u00e9tait toute sa vie. La pauvre femme est tr\u00e8s seule \u00e0 pr\u00e9sent.\n\nWang Miao r\u00e9pondit :\n\n\u2014 Tu en sais visiblement davantage que moi. Ne peux-tu pas m'en dire plus ? Crois-tu r\u00e9ellement que les lois de la physique sont variables dans le temps et l'espace ?\n\n\u2014 Je n'en sais rien. Une puissance inimaginable est en train d'assassiner la science.\n\n\u2014 Assassiner la science ? Qui ?\n\nDing Yi toisa Wang Miao un long moment, avant de l\u00e2cher :\n\n\u2014 Telle est la question.\n\nWang Miao savait qu'il ne faisait que terminer la phrase prononc\u00e9e par le colonel anglais : _\u00eatre ou ne pas \u00eatre_...\n\n* * *\n\n1 \"La boule de billard\", in _Le Robot qui r\u00eavait_ , J'ai lu, 2002.\n\n## 6\n\n## LE SNIPER ET LE FERMIER\n\nLe lendemain \u00e9tait un jour de week-end, mais Wang Miao se leva t\u00f4t. Il prit son appareil photo et partit faire une promenade \u00e0 v\u00e9lo. Amateur de photographie, son th\u00e8me favori \u00e9tait les espaces naturels o\u00f9 ne transparaissait pas l'empreinte humaine. Malheureusement, \u00e0 son \u00e2ge, il n'avait plus beaucoup de temps \u00e0 consacrer \u00e0 ce loisir de luxe. La plupart du temps, il devait se satisfaire de saisir des clich\u00e9s de paysages urbains. Volontairement ou non, il s\u00e9lectionnait les rares endroits de la ville o\u00f9 subsistaient quelques r\u00e9manences de nature, comme le fond du lac ass\u00e9ch\u00e9 d'un parc public, le sol fra\u00eechement exhum\u00e9 par les engins d'un chantier de construction, les mauvaises herbes poussant entre les fissures du b\u00e9ton... Pour \u00e9liminer des images les couleurs acidul\u00e9es de la ville, il n'utilisait que des pellicules en noir et blanc. De mani\u00e8re inattendue, ce style caract\u00e9ristique lui valait une petite r\u00e9putation dans le milieu. Ses \u0153uvres avaient \u00e9t\u00e9 pr\u00e9sent\u00e9es lors de deux grandes expositions nationales et il avait int\u00e9gr\u00e9 l'Association des photographes. Chaque fois qu'il sortait pour prendre des photos, il d\u00e9ambulait dans la ville \u00e0 v\u00e9lo, \u00e0 la recherche de l'inspiration et de la composition souhait\u00e9e. Cela lui prenait parfois une journ\u00e9e enti\u00e8re.\n\nCe jour-l\u00e0, Wang Miao se sentait un peu bizarre. Alors que d'ordinaire il se distinguait par des photographies \u00e9l\u00e9gantes, \u00e9quilibr\u00e9es \u2013 quoique assez classiques \u2013, il peinait aujourd'hui \u00e0 trouver la stabilit\u00e9 dont il avait besoin. Ce matin, la ville \u00e9mergeant de son sommeil lui paraissait b\u00e2tie sur un sol fait de sables mouvants, son \u00e9quilibre \u00e9tait illusoire. La nuit pr\u00e9c\u00e9dente, les deux boules de billard avaient pris possession de ses r\u00eaves, elles flottaient anarchiquement dans les t\u00e9n\u00e8bres de l'espace. Dans cette totale obscurit\u00e9, ce n'\u00e9tait que quand elle \u00e9clipsait la boule blanche que la boule noire r\u00e9v\u00e9lait sa pr\u00e9sence.\n\n _\u00c9tait-il possible que la nature fondamentale de la mati\u00e8re soit r\u00e9ellement sans loi ? \u00c9tait-il possible que la stabilit\u00e9 et l'ordre du monde ne soient qu'un \u00e9tat d'\u00e9quilibre \u00e9ph\u00e9m\u00e8re dans un coin de l'univers, un tourbillon fugace dans un torrent de chaos ?_\n\n _La science allait-elle \u00eatre assassin\u00e9e ?_\n\nSans s'en rendre compte, il \u00e9tait arriv\u00e9 au pied du nouveau b\u00e2timent de la China Central Television. Il posa pied \u00e0 terre et s'assit au bord de la route, puis leva les yeux vers la gigantesque tour en forme de A. Il essayait de retrouver un sentiment d'\u00e9quilibre. Son regard suivit la direction de la cime pointue de la tour qui scintillait sous les rayons du soleil, puis ses yeux s'ab\u00eem\u00e8rent dans le bleu infini du ciel. C'est alors que deux mots \u00e9merg\u00e8rent dans son esprit :\n\n _Sniper, fermier._\n\nDans leurs d\u00e9bats, les membres de la Soci\u00e9t\u00e9 des fronti\u00e8res de la science utilisaient souvent l'abr\u00e9viation SF, non pas pour science-fiction, mais parce qu'elle renvoyait aux initiales des mots \"sniper\" et \"fermier\". Ces mots faisaient r\u00e9f\u00e9rence \u00e0 deux hypoth\u00e8ses en lien avec la nature des lois de l'univers.\n\nL'hypoth\u00e8se du sniper \u00e9tait la suivante : un tireur d'\u00e9lite de g\u00e9nie s'amuse \u00e0 mitrailler une cible dans laquelle chaque impact de balle se situe \u00e0 une distance pr\u00e9cise de dix centim\u00e8tres l'un de l'autre. Imaginons que sur la surface m\u00eame de la cible vivent des cr\u00e9atures intelligentes en deux dimensions. Les honorables scientifiques de cette \u00e9trange esp\u00e8ce m\u00e8nent une \u00e9tude qui les conduit \u00e0 \u00e9dicter une loi fondamentale : dans l'univers, il y a un trou tous les dix centim\u00e8tres. Les cr\u00e9atures prennent la distraction du sniper pour un principe invariable de l'univers.\n\nL'hypoth\u00e8se du fermier, elle, tient du roman d'\u00e9pouvante : il \u00e9tait une fois des dindes vivant dans la basse-cour d'une ferme. Chaque jour \u00e0 11 heures du matin pr\u00e9cises, le fermier venait apporter le d\u00e9jeuner des volailles. Un scientifique de la soci\u00e9t\u00e9 des dindes \u00e9tudia ce ph\u00e9nom\u00e8ne et remarqua qu'il avait lieu de fa\u00e7on r\u00e9guli\u00e8re depuis pr\u00e8s d'un an, sans avoir jamais connu d'exception. Il en d\u00e9duisit donc qu'une loi fondamentale r\u00e9gissait l'univers : la nourriture arrive chaque matin \u00e0 11 heures. Il pr\u00e9senta cette loi le matin m\u00eame du jour de No\u00ebl \u00e0 ses compatriotes. Or ce jour-l\u00e0, \u00e0 11 heures, aucun d\u00e9jeuner n'arriva. Le fermier entra dans la basse-cour et \u00e9gorgea les dindes.\n\nWang Miao sentit le sol bouger sous ses pieds, comme s'il se trouvait sur des sables mouvants ; la tour en forme de A sembla vaciller, il reporta son regard sur la chauss\u00e9e pour \u00e9chapper \u00e0 ce sentiment de vertige et se for\u00e7a \u00e0 terminer sa pellicule. Il rentra chez lui sous le coup de midi. Son \u00e9pouse \u00e9tait sortie avec son fils et ils n'\u00e9taient pas revenus pour le d\u00e9jeuner. En temps ordinaire, Wang Miao aurait br\u00fbl\u00e9 d'impatience de d\u00e9velopper la pellicule, mais aujourd'hui, il n'en avait pas le go\u00fbt. Apr\u00e8s un repas frugal, comme il n'avait pas bien dormi la veille, il s'endormit pour ne se r\u00e9veiller qu'\u00e0 presque 17 heures. Il se souvint \u00e0 son r\u00e9veil des photos qu'il avait prises le matin m\u00eame, et il p\u00e9n\u00e9tra dans le placard mural qu'il avait transform\u00e9 en chambre noire.\n\nLe d\u00e9veloppement ne dura pas longtemps. Wang Miao commen\u00e7a \u00e0 trier celles qui m\u00e9ritaient d'\u00eatre agrandies. Cependant il d\u00e9couvrit quelque chose de curieux sur la premi\u00e8re photo. C'\u00e9tait un clich\u00e9 d'un morceau de pelouse \u00e0 l'ext\u00e9rieur d'un grand centre commercial. Sur l'image, il remarqua une ligne blanche qui, observ\u00e9e de plus pr\u00e8s, se r\u00e9v\u00e9la en fait \u00eatre une suite de chiffres :\n\n1200 : 00 : 00\n\nUne autre suite apparaissait sur la deuxi\u00e8me photo.\n\n1199 : 49 : 33\n\nDes petites suites de chiffres du m\u00eame genre se retrouvaient sur toutes les photos.\n\nLa troisi\u00e8me : 1199 : 40 : 18 ; la quatri\u00e8me : 1199 : 32 : 07 ; la cinqui\u00e8me : 1199 : 28 : 51 ; la sixi\u00e8me : 1199 : 15 : 44 ; la septi\u00e8me : 1199 : 07 : 38 ; la huiti\u00e8me : 1198 : 53 : 09... la trente-quatri\u00e8me : 1194 : 50 : 49 et la trente-cinqui\u00e8me \u2013 la derni\u00e8re photo : 1194 : 16 : 37.\n\nWang Miao pensa d'abord \u00e0 un probl\u00e8me de bobine. Il utilisait un Leica M2 fabriqu\u00e9 en 1988. Celui-ci \u00e9tait enti\u00e8rement manuel et sans la moindre fonction automatique. Il \u00e9tait donc impossible d'ajouter une date. La qualit\u00e9 hors pair de son objectif et de sa m\u00e9canique faisait du Leica M2 un appareil professionnel d'\u00e9lite, m\u00eame \u00e0 l'\u00e8re du num\u00e9rique.\n\nApr\u00e8s avoir \u00e0 nouveau \u00e9tudi\u00e9 les images, Wang Miao s'aper\u00e7ut d'un autre ph\u00e9nom\u00e8ne \u00e9trange concernant ces suites de chiffres : elles paraissaient s'adapter d'elles-m\u00eames au d\u00e9cor. Si le fond \u00e9tait noir, les num\u00e9ros apparaissaient en blanc et vice versa. Ils semblaient prendre la couleur permettant le plus grand contraste possible, afin d'\u00eatre bien visibles de l'observateur. \u00c0 la seizi\u00e8me photographie, le c\u0153ur de Wang Miao bondit dans sa poitrine. Dans la chambre noire, il sentit un frisson glacial remonter sa moelle \u00e9pini\u00e8re : la photo prise \u00e9tait celle d'un arbre sec devant un vieux mur tout pommel\u00e9. La logique voulait que la suite de chiffres \u2013 qu'elle soit de couleur blanche ou noire \u2013 ne puisse appara\u00eetre clairement sur le fond de l'image. Mais \u00e0 son grand \u00e9tonnement, les chiffres, de couleur blanche, \u00e9taient plac\u00e9s verticalement et rampaient comme un serpent le long des ondulations sinueuses du tronc.\n\nWang Miao se mit \u00e0 \u00e9tudier la relation math\u00e9matique entre les chiffres. Il songea en premier lieu \u00e0 des num\u00e9ros de s\u00e9rie, mais les intervalles entre les chiffres n'\u00e9taient pas les m\u00eames. Il comprit vite qu'ils repr\u00e9sentaient des unit\u00e9s d'heures, de minutes et de secondes. Il sortit son carnet de photographie sur lequel il consignait l'heure et la minute pr\u00e9cises \u00e0 laquelle il prenait ses photos. Il eut la confirmation que l'intervalle de temps compris entre chaque prise \u00e9tait identique \u00e0 celui affich\u00e9 entre les nombres. Apparemment, la pellicule avait d\u00e9compt\u00e9 \u00e0 l'envers le temps \u00e9coul\u00e9 dans la r\u00e9alit\u00e9. Wang Miao comprit aussit\u00f4t de quoi il s'agissait.\n\nUn compte \u00e0 rebours.\n\nLe compte \u00e0 rebours commen\u00e7ait \u00e0 1 200 heures. Il en restait maintenant 1 194. Maintenant ? Non, car la derni\u00e8re photo avait \u00e9t\u00e9 prise une heure avant. Le compte \u00e0 rebours \u00e9tait-il toujours en marche ?\n\nWang Miao sortit de la chambre noire. Il ins\u00e9ra une nouvelle bobine dans son Leica, puis prit quelques photos au hasard dans la pi\u00e8ce et depuis le balcon. Une fois la pellicule termin\u00e9e, il la sortit de l'appareil et alla la d\u00e9velopper dans la chambre noire. Les nombres r\u00e9apparurent comme des spectres sur chaque photo. La premi\u00e8re suite indiquait : 1187 : 27 : 39. Et de la suivante \u00e0 la derni\u00e8re, on retrouvait bien le m\u00eame intervalle de temps. La dur\u00e9e qui s\u00e9parait chaque suite \u00e9tait d'environ trois \u00e0 quatre secondes : 1187 : 27 : 35, 1187 : 27 : 31, 1187 : 27 : 27, 1187 : 27 : 24, c'est-\u00e0-dire la m\u00eame qu'entre les prises.\n\nLe compte \u00e0 rebours ne s'\u00e9tait pas arr\u00eat\u00e9.\n\nWang Miao ins\u00e9ra une nouvelle pellicule dans l'appareil et s'empressa de prendre des photos au hasard, y compris certaines avec le cache volontairement laiss\u00e9 sur l'appareil. Lorsqu'il eut fini de mitrailler la pi\u00e8ce, son \u00e9pouse et son fils rentr\u00e8rent \u00e0 la maison. Avant de s'empresser d'aller d\u00e9velopper les photos, Wang Miao mit une nouvelle pellicule dans son Leica, puis le tendit \u00e0 sa femme.\n\n\u2014 Tiens, finis la pellicule.\n\n\u2014 Que veux-tu que je prenne ? lui demanda-t-elle, en le regardant avec \u00e9tonnement.\n\nEn d'autres circonstances, il n'aurait jamais autoris\u00e9 quiconque \u00e0 utiliser son appareil, m\u00eame si, de toute fa\u00e7on, ni elle ni leur fils n'en avaient vraiment envie. \u00c0 leurs yeux, c'\u00e9tait une vieille antiquit\u00e9 achet\u00e9e plus de vingt mille yuans.\n\n\u2014 Peu importe, ce que tu veux.\n\nWang Miao fourra l'appareil dans la main de sa femme.\n\n\u2014 Dou-dou, viens ici que je te prenne en photo, dit-elle en pointant l'objectif vers son fils.\n\nDans l'esprit de Wang Miao apparut l'image du compte \u00e0 rebours fant\u00f4me traversant comme un n\u0153ud coulant le visage de son fils. Il ne put s'emp\u00eacher de frissonner \u00e0 cette \u00e9vocation.\n\n\u2014 Non, pas lui, n'importe quoi, mais pas lui.\n\nClic. Son \u00e9pouse prit la premi\u00e8re photo, puis elle lui lan\u00e7a :\n\n\u2014 Pourquoi \u00e7a ne marche plus quand j'appuie ?\n\nIl aida sa femme \u00e0 faire avancer la pellicule et dit :\n\n\u2014 Tu vois, comme \u00e7a. Maintenant tu peux prendre la prochaine photo.\n\nPuis il p\u00e9n\u00e9tra dans la chambre noire.\n\n\u2014 Qu'est-ce que c'est compliqu\u00e9 !\n\nSon \u00e9pouse, m\u00e9decin de son \u00e9tat, n'arrivait pas \u00e0 comprendre qu'on puisse s'amuser avec un jouet aussi ancien qui ne prenait que des photos en noir et blanc \u00e0 une \u00e9poque o\u00f9 on trouvait partout des appareils num\u00e9riques avec des dizaines de millions de m\u00e9gapixels.\n\nUne fois les pellicules d\u00e9velopp\u00e9es, il les passa sous la lumi\u00e8re rouge. Wang Miao d\u00e9couvrit que le compte \u00e0 rebours fant\u00f4me se prolongeait encore sur tous les n\u00e9gatifs. Sur toutes les photos qu'il avait prises, peu importe lesquelles et quand \u2013 y compris celles qu'il avait prises le cache ferm\u00e9 \u2013, on pouvait apercevoir distinctement : 1187 : 19 : 06, 1187 : 19 : 03 ; 1187 : 18 : 59, 1187 : 18 : 56...\n\nSa femme toqua deux fois \u00e0 la porte de la chambre noire et dit qu'elle avait termin\u00e9. Wang Miao ouvrit la porte et agrippa l'appareil. Ses mains tremblaient alors qu'il enlevait la pellicule. Il ne se pr\u00e9occupa pas du regard ahuri de son \u00e9pouse et retourna aussit\u00f4t dans la chambre noire, en fermant violemment la porte. Il agit avec pr\u00e9cipitation, jetant le r\u00e9v\u00e9lateur et le fixateur par terre. Il ne lui fallut pas longtemps pour terminer de d\u00e9velopper les images. Wang Miao ferma les yeux en priant :\n\n\u2014 Piti\u00e9, peu importe ce que c'est, pourvu que \u00e7a n'apparaisse pas. Pas maintenant, pas \u00e0 moi...\n\nIl regarda la pellicule encore humide \u00e0 travers l'agrandisseur : le compte \u00e0 rebours avait disparu. Il n'y avait sur les n\u00e9gatifs que les sc\u00e8nes d'int\u00e9rieur photographi\u00e9es par son \u00e9pouse. Elle avait utilis\u00e9 une faible vitesse d'obturation et l'opacit\u00e9 des images traduisait son amateurisme, mais Wang Miao trouva que c'\u00e9taient les photos les plus r\u00e9jouissantes qui lui aient jamais \u00e9t\u00e9 donn\u00e9es de voir.\n\nWang Miao sortit de la chambre noire en poussant un profond soupir de soulagement. Il remarqua qu'il \u00e9tait tremp\u00e9 de sueur. Son \u00e9pouse \u00e9tait all\u00e9e dans la cuisine pour pr\u00e9parer le repas et son fils \u00e9tait parti jouer dans sa chambre. Wang Miao s'affala seul sur le canap\u00e9 et commen\u00e7a \u00e0 r\u00e9fl\u00e9chir de fa\u00e7on plus rationnelle. Tout d'abord, ces nombres qui enregistraient pr\u00e9cis\u00e9ment l'\u00e9coulement du temps entre diff\u00e9rentes prises indiquaient des signes certains d'intelligence, ils n'avaient donc pas pu \u00eatre pr\u00e9imprim\u00e9s sur la pellicule. La seule possibilit\u00e9, c'\u00e9tait que quelque chose les avait projet\u00e9s par photosensibilit\u00e9. Mais quoi ? \u00c9tait-ce un d\u00e9faut de l'appareil ? Un m\u00e9canisme avait-il \u00e9t\u00e9 install\u00e9 \u00e0 son insu ? Il d\u00e9vissa l'objectif, d\u00e9monta l'appareil et examina les diff\u00e9rents composants avec l'agrandisseur. Ceux-ci n'\u00e9taient pas poussi\u00e9reux et ne semblaient rien pr\u00e9senter d'anormal. Soudain, il repensa aux sc\u00e8nes qu'il avait prises avec le cache sur l'objectif. La photosensibilit\u00e9 devait \u00eatre provoqu\u00e9e par des rayons ext\u00e9rieurs suffisamment puissants pour pouvoir traverser le cache. Or, c'\u00e9tait impossible d'un point de vue technique. D'o\u00f9 venaient les rayons et comment avaient-ils \u00e9t\u00e9 braqu\u00e9s sur l'appareil ?\n\nAu vu des technologies existantes, cette force apparaissait surnaturelle. Pour s'assurer que le compte \u00e0 rebours fant\u00f4me avait bien disparu, Wang Miao ins\u00e9ra une autre bobine dans son Leica et recommen\u00e7a \u00e0 prendre des photos au hasard, bien que plus lentement cette fois car il r\u00e9fl\u00e9chissait en m\u00eame temps. Au moment de d\u00e9velopper les photos, lui qui avait eu toutes les peines du monde \u00e0 retrouver son calme se sentit soudain une nouvelle fois pouss\u00e9 au bord du pr\u00e9cipice de la folie : le compte \u00e0 rebours \u00e9tait r\u00e9apparu et, si l'on en croyait les images, le temps ne s'\u00e9tait pas du tout arr\u00eat\u00e9. C'\u00e9tait simplement qu'il n'\u00e9tait pas apparu sur les photos prises par son \u00e9pouse.\n\n1186 : 34 : 13, 1186 : 34 : 02, 1186 : 33 : 46, 1186 : 33 : 35...\n\nWang Miao sortit pr\u00e9cipitamment de la chambre noire, puis de la maison, et frappa violemment \u00e0 la porte de son voisin. Zhang, professeur \u00e0 la retraite, lui ouvrit.\n\n\u2014 Mon vieux Zhang, as-tu un appareil photo chez toi ? Oh, pas un num\u00e9rique, un avec des pellicules !\n\n\u2014 C'est un grand photographe comme toi qui vient m'emprunter un appareil photo ? Le tien \u00e0 vingt mille yuans ne marche plus ? Je n'en ai qu'un num\u00e9rique... Tu te sens bien ? Tu as mauvaise mine.\n\n\u2014 Puis-je te l'emprunter ?\n\nLe Pr Zhang revint rapidement avec un Kodak parfaitement quelconque :\n\n\u2014 Tiens, il suffit d'effacer les photos...\n\n\u2014 Merci !\n\nWang Miao r\u00e9cup\u00e9ra l'appareil et retourna en trombe dans son appartement. Il poss\u00e9dait en r\u00e9alit\u00e9 encore trois appareils photo \u00e0 pellicules et un num\u00e9rique, mais il s'\u00e9tait dit qu'un appareil emprunt\u00e9 \u00e0 l'ext\u00e9rieur serait plus digne de confiance. Il observa les deux appareils et les quelques pellicules pos\u00e9s sur le canap\u00e9 et r\u00e9fl\u00e9chit quelques instants, avant d'ins\u00e9rer une pellicule couleur dans son pr\u00e9cieux Leica, et de tendre le num\u00e9rique \u00e0 son \u00e9pouse en train de faire la cuisine :\n\n\u2014 Vite ! Prends encore quelques photos, comme tout \u00e0 l'heure !\n\n\u2014 Mais qu'est-ce que c'est que cette histoire ? Regarde-toi... Qu'est-ce qui t'arrive ? demanda sa femme, affol\u00e9e.\n\n\u2014 Ne t'inqui\u00e8te pas, prends les photos !\n\nSon \u00e9pouse abandonna les assiettes qu'elle tenait dans les mains, puis elle s'approcha de son mari. \u00c0 l'affolement de tout \u00e0 l'heure s'ajoutait maintenant dans son regard une vive inqui\u00e9tude. Wang Miao d\u00e9tourna la t\u00eate et, abandonnant son \u00e9pouse, il pla\u00e7a le Kodak entre les mains de son fils de six ans :\n\n\u2014 Dou-dou, tu veux aider papa ? Appuie l\u00e0. Oui. Voil\u00e0. Encore une, tr\u00e8s bien. Encore, entra\u00eene-toi, prends ce qui te fait plaisir.\n\nIl ne fallut pas longtemps au gamin pour apprendre \u00e0 utiliser l'appareil, il trouva m\u00eame le jeu amusant et se mit \u00e0 prendre des photos \u00e0 toute vitesse. Wang Miao se retourna et ramassa le Leica pos\u00e9 sur le canap\u00e9 ; lui aussi recommen\u00e7a \u00e0 prendre des photos. P\u00e8re et fils mitraillaient l'appartement. Des larmes mont\u00e8rent aux yeux de son \u00e9pouse. Prise dans le feu des flashs, elle semblait compl\u00e8tement d\u00e9sempar\u00e9e.\n\n\u2014 Je sais que tu as beaucoup de stress au travail en ce moment, mais ne...\n\nWang Miao termina la pellicule du Leica avant de prendre l'appareil num\u00e9rique des mains de son fils. Il r\u00e9fl\u00e9chit un instant et, pour \u00e9viter d'\u00eatre d\u00e9rang\u00e9 par sa femme, il d\u00e9cida d'aller dans la chambre et prit encore quelques photos avec le num\u00e9rique. Il utilisa le mode viseur \u00e9lectronique et non l'\u00e9cran \u00e0 cristaux liquides. Il avait beau savoir qu'il finirait t\u00f4t ou tard par en prendre connaissance, il avait peur du r\u00e9sultat. Wang Miao emporta ensuite la bobine du Leica dans la chambre noire et s'empressa de bien refermer la porte. Apr\u00e8s avoir fini de d\u00e9velopper les photos, comme ses mains tremblaient, il dut tenir fermement l'agrandisseur. Le compte \u00e0 rebours continuait \u00e0 d\u00e9filer sur les images.\n\nWang Miao bondit hors de la chambre noire. Il commen\u00e7a par v\u00e9rifier les photos prises par l'appareil num\u00e9rique. Sur l'\u00e9cran \u00e0 cristaux liquides, on pouvait remarquer que les nombres n'apparaissaient pas sur les photos prises par son fils, tandis que sur les siennes en revanche, ils \u00e9taient bien visibles. L'\u00e9coulement du temps correspondait bien \u00e0 celui inscrit sur les n\u00e9gatifs du Leica.\n\nWang Miao s'\u00e9tait servi d'appareils diff\u00e9rents dans le but d'exclure l'hypoth\u00e8se du dysfonctionnement de l'appareil ou des bobines. Sans trop y r\u00e9fl\u00e9chir, il avait aussi demand\u00e9 \u00e0 son \u00e9pouse et \u00e0 son fils de prendre des photos, mais les r\u00e9sultats obtenus \u00e9taient encore plus \u00e9tranges : peu importe l'appareil, seules les photos prises par lui \u00e9taient marqu\u00e9es du compte \u00e0 rebours fant\u00f4me !\n\nDans un \u00e9lan de d\u00e9sespoir, Wang Miao ramassa la pile de pellicules, comme s'il s'agissait d'un nid de serpents ou d'un paquet de n\u0153uds impossibles \u00e0 d\u00e9m\u00ealer.\n\nIl savait qu'il ne parviendrait pas \u00e0 r\u00e9soudre cette \u00e9nigme en ne comptant que sur lui-m\u00eame. Il ne pouvait en parler aux coll\u00e8gues de son universit\u00e9 et de son laboratoire car, comme lui, ils raisonnaient en termes technologiques et son intuition lui disait que ce ph\u00e9nom\u00e8ne d\u00e9passait la technologie. Il pensa \u00e0 Ding Yi, mais il se dit que ce dernier \u00e9tait suffisamment d\u00e9stabilis\u00e9 par sa propre crise existentielle. Il songea enfin \u00e0 la Soci\u00e9t\u00e9 des fronti\u00e8res de la science. C'\u00e9tait un groupe de personnes avenantes et \u00e0 l'esprit ouvert. Il composa le num\u00e9ro de t\u00e9l\u00e9phone de Shen Yufei.\n\n\u2014 Docteur Shen, il m'est arriv\u00e9 quelque chose. Je dois t'en parler, l\u00e2cha h\u00e2tivement Wang Miao.\n\n\u2014 Viens, se contenta de dire Shen Yufei avant de raccrocher.\n\nWang Miao fut d\u00e9contenanc\u00e9. D'ordinaire, Shen Yufei \u00e9tait certes avare de paroles \u2013 si bien que les membres des Fronti\u00e8res de la science la taquinaient parfois en la surnommant \"Mme Hemingway\" \u2013 mais cette fois-ci, elle n'avait m\u00eame pas pris la peine de demander de quoi il retournait. Wang Miao ignorait s'il devait s'en sentir rassur\u00e9 ou encore plus inquiet.\n\nIl fourra la bobine dans une sacoche, emporta l'appareil photo num\u00e9rique et sortit de la maison sous les yeux m\u00e9dus\u00e9s de sa femme. Il aurait pu choisir de s'y rendre avec sa voiture mais, bien que la ville soit abondamment \u00e9clair\u00e9e, il ne voulait pas rouler seul et il appela un taxi.\n\nShen Yufei vivait dans un quartier hupp\u00e9 situ\u00e9 pr\u00e8s des nouveaux rails de banlieue. Ici, les lumi\u00e8res \u00e9taient bien plus t\u00e9nues qu'en centre-ville. Les villas \u00e9taient construites tout autour d'un lac artificiel o\u00f9 il \u00e9tait possible d'aller p\u00eacher. Le soir, une ambiance de village de campagne r\u00e9gnait sur le quartier. Shen Yufei \u00e9tait visiblement fortun\u00e9e, mais Wang Miao n'avait jamais vraiment su l'origine de sa richesse, car ni son ancien poste de chercheuse universitaire ni son emploi actuel au sein d'une compagnie priv\u00e9e ne lui auraient permis de gagner autant d'argent. Sa villa ne donnait toutefois pas l'impression d'un \u00e9talage de luxe. La maison servait d'espace de r\u00e9union aux Fronti\u00e8res de la science, dont les rencontres avaient lieu dans une petite biblioth\u00e8que qui faisait office de salle de conf\u00e9rences.\n\nDans le salon, Wang Miao croisa Wei Cheng, le mari de Shen Yufei. C'\u00e9tait un homme d'une quarantaine d'ann\u00e9es qui avait l'apparence d'un intellectuel sage et sinc\u00e8re. Wang Miao ne connaissait de lui que son nom, car c'\u00e9tait la seule chose qu'avait mentionn\u00e9e Shen Yufei quand elle le lui avait pr\u00e9sent\u00e9. Il n'avait apparemment pas de travail, restait \u00e0 la maison toute la journ\u00e9e et ne paraissait pas \u00e9prouver de v\u00e9ritable int\u00e9r\u00eat pour les activit\u00e9s des Fronti\u00e8res de la science. Les all\u00e9es et venues de scientifiques \u00e0 son domicile n'\u00e9taient pour lui qu'une banale routine. Mais il avait de quoi occuper ses journ\u00e9es : il semblait mener des recherches sur quelque chose et donnait l'impression d'\u00eatre toute la journ\u00e9e plong\u00e9 dans ses r\u00e9flexions. Il saluait distraitement les h\u00f4tes, puis il montait dans sa chambre \u00e0 l'\u00e9tage o\u00f9 il passait le plus clair de ses journ\u00e9es. Un jour, Wang Miao avait discr\u00e8tement jet\u00e9 un \u0153il dans la chambre dont la porte \u00e9tait entrouverte. Il y avait vu un d\u00e9cor surprenant : une station de travail HP tr\u00f4nait au milieu de la pi\u00e8ce. Il ne pouvait pas s'\u00eatre tromp\u00e9 car c'\u00e9tait le m\u00eame \u00e9quipement que celui de son centre de recherches : un bo\u00eetier gris fonc\u00e9 et un serveur RX8620 sorti quatre ans plus t\u00f4t. Il \u00e9tait pour le moins \u00e9tonnant qu'un particulier poss\u00e8de chez lui une machine d'une valeur d'un million de yuans. Que faisait Wei Cheng toute la journ\u00e9e avec cette machine ?\n\n\u2014 Yufei est occup\u00e9e \u00e0 l'\u00e9tage, merci d'attendre un moment ici, dit Wei Cheng avant de monter les escaliers.\n\nWang Miao essaya de contenir son impatience, mais il ne parvint pas \u00e0 rester en place et suivit finalement Wei Cheng \u00e0 l'\u00e9tage. Il le vit entrer dans la chambre o\u00f9 se trouvait la pr\u00e9cieuse station de travail. Wei Cheng ne parut pas contrari\u00e9 que Wang Miao le suive. Il lui indiqua la porte d'en face :\n\n\u2014 Elle est \u00e0 l'int\u00e9rieur.\n\nWang Miao frappa \u00e0 la porte, qui n'\u00e9tait pas compl\u00e8tement ferm\u00e9e. \u00c0 travers l'ouverture, il aper\u00e7ut Shen Yufei en train de jouer sur son ordinateur. Ce qui l'\u00e9tonna plus encore, c'\u00e9tait qu'elle portait une V-combinaison, la panoplie de r\u00e9alit\u00e9 virtuelle la plus \u00e0 la mode chez les joueurs de jeux vid\u00e9o. La panoplie se composait d'un casque \u00e0 vis\u00e9e panoramique et d'une combinaison haptique \u00e9quip\u00e9e de capteurs capables de reproduire les sensations du jeu : bagarres, coups de couteau, incendies... en jouant sur la temp\u00e9rature. Le costume permettait m\u00eame de simuler la sensation qu'on pouvait \u00e9prouver en plein milieu d'une temp\u00eate de pluie ou de neige. Wang Miao marcha jusqu'\u00e0 se retrouver derri\u00e8re elle mais, comme elle portait son casque, aucune image du jeu n'apparaissait sur l'ordinateur. \u00c0 cet instant, il se rappela que le commissaire Shi Qiang lui avait demand\u00e9 de noter les pages Web ou les adresses courriel des membres des Fronti\u00e8res de la science. Sans se faire remarquer, il jeta un coup d'\u0153il sur la barre URL du navigateur et n'eut aucun mal \u00e0 retenir l'adresse, \u00e0 la fois simple et intrigante.\n\nShen Yufei \u00f4ta son casque virtuel, puis retira sa combinaison. Elle chaussa sur son nez des lunettes qui paraissaient \u00e9normes sur son visage \u00e9maci\u00e9, puis, l'air indiff\u00e9rent, elle fit un signe de t\u00eate \u00e0 Wang Miao. Elle ne disait rien, elle attendait qu'il parle. Wang Miao sortit ses bobines de pellicules et commen\u00e7a le r\u00e9cit de sa myst\u00e9rieuse aventure. Shen Yufei l'\u00e9coutait avec attention. Quant aux pellicules, elle se contenta de les balayer du regard sans prendre la peine de les observer en d\u00e9tail, ce qui surprit Wang Miao et lui apporta la confirmation que Shen Yufei n'\u00e9tait pas totalement ignorante du ph\u00e9nom\u00e8ne auquel il faisait face. Il faillit couper son r\u00e9cit mais les signes de t\u00eate insistants de Shen Yufei le contraignirent \u00e0 raconter son histoire jusqu'au bout. Ce n'est qu'apr\u00e8s qu'il eut fini de parler que Shen Yufei pronon\u00e7a sa premi\u00e8re phrase de la soir\u00e9e :\n\n\u2014 O\u00f9 en est le projet sur les nanomat\u00e9riaux dont tu es responsable ?\n\nLa question d\u00e9stabilisa Wang Miao.\n\n\u2014 Notre projet sur les nanomat\u00e9riaux ? Mais quel est le rapport ? demanda-t-il en d\u00e9signant le tas de pellicules.\n\nShen Yufei ne r\u00e9pondit pas et continua \u00e0 le regarder calmement, attendant sa r\u00e9ponse. C'\u00e9tait sa mani\u00e8re \u00e0 elle de communiquer. Elle ne pronon\u00e7ait jamais un mot de trop.\n\n\u2014 Cesse tes recherches, l\u00e2cha Shen Yufei.\n\n\u2014 Comment ? (Wang Miao croyait avoir mal entendu.) Qu'as-tu dit ?\n\nShen Yufei garda le silence.\n\n\u2014 Cesser les recherches ? Mais c'est un projet national d'une importance capitale !\n\nShen Yufei demeura muette. Elle se bornait \u00e0 le fixer, le regard tranquille.\n\n\u2014 Essaie.\n\n\u2014 Mais que sais-tu \u00e0 la fin ? Dis-le-moi !\n\n\u2014 C'est tout ce que je peux te dire.\n\n\u2014 Mais je ne peux pas stopper ce projet, c'est tout simplement impossible !\n\n\u2014 Essaie.\n\nLa conversation sur le compte \u00e0 rebours fant\u00f4me s'arr\u00eata l\u00e0. Malgr\u00e9 les efforts de Wang Miao, Shen Yufei ne daigna plus rien ajouter \u00e0 ce sujet, hormis cette phrase :\n\n\u2014 Ou sinon, tes ennuis risquent d'\u00eatre encore plus graves.\n\n\u2014 Je comprends maintenant que la Soci\u00e9t\u00e9 des fronti\u00e8res de la science n'a rien d'un simple groupe de discussion autour de th\u00e9ories scientifiques fondamentales, comme vous le pr\u00e9tendez. Son rapport \u00e0 la r\u00e9alit\u00e9 est beaucoup plus complexe que ce que je croyais, dit Wang Miao.\n\n\u2014 Au contraire. Tu dois t'\u00f4ter cette id\u00e9e de l'esprit. Ce \u00e0 quoi touche la Soci\u00e9t\u00e9 des fronti\u00e8res de la science est quelque chose de bien plus fondamental que ce que tu imagines.\n\nD\u00e9pit\u00e9, Wang Miao se leva et partit sans m\u00eame un au revoir. Shen Yufei le raccompagna jusqu'\u00e0 la porte du jardin et l'observa monter dans son taxi. Au m\u00eame instant, une autre voiture d\u00e9boula \u00e0 toute allure et vint piler juste devant la porte. Un homme en sortit. Malgr\u00e9 ses lunettes de soleil, Wang Miao le reconnut d\u00e8s le premier coup d'\u0153il gr\u00e2ce \u00e0 la lueur filtrant \u00e0 travers la villa.\n\nC'\u00e9tait Pan Han, l'un des membres les plus c\u00e9l\u00e8bres de la Soci\u00e9t\u00e9 des fronti\u00e8res de la science. Biologiste de formation, il avait pr\u00e9dit avec succ\u00e8s les malformations g\u00e9n\u00e9tiques que causerait sur le long terme la consommation de produits agricoles g\u00e9n\u00e9tiquement modifi\u00e9s. De m\u00eame, il avait anticip\u00e9 les catastrophes \u00e9cologiques que provoqueraient les cultures d'OGM. Les pr\u00e9dictions de Pan Han \u00e9taient autrement plus concr\u00e8tes et d\u00e9taill\u00e9es que les proph\u00e9ties abstraites dont les oracles de la science abreuvaient d'ordinaire les m\u00e9dias. Toutes ses pr\u00e9visions s'\u00e9taient effectivement r\u00e9alis\u00e9es. Leur taux d'exactitude \u00e9tait d'ailleurs tel qu'une l\u00e9gende le disant venu du futur \u00e9tait n\u00e9e. Il devait aussi sa r\u00e9putation au fait d'avoir fond\u00e9 la premi\u00e8re communaut\u00e9 exp\u00e9rimentale en Chine. Contrairement aux tentatives de ces groupes occidentaux utopistes qui ne juraient que par \"le retour \u00e0 la nature\", la communaut\u00e9 appel\u00e9e \"Chine pastorale\" n'\u00e9tait pas bas\u00e9e en pleine nature, mais au milieu de la plus grande ville du pays. La communaut\u00e9 ne poss\u00e9dait rien. M\u00eame les aliments qu'elle consommait provenaient des d\u00e9chets de la ville. Contrairement \u00e0 ce que les gens avaient pronostiqu\u00e9, non seulement la \"Chine pastorale\" avait surv\u00e9cu, mais la communaut\u00e9 s'\u00e9tait rapidement \u00e9largie. Ses membres actifs d\u00e9passaient aujourd'hui les trois mille personnes, tandis que le nombre de ceux qui y restaient un temps plus court pour exp\u00e9rimenter ce nouveau mode de vie \u00e9tait incalculable. Forts de ces succ\u00e8s, les id\u00e9aux sociaux de Pan Han ne cess\u00e8rent de gagner du terrain. Il consid\u00e9rait que la r\u00e9volution technologique \u00e9tait une maladie des soci\u00e9t\u00e9s humaines. Il comparait la prolif\u00e9ration des technologies \u00e0 une propagation rapide de cellules canc\u00e9reuses causant l'extinction de tout nutriment organique, la destruction des organes, et en d\u00e9finitive la mort des organismes h\u00f4tes. Il plaidait pour l'\u00e9limination des \u00e9nergies dites \"agressives\" comme l'exploitation des combustibles fossiles et l'\u00e9nergie nucl\u00e9aire, qui devaient selon lui \u00eatre remplac\u00e9es par des \u00e9nergies \"douces\" comme le solaire ou la petite hydro\u00e9lectricit\u00e9. Il se pronon\u00e7ait en faveur de la dissolution progressive des villes et pour une r\u00e9partition des populations dans des villages afin de favoriser l'autosuffisance, avec comme fondement l'utilisation de technologies douces, le tout dans le but d'\u00e9tablir un nouveau type de soci\u00e9t\u00e9 agraire.\n\n\u2014 Il est \u00e0 l'int\u00e9rieur ? demanda Pan Han en pointant le premier \u00e9tage de la villa.\n\nShen Yufei ne r\u00e9pondit rien, se contentant de lui barrer la route en silence.\n\n\u2014 Je dois le pr\u00e9venir, et toi aussi. Ne nous forcez pas la main ! cria Pan Han en arrachant ses lunettes de soleil.\n\nShen Yufei ne lui r\u00e9pondit pas et se tourna vers Wang Miao qui venait de monter dans son taxi :\n\n\u2014 Tu peux partir, tout va bien.\n\nElle fit signe au chauffeur de d\u00e9marrer. Une fois le moteur mis en route, Wang Miao n'entendit plus rien de leur conversation. Apr\u00e8s quelques dizaines de m\u00e8tres, il tourna la t\u00eate et vit derri\u00e8re lui que Shen Yufei n'avait toujours pas fait entrer Pan Han dans la villa.\n\nQuand il rentra chez lui, il faisait d\u00e9j\u00e0 nuit noire. Le taxi d\u00e9posa Wang Miao \u00e0 l'entr\u00e9e de son quartier r\u00e9sidentiel. Soudain, une Santana noire vint freiner juste \u00e0 c\u00f4t\u00e9 de lui. La fen\u00eatre s'ouvrit et une bouff\u00e9e de cigarette s'en \u00e9chappa. C'\u00e9tait Shi Qiang. Sa carcasse massive \u00e9tait compress\u00e9e sur le si\u00e8ge conducteur.\n\n\u2014 \u00c7a alors ! Le Pr Wang ! L'acad\u00e9micien Wang ! Comment allez-vous depuis avant-hier ?\n\n\u2014 Vous me suiviez ? Vous n'avez rien d'autre \u00e0 faire ?\n\n\u2014 Ne le prenez pas comme \u00e7a. J'aurais tr\u00e8s bien pu m'en aller sans vous saluer. Je fais preuve de politesse et vous sortez les grands mots !\n\nShi Qiang se fendit de son inimitable sourire nigaud, et lui dit d'un ton espi\u00e8gle :\n\n\u2014 Avez-vous pu obtenir des informations l\u00e0-bas ? Avec qui \u00eates-vous entr\u00e9 en contact ?\n\n\u2014 Je vous ai d\u00e9j\u00e0 dit que je ne voulais pas avoir affaire avec vous. Cessez de me suivre !\n\n\u2014 Bien re\u00e7u..., l\u00e2cha Shi Qiang, avant de mettre le contact. Si vous croyez que \u00e7a me fait plaisir de faire des heures suppl\u00e9mentaires. Il y a un match de foot qui m'attend.\n\nQuand Wang Miao rentra chez lui, sa femme et son fils dormaient d\u00e9j\u00e0. Il pouvait entendre son \u00e9pouse remuer anxieusement dans le lit. Qui sait quels cauchemars avait fait na\u00eetre le comportement bizarre de son mari ? Wang Miao avala deux comprim\u00e9s d'anxiolytiques et s'allongea sur le lit, mais il s'\u00e9coula un long moment avant qu'il parvienne \u00e0 trouver le sommeil.\n\nSes r\u00eaves \u00e9taient tr\u00e8s d\u00e9cousus mais, comme il l'avait pr\u00e9vu, un \u00e9l\u00e9ment revenait sans cesse dans chacun d'entre eux : le compte \u00e0 rebours. Il s'attaquait au compteur qui lui apparaissait suspendu dans le ciel, il le pin\u00e7ait, le mordait, mais rien ne permettait d'en venir \u00e0 bout. Le compte \u00e0 rebours restait pendu au milieu du ciel et le temps continuait \u00e0 s'\u00e9grener. Et ce fut lorsque l'agacement de Wang Miao fut \u00e0 son comble qu'il se r\u00e9veilla. Il \u00e9carquilla les yeux et fixa le plafond, le regard encore trouble. Dehors, les lumi\u00e8res de la ville projetaient des ombres obscures sur les rideaux. Mais quelque chose l'avait suivi depuis le monde des r\u00eaves : le compte \u00e0 rebours fant\u00f4me. Celui-ci apparaissait maintenant devant ses yeux. Les nombres \u00e9taient fins mais tr\u00e8s lumineux, d'un blanc br\u00fblant.\n\n1185 : 11 : 34, 1185 : 11 : 33, 1185 : 11 : 32, 1185 : 11 : 31...\n\nWang Miao regarda autour de lui. Tout lui apparaissait flou dans la chambre. Il s'assura qu'il \u00e9tait bien r\u00e9veill\u00e9, mais le compte \u00e0 rebours \u00e9tait toujours l\u00e0. Il ferma les yeux, mais la suite de chiffres \u00e9mergeait encore dans cette totale obscurit\u00e9, comme du mercure coulant sur les plumes d'un cygne noir. Il ouvrit \u00e0 nouveau les yeux, les massa un peu, mais le compte \u00e0 rebours \u00e9tait toujours l\u00e0. Quelle que soit la direction dans laquelle il pointait son regard, la rang\u00e9e de nombres occupait imperturbablement le plein centre de son champ de vision.\n\nWang Miao s'assit brutalement. Il frissonnait de peur. Le compte \u00e0 rebours le suivait, quoi qu'il fasse. Il sauta du lit, tira les rideaux et ouvrit la fen\u00eatre. Dehors, la ville endormie brillait encore de mille feux. Le compte \u00e0 rebours s'affichait sur cette vaste toile nocturne comme les sous-titres sur un \u00e9cran de cin\u00e9ma.\n\nPendant un certain temps, Wang Miao se sentit comme \u00e9touff\u00e9 et il ne put s'emp\u00eacher de pousser un cri sourd. Devant les interrogations \u00e9pouvant\u00e9es de son \u00e9pouse \u2013 que le cri avait r\u00e9veill\u00e9e \u2013, il s'effor\u00e7a de retrouver son calme et la rassura en lui disant que tout allait bien. Puis il retourna s'allonger sur le lit, ferma les yeux, et ce fut sous la phosphorescence spectrale du compte \u00e0 rebours qu'il passa le restant de la nuit.\n\nLe matin, une fois lev\u00e9, Wang Miao lutta pour ne rien laisser para\u00eetre, mais son \u00e9pouse sentait bien qu'il n'\u00e9tait pas dans son \u00e9tat normal. Elle lui demanda si ses yeux allaient bien et s'il voyait distinctement.\n\nSon petit-d\u00e9jeuner aval\u00e9, Wang Miao demanda cong\u00e9 \u00e0 son centre de recherches et partit en voiture \u00e0 l'h\u00f4pital. Le long de la route, la barre horizontale du compte \u00e0 rebours fant\u00f4me le pr\u00e9c\u00e9dait inexorablement en s'adaptant d'elle-m\u00eame \u00e0 la luminosit\u00e9 et au d\u00e9cor. Wang Miao essaya m\u00eame de regarder fixement le lever de soleil, pour forcer le compte \u00e0 rebours \u00e0 dispara\u00eetre un moment, mais ses efforts furent vains, la suite diabolique s'afficha de fa\u00e7on effroyable sur l'astre solaire. Cette fois, elle n'eut pas besoin d'augmenter sa luminosit\u00e9, mais devint noire, comme si elle \u00e9tait projet\u00e9e sur l'orbe du soleil, ce qui la rendit encore plus mena\u00e7ante.\n\nIl \u00e9tait difficile d'obtenir un rendez-vous pour une consultation \u00e0 l'h\u00f4pital Tongren de P\u00e9kin, mais Wang Miao \u00e9tait pass\u00e9 directement par une ancienne camarade de sa femme, une ophtalmologue r\u00e9put\u00e9e. Il ne lui expliqua pas tout de suite ses sympt\u00f4mes, souhaitant d'abord que le m\u00e9decin examine ses yeux. Apr\u00e8s un examen pr\u00e9cis, celle-ci lui indiqua qu'il n'y avait aucune trace d'affection et que ses yeux fonctionnaient normalement.\n\n\u2014 J'ai sans cesse quelque chose devant les yeux. Peu importe o\u00f9 je porte mon regard, c'est toujours l\u00e0, dit Wang Miao tandis que le compte \u00e0 rebours s'affichait horizontalement sur le visage du m\u00e9decin.\n\n1175 : 11 : 34, 1175 : 11 : 33, 1175 : 11 : 32...\n\n\u2014 Le syndrome des mouches volantes, dit l'ophtalmologue, en sortant une ordonnance et en commen\u00e7ant \u00e0 \u00e9crire. C'est une maladie oculaire commune \u00e0 notre \u00e9poque. Elle est provoqu\u00e9e par l'opacit\u00e9 du cristallin, elle n'est pas facile \u00e0 traiter, mais il n'y a pas \u00e0 s'affoler, je vais vous prescrire des solutions d'iode et de la vitamine D qui pourront peut-\u00eatre les faire dispara\u00eetre. Il y a peu d'espoir, mais ce n'est pas un probl\u00e8me tr\u00e8s grave, il suffira de vous habituer \u00e0 ces impuret\u00e9s dans votre champ de vision. Elles n'ont pas d'impact sur la vue en tant que telle.\n\n\u2014 Dans le syndrome des mouches volantes dont vous parlez... \u00e0 quoi ressemblent ces impuret\u00e9s au juste ?\n\n\u2014 Il n'y a pas de r\u00e8gle, cela d\u00e9pend des personnes, souvent ce sont des petites taches noires, un peu comme des t\u00eatards.\n\n\u2014 Et lorsqu'on voit des nombres ?\n\nLe bruit du stylo du m\u00e9decin grattant sur le papier de l'ordonnance cessa :\n\n\u2014 Vous voyez des nombres ?\n\n\u2014 Oui, une rang\u00e9e horizontale de nombres que je n'arrive pas \u00e0 faire dispara\u00eetre.\n\nLe m\u00e9decin repoussa la feuille et le stylo et le regarda avec empathie :\n\n\u2014 Je l'ai su en vous voyant entrer dans le cabinet. Lors de notre dernier repas entre anciens camarades, Li Yao m'a racont\u00e9 que vous subissiez beaucoup de pression au travail. \u00c0 notre \u00e2ge, il faut faire attention \u00e0 notre sant\u00e9, nous n'avons plus vingt ans.\n\n\u2014 Vous voulez dire que c'est un probl\u00e8me d'origine psychologique ?\n\nLe m\u00e9decin hocha la t\u00eate :\n\n\u2014 Dans ce type de situation, je conseille d'ordinaire \u00e0 mes patients d'aller voir un psychologue, mais ce n'est pas indispensable, il n'y a rien de grave, c'est simplement un exc\u00e8s de fatigue. Reposez-vous quelques jours, prenez des vacances avec Li Yao et le petit \u2013 comment s'appelle-t-il d\u00e9j\u00e0, Dou-dou c'est \u00e7a ? Ne vous inqui\u00e9tez pas, vous irez bient\u00f4t mieux.\n\n1175 : 10 : 02, 1175 : 10 : 01, 1175 : 10 : 00, 1175 : 09 : 59...\n\n\u2014 Je vais vous dire ce que j'ai devant les yeux : un compte \u00e0 rebours ! Un compte \u00e0 rebours qui d\u00e9file seconde apr\u00e8s seconde, avec pr\u00e9cision ! Vous voulez dire que tout \u00e7a c'est dans ma t\u00eate ?\n\nLe m\u00e9decin lui sourit avec bienveillance :\n\n\u2014 Vous voulez savoir jusqu'\u00e0 quel degr\u00e9 des facteurs psychologiques peuvent affecter la vision ? Une jeune fille a \u00e9t\u00e9 admise dans le service le mois dernier \u2013 quel \u00e2ge avait-elle ? Quinze, seize ans ? Un jour, alors qu'elle \u00e9tait dans sa salle de classe, elle a soudain perdu la capacit\u00e9 de distinguer quoi que ce soit, elle a compl\u00e8tement perdu la vue ! Apr\u00e8s plusieurs examens, nous avons conclu que ses yeux \u00e9taient physiologiquement en bon \u00e9tat. Elle a suivi un traitement psychoth\u00e9rapeutique pendant un mois chez un psychiatre et, brusquement, ses yeux ont retrouv\u00e9 leur acuit\u00e9 visuelle !\n\nWang Miao avait conscience qu'il perdait son temps ici. Il se leva et glissa avant de partir :\n\n\u2014 D'accord, ne parlons plus de mes yeux. J'ai une derni\u00e8re question \u00e0 vous poser : savez-vous s'il peut exister une quelconque force externe qui serait capable de contr\u00f4ler \u00e0 distance notre vision ?\n\nLe m\u00e9decin r\u00e9fl\u00e9chit un instant.\n\n\u2014 Oui. Il y a quelque temps, j'ai eu l'occasion de faire partie de l'\u00e9quipe m\u00e9dicale du vaisseau spatial _Shenzhou 19_. Un jour, des astronautes ont rapport\u00e9 avoir vu des scintillements n'ayant aucune existence r\u00e9elle. Dans le pass\u00e9, des astronautes de la Station spatiale internationale ont aussi r\u00e9dig\u00e9 des rapports similaires alors qu'ils travaillaient \u00e0 l'ext\u00e9rieur du cockpit. Ce sont des \u00e9v\u00e9nements qui ont eu lieu lors d'activit\u00e9s intenses du soleil. Les particules \u00e9nerg\u00e9tiques solaires attaquent la r\u00e9tine et les astronautes ont cru voir briller quelque chose. Mais les nombres dont vous parlez \u2013 ou bien votre compte \u00e0 rebours \u2013 ne sont certainement pas li\u00e9s \u00e0 ce ph\u00e9nom\u00e8ne.\n\nWang Miao sortit de l'h\u00f4pital, troubl\u00e9, et le compte \u00e0 rebours toujours suspendu devant les yeux. Ce dernier semblait le talonner partout o\u00f9 il allait, comme un fant\u00f4me le hantant sans rel\u00e2che. Il acheta une paire de lunettes de soleil, afin d'\u00e9viter que quelqu'un ne remarque son regard perdu de somnambule.\n\nWang Miao entra dans le laboratoire principal du Centre de recherches en nanotechnologie. Il n'oublia pas d'enlever ses lunettes une fois \u00e0 l'int\u00e9rieur, mais cela n'emp\u00eacha pas les coll\u00e8gues qui le crois\u00e8rent de lui lancer des regards soucieux.\n\nAu centre de la salle d'exp\u00e9riences du laboratoire, Wang Miao vit que la cabine noire de r\u00e9action \u00e9tait toujours en activit\u00e9. Cet appareil imposant \u00e9tait constitu\u00e9 d'une sph\u00e8re reli\u00e9e \u00e0 de nombreux c\u00e2bles. Un nanomat\u00e9riau ultrar\u00e9sistant baptis\u00e9 \"poignard volant\" avait d\u00e9j\u00e0 \u00e9t\u00e9 produit. Mais cette nouvelle substance \u00e9tait tributaire d'une technique d'assemblage mol\u00e9culaire, c'est-\u00e0-dire que sa production n\u00e9cessitait l'utilisation d'une sonde permettant de faire s'agglom\u00e9rer des nanostructures mol\u00e9culaires et de les superposer brique par brique comme un mur. Ce processus de fabrication \u00e9tait tr\u00e8s gourmand en ressources, ce qui faisait du nanomat\u00e9riau produit \u00e0 l'arriv\u00e9e la pierre la plus pr\u00e9cieuse au monde. Il \u00e9tait inenvisageable d'en produire \u00e0 grande \u00e9chelle.\n\nLes chercheurs du laboratoire essayaient donc de provoquer une r\u00e9action catalys\u00e9e qui permettrait de remplacer les techniques d'assemblage mol\u00e9culaire, afin qu'une grande quantit\u00e9 de mol\u00e9cules s'assemblent d'elles-m\u00eames, comme un bloc de construction. Cette exp\u00e9rience avait lieu au c\u0153ur de la cabine noire de r\u00e9action. L'appareil pouvait simuler des exp\u00e9riences en utilisant un grand nombre de combinaisons mol\u00e9culaires diff\u00e9rentes. Si on s'\u00e9tait content\u00e9 d'avoir recours aux m\u00e9thodes artificielles traditionnelles, il aurait fallu plus d'un si\u00e8cle pour essayer ces diff\u00e9rentes combinaisons, tandis que la cabine noire de r\u00e9action permettait justement de tester ces combinaisons automatiquement et \u00e0 grande vitesse. Outre la capacit\u00e9 \u00e0 \u00e9prouver des r\u00e9actions r\u00e9elles, l'appareil permettait aussi d'effectuer des simulations analogiques. Quand une r\u00e9action atteignait un certain niveau, l'ordinateur reli\u00e9 \u00e0 la cabine utilisait les donn\u00e9es r\u00e9colt\u00e9es pour mettre en place un mod\u00e8le math\u00e9matique bas\u00e9 sur l'\u00e9tape interm\u00e9diaire de cette r\u00e9action et terminait l'exp\u00e9rience avec une simulation num\u00e9rique. Ce proc\u00e9d\u00e9 augmentait consid\u00e9rablement l'efficacit\u00e9 des exp\u00e9riences.\n\nLorsque le responsable de la branche exp\u00e9rimentale du laboratoire aper\u00e7ut Wang Miao, il accourut et commen\u00e7a \u00e0 lui faire le rapport de toute une s\u00e9rie de dysfonctionnements dans la cabine de r\u00e9action. Ces derniers temps, la sc\u00e8ne se r\u00e9p\u00e9tait quotidiennement, et ce d\u00e8s que Wang Miao arrivait au travail. Cela faisait plus d'un an que la cabine \u00e9tait en marche et des erreurs de mesure avaient impact\u00e9 la sensibilit\u00e9 des capteurs, si bien qu'il aurait fallu stopper la machine et mettre l'appareil en maintenance. Mais en tant que directeur scientifique du projet, Wang Miao insistait pour que la cabine ne soit pas arr\u00eat\u00e9e avant que la troisi\u00e8me s\u00e9rie de combinaisons synth\u00e9tiques soit termin\u00e9e. Les ing\u00e9nieurs n'avaient comme autre solution que de bricoler des appareils rectificatifs sur la machine. Mais \u00e0 pr\u00e9sent, ces appareils n\u00e9cessitaient \u00e0 leur tour d'autres appareils, ce qui avait le don d'exasp\u00e9rer l'\u00e9quipe. Toutefois le responsable prit des pincettes et n'\u00e9voqua pas l'hypoth\u00e8se d'un arr\u00eat des machines et d'une pause dans l'exp\u00e9rience, par peur que cela ne d\u00e9clenche chez Wang Miao les m\u00eames sautes d'humeur que les derni\u00e8res fois. Il exposa simplement les difficult\u00e9s rencontr\u00e9es, bien qu'on devine sans peine ce qu'il esp\u00e9rait au fond de lui.\n\nWang Miao leva la t\u00eate pour observer la cabine. Il lui trouvait l'apparence d'un ut\u00e9rus autour duquel les ing\u00e9nieurs, tels des chirurgiens, luttaient pour le maintenir en \u00e9tat de marche. Toute la sc\u00e8ne se d\u00e9roulait aur\u00e9ol\u00e9e du compte \u00e0 rebours.\n\n1174 : 21 : 11, 1174 : 21 : 10, 1174 : 21 : 09, 1174 : 21 : 08...\n\n _Cesse le projet. Essaie._ Wang Miao repensa tout \u00e0 coup aux paroles de Shen Yufei.\n\n\u2014 Combien de temps serait n\u00e9cessaire pour une mise \u00e0 jour compl\u00e8te des capteurs ? demanda-t-il.\n\n\u2014 Quatre \u00e0 cinq jours, probablement.\n\nLe responsable per\u00e7ut soudain comme une pointe d'espoir et s'empressa de rectifier :\n\n\u2014 Mais nous pouvons faire \u00e7a plus rapidement, trois jours ! Je m'y engage, professeur Wang !\n\n _Je n'ai pas c\u00e9d\u00e9_ , se dit Wang Miao int\u00e9rieurement. _La r\u00e9paration du mat\u00e9riel est r\u00e9ellement n\u00e9cessaire et pour cela, les exp\u00e9riences doivent \u00eatre suspendues. C'est tout._\n\nIl se tourna vers le responsable, qu'il regarda \u00e0 travers le compte \u00e0 rebours, et d\u00e9clara :\n\n\u2014 Arr\u00eatez l'exp\u00e9rience. Mettez la machine en maintenance, et suivez le calendrier que vous m'avez donn\u00e9.\n\n\u2014 Bien, professeur Wang. Je vous donnerai tr\u00e8s rapidement le programme complet de la mise \u00e0 jour. Nous pouvons arr\u00eater la machine cet apr\u00e8s-midi, r\u00e9pondit le responsable avec excitation.\n\n\u2014 Vous pouvez l'arr\u00eater d\u00e8s maintenant.\n\nLe responsable du laboratoire regarda Wang Miao d'un air incr\u00e9dule, mais il retrouva tr\u00e8s vite son enthousiasme, comme s'il craignait de laisser filer une occasion inesp\u00e9r\u00e9e. Il se saisit de son t\u00e9l\u00e9phone et donna l'ordre de stopper la machine. Les ing\u00e9nieurs et les chercheurs du groupe de travail, ext\u00e9nu\u00e9s, en furent naturellement ravis et commenc\u00e8rent aussit\u00f4t \u00e0 presser la centaine d'interrupteurs de l'appareil. Les \u00e9crans de contr\u00f4le s'\u00e9teignirent les uns apr\u00e8s les autres jusqu'\u00e0 ce qu'enfin le signal de l'arr\u00eat de la machine apparaisse sur l'\u00e9cran du moniteur principal.\n\nPresque simultan\u00e9ment, les chiffres qui d\u00e9filaient devant les yeux de Wang Miao se fig\u00e8rent. Le compte \u00e0 rebours affichait d\u00e9sormais le nombre 1174 : 10 : 07. Une poign\u00e9e de secondes plus tard, le compte \u00e0 rebours disparut apr\u00e8s quelques derniers clignotements.\n\nDe retour dans une r\u00e9alit\u00e9 d\u00e9barrass\u00e9e de nombres fant\u00f4mes, Wang Miao inspira profond\u00e9ment, comme si on l'avait tir\u00e9 de la noyade. Il s'assit, lessiv\u00e9, mais il r\u00e9alisa qu'on le regardait.\n\nIl dit au responsable :\n\n\u2014 La mise \u00e0 jour du syst\u00e8me est du ressort de l'unit\u00e9 d'\u00e9quipement. Vous autres membres de la branche exp\u00e9rimentale, prenez quelques jours de repos. Vous les avez bien m\u00e9rit\u00e9s.\n\n\u2014 Professeur Wang, vous aussi, vous \u00eates fatigu\u00e9. L'ing\u00e9nieur en chef Zhang va s'occuper de la supervision de la mise \u00e0 jour. Rentrez vous reposer.\n\n\u2014 Oui, fatigu\u00e9..., murmura faiblement Wang Miao.\n\nUne fois le responsable parti, il prit son t\u00e9l\u00e9phone et composa le num\u00e9ro de Shen Yufei. Celle-ci d\u00e9crocha apr\u00e8s une seule sonnerie :\n\n\u2014 Qui est derri\u00e8re tout \u00e7a ? demanda Wang Miao, en s'effor\u00e7ant en vain de rester calme.\n\nSilence.\n\n\u2014 Qu'y a-t-il \u00e0 la fin du compte \u00e0 rebours ?\n\nSilence.\n\n\u2014 Tu m'\u00e9coutes ?\n\n\u2014 Oui.\n\n\u2014 Pourquoi les nanomat\u00e9riaux ? Ce n'est pas un acc\u00e9l\u00e9rateur de particules, ce n'est que de la recherche appliqu\u00e9e ! Cela vaut-il la peine de vous en pr\u00e9occuper autant ?\n\n\u2014 Il ne nous appartient pas d'en juger.\n\n\u2014 La science moderne est une affaire collective. Pourquoi ne s'en prendre qu'\u00e0 moi ?\n\n\u2014 Les perc\u00e9es th\u00e9oriques les plus radicales ont avant tout \u00e9t\u00e9 accomplies par des individus. Nous attendons aussi de toi que tu uses de ton prestige et de ton influence pour induire cette \u00e9tude en erreur, pour l'amener dans une impasse.\n\n\u2014 Vous voulez rire ? Vous r\u00eavez !\n\n\u2014 Tu n'es pas en position de d\u00e9cider. Si tu crois \u00e0 l'existence de ce pouvoir, tu ne pourras pas continuer \u00e0 mener des r\u00e9flexions scientifiques.\n\n\u2014 Mais je n'y crois pas un seul instant ! beugla Wang Miao, dont la peur et le d\u00e9sespoir s'\u00e9taient soudain chang\u00e9s en rage folle. Vous croyez que votre petit tour de magie aura raison de moi ? Que vous pourrez stopper le progr\u00e8s de la technologie ? J'admets \u00eatre dans l'incapacit\u00e9 d'expliquer comment vous avez fait pour le moment, mais c'est parce que je n'ai pas encore trouv\u00e9 qui \u00e9tait l'odieux magicien aux commandes.\n\n\u2014 Comment pourrait-on te persuader ?\n\n\u2014 Peut-\"il\" faire \u00e9talage de son pouvoir \u00e0 une plus grande \u00e9chelle ?\n\n\u2014 Quelle \u00e9chelle ach\u00e8verait de te convaincre ?\n\nLes paroles de Shen Yufei laiss\u00e8rent Wang Miao abasourdi. Il ne s'\u00e9tait pas pr\u00e9par\u00e9 \u00e0 cette r\u00e9plique. Il se for\u00e7a \u00e0 garder son sang-froid pour ne pas tomber t\u00eate baiss\u00e9e dans le pi\u00e8ge :\n\n\u2014 Remballez votre malle \u00e0 magie. Grande \u00e9chelle ou non, cela reste des illusions. Vous pouvez tout \u00e0 fait projeter un hologramme dans le ciel, comme l'a fait l'Otan il y a deux ans, lors de la derni\u00e8re guerre. Il existe m\u00eame des lasers suffisamment puissants pour projeter une image sur toute la surface de la lune. Le sniper et le fermier peuvent jouer \u00e0 une \u00e9chelle inatteignable par les humains. Par exemple, le compte \u00e0 rebours peut-il appara\u00eetre sur la surface du soleil ?\n\nIl garda la bouche ouverte. \u00c0 sa propre surprise, il avait inconsciemment \u00e9voqu\u00e9 les deux hypoth\u00e8ses qui auraient d\u00fb rester taboues dans cette situation. Heureusement pour lui, il n'avait pas pouss\u00e9 plus loin cette r\u00e9flexion. Afin de reprendre l'initiative, il encha\u00eena :\n\n\u2014 Je viens de penser \u00e0 une option \u00e0 laquelle je n'avais pas encore song\u00e9. Qui sait, votre fameux magicien serait peut-\u00eatre capable d'atteindre l'\u00e9chelle du soleil. Pour que votre d\u00e9monstration soit vraiment convaincante, il faudrait pouvoir atteindre une \u00e9chelle plus grande encore.\n\n\u2014 La question est : en sortiras-tu indemne ? Nous sommes amis, je veux t'aider. Ne suis pas le destin de Yang Dong.\n\nEn entendant ce nom, Wang Miao ne put s'emp\u00eacher de frissonner mais, tr\u00e8s vite, sa col\u00e8re reprit le dessus :\n\n\u2014 Est-\"il\" pr\u00eat \u00e0 relever ce d\u00e9fi ?\n\n\u2014 Oui.\n\n\u2014 Que vas-tu faire ?\n\nLa voix de Wang Miao s'\u00e9tait affaiblie.\n\n\u2014 As-tu \u00e0 c\u00f4t\u00e9 de toi un ordinateur reli\u00e9 \u00e0 Internet ? Bien, entre l'adresse suivante : http:\/\/www.qsl.net\/bg3tt\/zl\/mesdm. htm. C'est fait ? Imprime la page et garde-la \u00e0 port\u00e9e de main.\n\nWang Miao vit qu'il n'y avait rien d'autre sur la page que l'alphabet morse.\n\n\u2014 Je ne comprends pas, c'est...\n\n\u2014 Dans les deux jours \u00e0 venir, essaie de trouver un endroit o\u00f9 tu pourras observer le fond diffus cosmologique. Pour plus de d\u00e9tails, tu consulteras le mail que je vais t'envoyer.\n\n\u2014 Ce... pourquoi ?\n\n\u2014 Je sais que ton projet de recherche en nanotechnologie a \u00e9t\u00e9 temporairement arr\u00eat\u00e9. As-tu l'intention de le red\u00e9marrer ?\n\n\u2014 Bien s\u00fbr, dans trois jours.\n\n\u2014 Dans ce cas, le compte \u00e0 rebours va recommencer.\n\n\u2014 \u00c0 quelle \u00e9chelle le verrai-je ?\n\nSa question fut suivie d'un long silence. Cette femme qui agissait comme le porte-parole d'une force d\u00e9passant tout entendement humain venait de lui fermer froidement toute issue de secours.\n\n\u2014 Dans trois jours, c'est-\u00e0-dire le 14, de 1 heure \u00e0 5heures du matin, c'est l'univers entier qui clignotera pour toi.\n\n## 7\n\n## _LES TROIS CORPS_ :\n\n## LE ROI WEN ET LA LONGUE NUIT\n\nWang Miao composa le num\u00e9ro de t\u00e9l\u00e9phone de Ding Yi. Ce ne fut que lorsque son interlocuteur d\u00e9crocha le combin\u00e9 qu'il r\u00e9alisa qu'il \u00e9tait d\u00e9j\u00e0 plus de 1 heure du matin.\n\n\u2014 C'est Wang Miao. Pardon de te d\u00e9ranger si tard.\n\n\u2014 \u00c7a ne fait rien, je n'arrivais pas \u00e0 dormir.\n\n\u2014 Je... il m'est arriv\u00e9 quelque chose, j'ai besoin de ton aide. Sais-tu s'il y a des lieux en Chine o\u00f9 il est possible d'observer le fond diffus cosmologique ?\n\nLe d\u00e9sir de raconter son histoire s'empara de Wang Miao, mais il le refr\u00e9na aussit\u00f4t en se disant qu'il valait mieux que peu de gens soient au fait de sa m\u00e9saventure.\n\n\u2014 Le fond diffus cosmologique ? D'o\u00f9 te vient cet int\u00e9r\u00eat soudain ? Effectivement, il a d\u00fb t'arriver quelque chose... Es-tu all\u00e9 rendre visite \u00e0 la m\u00e8re de Yang Dong ?\n\n\u2014 Ah... pardon, j'ai oubli\u00e9.\n\n\u2014 Ce n'est pas grave. De nos jours, dans le monde scientifique... il arrive des histoires \u00e0 beaucoup de gens... Tout le monde a l'esprit ailleurs. Mais je crois que tu devrais tout de m\u00eame aller la voir. Elle est \u00e2g\u00e9e maintenant et refuse d'engager une auxiliaire de vie. Si tu vois qu'elle a besoin d'un coup de main, aide-la, s'il te pla\u00eet. Oh, et \u00e0 propos du fond cosmologique, tu peux justement en parler \u00e0 la m\u00e8re de Yang Dong. Avant de prendre sa retraite, elle \u00e9tait professeur d'astrophysique. Elle conna\u00eet bien les observatoires astronomiques du pays.\n\n\u2014 Bien, bien, j'irai aujourd'hui, apr\u00e8s le travail.\n\n\u2014 Je te remercie par avance, je n'ai vraiment pas le courage de me retrouver face \u00e0 tout ce qui peut me rappeler Yang Dong.\n\nApr\u00e8s avoir raccroch\u00e9, Wang Miao s'assit devant son ordinateur et imprima le sobre alphabet morse. Il avait \u00e0 pr\u00e9sent repris son calme et r\u00e9ussi \u00e0 \u00e9carter de ses pens\u00e9es le mal\u00e9fique compte \u00e0 rebours. Il r\u00e9fl\u00e9chissait maintenant \u00e0 la Soci\u00e9t\u00e9 des fronti\u00e8res de la science, \u00e0 Shen Yufei et au jeu en ligne auquel il l'avait surprise en train de s'adonner. La seule chose qu'il pouvait affirmer au sujet de Shen Yufei, c'est qu'elle n'\u00e9tait pas une fan de jeux vid\u00e9o. Ses paroles \u00e9taient aussi condens\u00e9es et froides que des t\u00e9l\u00e9grammes, mais la froideur qu'elle d\u00e9gageait \u00e9tait diff\u00e9rente de celle de certaines femmes : ce n'\u00e9tait pas un masque, mais une rigidit\u00e9 qui recouvrait tout son \u00eatre.\n\nWang Miao avait auparavant fait une \u00e9trange analogie entre elle et DOS, ce syst\u00e8me d'exploitation informatique maintenant devenu obsol\u00e8te : un \u00e9cran noir, vide, avec une barre clignotante \"C : \\>\" toute simple. Vous saisissiez une formule et vous obteniez le r\u00e9sultat voulu. Jamais un mot de trop, jamais de changement de fond d'\u00e9cran. Mais d\u00e9sormais, il savait que derri\u00e8re le signal \"C : \\>\" se dissimulait un abysse sans fond. Se passionnait-elle vraiment pour les jeux en ligne au point d'acheter une combinaison virtuelle ? Elle n'avait pas d'enfants, elle avait forc\u00e9ment achet\u00e9 la V-combinaison pour son usage personnel. Wang Miao avait peine \u00e0 le croire.\n\nIl saisit l'adresse dans la barre de recherche : www.3body. net. Il n'avait eu aucun mal \u00e0 la m\u00e9moriser. Le site indiquait que le jeu n\u00e9cessitait l'utilisation d'une V-combinaison. Wang Miao se rappela qu'il y avait un accessoire de ce genre dans la salle de repos du centre de recherches. Il sortit du laboratoire d\u00e9sormais d\u00e9sert et se rendit dans le bureau du service de s\u00e9curit\u00e9 pour r\u00e9cup\u00e9rer la clef de la salle.\n\nIl longea les tables de billard et les appareils de fitness et trouva la V-combinaison \u00e0 c\u00f4t\u00e9 d'un ordinateur. Il parvint non sans mal \u00e0 enfiler la combinaison haptique, pla\u00e7a le casque \u00e0 vis\u00e9e panoramique sur son cr\u00e2ne et alluma l'ordinateur.\n\nUne fois le jeu lanc\u00e9, Wang Miao se retrouva au beau milieu d'une plaine d\u00e9sol\u00e9e baign\u00e9e par la clart\u00e9 de l'aube. Le sol de la plaine \u00e9tait d'un brun fonc\u00e9, et on peinait \u00e0 en distinguer les d\u00e9tails. Un petit \u00e9clat de lueur blanche pointait \u00e0 l'horizon, tandis qu'un ciel encore \u00e9toil\u00e9 composait le reste du paysage. Une explosion sourde retentit. Au loin, deux montagnes rutilantes s'effondr\u00e8rent sur le sol. Une lumi\u00e8re rouge\u00e2tre impr\u00e9gna le paysage. Lorsque les nuages de poussi\u00e8re se dispers\u00e8rent enfin, Wang Miao vit \u00e9clore entre terre et ciel les mots : _LES TROIS CORPS_.\n\nUne interface d'inscription apparut ensuite sur l'\u00e9cran. Wang Miao se choisit comme identifiant \"Hairen1\" et se connecta au jeu.\n\nIl se retrouva sur la m\u00eame plaine d\u00e9serte, mais il remarqua que les capteurs \u00e0 l'int\u00e9rieur de sa V-combinaison s'\u00e9taient mis en marche : il pouvait sentir un courant d'air glacial souffler sur son corps. Dans le paysage \u00e9merg\u00e8rent deux personnages en train de marcher qui, sous l'\u00e9clat de l'aube, lui apparaissaient comme deux silhouettes noires. Wang Miao d\u00e9pla\u00e7a sa souris pour les suivre. Il d\u00e9couvrit que les deux personnages \u00e9taient des hommes, v\u00eatus de longues robes en haillons, elles-m\u00eames envelopp\u00e9es dans des fourrures sales. Tous deux poss\u00e9daient une courte mais large \u00e9p\u00e9e de bronze. L'un d'eux portait sur le dos une longue malle en bois qui faisait la moiti\u00e9 de sa taille. Ce dernier se retourna et observa Wang Miao. Son visage paraissait aussi sale et rid\u00e9 que la fourrure qu'il portait sur le dos, mais ses yeux \u00e9taient vifs et refl\u00e9taient la lueur de l'aurore.\n\n\u2014 Il fait froid, dit-il.\n\n\u2014 Oui, vraiment froid, fit Wang Miao, en \u00e9cho.\n\n\u2014 Nous sommes \u00e0 l'\u00e9poque des Royaumes combattants2, je suis le roi Wen des Zhou3, dit l'homme qui portait la malle sur le dos.\n\n\u2014 Il me semble pourtant que le roi Wen n'est pas un personnage de la p\u00e9riode des Royaumes combattants, si ?\n\n\u2014 Il a surv\u00e9cu jusqu'\u00e0 aujourd'hui. Le roi Zhou des Shang est encore en vie, lui aussi, pr\u00e9cisa l'autre individu. Je suis le laquais du roi Wen. Mon identifiant est \"Laquais du roi Wen\". Le roi est un v\u00e9ritable g\u00e9nie.\n\n\u2014 Je suis Hairen. Que portez-vous sur le dos ? demanda Wang Miao.\n\nLe roi Wen posa \u00e0 terre sa longue malle rectangulaire et l'ouvrit verticalement, comme s'il s'agissait d'une porte, r\u00e9v\u00e9lant une structure compos\u00e9e de cinq compartiments. Gr\u00e2ce \u00e0 la lumi\u00e8re de l'aube, Wang Miao vit \u00e0 l'int\u00e9rieur que chaque \u00e9tage \u00e9tait couvert d'un petit tas de sable. Un filet de sable s'\u00e9coulait sur chaque compartiment \u00e0 travers un trou.\n\n\u2014 C'est un sablier. Le sable met en tout huit heures pour descendre jusqu'\u00e0 la base. Il faut le retourner trois fois pour faire une journ\u00e9e. Mais j'oublie souvent de le faire, et j'ai besoin de mon laquais pour me le rappeler, expliqua le roi Wen.\n\n\u2014 Vous avez apparemment un long voyage \u00e0 faire. Pourquoi vous encombrer d'une machine aussi lourde ?\n\n\u2014 Mais comment ferions-nous pour savoir l'heure ?\n\n\u2014 Un petit cadran solaire serait autrement plus commode. Ou alors il suffit de regarder la position du soleil pour avoir une id\u00e9e approximative de l'heure qu'il est.\n\nLe roi et son laquais s'interrog\u00e8rent du regard puis leurs visages se tourn\u00e8rent simultan\u00e9ment vers Wang Miao, comme s'il \u00e9tait idiot :\n\n\u2014 Le soleil ? Comment le soleil pourrait-il nous donner l'heure ? Nous sommes en pleine \u00e8re chaotique !\n\nAlors que Wang Miao s'appr\u00eatait \u00e0 les questionner sur ce curieux concept, le laquais se lamenta :\n\n\u2014 Qu'il fait froid ! Je crois que je vais y passer !\n\nWang Miao aussi trouvait qu'il faisait froid, mais il ne pouvait \u00f4ter sa combinaison car, dans la majorit\u00e9 des jeux, cela avait pour effet de vous d\u00e9connecter imm\u00e9diatement.\n\n\u2014 Il fera plus doux quand le soleil sera sorti, leur dit-il.\n\n\u2014 Vous vous prenez pour un proph\u00e8te ? M\u00eame le roi Wen n'est pas en mesure de faire de telles pr\u00e9dictions ! lui lan\u00e7a le laquais, indign\u00e9.\n\n\u2014 Pas besoin d'\u00eatre proph\u00e8te pour pr\u00e9dire que le soleil se l\u00e8vera dans une heure ou deux, dit Wang Miao en d\u00e9signant l'horizon.\n\n\u2014 C'est une \u00e8re chaotique ! reprit le laquais.\n\n\u2014 Qu'est-ce qu'une \u00e8re chaotique ?\n\n\u2014 Toutes les \u00e8res qui ne sont pas des \u00e8res r\u00e9guli\u00e8res, r\u00e9pondit le roi Wen, comme s'il parlait \u00e0 un enfant ignorant.\n\nEt effectivement, la lumi\u00e8re de l'aube commen\u00e7a \u00e0 s'obscurcir \u00e0 l'horizon, jusqu'\u00e0 bient\u00f4t dispara\u00eetre totalement. La plaine fut \u00e0 nouveau envelopp\u00e9e par le voile de la nuit brod\u00e9 d'\u00e9toiles scintillantes.\n\n\u2014 Nous \u00e9tions donc l'apr\u00e8s-midi et non le matin ? demanda Wang Miao.\n\n\u2014 C'\u00e9tait bien l'aube mais, le matin, le soleil ne se l\u00e8ve pas forc\u00e9ment. C'est l'\u00e8re chaotique.\n\nWang Miao trouvait le froid de plus en plus insupportable.\n\n\u2014 Il semble bien que le soleil ne revienne pas de sit\u00f4t, dit-il en frissonnant et en pointant la ligne d\u00e9sormais floue de l'horizon.\n\n\u2014 Voil\u00e0 que \u00e7a vous reprend ! Rien n'est certain, c'est l'\u00e8re chaotique, reprit le laquais, puis il se tourna vers le roi Wen : Seigneur Ji Chang, puis-je manger du poisson s\u00e9ch\u00e9 ?\n\n\u2014 Il n'en est pas question ! Je dois pouvoir manger \u00e0 ma faim. Il faut veiller \u00e0 ce que j'arrive sain et sauf \u00e0 la capitale Zhaoge4, quitte \u00e0 te sacrifier.\n\nAlors qu'ils parlaient, Wang Miao remarqua qu'un rayon de lumi\u00e8re avait point \u00e0 un autre endroit de la ligne de l'horizon. Il n'arrivait pas \u00e0 savoir quel \u00e9tait le point cardinal, mais il \u00e9tait s\u00fbr d'une chose : ce n'\u00e9tait pas celui par lequel l'aube venait de s'\u00e9clipser. Le ciel s'\u00e9claircit rapidement et, bient\u00f4t, le soleil se leva sur le monde. C'\u00e9tait un petit astre bleut\u00e9 qui ressemblait davantage \u00e0 une lune particuli\u00e8rement brillante, mais Wang Miao sentit tout de m\u00eame une pointe de chaleur, et put avoir une vision plus d\u00e9taill\u00e9e de la vaste plaine qui s'\u00e9tendait devant lui. Cependant cette vision fut fugace, car le soleil dessina un arc sur la ligne haute de l'horizon avant de dispara\u00eetre \u00e0 nouveau. La nuit et le froid drap\u00e8rent encore une fois la plaine.\n\nTous trois s'arr\u00eat\u00e8rent un moment sous un arbre mort. Le roi Wen et son laquais sortirent leurs \u00e9p\u00e9es de bronze et firent du petit bois, tandis que Wang Miao rassemblait les fagots. Le laquais sortit son briquet _huolian_5 et apr\u00e8s l'avoir frapp\u00e9 plusieurs fois, il parvint \u00e0 faire partir un feu. Wang Miao sentit que le bas de sa combinaison commen\u00e7ait \u00e0 se r\u00e9chauffer, mais son dos \u00e9tait toujours aussi glac\u00e9.\n\n\u2014 Nous devrions br\u00fbler quelques corps d\u00e9shydrat\u00e9s, le feu prendrait mieux, dit le laquais.\n\n\u2014 Tais-toi donc ! C'est bon pour un tyran comme le roi Zhou, ces choses-l\u00e0 !\n\n\u2014 De toute fa\u00e7on, les corps que nous avons vus \u00e9parpill\u00e9s le long de la route sont en trop mauvais \u00e9tat pour avoir un jour la chance d'\u00eatre r\u00e9hydrat\u00e9s. Si votre th\u00e9orie est juste, nous pourrions non seulement nous en servir pour faire du feu, mais \u00e9galement en manger quelques-uns. Que valent quelques vies en regard de l'importance de votre th\u00e9orie ?\n\n\u2014 Cesse de dire des balivernes ! Nous sommes des lettr\u00e9s !\n\nUne fois le bois enti\u00e8rement consum\u00e9, les trois compagnons reprirent la route. Comme ils parlaient peu entre eux, le syst\u00e8me acc\u00e9l\u00e9ra automatiquement la vitesse de d\u00e9roulement du jeu. Le roi Wen retourna \u00e0 six reprises le sablier qu'il portait dans le dos, et ce furent donc deux journ\u00e9es enti\u00e8res qui s'\u00e9coul\u00e8rent en un clin d'\u0153il. Pourtant, le soleil ne s'\u00e9tait pas lev\u00e9 une seule fois, Wang Miao n'avait pas vu le moindre soup\u00e7on de lumi\u00e8re \u00e0 l'horizon.\n\n\u2014 On dirait que le soleil ne se l\u00e8vera plus jamais, fit remarquer Wang Miao, tout en faisant appara\u00eetre le menu du jeu pour v\u00e9rifier o\u00f9 en \u00e9tait sa barre de vie. Celle-ci ne cessait de diminuer \u00e0 cause du froid.\n\n\u2014 Voil\u00e0 que tu te prends encore pour un grand proph\u00e8te..., dit le laquais, avant que Wang Miao et lui ne finissent en ch\u0153ur : C'est l'\u00e8re chaotique !\n\nPourtant, peu de temps apr\u00e8s avoir prononc\u00e9 ces mots, l'aurore pointa r\u00e9ellement \u00e0 l'horizon, le ciel s'\u00e9claircit rapidement et, bient\u00f4t, le soleil se leva. Wang Miao remarqua que l'astre cette fois \u00e9tait gigantesque. Seule la moiti\u00e9 \u00e9mergeait de derri\u00e8re l'horizon, et d\u00e9j\u00e0 son diam\u00e8tre occupait un cinqui\u00e8me du ciel. Wang Miao fut saisi par une vague de chaleur qui raviva et revigora son \u00e2me. Mais en tournant son regard vers le roi Wen et le laquais, il s'aper\u00e7ut qu'une expression d'horreur s'\u00e9tait dessin\u00e9e sur leurs visages, comme si le diable lui-m\u00eame s'appr\u00eatait \u00e0 sortir de terre.\n\n\u2014 Vite ! Trouvons un endroit ombrag\u00e9 ! hurla le laquais.\n\nWang Miao les suivit en courant jusqu'\u00e0 un rocher derri\u00e8re lequel ils s'accroupirent. L'ombre du rocher faiblissait petit \u00e0 petit. Le sol autour d'eux paraissait incandescent. Sous leurs pieds, le perg\u00e9lisol fondait \u00e0 toute allure. Cette surface \u00e0 l'origine aussi dure que le m\u00e9tal devenait maintenant un parterre de boue bouillonnant. Wang Miao transpirait \u00e0 grosses gouttes. Lorsque le soleil atteignit le z\u00e9nith, les trois hommes se couvrirent la t\u00eate avec leurs peaux de b\u00eates, mais les puissants rayons de lumi\u00e8re transper\u00e7aient comme des fl\u00e8ches les fentes et les trous des fourrures. Les trois compagnons se tra\u00een\u00e8rent jusque sous l'ombre qui venait de se former de l'autre c\u00f4t\u00e9 du rocher...\n\nLorsque le soleil se coucha, l'air \u00e9tait toujours empreint d'une chaleur moite et \u00e9touffante. Les trois hommes, encore tremp\u00e9s de sueur, s'assirent sur le rocher. Le laquais l\u00e2cha avec consternation :\n\n\u2014 C'est un v\u00e9ritable enfer de voyager pendant l'\u00e8re chaotique. Je ne peux plus le supporter ! Sans compter que je n'ai rien \u00e0 manger, car vous refusez de me donner du poisson s\u00e9ch\u00e9 et vous ne me laissez pas me nourrir de d\u00e9shydrat\u00e9s, alors...\n\n\u2014 Alors il ne te reste plus qu'\u00e0 te d\u00e9shydrater, l'interrompit le roi Wen en agitant sa peau de b\u00eate pour se faire un peu de vent.\n\n\u2014 Vous ne m'abandonnerez pas derri\u00e8re vous ?\n\n\u2014 Bien s\u00fbr que non, j'ai promis de t'emmener \u00e0 Zhaoge.\n\nLe laquais se d\u00e9fit de sa longue robe tremp\u00e9e de sueur, puis s'allongea nu sur le sable. Sous les derni\u00e8res lueurs du cr\u00e9puscule, Wang Miao vit soudain de la sueur suinter abondamment sur le corps du laquais. Il comprit vite que ce n'\u00e9tait pas de la transpiration mais toute l'eau de son corps qui s'\u00e9chappait. L'eau convergea en rigoles sur le sable, puis le corps du laquais se ratatina, comme une bougie fondue. Au bout de dix minutes, le corps \u00e9tait compl\u00e8tement ass\u00e9ch\u00e9, il ne restait plus qu'une peau flasque \u00e9tendue sur le sol et sur laquelle il \u00e9tait d\u00e9sormais difficile de distinguer les traits de son visage.\n\n\u2014 Est-il mort ? demanda Wang Miao.\n\nIl se rappela qu'il avait vu de nombreuses peaux \u00e0 formes humaines sur la route, dont certaines d\u00e9chiquet\u00e9es et incompl\u00e8tes. Il supposa que c'\u00e9taient les d\u00e9shydrat\u00e9s que le laquais voulait utiliser pour alimenter le feu.\n\n\u2014 Non, r\u00e9pondit le roi Wen.\n\nPuis il ramassa la peau flasque du laquais, la tapota un peu pour faire tomber le sable et l'enroula sur le rocher, comme on aurait enroul\u00e9 un ballon d\u00e9gonfl\u00e9.\n\n\u2014 Nous le mettrons \u00e0 tremper un moment dans l'eau et il retrouvera son aspect vivant initial. Un peu comme quand on fait tremper des champignons s\u00e9ch\u00e9s.\n\n\u2014 Ses os aussi ont ramolli ?\n\n\u2014 Oui. Ils sont devenus des fibres s\u00e8ches. C'est nettement plus pratique pour le transport.\n\n\u2014 Ici, tout le monde peut se d\u00e9shydrater de cette fa\u00e7on ?\n\n\u2014 Bien entendu. Toi aussi. Autrement, personne ne survivrait aux \u00e8res chaotiques.\n\nLe roi Wen tendit le rouleau de peau \u00e0 Wang Miao.\n\n\u2014 Prends-le. Si on l'abandonnait sur le bord du chemin, il serait br\u00fbl\u00e9 ou bien mang\u00e9.\n\nWang Miao r\u00e9cup\u00e9ra la peau du laquais, un petit rouleau tr\u00e8s l\u00e9ger. Il l'attacha autour de son avant-bras sans \u00e9prouver aucune sensation d'\u00e9tranget\u00e9.\n\nWang Miao, le laquais d\u00e9shydrat\u00e9 au bras, et le roi Wen avec son sablier sur le dos poursuivirent leur \u00e9pineux p\u00e9riple. Comme les jours pr\u00e9c\u00e9dents, le soleil de ce monde continua \u00e0 n'en faire qu'\u00e0 sa t\u00eate : une journ\u00e9e torride pouvait soudain succ\u00e9der \u00e0 une longue nuit glaciale, ou bien le contraire. Le roi Wen et Wang Miao comptaient l'un sur l'autre pour survivre. Ils allumaient des feux pour se r\u00e9chauffer et plongeaient dans des lacs pour \u00e9chapper \u00e0 la canicule. Fort heureusement, il \u00e9tait possible d'acc\u00e9l\u00e9rer le temps du jeu. Un mois entier passa en une demi-heure, ce qui rendit supportable cette odyss\u00e9e \u00e0 travers l'\u00e8re chaotique.\n\nUn jour, apr\u00e8s une interminable nuit \u2013 qui, \u00e0 en croire le sablier, dura une semaine enti\u00e8re \u2013, le roi Wen pointa le doigt vers le ciel et s'\u00e9cria gaiement :\n\n\u2014 Des \u00e9toiles volantes ! Deux \u00e9toiles volantes !\n\nWang Miao avait eu l'occasion de remarquer ces \u00e9tranges corps c\u00e9lestes, plus grands que des \u00e9toiles et se pr\u00e9sentant sous la forme de disques de la taille d'une balle de ping-pong. Ils se d\u00e9pla\u00e7aient dans la vo\u00fbte \u00e0 une vitesse telle qu'on pouvait d\u00e9tecter leur mouvement \u00e0 l'\u0153il nu. Mais c'\u00e9tait la premi\u00e8re fois qu'il en voyait deux en m\u00eame temps.\n\n\u2014 L'apparition de deux \u00e9toiles volantes signifie le d\u00e9but d'une \u00e8re r\u00e9guli\u00e8re ! expliqua le roi Wen.\n\n\u2014 Nous en avons d\u00e9j\u00e0 vu ces derniers temps.\n\n\u2014 Mais il n'y en avait qu'une chaque fois.\n\n\u2014 Deux est-il le maximum ?\n\n\u2014 Non, il peut parfois y en avoir trois, mais jamais plus.\n\n\u2014 Est-ce que l'apparition de ces trois \u00e9toiles volantes pr\u00e9sage une \u00e8re encore plus harmonieuse ?\n\nLe roi Wen le fixa avec des yeux \u00e9pouvant\u00e9s :\n\n\u2014 Que racontes-tu ? Trois \u00e9toiles volantes... Prie plut\u00f4t pour que ce jour n'arrive jamais.\n\nLe roi Wen avait vu juste. L'\u00e8re r\u00e9guli\u00e8re tant esp\u00e9r\u00e9e commen\u00e7a bient\u00f4t. Levers et couchers du soleil devinrent ponctuels et la nuit tombait \u00e0 heure fixe, vers 18 heures. Ce nouveau cycle jour\/nuit rendit le climat plus agr\u00e9able.\n\n\u2014 Combien de temps dure une \u00e8re r\u00e9guli\u00e8re ? demanda Wang Miao.\n\n\u2014 Un jour ou un si\u00e8cle, personne ne peut le pr\u00e9dire avec exactitude.\n\nLe roi Wen s'assit sur le sablier et leva les yeux vers le soleil de midi.\n\n\u2014 Selon les chroniques historiques, la dynastie des Zhou de l'Ouest6 a connu une \u00e8re r\u00e9guli\u00e8re qui a dur\u00e9 deux si\u00e8cles. Quelle chance ont eue ceux qui sont n\u00e9s \u00e0 cette \u00e9poque !\n\n\u2014 Et l'\u00e8re chaotique ?\n\n\u2014 Mais je te l'ai dit ! Toutes les \u00e8res qui ne sont pas r\u00e9guli\u00e8res sont chaotiques, l'une prend le relais de l'autre.\n\n\u2014 Cela veut donc dire que ce monde ne conna\u00eet aucune r\u00e8gle ?\n\n\u2014 Oui. La civilisation ne peut se d\u00e9velopper que sous les climats temp\u00e9r\u00e9s des \u00e8res r\u00e9guli\u00e8res. La majorit\u00e9 du temps, les humains doivent \u00eatre collectivement d\u00e9shydrat\u00e9s et, lorsque arrive une \u00e8re r\u00e9guli\u00e8re un peu plus longue, ils sont r\u00e9hydrat\u00e9s et peuvent \u00e0 nouveau produire et construire.\n\n\u2014 Comment fait-on pour pr\u00e9dire la longueur d'une \u00e8re r\u00e9guli\u00e8re ?\n\n\u2014 C'est impossible. Lorsque arrive une \u00e8re r\u00e9guli\u00e8re, la d\u00e9cision de savoir si les sujets d'un royaume doivent \u00eatre ou non r\u00e9hydrat\u00e9s repose sur l'intuition de son souverain. La plupart du temps, voici ce qui se passe : les d\u00e9shydrat\u00e9s reviennent \u00e0 la vie, des cultures sont plant\u00e9es, des villes commencent \u00e0 \u00eatre construites et alors m\u00eame que la vie vient tout juste de reprendre ses droits, l'\u00e8re r\u00e9guli\u00e8re prend fin, le froid et la chaleur r\u00e9duisent tout \u00e0 n\u00e9ant.\n\nLe roi d\u00e9signa alors Wang Miao du doigt et ses yeux s'illumin\u00e8rent :\n\n\u2014 Tu connais maintenant le but du jeu : t'appuyer sur nos connaissances et ton bon sens pour analyser ce ph\u00e9nom\u00e8ne, jusqu'\u00e0 ce que nous soyons capables de d\u00e9terminer les cycles du soleil. La survie de la civilisation en d\u00e9pend.\n\n\u2014 D'apr\u00e8s ce que j'ai pu voir, il semble que le mouvement du soleil ne soit soumis \u00e0 aucune r\u00e8gle.\n\n\u2014 C'est parce que tu n'as pas encore perc\u00e9 la nature fondamentale de ce monde.\n\n\u2014 Et vous ?\n\n\u2014 Oui. C'est la raison pour laquelle je me rends \u00e0 Zhaoge. Je vais faire cadeau au roi Zhou des Shang d'un calendrier pr\u00e9cis des activit\u00e9s du soleil.\n\n\u2014 Mais je ne vous ai pas vu l'utiliser durant notre p\u00e9riple.\n\n\u2014 La pr\u00e9diction des mouvements du soleil peut uniquement \u00eatre effectu\u00e9e \u00e0 Zhaoge, car c'est le lieu de rencontre du yin et du yang. Il n'y a que l\u00e0-bas que les oracles peuvent s'av\u00e9rer exacts.\n\nIls durent encore affronter une terrible \u00e8re chaotique, entrecoup\u00e9e d'une courte \u00e8re r\u00e9guli\u00e8re, avant d'atteindre enfin Zhaoge.\n\nQuand ils furent arriv\u00e9s dans la cit\u00e9, Wang Miao entendit un grondement continu qui ressemblait au tonnerre. Le bruit provenait d'objets curieux : d'\u00e9normes pendules, chacun de quelques dizaines de m\u00e8tres de hauteur. Les balanciers des pendules \u00e9taient constitu\u00e9s de gros blocs de pierre attach\u00e9s au bout d'une \u00e9paisse corde. Les pendules \u00e9taient reli\u00e9s entre eux par de minces passerelles en pierre. Les balanciers \u00e9taient quant \u00e0 eux secou\u00e9s par des soldats en armure post\u00e9s sous chaque pendule. Psalmodiant des chants \u00e9tranges, ils tiraient en rythme sur les cordes des balanciers pour maintenir le mouvement de ceux-ci. Wang Miao remarqua que tous les balanciers oscillaient de la m\u00eame mani\u00e8re. Vue de loin, la sc\u00e8ne \u00e9tait fascinante ; c'\u00e9tait comme si de gigantesques horloges mouvantes avaient \u00e9t\u00e9 \u00e9rig\u00e9es sur le sol ou bien comme si d'immenses symboles abstraits \u00e9taient tomb\u00e9s du ciel.\n\nAu sein du p\u00e9rim\u00e8tre d\u00e9limit\u00e9 par les pendule se dressait fi\u00e8rement une pyramide colossale qui, dans l'obscurit\u00e9 de la nuit, donnait l'impression d'\u00eatre une montagne noire. C'\u00e9tait le palais du roi Zhou. Wang Miao suivit le roi Wen et y entra par une porte basse situ\u00e9e \u00e0 la base de la pyramide, devant laquelle des sentinelles patrouillaient en silence tels des fant\u00f4mes dans la nuit. Ils long\u00e8rent un interminable corridor \u00e9troit et sombre, \u00e9clair\u00e9 tout du long par des torches accroch\u00e9es aux murs.\n\n\u2014 Durant les \u00e8res chaotiques, tout le pays se retrouve en \u00e9tat de d\u00e9shydratation, expliqua le roi Wen en marchant, mais le roi Zhou reste \u00e9veill\u00e9, r\u00e9gnant sur un royaume sans \u00e2mes. Pour survivre aux \u00e8res chaotiques, il faut vivre dans ce type d'architecture aux murs tr\u00e8s \u00e9pais, habiter sous terre, afin d'\u00e9chapper au froid et \u00e0 la chaleur.\n\nIls march\u00e8rent encore longtemps dans le corridor avant de p\u00e9n\u00e9trer enfin dans la grande salle d'audience du roi Zhou, situ\u00e9e au centre de la pyramide. Celle-ci n'avait en r\u00e9alit\u00e9 de grande que le nom et Wang Miao trouva qu'elle ressemblait davantage \u00e0 une grotte. Sous la lueur vacillante des torches, on pouvait deviner sans peine la silhouette du roi Zhou, assis sur une estrade en pierre et emmitoufl\u00e9 dans une fourrure de b\u00eate multicolore. Mais un autre homme attira l'attention de Wang Miao : un individu v\u00eatu d'une tunique noire qui se fondait dans l'obscurit\u00e9 de la caverne et dont le visage blanc p\u00e2le paraissait flotter dans l'air.\n\n\u2014 Je vous pr\u00e9sente Fu Xi7.\n\n\u00c0 peine \u00e9taient-ils entr\u00e9s dans la salle que le roi Zhou pr\u00e9senta l'homme en noir \u00e0 Wang Miao et au roi Wen, comme s'ils \u00e9taient l\u00e0 depuis le d\u00e9but et que c'\u00e9tait en r\u00e9alit\u00e9 l'homme en noir qui venait juste d'arriver.\n\n\u2014 Fu Xi croit que le soleil est un dieu au mauvais caract\u00e8re. Lors de ses p\u00e9riodes d'\u00e9veil, le dieu est tr\u00e8s \u00e9nerv\u00e9 : ce sont les \u00e8res chaotiques ; mais durant son sommeil, sa respiration est harmonieuse : ce sont les \u00e8res r\u00e9guli\u00e8res. Fu Xi a sugg\u00e9r\u00e9 de construire les grands pendules que vous avez vus \u00e0 l'ext\u00e9rieur de la pyramide et de les faire balancer de jour comme de nuit pour qu'ils bercent continuellement le dieu du Soleil et le maintiennent dans un long sommeil. Mais de ce que nous avons vu, cela n'emp\u00eache pas le soleil de se r\u00e9veiller. Le balancement des pendules lui fait tout au plus faire de courtes siestes.\n\nLe roi Zhou fit un geste de la main et quelqu'un lui apporta une cruche en terre cuite qui fut d\u00e9pos\u00e9e sur l'estrade en pierre devant Fu Xi. Wang Miao comprit plus tard qu'il s'agissait d'un bouillon d'assaisonnement. Fu Xi prit une profonde inspiration, puis il l'avala \u00e0 grandes goul\u00e9es. Le gargouillement du liquide dans sa gorge ressemblait aux battements d'un immense c\u0153ur tapi dans les t\u00e9n\u00e8bres. Apr\u00e8s en avoir bu la moiti\u00e9, il versa le restant du bouillon sur son corps, puis jeta la cruche avant de se diriger vers un tripode en bronze suspendu au-dessus de charbons ardents, dans un coin de la salle. Il grimpa sur le tripode et alors m\u00eame que le feu avait d\u00e9j\u00e0 enflamm\u00e9 sa tunique, il sauta \u00e0 l'int\u00e9rieur, provoquant un nuage de vapeur.\n\n\u2014 Assieds-toi, Ji Chang, le repas sera bient\u00f4t pr\u00eat, dit le roi Zhou en d\u00e9signant le tripode.\n\n\u2014 Sorcelleries idiotes, l\u00e2cha d\u00e9daigneusement le roi Wen en secouant la t\u00eate et en regardant le tripode.\n\n\u2014 As-tu appris quelque chose sur le soleil ? demanda le roi Zhou, dans les yeux duquel scintillaient les flammes des torches.\n\n\u2014 Le soleil n'est pas un dieu. Le soleil est yang, la nuit est yin, et le monde entier est r\u00e9gi par l'\u00e9quilibre du yin et du yang. Nous ne pouvons alt\u00e9rer cet \u00e9quilibre, mais nous pouvons l'anticiper, dit le roi Wen en sortant son \u00e9p\u00e9e de bronze de son fourreau.\n\nSur le sol \u00e9clair\u00e9 par la lumi\u00e8re faible des torches, il dessina les deux poissons yin et yang, puis \u00e0 une vitesse \u00e9tourdissante, il tra\u00e7a tout autour soixante-quatre hexagrammes semblables \u00e0 des anneaux de croissance.\n\n\u2014 Mon roi, voici le code de l'univers. Permettez-moi d'offrir \u00e0 votre dynastie le calendrier le plus pr\u00e9cis jamais con\u00e7u.\n\n\u2014 Ji Chang, j'ai besoin de savoir aujourd'hui quand aura lieu la prochaine longue \u00e8re r\u00e9guli\u00e8re.\n\n\u2014 Je vais r\u00e9aliser cette pr\u00e9diction d\u00e8s maintenant pour vous, dit le roi Wen.\n\nIl s'accroupit au centre du symbole des deux poissons yin et yang puis il leva la t\u00eate et regarda le plafond de la salle d'audience, comme si ses yeux pouvaient traverser le sommet de la pyramide et voir les \u00e9toiles. Il r\u00e9p\u00e9ta plusieurs fois les m\u00eames mouvements de doigts, semblant effectuer un calcul tr\u00e8s rapide. Dans le silence, on n'entendait que le fr\u00e9missement de la soupe dans le tripode, comme si le sorcier en train de mijoter \u00e9tait somniloque.\n\nLe roi Wen se releva et proclama :\n\n\u2014 La prochaine \u00e8re sera une \u00e8re chaotique qui durera quarante et un jours, puis lui succ\u00e9dera une \u00e8re r\u00e9guli\u00e8re de cinq jours, puis une \u00e8re chaotique de vingt-trois jours et une \u00e8re r\u00e9guli\u00e8re de dix-huit jours, et encore une \u00e8re chaotique de huit jours. Et quand cette \u00e8re prendra fin, mon roi, la longue \u00e8re r\u00e9guli\u00e8re tant esp\u00e9r\u00e9e arrivera. Elle se prolongera durant trois ans et neuf mois. Le climat sera temp\u00e9r\u00e9 et votre dynastie conna\u00eetra son \u00e2ge d'or.\n\n\u2014 Nous devons d'abord avoir une preuve que tes pr\u00e9dictions sont exactes, affirma le roi Zhou sans se laisser impressionner.\n\nWang Miao entendit un tremblement au-dessus de sa t\u00eate. Une dalle de pierre s'ouvrit au niveau du plafond de la salle d'audience, r\u00e9v\u00e9lant une ouverture de forme carr\u00e9e. Wang Miao changea de position et vit qu'elle donnait sur une profonde chemin\u00e9e de pierre qui menait jusqu'au sommet de la pyramide, au bout duquel il parvenait \u00e0 discerner quelques \u00e9toiles scintillantes.\n\nLe temps du jeu s'acc\u00e9l\u00e9ra, deux gardes vinrent retourner le sablier du roi Wen et, en une poign\u00e9e de secondes dans la r\u00e9alit\u00e9, ce furent huit heures qui s'\u00e9coul\u00e8rent dans le jeu. La fen\u00eatre du plafond se mit \u00e0 briller de fa\u00e7on irr\u00e9guli\u00e8re, laissant p\u00e9n\u00e9trer \u00e0 plusieurs reprises dans la salle d'audience les rayons de soleil d'une \u00e8re chaotique. Ceux-ci \u00e9taient parfois faibles, comme des rayons de lune, et d'autres fois d'un \u00e9clat si intense que les torches paraissaient bien p\u00e2les \u00e0 c\u00f4t\u00e9 de la zone carr\u00e9e incandescente projet\u00e9e sur le sol. Wang Miao compta le nombre de sabliers retourn\u00e9s par les soldats. Au bout de cent vingt sabliers, les rayons du soleil projet\u00e9s par la chemin\u00e9e se firent plus uniformes : la premi\u00e8re \u00e8re r\u00e9guli\u00e8re pr\u00e9dite par le roi Wen commen\u00e7a. Le sablier fut ensuite retourn\u00e9 vingt fois et les scintillements d\u00e9sordonn\u00e9s qui per\u00e7aient \u00e0 travers la chemin\u00e9e reprirent : c'\u00e9tait de nouveau une \u00e8re chaotique.\n\nUne autre \u00e8re r\u00e9guli\u00e8re passa, puis \u00e0 nouveau une \u00e8re chaotique. Si leur date de commencement et leur dur\u00e9e comportaient de l\u00e9g\u00e8res erreurs, elles correspondaient globalement aux pr\u00e9dictions du roi Wen. Au bout de huit jours de la derni\u00e8re \u00e8re chaotique d\u00e9buta la longue \u00e8re r\u00e9guli\u00e8re promise. Vingt jours pass\u00e8rent et les rais de lumi\u00e8re \u00e0 travers la chemin\u00e9e gardaient un rythme constant. Ce fut au bout de ces vingt jours que le temps du jeu ralentit et se r\u00e9adapta \u00e0 celui de la r\u00e9alit\u00e9.\n\nLe roi Zhou fit un signe de t\u00eate au roi Wen :\n\n\u2014 Ji Chang ! J'\u00e9rigerai un monument en ton honneur plus grand encore que cette salle !\n\nLe roi Wen s'inclina courtoisement :\n\n\u2014 Mon roi, r\u00e9veillez votre peuple et faites prosp\u00e9rer votre dynastie !\n\nLe roi Zhou se leva sur l'estrade, ouvrit les deux bras comme s'il voulait embrasser le monde et, avec une \u00e9trange voix qui ressemblait \u00e0 un chant, il clama :\n\n\u2014 R\u00e9hydratez !\n\nD\u00e8s l'annonce de cet ordre, tous les membres de la cour pr\u00e9sents dans la grande salle jaillirent hors de la caverne. Wang Miao suivit le roi Wen et sortit de la pyramide par le long corridor. Une fois dehors, il remarqua que le soleil \u00e9tait \u00e0 son z\u00e9nith et que la terre \u00e9tait impr\u00e9gn\u00e9e de sa chaleur. Il \u00e9tait envelopp\u00e9 d'une brise chaude et pouvait presque sentir le parfum du printemps. Le roi Wen et Wang Miao march\u00e8rent ensemble jusqu'\u00e0 un lac situ\u00e9 non loin de la pyramide. La couche de glace sur sa surface avait enti\u00e8rement fondu et les rayons du soleil chatoyaient au gr\u00e9 des rides de l'eau.\n\nLa premi\u00e8re troupe de soldats \u00e0 \u00eatre sortie de la pyramide criait :\n\n\u2014 R\u00e9hydratez ! R\u00e9hydratez !\n\nIls couraient tous dans la direction d'une grande b\u00e2tisse en pierre construite au bord du lac qui ressemblait \u00e0 un grenier \u00e0 c\u00e9r\u00e9ales. Durant leur voyage jusqu'\u00e0 Zhaoge, Wang Miao avait eu l'occasion de remarquer plusieurs de ces b\u00e2timents. Le roi Wen lui avait expliqu\u00e9 que c'\u00e9taient des \"silos de d\u00e9shydratation\", de grands hangars o\u00f9 \u00e9taient stock\u00e9s tous les individus d\u00e9shydrat\u00e9s. Les soldats ouvrirent la lourde porte en pierre du silo et en sortirent des rouleaux poussi\u00e9reux. Chaque soldat se dirigea vers le lac avec une s\u00e9rie de rouleaux sous les bras qu'il lan\u00e7a \u00e0 l'eau. D\u00e8s qu'ils entr\u00e8rent en contact avec l'eau, ils se d\u00e9ploy\u00e8rent et, au bout d'un certain temps, la surface du lac fut recouverte d'une kyrielle d'humains aplatis semblant avoir \u00e9t\u00e9 d\u00e9coup\u00e9s dans du papier. Chaque \"tranche d'humain\" se mit \u00e0 absorber de l'eau et, peu \u00e0 peu, les d\u00e9shydrat\u00e9s retrouv\u00e8rent la rondeur de leur chair et donn\u00e8rent bient\u00f4t signe de vie. Non sans peine, ils se mirent debout et sortirent du lac dont l'eau leur arrivait au nombril. Ils ouvrirent les paupi\u00e8res comme s'ils s'\u00e9veillaient d'un long r\u00eave en regardant ce monde baign\u00e9 de soleil et berc\u00e9 par la brise.\n\n\u2014 R\u00e9hydratez ! trompeta l'un d'eux et une explosion de cris joyeux lui fit aussit\u00f4t \u00e9cho.\n\n\u2014 R\u00e9hydratez ! R\u00e9hydratez !\n\nLes hommes s'extirp\u00e8rent du lac et coururent nus sur la rive. Puis ils fonc\u00e8rent vers les silos o\u00f9 ils aid\u00e8rent les soldats \u00e0 transporter davantage de rouleaux et \u00e0 les immerger dans le lac. Ce fut bient\u00f4t une foule de ressuscit\u00e9s qui jaillit du lac. La m\u00eame sc\u00e8ne avait lieu dans des lacs et des \u00e9tangs plus lointains. Le monde entier revenait \u00e0 la vie.\n\n\u2014 A\u00efe, eh ! Mon doigt...\n\nWang Miao regarda dans la direction d'o\u00f9 provenait le cri. Au milieu du lac, il vit un homme qui se tenait la main en g\u00e9missant. Il lui manquait un majeur et du sang gouttait dans l'eau du lac. Des vagues d'autres r\u00e9hydrat\u00e9s pass\u00e8rent \u00e0 c\u00f4t\u00e9 de l'estropi\u00e9, si press\u00e9s de rejoindre la rive qu'ils ne lui pr\u00eat\u00e8rent aucune attention.\n\n\u2014 Consid\u00e8re-toi comme chanceux ! lui lan\u00e7a un r\u00e9hydrat\u00e9. Certains ont perdu leurs bras ou leurs jambes, d'autres se sont fait grignoter la t\u00eate ! Si nous n'avions pas \u00e9t\u00e9 r\u00e9hydrat\u00e9s, nous aurions peut-\u00eatre fini par \u00eatre enti\u00e8rement bouff\u00e9s par les rats de l'\u00e8re chaotique.\n\n\u2014 Combien de temps sommes-nous rest\u00e9s d\u00e9shydrat\u00e9s ? demanda un autre ressuscit\u00e9.\n\n\u2014 On peut le d\u00e9duire \u00e0 la couche de poussi\u00e8re accumul\u00e9e sur le palais du roi. On raconte que le roi n'est pas le m\u00eame que celui du temps o\u00f9 nous avons \u00e9t\u00e9 d\u00e9shydrat\u00e9s. J'ignore si c'est son fils ou son petit-fils.\n\nLa r\u00e9hydratation ne se termina qu'au bout de huit jours. \u00c0 ce moment-l\u00e0, tous les d\u00e9shydrat\u00e9s avaient \u00e9t\u00e9 ressuscit\u00e9s et le monde s'\u00e9tait offert une nouvelle vie. Durant ces huit jours, les hommes purent profiter de cycles r\u00e9guliers du soleil de vingt heures chaque fois. Se d\u00e9lectant de ce climat printanier, tous les sujets du royaume, reconnaissants, chantaient les louanges du soleil et celles des dieux qui r\u00e9gissaient l'univers.\n\nLa nuit du huiti\u00e8me jour, les feux de camp sur la plaine furent plus denses que les \u00e9toiles, tandis que les villes et les cit\u00e9s abandonn\u00e9es lors des longues \u00e8res chaotiques retrouvaient leurs lumi\u00e8res et leur vacarme. Et comme lors des innombrables r\u00e9hydratations de masse de l'histoire, les hommes s'appr\u00eataient \u00e0 c\u00e9l\u00e9brer durant toute la nuit la nouvelle vie qui na\u00eetrait apr\u00e8s le prochain lever du soleil.\n\nMais le soleil ne se leva plus.\n\nTous les instruments de mesure du temps indiquaient que l'heure du lever du soleil \u00e9tait pass\u00e9e mais, partout \u00e0 l'horizon, tout \u00e9tait d'un noir de jais. Douze heures plus tard, on ne voyait toujours pas l'ombre du soleil, pas m\u00eame le pr\u00e9sage d'un rayon levant. Un jour entier passa, et l'interminable nuit se poursuivit ; deux jours et le froid nocturne \u00e9crasa la terre comme une main g\u00e9ante et terrifiante.\n\n\u2014 Mon grand roi, continuez \u00e0 m'accorder votre confiance. Cette p\u00e9riode est passag\u00e8re. J'ai vu toutes les \u00e9nergies yang se rassembler dans l'univers, le soleil va bient\u00f4t se lever. L'\u00e8re r\u00e9guli\u00e8re et le printemps reviendront ! implora \u00e0 genoux le roi Wen devant le roi Zhou, assis sur son tr\u00f4ne de pierre.\n\n\u2014 Allumez un feu sous le tripode, soupira le roi Zhou.\n\n\u2014 Majest\u00e9 ! Majest\u00e9 !\n\nUn ministre accourut et tr\u00e9bucha dans la salle d'audience, en criant d'une voix larmoyante :\n\n\u2014 Majest\u00e9 ! Dans le ciel... trois \u00e9toiles volantes !\n\nTous dans la salle furent saisis de stupeur. L'air semblait s'\u00eatre p\u00e9trifi\u00e9. Seul le roi Zhou parut ne pas perdre son calme. Il se tourna vers Wang Miao, \u00e0 qui il n'avait jusque-l\u00e0 pas daign\u00e9 adresser la parole :\n\n\u2014 Tu ignores ce que signifie l'apparition de trois \u00e9toiles volantes, n'est-ce pas ? Ji Chang, dis-lui.\n\n\u2014 Cela pr\u00e9sage une longue et terrible p\u00e9riode de froid, un froid si terrible qu'il est capable de changer les pierres en poussi\u00e8re, soupira le roi Wen.\n\n\u2014 D\u00e9shydratez..., ordonna le roi Zhou avec la m\u00eame \u00e9trange intonation chantante que lors de son pr\u00e9c\u00e9dent ordre.\n\nEn r\u00e9alit\u00e9, \u00e0 l'ext\u00e9rieur, tous avaient d\u00e9j\u00e0 commenc\u00e9 \u00e0 se d\u00e9shydrater, \u00e0 retrouver une texture s\u00e8che afin d'affronter la nuit interminable qui les attendait. Les plus chanceux eurent le temps d'\u00eatre port\u00e9s jusqu'aux silos de d\u00e9shydratation, mais la plupart des sujets furent laiss\u00e9s \u00e0 l'abandon sur la plaine.\n\nLe roi Wen se leva lentement et marcha vers le tripode. Il grimpa et s'arr\u00eata quelques instants au-dessus avant de plonger \u00e0 l'int\u00e9rieur. Peut-\u00eatre avait-il vu le visage bouilli de Fu Xi en train de lui sourire.\n\n\u2014 Faites mijoter \u00e0 feu doux, ordonna le roi Zhou sans conviction.\n\nPuis il s'adressa aux autres personnes pr\u00e9sentes dans la salle d'audience :\n\n\u2014 Vous pouvez cliquer sur _\"Exit\"_ si vous voulez quitter le jeu car, \u00e0 ce stade, il n'est plus tr\u00e8s int\u00e9ressant.\n\nL'indication _\"Exit\"_ apparut en lettres rouges sur la porte d'entr\u00e9e de la caverne. Une vague de joueurs se dirigea p\u00ealem\u00eale dans cette direction, suivie par Wang Miao. Une fois qu'il eut pass\u00e9 la porte et travers\u00e9 le long corridor jusqu'\u00e0 l'ext\u00e9rieur de la pyramide, il vit la neige tomber \u00e0 gros flocons dans le ciel nocturne. Il fut saisi d'un froid \u00e0 lui glacer les os. Dans un coin de l'\u00e9cran, la barre du temps du jeu augmenta rapidement.\n\nLa neige continua dix jours durant. Les flocons \u00e9taient lourds et \u00e9pais, ils tombaient sur la terre comme des gla\u00e7ons d'obscurit\u00e9. Quelqu'un murmura \u00e0 l'oreille de Wang Miao :\n\n\u2014 De la neige carbonique.\n\nWang Miao tourna la t\u00eate, c'\u00e9tait le laquais du roi Wen.\n\nDix jours encore s'\u00e9coul\u00e8rent. La neige tombait encore mais, cette fois, les flocons \u00e9taient plus fins et translucides. Sous la lueur faiblissante des torches qui filtrait depuis la caverne, les flocons chatoyaient d'une artificielle phosphorescence bleu p\u00e2le, comme des micas voltigeant dans le ciel.\n\n\u2014 Les flocons sont \u00e0 pr\u00e9sent compos\u00e9s d'oxyde d'azote. La temp\u00e9rature du ciel est proche du z\u00e9ro absolu, l'atmosph\u00e8re dispara\u00eet.\n\nPeu \u00e0 peu, la neige engloutit la pyramide. \u00c0 son pied se trouvait une couche de neige fondue, en son centre de la glace carbonique, et \u00e0 son sommet de la neige sous forme d'oxyde d'azote. La nuit devint anormalement claire et les \u00e9toiles rutil\u00e8rent comme des bougies aux flammes argent\u00e9es. Un texte s'afficha en plein milieu du ciel \u00e9toil\u00e9 :\n\n> Cette longue nuit a dur\u00e9 quarante-huit ans. La civilisation no 137 a \u00e9t\u00e9 compl\u00e8tement d\u00e9cim\u00e9e par le froid. Avant sa chute, cette civilisation avait atteint la p\u00e9riode des Royaumes combattants.\n> \n> Mais les graines de la civilisation ont surv\u00e9cu et une nouvelle civilisation germera des ruines du monde si impr\u00e9visible des _Trois Corps_. Nous vous invitons \u00e0 vous reconnecter dans le futur.\n\nAvant de quitter le jeu, Wang Miao observa une derni\u00e8re fois les trois \u00e9toiles volantes. Elles paraissaient s'entortiller les unes autour des autres, comme si elles entamaient dans les profondeurs de l'univers une danse aux secrets connus d'elles seules.\n\n* * *\n\n1 Litt\u00e9ralement \"Homme-mer\" ou \"Homme de la mer\" : jeu de mots bas\u00e9 sur le sens litt\u00e9ral des caract\u00e8res \"Wang\" et \"Miao\", qui peuvent signifier une \"vaste \u00e9tendue d'eau\". _(N.d.T.)_\n\n2 P\u00e9riode de l'histoire chinoise (481-221 av. J.-C.) durant laquelle le pays \u00e9tait morcel\u00e9 en principaut\u00e9s dont les querelles se soldaient par des guerres incessantes. _(N.d.T.)_\n\n3 Aussi connu sous son nom personnel de Ji Chang, Wen des Zhou (1152-1056 av. J.-C.) fut le p\u00e8re du fondateur de la dynastie Zhou, qui succ\u00e9da \u00e0 la dynastie Shang. La l\u00e9gende raconte que le roi Wen serait \u00e0 l'origine du deuxi\u00e8me plus ancien arrangement d'hexagrammes (figures divinatoires constitu\u00e9es de combinaisons de traits yin et yang), \u00e0 l'\u00e9poque o\u00f9 il \u00e9tait dans les cachots du tyrannique roi Zhou des Shang. _(N.d.T.)_\n\n4 Zhaoge fut la derni\u00e8re capitale de la dynastie Shang, qui r\u00e9gna sur la Chine de 1570 av. J.-C. \u00e0 1045 av. J.-C. _(N.d.T.)_\n\n5 Sorte de pierre \u00e0 feu en forme de faucille, utilis\u00e9e en Chine ancienne. _(N.d.T.)_\n\n6 La dynastie des Zhou de l'Ouest r\u00e9gna sur une vaste partie de la Chine d'environ 1046 \u00e0 771 av. J.-C. Lui succ\u00e9da la p\u00e9riode des Zhou de l'Est (771-256 av. J.-C.), qui voit un d\u00e9placement de la capitale des Zhou \u00e0 l'est et un affaiblissement progressif du r\u00e8gne de la dynastie. _(N.d.T.)_\n\n7 Fu Xi est un personnage important de la mythologie chinoise, suppos\u00e9 \u00eatre le fondateur de la m\u00e9decine traditionnelle chinoise et le cr\u00e9ateur des sinogrammes. On lui attribue les huit premiers hexagrammes de l'histoire. _(N.d.T.)_\n\n## 8\n\n## YE WENJIE\n\nWang Miao \u00f4ta sa combinaison et son casque de r\u00e9alit\u00e9 virtuelle. Alors m\u00eame que la fonction de refroidissement des capteurs avait tourn\u00e9 \u00e0 plein r\u00e9gime, il \u00e9tait tremp\u00e9 de sueur, comme s'il se r\u00e9veillait d'un cauchemar glacial. Il quitta le laboratoire, descendit chercher sa voiture et se rendit \u00e0 l'adresse de la m\u00e8re de Yang Dong que lui avait indiqu\u00e9e Ding Yi.\n\n _\u00c8re chaotique, \u00e8re chaotique, \u00e8re chaotique..._\n\nCe concept ne cessait de revenir en boucle dans sa t\u00eate. _Pourquoi les trajectoires du soleil de ce monde sont-elles si irr\u00e9guli\u00e8res ? L'orbite d'une plan\u00e8te peut \u00eatre plus ou moins circulaire ou elliptique, mais sa r\u00e9volution autour du soleil est n\u00e9cessairement p\u00e9riodique. Il est tout simplement impossible qu'une plan\u00e8te tourne autour d'un astre de fa\u00e7on compl\u00e8tement fantaisiste..._ Wang Miao s'aga\u00e7a. Il secoua la t\u00eate pour la d\u00e9barrasser de ces questions lancinantes. Ce n'\u00e9tait qu'un jeu apr\u00e8s tout. Mais il \u00e9choua \u00e0 les chasser de son esprit.\n\n _\u00c8re chaotique, \u00e8re chaotique, \u00e8re chaotique..._\n\n _N'importe quoi ! Pourquoi suis-je en train de penser \u00e0 cela ? Pourquoi sans cesse ? Pourquoi ?_\n\nTr\u00e8s vite, Wang Miao trouva la r\u00e9ponse. Il n'avait pas jou\u00e9 \u00e0 un jeu vid\u00e9o depuis fort longtemps : les logiciels et le mat\u00e9riel avaient manifestement \u00e9norm\u00e9ment \u00e9volu\u00e9 ces derni\u00e8res ann\u00e9es. Les sc\u00e8nes de r\u00e9alit\u00e9 virtuelle de m\u00eame que tous les effets sensoriels permis par les jeux d'aujourd'hui \u00e9taient sans commune mesure avec ceux des jeux auxquels il jouait quand il \u00e9tait \u00e9tudiant. Toutefois Wang Miao comprit que ce n'\u00e9tait pas \u00e0 son niveau de technologie que le jeu des _Trois Corps_ devait son r\u00e9alisme.\n\nIl se souvint qu'en troisi\u00e8me ann\u00e9e de licence, lors d'un cours de th\u00e9orie de l'information, le professeur avait apport\u00e9 en classe deux images : la premi\u00e8re \u00e9tait la c\u00e9l\u00e8bre peinture datant de la dynastie Song intitul\u00e9e _Le Jour de Qingming au bord de la rivi\u00e8re_ , c\u00e9l\u00e8bre pour ses d\u00e9tails saisissants ; l'autre, une photographie monochrome d'un ciel bleu, dans lequel n'apparaissait qu'un mince nuage blanc \u00e0 peine perceptible. Le professeur avait demand\u00e9 aux \u00e9tudiants laquelle des deux images contenait le plus d'informations. La r\u00e9ponse correcte donn\u00e9e par le professeur \u00e9tait la seconde image, dont l'ordre de grandeur des informations \u00e9tait un \u00e0 deux fois plus important.\n\nLe jeu des _Trois Corps_ partageait le m\u00eame paradoxe. Les donn\u00e9es et les informations les plus importantes \u00e9taient celles qu'il dissimulait le mieux. Wang Miao pouvait le sentir, mais il ne pouvait clairement l'exprimer. Il comprit soudain que l'originalit\u00e9 des _Trois Corps_ r\u00e9sidait dans le fait que son cr\u00e9ateur avait pris le parfait contrepied des jeux traditionnels. Alors que la plupart des \u00e9diteurs des jeux s'effor\u00e7aient d'introduire autant que possible un grand nombre de d\u00e9tails afin de produire un sentiment de r\u00e9alisme, ceux du jeu des _Trois Corps_ avaient comprim\u00e9 au maximum la taille des informations, pour dissimuler une r\u00e9alit\u00e9 encore plus complexe, exactement comme la photo du ciel qui pouvait para\u00eetre pauvre au premier coup d'\u0153il.\n\nWang Miao rel\u00e2cha la bride de ses pens\u00e9es qui retourn\u00e8rent donc vagabonder dans le monde des _Trois Corps_.\n\n _Des \u00e9toiles volantes ! La clef doit se trouver dans le secret de ces \u00e9toiles volantes en apparence insignifiantes. Une \u00e9toile volante, deux \u00e9toiles volantes, trois \u00e9toiles volantes... Que signifient ces combinaisons ?_\n\nEt alors qu'il r\u00e9fl\u00e9chissait \u00e0 cette question, sa voiture arriva \u00e0 l'entr\u00e9e du quartier o\u00f9 habitait la m\u00e8re de Yang Dong.\n\nDevant la porte d'entr\u00e9e de l'immeuble, Wang Miao vit une femme maigre, la soixantaine, les cheveux grisonnants et des lunettes sur le nez. Un gros panier de l\u00e9gumes dans les mains, elle semblait peiner \u00e0 monter les escaliers. Il devina qu'elle devait \u00eatre la personne \u00e0 qui il venait rendre visite.\n\nDes pr\u00e9sentations cordiales lui apprirent qu'elle \u00e9tait bien Ye Wenjie, la m\u00e8re de Yang Dong. Apr\u00e8s avoir compris la raison de la visite de Wang Miao, elle exprima sa profonde gratitude. Elle appartenait \u00e0 cette cat\u00e9gorie de vieux intellectuels que Wang Miao avait parfois l'occasion de rencontrer dans son travail. Les vicissitudes du temps avaient effac\u00e9 chez eux toute trace de rage et de rudesse. Il ne leur restait plus que la douceur de l'eau.\n\nWang Miao l'aida \u00e0 monter son panier jusqu'\u00e0 son appartement. \u00c0 peine la porte franchie, il d\u00e9couvrit qu'elle n'\u00e9tait pas aussi seule qu'il l'avait imagin\u00e9. Trois enfants \u00e9taient en train de jouer dans le salon. Le plus grand ne devait pas avoir plus de cinq ans, et le plus jeune semblait tout juste commencer \u00e0 marcher. Ye Wenjie expliqua que c'\u00e9taient les enfants des voisins.\n\n\u2014 Ils aiment venir jouer \u00e0 la maison. Le dimanche, leurs parents font des heures suppl\u00e9mentaires et ils me les laissent... Nan-nan, tu as fini ton dessin ? Oh, comme c'est joli ! Tu veux lui donner un titre ? \"Les petits canards sous le soleil\". Bravo ! Mamie va \u00e9crire le titre et la date : le 9 juin, par Nan-nan... Qu'est-ce que vous voulez manger \u00e0 midi ? Yang-yang ? Des aubergines frites ? D'accord, d'accord ; Nan-nan ? Des pois gourmands, comme hier ? D'accord, d'accord, et toi Mi-mi ? De la viande ? Non, maman a dit pas trop de viande, ce n'est pas facile \u00e0 dig\u00e9rer. On va manger du poisson, d'accord ? Regarde le gros poisson que mamie a achet\u00e9...\n\nEn voyant la m\u00e8re de Yang Dong absorb\u00e9e dans sa discussion avec les enfants, Wang Miao se dit qu'elle aurait certainement aim\u00e9 avoir des petits-enfants. Mais quand bien m\u00eame Yang Dong aurait encore \u00e9t\u00e9 en vie, aurait-elle voulu des enfants ?\n\nYe Wenjie apporta les courses dans la cuisine, puis elle ressortit en disant \u00e0 Wang Miao :\n\n\u2014 Miao, je vais d'abord faire tremper les l\u00e9gumes. On utilise trop de pesticides de nos jours, je dois les faire tremper au moins deux heures avant de les donner \u00e0 manger aux enfants... En attendant, tu peux aller faire un tour dans la chambre de Yang Dong.\n\nLa proposition de Ye Wenjie, qui paraissait en apparence \u00eatre une suggestion innocente et naturelle, rendit Wang Miao nerveux. Elle avait vu clair en lui et devin\u00e9 les raisons r\u00e9elles de sa visite. Elle repartit dans la cuisine sans m\u00eame un regard pour Wang Miao, comme pour ne pas \u00eatre t\u00e9moin de son embarras. Il \u00e9prouva un sentiment de gratitude pour cette r\u00e9action sciemment pens\u00e9e.\n\nIl passa devant les enfants qui jouaient gaiement dans le salon avant de se diriger vers la chambre que lui avait indiqu\u00e9e Ye Wenjie. Il s'arr\u00eata un moment devant la porte, brusquement envahi d'une sensation \u00e9trange, comme s'il \u00e9tait revenu au temps de son adolescence pleine de r\u00eaves. Une fragilit\u00e9 et une puret\u00e9 pareilles aux premi\u00e8res gouttes de la ros\u00e9e surgirent des tr\u00e9fonds de sa m\u00e9moire. Ses premiers chagrins et ses premi\u00e8res blessures, mais avec une teinte de rose.\n\nWang Miao poussa d\u00e9licatement la porte, il fut accueilli par un parfum l\u00e9ger auquel il ne s'attendait pas, l'odeur de la for\u00eat. C'\u00e9tait comme s'il avait p\u00e9n\u00e9tr\u00e9 dans la hutte d'un garde forestier au fond des bois. Les murs \u00e9taient tapiss\u00e9s d'une bande d'\u00e9corce brune. Trois tabourets pittoresques fabriqu\u00e9s \u00e0 partir de troncs d'arbres tr\u00f4naient dans la pi\u00e8ce, le bureau \u00e9tait con\u00e7u de trois branches \u00e9paisses et il y avait enfin ce lit, une simple natte qui semblait faite en carex des r\u00e9gions du Nord-Est. Le mobilier, sobre et grossier, ne semblait r\u00e9pondre \u00e0 aucun imp\u00e9ratif esth\u00e9tique. Au vu de sa position, Yang Dong devait avoir un salaire tr\u00e8s confortable. Elle aurait pu choisir d'acheter une maison dans une r\u00e9sidence de luxe \u00e0 la p\u00e9riph\u00e9rie de la ville, mais elle habitait encore ici avec sa m\u00e8re.\n\nWang Miao s'avan\u00e7a jusque devant le bureau. Celui-ci lui apparut tr\u00e8s aust\u00e8re. Rien ne rappelait son travail acad\u00e9mique, et il n'y vit pas la moindre touche de f\u00e9minit\u00e9. Il \u00e9tait possible que tous les objets qui se trouvaient autrefois sur le bureau aient \u00e9t\u00e9 d\u00e9plac\u00e9s, mais peut-\u00eatre aussi que rien n'y avait jamais \u00e9t\u00e9 pos\u00e9. Il remarqua une photo en noir et blanc ench\u00e2ss\u00e9e dans un cadre en bois, un portrait de la fille avec sa m\u00e8re. Yang Dong \u00e9tait encore une enfant, sa m\u00e8re \u00e9tait accroupie et faisait tout juste sa taille. Leurs cheveux s'entrem\u00ealaient, souffl\u00e9s par une bourrasque. Le fond de l'image \u00e9tait tr\u00e8s \u00e9trange, le ciel paraissait grillag\u00e9. Wang Miao examina en d\u00e9tail la structure m\u00e9tallique qui soutenait le grillage et il devina qu'il s'agissait d'une sorte d'antenne parabolique, si grande que ses extr\u00e9mit\u00e9s d\u00e9passaient le cadre de la photo.\n\nLes grands yeux de la petite Yang Dong laissaient filtrer une peur qui fit frissonner Wang Miao. C'\u00e9tait comme si l'enfant \u00e9tait terroris\u00e9e par le monde \u00e0 l'ext\u00e9rieur de l'image. Wang Miao remarqua \u00e9galement un carnet \u00e9pais pos\u00e9 sur un coin du bureau. La mati\u00e8re dans laquelle il \u00e9tait fait attira son attention. Sur la couverture, une main d'enfant avait \u00e9crit : \"Carnet en \u00e9corce de bouleau de Yang Dong\". Le mot \"bouleau\" n'avait pas \u00e9t\u00e9 \u00e9crit en caract\u00e8res, mais en transcription phon\u00e9tique. Le temps avait chang\u00e9 le blanc argent\u00e9 de l'\u00e9corce en jaune sombre. Il tendit la main pour toucher le carnet puis, apr\u00e8s un moment d'h\u00e9sitation, il se ravisa.\n\n\u2014 Tu peux y aller, c'est le carnet de dessins de Dong-dong quand elle \u00e9tait petite, lui glissa la m\u00e8re de Yang Dong qui se tenait \u00e0 la porte.\n\nWang Miao saisit le carnet et le feuilleta avec d\u00e9licatesse. Ye Wenjie avait \u00e9crit la date pour chaque dessin de sa fille, comme elle l'avait fait pour Nan-nan dans le salon. Wang Miao d\u00e9couvrit autre chose qui le laissa perplexe. Si on en croyait la date indiqu\u00e9e sur les dessins, Yang Dong devait avoir plus de trois ans, or les enfants de cet \u00e2ge parviennent normalement plus ou moins \u00e0 dessiner des traits de personnages ou d'objets. Pourtant les dessins de Yang Dong \u00e9taient des lignes d\u00e9sordonn\u00e9es et confuses. Wang Miao y voyait la col\u00e8re et le d\u00e9sespoir intenses d'une petite fille incapable d'exprimer quelque chose, un sentiment rare chez les enfants de cet \u00e2ge.\n\nYe Wenjie s'assit soigneusement au bord du lit, le regard riv\u00e9 sur le carnet en \u00e9corce de bouleau que Wang Miao tenait entre les mains. C'\u00e9tait ici m\u00eame que sa fille avait mis fin \u00e0 ses jours, pendant qu'elle dormait. Wang Miao s'assit \u00e0 ses c\u00f4t\u00e9s. Jamais il n'avait ressenti aussi intens\u00e9ment le d\u00e9sir de partager la douleur de quelqu'un d'autre.\n\nLa m\u00e8re de Yang Dong prit le carnet des mains de Wang Miao et le serra contre sa poitrine en murmurant :\n\n\u2014 Je n'ai pas \u00e9duqu\u00e9 Dong-dong comme j'aurais d\u00fb. Je l'ai expos\u00e9e trop t\u00f4t \u00e0 des choses trop abstraites et trop extr\u00eames. La premi\u00e8re fois qu'elle m'a fait part de son int\u00e9r\u00eat pour les th\u00e9ories abstraites, je lui ai r\u00e9pondu que ce n'\u00e9tait pas un monde pour les jeunes filles. Elle m'a r\u00e9torqu\u00e9 : \"Et Mme Curie ?\" Je lui ai r\u00e9pondu que Marie Curie n'avait jamais vraiment \u00e9t\u00e9 accept\u00e9e dans ce monde, que sa r\u00e9ussite venait de son d\u00e9vouement et de son travail acharn\u00e9 mais que, sans elle, quelqu'un d'autre aurait tout de m\u00eame termin\u00e9 le travail. Une scientifique comme Wu Chien-shiung \u00e9tait d'ailleurs all\u00e9e beaucoup plus loin. Non, ce n'\u00e9tait vraiment pas un monde fait pour les femmes. Je lui ai dit que les femmes ne pensaient pas comme les hommes, que ce n'\u00e9tait pas une diff\u00e9rence de valeur, que le monde avait besoin des deux.\n\nDong-dong ne m'a pas contredite. Plus tard, je me suis rendu compte qu'elle \u00e9tait vraiment diff\u00e9rente des autres. Par exemple, lorsque je lui enseignais une formule math\u00e9matique. D'autres enfants auraient peut-\u00eatre dit de la formule qu'elle \u00e9tait astucieuse ou autre chose, mais Dong-dong pr\u00e9tendait que la formule \u00e9tait belle, magnifique, elle en parlait comme si elle louait la beaut\u00e9 d'une fleur sauvage. Son p\u00e8re nous avait laiss\u00e9 une collection de disques : elle les a tous \u00e9cout\u00e9s, puis elle a fini par choisir un disque de Bach, qu'elle \u00e9coutait en boucle. Ce n'est pas le genre de musique qu'on fait d'habitude \u00e9couter aux enfants, encore moins aux petites filles. Au d\u00e9but, je me suis dit qu'elle avait choisi le disque au hasard, mais quand je lui ai demand\u00e9 ce qu'elle ressentait, Dong-dong m'a r\u00e9pondu qu'en entendant la m\u00e9lodie elle voyait un \u00eatre g\u00e9ant en train de fabriquer une immense et complexe maison. Une fois le morceau termin\u00e9, la maison \u00e9tait finie d'\u00eatre construite...\n\n\u2014 Vous avez tr\u00e8s bien \u00e9duqu\u00e9 votre fille, dit Wang Miao, sous le coup de l'\u00e9motion.\n\n\u2014 Non, \u00e7'a \u00e9t\u00e9 un \u00e9chec. Le monde dans lequel elle vivait \u00e9tait trop pur, il n'y avait rien d'autre que ces th\u00e9ories \u00e9th\u00e9r\u00e9es. Mais une fois ces fondations effondr\u00e9es, il ne lui est plus rien rest\u00e9 qui lui permette de se raccrocher \u00e0 la vie.\n\n\u2014 Professeur Ye, vous avez tort. Ce qui se passe en ce moment est quelque chose que personne n'avait jamais imagin\u00e9, un d\u00e9fi sans pr\u00e9c\u00e9dent lanc\u00e9 \u00e0 nos th\u00e9ories. Yang Dong n'est pas la seule scientifique \u00e0 avoir fait ce choix.\n\n\u2014 Mais c'est la seule femme. Les femmes devraient \u00eatre comme l'eau, couler encore et encore quelles que soient les tribulations de la vie !\n\nAu moment de partir, Wang Miao se rappela l'autre raison de sa visite. Il demanda \u00e0 Ye Wenjie si elle savait o\u00f9 il \u00e9tait possible d'observer le fond diffus cosmologique.\n\n\u2014 Ah, \u00e7a. Il y a deux endroits o\u00f9 on peut le voir en Chine. \u00c0 l'observatoire d'Urumqi \u2013 je crois que c'est un projet men\u00e9 par le Centre d'observations astronomiques spatial de l'Acad\u00e9mie des sciences \u2013 et dans un autre endroit, plus proche d'ici : un observatoire radioastronomique situ\u00e9 dans la banlieue de P\u00e9kin, sous la responsabilit\u00e9 conjointe de l'Acad\u00e9mie des sciences et du Centre de recherches en astrophysique de l'universit\u00e9 de P\u00e9kin. L'observatoire d'Urumqi permet une observation directe depuis le sol tandis que, dans celui de P\u00e9kin, ce ne sont que des donn\u00e9es envoy\u00e9es par des satellites. Les informations y sont n\u00e9anmoins plus pr\u00e9cises et plus compl\u00e8tes. J'ai un ancien \u00e9tudiant qui y travaille, je vais le contacter pour toi, dit Ye Wenjie, qui trouva le num\u00e9ro de t\u00e9l\u00e9phone et lui passa un coup de fil.\n\nLeur conversation parut tr\u00e8s bien se passer.\n\n\u2014 Pas de probl\u00e8me, je vais te donner l'adresse. Tu peux y aller d\u00e8s que tu le souhaites, mon \u00e9tudiant s'appelle Sha Ruishan, il est de service de nuit ce soir... Mais il me semble que ce n'est pas ton domaine de recherche, n'est-ce pas ? demanda Ye Wenjie en reposant le t\u00e9l\u00e9phone.\n\n\u2014 Non, je travaille dans les nanomat\u00e9riaux, mais cette fois je... il y a autre chose.\n\nWang Miao avait peur qu'elle ne continue \u00e0 l'interroger, mais elle n'en fit rien.\n\n\u2014 Miao ! Tu es bien p\u00e2le, est-ce que tu vas bien ? Tu n'as pas bonne mine, demanda Ye Wenjie avec sollicitude.\n\n\u2014 Ce n'est rien, tout va bien, mentit Wang Miao sans aucune autre pr\u00e9cision.\n\n\u2014 Attends un instant.\n\nYe Wenjie sortit un petit coffret en bois de l'armoire sur lequel Wang Miao vit qu'il \u00e9tait \u00e9crit \"Ginseng\".\n\n\u2014 Un vieil ami de la base est venu me voir avant-hier et il m'a donn\u00e9 \u00e7a. Prends-le. Non, non, prends-le. C'est du ginseng cultiv\u00e9 \u00e0 la main, rien de tr\u00e8s pr\u00e9cieux. De toute fa\u00e7on, j'ai de l'hypertension et je n'ai pas le droit d'en prendre. Coupe quelques tranches fines et fais-les infuser pour te pr\u00e9parer une tisane. \u00c0 voir comme tu es p\u00e2le, tu dois manquer de globules rouges. Tu es encore jeune, mais tu dois prendre soin de toi.\n\nUne bouff\u00e9e de chaleur enveloppa Wang Miao. Ses yeux \u00e9taient humides. C'\u00e9tait comme si son c\u0153ur, rudement mis \u00e0 l'\u00e9preuve ces deux derniers jours, venait d'\u00eatre d\u00e9pos\u00e9 dans un petit duvet douillet.\n\n\u2014 Professeur Ye, je viendrai vous voir souvent, dit-il en acceptant le coffret de ginseng.\n\n## 9\n\n## L'UNIVERS CLIGNOTE\n\nLa voiture de Wang Miao suivit la rue Jingmi jusqu'\u00e0 hauteur du district de Miyun, puis il prit la direction de Heilongtan et emprunta une petite route de montagne avant d'arriver \u00e0 l'observatoire radioastronomique du Centre d'observations astronomiques de l'Acad\u00e9mie nationale des sciences. Il remarqua une rang\u00e9e de vingt-huit antennes paraboliques, chacune de neuf m\u00e8tres de diam\u00e8tre, qui faisait penser \u00e0 une haie de majestueuses plantes en m\u00e9tal. Au bout de cette rang\u00e9e se dressaient deux hauts radiot\u00e9lescopes de cinquante m\u00e8tres de diam\u00e8tre construits en 2006. \u00c0 mesure qu'il se rapprochait, Wang Miao ne pouvait s'emp\u00eacher de repenser au fond de la photographie de Yang Dong et de sa m\u00e8re qu'il avait vue dans sa chambre.\n\nMais le travail de l'\u00e9tudiant de Ye Wenjie ne n\u00e9cessitait pas l'utilisation des antennes et des radiot\u00e9lescopes. Le laboratoire du Dr Sha Ruishan \u00e9tait principalement d\u00e9di\u00e9 \u00e0 la r\u00e9ception des donn\u00e9es d'observation de trois satellites : le Cobe (Cosmic Background Explorer), lanc\u00e9 en novembre 1989 et qui n'allait pas tarder \u00e0 \u00eatre d\u00e9mantel\u00e9, le WMAP (Wilkinson Microwave Anisotropy Probe) lanc\u00e9 en 2001 et enfin le satellite Planck, un observatoire spatial d\u00e9velopp\u00e9 par l'Agence spatiale europ\u00e9enne dont la mission \u00e9tait de cartographier les variations de temp\u00e9rature du fond diffus cosmologique.\n\nLe fond diffus cosmologique est un rayonnement \u00e9lectromagn\u00e9tique correspondant au rayonnement \u00e9mis par un corps noir stable \u00e0 une temp\u00e9rature de 2,726 K. Celui-ci est hautement isotrope, c'est-\u00e0-dire qu'il pr\u00e9sente les m\u00eames propri\u00e9t\u00e9s physiques dans toutes les directions. La fluctuation de temp\u00e9rature n'existe qu'\u00e0 l'ordre du cent milli\u00e8me de degr\u00e9. Le travail de Sha Ruishan consistait principalement \u00e0 cartographier de fa\u00e7on plus pr\u00e9cise le fond diffus cosmologique en s'appuyant sur les donn\u00e9es d'observation fournies par les satellites. Le laboratoire n'\u00e9tait pas tr\u00e8s grand. La salle principale regorgeait d'appareils dont la fonction \u00e9tait de r\u00e9colter les donn\u00e9es et trois terminaux informatiques y affichaient les informations respectives envoy\u00e9es par les trois satellites.\n\nEn apercevant Wang Miao, Sha Ruishan le traita avec une grande hospitalit\u00e9, propre \u00e0 ceux qui croisent un visiteur apr\u00e8s de longues p\u00e9riodes de travail solitaire. Il lui demanda quel type de donn\u00e9es il souhaitait pouvoir consulter.\n\n\u2014 Je souhaiterais pouvoir observer les variations de l'ensemble du fond diffus cosmologique.\n\n\u2014 Pouvez-vous... \u00eatre plus explicite ? demanda Sha Ruishan en regardant Wang Miao, intrigu\u00e9.\n\n\u2014 Eh bien, je souhaiterais observer les variations isotropes de l'ensemble du fond cosmologique, de un \u00e0 cinq pour cent.\n\nSha Ruishan lui sourit. Depuis le d\u00e9but du XXIe si\u00e8cle, l'observatoire astronomique de Miyun avait ouvert ses portes aux touristes et, pour arrondir ses fins de mois, Sha Ruishan servait souvent de guide ou de conf\u00e9rencier. Ce sourire, c'\u00e9tait sa mani\u00e8re de r\u00e9pondre aux questions fr\u00e9quentes des touristes \u2013 il avait fini par s'habituer \u00e0 leur effarant illettrisme scientifique.\n\n\u2014 Monsieur Wang, vous... n'\u00eates pas sp\u00e9cialiste de ce domaine, n'est-ce pas ?\n\n\u2014 Je travaille dans les nanotechnologies.\n\n\u2014 Ah, je comprends mieux. Mais vous avez probablement des connaissances de base concernant le fond diffus cosmologique ?\n\n\u2014 Pas beaucoup. Je sais que la th\u00e9orie actuelle de l'origine de l'univers estime sa naissance \u00e0 environ quatorze milliards d'ann\u00e9es avant notre \u00e8re. Aux premiers temps, la temp\u00e9rature de l'univers \u00e9tait tr\u00e8s \u00e9lev\u00e9e, puis elle a commenc\u00e9 \u00e0 refroidir et on appelle aujourd'hui fond diffus cosmologique les \"braises\" datant de cette \u00e9poque. Ce rayonnement fossile, diss\u00e9min\u00e9 dans l'univers tout entier, est observable dans sa longueur d'onde, exprim\u00e9e en centim\u00e8tres. Si je me souviens bien, je crois que c'est dans les ann\u00e9es 1960 que deux Am\u00e9ricains ont d\u00e9couvert par accident son existence alors qu'ils testaient l'antenne de r\u00e9ception d'un satellite de communication...\n\n\u2014 Vous en savez largement assez, l'interrompit Sha Ruishan d'un geste de la main. Vous devez donc \u00e9galement savoir que, contrairement aux petites variations locales que l'on peut parfois observer dans diff\u00e9rentes parties de l'univers, l'amplitude globale du rayonnement fossile est li\u00e9e \u00e0 l'expansion de l'univers. C'est une variation tr\u00e8s lente, mesurable \u00e0 l'\u00e9chelle de l'\u00e2ge de l'univers. Avec la pr\u00e9cision du satellite Planck, il est possible que, m\u00eame apr\u00e8s un million d'ann\u00e9es d'observation constante, cette variation n'ait toujours pas \u00e9t\u00e9 d\u00e9tect\u00e9e. Et vous voudriez constater d\u00e8s ce soir une fluctuation de l'ordre de cinq pour cent ? Savez-vous ce que cela signifierait ? Cela signifierait que l'univers se mettrait \u00e0 clignoter, comme un n\u00e9on sur le point de griller !\n\nEt de plus, qu'il se mettrait \u00e0 clignoter _pour moi_ , pensa Wang Miao.\n\n\u2014 Le Pr Ye doit me faire une blague, soupira Sha Ruishan en secouant la t\u00eate.\n\n\u2014 J'aimerais tant que ce ne soit qu'une blague, dit Wang Miao.\n\nIl \u00e9tait sur le point de lui r\u00e9v\u00e9ler ce qu'il n'avait pas os\u00e9 dire \u00e0 Ye Wenjie, mais il avait peur que Sha Ruishan ne refuse de l'aider. Il venait au moins de l\u00e2cher ce qu'il avait sur le c\u0153ur.\n\n\u2014 Bien, comme c'est le Pr Ye qui vous envoie, je vais vous laisser observer. Ce n'est pas bien compliqu\u00e9 ; pour une pr\u00e9cision de un pour cent, les donn\u00e9es envoy\u00e9es par l'antiquit\u00e9 qu'est Cobe seront suffisantes.\n\nEn m\u00eame temps qu'il parlait, Sha Ruishan s'activait sur le terminal. Tr\u00e8s vite, une ligne de couleur verte apparut sur l'\u00e9cran.\n\n\u2014 Cette courbe repr\u00e9sente l'\u00e9volution en temps r\u00e9el du fond diffus cosmologique. Oh, davantage qu'une courbe, il faudrait parler de ligne droite. La temp\u00e9rature est de 2,726 \u00b1 0,010 K. La marge d'erreur est la cons\u00e9quence, par effet Doppler, du mouvement de la Voie lact\u00e9e, qui a d\u00e9j\u00e0 \u00e9t\u00e9 d\u00e9compt\u00e9 du r\u00e9sultat. Si la fluctuation de un pour cent dont vous parliez devait avoir lieu, la ligne deviendrait rouge et se courberait sur l'\u00e9cran. Mais je suis pr\u00eat \u00e0 parier que la ligne restera verte jusqu'\u00e0 la fin des temps. Pour que cette variation soit visible \u00e0 l'\u0153il nu, il faudra attendre encore bien longtemps apr\u00e8s l'explosion du soleil.\n\n\u2014 Je ne vous d\u00e9range pas dans votre travail ?\n\n\u2014 Non, pas du tout. La pr\u00e9cision que vous demandez est tr\u00e8s faible. Vous pourrez consulter les donn\u00e9es de ce vieux tas de ferraille de Cobe. Bien, tout est pr\u00eat. Si, \u00e0 partir de maintenant, une fluctuation importante venait \u00e0 se produire, les donn\u00e9es la concernant seraient automatiquement sauvegard\u00e9es.\n\n\u2014 Il faudra peut-\u00eatre attendre 1 heure du matin.\n\n\u2014 Dites-moi, c'est tr\u00e8s pr\u00e9cis ! Pas de probl\u00e8me, je suis du service de nuit. Vous avez d\u00e9j\u00e0 d\u00een\u00e9 ? Bien, je vais vous faire visiter un peu.\n\nLa nuit \u00e9tait sans lune. Ils d\u00e9ambulaient le long de la rang\u00e9e d'antennes. Sha Ruishan les pointa du doigt en disant :\n\n\u2014 Impressionnant, n'est-ce pas ? Dommage que ce ne soient que des oreilles de sourd.\n\n\u2014 Pourquoi cela ?\n\n\u2014 Depuis leur construction, elles n'ont cess\u00e9 de subir des perturbations. Tout d'abord, la radiomessagerie \u00e0 la fin des ann\u00e9es 1980 et, aujourd'hui, l'explosion des communications mobiles. Ces radiot\u00e9lescopes ont plusieurs t\u00e2ches : faire des relev\u00e9s sur les objets c\u00e9lestes, d\u00e9tecter des sources radio variables, \u00e9tudier les r\u00e9manents de supernova... mais pour la majorit\u00e9 d'entre eux, ils sont inutilisables. Nous nous sommes plaints \u00e0 plusieurs reprises aupr\u00e8s de la Commission nationale de r\u00e9gulation de la radio, mais cela n'a servi \u00e0 rien. Comment lutter contre les compagnies China Mobile, China Unicom ou China Netcom ? Sans fric, les myst\u00e8res de l'univers, c'est du pipi de chat. Heureusement que mon projet d\u00e9pend des donn\u00e9es des satellites et que je n'ai aucun rapport avec ces \"attractions pour touristes\".\n\n\u2014 Ces derni\u00e8res ann\u00e9es, beaucoup de projets en recherche fondamentale ont r\u00e9ussi \u00e0 trouver des exploitations commerciales, comme pour le domaine de la physique des particules. Peut-\u00eatre aurait-il \u00e9t\u00e9 pr\u00e9f\u00e9rable de construire l'observatoire loin d'une grande ville.\n\n\u2014 Encore un probl\u00e8me d'argent. Aujourd'hui, notre seule voie de secours est de trouver des moyens techniques pour supprimer les perturbations. Dommage que le Pr Ye ne soit pas l\u00e0. Elle a obtenu des r\u00e9sultats remarquables dans ce domaine.\n\nLeur conversation s'attarda sur Ye Wenjie. Wang Miao apprit de la bouche de son \u00e9tudiant toutes les trag\u00e9dies qui avaient jalonn\u00e9 sa vie. Il \u00e9couta Sha Ruishan lui raconter comment elle avait assist\u00e9 \u00e0 la mort violente de son p\u00e8re pendant la R\u00e9volution culturelle, comment elle avait \u00e9t\u00e9 incrimin\u00e9e \u00e0 tort durant la p\u00e9riode o\u00f9 elle servait dans le Corps de production et de construction de Mongolie-Int\u00e9rieure et comment on avait perdu sa trace jusqu'\u00e0 sa r\u00e9apparition dans cette ville au d\u00e9but des ann\u00e9es 1990 pour enseigner l'astrophysique jusqu'\u00e0 sa retraite dans l'universit\u00e9 m\u00eame o\u00f9 avait travaill\u00e9 son p\u00e8re.\n\n\u2014 Je n'ai appris que tout r\u00e9cemment qu'elle avait pass\u00e9 plus de vingt ans dans la base de C\u00f4te Rouge.\n\n\u2014 C\u00f4te Rouge ? (Wang Miao, surpris, arr\u00eata de marcher.) Se pourrait-il que les rumeurs...\n\n\u2014 La plupart sont av\u00e9r\u00e9es. Un ancien chercheur qui a contribu\u00e9 au d\u00e9veloppement du syst\u00e8me de d\u00e9cryptage de C\u00f4te Rouge a \u00e9migr\u00e9 en Europe o\u00f9 il a \u00e9crit un livre qui a \u00e9t\u00e9 publi\u00e9 l'ann\u00e9e derni\u00e8re. La plupart des rumeurs dont vous parlez sont n\u00e9es avec ce livre. Pour ce que j'en sais, tout est cr\u00e9dible. Beaucoup de personnes ayant pris part au projet C\u00f4te Rouge sont encore en vie.\n\n\u2014 C'est vraiment... une l\u00e9gende fantastique !\n\n\u2014 Et encore plus pour cette \u00e9poque. C'est une l\u00e9gende dans la l\u00e9gende.\n\nIls discut\u00e8rent encore un moment, puis Sha Ruishan l'interrogea au sujet de son \u00e9trange requ\u00eate, mais Wang Miao \u00e9luda les questions et Sha Ruishan n'insista pas. Sa dignit\u00e9 de sp\u00e9cialiste ne lui permettait pas de poser trop de questions au sujet d'une requ\u00eate que ses connaissances professionnelles lui faisaient trouver non pertinente.\n\nIls se rendirent dans un bar pour touristes ouvert toute la nuit et y rest\u00e8rent durant deux heures. \u00c0 mesure qu'il enfilait les pintes de bi\u00e8re, Sha Ruishan devenait plus bavard. Mais Wang Miao \u00e9tait distrait depuis un long moment : la ligne verte du terminal occupait tout son esprit. Ce n'est qu'\u00e0 minuit cinquante que Sha Ruishan c\u00e9da enfin aux sollicitations insistantes de Wang Miao et accepta de retourner au laboratoire.\n\nLes projecteurs qui \u00e9clairaient les antennes avaient \u00e9t\u00e9 \u00e9teints et la rang\u00e9e d'antennes \u00e9tait devenue comme un dessin en deux dimensions avec le ciel nocturne pour d\u00e9cor, un chapelet de symboles abstraits tourn\u00e9s vers l'univers, dans l'attente d'une r\u00e9ponse. Cette sc\u00e8ne fit frissonner Wang Miao, qui se rappela les pendules du jeu des _Trois Corps_.\n\nIl \u00e9tait 1 heure pile lorsqu'ils revinrent dans le laboratoire. Alors qu'ils portaient leur regard sur l'\u00e9cran du terminal, une variation venait tout juste d'appara\u00eetre : la ligne droite \u00e9tait devenue une courbe, entrecoup\u00e9e \u00e0 intervalles irr\u00e9guliers de petits pics aigus. De verte elle \u00e9tait pass\u00e9e au rouge. On aurait dit un serpent se r\u00e9veillant apr\u00e8s une longue hibernation et rampant pour faire circuler son sang \u00e0 nouveau.\n\n\u2014 C'est certainement un d\u00e9faut de fonctionnement de Cobe, bredouilla Sha Ruishan en consid\u00e9rant la courbe avec \u00e9pouvante.\n\n\u2014 Ce n'est pas un dysfonctionnement, r\u00e9pondit calmement Wang Miao.\n\nArriv\u00e9 \u00e0 ce stade, il avait appris \u00e0 contr\u00f4ler ses \u00e9motions.\n\n\u2014 Nous allons tout de suite le savoir, dit Sha Ruishan.\n\nIl mit rapidement les deux autres terminaux en marche et il obtint bient\u00f4t les donn\u00e9es en temps r\u00e9el sur le fond diffus cosmologique fournies par WMAP et Planck. Il changea l'affichage en mode courbe. Les trois lignes affichaient les m\u00eames variations, absolument synchrones.\n\nSha Ruishan sortit son ordinateur portable qu'il d\u00e9marra les mains tremblantes. Il brancha un c\u00e2ble Ethernet et passa un coup de t\u00e9l\u00e9phone. Wang Miao comprit qu'il cherchait \u00e0 joindre l'observatoire d'Urumqi. Il attendit. Sha Ruishan ne lui donna aucune explication. Ses yeux \u00e9taient riv\u00e9s sur son appareil, comme si sa vie en d\u00e9pendait. Wang Miao pouvait entendre sa respiration haletante. Quelques minutes plus tard, une fen\u00eatre de navigation s'afficha sur l'\u00e9cran, puis une courbe rouge apparut. Elle suivait au d\u00e9tail pr\u00e8s les m\u00eames fluctuations que les trois autres.\n\nLes trois satellites et l'observatoire au sol confirmaient la m\u00eame r\u00e9alit\u00e9 : l'univers clignotait.\n\n\u2014 Peut-on imprimer ces courbes ? demanda Wang Miao.\n\nSha Ruishan essuya la sueur froide qui perlait \u00e0 son visage et hocha la t\u00eate. Il d\u00e9pla\u00e7a le curseur de la souris et cliqua sur la fonction impression. Wang Miao attendait avec impatience de saisir la premi\u00e8re feuille que recracherait l'imprimante laser et de tracer les courbes avec un crayon pour comparer les distances entre les pics de variation et l'alphabet morse.\n\n _Court-long-long-long-long, court-long-long-long-long, court-court-court-court-court, long-long-long-court-court, long-long-long-court-court-long-long, court-court-court-court-long, long-long-court-court-long-long, court-court-court-long-long, long-long-court-court-court_, c'est-\u00e0-dire 1108 : 21 : 37.\n\n _Court-long-long-long-long, court-long-long-long-long, court-court-court-court-court, long-long-long-court-court, long-long-long-court-court-long-long, court-court-court-court-long, long-long-court-court-long-long, court-court-court-long-long, long-court-court-court-court_, c'est-\u00e0-dire 1108 : 21 : 36.\n\n _Court-long-long-long-long, court-long-long-long-long, court-court-court-court-court, long-long-long-court-court, long-long-long-court-court-long-long, court-court-court-court-long, long-long-court-court-long-long, court-court-court-long-long, court-court-court-court-court_, c'est-\u00e0-dire 1108 : 21 : 35.\n\nLe compte \u00e0 rebours continuait mais, cette fois, \u00e0 l'\u00e9chelle de l'univers. Quatre-vingt-douze heures s'\u00e9taient d\u00e9j\u00e0 \u00e9coul\u00e9es, il n'en restait plus que mille cent huit.\n\nDebout \u00e0 c\u00f4t\u00e9 de lui, Sha Ruishan ne tenait plus en place. Il s'arr\u00eatait de temps en temps pour regarder derri\u00e8re l'\u00e9paule de Wang Miao qui tra\u00e7ait ses lignes de chiffres. N'y tenant plus, il hurla enfin :\n\n\u2014 Ne pouvez-vous pas me dire ce qui se passe ?\n\n\u2014 Docteur Sha, croyez-moi, je ne peux pas vous l'expliquer.\n\nWang Miao repoussa la feuille sur laquelle avaient \u00e9t\u00e9 imprim\u00e9es les courbes de variation du fond diffus cosmologique, puis il fixa les suites de chiffres du compte \u00e0 rebours en disant :\n\n\u2014 Peut-\u00eatre que les trois satellites et le t\u00e9lescope d'Urumqi ont tous eu une d\u00e9faillance au m\u00eame moment.\n\n\u2014 Vous savez bien que c'est impossible !\n\n\u2014 Et si quelqu'un avait sabot\u00e9 les syst\u00e8mes ?\n\n\u2014 Encore impossible ! Saboter en m\u00eame temps les donn\u00e9es de trois satellites et d'un observatoire au sol ? Ce serait l'\u0153uvre d'une force surnaturelle !\n\nWang Miao hocha la t\u00eate. Il pr\u00e9f\u00e9rait croire \u00e0 un acte surnaturel plut\u00f4t qu'\u00e0 un clignotement de l'univers. Mais Sha Ruishan lui \u00f4ta aussit\u00f4t ce dernier espoir.\n\n\u2014 Pour s'en assurer d\u00e9finitivement, c'est simple. La variation est telle qu'il est possible de la voir \u00e0 l'\u0153il nu.\n\n\u2014 Qu'est-ce que vous racontez ? Maintenant, c'est vous qui divaguez ! La longueur d'onde du rayonnement fossile est de sept centim\u00e8tres, c'est de sept ou huit ordres de grandeur sup\u00e9rieur \u00e0 la longueur d'onde de la lumi\u00e8re visible, comment pourrions-nous la voir \u00e0 l'\u0153il nu ?\n\n\u2014 Avec des lunettes 3K.\n\n\u2014 Des lunettes 3K ?\n\n\u2014 C'est un petit accessoire cr\u00e9\u00e9 par le plan\u00e9tarium de P\u00e9kin, dans un but de vulgarisation scientifique. Gr\u00e2ce aux technologies d'aujourd'hui, ils ont pu reproduire \u00e0 la taille des lunettes l'antenne cornet de Penzias et Wilson qui a permis la d\u00e9couverte du fond diffus cosmologique, il y a plus de quarante ans. Un convertisseur a \u00e9t\u00e9 install\u00e9 dans les verres pour compresser les longueurs d'onde de sept ordres de grandeur, de telle sorte que les longueurs d'onde de sept centim\u00e8tres sont visibles sous la forme de lumi\u00e8re rouge. Par cons\u00e9quent, la nuit, quand le public enfile les lunettes, il peut voir le rayonnement \u00e0 3K de ses propres yeux. Si les fluctuations sont telles qu'elles apparaissent sur le terminal, nous devrions voir l'univers clignoter \u00e0 travers les lunettes.\n\n\u2014 O\u00f9 se trouvent ces lunettes ?\n\n\u2014 Au plan\u00e9tarium, il y en a une vingtaine de paires.\n\n\u2014 J'ai besoin de pouvoir les utiliser avant 5 heures du matin.\n\nSha Ruishan passa un coup de t\u00e9l\u00e9phone. Son interlocuteur mit longtemps \u00e0 d\u00e9crocher et Sha Ruishan dut ensuite batailler ferme pour convaincre l'homme qu'il avait r\u00e9veill\u00e9 en pleine nuit d'attendre Wang Miao au plan\u00e9tarium dans une heure.\n\nAu moment de partir, Sha Ruishan dit \u00e0 Wang Miao :\n\n\u2014 Je ne viens pas avec vous. Ce soir, j'en ai d\u00e9j\u00e0 trop vu, je n'ai pas besoin de cette derni\u00e8re preuve. J'esp\u00e8re que vous pourrez me dire la v\u00e9rit\u00e9 le moment venu. Si ce ph\u00e9nom\u00e8ne produit des r\u00e9sultats de recherche particuliers, je ne vous oublierai pas.\n\n\u2014 Le clignotement cessera \u00e0 5 heures du matin. Ne perdez pas de temps \u00e0 essayer de chercher une raison \u00e0 ce ph\u00e9nom\u00e8ne, croyez-moi, vous ne trouverez rien d'int\u00e9ressant, lui confia Wang Miao en ouvrant la porti\u00e8re de la voiture.\n\nSha Ruishan fixa Wang Miao un long moment, puis il acquies\u00e7a :\n\n\u2014 Je comprends, il se passe des choses \u00e9tranges dans le monde de la science en ce moment...\n\n\u2014 Oui, r\u00e9pondit simplement Wang Miao en s'engouffrant dans la voiture.\n\nIl n'avait pas envie de prolonger cette discussion.\n\n\u2014 C'est notre tour ?\n\n\u2014 C'est au moins le mien, dit Wang Miao en mettant le moteur en marche.\n\nUne heure plus tard, Wang Miao arriva en centre-ville et se gara devant le plan\u00e9tarium. Les lumi\u00e8res de la ville filtraient \u00e0 travers les parois transparentes de cette \u00e9norme b\u00e2tisse en verre et laissaient vaguement deviner l'architecture int\u00e9rieure. Wang Miao se fit \u00e0 cet instant la r\u00e9flexion que si l'architecte avait voulu exprimer l'impression que lui inspirait l'univers, c'\u00e9tait r\u00e9ussi. Plus les choses sont transparentes, plus elles sont myst\u00e9rieuses. L'univers lui-m\u00eame est transparent : votre regard peut porter aussi loin que le permet votre instrument de vision. Mais plus vous regardez l'univers, plus il vous semble myst\u00e9rieux.\n\nL'employ\u00e9 du plan\u00e9tarium l'attendait devant l'entr\u00e9e, les yeux bouffis de sommeil. Il tendit une valise \u00e0 Wang Miao :\n\n\u2014 Il y a cinq paires de lunettes 3K \u00e0 l'int\u00e9rieur, elles sont toutes recharg\u00e9es. Le bouton de gauche, c'est pour allumer ou \u00e9teindre, le droit, pour r\u00e9gler la luminosit\u00e9. Il y a encore une dizaine de paires l\u00e0-haut, regardez ce que vous voulez, je retourne dormir, je suis dans la chambre \u00e0 c\u00f4t\u00e9 de l'entr\u00e9e. Quelque chose ne tourne pas rond chez ce Dr Sha.\n\nUne fois sa phrase achev\u00e9e, il rentra dans le b\u00e2timent obscur.\n\nWang Miao posa la valise sur le fauteuil de sa voiture, l'ouvrit et prit une paire de lunettes. L'accessoire ressemblait au casque \u00e0 vis\u00e9e panoramique de la V-combinaison. Il sortit du v\u00e9hicule et alla les essayer \u00e0 l'ext\u00e9rieur. Le paysage nocturne de la ville derri\u00e8re les verres n'avait pas chang\u00e9, il \u00e9tait seulement un peu plus sombre. Il se souvint soudain qu'il ne les avait pas allum\u00e9es.\n\nAussit\u00f4t, la ville lui apparut comme un amas de faisceaux brumeux. Il r\u00e9gla la luminosit\u00e9, mais certaines lumi\u00e8res continuaient \u00e0 clignoter ou \u00e0 scintiller. Il savait que ces sources d'\u00e9mission poss\u00e9daient des longueurs d'onde mesurables en centim\u00e8tres et qu'elles \u00e9taient converties en lumi\u00e8res visibles. \u00c0 chaque halo correspondait une source d'\u00e9mission mais, en raison de la taille de leur longueur d'onde, il \u00e9tait impossible d'en distinguer clairement la forme.\n\nIl leva la t\u00eate et vit un ciel miroitant de lumi\u00e8res rouge sombre. Voil\u00e0, il voyait \u00e0 pr\u00e9sent le fond diffus cosmologique, ce rayonnement vieux de plus de dix milliards d'ann\u00e9es, le vestige du Big Bang, les seules braises restantes du grand feu de la Gen\u00e8se. Il ne distinguait aucune \u00e9toile. Les lumi\u00e8res normalement visibles \u00e0 l'\u0153il nu ayant \u00e9t\u00e9 compress\u00e9es pour devenir invisibles, les \u00e9toiles auraient d\u00fb lui appara\u00eetre derri\u00e8re les lunettes sous la forme de taches noires, mais la diffraction des rayonnements aux longueurs d'onde d'ordre centim\u00e9trique submergeait toutes les formes et tous les d\u00e9tails.\n\nLorsque les yeux de Wang Miao se furent habitu\u00e9s, il vit la lumi\u00e8re rouge en arri\u00e8re-plan du ciel se mettre \u00e0 clignoter. C'\u00e9tait tout l'espace qui se mettait \u00e0 ciller de fa\u00e7on synchrone, comme si l'univers tout entier n'\u00e9tait qu'une lanterne solitaire dans le vent.\n\nDebout sous les lueurs vacillantes du ciel, Wang Miao eut soudain l'impression que l'univers \u00e9tait petit, si minuscule m\u00eame qu'il avait l'impression d'\u00eatre retenu \u00e0 l'int\u00e9rieur comme dans une prison. L'univers \u00e9tait un c\u0153ur ratatin\u00e9, un ut\u00e9rus \u00e9troit, parcouru de vaisseaux sanguins \u00e0 moiti\u00e9 translucides. En suspension dans le sang, il s'aper\u00e7ut que les lumi\u00e8res rouges clignotaient de fa\u00e7on erratique, comme si les pulsations de ce c\u0153ur, les contractions de cet ut\u00e9rus \u00e9taient irr\u00e9guli\u00e8res. Wang Miao sentit alors une pr\u00e9sence si d\u00e9mesur\u00e9e et si \u00e9trange qu'elle \u00e9tait destin\u00e9e \u00e0 ne jamais \u00eatre comprise par l'entendement humain.\n\nWang Miao \u00f4ta les lunettes 3K. Il s'assit f\u00e9brilement sur le sol, le dos appuy\u00e9 contre sa voiture. Dans ses yeux, la ville avait retrouv\u00e9 l'aspect r\u00e9aliste que les lumi\u00e8res visibles lui permettaient de d\u00e9crire, mais son regard vagabondait, comme s'il souhaitait capturer autre chose. En face, l'un des n\u00e9ons indiquant l'entr\u00e9e du zoo \u00e9tait presque grill\u00e9 et papillotait de fa\u00e7on irr\u00e9guli\u00e8re. Non loin, les feuilles d'un arbuste fr\u00e9missaient dans la brise nocturne en refl\u00e9tant la lumi\u00e8re des lampadaires qui, elle aussi, semblait scintiller de mani\u00e8re instable ; au loin, l'\u00e9toile \u00e0 cinq branches au sommet du b\u00e2timent d'architecture sovi\u00e9tique du Centre des expositions de P\u00e9kin refl\u00e9tait \u00e9galement les lumi\u00e8res des phares des voitures et clignotait par intermittence... Wang Miao s'\u00e9vertuait \u00e0 essayer de d\u00e9chiffrer le code en morse de ces clignotements. Il avait m\u00eame l'impression que les rides sur les drapeaux publicitaires volant au vent ou les ondulations \u00e0 la surface des flaques d'eau lui envoyaient des messages en morse. Tandis qu'il essayait de d\u00e9crypter ces signaux, le compte \u00e0 rebours fant\u00f4me continuait \u00e0 s'\u00e9couler devant ses yeux.\n\nAu bout d'un certain temps \u2013 il ignorait combien \u2013, l'employ\u00e9 du plan\u00e9tarium vint trouver Wang Miao pour lui demander s'il avait termin\u00e9. Quand il le vit dans cet \u00e9tat, son regard ensommeill\u00e9 s'\u00e9vanouit, il ramassa la valise contenant les lunettes, observa encore Wang Miao quelques secondes et retourna \u00e0 l'int\u00e9rieur en pressant le pas.\n\nWang Miao prit son t\u00e9l\u00e9phone et appela Shen Yufei. Elle d\u00e9crocha presque aussit\u00f4t. Peut-\u00eatre qu'elle non plus n'arrivait pas \u00e0 trouver le sommeil.\n\n\u2014 Que se passe-t-il \u00e0 la fin du compte \u00e0 rebours ? demanda Wang Miao, \u00e0 bout de forces.\n\n\u2014 Je ne sais pas, r\u00e9pondit-elle laconiquement avant de raccrocher.\n\nQu'est-ce que cela pouvait \u00eatre ? Peut-\u00eatre que c'\u00e9tait sa propre mort qui l'attendait, comme pour Yang Dong ? Peut-\u00eatre qu'une catastrophe allait avoir lieu, comme le grand tsunami de l'oc\u00e9an Indien quelques ann\u00e9es plus t\u00f4t ? Qui pouvait faire le lien avec son projet de recherche sur les nanotechnologies ? (Et ainsi, le fil de sa pens\u00e9e l'amena \u00e0 se demander si les grandes catastrophes du pass\u00e9, y compris les deux grandes guerres mondiales, n'avaient pas elles aussi \u00e9t\u00e9 les issues d'un compte \u00e0 rebours fant\u00f4me dont un \u00eatre insignifiant et insoup\u00e7onn\u00e9 comme lui devait assumer l'ultime responsabilit\u00e9.)\n\nPeut-\u00eatre que le monde allait imploser ? Dans cet univers monstrueux, ce serait peut-\u00eatre une d\u00e9livrance...\n\nUne chose \u00e9tait certaine. Quel que soit le terme du compte \u00e0 rebours, durant le millier d'heures qui lui restaient, ces sp\u00e9culations le tortureraient aussi cruellement que des d\u00e9mons et finiraient par l'an\u00e9antir mentalement.\n\nWang Miao retourna dans sa voiture et quitta le plan\u00e9tarium. Il roulait au hasard. \u00c0 cette heure, la chauss\u00e9e \u00e9tait d\u00e9serte, mais il n'osait pas conduire trop vite, comme si la vitesse de son v\u00e9hicule risquait d'acc\u00e9l\u00e9rer celle du compte \u00e0 rebours. Lorsque les premiers rayons du soleil point\u00e8rent \u00e0 l'est, il s'arr\u00eata au bord de la route et descendit pour marcher, toujours aveugl\u00e9ment. Dans son esprit vide ne demeurait que l'image du compte \u00e0 rebours suspendu devant le fond diffus cosmologique rouge sombre et rapetissant jusqu'\u00e0 prendre la taille d'un simple petit minuteur, un glas dont on ignorait pour qui il sonnait. Le jour se leva. Il \u00e9tait fatigu\u00e9 de marcher et s'assit sur un banc. Quand il leva la t\u00eate pour voir o\u00f9 l'avait men\u00e9 son subconscient, il ne put retenir un frisson.\n\nIl \u00e9tait assis devant l'\u00e9glise Saint-Joseph de Wangfujing. Sous la lueur encore p\u00e2le du matin, les trois clochers de style roman apparaissaient comme trois gigantesques doigts noirs d\u00e9signant quelque chose d'invisible dans le ciel.\n\nWang Miao se leva pour s'en aller, mais un chant sacr\u00e9 \u00e9manant de l'\u00e9glise le retint. Ce n'\u00e9tait pourtant pas dimanche. La chorale r\u00e9p\u00e9tait peut-\u00eatre en pr\u00e9vision de P\u00e2ques, car le refrain en question \u00e9tait celui d'un chant souvent utilis\u00e9 durant cette f\u00eate : \"Tenir une lampe allum\u00e9e\". Alors qu'il entendait ce chant solennel et lointain, Wang Miao sentit \u00e0 nouveau l'univers se r\u00e9tr\u00e9cir. Il \u00e9tait d\u00e9sormais devenu une \u00e9glise d\u00e9serte dont le d\u00f4me s'\u00e9clipsait derri\u00e8re la lumi\u00e8re rouge et clignotante du fond cosmologique, tandis que lui, fourmi insignifiante, rampait entre les fissures du plancher. Il sentait une colossale main invisible caresser son c\u0153ur tr\u00e9pidant. Tout \u00e0 coup, il redevenait un enfant faible et vuln\u00e9rable. Dans les profondeurs de sa conscience, quelque chose de dur s'\u00e9tait ramolli, comme de la cire br\u00fbl\u00e9e. Il prit son visage entre les mains et se mit \u00e0 pleurer.\n\n\u2014 Ha ha ha ha, encore un qui craque !\n\nUn \u00e9clat de rire interrompit les lamentations de Wang Miao.\n\nIl tourna la t\u00eate. Shi Qiang \u00e9tait debout derri\u00e8re lui, crachant la fum\u00e9e blanche de sa cigarette.\n\n## 10\n\n## SHI QIANG\n\nShi Qiang s'assit \u00e0 c\u00f4t\u00e9 de Wang Miao. Il lui tendit ses clefs de voiture :\n\n\u2014 Elle \u00e9tait gar\u00e9e au carrefour de Dongdan. Un peu plus et c'\u00e9tait la fourri\u00e8re.\n\n _Ah, Shi Qiang, vous savoir derri\u00e8re moi me r\u00e9conforte au moins un peu_ , pensa Wang Miao. Mais par dignit\u00e9, il ne pronon\u00e7a pas cette phrase \u00e0 haute voix. Il accepta la cigarette que lui proposait Shi Qiang. Sa premi\u00e8re bouff\u00e9e apr\u00e8s des ann\u00e9es de sevrage.\n\n\u2014 Que vous arrive-t-il, mon vieux ? Vous flanchez, hein ? Je vous avais pr\u00e9venu, mais vous n'avez voulu en faire qu'\u00e0 votre t\u00eate.\n\n\u2014 Vous ne comprendriez pas, dit Wang Miao en tirant furieusement sur la cigarette.\n\n\u2014 Votre probl\u00e8me \u00e0 vous, c'est que vous comprenez trop bien... Bon, allons casser une graine.\n\n\u2014 Je n'ai pas faim.\n\n\u2014 Bon, alors, allons boire un coup, c'est moi qui invite.\n\nWang Miao monta dans la voiture de Shi Qiang, qui le conduisit dans un petit restaurant en bord de mer, vide \u00e0 cette heure si matinale.\n\n\u2014 Deux portions de tripes de porc frites ! Deux bouteilles d' _erguotou_1 ! cria Shi Qiang sans lever la t\u00eate, il paraissait manifestement familier des lieux.\n\nEn voyant arriver deux assiettes remplies d'une pitance noir\u00e2tre, Wang Miao eut l'estomac retourn\u00e9, il manqua presque de vomir. Shi Qiang lui commanda un verre de lait de soja chaud et une portion de beignets frits dont Wang Miao avala \u00e0 contrec\u0153ur quelques bouch\u00e9es, puis ils commenc\u00e8rent \u00e0 boire verre sur verre. Il se sentait devenir plus l\u00e9ger et l'alcool aidant, il raconta au commissaire tout ce qui s'\u00e9tait pass\u00e9 ces trois derniers jours, bien qu'il soit conscient que celui-ci \u00e9tait probablement d\u00e9j\u00e0 au fait de tout \u2013 peut-\u00eatre m\u00eame en savait-il davantage que lui.\n\n\u2014 Vous voulez dire que l'univers vous faisait des clins d'\u0153il ? demanda Shi Qiang en levant la t\u00eate de son assiette et en aspirant les tripes comme des nouilles.\n\n\u2014 C'est une m\u00e9taphore appropri\u00e9e.\n\n\u2014 Conneries !\n\n\u2014 Votre sang-froid provient de votre ignorance.\n\n\u2014 Encore des conneries, allez, cul sec !\n\nWang Miao finit son verre. D\u00e9sormais, le monde tournait autour de lui comme un man\u00e8ge. Le seul \u00e0 garder un semblant d'\u00e9quilibre \u00e9tait Shi Qiang en train de d\u00e9vorer ses tripes en face de lui.\n\n\u2014 Shi Qiang ! Avez-vous d\u00e9j\u00e0... r\u00e9fl\u00e9chi \u00e0 des questions philosophiques ultimes ? Par exemple, d'o\u00f9 vient l'humanit\u00e9, o\u00f9 va-t-elle ? D'o\u00f9 vient l'univers, o\u00f9 va-t-il ? Ce genre de questions ?\n\n\u2014 Non.\n\n\u2014 Jamais ?\n\n\u2014 Jamais.\n\n\u2014 Vous regardez quand m\u00eame les \u00e9toiles, n'est-ce pas ? Elles ne vous ont jamais inspir\u00e9 de l'admiration ou de la curiosit\u00e9 ?\n\n\u2014 La nuit, je ne regarde pas le ciel.\n\n\u2014 Comment est-ce possible ? Je croyais que vous travailliez souvent la nuit ?\n\n\u2014 Mon vieux, si je levais la t\u00eate vers le ciel quand je bosse, qui attraperait les suspects ?\n\n\u2014 On ne peut vraiment pas avoir de discussion avec vous. Allez, cul sec !\n\n\u2014 En fait, m\u00eame si je regardais les \u00e9toiles, je ne me torturerais pas l'esprit avec ces questions philosophiques, j'ai d\u00e9j\u00e0 suffisamment de soucis comme \u00e7a : le remboursement de mon pr\u00eat immobilier, le gamin qui va entrer \u00e0 l'universit\u00e9, sans parler des enqu\u00eates qui n'en finissent pas... Je suis un homme simple et direct, on voit mon cul au fond de ma gorge. Naturellement, je me fous bien de savoir si mes chefs m'appr\u00e9cient. \u00c7a fait des ann\u00e9es que je suis comme \u00e7a, depuis que j'ai \u00e9t\u00e9 d\u00e9mobilis\u00e9 de l'arm\u00e9e... Si je n'\u00e9tais pas bon dans mon boulot, \u00e7a fait longtemps qu'on m'aurait foutu dehors... J'ai bien d'autres pr\u00e9occupations. Vous croyez que j'ai encore le temps de m'int\u00e9resser \u00e0 la philosophie ?\n\n\u2014 Vous avez raison. Trinquons.\n\n\u2014 Mais \u00e7a ne m'a pas emp\u00each\u00e9 d'inventer une th\u00e9orie ultime.\n\n\u2014 Dites-moi.\n\n\u2014 Quand il y a une embrouille, c'est forc\u00e9ment qu'il y a un loup.\n\n\u2014 Qu'est-ce que... c'est que cette th\u00e9orie \u00e0 la con ?\n\n\u2014 Quand je dis qu'il y a \"un loup\", je veux dire par l\u00e0 qu'il y a forc\u00e9ment quelqu'un derri\u00e8re qui tire les ficelles.\n\n\u2014 Si vous aviez un minimum de connaissances scientifiques, vous sauriez qu'aucune force n'est capable de r\u00e9aliser ces deux exploits. Surtout le dernier. Contr\u00f4ler les choses \u00e0 l'\u00e9chelle de l'univers ! Non seulement l'\u00e9tat des sciences actuel ne permet pas de l'expliquer, mais je n'arrive pas \u00e0 concevoir une autre explication, m\u00eame surnaturelle. \u00c7a va au-del\u00e0 de tout ce que je peux imaginer.\n\n\u2014 M\u00eame chose : conneries ! Des ph\u00e9nom\u00e8nes bizarres, j'en ai vu des tonnes !\n\n\u2014 Alors, dites-moi, quelle est ma prochaine \u00e9tape ?\n\n\u2014 Continuer \u00e0 boire et, quand vous aurez fini, dormir.\n\n\u2014 Bien.\n\nWang Miao ne savait pas comment il avait fait pour revenir dans sa voiture. Il s'allongea sur la banquette arri\u00e8re et sombra dans un sommeil profond et sans r\u00eave. Il avait l'impression qu'il n'avait pas dormi longtemps, mais quand il entrouvrit les yeux, il vit que le soleil allait bient\u00f4t se coucher \u00e0 l'ouest de la ville.\n\nIl descendit de voiture. Il \u00e9tait encore un peu faible \u00e0 cause de l'alcool ingurgit\u00e9 le matin m\u00eame, mais il se sentait bien mieux. Il remarqua qu'il se trouvait non loin d'un angle ext\u00e9rieur de la Cit\u00e9 interdite. Les rayons du cr\u00e9puscule formaient des rides dor\u00e9es \u00e0 la surface des douves de l'ancien palais. Dans ses yeux, le monde retrouva son antique \u00e9quilibre. Alors que le ciel commen\u00e7ait \u00e0 s'assombrir et qu'il profitait de ce moment de paix dont il n'avait pas joui depuis longtemps, une Santana noire famili\u00e8re \u00e9mergea soudain de la file de voitures et vint freiner juste \u00e0 c\u00f4t\u00e9 de lui. Shi Qiang en sortit.\n\n\u2014 Bien dormi ? ronchonna Shi Qiang.\n\n\u2014 Oui. Et maintenant, que dois-je faire ?\n\n\u2014 Qui ? Vous ? Aller manger, boire encore un peu, et dormir.\n\n\u2014 Et apr\u00e8s ?\n\n\u2014 Apr\u00e8s ? Il faudra bien retourner travailler, non ?\n\n\u2014 Le compte \u00e0 rebours en est d\u00e9j\u00e0 maintenant \u00e0... 1 091 heures.\n\n\u2014 Au diable le compte \u00e0 rebours ! Assurez-vous d'abord que vous \u00eates capable de garder l'\u00e9quilibre et de ne pas sombrer, nous parlerons du reste apr\u00e8s !\n\n\u2014 Shi Qiang, ne pourriez-vous pas me dire une partie de la v\u00e9rit\u00e9 ? Je vous en supplie.\n\nShi Qiang le fixa puis leva les yeux au ciel en riant :\n\n\u2014 Cette question, je ne sais pas combien de fois je l'ai pos\u00e9e \u00e0 Chang Weisi. Nous sommes dans la m\u00eame gal\u00e8re, je vous le dis en toute franchise : je n'ai pas la moindre foutue id\u00e9e de ce que si se passe. Je suis au bas de la hi\u00e9rarchie, ils ne me l\u00e2chent rien. Un vrai cauchemar !\n\n\u2014 Mais vous en savez certainement plus que moi.\n\n\u2014 Bon. Je vais vous raconter ce que je sais de plus que vous.\n\nShi Qiang indiqua la berge des douves, o\u00f9 ils all\u00e8rent s'asseoir ensemble. La nuit \u00e9tait tomb\u00e9e. Derri\u00e8re eux s'\u00e9coulait une rivi\u00e8re de phares de voitures. Ils pouvaient voir leurs silhouettes s'allonger et raccourcir \u00e0 la surface de l'eau.\n\n\u2014 Mon job, c'est de relier des choses qui n'ont \u00e0 premi\u00e8re vue pas grand-chose en commun. Une fois le puzzle reconstitu\u00e9, la v\u00e9rit\u00e9 appara\u00eet sous un nouveau jour. Il se passe beaucoup de choses bizarres ces derniers temps. Parmi les plus importantes, cette recrudescence sans pr\u00e9c\u00e9dent des attaques contre les institutions de recherche scientifique et le monde acad\u00e9mique. Bien s\u00fbr, vous avez entendu parler de l'affaire du bombardement de Liangxiang et du meurtre du laur\u00e9at du prix Nobel... Les motivations des criminels sont \u00e9tranges : ils ne tuent pas pour l'argent, ni par vengeance, ni m\u00eame pour des raisons politiques, mais pour semer la panique.\n\nIl y a aussi d'autres affaires qui ne sont pas des crimes. La vague de suicides des membres de la Soci\u00e9t\u00e9 des fronti\u00e8res de la science, par exemple. Les activit\u00e9s des militants \u00e9cologistes connaissent \u00e9galement depuis quelque temps un rebond \u00e9tonnant. Quand ils ne vont pas manifester sur le chantier d'un barrage hydro\u00e9lectrique ou d'une centrale nucl\u00e9aire, ils cr\u00e9ent je ne sais quelles soci\u00e9t\u00e9s exp\u00e9rimentales pour soi-disant \"retourner \u00e0 la nature\"... Et puis il y a des ph\u00e9nom\u00e8nes en apparence insignifiants : vous \u00eates all\u00e9 au cin\u00e9ma derni\u00e8rement ?\n\n\u2014 J'y vais rarement.\n\n\u2014 Beaucoup de superproductions r\u00e9centes ont des th\u00e8mes tr\u00e8s rustiques. Les histoires se d\u00e9roulent dans des montagnes verdoyantes ou au bord de ruisseaux d'eau claire o\u00f9 de beaux jeunes gens travaillent la terre, \u00e0 une \u00e9poque ind\u00e9termin\u00e9e o\u00f9 l'humanit\u00e9 vivait en harmonie avec la nature... Pour reprendre les termes des r\u00e9alisateurs, ces mondes \"repr\u00e9sentent la beaut\u00e9 de la vie telle qu'elle \u00e9tait avant son viol par la technologie\". Prenez par exemple _Le Jardin aux fleurs de p\u00eacher_ , c'est un film dont on savait d'avance que personne n'irait le voir, mais cela n'a pas emp\u00each\u00e9 d'investir des centaines de millions dedans. Et puis la cr\u00e9ation r\u00e9cente du grand prix du roman de science-fiction, dont le vainqueur peut gagner cinq millions de yuans ! Celui qui \u00e9crit l'avenir le plus pessimiste remporte le prix ! Et puis vous allez voir qu'ils vont encore claquer des millions pour l'adapter au cin\u00e9ma... Des sectes bizarro\u00efdes commencent \u00e0 pousser comme des champignons, leurs gourous s'en mettent plein les poches...\n\n\u2014 Et tout cela est li\u00e9 aux \u00e9v\u00e9nements dont vous avez parl\u00e9 au d\u00e9but ?\n\n\u2014 Il faut relier les ph\u00e9nom\u00e8nes et les observer ensemble. Bien entendu, je n'avais pas \u00e0 me pr\u00e9occuper de tout \u00e7a avant mais, depuis que j'ai \u00e9t\u00e9 transf\u00e9r\u00e9 de la brigade criminelle \u00e0 l'unit\u00e9 antiterroriste du Centre d'op\u00e9rations militaires, \u00e7a fait partie de mon job. Je dois connecter les \u00e9v\u00e9nements, je suis dou\u00e9 pour \u00e7a, m\u00eame le g\u00e9n\u00e9ral Chang Weisi est forc\u00e9 de le reconna\u00eetre.\n\n\u2014 Et votre conclusion ?\n\n\u2014 Tout est coordonn\u00e9 en coulisses par une seule et m\u00eame personne qui a un but pr\u00e9cis : d\u00e9truire compl\u00e8tement la recherche scientifique.\n\n\u2014 Qui ?\n\n\u2014 Je l'ignore, je l'ignore absolument, mais je peux sentir son plan. Un plan impressionnant, un plan global : an\u00e9antir les \u00e9quipements scientifiques, vous assassiner, vous, les chercheurs, ou bien vous contraindre au suicide, vous faire devenir fous, mais avant tout vous d\u00e9tourner de vos recherches, jusqu'\u00e0 ce que vous deveniez encore plus stupides que les gens ordinaires.\n\n\u2014 Merci pour la lucidit\u00e9 de ce jugement...\n\n\u2014 Et dans le m\u00eame temps, l'ennemi veut aussi jeter l'opprobre sur la science dans la soci\u00e9t\u00e9. Bien s\u00fbr, il y a toujours plus ou moins eu des gens pour le faire mais, cette fois-ci, je suis persuad\u00e9 que c'est organis\u00e9.\n\n\u2014 Je vous crois.\n\n\u2014 Ah, vous me croyez maintenant ! \u00c7a veut dire que moi qui ai arr\u00eat\u00e9 mes \u00e9tudes apr\u00e8s le lyc\u00e9e, j'ai r\u00e9ussi \u00e0 d\u00e9couvrir ce que vous autres grands scientifiques n'\u00eates pas parvenus \u00e0 voir ? Quand j'ai propos\u00e9 ma th\u00e9orie, ils \u00e9taient nombreux, les officiers et les scientifiques, \u00e0 se foutre de moi !\n\n\u2014 Si vous m'en aviez parl\u00e9 avant, je ne vous aurais pas ri au nez. Vous avez sans doute d\u00e9j\u00e0 entendu parler des pseudosciences. Vous savez qui sont ceux que les pseudoscientifiques craignent le plus ?\n\n\u2014 Les vrais scientifiques, non ?\n\n\u2014 Faux. Il y a dans le monde beaucoup de scientifiques de premier ordre qui se sont laiss\u00e9 berner par des pseudoscientifiques et qui leur ont m\u00eame servi de porte-drapeaux. Les adeptes de la pseudoscience craignent une cat\u00e9gorie de personnes qu'ils ont beaucoup de mal \u00e0 duper : les magiciens. La plupart des mensonges des pseudosciences ont d'ailleurs \u00e9t\u00e9 r\u00e9v\u00e9l\u00e9s par des magiciens. Par rapport \u00e0 nous autres rats de biblioth\u00e8que du monde acad\u00e9mique, vos nombreuses ann\u00e9es pass\u00e9es \u00e0 maintenir l'ordre dans la soci\u00e9t\u00e9 vous ont largement permis de mettre au jour ce genre de grand projet criminel.\n\n\u2014 Bah, des gens plus fut\u00e9s que moi, il y en a des tas. \u00c7a fait un moment que mes sup\u00e9rieurs ont compris ce qui se passait. S'ils se sont moqu\u00e9s de moi, c'est parce que je ne pr\u00e9sentais pas ma th\u00e9orie aux bons interlocuteurs, voil\u00e0 tout. Je les ai trouv\u00e9s plus tard, mais le g\u00e9n\u00e9ral m'a transf\u00e9r\u00e9 dans cette unit\u00e9 o\u00f9 je dois me contenter de faire des rondes... Bien, voil\u00e0 le peu de choses que je sais sur cette affaire.\n\n\u2014 Une question : pourquoi l'arm\u00e9e est-elle impliqu\u00e9e ?\n\n\u2014 Figurez-vous que je me suis aussi pos\u00e9 la question. Je leur ai demand\u00e9. Ils m'ont r\u00e9pondu que la guerre \u00e9tait d\u00e9clar\u00e9e et que, naturellement, la guerre relevait des comp\u00e9tences de l'arm\u00e9e. Comme vous, je me suis dit au d\u00e9but qu'ils d\u00e9liraient, mais je peux vous assurer qu'ils ne plaisantent vraiment pas. \u00c0 l'heure qu'il est, toutes les unit\u00e9s militaires sont mobilis\u00e9es. Il existe dans le monde plus d'une vingtaine de centres d'op\u00e9rations militaires comme celui dont je fais partie et il y a au-dessus d'eux une autre structure. Mais personne ne conna\u00eet les d\u00e9tails.\n\n\u2014 Qui est l'ennemi ?\n\n\u2014 Aucune id\u00e9e. Des officiers de l'Otan stationnent dans la salle de l'\u00e9tat-major de l'Arm\u00e9e populaire de lib\u00e9ration, et une unit\u00e9 chinoise a \u00e9t\u00e9 envoy\u00e9e au Pentagone. Qui peut avoir la foutue id\u00e9e de qui est l'ennemi ?\n\n\u2014 Si tout ce que vous dites est vrai, c'est parfaitement invraisemblable !\n\n\u2014 J'ai de vieilles connaissances du temps de la brigade qui sont maintenant devenues g\u00e9n\u00e9raux, je suis bien inform\u00e9.\n\n\u2014 Pourquoi les m\u00e9dias du monde entier n'ont-ils pas encore r\u00e9agi \u00e0 cette affaire ?\n\n\u2014 Voil\u00e0 encore quelque chose d'extraordinaire ! Tous les pays, sans exception, ont verrouill\u00e9 l'information, et avec quelle efficacit\u00e9 ! Je suis maintenant s\u00fbr d'une chose : l'ennemi est impitoyable. Ils ont peur en haut, je connais trop bien le g\u00e9n\u00e9ral Chang Weisi. C'est un homme qui d'habitude ne craint rien, pas m\u00eame que le ciel lui tombe sur la t\u00eate, mais maintenant j'ai comme l'impression que ce n'est peut-\u00eatre pas que le ciel qui s'appr\u00eate \u00e0 lui tomber dessus. Ils ont les p\u00e9toches, je vous dis, et ils ne sont absolument pas s\u00fbrs de pouvoir vaincre cet ennemi par la force.\n\n\u2014 Si c'est le cas, c'est tr\u00e8s angoissant.\n\n\u2014 Mais nous avons tous peur de quelque chose ! L'ennemi n'\u00e9chappe pas \u00e0 la r\u00e8gle. Et plus il est puissant, plus ce dont il a peur peut lui \u00eatre fatal.\n\n\u2014 De quoi a-t-il peur ?\n\n\u2014 Mais de vous, des scientifiques ! Et le plus bizarre, c'est que plus vos recherches sont th\u00e9oriques, plus elles sont abstraites ou d\u00e9brid\u00e9es comme celles de Yang Dong, plus il a peur ! Il a davantage peur des sciences th\u00e9oriques que vous quand vous voyez l'univers clignoter. C'est pour \u00e7a qu'il se montre si impitoyable. S'il avait jug\u00e9 utile de vous tuer, cela fait longtemps qu'il vous aurait massacr\u00e9s, mais la m\u00e9thode la plus efficace, c'est d'\u00e9branler vos certitudes. Quand quelqu'un meurt, un autre vient pour prendre sa place, mais quand la pens\u00e9e elle-m\u00eame part en vrille, c'est la fin de la science.\n\n\u2014 Vous voulez dire qu'il a peur des sciences fondamentales ?\n\n\u2014 Oui, des sciences fondamentales.\n\n\u2014 Mais les recherches que menait Yang Dong sont si \u00e9loign\u00e9es des miennes ! Et puis les nanotechnologies ne sont pas des sciences fondamentales, ce sont des projets de d\u00e9veloppement de mat\u00e9riaux \u00e0 haute r\u00e9sistance. En quoi cela constituerait-il une menace pour cette puissance ?\n\n\u2014 Votre cas est \u00e0 part. Jusqu'\u00e0 maintenant, il ne s'en \u00e9tait pas encore pris aux sciences appliqu\u00e9es, mais peut-\u00eatre que quelque chose l'effraie dans vos mat\u00e9riaux \u00e0 haute r\u00e9sistance.\n\n\u2014 Alors, que dois-je faire ?\n\n\u2014 Retourner bosser et continuer vos recherches, c'est le plus grand coup que vous puissiez lui porter. Ne vous pr\u00e9occupez pas de cette saloperie de compte \u00e0 rebours. Si vous voulez vous changer les id\u00e9es apr\u00e8s le travail, vous pouvez jouer \u00e0 ce jeu, l\u00e0. L'id\u00e9al serait de le terminer.\n\n\u2014 Le jeu ? _Les Trois Corps ?_ Il y aurait donc un lien ?\n\n\u2014 Il semblerait. Plusieurs sp\u00e9cialistes du Centre d'op\u00e9rations militaires y jouent aussi ; ce n'est pas un jeu comme les autres. Un analphab\u00e8te comme moi est incapable d'y jouer, il faut quelqu'un avec vos connaissances pour \u00e7a.\n\n\u2014 Ah... autre chose ?\n\n\u2014 Non, s'il y a du neuf, je vous appellerai. Laissez toujours votre portable allum\u00e9. Allez, mon vieux, gardez la t\u00eate froide ! Et si vous avez peur, rappelez-vous ma th\u00e9orie ultime !\n\nWang Miao n'eut m\u00eame pas le temps de le remercier que Shi Qiang \u00e9tait mont\u00e9 dans sa voiture et avait d\u00e9marr\u00e9 au quart de tour.\n\n* * *\n\n1 Alcool de sorgho g\u00e9n\u00e9ralement produit \u00e0 P\u00e9kin. _(N.d.T.)_\n\n## 11\n\n## _LES TROIS CORPS :_\n\n## MOZI ET LE FEU ARDENT\n\nAvant de rentrer chez lui, Wang Miao s'arr\u00eata en chemin dans un march\u00e9 de produits \u00e9lectroniques pour faire l'achat d'une V-combinaison. Son \u00e9pouse l'informa que des coll\u00e8gues avaient tent\u00e9 de le joindre toute la journ\u00e9e. Wang Miao alluma son t\u00e9l\u00e9phone rest\u00e9 \u00e9teint et constata qu'il avait re\u00e7u plusieurs appels de son centre de recherches. Il rappela en promettant qu'il serait vite sur pied et qu'il retournerait travailler le lendemain. Au repas, il suivit les conseils de Shi Qiang et but beaucoup de vin rouge, mais il ne trouva pas plus le sommeil. Une fois son \u00e9pouse et son fils endormis, il s'assit devant son ordinateur, enfila sa V-combinaison neuve et se connecta au jeu des _Trois Corps_.\n\nIl se retrouva sur la m\u00eame plaine d\u00e9sertique baign\u00e9e par les lueurs de l'aube. Il \u00e9tait devant la pyramide du roi Zhou, mais l'immense couche de neige dont elle avait \u00e9t\u00e9 recouverte avait disparu et les pierres soutenant l'\u00e9difice portaient les marques de l'\u00e9rosion du vent. Le sol avait chang\u00e9 de couleur. Au loin se dressaient de grands b\u00e2timents qu'il imagina \u00eatre des silos de d\u00e9shydratation. Cependant leur architecture n'\u00e9tait pas la m\u00eame que lors de sa derni\u00e8re connexion. Tout semblait indiquer qu'une longue p\u00e9riode de temps s'\u00e9tait \u00e9coul\u00e9e.\n\nProfitant de la lumi\u00e8re de l'aurore, Wang Miao chercha l'entr\u00e9e de la pyramide. \u00c0 l'emplacement o\u00f9 celle-ci se trouvait auparavant, il vit que le passage \u00e9tait condamn\u00e9 par un grand bloc de pierre. Mais \u00e0 c\u00f4t\u00e9 avait \u00e9t\u00e9 sculpt\u00e9 un long escalier qui menait jusqu'au point culminant de la pyramide. Il leva les yeux et remarqua que l'apex d'origine avait \u00e9t\u00e9 aplani pour faire place \u00e0 une plateforme. De pyramide \u00e9gyptienne le palais s'\u00e9tait transform\u00e9 en pyramide azt\u00e8que.\n\nWang Miao gravit l'escalier de pierre et atteignit le sommet. Il y vit comme une sorte d'ancien observatoire astronomique. Dans un coin de la plateforme se dressaient un t\u00e9lescope de plusieurs m\u00e8tres de hauteur et, \u00e0 c\u00f4t\u00e9 de lui, plusieurs autres mod\u00e8les, plus petits. De l'autre c\u00f4t\u00e9 de la plateforme, il remarqua des instruments \u00e9tranges qui lui rappel\u00e8rent les sph\u00e8res armillaires de la Chine ancienne. Mais ce qui capta le plus son attention \u00e9tait une grande boule en cuivre situ\u00e9e au centre de la plateforme. D'un diam\u00e8tre de deux m\u00e8tres environ, elle \u00e9tait install\u00e9e sur un ensemble sophistiqu\u00e9 de rouages de diff\u00e9rentes tailles qui la faisaient tourner lentement. Wang Miao s'aper\u00e7ut que son sens et sa vitesse de rotation \u00e9taient sans cesse changeants. Sous la machine se trouvait une fosse carr\u00e9e, dans laquelle Wang Miao aper\u00e7ut \u00e0 la lueur blafarde des torches des hommes \u2013 certainement des esclaves \u2013 en train de pousser une grande roue fournissant de l'\u00e9nergie au m\u00e9canisme du dessus.\n\nUn homme approcha. Comme lors de sa premi\u00e8re rencontre avec le roi Wen, celui-ci tournait le dos \u00e0 la lumi\u00e8re du soleil, et Wang Miao eut l'impression de voir deux yeux scintillants flotter au milieu de l'obscurit\u00e9. Il \u00e9tait grand, maigre et portait une large robe noire ; ses longs cheveux \u00e9taient n\u00e9gligemment nou\u00e9s au sommet de son cr\u00e2ne et de nombreuses m\u00e8ches volaient au vent.\n\n\u2014 Bonjour, dit l'homme. Je me pr\u00e9sente : Mozi1.\n\n\u2014 Bonjour, je suis Hairen.\n\n\u2014 Ah, mais je te connais ! dit Mozi en pointant Wang Miao du doigt. Tu as \u00e9t\u00e9 le compagnon du roi Wen des Zhou durant la civilisation no 137 !\n\n\u2014 C'est effectivement avec lui que je suis arriv\u00e9 jusqu'ici, mais je n'avais pas foi dans sa th\u00e9orie.\n\n\u2014 Et tu avais raison, reprit Mozi en opinant gravement de la t\u00eate.\n\nPuis il s'approcha et lui dit \u00e0 voix basse :\n\n\u2014 Durant les trois cent soixante-deux mille ann\u00e9es qui se sont \u00e9coul\u00e9es apr\u00e8s ton d\u00e9part, nous avons ressuscit\u00e9 \u00e0 quatre reprises. Les civilisations ont essay\u00e9 de prosp\u00e9rer tant bien que mal, au gr\u00e9 des alternances erratiques entre \u00e8res chaotiques et \u00e8res r\u00e9guli\u00e8res. La plus courte a disparu \u00e0 mi-chemin de l'\u00e2ge de pierre, mais la civilisation no 139 a battu un record, elle a atteint l'\u00e2ge de la machine \u00e0 vapeur !\n\n\u2014 Vous voulez dire que, dans cette civilisation, quelqu'un est parvenu \u00e0 comprendre les lois qui r\u00e9gissent les mouvements du soleil ?\n\nMozi partit d'un grand rire en secouant la t\u00eate :\n\n\u2014 Non, non ! C'\u00e9tait juste un coup de chance !\n\n\u2014 Mais les gens ont continu\u00e9 de chercher, n'est-ce pas ?\n\n\u2014 Bien s\u00fbr. Approchez, je vais vous montrer les efforts men\u00e9s durant la derni\u00e8re civilisation.\n\nMozi conduisit Wang Miao dans un coin de l'observatoire astronomique. La terre s'\u00e9tendait sous leurs pieds comme un vieux morceau de cuir us\u00e9. Mozi se saisit d'un petit t\u00e9lescope et le braqua sur le sol, puis il le tendit \u00e0 Wang Miao. Ce dernier plaqua son \u0153il sur l'oculaire du t\u00e9lescope et vit une sc\u00e8ne tr\u00e8s \u00e9trange : un squelette blanc comme la neige, dont l'\u00e9l\u00e9gance \u00e9tait perceptible, faiblement \u00e9clair\u00e9 par la clart\u00e9 de l'aube. Le plus \u00e9tonnant \u00e9tait que ce squelette se tenait debout. Sa posture semblait noble et distingu\u00e9e, une de ses mains soutenait sa m\u00e2choire, comme s'il caressait une barbe qui n'existait plus. Sa t\u00eate \u00e9tait inclin\u00e9e vers le haut, comme s'il voulait interroger le ciel.\n\n\u2014 Confucius, expliqua Mozi. Il \u00e9tait persuad\u00e9 que tout \u00e9tait li\u00e9 aux rites et que rien dans l'univers n'\u00e9chappait \u00e0 cette r\u00e8gle. Il a donc cr\u00e9\u00e9 un syst\u00e8me de rites pour l'univers en esp\u00e9rant pouvoir pr\u00e9dire le fonctionnement du soleil.\n\n\u2014 Je peux deviner le r\u00e9sultat.\n\n\u2014 En effet... Il avait la certitude que le soleil s'accorderait aux rites et a pr\u00e9dit une \u00e8re r\u00e9guli\u00e8re qui durerait cinq ans sans interruption. Nul besoin de le dire : cette pr\u00e9diction s'est av\u00e9r\u00e9e inexacte... L'\u00e8re r\u00e9guli\u00e8re n'a r\u00e9ellement dur\u00e9 qu'un mois.\n\n\u2014 Puis le soleil ne s'est plus lev\u00e9 ?\n\n\u2014 Oh si, le soleil s'est lev\u00e9, puis il est mont\u00e9 au milieu du ciel et il s'est soudain \u00e9teint.\n\n\u2014 Comment ? \u00c9teint ?\n\n\u2014 Oui. Il a commenc\u00e9 par s'obscurcir peu \u00e0 peu, par diminuer de volume, puis tout d'un coup, il s'est \u00e9teint ! La nuit est tomb\u00e9e et le froid... un froid terrible qui a congel\u00e9 Confucius sur place. Il est ici depuis ce jour.\n\n\u2014 Et puis plus rien ? Je veux dire, apr\u00e8s l'extinction du soleil ?\n\n\u2014 Une \u00e9toile volante est apparue \u00e0 son emplacement, comme l'\u00e2me du soleil mort.\n\n\u2014 \u00cates-vous s\u00fbr que le soleil s'est \u00e9teint et qu'une \u00e9toile volante a soudain pris sa place ?\n\n\u2014 Oui, certain, il s'est \u00e9teint puis l'\u00e9toile volante est apparue. Vous pouvez v\u00e9rifier dans les annales, c'est ce qui est \u00e9crit, pas de doute.\n\n\u2014 Oh...\n\nWang Miao m\u00e9dita longuement.\n\nIl avait \u00e9chafaud\u00e9 quelques hypoth\u00e8ses un peu vagues sur le myst\u00e8re du monde des _Trois Corps_ , mais les r\u00e9v\u00e9lations de Mozi venaient de les faire s'\u00e9crouler.\n\n\u2014 Pourquoi \"soudain\" ? grommela-t-il, contrari\u00e9.\n\n\u2014 Nous sommes maintenant dans la dynastie des Han \u2013 mais je ne sais pas vraiment si c'est celle des Han de l'Est ou de l'Ouest2.\n\n\u2014 Et vous avez pu survivre jusqu'\u00e0 aujourd'hui ?\n\n\u2014 J'ai une mission : pr\u00e9dire avec exactitude le m\u00e9canisme du soleil. Les sorciers, les m\u00e9taphysiciens et autres pr\u00eatres tao\u00efstes qui se sont succ\u00e9d\u00e9 avant moi \u00e9taient des bons \u00e0 rien. Des incapables tout juste bons \u00e0 imaginer des fantaisies mystiques, mais sans aucune connaissance pratique et sans volont\u00e9 de mener des exp\u00e9riences. Mais c'est d\u00e9sormais mon tour et j'agis diff\u00e9remment. Je suis capable de mener de v\u00e9ritables exp\u00e9riences, dit-il fi\u00e8rement en d\u00e9signant les instruments sur la plateforme.\n\n\u2014 Ces objets vous aideront-ils \u00e0 r\u00e9ussir votre mission ? demanda Wang Miao en d\u00e9signant non seulement les instruments, mais surtout la myst\u00e9rieuse boule en cuivre.\n\n\u2014 J'ai moi aussi ma th\u00e9orie, mais elle n'est pas d'inspiration mystique. Elle est le produit d'un grand nombre d'observations. Tout d'abord, savez-vous ce qu'est l'univers ? C'est une machine.\n\n\u2014 \u00c7a ne m'en dit pas plus.\n\n\u2014 Laissez-moi vous l'expliquer en termes plus concrets. L'univers est une grande sph\u00e8re creuse suspendue au-dessus d'un oc\u00e9an de flammes. La surface de cette sph\u00e8re est recouverte de petits trous et d'un trou de plus grande taille. La lumi\u00e8re \u00e9mise par l'oc\u00e9an de flammes passe par ces trous. Les petits trous sont les \u00e9toiles et le grand, c'est le soleil.\n\n\u2014 C'est un mod\u00e8le tr\u00e8s int\u00e9ressant, d\u00e9clara Wang Miao en regardant la grosse boule en cuivre.\n\nIl pensait \u00e0 pr\u00e9sent avoir compris ce qu'elle repr\u00e9sentait.\n\n\u2014 Mais votre th\u00e9orie comporte une lacune : quand le soleil se l\u00e8ve ou se couche, nous pouvons voir que le mouvement du soleil n'est pas synchrone avec celui des \u00e9toiles. Or dans votre mod\u00e8le, les trous sur la sph\u00e8re restent fixes dans leur position respective.\n\n\u2014 Excellente remarque ! C'est pourquoi j'ai revu mon mod\u00e8le. La sph\u00e8re de l'univers est en r\u00e9alit\u00e9 compos\u00e9e de deux sph\u00e8res, l'une ench\u00e2ss\u00e9e dans l'autre. Le ciel que nous pouvons voir est la sph\u00e8re int\u00e9rieure, elle est recouverte de petits trous, tandis que le grand trou se trouve sur la sph\u00e8re ext\u00e9rieure. La lumi\u00e8re qui filtre par le grand trou de la sph\u00e8re ext\u00e9rieure est refl\u00e9t\u00e9e et diss\u00e9min\u00e9e dans l'espace interm\u00e9diaire entre les deux sph\u00e8res. Puis \u00e0 son tour, cette lumi\u00e8re perce \u00e0 travers les petits trous de la sph\u00e8re int\u00e9rieure et c'est pourquoi nous voyons des \u00e9toiles.\n\n\u2014 Et le soleil ?\n\n\u2014 Le soleil est cette immense tache lumineuse projet\u00e9e depuis le trou de la sph\u00e8re ext\u00e9rieure sur la sph\u00e8re int\u00e9rieure. Cette projection est si lumineuse qu'elle parvient \u00e0 p\u00e9n\u00e9trer \u00e0 travers la sph\u00e8re int\u00e9rieure comme \u00e0 travers une coquille d'\u0153uf. Voil\u00e0 pourquoi nous voyons le soleil. Les rayons diss\u00e9min\u00e9s dans l'espace interm\u00e9diaire autour de cette tache de lumi\u00e8re sont eux aussi tr\u00e8s lumineux et traversent la coquille, voil\u00e0 comment nous pouvons y voir clair le jour.\n\n\u2014 Mais selon vous, quelle force contraint les deux sph\u00e8res \u00e0 effectuer des mouvements irr\u00e9guliers ?\n\n\u2014 Celle de l'oc\u00e9an de flammes situ\u00e9 hors de l'univers.\n\n\u2014 Pourtant, la taille du soleil et sa luminosit\u00e9 varient selon les p\u00e9riodes. Dans votre mod\u00e8le \u00e0 double sph\u00e8re, celles-ci doivent \u00eatre constantes. Quand bien m\u00eame les flammes seraient-elles irr\u00e9guli\u00e8res, la taille du trou, elle, ne change pas.\n\n\u2014 Ce mod\u00e8le est loin d'\u00eatre aussi simpliste que vous le pensez. La taille des sph\u00e8res augmente ou diminue au gr\u00e9 des variations de l'oc\u00e9an de flammes, d'o\u00f9 les changements de taille et de luminosit\u00e9 du soleil.\n\n\u2014 Et les \u00e9toiles volantes ?\n\n\u2014 Les \u00e9toiles volantes ? Mais qu'avez-vous donc avec ces \u00e9toiles volantes ? Ce sont juste des poussi\u00e8res soulev\u00e9es par hasard quelque part entre les deux sph\u00e8res de l'univers, rien de plus.\n\n\u2014 Non, il me semble que les \u00e9toiles volantes sont au contraire tr\u00e8s importantes. Par ailleurs, comment votre mod\u00e8le explique-t-il la soudaine extinction du soleil en plein ciel \u00e0 l'\u00e9poque de Confucius ?\n\n\u2014 C'est une exception, rare. C'est certainement une tache noire ou un nuage sombre issu de l'oc\u00e9an de flammes qui est pass\u00e9 juste derri\u00e8re la sph\u00e8re ext\u00e9rieure.\n\nWang Miao pointa la grosse boule en cuivre :\n\n\u2014 C'est certainement votre mod\u00e8le ?\n\n\u2014 Oui, j'ai cr\u00e9\u00e9 cette machine qui reproduit le m\u00e9canisme de l'univers. Admirez plut\u00f4t cet ensemble complexe de rouages qui fait tourner la boule, il simule les forces provenant de l'oc\u00e9an de flammes. Ces forces sont d\u00e9termin\u00e9es par la r\u00e9partition des flammes dans l'oc\u00e9an ext\u00e9rieur et des courants qui le traversent. C'est une conclusion \u00e0 laquelle je suis arriv\u00e9 apr\u00e8s plusieurs si\u00e8cles d'observation.\n\n\u2014 Cette boule peut-elle aussi s'agrandir et r\u00e9tr\u00e9cir ?\n\n\u2014 Bien s\u00fbr. Actuellement, elle est pr\u00e9cis\u00e9ment en train de r\u00e9tr\u00e9cir.\n\nWang Miao prit la balustrade de la plateforme comme point de rep\u00e8re fixe et sonda attentivement la boule. Mozi disait vrai.\n\n\u2014 Il y a donc aussi une sph\u00e8re \u00e0 l'int\u00e9rieur ?\n\n\u2014 \u00c9videmment ! Les mouvements des deux sph\u00e8res sont engendr\u00e9s par des m\u00e9canismes sophistiqu\u00e9s.\n\n\u2014 C'est une machine tr\u00e8s ing\u00e9nieuse ! admira Wang Miao avec sinc\u00e9rit\u00e9. Mais je ne vois pas de grand trou sur la surface de la sph\u00e8re ext\u00e9rieure pour projeter la lumi\u00e8re \u00e0 l'int\u00e9rieur.\n\n\u2014 Il n'y a pas de trou. Sur la paroi int\u00e9rieure de la sph\u00e8re ext\u00e9rieure, j'ai install\u00e9 une source de lumi\u00e8re pour simuler le grand trou. C'est une texture fluorescente fabriqu\u00e9e \u00e0 partir d'extraits de centaines de milliers de lucioles. Elle produit une lumi\u00e8re froide, car le pl\u00e2tre translucide utilis\u00e9 pour faire la coque int\u00e9rieure n'a pas une tr\u00e8s bonne conductivit\u00e9 thermique. J'\u00e9vite ainsi le risque d'une accumulation de chaleur dans la sph\u00e8re qu'impliquerait l'utilisation d'une source de lumi\u00e8re chaude. De cette mani\u00e8re, le scribe peut demeurer \u00e0 l'int\u00e9rieur une longue p\u00e9riode de temps.\n\n\u2014 Il y a donc quelqu'un dans la boule ?\n\n\u2014 Bien s\u00fbr. Le scribe est install\u00e9 debout sur le rayon d'une roue situ\u00e9e dans la partie inf\u00e9rieure du m\u00e9canisme. Cette position lui permet de se maintenir au centre de la sph\u00e8re. Lorsque nous serons parvenus \u00e0 faire correspondre notre mod\u00e8le d'univers avec l'univers r\u00e9el, les mouvements de la boule que j'ai con\u00e7ue nous permettront de pr\u00e9dire le futur de l'univers et donc, bien entendu, les mouvements \u00e0 venir du soleil. Le scribe notera tous les mouvements solaires et nous permettra de r\u00e9aliser un calendrier extr\u00eamement pr\u00e9cis. Vous avez devant vous le r\u00eave de centaines de civilisations pass\u00e9es !\n\nVous arrivez au bon moment. Mon mod\u00e8le d'univers vient tout juste d'indiquer qu'une \u00e8re r\u00e9guli\u00e8re de quatre ans est sur le point de commencer. Suivant ma pr\u00e9diction, l'empereur Wu des Han vient de donner l'ordre de r\u00e9hydrater l'empire3. Attendons ensemble que le soleil se l\u00e8ve !\n\nMozi fit appara\u00eetre l'interface sur l'\u00e9cran et acc\u00e9l\u00e9ra quelque peu le temps du jeu. Un soleil rouge apparut \u00e0 l'horizon, les lacs et les \u00e9tangs glac\u00e9s constellant la plaine se mirent \u00e0 fondre. La surface des lacs \u00e9tait jusqu'alors recouverte d'une \u00e9paisse couche de poussi\u00e8re qui les faisait se confondre avec la terre mais, \u00e0 pr\u00e9sent, ils devenaient peu \u00e0 peu des miroirs cristallins, comme si la terre avait ouvert une myriade d'yeux. De la hauteur o\u00f9 il se trouvait, Wang Miao ne distinguait pas les d\u00e9tails du processus de r\u00e9hydratation, mais il pouvait voir un nombre croissant d'individus au bord des lacs, comme des colonnes de fourmis sortant de leur fourmili\u00e8re aux premi\u00e8res chaleurs du printemps. Le monde ressuscitait une nouvelle fois.\n\n\u2014 Ne voudriez-vous pas descendre et participer \u00e0 cette fabuleuse renaissance ? Les femmes qui viennent de ressusciter sont celles qui d\u00e9sirent le plus ardemment l'amour, dit Mozi en d\u00e9signant \u00e0 Wang Miao cette terre en train de reprendre vie. Il ne vous sert \u00e0 rien de rester ici. Le jeu est termin\u00e9, je suis le vainqueur.\n\n\u2014 Votre machine de simulation de l'univers est ing\u00e9nieuse, c'est ind\u00e9niable. Mais quant \u00e0 ses pr\u00e9dictions... Oh, j'y pense, pourrais-je utiliser votre t\u00e9lescope un instant ?\n\n\u2014 Je vous en prie, dit Mozi en faisant un geste vers le grand t\u00e9lescope.\n\nWang Miao s'en approcha, mais il s'arr\u00eata aussit\u00f4t :\n\n\u2014 Comment l'utiliser pour observer le soleil ?\n\nMozi sortit un disque de verre noir d'un \u00e9tui en bois :\n\n\u2014 Utilisez ce filtre en verre fum\u00e9, dit-il en ins\u00e9rant l'objet devant l'oculaire du t\u00e9lescope.\n\nWang Miao pointa le t\u00e9lescope vers le soleil, qui \u00e9tait d\u00e9j\u00e0 plus haut que la moiti\u00e9 du ciel. Il ne put s'emp\u00eacher d'\u00e9prouver un sentiment d'admiration devant l'imagination de Mozi. Le soleil avait effectivement l'air d'un trou immense o\u00f9 venaient rouler les innombrables vagues d'un oc\u00e9an de feu, une infime partie d'une entit\u00e9 bien plus grande.\n\nMais apr\u00e8s un examen plus d\u00e9taill\u00e9, il remarqua que ce soleil \u00e9tait diff\u00e9rent du soleil de la r\u00e9alit\u00e9. Celui-ci semblait poss\u00e9der un petit noyau en son centre. Si on imaginait ce soleil comme un \u0153il, le noyau correspondait \u00e0 sa pupille. Le noyau n'\u00e9tait pas large, mais il \u00e9tait si lumineux et si dense que les couches qui l'entouraient paraissaient presque immat\u00e9rielles, erratiques et gazeuses. Le fait qu'il puisse voir le noyau \u00e0 travers les couches externes du soleil semblait indiquer que les couches \u00e9taient transparentes ou translucides, et que la lumi\u00e8re qu'elles diffusaient n'\u00e9tait peut-\u00eatre finalement que la lumi\u00e8re diss\u00e9min\u00e9e du noyau.\n\nWang Miao fut stup\u00e9fait du r\u00e9alisme frappant de l'image du soleil. Une nouvelle fois, il eut la confirmation que les concepteurs du jeu avaient intentionnellement dissimul\u00e9 derri\u00e8re des images en apparence simplistes une \u00e9norme quantit\u00e9 de d\u00e9tails qui n'attendaient que d'\u00eatre d\u00e9couverts par les joueurs.\n\nWang Miao se redressa puis r\u00e9fl\u00e9chit au sens de cette structure cach\u00e9e du soleil. Cela excitait son imagination. La vitesse du soleil avait augment\u00e9. Il avait d\u00e9j\u00e0 atteint l'ouest du ciel. Wang Miao ajusta le t\u00e9lescope pour l'observer et le suivit jusqu'\u00e0 ce qu'il passe sous la ligne d'horizon.\n\nLa nuit venue, les flammes des feux de camp refl\u00e9t\u00e8rent le ciel rempli d'\u00e9toiles. Wang Miao \u00f4ta le filtre du t\u00e9lescope et continua \u00e0 observer les cieux. Il se concentra sur les \u00e9toiles volantes et, tr\u00e8s vite, il parvint \u00e0 en trouver deux. Mais il n'eut le temps d'examiner que l'une d'entre elles, car le soleil se leva \u00e0 nouveau. Wang Miao repla\u00e7a le filtre sur l'oculaire... Il prolongea cette observation astronomique pendant plus de dix jours, jouissant du plaisir de la d\u00e9couverte. En r\u00e9alit\u00e9, la vitesse acc\u00e9l\u00e9r\u00e9e du jeu \u00e9tait propice \u00e0 ce type d'observation, les mouvements des corps c\u00e9lestes apparaissant ainsi de fa\u00e7on plus manifeste.\n\nLe dix-septi\u00e8me jour apr\u00e8s le d\u00e9but de l'\u00e8re r\u00e9guli\u00e8re, l'heure du lever du soleil avait d\u00e9j\u00e0 \u00e9t\u00e9 d\u00e9pass\u00e9e de cinq heures, or la terre \u00e9tait toujours envelopp\u00e9e par le voile de la nuit. Une mar\u00e9e de sujets s'\u00e9tait rassembl\u00e9e au pied de la pyramide, brandissant d'innombrables torches vacillant dans le vent glacial.\n\n\u2014 Le soleil ne se l\u00e8vera peut-\u00eatre pas, comme lors de la chute de la civilisation no 137, dit Wang Miao \u00e0 un Mozi en train de consulter son calendrier astronomique.\n\nMozi caressa sa barbe et arbora un sourire confiant :\n\n\u2014 Ne craignez rien, le soleil va bient\u00f4t se lever et l'\u00e8re r\u00e9guli\u00e8re va continuer. Je connais les principes de fonctionnement de mon mod\u00e8le d'univers. Mes pr\u00e9dictions ne peuvent \u00eatre fausses.\n\nEt c'est alors que, comme si elle avait voulu donner raison \u00e0 Mozi, la lueur du jour apparut \u00e0 l'horizon. Elle fut accueillie par une explosion de cris de joie de la foule qui s'\u00e9tait amass\u00e9e au pied de la pyramide.\n\n\u00c0 une vitesse inou\u00efe, la lumi\u00e8re argent\u00e9e du levant laissa place \u00e0 un \u00e9clat intense, comme si le soleil voulait rattraper le temps perdu. Bient\u00f4t, la clart\u00e9 de l'aube emplit la moiti\u00e9 du ciel et, bien que le soleil ne soit toujours pas sorti, il faisait sur la terre aussi clair qu'en plein jour.\n\nWang Miao regarda dans la direction de la source lumineuse et il remarqua qu'une clart\u00e9 \u00e9blouissante \u00e9mergeait de l'horizon. La ligne de l'horizon se courba comme un arc parfait allant d'un point \u00e0 l'autre du champ de vision. Wang Miao comprit bient\u00f4t que ce n'\u00e9tait pas l'horizon, mais les confins du soleil ! Un soleil gigantesque en train de se lever. Une fois que ses yeux se furent habitu\u00e9s \u00e0 la luminosit\u00e9, la ligne d'horizon r\u00e9apparut l\u00e0 o\u00f9 elle n'avait jamais cess\u00e9 de se trouver. Wang Miao vit s'\u00e9lever des colonnes de fum\u00e9e noire qui contrastaient de fa\u00e7on saisissante avec la lumi\u00e8re propag\u00e9e par le disque solaire. Ces nuages avaient \u00e9t\u00e9 occasionn\u00e9s par des feux ayant pris dans le lointain. Un cheval galopa jusqu'au pied de la pyramide depuis la direction o\u00f9 se levait le soleil, en soulevant une poussi\u00e8re telle qu'il laissa une tra\u00een\u00e9e blanche derri\u00e8re lui. La foule lui ouvrit la route et Wang Miao entendit le cavalier s'\u00e9poumoner :\n\n\u2014 D\u00e9shydratez ! D\u00e9shydratez !\n\nLe cavalier \u00e9tait suivi d'un troupeau de vaches et de chevaux, ainsi que d'autres animaux dont les corps en feu firent penser \u00e0 un gigantesque tapis brod\u00e9 de flammes glissant sur le sol. La moiti\u00e9 seulement de l'immense soleil avait \u00e9merg\u00e9 de la ligne de l'horizon, mais il occupait d\u00e9j\u00e0 la plus grande partie du ciel. La terre paraissait s'affaisser lentement le long d'une \u00e9norme et \u00e9blouissante muraille. Wang Miao pouvait maintenant voir en d\u00e9tail la surface du soleil : des vagues et des tourbillons se d\u00e9cha\u00eenaient au milieu d'un oc\u00e9an de flammes, des taches solaires planaient comme des fant\u00f4mes errants le long de routes incertaines et des couronnes solaires retroussaient paresseusement leurs longues manches d'or.\n\nSur terre, les hommes d\u00e9j\u00e0 d\u00e9shydrat\u00e9s comme ceux qui ne l'\u00e9taient pas encore s'embras\u00e8rent comme des allumettes jet\u00e9es dans une fournaise. Ils furent consum\u00e9s par des flammes d'une intensit\u00e9 plus grande encore que celle du charbon br\u00fblant, mais ils s'\u00e9teignirent rapidement.\n\nLe gigantesque soleil continua sa rapide progression et le ciel fut presque enti\u00e8rement recouvert lorsqu'il atteignit son z\u00e9nith. Wang Miao leva la t\u00eate. Un bouleversement extraordinaire s'\u00e9tait produit : ce qu'il voyait jusqu'alors au-dessous de lui \u00e9tait pass\u00e9 au-dessus. La surface du soleil \u00e9tait devenue une terre incandescente. Il eut la sensation de chuter dans cet enfer flamboyant.\n\nSur la terre, les lacs commenc\u00e8rent \u00e0 s'\u00e9vaporer, des nuages de vapeur blanche de la forme de champignons nucl\u00e9aires s'\u00e9lev\u00e8rent, avant de se disperser en recouvrant les amas de cendres humaines bordant les lacs.\n\n\u2014 L'\u00e8re r\u00e9guli\u00e8re va continuer, l'univers est une machine, j'ai cr\u00e9\u00e9 cette machine. L'\u00e8re r\u00e9guli\u00e8re va continuer, l'univers est...\n\nWang Miao tourna la t\u00eate. La voix \u00e9tait celle de Mozi, qui \u00e9tait en train de prendre feu. Son corps se retrouva prisonnier d'une longue colonne de feu orang\u00e9. Sa peau \u00e9tait d\u00e9j\u00e0 rid\u00e9e et carbonis\u00e9e, mais ses yeux \u00e9mettaient encore une lueur diff\u00e9rente de celle des flammes en train de l'engloutir. Ses mains, d\u00e9j\u00e0 chang\u00e9es en tiges de carbone br\u00fblantes, serraient encore un fragment de soie grise dont les cendres papillonnaient au vent : les restes du premier calendrier astronomique. Wang Miao br\u00fblait lui aussi. Il leva ses mains et vit deux torches enflamm\u00e9es.\n\nLe gigantesque soleil se d\u00e9pla\u00e7a rapidement vers l'ouest, laissant \u00e0 nouveau voir le ciel qu'il avait jusque-l\u00e0 dissimul\u00e9. Bient\u00f4t, il s'immergea derri\u00e8re l'horizon et la terre parut remonter le long de la muraille de feu. Les rayons \u00e9tincelants du cr\u00e9puscule disparurent alors et la nuit, telle une toile noire tir\u00e9e par des mains de g\u00e9ant, enveloppa le monde devenu cendres. La terre \u00e9mit un halo rouge sombre, comme une braise de charbon tout juste sortie d'un four. Pendant un bref instant, Wang Miao put voir quelques \u00e9toiles mais, tr\u00e8s vite, la vapeur d'eau et la brume envahirent le ciel et recouvrirent la terre rutilante. Le monde fut englouti par les t\u00e9n\u00e8bres du chaos. Un texte en caract\u00e8res rouges s'afficha sur l'\u00e9cran :\n\n> La civilisation no 141 a \u00e9t\u00e9 d\u00e9truite par les flammes, elle avait atteint la p\u00e9riode des Han de l'Est.\n> \n> Mais les graines de la civilisation ont surv\u00e9cu et une nouvelle civilisation germera des ruines du monde si impr\u00e9visible des _Trois Corps_. Nous vous invitons \u00e0 vous reconnecter dans le futur.\n\nWang Miao \u00f4ta sa V-combinaison. Apr\u00e8s avoir peu \u00e0 peu repris ses esprits, il eut une nouvelle fois le sentiment que, sous ses pr\u00e9tentions de fiction, le jeu des _Trois Corps_ faisait preuve d'un profond r\u00e9alisme et que, l'inverse, c'\u00e9tait le monde r\u00e9el devant ses yeux qui lui apparaissait comme la peinture _Le Jour de Qingming au bord de la rivi\u00e8re_ : complexe en surface mais si simple en v\u00e9rit\u00e9.\n\nLe lendemain, Wang Miao retourna travailler au Centre de recherches en nanotechnologie. En dehors de quelques probl\u00e8mes mineurs dus \u00e0 son absence de la veille, tout \u00e9tait en parfait \u00e9tat de marche. Il se rendit compte qu'une fois qu'il \u00e9tait absorb\u00e9 par le travail, celui-ci lui faisait l'effet d'un anesth\u00e9sique et ses cauchemars le laissaient provisoirement tranquille. Il s'effor\u00e7a de s'occuper la journ\u00e9e enti\u00e8re et ne quitta le laboratoire qu'une fois la nuit tomb\u00e9e.\n\nD\u00e8s qu'il eut pass\u00e9 le hall d'entr\u00e9e du centre, Wang Miao se retrouva avec la sensation d'\u00eatre \u00e0 nouveau pourchass\u00e9 par ses d\u00e9mons. Il avait l'impression que le ciel \u00e9toil\u00e9 \u00e9tait surmont\u00e9 d'une loupe et qu'il \u00e9tait un insecte nu sous la lentille de cette loupe, dans l'incapacit\u00e9 de lui \u00e9chapper. Il devait absolument trouver quelque chose \u00e0 faire et il se dit qu'il pourrait aller rendre visite \u00e0 la m\u00e8re de Yang Dong. Il conduisit donc jusqu'\u00e0 son domicile.\n\nYe Wenjie \u00e9tait seule. Lorsque Wang Miao entra chez elle, elle \u00e9tait en train de lire sur son canap\u00e9. Il s'aper\u00e7ut qu'elle \u00e9tait \u00e0 la fois myope et presbyte et qu'elle avait d\u00fb changer de lunettes pour lire. Elle fut ravie de voir Wang Miao, et fit remarquer qu'il avait bien meilleure mine que la derni\u00e8re fois.\n\n\u2014 Gr\u00e2ce \u00e0 votre ginseng ! sourit Wang Miao.\n\nLa m\u00e8re de Yang Dong secoua la t\u00eate :\n\n\u2014 Celui que je t'ai donn\u00e9 n'\u00e9tait pas de tr\u00e8s bonne qualit\u00e9. \u00c0 l'\u00e9poque je ramassais de l'excellent ginseng de montagne aux alentours de la base. Un jour j'ai trouv\u00e9 une racine grande comme \u00e7a... Je ne sais pas comment c'est maintenant l\u00e0-bas. J'ai entendu dire que c'\u00e9tait d\u00e9sert. Ah, je vieillis... Ces derniers temps, je n'arr\u00eate pas de penser au pass\u00e9.\n\n\u2014 On m'a racont\u00e9 que vous aviez subi beaucoup d'\u00e9preuves durant la R\u00e9volution culturelle.\n\n\u2014 C'est Ruishan qui t'en a parl\u00e9 ? (Ye Wenjie secoua les mains, comme si elle voulait chasser une toile d'araign\u00e9e devant ses yeux.) C'est du pass\u00e9, tout est du pass\u00e9... Hier soir, Ruishan m'a appel\u00e9. Il avait l'air paniqu\u00e9, je n'ai pas compris ce qu'il me racontait. J'ai seulement entendu qu'il t'\u00e9tait arriv\u00e9 quelque chose. Ah, quand tu auras mon \u00e2ge, tu comprendras que toutes ces choses que tu croyais importantes n'\u00e9taient en fait pas si extraordinaires.\n\n\u2014 Merci, dit Wang Miao.\n\nIl y avait bien longtemps qu'il n'avait pas \u00e9t\u00e9 parcouru par une telle vague de chaleur. \u00c0 pr\u00e9sent, cette vieille femme qui avait enfin trouv\u00e9 la paix apr\u00e8s tant d'\u00e9preuves et l'ignorant, le t\u00e9m\u00e9raire commissaire Shi Qiang \u00e9taient devenus les deux piliers qui emp\u00eachaient encore son esprit de tomber en ruine.\n\nYe Wenjie reprit :\n\n\u2014 En parlant de R\u00e9volution culturelle, j'ai quand m\u00eame eu beaucoup de chance. Au moment o\u00f9 je pensais que j'allais mourir, j'ai trouv\u00e9 un refuge o\u00f9 j'ai pu continuer \u00e0 vivre.\n\n\u2014 Vous voulez parler de la base de C\u00f4te Rouge ?\n\nYe Wenjie acquies\u00e7a.\n\n\u2014 C'\u00e9tait un projet extraordinaire, je croyais que ce n'\u00e9tait qu'une pure l\u00e9gende.\n\n\u2014 Ce n'est pas une l\u00e9gende. Si tu le souhaites, je peux te raconter ce que j'ai v\u00e9cu.\n\nWang Miao se sentit un peu nerveux :\n\n\u2014 Professeur Ye, je suis simplement curieux. Ne vous sentez pas forc\u00e9e de m'en parler.\n\n\u2014 Oh, ce n'est rien. C'est une bonne occasion de papoter, cela fait bien longtemps que je n'ai pas vraiment parl\u00e9 \u00e0 quelqu'un.\n\n\u2014 Vous devriez participer aux activit\u00e9s d'un cercle du troisi\u00e8me \u00e2ge. On se sent moins seul quand on est actif.\n\n\u2014 J'y retrouverais peut-\u00eatre de nombreux anciens coll\u00e8gues de l'universit\u00e9, mais je ne suis jamais arriv\u00e9e \u00e0 me fondre dans leur communaut\u00e9. Tout le monde aime ressasser ses souvenirs, mais seulement dans l'espoir qu'on les \u00e9coutera. Les souvenirs des autres les agacent. Il n'y a que toi que cette histoire de C\u00f4te Rouge peut bien int\u00e9resser.\n\n\u2014 Mais n'est-ce pas dangereux pour vous de parler de cela maintenant ?\n\n\u2014 C'est vrai, apr\u00e8s tout, le projet \u00e9tait confidentiel. Mais les langues se sont d\u00e9li\u00e9es apr\u00e8s la publication de ce livre. C'est devenu un secret de Polichinelle. L'auteur a vraiment \u00e9t\u00e9 irresponsable. Sans parler de ses intentions, beaucoup d'informations et de faits rapport\u00e9s dans l'ouvrage sont faux. Je me dois au moins d'y apporter des corrections.\n\nEt ainsi, Ye Wenjie commen\u00e7a \u00e0 raconter cette histoire encore vivace dans sa m\u00e9moire.\n\n* * *\n\n1 Mozi (479-392 av. J.-C.) est un philosophe chinois qui v\u00e9cut pendant les p\u00e9riodes des Printemps et Automnes et des Royaumes combattants. Il a cr\u00e9\u00e9 sa propre \u00e9cole philosophique : le mo\u00efsme, courant parfois pr\u00e9sent\u00e9 comme le premier en Chine \u00e0 donner naissance \u00e0 une v\u00e9ritable pens\u00e9e scientifique en privil\u00e9giant la logique et la relation de causalit\u00e9. _(N.d.T.)_\n\n2 Fond\u00e9e par Liu Bang, la dynastie Han r\u00e9gna sur la Chine de 202 av. J.-C. \u00e0 220 apr. J.-C. Elle est s\u00e9par\u00e9e en deux p\u00e9riodes : les Han de l'Ouest (dont la capitale \u00e9tait \u00e0 Chang'an) et les Han de l'Est (capitale \u00e0 Luoyang), entrecoup\u00e9es par l'\u00e9ph\u00e9m\u00e8re dynastie Xin (9-23 apr. J.-C.). _(N.d.T.)_\n\n3 Han Wudi, l'empereur Wu des Han (156-87 av. J.-C.), est le septi\u00e8me empereur de la dynastie Han. Son r\u00e8gne, l'un des plus longs de l'histoire imp\u00e9riale chinoise, dura cinquante-quatre ans. _(N.d.T.)_\n\n## 12\n\n## C\u00d4TE ROUGE II\n\n\u00c0 son entr\u00e9e dans la base de C\u00f4te Rouge, Ye Wenjie ne fut pas assign\u00e9e \u00e0 un poste fixe. Elle n'effectuait que quelques t\u00e2ches d'ordre purement technique et toujours sous la surveillance d'un gardien.\n\nDes ann\u00e9es plus t\u00f4t, alors qu'elle \u00e9tait en deuxi\u00e8me ann\u00e9e \u00e0 l'universit\u00e9, Ye Wenjie s'entendait bien avec le professeur qui \u00e9tait par la suite devenu son directeur de recherches en troisi\u00e8me cycle. Il lui avait signifi\u00e9 que, pour mener des recherches dans le domaine de l'astrophysique, \u00eatre excellent en th\u00e9orie \u00e9tait inutile sans connaissance des m\u00e9thodes exp\u00e9rimentales ni capacit\u00e9 d'observation \u2013 du moins, c'\u00e9tait le cas pour la Chine. C'\u00e9tait une vision qu'\u00e9tait loin de partager son p\u00e8re, mais Wenjie avait plut\u00f4t tendance \u00e0 donner raison \u00e0 son professeur : elle avait toujours trouv\u00e9 que son p\u00e8re \u00e9tait un homme trop th\u00e9orique.\n\nSon professeur avait \u00e9t\u00e9 l'un des pionniers de la radioastronomie en Chine, c'\u00e9tait sous son influence que Wenjie avait d\u00e9velopp\u00e9 un profond int\u00e9r\u00eat pour cette branche de l'astronomie, et qu'elle avait appris en autodidacte le g\u00e9nie \u00e9lectrique et les sciences de l'informatique \u2013 deux disciplines souvent regroup\u00e9es ensemble dans les universit\u00e9s de l'\u00e9poque et qui constituaient les fondements techniques des exp\u00e9riences et des observations scientifiques. Durant ses deux ann\u00e9es de troisi\u00e8me cycle, elle et son professeur avaient eu l'occasion de tester le premier petit radiot\u00e9lescope chinois. Elle avait ainsi accumul\u00e9 beaucoup d'exp\u00e9rience dans ce domaine.\n\nElle ne s'\u00e9tait pas imagin\u00e9 que ces connaissances trouveraient un jour leur utilit\u00e9 dans la base de C\u00f4te Rouge. Wenjie commen\u00e7a par \u00eatre affect\u00e9e aux t\u00e2ches d'entretien et de r\u00e9paration au sein de l'unit\u00e9 de transmission, mais tr\u00e8s vite, elle devint un \u00e9l\u00e9ment essentiel de toutes les op\u00e9rations men\u00e9es par l'unit\u00e9.\n\nCela la rendit tout d'abord quelque peu perplexe. Elle \u00e9tait la seule personne de la base \u00e0 ne pas porter d'uniforme militaire et, en raison de son identit\u00e9, les gens gardaient avec elle une certaine distance. Sa seule mani\u00e8re d'\u00e9chapper \u00e0 la solitude \u00e9tait de s'immerger corps et \u00e2me dans le travail. Mais cela ne suffisait pas \u00e0 expliquer pourquoi elle s'\u00e9tait rendue aussi vite indispensable. Apr\u00e8s tout, C\u00f4te Rouge \u00e9tait un projet clef de d\u00e9fense nationale. Comment se faisait-il que les ing\u00e9nieurs de la base soient m\u00e9diocres au point qu'une personne comme elle, non form\u00e9e en ing\u00e9nierie et sans exp\u00e9rience, arrive \u00e0 les remplacer si facilement ?\n\nElle en d\u00e9couvrit bient\u00f4t la raison. Contrairement \u00e0 ce qu'ils laissaient croire, les employ\u00e9s de la base \u00e9taient bien les meilleurs ing\u00e9nieurs du 2e corps d'artillerie. Une vie enti\u00e8re n'aurait pas suffi \u00e0 Wenjie pour emmagasiner toutes leurs connaissances en \u00e9lectrique et en informatique. Mais la base \u00e9tait loin de tout et les conditions de travail y \u00e9taient d\u00e9plorables. Les principales recherches sur le projet \u00e9tant achev\u00e9es, il ne restait plus que des travaux de maintenance et de r\u00e9paration, si bien que les opportunit\u00e9s d'accomplir une avanc\u00e9e technologique majeure \u00e9taient tr\u00e8s rares. Par cons\u00e9quent, la plupart des ing\u00e9nieurs s'effor\u00e7aient de ne pas para\u00eetre indispensables car, dans une base aussi secr\u00e8te, d\u00e8s que vous rejoigniez le c\u0153ur de la partie technique, il \u00e9tait quasiment impossible d'en sortir. Ils essayaient donc d\u00e9lib\u00e9r\u00e9ment de dissimuler leurs techniques, en se gardant toutefois de donner l'impression d'\u00eatre trop incomp\u00e9tents. Ainsi, lorsqu'un technicien sup\u00e9rieur leur disait d'aller \u00e0 l'est, ils s'obstinaient b\u00eatement \u00e0 aller \u00e0 l'ouest, de sorte que les responsables se disent : _Ce gar\u00e7on est pers\u00e9v\u00e9rant, mais il n'a pas les comp\u00e9tences requises, le garder serait contre-productif._\n\nGr\u00e2ce \u00e0 cette strat\u00e9gie, beaucoup avaient r\u00e9ussi \u00e0 \u00eatre mut\u00e9s ailleurs. Dans ces conditions, Ye Wenjie \u00e9tait, malgr\u00e9 elle, devenue une des techniciennes les plus importantes de la base. Mais une autre raison expliquait la rapidit\u00e9 avec laquelle elle avait acquis ce statut : de ce qu'elle pouvait en voir, la base de C\u00f4te Rouge ne poss\u00e9dait en fait aucune r\u00e9elle technologie de pointe.\n\nComme lors de son entr\u00e9e dans la base, Ye Wenjie continuait \u00e0 \u00eatre principalement affect\u00e9e \u00e0 l'unit\u00e9 de transmission, mais au fil du temps, les restrictions la concernant se rel\u00e2ch\u00e8rent progressivement, et le gardien qui veillait sur elle fut rappel\u00e9. Elle \u00e9tait \u00e0 pr\u00e9sent autoris\u00e9e \u00e0 avoir un acc\u00e8s direct \u00e0 la plupart des structures du syst\u00e8me d'armement de la base de C\u00f4te Rouge et \u00e0 lire les documents techniques connexes.\n\nNaturellement, il y avait encore des domaines qui lui \u00e9taient interdits, comme toute la plateforme de contr\u00f4le informatique. Mais Wenjie d\u00e9couvrit plus tard que cette branche n'avait pas l'importance qu'elle avait imagin\u00e9e. Elle en voulait pour preuve le fait que les ordinateurs de l'unit\u00e9 de transmission \u00e9taient encore plus anciens que les DJS 130 et poss\u00e9daient une m\u00e9moire \u00e0 tores magn\u00e9tiques et des rubans perfor\u00e9s. Le record du nombre d'heures pass\u00e9es sans dysfonctionnement des appareils ne d\u00e9passait pas quinze heures. Elle avait aussi d\u00e9couvert que la pr\u00e9cision du syst\u00e8me de ciblage des transmissions \u00e9tait tr\u00e8s limit\u00e9e, peut-\u00eatre m\u00eame pas celle d'un viseur de canon d'artillerie.\n\nCe jour-l\u00e0, le commissaire politique Lei Zhicheng vint trouver Wenjie pour discuter. Elle avait l'impression que Yang Weining et Lei Zhicheng avaient chang\u00e9 de r\u00f4le. Yang Weining n'\u00e9tait que l'ing\u00e9nieur en chef du projet, il n'avait pas de position politique tr\u00e8s \u00e9lev\u00e9e et, en dehors des domaines purement techniques, il n'avait aucune autorit\u00e9. Il devait se m\u00e9fier des techniciens et se montrer poli avec les soldats plac\u00e9s en sentinelle, car on l'aurait accus\u00e9 de trahir la Triple Alliance ou de freiner le progr\u00e8s de la pens\u00e9e. C'est ainsi que, d\u00e8s qu'il rencontrait des difficult\u00e9s, il se servait de Wenjie comme d'un punching-ball. \u00c0 l'inverse, l'importance croissante que celle-ci prenait dans les op\u00e9rations techniques avait modifi\u00e9 le regard que portait sur elle le commissaire Lei. Jadis froid et agressif, il faisait maintenant preuve de sollicitude \u00e0 son \u00e9gard.\n\n\u2014 Wenjie, depuis le temps que tu es ici, tu as pu te familiariser avec le syst\u00e8me de transmission. Il s'agit de la structure offensive de C\u00f4te Rouge, son unit\u00e9 principale. Pourrais-tu me donner ton avis sur le syst\u00e8me dans son ensemble ? demanda le commissaire politique Lei.\n\nIls \u00e9taient alors assis au bord de la falaise abrupte du pic du Radar. C'\u00e9tait l'endroit le plus paisible de toute la base, la falaise droite semblait donner sur un abysse sans fond. Ce panorama faisait peur au d\u00e9but \u00e0 Wenjie, mais elle s'y \u00e9tait habitu\u00e9e et, maintenant, elle aimait venir seule ici.\n\nWenjie ne savait pas vraiment comment r\u00e9pondre \u00e0 la question de Lei Zhicheng. Elle ne s'occupait que d'entretien et de r\u00e9paration et ignorait tout du syst\u00e8me global de la base militaire de C\u00f4te Rouge, y compris quelles \u00e9taient ses cibles et ses op\u00e9rations. Il ne lui \u00e9tait d'ailleurs m\u00eame pas permis d'assister aux transmissions ordinaires. Elle r\u00e9fl\u00e9chit, voulut dire quelque chose mais finit par s'abstenir.\n\n\u2014 Ne crains rien, dis ce que tu as sur le c\u0153ur, dit le commissaire en ramassant un brin d'herbe et en jouant avec distraitement.\n\n\u2014 Ce... ce n'est ni plus ni moins qu'un transmetteur radio.\n\n\u2014 Exactement, c'est un transmetteur radio, dit le commissaire en opinant de la t\u00eate d'un air satisfait. As-tu d\u00e9j\u00e0 entendu parler des fours micro-ondes ?\n\nWenjie fit non de la t\u00eate.\n\n\u2014 C'est un nouveau jouet de luxe des classes bourgeoises occidentales. Il cuit les aliments en utilisant l'effet chauffant g\u00e9n\u00e9r\u00e9 par l'action des micro-ondes sur les mol\u00e9cules d'eau. Dans mon ancien institut de recherches, nous en avions import\u00e9 un de l'\u00e9tranger pour calculer avec pr\u00e9cision l'usure \u00e0 haute temp\u00e9rature de certains composants. Apr\u00e8s le travail, nous l'utilisions pour r\u00e9chauffer nos pains \u00e0 la vapeur et pour faire cuire des pommes de terre. C'est tr\u00e8s int\u00e9ressant, les aliments sont chauds \u00e0 l'int\u00e9rieur alors qu'ils sont encore froids \u00e0 l'ext\u00e9rieur.\n\nLe commissaire politique Lei s'\u00e9tait lev\u00e9. Il continuait \u00e0 parler en marchant si pr\u00e8s du bord de la falaise qu'il rendait Wenjie anxieuse.\n\n\u2014 C\u00f4te Rouge est un four \u00e0 micro-ondes. Les cibles que nous voulons cuire, ce sont les engins spatiaux ennemis. Un simple rayonnement micro-onde de 0,1 \u00e0 1 watt par centim\u00e8tre carr\u00e9 suffirait \u00e0 mettre hors d'usage ou \u00e0 br\u00fbler les composants \u00e9lectroniques des satellites de communication, des radars ou des syst\u00e8mes de navigation.\n\nTout s'\u00e9clairait soudain dans l'esprit de Ye Wenjie. Le syst\u00e8me de C\u00f4te Rouge n'\u00e9tait certes qu'un transmetteur radio, mais cela ne voulait pas dire que c'\u00e9tait un transmetteur ordinaire. Ce qui la surprenait le plus, c'\u00e9tait sa puissance de transmission : elle pouvait atteindre 25 m\u00e9gawatts ! C'\u00e9tait non seulement beaucoup plus que tous les transmetteurs de communication existants, mais aussi que tous les transmetteurs radars. L'\u00e9nergie de transmission de C\u00f4te Rouge lui \u00e9tait fournie par un ensemble de gigantesques condensateurs. Du fait de sa puissance, ses circuits de transmission \u00e9taient bien diff\u00e9rents des circuits traditionnels. Wenjie comprenait maintenant l'utilisation de ces transmetteurs \u00e0 haute puissance. Cependant une autre question lui vint \u00e0 l'esprit :\n\n\u2014 Mais j'ai l'impression que les signaux sont d'abord modul\u00e9s avant d'\u00eatre transmis.\n\n\u2014 Effectivement, mais cette modulation est tr\u00e8s diff\u00e9rente de celle habituellement effectu\u00e9e pour des communications radio. Il ne s'agit pas de transmettre des informations, mais d'utiliser des fr\u00e9quences et des amplitudes variables destin\u00e9es \u00e0 percer les \u00e9ventuels syst\u00e8mes de protection pr\u00e9vus par l'ennemi. Bien entendu, tout cela en est encore au stade exp\u00e9rimental.\n\nWenjie hocha la t\u00eate. Beaucoup des questions qu'elle se posait venaient de trouver une r\u00e9ponse.\n\n\u2014 Nous avons r\u00e9cemment lanc\u00e9 \u00e0 Jiuquan deux satellites cibles que nous avons fait attaquer par le syst\u00e8me d'armement de C\u00f4te Rouge. L'op\u00e9ration a \u00e9t\u00e9 un succ\u00e8s, les cibles ont \u00e9t\u00e9 d\u00e9truites. L'int\u00e9rieur des satellites a \u00e9t\u00e9 chauff\u00e9 \u00e0 une temp\u00e9rature approchant 1 000 oC. Tous les instruments \u00e9lectroniques et les \u00e9quipements photographiques \u00e0 bord ont \u00e9t\u00e9 dissous. Dans les prochaines guerres, C\u00f4te Rouge sera capable d'attaquer avec une grande efficacit\u00e9 les satellites de communication et de reconnaissance ennemis, comme la s\u00e9rie de satellites espions am\u00e9ricains KH-8 et la future s\u00e9rie KH-9. Et c'est sans parler des satellites espions sovi\u00e9tiques, encore plus vuln\u00e9rables. Quand il le faudra, nous pourrons an\u00e9antir la station spatiale sovi\u00e9tique Saliout ou la station Skylab que les \u00c9tats-Unis vont lancer prochainement.\n\n\u2014 Commissaire Lei, de quoi parlez-vous ? dit quelqu'un dans le dos de Wenjie.\n\nElle se retourna. C'\u00e9tait Yang Weining, il fixait le commissaire politique avec un regard s\u00e9v\u00e8re.\n\n\u2014 C'est pour le travail, l\u00e2cha le commissaire avant de s'en aller.\n\nYang Weining regarda Wenjie sans rien dire et s'en alla \u00e0 son tour, laissant Wenjie seule.\n\n _C'est lui qui m'a fait venir ici mais, encore aujourd'hui, il ne me fait toujours pas confiance_ , se dit tristement Wenjie, tout en s'inqui\u00e9tant pour Lei Zhicheng. Dans la base, le pouvoir politique de Lei Zhicheng \u00e9tait certes plus important que celui de Yang Weining et c'\u00e9tait lui qui prenait toutes les d\u00e9cisions finales. Toutefois son d\u00e9part pr\u00e9cipit\u00e9 de tout \u00e0 l'heure signifiait clairement aux yeux de Wenjie qu'il avait commis une erreur devant l'ing\u00e9nieur en chef. Cela la confortait dans l'id\u00e9e qu'il lui avait r\u00e9v\u00e9l\u00e9 le v\u00e9ritable but de C\u00f4te Rouge.\n\n _Subirait-il les cons\u00e9quences de cette confiance ?_ Alors qu'elle regardait s'\u00e9loigner la robuste silhouette du commissaire politique, une vague de gratitude lui emplit le c\u0153ur. Ici, la confiance \u00e9tait \u00e0 n'en pas douter un luxe auquel elle n'aspirait d\u00e9j\u00e0 plus. Compar\u00e9 \u00e0 Yang Weining, Lei Zhicheng \u00e9tait plus proche de l'image qu'elle se faisait d'un militaire ; il en avait la franchise et la droiture, tandis que Yang Weining n'\u00e9tait rien d'autre que ce genre d'intellectuel quelconque comme il y en avait tant \u00e0 cette \u00e9poque : l\u00e2che et prudent, seulement \u00e0 la recherche de protection et de s\u00e9curit\u00e9. Wenjie pouvait le comprendre, mais la distance qui la s\u00e9parait de lui s'accrut.\n\nLe lendemain, Wenjie fut transf\u00e9r\u00e9e de l'unit\u00e9 de transmission \u00e0 l'unit\u00e9 de surveillance. Elle avait d'abord cru que cette d\u00e9cision \u00e9tait en lien avec ce qui s'\u00e9tait pass\u00e9 la veille, qu'on l'\u00e9loignait du secteur principal de C\u00f4te Rouge. Cependant, une fois arriv\u00e9e dans l'unit\u00e9 de surveillance, elle comprit que le centre de la base \u00e9tait bien l\u00e0. Les appareils des deux unit\u00e9s \u00e9taient certes les m\u00eames \u2013 on utilisait la m\u00eame antenne \u2013, mais le niveau de technologie de l'unit\u00e9 de surveillance \u00e9tait bien plus avanc\u00e9 que celui de l'unit\u00e9 de transmission.\n\nL'unit\u00e9 de surveillance poss\u00e9dait un syst\u00e8me de r\u00e9ception des ondes radio tr\u00e8s sensible et tr\u00e8s performant. Les signaux re\u00e7us par l'antenne \u00e9taient amplifi\u00e9s \u00e0 l'aide de masers \u00e0 ondes progressives en rubis et le noyau du syst\u00e8me de r\u00e9ception \u00e9tait immerg\u00e9 dans une cuve d'h\u00e9lium liquide \u00e0 \u2013 269 oC, de mani\u00e8re \u00e0 minimiser les fluctuations thermiques. Un h\u00e9licopt\u00e8re venait remplir la cuve \u00e0 intervalles r\u00e9guliers. Toute l'installation procurait au syst\u00e8me une tr\u00e8s haute sensibilit\u00e9 et il \u00e9tait capable de recevoir des signaux tr\u00e8s faibles. Wenjie ne pouvait s'emp\u00eacher de penser que si cet \u00e9quipement avait \u00e9t\u00e9 mis au service de ses recherches en radioastronomie, elle aurait obtenu des r\u00e9sultats fantastiques.\n\nLe syst\u00e8me informatique de l'unit\u00e9 de surveillance \u00e9tait lui aussi infiniment plus sophistiqu\u00e9 que celui de l'unit\u00e9 de transmission. La premi\u00e8re fois qu'elle \u00e9tait entr\u00e9e dans la salle informatique, elle avait vu une rang\u00e9e d'\u00e9crans \u00e0 tubes cathodiques. Elle avait d\u00e9couvert avec surprise que des lignes de codes de programmation d\u00e9filaient sur les \u00e9crans, c'est-\u00e0-dire que les programmes informatiques \u00e9taient susceptibles d'\u00eatre \u00e9dit\u00e9s et modul\u00e9s facilement gr\u00e2ce au clavier. Lorsqu'elle utilisait des ordinateurs du temps o\u00f9 elle \u00e9tait \u00e0 l'universit\u00e9, les codes \u00e9taient d'abord \u00e9crits \u00e0 la main sur des feuilles de programmation avant d'\u00eatre tap\u00e9s sur des rubans perfor\u00e9s via une machine \u00e0 \u00e9crire. Elle avait entendu dire qu'on pouvait maintenant faire de la programmation directement avec un clavier et un \u00e9cran, mais c'\u00e9tait la premi\u00e8re fois qu'elle voyait cela.\n\nCe qui la surprenait encore davantage, c'\u00e9taient les logiciels disponibles. Elle prit connaissance d'un logiciel nomm\u00e9 Fortran1 qui permettait d'utiliser un langage de programmation proche du langage naturel et m\u00eame d'entrer des \u00e9quations math\u00e9matiques ! La programmation avec le logiciel Fortran s'av\u00e9rait bien plus efficace que la programmation en langage machine. Il existait \u00e9galement une chose appel\u00e9e \"base de donn\u00e9es\" qui permettait d'emmagasiner et de manipuler \u00e0 son aise une grande quantit\u00e9 de donn\u00e9es.\n\nDeux jours plus tard, le commissaire politique Lei retourna voir Wenjie, mais la discussion eut cette fois-ci lieu dans la salle informatique de l'unit\u00e9 de surveillance, devant la rang\u00e9e d'\u00e9crans verts clignotants. Yang Weining s'assit non loin. Il ne participait pas \u00e0 leur conversation, mais ne semblait pas non plus dispos\u00e9 \u00e0 partir. Wenjie se sentait mal \u00e0 l'aise.\n\n\u2014 Wenjie, dit le commissaire politique Lei, je vais t'expliquer en quoi consiste principalement le travail de l'unit\u00e9 de surveillance. Pour le dire simplement, son r\u00f4le est de surveiller les mouvements ennemis dans l'espace, y compris les communications entre diff\u00e9rents engins spatiaux et la Terre et entre les engins eux-m\u00eames. L'unit\u00e9 de surveillance travaille en collaboration avec nos centres de contr\u00f4le a\u00e9rospatial pour observer la position des vaisseaux spatiaux \u00e9trangers en orbite dans l'espace. Elle fournit des donn\u00e9es au syst\u00e8me d'armement de la base. Pour le dire autrement, ce sont les yeux de C\u00f4te Rouge.\n\nYang Weining l'interrompit :\n\n\u2014 Commissaire Lei, je trouve que ce n'est pas une bonne id\u00e9e. Il n'y a aucune n\u00e9cessit\u00e9 \u00e0 lui raconter tout cela.\n\nWenjie observa Yang Weining assis un peu plus loin et murmura d'une voix inqui\u00e8te :\n\n\u2014 Commissaire, si ce n'est pas appropri\u00e9, je peux le comprendre et...\n\n\u2014 Non, non, Wenjie.\n\nLe commissaire Lei leva la main comme pour la faire taire, puis il se tourna vers Yang Weining et d\u00e9clara :\n\n\u2014 Toujours la m\u00eame phrase, ing\u00e9nieur en chef Yang : \"C'est pour le travail.\" Pour que Wenjie progresse, il faut lui dire ce qu'elle doit savoir.\n\nYang Weining se leva :\n\n\u2014 Je ferai un rapport.\n\n\u2014 C'est bien s\u00fbr ton droit, mais sois certain, ing\u00e9nieur en chef Yang, que j'en assumerai toute la responsabilit\u00e9, dit sereinement le commissaire Lei.\n\nFurieux, Yang Weining tourna les talons.\n\n\u2014 Ne t'en fais pas, Wenjie, l'ing\u00e9nieur en chef Yang est ainsi. Il est toujours trop prudent. Son travail l'\u00e9touffe parfois, sourit le commissaire politique en secouant la t\u00eate.\n\nPuis il regarda Wenjie dans les yeux et d\u00e9clara sur un ton solennel :\n\n\u2014 Wenjie, si nous t'avons fait venir \u00e0 la base, c'\u00e9tait au d\u00e9but pour une raison pr\u00e9cise : le syst\u00e8me de surveillance de C\u00f4te Rouge rencontre fr\u00e9quemment des perturbations dues \u00e0 des rayonnements \u00e9lectromagn\u00e9tiques provoqu\u00e9s par des \u00e9ruptions ou des taches solaires. Nous sommes par hasard tomb\u00e9s sur ton article et avons d\u00e9couvert que tu avais men\u00e9 des recherches approfondies sur les activit\u00e9s du soleil. Tu as \u00e9tabli ce qui est aujourd'hui en Chine le mod\u00e8le de pr\u00e9vision le plus pr\u00e9cis. Nous avons donc voulu faire appel \u00e0 tes comp\u00e9tences pour nous aider \u00e0 r\u00e9soudre ce probl\u00e8me.\n\nMais d\u00e8s ton arriv\u00e9e, tu as fait preuve de grandes comp\u00e9tences techniques et nous avons d\u00e9cid\u00e9 de te confier davantage de responsabilit\u00e9s. Voil\u00e0 quelle \u00e9tait mon id\u00e9e : t'affecter tout d'abord \u00e0 l'unit\u00e9 de transmission, puis \u00e0 celle de surveillance, pour que tu aies une compr\u00e9hension exhaustive du syst\u00e8me de C\u00f4te Rouge. Quel sera ton prochain poste ? Nous y r\u00e9fl\u00e9chirons en temps voulu. Bien entendu, comme tu l'as remarqu\u00e9, certains n'y \u00e9taient pas favorables, mais j'ai personnellement choisi de te faire confiance. Wenjie, je vais \u00eatre clair. Je suis aujourd'hui le seul \u00e0 t'accorder ma confiance. J'esp\u00e8re qu'en travaillant dur tu pourras gagner celle de toute l'organisation.\n\nLe commissaire politique Lei posa sa main robuste sur l'\u00e9paule de Wenjie. Elle sentit la force et la chaleur que celle-ci lui insufflait.\n\n\u2014 Wenjie, je vais te dire ce que je souhaite r\u00e9ellement : j'aimerais un jour pouvoir t'appeler \"camarade\".\n\nApr\u00e8s avoir prononc\u00e9 ces mots, le commissaire politique se leva et s'\u00e9loigna de son pas assur\u00e9 de militaire. Les yeux de Wenjie \u00e9taient gorg\u00e9s de larmes \u00e0 travers lesquelles elle voyait s'afficher sur les \u00e9crans une ronde de flammes dansantes. C'\u00e9tait la premi\u00e8re fois qu'elle pleurait depuis la mort de son p\u00e8re.\n\nUne fois rompue au travail dans l'unit\u00e9 de surveillance, elle s'aper\u00e7ut vite que son efficacit\u00e9 \u00e9tait loin de celle dont elle avait fait preuve au sein de l'unit\u00e9 de transmission. Ses connaissances en informatique \u00e9taient d\u00e9pass\u00e9es et elle devait souvent apprendre le fonctionnement des logiciels \u00e0 partir de z\u00e9ro. M\u00eame si le commissaire politique Lei lui faisait confiance, elle \u00e9tait toujours tr\u00e8s strictement encadr\u00e9e. Elle \u00e9tait autoris\u00e9e \u00e0 voir les codes sources des logiciels, mais ne pouvait avoir acc\u00e8s \u00e0 la base de donn\u00e9es.\n\nDurant la plupart de ses journ\u00e9es de travail, Wenjie \u00e9tait sous la tutelle de Yang Weining. Celui-ci \u00e9tait devenu encore plus irritable et se f\u00e2chait facilement, malgr\u00e9 les remontrances du commissaire Lei. C'\u00e9tait comme si, d\u00e8s qu'il voyait Wenjie, son c\u0153ur s'emplissait d'une indicible anxi\u00e9t\u00e9.\n\nPeu \u00e0 peu, elle d\u00e9couvrait au cours de son travail de plus en plus de choses qu'elle ne pouvait expliquer. Elle avait l'impression que les projets de la base de C\u00f4te Rouge \u00e9taient autrement plus complexes qu'elle ne se l'\u00e9tait imagin\u00e9.\n\nL'unit\u00e9 de surveillance intercepta un jour une transmission particuli\u00e8rement int\u00e9ressante qui, apr\u00e8s avoir \u00e9t\u00e9 d\u00e9crypt\u00e9e informatiquement, se r\u00e9v\u00e9la \u00eatre une s\u00e9rie de photographies prises par un satellite. Les images floues furent envoy\u00e9es au Bureau g\u00e9n\u00e9ral de la cartographie pour \u00eatre interpr\u00e9t\u00e9es.\n\nOn d\u00e9couvrit qu'il s'agissait de cibles militaires sensibles situ\u00e9es sur le territoire chinois, dont notamment le port militaire de Qingdao et quelques usines militaires importantes du projet Troisi\u00e8me Front2. Apr\u00e8s analyse, il fut \u00e9tabli que les photographies avaient \u00e9t\u00e9 prises par un satellite espion am\u00e9ricain KH-9.\n\nLe premier KH-9 venait tout juste d'\u00eatre envoy\u00e9 dans l'espace et son activit\u00e9 principale consistait principalement \u00e0 \u00e9jecter des capsules de films r\u00e9cup\u00e9rables. Mais il avait \u00e9galement pour objectif d'exp\u00e9rimenter une technologie plus avanc\u00e9e : le transfert de photographies num\u00e9riques. Cette technologie \u00e9tait toutefois encore loin d'\u00eatre aboutie \u00e0 l'\u00e9poque, et la fr\u00e9quence de transmission utilis\u00e9e plus basse. Les donn\u00e9es envoy\u00e9es num\u00e9riquement pr\u00e9sentaient donc davantage de risques d'\u00eatre intercept\u00e9es par le syst\u00e8me de C\u00f4te Rouge. Comme il s'agissait simplement d'envois exp\u00e9rimentaux, le cryptage des donn\u00e9es n'\u00e9tait pas tr\u00e8s \u00e9lev\u00e9 et elles pouvaient \u00eatre ais\u00e9ment d\u00e9chiffr\u00e9es. Ces transmissions \u00e9taient sans aucun doute les plus importantes \u00e0 surveiller, car il \u00e9tait rare d'avoir une telle opportunit\u00e9 d'obtenir des renseignements sur l'espionnage spatial am\u00e9ricain. Cependant, le surlendemain, l'ing\u00e9nieur en chef Yang ordonna inexplicablement de changer la fr\u00e9quence et la direction des appareils de surveillance. L'abandon de cette cible paraissait incompr\u00e9hensible pour Wenjie.\n\nElle fut t\u00e9moin d'un autre \u00e9v\u00e9nement qui l'intrigua. Bien qu'elle soit d\u00e9sormais affect\u00e9e \u00e0 l'unit\u00e9 de surveillance, on lui demandait parfois d'aller effectuer quelques t\u00e2ches dans l'unit\u00e9 de transmission. Une fois, elle assista par hasard au r\u00e9glage des fr\u00e9quences pour les futurs plans de transmission. Elle obtint la certitude que les fr\u00e9quences des transmissions 304, 318 et 325 qui avaient \u00e9t\u00e9 s\u00e9lectionn\u00e9es \u00e9taient plus basses que la gamme de fr\u00e9quences des micro-ondes et ne permettaient pas de provoquer d'effets thermiques sur les cibles.\n\nUn jour, Ye Wenjie fut soudain convoqu\u00e9e au quartier g\u00e9n\u00e9ral de la base. Au regard et \u00e0 la voix de l'officier, elle eut comme un mauvais pressentiment.\n\nDans le bureau l'attendait une sc\u00e8ne qu'elle ne connaissait que trop bien : les principaux responsables politiques de la base \u00e9taient l\u00e0, accompagn\u00e9s de deux officiers qu'elle n'avait jamais vus. D\u00e8s le premier coup d'\u0153il, on pouvait deviner qu'ils venaient de l'ext\u00e9rieur. Leurs regards se pos\u00e8rent froidement sur elle. Mais la sensibilit\u00e9 qu'elle avait d\u00e9velopp\u00e9e durant ces derni\u00e8res ann\u00e9es lui faisait dire qu'elle ne serait peut-\u00eatre pas la plus malchanceuse du jour. Elle n'\u00e9tait tout au plus qu'un objet fun\u00e9raire qu'on enterrerait avec le d\u00e9funt dans la tombe. Elle vit le commissaire politique Lei Zhicheng assis dans un coin de la pi\u00e8ce, l'air abattu.\n\n _Il va finalement payer le prix de la confiance qu'il m'a accord\u00e9e_ , pensa aussit\u00f4t Wenjie. D'embl\u00e9e, elle d\u00e9cida de prendre sur elle toute la responsabilit\u00e9, pour ne pas impliquer le commissaire politique. Elle ne s'emp\u00eacherait d'ailleurs pas de mentir. Mais ce fut le commissaire politique Lei qui prit le premier la parole, et ce qu'il expliqua fut pour le moins inattendu.\n\n\u2014 Ye Wenjie, avant tout, je tiens \u00e0 pr\u00e9ciser que je suis contre la raison qui te vaut d'\u00eatre aujourd'hui dans ce bureau. La d\u00e9cision qui va t'\u00eatre communiqu\u00e9e a \u00e9t\u00e9 prise par l'ing\u00e9nieur en chef Yang, apr\u00e8s qu'il l'a soumise \u00e0 l'approbation de la hi\u00e9rarchie. Il assumera l'enti\u00e8re responsabilit\u00e9 des cons\u00e9quences de cette d\u00e9cision.\n\nPuis il se tourna vers Yang Weining, qui opina solennellement de la t\u00eate.\n\n\u2014 Dans le but de mieux mettre \u00e0 profit tes comp\u00e9tences au sein de la base de C\u00f4te Rouge, l'ing\u00e9nieur en chef Yang a adress\u00e9 \u00e0 plusieurs reprises \u00e0 nos sup\u00e9rieurs une requ\u00eate te concernant. Nos camarades envoy\u00e9s par le d\u00e9partement politique g\u00e9n\u00e9ral de l'Arm\u00e9e populaire de lib\u00e9ration ont aussi pris connaissance de ta situation, poursuivit-il en d\u00e9signant les deux officiers qu'elle ne connaissait pas. Avec l'accord de nos sup\u00e9rieurs, nous allons te r\u00e9v\u00e9ler le contenu r\u00e9el du projet C\u00f4te Rouge.\n\nCe ne fut qu'au bout d'un certain temps que Wenjie comprit ce que signifiaient les paroles du commissaire politique : durant tout ce temps, il n'avait cess\u00e9 de lui mentir !\n\n\u2014 J'ose esp\u00e9rer que tu saisiras l'opportunit\u00e9 qui t'est offerte pour travailler dur et expier tes crimes. D\u00e9sormais, nous exigerons de toi que tu te comportes avec la plus grande probit\u00e9. Le moindre faux pas, le moindre soup\u00e7on d'une attitude r\u00e9actionnaire et la punition sera exemplaire !\n\nLa mani\u00e8re qu'avait le commissaire Lei de lui parler et de la d\u00e9visager donna la sensation \u00e0 Wenjie que ce n'\u00e9tait pas le m\u00eame individu qu'elle avait connu autrefois.\n\n\u2014 Me suis-je bien fait comprendre ? Bien, je vais donc demander \u00e0 l'ing\u00e9nieur en chef Yang de te donner les explications.\n\nLes autres quitt\u00e8rent la pi\u00e8ce. Ne rest\u00e8rent plus dans le bureau que Yang Weining et Ye Wenjie.\n\n\u2014 Si tu ne veux pas, il est encore temps de faire machine arri\u00e8re.\n\nWenjie connaissait le poids de ces mots et elle comprenait maintenant la nervosit\u00e9 de Yang Weining ces derniers jours \u00e0 son \u00e9gard. Pour pouvoir exploiter toutes ses comp\u00e9tences, elle devait \u00eatre mise au courant du but r\u00e9el de C\u00f4te Rouge, mais cela signifiait dans le m\u00eame temps que le dernier espoir de quitter un jour le pic du Radar dispara\u00eetrait \u00e0 jamais. C\u00f4te Rouge serait son tombeau.\n\n\u2014 J'accepte, dit Wenjie, lentement, mais avec d\u00e9termination.\n\nEt c'est ainsi que cet apr\u00e8s-midi du d\u00e9but de l'\u00e9t\u00e9, dans le mugissement de l'antenne parabolique et le bruissement des pins des montagnes du Grand Khingan, Yang Weining r\u00e9v\u00e9la \u00e0 Ye Wenjie le projet r\u00e9el de C\u00f4te Rouge.\n\nUne l\u00e9gende de cette \u00e9poque, bien plus incroyable que tous les mensonges de Lei Zhicheng.\n\n* * *\n\n1 Un langage de programmation de haut niveau pour la premi\u00e8re g\u00e9n\u00e9ration d'ordinateurs. _(N.d.A.)_\n\n2 Le projet Troisi\u00e8me Front \u00e9tait un vaste ensemble secret d'investissements militaro-industriels, lanc\u00e9 en 1964. _(N.d.T.)_\n\n## 13\n\n## C\u00d4TE ROUGE III\n\nS\u00e9lection de documents du projet C\u00f4te Rouge, d\u00e9classifi\u00e9s trois ans apr\u00e8s le r\u00e9cit de Ye Wenjie \u00e0 Wang Miao :\n\n### **I. Une question capitale n\u00e9glig\u00e9e par la majorit\u00e9 des recherches en sciences fondamentales du monde entier**\n\n(article originellement publi\u00e9 dans la revue confidentielle _R\u00e9f\u00e9rences internes_ , XX\/XX\/196X)\n\n **[R\u00e9sum\u00e9 de l'article]**\n\nAu regard de l'histoire moderne et contemporaine, les r\u00e9sultats de recherche en sciences fondamentales peuvent \u00eatre divis\u00e9s en deux modes : le mode progressif et le mode radical.\n\nMode progressif : les r\u00e9sultats sont th\u00e9oriques et fondamentaux avant de devenir graduellement des technologies appliqu\u00e9es. L'accumulation de ces technologies finit par provoquer une avanc\u00e9e capitale dans le domaine en question. Parmi les exemples r\u00e9cents : l'\u00e9volution des technologies spatiales.\n\nMode radical : les r\u00e9sultats th\u00e9oriques et fondamentaux deviennent rapidement des technologies appliqu\u00e9es et provoquent d\u00e8s leur mise en \u0153uvre des avanc\u00e9es capitales. On peut citer l'exemple r\u00e9cent des armes nucl\u00e9aires. Jusque dans les ann\u00e9es 1940, certains des physiciens les plus talentueux du monde estimaient encore qu'il ne serait jamais possible de lib\u00e9rer l'\u00e9nergie de l'atome. Pourtant l'arme atomique est apparue dans un laps de temps tr\u00e8s court. La transformation de la science fondamentale en technologie appliqu\u00e9e a fait un grand bond, et ce dans une dur\u00e9e extr\u00eamement br\u00e8ve. Ce mode op\u00e9ratoire est qualifi\u00e9 de mode radical.\n\nLes pays membres de l'Otan et ceux du pacte de Varsovie sont actuellement plus actifs que jamais dans la recherche en science fondamentale et n'h\u00e9sitent pas \u00e0 engager de lourds investissements. Des innovations technologiques peuvent donc survenir \u00e0 tout moment et cela repr\u00e9sente une menace majeure pour la planification strat\u00e9gique de notre pays.\n\nCet article d\u00e9fend l'id\u00e9e qu'au niveau national l'accent est aujourd'hui trop mis sur le mode progressif de la recherche scientifique et pas assez sur le mode radical qui permettrait pourtant des innovations technologiques capitales. En prenant du recul et en se pla\u00e7ant d'un point de vue strat\u00e9gique, il conviendrait d'\u00e9laborer une strat\u00e9gie globale et de mettre en place une s\u00e9rie de principes sur lesquels s'appuyer en cas d'innovation technologique.\n\nListe des domaines dans lesquels une innovation radicale est la plus envisageable :\n\n1. Physique : **[Sans objet]**\n\n2. Biologie : **[Sans objet]**\n\n3. Sciences informatiques : **[Sans objet]**\n\n4. Recherche de civilisations extraterrestres : Il s'agit du domaine dans lequel la probabilit\u00e9 d'une d\u00e9couverte technologique majeure est la plus grande. En cas d'innovation, son impact serait bien plus important que celui des innovations dans les trois autres domaines r\u00e9unis.\n\n[Texte int\u00e9gral] **[Sans objet]**\n\n _[Instructions] : Imprimer l'article et le distribuer aux personnes autoris\u00e9es pour des d\u00e9bats lors de groupes de discussion. Le point de vue de l'auteur n'est peut-\u00eatre pas du go\u00fbt de tous, mais il ne faut pas l'incriminer pour ses conclusions. L'importantest d'appr\u00e9cier la r\u00e9flexion \u00e0 long terme propos\u00e9e par l'auteur. Certains camarades semblent maintenant porter des \u0153ill\u00e8res, peut-\u00eatre en raison de l'environnement politique, mais aussi \u00e0 cause de l'arrogance de certains. Je le d\u00e9sapprouve. Une absence de vision strat\u00e9gique globale est dangereuse. J'estime que dans les quatre domaines d'innovation technologique possibles propos\u00e9s par l'auteur, le dernier cit\u00e9 est celui auquel nous pensons le moins, mais qui m\u00e9rite le plus notre attention. Nous devrions analyser cette question de fa\u00e7on plus syst\u00e9matique._\n\n _[Signature] XXX [Date] XX\/XX\/196X_\n\n### **II. Rapport d'\u00e9tude sur les possibilit\u00e9s d'innovations techniques dans le domaine de la recherche de civilisations extraterrestres**\n\n1. Les tendances de la recherche scientifique internationale **[Version r\u00e9sum\u00e9e]**\n\n(1) \u00c9tats-Unis et autres pays de l'Otan : les n\u00e9cessit\u00e9s strat\u00e9giques et scientifiques de la recherche d'intelligences extraterrestres sont d\u00e9j\u00e0 largement reconnues. C'est un domaine qui b\u00e9n\u00e9ficie du soutien du monde acad\u00e9mique.\n\nProjet Ozma : en 1960, \u00e0 l'Observatoire national de radioastronomie de Green Bank, en Virginie-Occidentale, une tentative de d\u00e9tection de signaux a \u00e9t\u00e9 r\u00e9alis\u00e9e au moyen d'un radiot\u00e9lescope de 26 m\u00e8tres de diam\u00e8tre \u00e9quip\u00e9 d'un r\u00e9cepteur monocanal. Le radiot\u00e9lescope a \u00e9t\u00e9 point\u00e9 dans la direction des \u00e9toiles Tau Ceti et Epsilon Eridani pendant environ 200 heures \u00e0 une fr\u00e9quence proche de 1 420 MHz. Un projet Ozma II est pr\u00e9vu pour 1972, avec un nombre \u00e9largi de cibles et des gammes de fr\u00e9quences plus hautes. Un autre projet est envisag\u00e9 pour la m\u00eame ann\u00e9e : celui du lancement des sondes spatiales Pionneer 10 et Pionneer 11. Toutes deux emporteront avec elle une plaque m\u00e9tallique contenant des informations sur la civilisation terrestre. En 1977, il est pr\u00e9vu de lancer les sondes spatiales Voyager 1 et 2, chacune porteuse d'un enregistrement sur un disque m\u00e9tallique.\n\nRadiot\u00e9lescope d'Arecibo \u00e0 Porto Rico : construit en 1963, c'est un instrument important pour la recherche de civilisations extraterrestres. Son antenne parabolique permet de couvrir une superficie de 20 acres, soit plus que la superficie totale couverte par tous les autres radiot\u00e9lescopes du monde additionn\u00e9s. Dot\u00e9 d'un syst\u00e8me informatique, le radiot\u00e9lescope est capable de surveiller simultan\u00e9ment 65 000 canaux, et poss\u00e8de \u00e9galement une fonction de transmission \u00e0 ultra-haute \u00e9nergie.\n\n(2) Union sovi\u00e9tique : les services de renseignement poss\u00e8dent moins d'informations, mais certains indices laissent penser que les Sovi\u00e9tiques ont eux aussi investi massivement dans ce domaine. En outre, en comparaison avec les pays de l'Otan, leurs projets de recherche semblent \u00eatre plus syst\u00e9matiques et davantage planifi\u00e9s sur le long terme.\n\nDes informations provenant de diff\u00e9rentes sources indiquent qu'un projet de construction est actuellement en cours : un syst\u00e8me \u00e0 port\u00e9e mondiale de radiot\u00e9lescopes bas\u00e9 sur les principes de la synth\u00e8se d'ouverture et de l'interf\u00e9rom\u00e9trie \u00e0 tr\u00e8s longue base. Une fois achev\u00e9, il deviendra le syst\u00e8me d'exploration de l'espace lointain le plus puissant au monde.\n\n2. Analyse pr\u00e9liminaire des soci\u00e9t\u00e9s extraterrestres \u00e0 la lumi\u00e8re de la conception mat\u00e9rialiste de l'histoire **[Sans objet]**\n\n3. Analyse pr\u00e9liminaire de l'influence des civilisations extraterrestres sur les tendances sociales et politiques des soci\u00e9t\u00e9s humaines **[Sans objet]**\n\n4. Analyse pr\u00e9liminaire des impacts sur le monde actuel g\u00e9n\u00e9r\u00e9s par un contact avec des civilisations extraterrestres\n\n(1) Contact unidirectionnel (simple r\u00e9ception de messages transmis par une civilisation extraterrestre) **[Sans objet]**\n\n(2) Contact bidirectionnel (\u00e9changes et contacts directs avec une civilisation extraterrestre) **[Sans objet]**\n\n5. Risques et cons\u00e9quences d'une situation de monopole d'une superpuissance \u00e9trang\u00e8re \u00e0 la suite d'un premier contact \u00e9tabli avec une civilisation extraterrestre\n\n(1) Risques et cons\u00e9quences d'une situation de monopole des \u00c9tats-Unis ou d'un autre pays de l'Otan \u00e0 la suite d'un premier contact \u00e9tabli avec une civilisation extraterrestre **[Non encore d\u00e9classifi\u00e9]**\n\n(2) Risques et cons\u00e9quences d'une situation de monopole de l'URSS ou d'un autre membre du pacte de Varsovie \u00e0 la suite d'un premier contact \u00e9tabli avec une civilisation extraterrestre **[Non encore d\u00e9classifi\u00e9]**\n\n _[Instructions] : J'ai pris connaissance du document. Des humains ont d\u00e9j\u00e0 envoy\u00e9 leurs messages dans l'espace. Il serait dangereux que les soci\u00e9t\u00e9s extraterrestres n'entendent qu'une seule voix. Nous devons faire entendre la n\u00f4tre. Elles pourront ainsi avoir un panorama plus complet des soci\u00e9t\u00e9s humaines. Il serait f\u00e2cheux qu'elles n'entendent qu'un seul son de cloche. Il faut le faire et il faut le faire vite._\n\n _[Signature] XXX [Date] XX\/XX\/196X_\n\n### **III. Rapport de recherche sur la phase initiale du lancement du projet C\u00f4te Rouge**\n\nXX\/XX\/196X\n\n **Top secret**\n\nNombre de copies : 2.\n\n **[R\u00e9sum\u00e9 du document]** Document central no XXXXXX envoy\u00e9 \u00e0 l'Administration d'\u00c9tat pour la science, la technologie et l'industrie de la D\u00e9fense nationale et \u00e0 l'Acad\u00e9mie chinoise des sciences, puis transf\u00e9r\u00e9 \u00e0 l'Administration d'\u00c9tat pour la planification centrale de la D\u00e9fense nationale, avant d'\u00eatre distribu\u00e9 lors des r\u00e9unions XXXXXX, XXXXXX et XXXXXX.\n\nNum\u00e9ro de code du projet : 3760.\n\nIdentifiant du projet : C\u00f4te Rouge.\n\n1. Principes g\u00e9n\u00e9raux **[Version r\u00e9sum\u00e9e]**\n\nChercher l'existence possible de civilisations extraterrestres et essayer d'\u00e9tablir des contacts et des \u00e9changes.\n\n2. Recherches th\u00e9oriques sur le projet C\u00f4te Rouge **[R\u00e9sum\u00e9]**\n\n(1) Exploration et surveillance\n\nGammes de fr\u00e9quences de la surveillance : 1 000 \u00e0 4 000 MHz.\n\nNombre de canaux de surveillance : 15 000.\n\nFr\u00e9quences importantes \u00e0 surveiller : fr\u00e9quence de r\u00e9sonance de l'atome d'hydrog\u00e8ne : 1 420 MHz ; fr\u00e9quence du groupe hydroxyle (un atome d'oxyg\u00e8ne et un atome d'hydrog\u00e8ne) : 1 667 MHz ; fr\u00e9quence de la mol\u00e9cule d'eau : 22 000 MHz.\n\nFourchette cibl\u00e9e pour la surveillance : un rayon de 1 000 ann\u00e9es-lumi\u00e8re, soit environ 20 millions d'\u00e9toiles. D\u00e9tails complets sur les cibles en annexe no 1.\n\n(2) Transmission de messages\n\nFr\u00e9quences de transmission : 2 800 MHz, 12 000 MHz, 22 000 MHz.\n\nPuissance de transmission : 10-25 MW.\n\nCibles de transmission : rayon de 200 ann\u00e9es-lumi\u00e8re autour de la Terre, soit environ cent mille \u00e9toiles. D\u00e9tails complets sur les cibles en annexe no 2.\n\n(3) D\u00e9veloppement du syst\u00e8me de code auto-interpr\u00e9tatif de C\u00f4te Rouge\n\nPrincipe directeur : utiliser les lois fondamentales et universelles des math\u00e9matiques et de la physique pour cr\u00e9er un code linguistique basique capable d'\u00eatre compris par n'importe quelle civilisation ma\u00eetrisant les bases de l'alg\u00e8bre, de la g\u00e9om\u00e9trie euclidienne et des lois de la m\u00e9canique newtonienne.\n\nAdjoindre au code linguistique des graphiques de basse r\u00e9solution, pour mettre progressivement sur pied un syst\u00e8me linguistique \u00e0 part enti\u00e8re. Langues support\u00e9es : chinois mandarin, esp\u00e9ranto.\n\nLe poids total du fichier contenant les informations est de 680 KB. Le temps de transmission pour des bandes de fr\u00e9quences de 2 800 MHz, 12 000 MHz et 22000 MHz est respectivement de 1183 minutes, 224 minutes et 132 minutes.\n\n3. Plan de mise en route du projet C\u00f4te Rouge\n\n(1) Plan de conception pr\u00e9liminaire des syst\u00e8mes d'exploration et de surveillance **[Non encore d\u00e9classifi\u00e9]**\n\n(2) Plan de conception pr\u00e9liminaire du syst\u00e8me de transmission **[Non encore d\u00e9classifi\u00e9]**\n\n(3) Plan pr\u00e9liminaire de s\u00e9lection du lieu pour la cr\u00e9ation de la base de surveillance et de transmission de C\u00f4te Rouge **[Sans objet]**\n\n(4) R\u00e9flexions pr\u00e9liminaires sur la constitution d'un corps militaire sp\u00e9cial pour C\u00f4te Rouge au sein du 2e corps d'artillerie de l'Arm\u00e9e populaire de lib\u00e9ration **[Non encore d\u00e9classifi\u00e9]**\n\n4. Contenu des informations transmises par C\u00f4te Rouge **[R\u00e9sum\u00e9]**\n\nAper\u00e7u g\u00e9n\u00e9ral de la plan\u00e8te Terre (3,1 KB), aper\u00e7u g\u00e9n\u00e9ral de la vie sur Terre (4,4 KB), aper\u00e7u g\u00e9n\u00e9ral sur les soci\u00e9t\u00e9s humaines (4,6 KB), informations de base sur l'histoire du monde (5,4 KB).\n\nPoids total de ces informations : 17,5 KB.\n\nApr\u00e8s auto-interpr\u00e9tation, les temps de transmission de ce message de 17,5 KB pour une fr\u00e9quence de 2 800 MHz, 12 000 et 22 000 MHz sont respectivement de 31 minutes, 7,5 minutes et 3,5 minutes. Avant d'\u00eatre envoy\u00e9e, l'information sera soumise \u00e0 un strict examen multidisciplinaire de fa\u00e7on \u00e0 s'assurer qu'elle ne comporte aucune indication sur les coordonn\u00e9es du syst\u00e8me solaire dans la Voie lact\u00e9e. La transmission pour les deux hautes fr\u00e9quences de 12 000 MHz et de 22 000 MHz devra \u00eatre minimis\u00e9e, afin de limiter le danger de d\u00e9voiler la localisation pr\u00e9cise de la source de transmission.\n\n### **IV. Message de salutation adress\u00e9 aux extraterrestres**\n\nPremi\u00e8re version **[Texte int\u00e9gral]**\n\n\u00c0 toutes les civilisations recevant ce message, attention ! Ce message a \u00e9t\u00e9 \u00e9mis par le pays terrien qui d\u00e9fend l'id\u00e9ologie de la r\u00e9volution ! Avant r\u00e9ception de ce message, vous avez peut-\u00eatre d\u00e9j\u00e0 re\u00e7u des informations provenant d'une autre source terrestre. Ces messages ont \u00e9t\u00e9 envoy\u00e9s par une puissance imp\u00e9rialiste, luttant avec une autre superpuissance pour le contr\u00f4le de la Terre, en essayant de replonger l'histoire de l'humanit\u00e9 dans ses heures les plus sombres. Nous esp\u00e9rons que vous n'\u00e9couterez pas leurs mensonges et que vous choisirez le camp de la justice, le camp de la r\u00e9volution !\n\n[Instructions] : _J'ai lu le document. C'est n'importe quoi ! Qu'on colle des_ dazibao1 _sur la surface de la Terre, passe encore, mais les envoyer dans l'espace... Les meneurs de la R\u00e9volution culturelle ne devraient pas \u00eatre impliqu\u00e9s dans le projet C\u00f4te Rouge. Le contenu d'un message aussi important doit faire l'objet d'une r\u00e9daction s\u00e9rieuse. Je recommande de cr\u00e9er un comit\u00e9 sp\u00e9cifique et de faire valider le contenu lors d'une r\u00e9union du Bureau politique._\n\n _[Signature] XXX [Date] XX\/XX\/196X_\n\nDeuxi\u00e8me version **[Sans objet]**\n\nTroisi\u00e8me version **[Sans objet]**\n\nQuatri\u00e8me version **[Texte int\u00e9gral]**\n\nSalutations amicales aux habitants du monde qui re\u00e7oit ce message.\n\nGr\u00e2ce aux informations ci-dessous, vous obtiendrez des connaissances de base sur la civilisation terrienne. Apr\u00e8s une longue histoire de labeur et de cr\u00e9ativit\u00e9, les humains ont b\u00e2ti une civilisation splendide et vu na\u00eetre une riche vari\u00e9t\u00e9 de cultures. Nous avons aussi commenc\u00e9 \u00e0 comprendre les lois de la nature et celles du d\u00e9veloppement de nos soci\u00e9t\u00e9s. Nous sommes tr\u00e8s fiers de ce que nous avons accompli jusqu'ici.\n\nMais notre monde est encore tr\u00e8s imparfait, il existe de la haine, des pr\u00e9jug\u00e9s et des guerres. En raison de la contradiction entre forces productives et rapports de production, il subsiste une grave in\u00e9galit\u00e9 de r\u00e9partition des richesses. Une partie importante des humains vit dans la mis\u00e8re et la pauvret\u00e9.\n\nLes soci\u00e9t\u00e9s humaines s'efforcent de r\u00e9soudre les difficult\u00e9s et les probl\u00e8mes qu'elles rencontrent et tentent de cr\u00e9er un avenir plus radieux pour la civilisation terrienne. Le pays \u00e0 l'origine de ce message est engag\u00e9 dans cet effort. Nous essayons de construire une soci\u00e9t\u00e9 id\u00e9ale dans laquelle le travail et la valeur de chaque \u00eatre humain seraient pleinement respect\u00e9s. Les besoins mat\u00e9riels et spirituels de chacun doivent \u00eatre pleinement satisfaits pour que la civilisation de la Terre soit encore plus parfaite.\n\nC'est avec la meilleure des intentions que nous esp\u00e9rons pouvoir cr\u00e9er un contact avec d'autres civilisations. Nous esp\u00e9rons que nous pourrons travailler ensemble \u00e0 b\u00e2tir une vie meilleure dans l'immensit\u00e9 de l'univers.\n\n### **V. Politiques et strat\u00e9gies**\n\n1. Politiques et strat\u00e9gies \u00e0 adopter apr\u00e8s r\u00e9ception d'un message par une civilisation extraterrestre **[Sans objet]**\n\n2. Politiques et strat\u00e9gies \u00e0 adopter apr\u00e8s \u00e9tablissement d'un contact avec une civilisation extraterrestre **[Sans objet]**\n\n _[Instructions] : Il est n\u00e9cessaire de prendre le temps d'investir dans d'autres projets malgr\u00e9 les probl\u00e8mes imm\u00e9diats auxquels la nation est confront\u00e9e. Ce projet nous a permis de r\u00e9fl\u00e9chir \u00e0 beaucoup de choses auxquelles nous n'avions jamais song\u00e9 avant. Nous ne pouvons trouver des solutions \u00e0 ces probl\u00e8mes qu'en prenant du recul et, sur ce point, C\u00f4te Rouge est d\u00e9j\u00e0 un grand succ\u00e8s. Ce serait merveilleux si d'autres cr\u00e9atures et soci\u00e9t\u00e9s existaient r\u00e9ellement dans l'univers ! Les spectateurs ont toujours une meilleure vue que les acteurs. La post\u00e9rit\u00e9 jugera si nous avons agi en bien ou en mal._\n\n _[Signature] XXX [Date] XX\/XX\/196X_\n\n* * *\n\n1 Les _dazibao_ (litt. \"journaux \u00e0 grands caract\u00e8res\") \u00e9taient des affiches non officielles r\u00e9dig\u00e9es par de simples citoyens qui pouvaient \u00eatre placard\u00e9es dans des lieux publics. Apparus dans l'\u00e9poque moderne au d\u00e9but de la R\u00e9volution culturelle en 1966, ils \u00e9taient souvent utilis\u00e9s dans des buts de propagande et de d\u00e9nonciation, mais certains de leurs auteurs s'en servaient pour attaquer les politiques men\u00e9es par les autorit\u00e9s. _(N.d.T.)_\n\n## 14\n\n## C\u00d4TE ROUGE IV\n\n\u2014 Professeur Ye, puis-je vous poser une question ? \u00c0 l'\u00e9poque, la recherche de civilisations extraterrestres \u00e9tait un domaine somme toute relativement marginal. Pourquoi le projet C\u00f4te Rouge b\u00e9n\u00e9ficiait-il d'un niveau de confidentialit\u00e9 si \u00e9lev\u00e9 ? demanda Wang Miao une fois que Ye Wenjie eut termin\u00e9 son r\u00e9cit.\n\n\u2014 Pour tout te dire, cette question n'a cess\u00e9 de se poser, depuis la phase initiale du projet jusqu'\u00e0 sa fin. Mais tu dois maintenant conna\u00eetre la r\u00e9ponse. Nous ne pouvons aujourd'hui qu'\u00eatre impressionn\u00e9s par la pr\u00e9voyance des d\u00e9cideurs de l'\u00e9poque.\n\n\u2014 Oui, ils ont vu les choses avant tout le monde, dit Wang Miao en hochant gravement la t\u00eate.\n\nComment et \u00e0 quel degr\u00e9 d'influence l'\u00e9tablissement d'un contact avec une civilisation extraterrestre allait-il impacter les soci\u00e9t\u00e9s humaines ? C'\u00e9tait une question tr\u00e8s s\u00e9rieuse, mais qui n'avait \u00e9t\u00e9 consid\u00e9r\u00e9e syst\u00e9matiquement que depuis quelques ann\u00e9es seulement. C'\u00e9tait d\u00e9sormais un sujet en vogue qui suscitait beaucoup d'\u00e9tudes, dont les conclusions pouvaient parfois \u00eatre d\u00e9routantes.\n\nLes projections id\u00e9alistes du pass\u00e9 avaient vol\u00e9 en \u00e9clats. Les sp\u00e9cialistes en \u00e9taient arriv\u00e9s \u00e0 la conclusion que, contrairement aux espoirs romantiques de la plupart des gens, il n'\u00e9tait pas souhaitable que la race humaine dans son ensemble entre en contact avec une civilisation extraterrestre. Au lieu d'unifier l'esp\u00e8ce humaine, ce contact aurait pour effet de la diviser et aggraverait plut\u00f4t qu'il ne r\u00e9duirait les conflits internes entre les diff\u00e9rentes civilisations de la Terre. En somme, les experts pr\u00e9voyaient que, d\u00e8s qu'un contact serait \u00e9tabli, les diff\u00e9rences culturelles entre les peuples seraient exacerb\u00e9es et que cette division au sein de l'esp\u00e8ce humaine m\u00e8nerait \u00e0 la catastrophe. La conclusion la plus effrayante \u00e9tait que ces impacts resteraient invariables, peu importe le degr\u00e9 ou la mani\u00e8re d'entrer en contact \u2013 unidirectionnel ou bidirectionnel \u2013 avec la civilisation extraterrestre, ou bien avec la forme ou le niveau d'\u00e9volution de cette civilisation.\n\nC'\u00e9tait l'id\u00e9e d\u00e9fendue par le sociologue Bill Matthews, du think tank de la RAND Corporation, dans son ouvrage intitul\u00e9 _Un rideau de fer de cent mille ann\u00e9es-lumi\u00e8re : une \u00e9tude sociologique de la recherche d'intelligence extraterrestre_. Il y proposait sa th\u00e9orie du \"contact comme symbole\". Selon lui, un contact avec une civilisation extraterrestre n'avait en fait qu'une fonction symbolique, ce n'\u00e9tait qu'un \u00e9l\u00e9ment d\u00e9clencheur : peu importe la nature du contact, l'effet serait identique. M\u00eame si, par exemple, ce contact ne faisait rien d'autre que prouver l'existence d'une intelligence extraterrestre dans l'univers, sans aucune autre information tangible \u2013 ce que Matthews appelait \"contact \u00e9l\u00e9mentaire\" \u2013, ses effets seraient amplifi\u00e9s par la psych\u00e9 collective et la culture de masse. Et par cons\u00e9quent, l'influence de ce contact \u00e9l\u00e9mentaire sur l'\u00e9volution de la civilisation humaine resterait immense. Si un pays ou une puissance politique venait \u00e0 monopoliser ce contact, il serait synonyme d'avantage d\u00e9cisif d'un point de vue tant \u00e9conomique que militaire.\n\n\u2014 Qu'ont donn\u00e9 les r\u00e9sultats de C\u00f4te Rouge ? demanda Wang Miao.\n\n\u2014 Tu peux s\u00fbrement l'imaginer.\n\nWang Miao hocha une nouvelle fois la t\u00eate. Il savait bien s\u00fbr que si le projet C\u00f4te Rouge avait \u00e9t\u00e9 couronn\u00e9 de succ\u00e8s, le monde ne serait pas celui qu'il \u00e9tait aujourd'hui. Mais il l\u00e2cha tout de m\u00eame une phrase qui se voulait r\u00e9confortante :\n\n\u2014 Nous ignorons encore aujourd'hui si le projet C\u00f4te Rouge est un \u00e9chec, car les ondes radio envoy\u00e9es par la base ne sont pas encore arriv\u00e9es tr\u00e8s loin dans l'univers.\n\nYe Wenjie secoua la t\u00eate :\n\n\u2014 Plus le signal s'\u00e9loigne, plus il s'affaiblit. Il existe de nombreuses perturbations dans l'espace. La probabilit\u00e9 pour qu'il soit re\u00e7u par une civilisation extraterrestre est tr\u00e8s mince. Des chercheurs ont montr\u00e9 que si nous souhaitions r\u00e9ussir une transmission radio dans l'espace, la puissance devait \u00eatre \u00e9quivalente au flux \u00e9nerg\u00e9tique d'une \u00e9toile de taille moyenne. L'astrophysicien sovi\u00e9tique Nikola\u00ef Kardachev a jadis propos\u00e9 de classer les civilisations extraterrestres en trois types, en fonction de l'\u00e9nergie qu'elles sont capables de mobiliser pour une communication \u00e0 l'\u00e9chelle interstellaire. Les civilisations de type I peuvent rassembler une \u00e9nergie \u00e9quivalente \u00e0 l'ensemble de l'\u00e9nergie re\u00e7ue par leur plan\u00e8te, provenant de l'\u00e9toile m\u00e8re : cette production \u00e9tait \u00e0 l'\u00e9poque \u00e9valu\u00e9e par Kardachev \u00e0 une fourchette entre 1015 et 1016 watts. Les civilisations de type II utiliseraient l'\u00e9quivalent de la production \u00e9nerg\u00e9tique d'une \u00e9toile classique, soit environ 1026 watts. Enfin, les civilisations de type III seraient capables d'utiliser une \u00e9nergie \u00e9quivalente \u00e0 1036 watts, correspondant \u00e0 la production \u00e9nerg\u00e9tique d'une galaxie. La plan\u00e8te Terre se trouverait actuellement \u00e0 0,7 sur l'\u00e9chelle de Kardachev, soit m\u00eame pas une civilisation de type I. \u00c0 titre de comparaison, l'\u00e9nergie mobilis\u00e9e pour les transmissions de C\u00f4te Rouge ne correspondait qu'\u00e0 un dixi\u00e8me de millioni\u00e8me de la production \u00e9nergique terrestre. Nos messages \u00e9taient des bourdonnements de moustique dans l'immensit\u00e9 du ciel. Personne n'a pu les entendre.\n\n\u2014 Si les civilisations de types II et III imagin\u00e9es par Kardachev existaient, nous pourrions entendre leurs voix.\n\n\u2014 En plus de vingt ans pass\u00e9s \u00e0 C\u00f4te Rouge, nous n'avons jamais rien entendu.\n\n\u2014 Dans ce cas, est-ce que C\u00f4te Rouge et tous les programmes occidentaux de recherche d'une intelligence extraterrestre ne prouvent pas que la Terre est en fin de compte la seule plan\u00e8te de l'univers \u00e0 abriter une forme de vie intelligente ?\n\nYe Wenjie poussa un l\u00e9ger soupir :\n\n\u2014 En th\u00e9orie, c'est une conclusion que nous ne pourrons jamais tirer avec certitude. Mais pour moi, ainsi que pour l'ensemble de ceux qui ont jadis travaill\u00e9 \u00e0 C\u00f4te Rouge, il n'y a aucun doute.\n\n\u2014 Quel dommage que la base ait \u00e9t\u00e9 d\u00e9mantel\u00e9e. Puisqu'elle \u00e9tait d\u00e9j\u00e0 construite, on aurait d\u00fb continuer \u00e0 la faire fonctionner. C'\u00e9tait une grande cause !\n\n\u2014 La fermeture de C\u00f4te Rouge s'est faite progressivement. Un grand projet de r\u00e9novation a \u00e9t\u00e9 entrepris dans les ann\u00e9es 1980, principalement pour moderniser les syst\u00e8mes informatiques des unit\u00e9s de transmission et de surveillance : le syst\u00e8me de transmission a \u00e9t\u00e9 automatis\u00e9 et l'unit\u00e9 de surveillance a \u00e9t\u00e9 \u00e9quip\u00e9e de deux mini-ordinateurs IBM. Cette r\u00e9novation a consid\u00e9rablement am\u00e9lior\u00e9 la capacit\u00e9 du traitement des donn\u00e9es. Nous pouvions d\u00e9sormais surveiller simultan\u00e9ment 40 000 canaux diff\u00e9rents.\n\nMais plus tard, les dirigeants ont pris conscience de l'extr\u00eame difficult\u00e9 des programmes de recherche de civilisations extraterrestres et ils ont progressivement perdu leur enthousiasme pour le projet. Le premier changement notable a \u00e9t\u00e9 la r\u00e9duction du degr\u00e9 de confidentialit\u00e9 de la base. La raison invoqu\u00e9e \u00e9tait que le niveau de s\u00e9curit\u00e9 de C\u00f4te Rouge \u00e9tait surestim\u00e9 ; l'\u00e9quipe de s\u00e9curit\u00e9 sur place a donc \u00e9t\u00e9 r\u00e9duite : d'une compagnie, elle est pass\u00e9e \u00e0 une escouade, puis plus tard, il n'est finalement rest\u00e9 qu'un petit groupe de cinq soldats en sentinelle. Apr\u00e8s la modernisation, m\u00eame si les employ\u00e9s de la base appartenaient toujours au 2e corps d'artillerie, la gestion des recherches scientifiques de C\u00f4te Rouge a \u00e9t\u00e9 confi\u00e9e \u00e0 l'Institut d'astronomie de l'Acad\u00e9mie des sciences. La base s'est vu confier des projets scientifiques n'\u00e9tant pas li\u00e9s \u00e0 la recherche de civilisations extraterrestres.\n\n\u2014 J'imagine que beaucoup de vos travaux scientifiques ont \u00e9t\u00e9 r\u00e9alis\u00e9s \u00e0 cette \u00e9poque-l\u00e0.\n\n\u2014 Au d\u00e9but, le syst\u00e8me de C\u00f4te Rouge a pris en charge des projets d'\u00e9tudes radioastronomiques, car la base poss\u00e9dait \u00e0 l'\u00e9poque le plus grand radiot\u00e9lescope de Chine. Mais par la suite, avec la construction d'autres bases d'observation radioastronomique, les recherches acad\u00e9miques de C\u00f4te Rouge se sont concentr\u00e9es sur l'observation et l'analyse de l'activit\u00e9 \u00e9lectromagn\u00e9tique du soleil. On y a m\u00eame install\u00e9 un t\u00e9lescope solaire. Les mod\u00e8les math\u00e9matiques de l'activit\u00e9 \u00e9lectromagn\u00e9tique solaire que nous avons contribu\u00e9 \u00e0 \u00e9tablir \u00e9taient pionniers \u00e0 l'\u00e9poque et ils ont trouv\u00e9 de nombreuses applications pratiques. Ces r\u00e9sultats post\u00e9rieurs compensaient enfin l'\u00e9norme budget investi dans C\u00f4te Rouge.\n\nLe m\u00e9rite en revenait beaucoup au commissaire politique Lei, bien qu'il y trouv\u00e2t naturellement son compte. Il avait pris conscience qu'\u00eatre un cadre politique au sein d'une unit\u00e9 technique n'\u00e9tait plus un statut aussi privil\u00e9gi\u00e9. Il avait fait des \u00e9tudes d'astrophysique avant de s'engager dans l'arm\u00e9e et il s'imaginait retourner dans le monde scientifique. Tous les projets ult\u00e9rieurs de C\u00f4te Rouge ne concernant pas la recherche de civilisations extraterrestres sont \u00e0 mettre \u00e0 son cr\u00e9dit.\n\n\u2014 \u00c9tait-il si facile pour lui de retourner \u00e0 la recherche scientifique apr\u00e8s tant d'ann\u00e9es ? Vous n'\u00e9tiez pas encore r\u00e9habilit\u00e9e politiquement \u00e0 l'\u00e9poque, n'est-ce pas ? Je suis plut\u00f4t port\u00e9 \u00e0 croire qu'il a sign\u00e9 de son nom la plupart de vos r\u00e9sultats de recherche. Je me trompe ?\n\nYe Wenjie eut un sourire bienveillant :\n\n\u2014 Sans le vieux Lei, C\u00f4te Rouge aurait \u00e9t\u00e9 ferm\u00e9e beaucoup plus t\u00f4t. Quand les activit\u00e9s scientifiques de la base sont pass\u00e9es sous la tutelle d'une institution civile, l'arm\u00e9e a abandonn\u00e9 le projet. L'Acad\u00e9mie des sciences n'est pas arriv\u00e9e \u00e0 maintenir le budget n\u00e9cessaire pour les frais de fonctionnement de la base et elle a \u00e9t\u00e9 ferm\u00e9e.\n\nC'est sur cette phrase que Ye Wenjie acheva son r\u00e9cit sur son exp\u00e9rience au sein de C\u00f4te Rouge. Wang Miao ne lui en demanda pas plus. Le reste, Sha Ruishan le lui avait racont\u00e9 dans l'observatoire astronomique de Miyun. La quatri\u00e8me ann\u00e9e apr\u00e8s son entr\u00e9e \u00e0 C\u00f4te Rouge, elle avait fond\u00e9 une famille avec Yang Weining. Tout s'\u00e9tait fait de fa\u00e7on naturelle, banale. Plus tard, Yang Weining et Lei Zhicheng avaient perdu la vie dans un accident. Yang Dong \u00e9tait n\u00e9e apr\u00e8s la mort de son p\u00e8re. M\u00e8re et fille \u00e9taient rest\u00e9es sur le pic du Radar jusqu'au d\u00e9but des ann\u00e9es 1980 avant la fermeture d\u00e9finitive de C\u00f4te Rouge. Ye Wenjie avait par la suite enseign\u00e9 l'astrophysique dans son universit\u00e9 d'origine jusqu'\u00e0 sa retraite.\n\n\u2014 La recherche de civilisations extraterrestres n'est pas une discipline ordinaire. Elle exerce une influence consid\u00e9rable sur la vie des chercheurs, dit Ye Wenjie en faisant tra\u00eener ses derni\u00e8res syllabes, comme si elle s'appr\u00eatait \u00e0 lire un conte pour enfants. La nuit, lorsque tout \u00e9tait calme, on pouvait entendre dans nos casques les bruits sans vie de l'univers. Ils \u00e9taient faibles, mais paraissaient plus immuables et plus \u00e9ternels que les \u00e9toiles. Parfois, j'avais au contraire l'impression qu'ils ressemblaient \u00e0 celui du vent glacial qui soufflait sans faiblir sur les montagnes du Grand Khingan. J'avais si froid, et cette solitude... Je ne saurais la d\u00e9crire. Quelquefois, apr\u00e8s le travail, je levais les yeux au ciel en m'imaginant que le tapis d'\u00e9toiles \u00e9tait un d\u00e9sert scintillant et que j'\u00e9tais une orpheline abandonn\u00e9e en plein milieu de ce d\u00e9sert... Comme je te l'ai d\u00e9j\u00e0 dit, j'avais le sentiment que la pr\u00e9sence de la vie sur Terre \u00e9tait le plus grand de tous les hasards de l'univers, que l'univers \u00e9tait un palais vide, habit\u00e9 par une seule petite fourmi : l'esp\u00e8ce humaine. Cette id\u00e9e m'a accompagn\u00e9e toute ma vie : je me dis parfois que la vie est pr\u00e9cieuse, qu'elle est ce qu'il y a de plus important au monde ; mais d'autres fois, je me dis que l'humain est si minuscule qu'il ne vaut finalement pas grand-chose. Toujours est-il que j'ai vieilli, jour apr\u00e8s jour et sans y prendre garde, toujours emplie de cette sensation \u00e9trange...\n\nWang Miao voulut glisser quelques mots de r\u00e9confort \u00e0 cette vieille femme qui avait consacr\u00e9 sa vie \u00e0 cette formidable cause solitaire, mais rien ne sortit de sa bouche, hormis une phrase qui le plongea lui-m\u00eame dans la solitude :\n\n\u2014 Professeur Ye, peut-\u00eatre qu'un jour je pourrais vous emmener faire un tour sur l'ancien site de C\u00f4te Rouge.\n\nYe Wenjie secoua lentement la t\u00eate :\n\n\u2014 Miao, nous sommes diff\u00e9rents, toi et moi. J'ai maintenant un certain \u00e2ge et je ne suis plus en tr\u00e8s bonne sant\u00e9. Il m'est difficile de pr\u00e9voir l'avenir, je vis au jour le jour.\n\nEn voyant ses cheveux blancs onduler sur son visage, Wang Miao sut qu'elle pensait de nouveau \u00e0 sa fille.\n\n## 15\n\n## _LES TROIS CORPS_ :\n\n## COPERNIC, LE RUGBY SPATIAL\n\n## ET LE CIEL \u00c0 TROIS SOLEILS\n\nApr\u00e8s avoir quitt\u00e9 le domicile de Ye Wenjie, Wang Miao n'arriva pas \u00e0 reprendre son calme. Ce qu'il avait v\u00e9cu ces deux derniers jours et l'histoire de C\u00f4te Rouge, ces deux choses _a priori_ sans rapport, s'enchev\u00eatraient dans son esprit et rendaient le monde \u00e9trangement nouveau \u00e0 ses yeux.\n\nUne fois rentr\u00e9 chez lui, pour chasser ces pens\u00e9es oppressantes, il alluma son ordinateur, enfila sa V-combinaison et se connecta pour la troisi\u00e8me fois au jeu des _Trois Corps_. Cette fa\u00e7on de se changer les id\u00e9es se r\u00e9v\u00e9la efficace, car l'interface de connexion \u00e0 peine apparue sur l'\u00e9cran, Wang Miao se sentit comme un homme nouveau ; il d\u00e9bordait d'une inexplicable excitation. Contrairement aux deux pr\u00e9c\u00e9dentes parties, il \u00e9tait cette fois investi d'une mission : percer les secrets de ce monde virtuel. Il cr\u00e9a donc un identifiant pour sa nouvelle identit\u00e9 : Copernic.\n\nWang Miao se retrouva une nouvelle fois sur la vaste plaine d\u00e9serte, face \u00e0 l'aube myst\u00e9rieuse du monde des _Trois Corps_. La gigantesque pyramide se dressait \u00e0 l'est, mais Wang Miao remarqua aussit\u00f4t qu'il ne s'agissait pas de la m\u00eame pyramide que celle du roi Zhou des Shang ou de Mozi. Son sommet \u00e9tait constitu\u00e9 d'une tour gothique qui pointait vers le ciel du matin. Elle lui fit penser \u00e0 l'\u00e9glise Saint-Joseph de Wangfujing devant laquelle il avait atterri la veille. Mais \u00e0 c\u00f4t\u00e9 de la pyramide, l'\u00e9glise Saint-Joseph aurait paru \u00eatre un simple vestibule. Il vit d'autres b\u00e2timents au loin, sans doute des silos de d\u00e9shydratation, mais construits cette fois dans un style gothique, avec de longs clochers effil\u00e9s, comme des \u00e9pines ayant pouss\u00e9 sur la plaine.\n\nSur une des faces de la pyramide, Wang Miao aper\u00e7ut une porte qui laissait s'\u00e9chapper une lueur vacillante. Il marcha vers la porte et, une fois entr\u00e9, il vit tout le long du corridor des sculptures noircies de dieux grecs arm\u00e9s de torches. Il p\u00e9n\u00e9tra dans la grande salle d'audience et d\u00e9couvrit qu'il y faisait encore plus sombre qu'\u00e0 l'entr\u00e9e de la pyramide. Elle n'\u00e9tait \u00e9clair\u00e9e que par le halo de deux chandeliers en argent pos\u00e9s sur une longue table en pierre.\n\nQuelques hommes \u00e9taient assis autour de la table. Malgr\u00e9 la lumi\u00e8re tamis\u00e9e, Wang Miao put distinguer qu'il s'agissait d'Occidentaux. Leurs yeux se terraient dans la p\u00e9nombre de leurs orbites, mais Wang Miao put sentir les regards se tourner vers lui. Ils semblaient porter des robes m\u00e9di\u00e9vales, mais en observant mieux, il remarqua que deux d'entre eux \u00e9taient v\u00eatus de robes plus simples, comme des toges grecques. Au bout de la table tr\u00f4nait un homme grand et maigre qui portait une tiare d'or sur la t\u00eate, unique autre objet brillant dans la pi\u00e8ce en dehors des bougies sur les chandeliers. Wang Miao parvint tant bien que mal \u00e0 voir qu'il portait une robe de couleur diff\u00e9rente, une robe rouge.\n\nWang Miao eut la confirmation d'une chose : le jeu \u00e9voluait diff\u00e9remment pour chaque joueur. Il avait l'impression d'\u00eatre dans un monde inspir\u00e9 du Moyen \u00c2ge europ\u00e9en, que le logiciel avait d\u00e9termin\u00e9 en fonction de son identifiant.\n\n\u2014 Vous \u00eates en retard. La r\u00e9union a commenc\u00e9 depuis longtemps, dit l'homme \u00e0 la tiare d'or et \u00e0 la robe rouge. Je suis le pape Gr\u00e9goire.\n\nWang Miao essaya de se rappeler l'histoire du Moyen \u00c2ge occidental \u2013 dont il n'\u00e9tait gu\u00e8re familier \u2013 pour en d\u00e9duire le niveau d'avancement de la civilisation. Mais il se fit la r\u00e9flexion que le monde des _Trois Corps_ ne se pr\u00e9occupait gu\u00e8re de coh\u00e9rence historique et estima que cet effort de pens\u00e9e \u00e9tait superflu.\n\n\u2014 Vous avez chang\u00e9 d'identifiant, mais nous vous avons reconnu. Lors des deux derni\u00e8res civilisations, vous avez voyag\u00e9 en Orient. Oh, je me pr\u00e9sente : Aristote, dit un homme aux cheveux blancs boucl\u00e9s, v\u00eatu d'une toge grecque.\n\n\u2014 Oui, approuva Wang Miao de la t\u00eate, j'ai \u00e9t\u00e9 t\u00e9moin de l'an\u00e9antissement de deux civilisations : la premi\u00e8re \u00e0 cause d'un froid mortel et la deuxi\u00e8me \u00e0 cause d'un soleil torride. J'ai aussi assist\u00e9 aux efforts des savants orientaux pour comprendre les lois qui r\u00e9gissent les mouvements du soleil.\n\n\u2014 Ha !\n\nLa voix \u00e9tait celle d'un homme cach\u00e9 dans l'obscurit\u00e9, plus maigre que le pape, avec une barbichette.\n\n\u2014 Ces savants d'Orient tentent de comprendre les secrets du soleil par la m\u00e9ditation, la r\u00e9v\u00e9lation divine et m\u00eame les r\u00eaves. C'est d'un ridicule !\n\n\u2014 Je vous pr\u00e9sente Galil\u00e9e, dit Aristote. Il pr\u00f4ne une connaissance du monde \u00e0 travers l'exp\u00e9rience et l'observation. C'est une pens\u00e9e encore artisanale, mais les r\u00e9sultats qu'il a obtenus plaident pour lui.\n\n\u2014 Mozi aussi croyait dans l'exp\u00e9rience et l'observation, r\u00e9pliqua Wang Miao.\n\nGalil\u00e9e pouffa une nouvelle fois :\n\n\u2014 Mais la pens\u00e9e de Mozi reste orientale ! Il ne s'agit ni plus ni moins que de mysticisme maquill\u00e9 en science, il n'a jamais r\u00e9ellement pr\u00eat\u00e9 attention aux r\u00e9sultats de ses observations. Il s'est seulement appuy\u00e9 sur ses hypoth\u00e8ses subjectives pour \u00e9tablir un mod\u00e8le analogique complet de l'univers, c'est risible ! Dommage pour ses appareils, plut\u00f4t ing\u00e9nieux, je dois l'avouer. Mais je suis diff\u00e9rent. Je me base sur un grand nombre d'observations et d'exp\u00e9riences pour proposer des th\u00e9ories solides et \u00e9tablir un mod\u00e8le de l'univers. Et je retourne ensuite \u00e0 l'exp\u00e9rience et \u00e0 l'observation pour prouver leur validit\u00e9.\n\n\u2014 C'est la bonne d\u00e9marche, approuva Wang Miao. C'est aussi ma mani\u00e8re de fonctionner.\n\n\u2014 Vous nous avez donc apport\u00e9 un calendrier ? ironisa le pape.\n\n\u2014 Je n'ai pas de calendrier, seulement des donn\u00e9es d'observation qui m'ont permis de constituer un mod\u00e8le d'univers. Cependant je dois pr\u00e9ciser que m\u00eame si ce mod\u00e8le s'av\u00e8re exact, je ne pourrai peut-\u00eatre pas ma\u00eetriser avec certitude les r\u00e8gles de fonctionnement du soleil, ni m\u00eame \u00e9tablir un calendrier. Cependant c'est une premi\u00e8re \u00e9tape n\u00e9cessaire.\n\nIl entendit l'\u00e9cho de quelques applaudissements solitaires, ceux de Galil\u00e9e :\n\n\u2014 Excellent, Copernic, excellent. Votre approche pragmatique et bas\u00e9e sur l'exp\u00e9rience scientifique est un exemple pour beaucoup de chercheurs. Rien que pour cela, votre th\u00e9orie m\u00e9rite d'\u00eatre entendue.\n\nLe pape fit un signe de t\u00eate \u00e0 Wang Miao :\n\n\u2014 Nous vous \u00e9coutons, Copernic.\n\nWang Miao s'avan\u00e7a jusqu'\u00e0 l'extr\u00e9mit\u00e9 de la table et commen\u00e7a son explication :\n\n\u2014 C'est assez simple en r\u00e9alit\u00e9. Si les mouvements du soleil semblent si irr\u00e9guliers, c'est que notre monde poss\u00e8de en fait trois soleils. Sous l'influence de leurs interactions gravitationnelles, ils donnent naissance \u00e0 ce mouvement impr\u00e9visible que nous appelons le probl\u00e8me \u00e0 trois corps. Lorsque notre plan\u00e8te tourne autour d'un des soleils sur une orbite stable, nous nous trouvons dans une \u00e8re r\u00e9guli\u00e8re. Mais lorsqu'un ou deux autres soleils s'approchent \u00e0 une certaine distance, leur attraction gravitationnelle d\u00e9vie la plan\u00e8te de son orbite originelle et celle-ci se retrouve \u00e0 errer de fa\u00e7on instable dans les champs gravitationnels des trois soleils. Nous sommes alors dans une \u00e8re chaotique. Apr\u00e8s une dur\u00e9e de temps incertaine, quand notre plan\u00e8te est \u00e0 nouveau captur\u00e9e par un des trois soleils, une orbite stable peut \u00eatre temporairement r\u00e9tablie et c'est une nouvelle \u00e8re r\u00e9guli\u00e8re qui commence. C'est un match de rugby \u00e0 l'\u00e9chelle de l'univers. Les joueurs sont les trois soleils, et le ballon, c'est notre plan\u00e8te.\n\nDes rires fus\u00e8rent dans l'obscurit\u00e9 de la salle d'audience.\n\n\u2014 Br\u00fblez-le, l\u00e2cha le pape, sans autre forme de proc\u00e8s.\n\nAussit\u00f4t, les deux sentinelles aux lourdes armures rouill\u00e9es s'approch\u00e8rent de Wang Miao avec une d\u00e9marche de robots maladroits.\n\n\u2014 Oui, br\u00fblez-le, soupira Galil\u00e9e en secouant la main. Vous m'aviez donn\u00e9 un peu d'espoir, mais vous n'\u00eates finalement qu'un mystique ou un sorcier, comme les autres.\n\n\u2014 Ce genre de savants est devenu une nuisance publique de nos jours, rench\u00e9rit Aristote.\n\n\u2014 Laissez-moi au moins finir, dit Wang Miao en repoussant les gants de fer des deux gardes.\n\n\u2014 Avez-vous d\u00e9j\u00e0 vu les trois soleils ? Quelqu'un les a-t-il d\u00e9j\u00e0 vus ? demanda Galil\u00e9e en inclinant la t\u00eate.\n\n\u2014 Tout le monde les a vus.\n\n\u2014 Eh bien, en dehors de celui qui appara\u00eet aussi bien durant les \u00e8res chaotiques que r\u00e9guli\u00e8res, o\u00f9 sont les deux autres ?\n\n\u2014 Il faut d'abord expliquer une chose : le soleil que nous voyons \u00e0 des moments diff\u00e9rents n'est peut-\u00eatre pas le m\u00eame. Lorsque nous ne voyons qu'un seul soleil, les deux autres nous apparaissent sous la forme d'\u00e9toiles volantes.\n\n\u2014 Votre ignorance de la science est consternante, dit Galil\u00e9e en secouant la t\u00eate d'un air d\u00e9sapprobateur. Afin de s'\u00e9loigner, le soleil doit suivre un mouvement continu. Il ne peut pas faire des bonds dans l'espace. Ainsi, selon votre hypoth\u00e8se, il devrait y avoir un cas interm\u00e9diaire o\u00f9 il appara\u00eet certes plus petit que la normale mais est tout de m\u00eame plus grand qu'une \u00e9toile volante, jusqu'\u00e0 devenir de la taille d'une \u00e9toile volante \u00e0 mesure qu'il s'\u00e9loigne. Mais nous n'avons jamais vu de tel soleil.\n\n\u2014 Vous qui avez re\u00e7u une \u00e9ducation scientifique, vous devriez avoir quelques connaissances de base sur la structure du soleil.\n\n\u2014 C'est la d\u00e9couverte dont je suis le plus fier : le soleil est constitu\u00e9 d'une couche externe gazeuse \u00e9paisse mais clairsem\u00e9e et d'un noyau dense et br\u00fblant.\n\n\u2014 C'est exact, reprit Wang Miao, mais il semble que vous n'ayez pas encore d\u00e9couvert les effets optiques singuliers existant entre cette couche de gaz et l'atmosph\u00e8re de notre plan\u00e8te. C'est un ph\u00e9nom\u00e8ne similaire au filtrage de polarisation qui fait qu'au-del\u00e0 d'une certaine distance avec le soleil, la couche gazeuse du soleil ne nous appara\u00eet pas visible depuis l'atmosph\u00e8re. Nous ne voyons plus que son noyau brillant. C'est pourquoi le soleil se r\u00e9duit d'un coup dans notre champ de vision \u00e0 la taille de son noyau et devient ce que nous appelons une \u00e9toile volante.\n\nC'est ce ph\u00e9nom\u00e8ne qui a tromp\u00e9 les chercheurs de toutes les derni\u00e8res civilisations et qui les a emp\u00each\u00e9s de prendre conscience de l'existence des trois soleils. Vous comprenez \u00e0 pr\u00e9sent pourquoi l'apparition dans le ciel de trois \u00e9toiles volantes est pr\u00e9sage d'un hiver sans fin, car \u00e0 cet instant les trois soleils se trouvent \u00e0 une distance extr\u00eamement \u00e9loign\u00e9e de la plan\u00e8te.\n\nUn bref silence s'empara de la salle. Tous semblaient m\u00e9diter les paroles de Wang Miao. Ce fut Aristote qui parla en premier :\n\n\u2014 Votre raisonnement n'est pas logique. Il est certes possible de voir appara\u00eetre trois \u00e9toiles volantes et cette apparition s'accompagne en effet d'un hiver destructeur. Mais selon votre th\u00e9orie, nous devrions aussi pouvoir voir trois soleils de taille normale, ce qui n'est jamais arriv\u00e9. Dans toutes les annales de l'histoire, personne n'en a jamais fait mention !\n\n\u2014 Attendez !\n\nUn homme barbu et chauss\u00e9 d'une \u00e9trange coiffe se leva pour prendre la parole :\n\n\u2014 Je crois que l'histoire en fait mention. Une civilisation a jadis eu l'occasion de voir deux soleils dans le ciel, mais elle a \u00e9t\u00e9 an\u00e9antie par leur chaleur. Les chroniques qui nous sont parvenues sont presque ind\u00e9chiffrables. Je me pr\u00e9sente, L\u00e9onard de Vinci.\n\n\u2014 Nous parlons de trois soleils, pas de deux ! \u00e9ructa Galil\u00e9e. Selon la th\u00e9orie de Copernic, nous devrions pouvoir voir appara\u00eetre trois soleils proches de la plan\u00e8te, le m\u00eame nombre que les \u00e9toiles volantes !\n\n\u2014 Les trois soleils sont jadis apparus \u00e0 une distance proche de la plan\u00e8te, r\u00e9pondit calmement Wang Miao. Des \u00eatres vivants ont assist\u00e9 \u00e0 ce ph\u00e9nom\u00e8ne, mais aucun d'entre eux n'a pu survivre suffisamment longtemps \u00e0 ce spectacle apocalyptique pour transmettre cette information. Un ciel \u00e0 trois soleils est la catastrophe la plus effroyable qui puisse arriver \u00e0 notre monde. \u00c0 cet instant pr\u00e9cis, la surface de la plan\u00e8te se change aussit\u00f4t en four en fusion, \u00e0 une temp\u00e9rature si \u00e9lev\u00e9e qu'elle peut faire fondre la roche. Dans un monde o\u00f9 tout a \u00e9t\u00e9 d\u00e9truit par un ciel \u00e0 trois soleils, il faut attendre longtemps avant que la vie et la civilisation ne renaissent. C'est pourquoi il n'y a jamais eu aucun r\u00e9cit de l'apparition simultan\u00e9e de trois soleils.\n\nSilence. Tout le monde observait le pape.\n\n\u2014 Br\u00fblez-le, r\u00e9p\u00e9ta doucement le pape, avec un sourire qui parut familier \u00e0 Wang Miao.\n\nC'\u00e9tait celui du roi Zhou.\n\n\u00c0 l'int\u00e9rieur de la grande salle, tous s'agit\u00e8rent soudain, comme s'ils s'appr\u00eataient \u00e0 c\u00e9l\u00e9brer un joyeux \u00e9v\u00e9nement. Galil\u00e9e et d'autres sortirent avec enthousiasme d'un coin de la pi\u00e8ce une grande croix et se pr\u00e9par\u00e8rent \u00e0 y mettre le feu. Ils d\u00e9tach\u00e8rent le corps carbonis\u00e9 encore accroch\u00e9 et le jet\u00e8rent dans un coin puis ils redress\u00e8rent la croix. Un autre petit groupe rassemblait fr\u00e9n\u00e9tiquement des b\u00fbches. Seul de Vinci semblait rester compl\u00e8tement indiff\u00e9rent \u00e0 ce spectacle. Il s'assit au bord de la table pour r\u00e9fl\u00e9chir et griffonner des calculs sur un bout de papier.\n\n\u2014 Giordano Bruno, dit Aristote en d\u00e9signant le corps calcin\u00e9. Comme vous, il a \u00e9t\u00e9 br\u00fbl\u00e9 pour avoir racont\u00e9 des mensonges.\n\n\u2014 \u00c0 petit feu, pr\u00e9cisa le pape \u00e0 l'endroit des bourreaux.\n\nLes deux gardes commenc\u00e8rent \u00e0 ligoter Wang Miao \u00e0 la croix avec des cordes en amiante r\u00e9sistantes au feu. Wang Miao pointa sa main encore libre en direction du pape en criant :\n\n\u2014 Vous n'\u00eates qu'un programme ! Et vous autres, si vous n'\u00eates pas des programmes, vous \u00eates des imb\u00e9ciles. Je me reconnecterai !\n\n\u2014 Vous ne reviendrez pas, vous dispara\u00eetrez \u00e0 jamais du monde des _Trois Corps_ , dit Galil\u00e9e en partant d'un rire \u00e9trange.\n\n\u2014 Vous aussi vous devez \u00eatre un programme, un vrai joueur saurait parfaitement que m\u00eame si mon adresse IP est enregistr\u00e9e, il me suffit de changer d'ordinateur et d'identifiant pour revenir et annoncer mon retour.\n\n\u2014 Le syst\u00e8me a fait un scan de votre r\u00e9tine, par le biais du casque de r\u00e9alit\u00e9 virtuelle, dit de Vinci en jetant un regard sur Wang Miao, avant de replonger dans ses calculs.\n\nWang Miao fut saisi de frayeur. Il se mit \u00e0 hurler :\n\n\u2014 Vous ne pouvez pas faire \u00e7a ! L\u00e2chez-moi ! Je dis la v\u00e9rit\u00e9 !\n\n\u2014 Si vous dites la v\u00e9rit\u00e9, vous ne mourrez pas sur le b\u00fbcher. Le jeu r\u00e9compense ceux qui font route dans la bonne direction, dit Aristote en arborant un sourire cruel.\n\nIl sortit un briquet Zippo, le fit tourner entre ses doigts avec un mouvement complexe avant de l'allumer. Au moment o\u00f9 il tendit la main pour enflammer les b\u00fbches sous la croix, un faisceau de lumi\u00e8re rouge jaillit \u00e0 travers la porte du corridor. Il fut aussit\u00f4t suivi d'un courant d'air chaud charg\u00e9 de fum\u00e9e. Un cheval surgit \u00e0 travers la lumi\u00e8re. Le corps de l'animal \u00e9tait d\u00e9j\u00e0 en feu. Tandis qu'il galopait, il se transforma dans un sifflement en boule de feu. L'armure et le heaume du cavalier, d'o\u00f9 s'\u00e9chappait un nuage de fum\u00e9e blanche, \u00e9taient rougis par les flammes.\n\n\u2014 La fin du monde !! La fin du monde !! D\u00e9shydratez ! D\u00e9shydratez !\n\nAlors que le cavalier hurlait, le cheval s'\u00e9croula sur le sol et s'enflamma comme un v\u00e9ritable feu de joie. \u00c9ject\u00e9 de sa monture, le cavalier alla rouler jusque sous la croix. Son heaume ne bougeait plus, mais l'\u00e9paisse fum\u00e9e blanche continuait \u00e0 s'\u00e9chapper des fentes de son armure. La graisse suintante de l'homme se propageait sur le sol, parant son armure d'une paire d'ailes en feu.\n\nTous ceux qui se trouvaient dans la salle d'audience se ru\u00e8rent vers la sortie, avant de dispara\u00eetre rapidement dans la lumi\u00e8re rouge qui per\u00e7ait \u00e0 travers la porte. Wang Miao se d\u00e9battit pour essayer de se d\u00e9faire de ses liens, mais ce fut la croix qui le retenait qui finit par tomber sur le sol, faute de socle pour la maintenir droite. En se tortillant, il parvint tant bien que mal \u00e0 se lib\u00e9rer. Il contourna le cavalier et son cheval br\u00fbl\u00e9s, traversa la salle d\u00e9sormais d\u00e9serte, courut en suffoquant le long du corridor et arriva \u00e0 l'ext\u00e9rieur.\n\nLa terre \u00e9tait aussi rouge que du m\u00e9tal dans le four d'un forgeron. Des rivi\u00e8res de lave \u00e9blouissantes s'\u00e9coulaient sur le sol, comme les mailles d'un terrifiant filet de feu s'\u00e9tirant vers l'horizon. De minces geysers paraissaient jaillir de terre, c'\u00e9taient les silos de d\u00e9shydratation qui flambaient. \u00c0 l'int\u00e9rieur, les corps embras\u00e9s des d\u00e9shydrat\u00e9s donnaient aux geysers d'\u00e9tranges teintes bleut\u00e9es.\n\nNon loin, Wang Miao aper\u00e7ut une dizaine de colonnes de feu de la m\u00eame couleur. C'\u00e9taient les hommes qui venaient tout juste de s'\u00e9chapper en courant de la pyramide : le pape Gr\u00e9goire, Galil\u00e9e, Aristote, de Vinci... Les flammes qui les retenaient prisonniers \u00e9taient d'un bleu translucide et laissaient voir leurs visages et leurs corps d\u00e9sarticul\u00e9s par le feu. Leurs regards converg\u00e8rent vers Wang Miao qui venait de sortir. Tous \u00e9taient comme p\u00e9trifi\u00e9s dans la m\u00eame position, les deux bras enflamm\u00e9s point\u00e9s vers le ciel, criant comme s'ils chantaient \u00e0 l'unisson :\n\n\u2014 Un ciel \u00e0 trois soleils !\n\nWang Miao leva les yeux et vit trois \u00e9normes soleils qui tournaient lentement autour d'un axe visible, comme les pales d'un gigantesque ventilateur faisant souffler sur la terre une bourrasque mortelle.\n\n\u00c0 eux trois, les soleils occupaient presque la totalit\u00e9 du firmament et, alors qu'ils se d\u00e9calaient vers l'ouest, la moiti\u00e9 disparut sous la ligne de l'horizon. Cependant le \"ventilateur\" tournait encore. Une de ses lames irradiantes \u00e9mergeait parfois au-dessus de l'horizon, comme s'il voulait offrir \u00e0 ce monde ravag\u00e9 une alternance de levers et de couchers de soleil. Quand les soleils s'\u00e9vanouissaient derri\u00e8re l'horizon, la terre br\u00fbl\u00e9e continuait \u00e0 scintiller d'une couleur rouge sombre. Mais le lever soudain des soleils l'arrosait \u00e0 nouveau de ses rayons aveuglants.\n\nUne fois que l'ensemble des trois soleils eut totalement disparu sous l'horizon, des nuages de vapeur se form\u00e8rent en surface et diss\u00e9min\u00e8rent la lumi\u00e8re rouge du ciel qui br\u00fblait avec la beaut\u00e9 folle de l'enfer.\n\n> La civilisation no 183 a \u00e9t\u00e9 an\u00e9antie par un ciel \u00e0 trois soleils. La civilisation avait atteint le Moyen \u00c2ge. Dans bien longtemps, une nouvelle civilisation germera des ruines du monde si impr\u00e9visible des _Trois Corps_.\n> \n> Mais dans cette civilisation, Copernic a r\u00e9v\u00e9l\u00e9 avec succ\u00e8s la structure basique de l'univers. La civilisation des _Trois Corps_ a connu son premier grand bond. Vous passez maintenant au deuxi\u00e8me niveau du jeu. Nous vous invitons \u00e0 vous reconnecter dans le futur.\n\n## 16\n\n## LE PROBL\u00c8ME \u00c0 TROIS CORPS\n\nWang Miao venait tout juste de quitter le jeu lorsque le t\u00e9l\u00e9phone sonna.\n\nC'\u00e9tait Shi Qiang. Il avait quelque chose d'urgent \u00e0 lui dire et lui intima de venir imm\u00e9diatement le rejoindre au commissariat. Wang Miao regarda sa montre. Il \u00e9tait plus de 3 heures du matin.\n\nLe bureau de Shi Qiang \u00e9tait un vrai capharna\u00fcm engorg\u00e9 de fum\u00e9e de cigarette, ce qui obligeait la jeune offici\u00e8re de police qui partageait l'espace \u00e0 se ventiler sans cesse le nez avec son registre. Shi Qiang la pr\u00e9senta : Xu Bingbing, experte en informatique du d\u00e9partement de s\u00e9curit\u00e9 de l'information.\n\nWang Miao ne put contenir sa surprise lorsqu'il d\u00e9couvrit quelle \u00e9tait l'identit\u00e9 de la troisi\u00e8me personne \u00e0 se trouver dans le bureau : Wei Cheng, le mari de Shen Yufei. Les cheveux en bataille, il leva les yeux pour regarder Wang Miao. Il paraissait avoir oubli\u00e9 qu'ils s'\u00e9taient d\u00e9j\u00e0 rencontr\u00e9s par le pass\u00e9.\n\n\u2014 Navr\u00e9 de vous avoir d\u00e9rang\u00e9 mais, apparemment, vous ne dormiez pas. Quelque chose est arriv\u00e9, mais je n'en ai pas encore inform\u00e9 le Centre d'op\u00e9rations militaires. J'avais besoin de vos conseils.\n\nPuis il se tourna vers Wei Cheng :\n\n\u2014 Racontez-lui.\n\n\u2014 Je vous l'ai d\u00e9j\u00e0 dit, je suis en danger de mort, dit Wei Cheng, avec une mine abasourdie.\n\n\u2014 Commencez depuis le d\u00e9but.\n\n\u2014 D'accord, depuis le d\u00e9but. Mais ne vous plaignez pas si \u00e7a tra\u00eene en longueur. \u00c7a fait quelque temps que j'avais besoin de raconter cette histoire \u00e0 quelqu'un.\n\nIl se tourna vers Xu Bingbing :\n\n\u2014 Vous n'avez pas besoin de prendre des notes ?\n\n\u2014 Ce n'est pas utile maintenant. Vous n'aviez personne \u00e0 qui parler ? demanda Shi Qiang, qui ne voulait laisser passer aucune occasion d'obtenir des informations.\n\n\u2014 Non, ce n'est pas \u00e7a. Mais j'\u00e9tais trop paresseux pour en parler. J'ai toujours \u00e9t\u00e9 tr\u00e8s paresseux.\n\n### R\u00c9CIT DE WEI CHENG\n\n_Je suis un feignant. Depuis tout petit. Quand je vivais dans le dortoir de l'\u00e9cole, je ne lavais jamais mon bol, je ne pliais jamais mes draps, je ne m'int\u00e9ressais \u00e0 rien. J'avais la paresse d'\u00e9tudier, la paresse de jouer et les jours passaient sans que j'aie vraiment de but dans la vie. Pourtant, je savais que j'avais certains dons que les autres enfants n'avaient pas. Par exemple, si vous traciez une ligne droite, j'\u00e9tais capable de tracer une autre ligne partant de celle-ci, dont vous pouviez \u00eatre s\u00fbr que le rapport \u00e9tait \u00e9gal au nombre d'or : 1,618. Mes camarades disaient que je ferais un bon charpentier, mais je savais que mon talent ne s'arr\u00eatait pas l\u00e0, j'avais une sorte de don pour les chiffres et les figures. Toutefois, mes notes en math\u00e9matiques \u00e9taient aussi catastrophiques que dans les autres mati\u00e8res. Durant mes examens de maths, j'\u00e9tais si feignant que je ne prenais pas la peine de r\u00e9diger la d\u00e9monstration qui m'avait permis de donner le bon r\u00e9sultat. Je devinais les r\u00e9ponses, qui s'av\u00e9raient justes dans quatre-vingt-dix-neuf pour cent des cas, mais j'avais quand m\u00eame de mauvaises notes._\n\n _En deuxi\u00e8me ann\u00e9e de lyc\u00e9e, un professeur de math\u00e9matiques m'a rep\u00e9r\u00e9. \u00c0 l'\u00e9poque, un certain nombre d'enseignants du secondaire \u00e9taient des g\u00e9nies de la science : lors de la R\u00e9volution culturelle, beaucoup de grands scientifiques avaient \u00e9t\u00e9 charg\u00e9s d'enseigner dans les coll\u00e8ges et les lyc\u00e9es. Mon professeur \u00e9tait l'un d'entre eux._\n\n _Un jour, apr\u00e8s les cours, il m'a demand\u00e9 de rester dans la classe et a \u00e9crit une dizaine de suites arithm\u00e9tiques sur le tableau noir.Puis il m'a demand\u00e9 de donner directement leurs sommes. J'en ai rapidement donn\u00e9 un certain nombre et j'ai pu dire au premier coup d'\u0153il lesquelles \u00e9taient divergentes. Le professeur a sorti un livre : une anthologie des enqu\u00eates de Sherlock Holmes. Il l'a feuillet\u00e9 et s'est arr\u00eat\u00e9 \u00e0 une page \u2013 je crois bien que c'\u00e9tait dans_ Une \u00e9tude en rouge. _La sc\u00e8ne de l'histoire \u00e9tait \u00e0 peu pr\u00e8s celle-ci : Watson remarque un individu lambda qui tient une enveloppe \u00e0 la main ; il le d\u00e9signe \u00e0 Holmes qui lui dit : \"Vous parlez de ce sergent d'infanterie de marine \u00e0 la retraite ?\"Watson est tr\u00e8s intrigu\u00e9 et demande \u00e0 savoir comment Holmes s'y est pris pour deviner l'identit\u00e9 de l'individu. Holmes lui-m\u00eame n'est pas tr\u00e8s s\u00fbr, il r\u00e9fl\u00e9chit un bon moment avant d'\u00e9num\u00e9rer les d\u00e9tails qui lui ont permis d'en arriver \u00e0 cette d\u00e9duction : la main de l'homme, sa d\u00e9marche... Holmes ajoute que sa difficult\u00e9 \u00e0 se justifier n'est pas si \u00e9trange, on est bien embarrass\u00e9 quand il s'agit de prouver que deux et deux font quatre._\n\n _Le professeur a referm\u00e9 le livre et m'a dit : \"Tu es comme Sherlock Holmes. Tes d\u00e9ductions sont si rapides et si instinctives que tu as du mal \u00e0 expliquer comment tu t'y es pris.\" Puis il m'a demand\u00e9 : \"Que ressens-tu quand tu vois une suite arithm\u00e9tique ? Je parle bien de tes sentiments.\" Je lui ai r\u00e9pondu que n'importe quelle combinaison de chiffres m'apparaissait comme une figure en trois dimensions. Bien s\u00fbr, je n'arrivais pas \u00e0 dire quelle \u00e9tait la forme de la figure en question, mais j'\u00e9tais s\u00fbr qu'elle en avait une. \"Et quand tu vois des figures g\u00e9om\u00e9triques ?\" J'ai r\u00e9pondu que c'\u00e9tait le contraire : je me les repr\u00e9sentais mentalement sous la forme non pas de figures, mais de nombres. Un peu comme quand vous regardez les photos d'un journal de tr\u00e8s pr\u00e8s et que vous voyez comme des petits points (bien s\u00fbr, de nos jours, les photos des journaux ne font plus cet effet-l\u00e0)._\n\n _\"Tu as vraiment un don pour les math\u00e9matiques, m'a dit le professeur, mais, mais...\" Il a encore ajout\u00e9 beaucoup de \"mais\" et a tourn\u00e9 en rond au milieu de la salle de classe, comme si mon cas \u00e9tait tr\u00e8s difficile \u00e0 traiter et qu'il ne savait pas quoi faire de moi. \"Mais les gens comme toi ne mettent pas assez leur don en valeur.\" Il a r\u00e9fl\u00e9chi pendant longtemps et a eu l'air de vouloir laisser tomber, mais il a fini par l\u00e2cher : \"Et si tu allais participer au concours de math\u00e9matiques du district le mois prochain ? Jene t'enseignerai plus rien \u00e0 partir d'aujourd'hui. Pour quelqu'un comme toi, c'est une perte de temps. Souviens-toi simplement de bien \u00e9crire les d\u00e9monstrations par lesquelles tu es arriv\u00e9 \u00e0 r\u00e9soudre les probl\u00e8mes.\"_\n\n _J'ai donc particip\u00e9 au concours, celui du district d'abord, puis tous les suivants jusqu'aux Olympiades internationales de math\u00e9matiques de Budapest o\u00f9 j'ai remport\u00e9 toutes les \u00e9preuves. \u00c0 mon retour, j'ai int\u00e9gr\u00e9 directement le d\u00e9partement de math\u00e9matiques d'une prestigieuse universit\u00e9 en \u00e9tant dispens\u00e9 d'examen d'entr\u00e9e..._\n\n _\u00c7a ne vous ennuie pas, ce que je raconte ? Ah, tant mieux, il \u00e9tait n\u00e9cessaire de dire tout cela avant de raconter la suite. Mon professeur de lyc\u00e9e avait vu juste, je ne mettais pas suffisamment mes talents en valeur. Licence, ma\u00eetrise, doctorat, j'ai tout r\u00e9ussi, mais en dilettante. Ce n'est qu'une fois confront\u00e9 au monde du travail que j'ai d\u00e9couvert que je n'\u00e9tais qu'un bon \u00e0 rien et que je ne savais rien faire d'autre que des math\u00e9matiques. Je me mettais en \u00e9tat de semi-hibernation d\u00e8s que des rapports avec d'autres personnes devenaient plus complexes. Plus j'essayais de faire des efforts et plus je m'enfon\u00e7ais. J'ai fini par devenir professeur \u00e0 l'universit\u00e9, mais sans grand succ\u00e8s. Je ne pr\u00e9parais pas s\u00e9rieusement mes cours et me contentais d'\u00e9crire \"Facile \u00e0 d\u00e9montrer\" sur le tableau. Mes \u00e9tudiants devaient se d\u00e9brouiller avec \u00e7a pendant des heures. Plus tard, quand a \u00e9t\u00e9 mis en place le syst\u00e8me d'\u00e9valuation des professeurs par les \u00e9l\u00e8ves, on ne m'a plus jamais confi\u00e9 aucun cours._\n\n _\u00c0 partir de ce moment-l\u00e0, j'ai commenc\u00e9 \u00e0 \u00eatre d\u00e9go\u00fbt\u00e9 de tout. Je suis parti avec un seul bagage dans le Sud, dans un temple enfoui dans la montagne. Oh, non, je ne suis pas devenu moine, j'\u00e9tais bien trop feignant pour \u00e7a, j'avais juste envie de vivre un moment dans un endroit vraiment pur. Le vieil abb\u00e9 qui dirigeait le monast\u00e8re \u00e9tait un ami de mon p\u00e8re. C'\u00e9tait un homme d'une grande \u00e9rudition qui, arriv\u00e9 \u00e0 un certain \u00e2ge, avait choisi de tout abandonner pour entrer dans les ordres. Comme le disait mon p\u00e8re, \u00e0 son niveau, c'\u00e9tait le seul chemin possible. L'abb\u00e9 a accept\u00e9 que je reste. Je lui ai dit que je cherchais un endroit paisible pour passer le restant de mes jours. Toutefois l'abb\u00e9 m'a r\u00e9pondu que le temple n'\u00e9tait pas particuli\u00e8rement tranquille, qu'il \u00e9tait truff\u00e9 de touristes et de p\u00e8lerins. La vraie paix pouvait \u00eatre trouv\u00e9e dans les villes les plus tumultueuses, mais pour y parvenir, il fallaitavant tout faire le vide en soi. J'ai r\u00e9pondu que j'\u00e9tais suffisamment vide, que je me sentais aussi l\u00e9ger qu'un nuage et que tous les moines du temple \u00e9taient plus mat\u00e9rialistes que moi. L'abb\u00e9 a secou\u00e9 la t\u00eate en disant : \"Le vide n'est pas le n\u00e9ant, le vide est une forme d'existence. Tu dois te remplir de cette existence du vide.\" Ces paroles m'ont inspir\u00e9. Plus tard en y repensant, je me suis fait la r\u00e9flexion que ce n'\u00e9tait pas du tout un pr\u00e9cepte bouddhiste, mais plut\u00f4t une th\u00e9orie physique moderne. L'abb\u00e9 a ajout\u00e9 que nous ne parlerions pas de bouddhisme ensemble, pour la m\u00eame raison que celle invoqu\u00e9e par mon professeur de lyc\u00e9e : pour quelqu'un comme moi, c'\u00e9tait une perte de temps._\n\n _Le premier soir, je n'ai pas r\u00e9ussi \u00e0 trouver le sommeil dans ma petite chambre du dortoir de la cour du temple. Je n'avais pas imagin\u00e9 que ce refuge hors du monde pouvait \u00eatre si inconfortable : ma couverture et mon matelas \u00e9taient humides \u00e0 cause de la brume et mon lit \u00e9tait dur comme de la pierre. Aussi, pour m'endormir, j'ai essay\u00e9 de faire comme me l'avait dit l'abb\u00e9 : me remplir de \"vide\"._\n\n _Le premier vide que j'ai cr\u00e9\u00e9 dans ma conscience a \u00e9t\u00e9 l'infinit\u00e9 de l'espace. Un espace sans rien dedans, pas m\u00eame de la lumi\u00e8re, un ciel enti\u00e8rement vide. Mais bient\u00f4t, j'ai d\u00e9couvert que cet univers sans fin ne parvenait pas \u00e0 m'apaiser. Me retrouver au milieu de cet univers me mettait au contraire mal \u00e0 l'aise, comme un noy\u00e9 n'arrivant pas \u00e0 trouver une prise pour remonter \u00e0 la surface._\n\n _J'ai donc cr\u00e9\u00e9 une sph\u00e8re au c\u0153ur de cet univers, pas trop grande, une sph\u00e8re poss\u00e9dant une masse. Mais je ne me suis pas senti beaucoup mieux. La sph\u00e8re flottait en plein milieu du vide \u2013 bien qu'en fin de compte, dans un espace infini, le \"milieu\" soit partout. Au sein de cet univers, rien ne pouvait agir sur cette sph\u00e8re et elle-m\u00eame ne pouvait agir sur rien. Elle \u00e9tait seulement l\u00e0, suspendue, sans jamais faire le moindre mouvement, sans jamais conna\u00eetre le moindre changement. La m\u00e9taphore parfaite de la mort._\n\n _Alors, j'ai cr\u00e9\u00e9 une deuxi\u00e8me sph\u00e8re, \u00e0 peu pr\u00e8s avec la m\u00eame masse que la premi\u00e8re. Leurs surfaces \u00e9taient compos\u00e9es d'une mati\u00e8re miroitante et elles se renvoyaient leurs reflets, seules entit\u00e9s existantes dans tout l'univers. Mais la situation ne s'est pas am\u00e9lior\u00e9e, car si chacune des deux sph\u00e8res n'avait pas de mouvement initial \u2013 c'est-\u00e0-dire sans pouss\u00e9e de ma part \u2013, elles seraientcertainement attir\u00e9es l'une contre l'autre par leur champ gravitationnel. Puis les deux sph\u00e8res resteraient suspendues ensemble, immobiles, comme un symbole de mort. Si elles avaient un mouvement initial et n'entraient pas en collision, elles finiraient par faire leur r\u00e9volution chacune autour de l'autre sous l'influence de leur gravit\u00e9. Peu importe le mouvement d'impulsion initial, leurs cycles de r\u00e9volution finiraient par se stabiliser et devenir \u00e0 jamais inchang\u00e9s, comme une danse de mort._\n\n _Alors, j'ai introduit une troisi\u00e8me sph\u00e8re, et il s'est produit un changement \u00e9tonnant. Comme je vous l'ai dit, toutes les figures g\u00e9om\u00e9triques apparaissent dans mon esprit sous la forme de chiffres. Les univers pr\u00e9c\u00e9dents, ceux sans sph\u00e8re, avec une ou deux sph\u00e8res, s'\u00e9taient manifest\u00e9s sous la forme d'une ou plusieurs \u00e9quations, telles les feuilles d'arbre solitaires d'un automne tardif. Mais l'ajout de la troisi\u00e8me sph\u00e8re a fait s'incarner quelque chose de nouveau : le \"vide\". D\u00e8s les mouvements initiaux donn\u00e9s aux sph\u00e8res, celles-ci ont commenc\u00e9 \u00e0 se mouvoir de fa\u00e7on complexe dans l'espace, comme une chor\u00e9graphie chaque fois in\u00e9dite. Les \u00e9quations descriptives se sont cette fois mises \u00e0 pleuvoir comme une averse sans fin._\n\n _C'est ainsi que j'ai r\u00e9ussi \u00e0 m'endormir. Les trois sph\u00e8res continuaient \u00e0 danser dans mes r\u00eaves, c'\u00e9tait une valse d\u00e9r\u00e9gl\u00e9e, \u00e9ternellement changeante. Pourtant, au fond de moi, cette valse avait trouv\u00e9 un rythme, c'\u00e9tait simplement que sa cadence \u00e9tait infiniment longue. J'\u00e9tais fascin\u00e9, je voulais chercher \u00e0 d\u00e9crire l'int\u00e9gralit\u00e9 ou du moins une partie de cette cadence._\n\n _Le lendemain, je ne pouvais m'\u00f4ter de l'esprit ces trois sph\u00e8res dansant dans le \"vide\". Jamais mon cerveau n'avait autant carbur\u00e9, si bien que des moines ont demand\u00e9 \u00e0 l'abb\u00e9 si je n'avais pas des probl\u00e8mes mentaux. L'abb\u00e9 a r\u00e9pondu en riant que j'allais bien, que j'avais trouv\u00e9 le vide. Oui, j'avais trouv\u00e9 le vide. D\u00e9sormais, m\u00eame au beau milieu de la foule grouillante d'une ville surpeupl\u00e9e, mon c\u0153ur serait paisible. Pour la premi\u00e8re fois, les math\u00e9matiques m'amusaient. Les principes physiques du probl\u00e8me \u00e0 trois corps \u00e9taient en r\u00e9alit\u00e9 tr\u00e8s simples, c'\u00e9tait un banal probl\u00e8me math\u00e9matique_1. _Je me suis senti comme un homme quiaurait couru les jupons la moiti\u00e9 de sa vie et qui tomberait soudain amoureux pour la premi\u00e8re fois._\n\n\u2014 N'aviez-vous jamais entendu parler de Poincar\u00e92 ? l'interrompit Wang Miao.\n\n_Non, pas \u00e0 l'\u00e9poque. Je sais bien qu'un \u00e9tudiant en math\u00e9matiques se doit d'\u00eatre familier des th\u00e9ories de Poincar\u00e9, mais je ne r\u00e9v\u00e9rais aucun grand ma\u00eetre et je ne voulais pas non plus en devenir un moi-m\u00eame. Mais je crois qu'en ce temps-l\u00e0, m\u00eame si j'avais connu les conclusions de Poincar\u00e9, j'aurais continu\u00e9 \u00e0 travailler sur le probl\u00e8me \u00e0 trois corps. Tout le monde croit que Poincar\u00e9 a prouv\u00e9 que le probl\u00e8me \u00e0 trois corps est insoluble mais, \u00e0 mon avis, c'est un malentendu. Il n'a fait que montrer la sensibilit\u00e9 aux conditions initiales et prouver qu'un syst\u00e8me \u00e0 trois corps n'\u00e9tait pas un syst\u00e8me int\u00e9grable. Mais \"sensibilit\u00e9\" ne veut pas dire pour autant incertitude totale, c'est simplement que cette incertitude implique un plus grand nombre de solutions admissibles. Ce qu'il fallait, c'\u00e9tait un nouvel algorithme._\n\n _J'ai tout de suite pens\u00e9 \u00e0 quelque chose. Avez-vous d\u00e9j\u00e0 entendu parler de la m\u00e9thode de Monte-Carlo ? Non ? Il s'agit d'un algorithme informatique souvent utilis\u00e9 pour calculer l'aire d'une forme g\u00e9om\u00e9trique irr\u00e9guli\u00e8re. Le principe est le suivant : le logiciel ins\u00e8re la forme g\u00e9om\u00e9trique dont on veut conna\u00eetre l'aire \u00e0 l'int\u00e9rieur d'une autre forme dont l'aire est connue, puis il mitraille l'ensemble de petites billes, en veillant \u00e0 ce que chaque bille n'atteigne jamais la m\u00eame cible. Ces tirs al\u00e9atoires se poursuivent durant un certain temps, jusqu'\u00e0 ce que chaque espace ait \u00e9t\u00e9 touch\u00e9 au moins une fois. Arriv\u00e9 \u00e0 ce stade, le logiciel propose un graphiqueindiquant combien de billes ont atteint l'aire de la forme g\u00e9om\u00e9trique connue et combien ont touch\u00e9 celle recherch\u00e9e. \u00c0 partir de l\u00e0, il est possible d'estimer l'aire de la surface irr\u00e9guli\u00e8re. Bien entendu, plus les billes sont petites, plus le r\u00e9sultat est pr\u00e9cis._\n\n _Cette m\u00e9thode est certes simple, mais elle montre comment, en math\u00e9matiques, la rudesse de l'al\u00e9atoire peut triompher de la logique et de la pr\u00e9cision. Autrement dit, c'est une approche de calcul qui utilise la quantit\u00e9 pour arriver \u00e0 la qualit\u00e9. C'\u00e9tait la strat\u00e9gie que j'avais choisie pour r\u00e9soudre le probl\u00e8me \u00e0 trois corps. J'ai donc \u00e9tudi\u00e9 le mouvement des trois sph\u00e8res section de temps apr\u00e8s section de temps. \u00c0 chaque section de temps, il existait des combinaisons infinies de vecteurs de mouvement. Je traitais chacune de ces combinaisons comme des entit\u00e9s vivantes. L'important \u00e9tait d'\u00e9tablir un ensemble de r\u00e8gles : quelles combinaisons de vecteurs de mouvement \u00e9taient \"saines\" et \"b\u00e9n\u00e9fiques\" et lesquelles \u00e9taient \"nuisibles\" et \"n\u00e9fastes\" ? La premi\u00e8re cat\u00e9gorie avait plus de chances de survivre, tandis que la seconde \u00e9tait d\u00e9savantag\u00e9e. Ce type de calcul m'amenait \u00e0 proc\u00e9der par \u00e9limination : je conservais les combinaisons saines et b\u00e9n\u00e9fiques et j'\u00e9liminais celles que j'estimais nuisibles et n\u00e9fastes. Les combinaisons restantes \u00e9taient les pr\u00e9dictions exactes de la prochaine configuration du syst\u00e8me des trois corps._\n\n\u2014 Un algorithme \u00e9volutionnaire, en somme, dit Wang Miao.\n\n\u2014 Je crois que j'ai bien fait de vous faire venir, dit Shi Qiang en hochant la t\u00eate.\n\n_Oui, mais ce n'est que bien plus tard que j'ai eu connaissance de ce terme. Mon algorithme se caract\u00e9rise par la masse immense de calculs \u00e0 effectuer. Pour r\u00e9soudre de cette mani\u00e8re le probl\u00e8me \u00e0 trois corps, les ordinateurs existant de nos jours ne suffisent pas. Imaginez qu'\u00e0 l'\u00e9poque, dans le temple, je ne disposais m\u00eame pas d'une calculatrice, je n'avais qu'un cahier blanc et un stylo r\u00e9cup\u00e9r\u00e9s aupr\u00e8s du bureau de comptabilit\u00e9 du temple. J'ai commenc\u00e9 \u00e0 construire mon mod\u00e8le math\u00e9matique sur le cahier, mais cela demandait beaucoup de travail et, tr\u00e8s vite, j'ai fini plus d'une dizaine de cahiers, ce qui faisait enrager les moines tr\u00e9soriers. Mais sur injonction de l'abb\u00e9, ils acceptaient \u00e0 contrec\u0153ur de me donner d'autres cahiers et d'autres crayons pour poursuivre mes calculs.Je mettais les r\u00e9sultats sous mon oreiller et je jetais les brouillons dans le br\u00fbloir \u00e0 encens de la cour du temple._\n\n _Un soir, une jeune femme a soudain fait irruption dans ma chambre. C'\u00e9tait la premi\u00e8re fois qu'une femme entrait chez moi ; elle tenait dans la main des morceaux de papier aux extr\u00e9mit\u00e9s br\u00fbl\u00e9es : les brouillons que j'avais jet\u00e9s dans le br\u00fbloir._\n\n _\u2014 Ils disent que c'est \u00e0 toi. Tu \u00e9tudies le probl\u00e8me \u00e0 trois corps ? demanda-t-elle avec pr\u00e9cipitation._\n\n _Derri\u00e8re ses grosses lunettes, ses yeux semblaient en feu. J'\u00e9tais tr\u00e8s surpris par la question de cette femme. Mes calculs math\u00e9matiques \u00e9taient tr\u00e8s peu conventionnels et je ne prenais pas la peine de donner le d\u00e9tail de mes d\u00e9ductions. Pourtant, chose inou\u00efe, elle avait pu deviner rien qu'\u00e0 mes brouillons quel \u00e9tait l'objet de ma recherche. Elle devait avoir un don rare pour les math\u00e9matiques et je pouvais presque \u00eatre s\u00fbr que, comme moi, elle \u00e9tait plong\u00e9e dans l'\u00e9tude du probl\u00e8me \u00e0 trois corps._\n\n _Les touristes et les p\u00e8lerins qui venaient ici ne me faisaient pas une bonne impression. Les touristes ignoraient totalement ce qu'ils \u00e9taient venus chercher ; ils se contentaient de courir \u00e0 droite et \u00e0 gauche pour prendre des photos. Quant aux p\u00e8lerins, ils \u00e9taient visiblement plus pauvres que les touristes et paraissaient tous \u00eatre dans un \u00e9tat de torpeur et d'inhibition intellectuelle. Mais cette jeune femme \u00e9tait diff\u00e9rente. Elle avait l'apparence d'une scientifique. J'ai su par la suite qu'elle \u00e9tait venue ici avec un groupe de touristes japonais._\n\n _Sans m\u00eame attendre ma r\u00e9ponse, elle a poursuivi :_\n\n _\u2014 Votre approche est brillante. Cela faisait longtemps que nous cherchions une m\u00e9thode capable de changer l'\u00e9pineux probl\u00e8me \u00e0 trois corps en une question capable d'\u00eatre trait\u00e9e par une masse, m\u00eame consid\u00e9rable, de calculs. Mais il est n\u00e9cessaire d'avoir un ordinateur suffisamment puissant._\n\n _Je lui ai r\u00e9pondu sinc\u00e8rement :_\n\n _\u2014 Tous les ordinateurs du monde r\u00e9unis n'y parviendraient pas._\n\n _\u2014 Mais vous avez tout de m\u00eame besoin d'un environnement de recherche appropri\u00e9, ce dont vous ne disposez pas ici. Je peux vous offrir la chance d'utiliser un ordinateur tr\u00e8s puissant et je vous donnerai aussi un mini-ordinateur. Nous partons demain matin._\n\n _C'\u00e9tait Shen Yufei. D\u00e9j\u00e0 \u00e0 l'\u00e9poque elle \u00e9tait tranchante et autoritaire, mais elle \u00e9tait plus s\u00e9duisante qu'aujourd'hui. J'\u00e9tais d'une nature froide et j'avais encore moins d'int\u00e9r\u00eat pour les femmes que pour les moines qui gravitaient autour de moi. Mais Shen Yufei avait quelque chose de sp\u00e9cial. C'\u00e9tait pr\u00e9cis\u00e9ment son absence de f\u00e9minit\u00e9 qui m'attirait le plus chez elle. Quoi qu'il en soit, je n'avais rien d'autre \u00e0 faire, alors j'ai accept\u00e9 sa proposition._\n\n _La nuit, je ne suis pas arriv\u00e9 \u00e0 trouver le sommeil, j'ai enfil\u00e9 un maillot et je suis sorti dans la cour. Au loin, dans l'obscurit\u00e9 de la salle principale du temple, j'ai aper\u00e7u la silhouette de Shen Yufei. Elle br\u00fblait un b\u00e2ton d'encens devant une statue du Bouddha et ses gestes paraissaient empreints de d\u00e9votion. Je me suis approch\u00e9 sans bruit de la porte d'entr\u00e9e du temple, et je l'ai entendue psalmodier cette pri\u00e8re :_\n\n _\u2014 Bouddha, b\u00e9nis mes dieux, sauve-les de leur mer de souffrance \u00e9ternelle._\n\n _Je pensais avoir mal entendu, mais elle a murmur\u00e9 une nouvelle fois :_\n\n _\u2014 Bouddha, b\u00e9nis mes dieux, sauve-les de leur mer de souffrance \u00e9ternelle._\n\n _Je ne m'y entendais pas en religion et je n'y portais de toute fa\u00e7on aucun int\u00e9r\u00eat. Mais je n'aurais jamais imagin\u00e9 qu'il puisse exister des pri\u00e8res aussi \u00e9tranges. Je n'ai pu m'emp\u00eacher de lui demander :_\n\n _\u2014 Que dites-vous ?_\n\n _Shen Yufei n'a pas sembl\u00e9 se pr\u00e9occuper de ma pr\u00e9sence. Elle a gard\u00e9 les yeux ferm\u00e9s et les mains jointes, comme si elle attendait que sa pri\u00e8re monte vers le Bouddha en m\u00eame temps que la fum\u00e9e de l'encens. Elle n'a ouvert les yeux qu'apr\u00e8s un bon moment et s'est tourn\u00e9e vers moi._\n\n _\u2014 Allez dormir, nous partons t\u00f4t demain matin, a-t-elle dit, sans me regarder._\n\n _\u2014 Les dieux dont vous avez parl\u00e9, est-ce que ce sont des divinit\u00e9s bouddhistes ? ai-je demand\u00e9._\n\n _\u2014 Non._\n\n _\u2014 Mais alors..._\n\n _Shen Yufei est partie sans un mot et a acc\u00e9l\u00e9r\u00e9 le pas. Je n'ai pas eu le temps de l'interroger davantage. J'ai r\u00e9p\u00e9t\u00e9 la pri\u00e8re au fond de moi, je la trouvais de plus en plus myst\u00e9rieuse. Puis, une peurinsens\u00e9e s'est insinu\u00e9e en moi, alors je me suis h\u00e2t\u00e9 d'aller frapper \u00e0 la porte de la chambre de l'abb\u00e9._\n\n _\u2014 Qu'est-ce que cela signifie lorsque quelqu'un prie un dieu de prot\u00e9ger d'autres dieux ? ai-je demand\u00e9, en lui rapportant en d\u00e9tail la sc\u00e8ne \u00e0 laquelle j'avais assist\u00e9._\n\n _L'abb\u00e9 regardait le livre qu'il tenait entre les mains, mais il \u00e9tait \u00e9vident qu'il ne lisait pas. Il \u00e9coutait mon r\u00e9cit. Il a fini par me dire :_\n\n _\u2014 Sors d'abord un moment, laisse-moi r\u00e9fl\u00e9chir._\n\n _J'ai quitt\u00e9 la chambre, conscient qu'il \u00e9tait inhabituel pour l'abb\u00e9 d'agir ainsi. C'\u00e9tait un homme d'une grande sagesse qui pouvait r\u00e9pondre \u00e0 des questions sur la religion, l'histoire ou la culture sans m\u00eame avoir besoin de r\u00e9fl\u00e9chir. J'ai attendu derri\u00e8re la porte le temps d'une cigarette et l'abb\u00e9 m'a rappel\u00e9._\n\n _\u2014 Je ne vois qu'une seule possibilit\u00e9, dit-il avec une mine sinistre._\n\n _\u2014 Laquelle ? Quelle possibilit\u00e9 ? Cette religion existe-t-elle ? Une religion dont les dieux ont besoin que leurs fid\u00e8les prient un autre dieu de les sauver ?_\n\n _\u2014 Ses dieux, ils existent r\u00e9ellement._\n\n _Cette phrase m'a rendu perplexe :_\n\n _\u2014 Mais... le Bouddha n'existe pas r\u00e9ellement ?_\n\n _Aussit\u00f4t apr\u00e8s avoir prononc\u00e9 cette phrase, je me suis rendu compte de mon blasph\u00e8me et je me suis platement excus\u00e9._\n\n _L'abb\u00e9 a doucement secou\u00e9 les mains :_\n\n _\u2014 Je te l'ai d\u00e9j\u00e0 dit, rien ne sert de parler de bouddhisme entre nous. Le Bouddha existe d'une mani\u00e8re qu'il te serait impossible d'appr\u00e9hender. Les dieux dont parle cette femme, eux, existent d'une mani\u00e8re qu'il t'est possible de comprendre. Je ne peux pas t'en dire plus, mais si je peux te donner un conseil, ne pars pas avec elle..._\n\n _\u2014 Pourquoi ?_\n\n _\u2014 Ce n'est qu'une intuition, j'ai l'impression que derri\u00e8re elle se cachent des choses que ni toi ni moi ne sommes en mesure d'imaginer._\n\n _Je suis sorti de la chambre de l'abb\u00e9, et j'ai travers\u00e9 la cour du temple jusqu'\u00e0 ma chambre. C'\u00e9tait la pleine lune. J'ai lev\u00e9 la t\u00eate et je me suis senti observ\u00e9 par un \u00e9trange \u0153il argent\u00e9. La lueur de la lune \u00e9tait froide et angoissante._\n\n _Le lendemain, je suis parti avec Shen Yufei. Je ne pouvais de toute fa\u00e7on pas rester ind\u00e9finiment dans le temple. Mais je ne m'\u00e9tais pas attendu \u00e0 vivre les ann\u00e9es qui ont suivi comme dans un r\u00eave. Shen Yufei avait tenu sa promesse, j'avais d\u00e9sormais en ma possession un mini-ordinateur et pouvais b\u00e9n\u00e9ficier d'un environnement agr\u00e9able pour faire mes calculs. Nous nous sommes m\u00eame rendus plusieurs fois \u00e0 l'\u00e9tranger pour utiliser un ordinateur tr\u00e8s performant. Je n'avais pas besoin de partager son temps d'utilisation, et toutes les capacit\u00e9s de l'unit\u00e9 centrale de traitement \u00e9taient consacr\u00e9es \u00e0 mes seuls calculs. Elle avait \u00e9norm\u00e9ment d'argent dont j'ignorais la provenance._\n\n _Plus tard, nous nous sommes mari\u00e9s, moins par amour ou par passion que par commodit\u00e9. Nous avions chacun une mission \u00e0 accomplir. Je peux dire que les ann\u00e9es qui ont suivi m'ont sembl\u00e9 passer aussi vite qu'une simple journ\u00e9e. Je vivais paisiblement. Dans cette villa, je n'avais qu'\u00e0 tendre la main pour obtenir \u00e0 manger. Je pouvais me consacrer enti\u00e8rement \u00e0 la recherche du probl\u00e8me \u00e0 trois corps. Yufei n'interf\u00e9rait jamais dans ma vie, j'avais une voiture dans le garage que je pouvais prendre pour aller o\u00f9 je voulais. J'ose m\u00eame affirmer que si j'avais ramen\u00e9 une fille \u00e0 la maison, elle ne m'en aurait pas tenu rigueur. Elle ne se pr\u00e9occupait que de ma recherche et les seules discussions que nous avions quotidiennement avaient pour sujet le probl\u00e8me \u00e0 trois corps. Elle voulait savoir chaque jour l'\u00e9tat d'avancement de mes calculs._\n\n\u2014 Connaissez-vous les autres activit\u00e9s de Shen Yufei ? demanda Shi Qiang.\n\n\u2014 Vous voulez parler des Fronti\u00e8res de la science ? Elle y passe ses journ\u00e9es. Des gens viennent tous les jours \u00e0 la maison.\n\n\u2014 Elle ne vous a jamais propos\u00e9 d'en devenir membre ?\n\n\u2014 Jamais. Pour tout dire, elle est tr\u00e8s discr\u00e8te \u00e0 ce sujet et \u00e7a me va tr\u00e8s bien. Je suis comme \u00e7a, je me fiche de pas mal de choses. Elle conna\u00eet parfaitement cet aspect de ma personnalit\u00e9, elle dit que je suis un feignant sans aucun sens de la mission, que la Soci\u00e9t\u00e9 n'est pas faite pour moi et qu'au contraire cela perturberait mes recherches.\n\n\u2014 Avez-vous avanc\u00e9 dans vos recherches sur le probl\u00e8me \u00e0 trois corps ? demanda Wang Miao.\n\n_Au regard de l'\u00e9tat actuel des recherches, je peux dire que mes progr\u00e8s constituent une grande avanc\u00e9e. Il y a quelques ann\u00e9es, Richard Montgomery, de l'universit\u00e9 de Californie \u00e0 Santa Cruz, et Alain Chenciner, de l'universit\u00e9 Paris-Diderot, avec l'aide de chercheurs de l'Institut fran\u00e7ais de m\u00e9canique c\u00e9leste, ont d\u00e9couvert gr\u00e2ce \u00e0 une m\u00e9thode variationnelle la possibilit\u00e9 d'une stabilit\u00e9 du mouvement des trois corps. Dans des conditions initiales particuli\u00e8res, les trois corps se d\u00e9placent sur une m\u00eame orbite en forme de 8. Apr\u00e8s cette d\u00e9couverte, beaucoup d'autres scientifiques se sont acharn\u00e9s \u00e0 chercher des configurations stables, dans le cas de conditions initiales sp\u00e9cifiques. Chaque nouvelle d\u00e9couverte a \u00e9t\u00e9 c\u00e9l\u00e9br\u00e9e en grande pompe. \u00c0 l'heure actuelle, seules trois ou quatre configurations de ce type ont \u00e9t\u00e9 d\u00e9couvertes._\n\n _Mais gr\u00e2ce \u00e0 mon algorithme \u00e9volutionnaire, j'en ai pour ma part d\u00e9j\u00e0 trouv\u00e9 une centaine. Avec les dessins de ces orbites, je crois bien que je pourrais ouvrir une galerie d'art postmoderne. Mais ce n'est pas mon but. Pour arriver \u00e0 r\u00e9soudre r\u00e9ellement le probl\u00e8me \u00e0 trois corps, il faut cr\u00e9er un mod\u00e8le math\u00e9matique qui, du moment que l'on conna\u00eet les vecteurs de mouvement initiaux, puisse nous permettre de pr\u00e9dire avec certitude les prochains mouvements des trois corps. C'est aussi le but recherch\u00e9 par Shen Yufei._\n\n _Mais cette vie paisible s'est achev\u00e9e hier. Mes ennuis ont commenc\u00e9._\n\n\u2014 Est-ce la plainte que vous \u00eates venue d\u00e9poser ? demanda Shi Qiang.\n\n\u2014 Oui. Hier, un homme a t\u00e9l\u00e9phon\u00e9. Il m'a dit que si je ne cessais pas imm\u00e9diatement mes recherches, je serais assassin\u00e9.\n\n\u2014 Qui \u00e9tait cet homme ?\n\n\u2014 Je ne sais pas.\n\n\u2014 Son num\u00e9ro de t\u00e9l\u00e9phone ?\n\n\u2014 Je ne sais pas, l'appel \u00e9tait masqu\u00e9.\n\n\u2014 D'autres informations en lien avec cette menace ?\n\n\u2014 Je ne sais pas.\n\nShi Qiang ricana et jeta son m\u00e9got de cigarette :\n\n\u2014 Vous nous avez racont\u00e9 votre vie et votre plainte tient \u00e0 la fin en une phrase et en une s\u00e9rie de \"je ne sais pas\" ?\n\n\u2014 Si je ne vous avais pas racont\u00e9 ma vie, comme vous dites, auriez-vous compris ? Et puis, s'il n'y avait eu que \u00e7a, je ne serais pas venu. Rappelez-vous bien que je suis paresseux par nature. Hier soir \u2013 enfin, au milieu de la nuit, je ne sais pas si c'\u00e9tait hier ou aujourd'hui \u2013 bref, pendant que je dormais, j'ai senti vaguement comme quelque chose de froid qui bougeait sur mon visage. J'ai ouvert les yeux et j'ai vu Yufei. J'ai eu une de ces frousses !\n\n\u2014 Qu'y a-t-il d'effrayant \u00e0 voir sa femme dans son lit au milieu de la nuit ?\n\n\u2014 Son regard... Un regard qu'elle n'avait jamais eu auparavant. La lumi\u00e8re bleut\u00e9e des lampadaires du jardin se refl\u00e9tait sur son visage. On aurait dit un fant\u00f4me. Elle tenait quelque chose dans sa main : un pistolet ! Elle fr\u00f4lait mon visage avec le canon du pistolet en me disant que je devais continuer mes recherches sur le probl\u00e8me \u00e0 trois corps, que sinon elle me tuerait.\n\n\u2014 Mmh, voil\u00e0 qui devient un peu plus int\u00e9ressant, dit Shi Qiang en hochant la t\u00eate avec satisfaction et en s'allumant une nouvelle cigarette.\n\n\u2014 \"Int\u00e9ressant\" ? \u00c9coutez, je n'avais nulle part o\u00f9 aller, c'est pour \u00e7a que je suis venu ici !\n\n\u2014 R\u00e9p\u00e9tez au mot pr\u00e8s ce qu'elle vous a dit.\n\n\u2014 Elle a dit : \"Si tu r\u00e9ussis \u00e0 r\u00e9soudre le probl\u00e8me \u00e0 trois corps, tu deviendras le sauveur du monde. Mais si tu t'arr\u00eates maintenant, tu seras consid\u00e9r\u00e9 comme un profanateur. Si quelqu'un pouvait choisir entre sauver l'humanit\u00e9 ou la d\u00e9truire, ton choix \u00e0 toi vaudrait le double.\"\n\nShi Qiang cracha une \u00e9paisse fum\u00e9e, il fixa Wei Cheng pendant un moment, jusqu'\u00e0 ce que ce dernier se sente troubl\u00e9. Puis il saisit un cahier dans le fourbi de sa table et prit un stylo.\n\n\u2014 Vous vouliez qu'on prenne des notes, n'est-ce pas ? R\u00e9p\u00e9tez-moi ce que vous venez de dire.\n\nWei Cheng r\u00e9p\u00e9ta une nouvelle fois les paroles de Shen Yufei. Wang Miao intervint :\n\n\u2014 Cette phrase est tr\u00e8s \u00e9trange. Pourquoi \"le double\" ?\n\nWei Cheng cligna des yeux et dit \u00e0 Shi Qiang :\n\n\u2014 C'est s\u00e9rieux, n'est-ce pas ? D\u00e8s que je suis arriv\u00e9 ici, le policier de garde m'a envoy\u00e9 vers vous. On dirait que vous nous aviez d\u00e9j\u00e0 \u00e0 l'\u0153il.\n\nShi Qiang hocha la t\u00eate :\n\n\u2014 Encore une question : pensez-vous que le pistolet de votre femme \u00e9tait un vrai ?\n\nVoyant que Wei Cheng ne savait comment r\u00e9pondre, il ajouta :\n\n\u2014 Le pistolet avait-il une odeur d'huile ?\n\n\u2014 Oui. J'en ai la certitude, il sentait l'huile !\n\n\u2014 Bien, dit Shi Qiang qui sauta de la table o\u00f9 il \u00e9tait assis. On a au moins \u00e7a : susception de port ill\u00e9gal d'arme \u00e0 feu. \u00c7a suffira pour une perquisition, nous remplirons les papiers demain, nous devons agir imm\u00e9diatement.\n\nIl se tourna vers Wang Miao :\n\n\u2014 Mon vieux, je vais avoir besoin de vous.\n\nPuis il lan\u00e7a \u00e0 Xu Bingbing qui n'avait pas encore parl\u00e9 :\n\n\u2014 Bingbing, je n'ai que deux gars avec moi en service cette nuit, \u00e7a ne suffit pas. Je sais bien que les membres du d\u00e9partement de s\u00e9curit\u00e9 de l'information pr\u00e9f\u00e8rent leur bureau douillet au terrain mais, aujourd'hui, je vais avoir besoin d'une pro comme toi avec nous.\n\nXu Bingbing fit rapidement oui de la t\u00eate, trop heureuse d'avoir enfin l'opportunit\u00e9 de quitter cet espace enfum\u00e9.\n\nOutre Shi Qiang et Xu Bingbing, l'\u00e9quipe charg\u00e9e de l'investigation \u00e9tait aussi compos\u00e9e de deux autres officiers de la brigade criminelle. Avec Wang Miao et Wei Cheng, ce fut donc un groupe de six personnes qui se r\u00e9partit dans deux voitures de police et, dans l'obscurit\u00e9 pr\u00e9c\u00e9dant l'aube, partit en direction des quartiers r\u00e9sidentiels \u00e0 la p\u00e9riph\u00e9rie de la ville.\n\nXu Bingbing et Wang Miao \u00e9taient assis \u00e0 l'arri\u00e8re. La voiture avait \u00e0 peine d\u00e9marr\u00e9 que la jeune femme glissa \u00e0 voix basse \u00e0 Wang Miao :\n\n\u2014 Professeur Wang, vous \u00eates c\u00e9l\u00e8bre dans le monde des _Trois Corps_.\n\nWang Miao se trouva d\u00e9sorient\u00e9 d'entendre quelqu'un mentionner les _Trois Corps_ dans le monde r\u00e9el. Il eut l'impression que la distance entre lui et cette jeune femme en uniforme s'\u00e9tait estomp\u00e9e.\n\n\u2014 Vous y jouez ?\n\n\u2014 J'ai pour mission de surveiller ce qui se passe dans le jeu, ce n'est pas une t\u00e2che amusante.\n\nWang Miao s'empressa de l'interroger :\n\n\u2014 Pouvez-vous me dire ce qui se cache derri\u00e8re ce jeu ? Je meurs d'envie de savoir.\n\nGr\u00e2ce \u00e0 la faible lueur des r\u00e9verb\u00e8res filtrant \u00e0 travers la fen\u00eatre des v\u00e9hicules, Wang Miao vit se dessiner sur les l\u00e8vres de Xu Bingbing un myst\u00e9rieux sourire :\n\n\u2014 Nous aimerions le savoir nous-m\u00eames. Mais tous leurs serveurs sont bas\u00e9s \u00e0 l'\u00e9tranger. Le syst\u00e8me et le pare-feu sont tr\u00e8s s\u00e9curis\u00e9s et c'est dur d'y entrer. Nous savons peu de choses, mais nous sommes s\u00fbrs que le jeu n'a pas \u00e9t\u00e9 cr\u00e9\u00e9 dans un but lucratif. Le niveau du logiciel est \u00e9lev\u00e9, je dirais m\u00eame anormalement \u00e9lev\u00e9. La somme des informations contenues elle aussi est, comme vous le savez, peu commune pour un jeu en ligne.\n\n\u2014 Avez-vous... (Wang Miao s'attacha \u00e0 choisir ses mots.) Avez-vous trouv\u00e9 des traces de ph\u00e9nom\u00e8nes surnaturels ?\n\n\u2014 Ce n'est pas l'hypoth\u00e8se que nous retenons pour l'instant. Beaucoup de gens participent au d\u00e9veloppement du jeu, partout dans le monde. Ces m\u00e9thodes de collaboration ressemblent un peu \u00e0 celles utilis\u00e9es pour le d\u00e9veloppement du syst\u00e8me Linux. Mais cette fois, ils utilisent sans aucun doute des outils de d\u00e9veloppement de pointe. Quant aux informations stock\u00e9es, nous n'avons pas la moindre id\u00e9e de leur provenance. Il y a quelque chose d'un peu... \"surnaturel\", comme vous dites. Mais nous croyons le c\u00e9l\u00e8bre pr\u00e9cepte du commissaire Shi : il y a forc\u00e9ment des humains derri\u00e8re tout \u00e7a. Nos filatures sont efficaces, nous obtiendrons bient\u00f4t des r\u00e9sultats.\n\nLa jeune fille ne savait pas mentir, sa derni\u00e8re phrase trahissait le fait qu'elle lui cachait encore beaucoup d'informations.\n\n\u2014 C'est un pr\u00e9cepte devenu c\u00e9l\u00e8bre, alors, dit Wang Miao en regardant Shi Qiang qui \u00e9tait au volant.\n\nIl ne faisait pas encore jour lorsqu'ils arriv\u00e8rent \u00e0 la villa. Une chambre \u00e9tait allum\u00e9e \u00e0 l'\u00e9tage, mais le reste de la b\u00e2tisse \u00e9tait plong\u00e9 dans le noir.\n\nD\u00e8s que Wang Miao fut descendu de voiture, il entendit une s\u00e9rie de bruits venant de l'\u00e9tage, comme si quelque chose heurtait le mur. Shi Qiang, qui venait de descendre du v\u00e9hicule, fut aussit\u00f4t sur le qui-vive. Il ouvrit la porte de la cour d'un coup de pied et, pr\u00e9c\u00e9dant ses trois coll\u00e8gues, il se rua dans la villa avec une agilit\u00e9 surprenante pour son corps massif.\n\nWang Miao et Wei Cheng les suivirent dans la maison. Ils mont\u00e8rent \u00e0 l'\u00e9tage par l'escalier du salon et entr\u00e8rent dans la chambre dont la lumi\u00e8re \u00e9tait allum\u00e9e. Leurs pieds pataugeaient dans une mare de sang. C'\u00e9tait la m\u00eame heure que lorsque Wang Miao avait d\u00e9couvert l'autre nuit Shen Yufei en train de jouer aux _Trois Corps_. \u00c0 pr\u00e9sent, celle-ci \u00e9tait \u00e9tendue au milieu de la pi\u00e8ce et du sang coulait des impacts de balles qui avaient perc\u00e9 sa poitrine. Une troisi\u00e8me balle avait transperc\u00e9 son sourcil gauche et son visage \u00e9tait macul\u00e9 de sang. Non loin d'elle, un pistolet baignait dans une flaque de sang.\n\nAu moment m\u00eame o\u00f9 Wang Miao entra, Shi Qiang et un de ses coll\u00e8gues s'engouffr\u00e8rent dans la chambre sombre d'en face. Sa fen\u00eatre \u00e9tait grande ouverte. Wang Miao entendit un bruit de voiture en train de d\u00e9marrer. L'officier qui se trouvait dans la chambre de Shen Yufei passa un coup de t\u00e9l\u00e9phone, tandis que Xu Bingbing observait avec anxi\u00e9t\u00e9 la sc\u00e8ne d'un peu plus loin. Comme Wang Miao et Wei Cheng, c'\u00e9tait sans doute la premi\u00e8re fois qu'elle assistait \u00e0 un pareil tableau. Shi Qiang revint rapidement et, tout en rangeant son pistolet dans le holster de sa poitrine, il dit au policier qui t\u00e9l\u00e9phonait :\n\n\u2014 Une Santana, noire. Un seul suspect, plaque ind\u00e9termin\u00e9e, faites fermer toutes les entr\u00e9es du cinqui\u00e8me p\u00e9riph\u00e9rique. Putain. On l'a peut-\u00eatre laiss\u00e9 filer.\n\nShi Qiang sonda la pi\u00e8ce et remarqua des impacts de balles sur le mur. Il \u00e9tudia les cartouches tomb\u00e9es au sol :\n\n\u2014 Le tueur a tir\u00e9 cinq balles, il l'a touch\u00e9e par trois fois ; elle a tir\u00e9 deux fois mais n'a pas atteint sa cible.\n\nPuis lui et un de ses coll\u00e8gues se baiss\u00e8rent pour examiner le corps. Xu Bingbing \u00e9tait toujours debout un peu plus loin, regardant discr\u00e8tement Wei Cheng. Shi Qiang leva lui aussi la t\u00eate pour voir sa r\u00e9action.\n\nSur son visage, on lisait un reste de choc et de tristesse, mais pas davantage. Il ne s'\u00e9tait pas d\u00e9parti de son c\u00f4t\u00e9 empot\u00e9 et semblait tr\u00e8s serein en comparaison de Wang Miao.\n\n\u2014 Vous n'avez pas l'air pas tr\u00e8s perturb\u00e9. L'assassin \u00e9tait peut-\u00eatre venu pour vous, dit Shi Qiang \u00e0 Wei Cheng.\n\nWei Cheng eut un rire d\u00e9sesp\u00e9r\u00e9 :\n\n\u2014 Que voulez-vous que je fasse ? Jusqu'\u00e0 aujourd'hui, je ne connaissais rien d'elle. Plus d'une fois, je lui ai conseill\u00e9 de mener une vie plus simple, mais... Ah, \u00e7a me fait penser \u00e0 ce que m'avait dit l'abb\u00e9 cette nuit-l\u00e0.\n\nShi Qiang se leva, il marcha jusqu'\u00e0 se retrouver en face de Wei Cheng et sortit une cigarette.\n\n\u2014 Je crois que vous avez encore des choses \u00e0 nous dire.\n\n\u2014 Oui, quelques-unes. Mais j'\u00e9tais trop feignant pour vous les raconter.\n\n\u2014 Je vous conseille de parler, et vite.\n\nWei Cheng r\u00e9fl\u00e9chit, puis dit :\n\n\u2014 Cet apr\u00e8s-midi \u2013 oh non, hier apr\u00e8s-midi \u2013, Yufei s'est disput\u00e9e avec quelqu'un dans le salon. C'\u00e9tait ce Pan Han, le c\u00e9l\u00e8bre \u00e9cologiste. Ils s'\u00e9taient d\u00e9j\u00e0 disput\u00e9s plusieurs fois dans le pass\u00e9, mais ils le faisaient d'habitude en japonais, peut-\u00eatre par crainte que je les comprenne. Mais hier, ils ne se sont pas donn\u00e9 cette peine, ils ont parl\u00e9 en chinois et j'ai entendu quelques bribes de leur dispute.\n\n\u2014 Essayez de nous rapporter le plus pr\u00e9cis\u00e9ment possible le contenu de leur conversation.\n\n\u2014 Bien. Pan Han a dit : \"En apparence, nous marchons ensemble mais, en r\u00e9alit\u00e9, nous sommes des ennemis irr\u00e9conciliables !\"Yufei a r\u00e9pondu : \"Oui, vous utilisez le pouvoir divin contre l'esp\u00e8ce humaine.\" Pan Han : \"Ton jugement n'est pas tout \u00e0 fait faux. Nous avons besoin de la venue des dieux sur Terre pour nous punir de nos p\u00e9ch\u00e9s. Mais vous autres vous voulez emp\u00eacher cette venue. Nous ne pouvons le tol\u00e9rer. Si vous n'arr\u00eatez pas de vous-m\u00eames, nous n'aurons d'autre choix que de vous forcer la main !\" Shen Yufei : \"Le Guide s'est laiss\u00e9 aveugler lorsqu'il a autoris\u00e9 une bande de d\u00e9mons tels que vous \u00e0 rejoindre l'Organisation !\" Pan Han : \"En parlant du Guide, \u00e0 ton avis, dans quel camp est-il ? Celui des adventistes ou celui des r\u00e9demptoristes ? Es-tu capable de le dire avec certitude ?\" Ces paroles ont plong\u00e9 Shen Yufei dans un long silence. Puis leurs voix ont baiss\u00e9 de volume, et je n'ai pas entendu la suite.\n\n\u2014 Et la voix qui vous a menac\u00e9 par t\u00e9l\u00e9phone ? \u00c0 quoi ressemblait-elle ?\n\n\u2014 Vous voulez savoir si c'\u00e9tait celle de Pan Han ? Je l'ignore, la voix \u00e9tait faible, je n'ai pas bien entendu.\n\nQuelques voitures de police, gyrophares allum\u00e9s, s'arr\u00eat\u00e8rent \u00e0 l'ext\u00e9rieur de la villa. Des policiers munis de gants blancs et d'appareils photo mont\u00e8rent \u00e0 l'\u00e9tage et pass\u00e8rent la maison au crible. Shi Qiang dit \u00e0 Wang Miao de rentrer chez lui se reposer. Mais avant de partir, celui-ci alla trouver Wei Cheng dans la chambre o\u00f9 se trouvait le mini-ordinateur.\n\n\u2014 Pourriez-vous me donner un r\u00e9sum\u00e9 de votre algorithme \u00e9volutionnaire des trois corps ? Je souhaiterais... le montrer \u00e0 quelques personnes. C'est une demande un peu soudaine, je comprendrais bien entendu que vous refusiez.\n\nWei Cheng sortit un CD et le tendit \u00e0 Wang Miao :\n\n\u2014 Tout est l\u00e0-dedans, tous les mod\u00e8les et toutes les annexes. Soyez gentil, publiez tout sous votre propre nom. \u00c7a m'aiderait beaucoup.\n\n\u2014 Non, non, je ne peux pas faire \u00e7a !\n\n\u2014 Professeur Wang, je me souviens que vous \u00eates d\u00e9j\u00e0 venu ici. Vous \u00eates un gars bien. Responsable. Si je peux vous donner un conseil, \u00e9loignez-vous de tout \u00e7a. Le monde est \u00e0 la veille d'un grand bouleversement. Chacun devrait s'efforcer de passer le restant de sa vie en s\u00e9curit\u00e9, ce serait d\u00e9j\u00e0 une chance. Ne r\u00e9fl\u00e9chissez pas trop. De toute fa\u00e7on, \u00e7a ne sert \u00e0 rien.\n\n\u2014 Vous avez l'air d'en savoir plus que ce que vous dites ?\n\n\u2014 Je vivais avec elle au quotidien. J'ai forc\u00e9ment une petite id\u00e9e...\n\n\u2014 Pourquoi n'avez-vous pas pr\u00e9venu la police ?\n\nWei Cheng sourit avec d\u00e9dain :\n\n\u2014 Les policiers sont des bons \u00e0 rien. Dieu lui-m\u00eame est impuissant. L'humanit\u00e9 est aujourd'hui arriv\u00e9e \u00e0 un stade o\u00f9 plus personne n'entend ses pri\u00e8res.\n\nWei Cheng se tenait pr\u00e8s de la fen\u00eatre donnant sur le versant est. Au loin, derri\u00e8re les gratte-ciel de la ville, on voyait percer les premi\u00e8res lueurs de l'aube. Curieusement, ce spectacle rappela \u00e0 Wang Miao l'aube myst\u00e9rieuse qui l'accueillait \u00e0 chaque entr\u00e9e dans le monde des _Trois Corps_.\n\n\u2014 Je ne suis pas aussi indiff\u00e9rent que vous le pensez. Ces derniers temps, je n'arrive pas \u00e0 m'endormir la nuit. Le matin, quand je vois le lever de soleil, j'ai toujours cette impression \u00e9trange qu'il est en fait en train de se coucher.\n\nIl se tourna vers Wang Miao et, apr\u00e8s un long silence, il l\u00e2cha :\n\n\u2014 Et tout \u00e7a parce que ses bienfaiteurs, ou ses \"dieux\", comme elle les appelle, sont incapables de se prot\u00e9ger eux-m\u00eames.\n\n* * *\n\n1 La description des mouvements de trois astres aux conditions semblables ou proches s'attirant les uns les autres sous l'effet de la gravitation est un probl\u00e8me classique des sciences physiques. C'est une question importante dans le domaine de la m\u00e9canique c\u00e9leste. Le probl\u00e8me a re\u00e7u une grande attention depuis le XVIe si\u00e8cle. Les math\u00e9maticiens suisse et fran\u00e7ais Euler et Lagrange, puis plus r\u00e9cemment d'autres chercheurs, \u00e0 l'aide de simulations sur ordinateur, ont trouv\u00e9 des solutions particuli\u00e8res au probl\u00e8me \u00e0 trois corps. _(N.d.A.)_\n\n2 Le math\u00e9maticien fran\u00e7ais prouva que la trajectoire des trois corps ne pouvait \u00eatre d\u00e9termin\u00e9e avec certitude par les math\u00e9matiques. L'\u00e9quation diff\u00e9rentielle du probl\u00e8me \u00e0 trois corps lui inspira une nouvelle m\u00e9thode math\u00e9matique. _(N.d.A.)_\n\n## 17\n\n## _LES TROIS CORPS_ :\n\n## NEWTON, VON NEUMANN, QIN SHI HUANG\n\n## ET LA SYZYGIE DES TROIS SOLEILS\n\nLe premier paysage du deuxi\u00e8me niveau des _Trois Corps_ n'\u00e9tait pas tr\u00e8s diff\u00e9rent de celui du premier. C'\u00e9tait toujours la m\u00eame plaine baign\u00e9e par la m\u00eame aube myst\u00e9rieuse et sur laquelle se dressait la m\u00eame pyramide. Mais cette fois, le monument avait retrouv\u00e9 une architecture plus orientale.\n\nWang Miao entendit le son de m\u00e9taux s'entrechoquant, mais cette dissonance faisait paradoxalement ressortir le silence qui planait au-dessus de la plaine glaciale. Il tenta de remonter \u00e0 la source du bruit et aper\u00e7ut deux silhouettes noires scintiller au pied de la pyramide. Dans la p\u00e9nombre du matin, il vit des reflets froids d'objets en acier fendant l'air : deux hommes se livraient \u00e0 un duel \u00e0 l'\u00e9p\u00e9e.\n\nUne fois ses yeux habitu\u00e9s \u00e0 l'obscurit\u00e9, Wang Miao put distinguer plus clairement les deux individus qui croisaient le fer. En se basant uniquement sur la forme architecturale de la pyramide, on se serait facilement imagin\u00e9 dans un pays d'Asie. Cependant les duellistes \u00e9taient deux Europ\u00e9ens v\u00eatus de costumes d'\u00e9poque XVIe ou XVIIe si\u00e8cle. Le plus petit des deux baissa la t\u00eate pour esquiver un coup d'\u00e9p\u00e9e, mais sa perruque grise tomba sur le sol. Ils \u00e9chang\u00e8rent encore quelques estocades puis un autre homme surgit de derri\u00e8re la pyramide, en les conjurant de cesser le combat. Mais les lames continuaient \u00e0 siffler dans l'air et il n'eut pas le courage de s'interposer. Il cria :\n\n\u2014 Messieurs, cessez donc ! N'avez-vous rien de mieux \u00e0 faire que de vous donner en spectacle ? O\u00f9 est donc votre sens de la mesure ? Si notre civilisation n'a aucun futur, quelle gloire pourrait bien vous apporter une victoire \u00e9ph\u00e9m\u00e8re ?\n\nLes deux combattants l'ignoraient, trop absorb\u00e9s par le duel. Le plus grand des deux poussa soudain un cri de douleur et son \u00e9p\u00e9e tomba sur le sol dans un bruit m\u00e9tallique. Se tenant le bras, il se mit \u00e0 s'enfuir en courant. Son adversaire le pourchassa sur quelques m\u00e8tres, avant de s'arr\u00eater et de cracher avec m\u00e9pris dans sa direction.\n\n\u2014 Ah, le l\u00e2che !\n\nIl se baissa pour ramasser sa perruque et remarqua la pr\u00e9sence de Wang Miao. Il pointa son \u00e9p\u00e9e vers l'homme en fuite et dit :\n\n\u2014 Ce faquin a eu l'outrecuidance d'affirmer qu'il avait invent\u00e9 le calcul infinit\u00e9simal !\n\nPuis, tout en r\u00e9ajustant sa perruque, il se frappa la poitrine et fit une r\u00e9v\u00e9rence :\n\n\u2014 Isaac Newton, pour vous servir.\n\n\u2014 Votre adversaire en fuite serait donc Leibniz ? demanda Wang Miao.\n\n\u2014 Lui-m\u00eame, monsieur ! Ah, le mis\u00e9rable ! Je n'aurais pas d\u00fb lui faire l'honneur de me battre pour une petite renomm\u00e9e suppl\u00e9mentaire. Avoir invent\u00e9 les trois lois de la m\u00e9canique a suffi \u00e0 faire de moi le plus grand apr\u00e8s Dieu. Du mouvement des plan\u00e8tes \u00e0 la division cellulaire, tous les ph\u00e9nom\u00e8nes de l'existence sont soumis \u00e0 ces trois grandes lois. Et \u00e0 pr\u00e9sent, gr\u00e2ce \u00e0 ce puissant outil math\u00e9matique qu'est le calcul infinit\u00e9simal, ce n'est qu'une question de temps avant que nous puissions \u00eatre capables de d\u00e9terminer les mouvements des trois soleils.\n\n\u2014 Ce n'est pas aussi simple, dit l'homme qui avait essay\u00e9 d'arr\u00eater le combat. Avez-vous r\u00e9fl\u00e9chi au nombre de calculs que cela demanderait ? J'ai vu la liste d'\u00e9quations diff\u00e9rentielles que vous avez dress\u00e9e, il me semble tout bonnement impossible de trouver une solution analytique, vous ne pourrez en tirer tout au plus qu'une solution num\u00e9rique. Pour mener \u00e0 bien tous ces calculs, il faudrait r\u00e9unir tous les math\u00e9maticiens du monde et les faire travailler sans rel\u00e2che. Et encore, il est tr\u00e8s probable qu'ils n'aient toujours pas fini quand viendra le jour du Jugement dernier. D'autant que si nous ne trouvons pas rapidement un moyen de contr\u00f4ler la m\u00e9canique des soleils, ce jour est pour bient\u00f4t.\n\nL'homme inclina ensuite la t\u00eate, dans un style plus moderne :\n\n\u2014 John von Neumann. Enchant\u00e9.\n\n\u2014 N'est-ce pas pr\u00e9cis\u00e9ment la raison pour laquelle nous avons parcouru des milliers de kilom\u00e8tres \u00e0 l'est ? Pour r\u00e9soudre le probl\u00e8me du calcul de ces \u00e9quations ? demanda Newton, avant de se tourner vers Wang Miao. Nous ne sommes pas partis seuls, Norbert Wiener et le maraud que vous avez vu fuir tout \u00e0 l'heure ont \u00e9t\u00e9 du voyage. Nous avons rencontr\u00e9 des pirates au large de Madagascar. Wiener s'est sacrifi\u00e9 pour que nous puissions nous enfuir. Il est mort en h\u00e9ros.\n\n\u2014 Pourquoi \u00e9tait-il n\u00e9cessaire de venir en Orient pour construire un ordinateur ?\n\nVon Neumann et Newton \u00e9chang\u00e8rent un regard surpris :\n\n\u2014 Un _ordinateur_ ? Vous voulez dire une machine \u00e0 calculer ? Est-ce qu'un tel outil existe ?\n\n\u2014 Vous ne connaissez pas les ordinateurs ? Mais comment comptez-vous vous y prendre pour effectuer ce nombre incroyable de calculs ?\n\nVon Neumann regarda Wang Miao les yeux \u00e9carquill\u00e9s, comme s'il ne comprenait pas du tout sa question :\n\n\u2014 Comment ? Mais avec des hommes bien s\u00fbr ! Dans ce monde, qui d'autre pourrait bien faire des calculs ?\n\n\u2014 Mais vous avez dit \u00e0 l'instant que m\u00eame si vous arriviez \u00e0 r\u00e9unir tous les math\u00e9maticiens du monde, ils n'y parviendraient pas.\n\n\u2014 C'est pourquoi nous ne ferons pas appel \u00e0 des math\u00e9maticiens, mais \u00e0 des gens du peuple ! \u00c0 des travailleurs ordinaires ! Mais il nous en faut beaucoup, au moins trente millions ! Nous ferons des math\u00e9matiques en nous fondant sur la tactique militaire de la mar\u00e9e humaine !\n\n\u2014 Des gens du peuple ? Trente millions ? demanda Wang Miao au comble de l'\u00e9tonnement. Si je comprends bien, \u00e0 une \u00e9poque o\u00f9 neuf hommes sur dix sont analphab\u00e8tes, vous pensez pouvoir trouver trente millions de personnes qui comprennent le calcul infinit\u00e9simal ?\n\n\u2014 Connaissez-vous cette blague au sujet de l'arm\u00e9e du Sichuan ? demanda von Neumann en sortant un \u00e9pais cigare. (Il en mordit l'extr\u00e9mit\u00e9 et l'alluma.) Les soldats de l'arm\u00e9e du Sichuan devaient s'entra\u00eener \u00e0 la marche militaire, mais leur niveau d'\u00e9ducation \u00e9tait si faible qu'ils ne comprenaient pas l'officier quand il disait : \"une-deux une-deux\". L'officier eut une id\u00e9e : il fit enfiler aux pieds de chaque soldat une botte en paille \u00e0 gauche et une botte en toile \u00e0 droite. Et lors de l'exercice, il criait en sichuanais : \"pied d'paille-pied d'toile pied d'paille-pied d'toile\". C'est ce genre de soldats dont nous avons besoin, mais il nous en faut trente millions.\n\nEn entendant cette plaisanterie moderne, Wang Miao comprit imm\u00e9diatement que le personnage auquel il faisait face n'\u00e9tait pas un programme informatique, mais un homme, et tr\u00e8s certainement chinois.\n\n\u2014 Une si grande arm\u00e9e est difficile \u00e0 imaginer, dit Wang Miao en secouant la t\u00eate.\n\n\u2014 C'est pourquoi nous sommes venus demander l'aide de l'empereur Qin Shi Huang1, dit Newton en d\u00e9signant la pyramide.\n\n\u2014 C'est encore lui qui gouverne ici ? demanda Wang Miao en regardant les alentours.\n\nEt en effet, les soldats gardant l'entr\u00e9e de la pyramide \u00e9taient v\u00eatus de simples cottes de mailles et arm\u00e9s de hallebardes de l'\u00e9poque des Qin. Ces anachronismes n'\u00e9tonnaient plus Wang Miao tant le monde des _Trois Corps_ en \u00e9tait parsem\u00e9.\n\n\u2014 Il est le ma\u00eetre du monde. Il commande une arm\u00e9e de trente millions d'hommes, pr\u00eats \u00e0 conqu\u00e9rir l'Europe. Bien, allons le voir, dit von Neumann en montrant l'entr\u00e9e de la pyramide, puis il se tourna vers Newton : L\u00e2che cette \u00e9p\u00e9e !\n\nNewton laissa tomber son \u00e9p\u00e9e par terre et tous trois p\u00e9n\u00e9tr\u00e8rent dans la pyramide. Au bout du corridor, au moment d'entrer dans la salle d'audience, un garde insista pour qu'ils enl\u00e8vent leurs v\u00eatements. Newton protesta de fa\u00e7on virulente en disant qu'ils \u00e9taient des savants et qu'ils ne dissimulaient aucune arme ! Tandis que les deux parties se querellaient, une voix grave retentit de l'int\u00e9rieur de la salle :\n\n\u2014 C'est cet Europ\u00e9en qui a d\u00e9couvert les trois lois de la m\u00e9canique ? Laisse-les entrer, lui et ses compagnons.\n\nIls entr\u00e8rent dans la salle d'audience. L'empereur tr\u00e9pignait au milieu de la salle. Les pans de sa robe, ainsi que sa c\u00e9l\u00e8bre \u00e9p\u00e9e de c\u00e9r\u00e9monie, tra\u00eenaient sur le sol. Il se retourna et d\u00e9visagea les trois savants ; Wang Miao remarqua aussit\u00f4t que ses yeux \u00e9taient les m\u00eames que ceux du roi Zhou et du pape Gr\u00e9goire.\n\n\u2014 Je connais la raison de votre visite. Vous \u00eates europ\u00e9ens, pourquoi ne pas avoir simplement adress\u00e9 votre requ\u00eate \u00e0 C\u00e9sar ? Son empire est immense, il aurait s\u00fbrement pu r\u00e9unir une arm\u00e9e de trente millions d'hommes.\n\n\u2014 Mais Votre Majest\u00e9 imp\u00e9riale, connaissez-vous cette arm\u00e9e ? Savez-vous dans quel \u00e9tat est aujourd'hui l'empire de C\u00e9sar ? Entre les murs de l'\u00e9ternelle cit\u00e9 de Rome, les cours d'eau sont gravement pollu\u00e9s. Savez-vous pourquoi ?\n\n\u2014 Les industries militaires ?\n\n\u2014 Non, non, Votre Grandeur, le vomi des Romains apr\u00e8s leurs orgies ! Ces noblaillons vont m\u00eame jusqu'\u00e0 am\u00e9nager des civi\u00e8res sous les tables de leurs banquets, de telle sorte que lorsqu'ils ne peuvent plus marcher \u00e0 force d'avoir trop ripaill\u00e9, leurs serviteurs les emportent sur les civi\u00e8res. L'empire tout entier est emp\u00eatr\u00e9 dans un bourbier de d\u00e9bauche duquel il ne peut se sortir. M\u00eame si nous parvenions \u00e0 r\u00e9unir une arm\u00e9e de trente millions de soldats, ceux-ci n'auraient ni les qualit\u00e9s mentales ni la force physique de mener \u00e0 bien un si immense calcul.\n\n\u2014 Cela je le sais, dit Qin Shi Huang, mais C\u00e9sar est en train de se r\u00e9veiller, il refonde ses arm\u00e9es. Je crains la sagesse des Europ\u00e9ens. Vous n'\u00eates pas plus intelligents que nous autres Orientaux, mais vous savez trouver le bon chemin. Copernic a su pr\u00e9dire l'existence des trois soleils et vous avez compris les trois lois de la m\u00e9canique. Ce sont des d\u00e9couvertes extraordinaires que nous n'avons pas \u00e9t\u00e9 capables de faire ici. Je n'ai pas encore la force de frappe qui me permettra de conqu\u00e9rir l'Europe, mes navires sont en nombre insuffisant et nous ne pouvons pas passer par les routes terrestres : nous n'aurons jamais assez de ravitaillement pour toute la travers\u00e9e.\n\n\u2014 Voil\u00e0 pourquoi, Votre Majest\u00e9 imp\u00e9riale, votre glorieux empire doit continuer \u00e0 se d\u00e9velopper ! dit von Neumann, sautant sur l'occasion. Si vous parvenez \u00e0 d\u00e9terminer les mouvements des soleils, vous pourrez utiliser chaque \u00e8re r\u00e9guli\u00e8re \u00e0 votre profit et \u00e9chapper aux d\u00e9sastres caus\u00e9s par les \u00e8res chaotiques. Vous progresserez \u00e0 une vitesse beaucoup plus grande que l'Europe. Faites-nous confiance, nous sommes des savants, nous mettrons \u00e0 votre service les trois lois de la m\u00e9canique et le calcul infinit\u00e9simal pour pr\u00e9dire avec exactitude la m\u00e9canique des soleils. Nous ne nous pr\u00e9occupons gu\u00e8re de savoir qui gouverne le monde.\n\n\u2014 Bien s\u00fbr, j'ai besoin de pr\u00e9dire les mouvements des soleils. Mais avant que je rassemble une arm\u00e9e de trente millions d'hommes, vous devez au moins me faire une d\u00e9monstration de vos calculs.\n\n\u2014 Votre Majest\u00e9, donnez-moi trois hommes et je vous ferai cette d\u00e9monstration, r\u00e9pondit von Neumann, enthousiaste.\n\n\u2014 Trois hommes ? Trois seulement ? Il ne me serait pas bien difficile de vous en fournir trois mille.\n\nL'empereur d\u00e9visageait von Neumann avec un regard incr\u00e9dule.\n\n\u2014 Votre Majest\u00e9, vous venez d'\u00e9voquer les d\u00e9faillances de la pens\u00e9e scientifique orientale. En voil\u00e0 un parfait exemple. Vous n'avez pas encore pris conscience que la structure complexe de l'univers repose en r\u00e9alit\u00e9 sur des \u00e9l\u00e9ments simples. Je n'aurai besoin que de trois hommes, Votre Altesse.\n\nQin Shi Huang fit venir trois soldats, tous tr\u00e8s jeunes. Comme tous les autres soldats de l'arm\u00e9e des Qin, ils \u00e9taient aussi disciplin\u00e9s que des machines.\n\n\u2014 Je ne connais pas vos noms, dit von Neumann en tapant sur l'\u00e9paule des deux premiers soldats. Vous serez tous deux responsables d'entrer les signaux : je vous appellerai donc \"Entr\u00e9e 1\" et \"Entr\u00e9e 2\".\n\nIl d\u00e9signa le troisi\u00e8me soldat :\n\n\u2014 Toi, tu t'occuperas de la sortie des signaux. Je t'appellerai donc \"Sortie\".\n\nDe la main, il fit signe aux soldats de se placer selon un ordre pr\u00e9cis.\n\n\u2014 Voil\u00e0, vous formez maintenant un triangle. Sortie se situe au niveau du sommet et Entr\u00e9e 1 et Entr\u00e9e 2 \u00e0 la base.\n\n\u2014 Vous auriez simplement pu leur demander de se mettre en formation de triangle ! cracha Qin Shi Huang en regardant von Neumann avec m\u00e9pris.\n\nNewton sortit six petits drapeaux : trois blancs et trois noirs. Il les donna \u00e0 von Neumann qui les distribua \u00e0 son tour aux soldats, un blanc et un noir \u00e0 chacun.\n\n\u2014 Le drapeau blanc repr\u00e9sente 0 ; le noir repr\u00e9sente 1. Bien. Maintenant, \u00e9coutez-moi attentivement. Sortie, tourne-toi et observe bien Entr\u00e9e 1 et Entr\u00e9e 2. Si tous deux l\u00e8vent leurs drapeaux noirs, tu l\u00e8veras aussi ton drapeau noir. Dans toutes les autres situations, tu l\u00e8veras ton drapeau blanc, c'est-\u00e0-dire : blanc-noir, noir-blanc, blanc-blanc.\n\n\u2014 Je pense que vous devriez choisir une autre couleur, dit Qin Shi Huang, le drapeau blanc, c'est pour d\u00e9poser les armes.\n\nEmport\u00e9 par l'enthousiasme, von Neumann ne prit pas la peine de r\u00e9pondre. Il ordonna en criant aux trois soldats :\n\n\u2014 Commencez l'op\u00e9ration ! Entr\u00e9e 1 et Entr\u00e9e 2, levez le drapeau que vous voulez. Bien ! Levez ! Bien ! Levez encore ! Levez !\n\nEntr\u00e9e 1 et Entr\u00e9e 2 avaient renouvel\u00e9 l'op\u00e9ration trois fois de suite. La premi\u00e8re avait donn\u00e9 : noir-noir, la deuxi\u00e8me blanc-noir, et la troisi\u00e8me noir-blanc. Sortie eut chaque fois la bonne r\u00e9action en levant respectivement une fois son drapeau noir et deux fois son drapeau blanc.\n\n\u2014 Parfait ! Tout a \u00e9t\u00e9 parfait. Votre Altesse, vos soldats sont tr\u00e8s intelligents !\n\n\u2014 C'est une chose dont m\u00eame de parfaits idiots seraient capables ! Pouvez-vous m'expliquer ce que vous \u00eates en train de faire ?\n\n\u2014 Ces trois hommes forment l'une des composantes d'un immense syst\u00e8me de calcul. Une sorte de porte, une porte \"ET\".\n\nVon Neumann fit une pause afin de bien laisser \u00e0 l'empereur le temps de comprendre ces paroles.\n\n\u2014 Je ne suis pas convaincu. Mais soit, poursuivez, dit Qin Shi Huang, sans laisser filtrer la moindre \u00e9motion.\n\nVon Neumann se retourna vers le triangle de soldats :\n\n\u2014 Nous allons maintenant cr\u00e9er la prochaine porte : toi, Sortie, il suffit qu'Entr\u00e9e 1 ou Entr\u00e9e 2 l\u00e8vent au moins un drapeau noir pour que tu l\u00e8ves aussi le tien. C'est-\u00e0-dire trois situations : noir-noir, blanc-noir ou noir-blanc ; il en reste une : blanc-blanc pour laquelle tu l\u00e8veras le drapeau blanc. As-tu compris ? Brave gar\u00e7on, tu es intelligent. Tu es l'\u00e9l\u00e9ment essentiel du fonctionnement de cette porte. Agis bien et ton empereur te r\u00e9compensera ! Commen\u00e7ons : levez ! Bien, levez encore ! Tr\u00e8s bien ! Tout se passe \u00e0 merveille, Votre Altesse. Cette porte est appel\u00e9e porte \"OU\".\n\nPar la suite, von Neumann fit encore former aux trois soldats les portes \"NON-ET\", \"OU exclusif\", \"NON-OU exclusif\" et une \"porte \u00e0 trois \u00e9tats\". Puis, pour finir, il se servit de seulement deux hommes pour mettre en place la porte la plus simple, la porte \"NON\", en demandant \u00e0 Sortie de lever le drapeau de la couleur contraire \u00e0 celle d'Entr\u00e9e 1.\n\nVon Neumann s'inclina devant l'empereur et lui dit :\n\n\u2014 Votre Altesse, nous avons maintenant termin\u00e9 de faire la d\u00e9monstration de toutes les portes logiques existantes. C'est simple, n'est-ce pas ? N'importe quel groupe de trois soldats pourra \u00eatre en mesure de comprendre leur fonctionnement en moins d'une heure.\n\n\u2014 Ils n'auront rien besoin d'apprendre de plus ? demanda Qin Shi Huang.\n\n\u2014 Non. Nous allons constituer dix millions de portes de ce type, puis les assembler en syst\u00e8me. Ce syst\u00e8me sera capable de mener tous les calculs n\u00e9cessaires \u00e0 la r\u00e9solution des \u00e9quations diff\u00e9rentielles qui nous permettront de pr\u00e9dire les mouvements des trois soleils. Nous appelons ce syst\u00e8me un... euh, un...\n\n\u2014 Un ordinateur, souffla Wang Miao.\n\n\u2014 Voil\u00e0, un ordinateur, fit von Neumann en levant son pouce vers Wang Miao. Un \"ordinateur\", ce nom est tr\u00e8s bien. Tout ce syst\u00e8me formera une gigantesque machine, la plus sophistiqu\u00e9e ayant jamais exist\u00e9 dans l'histoire du monde.\n\nLe temps du jeu s'acc\u00e9l\u00e9ra. Trois mois s'\u00e9coul\u00e8rent.\n\nQin Shi Huang, Newton, von Neumann et Wang Miao \u00e9taient debout sur la plateforme au sommet de la pyramide. C'\u00e9tait la m\u00eame plateforme que lors de sa rencontre avec Mozi. Elle \u00e9tait jonch\u00e9e d'un grand nombre d'instruments astronomiques, dont certains dataient de l'\u00e9poque moderne. Au pied de la pyramide, les trente millions d'unit\u00e9s d'infanterie de la grande arm\u00e9e des Qin avaient \u00e9t\u00e9 r\u00e9parties dans un carr\u00e9 d'un p\u00e9rim\u00e8tre de trente-six kilom\u00e8tres. Lorsque le soleil se leva, les soldats se fig\u00e8rent, comme un tapis g\u00e9ant compos\u00e9 de trente millions de soldats en terre cuite. Mais lorsqu'une nu\u00e9e d'oiseaux ayant pris l'arm\u00e9e pour un v\u00e9ritable tapis vint voler \u00e0 sa hauteur, les volatiles sentirent aussit\u00f4t la puissante odeur de mort qui se d\u00e9gageait de sous leurs ailes. Saisis d'effroi, ils reprirent de la hauteur et tournoy\u00e8rent autour.\n\nWang Miao fit quelques calculs mentaux. Si toute la population de la Terre devait un jour \u00eatre ainsi r\u00e9unie, elle n'occuperait pas une surface plus grande que le district de Pudong \u00e0 Shanghai. Au-del\u00e0 de cette d\u00e9monstration de puissance, cette arm\u00e9e r\u00e9v\u00e9lait aussi toute la fragilit\u00e9 de la civilisation.\n\n\u2014 Votre Imp\u00e9riale Grandeur, votre arm\u00e9e est sans rivale ! Dans un temps aussi court, nous avons pu achever de tels exercices, dit von Neumann au souverain sur un ton flatteur.\n\n\u2014 M\u00eame si le syst\u00e8me semble complexe, la mission de chaque soldat est tr\u00e8s simple, surtout en comparaison des exercices qu'ils ont d\u00fb effectuer avant notre bataille contre les phalanges mac\u00e9doniennes, r\u00e9pliqua Qin Shi Huang en empoignant le manche de sa longue \u00e9p\u00e9e.\n\n\u2014 Dieu soit lou\u00e9, nous avons connu deux p\u00e9riodes cons\u00e9cutives de longues \u00e8res r\u00e9guli\u00e8res, dit Newton.\n\n\u2014 Mon arm\u00e9e peut s'entra\u00eener m\u00eame en p\u00e9riode d'\u00e8re chaotique. \u00c8re r\u00e9guli\u00e8re ou non, ils m\u00e8neront \u00e0 bien vos calculs, dit Qin Shi Huang en embrassant fi\u00e8rement son arm\u00e9e du regard.\n\n\u2014 Dans ce cas, que Votre Altesse imp\u00e9riale donne le grand ordre ! dit von Neumann d'une voix tremblante d'excitation.\n\nQin Shi Huang hocha la t\u00eate. Un garde accourut \u00e0 sa hauteur, tira le manche de l'\u00e9p\u00e9e de son empereur et recula de quelques pas. L'\u00e9p\u00e9e de bronze de Qin Shi Huang \u00e9tait en effet si longue qu'il ne pouvait la retirer seul de son fourreau. Le garde s'agenouilla et tendit l'\u00e9p\u00e9e \u00e0 son empereur qui la saisit et la brandit en l'air en clamant d'une voix puissante :\n\n\u2014 En formation d'ordinateur !\n\nAux quatre coins de la pyramide, des tripodes en bronze s'enflamm\u00e8rent simultan\u00e9ment. Une troupe de soldats post\u00e9e devant la pyramide et faisant face \u00e0 l'arm\u00e9e reporta l'ordre du souverain en chantant en ch\u0153ur :\n\n\u2014 En formation d'ordinateur !\n\nAu sol, les couleurs des unit\u00e9s d'infanterie commenc\u00e8rent \u00e0 s'agiter. On vit \u00e9merger des motifs sophistiqu\u00e9s. Dix minutes plus tard, l'arm\u00e9e avait cr\u00e9\u00e9 une carte m\u00e8re de trente-six kilom\u00e8tres carr\u00e9s.\n\nEn pointant ce gigantesque circuit humain du doigt, von Neumann commen\u00e7a son explication :\n\n\u2014 Votre Majest\u00e9, nous avons donn\u00e9 \u00e0 cette machine \u00e0 calculer le nom de \"Qin I\". Voyez, au centre, se trouve l'unit\u00e9 centrale de traitement, la composante clef de tout le syst\u00e8me, constitu\u00e9e par vos cinq meilleurs r\u00e9giments. En suivant ce sch\u00e9ma de r\u00e9f\u00e9rence, vous pouvez distinguer les additionneurs, les registres, la pile, et ainsi de suite. Toute la partie autour, qui para\u00eet r\u00e9guli\u00e8re, constitue la m\u00e9moire. Au moment de composer cet \u00e9l\u00e9ment, nous nous sommes rendu compte qu'il ne restait plus assez d'hommes. Mais heureusement, les actions effectu\u00e9es au sein de ce module sont les plus faciles et nous avons entra\u00een\u00e9 les soldats \u00e0 utiliser d'autres drapeaux de couleur. Chaque homme est maintenant capable d'effectuer une action qui en aurait initialement demand\u00e9 vingt. Cela nous a permis d'augmenter la capacit\u00e9 de m\u00e9moire minimale pour faire fonctionner le syst\u00e8me d'exploitation \"Qin 1.0\". Voyez aussi cette all\u00e9e qui passe \u00e0 travers toutes les unit\u00e9s et cette cavalerie l\u00e9g\u00e8re qui y attend nos ordres. C'est le canal de communication du syst\u00e8me, la cavalerie est charg\u00e9e de transmettre les informations \u00e0 toutes les composantes du syst\u00e8me.\n\nCe canal de communication est une grande invention. Chaque nouveau module d'extension \u2013 qui peut \u00eatre cr\u00e9\u00e9 pour un maximum de dix r\u00e9giments \u2013 sera rapidement ajout\u00e9 par le biais du canal. Cette composante permettra une expansion et une mise \u00e0 jour facile de \"Qin 1.0\". Un peu plus loin, de ce c\u00f4t\u00e9 \u2013 peut-\u00eatre vous faut-il utiliser les jumelles \u2013, c'est la m\u00e9moire externe, que nous avons appel\u00e9e \"disque dur\", sur la suggestion de Copernic. Cette composante est constitu\u00e9e des trois millions de soldats les plus cultiv\u00e9s de votre arm\u00e9e. Il faut croire que vous avez eu raison d'\u00e9pargner ceux-ci lors de l'\u00e9pisode de l'enterrement des lettr\u00e9s2. Ils sont \u00e9quip\u00e9s d'un crayon et d'un carnet sur lequel ils sont charg\u00e9s de consigner les r\u00e9sultats des calculs. Bien entendu, l'essentiel de leur travail consiste \u00e0 agir comme une m\u00e9moire virtuelle en stockant des r\u00e9sultats interm\u00e9diaires. C'est dans ces composantes que s'exprime toute la difficult\u00e9 de la vitesse du calcul. Plus pr\u00e8s de nous, ici, c'est le module d'affichage. Il a pour r\u00f4le de nous montrer en temps r\u00e9el les principaux param\u00e8tres de l'ordinateur.\n\nVon Neumann et Newton offrirent \u00e0 Qin Shi Huang un immense rouleau de parchemin, de la taille d'un homme. Quand le parchemin fut enti\u00e8rement d\u00e9roul\u00e9, Wang Miao poussa un soupir de soulagement car, contrairement \u00e0 la l\u00e9gende de l'empereur et de l'assassin, aucun poignard n'y avait \u00e9t\u00e9 dissimul\u00e9. Devant leurs yeux s'\u00e9tendait seulement un parchemin immens\u00e9ment long recouvert de symboles de la taille d'une mouche et si denses qu'ils semblaient aussi \u00e9tourdissants que les formations au pied de la pyramide.\n\n\u2014 Votre Majest\u00e9, je vous pr\u00e9sente le syst\u00e8me d'exploitation \"Qin 1.0\", que nous avons mis au point. Que Sa Majest\u00e9 veuille bien regarder. (Von Neumann lui montra les formations de l'ordinateur humain.) Vos troupes constituent le mat\u00e9riel de ce syst\u00e8me, tandis que ce parchemin en est le logiciel. La relation qui unit logiciel et mat\u00e9riel est la m\u00eame que celle entre une cithare _qin_ et ses partitions.\n\nTandis qu'il parlait, Newton et lui \u00e9tal\u00e8rent sur le sol un nouveau rouleau de parchemin, aussi grand que le premier.\n\n\u2014 Votre Majest\u00e9, ce logiciel utilise des m\u00e9thodes num\u00e9riques pour r\u00e9soudre les \u00e9quations diff\u00e9rentielles. Une fois entr\u00e9 les vecteurs de mouvement des trois soleils \u00e0 un moment particulier tels qu'ils ont \u00e9t\u00e9 obtenus par des observations astronomiques, le logiciel va nous permettre de pr\u00e9dire les prochains mouvements des soleils \u00e0 n'importe quel moment futur. Nos premiers calculs nous permettront de proph\u00e9tiser les mouvements du soleil durant les deux prochaines ann\u00e9es. L'intervalle de temps entre chaque pr\u00e9diction sera de cent vingt heures.\n\nL'empereur hocha la t\u00eate :\n\n\u2014 Bien. Commencez.\n\nVon Neumann leva les deux mains au-dessus de sa t\u00eate et cria c\u00e9r\u00e9monieusement :\n\n\u2014 Sur ordre de Son Altesse imp\u00e9riale, allumez l'ordinateur ! Autotest du syst\u00e8me !\n\nPerch\u00e9s \u00e0 mi-hauteur de la pyramide, des soldats agit\u00e8rent des drapeaux pour transmettre l'ordre qui venait d'\u00eatre donn\u00e9. Apr\u00e8s un moment, l'immense carte m\u00e8re form\u00e9e par les trente millions de soldats de l'empire des Qin parut se liqu\u00e9fier et p\u00e9tiller de lumi\u00e8res vacillantes, des millions de drapeaux s'\u00e9taient mis \u00e0 fr\u00e9tiller. Au sein du module d'affichage, situ\u00e9 non loin de la pyramide, une barre de progr\u00e8s constitu\u00e9e d'innombrables drapeaux verts se remplit progressivement pour indiquer l'\u00e9tat d'avancement de l'autotest. Dix minutes plus tard, la barre de progr\u00e8s \u00e9tait pleine.\n\n\u2014 Autotest termin\u00e9 ! Amor\u00e7age ! Chargement du syst\u00e8me d'exploitation !\n\nSur le sol, la cavalerie l\u00e9g\u00e8re traversa les groupes en formation comme un cours d'eau tumultueux qui se ramifia en minces et innombrables affluents pour s'insinuer au c\u0153ur de chaque module. Rapidement, les faibles ondulations des drapeaux noirs et blancs se chang\u00e8rent en vagues furieuses qui d\u00e9ferl\u00e8rent \u00e0 la surface de la carte m\u00e8re. La zone occup\u00e9e par l'unit\u00e9 centrale paraissait la plus houleuse, comme si on avait mis le feu \u00e0 une tra\u00een\u00e9e de poudre.\n\nMais soudain, le tourbillon de flammes temp\u00e9tueuses de l'unit\u00e9 centrale sembla s'apaiser et finit m\u00eame par \u00eatre totalement \u00e9touff\u00e9. Partant de l'unit\u00e9 centrale, ce calme soudain se r\u00e9pandit sur l'ensemble comme la surface d'une mer gelant \u00e0 toute vitesse. Bient\u00f4t ce fut toute la carte m\u00e8re qui s'immobilisa, hormis quelques r\u00e9manences sans vie de composantes continuant \u00e0 clignoter en boucle. Le module d'affichage clignotait en rouge.\n\n\u2014 Verrouillage du syst\u00e8me ! cria l'officier en charge des signaux.\n\nOn d\u00e9termina vite la raison du dysfonctionnement : le registre de l'unit\u00e9 centrale indiquait qu'une erreur avait \u00e9t\u00e9 commise lors de l'op\u00e9ration d'une des portes.\n\n\u2014 Red\u00e9marrage du syst\u00e8me ! ordonna von Neumann avec assurance.\n\n\u2014 Attendez ! s'\u00e9cria Newton, en faisant signe \u00e0 l'officier en charge des signaux.\n\nPuis il se tourna avec un air perfide vers l'empereur :\n\n\u2014 Votre Altesse, je crois que pour am\u00e9liorer la stabilit\u00e9 du syst\u00e8me, vous devriez proc\u00e9der \u00e0 une mesure de remplacement des composantes d\u00e9fectueuses.\n\nQin Shi Huang empoigna son \u00e9p\u00e9e :\n\n\u2014 Remplacez la porte ayant pr\u00e9sent\u00e9 des signes de dysfonctionnement et tranchez la t\u00eate \u00e0 ses soldats ! Dor\u00e9navant, qu'on se le dise, toutes les portes \u00e0 l'origine d'une erreur conna\u00eetront le m\u00eame sort !\n\nVon Neumann toisa Newton avec un regard de d\u00e9go\u00fbt. Ils virent des cavaliers arm\u00e9s se faufiler au c\u0153ur de la carte m\u00e8re et \"r\u00e9parer\" la composante d\u00e9fectueuse d'un coup d'\u00e9p\u00e9e. On donna l'ordre de red\u00e9marrer le syst\u00e8me. Cette fois, tout se passa sans obstacle. Vingt minutes plus tard, le syst\u00e8me \"Qin 1.0\", cet ordinateur humain imagin\u00e9 par von Neumann dans le monde des _Trois Corps_ , \u00e9tait op\u00e9rationnel.\n\n\u2014 Lancement du logiciel de calcul de l'orbite solaire \"Trois Corps 1.0\" ! cria von Neumann d'une voix vibrante. Pr\u00e9paration des commandes de calcul ! Chargement du module de calcul diff\u00e9rentiel ! Chargement du module d'analyse des \u00e9l\u00e9ments finis ! Chargement du module d'analyse spectrale ! D\u00e9finition des param\u00e8tres de conditions initiales ! D\u00e9marrage du calcul !\n\nLa carte m\u00e8re se remit \u00e0 onduler, tandis que les signaux de couleur du module d'affichage commenc\u00e8rent \u00e0 clignoter. L'ordinateur humain entra dans une p\u00e9riode de calcul qui durerait longtemps.\n\n\u2014 Tout cela est tr\u00e8s int\u00e9ressant, dit Qin Shi Huang en pointant du doigt cette machine spectaculaire. Chaque unit\u00e9 a une action tr\u00e8s simple \u00e0 r\u00e9aliser, mais l'ensemble est d'une tr\u00e8s grande complexit\u00e9 ! Les Europ\u00e9ens aiment me critiquer en disant que mon r\u00e8gne est tyrannique, que je brise tout \u00e9lan de cr\u00e9ativit\u00e9. Mais je vous le dis, une foule d'individus sous le joug d'un r\u00e9gime strict peut faire preuve, une fois r\u00e9unie, d'une grande sagesse !\n\n\u2014 Votre Imp\u00e9riale Grandeur, ce n'est pas de la sagesse, vos soldats agissent comme des machines. Individuellement, ce sont tous des 0, ce n'est que lorsqu'ils sont guid\u00e9s par un 1 comme Votre Majest\u00e9 que la totalit\u00e9 prend son sens, susurra Newton avec un sourire flagorneur.\n\n\u2014 Une philosophie ignoble, l\u00e2cha von Neumann en jetant un regard sur Newton. Si, \u00e0 la fin, les pr\u00e9dictions obtenues gr\u00e2ce \u00e0 votre th\u00e9orie et \u00e0 votre mod\u00e8le math\u00e9matique ne s'accordent pas avec la r\u00e9alit\u00e9, vous et moi nous serons moins que des 0.\n\n\u2014 En effet. Si tel \u00e9tait le cas, vous seriez des bons \u00e0 rien, dit Qin Shi Huang qui s'\u00e9loigna dans un geste de manche.\n\nLe temps passa \u00e0 toute vitesse, l'ordinateur humain fonctionna durant un an et quatre mois. En faisant la soustraction du temps qui avait \u00e9t\u00e9 n\u00e9cessaire pour param\u00e9trer le programme, le temps de calcul r\u00e9el fut d'un an et deux mois. Pendant cette p\u00e9riode, l'op\u00e9ration fut interrompue \u00e0 seulement deux reprises par le d\u00e9r\u00e8glement du climat lors d'une \u00e8re chaotique, mais le registre avait pu sauvegarder les donn\u00e9es recueillies avant chaque interruption et le calcul put reprendre sans encombre lors du retour \u00e0 une \u00e8re r\u00e9guli\u00e8re. Quand Qin Shi Huang et les savants europ\u00e9ens se retrouv\u00e8rent \u00e0 nouveau sur la plateforme de la pyramide, la premi\u00e8re \u00e9tape du calcul \u00e9tait achev\u00e9e. Les donn\u00e9es r\u00e9colt\u00e9es allaient permettre de pr\u00e9dire avec pr\u00e9cision les orbites solaires pour les deux prochaines ann\u00e9es.\n\nL'aube \u00e9tait glaciale. Les myriades de torches qui avaient \u00e9clair\u00e9 la gigantesque carte m\u00e8re durant des mois \u00e9taient maintenant \u00e9teintes. Son calcul termin\u00e9, \"Qin 1.0\" se mit en veille. Les vagues d\u00e9ferlant \u00e0 sa surface n'\u00e9taient plus d\u00e9sormais que de l\u00e9gers clapotis.\n\nVon Neumann et Newton offrirent au souverain le rouleau de parchemin consignant les r\u00e9sultats des calculs. Newton prit la parole :\n\n\u2014 Votre Altesse imp\u00e9riale, les calculs se sont termin\u00e9s il y a trois jours. Nous avons attendu jusqu'\u00e0 maintenant pour vous pr\u00e9senter les r\u00e9sultats car ils ont r\u00e9v\u00e9l\u00e9 que cette interminable nuit froide allait bient\u00f4t prendre fin. Nous accueillerons d'ici peu le premier lever de soleil d'une longue \u00e8re r\u00e9guli\u00e8re. Cette \u00e8re durera toute une ann\u00e9e et, \u00e0 en juger par les param\u00e8tres orbitaux des soleils, le climat sera doux et agr\u00e9able. Que Votre Majest\u00e9 donne l'ordre de r\u00e9hydrater les sujets de votre empire !\n\n\u2014 Depuis le d\u00e9but de ce calcul, aucun sujet de mon empire n'a \u00e9t\u00e9 d\u00e9shydrat\u00e9, grogna Qin Shi Huang en se saisissant du parchemin. Toute la population de l'empire a \u00e9t\u00e9 employ\u00e9e \u00e0 maintenir le bon fonctionnement de l'ordinateur humain. Les caisses sont vides. Un grand nombre de mes sujets sont morts de faim, de fatigue et de froid pour cet ordinateur, rajouta l'empereur en pointant le lointain avec le rouleau de parchemin.\n\nDans la p\u00e2le lumi\u00e8re de l'aube, on pouvait distinguer aux marges de la carte m\u00e8re des dizaines de lignes blanches rayonnantes qui partaient dans toutes les directions et disparaissaient \u00e0 l'horizon : les routes d'approvisionnement de la carte m\u00e8re qui reliaient chaque r\u00e9gion de l'empire.\n\n\u2014 Votre Altesse, vous constaterez bient\u00f4t que ces sacrifices n'auront pas \u00e9t\u00e9 vains, dit von Neumann. Lorsque vous aurez ma\u00eetris\u00e9 les orbites des soleils, l'empire des Qin conna\u00eetra un d\u00e9veloppement rapide et florissant et prosp\u00e9rera bien plus rapidement qu'avant le calcul.\n\n\u2014 Si l'on en croit les calculs, le soleil ne devrait pas tarder \u00e0 se lever \u00e0 pr\u00e9sent. Votre Grandeur, pr\u00e9parez-vous \u00e0 rayonner de gloire !\n\nComme en r\u00e9ponse aux paroles de Newton, l'\u00e9clat d'un soleil rouge surgit \u00e0 l'horizon et baigna la pyramide et l'ordinateur humain d'une lumi\u00e8re d'or. Des vivats mont\u00e8rent de la carte m\u00e8re.\n\nC'est alors qu'un homme accourut \u00e0 leur rencontre. Il devait \u00eatre totalement \u00e9puis\u00e9 car, au moment de s'agenouiller devant son souverain, il s'affala sur le sol et peina \u00e0 reprendre sa respiration. C'\u00e9tait le ministre de l'Astronomie de l'empire des Qin.\n\n\u2014 Sire, c'est une catastrophe, les calculs sont faux ! Nous allons au-devant d'un d\u00e9sastre ! g\u00e9mit-il.\n\n\u2014 Que racontes-tu donc, ministre ?\n\nSans m\u00eame attendre la r\u00e9ponse de l'empereur, Newton faucha le ministre d'un coup de pied :\n\n\u2014 Eh bien, maraud, n'as-tu donc pas vu que le soleil se levait, exactement comme l'ont pr\u00e9dit les r\u00e9sultats de nos calculs ?\n\n\u2014 Mais... (Le ministre se releva \u00e0 moiti\u00e9, sa main point\u00e9e vers le soleil.) Combien de soleils voyez-vous ?\n\nTous lev\u00e8rent aussit\u00f4t les yeux, d\u00e9sorient\u00e9s.\n\n\u2014 Allons, monsieur le ministre, vous avez re\u00e7u une \u00e9ducation occidentale et vous \u00eates m\u00eame titulaire d'un doctorat de l'universit\u00e9 de Cambridge, vous n'\u00eates pas idiot au point de ne pas savoir compter ! Vous voyez bien qu'il n'y a qu'un seul soleil. Et la temp\u00e9rature est tr\u00e8s agr\u00e9able, temp\u00e9ra von Neumann.\n\n\u2014 Non, il y en a trois ! sanglota le ministre. Les deux autres sont derri\u00e8re le premier !\n\nTous regard\u00e8rent une nouvelle fois le soleil, encore dubitatifs.\n\n\u2014 La conclusion de l'Observatoire imp\u00e9rial est sans \u00e9quivoque. Nous assistons en ce moment m\u00eame \u00e0 un ph\u00e9nom\u00e8ne rarissime : une syzygie trisolaire. Cela signifie que les trois soleils sont parfaitement align\u00e9s et se meuvent autour de la Terre avec la m\u00eame vitesse angulaire ! Notre plan\u00e8te et les trois soleils se retrouvent ainsi sur une m\u00eame ligne droite ! Et notre plan\u00e8te est au bout de cette ligne !\n\n\u2014 Es-tu certain de n'avoir fait aucune erreur ? demanda Newton en empoignant le ministre par le col.\n\n\u2014 Absolument certain ! L'observation a \u00e9t\u00e9 men\u00e9e par des astronomes occidentaux de l'Observatoire imp\u00e9rial, dont Johannes Kepler et William Herschel. Ils ont utilis\u00e9 le plus grand t\u00e9lescope du monde, directement import\u00e9 d'Europe.\n\nNewton rel\u00e2cha son \u00e9treinte sur le ministre qui put se relever. Wang Miao remarqua que le visage du savant avait bl\u00eami, mais il avait l'air de ne pas avoir perdu son entrain. Il croisa les bras sur son torse et s'adressa \u00e0 Qin Shi Huang :\n\n\u2014 Grande et Honorable Majest\u00e9, c'est le plus favorable de tous les pr\u00e9sages ! Voici que les trois soleils sont en orbite autour de notre plan\u00e8te ! Votre empire est devenu le centre de l'univers ! C'est Dieu qui nous r\u00e9compense de nos efforts ! Donnez-moi encore un peu de temps pour revoir les r\u00e9sultats de nos calculs et j'en donnerai la preuve !\n\nNewton profita de la sid\u00e9ration g\u00e9n\u00e9rale pour filer en douce. Plus tard, on annon\u00e7a \u00e0 l'empereur que Sir Isaac avait vol\u00e9 un cheval et s'\u00e9tait enfui sans demander son reste.\n\nWang Miao laissa passer un silence empreint de nervosit\u00e9, puis il prit la parole :\n\n\u2014 Majest\u00e9, d\u00e9gainez votre \u00e9p\u00e9e.\n\n\u2014 Pourquoi donc ? demanda Qin Shi Huang, perplexe.\n\nMais il fit tout de m\u00eame signe \u00e0 son serviteur de l'aider \u00e0 sortir son arme et celui-ci s'ex\u00e9cuta aussit\u00f4t.\n\n\u2014 Faites-la tournoyer en l'air, dit Wang Miao.\n\nQin Shi Huang prit l'\u00e9p\u00e9e et fit quelques moulinets. Une expression de surprise envahit ses traits.\n\n\u2014 \u00c7a alors, pourquoi est-elle si l\u00e9g\u00e8re ?\n\n\u2014 Les capteurs de nos V-combinaison ne permettent pas de ressentir l'apesanteur, sinon nous nous sentirions beaucoup plus l\u00e9gers.\n\n\u2014 Regardez ! En bas ! Regardez ces chevaux, regardez ces hommes ! cria quelqu'un.\n\nIls se pr\u00e9cipit\u00e8rent au bord de la plateforme pour voir ce qui se passait. Au pied de la pyramide, ils virent un r\u00e9giment de cavalerie au galop dont les montures paraissaient flotter dans les airs. Les chevaux parcoururent une longue distance en apesanteur avant de retomber les fers sur le sol. Des hommes en train de courir firent des bonds de plus de dix m\u00e8tres, avant de retomber lentement sur leurs pieds. Sur la pyramide, un garde essaya de sauter et put atteindre sans peine une hauteur de trois m\u00e8tres.\n\n\u2014 Quel est ce mal\u00e9fice ? dit l'empereur en observant avec frayeur le garde retomber apr\u00e8s un bond vertigineux.\n\n\u2014 Sire, les trois soleils forment maintenant une ligne droite avec notre plan\u00e8te. Par cons\u00e9quent, leurs attractions gravitationnelles s'ajoutent les unes aux autres dans la m\u00eame direction..., expliqua le ministre de l'Astronomie, en s'apercevant que ses deux pieds avaient d\u00e9j\u00e0 quitt\u00e9 le sol.\n\nIl se trouvait maintenant \u00e0 l'horizontale dans l'air.\n\nLes autres flottaient aussi dans l'atmosph\u00e8re, inclin\u00e9s selon des angles diff\u00e9rents. Plus personne n'avait les pieds sur la terre ferme, on aurait cru voir un groupe de noy\u00e9s ne sachant pas nager. Ils agitaient maladroitement leurs quatre membres pour trouver un \u00e9quilibre mais se cognaient les uns aux autres.\n\n\u00c0 cet instant pr\u00e9cis, le sol qu'ils venaient tout juste de quitter se fissura comme une toile d'araign\u00e9e. Les fissures s'\u00e9largirent rapidement et, accompagn\u00e9e d'un brouillard de mortier et du terrible fracas d'un effondrement, la pyramide se disloqua en d'innombrables blocs de pierre qui avaient jadis servi \u00e0 la b\u00e2tir. Au travers de ces amas pierreux qui s'\u00e9taient mis \u00e0 flotter dans les airs, Wang Miao put voir la salle d'audience en train de se d\u00e9sagr\u00e9ger. Le tripode qui avait nagu\u00e8re servi \u00e0 cuire Fu Xi et la croix sur laquelle il avait \u00e9t\u00e9 ligot\u00e9 flottaient en plein milieu de la salle.\n\nLe soleil atteignit son z\u00e9nith. Tout ce qui flottait : les hommes, les \u00e9normes blocs de pierre, les instruments astronomiques, les tripodes en bronze, commen\u00e7a \u00e0 s'\u00e9lever lentement dans les airs puis leur allure acc\u00e9l\u00e9ra rapidement. Par hasard, Wang Miao jeta un \u0153il \u00e0 l'ordinateur humain et vit une sc\u00e8ne de cauchemar : les trente millions d'hommes qui composaient la carte m\u00e8re l\u00e9vitaient dans les airs, comme une colonie de fourmis aspir\u00e9e par un tuyau d'aspirateur. Sur la surface apparaissaient clairement les empreintes des circuits de la carte m\u00e8re. Cette s\u00e9rie de motifs fins et complexes ne pouvait \u00eatre int\u00e9gralement observ\u00e9e qu'\u00e0 une certaine hauteur et deviendrait pour la prochaine civilisation des _Trois Corps_ qui na\u00eetrait bien plus tard un \u00e9nigmatique site arch\u00e9ologique.\n\nWang Miao leva les yeux. Le ciel \u00e9tait couvert d'une \u00e9trange couche de nuages marbr\u00e9s, des nuages compos\u00e9s de poussi\u00e8re, de pierres ou encore de corps humains, derri\u00e8re lesquels resplendissait le soleil.\n\nAu loin, Wang Miao vit encore une longue cha\u00eene de montagnes transparentes en train de s'\u00e9lever dans les cieux ; elles \u00e9taient aussi claires que du cristal, et elles changeaient de forme, baign\u00e9es d'une lumi\u00e8re chatoyante. C'\u00e9tait en r\u00e9alit\u00e9 l'oc\u00e9an qui se faisait aspirer par l'espace !\n\nTout ce qui se trouvait \u00e0 la surface du monde des _Trois Corps_ \u00e9tait aspir\u00e9 par le soleil.\n\nWang Miao regarda autour de lui. Il vit von Neumann et Qin Shi Huang. Von Neumann, qui flottait dans les airs, cria quelque chose \u00e0 l'empereur, mais aucun son ne sortait de sa bouche. Des sous-titres apparurent \u00e0 l'\u00e9cran :\n\n> J'ai une id\u00e9e ! Des composants \u00e9lectroniques ! Nous devrions utiliser des composants \u00e9lectroniques pour faire nos portes logiques et les combiner en ordinateur ! De tels ordinateurs seraient infiniment plus rapides et prendraient bien moins d'espace, j'estime qu'un petit b\u00e2timent devrait suffire \u00e0 les entreposer... Votre Altesse, m'\u00e9coutez-vous ?\n\nQin Shi Huang fit tournoyer son \u00e9p\u00e9e en direction de von Neumann. Ce dernier donna un coup de pied dans un bloc de pierre qui passa \u00e0 c\u00f4t\u00e9 de lui et para le coup. L'arme heurta la pierre, qui se brisa en deux et fit des \u00e9tincelles. Apr\u00e8s quoi, la pierre entra en collision avec une autre et l'empereur se retrouva pris en sandwich entre eux deux. Ils assist\u00e8rent au monstrueux spectacle de gravats et de sang se r\u00e9pandant \u00e0 l'horizontale.\n\nCurieusement, Wang n'entendit pas le bruit de l'impact. Autour de lui, tout \u00e9tait d'un silence de mort. L'atmosph\u00e8re ayant disparu, il n'entendait plus aucun son. Sous l'effet de la pression nulle, le sang des hommes se mit \u00e0 bouillir et ils crach\u00e8rent leurs visc\u00e8res. Bient\u00f4t, ils ne furent plus que d'\u00e9tranges masses indistinctes surplomb\u00e9es de nuages cristallins form\u00e9s par le liquide qui s'\u00e9tait \u00e9chapp\u00e9 de leurs corps. La disparition de l'atmosph\u00e8re avait obscurci le ciel. Tout ce qui flottait maintenant dans l'espace refl\u00e9tait la lumi\u00e8re du soleil et composait une sorte de n\u00e9buleuse brillante. Cette n\u00e9buleuse se transforma en un \u00e9norme tourbillon qui s'\u00e9leva en spirale jusqu'\u00e0 son ultime demeure : le soleil.\n\nWang Miao s'aper\u00e7ut que ce dernier changeait \u00e0 pr\u00e9sent de forme. Il comprit aussit\u00f4t qu'il voyait en r\u00e9alit\u00e9 les deux autres soleils, qui r\u00e9v\u00e9laient derri\u00e8re le premier une partie de leur silhouette. D'o\u00f9 il se trouvait, les trois soleils align\u00e9s figuraient l'\u0153il flamboyant de l'univers.\n\nUn texte apparut devant le d\u00e9cor de la syzygie des trois soleils :\n\n> La civilisation no 184 a \u00e9t\u00e9 d\u00e9truite sous l'accumulation des attractions gravitationnelles d'une syzygie trisolaire. Cette civilisation avait atteint l'\u00e8re de la r\u00e9volution scientifique et de la r\u00e9volution industrielle. Dans cette civilisation, Newton a formul\u00e9 les principes de la m\u00e9canique classique d\u00e9crivant les mouvements des corps se d\u00e9pla\u00e7ant \u00e0 des vitesses faibles. Dans le m\u00eame temps, gr\u00e2ce \u00e0 l'invention du calcul infinit\u00e9simal et celle de l'architecture informatique par von Neumann, ont \u00e9t\u00e9 pos\u00e9es les bases d'une analyse math\u00e9matique quantitative du mouvement des trois corps.\n> \n> Apr\u00e8s une longue p\u00e9riode, les graines de la civilisation seront replant\u00e9es et, une fois encore, le monde si impr\u00e9visible des _Trois Corps_ verra se poursuivre son \u00e9trange destin\u00e9e. Nous vous invitons \u00e0 vous reconnecter dans le futur.\n\nWang Miao venait tout juste de quitter le jeu lorsqu'il re\u00e7ut un appel inconnu. Il entendit une voix masculine, envo\u00fbtante :\n\n\u2014 Bonsoir. Tout d'abord je vous remercie d'avoir indiqu\u00e9 votre num\u00e9ro de t\u00e9l\u00e9phone r\u00e9el. Je suis l'administrateur syst\u00e8me du jeu _Les Trois Corps_.\n\nLe c\u0153ur de Wang Miao se mit \u00e0 battre d'excitation et de nervosit\u00e9.\n\n\u2014 Puis-je vous demander votre \u00e2ge, votre formation scolaire, votre employeur et votre m\u00e9tier ? Ces informations n'apparaissent pas dans le formulaire d'inscription que vous avez rempli.\n\n\u2014 Qu'est-ce que cela a \u00e0 faire avec le jeu ?\n\n\u2014 Arriv\u00e9 \u00e0 ce niveau, vous devez fournir ces informations. En cas de refus, nous cl\u00f4turerons d\u00e9finitivement votre compte.\n\nWang Miao r\u00e9pondit avec sinc\u00e9rit\u00e9 aux questions de l'administrateur.\n\n\u2014 Parfait, professeur Wang. Vous remplissez les conditions pour continuer votre exploration du monde des _Trois Corps_.\n\n\u2014 Merci, puis-je vous poser quelques questions ? demanda Wang Miao \u00e0 la h\u00e2te.\n\n\u2014 Non. Mais demain soir a lieu une r\u00e9union de joueurs des _Trois Corps_ , vous \u00eates le bienvenu.\n\nWang Miao nota l'adresse.\n\n* * *\n\n1 \u00c0 la t\u00eate d'une puissante arm\u00e9e, le roi de Qin (259-210 av. J.-C.) conquit un \u00e0 un l'ensemble des Royaumes combattants et fonda la dynastie Qin (221-207 av. J.-C.), dont il devint l'empereur en prenant le nom de Shi Huangdi (\"Premier Auguste Souverain\"). Souvent consid\u00e9r\u00e9 comme le premier grand unificateur de la Chine imp\u00e9riale, il entreprit une standardisation de l'\u00e9criture, de la langue, des mesures ou encore de la monnaie, mais resta dans l'histoire comme un souverain sanguinaire. _(N.d.T.)_\n\n2 R\u00e9f\u00e9rence faite ici au grand autodaf\u00e9 et \u00e0 l'enterrement vivant de quatre cent soixante lettr\u00e9s confuc\u00e9ens qu'aurait ordonn\u00e9s Qin Shi Huang durant son r\u00e8gne, afin de mater toute contestation politique. _(N.d.T.)_\n\n## 18\n\n## LA R\u00c9UNION DES JOUEURS\n\nLa r\u00e9union des joueurs des _Trois Corps_ avait lieu dans un petit caf\u00e9 isol\u00e9, loin du centre-ville. Wang Miao s'imaginait que les assembl\u00e9es de joueurs de jeux vid\u00e9o \u00e9taient des repas anim\u00e9s et pleins de monde mais, ici, ils n'\u00e9taient que sept, lui compris. Comme lui, les six autres joueurs n'avaient absolument pas l'air d'\u00eatre des fans de jeux vid\u00e9o. Seuls deux d'entre eux paraissaient relativement jeunes. Trois autres \u2013 dont une femme \u2013 \u00e9taient d'\u00e2ge moyen. Le dernier, un homme, semblait avoir soixante ou soixante-dix ans.\n\nWang Miao s'\u00e9tait figur\u00e9 que, d\u00e8s qu'ils se verraient, ils se lanceraient aussit\u00f4t dans une discussion passionn\u00e9e autour des _Trois Corps_ , mais il dut se rendre \u00e0 l'\u00e9vidence : il s'\u00e9tait tromp\u00e9. La complexit\u00e9, tout comme l'\u00e9tranget\u00e9 des _Trois Corps_ , exer\u00e7ait un tel impact psychologique sur les joueurs que tous \u2013 et Wang Miao ne faisait pas exception \u2013 \u00e9taient incapables d'amener facilement le sujet sur la table. Ils se content\u00e8rent donc dans un premier temps de br\u00e8ves pr\u00e9sentations. Le vieil homme sortit une pipe exquis\u00e9ment ouvrag\u00e9e qu'il remplit avec des feuilles de tabac. Il parcourut la pi\u00e8ce en fumant et en admirant les peintures accroch\u00e9es au mur. Les autres joueurs s'assirent silencieusement en attendant la venue de l'organisateur. Ils \u00e9taient tous arriv\u00e9s en avance.\n\nWang Miao connaissait en fait deux personnes sur les six. Le vieil homme aux cheveux blancs \u00e9tait un intellectuel c\u00e9l\u00e8bre qui s'\u00e9tait fait conna\u00eetre en croisant la philosophie orientale avec des r\u00e9flexions issues de la science moderne. La femme \u00e0 l'accoutrement saugrenu \u00e9tait un \u00e9crivain c\u00e9l\u00e8bre, l'une des rares romanci\u00e8res exp\u00e9rimentales contemporaines b\u00e9n\u00e9ficiant d'un nombre \u00e9lev\u00e9 de lecteurs. L'une des particularit\u00e9s de ses livres \u00e9tait qu'on pouvait les lire \u00e0 partir de n'importe quelle page. Les deux hommes d'\u00e2ge moyen \u00e9taient respectivement vice-pr\u00e9sident de la plus grande entreprise de logiciels informatiques du pays \u2013 m\u00eame si ses v\u00eatements n\u00e9glig\u00e9s ne le laissaient pas deviner \u2013 et cadre sup\u00e9rieur dans la Compagnie nationale d'\u00e9lectricit\u00e9. Quant aux deux jeunes, l'un \u00e9tait journaliste dans une grande gazette nationale, et l'autre doctorant en sciences naturelles. Wang se fit la remarque qu'une part importante des joueurs des _Trois Corps_ devait comme eux appartenir \u00e0 l'\u00e9lite intellectuelle du pays.\n\nL'organisateur de la r\u00e9union arriva peu apr\u00e8s. Quand Wang Miao le vit, son c\u0153ur se mit \u00e0 battre \u00e0 tout rompre. Il le connaissait. C'\u00e9tait Pan Han, le principal suspect du meurtre de Shen Yufei. Wang Miao sortit son t\u00e9l\u00e9phone, le posa sur la table et envoya discr\u00e8tement un message \u00e0 Shi Qiang.\n\n\u2014 Ah ah, tr\u00e8s bien, je vois que tout le monde est arriv\u00e9 en avance ! les salua Pan Han de mani\u00e8re parfaitement d\u00e9tendue, comme si aucun bouleversement n'\u00e9tait r\u00e9cemment intervenu dans sa vie.\n\nIl avait d\u00e9laiss\u00e9 les v\u00eatements sales de vagabond qu'il arborait dans les m\u00e9dias pour un \u00e9l\u00e9gant costume deux pi\u00e8ces et des chaussures en cuir.\n\n\u2014 Vous \u00eates exactement comme je vous avais imagin\u00e9s : d'une intelligence sup\u00e9rieure. Le jeu des _Trois Corps_ a en effet \u00e9t\u00e9 con\u00e7u pour des personnes de votre calibre, capables d'appr\u00e9cier pleinement son contenu et son sens profond. Y jouer exige des connaissances pointues qui ne sont pas \u00e0 la port\u00e9e du premier venu.\n\nWang Miao envoya son message : _\"Ai localis\u00e9 Pan Han. Caf\u00e9 Rivi\u00e8re des Nuages \/ district de Xicheng.\"_\n\nPan Han poursuivit :\n\n\u2014 Vous \u00eates tous des joueurs de premier plan. Vous avez obtenu des scores impressionnants, et vous \u00eates tr\u00e8s impliqu\u00e9s dans le jeu. Si je ne me trompe pas, le jeu des _Trois Corps_ est devenu pour chacun d'entre vous une partie de votre vie.\n\n\u2014 Absolument, il fait partie int\u00e9grante de ma vie, confirma le jeune doctorant.\n\n\u2014 Je l'ai d\u00e9couvert par hasard sur l'ordinateur de mon petit-fils, ajouta le vieux philosophe en caressant le manche de sa pipe. Le gamin a abandonn\u00e9 au bout de quelques parties en disant que c'\u00e9tait trop abstrait. Mais moi je m'y suis laiss\u00e9 prendre. Ce monde imaginaire complexe, myst\u00e9rieux, effrayant et \u00e0 la fois plein de po\u00e9sie... Sa logique, son s\u00e9rieux, cette masse d'informations et de d\u00e9tails cach\u00e9s derri\u00e8re son apparence toute simple... Tout \u00e7a m'a fascin\u00e9.\n\nTout le monde, y compris Wang Miao, approuva d'un signe de t\u00eate.\n\nWang Miao re\u00e7ut la r\u00e9ponse de Shi Qiang : _\"Nous sommes au courant. Pas de panique. Faites ce qu'on vous demande. Jouez au fanatique, mais n'en faites pas trop pour ne pas vous faire pincer.\"_\n\n\u2014 Oui, dit l'\u00e9crivain. D'un point de vue litt\u00e9raire aussi, _Les Trois Corps_ est remarquable. L'histoire de l'expansion et du d\u00e9clin de ces deux cent trois civilisations est digne des plus belles \u00e9pop\u00e9es.\n\nElle avait \u00e9voqu\u00e9 deux cent trois civilisations, mais Wang Miao n'en \u00e9tait qu'\u00e0 cent quatre-vingt-quatre. Cela lui apporta la confirmation que la progression dans le jeu des _Trois Corps_ diff\u00e9rait selon le profil du joueur.\n\n\u2014 Je suis fatigu\u00e9 du monde r\u00e9el. Le monde des _Trois Corps_ est devenu ma deuxi\u00e8me r\u00e9alit\u00e9, dit le jeune journaliste.\n\n\u2014 Vraiment ? l'interrompit Pan Han, int\u00e9ress\u00e9.\n\n\u2014 Pour moi aussi. Compar\u00e9e aux _Trois Corps_ , je trouve la r\u00e9alit\u00e9 banale et fade, ajouta le vice-pr\u00e9sident des logiciels IT.\n\n\u2014 Quel dommage que ce ne soit qu'un jeu, rench\u00e9rit le cadre sup\u00e9rieur.\n\n\u2014 Tr\u00e8s int\u00e9ressant..., dit Pan Han, en hochant la t\u00eate.\n\nWang Miao remarqua que ses yeux p\u00e9tillaient d'excitation.\n\n\u2014 J'ai une question, dont je suis s\u00fbr que tout le monde ici se la pose, dit Wang Miao.\n\n\u2014 Je la connais. Mais posez-la tout de m\u00eame.\n\n\u2014 _Les Trois Corps_ n'est-il qu'un jeu ?\n\nLes autres joueurs acquiesc\u00e8rent tous. Manifestement, cette question leur br\u00fblait aussi les l\u00e8vres.\n\nPan Han se leva et annon\u00e7a avec gravit\u00e9 :\n\n\u2014 Le monde des _Trois Corps_ \u2013 Trisolaris \u2013 existe r\u00e9ellement.\n\n\u2014 O\u00f9 est-il ? demand\u00e8rent d'une m\u00eame voix \u00e9bahie plusieurs joueurs.\n\nPan Han s'assit et laissa passer un long silence avant de reprendre la parole :\n\n\u2014 Je peux r\u00e9pondre \u00e0 certaines de vos questions, mais pas \u00e0 toutes. Si votre destin est li\u00e9 \u00e0 celui de Trisolaris, un jour viendra forc\u00e9ment o\u00f9 vous obtiendrez les r\u00e9ponses \u00e0 toutes vos interrogations.\n\n\u2014 Mais alors, le jeu est-il une repr\u00e9sentation r\u00e9aliste de ce monde de Trisolaris ? demanda le journaliste.\n\n\u2014 Premi\u00e8rement, la capacit\u00e9 des Trisolariens \u00e0 se d\u00e9shydrater, comme vous avez pu l'observer dans beaucoup de civilisations, est r\u00e9elle. C'est gr\u00e2ce \u00e0 cette technique qu'ils peuvent s'adapter \u00e0 un environnement naturel impr\u00e9visible. Ils sont \u00e0 tout moment capables de se vider de l'eau de leurs corps et de devenir des entit\u00e9s s\u00e8ches et fibreuses pour \u00e9chapper \u00e0 un climat hostile.\n\n\u2014 \u00c0 quoi ressemblent les Trisolariens ?\n\nPan Han secoua la t\u00eate :\n\n\u2014 Je l'ignore. Je l'ignore r\u00e9ellement. L'apparence des Trisolariens change \u00e0 chaque nouveau cycle de civilisation. Par ailleurs, le jeu refl\u00e8te aussi un autre \u00e9l\u00e9ment r\u00e9el de Trisolaris : le fameux ordinateur \u00e0 grande \u00e9chelle.\n\n\u2014 Ha ! ha ! Je trouvais au contraire que c'\u00e9tait ce qu'il y avait de moins r\u00e9aliste ! s'exclama le vice-pr\u00e9sident d'IT. J'ai effectu\u00e9 un petit test dans mon entreprise en faisant participer plus d'une centaine de mes employ\u00e9s. M\u00eame si l'id\u00e9e para\u00eet r\u00e9alisable, la vitesse de calcul d'un ordinateur humain est encore plus lente qu'un banal calcul mental !\n\nPan Han r\u00e9pondit par un sourire \u00e9nigmatique :\n\n\u2014 Vous avez raison. Mais imaginez \u00e0 pr\u00e9sent que les trente millions de soldats formant cet ordinateur soient capables d'agiter leurs drapeaux noirs et blancs cent mille fois par seconde. Supposez encore que la cavalerie qui sert de canal de communication puisse galoper \u00e0 une vitesse bien plus grande que la vitesse du son, voire plus rapidement encore. Les r\u00e9sultats ne seraient pas les m\u00eames.\n\nVous venez de m'interroger sur l'apparence des Trisolariens. Des indices portent \u00e0 croire que le corps des Trisolariens qui composent l'ordinateur est couvert d'une surface purement r\u00e9flexive, comme un miroir. Cette particularit\u00e9 est peut-\u00eatre une \u00e9volution naturelle leur permettant de survivre \u00e0 des conditions solaires extr\u00eames. Cette surface en miroir peut prendre des formes diff\u00e9rentes et les Trisolariens parviennent \u00e0 communiquer entre eux gr\u00e2ce \u00e0 la focalisation de la lumi\u00e8re sur leurs corps. Ce langage bas\u00e9 sur la lumi\u00e8re leur permet de transmettre des informations \u00e0 une vitesse exceptionnelle et c'est cette singularit\u00e9 qui garantit la grande efficacit\u00e9 de l'ordinateur. Bien entendu, ce dispositif reste tr\u00e8s imparfait, mais il a effectivement \u00e9t\u00e9 capable de mener \u00e0 bien des calculs qui auraient \u00e9t\u00e9 impossibles \u00e0 r\u00e9aliser manuellement. Dans le monde trisolarien, les ordinateurs sont d'abord apparus sous la forme de formations collectives avant de devenir m\u00e9caniques, puis \u00e9lectroniques.\n\nPan Han se leva. Il arpenta la pi\u00e8ce et passa derri\u00e8re les joueurs :\n\n\u2014 Voici ce que je peux vous dire : en tant que jeu, _Les Trois Corps_ utilise le cadre de l'histoire humaine pour simuler le d\u00e9veloppement du monde de Trisolaris. Si le jeu a \u00e9t\u00e9 ainsi imagin\u00e9, c'est pour que les joueurs puissent appr\u00e9hender son environnement de fa\u00e7on plus famili\u00e8re. Les diff\u00e9rences entre Trisolaris et le monde des _Trois Corps_ que vous avez d\u00e9couvert dans le jeu sont en r\u00e9alit\u00e9 immenses. Mais comme son nom l'indique, Trisolaris est bien dot\u00e9 de trois soleils. C'est la nature fondamentale de ce monde.\n\n\u2014 Le d\u00e9veloppement du jeu a d\u00fb demander beaucoup d'efforts. Mais son objectif n'est clairement pas celui de la rentabilit\u00e9, fit remarquer le vice-pr\u00e9sident.\n\n\u2014 Le but du jeu des _Trois Corps_ est simple : r\u00e9unir ceux qui, comme nous, nourrissent des id\u00e9aux communs, r\u00e9pondit Pan Han.\n\n\u2014 Quels id\u00e9aux ? Qu'avons-nous en commun ? demanda Wang Miao avant de regretter aussit\u00f4t sa question.\n\nIl se demanda si le ton de sa voix avait trahi une quelconque animosit\u00e9.\n\nComme il l'avait pr\u00e9vu, sa question plongea Pan Han dans le silence. Ce dernier examina du regard tous les joueurs, puis il s'adressa \u00e0 eux d'une voix douce :\n\n\u2014 Quelle serait votre r\u00e9action si la civilisation trisolarienne devait un jour p\u00e9n\u00e9trer dans notre monde ?\n\n\u2014 J'en serais ravi, r\u00e9pondit le jeune journaliste. Les \u00e9v\u00e9nements de ces derni\u00e8res ann\u00e9es m'ont fait perdre tout espoir dans l'esp\u00e8ce humaine. Les soci\u00e9t\u00e9s humaines sont dans l'incapacit\u00e9 de progresser d'elles-m\u00eames, elles ont besoin de l'intervention d'une force ext\u00e9rieure.\n\n\u2014 Je suis d'accord ! cria l'\u00e9crivain \u00e0 haute voix et avec une \u00e9motion manifeste, comme si elle trouvait enfin une occasion d'\u00e9pancher librement ses sentiments. Qu'est-ce que l'humanit\u00e9 ? Une race abjecte ! J'ai pass\u00e9 la moiti\u00e9 de ma vie \u00e0 utiliser la litt\u00e9rature comme scalpel pour diss\u00e9quer cette abjection. Je suis fatigu\u00e9e \u00e0 pr\u00e9sent. Je place tous mes espoirs dans la civilisation trisolarienne pour apporter la vraie beaut\u00e9 \u00e0 ce monde.\n\nPan Han n'avait encore rien dit, mais le p\u00e9tillement de ses yeux traduisait la m\u00eame excitation.\n\nLe vieux philosophe agita sa pipe d\u00e9j\u00e0 \u00e9teinte et d\u00e9clara d'un ton grave :\n\n\u2014 Essayons de r\u00e9fl\u00e9chir plus profond\u00e9ment \u00e0 la question : vous avez sans doute d\u00e9j\u00e0 entendu parler de la civilisation azt\u00e8que. Quelles impressions avez-vous de celle-ci ?\n\n\u2014 Obscure et violente. Des pyramides d\u00e9goulinant de sang \u00e9clair\u00e9es \u00e0 la lueur de torches dans une for\u00eat sinistre, r\u00e9pondit l'\u00e9crivain.\n\nLe philosophe hocha la t\u00eate :\n\n\u2014 Tr\u00e8s bien. Imaginons maintenant que cette civilisation n'ait jamais \u00e9t\u00e9 envahie par les Conquistadors espagnols. Quelle aurait \u00e9t\u00e9 son influence sur le cours de l'humanit\u00e9 ?\n\n\u2014 Vous refaites l'histoire ! dit le vice-pr\u00e9sident au philosophe. \u00c0 cette \u00e9poque, les Espagnols ayant envahi l'Am\u00e9rique \u00e9taient des pilleurs et des assassins !\n\n\u2014 Quand bien m\u00eame, ils ont au moins eu le m\u00e9rite de mettre un terme \u00e0 l'expansion de la civilisation azt\u00e8que, qui aurait fini par transformer l'Am\u00e9rique en un titanesque empire sanglant et t\u00e9n\u00e9breux. La civilisation et la d\u00e9mocratie telles que nous les connaissons aujourd'hui en Am\u00e9rique ne seraient certainement jamais apparues. Ce que je veux dire, c'est que, selon moi, peu importe la nature de ces Trisolariens, leur arriv\u00e9e sur Terre serait un signe de bonne nouvelle pour \u00e9radiquer le fl\u00e9au contagieux qu'est l'humanit\u00e9.\n\n\u2014 Mais avez-vous oubli\u00e9 que les Azt\u00e8ques et leur civilisation ont \u00e9t\u00e9 massacr\u00e9s par les envahisseurs occidentaux ? demanda le cadre sup\u00e9rieur. (Il fit suivre sa remarque d'un regard circulaire, comme s'il voyait les autres joueurs pour la premi\u00e8re fois.) Les pens\u00e9es qui sont exprim\u00e9es ici sont dangereuses !\n\n\u2014 Profondes, vous voulez dire ! r\u00e9torqua le jeune \u00e9tudiant en levant le doigt et en hochant vigoureusement la t\u00eate en direction du philosophe. Je pense comme vous, mais je ne savais pas comment l'exprimer. Vous l'avez si bien dit !\n\nApr\u00e8s un moment de silence, Pan Han se tourna vers Wang Miao :\n\n\u2014 Les six autres joueurs ont exprim\u00e9 leurs sentiments dans l'hypoth\u00e8se d'une arriv\u00e9e sur Terre des Trisolariens. Qu'en pensez-vous ?\n\n\u2014 Je me range de leur c\u00f4t\u00e9, dit Wang Miao en d\u00e9signant le philosophe et les autres.\n\nIl se contenta de cette phrase simple, sachant qu'en dire plus pouvait lui en co\u00fbter.\n\n\u2014 Parfait, dit Pan Han ; puis il pivota vers le vice-pr\u00e9sident d'IT et le cadre sup\u00e9rieur : Vous n'\u00eates d\u00e9sormais plus les bienvenus dans cette r\u00e9union ni dans le jeu des _Trois Corps_. Vos identifiants seront supprim\u00e9s. Veuillez sortir \u00e0 pr\u00e9sent. Merci.\n\nLes deux hommes se lev\u00e8rent et \u00e9chang\u00e8rent un regard. D\u00e9concert\u00e9s, ils jet\u00e8rent un dernier coup d'\u0153il autour d'eux puis sortirent du caf\u00e9.\n\nPan Han tendit la main aux cinq joueurs restants et la leur serra \u00e9nergiquement, puis il leur annon\u00e7a sur un ton solennel :\n\n\u2014 \u00c0 pr\u00e9sent, nous sommes camarades.\n\n## 19\n\n## _LES TROIS CORPS_ : EINSTEIN, LE PENDULE\n\n## ET LA GRANDE D\u00c9CHIRURE\n\nC'\u00e9tait la cinqui\u00e8me fois que Wang Miao se connectait au jeu des _Trois Corps_. Le monde baign\u00e9 par l'aube auquel il s'\u00e9tait habitu\u00e9 n'avait plus du tout le m\u00eame aspect. La pyramide qui \u00e9tait apparue lors de ses quatre derni\u00e8res parties avait \u00e9t\u00e9 d\u00e9truite lors de la syzygie des trois soleils. \u00c0 sa place se dressait maintenant une grande tour moderne gris fonc\u00e9, qui rappela \u00e0 Wang Miao l'architecture du si\u00e8ge des Nations unies. Au loin, d'autres tours parsemaient la plaine, probablement des silos de d\u00e9shydratation. Les silos \u00e9taient rev\u00eatus d'une surface miroitante qui, sous la lueur du matin, leur donnait l'apparence de gigantesques plantes de cristal qui auraient jailli du sol.\n\nWang Miao entendit le son d'un violon qui jouait un morceau de Mozart. Le violoniste ne donnait pas l'air d'\u00eatre un musicien professionnel, mais il d\u00e9gageait un charme singulier donnant l'impression qu'il jouait pour lui-m\u00eame et que cela suffisait \u00e0 son bonheur. La musique provenait du perron situ\u00e9 devant la grande porte de la tour. Le joueur \u00e9tait un vieux clochard dont l'\u00e9paisse chevelure argent\u00e9e volait au vent. \u00c0 ses pieds, un haut-de-forme d\u00e9fra\u00eechi contenait quelques pi\u00e8ces.\n\nWang Miao remarqua soudain que le soleil s'\u00e9tait lev\u00e9, mais dans la direction oppos\u00e9e de l'horizon d'o\u00f9 pointait la lumi\u00e8re de l'aube. Au m\u00eame endroit, quelques minutes plus t\u00f4t, le ciel \u00e9tait compl\u00e8tement noir. Le soleil \u00e9tait immense. La moiti\u00e9 seulement de sa circonf\u00e9rence occupait d\u00e9j\u00e0 un tiers de l'horizon visible. Le c\u0153ur de Wang Miao se mit \u00e0 battre plus vite. Un si grand soleil \u00e9tait forc\u00e9ment le pr\u00e9sage d'une nouvelle catastrophe. Mais lorsqu'il tourna la t\u00eate, il vit que le vieux clochard continuait \u00e0 jouer sereinement de son instrument comme si rien ne s'\u00e9tait pass\u00e9. Ses cheveux d'argent parurent s'embraser sous l'effet des rayons du soleil.\n\nLe soleil \u00e9tait argent\u00e9, de la m\u00eame couleur que les cheveux du violoniste. Il irradia le sol de rais cristallins, mais Wang Miao n'\u00e9prouva \u00e9trangement aucune sensation de chaleur. Il observa l'astre qui \u00e9tait maintenant enti\u00e8rement sorti au-dessus de la ligne de l'horizon. \u00c0 la surface de ce gigantesque disque d'argent, il put apercevoir avec clart\u00e9 des motifs marbr\u00e9s : des cha\u00eenes de montagnes, \u00e0 l'\u00e9tat solide.\n\nWang Miao remarqua que le disque n'\u00e9mettait aucune lumi\u00e8re et ne faisait que refl\u00e9ter les rayons provenant du vrai soleil qui se levait de l'autre c\u00f4t\u00e9. Ce n'\u00e9tait pas un soleil, mais une lune g\u00e9ante ! Elle se d\u00e9pla\u00e7ait \u00e0 une vitesse ahurissante, si rapidement qu'on pouvait d\u00e9tecter son trajet dans le ciel \u00e0 l'\u0153il nu. Durant son parcours, elle diminua peu \u00e0 peu de volume jusqu'\u00e0 prendre la forme d'une demi-lune, puis d'un croissant. L'air froid du matin paraissait berc\u00e9 par le son agr\u00e9able du violon du clochard. C'\u00e9tait comme si toute la splendeur de l'univers s'incarnait dans sa musique. Wang Miao \u00e9prouva une merveilleuse sensation de bien-\u00eatre.\n\nLe croissant de lune fut bient\u00f4t \u00e9clips\u00e9 par l'intensit\u00e9 grandissante du lever du jour. Lorsque ne rest\u00e8rent plus que deux pointes de rayons argent\u00e9s au-dessus de la ligne d'horizon, Wang Miao fut soudain habit\u00e9 par l'image de deux cornes appartenant \u00e0 un taureau g\u00e9ant en train de charger le soleil.\n\n\u2014 Honorable Copernic, faites donc halte quelques instants. Profitez d'un morceau de Mozart, et peut-\u00eatre m'offrirez-vous de quoi me payer un d\u00e9jeuner, fit le clochard, une fois que la lune eut totalement disparu derri\u00e8re l'horizon.\n\n\u2014 Si je ne m'abuse, vous \u00eates..., commen\u00e7a Wang Miao en contemplant ce visage dont les rides harmonieuses n'affectaient en rien la gr\u00e2ce\n\n\u2014 Vous ne vous trompez pas. Je suis Einstein. Pauvre enfant de Dieu abandonn\u00e9 par Lui.\n\n\u2014 Quelle est cette \u00e9norme lune ? Je ne l'ai jamais vue lors de mes pr\u00e9c\u00e9dentes venues.\n\n\u2014 Elle est refroidie maintenant.\n\n\u2014 Quoi donc ?\n\n\u2014 La lune bien s\u00fbr ! Lorsque j'\u00e9tais enfant, elle \u00e9tait encore chaude. Quand elle s'\u00e9levait au milieu du ciel, on pouvait voir son c\u0153ur rougeoyant sur ses plaines centrales, mais elle est refroidie \u00e0 pr\u00e9sent... Vous n'avez jamais entendu parler de la grande d\u00e9chirure ?\n\n\u2014 Non, qu'est-ce que c'est ?\n\nEinstein poussa un soupir et secoua la t\u00eate :\n\n\u2014 N'en parlons pas. Le pass\u00e9 est trop douloureux. Mon pass\u00e9, le pass\u00e9 de la civilisation, le pass\u00e9 de l'univers... tout est trop douloureux.\n\n\u2014 Comment en \u00eates-vous arriv\u00e9 l\u00e0 ?\n\nWang Miao fouilla dans sa poche et y d\u00e9couvrit quelques pi\u00e8ces de monnaie. Il se baissa et jeta les pi\u00e8ces dans le chapeau.\n\n\u2014 Merci Copernic. Prions pour que Dieu ne vous abandonne pas non plus. Mais j'en doute fort. J'avais le sentiment que le mod\u00e8le d'ordinateur que vous, Newton et les autres aviez cr\u00e9\u00e9 en Orient \u00e9tait tr\u00e8s proche de l'exactitude. Mais voil\u00e0, une seule petite erreur et ce fut un infranchissable gouffre pour Newton et sa clique. J'ai toujours consid\u00e9r\u00e9 que m\u00eame si je n'avais pas \u00e9t\u00e9 l\u00e0, quelqu'un d'autre aurait fini par d\u00e9couvrir la relativit\u00e9 restreinte. L'erreur de Newton, c'est d'avoir n\u00e9glig\u00e9 la courbure de l'espace-temps provoqu\u00e9e par les orbites plan\u00e9taires, telle que la d\u00e9crit la th\u00e9orie de la relativit\u00e9 g\u00e9n\u00e9rale. C'est un oubli certes minime, mais qui s'est av\u00e9r\u00e9 fatal pour les r\u00e9sultats de ses calculs. Il aurait fallu ajouter la courbure de l'espace-temps dans l'\u00e9quation classique pour obtenir un mod\u00e8le math\u00e9matique correct. La somme des calculs \u00e0 effectuer \u00e9tait en r\u00e9alit\u00e9 beaucoup plus importante que ce que vous aviez tent\u00e9 de faire en Orient. Mais, \u00e0 l'aide des ordinateurs modernes, cela ne pose pas un grand probl\u00e8me.\n\n\u2014 Les r\u00e9sultats des calculs ont-ils \u00e9t\u00e9 confirm\u00e9s par des observations astronomiques ?\n\n\u2014 Si tel avait \u00e9t\u00e9 le cas, pensez-vous que je serais ici ? Mais d'un point de vue esth\u00e9tique, ce n'est pas moi qui suis en tort, c'est l'univers. Dieu m'a abandonn\u00e9, puis les autres hommes m'ont abandonn\u00e9 \u00e0 leur tour. Je suis devenu un paria. L'universit\u00e9 de Princeton m'a d\u00e9mis de mes fonctions de professeur, l'Unesco ne m'a plus propos\u00e9 de poste de consultant scientifique alors qu'ils m'imploraient \u00e0 genoux quelques mois plus t\u00f4t. J'ai m\u00eame pens\u00e9 \u00e0 finalement accepter de devenir pr\u00e9sident d'Isra\u00ebl, mais ils m'ont dit avoir chang\u00e9 d'id\u00e9e, que je n'\u00e9tais qu'un imposteur...\n\nEinstein se remit \u00e0 jouer du violon, reprenant le morceau pr\u00e9cis\u00e9ment l\u00e0 o\u00f9 il s'\u00e9tait arr\u00eat\u00e9 quelques minutes plus t\u00f4t. Wang Miao l'\u00e9couta encore un instant et se dirigea vers la porte du si\u00e8ge des Nations unies.\n\n\u2014 Il n'y a plus personne \u00e0 l'int\u00e9rieur, lui dit Einstein en continuant \u00e0 jouer. Tous les participants \u00e0 la session de l'Assembl\u00e9e g\u00e9n\u00e9rale sont derri\u00e8re la tour. Ils assistent \u00e0 la c\u00e9r\u00e9monie d'inauguration du m\u00e9morial du Pendule.\n\nWang Miao fit le tour et arriva derri\u00e8re le b\u00e2timent. Il se retrouva face \u00e0 un tableau incroyable : devant lui se dressait un pendule gigantesque qui semblait relier la terre au ciel. Il avait aper\u00e7u plus t\u00f4t une partie du pendule derri\u00e8re la tour, mais il ignorait alors ce que c'\u00e9tait.\n\nC'\u00e9tait le m\u00eame genre de pendule que ceux cr\u00e9\u00e9s par Fu Xi pour bercer le dieu du Soleil lors de sa premi\u00e8re connexion au monde des _Trois Corps_ , \u00e0 l'\u00e9poque des Royaumes combattants. Mais le pendule qu'il avait \u00e0 pr\u00e9sent devant les yeux \u00e9tait de facture moderne. Les deux piliers soutenant l'instrument \u00e9taient en m\u00e9tal et chacun \u00e9tait aussi haut que la tour Eiffel. Le poids du pendule \u00e9tait en m\u00e9tal fusel\u00e9 et sa surface miroitante \u00e9lectroplaqu\u00e9e \u00e9tait lisse. La tige, faite dans un mat\u00e9riel ultrar\u00e9sistant, \u00e9tait si fine qu'on ne la voyait presque pas, ce qui donnait l'impression que le poids du pendule flottait dans l'air entre deux immenses tours.\n\nUn attroupement d'individus en costume s'\u00e9tait form\u00e9 sous le pendule, sans doute les dirigeants des diff\u00e9rents pays participant \u00e0 la session de l'Assembl\u00e9e g\u00e9n\u00e9rale. Ils \u00e9taient divis\u00e9s en petits groupes et bavardaient tranquillement, comme s'ils attendaient quelque chose.\n\n\u2014 Ah, Copernic ! L'homme qui a travers\u00e9 quatre \u00e9poques ! cria quelqu'un \u00e0 haute voix, et tous lui firent un signe de bienvenue.\n\n\u2014 Vous \u00eates donc celui qui a vu de ses propres yeux les pendules de l'\u00e9poque des Royaumes combattants ! lui dit un homme affable \u00e0 la peau noire en lui serrant la main.\n\nOn le lui pr\u00e9senta comme le secr\u00e9taire g\u00e9n\u00e9ral des Nations unies.\n\n\u2014 Oui, en effet, mais pourquoi en avoir reconstruit un aujourd'hui ? demanda Wang Miao.\n\n\u2014 C'est un monument comm\u00e9moratif pour Trisolaris, ainsi qu'une st\u00e8le fun\u00e9raire, expliqua le secr\u00e9taire g\u00e9n\u00e9ral en montrant le pendule. D'ici, il a l'air aussi grand qu'un sous-marin.\n\n\u2014 Une st\u00e8le fun\u00e9raire ? Pour qui ?\n\n\u2014 Pour un effort. Un effort poursuivi durant pr\u00e8s de deux cents civilisations. Celui de r\u00e9soudre le probl\u00e8me \u00e0 trois corps, celui de comprendre le mouvement des trois soleils.\n\n\u2014 Cet effort a donc pris fin ?\n\n\u2014 Aujourd'hui, oui. Maintenant, il n'a plus de raison d'\u00eatre.\n\nWang Miao h\u00e9sita un instant puis sortit une pile de documents : le mod\u00e8le math\u00e9matique des trois corps qu'avait r\u00e9alis\u00e9 Wei Cheng.\n\n\u2014 Je... je suis revenu pour cela. Je suis venu vous apporter un mod\u00e8le math\u00e9matique capable de r\u00e9soudre le probl\u00e8me \u00e0 trois corps. J'ai toutes les raisons de croire qu'il peut marcher.\n\nMais \u00e0 peine Wang Miao eut-il prononc\u00e9 ces mots qu'il s'aper\u00e7ut que tous ceux qui l'entouraient s'\u00e9taient d\u00e9sint\u00e9ress\u00e9s de lui et \u00e9taient repartis dans les petits cercles de discussion qu'ils avaient quitt\u00e9s quelques instants plus t\u00f4t pour le saluer. Wang Miao remarqua m\u00eame que certains s'\u00e9taient \u00e9loign\u00e9s en gloussant et en secouant la t\u00eate. Le secr\u00e9taire prit ses documents et, sans m\u00eame y jeter un \u0153il, il les transmit \u00e0 un grand homme maigre \u00e0 lunettes qui se tenait \u00e0 c\u00f4t\u00e9 de lui.\n\n\u2014 Par respect pour votre immense r\u00e9putation, je demanderai \u00e0 mon consultant scientifique d'examiner ce document. Ici, tout le monde a un grand respect pour vous. Si n'importe qui d'autre avait dit ce que vous venez de dire, on se serait aussit\u00f4t moqu\u00e9 de lui.\n\nLe consultant scientifique feuilleta le document :\n\n\u2014 Un algorithme \u00e9volutionnaire ? Copernic, vous \u00eates un g\u00e9nie. Tous ceux qui sont en mesure de r\u00e9aliser ce type d'algorithme sont des g\u00e9nies. Cela n\u00e9cessite non seulement un don ind\u00e9niable pour les math\u00e9matiques, mais aussi beaucoup d'imagination.\n\n\u2014 \u00c0 vous entendre, il semblerait que quelqu'un ait d\u00e9j\u00e0 cr\u00e9\u00e9 ici un tel mod\u00e8le math\u00e9matique ?\n\n\u2014 Oui. Il en existe en r\u00e9alit\u00e9 plus d'une dizaine, dont plus de la moiti\u00e9 sont encore plus avanc\u00e9s que le v\u00f4tre. Ils ont \u00e9t\u00e9 cr\u00e9\u00e9s et mis en application par des ordinateurs. Durant ces deux derniers si\u00e8cles, ces immenses calculs ont constitu\u00e9 l'activit\u00e9 principale de notre monde. Nous attendions les r\u00e9sultats comme si c'\u00e9tait le jour du Jugement dernier.\n\n\u2014 Et ?\n\n\u2014 Nous avons d\u00e9finitivement prouv\u00e9 que le probl\u00e8me \u00e0 trois corps ne pouvait \u00eatre r\u00e9solu.\n\nWang Miao leva les yeux vers le gigantesque pendule. Dans la lueur de l'aube, il brillait comme du cristal. Sa surface miroitante d\u00e9form\u00e9e refl\u00e9tait les alentours comme s'il \u00e9tait l'\u0153il du monde. Sur cette m\u00eame plaine, bien des civilisations plus t\u00f4t, lui et le roi Wen \u00e9taient pass\u00e9s devant une rang\u00e9e de pendules g\u00e9ants avant d'atteindre le palais du roi Zhou. Et ainsi, l'histoire avait trac\u00e9 une grande boucle et \u00e9tait retourn\u00e9e \u00e0 son point de d\u00e9part.\n\n\u2014 Comme nous l'avions pr\u00e9dit il y a longtemps, le syst\u00e8me des trois corps est un syst\u00e8me chaotique. D'infimes perturbations peuvent l'amplifier sans fin. Son mod\u00e8le de mouvements ne peut par d\u00e9finition pas \u00eatre pr\u00e9dit, expliqua le consultant scientifique.\n\nL'espace d'un instant, toutes les connaissances scientifiques de Wang Miao lui parurent floues. \u00c0 leur place \u00e9mergeait une confusion qu'il n'avait jamais \u00e9prouv\u00e9e :\n\n\u2014 Mais alors, si m\u00eame un syst\u00e8me aussi simple que celui des trois corps n'est qu'un impr\u00e9visible chaos, comment garder foi dans la d\u00e9couverte un jour futur des lois qui r\u00e9gissent la complexit\u00e9 de notre univers ?\n\n\u2014 Dieu est sans piti\u00e9. Il joue aux d\u00e9s. Il nous a abandonn\u00e9s ! dit soudain Einstein, qu'il n'avait pas vu arriver.\n\nLe secr\u00e9taire g\u00e9n\u00e9ral opina lentement de la t\u00eate :\n\n\u2014 Oui. Dieu est un joueur de d\u00e9s. Et notre seul espoir est de lancer le d\u00e9 \u00e0 notre tour.\n\nC'est alors que la lune g\u00e9ante r\u00e9apparut du c\u00f4t\u00e9 obscur de l'horizon. Son image argent\u00e9e se refl\u00e9ta sur la surface du poids du pendule. Sa lumi\u00e8re se mit \u00e0 vaciller \u00e9trangement, comme si un myst\u00e9rieux contact t\u00e9l\u00e9pathique s'\u00e9tait \u00e9tabli entre elle et le pendule.\n\n\u2014 En parlant de civilisation, la v\u00f4tre semble d\u00e9j\u00e0 avoir atteint un niveau tr\u00e8s avanc\u00e9, dit Wang Miao.\n\n\u2014 Oui. Nous ma\u00eetrisons d\u00e9sormais l'\u00e9nergie nucl\u00e9aire et nous avons atteint l'\u00e2ge de l'information, r\u00e9pondit le secr\u00e9taire g\u00e9n\u00e9ral, sans pour autant faire montre de fiert\u00e9.\n\n\u2014 Il existe peut-\u00eatre un espoir : la civilisation peut continuer \u00e0 progresser jusqu'\u00e0 atteindre un niveau o\u00f9, m\u00eame sans conna\u00eetre la m\u00e9canique des soleils, elle pourra survivre aux \u00e8res chaotiques et r\u00e9sister aux ravages caus\u00e9s par les mouvements irr\u00e9guliers des soleils.\n\n\u2014 C'est ce que croyaient les gens autrefois. C'\u00e9tait l'une des raisons qui ont jadis pouss\u00e9 les civilisations de Trisolaris \u00e0 rena\u00eetre encore et toujours. Mais nous avons pris conscience de la na\u00efvet\u00e9 de cette croyance, dit le secr\u00e9taire g\u00e9n\u00e9ral en pointant du doigt la lune g\u00e9ante. Vous voyez peut-\u00eatre cette lune gigantesque pour la premi\u00e8re fois. Elle fait environ un quart de la taille de notre plan\u00e8te. Ce n'est d\u00e9j\u00e0 plus une lune, mais plut\u00f4t une autre plan\u00e8te dans un syst\u00e8me binaire, le produit de la grande d\u00e9chirure.\n\n\u2014 La grande d\u00e9chirure ?\n\n\u2014 La catastrophe qui a an\u00e9anti la derni\u00e8re civilisation. Par comparaison avec les cataclysmes ayant d\u00e9truit les civilisations plus anciennes, nous avons \u00e9t\u00e9 avertis bien plus t\u00f4t de l'imminence de cette catastrophe. Les chroniques datant de cette \u00e9poque indiquent que les astronomes de la civilisation no 191 avaient observ\u00e9 depuis longtemps une \u00e9toile volante fig\u00e9e.\n\nEn entendant cette derni\u00e8re phrase, Wang Miao sentit son c\u0153ur se nouer. Une \u00e9toile volante fig\u00e9e repr\u00e9sentait le plus terrible des pr\u00e9sages. Quand une \u00e9toile volante \u2013 ou plut\u00f4t un soleil \u00e9loign\u00e9 \u2013 paraissait, vue de la surface, s'\u00eatre fig\u00e9e sur la toile de l'univers, cela ne pouvait signifier qu'une chose : le soleil et la plan\u00e8te \u00e9taient align\u00e9s. Cette situation pouvait donner lieu \u00e0 trois interpr\u00e9tations possibles : le soleil et la plan\u00e8te se d\u00e9pla\u00e7aient \u00e0 la m\u00eame vitesse et dans la m\u00eame direction ; le soleil s'\u00e9loignait de la plan\u00e8te et, enfin, le soleil se rapprochait de la plan\u00e8te. Avant la civilisation no 191, cette derni\u00e8re interpr\u00e9tation n'\u00e9tait qu'une catastrophe th\u00e9orique, car elle n'avait jamais eu lieu. Cependant la peur et l'appr\u00e9hension \u00e9prouv\u00e9es face \u00e0 ce ph\u00e9nom\u00e8ne tant redout\u00e9 \u00e9taient omnipr\u00e9sentes dans la population. La simple \u00e9vocation d'\"\u00e9toile volante fig\u00e9e\" portait malheur \u00e0 Trisolaris. Une seule \u00e9toile volante fig\u00e9e suffisait \u00e0 faire frissonner chacun de peur.\n\n\u2014 Mais ce jour-l\u00e0, ce furent trois \u00e9toiles volantes qui se fig\u00e8rent simultan\u00e9ment. Le peuple de la civilisation no 191 \u00e9tait debout sur la terre, le regard impuissant tourn\u00e9 vers les trois \u00e9toiles volantes suspendues dans la vo\u00fbte c\u00e9leste, observant les trois soleils s'avancer dans la direction de leur plan\u00e8te. Au bout de quelques jours, l'un des soleils s'est trouv\u00e9 \u00e0 une distance si proche qu'on a pu voir sa couche externe gazeuse. Dans la tranquillit\u00e9 de la nuit, l'\u00e9toile volante s'est m\u00e9tamorphos\u00e9e en soleil ardent et, \u00e0 des intervalles d'un peu plus de trente heures, les deux autres soleils lui ont succ\u00e9d\u00e9. Ce n'\u00e9tait pas un ciel normal \u00e0 trois soleils car, lorsque la derni\u00e8re \u00e9toile volante a eu pris l'apparence d'un soleil, le premier avait d\u00e9j\u00e0 d\u00e9pass\u00e9 la plan\u00e8te \u00e0 une distance extr\u00eamement proche. Puis \u00e7'a \u00e9t\u00e9 au tour des deux autres soleils de fr\u00f4ler la plan\u00e8te d'encore plus pr\u00e8s. Les trois soleils ont franchi la limite de Roche1, si bien que la force des mar\u00e9es provoqu\u00e9e par les trois astres a d\u00e9pass\u00e9 l'autoattraction de la plan\u00e8te. En rasant la plan\u00e8te, le premier soleil a d\u00e9plac\u00e9 ses couches g\u00e9ologiques les plus profondes. Le deuxi\u00e8me a provoqu\u00e9 une fissure qui s'est prolong\u00e9e jusqu'\u00e0 son noyau. Et le troisi\u00e8me a d\u00e9chir\u00e9 la plan\u00e8te en deux.\n\nLe secr\u00e9taire g\u00e9n\u00e9ral montra du doigt la lune g\u00e9ante dans le ciel :\n\n\u2014 Vous voyez ici la moiti\u00e9 la plus petite. L\u00e0-bas se trouvent les ruines de la civilisation no 191. Mais plus aucune cr\u00e9ature vivante ne peuple ce monde. C'est le plus terrible d\u00e9sastre de toute l'histoire de Trisolaris. Apr\u00e8s la fissure de la plan\u00e8te, les deux morceaux, de forme irr\u00e9guli\u00e8re, ont retrouv\u00e9 leur apparence sph\u00e9rique sous l'influence de leur propre gravit\u00e9. La mati\u00e8re en fusion du noyau de la plan\u00e8te a jailli \u00e0 la surface et les oc\u00e9ans se sont mis \u00e0 bouillir ; les continents ont d\u00e9riv\u00e9 comme des icebergs sur ce magma. Apr\u00e8s leur collision, le sol est devenu aussi liquide que les oc\u00e9ans, des cha\u00eenes de montagnes immenses de plusieurs dizaines de milliers de m\u00e8tres se sont form\u00e9es en moins d'une heure et ont disparu aussit\u00f4t.\n\nDurant un certain temps, les deux fragments fissur\u00e9s de la plan\u00e8te ont pu \u00eatre reli\u00e9s entre eux par une sorte de pont dans l'espace, form\u00e9 par une rivi\u00e8re de lave en fusion. La lave a ensuite refroidi dans l'espace et des anneaux se sont form\u00e9s autour de la plan\u00e8te, mais les anneaux \u00e9taient instables \u00e0 cause de la perturbation gravitationnelle des deux plan\u00e8tes. Les rochers de lave sont donc retomb\u00e9s confus\u00e9ment sur le sol des plan\u00e8tes. Cette pluie de m\u00e9t\u00e9orites a dur\u00e9 pendant des si\u00e8cles... Pouvez-vous imaginer cet enfer ? Ce fut la catastrophe \u00e9cologique la plus terrible de l'histoire. La vie a totalement disparu de notre plan\u00e8te double, et notre plan\u00e8te m\u00e8re, celle sur laquelle nous nous trouvons, a bien failli conna\u00eetre le m\u00eame sort. Mais les graines de la vie ont repouss\u00e9 ici et, lorsque la plan\u00e8te est parvenue \u00e0 trouver un \u00e9tat g\u00e9ologique stable, l'\u00e9volution a repris \u00e0 un rythme effr\u00e9n\u00e9 sur des continents et des oc\u00e9ans nouveaux, jusqu'\u00e0 ce que la civilisation renaisse pour la cent quatre-vingt-douzi\u00e8me fois. Ce processus a pris quatre-vingt-douze millions d'ann\u00e9es.\n\nL'univers dans lequel se trouve le monde de Trisolaris est encore plus impitoyable que nous l'imaginions. Quand verrons-nous la prochaine \u00e9toile volante fig\u00e9e ? Il y a de grands risques que notre plan\u00e8te ne fasse pas que fr\u00f4ler les soleils, mais se retrouve happ\u00e9e par leurs oc\u00e9ans de flammes. Plus le temps passe, plus ce danger se rapproche.\n\nCe n'\u00e9tait autrefois qu'une supposition effrayante, mais des observations astronomiques r\u00e9centes nous ont fait perdre tout espoir dans la destin\u00e9e de notre monde. En \u00e9tudiant des r\u00e9sidus datant de la formation de notre syst\u00e8me solaire, les scientifiques avaient pour objectif de comprendre l'histoire de la formation des soleils et des plan\u00e8tes. Au lieu de cela, ils ont fait une terrible d\u00e9couverte : il y a bien longtemps, notre syst\u00e8me trisolarien comptait douze plan\u00e8tes ! Il n'en reste maintenant qu'une, la n\u00f4tre.\n\nUne seule explication est possible : \u00e0 l'\u00e9chelle du temps d\u00e9mesur\u00e9 de l'astronomie, les onze autres plan\u00e8tes ont \u00e9t\u00e9 consum\u00e9es par les trois soleils ! Notre monde n'est ni plus ni moins que le dernier survivant d'une grande chasse \u00e0 l'\u00e9chelle de l'univers. C'est uniquement par chance que la civilisation a pu rena\u00eetre \u00e0 cent quatre-vingt-douze reprises. Apr\u00e8s des recherches encore plus pouss\u00e9es, nous avons d\u00e9couvert le ph\u00e9nom\u00e8ne de \"respiration\" des trois \u00e9toiles.\n\n\u2014 La \"respiration\" des \u00e9toiles ?\n\n\u2014 Ce n'est qu'une image. Vous avez d\u00e9couvert la couche gazeuse externe entourant les soleils, mais vous ignorez que cette couche gazeuse conna\u00eet de longs cycles de dilatation et de contraction, comme si elle respirait. Lorsque la couche gazeuse se dilate, elle peut devenir dix fois plus \u00e9paisse, ce qui augmente bien s\u00fbr consid\u00e9rablement le diam\u00e8tre de l'\u00e9toile. Elle devient alors comme une main g\u00e9ante capable d'empoigner les plan\u00e8tes \u00e0 son aise. Quand une plan\u00e8te et un soleil se trouvent \u00e0 une certaine distance, la plan\u00e8te p\u00e9n\u00e8tre dans la couche gazeuse, et le frottement intense entre les deux corps lui fait perdre de la vitesse. Pour finir, comme la longue queue de feu d'une com\u00e8te, elle s'ab\u00eeme dans l'oc\u00e9an de flammes du soleil.\n\nLes r\u00e9sultats de ces travaux ont montr\u00e9 que, durant la longue histoire du syst\u00e8me trisolarien, chaque fois que la couche gazeuse d'un soleil est entr\u00e9e en p\u00e9riode de dilatation, celui-ci a englouti une \u00e0 deux plan\u00e8tes. Les onze plan\u00e8tes ont en effet sombr\u00e9 dans la mer de flammes d'un soleil durant la p\u00e9riode o\u00f9 la dilatation de sa couche gazeuse \u00e9tait la plus grande. Aujourd'hui, les couches gazeuses des soleils sont dans un cycle de contraction. Autrement, les soleils n'auraient pas fait que fr\u00f4ler notre plan\u00e8te il y a quatre-vingt-douze millions d'ann\u00e9es. Mais selon les pr\u00e9dictions des chercheurs, la prochaine p\u00e9riode de dilatation aura lieu dans cent cinquante \u00e0 deux cent mille ans.\n\n\u2014 Nous ne pouvons plus rester dans cet endroit maudit, conclut Einstein, qui s'\u00e9tait allong\u00e9 en travers du sol, le violon entre les bras, comme un vieux mendiant.\n\nLe secr\u00e9taire g\u00e9n\u00e9ral opina de la t\u00eate et reprit :\n\n\u2014 Oui, nous devons partir, et vite. Notre derni\u00e8re porte de sortie, c'est de faire un pari avec l'univers.\n\n\u2014 Quel genre de pari ? demanda Wang Miao.\n\n\u2014 Sortir de ce syst\u00e8me stellaire, voguer dans le vaste oc\u00e9an d'\u00e9toiles et chercher dans la galaxie un nouveau monde o\u00f9 \u00e9migrer !\n\n\u00c0 cet instant, Wang Miao entendit un grincement sourd. Il vit le poids g\u00e9ant du pendule \u00eatre hiss\u00e9 par un mince c\u00e2ble attach\u00e9 \u00e0 un treuil plac\u00e9 en hauteur. Quand il fut hiss\u00e9 jusqu'en haut, un \u00e9norme fragment de lune descendant dans la lumi\u00e8re de l'aube \u00e9mergea derri\u00e8re lui, dans le fond du ciel.\n\nLe secr\u00e9taire g\u00e9n\u00e9ral annon\u00e7a solennellement :\n\n\u2014 Lancez le pendule !\n\nLe treuil rel\u00e2cha le c\u00e2ble et le poids du pendule tomba silencieusement le long d'une trajectoire lisse et arqu\u00e9e. Son mouvement commen\u00e7a tout d'abord par \u00eatre lent, puis il prit de la vitesse et atteignit sa vitesse maximale en bas de l'arc. Quand il d\u00e9chira l'air, on entendit le bruit lourd et sifflant du vent, puis le pendule remonta le long de l'arc jusqu'\u00e0 son point culminant. Il se figea quelques instants, puis red\u00e9marra son mouvement de balancier.\n\nWang Miao pouvait sentir l'immense force g\u00e9n\u00e9r\u00e9e par le mouvement du pendule. Le sol semblait comme secou\u00e9 par son balancement. Contrairement aux pendules du monde r\u00e9el, le temps de balancement du pendule n'\u00e9tait pas r\u00e9gulier, il ne cessait de changer, une particularit\u00e9 due \u00e0 l'attraction gravitationnelle variable de la lune g\u00e9ante. Lorsque la lune g\u00e9ante se trouvait de ce c\u00f4t\u00e9 de la plan\u00e8te, sa gravit\u00e9 et celle de la plan\u00e8te s'annulaient mutuellement et le pendule perdait du poids. \u00c0 l'inverse, lorsque la lune passait de l'autre c\u00f4t\u00e9, sa gravit\u00e9 s'ajoutait \u00e0 celle de la plan\u00e8te et le poids du pendule augmentait jusqu'\u00e0 revenir quasiment \u00e0 celui qui \u00e9tait le sien avant la grande d\u00e9chirure.\n\nEn levant les yeux pour observer le mouvement impressionnant de ce monument en l'honneur de Trisolaris, Wang Miao se demanda si le pendule comm\u00e9morait l'espoir d'un ordre ou bien la soumission au chaos. Il s'imagina le pendule comme un gigantesque poing de fer \u00e9ternellement brandi contre l'univers, comme si Trisolaris clamait en silence qu'elle resterait \u00e0 jamais insoumise... Tandis que les yeux de Wang Miao se brouillaient de larmes, il vit un texte s'afficher devant le pendule :\n\n> Quatre cent cinquante et un ans plus tard, la civilisation no 192 a \u00e9t\u00e9 d\u00e9truite par les flammes d'un ciel \u00e0 deux soleils. La civilisation avait atteint l'\u00e2ge de l'atome et l'\u00e2ge de l'information.\n> \n> La civilisation no 192 a pos\u00e9 un jalon important dans l'histoire de la civilisation trisolarienne. Elle a enfin prouv\u00e9 que le probl\u00e8me \u00e0 trois corps ne poss\u00e9dait aucune solution. C'est elle qui a choisi d'abandonner les efforts vains des cent quatre-vingt-onze civilisations l'ayant pr\u00e9c\u00e9d\u00e9e. Elle a d\u00e9termin\u00e9 une nouvelle trajectoire pour les civilisations futures. D\u00e9sormais, le nouveau but ultime du jeu des _Trois Corps_ est le suivant :\n> \n> Partir dans l'univers, chercher une nouvelle demeure.\n> \n> Nous vous invitons \u00e0 vous reconnecter dans le futur.\n\nApr\u00e8s avoir quitt\u00e9 le jeu des _Trois Corps_ , Wang Miao se sentit lessiv\u00e9. Mais malgr\u00e9 sa fatigue, il ne se reposa cette fois-ci qu'une petite demi-heure avant de se reconnecter. Sur le fond noir s'afficha une information inattendue :\n\n> Urgent ! Les serveurs des _Trois Corps_ vont bient\u00f4t fermer. Connectez-vous librement pendant le temps restant. Le jeu des _Trois Corps_ va maintenant passer directement \u00e0 la sc\u00e8ne finale.\n\n* * *\n\n1 L'astronome fran\u00e7ais \u00c9douard Roche prouva que n'importe quel corps c\u00e9leste solide, lorsqu'il s'approche d'un corps c\u00e9leste bien plus gros que lui, per\u00e7oit les effets des forces des mar\u00e9es du gros corps c\u00e9leste, ce qui m\u00e8ne \u00e0 sa d\u00e9sint\u00e9gration. La limite \u00e0 laquelle le petit corps c\u00e9leste se fait d\u00e9sint\u00e9grer est ainsi appel\u00e9e \"limite de Roche\", elle est g\u00e9n\u00e9ralement estim\u00e9e pour deux corps de m\u00eame densit\u00e9 \u00e0 2,44 fois le rayon du grand corps. _(N.d.A.)_\n\n## 20\n\n## _LES TROIS CORPS_ : L'EXP\u00c9DITION\n\nLa plaine baign\u00e9e par les lueurs glaciales de l'aube \u00e9tait \u00e0 pr\u00e9sent totalement d\u00e9serte. Ni pyramide ni si\u00e8ge des Nations unies. Le m\u00e9morial du Pendule avait disparu. Il ne restait plus que ce d\u00e9sert de Gobi \u00e0 perte de vue, comme la premi\u00e8re fois qu'il avait mis le pied dans ce monde.\n\nWang Miao s'aper\u00e7ut cependant tr\u00e8s vite que c'\u00e9tait une illusion. Ce qu'il avait pris pour des cailloux jonchant le sol du d\u00e9sert \u00e9taient en r\u00e9alit\u00e9 des t\u00eates humaines ! Une foule immense recouvrait la terre. En les regardant, perch\u00e9 au sommet d'une petite colline, Wang Miao essaya de calculer le nombre de cette improbable mar\u00e9e humaine. Jusqu'o\u00f9 pouvait porter son regard, il les estima \u00e0 plusieurs centaines de millions ! Il comprit que tous les Trisolariens s'\u00e9taient donn\u00e9 rendez-vous ici.\n\nLe silence de ces centaines de millions de personnes provoquait une \u00e9tranget\u00e9 suffocante. Cette foule mass\u00e9e ici attendait quelque chose. Wang Miao sonda les alentours et remarqua que tous levaient la t\u00eate au ciel.\n\nWang Miao les imita et d\u00e9couvrit dans le ciel \u00e9toil\u00e9 un incroyable spectacle : les \u00e9toiles s'\u00e9taient d\u00e9ploy\u00e9es en formation carr\u00e9e ! Bient\u00f4t, il r\u00e9alisa que les \u00e9toiles qu'il avait imagin\u00e9es en formation se trouvaient simplement sur une orbite synchrone au-dessus de la plan\u00e8te. Derri\u00e8re cet \u00e9trange cort\u00e8ge, le vaste oc\u00e9an d'\u00e9toiles de la Voie lact\u00e9e paraissait par contraste tout \u00e0 fait immobile.\n\nParmi les \u00e9toiles en formation, les plus proches du c\u00f4t\u00e9 o\u00f9 l'aube se levait \u00e9taient aussi les plus lumineuses. Elles brillaient d'une lumi\u00e8re argent\u00e9e qui projetait sur le sol les ombres de la foule. Plus on s'\u00e9loignait de l'aube, plus la luminosit\u00e9 d\u00e9clinait. Wang Miao compta plus de trente \u00e9toiles de chaque c\u00f4t\u00e9 de la formation ; il devait donc y en avoir plus de mille en tout. Le mouvement lent et simultan\u00e9 de cette formation, manifestement d'origine artificielle, sur le fond \u00e9toil\u00e9 du ciel d\u00e9gageait une puissance solennelle.\n\nC'est alors qu'un homme qui se tenait \u00e0 ses c\u00f4t\u00e9s lui donna un petit coup de coude et lui susurra \u00e0 voix basse :\n\n\u2014 Ah, v\u00e9n\u00e9rable Copernic, pourquoi arrivez-vous si tard ? Trois civilisations ont pass\u00e9. Vous avez manqu\u00e9 beaucoup de grands accomplissements !\n\n\u2014 Qu'est-ce que c'est ? demanda Wang Miao en pointant du doigt la formation d'\u00e9toiles.\n\n\u2014 La glorieuse flotte trisolarienne, pr\u00eate \u00e0 commencer son exp\u00e9dition !\n\n\u2014 Vous voulez dire que la civilisation a atteint l'\u00e2ge des voyages interstellaires ?\n\n\u2014 Oui. Ces magnifiques vaisseaux peuvent atteindre une vitesse \u00e9quivalant \u00e0 un dixi\u00e8me de la vitesse de la lumi\u00e8re.\n\n\u2014 Un dixi\u00e8me de la vitesse de la lumi\u00e8re ? C'est une avanc\u00e9e \u00e9norme, du moins pour ce que j'en sais. Mais cela semble quand m\u00eame assez lent pour un vaisseau interstellaire.\n\n\u2014 C'est la premi\u00e8re \u00e9tape d'un tr\u00e8s long voyage, r\u00e9pondit l'homme. La clef est de trouver la bonne cible.\n\n\u2014 Quelle est la destination de la flotte ?\n\n\u2014 Un syst\u00e8me stellaire situ\u00e9 \u00e0 environ quatre ann\u00e9es-lumi\u00e8re d'ici \u2013 l'\u00e9toile la plus proche du syst\u00e8me de Trisolaris.\n\nWang Miao fut surpris :\n\n\u2014 L'\u00e9toile la plus proche de notre syst\u00e8me est aussi situ\u00e9e \u00e0 cette distance de quatre ann\u00e9es-lumi\u00e8re.\n\n\u2014 Votre \u00e9toile ?\n\n\u2014 Celle de la plan\u00e8te Terre.\n\n\u2014 Oh, il n'y a rien d'\u00e9tonnant. Dans cette grande r\u00e9gion qu'est la Voie lact\u00e9e, la densit\u00e9 des \u00e9toiles est relativement homog\u00e8ne. C'est le r\u00e9sultat du lent conditionnement de la gravit\u00e9 des \u00e9toiles. La distance entre la majorit\u00e9 des \u00e9toiles de la galaxie est d'environ trois \u00e0 six ann\u00e9es-lumi\u00e8re.\n\nUne gigantesque clameur de joie retentit soudain. Wang Miao leva la t\u00eate et vit que la luminosit\u00e9 de chaque \u00e9toile de la formation s'intensifiait. C'\u00e9tait la lumi\u00e8re \u00e9mise par les vaisseaux eux-m\u00eames. Rapidement, l'aube fut inond\u00e9e de cette clart\u00e9 et les mille \u00e9toiles se transform\u00e8rent en mille petits soleils. Un jour glorieux se levait sur Trisolaris. Sur terre, la foule leva les mains au ciel, esquissant une prairie infinie de bras lev\u00e9s.\n\nLa flotte trisolarienne commen\u00e7a \u00e0 acc\u00e9l\u00e9rer et se d\u00e9pla\u00e7a en grande pompe \u00e0 travers le d\u00f4me c\u00e9leste. Elle rasa le sommet de la lune g\u00e9ante qui venait tout juste de se lever, projetant un halo bleut\u00e9 sur les montagnes et les vall\u00e9es de sa surface.\n\nLa clameur se tut. D\u00e9sormais muet, le peuple de Trisolaris regarda son espoir traverser l'ouest du ciel. Ils ne conna\u00eetraient jamais de leur vivant l'aboutissement de cette odyss\u00e9e, mais dans quatre ou cinq si\u00e8cles, leurs descendants recevraient des nouvelles du succ\u00e8s de la flotte depuis un autre monde, le d\u00e9but d'une nouvelle vie pour la civilisation trisolarienne. Wang Miao se tenait avec eux, observant silencieusement la flotte s'\u00e9loigner jusqu'\u00e0 ce que les mille \u00e9toiles soient enfin r\u00e9duites \u00e0 la taille d'une seule et que celle-ci disparaisse \u00e0 l'ouest, dans le ciel nocturne. Puis un texte s'afficha :\n\n> L'exp\u00e9dition de la civilisation trisolarienne vers le nouveau monde a commenc\u00e9. La flotte est en route...\n> \n> Le jeu des _Trois Corps_ est maintenant termin\u00e9. Lorsque vous retournerez dans le monde de la r\u00e9alit\u00e9, si vous \u00eates fid\u00e8le \u00e0 votre promesse, veuillez vous connecter \u00e0 la page Web dont le lien vous sera envoy\u00e9 par courriel et rendez-vous \u00e0 l'assembl\u00e9e de l'Organisation Terre-Trisolaris.\n\n# II\n\n# LE CR\u00c9PUSCULE DES HOMMES\n\n## 21\n\n## LES REBELLES DE TERRE-TRISOLARIS\n\nIl y avait bien plus de monde \u00e0 cette assembl\u00e9e que lors de la derni\u00e8re r\u00e9union de joueurs des _Trois Corps_. Elle avait lieu dans la caf\u00e9t\u00e9ria d'une usine chimique. L'entreprise avait \u00e9t\u00e9 d\u00e9localis\u00e9e et le b\u00e2timent, v\u00e9tuste, allait bient\u00f4t \u00eatre d\u00e9truit. Trois cents personnes \u00e9taient r\u00e9unies dans la vaste salle. Wang Miao aper\u00e7ut beaucoup de visages familiers. Des c\u00e9l\u00e9brit\u00e9s et des experts de toutes sortes : scientifiques, \u00e9crivains, hommes politiques...\n\nLa premi\u00e8re chose qui attira son attention \u00e9tait une installation insolite, plac\u00e9e au centre de la caf\u00e9t\u00e9ria : trois sph\u00e8res argent\u00e9es, chacune un peu plus petite qu'une boule de bowling, l\u00e9vitaient en tourbillonnant au-dessus d'un socle en m\u00e9tal. Wang supposa que l'ensemble fonctionnait gr\u00e2ce au principe de la sustentation \u00e9lectromagn\u00e9tique. L'orbite suivie par les sph\u00e8res paraissant absolument al\u00e9atoire, Wang Miao y vit une reproduction r\u00e9elle du mouvement des trois corps.\n\nLes autres convives ne semblaient pas pr\u00eater une grande attention \u00e0 cette \u0153uvre d'art. Leur attention \u00e9tait riv\u00e9e sur un individu qui se trouvait debout sur une table cass\u00e9e au centre de la salle. Pan Han.\n\n\u2014 As-tu assassin\u00e9 la camarade Shen Yufei ? l'interrogea quelqu'un.\n\n\u2014 Oui, r\u00e9pondit Pan Han avec sang-froid. L'Organisation fait aujourd'hui face \u00e0 une crise provoqu\u00e9e par des tra\u00eetres qui, comme elle, ont infiltr\u00e9 les rangs adventistes.\n\n\u2014 De quel droit as-tu os\u00e9 ?\n\n\u2014 C'\u00e9tait mon devoir, pour le bien de l'Organisation !\n\n\u2014 Ton devoir ? Je crois plut\u00f4t que tes intentions \u00e9taient mauvaises d\u00e8s le d\u00e9but !\n\n\u2014 Qu'insinues-tu ?\n\n\u2014 Qu'a accompli la section environnementale de l'Organisation depuis que tu en as pris la responsabilit\u00e9 ? Votre mission \u00e9tait de vous servir des probl\u00e8mes environnementaux existants ou d'en cr\u00e9er de nouveaux pour susciter au sein de la soci\u00e9t\u00e9 la haine de la science et de l'industrie modernes. Et tout ce que tu as fait, c'est utiliser les technologies et les pr\u00e9dictions divines pour servir tes propres int\u00e9r\u00eats, pour accro\u00eetre ta r\u00e9putation et t'enrichir !\n\n\u2014 Crois-tu vraiment que ma r\u00e9putation m'importe ? \u00c0 mes yeux, l'humanit\u00e9 tout enti\u00e8re n'est qu'un tas d'immondices, pourquoi voudrais-je sa reconnaissance ? Mais si je n'\u00e9tais pas c\u00e9l\u00e8bre, penses-tu que je pourrais influencer les esprits ?\n\n\u2014 Tu as toujours choisi la voie de la facilit\u00e9 ! Tout ce que tu as fait aurait tr\u00e8s bien pu \u00eatre accompli par des militants \u00e9cologistes issus de la soci\u00e9t\u00e9 civile. Au moins, ils \u00e9taient bien plus sinc\u00e8res et passionn\u00e9s que tu ne l'es. Il aurait suffi de les orienter un peu et nous aurions pu utiliser leurs combats \u00e0 notre profit ! La t\u00e2che principale de la section environnementale, c'est de provoquer des catastrophes \u00e9cologiques que nous pourrons ensuite exploiter : empoisonner des r\u00e9servoirs d'eau, provoquer des fuites dans des usines chimiques... Mais qu'avez-vous fait ? Rien de tout \u00e7a !\n\n\u2014 Nous avions r\u00e9fl\u00e9chi \u00e0 de nombreux programmes et plans d'action, mais tous ont \u00e9t\u00e9 refus\u00e9s par le Guide. De toute fa\u00e7on, ces actions auraient \u00e9t\u00e9 stupides, du moins jusqu'\u00e0 aujourd'hui. Il y a quelque temps, la section biologique et m\u00e9dicale a tent\u00e9 de cr\u00e9er une catastrophe en provoquant une overdose d'antibiotiques mais, comme vous le savez, leur plan a \u00e9t\u00e9 rapidement d\u00e9couvert ! Sans parler de la filiale europ\u00e9enne qui a bien failli nous faire rep\u00e9rer !\n\n\u2014 C'est toi qui parles de nous faire rep\u00e9rer, mais tu as commis un meurtre !\n\n\u2014 Camarades, \u00e9coutez-moi ! T\u00f4t ou tard, c'\u00e9tait in\u00e9vitable. Comme vous le savez d\u00e9j\u00e0, m\u00eame si nous n'avons pas encore \u00e9t\u00e9 clairement identifi\u00e9s, nombreux sont les gouvernements qui ont d\u00e9cr\u00e9t\u00e9 l'\u00e9tat de guerre. En Europe et en Am\u00e9rique du Nord, ils ont d\u00e9j\u00e0 proc\u00e9d\u00e9 \u00e0 la capture de certains membres de l'Organisation. D\u00e8s que les arrestations commenceront ici, je ne doute pas un instant que les r\u00e9demptoristes se rangeront du c\u00f4t\u00e9 des autorit\u00e9s. Nous devons agir avant que cela n'arrive et purger l'Organisation de tous les tra\u00eetres r\u00e9demptoristes !\n\n\u2014 Tu n'as pas cette autorit\u00e9.\n\n\u2014 Bien s\u00fbr, ce sera au Guide d'en d\u00e9cider. Mais, camarades, il est de mon devoir de vous informer que le Guide est adventiste !\n\n\u2014 Tu affabules ! Tout le monde ici conna\u00eet l'autorit\u00e9 du Guide. Si la situation \u00e9tait telle que tu le pr\u00e9tends, cela fait bien longtemps que les r\u00e9demptoristes auraient \u00e9t\u00e9 exclus de l'Organisation.\n\n\u2014 Le Guide a sans doute ses raisons. Peut-\u00eatre m\u00eame que c'est le but de l'assembl\u00e9e d'aujourd'hui.\n\nCes paroles prononc\u00e9es, l'attention de la foule se d\u00e9tourna de Pan Han et tous commenc\u00e8rent \u00e0 discuter bruyamment de la crise \u00e0 laquelle faisait face l'Organisation. Un scientifique c\u00e9l\u00e8bre ayant jadis remport\u00e9 le prix Turing sauta sur la table, brandit le poing et lan\u00e7a :\n\n\u2014 Camarades, que croyez-vous qu'il est le moment de faire ?\n\n\u2014 La r\u00e9bellion mondiale !\n\n\u2014 C'est du suicide !\n\n\u2014 Longue vie \u00e0 l'esprit de Trisolaris ! Nous rena\u00eetrons, comme des graines tenaces qui repoussent apr\u00e8s un incendie de for\u00eat !\n\n\u2014 Gr\u00e2ce \u00e0 notre r\u00e9bellion, le monde conna\u00eetra enfin notre visage ! Ce sera la premi\u00e8re fois que l'Organisation entrera sur la sc\u00e8ne de l'histoire. Si notre programme est coh\u00e9rent, il trouvera un vaste soutien de par le monde !\n\nCette derni\u00e8re phrase, prononc\u00e9e par Pan Han, souleva des applaudissements.\n\nQuelqu'un cria :\n\n\u2014 Le Guide !\n\nLa foule s'\u00e9carta pour lui frayer un passage. Wang Miao se sentit pris de vertige. Le monde apparut devant ses yeux comme un film en noir et blanc, avec un seul personnage en couleurs : celle qui venait juste de faire son apparition. Escort\u00e9 par des gardes du corps s'avan\u00e7ait \u00e0 pas d\u00e9cid\u00e9s le Guide des rebelles de l'Organisation Terre-Trisolaris : Ye Wenjie.\n\nElle vint se placer au milieu d'un cercle qu'on avait lib\u00e9r\u00e9 pour elle. Elle leva un poing maigre et, avec une voix puissante et d\u00e9termin\u00e9e que Wang Miao ne lui soup\u00e7onnait pas, elle clama :\n\n\u2014 Mettons fin \u00e0 la tyrannie humaine !\n\nLa foule de rebelles lui r\u00e9pondit par un slogan qui semblait avoir \u00e9t\u00e9 maintes fois r\u00e9p\u00e9t\u00e9 :\n\n\u2014 Le monde appartient \u00e0 Trisolaris !\n\n\u2014 Chers camarades, commen\u00e7a Ye Wenjie.\n\nSa voix avait retrouv\u00e9 la douceur et la l\u00e9g\u00e8ret\u00e9 que lui connaissait Wang Miao, si bien que ce ne fut en fin de compte qu'\u00e0 cet instant qu'il put avoir l'assurance que cette femme \u00e9tait bien le Pr Ye.\n\n\u2014 J'ai \u00e9t\u00e9 quelque peu malade ces derniers temps et je n'ai pas pu \u00eatre aussi pr\u00e9sente \u00e0 vos c\u00f4t\u00e9s que je l'aurais souhait\u00e9. La situation est grave. Je sais que chacun de vous est soumis \u00e0 une pression insupportable, c'est pourquoi j'ai d\u00e9cid\u00e9 de venir vous voir.\n\n\u2014 Guide, prenez soin de vous..., lanc\u00e8rent quelques personnes dans la foule.\n\nWang Miao reconnut \u00e0 leur voix que leurs paroles venaient du fond du c\u0153ur.\n\n\u2014 Avant de passer aux choses importantes, nous devons r\u00e9gler un point de d\u00e9tail : Pan Han ! dit Ye Wenjie, les yeux toujours riv\u00e9s sur la table bien qu'elle l'ait appel\u00e9 par son nom.\n\n\u2014 Me voici, Guide.\n\nPan Han sortit de la foule au sein de laquelle il avait, plus t\u00f4t, tent\u00e9 de se fondre. Il se voulait serein, mais la peur qui l'habitait \u00e9tait flagrante. Le Guide ne l'avait pas appel\u00e9 \"camarade\", c'\u00e9tait un mauvais pr\u00e9sage.\n\n\u2014 Tu as gravement transgress\u00e9 le code de l'Organisation, continua Ye Wenjie, toujours sans le regarder, avec la voix douce d'une m\u00e8re devant un enfant ayant fait une b\u00eatise.\n\n\u2014 Guide, l'Organisation fait face \u00e0 une crise qui pourrait la mener \u00e0 l'extinction. Si nous ne prenons pas des mesures radicales pour \u00e9liminer les dissidents et les ennemis qui sont parmi nous, nous risquons de tout perdre !\n\nYe Wenjie leva la t\u00eate pour observer Pan Han. Son regard \u00e9tait tendre, mais il coupa la respiration de l'activiste pendant une poign\u00e9e de secondes.\n\n\u2014 L'id\u00e9al et l'objectif ultimes de l'organisation sont pr\u00e9cis\u00e9ment de tout perdre. Perdre tout ce qui appartient \u00e0 l'esp\u00e8ce humaine, nous y compris.\n\n\u2014 Vous \u00eates donc adventiste ! Guide, pourriez-vous le d\u00e9clarer publiquement ? C'est important pour nous autres, n'est-ce pas, camarades ? Tr\u00e8s important ! cria-t-il en levant le bras.\n\nIl regarda tout autour de lui, mais la foule \u00e9tait toujours plong\u00e9e dans le silence. Personne ne lui avait fait \u00e9cho.\n\n\u2014 Ce n'est pas \u00e0 toi qu'il revient d'adresser cette requ\u00eate. Tu as enfreint le code. Si tu souhaites faire appel, tu peux le faire maintenant ou bien assumer la responsabilit\u00e9 de ton acte.\n\nYe Wenjie avait parl\u00e9 lentement, en d\u00e9tachant bien chaque syllabe, comme si elle craignait de ne pas \u00eatre comprise de l'enfant qu'elle \u00e9tait en train de sermonner.\n\n\u2014 Mon but \u00e9tait de nous d\u00e9barrasser du math\u00e9maticien surdou\u00e9. La d\u00e9cision a \u00e9t\u00e9 prise par le camarade Evans et adopt\u00e9e \u00e0 l'unanimit\u00e9 lors de la s\u00e9ance pl\u00e9ni\u00e8re du comit\u00e9. Si le math\u00e9maticien arrivait vraiment \u00e0 cr\u00e9er un mod\u00e8le math\u00e9matique exact du probl\u00e8me \u00e0 trois corps, les dieux ne viendraient jamais sur Terre, et le grand projet de l'Organisation serait r\u00e9duit \u00e0 n\u00e9ant. J'ai agi en \u00e9tat de l\u00e9gitime d\u00e9fense, c'est Shen Yufei qui a tir\u00e9 la premi\u00e8re.\n\nYe Wenjie hocha la t\u00eate :\n\n\u2014 Disons que pour cette fois nous te croyons. De toute fa\u00e7on, ce n'est pas le sujet qui nous pr\u00e9occupe le plus aujourd'hui. J'esp\u00e8re que nous pourrons continuer \u00e0 te faire confiance. Maintenant, r\u00e9p\u00e8te une nouvelle fois la requ\u00eate que tu viens d'adresser.\n\nPan Han resta un moment abasourdi. Il ne paraissait pas avoir retrouv\u00e9 son calme :\n\n\u2014 Je... Pourriez-vous d\u00e9clarer publiquement que vous \u00eates adventiste ? Apr\u00e8s tout, le plan d'action des adventistes est conforme \u00e0 votre id\u00e9al.\n\n\u2014 Eh bien, r\u00e9p\u00e8te donc ce fameux plan d'action.\n\n\u2014 L'humanit\u00e9 ne peut plus compter sur elle-m\u00eame pour r\u00e9soudre ses probl\u00e8mes. Elle ne peut plus compter sur ses propres forces pour mettre un terme \u00e0 sa folie. C'est pourquoi nous demandons aux dieux de venir dans notre monde afin que, gr\u00e2ce \u00e0 leur pouvoir, nous puissions surveiller et transformer par la force les soci\u00e9t\u00e9s humaines, et b\u00e2tir une nouvelle civilisation, parfaite et \u00e9clair\u00e9e.\n\n\u2014 Les adventistes sont-ils loyaux \u00e0 ce plan ?\n\n\u2014 Bien s\u00fbr ! Guide, ne croyez pas les fausses rumeurs qui circulent.\n\n\u2014 Ce ne sont pas des rumeurs ! cria un homme en se frayant un chemin dans la foule. Je m'appelle Rafael, je suis isra\u00e9lien. Il y a trois ans, mon fils de quatorze ans est mort dans un accident de voiture. J'ai fait don de son rein \u00e0 une petite Palestinienne qui souffrait d'ur\u00e9mie. Par cet acte, je voulais exprimer mon v\u0153u que les deux peuples puissent coexister en paix. J'\u00e9tais pr\u00eat \u00e0 donner ma vie pour ce r\u00eave. Beaucoup d'Isra\u00e9liens et de Palestiniens font le m\u00eame genre d'effort, avec toute la sinc\u00e9rit\u00e9 du monde. Mais tout cela est inutile. Nos territoires s'enlisent plus profond\u00e9ment chaque jour dans le bourbier de la haine et de la vengeance. J'ai perdu toute foi dans l'humanit\u00e9, et c'est ainsi que j'ai int\u00e9gr\u00e9 l'Organisation Terre-Trisolaris. D'un artisan de la paix, le d\u00e9sespoir a fait de moi un extr\u00e9miste. Gr\u00e2ce \u00e0 mes dons financiers importants \u00e0 l'Organisation, j'ai pu int\u00e9grer le noyau du courant adventiste.\n\nJe vous le dis aujourd'hui : les adventistes poss\u00e8dent leurs propres principes. Pour eux, l'humanit\u00e9 est une esp\u00e8ce mauvaise par nature. La civilisation humaine a commis trop de crimes odieux sur Terre et elle doit \u00eatre punie \u00e0 la hauteur de ses m\u00e9faits. L'objectif ultime des adventistes est de demander aux dieux d'infliger \u00e0 l'humanit\u00e9 le ch\u00e2timent qu'elle m\u00e9rite : sa destruction totale !\n\n\u2014 Le programme r\u00e9el des adventistes n'est un secret pour personne, cria quelqu'un.\n\n\u2014 Mais ce que vous ignorez, c'est que ce programme n'a pas \u00e9volu\u00e9 avec le temps, il a \u00e9t\u00e9 d\u00e9termin\u00e9 d\u00e8s le d\u00e9but ! \u00c7a a toujours \u00e9t\u00e9 le r\u00eave d'Evans, le cerveau du mouvement adventiste ! Il a tromp\u00e9 l'Organisation et a bern\u00e9 tout le monde, y compris le Guide ! Evans n'a cess\u00e9 de travailler \u00e0 atteindre cet objectif. C'est lui qui a transform\u00e9 le mouvement adventiste en royaume de la terreur, peupl\u00e9 d'\u00e9cologistes extr\u00e9mistes et de fous qui n'ont que de la haine pour l'humanit\u00e9 !\n\n\u2014 Je n'ai d\u00e9couvert les intentions r\u00e9elles d'Evans que tardivement, admit Ye Wenjie. Cependant, j'ai essay\u00e9 de recoller les morceaux pour que l'Organisation reste soud\u00e9e, mais certains actes commis par les adventistes ont r\u00e9duit ces efforts \u00e0 n\u00e9ant.\n\nPan Han reprit la parole :\n\n\u2014 Guide, le mouvement adventiste est le c\u0153ur m\u00eame de l'OTT. Sans nous, il n'y aurait jamais eu de mouvement !\n\n\u2014 Ce n'est pas une raison suffisante pour monopoliser toutes les communications entre les dieux et l'Organisation.\n\n\u2014 Nous avons construit C\u00f4te Rouge 2 ! C'est naturellement \u00e0 nous qu'il revient de la faire fonctionner !\n\n\u2014 Les adventistes ont profit\u00e9 de cet avantage pour se rendre coupables d'un acte de trahison impardonnable : vous avez intercept\u00e9 les messages que les dieux ont envoy\u00e9s \u00e0 l'Organisation et n'en avez r\u00e9v\u00e9l\u00e9 qu'une infime partie. Pire, vous en avez m\u00eame falsifi\u00e9 le contenu. Par l'interm\u00e9diaire de C\u00f4te Rouge 2, vous avez transmis une grande quantit\u00e9 d'informations aux dieux, sans avoir consult\u00e9 l'Organisation.\n\nLe silence descendit sur la salle comme une chape de plomb, provoquant des picotements sur le cr\u00e2ne de Wang Miao. Pan Han ne r\u00e9pondit rien. Son regard \u00e9tait devenu froid, comme s'il voulait dire : _Enfin, nous y sommes._\n\n\u2014 Nous avons d\u00e9j\u00e0 accumul\u00e9 beaucoup de preuves de la trahison des adventistes. La camarade Shen Yufei nous en a fourni un certain nombre. Elle appartenait autrefois au noyau dur du mouvement adventiste mais, au plus profond d'elle-m\u00eame, c'\u00e9tait une fervente r\u00e9demptoriste. Vous l'avez d\u00e9couvert trop tard, elle en savait d\u00e9j\u00e0 trop. Evans ne t'a pas envoy\u00e9 tuer une seule personne mais bien deux.\n\nPan Han jeta un \u0153il tout autour de lui, il semblait vouloir \u00e9valuer la situation. Ye Wenjie remarqua son attitude.\n\n\u2014 Tu peux voir que la plupart des camarades ici pr\u00e9sents sont des r\u00e9demptoristes. Je suis persuad\u00e9e que la minorit\u00e9 adventiste ici pr\u00e9sente respectera les d\u00e9cisions de l'Organisation. Mais des gens comme Evans et toi sont irr\u00e9cup\u00e9rables. Pour prot\u00e9ger notre plan et notre id\u00e9al, il nous faut r\u00e9soudre en profondeur le probl\u00e8me des adventistes.\n\nLe silence tomba une nouvelle fois sur la salle. Deux ou trois minutes plus tard, l'une des gardes du corps de Wenjie, une svelte et jolie jeune femme, eut un rictus. Tous port\u00e8rent leurs regards vers elle. D'un pas f\u00e9lin, elle s'approcha de Pan Han.\n\nLe visage de ce dernier changea de couleur, il ins\u00e9ra sa main dans la poche int\u00e9rieure de sa veste, mais en un \u00e9clair, la jeune femme bondit sur lui. Si vite que personne ne la vit faire, elle passa un bras d\u00e9licat mais aussi souple que du lierre de printemps autour du cou de Pan Han. Elle pla\u00e7a son autre main sur le sommet de son cr\u00e2ne et, en pressant \u00e0 un angle pr\u00e9cis avec une force inou\u00efe et une expertise certaine, elle fit pivoter sa t\u00eate de cent quatre-vingts degr\u00e9s. Dans le silence de la salle, on entendit le craquement de sa vert\u00e8bre cervicale. La jeune femme \u00f4ta alors rapidement ses mains, comme si le cr\u00e2ne \u00e9tait br\u00fblant. Pan Han s'\u00e9croula sur le sol. Le pistolet qui avait servi \u00e0 tuer Shen Yufei roula sous la table. Son corps \u00e9tait encore pris de convulsions, mais ses yeux exorbit\u00e9s et sa langue pendante \u00e9taient immobiles, comme s'ils n'avaient jamais appartenu au reste de son corps. Quelqu'un approcha pour emporter la carcasse de Pan Han. Un filet de sang se r\u00e9pandit tout le long du chemin o\u00f9 il fut tra\u00een\u00e9.\n\n\u2014 Ah, Miao, tu es venu aussi, bonjour.\n\nYe Wenjie lui fit un signe de t\u00eate et lui sourit affectueusement.\n\nPuis, se tournant vers les autres, elle expliqua :\n\n\u2014 Je vous pr\u00e9sente mon ami le Pr Wang Miao, de l'Acad\u00e9mie nationale des sciences. Il travaille sur les nanomat\u00e9riaux, la premi\u00e8re technologie terrestre que les dieux souhaitent an\u00e9antir.\n\nPersonne ne regarda Wang Miao, et ce dernier n'eut pas la force de s'exprimer. Il dut attraper la manche d'une personne qui se trouvait \u00e0 c\u00f4t\u00e9 de lui pour ne pas perdre l'\u00e9quilibre, mais celle-ci le repoussa d\u00e9licatement.\n\n\u2014 Miao, je vais \u00e0 pr\u00e9sent poursuivre l'histoire de C\u00f4te Rouge que j'ai commenc\u00e9 \u00e0 te raconter lors de notre derni\u00e8re rencontre. Camarades, \u00e9coutez attentivement. Ce n'est pas une perte de temps. En ces heures extraordinaires, il nous faut revenir \u00e0 la gen\u00e8se de l'Organisation.\n\n\u2014 La base de C\u00f4te Rouge ? Vous n'aviez donc pas termin\u00e9 votre histoire ? demanda maladroitement Wang Miao.\n\nYe Wenjie s'approcha \u00e0 pas lents de la sculpture des trois corps, comme envo\u00fbt\u00e9e par les sph\u00e8res en l\u00e9vitation. \u00c0 travers la fen\u00eatre bris\u00e9e de la caf\u00e9t\u00e9ria, les rayons du cr\u00e9puscule \u00e9clairaient l'\u0153uvre d'art et les sph\u00e8res dansantes refl\u00e9taient de fa\u00e7on irr\u00e9guli\u00e8re la lumi\u00e8re du soleil sur le Guide des rebelles, comme si des flammes chatoyantes dansaient sur son corps.\n\n\u2014 Non. Ce n'\u00e9tait que le d\u00e9but, dit Ye Wenjie \u00e0 voix basse.\n\n## 22\n\n## C\u00d4TE ROUGE V\n\n\u00c0 son entr\u00e9e dans la base, Ye Wenjie ne s'\u00e9tait jamais imagin\u00e9 qu'elle pourrait en sortir un jour. Apr\u00e8s avoir pris connaissance de l'objectif r\u00e9el du projet C\u00f4te Rouge \u2013 une information si secr\u00e8te que de nombreux cadres interm\u00e9diaires de la base l'ignoraient eux-m\u00eames \u2013, elle avait mentalement coup\u00e9 tout contact avec le monde ext\u00e9rieur et s'\u00e9tait totalement abandonn\u00e9e \u00e0 son travail. D\u00e8s lors, elle avait eu l'occasion de p\u00e9n\u00e9trer plus profond\u00e9ment encore au c\u0153ur de la technologie de C\u00f4te Rouge et avait commenc\u00e9 \u00e0 assumer de plus grandes responsabilit\u00e9s en termes de recherche.\n\nLei Zhicheng en voulait \u00e0 Yang Weining d'avoir fait confiance \u00e0 Ye Wenjie, mais il acceptait tout de m\u00eame volontiers de confier \u00e0 celle-ci des projets de recherche plus importants. \u00c9tant donn\u00e9 son statut, Ye Wenjie ne poss\u00e9dait de toute fa\u00e7on aucun droit sur les r\u00e9sultats de ses recherches. Seul autre dipl\u00f4m\u00e9 d'astrophysique de toute la base, Lei Zhicheng \u00e9tait \u00e0 l'\u00e9poque l'un des rares cadres politiques \u00e0 \u00eatre issus du monde intellectuel. Aussi, il signait de son nom les r\u00e9sultats et les articles scientifiques de Wenjie, ce qui faisait de lui un cadre politique mod\u00e8le, \u00e0 la fois fervent r\u00e9volutionnaire et savant technicien.\n\nLa raison premi\u00e8re qui avait permis \u00e0 Ye Wenjie de rejoindre la base, c'\u00e9tait une \u00e9tude math\u00e9matique autour du soleil qu'elle avait publi\u00e9e dans _The Journal of Astrophysics_ , lorsqu'elle \u00e9tait encore \u00e9tudiante de deuxi\u00e8me cycle. Par comparaison avec la Terre, le soleil \u00e9tait un syst\u00e8me physique autrement plus simple : il \u00e9tait presque enti\u00e8rement compos\u00e9 de deux \u00e9l\u00e9ments, l'hydrog\u00e8ne et l'h\u00e9lium. M\u00eame si les processus physiques y \u00e9taient complexes, il ne s'agissait que de variations de la fusion de l'hydrog\u00e8ne en h\u00e9lium. Il \u00e9tait donc possible de construire un mod\u00e8le math\u00e9matique susceptible de d\u00e9crire plus pr\u00e9cis\u00e9ment la nature du soleil. Son article \u00e9tait assez basique, mais Yang Weining et Lei Zhicheng y avaient vu un espoir de r\u00e9soudre les probl\u00e8mes techniques auxquels faisait face le syst\u00e8me de surveillance de C\u00f4te Rouge.\n\nLes pannes solaires constituaient par exemple des obstacles r\u00e9guliers au bon fonctionnement du syst\u00e8me. Cette expression de \"panne solaire\" \u00e9tait n\u00e9e en m\u00eame temps que la technologie des satellites de communication, toute r\u00e9cente. Lorsque la Terre, le satellite et le soleil \u00e9taient align\u00e9s, les antennes de r\u00e9ception au sol avaient le soleil pour fond. Celui-ci \u00e9tant une immense source de rayonnement \u00e9lectromagn\u00e9tique, les transmissions par ondes des satellites \u00e9taient soumises \u00e0 de fortes perturbations. Vingt ans plus tard, au XXIe si\u00e8cle, ce probl\u00e8me n'avait toujours pas \u00e9t\u00e9 r\u00e9solu.\n\nC\u00f4te Rouge faisait face \u00e0 des perturbations de ce type, \u00e0 la diff\u00e9rence pr\u00e8s que la source des perturbations (le soleil) se situait entre la source des transmissions (l'espace) et l'antenne de r\u00e9ception au sol. En comparaison avec des satellites de communication, les pannes solaires dont \u00e9tait victime C\u00f4te Rouge \u00e9taient plus fr\u00e9quentes et plus graves. Les infrastructures de la base \u00e9taient en r\u00e9alit\u00e9 bien plus modestes que ce qui avait \u00e9t\u00e9 imagin\u00e9 lors de sa conception. Les unit\u00e9s de surveillance et de transmission partageaient par exemple une seule et m\u00eame antenne. Le temps consacr\u00e9 aux activit\u00e9s de surveillance \u00e9tait donc d'autant plus pr\u00e9cieux, et les pannes solaires d'autant plus dommageables.\n\nL'id\u00e9e de Yang Weining et de Lei Zhicheng \u00e9tait simple : tout d'abord, d\u00e9terminer la fr\u00e9quence et les particularit\u00e9s des rayonnements \u00e9lectromagn\u00e9tiques solaires qui perturbaient le syst\u00e8me de r\u00e9ception et, ensuite, les filtrer num\u00e9riquement. Ils \u00e9taient tous deux techniciens, ce qui \u00e9tait certes tr\u00e8s appr\u00e9ciable pour l'\u00e9poque. Cependant, Yang Weining n'\u00e9tait pas sp\u00e9cialiste d'astrophysique et Lei Zhicheng s'\u00e9tant destin\u00e9 \u00e0 une carri\u00e8re politique, ses connaissances dans ce domaine n'\u00e9taient pas aussi expertes. En r\u00e9alit\u00e9, le spectre \u00e9lectromagn\u00e9tique du soleil \u00e9tait stable uniquement entre le domaine de rayonnement ultraviolet proche et le domaine de rayonnement infrarouge moyen \u2013 ce qui incluait ainsi le domaine de la lumi\u00e8re visible. Pour tous les autres domaines, le rayonnement du soleil s'av\u00e9rait instable.\n\nYe Wenjie avait brillamment montr\u00e9 dans son premier rapport de recherche que, durant les intenses p\u00e9riodes d'activit\u00e9 solaire \u2013 taches solaires, \u00e9ruptions solaires, \u00e9jections de masse coronale, etc. \u2013, il \u00e9tait impossible de supprimer les perturbations solaires. L'objet de ses recherches se limitait donc aux rayonnements \u00e9lectromagn\u00e9tiques situ\u00e9s dans les gammes de fr\u00e9quences surveill\u00e9es par C\u00f4te Rouge et durant les phases normales de l'activit\u00e9 solaire.\n\nLa base b\u00e9n\u00e9ficiait de conditions de recherche satisfaisantes. La biblioth\u00e8que disposait d'une collection assez compl\u00e8te d'ouvrages de r\u00e9f\u00e9rence en langue \u00e9trang\u00e8re qu'il lui \u00e9tait possible de consulter et elle recevait les derniers num\u00e9ros des revues universitaires am\u00e9ricaines et europ\u00e9ennes \u2013 une chose rare \u00e0 l'\u00e9poque. Ye Wenjie pouvait \u00e9galement utiliser la ligne t\u00e9l\u00e9phonique militaire pour contacter deux \u00e9quipes de l'Acad\u00e9mie chinoise des sciences qui menaient des recherches sur les ph\u00e9nom\u00e8nes solaires. Elle recevait r\u00e9guli\u00e8rement par fax leurs donn\u00e9es et leurs observations.\n\nLa recherche de Ye Wenjie se poursuivit durant six mois, mais elle n'entrevoyait aucun espoir de r\u00e9ussite. Elle avait rapidement d\u00e9couvert que, dans les gammes de fr\u00e9quences surveill\u00e9es par C\u00f4te Rouge, les rayonnements solaires \u00e9taient impr\u00e9visibles. Apr\u00e8s une analyse d'une grande somme de donn\u00e9es d'observation, elle d\u00e9couvrit un myst\u00e8re d\u00e9concertant. Parfois, alors qu'avait lieu une soudaine variation du rayonnement solaire, l'activit\u00e9 \u00e0 la surface du soleil apparaissait paradoxalement calme. Ce n'\u00e9tait pas un cas isol\u00e9, car un bon millier de donn\u00e9es d'observations documentaient le m\u00eame ph\u00e9nom\u00e8ne. Les rayonnements micro-ondes et d'ondes courtes issus du noyau du soleil ne pouvant traverser les centaines de milliers de kilom\u00e8tres constituant la couche du soleil, ces rayonnements devaient par cons\u00e9quent provenir de sa surface. Une activit\u00e9 solaire particuli\u00e8re aurait d\u00fb \u00eatre observ\u00e9e lors du d\u00e9clenchement de ces perturbations. Or, aucune activit\u00e9 correspondante n'apparaissait \u00e0 la surface du soleil. Quel ph\u00e9nom\u00e8ne provoquait donc les variations soudaines de ces \u00e9troites gammes de fr\u00e9quences ? Plus elle y pensait, plus ce myst\u00e8re l'intriguait.\n\nApr\u00e8s avoir \u00e9puis\u00e9 toutes les hypoth\u00e8ses, Ye Wenjie d\u00e9cida d'abandonner. Dans le dernier rapport qu'elle avait r\u00e9dig\u00e9, elle avouait son incomp\u00e9tence \u00e0 r\u00e9soudre le probl\u00e8me. L'\u00e9chec ne serait pas mal per\u00e7u, car les deux \u00e9quipes universitaires jadis consult\u00e9es par les militaires s'\u00e9taient retrouv\u00e9es dans la m\u00eame impasse. Yang Weining avait simplement voulu essayer une nouvelle fois, en faisant appel \u00e0 l'expertise de Ye Wenjie. Le but de Lei Zhicheng \u00e9tait encore plus simple : il voulait l'article de Wenjie. Ce sujet de recherche \u00e9tait tr\u00e8s th\u00e9orique et lui permettrait de faire valoir son niveau scientifique. La terrible fi\u00e8vre de la R\u00e9volution culturelle commen\u00e7ait d\u00e9sormais \u00e0 retomber et les exigences envers les cadres du Parti \u00e9taient en train de changer. Sa maturit\u00e9 politique et ses insignes succ\u00e8s acad\u00e9miques lui conf\u00e9raient un profil rare et le destinaient \u00e0 une carri\u00e8re prometteuse. Il se pr\u00e9occupait donc peu de savoir si le probl\u00e8me des pannes solaires pouvait ou non \u00eatre r\u00e9solu.\n\nMais finalement, Ye Wenjie ne rendit aucun rapport. Elle s'\u00e9tait dit que si elle d\u00e9clarait avoir termin\u00e9 ses recherches, la biblioth\u00e8que cesserait ses commandes d'ouvrages et de revues sur ce th\u00e8me. Elle n'aurait plus jamais acc\u00e8s \u00e0 ces riches collections relatives \u00e0 l'astrophysique. Aussi \u2013 du moins officiellement \u2013 poursuivait-elle ses recherches, tandis qu'elle concentrait d\u00e9sormais ses efforts sur la conception de son mod\u00e8le math\u00e9matique solaire.\n\nCe soir-l\u00e0, Ye Wenjie \u00e9tait, comme d'ordinaire, la seule personne \u00e0 se trouver dans la glaciale salle de lecture de la biblioth\u00e8que. Sur la longue table devant laquelle elle \u00e9tait assise \u00e9tait pos\u00e9e une pile de livres et de revues. Apr\u00e8s avoir achev\u00e9 les calculs d'une matrice complexe, elle se frotta les mains pour les r\u00e9chauffer et saisit le dernier num\u00e9ro de la revue _Journal of Astrophysic_ s, pour prendre une pause. Alors qu'elle le feuilletait, une note portant sur Jupiter attira son attention :\n\n> Dans l'article \"Une nouvelle et puissante source de rayonnement dans le syst\u00e8me solaire\" paru dans notre dernier num\u00e9ro, notre confr\u00e8re, le Dr Harry Petterson de l'observatoire du mont Wilson, a publi\u00e9 une liste de donn\u00e9es obtenues lors de l'observation de la pr\u00e9cession de Jupiter, entre le 12 juin et le 2 juillet. Durant cette observation, le Dr Petterson a fait la d\u00e9couverte fortuite de deux intenses rayonnements \u00e9lectromagn\u00e9tiques provenant de Jupiter elle-m\u00eame. La dur\u00e9e de chaque rayonnement a respectivement \u00e9t\u00e9 de 81 et 76 secondes. Les donn\u00e9es publi\u00e9es par Petterson comprennent aussi les gammes de fr\u00e9quences, ainsi que d'autres param\u00e8tres. Au cours de ces sursauts radio, Petterson a not\u00e9 des variations de la Grande Tache rouge de Jupiter. Les sursauts radio jupit\u00e9riens ont suscit\u00e9 depuis un grand int\u00e9r\u00eat dans la communaut\u00e9 des plan\u00e9tologues. Dans ce num\u00e9ro, G. Mackenzie revient dans son article sur cette d\u00e9couverte en proposant l'hypoth\u00e8se qu'il pourrait s'agir d'un signe avant-coureur du d\u00e9clenchement d'une fusion nucl\u00e9aire sur Jupiter. Dans notre prochain num\u00e9ro, nous publierons un article de Kumoishi Inoue qui attribue pour sa part les sursauts radio jupit\u00e9riens \u00e0 un m\u00e9canisme plus complexe : les mouvements internes de plaques d'hydrog\u00e8ne m\u00e9tallique. Il en donnera une description math\u00e9matique compl\u00e8te.\n\nYe Wenjie se rappelait avec nettet\u00e9 les deux dates dont il \u00e9tait question. Ces jours-l\u00e0, le syst\u00e8me de surveillance de C\u00f4te Rouge avait \u00e9t\u00e9 soumis \u00e0 de fortes perturbations provoqu\u00e9es par des pannes solaires. Elle consulta le journal des op\u00e9rations de l'unit\u00e9, qui ne fit que confirmer ses souvenirs. Les dates \u00e9taient proches, \u00e0 la diff\u00e9rence pr\u00e8s que les pannes solaires avaient eu lieu sur Terre six minutes et quarante-deux secondes apr\u00e8s les sursauts radio de Jupiter. Ces six minutes et quarante-deux secondes \u00e9taient la clef ! Wenjie essaya de calmer les palpitations furieuses de son c\u0153ur. Elle demanda au biblioth\u00e9caire de contacter l'Observatoire national pour obtenir les \u00e9ph\u00e9m\u00e9rides de Jupiter et de la Terre correspondant \u00e0 ces deux p\u00e9riodes.\n\nElle tra\u00e7a sur le tableau noir un grand triangle aux sommets duquel elle pla\u00e7a le soleil, la Terre et Jupiter. Elle inscrivit les distances le long des trois c\u00f4t\u00e9s, et \u00e9crivit \u00e0 c\u00f4t\u00e9 du sommet \"Terre\" les heures auxquelles elle avait observ\u00e9 les perturbations. Il \u00e9tait facile de calculer le temps du trajet mis par les rayonnements \u00e9lectromagn\u00e9tiques entre Jupiter et la Terre. Elle calcula ensuite le temps qu'il avait fallu aux rayonnements pour relier Jupiter au soleil et le soleil \u00e0 la Terre. La diff\u00e9rence entre les deux \u00e9tait d'exactement six minutes et quarante-deux secondes.\n\nYe Wenjie se reporta \u00e0 son mod\u00e8le math\u00e9matique de la structure solaire et essaya de chercher le moindre indice d'une th\u00e9orie pouvant expliquer le ph\u00e9nom\u00e8ne. Ses yeux se riv\u00e8rent sur ce qu'elle avait jadis appel\u00e9 dans son travail les \"miroirs d'\u00e9nergie\" de la zone radiative du soleil.\n\nL'\u00e9nergie lib\u00e9r\u00e9e lors des r\u00e9actions de fusion appara\u00eet en premier lieu sous la forme de rayons gamma de haute \u00e9nergie. La zone radiative absorbe ces photons de haute \u00e9nergie et les \u00e9met \u00e0 nouveau mais, cette fois, \u00e0 une \u00e9nergie plus basse. Apr\u00e8s un long processus r\u00e9p\u00e9t\u00e9 d'absorption et de r\u00e9\u00e9mission (un photon peut parfois mettre mille ans avant de quitter le soleil), les rayons gamma deviennent des rayons X, des rayons ultraviolets extr\u00eames, puis des rayons ultraviolets, avant de se transformer petit \u00e0 petit en lumi\u00e8re visible et en d'autres formes de rayonnement.\n\nCe ph\u00e9nom\u00e8ne \u00e9tait d\u00e9j\u00e0 bien connu des chercheurs travaillant sur les activit\u00e9s du soleil. Le nouveau r\u00e9sultat produit par le mod\u00e8le math\u00e9matique de Ye Wenjie \u00e9tait le suivant : pendant le parcours du rayonnement solaire dans la zone radiative, celui-ci passe entre des couches \u00e0 fr\u00e9quences diff\u00e9rentes. Entre deux couches adjacentes existent des interfaces et, chaque fois qu'une de ces interfaces est franchie, la fr\u00e9quence de rayonnement diminue d'un niveau.\n\nCette \u00e9laboration allait \u00e0 l'encontre de la conception traditionnelle consid\u00e9rant que la fr\u00e9quence dans la zone de radiation diminue graduellement. Ses calculs montraient que ces fameuses interfaces refl\u00e9taient les rayonnements de fr\u00e9quence faible. C'est ce qui lui avait inspir\u00e9 l'appellation de \"miroirs d'\u00e9nergie\".\n\nWenjie avait examin\u00e9 de pr\u00e8s ces fines membranes instables en suspension au-dessus de l'oc\u00e9an de plasma du soleil. Elle avait d\u00e9couvert que ces substances poss\u00e9daient des caract\u00e9ristiques exceptionnelles. L'une des plus incroyables \u00e9tait leur \"sur-r\u00e9flexivit\u00e9\". Mais cette derni\u00e8re singularit\u00e9 paraissait si extraordinaire qu'elle \u00e9tait difficile \u00e0 prouver. Ye Wenjie elle-m\u00eame avait peine \u00e0 croire qu'elle soit r\u00e9elle. Il lui semblait plus probable d'avoir commis des erreurs durant ses complexes et vertigineux calculs.\n\nMais \u00e0 pr\u00e9sent, Ye Wenjie avait fait un premier pas corroborant l'hypoth\u00e8se de la sur-r\u00e9flexivit\u00e9 des miroirs d'\u00e9nergie solaires : ces miroirs ne se contentaient pas simplement de refl\u00e9ter les rayonnements de fr\u00e9quence faible, ils les amplifiaient ! Toutes les myst\u00e9rieuses et soudaines variations \u00e0 l'int\u00e9rieur de gammes de fr\u00e9quences \u00e9troites qu'elle avait pu observer par le pass\u00e9 \u00e9taient donc le r\u00e9sultat de l'amplification par ces miroirs de rayonnements provenant de l'espace. Voil\u00e0 pourquoi aucune perturbation li\u00e9e \u00e0 ces rayonnements n'\u00e9tait visible \u00e0 la surface du soleil.\n\nCes deux fois-l\u00e0, il \u00e9tait possible que les sursauts radio re\u00e7us par le soleil en provenance de Jupiter aient \u00e9t\u00e9 \u00e9mis \u00e0 nouveau, apr\u00e8s avoir \u00e9t\u00e9 amplifi\u00e9s pr\u00e8s d'une centaine de millions de fois. Quant \u00e0 la Terre, elle avait re\u00e7u les deux rayonnements, avant et apr\u00e8s leur amplification, s\u00e9par\u00e9s par un intervalle de six minutes et quarante-deux secondes.\n\nLe soleil \u00e9tait un amplificateur d'ondes radio !\n\nMais une question se posait. Le soleil devait certainement recevoir chaque seconde des rayonnements \u00e9lectromagn\u00e9tiques provenant de l'espace, y compris les ondes radio \u00e9mises par la Terre. Pourquoi d\u00e8s lors n'en amplifiait-il qu'une seule partie ? La r\u00e9ponse \u00e9tait simple. Non seulement toutes les fr\u00e9quences n'\u00e9taient pas forc\u00e9ment refl\u00e9t\u00e9es par les miroirs d'\u00e9nergie mais, surtout, la zone convective du soleil jouait le r\u00f4le de bouclier protecteur. Sans cesse bouillonnante et visible sous la forme de granulations solaires, elle se situait au-dessus de la zone radiative. C'\u00e9tait la couche la plus externe du soleil. Les ondes radio venues de l'espace devaient donc d'abord p\u00e9n\u00e9trer cette zone de convection avant d'atteindre les miroirs d'\u00e9nergie de la zone de radiation, et \u00eatre ensuite amplifi\u00e9es avant d'\u00eatre \u00e9mises \u00e0 nouveau. Cela n\u00e9cessitait des ondes radio entrantes qu'elles d\u00e9passent un certain seuil, que la plupart des ondes radio \u00e9mises par la Terre \u00e9taient incapables d'atteindre, au contraire des sursauts radio de Jupiter.\n\nLa puissance maximale de C\u00f4te Rouge permettait n\u00e9anmoins de d\u00e9passer ce seuil.\n\nSi le probl\u00e8me des pannes solaires n'avait pas \u00e9t\u00e9 r\u00e9solu, une perspective excitante en d\u00e9coulait : les humains \u00e9taient capables d'utiliser le soleil comme une super-antenne pour transmettre des ondes radio dans l'univers. Ces ondes seraient envoy\u00e9es gr\u00e2ce \u00e0 l'\u00e9nergie fournie par le soleil et leur puissance d'\u00e9mission serait des centaines de millions de fois sup\u00e9rieure \u00e0 la totalit\u00e9 de l'\u00e9nergie terrestre.\n\nLa civilisation terrienne avait la capacit\u00e9 de transmission de communication d'une civilisation de type II sur l'\u00e9chelle de Kardachev !\n\nLa prochaine \u00e9tape consistait \u00e0 comparer les formes d'ondes des deux sursauts radio de Jupiter avec celles des pannes solaires dont C\u00f4te Rouge \u00e9tait victime. Si celles-ci correspondaient, un pas \u00e9norme serait fait dans la v\u00e9rification de cette hypoth\u00e8se.\n\nYe Wenjie demanda aux responsables de la base de pouvoir contacter le Dr Harry Petterson pour obtenir ses donn\u00e9es sur les formes d'ondes des sursauts radio jupit\u00e9riens. Ce n'\u00e9tait pas chose facile, car il \u00e9tait difficile de trouver les bons r\u00e9seaux et il fallait faire un nombre consid\u00e9rable de d\u00e9marches administratives aupr\u00e8s de diff\u00e9rents bureaux. La moindre erreur, et elle pouvait \u00eatre suspect\u00e9e d'espionnage. Wenjie fit ce qu'elle avait de mieux \u00e0 faire : attendre.\n\nCependant il existait un moyen bien plus direct d'\u00e9prouver son hypoth\u00e8se : envoyer, depuis C\u00f4te Rouge, une transmission d\u00e9passant le fameux seuil vers le soleil.\n\nYe Wenjie alla trouva les responsables et exprima sa requ\u00eate. Elle n'osa pas donner le v\u00e9ritable motif de cette demande \u2013 c'\u00e9tait bien trop extraordinaire \u2013 et se contenta d'indiquer que l'exp\u00e9rience s'inscrivait dans le cadre de ses recherches sur le soleil : le syst\u00e8me de transmission de C\u00f4te Rouge servirait de radar d'exploration et les \u00e9chos que recevrait la base seraient utilis\u00e9s comme donn\u00e9es pour analyser les rayonnements \u00e9lectromagn\u00e9tiques solaires. Lei Zhicheng et Yang Weining avaient tous deux un pass\u00e9 de techniciens et il n'\u00e9tait pas facile de les berner. N\u00e9anmoins, l'exp\u00e9rience d\u00e9crite par Ye Wenjie avait des ant\u00e9c\u00e9dents en Occident. De plus, son projet de radar d'exploration solaire \u00e9tait techniquement plus simple que les radars utilis\u00e9s pour explorer d'autres plan\u00e8tes telluriques.\n\n\u2014 Wenjie, tes demandes sont de plus en plus d\u00e9mesur\u00e9es. Tu devais limiter tes recherches \u00e0 des questions th\u00e9oriques. Est-il vraiment si n\u00e9cessaire de provoquer tout ce remue-m\u00e9nage ? dit Lei Zhicheng en secouant la t\u00eate.\n\n\u2014 Commissaire Lei, j'ai peut-\u00eatre fait une grande d\u00e9couverte, je dois absolument mener cette exp\u00e9rience. Juste une fois, je vous en prie, l'implora Wenjie.\n\nYang Weining lui apporta son soutien :\n\n\u2014 Commissaire, peut-\u00eatre pourrions-nous lui accorder cette exp\u00e9rience ? Apr\u00e8s tout, ce n'est techniquement pas si difficile, le temps de r\u00e9ception des \u00e9chos apr\u00e8s transmission est tout au plus de...\n\n\u2014 Dix minutes, compl\u00e9ta Lei Zhicheng.\n\n\u2014 C'est tout juste le temps dont a besoin le syst\u00e8me de C\u00f4te Rouge pour passer du mode de transmission au mode de surveillance.\n\nLei Zhicheng secoua une nouvelle fois la t\u00eate :\n\n\u2014 Je sais bien que cela ne pose pas de probl\u00e8me ni en termes techniques ni en termes de charge de travail, mais... ah, ing\u00e9nieur Yang ! O\u00f9 as-tu la t\u00eate ? \u00c9mettre des ondes radio surpuissantes vers le soleil, tu n'as donc pas r\u00e9fl\u00e9chi au symbole politique que cela repr\u00e9sente1 ?\n\nYang Weining et Ye Wenjie \u00e9chang\u00e8rent un regard. Ils \u00e9taient abasourdis, non parce qu'ils trouvaient cet argument ridicule, mais parce qu'ils \u00e9taient effray\u00e9s \u00e0 l'id\u00e9e de ne pas y avoir pens\u00e9. \u00c0 l'\u00e9poque, l'obsession absurde qui consistait \u00e0 trouver partout des m\u00e9taphores politiques touchait \u00e0 son paroxysme. Certains gardes rouges avaient par exemple d\u00e9cr\u00e9t\u00e9 qu'il serait d\u00e9sormais interdit de tourner \u00e0 droite, il fallait uniquement tourner \u00e0 gauche. Les feux pi\u00e9tons avaient \u00e9t\u00e9 modifi\u00e9s : on devait dor\u00e9navant s'arr\u00eater au vert et passer au rouge (une r\u00e8gle \u00e0 laquelle le Premier ministre Zhou Enlai mit plus tard un terme). Une pi\u00e8ce de monnaie avait jadis \u00e9t\u00e9 frapp\u00e9e \u00e0 l'effigie d'une foule de paysans, parmi lesquels deux portaient une pelle et une houe, or les syllabes signifiant \"pelle\" et \"houe\" \u00e9taient en chinois homophones du mot \"\u00e9liminer\" : des gardes rouges trop z\u00e9l\u00e9s y avaient vu un message contre-r\u00e9volutionnaire pr\u00f4nant l'\u00e9limination de la classe paysanne. L'individu \u00e0 l'origine de la conception de la pi\u00e8ce avait m\u00eame \u00e9t\u00e9 s\u00e9v\u00e8rement ch\u00e2ti\u00e9. On racontait qu'un autre individu avait accroch\u00e9 dans sa maison un portrait de Mao Zedong, qu'il avait lui-m\u00eame encadr\u00e9 : cet outrage lui avait valu pr\u00e8s de dix ans de prison...\n\nAu d\u00e9but, lorsque Ye Wenjie transmettait ses rapports, ceux-ci devaient \u00eatre minutieusement relus par Lei Zhicheng, et des termes techniques li\u00e9s au soleil devaient \u00eatre fr\u00e9quemment modifi\u00e9s, comme par exemple celui de \"taches solaires2\". Elle pouvait bien avoir mille raisons valables d'effectuer une exp\u00e9rience de transmission d'ondes radio vers le soleil, aussi longtemps qu'une interpr\u00e9tation politique \u00e9tait possible, le danger existait. La raison invoqu\u00e9e par Lei Zhicheng pour rejeter l'exp\u00e9rience ne pouvait pas \u00eatre discut\u00e9e.\n\nYe Wenjie ne l\u00e2cha pas l'affaire pour autant. En fait, tant qu'elle ne prenait pas de risque inconsid\u00e9r\u00e9, l'exp\u00e9rience ne s'av\u00e9rait pas si difficile \u00e0 r\u00e9aliser. L'appareil de transmission de C\u00f4te Rouge \u00e9tait tr\u00e8s puissant, mais tous ses composants \u00e9taient produits sur le territoire national, \u00e0 l'\u00e9poque de la R\u00e9volution culturelle. Comme ils \u00e9taient de qualit\u00e9 bien inf\u00e9rieure \u00e0 la moyenne, leur taux de dysfonctionnement \u00e9tait \u00e9lev\u00e9. Les techniciens \u00e9taient donc contraints de proc\u00e9der \u00e0 une r\u00e9vision int\u00e9grale des appareils toutes les quinze transmissions, puis, apr\u00e8s chaque r\u00e9vision, d'effectuer un essai de transmission. Peu de gens participaient \u00e0 cet essai, et les cibles ainsi que les param\u00e8tres de transmission \u00e9taient d\u00e9termin\u00e9s arbitrairement.\n\nCe jour-l\u00e0, Wenjie avait \u00e9t\u00e9 d\u00e9sign\u00e9e pour proc\u00e9der \u00e0 l'essai apr\u00e8s la r\u00e9vision du syst\u00e8me. L'essai de transmission \u00e9tant exempt\u00e9 de toutes les \u00e9tapes habituelles, il n'y avait en dehors de Wenjie que cinq employ\u00e9s dans la pi\u00e8ce. Trois d'entre eux n'\u00e9taient que des op\u00e9rateurs qui, ne comprenant pas r\u00e9ellement les principes de fonctionnement de l'appareil, se cantonnaient d'ordinaire \u00e0 l'ex\u00e9cution des ordres de transmission. Les deux autres \u00e9taient respectivement technicien et ing\u00e9nieur. \u00c9reint\u00e9s apr\u00e8s deux jours pass\u00e9s \u00e0 s'occuper de la r\u00e9vision du mat\u00e9riel, ils n'avaient pas la t\u00eate \u00e0 \u00e7a. Wenjie commen\u00e7a par r\u00e9gler la puissance de transmission juste au-dessus du seuil d\u00e9fini par sa th\u00e9orie sur la sur-r\u00e9flexivit\u00e9 du soleil \u2013 ce qui \u00e9tait d\u00e9j\u00e0 la puissance maximale de la base de C\u00f4te Rouge. Elle r\u00e9gla ensuite la fr\u00e9quence de mani\u00e8re qu'elle ait la plus grande probabilit\u00e9 d'\u00eatre amplifi\u00e9e par les miroirs d'\u00e9nergie. Au pr\u00e9texte de vouloir tester les propri\u00e9t\u00e9s m\u00e9caniques de l'antenne, elle s\u00e9lectionna comme cible de transmission le soleil, qui se trouvait alors \u00e0 l'ouest du ciel. Elle d\u00e9marra ensuite la transmission, avec le m\u00eame contenu que lors des envois habituels.\n\nC'\u00e9tait un apr\u00e8s-midi clair de l'automne 1971. Plus tard, Ye Wenjie y repenserait souvent mais, \u00e0 cet instant, elle n'avait aucune sensation particuli\u00e8re, hormis un peu d'anxi\u00e9t\u00e9, et l'envie que tout soit rapidement termin\u00e9. Tout d'abord, elle avait peur de se faire remarquer par ses coll\u00e8gues de la base. Bien qu'elle ait d\u00e9j\u00e0 song\u00e9 \u00e0 des excuses, l'utilisation de la puissance maximale de transmission, qui d\u00e9t\u00e9riorait les composants, \u00e9tait anormale lors d'un test. Deuxi\u00e8mement, le syst\u00e8me de transmission de C\u00f4te Rouge n'\u00e9tait pas pr\u00e9vu pour cibler le soleil. Ye Wenjie pouvait sentir rien qu'en le touchant que le syst\u00e8me optique \u00e9tait en train de chauffer ; si le tout prenait feu, elle aurait de graves probl\u00e8mes.\n\n\u00c0 l'ouest, le soleil \u00e9tait peu \u00e0 peu en train de se coucher. Le syst\u00e8me de suivi automatique des cibles de transmission \u00e9tant hors service, Wenjie dut suivre le signal manuellement. L'antenne de C\u00f4te Rouge se tourna et pivota lentement dans la direction du coucher du soleil, comme un immense tournesol. Lorsque le signal de la lumi\u00e8re rouge indiquant la fin de la transmission s'\u00e9claira, elle \u00e9tait tremp\u00e9e de sueur.\n\nElle jeta un coup d'\u0153il autour d'elle. Les trois op\u00e9rateurs \u00e9taient en train d'\u00e9teindre les appareils les uns apr\u00e8s les autres, en suivant les instructions d'un manuel technique qu'ils tenaient entre les mains. L'ing\u00e9nieur buvait un verre d'eau dans la console de contr\u00f4le. Le technicien, lui, s'\u00e9tait assoupi sur sa chaise. Les historiens et les \u00e9crivains gratifi\u00e8rent plus tard cette sc\u00e8ne de descriptions \u00e9piques mais, ce jour-l\u00e0, tout avait \u00e9t\u00e9 d'une parfaite banalit\u00e9.\n\nLa transmission termin\u00e9e, Wenjie sortit en h\u00e2te de la console de contr\u00f4le et courut jusqu'au bureau de Yang Weining.\n\n\u2014 Vite ! Donnez \u00e0 la station radio de la base l'ordre de se r\u00e9gler sur la bande de fr\u00e9quences 12 000 MHz ! lui dit-elle, le souffle court.\n\n\u2014 Que recevons-nous ?\n\nL'ing\u00e9nieur en chef Yang regarda avec \u00e9tonnement Ye Wenjie, dont les cheveux collaient au visage \u00e0 cause de la transpiration. Par comparaison avec le syst\u00e8me de r\u00e9ception ultrasensible de C\u00f4te Rouge, utiliser la radio militaire ordinaire \u2013 cantonn\u00e9e habituellement aux communications avec l'ext\u00e9rieur de la base \u2013 relevait du jeu d'enfant.\n\n\u2014 Nous allons peut-\u00eatre recevoir quelque chose. Nous n'avons pas le temps de passer les syst\u00e8mes de C\u00f4te Rouge sur le mode surveillance ! dit Wenjie.\n\nEn temps normal, la mise en chauffe du syst\u00e8me de surveillance n'aurait pris qu'une dizaine de minutes, mais celui-ci \u00e9tait aussi en r\u00e9vision et de nombreux composants n'avaient pas encore \u00e9t\u00e9 r\u00e9assembl\u00e9s.\n\nYang Weining observa Wenjie quelques secondes, puis il prit son t\u00e9l\u00e9phone et donna au bureau des communications les instructions souhait\u00e9es par Wenjie.\n\n\u2014 Avec le degr\u00e9 de sensibilit\u00e9 de cette radio, nous arriverons tout au plus \u00e0 capter des signaux d'extraterrestres vivant sur la Lune.\n\n\u2014 Le signal viendra du soleil, dit Wenjie.\n\nDerri\u00e8re la fen\u00eatre, les lisi\u00e8res du soleil avaient d\u00e9j\u00e0 atteint les sommets rouge sang des montagnes de l'horizon.\n\n\u2014 Tu as utilis\u00e9 le syst\u00e8me de transmission de C\u00f4te Rouge pour envoyer un signal vers le soleil ? demanda Yang Weining, inquiet.\n\nWenjie hocha la t\u00eate.\n\n\u2014 Personne ne doit \u00eatre mis au courant. Cela ne doit plus jamais arriver, tu m'entends ? Jamais ! souffla Yang Weining en regardant la porte avec prudence.\n\nWenjie hocha \u00e0 nouveau la t\u00eate.\n\n\u2014 Mais que cherches-tu ? Les \u00e9chos sont extr\u00eamement faibles, une radio conventionnelle n'a pas la sensibilit\u00e9 suffisante pour les entendre.\n\n\u2014 Non. Si mes pr\u00e9visions sont exactes, nous recevrons un \u00e9cho puissant comme... un \u00e9cho inimaginable. Si la puissance de transmission d\u00e9passe un certain seuil, le soleil... le soleil peut amplifier les ondes radio une centaine de millions de fois !\n\nYang Weining regarda une nouvelle fois Wenjie avec surprise. Elle demeura silencieuse. Ils attendirent ensemble sans rien se dire. Yang Weining pouvait entendre la respiration et les battements saccad\u00e9s du c\u0153ur de Wenjie. Il n'avait pas vraiment fait attention \u00e0 ce qu'elle avait dit, mais les sentiments qu'il avait dissimul\u00e9s en lui depuis tant d'ann\u00e9es refaisaient maintenant surface. Il ne pouvait que les refr\u00e9ner et attendre. Vingt minutes plus tard, il prit le t\u00e9l\u00e9phone, appela le bureau des communications, et posa quelques questions simples.\n\n\u2014 Ils n'ont rien re\u00e7u, dit Yang Weining, en raccrochant le combin\u00e9.\n\nWenjie poussa un grand soupir. Ce ne fut qu'apr\u00e8s un long moment qu'elle hocha finalement la t\u00eate.\n\n\u2014 L'astronome am\u00e9ricain, lui, a r\u00e9pondu, dit Yang Weining en sortant une \u00e9paisse enveloppe qu'il tendit \u00e0 Wenjie.\n\nL'enveloppe \u00e9tait couverte de tampons de douane. Impatiente, Wenjie la d\u00e9cacheta aussit\u00f4t et balaya son contenu des yeux. Dans sa lettre, Harry Petterson expliquait qu'il n'avait pas imagin\u00e9 que des coll\u00e8gues chinois \u00e9tudiaient aussi les rayonnements \u00e9lectromagn\u00e9tiques des plan\u00e8tes. Il se disait tout dispos\u00e9 \u00e0 \u00e9changer et \u00e0 collaborer. Petterson avait \u00e9galement envoy\u00e9 deux piles de papiers : les rapports complets des formes d'ondes observ\u00e9es lors des deux sursauts de Jupiter. Les formes d'ondes avaient apparemment \u00e9t\u00e9 photocopi\u00e9es \u00e0 partir d'un seul long document et il fallait \u00e0 pr\u00e9sent les assembler. Peu de Chinois de l'\u00e9poque avaient eu l'occasion de voir un photocopieur \u00e0 l'\u0153uvre. Wenjie se saisit de la dizaine de photocopies et les disposa \u00e0 m\u00eame le sol en commen\u00e7ant \u00e0 les aligner en deux rang\u00e9es. Mais \u00e0 peine avait-elle reconstitu\u00e9 la moiti\u00e9 du signal qu'elle avait d\u00e9j\u00e0 perdu tout espoir. Elle connaissait trop bien les formes d'onde constat\u00e9es lors des deux pannes solaires dont C\u00f4te Rouge avait \u00e9t\u00e9 victime. Elles ne correspondaient pas \u00e0 celles de Petterson.\n\nWenjie ramassa d\u00e9licatement les feuilles qu'elle avait \u00e9tal\u00e9es sur le sol. Yang Weining s'accroupit pour l'aider. Quand il tendit un paquet de feuilles \u00e0 cette fille qu'il avait jadis profond\u00e9ment aim\u00e9e, il la vit sourire en secouant la t\u00eate. Un sourire si triste qu'il le fit frissonner.\n\n\u2014 Que se passe-t-il ? demanda-t-il, sans se rendre compte qu'il ne s'\u00e9tait jamais adress\u00e9 \u00e0 elle avec une voix aussi douce.\n\n\u2014 Rien. Rien d'autre qu'un r\u00eave dont je viens de me r\u00e9veiller.\n\nElle lui sourit et sortit du bureau avec le paquet de photocopies et l'enveloppe sous le bras. Sur le chemin de sa chambre, elle r\u00e9cup\u00e9ra au passage son panier-repas. Elle d\u00e9couvrit qu'il ne restait plus qu'un pain \u00e0 la vapeur et quelques l\u00e9gumes marin\u00e9s dans le vinaigre. L'employ\u00e9 de la caf\u00e9t\u00e9ria lui dit d'un air peu aimable qu'il allait fermer. Elle sortit et alla s'asseoir sur l'herbe, au bord de la falaise, pour manger son pain \u00e0 la vapeur froid.\n\nLe soleil venait de se coucher. Les monts du Grand Khingan \u00e9taient brumeux, comme la vie de Ye Wenjie. Un r\u00eave \u00e9tait apparu dans cette grisaille, color\u00e9 et \u00e9tincelant, mais, comme toujours, elle avait fini par se r\u00e9veiller, comme le soleil qui finit toujours par se lever, sans aucun nouvel espoir. Wenjie s'imagina soudain le reste de sa vie, d'un gris sans fin. Les larmes aux yeux, elle sourit encore, puis croqua dans son pain \u00e0 la vapeur.\n\nElle ignorait alors qu'au m\u00eame moment le premier pleur humain qui ait jamais \u00e9t\u00e9 entendu dans l'espace avait atteint le soleil, d'o\u00f9 il \u00e9tait reparti vers l'univers entier \u00e0 la vitesse de la lumi\u00e8re. Une onde radio d'une puissance stellaire, telle une somptueuse vague, \u00e9tait d\u00e9j\u00e0 au large de l'orbite de Jupiter.\n\n\u00c0 cet instant pr\u00e9cis, \u00e0 une fr\u00e9quence de 12 000 MHz, le soleil \u00e9tait l'\u00e9toile la plus brillante de toute la Voie lact\u00e9e.\n\n* * *\n\n1 Mao Zedong \u00e9tait souvent d\u00e9sign\u00e9 le \"Soleil rouge\" dans les discours de propagande. _(N.d.T.)_\n\n2 Litt\u00e9ralement : \"points noirs du soleil\", le noir pouvant \u00eatre interpr\u00e9t\u00e9 comme une r\u00e9f\u00e9rence \u00e0 la couleur des contre-r\u00e9volutionnaires. _(N.d.T.)_\n\n## 23\n\n## C\u00d4TE ROUGE VI\n\nLes huit ann\u00e9es qui suivirent furent les plus paisibles de la vie de Ye Wenjie. La terreur n\u00e9e de son exp\u00e9rience de la R\u00e9volution culturelle s'\u00e9tait peu \u00e0 peu estomp\u00e9e et elle pouvait enfin reposer son esprit. Le projet C\u00f4te Rouge avait achev\u00e9 ses phases d'exp\u00e9rimentation et de rodage, et les op\u00e9rations \u00e9taient devenues un travail de routine. Les dysfonctionnements s'\u00e9taient faits de moins en moins nombreux, et son travail comme sa vie avaient trouv\u00e9 une certaine stabilit\u00e9.\n\nD\u00e9sormais en paix, la m\u00e9moire qui avait jusque-l\u00e0 \u00e9t\u00e9 comprim\u00e9e par l'angoisse et la peur commen\u00e7a \u00e0 se r\u00e9veiller. Wenjie d\u00e9couvrit que ses souffrances venaient en r\u00e9alit\u00e9 tout juste de commencer. Des souvenirs cauchemardesques refaisaient surface avec de plus en plus de vigueur comme un feu n\u00e9 d'un parterre de cendres br\u00fblant enflammant son \u00e2me. Chez une personne ordinaire, le temps aurait peut-\u00eatre peu \u00e0 peu pans\u00e9 ces blessures \u2013 apr\u00e8s tout, nombreuses \u00e9taient les jeunes femmes \u00e0 avoir connu de tels drames durant la R\u00e9volution culturelle et, par comparaison avec beaucoup d'autres, elle s'en \u00e9tait finalement bien tir\u00e9e. Mais Ye Wenjie avait un esprit scientifique, elle refusait d'oublier. Elle observait la folie et la haine qui l'avaient tant bless\u00e9e avec le regard de la raison.\n\nEn r\u00e9alit\u00e9, cet examen rationnel du mal de l'humanit\u00e9 avait d\u00e9but\u00e9 d\u00e8s l'\u00e9poque de sa lecture de _Printemps silencieux_. Gr\u00e2ce \u00e0 l'intimit\u00e9 de plus en plus grande qu'elle entretenait avec Yang Weining, elle avait fait acheter par la base \u2013 gr\u00e2ce au concours de Yang et sous pr\u00e9texte de collecter des donn\u00e9es techniques indispensables \u2013 de nombreux ouvrages \u00e9trangers de philosophie et d'histoire. Elle fr\u00e9missait en d\u00e9couvrant l'histoire sanglante de l'humanit\u00e9, et les remarquables r\u00e9flexions des philosophes l'aidaient \u00e0 comprendre l'essence et le myst\u00e8re de la nature humaine.\n\nM\u00eame dans un endroit comme le pic du Radar, semblable \u00e0 une source aux fleurs de p\u00eacher1, elle \u00e9tait chaque jour t\u00e9moin de l'irrationalit\u00e9 et de la folie humaines. Au pied du pic, elle voyait toujours ses anciens compagnons raser furieusement la for\u00eat ; les surfaces nues s'\u00e9largissaient de jour en jour. C'\u00e9tait comme s'ils \u00e9corchaient la peau des montagnes du Grand Khingan. Lorsque ces espaces vierges devenaient de vraies plaines, les quelques arbres ayant surv\u00e9cu au carnage avaient l'air d'\u00e9tranges intrus. Ils faisaient partir des feux pour finir de d\u00e9boiser ces montagnes d\u00e9j\u00e0 presque chauves, et le pic du Radar \u00e9tait devenu le dernier repaire d'oiseaux fuyant une mer de flammes. Tandis que les incendies faisaient rage, les oiseaux de la base aux plumes noircies chantaient d'incessantes complaintes de d\u00e9sespoir.\n\nPlus loin d'ici, la folie humaine atteignait son sommum dans toute l'histoire de la civilisation. La guerre froide \u00e9tait \u00e0 son comble. Dans les innombrables silos \u00e0 missiles diss\u00e9min\u00e9s sur les deux continents ou sur les sous-marins nucl\u00e9aires qui se terraient comme des fant\u00f4mes dans les profondeurs de l'eau, des armes capables de d\u00e9truire dix fois la Terre pouvaient \u00eatre activ\u00e9es \u00e0 tout moment. Un seul sous-marin nucl\u00e9aire de classe Typhoon ou Ethan Allen \u00e9tait en mesure d'an\u00e9antir des centaines de villes, et de tuer des millions de personnes. Mais la plupart des gens continuaient \u00e0 vivre le sourire aux l\u00e8vres, comme si tout cela ne les atteignait pas.\n\nEn tant qu'astrophysicienne, Ye Wenjie \u00e9tait particuli\u00e8rement sensible au potentiel de destruction des armes nucl\u00e9aires : elle savait que c'\u00e9tait un pouvoir que seules les \u00e9toiles devaient poss\u00e9der. Elle \u00e9tait consciente qu'il existait des forces encore plus terrifiantes dans l'univers : les trous noirs, la mati\u00e8re noire... En comparaison avec ces forces, une bombe atomique n'\u00e9tait qu'une petite chandelle. Si les humains parvenaient un jour \u00e0 contr\u00f4ler ces forces, le monde pourrait se retrouver pulv\u00e9ris\u00e9 \u00e0 tout instant. La raison \u00e9tait impuissante face \u00e0 la folie.\n\nQuatre ans apr\u00e8s son arriv\u00e9e dans la base, Ye Wenjie et Yang Weining se mari\u00e8rent. Yang Weining l'aimait d'un amour sinc\u00e8re : pour elle, il avait compromis son avenir. La p\u00e9riode la plus br\u00fblante de la R\u00e9volution culturelle \u00e9tait derri\u00e8re eux et le climat politique s'\u00e9tait un peu assoupli. Yang Weining ne fut pas particuli\u00e8rement pers\u00e9cut\u00e9 apr\u00e8s son mariage, mais \u00e9pouser une femme catalogu\u00e9e comme contre-r\u00e9volutionnaire \u00e9tait consid\u00e9r\u00e9 comme un manque de maturit\u00e9 politique et on lui retira son poste d'ing\u00e9nieur en chef. Si son \u00e9pouse et lui purent rester \u00e0 la base avec le statut de techniciens ordinaires, c'\u00e9tait parce qu'on aurait difficilement pu se passer de leurs comp\u00e9tences. Quant \u00e0 Ye Wenjie, son mariage avec Yang Weining \u00e9tait davantage une mani\u00e8re de lui exprimer sa gratitude. Au plein c\u0153ur de la temp\u00eate, il l'avait secourue et abrit\u00e9e dans ce port, lui permettant de couper avec le monde, sans quoi elle ne serait sans doute plus en vie. Yang Weining \u00e9tait un homme talentueux, cultiv\u00e9 et avec du savoir-vivre. Elle avait pour lui de l'affection, mais son c\u0153ur n'\u00e9tait plus qu'un tas de cendres froides sur lequel le feu de l'amour ne pouvait \u00eatre raviv\u00e9.\n\nSes r\u00e9flexions sur la nature et la condition de l'homme plong\u00e8rent Ye Wenjie dans une profonde crise spirituelle. Ce qu'elle dut d'abord affronter, c'\u00e9tait l'absence d'un but dans lequel s'investir corps et \u00e2me. Elle avait \u00e9t\u00e9 id\u00e9aliste autrefois, elle avait voulu mettre son talent au service d'une grande cause, mais elle avait compris que tout ce qu'elle avait r\u00e9alis\u00e9 jusque-l\u00e0 n'avait eu aucun sens et que l'avenir, lui non plus, ne lui offrait pas d'objectif \u00e0 poursuivre. Cet \u00e9tat psychologique perdura et, peu \u00e0 peu, elle \u00e9prouva \u00e0 l'\u00e9gard du monde un sentiment d'\u00e9tranget\u00e9 : elle n'avait pas l'impression de lui appartenir. Cette sensation d'errance spirituelle la tourmentait cruellement et, apr\u00e8s son mariage avec Yang Weining, son \u00e2me devint en r\u00e9alit\u00e9 plus solitaire que jamais.\n\nYe Wenjie \u00e9tait de service de nuit ce soir-l\u00e0. C'\u00e9tait un de ces moments durant lesquels elle se sentait le plus seule. Dans le silence de minuit, l'univers apparaissait \u00e0 ceux qui tendaient l'oreille vers lui comme une vaste plaine d\u00e9sertique. Wenjie d\u00e9testait voir ces courbes oscillant l\u00e9g\u00e8rement sur l'\u00e9cran de surveillance. C'\u00e9taient les ondes radio de l'univers que recevait C\u00f4te Rouge : un bruit de fond sans aucune signification. Wenjie voyait dans ces lignes sans fin comme une repr\u00e9sentation abstraite de l'univers : une extr\u00e9mit\u00e9 reli\u00e9e \u00e0 un pass\u00e9 infini, et l'autre \u00e0 un futur infini. Et, au milieu, une courbe al\u00e9atoire sans r\u00e8gle et sans vie travers\u00e9e de pics et de vall\u00e9es, comme un d\u00e9sert unidimensionnel tapiss\u00e9 de grains de sable de diff\u00e9rentes tailles. Des rang\u00e9es de grains de sable solitaires, d\u00e9sol\u00e9es, si longues qu'elles en \u00e9taient insupportables. On pouvait longer ce d\u00e9sert d'un bout \u00e0 l'autre, sans jamais en trouver la fin.\n\nCette nuit-l\u00e0 cependant, en laissant tra\u00eener son regard sur l'\u00e9cran, Wenjie aper\u00e7ut quelque chose d'anormal. M\u00eame des experts auraient eu du mal \u00e0 dire, \u00e0 l'\u0153il nu, si la forme d'onde qui venait d'appara\u00eetre \u00e9tait porteuse d'une information quelconque. Mais Wenjie connaissait trop bien le bruit de fond de l'univers. La forme d'onde qui s'\u00e9tait dessin\u00e9e devant ses yeux avait quelque chose de plus, quelque chose d'inexprimable. Cette courbe montante et descendante paraissait avoir une \u00e2me. Elle \u00e9tait certaine que cette onde radio avait \u00e9t\u00e9 cr\u00e9\u00e9e par une forme d'intelligence.\n\nWenjie se pr\u00e9cipita devant un autre terminal et v\u00e9rifia le niveau de confiance du contenu du signal estim\u00e9 par l'ordinateur : AAAAA ! Aucune des ondes radio re\u00e7ues jusqu'\u00e0 ce jour n'avait d\u00e9pass\u00e9 un niveau de confiance de C. Si un signal atteignait A, cela signifiait que la probabilit\u00e9 que la transmission contienne une information intelligente \u00e9tait sup\u00e9rieure \u00e0 quatre-vingt-dix pour cent. Cinq A d'affil\u00e9e \u00e9tait une situation rarissime : ils indiquaient que l'information re\u00e7ue utilisait le m\u00eame code de langage que celui des transmissions de C\u00f4te Rouge !\n\nWenjie d\u00e9marra le syst\u00e8me de d\u00e9chiffrage de C\u00f4te Rouge. Le logiciel pouvait tenter de d\u00e9crypter n'importe quelle information dont le niveau de confiance \u00e9tait sup\u00e9rieur \u00e0 B. Depuis le lancement du projet C\u00f4te Rouge, celui-ci n'avait jamais r\u00e9ellement fonctionn\u00e9. Sur la base des essais effectu\u00e9s par le logiciel, le d\u00e9chiffrage d'un message intelligent pouvait prendre plusieurs jours, voire plusieurs mois. Sans compter que la majorit\u00e9 du temps, le logiciel \u00e9chouait \u00e0 d\u00e9crypter l'information. Mais cette fois-ci, \u00e0 peine le signal avait-il \u00e9t\u00e9 transmis qu'un message indiquant le succ\u00e8s du d\u00e9chiffrage s'afficha sur l'\u00e9cran.\n\nWenjie ouvrit le fichier et, pour la premi\u00e8re fois, un \u00eatre humain lut un message provenant d'un autre monde dans l'univers. Son contenu d\u00e9passait tout ce que l'on aurait pu imaginer. C'\u00e9tait un avertissement, r\u00e9p\u00e9t\u00e9 trois fois :\n\n_Ne r\u00e9pondez pas !_\n\n _Ne r\u00e9pondez pas !!_\n\n _Ne r\u00e9pondez pas !!!_\n\nSaisie d'une excitation \u00e9tourdissante, Wenjie d\u00e9chiffra un second message :\n\n_Ce monde a bien re\u00e7u votre message._\n\n _Ici, je suis un pacifiste. Vous avez eu de la chance que je sois le premier \u00e0 intercepter votre message. Je vous avertis : Ne r\u00e9pondez pas ! Ne r\u00e9pondez pas !! Ne r\u00e9pondez pas !!!_\n\n _Il existe des dizaines de millions d'\u00e9toiles dans votre direction. Tant que vous ne r\u00e9pondrez pas, notre monde n'aura aucun moyen de localiser la source de votre transmission._\n\n _Mais si vous r\u00e9pondez, vous d\u00e9voilerez votre position. Votre syst\u00e8me stellaire sera envahi, et votre monde occup\u00e9 !_\n\n _Ne r\u00e9pondez pas ! Ne r\u00e9pondez pas !! Ne r\u00e9pondez pas !!!_\n\n\u00c0 mesure qu'elle lisait les lignes vertes du texte affich\u00e9 sur l'\u00e9cran, Wenjie se trouvait dans l'incapacit\u00e9 de r\u00e9fl\u00e9chir calmement. Son esprit, paralys\u00e9 par l'ivresse et le choc, ne voyait plus qu'une seule chose : moins de neuf ann\u00e9es s'\u00e9taient \u00e9coul\u00e9es depuis qu'elle avait envoy\u00e9 le message vers le soleil. Cela signifiait donc que la source de transmission se situait \u00e0 environ quatre ann\u00e9es-lumi\u00e8re. Elle ne pouvait provenir que d'un seul syst\u00e8me, le plus proche de la Terre, Alpha du Centaure !\n\nL'univers n'\u00e9tait pas d\u00e9sert. L'univers n'\u00e9tait pas vide. L'univers \u00e9tait plein de vie ! L'homme dirigeait son regard aux confins de l'univers, sans jamais avoir soup\u00e7onn\u00e9 que des formes de vie intelligentes peuplaient le syst\u00e8me stellaire le plus proche du sien !\n\nWenjie observa les formes d'ondes sur l'\u00e9cran : le signal continuait \u00e0 p\u00e9n\u00e9trer par l'antenne de C\u00f4te Rouge. Elle ouvrit une autre interface et mit en route le d\u00e9chiffrage instantan\u00e9. Le contenu des messages s'afficha aussit\u00f4t. Durant les quatre heures qui suivirent, Wenjie apprit l'existence de Trisolaris, de cette civilisation sans cesse en train de rena\u00eetre et de sa tentative de migration vers les \u00e9toiles.\n\n\u00c0 4 heures du matin, la transmission depuis Alpha du Centaure se termina. Le syst\u00e8me de d\u00e9chiffrage se mit \u00e0 tourner dans le vide et \u00e0 afficher des messages d'\u00e9chec. Le syst\u00e8me de surveillance de C\u00f4te Rouge entendait \u00e0 nouveau le bruit de fond d'un univers d\u00e9sol\u00e9.\n\nMais Ye Wenjie \u00e9tait certaine de ne pas avoir r\u00eav\u00e9.\n\nSi le soleil \u00e9tait r\u00e9ellement une antenne g\u00e9ante, pourquoi donc n'avait-elle jamais re\u00e7u les \u00e9chos de l'exp\u00e9rience men\u00e9e huit ans plus t\u00f4t ? Pourquoi les formes d'ondes des sursauts radio de Jupiter et des rayonnements solaires n'avaient-elles pas correspondu ? Wenjie r\u00e9ussirait plus tard \u00e0 trouver des explications. Peut-\u00eatre que la station radio de la base ne pouvait tout simplement pas recevoir des ondes radio \u00e0 cette fr\u00e9quence, ou peut-\u00eatre que le bureau des communications avait bien re\u00e7u un \u00e9cho, mais sous la forme de bruit de fond, et avait consid\u00e9r\u00e9 n'avoir rien re\u00e7u. Quant aux formes d'ondes, il \u00e9tait tr\u00e8s probable que, lors de l'amplification des ondes radio par le soleil, celui-ci ait aussi ajout\u00e9 une nouvelle forme. Il n'avait pas \u00e9t\u00e9 tr\u00e8s compliqu\u00e9 pour le syst\u00e8me de d\u00e9cryptage de la civilisation extraterrestre de la filtrer, tandis que les formes d'ondes de Jupiter et du soleil \u00e9taient apparues tr\u00e8s diff\u00e9rentes \u00e0 ses yeux. Elle obtiendrait plus tard confirmation de ce point : le soleil avait ajout\u00e9 une onde sinuso\u00efdale.\n\nPrudente, elle regarda tout autour d'elle dans la console de contr\u00f4le. Trois autres employ\u00e9s \u00e9taient de service de nuit. Deux d'entre eux discutaient dans un coin et un dernier somnolait devant un terminal. Dans la section d'analyse de donn\u00e9es de l'unit\u00e9 de surveillance, seuls les deux terminaux devant lesquels elle se trouvait permettaient de v\u00e9rifier le niveau de confiance d'un signal et d'acc\u00e9der au logiciel de d\u00e9cryptage. Elle agit rapidement, sans bruit et sans bouger : elle enregistra tout d'abord l'ensemble des signaux re\u00e7us dans un r\u00e9pertoire cach\u00e9 et plusieurs fois encrypt\u00e9, puis rempla\u00e7a leur contenu par un enregistrement de bruit de fond datant d'un an plus t\u00f4t.\n\nEnfin, elle entra un court signal dans la m\u00e9moire du syst\u00e8me de transmission de C\u00f4te Rouge.\n\nYe Wenjie se leva et sortit par la porte principale de la console de contr\u00f4le de l'unit\u00e9 de surveillance. Une brise glaciale souffla sur son visage bouillant. \u00c9clair\u00e9e par les faibles lueurs de l'aube qui se levait \u00e0 l'est, elle marcha sur le sentier pav\u00e9, dans la direction de la console de contr\u00f4le de l'unit\u00e9 de transmission. Au-dessus de sa t\u00eate, l'immense main de l'antenne s'ouvrait silencieusement vers l'univers. La lumi\u00e8re du matin illumina les silhouettes d\u00e9coup\u00e9es des soldats en faction devant la porte. Comme \u00e0 l'accoutum\u00e9e, Wenjie ne fit pas attention \u00e0 eux et entra directement.\n\nLa console de contr\u00f4le de l'unit\u00e9 de transmission \u00e9tait beaucoup plus sombre que celle de l'unit\u00e9 de surveillance. Wenjie longea des rang\u00e9es de placards et se dirigea droit vers la plateforme de contr\u00f4le. Avec aisance, elle alluma une dizaine d'interrupteurs et d\u00e9marra la mise en chauffe du syst\u00e8me. Les deux op\u00e9rateurs assis \u00e0 c\u00f4t\u00e9 de la plateforme l'observ\u00e8rent avec un regard fatigu\u00e9. L'un d'eux tourna la t\u00eate pour regarder l'horloge sur le mur, puis se remit \u00e0 somnoler. L'autre feuilletait un journal qu'il devait avoir lu plusieurs fois. Au sein de la base, Ye Wenjie n'avait naturellement aucune position politique, mais elle b\u00e9n\u00e9ficiait d'une certaine libert\u00e9 sur le plan technique, et \u00e9tait souvent charg\u00e9e de v\u00e9rifier le mat\u00e9riel avant l'op\u00e9ration de transmission. Bien qu'il soit peut-\u00eatre un peu t\u00f4t \u2013 les premi\u00e8res transmissions ne commen\u00e7aient que trois heures plus tard \u2013, il n'\u00e9tait pas si \u00e9trange de voir Ye Wenjie mettre le syst\u00e8me en chauffe.\n\nUne demi-heure interminable s'\u00e9coula. Durant ce temps, Wenjie param\u00e9tra la fr\u00e9quence de transmission afin qu'elle soit le plus optimale pour \u00eatre amplifi\u00e9e par les miroirs d'\u00e9nergie solaire, puis elle r\u00e9gla la puissance de transmission \u00e0 son maximum. Elle approcha son \u0153il de l'oculaire du syst\u00e8me de vis\u00e9e optique et regarda l'astre du jour se lever \u00e0 l'horizon. Elle d\u00e9marra le syst\u00e8me de vis\u00e9e de l'antenne et l'aligna lentement pour la faire se retrouver face au soleil. Alors que la gigantesque antenne \u00e9tait en train de pivoter, un bourdonnement se fit entendre dans la console de contr\u00f4le. Un des op\u00e9rateurs jeta un \u0153il sur Wenjie, mais sans rien dire.\n\nLe soleil avait maintenant compl\u00e8tement d\u00e9pass\u00e9 les montagnes. L'extr\u00e9mit\u00e9 haute de l'astre \u00e9tait au centre du r\u00e9ticule de vis\u00e9e de l'antenne, de mani\u00e8re que soit pris en compte le temps que les ondes radio mettent \u00e0 rejoindre le soleil. Le syst\u00e8me de transmission \u00e9tait pr\u00eat. Le bouton de transmission \u00e9tait un long rectangle, de la forme d'une barre d'espace sur un clavier d'ordinateur, mais de couleur rouge. Les doigts de Ye Wenjie \u00e9taient suspendus deux centim\u00e8tres au-dessus.\n\nLe destin de l'humanit\u00e9 tout enti\u00e8re tenait \u00e0 pr\u00e9sent \u00e0 ces doigts fins.\n\nSans la moindre h\u00e9sitation, Ye Wenjie pressa le bouton.\n\n\u2014 Que fais-tu ? demanda l'homme en service, encore endormi.\n\nYe Wenjie lui sourit, mais sans lui r\u00e9pondre. Elle pressa le bouton jaune pour cesser la transmission et manipula le levier de commande pour changer la direction dans laquelle pointait l'antenne, avant de quitter la console de contr\u00f4le, pour aller marcher \u00e0 l'ext\u00e9rieur.\n\nL'op\u00e9rateur jeta un \u0153il sur sa montre. Il avait termin\u00e9 son travail. Il prit le journal de transmissions et pensa consigner l'op\u00e9ration de d\u00e9marrage du syst\u00e8me de transmission que venait d'effectuer Ye Wenjie. Apr\u00e8s tout, ce n'\u00e9tait pas la proc\u00e9dure habituelle. Mais il examina la bande de papier o\u00f9 \u00e9taient enregistr\u00e9es les donn\u00e9es et vit que la transmission avait dur\u00e9 \u00e0 peine trois secondes. Il reposa le journal \u00e0 sa place, b\u00e2illa, enfila sa casquette et sortit.\n\nLe message qui \u00e9tait en chemin vers le soleil \u00e9tait celui-ci :\n\n> Venez ! Je vous aiderai \u00e0 conqu\u00e9rir ce monde. Notre civilisation est incapable de r\u00e9soudre ses propres probl\u00e8mes. Nous avons besoin de votre intervention.\n\nLe soleil qui venait de se lever \u00e9blouit les yeux de Ye Wenjie. Elle ne marcha pas longtemps et s'\u00e9vanouit soudain sur l'herbe.\n\nQuand elle se r\u00e9veilla, elle d\u00e9couvrit qu'elle \u00e9tait dans l'infirmerie de la base. Yang Weining \u00e9tait assis \u00e0 ses c\u00f4t\u00e9s et la regardait avec sollicitude. C'\u00e9tait le m\u00eame regard qu'il avait port\u00e9 sur elle dans l'h\u00e9licopt\u00e8re, bien des ann\u00e9es plus t\u00f4t. Le docteur dit \u00e0 Ye Wenjie de faire attention et de bien se reposer.\n\nElle \u00e9tait enceinte.\n\n* * *\n\n1 R\u00e9f\u00e9rence \u00e0 un c\u00e9l\u00e8bre po\u00e8me en prose chinois, compos\u00e9 au d\u00e9but du Ve si\u00e8cle par Tao Yuanming, dans lequel un p\u00eacheur d\u00e9couvre un lieu utopique, comme hors du monde : la source aux fleurs de p\u00eacher. _(N.d.T.)_\n\n## 24\n\n## LA R\u00c9BELLION\n\nLorsque Ye Wenjie termina son r\u00e9cit, l'assembl\u00e9e plongea dans le silence. Manifestement, beaucoup ici l'entendaient pour la toute premi\u00e8re fois. Wang Miao \u00e9tait si absorb\u00e9 qu'il en oublia le danger et la peur. Il ne put s'emp\u00eacher de demander :\n\n\u2014 Comment l'Organisation s'est-elle d\u00e9velopp\u00e9e pour prendre une telle ampleur ?\n\nYe Wenjie r\u00e9pondit :\n\n\u2014 Je devrais commencer par te raconter ma premi\u00e8re rencontre avec Evans... Mais les camarades ici pr\u00e9sents connaissent bien cette partie de l'histoire. Ne perdons pas de temps, j'aurai l'occasion de te la raconter plus tard, quand nous serons seuls. Mais aurons-nous cette chance ? C'est \u00e0 toi qu'il revient d'en d\u00e9cider... Miao, parlons plut\u00f4t de tes nanomat\u00e9riaux.\n\n\u2014 Ces... dieux dont vous parlez, pourquoi craignent-ils tant les nanomat\u00e9riaux ?\n\n\u2014 Parce qu'ils permettraient aux humains de se lib\u00e9rer de la gravit\u00e9 terrestre et d'investir l'espace.\n\n\u2014 L'ascenseur spatial ? songea aussit\u00f4t Wang Miao.\n\n\u2014 Oui. D\u00e8s que des nanomat\u00e9riaux ultrar\u00e9sistants pourront \u00eatre cr\u00e9\u00e9s en masse, il sera possible d'envisager la construction d'un ascenseur spatial reliant directement la surface de la Terre \u00e0 une orbite g\u00e9ostationnaire. Pour nos dieux, cela n'est qu'une avanc\u00e9e minime, mais sa port\u00e9e serait capitale \u00e0 l'\u00e9chelle humaine. Gr\u00e2ce \u00e0 cette technologie, les humains pourraient entrer facilement dans l'orbite terrestre basse et construire des syst\u00e8mes de d\u00e9fense de grande envergure. C'est pourquoi nos dieux ordonnent que cette technologie soit annihil\u00e9e.\n\n\u2014 Qu'y a-t-il \u00e0 la fin du compte \u00e0 rebours ? demanda Wang Miao que cette question effrayait depuis le d\u00e9but.\n\nYe Wenjie sourit :\n\n\u2014 Je ne sais pas.\n\n\u2014 Il est inutile de vouloir m'arr\u00eater ! Je ne fais pas de recherche fondamentale ! \u00c0 l'heure actuelle, n'importe qui est capable de cr\u00e9er des nanomat\u00e9riaux ! cria Wang Miao, nerveux.\n\n\u2014 Oui, c'est assez inutile. Il est beaucoup plus efficace de perturber les syst\u00e8mes de pens\u00e9e des chercheurs ; nous n'avons pas toujours \u00e9t\u00e9 aussi performants que nous l'aurions voulu. Comme tu l'as dit toi-m\u00eame, tu fais des recherches appliqu\u00e9es, contre lesquelles nous sommes loin d'avoir la m\u00eame efficacit\u00e9 que contre les recherches fondamentales...\n\n\u2014 En parlant de recherches fondamentales, comment est vraiment morte votre fille ?\n\nCette question plongea Ye Wenjie dans un silence qui dura plusieurs secondes. Wang Miao remarqua que son regard s'\u00e9tait assombri, quoique de mani\u00e8re presque imperceptible. Elle reprit aussit\u00f4t le fil de la conversation :\n\n\u2014 Pour tout dire, rien de ce que nous faisons n'a de sens en comparaison de la toute-puissance des dieux. Nous faisons simplement ce que nous sommes en mesure de faire.\n\nAlors que Ye Wenjie venait de prononcer ces mots, on entendit des d\u00e9tonations. Les deux portes de la caf\u00e9t\u00e9ria vol\u00e8rent en \u00e9clats. Une troupe de soldats entra dans la caf\u00e9t\u00e9ria, mitraillettes \u00e0 la main. Wang Miao nota qu'il ne s'agissait pas de forces de police mais bien de militaires. Les soldats se d\u00e9ploy\u00e8rent sans bruit dans la salle en longeant les murs. Bien vite, les rebelles de l'OTT furent encercl\u00e9s. Shi Qiang fut le dernier \u00e0 entrer. Sa veste en cuir \u00e9tait ouverte sur sa poitrine, et il tenait dans la main le canon de son arme, dont le manche pendait comme celui d'un marteau. Il inspecta nonchalamment la pi\u00e8ce puis, soudain, il fon\u00e7a droit devant lui, fit pivoter le pistolet qu'il tenait dans la main et on entendit l'impact d'une balle sur un cr\u00e2ne. Un rebelle de l'OTT tomba \u00e0 terre et le pistolet qu'il n'avait pas eu le temps de sortir alla rouler plus loin. Quelques soldats tir\u00e8rent en l'air, et de la poussi\u00e8re tomba du plafond. Quelqu'un attrapa Wang Miao et l'exfiltra \u00e0 toute vitesse hors du groupe des rebelles pour le mettre en s\u00e9curit\u00e9 derri\u00e8re une rang\u00e9e de militaires.\n\n\u2014 Jetez vos armes sur la table ! Le premier qui bouge, je le cr\u00e8ve ! cria Shi Qiang, puis il d\u00e9signa la rang\u00e9e de militaires arm\u00e9s dans son dos. Je sais bien que vous n'avez pas peur de mourir, mais nous non plus. Je vais \u00eatre franc : les proc\u00e9dures l\u00e9gales habituelles ne s'appliquent pas \u00e0 vous. D'ailleurs, m\u00eame les r\u00e8gles humaines fondamentales de la guerre ne s'appliquent pas non plus ! Comme vous avez d\u00e9cid\u00e9 de faire de l'ensemble de l'esp\u00e8ce humaine votre ennemi, plus rien ne nous retient.\n\nUne agitation parcourut les rangs des rebelles de l'Organisation, mais aucune vague de panique. Ye Wenjie, elle, demeurait de marbre. Trois individus sortirent soudain du groupe, dont la jeune femme qui avait bris\u00e9 le cou de Pan Han. Ils coururent en direction de l'\u0153uvre artistique des trois corps toujours en l\u00e9vitation au-dessus de son socle. Chacun d'entre eux se saisit d'une des trois sph\u00e8res en m\u00e9tal et la serra contre sa poitrine.\n\nLa jeune femme brandit la sph\u00e8re \u00e9tincelante comme si elle s'appr\u00eatait \u00e0 faire une d\u00e9monstration de gymnastique, puis se fendit d'un \u00e9trange sourire et d\u00e9clara d'une voix doucereuse :\n\n\u2014 Messieurs les officiers, nous avons en notre possession trois bombes nucl\u00e9aires, chacune poss\u00e9dant un \u00e9quivalent en TNT de 1,5 kilotonne. Ce n'est pas tr\u00e8s gros, nous aimons les petits jouets. Voici le d\u00e9tonateur.\n\nL'atmosph\u00e8re de la salle se figea. Le seul qui osa bouger fut Shi Qiang. Il repla\u00e7a son arme dans l'\u00e9tui sous son aisselle gauche, puis il applaudit pour d\u00e9tendre l'atmosph\u00e8re.\n\n\u2014 Notre revendication est simple : laissez partir le Guide. Ensuite nous nous amuserons ensemble, dit la fille sur un ton fac\u00e9tieux.\n\n\u2014 Je reste ici avec les camarades, dit Ye Wenjie avec sang-froid.\n\n\u2014 Pouvez-vous confirmer que ce sont des bombes ? demanda Shi Qiang \u00e0 voix basse \u00e0 un officier, probablement un expert en explosifs.\n\nL'expert lan\u00e7a un sac plastique aux trois membres de l'OTT qui tenaient une sph\u00e8re dans les mains. Le sac contenait un dynamom\u00e8tre. L'un des trois individus se baissa pour le ramasser, il pla\u00e7a la sph\u00e8re dans le sac et accrocha celui-ci au crochet du dynamom\u00e8tre. Puis il le leva et le secoua en l'air, avant de sortir la sph\u00e8re du sac et de la jeter au sol. La jeune femme partit d'un \u00e9clat de rire. L'expert, lui, eut un ricanement de m\u00e9pris. Le deuxi\u00e8me rebelle fit de m\u00eame et rel\u00e2cha la sph\u00e8re. La fille rit encore une fois, et finit par placer la sienne dans le sac qu'elle accrocha au dynamom\u00e8tre. Mais cette fois-ci, la jauge de l'appareil atteignit son maximum.\n\nLe sourire sur le visage de l'expert se figea, il murmura \u00e0 Shi Qiang :\n\n\u2014 C'en est une.\n\nShi Qiang resta impassible.\n\n\u2014 Nous pouvons au moins \u00eatre s\u00fbrs d'une chose : la sph\u00e8re contient des \u00e9l\u00e9ments lourds pouvant subir une fission nucl\u00e9aire. Mais nous ignorons si le m\u00e9canisme de d\u00e9tonation fonctionne, reprit l'expert.\n\nLes lampes torches des mitraillettes mirent la jeune femme en joue. Entre les mains de cette belle fleur de la mort se trouvait une bombe de 1,5 kilotonne. Elle sourit, comme si elle attendait les applaudissements et les louanges de la foule sous les projecteurs d'une sc\u00e8ne de th\u00e9\u00e2tre.\n\n\u2014 Il y a une chance : il faut tirer sur la sph\u00e8re, dit l'expert \u00e0 l'oreille de Shi Qiang.\n\n\u2014 Cela ne risque pas de d\u00e9clencher la bombe ?\n\n\u2014 Il y aura quelques d\u00e9flagrations, mais elles seront dispers\u00e9es et, surtout, cela n'entra\u00eenera pas la compression pr\u00e9cise du mat\u00e9riel de fission au centre de la sph\u00e8re, r\u00e9action indispensable pour que se produise l'explosion nucl\u00e9aire.\n\nShi Qiang fixa la fille \u00e0 la bombe sans rien dire.\n\n\u2014 Un tireur d'\u00e9lite ? sugg\u00e9ra l'expert.\n\nShi Qiang secoua presque imperceptiblement la t\u00eate :\n\n\u2014 Il n'y a pas de position ad\u00e9quate. Elle est fut\u00e9e, cette diablesse. Elle saura d\u00e8s qu'un sniper l'aura au bout de son viseur.\n\nShi Qiang s'avan\u00e7a alors. Il \u00e9carta la foule et vint se placer dans l'espace vide au centre de la caf\u00e9t\u00e9ria.\n\n\u2014 Stop ! l'avertit la fille en lui jetant un regard espi\u00e8gle.\n\nElle maintenait son pouce sur le bouton du d\u00e9tonateur et son vernis \u00e0 ongles scintillait sous la lumi\u00e8re des lampes torches des armes.\n\n\u2014 Du calme, petite, du calme. J'ai ici une information qui va t'int\u00e9resser.\n\nShi Qiang se tenait maintenant \u00e0 une distance de sept ou huit m\u00e8tres de la fille. Il sortit une enveloppe de sa veste en cuir.\n\n\u2014 On a retrouv\u00e9 ta m\u00e8re.\n\nLes yeux fanatiques de la fille p\u00e2lirent tout \u00e0 coup et, un court instant, devinrent une fen\u00eatre sur son \u00e2me.\n\nShi Qiang profita de ce moment de flottement pour avancer de deux pas. La distance qui les s\u00e9parait n'\u00e9tait plus d\u00e9sormais que de cinq m\u00e8tres environ. La fille, alerte, brandit la bombe et lui intima d'un regard de s'arr\u00eater, mais elle avait d\u00e9j\u00e0 l'esprit ailleurs. Un des deux rebelles \u00e0 avoir rel\u00e2ch\u00e9 une fausse bombe marcha vers Shi Qiang et tendit la main pour prendre l'enveloppe. En un \u00e9clair, le commissaire d\u00e9gaina son arme. Son mouvement ayant \u00e9t\u00e9 dissimul\u00e9 par le d\u00e9placement de l'homme, la fille ne vit rien d'autre qu'un flash passer \u00e0 c\u00f4t\u00e9 de l'oreille de son comparse et la bombe qu'elle tenait dans sa main explosa.\n\nIl y eut un souffle sourd et Wang Miao ne vit tout \u00e0 coup plus rien qu'une chape d'obscurit\u00e9. Quelqu'un le tira hors de la caf\u00e9t\u00e9ria. Une \u00e9paisse fum\u00e9e jaune s'\u00e9chappait par la porte. \u00c0 l'int\u00e9rieur de la salle, on entendait une cacophonie de tirs et de cris. Des individus fon\u00e7aient \u00e0 travers la fum\u00e9e pour sortir de la caf\u00e9t\u00e9ria... Wang Miao voulut s'\u00e9lancer pour retourner dans la salle, mais l'expert en explosifs lui bloqua la route en l'attrapant par la taille :\n\n\u2014 Attention ! C'est bourr\u00e9 de radiations !\n\nLe chaos s'apaisa rapidement. Une bonne dizaine de combattants de l'OTT avaient \u00e9t\u00e9 abattus dans les \u00e9changes de tirs. Plus de deux cents personnes, dont Ye Wenjie elle-m\u00eame, furent arr\u00eat\u00e9es. Le corps de la jeune fille avait \u00e9t\u00e9 d\u00e9chiquet\u00e9 par la bombe, mais elle avait \u00e9t\u00e9 la seule victime de l'explosion. L'homme venu saisir la lettre de Shi Qiang \u00e9tait gri\u00e8vement bless\u00e9 : il avait servi de bouclier humain au commissaire et les blessures de ce dernier n'\u00e9taient que l\u00e9g\u00e8res. Toutefois, comme tous ceux qui \u00e9taient rest\u00e9s dans la caf\u00e9t\u00e9ria apr\u00e8s l'explosion, il avait \u00e9t\u00e9 gravement expos\u00e9 \u00e0 la contamination radioactive.\n\nWang Miao aper\u00e7ut Shi Qiang \u00e0 travers la fen\u00eatre de l'ambulance. Une profonde entaille sur son cr\u00e2ne continuait \u00e0 saigner. L'infirmi\u00e8re qui lui faisait son bandage portait une combinaison antiradiation. Ils durent se parler par t\u00e9l\u00e9phone.\n\n\u2014 Qui \u00e9tait la fille ? demanda Wang Miao.\n\n\u2014 Je n'en ai pas la foutue moindre id\u00e9e, ricana Shi Qiang. J'ai tap\u00e9 au hasard. Une fille comme \u00e7a, \u00e7a fait longtemps qu'elle n'avait pas d\u00fb voir sa m\u00e8re. Il y a vingt ans que je fais ce job, j'ai appris \u00e0 comprendre les gens.\n\n\u2014 Vous aviez vu juste. Il y avait bien un loup, dit Wang Miao en se for\u00e7ant \u00e0 sourire, esp\u00e9rant que Shi Qiang pourrait le voir \u00e0 travers la vitre du v\u00e9hicule.\n\n\u2014 C'est vous qui aviez vu juste, mon vieux, sourit Shi Qiang en secouant la t\u00eate. Putain, je n'aurais jamais pari\u00e9 que les extraterrestres aussi \u00e9taient dans le coup !\n\n## 25\n\n## LA MORT DE LEI ZHICHENG\n\n## ET DE YANG WEINING\n\nL'ENQU\u00caTEUR\n\nNom et pr\u00e9nom ?\n\nYE WENJIE\n\nYe, Wenjie.\n\nL'ENQU\u00caTEUR\n\nDate de naissance ?\n\nYE WENJIE\n\nJuin 1947.\n\nL'ENQU\u00caTEUR\n\nProfession ?\n\nYE WENJIE\n\nAncienne professeur d'astrophysique dans le d\u00e9partement de physique de l'universit\u00e9 XX, \u00e0 la retraite depuis 2004.\n\nL'ENQU\u00caTEUR\n\nEn consid\u00e9ration de votre \u00e9tat de sant\u00e9, vous avez l'autorisation de demander des pauses durant cet interrogatoire.\n\nYE WENJIE\n\nMerci, ce ne sera pas utile.\n\nL'ENQU\u00caTEUR\n\nCet interrogatoire est effectu\u00e9 dans le cadre d'une proc\u00e9dure p\u00e9nale ordinaire. Nous n'entrerons pas aujourd'hui en d\u00e9tail dans les affaires graves vous concernant. Ce n'est pas l'\u00e9l\u00e9ment principal de notre enqu\u00eate et nous esp\u00e9rons pouvoir terminer rapidement. Nous comptons sur votre collaboration.\n\nYE WENJIE\n\nJe sais \u00e0 quoi vous faites r\u00e9f\u00e9rence. Je collaborerai.\n\nL'ENQU\u00caTEUR\n\nNotre investigation a r\u00e9v\u00e9l\u00e9 que vous avez \u00e9t\u00e9 suspect\u00e9e de meurtre, durant la p\u00e9riode o\u00f9 vous travailliez \u00e0 la base de C\u00f4te Rouge.\n\nYE WENJIE\n\nJ'ai tu\u00e9 deux personnes.\n\nL'ENQU\u00caTEUR\n\nDate ?\n\nYE WENJIE\n\nLe 21 octobre 1979.\n\nL'ENQU\u00caTEUR\n\nNoms et pr\u00e9noms des victimes ?\n\nYE WENJIE\n\nLe commissaire politique Lei Zhicheng et mon \u00e9poux, l'ing\u00e9nieur Yang Weining.\n\nL'ENQU\u00caTEUR\n\nVeuillez nous exposer le mobile de ces meurtres.\n\nYE WENJIE\n\nJe... Je suppose que vous avez \u00e9t\u00e9 inform\u00e9 du contexte ?\n\nL'ENQU\u00caTEUR\n\nOui. Si j'ai des questions, je vous interromprai.\n\nYE WENJIE\n\nBien. Le jour m\u00eame o\u00f9 j'ai re\u00e7u le message extraterrestre, puis y ai r\u00e9pondu, j'ai appris que je n'avais pas \u00e9t\u00e9 la seule \u00e0 lire ce message. Le commissaire politique Lei Zhicheng l'avait lui aussi intercept\u00e9. Le commissaire Lei \u00e9tait un cadre typique. Il avait un sens aigu de la politique et, pour utiliser une expression de l'\u00e9poque, il envisageait tout \u00e0 la lumi\u00e8re de l'id\u00e9ologie de la lutte des classes. \u00c0 l'insu de la plupart des techniciens de C\u00f4te Rouge, il avait install\u00e9 un programme dans l'ordinateur principal de la plateforme de contr\u00f4le. Ce programme lisait la m\u00e9moire tampon des signaux transmis et re\u00e7us et stockait le contenu de ces informations dans un dossier cach\u00e9 et crypt\u00e9. Il existait par cons\u00e9quent une copie de tout ce que C\u00f4te Rouge transmettait et recevait, que lui seul \u00e9tait en mesure de consulter. C'est gr\u00e2ce \u00e0 cette copie qu'il a \u00e9t\u00e9 inform\u00e9 du message provenant de la civilisation extraterrestre. L'apr\u00e8s-midi du jour o\u00f9 j'ai envoy\u00e9 mon message vers le soleil, ce jour m\u00eame o\u00f9 je venais d'apprendre que j'\u00e9tais enceinte, Lei Zhicheng m'a convoqu\u00e9e dans son bureau. J'ai eu l'horreur de d\u00e9couvrir sur son \u00e9cran le message de Trisolaris que j'avais re\u00e7u la veille au soir.\n\n\u2014 Il s'est \u00e9coul\u00e9 huit heures depuis la r\u00e9ception du premier message. Au lieu de faire un rapport, tu as supprim\u00e9 ce message et tu en as gard\u00e9 une copie, n'est-ce pas ? m'a dit Lei Zhicheng.\n\nJ'ai baiss\u00e9 la t\u00eate, sans rien r\u00e9pondre.\n\n\u2014 Je sais tr\u00e8s bien quelle est la prochaine \u00e9tape de ton plan : tu as l'intention de r\u00e9pondre. Si je ne t'avais pas d\u00e9couverte \u00e0 temps, tu aurais sign\u00e9 de tes propres mains la fin de la civilisation humaine ! Bien entendu, nous ne craignons pas une invasion extraterrestre. M\u00eame si cela devait arriver, les troupes ennemies seraient submerg\u00e9es par l'oc\u00e9an arm\u00e9 du peuple !\n\nJ'ai compris qu'il ignorait que j'avais r\u00e9pondu \u00e0 ce message. Lorsque j'avais enregistr\u00e9 ma r\u00e9ponse dans la m\u00e9moire du syst\u00e8me de transmission, je n'avais pas utilis\u00e9 l'interface ordinaire. Sans le vouloir, j'avais \u00e9chapp\u00e9 \u00e0 sa surveillance.\n\n\u2014 Ye Wenjie, je savais que tu serais capable d'un acte aussi malveillant. La haine de la r\u00e9volution et du prol\u00e9tariat est profond\u00e9ment grav\u00e9e dans tes os. Tu ne laisseras jamais passer une occasion de te venger. Connais-tu les cons\u00e9quences de ton acte ?\n\nBien entendu, je les connaissais. J'ai donc hoch\u00e9 la t\u00eate. Lei Zhicheng est rest\u00e9 un moment silencieux, mais ce qu'il a dit ensuite a \u00e9t\u00e9 totalement inattendu :\n\n\u2014 Ye Wenjie, je ne t'ai jamais port\u00e9e dans mon c\u0153ur. Je t'ai toujours consid\u00e9r\u00e9e comme une ennemie du peuple. Mais Yang et moi nous avons servi ensemble pendant des ann\u00e9es. Je refuse de le voir an\u00e9anti en m\u00eame temps que toi. Je ne peux laisser votre futur enfant \u00eatre d\u00e9truit. Tu es enceinte, n'est-ce pas ?\n\nIl ne disait pas cela au hasard. \u00c0 l'\u00e9poque, la gravit\u00e9 de mes actes aurait \u00e9galement condamn\u00e9 mon \u00e9poux, quand bien m\u00eame il n'avait aucune responsabilit\u00e9 dans cette affaire. Et bien entendu, notre enfant aussi aurait \u00e9t\u00e9 touch\u00e9.\n\nLei Zhicheng a baiss\u00e9 le ton de sa voix et m'a dit :\n\n\u2014 \u00c0 pr\u00e9sent, nous sommes les deux seules personnes \u00e0 savoir ce qui s'est pass\u00e9. Nous devons tout faire pour minimiser les cons\u00e9quences de tes actes. Dor\u00e9navant, tu ne t'occuperas de rien, tu pr\u00e9tendras que rien n'est jamais arriv\u00e9, tu n'en parleras jamais \u00e0 personne, pas m\u00eame \u00e0 Yang Weining. Je me chargerai du reste. Wenjie, tant que tu coop\u00e9reras, je pourrai vous \u00e9viter le pire.\n\nJ'avais tout de suite compris ce que recherchait Lei Zhicheng : il voulait devenir le premier homme \u00e0 avoir d\u00e9couvert une intelligence extraterrestre. C'\u00e9tait pour lui une occasion unique de voir son nom \u00e9crit dans les manuels d'histoire.\n\nJ'ai promis, puis j'ai quitt\u00e9 le bureau. Mais je savais d\u00e9j\u00e0 ce que j'allais faire.\n\nJ'ai pris une petite clef \u00e0 molette et je suis entr\u00e9e dans le local o\u00f9 se trouvait le processeur frontal du syst\u00e8me de r\u00e9ception de la base. J'ai ouvert l'armoire et j'ai desserr\u00e9 les boulons qui maintenaient le fil de terre. Comme je venais fr\u00e9quemment v\u00e9rifier le mat\u00e9riel, personne n'a remarqu\u00e9 ce que je faisais. La valeur de la r\u00e9sistance de terre est alors aussit\u00f4t pass\u00e9e de 0,6 ohm \u00e0 5 ohms, ce qui a provoqu\u00e9 des perturbations dans le syst\u00e8me.\n\nLe technicien en service a tout de suite compris que le probl\u00e8me venait d'un dysfonctionnement du fil de terre, car ce genre d'incident \u00e9tait arriv\u00e9 plusieurs fois dans le pass\u00e9. C'\u00e9tait un probl\u00e8me facile \u00e0 diagnostiquer, mais jamais il n'aurait pu imaginer qu'il provenait de l'extr\u00e9mit\u00e9 du fil. Celui-ci \u00e9tait maintenu fermement et personne ne le touchait jamais, d'autant que je venais de lui annoncer que j'avais tout v\u00e9rifi\u00e9.\n\nLe sol du pic du Radar \u00e9tait recouvert d'une couche d'argile de quelques dizaines de m\u00e8tres d'\u00e9paisseur dont la conductivit\u00e9 \u00e9tait tr\u00e8s faible. Apr\u00e8s avoir enterr\u00e9 les fils, on s'\u00e9tait rendu compte que les r\u00e9sistances de terre ne parvenaient pas \u00e0 fonctionner correctement. Les fils de terre ne pouvaient pas \u00eatre enterr\u00e9s trop profond\u00e9ment, car la couche d'argile avait un effet tr\u00e8s corrosif sur les fils qui finirait, au bout d'un certain temps, par corroder la section centrale du fil de terre. On s'\u00e9tait finalement r\u00e9sign\u00e9 \u00e0 tendre le fil depuis le sommet de la falaise et \u00e0 le faire longer la paroi jusqu'\u00e0 passer en dessous de la couche d'argile. On avait donc enterr\u00e9 le p\u00f4le \u00e9lectrique \u00e0 cette hauteur de la falaise. Mais m\u00eame ainsi, le fil n'\u00e9tait pas toujours tr\u00e8s stable et la valeur de la r\u00e9sistance d\u00e9passait souvent la valeur autoris\u00e9e. Les probl\u00e8mes venaient principalement de la partie du fil suspendu le long de falaise. Par cons\u00e9quent, les techniciens de r\u00e9paration devaient descendre en rappel pour r\u00e9gler le dysfonctionnement.\n\nLe technicien a appel\u00e9 l'\u00e9quipe de maintenance et un soldat volontaire a accroch\u00e9 une corde \u00e0 un pilier de fer avant de descendre en rappel le long de la paroi. Il est rest\u00e9 en bas environ une demi-heure avant de remonter le visage plein de sueur en disant qu'il n'avait pas trouv\u00e9 l'origine du probl\u00e8me. La prochaine op\u00e9ration de surveillance allait \u00eatre impact\u00e9e et le technicien a d\u00fb pr\u00e9venir le centre de commandes de la base. De mon c\u00f4t\u00e9, j'attendais pr\u00e8s de la falaise, \u00e0 c\u00f4t\u00e9 du pilier o\u00f9 avait \u00e9t\u00e9 accroch\u00e9e la corde. Lei Zhicheng est revenu avec le soldat : tout se passait comme je l'avais pr\u00e9vu.\n\nJe dois pr\u00e9ciser que Lei Zhicheng \u00e9tait quelqu'un de tr\u00e8s d\u00e9vou\u00e9 aux instructions politiques. Il suivait pieusement ce qui \u00e9tait exig\u00e9 par l'id\u00e9ologie de l'\u00e9poque : il n'h\u00e9sitait pas \u00e0 se m\u00ealer \u00e0 la masse et \u00e0 se placer en premi\u00e8re ligne. C'\u00e9tait peut-\u00eatre hypocrite, mais il le faisait tr\u00e8s bien. D\u00e8s qu'une t\u00e2che s'av\u00e9rait difficile ou p\u00e9rilleuse, il acceptait de se porter volontaire. Bien souvent, il effectuait des actions dangereuses et fatigantes, comme, entre autres, l'entretien des fils de terre de la falaise. Ce travail ne requ\u00e9rait pas de connaissances techniques particuli\u00e8res mais n\u00e9cessitait une certaine exp\u00e9rience, car le dysfonctionnement pouvait provenir d'un faux contact difficile \u00e0 d\u00e9tecter, provoqu\u00e9 par l'exposition du fil \u00e0 l'air libre, ou peut-\u00eatre encore que l'endroit o\u00f9 \u00e9tait enterr\u00e9 le p\u00f4le \u00e9lectrique \u00e9tait trop sec. Les soldats volontaires qui formaient l'\u00e9quipe de r\u00e9paration \u00e9taient nouveaux \u00e0 la base et n'avaient pas une grande exp\u00e9rience. Je savais que Lei Zhicheng aurait de grandes chances de prendre les choses en mains.\n\nIl a attach\u00e9 son harnais avant de descendre le long de la paroi, faisant comme si je n'existais pas. J'ai trouv\u00e9 un pr\u00e9texte pour faire partir le soldat et pouvoir rester seule au bord de la falaise. J'ai sorti une petite scie de la poche de ma veste. C'\u00e9tait en fait une longue scie, mais constitu\u00e9e de trois lames r\u00e9tractables aux dents si grossi\u00e8res que leurs entailles sur une corde ne laisseraient jamais penser \u00e0 un sabotage volontaire.\n\nC'est \u00e0 ce moment-l\u00e0 qu'est arriv\u00e9 Yang Weining, mon \u00e9poux.\n\nApr\u00e8s que je lui ai expliqu\u00e9 la situation, il a regard\u00e9 en bas de la falaise, et a dit qu'il fallait creuser la roche pour v\u00e9rifier l'\u00e9tat du p\u00f4le \u00e9lectrique. Le vieux Lei n'y arriverait pas tout seul. Il a d\u00e9cid\u00e9 de descendre lui pr\u00eater main-forte. Il a alors enfil\u00e9 le harnais laiss\u00e9 par le soldat. Je lui ai recommand\u00e9 de prendre une autre corde, mais il m'a dit que ce n'\u00e9tait pas n\u00e9cessaire, celle-ci \u00e9tait suffisamment \u00e9paisse et r\u00e9sistante pour supporter sans probl\u00e8me le poids de deux hommes. J'ai insist\u00e9 et il m'a envoy\u00e9 chercher une autre corde. Mais alors que je revenais en toute h\u00e2te vers la falaise avec une nouvelle corde, il \u00e9tait d\u00e9j\u00e0 descendu le long de celle utilis\u00e9e par Lei Zhicheng. En baissant la t\u00eate, je les ai vus tous les deux : ils venaient de terminer la v\u00e9rification et \u00e9taient en train de remonter. Lei Zhicheng \u00e9tait devant.\n\nJe n'aurais jamais eu d'autre occasion. J'ai sorti mon outil et j'ai sci\u00e9 la corde.\n\nL'ENQU\u00caTEUR\n\nJe souhaiterais vous poser une question, mais votre r\u00e9ponse ne figurera pas dans le proc\u00e8s-verbal. Quel sentiment avez-vous eu \u00e0 ce moment-l\u00e0 ?\n\nYE WENJIE\n\nAucun. J'ai agi avec sang-froid. J'avais enfin trouv\u00e9 une mission \u00e0 laquelle je pouvais me d\u00e9vouer enti\u00e8rement, peu importe le tribut \u00e0 payer. Je savais de toute fa\u00e7on que c'\u00e9tait l'humanit\u00e9 tout enti\u00e8re qui en paierait le prix. Ce premier sacrifice n'\u00e9tait qu'une bagatelle.\n\nL'ENQU\u00caTEUR\n\nBien. Poursuivez.\n\nYE WENJIE\n\nJ'ai entendu deux ou trois cris de surprise, puis le bruit de corps se fracassant contre les rochers. J'ai attendu encore un moment et j'ai vu que le ruisseau qui coulait au pied de la falaise s'\u00e9tait empourpr\u00e9... Voil\u00e0 ce que j'avais \u00e0 dire sur cette affaire.\n\nL'ENQU\u00caTEUR\n\nEntendu. Voici le proc\u00e8s-verbal, relisez-le attentivement. S'il ne comporte aucune erreur, merci d'y apposer votre signature.\n\n## 26\n\n## AUCUNE REPENTANCE\n\nLa disparition de Lei Zhicheng et de Yang Weining fut class\u00e9e comme accident de travail par les responsables de C\u00f4te Rouge. Tout le monde savait que Ye Wenjie et Yang Weining formaient un couple heureux et personne ne songea jamais \u00e0 la suspecter. Un nouveau commissaire politique fut nomm\u00e9 et la vie reprit son cours. Le petit \u00eatre vivant dans le ventre de Wenjie grandissait jour apr\u00e8s jour. Elle sentait que le monde changeait autour d'elle.\n\nUn jour, le chef de la s\u00e9curit\u00e9 demanda \u00e0 Ye Wenjie de se rendre dans la gu\u00e9rite situ\u00e9e \u00e0 l'entr\u00e9e de la base. En entrant, quelle ne fut pas sa surprise quand elle se retrouva nez \u00e0 nez avec deux gar\u00e7ons et une fille, d'environ quinze ou seize ans. Ils \u00e9taient v\u00eatus de vieux manteaux trou\u00e9s et portaient sur le cr\u00e2ne des bonnets de fourrure. Ils semblaient \u00eatre de la r\u00e9gion. Le garde lui indiqua qu'ils venaient du village de Qijia. Les adolescents avaient entendu dire que des savants se trouvaient sur le pic du Radar et voulaient leur poser des questions relatives \u00e0 leurs \u00e9tudes.\n\nYe Wenjie se demandait comment ils avaient os\u00e9 venir ici. Le pic du Radar \u00e9tait une zone militaire interdite aux civils et les soldats avaient ordre de tirer apr\u00e8s un seul avertissement. Le garde en faction dans la gu\u00e9rite, qui avait remarqu\u00e9 la surprise de Wenjie, lui pr\u00e9cisa qu'il avait r\u00e9cemment re\u00e7u l'ordre de rel\u00e2cher d'un niveau la vigilance sur C\u00f4te Rouge. Tant que les habitants de la r\u00e9gion ne p\u00e9n\u00e9traient pas dans la base, ils pouvaient se rendre \u00e0 leur aise sur le pic du Radar. Des paysans du village \u00e9taient d'ailleurs venus la veille leur offrir des l\u00e9gumes.\n\nUn des adolescents sortit un livre de physique en piteux \u00e9tat. Ses mains \u00e9taient noires et pleines de ger\u00e7ures, comme l'\u00e9corce d'un arbre. Avec un fort accent du Nord-Est, il posa \u00e0 Wenjie une question de physique, de niveau coll\u00e8ge : le manuel disait qu'un corps en chute libre commence par acc\u00e9l\u00e9rer, puis finit par atteindre une vitesse constante. Ils y avaient r\u00e9fl\u00e9chi pendant plusieurs nuits, mais n'\u00e9taient pas arriv\u00e9s \u00e0 comprendre pourquoi.\n\n\u2014 Vous \u00eates venus de si loin juste pour me poser cette question ? demanda Wenjie.\n\n\u2014 Professeur Ye, vous n'\u00eates pas au courant ? Les examens de fin de lyc\u00e9e ont repris, dit la jeune fille, pleine d'entrain.\n\n\u2014 Les examens de fin de lyc\u00e9e ?\n\n\u2014 Oui, pour entrer \u00e0 l'universit\u00e9 ! Seuls les \u00e9tudiants avec les meilleures notes peuvent y aller !\n\n\u2014 Plus besoin d'\u00eatre recommand\u00e9 ?\n\n\u2014 Non, tout le monde peut passer les examens ! M\u00eame les enfants des cinq cat\u00e9gories noires1 !\n\nWenjie resta un moment abasourdie. Elle \u00e9tait profond\u00e9ment touch\u00e9e par ce bouleversement. De longues minutes pass\u00e8rent avant qu'elle ne se souvienne que les enfants l'attendaient, le livre ouvert. Elle s'empressa de leur r\u00e9pondre en expliquant que ce ph\u00e9nom\u00e8ne \u00e9tait d\u00fb \u00e0 l'\u00e9quilibre \u00e9tabli entre la r\u00e9sistance de l'air et la gravit\u00e9. Elle leur fit promettre que s'ils avaient \u00e0 l'avenir d'autres questions concernant leurs \u00e9tudes, ils ne devraient pas h\u00e9siter \u00e0 revenir la voir.\n\nTrois jours plus tard, ce furent sept enfants qui vinrent cette fois poser leurs questions \u00e0 Ye Wenjie. En plus des trois de la derni\u00e8re fois, quatre autres venaient de villages et de bourgades encore plus \u00e9loign\u00e9s. La troisi\u00e8me fois, ce furent quinze \u00e9l\u00e8ves et leur professeur de coll\u00e8ge qui mont\u00e8rent jusqu'au pic. Les \u00e9coles manquant de personnel, le professeur enseignait \u00e0 la fois la physique, les math\u00e9matiques et la chimie. Il \u00e9tait venu demander des conseils p\u00e9dagogiques \u00e0 Wenjie. L'homme avait d\u00e9j\u00e0 plus de la cinquantaine et son visage \u00e9tait marqu\u00e9 par les trag\u00e9dies de l'histoire. Devant Wenjie, il ne savait plus o\u00f9 donner de la t\u00eate, et avait recouvert le sol de livres. Plus tard, en sortant de la gu\u00e9rite, Wenjie l'entendit glisser aux \u00e9l\u00e8ves :\n\n\u2014 Les enfants, vous venez de voir une scientifique. Une vraie scientifique !\n\nD\u00e9sormais, des \u00e9l\u00e8ves venaient la trouver presque tous les jours. Ceux-ci \u00e9taient parfois si nombreux qu'ils ne tenaient pas tous dans la gu\u00e9rite. Avec l'accord du chef de la s\u00e9curit\u00e9, les gardes les escortaient jusque dans la cantine de la base, o\u00f9 Wenjie leur faisait cours sur un tableau noir.\n\nLa veille du Nouvel An chinois de 1978, il faisait nuit lorsque Ye Wenjie termina son travail. La plupart des employ\u00e9s de la base avaient pris trois jours de vacances et \u00e9taient descendus de la montagne. La base \u00e9tait silencieuse et d\u00e9serte. Wenjie retourna dans sa chambre, ce lieu qu'elle partageait autrefois avec Yang Weining. Tout \u00e9tait vide \u00e0 pr\u00e9sent, elle \u00e9tait seulement accompagn\u00e9e de l'enfant qu'elle portait dans son ventre. Dehors, il faisait un froid glacial, on entendait dans les rugissements du vent soufflant sur les montagnes du Grand Khingan les bruits indistincts des p\u00e9tards qui montaient du village de Qijia. Wenjie \u00e9tait oppress\u00e9e par la solitude, elle sentait sur elle comme une main \u00e9norme qui l'\u00e9crasait jusqu'\u00e0 la faire dispara\u00eetre dans un minuscule coin de l'univers...\n\nElle entendit soudain frapper \u00e0 la porte. Quand elle ouvrit, elle vit tout d'abord le garde de la gu\u00e9rite et, derri\u00e8re lui, des torches de pin vaciller dans le vent glacial. Les torches \u00e9taient tenues par un groupe d'enfants aux visages rougis par le froid et aux bonnets de fourrure couverts de givre. En entrant dans la chambre, ils apport\u00e8rent avec eux une vague de froid. Deux des gar\u00e7ons paraissaient particuli\u00e8rement frigorifi\u00e9s. Ils \u00e9taient l\u00e9g\u00e8rement v\u00eatus car ils tenaient entre les mains une chose qu'ils avaient envelopp\u00e9e dans leurs \u00e9pais manteaux. Ils ouvrirent les manteaux et r\u00e9v\u00e9l\u00e8rent une grande marmite en porcelaine dans laquelle se trouvaient des raviolis au chou et au porc encore fumants.\n\nCette ann\u00e9e-l\u00e0, huit mois apr\u00e8s avoir envoy\u00e9 un signal au soleil, Ye Wenjie commen\u00e7a \u00e0 avoir des contractions mais, le f\u0153tus \u00e9tant mal positionn\u00e9 et son corps tr\u00e8s faible, la clinique de la base n'\u00e9tait pas en mesure de la prendre en charge et dut la transf\u00e9rer dans l'h\u00f4pital de bourgade le plus proche.\n\nL'accouchement de Wenjie fut un enfer. En proie \u00e0 de grandes douleurs et perdant beaucoup de sang, elle eut m\u00eame une syncope. Dans son coma, elle crut voir indistinctement trois soleils aveuglants tourner lentement en orbite autour d'elle, consumant son corps. Cette vision se poursuivit longtemps, et dans le flou de la conscience, elle se dit que c'\u00e9tait peut-\u00eatre ici que tout allait se terminer, son enfer \u00e0 elle, fait de trois soleils \u00e0 jamais en train de la r\u00f4tir. C'\u00e9tait le ch\u00e2timent inflig\u00e9 pour sa terrible trahison. Elle sombra dans une peur intense, pas pour elle, mais pour son enfant. \u00c9tait-il encore dans son ventre ? Le suivrait-il dans cet enfer, serait-il lui aussi condamn\u00e9 \u00e0 souffrir pour l'\u00e9ternit\u00e9 ? Elle ne sut combien de temps s'\u00e9tait \u00e9coul\u00e9 lorsque les trois soleils s'\u00e9cart\u00e8rent. Ils s'\u00e9loign\u00e8rent jusqu'\u00e0 une certaine distance avant de rapetisser brusquement et de se transformer en trois \u00e9tincelantes \u00e9toiles volantes. L'air autour d'elle s'\u00e9tait rafra\u00eechi, sa douleur \u00e9tait moins vive. Elle se r\u00e9veilla enfin.\n\nYe Wenjie entendit des pleurs pr\u00e8s de son oreille. Elle eut toutes les peines du monde \u00e0 tourner la t\u00eate, mais put voir un petit visage d'enfant rose, mouill\u00e9 et chiffonn\u00e9.\n\nLe m\u00e9decin lui expliqua qu'elle avait perdu plus de 2 000 millilitres de sang. Des villageois de Qijia \u00e9taient venus pour donner le leur. Certains d'entre eux avaient des enfants qui suivaient les cours de Wenjie, mais beaucoup \u00e9taient de simples villageois et avaient simplement entendu parler d'elle par les adolescents et leurs parents. Sans eux, elle serait certainement morte.\n\nLes jours qui suivirent l'accouchement furent difficiles pour Ye Wenjie. Elle n'avait ni famille ni amis. Un vieux couple de villageois vint un jour au sommet du pic. Ils expliqu\u00e8rent aux responsables de la base qu'ils pouvaient ramener Wenjie et sa fille dans leur village et s'occuper d'elles. L'homme \u00e9tait \u00e0 l'origine chasseur et cueilleur d'herbes m\u00e9dicinales, mais les for\u00eats de la r\u00e9gion \u00e9tant maintenant r\u00e9duites \u00e0 peau de chagrin, il avait commenc\u00e9 \u00e0 cultiver son propre champ. Les gens continuaient par habitude \u00e0 l'appeler \"Qi le chasseur\". Wenjie n'avait jamais donn\u00e9 de cours \u00e0 leurs enfants. Ils avaient deux fils et deux filles. Les filles s'\u00e9taient mari\u00e9es et \u00e9taient parties vivre ailleurs. Un de leurs fils \u00e9tait \u00e0 l'arm\u00e9e. L'autre, d\u00e9j\u00e0 mari\u00e9, vivait avec eux et son \u00e9pouse, qui venait tout juste d'avoir un enfant.\n\nWenjie n'avait pas encore \u00e9t\u00e9 r\u00e9habilit\u00e9e et les responsables de la base se trouvaient dans une situation d\u00e9licate. Mais ils finirent par accepter que le couple ram\u00e8ne Wenjie et sa fille en tra\u00eeneau depuis l'h\u00f4pital jusqu'\u00e0 leur domicile.\n\nYe Wenjie v\u00e9cut plus de six mois dans cette famille de paysans des montages du Grand Khingan. Elle avait \u00e9t\u00e9 si faible apr\u00e8s son accouchement qu'elle n'avait pas eu de mont\u00e9e de lait. Pendant ce temps, la petite Yang Dong fut allait\u00e9e par les femmes du village. Celle qui lui donnait le plus souvent le sein \u00e9tait la belle-fille de la famille de Qi le chasseur, Dafeng. C'\u00e9tait une jeune femme robuste du Nord-Est qui se nourrissait chaque jour de bouillie de sorgho. Ses seins restaient toujours gorg\u00e9s de lait, m\u00eame si elle faisait t\u00e9ter deux nourrissons. D'autres jeunes m\u00e8res du village venaient elles aussi pour allaiter Yang Dong. Elles l'aimaient beaucoup et disaient que la petite semblait avoir h\u00e9rit\u00e9 de la vivacit\u00e9 d'esprit de sa m\u00e8re. Peu \u00e0 peu, la maison de Qi le chasseur devint un lieu de rendez-vous des femmes du village, les plus jeunes comme les plus \u00e2g\u00e9es, les \u00e9pouses comme les jeunes c\u00e9libataires. Elles venaient ici quand elles n'avaient rien \u00e0 faire, pleines d'admiration et de curiosit\u00e9 pour Ye Wenjie. Cette derni\u00e8re se rendit compte qu'elle avait de son c\u00f4t\u00e9 beaucoup de choses \u00e0 leur raconter. Durant les nombreuses journ\u00e9es ensoleill\u00e9es, Wenjie prenait Yang Dong dans ses bras et allait s'asseoir avec les autres femmes dans la cour bord\u00e9e par des bouleaux de Mandchourie. Les enfants jouaient \u00e0 c\u00f4t\u00e9 d'elles, tandis qu'un grand chien noir paresseux \u00e9tait allong\u00e9 sur le sol de la cour, baign\u00e9e par la douce chaleur de l'astre du jour. Elle aimait particuli\u00e8rement observer les femmes qui fumaient du tabac dans des pipes en cuivre et recrachaient nonchalamment des volutes de fum\u00e9e emplies de rayons de soleil doux et argent\u00e9s, comme les poils qui couraient le long de leurs membres dodus. Un jour, l'une d'entre elles lui tendit sa longue pipe blanche en cuivre, pour lui faire \u2013 disait-elle \u2013 \"oublier ses soucis\". Apr\u00e8s seulement deux bouff\u00e9es, Wenjie se sentit prise de vertige, ce qui fit bien rire tout le monde durant plusieurs jours.\n\nQuant aux hommes, Wenjie n'avait pas grand-chose \u00e0 leur dire. Elle avait du mal comprendre ce qui les occupait chaque jour. Elle avait plus ou moins compris qu'ils projetaient de profiter de la d\u00e9tente politique pour gagner de l'argent en cultivant du ginseng, mais ils n'avaient pas le courage de se lancer. Ils traitaient Wenjie avec beaucoup de respect et \u00e9taient avec elle d'une grande courtoisie. Elle n'y avait pas vraiment accord\u00e9 d'importance au d\u00e9but mais, les jours passant, en voyant les m\u00eames gaillards battre leurs \u00e9pouses et faire des avances aux veuves du village, en les entendant raconter des histoires graveleuses dont la moiti\u00e9 seulement suffisait \u00e0 la faire rougir, elle comprit tout ce que leurs \u00e9gards avaient de rare et de pr\u00e9cieux. Tous les deux ou trois jours, quelqu'un apportait un li\u00e8vre ou un faisan \u00e0 la maison de Qi le chasseur et offrait \u00e0 la petite Yang Dong des jouets curieux et d\u00e9suets qu'ils avaient fabriqu\u00e9s eux-m\u00eames.\n\nDans la m\u00e9moire de Wenjie, ces jours ne semblaient pas lui avoir appartenu. C'\u00e9tait comme s'ils avaient \u00e9t\u00e9 les lambeaux \u00e9pars de la vie d'une inconnue tomb\u00e9s sur elle comme des plumes souffl\u00e9es par le vent. Les souvenirs de cette p\u00e9riode se condensaient dans son esprit sous la forme de peintures classiques. \u00c9trangement, elle ne voyait pas des peintures chinoises traditionnelles, mais bien des peintures \u00e0 l'huile occidentales. Il y avait trop de vide dans la peinture chinoise, alors que ces heures pass\u00e9es au village de Qijia n'en comportaient pas, elles avaient la richesse et la densit\u00e9 des couleurs des peintures \u00e0 l'huile. Tout n'\u00e9tait que chaleur : les lits chauffants sous lesquels on faisait br\u00fbler d'\u00e9paisses couches de carex, le tabac _guandong_ et _mohe_ dans les pipes en cuivre, les nourrissants repas au sorgho, l'alcool de sorgho \u00e0 65 degr\u00e9s... toute cette vie s'\u00e9coulait silencieusement et paisiblement, comme le ruisseau bordant le village.\n\nYe Wenjie gardait un souvenir inoubliable des veill\u00e9es. Le fils du chef de Qi le chasseur \u00e9tait parti en ville pour vendre des champignons. Il avait \u00e9t\u00e9 le premier villageois \u00e0 quitter Qijia pour gagner sa vie. Wenjie partageait donc la chambre avec son \u00e9pouse Dafeng. Il n'y avait pas d'\u00e9lectricit\u00e9 au village et, tous les soirs, elles s'\u00e9clairaient \u00e0 la lumi\u00e8re d'une lampe \u00e0 huile ; Wenjie lisait, tandis que Dafeng faisait de la broderie. Il arrivait parfois que, sans le vouloir, Wenjie approche son livre et ses yeux trop pr\u00e8s de la lampe, si bien qu'elle faisait roussir sa frange et poussait un petit cri. Toutes deux levaient la t\u00eate et \u00e9changeaient un sourire. Cette m\u00e9saventure n'arrivait bien s\u00fbr jamais \u00e0 Dafeng, qui avait de tr\u00e8s bons yeux et arrivait \u00e0 r\u00e9aliser des ouvrages d\u00e9licats rien qu'\u00e0 la faible lueur des braises. Leurs deux enfants, \u00e2g\u00e9s d'\u00e0 peine six mois, dormaient c\u00f4te \u00e0 c\u00f4te dans un lit chauffant. Wenjie aimait les regarder dormir. Dans toute la maison, on n'entendait que le bruit r\u00e9gulier de leur respiration.\n\nAu d\u00e9but, Wenjie avait du mal \u00e0 dormir sur le lit chauffant, elle avait fr\u00e9quemment des coups de chaud, mais elle finit peu \u00e0 peu par s'y habituer. Dans ses r\u00eaves, elle devenait quelquefois son propre enfant et se blottissait entre de grands bras chauds. Cette sensation \u00e9tait si r\u00e9aliste qu'elle se r\u00e9veillait souvent en larmes. Celui qui la tenait dans ses bras n'\u00e9tait ni son p\u00e8re ni sa m\u00e8re, ni non plus son mari d\u00e9c\u00e9d\u00e9, mais quelqu'un dont elle ignorait l'identit\u00e9.\n\nUn soir, elle posa son livre et vit Dafeng, les semelles qu'elle \u00e9tait en train de rapi\u00e9cer pos\u00e9es sur ses genoux. Elle fixait la lampe, immobile. Quand elle s'aper\u00e7ut que Wenjie la regardait, elle lui demanda soudain :\n\n\u2014 Grande s\u0153ur Wenjie, est-ce que les \u00e9toiles ne vont pas tomber du ciel ?\n\nWenjie examina Dafeng. La lampe \u00e0 huile \u00e9tait une artiste formidable, elle esquissait dans la chambre une fresque aux lignes et aux couleurs chatoyantes : son manteau en laine pos\u00e9 sur ses \u00e9paules, Dafeng exhibait, sous sa nuisette rouge, un bras robuste mais gracieux. La lumi\u00e8re de la lampe \u00e0 huile mettait en valeur la plus belle partie de son corps en la teintant de la couleur la plus chaude et en plongeant le reste de la pi\u00e8ce dans une sombre volupt\u00e9. Le d\u00e9cor paraissait s'\u00eatre fondu dans la douceur de l'obscurit\u00e9 mais, en regardant bien, on pouvait voir un halo rouge sombre qui ne provenait pas de la lampe \u00e0 huile mais des braises de charbon sur le sol. Dehors, le froid de l'hiver sculptait de magnifiques motifs sur les carreaux des fen\u00eatres gr\u00e2ce \u00e0 l'air chaud et humide de la maisonn\u00e9e.\n\n\u2014 Tu as peur que les \u00e9toiles tombent ? demanda calmement Wenjie.\n\n\u2014 Pourquoi avoir peur ? rit Dafeng en secouant la t\u00eate. Elles sont si petites !\n\nWenjie ne lui donna pas la r\u00e9ponse d'une astrophysicienne. Elle lui dit simplement :\n\n\u2014 Elles sont tr\u00e8s, tr\u00e8s loin, elles ne risquent pas de tomber.\n\nDafeng se satisfit de cette explication et replongea dans son ouvrage. Mais Wenjie ne parvint pas \u00e0 rester tranquille, elle abandonna son livre et s'allongea sur la surface chaude du lit. Elle ferma les yeux et, dans son imagination, elle fit dispara\u00eetre tout l'univers autour de la chaumi\u00e8re, comme la lampe \u00e0 huile faisait se fondre la plus grande partie de la chambre dans l'obscurit\u00e9. Puis elle rempla\u00e7a l'univers r\u00e9el par celui de Dafeng. \u00c0 cet instant, le ciel nocturne devint un gigantesque d\u00f4me noir dont la taille permettait tout juste d'envelopper le monde. Sur le d\u00f4me \u00e9taient incrust\u00e9es une infinit\u00e9 d'\u00e9toiles d'une phosphorescence argent\u00e9e. Chaque \u00e9toile n'\u00e9tait pas plus grosse que le miroir rond pos\u00e9 sur la vieille table de chevet en bois de la chambre. Le monde \u00e9tait plat et s'\u00e9tendait loin dans chaque direction, mais finissait par s'arr\u00eater aux extr\u00e9mit\u00e9s du d\u00f4me. Cette surface plane \u00e9tait couverte de cha\u00eenes de montagnes comme celles du Grand Khingan et de for\u00eats auxquelles \u00e9taient adoss\u00e9s de petits villages comme Qijia... Cet univers en forme de coffre \u00e0 jouets r\u00e9conforta Wenjie et, peu \u00e0 peu, cette image mentale se changea en r\u00eave.\n\nDans ce minuscule hameau cach\u00e9 au plus profond des montagnes du Grand Khingan, quelque chose dans le c\u0153ur de Ye Wenjie fondait peu \u00e0 peu. Sur le champ de glace qu'\u00e9tait son \u00e2me \u00e9tait apparu un lac d'eau claire.\n\nApr\u00e8s la naissance de sa fille, Wenjie passa encore deux ans dans la base de C\u00f4te Rouge, partag\u00e9e entre paix et anxi\u00e9t\u00e9. Elle fut inform\u00e9e que son p\u00e8re et elle avaient \u00e9t\u00e9 politiquement r\u00e9habilit\u00e9s. Elle re\u00e7ut bient\u00f4t une lettre de son ancienne universit\u00e9 lui indiquant qu'elle pouvait reprendre son poste d\u00e8s qu'elle le souhaitait. La lettre \u00e9tait accompagn\u00e9e d'un mandat postal : le salaire d\u00fb \u00e0 son p\u00e8re apr\u00e8s sa r\u00e9habilitation. Lors d'une r\u00e9union g\u00e9n\u00e9rale de la base, les responsables de Wenjie l'appel\u00e8rent enfin \"camarade\".\n\nWenjie accueillit ces bouleversements avec beaucoup de calme ; elle n'\u00e9tait ni \u00e9mue ni euphorique. Le monde ext\u00e9rieur ne lui inspirait aucun int\u00e9r\u00eat. Elle pr\u00e9f\u00e9rait rester \u00e0 C\u00f4te Rouge, loin de tout. Mais pour l'\u00e9ducation de sa fille, elle finit par partir de cette base o\u00f9 elle avait jadis cru qu'elle terminerait sa vie, et retourna dans son universit\u00e9 d'origine.\n\nApr\u00e8s avoir quitt\u00e9 la montagne, Ye Wenjie sentit le printemps tout autour d'elle. Le terrible hiver de la R\u00e9volution culturelle \u00e9tait fini, tout \u00e9tait en train de bourgeonner de nouveau. M\u00eame si la catastrophe venait tout juste de se terminer et que, partout o\u00f9 portait le regard, tout n'\u00e9tait que ruines et cicatrices que d'innombrables personnes pansaient en silence, l'aube d'une nouvelle vie \u00e9tait bel et bien apparue dans les yeux des gens. Des \u00e9tudiants et leurs enfants r\u00e9investissaient les universit\u00e9s ; dans les librairies, on s'arrachait des ouvrages de litt\u00e9rature ; les innovations technologiques et industrielles connaissaient des avanc\u00e9es incroyables et la recherche scientifique se retrouvait \u00e0 nouveau aur\u00e9ol\u00e9e d'un prestige sacr\u00e9. La science et la technologie \u00e9taient les seules clefs qui permettaient d'ouvrir le verrou de l'avenir, si bien que la population accueillait maintenant la science avec la m\u00eame candeur que de jeunes \u00e9l\u00e8ves de primaire. Et quoique leur enthousiasme soit na\u00eff, ils gardaient les pieds sur terre. Lors de la premi\u00e8re Conf\u00e9rence nationale sur les sciences, Guo Moruo, \u00e9crivain et pr\u00e9sident de l'Acad\u00e9mie chinoise des sciences, proclama le d\u00e9but du \"printemps de la science\".\n\n\u00c9tait-ce la fin de la folie ? Le retour du r\u00e8gne de la science et de la raison ? Ye Wenjie se posait sans cesse cette question.\n\nJusqu'\u00e0 son d\u00e9part de C\u00f4te Rouge, Ye Wenjie ne re\u00e7ut plus aucun message provenant de Trisolaris. Elle savait qu'elle devait attendre au moins huit ans avant d'avoir une r\u00e9ponse mais, apr\u00e8s son d\u00e9part de la base, elle ne b\u00e9n\u00e9ficiait plus du mat\u00e9riel n\u00e9cessaire.\n\nCe secret \u00e9tait lourd \u00e0 porter et elle avait agi seule. Cela lui donnait un sentiment d'irr\u00e9alit\u00e9, et plus le temps passait, plus ce sentiment \u00e9tait fort. De plus en plus, elle avait la sensation que tout n'avait \u00e9t\u00e9 qu'une illusion, un r\u00eave. Le soleil avait-il vraiment le pouvoir d'amplifier des ondes radio ? L'avait-elle vraiment utilis\u00e9 comme une antenne pour envoyer un message dans l'univers ? Avait-elle vraiment re\u00e7u la r\u00e9ponse d'une civilisation extraterrestre ? Ce matin sanglant o\u00f9 elle avait trahi l'humanit\u00e9 tout enti\u00e8re avait-il exist\u00e9 ? Et ce double meurtre...\n\nWenjie essaya de s'ab\u00eemer dans son travail pour mieux tout oublier. Et elle faillit presque r\u00e9ussir : un \u00e9trange instinct protecteur inhibait ses souvenirs, notamment ceux de ce premier contact avec une civilisation extraterrestre. Sa vie s'\u00e9coulait paisiblement, jour apr\u00e8s jour.\n\nUn certain temps apr\u00e8s \u00eatre retourn\u00e9e dans son universit\u00e9, Wenjie emmena Dong-dong rendre visite \u00e0 sa grand-m\u00e8re, Shao Lin. Apr\u00e8s la mort tragique de son \u00e9poux, Shao Lin \u00e9tait parvenue \u00e0 sortir de sa d\u00e9mence et avait continu\u00e9 \u00e0 survivre entre les fissures de la politique. \u00c0 force de suivre le vent tournant et de hurler avec les loups, elle avait enfin obtenu une modeste r\u00e9compense : \u00e0 l'occasion du mouvement \"Retourner en classe pour faire la r\u00e9volution\", on lui avait propos\u00e9 un poste de professeur2.\n\nCependant elle avait ensuite pris un chemin inattendu : elle s'\u00e9tait remari\u00e9e avec un ancien haut cadre du minist\u00e8re de l'\u00c9ducation, pers\u00e9cut\u00e9 durant la R\u00e9volution culturelle. Quand elle l'avait \u00e9pous\u00e9, le cadre se trouvait encore emprisonn\u00e9 dans une \"\u00e9table3\". Cet acte pouvait para\u00eetre d\u00e9concertant, venant d'une femme qui pesait, au d\u00e9tail pr\u00e8s, chacune de ses d\u00e9cisions. Mais son esprit \u00e9tait tr\u00e8s clair, elle avait tout pr\u00e9vu. Elle savait que le chaos de la R\u00e9volution culturelle ne durerait pas longtemps et que les jeunes rebelles qui tentaient de prendre le pouvoir n'avaient pas l'exp\u00e9rience n\u00e9cessaire pour gouverner le pays. T\u00f4t ou tard, les vieux cadres ayant \u00e9t\u00e9 pers\u00e9cut\u00e9s ou marginalis\u00e9s reviendraient aux affaires. La suite montra que son pari \u00e9tait le bon. Alors que la R\u00e9volution culturelle n'\u00e9tait m\u00eame pas encore termin\u00e9e, son mari avait r\u00e9cup\u00e9r\u00e9 son ancien poste, et au lendemain de la troisi\u00e8me session pl\u00e9ni\u00e8re du XIe Congr\u00e8s du Parti communiste chinois en 19784, il acc\u00e9da m\u00eame \u00e0 un poste de vice-ministre. Dans ce contexte o\u00f9 les intellectuels retrouvaient les faveurs des dirigeants, la carri\u00e8re de Shao Lin monta en fl\u00e8che. Apr\u00e8s \u00eatre devenue membre de l'Acad\u00e9mie chinoise des sciences, elle quitta habilement son ancienne \u00e9cole et fut promue vice-pr\u00e9sidente d'une universit\u00e9 prestigieuse.\n\nQuand Ye Wenjie rencontra sa m\u00e8re, elle vit une femme \u00e9duqu\u00e9e, prenant grand soin de son apparence. Elle ne semblait pas porter le moindre stigmate de ses pers\u00e9cutions pass\u00e9es. Elle accueillit chaleureusement Wenjie et sa petite-fille et leur demanda avec empathie comment elles avaient pass\u00e9 ces derni\u00e8res ann\u00e9es. Elle s'extasia devant la petite Dong-dong \u2013 si maligne et si mignonne \u2013 et d\u00e9tailla \u00e0 sa cuisini\u00e8re les plats pr\u00e9f\u00e9r\u00e9s de Wenjie... Tout \u00e9tait fait avec finesse, habilet\u00e9 et justesse. Mais Ye Wenjie pouvait sentir le foss\u00e9 qui s'\u00e9tait creus\u00e9 entre elles. Toutes deux \u00e9vit\u00e8rent de parler de sujets sensibles et n'\u00e9voqu\u00e8rent jamais son p\u00e8re.\n\nApr\u00e8s le d\u00e9jeuner, Shao Lin et son \u00e9poux raccompagn\u00e8rent Wenjie et sa fille dans la rue pour leur dire au revoir. Le vice-ministre demanda \u00e0 parler \u00e0 Wenjie et Shao Lin rentra la premi\u00e8re. Aussit\u00f4t, le sourire chaleureux du vice-ministre se fit de glace, comme s'il avait enfin l'occasion d'\u00f4ter son masque :\n\n\u2014 Toi et ta fille \u00eates les bienvenues ici, mais \u00e0 une condition : n'essaie jamais de remuer de vieilles histoires. Ta m\u00e8re n'a aucune responsabilit\u00e9 dans la mort de ton p\u00e8re, elle aussi est une victime. C'est plut\u00f4t ton p\u00e8re qui est pathologiquement rest\u00e9 but\u00e9 dans ses convictions. Il a lui-m\u00eame choisi le chemin obscur. Il a abandonn\u00e9 sa responsabilit\u00e9 de p\u00e8re de famille et vous a laiss\u00e9es, m\u00e8re et filles, endurer de grandes souffrances.\n\n\u2014 De quel droit me parlez-vous de mon p\u00e8re ? riposta Wenjie, furieuse. C'est une histoire entre ma m\u00e8re et moi, cela n'a rien \u00e0 voir avec vous.\n\n\u2014 En effet, dit froidement l'\u00e9poux de Shao Lin en opinant de la t\u00eate, je ne fais que transmettre le message de ta m\u00e8re.\n\nWenjie leva les yeux vers la r\u00e9sidence avec jardin r\u00e9serv\u00e9e aux hauts cadres du Parti. Shao Lin avait tir\u00e9 les rideaux et les \u00e9piait depuis un coin de la maison. Sans un mot, Wenjie partit avec Dong-dong dans les bras. Elle ne revint plus jamais.\n\nYe Wenjie partit \u00e0 la recherche des quatre jeunes filles gardes rouges qui avaient tu\u00e9 son p\u00e8re cette ann\u00e9e-l\u00e0. Apr\u00e8s avoir retourn\u00e9 ciel et terre, elle parvint \u00e0 retrouver trois d'entre elles. Jadis envoy\u00e9es \u00e0 la campagne comme jeunes instruites, elles \u00e9taient maintenant de retour en ville et se trouvaient actuellement sans emploi. Ayant localis\u00e9 leurs adresses, Ye Wenjie \u00e9crivit une lettre toute simple \u00e0 chacune en leur donnant rendez-vous \u00e0 l'endroit m\u00eame o\u00f9 son p\u00e8re \u00e9tait mort, simplement pour parler.\n\nYe Wenjie n'avait pas l'intention de se venger. Ce matin-l\u00e0, \u00e0 C\u00f4te Rouge, elle avait d\u00e9j\u00e0 pris sa revanche sur l'humanit\u00e9 tout enti\u00e8re, y compris les jeunes gardes rouges. Elle voulait seulement les entendre se repentir du meurtre de leur p\u00e8re et voir si elles avaient retrouv\u00e9 un semblant d'humanit\u00e9.\n\nCet apr\u00e8s-midi, apr\u00e8s ses cours, Ye Wenjie les attendit pr\u00e8s du terrain de sport. Elle n'entretenait pas beaucoup d'espoir de les voir, elle \u00e9tait m\u00eame presque s\u00fbre qu'elles ne viendraient pas. Mais \u00e0 l'heure du rendez-vous, les trois anciennes gardes rouges arriv\u00e8rent.\n\nYe Wenjie les reconnut de loin. Elles \u00e9taient encore v\u00eatues d'uniformes verts de gardes rouges que l'on ne voyait plus que rarement ces jours-ci. Quand elles furent plus proches, Wenjie r\u00e9alisa qu'il y avait de fortes chances que ces uniformes soient d'ailleurs ceux qu'elles avaient port\u00e9s ce jour-l\u00e0, lors de la s\u00e9ance de critique publique. Leur couleur avait terni \u00e0 force de lavages, et ils avaient visiblement \u00e9t\u00e9 rapi\u00e9c\u00e9s. En dehors de ces uniformes, les trois femmes, qui avaient maintenant la trentaine, n'avaient plus rien de commun avec les trois jeunes et vaillantes gardes rouges de l'\u00e9poque. Elles ne semblaient pas seulement avoir perdu leur jeunesse, mais aussi beaucoup d'autres choses.\n\nLa premi\u00e8re impression de Ye Wenjie \u00e9tait que, bien qu'elles aient \u00e9t\u00e9 format\u00e9es dans le m\u00eame moule, il existait aujourd'hui de grandes diff\u00e9rences entre elles. L'une d'elles \u00e9tait devenue petite et osseuse, son vieil uniforme paraissait m\u00eame presque trop grand sur elle, son dos \u00e9tait vo\u00fbt\u00e9, ses cheveux avaient jauni et elle avait l'apparence d'une vieillarde. La deuxi\u00e8me, elle, \u00e9tait devenue beaucoup plus robuste et n'\u00e9tait pas parvenue \u00e0 boutonner la veste de son uniforme. Elle avait une grosse tignasse sur la t\u00eate et son visage \u00e9tait sombre, comme si les vicissitudes de la vie lui avaient d\u00e9rob\u00e9 toute \u00e9l\u00e9gance f\u00e9minine en ne lui laissant que torpeur et rudesse. La troisi\u00e8me avait gard\u00e9 une silhouette jeune, mais l'une des manches de son uniforme \u00e9tait vide, et elle marchait en boitillant.\n\nLes trois anciennes gardes rouges march\u00e8rent \u00e0 la rencontre de Ye Wenjie et s'align\u00e8rent devant elle, comme elles l'avaient fait cette ann\u00e9e-l\u00e0 devant Ye Zhetai. Mais quand bien m\u00eame il se serait agi d'une tentative de leur part pour retrouver une dignit\u00e9 longtemps perdue, l'ardeur diabolique qui avait \u00e9t\u00e9 la leur cette ann\u00e9e-l\u00e0 n'existait plus. Le visage de la plus maigre arborait une expression de souris, celui de la femme robuste ne montrait que de la torpeur, tandis que les deux yeux de l'estropi\u00e9e regardaient le ciel.\n\n\u2014 Vous pensiez que nous n'aurions pas le cran de venir ? demanda la femme robuste sur un ton de d\u00e9fi.\n\n\u2014 Je pensais simplement que nous devions nous voir. T\u00f4t ou tard, il fallait tourner la page du pass\u00e9, dit Ye Wenjie.\n\n\u2014 Oui, c'est du pass\u00e9, vous devriez le savoir, dit \u00e0 son tour la plus maigre, avec une voix per\u00e7ante, comme si elle \u00e9tait effray\u00e9e par quelque chose d'inconnu.\n\n\u2014 Je veux dire, tourner la page, spirituellement.\n\n\u2014 Vous voulez nous entendre nous repentir ? demanda la femme robuste.\n\n\u2014 Ne croyez-vous pas que vous devriez ?\n\n\u2014 Et qui se repentira aupr\u00e8s de nous ? r\u00e9torqua l'estropi\u00e9e, qui \u00e9tait jusqu'ici rest\u00e9e silencieuse.\n\nLa femme robuste prit la suite :\n\n\u2014 De nous quatre, trois ont \u00e0 l'\u00e9poque sign\u00e9 le _dazibao_ du lyc\u00e9e affili\u00e9 \u00e0 l'universit\u00e9 Qinghua5. Nous avons \u00e9t\u00e9 de tous les combats : les campagnes de ralliement sur la place Tian'anmen, les cent jours de combat arm\u00e9 de Qinghua, le premier bataillon des gardes rouges, le deuxi\u00e8me bataillon, le troisi\u00e8me, le Comit\u00e9 d'action unie, les palissades de l'Ouest, les palissades de l'Est, la Grande Commune de la Nouvelle Universit\u00e9 de P\u00e9kin, l'\u00c9quipe de combat du drapeau rouge, le Groupe de l'Orient rouge \u2013 nous avons particip\u00e9 \u00e0 toute l'\u00e9pop\u00e9e des gardes rouges, de leur naissance jusqu'\u00e0 leur mort.\n\nL'estropi\u00e9e continua :\n\n\u2014 Sur le campus de Qinghua, pendant les cent jours de combat arm\u00e9, deux d'entre nous appartenaient \u00e0 la faction du mont Jinggang et les deux autres \u00e0 la faction du 14 Avril. J'ai attaqu\u00e9 un tank artisanal du mont Jinggang avec une grenade, mais j'ai eu le bras \u00e9cras\u00e9 par les chenilles du tank. Ma chair et mes os n'ont fait qu'un avec la boue et le sol... J'avais \u00e0 peine quinze ans !\n\n\u2014 On nous a ensuite envoy\u00e9es en pleine campagne ! dit la femme robuste en levant les mains. Deux d'entre nous sont parties dans la province du Shaanxi et les deux autres dans le Henan, les endroits les plus pauvres et les plus perdus de tous ! \u00c0 notre d\u00e9part, nous \u00e9tions encore pleines d'ardeur id\u00e9aliste, mais \u00e7a n'a pas dur\u00e9. \u00c0 la fin d'une journ\u00e9e de labeur dans les champs, nous \u00e9tions si fatigu\u00e9es que nous n'avions m\u00eame pas la force de laver nos habits. Nous nous \u00e9tendions dans nos chaumi\u00e8res aux toits fuyants et entendions au loin les hurlements des loups, qui nous arrachaient peu \u00e0 peu \u00e0 nos r\u00eaves. Nous \u00e9tions coinc\u00e9es dans des villages oubli\u00e9s et personne ne pouvait r\u00e9pondre \u00e0 nos suppliques.\n\nLes yeux vides, baiss\u00e9s sur le sol, l'estropi\u00e9e poursuivit :\n\n\u2014 Parfois, sur des sentiers perdus de montagnes arides, nous rencontrions d'anciens camarades ou ennemis gardes rouges. Nous \u00e9changions un regard : les m\u00eames uniformes en lambeaux, les m\u00eames corps couverts de poussi\u00e8re et de bouse de vache. Nous ne savions pas quoi nous dire.\n\n\u2014 Tang Hongjing, dit la femme robuste \u00e0 Ye Wenjie. C'est elle qui a donn\u00e9 le coup fatal \u00e0 ton p\u00e8re avec la boucle de sa ceinture. Elle est morte noy\u00e9e dans le fleuve Jaune. Ce jour-l\u00e0, un torrent avait emport\u00e9 des moutons appartenant au r\u00e9giment. Le cadre local du Parti a lanc\u00e9 aux jeunes instruits : \"Petits g\u00e9n\u00e9raux de la r\u00e9volution ! C'est le moment de vous mettre \u00e0 l'\u00e9preuve !\" Tang Hongjing et trois autres jeunes ont alors saut\u00e9 dans l'eau pour r\u00e9cup\u00e9rer les moutons, mais il faisait terriblement froid, l'eau \u00e9tait m\u00eame couverte d'une couche de glace ! Les quatre sont morts, nous n'avons jamais su si c'\u00e9tait de froid ou de noyade. Quand j'ai vu leurs cadavres... Je... Oh, merde, je n'y arrive pas...\n\nElle se prit le visage entre les mains et se mit \u00e0 pleurer. L'estropi\u00e9e continua :\n\n\u2014 Les cadavres de Hongjing et des autres ont \u00e9t\u00e9 stock\u00e9s dans le grenier du r\u00e9giment, comme si c'\u00e9taient de vulgaires b\u00fbches pour le feu, \u00e0 c\u00f4t\u00e9 des choux et des patates, avec... avec les moutons morts qu'ils avaient r\u00e9ussi \u00e0 rep\u00eacher.\n\nLa femme maigre poussa un soupir, elle avait les larmes aux yeux :\n\n\u2014 Nous sommes maintenant de retour en ville, mais \u00e0 quoi bon ? Nous n'avons rien, les conditions de vie des jeunes instruits de retour de la campagne ne sont pas faciles. M\u00eame pour les travaux les plus d\u00e9gradants, on ne veut pas de nous ! Nous sommes sans travail, sans argent, sans avenir. Nous n'avons rien.\n\nYe Wenjie \u00e9tait interloqu\u00e9e.\n\n\u2014 Il y a un film sorti r\u00e9cemment, dont le titre est _\u00c9rable_ , je ne sais pas si tu l'as vu. \u00c0 la fin, un adulte et un enfant se tiennent devant la tombe d'un garde rouge mort durant les combats civils entre diff\u00e9rentes factions. L'enfant demande \u00e0 l'adulte : \"Sont-ils des h\u00e9ros ?\" L'adulte r\u00e9pond que non ; alors l'enfant demande : \"Sont-ils des ennemis ?\" L'adulte r\u00e9pond encore une fois que non. L'enfant demande : \"Alors que sont-ils ?\" Et l'adulte r\u00e9pond : \"De l'histoire ancienne.\"\n\n\u2014 Tu as entendu ? De l'histoire ancienne ! De l'histoire ? cria fr\u00e9n\u00e9tiquement la femme robuste en agitant une main devant Ye Wenjie. C'est maintenant le d\u00e9but d'une nouvelle \u00e8re ! Qui se souviendra de nous ? Qui croira que tu as exist\u00e9, que nous avons exist\u00e9 ? Tout le monde aura bient\u00f4t oubli\u00e9 !\n\nLes trois vieilles gardes rouges partirent, laissant Wenjie seule sur le terrain de sport. Plus de dix ans auparavant, lors d'un apr\u00e8s-midi pluvieux, elle s'\u00e9tait aussi trouv\u00e9e l\u00e0, seule et droite, en regardant le cadavre de son p\u00e8re. Les paroles de la garde rouge r\u00e9sonnaient en elle : \"Tout le monde aura bient\u00f4t oubli\u00e9...\"\n\nLes rayons du cr\u00e9puscule allong\u00e8rent la silhouette fr\u00eale de Ye Wenjie. Le petit sursaut d'espoir pour la soci\u00e9t\u00e9 qui avait \u00e9merg\u00e9 dans son esprit s'\u00e9tait \u00e9vapor\u00e9 comme une goutte d'eau sous un soleil br\u00fblant. Le doute qui s'\u00e9tait empar\u00e9 d'elle apr\u00e8s son acte de trahison avait disparu sans laisser de trace. D\u00e9sormais, son nouvel id\u00e9al serait in\u00e9branlable : permettre \u00e0 une civilisation sup\u00e9rieure de s'infiltrer dans le monde des humains.\n\n* * *\n\n1 Les \u00e9l\u00e9ments appartenant aux cat\u00e9gories noires \u2013 les propri\u00e9taires fonciers, les paysans riches, les mauvais \u00e9l\u00e9ments, les contre-r\u00e9volutionnaires et les droitiers \u2013 et leurs familles \u00e9taient trait\u00e9s comme des parias lors de la R\u00e9volution culturelle. _(N.d.T.)_\n\n2 Au d\u00e9but de la R\u00e9volution culturelle en 1966, les \u00e9coles furent ferm\u00e9es pour permettre aux \u00e9l\u00e8ves et aux \u00e9tudiants de \"faire la r\u00e9volution\". L'\u00e9cole des gardes rouges devenait la soci\u00e9t\u00e9 tout enti\u00e8re. Mais en 1967, les autorit\u00e9s, d\u00e9bord\u00e9es par la situation, revinrent sur leur d\u00e9cision en demandant aux \u00e9l\u00e8ves de retourner dans les \u00e9coles. _(N.d.T.)_\n\n3 Durant la R\u00e9volution culturelle, les \"ennemis du peuple\" \u00e9taient repr\u00e9sent\u00e9s par des d\u00e9mons \u00e0 t\u00eate de b\u0153uf. Pour filer la m\u00e9taphore, les cachots ou les \"camps de r\u00e9\u00e9ducation par le travail\" dans lesquels ils \u00e9taient retenus \u00e9taient appel\u00e9s \"\u00e9tables\". _(N.d.T.)_\n\n4 Cet \u00e9v\u00e9nement marque un tournant important dans l'histoire de la Chine contemporaine. Sous l'impulsion de Deng Xiaoping, le Parti communiste chinois d\u00e9cide de mettre en place des r\u00e9formes d'ouverture politiques \u2013 qui s'av\u00e9reront n\u00e9anmoins vite restreintes \u2013 et \u00e9conomiques. _(N.d.T.)_\n\n5 Le 29 mai 1966, des lyc\u00e9ens coll\u00e8rent un _dazibao_ ayant pour titre \"Gardes rouges\" sur un des murs de l'universit\u00e9 Qinghua. Il fut sign\u00e9 par plus de cent \u00e9tudiants. Cet \u00e9v\u00e9nement marqua la premi\u00e8re apparition de l'expression \"gardes rouges\". Les sp\u00e9cialistes de la R\u00e9volution culturelle consid\u00e8rent que c'est ce jour-l\u00e0 que naquirent les brigades des gardes rouges. _(N.d.A.)_\n\n## 27\n\n## EVANS\n\nSix mois apr\u00e8s son retour \u00e0 l'universit\u00e9, on confia \u00e0 Ye Wenjie une t\u00e2che importante : la conception d'un observatoire radioastronomique de grande envergure. Elle et son \u00e9quipe de conception parcoururent le pays en qu\u00eate d'un site appropri\u00e9 o\u00f9 \u00e9tablir la base. Au d\u00e9but, leurs pr\u00e9occupations \u00e9taient purement techniques. Contrairement \u00e0 la discipline astronomique traditionnelle, les exigences de la radioastronomie au sujet de la qualit\u00e9 de l'air et de la lumi\u00e8re visible n'\u00e9taient pas si \u00e9lev\u00e9es, mais il fallait \u00e0 tout prix \u00e9viter les perturbations \u00e9lectromagn\u00e9tiques agissant au niveau des spectres non visibles. Ils sillonn\u00e8rent de nombreuses r\u00e9gions avant de choisir un lieu o\u00f9 l'environnement \u00e9lectromagn\u00e9tique s'av\u00e9rait le plus favorable : une r\u00e9gion vallonn\u00e9e et isol\u00e9e du Nord-Ouest.\n\nAucune v\u00e9g\u00e9tation ou presque ne poussait sur ces collines de l\u0153ss. Les vall\u00e9es n\u00e9es de l'\u00e9rosion des sols donnaient aux collines l'apparence de vieux visages couverts de rides. Une fois \u00e9tabli une premi\u00e8re s\u00e9lection de sites susceptibles d'accueillir l'observatoire, les v\u00e9hicules de l'\u00e9quipe de conception s'arr\u00eat\u00e8rent un moment dans un village o\u00f9 les gens vivaient presque tous dans des habitations troglodytes. Quand le responsable de l'unit\u00e9 de production du village apprit que Ye Wenjie avait un certain niveau d'\u00e9ducation, il lui demanda si elle parlait les langues \u00e9trang\u00e8res. Elle lui demanda de pr\u00e9ciser quelle langue, mais le responsable ne sut pas lui r\u00e9pondre. Si elle \u00e9tait capable de parler une langue \u00e9trang\u00e8re, il enverrait un villageois sur la colline pour aller chercher Bethune, car l'unit\u00e9 de production devait s'entretenir avec lui.\n\n\u2014 Bethune1 ? demanda-t-elle, surprise.\n\n\u2014 Nous ne connaissons pas son vrai nom, c'est comme \u00e7a que nous l'appelons.\n\n\u2014 Il est m\u00e9decin ?\n\n\u2014 Non. Il plante des arbres derri\u00e8re la colline, il est ici depuis trente ans.\n\n\u2014 Il plante des arbres ? Pourquoi ?\n\n\u2014 Il dit que c'est pour les oiseaux. Une esp\u00e8ce d'oiseau dont il dit qu'elle va bient\u00f4t dispara\u00eetre.\n\nWenjie et ses coll\u00e8gues furent tr\u00e8s surpris et ils demand\u00e8rent au responsable de l'unit\u00e9 de production de les conduire jusqu'\u00e0 ce Bethune. Ils suivirent un sentier et arriv\u00e8rent au sommet d'un tertre. Le responsable leur montra quelque chose de la main. Les yeux de Ye Wenjie s'illumin\u00e8rent : sur le flanc d'une colline de l\u0153ss aride s'\u00e9tendait une for\u00eat verdoyante, comme une \u00e9claboussure de peinture verte sur une vieille toile jaunie.\n\nYe Wenjie et ses compagnons aper\u00e7urent l'\u00e9tranger. En dehors de ses cheveux blonds et de ses yeux verts, de son pantalon et de sa veste en jean ab\u00eem\u00e9s, il n'avait pas l'air si diff\u00e9rent d'un paysan du coin ayant pass\u00e9 sa vie \u00e0 travailler la terre. Sa peau \u00e9tait aussi noircie par le soleil que celle des habitants de la r\u00e9gion. Il ne se montra pas tr\u00e8s int\u00e9ress\u00e9 par la pr\u00e9sence de ces nouveaux visiteurs. Il se pr\u00e9senta : Mike Evans. Il ne d\u00e9clina pas sa nationalit\u00e9, mais son anglais \u00e9tait empreint d'un fort accent am\u00e9ricain. Il habitait dans une simple cabane en adobe, \u00e0 la lisi\u00e8re de la for\u00eat. Celle-ci \u00e9tait jonch\u00e9e d'outils qu'il utilisait pour planter des arbres : houes, pelles, petites scies pour \u00e9laguer les branches... tous rudimentaires et fabriqu\u00e9s dans la r\u00e9gion. Le sable du Nord-Ouest s'\u00e9tait invit\u00e9 dans sa cabane \u00e0 deux pi\u00e8ces et recouvrait son lit grossier et ses simples ustensiles de cuisine. Une pile de livres \u00e9tait \u00e9tal\u00e9e sur le lit. C'\u00e9tait pour la plupart des ouvrages de biologie, mais Ye Wenjie remarqua aussi _La Lib\u00e9ration animale_ , de Peter Singer. Le seul signe apparent de modernit\u00e9 \u00e9tait un petit poste radio \u00e0 piles AA, qu'il avait \u00e9puis\u00e9es et remplac\u00e9es par des piles D. Il y avait \u00e9galement de vieilles jumelles. Evans s'excusa en disant qu'il n'avait rien \u00e0 leur offrir \u00e0 boire. Il n'avait pas bu de caf\u00e9 depuis longtemps. Il avait bien de l'eau, mais n'avait qu'un seul verre.\n\n\u2014 Que faites-vous ici ? demanda un des coll\u00e8gues de Wenjie.\n\n\u2014 Je sauve le monde.\n\n\u2014 Vous sauvez... Vous sauvez les habitants de la r\u00e9gion ? C'est vrai que l'environnement ici est vraiment...\n\n\u2014 Vous \u00eates donc tous les m\u00eames ! explosa soudain Evans de col\u00e8re. Quand on vous parle de sauver le monde, vous ne pensez qu'aux humains ! Et sauver les autres esp\u00e8ces, c'est d\u00e9risoire ? Qui a donn\u00e9 aux humains une telle place d'honneur sur la plan\u00e8te ? Non, les humains n'ont pas besoin d'\u00eatre sauv\u00e9s. Ils vivent d\u00e9j\u00e0 bien mieux qu'ils ne le m\u00e9ritent.\n\n\u2014 On nous a dit que vous tentiez de sauver une esp\u00e8ce d'oiseau ?\n\n\u2014 Oui, un p\u00e9trel, de la sous-esp\u00e8ce des p\u00e9trels bruns du Nord-Ouest. Je vous \u00e9pargne son nom latin, il est trop long. Tous les printemps, ils reviennent du sud par les m\u00eames routes migratoires depuis la nuit des temps. Ils ne nichent que dans cette r\u00e9gion. Mais, ici, les for\u00eats disparaissent, ann\u00e9e apr\u00e8s ann\u00e9e. Les oiseaux ne trouvent d\u00e9j\u00e0 plus d'arbres o\u00f9 vivre et faire leurs nids. Quand je les ai d\u00e9couverts, ils n'\u00e9taient plus que dix mille individus. Si rien ne change, dans cinq ans, l'esp\u00e8ce se sera totalement \u00e9teinte. Les arbres que je plante servent d'habitat pour une partie d'entre eux. Leur population a commenc\u00e9 \u00e0 augmenter \u00e0 nouveau. Bien entendu, je dois planter encore plus d'arbres et \u00e9tendre la surface de ce jardin d'\u00c9den.\n\nEvans permit \u00e0 Wenjie et \u00e0 ses coll\u00e8gues d'utiliser ses jumelles. Sur ses indications, ils observ\u00e8rent le paysage un long moment avant d'apercevoir quelques oiseaux \u00e0 la couleur gris\u00e2tre \u00e9merger des arbustes.\n\n\u2014 Ils n'ont rien de resplendissant, n'est-ce pas ? C'est certain, ils n'attirent pas la m\u00eame sympathie que les pandas. Chaque jour dans le monde, des esp\u00e8ces dont personne ne se soucie disparaissent.\n\n\u2014 Avez-vous plant\u00e9 ces arbres seul ?\n\n\u2014 La majorit\u00e9, oui. \u00c0 l'origine, j'ai employ\u00e9 quelques paysans de la r\u00e9gion, mais je suis vite tomb\u00e9 \u00e0 court de ressources. Acheter des jeunes plants et irriguer co\u00fbte cher... Cependant, vous ne le devinerez jamais. Mon p\u00e8re est milliardaire, il dirige une compagnie multinationale sp\u00e9cialis\u00e9e dans le p\u00e9trole. Mais il m'a coup\u00e9 les fonds et, de toute fa\u00e7on, je ne veux plus utiliser son argent.\n\nLa bo\u00eete \u00e0 paroles d'Evans avait \u00e9t\u00e9 ouverte et il semblait ne plus pouvoir s'arr\u00eater :\n\n\u2014 \u00c0 l'\u00e2ge de mes douze ans, un p\u00e9trolier de trente mille tonnes appartenant \u00e0 mon p\u00e8re a heurt\u00e9 un r\u00e9cif sur la c\u00f4te atlantique. Plus de vingt mille tonnes de p\u00e9trole brut se sont d\u00e9vers\u00e9es dans l'oc\u00e9an. \u00c0 ce moment-l\u00e0, ma famille et moi nous passions des vacances dans une villa au bord de la mer, situ\u00e9e pas tr\u00e8s loin de l'accident. Quand mon p\u00e8re a appris la nouvelle, sa premi\u00e8re pens\u00e9e a \u00e9t\u00e9 de savoir comment il allait se d\u00e9charger de cette responsabilit\u00e9 et diminuer les pertes de son entreprise.\n\nL'apr\u00e8s-midi m\u00eame, je suis all\u00e9 voir les ravages sur la c\u00f4te. La mer avait vir\u00e9 au noir. Emprisonn\u00e9es dans des membranes huileuses, les vagues paraissaient lisses et sans force. La plage aussi \u00e9tait recouverte d'une couche de mazout. Avec quelques b\u00e9n\u00e9voles, nous avons tent\u00e9 de sauver les oiseaux aquatiques encore vivants qui s'\u00e9taient retrouv\u00e9s prisonniers de cette mar\u00e9e noire. Ils se d\u00e9battaient dans le p\u00e9trole, on aurait dit des sculptures de bitume. Seuls leurs yeux prouvaient encore qu'ils appartenaient au rang des cr\u00e9atures vivantes. Ces regards, prisonniers d'une nappe de mazout, ont continu\u00e9 \u00e0 hanter mes r\u00eaves durant des ann\u00e9es. Nous avons immerg\u00e9 les oiseaux de mer dans du d\u00e9tergent pour laver leur plumage, mais c'\u00e9tait tr\u00e8s dur, le p\u00e9trole brut \u00e9tait profond\u00e9ment incrust\u00e9 \u00e0 leur duvet, et si nous forcions trop, les plumes risquaient de partir avec le p\u00e9trole... Le soir venu, la plupart des oiseaux \u00e9taient morts.\n\nLe corps couvert de mazout, \u00e9puis\u00e9, je me suis assis sur la plage noire en regardant le soleil se coucher derri\u00e8re une mer obscure. J'avais l'impression que c'\u00e9tait la fin du monde. Mon p\u00e8re se tenait derri\u00e8re moi, j'ignorais depuis combien de temps il \u00e9tait l\u00e0. Il m'a demand\u00e9 si je me souvenais du petit squelette de dinosaure. Bien s\u00fbr que je m'en souvenais, il avait \u00e9t\u00e9 d\u00e9couvert lors d'une op\u00e9ration de prospection p\u00e9troli\u00e8re, il \u00e9tait en tr\u00e8s bon \u00e9tat, mon p\u00e8re avait pay\u00e9 tr\u00e8s cher pour l'acqu\u00e9rir, puis il l'avait install\u00e9 dans le jardin de mon grand-p\u00e8re.\n\nMon p\u00e8re m'a alors dit : \"Mike, je vais te raconter comment les dinosaures se sont \u00e9teints. Un ast\u00e9ro\u00efde a percut\u00e9 la Terre, le monde est devenu un oc\u00e9an de flammes, puis a sombr\u00e9 dans une tr\u00e8s longue obscurit\u00e9 glaciale... Une nuit, tu t'es r\u00e9veill\u00e9 d'un cauchemar, tu m'as racont\u00e9 que tu avais r\u00eav\u00e9 que tu vivais \u00e0 cette \u00e9poque terrifiante. Je vais te dire maintenant ce que j'aurais voulu te dire \u00e0 l'\u00e9poque : tu aurais \u00e9t\u00e9 chanceux de vivre \u00e0 la fin du cr\u00e9tac\u00e9, car notre \u00e9poque actuelle est encore plus effrayante, les esp\u00e8ces vivantes terrestres disparaissent bien plus vite que pendant le cr\u00e9tac\u00e9 tardif. La vraie extinction de masse, elle a lieu maintenant ! Fiston, ce que tu as vu aujourd'hui n'est rien, ce n'est qu'un \u00e9pisode minime dans une gigantesque trag\u00e9die, nous pouvons nous passer d'oiseaux de mer, mais nous ne pouvons pas nous passer du p\u00e9trole. Peux-tu seulement imaginer un monde sans p\u00e9trole ? Tu vois cette belle Ferrari que je t'ai offerte en cadeau d'anniversaire l'an dernier et que je t'ai promis que tu pourrais conduire apr\u00e8s tes quinze ans ? Sans p\u00e9trole, ce n'est qu'un vulgaire tas de ferraille, tu ne pourrais jamais la conduire. Si tu veux rendre visite \u00e0 ton grand-p\u00e8re, il te suffit de prendre mon jet priv\u00e9 et tu traverseras l'oc\u00e9an en une petite dizaine d'heures seulement. Sans p\u00e9trole, tu devrais prendre un voilier et tanguer sur les flots pendant un mois... Ce sont les r\u00e8gles du jeu de la civilisation : s'assurer tout d'abord de la survie de l'esp\u00e8ce humaine et lui garantir une vie confortable. Tout le reste est secondaire.\"\n\nMon p\u00e8re avait plac\u00e9 de grands espoirs en moi, mais il n'est finalement pas parvenu \u00e0 me fa\u00e7onner comme il l'esp\u00e9rait. Les jours qui ont suivi, les regards de ces oiseaux mourants n'ont cess\u00e9 de me poursuivre et ils ont fini par d\u00e9terminer mon destin. L'anniversaire de mes treize ans, mon p\u00e8re m'a demand\u00e9 ce que je comptais faire plus tard. Je n'ai rien r\u00e9pondu, je voulais seulement sauver le monde. Ce n'\u00e9tait pas un grand r\u00eave, ce que je voulais, c'\u00e9tait sauver une esp\u00e8ce en voie de disparition, cela pouvait \u00eatre un oiseau pas tr\u00e8s beau, un papillon tout gris ou un modeste scarab\u00e9e. J'ai fait des \u00e9tudes de biologie et je suis devenu sp\u00e9cialiste des oiseaux et des insectes. \u00c0 mes yeux, mon id\u00e9al est le plus digne qui soit : sauver une esp\u00e8ce d'oiseau ou d'insecte n'est pas diff\u00e9rent de sauver l'humanit\u00e9. Toutes les vies se valent, c'est le principe de base du communisme antisp\u00e9ciste.\n\n\u2014 Le quoi ?\n\nYe Wenjie n'\u00e9tait pas s\u00fbre d'avoir bien entendu le mot.\n\n\u2014 Le communisme antisp\u00e9ciste. C'est un courant que j'ai cr\u00e9\u00e9. On pourrait aussi dire que c'est une profession de foi, dont le credo est que toutes les cr\u00e9atures vivantes de la plan\u00e8te naissent \u00e9gales.\n\n\u2014 C'est id\u00e9aliste, ce n'est pas pragmatique. Notre agriculture aussi fait pousser des esp\u00e8ces vivantes. C'est une \u00e9galit\u00e9 qui ne pourra jamais \u00eatre atteinte.\n\n\u2014 Dans une \u00e9poque lointaine, les seigneurs avaient la m\u00eame conception de leurs esclaves. N'oubliez pas que la technologie avance. Un jour viendra o\u00f9 l'humanit\u00e9 pourra synth\u00e9tiser sa nourriture. Mais, avant cela, nous devons pr\u00e9parer le terrain, th\u00e9oriquement et id\u00e9ologiquement. En fait, le communisme antisp\u00e9ciste n'est ni plus ni moins que le prolongement naturel de la D\u00e9claration universelle des droits de l'homme. La R\u00e9volution fran\u00e7aise date d'il y a deux si\u00e8cles, et nous en sommes toujours au m\u00eame point. On reconna\u00eet bien l\u00e0 l'\u00e9go\u00efsme et l'hypocrisie de l'esp\u00e8ce humaine.\n\n\u2014 Combien de temps pensez-vous rester ici ?\n\n\u2014 Je l'ignore. Sauver le monde demande bien une vie enti\u00e8re. C'est un sentiment magnifique, vous savez. Mais, bien s\u00fbr, je ne m'attends pas \u00e0 ce que vous me compreniez.\n\nQuand il eut fini de parler, Evans retrouva sa mine terne. Il dit qu'il allait travailler. Il prit une pelle, une scie et quitta la cabane. Au moment de leur dire adieu, il fixa son regard un peu plus longuement sur Ye Wenjie ; il avait cru voir quelque chose de diff\u00e9rent chez cette femme.\n\n\u2014 \"C'est un homme aux sentiments nobles, int\u00e8gre, un homme d'une haute moralit\u00e9, d\u00e9tach\u00e9 des int\u00e9r\u00eats mesquins.\"\n\nSur le chemin du retour, un coll\u00e8gue de Ye Wenjie r\u00e9p\u00e9ta cette phrase extraite d' _\u00c0 la m\u00e9moire de Norman Bethune_. Puis il s'exclama :\n\n\u2014 Je ne savais pas qu'on pouvait encore vivre comme \u00e7a !\n\nTous les autres approuv\u00e8rent avec \u00e9motion.\n\n\u2014 Si seulement il y avait plus d'hommes comme lui, m\u00eame quelques-uns seulement, les choses se seraient peut-\u00eatre pass\u00e9es tr\u00e8s diff\u00e9remment, l\u00e2cha Ye Wenjie, comme si elle se parlait \u00e0 elle-m\u00eame.\n\nNaturellement, personne ne comprit le sens v\u00e9ritable de ces paroles.\n\nLe chef de l'\u00e9quipe de conception ramena la conversation sur la raison de leur venue dans la r\u00e9gion :\n\n\u2014 Je crois que nous ne pourrons pas construire l'observatoire ici. Ce ne sera pas approuv\u00e9 par les autorit\u00e9s.\n\n\u2014 Pourquoi pas ? Des quatre sites possibles, c'est celui dont l'environnement \u00e9lectromagn\u00e9tique est le plus pur !\n\n\u2014 Oui, mais l'environnement humain ? Camarades, oublions un moment la technologie. Regardez comme cet endroit est pauvre. Vous le savez, plus les r\u00e9gions sont pauvres, plus leurs habitants sont pr\u00eats \u00e0 tout. Je crains que nous n'ayons beaucoup de probl\u00e8mes avec les gens d'ici. Pour eux, un observatoire radioastronomique, ce ne sera ni plus ni moins qu'un morceau de viande qu'ils n'h\u00e9siteront pas \u00e0 d\u00e9vorer sans vergogne.\n\nSans surprise, le choix de ce site ne fut pas approuv\u00e9 par les autorit\u00e9s, pr\u00e9cis\u00e9ment \u00e0 cause de la raison invoqu\u00e9e par le chef de l'\u00e9quipe de conception.\n\nTrois ann\u00e9es pass\u00e8rent, durant lesquelles Ye Wenjie n'entendit plus parler d'Evans.\n\nMais un jour de printemps, elle re\u00e7ut une carte postale envoy\u00e9e par Evans sur laquelle il \u00e9tait \u00e9crit une simple phrase : \"Venez et dites-moi comment continuer \u00e0 vivre.\"\n\nYe Wenjie prit le train pendant un jour et une nuit, puis elle loua une voiture et roula encore quelques heures avant d'arriver dans le village recul\u00e9 de cette r\u00e9gion du Nord-Ouest.\n\nQuand elle parvint au sommet de la petite colline, elle vit la for\u00eat verdoyante. Sa superficie ne semblait gu\u00e8re avoir chang\u00e9 en trois ans, mais les arbres avaient grandi et elle avait l'air beaucoup plus dense. Elle d\u00e9couvrit toutefois rapidement que l'\u00e9tendue de la for\u00eat \u00e9tait en r\u00e9alit\u00e9 plus petite que la derni\u00e8re fois. Un grand nombre d'arbres avaient \u00e9t\u00e9 coup\u00e9s. La d\u00e9forestation \u00e9tait d'ailleurs en cours : partout des arbres \u00e9taient en train de tomber, comme des feuilles vertes d\u00e9vor\u00e9es par des pucerons. \u00c0 ce rythme-l\u00e0, la for\u00eat n'allait pas tarder \u00e0 dispara\u00eetre. Les b\u00fbcherons \u00e9taient originaires de deux villages situ\u00e9s non loin de l\u00e0. Ils tranchaient les jeunes arbustes \u00e0 coups de hache et de scie puis, avec des tracteurs et des chars \u00e0 b\u0153ufs, ils les transportaient jusqu'au pied de la colline. Les b\u00fbcherons \u00e9taient nombreux et de f\u00e9roces disputes \u00e9clataient entre eux.\n\nM\u00eame si les arbustes ne faisaient pas grand bruit en tombant sur le sol et m\u00eame si elle n'entendait pas non plus le ronflement des tron\u00e7onneuses, cette sc\u00e8ne parut tristement famili\u00e8re \u00e0 Ye Wenjie. Son c\u0153ur se gla\u00e7a.\n\nQuelqu'un la salua. C'\u00e9tait le responsable de l'unit\u00e9 de production, maintenant devenu maire du village. Il avait reconnu Ye Wenjie. Cette derni\u00e8re lui demanda pourquoi ils rasaient ainsi la for\u00eat.\n\n\u2014 Cette for\u00eat n'est pas prot\u00e9g\u00e9e par la loi, r\u00e9pondit-il.\n\n\u2014 Comment pouvez-vous dire cela alors que la loi sur les for\u00eats vient tout juste d'\u00eatre promulgu\u00e9e ?\n\n\u2014 Qui a autoris\u00e9 Bethune \u00e0 planter des arbres ici ? Un \u00e9tranger qui vient faire pousser sans autorisation des arbres sur le flanc d'une colline chinoise n'est pas prot\u00e9g\u00e9 par la loi !\n\n\u2014 Mais enfin, il les a plant\u00e9s sur des collines arides ! D'ailleurs, quand il a commenc\u00e9 il y a quelques ann\u00e9es, vous ne l'en avez pas emp\u00each\u00e9 !\n\n\u2014 C'est vrai. Le comt\u00e9 lui avait m\u00eame octroy\u00e9 une subvention pour son projet de reboisement. Nous voulions d'abord attendre encore quelques ann\u00e9es avant de couper le bois de la for\u00eat \u2013 il vaut mieux attendre que le porc soit gras avant de le tuer, n'est-ce pas ? Mais des gars du village de Nange n'ont pas eu cette patience et ont attaqu\u00e9 avant nous ! Si nous ne nous y mettons pas \u00e0 notre tour, il ne nous restera rien !\n\n\u2014 Cessez imm\u00e9diatement ! Je ferai un rapport aux autorit\u00e9s !\n\n\u2014 C'est inutile, dit le maire en allumant une cigarette, puis il d\u00e9signa plus loin le camion dans lequel on chargeait des arbres coup\u00e9s. Vous voyez ce v\u00e9hicule ? C'est celui du directeur adjoint du Bureau des for\u00eats du comt\u00e9. Il y a aussi des membres de la police municipale. Ceux-l\u00e0 ont m\u00eame emport\u00e9 plus d'arbres que n'importe qui d'autre ! Comme je vous l'ai dit, cette for\u00eat n'a aucun statut, elle n'est \u00e0 personne et n'est pas prot\u00e9g\u00e9e par la loi. Vous pouvez bien aller voir qui vous voulez, ce sera inutile. Et puis, camarade Ye, vous \u00eates professeur \u00e0 l'universit\u00e9, n'est-ce pas ? Qu'est-ce que tout cela peut bien vous faire ?\n\nLa cabane en adobe \u00e9tait indemne, mais Evans n'\u00e9tait pas \u00e0 l'int\u00e9rieur. Wenjie le trouva dans la for\u00eat, il mettait tout son c\u0153ur \u00e0 \u00e9laguer des branches avec sa hache. Cela faisait manifestement longtemps qu'il travaillait, son visage \u00e9tait fatigu\u00e9.\n\n\u2014 Peu importe que cela n'ait plus aucun sens. Je ne peux pas arr\u00eater, si j'arr\u00eate, je vais exploser, grommela Evans en \u00e9laguant une branche tordue d'une main experte.\n\n\u2014 Allons voir ensemble le gouvernement du comt\u00e9 et, si \u00e7a ne suffit pas, allons \u00e0 la capitale provinciale, il y aura forc\u00e9ment quelqu'un pour les arr\u00eater, dit Wenjie en le regardant avec sollicitude.\n\nEvans s'arr\u00eata et fixa Wenjie avec un regard surpris. Les rayons du cr\u00e9puscule per\u00e7aient \u00e0 travers les arbres. Une lueur passa dans ses yeux :\n\n\u2014 Croyez-vous vraiment que je fais \u00e7a pour ce morceau de for\u00eat ?\n\nIl rit et secoua la t\u00eate, puis il jeta la hache qu'il tenait dans les mains et s'assit contre un arbre.\n\n\u2014 Les arr\u00eater maintenant serait un jeu d'enfant.\n\nIl posa sa sacoche \u00e0 outils vide sur le sol et invita Wenjie \u00e0 s'asseoir, puis il continua :\n\n\u2014 Je viens tout juste de revenir des \u00c9tats-Unis, mon p\u00e8re est mort il y a deux mois, j'ai h\u00e9rit\u00e9 de la majeure partie de son patrimoine. Mon fr\u00e8re et ma s\u0153ur n'ont eu que cinq millions de dollars chacun. Je ne m'y attendais pas. Peut-\u00eatre qu'au fond de lui il avait du respect pour moi, ou pour mes id\u00e9aux. Sans compter les actifs immobilis\u00e9s, savez-vous combien je poss\u00e8de maintenant ? Pr\u00e8s de 4,5 milliards de dollars. Je pourrais facilement les emp\u00eacher de couper les arbres de cette for\u00eat, et les leur faire replanter jusqu'\u00e0 ce que ces collines de l\u0153ss soient recouvertes \u00e0 perte de vue de for\u00eats luxuriantes. Mais \u00e0 quoi bon ? Tout ce que vous voyez ici est le r\u00e9sultat de la pauvret\u00e9. Mais est-ce beaucoup mieux dans les pays riches ? Ils prennent bien soin de leurs beaux environnements, mais d\u00e9localisent leurs industries polluantes dans les pays pauvres. Vous \u00eates peut-\u00eatre au courant que le gouvernement am\u00e9ricain vient de refuser de ratifier le protocole de Kyoto... C'est dans la nature de l'humain. Si la civilisation continue \u00e0 se d\u00e9velopper, les p\u00e9trels que je voulais sauver et tous les autres oiseaux dispara\u00eetront. Ce n'est qu'une question de temps.\n\nWenjie \u00e9tait assise, silencieuse. Elle regardait les rayons de lumi\u00e8re projet\u00e9s sur les arbres par le soleil couchant et entendait le raffut lointain des b\u00fbcherons. Ses pens\u00e9es la ramen\u00e8rent vingt ans en arri\u00e8re, dans les for\u00eats des montagnes du Grand Khingan, \u00e0 la conversation similaire qu'elle avait eue avec un autre homme.\n\n\u2014 Savez-vous pourquoi je suis venu ici ? reprit Evans. Mon communisme antisp\u00e9ciste prend racine dans l'Orient ancien.\n\n\u2014 Vous voulez parler du bouddhisme ?\n\n\u2014 Oui. Le christianisme ne s'est jamais int\u00e9ress\u00e9 qu'\u00e0 l'homme et, malgr\u00e9 l'arche de No\u00e9, la religion n'a jamais donn\u00e9 aux autres esp\u00e8ces vivantes la m\u00eame valeur qu'\u00e0 l'esp\u00e8ce humaine. Au contraire, le bouddhisme a pour id\u00e9e de d\u00e9livrer toutes les cr\u00e9atures vivantes. C'est ce qui avait motiv\u00e9 mon d\u00e9part pour l'Asie, mais... je crois que c'est maintenant partout pareil.\n\n\u2014 Oui, c'est partout pareil. Partout, l'humanit\u00e9 est la m\u00eame.\n\n\u2014 Que puis-je faire maintenant ? Quel est le sens de ma vie ? Je poss\u00e8de 4,5 milliards de dollars et une entreprise multinationale de p\u00e9trole. Et apr\u00e8s ? Les hommes ont d\u00fb investir dans le pass\u00e9 plus de 45 milliards pour sauver des animaux en voie d'extinction et plus de 450 milliards pour essayer de sauver l'environnement, mais \u00e0 quoi cela a-t-il servi ? La civilisation marche toujours sur le m\u00eame sentier, celui de la destruction de toute vie sur Terre en dehors de la sienne. 4,5 milliards de dollars suffiraient \u00e0 construire un porte-avions mais, m\u00eame si on en construisait mille, il serait impossible d'arr\u00eater la folie de l'humanit\u00e9.\n\n\u2014 L'humanit\u00e9 ne peut plus compter sur elle-m\u00eame pour r\u00e9soudre ses probl\u00e8mes.\n\n\u2014 Mais y a-t-il une autre force en dehors de la sienne ? Si Dieu existe, il y a bien longtemps qu'il est mort.\n\n\u2014 Oui. Il existe une autre force.\n\nLe soleil s'\u00e9tait couch\u00e9 et les b\u00fbcherons \u00e9taient partis. Le silence enveloppait les collines de l\u0153ss qui enclavaient la for\u00eat.\n\nYe Wenjie raconta \u00e0 Evans l'histoire de la base de C\u00f4te Rouge et du monde de Trisolaris. Il l'\u00e9couta calmement. C'\u00e9tait comme s'il n'\u00e9tait pas son seul auditeur et que la for\u00eat et les plaines de l\u0153ss alentour, baign\u00e9es par la lumi\u00e8re du cr\u00e9puscule, tendaient aussi l'oreille pour entendre le r\u00e9cit de Wenjie. Lorsqu'elle eut fini, la pleine lune se leva \u00e0 l'est du ciel, projetant des ombres mouchet\u00e9es sur le sol.\n\nEvans prit la parole :\n\n\u2014 J'ai encore du mal \u00e0 croire ce que vous racontez. C'est trop fantastique. Mais, par chance, je poss\u00e8de les ressources n\u00e9cessaires pour prouver vos dires. Si tout est vrai, alors... (il lui tendit la main et pronon\u00e7a les mots qui devinrent plus tard ceux adress\u00e9s \u00e0 tous les nouveaux membres de l'Organisation Terre-Trisolaris) _nous sommes camarades_.\n\n* * *\n\n1 Norman Bethune (1890-1939) \u00e9tait un m\u00e9decin canadien ayant travaill\u00e9 en Chine durant la guerre sino-japonaise (1937-1945). Mao Zedong a rendu hommage \u00e0 ce compagnon de route du Parti communiste chinois dans un essai intitul\u00e9 _\u00c0 la m\u00e9moire de Norman Bethune_. _(N.d.T.)_\n\n## 28\n\n## LA DEUXI\u00c8ME BASE DE C\u00d4TE ROUGE\n\nTrois ans de plus s'\u00e9coul\u00e8rent. Evans avait sembl\u00e9 dispara\u00eetre, Ye Wenjie \u00e9tait sans nouvelles de lui. Elle ignorait s'il vivait encore quelque part dans un coin du monde, en train de v\u00e9rifier si elle avait dit vrai, ne sachant d'ailleurs pas comment il allait s'y prendre. M\u00eame \u00e0 l'\u00e9chelle de l'univers, une distance de quatre ann\u00e9es-lumi\u00e8re reste incroyablement lointaine pour une petite vie fragile. Les deux mondes \u00e9taient comme la source et l'embouchure d'une rivi\u00e8re spatiale. Tout contact entre les deux \u00e9tait aussi subtil qu'un cheveu d'ange.\n\nMais un hiver, Ye Wenjie re\u00e7ut soudain de la part d'une obscure universit\u00e9 d'Europe de l'Ouest une invitation \u00e0 devenir professeur invit\u00e9e pendant un semestre. Elle accepta. \u00c0 son atterrissage \u00e0 l'a\u00e9roport de Londres-Heathrow, elle fut accueillie par un jeune homme. Alors m\u00eame qu'ils n'avaient pas encore quitt\u00e9 le hall de l'a\u00e9roport, celui-ci l'emmena sur la piste de d\u00e9collage. Le jeune homme la fit monter dans un h\u00e9licopt\u00e8re. Lorsque l'appareil d\u00e9colla en vrombissant au-dessus du ciel brumeux de Londres, Wenjie crut remonter le temps. Elle avait l'impression d'avoir d\u00e9j\u00e0 v\u00e9cu cette sc\u00e8ne. Bien des ann\u00e9es plus t\u00f4t, son premier voyage en h\u00e9licopt\u00e8re avait boulevers\u00e9 son destin. Que lui r\u00e9servait cette fois-ci la providence ?\n\n\u2014 Nous allons \u00e0 la deuxi\u00e8me base de C\u00f4te Rouge, dit le jeune homme.\n\nL'h\u00e9licopt\u00e8re passa la c\u00f4te et continua vers l'Atlantique. Ils survol\u00e8rent environ une demi-heure l'oc\u00e9an avant que l'appareil se pose sur un immense bateau. D\u00e8s le premier coup d'\u0153il, Wenjie pensa au pic du Radar ; elle ne r\u00e9alisa qu'\u00e0 cet instant que le pic avait la forme de la proue d'un gigantesque navire. L'Atlantique, lui, \u00e9tait comme les for\u00eats des montagnes du Grand Khingan. Mais ce qui lui fit surtout penser \u00e0 C\u00f4te Rouge, c'\u00e9tait la grande antenne radar qui se dressait au centre du pont, comme un colossal m\u00e2t cylindrique. Le navire avait \u00e9t\u00e9 con\u00e7u sur les fondations d'un p\u00e9trolier de soixante mille tonnes, c'\u00e9tait comme une \u00eele flottante en acier. Evans avait construit sa base sur le navire sans doute parce qu'il voulait b\u00e9n\u00e9ficier du meilleur site pour ses activit\u00e9s de transmission et de r\u00e9ception, mais peut-\u00eatre aussi pour rester cach\u00e9. Elle apprit plus tard que le bateau avait \u00e9t\u00e9 baptis\u00e9 _Jugement Dernier_.\n\nEn descendant de l'h\u00e9licopt\u00e8re, Wenjie entendit un bourdonnement familier, celui de la gigantesque antenne qui vibrait dans le vent marin. Ce bruit la ramena plus profond\u00e9ment encore dans le pass\u00e9. Sur le pont se pressait une foule de deux mille hommes.\n\nEvans vint \u00e0 sa rencontre et lui dit avec gravit\u00e9 :\n\n\u2014 Gr\u00e2ce \u00e0 la fr\u00e9quence et aux coordonn\u00e9es que vous m'avez indiqu\u00e9es, nous avons re\u00e7u un message de Trisolaris. Tout ce que vous avez dit est vrai.\n\nYe Wenjie hocha la t\u00eate calmement.\n\n\u2014 La grande flotte trisolarienne est en route. Sa destination est le syst\u00e8me solaire, elle arrivera dans quatre cent cinquante ans.\n\nWenjie resta de marbre. Plus rien ne pouvait la surprendre maintenant.\n\nEvans lui montra la foule amass\u00e9e sous l'antenne :\n\n\u2014 Ceux que vous voyez ici sont les premiers membres de l'Organisation Terre-Trisolaris. Notre id\u00e9al est d'inviter la civilisation trisolarienne \u00e0 r\u00e9former l'humanit\u00e9, \u00e0 mettre un terme \u00e0 sa folie, pour que la plan\u00e8te Terre revienne \u00e0 ses heures glorieuses durant lesquelles elle vivait sans p\u00e9ch\u00e9, en harmonie avec son environnement. De plus en plus de gens se reconnaissent dans notre id\u00e9al, notre organisation est en augmentation constante. Nous avons maintenant des membres dans le monde entier.\n\n\u2014 Que puis-je faire ? demanda Ye Wenjie \u00e0 voix basse.\n\n\u2014 Devenir le Guide de l'Organisation. Les guerriers de l'OTT reconnaissent tous votre l\u00e9gitimit\u00e9 !\n\nYe Wenjie resta silencieuse quelques secondes, puis elle hocha lentement la t\u00eate :\n\n\u2014 Je ferai de mon mieux.\n\nEvans leva le poing en l'air et cria \u00e0 la foule :\n\n\u2014 Mettons fin \u00e0 la tyrannie humaine !\n\nAccompagn\u00e9s du bruit des vagues se fracassant contre la coque du navire et du bourdonnement de l'antenne dans le vent, les guerriers de l'OTT reprirent en ch\u0153ur :\n\n\u2014 Le monde appartient \u00e0 Trisolaris !\n\nCe jour marqua la naissance officielle du mouvement Terre-Trisolaris.\n\n## 29\n\n## LE MOUVEMENT TERRE-TRISOLARIS\n\nL'aspect le plus \u00e9tonnant du mouvement Terre-Trisolaris \u00e9tait que tant de gens aient perdu tout espoir dans la civilisation humaine et que, le c\u0153ur plein de haine, ils soient pr\u00eats \u00e0 trahir leur propre esp\u00e8ce. Tant de gens dont le plus grand r\u00eave \u00e9tait d'\u00e9liminer l'humanit\u00e9 enti\u00e8re, y compris eux-m\u00eames et leurs descendants.\n\nL'OTT \u00e9tait une organisation de bourgeois intellectuels. La majorit\u00e9 de ses membres \u00e9tait issue de l'intelligentsia, et comptait un nombre non n\u00e9gligeable des repr\u00e9sentants des \u00e9lites politiques et \u00e9conomiques. L'Organisation avait tent\u00e9 de s'\u00e9largir \u00e0 un plus grand public, mais ces efforts s'\u00e9taient r\u00e9v\u00e9l\u00e9s vains. Peut-\u00eatre les gens ordinaires n'avaient-ils pas une compr\u00e9hension aussi fine que l'intelligentsia du mal profond de l'humanit\u00e9. Plus important encore, leur pens\u00e9e \u00e9tant moins influenc\u00e9e par la science et la philosophie modernes, ils \u00e9prouvaient un \u00e9crasant sentiment d'identification \u00e0 l'\u00e9gard de leur esp\u00e8ce. Trahir la race humaine dans son ensemble leur \u00e9tait inconcevable. Au contraire, beaucoup de membres de l'\u00e9lite intellectuelle tentaient de d\u00e9passer les simples perspectives anthropocentr\u00e9es. La civilisation humaine avait enfin donn\u00e9 naissance en son propre sein \u00e0 une puissante force ali\u00e9n\u00e9e.\n\nSi une masse incroyablement croissante de membres rejoignait chaque jour les rangs de l'OTT, ce n'\u00e9tait pas vraiment \u00e0 leur nombre qu'il fallait juger la puissance de l'Organisation, car beaucoup d'entre eux tenaient des postes clefs au sein de la soci\u00e9t\u00e9, et leur pouvoir comme leur influence \u00e9taient \u00e9normes.\n\nEn tant que Guide des rebelles de Terre-Trisolaris, Ye Wenjie se contentait d'un r\u00f4le de leader spirituel. Elle ne participait jamais aux actions concr\u00e8tes de l'Organisation et ignorait comment celle-ci avait pu atteindre une telle envergure. Elle ne savait m\u00eame pas le nombre exact de ses membres.\n\nLes gouvernements des diff\u00e9rents pays n'avaient pas suffisamment pr\u00eat\u00e9 attention \u00e0 la mont\u00e9e en puissance des rebelles de Terre-Trisolaris. Afin de gagner davantage de partisans, les activit\u00e9s de l'Organisation \u00e9taient en partie publiques. Ils savaient qu'ils seraient naturellement prot\u00e9g\u00e9s par le conservatisme et le manque d'imagination des gouvernants. Et de fait, dans les organes du pouvoir, personne ne croyait \u00e0 leurs histoires, qu'on prenait pour du charabia de gourous sectaires, quoique, en raison du statut social de plusieurs membres de l'Organisation, on gard\u00e2t tout de m\u00eame un \u0153il sur leurs activit\u00e9s. Mais ce ne fut v\u00e9ritablement que lorsque l'OTT commen\u00e7a \u00e0 d\u00e9velopper son propre syst\u00e8me d'armement que les organes de s\u00e9curit\u00e9 gouvernementaux commenc\u00e8rent \u00e0 la prendre au s\u00e9rieux et s'aper\u00e7urent peu \u00e0 peu du caract\u00e8re tr\u00e8s inhabituel du mouvement. Quant aux attaques concr\u00e8tes contre l'OTT, elles n'avaient d\u00e9but\u00e9 que depuis deux ans \u00e0 peine.\n\nL'OTT n'\u00e9tait pas une organisation homog\u00e8ne. Elle se divisait en plusieurs chapelles id\u00e9ologiques, dont les deux principales \u00e9taient celles des adventistes et des r\u00e9demptoristes.\n\nLes adventistes \u00e9taient les premiers et les membres les plus fondamentalistes de Terre-Trisolaris. Cette faction \u00e9tait principalement compos\u00e9e des disciples du communisme antisp\u00e9ciste d'Evans. Leur id\u00e9al \u00e9tait simple : an\u00e9antir la civilisation humaine. Les adventistes avaient perdu tout espoir dans la nature de l'homme, et ce d\u00e9sespoir prenait sa source dans la destruction de masse de toutes les autres esp\u00e8ces vivantes par la civilisation moderne. Evans en \u00e9tait la figure majeure. Plus tard, la haine des adventistes pour l'humanit\u00e9 s'\u00e9tendit \u00e0 d'autres sph\u00e8res. Elle ne se limitait plus aux questions d'\u00e9cologie ou de guerre, mais s'\u00e9levait \u00e0 un niveau plus abstrait et plus philosophique. Contrairement \u00e0 ce qui fut imagin\u00e9 bien plus tard, les adventistes \u00e9taient des pragmatiques. Ils ne pla\u00e7aient pas d'espoir immense dans la venue de la civilisation extraterrestre qu'ils avaient d\u00e9cid\u00e9 de servir. Leur trahison se fondait uniquement sur leur d\u00e9sespoir et leur aversion pour l'humanit\u00e9. Une phrase d'Evans \u00e9tait devenue leur devise :\n\n_Nous ne connaissons pas la civilisation extraterrestre, mais nous connaissons la civilisation humaine._\n\nLe r\u00e9demptorisme, pour sa part, \u00e9tait un courant n\u00e9 bien apr\u00e8s la fondation de l'OTT. C'\u00e9tait une faction de nature religieuse, form\u00e9e par les disciples du trisolarisme.\n\nLes classes intellectuelles \u00e9prouvaient naturellement un grand attrait pour cette civilisation extrahumaine et il \u00e9tait facile de d\u00e9velopper toute une mythologie autour de celle-ci. Pour prendre une m\u00e9taphore quelque peu inappropri\u00e9e : l'humanit\u00e9 \u00e9tait un \u00eatre ignorant marchant seul dans le d\u00e9sert de l'univers et qui venait tout juste de d\u00e9couvrir une personne d'un autre sexe. Bien qu'il (ou elle) ne v\u00eet ni son visage ni sa silhouette, il (elle) savait qu'il (elle) \u00e9tait l\u00e0, quelque part dans le lointain, et le fantasme entourant cette pr\u00e9sence faisait sur lui (elle) l'effet d'un feu ardent. Peu \u00e0 peu, cette civilisation \u00e9loign\u00e9e fit germer dans les esprits des r\u00e9demptoristes des id\u00e9es foisonnantes. Ils se mirent \u00e0 \u00e9prouver pour la civilisation trisolarienne des sentiments religieux. Alpha du Centaure devint un mont Olympe cosmique, une demeure divine, et c'est ainsi que la religion trisolariste vit le jour. Elle diff\u00e9rait des autres religions humaines en cela que les trisolaristes priaient quelque chose de tangible ; et au contraire des autres formes de spiritualit\u00e9, c'\u00e9taient leurs dieux qui se trouvaient dans une situation de d\u00e9tresse et c'\u00e9taient leurs disciples qui \u00e9taient responsables de leur salut.\n\nLes informations sur la culture trisolarienne \u00e9taient propag\u00e9es par le biais du jeu des _Trois Corps_. Cet ambitieux logiciel avait \u00e9t\u00e9 d\u00e9velopp\u00e9 gr\u00e2ce aux efforts importants des rebelles de Terre-Trisolaris. Les objectifs initiaux \u00e9taient tout d'abord de convertir un grand nombre de m\u00e9cr\u00e9ants au trisolarisme puis d'\u00e9tendre les tentacules de l'Organisation pour ne pas la restreindre aux hautes sph\u00e8res intellectuelles de la soci\u00e9t\u00e9 en recrutant des membres plus jeunes, issus de classes moins favoris\u00e9es.\n\nLe jeu utilisait comme carapace des \u00e9l\u00e9ments emprunt\u00e9s \u00e0 l'histoire et \u00e0 la soci\u00e9t\u00e9 humaines pour narrer celles de Trisolaris et \u00e9viter que les joueurs d\u00e9butants ne soient trop d\u00e9stabilis\u00e9s par leur \u00e9tranget\u00e9. Lorsque le joueur avait atteint un certain niveau du jeu et qu'il avait commenc\u00e9 \u00e0 appr\u00e9cier le charme de la civilisation trisolarienne, l'OTT le contactait directement, testait son degr\u00e9 de sympathie envers Trisolaris et, s'il \u00e9tait retenu, le joueur devenait finalement membre de l'Organisation. Cependant, le jeu des _Trois Corps_ peinait \u00e0 renforcer la visibilit\u00e9 de l'Organisation, car il n\u00e9cessitait d'avoir des pr\u00e9-acquis importants en mati\u00e8re de science et une capacit\u00e9 de r\u00e9flexion pouss\u00e9e. Les jeunes joueurs n'avaient ni les comp\u00e9tences ni la patience n\u00e9cessaires pour d\u00e9couvrir la v\u00e9rit\u00e9 troublante se dissimulant derri\u00e8re le voile en apparence simpliste de ce monde virtuel. Ainsi, la plupart des joueurs des _Trois Corps_ \u00e9taient-ils eux aussi des intellectuels.\n\nUne grande partie des r\u00e9demptoristes avaient d\u00e9couvert la civilisation de Trisolaris par l'interm\u00e9diaire des _Trois Corps_. On pouvait dire que le jeu \u00e9tait le berceau de ce courant.\n\nAlors que les r\u00e9demptoristes avaient d\u00e9velopp\u00e9 pour la civilisation trisolarienne des sentiments religieux, leur attitude envers l'humanit\u00e9 n'\u00e9tait pas aussi radicale que celle des adventistes. Leur id\u00e9al ultime \u00e9tait de sauver les dieux et, pour atteindre ce but, une portion du monde humain pouvait \u00e9ventuellement \u00eatre sacrifi\u00e9e. Mais beaucoup d'entre eux se disaient que si les Trisolariens arrivaient \u00e0 survivre dans leur propre syst\u00e8me d'Alpha du Centaure, ils n'auraient plus besoin d'envahir le syst\u00e8me solaire, et l'issue serait id\u00e9ale pour les deux mondes. Ils avaient la na\u00efvet\u00e9 de croire que la r\u00e9solution de l'\u00e9nigme physique du probl\u00e8me \u00e0 trois corps permettrait d'atteindre ce but, en sauvant les deux mondes. Cette id\u00e9e n'\u00e9tait en r\u00e9alit\u00e9 pas aussi candide qu'elle en avait l'air car, bien longtemps, cette issue avait \u00e9t\u00e9 celle envisag\u00e9e par la civilisation trisolarienne. Les efforts pour r\u00e9soudre le probl\u00e8me \u00e0 trois corps les avaient occup\u00e9s durant plusieurs centaines de cycles de civilisation. Les adeptes r\u00e9demptoristes comptaient dans leurs rangs beaucoup de physiciens et de math\u00e9maticiens qui s'\u00e9taient donn\u00e9 pour mission de trouver une solution \u00e0 cette \u00e9nigme. M\u00eame apr\u00e8s avoir appris que le probl\u00e8me \u00e0 trois corps \u00e9tait par essence impossible \u00e0 r\u00e9soudre math\u00e9matiquement, ils n'avaient pas cess\u00e9 leurs efforts. Les tentatives de r\u00e9solution du probl\u00e8me \u00e0 trois corps \u00e9taient devenues une sorte de rituel de la religion trisolariste. Mais malgr\u00e9 l'investissement de physiciens et de math\u00e9maticiens de haut rang, aucune avanc\u00e9e majeure n'avait \u00e9t\u00e9 accomplie et, \u00e0 l'inverse, c'\u00e9tait le g\u00e9nial math\u00e9maticien Wei Cheng, qui n'avait rien \u00e0 faire avec l'OTT et le trisolarisme, qui \u00e9tait parvenu \u00e0 effectuer \u2013 sans vraiment en \u00eatre conscient \u2013 le bond le plus marquant dans ce champ de recherche o\u00f9 les r\u00e9demptoristes pla\u00e7aient tous leurs espoirs.\n\nLes deux groupes \u00e9taient dans une situation de conflit permanent. Les adventistes estimaient que les r\u00e9demptoristes constituaient une menace pour le mouvement Terre-Trisolaris. Ce n'\u00e9tait d'ailleurs pas une consid\u00e9ration d\u00e9nu\u00e9e de fondement : en effet, c'\u00e9tait par le biais de personnalit\u00e9s ayant de grandes responsabilit\u00e9s au sein de la frange r\u00e9demptoriste que les gouvernements nationaux d\u00e9couvraient peu \u00e0 peu la terrifiante v\u00e9rit\u00e9 que tentait de cacher l'Organisation. Chaque faction poss\u00e9dait une force sensiblement \u00e9gale au sein de l'Organisation ; chacune d'entre elles en \u00e9tait d'ailleurs arriv\u00e9e \u00e0 d\u00e9velopper ses propres milices et la guerre interne couvait. Ye Wenjie faisait tout ce qu'elle pouvait pour combler le foss\u00e9 existant entre les deux camps, en usant de son prestige et de son autorit\u00e9, mais avec peu d'effet.\n\nAu fur et \u00e0 mesure de l'\u00e9volution du mouvement, une troisi\u00e8me force apparut au sein des rebelles de l'OTT : les survivalistes. Lorsque la preuve fut apport\u00e9e qu'une flotte extraterrestre s'appr\u00eatait bel et bien \u00e0 envahir le syst\u00e8me solaire, survivre \u00e0 cette guerre qui s'annon\u00e7ait \u00e9tait devenu le d\u00e9sir humain le plus naturel. Bien entendu, la guerre aurait lieu dans quatre cent cinquante ans et le monde n'aurait rien de commun avec celui d'aujourd'hui, mais beaucoup esp\u00e9raient que, si l'humanit\u00e9 devait perdre la guerre dans quatre si\u00e8cles, leurs descendants pourraient implorer leur survie s'ils se mettaient d\u00e8s maintenant au service des envahisseurs. Par comparaison avec les deux autres factions majoritaires, les survivalistes \u00e9taient compos\u00e9s en grande partie d'\u00e9l\u00e9ments issus de classes plus basses, et les Asiatiques \u2013 en particulier des Chinois \u2013 y \u00e9taient nombreux. Ils ne repr\u00e9sentaient pour l'instant qu'une portion minime dans l'Organisation, mais leur nombre ne cessait de cro\u00eetre. \u00c0 mesure que la culture trisolarienne se r\u00e9pandrait de par le monde, ils constitueraient peu \u00e0 peu une faction incontournable.\n\nUne force ali\u00e9n\u00e9e n\u00e9e des carences de l'humanit\u00e9, le fantasme et l'adoration d'une civilisation sup\u00e9rieure et enfin le d\u00e9sir supr\u00eame de voir ses descendants survivre \u00e0 une guerre ultime : ces trois puissantes forces motrices donnaient au mouvement Terre-Trisolaris un \u00e9lan fantastique, si bien que, quand les autorit\u00e9s commenc\u00e8rent \u00e0 mettre le nez dans leurs affaires, le feu avait d\u00e9j\u00e0 pris sur la plaine.\n\nAu m\u00eame moment, la civilisation extraterrestre se trouvait encore dans les profondeurs de l'espace, \u00e0 quatre ann\u00e9es-lumi\u00e8re de l\u00e0, s\u00e9par\u00e9e du monde humain par un long p\u00e9riple qui durerait quatre si\u00e8cles et demi. Et la seule chose qui avait \u00e9t\u00e9 envoy\u00e9e sur la Terre, c'\u00e9tait une s\u00e9rie d'ondes radio.\n\nLa th\u00e9orie du \"contact comme symbole\" de Bill Matthews recevait l\u00e0 une parfaite mais terrifiante d\u00e9monstration.\n\n## 30\n\n## DEUX PROTONS\n\nL'ENQU\u00caTEUR\n\nNous allons commencer l'interrogatoire du jour. J'esp\u00e8re que vous vous montrerez aussi collaborative que la derni\u00e8re fois.\n\nYE WENJIE\n\nJe vous ai d\u00e9j\u00e0 dit tout ce que je savais. En r\u00e9alit\u00e9, c'est moi qui ai besoin d'en apprendre de votre bouche.\n\nL'ENQU\u00caTEUR\n\nJe ne crois pas que vous nous ayez tout dit. Tout d'abord, ce que nous avons besoin de conna\u00eetre, c'est le contenu des informations transmises par Trisolaris \u00e0 la Terre et qui ont \u00e9t\u00e9 intercept\u00e9es par les adventistes.\n\nYE WENJIE\n\nJe l'ignore. Leur organisation est tr\u00e8s secr\u00e8te, je sais uniquement qu'ils ont dissimul\u00e9 certaines transmissions.\n\nL'ENQU\u00caTEUR\n\nBien, passons \u00e0 un autre sujet. Apr\u00e8s que les \u00e9changes avec Trisolaris sont tomb\u00e9s sous le monopole des adventistes, avez-vous construit une troisi\u00e8me C\u00f4te Rouge ?\n\nYE WENJIE\n\nNous en avons eu l'id\u00e9e, mais nous n'avons finalement con\u00e7u qu'un appareil de r\u00e9ception et la construction a \u00e9t\u00e9 stopp\u00e9e. L'\u00e9quipement et la base ont \u00e9t\u00e9 d\u00e9mantel\u00e9s.\n\nL'ENQU\u00caTEUR\n\nPour quelle raison ?\n\nYE WENJIE\n\nParce que aucun message ne provenait plus d'Alpha du Centaure. Il n'y avait plus rien \u00e0 cette fr\u00e9quence. Je pense que vous avez d\u00e9j\u00e0 eu l'occasion de le v\u00e9rifier.\n\nL'ENQU\u00caTEUR\n\nC'est exact. Vous voulez donc dire que, au moins durant ces quatre derni\u00e8res ann\u00e9es, Trisolaris aurait cess\u00e9 de communiquer avec la Terre ? Cela rend d'autant plus important le contenu des messages d\u00e9tourn\u00e9s par les adventistes.\n\nYE WENJIE\n\nEn effet. Mais je ne peux vraiment rien vous dire de plus \u00e0 ce sujet.\n\nL'ENQU\u00caTEUR _(marquant une pause de quelques secondes)_\n\nDans ce cas, discutons d'autre chose. Mike Evans vous a menti, n'est-ce pas ?\n\nYE WENJIE\n\nOn peut le dire comme \u00e7a. Il ne m'a jamais v\u00e9ritablement r\u00e9v\u00e9l\u00e9 le fond de sa pens\u00e9e, il m'a seulement parl\u00e9 du sens du devoir qu'il ressentait vis-\u00e0-vis des autres cr\u00e9atures vivantes de la Terre. Je n'ai jamais pens\u00e9 que ce sentiment avait fait na\u00eetre chez lui un d\u00e9go\u00fbt de l'humanit\u00e9 si extr\u00eame qu'il \u00e9tait pr\u00eat \u00e0 faire de la destruction de l'homme son id\u00e9al ultime.\n\nL'ENQU\u00caTEUR\n\nObservons la situation actuelle de l'Organisation Terre-Trisolaris. Les adventistes veulent se servir d'une force extraterrestre pour d\u00e9truire l'humanit\u00e9 ; le camp r\u00e9demptoriste v\u00e9n\u00e8re cette civilisation, qu'il consid\u00e8re comme divine, et l'objectif des survivalistes est de trahir l'humanit\u00e9 en \u00e9change du salut de leurs descendants. Aucune de ces tendances n'est en fait conforme \u00e0 votre id\u00e9al premier de r\u00e9former l'humanit\u00e9 gr\u00e2ce \u00e0 la civilisation extraterrestre.\n\nYE WENJIE\n\nJ'ai allum\u00e9 un feu, mais je n'ai pas \u00e9t\u00e9 en mesure de contr\u00f4ler l'incendie.\n\nL'ENQU\u00caTEUR\n\nVous avez projet\u00e9 d'\u00e9liminer les adventistes de l'Organisation, et vous avez m\u00eame commenc\u00e9 \u00e0 mettre ce plan en ex\u00e9cution. Le quartier g\u00e9n\u00e9ral et le centre de commandement adventistes sont bas\u00e9s \u00e0 bord du _Jugement Dernier_. Mike Evans et les autres leaders de son camp s'y trouvent la plupart du temps. Pourquoi ne pas avoir tout simplement attaqu\u00e9 le navire ? Les forces arm\u00e9es r\u00e9demptoristes vous sont presque enti\u00e8rement acquises. Vous auriez pu couler le bateau, ou le capturer.\n\nYE WENJIE\n\n\u00c0 cause des messages des dieux que les adventistes ont intercept\u00e9s. Ces informations sont toutes sauvegard\u00e9es dans le syst\u00e8me de la seconde base de C\u00f4te Rouge, c'est-\u00e0-dire sur un ordinateur \u00e0 bord du _Jugement Dernier_. Si nous avions attaqu\u00e9 le bateau, les adventistes auraient pu sentir que leur fin \u00e9tait proche et supprimer ces informations. Ces messages sont trop importants pour que nous prenions le risque de les perdre. Pour les r\u00e9demptoristes, perdre ces messages, c'est comme perdre la Bible pour les chr\u00e9tiens ou le Coran pour les musulmans. J'imagine que vous faites face au m\u00eame probl\u00e8me. Les adventistes gardent ces messages divins en otages et c'est la raison pour laquelle le _Jugement Dernier_ est encore intact.\n\nL'ENQU\u00caTEUR\n\nAvez-vous des conseils \u00e0 nous donner \u00e0 ce sujet ?\n\nYE WENJIE\n\nNon.\n\nL'ENQU\u00caTEUR\n\nVous aussi, vous parlez de Trisolaris en termes divins. Cela veut-il dire que, comme les r\u00e9demptoristes, vous avez d\u00e9velopp\u00e9 une foi religieuse pour le monde trisolarien ? Ou que vous vous \u00eates convertie au trisolarisme ?\n\nYE WENJIE\n\nNon, ce n'est qu'une question d'habitude... Je ne veux plus parler de \u00e7a.\n\nL'ENQU\u00caTEUR\n\nDans ce cas, revenons aux informations intercept\u00e9es. Vous en ignorez peut-\u00eatre le contenu pr\u00e9cis, mais vous avez certainement une id\u00e9e g\u00e9n\u00e9rale de leur teneur ?\n\nYE WENJIE\n\nCe ne sont peut-\u00eatre que des rumeurs.\n\nL'ENQU\u00caTEUR\n\nPar exemple ?\n\nYE WENJIE\n\n _(Silence.)_\n\nL'ENQU\u00caTEUR\n\nTrisolaris a-t-elle transf\u00e9r\u00e9 aux adventistes des technologies plus avanc\u00e9es que les technologies humaines actuelles ?\n\nYE WENJIE\n\n\u00c7a m'\u00e9tonnerait beaucoup. Car ce serait courir le risque que ces technologies tombent entre vos mains.\n\nL'ENQU\u00caTEUR\n\nUne derni\u00e8re question, et la plus importante : jusqu'\u00e0 aujourd'hui, Trisolaris a-t-elle seulement envoy\u00e9 sur Terre des transmissions radio ?\n\nYE WENJIE\n\nPresque.\n\nL'ENQU\u00caTEUR\n\nPresque ?\n\nYE WENJIE\n\nLa civilisation trisolarienne actuelle est capable de voyages interstellaires \u00e0 une vitesse \u00e9quivalente \u00e0 un dixi\u00e8me de la vitesse de la lumi\u00e8re. Cette avanc\u00e9e technologique date d'il y a quelques dizaines d'ann\u00e9es terrestres. Auparavant, la vitesse maximale de leurs vaisseaux stagnait aux alentours d'un milli\u00e8me de la vitesse de la lumi\u00e8re. Leurs minuscules sondes spatiales n'ont m\u00eame pas encore parcouru un centi\u00e8me du voyage qui les m\u00e8nera de leur plan\u00e8te jusqu'\u00e0 la n\u00f4tre.\n\nL'ENQU\u00caTEUR\n\nUne question : si la flotte trisolarienne est d\u00e9j\u00e0 en route et qu'elle voyage \u00e0 une vitesse correspondant \u00e0 un dixi\u00e8me de la vitesse de la lumi\u00e8re, elle devrait mettre seulement quarante ann\u00e9es avant d'arriver dans le syst\u00e8me solaire ! Pourquoi parlez-vous de quatre cents ans ?\n\nYE WENJIE\n\nEn th\u00e9orie, vous avez raison. Mais la flotte trisolarienne est compos\u00e9e de vaisseaux spatiaux de tr\u00e8s grande taille, dont le processus d'acc\u00e9l\u00e9ration est tr\u00e8s lent. Un dixi\u00e8me de la vitesse de la lumi\u00e8re est la vitesse maximale qu'ils peuvent atteindre, mais ils ne peuvent rester \u00e0 cette vitesse de croisi\u00e8re qu'un tr\u00e8s court laps de temps, car ils doivent commencer \u00e0 d\u00e9c\u00e9l\u00e9rer \u00e0 l'approche de la Terre. Par ailleurs, un champ magn\u00e9tique puissant se trouve en permanence devant le vaisseau. Celui-ci forme une sorte d'entonnoir magn\u00e9tique destin\u00e9 \u00e0 collecter des particules d'antimati\u00e8re dans l'espace. Leur annihilation par des particules de mati\u00e8re ordinaire fournit en effet la propulsion de leurs vaisseaux. Or, ce processus de collecte est tr\u00e8s lent, ce n'est qu'apr\u00e8s une assez longue p\u00e9riode de temps qu'il est possible d'amasser une quantit\u00e9 suffisante d'antimati\u00e8re pour permettre au vaisseau d'acc\u00e9l\u00e9rer, de sorte que l'acc\u00e9l\u00e9ration de la flotte est intermittente. C'est pourquoi le temps que mettra r\u00e9ellement la flotte trisolarienne avant d'atteindre le syst\u00e8me solaire est dix fois plus long que celui d'une petite sonde.\n\nL'ENQU\u00caTEUR\n\nQue signifie le \"presque\" que vous venez d'utiliser ?\n\nYE WENJIE\n\nLorsqu'on parle de la vitesse d'un voyage spatial, il faut avant tout se mettre d'accord sur l'\u00e9chelle. Il existe une \u00e9chelle o\u00f9 m\u00eame la civilisation arri\u00e9r\u00e9e des humains est capable de faire acc\u00e9l\u00e9rer des entit\u00e9s de mati\u00e8re \u00e0 une vitesse proche de celle de la lumi\u00e8re.\n\nL'ENQU\u00caTEUR _(marquant une pause)_\n\nVous voulez parler des \u00e9chelles macroscopique et microscopique, n'est-ce pas ? \u00c0 une \u00e9chelle microscopique, les hommes sont d\u00e9j\u00e0 capables d'utiliser des acc\u00e9l\u00e9rateurs \u00e0 haute \u00e9nergie pour permettre \u00e0 des particules subatomiques d'atteindre une vitesse proche de celle de la lumi\u00e8re. Ces particules, ce sont les \"entit\u00e9s de mati\u00e8re\" dont vous parlez ?\n\nYE WENJIE\n\nVous \u00eates tr\u00e8s intelligent.\n\nL'ENQU\u00caTEUR _(pointant son casque)_\n\nJ'ai les plus grands sp\u00e9cialistes au monde derri\u00e8re moi.\n\nYE WENJIE\n\nOui, je veux bien parler de particules subatomiques. Il y a six ans, dans le lointain syst\u00e8me stellaire d'Alpha du Centaure, le monde de Trisolaris a utilis\u00e9 un acc\u00e9l\u00e9rateur de particules pour envoyer vers notre syst\u00e8me deux noyaux d'atomes d'hydrog\u00e8ne, \u00e0 une vitesse proche de celle de la lumi\u00e8re. Ces deux noyaux d'hydrog\u00e8ne \u2013 ou protons \u2013 ont atteint le syst\u00e8me solaire il y a deux ans, avant d'arriver sur Terre.\n\nL'ENQU\u00caTEUR\n\nDeux protons ? Ils n'ont envoy\u00e9 que deux protons ? Autant dire qu'ils n'ont presque rien envoy\u00e9 !\n\nYE WENJIE _(dans un sourire)_\n\n\u00c0 votre tour d'utiliser le mot \"presque\". C'est le maximum que puisse faire Trisolaris : envoyer un objet de la taille d'un proton \u00e0 une vitesse proche de la lumi\u00e8re. C'est pourquoi, \u00e0 une distance de quatre ann\u00e9es-lumi\u00e8re, ils ne peuvent envoyer que deux protons.\n\nL'ENQU\u00caTEUR\n\n\u00c0 une \u00e9chelle macroscopique, ces deux protons ne sont rien. Un seul cheveu de bact\u00e9rie contient \u00e0 lui seul des milliards de protons. \u00c0 quoi servirait un proton ?\n\nYE WENJIE\n\nC'est un verrou.\n\nL'ENQU\u00caTEUR\n\nUn verrou ? Que veulent-ils verrouiller ?\n\nYE WENJIE\n\nLes progr\u00e8s de la science. \u00c0 cause de l'existence de ces deux protons, l'humanit\u00e9 sera incapable de faire la moindre avanc\u00e9e scientifique durant les quatre si\u00e8cles et demi que mettra la flotte trisolarienne \u00e0 arriver sur Terre. Si on en croit Evans, le jour o\u00f9 ces protons sont arriv\u00e9s sur Terre, a \u00e9t\u00e9 sign\u00e9e la mort de la science.\n\nL'ENQU\u00caTEUR\n\nC'est... un peu surr\u00e9aliste. Comment cela est-il possible ?\n\nYE WENJIE\n\nJe l'ignore. Vraiment. \u00c0 c\u00f4t\u00e9 des Trisolariens, nous ne sommes peut-\u00eatre m\u00eame pas des barbares primitifs. Nous ne sommes que des vermines grouillantes.\n\nIl \u00e9tait d\u00e9j\u00e0 pr\u00e8s de minuit lorsque Wang Miao et Ding Yi sortirent du Centre d'op\u00e9rations militaires. Ils avaient \u00e9cout\u00e9 en direct l'interrogatoire de Ye Wenjie.\n\n\u2014 Crois-tu ce qu'a racont\u00e9 Ye Wenjie ? demanda Wang Miao.\n\n\u2014 Et toi ?\n\n\u2014 Il s'est pass\u00e9 tellement de choses incroyables ces derniers temps... Mais deux protons capables de verrouiller tous les progr\u00e8s scientifiques humains ? C'est un peu...\n\n\u2014 Concentrons-nous d'abord sur une chose : Trisolaris a envoy\u00e9 depuis le syst\u00e8me d'Alpha du Centaure deux protons vers la Terre, \u00e0 quatre ann\u00e9es-lumi\u00e8re d'ici, et ils ont atteint leur cible ! Ce degr\u00e9 d'exactitude est extraordinaire ! Le chemin jusqu'\u00e0 la Terre est sem\u00e9 d'emb\u00fbches, la poussi\u00e8re interstellaire, par exemple. Sans compter que le syst\u00e8me solaire et la Terre sont sans cesse en mouvement. Cela demande une pr\u00e9cision plus grande encore que d'abattre un moustique depuis Pluton. Les talents du sniper qui a r\u00e9ussi ce coup doivent \u00eatre inou\u00efs.\n\nEn entendant \"sniper\", le c\u0153ur de Wang Miao s'emballa.\n\n\u2014 Qu'est-ce que cela explique ?\n\n\u2014 Je ne sais pas. D'apr\u00e8s toi, \u00e0 quoi ressemble une particule subatomique comme un proton ou un neutron ?\n\n\u2014 Je dirais \u00e0 un point. Mais un point avec une structure interne, bien s\u00fbr.\n\n\u2014 Par chance, la repr\u00e9sentation que je m'en fais est plus r\u00e9aliste que la tienne, dit Ding Yi, avant de jeter le m\u00e9got de sa cigarette. Qu'est-ce que c'est ? demanda-t-il \u00e0 Wang Miao en pointant le m\u00e9got.\n\n\u2014 Un filtre de cigarette.\n\n\u2014 Bien. En le regardant d'ici, comment le d\u00e9crirais-tu ?\n\n\u2014 Un simple petit point.\n\n\u2014 Pr\u00e9cis\u00e9ment.\n\nDing Yi marcha pour ramasser le m\u00e9got, puis il l'ouvrit devant les yeux de Wang Miao, r\u00e9v\u00e9lant ses minuscules filaments spongieux jaunis.\n\nWang Miao sentit une odeur de goudron br\u00fbl\u00e9. Ding Yi poursuivit :\n\n\u2014 Comme tu le vois, un si petit machin, une fois ouvert, poss\u00e8de une surface d'adsorption aussi grande que celle d'un salon.\n\nIl jeta \u00e0 nouveau le m\u00e9got.\n\n\u2014 Tu fumes la pipe ? demanda Ding Yi.\n\n\u2014 Je ne fume plus rien.\n\n\u2014 Les pipes utilisent un autre type de filtre, plus avanc\u00e9, \u00e0 trois yuans le filtre. Leur diam\u00e8tre est \u00e0 peu pr\u00e8s le m\u00eame que celui des filtres de cigarette, mais un peu plus long. C'est un tout petit tube en papier rempli de charbon actif. Si tu enl\u00e8ves le charbon actif, il te reste des petites particules de noir de carbone, comme des crottes de souris. Mais \u00e0 l'int\u00e9rieur, la surface d'adsorption form\u00e9e par les minuscules petits trous est aussi vaste qu'un terrain de tennis. C'est la raison pour laquelle le charbon actif a un remarquable pouvoir adsorbant.\n\n\u2014 O\u00f9 veux-tu en venir ? demanda Wang Miao, qui l'\u00e9coutait tr\u00e8s attentivement.\n\n\u2014 L'\u00e9ponge ou le charbon actif \u00e0 l'int\u00e9rieur des filtres sont en trois dimensions, mais leur surface d'adsorption, elle, est en deux dimensions. Ce qu'on peut en d\u00e9duire, c'est qu'une minuscule structure de haute dimension peut contenir une immense structure de plus faible dimension. \u00c0 l'\u00e9chelle macroscopique, la capacit\u00e9 d'un espace de haute dimension \u00e0 contenir un espace de plus faible dimension s'arr\u00eate \u00e0 cette limite. Car Dieu est radin : lors du Big Bang, il n'a donn\u00e9 que trois dimensions \u00e0 notre monde macroscopique. Mais cela ne veut pas dire pour autant que d'autres dimensions, plus grandes, n'existent pas. Huit autres dimensions suppl\u00e9mentaires sont emprisonn\u00e9es dans l'\u00e9chelle microscopique. Si on les ajoute aux trois dimensions du monde macroscopique, il existe donc en r\u00e9alit\u00e9 onze dimensions de particules \u00e9l\u00e9mentaires.\n\n\u2014 Et alors ?\n\n\u2014 Je voulais simplement expliquer un fait : dans l'univers, un marqueur important pour jauger le niveau de technologie d'une civilisation est sa capacit\u00e9 \u00e0 contr\u00f4ler et \u00e0 utiliser ces dimensions microscopiques. Pour ce qui est des particules \u00e9l\u00e9mentaires, nos anc\u00eatres nus et poilus ont commenc\u00e9 \u00e0 les utiliser lorsqu'ils allumaient des feux dans leurs grottes. Contr\u00f4ler une r\u00e9action chimique revient ni plus ni moins \u00e0 manipuler des particules microscopiques. Bien entendu, ce genre de contr\u00f4le a \u00e9volu\u00e9 depuis l'\u00e9poque primitive, nous sommes pass\u00e9s des feux de camp aux machines \u00e0 vapeur, et des machines \u00e0 vapeur aux g\u00e9n\u00e9rateurs \u00e9lectriques ; \u00e0 pr\u00e9sent, le niveau de contr\u00f4le des particules microscopiques \u00e0 une dimension macroscopique a atteint son apog\u00e9e. Nous avons des ordinateurs, et aussi vos nanomat\u00e9riaux. Mais tout cela est encore limit\u00e9 \u00e0 un contr\u00f4le des particules \u00e0 une \u00e9chelle macroscopique. Aux yeux de civilisations plus avanc\u00e9es dans l'univers, le feu, les ordinateurs et les nanomat\u00e9riaux sont de m\u00eame nature : ils appartiennent tous au m\u00eame ordre. C'est la raison pour laquelle les Trisolariens consid\u00e8rent les humains comme de la vermine. Et h\u00e9las pour nous, ils ont raison.\n\n\u2014 Peux-tu \u00eatre plus pr\u00e9cis ? Qu'est-ce que tout cela a \u00e0 voir avec les deux protons ? Que sont-ils capables de faire, une fois sur Terre ? Comme vient de le dire l'enqu\u00eateur, un minuscule cheveu d'une bact\u00e9rie peut contenir \u00e0 lui seul des milliards de protons. M\u00eame si ces deux protons se transformaient enti\u00e8rement en \u00e9nergie sur le bout d'un de mes doigts, je ne sentirais peut-\u00eatre rien d'autre que la sensation d'une piq\u00fbre d'\u00e9pingle.\n\n\u2014 Tu ne sentirais rien du tout. M\u00eame s'ils se convertissaient en \u00e9nergie dans une bact\u00e9rie, la bact\u00e9rie ne sentirait rien.\n\n\u2014 Mais alors, que veux-tu dire ?\n\n\u2014 Rien. Je ne sais rien. Que peut savoir une vermine ?\n\n\u2014 Mais tu es une vermine physicienne, tu en sais forc\u00e9ment plus que moi. Disons qu'au moins tu n'es pas aussi perdu que moi face \u00e0 ces \u00e9v\u00e9nements. Explique-moi, fais comme si je te suppliais, sinon, je crois que je n'arriverai pas \u00e0 dormir cette nuit.\n\n\u2014 Si je t'en dis davantage, tu auras encore plus de mal \u00e0 t'endormir. Laisse tomber. Pourquoi se faire du souci pour tout \u00e7a ? Nous devrions suivre la philosophie de Wei Cheng et Shi Qiang : nous attacher \u00e0 faire du mieux que nous pouvons. Viens, allons boire un verre, puis retournons \u00e0 notre sommeil de vermines !\n\n## 31\n\n## OP\u00c9RATION CITHARE\n\n\u2014 Ne vous inqui\u00e9tez pas, mon vieux, je ne suis plus radioactif, dit Shi Qiang \u00e0 Wang Miao, qui \u00e9tait assis \u00e0 c\u00f4t\u00e9 de lui. Ces derniers jours, ils m'ont lav\u00e9 \u00e0 l'int\u00e9rieur et \u00e0 l'ext\u00e9rieur, comme un sac de farine. Votre pr\u00e9sence \u00e0 la r\u00e9union n'\u00e9tait pas pr\u00e9vue, mais j'ai insist\u00e9. Quelque chose me dit qu'aujourd'hui nous allons voler la vedette.\n\nTandis qu'il parlait, Shi Qiang ramassa un bout de cigare dans un des cendriers pos\u00e9s sur la table de la r\u00e9union, l'alluma, tira une bouff\u00e9e, hocha la t\u00eate d'un air satisfait puis, visiblement d\u00e9tendu, il cracha la fum\u00e9e vers les visages des participants situ\u00e9s \u00e0 l'autre bout de la table. L'un d'entre eux \u00e9tait le propri\u00e9taire originel du cigare : le colonel Stanton, du corps des marines des \u00c9tats-Unis. Il jeta \u00e0 Shi Qiang un regard m\u00e9prisant.\n\nIl y avait \u00e0 cette r\u00e9union beaucoup plus d'officiers militaires en uniforme que lors de la derni\u00e8re. C'\u00e9tait la premi\u00e8re fois dans l'histoire de l'humanit\u00e9 que les forces arm\u00e9es du monde entier devaient faire face \u00e0 un ennemi commun.\n\nLe g\u00e9n\u00e9ral Chang Weisi prit la parole :\n\n\u2014 Camarades, tout le monde ici a maintenant la m\u00eame compr\u00e9hension de base de la situation. Pour reprendre une expression de notre cher commissaire Shi Qiang, nos informations sont maintenant sur un pied d'\u00e9galit\u00e9. La guerre entre les envahisseurs extraterrestres et l'humanit\u00e9 a commenc\u00e9. Ce sont toutefois nos descendants qui affronteront dans quatre si\u00e8cles et demi les forces venues de Trisolaris. Nos ennemis sont pour l'heure humains et, par d\u00e9finition, ils doivent \u00eatre trait\u00e9s comme des tra\u00eetres \u00e0 l'esp\u00e8ce humaine et consid\u00e9r\u00e9s comme hors de la civilisation. C'est la premi\u00e8re fois qu'il nous faut faire face \u00e0 ce type d'adversaires. La suite des op\u00e9rations est claire : nous devons capturer les messages trisolariens intercept\u00e9s \u00e0 bord du _Jugement Dernier_. Ces messages peuvent \u00eatre d'une importance capitale pour la survie de notre civilisation.\n\nNous n'avons jusqu'\u00e0 maintenant jamais contrari\u00e9 la bonne marche du _Jugement Dernier_. Le navire jouit d'un statut qui lui permet de naviguer l\u00e9galement dans l'Atlantique. Nous avons \u00e9t\u00e9 inform\u00e9s qu'il a d\u00e9pos\u00e9 une demande aupr\u00e8s de l'Autorit\u00e9 du canal de Panam\u00e1 pour pouvoir traverser le canal dans quatre jours. C'est une occasion unique pour nous de passer \u00e0 l'action. Au regard de l'\u00e9volution de la situation, c'est une chance qui ne se repr\u00e9sentera peut-\u00eatre plus. Les diff\u00e9rents centres d'op\u00e9rations militaires du globe sont en ce moment m\u00eame en train d'\u00e9laborer des plans d'action. L'\u00e9tat-major central \u00e9tudiera ces plans et s\u00e9lectionnera d'ici moins de dix heures celui qu'il estimera le plus efficace. Le but de cette r\u00e9union est de r\u00e9fl\u00e9chir ensemble \u00e0 des plans possibles et d'en soumettre un \u00e0 trois \u00e0 l'\u00e9tat-major. Camarades, le temps nous est compt\u00e9, nous devons faire preuve d'efficacit\u00e9.\n\nVeuillez noter que ces plans doivent ne pas perdre de vue une chose : il faut s'emparer de fa\u00e7on s\u00e9curis\u00e9e des informations contenues dans le syst\u00e8me de _Jugement Dernier_. Le navire a \u00e9t\u00e9 construit sur les bases d'un ancien p\u00e9trolier, la structure des coques ext\u00e9rieure et int\u00e9rieure a \u00e9t\u00e9 renforc\u00e9e et complexifi\u00e9e. M\u00eame les membres de l'\u00e9quipage doivent s'aider d'un plan pour retrouver leur chemin lorsqu'ils p\u00e9n\u00e8trent dans des zones dont ils ne sont pas familiers. Alors imaginez la compr\u00e9hension que nous pouvons avoir de l'architecture du bateau. \u00c0 l'heure actuelle, nous n'avons m\u00eame pas r\u00e9ussi \u00e0 localiser l'emplacement exact du centre informatique, et nous ne savons pas si les messages de Trisolaris sont stock\u00e9s sur le serveur central, ni combien il existe de sauvegardes. La seule fa\u00e7on d'atteindre notre objectif est de prendre le contr\u00f4le du _Jugement Dernier_. Toute la difficult\u00e9 est d'\u00e9viter que l'ennemi ne supprime les donn\u00e9es lors de cette op\u00e9ration. Il serait tr\u00e8s facile pour eux de les effacer. En cas d'assaut, il est peu probable que nos ennemis suppriment les messages en utilisant des m\u00e9thodes conventionnelles car, avec les technologies \u00e0 notre disposition aujourd'hui, nous aurions t\u00f4t fait de les r\u00e9cup\u00e9rer. Cependant, s'ils vident un chargeur de balles sur le disque dur du serveur ou tout autre p\u00e9riph\u00e9rique de stockage, c'est foutu. Et \u00e7a ne leur aura m\u00eame pas pris dix secondes. C'est-\u00e0-dire que dix secondes avant que notre plan d'action soit d\u00e9couvert, il faut \u00eatre suffisamment proche du dispositif de stockage pour emp\u00eacher l'ennemi d'agir. Comme nous ignorons son emplacement pr\u00e9cis et le nombre de sauvegardes, nous devons \u00e9liminer la totalit\u00e9 des ennemis du _Jugement Dernier_ dans un temps extr\u00eamement court avant que la cible ne soit alert\u00e9e. Et dans le m\u00eame temps, nous ne pouvons pas causer de dommages trop lourds \u00e0 leurs installations, en particulier \u00e0 leur syst\u00e8me informatique. Cette mission est donc extr\u00eamement d\u00e9licate. Certains camarades pr\u00e9tendent m\u00eame qu'elle est impossible.\n\nUn officier des forces japonaises d'autod\u00e9fense dit :\n\n\u2014 \u00c0 notre avis, la seule chance de succ\u00e8s de cette mission est d'utiliser nos espions sur le _Jugement Dernier_. S'ils peuvent se familiariser avec le syst\u00e8me de stockage des messages de Trisolaris, ils seront \u00e0 m\u00eame de prendre le contr\u00f4le du centre informatique ou bien de d\u00e9placer le mat\u00e9riel de stockage avant que nous passions \u00e0 l'action.\n\nQuelqu'un demanda :\n\n\u2014 La surveillance du _Jugement Dernier_ est sous la responsabilit\u00e9 des services de renseignement de l'Otan et de la CIA. A-t-on seulement des espions infiltr\u00e9s ?\n\n\u2014 Aucun, r\u00e9pondit un coordinateur de l'Otan.\n\n\u2014 Alors votre id\u00e9e, c'est du pipi de chat ! glissa Shi Qiang, qui re\u00e7ut aussit\u00f4t une salve de regards d\u00e9daigneux.\n\n\u2014 \u00c9liminer tout le personnel d'une structure ferm\u00e9e, sans en m\u00eame temps ab\u00eemer le mat\u00e9riel, je pense tout de suite \u00e0 utiliser la foudre en boule, lan\u00e7a le colonel Stanton.\n\nDing Yi secoua la t\u00eate :\n\n\u2014 Impossible. Cette arme est d\u00e9sormais connue de tous. Nous ignorons si le bateau a \u00e9t\u00e9 \u00e9quip\u00e9 de parois magn\u00e9tiques de protection contre la foudre en boule. Et m\u00eame s'il n'y en a pas, si la foudre en boule peut en effet tuer tout l'\u00e9quipage, elle ne peut pas le faire simultan\u00e9ment. De plus, une fois que la foudre en boule sera entr\u00e9e dans le bateau, elle planera peut-\u00eatre un moment dans l'air au-dessus du pont avant de lib\u00e9rer son \u00e9nergie. Cela peut prendre quelques dizaines de secondes, parfois m\u00eame jusqu'\u00e0 une minute ! L'ennemi aurait tout \u00e0 fait le temps de d\u00e9couvrir qu'il est attaqu\u00e9 et de d\u00e9truire les donn\u00e9es.\n\nLe colonel Stanton reprit :\n\n\u2014 Et une bombe \u00e0 neutrons ?\n\n\u2014 Colonel, vous savez bien que c'est impossible, r\u00e9pliqua un officier russe. Les radiations d'une bombe \u00e0 neutrons ne tuent pas imm\u00e9diatement les victimes. Une fois qu'elle sera amorc\u00e9e, les ennemis auront le temps de tenir une r\u00e9union comme la n\u00f4tre.\n\nCe fut au tour d'un officier de l'Otan de prendre la parole :\n\n\u2014 Nous pourrions utiliser du gaz innervant, mais le temps qu'il se propage sur le bateau, nous ne serons pas parvenus \u00e0 atteindre l'objectif \u00e9voqu\u00e9 par le g\u00e9n\u00e9ral Chang Weisi.\n\n\u2014 Il ne nous reste qu'une seule solution : utiliser des grenades incapacitantes et des infrasons, dit le colonel Stanton.\n\nTout le monde attendit la suite, mais le colonel s'arr\u00eata l\u00e0. Ce fut Shi Qiang qui r\u00e9pondit :\n\n\u2014 Les grenades incapacitantes sont des jouets que nous utilisons souvent dans la police. Si elles peuvent effectivement \u00e9tourdir des personnes \u00e0 l'int\u00e9rieur d'un b\u00e2timent, elles sont tout justes bonnes \u00e0 \u00eatre utilis\u00e9es pour une chambre ou deux. Avez-vous des grenades incapacitantes utilisables \u00e0 l'\u00e9chelle d'un p\u00e9trolier ?\n\nLe colonel Stanton secoua la t\u00eate :\n\n\u2014 Non, et si de tels explosifs existaient, je crois bien que nous d\u00e9truirions aussi les infrastructures internes.\n\n\u2014 Et des armes infrasoniques ? demanda quelqu'un.\n\n\u2014 Nous sommes encore au stade exp\u00e9rimental. Nous ne pensons pas pouvoir les utiliser en combat r\u00e9el, d'autant que le bateau est immense. \u00c9tant donn\u00e9 l'efficacit\u00e9 actuelle des armes infrasoniques \u00e0 l'essai, si nous menions un assaut majeur contre le _Jugement Dernier_ , elles n'auraient peut-\u00eatre comme seul effet que de leur donner des maux de t\u00eate et la naus\u00e9e.\n\n\u2014 Ha ! (Shi Qiang \u00e9teignit son m\u00e9got de cigare, maintenant de la taille d'une cacahu\u00e8te.) C'est ce que j'ai dit tout \u00e0 l'heure : nous perdons du temps \u00e0 discuter de pipi de chat. Souvenez-vous de ce que le g\u00e9n\u00e9ral a dit : le temps nous est compt\u00e9 !\n\nPuis il se tourna vers l'interpr\u00e8te, une jolie lieutenante qui n'avait pas l'air tr\u00e8s \u00e0 l'aise.\n\n\u2014 Pas facile \u00e0 traduire, hein, camarade ? Donnez-leur juste le sens g\u00e9n\u00e9ral, \u00e7a suffira !\n\nLe colonel Stanton sembla avoir compris ce qu'il disait. Il pointa le nouveau cigare qu'il venait de sortir de sa bo\u00eete sur Shi Qiang et lui lan\u00e7a :\n\n\u2014 Pour qui se prend-il, ce policier, pour nous parler comme \u00e7a ?\n\n\u2014 Et vous \u00eates ? riposta Shi Qiang.\n\n\u2014 Le colonel Stanton est un expert en op\u00e9rations sp\u00e9ciales, d\u00e9clara un officier de l'Otan, il a particip\u00e9 \u00e0 toutes les grandes op\u00e9rations militaires depuis le Vietnam.\n\n\u2014 Je vais vous dire qui je suis. Il y a plus de trente ans, j'ai fait partie d'un peloton de reconnaissance qui a creus\u00e9 des dizaines de kilom\u00e8tres en dessous des lignes vietnamiennes ; nous avons pris d'assaut une station hydrologique sous haute protection et nous avons d\u00e9jou\u00e9 le projet des Vietnamiens de faire exploser le barrage pour bloquer la route \u00e0 notre arm\u00e9e. Voil\u00e0 mes qualifications : j'ai vaincu l'ennemi qui vous a vaincu.\n\n\u2014 Shi Qiang, \u00e7a suffit ! lan\u00e7a le g\u00e9n\u00e9ral Chang Weisi en tapant sur la table. Ne nous \u00e9cartons pas de nos pr\u00e9occupations. Dis-nous plut\u00f4t quel est ton plan.\n\n\u2014 Je crois que nous ne devrions pas perdre de temps avec ce policier, l\u00e2cha avec m\u00e9pris le colonel Stanton en allumant son cigare.\n\nSans m\u00eame attendre la traduction de l'interpr\u00e8te, Shi Qiang bondit de sa chaise :\n\n\u2014 _Pao-li-si_ , _pao-li-si_ , \u00e7a fait la deuxi\u00e8me fois que j'entends ce mot ! Quoi ? Vous snobez la police ? S'il s'agissait de balancer un tas de bombes pour faire p\u00e9ter le bateau en mille morceaux, ce serait un job pour vous, mais pour ce qui est de r\u00e9cup\u00e9rer quelque chose sans causer aucun dommage, m'est avis que vos petites \u00e9toiles sur les \u00e9paules ne vous serviront pas \u00e0 grand-chose, un cambrioleur ferait \u00e7a bien mieux que vous. Pour ce genre de mission, il faut \u00eatre roublard. Et quand je dis roublard, je parle de vraie roublardise ! Et pour \u00e7a, vous ne serez jamais aussi bons que les criminels, ce sont eux les grands ma\u00eetres de la roublardise !\n\nVous savez jusqu'\u00e0 quel point ils peuvent \u00eatre roublards ? J'ai enqu\u00eat\u00e9 sur un vol une fois, les mecs avaient r\u00e9ussi \u00e0 voler une bagnole sur un train en marche. Les voitures de devant et de derri\u00e8re n'avaient pas boug\u00e9, mais le train est arriv\u00e9 \u00e0 destination avec une voiture manquante. Leurs outils ? Des c\u00e2bles et quelques crochets en fer. \u00c7a, c'est ce que j'appelle des experts en op\u00e9rations sp\u00e9ciales ! Et un policier de la brigade criminelle comme moi, qui ai pourchass\u00e9 pendant plus de dix ans des roublards dans ce genre, est form\u00e9 \u00e0 la meilleure \u00e9cole.\n\n\u2014 Expose ton plan ou tais-toi ! tonna Chang Weisi \u00e0 l'encontre de Shi Qiang.\n\n\u2014 Il y a tellement de grosses pointures ici que j'aie eu peur que ce ne soit jamais mon tour et que, mon g\u00e9n\u00e9ral, vous alliez encore me dire que j'\u00e9tais malpoli.\n\n\u2014 Tu as d\u00e9j\u00e0 atteint le comble de l'impolitesse ! Maintenant, d\u00e9p\u00eache-toi, parle-nous de ta \"roublardise\" !\n\nShi Qiang prit un stylo et se mit \u00e0 dessiner deux courbes sur la table : \"le canal\". Puis il prit un cendrier et le posa entre les deux courbes. \"le _Jugement Dernier_ \". Il se rendit ensuite au bout de la table et arracha des mains du colonel Stanton le cigare qu'il venait d'allumer.\n\n\u2014 Cet abruti d\u00e9passe les bornes ! cria le colonel en se redressant.\n\n\u2014 Shi Qiang, dehors, dit Chang Weisi avec une voix s\u00e9v\u00e8re.\n\n\u2014 Attendez que j'aie fini. Donnez-moi encore une minute.\n\nIl tendit la main vers le colonel Stanton.\n\n\u2014 Quoi encore ? demanda le colonel, m\u00e9dus\u00e9.\n\n\u2014 Donnez-m'en un autre !\n\nLe colonel h\u00e9sita un instant, et finit par sortir de sa jolie bo\u00eete un autre cigare qu'il tendit \u00e0 Shi Qiang. Ce dernier planta le premier cigare, bout allum\u00e9 vers le haut, sur la table, de mani\u00e8re qu'il se dresse \u00e0 la verticale sur la rive du canal qu'il venait de tracer. Il pla\u00e7a ensuite le second cigare de la m\u00eame mani\u00e8re, mais sur l'autre rive.\n\n\u2014 Nous dressons deux piliers sur les rives du canal. Entre les deux piliers, nous tirons de nombreux filaments minces et parall\u00e8les que nous espa\u00e7ons d'un demi-m\u00e8tre environ. Ces filaments seront faits avec des \"poignards volants\", les nanomat\u00e9riaux cr\u00e9\u00e9s par le Pr Wang.\n\nUne fois que Shi Qiang eut fini de parler, il attendit debout quelques secondes, haussa les bras en direction des participants qui n'avaient pas encore eu le temps de r\u00e9agir, avant de l\u00e2cher :\n\n\u2014 Voil\u00e0, c'est tout.\n\nPuis il tourna les talons et quitta la pi\u00e8ce.\n\nL'air semblait s'\u00eatre fig\u00e9 dans la salle, comme si tous les participants s'\u00e9taient transform\u00e9s en pierre ; m\u00eame le bourdonnement des ordinateurs autour paraissait s'\u00eatre tu. Ce ne fut qu'au bout d'un certain temps que quelqu'un rompit timidement le silence :\n\n\u2014 Professeur Wang, ces \"poignards volants\" se pr\u00e9sentent vraiment sous la forme de filaments ?\n\nWang Miao acquies\u00e7a de la t\u00eate :\n\n\u2014 Avec les techniques actuelles d'architecture mol\u00e9culaire, nous ne pouvons en effet produire qu'un mat\u00e9riau filiforme, \u00e0 peu pr\u00e8s \u00e9quivalent \u00e0 un dixi\u00e8me de l'\u00e9paisseur d'un cheveu humain... Le commissaire Shi Qiang m'avait demand\u00e9 ces informations avant le d\u00e9but de la r\u00e9union.\n\n\u2014 En avez-vous une quantit\u00e9 suffisante ?\n\n\u2014 Quelle est la largeur du canal ? Et la hauteur du bateau ?\n\n\u2014 Le point le plus \u00e9troit du canal est de cent cinquante m\u00e8tres. Le _Jugement Dernier_ fait trente et un m\u00e8tres de hauteur, avec un tirant d'eau d'environ huit m\u00e8tres.\n\nWang Miao regarda les cigares sur la table et fit un calcul rapide :\n\n\u2014 \u00c7a devrait le faire.\n\nSuivit un autre long silence. Tous essayaient de se remettre de leur choc.\n\n\u2014 Et si jamais le mat\u00e9riel de stockage des messages trisolariens : disques durs, disques de sauvegarde ou autres... \u00e9tait lui aussi d\u00e9coup\u00e9 ? demanda quelqu'un.\n\n\u2014 Les risques sont faibles.\n\n\u2014 M\u00eame si c'\u00e9tait le cas, \u00e7a ne poserait pas un gros probl\u00e8me, expliqua un expert en informatique. Je suppose que ce genre de filaments doit \u00eatre suffisamment aiguis\u00e9 pour faire des coupes homog\u00e8nes. Que ce soit un disque dur, un CD-Rom ou m\u00eame une m\u00e9moire sous forme de circuit int\u00e9gr\u00e9, la plupart des informations pourront \u00eatre r\u00e9cup\u00e9r\u00e9es.\n\n\u2014 Quelqu'un a-t-il une meilleure id\u00e9e ? demanda Chang Weisi en regardant les participants.\n\nComme personne ne r\u00e9pondait, il poursuivit :\n\n\u2014 Bien. Nous allons d\u00e9sormais nous concentrer sur ce plan et en \u00e9tudier les d\u00e9tails.\n\nLe colonel Stanton, qui \u00e9tait rest\u00e9 muet jusque-l\u00e0, se leva :\n\n\u2014 Je vais chercher le policier.\n\nChang Weisi lui fit signe de rester, puis il cria d'une voix sonore :\n\n\u2014 Shi Qiang !\n\nCe dernier rentra dans la salle et d\u00e9visagea les participants avec une mine goguenarde, puis il s'assit en ramassant les deux cigares plant\u00e9s \u00e0 c\u00f4t\u00e9 du canal. Il enfourna le cigare allum\u00e9 dans son bec et fourra l'autre dans sa poche.\n\nQuelqu'un demanda :\n\n\u2014 Quand le _Jugement Dernier_ passera entre les piliers, ceux-ci pourront-ils soutenir la force exerc\u00e9e par les \"poignards volants\" ? Et si c'\u00e9taient les piliers qui venaient \u00e0 c\u00e9der en premier ?\n\n\u2014 C'est facile \u00e0 r\u00e9soudre. Nous avons produit une petite quantit\u00e9 de \"poignards volants\" se pr\u00e9sentant sous forme plate, nous pourrons les utiliser pour renforcer les endroits o\u00f9 sont fix\u00e9s les nanofilaments.\n\nUne discussion d\u00e9buta ensuite entre les officiers de marine et les sp\u00e9cialistes de la navigation navale.\n\n\u2014 Le _Jugement Dernier_ sera peut-\u00eatre le bateau au plus grand tonnage \u00e0 jamais passer par le canal de Panam\u00e1. Son tirant d'eau est tr\u00e8s profond, il faudra aussi songer \u00e0 installer des nanofilaments sous l'eau.\n\n\u2014 Ce sera une t\u00e2che autrement plus d\u00e9licate. Si le temps nous manque, je sugg\u00e8re de laisser tomber l'id\u00e9e d'installer des \"poignards volants\" sous l'eau. La partie immerg\u00e9e d'un bateau, c'est g\u00e9n\u00e9ralement l'endroit o\u00f9 se trouvent les moteurs, le carburant et le ballast. Le bruit, les vibrations et les interf\u00e9rences y sont importants, l'air y est mauvais, ce n'est vraiment pas l'endroit id\u00e9al pour installer un centre informatique ou ce genre d'appareils. En revanche, en surface, si on pouvait diminuer quelque peu la distance entre les nanofilaments, le r\u00e9sultat serait encore plus efficace.\n\n\u2014 L'id\u00e9al serait donc d'agir \u00e0 la hauteur de l'une des trois \u00e9cluses du canal. Le _Jugement Dernier_ est class\u00e9 comme Panamax, il a \u00e9t\u00e9 construit pour pouvoir passer tout juste l'\u00e9cluse1. Nous aurons simplement besoin de filaments de trente-deux m\u00e8tres de long. Il sera aussi plus facile d'\u00e9riger des piliers et de relier les nanofilaments entre eux, sp\u00e9cialement pour la partie sous-marine.\n\n\u2014 Non. Le passage des \u00e9cluses est une situation complexe, les navires sont tir\u00e9s par quatre locomotives \u00e9lectriques \u00e0 une vitesse tr\u00e8s lente. Ce sera certainement le moment o\u00f9 l'\u00e9quipage du _Jugement Dernier_ sera le plus vigilant. Nous avons de grandes chances d'\u00eatre d\u00e9couverts au moment de la d\u00e9coupe.\n\n\u2014 Que pensez-vous du pont des Am\u00e9riques, juste apr\u00e8s les \u00e9cluses de Miraflores ? Les piliers du pont pourraient \u00eatre utilis\u00e9s pour tendre les filaments.\n\n\u2014 Non. La distance entre les piliers est trop importante. Nous n'aurons jamais assez de \"poignards volants\".\n\n\u2014 Alors c'est d\u00e9cid\u00e9, l'op\u00e9ration aura lieu au niveau le plus \u00e9troit de la coupe Gaillard : cent cinquante m\u00e8tres de large. Disons cent soixante-dix, en comptant un peu de marge pour les piliers2.\n\n\u2014 Si c'est comme \u00e7a, la distance minimale entre les filaments devra \u00eatre de cinquante centim\u00e8tres, car nous n'en avons pas en quantit\u00e9 suffisante, dit Wang.\n\n\u2014 Ce qui veut dire..., dit Shi Qiang en crachant une bouff\u00e9e de fum\u00e9e, qu'il faut agir en plein jour.\n\n\u2014 Pourquoi ?\n\n\u2014 Mais parce que l'\u00e9quipage dort la nuit ! Ils seront couch\u00e9s, et un espace de cinquante centim\u00e8tres sera insuffisant. La journ\u00e9e, m\u00eame s'ils sont assis ou accroupis, \u00e7a ira largement.\n\nQuelques \u00e9clats de rire fus\u00e8rent. Cette pr\u00e9cision macabre apporta un peu de d\u00e9tente dans l'atmosph\u00e8re pesante de la salle.\n\n\u2014 Vous \u00eates vraiment diabolique ! dit une repr\u00e9sentante de l'ONU \u00e0 ses c\u00f4t\u00e9s.\n\n\u2014 Des innocents risquent-ils d'\u00eatre bless\u00e9s ? demanda Wang Miao d'une voix tremblante.\n\nCe fut un officier de marine qui lui r\u00e9pondit :\n\n\u2014 Au moment de passer les \u00e9cluses, il y aura \u00e0 bord une dizaine d'employ\u00e9s de c\u00e2blage, mais ils seront descendus avant la coupe Gaillard. En revanche, le pilote du canal qui doit les accompagner durant les quatre-vingt-deux kilom\u00e8tres de leur travers\u00e9e sera, lui, sacrifi\u00e9 !\n\nUn agent de la CIA ajouta :\n\n\u2014 De m\u00eame qu'une partie de l'\u00e9quipage du _Jugement Dernier_ qui ignore probablement tout du r\u00f4le r\u00e9el du bateau sur lequel ils ont embarqu\u00e9.\n\n\u2014 Professeur, ne pensons pas \u00e0 tout cela. Ce ne sont pas des choses dont il faut se soucier. Nous avons besoin de r\u00e9cup\u00e9rer ces informations, la survie de l'humanit\u00e9 est en jeu. Quelqu'un d'autre aura la responsabilit\u00e9 de prendre la d\u00e9cision finale, dit Chang Weisi.\n\n\u00c0 la fin de la r\u00e9union, le colonel Stanton poussa sa magnifique bo\u00eete en bois remplie de cigares vers Shi Qiang :\n\n\u2014 Commissaire, des havanes de premi\u00e8re qualit\u00e9. Je vous les offre.\n\n_Quatre jours plus tard, \u00e0 la coupe Gaillard._\n\nWang Miao n'avait pas la sensation de se trouver en pays \u00e9tranger. Il savait pourtant que pas tr\u00e8s loin \u00e0 l'ouest s'\u00e9tendaient le magnifique lac Gat\u00fan et, \u00e0 l'est, le splendide pont des Am\u00e9riques et la ville de Panam\u00e1. Mais il n'aurait aucune chance de les voir. Deux jours plus t\u00f4t, ils avaient pris un vol direct depuis la Chine jusqu'\u00e0 l'a\u00e9roport international de Tocumen, pas loin de la ville de Panam\u00e1, avant de monter dans un h\u00e9licopt\u00e8re qui les avait amen\u00e9s jusqu'ici.\n\nLe paysage devant ses yeux n'avait rien d'extraordinaire. Des ouvriers travaillant \u00e0 \u00e9largir le canal \u00e9tiraient et rasaient la for\u00eat tropicale sur chaque rive, r\u00e9v\u00e9lant des \u00e9tendues de terre jaun\u00e2tre, une couleur dont Wang Miao \u00e9tait familier. De l\u00e0 o\u00f9 il \u00e9tait, le canal n'apparaissait pas si grandiose, peut-\u00eatre parce qu'il se situait dans une portion tr\u00e8s \u00e9troite. Ce passage avait pourtant \u00e9t\u00e9 creus\u00e9 par cent mille personnes au d\u00e9but du si\u00e8cle pr\u00e9c\u00e9dent.\n\nWang Miao et le colonel Stanton \u00e9taient assis sur des chaises longues, sous un kiosque, \u00e0 mi-flanc d'une colline avoisinante. Tous deux portaient d'amples chemises \u00e0 fleurs, et de grands chapeaux de paille folkloriques \u00e9taient pos\u00e9s \u00e0 c\u00f4t\u00e9 d'eux. Ils avaient l'air de deux touristes ordinaires. De cette position, ils avaient une vue panoramique sur le canal.\n\nSous leurs pieds, sur chacune des rives du canal, \u00e9taient couch\u00e9s \u00e0 plat deux piliers d'acier, longs de vingt-quatre m\u00e8tres, ainsi que cinquante c\u00e2bles de cent soixante m\u00e8tres de nanofilaments ultrar\u00e9sistants, reli\u00e9s aux deux piliers et espac\u00e9s par des intervalles de cinquante centim\u00e8tres. Au bout de la rive, les nanofilaments \u00e9taient reli\u00e9s \u00e0 du fil d'acier ordinaire, ce qui permettait aux nanomat\u00e9riaux d'\u00eatre immerg\u00e9s sous l'eau gr\u00e2ce \u00e0 des poids. Tout avait \u00e9t\u00e9 pr\u00e9vu pour laisser passer les autres navires. Fort heureusement, le trafic n'\u00e9tait pas aussi intense que l'avait craint Wang Miao. Chaque jour, une moyenne de quarante navires passaient par le canal. La position des deux piliers en acier devait prendre en compte les activit\u00e9s \u00e0 la surface : il fallait attendre que le dernier bateau pr\u00e9c\u00e9dant le _Jugement Dernier_ soit pass\u00e9 avant de les redresser. Le code de l'op\u00e9ration \u00e9tait _guzheng_ , du nom d'une cithare sur table chinoise. La comparaison de la structure avec cet instrument \u00e0 cordes avait \u00e9t\u00e9 naturelle, tandis que le filet coupant de nanomat\u00e9riaux avait, lui, \u00e9t\u00e9 surnomm\u00e9 le _qin_ , un autre genre de cithare.\n\nLe _Jugement Dernier_ \u00e9tait entr\u00e9 dans la coupe Gaillard une heure plus t\u00f4t par le lac Gat\u00fan.\n\nLe colonel Stanton demanda \u00e0 Wang Miao s'il \u00e9tait d\u00e9j\u00e0 venu au Panam\u00e1. Ce dernier r\u00e9pondit par la n\u00e9gative.\n\n\u2014 Je suis venu en 1990, dit le colonel.\n\n\u2014 Lors de la guerre ?\n\n\u2014 Oui. Mais ce n'est pas une guerre dont je garde beaucoup de souvenirs. Je me souviens seulement d'avoir \u00e9t\u00e9 l\u00e0, en face de l'ambassade du Vatican o\u00f9 s'\u00e9tait r\u00e9fugi\u00e9 l'ancien pr\u00e9sident Noriega, quand nous avons diffus\u00e9 le morceau de rock _Nowhere to Run_ de Martha and the Vandellas. C'\u00e9tait mon id\u00e9e.\n\nEn bas, sur le canal, un navire de croisi\u00e8re fran\u00e7ais avan\u00e7ait lentement. Son pont \u00e9tait rev\u00eatu d'une moquette verte sur laquelle se promenaient des touristes aux tenues bariol\u00e9es.\n\n\u2014 Second poste d'observation au rapport. Aucun autre bateau en vue droit devant, r\u00e9sonna le talkie-walkie.\n\n\u2014 Levez la cithare, ordonna Stanton.\n\nDes hommes coiff\u00e9s de casques d'ouvriers apparurent sur les deux rives, Wang Miao se leva mais le colonel le tira pour qu'il se rasseye.\n\n\u2014 Ne vous inqui\u00e9tez pas, professeur, ils savent ce qu'ils ont \u00e0 faire.\n\nWang Miao aper\u00e7ut sur la rive droite les hommes en train de remonter soigneusement les \"poignards volants\" et de les fixer, une fois tendus, aux piliers, s\u00e9curis\u00e9s par des plaques elles-m\u00eames faites en nanofilaments. Sur les deux rives, les hommes tir\u00e8rent ensuite sur un long c\u00e2ble, faisant d\u00e9licatement se dresser les deux colonnes en acier. En guise de camouflage, on avait accroch\u00e9 sur les deux piliers des panneaux d'aide \u00e0 la navigation et des indications sur la profondeur de l'eau. Les ouvriers agissaient avec une grande s\u00e9r\u00e9nit\u00e9. Certains avaient m\u00eame l'air de se comporter en dilettantes, comme s'ils n'effectuaient l\u00e0 qu'un insipide travail de routine. Wang Miao observa l'espace entre les deux piliers. Rien ne semblait s'y trouver, mais la cithare de mort \u00e9tait bien en place.\n\n\u2014 Cible \u00e0 quatre kilom\u00e8tres de la cithare ! dit la voix dans le talkie-walkie.\n\nStanton posa son talkie et continua la discussion :\n\n\u2014 La deuxi\u00e8me fois que je suis venu au Panam\u00e1, c'\u00e9tait en 1999, pour la c\u00e9r\u00e9monie de restitution du canal \u00e0 l'\u00c9tat du Panam\u00e1. \u00c9trangement, quand nous sommes arriv\u00e9s devant l'\u00e9difice de contr\u00f4le du canal, j'ai vu que le drapeau \u00e9toil\u00e9 avait d\u00e9j\u00e0 \u00e9t\u00e9 enlev\u00e9. On a pr\u00e9tendu que c'\u00e9tait le gouvernement des \u00c9tats-Unis qui avait demand\u00e9 \u00e0 \u00f4ter le drapeau un jour plus t\u00f4t, pour \u00e9viter que son abaissement ne se fasse devant une foule nombreuse... Je croyais \u00e0 l'\u00e9poque que j'assistais \u00e0 un moment historique, mais maintenant que j'y pense, toutes ces choses-l\u00e0 paraissent bien insignifiantes.\n\n\u2014 Cible \u00e0 trois kilom\u00e8tres !\n\n\u2014 Oui, insignifiantes, reprit Wang Miao en \u00e9cho.\n\nIl n'avait en r\u00e9alit\u00e9 pas \u00e9cout\u00e9 ce que racontait Stanton. En cet instant pr\u00e9cis, le reste du monde avait d\u00e9j\u00e0 cess\u00e9 d'exister. Toutes ses pens\u00e9es \u00e9taient tourn\u00e9es vers ce _Jugement Dernier_ qui n'\u00e9tait pas encore apparu dans leur champ de vision. Le soleil qui s'\u00e9tait lev\u00e9 sur le Pacifique s'appr\u00eatait \u00e0 se coucher sur l'Atlantique et les eaux du canal miroitaient d'or. Plus pr\u00e8s, sous leurs pieds, la cithare de mort attendait en silence. Les deux piliers d'acier \u00e9taient sombres et ne refl\u00e9taient aucun rayon du soleil. Ils avaient l'air encore plus anciens que le canal s'\u00e9coulant au milieu.\n\n\u2014 Cible \u00e0 deux kilom\u00e8tres !\n\nStanton ne semblait pas avoir \u00e9tendu la voix dans son talkie-walkie. Il continuait \u00e0 parler pour lui-m\u00eame :\n\n\u2014 Depuis que je sais qu'une flotte extraterrestre se dirige vers la Terre, je deviens amn\u00e9sique. C'est tr\u00e8s \u00e9trange. Je ne me souviens pas des d\u00e9tails des guerres que j'ai v\u00e9cues. Comme je viens de le dire, elles me paraissent \u00e0 pr\u00e9sent insignifiantes. Quand j'ai appris cette nouvelle, chaque individu est devenu dans mon esprit comme une nouvelle personne, et le monde comme un nouveau monde. Je me pose sans cesse cette question : si deux mille ans plus t\u00f4t \u2013 ou il y a encore plus longtemps \u2013 les hommes avaient su qu'une flotte d'extraterrestres allait envahir le monde des milliers d'ann\u00e9es plus tard, \u00e0 quoi ressemblerait la civilisation humaine aujourd'hui ? Vous pouvez l'imaginer, professeur ?\n\n\u2014 Oh, non, je ne peux pas..., r\u00e9pondit superficiellement Wang Miao, l'esprit ailleurs.\n\n\u2014 Cible \u00e0 un kilom\u00e8tre et demi !\n\n\u2014 Professeur, je crois que vous \u00eates le Gaillard de ce nouveau si\u00e8cle. Nous attendons la construction de votre canal de Panam\u00e1. Quand on y pense, un ascenseur spatial, c'est une sorte de canal. Le canal de Panam\u00e1 peut relier deux oc\u00e9ans, et l'ascenseur peut connecter la Terre \u00e0 l'espace...\n\nWang Miao comprit \u00e0 pr\u00e9sent que les bavardages du colonel \u00e9taient en fait destin\u00e9s \u00e0 l'aider \u00e0 traverser cette \u00e9preuve difficile. Il lui en \u00e9tait reconnaissant, mais ils n'avaient pas vraiment fait leur effet.\n\n\u2014 Cible \u00e0 un kilom\u00e8tre !\n\nAu coude ouest du canal apparut le _Jugement Dernier_ , dos au soleil. Une silhouette noire fusant sur les ondes dor\u00e9es de l'eau.\n\nCe b\u00e2timent de soixante mille tonnes \u00e9tait beaucoup plus massif que ne l'avait imagin\u00e9 Wang Miao. Quand il fit son apparition, ce fut comme si une montagne s'\u00e9tait soudain lev\u00e9e \u00e0 l'ouest et, bien que Wang Miao sache que le canal pouvait laisser passer des navires allant jusqu'\u00e0 soixante-dix mille tonnes, il trouva que le passage d'un tel navire dans un bras d'eau si \u00e9troit avait quelque chose de stup\u00e9fiant. \u00c9tant donn\u00e9 sa taille immense, le canal sur lequel il voguait paraissait presque ne pas exister, comme une montagne glissant sur la terre. Apr\u00e8s s'\u00eatre habitu\u00e9 aux rayons \u00e9blouissants du couchant, il vit que la coque du _Jugement Dernier_ \u00e9tait de couleur noire, et les infrastructures sur le pont de couleur blanche. La gigantesque antenne avait \u00e9t\u00e9 d\u00e9truite. Il pouvait entendre le bourdon des moteurs, accompagn\u00e9 par le grondement des vagues rejet\u00e9es sur la rive par la proue fendant les eaux.\n\n\u00c0 mesure que la distance entre le _Jugement Dernier_ et la cithare de mort s'amenuisait, le rythme cardiaque de Wang Miao augmentait, sa respiration devenait saccad\u00e9e. Il eut envie de fuir, mais un exc\u00e8s de faiblesse l'emp\u00eacha d'\u00eatre ma\u00eetre de son propre corps. Une soudaine vague de haine envers Shi Qiang le submergea. Comment ce salaud avait-il pu avoir une telle id\u00e9e ? Comme l'avait dit la repr\u00e9sentante de l'ONU, c'\u00e9tait le diable ! Mais ce sentiment s'\u00e9vanouit aussit\u00f4t. Il se dit que s'il se trouvait \u00e0 pr\u00e9sent \u00e0 c\u00f4t\u00e9 de Shi Qiang, il se sentirait bien plus rassur\u00e9. Le colonel Stanton avait invit\u00e9 le commissaire \u00e0 participer \u00e0 l'op\u00e9ration, mais Chang Weisi s'y \u00e9tait oppos\u00e9. On avait besoin de lui l\u00e0 o\u00f9 il \u00e9tait. Wang Miao sentit dans son dos une tape amicale du colonel.\n\n\u2014 Professeur, \u00e7a va passer.\n\nLe _Jugement Dernier_ \u00e9tait en train d'avancer, il traversait la cithare de mort. L'air semblait vide quand la proue p\u00e9n\u00e9tra dans l'espace situ\u00e9 entre les deux piliers. Wang Miao eut un frisson. Mais rien ne se passa, l'\u00e9norme coque du navire glissa lentement entre les piliers. Alors que la moiti\u00e9 du bateau \u00e9tait sortie, Wang Miao se prit m\u00eame \u00e0 douter de l'existence des nanofilaments.\n\nMais un petit indice mit fin \u00e0 ses soup\u00e7ons ; il remarqua tout en haut d'un \u00e9difice b\u00e2ti sur le pont qu'une antenne avait \u00e9t\u00e9 sectionn\u00e9e et que son sommet avait roul\u00e9 sur le sol. Tr\u00e8s vite, un deuxi\u00e8me signe r\u00e9v\u00e9la la pr\u00e9sence des nanofilaments et manqua de provoquer l'effondrement total de Wang Miao. Le vaste pont du _Jugement Dernier_ \u00e9tait d\u00e9sert, un seul matelot se trouvait sur le pont arri\u00e8re en train de nettoyer au jet d'eau les bollards d'amarrage. De sa hauteur, Wang Miao vit le corps de l'homme se raidir subitement lorsque cette partie du bateau passa entre les piliers. Le tuyau glissa de ses mains, le raccord en caoutchouc se fendit en deux et de l'eau se mit \u00e0 jaillir. L'homme resta debout encore quelques secondes, puis tomba. Au contact du sol du pont, son corps se scinda en deux et l'eau qui sortait du robinet se teinta de rouge. La moiti\u00e9 sup\u00e9rieure du corps rampa un instant dans une mare sanguinolente, mais l'homme ne put s'aider que de deux moignons, car ses bras avaient \u00e9t\u00e9 tranch\u00e9s.\n\nLorsque ce fut au tour de la poupe de passer entre les deux piliers, le _Jugement Dernier_ avan\u00e7ait encore \u00e0 une vitesse r\u00e9guli\u00e8re. Tout paraissait calme \u00e0 bord. Mais Wang Miao entendit brusquement une disharmonie dans la symphonie des moteurs, suivie par un bruit de chaos. C'\u00e9tait comme si une clef de serrage avait \u00e9t\u00e9 jet\u00e9e dans les h\u00e9lices du moteur \u2013 ou plut\u00f4t non, beaucoup de clefs. Wang Miao devina que la partie permettant la rotation du moteur avait \u00e9t\u00e9 tranch\u00e9e. Apr\u00e8s un assourdissant fracas, un trou apparut \u00e0 la poupe du navire, provoqu\u00e9 par la fissure de la coque par un \u00e9norme \u00e9l\u00e9ment m\u00e9tallique. L'\u00e9l\u00e9ment tomba dans le canal et souleva une gigantesque colonne d'eau. Au moment o\u00f9 il s'\u00e9leva dans les airs, Wang Miao put voir qu'il s'agissait d'un vilebrequin du moteur. Une \u00e9paisse fum\u00e9e monta du trou ; le _Jugement Dernier_ , qui naviguait jusqu'alors en longeant la rive droite, changea de direction, la poupe en feu, avant de jaillir hors de l'eau et de tamponner la rive gauche.\n\nWang Miao vit la colossale proue se d\u00e9former et fendre une colline comme si c'\u00e9tait de l'eau, provoquant des vagues de terre d\u00e9cha\u00een\u00e9es. C'est alors que le _Jugement Dernier_ se scinda en une quarantaine de plaques d'une \u00e9paisseur de cinquante centim\u00e8tres. De cette distance, on vit le navire s'ouvrir comme un ch\u00e2teau de cartes, celles du haut tombant plus vite que celles du bas. Ces quarante et quelques plaques gliss\u00e8rent sur le sol et se t\u00e9lescop\u00e8rent dans un crissement, comme si d'innombrables doigts g\u00e9ants raclaient un tableau noir.\n\nCe son insupportable disparut enfin. Le _Jugement Dernier_ n'\u00e9tait d\u00e9j\u00e0 plus qu'un amas de plaques sur la rive, comme une pile d'assiettes renvers\u00e9e par un serveur maladroit. Les assiettes les plus en hauteur \u00e9taient celles qui avaient vol\u00e9 le plus loin. Les plaques avaient l'air aussi souples que des fragments de tissu, elles se disloqu\u00e8rent bient\u00f4t pour prendre des formes complexes, et il \u00e9tait devenu impossible d'imaginer qu'elles avaient un jour \u00e9t\u00e9 les morceaux d'un bateau.\n\nDes troupes de militaires jaillirent de derri\u00e8re la colline et se pr\u00e9cipit\u00e8rent vers la rive. Wang Miao fut surpris de constater que tant d'hommes se cachaient si pr\u00e8s depuis il ne savait combien de temps. Une flotte d'h\u00e9licopt\u00e8res vrombissants vint planer au-dessus du canal, elle survola la surface de l'eau, maintenant couverte d'une couche d'huile iridescente. Les h\u00e9licopt\u00e8res s'arr\u00eat\u00e8rent \u00e0 hauteur de l'\u00e9pave du _Jugement Dernier_ et l'asperg\u00e8rent d'une grande quantit\u00e9 de mousse extinctrice. Ils furent bient\u00f4t en mesure de contr\u00f4ler l'incendie qui couvait sur l'\u00e9pave, tandis que trois autres h\u00e9licopt\u00e8res sortirent des c\u00e2bles pour faire descendre en rappel des enqu\u00eateurs. Surmontant ses frissons, Wang Miao regarda le _Jugement Dernier_ d\u00e9coup\u00e9 en plus de quarante morceaux. Il \u00e9tait d\u00e9j\u00e0 \u00e0 moiti\u00e9 recouvert de mousse. Mais sur les parties qui restaient encore visibles, Wang Miao put voir leurs surfaces, dont la d\u00e9coupe \u00e9tait aussi lisse que celle d'un miroir et qui refl\u00e9taient sans le d\u00e9former le soleil rouge feu du cr\u00e9puscule. Il vit \u00e9galement un grand cercle \u00e9carlate sur les miroirs, dont il ignorait si c'\u00e9tait du sang.\n\n_Trois jours plus tard._\n\nL'ENQU\u00caTEUR\n\n\u00cates-vous famili\u00e8re de la civilisation trisolarienne ?\n\nYE WENJIE\n\nNon. Nous n'avons eu acc\u00e8s qu'\u00e0 des informations tr\u00e8s limit\u00e9es. En fait, hormis Mike Evans et les autres leaders adventistes qui ont intercept\u00e9 les messages trisolariens, personne ne poss\u00e8de de d\u00e9tails r\u00e9els et pr\u00e9cis sur leur civilisation.\n\nL'ENQU\u00caTEUR\n\nAlors d'o\u00f9 vous vient votre espoir ? Comment pouvez-vous savoir qu'ils seront en mesure de r\u00e9former et d'am\u00e9liorer la soci\u00e9t\u00e9 humaine ?\n\nYE WENJIE\n\nS'ils sont capables de traverser les \u00e9toiles pour venir jusque dans notre monde, cela indique que leur science a d\u00e9j\u00e0 atteint un niveau tr\u00e8s avanc\u00e9. Une soci\u00e9t\u00e9 avec un tel niveau scientifique doit avoir de hauts standards moraux.\n\nL'ENQU\u00caTEUR\n\nPensez-vous que cette conclusion est elle-m\u00eame scientifique ?\n\nYE WENJIE\n\n _(Silence.)_\n\nL'ENQU\u00caTEUR\n\nPermettez-moi de deviner : votre grand-p\u00e8re \u00e9tait persuad\u00e9 que la science pouvait sauver la Chine, il a tr\u00e8s profond\u00e9ment influenc\u00e9 votre p\u00e8re, qui vous a influenc\u00e9e \u00e0 son tour.\n\nYE WENJIE _(poussant un soupir silencieux)_\n\nJe ne sais pas.\n\nL'ENQU\u00caTEUR\n\nJe vous informe que nous avons r\u00e9cup\u00e9r\u00e9 l'int\u00e9gralit\u00e9 des messages trisolariens intercept\u00e9s par la frange adventiste.\n\nYE WENJIE\n\nOh... Qu'est-il arriv\u00e9 \u00e0 Evans ?\n\nL'ENQU\u00caTEUR\n\nIl est mort durant l'op\u00e9ration de capture du _Jugement Dernier_. _(Evans avait \u00e9t\u00e9 d\u00e9coup\u00e9 en trois morceaux par les \"poignards volants\". Il se trouvait alors dans le centre de commandement du navire. Son tronc avait ramp\u00e9 sur pr\u00e8s d'un m\u00e8tre. Lorsqu'il fut d\u00e9couvert, ses deux yeux \u00e9taient point\u00e9s dans la direction d'un ordinateur, celui dans lequel on avait trouv\u00e9 les messages trisolariens.)_\n\nYE WENJIE\n\nY a-t-il beaucoup de messages ?\n\nL'ENQU\u00caTEUR\n\nBeaucoup. Environ 28 gigaoctets.\n\nYE WENJIE\n\nC'est impossible. Les bandes de fr\u00e9quences des communications interstellaires rendent leur vitesse de transmission tr\u00e8s faible. Comment auraient-ils pu envoyer une somme si importante de donn\u00e9es ?\n\nL'ENQU\u00caTEUR\n\nCela nous a aussi intrigu\u00e9s au d\u00e9but. Mais la r\u00e9alit\u00e9 est bien plus extraordinaire que ce que nous pensions \u2013 m\u00eame dans notre imagination la plus fantaisiste ou la plus d\u00e9brid\u00e9e. Je vous propose de lire une partie de ces informations. Vous verrez que la r\u00e9alit\u00e9 de la civilisation trisolarienne est bien diff\u00e9rente de celle dont vous avez r\u00eav\u00e9.\n\n* * *\n\n1 Pour passer les \u00e9cluses du canal de Panam\u00e1 qui font trente-deux m\u00e8tres de large, la plupart des bateaux class\u00e9s comme Panamax sont con\u00e7us avec une largeur de trente et un m\u00e8tres au maximum. _(N.d.A.)_\n\n2 La coupe Gaillard est une vall\u00e9e artificielle qui fait partie du canal de Panam\u00e1. _(N.d.A.)_\n\n## 32\n\n## LE GUETTEUR\n\n(Les donn\u00e9es captur\u00e9es ne comprenaient aucune description de l'apparence biologique des Trisolariens. Les humains devraient attendre plus de quatre si\u00e8cles avant de les d\u00e9couvrir de leurs propres yeux. \u00c0 la lecture de ces documents, Ye Wenjie se r\u00e9signa donc \u00e0 les imaginer comme des cr\u00e9atures humano\u00efdes.)\n\nLe poste de guet 1379 existait depuis d\u00e9j\u00e0 plus d'un mill\u00e9naire. Il y en avait plusieurs milliers du m\u00eame genre sur Trisolaris, tous enti\u00e8rement d\u00e9di\u00e9s \u00e0 l'\u00e9coute de possibles signes d'une civilisation intelligente ailleurs dans l'univers.\n\n\u00c0 l'origine, chaque poste comptait plus de cent guetteurs mais, avec le d\u00e9veloppement de la technologie, la t\u00e2che pouvait maintenant \u00eatre effectu\u00e9e par un seul individu. Guetteur \u00e9tait un m\u00e9tier modeste : m\u00eame si le syst\u00e8me de thermostat \u00e0 temp\u00e9rature constante des postes de guet leur permettait de survivre aux \u00e8res chaotiques sans avoir besoin de se d\u00e9shydrater, les guetteurs devaient toutefois passer leur vie enti\u00e8re dans un minuscule espace et ne go\u00fbtaient que tr\u00e8s peu \u00e0 la joie \u00e9prouv\u00e9e par leurs semblables lors du retour des \u00e8res r\u00e9guli\u00e8res.\n\nLe guetteur du poste 1379 regarda \u00e0 travers sa petite fen\u00eatre le monde de Trisolaris. C'\u00e9tait la nuit sombre d'une \u00e8re chaotique. La lune g\u00e9ante ne s'\u00e9tait pas encore lev\u00e9e et la plupart de ses cong\u00e9n\u00e8res \u00e9taient en \u00e9tat d'hibernation d\u00e9shydrat\u00e9e. M\u00eame les v\u00e9g\u00e9taux de Trisolaris avaient d\u00e9velopp\u00e9 avec le temps un instinct de d\u00e9shydratation : lors des \u00e8res chaotiques, ils devenaient des bouquets sans vie de fibres s\u00e9ch\u00e9es. Sous les rayons des \u00e9toiles, la terre paraissait \u00eatre un bloc de m\u00e9tal froid.\n\nC'\u00e9tait un de ces moments durant lesquels il se sentait le plus seul. Dans le silence de minuit, l'univers apparaissait \u00e0 ceux qui tendaient l'oreille vers lui comme une vaste plaine d\u00e9sertique. Le guetteur du poste 1379 d\u00e9testait voir ces courbes oscillant l\u00e9g\u00e8rement sur l'\u00e9cran de surveillance. C'\u00e9taient les ondes radio de l'univers que recevait le poste de guet : un bruit de fond sans aucune signification.\n\nLe guetteur voyait dans ces lignes sans fin comme une repr\u00e9sentation abstraite de l'univers : une extr\u00e9mit\u00e9 reli\u00e9e \u00e0 un pass\u00e9 infini, et l'autre \u00e0 un futur infini. Et, au milieu, une courbe al\u00e9atoire sans r\u00e8gle et sans vie travers\u00e9e de pics et de vall\u00e9es, comme un d\u00e9sert unidimensionnel tapiss\u00e9 de grains de sable de diff\u00e9rentes tailles. Des rang\u00e9es de grains de sable solitaires, d\u00e9sol\u00e9es, si longues qu'elles en \u00e9taient insupportables. On pouvait longer ce d\u00e9sert d'un bout \u00e0 l'autre, sans jamais en trouver la fin.\n\nCette nuit-l\u00e0 cependant, en laissant tra\u00eener son regard sur l'\u00e9cran, le guetteur aper\u00e7ut quelque chose d'anormal. M\u00eame des experts auraient eu du mal \u00e0 dire, \u00e0 l'\u0153il nu, si la forme d'onde qui venait d'appara\u00eetre \u00e9tait porteuse d'une information quelconque. Mais le guetteur connaissait trop bien le bruit de fond de l'univers ; la forme d'onde qui s'\u00e9tait dessin\u00e9e devant ses yeux avait quelque chose en plus, quelque chose d'inexprimable. Cette courbe montante et descendante paraissait avoir une \u00e2me. Il \u00e9tait certain que cette onde radio avait \u00e9t\u00e9 cr\u00e9\u00e9e par une forme d'intelligence.\n\nLe guetteur se pr\u00e9cipita devant un autre terminal et v\u00e9rifia le niveau de confiance du contenu du signal estim\u00e9 par l'ordinateur : Rouge 10 ! Aucune des ondes radio re\u00e7ues jusqu'\u00e0 ce jour n'avait d\u00e9pass\u00e9 un niveau de confiance de Bleu 2. Si un signal atteignait le niveau rouge, cela signifiait que la probabilit\u00e9 que la transmission contienne une information intelligente \u00e9tait sup\u00e9rieure \u00e0 quatre-vingt-dix pour cent. Un Rouge 10 indiquait que le message re\u00e7u utilisait un syst\u00e8me de code auto-interpr\u00e9tatif. Son ordinateur de d\u00e9chiffrage se mit en marche et tourna \u00e0 plein r\u00e9gime. La machine d\u00e9crypta le code et un message indiquant le succ\u00e8s de l'op\u00e9ration s'afficha sur l'\u00e9cran. Le guetteur ouvrit le fichier et, pour la premi\u00e8re fois, un \u00eatre trisolarien lut une information en provenance d'un autre monde de l'univers :\n\n_Salutations amicales aux habitants du monde qui re\u00e7oit ce message._\n\n _Gr\u00e2ce aux informations ci-dessous, vous obtiendrez des connaissances de base sur la civilisation terrienne. Apr\u00e8s une longue histoire de labeur et de cr\u00e9ativit\u00e9, les humains ont b\u00e2ti une civilisation splendide et vu na\u00eetre une riche vari\u00e9t\u00e9 de cultures. Nous avons aussi commenc\u00e9 \u00e0 comprendre les lois de la nature et celles du d\u00e9veloppement de nos soci\u00e9t\u00e9s. Nous sommes tr\u00e8s fiers de ce que nous avons accompli jusqu'ici._\n\n _Mais notre monde est encore tr\u00e8s imparfait, il existe de la haine, des pr\u00e9jug\u00e9s et des guerres. En raison de la contradiction entre forces productives et rapports de production, il subsiste une grave in\u00e9galit\u00e9 de r\u00e9partition des richesses. Une partie importante des humains vit dans la mis\u00e8re et la pauvret\u00e9._\n\n _Les soci\u00e9t\u00e9s humaines s'efforcent de r\u00e9soudre les difficult\u00e9s et les probl\u00e8mes qu'elles rencontrent et tentent de cr\u00e9er un avenir plus radieux pour la civilisation terrienne. Le pays \u00e0 l'origine de ce message est engag\u00e9 dans cet effort. Nous essayons de construire une soci\u00e9t\u00e9 id\u00e9ale dans laquelle le travail et la valeur de chaque \u00eatre humain seraient pleinement respect\u00e9s. Les besoins mat\u00e9riels et spirituels de chacun doivent \u00eatre pleinement satisfaits pour que la civilisation de la Terre soit encore plus parfaite._\n\n _C'est avec la meilleure des intentions que nous esp\u00e9rons pouvoir cr\u00e9er un contact avec d'autres civilisations. Nous esp\u00e9rons que nous pourrons travailler ensemble \u00e0 b\u00e2tir une vie meilleure dans l'immensit\u00e9 de l'univers._\n\nSaisi d'une excitation \u00e9tourdissante, le guetteur observa les formes d'ondes radio s'afficher sur l'\u00e9cran. L'information continuait \u00e0 abonder sans cesse de l'espace en p\u00e9n\u00e9trant par l'antenne. Gr\u00e2ce \u00e0 l'existence du code auto-interpr\u00e9tatif, l'ordinateur \u00e9tait capable de traduire et d'afficher en simultan\u00e9 le contenu des informations re\u00e7ues. Durant les deux heures trisolariennes qui suivirent, le guetteur apprit l'existence du monde terrien et de son soleil unique. Il apprit que la Terre vivait dans une \u00e8re r\u00e9guli\u00e8re \u00e9ternelle et que ce paradis au climat toujours stable avait donn\u00e9 naissance \u00e0 la civilisation humaine.\n\nLa transmission depuis le syst\u00e8me solaire se termina. L'ordinateur de d\u00e9chiffrage se mit \u00e0 tourner dans le vide. Le syst\u00e8me de surveillance du poste 1379 entendait \u00e0 nouveau le bruit de fond d'un univers d\u00e9sol\u00e9.\n\nMais le guetteur \u00e9tait certain de ne pas avoir r\u00eav\u00e9.\n\nIl savait que les milliers de postes de guet diss\u00e9min\u00e9s dans le monde avaient eux aussi re\u00e7u ce message que tous attendaient depuis des millions d'ann\u00e9es. Les deux cents civilisations ayant jadis ramp\u00e9 dans un tunnel obscur voyaient enfin un filet de lumi\u00e8re au bout de ce tunnel.\n\nLe guetteur relut une nouvelle fois le message venu de la Terre. Ses pens\u00e9es s'\u00e9gar\u00e8rent dans ces oc\u00e9ans bleus qui ne gelaient jamais, dans ces champs et ces for\u00eats vert \u00e9meraude o\u00f9 l'on sentait la chaleur des rayons du soleil et la caresse fra\u00eeche de la brise. Quel monde merveilleux ! Le paradis dont avaient r\u00eav\u00e9 deux cents civilisations existait r\u00e9ellement !\n\nSon excitation et son enthousiasme furent toutefois vite refroidis. Tout ce qui lui restait, c'\u00e9tait une sensation de perte et d'amertume. Durant ses longues heures de solitude, le guetteur s'\u00e9tait demand\u00e9 plus d'une fois : s'il recevait vraiment un jour une information provenant d'un monde extratrisolarien, sa vie serait-elle boulevers\u00e9e pour autant ? Ce paradis ne lui appartiendrait pas, et sa modeste existence de guetteur solitaire ne conna\u00eetrait pas le moindre changement.\n\nMais le paradis pouvait au moins lui appartenir en r\u00eave ! Et alors qu'il \u00e9tait perdu dans ses pens\u00e9es, le guetteur se mit en \u00e9tat de sommeil. Dans l'environnement inhospitalier de leur plan\u00e8te, les Trisolariens avaient \u00e9volu\u00e9 de mani\u00e8re \u00e0 pouvoir contr\u00f4ler leur entr\u00e9e dans le sommeil et leur sortie du sommeil : ils pouvaient s'endormir en quelques secondes seulement.\n\nMais il n'obtint pas du tout le r\u00eave souhait\u00e9. La Terre bleue lui apparut bien en songe, mais sous les feux nourris d'une immense flotte interstellaire. Les si beaux continents terriens commen\u00e7aient \u00e0 s'embraser, l'eau bleu marine des oc\u00e9ans se mettait \u00e0 bouillir et \u00e0 s'\u00e9vaporer \u00e0 cause de la chaleur...\n\nLorsqu'il s'\u00e9veilla de son r\u00eave, le guetteur vit passer \u00e0 travers sa fen\u00eatre un rayon froid appartenant \u00e0 la lune g\u00e9ante. Il regarda dehors la terre glac\u00e9e et repensa \u00e0 sa vie de solitude. Il avait d\u00e9j\u00e0 v\u00e9cu six cent mille heures trisolariennes ; l'esp\u00e9rance de vie moyenne des Trisolariens \u00e9tait de sept \u00e0 huit cent mille heures. Cependant, la plupart perdaient toute productivit\u00e9 dans leur travail bien plus t\u00f4t. Ils \u00e9taient alors forc\u00e9s de se d\u00e9shydrater et les fibres s\u00e8ches de leurs corps \u00e9taient br\u00fbl\u00e9es. La soci\u00e9t\u00e9 de Trisolaris n'avait que faire des oisifs.\n\nSoudain, le guetteur songea \u00e0 une autre possibilit\u00e9. Il n'\u00e9tait pas tout \u00e0 fait exact de dire que la r\u00e9ception du message extratrisolarien n'aurait aucune cons\u00e9quence sur sa vie. Apr\u00e8s confirmation de la cible, le monde de Trisolaris r\u00e9duirait certainement le nombre de ses postes de guet. Un poste comme le sien ferait sans doute partie des premiers ferm\u00e9s. Il se retrouverait alors sans emploi, et les talents d'un guetteur \u00e9tant limit\u00e9s \u2013 ils consistaient en de simples op\u00e9rations et protections routini\u00e8res \u2013, il lui serait difficile de trouver un autre emploi. S'il \u00e9tait incapable de trouver un nouveau travail dans les cinq mille heures trisolariennes \u00e0 venir, il serait \u00e0 son tour contraint de se d\u00e9shydrater, puis jet\u00e9 au feu.\n\nLa seule voie pour \u00e9chapper \u00e0 ce destin \u00e9tait de s'unir avec un membre du sexe oppos\u00e9. Quand ce genre d'\u00e9v\u00e9nement avait lieu, la substance organique des deux corps fusionnait en une seule entit\u00e9. Intervenait alors un processus complexe : les deux tiers de cette entit\u00e9 \u00e9taient utilis\u00e9s pour d\u00e9clencher une r\u00e9action biochimique entra\u00eenant le renouvellement int\u00e9gral des cellules du tiers restant, ce qui faisait na\u00eetre un \u00eatre nouveau. Cet \u00eatre donnait \u00e0 son tour naissance \u00e0 de nouvelles vies, de trois \u00e0 cinq, selon les cas : ses enfants. Ceux-ci h\u00e9ritaient d'une partie de la m\u00e9moire de leurs parents, dont ils devenaient en quelque sorte une prolongation, en commen\u00e7ant un nouveau cycle de vie. Mais au vu du statut social du guetteur, vivant seul et confin\u00e9 dans son espace de travail, et, qui plus est, \u00e0 son \u00e2ge, o\u00f9 pourrait-il trouver un membre de sexe oppos\u00e9 int\u00e9ress\u00e9 pour fusionner ?\n\nDans ses vieilles ann\u00e9es, le guetteur s'\u00e9tait pos\u00e9 des millions de fois la m\u00eame question : est-ce ainsi qu'est ma vie ? Et des millions de fois, il se r\u00e9pondait \u00e0 lui-m\u00eame : oui, c'est ainsi qu'est ta vie. Tout ce que tu poss\u00e8des, c'est cette solitude infinie dans l'espace \u00e9troit de ton poste de guet.\n\nIl ne pouvait pas perdre ce paradis lointain, m\u00eame s'il n'existait que dans ses r\u00eaves.\n\nLe guetteur savait qu'\u00e0 l'\u00e9chelle de l'univers il n'existait pas de mesure de r\u00e9f\u00e9rence suffisamment longue pour localiser la source d'ondes radio de basse fr\u00e9quence provenant de l'espace. On ne pouvait d\u00e9terminer que la direction depuis laquelle avait \u00e9t\u00e9 transmis le message, mais pas sa distance. La source pouvait avoir transmis un message \u00e0 haute fr\u00e9quence, mais se situer tr\u00e8s loin ou, au contraire, avoir transmis \u00e0 faible fr\u00e9quence, mais se trouver tr\u00e8s proche. Dans la direction d'o\u00f9 provenait le signal existaient des milliards d'\u00e9toiles scintillant devant un fond constitu\u00e9 d'un oc\u00e9an d'autres \u00e9toiles. Sans conna\u00eetre la distance de la source de transmission, il \u00e9tait impossible de conna\u00eetre ses coordonn\u00e9es exactes.\n\n _La distance. La clef, c'\u00e9tait la distance._\n\nEn r\u00e9alit\u00e9, il y avait un moyen simple de conna\u00eetre cette distance. Il fallait tout simplement renvoyer un message et que l'interlocuteur y r\u00e9ponde rapidement, afin qu'on puisse d\u00e9terminer sa distance gr\u00e2ce au temps mis pour faire l'aller-retour entre transmission et r\u00e9ception, et ce \u00e0 la vitesse de la lumi\u00e8re. Toute la question \u00e9tait de savoir si l'autre r\u00e9pondrait. Ou alors peut-\u00eatre mettrait-il un certain temps \u00e0 r\u00e9pondre, ne permettant pas aux Trisolariens de d\u00e9terminer le temps mis par le trajet des ondes radio.\n\nLa source de cette transmission avait cependant volontairement envoy\u00e9 cet appel vers l'univers, il y avait donc de grandes chances qu'ils r\u00e9pondent apr\u00e8s r\u00e9ception d'un message de Trisolaris. Le guetteur \u00e9tait s\u00fbr que le gouvernement trisolarien avait d\u00e9j\u00e0 donn\u00e9 l'ordre d'envoyer un message \u00e0 ce monde \u00e9loign\u00e9 pour les app\u00e2ter et leur donner envie de r\u00e9pondre. Peut-\u00eatre le message avait-il \u00e9t\u00e9 transmis, peut-\u00eatre que non. Dans le second cas, il avait enfin la chance d'allumer une flamm\u00e8che qui illuminerait le quotidien de sa vie.\n\nComme pour la base de C\u00f4te Rouge sur Terre, une partie des postes de guet du monde de Trisolaris envoyaient \u00e9galement des messages dans l'univers, dans l'espoir que des civilisations extratrisolariennes leur r\u00e9pondent. Les scientifiques trisolariens s'\u00e9taient tr\u00e8s t\u00f4t rendu compte que les \u00e9toiles pouvaient amplifier les ondes radio. Mais \u00e0 leur grand regret, la structure des trois soleils d'Alpha du Centaure \u00e9tait tr\u00e8s diff\u00e9rente du soleil terrien et il existait autour de la plan\u00e8te de Trisolaris une grande plasmasph\u00e8re \u2013 cette couche de plasma responsable de la transformation soudaine d'un soleil en \u00e9toile volante \u00e0 une certaine distance de la plan\u00e8te ou de l'apparition des soleils sous forme d'\u00e9toiles volantes. Cette plasmasph\u00e8re faisait l'effet d'un bouclier anti-ondes \u00e9lectromagn\u00e9tiques, de sorte que le seuil \u00e0 partir duquel les ondes radio pouvaient atteindre les miroirs solaires \u00e9tait bien plus \u00e9lev\u00e9 que celui du syst\u00e8me solaire. Cela rendait impossible l'utilisation du soleil comme antenne amplificatrice. Il leur \u00e9tait simplement possible de transmettre des signaux directement, depuis une antenne au sol. Dans le cas contraire, les humains auraient d\u00e9j\u00e0 eu depuis longtemps connaissance de l'existence de la civilisation de Trisolaris.\n\nLe guetteur se pr\u00e9cipita devant l'\u00e9cran de l'ordinateur d'ex\u00e9cution, il composa un court et simple message, puis ordonna \u00e0 la machine de traduire le message dans le m\u00eame langage que celui qu'elle avait re\u00e7u de la Terre. Il pointa ensuite l'antenne du poste de guet vers la direction de la source de transmission. Le bouton de transmission avait la forme d'un rectangle rouge. Les doigts du guetteur \u00e9taient suspendus deux centim\u00e8tres au-dessus.\n\nLe destin de Trisolaris tenait \u00e0 pr\u00e9sent \u00e0 ces doigts fins.\n\nSans la moindre h\u00e9sitation, le guetteur pressa le bouton. Ces ondes radio transmises \u00e0 haute fr\u00e9quence, porteuses d'un message tr\u00e8s court mais capable de sauver une autre civilisation, s'envol\u00e8rent dans l'obscurit\u00e9 de l'univers :\n\n_Ce monde a bien re\u00e7u votre message._\n\n _Ici, je suis un pacifiste. Vous avez de la chance que je sois le premier \u00e0 intercepter votre message. Je vous avertis : Ne r\u00e9pondez pas ! Ne r\u00e9pondez pas !! Ne r\u00e9pondez pas !!!_\n\n _Il existe des dizaines de millions d'\u00e9toiles dans votre direction. Tant que vous ne r\u00e9pondrez pas, ce monde n'aura aucun moyen de localiser la source de votre transmission._\n\n _Mais si vous r\u00e9pondez, vous d\u00e9voilerez votre position. Votre syst\u00e8me stellaire sera envahi, et votre monde occup\u00e9 !_\n\n _Ne r\u00e9pondez pas ! Ne r\u00e9pondez pas !! Ne r\u00e9pondez pas !!!_\n\n(Nous ignorons \u00e0 quoi ressemble le palais du chancelier de Trisolaris, mais nous avons la certitude qu'il est s\u00e9par\u00e9 du monde ext\u00e9rieur par des parois tr\u00e8s \u00e9paisses qui le prot\u00e8gent du climat rigoureux de la plan\u00e8te. La pyramide du jeu des _Trois Corps_ est une hypoth\u00e8se parmi d'autres. Il existe une th\u00e9orie disant que l'\u00e9difice est souterrain.)\n\nCinq heures trisolariennes plus t\u00f4t, le chancelier avait re\u00e7u un rapport sur le message extratrisolarien. Deux heures trisolariennes plus t\u00f4t, il avait re\u00e7u un rapport lui indiquant cette fois qu'un message d'avertissement \u00e9tait parti du poste de guet 1379.\n\nLe premier rapport ne l'avait pas rendu fou de joie ni le second empli de d\u00e9sespoir. Il n'\u00e9prouvait pas non plus de haine envers le guetteur du poste 1379. Ces \u00e9motions \u2013 tout comme d'ailleurs toutes les autres : peur, tristesse, bonheur, go\u00fbt pour la beaut\u00e9, etc. \u2013 \u00e9taient des sentiments que la civilisation trisolarienne s'effor\u00e7ait de bannir et de proscrire. Car ils affaiblissaient spirituellement les individus et la soci\u00e9t\u00e9, et n'\u00e9taient d'aucune utilit\u00e9 pour survivre dans leur terrible environnement. Les sentiments n\u00e9cessaires dans le monde de Trisolaris \u00e9taient le sang-froid et l'insensibilit\u00e9. L'histoire des deux cents cycles pass\u00e9s leur avait appris que les civilisations ayant mis ces deux inclinations au centre de leur spiritualit\u00e9 avaient \u00e9t\u00e9 celles dont le pouvoir de survie avait \u00e9t\u00e9 le plus puissant.\n\n\u2014 Pourquoi as-tu agi ainsi ? demanda le chancelier au guetteur du poste 1379 qu'il avait convoqu\u00e9 devant lui.\n\n\u2014 Pour ne pas avoir v\u00e9cu pour rien, r\u00e9pondit calmement le guetteur.\n\n\u2014 Tu as envoy\u00e9 un signal d'avertissement. Tu as peut-\u00eatre fait perdre \u00e0 la civilisation trisolarienne sa seule chance de survie.\n\n\u2014 Mais j'ai accord\u00e9 une chance \u00e0 la civilisation terrienne. Chancelier, permettez que je vous raconte une histoire : durant une \u00e8re chaotique d'il y a environ dix mille heures trisolariennes, le fourgon de ravitaillement a oubli\u00e9 d'approvisionner le poste de guet 1379 lors de sa tourn\u00e9e. Pendant les cent heures trisolariennes qui ont suivi, j'ai donc \u00e9t\u00e9 priv\u00e9 de nourriture. J'ai mang\u00e9 tout ce qu'il \u00e9tait possible d'avaler dans mon poste de guet, y compris mes propres v\u00eatements. Malgr\u00e9 tout, lorsque le fourgon est revenu, j'\u00e9tais d\u00e9j\u00e0 presque mort de faim. Mes sup\u00e9rieurs m'ont accord\u00e9 la plus grande p\u00e9riode de repos de toute ma carri\u00e8re. Je suis retourn\u00e9 en ville \u00e0 bord du fourgon de ravitaillement et, durant le chemin, j'ai \u00e9t\u00e9 pris d'un puissant d\u00e9sir de poss\u00e9der toute la nourriture du v\u00e9hicule.\n\nD\u00e8s que j'apercevais quelqu'un manger quelque chose \u00e0 bord, mon c\u0153ur s'emplissait de fureur, j'avais envie de le tuer ! Je ne cessais de voler des aliments, je les cachais dans les plis de mes v\u00eatements et sous mon si\u00e8ge. Les employ\u00e9s du fourgon ont trouv\u00e9 mon attitude fascinante et m'ont offert de la nourriture en cadeau. Lorsque je suis arriv\u00e9 en ville et que je suis descendu du fourgon, je portais sur le dos une quantit\u00e9 de nourriture bien sup\u00e9rieure \u00e0 mon propre poids.\n\nBien s\u00fbr, j'ai fini par retrouver mes esprits, mais cet intense d\u00e9sir de possession est rest\u00e9 grav\u00e9 en moi. La civilisation de Trisolaris se trouve elle aussi face \u00e0 une situation de crise pour sa survie. Son d\u00e9sir de poss\u00e9der un nouvel espace de vie est au moins aussi intense et insatiable que l'a \u00e9t\u00e9 mon d\u00e9sir de poss\u00e9der la nourriture. Elle ne pourra jamais partager ce monde avec les Terriens. Nous d\u00e9truirons sans h\u00e9siter la civilisation terrienne et entrerons en possession de tout le syst\u00e8me solaire... J'ai raison, n'est-ce pas ?\n\n\u2014 Oui. Mais il existe une autre raison de d\u00e9truire la civilisation terrienne : c'est une race belliqueuse qui peut se montrer tr\u00e8s dangereuse. Si nous essayions de coexister, ils apprendraient rapidement notre technologie et les deux civilisations finiraient par s'affronter. La politique que nous avons d\u00e9cid\u00e9e est celle-ci : apr\u00e8s la prise de contr\u00f4le du syst\u00e8me solaire et de la Terre par la flotte trisolarienne, nous n'interviendrons pas dans le cours de la vie des Terriens. Ils pourront continuer \u00e0 vivre comme avant, comme si les occupants trisolariens n'existaient pas. Mais il leur sera interdit de se reproduire. Maintenant, laisse-moi te poser une question : tu voudrais \u00eatre le sauveur de la Terre, mais ne te sens-tu pas responsable de ta propre civilisation ?\n\n\u2014 Je suis fatigu\u00e9 du monde de Trisolaris. Nous n'avons rien dans nos vies et nos esprits que la lutte pour la survie.\n\n\u2014 O\u00f9 est le mal ?\n\n\u2014 Il n'y a pas de mal. La survie est la pr\u00e9misse de tout le reste mais, chancelier, regardez nos vies : tout est d\u00e9di\u00e9 \u00e0 la survie de la civilisation. Afin de permettre \u00e0 la civilisation de subsister, le respect de l'individu est presque inexistant. Ceux qui ne peuvent plus travailler doivent mourir. La soci\u00e9t\u00e9 trisolarienne est extr\u00eamement totalitaire. La loi fonctionne selon deux principes uniques : coupable et non-coupable. Les coupables doivent p\u00e9rir et les non-coupables sont rel\u00e2ch\u00e9s. Mais le plus intol\u00e9rable \u00e0 mes yeux, c'est l'uniformit\u00e9 et l'ass\u00e8chement de notre vie spirituelle. Tout ce qui peut conduire \u00e0 une faiblesse de l'esprit est d\u00e9cr\u00e9t\u00e9 pernicieux. Nous n'avons ni litt\u00e9rature ni arts, nous ne cherchons ni la beaut\u00e9 ni l'extase. Nous n'avons m\u00eame pas de mots pour parler d'amour... Chancelier, cette vie-l\u00e0 a-t-elle un sens ?\n\n\u2014 Le genre de civilisation \u00e0 laquelle tu aspires a nagu\u00e8re exist\u00e9 sur Trisolaris. C'\u00e9taient des soci\u00e9t\u00e9s libres, d\u00e9mocratiques et elles ont laiss\u00e9 derri\u00e8re elles de riches h\u00e9ritages culturels. Seule une infime partie de cet h\u00e9ritage est autoris\u00e9e au public, la plupart de leurs accomplissements sont scell\u00e9s et interdits d'acc\u00e8s. Mais de tous les cycles de civilisations trisolariennes, ce sont ces civilisations qui ont \u00e9t\u00e9 les plus faibles et les plus \u00e9ph\u00e9m\u00e8res. Elles ont disparu d\u00e8s la premi\u00e8re petite catastrophe d'une \u00e8re chaotique. Observe une nouvelle fois cette civilisation terrienne que tu veux sauver, cette soci\u00e9t\u00e9 n\u00e9e et \u00e9lev\u00e9e sous une sph\u00e8re au printemps \u00e9ternel. Si on la transplantait sur Trisolaris, elle ne survivrait certainement pas plus d'un million d'heures trisolariennes.\n\n\u2014 C'est une fleur fragile, mais elle est d'une splendeur sans \u00e9gale. Dans la tranquillit\u00e9 de son paradis, elle sent le parfum de la libert\u00e9 et de la beaut\u00e9.\n\n\u2014 Si la civilisation trisolarienne entrait en possession de ce monde, nous pourrions nous aussi avoir une telle vie.\n\n\u2014 Chancelier, j'en doute. L'esprit des Trisolariens est aussi dur que du m\u00e9tal, il s'est solidifi\u00e9 dans chacune de nos cellules. Croyez-vous que cet esprit fondra un jour ? Je ne suis qu'un \u00eatre insignifiant, j'appartiens aux couches les plus basses de la soci\u00e9t\u00e9. Personne ne pr\u00eate attention \u00e0 moi, je vis seul, sans richesses, sans position, sans amour et sans espoir. En sauvant ce merveilleux monde lointain dont je suis tomb\u00e9 amoureux, je n'aurais au moins pas v\u00e9cu tout ce temps pour rien. Bien entendu, chancelier, cela m'a \u00e9galement donn\u00e9 la chance de vous rencontrer. Sans cet acte, un homme comme moi n'aurait jamais pu vous admirer autrement que par l'interm\u00e9diaire d'un \u00e9cran. Permettez-moi de vous dire que je suis honor\u00e9.\n\n\u2014 Tu es coupable, sans aucun doute. Tu es le plus grand criminel de toute l'histoire des cycles de civilisation de Trisolaris. Mais nous ferons une exception dans la loi : tu es libre.\n\n\u2014 Chancelier, comment cela est-il possible ?\n\n\u2014 La d\u00e9shydratation et le b\u00fbcher ne seraient pour toi qu'une m\u00e9diocre punition. Tu es \u00e2g\u00e9 et tu ne verras jamais la destruction finale de la civilisation terrestre. Mais je veux au moins \u00eatre s\u00fbr que tu saches que tu ne la sauveras pas. Je veux que tu vives jusqu'au jour o\u00f9 elle aura perdu tout espoir. Bien, tu peux te retirer \u00e0 pr\u00e9sent.\n\nUne fois le guetteur du poste 1379 parti, le chancelier appela le consul en charge des postes de guet. Le chancelier \u00e9vita aussi de s'\u00e9nerver contre lui, ce n'\u00e9tait qu'une affaire de routine.\n\n\u2014 Comment as-tu pu laisser un \u00eatre si faible et si d\u00e9moniaque devenir guetteur ?\n\n\u2014 Chancelier, nous employons plus de cent mille guetteurs. Il est difficile de d\u00e9pister ce genre d'individus, d'autant que 1379 n'avait jusqu'alors jamais commis la moindre effraction. Naturellement, c'est une faute grave, dont j'assume l'enti\u00e8re responsabilit\u00e9.\n\n\u2014 Combien d'autres individus partagent cette responsabilit\u00e9 dans le syst\u00e8me de surveillance ?\n\n\u2014 J'ai men\u00e9 une premi\u00e8re investigation. \u00c0 tous les niveaux, je dirais six mille personnes.\n\n\u2014 Ils sont tous coupables.\n\n\u2014 Oui.\n\n\u2014 Que l'on d\u00e9shydrate ces six mille personnes, puis qu'on les br\u00fble sur la place de la capitale. Quant \u00e0 toi, tu serviras \u00e0 faire d\u00e9marrer le feu.\n\n\u2014 Merci chancelier, cela apaisera notre conscience.\n\n\u2014 Mais avant cela, j'ai encore une question : quelle est la distance maximale que peut atteindre ce message d'avertissement ?\n\n\u2014 Le poste 1379 est un poste de guet de petite envergure, sa capacit\u00e9 de transmission n'est pas tr\u00e8s grande. Je dirais que la distance maximale est d'environ douze millions d'heures trisolariennes, soit mille deux cents ann\u00e9es-lumi\u00e8re.\n\n\u2014 C'est d\u00e9j\u00e0 suffisamment loin. As-tu un conseil \u00e0 nous donner sur la prochaine \u00e9tape \u00e0 mener ?\n\n\u2014 Et si nous envoyions \u00e0 ce monde un autre message, pour les s\u00e9duire et leur donner envie de r\u00e9pondre ?\n\n\u2014 Non, cela ne ferait qu'empirer les choses. Par chance, le contenu du message d'avertissement est tr\u00e8s court, esp\u00e9rons que les Terriens n'y feront pas attention ou qu'ils l'interpr\u00e9teront de travers. Bien... Tu peux te retirer \u00e0 pr\u00e9sent.\n\nUne fois le consul parti, le chancelier appela cette fois l'amiral de la flotte trisolarienne.\n\n\u2014 Combien de temps est-il n\u00e9cessaire pour pr\u00e9parer le d\u00e9part de la premi\u00e8re flotte ?\n\n\u2014 Chancelier, la flotte en est encore \u00e0 sa derni\u00e8re phase de construction. Nous avons besoin d'au moins soixante mille heures trisolariennes avant qu'elle soit pleinement op\u00e9rationnelle.\n\n\u2014 Je pr\u00e9senterai bient\u00f4t mon plan \u00e0 l'Assembl\u00e9e des consuls : d\u00e8s que la flotte sera pr\u00eate, elle s'envolera dans cette direction.\n\n\u2014 Chancelier, \u00e9tant donn\u00e9 la fr\u00e9quence de la transmission, nous ne pouvons pas m\u00eame \u00eatre s\u00fbrs avec exactitude de la direction de la source. Vous devez savoir que notre flotte sera incapable d'effectuer une exploration large dans cette direction. Si nous ignorons la distance de la cible, la flotte finira par chuter dans les abysses de l'espace.\n\n\u2014 Mais regarde nos trois soleils. La couche externe de l'un d'entre eux peut \u00e0 tout moment entrer dans un cycle de dilatation et avaler la derni\u00e8re plan\u00e8te du syst\u00e8me, la n\u00f4tre. Nous n'avons pas d'autre choix. Nous devons prendre ce risque.\n\n## 33\n\n## LES INTELLECTRONS\n\n_Quatre-vingt-cinq mille heures trisolariennes_\n\n_(environ 8,6 ann\u00e9es terrestres) plus tard._\n\nLe chancelier convoqua une r\u00e9union de crise de tous les consuls de Trisolaris. C'\u00e9tait tr\u00e8s inhabituel, il devait s'\u00eatre pass\u00e9 quelque chose de grave.\n\nVingt mille heures trisolariennes plus t\u00f4t, la flotte trisolarienne avait mis les voiles. Les vaisseaux avaient une id\u00e9e approximative de la direction dans laquelle se trouvait leur destination, mais ils ignoraient sa distance. Il se pouvait que cette cible soit \u00e0 des millions d'ann\u00e9es-lumi\u00e8re de l\u00e0, peut-\u00eatre m\u00eame \u00e0 l'autre extr\u00e9mit\u00e9 de la galaxie. Face \u00e0 cette mer infinie d'\u00e9toiles, les chances de succ\u00e8s de l'exp\u00e9dition \u00e9taient tr\u00e8s faibles.\n\nLa r\u00e9union des consuls se tenait sous le m\u00e9morial du Pendule. (\u00c0 la lecture de ce passage du rapport, Wang Miao ne put s'emp\u00eacher de s'imaginer mentalement le si\u00e8ge des Nations unies dans _Les Trois Corps_. Le m\u00e9morial du Pendule \u00e9tait l'un des rares \u00e9l\u00e9ments du jeu \u00e0 exister r\u00e9ellement sur Trisolaris.)\n\nLe choix du lieu pour cette r\u00e9union avait plong\u00e9 la plupart des consuls dans la perplexit\u00e9. L'\u00e8re chaotique n'\u00e9tait pas encore termin\u00e9e, un petit soleil venait tout juste de se lever \u00e0 l'horizon, mais il pouvait se recoucher \u00e0 tout moment. Il faisait monstrueusement froid, si bien que tous les participants \u00e9taient contraints de porter des combinaisons \u00e9lectrothermiques imperm\u00e9ables. L'\u00e9norme balancier m\u00e9tallique du pendule oscillait majestueusement, fendant l'air glacial, tandis que le petit soleil projetait de longues ombres sur le sol, comme si un g\u00e9ant \u00e0 la silhouette imposante \u00e9tait en train de le fouler \u00e0 grandes enjamb\u00e9es. Sous les yeux de la foule, le chancelier marcha jusqu'au socle du pendule, pressa un interrupteur rouge, puis il se tourna vers les consuls :\n\n\u2014 Je viens d'\u00e9teindre l'alimentation qui permettait le mouvement du pendule. Il s'arr\u00eatera lentement sous l'effet de la r\u00e9sistance de l'air.\n\n\u2014 Chancelier, pourquoi ? demanda un consul.\n\n\u2014 Nous connaissons tous le r\u00f4le historique du pendule. Il \u00e9tait utilis\u00e9 pour endormir Dieu. Mais nous savons d\u00e9sormais qu'il est pr\u00e9f\u00e9rable pour la civilisation trisolarienne que Dieu soit \u00e9veill\u00e9, car Il est maintenant avec nous.\n\nTout le monde garda le silence, m\u00e9ditant sur le sens des paroles du chancelier. Apr\u00e8s encore trois mouvements de balancier du pendule, quelqu'un demanda :\n\n\u2014 La Terre a-t-elle r\u00e9pondu ?\n\nLe chancelier hocha la t\u00eate :\n\n\u2014 Oui. J'ai re\u00e7u le rapport il y a une demi-heure. Une r\u00e9ponse au message d'avertissement.\n\n\u2014 Si vite ? Cela fait \u00e0 peine plus de quatre-vingt mille heures que l'avertissement a \u00e9t\u00e9 envoy\u00e9. Ce qui veut dire... ce qui veut dire...\n\n\u2014 Ce qui veut dire que la Terre se trouve \u00e0 une distance de quarante mille heures-lumi\u00e8re de Trisolaris.\n\n\u2014 Mais n'est-ce pas la distance qui nous s\u00e9pare de l'\u00e9toile la plus proche ?\n\n\u2014 Oui. C'est ce que je disais : Dieu est avec Trisolaris.\n\nUne euphorie s'empara de la foule, mais aucun ne pouvait l'exprimer et tous durent refouler cette \u00e9ruption de joie. Le chancelier savait qu'il \u00e9tait dangereux de laisser d\u00e9border ces \u00e9motions, c'est pourquoi il s'empressa d'asperger ce volcan avec un seau d'eau glac\u00e9e :\n\n\u2014 J'ai donn\u00e9 l'ordre \u00e0 la flotte de faire route vers cette \u00e9toile. Mais je ne suis pas aussi optimiste que vous le croyez. D'apr\u00e8s les informations dont nous disposons, notre flotte vogue en r\u00e9alit\u00e9 vers sa propre tombe.\n\nLes paroles du chancelier temp\u00e9r\u00e8rent aussit\u00f4t l'enthousiasme des consuls.\n\n\u2014 L'un d'entre vous comprend-il la raison de cette r\u00e9serve ?\n\n\u2014 Je comprends, dit le consul de la science. Nous avons \u00e9tudi\u00e9 en d\u00e9tail les premiers messages envoy\u00e9s par la Terre. Une section a particuli\u00e8rement retenu notre attention : son histoire. Voil\u00e0 quels sont les faits : les humains ont mis quelques dizaines de milliers d'ann\u00e9es terrestres pour passer de l'\u00e2ge des chasseurs-cueilleurs \u00e0 l'\u00e2ge de l'agriculture ; quelques milliers d'ann\u00e9es, de l'\u00e2ge de l'agriculture \u00e0 celui de l'industrie ; et seulement deux cents ann\u00e9es terrestres pour passer de l'\u00e2ge industriel \u00e0 l'\u00e2ge de l'atome. Enfin, ils ont atteint l'\u00e2ge de l'information en seulement quelques dizaines d'ann\u00e9es terrestres. Cette civilisation poss\u00e8de le terrifiant pouvoir d'acc\u00e9l\u00e9rer son \u00e9volution ! Sur Trisolaris, aucune de nos deux cents civilisations \u2013 y compris la n\u00f4tre \u2013 n'a jamais connu un d\u00e9veloppement aussi rapide. Tous les progr\u00e8s scientifiques et \u00e9conomiques de la civilisation trisolarienne se sont faits \u00e0 un rythme constant, voire en ralentissant peu \u00e0 peu. Dans notre monde, chaque nouvel \u00e2ge technologique n\u00e9cessite de s'inscrire sur un temps d'\u00e9volution long.\n\nLe chancelier acquies\u00e7a :\n\n\u2014 La r\u00e9alit\u00e9 est que, dans quatre millions et cinq cent mille heures, lorsque la flotte trisolarienne atteindra le syst\u00e8me solaire, le niveau de technologie de la civilisation terrienne aura largement surpass\u00e9 le n\u00f4tre ! Durant son long p\u00e9riple, notre flotte devra traverser deux ceintures de poussi\u00e8re interstellaire. Nous aurons s\u00fbrement de grandes pertes \u00e0 d\u00e9plorer et nous craignons que seule la moiti\u00e9 de la flotte ne parvienne \u00e0 rejoindre le syst\u00e8me solaire saine et sauve. Le reste aura sombr\u00e9 durant la travers\u00e9e. La flotte sera alors \u00e0 la merci des ripostes terriennes. Nous ne partons pas en conqu\u00eate, nous naviguons vers notre propre mort !\n\n\u2014 Si cela est vrai, chancelier, il y a encore plus effrayant..., commen\u00e7a le consul de la guerre.\n\n\u2014 Oui. C'est simple \u00e0 imaginer. Nous avons r\u00e9v\u00e9l\u00e9 notre position. Pour parer \u00e0 des menaces futures, la flotte terrienne contre-attaquera. Et peut-\u00eatre qu'avant m\u00eame que notre plan\u00e8te soit engloutie par un soleil dilat\u00e9, les Terriens auront d\u00e9j\u00e0 r\u00e9duit la civilisation trisolarienne en cendres.\n\nL'avenir si brillant qui s'\u00e9tait ouvert quelques instants plus t\u00f4t s'\u00e9tait tout \u00e0 coup assombri, plongeant l'assembl\u00e9e dans un long silence.\n\n\u2014 Voici ce que nous devons faire : freiner le progr\u00e8s de la science sur Terre. D\u00e8s la r\u00e9ception de la premi\u00e8re s\u00e9rie de messages terriens, nous avons imm\u00e9diatement commenc\u00e9 \u00e0 \u00e9laborer des plans allant dans ce sens. Les conditions sont maintenant favorables \u00e0 la mise en \u0153uvre de ces plans. La r\u00e9ponse que nous venons de recevoir a \u00e9t\u00e9 envoy\u00e9e par un tra\u00eetre terrien. Nous avons des raisons de croire qu'il existe dans la civilisation terrienne des forces ali\u00e9n\u00e9es qu'il nous faut exploiter.\n\n\u2014 Chancelier, c'est plus facile \u00e0 dire qu'\u00e0 faire. Nos communications avec la Terre ne tiennent qu'\u00e0 un fil. Il est n\u00e9cessaire d'attendre quatre-vingt mille heures avant que l'un r\u00e9ponde \u00e0 la question de l'autre.\n\n\u2014 Voyons plus loin. Lorsque le monde terrien apprendra l'existence d'une autre civilisation dans l'univers, l'impact sur la soci\u00e9t\u00e9 dans son ensemble sera \u00e9norme et laissera des marques profondes. Nous pensons que les forces ali\u00e9n\u00e9es de la Terre s'uniront et cro\u00eetront au fil du temps.\n\n\u2014 Que seront-elles en mesure de faire ? Des actes de sabotage ?\n\n\u2014 \u00c9tant donn\u00e9 les quarante mille heures qui nous s\u00e9parent, les strat\u00e9gies de guerre ou les actes terroristes traditionnels n'auraient que des cons\u00e9quences limit\u00e9es, car les humains pourraient facilement s'en remettre. Sur un temps long comme celui-ci, il est n\u00e9cessaire d'enrayer efficacement le progr\u00e8s de la civilisation humaine, et de les d\u00e9sarmer pour de bon. Pour cela, il n'existe qu'une seule solution : tuer leur science. J'invite le consul de la science \u00e0 vous pr\u00e9senter dans les grandes lignes les trois plans que nous avons choisi d'adopter.\n\n\u2014 Le premier de ces trois plans a pour nom de code \"Feuillage\", expliqua le consul de la science. Nous exploiterons les contre-effets n\u00e9fastes de la technologie pour susciter au sein de la population humaine une peur et une haine de la science. En effet, sur Terre aussi, l'\u00e9volution technologique s'accompagne de graves probl\u00e8mes environnementaux. Le deuxi\u00e8me plan, \"Miracle\", consiste \u00e0 cr\u00e9er sur Terre des miracles qui appara\u00eetront surnaturels aux Terriens. \u00c0 travers cette s\u00e9rie de prodiges, l'objectif est de fabriquer de toutes pi\u00e8ces un univers dont la science humaine sera incapable d'expliquer la logique. Lorsque ces illusions se seront bien enracin\u00e9es, la civilisation trisolarienne y deviendra peut-\u00eatre l'objet d'un d\u00e9vouement religieux. Et alors, les m\u00e9thodes de pens\u00e9e non scientifiques prendront le pas sur les r\u00e9flexions de la science, entra\u00eenant l'implosion de tout le syst\u00e8me de pens\u00e9e scientifique \u00e9tabli.\n\n\u2014 Comment cr\u00e9erons-nous des miracles ?\n\n\u2014 La clef de la r\u00e9ussite de ces miracles est que les Terriens ne soient pas en mesure de les comprendre rationnellement. Il nous sera peut-\u00eatre n\u00e9cessaire de transf\u00e9rer certaines de nos technologies avanc\u00e9es aux forces humaines ali\u00e9n\u00e9es.\n\n\u2014 C'est bien trop dangereux. Et si ces technologies tombaient entre de mauvaises mains ? C'est jouer avec le feu !\n\n\u2014 Naturellement, une \u00e9tude pr\u00e9liminaire sera men\u00e9e pour \u00e9valuer le niveau de technologie qu'il sera possible de transf\u00e9rer...\n\n\u2014 Que le consul de la science s'arr\u00eate un instant ! dit le consul de la guerre, en se levant devant ses pairs. Chancelier, \u00e0 mon humble avis, ces deux plans ne seront d'une efficacit\u00e9 que tr\u00e8s limit\u00e9e si nous souhaitons causer la mort de la science humaine.\n\n\u2014 Certes, mais c'est toujours mieux que rien, approuva le consul de la science, avant m\u00eame que le chancelier ait pu r\u00e9pondre.\n\n\u2014 Si peu..., laissa tomber le consul de la guerre sur un ton d\u00e9daigneux.\n\n\u2014 Je suis d'accord avec vous, dit le chancelier. Les plans \"Feuillage\" et \"Miracle\" ne provoqueront que quelques interf\u00e9rences minimes dans le d\u00e9veloppement de la science terrienne.\n\nPuis il se tourna vers ses autres coll\u00e8gues :\n\n\u2014 Ce dont nous avons besoin en priorit\u00e9, c'est d'une action forte qui \u00e9touffera toute la science sur la plan\u00e8te Terre et la verrouillera \u00e0 son niveau actuel. Concentrons-nous maintenant sur l'\u00e9l\u00e9ment clef de cette action : le d\u00e9veloppement global de la technologie repose sur le progr\u00e8s des sciences fondamentales, et la base des sciences fondamentales est l'exploration de la structure profonde de la mati\u00e8re. Sans avanc\u00e9e dans ce domaine, ni la science ni la technologie ne conna\u00eetront de bond d\u00e9cisif. Ce n'est d'ailleurs pas seulement vrai pour la civilisation sur Terre, cela s'applique \u00e0 toutes les autres cibles que la civilisation trisolarienne souhaiterait un jour conqu\u00e9rir. Avant m\u00eame d'avoir re\u00e7u le premier message terrien, nos scientifiques avaient d\u00e9j\u00e0 travaill\u00e9 sur cette question. Nous nous sommes r\u00e9cemment content\u00e9s d'acc\u00e9l\u00e9rer ces efforts. \u00c0 pr\u00e9sent, levez les yeux. Que voyez-vous ?\n\nLe chancelier pointa le ciel. Les consuls virent dans l'espace un anneau sur lequel le soleil faisait briller des reflets m\u00e9talliques.\n\n\u2014 N'est-ce pas l'atelier dans lequel est en train d'\u00eatre fabriqu\u00e9e la deuxi\u00e8me flotte spatiale ?\n\n\u2014 Non. C'est un acc\u00e9l\u00e9rateur de particules g\u00e9ant en construction. Nous avons annul\u00e9 le projet de construction d'une deuxi\u00e8me flotte et allou\u00e9 son budget au projet Intellectron.\n\n\u2014 Le projet Intellectron ?\n\n\u2014 Oui. Ce plan a \u00e9t\u00e9 tenu secret, c'est pourquoi la plupart d'entre vous n'ont pas \u00e9t\u00e9 mis au courant. Le consul de la science va vous en pr\u00e9senter la teneur.\n\n\u2014 J'ai eu vent de ce projet, mais je n'imaginais pas qu'il \u00e9tait aussi avanc\u00e9, dit le consul de l'industrie.\n\n\u2014 J'en avais vaguement entendu parler, mais je pensais que c'\u00e9tait une l\u00e9gende, ajouta le consul de la culture et de l'\u00e9ducation.\n\n\u2014 Pour le r\u00e9sumer en quelques mots, le projet Intellectron consiste \u00e0 transformer un proton en un ordinateur super-intelligent, commen\u00e7a le consul de la science.\n\n\u2014 C'est un fantasme bien connu de la science, fit remarquer le consul de l'agriculture. Je suis \u00e9tonn\u00e9 d'apprendre si brusquement qu'il est sur le point de se r\u00e9aliser. Je sais que nos physiciens sont maintenant en mesure de manipuler neuf des onze dimensions du monde microscopique, mais il est difficile d'imaginer qu'ils puissent ins\u00e9rer une pincette dans un proton pour y graver des circuits int\u00e9gr\u00e9s !\n\n\u2014 C'est bien entendu impossible. Graver des microcircuits int\u00e9gr\u00e9s ne peut se faire qu'\u00e0 une macro-\u00e9chelle, et uniquement sur un plan macroscopique \u00e0 deux dimensions. Pour cela, nous devons d\u00e9ployer le proton en deux dimensions.\n\n\u2014 D\u00e9ployer une structure de neuf dimensions en deux dimensions ? Mais de quelle taille serait sa surface ?\n\n\u2014 Une surface gigantesque, comme vous allez le voir, dit le consul de la science dans un sourire.\n\nLe temps fila. Six cent mille heures trisolariennes s'\u00e9coul\u00e8rent. Deux cent mille heures apr\u00e8s que le grand acc\u00e9l\u00e9rateur spatial de particules fut achev\u00e9, le proton allait \u00eatre d\u00e9ploy\u00e9 sur une orbite synchrone autour de Trisolaris.\n\nC'\u00e9tait une journ\u00e9e ensoleill\u00e9e d'une \u00e8re r\u00e9guli\u00e8re, le ciel \u00e9tait particuli\u00e8rement clair. Comme quatre-vingt-trois mille heures plus t\u00f4t, lors du d\u00e9part de la premi\u00e8re flotte, toute la population de Trisolaris se trouvait sur la plaine. Elle regardait, les yeux en l'air, l'immense anneau spatial. Le chancelier et les consuls se r\u00e9unirent une nouvelle fois sous le m\u00e9morial du Pendule. Celui-ci avait cess\u00e9 de se balancer depuis longtemps, et son poids ressemblait \u00e0 un grand bloc de roche fermement maintenu entre deux supports de fixation, comme un symbole de stabilit\u00e9. Il \u00e9tait difficile de croire qu'il avait un jour \u00e9t\u00e9 en mouvement.\n\nLe consul de la science donna l'ordre de commencer le d\u00e9ploiement en deux dimensions. Dans l'espace, trois cubes planaient autour de l'anneau : les trois g\u00e9n\u00e9rateurs \u00e9lectriques fournissant l'\u00e9nergie \u00e0 l'acc\u00e9l\u00e9rateur. Leurs dissipateurs thermiques en forme d'ailes se mirent peu \u00e0 peu \u00e0 briller d'une lueur rouge\u00e2tre. Le consul de la science informa le chancelier que le processus de d\u00e9ploiement avait d\u00e9but\u00e9. La foule scrutait avec anxi\u00e9t\u00e9 l'acc\u00e9l\u00e9rateur spatial, mais rien ne semblait se passer.\n\nUn dixi\u00e8me d'heure trisolarienne plus tard, le consul de la science mit un casque sur ses oreilles, puis il dit :\n\n\u2014 Chancelier, veuillez m'excuser, je crains que le d\u00e9ploiement n'ait \u00e9chou\u00e9. Le proton a \u00e9t\u00e9 r\u00e9duit \u00e0 une dimension de trop. Il est maintenant unidimensionnel.\n\n\u2014 Unidimensionnel ? C'est une ligne ?\n\n\u2014 Oui. Une ligne filiforme et infinie. Th\u00e9oriquement, sa longueur doit \u00eatre de 1,5 ann\u00e9e-lumi\u00e8re.\n\n\u2014 Quelle gabegie..., grommela le consul de la guerre. Nous avons d\u00e9pens\u00e9 le budget d'une flotte spatiale pour... \u00e7a ?\n\n\u2014 Dans une exp\u00e9rience scientifique, un processus d'adaptation est toujours n\u00e9cessaire. Apr\u00e8s tout, ce n'est que notre premi\u00e8re tentative de d\u00e9ploiement.\n\nLa foule se dispersa, d\u00e9\u00e7ue. Cependant, l'exp\u00e9rience n'\u00e9tait pas termin\u00e9e. On imaginait au d\u00e9but que le proton unidimensionnel tournerait \u00e0 jamais en orbite synchrone autour de la plan\u00e8te, or la r\u00e9sistance g\u00e9n\u00e9r\u00e9e par une \u00e9ruption solaire avait ralenti son mouvement et des morceaux de la ligne chut\u00e8rent dans l'atmosph\u00e8re. Six heures trisolariennes plus tard, les Trisolariens qui se trouvaient \u00e0 l'ext\u00e9rieur remarqu\u00e8rent d'\u00e9tranges bandeaux de lumi\u00e8re autour de l'anneau, des \u00e9clairs filamenteux et fugaces qui disparaissaient aussi vite qu'ils \u00e9taient apparus.\n\nIls apprirent rapidement dans les journaux que c'\u00e9tait le proton unidimensionnel qui, sous l'effet de la gravit\u00e9, \u00e9tait tomb\u00e9 sur la surface. La ligne avait beau \u00eatre infiniment fine, elle produisait un champ d'\u00e9nergie capable de r\u00e9fl\u00e9chir la lumi\u00e8re visible, et pouvait donc \u00eatre per\u00e7ue. C'\u00e9tait la premi\u00e8re fois qu'il \u00e9tait possible de voir \u00e0 l'\u0153il nu une entit\u00e9 de mati\u00e8re n'\u00e9tant pas constitu\u00e9e d'atomes \u2013 les minuscules parties d'un seul et m\u00eame proton.\n\n\u2014 Ces petites choses m'agacent, dit le chancelier en agitant sa main devant son visage.\n\nLe consul de la science et lui se tenaient debout sur les larges marches du palais gouvernemental.\n\n\u2014 J'ai sans cesse une impression de picotements sur le visage.\n\n\u2014 Chancelier, c'est l'effet de votre imagination car la somme de toutes ces cordes a la masse d'un seul proton. Celles-ci n'ont absolument aucun effet sur le monde macroscopique. Elles sont tout \u00e0 fait inoffensives, c'est comme si elles n'existaient pas.\n\nCependant, les morceaux de proton qui tombaient du ciel se faisaient de plus en plus denses. Non loin de la surface, l'air \u00e9tait gorg\u00e9 de minuscules cordelettes lumineuses, si bien que le soleil et les \u00e9toiles paraissaient entour\u00e9s de cercles en velours argent\u00e9. Les morceaux de proton unidimensionnel recouvraient le corps des Trisolariens qui se trouvaient \u00e0 l'ext\u00e9rieur, et ceux-ci tra\u00eenaient en marchant de petites lumi\u00e8res derri\u00e8re eux. Quand ils retournaient chez eux, les cordes se mettaient \u00e0 scintiller sous les sources de lumi\u00e8re artificielle. \u00c0 peine bougeaient-ils que le reflet des cordes r\u00e9v\u00e9lait autour d'eux les courants d'air perturb\u00e9s par leurs mouvements. Bien que ces cordes soient sans danger et ne puissent \u00eatre d\u00e9tect\u00e9es qu'\u00e0 la lumi\u00e8re visible, elles suscit\u00e8rent rapidement l'agacement de la population.\n\nLa pluie de cordes unidimensionnelles dura plus de vingt heures trisolariennes avant de s'arr\u00eater, mais pas parce qu'elles \u00e9taient toutes tomb\u00e9es sur le sol. Bien que leur masse soit incroyablement faible, elles en avaient tout de m\u00eame une et leur acc\u00e9l\u00e9ration sous la force de gravit\u00e9 \u00e9tait la m\u00eame que pour des objets normaux. Cependant, en entrant dans l'atmosph\u00e8re, elles se retrouvaient \u00e0 la merci des courants d'air et ne tombaient plus sur le sol. Apr\u00e8s avoir \u00e9t\u00e9 d\u00e9ploy\u00e9es en une dimension, les interactions fortes \u00e0 l'int\u00e9rieur du proton s'\u00e9taient grandement affaiblies, fragilisant les cordes et les brisant progressivement en petits fragments, si bien que la lumi\u00e8re qu'elles refl\u00e9taient n'\u00e9tait plus visible. La population avait l'impression qu'elles avaient disparu, mais la poussi\u00e8re de cordes unidimensionnelles flotterait dans l'air de Trisolaris pour l'\u00e9ternit\u00e9.\n\nCinquante heures trisolariennes plus tard eut lieu une deuxi\u00e8me tentative de d\u00e9ploiement du proton en deux dimensions. Cette fois, la foule rassembl\u00e9e sur la plaine distingua quelque chose. Lorsque les dissipateurs thermiques commenc\u00e8rent \u00e0 rougir, surgirent soudain pr\u00e8s de l'acc\u00e9l\u00e9rateur des objets gigantesques \u00e0 l'apparence de formes g\u00e9om\u00e9triques tr\u00e8s r\u00e9guli\u00e8res : sph\u00e8res, t\u00e9tra\u00e8dres, cubes, c\u00f4nes, etc. Leurs surfaces poss\u00e9daient une palette complexe de couleurs. Toutefois, en les observant bien, on pouvait remarquer que celles-ci \u00e9taient en fait incolores, mais qu'elles \u00e9taient en revanche totalement r\u00e9flexives. Ce que l'on y voyait \u00e9taient les images d\u00e9form\u00e9es du monde de Trisolaris telles que les refl\u00e9taient les formes g\u00e9om\u00e9triques.\n\n\u2014 Avez-vous r\u00e9ussi cette fois ? demanda le chancelier. Le proton a-t-il \u00e9t\u00e9 d\u00e9ploy\u00e9 en deux dimensions ?\n\n\u2014 Malheureusement, chancelier, c'est encore un \u00e9chec. Nous venons de recevoir un rapport du centre de contr\u00f4le de l'acc\u00e9l\u00e9rateur. Il y a cette fois une dimension en moins. Le proton a \u00e9t\u00e9 d\u00e9ploy\u00e9 en trois dimensions.\n\nLes gigantesques formes g\u00e9om\u00e9triques r\u00e9flexives continuaient \u00e0 jaillir de l'acc\u00e9l\u00e9rateur \u00e0 un rythme rapide et s'en \u00e9loignaient en planant. Les formes \u00e9taient de plus en plus vari\u00e9es, on voyait maintenait des tores et des croix en trois dimensions, et m\u00eame un objet qui ressemblait \u00e0 un ruban de M\u00f6bius. Environ une heure et demie plus tard, les formes avaient compl\u00e8tement envahi le ciel, comme si un enfant de g\u00e9ant avait renvers\u00e9 des blocs de construction sur la vo\u00fbte c\u00e9leste. Les reflets du soleil sur les formes g\u00e9om\u00e9triques doubl\u00e8rent l'intensit\u00e9 lumineuse de la surface et elles se mirent \u00e0 scintiller. L'ombre colossale du pendule apparaissait et disparaissait sans cesse, clignotant un coup \u00e0 gauche, un coup \u00e0 droite.\n\nPuis, tous les objets commenc\u00e8rent \u00e0 changer de formes, celles-ci perdirent de leur r\u00e9gularit\u00e9, on aurait dit qu'elles avaient fondu au soleil. Ces mutations devinrent de plus en plus soudaines, et les formes de plus en plus sophistiqu\u00e9es. Ces objets c\u00e9lestes ne faisaient d\u00e9sormais plus penser \u00e0 des blocs de construction, mais plut\u00f4t aux membres d\u00e9pec\u00e9s et aux visc\u00e8res \u00e9ventr\u00e9s d'un g\u00e9ant. En raison de l'irr\u00e9gularit\u00e9 des formes, la lumi\u00e8re qu'elles projetaient maintenant sur le sol s'\u00e9tait adoucie, mais les couleurs \u00e0 leur surface \u00e9taient devenues encore plus \u00e9tranges et impr\u00e9visibles.\n\nDans ce ciel chaotique gorg\u00e9 de formes g\u00e9om\u00e9triques tridimensionnelles, les observateurs au sol firent particuli\u00e8rement attention \u00e0 certaines d'entre elles : tout d'abord parce qu'elles \u00e9taient tr\u00e8s similaires puis, apr\u00e8s un examen plus minutieux, on comprit ce qu'elles repr\u00e9sentaient. Un frisson d'horreur parcourut le monde de Trisolaris.\n\nC'\u00e9taient des yeux.\n\n(Nous ignorons \u00e0 quoi ressemblent les yeux des Trisolariens, mais nous pouvons \u00eatre s\u00fbrs d'une chose : n'importe quelle entit\u00e9 intelligente est sensible \u00e0 la repr\u00e9sentation des yeux.)\n\nLe chancelier \u00e9tait l'un des seuls \u00e0 garder v\u00e9ritablement son calme. Il demanda au consul de la science :\n\n\u2014 Quel degr\u00e9 de complexit\u00e9 peut atteindre la structure interne d'une particule subatomique ?\n\n\u2014 Tout d\u00e9pend le nombre de dimensions de notre perspective. Si on observe une particule subatomique avec une perspective unidimensionnelle \u2013 ce qui est le cas de la plupart des gens \u2013, ce n'est qu'un point ; \u00e0 une perspective bidimensionnelle ou tridimensionnelle, la particule commence \u00e0 d\u00e9voiler sa structure interne ; \u00e0 une perspective quadridimensionnelle, la particule subatomique est d\u00e9j\u00e0 un monde immense.\n\n\u2014 Immense ? Incroyable qu'on puisse utiliser un tel mot pour qualifier une particule aussi petite que le proton, dit le chancelier.\n\nLe consul de la science ignora le chancelier et continua \u00e0 parler, comme pour lui-m\u00eame :\n\n\u2014 Si nous passons \u00e0 une dimension plus haute, le degr\u00e9 de complexit\u00e9 et le nombre de structures internes dans la particule augmenteront consid\u00e9rablement. Cette comparaison est tr\u00e8s impr\u00e9cise, mais voici une id\u00e9e de ces diff\u00e9rences d'\u00e9chelles : une particule observ\u00e9e \u00e0 une perspective en sept dimensions poss\u00e8de la complexit\u00e9 du syst\u00e8me stellaire trisolarien en trois dimensions ; \u00e0 une perspective en huit dimensions, la particule est aussi vaste que la Voie lact\u00e9e ; \u00e0 neuf dimensions, la complexit\u00e9 et le nombre de structures internes sont presque \u00e9quivalents \u00e0 l'univers tout entier. Et pour ce qui est des dimensions sup\u00e9rieures, nos sciences physiques ne sont pas encore en mesure de les appr\u00e9hender, et nous ne pouvons pas encore mettre de mots sur leur complexit\u00e9.\n\nLe chancelier pointa les yeux immenses qui flottaient dans le ciel :\n\n\u2014 Ce spectacle ne prouve-t-il pas, une fois d\u00e9ploy\u00e9, que l'univers microscopique d'un proton abrite une vie intelligente ?\n\n\u2014 Notre conception de la vie dans le monde macroscopique n'est peut-\u00eatre pas applicable au monde microscopique. Pour \u00eatre plus exact, nous pouvons simplement dire qu'il existe de l'intelligence ou de la sagesse dans cet univers. C'est peut-\u00eatre ce que les scientifiques avaient pr\u00e9dit depuis longtemps : il serait \u00e9tonnant qu'aucune forme d'intelligence n'ait \u00e9merg\u00e9 dans un univers aussi vaste et complexe.\n\n\u2014 Pourquoi des yeux qui nous regardent ?\n\nLe chancelier leva la t\u00eate, les yeux \u00e9taient de superbes sculptures, belles et r\u00e9alistes \u00e0 la fois. Ils observaient la plan\u00e8te d'en dessous avec un regard myst\u00e9rieux.\n\n\u2014 Peut-\u00eatre simplement pour nous indiquer leur existence.\n\n\u2014 Ces choses vont-elles nous tomber dessus ?\n\n\u2014 Non, rassurez-vous, chancelier. Et m\u00eame si elles venaient \u00e0 tomber, tout comme les cordes unidimensionnelles, la masse globale de ces \u00e9normes objets est la m\u00eame que celle d'un proton. Leur chute n'aurait aucun impact sur nous. Il faudra simplement que la population s'habitue \u00e0 cet \u00e9trange paysage.\n\nMais cette fois-ci, le consul de la science se trompait.\n\nLa foule avait remarqu\u00e9 que, parmi tous les objets en trois dimensions qui emplissaient le ciel, ceux sous forme d'yeux bougeaient \u00e0 une vitesse nettement plus rapide que les autres, et ils convergeaient de plus vers un m\u00eame point. Tr\u00e8s vite, deux yeux se t\u00e9lescop\u00e8rent et ne firent plus qu'un. La fusion des deux formes avait toujours celle d'un \u0153il, mais son volume avait augment\u00e9. D'autres yeux fusionn\u00e8rent de la m\u00eame mani\u00e8re avec cet \u0153il, augmentant sans cesse son volume. Enfin, tous les yeux s'agglutin\u00e8rent pour n'en former qu'un seul, si \u00e9norme qu'il semblait repr\u00e9senter l'univers lui-m\u00eame en train d'observer Trisolaris. Sa pupille \u00e9tait claire et lumineuse, et refl\u00e9tait en son centre un soleil, tandis que sur sa vaste corn\u00e9e tombait un d\u00e9luge de couleurs diff\u00e9rentes. Bient\u00f4t, les contours de la pupille commenc\u00e8rent \u00e0 s'effacer, puis finirent par dispara\u00eetre. L'immense \u0153il devint alors un \u0153il aveugle et sans pupille. Il se mit \u00e0 changer d'aspect jusqu'\u00e0 perdre l'apparence d'un \u0153il et devenir un cercle parfait. Quand ce cercle commen\u00e7a \u00e0 bouger, la foule d\u00e9couvrit que celui-ci n'\u00e9tait pas plat, mais parabolique, comme un morceau coup\u00e9 dans une \u00e9norme sph\u00e8re.\n\nLe consul de la guerre regarda l'objet se d\u00e9placer lentement. Puis une lumi\u00e8re traversa son esprit et il s'\u00e9cria :\n\n\u2014 Chancelier, et vous tous, vite, rentrez dans vos abris souterrains !\n\nIl pointa le ciel en disant :\n\n\u2014 C'est...\n\n\u2014... un miroir, conclut calmement le chancelier. Donnez l'ordre \u00e0 l'unit\u00e9 de d\u00e9fense a\u00e9rienne de le d\u00e9truire. Nous resterons ici pour l'observer.\n\nLe miroir parabolique avait commenc\u00e9 \u00e0 refl\u00e9ter l'\u00e9clat des rayons solaires sur la surface de Trisolaris. Au d\u00e9but, la surface sur laquelle \u00e9tait projet\u00e9e la lumi\u00e8re \u00e9tait tr\u00e8s large et la chaleur qu'elle g\u00e9n\u00e9rait n'\u00e9tait pas encore mortelle. Mais cette tache lumineuse se d\u00e9pla\u00e7a, \u00e0 la recherche de sa proie. Le miroir parabolique avait visiblement choisi la capitale, la plus grande ville de Trisolaris, car la tache de lumi\u00e8re s'en approcha et, bient\u00f4t, l'engloutit sous ses rayons. Sous le m\u00e9morial du Pendule, la foule ne vit dans le ciel qu'une effusion de lumi\u00e8re, si intense qu'elle semblait tout avaler dans l'atmosph\u00e8re. Ils furent submerg\u00e9s par une vague d'une extr\u00eame chaleur. La tache de lumi\u00e8re qui enveloppait la ville diminua de volume : le miroir parabolique concentrait encore davantage les rayons du soleil ; dans l'espace, la luminosit\u00e9 se renfor\u00e7ait et emp\u00eachait la foule de lever la t\u00eate. Ceux qui se tenaient sous cette tache sentirent la temp\u00e9rature s'\u00e9lever. Au moment m\u00eame o\u00f9 la chaleur devenait insupportable, l'extr\u00e9mit\u00e9 de la tache de lumi\u00e8re passa au-dessus du m\u00e9morial du Pendule, et tout s'obscurcit brusquement. Les gens mirent un certain temps avant de r\u00e9adapter leurs yeux \u00e0 la luminosit\u00e9 normale. En levant la t\u00eate, ils virent un pilier irradiant qui partait du sol et montait jusqu'au ciel, comme un c\u00f4ne invers\u00e9 dont la base \u00e9tait constitu\u00e9e par le miroir parabolique, tandis que son sommet pointait vers le centre de la capitale, qui se retrouva baign\u00e9e dans une blancheur incandescente. Des volutes de fum\u00e9e tourbillonnantes s'\u00e9levaient dans les airs ; des tornades, provoqu\u00e9es par la chaleur in\u00e9gale du c\u00f4ne de lumi\u00e8re, firent se dresser des colonnes de poussi\u00e8re qui ralli\u00e8rent le ciel en s'enroulant et en dansant autour du c\u00f4ne...\n\nDes boules de feu \u00e9blouissantes apparurent en diff\u00e9rents endroits du miroir. Leur couleur bleue diff\u00e9rait de celles refl\u00e9t\u00e9es par le miroir. C'\u00e9taient les ogives nucl\u00e9aires envoy\u00e9es par les forces de d\u00e9fense a\u00e9rienne trisolariennes qui atteignaient leur cible. Les explosions ayant lieu au-del\u00e0 de l'atmosph\u00e8re, on n'entendait pas leur bruit. Quand les boules de feu s'\u00e9teignirent, de larges trous apparurent sur le miroir, puis sa surface commen\u00e7a \u00e0 se fissurer, jusqu'\u00e0 ce qu'il \u00e9clate en une dizaine de morceaux. Au m\u00eame moment, le mortel c\u00f4ne de lumi\u00e8re disparut et le monde retrouva sa luminosit\u00e9 d'origine. L'espace d'un instant, on se crut au clair de lune. Les fragments bris\u00e9s du miroir, d\u00e9sormais amput\u00e9s de toute intelligence, continuaient \u00e0 se distordre. Tr\u00e8s vite, ils fusionn\u00e8rent avec les autres objets spatiaux en trois dimensions et on ne les distingua plus.\n\n\u2014 \u00c0 quoi aurons-nous droit lors de votre prochaine exp\u00e9rience ? demanda d'un air ironique le chancelier au consul de la science. Un proton d\u00e9ploy\u00e9 en quatre dimensions ?\n\n\u2014 Chancelier, m\u00eame si cela devait arriver, ce ne serait pas un grand probl\u00e8me. Car s'il se d\u00e9ploie en quatre dimensions, le volume du proton sera bien plus petit. Si l'unit\u00e9 de d\u00e9fense a\u00e9rienne est pr\u00eate \u00e0 passer \u00e0 l'attaque, elle pourra d\u00e9truire de la m\u00eame mani\u00e8re sa projection dans le monde en trois dimensions.\n\n\u2014 Vous osez duper le chancelier ! lan\u00e7a avec col\u00e8re le consul de la guerre au consul de la science. Vous lui cachez le v\u00e9ritable danger ! Que se passera-t-il si le proton se d\u00e9ploie en z\u00e9ro dimension ?\n\n\u2014 En z\u00e9ro dimension ? demanda le chancelier avec curiosit\u00e9. Un point, mais sans taille ?\n\n\u2014 Oui, une singularit\u00e9 ! M\u00eame un proton para\u00eetrait infiniment grand en comparaison. Toute la masse du proton serait d\u00e8s lors contenue dans cette singularit\u00e9, et sa densit\u00e9 serait prodigieusement grande. Chancelier, vous pouvez certainement imaginer ce qu'il deviendrait.\n\n\u2014 Un trou noir ?\n\n\u2014 Oui.\n\n\u2014 Chancelier, expliqua le consul de la science, si nous avons choisi de d\u00e9ployer un proton et non un neutron, c'est pr\u00e9cis\u00e9ment pour \u00e9viter ce genre de risque. Si un d\u00e9ploiement en z\u00e9ro dimension venait vraiment \u00e0 se produire, la charge \u00e9lectrique du proton serait transf\u00e9r\u00e9e au trou noir que le d\u00e9ploiement aurait cr\u00e9\u00e9, et nous serions en mesure de le capturer et de le contr\u00f4ler gr\u00e2ce \u00e0 la force \u00e9lectromagn\u00e9tique.\n\n\u2014 Et si vous ne parvenez pas \u00e0 le trouver ou \u00e0 le contr\u00f4ler ? r\u00e9torqua le consul de la guerre. Il tomberait \u00e0 la surface et augmenterait sa masse \u00e0 mesure qu'il absorberait de la mati\u00e8re, puis il sombrerait jusqu'au noyau de la plan\u00e8te et avalerait pour finir la totalit\u00e9 de notre monde.\n\n\u2014 Cela n'arrivera pas, je vous en donne ma parole ! Pourquoi \u00eates-vous sans cesse en train de me provoquer ? Je vous l'ai dit, c'est une exp\u00e9rience scientifique...\n\n\u2014 Il suffit, dit le chancelier. Quelles sont les chances de succ\u00e8s de la prochaine exp\u00e9rience ?\n\n\u2014 Pr\u00e8s de cent pour cent, chancelier, veuillez me croire. Gr\u00e2ce \u00e0 ces deux \u00e9checs, nous avons maintenant compris comment ma\u00eetriser les principes de d\u00e9ploiement d'une particule subatomique \u00e0 une dimension plus faible dans le monde macroscopique.\n\n\u2014 Soit. Pour la survie de Trisolaris, nous devons prendre ce risque.\n\n\u2014 Merci, chancelier !\n\n\u2014 Cependant, sache que si tu \u00e9choues la prochaine fois, toi et tous les scientifiques du projet Intellectron, vous serez coupables.\n\n\u2014 Oui, bien s\u00fbr, nous serons coupables.\n\nSi les Trisolariens avaient transpir\u00e9, le consul de la science aurait certainement pass\u00e9 sa main sur le front.\n\nIl fut nettement plus facile de d\u00e9blayer les restes du proton tridimensionnel qui se trouvaient maintenant en orbite synchrone autour de Trisolaris que \u00e7a n'avait \u00e9t\u00e9 le cas pour le proton unidimensionnel. Des petits vaisseaux les tract\u00e8rent jusqu'\u00e0 une distance suffisamment \u00e9loign\u00e9e de la plan\u00e8te pour \u00e9viter que les formes ne p\u00e9n\u00e8trent dans l'atmosph\u00e8re. Ces substances, aussi grandes que des cha\u00eenes de montagnes, n'avaient presque aucune masse, comme d'\u00e9normes ombres argent\u00e9es que m\u00eame un nouveau-n\u00e9 n'aurait eu aucune peine \u00e0 d\u00e9placer.\n\nLe chancelier demanda plus tard au consul de la science :\n\n\u2014 Lors de cette exp\u00e9rience, n'avons-nous pas d\u00e9truit la civilisation d'un univers microscopique ?\n\n\u2014 Sans doute au moins une entit\u00e9 intelligente, oui. De plus, chancelier, nous avons an\u00e9anti un univers entier. Un univers immense \u00e0 l'\u00e9chelle des dimensions les plus hautes, o\u00f9 il n'existait probablement pas qu'une seule forme d'intelligence ou de civilisation, mais celles-ci n'ont simplement pas eu l'occasion de se r\u00e9v\u00e9ler dans le monde macroscopique. Bien entendu, \u00e0 des dimensions aussi importantes et \u00e0 des \u00e9chelles si minuscules, il nous est absolument impossible d'imaginer la forme que prennent ces intelligences et ces civilisations. Elles n'ont rien en commun avec ce que nous connaissons. Et ce genre de destruction ne s'est pas produit pour la premi\u00e8re fois.\n\n\u2014 Ah ?\n\n\u2014 Dans la longue histoire de la science, combien de protons sont d\u00e9j\u00e0 entr\u00e9s en collision dans les acc\u00e9l\u00e9rateurs des physiciens ? Combien de neutrons et d'\u00e9lectrons ? Peut-\u00eatre au moins une centaine de millions. Chaque collision s'est av\u00e9r\u00e9e \u00eatre le g\u00e9nocide d'une intelligence ou d'une civilisation dans l'univers qui les contenait. En r\u00e9alit\u00e9, m\u00eame dans la nature, est orchestr\u00e9 \u00e0 chaque instant ce genre de mises \u00e0 mort \u2013 la d\u00e9sint\u00e9gration des protons et des neutrons, par exemple, ou bien l'entr\u00e9e dans l'atmosph\u00e8re de rayons cosmiques de haute \u00e9nergie qui peuvent d\u00e9truire des milliers de micro-univers... Seriez-vous donc tristes pour elles ?\n\n\u2014 Ce que tu dis est amusant. Je vais de ce pas rapporter cette information au consul de la propagande, en demandant qu'il r\u00e9pande cette v\u00e9rit\u00e9 scientifique dans la population, afin de bien faire comprendre au peuple que des destructions de civilisations sont des \u00e9v\u00e9nements somme toute ordinaires ayant lieu \u00e0 chaque instant de la vie.\n\n\u2014 Pourquoi donc, chancelier ? Serait-ce pour que le peuple affronte avec courage la destruction prochaine de la civilisation trisolarienne ?\n\n\u2014 Non, pour qu'ils affrontent celle des humains. Comme tu le sais, lorsque nous avons annonc\u00e9 ce que nous comptions faire de la civilisation terrienne, cela a suscit\u00e9 des sentiments pacifistes extr\u00eamement dangereux. Ce n'est qu'alors que nous avons d\u00e9couvert qu'il existait en notre sein de nombreux individus comme le guetteur 1379. Nous devons contr\u00f4ler et \u00e9liminer ces \u00e9motions vuln\u00e9rables.\n\n\u2014 Chancelier, ce genre de sentiment a surtout \u00e9t\u00e9 inspir\u00e9 par la r\u00e9v\u00e9lation du contenu des r\u00e9cents messages re\u00e7us. Votre pr\u00e9diction s'est r\u00e9alis\u00e9e : les forces ali\u00e9n\u00e9es terriennes gagnent du terrain. Elles ont maintenant construit leur propre site de transmission, enti\u00e8rement autonome, et ont commenc\u00e9 \u00e0 nous envoyer une grande quantit\u00e9 d'informations sur la civilisation terrienne. Je dois admettre que cette culture poss\u00e8de un pouvoir de s\u00e9duction dangereux. Notre peuple croit entendre un chant sacr\u00e9 venu du paradis. Les id\u00e9es humanistes des Terriens risquent d'\u00e9garer les esprits de beaucoup d'entre nous. Sur Terre, notre civilisation a donn\u00e9 naissance \u00e0 une sorte de religion. Mais le risque existe aussi que cela se produise dans le sens inverse sur Trisolaris.\n\n\u2014 Tu pointes l\u00e0 un grand danger. Nous devons d\u00e9sormais veiller \u00e0 ce que les informations concernant la Terre ne se propagent plus, et en particulier celles concernant sa culture.\n\nLa troisi\u00e8me tentative de d\u00e9ploiement en deux dimensions d'un proton eut lieu trente-trois heures trisolariennes plus tard, durant la nuit. En observant l'acc\u00e9l\u00e9rateur de particules spatial depuis la surface, on ne distinguait que le halo rouge des dissipateurs thermiques des g\u00e9n\u00e9rateurs \u00e9lectriques, qui permettaient de r\u00e9v\u00e9ler sa position. Peu de temps apr\u00e8s la mise en route de l'acc\u00e9l\u00e9rateur, le consul de la science annon\u00e7a la r\u00e9ussite de l'op\u00e9ration.\n\nLa foule leva les yeux au ciel. Elle ne vit tout d'abord rien mais, tr\u00e8s vite, elle eut une vision prodigieuse : le ciel s'\u00e9tait scind\u00e9 en deux. Entre les deux morceaux, les motifs des \u00e9toiles ne correspondaient pas, on aurait dit deux photographies du ciel plac\u00e9es l'une sur l'autre, la plus petite sur la plus grande. La Voie lact\u00e9e \u00e9tait coup\u00e9e au niveau de leur fronti\u00e8re commune. Un des morceaux du ciel \u00e9tait circulaire et s'\u00e9tendait rapidement sur le fond \u00e9toil\u00e9 du firmament ordinaire.\n\n\u2014 Cette constellation est celle que l'on peut observer dans l'h\u00e9misph\u00e8re Sud ! s'exclama le consul de la culture et de l'\u00e9ducation en pointant du doigt le morceau circulaire du ciel en train de s'agrandir.\n\nAlors que la foule tentait de faire marcher son imagination en essayant de comprendre comment les \u00e9toiles de l'h\u00e9misph\u00e8re Sud pouvaient maintenant se superposer \u00e0 celles de l'h\u00e9misph\u00e8re Nord, elle assista \u00e0 un spectacle encore plus \u00e9tonnant : aux extr\u00e9mit\u00e9s du morceau de ciel de l'h\u00e9misph\u00e8re Sud apparut le fragment d'un \u00e9norme globe. De couleur brune, le globe se d\u00e9voilait \u00e0 une vitesse tr\u00e8s lente, comme une image num\u00e9ris\u00e9e sur un \u00e9cran. Tous connaissaient ce globe, car ils avaient reconnu \u00e0 sa surface des motifs familiers. Lorsque l'image du globe fut enti\u00e8rement r\u00e9v\u00e9l\u00e9e, celui-ci occupait d\u00e9j\u00e0 un tiers du ciel et les d\u00e9tails de sa surface \u00e9taient maintenant limpides : on voyait des continents bruns travers\u00e9s de cha\u00eenes de montagnes, elles-m\u00eames surmont\u00e9es de nuages blancs, comme une couche de neige \u00e9ternelle... C'est alors que quelqu'un l\u00e2cha enfin :\n\n\u2014 C'est notre plan\u00e8te !\n\nOui, un autre monde de Trisolaris \u00e9tait apparu dans l'espace.\n\nPuis le ciel s'\u00e9claircit. Pr\u00e8s du second Trisolaris, dans l'espace, \u00e9mergea sur la constellation d'\u00e9toiles de l'h\u00e9misph\u00e8re Sud l'image d'un soleil, manifestement le m\u00eame astre qui \u00e9clairait en ce moment l'h\u00e9misph\u00e8re Sud, bien que celui-ci ne fasse qu'environ la moiti\u00e9 de sa taille.\n\nQuelqu'un per\u00e7a enfin le myst\u00e8re de ce ph\u00e9nom\u00e8ne :\n\n\u2014 C'est un miroir !\n\nL'immense miroir qui \u00e9tait apparu au-dessus de Trisolaris \u00e9tait le proton en train de se d\u00e9ployer en deux dimensions, c'\u00e9tait maintenant une surface g\u00e9om\u00e9trique parfaitement plane, sans aucune \u00e9paisseur.\n\nLorsque le d\u00e9ploiement en deux dimensions fut achev\u00e9, le firmament \u00e9tait enti\u00e8rement enrob\u00e9 par le reflet du ciel de l'h\u00e9misph\u00e8re Sud. Au z\u00e9nith de celui-ci se trouvaient les reflets de la plan\u00e8te Trisolaris et d'un soleil. Puis, le ciel autour de l'horizon se d\u00e9forma et le reflet des \u00e9toiles s'\u00e9tira et se distordit, comme si elles fondaient goutte par goutte. Cette d\u00e9formation se prolongea petit \u00e0 petit depuis les extr\u00e9mit\u00e9s de la vo\u00fbte jusqu'\u00e0 son centre.\n\n\u2014 Chancelier, la surface du proton se d\u00e9forme sous l'influence de la gravit\u00e9 de notre plan\u00e8te, expliqua le consul de la science, puis il d\u00e9signa les nombreuses lumi\u00e8res qui venaient d'appara\u00eetre dans le ciel, comme si des individus agitaient des torches pour \u00e9clairer le sommet de la vo\u00fbte. Ce sont des rayons \u00e9lectromagn\u00e9tiques artificiels \u00e9mis \u00e0 partir du sol dont l'objectif est d'ajuster la courbure de la surface d\u00e9form\u00e9e par la gravit\u00e9. Ainsi, la surface plane du proton pourra compl\u00e8tement envelopper notre plan\u00e8te. Apr\u00e8s quoi, l'\u00e9mission des rayons \u00e9lectromagn\u00e9tiques se prolongera pour continuer \u00e0 maintenir et \u00e0 stabiliser cette sph\u00e8re, comme les rayons d'une roue. La plan\u00e8te Trisolaris devient un \u00e9tabli permettant de pr\u00e9server l'\u00e9quilibre de ce proton bidimensionnel. La gravure de circuits int\u00e9gr\u00e9s sur sa surface pourra alors commencer.\n\nUne longue p\u00e9riode s'\u00e9coula avant que le proton bidimensionnel finisse par envelopper la plan\u00e8te. Lorsque la d\u00e9formation du reflet de la plan\u00e8te Trisolaris atteignit le z\u00e9nith du ciel, les \u00e9toiles avaient compl\u00e8tement disparu, car la surface plane du proton \u2013 qui s'\u00e9tait maintenant courb\u00e9e jusqu'\u00e0 l'autre c\u00f4t\u00e9 de la plan\u00e8te \u2013 avait tout recouvert. Des rayons solaires p\u00e9n\u00e9tr\u00e8rent encore un certain temps \u00e0 l'int\u00e9rieur de la courbe plane du proton et on ne reconnaissait d\u00e9j\u00e0 plus le monde de Trisolaris sur le miroir d\u00e9formant de l'espace. Mais lorsque le dernier rayon disparut, tout se retrouva plong\u00e9 dans une obscurit\u00e9 infinie. Ce fut la nuit la plus noire de toute l'histoire de Trisolaris. Sous l'\u00e9quilibre de la gravit\u00e9 de la plan\u00e8te et gr\u00e2ce au maintien des rayons \u00e9lectromagn\u00e9tiques, la surface plane du proton bidimensionnel forma une gigantesque coquille sph\u00e9rique autour de Trisolaris, avec laquelle elle \u00e9tait en orbite synchrone.\n\nUn froid glacial s'abattit sur le monde. La surface parfaitement r\u00e9flexive du proton d\u00e9via plus loin dans l'espace les rayons chauds des soleils qui descendaient vers la plan\u00e8te. La temp\u00e9rature sur Trisolaris chuta \u00e0 un degr\u00e9 comparable \u00e0 celui correspondant \u00e0 l'apparition de trois \u00e9toiles volantes \u2013 ph\u00e9nom\u00e8ne qui avait d\u00e9truit nombre de civilisations par le pass\u00e9. La population trisolarienne fut d\u00e9shydrat\u00e9e et entrepos\u00e9e dans les silos de d\u00e9shydratation. Un silence de mort enveloppa les t\u00e9n\u00e8bres. Dans l'espace ne restaient plus que les faibles lueurs vacillantes des rayons \u00e9lectromagn\u00e9tiques, dont la t\u00e2che \u00e9tait de maintenir stable la gigantesque membrane. On pouvait voir de temps \u00e0 autre quelques lumi\u00e8res en orbite : celles des vaisseaux en train de graver des circuits int\u00e9gr\u00e9s \u00e0 la surface du proton.\n\nLes principes r\u00e9gissant les circuits int\u00e9gr\u00e9s microscopiques n'\u00e9taient pas les m\u00eames que ceux des circuits traditionnels car, la surface sur laquelle ils \u00e9taient grav\u00e9s \u00e9tant celle d'un proton, le mat\u00e9riel utilis\u00e9 ne pouvait pas \u00eatre constitu\u00e9 d'atomes. Les jonctions \"p-n\" des circuits \u00e9taient cr\u00e9\u00e9es gr\u00e2ce \u00e0 une d\u00e9formation des interactions fortes de sa surface, et les fils conducteurs \u00e9taient obtenus gr\u00e2ce aux m\u00e9sons transportant la force nucl\u00e9aire. La surface sur laquelle devait \u00eatre grav\u00e9 le circuit int\u00e9gr\u00e9 \u00e9tant immense, sa taille \u00e0 l'\u00e9chelle macroscopique \u00e9tait elle aussi d\u00e9mesur\u00e9e. Les lignes des circuits avaient l'\u00e9paisseur de cheveux et il \u00e9tait possible de les voir \u00e0 l'\u0153il nu en s'en approchant. En survolant de pr\u00e8s la surface du proton, on pouvait d\u00e9couvrir une vaste plaine sillonn\u00e9e de gracieux et complexes circuits int\u00e9gr\u00e9s. La surface totale occup\u00e9e par les circuits \u00e9tait des dizaines de fois plus grande que la surface totale des continents de Trisolaris qu'elle enveloppait.\n\nLa gravure des circuits int\u00e9gr\u00e9s sur le proton \u00e9tait un projet titanesque. Plus de mille vaisseaux travaill\u00e8rent d'arrache-pied durant quinze mille heures trisolariennes avant d'achever leur t\u00e2che. La mise au point des logiciels dura, elle, encore cinq mille heures trisolariennes suppl\u00e9mentaires. Et enfin vint le moment de tester pour la premi\u00e8re fois l'intellectron.\n\nDans la salle de contr\u00f4le souterraine de l'intellectron, l'\u00e9cran afficha la fin de la s\u00e9quence d'autotest, puis la barre de progr\u00e8s du processus du chargement. Enfin, sur le terminal bleu vide, apparut cette phrase :\n\n _Micro-intelligence 2.10 charg\u00e9e. Intellectron 1 en attente des instructions._\n\nLe consul de la science claironna :\n\n\u2014 Un intellectron est n\u00e9. Un proton que nous avons dot\u00e9 d'intelligence. C'est l'entit\u00e9 d'intelligence artificielle la plus petite que nous puissions cr\u00e9er.\n\n\u2014 Mais de ce que nous en voyons maintenant, c'est plut\u00f4t la plus grande que nous ayons cr\u00e9\u00e9e, dit le chancelier.\n\n\u2014 Chancelier, nous allons bient\u00f4t augmenter la dimension de ce proton, il deviendra rapidement microscopique.\n\nLe consul de la science s'installa devant le terminal et saisit une question :\n\n _\u2014 Intellectron 1, le programme d'ajustement dimensionnel est-il op\u00e9rationnel ?_\n\n _\u2014 Parfaitement op\u00e9rationnel. Intellectron 1 peut d\u00e9marrer \u00e0 tout moment le programme._\n\n _\u2014 Ajustez en trois dimensions._\n\nUne fois l'instruction donn\u00e9e, la membrane du proton bidimensionnel se contracta, comme si une main g\u00e9ante d\u00e9chirait le voile du monde, et, presque en un instant, les rayons de soleil arros\u00e8rent la terre. Le proton se d\u00e9ploya en trois dimensions et devint une gigantesque sph\u00e8re en orbite synchrone autour de Trisolaris. Elle paraissait de la taille de la lune g\u00e9ante. Le proton se situait sur la face nocturne de la plan\u00e8te, mais les reflets des rayons du soleil sur sa surface miroitante chang\u00e8rent la nuit en jour. Il faisait encore extr\u00eamement froid, et les Trisolariens \u00e9veill\u00e9s ne pouvaient contempler la sc\u00e8ne que depuis un \u00e9cran de la salle de contr\u00f4le.\n\n _\u2014 Ajustement en quatre dimensions termin\u00e9. Intellectron 1 en attente des instructions._\n\n _\u2014 Ajustez en quatre dimensions._\n\nDans l'espace, la sph\u00e8re g\u00e9ante se mit \u00e0 r\u00e9tr\u00e9cir jusqu'\u00e0 prendre la taille d'une \u00e9toile volante. La nuit tomba une nouvelle fois sur la plan\u00e8te.\n\n\u2014 Chancelier, la sph\u00e8re que nous voyons \u00e0 pr\u00e9sent n'est pas le vrai intellectron, il s'agit simplement de sa projection dans le monde tridimensionnel. Dans la quatri\u00e8me dimension, c'est un g\u00e9ant, tandis que notre monde est une feuille de papier fin en trois dimensions. Le g\u00e9ant se tient debout sur cette feuille et nous ne sommes en mesure de voir que l'endroit o\u00f9 son pied touche le papier.\n\n _\u2014 Ajustement en quatre dimensions termin\u00e9. Intellectron 1 en attente des instructions._\n\n _\u2014 Continuez l'ajustement jusqu'en six dimensions._\n\nLa sph\u00e8re s'\u00e9vanouit dans le ciel.\n\n\u2014 Quelle est la taille de l'intellectron \u00e0 six dimensions ? demanda le chancelier.\n\n\u2014 Il a un rayon d'environ cinq centim\u00e8tres, r\u00e9pondit le consul de la science.\n\n _\u2014 Ajustement termin\u00e9. Intellectron 1 en attente des instructions._\n\n _\u2014 Intellectron 1, pouvez-vous nous voir ?_\n\n _\u2014 Oui, je peux voir la salle de contr\u00f4le et chaque personne \u00e0 l'int\u00e9rieur. Je peux aussi voir les organes de chacun d'entre vous, et m\u00eame les organes \u00e0 l'int\u00e9rieur de vos organes._\n\n\u2014 Que dit-il ? demanda le chancelier, intrigu\u00e9.\n\n\u2014 Quand nous observons un espace tridimensionnel depuis la sixi\u00e8me dimension, c'est comme si nous contemplions une peinture en deux dimensions. Il peut naturellement voir \u00e0 l'int\u00e9rieur de nous. Intellectron 1, entrez dans la salle de contr\u00f4le.\n\n\u2014 Il peut passer \u00e0 travers le sol ?\n\n\u2014 Pas \"\u00e0 travers\", chancelier, c'est plut\u00f4t comme s'il entrait dans notre dimension depuis une dimension plus \u00e9lev\u00e9e. Il est capable de p\u00e9n\u00e9trer dans n'importe quel espace clos de notre monde. C'est encore une fois la m\u00eame relation qu'entre notre monde tridimensionnel et une surface en deux dimensions : nous pouvons tout \u00e0 fait entrer dans un cercle trac\u00e9 sur une feuille de papier, tandis qu'un objet bidimensionnel ne le pourra jamais, sauf en trouant ce cercle.\n\nQuand le consul de la science eut termin\u00e9, une petite sph\u00e8re \u00e0 la surface r\u00e9flexive apparut au milieu de la salle de contr\u00f4le, flottant dans l'air. Le chancelier s'approcha et vit \u00e0 sa surface son reflet d\u00e9form\u00e9.\n\n\u2014 Ceci est donc un intellectron ? demanda-t-il, impressionn\u00e9.\n\n\u2014 Chancelier, ce n'est que la projection dans un espace tridimensionnel d'un proton \u00e0 six dimensions.\n\nLe chancelier tendit la main et, comme il vit que le consul de la science ne l'en emp\u00eachait pas, il toucha l'intellectron. \u00c0 peine l'eut-il effleur\u00e9 que l'intellectron fut repouss\u00e9 \u00e0 une certaine distance.\n\nLe chancelier \u00e9tait d\u00e9concert\u00e9 :\n\n\u2014 \u00c7a a l'air tr\u00e8s lisse. Il n'a que la masse d'un proton, mais bizarrement, j'ai pu sentir une r\u00e9sistance sur ma main.\n\n\u2014 Cette impression est due \u00e0 l'effet de la r\u00e9sistance de l'air sur la sph\u00e8re.\n\n\u2014 S'il passait \u00e0 la onzi\u00e8me dimension, retrouverait-il sa taille de proton ?\n\nLe chancelier n'eut m\u00eame pas le temps de finir sa phrase que le consul de la science cria avec angoisse \u00e0 l'encontre de l'intellectron :\n\n _\u2014 Attention, ceci n'est pas un ordre !_\n\n _\u2014 Intellectron 1 a compris._\n\n\u2014 Chancelier, si nous le faisions passer en onzi\u00e8me dimension, nous le perdrions \u00e0 jamais. Si l'intellectron passe \u00e0 la taille d'une particule subatomique ordinaire, ses senseurs internes et ses p\u00e9riph\u00e9riques d'entr\u00e9e et de sortie deviendront plus petits que la longueur d'onde du spectre \u00e9lectromagn\u00e9tique, ce qui voudrait dire qu'il ne serait plus capable de percevoir le monde macroscopique et donc de recevoir nos ordres.\n\n\u2014 Mais il faudra bien, en fin de compte, lui redonner son apparence de particule subatomique !\n\n\u2014 Certes, cependant, il nous faudra attendre la fabrication d'Intellectron 2, 3 et 4. Plusieurs intellectrons peuvent, ensemble, gr\u00e2ce au principe de l'intrication quantique, structurer un syst\u00e8me dans lequel ils seront capables de ressentir le monde macroscopique. Permettez que je prenne un exemple : imaginons un noyau atomique constitu\u00e9 de deux protons. Tous deux interagissent et se meuvent en suivant des r\u00e8gles bien d\u00e9finies. Si par exemple les protons tournent sur eux-m\u00eames, ils doivent le faire dans un sens oppos\u00e9 l'un \u00e0 l'autre. Lorsque ces deux protons sont extraits du noyau, peu importe la distance \u00e0 laquelle ils se trouvent s\u00e9par\u00e9s, les principes r\u00e9gissant leurs interactions demeurent les m\u00eames. Si l'on change le sens de rotation d'un proton, l'autre changera imm\u00e9diatement le sien pour r\u00e9pondre en cons\u00e9quence, o\u00f9 qu'il soit. Quand deux protons sont transform\u00e9s en intellectrons, ils forment donc une seule entit\u00e9 intriqu\u00e9e. Un ensemble plus vaste d'intellectrons forme quant \u00e0 lui une formation intriqu\u00e9e. L'\u00e9chelle de cette formation peut \u00eatre ajust\u00e9e \u00e0 n'importe quelle taille. Elle peut donc recevoir les ondes \u00e9lectromagn\u00e9tiques de n'importe quelle fr\u00e9quence et percevoir l'ensemble du monde macroscopique. Bien entendu, les ph\u00e9nom\u00e8nes quantiques permettant de structurer de telles formations d'intellectrons sont extr\u00eamement complexes. Mon explication n'est qu'une simple analogie.\n\nLes d\u00e9ploiements des trois protons suivants furent un succ\u00e8s d\u00e8s leurs premi\u00e8res tentatives. Le temps utilis\u00e9 pour fabriquer ces intellectrons fut moiti\u00e9 moindre que pour Intellectron 1. Apr\u00e8s la cr\u00e9ation d'Intellectron 2, d'Intellectron 3 et d'Intellectron 4, on parvint avec le m\u00eame succ\u00e8s \u00e0 reproduire les intrications quantiques \u00e9voqu\u00e9es par le consul de la science.\n\nLe chancelier et l'ensemble des consuls se r\u00e9unirent une nouvelle fois au pied du m\u00e9morial du Pendule. Au-dessus de leurs t\u00eates flottaient les quatre intellectrons \u00e0 six dimensions. Leurs surfaces miroitantes et cristallines refl\u00e9taient le soleil levant. Cette sc\u00e8ne rappela \u00e0 ceux qui \u00e9taient pr\u00e9sents les yeux tridimensionnels qui \u00e9taient apparus jadis dans le ciel de Trisolaris.\n\n _\u2014 Formation Intellectron, ajustez en onzi\u00e8me dimension._\n\nLes quatre sph\u00e8res disparurent.\n\n _\u2014_ Chancelier, expliqua le consul de la science, Intellectron 1 et Intellectron 2 sont partis pour la Terre. Gr\u00e2ce \u00e0 la base de donn\u00e9es que nous avons stock\u00e9e dans leurs circuits \u00e9lectroniques microscopiques, ils connaissent la nature de l'espace sur le bout des doigts. Ils sont capables de puiser de l'\u00e9nergie dans le vide et, dans un temps tr\u00e8s court, de se transformer en particules \u00e0 haute \u00e9nergie pour voguer \u00e0 une vitesse proche de celle de la lumi\u00e8re. Cela para\u00eet violer la loi de conservation de l'\u00e9nergie, mais les intellectrons \"empruntent\" de l'\u00e9nergie \u00e0 la structure du vide. Cependant, le temps que ceux-ci mettront pour \"rendre\" l'\u00e9nergie sera infiniment long, il faudra attendre la d\u00e9sint\u00e9gration du proton et nous ne serons pas loin, ce jour-l\u00e0, de la fin de l'univers.\n\nUne fois sur Terre, la premi\u00e8re mission des deux \u00e9lectrons sera de localiser les acc\u00e9l\u00e9rateurs de particules \u00e0 haute \u00e9nergie utilis\u00e9s par les physiciens humains, et de s'y dissimuler. Au niveau scientifique actuel des Terriens, la m\u00e9thode basique d'exploration de la structure profonde de la mati\u00e8re est de faire entrer en collision des particules de haute \u00e9nergie avec des particules cibles \u00e0 l'int\u00e9rieur de ces acc\u00e9l\u00e9rateurs. Lorsque ces particules se sont t\u00e9lescop\u00e9es, des analyses sont men\u00e9es sur les r\u00e9sultats de l'exp\u00e9rience, et les scientifiques tentent d'en d\u00e9duire des informations sur la structure profonde de la mati\u00e8re.\n\nDans leurs exp\u00e9riences r\u00e9elles, les particules cibles sont contenues dans des noyaux d'atomes. Or, un atome est quasiment vide : si un atome \u00e9tait grand comme une salle d'op\u00e9ra, son noyau serait une noix suspendue au milieu de la salle. Les collisions sont donc rares. La plupart du temps, une grande quantit\u00e9 de particules \u00e0 haute \u00e9nergie est dirig\u00e9e pendant une longue p\u00e9riode de temps vers la substance servant de cible avant que la collision se produise. C'est un peu comme si on essayait de trouver une goutte de pluie d'une couleur l\u00e9g\u00e8rement diff\u00e9rente dans un orage d'\u00e9t\u00e9.\n\nC'est une occasion donn\u00e9e \u00e0 l'intellectron de remplacer les particules cibles cens\u00e9es recevoir la collision. L'intellectron \u00e9tant dot\u00e9 d'une grande intelligence, il peut, gr\u00e2ce aux intrications quantiques de sa formation, d\u00e9terminer en un instant tr\u00e8s court le chemin que vont suivre les particules acc\u00e9l\u00e9r\u00e9es, puis se placer \u00e0 la position appropri\u00e9e. Par cons\u00e9quent, le degr\u00e9 de r\u00e9ussite d'une collision avec un intellectron est des millions de fois sup\u00e9rieur \u00e0 celui d'une collision avec une particule cible habituelle. Lorsque l'intellectron sera touch\u00e9, il fournira des r\u00e9sultats faux et chaotiques. Et m\u00eame s'il arrive de temps en temps que la vraie particule cible soit touch\u00e9e, les physiciens de la Terre ne pourront pas distinguer les r\u00e9sultats corrects de ceux manipul\u00e9s par l'intellectron.\n\n _\u2014_ Mais ainsi, l'intellectron ne risque-t-il pas d'\u00eatre d\u00e9truit ?\n\n _\u2014_ Non. L'intellectron est d\u00e9j\u00e0 un \u00e9l\u00e9ment de base structurant la mati\u00e8re, ce qui diff\u00e8re fondamentalement de la mati\u00e8re du monde macroscopique. Un intellectron peut \u00eatre fragment\u00e9, mais il ne peut pas \u00eatre d\u00e9truit. Quand un intellectron est bris\u00e9 en plusieurs morceaux, il produit en r\u00e9alit\u00e9 d'autres intellectrons. Et ceux-ci continuent d'avoir entre eux les m\u00eames intrications quantiques. C'est comme si vous coupiez un aimant en deux, vous en obtiendriez deux au lieu d'un. Si ces fragments d'intellectrons sont bien moins performants que l'intellectron d'origine, sous le commandement du logiciel de r\u00e9paration int\u00e9gr\u00e9 \u00e0 celui-ci, les fragments se r\u00e9assembleront en une seule entit\u00e9 pour redevenir le m\u00eame intellectron ayant pr\u00e9c\u00e9d\u00e9 l'impact. Ce processus, qui ne n\u00e9cessite qu'un centi\u00e8me de seconde, aura lieu apr\u00e8s la collision dans l'acc\u00e9l\u00e9rateur et apr\u00e8s que les intellectrons auront envoy\u00e9 des r\u00e9sultats erron\u00e9s, soit dans la chambre \u00e0 bulles, soit sur les pellicules d'appareil photo utilis\u00e9es dans les acc\u00e9l\u00e9rateurs.\n\nQuelqu'un demanda encore :\n\n _\u2014_ Est-il possible que les Terriens parviennent, d'une mani\u00e8re ou une autre, \u00e0 identifier l'intellectron, puis \u00e0 le capturer gr\u00e2ce \u00e0 un puissant champ \u00e9lectromagn\u00e9tique ? Apr\u00e8s tout, les protons ont une charge positive !\n\n _\u2014_ C'est impossible. Pour identifier les intellectrons, il faudrait que la science terrienne connaisse une soudaine et exceptionnelle avanc\u00e9e dans le domaine de la recherche sur la structure profonde de la mati\u00e8re. Or, si les acc\u00e9l\u00e9rateurs ont \u00e9t\u00e9 r\u00e9duits \u00e0 l'\u00e9tat de ferraille par l'action de nos intellectrons, ces recherches ne pourront \u00eatre men\u00e9es \u00e0 bien. Les yeux des chasseurs seront d'abord aveugl\u00e9s par les proies qu'ils cherchent \u00e0 traquer.\n\n _\u2014_ Les humains pourraient utiliser une technique plus grossi\u00e8re, dit le consul de l'industrie, construire un grand nombre d'acc\u00e9l\u00e9rateurs, plus rapidement que nous ne fabriquons des intellectrons, et alors il finira par exister des acc\u00e9l\u00e9rateurs sans intellectrons, capables de fournir des r\u00e9sultats corrects.\n\n _\u2014_ C'est certainement l'aspect le plus int\u00e9ressant du projet Intellectron ! fit le consul de la science, que cette remarque avait r\u00e9joui. Monsieur le consul de l'industrie, ne vous inqui\u00e9tez pas, la fabrication de masse des intellectrons ne ruinera pas l'\u00e9conomie du monde de Trisolaris, car nous n'aurons pas besoin d'agir ainsi. Nous cr\u00e9erons peut-\u00eatre encore quelques intellectrons, mais peu. En r\u00e9alit\u00e9, deux seuls suffisent, car les intellectrons sont multit\u00e2ches.\n\n _\u2014_ Multit\u00e2ches ?\n\n _\u2014_ C'est un vieux terme technique appartenant au vocabulaire des anciennes s\u00e9ries d'ordinateurs. \u00c0 l'\u00e9poque, l'unit\u00e9 centrale de traitement d'un ordinateur \u00e9tait uniquement capable de r\u00e9aliser une seule t\u00e2che \u00e0 la fois mais, en raison de sa rapidit\u00e9 d'ex\u00e9cution et des interruptions de l'ordonnanceur, notre perspective limit\u00e9e nous donnait l'impression que l'ordinateur menait en m\u00eame temps de front plusieurs programmes. Comme vous le savez, les intellectrons se d\u00e9placent \u00e0 une vitesse proche de celle de la lumi\u00e8re. \u00c0 cette \u00e9chelle, la Terre est en fait minuscule. Si les intellectrons sont capables de patrouiller \u00e0 cette vitesse dans les diff\u00e9rents acc\u00e9l\u00e9rateurs de particules de la plan\u00e8te, c'est, vu de la perspective des Terriens, comme s'ils se trouvaient en m\u00eame temps dans chaque acc\u00e9l\u00e9rateur et qu'ils pouvaient provoquer simultan\u00e9ment des r\u00e9sultats erron\u00e9s dans chacun des appareils.\n\nNous avons calcul\u00e9 que chaque intellectron pouvait \u00eatre capable de contr\u00f4ler parall\u00e8lement jusqu'\u00e0 dix mille acc\u00e9l\u00e9rateurs de haute \u00e9nergie. \u00c0 l'oppos\u00e9, les humains ont besoin \u00e0 l'heure actuelle de quatre ou cinq ann\u00e9es avant de terminer la construction d'un acc\u00e9l\u00e9rateur de ce type. De plus, \u00e9tant donn\u00e9 la situation \u00e9conomique et les ressources de leur plan\u00e8te, il est absolument inenvisageable qu'une construction massive d'acc\u00e9l\u00e9rateurs soit mise en route. Bien entendu, ils pourraient \u00e9largir la distance existant entre leurs acc\u00e9l\u00e9rateurs, en en installant sur les autres plan\u00e8tes de leur syst\u00e8me stellaire. Cela perturberait effectivement le fonctionnement multit\u00e2che des intellectrons, mais cela leur prendrait \u00e9norm\u00e9ment de temps, et nous aurions le temps de cr\u00e9er dix autres intellectrons, voire plus.\n\nDe plus en plus d'intellectrons se prom\u00e8neront donc dans le syst\u00e8me stellaire, qui, m\u00eame assembl\u00e9s, ne feront pas la taille d'un milliardi\u00e8me de bact\u00e9rie. Ceux-ci emp\u00eacheront pourtant \u00e0 jamais les physiciens terriens de pouvoir ne serait-ce qu'entrevoir les secrets de la structure profonde de la mati\u00e8re. Que ce soit dans quatre cent cinquante ans ou dans quatre cent cinquante millions d'ann\u00e9es, la ma\u00eetrise humaine du monde microscopique se limitera donc aux objets de moins de cinq dimensions. La technologie et la science de la civilisation terrestre n'arriveront jamais \u00e0 faire de perc\u00e9e fondamentale dans ce domaine et ils stagneront \u00e0 un stade primitif de la science. La science sur Terre sera compl\u00e8tement verrouill\u00e9e, et ce verrou sera si r\u00e9sistant que les humains ne seront jamais capables de le desserrer par leurs propres moyens.\n\n _\u2014_ Voil\u00e0 qui est extraordinaire ! Je vous demande pardon d'avoir mis en doute le projet Intellectron, s'excusa platement le consul de la guerre.\n\n _\u2014_ Dans les faits, il n'existe actuellement sur Terre que trois acc\u00e9l\u00e9rateurs permettant d'obtenir des r\u00e9sultats significatifs dans ce domaine. Apr\u00e8s leur arriv\u00e9e sur Terre, Intellectron 1 et Intellectron 2 seront donc presque inactifs. Mais pour pouvoir profiter pleinement de leurs capacit\u00e9s, nous leur assignerons d'autres t\u00e2ches que la simple perturbation des r\u00e9sultats des exp\u00e9riences de collision dans les acc\u00e9l\u00e9rateurs. Ils deviendront par exemple nos principaux assistants techniques pour la r\u00e9alisation du plan \"Miracle\".\n\n _\u2014_ Les intellectrons pourront produire des miracles ?\n\n _\u2014_ C'est en tout cas comme cela que le percevront les humains. Tout le monde sait que les acc\u00e9l\u00e9rateurs \u00e0 haute \u00e9nergie utilisent parfois des pellicules photo sur lesquelles les particules laissent des traces lorsqu'elles les traversent. C'est l'une des m\u00e9thodes qui a autrefois permis aux archa\u00efques acc\u00e9l\u00e9rateurs humains de d\u00e9tecter des particules individuelles. Quand un intellectron passera \u00e0 travers la pellicule, il laissera sur celle-ci un point lumineux. Si les intellectrons passent et repassent \u00e0 travers cette pellicule, ils pourront connecter ces points et former des lettres ou des nombres \u2013 voire des images \u2013 comme s'ils faisaient de la broderie. Ce processus peut \u00eatre rapide, bien plus rapide que la vitesse du d\u00e9veloppement de la pellicule d'un appareil photo humain. Par ailleurs, les r\u00e9tines humaines \u00e9tant sensiblement les m\u00eames que celles des Trisolariens, nos intellectrons pourront utiliser la m\u00eame technique pour projeter sur leurs r\u00e9tines des lettres, des chiffres ou des images... Si ces banals petits artifices ont d\u00e9j\u00e0 pour effet de d\u00e9stabiliser et d'effrayer les humains, un grand miracle suffira \u00e0 faire mourir de peur ces mis\u00e9rables vermines : les intellectrons auront le pouvoir de faire clignoter devant leurs yeux le fond diffus cosmologique !\n\n _\u2014_ Ce prodige ferait m\u00eame frissonner mes scientifiques, comment est-ce possible ?\n\n\u2014 C'est simple. Nous avons programm\u00e9 sur la surface de l'intellectron un logiciel de d\u00e9ploiement bidimensionnel. Une fois pass\u00e9 en deux dimensions, l'intellectron pourra envelopper la plan\u00e8te Terre comme une membrane ; ce logiciel sera \u00e9galement en mesure de rendre cette membrane transparente. Mais le degr\u00e9 de transparence pourra \u00eatre r\u00e9gl\u00e9 sur les fr\u00e9quences du fond diffus cosmologique. Naturellement, l'intellectron pouvant se d\u00e9ployer dans toutes les dimensions existantes, il pourra dans le futur \u00eatre \u00e0 l'origine de miracles encore plus impressionnants. Nous sommes d'ailleurs en train de d\u00e9velopper un logiciel dans ce sens. Mais les miracles dont je viens de parler suffiront largement \u00e0 cr\u00e9er une atmosph\u00e8re vici\u00e9e dans les syst\u00e8mes de pens\u00e9e scientifique humains. Ainsi, nous nous servirons du plan \"Miracle\" comme moyen de dissuasion pour stopper les autres progr\u00e8s scientifiques terriens n'appartenant pas au domaine de la physique.\n\n\u2014 Une derni\u00e8re question : pourquoi ne pas avoir envoy\u00e9 ensemble les quatre intellectrons vers la Terre ?\n\n\u2014 Les ph\u00e9nom\u00e8nes quantiques restent valables m\u00eame si les quatre intellectrons sont aux deux extr\u00e9mit\u00e9s de l'univers. Leurs corr\u00e9lations restent op\u00e9rationnelles. Si nous avons conserv\u00e9 Intellectron 3 et Intellectron 4 ici, c'est parce qu'ils pourront recevoir instantan\u00e9ment les informations que leur enverront depuis la Terre Intellectron 1 et Intellectron 2. Nous pourrons donc ainsi mener une surveillance en temps r\u00e9el de la Terre. En outre, la formation d'intellectrons nous servira pour communiquer sans d\u00e9calage avec les membres des forces ali\u00e9n\u00e9es de la civilisation terrienne.\n\n\u2014 Vous pointez quelque chose d'important, dit le chancelier en lui coupant la parole. Avec ces intellectrons, le monde de Trisolaris r\u00e9v\u00e9lera aux humains ses v\u00e9ritables intentions \u00e0 l'\u00e9gard de la civilisation terrienne.\n\n\u2014 Nous allons donc r\u00e9v\u00e9ler aux humains que la flotte trisolarienne les emp\u00eachera de procr\u00e9er et, en d\u00e9finitive, les fera dispara\u00eetre de la surface de la Terre ?\n\n\u2014 Oui. Une telle strat\u00e9gie pourra avoir deux cons\u00e9quences possibles : la premi\u00e8re, c'est que les Terriens abandonneront tout et se pr\u00e9pareront \u00e0 une guerre ultime ; la deuxi\u00e8me, c'est que la peur et le d\u00e9sespoir plongeront leurs soci\u00e9t\u00e9s dans le d\u00e9clin et l'effondrement. Au vu des analyses qui ressortent des documents re\u00e7us jusqu'\u00e0 ce jour concernant la civilisation terrienne, je crois que la seconde des hypoth\u00e8ses est la plus probable.\n\nSans que personne s'en soit rendu compte, le soleil, qui venait tout juste de se lever, avait de nouveau disparu derri\u00e8re l'horizon. L'aube \u00e9tait devenue le cr\u00e9puscule. Une nouvelle \u00e8re chaotique commen\u00e7ait dans le monde de Trisolaris.\n\nTandis que Ye Wenjie lisait ces informations sur la civilisation trisolarienne, une r\u00e9union importante se tenait au m\u00eame moment au Centre d'op\u00e9rations militaires, o\u00f9 une investigation pr\u00e9liminaire \u00e9tait men\u00e9e sur le contenu des documents captur\u00e9s lors de l'op\u00e9ration Cithare. Avant que la r\u00e9union commence, le g\u00e9n\u00e9ral Chang Weisi annon\u00e7a :\n\n\u2014 Camarades, je vous demande votre attention. La r\u00e9union qui va se tenir ici est peut-\u00eatre surveill\u00e9e par des intellectrons. \u00c0 partir de maintenant, nous n'aurons plus aucun secret.\n\nTandis qu'il pronon\u00e7ait ces mots, le monde autour d'eux paraissait le m\u00eame que ce qu'il avait toujours \u00e9t\u00e9 ; derri\u00e8re les rideaux dansaient les ombres des arbres d'\u00e9t\u00e9. Mais dans les yeux de tous ceux qui \u00e9taient pr\u00e9sents, le monde n'\u00e9tait plus comme avant ; ils sentaient maintenant une paire d'yeux braqu\u00e9e sur eux o\u00f9 qu'ils aillent et, sous ce regard, le monde n'avait plus nulle part o\u00f9 se cacher. Ce sentiment les suivrait toute leur vie et m\u00eame leurs descendants ne pourraient s'en d\u00e9barrasser. Les hommes allaient vivre de longues ann\u00e9es avant que leurs esprits soient en mesure de s'adapter \u00e0 cette \u00e9trange r\u00e9alit\u00e9.\n\nTrois minutes pr\u00e9cis\u00e9ment apr\u00e8s la prise de parole du g\u00e9n\u00e9ral Chang Weisi, le monde de Trisolaris communiqua pour la premi\u00e8re fois avec des humains n'appartenant pas au groupe rebelle de l'OTT. Apr\u00e8s quoi, ils coup\u00e8rent toutes les communications qu'ils avaient avec les adventistes et, pour le restant de la vie de tous les participants \u00e0 cette r\u00e9union, Trisolaris n'envoya plus jamais d'autre message \u00e0 la Terre. Tous ceux qui se trouvaient dans le Centre d'op\u00e9rations militaires virent s'afficher le message devant leur r\u00e9tine, comme Wang Miao avait vu s'afficher le compte \u00e0 rebours. Le message clignota \u00e0 peine deux secondes avant de s'effacer, mais tous avaient pu lire clairement ce qui \u00e9tait \u00e9crit, une phrase toute simple :\n\n_Vous \u00eates de la vermine !_\n\n## 34\n\n## LA VERMINE\n\n\u2014 Tout \u00e7a doit s\u00fbrement te faire penser \u00e0 ton fait de gloire : ta d\u00e9couverte du macro-atome, il y a trois ans, lors de tes recherches sur la foudre en boule, dit Wang Miao \u00e0 Ding Yi.\n\nIls se trouvaient \u00e0 pr\u00e9sent tous deux dans le salon spacieux de Ding Yi, assis le dos contre la table de billard.\n\n\u2014 Oui. Je n'ai jamais vraiment cess\u00e9 de travailler sur ma th\u00e9orie du macro-atome, et les \u00e9v\u00e9nements r\u00e9cents m'ont inspir\u00e9 de nouveaux d\u00e9veloppements : le macro-atome n'est peut-\u00eatre en r\u00e9alit\u00e9 qu'un atome ordinaire, mais d\u00e9ploy\u00e9 dans une plus petite dimension. Ce d\u00e9ploiement se serait produit gr\u00e2ce \u00e0 une force naturelle que nous ne connaissons pas encore, peut-\u00eatre au lendemain du Big Bang... Ou peut-\u00eatre qu'il se produit ici et maintenant, en permanence... Si \u00e7a se trouve, tous les atomes de l'univers finissent avec le temps par se d\u00e9ployer dans une plus petite dimension. Et si l'ultime destin de l'univers \u00e9tait de devenir un macro-univers, fait d'atomes de plus petite dimension ? Peut-\u00eatre pourrait-on consid\u00e9rer cette trajectoire comme un processus d'expansion entropique... Je croyais \u00e0 l'\u00e9poque que la d\u00e9couverte du macro-atome allait marquer une avanc\u00e9e majeure de la physique. Il semblerait bien que ce ne soit plus le cas aujourd'hui, l\u00e2cha Ding Yi, avant de se lever et d'aller chercher quelque chose dans sa biblioth\u00e8que.\n\n\u2014 Pourquoi ? Nous sommes d\u00e9j\u00e0 en mesure de capturer le macro-atome : pourquoi ne pourrions-nous pas nous passer des acc\u00e9l\u00e9rateurs de particules et extraire directement des \u00e9l\u00e9ments du macro-atome pour \u00e9tudier les structures profondes de la mati\u00e8re ?\n\n\u2014 C'est ce que j'ai pens\u00e9 au d\u00e9but. (Ding Yi revint de sa biblioth\u00e8que, un cadre photo aux bords argent\u00e9s dans les mains). Mais \u00e7a me para\u00eet ridicule \u00e0 pr\u00e9sent. (Il se baissa pour ramasser sur le sol sale un m\u00e9got de cigarette.) Regarde ce filtre, nous avions dit qu'une fois d\u00e9ploy\u00e9 en deux dimensions, il avait la m\u00eame surface que le salon, mais quand il sera d\u00e9ploy\u00e9, crois-tu vraiment qu'on pourra se baser sur cette surface en deux dimensions pour comprendre la nature du filtre en trois dimensions ? C'est \u00e9videmment impossible. Les informations contenues dans sa structure tridimensionnelle auront disparu d\u00e8s son d\u00e9ploiement. Imagine un verre cass\u00e9. Jamais il ne pourra retrouver son aspect original. Le d\u00e9ploiement d'un atome naturel est un processus irr\u00e9versible. Si nous voulons comprendre la structure profonde de la mati\u00e8re, il nous faut d'abord explorer le monde microscopique \u00e0 onze dimensions. Et pour cela, nous avons besoin des acc\u00e9l\u00e9rateurs. Les acc\u00e9l\u00e9rateurs sont notre boulier et notre r\u00e8gle \u00e0 calcul. Sans eux, nous n'aurions pas pu inventer l'ordinateur.\n\nDing Yi laissa voir \u00e0 Wang Miao la photo dans le cadre. Sur l'image, une jeune et jolie offici\u00e8re militaire se tenait debout, au milieu d'un groupe d'enfants. Elle avait le regard pur et souriait tendrement en fixant l'objectif. Les enfants et elle posaient sur une pelouse verte bien taill\u00e9e, sur laquelle gambadaient de petits animaux blancs. Derri\u00e8re eux se dressait un immense b\u00e2timent, semblable \u00e0 une usine. Ses murs \u00e9taient satur\u00e9s de dessins d'animaux multicolores, de ballons de baudruche, de fleurs, etc.\n\n\u2014 Tu l'as connue avant Yang Dong ? Tu as eu une vie mouvement\u00e9e, sourit Wang Miao en regardant la photo.\n\n\u2014 Elle s'appelle Lin Yun, elle a apport\u00e9 une contribution majeure aux \u00e9tudes sur la foudre en boule et \u00e0 la d\u00e9couverte du macro-atome. Sans elle, je n'aurais jamais fait cette d\u00e9couverte.\n\n\u2014 Je n'ai jamais entendu parler d'elle.\n\n\u2014 Oui, pour des raisons dont tu n'as jamais entendu parler non plus. Mais c'est injuste.\n\n\u2014 O\u00f9 est-elle maintenant ?\n\n\u2014 Dans... dans un endroit, ou peut-\u00eatre _des_ endroits... Ah, si seulement elle pouvait \u00eatre ici aujourd'hui...\n\nWang Miao ne fit pas attention \u00e0 son \u00e9trange r\u00e9ponse, il n'\u00e9tait pas int\u00e9ress\u00e9 par la photo. Il rendit le cadre \u00e0 Ding Yi et fit un geste de la main :\n\n\u2014 \u00c0 quoi bon ? Plus rien n'a d'importance.\n\n\u2014 Oui, plus rien n'a d'importance.\n\nDing Yi reposa d\u00e9licatement le cadre photo sur la table de billard, le regarda encore puis tendit la main vers un coin de la table pour saisir une bouteille d'alcool.\n\nLorsque Shi Qiang poussa la porte du salon, Ding Yi et Wang Miao \u00e9taient d\u00e9j\u00e0 compl\u00e8tement saouls. Ils semblaient enchant\u00e9s que Shi Qiang les rejoigne. Wang Miao se leva et attrapa le commissaire par les \u00e9paules :\n\n\u2014 Ah, ce vieux Shi, ce vieux commissaire...\n\nEn titubant, Ding Yi alla chercher un verre, le posa sur la table et le remplit d'alcool.\n\n\u2014 \u00c7a valait bien la peine, votre plan diabolique ! Qu'on ait ou non trouv\u00e9 ces informations, le r\u00e9sultat dans quatre si\u00e8cles sera le m\u00eame !\n\nShi Qiang s'assit devant la table de billard, en les d\u00e9visageant d'un air retors :\n\n\u2014 Alors c'est comme \u00e7a ? Tout est fini ?\n\n\u2014 Bien s\u00fbr, tout est fini.\n\n\u2014 Vous ne pouvez plus utiliser vos acc\u00e9l\u00e9rateurs ni explorer la structure profonde de la mati\u00e8re, et tout est fini ?\n\n\u2014 Mmh... qu'en pensez-vous ?\n\n\u2014 La technologie fait encore des progr\u00e8s, non ? Et notre acad\u00e9micien Wang et ses amis nous cr\u00e9eront des nanomat\u00e9riaux...\n\n\u2014 Imaginez-vous un royaume de l'ancien temps. Leur technologie progresse, elle aussi. Ils inventent pour leurs soldats de meilleures lances, de meilleurs boucliers... Ils sont m\u00eame capables de cr\u00e9er un arc avec des fl\u00e8ches pouvant partir en rafale, comme une mitrailleuse. Cependant...\n\nShi Qiang hocha la t\u00eate. Il comprenait :\n\n\u2014... cependant, s'ils ignorent que la mati\u00e8re est faite d'atomes et de mol\u00e9cules, ils ne pourront jamais cr\u00e9er de missiles ni de satellites, et ils seront \u00e9ternellement limit\u00e9s par leur niveau scientifique.\n\nDing Yi tapa sur l'\u00e9paule de Shi Qiang :\n\n\u2014 D\u00e8s le d\u00e9but, j'ai su que notre commissaire \u00e9tait un homme intelligent...\n\nWang Miao continua :\n\n\u2014 Les recherches sur la structure profonde de la mati\u00e8re constituent la base de toutes les sciences fondamentales. Sans avanc\u00e9e dans ce domaine, tout le reste, c'est \u2013 permettez-moi de vous citer \u2013 du pipi de chat !\n\nDing Yi pointa Wang Miao du doigt :\n\n\u2014 Mais notre acad\u00e9micien Wang aura du travail pour le restant de sa vie ! Il continuera \u00e0 am\u00e9liorer nos poignards, nos \u00e9p\u00e9es et nos boucliers en nanomat\u00e9riaux. Et moi, qu'est-ce que je vais bien pouvoir foutre ? Dieu seul le sait ! dit-il en renversant la bouteille sur la table et en essayant de la tamponner avec une boule de billard.\n\n\u2014 Bravo ! dit Wang Miao en levant son verre. Il faudra bien que nous vivions quand m\u00eame ! D\u00e9sormais, la d\u00e9cadence et la d\u00e9bauche auront de bonnes raisons d'\u00eatre ! Nous sommes de la vermine ! De la vermine en voie d'extinction, ha, ha...\n\n\u2014 Bien parl\u00e9 ! dit Ding Yi en levant son verre. Trinquons \u00e0 la vermine ! Je n'aurais jamais pens\u00e9 que la fin du monde serait si gaie ! Vive la vermine ! Vive les intellectrons ! Vive la fin du monde !\n\nShi Qiang secoua la t\u00eate, il avala son verre d'une traite, puis secoua encore une fois la t\u00eate :\n\n\u2014 Pff, regardez-vous.\n\n\u2014 Qu'est-ce que vous voulez ? dit Ding Yi en fixant Shi Qiang, les yeux embu\u00e9s d'alcool. Vous esp\u00e9rez pouvoir nous redonner du courage ?\n\nShi Qiang se leva :\n\n\u2014 Allons-y.\n\n\u2014 O\u00f9 \u00e7a ?\n\n\u2014 Chercher du courage !\n\n\u2014 Allons, vieux fr\u00e8re, asseyez-vous, buvez.\n\nShi Qiang leur attrapa les bras et les mit debout.\n\n\u2014 Allons-y. Prenez votre alcool, si c'est n\u00e9cessaire.\n\nApr\u00e8s avoir descendu l'\u00e9tage, tous trois grimp\u00e8rent dans la voiture de Shi Qiang. Une fois le v\u00e9hicule mis en marche, Wang Miao demanda d'une voix faible o\u00f9 il les emmenait.\n\n\u2014 Dans ma r\u00e9gion natale, pas tr\u00e8s loin d'ici.\n\nLa voiture quitta la ville en empruntant l'autoroute P\u00e9kin-Shijiazhuang en direction de l'ouest. \u00c0 peine entr\u00e9s dans la province du Hebei, ils sortirent de l'autoroute. Shi Qiang arr\u00eata la voiture et fit sortir ses deux passagers. Les rayons brillants du soleil de l'apr\u00e8s-midi les \u00e9blouirent. Devant eux s'\u00e9tendait la Grande Plaine de Chine du Nord, recouverte de champs cultiv\u00e9s.\n\n\u2014 Pourquoi nous avoir emmen\u00e9s ici ? demanda Wang Miao.\n\n\u2014 Pour observer la vermine.\n\nShi Qiang alluma un cigare que lui avait offert le colonel Stanton, et d\u00e9signa la plaine devant eux.\n\nCe n'est alors que Wang Miao et Ding Yi remarqu\u00e8rent que les champs \u00e9taient envahis d'une \u00e9paisse nu\u00e9e de criquets locustes. Chaque brin de paille \u00e9tait occup\u00e9 par plusieurs insectes et un nombre encore plus important de locustes se tortillait sur le sol. L'ensemble ressemblait \u00e0 une sorte de liquide visqueux.\n\n\u2014 Il y a eu une invasion de locustes, ici ?\n\nWang Miao chassa un essaim d'insectes au bord d'un champ et s'assit.\n\n\u2014 Comme une temp\u00eate de sable ! Il y en avait d\u00e9j\u00e0 eu il y a dix ans, mais cette ann\u00e9e, ils sont encore plus d\u00e9vastateurs.\n\n\u2014 Et alors quoi ? Quelle importance ? demanda Ding Yi, d'une voix encore enrou\u00e9e par l'alcool.\n\n\u2014 Je voulais seulement que vous r\u00e9fl\u00e9chissiez tous deux \u00e0 cette question : la diff\u00e9rence de niveau technologique entre les Terriens et les Trisolariens n'est-elle pas la m\u00eame qu'entre les locustes et nous ?\n\nCette question fit sur la t\u00eate des deux scientifiques ivres l'effet d'une carafe d'eau glac\u00e9e. Ils examin\u00e8rent du regard les tas d'insectes qu'ils avaient devant eux. Leurs mines se firent peu \u00e0 peu plus graves. Ils avaient vite compris ce que voulait dire Shi Qiang.\n\n\u2014 Regardez-la, cette vermine, la diff\u00e9rence de technologie entre nous et eux est de loin plus grande que celle existant entre les Trisolariens et nous. Les humains ont tout tent\u00e9 pour les \u00e9liminer : ils les ont bombard\u00e9s de toutes sortes de poisons, les ont arros\u00e9s par avion, ont essay\u00e9 d'introduire et d'\u00e9lever leurs pr\u00e9dateurs naturels, de trouver et de d\u00e9truire leurs \u0153ufs, de les modifier g\u00e9n\u00e9tiquement pour les st\u00e9riliser, ils les ont br\u00fbl\u00e9s, noy\u00e9s... Toutes les familles de la r\u00e9gion poss\u00e8dent chez elles des vaporisateurs insecticides de la marque Miehailing. Sous chaque table se trouvent des tapettes \u00e0 mouches... Mais cette guerre dure depuis le d\u00e9but de l'histoire et les hommes n'ont toujours pas remport\u00e9 la victoire. La vermine n'a pas \u00e9t\u00e9 \u00e9limin\u00e9e, elle continue d'aller et venir fi\u00e8rement sur la terre et son nombre n'est pas forc\u00e9ment moindre qu'avant l'apparition des hommes. En consid\u00e9rant les humains comme de la vermine, les Trisolariens semblent avoir oubli\u00e9 quelque chose : personne n'a jamais triomph\u00e9 de la vermine.\n\nUn petit nuage noir cacha le soleil et projeta une ombre mouvante sur le sol. Mais ce n'\u00e9tait pas un nuage, c'\u00e9tait un essaim de criquets qui venait d'arriver. Alors que l'essaim atterrissait dans le champ, la temp\u00eate de la vie paraissait avoir rinc\u00e9 les trois hommes. Ils ressentaient maintenant toute la dignit\u00e9 de la vie sur Terre. Ding Yi et Wang Miao renvers\u00e8rent leurs deux bouteilles d'alcool \u00e0 leurs pieds et, sur la Grande Plaine de la Chine du Nord, ils port\u00e8rent un toast \u00e0 la vermine.\n\n\u2014 Mon vieux Shi, merci, dit Wang Miao en tendant la main \u00e0 Shi Qiang.\n\n\u2014 Je vous remercie moi aussi, ajouta Ding Yi en lui attrapant l'autre main.\n\n\u2014 Rentrons, dit Wang Miao, nous avons du pain sur la planche.\n\n## 35\n\n## LES RUINES\n\nPersonne n'aurait cru que Wenjie puisse gravir le pic du Radar par elle-m\u00eame. Mais elle y \u00e9tait finalement parvenue, et personne ne l'avait aid\u00e9e en chemin. Elle s'\u00e9tait seulement repos\u00e9e \u00e0 deux reprises dans des petites gu\u00e9rites abandonn\u00e9es sur le flanc du pic. Elle semblait avoir \u00e9puis\u00e9 sans le moindre scrupule une vitalit\u00e9 qui ne se renouvellerait pas.\n\nDepuis qu'elle avait appris la v\u00e9rit\u00e9 sur la civilisation trisolarienne, Ye Wenjie s'\u00e9tait mur\u00e9e dans le silence. Elle ne parlait presque plus et n'avait \u00e9mis qu'une seule requ\u00eate : retourner voir les ruines de la base de C\u00f4te Rouge.\n\nQuand le petit groupe atteignit le sommet, celui-ci venait \u00e0 peine d'\u00e9merger des nuages. Ils avaient march\u00e9 un jour entier dans la brume et le brouillard, et lorsqu'ils virent le soleil lumineux \u00e0 l'ouest et la couleur bleue du ciel, ce fut comme s'ils \u00e9taient arriv\u00e9s dans un autre monde. Depuis le sommet du pic, les nuages formaient une mer blanche et argent\u00e9e, dont le balancement des vagues \u00e9tait comme une repr\u00e9sentation abstraite des montagnes du Grand Khingan en dessous.\n\nIl n'y avait aucune trace des ruines qu'ils s'\u00e9taient imagin\u00e9es. La base avait \u00e9t\u00e9 int\u00e9gralement ras\u00e9e. Il ne restait plus sur le sommet qu'une bande de terre d\u00e9serte. Les fondations des b\u00e2timents, ainsi que la route, avaient \u00e9t\u00e9 ensevelies, et tout ne ressemblait plus qu'\u00e0 une \u00e9tendue vierge, comme si tout ce qui \u00e9tait arriv\u00e9 \u00e0 C\u00f4te Rouge n'avait jamais eu lieu. Mais Ye Wenjie remarqua un vestige. Elle marcha jusqu'\u00e0 un grand rocher et ouvrit les lierres qui le recouvraient, r\u00e9v\u00e9lant une surface tachet\u00e9e de rouille. Ses compagnons d\u00e9couvrirent que le rocher \u00e9tait en fait un large socle en m\u00e9tal.\n\n\u2014 C'\u00e9tait le socle sur lequel reposait l'antenne, dit Ye Wenjie.\n\nLe premier appel de la Terre jamais entendu par une civilisation extraterrestre \u00e9tait pass\u00e9 par cette antenne avant de partir vers le soleil et, du soleil, il avait \u00e9t\u00e9 amplifi\u00e9 jusqu'\u00e0 s'\u00e9tendre \u00e0 tout l'univers.\n\nIls remarqu\u00e8rent une petite plaque \u00e0 c\u00f4t\u00e9 du socle, presque enti\u00e8rement recouverte de mauvaises herbes. Il y \u00e9tait \u00e9crit :\n\n>  _Site de la base de C\u00f4te Rouge (1968-1987)_\n> \n> _Acad\u00e9mie chinoise des sciences 21.03.1989_\n\nLa plaque \u00e9tait minuscule, comme si on avait cherch\u00e9 \u00e0 faire oublier l'ancienne existence de la base plut\u00f4t que de la comm\u00e9morer.\n\nYe Wenjie marcha jusqu'au bord de la falaise, \u00e0 l'endroit m\u00eame o\u00f9 elle avait mis fin \u00e0 la vie de deux officiers. Elle ne contempla pas, comme ses compagnons, la lointaine mer de nuages. Son regard se porta dans une autre direction, sous la couche de nuages. Un petit village appel\u00e9 Qijia.\n\nLe c\u0153ur de Ye Wenjie battait furieusement, comme la corde d'un luth mena\u00e7ant de rompre. Un brouillard noir commen\u00e7a \u00e0 voiler ses yeux. Elle utilisa toute l'\u00e9nergie qui lui restait pour demeurer en vie encore quelques instants. Avant que tout devienne noir \u00e0 jamais, elle voulait voir une derni\u00e8re fois le coucher du soleil sur C\u00f4te Rouge.\n\n\u00c0 l'ouest, alors que les rayons du jour semblaient se fondre dans la mer argent\u00e9e, le soleil fissura les nuages et sa clart\u00e9 se r\u00e9pandit dans le ciel, l'illuminant d'une magnifique couleur rouge sang.\n\n\u2014 Le cr\u00e9puscule des hommes, murmura faiblement Ye Wenjie.\nLe traducteur remercie Jo\u00eblle Gaffric pour sa relecture attentive et passionn\u00e9e, et Lin Chieh-an, docteur en astrophysique au centre CEA de Saclay, pour ses pr\u00e9cieuses corrections et suggestions.\nOuvrage r\u00e9alis\u00e9\n\npar le Studio Actes Sud\n\nCe livre num\u00e9rique a \u00e9t\u00e9 converti initialement au format EPUB par Isako www.isako.com \u00e0 partir de l'\u00e9dition papier du m\u00eame ouvrage.\n\n# Sommaire\n\nCouverture\n\nLe point de vue des \u00e9diteurs\n\nLiu Cixin\n\nLe probl\u00e8me \u00e0 trois corps\n\nI - UN PRINTEMPS SILENCIEUX\n\n1 - LES ANN\u00c9ES DE FOLIE\n\n2 - LE PRINTEMPS SILENCIEUX\n\n3 - C\u00d4TE ROUGE I\n\n4 - LES FRONTI\u00c8RES DE LA SCIENCE\n\n5 - LE BILLARD\n\n6 - LE SNIPER ET LE FERMIER\n\n7 - LES TROIS CORPS : LE ROI WEN ET LA LONGUE NUIT\n\n8 - YE WENJIE\n\n9 - L'UNIVERS CLIGNOTE\n\n10 - SHI QIANG\n\n11 - LES TROIS CORPS : MOZI ET LE FEU ARDENT\n\n12 - C\u00d4TE ROUGE II\n\n13 - C\u00d4TE ROUGE III\n\n14 - C\u00d4TE ROUGE IV\n\n15 - LES TROIS CORPS : COPERNIC, LE RUGBY SPATIAL ET LE CIEL \u00c0 TROIS SOLEILS\n\n16 - LE PROBL\u00c8ME \u00c0 TROIS CORPS\n\n17 - LES TROIS CORPS : NEWTON, VON NEUMANN, QIN SHI HUANG ET LA SYZYGIE DES TROIS SOLEILS\n\n18 - LA R\u00c9UNION DES JOUEURS\n\n19 - LES TROIS CORPS : EINSTEIN, LE PENDULE ET LA GRANDE D\u00c9CHIRURE\n\n20 - LES TROIS CORPS : L'EXP\u00c9DITION\n\nII - LE CR\u00c9PUSCULE DES HOMMES\n\n21 - LES REBELLES DE TERRE-TRISOLARIS\n\n22 - C\u00d4TE ROUGE V\n\n23 - C\u00d4TE ROUGE VI\n\n24 - LA R\u00c9BELLION\n\n25 - LA MORT DE LEI ZHICHENG ET DE YANG WEINING\n\n26 - AUCUNE REPENTANCE\n\n27 - EVANS\n\n28 - LA DEUXI\u00c8ME BASE DE C\u00d4TE ROUGE\n\n29 - LE MOUVEMENT TERRE-TRISOLARIS\n\n30 - DEUX PROTONS\n\n31 - OP\u00c9RATION CITHARE\n\n32 - LE GUETTEUR\n\n33 - LES INTELLECTRONS\n\n34 - LA VERMINE\n\n35 - LES RUINES\n\nRemerciements\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"Guide To\n\nCaesar's Civil War\n\n49\u201344 BC\n\nAdrian Goldsworthy\n\nContents\n\nIntroduction\n\nChronology\n\nBackground to war\n\nThe First Triumvirate\n\nWarring sides\n\nLegion against legion\n\nOutbreak\n\nCrossing the Rubicon\n\nThe fighting\n\nCivil War\n\nPortrait of a soldier\n\nCaesar's centurions\n\nThe world around war\n\nA Mediterranean war\n\nPortrait of a civilian\n\nCicero and the Civil War\n\nHow the war ended\n\nThe Ides of March\n\nConclusion and consequences\n\nCivil wars and the end of the Republic\n\nGlossary\n\nFurther reading\nIntroduction\n\nThe Roman Republic and its growing problems\n\nAlthough originally a monarchy, Rome had become a Republic near the end of the sixth century BC. Such political revolutions were commonplace in the city-states of the ancient world, but after this Rome proved remarkably stable, free from the often violent internal disputes that constantly beset other communities. Gradually at first, the Romans expanded their territory, and by the beginning of the third century BC they controlled virtually all of the Italian peninsula. Conflict with Carthage, which began in 265 (all dates are BC unless stated otherwise) and continued sporadically until that city was utterly destroyed in 146, resulted in the acquisition of overseas provinces. By this time Rome dominated the entire Mediterranean world, having defeated with ease the successor kingdoms which had emerged from the break-up of Alexander the Great's empire.\n\nRoman expansion continued, and time and time again her legions were successful in foreign wars, never losing a conflict even if they sometimes suffered defeat in individual battles. Yet, the stability and unity of purpose which had so characterised Roman political life for centuries began to break down. Politicians started to employ violent means to achieve their ends, the disputes escalating until they became civil wars fought on a massive scale. A succession of charismatic military leaders emerged, men able to persuade their soldiers to fight other Romans. In 49 Julius Caesar, faced with the choice between being forced out of politics altogether or starting a civil war, invaded Italy. His success effectively tolled the death knell of the Republican political system, for after his victory he established himself as sole ruler of the Roman world. Caesar was murdered because his power was too blatant, and the assassination returned Rome to another period of civil war, which ended only when Caesar's nephew and adopted son Octavian defeated his last rival in 31. It was left to Octavian, later given the name Augustus, to create the regime known as the Principate, a monarchy in all but name, returning stability to Rome and its empire at the cost of a loss of political freedom.\n\nThe Roman republican system was intended to prevent any individual or group within the state from gaining overwhelming and permanent power. The Republic's senior executive officers or magistrates, the most senior of whom were the two consuls, held power ( _imperium_ ) for a single year, after which they returned to civilian life. A mixture of custom and law prevented any individual being elected to the same office in successive years, or at a young age, and in fact it was rare for the consulship to be held more than twice by any man. Former magistrates, and the pick of the wealthiest citizens in the state formed the Senate, a permanent council which advised the magistrates and also supervised much of the business of government, for instance, despatching and receiving embassies. The Senate also chose the province (which at this period still meant sphere of responsibility and only gradually was acquiring fixed geographical associations) to be allocated to each magistrate, and could extend the _imperium_ of a man within the same province for several years.\n\nRoman politics was fiercely competitive, as senators pursued a career that brought them both civil and military responsibilities, sometimes simultaneously. It was very rare for men standing for election to advocate any specific policies, and there was nothing in any way equivalent to modern political parties within the Senate. Each aristocrat instead tried to represent himself as a capable man, ready to cope with whatever task the Republic required of him, be it leading an army or building an aqueduct. Men paraded their past achievements and \u2013 since often before election they personally had done little \u2013 the achievements of past generations of their family. Vast sums of money were lavished on the electorate, especially in the form of games, gladiator shows, feasts and the building of great monuments. This gave great advantages to a small core of established and exceptionally wealthy families who as a result tended to dominate the senior magistracies. In the first century there were eight praetorships (senior magistracies of lower ranking than consulships), and even more of the less senior posts, but still only ever two consulships. This meant that the majority of the 600 senators would never achieve this office. The higher magistracies and most of all the consulship offered the opportunity for the greatest responsibilities and therefore allowed men to achieve the greatest glory, which enhanced their family name for the future. The consuls commanded in the most important wars, and in Rome military glory always counted for more than any other achievement. The victor in a great war was also likely to profit from it financially, taking a large share of the booty and the profits from the mass enslavement of captured enemies. Each senator strove to serve the Republic in a greater capacity than all his contemporaries. The propaganda of the Roman \u00e9lite is filled with superlatives, each man striving to achieve bigger and better deeds than anyone else, and special credit was attached to being the first person to perform an act or defeat a new enemy. Aristocratic competition worked to the Republic's advantage for many generations, for it provided a constant supply of magistrates eager to win glory on the state's behalf.\n\nHowever, in the late second century BC the system began to break down. Rome had expanded rapidly, but the huge profits of conquest had not been distributed evenly, so a few families benefited enormously. The gap between the richest and poorest in the Senate widened, and the most wealthy were able to spend lavishly to promote their own and their family's electoral success. It became increasingly expensive to pursue a political career, a burden felt as much by members of very old but now modestly wealthy families as by those outside the political \u00e9lite. Such men could only succeed by borrowing vast sums of money, hoping to repay these debts once they achieved the highest offices. The risk of failure, which would thus bring financial as well as political ruin, could make such men desperate. At the same time men from the richest and most prestigious families saw opportunities to have even more distinguished careers than their ancestors by flouting convention and trying to build up massive blocks of supporters. Both types were inclined to act as _populares_ , an abusive term employed by critics to signify men who appealed to the poorer citizens for support by promising them entertainment, subsidised or free food, or grants of land. The _popularis_ was an outsider, operating beyond the bounds of and with methods unattractive to the well-established senators. It was a very risky style of politics, but one which potentially offered great opportunities. In 133 a radical tribune \u2013 the ten tribunes of the plebs were magistrates without military responsibilities who were supposed to protect the interests of the people \u2013 from one of the most prestigious families, Tiberius Sempronius Gracchus, was lynched by a mob of senators when he tried to gain re-election to a second year of office. In 121, his brother Caius, who pursued an even more radical agenda, was killed by his opponents in something that came close to open fighting in the very centre of Rome. Yet a small number of men began to have previously unimaginable electoral success, as many of the old precedents restricting careers were broken. From 104 to 100, a successful general named Caius Marius was elected to five successive consulships.\n\nIn the same period the conversion of the Roman army into a professional force fundamentally altered its relationship with the rest of society. Until this time the legions had been militia forces, all citizens who possessed a certain property qualification being obliged to serve when called upon by the Republic. The wealthiest, able to provide themselves with a horse and the necessary equipment, served as cavalry, the moderately well off as heavy infantry, the poorer as light infantry and the poorest rarely served at all. In a real sense the army represented a cross-section of Roman society under arms. For these men service in the army was not a career but a duty to the Republic. As men of property \u2013 most were farmers \u2013 they easily returned to civilian life after each period of service. However, as the empire expanded, wars tended to last longer and be fought further away, while there was a growing need for permanent garrisons to protect conquered territory. A decade of service in a garrison in one of the Spanish provinces could well mean ruination for the owner of a small farm. Service became increasingly unpopular and the eventual solution was to turn to men willing to make the army their profession. A soldier's pay was low, the conditions of his service extremely harsh, and a military career tended only to be attractive to the poorest citizens, who in the past had not been obliged to serve. Such men proved excellent soldiers, but when the war ended and their legion was disbanded they had nothing to return to in civilian life. The Senate refused to acknowledge this change, maintaining that military service was a duty requiring no formal reward, and made no provision to provide for discharged soldiers. Individual commanders began to demand land for their veteran soldiers, wanting to settle them in colonies on conquered territory. Soldiers started to become more loyal to generals who offered such rewards than to the Republic which neglected them.\n\nThe rise of the professional army was probably the most important of the problems besetting the Republic with which the Senate failed to deal, but it was by no means the only one. Italy's economy and society had been profoundly changed by Roman expansion and the influx of huge numbers of slaves. The population of Rome itself had swollen to 1,000,000 by the end of the 1st century BC, a high proportion of them without steady employment. _Popularis_ politicians who tried to address the problems of dispossessed farmers or the urban or rural poor were sure of winning support. All of these factors produced a dangerous instability. In 88 the consul Publius Cornelius Sulla led his legions to seize power in Rome when Marius tried to seize the command allocated to him. Civil war followed, leading to Sulla eventually becoming dictator for more than a year. After this, stability never really returned to the Republic for more than very brief periods, as attempted coups, political violence and civil war followed each other with monotonous regularity. Sulla was a member of an old aristocratic family that had fallen on hard times, and had to use extreme methods to achieve the distinguished position within the Republic that he felt his birth warranted. There were several other men from similar backgrounds who acted in a similar way, and the most successful of these was Caius Julius Caesar.\n\nChronology\n\n**106** Birth of Pompey and Cicero\n\n**105** Romans suffer massive defeat at Arausio at the hands of migrating German tribes\n\n**104\u2013100** Marius elected to five successive consulships to deal with German threat\n\n**100** Birth of Julius Caesar\n\n**91\u201389** The Social War, a widespread rebellion of Rome's Italian allies. Defeated only after heavy Roman losses. Roman citizenship is extended to nearly all the peoples of Italy\n\n**88** Marius attempts to take the eastern command away from Sulla. Sulla marches with his army on Rome, the first time any Roman commander has done this\n\n**87** Marius and his ally Cinna seize power in Rome, massacring their opponents. Marius dies of natural causes\n\n**83\u201380** Sulla lands in Italy and is joined by Pompey. Sulla defeats his opponents and wins the civil war. Sulla becomes dictator and publishes the proscriptions, lists of names of men who are to be executed. He attempts to reform the state, rebuilding the Senate's authority\n\n**79** Sulla retires\n\n**78** One of the consuls, Lepidus, stages a coup. The Senate uses Pompey to defeat him\n\n**73\u201371** An escaped gladiator called Spartacus rebels and forms a huge army of slaves. He defeats successive Roman armies and devastates much of Italy before he is finally defeated by Crassus\n\n**71** Pompey and Crassus camp with their armies outside Rome and demand the right to stand for election to the consulship\n\n**70** Consulship of Pompey and Crassus\n\n**67** Pompey given extraordinary command against the pirates\n\n**66** Pompey given extraordinary command against Mithridates of Pontus\n\n**63** The consul Cicero defeats the attempted coup of Catiline\n\n**62** Pompey returns from the east but fails to secure land for his veterans or the ratification of his Eastern Settlement\n\n**61** Caesar becomes propraetorian governor of Further Spain\n\n**60** Caesar returns and forms the 'first triumvirate' with Pompey and Crassus\n\n**59** Caesar's consulship\n\n**58\u201350** Caesar's campaigns in Gaul\n\n**58** Clodius forces Cicero into exile\n\n**57** Serious rioting in Rome. Pompey called upon to supervise corn supply\n\n**56** Crisis in the triumvirate averted by meeting of Pompey, Crassus and Caesar at Luca\n\n**55** Second consulship of Pompey and Crassus\n\n**54** Serious rioting in Rome. Death of Julia. Crassus invades Parthia\n\n**53** Crassus defeated and killed by Parthians at Carrhae\n\n**52** Milo's gang kills Clodius\n\n**51** Repeated attacks on Caesar's position in the Senate. Pompey passes law requiring a five-year interval between holding a magistracy and being appointed to a province. Cicero sent to Cilicia\n\n**50** Curio acts on Caesar's behalf in the Senate. However, Cato and other prominent senators struggle to ensure that Caesar will not be permitted to stand for the consulship without laying down his command. Pompey's position unclear for much of the year\n\n**49** The tribunes flee from Rome. Caesar crosses the Rubicon and the Civil War begins. Pompey chased out of Italy, but able to sail with most of his troops from Brundisium to Macedonia. Caesar goes to Spain and defeats Afranius and Petreius in the Ilerda campaign. Curio defeated and killed in Africa\n\n**48** Caesar crosses to Macedonia. Prolonged stalemate at Dyrrachium eventually broken when Caesar retreats. Pompey brought to battle at Pharsalus and utterly defeated. Pompey flees to Egypt and is murdered. Caesar pursues him and is besieged in Alexandria. Beginning of affair between Caesar and Cleopatra\n\n**47** Reinforcements arrive and Caesar is able to break the siege and defeat the Egyptian army. Later in the year he moves to Asia and defeats Pharnaces at Zela. Caesar returns to Rome and prepares to campaign against the Pompeian army mustering in Africa under Scipio, Cato and Juba\n\n**46** African war ended by Caesar's victory at Thapsus. Cato and Juba commit suicide, and Scipio is drowned. Caesar returns to Rome and celebrates triumphs, but departs for Spain in the autumn\n\n**45** Spanish War ended by Caesar's victory at Munda. Labienus, Pompey's eldest son killed. Caesar returns to Rome and establishes dictatorship\n\n**44** Caesar planning major Parthian expedition. However, on 15 March he is murdered by a conspiracy led by Brutus and Cassius. Octavian arrives in Rome and rallies support from Caesar's veterans. Antony given command in Cisalpine Gaul\n\n**43** Octavian initially fights Antony on the Senate's behalf, but later in the year they are joined by Lepidus to form the Second Triumvirate. They capture Rome and reintroduce the proscriptions, executing large numbers of prominent Romans, including Cicero\n\n**42** Brutus and Cassius defeated at Philippi\n\n**41** Antony visits Cleopatra in Alexandria and their affair begins (or at least becomes publicly known)\n\n**40** Antony marries Octavia\n\n**40\u201336** Antony's Parthian War\n\n**38** Sextus Pompeius wins naval victories over Octavian\n\n**37** Antony publicly 'marries' Cleopatra\n\n**36** Sextus Pompeius defeated at Naulochus near Sicily\n\n**32** Octavia openly divorced by Antony. Open civil war between Antony and Octavian\n\n**31** Octavian defeats Antony at Actium. Antony and Cleopatra escape, but commit suicide. Octavian undisputed master of the Roman world. In stages he creates the Principate, a veiled form of monarchy that will endure for over 300 years.\nBackground to war\n\nThe First Triumvirate\n\nFor nearly two years Sulla ruled as dictator with absolute power and only laid this down when he went into voluntary retirement. Before he did so, Sulla attempted to restore the Senate's position within the Republic, confirming its traditional powers and filling it with his supporters. He passed a law that was intended to prevent army commanders from following his own example and using their legions outside their own provinces without permission. The career pattern ( _cursus honorum_ ) followed by Roman senators was also to be regulated more closely. The Republic was not to be dominated by a few individuals, but guided by the collective wisdom of the 600 senators.\n\nSulla's reforms were reactionary, impractical and weakened by the example of his own rise to power, so that many Romans did not consider them to be legitimate. Most importantly Sulla had failed to do anything to cater for the demands of the army on a permanent basis, so that discharged soldiers continued to have no source of livelihood and were therefore still inclined to follow any commander who promised them land. The chaos of the civil war and the rapid collapse of the Sullan constitution fostered a continuation of political disorder and eventually the renewal of open war in 49. This period also had a profound influence on the careers and attitudes of the main protagonists in 49\u201345. Caesar himself first rose to prominence during Sulla's dictatorship, narrowly avoiding execution by the dictator when he publicly celebrated his relation by marriage to Marius at a family funeral.\n\nHowever, a far more dramatic role was played by Cnaeus Pompey, who in 83 came to the support of Sulla at the head of three legions raised from his family's estates and veterans who had served under his late father, Pompeius Strabo ('squinty'). At the time Pompey was only 23 and, having never held public office, had no legal authority on which to base his power. Fighting with distinction in Italy, Sicily and north Africa, Pompey was granted the title Magnus ('The Great') by Sulla, though this may have been more than a little ironic. After Sulla's retirement, the Senate continued to employ the services of this private citizen and his personal army to suppress an attempted coup in Italy in 78 and then to fight the last of Marius' adherents in Spain. Employing Pompey, rather than a legally appointed magistrate under their control, set an exceptionally bad precedent. Probably the Senate felt that, since Pompey and his legions existed, it was better to use him than risk his turning against them.\n\nIn 71 Pompey returned victorious from Spain, and decided to stand for the consulship for the following year. He was too young, and had held none of the normally required junior magistracies, but he kept his legions outside the city as a scarcely veiled threat. Marcus Licinius Crassus, who had just returned from suppressing Spartacus's slave rebellion, took the opportunity to retain his own army and in turn declared himself a candidate for the consulship. Crassus was exceptionally wealthy, his fortune based originally on property confiscated from Sulla's executed opponents. The Senate was forced to permit their candidature and the Roman people, who were on the whole well disposed to both men after their successes, duly elected Pompey and Crassus as consuls for 70. Thus Pompey at the age of 36 entered the Senate directly as a consul, an utterly unprecedented action. His military record was already spectacular, but, given his age, he clearly expected to be given further important tasks.\n\nSince Sulla's reforms, a magistrate remained in Rome itself during his year of office. He was then appointed as a promagistrate to govern a province. Former consuls, or proconsuls, were sent to the most important provinces while former praetors, or propraetors, went to the less significant areas. The appointment as governor was normally made for a year, but could, if the Senate chose, be renewed for additional 12-month periods at the end of this time. As governor, the promagistrate possessed supreme military and civil power within his province, dispensing legal decisions or leading an army as the situation required. He could not be recalled or prosecuted until his term of office expired. A governor's powers (or _imperium_ ), lapsed as soon as he re-entered Rome and he became a private citizen again, simply one senator among many.\n\nThe Senate had traditionally chosen the provinces for each new political year, although individual magistrates were then normally allocated a task by lot. In 88 Marius had arranged for a popular vote giving him the command in the Asian War, a move which prompted Sulla's march on Rome. In 67 Pompey employed the same method of a vote in the People's Assembly ( _concilium plebis_ ) to give him a wide-ranging command against the pirates plaguing the Mediterranean. A combination of careful organisation, massive resources, and a willingness to accept the surrender of pirate communities and resettle them elsewhere, allowed Pompey to achieve victory in under two months. In 66 another law was passed by the people sending Pompey to Asia to fight against Mithridates of Pontus. This meant that the existing commander in this war, Lucullus, who had achieved great success, was replaced in spite of the Senate's desire to leave him in charge. Since the war was virtually over before he arrived, it took little time for Pompey to complete the defeat of Mithridates, who committed suicide when his own son turned against him. Pompey then proceeded to campaign throughout the near east, for instance, intervening in a domestic squabble between the kings of Judaea. After a three-month siege, Pompey took Jerusalem. He and his officers went into the Holy of Holies in the Great Temple, although they declined to take any of its treasures. This was a great propaganda success, the Roman aristocracy always striving to be the first to do any spectacular deed. As well as his military operations, Pompey carried out extensive administrative reform of the east. Provincial boundaries were altered, cities founded or refounded with new constitutions and relations with client kingdoms regulated. Many aspects of his settlement would endure for over 500 years.\n\nPompey had acquired so much glory and plunder on his campaigns that he had no serious rival within the Senate, and there was growing fear of what he would do when he returned to Italy. Many wondered whether he might copy Sulla and seize absolute power by force. In fact, Pompey behaved in a manner that was scrupulously correct, disbanding his army almost as soon as it had landed in Brundisium, and returning to Rome to celebrate an especially lavish triumph. He seems to have simply wanted to take his place as one of the Senate's most important members, but he also had two immediate political objectives. The first was to gain formal approval for all of his reforms in the eastern provinces. The second was to secure grants of land for the soldiers who had served him so well. In spite of his tremendous prestige, and in part because he had spent so much time on campaign and so little at Rome, Pompey was a poor politician. His speech in the Senate fell flat, and he did not seem to know how to use his great reputation and wealth to achieve his ends. He was opposed, most notably by Crassus, who was jealous of Pompey's prestige; Lucullus, who resented having been superseded in the command against Mithridates; and Cato the Younger, who disliked the revolutionary nature of Pompey's career and was reluctant to see him prosper. Time and again this opposition thwarted any attempt to ratify Pompey's settlement or grant land to his veterans. The impasse dragged on for nearly two years and was finally resolved in a manner that astounded most senators.\n\nIn 60 Julius Caesar returned from Further Spain, where he had served as a propraetorian governor and campaigned with success against local tribes. Six years younger than Pompey, Caesar's career had been fairly conventional up to this point, although his lavish spending on games and public feasting, combined with his rakish lifestyle, had won him numerous political enemies. Having won the right to celebrate a triumph, Caesar hoped this honour would permit him to win the consulship for 59. However, candidates had to present themselves for election in the city itself, and a general, still in command of the troops who would march in procession behind his chariot during the triumph, was not permitted to enter Rome until the day of the ceremony. Unable to gain an exemption, Caesar gave up his right to a triumph, dismissed his troops, and entered the city as a civilian. Thwarted, his opponents arranged for one of the consular provinces for the next year to be the supervision of the forests and country paths of Italy, a command without any troops or opportunities for profit and glory.\n\nAround this time Caesar made approaches to both Crassus and Pompey and managed to reconcile them. Together the three men formed a secret political alliance, which is known by historians as the First Triumvirate. To cement the alliance, Pompey married Caesar's daughter Julia, a union which, for all its political inspiration, proved to be a remarkably happy one. In return for supporting his candidature, Caesar undertook to gain land for Pompey's veterans and to secure the ratification of his Eastern Settlement. Crassus paid off the massive debts Caesar had incurred in the promotion of his career, and gained a secure place as one of the most powerful men in the state. Caesar won the election and during his year of office was able to override his consular colleague, Lucius Calpurnius Bibulus. On several occasions large numbers of Pompey's veterans packed into the forum and voting assemblies, using threats or actual force to control the voting. One common joke at the time was that this year Rome had two consuls \u2013 Julius and Caesar. Together the three members of the triumvirate possessed massive patronage. Many senators owed them money, especially Crassus, who was highly skilled in using his fortune to win influence, and all had to go to the triumvirs if they wished to secure an appointment to any of the more senior positions in the army or government. Both Crassus and Pompey were highly satisfied and, in return, Caesar was granted a far more important province by popular vote. A special command consisting of three normal provinces, Illyricum, Cisalpine Gaul and Transalpine Gaul (modern-day Provence in southern France) was allocated to him for five years.\n\nCaesar departed for his province in 58, never to return to Italy until the beginning of the Civil War. Crassus had covered his immediate debts, but Caesar was in great need of money to further his career. Very much the junior partner in the triumvirate, he also needed military glory to rival Crassus and, especially, Pompey. At first he appears to have contemplated a Balkan war against the Dacian King Burebista, but the news of the migration of a Gallic tribe towards Transalpine Gaul shifted his focus away from Illyricum. Over the next years Caesar campaigned throughout Gaul, twice bridged the Rhine and marched into Germany, and led two expeditions across the sea to Britain. That island remained mysterious to the Romans, and the euphoria over Caesar's expeditions could be compared to the excitement that greeted the moon landing in 1969. Caesar won massive glory during his Gallic campaigns, and produced his _Commentaries_ , probably published in annual instalments, to celebrate his achievements. As well as gaining glory, Caesar became one of the wealthiest men in the world from plunder and sale of slaves, hundreds of thousands of whom were captured during the conflict.\n\nThough unable to leave his province without also laying down his command, Caesar took care to keep a close eye on affairs in Rome, and spent every winter as close as possible, overseeing the administration of Cisalpine Gaul. He supported a radical politician, Publius Clodius, a demagogue who employed a gang of political thugs to force his legislation through. Rome had no police force, nor was it permitted for troops to be stationed within the city, so the state had no force with which to combat this violence. Clodius passed laws that complemented the legislation of Caesar's consulship, but which also attacked prominent figures within the Senate. In 58 Cicero was forced into exile, a success Clodius celebrated by leading a riot which burned down his house. Next Clodius turned his attention to Pompey, a move that presumably was not sanctioned by Caesar. Pompey responded by backing another gang of thugs led by Titus Annius Milo. Running battles were fought in and around Rome as the city descended into chaos. In 57 Pompey sponsored a law recalling Cicero. Three days after Cicero's return, that is on 7 September, Pompey was given the major responsibility of overseeing the City's corn supply and once again displayed his considerable talents for organisation in rapidly remedying the situation. His return to the public eye provoked a renewal of rivalry with Crassus and it was clear that the triumvirate was coming under strain.\n\nCrassus went to consult Caesar in his province and, after some cajoling, Pompey travelled to join them in April 56. In the town of Luca the triumvirs, along with a hundred or so senators who had accompanied them to show their goodwill, held a conference in which the alliance was patched up. Pompey and Crassus would both stand for the consulship in 55 and, since both their fame and the presence of a considerable body of Caesar's soldiers on leave ensured success, they were able to arrange matters to the benefit of all three. Caesar's command was extended for five years, although there is some doubt as to precisely when in late 50 or early 49 it was to expire. Pompey received both the Spanish provinces, but in an unprecedented move was allowed to remain in Rome and command through subordinates. Crassus was given Syria, from which he planned to lead an invasion of Parthia, for it seems that he felt the need to rival the conquests of his colleagues. Aged almost 60, he was considered rather old for active command by Roman standards and there were doubts about the legitimacy of a war with Parthia, but the triumvirs were too strong for any opposition to stand much chance.\n\nIn 54 Julia died in childbirth, and Crassus left to join the army in Syria. The following year he was defeated by the Parthians at Carrhae, and then killed when his army was forced to retreat. In spite of these blows, Pompey appeared still to consider himself bound to Caesar and in 53 sent one of his Spanish legions to reinforce Caesar's army in Gaul. Rome continued to be plagued by political violence, as Clodius' and Milo's followers clashed with renewed frenzy. In 52 Clodius was killed and his supporters carried his body into the Senate House, where they cremated it, burning the building to the ground in the process. In the face of anarchy, the Senate appointed Pompey sole consul and charged him with restoring order, for the first time permitting troops to guard Rome itself. Milo was put on trial and forced into exile as order was restored.\n\nCaesar knew that he had many opponents in the Senate, chief among them Cato the Younger. In spite of his new wealth and the freedom with which he had tried to buy support, Caesar knew that a good number of influential men loathed him, and would not forgive him for his actions in 59. As a serving magistrate he was not subject to prosecution, but as soon as his office expired and he returned to civilian life this protection was withdrawn. He did not believe that he would receive a fair trial. During the Gallic campaigns Cato had even once suggested that he ought to be handed over to the Germans for war crimes. Defeat would mean exile and the end of his political career. To avoid this, Caesar wanted to go straight into a second consulship, after which he would be given another military command, perhaps against the Parthians. In this way he could continue to serve the Republic in a distinguished capacity.\n\nIn 52 Pompey passed a law which stipulated a five-year interval between a magistracy and a provincial command, although he specifically exempted Caesar in a clause apparently written in his own handwriting. However, around the same time he married the daughter of Publius Metellus Scipio, a known opponent of Caesar. Pressure on Caesar mounted, as incoming consuls lobbied to have him replaced in his province, since the war in Gaul appeared to be over. Pompey's attitude appeared increasingly ambivalent and the extension of his Spanish command gave him military might to match against Caesar. The latter was being forced into a corner. He had either to give up his command and trust to Pompey to protect him from the inevitable prosecution or to fight.\nWarring sides\n\nLegion against legion\n\nRome's civil wars split the state into factions, and the army with it. Since there were no ethnic, ideological or social differences between the rival sides, it was inevitable \u2013 even more than in any other civil war \u2013 that the organisation, tactical doctrine and equipment of their armies was virtually identical. The main strength of the Roman army lay in the legions, units with a paper strength of about 5,000. In theory the legions were recruited only from Roman citizens, but during the civil wars many non-citizens were enlisted to bolster numbers. In his _Commentaries_ , Caesar frequently emphasised the heterogeneous nature of the enemy armies, but he had himself formed an entire legion, Legio V Alaudae, from Gauls, only later giving them the franchise as a reward for distinguished service. Given the dominance of the Roman military system, some allied kings had remodelled their armies after the Roman style. King Juba of Numidia included four legions in his large army, while Deiotarus of Galatia formed two which would later be amalgamated and formed into Legio XXII Deiotariana as a fully fledged part of the Roman army.\n\nIn this period a legion consisted entirely of heavy infantry. It had no permanent commander, but the practice had evolved of appointing an officer, usually one of the general's representatives, or legates, to fulfil this role. Much of the unit's administration was overseen by the six military tribunes, probably assisted by a small staff. These were largely equestrians (the class immediately below the Senate and possessing similar property) and at that time many were career soldiers of considerable experience. The basic tactical unit of the legion was the cohort of some 480 men. There were ten of these in each legion, and the cohort in turn was subdivided into six centuries of 80. The century was led by a centurion, supported by an _optio_ (second-in-command), _signifer_ (standard-bearer), and _tesserarius_ (guard commander). Centurion represented a grade of officer rather than a specific rank and these men differed greatly in seniority. On several occasions Caesar mentions rewarding brave centurions by promoting them to a higher grade, often in a newly formed legion that would benefit from having experienced officers. One of the six centurions probably acted as commander of the cohort, either the man with longest service or the centurion of the senior century, the _pilus prior_.\n\nAll legionaries were equipped with the same basic defensive gear, consisting of a bronze helmet (most often of Montefortino or Coolus patterns), cuirass (usually mail but sometimes of scale), and a large semi-cylindrical bodyshield constructed from three layers of plywood to give it both flexibility and strength. The latter seem most often to have been oval in shape, but it is possible that the transition to a more rectangular shape was already underway. Such shields were heavy \u2013 reconstructed examples weighing in at 22lbs \u2013 but offered good protection. They could also be used offensively, the soldier punching forward with all his body weight behind the shield's bronze boss. We are told that one of Caesar's soldiers, in spite of having his right hand chopped off almost as soon as he had boarded a warship, was able to clear the deck of enemies by knocking them down with his shield during the fighting off Massila. A soldier's other offensive equipment consisted of a short sword, the famed _gladius_ , sometimes a dagger, and a heavy throwing javelin known as the _pilum_. The _pilum_ consisted of a wooden haft about 4 feet long, topped by a narrow iron shank 2 feet in length and ending in a pyramid-shaped point. When thrown, all of its great weight was concentrated behind this small tip, giving it formidable penetrative power. It was designed so that once it punched through an enemy's shield, the slim iron shank would slide easily through the hole made by the point and had the reach to wound the man behind. Soldiers may have carried two _pila_ on campaign, but only one on the day of battle itself. The doctrine of the period was to deliver a massed volley at very short range \u2013 some 15 yards or so \u2013 and follow this up with a charge, sword in hand.\n\nRoman legionaries were not simply soldiers, for many were trained as engineers or artillerymen. Such men remained with their cohorts until required, and were then formed in temporary units to complete a task. The Civil War would be marked by many remarkable feats of engineering.\n\nIn battle a legion most often formed in three lines, four cohorts in the first line and three each in the second and third. Intervals were maintained between each unit and the cohorts from the next line stationed to cover these gaps, creating something resembling a checkerboard formation. However, since all cohorts were armed uniformly, the legion was perfectly capable of fighting effectively in other formations, and we also hear of armies in four or two lines, although a single line was considered too brittle to be employed save in dire need. The legion was a very flexible force. Its structure and size made it an important subunit within the battle line, but one or several cohorts could as easily be detached for smaller operations. As with all armies throughout history, theoretical unit sizes were rarely reflected in the field. At Pharsalus the cohorts of Pompey's legions averaged around 400 men apiece, while Caesar's force was little more than half that size. Campaign attrition reduced one of Caesar's legions to less than 1,000 men during the Egyptian campaign.\n\nThe legions were the mainstay of any army, especially decisive in pitched battles, but both sides supplemented their numbers with allied soldiers or auxiliaries, fighting in their own traditional style. Such troops were especially useful in providing cavalry and light infantry. In most cases they were locally recruited and led by their own native chieftains. At first Caesar's auxiliaries came primarily from the Gallic and German tribes, and Pompey's from his provinces in Spain and his many clients in the east, but as the war progressed, troops were recruited wherever possible and the pattern became more complex.\n\nBy the end of the Gallic campaigns, Caesar commanded ten legions (numbered V to XIV). Two more, XV and I, the latter on loan from Pompey's Spanish armies, had been withdrawn earlier in 50 to be sent against the Parthians. The majority of these troops were seasoned veterans, utterly devoted to Caesar and confident in their own and their commander's ability. In support were bands of excellent Gallic and German cavalry. To match against this Pompey had seven legions garrisoning his Spanish provinces, although these had little actual combat experience. There were also the I and the XV which had not yet left for the east and were still in Italy, but as both had recently served under Caesar their loyalty appeared questionable. However, he boasted that he had only to stamp his foot in Italy for more legions to appear, and was also sure of the loyalty of the eastern provinces which he had reorganised just over a decade before. In the long term, Pompey could probably claim greater resources than Caesar, but it would take time to mobilise these into field armies.\n\nIn 49 Pompey was almost 58, but remained an extremely fit and active man, and others marvelled at the energy he showed in joining the training exercises of his soldiers. His military record was extremely good, even if he had made something of a habit of arriving in the last stages of a conflict to claim the credit largely won by someone else. He was certainly a brilliant organiser, as the campaign against the pirates, as well as, more recently, his supervision of Rome's corn supply, had shown. In his youth he had been a bold commander, on several occasions leading charges in person, but his aggression, in a properly Roman way, had always been based on sound preparation. However, although he was only six years older than Caesar, Pompey had spent the last decade in Rome and had not served on campaign since 62. His performance during the Civil War would suggest that he was past his best as a general. He was not helped by the presence of so many distinguished senators in his camp. Unlike Caesar, whose followers were undistinguished and whose authority was unchallenged, Pompey was always under pressure to alter his plans. Most of the senators who flocked to his cause had more prestige than ability, and on more than a few occasions proved a positive hindrance. The ablest of his subordinates, Titus Atius Labienus, had served with Caesar throughout the Gallic campaigns. It is probable that he had a prior connection with Pompey, for he defected from Caesar's camp at the beginning of the war. On hearing of this, the latter ordered his baggage to be sent on after him.\n\nCaesar failed to attract any distinguished supporters from the senior members of the Senate. Now in his early 50s, he was still very much at the peak of his ability, and was fresh from a decade of successful fighting in Gaul. His strategy during the Civil War, as in Gaul, was based on rapid offensives, sometimes in the face of great odds. Though often criticised for recklessness by modern commentators, it is important to emphasise that such boldness was characteristically Roman, and should not conceal that much preparation underlay these enterprises. Although subject to occasional epileptic fits, he was in other respects an extremely healthy and active man, capable of massive effort and rapid long-distance travel. Caesar promoted and lavishly rewarded any soldiers who distinguished themselves, but even more than this it was his remarkable charisma that ensured that his soldiers were devoted to him. Throughout the war, desertions from the Pompeian forces were common, but all of our sources claim that there were no defections in the other direction. Fighting a war to protect his own honour and status, Caesar's objective was clear and obvious, giving the Caesarian war effort a unity of purpose not displayed by the other side. Yet it also meant that it was much easier for him to lose. If Caesar were killed, or his army defeated so heavily that he was discredited, then the war would effectively have been over. Only the Pompeians could suffer defeat after defeat and still prolong the struggle.\n\nIt is hard now to say whether Pompey or Caesar was the better general. The vast bulk of our evidence comes, directly or indirectly, from Caesar's version of events. His _Commentaries_ obviously present his own actions in a favourable light, while dismissing those of the enemy. However, they also provide evidence that allows the wisdom of some of Caesar's decisions to be questioned. Yet, for the Romans the answer was obvious, for the most important attribute of a great general was that he won his wars. Caesar defeated Pompey, and in the end there was no more to be said.\nOutbreak\n\nCrossing the Rubicon\n\n'They wanted it. Were it not for the support of my army they would have passed judgement upon me in spite of my achievements.' (Caesar looking at the bodies of dead senators after Pharsalus)\n\nBy 50 the mood in Rome was increasingly tense. The fear was similar to that in anticipation of Pompey's return in 62, but probably even worse, for Caesar was perceived now as a more open revolutionary, and his province, with its large, veteran army, lay on Italy's own border. Many Romans feared that this force would be turned against the state in a bid for dictatorship. A much smaller group of senators, led by Cato and including many of the House's most influential members, was determined that Caesar should not be allowed to return to normal politics, since his new-found wealth and prestige would make it difficult to oppose him. Were he allowed a second consulship, it was feared that his behaviour this time would be even worse than in 59. Everyone realised that Pompey's attitude would be decisive, but his intentions remained unclear. Stopping Caesar from arranging to stand _in absentia_ (and so retaining his army) for the consulship required at the very least Pompey's inaction, while if it came to a war, he was the only one capable of matching Caesar's military might. Yet if Caesar was defeated and killed or exiled, this would remove Pompey's last serious rival, leaving him with massively greater power, influence and wealth than anyone else within the Republic. This in itself threatened monarchy, but Cato and his supporters clearly believed this to be the lesser of two evils. At worst Pompey was a less skilful politician than Caesar and so would have greater difficulty in exploiting his position, but it seems likely that they hoped in some way to negate him. Perhaps the only real chance for the Republic would have been to accept Caesar's return and continue to have two leading senators or _principes_ far outstripping their fellows and so balancing each other's power. Even if this had occurred, there was always the risk that the two would fall out at a later date and that a war would result. In the event, intransigence on both sides prevented any compromise.\n\nIn 51 Caesar had tried to have his command extended until the very end of 49, presumably so that he could then move directly into the consulship for 48, but the measure was successfully opposed in the Senate, in part because Pompey failed to support it. This was followed by several attempts to have Caesar recalled immediately, using the argument that the war in Gaul had already been completed. Pompey opposed these moves, and in March 50 made it clear that Caesar ought to be permitted the original extent of his governorship, no more and no less. The failure to support his old ally more fully encouraged the belief that there was a split between the two.\n\nIn the meantime Caesar had been employing the profits of his campaigns to buy influence and friends at Rome. One of the consuls of 50, Lucius Aemilius Paullus, allegedly received 36,000,000 sesterces (as a guide an ordinary soldier was paid 1,000 sesterces a year), enough to cover the great debts he had incurred in restoring the Basilica Aemilia (originally built by an ancestor) in the forum. Paullus did not support his colleague Marcellus in his attacks on Caesar. More active support was purchased from the tribune of the plebs Caius Scribonius Curio, at the cost of 10,000,000, which also went mainly to his many creditors. Curio was highly talented, but unreliable, having been involved with many of the scandals of the last decade, and had previously been vocal in his condemnation of Caesar. Now he proved vigorous in his support of Caesar's objectives throughout his year as tribune. In the Senate he argued that Pompey's Spanish command should end at the same time as Caesar's post, although it in fact had several years still to run. More than a few senators responded favourably to this idea, hoping that in this way open war between the two men could be averted. When the Senate finally voted on this proposal on 1 December 50, it was carried by 370 votes to 22. However, in the same session Marcellus arranged for them also to vote separately on whether Caesar and Pompey should be removed from their commands. A large majority was in favour of Caesar laying down his governorship, and as big a majority against forcing Pompey to do so. Rejecting the Senate's decision that both should give up their armies, Marcellus and a group of supporters went to call on Pompey. Giving him a sword, they requested that he take action to preserve the Republic. Pompey neither accepted nor declined this task, and the city's uncertainty deepened. Then, a few days later he seemed to have declared himself openly, and took command of two legions, I and XV. Veterans from his old armies were summoned to Rome.\n\nCurio's term as tribune expired later in the month, but another of Caesar's supporters, Mark Antony, had been elected to the office and continued his work. In the meantime Caesar had written to the Senate, recounting his many victories won on Rome's behalf and his other services to the state, reminding them that he had already been granted the right to stand for consulship _in absentia_ , and offering to lay down his command, provided that Pompey did the same thing. If he did not, then Caesar felt that he was obliged to retain his legions as protection against the faction opposed to him. The letter included the scarcely veiled threat that he was also willing to free Rome from the tyranny of this faction. On the same day that this was read in the Senate, Scipio, Pompey's father-in-law, proposed issuing a decree that Caesar must hand over his legions by a set date \u2013 probably some time in the summer \u2013 but the measure was vetoed by the tribunes. Another group of senators, this time headed by Caesar's father-in-law, Calpurnius Piso, asked leave of absence to visit Caesar in his province and negotiate with him, but this was refused. Curio acted as Caesar's representative and proposed various compromises, at first that Caesar would give up the main military province, Transalpine Gaul, later extended to Cisalpine Gaul. He would remain governor of Illyricum with command of just one legion, but must be allowed to stand _in absentia_ for the consulship. If the offer was serious, and we have no reason to doubt that it was, this would have made it virtually impossible for Caesar to fight a civil war and seize power by force. Pompey seems to have been tempted, but Cato and his associates so detested Caesar that they simply would not accept his standing for election without first becoming a private citizen, and therefore subject to prosecution. Another suggestion, supported by Cicero, was that at the same time Pompey should leave Italy and actually go to govern his Spanish provinces. One of the consuls for 49, Lucius Lentulus Cornelius Crus was violently opposed to any compromise, and continually insulted both Antony and Curio.\n\nOn 7 January 49 the Senate met and passed its ultimate decree, the _senatus consultum ultimum_ , which called on the magistrates to use any means to defend the state. Caesar's supporters among the tribunes felt threatened with physical assault if they remained in the city. Disguised as slaves, they were hidden in carts and fled north to join Caesar, as did Curio. In the coming months Caesar's propaganda would exploit the threats made to the tribunes of the plebs, for this office was held in particular respect and affection by the population as a whole. In the days to come Pompey and the Senate began to prepare the war effort against Caesar. Scipio was given command of Syria and Lucius Domitius Ahenobarbus, consul in 54 and a long-time opponent of Caesar, received the Gallic provinces.\n\nThe news reached Caesar at Ravenna on 10 January. He spent the day watching gladiators training and held a previously arranged dinner in the evening, but secretly issued orders for several parties of soldiers to travel in civilian clothes carrying concealed weapons to Ariminum (modern Rimini), the nearest town in Italy. With him he had only a single legion, XIII, and apparently some 300 cavalrymen. Late in the evening he excused himself to his guests, and then departed for Ariminum in a carriage drawn by two mules. One tradition claims that in the night they lost their way, and it was only after they had found a local guide that they returned to the right road and reached the river Rubicon, which marked the boundary of his province. Commanders were barred by law from leading troops outside their province without the Senate's express permission, so crossing the river would turn Caesar into a rebel. In some versions Caesar paused uncertainly for some time, discussing with his officers what he should do. One of these was Asinius Pollio, who later wrote a history of the war (now sadly lost), so it is possible that he was truly indecisive and that these accounts are not simply inventions intended to heighten dramatic tension. Less likely is the story that they were confronted by the vision of a god playing pipes. However long it took, Caesar crossed the Rubicon, uttering the famous line 'the die is cast' \u2013 _alea iacta est_ in Latin, although some versions claim that he spoke in Greek.\n\nCaesar and his men occupied Ariminum without a fight and were soon joined by the tribunes. There was widespread fear of what he would do next. Cicero's correspondence from the early weeks of 49 is filled with gloomy forecasts of the bloodshed which everyone was sure would accompany the advance of the Caesarian army. In Gaul, Caesar and his legions had fought very aggressively and often with extreme brutality, some sources claiming that over a million people had been killed in less than a decade. Perhaps, as some modern commentators claim, many expected the legions to behave in no less harsh a manner now that they had burst into Italy, and Cicero on one occasion even wondered whether Caesar would not prove more like Hannibal than a Roman general. Yet we should remember that nearly all Romans, including Cicero and many opponents, had revelled in Caesar's victories over foreign enemies. Cato had wanted to have Caesar prosecuted for breach of faith when he attacked during a truce, and was not primarily concerned with the massacre of barbarian tribesmen. Far more worrying to contemporaries was the precedent of every civil war and rebellion fought in the previous 40 years. Marius had massacred any opponents he could catch when he seized Rome in 87. Sulla had made the process more formal, with the proscriptions, long lists of names posted in the forum. Any citizen proscribed lost all his legal rights; this made it legitimate for anyone to kill them and in doing so they would gain a share of the victim's property that would otherwise go to the state. Rome's civil wars were not fought between rival political ideologies but rival individuals, and normally ended in the death of all those on the losing side. There was no reason to suspect that Caesar would be any different, and the political violence he had employed during his consulship only seemed to make this more likely.\n\nIn fact the war did not begin as anyone had expected. Caesar moved quickly, seizing towns with the limited forces at his disposal rather than waiting to gather his legions. He was largely unopposed, but the advance of his army was not accompanied by massacre or atrocity and his soldiers were under strict orders not to loot. There was a strange, phoney war quality to the first few weeks. Caesar in particular, was trying to show that he was still willing to compromise. Messages went back and forth as he suggested various compromises. Pompey and his allies replied by saying that they could not negotiate while Caesar commanded troops on Italian soil, and that he must return to Cisalpine Gaul before anything could be discussed. Yet Pompey did offer to leave for Spain once Caesar had laid down his command. Caesar refused the offer, perhaps not trusting the Senate or maybe feeling that he had gone too far to withdraw at this stage. Even so, both sides continued to claim publicly that they still hoped for a negotiated settlement, and were only thwarted by the enemy's intransigence.\nThe fighting\n\nCivil War\n\n'Blitzkrieg' \u2013 the Italian campaign \u2013 January\u2013March 49\n\nThe suddenness of Caesar's advance surprised and unnerved his opponents, just as he had intended. Pompey left Rome in the second half of January, declaring that it could not be defended. He was followed by most of the magistrates, including the consuls, who left in such haste that it suggested panic. Many Romans were still uncertain about just how firmly committed each side was to fighting, and this open admission of military weakness made many wonder whether Pompey could really be relied on to defend the Republic.\n\nThe first clash came at Corfinum, where Domitius Ahenobarbus had mustered some 30 cohorts of new recruits and planned to hold the city. This was in spite of Pompey's repeated pleas for Domitius to bring his men south to join his two legions (I and XV) at Capua. It was the first sign of the great divisions between the commanders opposing Caesar. Caesar's army now mustered two legions, for he had been joined by the XII, plus some Gallic cavalry, although some detachments from these units were probably still scattered as garrisons in the towns already occupied. In February Corfinum was surrounded and Ahenobarbus learned that Pompey had no intention of coming to his aid. Caesar tells us that Ahenobarbus then panicked and planned to slip out of the city and escape, abandoning his soldiers to their fate. Hearing of this, the garrison mutinied and surrendered themselves and the town. Ahenobarbus and the other senators were brought before Caesar, who made a great display of formally telling them of the grievances that had prompted him to march into Italy, and then allowing all who wished to, to go free. This was the first important display of Caesar's clemency ( _clementia_ ), a policy to which he would adhere throughout the conflict, in marked contrast to his opponents, who employed the more brutal methods normal in past civil wars. The ordinary soldiers and most of their officers took an oath of allegiance to Caesar and were taken into his own army.\n\nPompey had probably already decided to abandon Italy altogether and cross the Adriatic to Greece. In the eastern provinces he had many clients and allies who could provide him with soldiers. The only properly trained troops in Italy had until recently been serving under Caesar, and Pompey was reluctant to test their loyalty so soon by advancing on the enemy. With Domitius's army gone, he was in any case outnumbered. He would go east, recruit and train a vast army and fleet and deal with Caesar in due course, returning to invade Italy by sea. Often he is supposed to have declared that 'Sulla did it, why shouldn't I?', referring to the dictator's successful return from the east in 83. Militarily this plan made perfect sense, but politically it was very damaging. Many Romans felt abandoned, and one senator sarcastically reminded Pompey that if he was short of soldiers to fight Caesar then perhaps he ought to start stamping his feet.\n\nPompey retreated south to Brundisium. Caesar followed in pursuit as quickly as he could, but a large number of recruits and senators had already been shipped across the Adriatic before he arrived outside the city. Pompey remained with about two legions, waiting for the ships to return and complete the evacuation. Caesar's engineers began to supervise the construction of a huge mole, intended to block access to the harbour. At first a solid breakwater was built out from the shore, but in deeper water this was continually swept away, so instead they began adding large rafts to the structure. Pompey used merchant vessels mounting hastily constructed three-storey towers and equipped with light artillery to attack and hinder construction. Pompey's fleet returned before the mole was complete and were able to enter the harbour and embark the garrison. The city's gates were blocked and a small force left as rearguard to allow a smooth evacuation, but the citizens of Brundisium, either through hostility towards Pompey or fear of Caesar's men, helped the attackers in and the retreat became more hasty than planned.\n\nIn less than two months Caesar had seized control in Italy. Pompey had escaped, with the best of his soldiers, and many leading senators. At present Caesar lacked a fleet and was in no position to follow him. On 18 March he was back in Rome, trying to persuade as many senators as possible to convene. His clemency had surprised and relieved many who were neutral or wavering, though some were still convinced that this was simply a ploy and that, in time, Caesar's own cruel nature or that of his disreputable followers would prevail. When he led his soldiers to seize the state Treasury held in the Temple of Saturn, and Caesar threatened to execute the tribune Metellus who stood in his way, such fears seemed confirmed.\n\nCurio was sent with two legions to secure Sardinia and then Africa. Caesar himself decided to set out for Spain overland and defeat Pompey's legions there. These were the best of the enemy troops then in existence, but their strategic role was unclear now that Pompey had shifted his main focus to the eastern Mediterranean. Caesar is supposed to have claimed that he went first to fight an army without a leader, before going to fight a leader without an army.\n\n'An army without a leader' \u2013 the Spanish campaign \u2013 April\u2013August 49\n\nThe main Pompeian army in Spain was at Ilerda (modern L\u00e9rida) commanded by Lucius Afranius and Marcus Petrius. Between them they had five legions, 80 cohorts of Spanish auxiliaries \u2013 a mixture of both close- and open-order infantry \u2013 and 5,000 cavalry. The other two legions, again supported by auxiliaries, remained far to the west in Further Spain under the command of Marcus Terentius Varro. To face the force at Ilerda, Caesar was able to muster six legions, along with 3,000 cavalry of various nationalities which had served with his army throughout the Gallic campaigns, and the same number of recently recruited Gauls. Also mentioned is a force of 900 horsemen kept as his personal bodyguard, but it is not entirely clear whether these were included in the above total. Most of this force was sent on in advance under the command of Caius Fabius. As Caesar marched to join them, his tribunes and centurions offered to loan him money with which to pay the soldiers. Gratefully accepting this gesture, their commander felt that this commitment helped bond both officers and legionaries to his cause.\n\nIlerda lies on a ridge to the west of the river Sicoris (modern Segre). A small force covered the bridge outside the town, but the main Pompeian camp was situated further south on the same high ground, where they had gathered considerable store of provisions from the surrounding area. On arrival Fabius built two bridges across the river and crossed to camp on the west bank. Finding it difficult to gather food and fodder, he sent a foraging expedition of two legions back across to the eastern side, but this was threatened when one of the bridges unexpectedly collapsed and only rescued when a force was sent to its aid.\n\nCaesar arrived in June, and immediately advanced and offered battle at the foot of the ridge. When Afranius refused to be drawn, Caesar had the third of his three lines dig a deep trench \u2013 a rampart would have been more visible and would therefore have invited attack \u2013 and then camped behind it. It took several days to complete the defences of this camp to Caesar's satisfaction, and after that he attempted to seize the hill between Ilerda and the enemy camp. Three legions formed for battle, and then the front line of one advanced to capture the height. The Pompeians responded quickly, and, moving swiftly and operating in a looser order learned in fighting the tribes of Lusitania (roughly the area of modern-day Portugal), occupied and defended the hilltop. Both sides fed in reinforcements throughout the day, although the narrow slope only allowed three cohorts in the fighting line at any one time. In the end, Legio IX charged uphill and drove the enemy back for sufficient time to permit Caesar's army to withdraw. They had lost 70 dead, including the senior centurion ( _primus pilus_ ) of Legio XIV, and over 600 wounded, while the enemy had suffered 200 fatalities including five centurions.\n\nTwo days later heavy rainfall raised the level and power of the river and swept away the bridges used by Caesar's army, largely cutting them off from supplies. Coming under increasing pressure, Caesar's soldiers built boats of the type they had seen in Britain. Ferrying a legion across the Sicoris at night, they secured a bridgehead and permitted the construction of a new bridge, allowing the army once again to operate effectively on the eastern bank as well as the west, winning several small engagements. This success encouraged many local communities to join Caesar, providing new sources of food. His engineers then dug canals off from the main river, lowering its level and creating a serviceable ford for men on horseback much nearer to the army's camp than the bridge. More Gallic cavalry had arrived, giving Caesar's horsemen a considerable numerical advantage over the Pompeians which soon made it increasingly difficult for their foraging parties to operate.\n\nAfranius and Petreius decided that they could no longer maintain their position and so, using barges gathered from along the river as pontoons, they threw another bridge across the Sicoris and under cover of darkness crossed to the east bank, before turning south. Caesar's scouts reported this activity and in response he sent his cavalry to harass the enemy retreat, the rest of the army following at dawn. His men waded through the rudimentary ford, the infantry moving between two lines of pack animals to slow the flow of water. In the next days, Caesar's patrols discovered a more direct path than the enemy and by hard marching he was able to get in front of them. The enemy were now at a severe disadvantage, but Caesar refused the pleas of his officers and men to fight a battle, saying that he hoped to win with as little loss of Roman life as possible. Camped close together, the legionaries on both sides began to fraternise. Nervous of their men's loyalty, Afranius and Petreius massacred all of Caesar's men they could catch, though their opponent released all the Pompeian soldiers in his camp unharmed. They then made their officers and men swear an oath not to abandon the cause. In spite of this, within a short time, the generals, their troops cut off from all supplies, capitulated. The commanders were pardoned and allowed to go free, some of the army incorporated into Caesar's own and the rest disbanded. Varro and the two legions in Further Spain surrendered immediately on the approach of Caesar's forces. In a matter of months, and through a mixture of boldness and skilful manoeuvre, Caesar had overrun Spain at minimal loss to himself. Appointing Quintus Cassius Longinus as governor \u2013 an extraordinary appointment for a man serving as tribune of the plebs \u2013 he returned to Italy.\n\nMassilia \u2013 spring\u2013summer 49\n\nEn route to Spain, Caesar had been refused admission to the great trading city of Massilia (modern Marseilles) by the authorities there, who claimed that they wished to remain neutral. The claim was immediately discredited when Domitius Ahenobarbus sailed into the harbour and took command of the city. Leaving Caius Trebonius with three legions and Decimus Brutus with a fleet to prosecute the siege, he had then moved on to Spain. The defenders proved very active, displaying especial skill in the small but fierce naval actions fought outside the harbour. Yet the attackers persisted in spite of heavy damage inflicted on their siege works by enemy sorties, and in the end the Massiliotes decided to surrender. Ahenobarbus learned of this, and fled by sea. Caesar, returning from Spain in late September, was present to accept the surrender, installing a garrison of two legions, but generally being lenient to the city.\n\nCurio in Africa \u2013 spring\u2013summer 49\n\nCurio occupied Sicily without fighting, and then crossed with three legions to Africa, where the governor, Publius Attius Varus had declared against Caesar. The latter was supported by the Numidian King Juba, who commanded a large, if sometimes unreliable army. Curio had little military experience \u2013 none at all of high command \u2013 and was considered brilliant but unreliable by most contemporaries. His army consisted of troops originally raised by the Pompeians who had taken the oath to Caesar after their surrender at Corfinum. Curio landed successfully near Utica, surprising the enemy, and soon came into contact with Varus' army. The two sides formed for battle on either side of a steep valley. Varus' brother, Sextus, had been at Corfinum and appealed to Curio's legions to desert and return to their original loyalty. However, the soldiers refused and, after a success in a cavalry skirmish, Curio led them in a bold, uphill attack which swiftly routed Varus' army. Encouraged by this success, Curio acted on what proved to be faulty intelligence, and attacked what he believed was a detachment of Juba's army. In fact, the bulk of the king's forces was there and, after an initial success, the Romans were ambushed and virtually annihilated. Curio was surrounded with the remnants of his troops on a hilltop and died fighting. Only a small fraction of the army, including the historian Asinius Pollio, escaped to Sicily.\n\n'A leader without an army' \u2013 Greece \u2013 January\u2013August 48\n\nThe report of Curio's defeat was not the only bad news reaching Caesar in late 49, for Mark Antony had suffered a lesser defeat in Illyricum. Even more serious was a mutiny involving four of his legions, and in particular Legio IX, at Placentia (modern Piacenza) on the river Po. The troops complained that many who had served throughout the Gallic campaigns were now long overdue for discharge, and that no one had yet received the donative of 500 denarii per man (double their annual salary) promised by Caesar at Brundisium some months before. Caesar's response was harsh, berating them for their impatience and declaring that he would decimate Legio IX, which meant executing one man in ten. In the end he relented, and only had the 12 ringleaders killed.\n\n**Dyrrachium**\n\nCaesar spent a short time at Rome, having been appointed dictator before he arrived and held the post for 11 days, using his powers to hold elections in which he was voted to the consulship. He was eager to move against Pompey, and near the end of the year went to join the army of some 12 legions along with 1,000 cavalry which had been assembled at Brundisium. Attrition meant that it was unlikely any of the legions mustered more than 3,000 men, and some units were closer to the 2,000 mark. Nevertheless, this was still a formidable total to ship across the Adriatic and to supply once there. Caesar exhorted his soldiers by saying that this next campaign would be the culmination of their labours, and then told them to carry only the absolutely essential baggage and to leave nearly all servants, slaves and families behind. On 4 January 48, there were sufficient ships to embark seven legions and 500 cavalrymen. The crossing was a great gamble, for Caesar had no significant naval force with which to oppose the vast Pompeian fleet, currently commanded by his old enemy and consular colleague in 59, Bibulus. Yet the enemy did not expect him to move in winter when the weather was poor, and Caesar landed without opposition at Paeleste in Epirus. Bibulus was alerted by the time the transport ships headed back to Brundisium and intercepted some of them. For the moment it proved impossible for Mark Antony to run the blockade and bring the remainder of the army across to join Caesar.\n\nCaesar was isolated and severely outnumbered by the enemy. Pompey had had more than nine months to muster his forces, and by this time they amounted to nine legions, supported by over 5,000 light infantry and 7,000 cavalry. A further two legions under Scipio were on their way from Syria. Pompey, always a great organiser, had taken care to gather plenty of food and fodder to supply his troops even in the winter months. Caesar's men had to make do with the little they had brought with them and whatever could be gathered from local communities. The situation was increasingly desperate, but Caesar was not really strong enough to open a full-scale offensive. Some manoeuvring took place, along with further attempts at negotiation, but there was no serious fighting. At one point he put to sea in a small ship during appalling weather, hoping to reach Brundisium and hurry his reinforcements over, but the weather proved so bad that he was forced to return to the shore. In was not until 10 April that Mark Antony managed to bring the remaining legions across the Adriatic. Pompey responded too slowly and failed to prevent the union of the two forces.\n\nCaesar had all 11 legions, but was still outnumbered and continued to have supply problems. Nevertheless he immediately resolved to make a bold attack on one of the enemy's major supply dumps at the port city of Dyrrachium. Outmarching the enemy, he managed to get between Pompey and the city, although he was not able to seize the latter. Pompey camped on the coast on a hill called Petra, overlooking a natural harbour which continued to allow him to receive supply shipments. Caesar's main camp was on a hill further north, but he continued to have supply problems as the harvest was not yet ripe and the region had been thoroughly plundered by the enemy. A line of hills ran around Pompey's camp and Caesar began construction of a line of forts connected by a ditch and wall, which were intended eventually to surround the enemy completely. This provided some protection for his patrols and foraging parties from the more numerous enemy cavalry, although in part the rugged ground made the operations of the enemy cavalry less effective. More importantly the willingness of Pompey, with a numerically superior army, to be hemmed in by the enemy would be a public humiliation, perhaps weakening the loyalty of his allies. Pompey replied by beginning his own line of fortifications parallel to Caesar's. There was considerable skirmishing between the two sides as they fought for possession of key positions or simply to hinder each other's progress. Pompey's more numerous army had the advantage of being on the inside and so having to build a shorter line, which eventually measured only some 15 miles to the more than 17 miles of the Caesarean works. In effect two armies were conducting a siege instead of the more normal forms of open warfare which, for Caesar's forces, were reminiscent of some of their conflict in Gaul.\n\nBoth armies, and especially the Caesareans, were on very short rations, but Pompey's army had a very large number of animals, both cavalry mounts and baggage animals, which began to suffer. Priority went to the cavalry and soon pack and draught animals were dying in great numbers. Caesar managed to dam the streams that carried water into the enemy positions. For a while Pompey's men survived by digging wells, but these did not really provide sufficient quantities and after a while the bulk of his cavalry and their mounts were shipped out. In the meantime, Caesar's legionaries dreamed of ripening crops and survived by eating barley rather than wheat, consuming far more meat than usual, and using a local root called charax to make a kind of bread. On seeing an example of this bread, Pompey is said to have declared that they were fighting beasts and not men.\n\nThe work went on as each side extended its fortifications further and further south. Pompey's army mounted a heavy attack which was easily repulsed by the troops led by Publius Sulla. The enemy retreat was so precipitate that some of Caesar's officers felt that an immediate all-out attack might have won the war there and then. However, Caesar was not present and praised Sulla for not going beyond his orders, feeling that such an important change of plan was the responsibility of the commander, not that of a subordinate or legate. On the same day a series of diversionary assaults were also made in some force. All failed at the cost of some 2,000 casualties, but especially heavy fighting occurred around a small fort held by three of Caesar's cohorts under Volcatius Tullus. Heavily outnumbered, the defenders suffered many wounds from the vast number of missiles shot into the camp by the hordes of slingers and archers supporting the attacking legion. Nearly all the defenders were wounded, four out of six centurions in one cohort losing their eyes, but somehow they held on. Caesar rewarded his officers and men lavishly, and they were granted extra rations, which at the time may have seemed even more satisfying than promotions and medals.\n\nSoon after this success, two Gallic chieftains, Roucillus and Egus, defected to Pompey, along with their closest followers. Caesar claims that they had been caught claiming pay for non-existent cavalrymen and feared punishment. The desertion raised enemy morale and provided Pompey with considerable intelligence about Caesar's dispositions. Using this information, he planned a powerful attack on an incomplete section at the southern end of Caesar's fortifications, the main body striking from his own lines, while detachments of light troops were taken by sea and landed behind the enemy. The attack achieved some initial success, but as Antony and then Caesar himself led up reserves the tide was turned and the enemy driven back. To regain the initiative, Caesar replied with a heavy counter-attack against a camp originally built by Legio IX, subsequently abandoned and now occupied by the enemy. His troops moved through dead ground and woodland and achieved initial surprise, breaking into the camp, but then things began to go wrong. One of the attacking columns got lost, mistaking another wall for the rampart of the camp and following it towards the sea. Pompey shifted reserves to the area. The leading attackers began to flee and the panic spread rapidly as most of the 33 cohorts involved dissolved into rout. Caesar tried to stop standard bearers as they fled past him \u2013 a common gesture used by Roman commanders to rally their men \u2013 but all rushed on, some leaving the standard in his hands, and one even trying to stab him with its butt-spike, prompting a bodyguard to slice off the man's arm. Losses amounted to over 960 men and 32 tribunes and centurions killed and more taken prisoner. Fortunately, Pompey failed to follow up his advantage so soon after the failure of his own attack, prompting Caesar to declare that he would have lost if only the enemy commander had known how to win. Labienus was allowed to take charge of the prisoners and had them all executed. Parading his army, Caesar publicly punished several of the standard-bearers and tried to inspire the rest. Judging that their morale was at a low ebb, he decided that they needed to be encouraged before he risked a major action. Evacuating his sick and wounded, Caesar decided to withdraw, sending his baggage train out of camp at night to conceal his intention from the enemy. The main column was then able to withdraw with little hindrance. Only a few Pompeian cavalry managed to catch up with the retreating army and these were defeated by Caesar's cavalry, closely supported by a picked unit of 400 infantry.\n\n**Pharsalus**\n\nCaesar headed into Thessaly, hoping to join up with a detachment under Domitius Calvinus which he had sent to intercept Scipio and his two legions. His army began to recover its strength as they passed through unplundered land and were able to harvest the now ripening grain. However, the reverse at Dyrrachium made some communities doubt Caesar's prospects of victory and the city of Gomphi refused to admit him or provide food. Caesar stormed the place and, for one of the very few times during the Civil War, allowed his men to sack the town. Some sources claim that the next day's march was more like a drunken revel, but also that the overindulgence appeared to cure much of the sickness from which many soldiers were suffering.\n\nPompey now had several options. One would have been to use his fleet to cross to Italy, now largely unprotected, but this would still mean that Caesar had to be defeated at some future date, and might be seen as running from his opponent. His personal belief was that they ought to shadow Caesar's army, but avoid open confrontation, hoping to wear him down by depriving him of supplies. This was a well-recognised Roman strategy, often known by the nickname of 'kicking the enemy in the belly'. However, there was massive pressure from the senators with the army to bring matters to a swift conclusion by bringing the enemy to battle. In early August the two armies camped near each other on the plains of Pharsalus. Several days were spent in the manoeuvring and formal challenges to battle that so often preceded the battles of this period. The pressure on Pompey to fight grew stronger and stronger. Many of the senators were so confident that arguments broke out over who should receive Caesar's post of Pontifex Maximus, one of the senior priesthoods in Rome, as well as what punishment was appropriate for those who had supported him or tried to remain neutral.\n\nOn the morning of 9 August Caesar was preparing to move his camp to another position where the army could more easily find food, when he noticed that the Pompeian army had advanced much further from the rampart of their camp than was usual, and had come fully onto the level ground by the river Enipeus. Quickly, the order was passed for Caesar's men to take off their packs and then re-form in columns, wearing only the equipment necessary for battle. Then the army marched out and formed up facing the enemy. Altogether Caesar had 80 cohorts totalling 22,000 men and 1,000 cavalry. He formed the legions into the usual three lines, with the most experienced units on the flanks. Legio IX had suffered heavily at Dyrrachium, so it was combined with the almost equally depleted but veteran Legio VIII into a single command and placed on the left, next to the river. On the right flank was Caesar's favourite unit, Legio X. The entire army was split into three commands, Mark Antony on the left, Cnaeus Domitiius Calvinus in the centre and Sulla on the right. Caesar was free to move to wherever a crisis developed, but in fact was to spend nearly all the battle with Legio X. The cavalry were all massed on the right.\n\nPompey's army was significantly larger, with the 110 cohorts in its three lines totalling 45,000 men and an enormous force of 7,000 cavalry on the left flank, supported by significant numbers of archers and slingers. Next to the cavalry were the two legions that had once served with Caesar, I and XV (now renumbered III). In the centre were the legions from Syria and on the right nearest the river the legions from Cilicia, plus some troops from Spain. The army was also divided into three commands, with Ahenobarbus on the left, Scipio in the centre and Afranius on the right. The main line was formed very deep by Roman standards, with each cohort deploying in ten ranks. Caesar's men must have been in four or five ranks, much closer to the Roman norm. Pompey had also given his infantry an unusual order, telling them not to advance to meet the enemy, but to remain stationary and throw their _pila_ as soon as the enemy came within range. Both of these decisions suggest that Pompey doubted the effectiveness of his own legionaries compared with Caesar's more experienced soldiers. Instead of relying on them, he planned to win the battle with his cavalry. Concentrated on the left flank, they outnumbered Caesar's horsemen by around seven to one. They would advance to open the battle, smashing their opponents and then wheeling round to take Caesar's infantry from the flank and rear. Labienus was in charge of this attack, and may even have devised the plan.\n\nThe enemy's deployment made the massive superiority of their cavalry obvious. To counter this, Caesar took a single cohort from the third line of each of his legions and stationed this force behind his own cavalry, probably echeloned back from the infantry line. This fourth line was concealed behind the horsemen and Legio X and not observed by the enemy. Both armies were now ready for battle, though some delay may well have elapsed as the commanders encouraged their men. For the battle Caesar's army were given the password 'Venus, Bringer of Victory' and Pompey's men 'Hercules, Unconquered'.\n\nThe battle began with an advance all along Caesar's line. Most of the Pompeians remained in position, but their cavalry surged forward against Caesar's horse, which gave way. During the charge and subsequent combat the Pompeians seem to have fallen into some disorder, the individual squadrons losing formation and merging into one great mass. Many of these horsemen were relatively recent recruits and neither officers nor men had much experience of operating in such large numbers, but there was always a tendency for this to happen if cavalry became too crowded. Suddenly, Caesar gave the signal for his fourth line to attack. The legionaries charged forward, yelling their battle cry and sounding their trumpets, and then using the _pila_ as spears. The result was almost immediate panic, which spread throughout the mass of enemy cavalry until there was a great stampede to the rear. The supporting light infantry were abandoned and massacred or dispersed by the legionaries. Pompey's main attack had failed.\n\nIn the meantime the main infantry lines had come into contact. Caesar's men had begun their charge at the usual distance, turning their steady forward march into a run preparatory to throwing their _pila_ , but had then noticed that the enemy was not moving. To prevent the cohorts from running too far, losing their formation and maybe wasting their missiles, the centurions halted the line. The nonchalance with which Caesar's men paused and redressed their ranks such a short distance from the Pompeians was another indication of their superb discipline. Reformed, the attack went in, the legionaries waiting until they were within 50 feet or so of the enemy before throwing a volley of missiles and charging sword in hand into contact. The Pompeians replied with their own _pila_ \u2013 though it is doubtful if the men in the rear ranks of each cohort can have thrown their weapons effectively. A hard struggle developed, the second line of each army, which always acted as close supports to the first line, soon being drawn in. However, the fourth line followed up its success by attacking the exposed left flank of the Pompeian infantry, throwing that section of the line into disorder. Caesar gave the signal for his third line to advance and renew forward momentum in the fighting line. The pressure on the enemy was too much. At first the Pompeians went back slowly, but more and more units began to dissolve into rout. Caesar sent officers out to ensure that enemy legionaries were permitted to surrender, although his men were allowed to massacre the foreign auxiliaries.\n\nPompey had left the battlefield almost as soon as his cavalry had been swept away. He had ridden back to his camp, instructing the guards there to maintain a careful watch, and then gone to his tent. Later, as the rout of his army became obvious, he laid aside his general's cloak and left for the coast. If these accounts of his behaviour are accurate \u2013 and there must be some doubt, as they all come from hostile sources \u2013 his command at Pharsalus was remarkably spiritless, and his behaviour, being the first rather than the last to despair, utterly inappropriate for a Roman general. Caesar also claims that his men were astounded by the luxuries that they discovered in the Pompeian camp, items more suitable for effete Orientals than true Romans, although again this could well be propaganda.\n\nPompeian prisoners numbered 24,000, with supposedly another 15,000 killed. Nine eagles (the standard of an entire legion) and 180 signa (the standard of a century) were among the trophies. Once again most Pompeians were pardoned by Caesar, and he is supposed to have been especially pleased when his men brought in Marcus Brutus, son of one of his former mistresses and later leader of the conspirators who would murder him. Many other Pompeians escaped, but Ahenobarbus died in the pursuit. Some fugitives went to north Africa, but Pompey travelled to Egypt. Caesar's own loss was comparatively light at 200 men, along with 30 centurions. Such a disproportionately high loss among these officers was not uncommon, the result of their aggressive and inevitably dangerous style of leadership.\n\nEgypt \u2013 September 48\u2013August 47\n\nCaesar rested only for a very short time after the victory. Mark Antony was sent back to Italy, while Domitius Calvinus went with three legions, mainly consisting of former Pompeians, to Syria. Caesar himself took Legio VI, now reduced to a mere 1,000 men, another legion mustering some 1,400 men, and 800 cavalry and rushed in pursuit of Pompey; until he had been taken or killed there could be no end to the war. News arrived that Pompey had gone to Rhodes and then taken ship for Egypt, hoping to receive aid in rebuilding an army.\n\nEgypt was wracked by its own civil war at this time, for the old King Ptolemy XI Auletes (or flute-player) had left the throne jointly to his son Ptolemy XII \u2013 a boy of about 14 \u2013 and his eldest daughter Cleopatra. The boy king was dominated by his advisers, Pothinus the eunuch and Achillas the commander of his army, a force that effectively included two Roman legions which had been in the province since 55 and had largely 'gone native'. Pompey's ship arrived on the coast near Ptolemy's camp and he appealed to the young king for aid. Since the king was unwilling to support a loser and eager to win favour with the victor, Pompey was lured ashore and murdered, the first blow being struck by a centurion who had served under him during his Asian campaigns.\n\nCaesar landed at Alexandria on 2 October 48, and was met by a deputation from Ptolemy that presented him with Pompey's head and signet ring. Caesar is supposed to have wept, distraught at the loss of his former friend and missing the opportunity of pardoning him. This emotion may have been genuine, as indeed may his alleged desire to spare Pompey, but it is equally possible that he simply wished to distance himself from the cruelty of an act from which he derived political benefit. Nevertheless he gave honourable burial to Pompey's remains, the tomb surviving to be desecrated by Jewish rebels in the second century AD. Preceded by his lictors \u2013 the attendants carrying the fasces, the bundle of rods and an axe that symbolised the power of Roman magistrates \u2013 Caesar marched in great pomp to the palace. This display enraged the volatile Alexandrians and provoked some rioting. Caesar's soldiers responded with force and, since the late king had recommended his children to Rome, declared that both sides in the Civil War should disarm and submit to his arbitration. Some time in the next few days Cleopatra visited Caesar. The most famous story is that she was wrapped up in a carpet or blanket and carried secretly into the palace by a faithful Greek attendant, before being unrolled in front of a mesmerised Caesar. Cleopatra was 21 \u2013 more than 30 years younger than Caesar \u2013 exceptionally attractive if not quite flawlessly beautiful, highly educated, intelligent, and with a fascinating personality. Thus began one of the most famous romances in history.\n\nIt was not long before Ptolemy's advisers felt that their cause could not compete with his sister's for Caesar's favour. Leading their army to support the mob of Alexandria, they besieged the palace, blockading Caesar's men for six months. His soldiers were close to panic when the water supply was cut off, but new wells were dug inside the compound and the crisis averted. Reinforced by Legio XXXVII, composed of former Pompeians, Caesar became bolder and attempted to seize the whole of the Pharos Island, on which the great lighthouse, one of the Seven Wonders of the World, was built. The skill of his Rhodian captains and sailors prevailed in a naval action fought within the Great Harbour, and allowed Caesar to land troops on the mole joining the Island. However, things began to go wrong as the enemy rushed reserves to the spot. Panic began with the crews of some ships who had landed to plunder and then spread to the legionaries. The boat carrying Caesar away was swamped by fugitives, forcing him to dive into the water and swim to safety, at the cost of abandoning his general's cloak.\n\nPtolemy had been held hostage by Caesar from early in the siege, and after this reverse Caesar decided to release him. The lad claimed to be reluctant to go, then promised to end the war, but, once he joined the army, promptly led it back to fight the Romans. The balance of power had shifted in his court by this time; Pothinus, assisted by Ptolemy's other sister Arsino\u00eb, had murdered Achillas and these two were the real powers behind the throne. In the meantime an army led by Caesar's ally, King Mithridates of Pergamum, had marched overland from Asia Minor to Egypt. Leaving only a small garrison, Caesar took the bulk of his 5,000 or so men, and sailed out of Alexandria's harbour to join his ally. Ptolemy's forces heard of this and attempted to prevent their juncture, but failed. In open manoeuvring, Caesar showed the superiority of his men over the enemy and in a rapid campaign trounced the Egyptian army. Ptolemy fled but drowned when the boat carrying him to safety capsized. Arsino\u00eb was exiled to Italy. Caesar returned to relieve Alexandria.\n\nThe war in Egypt was over, but for more than half a year Caesar had been out of contact with the rest of the world. The surviving Pompeians had had time to regroup and the Civil War would drag on. Yet, even though the war in Egypt was now complete, Caesar remained there for two months, allegedly spending his time feasting with Cleopatra. At one stage the Queen is supposed to have taken him on a luxurious cruise down the Nile. Militarily and politically, Caesar's inaction for this long period makes no sense. Perhaps he had never had a clear plan for what he should do once he had won the Civil War, or perhaps he was simply exhausted and could not resist a time of rest in fascinating company.\n\n_Veni, vidi, vici_ \u2013 the Zela campaign\n\nIt was not until late May or early June that Caesar finally stirred himself to move. There was bad news from Syria, and he sailed there with Legio VI, leaving the rest of his army to garrison Egypt. After the suicide of Mithridates of Pontus, his son Pharnaces had been left with only a small fraction of the old kingdom of Pontus. Seeing the disorder caused within the empire by the Civil War, Pharnaces decided to seize once more the lost territory, and invaded the old heartland of Pontus. Caesar's legate Domitius Calvinus had marched to oppose him, but suffered defeat. Pharnaces celebrated his victory in brutal manner, torturing and executing his prisoners, and castrating large numbers of young Romans who fell into his hands.\n\nThe forces at Caesar's disposal were small, consisting of the greatly reduced but veteran Legio VI, along with the survivors of Domitius's army. These included a legion of Deiotarus's Galatians which had fled before contact, another raised in Pontus, and Legio XXXVI which, although composed of former Pompeians, had fought well. Though outnumbered, Caesar characteristically chose to advance on Pharnaces, stopping five miles away from the enemy camp outside the town of Zela. In the night Caesar suddenly marched out and began to build a new camp on the opposite side of a valley to the Pontic army. On the next morning, 2 August 47, Pharnaces drew up his army in battle order. However, because the ravine separating them was steep, offering very bad going to any attacker trying to climb it, Caesar thought that this was simply a gesture of confidence, of the type commonly made by armies in this period, and so allowed his men to continue constructing the camp. He was amazed when Pharnaces led his troops down across the valley in a full-scale attack. The Romans were unprepared and hastily tried to put together a fighting line. Scythed chariots \u2013 all but useless against steady and properly formed troops \u2013 caused some losses among the dispersed Romans, before their teams were shot down with missiles. The fighting was long and bitter, but eventually Legio VI on the right flank punched through the enemy line and exploited the success to threaten the remainder of their army in the flank. Finally, the Pontic army dissolved into rout and the fleeing men were massacred by the vengeful Romans. The legionaries were so exhilarated that they crossed the valley and stormed the enemy camp, in spite of the resistance of its garrison.\n\nAlthough the battle of Zela proved hard-fought, it decided the war within days of the beginning of the campaign. Caesar is said to have commented on how lucky Pompey had been to make his reputation as a commander fighting such opponents. Later, when he celebrated his triumph over Pontus, the procession included placards bearing just three Latin words: ' _Veni, vidi, vici_ ' ('I came, I saw, I conquered').\n\nAfrica \u2013 December 47\u2013April 46\n\nAlthough the eastern Mediterranean was now settled, many problems had developed elsewhere during Caesar's absence. Cassius's behaviour in Spain had provoked rebellion, while in Africa, Scipio, Afranius, Labienus, Cato and many other die-hard senators had raised an enormous army supported by King Juba. There were also difficulties in Italy, made worse by the lack of communication from Caesar while he was in Egypt. Several of his supporters, notably the tribune Publius Cornelius Dolabella and Cicero's friend Caelius Rufus, had tried to rally support by advocating the abolition of debt and had had to be suppressed by Mark Antony, who was suspected of having acted too harshly and was replaced as the dictator's subordinate or Master of Horse ( _Magister Equitum_ ) by Marcus Lepidus.\n\nThere was also another mutiny among Caesar's veterans, news made all the more bitter because the ringleaders this time came from his own favourite, Legio X. The older soldiers wanted to be discharged and others complained that they had not received the rewards promised once their labours were at an end. These were their public grievances, but boredom may have played as big a part in provoking the outbreak, for throughout history armies have been more prone to mutiny when they are inactive. Caesar arrived back in Italy just as the mutineers were gearing themselves up to march on Rome. His behaviour amazed them when he rode into their camp and addressed them and asked what they wanted. When they shouted out that they wished to be demobilised, Caesar declared that they were discharged and informed them that he would let them have all that he had promised once he had won the war in Africa with other troops. Already stunned, the veterans were horrified when he addressed them as Quirites \u2013 civilians rather than soldiers \u2013 instead of comrades. It was an incredible display of Caesar's charisma and self-assurance, for soon the legionaries and especially Legio X were begging him to decimate them and take them back into his service.\n\nCaesar was impatient to embark on the African campaign, and spent the bare minimum of time in Rome before hurrying across to Sicily. Reaching the port of Lilybaeum with just one legion, only bad weather prevented him immediately embarking for Africa. Frustrated, he ordered his tent to be pitched on the beach as a public demonstration of his eagerness and confidence. This was in spite of reliable reports stating that Scipio had formed ten legions, supported by four of Juba's as well as many auxiliaries and 120 war elephants. Even when he finally did set sail on 25 December 47, Caesar still had only six legions and 2,000 cavalry. The operation was not well planned, the ships' captains not having been briefed as to where they should land, and this, combined with unfavourable winds, resulted in the fleet becoming scattered. When Caesar disembarked near Hadrumentum, he had only 3,000 legionaries and 150 horsemen. Perhaps his instinctive \u2013 and characteristically Roman \u2013 boldness was now verging on recklessness, or maybe the stress and exhaustion of so many years of command were taking their toll.\n\nYet Caesar proved just as capable at improvisation as he always had in the past. Messengers went back to Sicily and Sardinia and soon he had gathered most of his troops. On 3 January 46 he shifted the army's main camp to Ruspina, deposited his baggage there under guard and sent the rest of the troops out foraging. A few days later he led another similar expedition to gather food, taking 30 cohorts, 400 cavalry and 150 archers. The legionaries marched _expedita_ , that is, without packs and ready for battle, although the term is often mistranslated as 'lightly armed'. This time they were intercepted by a strong force of enemy cavalry and light infantry led by Labienus, which was later joined by another force under Petreius. Time and again the Numidian light cavalry swooped down on Caesar's line, throwing javelins before they swung round and retired. The legionaries charged forward to catch their attackers, but the Numidians easily evaded the men on foot. Whenever a cohort attacked it was exposed to more missiles from the infantry skirmishers, especially against the men's unshielded right flank. Casualties slowly mounted and progress was slow as the column moved across the open plain. Worse was the effect on the morale of the mostly young soldiers in the army. Their enemy was wearing them down and they were unable to strike back; some began to lose heart. One story, which probably refers to this fight, tells of Caesar grabbing hold of a standard bearer who was beginning to run, turning him around, and saying: 'Look, that's where the enemy are!'\n\nIn the meantime Labienus was riding up and down along the line, hurling abuse at Caesar's soldiers in the rough jargon of the camp. An experienced soldier, who had once served with Legio X but was now with another unit, managed to bring down Labienus's horse with his _pilum_ , and Caesar's old subordinate was carried from the field. The situation remained desperate. Now surrounded, Caesar stretched his cohorts out into a single line \u2013 a rare formation for a Roman army \u2013 and had alternate units face about, so that they could throw missiles or charge in either direction. They then charged and drove the enemy back for some distance. Quickly disengaging, the Caesareans used this temporary advantage to march on towards Ruspina, until they were again attacked by a new force of the enemy. Going round the line, Caesar urged his men to a last effort. As darkness was falling they launched an attack against an enemy-held hill blocking their line of march, driving the Numidians off and giving the whole army time to withdraw. Another version claims that Petreius withdrew because he wanted his commander-in-chief Scipio to gain the glory of defeating Caesar.\n\nCaesar became more cautious after this display of enemy strength, sending to Italy and Sicily for supplies and reinforcements. Later in January two legions, XIII and XIV, arrived, along with 800 Gallic cavalry and 1,000 archers and slingers.\n\nCaesar now moved forward to besiege Uzitta, forcing Scipio to come to its support. In spite of Pompeian naval activity, two more legions \u2013 VIII and X \u2013 arrived and Caesar felt confident enough to offer battle outside the town, although the enemy declined. He also took the opportunity of making an example of one of the tribunes of Legio X who had been heavily involved in last year's mutiny. This man, Avienus, had brought so many servants and horses, along with copious amounts of personal baggage, that he had required an entire ship simply to transport his household. Such extravagance was shocking at a time when Caesar needed every transport to be crammed with soldiers or supplies, and so he publicly rebuked Avienus and dismissed him from the army, along with another tribune and several centurions who were known to have been ringleaders in the mutiny.\n\nCaesar was still having supply difficulties and sent out several expeditions, one of which provoked a large-scale cavalry action in which Labienus was again defeated. The enemy was also sending out detachments to gather food, and Caesar attempted to intercept two legions isolated from the rest. The operation failed and as it retired his army was harassed by Labienus and a strong force of Numidian light cavalry and infantry skirmishers. Once again his own shortage of cavalry and light infantry made it difficult for Caesar to deal with such attacks. He gave orders that in future 300 soldiers in each legion should march _expedita_ , without their packs and ready for action. These troops operated in close support of the cavalry, forming a dense block behind which the horsemen could rally, rest and reform after a charge and so prepare to advance again. This tactic surprised the enemy and caused them to become more cautious. After a period of manoeuvring failed to provoke the enemy to battle and left Caesar camped in an area without an adequate supply of water, on the night of 4 April he led his army out and marched back to Thapsus. The town was still held by the enemy, and the threat prompted Scipio to come to their support, dividing his army into two camps some eight miles from the town.\n\nThapsus was not easy for an enemy to approach because a wide salt lake permitted access only across a relatively narrow plain from the west or south. Such restricted battlefields offered an effective counter to the numerous and fast-moving Numidian cavalry which might otherwise slip round the flanks of Caesar's army. Even so, when Caesar observed that Scipio had deployed his army with elephants in front of each wing, he took care to strengthen his own flanks. The legions deployed in the usual three lines, with the veteran II and X on the right and equally experienced VIII and IX on the left. He then divided Legio V Alaudae (or 'Larks'), the legion recruited from Gauls, into two sections of five cohorts apiece and stationed each group in a fourth line behind the flanks. His cavalry and light troops were divided into two and stationed on the wings.\n\nThe enemy advance was sudden, and Caesar busily rode around marshalling his army and encouraging the soldiers. They could see that the enemy army appeared confused, and the more experienced soldiers urged Caesar to attack immediately, confident that the Pompeians would not stand. Reluctant to enter a battle before his army was ready, Caesar rebuked them for their impertinence and tried to finish drawing up the lines. However, a trumpeter on the right with the veteran legions gave in to the soldiers and sounded the advance. The call was quickly taken up by the musicians in the other cohorts and the whole line began to surge forward. Centurions desperately turned about and tried to restrain the legionaries, but Caesar quickly realised that it was now too late, so he gave the watchword for the day \u2013 'Good Luck' ( _Felicitas_ ) \u2013 and spurred his horse towards the enemy. This at least is the version presented by whichever of Caesar's officers wrote the _African War_. Another tradition claims that Caesar began to suffer an epileptic fit and this was the reason why the battle started in such a disorganised way.\n\nHowever the battle began, it proved to be one of the swiftest of Caesar's victories. The enemy elephants were specially targeted by his archers and slingers, panicking the animals who fled, trampling their own troops. Elsewhere the Pompeian legions gave way with very little fighting. The attack of the Caesarean legionaries was ferocious and they mercilessly cut down even those enemies who tried to surrender. The veterans seemed determined to end the war once and for all. Caesar's casualties were very slight, compared to an enemy loss of many thousands. Cato committed suicide, as did Juba after he had fought a gladiatorial bout with and killed Petreius in a strange suicide pact. Scipio fled by sea, but drowned when his ship sank. Afranius, pardoned once before, this time was captured and executed. However, Labienus and Pompey's two sons escaped to Spain to continue the struggle.\n\nCaesar went back to Rome. In the past he had held the dictatorship for long enough to hold consular elections, but now the Senate voted him into the office for ten years. He held four triumphs over the Gauls, Egyptians, Pharnaces and Juba respectively. Yet, in November 46, he had to leave for Spain to fight the final campaign of the Civil War.\n\nSpain \u2013 November 46 \u2013 September 45\n\nCassius had proved both corrupt and incompetent as governor of Spain, alienating both his own troops and the local population. By the time he was replaced by Caius Trebonius, the situation was almost beyond redemption and the new governor was expelled by mutinous soldiers. Pompey's elder son Cnaeus arrived and was rapidly acclaimed as commander of the rebellious legions. He was soon joined by other Pompeians, including his brother Sextus and Labienus. A huge army of 13 legions and many auxiliaries was raised, although the quality of most of the new units was highly questionable.\n\nCaesar travelled rapidly as was his wont, covering the 1,500 miles to Corduba in just 27 days, and whiling away the trip by composing a long poem, _The Journey_. He had eight legions \u2013 the best probably being Legio V Alaudae which was experienced but still eager \u2013 and the old soldiers of Legio X, and 8,000 cavalry. The early stages of the fighting included a number of fierce skirmishes, but Cnaeus Pompey was reluctant to risk a battle. It was already proving the most brutal campaign of the entire conflict.\n\nThe Pompeians were suffering a continual trickle of deserters. Men accused of publicly stating that they thought Caesar would win were arrested. Of these soldiers 74 were executed and the remainder imprisoned. In the middle of March Pompey reached the hilltop town of Munda. Caesar followed in pursuit and camped nearby. The next morning, 17 March 45, he prepared to march after the enemy, but then saw that they were forming up in battle order on the high ground. Pompey had the bulk of 13 legions, a strong force of cavalry, and some 12,000 Spanish auxiliaries, half of them skirmishers. There was a level plain between this rise and the hill on which the Caesarean camp was located. His army marched out to deploy in the usual three lines, Legio X on the right and III and V Alaudae on the left, each flank guarded by cavalry. Once formed, the Caesareans marched down onto the open ground, expecting the enemy to do the same. The Pompeians did not move, keeping to the high ground so that the enemy would have to attack uphill. Caesar's men were as eager as they had been at Thapsus and he attacked anyway, in spite of the disadvantage. The fighting was fierce and determined. One tradition claims that the Caesareans began to waver and that he dismounted and charged alone against the enemy, rallying first his officers and then the remainder of the army. In the end, the veterans of Legio X started to drive back the enemy left. Pompey tried to shift troops from his right to plug the gap, but Caesar's cavalry renewed the pressure on that flank and pinned the legions there. At last the Pompeians broke and were slaughtered in great numbers. All 13 eagles were taken, and most of the enemy leaders, including Cnaeus Pompey and Labienus, were killed in the next few days. Some 1,000 Caesareans had fallen, a heavier loss than in any of the earlier victories and testimony to hard fighting. Munda was blockaded, the legionaries grimly fixing the severed heads of Pompeians to spikes topping their rampart. The mopping-up took several months. Caesar had won the Civil War, but now it remained to be seen whether he could win the peace.\n\nPortrait of a soldier\n\nCaesar's centurions\n\nNo personal account written by an ordinary soldier or junior officer survives for the Civil War. In the surviving narratives only a handful of men from the ranks are even mentioned by name, usually because they performed some conspicuous act of heroism. We know that soldiers were primarily recruited from the poorer classes. In normal circumstances most, if not all, were volunteers, but during civil wars many were probably unwilling conscripts. Soldiering had become a career, but the wages were low, lower than a man could earn as a labourer on the land or as a casual worker in the city. When Caesar doubled the pay of his soldiers, an ordinary legionary still received only 225 denarii (1,000 sesterces) a year. We do not know whether or not there were fixed terms of service, and the traditional maximum of 16 years may still have been in force, although during the civil wars some men served for more than two decades. Active campaigning, especially in a prosperous area, might bring greater rewards in the form of plunder, either taken individually or as the soldier's share in the booty acquired by the entire army. The most successful generals rewarded their soldiers lavishly. Conditions in the army were basic and the discipline brutal. At the whim of his centurion a man could be flogged, and many other crimes were punishable by death. At the end of their service, soldiers hoped to be provided with some source of livelihood. Usually this meant the grant of a plot of land, which suggests that many recruits were still coming from rural areas.\n\nLegionaries were men who had received little from the Republic or the Senate and had small interest in maintaining it. They identified far more strongly with their legion, the community in which they lived for many years. The legions of this period had a far greater sense of corporate identity than the temporary militia legions of the Middle Republic. They kept a number, and sometimes a name, for many years, marching under their eagle standard. Even at Munda, and in spite of earlier mutiny and indiscipline, it was the depleted veterans of Legio X who made the critical breakthrough. In fact these men would go on to play a prominent role in the civil war after Caesar's death. A good commander could inspire incredible devotion in his men. Caesar had been lavish with his plunder in Gaul, although during the Civil War he was less willing to permit his men to profit from defeating fellow citizens. Yet it was not simply financial self-interest that bonded his soldiers so closely to him. Caesar's army developed almost a cult of heroism, especially bravery performed in the sight of all. Their commander shared the rigours of campaign with his men, leading them on marches and in battle. In his speeches he called them comrades and spoke of their shared efforts. Even if our sources exaggerate a little when they claim that no Caesarean soldier ever deserted to the enemy, it is certainly true that defections were far more common from Pompey's legions. In his _Commentaries_ , Caesar repeatedly claims that he explained to his soldiers the wrongs done to him and the ill-treatment of the tribunes of the plebs. It is hard to know how far such concerns swayed his soldiers, and whether their prominence in his accounts has more to do with the intended audience for these works.\n\nBy no means all the soldiers fighting in the Civil War were Romans. Many non-citizens were swept into the legions, although each side tended to accuse the other of doing this. There were also large numbers of foreign auxiliaries, often serving under their own tribal leaders. The chieftains of Gallic, German and other tribal peoples displayed their status by the number of warriors in their personal following. They supported these men, some of whom might come from outside the tribe, feeding them and rewarding their courage, and in turn the warriors were obliged to fight for them. Their loyalty was to the person of the chieftain, rather than a tribe, nation or cause. Therefore the warriors fought for one of the sides in the Civil War simply because their chief had chosen to do so, and if the leader chose to stop fighting or change sides, then his warriors followed. When the Allobrogian brothers Roucillus and Egus defected, the warriors of their household automatically went with them. The bond between chieftain and warriors was exceptionally close and similar relations could develop even if the commander was not a native chieftain. Caesar had a bodyguard of 900 German and Gallic cavalry, and Labienus was followed by another unit of tribesmen who proved just as loyal to him. Later Cleopatra received a bodyguard of Gauls.\n\nWhile the ordinary legionaries receive little individual mention in our sources, a far more prominent role is reserved for the centurions. Time and again Caesar explains the success of his legions as being due to the courage and leadership of these officers. Even in disasters, the heroism of his centurions often provided the one bright note. It has often been assumed that the vast majority of centurions were promoted from amongst the ranks of the ordinary soldiers after long and distinguished service. Sometimes they are compared to the sergeant-majors who provide the backbone of many modern armies. There is virtually no evidence for this view, and certainly Caesar never once mentions a centurion promoted from the ranks, although he frequently refers to centurions being promoted to higher grades. It is far more likely that many centurions entered the army in that capacity, or as one of the junior officers within the century, and that they were recruited from the better-educated and reasonably well-off classes, rather than from the very poor who provided the mass of the ordinary legionaries. Both in Cicero and Caesar centurions appear as far more politically significant than the ordinary soldiers. Centurions were professional officers, rather than professional soldiers.\n\nThe deeds of several centurions are recorded in some detail in our sources. One such man was Scaeva, who made a name for himself at Dyrrachium. He was a centurion in one of the three cohorts holding an isolated fort that was attacked by an entire enemy legion supported by many archers and slingers. Fighting ferociously, Scaeva's shield is supposed to have been hit by 120 missiles. In the end, like many of his colleagues, he was struck by an arrow in the eye. Wounded, he called out to the enemy as if to surrender. When two men sprang forward to take such a distinguished prisoner, Scaeva killed one and sliced the arm of the other. His stubbornness inspired his men to continue the struggle. Caesar promoted him to the post of _primus pilus_ , along with a bounty of 50,000 denarii, and publicly praised him. Although he is not mentioned again, there is some evidence to suggest that Scaeva continued to have a distinguished military career. The tombstone of a former cavalry trooper from Gallia Narbonensis, which probably dates to the 30s BC, records his unit as the Ala Scaevae (cavalry regiment of Scaeva). Thus it seems that Scaeva went on to command a unit of Gallic auxiliary cavalry. The bold centurion described by Caesar would certainly seem an ideal candidate to act as chieftain for a group of such warriors.\n\nA former _primus pilus_ of Legio X figures prominently in Caesar's account of Pharsalus. This man, one Crastinus, had rejoined the army and now commanded a unit of 120 other veterans who had returned to service. He is supposed to have addressed the men, telling them that this battle would win back Caesar's position and allow them to retire again, their duty to him fulfilled. Turning to Caesar as he rode past, marshalling the army, he called out: 'Today, general, I shall earn your thanks whether I live or die.' Crastinus and his men led the charge, hacking their way into the enemy ranks. He was finally killed when a sword was thrust into his mouth and came out at the back of his neck. Caesar ordered men to search for him after the battle and after laying military decorations on his corpse \u2013 a rare thing, for the Romans did not usually give posthumous awards \u2013 buried him in a special tomb away from the massed grave of the other casualties.\n\nIn the army of the imperial period, it became very common for soldiers to erect engraved monuments recording the details of their military career, but this practice was only just beginning in the Late Republic. One of Pompey's centurions, a man called Granonius, from Luceria in Italy, died at Athens and was commemorated on a tombstone, probably sometime in 49\u201348. One man who may have served at the very beginning of the Civil War and had certainly fought under Caesar in Gaul, was Caius Canuleius of Legio VII. He was commemorated by his father on a monument, along with his brother who had been killed in Gaul at the age of 18 while serving with Legio VII. Some other men who are commemorated on tombstones and memorials from the late first century BC may well have fought in the Civil War, but little or no detail is given as to their length of service, so such information must remain conjecture.\nThe world around war\n\nA Mediterranean war\n\nThe Civil War affected the entire Roman world and ultimately destroyed a centuries-old political system. Many ordinary citizens were swept up into the armies, to serve in brutal conditions. Some died in battle, many more probably through disease and privations, while others were permanently crippled. At least some soldiers, especially auxiliary troops such as Gauls, were accompanied on campaign by their wives and children, who in turn suffered from the rigours of hard journeys and poor food. Yet it was not just soldiers and their families who were caught up in the conflict, for many civilian communities also suffered. An extreme case was a town like Gomphi in Thessaly, which Caesar permitted his soldiers to sack in an effort to restore their spirits after the retreat from Dyrrachium. In such circumstances Roman soldiers were extremely brutal, impossible for their officers to restrain even had they wished to do so. Caesar deliberately did not march into Corfinum at night in 49 because he did not trust his men to keep their discipline once they slipped off into the dark streets, and he did not wish to begin the campaign by plundering an Italian city. In 46, it was also considered a considerable achievement when Caesar's men were camped outside Hadrumentum and he was able to prevent them from plundering it.\n\nCommunities within the area of any of the campaigns were likely to suffer even if they were not subjected to a sack. The armies needed food in vast quantities. Most hoped to gain as much of this as possible from willing allies or to ship supplies in from elsewhere, but this was not always possible, especially as operations became more protracted. The needs of the local population mattered little when the armies sent out foraging expeditions to gather up all the grain and cattle they could find. There were also cases when the armies clashed within urban areas, often causing damage. The siege of Caesar's small force in Alexandria brought considerable destruction on the city, as buildings were demolished or set on fire.\n\nThe Civil War pitted legion against legion and made Rome vulnerable to foreign enemies. The most spectacular success was enjoyed by Pharnaces, until his army was destroyed at Zela. Other threats failed to materialise. Parthia had already invaded Syria once after its victory over Crassus, and seemed on the brink of doing so once again in late 50. Internal problems, during which the victorious commander at Carrhae was executed by the king as a potential rival, absorbed Parthia and delayed a new offensive. In the west, Caesar's conquest of Gaul had been quick and spectacularly successful, but the new province had not yet been properly consolidated. In fact there were no repeats of the rebellions that had broken out between 54 and 51 during Caesar's lifetime, but it did take another generation and further unrest before the province was fully pacified. Other allied countries sought to benefit from involvement in the Civil War. Deiotarus sent troops to aid Pompey in part because he was his client but also in the hope of securing his kingdom. Juba's attitude was similar. In Egypt the rivals in their own civil war tried to win favour from victors in the Roman conflict, Cleopatra gaining greatly from this and preserving some measure of independence for an area that had been in something of an anomalous position between ally and province for some time.\n\n'Sulla did not know his political alphabet' \u2013 Caesar's dictatorship\n\nIn the first years of the Civil War Caesar spent little time in Rome. In 46 he spent the greater part of the year there, having just been appointed dictator for ten years, and then after his return from Spain in October 45 remained there until his assassination on 15 March 44. The rest of the time Caesar was busy on campaign and ruled Rome through deputies. He was planning to leave once again to fight a war in the Balkans against Dacia and then to move east and confront Parthia in spring 44, a task which at the very least would have kept him away from Italy for several years. Yet, although he spent little time in the city, the period of Caesar's rule profoundly and permanently changed the nature of Roman political life. Caesar is supposed to have declared frequently that Sulla showed his political illiteracy when he resigned his dictatorship. Evidently, Caesar did not plan to withdraw from politics and so the Republic would inevitably be dominated by a single all-powerful individual on a permanent basis, the very thing Rome's constitution was supposed to avoid.\n\nHowever, it is much harder to say precisely what sort of position Caesar envisaged for himself in the long term. Civil war was renewed within months of his death, as his supporters sought vengeance against his murderers, and it was obviously in the interests of both sides to distort the record of what Caesar had done and, even more, what he planned to do. The conspirators needed to show that Caesar was bent on becoming an autocratic monarch who would have denied freedom to the rest of the Senate. Therefore he had to be killed because of what he would become. Caesar's heirs and supporters maintained the opposite view, pointing out that Caesar's rule had been and would have continued to be benevolent and that he had not wanted to become anything as un-Roman as a king. We have very few contemporary sources for details of the last months of Caesar's life, since most of Cicero's letters from that period were not preserved, and virtually all our accounts are later and inevitably influenced by the propaganda of both sides. Eventually Caesar's adopted son Octavian (Augustus) would make himself Rome's first emperor after his defeat of Mark Antony and Cleopatra in 31. As part of the adoption, his name became Caius Julius Caesar Octavianus. Therefore if either Caesar himself or Octavian passed a law, it would be recorded as a Julian law ( _lex Julia_ ), and if they founded a colony it would be a Julian colony ( _colonia Julia_ ). This makes it very easy to confuse the actions of the two men, and it is not always clear when Augustus actually implemented a measure that Caesar had planned.\n\nInitially Caesar was given a ten-year dictatorship with the additional title of _praefectus morum_ (prefect of morals). This was an invention that seems to have given him most of the powers of the censors, in particular the ability to appoint and expel senators and add names to the roll of citizens. In effect his power was greater than the consuls and, not only did he sit among the magistrates, but his opinion was always called for first. Even more importantly, he appointed all the significant magistrates. This provided him not only with control of the important offices of state, but also with the ability to reward the loyalty of his followers during the Civil War. The Senate grew enormously in size as Caesar's partisans were rewarded. By his death it had almost 1,000 members, compared with the 600 established by Sulla. Many of the new appointments were considered utterly unsuitable by traditionalists, for throughout the war Caesar's party had been seen as a haven for all the disreputable men and failures in the state. As Cicero joked grimly, how could you expect such men to guide the Republic when they could not even manage their own fortunes for a couple of months? Rumours circulated that in former years Caesar had gone round recruiting such wastrels by telling them that what they really needed was a civil war to restore their fortunes. Other appointments were unpopular, less because of the reputation of the individuals in question, but because of their nationality, for there were many aristocrats from the towns of Italy. In addition, Caesar, who had as consul granted citizenship to Cisalpine Gaul and during his campaigns extended the franchise to many noblemen north of the Alps, appointed several Gallic senators. This move again prompted caustic comments about barbarians, only recently bereft of their trousers and put into a toga, wandering aimlessly around the city as they failed to find the forum. Throughout his career Caesar had taken pride in always rewarding those who assisted him, and his actions as dictator simply confirmed this.\n\nSulla had filled his enlarged Senate with his partisans. Caesar to a great extent did the same thing. However, although most of the older and more famous senators who had sided against him were now dead, other, younger, men who had received Caesar's clemency were taken back and given honours and magistracies. Brutus and Cassius, the two main leaders in the conspiracy that would murder Caesar, had both fought against him in 49\u201348. Caesar appointed both men to the praetorship in 45. Yet all such honours were devalued by the freedom with which he doled them out to his supporters. The number of praetorships was increased to accommodate the number of men whose loyalty demanded a magistracy. The number of consuls was not increased from two, but Caesar encouraged men to resign before their year of office, allowing him to appoint others as suffect or replacement consuls. Such men were entitled to all the insignia of the full rank. An extreme case was Caius Caninius Rebilus, who had served as an officer with Caesar throughout much of the Civil War. When one of the consuls for 45 died on the last day of his office, Caesar appointed Caninius to hold a suffect consulship for less than 24 hours. Cicero quipped that he proved one of the Republic's most dedicated magistrates, never sleeping a wink while he held power, and that everyone needed to rush and congratulate the new consul before he had to relinquish it. In spite of this he, and many other senators, resented Caesar's disdainful treatment of the hallowed offices of the Republic.\n\nAt first Caesar was assisted by the dictator's traditional subordinate, the Master of Horse ( _Magister Equitum_ ), a post held by Antony and later Lepidus. In 45 eight prefects were appointed to aid the Master of Horse, marking another stage between Pompey's indirect rule of his Spanish provinces and the use of legates and prefects by the later emperors to govern the empire. Though large, the Senate lacked any real freedom of debate and was becoming distanced from the main decision-making processes that tended to occur in private and involved only Caesar and his trusted advisers. If senators had, in 59 and afterwards, resented the need to go to one of the triumvirs if they wanted to secure any post or favour for a client, now the situation was much worse. It is possible that Caesar had become too accustomed to supreme military command to adapt his style of leadership to the more tactful needs of political life. He had spent most of his life for more than a decade issuing orders, which had always ultimately resulted in success. Caesar knew his own abilities, trusting them far more than he trusted the capacity of anyone else. His manner often suggested impatience with display and the feelings of others whom he did not respect.\n\nOn one occasion he caused offence when he failed to rise from his work and greet the consuls as soon as they came into his presence. The people loved the lavish games and spectacles he staged in Rome, but did not like his habit of listening to and answering correspondence while he sat in his box.\n\nCaesar certainly displayed all the energy he had shown as an army commander during his dictatorship and the range of reforms he initiated during such a short time is truly remarkable. In some cases this consisted of tidying up an existing situation, as when he reformed the constitutions of the towns in Italy. The provinces too were affected, with renewal of the taxation system, usually in favour of the provincials. Plans were drawn up for a massive programme of colonisation throughout the provinces. This was to include not only the vast number of soldiers enrolled during the Civil War and now nearing retirement or no longer required, but also a significant number of the urban poor. Caesar had resisted pressure to abolish debt completely, the habitual desire of many citizens who lacked regular employment and lived in rented apartments, and arranged a more equitable system of repayment, but this measure would have eased the plight of many as well as adding to the number of prosperous citizens. It is uncertain just how many of these colonies were actually founded, for as mentioned above they are easily confused with the more numerous Augustan foundations. However, the programme was certainly already underway in Transalpine Gaul by the time of Caesar's death and had probably also begun elsewhere. Perhaps as many as 100,000 colonists were settled in Spain, Gaul and Africa.\n\nThe removal of part of Rome's population, which by this time was close to the one million mark, helped to relieve some of the city's problems. Laws were passed banning the trade guilds which men like Clodius and Milo had turned into gangs of thugs, although this measure may not have achieved much, as it still allowed 'legitimate' organisations. The number of poor citizens receiving a handout of wheat purchased and distributed by the state was reduced by more than half to only 150,000 recipients, many of those removed from the list being sent out as colonists. As a further measure to improve the food supply to the city, Caesar ordered the construction of a massive new harbour at Ostia, but this does not appear to have moved beyond the planning stage. A major building programme was begun in Rome, with older temples and public buildings being heavily restored and new monuments built. An entire forum complex, the Forum Julium, containing a new Senate House or curia to replace the one burnt down by Clodius' supporters, was begun. Occupying a prominent place within the complex was a grand temple dedicated to his divine ancestor, Venus Genetrix. Apart from making the city more splendid, projects such as this provided work for large numbers of the urban poor who might otherwise be a source of instability.\n\nApart from the grand parades that marked Caesar's triumphs in 46 and 45, he held many spectacular public entertainments, and in particular gladiatorial displays. On a more cultural note, a large public library was planned, supervised by the polymath (wide-ranging scholar) Varro. Caesar seems to have interested himself in practically everything, and probably his most enduring measure was the reform of the calendar. The traditional Roman calendar required constant attention from the priests, and had long since ceased to conform to the natural season, creating problems for Rome's political year. Caesar replaced this with a calendar of twelve 30-day solar months. In 46, 67 days were added to the year so that the new calendar would begin at the right time. The modern Gregorian calendar, created in the late sixteenth century and slowly adopted throughout the world, modified Caesar's system but left it substantially intact. The month of July still bears the name of the Roman dictator.\nPortrait of a civilian\n\nCicero and the Civil War\n\nThe Civil War presented the vast majority of Romans with a dilemma, for it was clear that joining either side or remaining inactive all had their perils. As we have seen, only a minority even among the Senate actually wanted war. The letters written and received by the great orator Cicero during these last months of peace and the years of war provide us with a remarkable insight into these times and the impact of the war on one man, his family and friends. The majority of these letters were to his long-time friend and correspondent Atticus, an equestrian who remained outside formal politics and yet seemed to know, and have friendly relations with, every prominent Roman in this period.\n\nMarcus Tullius Cicero was a 'new man', the first in his family to reach the consulship. His rise was almost entirely due to his skill as an orator, for his fame came more from winning famous cases in the courts than military achievements. Almost an exact contemporary of Pompey, Cicero had advanced his career through the same turbulent decades of civil war, dictatorship and attempted coups and revolutions. His great moment came as consul in 63, when he presided over the defeat and punishment of Catiline's conspirators. In spite of his fame, Cicero did not have the wealth, influence and client-base of a Pompey, Crassus or Caesar and would never be more than one of a number of distinguished senators. His vulnerability had been made all too clear in 58, when Clodius had forced him into exile for alleged illegal behaviour during his consulship. Although this was only temporary, it had proved that he could not rely on the support and protection of men like Pompey.\n\nOn 24 November 50 as tension grew in Rome, Cicero arrived back at Brundisium after a year-long tenure as proconsul of Cilicia in Asia Minor. This in itself was an indirect consequence of the machinations of Caesar's opponents, for the law decreeing a five-year interval between magistracy and governorship had created a shortage of provincial governors. As a result, men like Cicero, who had been consul over a decade before and had no real ambition to go to a province, were required to fulfil their obligations. In Cilicia he did his best to govern well, preparing the defences in case the Parthians, flushed with their success at Carrhae, launched the expected invasion. When this did not materialise he conducted a minor campaign against the tribesmen of Mount Amanus, for which he hoped to receive a triumph. In spite of the continuing Parthian threat, Cicero left at the first legal opportunity, arriving back in Rome just in the last period of peace. As a governor he still had _imperium_ , which he could not lay down if he wanted to be granted a triumph. In fact in the end he was only granted the lesser honour of a 'supplication', which was probably more in keeping with the scale of his success.\n\nCicero's correspondents had kept him well informed about the impending crisis. He had always been closer to Pompey than Caesar, though Pompey's failure to protect him from Clodius still rankled. When Pompey had been allied with Crassus and Caesar, Cicero had aided them, for instance, delivering a powerful speech in favour of extending Caesar's initial command in Gaul, while his brother Quintus had served as one of Caesar's legates in Gaul. Even before he reached Rome, Cicero was writing to Atticus saying that, publicly, he would vote with Pompey, although in private he would urge him to strive for peace. Caesar's supporters he saw as wastrels, most of them young and already associated with criminal activity. Yet he realised their strength, claiming that the only thing Caesar's side 'lacked was a good cause, since they had everything else in abundance'. Yet already Cicero could not help wondering why this situation had been allowed to occur. Would it not have been better to have opposed Caesar when he was weaker, rather than waiting until the Senate itself had granted him honours and power, making him a far more dangerous opponent. Caesar had been allowed to win office because the Senate had not effectively opposed him when he was weak and vulnerable.\n\nBy the middle of December Cicero was outside Rome and began to realise just how divided the Senate was over the issue. The vast majority, both in the Senate and the equestrian order, wanted peace. Most were also dubious about Pompey's intentions and what would happen once he defeated Caesar. As Cicero put it, 'from victory (for Pompey and the Senate) will come many evils, and certainly a tyrant.' Seeing just how strong was the desire for peace, he again wondered why they had allowed Caesar so much power if they were only going to fight him in the long run. When war finally came, Cicero's correspondence became filled with the rumours that circulated among the nervous citizens. He could not understand Pompey's decision to leave Rome and then not to make any effort to defend the city, seeing this as an open admission of weakness. He remained there for a few days, before retiring to the country. He corresponded with Pompey and Caesar as well as many other friends, most of whom urged him to declare himself more openly.\n\nMarcus Caelius Rufus was one of that wild, irresponsible generation that figured so heavily in the radical politics of the Late Republic, but he remained very friendly with Cicero, who had successfully defended him in court. During his tenure in Cilicia, Caelius had sent a series of gossipy letters packed with news and scandal from the city. Now he had joined Caesar, feeling that even if Pompey might have the more honourable cause, then Caesar certainly had the better army, which was what counted as soon as a political dispute spilled over into open war.\n\nPompey's decision to abandon Italy and instead build up his power in the east dismayed Cicero along with many others. It also added to his own uncertainty, made worse because he had still not laid down his _imperium_ as proconsul and therefore had the power to command troops. During these months his letters to Atticus were probably more frequent than at any other time in his life, and on some days he wrote more than one. Pompey's strategy seemed misguided, and yet still he felt a loyalty to him and gratitude, even if he did not really believe in his cause. Both Caesar and several of his associates begged Cicero to return to Rome, for he wanted to summon a legitimate Senate, and the presence of a distinguished ex-consul would add greatly to its authority. There is a strange, almost unreal quality about some of these letters, as Caesar's associates quote their commander's letters reporting that his army has cornered Pompey in Brundisium and telling of the progress of the siege. At the end of March, as Caesar returned to Rome, he called on Cicero and in person assured him of his respect and tried to persuade him to go back to Rome. Cicero said that if he came, he would say that the Senate could not approve of Caesar taking his legions to fight in Spain or Greece, and then lament Pompey's fate. When Caesar replied that he did not want such sentiments expressed publicly, Cicero explained that he could not go to Rome and speak under any other circumstances, which was why he chose to remain in the country.\n\nCaesar left soon afterwards for the Spanish campaign, and Cicero began to wonder about belatedly following Pompey, or perhaps travelling simply to stay out of the conflict. Caelius marched with Caesar and in April wrote to Cicero during the march, telling him that he ought not to join the enemy, for Caesar was already gaining a marked advantage. Around the same time Curio stopped at Cicero's villa en route to Sicily. Cicero found him as boastful and unrestrained in his speech as ever, and was disturbed to hear that Curio also believed that Caesar's clemency was a temporary ploy and that his true nature would eventually assert itself. In the end, after continued heart-searching, he decided to embark for Macedonia and join Pompey's army. His teenage son, also called Marcus, was already there, having volunteered to serve as a cavalry officer. What Cicero found in the Pompeian camp dismayed him, for the senators had become increasingly extreme, and spoke of extreme punishments not simply for Caesar's partisans, but also for anyone who had remained neutral. Pompey seemed to lack his old confidence and purpose and there was little sense of unity among the commanders. Illness kept him from the field at Pharsalus where the defeat confirmed his low opinion of the army. In the aftermath as an ex-consul still possessed of proconsular _imperium_ , Cato is supposed to have offered him command of the survivors, but Cicero declined and returned to Italy.\n\nCaesar's long stay in Egypt, and the lack of communication from him for months on end were incomprehensible. All Cicero wanted was for the war to end, and for at least some semblance of normality to return to Roman politics, but now that Caesar had won the war, he failed to end it utterly. Cicero waited near Brundisium for the victor to return, nervously wondering how he would be treated. In the event, Caesar proved extremely friendly, but even so Cicero spent increasingly little time in formal politics and more writing philosophical works. Part of him hoped that Caesar could guide the state and allow a gradual return to the proper institutions of the Republic. Yet the reality of Caesar's supremacy, and the dictator's continued reliance on the dubious individuals who had proved loyal to him in the past, steadily alienated him. Cicero was not involved in the conspiracy, although since the letters to Atticus for the months before Caesar's death were not published it is possible that his friend was implicated in some small way, but wished to conceal this by the time that the letters were published. He had high hopes of better things after the deed and, for almost a year, once again took a leading role in politics. His respect for Brutus was considerable, even though he had seen the ruthless and unscrupulous side of his character during his own governorship of Cilicia, where Brutus' agents had demanded four times the legal rate of interest on a loan given to the city of Salamis. Even so, he was not persuaded by Brutus when the latter argued that he should not encourage Octavian's ambitions, lest they raise up another Caesar. Cicero saw Antony as the real enemy, and was willing to deploy any means to destroy him. He failed, and himself perished, leaving his letters, speeches and philosophical books as a permanent memorial.\nHow the war ended\n\nThe Ides of March\n\nOn 15 February 44, Caesar's dictatorship and other powers were extended for life. A month later he was stabbed to death by a group of senators that included men who had served him for years, as well as pardoned Pompeians. Before discussing why the conspirators acted in this way, we must consider the difficult question of Caesar's own long-term aims \u2013 a subject of continuing scholarly debate and little agreement. It has often been stated that the Roman Republic failed and was replaced by the rule of emperors because the system, designed to regulate the public affairs of a city-state, could not cope with the changed circumstances of governing a world empire. There is some truth in this as we have seen, for during the last years of the Republic it became increasingly difficult to accommodate and regulate the competition between a few overwhelmingly powerful individuals. At the same time the Senate failed to acknowledge the emergence of a professional army or to do anything to provide for discharged soldiers who were no longer men of property, encouraging them to a closer bond with generals who offered them more. Yet, even under the empire, the institutions of Rome were to a great extent those of a city-state, but the emperors imposed more control of the system and encouraged the integration of first Italy and then the provinces. Institutions developed to support a permanent army, kept loyal to the emperor alone. Senators still held most of the senior positions in imperial government, although usually with authority delegated from the emperor, but the number of people, both citizen and non-citizen, benefiting from the regime was greatly increased. The empire, or Principate as it is more often known, gave Rome and the provinces a remarkable level of stability, broken only twice by civil war in the first two centuries of its existence, in comparison to the period from 133 BC to 31 BC.\n\nAugustus was Caesar's adopted heir, had risen to power as his father's avenger, and copied some of Caesar's innovations to create the Principate, although in other respects he learned from the dictator's mistakes and did things very differently. The Roman Republic faced many political, social, economic and military problems in the first century, and it is worth considering to what extent Caesar was aware of these. He had fought the Civil War to maintain his own honour and political status. Had a compromise been reached that permitted him to stand for a second consulship and go on to a further provincial command, the future would have been very different, with Rome dominated by two great men, Caesar and Pompey, instead of just one. This did not occur and, whether or not Caesar had long aimed at supreme power, he did achieve it through his victory. His reforms as dictator were wide ranging, but did they have the coherence of a clear plan to solve Rome's problems, whether or not the solutions in themselves were practical?\n\nThere are essentially two ways of viewing Caesar. The first is to see him as a man perceptive enough to understand that the Republican constitution could no longer function. Throughout his career he had taken considerable interest in the conditions of the poor in Rome and the native population in Rome's provinces, and realised that the territories could not be run simply for the selfish benefit of a tiny \u00e9lite in Rome. Observing the incompetence and weakness of the Senate as a group and of individual senators, and contrasting this with his own abilities, Caesar knew that the state needed to be guided by a single individual who could discern the general good and act accordingly. In this way he tried to bring to Rome the stability it would gain from the Principate, and failed only because the Romans were not yet ready for this revolution; and perhaps because he let his impatience show. If Caesar thought in this way, then he may not have been entirely unique. On several occasions Cicero had talked of the need for the Republic to have a rector, a powerful leader who would help to guide the Senate and magistrates in making decisions for the common good. He had hoped that Pompey might fulfil this role, but even at his most optimistic had lower expectations for Caesar. Cicero's rector was certainly a less powerful figure than the dictator Caesar had become.\n\nThe alternative is to see Caesar more as an aristocrat steeped in the traditions of the Senate than as a visionary. Caesar, like all men of his class and upbringing, wished to have a distinguished public career, holding high office and winning fame and glory on the state's behalf. Perhaps because his family had for generations been removed from the inner circle of the Senate, or perhaps just because he was aware of his own great gifts, his ambition was especially great, and he not only wished to succeed, but to achieve more than anyone else. This is the man who is supposed to have said that it would be harder to push him down from first to second place in the state than from second to last. He pursued his ambition with relentless purpose, adopting any radical measure to achieve his ends, even to the extent of fighting a civil war. By 45 Caesar had achieved his objective, for all potential rivals were dead and he was able to celebrate more and greater triumphs than any other Roman, permanently commemorating his achievements in a massive building programme. That he now had supreme authority in the state and the ability to reform the Republic were largely incidental. There was no grand plan for solving Rome's problems, for Caesar was either unaware of them or could not think of any way of solving them. Instead he wasted his energy with huge numbers of unconnected initiatives and reforms, tinkering with minor problems rather than confronting the real issues. It was not long before he wished again for the simple objectives of an army commander, and so decided to leave his political problems behind and instead to go and fight long wars in Dacia and Parthia.\n\nWe do not know what Caesar's long-term plans were, and the contradictory propaganda of both sides after his death will probably forever make these uncertain. Perhaps he was a mixture of the two extremes. Certainly there is no evidence that he planned ever to resign his considerable powers, but whether he thought of these as personal, or planned to create a permanent position of dictator, emperor or king and to pass this on to a successor is impossible to state with certainty. At the time many Romans certainly feared that this would happen, and at least some of the conspirators thought that they were striking a blow for liberty, in the sense of desiring that Roman aristocrats had freedom to pursue their political careers without the supervision of one all-powerful individual.\n\nBrutus, one of the leaders of the conspiracy and the man felt by all to have had the most altruistic motives, certainly acted because he feared what Caesar might become. Married to the daughter of Cato, and a learned and serious student of philosophy himself, Brutus objected to the idea of a dictator or king, but did not hate Caesar himself. Caesar was indeed very fond of Brutus, and had once had an affair with his mother, the lively and intelligent Servilia, whom he appears to have regarded far more highly than any of his other mistresses with the exception of Cleopatra. This relationship prompted rumours, undoubtedly false but no less persistent for all that, that Brutus was in fact Caesar's illegitimate son. Others among the conspiracy acted more from personal hatred, or in the case of his former officers, who included Decimus Brutus and Caius Trebonius, disappointment with their rewards.\n\nCaesar had already been voted many exceptional honours, not unprecedented but usually on a grander scale than any of his predecessors. Caesar's link with the goddess Venus seemed to be more public and was represented as closer than the claims of past commanders, such as Sulla and Pompey, to be especially favoured by the gods. A temple was dedicated to Caesar's Clementia, the clemency with which he had pardoned so many of his bitter enemies. In public Caesar was granted the right always to wear a laurel wreath \u2013 an honour which is said to have especially pleased him for he was concerned over his growing baldness \u2013 as well as the other robes of a triumphing general, and he sat on a gilded chair instead of the magistrate's simpler seat. Rumours abounded that he wished to go a stage further and become a king, perhaps after the model of the Hellenistic world where the monarch was considered to be a god. When a crowd hailed him as rex (king), he replied that he was 'Not Rex, but Caesar' for Rex was also a family name in Rome. Later he made great show of refusing a crown offered to him by the mob.\n\nYet his behaviour gave sufficient grounds for doubting his long-term intentions. He dressed in the long-sleeved tunic and high boots of the kings of Alba Longa, a long-vanished city that had been a rival of early Rome, and from the royal family of which the Julii Caesares claimed descent. Caesar, having lost his only legitimate child, Julia, had already adopted his nephew Octavian as his heir, sending the teenager to Greece to prepare for the eastern expedition, but it was not clear whether he was to inherit just his private possessions or also his position. Even more worryingly, Cleopatra had come to Italy and been openly installed in a big house as Caesar's mistress. A statue of the Egyptian queen was placed next to that of the goddess in Caesar's great temple to Venus. Wild rumours circulated about special legislation being planned to permit Caesar to marry her. After Caesar's death a boy was produced by the queen and Mark Antony, who claimed that he was Caesar's illegitimate son and named him Caesarion, claiming that the dictator had acknowledged him. There is no contemporary record of the child dating to before March 44, and considerable doubt must exist as to his actual paternity. Another rumour current at the time spoke of an ancient prophecy, part of the Sybilline Books that had often guided the Republic at times of crisis, which declared that Parthia would only be conquered by a king. The Senate is supposed to have been planning a decree that would grant Caesar the title throughout the empire but not in Italy itself. Yet, whatever his ambitions Caesar made no attempt to rule by force, dismissing his personal bodyguard and travelling through the streets of Rome just like any other senator.\n\nCaesar planned to leave Rome on 18 March 44 and, given the scale of his planned campaigns, would be most unlikely to return for several years. Brutus, Cassius and the more than 60 other conspirators decided that they must act. They were a disparate group, but had preserved their secret for several months. On the morning of 15 March (a date known as the Ides) there was some dismay when Caesar did not arrive at the Senate on time. Eventually he came and the Senate rose to greet him. The conspirators clustered round his chair, using the excuse of pleading for the recall of Publius Cimber. For a while the charade went on, but when Caesar stood to leave and tried to shake them off, the conspirators drew their knives, Casca striking the first blow from behind. Caesar died of multiple stab wounds. There was a final irony about his death, for Caesar's own Senate House had not been completed and the old _curia_ still lay in ruins from its destruction by Clodius's men. As a result, the Senate had assembled in a temple attached to Pompey's theatre complex. When Caesar fell, his body lay at the foot of a statue of Pompey.\nConclusions and consequences\n\nCivil wars and the end of the Republic\n\nAt Brutus' insistence the conspirators killed only Caesar. Mark Antony threw off his senator's toga to escape, not realising that he was not in danger, mingling with the crowd as the senators fled in panic. No one seems to have had much idea of what was going to happen next. Slowly and cautiously, apparently realising that there were not gangs of supporters bent on revolution and pillage, the Senate went back to the Capitol and spoke to the conspirators. The value of Brutus' reputation to the conspirators was now proved, for the vast majority of the senators were ready to listen to him. The more distinguished members, including Cicero, stood with the conspirators, signifying their support and after a few hours even Antony and Lepidus, Caesar's most important subordinates, appeared to be reconciled to the deed. The reaction of the population as a whole was less certain, for Caesar had always been popular with the poorer citizens, and there was some open protest when Brutus made a public speech explaining their motives.\n\nPerhaps the conspirators simply expected everything to return to normal. The dictator was dead, so the Senate and properly elected magistrates could resume their guidance of the state. The problem was that virtually no one could remember a time when the traditional institutions of the Republic had functioned properly. Even the oldest, and there were few enough of these left after the Civil War, had grown up with a world of dictators like Sulla and Caesar, the dominance of informal triumvirs and the ever present threat of revolution from men like Lepidus, Catiline and Clodius. Caesar's former supporters seemed willing to agree to a general amnesty for the conspirators. The latter made no demands for personal power, and although the leaders were soon given provinces, this was no more than their due as ex-magistrates. Brutus even granted Antony's request to hold a public funeral for Caesar. At this ceremony Antony read out Caesar's will, which included sizeable benefactions to the ordinary citizens, and, sensing their growing hostility to the conspirators, roused the mob to demand vengeance against the murderers of their hero. Some of Caesar's soldiers were making similar demands and most turned to Antony or Lepidus to lead them. The uncertain truce between the two sides continued for some months.\n\nA new factor arose when Octavian, formally taking the name Caius Julius Caesar Octavianus now that his adoption had been confirmed in the will, returned to Italy. Moving from Brundisium to Rome, he rallied a few of Caesar's veterans. He was just 19, but incredibly self-confident. Mark Antony failed to take him seriously, and anyway saw him as a rival for the loyalty of Caesar's supporters rather than as a useful ally. It was round about this time that he and Cleopatra brought the child Caesarion into the public eye, presenting an actual son of Caesar to counter the adopted heir. Antony soon left for Cisalpine Gaul, taking charge of an enlarged army \u2013 for part of the garrison of Macedonia was posted there \u2013 with which he was in a position to threaten Rome. To those senators who hoped for a return to peace and stability and were broadly sympathetic to the conspirators, Antony was clearly the greatest threat to peace, for Lepidus was cautious by nature and unlikely to act of his own accord, even though he had command of the legions in Transalpine Gaul and Nearer Spain. Cicero had his last great moment of glory, emerging as one of the most distinguished of the surviving senators to dominate the debates in the House. At this time he delivered a series of speeches attacking Antony in a way that was vitriolic even by the standards of Roman politics. The speeches were known as the Philippics, for he modelled them on the tirades directed at Philip II of Macedon (Alexander's father) by the great Athenian orator Demosthenes. Octavian was seen as a useful figurehead, who would help to draw support away from Antony. Cicero is supposed to have said that they would 'praise the young man, decorate him, and discard him' ( _laudanum aduluscentem, ornandum, tollendum_ \u2013 there is a rhythm and double meaning to the Latin which does not easily translate). Yet Octavian was building up his power and rallied a force of veterans from Legio VII and VIII, and was soon joined by two more legions which were nominally under Antony's command but answered the call of Caesar's heir.\n\nAt the beginning of 43 Antony reached Cisalpine Gaul, but was resisted by the governor from the previous year and one of the conspirators, Decimus Brutus. Antony's army was superior in both numbers and quality and Brutus was soon besieged at Mutina. The Senate resolved to send the two new consuls for 43, Hirtius \u2013 one of Caesar's old officers and the man who had completed his Gallic _Commentaries_ and possibly also written some of the books continuing the _Civil War Commentaries_ \u2013 and Pansa to relieve Brutus. Cicero and the other senators decided to employ Octavian and his legions to aid them, giving the youngster, who was not even a senator, proconsular _imperium_ , just as an earlier Senate had chosen to make use of Pompey and his private army in the 70s. The armies clashed in a confused battle at Forum Gallorum on 14 April 43, and after a hard struggle the arrival of fresh units forced Antony back with the loss of two eagles. Pansa was wounded by a missile during the fighting and died some days later. The army moved on to Mutina and attacked Antony's camp. At first things went well and they broke in, but then Hirtius was killed in the fighting near Antony's tent and Octavian and his men forced to retreat. Decimus Brutus was released from siege, but Octavian had no desire to welcome one of his father's murderers. Brutus began to journey to join the other conspirators in the east, but was killed during the journey.\n\nBoth consuls had died within a matter of days, and Octavian was now, effectively, in control of three armies, altogether some eight legions, plus cavalry and other auxiliaries. Rumours circulated at the time and later claiming that Octavian had had a hand in the deaths of both his colleagues. He moved south and stood successfully for election to the consulship for the next year, though he was probably very aware that the Senate was attempting to use him only as a short-term measure. Now that Antony was for the moment checked, they could begin to discard him, and rely instead on the conspirators. Letters began to pass between Antony, Lepidus and Octavian. After a while the first two joined forces, and later in the year all three met at Bononia. Together, at the head of a huge army \u2013 altogether nearly 43 legions, though not all were present \u2013 they seized Rome and on 27 November 43 had a tribune pass a law by which they became triumvirs with consular power to restore the state ( _triumviri rei publicae constituendae consulari potestate_ ) for five years. The wording was almost the same as the dictatorships adopted by Sulla and Caesar, save that this time there were three men instead of one. The need to avenge Caesar figured heavily in their propaganda, and the dead dictator was formally deified and a temple constructed for his cult. A comet seen in 44 was proclaimed as a clear sign that Caesar had ascended to heaven after his murder, and from now on Octavian was regarded as the son of a god.\n\nThere was far more of Sulla than Caesar about the triumvirs' behaviour, for this time there was no talk of clemency, and the lists of the proscribed were again posted. Some 200\u2013300 senators and several thousand equestrians suffered death as a result. Among them was Cicero, caught by Antony's horsemen as he fled in his carriage. His head, along with the hand that had penned the Philippics, was nailed to the speaker's platform in the forum. Many of these men were killed for political reasons, but the triumvirs needed money to support their huge war effort and plenty of names were added to the list simply to confiscate their property. In spite of this they still had to levy extraordinary taxation. What was left of the Senate was packed with the triumvirs' supporters and simply confirmed, often in advance, their actions. Preparing for war, and also keen to cement their own power, they took provinces. Antony received Gallia Comata (long-haired Gaul), the area conquered by Caesar, Lepidus had Transalpine Gaul and Spain, and Octavian was given Sicily, Sardinia and Africa. Octavian was also betrothed to Antony's step-daughter Claudia.\n\nMeanwhile, Brutus and Cassius had had time to prepare a large army, drawing on the provinces around the eastern Mediterranean just as Pompey had done. In the end they amassed some 17 legions, including some such as Legio XXXVI which had fought first for Pompey and then for Caesar in the last Civil War, and would now fight against Caesar's heirs. Cicero's son was with them, serving as a cavalry officer. Antony and Octavian brought 22 legions to oppose them in the summer of 42. They met in the twin battles of Philippi. In the first Antony routed the wing commanded by Cassius, who committed suicide without realising that Brutus had in turn smashed Octavian's legions. Various stories claimed that the latter had either fled in terror or been ill in his tent during this battle. A few weeks later the second battle was fought and, on this occasion, the Caesarean cause won an outright victory, Brutus emulating the action of his colleague. Most of the credit for the victory went, probably rightly, to Antony.\n\nAfter this victory Octavian and Mark Antony began gradually to ease out Lepidus, who was transferred to the province of Africa, while Octavian took Spain and Antony Gaul. Afterwards Antony went to the east to ensure the loyalty of the region and to secure provinces still threatened by the Parthians, who had begun to become more aggressive again. The son of Labienus had gone into exile at the king's court, and led a band of followers as part of a Parthian invasion of Syria. At the same time Pompey's younger son Sextus, who had escaped after Munda, had built up a considerable fleet in Sicily and was actively opposing the triumvirs. He was a problem most of all for Octavian, whose task it was to supervise Italy. One of Octavian's greatest tasks was to arrange the demobilisation of nearly 100,000 soldiers, a mixture of captured enemies and men whose service was up or who were no longer needed after the victory at Philippi. In 41 he began confiscating land throughout Italy to provide farms for these veterans, evicting many farmers, including the poet Virgil. Capitalising on the resentment this caused, while at the same time hoping to win over as many veterans as they could, Antony's formidable wife Fulvia and his opportunistic brother Lucius publicly rallied support against Octavian. In the autumn they raised an army, but were besieged at the town of Perusia. Excavations on the site have produced many moulded-lead sling bullets fired by both sides, which often contain political slogans and even more frequently extremely crude insults. It was not until the beginning of the next year that Lucius was forced to surrender, but during this time most of Antony's commanders in the west showed their allegiance to Antony.\n\nIt looked as if an open breach had occurred between the triumvirs which could only be solved by yet another civil war. Fighting began at Brundisium, but at the last minute the two leaders patched up their alliance. Fulvia had died of disease, so Antony married Octavian's sister Octavia. They confirmed the division of the empire, so that effectively Antony controlled the eastern Mediterranean and Octavian the west. A short-lived treaty was agreed at Misenum with Sextus Pompey, granting him pardon and acknowledging his power, but this was soon in ruins, since neither Sextus nor Octavian adhered to its terms. Antony busied himself with a Parthian expedition, while Octavian built up his naval power to confront Sextus. In 36, aided by squadrons sent by Antony, Octavian's admiral and close friend Marcus Vipsanius Agrippa defeated the Pompeian fleet at the battle of Naulochos fought off the coast of Sicily. Sextus fled to the east, where he was captured by one of Antony's officers and executed.\n\nOctavian's military resources had been built up considerably to undertake this conflict and were now markedly superior to Antony's. An abortive rising by Lepidus in Italy was swiftly defeated, and Octavian for once emulated his adoptive father's clemency. Lepidus was spared and allowed to live out the rest of his life in comfortable retirement, retaining his post as _Pontifex Maximus_ , Rome's senior priest. In the meantime Antony had launched a major invasion of Parthia, beginning the war which Caesar had planned. Despite initial success, his offensive bogged down as the enemy harassed his supply lines. During the subsequent retreat the Romans suffered heavy casualties. The war had been a costly failure, but Antony refused the aid sent to him by his wife Octavia, and instead publicly praised Cleopatra for her assistance. His affair with the Egyptian Queen became more open, and they paraded both Caesarion and their own children. Over the next years the fragile alliance between Octavian and Antony broke down altogether. Antony's obsession with Cleopatra made it easy for Octavian's propagandists to depict him as a man so dominated by a sinister eastern seductress that he had betrayed his Roman origins. His scornful treatment of the respectable, and Roman, matron Octavia only made this task easier. Octavian portrayed himself as the champion of all Italy ( _tota Italia_ ) against the eastern menace. War finally came in 31, and culminated in Antony's defeat at the naval battle of Actium. He and Cleopatra both escaped to Egypt, and commited suicide shortly afterwards.\n\nOctavian was now unrivalled master of the Roman world, commanding an enormous army of some 60 legions. Militarily, he was more secure than either Sulla or Caesar, but his actions soon showed that he had learned from the failures of both. When he returned to Rome in 29 he formally laid down his powers, dissolving the triumvirate. Eventually he created the system known as the Principate, but this evolved gradually and there were more than a few false starts along the way. At first his power was still too blatant, for he held the consulship each year, and there was resentment, especially whenever he left the city. It was at this time that he appears to have been planning to build an enormous palace on the Palatine Hill, with a monumental entrance approached along a new road from the opposite side of the hill to the forum and Rome's political centre. In time, Octavian's public position was made to seem less monarchic. He made considerable effort to disassociate himself from Octavian the triumvir, the man responsible for the proscriptions and other cruel and violent acts. Eventually he became, instead, Augustus, a name with deeply traditional associations, and the Father of his Country ( _pater patriae_ ). When it was finally built, his palace was less grand, in appearance more like an ordinary aristocratic house, and was approached through the forum along a road lined with the houses of other senators. To all intents and purposes Augustus was a monarch, for his power could not be opposed by any constitutional means. From the beginning the Greek-speaking eastern half of the empire referred to him as king ( _basileus_ ). Yet he managed to maintain the illusion that he was not the master of the state, but its servant, a magistrate like all other magistrates save that his authority, and his continued services to the state, were greater.\n\nIn its final form Augustus' powers rested on two chief elements. The most important, though the least public, was his 'power greater than any proconsul' ( _maius imperium proconsulare_ ). Pompey had enjoyed similar, though not quite as extensive, power during his brief command against the pirates in 67. During his second consulship he had been granted a massive province embracing all of Spain and yet been allowed to remain in Rome and govern through representatives. Augustus was granted the same privilege, but his province was truly vast, including most importantly Syria, Egypt and the frontier zones on the Rhine and Danube. Like Pompey and others who had dominated the state, Augustus's power was ostensibly given to him by the Senate, and renewed every five or ten years, but there was clearly never any possibility of its being withdrawn. Every province garrisoned by a legion, with the sole exception of Africa, formed part of the Emperor's province and was governed by his representative or legate. In most cases these were senators, but Egypt, the supplier of a high proportion of the grain consumed by the city of Rome, was governed by an equestrian, for it was too risky to grant such a command to a potential rival. A new senatorial career \u2013 and soon also an equestrian one \u2013 emerged in which traditional magistracies, which remained prestigious even if they lacked real power, were mingled with posts such as the emperor's legate. Like Caesar before him, Augustus effectively controlled elections to all significant posts.\n\nThe other, far more public, element of Augustus' formal power was the 'power of the tribunate' ( _tribunicia potestas_ ). The Roman people, especially the poorer citizens, had strong emotional attachment to the tribunes of the plebs, who had originally been created to defend them from the misuse of power by other magistrates. In this guise Augustus' was the people's champion. Through it he was able to summon the Senate or the Popular Assemblies and could impose his veto. In fact Augustus made little use of these powers, but he referred to them frequently, even numbering the years of his reign from the time this title was granted to him.\n\nUltimately Augustus' powers rested on military force. For the first time Rome received a permanent garrison. The emperor had his Praetorian Guard, and also formed a police force (the Urban Cohorts) and fire brigade (the Vigiles). All of these troops were kept directly under his personal control. He also took great care to ensure the loyalty of the army. Service conditions were fixed, as were the soldiers' legal status and rights. On honourable discharge each soldier was entitled either to a plot of land or a lump sum of money. This, along with the soldiers' pay, was funded by a special Military Treasury ( _aerarium militare_ ) which was supervised, and often subsidised, by Augustus. The problem of veterans looking to their commanders to provide them with some form of livelihood was at long last averted, and Augustus also took care that the legionaries' loyalty was focused on him and no one else. The men were paid by the emperor, swore an oath of loyalty to him, and, when they performed any feat of gallantry, received medals awarded by him.\n\nMilitary power lay behind the Augustan regime, but attention was rarely drawn to this. Most of the Republic's institutions persisted. The Senate was reformed and reduced in size to remove many of the less suitable men who had been enrolled in reward for dubious favours to the various sides in the civil wars. More Italians were included and in time senators would come from the aristocratic families of many of the provinces. Augustus attended the Senate as simply another member, if a highly distinguished one, pretending to be merely the 'first in the senate' ( _princeps senatus_ ) an old and thoroughly republican title. He encouraged the members to debate freely and to vote with their conscience. Augustus may have genuinely desired them to do this, but in practice this was a sham. Every senator knew that his future career depended on the emperor's favour, and so the vast majority said what they felt he wanted them to say. Both the senators and emperor wished publicly to pretend that Rome had not become a monarchy, politely ignoring the obvious reality. From early in his reign Augustus began to groom a successor, although the appallingly high mortality rate within the imperial family meant that quite a few individuals filled this role. When Augustus finally died in AD 14, his successor, Tiberius, had his powers formally voted to him by the Senate and at first feigned reluctance to take on the role. By this time scarcely anyone could conceive of, or remember, life without an emperor.\n\nAugustus succeeded where Caesar had failed. He had learned from his father's murder and tried to veil his power behind more acceptable titles. By 31 the population of all classes was also far more willing to accept the rule of anyone who could put an end to the chaos of continuing civil war. The Augustan regime was a very Roman form of monarchy. Through the success of his adopted son, Rome was to be ruled by 'Caesars' for centuries, for the name became synonymous with supreme power. Even at the beginning of the twentieth century there was a tsar in Russia and a Kaiser in Germany, emperors whose titles derived from the family name of a Roman aristocrat who had made himself dictator and was murdered in 44 BC.\nGlossary\n\n**Afranius, Lucius:** one of Pompey's officers who fought for him in Spain, Macedonia and Africa.\n\n**Ahenobarbus, Lucius Domitius:** consul in 54 and a leading opponent of Caesar in the build-up to the Civil War. Defeated at Corfinum and Massilia, and finally killed in the aftermath of Pharsalus.\n\n**Antony, Mark (c. 81\u201330):** one of Caesar's subordinate officers, he was given both administrative and military posts. Emerged as one of the main leaders of Caesar's supporters after his assassination.\n\n**Brutus, Marcus Junius (c. 85\u201342):** influential younger member of the Senate who fought against Caesar in 49\u201348. Captured and pardoned, he was one of the leaders who led the conspiracy against him.\n\n**Caelius Rufus, Marcus:** friend of Cicero, but sided with Caesar during the Civil War. The unstable Caelius Rufus then rebelled against him and was killed.\n\n**Caesar, Caius Julius (100\u201344):** maverick politician and brilliant commander, Caesar rose through Civil War to establish himself as dictator. Murdered by a conspiracy of senators, Caesar's fame has nevertheless endured to the present day.\n\n**Cassius Longinus, Caius (c. 85\u201342):** having won a name for himself by defending Syria after the death of Crassus, Cassius sided with Pompey during the Civil War. Captured and pardoned he and Brutus led the conspiracy against the dictator.\n\n**Cicero, Marcus Tullius (106\u201343):** the greatest orator of his day, Cicero was more a politician than soldier. He survived the Civil War only to be executed on Mark Antony's orders. Cicero's Correspondence and other writings provide a mass of information about the period.\n\n**Cleopatra (c. 69\u201330):** queen of Egypt and subsequently mistress of first Caesar and then Antony.\n\n**Crassus, Marcus Licinius (85\u201353):** the man who suppressed Spartacus' rebellion and later one of the triumvirate with Pompey and Caesar. Crassus mounted an invasion of Parthia in 54 and was killed the next year after the defeat at Carrhae.\n\n**Curio, Caius Scribonius:** reckless young senator who was bribed to join Caesar and defended his interests as tribune of the plebs in 50. Killed in Africa the following year.\n\n**Domitius Calvinus, Cnaeus:** one of Caesar's subordinate officers, elected to the consulship in 53 and 42.\n\n**Juba, King:** ruler of Numidia and Gaetulia, he sided with Pompey, but took his own life after the defeat in 46.\n\n**Marius, Caius (c. 157\u201387):** a man of humble background, but great military talent, Marius reformed the Roman army and had a spectacular career, but also provoked Rome's first Civil War in 88.\n\n**Octavia:** sister of Octavian, she was married to Antony to cement their political alliance. However, he subsequently discarded her for Cleopatra.\n\n**Octavian\/Augustus (Caius Julius Caesar Octavianus, 63\u2013AD 14):** Caesar's nephew and adopted son. His rise to power and eventual defeat of all rivals led to the creation of a form of monarchy known as the Principate.\n\n**Petreius, Marcus:** one of Pompey's senior subordinates, he commanded large forces in Spain, Macedonia and Africa, but committed suicide after the defeat at Thapsus.\n\n**Pompey the Great (Pompeius Magnus), Cnaeus (106\u201348):** Pompey rose to fame at a young age during the Sullan Civil War, forging a career which was as spectacular as it was unconstitutional. Joined with Crassus and Caesar to form the First Triumvirate, but, after the death of Crassus, relations with Caesar broke down and led to Civil War.\n\n**Pompeius, Cnaeus:** elder son of Pompey, who fought against Caesar in Spain and was defeated and killed at Munda in 45.\n\n**Pompeius, Sextus:** younger son of Pompey and a gifted naval commander, he fought with success against Octavian until his final defeat in 36.\n\n**Scipio, Quintus Metellus Pius Nasica:** one of Caesar's main opponents in the Senate, he proved an inept commander and was defeated at Thapsus in 46.\n\n**Sulla, Publius Cornelius the dictator (138\u201378):** the first man to lead a Roman army against Rome and the victor in its first Civil War, he became dictator and attempted to reform the Roman state.\n\n**Sulla, Pubius Cornelius:** nephew of the dictator, he served as one of Caesar's officers, but died in 45.\nFurther reading\n\nPrimary sources\n\nThe Civil War is well documented by the standards of ancient conflicts, but there remain many gaps in our knowledge. The best account is provided by Caesar's _War Commentaries_ in three books covering 49\u201348, supplemented by separate accounts of the _Alexandrian War_ , _African War_ and _Spanish War_ written by his continuators. The identities of the latter are unknown, but all appear to have been officers who served with Caesar and witnessed at least some of the events they described. All of these accounts are inevitably highly favourable to Caesar, even though in his own narrative he writes about himself in the third person, but they are very detailed. Another of Caesar's officers, Caius Asinius Pollio, claimed that Caesar did not always bother to check the details of events which he did not witness. Pollio's own account, written after Caesar's death, has not survived, although it is referred to in some of our other sources, such as the early-second-century AD _Roman History_ of Appian. Roughly contemporary with the latter were Plutarch's _Parallel Lives_ , which include biographies of many of the main protagonists in the Civil War. Less useful for military detail, but invaluable for the political background and a vivid impression of the time, are the speeches, and particularly the correspondence, of Cicero.\n\nCaesar (translated by J. Gardner), _The Civil War_ , 1967.\n\nAppian (translated by J. Carter), _The Civil Wars_ , 1996.\n\nPlutarch (translated by R. Warner), _Fall of the Republic_ , includes _Lives_ of Crassus, Pompey, Caesar and Cicero, revised, 1972.\n\nPlutarch (translated by I. Scott-Kilvert), _The Makers of Rome_ , includes _Lives_ of Brutus and Mark Antony, 1965.\n\nSuetonius (translated R. Graves), _The Twelve Caesars_ , revised, 1979.\n\nAll of the above are also available as parallel-text volumes (i.e. with the Latin or Greek on one page and the translation facing it) in the Loeb Series. They also provide the most convenient version of Cicero's _Letters to Atticus_ and _Letters to his Friends_.\n\nSecondary sources\n\nAdcock, F., _The Roman Art of War under the Republic_ , 1940.\n\nBishop, M., and J. Coulston, _Roman Military Equipment_ , 1993.\n\nLe Bohec, Y., _The Imperial Roman Army_ , 1994.\n\nBrunt, P., _The Fall of the Roman Republic_ , 1988.\n\nDupuy, T., _The Military Life of Julius Caesar_ , 1969.\n\nFuller, J., _Julius Caesar: Man, Soldier and Tyrant_ , 1965.\n\nGelzer, M., _Caesar, Politician and Statesman_ , 1968.\n\nGilliver, C., _The Roman Art of War_ , 1999.\n\nGoldsworthy, A., _The Roman Army at War, 100 BC\u2013AD 200_ , 1996.\n\nGoldsworthy, A., _Roman Warfare_ , 2000.\n\nGruen, E. S., _The Last Generation of the Roman Republic_ , 1974.\n\nKeppie, L., _The Making of the Roman Army_ , 1984.\n\nMeier, C., (translated by D. McLintock) _Caesar_ , 1995.\n\nParker, H., _The Roman Legions_ , 1928.\n\nSaddington, D., _The Development of the Roman Auxiliary Forces from Caesar to Vespasian_ , 1982.\n\nSeager, R. _Pompey, A Political Biography_ , 1979.\n\nSyme, I., _The Roman Revolution_ , 1939.\n\nWelch, K., and A. Powell, (eds), _Julius Caesar as Artful Reporter. The War Commentaries as Political Instruments_ , Swansea, 1998.\n\nWeinstock, S., _Divus Iulius_ , 1971.\n\nYavetz, Z., _Julius Caesar and his Public Image_ , 1983.\nFirst published in Great Britain in 2002 by Osprey Publishing,\n\nPO Box 883, Oxford, OX1 9PL, UK\n\nPO Box 3985, New York, NY 10185-3985, USA\n\nEmail: info@ospreypublishing.com\n\nOsprey Publishing is part of the Osprey Group.\n\n\u00a9 2002 Osprey Publishing Ltd.\n\nAll rights reserved. Apart from any fair dealing for the purpose of private study, research, criticism or review, as permitted under the Copyright, Designs and Patents Act, 1988, no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, electrical, chemical, mechanical, optical, photocopying, recording or otherwise, without the prior written permission of the copyright owner. Enquiries should be addressed to the Publishers.\n\nText first published as part of the _Essential Histories_ series\n\nISBN: 978 1 4728 0988 9\n\nEditorial by Rebecca Cullen\n\nCartography by The Map Studio\n\n\u00a9 Osprey Publishing. Access to this book is not digitally restricted. In return, we ask you that you use it for personal, non-commercial purposes only. Please don't upload this ebook to a peer-to-peer site, email it to everyone you know, or resell it. Osprey Publishing reserves all rights to its digital content and no part of these products may be copied, stored in a retrieval system or transmitted in any form by any means, electronic, mechanical, recording or otherwise (except as permitted here), without the written permission of the publisher. Please support our continuing book publishing programme by using this e-book responsibly.\n\n**www.ospreypublishing.com**\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# ALL\n\n# THE\n\n# LIGHTS\n\n# ON\n\n# ALL\n\n# THE\n\n# LIGHTS\n\n# ON\n\n## REIMAGINING THEATER WITH TEN THOUSAND THINGS\n\nThe publication of this book was supported through a generous grant from the Atherton and Winifred (Wollaeger) Bean Fund for Business History\n\nText \u00a92014 by Michelle Hensley. All rights reserved. No part of this book may be used or reproduced in any manner whatsoever without written permission except in the case of brief quotations embodied in critical articles and reviews. For information, write to the Minnesota Historical Society Press, 345 Kellogg Blvd. W., St. Paul, MN 55102\u20131906.\n\nwww.mnhspress.org\n\nThe Minnesota Historical Society Press is a member of the Association of American University Presses.\n\nManufactured in the United States of America\n\n10 9 8 7 6 5 4 3 2 1\n\n  The paper used in this publication meets the minimum requirements of the American National Standard for Information Sciences\u2014Permanence for Printed Library Materials, ANSI Z39.48\u20131984.\n\nInternational Standard Book Number\n\nISBN: 978-0-87351-983-0 (paper)\n\nISBN: 978-0-87351-984-7 (e-book)\n\nLibrary of Congress Cataloging-in-Publication Data\n\nHensley, Michelle. All the lights on : reimagining theater with Ten Thousand Things \/Michelle Hensley.\n\npages cm\n\nSummary: \"A history of the Twin Cities' theater company Ten Thousand Things, which for more than twenty years has been bringing intelligent, lively theater to nontraditional audiences as well as the general public\"\u2014Provided by publisher.\n\nISBN 978-0-87351-983-0 (paperback)\u2014ISBN 978-0-87351-984-7 (ebook)\n\n1. Ten Thousand Things (Theater company)\u2014History. I. Title.\n\nPN2297.T46H46 2015\n\n792.09776'579\u2014dc23\n\n2014043270\n\nThis and other Minnesota Historical Society Press books are available from popular e-book vendors.\n\nCover: Sally Wingert and Isabell Monk O'Connor in Twelfth Night, Dorothy Day Center, St. Paul. Insets: Barbara Kingsley in The Good Person of Szechwan, women's correctional facility; audience at men's correctional facility; Peter Vitale plays music for Twelfth Night, Dorothy Day Center. All photos by Paula Keller.\n\nInterior: Photos pages 20, 120, 156, 194 by Tom Wallace. Photos pages 40, 98, 154, 196 by Keri Pickett. All other photos by Paula Keller.\nTo the audiences and artists of TEN THOUSAND THINGS\n\n# CONTENTS\n\nIntroduction: A Little Bag of Silver\n\n1 Radiance: The Heart\n\n2 Imagining the Audience, Right or Wrong: Beginnings\n\n3 Fairy Tales\u2014Big Enough Stories: Choosing a Script\n\n4 Audiences at Life's Extremes\u2014The Greeks: Fairy Tale Group #1\n\n5 \"Just Tell the Story\"\u2014Shakespeare: Fairy Tale Group #2\n\n6 Abbondanza! The Wealth of Very Few Things: Production Values\n\n7 Everybody Needs to Sing\u2014American Musicals: Fairy Tale Group #3\n\n8 Concept-free Directing\u2014Originality from Necessity: Preparation, Rehearsals\n\n9 No Place to Hide: Performances\n\n10 The Power of Unexpected Connection\u2014Political Theater: Going into the World\n\n11 All the Lights On\u2014Facing Misogyny and Reinventing Dead Systems: Organic Organization\n\nEpilogue. What This Book Is For: Ending Words, Beginning Words\n\nAppendix: Ten Thousand Things Production History\n\nAcknowledgments\n\n# ALL\n\n# THE\n\n# LIGHTS\n\n# ON\n\n# INTRODUCTION\n\n## A LITTLE BAG OF SILVER\n\nMY FAVORITE PLAY is The Good Person of Szechwan by Bertolt Brecht. In it, a poor woman named Shen Te gets a gift\u2014a bag of silver from the gods. She has to figure out how to spend it wisely. It's all she has.\n\nWhen I was in my early twenties, I knew I loved theater and thought it might be what I wanted to spend my life on. But I also knew I had only one life. I wanted to be sure I could use it to give something back to the world. I wrestled with this almost every day throughout my twenties, through college and grad school and in between. With so many overwhelming problems on the planet\u2014most humans living in desperate poverty, struggling each day with violence and hunger, pollution threatening our very existence\u2014I didn't see how theater could help very much.\n\nAlso, I honestly couldn't imagine myself going down the recommended path for aspiring young directors, climbing the ladder at a regional theater. I sensed I would need to act a \"part\" and squeeze myself into some kind of mold of authority and confidence in order to prove that I had \"the stuff\" to be a director. If I worked hard to ingratiate myself with those in charge, I might eventually be allowed to direct a reading of a script I probably wouldn't even like, then after more time, perhaps a staged reading of another script I didn't really care about\u2014all for very little if any pay. I knew that if I couldn't find the space to truly be myself and work on things I loved, my art wouldn't be very good.\n\nMost important, some things about theater itself made me uncomfortable. Looking around at the audience before the lights went down, I saw mostly well-off, educated white people. Lots of other people seemed left out. I thought of my grandfather, an Iowa farmer who lost his land in the Depression. He usually wore striped engineer overalls and a seed corn cap. He was very intelligent but never went to college. I imagined he would feel out of place walking into most of the theaters I'd been to, both the scruffy little black boxes and the big theaters with their fancy lobbies. And he probably wouldn't be interested in much of what was happening up on the stage, certainly not most nonlinear experimental theater and performance art (much of which I personally enjoyed) but also not even many of the classic plays done on big regional stages, where the audience had to sit so far away. I imagined much of it would feel pretentious to him, or at least very puzzling, and certainly remote. I really didn't want to spend my life on an art form that seemed set up to leave so many people out.\n\nI grew up in the very white, very middle-class world of Des Moines, Iowa, in the sixties and early seventies, in a period of American history that now is starting to look like an aberrant bubble of relative income equality. The wealth then was distributed in such a way that taxes supported a very good public school system and other social safety nets, and employers generally shared more generously with their employees, through salaries, good health care, and pension plans. My late childhood was affected by the protests against the Vietnam War and the general 1960s' expansion of consciousness, all of which I could observe from a certain distance. Not old enough to personally experience the painfulness of the draft or drugs or discrimination, I instead absorbed the era's more positive messages. It all gave me a sense of hopefulness about the possibilities for change in the world. (In sixth grade, I wore pants to school for several days, joining a protest against the girls' dress code in public schools\u2014and felt the success of getting those rules changed!) My parents were both very liberal, and our family talked about politics a lot at the dinner table, giving me a keen sensitivity to unfairness and injustice. The notion that it was our job to try to make the world at least a little less unfair took deep root. This gave a kind of ferocity, I think, to my searching for how to best spend the \"little bag of silver\" of my own life, and how, if I were to make theater my profession, I really wanted to do it.\n\nBoth my parents grew up during the Depression, and through their storytelling I was very aware that the material comforts I enjoyed were not part of their childhoods, although they also conveyed that they had been happy enough without having so much. My father always liked to remind me that before World War II, America was pretty much a third-world country, with most of its population living in rural poverty. But he also stressed that because most people were in the same boat, it didn't seem to matter all that much, especially as a child. These stories, along with the anti-materialism that sprung up in the sixties, gave me a strong sense that there was happiness beyond material things and, indeed, that too many things could actually get in the way of happiness. My father in those times was able to save up enough by working as a lawyer at an insurance company to send me to an Ivy League school without taking on any debt (tuition at the time was around five thousand dollars). It was still a time when college was not only affordable for the middle class but seen as a place to learn and explore, rather than to acquire specific training for specific employment.\n\nAnd in my after-college years in the early eighties, I continued in what now seems like a luxurious bubble of exploring and not knowing. I headed out to San Francisco with the sense that it was okay to take some time to really try to figure out exactly what my passion might be. I worked in coffee shops and restaurants and a nursery school, I got a job as a government-paid CETA puppeteer at the elementary school in Haight-Ashbury. Most of the hippies were gone from the neighborhood by then, but the city still afforded a leisurely sense of openness and questioning. I was able to patch together ways to support myself, paying my quite manageable rent, and I never felt the need to rush toward leading a life that would bring bigger bills. As I wandered from job to job, I read all I could by Peter Brook and Grotowski and Artaud and Joseph Chaikin, people who asked big questions about the possibilities of theater. I am so grateful that the culture back then permitted me this time for searching. Four years later, I went to graduate directing school in Los Angeles, in large part because the school paid my way. I don't need to say too much about graduate school except that I came away with at least a very clear sense of what kind of theater I didn't want to make.\n\nIn addition to noticing how many people theater excluded, I sensed that so many people who actually did come to theater didn't want to be there. Especially in Los Angeles. Many came out of a sense of obligation\u2014they'd been invited by acquaintances in the cast who were desperate to be seen by people in the TV and film industry or who were just desperate for an audience. These people had seen lots of theater before and kept their distance. They liked to judge: how did this interpretation compare to others they had seen; how interesting was the set concept, how authentic was the dialogue, how good was the lighting design, how striking were the costumes? But they didn't seem moved by the story. If you could see such people's \"theater bodies,\" I imagined, they would be bloated and fat because they had seen so much theater already. No one seemed hungry for the actual story.\n\nWhen I thought about doing The Good Person of Szechwan, I really didn't want to do it for audiences like that. I loved the story too much. I loved how the play, told from the rare point of view of a female character, paid attention to something I hadn't really seen onstage before: our fundamental human urge to be kind. I loved the way the play explored, with so much humor and irony, how hard it is to act on this deep desire to help others, particularly in our very materialistic world. It acknowledged the profound ache that comes from having to shut down your kindness when the large systems around you force you to use all your energy just to survive. I didn't think the audiences I knew would let themselves care enough about all this. I didn't want them to sit there and judge the production values, missing altogether the way the story could connect with their lives. This really is the core of everything that followed: I just wanted to find an audience who loved the play as much as I did, an audience for whom the story really mattered.\n\nIn the play, Shen Te lives in a shack in the poorest part of the city, where she earns a meager living as a prostitute. She gets the bag of silver from three gods who visit her in disguise, as a reward for her kindness in giving them shelter for the night when no one else would. They instruct her to use the money to \"be good.\" Most of the play is about Shen Te trying to help out her poor friends and neighbors but becoming overwhelmed by the enormous difficulty of it all. Their needs are so vast, their demands so endless, her little bag so small. And she needs to take care of herself\u2014and, later, her child\u2014at the same time.\n\nWho else would really care about this story? Was there an audience who would really, truly understand the depth and desperation of Shen Te's struggle to be good? At one point she explodes, \"How can I be good when everything is so expensive!\" The response of mild chuckles from the comfortable audiences I knew would be about the \"difficulty\" of choosing between writing a hundred-dollar check to a charity or going out to dinner at a nice restaurant. For most of us with a middle-class existence, being taken advantage of when trying to \"do good\" is inconvenient and sometimes frustrating, but it doesn't leave us tottering on the brink of survival, like everyone in Shen Te's world. It would be very different to watch the play, I imagined, if you were also truly living on the edge, knowing that if you give away a bowl of rice, it could be the last you'd have for many days. \"Expensive\" would have a very different meaning.\n\nThinking about an audience of people struggling daily just to survive suddenly opened up all kinds of new questions about the play. I had to admit there was probably a lot that I didn't understand, either: yes, I was barely getting by month to month with my temp jobs, trying to be a theater artist, living in an ugly little rental duplex, but should something go terribly wrong, in the back of my mind I knew that my middle-class parents in Des Moines would always be able to help. The knowledge of that safety net allowed me to take some risks. The value of this net should not be underestimated.\n\nSo I became deeply curious to know what an audience without any cushion at all would make of Shen Te's struggles. Did they have the same ones? Would they share her sincere desire to help others? And recognize the way this desire was always getting trampled on, often by the very need and greediness of those she wanted to help? In my little living room in Venice, California, one afternoon, I gathered together a group of talented actor friends, some from grad school, all anxious to practice their craft. All of us shared a sense that theater was leaving a lot of people out. I think most theater artists actually share this ache at some level. My friends agreed that these were worthy questions and that we should take the play, jump in, and try to connect.\n\nBut we couldn't imagine how to get such an audience to come to the theater. The forces against it felt enormous: there was the ticket price of course, but also the fact that this audience, like my grandpa, probably wouldn't feel comfortable. Seeing it through their eyes, the theater \"scene\" felt daunting: people were all dressed up, they'd been to college, they knew some mysterious code of behavior that told them when to applaud and when to be quiet. The plush lobby and seats seemed foreign and unwelcoming. My audience would never come. I realized the only way to reach my imagined, caring audience would be to go to them.\n\nIt was 1989, and the sunny beach communities of Venice and Santa Monica were full of people who could not provide for themselves, including many of the mentally ill that Reagan administration policies had turned out of group homes and onto the streets. A homeless shelter might provide us with an audience who could relate to the struggles of the play. I found a drop-in center on an unglamorous stretch of warehouses near the freeway. The center didn't have beds, but it did offer a safe place to hang out during the day, along with some social services and a cup of coffee. It had a noisy central room with worn blue carpet, surrounded by a balcony of little offices buzzing with loud voices and ringing phones. When I approached a staff member and offered a performance of a play by Bertolt Brecht, she seemed puzzled and too busy and harried to think about it very much. However, she said I could put up some flyers announcing a date and time.\n\nWe set to work. Seven excellent actors but playing thirty-five characters. A play more than two hours long, plus an intermission. We were anxious, to say the least. What would our audience think? Would they stay with the complicated plot, follow all the character changes? More important, could we bring enough truth to what we were doing to make them want to stay and watch the whole thing? Most daunting of all\u2014how could we of the middle class possibly think that we had anything to say about poverty to people who lived with it every day? We were, at least, an ethnically inclusive group that included an African American, Asian American, Hispanic American, and several European Americans, plus an Australian (it isn't hard to assemble a diverse cast in Los Angeles!). Still, the gulf between our life experiences and those of our audience felt huge.\n\nAs we rehearsed, we kept wondering about these homeless people, trying to imagine how they might see things. Our wondering informed the choices we made in rehearsal. We weren't trying to be \"original\" or \"innovative.\" We weren't thinking about how brilliant choices might enhance our careers or please the critics. We were just trying to figure out how we could best tell the story so that people would connect it with their lives.\n\nOne thing I vowed\u2014we would not have the slightest whiff of condescension to our work. I imagined that people in shelters were always getting preached to: \"Here's what's wrong with you; here's how you can improve yourself; here's how you can solve your problems. You probably won't be able to understand anything difficult, so we'll keep it really simple. We know how to help you; let us help you so we can feel better about ourselves.\" We approached our audience with enormous respect for their hard-won life experiences. It wasn't difficult. One step inside a shelter and humility is pretty inescapable.\n\nAlthough the play takes place in a slum, it felt very important to stress that this was a \"fairy tale\" world and not a realistic depiction of poverty. I wanted to be very clear from the start that we were not going to try to pretend to be experts in those physical details. It seemed essential to make up a world, set in another time and another place, where no one could be an expert. Imaginative distance was crucial. Because the script had so much humor and irony, it seemed that whatever set we made should be playful, too.\n\nWe didn't have any money. But here that invisible safety net of a middle-class family\u2014and a caring, supportive parent\u2014came into play. I got my own little bag of silver from my mother without even asking: five hundred dollars to pay for the set. It felt wrong not to pay my actor friends for their hard work, and I vowed this would be the last time that ever happened. But it was a start.\n\nWe bought fabric and dye and paint. I was thinking of the colors of the sunset on the beach, where many homeless people spent a lot of time. We dyed cloth and cut out shapes to hang on a clothesline\u2014red and pink doors, yellow shop windows, purple and orange trees, green factory signs. We set four poles in small cardboard barrels to form the four corners of our little stage and stretched rope between to hang the cutouts on with clothespins. We filled plastic shopping bags with sand from the beach to weigh down the poles in their barrels, making sure they would stand up. It took a lot of heavy bags, so everyone had to carry piles of them in their car trunks. Then just a few boards for benches and shelves, a bowl of rice, some cigars made from cardboard tubes, not too much else. We rehearsed over several months, whenever we could find spare time.\n\nOn the afternoon of the performance, I remember shouting across the parking lot of the shelter to greet the actress playing Shen Te, Christi Mays, both of us lugging our bags of sand. Though we were smiling, we were terrified. My head was pounding: \"What are you doing? This is going to be a disaster. Why are you doing a play here? This is absolutely ridiculous. They are going to think you're stupid and weird and laugh in your face; they're going to hate it. Put the sand bags back in the trunk and get out of here.\" It was my first experience of a moment that has happened at least once in every single production since: a sharp, clear realization, like the chime of a bell, that I am out of my mind. But as also always happens, the energy of sheer necessity took over: we'd said we'd do the show, we'd rehearsed it, the actors were arriving with costumes and set pieces, of course we had to go ahead and do it.\n\nWe walked into the space. Sunshine streamed through the windows\u2014the beautiful weather outside seemed much more tempting than the prospect of staying inside to watch some little play. There actually weren't very many homeless people around: a weathered-looking young man wrapped in a blanket, a few women with shopping bags, a group of older black men in the corner drinking coffee. But we set up the poles, tied a rope between the back two, and stretched a purple cloth between the front two to make a curtain, using paper clips for hooks. We had brought extra cardboard barrels and boards and used them to make little benches for the audience, setting them up hopefully in front of the stage.\n\nIt didn't seem that anyone had noticed the flyers. No one\u2014including the staff\u2014seemed to have any idea that we were doing a Brecht play there that afternoon. The mood in the room among the twenty or so homeless people ranged from skepticism to puzzlement to complete indifference as we put up the set. It felt like a ridiculous place to attempt to perform a two-and-a-half-hour play. My \"you've got to be out of your mind\" chime kept getting louder.\n\nWe'd photocopied some programs, so I went around the room offering them, inviting people to take a seat on a bench and try the play out. Most people just looked at me suspiciously. Finally one or two women reluctantly moved over to the benches; a minute or two later, a few more followed. Most still hung back, arms folded, probably in anticipation of a blast of condescension headed their way from the \"do-gooders.\"\n\nWe began. Wang the Water Seller came out in front of his sewer pipe home, painted on the purple curtain, to sing about how hard it is to sell water when it's raining. We wrote the song to be very minimal, half-spoken, anticipating how strange the idea of someone just starting to sing in the middle of such a room might be, but it still seemed very strange. The audience kept their distance. The three flimsily disguised gods entered, demanding that Wang find them lodging for the night. I sensed a slight tension in the room and felt myself go on alert: of course, any presentation of \"gods\" onstage smelled of trouble\u2014was this weary audience worried we might be trying to preach to them about how to be good?\n\nThe three gods: Bill Burns, Craig Fernandez, Marilyn Henkus, The Good Person of Szechwan, Venice soup kitchen, 1990.\n\nWhen Shen Te answered her door, I thought I sensed another tiny audience shift. Shen Te was known throughout the slum for her kind heart, but one look at the gods and she responded with the same suspicion the audience was feeling; she indicated she was not particularly interested in pleasing such ridiculous beings. She finally agreed, with more than a little reluctance, to put them up, but only out of loyalty to her good friend Wang, who was terrified of what the gods might do if he failed to find them a room. The audience seemed to be paying a little more attention; perhaps they were approving of Shen Te's honesty?\n\nFinally, about ten minutes into the play, we got a signal from the audience that they were connecting. The gods were showering Shen Te with praise for her good deed, proclaiming her the last truly good person on earth, when she interrupted them with a very inconvenient fact of her life: she sold herself for a living. \"I'd like to keep the commandments, to honor my mother and father and always tell the truth,\" she protested. \"I'd be overjoyed not to covet my neighbor's house, and I'd be delighted to stay with just one man. But how? How is it done? Even by breaking a few commandments, I barely get by. How can I be good when everything is so expensive?\" The second god responded sharply, \"There's nothing we can do about that. We never meddle in economics.\" At this moment, there was a chuckle. A few more, and then a guffaw from the back. The audience began to catch on that we were not on the side of the gods and their endless exhortations to \"be good.\" The play was on Shen Te's side\u2014and on their side. We were not going to lecture them, like the gods, and tell them how to be better.\n\nThat first little chuckle was a whiff of cool ocean air, allowing us all a big internal sigh of relief. Someone was connecting, perhaps in spite of herself, someone was deciding to open up to the strange event occurring in this very unlikely place. The skepticism had started to melt away. When the gods, after a quick conference, decided to \"pay\" Shen Te for their lodging, giving her a small bag containing a thousand silver dollars\u2014along with strict instructions to use it to \"do good\"\u2014her delighted yelps of joy seemed to win over the audience completely. It was a fantasy everyone could relate to, and the audience was with Shen Te for the rest of the ride.\n\nTheir pleasure in Shen Te continued as she used the money to buy her own tobacco shop. They seemed also to approve wholeheartedly when Shen Te, overwhelmed by the need and greediness of her neighbors in the slum, found she had to create an alter ego for herself, a \"cousin,\" the ruthless businessman Mr. Shui Ta. Donning a hat and suit coat and lowering her voice, she kicked the neighbors out and order was restored. The audience shifted back and forth easily with Shen Te between her two personas, the kind-hearted one and the ruthless one, and began to vocalize their approval. At one point, dressed as Mr. Shui Ta, Shen Te turned away in the middle of making a deal and took off her bowler hat to exclaim, \"I am horrified to see how much luck is needed to keep from being crushed! How many brilliant ideas! How many good friends!\" The audience responded with murmurs of support, \"Uh-huh!\" and \"You got that right!\" Mr. Shui Ta then turned back to seal the deal with a cigar, and Christi, the actress playing him, quietly glowed from the support her character had just received.\n\nAt other points the listening became very intense and focused, as when Shen Te sat on a park bench with a young pilot who had tried to hang himself because he hadn't been able to come up with a bribe to get a flying license. She told a story of a crane with a broken wing she had known as a child. The crane was good natured but got very restless in the spring and fall when big flocks of birds flew overhead. The quiet in the room was electric. For a moment there was a shared understanding among thirty people in a homeless shelter, including a janitor and some staff who had paused in their work, of what it feels like to have a broken wing, to be unable to fly.\n\nThe vocal responses grew. When Shen Te paused on the way to her wedding to the young pilot to express doubts about whether her groom really loved her or was just after her tobacco shop money, some women in the audience shouted out warnings: \"That's right, honey, you stay away from him! He's bad news!\" Christi discovered that her \"monologue\" was actually a conversation with the audience. She spoke her next lines a little more defiantly, \"But I want to go with the man I love. I don't want to count the cost. I don't want to think about whether it's right. I don't want to know whether he loves me. I want to go with the man I love!\" The women in the audience shifted their sympathies once again. \"Yes. Yes. Mmmm-hmmm. You go girl,\" they murmured in knowing, supportive tones. I was speechless with surprise and delight, and I knew that Christi was, too. This audience had no problem completely embracing both sides.\n\nWe were also surprised by responses we had not imagined. At one point, Shen Te was outside hanging up her laundry, having decided not to go through with her marriage. As she bent over her basket, she suddenly felt a little dizzy. There was a quiet chorus of \"uh-ohs.\" The audience instantly knew. No words needed. A baby was coming. And a baby can bring a lot of trouble when you're living on the edge.\n\nBut Shen Te ignored the audience's alarm and went on to express joy at the discovery. She proceeded to take her imagined child on a tour of the neighborhood, holding his little invisible hand, showing him the wonders of the world. I had always greatly loved this part; I had never really seen the anticipation of the delights of having a child expressed so well in a script. But the audience didn't seem to share these feelings quite so much. We were surprised\u2014and informed\u2014by the audience's different response. For many, the pleasant anticipation of a child might be a luxury, outweighed by the difficult realities a single mother would face raising her child in a very harsh world. Though when Shen Te paused in her tour to point out a magnificent cherry tree in a wealthy neighbor's yard, everyone expressed great pleasure and approval as she then taught her imagined child how to steal the cherries!\n\nThe reverie ended abruptly when Shen Te stumbled on the sight of an actual child fishing for food in a garbage can. At this moment, her feelings swung back in line with the audience's. Shaken to her core, she determined to change herself back into Mr. Shui Ta one last time, vowing to be good to her child but a wild beast to all others. There seemed to be little question in the audience that Shen Te was doing the right thing by completely giving up any aspiration to do good. The absolute ferocity and ruthlessness needed to protect your child in a brutal, cutthroat world was something everyone clearly understood.\n\nAs the play built to a close, the honest and easy shifting of the audience's sympathies continued. They seemed glad to have Mr. Shui Ta (growing ever plumper!) take matters firmly in hand once again and open up a tobacco factory. But they also sided with Shen Te's neighbors, who, with no other jobs around, were forced to work at the factory for extremely low pay. Like the workers, the audience then began to miss Shen Te, who had not been seen for many days. In any case, most seemed pleased with the irony when Mr. Shui Ta was arrested for Shen Te's kidnapping and found himself in court facing a tribunal of three judges who looked suspiciously like the three gods.\n\nThe tobacco factory, (left to right) Christianne Mays, Suzanne Kato, Kate Cherry, Bill Burns, Marilyn Henkus, Kenneth Ransom, The Good Person of Szechwan, Los Angeles Women's Shelter, 1990.\n\nI had found myself an audience who deeply understood both sides of Shen Te's very human dilemma, appreciating it with all its pain and humor and irony. Thank goodness Brecht, in ending the play, did not choose a side or offer any easy answers. As Mr. Shui Ta began to remove his coat and tie, revealing to the judges that he was actually the same person as the now quite pregnant Shen Te, he\/she cried out the heartbreak the audience understood deeply in their bones:\n\nYour order long ago\n\nTo be good and yet to live\n\nWas a thunderbolt:\n\nIt has torn me in two halves.\n\nI don't know how it happened\n\nBut to be good to others\n\nand to myself at the same time\n\nI could not do it.\n\nYour world is not an easy one, Enlightened Ones!...\n\nAnd yet to give was a passion with me.\n\nA happy face and I walked on clouds.\n\nCondemn me: all my crimes\n\nwere committed to help my neighbor,\n\nto love my lover\n\nand to keep my little one from want.\n\nFor your great, godly deeds,\n\nI was too poor, so small.\n\nAnd, of course, to the audience's great relish, the gods, as they had done all along, ignored Shen Te's honesty. \"Just be good and everything will turn out all right,\" they reassured her as they stepped onto a little cluster of rose-colored clouds (cardboard cutouts, of course), hurrying to get back to heaven as quickly as possible. As Shen Te ran after them, she pleaded for the use of her ruthless cousin. The gods paused just long enough to rule that she could be Shui Ta, but only once a month. Then they happily floated away, barely able to hear the last word of the play, a guttural cry from the woman whose belly was swollen with child, \"Help!\"\n\nThe gods had left Shen Te\u2014and us\u2014behind, torn in two, without any answers at all.\n\nThere was hearty applause from the little audience of thirty. The little audience that remains the biggest one of my life. The actors came out, wild eyed with relief, to take several bows in front of our small purple curtain. They stepped into the audience to mingle.\n\nIt had not been a completely smooth ride. During the performance, people sometimes got up to leave. Whenever someone did, our hearts sank: we were sure they must not have liked the show. In this very intimate performance situation, there was time to make a personal connection with almost every audience member. The older men around the edges of the room would often talk during the play, though sometimes they'd stop to tune in. Social workers would scurry by without even a glance. Phones rang and sometimes the conversations were really loud. One person fell asleep for part of the show (though he did seem to enjoy it again once he woke up).\n\nThe Good Person of Szechwan, curtain call, Venice soup kitchen, 1990.\n\nAs the director, standing behind the audience, or sometimes crouching when things got really intense, I experienced the moments of disconnect more keenly than the actors, since I could only watch. For me, it was a stomach-lurching roller coaster ride. At the time, my only explanation for those moments was, \"They must not like the show!\" Only later would we come to understand that people, no matter how much they might be enjoying a performance, have to leave to meet with their social workers or for a job interview. Or that someone might have been really looking forward to a cup of coffee and the chance to talk to others after a lonely night on the street. Or that drop-in centers often provide the only safe, warm place to sleep, even if it's in the daytime. Even if there's a play going on right in front of you. Life is more important than theater. I understood this intellectually, but it would take me a long time to really absorb it and just be able to relax and let go when such things happened.\n\nBut despite all this, we really had succeeded in connecting. Of course, I couldn't tell exactly what was going through any one person's mind. But, as I think any director does observing an audience, I'd put together a narrative from the moments where connection\u2014or the lack of it\u2014was tangible. After the show, the audience was effusive, shaking the actors' hands, praising them, thanking them. We had also put out a little notebook where people could write comments, so I still have some exact words: \"I enjoyed this play\u2014boy was it real good!\" \"Terrific, well-acted out\u2014one of the most funniest performances!\" \"The most extraordinary performance I've ever seen.\"\n\nMore important, the play had spoken to people's lives. \"I think someone wrote my unauthorized biography!\" wrote one woman. \"It was moving and a release,\" wrote another, \"Wow! It helped me get a new perspective on a lot of difficult things in my life. It was fun and heartful.\" Opinions on the characters' behavior were offered, including one of my all-time favorites: \"I think this dual personality chick should just take the money and say fuck you.\" And it was very clear that for some, at least, the story truly did matter. As another man wrote, \"Things like this make me really want to get my life in order. It makes me see life really has a meaning. Thank you.\" No, thank you, I thought. For this audience anyway, theater was about life, not just aesthetic choices. And they had enabled me to see so clearly that my success as an artist came from having been able to engage and connect, in a deep and meaningful way, with them\u2014with people whom many would think I had little or no connection at all.\n\nThe best comment that day came from the janitor standing in the back, pausing to watch whenever he could between chores. He came right up to me and said, \"Thank you for treating us like we have brains in our heads.\" A light from his eyes shone directly into mine. It confirmed all my instincts. At that moment I renewed that vow to always treat my audiences with the utmost respect\u2014for their intelligence, for their imaginations, and for their hard-won life experiences. This would always be my starting point.\n\nThat very same janitor made another comment that afternoon that confirmed our intuitions about how we approached Brecht. He presented us with another discovery when he said, \"I liked how you could sit back and think about what was going on at the same time you were really enjoying the story.\" The audience had thought about the story as they watched it, commenting on it as it went on. And all just because it was there in the writing. At certain moments, Brecht simply had a character turn to the audience and ask them what they thought about the situation. This audience answered back, offering their opinions out loud. The \"alienation effect.\" That's really all it took. There was no need for special acting techniques or angular movements or pale makeup. Just characters who directly asked the audience questions\u2014and an audience unafraid to answer back. It was the first of many, many discoveries that our first-time audiences would help us make.\n\nAnd now, more than twenty years later, here I am. Here is Minneapolis. Soon after my daughter was born, my then husband and I moved from Los Angeles to Minneapolis. Another city with a large pool of very talented actors, but one with deeply committed theatergoers, arts advocates, and philanthropists as well. A place where the unequal distribution of wealth seems a little less severe. Here I have raised my daughter and a theater company. Through an inexplicable process of groping and stumbling, listening and focusing, lugging large objects up stairways and down hallways, plus bumping into both lucky good fortune and the bounteous generosity and kindnesses of so many, we have made a theater company. Ten Thousand Things brings the best possible theater, plays of Shakespeare and Aeschylus and Beckett, to audiences who have seen little of it before, those in prisons and shelters and adult education centers and rural towns and housing projects and Indian reservations and chemical dependency treatment centers, as well as to enthusiastic veteran theatergoers in consistently sold-out performances for the general public, all performed in large bare rooms, with no stage, just right on the floor inside a small circle of folding chairs, with all the fluorescent lights in the room turned on. Our budget is modest, we don't need our own building, our set supply budget is little more than that of our very first show, but we pay our highly skilled artists on a par with the largest theater companies in town. We have even become Johnny Appleseeds of a sort, taking this unique model to other theaters around the country who are also eager to find ways to reach outside their buildings with excellent work. And all along this journey, the honest, openhearted encounters of our first-time audiences with our first-rate artists have led us to make wonderful discoveries about theater\u2014pinpointing just what makes it thrive and flourish.\n\nBut as I dragged the rattling poles back to my hatchback, I couldn't possibly understand the implications of what had just happened. What I did know for certain then was that this audience who had seen little if any theater before, who did not take it for granted, had opened their hearts to us\u2014and to the story. They allowed the play to resonate with their very lives. They had actually given us more than we had given them. They gave us delight and amazement and affirmation through the support and encouragement and advice they had offered to the characters during the play. And they also gave us new depths of understanding into the play itself. These were all people who lived on the edge financially, every day, just like Shen Te. They understood, in a way a paying audience simply couldn't, the enormous difficulty and risk of Shen Te's efforts to do good. They had made us aware, through their imagined presence in rehearsals and their actual presence during the show, of the high stakes involved in every choice and decision Shen Te has to make. And by opening their hearts to us, they had allowed us to connect with people we had somehow been taught to think we might have nothing in common with at all.\n\nOne last thing I did know then, as I swung the last sandbag into the trunk, was that I was going to have to do this again.\n\n# 1\n\n# RADIANCE\n\n## The Heart\n\nTHAT VERY FIRST CHUCKLE from the audience in the Santa Monica shelter was a tiny burst of treasure that I have been searching ardently for more of ever since. The chuckle was a sign that we as theater artists somehow had emitted something that had touched one homeless woman's experience of the world; her small breath of laughter had in turn fed us, easing our enormous insecurity that we might fail to touch on anything we had in common. As the performance went on, the audience began to trust us still more, letting down their guard, first murmuring and then talking to the characters; they encouraged the actors to adjust and deepen, and then to reach further out and try for more. The energies that were released and exchanged grew, until at times it seemed like small suns within each of us had been uncovered.\n\nTHE PURSUIT OF RADIANCE This ineffable feeling of deep exchange with an audience is one I would venture to say most theater practitioners long for yet do not experience as often as they would like in their careers. I've just recently thought of a name for these bursts, which come from humans sharing with each other, even for just a moment, their most profound and honest selves: radiance. Radiance is more than a \"connection,\" which suggests energy flowing just two ways, back and forth, from actor to audience and back again. Because radiance comes from such a deep place, with minds, hearts, and imaginations all engaged at once, it spills out into the world in all directions, in surprising and unexpected ways. Theater is uniquely suited to create radiance because of its collective aliveness, because we meet in one space and imagine new worlds together in the moment.\n\nSteven Epp, Man of La Mancha, women's correctional facility, 2011.\n\nThe pursuit of radiance has shaped everything I've done at Ten Thousand Things. I loved the first experience of it so much that I wanted to figure out everything else I could possibly do to encourage it. My thirst for radiance came, of course, from wanting people to be blown away by their very first encounter with theater. But it also came from craving the way our work as artists was clarified and deepened by the audiences' refreshing responses. For many years, I was compelled to sit and watch each and every performance from beginning to end, observing the audience and the actors intently, obsessively, hoping for moments of engagement, basking in them if they happened, writhing in quiet agony if there was confusion or disinterest instead. Every moment got etched into my being, and all the way through each tour I worked on figuring out how we could make it work better.\n\nAnd, yet, as ineffable as radiance is, I believe there are many factors that are in fact quite technical that can help to foster it. Going to the \"regular\" theater, where I very rarely felt this quality, I began to notice so many obstacles and barriers that were put in the way. I began to realize that a lot of what Ten Thousand Things did to increase radiance was simply to remove those obstacles and barriers. If a small sun is hidden in each of us, then a big part of allowing it to shine comes from taking away whatever keeps it hidden.\n\nBARRIERS TO RADIANCE Many of the barriers to radiance are physical. When we removed them, we did so without really thinking about it. We removed them because we had to. Because Ten Thousand Things goes directly to its audiences, instead of expecting them to come to us, we had to make do without a lot of \"necessities.\"\n\nStages, for example. To reach our audiences, wherever they might be, we created the play so that we only needed a large room and some chairs. Insisting on a stage would have drastically limited the number of places we could perform. Once, early on, we happened to perform in a church space that had a small stage, so we thought we might as well use it. We were doing the seventeenth-century Spanish drama Life's a Dream by Pedro Calder\u00f3n de la Barca. We discovered, oh so painfully, that the distance that stage created between us and the audience, with the actors raised up on a platform, was deadly. The audience of inner-city high school kids was restless and bored with a story that had been riveting to other groups of youth at risk. We realized it was necessary to be on the same level as our audience, just a few feet or, even better, just a few inches away. Certainly never more than three rows back. The distance presented by a stage was just too big an obstacle for us to overcome.\n\nWe didn't bring lighting instruments along, of course; we just turned on whatever fluorescent lights were in the room. Again, we didn't think much about it; it's just what we had to do. It wasn't until about our seventh production that an actress, standing on the sidelines waiting to go on, whispered to me, \"I just love how we can see the audience!\" I started with surprise\u2014this had never occurred to me before. I was flooded with all the ramifications of this obvious fact. The actors could see the audience all the time, onstage or off, and so could receive every detail of their reactions full force, fine-tuning and adjusting all the while. And of course all those lights explained both my ability to make such detailed observations of the audience and the intensity of my elation and agony\u2014greater, I think, than when a director sits in the back row of a dark house, only able to listen to a muffled audience. Dark houses, we discovered, actually dimmed the intensity of the exchange between audience and actors. We were much better off without them.\n\nMAKING THINGS UP TOGETHER We began to see how our inability to have elaborate sets and oodles of props boosted the intensity of our work. Personally, I had always preferred empty stages, with lots of spaces for imagination to fill in; I intuited that this was how theater worked best. But now we were entering places where people were without expectations and used to making do with very little, where the rooms we had to perform in were already very spare\u2014cinderblock walls, linoleum floors, maybe industrial carpeting. As we held up a Hula-Hoop and asked the audience to imagine it was the moon, we could viscerally feel the pleasure they took in the invitation. They would giggle, resisting for a bit, of course, but then take the plunge. We were offering them a chance to exercise the muscles of their imaginations and participate in the creation of an escape from their barren surroundings, at least for a few hours. We were helping them remember that they could still pretend, just like kids do. There is a palpable energy when everyone in a room is making things up together. It most definitely adds to radiance.\n\nNEGLECTED STORYTELLING Many of the barriers we concentrated on removing were not physical ones. All the moments where the story wasn't clear. Or where there was a lack of urgency in what was happening onstage. Or where actors were not reaching out to directly connect with the audience right in front of their noses. Those were moments I could \"fix\" myself, as a director. Quite honestly, I mostly wanted to eliminate the pain I felt whenever first-time audiences didn't connect, when they became restless, or when it seemed we were actually confirming, even for a moment, their suspicions that theater was boring after all. I didn't want to have to sit through any more experiences of that. My directing muscles actually got quite a workout, since I had very few tools beyond actors' bodies, sound, and the text. But any time the story was clear and urgent and lively, I noticed most of the audience leaning forward, so I kept trying everything within my power to make them lean forward all the time.\n\nAUDIENCE RESISTANCE As we were noticing how certain physical \"requirements\" for theater actually got in the way, we were also encountering the enormous barriers presented by the initial attitudes of our audience members. We had sort of guessed, but didn't fully understand until we started, that most people who have never seen theater before really don't want to see it. So many have picked up the impression that it's just for rich people who have college degrees, that it's boring and hard to understand. The audiences we encounter can be defensive at first, and this sometimes comes off as contempt. I remember one homeless woman who, upon walking into a shelter and seeing our little setup, exclaimed, \"I don't wanna see no Shakespeare shit!\" She turned right back around and exited out the door. (We weren't actually doing Shakespeare that time.) On top of these learned attitudes, there was also usually well-founded suspicion and distrust of strangers who came into their space: so often they were just \"do-gooders\" who wanted to feel better about themselves by \"helping.\" As we began almost every show, we had to confront these very thick walls of suspicion, contempt, and indifference head on. As performers, we had no choice but to learn to blast right through it with the only tools we had\u2014honesty, humility, focus, and playfulness. Such attitudes took audiences by surprise. Over all these years, I have come to greatly appreciate the enormous importance of the element of surprise in cracking through people's barriers. Almost always, after ten minutes or so of strong doses of these unexpected demonstrations of respect toward people who are usually afforded so little of it, the walls started to fall. Radiance would start to seep through.\n\nAudience at women's correctional facility, The Good Person of Szechwan, 2005.\n\nNow, more than twenty years in, half of the performances we do are for members of the paying public, always in the exact same bare-bones manner in which we perform for our other audiences. I want to be very clear that audiences who pay for their tickets and have seen a lot of theater also put up barriers. Indeed, their attitudes were the very barriers that had moved me to seek other audiences in the first place. Quite honestly, we almost always find that the walls of judgment and critical distance erected by experienced theatergoers are much harder to blast through than anything nontraditional audiences offer. And yet, all the ways we've had to shape our theater in order to reach first-time audiences\u2014focus on clarity, urgency, depth, and honesty, with no darkness in which to hide, plus insistence on all imaginations having to work full throttle\u2014all this has also worked to shake up those jaded paying audiences. We surprise them, too. Eventually (though it usually takes about thirty minutes instead of ten), we are able to break through the walls put up by people with wealth and watch them begin to allow their own radiance to burst through.\n\nHere, I'd like to highlight the potential for radiance that all people bring with them to the meeting place of the theater. While I truly believe all humans have light inside them, at Ten Thousand Things we've noticed over the years that some people just seem to have easier access to this light. With many of our nontraditional audiences, despite all the hardships they've experienced, there is not as much \"crust\" to remove. In a funny way, both Ten Thousand Things actors and nontraditional audiences have a radiance that is perhaps a little closer to the surface than usual. (I've also noticed that over time the veteran theatergoers who keep returning to our barebones performances seem to have easier access to that radiance as well.)\n\nLuverne Seifert and Jim Lichtscheidl, At Your Service by Kevin Kling, at Open Book, performing for a traditional audience, 2004.\n\nThe theatrical work we do also attracts a certain kind of artist. It doesn't really appeal to people who are primarily interested in bolstering their r\u00e9sum\u00e9s and advancing in their careers. Indeed, many actors have told me that our work provides a welcome relief from those kinds of pressures because it calls on something much more powerful: a desire to reach out for something bigger than oneself, to find a way to connect with other humans who on the surface appear very different. I think the work actually attracts, if I may call them this, radiant artists. Artists with easy access to the honesty, humility, humor, and generosity that are necessary to break through audience barriers.\n\nH. Adam Harris and Sun Mee Chomet, Dirt Sticks by Kira Obolensky, Avalon School, 2014.\n\nAnd our audiences in prisons and shelters, in turn, often bring that same kind of easy access to the radiance in their beings\u2014once we have earned their trust. To be treated with respect for their intelligence, imaginations, and life experience is rare\u2014and when it actually happens, they permit their radiance to burst through. They have often experienced a long drought of respect, and once they feel it, they are eager to give back in return.\n\nOf course, every good theater performance in any venue can bring about connections and exchanges between actors and audience. But the extremes of our performance conditions can cause an intensity so strong, a connection that feels so immediate and charged that I think radiance occurs more often. Truthfully, I haven't experienced a lot of radiance in much theater I've attended in buildings where we sit far away in the dark, watching a story unfold on a stage laden with detailed sets that fill in every empty space with every conceivable prop to make things easier on our imaginations. There is radiance sometimes, to be sure\u2014but I think more of it in theater couldn't hurt.\n\nIn the chapters that follow, I want to share in more detail the discoveries we've made about what creates more radiant theater. As I talk about these discoveries, I'll follow a path that roughly corresponds to the way a production is created and born\u2014through the first imaginings of an audience, the choice of a play, the imagining of production values, rehearsing, and performing. I also want to discuss the political consequences of this way of doing theater in the big world, as well as describe the organization that has organically emerged to support this work. I hope simply to spark your own questionings of theater today and encourage your own reimagining of how else it might be. I hope you will take for yourself anything you find provocative, inspiring, or useful and ignore anything you don't.\n\nStorm-tossed ship at sea, The Tempest, Hubbs Center for Lifelong Learning, 1999.\n\nMost of these discoveries were made through absolute necessity. They were made because of the inescapable conditions that come as part of wanting to connect plays to audiences who might find their stories urgent. Indeed, it strikes me that theater is perhaps at its most radiant when everything about it is utterly necessary. When there is nothing that is superfluous. I think of the simple circle of chairs in which we perform, with actors and rings of audience radiating their energies from the center. I see it from an aerial view. While this book is in no way meant to be an argument to urge others to do theater exactly the way we do it, I start to zoom out, getting a view of people pulling up chairs to make other small circles in other places, each generating their own light, dotting the land with small radiant fires. It does seem like something we could use more of.\n\n# 2\n\n# IMAGINING THE AUDIENCE, RIGHT OR WRONG\n\n## Beginnings\n\nIN MY TWENTIES, I had my first \"professional\" job, assisting a famous director at a large theater in Los Angeles. We were working on a new play, and the two of us seemed to have a fun, easygoing rapport. I commented once during rehearsals, \"Oh, the audience will probably really like this part.\" The director snapped back at me, \"How do you know? You can't ever predict what an audience will like or how they will respond! Don't even try.\"\n\nThis harsh admonition to leave the audience out of it startled me. Of course, in one way he was right: you can't ever predict how an audience will respond, and if you try, you'll most likely be wrong. But at the time I was in the middle of trying to prepare for our first Good Person rehearsals, working on the script in my free time between the assistant directing job and another job as a proofreader. As I read the play, I felt absolutely compelled to try to imagine the responses of the audience of homeless people, whose experiences of life were probably different from mine in many ways. Even though I knew I might be wrong in my imagining, I just felt I had to try. I didn't want to risk selling short the experiences of people who had lived very hard lives. I wanted to be absolutely sure we connected with them. I wanted them to be hit by truth in their first contact with theater, perhaps even experience joy from seeing their truth reflected by others onstage. Necessity dictated that I try to imagine how the audience might see things. To avoid doing so felt arrogant and foolish.\n\nI've come to see that this simple action\u2014of consciously imagining how an audience might see the events and characters in a play\u2014is actually not common in most of our theaters. For example, when inviting directors to work with the company, I always ask them to propose their own projects\u2014I want them to work on plays they personally find compelling. But I am always surprised at how often their proposals stop with their own enthusiasm for a script. I often have to prod them to take the next step: \"But why would anyone else want to watch this?\" And of course, with Ten Thousand Things, the word \"anyone\" has the force of its full meaning: immigrants and the homeless and inmates as well as seasoned theatergoers, encompassing almost all ethnicities and income levels. Many directors report that they've never really thought about this question before. The assumption seems to be that if the artist finds a play compelling, that's enough. If an artist is interested, an audience will be, too.\n\nMaggie Chestovich and Kiseung Rhee, Life's a Dream, Dorothy Day Center, 2010.\n\nFEARS AND HAZARDS Of course, imagining the answer to the question, \"Why would anyone else want to watch this?\" contains some hazards. It requires that you make some assumptions about a group of people, and that could lead to errors and even stereotyping. But I would argue very strongly that the real danger lies in not trying to imagine the audience. When you don't try to imagine them, you are making assumptions about them anyway\u2014without being aware of it. Most likely you're assuming that the audience will be the people who usually do come to theater\u2014white, upper-middle-class, and over fifty. And this group of people, speaking in broad strokes, has a specific set of circumstances that informs their perceptions. They are sitting on a cushion of wealth. So this means some things are more important or interesting to them than to people who do not have such cushions. Such financially comfortable people are actually a very small fraction of the human population. And yet that is to whom most of our theater, without really being aware of it, is shaped to speak. We assume that everyone is like this small segment. It's no wonder so many people don't bother to go to theater.\n\nThe fear of making assumptions about an audience is understandable; on one level, it does seem ridiculous to try. You can't possibly know the astonishing variety of specific experiences of individuals in any group. And of course, as we all know, the responses of each audience depend on so many intangibles. The day of the week, for example: Friday-night audiences are often quieter, perhaps because everyone has just finished the workweek and is tired; Saturday-night audiences are often louder and livelier. And yet, just now, in talking about Fridays and Saturdays, I've made assumptions. These assumptions turn out to be correct, more often than not. Of course we've all experienced a lively Friday-night crowd and a quiet Saturday crowd\u2014there are always exceptions and mistakes. But we seem to need to make assumptions anyway. We seem compelled to lump each audience together and talk of it as one being. Why?\n\nA CONVERSATION WITH THE AUDIENCE Because theater is at heart a conversation. When you talk to someone, you're always trying to imagine what is going on inside their heads, since you can't see their actual thoughts. You scan their faces for twitches, listen to the intensity of their breath, their tones, inflections, and pauses, searching far beyond just the content of the words they use to reply. Using all the visual and aural cues you have, you still can't exactly know what's really going on inside\u2014so you make assumptions. As theater artists, we do this all the time in performance: if most of the audience seemed to laugh at a particular line on a particular night, an actor adjusts his next line accordingly, even if not every person was laughing. And after that particular performance, we talk about the experience of it by grouping the audience together as one animal: \"Boy that was a lively crowd tonight; they really liked it!\"\u2014even though many people in the audience might not have been laughing or enjoying the play at all.\n\nSimilarly, when you prepare for an important conversation, you use all the information you have about your intended listener to try to imagine how they might respond, to help you plan what to say and how best to say it. You make assumptions based on whatever you know of their experiences. Sometimes those assumptions are correct and the conversation goes smoothly\u2014and sometimes the assumptions are wrong, and the listener explodes with a response you couldn't possibly have imagined. So you jump around and readjust. That's how it goes. Making assumptions about your listener is an absolutely essential and inescapable part of preparing for any conversation. So why wouldn't we do this when preparing to perform a play?\n\nONE-TO-ONE MATCHING After that first production of The Good Person of Szechwan for the homeless shelter in Los Angeles, I continued to try to match a play to a specific audience. Our next endeavor, Electra by Sophocles, I matched, somewhat obviously, to girls in juvenile prison. I imagined they might also share Electra's experience of being deeply betrayed and craving revenge. This simple match up of main character to specific audience worked well, so I continued the process: Calder\u00f3n's Life's a Dream, a seventeenth-century Spanish drama about a prince whom the stars predict will grow up a monster, suggested audiences of youth at risk. Mud by Mar\u00eda Irene Forn\u00e9s, about a rural woman struggling to learn to read while supporting her very dependent brother, suggested audiences of adult literacy students.\n\nBut then it started to become clear that we needed to reach beyond these easy one-to-one match ups. For one thing, there weren't that many juvenile correctional facilities or adult literacy centers in our area to support an entire tour. The actors needed more performances to give them a chance to really dig in, and after putting in so much rehearsal time, I wanted more people to see each show. I began to see I could become more universal in my imagining, that the stories we told had large struggles that could appeal to all kinds of marginalized audiences as well as paying ones. Inmates would of course have a strong sense of imprisonment and confinement as well as an experience of the justice system and how it feels to be judged. Elderly people in nursing homes could probably relate to experiences of confinement and judgment as well, in addition to feelings of abandonment. And many inmates would probably deeply understand what it feels like to be abandoned; homeless people might also, or those in chemical dependency treatment centers. The process of looking at stories through the eyes of different groups of people revealed for me the common struggles that we as human beings all share.\n\nA MULTIPLICITY OF AUDIENCES Our audiences now have broadened beyond homeless shelters to include an array of groups of people living on the edge\u2014inmates, immigrants, the elderly, youth at risk, people in psychiatric hospitals, women in chemical dependency treatment, students at tribal colleges. Each time we add a new group to a tour, I feel compelled to try to reach out to imagine them, though my imaginings are admittedly very rough and often based in ignorance. I use just a very few broad brushstrokes to help me try to understand how they might perceive the play. My brushstrokes come from making one or two assumptions about experiences that they probably share, based on a few obvious factual circumstances of their lives. Low-income people, for example, do not have access to much money. That simple fact affects how they experience the world. I try to imagine how they might see certain moments, certain characters, certain scenes. And as I try to see the play through their eyes, some moments become more important. I feel new empathy for certain characters that I might have been more dismissive of before. It feels more important than ever to give all characters equal weight and compassion and create a level playing field. The stakes of almost everything that happens in the story become much higher. New possibilities, new choices open up to me, as I imagine audiences different from the one we are all most accustomed to.\n\nA TREASURE CHEST OF LENSES The kind of imagining of audience groups we do as artists at Ten Thousand Things is quite different from the act of thinking in stereotypes. Our imagining centers on the possible shared experiences of a group of people. Unlike stereotypes, these assumptions have nothing to do with people's intelligence, imagination, sensitivity, or worth as human beings. In fact, with our artistic work, we shatter such hurtful stereotypes: we assume from the beginning that marginalized people have enormous quantities of all these positive qualities. But we need a place to begin the conversation. So I start by imagining through a lens constructed from a few inescapable facts of a group's economic and physical circumstances. This new lens can, in turn, offer startling new perspectives on well-worn theatrical territory and become a wellspring of creative ideas.\n\nIndeed, as the wildly different groups we took each play to expanded, I found myself collecting a treasure chest of different lenses, which has truly become a fantastic source of wealth for me as an artist. As we returned again and again to certain venues, to juvenile facilities or immigrant centers or veterans' homes, certain general characteristics of each audience became clearer, and the basis on which we made our assumptions became more grounded in actual experience. As the array of audiences we performed for grew, the conversations I had in my head with different groups at different moments of the play just became a natural part of working on any script. I found myself going through the script with a kind of ceaseless dialogue in my head: The female inmates will probably really laugh hard at this part; the men not so much. Homeless people might really connect with this character, especially if he is taken very seriously. I think this part of the play is really boring; I imagine that most audience members will think so, too. Nothing important is really happening here, so why don't I just cut it? Being able to imagine a play through so many different lenses of experience has both multiplied and clarified the choices I can make as a director; it's now become a wonderful bubbling font of creative ideas.\n\nBEING WRONG IS PART OF IT ALL The other major danger of making assumptions about an audience is being wrong. And indeed, when you try to imagine through the eyes of others, you will very often be wrong. At Ten Thousand Things, we are so often wrong that we've come not only to accept error as necessary but to embrace it as a vital part of our work. Errors are just opportunities to learn and make discoveries. We like to make discoveries; we make a lot of them. The countless moments of being proven wrong in performance about the assumptions we've made could easily fill several volumes.\n\nOur fourth tour, when we performed Mar\u00eda Irene Forn\u00e9s's Mud at our very first women's prison, I worried a lot about how the women would receive the ending, where the main character, Mae, who has been caring for her mentally impaired brother, Lloyd, is shot and killed by him. I worried that the women would see themselves in Mae, and the ending would be too tragic and sad to bear. But to our great surprise, after the gunshot, the women began to cry out, \"Lloyd! Lloyd!\" as, like a great child, he picked up his dead sister in his arms, not understanding at all what he had done. The women's sympathies instantly went to him. He was left alone, abandoned, without anyone to care for him. And there I learned the great lesson that for most women in prison, the constant black hole in their hearts is the children they have had to leave behind. Whenever a story offers the tiniest chance to release this anxiety, the women's emotions will instantly go there. They weren't that concerned with Mae's fate at all.\n\nWhen we did The Ballad of the Sad Cafe (by Edward Albee, based on Carson McCullers's novella), I was certain that female inmates would love the main character, Miss Amelia, a tough loner in a small town in the south who wore overalls and ran the town's only general store. It turns out they did not like her very much at all. Their dislike intensified when, at a crucial moment, Miss Amelia wasn't able to make herself vulnerable. She was unable to receive a very sincere, heartfelt confession of love from Marvin Macy, an appealing ex-convict who was visiting town. Marvin got down on his knees and begged her to love him, but she said no. The women simply couldn't forgive such coldness. They gave the actress playing Miss Amelia very faint applause during the curtain call.\n\nAnother example of having our original assumption proved wrong occurred when we performed a new play based on ancient comic Kyogen tales, At Your Service, written for us by local storyteller Kevin Kling. Though double- and triple-casting was a hallmark of Ten Thousand Things performance style, I hadn't yet ventured into male gender-crossing, asking men to play women's parts, in part because I feel like men generally have enough good roles in theater as it is but also in part because I was more than a little worried how the male inmates might receive such casting. Days before rehearsal began, however, the only woman in this three-actor play had to drop out, and the only available actor with enough comic inventiveness to match hers was a man. He would have to play the female characters as well as some that could be either sex.\n\nNecessity once again brought us to the brink. Setting up in the gym of the first men's prison of the tour, we were all more than a little terrified, fearing the men might start mocking the actor and the play. In my short introduction, I made it a point to say that a male actor would be playing the women's roles, just as was done in ancient Japan, when women were not allowed onstage. I noticed a shift in the men at that point; this seemed to make an impression on them, appealing to their own sense of importance (though perhaps not for quite the right reasons\u2014but we went with it). When the first female character appeared onstage, the men seemed very receptive, even approving, laughing heartily at the humor in the scene but not at all in a meanspirited way. Our assumptions had been misguided; our relief was profound.\n\nBradley Greenwald and Jim Lichtscheidl, At Your Service, Hubbs Center for Lifelong Learning, 2005.\n\nTHE LENS OF THE TRADITIONAL AUDIENCE And now we can see that our paying audience is also just one small group, with its own specific experiences that cause it to behave and respond in certain fairly predictable ways. We can make assumptions about them based on our experiences with them, aware of their specificities because of our ability to compare them with other groups. They are generally much quieter. They don't laugh as much at bawdy humor. They notice different details of the play because of their educational backgrounds. It is sometimes challenging for the actors, who have become accustomed to liveliness and open honesty from our other audiences, to make the shift, but we have found aspects of the listening of our more financially comfortable audiences that we appreciate as well.\n\nMore important, our paying audiences can surprise us, too, breaking out of the mold. The Seven, a hip-hop retelling of Aeschylus's The Seven Against Thebes, by playwright Will Power, gave our expectations a delightful jolt. Our main anxiety in presenting this play, which tells the story of two brothers trying to overcome their father's curse and peacefully share power, lay with our traditional audiences' willingness to venture into this musical world fraught with the n-word and the f-bomb. And at the beginning of each paying show, I reminded people that we often ask our nontraditional audiences to step onto unfamiliar linguistic turf, as when we do Shakespeare or Beckett, but that this time, we were asking some of them to step into unfamiliar territory instead, and we hoped they would keep open minds and enjoy the ride. Though each audience seemed a bit taken aback in the opening minutes, it didn't take long for these generally over-sixty, predominantly upper-middle-class audiences to start to bop along with the actors, surprising us and often themselves, sometimes even raising their hands above their heads and swaying their arms to the music. Many approached us afterward to tell us how deeply the story had moved them\u2014and how much they had enjoyed hip-hop. Paying audiences, too, love being surprised.\n\nImagining an audience means making assumptions about a group of people so that you can begin a conversation with them. What is crucial is that you become as aware as you possibly can of these assumptions and know that the audience will tell you when you are wrong. At Ten Thousand Things we now heartily, and with great pleasure, embrace the strong possibility that many of our assumptions will be wrong. Our mistakes, sometimes painful, sometimes delightful, are our opportunities. Holding them thus, lightly, in awareness lets us open ourselves in humility to keep making discoveries about the boundless capacities of human beings. This conversation seems a fine thing for theater to strive for.\n\nIn a strange way, this really is all just a way to make myself bigger. Like most directors, when I decide to work on a script, I start with me, with my own likes and dislikes of the material. Performing for so many different groups of people has given me an amazing opportunity to broaden and extend myself as I try to understand how others might see it, too. Such imagining has made me a bigger human being, and as a result, I firmly believe, a better artist. This simple act of trying to imagine how others experience the world is really the essence of theater. All actors, directors, designers, and playwrights try to do so through the eyes of different characters in a play. It's strange how rarely we try to do this through the eyes of our audience. But I think that's because our audiences are right now, for the most part, all the same. This is a sad, self-imposed poverty for the theater, especially as we grasp the overwhelming wealth of experience that truly varied and diverse audiences can bring to us as theater makers.\n\nBruce A. Young as Oedipus, The Seven, at Open Book, 2012.\n\nAimee K. Bryant, Ragtime, women's correctional facility, 2005.\n\n# 3\n\n# FAIRY TALES\u2014BIG ENOUGH STORIES\n\n## Choosing a Script\n\nPEOPLE OFTEN ASK how I choose plays for Ten Thousand Things, and my first answer is that it's always been very intuitive. When reading scripts, I wait for a pull in my gut, one that alerts me not only that I am finding this play very interesting but that our diverse audiences probably will, too. That's it. That's how I choose. At first glance, the list of plays we've done over the years seems somewhat puzzling and random: King Lear, The Unsinkable Molly Brown, The Ballad of the Sad Caf\u00e9, Antigone, Blood Wedding, Ragtime, Doubt, Endgame. But stepping back, I can see that these plays all actually share a great deal in common and can better articulate what goes into causing me to feel that pull.\n\nPLAYS FOR AUDIENCES WITH WEALTH From the treasure box of the lenses of different audience perspectives, I almost always started by choosing the lens that let me see a story through the eyes of those without wealth. Looking through this lens, I was astonished at how many plays in Western dramatic literature really seemed to be written for audiences with a good deal of money. So many plays from centuries past are about wealthy young couples foiled by their parents in attempts to marry their true loves. So many contemporary plays are about relationships or marriages of well-off couples gone sour or careers not going as planned. A friend once dubbed this category of contemporary drama, currently quite popular on Broadway and in the regional theater, \"Rich People Being Mean to Each Other.\" The struggles of such stories felt small, at least when compared to the prospect of not being able to find food or shelter for your children, or not being able to find a job because you can't read, or being thrown into prison. I didn't think my nontraditional audiences would be that interested in these more cushioned \"problems.\"\n\nAnd yet my intuition also told me that contemporary, realistic plays about living in poverty could be disastrous. In the first place, our low-income audiences would know far more about those situations than we, as mostly middle-class artists, ever could. We could meet in a \"make-believe\" slum, as we had in The Good Person, but not in one realistically re-created around detailed knowledge of poverty today. It felt arrogant to think we would have anything to contribute: we would most likely appear ignorant and clumsy. To top it off, living in poverty was so often painful, why would anyone actually living in it want to experience more of it? It made me realize that many plays about contemporary poverty are probably written for wealthy audiences as well, to help them better understand existences different from their own. This is not a bad thing, but it's not what we were out to do here.\n\nONCE UPON A TIME Looking at scripts from this vantage point helped to quickly eliminate a great many in my stack. I approached the remaining ones once again with my intuition; I began to see similarities in the ones that pulled at me. I like to call these types of plays \"fairy tales.\" A fairy tale is a story that lets us all, no matter what our background or economic status, enter its world together as equals. Fairy tales are also, quite conveniently, the kinds of plays that I have always most enjoyed. They are the stories I actually think theater tells the best, better than any other art form.\n\nFirst and foremost, fairy tales wrestle with very deep struggles, ones that every human being knows in the bones. Stories of the unrelenting hold of jealousy, the pain of great betrayal, the violent urge for revenge. They offer no easy answers, and thus are stories that respect their audiences, knowing that all of us have been wrestling with these huge issues for centuries, without much success. No one of us really knows, better than anyone else, how to answer the questions a fairy tale raises.\n\nSetting has a great deal to do with the fairy tale's broad appeal. These are stories that happen in another time and another place, usually, though not always, outside the scope of actual human history, places and times where, again, none of us can be expert. In our contemporary, realistic world, some of us are experts in the details of poverty; others are experts in the details of middle-class comfort or upscale wealth. But none of us really knows what it is to be a king or a queen. We all like to imagine ourselves as royalty, at the center of the world, as someone of supreme importance and worthy of being loved. But we all enter this terrain as novices. So through its setting, too, a fairy tale creates a level playing field on which we all are equals.\n\nMULTIPLICITY OF CHARACTERS Because they are such big stories, fairy tales usually have lots of characters, from lots of different classes. Such teeming populations are hard to find in the economically restricted storytelling required of contemporary playwrights by so many theaters today, theaters worried about cast sizes and budgets. But having many different characters onstage provides access to many more different points of view. Because we have created a theater that performs for people of so many experiences, particularly economic ones, often seated together at the same performance, the ability to consider many points of view in one story is crucial.\n\nIn a fairy tale world, anyone can be anything. These nonrealistic worlds, unbound by the painful histories of oppression in our own, allow for creative and diverse casting. It's no problem to have Puerto Rican kings and African American queens with a Korean American son for the prince, alongside female warriors and jesters. There is room for anyone to be almost anything in a fairy tale world, and this allows us to extend the invitation to enter the story even more compellingly to all. And because so many of the places where we perform are not only very racially mixed but often racially charged and tense (as in many prisons), it has become important for us to create, for an hour or two anyway, a world where race doesn't matter so much. The relief of an audience, once they've realized that for a little while they can let down their guard about race, is palpable; entering such a fairy tale world becomes part of their ability to relax and enjoy.\n\nPERSPECTIVE ON PAIN A fairy tale's far-away setting can provide imaginative distance from deep pain in a way that contemporary naturalistic plays cannot. In our early days, we worked with a playwright who, though he tried to comply with my exhortations to set his story in a made-up place, ended up locating much of it in a very realistically detailed contemporary urban world. The story concerned a mother and a teenage daughter, both drug addicts, who each ended up losing her child to foster care. There is probably nothing more painful than losing a child. I remember a performance at a shelter where droves of women got up to leave in the middle of the play. The events were simply too painful to watch. Why would anyone who had experienced such a wrenching separation ever want to go through it again? Such a realistic story was not wanted or needed; it just hit too close to home. It was probably our hardest Ten Thousand Things tour ever. I vowed it would never happen again and have remained fiercely devoted to fairy tales ever since.\n\nAnd indeed, through fairy tales we have been able to do plays about losing a child that people can bear. Brecht's The Caucasian Chalk Circle tells of a child, born of a wealthy woman but abandoned in war, found and cared for by a poor kitchen maid, whose true mother is uncertain. Near the play's end, the child is placed in a circle in a contest between the two women, the winner being the one who can pull hard enough to drag him out. Grusha, the adoptive mother, doesn't want to cause injury, so of course she lets go. She then has to watch her child walk away with someone else. Set in a made-up time and place, though, the pain of this sight was much easier for our audiences to watch; the fairy tale setting granted distance and perspective.\n\nOf course Grusha gets her child back in the end. But even with the Greek tragedies which do not end so happily, like Iphigenia, where Clytemnestra's child is killed in a sacrifice to the gods, the loss is easier to watch because the story takes place so long ago and far away. There is comfort in knowing that you are not alone, that others before you have also suffered so unfairly. Because of imaginative distance, male and female inmates, all separated from their children, were able to sit and watch these plays straight through, gaining perspective on and solace for their own situations.\n\nOPEN SPACE INVITES ACTIVE IMAGINATIONS Fairy tales contain so many other enticements to audiences: most wonderfully, they ask everyone to jump in and use their imaginations full blast. Because no one can be an expert at what a fairy tale world is actually like, the stories contain lots of space for us to fill in, especially in a Ten Thousand Things performance situation where we have almost no set. A homeless woman can imagine what a palace looks like just as well as a suburban housewife. With fairy tales we need fewer \"things\" to help us imagine the world. Blenders and toasters and large-screen TVs placed on the stylish shelves of the elaborately reconstructed walls of a split-level suburban home can serve as a barrier to people who are constantly reminded, every time they look at the set, of what they do not have. But a sword or a scepter or a scroll, a simple strange object that almost no one these days actually possesses, can invite everyone to fill in the space around it.\n\nFairy tales free up all of us to concentrate on more important things than \"the correctness of the objects.\" So often, when I go to realistic contemporary plays, I find myself thinking about whether the props or costumes are right, whether that kind of person would actually have that kind of armchair in their living room or that kind of can opener in their kitchen or wear that particular style of skirt or those particular shoes. Since no one can be an expert in a made-up world, we are all free to concentrate on other matters: whether emotional depths ring true, whether the behavior of the characters seems ethical or just. Fairy tales point our attentions more firmly toward the questions that unite us rather than the material things that so often divide and create hierarchy.\n\nROOM FOR MOVEMENT A fairy tale also invites lots of movement. You don't usually find sofas or armchairs there; fairy tales usually don't require large furniture pieces that encourage actors to sit down and talk and talk and create sinkholes of static energy onstage. The invented worlds of fairy tales are filled with so many more possibilities for movement: Characters can break the confines of the small gestures to which we usually limit ourselves in everyday life. People can skip and spin and leap. A sudden spurt across the stage, a sharp turn, or an abrupt halt, each draws our focus and helps to physicalize what is going on underneath all the words. If a play is set in another time and place but requires people just to sit and talk for lengthy periods, it probably is not the right play for the diversity of our audiences. Surprising physical movement helps so much to clarify and intensify the story.\n\nThe large, open spaces of fairy tales, I firmly believe, make them the kinds of stories that theater tells best. We can tell them far better than film or television. I so often wish that theaters would acknowledge this and stop trying to be like TV or movies. Neither of those dramatic storytelling mediums can have any empty space. The camera must record the reality in front of it, and this guarantees that all the backgrounds will be filled in: all the bricks in the wall, all the buildings along the street, all the trees on the mountains inescapably fill the frames. When a film tells a story about a world none of us knows, it still does all the imagining for us, filling in exactly what the boarding school for young wizards looks like or presenting a flying dragon in the flesh through special effects. Science fiction and fantasy worlds must be presented in full \"realistic\" detail. There is no empty space for our imaginations to participate in their creation, and as a result the worlds are flattened: the details are already complete.\n\nALL KINDS OF PLAYS QUALIFY Of course all kinds of stories can be fairy tales\u2014they don't have to take place in make-believe kingdoms with palaces. There are endless ways to create the imaginative distance of \"fairy.\" A story set in an actual historical period, one different from our own, like the musical Ragtime or Strindberg's Miss Julie, can easily be interpreted as a fairy tale. Many in our audience have little knowledge of history anyway, so this frees us from having to worry about re-creating the actual historical world in accurate detail. Authentic period costumes like ruffs and stockings for Shakespeare or elaborate and accurately flounced and frilled gowns would most likely serve as a barrier to many nontraditional audiences, who immediately associate such trappings with \"rich, white people stories\" anyway, so a fairy tale perspective frees us from all that. Even a more contemporary story like John Patrick Shanley's Doubt, which tells of the struggle between a nun and a priest, can be seen from a fairy tale perspective. Though many in our audiences have had direct experience of being abused as a child or being falsely accused of a crime, the world of a 1960s Catholic monastery, with the strange robes and caps of its inhabitants, was exotic enough to provide distance from the pain of the issues. It just has to be \"another time, another place.\" We can invent the rest together.\n\nRegina Williams, Doubt, senior center, 2011.\n\nBut I have watched Ten Thousand Things audiences enough now to confidently assert that although there are many ways to bring make-believe distance onto the stage without having actual \"fairies,\" there must be a tale. I myself am often attracted to nonlinear, experimental plays; I enjoy plunging in and trying to make my own sense of it all\u2014but I have had practice and training over the years to help me become familiar and comfortable with such experiences. Many in our first-time theater audiences have not been afforded this luxury, and as a company we do not have time to educate. We come in to perform for what are often very transient populations, groups that fluctuate dramatically day to day. Story, however, is something we all can grasp right away. We need no explanation or education around it; every human has deep experience with it. Story is the thread that we all, across the entire spectrum of humanity, can hold onto, together; story will pull us all through from beginning to end.\n\nA SCRAP OF HOPE, A LIGHT TO SHARE There's one more requirement to choosing a story for Ten Thousand Things, however. From these many years of performing for people who have led very difficult lives, I've come to believe that, no matter how dark the telling, no matter how deep the probing of human misdeeds, the tales we choose to tell must contain a tiny shard of hope. We all need just a little something to seize onto, some tiny thread that allows us to keep moving forward a little, that allows for some possibility of change. I have come to feel morally responsible for finding this tiny shard in every story. If it's not apparent, I dig hard to find at least a bit of it. This has nothing to do with happy endings, of course, which we all know are impossibly rare. But even to have been able to perceive that for one character at one moment another choice was possible, even if that choice was not ultimately made\u2014even just that can provide a scrap of hope.\n\nI have come to see that choosing to tell a story without any hope, as sometimes happens in contemporary plays, is an indulgence. It feels to me, ironically, a possibility born of the luxury of having a comfortable cushion. Telling a story without hope is a choice artists make because none of us knows for sure whether there is or is not hope for humanity. I know that I need hope to function. These days, as temperatures on our planet continue to rise unabated, as the economic gap continues to widen, as our country is divided again and again along racial lines, I spend lots of time searching for hope myself. It's hard work. I want to come by any hope honestly, of course, by looking reality squarely in the face. But I still need to see things in such a way that I won't be paralyzed with despair. A scrap of hope allows us to try to change until the last possible moment. We have none of us reached that last moment, so I believe we must search hard to find a way to honestly tell our stories with hope. As artists now, I believe a big part of our job is to work mightily to find some bit of possible light to share.\n\nACCESS TO NA\u00cfVET\u00c9 And then there is that word \"fairy,\" which, in the way we use it, has almost nothing to do with actual fairies but a great deal to do with na\u00efvet\u00e9. Na\u00efvet\u00e9 is another great leveler. It comes from the place of the child in each of us, the child who has participated in making things up and believing, at least for a while, that what she has created is real. Theater asks us, as adults, to behave na\u00efvely in each other's presence: to believe that the aluminum pole is a tree, that the actor with long furry ears is a donkey, that the actress whose hands are covered with red dye has just murdered someone. Such full-throttle make-believe is not something most of us are comfortable doing publicly in our regular adult lives. But if the pull of the story is strong enough, we all end up doing it anyway.\n\nAnd with Ten Thousand Things, the adults do it, not hidden in the dark, but in full fluorescent light, in a circle where everyone can be seen. The moment when everyone looks up at once to discover that we've all been engaged in this ridiculous activity together is a wonderful equalizer. With so little onstage, it's very clear that the world is only held together by our pretending. We share a moment of embarrassed laughter. And we are reminded in a funny way that we were all once children. This moment is a crucial equalizer, and I always try to make sure that the first such moment happens fairly early on in the show. We acknowledge the ridiculousness of it all together, and then we can all jump back into the story and get on with it.\n\nAnd thus, the heightened na\u00efvet\u00e9 required by a fairy tale also guarantees that the audience will always be able to find laughter. In fairy tale worlds it's so much easier to step back and see that we're just making it all up. This is critical in the challenging worlds that these stories so often explore. Laughter is absolutely necessary to survive darkness onstage or off. It's actually profoundly important practice for people who are afforded little occasion to laugh in their daily lives. Indeed, the act of laughter almost always offers its own shard of hope. The darker the world, the more vital flashes of humor become to allow us to truly see in the darkness.\n\nIt is often challenging to find contemporary plays that are written with these kinds of attributes. That's why I was extremely lucky to discover, living just twelve blocks away from me in Minneapolis, a fantastic playwright, Kira Obolensky, who writes naturally from the perspective of \"fairy tales.\" Kira loves to create, with great humor, worlds that are different from the ones we inhabit in our everyday lives; her language is simple but playful and heightened just enough to lift us out of our own time and place. Because she lives so close by, she's been able, over the years, to attend performances for almost every kind of audience group, and she now carries these audiences around in her head as she writes. When writers start to imagine what stories they would choose to tell if they knew that the true diversity of humanity would be sitting in their audience and not just the small homogenous sliver they are most familiar with, their stories just naturally start to expand. Kira has been surprised and delighted to discover what has happened to her writing as she has continued to do such imagining of the audience. It is exciting to think what might happen to the American theater if more playwrights could similarly experience and began to write for audiences containing many classes, cultures, and lives outside the \"traditional\" spectrum.\n\nIt's important to note that all these \"guidelines\" for choosing a play should be broken if your intuition tells you a different kind of story might connect. Many wonderful guest directors have proposed plays that at first seemed to me to fall outside my experience of what would engage our hugely diverse populations. When Peter Rothstein wanted to do Doubt or Lear de Bessonet proposed Streetcar, I worried the stories might be too contemporary and realistic, too narrow in scope, or without enough distance from pain. I was delighted to have my fears allayed. Both directors understood the strangeness of the plays' worlds and the deep universal resonance of the stories. Each one actually found the \"fairy tale\" in plays that at first glance seemed too small and specific. I was delighted to see how well each of these seemingly unlikely plays could work.\n\nAnd it could very well be that a group of actors and a director with great understanding and experience of the contemporary world of poverty could do a realistic story about the here and now with great success. Part of my reluctance to do so certainly has to do with my relative ignorance about such lives and is an honest acknowledgment of my own limitations and those of most of the actors I work with, though part of it also has to do with my strong sense that these realistic plays do not provide enough imaginative distance from pain. But I certainly could be wrong.\n\nI do believe, however, that \"fairy tales\" are still the best meeting place for all of us, allowing theater to work full force and generating maximum radiance. Their wide-open spaces contain enough room to invite in every kind of human to use their imaginations full throttle, to become na\u00efve and to pretend, with easy access to humor, to laugh at the ridiculousness of it all. Because there's so little clutter, it's easier to see connections between things. Against an empty backdrop, each object, each person, each situation becomes more heightened and resonant. Our imaginations have room to caress what is there and can more easily spot connections and metaphors. Fairy tales combined with the empty spaces of theater allow us the room to more easily find food for our souls, which are always searching for meaning and connection with each other. For the wide diversity of our audiences, they are truly, I believe, the stories that theater is best suited to tell.\n\nKaren Wiese-Thompson as a ghost and Kris Nelson as Raskolnikov in Kira Obolensky's Raskol, House of Charity, 2009.\n\nGreta Oglesby as Clytemnestra in The Furies, Dorothy Day Center, 2001.\n\n# 4\n\n# AUDIENCES AT LIFE'S EXTREMES\u2014THE GREEKS\n\n## Fairy Tale Group #1\n\nMY ONLY EXPERIENCE watching Greek tragedies before Ten Thousand Things came in my early twenties. I'd be seated in a large theater, usually toward the back, a long distance from the actors who were emoting full force onstage. I was working nine-to-five temp jobs, and like many people around me, I was tired from the day, struggling to keep my eyes open and to find a way to connect to people who were murdering their family members or suffering from enormous loss of loved ones who'd been killed. (In fact, one of my all-time favorite audience comments came from a women's prison where we were doing Iphigenia. An inmate sitting down before the show picked up the program we had placed on her chair and declared, \"I don't wanna see no damn Greek pathology!\")\n\nDespite these experiences, after the first Good Person tour, while I was casting my nets far and wide to find the next play, my intuition drew me to Greek tragedies. I paged through stories from long ago, brutal worlds of kings and queens and their offspring. The plays addressed huge questions that humans haven't stopped struggling with, so often centering on our almost compulsive need for vengeance. The struggle between the pain of being wronged and the need to ease that pain by doing wrong in return seems one of the most eternal of all. All signs pointed to a \"fairy tale,\" a big story in another time and place that might somehow connect with its exploration of enormous life struggles.\n\nAt the same time, I dreamed about other places we might go to find first-time audiences. I kept returning to one experience on the Good Person tour where we'd performed at a shelter just for women. The freedom and emotional responsiveness of that particular audience remained with me. That site had provided our biggest burst of radiance, and it seemed promising territory to continue to explore. How often do we get to perform for audiences of only women? Such audiences would have a hunger for complex, intelligent, and powerful female characters that matched my own. All of this was swirling around inside me when I stopped at Electra by Sophocles. Once again matching an audience somewhat simply and obviously to the play's main character, I tried to think of where we might find young women, like Electra, who were familiar with painful cycles of betrayal and revenge. I decided we would try to perform this tragedy for girls in detention.\n\nELECTRA: THE PRIMAL URGE FOR VENGEANCE The events of the play felt suddenly relevant to me as I knew that many juveniles in prison have been \"betrayed\" by their families: abused physically or emotionally, neglected, or abandoned. I thought at least the rough contours of the situation would resonate. At the beginning of the story, Electra's mother has murdered her father. Agamemnon had just returned from battle and was cleaning his wounds in the bath when Clytemnestra stabbed him with a knife, in vengeance for his murder of their older daughter; he'd sacrificed Iphigenia's blood to the gods so he could win the war. But Clytemnestra's motives don't concern Electra. All she knows is that she deeply loved her father and her mother has betrayed her. Electra is consumed with hatred, finding herself a servant in her own home, ignored and dismissed by her mother and a new stepfather.\n\nIn my rough imaginings, one particular line of Electra's leapt out: \"Hope is dying inside me. With evil all around me, how can I keep myself sane and whole and moderate?\" It seemed that question might really resonate with a young woman finding herself alone in prison.\n\nThis Greek \"fairy tale\" offered no easy answers. The tension of Electra comes from not being able to say whether she is a victim of the vicious cycle of her family's violence or whether she is evil and out of control for helping her brother murder their mother. Of course I did not want to present Electra as just a victim, nor did I want to view the audience simply as victims. I imagined that at least some of the young women found themselves in prison for truly doing wrong, no matter how badly they themselves had been treated. The complicated reality of my imagined audience again urged me to look at the complexity of the story; it made me want to heighten the play's tensions and hard edges, giving all sides a fair hearing, all characters their due. The Chorus, with their periodic questions and responses, seemed like an excellent vehicle to help the audience process this complexity. Most important, the play ended with a big question, just as The Good Person had, with the Chorus asking the audience, \"What shall we say of this? Are we free from Evil? Or caught in more Evil? What is Good and Evil? I do not know.\"\n\nI knew that girls of all racial backgrounds would be in the audience, and the story's fairy tale setting allowed me space to create a world where anyone could be anything. I was able to cast a wide array of excellent, racially diverse actors, something easy to do in Los Angeles. In such a made-up world, a Puerto Rican queen could be mother to a white daughter, a Chicana daughter, and an African American son. In addition to offering the pleasure of being able to see oneself onstage, such casting felt healing somehow. If race didn't matter to the family, it probably wouldn't have to matter so much to the audience, either. I imagined that prisons could be very racially charged situations, and such families might allow the audience to brush racial politics aside at least for a while, almost in relief. At one level, I hoped, this casting might confirm our deepest instincts that such \"fairy tale families\" should really be possible.\n\nWe did only three performances of the play. It was all still so new, and I wanted to be sure to pay people fairly this time. Since The Good Person, I had figured out how to become a nonprofit using some DIY software and had managed to obtain a small grant for the project from the city of Los Angeles.\n\nOur most powerful performance took place in an LA County juvenile prison. It was a shockingly grim place where both young women and men were sent. The old brick buildings were gloomy and foreboding. My heart skipped several beats as we walked past rows of young men in bright orange jumpsuits lined up in a courtyard, hands clasped behind their backs, a few at the ends of the lines so small that they couldn't have been older than ten or twelve.\n\nThere were two dingy common rooms for the girls, one with cafeteria tables and Saturday-morning cartoons blaring from television sets suspended from the ceiling, the other, our performance space, a small room of gray cinderblock with several wide steplike ledges on three sides on which the girls could sit, dim light filtering through grated windows. A few girls elected to stay in the other room and watch cartoons, but most, probably about thirty, decided to see what we were up to. Two girls in their mid-teens sat on the top ledge, both of them sucking their thumbs, making it clear that many here were really, at bottom, in need of comfort. Who could imagine they might find it in a Greek tragedy?\n\nAs the musician began to drum, this small group of young women entered the turbulent world of the House of Agamemnon effortlessly. Although the actors were plumbing deep, complex emotions just inches away, the girls were at ease with the intensity; it seemed these extremes were comfortable and familiar. There was no need for them to strain to connect. They absorbed it all: Electra's enormous outpourings of grief at her father's murder, her arguments with her quieter and more pragmatic sister, her ragings at her mother, her joy at the return of her long-absent brother Orestes. They sat intently as the plot unfolded and followed with acceptance and ease each twist and turn, from Electra's successful enlistment of her brother in the plan to kill their mother to her deep satisfaction as her stepfather unrolled the sheets to discover Clytemnestra's corpse. The sudden shifting extremes of Greek tragedy seemed a familiar reflection of the emotional roller coasters of their own adolescent lives. They sat quite still, taking it all in. I felt again that the distance of this long-ago \"fairy tale\" world made their ease possible in a way a contemporary retelling would not allow.\n\nOnce the story was done, they eagerly broke their silence. Without any prompting from us, they began to ask questions. It was the most seamless transition from play to discussion I have ever witnessed. \"Did they kill people like that in the old days?\" one voice asked. \"Is this a true story?\" \"It's a myth,\" an actress replied. \"It represents feelings we all have inside us. All of us have the potential to love, to hate, to want revenge, to kill, don't you think?\" The girls nodded their assent. \"How does it feel to act out murdering people?\" one girl asked Orestes. \"I'm still not sure I really know,\" replied the actor quite honestly. \"I know how it feels to want to kill someone.\"\n\nThen a quiet voice spoke from the top ledge: \"My mother killed my father, so I know how Electra feels. It stays with you always.\" All of us onstage took deep breaths, letting the reality of this statement sink in. We felt the shock of realizing that we were not just telling a story or a myth; we were treading very real and difficult paths that many in our audience knew only too well. As artists we could never let ourselves pretend we really understood those paths; we could only keep searching for a deeper understanding. Once again we were overwhelmed with a profound sense of humility. In case any of us had forgotten, we were reminded that this state of mind was absolutely necessary for our own artistic work.\n\nThe girls spoke to the characters more often than the actors. \"How does it feel to be a servant in your own mother's house?\" they asked Electra. \"How could you treat your own daughter like a servant?\" they demanded of Clytemnestra. She replied, \"Electra reminds me of my husband, who killed my daughter Iphigenia. Every time I look at her, I'm reminded of this man I hate. So I don't mind having her out of my sight. And I'm under the influence of my new husband.\" Aegisthus, Clytemnestra's lover, chimed in, \"Doesn't bother me at all. Electra could live in a pigsty for all I care.\" The girls hooted.\n\nNext came a debate about whether it was better to be outspoken and brash, like Electra, or more quiet and politic like her sister Chrysothemis. \"Chrysothemis was a hypocrite!\" cried one girl. \"No, no way, Electra was an asshole!\" averred another. The actress playing Chrysothemis chimed in, \"From my point of view, Electra's ranting and raving weren't doing any good. She was still weak, even though she acted so tough.\" \"Yes, yes!\" agreed about half of the girls.\n\nThe young women identified keenly with the arguments between Electra and her mother. \"I was crying, you reminded me of my mother so much!\" one girl explained. The actress playing Clytemnestra later confided that whenever she spoke to Electra, she had only to speak, in her mind, to the girls in the audience, and her heart instantly opened and was torn in two, as was Clytemnestra's. The audience, once again, made our artistry better.\n\nAll in the audience could clearly see the cycle of violence and revenge in this ancient tale, easily connecting it to forces at play in their own communities. They saw the stupidity of the murders and yet seemed to understand how easily one can be swept up into the machinations of that very powerful urge for vengeance. There was even sympathy for the callous stepfather Aegisthus, who, when faced with his own death by Orestes's hand, had cried out,\n\nWait. Think. Both of you.\n\nHow will it end?\n\nI helped kill your father.\n\nYou helped your brother to kill me.\n\nSo someone of my kin\n\nWill have to kill you both\n\nAnd it will go on forever.\n\nOur house is accursed.\n\nSo... what will you do?\n\nGo on doing Evil\n\nand saying you do Justice?\n\nWhat does that gain? Tell me.\n\nAt that moment, most of the girls were on his side.\n\nThe girls were interested in the extreme racial mix of the family and expressed their appreciation of it. Not only did this confirm my instincts that diverse racial casting around family groupings could actually help to set aside race as an issue, allowing everyone to move more quickly into the story, but I discovered that for many, such a family makeup was not a \"fairy tale.\" Some of the girls offered that they actually came from such racially diverse families themselves. It was gratifying for them to see the reality of their families\u2014often composed of stepparents and stepsiblings of different races, families breaking apart and reassembling\u2014reflected onstage.\n\nI was happy when one girl asked, \"Why did you decide to do this play for us?\" This is always a key question for an audience to whom we bring a play, rightly suspicious of strangers and their \"do-gooder\" motives. Though I always try to make very clear that we are not preaching in the presentation of the actual play itself, I was glad to have a chance to answer the question directly and emphasize that we were not passing judgment on any characters. \"We wanted to explore feelings of revenge, which we all have,\" I said, \"and to look at what happens when you act on them.\" The actress playing Clytemnestra added, \"And to see whether, if you can see the history, the cycle, the pattern, then perhaps you can find a way to break out of it.\" The girls murmured in understanding.\n\nThey offered up unsolicited praise: \"It took lots of guts to concentrate like that with all of us in front of you.\" The fundamentals of theater, which we completely took for granted, were wondrously strange to them. \"I like how you guys were really serious and it's like the seven of you didn't even realize we were right in front of you.\" Perhaps most gratifying of all was the comment, \"I really appreciate you taking this really old play and finding things in it that have to do with what's going down now.\" Once again, we received confirmation of our intuition that the imaginative distance of a story set in \"another time, another place\" would actually allow deep connections to occur.\n\nAs we carried our set pieces back to our cars that afternoon, we felt elated. We were all moved by how the girls had taken things so personally, causing us once again to wonder at how rarely paying audiences allow themselves to admit that they have been personally affected by what they see onstage, how rarely they talk about a play's speaking directly to their own lives. The actor playing Orestes, carrying a drum and keyboards, turned to me to ask, \"Is that what you had in mind?\" Oh, yes, and ever so much more. By taking a big story of theater to people who had never seen it before, we had uncovered radiance: moments where the audience's response to the play was intense, profound, and even transcendent.\n\nINSURMOUNTABLE OBSTACLES: PERFORMANCE CONDITIONS Every Ten Thousand Things \"tour\" is an unpredictable adventure, and although the Electra tour was short, in addition to this exhilarating performance, in keeping with the emotional roller coaster of Greek tragedy, we also experienced our first excruciating show, one where we could make almost no connection with the audience at all. And though at the time we were quite shaken, in hindsight I now can clearly see that it was the conditions of the performance that made it all so awful, rather than the actual performance itself.\n\nI had heard that there was going to be a conference of Chicano youth at risk around the same time as our tour. That seemed a promising audience, so I contacted staff at the hosting organization and they were willing to try a performance. But when we arrived at the meeting space, we found ourselves carrying our urns and painted flats into a cavernous gymnasium. Assembled within was a teenage audience of both sexes. A mixture which, we quickly discovered, unleashes its own large and very different dynamic: the presence of the boys seemed to cause the girls to become quiet and subdued. In addition, the teens had been required by adults to attend as part of the conference. This created a hostile barrier that I have since learned is almost always impenetrable. All the lights in the gym were turned off and we were lit with a few very bright spotlights so that we could not see the audience at all, except for a few scowling faces in the very front row. It was quite clear no one wanted to be there, and we weren't able to draw them in\u2014the barriers of darkness and mixed-sex teenagers not present by their own choice were just too much. We left feeling defeated. And yet, as has happened on every tour since, we clung tightly to our exhilarating experience at the girls' correctional facility. When the performance conditions had been right, the play really had connected.\n\nTHE FURIES: AN ENDING REVISIONED We've returned to the well of the Greeks many times since. And it's no accident that the stories work. Scholars seem to have reached a general consensus that audiences for Greek tragedies were huge, perhaps in the tens of thousands, and contained citizens of almost all classes, including prisoners who were released for the performance. This broad swath of humanity in the audience probably goes a long way to explain why the Greek playwrights work so well for Ten Thousand Things. But it is very important to note that while the Greek tragedies feature a host of complex and powerful female roles, Greek audiences included very few women. The plays themselves often reflect this misogynistic view. As is true of most of the theatrical canon, my creativity is called upon again by necessity because I am a woman presenting plays to audiences who are sometimes composed entirely of women. I have to make adjustments either in casting or in editing and interpretation to be certain that truly everyone is welcomed into the world of the play.\n\nRon Menzel as Orestes fleeing outside the audience circle, The Furies, Dorothy Day Center, 2001.\n\nYears later, poring through the plays, I was especially attracted to The Eumenides, once again because of its prominent female characters, the dark and mysterious Furies. The play grapples with fundamental issues of justice, venturing beyond the personal world of Electra, taking family struggles into the bigger political world of government systems\u2014and the gods. But reading the play with contemporary eyes\u2014my own and those of my all-female audiences especially\u2014the gulf between the play's original world view, in which the female characters are treated with contempt, and our own, seemed almost unfathomable.\n\nThe story of The Eumenides picks up where Electra leaves off, following Orestes after he has killed his mother and stepfather. I imagined lots of places where the story might resonate with our now very diverse nontraditional audiences. I thought many could identify with Orestes, fleeing to escape the wrath of the Furies, who want vengeance for his murder of Clytemnestra. I thought many might understand what it's like to run from feelings of guilt and blame and self-hatred for wrongdoings. I imagined many, particularly in our all-female audiences, might also identify with the rage of the Furies over a mother who was murdered by her own son. Indeed, I imagined that many could identify with both Orestes and the Furies and their need to throw themselves down to beg for mercy, whether of the court or of the gods. I imagined that the need to find some kind of relief from overwhelming, devouring feelings of vengeance or guilty torment would resonate; I felt most could connect with the deep desire to receive, through a judge's decree, at least some restoration of sanity.\n\nThe part of the play that most deeply concerned me was its \"happy ending.\" Against all reason, Athena and the gods decide to rule in favor of Orestes, finding him innocent of the murder of his mother. Athena breaks a tie in the jury and casts the deciding vote for Orestes, using \"the prerogative of the male\" as her justification. In essence she rules in favor of Orestes because men are more important. Realizing that the Furies might be more than a little disturbed by this, she acts quickly to appease their wrath, promising them a \"partnership\" in ruling the city if only they will retreat into their cave. Within minutes, the Furies begin to sing praises of the goddess and her wisdom and fairness, blessing Thebes and retreating happily into their cave, their names being changed to the Eumenides, or the Benevolent Ones.\n\nThat this is considered a plausible ending, let alone a happy one, seems inconceivable, especially with such a judgment coming from a female goddess! Thinking of all the women in our audiences struggling with their own sense of self-worth, the contempt with which the Furies and Clytemnestra's ghost are treated was problematic, to say the least.\n\nBut I was drawn to the darkness of the injustice and decided to embrace all these \"problems,\" those places where I did not respond at all favorably to what the play seemed to be saying, as an excellent opportunity to explore a fundamental question, \"How does one conduct one's life in the face of injustice?\" I knew this was a challenge faced by many in our audience, who had often been the recipients of life's harshest injustices in many ways. Is there a way to live with dignity despite deep unfairness, without shutting down one's gifts and powers but instead using them, along with anger, to be a creative force for change? I really wanted to see if indeed there were any possible truthful way the Furies could transform so quickly into the Eumenides, knowing that so many in our audiences were also asked to quickly and cheerfully accept their unjust fates.\n\nBecause the scales were tipped so strongly in favor of the men, I began to consider ways I might reshape the play, to make it palatable for anyone today to watch. I tried to preserve as much of the original text as possible, as always looking for the simplest and sparest translation, but made several choice cuts where the misogyny was almost unbearable. I then enlisted the services of playwright Lisa D'Amour to help craft new language for the play's introduction. The original play leaves out the reason for Clytemnestra's murder of her husband\u2014he did, after all, kill their child! We wanted the audience to understand that Clytemnestra's ghost, who appears in the play to urge the Furies on, has reason for her anger, that she was not just \"another crazy woman.\" I wanted to be sure the complexity of the characters and situations was clear, something with which many other Greek tragedies were successful.\n\nThen the world entered into our storytelling in a way we could not have anticipated. We began rehearsals just a few days after 9\/11; Lisa had been in New York City on the day of the attack and had looked up at the giant smoking towers. Suddenly, the Furies and the urgency and difficulty of calming the forces of vengeance became very real, and we found ourselves slightly more sympathetic to Athena's plight. The unjust decision of the 2000 elections, with a tie in the electoral vote and the Supreme Court ruling that tipped the scales, was also quite present in our minds. (Though I cannot count on our nontraditional audiences to have detailed knowledge of politics, I ventured that this would be one story everyone knew.) I decided to invite them into the play a little more strongly than usual, offering them an opportunity to participate in the wielding of justice. Athena would choose a jury by offering twelve audience members a gold scarf to hang around their necks. This jury would decide whether Orestes or Clytemnestra and the Furies would win the trial. We didn't ask anyone to get out of their seats, understanding that too much audience participation can be uncomfortable. Athena's assistant handed each a cardboard ballot from which the juror could punch out a square for Orestes or Clytemnestra\u2014complete with hanging chad. Then, of course, no matter what the actual audience vote was, Athena would always rule in favor of Orestes. We created our own rigged election, our own rigged system of justice, and audience members took great delight in seeing the connection with our current world. (We were fascinated to see how the actual punch card ballots were usually fairly evenly divided between the Furies and Orestes, no matter what kind of audience.)\n\nAs I explored the difficult ending of the play\u2014the instant transformation of the Furies into the Eumenides\u2014I decided I wanted to open up ways for them to express their wrath when first presented with Athena's decision. Imagining that our all-male audiences in prisons might be more dismissive of the Furies if they were only females, I cast a man as well as three women in the roles. In rehearsals I challenged them all to find the fastest way they could accept the incredibly unjust ruling, asking them to manage to say at least a few of the lines of blessing given by the Furies at the play's end. But the emotions at this juncture were so intense that many could not even form words. I thought that music might help; we turned to shape note singing as a way to invoke sounds to help with some kind of transformation. It was incredibly painful. Only one or two actors could get to the point of saying words. One was finally able to say the first part of one line, \"Let no wind blow to tear the trees,\" while another repeated the end, \"Singing for some, for others a life of tears.\" The others just made sounds. They all ended in silence, glowering.\n\nThe transformation of the Furies into the Eumenides remained incomplete at best. Watching the ending, one felt that it would take years before any of them got to any place of acceptance. The play became a powerful portrait of the embers of rage and pain that remain when injustice is thrust upon people. Many in our audiences watched the ending in tears. The feelings of being forced to live with enormous injustice were extremely familiar, though our nontraditional audiences usually related them to their own personal experiences within the criminal justice or social service systems, while our paying audiences tended to connect them to their pain over the politics of the recent presidential election.\n\nANTIGONE: IN THE BUSH YEARS Each time we return to the Greeks, our audiences continue to shed light on the plays and help us discover new ways to perform them. During the Bush years, I was attracted to the play Antigone, with its exploration of righteousness and stubbornness and the problems these attitudes can create for humans. I invited playwright Emily Mann, who had been interested in our work, to adapt this story, both of us aware of how righteousness seemed to be strangling our very democracy. But as with any attempt to raise our audience's eyes to consider the larger political world, we had to start from what is very personal, which the Greeks do so well. I imagined that stubbornness and self-righteousness had gotten many in our audience into difficult situations in their lives and felt Antigone's story would resonate on a personal level with them as well.\n\nBob Davis as Creon, Antigone, Cornerstone Women's Shelter, 2005.\n\nAntigone has a Chorus, usually a challenge because a group of people speaking the same thing at the same time seems extremely stiff and unnatural. But when we came to consider what this might be, in light of what audiences had taught us over the years, it did not seem a problem at all. Our audiences, who do not sit quietly and politely in the dark but who instead often comment continuously on the action of the play, judging the characters and situations, inspired us to see the Chorus in a completely new light. We decided to seat the members of the Chorus in the circle of the audience as well. The actors got up from their chairs to play certain characters when the story required and then sat back down when they were \"offstage,\" becoming chorus members, always participating in the storytelling, commenting on what they saw. This further encouraged our audience to make their comments known, and Antigone became a livelier event (though I do not recall that we were successful in getting any of our paying audiences to make vocal comments; traditional theater audiences have been too well trained to follow the rules of theatergoing).\n\nAfter one performance at a women's prison, Kate Eifrig, a tall and powerful white actress who played the fierce Antigone (matched in our production by the equally fierce Ismene, played with passionate intensity by the petite African American actress Sonja Parks), encountered two inmates, a large, gregarious white woman and her petite and quiet African American companion. The large woman asked if Kate had heard about a recent shooting in North Minneapolis. \"Because that was her brother who was shot,\" she continued, looking at her quiet friend. \"And she's worried right now that she won't be given a leave for her brother's funeral. So we were kind of nervous when we heard in the introduction what this play was about.\" She paused again to look at her friend and then continued, \"But it was a real good story. You guys did a really good job. We're glad we came after all.\" The small black woman remained quiet. We were never really quite sure what she made of the experience. We can only hope that she found some comfort in knowing that she was not alone in her sufferings. But her silence served as a reminder of how fine a line we sometimes walk when we do plays about life's extremes for people who often experience those extremes in their daily lives. Most of the time, the imaginative distance of the stories seems to make it approachable. But that distance is vital, and even with it we sometimes are unsure about our impact on individual audience members, as is true of all theater, I suppose.\n\nThe Greeks, writing plays more than two thousand years ago, have provided a powerful stream of fairy tales for the diverse audiences of Ten Thousand Things. Our audiences, in turn, have helped us to breathe new life into these very old stories, whether by helping us to better plumb at the depths of the characters' emotions, or inspiring us to reimagine the Chorus, or compelling us to make room for women's experiences of the world. I needed to find more such stories, with breadth and depth and power and imaginative distance. They were hard to find. Out of desperation, almost\u2014out of necessity, certainly\u2014I turned my gaze to a world I had never approached as a director before: the seemingly formidable works of a certain British playwright in the time of the first Queen Elizabeth.\n\nKate Eifrig as Antigone and Sonja Parks as Ismene, Antigone, Cornerstone Women's Shelter, 2005.\n\nSuzanne Warmanen as the Duke and Sonja Parks as Isabella, Measure for Measure, Project Success, 2012.\n\n# 5\n\n# \"JUST TELL THE STORY\"\u2014SHAKESPEARE\n\n## Fairy Tale Group #2\n\nA FEW YEARS INTO my casting about for big enough stories, I stumbled upon a highly intriguing one that wrestled with that very big question of what makes for good justice. In it, a duke puts one of his deputies in charge and goes into hiding so he can spy on what happens to the realm's justice system in his absence (or as a homeless man would later so aptly describe it, \"It's like an episode of that TV show Undercover Boss\"). The deputy unleashes a fierce moral righteousness: he immediately dusts off some old laws that call for anyone who has sex out of wedlock to be executed. The story takes us to the courts, to the streets, and into prison, with a huge swath of characters from all classes\u2014prostitutes and barkeeps, legal professionals, religious clergy, and fallen nobility\u2014all wrestling with the question of what is good justice or, even more precisely, what is it like to both judge others and be judged. The play was funny and dark, looking at sex and morality and hypocrisy, and presented no easy answers to the questions it raised. I felt like our audiences would love it. The only problem was\u2014I had never directed Shakespeare before.\n\nI was intimidated, to say the least. Somewhere in my education I had picked up the message that Shakespeare should only be directed by experts, those who had studied him in graduate school, who understood all the historical background and scholarly controversies as well as all the intricacies of meter and scansion. There was an aura of mystery and erudition around Shakespeare, and I had never even directed a scene from one of his plays. I felt unworthy of the task.\n\nMEASURE FOR MEASURE: CLEAR, URGENT, LIVELY I began with the simple awareness of something that turned out to be very complicated\u2014I would just have to make the story of Measure for Measure very clear. My limited experiences of watching Shakespeare were ones of straining to piece together what was happening onstage. I couldn't actually understand much of what people were saying, though they might be speaking in beautiful, rotund, oratorical tones. Measure for Measure had a complicated plot, not to mention elevated language. I was terrified that I might not be able to do it.\n\nAs part of my strong commitment to avoid condescension, the thought of altering the language never crossed my mind. I was fairly certain that if the story were clear, audiences would love its music and playfulness and power. I also wanted to keep the entire story, every twist and turn of the plot, intact while being aware that I myself would have a difficult time sitting through anything much longer than two hours. In addition, I knew that the necessary strictures of prison schedules, with periodic counts of inmates throughout the day, wouldn't allow for a longer show. It did seem like there was a lot of repetition that could be cut. Sometimes characters would try to explain a metaphor they were using with a complex analogy that only made it more difficult to understand what they were saying. If I cut that extra explanation, suddenly the original, simple metaphor sprang much more clearly into focus, and we could actually better digest and enjoy it. There were also cuts that would help shed the play of archaic phrases that I felt any audience, no matter their economic status or educational background, would be grateful not to have to spend time trying to decipher. I would also try to find ways to replace a necessary but archaic word with a clearer one.\n\nI knew I would have to work with the actors to make the stakes for each character very high so that the story had a fierce urgency and momentum. I knew that, as with all the other shows I'd done thus far, I would have to work to make every single moment lively and engaging so no one would ever be able to sit back in their chair and space out. Clear, urgent, lively. Other than that\u2014I really didn't know. While that sense of not knowing felt pretty terrifying at first, I've since come to understand that coming from a place of not knowing, rather than expertise, is a very good way to approach Shakespeare's work. I took heart from a line I found in the writings of the well-known British Shakespearean director Declan Donnelan: \"The most important thing to remember about Shakespeare is that he's dead\u2014and we're alive. It is an odd thing to perform a play that is 400 years old.\" We fondly remember these words at the start of rehearsal for every Shakespeare play.\n\nROOM FOR WOMEN I began, as I do with all the plays I direct, with my own responses to the script. And again, as with almost any classic drama, I didn't like the amount of space I found for myself as a woman in Measure for Measure. The few female characters included a nun, a prostitute, and Julietta, a fallen young noblewoman with about three lines (she's fallen into disgrace because she is pregnant and unmarried). There was one other small, troubling role: Mariana, a rejected woman, living in isolation and in constant mourning for a man who had treated her badly, appearing briefly in only a few scenes near the end of the play. Women needed more space; my all-female prison audiences only added reinforcement to my own desires.\n\nI determined that more of the characters needed to be women. I noticed it would actually be quite easy to cast a woman as the Duke. Her trusted advisor, Escalus, could be a woman, too. Rather than cast the usual elderly man in that role, a younger woman, smart and wise and more than a little ambitious, anxious to win her mentor's favor and climb up the ladder, would be a much more fun and interesting Escalus. This would give excellent actresses a chance to explore the wonderful complexities of these parts. Even more important, it would help open up the play to our own times, when women actually sometimes do get to be in charge. It would give all our audiences a chance to experience more aspects of the play from a fresher, female point of view.\n\nSuch casting would also help to alleviate the extremely disturbing experience I had when reading the play's ending. The Duke, who has disguised himself as a friar, returns to interfere in his underlings' bad decisions, throwing off his monk's robes and revealing himself to everyone in a dramatic coup. As part of it all, he proposes marriage to Isabella, the young nun-to-be. Throughout the play he has spied on Isabella; he has manipulated and deceived her into believing her brother has been killed. The play's \"happy ending\" with his marriage proposal to a woman he has greatly abused felt very disturbing to me. Keeping the women's prison audiences very close to my heart, I did not want the play to end with the prospect of such a troubling relationship, in which a woman puts aside all her questions and qualms about a man, as well as her religious convictions, and agrees to marriage. A female Duke, however, could open up new possibilities when she, at the end of the play, says simply, \"Dear Isabella, I have a motion much imports your good, whereto if you'll a willing ear incline, what's mine is yours and what's yours is mine.\" Isabella is given no words to respond. The Duke just keeps talking, urging everyone quickly into the palace, assuring them all their questions will be answered.\n\nA female Duke wouldn't need to be making a marriage proposal. If she just extended her gavel, we could think she was offering Isabella a partnership administering justice in the realm, having been much impressed with the young woman's ability to fight for her convictions as well as her astonishing ability to be merciful and forgive. Isabella's mind could still be full of lots of questions\u2014as would our minds, too\u2014but at least one of the questions wouldn't have to be about a marriage proposal from a somewhat creepy old man. I was quite weary of those scenarios, and I felt certain most women\u2014in all our audiences\u2014would likely be tired of them, too. I am always looking for ways to tell a different story than \"Look! Men oppress women!\" That story is all too familiar.\n\nA NEW LENS Having the main character now be a woman, I looked at the play differently. I noticed a very short scene at the top of Act IV, one that is usually cut because it involves a child. This child sings for a moment and then leaves; his relationship to anyone else in the play is never explained. He sings to Mariana, the jilted woman, who also has appeared suddenly very late in the story. Since the Child doesn't seem to add anything to the plot, I might have cut him, too\u2014except that I was now seeing the play through a woman's eyes.\n\nI started to pay attention to all the references to pregnancy and children in the play, as well as all the times characters use wombs and full bellies as metaphors. Julietta, the fallen noble woman with only a few lines, is paraded through town with her large belly (silently, of course). We also learn peripherally that Lucio, a street character, has fathered a child with a prostitute, Kate Keepdown, who is herself unseen in the play. Children without fathers are a result of having sex out of wedlock. The play is filled with references to punishing people for having sex without being married, which seems ridiculous to us, of course. But when you look through the eyes of female characters, it's easy to see that such profligate behavior does have troubling consequences: it lets fathers off the hook, especially for assuming their fair share of responsibility for their child's upbringing, financial and otherwise. A female Duke might be very interested in this problem, as might many of the single mothers in our audience. Suddenly, an issue we might be dismissive of today, having sex out of wedlock, took on new importance.\n\nAnd, of course, we learn that Mariana, the woman the Child sings to, was in fact jilted by Lord Angelo. The righteous deputy now in charge is guilty of breaching a promise of marriage. Would it be that far off base to assume that the mysterious Child in the scene was Angelo's, born out of wedlock? In a play focused on exploring hypocrisy of those who judge, could Angelo be lashing out at others for having sex out of wedlock because he himself had done the same thing? This is certainly a very human behavior pattern, especially among people with power.\n\nI decided to make all the \"invisible\" children in this story very visible and force their fathers to deal with them. At the end of the play, I decided the Child would reappear with Mariana; my own daughter, an eager six-year-old, could play the role and add another female to the cast, though in prisons we'd have to use a large doll. And our female stage manager could appear for a moment as Kate Keepdown, Lucio's prostitute, holding their toddler (also a puppet), and Julietta could appear holding out the bundle of her newborn child to its father, Claudio. As an added bonus, such strong visuals would also help to make the details of this complicated story clearer.\n\nIn addition to wanting to honor my all-female audiences, my imaginings about the script were also affected by knowing that we would be performing at our first men's prison. With Measure for Measure I'd decided it was time to take this plunge. As always, my preconceptions of a brand-new audience were in some part based on my initial ignorance and fears. Performing a play that explored sexual power in a men's prison felt a little dangerous. I imagined the men might be open and honest about their misogyny\u2014simply because honesty had been an inherent part of all our nontraditional audiences thus far. This made me want to advocate for the female characters in the play even more strongly; I felt protective of them and the actresses playing them.\n\nIn particular, I wondered what the men would make of Lord Angelo's sexual advances on Isabella. I worried that they might take his side and cheer him on. I also worried that by the time we got to the scene where Isabella tells her brother Claudio that it's better that he be executed than she give in and have sex with Angelo, the men would truly despise her. Our Isabella was going to be fierce, intelligent, and opinionated, choosing to enter a nunnery in part to escape men, freeing herself from their condescension and constant sexual advances. But her assertion that \"More than our brother is our chastity\" was certainly not an easy one to stomach, especially, I imagined, for an audience of all men.\n\nAnd, of course, imagining the play through the eyes of women did not by any means make me want to gloss over the imperfections, difficulties, and blind spots of the female characters. One of the many blessings of knowing that your audience will truly contain everyone\u2014people of all races, ages, and especially class\u2014is that you can't take sides. Contemplating such an array of audiences makes it clearer than ever that Shakespeare doesn't take sides either\u2014or at least his writing makes it possible not to take sides. His writing is like a crystal, enabling us to see the world through the multiple lenses equally. Or as British theater and film director Peter Brook so elegantly put it, \"Shakespeare has no authorship or personal expression; his writing is not a series of messages, but a series of impulses that can produce many understandings.\"\n\nIn Measure for Measure especially, there are no complete heroes or complete villains; every character has his or her flaws. In court, Isabella constantly admonishes Angelo to put himself in another's place before he passes judgment: \"I would to heaven I had your potency and you were Isabella. Should it then be thus? No, I would tell what 'twere to be a judge, and what a prisoner.\" But when push comes to shove, Isabella is completely unable to put herself in her brother Claudio's place when he pleads with her to imagine what it feels like to be condemned to die.\n\nGiving all characters equal weight, as I gave all my audiences equal weight, meant that Angelo, of course, could not simply be a villain. I wanted to be fair and sympathetic to him, too, knowing that many men would have found themselves in the position of lusting after someone forbidden, acting on their impulses and regretting it. Angelo's remorse at the end of the play had to be utterly sincere and repentant, opening up this possibility for the men in the audience, even if they hadn't felt repentant before.\n\nPARING DOWN TO THE ESSENCE Angelo hated the sexual part of himself so much that he completely shut himself off from the memory of what he had done with Mariana. Thus closed down, he could not empathize with Claudio or anyone else who had had sex outside of marriage. I started to see that the inability to put oneself in another's place was a problem faced by almost every single character in the play. No one, not Isabella, not the Duke, and certainly not Angelo, seemed to be very good at it. I realized this could be the \"spine\" of the play\u2014every character at one point or another is asked to put himself in another's place, with varying degrees of success. It is crucial to me when working on a script to pare it down to its essence, which in theater is best expressed as an action. Some also call it a \"spine.\" It's a common action that every single character seems to be trying to do, to one degree or another. A strong spine is another affirmation of our common humanity. Paring a story down to find the spine gives the play a shape that the audience can somehow more easily grasp. The most potent spine is an action that is not only what all the characters in the play are trying to do but what you sense the audience is also grappling with in their actual lives at the time.\n\nDiscovering this particular spine in the play was exciting to me because putting yourself in someone else's place is the fundamental activity of theater. It's what we ask actors to do when they work on characters and what we ask our audiences to do as they watch a play. And, of course, I tried to keep doing it with my audiences as I worked on the script, also with varying degrees of success.\n\nALL CLASSES EQUAL Knowing that people of all classes would be in my audience also helped me to approach each and every character with as much humanity and respect as possible when making choices about them. Again, I found that Shakespeare's writing usually made my task easier. It certainly was not an option to mock or make fun of lower-class characters any more than wealthier characters could be completely vilified for having money and being shortsighted. Prostitutes and barkeeps must have a certain dignity as working people just trying to find a way to get by. Mistress Overdone could not be a silly floozy, despite her name. She was simply a businesswoman, trying to make a living in a world where that is still a hard thing for a woman on her own to do. Lucio, whose witty and honest commentary I imagined the audience would really appreciate, is described as \"a fantastic\" in the character list, but I chose to make him of the lower classes, earthily rather than extravagantly dressed. I felt it would make him still more accessible to nontraditional audiences, a guy scraping by on the streets who had seen it all and wasn't afraid of anything. And of course we had to give upper-class characters\u2014the Duke, Angelo, and Escalus\u2014the same humanity and respect, not only because we'd be performing for the general public but because the point is to make it as hard as possible to take sides in order that the questions in the play be kept alive. That constant tension of starting to side with one character and then running into a flaw is what makes \"the inert lump of coal\" that is Shakespeare's writing give off heat and light. Much of my script work is about a kind of constant balancing and rebalancing so that we can see the positives and negatives of every character, not just to be fair but because it makes everything so much more interesting.\n\nENTERING INTO UNKNOWING As we headed into rehearsal, I decided to, well, not exactly lie, but at least not admit to the cast that this was my first time directing Shakespeare. I wanted to seem to be an authority; I wanted the actors to trust and have confidence in me. Ever since, though, I have always readily admitted my ignorance and lack of expertise to my casts. I've become confident that not knowing is the best place from which to approach Shakespeare. Not only does it help to put the actors at ease, allowing us all to make mistakes and be foolish, but most important it helps us put ourselves in the audience's place of not understanding anything that is going on.\n\nI knew that my first challenge in rehearsal would be to help the actors find stakes for their characters that were high enough. I had learned from past productions that a lot of actors tended to make less urgent, more moderate choices, thinking they were somehow \"more interesting.\" Again, our imagined audiences helped us to quickly get to the high stakes of almost any situation. We knew that many might themselves be prostitutes or barkeeps or pimps or prison guards or even judges. For example, we imagined much more fully the utter desperation of Mistress Overdone when she learns that Angelo has decreed that her business, the brothel, will have to be shut down. We could better imagine the frightening reality of suddenly having no means of support. We found new urgency and depth in the scene where Pompey, her barkeep, tries to reassure her that all will be well. None of this \"seriousness\" in any way detracted from the characters' humor\u2014people in desperate straits often turn to quite dark humor as a release. Finding the highest stakes possible doesn't make everything grim; it just intensifies everything, including the humor, driving the momentum of each scene.\n\nThen it came time to \"not know.\" After working with the actors for several weeks on the stakes of their situations and figuring out the strongest possible actions they could try to take in the light of such dire circumstances, there came a point where I needed to back away. I had to clear my mind of everything and sit in the audience as someone who knew nothing about what was going on. As I watched scenes from this position of complete ignorance, my mind filled with questions. Who is that guy who just walked onstage? What is everyone blathering on about? What just happened? Why did that guy make the other guys leave the stage and where are they going? Where did that woman come from? Wait, why did that guy just get angry? Shakespeare, for all his splendor, is not always the clearest storyteller.\n\nSome storytelling problems were fairly easily solved. Characters often used pronouns instead of names when they were talking, so I didn't really understand who they were talking about. Sometimes there were two or three different pronouns in the same sentence, each one referring to a different person! This of course was easily fixed by inserting actual names. Sometimes this didn't quite fit with the scansion, but I figured understanding the story was more important.\n\nI also noticed that characters often talked about people or events offstage that I couldn't see. Without some kind of visual anchor in the sea of floating, unattached words, it was actually very hard to understand who or what was being talked about. This was often simply solved by having characters gesture or point to where the character had last been seen or where we understood the event might be taking place. I am not a fan of actors who make heavy use of hand gestures to act out words; I find it condescending and often ultimately confusing. But pointing to help us understand offstage events can be such a huge and welcome relief. One simple gesture can unlock a tumble of words; one pointing finger and, amazingly, all becomes clear. We've now embraced this concept thoroughly, referring to it as \"The Ten Thousand Things Pointing School of Acting.\"\n\nTHE NEED FOR LOTS OF MOVEMENT And even when the story was fairly clear when reading the actual words of the text, when the actors spoke onstage, I still often just heard a bunch of words without having any real idea what anyone was actually saying. I began to realize that when words washed over me, more often than not it was because the action underlying the words of the character was not strong and clear. It's such a basic directing rule, hammered away at incessantly in directing school\u2014every time a character speaks, he or she is trying to do something to the person they are talking to. We are all taught that there's always a very active subtext underneath anyone's words. No matter what a character is saying, we were taught to look for what he or she is really doing, whether it's dismissing, or embracing, or pushing away, or seducing. If an action is strong and clear, you can feel it viscerally, no matter the language. So it was surprising to me how often, when I watched other Shakespeare productions, I couldn't feel the actions of the characters underneath their speeches. Feeling those actions helps so much to make sense of the words. With Measure for Measure, I realized I needed this play of amazing language, in a funny way, to be \"language proof\"\u2014so you could understand it even if you couldn't understand the words.\n\nI also noticed that it became harder for me to understand what characters were saying whenever they stood still for a long time. Stillness seemed to be a big part of Shakespeare, especially during long speeches. I saw quickly how movement could help so much to understand a character's words, especially if it mirrored the action underneath. If the Duke at the beginning of the story was trying to throw all her colleagues off balance, surprise them, and get out of town as quickly as she could, her movements as she spoke could reflect this. She could head toward the door and then suddenly come back in for a bit of further instruction before heading out the door again, darting back and forth quickly as she spoke. Movement could help us understand how a character's thinking was unfolding, too, pacing as she worked to figure out a problem, stopping or starting suddenly as a new idea occurred. If Angelo, talking alone onstage, was trying to figure out a way to handle his lust, his movements could reflect his thinking, with a change of direction every time he changed the direction of his thoughts. The stops, starts, and changes of direction could break up those long speeches, helping us physically to understand his thought processes and discoveries of new ideas.\n\nSometimes veteran Shakespearean actors new to the company have resisted this kind of movement while speaking, worrying that it will be too \"illustrative\" or perhaps condescending. But once they are plunged into the heat of the immediacy of a Ten Thousand Things performance, that concern rapidly vanishes. All anyone wants is to have the story be clear. I have never encountered an audience, whether first timers or veterans, who felt that this kind of movement was too simplistic. Rather, all have seemed to find whatever help they can get in understanding the onslaught of words rushing toward them as a relief. They can spend their energies entering into the depths of the play rather than have to waste them by trying to clear away their confusion.\n\nI began to see how, with so little else to fall back on in terms of set, lighting, or props, movement was actually our best tool for creating clarity, helping the audience to understand the actions under the words as characters spoke, approached, or retreated, underscoring their shifting needs toward one another. Almost never in a Ten Thousand Things production of any kind does a character stand still for more than a minute (and almost never does one sit), as it robs us of the chance for movement, the chance to get some further understanding into what is really going on onstage. Audiences quickly grow restless when actors stand still and declaim\u2014or wallow.\n\nI had noticed during our previous tours that audiences became quite restless whenever a character paused to emote. Certainly many of their own lives, so often focused around daily survival, afforded little time for emotional wallowing. If the stakes of a situation are truly urgent, people don't have the luxury of time to sit around and feel things. Action needs to be taken. Action, not emotion, is the building material of theater for good reason, and the restlessness of our audiences whenever a character took too long a pause in what he or she was doing to just \"feel things\" helped to strongly underscore this. We needed to keep things moving.\n\nTHE LANGUAGE COMES LAST Then, finally in rehearsals, I turned my attention to the language. I know this seems strange, perhaps even heretical to some Shakespeare artists, who so often start with the language. Certainly the actors, many skilled in scansion and meter, had been aware of and working with it as a tool all along. But I only paid attention to the language once the stakes and the actions were very strong and clear. It was like I was coming up for air, having watched everything underwater to be sure I could feel the actions in my gut. Only then did I come up to listen to the actual words. When I still couldn't understand exactly what people were saying, I often found that all that was needed was a little tweaking, perhaps just a little stronger emphasis on a certain word, and then the whole sentence would become clear. I didn't apply any external formulas or rules about how Shakespeare was supposed to be spoken. I just listened for what I needed to know to understand the story and made little adjustments to emphasis or rhythm or speed to bring clarity.\n\nAt this point, I'd also notice what words didn't seem \"full enough.\" It really stood out if an actor was rushing through the words, taking them all for granted as he spoke. It was also a signal that the actor might not have gone deeply enough into the stakes of his situation so that he could truly fill up the words. In addition to urging more depth, I challenged the actors to grapple with how the strange and beautiful words were forged in the belly of necessity, each particular one arising because it was absolutely needed at that moment to help the character do whatever he or she was trying to do. It was a gift to be given the ability as a human being to find the most powerful words. Finally, at the very end of the rehearsal process, I could embrace the joy and delight of the language, helping the actors, in the places they hadn't been able to yet, to do the same.\n\nA SHAKESPEARE EPIPHANY And thus, armed with our attempts to imagine the characters and story through our audience's eyes\u2014the high stakes, the strong actions, our clear and urgent story, the amazing and necessary words\u2014we set out to meet our first audience. We made a circle of chairs at the Dorothy Day Center in downtown St. Paul, Minnesota, an enormous space for the homeless to drop in during the day, to hang out, stay warm, have a cup of coffee and conversation. Gradually, about sixty people filled the two rows. We were all terrified. I honestly did not know if they would be able to follow this complex story with its unfamiliar language. But all we could do at that point was take the plunge. And at this very first performance, we had what we now refer to as \"our Shakespeare epiphany.\"\n\nI began by making a brief introduction to the story, setting the stage so it would be easier to enter. I also took a moment to reassure everyone that even people who have seen lots of Shakespeare before always find it challenging to understand what's going on at first because the language is so different from what we're used to hearing, but that after five or ten minutes, everything should become clear. After the first few scenes, the audience seemed to be pretty engaged, following the story, as best we could tell. Indeed, as time passed, I got my first clear sense that the spareness of our staging, the lack of objects and set pieces on the stage, actually made it much easier for everyone to focus on the language, heightening it and allowing all to take pleasure in it.\n\nThen we came to the courtroom scene where Isabella pleads with the rigid, puritanical deputy Angelo to spare her brother's life, unsure of herself at first but coming up with an arsenal of fierce arguments as the scene progresses. Angelo, somehow moved, asks her to leave and return the next day for an answer. He is left alone onstage to wrestle with the feelings that Isabella has produced in him. \"What's this? What's this? The tempted or the temptress, who sins most?\" he asks.\n\nAnd at that moment, an older woman sitting less than a foot away from Angelo, played by the very experienced Shakespearean actor Steve Hendrickson in his first ever Ten Thousand Things performance, looked him directly in the eyes and said, \"I think it's your fault, shithead.\" Everyone seated around her began to laugh. Then a man standing at the back of the room shouted out, \"Oh, go ahead and fuck her!\" The entire audience burst into hoots and guffaws. Steve began to panic. He told us later that he felt like he had completely lost the crowd and didn't know how to get them back. Then he thought of his next line. All he had to do was say it. \"Not she, nor does she tempt, but tis I.\" A conversation with the audience was born.\n\nWe suddenly remembered the groundlings, the people who paid a penny for a ticket, who stood in front of the stage at the Globe and shouted remarks at the actors. Shakespeare wrote his plays expecting them to shout back to his characters. A \"soliloquy\" was not a quiet solitary moment for a character to speak his thoughts aloud. It was an active conversation with the audience, with people shouting out advice and judgments\u2014just as they had at our first performance of The Good Person and at so many other performances since then. That's when we truly realized that Shakespeare did write for everyone, not just the polite, quiet, wealthy, and well-educated patrons sitting in box seats. To write such conversations he must have known his audience, both rich and poor, well enough to imagine what they would say in response. Our performance at the homeless drop-in center was a visceral confirmation of this simple fact. He, like us, must have been holding his audience in mind as he wrote.\n\nOur audience at the Hennepin County Women's Correctional Facility was even livelier; the women openly booed and cheered on the characters. These women gave us the answer to that problem that had always troubled me in Measure for Measure. They taught us how to understand Isabella's choice of her \"chastity\" over her brother's life. This audience was all too familiar with Isabella's situation, completely alone and at the mercy of a powerful man. They immediately knew what Angelo was up to with his request for Isabella to return the next day; they began to hiss and catcall. They were fiercely on Isabella's side. They understood completely what was at stake in this scene, and religious conviction was the least of it. What Angelo was asking for wasn't \"just sex.\" It was rape. Pure and simple. It was a horrible violation, an act of brute force, no matter how politely it was couched. The cloaking of Isabella's feelings in religious garb and the use of the word \"chastity\" somehow served to diminish the violence of it all. Seeing this situation through the eyes of women who had been there themselves suddenly made the balance between the horror of Claudio's looming execution and Isabella's looming rape much more balanced. These women finally showed us how to take Isabella's situation in all its full seriousness, in a way we had not been able to imagine before.\n\nSteve Hendrickson played Angelo in Measure for Measure at the Dorothy Day Center in 1998.\n\nIn the years following, I have had the chance to bring Measure for Measure on several other tours (including one for the Public Theater in New York City, which wanted to experience our model). The reactions to the Angelo\/Isabella scenes are always visceral in the women's prisons, but they aren't always quite the same. There are differences even according to the type of prison. Isabella has a \"soliloquy moment\" in the play, too, when she returns to court the next day to receive the ruling about her brother. Angelo presents her with a stark choice\u2014the only way to spare your brother is to let me rape you. Left alone to process this impossible situation, Isabella asks, \"To whom should I complain? Should I tell this, who would believe me?\" In a minimum-security jail in Manhattan, the women indignantly shouted out advice: \"Call the police! That guy's an asshole!\" But in a maximum-security prison outside the city, complete silence greeted Isabella's question. The silence was very full, and the actress let it linger. After a moment, a sole voice spoke out quietly: \"That's what it's like. No one to listen.\"\n\nI don't know how Shakespeare intended this scene to be played. I don't think it's possible to know. I was able to see a production by London's Globe Theater, with an all-male cast, just as it was cast in Shakespeare's time. Isabella was played by a lanky young man, and both she and Angelo came across as ridiculous and uptight prudes. That certainly might have resonated with Shakespeare's audiences, and it also seemed to do so with the largely liberal upper-middle-class audience who watched the play, resonating as a way to skewer the hypocrisy of many right-wing Christian fundamentalists. All I know is that with our larger audiences who have lived at the edges, many of whom haven't the time to be involved in the political discussion of the day, we have to take a different approach to mirror the truth of their lives. We must approach things first and foremost at a deeply personal level. Artistic choices made out of necessity, once again.\n\nNathan Barlow as Claudio, Measure for Measure, Project Success, 2012.\n\nALL-MALE AUDIENCES Turning back to that very first tour, we finally came to our anxiously anticipated, first-ever men's prison audience. As I looked out at the rings of chairs filled by very large men in gray sweats and black Converse sneakers, slouched in their seats with arms folded, defying us to entertain them, I experienced yet another \"you must be out of your mind\" moment. I looked at the actress playing Isabella, Signe Albertson. She was clearly feeling the same way.\n\nThe first unexpected surprise was the audience's complete embrace of Shakespeare's bawdy humor. I had never heard an audience laugh so hard at Shakespeare. No glossary or footnotes were needed for the men to pick up the sexual innuendo contained in the lines between Pompey and Mistress Overdone and Lucio. They even caught a few that we had missed! The laughter was deep and hearty, pure enjoyment of the ability to hear such things said in a prison, under the watch of the guards (who were themselves smiling), and appreciation of the actors for being so honest. Any audience who has been deprived of sex for many months, we quickly learned, will thoroughly relish the slightest hint of it. Our prison audiences have helped us to understand that most theater is really in some way about sexual energy, about the tensions produced by live bodies on display, interacting onstage.\n\nThen came the dreaded scenes between Isabella and Angelo. Talking about sexuality and a possible rape felt dangerous in that gym full of men. But our fears that the men would side with Angelo and just see Isabella as an object of lust were soon put to rest. Most of the men did quickly recognize Angelo's intentions, smiling at the subtext they could feel beneath his polite arguments and language. But we also started to sense a protectiveness toward Isabella. It became clear that many of the men were identifying with her as they might with a younger sister who found herself in a vulnerable situation. It was actually easier for them to take her side than the side of a white male authority figure, a judge perhaps not unlike the ones who had sentenced them to jail. In the next scene, where Claudio begs his sister to imagine what he must feel like, being condemned to death, the men were clearly sympathetic to Claudio and didn't view the plights as equal. But they did not seem to reject or ridicule Isabella for her passionate defense of her chastity. Our fears had been our own. Our assumptions had not at all been correct.\n\nAnd then, in the final scene of the play, when Claudio miraculously appears to Isabella, who has been led to believe that her brother has been killed, I looked around the audience and saw tears in the eyes of some of the inmates. I was startled. The men seemed to be deeply identifying with Claudio, longing for their own moment of reunion with a family member or loved one. I could never remember seeing tears in the eyes of any audience watching Shakespeare, ever. This story was the story of the audience's lives.\n\nAfterward, we were lucky to be able to have a conversation with some of the inmates (this is not always possible in jail). One man commented, \"I'm getting out tomorrow. I didn't know theater was like this. I'm going to take my kid.\" Another inmate turned out to be a Shakespearean scholar from South Africa who had enjoyed the performance very much and appreciated our casting of a woman as the Duke. And one inmate eloquently offered, \"This is my first taste of Shakespeare and I hunger for more.\"\n\nLuverne Seifert as Pompey, Measure for Measure, Project Success, 2012.\n\nFinally, we brought the show to paying customers for three performances at a somewhat rickety dance space on the second floor of a building in downtown Minneapolis. Perhaps almost as gratifying as the response of our nontraditional audiences to our Shakespearean work has been the response of those who pay for their tickets. We now do four weekends, or sixteen performances of every show, for the general public, usually selling out quickly. It seems that this stripped-down, fierce, clear, and intimate style of doing Shakespeare is speaking to people who have seen a lot of theater. So many people have told us it's the best Shakespeare they've ever seen, that for the first time they have been truly able to understand and connect with the stories going on. I believe audiences now, used to close-ups of actors' faces on television and film, are hungering for the same closeness in theater so they can read the details of actors' facial movements. They want to see actors who don't have to strain to project emotions to the back of large auditoriums. They hunger for the jarring intimacy you feel close to live bodies radiating enormous and truthful emotions. Because these veteran audiences seem so grateful that the story is clear, the language is accessible, the momentum is rapid, the depth and truth of the acting so profound, and the details so easily visible, it makes me think that perhaps some of those things have not been available to them in the Shakespeare they've seen elsewhere. Creating Shakespeare by holding in mind an audience that includes every sort of human being seems to enrich and enliven the experience for all.\n\nFor most of the seasons following this first experience, we have included a Shakespeare play, ranging from an all-male Richard III to an all-female Twelfth Night. We've performed his comedies and tragedies and each time used the multifaceted lenses of our many audiences with which to view the multifaceted dimensions of his plays. We don't need to have a \"concept\" or choose a historical setting like 1930s Cuba or 1960s sitcoms to make the plays come to life. Each time, through necessity, the simple act of imagining our audiences makes most of the artistic choices for us. We often just look for one large and simple territory to explore with each play, as a focus for our work but also because many of the plays, for all their complexity, seem to have been written that way. Richard III is a play about being an outcast in the world, and certainly a situation most in our nontraditional audiences can relate to. The Tempest becomes a story where every character struggles with the very human need for revenge. Othello is a play whose characters each wrestle with jealousy on a spectrum. Yes, these one-word emotional territories seem simple and obvious\u2014but jealousy or revenge or judgment each provide an endlessly fertile, deep, and complex terrain to explore. Such a clear and simple demarcation lends a focus to our storytelling and allows the audience to consider these fundamental human conditions from all angles, which can be exhilarating. You don't need to get much fancier.\n\nShakespeare wrote plays knowing that everyone would be in his audience. It seems the most pressing Shakespearean work we can do today is to attempt to tell the stories again, clearly and deeply, to an audience that also truly contains everyone. Though to many this may seem daunting, it is an exciting charge, fraught with creative possibility, and I believe necessary for those who hope to keep Shakespeare's work living into the future.\n\nBubbles for The Emperor of the Moon finale, Dorothy Day Center, 1997.\n\n# 6\n\n# ABBONDANZA! THE WEALTH OF VERY FEW THINGS\n\n## Production Values\n\nMORE THAN TWENTY YEARS after making the Good Person set from cloth cutouts, cardboard barrels, two ropes, and a few boards, using that little bag of silver from my mom, we still have never spent more than one thousand dollars on set supplies for any show. There are several very good reasons for this, but the most important is the primary principle of Ten Thousand Things design: \"Yeah, but do you wanna carry it?\" Our sets must be carried down long hallways, lugged up long staircases, crammed into elevators, and heaved in and out of our small rented cargo van. Out of such necessity, we have our defining aesthetic.\n\nWe are certainly not the only theater company with an aesthetic of using minimal set pieces and very few props. Many other wonderful theater artists choose \"empty space\" over a detailed rendering of time and place when creating their work. But working in a bare-bones style for so many years because we must, as well as performing for so many audiences who also find themselves with very few material possessions has given us a unique perspective from which to see the abundance that comes from having very few things.\n\nAt first, the cinderblock walls and linoleum floors or industrial carpeting, in hues of beige and gray, that make up the world in which many in our audiences spend a great deal of their time seemed like something to try to overcome. I believe that all of us have a hunger for beauty, and I wanted to try to use our sets to transform these dreary spaces. For Electra, our second show, I worked with the designer to make some simple, light canvas flats that glowed with mysterious blues and greens, hoping they would somehow cover up whatever drab room in the juvenile prisons we might find ourselves in. But the painted flats actually just heightened the surrounding dinginess. The cold, industrial rooms overwhelmed the little set and made things somehow even bleaker.\n\nAs with everything else, it took me a while to process all this. With our next show, Life's a Dream (still performed proscenium style in front of rows of chairs), I again tried to blot out the room. We hung large, brightly colored cloths on a rod frame behind the actors to make a backdrop. And, once again, this didn't transform anything; it all just looked sorry when viewed against the bleak space. I finally started to understand that the only way to transform a room would be through the imagination of the audience\u2014not through covering it up with things.\n\nOur first foray into comedy came soon after. The raucous laughter of our audiences at comic moments in the plays we'd done so far helped me understand how rare and precious a thing laughter can be for people who are living very hard lives, so I thought we'd try a whole play devoted to it. Carlo Gozzi's seventeenth-century fantasy The King Stag attracted my attention. In an early attempt to wrestle with the horrible sexism of this particular classical play, I tried reversing the gender of all the roles (I would now recommend against this!) and retitled it The Queen Stag. I still wanted the play to grapple with a struggle in the lives of our audiences somehow; the play seemed to be wrestling with the way we so often confuse people's appearance with the nature of their souls. I thought that might resonate with many in our audience who are so often judged on their appearance. I honestly don't know if anyone connected with that issue while they watched the play\u2014but they certainly all did end up laughing a lot.\n\nI was also attracted to this wildly fanciful story of transmigrating souls because of the challenge of its scenic \"requirements.\" The play takes place in magical forests and palaces; to do so \"realistically,\" by filling in all the empty space with actual full-size forests and castles, would be impossible, of course, so I went to the other extreme. To set the scene, an actor just ran across the playing space trailing a long green piece of fabric, whispering \"Forest, forest, forest.\" And when the setting changed, a long gold piece of cloth was whisked across, with another whispered prompt, \"Palace, palace, palace.\" The audience was delighted with the invitation and willingly jumped right in to imagine. The forests and palaces they created themselves were likely much more wonderful than anything we could have made by ourselves. (That set remains one of my very favorites because it was the lightest; just pieces of cloth we folded up and carried effortlessly to our cars.)\n\nFrom that point on, I heartily embraced the limitations of Ten Thousand Things staging. I had always instinctively been drawn toward the theater of empty space anyway, so it was an easy match. Instead of trying to create a world with set pieces and objects, I wanted to find the least we would need to prompt the audience to create their own worlds. I began to relish tackling seemingly impossible staging challenges with as little as possible.\n\nSETS I knew that with at least some of the plays we did, I wanted to continue to explore the territory of joy. I was on the lookout for stories about wily servants deceiving their masters, stories that in some way at least made room for different perspectives of class, and, always, a female point of view. All these elements came together when I stumbled on The Emperor of the Moon, by England's first woman playwright, Aphra Behn. This wildly ridiculous story, about using trickery and subterfuge against the powers that be to claim one's own true love, has immense staging challenges. In the finale, when all the loving pairs are correctly united at last, Behn calls for\n\nThe Hill of Parnassus; a noble large walk of trees leading to it with eight or ten Negroes upon Pedestals, ranged on each side. Next Keplair and Gailieus descend on each side in Chariots with Perspectives in their hands. Next the Zodiack descends, a Symphony playing all the while; when it is landed it delivers the twelve Signs: then the Song, the Persons of the Zodiack being the singers. After which the Globe of the Moon appears, first like a new Moon, as it moves forward it increases till it comes to the Full. When it is descended it opens and shews the Emperor and the Prince, who come forth with all their Train born by Cupids, the Flutes playing a Symphony before them.\n\nI relished the challenge. I needed to translate such excess into something simple and expressive of the play's joyous spirit. The spheres and globes of the planets and moons suggested bubbles. I discovered I could rent three bubble machines for a few weeks for less than a hundred dollars. We hid them in painted boxes, and at the moment of the grand finale, we switched them on. Hundreds of bubbles filled the tired spaces of shelters and prisons, floating over the audience's heads and popping into their faces, all to the sound of a wailing saxophone. So much joy and delight from a bubble, much more, I think, than any elaborate set could ever have created.\n\nWith more productions under my belt, I began to work with a sculptor whose ways of thinking about materials and objects were much different from those of many set designers. Theatrical designers are accustomed to making \"fake things\" and transforming them with various tricks of paint and lighting (lighting, of course, that we didn't have). I wanted the set pieces we brought into our production spaces to be beautiful in and of themselves; I felt they could also be more abstract as well as more artful. I was drawn to a sculptor, Stephen Mohring, who worked with bare wood and metal. Though I couldn't have articulated this at the time, it felt better to just be honest about where we were and what little we had. I thought we would draw in the audience even more strongly by giving them the sparest suggestions of place, without paint or much color, inviting them to transform a few objects that in their starkness were actually quite beautiful.\n\nMeasure for Measure, in 1998, was the first play Stephen and I started working on together to develop this aesthetic. Always interested in economy and carrying less, we began to explore the possibilities of having each set piece function as many different things. I craved simple shapes, a world made of just circles or just rectangles. With Measure for Measure, the scales of justice came to mind, and their triangular shape permeated my thinking. Black steel tubing could form the outlines of triangles: waist-high triangles could be gates in the prison or podiums in the court, a few triangular-peaked archways on wheels could be entrances to court or to a monastery, a triangular metal stool for a garden bench or a seat in the prison waiting room. The sounds of clanging metal could help to shape the percussive pace of the play. Using one shape to become many different things helped to create a separate world that was very different from our everyday one, or any historically accurate one, and it created a coherence to the made-up world. Even more important, the single shape allowed us to start to see connections between otherwise very separate worlds of prisons, churches, and courts.\n\nMeasure for Measure set pieces by Stephen Mohring.\n\nWHAT TO WEAR With Measure for Measure, we started refining our ideas about costuming, too. Because set pieces take up so little space, costumes are really what create the fullness of our worlds, their color and texture. I found myself drawn away from theatrically trained costume designers, who so often are oriented to creating an individuality for each character, giving each one a distinctive color and shape to differentiate them from everyone else. I wanted to make up a coherent world, and that meant exploring what the characters shared in common. My interest turned toward fiber artists, who easily plunged into the spirit of such work.\n\nWe sensed that by strictly limiting the palette of color used for all the costumes, we could create a strong visual unity. The harsh world of Measure for Measure drew us to blacks and grays, matching the steel tubing, with hints of tan and brown to match the few pieces of bare wood in the set. The textile artist, Ellen Hutchinson, and I went to a large fabric warehouse looking for cloth that seemed to belong in the world of the play. We pulled bolts from the shelves with glee and spread them on a big table to consider. We relished being able to really make use of texture. Because our audience sits so close with no lighting to get in the way, the actual textures of the fabrics can really be seen and appreciated. Within our strict color palette, with everyone dressed in the same colors, texture would be the way to distinguish the economic circumstances of a character\u2014coarser, rougher cloth for those of lower classes; shinier, smoother embossed fabric for those of means. Yet I didn't want such distinctions to be strict. We treated the array of bolts we had chosen as the universe of available fabrics in the world. Characters might share some fabrics\u2014one might have a vest and another a skirt made of the same cloth. Sometimes a small piece of coarse fabric might appear in the costume of a wealthier person, and a poorer character might have a scrap of shiny fabric he'd patched into his sleeve. As with the beauty of the sculpted bare wood and metal set pieces, texture in the fabric of the costumes was a way we could pull in the audience to notice the beauty of the simple materials directly in front of them.\n\nWe also looked for coherence in the shapes of the clothes. Ellen thought nothing of sewing all the pieces from scratch, so we were able to choose one or two strange shapes, perhaps balloon pants and long tunics, that most everyone could wear, no matter their economic circumstances. Unusual shape, unconnected to any specific historical period, along with a strictly limited color palette and a universe of fabrics shared among all characters became the simplest way to make our fairy tale worlds. Already limited by what we could carry, we chose to restrict ourselves even more, by shape, color, and texture. Out of these limitations we could invite our audience in to help us create worlds from times and places no one had ever experienced before, worlds in which no one was the expert.\n\nSuch strong visual coherence helps to satisfy the other absolute necessity of Ten Thousand Things performance situations: creating focus. Remember that we have no elevated stage, no stage lighting, no spotlights, no dark house\u2014none of the usual theatrical tools for creating focus. The spaces are often large and chaotic, sometimes with lines of other people in the room waiting for health care or for food. Soda machines hum, bulletin boards and brightly colored posters adorn the walls, phones ring and announcements come suddenly over loudspeakers. Our audiences are often skeptical or uncertain, distracted by larger issues in their lives. As artists, we must strive with all our might to create the strongest focus we can, for each and every minute of the play. A powerful visual coherence to our sets and costumes is one way to help with this. And the strong invitation to the audience to help create the world with us also harnesses otherwise scattered energy.\n\nTHE CIRCLE OF AUDIENCE Having the audience define the edges of our stage has become another crucial way to create focus, but again, it took me five or six productions to discover this. Our one disastrous experience with an actual stage when we had to do Life's a Dream for the youth at risk at the church showed us what an enormous barrier a stage's elevation can be for people who have never experienced theater before. After that I vowed to always perform on the floor, on the same level as our audiences, even if a stage was available. But for a few more shows, we continued to play proscenium style in front of rows of chairs. It wasn't until Brecht's The Caucasian Chalk Circle that I stumbled upon the idea of performing in the round. A circle. It's in the title. Once again the discovery came out of necessity. The story of Grusha, the palace kitchen maid, and the abandoned child she decides to care for, requires them to journey all over the war-torn countryside. The thought of conveying all this traveling back and forth across the straight lines of a proscenium stage seemed very tiresome. I finally realized that if we performed in the round, with a few aisles radiating from the center, Grusha could move around the outside of the circle behind the audience. The focus on the story would also be sustained if the audience could watch her travel around between the scenes.\n\nRagtime curtain at women's correctional center, 2005.\n\nA circle offered many other benefits. The edges of the whole playing area would now be completely defined by the audience or, even more precisely, the tension between the actors and the audience. The actors could break that tension by traveling into the four radiating aisles in the \"corners\" of the circle as well as the space behind the chairs. At times the play could completely surround the audience. The circle could always be the same size, so our stage could always be twelve to fifteen feet across, no matter what the size of the room. We almost never have more than three rows of chairs ringing the stage. Actors might be inches away from an audience member, almost never further than a dozen feet. In addition, the circle of audience members would all be able to see each other's faces, fully lit, not just dark silhouettes of the backs of heads, as is the case in a conventional theater. In-the-round staging also encourages a multiplicity of viewpoints and perspectives. Sometimes an actor's back is to us and we must imagine her facial expression, or sometimes when our view is temporarily blocked by an actor we can look at an audience member seated across the stage and imagine how things look from her point of view. Plus, practically speaking, this kind of staging allowed us to quadruple the number of people who could sit in the front row. All of these factors greatly intensified the quality of the audience's attention. We've performed in a circle or a square ever since.\n\nBut all of these efforts to create focus\u2014strong visual coherence from limited palettes of shape and color, inviting the audience in to help create the world with their imaginations, completely surrounding the play with the audience and the audience with the play\u2014are still not enough for us to be able to tell a story clearly.\n\nYOU MUST HAVE SOUND Live sound is as critical to creating focus in our work as lighting instruments are in the regular theater. We always have live sound for every production. A gong, a chime, a drum, a dissonant chord, or a ukulele twang can create a spotlight, a color, a transition, or a blackout\u2014in full fluorescent light. And as the multitalented actor\/musician\/singer\/composer Peter Vitale and I have learned over the years, to create focus with such \"lighting,\" sound is preferable to music. Musical interludes often have a precise emotional content and so tend to shape our feelings. Percussion and unfamiliar, unspecific sounds allow listeners more emotional space and flexibility. This important difference sets Peter's work apart from that of more traditional \"sound designers.\" Rather than highlighting the mood or emotion of a scene, our sound tends to punctuate the action onstage. We use it to draw the audience's attention to exits, to entrances, to key moments, helping to make the story clear rather than telling them what to feel.\n\nAND IT HAS TO BE LIVE Live sound creates a vital energy that prerecorded sounds can never approach. Taped sounds, as I learned frantically punching the buttons on a cassette player in one painful early experience before I met Peter, are deadly. Again, abstract live sounds work best. Rather than actual sounds of doorbells ringing or dogs barking, imaginations are more engaged by strange sounds that suggest these things. We invite the audience to fill in the blanks themselves because, once again, this intensifies their engagement. Peter always sits at the end of one of the four aisles, barely removed from the playing area, surrounded by a dizzying array of acoustic and often invented instruments: drums, wind chimes, whistles, gongs, and mallets, plus strangely welded pieces of metal, with a ukulele, a clarinet, a harp and\/or a saxophone thrown in for good measure, depending on the needs of each story. He's become adept at playing two or three sounds at once, or four or five in rapid succession. Occasionally Peter will use the sounds available from an electric keyboard, but we always try at first to see if we can do without because we all hate to carry the amp and electrical cords. We layer the sounds in over the entire course of the rehearsal period, adding and eliminating instruments as we go; the actors love the clarity, depth, and focus these sounds lend and come to depend on them. Peter's live presence in the room, punctuating and heightening the story in the moment with the actors, creates still more energy, immediacy, and edge that compel the audience to enter into the story even more deeply. Our nontraditional audiences often gather with great curiosity around Peter, hanging out before and after performances. They are hungry for live music and sound, some of them even quite literally starved for it.\n\nTHE REVELATIONS OF METAPHOR Already limited by what we can carry and the wildly varied rooms in which we perform, we've learned to embrace still more limits, restricting what little we have even further, in part to create supremely necessary focus. And with this embrace of more limitations, we've uncovered another enormous blessing\u2014the revelations of metaphor. Metaphors allow us to see the connections between things that seem unrelated. In real life, with all its clutter, it's often hard to see these connections. But when theater strips everything down, the connections are all much easier to see.\n\nTake Richard III, for example, which we performed with an all-male cast. Hoops were the shape of our few set pieces, props, and costumes\u2014black hoops that seemed to have been pounded out of iron. Five small hoops formed the cross sections of a small skeletal tower\u2014the Tower of London\u2014that could be wheeled into a corner of the circular playing space as a reminder whenever the dreadful prison appeared in the story. Dark hoops were the structure of the female characters' skeletal skirts and corsets, allowing the men's bodies and pants to appear through the frames; such women's clothing suggested how the female characters were also imprisoned and constrained by their roles. Metal hoops formed the structure of the men's armor as well, for warriors are prisoner to the demands of the battlefield. And the crown, another black metal hoop, had five spikes on top of it, just as the top hoop of the Tower did, allowing us to see the connection between a king's role and a prison's constraints. All we offered our audiences to look at were dark metal hoops, but because there was so little, we could more easily see the connections between unrelated things. We could literally see the metaphors, feeling the harshness of the black iron constraints everywhere.\n\nWith The Little Shop of Horrors, it was easy to see how the story of the struggles of Seymour, the hapless flower shop clerk on skid row, might connect with our audience members. But since we artists were relatively ignorant of the realities of such places, we decided to limit the set materials to rebar and tinfoil, creating an abstract skid row shop that lifted the play out of its reality into more of a fairy tale realm. We were freed from having to replicate the details of skid row, and our audiences, in turn, were freed from having to cringe over how we'd gotten the details so wrong. The audience also got some distance from any actual painful skid row experiences they might have had, and everyone could focus on the story at hand instead.\n\nLuverne Seifert, Richard Ooms, and Craig Johnson as the Queens, costumes by Kathy Kohl, Richard III, Minnesota Opera Center, 2007.\n\nThen there was that giant man-eating plant. To try to believably operate and bring to life\u2014let alone transport\u2014some enormous gaudy puppet big enough to swallow people, especially without the help of any stage lighting or curtains, was clearly ridiculous and impossible. This was, of course, an irresistible challenge. My mind traveled to the very opposite end of the puppet spectrum. A bare hand. Four fingers opening and closing on top of the thumb in a talking beak, the kind every kid makes. We could make a flowerpot out of which grew a metal stem with a loop on top, a loop big enough to stick a hand through. The fabulously physically inventive actor playing Seymour, Jim Lichtscheidl, could just lower his voice and make his bare hand talk whenever the plant did. I knew Jim would relish singing both parts of Seymour's duets with the plant. Every time the plant grew, we could raise the stem higher or move to a bigger pot with a taller stem and bigger loop. A piece of black cloth could cover those who got eaten as the plant grew, and I knew that our particular Seymour would take great delight in figuring out how to get his hand to eat himself at the end.\n\nWithout that big gaudy prop of a flower puppet in the way, a metaphor was revealed. Wasn't that big hungry plant just an extension of shy, uncertain Seymour? His suppressed hunger for love and attention an alter ego manifesting itself in a flowerpot? The connection between Seymour and the ravenous man-eating plant became clear, and the childlike theatrical trick delighted the audiences.\n\nMusicals always demand the most in terms of set and props, no matter how much we try to pare things down. We let ourselves go a little wild on such shows, though we still remain, as an actress once declared in a moment of exasperation after being denied a request for a prop, \"The Amish Theater Company!\" The Most Happy Fella tells the story of an unlikely romance growing out of correspondence between an elderly Italian immigrant vineyard owner and a young (in our cast, African American) waitress. Unlikely romance is something we all yearn for, and in our production I wanted to focus on the fears that kept getting in the way of love's blossoming generosity.\n\nMost of the play takes place on a Napa Valley grape ranch. I was drawn to the Y shape of the wooden supports of the vines. They seemed like two generous outspread arms, echoing our exploration of abundance and giving without fear. In addition to supporting vines, the bare wood Y-shapes could support little round tabletops in Rosabella's restaurant as well as the street sign on the town corner where ranch hands stood \"watchin' all the girls go by.\" To create the rows of the vineyard, we made nine wooden Ys and stuck them in bare wood boxes, or planters, on wheels, three in each row across the playing square. Against all this bare wood, we restricted the costumes to a palette of greens and purples, echoing the colors of a vineyard.\n\nThe bare wood focused our attention on the vines. We just made them out of bright green string: the vines could \"grow\" by having actors lift them out of the boxes and drape them over the Y supports. During times of abundance and love, actors could lift out vines growing not only bunches of grapes but little plastic bottles of wine and rounds of cheese and pieces of cake and even\u2014as the pinnacle of our special effects\u2014little round paper lanterns, sparkling with intertwined Christmas lights, all shining with glorious color. In times of fear and deception in the couple's story, actors would cause the vines to shrivel and disappear back into the wooden boxes. Trying to mend their relationship after weeks of misunderstanding, Rosabella and Tony working together in the vineyard might pull up little green shoots of vines from the boxes, helping them to begin to grow on the Y supports once again. Wooden Ys and green strings were all it took to get us to see clearly how the objects in the play, pared down to what was only most essential, could serve as metaphors for the emotions and relationships growing and changing onstage.\n\nBack in the early days, when I was trying to come up with a name for our work, I wanted to avoid anything obvious or condescending like \"The Poor People's Theater Company.\" At the time I was reading a novel about a mysterious island in Indonesia, a strange and beautiful tale called The Ten Thousand Things by Maria Dermout. I liked the breadth of imaginative possibility this name presented; it suggested that we could do any kind of theater for any kind of audience. I snatched it up.\n\nJim Lichtscheidl as Seymour with his hand as the man-eating plant, Little Shop of Horrors, St. Paul low-income housing, 2007.\n\nSeveral years later I stumbled upon the phrase again, this time in reference to a Buddhist concept, a way to describe \"all the material things in the world.\" I drew a sharp breath of recognition. I loved how the name corralled all the innumerable material things in the world into a limited space. There were only ten thousand, and no more. I was immediately able to see how most people in our audiences\u2014and we as a theater company\u2014possessed very few of those ten thousand things. Yet we did not think of ourselves as poor. Placing limits around material objects lets us begin to think about what else there is besides those things. Imagination and spirit, fears and hungers, kindness and heart and laughter: these are what connect us all as human beings. When there is not a lot of stuff in the way, somehow these things get easier to see.\n\nSo much of my work on a play at Ten Thousand Things is about paring down to the essence. When I'm restricted, by necessity, in the number of things I can put onstage, this also helps to get to essences, to what is truly most important. Having so few things onstage helps us to see the essence of the objects, too: a crown is just a circle of metal, and a prison tower is really just the same. With essences revealed, it becomes easier to see the connection between unlike things. I believe this practice makes us better at seeing what we all, as human beings, share in common as well. It's all part of the work of getting to radiance, work made easier by not having too many things in the way.\n\nLarissa Kokernot as Molly Brown on brass bed, The Unsinkable Molly Brown, Hubbs Center for Lifelong Learning, 1999.\n\n# 7\n\n# EVERYBODY NEEDS TO SING\u2014AMERICAN MUSICALS\n\n## Fairy Tale Group #3\n\nTHE WAY AMERICAN MUSICALS are usually presented\u2014in big theaters with elaborate and extremely expensive production values\u2014has not appealed to me for most of my adult life. I feel so far away from the emotions onstage, in large part because the acting so often feels shallow and superficial, part of a big production machine that can't slow down for much real truth. In the first few years of Ten Thousand Things, I couldn't imagine us ever doing a musical. I wondered how our audiences would deal with actors suddenly bursting into song using big booming Broadway-style voices, so often seeming intended to show off the singer rather than to help tell the story.\n\nAs we continued to tour the plays of Brecht, we couldn't help but notice how the fairly unobtrusive songs in his stories created a powerful attraction. People often gathered with great curiosity around Peter Vitale, our sound maker, wanting to touch the instruments. And whenever a character eased into a song, the audience leaned forward, their attention focused even more keenly on what was happening onstage.\n\nSo, despite my reservations, we decided to try a musical, even though we'd only have a one-man orchestra and a cast of no more than eight. I started to think that there was a way that a musical, if set in another time and place, could be a kind of fairy tale, too. I felt there must be a way to get at the essence of a musical, strip it down to its fundamental story line, and truly explore the complexities of the situations through good acting. Peter was a huge fan of the genre and brought me a tall stack of scripts and CDs. I read them all, but when I excitedly brought Peter my choice, the story that had tugged at me the most, he was horrified. In my overall musical ignorance, focusing only on the stories and text, out of the stack of classics I had chosen the one Peter, musically at least, detested.\n\nTHE UNSINKABLE MOLLY BROWN The Unsinkable Molly Brown, though based on the true story of a young woman born into poverty who ends up striking it rich, is a fairy tale. Living in another time and another place, Colorado in the late 1800s, Molly, with a cyclone of desires swirling inside her, sets off into the world to try to satisfy her enormous appetites. As she leaves her family's little Colorado mining shack, her father asks her, \"What would be enough, Molly?\" Molly doesn't really know\u2014she has an endless list\u2014but she answers very honestly, \"I ain't settlin' for happiness.\"\n\nMolly ends up getting hired at a saloon and falling in love with a poor young miner, Johnny Brown. But she makes it very clear from the start that love isn't enough\u2014Johnny needs to be rich before she'll marry him. Luckily, Johnny soon strikes gold, and Molly quickly becomes one of the wealthiest women in the land, living in a mansion in Denver, traveling to Paris to meet painters and poets. Johnny Brown, though, is not comfortable with wealth and is made unhappy by their lavish lifestyle. Molly finds herself having to choose between his deep love and her fabulous riches.\n\nMolly doesn't back away from her fortune. As she tells her father at the beginning of the play, \"It ain't the money I love, Pa. It's the not having it I hate.\" The story actually seemed a fun but important exploration of that age-old question of love or money, a question that we in the middle and upper classes answer all too easily, spouting back with truisms, \"Why, love is all you need, of course!\" Molly, born into poverty, seemed to be taking a much more honest look at that question, and I sensed that our nontraditional audiences, living with the reality of not having enough money, would also give more weight to the difficulty of the choice. I relished the chance to explore it all with them.\n\nFor Johnny Brown, love truly is enough. \"You're all I want in this life,\" he says to Molly, purely and simply, and he actually means it. With his character, I wanted to take a hard look at how challenging that would really be. At one point, Johnny sings a song to Molly, \"I'll Never Say No to You.\" When I was reading the script, I considered this not as a \"song\" but as text the character was speaking; I marveled at the difficulty involved in really meaning those words. To never say no? The fact that a male character was truly committing to such words made it all the more interesting. It would take enormous trust and enormous discipline; it would be so impossibly hard not to say no at least sometimes to the requests of the one you loved. But Johnny really did seem to be one of those rare men who is willing to accept a woman with all her complexities and difficulties. After giving Molly all the mansions and gowns and jewels and parties that she had asked for, Johnny says to her quietly, \"You don't know what it is you do want. But I guess you gotta keep looking.\" Such acceptance of another person's true and very different self was quite moving.\n\nAnd the miracle of this rare acceptance finally gets through to Molly, too, though almost too late. Johnny, drowning in the unhappiness of all their material wealth, does finally have to say no to one of Molly's desires: her request for a divorce. He can't allow it, but he lets her go back to Paris, the place where she had felt so happy, amusing the nobility there. Once back in Europe, Molly takes off her wedding ring and discovers an inscription on the inside that Johnny made long ago, \"Always remember I love you\u2014and the name of the bank.\" This honest acknowledgment of the truth of who Molly really is, someone who will always need both love and money, is what makes her go rushing back across the sea, into Johnny's arms. So at the play's ending, Molly again has both wealth and love in abundance in her life\u2014and we are still not sure which is more important. Money or love. I loved the honesty of this \"fairy tale\" ending, a very happy one, but with its big question still intact.\n\nThe Unsinkable Molly Brown wasn't just a play, of course; it was a musical. Peter's dislike of the play was due entirely to the music, which he felt was at best unmemorable and at worst mind-numbingly simple. I hadn't listened to the CD; I'd only read the story. And of course Molly Brown had scenes in mansions in Denver and palaces in Paris, with ball gowns and jewels and fancy furniture; it seemed like it was going to need a lot of things. We couldn't, of course, have an orchestra, just Peter's one-man band. But I had also never seen the movie with Debbie Reynolds, and my ignorance was once again an asset. I clung to my conviction that as long as the story was really interesting and compelling, inviting everyone in, we could make it work. Even though, as usual, we started out not knowing how exactly to do that.\n\nI did know that, as always, we would need excellent actors to bring out the truths and complexities of the story. I'd need an exceptionally special actor to play Johnny, one who could give body to the truth of the difficult work required to \"never say no.\" I needed someone who already knew about the discipline and patience behind such love. At auditions, a tall, lanky actor finally came into the room, radiating a kind of quiet white light, and I knew at once that we had found our Johnny. The only problem, as Peter patiently pointed out, was that he couldn't sing. Well, he could sing a little, I countered; his quavering voice could actually hit most of the required notes. Peter, demonstrating a kind of Johnny Brown\u2013like patience and trust himself, sighed and said he would try to make it work.\n\nNot only was I doing a play with music Peter didn't like, I wasn't approaching any of it as he thought it should be approached. Peter, in addition to being an actor, was formally trained in voice for opera as well as musical theater and could play an astonishing array of instruments. But Peter was also a very kind man and a very good listener. He had worked with Ten Thousand Things' nontraditional audiences enough to know that they would probably be quite skeptical of, and even resistant to, a character who just burst into big, full-voiced song. He shared with me an interest in discovering just why\u2014and how\u2014a character would start to sing, knowing that the transition would have to be made in a clear and sensible way. I wanted to pare everything way back, strip it all down so we could see what was going on. Every person at some time in their lives, even if only as a child, has felt the need to burst spontaneously into song. The impulse often gets beaten out of us fairly early on, and as adults most of us rarely indulge it. If our audiences were going to buy that this could really happen, the transitions would be the key. Our actors would really have to earn their right to sing, each and every time. We would have to thoroughly explore every transition.\n\nIf I could have had my way, I would have had all the characters sing everything a cappella. I wanted to hear all the nuances of the honesty and vulnerability and complexity while they sang. I didn't want any of it covered up by other instruments, and a cappella would make the transition into the song clearest and most transparent. But Peter recognized what a disaster that could be for untrained singers and found ways to gently and sparely support the actors with a guitar strum or a few plunked notes on a toy piano. He was brilliant at pulling sounds out of the actual environment to become musical accompaniment. He helped me, who only knew how to think theatrically, not musically, find a way to get at the musical truth of the story.\n\nAnd as for all the fancy sets and props this musical about becoming rich seemed to require, we again found that stripping away and paring down brought both great pleasure and truth. Toy furniture would do the trick. The richer Molly got, the smaller her furniture would become: the piano in the saloon where Molly first stopped on her journey was toddler sized, perhaps three or four feet high; the one in Mrs. McGloin's Denver mansion was only about eight inches tall (though still playable\u2014we made a splurge purchase from the American Girl doll catalog). Always aware of the limitations in sight lines that come with no risers, we raised the little piano and other toy red velvet furniture up, mounting it on poles at the actors' eye level so that audience members seated two or three rows back in the circle could see. For the surprise of a curtain rising to show the new Denver digs, we wheeled in the pole tops covered with pieces of cloth, and Molly unveiled each little piece of furniture as the scene began. Finally, in her Paris apartment, Molly unveiled the objects on the poles to reveal, to the approval of her high-society guests, tiny gold velvet doll house sofas and chairs, just a few inches tall; there was great hilarity to be had at the seriousness with which the nobility handled the teeny, thimble-size champagne glasses and rested their fingers on the little divans, the smallness heightening the silliness of all the elaborate material display.\n\nFor costumes, we again created a universe of fabric. This time the characters shared coarse woven browns for the Colorado mining world, with a few hints of red in smoother, shinier cloth, bringing in more fancy red fabric as Molly scaled society's heights. We brought in shiny golds when Molly reached Paris\u2014but never completely covered the coarse brown cloth underneath. Something about the form of the musical caused us to become a little extravagant; our designer, the multitalented actress Signe Albertson, had great fun making elaborate and ridiculous hats for the high-society folks.\n\nAs exciting as we felt all our theatrical choices to be, the ultimate moment of truth in any Ten Thousand Things show comes when we step in front of the audience. I experienced one of my strongest \"you must be out of your mind\" moments ever on this tour, standing at the back of the men's prison gym, watching the guys file in. I was doing The Unsinkable Molly Brown? For male inmates? What in God's name was I thinking? Experience had taught me that men were usually not that interested in stories about women. They were generally not fans of musicals. And yet. Molly had started out poor and rough and tumble, just as most of them had, though her Rocky Mountain shack provided welcome imaginative distance from the urban poverty many knew so well. As she wrestled with her brothers in the Colorado dirt, she shouted out the words to a song:\n\nThere'll come a time cause\n\nNothin [slam] ner\n\nNobody wants me\n\nDown like I wants me up [slam]\n\nUp where the people are\n\nUp where the talking is\n\nUp where the joke's goin\n\nOn! Now look-a here\n\nI am important to\n\nMe [slam] I ain't no\n\nBottom to no pile!\n\nI could feel the men start to open up. They were rooting for Molly. They didn't even seem to think her anthem \"I'm gonna learn to read and write\" was corny. They seemed to accept the ferocity of her desires wholeheartedly.\n\nThis story was also very much about class, for Molly, despite all the gold that Johnny had found, never really fit in with the old money wealth. Her generosity and honesty were just too much for the society people of Denver to handle. Molly never took her wealth for granted or felt that it should be used as a way to exclude others. \"Already took ten million outta God's good earth. Guess he deserves a little kickback,\" she exclaimed as she wrote a ridiculously generous check to a Denver priest trying to raise funds for charity. The male inmates took to Molly's out-of-place honesty so much that by the end of the play they were standing and chanting \"Mol-ly! Mol-ly!\" during curtain call. We were stunned. Another monumental failure on our part to imagine the audience. We could not have dreamed that these men would care so much about a musical about a girl.\n\nI hadn't had any doubt that the female inmates would love Molly. This was one of the first shows we did in both men and women's prisons, and it was truly fascinating to see how starkly different the reactions were at some parts of the story. The musical is certainly about class and wealth, but it's also about romance and sex, and each gender brought a very different perspective to those scenes. For many women, Johnny's undying love for Molly touched a powerful chord, reminding them of their own lonely aches and the absence of sweetness and tenderness in their lives. Many of the women sobbed openly, at times so hard that one or two needed to run across the stage into the bathroom to get tissue.\n\nThen there was the \"gift-giving scene.\" Johnny, having just struck the vein of gold, returns to their little mountain cabin to present Molly with a series of presents. He unveils several toddler-size pieces of furniture, including a new wood-burning stove (made from a trash can we'd turned over and painted black). He then comes to the final brown cloth and pulls it off to reveal a toddler-size brass bed, entwined with roses about its frame. The women burst into ooohs and ahhs, so touched at the loveliness and thoughtfulness of the gift.\n\nThe men's reaction to the unveiling of the brass bed was quite different. When the brown cloth was pulled off, they burst into loud whoops and hollers of delight. To their way of thinking, Molly had just received a bunch of presents from Johnny, and now it was her turn to give some payback. The brass bed meant only one thing to this audience\u2014sex. The actor playing Johnny Brown, Ron Menzel, however, seized on the men's honesty as a chance to have a \"conversation\" with them. As an actor, he felt able to harness their energies by using his next lines like a lasso, pulling them away from the idea of sex and guiding them over toward still more romance. When Molly exclaimed at his gift, \"You got yourself a bed, I'll say that for you, Leadville,\" Johnny gave his next lines a stronger than usual subtext. Underneath his actual words to Molly, \"Yeah, and at night the moon comes a-shinin' in and it's all starry like,\" Ron was also saying to the men, \"No guys, we're not going to demand sex now, we're gonna give her still more, we're going to recite her a sonnet. Giving, without asking for anything in return. That's what love is.\" The men's surprise was visceral. Though some resisted, making disdainful sneers, you could actually feel most of the others deciding to go along with Johnny to see what would happen. When an audience is so open and honest, true conversations can happen during the performance without any change to the actual lines of the script.\n\nMolly and Johnny's happy ending in front of their Lincoln Log palace, Hubbs Center for Lifelong Learning, 1999.\n\nMolly Brown brought us another of our most radiant performance moments, this time again at the Dorothy Day Center for the homeless in downtown St. Paul. For a while when I was contemplating doing the play, I worried about its \"happy ending.\" Perhaps the story was too much of a fairy tale. People in our audiences clearly knew that life almost never ended up like Molly's, leaping from poverty to enormous wealth, and I didn't want anyone to think that we were suggesting this was easy or even remotely possible. But Molly Brown wasn't a typical American rags-to-riches story, either. Molly and Johnny didn't end up with what they had through hard work\u2014Johnny discovered that vein of gold through sheer luck. Both characters are always very clear about this. And truth be told, I thought, we would all be better off if wealthy people were so honest. So many end up with their money through accidents of geography or birth, with strokes of good timing, fortuitous connections, and the help of others all playing much more important roles than any one individual's effort. Luck has so much to do with why some people get rich and others don't. It's a relief to be reminded of this sometimes. And with Molly's journey, because there had been so much laughter and delight and silliness along the way, it became clear, after several performances, that our audiences wanted it to end well for her, wishing the same good fortune for themselves.\n\nBut even I could not bring myself to end the play with everyone singing one more round of Molly's anthem, \"I'm gonna learn to read and write.\" The reprise seemed tired and trivial by that point. At the time, the pop group Bare-naked Ladies was quite popular, and in rehearsal someone started singing their song, \"If I Had a Million Dollars.\" It became instantly clear to all of us that this was the perfect song to end with. As we drew near the end of our performance at the Dorothy Day Center, Molly and Johnny started to waltz in front of their little log cabin\u2014now fancier with additions (we'd built one out of Lincoln logs and put it up on a pole); the rest of the cast started to sing, \"If I had a million dollars, I would buy you a house.\" A few members of the audience started to chime in. It seemed a lot of homeless people were quite tuned in to popular culture. Soon, almost everyone in the very large room was singing, \"If I had a million dollars, we wouldn't have to eat Kraft dinner (but we would eat Kraft dinner, of course we would, we'd just eat more)... If I had a million dollars, I'd buy you a monkey (haven't you always wanted a monkey?).\" So much joy and warmth among people, people from all kinds of different economic circumstances, all united for a few moments by acknowledging, with humor and heart, that in our world, both love and money\u2014or at least our fantasies about what money can buy\u2014bring enormous pleasure.\n\nTHE MOST HAPPY FELLA After The Unsinkable Molly Brown, we all decided we loved doing musicals. Musicals just made everyone happy, from the artists who worked on them in fun, music-filled rehearsal processes to the audiences who experienced them. Working within the necessary strictures of Ten Thousand Things performance conditions and our need to imagine a first-time audience's reception of people who suddenly burst into song actually just made it all even more fun. We next turned to The Most Happy Fella, the story about the unlikely romance between the elderly Napa Valley vineyard owner Tony and the young waitress Rosabella. The idea of an unlikely love match was often echoed in the actual performance itself, particularly one very memorable show at a men's prison, all through the medium of one very implausible and reimagined song.\n\nIn addition to the tricky romance of Rosabella and Tony, the musical offers several other unusual love matches, including one between Herman, a sweet, shy ranch hand, and Cleo, another waitress, but a brash, outspoken one from Texas. We had the enormous good fortune to be able to cast a former Minnesota Vikings noseguard in the role of Herman. Esera Tuaolo is of Samoan descent, and as his former job description suggests, he is an enormous, powerfully muscular man, but one whose inner being perfectly matches Herman's sweetness and kindness. And he can sing with all his soul.\n\nThere is a moment near the end of the play where Cleo gets in a fight and is roughly handled by another ranch hand. Herman's protective instincts spring into action, and, completely out of character, he punches the accoster. He is so surprised by this action that he finds he has to sing. In the original 1950s version, Herman's song, \"I Made a Fist,\" is a triumphant celebration of his masculinity, complete with marching brass band. Peter and I knew we could never go into a men's prison\u2014a place where so many found themselves trapped because of acts of violence\u2014and sing such a song. But once again, our necessarily spare approach to the material came to the rescue. Peter noticed that when the brass band accompaniment was stripped away, the melody of \"I Made a Fist\" was actually somewhat melancholy. Esera began the song very quietly, with only one spare piano note as accompaniment, and the song became not a celebration but a sad realization of his discovery of his ability to hurt people.\n\nWhen we first walked into the men's prison gym, many of the guys quickly recognized Esera from his Vikings days and gathered around to admire the tattoos on his astonishing biceps as we set up. At this time, the prison would not allow chairs in the gym, for fear they might scratch the floor, so the men sat on mats that ringed the playing space. When Esera began to sing his first song, \"I Like Everybody That I've Ever Met,\" tossing in a little dancing as well, you could clearly see the shock on most of the men's faces. This astonishment transformed into a sort of open-jawed wonder as the story exerted its pull and the men, seemingly without realizing it, dropped their defenses and entered the fairy tale world with the requisite na\u00efvet\u00e9.\n\nI especially loved how this primarily African American male audience (an enraging reality of male incarceration, but that's for another book) seemed to completely empathize with the plight of vineyard owner Tony, who was quite tall, thin, elderly, and very white. But Tony was also an immigrant, an outsider who couldn't speak English well and who felt ashamed of himself. Tony's vulnerability about these shortcomings provided the men a strong connection to the character, so that neither his wealth nor his skin color got in the way.\n\nEsera Tuaolo as Herman in The Most Happy Fella, 2001.\n\nBy the time Herman got to singing about the punch he had just thrown, the men had opened up completely. As Esera sang about the pain of discovering that as a large strong man, he had the power to hurt others, I looked around the circle of audience and saw tears glimmering in many of the men's eyes. Men in prison usually need to be cautious and guarded, but this was one of the most open men's prison audiences I have ever experienced. The Broadway musical of the 1950s had met with the Minnesota Vikings and inmates in a men's prison through the artists of Ten Thousand Things. Our spare reimagining of a story\u2014and an unlikely romance between all these partners\u2014blossomed in that prison's gym.\n\nReimagining the stories of musicals through the eyes of our nontraditional audiences has also served to delight our veteran theater audiences, who take great pleasure in the way the stories are refreshed. We can explore what happens to My Fair Lady if Eliza Doolittle, not Henry Higgins, truly becomes the center of the story; or what happens to Ragtime if there are only eight actors, plus a few puppets and props to tell the epic historic tale; or what happens to Man of La Mancha if the whole story, which is told by inmates in a prison, actually only uses the objects that can fit in Cervantes's battered trunk. In all of these, the stakes become very high, the situations have new urgency, the lower-income characters receive much more attention, and the humanity of the stories shines even more brightly when all the glitz is stripped away. This seems to delight experienced and perhaps somewhat jaded audiences who have seen many productions of the same shows before.\n\nCAROUSEL Once in a while, our traditional audiences have been unable to put aside their own expectations of what a musical should be. I was attracted to Rodgers and Hammerstein's Carousel as a story about working-class people. The characters were all trying to scrape through a hardscrabble existence, working in factories and fishing on rough seas. But it was all still approachable as a fairy tale because of its setting in an unfamiliar time and place. Peter had wanted to do this musical for a long time, but I had resisted because I found the misogyny so relentless. I finally saw a way in, though, when I thought of casting a particular actress, Carolyn Goelzer, as Julie. Julie would need to be tough, to stand up as fiercely as she could to the misogynistic forces around her. Carolyn was large and strong, with a deep raspy voice, and, although very attractive in her strength, not conventionally so, certainly nothing like Shirley Jones in the movie. The actor playing Billy Bigelow, Terry Hempleman, was not the most accomplished singer, but he also brought a knowledge of working-class life in his very being that lent great credibility to the story.\n\nRegina Williams as Aldonza, Man of La Mancha, women's correctional facility, 2011.\n\nIn the men's prison, overcome with a need to sing about his love, Billy's rendition of \"If I Loved You\" was tentative and awkward; he was not a man who was used to singing. The inmates seemed not only to accept this but to welcome it. Billy was singing like they would sing if they ever had to. Precisely because of Billy's untrained voice, the men were able to accept the strangeness of the musical form. After that particular performance, all of us in the cast and crew were quite moved and inspired by the connections the story had made with the inmates. But our traditional audiences just did not seem to be able to shake John Raitt and his big booming operatic voice from their minds, nor to let go of their idea that Julie should be petite and pretty. Though female inmates had screamed with laughter at the antics of the shifty-eyed Jigger, even chanting his name every time he appeared, our paying audiences did not find him funny at all. It was probably our most schizophrenic tour, in terms of the completely different responses of traditional and nontraditional audiences.\n\nMY FAIR LADY We are sometimes able to cast excellent actors who are also fantastic singers, though acting excellence is always the first requirement. With My Fair Lady, guest directed by Lear deBessonet, we were excited to work with one of the best singers in town, a wonderful, kind, generous man who was also an amazing actor, blessed with great comic physical and improvisatory skills. In addition to playing the housekeeper Mrs. Pearce, Bradley Greenwald played the part of Freddy, the young upper-class man who becomes smitten with Eliza. This meant he got to sing \"On the Street Where You Live\" with his beautiful, classically trained tenor voice, a moment which always brought all of us in the cast, and in our paying audiences, to tears. In the prisons, not so much. As soon as his voice started to boom out, large and full, we could sense a barrier of suspicion go up among the audience. Music from different cultures often has politics attached to it\u2014not every kind of music is equally appealing to everyone, and musical theater music is no exception. Because Bradley was such a wonderful actor and able to sing with such sincerity and heart, he eventually won the nontraditional audiences over\u2014but it wasn't an easy victory. It often took until almost the end of the song for audiences unfamiliar with the form to get over their suspicions and work around the initial obstacle that Bradley's well-trained voice presented. With Ten Thousand Things, we have to find an approachable, egalitarian playing field for musical styles, too, inviting everyone in as best we can, whether it's an invitation to nontraditional audiences to enter a Broadway show tune or one to our older, whiter traditional audiences to try out some hip-hop, as with Will Power's The Seven.\n\nWe now do a musical almost every season. They make us happy, and their fairy tale qualities, in addition to their songs, really do deeply engage our audience. I don't think our approach to the musical is unique. Many other theaters, especially in recent years, have discovered the joys and blessings of doing musicals in a pared-down style. But at Ten Thousand Things it's all out of necessity. In our fifteen-foot performance space, three people can easily make a chorus; a dance is just several people moving to music together, usually in a very pedestrian way. But it all has an enormous impact because the audience is so close. The necessity of imagining the lives of very different audiences gives us fresh perspective.\n\nWhen you've never seen live theater before, let alone a musical, it is truly bizarre when people just start to sing in the middle of a scene. Many in our audiences have suspicions about musicals to begin with, just as I did. But because we are always trying to be honest about the need to sing, not trying to impress with how good the voices are, these people can enter the world of musicals, too. They can connect because once in a while they too have felt the need to sing, in the shower or perhaps even while walking down the street, when no one else is around. It's a simple human urge.\n\nSeveral years after The Unsinkable Molly Brown tour, I was walking down the hall of that same men's prison where we first opened, preparing to do another play. The hall was empty at the time, except for an approaching inmate. I was a little nervous, but as he came nearer, I could tell he was singing. He was singing \"Belly Up to the Bar, Boys.\" He gave me a big grin and waved as we passed. Not only had a musical connected with an unlikely audience in the moment\u2014it had stayed around for quite a long time.\n\nA Midsummer Night's Dream cast in pajamas at curtain call, Avalon School, 2013.\n\n# 8\n\n# CONCEPT-FREE DIRECTING\u2014ORIGINALITY FROM NECESSITY\n\n## Preparation, Rehearsals\n\nAS A DIRECTOR, I never worry about how to make my work \"original.\" In the Twin Cities, theater audiences might choose between a Shakespeare comedy set in the Roaring Twenties and one set on a cruise ship; the Ten Thousand Things version of the play will be focused on telling the story with as much clarity and as few objects as possible. I do want freshness and originality to be abundant in the production, but this just happens out of necessity, without any conscious effort to \"come up with a concept.\" If I just do what I have to, viewing the story through the eyes of all our different audiences and solving problems within our bare-bones performance constraints, then a truly unique perspective, full of all kinds of interesting and unusual choices, emerges quite naturally, without my really having to try.\n\nAt the moment, I'm working on A Midsummer Night's Dream. I'll use it as an example to go through my working process, some of which is probably not so different from what many directors do. Although Midsummer is most certainly a fairy tale in the true sense of the word, I resisted it for a long time. The productions I had seen, often in school or community theater settings, had been so annoying, with most characters presented superficially for cheap laughs. It didn't feel like anything crucial was at stake, beyond some mixed-up love matches. But the working-class characters known as the Rude Mechanicals broadened the world of the play to include people who weren't just of the nobility. And it certainly was set in another time and place.\n\nAs often happens, the prison audiences leapt to the forefront of my imaginings; this time it was the men. With their misogyny often openly and honestly expressed, in my imagination male inmates can present the most formidable resistance to entering a lot of plays, resistance for me to try to figure out how to overcome. My instincts immediately told me that we would need to use Shakespeare's term \"goblin\" much more often than \"fairy,\" as the guys in prison would most likely not be too enthused about delicate gauzy sprites with gossamer wings. Okay, we could have goblins.\n\nI read the script again, and as I did, I felt that intuitive tug, the pull that told me something important was going on in the story that might be of vital interest to all our audiences. I found myself being drawn to one aspect of the behavior of the indistinguishable lovers, Helena, Hermia, Lysander, and Demetrius\u2014their extreme shifts between complete love for and absolute hatred of each other. Those sudden changes in passion would probably resonate. Everyone has been through violent, horrible breakups, discovering they suddenly despise the person they once intensely desired. I decided to plunge in and keep trying to figure out just what was going on.\n\nEIGHT ACTORS But first, I put aside the search to find the play's common struggle, one shared by all the characters (and the audience as well, of course), and turned, as I often do, to thinking about the practical issues involved in staging it, just to make sure it was feasible. Thinking through the necessities of our performance constraints often helps me shed light on the heart of the play. There were over twenty roles, so how many actors would we need? Eight is usually the maximum number I cast in a large play. Double- and even triple-casting is a necessity for us, in part because of the budgetary benefits, but even more because such casting ensures that everyone will be able to be onstage as much as possible, something both actors and audiences enjoy. Very large casts, in which some actors have only one small part with a few lines and spend a great deal of time sitting around waiting to go on, often feel sluggish, both in rehearsals and in performances, where audiences sense how little of the actors' energies are being used. Most actors, at least the kind of generous, playful actors I like to work with, are happiest when they're onstage a lot, and audiences quickly pick up on the fun they're having.\n\nMidsummer has three groups of characters: the nobility, the workers, and the goblins. I started to make charts of the scenes. If each actor played one character in each of those three groups, it might work. It would mean some lightning-fast costume changes, though, so the costumes would have to be really simple, with one well-defined piece for each character. I reasoned that there would need to be a base costume that the other pieces could be thrown over. Pajamas popped to mind. It's a midsummer night's dream, after all. The pajamas could all be different shapes and textures, so it wouldn't be boring to look at, but if they all shared the same color, some sort of dark midnight blue, with maybe some flashes of silvery gray, those colors could help create a world of the night sky, of the darkness and mystery of the forest. I also liked how having that base costume equalized all three groups of characters, so you couldn't say which was more important\u2014the nobles, the workers, or the goblins. You also couldn't really tell which was the most \"real.\" The actors in pajamas were really dreamers, stepping into different lives for a little bit, as we all do when we dream, when we watch a play\u2014or when we empathize with someone who is different.\n\nMISOGYNY ONCE AGAIN It's not unusual for me to be irritated by at least some parts of almost every play I choose to work on; irritation can serve as an irresistible challenge to figure out how to overcome. One of my major sources of irritation with A Midsummer Night's Dream is, of course, its overwhelming sexism. Once again, even as I loved the bigness of the world of this classical play, I was immediately wearied by the narrow treatment of its few female characters. How could anyone believe that Hippolyta, the Amazon queen whom Theseus had captured and was now forcing into marriage, could ever become a willing partner to such a wedding? And this play had the old tired drama around a father, Egeus, forcing a daughter, Hermia, to marry someone she doesn't want to. This kind of conflict has so little relevance today, at least in most Western cultures. I was bored by it. In the goblin world we had the cruelty of Oberon, tricking and even drugging his queen, Titania, so she would give up the child of a dear friend who had died in childbirth, a child she had promised to raise as her own. I was just really tired of all these situations, and I had no idea how anyone could view them as palatable, let alone enjoyable, as most of the other mind-numbing productions I had seen had asked me to do.\n\nCASTING These irritations immediately caused me once again to turn my attention to casting as one way to ease some of my annoyance. There is that old directing school adage that \"casting is 90 percent of directing.\" I actually believe this to be fairly true. But it doesn't seem that most directors put 90 percent of their creative energies into casting. I've heard of other directors placing a lot of energy into finding the right \"type\" for a part. But viewing actors and characters as \"types\" does not seem to me to require much creativity.\n\nFor me, casting is where my creativity truly begins. It stems from the simple notion that because I know that every kind of human will be in my audience, I need to find room in the play for as many kinds of humans as possible, putting as many races, ages, shapes, and sizes onstage as I can. These are wonderful colors and textures to paint any story with. I don't call this color-\"blind\" casting, of course, because great attention and care must be paid to the way actors of color are used, especially taking into consideration the history of oppression, discrimination, and hurtful portrayals each race and gender has had to bear. I especially, as you know by now, want to find room for women, who always make up more than half my audience, with some audiences entirely female. Thus, if at all possible, I want women to have at least 50 percent of the space in any story we tell. Smart, imaginative casting can so easily bring fresh perspectives and new insights to any play. It is astonishing to me how much fun it is to do\u2014but how few directors really do it.\n\nPerhaps the general lack of creativity on the part of directors when they cast is partly because many contemporary plays seem too \"realistic\" for creative casting. But I believe that need not always be the case. A Streetcar Named Desire, for example, at first seems not to be a good fit for Ten Thousand Things, in part because it is confined to a small, naturalistic 1940s New Orleans apartment with all white characters. But the story of family dysfunction was compelling and universal. We just needed to find a way to open up the world to more of our audiences. We did cast a white man as Stanley, but we also cast an African American woman as Blanche, a Dominican American woman as Stella, and an Asian American man as Mitch. It was completely believable. New Orleans has always been a place of great diversity. More important, the casting invited our audience into the story in a way that an all-white cast simply could not have done. I like to call this color-ful casting, casting actors of all colors with great thought and care. Or, even better, \"creative\" casting, as opposed to standard, unimaginative \"typecasting,\" which can actually feed into stereotyping.\n\nI am actually very rarely interested in plays that \"require\" me to cast by type. Plays that demand a very specific age and look, a thirty-year-old woman with blonde hair, slim waist, and attractive appearance, for example, are usually not fairy tales. They almost always leave lots of people out. I also don't like plays with such instructions because they limit my creativity as a director. In fact, these days, when I see that a play has an all-white cast my immediate response is to feel much less interested in attending. Not only do such plays leave others out; they are usually blissfully unaware that they are doing so. I find it wearying and sometimes even painful to sit in their audiences.\n\nOf course, you can't cast just anyone as anything. Each role does have certain contours and confines that a director needs to come to understand. But if a play does seem to demand that someone of a specific race or gender be cast in a part, then I want to take a hard look at why this should be. Sometimes it's just the demands of a heterosexual romance, but there will often be racism or sexism involved. I love to challenge myself to think as expansively about those demands as I can, which can be a great deal of fun.\n\nReturning to Midsummer, clear about the need for each actor to play two or three parts, I started looking for more roles for women. A woman could be Puck, of course; there's nothing unusual in that idea, and I realized a woman could also easily play Bottom. All the Rude Mechanicals could be women, really. Indeed, when the usually all-male Rude Mechanicals put on their play, they cast Flute as a woman in the role of Thisbe. So we could certainly do some gender stretching, too. But if each Mechanical also had to be a member of the nobility, how could we do all the romances? Well, there actually isn't any reason why one of the romances couldn't be between two women. Lysander could be a woman, easily. I couldn't imagine much resistance to such a same-sex match up among the female inmates, or among the male inmates either, for that matter (a male couple, we'd learned from The Seven, was more challenging for the men). And Egeus could be a dominating mother, rather than a father trying to control a daughter's marriage\u2014that might freshen up that story a bit. Actually, we could probably do the whole play with only two men: one to play King Theseus, and one to play Demetrius in the confusing foursome of noble lovers. I started to imagine what other roles I might cast these two men in.\n\nIn dreams and in plays, we can escape the confines of our gender. What if King Theseus, in a dream of the goblin world, found himself having to be Titania, the goblin Queen? And the captive Amazon Queen Hippolyta might dream herself to be the goblin King, Oberon? That would help address some of the gender imbalance. And of course the actor playing Demetrius, when he entered the Rude Mechanical world, would have to be Flute, the one that gets cast as a woman in the play that the workers put on. What fun!\n\nIt's all about opening things up. This is actually the biggest reason I love to throw so much of my creative energies into casting. Creative casting opens up possibilities for audiences that have often felt trapped by the perceptions of others. Such casting opens up exciting new possibilities for actors, too, who are so often boxed in and shut down as they are cast over and over again by type. It allows them instead to stretch their muscles with many varied characters. It lets audiences take pleasure in how one actor can become so many different people, and thus illuminates the humanity common to all of those characters, whether upper or lower class, male or female. Opening up possibilities means that new energy and radiance is released, both in rehearsals and in performance. There is so much joy and delight in being able to escape from the box, to see and be seen in new light.\n\nConsidering the possibilities of casting opened up more depths of the play for me. As the same actors shifted back and forth between the worlds of the nobles, the workers, and the goblins, transforming as we do in our dreams, what if a faint trace of the previous reality sometimes remained? What if King Theseus, suddenly finding himself a fairy Queen, had to experience what it felt like to be dominated by a powerful husband, tricked into doing things he didn't really want to do? What if the captive Amazon Queen Hippolyta suddenly found herself powerful again as a goblin King? Would she experience such power differently, with perhaps some faint memory, as from a dream, of once having been captured? With each actor taking part in all three of Midsummer's worlds, would there be moments when the dreamlike traces of one world might intrude on another? I had no idea, but it seemed like such an interesting idea to explore.\n\nAnd thus, by thinking about the necessities of casting\u2014and costuming\u2014the terrain of the play was now illuminated for me. I returned to the source of my initial tug about what the common struggle might be, pointed to by the lovers' sudden changes of heart. The leap from loving to hating someone is an extreme shift of reality, one baffling and difficult for anyone to explain. I started to think about how so often in life our realities shift without our even noticing it, except for those few moments when we \"wake up\" to marvel at the change and wonder how it could have happened. I imagined a woman one day waking up to find herself in a jail cell, having no idea how it had happened. I imagined a man suddenly waking up to find himself living on the streets, wondering just how he had come to be there. A woman looking up from her desk to suddenly notice she had just spent years of her life in a little office in a big law firm, married to a man she was no longer interested in, the mother of resentful, rebellious teenagers, having no idea how such creatures had come to live with her. I began to notice the sudden shifts of reality that most of the characters in Midsummer find themselves experiencing, whether from a loving to a hateful relationship, from being a queen to a captive, from being a worker to the lover of a fairy queen.\n\nSLEEPING WHILE STANDING The characters in Midsummer fall asleep onstage a lot. Thinking practically again, I knew anyone lying down on the floor would be a problem because of sight lines. At Ten Thousand Things only people seated in the first row can see what's happening on the ground. (And, it should be noted, the floors at some facilities don't make for pleasant\u2014or sanitary\u2014reclining.) The characters were just going to have to fall asleep standing up. Suddenly, I saw a way the costume changes, the transitions between one world of characters and the next, could work to illuminate the central experience of the play. Sleepwalking. We all sleepwalk through much of our lives, most of us waking up only occasionally to be astonished at how things have changed without our understanding how or why.\n\nIf this were a world where people fell asleep standing up, it would be an easy transition to start walking while still sleeping. The pajama-clad actors could do so through the changes, smoothly dropping one costume piece and putting on the next without even being aware. Later in the next scene, we might find a moment where one or two of the actors could briefly \"wake up\"; they could have a sharp sudden sense that things had been different not too long ago. It would be momentarily bewildering, but then they would just have to carry on with the scene. I knew that too many moments of bewildered walking around could get tedious, but a few moments of recognition of the mysterious shifts would underline all the other places in the play where the characters are suddenly astonished by what seems to have just happened.\n\nAs a director, I have certain emotional terrains that I especially love. Midsummer would allow me to explore one of my favorites\u2014bewilderment. Those times when we must drop all our pretenses and acknowledge that we are confused and overwhelmed, that we really do not understand at all what is going on. Bewilderment is a great leveler; it is a state of mind all humans\u2014all classes, races, and genders\u2014experience. At bottom, none of us really understands what is going on. I have noticed that when we can acknowledge this together, usually through laughter, when we can admit together publicly that we're really all just pretending to understand, it can be a great source of connection.\n\nBewilderment is also akin to the states of astonishment and wonder, which express the enormous mystery of life. I love to watch these feelings onstage, but I find they are rarely portrayed, especially in naturalistic theater. I'm not sure why, though it is certainly interesting to think about what this says about our world right now. In any case, Midsummer is full of moments of bewilderment, astonishment, and wonder, and I wanted to give them time to live and breathe, time for all of us to recognize such feelings together. Suddenly waking up to discover, in bewilderment, that your life has drastically changed is an experience that every audience member could probably relate to. It's a very interesting territory to explore, and as a director it helped me give the play a shape that would help audiences, in turn, to better grasp its complex story.\n\nSCRIPT WORK With this much clearer sense of a central human experience of the play, I then turn to more mundane script work tasks, more of those directing school fundamentals, which I approach with renewed attention because of Ten Thousand Things' audiences. I work, as most directors do, on finding the actions underlying the characters' lines, trying to understand what each is really trying to do to others when they speak, knowing that the ability to feel the visceral actions underneath the fancy words is the way we all, in the audience, best understand what is going on. Finding the actions while still just reading the script is always a guess at best. I know I won't really understand what's going on until we start to rehearse, but I take an initial stab at it.\n\nAfter searching for the actions of individual characters, I then try to sense the accumulated impact of all those actions to find the spine\u2014the one central action every character is trying to accomplish, in one way or another. Again, finding a spine is basic directing preparation, but it is particularly crucial in the spare and urgent world of Ten Thousand Things performances, where we all need to be able to easily get at the essence of a play. It's another important way to pare down, to get at what is most important and figure out what we might all share. As I worked on Midsummer, it became clear that every character, in one way or another, was trying to somehow pull someone else into their vision of love. That seemed like a common human action that most everyone in our audiences could relate to. We've all struggled with ways to try to make others love us back as much as we love them. It's what each of the lovers is trying to do, of course, as they try to pull each other into marriage. It's what each of the Rude Mechanicals wants to do as they try to make a moving play into which they can pull the lords and ladies in their audience. And it's what the goblins are trying to do as they apply their herbs and drugs and pull their unsuspecting partners into their forced visions.\n\nSome characters are more successful at this than others. I always like to put all the characters on a spectrum to help me understand a play's different points of view. Hermia and Lysander (in my version, the lesbian couple) seem the most successful, as they both willingly fall in love and marry each other. The goblin king Oberon is also \"successful,\" though he has to use magic drugs on Titania. Demetrius also has to be drugged to make him want to return to his original love, Helena. Both those couplings are more than a little unsettling. On the least successful side of the spectrum, to me, it wasn't at all clear that Hippolyta would stay willingly in Theseus's fantasy of a love nest if she weren't forced to.\n\nI decided to put the Rude Mechanicals at the more successful end of the spectrum. Though at first glance they don't seem to have been very successful at getting the nobles to enter the world of their play through their vision of the love story of Pyramus and Thisbe, I knew that in our version the audiences would have to be touched and moved. The lords and ladies certainly make a lot of sarcastic comments about the crudeness of the production. But the Rude Mechanicals themselves believe in their characters wholeheartedly, complete with a terrifying lion and a tragic death scene. They have poured so much love and care into their vision\u2014I knew their play would have to have an impact, just like Ten Thousand Things enlists our audiences' imaginations with a minimum of props, using a great deal of heart and spirit to break down the audience's resistance to seeing the world with na\u00efvet\u00e9. So, yes, in our version, the Rude Mechanicals, though their production might be awful, would have so much heart and spirit that they would be a success.\n\nWhen every character in the play is trying to pull someone into their own vision of love, whether it's Helena trying to enchant Demetrius or Quince trying to corral her unruly actors into the reality of her script or Titania trying to seduce Bottom, we get to see this one same situation from many points of view. Given the diversity of our audiences' experiences, I always want to look at the central issues of a play from as many angles and perspectives as I can, raising many questions, suggesting many possible answers, so the experience feels sharp and multifaceted. Shakespeare, with his wide swath of characters, is the very best at providing this for us all.\n\nCHARACTER QUESTIONS One of the other things I do before I get into a rehearsal room, again certainly not at all unusually, is to come up with as many questions as I can about each character. Knowing that many in my audiences will be experiencing theater for the very first time makes me feel comfortable asking very obvious questions. It is crucial not to take anything for granted\u2014because I know that my audiences won't. And such obvious questioning often leads to new discoveries. Holding in mind the breadth and diversity of my audiences causes me to work hard to ask questions about characters to ferret out their complexities. No character can be dismissed as unimportant or one-dimensional. Whether the character is rich or poor, \"good guy\" or criminal, onstage a lot or hardly at all, each must be seen as fully human, complicated, with both strengths and flaws. I don't try to answer the questions I raise about the characters, or if I do come up with an answer I just let it float, because I always want to give the actors a chance to answer the questions for themselves, always holding open the possibility that they will develop a much more interesting answer than I have.\n\nGavin Lawrence as Titania and Elise Langer as Bottom, A Midsummer Night's Dream, Avalon School, 2013.\n\nSET DESIGN I completed my pre-rehearsal imagining of the world of Midsummer by trying to figure out what the set would be, which basically means figuring out the fewest number of set pieces we would need. The pieces would probably be made out of aluminum, which has become our favorite raw material as it is incredibly light and easy to carry. I had been drawn to Titania's description in Act II of the world as a place at the mercy of seasons, seasons which had become chaotic and unruly with storms and floods, leaving the forest rotted and bare. It reminded me so much of the turbulence that global warming is beginning to unleash on our own world. I hadn't seen Midsummer sets that reflected her description (though there certainly may have been some). Those I'd seen were usually lush and magical rather than barren and spare. A barren and spare set, of course, felt right to me! As with all Ten Thousand Things sets, the audience would have to make it lush with their own imaginations. This, after all, was what the characters in the play were so often doing when they transformed a seemingly bleak and hopeless \"love landscape\" with their own surreal visions.\n\nWe needed to create, in our little fifteen-foot-square space, a \"forest\" of some sort, five or six pieces to make a maze in which characters could become confused and lost. I found a photo of some strange metal sculptures that seemed dripping and rotted, as if the metal had hardened when melting. They were unsettling, and mostly black, but glints of silver metal offered a hint of a strange beauty. I offered the photos to our designer Stephen Mohring as inspiration. We could blacken our light and portable aluminum and then polish bits to let silver shine through. A couple of the same \"tree trunk\" structures would also need to serve as pillars or columns of some sort when we were outside the royal palace. I also wanted a little height in the world, to reflect the play's sense of hierarchy as well as provide relief to the eye. It's funny how someone stepping up even just a few inches can create intense focus. Theseus could stand on a little aluminum step of some kind outside the palace; when moved to another spot onstage, this piece could also serve as a little hillock in the forest and a place for Titania's bower. As I thought about how we would move the hillock around, I realized that the actors could lift it over their heads and carry it across stage, so its circular bottom turned toward the audience could perhaps suggest the crescent moon, which offers only a sliver of light in the dark play. Stephen took in all the requirements and created beautiful and strange objects\u2014narrow, twisted columnar structures from winding strips of aluminum that allowed the audience to see through them, preventing the trunks from being visual obstacles. The hillock was just a skeleton structure, too, with a pounded aluminum crescent moon shape on its bottom.\n\nWhat was most interesting to me about the bleak world of Midsummer was that it was caused by the jealous quarreling of the goblin king and queen. Their wild, impulsive emotions were at the root of all the destruction and chaos. That's what the goblins must be: impulsive, emotional, irrational, and thus often, though not always, destructive. They were the lunatics, I thought, really. Theseus's famous speech dismissing nobles, lunatics, and lovers came to mind. The nobles were the lovers. And the Rude Mechanicals were the poets when they created and performed their play. All three groups dreamed up realities and tried to impose them on others. And all three experienced prolonged moments of bewilderment, moments in between worlds, when they realized that their realities had changed, not really knowing how. Each group really could be said to be lunatics, lovers, and poets, all three at once, and each actor in pajamas would play parts in all three worlds as well.\n\nI write about all this in such detail not because I think my ideas about Midsummer are astonishing or brilliant. There have been many deeply original productions of this story. I just wanted to take you through my particular preparation process as a director to help you see how simply imagining all my different audiences as I read, from the skeptical male inmates to the women who make up the majority of my audience (and of almost every theater audience) and who are tired of seeing such limited portrayals of themselves, causes me to reimagine casting. This imagining, in addition to wrestling with the necessary constraints of our spare productions, causes me to fairly effortlessly come up with an approach into the play. The need for the eight actors who have to play all the parts to have a base costume suggests, quite obviously, pajamas. The need for actors to remain standing so that the whole audience can see them means they will have to sleep standing up, which easily leads to sleepwalking between scenes. It's really not much fancier than that. A framework has been uncovered. Now on to see what happens in rehearsals.\n\nREHEARSALS Rehearsals are for me a sacred space. They are a place for me to practice being the best human being that I can possibly be. They provide a place for me to practice treating others with kindness, generosity, and honesty, creating as much room as possible for playfulness, hilarity, and delight. From the very first day I try to make rehearsals a place where we all feel safe to profess our ignorance, to say \"I don't know\" and \"I don't understand.\" It must be a place to take risks and make a fool of oneself. These are the conditions that make true discovery possible and that allow the radiance of the actor to begin to shine.\n\nMaking rehearsals into a radiant space also includes being respectful of people's time. I try to structure rehearsals so that people have to be there as little as possible. Just as it is an energy drag during performances to have actors sitting around offstage for long periods, the same holds true in rehearsals. Everyone leads such busy lives that if I can release people to attend to other matters when they aren't really needed, I find their attention and focus to be even stronger and clearer when they are present. I try to give out rehearsal schedules a week ahead of time so people can plan around it. We just rehearse five days a week, during the daytime, with weekends off, which not only makes life so much easier for actors with children and families but, I am convinced, also creates a level of relaxation that allows for more \"simmering on the back burner\" and a deeper creativity than is possible with just one day off a week. I love time off during rehearsals, so I've learned to become efficient at solving problems. Sometimes I even end rehearsal early, having accomplished everything I wanted for the day. This respect for time is returned with gratitude and renewed focus on the part of the actors, making the rehearsal room that much more pleasant and productive.\n\nFIRST DAY On the first day of rehearsal, we sit in a circle, without a table, open and available to each other without any kind of barrier, and begin by going through a little ritual. Everyone offers a story about a memorable Ten Thousand Things performance moment. From that very first day, all of us, both veterans and newcomers, are reminded of our priorities. For our work, the audience must be at the center. Not the playwright's intent, not the director's vision, not the actor's brilliant choices. For us, the necessity of connecting with the audience is always the determining factor in our decision making. And this audience is very different from the one taken for granted in other rehearsal spaces. We try to imagine how the story we want to tell will be of interest. On that first day, especially after hearing a few excruciating tales of past shows, it becomes very clear that if the audience isn't engaged, disaster lurks. That prospect is a huge motivator in helping everyone to focus in right away. Fear isn't the only motivator, of course. The exhilarating stories provide the sense that if we can connect this story to our audiences' lives, great reward and satisfaction await. The awareness that we only have bodies, live sounds, and the words of the play with which to tell the story also enters in. The stakes on the first day of rehearsal become very high, and not because anyone is worried about what critics might think.\n\nNext, I offer what I think I've figured out so far about the play, a sort of framework for us to focus in on as we work. For Midsummer, I offered much of what I've written about so far in this chapter as our point of departure. This part of the day's schedule gets shorter and shorter the older I get. I've become much more efficient at putting up a framework for our play. Then it's time to let my preparatory work go and see what the other talented people in the room might bring to the process, what surprising discoveries we all might make together. A lot of my ideas don't pan out at all, but, surprisingly, quite a few do manage to survive. I keep checking back in with my main ideas at different points in rehearsal to make sure they aren't getting buried or lost, to make sure the play still has a shape, though often a somewhat different shape than I'd envisioned when everything was just flat on paper and in my head.\n\nQUICK JUMP TO THE PHYSICAL The urgency of all this brings us quickly to our feet. We don't really do \"table work\" at Ten Thousand Things; we don't spend days sitting and talking about the play. We take a little time when first starting to work on a scene to ask a lot of questions. The imagined audiences inform our questioning, encouraging us to ask obvious questions and intensifying our imagining of the depths, complexities, and urgency of the characters' situations. Actors in the cast with past Ten Thousand Things experience are easily able to summon up an audience of female inmates or the crowd at Dorothy Day to help imagine a moment; they can easily go deeper when they realize they will be held accountable by a male inmate who has lived the very experience they are trying to create onstage. Our audiences are a vast resource in the rehearsal room as we begin to ask questions. We feel no pressure to answer any of them right away. I put my ignorance front and center, sprinkling many \"I don't knows\" into the question session, encouraging the actors to do the same.\n\nAfter about an hour or so of asking and floating questions about a scene, we get on our feet. We've found we can only really begin to understand our work when it begins to enter our bodies. Experiencing the reality of each situation through the body, we've found, just gets at the truth much more quickly than sitting around, talking and talking. The work of a Ten Thousand Things actor is very physical, and I want to start tapping into the energy and knowledge of each actor's body as soon as possible. As the actors begin to feel the scene out on their feet, I find myself getting up and moving around, too, trying to feel things in my body as well. Moving, I like to watch what happens to actors' bodies and how they position themselves in relation to one another. I rely very much on their physical instincts and impulses, often just adjusting here and there to help make the storytelling clearer.\n\nHIGH STAKES We worked to make the urgency of the first scene of Midsummer, when the lovers must make their cases before the Duke, very high. Our imagined audiences, living at life's extremes, helped. When the Duke pronounced that Hermia's penalty for disobeying her mother's wishes would be death, the actors were able to jump very quickly to the severity of this reality. Having Lysander and Hermia be a same-sex couple also helped to raise the stakes, so the whole situation of facing severe punishment simply for falling in love with the wrong person became more believable. We decided that homosexuality would not be an issue in this world, that Egeus's objections to Lysander marrying her daughter had nothing to do with the fact that they were both women\u2014she just preferred Demetrius. But the echoes and resonance with our own world, where same-sex couples still face dangers, were inescapable; the whole situation became more real and more heartbreaking. Gay marriage had just won a triumphant victory in the state of Minnesota, so the issue was very timely.\n\nIn that first scene, we also tried to take Helena, a character who is often trivialized and ridiculed, seriously. Helena has been betrayed by Demetrius, who has suddenly asked Hermia, her best friend, to marry him. We chose to highlight how Helena's deep friendship with Hermia helps her to overcome any jealous feelings she might have toward her friend. We thought it more interesting to watch her struggle with her feelings and finally push them aside, rather than easily and thoughtlessly venting her jealousy. This portrait of the possibilities for depth in women's friendship also proved refreshing.\n\nOnce we start to grasp the emotional depths of the situations in a scene, our audiences next demand we turn our attention to the storytelling. The experienced actors in the room know the painful consequences of a story not being clear. Brightly lit faces full of boredom, restlessness, and confusion do not make good acting partners.\n\nCREATING FOCUS The most important part of clear storytelling is actually about creating focus. I pay a great deal of attention to exactly what I want the audience to focus on every moment. We have very few resources with which to create focus: actors' bodies, the occasional prop, as well as strong actions that can be felt underneath the words. Sounds, of course, also help to create focus, and Peter and I, once a scene starts taking shape, begin to layer in the sounds that will cause audiences to focus their attention: on exits and entrances, of course, but also on words we really want them to hear, gestures we really want them to see, lines we really want to resonate. For Midsummer, with all its magic, Peter had a real workout. Sound is really the only way we have to suggest magic is happening on our little bare stage: a stroke on a gong, a crotales, or a cymbal, a run on the high wind chimes. I love to watch an audience to see how little it takes to make them believe that a spell has been cast, a dream has been entered, magic has occurred. Most of the time the entranced audience is completely unaware of Peter's acrobatic efforts just off to the side.\n\nEven actors' eyes can be used as spotlights to create focus. I'm serious. Sometimes during a particularly challenging line, I find I have nothing else at my disposal to help the audience understand what is going on than the eyes of the actor who is listening to the line. Having his eyes shift on a certain word can cause our very close audience to focus on what I want them to hear. Creating focus is all about causing people to pay attention\u2014and paying attention is the essence of feeling alive.\n\nOne of the most important discoveries we've made about storytelling is that just because a character says something doesn't mean anyone in the audience has heard it. A playwright may weave a telling detail into a character's speech, and the character may say it, but that still doesn't mean it's been heard. Sound, emphasis, speed, and gestures are the tools we have to make sure that audiences hear the important things we want them to hear.\n\nAnother important part of storytelling is momentum and pacing. Nothing extraordinary about this either, though I think the extreme life situations of our audiences make us even more aware of the importance of moving things along. Experienced Ten Thousand Things actors know that, in the high-stakes world we inhabit, there is little room for emotional indulgence. We have felt too many audiences become restless and bored when an actor takes the time to pause and cry for even ten seconds. No one really wants to watch someone cry for very long. There are too many more important things to be done.\n\nAnd I must say, using all these elements, I feel we were able to make the storytelling in Midsummer crystal clear, particularly in the often very confusing first scene. We replaced pronouns with names, actors gestured toward the person being spoken about. Actors found the right times to move toward or away from each other, punctuated with sharp sound, all bringing focus to who and what was being discussed. The high stakes added to the focus, of course, and demanded that everything move right along, giving no one time to indulge. I honestly felt we were able to make that first scene one of the clearest openings of a Shakespeare play we'd ever done. In performance, the rapt attention of almost every audience during this scene, bodies leaning forward, faces deeply concerned, really confirmed that the focus was working.\n\nSTORYTELLING BETWEEN SCENES As the four and a half weeks of rehearsal passed, I kept checking back to the original common human experiences I hoped we would illuminate through the play. And ironically, given the title of this chapter, I must confess that during rehearsals I got a little too caught up in one of my ideas, even though it had come from necessity. My desire to highlight the experience of how we \"sleepwalk through enormous changes in our lives\" grew a little cumbersome. I had originally imagined that for every scene change we would watch all the pajamaed actors drop the costume pieces from one world and put on new ones in a sleeplike trance, standing around outside the circle as they changed. We spent quite a bit of time trying to make this happen, hammering away with sounds to punctuate, but it just wasn't working. It all felt tedious and slow. I finally realized, by way of an actor's helpful comment, that it was pretty much impossible to create focus around the entire outside of the circle since the audience can only see part of it at any one time. People just naturally look to the center of the circle instead. So I dropped that idea. Instead, in between scenes, we told one little sleepwalking story inside the playing area while the other actors changed their costumes quickly outside the circle, unnoticed. With no blackouts, scene changes become one of the most important resources we have to make the storytelling clear.\n\nWe didn't have to toss out all the work we'd done. The play still began with actors sleepwalking into the circular playing area\u2014with dreamy chimes and sounds\u2014in their eclectic dark blue and gray pajamas. Before the show, we placed the red and purple satin robes of the nobles and lovers on the floor of the circle. One actor stopped suddenly (with a drum thump) when he was about to step on a crown. The actor looked down, saw it, and slowly bent to pick it up, feeling the transformation that occurred when he put it on his head. He became Theseus. The other actors did the same with the robes they stopped in front of, bending to slip them on, transforming into the characters of the nobility. Couples reached out for one another and drifted offstage in pairs; the older actress who had become Egeus, Hermia's mother, followed Hermia and her female lover offstage with concern. The actress playing Hippolyta started to head offstage alone, but Theseus, still in slow motion, took hold of her arm and pulled her back across the stage through the palace gates, giving a clear sense that she was an unwilling partner. One last actress entered, late, in her pajamas, carrying the silver moon\/hillock across the stage, then placing it down in between the aluminum pillars to become the steps into the palace. She turned to discover that only one costume was left on the floor: a servant's jacket and hat. Slowly and with great disappointment, she put it on, becoming Philostrate, heading up the steps into the palace with a sigh.\n\nThis storytelling about sleepwalking through big changes continued in the transitions. We focused on just one or two actors at a time instead of all of them. At the end of the first scene, we saw the actor who had been stuck with the role of Philostrate the servant step back out on the palace steps to look at the moon, filled with a yearning to play a bigger part. She spotted a roughly woven brown vest and hat in one of the aisles and crossed to change costumes to become Bottom. As Bottom, she could play all the good parts! In the next transition, as all the Rude Mechanicals scurried away, Hippolyta stepped out of the palace to gaze at the moon. She too was filled with a yearning to play a part other than that of a captive bride. Puck scurried across the stage and dropped Oberon's black robes in the center, and the actress, surrounded by dream chimes, changed costumes and roles with pleasure and relief. The empty yard outside the palace transformed into the strange and beleaguered forest as goblins scampered in with the trees and hillocks, all to eerie night sounds.\n\nThus we proceeded through the scene changes of the play, sometimes telling the story of an actor sleepwalking into a new reality and sometimes just telling the actual story of the play more clearly. For example, we used a doll puppet as the changeling child, to help to make that difficult story clearer. In a later transition, we were able to watch Oberon steal the changeling from Titania, unnoticed, as she and Bottom passed by completely enamored with one another.\n\nI think all our work on the transitions paid off, not only in clarifying the story but in allowing the audience to consider \"sleepwalking through life.\" One of my favorite comments came after a performance at a detox center, when a woman came up to us and said, \"I liked how the actors played lots of parts. I've played lots of parts in my life, too, most of them really shitty ones.\" Sometimes we heard audience members making comments like, \"Uh-oh, that one's going to fall asleep again!\" It seemed to attune everyone more clearly to the times when characters in the actual story, like Bottom waking up from her dream, for example, or Titania from hers, were making transitions between realities of their own.\n\nSun Mee Chomet as Oberon with tree trunk, A Midsummer Night's Dream, Avalon School, 2013.\n\nThere was one transition I do wish I could have figured out how to make clearer. In Act IV, when our King Oberon stepped out of the goblin world and went back to being the captive bride Hippolyta while Queen Titania returned to being Theseus, I wanted to make it clear how each had been affected by their experiences in the goblin world as a member of the opposite sex. Each had suddenly experienced having a lot more or a lot less power. Our double-casting required a lightning-fast change because one second after Oberon and Titania leave the slumbering lovers in the forest, Theseus and Hippolyta are supposed to enter the same spot. We would have to watch the costume change, so I decided to have it happen as they sleepwalked around the outside of the circle, dropping their goblin clothes and picking up their royal ones as they exited the forest on one side and reentered from the other. It never worked. The audience never paid attention to the actors walking outside the circle, and the impact of this important transition was almost always lost. I can now see it would have been so much better just to have them change their costumes in the center of the stage, right in the middle of the forest full of sleeping lovers. We could have watched their wonder at their transformation so clearly then. But this became just another of the many mistakes I've made over all these years, discovering what would work better from watching audiences and shows over and over again.\n\nAD LIBS When I cast a Ten Thousand Things show, in addition to finding the very best and most diverse actors, I place enormous weight on assembling a group of people who are also kind, generous, and playful. The tour can be unpredictable and challenging, and the actors need to be able to approach each performance with complete trust of one another. Even one selfish or unaware actor can make the tour almost unbearably hard. With Midsummer, I had certainly assembled such a kind and generous cast\u2014and one that was even more playful than usual.\n\nOne of the most unexpected discoveries of rehearsals was the surprising ad libs and asides the actors came up with. This hadn't ever really happened in our Shakespeare work before, but something about the strangeness of this play and its story, along with the camaraderie of the cast, seemed to inspire it. Several of the actors had worked with Ten Thousand Things many times before, and being very familiar with our audiences, in rehearsal they just naturally started making little comments to the empty seats. Karen Wiese-Thompson, a large, middle-aged woman playing Puck, modeled the goblin on the behavior of her son as a teenager and began to mutter lots of comments under her breath, questioning Oberon's commands with, \"Seriously?\" \"Are you kidding me?\" Sun Mee Chomet, new to Ten Thousand Things and playing both Hippolyta and Oberon, was eager to connect with our audiences and, inspired by Karen, began to comment occasionally to the audience as well. I recall one moment in particular when Sun Mee, who had been struggling a bit with the goblin King role, decided in the middle of one of Oberon's tirades to pick up one of the aluminum tree trunks and start to thrust with it as a phallus. This action seemed to release a huge energy, both in herself and in the rest of the cast, a sexual energy to be sure, but also an enormously playful one that never left rehearsals or performances thereafter.\n\nElise Langer, a younger, petite woman playing Bottom, brought all her clowning skills front and center, inspiring all the other Rude Mechanicals to make lively little comments as they worked on their play about Pyramus and Thisbe. Kurt Kwan, who played Demetrius, Mustardseed, and Flute, the lone male of our Rude Mechanicals who finds himself cast as the heroine Thisbe, was an astonishing source of inventiveness throughout rehearsals, delighting us every day with something new. Early on, he came up to me privately and said, \"Flute doesn't want to play a woman, but once he decides to accept the part, he decides he's going to be the best damn woman he can be.\" Kurt had Flute draw on whatever sparse ideas of women he had, largely from pop culture. Flute began to incorporate the movements of Beyonc\u00e9 into his dialogue, as well as twerking at various key moments.\n\nI was a little concerned about the number of ad libs and asides and sometimes reined them in a bit, not wanting them to overwhelm the play. But a playful spirit had clearly been set loose, and after our first few performances, where our audiences of female inmates were almost riotous with laughter, I really could see how the ad libs, rather than detracting, opened up the world of the play still wider and allowed the audience to feel connected to the characters and enter in more completely. The ad libs wouldn't have worked if the audience hadn't been following the story clearly. A few of the critics who attended our paying shows seemed to be a bit taken aback by the unorthodoxy of it (we often have great fun watching the critics' faces in our fully lit performances, something they are not used to). We could see a few of them resist joining in the merriment, and one or two voiced their disapproval in reviews. I don't think I would have let the actors ad lib so much with most other Shakespeare plays. But Midsummer was a strange beast, and over and over again our experience was that the little off-hand remarks only served to welcome and pull the audience, traditional and nontraditional alike, even more deeply into this very strange world. When an inmate came up to us afterward and proclaimed, \"That was dope as hell!\" I couldn't help but think that Shakespeare, whose plays were full of asides, and his actors, quite used to the open comments of the groundlings, would have approved.\n\nNO TECH Though this chapter was meant to focus on preparation and rehearsals, it's hard to write about a production without including at least some of the responses of our audiences. Seeing Midsummer afresh through the eyes of our delighted first-time audiences helped us all to discover what a very weird and strange story Shakespeare decided to tell. But I'd like to end the chapter in much the same place we end all our rehearsals. After three and a half weeks of working on scenes and piecing them together, we spend another week running the show five times, once a day. Although I usually refer to this as our \"birthing week,\" where concentration has to be particularly focused and intense, what is especially delightful about these final run-throughs is that we have no need for technical rehearsals. There is no interruption to the flow of the acting discoveries by taking long days to work on lighting cues. Our lighting\u2014the sound\u2014is already in place (though sometimes we take part of a rehearsal to go through all the sound cues). Because we have no tech, we are able to maintain our focus on deepening the acting and clarifying the storytelling.\n\nAfter our final run-through, we load up the van. We all look at each other. We are teetering on a brink, with our play halfway completed. People in any production feel this way before opening night, of course. But at Ten Thousand Things, this sense of teetering on the brink of the unknown is perhaps even more intense. Our nontraditional audiences generally know more about the world of the play than we do. It's why humility is a keystone of the rehearsal process. We stand on the precipice of possibly the most humbling moment of all. We've tried the best we can to imagine the audience. Now we set off to meet them face to face and see how well we've done.\n\nSarah Agnew in Carousel, Skyline Housing, 2004.\n\n# 9\n\n# NO PLACE TO HIDE\n\n## Performances\n\nON THE FIRST DAY OF REHEARSAL, we begin our focus on the audience with veteran performers sharing tales of favorite exhilarating and excruciating performance moments. We have an endless supply:\n\nTALES OF PAIN... There are always plenty of excruciating stories. In fact, many actors describe these quite painful experiences with a kind of glee, celebrating them as survivors celebrate battle scars, relishing the looks of consternation that often appear on the faces of the novices. And every such story still brings up a tiny clench in my muscles, making me want to hunch over and avert my eyes, just as I most likely did at the time of the incident. The time, for example, with Measure for Measure in a women's prison, when we found ourselves with eight minutes left of the show but only four minutes away from \"count\": the exact moment when, no matter what, the bell would ring, the show would have to stop, and the audience would immediately be herded back to their cells by the guards. Miraculously, the actors managed to barrel through to the end with seconds to spare. Or showing up to perform Richard III in a men's shelter, we discovered that the empty and peaceful large room we had previewed during the day became a hotbed of personal territorial battles at night, with each man fiercely marking out his regular squares of linoleum for his sleeping bag and few belongings. Many of the men were very unhappy, of course, that we were occupying their spaces with our stage area and our props. We learned there were no chairs for the audience to sit on and there was no other separate room for men who didn't want to watch the show to retreat to. Many had food to heat in the corner microwave. Shouts of \"Hey, that's my mac and cheese!\" punctuated the show. Or the time when we took Red Noses to a juvenile correctional facility and, facing a restless, overcrowded room of kids just moments before the time the show absolutely had to begin so it could end in time to avoid a mass exodus at the very loud lunch bell, we learned that the van with our set, props, and musical instruments was stuck in gridlock traffic on a highway several miles away. We had no choice but to plunge in anyway, improvising props from car keys and pencils (slowly sneaking set pieces onto the stage once the van finally arrived). Or most recently on our Music Man tour, the cast was heartbroken when the production, which had wildly delighted audiences of both male and female inmates, met with a cold, almost hostile reception at the final women's prison on the tour. The performance was so painful to plow through, having to constantly look at the rows of unfriendly faces. Near the end, we were able to figure out that the presence of a male videographer we had hired to record the show had made the women very suspicious, defensive, and even angry. That won't happen again. The excruciating performances we've had on our tours seem truly endless, and the stories could easily fill a volume of their own. But despite the pain, these experiences have also been blessings, each one teaching us something important about what works and what to avoid.\n\nEXTERNAL CONDITIONS Many, if not most, of our excruciating performances are caused by actual performance conditions at the sites, which are often beyond our control. In the early days of the company, when I was doing everything myself\u2014grant writing, budgeting, tour planning, as well as directing and raising a toddler\u2014I would hear of a promising place for a performance and make arrangements by phone, trusting that everything would work out. Then we showed up at a women's shelter to discover we were expected to perform The Emperor of the Moon in the kitchen, a space no more than eight feet wide lined with enormous refrigerators and stoves. That was one of the few shows we actually had to cancel.\n\nSCOUTING THE SITE Over the years, we have learned ways to try to minimize the chance of a disaster. First, we scout each site. We have to scope it out to make sure there's a room that is large enough for our fifteen-by-fifteen-foot square or circular playing area, the rings of chairs, and an aisle behind the last row for the actors to travel around. Large conference rooms work wonderfully well; gyms are challenging acoustically, though they are often the only available option. In cafeterias, we have to be sure pillars won't get in the way too much. But if we want to reach a particular audience badly enough, we can be creative in how we adapt.\n\nBUILT-IN AUDIENCES Next, we have to make sure there will actually be an audience. This means we have to be sure that the audience will already be there. We very quickly learned that putting up flyers announcing a performance, with the expectation that people will make a special trip to the site at a special time, never works. Never. People who haven't seen theater before are very reluctant to go out of their way to do so. Even crossing a parking lot from the residential building of a homeless shelter to the cafeteria in a separate building is too big a distance to travel. We have to \"capture\" our audience when they are already at the space, at a time when there isn't too much else to do.\n\nVISIBLE EMPTY SEATS And we want to have an audience because, quite simply, with all the lights on, we can see all the empty seats, and it hurts. Audience members are our acting partners, and unfilled chairs mean our acting partners haven't shown up. We can't mute their absence as actors performing to a darkened house can. It quickly becomes very clear how profoundly actors need the pushback of an audience's energies; the glaringly obvious empty seats feel like black holes. The absence of the audience is a primal experience.\n\nLuckily, the fluid \"architectural design\" of our theater space allows us to alleviate some of the pain of empty seats. We can simply remove all the empty chairs. As long as the front row is filled (usually about twenty-four to twenty-eight people fit around the playing space, six or seven to a side), we can remove chairs in the second or third rows. The front row is crucial. If most people decide to sit in the second or third row but the front is largely empty, the energy drain is very noticeable. We have become masters at harnessing the audience's energy through seating arrangements, with lots of encouragement as they enter the space: \"Front row seats are the best! Try them: you can see everything from there!\" If we sense that despite our best efforts a particular audience will want to \"hide\" in the second or third row\u2014or that more than a few dozen people may not show up, we only put out a front row, forcing everyone to sit up front, adding seats only as additional bodies enter the room. No matter what, though, we do need enough bodies to fill up that first row, and there have been times when even two dozen people don't show.\n\nWomen's correctional facility, Ragtime, 2005.\n\nWe once set up in the lobby of a low-income senior center for a 7 PM show, only to learn that no one was coming because everyone had already gone to bed. We planned for an afternoon show at a men's shelter and not one homeless person showed up. We learned that, although early Sunday afternoon had been identified as the time most likely to draw the biggest audience, the staff had forgotten that the NCAA playoffs were on TV; there was just no way any of the men were crossing the street to the church gym to see Cymbeline. Once at a homeless shelter, we were about three-quarters of the way through the play and a horde of people started to stream into the room for their dinner, making it almost impossible for the actors to shout above the din to finish the play. We quickly learned we needed to schedule shows there so that they were done by 4 PM. Clearly, knowledge of the intricacies of just how each site actually works, including the exact time of day and the exact day of the week when a potential audience will actually be present, is essential to properly filling the chairs.\n\nSTAFF AS ALLIES This is where the next crucial component of a successful performance comes into play\u2014one enthusiastic staff member. When we're scouting, checking for that large-enough room, we always have our radar out for that one person on staff who \"gets it,\" who truly understands how rewarding the experience could be for those with whom they work. Someone who is willing to put in some \"coaxing time,\" encouraging people on an individual basis: \"Come on, you should try it. I promise you it won't be boring. These guys are great, such good actors. You have to give it a chance!\" Talking it up every day, often for a period of several weeks before the show, takes a lot of persistence, patience, and energy from staff who are already often stressed from the demands of their thinly funded low-income centers.\n\nLuverne Seifert, Man of La Mancha, women's correctional facility, 2011.\n\nIt is vital to always remember how strong the resistance to seeing theater is among most people who've never experienced it. This took me a good while to truly understand. In the early days, before I'd really realized what a vital ally the right staff person could be, I tried to drum up enthusiasm among resistant potential audience members myself, coaxing and pleading, marching up and down empty hallways to knock on doors before the show began. Once at a soup kitchen we were confident that we had found the right time of day to perform, just after noon, when the cafeteria was still full. But when it was announced with a microphone that \"There will be a play after lunch,\" people looked up in consternation, gobbled down their meals, and fled. (It turned out that, because it was a Catholic soup kitchen, people were afraid it would be a bad reenactment of some kind of religious tract.) During our very first women's prison show, a weary voice announced over the public address system, \"You can come see a play in the cafeteria now, or you can stay in your cell.\" The \"enthusiasm\" of this announcement managed to motivate about fifteen women to straggle down the halls. (Those fifteen women by the way, turned out to be one of our all-time favorite audiences, with their unrestrained responses to Forn\u00e9s's Mud.)\n\nWe relearn the value of that staff advocate whenever one is absent. When the beloved volunteer coordinator at a county prison system was sick, we showed up to do Carousel at the men's facility, which at the time allowed no chairs in the gym, only mats on the floor. Only ten men showed up for the performance. We later learned that an unsympathetic guard had arbitrarily decided his entire cell block would be unable to attend. Our volunteer coordinator would ordinarily have known how to handle this situation and work around the guard. We carried on, more people in the cast than in the audience, the men lying on the mats like it was preschool nap time, though usually awake and thoroughly enjoying themselves. We enjoyed ourselves, too, once we got over the initial surprise and relaxed. Somehow the bodies stretched out on the mats filled the empty space\u2014and it was weirdly fun to have to perform for such a low eye level. Another time a beloved contact at a state maximum-security facility died suddenly, and we learned that we would no longer be allowed into the prison. Only then did we fully realize how crucial this one man had been in navigating the difficult prison bureaucracy. It took us several years to find another person on staff who \"got it\" and would be able to work the system to allow us back in.\n\nPeople often ask us how we \"get into prison.\" In Minnesota it hasn't been that hard, especially at the county level. When I first approach program officers or volunteer coordinators with the idea of bringing in a play, many are puzzled but willing to give it a try, especially because it is free. Usually, once they see the enthusiasm of inmates for the work, they become allies. Now we even have guards at some prisons who schedule their shifts so they will be able to stand and watch the show while they're \"on duty.\"\n\nMIXED AUDIENCES We've also found another way to hedge against the possibility of empty seats. In case a nontraditional audience just doesn't show up despite our and the staff members' best efforts, we invite members of the regular theatergoing public to come to these community sites as well. Many of our paying audience members are very curious to see what happens at a nontraditional site and willingly travel to unfamiliar locations. These shows are free, so it's easy to guarantee that at least several dozen seats at every location will be filled. It's discouraging, of course, when very few from the target low-income community show up and we find ourselves just performing for people who could have afforded to buy a ticket, but at least we've made sure that front row won't be empty. Indeed, at one point the danger became that regular theatergoers might overwhelm our community sites. We were doing Ragtime at a battered women's shelter, and over ninety regular patrons showed up. When the dozen or so residents of the shelter came down on the elevator to the performance room, they were startled to see so many strange faces and they quickly pushed the buttons to close the elevator doors and go back up to their rooms. Another kind of excruciating performance situation was born, when the people we wanted to perform for didn't feel welcome or comfortable in their own space. So we began to limit the number of free seats we made available to the general public, taking reservations beforehand. Those seats are now all snapped up right away.\n\nWith any venue, it takes time to figure out the details. Often, our first performance at a site doesn't work because, despite our best efforts, we don't hit on the right time or the staff member wasn't able to work up the enthusiasm she'd hoped. Very few spaces are perfect, so it's always a balancing act, taking into account our desire to reach a potentially very interesting audience and balancing it with the amount of uncontrollable chaos that seems to be inherent in the space.\n\nUNTAMABLE CHAOS Sometimes the chaos is just too much. Indeed, once we start performing, I often find myself shifting from being concerned about the audience to being concerned about the performers. I feel fiercely protective of them, wanting to do whatever I can to make the conditions as friendly toward their efforts as possible. And we certainly have encountered situations that we all agree are just too challenging, times when we decide we need to sever the relationship and give up. The actors often surprise me by how much they are willing to endure. We once did a performance of Stones in His Pocket at a homeless shelter in the winter. Because of the cold weather, the actors in this small two-man show found themselves acting in a huge room in the midst of hundreds of people who were there only because of the temperature outside, not because they wanted to see the play. I could barely watch. Whenever I did manage to raise my head and peek out, I could at least spot a few people sitting in the inner rings who seemed quite delighted and caught up in the story. I approached the actors after the show full of apologies, but one of them, the always astonishing Steve Epp, interrupted me, almost panting with pride, exclaiming, \"It was like performing in the Theater Olympics! The amount of concentration and focus we had to give each other, I loved it!\" Nevertheless, we will not perform at that shelter in the winter again, and we will only perform shows with large casts, who can counterbalance the energies of the always somewhat chaotic room.\n\nJim Lichtscheidl, At Your Service, Hubbs Center for Lifelong Learning, 2005.\n\nWe also learned that doing two shows in one day is too much. We were performing In a Garden, two mini-operettas by Gertrude Stein and one by local playwright Kevin Kling. The pieces blended together to create a gleeful celebration of childlike playfulness and mischief, something I had noticed all of our audiences hunger for at some level. I had cast three extremely jubilant, inventive actors who somehow made it all work. But one afternoon the actors performed for an audience of over a hundred men at a maximum-security prison. Then, just a few hours later, they found themselves performing at a women's shelter for an audience of toddlers, dropped off after dinner by their mothers who had thought it was a children's play. Maximum-security prisoners and toddlers in one day. The inmates, of course, took quite a bit of coaxing to get to any kind of playful, silly place, while the toddlers became frightened at some of the adult matters in the operetta, upsetting their mothers as well. It was all much too much. The three actors now proudly tell this battle tale with great hilarity, but this will never be a company that asks its actors to do eight shows a week.\n\nNOT EVERY SHOW FOR EVERY AUDIENCE There are also excruciating shows caused not just by challenging performance conditions but by a mismatch of material and audience. We know, of course, despite our best efforts to find \"big enough fairy tales,\" that certain stories just aren't going to interest teen audiences and others probably won't interest senior citizens. We've decided that Shakespeare is just too much for English language learners (though we've had people come up to us after such performances and tell us that although they couldn't understand much, they could hear the beauty of the poetry). We now have a roster of over fifty different possible venues in the Twin Cities, five or six of which we return to with almost every show, like the county prisons or a few large homeless shelters, because their populations are so transient, it's largely a new crowd each time. From the rest of the roster, we try to choose other groups that might respond especially strongly to each particular play, and we now add two or three shows in a rural community to each tour as well.\n\nBut sometimes there is still a big misfire. In our earliest years, we performed Mar\u00eda Irene Forn\u00e9s's Mud at an adult education center. In more than one scene, the character Lloyd talks about how he likes to fuck pigs. The adult audience clearly relished the language, along with the respect it conveyed to them as being adults who could, of course, handle such material. But the teachers were livid. Many came up afterward to tell us how upset the language had made them; one teacher hissed, \"What's the name of your company again? Ten Thousand Swear Words?\" Since then I've had ample time to consider that we are intruders\u2014we are always performing in someone else's space, and it is important to be respectful of this. Though we will never censor any material, we now are mindful of how some institutions, especially those that work with youth, are trying to encourage respectful language in their spaces. So, before agreeing to a performance, we warn staff that the play contains challenging language and allow them to make their own decision about whether they want us to come.\n\nVOLUNTARY AUDIENCES It is also critical to note that we only perform for voluntary audiences. Even in prisons, we are clear that we do not want to perform for people who have been compelled to attend. Usually there is an advance sign-up sheet for inmates who are interested, and sometimes attending a play becomes a kind of privilege. Once or twice, staff have either forgotten or ignored these requirements, and the result is disastrously painful for everyone. The audience is suspicious and hostile, but the actors have no choice but to perform for them. For this reason, we don't do \"school shows,\" where students are required by their teachers to come, although sometimes we will perform at a school during after hours. We usually only perform for one or two youth groups per tour, however. The work of Ten Thousand Things is primarily for adults, as the plays we regularly choose to do are based on intensely adult experiences of life. And we've learned that already self-conscious teens often don't respond well to the fully lit intimacy of our performances.\n\nTHE INTRODUCTION A little advance warning right before the play begins can go a long way in easing the bumps along the road. One of the most crucial components of any performance, be it for a nontraditional or a paying audience, has become \"The Introduction.\" It's usually just a few sentences. In my love of getting to the essence, I put a great deal of care into finding the fewest words needed to give any audience member the right context to have it all make sense. Most of our nontraditional audiences show up at a performance without having any idea of what the play is about, unlike paying audiences who have usually had a chance to read about a show in the paper and choose among shows to find the one that most appeals. Program notes don't work. Many in our audiences don't know how to read, and those who can often don't know that information about time and setting can be found in the program. Besides such basics, I try to provide one hook or one question that might interest people in what is to come. I usually don't know what these sentences are until a day or two before we open, and it usually takes one or two performances to refine it perfectly. For The Seven, Will Power's retelling of Aeschylus's The Seven Against Thebes, for example, I simply said, \"This is the story of two brothers who take over a kingdom, vowing to share it and rule it peacefully together, and all the things that get in the way of their doing this.\" I added that it was a story that was more than two thousand years old, originally from ancient Greece, and was now being retold by a living writer in hip-hop style. Just those two sentences were enough to orient the audience and spark their interest. For Measure for Measure, I ended up just giving the brilliant summary presented to me by a homeless man at intermission during a shelter show: \"This is really a lot like that TV show Undercover Boss!\" I added that in the play, a duke decides to put one of her deputies in charge and go into hiding, disguising herself so she can spy on her employees to see what happens in her absence. Those few sentences were all it took.\n\nThe Introduction can also be very effective at calming and centering an audience, allaying their suspicions and fears. With Shakespeare, for example, I always note that it takes any audience, no matter how many times they've seen Shakespeare before, five minutes or so to get used to how the characters talk, but I promise that if people hang on, everything will soon become very clear. The relief is palpable; our veteran audiences breathe more easily as well.\n\nThe Introduction can also help to shape an audience's energies\u2014again, with just a sentence or two\u2014even displacing the misogyny we so often wrestle with in men's prisons, especially in plays with sexual content. One of our most excruciating moments came during a performance of Othello at a men's county jail, where the men started cheering as Othello strangled Desdemona. Chances are that's what the groundlings did, too, during Shakespeare's time, but it was profoundly upsetting to all of us, especially the actress playing Desdemona. Usually, the misogyny isn't quite as overt, but when it's there, it always puts us on edge.\n\nOver time, we've come to see such misogyny as honesty about feelings that many men have but which some have just learned to hide better than others. Although we much prefer the men's prison shows where it isn't present, its open expression gives us a chance for a conversation about it through our performance. Just recently, with our production of A Streetcar Named Desire, I think I finally hit upon the right words in the Introduction to rein it in. Streetcar is replete with moments of a man being cruel and abusive to his wife and his sister-in-law. I started out noting that we did three plays a year at the facility and thanked the men for always being a good audience. (I've learned not to ask how many have seen a show before, as many are reluctant to admit how long they've been in prison.) Then I mentioned that I was sure they knew how much concentration and focus acting takes, especially with the audience so close up, and added that there were several scenes in this play that were particularly challenging for the two actresses. Not only did this perk the men's interest, but I could feel their energy shift from being dismissive of the female characters to being protective of them. They were attentive and respectful throughout the entire play. We were able to have a great discussion with them afterward, too, about Stanley's slapping of Stella, getting their thoughts on why she would want to go back to him, as well as their judgments of whether Blanche should have been sent to a mental institution (many of them saw her as quite strong and resilient). We'll see if this Introduction solution holds up as time goes on.\n\nThe Introduction is also a supremely useful tool for preparing experienced, paying audiences for what they are about to see. As discussed, these audiences bring their own expectations, which can serve as barriers. One thing we finally figured out, after several years of finding paying audiences to be so much stiffer than nontraditional ones, is that they expect to sit in the dark. It makes them really uncomfortable to be so exposed, seated in a circle with all the lights on. Ever since, I always mention that we're going to leave the lights on and point out that not only will the actors be able to see them (noting how much the actors enjoy this, compared to playing to a dark house) but that everyone else will be able to see them and that this awareness may make them uncomfortable for a bit but that they will quickly forget about it and come to enjoy watching everyone else as part of the show.\n\nMore important, I use these introductions as a way to help paying audiences to understand our work, listing all the different places we have performed and describing some of the responses of our nontraditional audiences. This not only calms and centers the paying audience, but you can see, during the performance, that they start to imagine the play through the eyes of others, others who are from quite different circumstances. You can almost hear them start to ask, \"What would a homeless woman think of that scene? How would an inmate have reacted to that line?\" Seeing the world through the eyes of others is the principle act of theater. And this act can be multiplied geometrically with our paying audiences, all because of a few simple sentences in the Introduction.\n\nDESPITE OUR BEST EFFORTS Excruciating performances still happen. On every tour. Less frequently than in the early years, of course, but frequently enough so that we have all learned to embrace them as a necessary part of the work. It helps us to realize that, often, what has felt excruciating to us hasn't felt that way to many in the audience. Many people get sucked into the world of the play, no matter what the surroundings. And I always remind actors that ordinary dark houses shield them from so much. Were you to shine a spotlight on any audience at any big regional theater at any moment, you would always see restless bodies, hanging heads, yawning, tapping toes, bored looks.\n\nWith all the lights on, we must remember that outside appearances are so often deceptive. Take the audience of inmates who seemed so restless during Antigone but who leapt up into a three-minute standing ovation after the final line. Or our very first time at a mental health facility where we were to perform The Most Happy Fella. We were full of nerves beforehand, imagining that people might be erratic and irrational, jumping up to shout at unpredictable moments. In fact, it was like performing in the midst of a circle of absolutely motionless statues seated in wheelchairs; the audience was so highly medicated that their facial muscles didn't twitch. Our acting partners felt absent, and we experienced it as a very hard show. And yet afterward, as we were packing up, staff members rushed up to us with beaming faces, exclaiming that they had never seen their clients so engaged. They were clearly able to detect muscle movements that we were not\u2014but their assurances were so surprised and heartfelt that we had to believe them. We probably won't do Shakespeare for such audiences, but now that we know what to prepare ourselves for, upcoming performances won't be so hard.\n\nIn every excruciating show, even those where most of the audience really is disengaged for whatever reason, even in those performances, no matter what, there are always at least a few people whom we touch profoundly. Actors can always look around the circle and find at least a few open and friendly faces to connect with for support. There are always a few who rush up to us after the show, exclaiming their surprise over how much they loved it, wondering how they might be able to take their families next time, almost always at least one comment like, \"This is the best TV I ever saw,\" that can make us feel that even the excruciating show was somehow worth it.\n\nStill, no matter what we learn later about how much seemingly disengaged audiences have actually loved the performance, in the moment such performances are really hard on the actors and on me. So we have invented the Cocktail Coupon. During each tour, if a very difficult show makes it feel necessary, Ten Thousand Things will take the whole cast out for drinks. At least there we can laugh and commiserate, and soon the whole painful ordeal retreats into our collective pantheon of good stories to share.\n\nTHE ONES THAT MAKE IT ALL WORTHWHILE For every excruciating performance there is an exhilarating one, the shows where the connection with the audience is profound. We usually have two or three of these on every tour as well, and, again, after so many years of touring, the stories could fill their own volume. But it is safe to say we will always remember the performance of Ragtime at a women's facility where the inmates, so many of them separated from their children, so completely identified with the character Sarah, the African American maid who ends up bearing the child of the jazz musician Coalhouse Walker. The women spoke out words of comfort when Sarah, abandoned by Coalhouse and with no land of her own, is caught burying her baby in the yard of a wealthy white woman. They sobbed and cried out her name when Sarah is shot by the police while trying to defend Coalhouse from his own unjust arrest, and they literally screamed with joy when the two reunite in the play's reprise of the musical number \"Wheels of a Dream.\" It was a group catharsis the likes of which none of us had ever witnessed before. We will always remember the moment when, in King Lear, the usually humming people at the edge of the large homeless shelter room fell dead silent as Lear, in rags on the open heath, began to rant and rave at the cruelty of the passing storm. We will always remember the performance of the seventeenth-century commedia piece by Carlo Goldoni, Il Campiello, in a men's prison where almost every hardened face burst open into glee and then wonder, where men lined up afterward to shake the actors' hands, exclaiming that it was \"the best damn shit they'd ever seen.\"\n\nThese shows are a huge part of the reason why it's so easy, despite all the challenge and difficulty and often pain of this work, to come back and do it again. In fact, it's hard to go back to \"real theater\" once you've experienced the intensity of the deep, unexpected connections between the lives of the audience and the life of the play. It is as \"real\" as theater gets.\n\nRandy Reyes, Il Campiello, men's correctional facility, 2011.\n\n# 10\n\n# THE POWER OF UNEXPECTED CONNECTION\u2014POLITICAL THEATER\n\n## Going into the World\n\nI AM THE FIRST TO ADMIT that I bristle at the labels \"political\" or \"social justice\" or \"educational\" when applied to theater. Because of my extreme sensitivity to hints of condescension, I will openly defy anyone who asks that Ten Thousand Things demonstrate how we \"change people's lives.\" I think it is extremely difficult to change anyone, and I'm not sure I've ever seen any theater performance change someone in any significant or measureable way. Part of my ferocity comes from knowing that larger theaters with primarily upper-middle-class audiences are generally not asked to quantify how they change lives. They are not asked to provide data on how many people, as the result of watching a play, stopped drinking too much or beating their wives or robbing workers of fair pay or deceiving the public about the amount of pollution their businesses pour into the air. I resist any attempts to make Ten Thousand Things show how our work changes the behavior of low-income people.\n\nI think theater becomes flattened and dead when plays try to change, to educate, or to deliver overtly political messages. I have never seen such \"message theater\" be very successful; I don't recall ever witnessing it directly change anyone's mind about an issue. Such theater, in my experience anyway, usually delivers its messages to people who already agree. Probably the best political theater I've ever experienced is comic, like that of the San Francisco Mime Troupe, which delivers its messages to large like-minded crowds in Bay Area parks with large doses of humor and satire. I would be surprised to learn the skits have ever changed anyone's mind, but at least they energize the base, giving everyone a good laugh. Whenever political message theater tries to be \"serious,\" it makes me feel trapped and smaller; sometimes the hammering away at messages makes me feel so confined that I actually get a headache. I don't like to be preached to; I like to figure things out for myself. And I firmly believe, from years of having to work through the initial suspicions and resistance of marginalized people, that most of them feel the same way. They don't want to be told what to think; they know in their bones that life has no easy answers. Such experiences have led me to passionately believe that theater works best when it is a place we go to ask questions, to experience issues from multiple perspectives and be challenged to broaden our sense of the possibilities of the world.\n\nWomen's correctional facility, The Good Person of Szechwan, 2003.\n\nSPREADING THE WEALTH OF THEATER With all that said, I do think that Ten Thousand Things is political. Just not in the ordinary sense of the word. We are not political through the content of our plays. Our work is political first and foremost because of the way it is distributed. For me, the most pressing political issue facing our country and, indeed, our world is its hugely unequal distribution of wealth. Theater is a wealth. Very few people have access to it. It is primarily available only to those with higher incomes, in addition to the education about how to access it\u2014how to find out where theaters are, what shows are playing, whether those plays are done well or would even be of interest. Add in access to transportation, to child care, and to free time, along with schooling in how to dress for the theater and how to behave as an audience member. It is accessible to very few.\n\nMen's correctional facility, Il Campiello, 2011.\n\nTen Thousand Things radically alters the system of distribution of theater's wealth. We bring the highest excellence theater can offer directly to people, in spaces where they are already comfortable, instead of expecting them to come to us. When you do this, everything changes. Our politics are really as simple\u2014and profound\u2014as that. If all the arts were to think deeply about their distribution systems and then work hard to actually change them, the political ramifications would be profound.\n\nTHEATER OF EMPATHY I also believe that much of what theater has to offer is fundamentally political but, again, not because of any messages in its plays. The wealth of this live experience we are distributing comes with what theater asks us to do, not what it says. Theater is inherently political because it asks you to practice one simple, fundamental activity: putting yourself in another's place. All theater asks this of us at some level. Whenever we engage with a character, we are seeing things through her eyes, not just our own. Sometimes this empathy work is not too demanding, and we find ourselves primarily identifying with just one character, the one who is most like us. Sometimes, in the best plays, we are compelled to look through the eyes of more than one character, many of whom are different from ourselves. When we experience some sort of connection with a character, we experience a momentary sense that we share something in common. The visceral experience of sharing something in common with someone we thought was entirely different is surprising. And surprise can start a shift. The hoe hits the hardened soil of our thought patterns, breaking it up a bit. The more surprising this discovery of the connection, the stronger the force of the hoe, the looser the soil becomes. Loose soil is more receptive to seeds. Perhaps later down the road, a seed will drop into it; perhaps still later, something new will sprout.\n\nOr perhaps not. I think it is important to be very honest about the actual possibilities of theater. I agree with the British director Declan Donnellan, who once said, \"The only social service theater provides is to make people bigger.\" This is theater's limitation\u2014and its profound wealth. \"Making people bigger,\" making a little more room inside ourselves to incorporate the experience of another, is really the most theater can do to be political\u2014but this is actually a lot.\n\nTHE THEATER OF UNEXPECTED CONNECTIONS In Ten Thousand Things productions, the power of discovering unexpected connections with those who seem unlike us, the power of radiant energy, is multiplied and magnified. Not only do we have a chance to experience the multiple viewpoints of many characters in the play but, with all the lights on, we are able to consider the differing viewpoints of the other audience members seated around the circle. Some of our very favorite performances are those \"mixed\" ones at social service centers, where people from the general public can also attend. There, people from very different economic classes sit next to each other, and this \"audience viewpoint effect\" gets magnified even more strongly. A corporate manager who is a veteran theatergoer might find herself sitting across from a homeless man. Their different reactions to different moments of the play force each other to consider different perspectives. The homeless man may laugh at a particular moment, and the manager, after considering it for a bit, discovers the humor, too. She becomes a little bigger. Then she may laugh at something, causing the homeless man to see the situation in a new light; he finds himself laughing, too. They might laugh together at certain moments. This similarity might also cause surprise, making each one open up a bit more. That the connections are unexpected make them that much more powerful. There is so much power in surprise. Indeed, it is a rare experience in our world when people from vastly different economic backgrounds sit across from or next to each other as equals. I have always noticed a kind of sweetness to these performances for \"mixed\" audiences, a kind of relaxation and openness that perhaps the common meeting place of the story and the imaginary world of the play provide.\n\nEven with our homogenous audiences, where mixing of economic perspectives is hard to come by, whether at paying shows or in the prisons, there is the perspective of the \"imagined other audience.\" With the help of a little prodding from my introduction, our paying audiences find themselves trying to imagine the play not only from the viewpoints of the faces immediately around them but, through imagination, from the perspective of someone who is an inmate or living in poverty. In a prison, inmates often ask us if the same play they are experiencing has been seen by people who have paid for their own tickets. They are impressed to know that those audiences are also experiencing it in exactly the same way, with no stage and all the lights on. They imagine why those who can pay might have wanted to come watch this show. The hoe strikes again and again; the soil is further loosened. Through unexpected connection, we create at least the potential for change.\n\nActually, all of this \"political activity\" is very personal. Being moved to step outside oneself, to experience a small surprising moment of personal connection with another who is different\u2014this is not how we are normally taught to think of politics. But our audiences have taught me that this very personal interaction is actually at the heart and core of any true political change. Many people who have led hardscrabble lives, trying each day to survive, do not bring political lenses to the table. They often haven't had time to learn to think in terms of labels or frameworks like \"capitalism\" or \"socialism,\" \"Republican\" or \"Democrat,\" \"left\" or \"right.\" They just experience and respond to situations directly in the moment, as individuals, without thinking of political frames.\n\nIf there is an imbalance of power in a scene, they experience it personally, as the characters in the play do. In Measure for Measure, when Lord Angelo tells Isabella she must sleep with him or he will execute her brother, everyone in a prison feels the abuse of power. Women quickly put themselves in Isabella's place. For many, it's very easy to do because they have actually been in that place themselves. Some male inmates, however, resist making the leap into Isabella's perspective, preferring to enjoy the power of Lord Angelo. When Angelo continues to press, though, some men become a little uncomfortable because they too have experienced abuses of power from those above. Some men start to imagine their sisters in such a situation. Some of the men are surprised into finding they understand how things feel to Isabella.\n\nSeveral scenes later when Claudio asks Isabella to put herself in his place and try to imagine what it feels like to be facing execution and sudden death, she refuses to do so. The perspective of some of the women may then start to shift a bit\u2014they had been fully on Isabella's side when she felt trapped, but they now have been asked to consider how Claudio feels, and they understand his desperation. Isabella feels a little heartless to them now because she can't understand her brother's plight. The connection of one's heart gets shifted around a bit more. Firm judgments one moment become less certain, a little looser still a few moments later. We've been forced to stretch, to become a little bigger in our understanding than we were just moments ago.\n\nReal political change begins not from absorbing messages but from experiencing directly how things feel to someone different from oneself. We begin with respect for those in our audience who rarely experience respect. This makes them feel listened to. When people feel listened to by those who usually don't listen, they are more likely to reach out and try to listen to others who are different. When a character speaks directly from the heart, it becomes more likely that the listener will see a connection. The power of Ten Thousand Things' theatrical work is that much greater because it is all so unexpected. Our nontraditional audiences, who so often feel excluded from the rest of the world, feel included for a moment or two. They feel respected because they are not being preached to. It surprises them, and they become more open. And our paying audiences, who usually feel quite included in the world, are surprised to find that others whom they had imagined to be quite different from themselves have actually greatly enjoyed the excellent theater which they enjoy as well\u2014indeed, that these different audiences have perhaps even enjoyed it more keenly and intensely. Those of means are surprised to find they have something in common with those less fortunate. Everyone's size is increased just a bit.\n\nAs I discussed before, Ten Thousand Things' bare-bones production style can also give people practice at seeing unexpected connections. When there are few objects onstage, abstracted from their usual realistic settings, it becomes easier to spot their similarities, easier to see surprising connections between unrelated things. This practice can extend to seeing connections between people who are seemingly completely unalike. Our hallmark double- and triple-casting, when the same actor is seen as an imperious judge one moment and, with only a minor costume change of hat and vest, a goofy barkeep the next, can serve as a visceral reminder of the common humanity that both characters share. It can strengthen our intuitive understanding that in acting to help others, we are actually helping ourselves.\n\nREALIGNING PRIORITIES There is yet another way that the work of Ten Thousand Things is political. It has to do with the way, within our own organization, we have reimagined how to distribute wealth and power. As an organization, our priorities for how we spend our money are different from those of many other theaters. We don't need a building, so we don't have to spend money on that\u2014just a dozen or so weeks in a church basement for rehearsals at fifty dollars a week, plus renting the large room in a literary center for sixteen weekends for our paying audiences. We obviously don't have to spend money on lighting equipment, and our sets still come in under eight hundred dollars. With the list of expenses narrowed, it becomes easier to name our priorities. We don't need many of the other trappings of theater, but we do need the very best artists, those at the peak of their craft. Our priorities thus become very clear\u2014we will spend our money on human creative energy, on people, and particularly on artists. We openly state this as our priority, and the radiance of this decision comes across onstage as artists feel respected and valued.\n\nAll of this \"political activity\" once again starts from necessity. Springing from the simple desire to find audiences who will deeply connect to stories, we are compelled to bring the work to them, thus redistributing the wealth of theater. Because we have so many different kinds of people in our audiences, our plays must reflect a multiplicity of perspectives that audiences are compelled to consider. Because we have no stage, the audience must define it by sitting in a circle or square, and because we have all the lights on, the audience is compelled to consider each other's perspectives. Because we cause people of very different life experiences to sit next to each other, they are forced to consider things from even more radically different perspectives than they would if they were sitting in a more homogenous group. Because we can't carry a lot of stuff, the few objects onstage must serve different purposes and cause people to more easily see metaphors and connections between unlikely objects. And because we must strip down to essentials in doing our work, really only needing excellent artists, our spending priorities become very clear. The political work of Ten Thousand Things all centers on redistributing wealth, both the actual financial wealth of our organization and the wealth our art form affords. Through theater, we share more fairly the wealth of the opportunity to consider a situation through the eyes of others. We offer more fairly the opportunity to experience theater's radiant energy and its loosening up of hardened soils within, the ways it can make one become a little \"bigger.\" All of this is political work. It all serves to increase, at least a little, the radiance of our world.\n\nThe author and her daughter.\n\n# 11\n\n# ALL THE LIGHTS ON\u2014FACING MISOGYNY AND REINVENTING DEAD SYSTEMS\n\n## Organic Organization\n\nWHEN I WAS SURVEYING the very male world of directing in the 1990s, I could sense fairly quickly not only that I was a misfit but that it would be disastrous for me to even try to fit in. The few times I put my foot on one of the paths available to young directors, it almost always pointed in the direction of big regional theaters. I began to feel a tightness in my chest and a fluttery panic. There was a hierarchy, there was a ladder, and in order to climb it I would have to win the approval of \"the boys.\" They would be looking over my shoulder, testing me, making sure I was \"tough enough,\" in the way they were quite certain that directors needed to be \"tough.\"\n\nIn my childhood growing up in the sixties, I never had the thought that being a girl could get in the way of my doing exactly whatever I wanted to do. I do believe there is a special kind of girl who has directing impulses from a very early age. We are called \"bossy girls.\" We want to shape reality to be like we think it should be. We like to organize play; we like to make events for others to come see; we like to be in charge of telling the story. We tell the other kids we're playing with that they should be this way, not that way, that they're doing it wrong, that they should be angrier or sadder or funnier. I recognize little girls like this all the time, and I deeply love them.\n\nMy first real urges to direct a play came on a trip to Chicago in fifth grade, when my parents took me and my brother to see You're a Good Man, Charlie Brown. I was deeply affected by the \"truth\" I felt was being reflected onstage. When I got home, I ordered a copy of the script from our local book-store and organized some friends to be in the cast. I assigned all the parts, which included casting myself as both Lucy and Snoopy, the two best parts. Not long after, the interest of the other girls quickly dwindled; my friends stopped coming to play practice. I started to figure out then that when I was creating these events, I had to share. Girls, I think, are taught much more quickly than boys that there are limits to how bossy you get to be. And I think, in general, that this is a good thing. I think more boys need to learn this lesson about sharing; they need to learn it early and often. The world would be much better off.\n\nAnd yet, like most girls entering their teens, I learned this lesson too well. My girlhood confidence and \"bossiness\" certainly weakened in adolescence. I absorbed the message about sharing so well that by the time I entered the first \"all female\" directing class at graduate school (there were two of us), I was startled to find that my male professors and all the male directors in other classes just assumed they should be in charge and make all the decisions. They had no questions about why they should be the authorities. Director was a role they assumed with confidence and ease. This made no sense to me. My female directing compatriot and I felt out of place with all the assumptions in this very male world.\n\nI remember one of my professors, whose claim to fame was from directing a few successful Hollywood films in the fifties, telling me in confidence a story he thought would be helpful. He told of a time when he was directing Kirk Douglas in a movie. They had a disagreement, and his impulse was to want to \"punch Kirk out.\" But then, he went on, he managed to rein in these impulses and instead put his arm around Kirk, saying in a friendly way, \"Let's see if we can talk through this and find a compromise.\"\n\nTo me, this story epitomized the huge gulf between me and the all-male directing faculty. I have never had an impulse to want to \"punch out\" an actor in my entire life. My problem was quite the opposite, especially in the beginning\u2014to find the confidence I needed to give voice to my perceptions and ideas so they could at least be on level ground with the ideas of the actors and other artists in the room. It's why I felt so strongly that I needed to make my own space so I could figure it out and find my voice organically, in my own way. I thought often of the story of Georgia O'Keeffe in her early years, shutting herself away in a room and drawing a line, over and over again, for days and days, until she felt that that line was truly her own, coming from her own authentic self.\n\nI admire women who decide to take on the hierarchy directly, fiercely proving themselves against the standards of the dominant culture, and who, once inside, set out to change the system as best they can. I just knew fairly quickly that I could never do that. I simply wouldn't survive as an artist. I knew that as I tried to withstand the pressures to conform to this male-dominated world, taking the test punches they gave out and finding ways to get back up on my feet, my authentic creativity would almost surely wither. I'd had very strong tastes of how it all worked in graduate school, with all my male professors, and I could easily tell that, with a few notable exceptions, the \"are-you-man-enough test system\" was even more strongly in place in the actual theater world.\n\nMAKING MY OWN PLACE It all made me feel tired. Not only would the structures of most theaters keep me from reaching the audiences I yearned for, but they would also prevent me from being myself as an artist. All of this punching and testing had nothing to do with anything that really mattered to me. I needed to make my own place. Without really realizing what I was doing, I set out to create a place where I would feel safe, a place where I felt unjudged and unconstrained, free to let whomever I was as an artist blossom. For me, the strongest way to face down the misogyny of the theater world was to turn around and not face it at all. By necessity, I began to make up another world for myself.\n\nThis decision meant reinventing everything from scratch. It was slow and plodding\u2014often comical\u2014but for me it was necessary. It's actually, for better or for worse, how I've approached most projects in my life. I have a mistrust of big systems; I need to understand why something is done before I commit to doing it. I wanted to carefully feel out each step of the way as I built the infrastructure for Ten Thousand Things, determining whether it was truly needed. Once again, having never really worked much in conventional theaters beyond a few assistant director jobs in Los Angeles, I started from a place of ignorance; I didn't really know how things were usually \"done.\" I figured out how to set up my own nonprofit with software from a legal press. (I did latch on to the nonprofit model; I didn't have time to think about reinventing that wheel. Though perhaps today if I were starting up a company I'd rethink that, too, now having educated myself a bit about the roots and origins of that particular system.) I learned how to write grant proposals, puzzled out what a budget was and how to write one. The \"how to create your own nonprofit\" instruction book I was following at the time said I needed a board of directors of at least three people to meet at least once a year. So for a few years I had three friends over for lunch once every summer, really having no idea what we should be doing.\n\nTHE PURSUIT OF MOTHERHOOD AND ART In my early thirties, still living in Los Angeles, I also wanted to have a child. I wanted to have one as fiercely as I wanted to find work that let me give back with all my being. The problem was that the world had never presented women with any good ways to balance the desires for work and for children. I didn't have any role models. My mom and most of her friends graduating from college in the fifties generally seemed to have given up their career dreams to become mothers. Sometime after the first Good Person tour, I discovered I was pregnant. I found a tiny, jeweled box and inside it I put a small piece of paper on which I wrote a promise to myself not to give one up for the other. I knew ultimately the two were not in competition with each other but intertwined. I wrapped the little box in a beautiful silk cloth and kept it in my top dresser drawer, a visual reminder of that promise every time the drawer was opened. I knew, deep down, that my child would be so much better off with a mother who could live out her dreams, and I knew that as an artist, my child would inspire me. I think Shen Te's ferocious promises to her unborn child were also ringing in my mind, but unlike Shen Te, I didn't want to have to choose between two sides. I wanted a life that would let me give back through my work with kindness and generosity and intelligence and imagination\u2014and I wanted a child.\n\nThe main challenge, I knew, would be balancing time. To allow my art and my child to nourish each other, I needed to keep time firmly under my control, making sure one set of demands didn't take much away from the other. Of course at this point, sustaining myself financially as an artist was impossible; I needed other jobs that could be done at home in my own time. I proofread and summarized depositions for law firms; I taught ESL classes at night when my baby was sleeping. I was always on the lookout for flexible work that took very little of my own energy. (My husband at the time was a freelance writer. Our marriage was ultimately unable to survive, in part because of the tensions and struggles over allocation of time and money. Ten years later we would divorce.) Throughout that first year of pregnancy and infancy, I kept trying to learn and plan for the next project, the Electra tour. Blurry eyed, I worked on the script, wrote grant proposals, and figured out budgets during Molly's naps.\n\nWhen Molly was nine months old, she began to walk. I had taken her to visit a small cooperative day-care center in our little Venice neighborhood. Luckily, this child was also an early talker, and soon began to pound on the door, chanting one word, \"School! School!\"\u2014most likely having sensed how much more fun things could be among other small people her own age. I gladly obliged. Day care, even just three or four half-days a week, suddenly allowed me room to breathe and carved out a space for me to build up my powers of concentration for my work.\n\nBy then I'd rounded up a few small grants and arranged a small tour of Electra to juvenile correctional facilities and youth groups. Somehow, I stumbled through rehearsals. I was working with many of the same actors who had worked on The Good Person; this perhaps explains their generosity toward my directing work, which at the time felt very inadequate. It was so much harder than I'd imagined to come up with the energy required to nurture a one-year-old and to direct a play at the same time. I kept eyeing that little bundle in my dresser, drawing from it, as well as from Shen Te, as much strength and determination as I could.\n\nThe tour, despite my real or imagined inadequacies, still offered exhilarating moments of connection, both with the girls in juvenile detention and with our one paying audience.\n\nAnd then we felt it might be time to move. As two freelancers, my then husband and I found the prospect of raising a child in Los Angeles daunting. We needed a city where we could afford a little house and could use the public schools, as well as one full of excellent actors. I also needed a city (unlike Los Angeles) with a strong and vibrant culture of support for theater. That narrowed it down to Minneapolis. It still is shocking to me to think that our nation affords only one or two cities where such a life is possible.\n\nOnce we'd landed in our affordable little Twin Cities house, complete with a picket fence, it took another year to get the company going again. I had to find new grant sources, learn about the community, meet actors, and find new low-income performance venues. I discovered a rehearsal space in a Quaker meeting house I occasionally attended, a bare room where on Sundays people sat in a circle of folding chairs as they silently worshiped. (This circular configuration must have seeped its way into my brain to help inspire the idea of performing in the round a few years down the road. And, of course, I liked going to Meeting sometimes because of its bare-bones worship service, with nothing to look at except the other people seated around the circle engaging in their own private forms of worship. I thought it was theater of the very best kind.)\n\nI took a short class about starting arts nonprofits. I finally learned what a board of directors could do: raise money. Out of necessity, though, I again started the board with a few good friends, none of whom were wealthy. I was temping and teaching ESL classes, too, all the while chasing after an active two-year-old. We had found a good parent co-op day care nearby where Molly spent part of the day. I really wanted to spend half of my daytime with this fun little creature, who shared with me her wonder and joy at the world, inspiring and weirdly energizing me, although I was always exhausted by day's end. I managed to get theater work done in the mornings when my brain worked best.\n\nMAKING IT UP AS I WENT ALONG I can now see that I found the energy and drive to keep doing all the paperwork and running around necessary to get a small theater going at the same time I was raising a toddler because I was making everything up myself. In particular, this meant that I was able to choose the values I wanted to shape the organization around rather than try to fit my work into some preexisting structure. I was able to choose the values that energized and motivated me, values that were also, of course, organic to the work. As was my habit, I pared everything down to the essence. The values such work demanded seemed to all come down to two things: gifts and good pay for artists.\n\nWhatever \"plan\" I had in mind for the company's finances had always been centered on gifts. The organization would receive gifts from others, which it would use to pay artists as much as possible, enabling them to give the gift of theater to those who had never had a chance to receive such a gift before. The thought of asking low-income centers to pay for a performance never once occurred to me. That would truly be robbing Peter to pay Paul. The performances would be a gift, but these new audiences in turn would give us as artists the gift of their rich life experiences and their honest responses.\n\nSo, of course, I began with the most obvious gifts, those that came from grant-making organizations: the government, private foundations, and sometimes corporations. And here I must stress that I consider it a great blessing, despite all the energy it took, that when the company started out, I was able at first to be my own \"arts administrator.\" I learned so much. Writing grant proposals helped me to clearly articulate exactly what I was trying to do. And I think that the proposals I wrote were more vivid and interesting to read because I wrote authentically as an artist. Crafting my own budgets made me very aware of every dollar that was spent, being extremely careful at first never to spend money that I didn't know I already had in the bank. It all felt organic because I could connect the paperwork directly to the human and artistic values that motivated me.\n\nBut sometimes my plodding reinvention of everything was indeed comical. Although I had seen lists of individual donors in theater programs, of course, I hadn't really conceived of the possibility that individuals, as well as foundations, might want to give money to our work. It wasn't until I'd been in the Twin Cities for a few years that the idea occurred to me, probably as I wracked my brain for ways to come up with more money to pay people more. It was scary, but I wrote personal letters out by longhand\u2014each several pages, to friends around the country\u2014and was truly amazed when checks were sent back, some of them quite large (at the time, this meant over a hundred dollars). Though it was hard to ask people I knew personally for money, the thought of the brave and heartfelt efforts of the artists I worked with and their struggles to make a barely decent living fueled me. In some ways, it motivated me even more powerfully than the thought of the underserved audiences the money could help us reach, I suppose because it was even more personal\u2014the actors were my friends. The simplicity of our production values, not needing to pay for a building, or lights, or elaborate trappings, allowed me to truly focus on artists' pay. I could become passionate about fund-raising because of this focus. In addition to wanting to make life a little easier for my friends, I did not want theater to become a profession where only those with preexisting wealth could afford to practice it.\n\nBRINGING IN THE CRITICS The biggest obstacles to my fund-raising, actually, were knee-jerk responses that came from many people when they first learned that we \"did plays in prisons.\" Most people automatically assumed that our work was therefore mediocre and uninteresting, that we must just put on \"bad skits.\" I needed a concrete way to quickly establish that our work was artistically excellent. And for this, I confess, I turned to critics. I needed their quotes so I could use them in grant proposals. It was as simple as that. I tried for several years without success to get critics to come see a show, but finally a young woman from a weekly alternative paper came to see The Caucasian Chalk Circle in a shelter. Amazement and surprise were expressed and praises were sung. I got another critic to come see The Tempest at a women's prison. More glowing reviews followed; more quotes were generated to put in the grant proposals. And slowly more regular theatergoers started to come see the shows.\n\nAlso perhaps somewhat comically, as part of my plan of \"gifts from gifts,\" I had really never considered the possibility of earning much income from ticket sales. I had always wanted to perform once or twice for the general public just to be sure the work spoke to experienced theatergoers as well, but in my mind that really just meant a few performances for friends. Again, part of the entire genesis of the work was to avoid having to plead desperately to get reluctant audiences to buy tickets. But as word of mouth in Minneapolis grew, along with critical acclaim, I began to see that an enthusiastic paying audience could be a door to more gifts. We slowly began to add more shows where people paid for tickets to help support the free performances. And then those people, excited by the quality of the work they saw as well as our mission, became individual donors, too.\n\nWORK AND CHILD INTERTWINED The organization of support for the work took shape around the values of including everyone, gifts, and artist pay. I still wasn't making much money from it, but it was exactly the work I wanted to be doing, and my time was still very much under my control, allowing me to be there for my child. I could have rehearsals during the day while she was at school, and I could schedule many of our performances during the day, too, making sure to be done by the time she got off the school bus. (Hilariously, I used to start rehearsals at 9 AM because Molly was off to school by then. This went on until some desperately tired actors, with the active late-night lives of most performers, begged for us not to start until 10.) My time was my own, and that has always been worth tens of thousands of dollars to me.\n\nMolly became old enough to come to some rehearsals, although I was careful never to use them as convenient day care. I wanted them to be a special occasion, where she could be surprised and delighted by what she saw. I have noticed that children who grow up around theater artists are often pretty great. Perhaps it's because they get to hang out with adults, especially actors, who treat children with great respect, as equals in many ways. Actors are adults who take \"play\" as seriously as children do; children experience enormous validation from this. These same children, I've observed, seem emotionally healthy and comfortable somehow\u2014perhaps because they've been exposed to the wide range of human emotions as they watch rehearsals, with the distance and perspective that comes from knowing the emotions aren't \"really real.\" They seem relaxed around emotional extremes and their rapid fluctuation; they've had practice observing them all. I remember in particular Molly walking into a rehearsal of The Ballad of the Sad Cafe during the scene where Miss Amelia and Marvin Macy were having a wrestling match. When Molly saw Miss Amelia punch Marvin in the face, she burst into tears and ran to hide behind a table. The actors, Carolyn Goelzer and Terry Hempleman, stopped what they were doing and very kindly and gently showed Molly just how a stage punch worked, with its sound coming from a slap of the thigh by the hand not throwing the punch. Molly relaxed, her sniffles turning into giggles, and she sat happily to watch the entire scene as we rehearsed.\n\nAlthough Molly at age ten firmly declared to me that, while she enjoyed theater, it wasn't her \"passion\" (writing was), occasionally I would find a small part for her to play in at least some of the performances on tour (with a child-sized puppet standing in for her when she couldn't be in a particular performance). She loved being included, her imagination was stretched as she was asked to inhabit a character, and she gained confidence and pride in her own voice as an artist.\n\nAnd as much richness as theater artists provided Molly's life, Molly gave back many things that were equally as important. She gave me, as a theater artist, a deep sense of connection to all the other humans before me and around me who had themselves raised children. And because I was so determined to be sure she received her fair share of my time, she brought a wonderful economy to the organization, making me focus fiercely on the mission, avoiding at all costs any unnecessary work or distractions. She kept Ten Thousand Things' focus clear and tight.\n\nORGANIC GROWTH The growth of the company in the first dozen years was slow but organic. Until Molly was ten, I still did all the fund-raising and administrative work myself. Because of my focus on artist pay, I was always reluctant to spend money on any other staff. But finally, as the board of directors attracted people with more actual experience with nonprofits, they convinced me to get someone to help out. In the bounteous world of Minneapolis arts philanthropy, we found a grant for hiring new part-time staff, and we found a wonderful woman (her name also Michelle) of great intelligence and kindness who was also a young mother. Like so many such remarkable women at that age, she had a desire for flexible time and the ability to work from home. Our \"organizational structure\" could easily accommodate that. E-mail allowed us to communicate, so no office rent was required. One more staff person made a noticeable difference in helping our income grow more quickly so we could pay artists better.\n\nThe author's daughter, Molly, at the curtain call for Carousel, Skyline Housing, 2003.\n\nAs someone else stepped in to take over more grant writing and budget tracking, I found myself with more time to dig into my artistic work\u2014and with my child now in school full time, I do believe the plays became even richer and fuller. Several years later, it became clear that Michelle Woster, our managing director, could do better work if someone could help out with monitoring individual donations and keeping track of the box office, and so we found a grant to bring in another intelligent and kind woman to help, Amy Tang, another young mother who wanted flexible time, working from home.\n\nWHO WILL DRIVE THE VAN? Because of my unfamiliarity with the ways of conventional theater, I had never understood exactly what a stage manager did, especially because our work had no lighting or sound cues. At the beginning, I tried to do everything myself, including setting up the stage. At one point, however, I remember that an ironing board I'd set up for an actress to use in a scene collapsed during the show, and I acknowledged that it really was too difficult to both direct and focus properly on the mechanical details necessary to make the production run smoothly. With at least this acknowledgment of my own limitations, I then managed to pay someone a little bit to help out with gathering props and being on book during rehearsals to help the actors learn their lines. There also were some jobs around production that I just didn't want to do. When we discovered we had enough money to rent a van for the tour so that actors didn't have to haul set pieces in their cars, I knew I didn't want to have to drive the van. I drew a line. Magically, around that time another amazing young woman emerged (not a mother) who not only truly liked to do all those things but who could weld and hammer as well as sew, making repairs when needed, the marvelous Nancy Waldoch. She was wonderful at interacting with people, too, with great patience and kindness\u2014and thus, a production manager position became apparent and sprung into place. Gradually, we passed over to her the duties of setting up and arranging the tours. I found myself with still more time to do artistic work, planning and dreaming. But because, for a while at least, I'd done most everything myself, I had a personal understanding of how most everything worked, knowledge I still greatly value.\n\nNow, more than twenty years after we began, it feels like we're pretty much the right size, almost just big enough. We are a staff of just six, three of whom are artists (myself, Peter Vitale, our music director, and Kira Obolensky, our playwright-in-residence with full salary and benefits thanks to the Andrew W. Mellon Foundation). We still have no need of an office. We just begin every Monday by e-mailing each other our lists of what we plan to do over the course of the week and trust each other to get it done. Through such trust and friendship, we've been able to create an organization that supports the artistic work with the least amount of infrastructure necessary. It's enough. We do just three productions a year. There is no burning desire on anyone's part to do more. No one wants to do theater in the summer, the only three months of good weather in Minnesota. No one wants to work so hard that they burn out. Life is rich with things other than theater.\n\nThis sense that there is more to life than theater gives a quality to the Twin Cities acting community as well that I think is fairly unique, and for which I am most grateful: playfulness. It's almost as if because no one here seriously entertains the idea that they will someday become rich or famous through theater, let alone film or TV, that each actor is actually more focused on her or his craft, and thus able to embrace much more fully that the heart of all truly excellent acting is openness: playing, exploring, taking chances and making a fool of oneself. No one here has an agent for theatrical work; we deal with each other as human beings. And, though challenging, it is possible as an actor here to have a family and even raise children, which also helps to deflate any tendencies to self-obsession. The acting community here feels generous and supportive of each other. I consider many of its members to be my very best friends.\n\nI am probably one of the few people from my Ivy League class who is still nowhere near making a six-figure income. But I have enough. And I am happy. My work offers its own kind of riches and rewards, not the least of which is that my time is pretty much my own with which to do whatever I want whenever I want. And I would feel profoundly uncomfortable earning, on an hourly basis at least, a whole lot more than the most experienced artists and staff of Ten Thousand Things. I certainly would not be happy making multiple times more per hour than the people I work with. I fail to understand artistic leaders who are. At Ten Thousand Things, perhaps in part because of our connection to our audiences who have very few things, we are committed to the notion that all our boats need to rise together.\n\nSo no matter what, though we feel \"big enough,\" Ten Thousand Things will always keep trying to raise artist pay, which will then make us feel more comfortable with raising our own pay as staff. This is one way we have come to think of \"growth.\" And, like Johnny Appleseed, we have started to spread this unique model for doing theater that we have created by taking it to other communities and theaters that want to try it out for themselves. It's another way we've come to look at \"growing.\"\n\nSometimes Ten Thousand Things feels more like an organism than an organization to me. A living and breathing being of its own right. The little seedling grew organically, supported by the strong stake of the values of inclusion, generosity, gifts, and artist pay; other shoots and tendrils naturally arose as needed to wind themselves around the stake. We created what was necessary, adapting some old structures and inventing new forms and practices to fit our very specific circumstances. We've managed to keep everything at a very human scale. There is almost no hierarchy; we communicate on a level playing field with humor and generosity. We have wine at our staff and board meetings. I still cook all the food for cast parties and donor gatherings at my home (though sometimes I do long for a caterer. But there could be worse problems). I think everyone feels respected and valued, with room to be funny and playful and honest, both at work and in their other lives.\n\nSometimes I hear stories of behind-the-scenes personal dramas and politics and backstabbing taking place at larger theaters. It's made me see that Ten Thousand Things' focus on connecting with our audiences from very different worlds has really helped to keep egos at bay. When we step into a room full of inmates, the anxieties and whinings of the ego seem petty and puling. I am grateful for the intensity of this focus, the clarity it gives to our priorities, and the ethical behavior it encourages. This focus has helped to create a radiant organization that supports and amplifies the radiance of the actors onstage and the radiance of the audience in their honesty and surprise, which causes us to want to focus on them still more, making one big circle. It really does feel like, as much as possible, all the lights are on.\n\nThe author introduces Man of La Mancha at a women's correctional facility, 2011.\n\n# EPILOGUE\n\n# WHAT THIS BOOK IS FOR\n\n## Ending Words, Beginning Words\n\nI'M WRITING THIS BOOK in the hopes that it will provoke you and perhaps inspire you. I am not trying to argue that everyone should do theater the way Ten Thousand Things does it\u2014there is room for all kinds of theater in this world, of course. I honestly do not want this to be a \"How To Do Theater This Way\" manual. I am quite protective of first-time audiences. I want them to experience only the very best that theater has to offer. Ten Thousand Things' model is for professional theater artists. If this book were to cause amateur groups or high school and college theater classes to \"go do a play for prisoners\" because it \"would be a neat experience for the performers,\" I would actually be mortified. Please, please do not try to foist mediocre theater upon those who are already so often condescended to. Please be deeply honest with yourself about whether you really do have the experience and skills to give first-time audiences the absolute best, matching the truth of their hard-won experiences with your own artistry. Don't do more damage by confirming their already firmly rooted bias that theater is boring and pretentious. If, as a professional artist, however, you feel confident of your skills and talents and you feel deeply moved to try to bring theater to marginalized audiences, then of course please do, with my blessings. Just be very sure.\n\nI do hope I have provoked you to think about \"audience\" differently. I hope you will no longer take them for granted. I hope you will think honestly about who you would like your audience to be and how you might better reach them. Even more importantly, I hope you will think honestly and realistically about just who your audience actually is. Who is actually coming to your theater right now, in terms of class and race? And why is that? If you're going to do a play about an upper-middle-class white family, that's fine, but please just be honest about who actually will or won't be interested in that story. If you're doing a whole season of work about contemporary people who are financially comfortable, be aware of that and be clear about who will really be interested in the stories you're choosing to tell. And I hope you will find ways to keep your audience present, at the front and center of your attention, during your planning work and during rehearsals, and that you will pay even greater attention to them during the performances, even if they are sitting in the dark.\n\nI want to have provoked you to think about the hugely unequal distribution of wealth in our world and, in particular, the unequal distribution of the wealth of the arts. I hope I have drawn your attention to the poverty that results. Not only the obvious material poverty of the \"have-nots\" but the dimness and narrowness of the world of the \"haves.\" The \"poverty\" of those who have actually become glutted with too much theater. I began Ten Thousand Things precisely because many theater audiences seemed as if they had seen too much; their responses had actually become muted and dulled. I didn't enjoy performing for those audiences. They were so used to seeing theater through a critical lens, which allowed them to make sure they were \"getting what they paid for\" in terms of aesthetic quality, that they had forgotten how to connect the work to their own lives. They came to the theater not out of hunger but out of obligation.\n\nWomen's correctional facility, Ragtime, 2005.\n\nHousing project, Carousel, 2003.\n\nKaren Wiese-Thompson, Nathan Keepers, Brian Curtis James, and Kimberly Richardson, Il Campiello, men's correctional facility, 2005.\n\nWhat became clear to me, as I began to take theater around to those who did not have easy access, comparing and contrasting, is that the work of the theater that only served the audiences of the \"haves\" had itself become impoverished. The stories it told were cushioned by the wealth of the characters. The questions the plays explored weren't that deep. The stages were groaning under enormous and extravagant sets and elaborate special effects at the same time its artists were straining just to make a living. The artists themselves were often motivated by career concerns, wanting to impress critics and get good reviews, hoping to land the next good job. The theater experience also just wasn't as much fun, with everyone trained to be so polite and constrained. Characters onstage and audiences came from the same kinds of worlds, and their responses were usually very similar to those of their neighbors, all seated in the darkness, unable to see each other and the artists unable to see them.\n\nWhat I feel so blessed to have discovered, through my particular work, groping and stumbling along the way, is that by including everyone in the audience and telling stories that engage all of us intensely, we uncover the true wealth of humanity, which is what we share in common. Wealth increases as it is spread around.\n\nI want to provoke you to think about how the wealth of theater is distributed. I want to spark everyone who does theater to think deeply for themselves about what isn't working in our art form. Think about what you don't like about how theater is made now. Once you've located whatever that might be for you, I hope you will think with your heart and soul about how to fix it, even if it means starting over from scratch.\n\nI do hope that somehow all this will inspire you to invent new ways and new structures that could allow the kind of theater you've been provoked to imagine to happen. I firmly believe that this is now an integral part of the work of any theater artist\u2014not just to create the work that appears onstage but to challenge the hierarchical structures that exist and to invent new structures that will allow it to blossom and be better supported. Our work is to question every convention\u2014and, if need be, to make it up fresh and new. There are so many wonderful theater artists in this country already engaged in this reimagining. I am just adding my voice to the many others already out there, theaters like Cornerstone and the Foundry Theater, Sojourn Theater, and, right here in Minneapolis, Mixed Blood and Pillsbury House, Bedlam, Sod House, Wonderlust, and Theater Mu, just to name a few. Even larger institutions like the Public Theater in New York City or California Shakespeare Theater or Oregon Shakespeare Festival or the Old Globe are all trying to stretch their wings and reach out beyond their well-established confines, all trying out their own ideas, inventing, experimenting, sometimes failing, sometimes succeeding.\n\nI can now see that for me and Ten Thousand Things, the focus has actually been on allowing the lights in all human beings to shine as brightly as possible through theater. I am excited to see what happens as you set off into the world to make or participate in theater that feels bright and alive to you.\n\n# APPENDIX\n\n## TEN THOUSAND THINGS PRODUCTION HISTORY\n\n## 2014\n\n### DIRT STICKS BY KIRA OBOLENSKY\n\nDirected by Michelle Hensley\n\nWith Stephen Cartmell, Sun Mee Chomet, H. Adam Harris, Thomasina Petrus, and Kimberly Richardson\n\nSets by Irve Dell\n\nCostumes by Sonya Berlovitz and Samuel Forney\n\nMusic and Sound by Peter Vitale\n\n### THE MUSIC MAN BY MEREDITH WILLSON\n\nDirected by Lear deBessonet\n\nWith Luverne Seifert as Harold Hill and Sarah Agnew, Aimee K. Bryant, Bradley Greenwald, Jim Lichtscheidl, Dennis Spears, Kimberly Richardson, and Ricardo Vazquez\n\nMusic Direction by Peter Vitale\n\nChoreography by Jim Lichtscheidl\n\nSets by Joel Sass\n\nCostumes by Mary Anna Culligan\n\nKeyboards by Jake Endres\n\n## 2013\n\n### A MIDSUMMER NIGHT'S DREAM BY WILLIAM SHAKESPEARE\n\nDirected by Michelle Hensley\n\nWith Brittany Bradford, Sun Mee Chomet, Kurt Kwan, Elise Langer, Gavin Lawrence, Mo Perry, Anna Sundberg, and Karen Wiese-Thompson\n\nSets by Stephen Mohring\n\nCostumes by Sonya Berlovitz\n\nSound by Peter Vitale\n\n### A STREETCAR NAMED DESIRE BY TENNESSEE WILLIAMS\n\nDirected by Randy Reyes with Lear deBessonet\n\nWith Elizabeth Grullon, Kurt Kwan, Kris Nelson, and Austene Van\n\nSets by Dean Holzman\n\nCostumes by Mary Anna Culligan\n\nSound by Peter Vitale\n\n### THE SEVEN BY WILL POWER\n\nDirected by Sarah Rasmussen\n\nWith Katie Bradley, Aimee K. Bryant, H. Adam Harris, Kinaundrae Lee, Brian Sostek, Ricardo Vazquez, Joetta Wright, and Bruce A. Young\n\nMusic Direction by Peter Vitale\n\nChoreography by Kahlil Queen and Aimee K. Bryant\n\nSets by Stephen Mohring\n\nCostumes by Annie Cady\n\n## 2012\n\n### MEASURE FOR MEASURE BY WILLIAM SHAKESPEARE\n\nDirected by Michelle Hensley\n\nWith Nathan Barlow, Zach Curtis, India Gurley, Kurt Kwan, Sonja Parks, Luverne Seifert, Suzanne Warmanen, and Karen Wiese-Thompson\n\nSets by Stephen Mohring\n\nCostumes by Amelia Cheever\n\nSound by Peter Vitale\n\n### VASA LISA BY KIRA OBOLENSKY\n\nDirected by Michelle Hensley\n\nOriginal Music by Peter Vitale\n\nChoreography by Jim Lichtscheidl\n\nWith Elise Langer, Jim Lichtscheidl, Tracey Maloney, Luverne Seifert, and Sally Wingert\n\nSets by Irve Dell\n\nCostumes by Sonya Berlovitz\n\nAdditional instruments by Heather Barringer and Annie Enneking\n\n### AS YOU LIKE IT BY WILLIAM SHAKESPEARE\n\nDirected by Lear deBessonet\n\nWith Aimee K. Bryant, Pearce Bunting, Maggie Chestovich, Bradley Greenwald, Randy Reyes, and Kimberly Richardson\n\nSets by Stephen Mohring\n\nCostumes by Mary Anna Culligan\n\nSound by Peter Vitale\n\n## 2011\n\n### IL CAMPIELLO BY CARLO GOLDONI, ADAPTATION BY STEVEN EPP\n\nDirected by Michelle Hensley\n\nWith Sarah Agnew, Christiana Clark, Brian Curtis James, Nathan Keepers, Elise Langer, Thomasina Petrus, Randy Reyes, Kimberly Richardson, and Karen Wiese-Thompson\n\nSets by Stephen Mohring\n\nCostumes by Amelia Cheever\n\nSound by Peter Vitale\n\n### MAN OF LA MANCHA BY DALE WASSERMAN, JOE DARION, MITCH LEIGH\n\nDirected by Michelle Hensley\n\nMusical Direction by Peter Vitale\n\nWith Steven Epp, Matt Guidry, Tracey Maloney, T. Mychael Rambo, Luverne Siefert, and Regina Marie Williams\n\nSets by Stephen Mohring\n\nCostumes by Kelsey Glasener\n\nKeyboards by Michael Pearce Donley\n\n### DOUBT: A PARABLE BY JOHN PATRICK SHANLEY\n\nDirected by Peter Rothstein\n\nWith Jane Froiland, Kris Nelson, Regina Marie Williams, and Sally Wingert\n\nSets by Dean Holzman\n\nCostumes by Amelia Cheever\n\nMusic by Peter Vitale\n\n## 2010\n\n### LIFE'S A DREAM BY PEDRO CALDER\u00d3N DE LA BARCA\n\nDirected by Michelle Hensley\n\nWith Maggie Chestovich, Stephen D'Ambrose, Celeste Jones, Elise Langer, Kiseung Rhee, Namir Smallwood, and Dennis Spears\n\nCostumes by Kelsey Glasener\n\nProps by Erica Zaffarano\n\nSound by Peter Vitale\n\n### MY FAIR LADY BY ALAN JAY LERNER AND FREDERICK LOEWE\n\nDirected by Lear deBessonet\n\nMusic Direction by Peter Vitale\n\nWith Kate Eifrig, Bradley Greenwald, Steve Hendrickson, Kimberly Richardson, and Luverne Seifert\n\nChoreography by Jim Lichtscheidl\n\nSets by Jeremy Wilhelm\n\nCostumes by Kathy Kohl\n\n### STONES IN HIS POCKETS BY MARIE JONES\n\nDirected by Michelle Hensley\n\nWith Steven Epp and Jim Lichtscheidl\n\nCostumes by Kelsey Glasener\n\nSound by Eric Jensen\n\n## 2009\n\n### OTHELLO BY WILLIAM SHAKESPEARE\n\nDirected by Sonja Parks and Michelle Hensley\n\nWith Ansa Akyea, Christiana Clark, Peter Hansen, Tracey Maloney, Kimberly Richardson, Matt Sciple, and Luverne Seifert\n\nSet by Stephen Mohring\n\nCostumes by Mary Anna Culligan\n\nSound by Peter Vitale\n\n### RASKOL BY KIRA OBOLENSKY\n\nDirected by Michelle Hensley\n\nOriginal Music by Peter Vitale\n\nWith Lisa Clair, Craig Johnson, Tracey Maloney, Kris Nelson, Charles Schuminski, Luverne Seifert, and Karen Wiese-Thompson\n\nSet by Irve Dell\n\nCostumes by Kelsey Glasener\n\nAdditional instruments by Nathan Hanson and Chris Bates\n\n### ENDGAME BY SAMUEL BECKETT\n\nDirected by Marion McClinton\n\nMusic by Heather Barringer\n\nWith Terry Bellamy, Barbara Berlovitz, Christiana Clark, and Steve Hendrickson\n\nSet by Stephen Mohring\n\nCostumes by Vera Mariner\n\nSound by Heather Barringer\n\n## 2008\n\n### TWELFTH NIGHT BY WILLIAM SHAKESPEARE (ALL FEMALE)\n\nDirected by Michelle Hensley\n\nWith Maggie Chestovich, Kate Eifrig, Barbara Kingsley, Isabell Monk O'Connor, Sonja Parks, Kimberly Richardson, and Sally Wingert\n\nSet by Erica Zaffarano\n\nCostumes by Sonya Berlovitz\n\nMusic and Sound by Peter Vitale\n\n### ONCE ON THIS ISLAND BY LYNN AHRENS AND STEPHEN FLAHERTY\n\nDirected by Peter Rothstein\n\nMusical direction by Peter Vitale\n\nChoreography by Aimee K. Bryant\n\nWith Eric Avery, Michelle Carter, Celeste Jones, Kinaundrae Lee, Kahlil Queen, Greta Oglesby, Dennis Spears, and Regina Marie Williams\n\nSet by Erica Zaffarano\n\nCostumes by Willene Mangham\n\nKeyboards by Michael Donley\n\n### EURYDICE BY SARAH RUHL\n\nDirected by Larissa Kokernot\n\nWith Lisa Clair, Marc Halsey, Steve Hendrickson, Leif Jurgenson, Vera Mariner, and Sonja Parks\n\nSet by Joe Stanley\n\nCostumes by Sonya Berlovitz\n\nSound by Peter Vitale\n\n## 2007\n\n### RICHARD III BY WILLIAM SHAKESPEARE (ALL MALE)\n\nDirected by Michelle Hensley\n\nWith Bob Davis, Shawn Hamilton, Craig Johnson, Darien Johnson, Jim Lichtscheidl, Richard Ooms, and Luverne Seifert\n\nSets by Stephen Mohring\n\nCostumes by Kathy Kohl\n\nSound by Heather Barringer\n\n### LITTLE SHOP OF HORRORS BY HOWARD ASHMAN\n\nMusic by Alan Menken\n\nDirected by Michelle Hensley\n\nMusic Direction by Peter Vitale\n\nChoreography by Jim Lichtscheidl\n\nWith Kate Eifrig, Jim Lichtscheidl, Sonja Parks, Thomasina Petrus, Luverne Seifert, Harry Waters Jr., and Karen Wiese-Thompson\n\nSets by Stephen Mohring\n\nCostumes by Vera Mariner\n\n### BLOOD WEDDING BY FEDERICO LORCA\n\nDirected by Juliette Carrillo\n\nWith Barbara Berlovitz, Sha Cage, Maggie Chestovich, Kurt Kwan, and Danny Salmen\n\nPuppets and Set by Allison Heimstead\n\nCostumes by Emily Pepper\n\nMusic by Peter Vitale\n\n## 2006\n\n### THE MERCHANT OF VENICE BY WILLIAM SHAKESPEARE\n\nDirected by Michelle Hensley\n\nWith Lisa Clair, Matt Guidry, Steve Hendrickson, Catherine Johnson, Darien Johnson, Kiseung Rhee, Stacia Rice, and David Wiles\n\nSets by Stephen Mohring\n\nCostumes by Kathy Kohl\n\nSound by Peter Vitale\n\n### RED NOSES BY PETER BARNES\n\nDirected by Larissa Kokernot\n\nMusic by Peter Vitale\n\nWith Lisa Clair, Kate Eifrig, Dylan Fresco, Alayne Hopkins, Kris L. Nelson, and Lee Mark Nelson\n\nSet by Stephen Mohring\n\nCostumes by Kathy Kohl\n\n### IN A GARDEN BY GERTRUDE STEIN AND KEVIN KLING\n\nDirected by Michelle Hensley\n\nWith Jim Lichtscheidl, Tracey Maloney, and Luverne Seifert\n\nMusic by Meyer Kupferman and Peter Vitale\n\nSets by Michael Sommers\n\nCostumes by Vera Mariner\n\n## 2005\n\n### ANTIGONE BY SOPHOCLES, ADAPTATION BY EMILY MANN\n\nDirected by Michelle Hensley\n\nWith Bob Davis, Kate Eifrig, Darien Johnson, Ron Menzel, Carla Noack, Sonja Parks, Luverne Seifert, and Patti Shaw\n\nSet by Stephen Mohring\n\nCostumes by Vera Mariner\n\nSound by Heather Barringer\n\n### RAGTIME BY TERENCE McNALLY\n\nDirected by Michelle Hensley\n\nMusical Direction by Marya Hart\n\nChoreography by Jim Lichtscheidl\n\nWith Aimee K. Bryant, Bob Davis, Shawn Hamilton, Darien Johnson, Jim Lichtscheidl, Vera Mariner, Matt Sciple, and Sally Wingert\n\nSets by Julian McFaul\n\nCostumes by Ellen Hutchinson\n\n### IPHIGENIA BY EURIPIDES\n\nDirected by Theodora Skipitares\n\nWith Carena Crowell, Carolyn Goelzer, Casey Grieg, Kris Nelson, and Luverne Seifert\n\nPuppets by Theodora Skipitares and Cecilia Schiller\n\nCostumes by Kathy Kohl\n\nMusic by Mike Croswell\n\n## 2004\n\n### AT YOUR SERVICE! BY KEVIN KLING\n\nDirected by Michelle Hensley\n\nWith Bradley Greenwald, Jim Lichtscheidl, and Luverne Seifert\n\nSets by Stephen Mohring\n\nCostumes by Ellen Hutchinson\n\nSound by Heather Barringer\n\n### CYRANO DE BERGERAC BY EDMOND ROSTAND\n\nDirected by Michelle Hensley\n\nWith Steve Hendrickson, Terry Hempleman, Darien Johnson, Ron Menzel, Sonja Parks, Matt Sciple, and Maren Ward\n\nSets by Stephen Mohring\n\nCostumes by Ellen Hutchinson\n\nSound by Peter Vitale\n\n### THE WINTER'S TALE BY WILLIAM SHAKESPEARE\n\nDirected by Tracy Young\n\nWith Aimee K. Bryant, Nathan Christopher, Kate Eifrig, Matt Guidry, Steve Hendrickson, and Marie-Francoise Theodore\n\nSets by Stephen Mohring\n\nCostumes by Sonya Berlovitz\n\nSound by Peter Vitale\n\n## 2003\n\n### THE GOOD PERSON OF SZECHWAN BY BERTROLT BRECHT\n\nDirected by Michelle Hensley\n\nWith Nathan Christopher, Stephen D'Ambrose, Kate Eifrig, Matt Guidry, Barbara Kingsley, Sonja Parks, Harry Waters Jr., and Karen Wiese-Thompson\n\nSets by Stephen Mohring\n\nCostumes by Vera Mariner\n\nMusic by Peter Vitale\n\n### CAROUSEL BY RODGERS AND HAMMERSTEIN\n\nDirected by Michelle Hensley\n\nMusical Direction by Peter Vitale\n\nWith Sarah Agnew, Maggie Chestovich, Carolyn Goelzer, Matt Guidry, Terry Hempleman, Tyrone Lewis, Jason Little, Ruth MacKenzie, Vera Mariner, and Matt Sciple\n\nSets by Joel Sass\n\nCostumes by Amelia Cheever\n\nMovement by Matt Guidry\n\nFiddle by Molly Sue McDonald\n\n### THE ISLAND BY ATHOL FUGARD\n\nDirected by Carol MacVey\n\nWith James Austin Williams and James Young\n\nSet by Stephen Mohring\n\nCostumes by Kathleen Richert\n\n## 2002\n\n### KING LEAR BY WILLIAM SHAKESPEARE\n\nDirected by Michelle Hensley\n\nWith Stephen D'Ambrose, Sam Chase, Barbara Kingsley, Jim Lichtscheidl, Greta Oglesby, Kim Schultz, Charles Schuminski, Matt Sciple, Luverne Seifert, Sandra Struthers, and Marie-Francoise Theodore\n\nSets by Stephen Mohring\n\nCostumes by Kathleen Richert\n\nMusic by Heather Barringer\n\n### ANNA BELLA EEMA BY LISA D'AMOUR\n\nMusic by Chris Sidorsky\n\nDirected by Michelle Hensley\n\nMusical Direction by Peter Vitale\n\nWith Larissa Kokernot, Ruth MacKenzie, and Vera Mariner\n\nSets by Stephen Mohring\n\nCostumes by Signe Albertson\n\n### MISS JULIE BY AUGUST STRINDBERG\n\nDirected by Peter Rothstein\n\nWith Norah Long, Terry Hempleman, and Carla Noack\n\nSets by Stephen Mohring\n\nCostumes by Kathleen Richert\n\nMusic by George Mauer\n\n## 2001\n\n### THE FURIES BY AESCHYLUS, WITH TEXT BY LISA D'AMOUR\n\nDirected by Michelle Hensley\n\nWith Jennifer Blagen, Carolyn Goelzer, Jodi Kellogg, Barbara Kingsley, Ron Menzel, Greta Oglesby, Matt Sciple, Luverne Seifert, and Marie-Francoise Theodore\n\nSet and Costumes by Sandy Spieler\n\nMusic by Becky Dale\n\nPercussion by Heather Barringer\n\n### THE MOST HAPPY FELLA BY FRANK LOESSER\n\nDirected by Michelle Hensley\n\nMusic Direction by Peter Vitale\n\nWith Signe Albertson, Aimee K. Bryant, Stephen D'Ambrose, Terry Hempleman, Vera Mariner, Matt Sciple, Marquetta Senters, and Esera Tuaolo\n\nSets by Stephen Mohring\n\nCostumes by Kathleen Richert\n\n### WAITING FOR GODOT BY SAMUEL BECKETT\n\nDirected by Matt Sciple\n\nWith Adena Brumer, Zach Curtis, Jim Lichtscheidl, Dale Pfeilsticker, and Karen Wiese-Thompson\n\nSet by Mark Abel Garcia and Matt Sciple\n\nCostumes by Ellen Hutchinson\n\nSound by Dale Pfeilsticker\n\n## 2000\n\n### CYMBELINE BY WILLIAM SHAKESPEARE\n\nDirected by Michelle Hensley\n\nWith Aimee K. Bryant, Warren Bowles, Zach Curtis, Steve Hendrickson, Kelly Hilliard, Barbara Kingsley, Norah Long, and Charles Schuminski\n\nSet by Stephen Mohring\n\nCostumes by Ellen Hutchinson\n\nMusic by Mike Croswell\n\n### THE THREE LIVES OF LUCIE CABROL BY JOHN BERGER\n\nDirected by Michelle Hensley\n\nWith Brian Baumgartner, Pete Colburn, Matt Guidry, Terry Hempleman, Jodi Kellogg, Larissa Kokernot, and Matt Sciple\n\nSet by Stephen Mohring\n\nCostumes by Ellen Hutchinson\n\nMusic by Peter Vitale\n\nFiddle by Molly Sue McDonald\n\n## 1999\n\n### THE TEMPEST BY WILLIAM SHAKESPEARE\n\nDirected by Michelle Hensley\n\nWith Signe Albertson, Aimee K. Bryant, Zach Curtis, Stephen D'Ambrose, Steve Hendrickson, Jodi Kellogg, Barbara Kingsley, and Ron Menzel\n\nSet by Stephen Mohring\n\nCostumes by Ellen Hutchinson\n\nMusic by Mike Croswell\n\n### THE UNSINKABLE MOLLY BROWN BY MEREDITH WILLSON\n\nDirected by Michelle Hensley\n\nMusic Direction by Peter Vitale\n\nWith Jodi Kellogg, Larissa Kokernot, Ron Menzel, Dale Pfeilsticker, Grant Richey, Carolyn Pool, Matt Sciple, and Ruth Williams\n\nSet by Stephen Mohring\n\nCostumes by Signe Albertson and Ellen Hutchinson\n\nFiddle by Char Bostrom\n\n## 1998\n\n### MEASURE FOR MEASURE BY WILLIAM SHAKESPEARE\n\nDirected by Michelle Hensley\n\nWith Signe Albertson, Zach Curtis, Mary Alette Davis, Steve Hendrickson, Barbara Kingsley, Matt Sciple, Elizabeth Teefy, Peter Vitale, and Molly Rose Hensley-Clancy\n\nSets by Stephen Mohring\n\nCostumes by Ellen Hutchinson\n\nMusic by Peter Vitale\n\n### THE BALLAD OF THE SAD CAFE BY EDWARD ALBEE AND CARSON McCULLERS AND DROWNING BY MAR\u00cdA IRENE FORN\u00c9S\n\nDirected by Michelle Hensley\n\nWith Brian Baumgartner, Carolyn Goelzer, Terry Hempleman, Jon Micheels Leiseth, Carolyn Pool, and Matt Sciple\n\nSet and Masks by Stephen Mohring\n\nCostumes by Ellen Hutchinson\n\nMusic by Eric Mohring and Brian Barnes\n\n## 1997\n\n### THE EMPEROR OF THE MOON BY APHRA BEHN\n\nDirected by Michelle Hensley\n\nWith Signe Albertson, Greg Arata, Aimee K. Bryant, Barbara Kingsley, Larissa Kokernot, Dale Pfeilsticker, Matt Sciple, and Molly Hensley-Clancy\n\nSet by Joe Stanley\n\nCostumes by Ellen Hutchinson\n\nMusic by Peter Vitale\n\n### DAYS ARE SILVER NIGHTS ARE GOLD BY ERIK EHN\n\nDirected by Michelle Hensley\n\nWith Signe Albertson, Aimee K. Bryant, Christopher Edwards, and Molly Hensley-Clancy\n\nSet and Costumes by Julie Archer\n\nMusic by John Banks\n\n## 1996\n\n### THE CAUCASIAN CHALK CIRCLE BY BERTOLT BRECHT\n\nDirected by Michelle Hensley\n\nWith Signe Albertson, Barbara Kingsley, Larissa Kokernot, Ruth McKenzie, Alexander Parker, Dale Pfeilsticker, Carolyn Pool, Matt Sciple, Peter Vitale, and Molly Hensley-Clancy\n\nSet and Costumes by Ellen Hutchinson\n\nMusic by Scott Spencer\n\n### MUD BY MAR\u00cdA IRENE FORN\u00c9S\n\nDirected by Michelle Hensley\n\nWith Mike McGowan, Maren Perry, and Luverne Seifert\n\nSets and Costumes by Ellen Hutchinson\n\nMusic by Lisa Fuglie\n\n## 1995\n\n### THE KING STAG (THE QUEEN STAG) BY CARLO GOZZI\n\nDirected by Michelle Hensley\n\nWith George Keller, Larissa Kokernot, Rhonda Lund, Mike McGowan, Maren Perry, and Luu Pham\n\nSets and Costumes by Ellen Hutchinson\n\n## 1994\n\n### LIFE'S A DREAM BY PEDRO CALDER\u00d3N DE LA BARCA\n\nDirected by Michelle Hensley\n\nWith Brian Chapman-Evans, Don Cosgrove, Dana Farner, Mike McGowan, Maren Perry, William Bird Wilkens, and Ahanti Young\n\nSets and Costumes by Ellen Hutchinson\n\nMusic by Bob Hughes\n\n## 1992\n\n### ELECTRA BY SOPHOCLES\n\nDirected by Michelle Hensley\n\nWith Gayle Harbor, Kenneth Ransom, Lucy Rodriguez, Ivonne Coll, Bill Burns, and Christianne Mays\n\nSets and Costumes by Elizabeth Luce\n\nMusic by Gary Johnson\n\n## 1991\n\n### THE GOOD PERSON OF SZECHWAN BY BERTOLT BRECHT\n\nDirected by Michelle Hensley\n\nWith Bill Burns, Kate Cherry, Craig Fernandez, Marilyn Henkus, Suzanne Kato, Christianne Mays, and Kenneth Ransom\n\nSets and Costumes by Norma Bowles\n\nMusic by Marilyn Henkus\n\n# ACKNOWLEDGMENTS\n\nSO MANY ARTISTS, colleagues, and audience members have been crucial to the way Ten Thousand Things has taken shape today. It is impossible to even begin to name and thank them adequately for all their contributions. But I would like to thank these especially formative people in my life and this creation process:\n\nCarol MacVey, for demonstrating the possibilities for playfulness, humor, and joy in the rehearsal room; Polly Carl, a longtime friend, former Ten Thousand Things board member, and creator of HowlRound, for her steadfast support and for being the first to step up and offer to publish this book; Kira Obolensky, Ten Thousand Things' resident playwright, for our wonderful friendship, for her collaboration in editing this book, and for her astonishing imagination, curiosity, and \"soul mate\" understanding of the kinds of plays and storytelling Ten Thousand Things needs; Lynette D'Amico for her editing, followed by the enthusiastic editors and staff at the Minnesota Historical Society Press; my mom for giving me wings, and my dad for giving me roots; Molly for bringing unimaginable joy and delight into my life, and Bill for being there, always, with his patience, wisdom, humor, and kindness.\n\nMICHELLE HENSLEY is founder and artistic director of Ten Thousand Things in Minneapolis, where she has brought over fifty tours of award-winning drama to nontraditional audiences in prisons, shelters, and housing projects, as well as the general public.\n\nAll the Lights On was designed and set in type by Judy Gilats in St. Paul, Minnesota. The display face (borrowed from Jim Davis's cover design) is Goshen, designed by Chank Diesel. The text face is Kievit Slab, designed by Mike Abbink. The book was printed at Sheridan in Ann Arbor, Michigan.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nLucy MacDonald is a Quebec-based motivational speaker with an academic background in psychology and counselling. Her clients include Pfizer Canada and McGill University.\nTo my father, Bernard Filliol,  \nwho taught me my first  \nand most important lesson  \nin time management,\n\nand\n\nto my mother, Aline Filliol, who taught me the  \nimportance of making  \ntime for myself.\n\n## contents\n\nIntroduction\n\n**Understanding time**\n\nForms of time\n\nWhat is time management?\n\nThe benefits of time management\n\n_Exercise one: map your time_\n\nTime urgency\n\n_Exercise two: find your hurry zone_\n\nThe Pareto Principle\n\n_Exercise three: keep a time-use diary_\n\n**What's stopping you?**\n\nProcrastination\n\n_Exercise four: stop procrastinating now_\n\nPerfectionism\n\n_Exercise five: create a collage of your aspirations_\n\nFear of failure\n\nFear of success\n\n_Exercise six: find your own definition of success_\n\nWorkaholism\n\n_Exercise seven: assess your attitude to work_\n\n_Exercise eight: learn to set limits_\n\n**Quality time**\n\nWhat is important to you?\n\n_Exercise nine: identify your values_\n\nThe purpose of your life\n\n_Exercise ten: write your personal \"mission statement\"_\n\n_Exercise eleven: leave your own legacy_\n\n**Time to plan**\n\nChoosing and using your planner\n\nPlanning your week\n\n_Exercise twelve: analyze your time budget_\n\nYour daily schedule\n\n_Exercise thirteen: plan your day the easy way_\n\nLong-term planning\n\n_Exercise fourteen: set your five-year goal_\n\n**Time management at work**\n\nManaging your business contacts\n\nPaper management\n\n_Exercise fifteen: climb the paper mountain_\n\nDe-clutter your office\n\nTime-saving technology\n\nManaging email\n\nPhone \u2013 slave or master?\n\n_Exercise sixteen: track your phone time_\n\nProject management\n\n**Time-wasters and time-savers**\n\nInformation overload\n\nThe management of meetings\n\n_Exercise seventeen: make your meetings meaningful_\n\nDealing with interrruptions\n\nLearn to delegate\n\n_Exercise eighteen: assess your workload_\n\nHow to say \"no\"\n\nMaking use of waiting time\n\n**Time management for everyone**\n\nTime management for children\n\n_Exercise nineteen: make a countdown calendar for kids_\n\n_Exercise twenty: make time \"pizzas\"_\n\nTime management for students\n\n_Exercise twenty-one: create a semester schedule_\n\n_Exercise twenty-two: make a weekly schedule_\n\n_Exercise twenty-three: create a study schedule for exams_\n\nTime management for parents\n\n_Exercise twenty-four: create a \"command center\"_\n\nWorking from home\n\nTime management for creative people\n\n_Exercise twenty-five: take advantage of your creative time_\n\nManaging time in retirement\n\n_Exercise twenty-six: map your retirement_\n\n_Exercise twenty-seven: visualize a positive retirement_\n\nConclusion\n\nFurther reading\n\nStudies and Papers\n\nAcknowledgments\n\n## introduction\n\nWe've all been there. Running through the house looking for lost car keys, late for an appointment, thinking about the mess we're leaving behind, wondering when we will have time to drop off the dry cleaning, worrying that our current work project will not get done on time, angry that the person in front of us is moving too slowly ... . Hurry up, hurry up, we tell ourselves and those around us. The whole world is in our way. We feel stressed, upset, and resentful.\n\nFrom the time we get up in the morning until we put our heads on the pillow at the end of the day, we think about what we have to do, and we worry about what's not getting done. We wake up in the middle of the night worrying about when we will find the time we need to complete the never-ending list of \"To Dos\". If something goes off schedule we just give up, despairing that we will ever be able to catch up.\n\nIf this sounds like you, you are not alone. We are living in a period of the time-famished; we are hurried and harried with no end in sight. People are exhausted and overwhelmed, with no time for personal conversations, for taking a leisurely walk, for waiting patiently for a child to finish their dinner.\n\nHaving enough time is the luxury item of the moment. Today men and women \u2013 and even children \u2013 all around the world are experiencing increasing levels of time stress. This chronic stress is linked to poor physical and psychological health.\n\nWe all have the same 24 hours in the daily time bank. Time is the constant, so what needs to change are the choices that we make about how we use our time. Managing time is about making decisions concerning how you want to live your life. You have the life you have because of the decisions you make regarding how you spend your time. If you want a different life, you must make different decisions. If you are not convinced about this, look at the successful people you know and pay attention to how they spend their time.\n\nMy own life is an example of how managing time has enabled someone to achieve their goals. My earliest recollection of the importance of time management goes back to when I was five years old and my youngest brother, Robbie, was born. The long-awaited day when he was coming home arrived, and I so wanted to be part of the welcoming party. My father told me to hurry up and get ready. Excited, I took immense care and a long time to choose the perfect outfit for this important event \u2013 only to discover that the welcoming party had left without me. I was devastated. This incident made such an impression of me that, to this date, I hate being late for anything.\n\nMy mother was also instrumental in teaching me how to manage my time. Despite having five children, she was employed in an era when most mothers did not work outside the home. She was able to keep things running fairly smoothly by maintaining a routine and being an expert in delegation, enlisting all five of us to help with daily chores.\n\nI've acquired my time-management skills through trial and error. Each stage of my life has presented different challenges when it comes to time, and I have had to make adjustments. I learned basic time-management skills when I became the mother of four children whose ages ranged over ten years from oldest to youngest. I quickly realized that unless I kept a very firm grip on the daily schedule, the family descended into chaos. I also discovered that it was essential to put into practice my mother's one and only piece of parenting advice: \"When you need time for yourself, take it, because no one will come and offer it to you.\"\n\nIt wasn't until I returned to university at the age of thirty nine, when my children were aged between 8 and 18, that my time-management skills became finely honed. I attended university for six years \u2013 winter, spring, summer, and fall \u2013 and completed a BA in Psychology and a Masters in Education \u2013 Counseling Psychology. In order to graduate I used all the main time-management techniques outlined in this book. I had a vision of what I wanted to accomplish, I set myself goals, I created a plan for each semester, and I lived according to a weekly and daily schedule. I learned how to avoid procrastination, and how to deal with disruptions and time-wasters. I used my daily planner as a road map to achieve my goals \u2013 much the same as I do today.\n\nAfter graduation I started a private counseling practice and seminar business, and once again my ability to manage my time effectively was, and is, a tremendous advantage. As I am a self-employed businesswoman, time is the framework within which I earn a living \u2013 a poor grasp of time would be reflected on my business balance sheet.\n\nHowever, I wouldn't want you to get the impression that I always manage my time well. That is simply not the case. I continue to be challenged by new situations, but I am motivated to get back on track because I believe that by wasting my time I am wasting my life. Whenever I hit difficulties, I am able to renew my commitment to use my time wisely by revisiting my life purpose and goals.\n\nTime is an extremely precious resource, and managing it well can help you to achieve your goals and your dreams. _You Can Manage Your Time Better_ is a straightforward, step-by-step guide that can be of benefit to anyone, anywhere \u2013 from a child to a retired person, and at school, at home, or in the workplace.\n\nI hope that you will use this book to set yourself free: free from the chronic hurriedness of everyday living, free from the fear of wasting time, and free from the tyranny of having too much to do and not enough time in which to do it.\n\nUltimately, how you spend your time is how you live your life \u2013 so spend it wisely.\n\nLucy MacDonald\n\n## understanding time\n\n\"I wish I had more time,\" is a common complaint. Although we can't increase the number of hours in the day, we can decide what we are going to do with them. The first step to gaining control of your life is to understand what effective time management is and how it will help you to reduce time-related stress.\n\nIn this chapter you will discover everything you need to know to start managing your time. You will learn different ways to track how you spend your time, and how to focus your energies on the most productive and fulfilling activities. You can use the Hurry Zone quiz to get an idea of your level of \"hurry sickness\", then follow the practical suggestions to help you slow down and enjoy life a little more.\n\n## forms of time\n\nIf I were to stop you in the street and ask you what time it is, chances are you would look at your wristwatch and tell me. But what if I were to ask you \"What is time?\" If you are not able to answer readily, you are in good company. The concept of time has tested the greatest thinkers throughout history.\n\nIn the 17th century Isaac Newton proposed the idea that time is absolute \u2013 a universal, steady process that everyone everywhere experiences in the same way. Albert Einstein challenged Newton's idea of time with his general theory of relativity \u2013 that time is relative, dependent upon how fast we are traveling.\n\n_\"What, then, is time? If no one asks me, I know what it is. If I wish to explain it to him who asks, I do not know.\"_\n\nST. AUGUSTINE (354\u2013386)\n\nAnother way to try to understand time is to take a brief look at horology, the study of time measurement. Throughout history civilizations observed the stars, the cycle of the seasons and the movement of day into night, devising ingenious ways to track of the passage of time. The earliest Egyptian calendars were based on the moon's cycles and evolved into a 365-day calendar around 4300bc. Starting in 1582 the Gregorian calendar was adopted by most of the West and is the most commonly used civil calendar today.\n\nOriginating in the 11th century, the hourglass was used to gauge everything, from the speed of a ship to the length of a preacher's sermon. In the 1300s mechanical clocks using weights or springs were built in Europe. The advent of the mechanical clock marked the shift from observing the passage of time to providing a precise calibration. Clocks were further refined with the development of the pendulum clock in the 1600s. With each generation of clocks, the measurement of time became more accurate. The development of the wristwatch made the measurement of time portable, while the mass production of watches made it widely accessible. Today we have the cesium fountain atomic clock that keeps almost perfect time. Out by only a second every 20 million years, this clock is far more precise than most of us will ever require in our day-to-day lives!\n\n#### How we experience time\n\nThe ancient Greeks conceptualized time as either chronos or kyros. Chronos is clock time measured in seconds, minutes, and hours, and it is the root of words such as \"chronology\" and \"chronological.\" Kyros is timelessness, living in the here and now, synchronicity and carpe diem \u2013 seizing the moment. Chronos and kyros represent quantity and quality, which reflect what our struggle with time is all about. While chronos is fixed at an objective rate, kyros varies considerably because it is subjective and can be a positive or negative experience. To most people 45 minutes spent in the dentist's chair will feel longer than the same period of time spent reading a fascinating book.\n\nHow can we experience more positive kyros time and less chronos? Doing one thing at a time, refusing to live an over-scheduled life, and making time for the important people and activities in our lives are key ways to help us balance the quantity and the quality of our time.\n\n## what is time management?\n\nTime management is really a misnomer. We can't actually manage time because it moves on regardless of what we do, or don't do. The objective is to take control of the time that we do have, so that we can fulfill our needs and responsibilities. You want to make sure that your intended use of time equals your actual time use. Chronically mismanaging your time can leave you feeling hurried and harried. Time management can be described more accurately as self-management. We can all make better choices about what we do with our time. Think of someone you know who is a good time manager. How would you describe them? Organized, productive, competent, happy, calm? That is what managing time can do for you.\n\nDay-to-day time management is about finding a system that helps you to gain more control over your 24-hour day so that you can balance roles, spending your time doing the things you want to do and have to do. There are many time-management systems, from simple paper planners to state-of-the-art computers and electronic handheld organizers (we shall explore these later in more depth, see pp.68\u201371). Your might prefer one system or another, depending on your circumstances and stage of life, so don't be afraid to let go of a method that no longer works for you and experiment with another approach as your needs change.\n\nThere is no one, right way to manage time. Your approach depends on what is important to you: This book will help you identify these values and goals, in both the long and short term. But regardless of what type of time-management system you choose, the key is to use it!\n\nSuccessful people recognize that time management is crucial to help them to reach their goals, both personal and professional. Learning to manage your time effectively and efficiently is an important life tool that can help you at home and at work to achieve whatever is important to you, and to live the life of your dreams.\n\n## the benefits of time management\n\nBy managing your time you can improve your quality of life. Imagine waking up knowing that you will be able to accomplish the day's tasks with a minimum of stress. Who wouldn't want that?\n\n**Reduce stress and anxiety** Much of our daily stress is related to the time crunch: too much to do in too little time. Structuring your time will help relieve chronic stress, which, over weeks, months, and years, becomes a slippery slope toward burnout and depression. Reduce your stress and anxiety levels by including some relaxation in your schedule.\n\n**Increase personal satisfaction** When was the last time you had that \"job-well-done\" feeling? Not using our time effectively leaves us feeling dissatisfied with ourselves, resulting in us putting ourselves down and thinking negatively. Schedule enough time to get the job done by adding some buffer time to take into account unforeseen interruptions and delays.\n\n**You are in control of your life** This is your life and you get only one go (as far as I know). Taking control of how you spend your time is permitting yourself to live the life you choose. Make a three-column list: \"Must Do,\" \"Should Do,\" and \"Would Like to Do.\" Your \"Must Do\" list includes non-negotiable things, such as making a living and looking after your children. Evaluate your \"Should Do\" list and see how many \"Should Dos\" you could stop doing. Now add some \"Would Like to Dos\" into your schedule.\n\n##### **misconceptions about time management**\n\nPerhaps you've never tried managing your time because you don't think you have the time to do so. Perhaps you've tried managing your time, but wound up using a system that was cumbersome and time-consuming. The truth is that time management does require an initial investment of your time \u2013 to draw up a plan, to get organized, and to determine how you wish to spend your time. But once you've laid the groundwork, you need less than ten minutes per day to put your plan into action.\n\nAlthough the best time-management plans are often the simplest, time management is not necessarily mere common sense. For example, are you truly achieving your goals? What? No goals? Then identifying your goals would be a great place to start. Are you spending your time wisely or is your time account on the verge of going into the red? What is required beyond a plan is the daily self-management or discipline to use that plan.\n\nHave you ever heard anyone say, \"Time management takes away the edge I get from working under pressure\"? There is a difference between having enough pressure to motivate ourselves to get things done and being in a panic because we're in danger of missing the deadline. Procrastinating is never a good thing, and the chronic stress of having to work feverishly on tasks to make up for lost time as the due date approaches is simply unhealthy.\n\nIf you are worried that learning to manage your time well is going to make you inflexible and take all the fun and spontaneity out of your life, think again. Structuring your time means including time to go to the movies, or take a walk, or read that bestseller. Want more time and more spontaneity? Block off a few hours each month with no set plan or goal and see where the free time takes you.\n\n_\"Start by doing what is necessary; then do what is possible; and suddenly you are doing the impossible.\"_\n\nST. FRANCIS OF ASSISI (1181\/2\u20131226)\n\n**Create work\u2013life balance** Scheduling your time helps you create balance between your work and personal life. Making an appointment with yourself emphasizes that personal time is just as important as work time.\n\n**Be more productive** Organizing your time helps you to accomplish more in the allotted time because you are able to focus on the job at hand without worrying that you should be doing something else.\n\nAchieve more of what is important to you Using a good time-management system helps you not only to get things done but also allows you to examine why you are doing certain chores\/jobs and thus focus on what is meaningful and personally rewarding to you.\n\n### _map your time_\n\n### **EXERCISE ONE**\n\nSome days it can feel like you don't accomplish very much. To discover exactly what you do with your time, try this exercise, which will help you to list all your various activities, tasks, and responsibilities over the course of one month.\n\n_**1.** Taking a sheet of paper and a pen, jot down the various roles you play in your life \u2013 for example: husband, father, daughter, employee, volunteer, friend, and so on._\n\n_**2.** Now take another sheet of paper and divide it into columns, writing one role at the top of each. Under each role list the activities associated with it. For example, under \"father\" you might have \"reading the children a bedtime story\" and \"taking the boys to soccer practice.\" Make sure to include socializing, such as meeting with friends, as this also requires time._\n\n_**3.** Update your list daily, but if this is not practical, do it at least twice a week._\n\n_**4.** After a month review the list and assign a number from 1 to 5 to each activity, task, and responsibility, 1 being most important and 5 the least._\n\n_**5.** Analyze your results. How many 1s do you have? How much time do you spend doing them? And how many 5s? How much time do you spend doing them?_\n\n_**6.** Starting with your least-important activities, tasks, and responsibilities, assess whether they are all really necessary. What would happen if you stopped doing them? This will help you weed out unimportant activities and give you more time to spend doing important things._\n\n## time urgency\n\nEveryone, it seems, is in a hurry. Wherever we go we can witness the effects of \"hurry sickness\": people getting upset because they have to wait in line, motorists honking horns because the driver in front of them is not going fast enough. You might succumb to \"hurry sickness\" if: you regularly find that you are short of time; you have too much to do so you do everything as quickly as possible; or, a delay of any kind upsets you.\n\nIn the 1950s, doctors Meyer Friedman and Ray Rosenman coined the term \"hurry sickness\" while studying personality types. As a result they defined the \"Type A personality,\" which is characterized by impatience, hostility, competitiveness, tension, and aggression.\n\nExpanding on the work of Freidman and Rosenman, Dr. Li Jing L. Yan zeroed in on a Type A behaviour pattern that has three main features: time urgency\/impatience, hostility, and competitive drive. In 2002 Dr. Li Jing L. Yan found that of the 3,000 men and women he studied, those who spent their 20s hurrying from one thing to another and feeling impatient were twice as likely to have high blood pressure 15 years later than the people who took things at a slower pace. High blood pressure is an important risk factor for developing heart disease and\/or having a stroke.\n\nLiving in a state of chronic time stress has an adverse effect on your health. Feeling constantly pressured for time can weaken your immune system; cause muscle pain, insomnia, and headaches; and increase your risk of depression and anxiety. Stress as a result of time urgency and hostility can decrease the supply of blood to the muscles of the heart, and studies have shown that these time-pressure episodes can be a precursor to a heart attack.\n\nAs with many problems, we can modify and manage the problem of time urgency. Take time to do activities that require you to slow down. Visit an art gallery; listen to someone without interrupting them; do only one thing at a time; or take up meditation. Practice patience when waiting in line \u2013 the proverbial \"counting to ten\" technique is quick and easy to use in such situations. Of course, it is important to make sure that you leave yourself enough time to get your errands done in the first place. Estimate how long each task will take you and then factor in extra time for delays, such as traffic jams. Walk instead of driving whenever possible \u2013 walking is good exercise and it will help to release stress.\n\n##### **the relaxed personality**\n\nFriedman and Rosenman also developed a personality profile called the Type B personality. Type B people are able to work without becoming anxious and agitated, be patient with delays, and relax without feeling guilty.\n\nHere are some ways to develop Type B characteristics in yourself:\n\n\u2022 Concentrate on speaking slowly and pause briefly between sentences \u2013 people will experience you as more thoughtful and less aggressive.\n\n\u2022 Participate in activities involving slow movement, such as tai chi or drawing, to help you appreciate unhurried, deliberate action.\n\n\u2022 Don't take delays personally. Delays are part of life and most often have nothing to do with us (unless, of course, we've not allowed ourselves enough time to get things done in the first place).\n\n_\"Though I am always in haste, I am never in a hurry.\"_\n\nJOHN WESLEY (1703\u201391)\n\nAbdominal breathing using your diaphragm is another great way to slow down. To get a sense of abdominal breathing, try placing your right hand on your abdomen and your left hand on your chest. Inhale slowly through your nose, allowing your abdomen to expand first. Your right hand should move outward as you breathe in and your left hand should stay relatively still. Now practice it without using your hands. This is an excellent way for you to reduce stress in almost any situation. You might like to try this the next time you have to wait in line. This is also a wonderful relaxation technique to teach to children.\n\n### _find your hurry zone_\n\n### **EXERCISE TWO**\n\nTake the Hurry Zone quiz to discover if you are in the green, yellow, or red hurry zone. Answer each question and award yourself points according to the following scale: 0 = never 1 = not often 2 = most of the time 3 = always.\n\nThen add up your total score and discover which zone you are in.\n\n\u2022 _I walk fast._\n\n\u2022 _I talk fast._\n\n\u2022 _I'm impatient when people speak slowly._\n\n\u2022 _I interrupt people when they speak._\n\n\u2022 _I like being first in line._\n\n\u2022 _I'm impatient if I have to wait in line or in traffic._\n\n\u2022 _I eat my meals quickly and often don't take time to enjoy what I'm eating._\n\n\u2022 _I do more that one thing at a time._\n\n\u2022 _I'm fidgety._\n\n\u2022 _I get my work done quickly._\n\n\u2022 _I get impatient with office machinery._\n\n\u2022 _I charge around when I'm shopping and often forget things in the process._\n\n**_Your score_**\n\n**_Green Zone = 0 to 12_**\n\n_You are in the \"no hurry\" zone. But if you scored any 2s or 3s, look for ways to slow yourself down in such situations._\n\n**_Yellow Zone = 13 to 24_**\n\n_Proceed with caution! Try to become hurry-savvy; that is, allow more time to do things and make an effort to slow down._\n\n**_Red Zone = 25 or more_**\n\n_Stop! You are in hurry overload and you can't continue without serious consequences to your well-being. You must take stock and make yourself slow down._\n\n## the pareto principle\n\nHave you ever noticed that approximately 80 per cent of decisions come from 20 per cent of meeting time; that 80 per cent of the phone calls you make are to 20 per cent of the people in your life; and so on. In time management the Pareto Principle (named after its inventor Vilfredo Pareto), also known as the 80\/20 rule, is the idea that 20 per cent of time and effort generate 80 per cent of results. Of course, the corollary is also true: 20 per cent of results take up 80 per cent of time and effort.\n\nThe quality management expert J.M. Juran described the 20 per cent of time and effort as the \"vital few\" and the 80 per cent as the \"trivial or useful many.\" When you are functioning in the \"trivial many\" (the 80 per cent zone), you may complain about how little time you have, be doing tasks that you are not interested in or not particularly good at, or be doing things that are not connected to either your short- or long-term goals. You are functioning in the \"vital few\" (the 20 per cent zone) when your interest is high, you are good at what you are doing, your creativity is being put to good use, and you are contributing to your short- and long-term goals.\n\nFocusing on the 20 per cent that matters The 80\/20 rule helps us to shift the focus away from being busy for its own sake to concentrating on results. Use a list of your \"vital few\" activities (see Exercise three, opposite) to remind yourself when you get sidetracked by \"trivial-many\" items, or when there is a crisis. After a crisis use the list to make sure that you don't waste any precious time in the 80 per cent zone.\n\n### _keep a time-use diary_\n\n### **EXERCISE THREE**\n\nThis exercise is specifically designed to help you evaluate where your 80\/20 time split falls by recording and assessing the relative importance of every activity you perform over a seven-day period. You will need a diary or small notebook, a pen, and a sheet of paper.\n\n_**1.** Taking your diary or notebook and pen, mark out your waking time for the folowing seven days in fifteen-minute periods._\n\n_**2.** Each evening fill in how you spent your time that day. Make sure that you include everything, no matter how trivial it might seem. Do this until you have a record for the entire week._\n\n_**3.** Review where your time went. Take the sheet of paper and draw two columns. Name one \"Personal\" and the other \"Work\" and allocate your week's activities accordingly. (Whether you work in an office, from home, or are a full-time parent \u2013 you work.)_\n\n_**4.** Next, decide which activities fall into the 80 per cent zone. Then analyze which activities are in the 20 per cent zone. For example, if you sell gift baskets from home but spend two hours per day filing, are those two hours developing your business? No. So such filing is a \"trivial-many\" activity. A \"vital-few\", work-related activity might be contacting customers to ensure that they were satisfied with their order._\n\n_**5.** Once you have a clear idea of where your time is going, resolve to focus more on the important \"vital-few\" activities and begin to cut down on the less productive \"trivial-many\" things that you do._\n\n## what's stopping you?\n\nSome of the biggest and most common obstacles to managing time efficiently are procrastination (putting things off, and perhaps never doing them at all), perfectionism, fear of failure, fear of success, and workaholism. These obstacles all have one thing in common \u2013 they originate in our thoughts, beliefs, and our self-concept.\n\nIn this chapter you will explore how to stop the cycle of procrastination and how to break the drive for perfection, which holds you back from accomplishing new things. You can also take a quiz to find out if you are working too hard. You will be given all the tools you need to change your negative thinking and to discover what is holding you back from becoming an expert time manager.\n\n## procrastination\n\nIt is 3am and Janet is sitting at her desk, staring in to space. She is feeling very anxious because she needs to come up with some clever and innovative ideas for the company's latest sales drive, and she has to present them to her boss tomorrow morning. She knew three weeks ago that the deadline was tomorrow, yet she didn't start working on the project until yesterday. Was she too busy? Perhaps the task was too difficult for her or she wasn't briefed properly? None of the above: Janet is struggling with procrastination, and it's causing her a great deal of stress.\n\nProcrastination \u2013 from the Latin pro cras meaning \"for tomorrow\" \u2013 means to defer a task until the last moment, or even to fail to do it at all. At a simple level, procrastination can result in missed deadlines, but it can also leave us feeling guilty, inadequate, self-loathing, and even depressed.\n\nIf people feel so bad because they leave things to the last minute, why do they do it? Here are some of the reasons why we procrastinate.\n\n**Fear of failure** (see pp.38\u201340) \"I just know I can't do a good job of this. Better not to do it at all than to show them that I can't do it.\"\n\n**Fear of success** (see pp.41\u20133) \"If I make a good job of this project, they'll continue to expect more and more from me.\" Or, \"I don't deserve praise \u2013 recognition makes me feel uncomfortable.\"\n\n**Negative self-beliefs** If you tell yourself over and over again \"I'm not good at this,\" you will put off the anxiety-producing task.\n\n**Being too busy** \"I am much too busy to get anything else done.\"\n\n**Being disorganized** \"I need to get myself organized before I can write that report.\"\n\n##### **the cycle of procrastination**\n\nBetween tasks we start the procrastination cycle by thinking \"Next time, I'll start earlier.\" Then, when we have our next task, we think \"I'll start soon.\" Anxiety builds as we begin an internal chain of thoughts along the lines of:\n\n\u2022 What will happen if I don't start?\n\n\u2022 I'm doing everything except what I should be doing.\n\n\u2022 I can't enjoy doing anything else while this is waiting to be done.\n\n\u2022 Why didn't I start sooner?\n\n\u2022 What if someone finds out that I'm in this mess?\n\nBut the procrastinator thinks, \"There's still time for me to get it done.\" However, then self-criticism often sets in and they lose confidence, asking \"Do I get the job done or do I abandon ship?\" Having failed to complete the task and feeling very negative, the procrastinator vows never to put things off again. But after the memory of the anxiety has faded, the cycle begins again.\n\n**Poor planning and time management** \"I should not have spent so much time with my friends at the weekend when I knew that I had this job to do.\"\n\n**Getting easily frustrated** \"I just can't stand it when I try to work and it doesn't go smoothly or easily. I quit.\"\n\n**Depression** \"There's no point even trying to do this. What's the use?\"\n\n#### Overcoming procrastination\n\nReflect on the price you are paying for procrastinating: anxiety, depression, and stress. Also, be realistic about what it has cost you in grades, employment opportunities, or career advancement, and the toll it has taken on your relationships. Now resolve to do something about it. The earlier in the cycle you can stop yourself, the easier it will be to get back on track. Then, break down the task into small, manageable steps. Write a schedule, setting a date by which each step will be completed. Now comes the crunch \u2013 getting started. This is often the biggest hurdle, but start you must. After you complete a step, give yourself a little reward. Then, following your schedule, start on the next one. As you complete each step you will gain in confidence, and before you know it, you will have reached the end of the project. Congratulate yourself!\n\n### _Stop procrastinating now_\n\n### **EXERCISE FOUR**\n\nIf you are currently delaying doing something that you know needs to get done, here is a way to talk yourself into meeting that important deadline.\n\n_**1.** Write down the task that you are avoiding. \"Study more\" is too vague; something more specific, such as \"Write essay on Shakespeare's sonnets,\" is much better._\n\n_**2.** Ask yourself why you are avoiding it. Don't you like the task? Are you afraid that you can't do it well? Do you feel you are too disorganized? If you are having difficulty discovering why you procrastinate, ask a friend or colleague the reasons why they might put things off._\n\n_**3.** List reasons why delaying might benefit you. For example, you can do other things that you prefer instead; you don't actually have to get down to work; you don't have to face anxieties about the task; and so on._\n\n_**4.** List reasons why delaying might hurt you. For example, you won't get a good exam grade; or your children will be too embarrassed to have their friends over if you continue to avoid cleaning the house._\n\n_**5.** Think positively about yourself and the task before you. Break the task down into smaller, more manageable chunks, and set yourself mini-deadlines by which you need to have completed each part of the task. Aim for completion, not perfection._\n\n_**6.** Ask a friend or colleague to help you by making you accountable to them. Tell them the date by which the task has to be completed and get them to check on your progress from time to time._\n\n## perfectionism\n\nWhen we have a task to do, most of us strive to do a good job. However, for some people, this is not enough \u2013 they want to do a perfect job. Perfectionists live in a world of self-imposed, unrealistic standards and expectations that cause them to strive continually for the unobtainable. They don't tolerate mistakes and they become bogged down in the smallest details, often starting over again and again in a bid to get the job done absolutely right.\n\nIn time management, perfectionism and procrastination are two sides of the same coin because both prevent us from using our time wisely and completing the task. A perfectionist seldom accomplishes anything that is acceptable the first time around. They may delay starting a task because they know that it will take a huge amount of energy and they will become frustrated, angry, and anxious as time passes.\n\nPerfectionism often starts in childhood as a desire to please others. Gifted children can struggle with the high standards imposed by adults and take on these standards as their own. They may mistakenly internalize the message that their value is equal to their performance. The child wants to please their parents and teachers, and over time their desire to please others turns into a fear of rejection. Often motivated more by fear than the desire to succeed, the perfectionist may avoid new situations or refuse a job promotion because they are afraid that they cannot perform to their own high standards.\n\nThe perfectionist lives by the rule of the \"shoulds\": I should be perfect and everything I do should be perfect; I shouldn't make mistakes; I should always give 150 per cent regardless of the circumstances; I shouldn't attempt to do things if I know ahead of time that they won't be perfect.\n\nBeing a perfectionist can be a lonely experience. Low self-esteem and lack of confidence are obstacles that perfectionists struggle with. Unwilling to try new things for fear that they can't do them well, the perfectionist may become stuck in rigid thinking about what they can and cannot do. Perfectionists are at risk of pessimistic thinking and depression. All-or-nothing thinking is a forerunner to pessimism and depression in the perfectionist's world. Anything less than 100 per cent success is a total failure. \"If I can't do it perfectly, I won't do it at all.\"\n\n#### Overcoming perfectionism\n\nAccept that making mistakes is a human trait. You will learn more about yourself through failure than success, but nothing is a complete failure if you've learned something.\n\nStrive for excellence instead of perfection. Block out the time required, do the preparation, get the job done in the time allotted, and feel good about it. Use the Pareto Principle (see pp.26\u20137) to help you learn where to put your time and energy. In this case 20 per cent of your effort produces 80 per cent of the quality. Doing an adequate or satisfactory job (the 80 per cent) is much better than getting nothing done at all.\n\nLearn to differentiate between doing something right and doing the right thing by asking yourself if the details that you are fussing over are important to the overall project. Give yourself a reasonable schedule and stick to it. Use a small alarm clock or timer to remind you when the time is up. Then force yourself to move on to the next task. Factor in time at the end to review and tweak, but set a time limit on that as well.\n\nBe patient with yourself as you practice being imperfect. Perfectionism is a trap that is worth escaping \u2013 when you do, you will like yourself more, enjoy the process of doing things rather than focus solely on the result, expand your repertoire of skills, and, most importantly, you will be able to let go of the stress and anxiety that are intrinsic to being a perfectionist.\n\n### _create a collage of your aspirations_\n\n### **EXERCISE FIVE**\n\nWhat would you do if you knew that you didn't have to do it perfectly? This exercise helps you discover what you have always wanted to try but were afraid to, because your inner perfectionist is holding you back.\n\n_**1.** Assemble the following: a collection of different types of magazines (lifestyle, travel, business, gardening, and so on), a large sheet of art paper, a pair of scissors, and some glue._\n\n_**2.** Block off one hour of time, play some of your favourite music, light a scented candle, and take a moment to relax. Flip through the magazines, one page at a time, and continually ask yourself the question, \"What would I like to do if I knew I didn't have to do it perfectly?\"_\n\n_**3.** Follow your impulse and tear out any pictures that represent something you'd like to try but haven't because of your need to be perfect in all things. Don't censor yourself or question your abilities. Just follow your intuition._\n\n_**4.** After about 30 minutes, use the scissors to trim the images and paste them onto the art paper to create a collage._\n\n_**5.** Reserve the last 15 minutes to think about which activity you would like to try most. How would you begin? For example, say you'd like to make over your garden but have been intimidated by your own high standards and the size of the task. You might start by planting a small window box. Whatever you choose, the pressure is off when you no longer need to be perfect._\n\n## fear of failure\n\nNo one likes to fail at anything and a fear of failure is one of the most common anxieties we face. Everyone has doubts about their abilities from time to time, which can cause them to hesitate to try something new or something they know they are not particularly good at. A fear of failure is linked with a fear of being criticized or rejected, but not all fears are bad \u2013 our innate survival instinct, which is based on fear, protects us from harm and warns us of danger. However, fear becomes a problem when it limits our capabilities and diminishes our quality of life.\n\nThe fear of failure can spiral into a vicious cycle. The anticipation of the worst-case scenario leaves you feeling anxious and stressed. You withdraw from the task at hand and\/or from others because you feel embarrassed and don't want to ask for help. You abandon what you are doing, or keep putting it off, or perhaps you never even start. Your self-confidence plummets and you generally feel bad about yourself, thinking, \"I'm a loser, I can't do anything.\" You feel like a failure in all areas of life; your anxiety increases.\n\nYet people who accomplish their goals know that failure is a natural by-product of trying. If you don't try, you will certainly never fail \u2013 but you won't succeed either.\n\nAsk yourself what exactly you are afraid of. Being judged? Looking silly? The possibility that friends or colleagues won't like you? Is there any evidence to support these fears? What's the worst that can happen if someone judges you? Often, once we have accepted what the worst outcome could be, our fears lessen and we feel more positive about trying.\n\nTry not to let your emotions control what you are going to do and when you are going to do it. Take action first and the good feelings will follow. Similarly, don't wait until you get in the right mood before starting; instead, just get going \u2013 your mood will improve because you are doing something and you will feel better about yourself because you are trying.\n\n_What may be done at any time will be done at no time._\n\nSCOTTISH PROVERB\n\nAvoid undermining yourself and asking yourself \"what if\" questions, such as: \"What if I can't do it well?\" and \"What if I fail?\" This will only drain your confidence. Focus on your effort \u2013 that is the most important part \u2013 and congratulate yourself for trying.\n\nDevelop persistence. Follow the advice of the American humorist Josh Billings (1818\u201385): \"Be like a postage stamp \u2013 stick to one thing until you get there.\" Keep trying and when you get anxious or frustrated give yourself ten more minutes to work on the task and then take a short break. Then, return to work with renewed determination to see the task through.\n\nInevitably, everyone experiences failure at some time or another, and when you do you will realize that missing a deadline or not managing to complete a task is not as terrible as you had thought \u2013 it doesn't mean you lack character and it doesn't reflect on your value as a person. Instead of chastising yourself, try to use the experience as an opportunity to grow and develop.\n\nIt can also help to review a situation where you failed to complete something because you were afraid of failing. Identify where you got stuck. Was it before you started the task, after you started it, or toward the end? Why do you think you stopped when you did? Were you unprepared to do a certain aspect of the task? Did it take longer than you thought and you panicked at the idea of running out of time? And so on. Once you have some answers, you can approach your next task differently so that the fear of failure does not prevent you from completing the job.\n\nFinally, watch your thoughts. Be aware of your internal dialogue when things go wrong. \"It didn't work out\" is quite different from saying \"I am a failure.\" Changing your perspective and reframing the failure as a temporary setback will allow you to learn from the experience and move on.\n\n## fear of success\n\nWhile the fear of failure is the fear of making mistakes and looking foolish, the fear of success is the fear of achievement and the recognition that success brings. People who struggle with the fear of success often do not understand why they have problems making decisions, lack motivation to achieve their goals, and are chronic underachievers. If they do complete something successfully, they denigrate their achievements, put themselves down, and say that it was just a stroke of luck.\n\nThose who fear success are masters of self-sabotage. They set things up to fail: They hand in the report late or incomplete, or they don't turn up for their creative writing workshop even though their work is highly regarded by the teacher. But why? Perhaps being successful does not fit with their self-concept. They view themselves as undeserving of success.\n\nHow can the fear of success sabotage effective time management? Self-sabotaging your time-management efforts can occur in subtle ways: not preparing a schedule when you know that this is the best way for you to operate; not using your calendar even though you carry it around with you; letting people interrupt you and take up your valuable time when you are trying to finish a project; or waiting until the end of the day when your energy is low to work on important projects.\n\nIt's a good idea to analyze what would be different in your life if you stopped sabotaging yourself and became successful. What are the pros and cons of success? There must be some benefits from self-sabotage or you wouldn't be doing it. Perhaps you don't want to deal with the consequences \u2013 such as promotion or moving house \u2013 that success might bring? Once you pinpoint why you are sabotaging yourself, you can start to address the problem.\n\nAnother way to push past your fears to success is to define your goals, both long- and short-term. While the fear of success is in the present, your goals are future-oriented. Use them to remind yourself why you need to overcome your fears.\n\nA little positive brainwashing can also go a long way to helping you overcome a fear of success. Repeat to yourself, \"I deserve to be successful,\" at least ten times a day. Also, write out this positive affirmation on small, adhesive notes and put them in places where you often see them, such as on your bedroom mirror, on your computer monitor, on the dashboard of your car, and so on.\n\n### _Find your own definition of success_\n\n### **EXERCISE SIX**\n\nHow will you know when you are successful? What does being successful mean? This exercise will help you to discover your own, personal definition. You will need a paper plate or a sheet of paper with a circle drawn on it, and a pen.\n\n_**1.** Divide the paper plate or circle into segments representing the areas of life that are important to you (see diagram below). Label each segment and mark on the numbers as shown. Now indicate your satisfaction with each area of your life on a scale from 0 to 10 (where 0 = not satisfied at all and 10 = completely satisfied). Draw a line across each segment at the appropriate point and shade in the area produced._\n\n_**2.** Join up the points on the segment lines. Chances are, you are closer to your definition of success in some areas than others. Reflect on how you have achieved success in these areas._\n\n_**3.** In the segments where you rated your level of success as low, answer the question: \"How will I know when I have achieved success in this area?\" For example, in Physical, your answer might be: \"When I can run a half marathon.\"_\n\n_**4.** Schedule time in the areas you need to improve to make your answers a reality._\n\n## workaholism\n\nBeing a workaholic is a socially acceptable addiction \u2013 no one looks down on hard workers: We encourage and reward them. There is nothing wrong with loving your work, feeling satisfied when you have put in a long day, and going the extra mile to make sure a project is completed on time. The difference between a hard worker and a workaholic is control. The hard worker is in control of when and how hard they work, and there is a balance between work and the rest of their lives. The workaholic, on the other hand, feels anxious when not working, finds it almost impossible to relax, and sometimes resents time spent with family and friends. The key indicator that a person is out of control and a workaholic is the bad state of their personal relationships. Spouses and children often suffer most because their husband\/wife\/father\/mother is spending so little time at home. Workaholism can be an important factor in divorce, and children of a workaholic parent often complain about the lack of time spent together.\n\nWorkaholics can be found in virtually every profession and work setting: doctors, lawyers, carpenters, teachers, social workers, artists \u2013 no occupation is exempt. People in professions that use hourly billing, or work according to a corporate culture that rewards those who put in extra-long hours, or entrepreneurs and the self-employed are especially at risk.\n\nWork addiction takes both a physical and a mental toll. Stress caused by burning the candle at both ends can result in many symptoms, such as: high blood pressure, anxiety, skin rashes, a depressed immune system, insomnia, bouts of anger, impatience, nausea, and back and joint pain. And if the workaholic does not slow down and learn to enjoy time away from work, he or she is at risk of burn-out.\n\nAlthough burn-out is not recognized as a medical condition, it can be very debilitating. Burn-out related to work addiction is a type of depression that develops as a response to work-related stress \u2013 in this case caused by spending too much time at work. It develops over years and is characterized by physical exhaustion, sadness or depression, taking longer to complete work responsibilities, shame that you can't work as hard as you used to, poor concentration, and an inability to make decisions. Many workaholics refuse to acknowledge what is happening until their symptoms are so severe that they are physically unable to address the problem. At that point, we need rest and recovery. Burn-out often forces us to re-assess who we are in relation to work, why we do a particular kind of work, and what needs to change so that we can find a better balance between work and the rest of our lives. Returning to work gradually, with a renewed sense of purpose and a more flexible attitude, will help us to become happier and healthier.\n\nThe road back from workaholism and burn-out can sometimes feel slow and painful, but it can be done. A good start is to let go of the guilt and anxiety of not working 24\/7. Remind yourself that you are a better employer\/colleague\/entrepreneur\/student if you take time to relax and revitalize yourself. You will be more efficient, creative, and productive.\n\nOnce back at work learn to delegate. Relieve yourself of some of your workload by delegating chores to a trusted employee. Practice trusting others to do the job as well as you would. If you don't have an employee or you work at home, consider hiring someone to do the mundane tasks to free up some of your time. Or try trading tasks with a friend.\n\nMake sure that you give the important people in your life a chunk of your prime time. Stop giving your family and friends the left-over time at the end of the day and at the weekends and spend some time with them when you are in top form. Perhaps you could schedule some fun time for a Saturday morning when you would normally go to work, or set up (and keep) a date night once a month with your partner or with friends.\n\nTake time to take care of your physical health. Being in good physical shape has many well-known benefits, but for the workaholic there is an additional one: It requires taking time away from work. Schedule time to run, to go to the gym, or to use exercise equipment at home \u2013 whatever you can do to help you get, and stay, in shape. Exercise helps to manage anxiety and stress and produces mood-enhancing endorphins. Exercising with others, such as your children or friends, will give you the added bonus of re-building your relationships.\n\nAs you incorporate changes into your work routine and work less hours, be aware that you may experience withdrawal symptoms. You might find yourself feeling down, especially if you liked the adrenaline rush associated with a fast-paced, all-consuming work life. You might also feel agitated and impatient. These feelings are normal and common when lessening the grip of the amount of time and energy that your work consumes.\n\nTime management is not a method for you to work harder and longer. Effective time management incorporates all of the areas of your life from work to family to leisure.\n\n### _assess your attitude to work_\n\n### **EXERCISE SEVEN**\n\nThis exercise is adapted from _How do I know if I am a workaholic?_ , published by Workaholics Anonymous.\n\nAnswer the following questions with a simple \"true\" or \"false\" response:\n\n\u2022 _I get more excited about my work than my family or anything else._\n\n\u2022 _I take work with me on vacation._\n\n\u2022 _Work is the activity I like to do best and talk about most._\n\n\u2022 _I work more than 55 hours per week._\n\n\u2022 _My family and\/or friends do not expect me to be on time._\n\n\u2022 _I believe it is okay to work long hours if I love what I am doing._\n\n\u2022 _I get impatient with people who have other priorities beside work._\n\n\u2022 _I am afraid that if I don't work extra hard I will lose my job or be a failure._\n\n\u2022 _I worry about the future even when things are going very well._\n\n\u2022 _I get irritated when people ask me to stop doing my work in order to do something else._\n\n\u2022 _My family complains about the amount of hours I work and that I am rarely home._\n\n\u2022 _I think about work most of the time: while driving, when falling asleep, when I wake up during the night, when others are talking to me._\n\n\u2022 _I work or read work-related material during meals._\n\n_The more \"true\" answers you have given, the closer you are to being a workaholic. An awareness that you might have a problem is the first step toward changing. If you feel that you might need professional help, contact an organization such as Workaholics Anonymous or seek advice from a mental health professional._\n\n### _learn to set limits_\n\n### **EXERCISE EIGHT**\n\nThere are only 24 hours in a day and it is neither healthy nor desirable to spend more than one third of this time working. This exercise will teach you how to cut back on long working hours and regain a balance in your life.\n\n_**1.** First you need to find out exactly how you are spending your time, so get yourself a notebook and pen. For the next two weeks you are going to record everything you do. Each evening write down what you did that day and how long each thing took. List everything, including everyday activities such as sleeping, eating, and washing._\n\n_**2.** Analyze your time record, concentrating on the time you spent working over and above your required hours (in most jobs a maximum of eight hours per day). Look for ways gradually to reduce the extra hours._\n\n_**3.** Start by making small changes. (If you try to make too many changes too quickly they won't last.) For example, try finishing work half an hour earlier each day until you leave at the designated time. Or, take 15 minutes every lunchtime to go for a walk._\n\n_**4.** Learn to say \"no,\" both to yourself and to others who make demands on your time. Don't add anything to your schedule without removing something that requires the same amount of time and energy._\n\n_**5.** Find a mentor to support you in your quest to reach a better work\u2013 life balance, someone you can trust: your partner, a friend, or even a counselor. Set up a system whereby they can alert you, or you can go to them, if you veer off track._\n\n## quality time\n\nDo you know where you are going in your life? If you don't, how will you know when you arrive? You need a map to give you direction, motivation, and a time frame. How you spend your time is a reflection of what is important to you. Identifying your values will give you an understanding of what you are trying to accomplish with your life.\n\nThis chapter provides you with the necessary guidelines for writing a personal statement of purpose, or a \"mission statement,\" shows you how to set goals, and gives suggestions about how to reach a balance between your work life and your personal life.\n\n## what is important to you?\n\nIf someone were to follow you around for a week and record how you spend your time, would they be able to extrapolate what is important to you? Are you happy with the way you spend your time? At the end of most days, do you feel satisfied that you've spent the bulk of your time on the things that are important to you?\n\nTime management is more than creating \"To Do\" lists, using detailed diaries or studying time-use grids. Organizing your time isn't a way in which to get everything done, because that is simply impossible. If you approach life with an attitude of, \"If I hurry up, if I work harder, if I work faster, I will be able to accomplish it all,\" you will end up feeling frustrated and discouraged, not to mention exhausted. What is possible is to accomplish the things that are most vital to your personal and professional well-being.\n\nLet's now look at the Pyramid Principle and the Priority Principle \u2013 two ways to evaluate your time and to make sure that you are using your time doing what is most important to you.\n\n#### The Pyramid Principle\n\nCharles R. Hobbs was influential in the 1970s and 1980s for moving time management away from techniques to increase productivity and toward developing a personal time-management approach grounded in core values. Hobbs felt that it was important to have a cohesive system that incorporated individual values, goals, and daily planning. This he did by creating what he called a \"productivity pyramid.\" The first step in building a productivity pyramid is to define your values and write a general action statement about each \u2013 for example \"Show compassion\" or \"Have faith in others.\" These Hobbs called the \"unifying principles.\" You then use these unifying principles as a basis from which to formulate your long-term goals. From your long-term goals you set intermediate ones, and then base your daily goals on your intermediate goals. The pyramid is thus formed with your unifying principles at the base, your long-term goals in the next layer, your intermediate goals above that, and your daily goals at the top. This fits in with Hobbs' philosophy that any time-management system that you are actually going to use, must involve a synthesis between what you believe and what you do.\n\nIn _The Seven Habits of Highly Successful People_ , Stephen Covey proposes that we sometimes go through a personal evolution of four \"generations\" of time management. Each generation builds on the next and becomes more effective and useful.\n\nAccording to Covey most people who are interested in learning about time management use the \"get organized\" approach (first generation) that runs on reminders and \"To Do\" lists. There is no indication of priorities, no specific time allotted to specific tasks, and what they don't get done today, they put on tomorrow's list.\n\nThe second generation of time management is focused on planning and preparation. Calendars and agendas or diaries are the main tools we use; people set goals, make plans, schedule events. Typically, the only things that they schedule are work-related items, medical appointments, meetings with teachers, and so on. The schedule rules, and it does not tend to include the time we need to maintain personal relationships and to develop personal goals.\n\nThe third generation moves beyond planning and preparing, and adds prioritizing and controlling. People functioning in this level of time management spend time articulating values and priorities. They divide goals into the short- and long-term, based on their own values. They use planners and organizers (paper or electronic), and often use highly detailed forms for planning their day. According to Covey what is missing at this level is the concept of spending time on what is actually most important.\n\n_\"Never leave until tomorrow that which can be done today.\"_\n\nBENJAMIN FRANKLIN (1706\u201390)\n\nCovey outlines the need for a fourth generation of management that moves beyond managing time to managing yourself and your life. It includes all the strengths of the previous generations and adds a quality-of-life component. It helps us to create and maintain meaningful relationships and peace of mind, and engenders confidence that not only are we doing what matters most, we are doing it well.\n\n#### The Priority Principle\n\nValue-based time management requires an understanding of what our personal and professional priorities are and our making time-use choices based on those priorities.\n\nCharles R. Hobbs and Stephen Covey determined that our time use can be divided into four categories: 1. Important\/Not Urgent; 2. Important\/Urgent; 3. Not Important\/Urgent; and 4. Not Important\/Not Urgent.\n\n##### **the four categories of time use**\n\n**Category 1: Important and Not Urgent \u2013 Priority Principle**\n\n\u2022 Managing your time as an extension of your values and priorities.\n\n\u2022 Being able to focus on what you are doing without the fear that you should be doing something else.\n\n\u2022 Including time for relationships.\n\n\u2022 Examples: balancing work and personal time; scheduling time for relaxation, physical fitness, personal growth and development.\n\n**Category 2: Important and Urgent \u2013 Crisis Management**\n\n\u2022 Giving your immediate attention and dropping everything else.\n\n\u2022 Examples: medical emergencies, deadlines, family or relationship crises.\n\n**Category 3: Not Important and Urgent \u2013 Distractions as Denial**\n\n\u2022 Responding immediately to all distractions and interruptions; attending all meetings.\n\n\u2022 Examples: answering all emails and phone calls as they come in, regardless of who they are from; going out for dinner three or four nights a week with friends\/colleagues.\n\n**Category 4: Not Important and Not Urgent \u2013 Busy with the \"Trivial Many\"**\n\n\u2022 Spending your time on things that make little or no contribution to your quality of life.\n\n\u2022 Examples: playing card games on your computer; doing work-related activities, such as filing, that don't help to build your career or your business; answering personal phone calls when you are working at home; allowing friends or family to drop in and distract you when you are working in a home-based business.\n\n##### **what do other people value?**\n\nThink of five people you know who you particularly respect and admire, and arrange to interview them. In order to access as wide a variety of perspectives as possible, choose individuals from a range of ages (perhaps including a grandparent and a teenager), and diverse backgrounds, levels of education, professions, and so on.\n\nAsk each interviewee to list the five values that they hold most important and adhere to in order to live a good life. Then find out why their chosen values are so important to them. Use the information gleaned from the interviews as a starting point to help you to determine your own values.\n\nThe objective is to focus on creating a life that is based on the foundation of Category 1 principles, to learn to be an effective Category 2 crisis manager, and to avoid useless distractions (Category 3), and being busy for the sake of being busy (Category 4).\n\nLiving your life and using your time based on your principles and values will help you to accomplish your life goals, and create order in and balance between the different areas of your life. You will be in control of your life and, because you are managing your time effectively, you will have many fewer crises to deal with. Take a pen and a sheet of paper, and divide it into four sections: top left, Important\/Not Urgent; top right, Important\/Urgent; bottom left, Not Important\/Urgent; and bottom right, Not Important\/Not Urgent.\n\nNow discover which category you are currently in. Take your calendar or agenda (or rely on your memory, if you must) and allocate a month's worth of your activities among the four categories, according to the criteria described opposite. The next step is one of assessment and reflection. Ask yourself: Which category was I in most of the time? How much time did I spend doing activities that were Important and Not Urgent? This should show you clearly whether or not you are spending your time wisely and focused on priority activities.\n\n#### What are your values?\n\nTo make the shift to living with your priorities in mind, you first have to determine what your priorities and values are. Your values are the beliefs that reflect your definition of a meaningful life \u2013 they drive your behaviour. You can use your definition of success from Chapter 2 (see exercise six, p.43) as a basis for determining your values.\n\nOnce you have ascertained exactly what your values are, the task of allocating your time becomes easier. As you learn to live your life and take charge of your time based on your values, you become sensitized to the good feelings that emerge as a result. Conversely, you will experience psychological discomfort when you spent time on unimportant tasks. If you find yourself doing this often, you need to re-assess what you are doing and why and then make the necessary adjustments to get your value-based time management approach back on track.\n\n### _identify your values_\n\n### **EXERCISE NINE**\n\nImagine that you have been asked to divulge to future generations the secret of living a meaningful life. You first need to compile a list of core values that you feel are essential to you personally, and which reflect what you consider to be most worth upholding and defending.\n\n_**1.** Taking a sheet of paper and a pen, list all the values that matter to you down the left-hand side in a column. These might include: Integrity; Honesty; Intimacy; Friendship; Family happiness; Comfortable living; Hard work; Learning; Physical and mental health; Spirituality; Personal growth; and Independence or autonomy. Try to narrow down the list to your ten most important values._\n\n_**2.** Write a time-related statement of principle next to each value. For example, if friendship features on your list, you might put, \"I will plan and schedule time to spend with friends.\" Or for personal growth, \"Each year I will take one course that will help me to develop into a well-rounded human being.\" Keep the completed list and statements in your agenda and read them every day to inspire you._\n\n_**3.** Review your list and statements on an ongoing basis. Circumstances change, and so will the relative importance of some of your values. For example, if you become a parent, family happiness will probably become more important than independence. Hone your list and statements so that they give a full and accurate reflection of your current values._\n\n## the purpose of your life\n\nPeople can be divided into the \"haves\" and the \"have-nots.\" But I am not talking about their abundance or lack of financial success, being with or without the perfect partner, having that envied career or putting a career on hold while you focus on raising children. There are those who have a definite sense of purpose; others who simply seem to wander aimlessly throughout their lives without any rhyme or reason. Some people who appear on the surface to have it all, in reality have a poor quality of life because they have no idea what their life purpose is \u2013 they are just putting in time.\n\n_\"The more we do, the more we can do; the more busy we are, the more leisure we have.\"_\n\nWILLIAM HAZLITT (1778\u20131830)\n\nLiving your life purpose is at the heart of who you are \u2013 the very essence of your being. Living a purposeful life provides you with a sense of direction and mission. When you know what your life purpose is, you start every day with that in mind and plan accordingly; it shapes how you think, what you do, how you spend your time and how you approach life.\n\nYour life purpose is not your job, or your roles (daughter\/son, mother\/father, and so on). These are vehicles through which you deliver your purpose. Your life purpose comes from within, from your values, and it should infuse you with joy and contentment.\n\nKnowing what your life purpose is creates within you a desire to live with passion. This passion fuels your daily activities. When you are living purposefully you feel connected to a greater whole; you have an understanding that your purpose is part of a greater scheme of things, however you choose to define that.\n\nWriting a personal statement of purpose or \"mission statement,\" will clarify the direction you should take in life. It will help you to stay focused when your enthusiasm wanes and at any time when you end up using your precious time on inconsequential trivia. Your mission statement is also a useful measure, on a monthly or yearly basis, of where you have been, where you are now, and where you are headed.\n\nOnce you have a clear idea of what your life purpose is, write it down. You will be able to use your statement of purpose as a guide in making both small and large decisions, such as which university to attend; whether or not you need to move house or job; and whether you should pursue a particular business venture.\n\nA statement of purpose prepares you for stability in a world of constant change. The ever-changing nature of work, leisure (or lack of it), family, and social interactions has a tremendous impact on our daily lives. Having a statement of purpose can be a constant in your life while all things around you are in flux.\n\nEach one of us has a unique purpose. If you do not fulfill your life purpose the world is forever deprived of what you have to offer.\n\n#### Creating goals based on your values\n\nIt is a good idea to write down your goals, as people who do so are more likely to accomplish them than people who don't. You can use your list of values (Exercise nine, p.59) or your personal mission statement (Exercise ten, p.63) to help you.\n\nStart by making a list of what you want to accomplish this year. Then put the goals in order of importance. Starting with the most important goal, work out what steps you would need to take to accomplish it. For example, say that you want to improve your education. You will need to find the right course. Would an evening class be best or can you talk to your employer and arrange to work fewer hours so that you can attend in the day? How will it affect your other commitments, such as looking after your children? Will there be financial implications? If so, do you need to set aside some money beforehand? Do you have to apply for the course by a certain time? And so on.\n\n##### **make your goals SMART**\n\nWhether short-term or long-term, well-formulated goals are SMART. That is:\n\nSpecific \u2013 what you wish to accomplish, precisely detailed.\n\nMeasurable \u2013 possible to assess in terms of progress.\n\nAttainable \u2013 within reach given your circumstances.\n\nRealistic \u2013 within your personal capability.\n\nTime-lined \u2013 possible to accomplish by a certain date.\n\nLet's use the goal of physical fitness as an example.\n\nSpecific \u2013 complete a half-marathon.\n\nMeasurable \u2013 keep a weekly running log.\n\nAttainable \u2013 already a recreational runner.\n\nRealistic \u2013 eight months to train.\n\nTime-lined \u2013 entered for the half-marathon on September 17.\n\nIt is important to review your progress regularly and make the necessary adjustments so that your goals become a reality.\n\n### _write your personal \"mission statement\"_\n\n### **EXERCISE TEN**\n\nYour life purpose = your passion + its benefit to others. Once you determine your life purpose, you can encapsulate it in a personal mission statement. This will always inspire you and remind you of what is important in your life.\n\n_**1.** Let's start with your passion. It could be anything \u2013 from singing opera to D-I-Y to bungee-jumping. If you are unsure what it is, first take another look at the list of values you created in Exercise nine, p.59. Now ask yourself the following questions and write down your answers. What do my friends and family ask my advice about? Do my friends see me as having expertise in a specific area? When I enter a bookstore, do I always gravitate to a specific section like gardening, physical fitness, or travel?_\n\n_**2.** Next, consider who will benefit from your passion. Are you interested in sharing it with a specific group or profession, such as seniors, dentists, or single parents, or with people in general? Finally, how will others benefit? For example, if your passion is showing others how to find their work\u2013life balance, a possible benefit will be that they get to enjoy more time with their children._\n\n_**3.** Look at your answers and then write a clear and concise mission statement. based on the following sentence stems:_\n\n\u2022 _My passion is ..._\n\n\u2022 _The main beneficiaries of my sharing of this passion will be ..._\n\n\u2022 _They will benefit by ..._\n\n**_4._** _Refer to your statement whenever you need to remind yourself what is meaningful in your life._\n\nThink about each of the goals on your list in the same way, and break down each one into small steps that you can gradually achieve.\n\n### _leave your own legacy_\n\n### **EXERCISE ELEVEN**\n\nUse this exercise to help you envision the kind of life you want to lead and to help you to put in place some long-term goals.\n\n_**1.** Imagine that you are listening in to your family and friends eulogizing at your funeral. What would you hope they would say about your legacy \u2013 the contribution you made to your family, your friends, and the community?_\n\n_**2.** Be as specific as possible when formulating your personal legacy. For example:_\n\n_I would like to be remembered as:_\n\n\u2022 _A partner who..._\n\n\u2022 _A parent who..._\n\n\u2022 _A son\/daughter who..._\n\n\u2022 _A sister\/brother who ..._\n\n\u2022 _A person who ... (fill in the blank) in my community_\n\n\u2022 _A co-worker who..._\n\n\u2022 _A neighbour who..._\n\n_**3.** Once you have written your personal legacy, you can think about ways to turn this into a reality. For example, if you want to be remembered as a helpful neighbour, what do you need to do to become one? Perhaps you can set aside one Saturday per month on which to do errands for the elderly couple who live across from you. Write down on a sheet of paper for each \"I would like to be remembered as ... \" statement at least one goal that you can incorporate into your agenda or planner as a reminder._\n\n_**4.** Take your agenda and schedule your life-affirming goals over a six-month period. Remind yourself that every time you reach one of them, you are building your legacy._\n\n## time to plan\n\nWe can remedy many of our poor time management habits through good planning. Choosing a planner is an important first step and you can find one that suits your lifestyle, whether you are a student, or a professional, or you are at home parenting full-time. Planning, both short- and long-term, will give you peace of mind, a purpose in life, and a sense of accomplishment.\n\nIn this chapter you will learn the value of planning each day, as well as the advantages of planning ahead several years. You can also have fun creating a list of 100 things you would like to do in your lifetime, and learn how to turn these dreams into a reality.\n\n## choosing and using your planner\n\nOnce you have made a commitment to become a good time manager, the next step is to find the right time-management tool that will help you keep track of your goals, and schedule appointments, and provide you with easy access to important information.\n\nThere is no shortage of choices when it comes to planners: from calendars, paper and pencil planners, and agendas to computer-based software programmes and smartphones. Planners are available for professionals, such as teachers, lawyers, and salespeople, and for specific-interest groups, such as gardeners and dog-lovers.\n\nThe biggest dilemma when choosing a time-management tool is which format to use \u2013 paper, computer, or smartphone. So let's consider the advantages and disadvantages of each in turn.\n\nPaper planners come in a variety of sizes and colours, and cover either the calendar or the academic year. They generally show either a day, a week, or a month per page. If you are uncomfortable with technology, like to be able to view your week or month on one page, feel more secure using a paper planner, and prefer writing to using a keyboard, you will probably find that a paper planner is your best option.\n\nThe week- or month-at-a-glance feature is handy for people who like to see their week or month in one place. But if you need to schedule appointments on a daily basis, the daily-view planner with plenty of space to write down appointments is a good choice. Also consider the size of your handwriting \u2013 the bigger your writing, the bigger page size you will need. There are benefits to using a paper planner: there are no batteries to charge; it is low in cost compared to electronic versions; and you can use it pretty much anywhere, except when you are driving! The disadvantages of a paper planner include the limited capacity to store information, the need to transfer annually-recurring information, such as birthdays and anniversaries, and (unless you keep a duplicate copy) nowhere to back up the information \u2013 so if you lose it you are sunk!\n\nComputer-based programmes can be good tools to help you manage your time, particularly if you use a desk- or laptop computer most of the time (either for work or for personal use), you are comfortable using computer software, you like to be reminded of scheduled events visually or with a sound, and you like to be able to store more information than a paper planner can handle.\n\nAlmost anyone using a desk- or laptop computer will have a calendar and contact-management system as part of their email programme. You can also opt to use software dedicated solely to time management. A computer-based system allows you to sort, store, and organize almost limitless amounts of information. You can set up the programme to remind you \u2013 with a \"popup\" on your computer screen, along with a sound alert \u2013 of impending appointments, phone calls, and other items that you have scheduled. Co-workers, employers, and\/or assistants can also access these calendars. The disadvantage with a desktop computer is that it isn't portable; and if something goes wrong and you lose your information, often it can be difficult or impossible to retrieve it, unless you have previously backed it up.\n\nThe third type of planner available is a smartphone. Smartphones have become one of the most popular and convenient ways to keep on time due to the vast amount of free apps available to download, from to do list apps that have brought list-making into the Internet era to apps that can track how you're making use of your digital time and monitoring your productivity. If you travel a fair amount, need something small and portable, and need to be able to sync information electronically between your and other colleagues' home\/work computers and time-management tools, a smartphone is probably your best bet. Smartphone's have most of the same advantages as computers and when you add to that their portability, you have the road-warrior's time-management dream. Although it can sometimes take longer to set up a smartphone and to find the right place in which to input your data compared to making a quick note in a paper planner, the advantage of being able to store vast amounts of information makes it infinitely superior. If you are prepared to keep the smartphone charged and back up your data, then this may be the right option for you.\n\n_Ordinary people think merely of spending time. Great people think of using it._\n\nANONYMOUS\n\nHere are some other guidelines to keep in mind when selecting your planner:\n\n\u2022 Choose one and only one planner, and use it everywhere and for everything that requires scheduling. Store all contacts, phone numbers, and schedules in one place.\n\n\u2022 Keep your planner in the same place for easy access \u2013 perhaps near the phone or on your desk. Don't forget to take it with you when you head out the door.\n\n\u2022 If you want to include your long- and short-term goals, choose a planner that allows you to record and track your progress.\n\n\u2022 Don't become a slave to your time-management system. Creating extensive \"To Do\" lists (and lists of your lists), and scheduling everything down to the last minute, will take the joy out of life and will actually be counterproductive.\n\n\u2022 Make sure that there is a good fit between the way you work and your time-management system. Adapt the system to fit your style and the type of work you do \u2013 your time-management system works for you, not the other way around.\n\n## planning your week\n\nPlanning your life one week at a time offers a convenient time frame within which to operate because it includes both work time and personal time. Weekly planning covers a manageable chunk of time and is a good way to approach the bigger picture of your life.\n\nChoose a specific time to plan for the upcoming week. Some people choose the end of the workday on Friday; others choose Sunday evening. Setting up your weekly planning session in a specific location, such as your desk at home or at the office, will help you be consistent with your weekly planning. Remind yourself to set this time aside by marking it down in your planner.\n\nStart by quickly reviewing your \"mission statement\" (Exercise ten, p.63), your value statements (Exercise nine, p.59), and your long-term and short-term goals (pp.61\u20132). It will help you to stay on track if these are at the forefront of your mind as you plan your week. Then, review your previous week and measure your progress against your planned activities. Were you able to get your important activities done? If not, why not? This information will be particularly helpful as you become more proficient and accurate in scheduling your time on a weekly basis. Your next step is to focus on the coming week. What would you like to achieve this week? Are these expectations realistic? Don't try to do too many things or you will become frustrated, stressed, and exhausted. But equally, if you don't do enough, you will be bored and not accomplish very much. Gradually, you will get a feel for how much you can comfortably get done in one week.\n\n#### How to schedule\n\nPlanning on a weekly basis will help you move from crisis-management to a more proactive, goal-oriented, and realistic time frame.\n\n\u2022 Schedule your priority tasks at the beginning of the week to ensure that you will have sufficient time to get them done.\n\n\u2022 Be aware of deadlines. Make a habit of scheduling tasks to be completed ahead of the deadline date. This will give you a comfortable time cushion in which to deal with emergencies or unforeseen events.\n\n\u2022 Overestimate rather than underestimate how long it will take you to complete a project. Doing this will help you to get your work finished even if you are faced with interruptions and delays, and it will also allow you to work in a more relaxed, less stressed manner.\n\n\u2022 If you are calling a meeting, schedule a set time for the meeting to end.\n\n\u2022 Make appointments back to back. If you have two appointments in the same building (or in your home office), give yourself only a five-minute break between them.\n\n\u2022 Schedule your work as an appointment with yourself.\n\n\u2022 If you must interrupt your scheduled task to take care of an emergency, be sure to reschedule your task for as soon as possible.\n\n\u2022 Once you have scheduled your priorities you can fill in some of your schedule with other, less important activities.\n\n##### **how long will it take?**\n\nIf you are not quite sure how long it will take to get a certain task done, ask yourself the following questions:\n\n\u2022 What do I need to do?\n\n\u2022 What materials do I need? How long will it take to get them together?\n\n\u2022 When will I do it?\n\n\u2022 Can I do it all at once or will I divide it into smaller chunks of time?\n\n\u2022 Can I do it by myself, or do I need help? If so, when will this be available?\n\nAfter you've completed the task, assess if your estimate of the time required was close to the reality. If not, look at why not, and use that information next time you need to estimate how long something will take.\n\n### _analyze your time budget_\n\n### **EXERCISE TWELVE**\n\nHow can you manage your weekly time budget if you are not sure how you are spending it? Use this exercise to help you to budget the number of hours in a week against the amount of time you need to get your work and activities done.\n\n_**1.** Taking a sheet of paper and a pen, draw two columns. Name the left-hand one \"Activities\" and the right-hand one \"Hours spent.\" Under \"Activities\" list everything you did during the last week: sleeping, eating, working, commuting, doing chores and errands, socializing, watching TV, and so on. Don't leave anything out. Then, using your diary or the time record you made in Exercise eight (p.48) and bearing in mind that there are 168 hours in a week, write in the right-hand column how long you spent during the last week on each activity._\n\n_**2.** Total up the number of hours you spent on everything. Now subtract this figure from 168. Was there any time left over? Or were you in time deficit? (This is not as strange as it sounds, because sometimes we do more than one thing at a time.)_\n\n_**3.** Analyze your results. If you had any free time left over, look at your lists of long- and short-term goals, and allocate some more time to achieving these. But if you were in time deficit, you need to re-examine your time budget to find yourself more time to spend on important activities. You need time to sleep well, to eat healthily, and to work. But do you really need to watch TV? Can you find a better use for that time? Try reallocating your time, until you are back within your time budget. With deft \"pruning\" you can even find some free time._\n\n## your daily schedule\n\nGreat time managers use three basic tools of daily planning: a master list, a \"To Do\" list, and a schedule of appointments and meetings. A master list is simply a running list of everything you have to get done, and it is kept separate from your \"To Do\" list. Your master list is not prioritized in any way: Its purpose is simply to list your \"To Dos\" as they come up. Having a place to keep an ongoing list frees you from the worry of forgetting to do something important. You can then refer to your master list to create your daily \"To Do\" list.\n\nA \"To Do\" list can help you to stay focused on the activities and priorities for a particular day. You are more likely to complete things if you have a list and write down your plans for the day. It makes your responsibilities appear more manageable and less overwhelming. Check your list mid-morning and again at lunchtime to make sure that you are getting your priorities done first. Make sure that the planner you are using, whether it is paper, computer, or handheld, allows you space to create and update your \"To Do\" list fairly easily.\n\nAnother suggestion to keep you focused and motivated on a daily basis is, first thing each morning, to pull out your list of short- and long-term goals and read them \u2013 you could even write them out again if you have time. This is a great way to keep them fresh and uppermost in your mind.\n\nAt the end of your work or school day, take five minutes to plan the next day. If you wait until the morning to plan that day, too many things could interrupt you and you might find yourself with no plan at all. The first step is to review any commitments that have specific start times and to check that you've allotted them sufficient time. Don't forget to include travel time to and from meetings if you are leaving home or your office, especially if you live in a big city where traffic jams and problems with public transport can delay you.\n\nUsing your time-management planner and anything left over from today's \"To Do\" list, write down everything that you would like to accomplish tomorrow. Now that you have tomorow's \"To Do\" list, remove as many tasks as possible. Ask yourself whether or not you, personally, have to do everything on the list. Are there any tasks that you can delegate to your partner\/children\/friend\/colleague? If so, ask for their help.\n\nDraw three columns on a sheet of paper with numbers down the left-hand side. Name the three columns \"Must Do,\" \"Would Like to Do,\" and \"Re-schedule.\" Before you decide which tasks should go into which column, ask yourself the following questions about each task:\n\n\u2022 Is it crucial that I do this task today?\n\n\u2022 Can I re-schedule this task without any penalty to myself?\n\n\u2022 Does this task help me reach my weekly goal? My monthly goal?\n\n\u2022 Can I eliminate this task without any penalty to myself?\n\nNow divide the tasks on your \"To Do\" list among the three columns. Schedule your \"Must Dos\" first, then add in items from the \"Would Like to Do\" list. Use the \"Re-schedule\" column to do just that, using your planner to write down the new time and date you will finish that task. If you have re-scheduled something more than three times, evaluate why it is on your list at all and perhaps delegate it or remove it from the list altogether.\n\nProtect your daily schedule. Deal with non-urgent interruptions by letting people know that you are working to a deadline and scheduling a specific time to talk. Checking phone and email messages at particular times during the day, such as mid-morning and mid-afternoon, will help you stick to your agenda. Keep your work space organized by having a folder for each project or task. Once you've completed as much as possible on one task, put the folder away before starting on another project.\n\nAt the end of the day evaluate your achievements and be proud of what you've managed to get done. Create a new \"To Do\" list for the next day so that you can leave your work behind you and move into your personal time. Clear your desk, and leave work with the peace of mind of knowing that you've already taken control of tomorrow's time.\n\n### _plan your day the easy way_\n\n### **EXERCISE THIRTEEN**\n\nWe all feel swamped sometimes when we think about how much we need to get done in a day. This exercise teaches a streamlined method of daily planning that is a quick and easy introduction to the concept.\n\n_**1.** At the end of your day make a list of everything you hope to do tomorrow._\n\n_**2.** Choose the six most important tasks on your list and number them one to six._\n\n_**3.** When you begin your day, start with the number one task on your list and keep working on it until you have completed it or you've taken it as far as you possibly can._\n\n_**4.** Move on to the second task and approach it the same way. Don't start the third task until you've finished the second._\n\n_**5.** If you are called upon to deal with other tasks during the day, deal with them only if they are of greater importance than the task at hand. If a new item is not more important, simply add it to the bottom of your list of tasks to be prioritized when you revise the list at the end of the day._\n\n_**6.** When you have successfully completed the six items on your list or have done as much work on them as possible today, start compiling a fresh, prioritized list of tasks for tomorrow._\n\n_**7.** Even if you were not able to complete all six tasks that you had on your list by the end of the day, you have the satisfaction of knowing that you completed the most important ones._\n\n## long-term planning\n\nIf you don't know where you are headed, then any road will take you there. Exercises nine to eleven should already have helped you to identify some medium- and long-term goals. Now you are going to take a longer view.\n\n#### 100 things to do in your lifetime\n\nThis is a great activity to do together as a family or group of friends. Children also love this activity and it is great fun to see how their list changes over time. Purchase some special notebooks or journals and give one to each person to write their list in and use over and over again as part of a yearly review.\n\nThe objective is to make a list of 100 things that you would like to do in your lifetime. Think of as many as you can to get started and then add to your list as and when you think of new things, dating each entry for future reference. Don't restrict yourself by worrying about how much things cost or the likelihood of achieving something. Simply write down your dreams and wishes as they come to you. For example, your list might include: to travel round the world; to run a marathon; to write a book; or to climb Mount Everest. At the end of each year, take the time to review your list, to strike off what you have accomplished, and to add new items.\n\n### _Set your five-year goal_\n\n### **EXERCISE FOURTEEN**\n\nMaking a five-year plan is no guarantee that you will achieve everything you wish to do in this period. But what it does do is increase your chances of doing so. You can focus this exercise either toward your personal life, or toward developing your career or business.\n\n_**1.** Set aside some time, preferably a half day or even a weekend, to hold a five-year planning retreat. You can do this in your own home or go somewhere peaceful to create a greater sense of separation from your daily routine._\n\n_**2.** Give yourself time to relax and then start to contemplate where you want to be in five years' time. If you've not yet done so, write your personal \"mission statement.\" (see Exercise ten, p.63). Answer the question \"What is the purpose of my life?\"_\n\n_**3.** Ask yourself these questions too. What would be your ideal life five years from now? Where do you want to live? Do you want to live with someone or alone? How do you want to spend your days? Perhaps you want to be a parent and you see children in your life five years from now. Where would you like to see your business? Where do you see yourself on the career ladder? And so on ..._\n\n_**4.** Plan what proactive steps you need to take to make your dream a reality. Do you need to save a certain amount of money per year? Do you need to take a course? Do you need to meet new people? Set yourself yearly targets that will build up toward your five-year goal._\n\n## time management at work\n\nWhatever type of work we do, we all have the same objective, which is to get as much work done as possible within the time allotted. One way to save time and be more efficient is to streamline the way we work and make better use of the tools available to us.\n\nThis chapter will help you manage your business contacts, show you how to de-clutter your office, teach you how to get the mounds of paper under control, and give you all the information you need to be able to take advantage of the latest time-saving technology. There are tips, too, on how to become the master of your telephone and your email so that you increase your productivity \u2013 and get home sooner.\n\n## managing your business contacts\n\nOnce upon a time, if you wanted to reach someone, you called them on the phone either at home or at work, or if the matter could wait, you wrote to them. Today, we are overwhelmed with ways in which to contact people \u2013 there are home phones, office phones, cell phones, email, social media, as well as plain, old-fashioned letters. Keeping our contact information up-to-date has never been more time-consuming.\n\nAccording to a recent Data Warehousing Institute study, 25 to 35 per cent of the average database becomes outdated within a year. Obsolete contact information can waste your time and may also result in lost business. It is therefore imperative to do periodic updates \u2013 say, every year \u2013 and make sure that your staff and\/or co-workers all have access to the current contact information.\n\nThe professions or careers of people such as consultants, recruiters, health-care professionals, lawyers, and salespeople depend on building and maintaining relationships with clients and customers. As it is virtually impossible to rely on memory when it comes to such matters, we can manage these relationships or contacts most easily and efficiently using contact-management software packages. Such programmes create databases of information to help you keep track of the basic details. Depending on the nature of your work, you could divide your business contacts into various categories, such as clients or customers, leads or potential clients or customers, vendors and suppliers, and key colleagues. Most programmes for managing contacts allow you to collect detailed information. For example, you can find out when you had your last appointment with a particular person, as well as a list of the dates of your phone conversations, including your notes on what was said, and any emails and letters you sent and received. If you are selling products and\/or services, your contact-management software can also track your marketing and sales efforts, and results.\n\nHere are some guidelines to follow when you choose your contact-management software.\n\n### _\"In truth, people can generally make time for what they choose to do; it's not really the time but the will that is lacking.\"_\n\nSIR JOHN LUBBOCK (1834\u20131913)\n\n**Recommendations** Ask friends and colleagues which software programme they use and inquire about its strengths and weaknesses. **Easy to use** Look for software that you can start using right away \u2013 it's no good choosing a programme that will require you to undergo intensive training to understand how it functions. Having easy access to technical support if you have questions about or problems with your software is another important consideration.\n\n**Synchronization** Try to choose a programme that saves you time by integrating easily with your word-processing and\/or email software.\n\n**Good fit** To find out if there are any programmes designed specifically for your profession, ask colleagues or your professional association.\n\nKeep your business contacts as current as possible by updating the information as it comes across your desk. Don't forget to print out a copy of your contacts or back them up electronically.\n\n#### Managing your business cards\n\nYou head back to the office after a networking breakfast with a pocketful of business cards from new contacts. But then if you just add them to the ever-growing pile that you've accumulated over the past few months, you won't be able to follow up the new contacts easily. Try using the following tips to help you save time by managing your business cards efficiently.\n\n\u2022 When you return from a business trip, trade show, or meeting, make organizing the business cards you accumulated one of your first tasks.\n\n\u2022 Sort through the cards and keep only those that you know will be useful to you. Ask yourself if you are likely to contact this person again. If so, why and when? Discarding any that are unlikely to be useful to you.\n\n\u2022 Write a brief note on the back of each card that you are going to keep to remind you where and when you met the person. You could do this as or immediately after you meet the person, so there is no danger of forgetting. Include some personal information they told you, such as where they went to school, or how many children they have, so that you connect well with them the next time you meet or contact them.\n\n\u2022 Do an annual purge of your contacts. If you have not had any contact with someone for a year, send them an email, and ask them to let you know if they would like to be contacted by you in the future. If they don't respond, simply remove them from your list.\n\n#### Storing your business cards\n\nOnce you have selected the cards you want to keep, use a rotary-type file with alphabetical tab dividers to organize the business cards by category, such as accountant, plumber, and so on. Or, if handling hundreds of business cards is part of your work, consider purchasing a time-saving card scanner that connects to your computer and inputs all the information directly into your database.\n\n## paper management\n\nThe advent of computer-based technology predicted the paperless office. However, you have only to look around most offices to know that this simply hasn't happened. A survey conducted in 2002 by the University of California found that offices around the world used 43 per cent more paper than they did in 1999. It is estimated that 90 per cent of all business information is captured on paper. Some experts contend that the average person spends 150 hours per year searching for lost or misplaced information from cluttered desks and files.\n\nManaging the flow of paper into your office allows you to set up a system so you can process, store, and find important information. Get started by dealing with today's papers first; otherwise you will simply be creating another pile to deal with later on.\n\n#### The daily paper flow\n\n\u2022 Deal with paper as it comes across your desk, then file it as soon as possible.\n\n\u2022 When filing, place most recent papers on top in the folder so you can see the most recent correspondence every time you open the file.\n\n\u2022 Designate an \"In\" tray for your daily incoming materials, and choose a routine time to sort and process these.\n\n\u2022 Create a master list of all of your folders, which you can use as a quick reference guide. Your master list will also be helpful if a colleague needs to find certain paperwork when you are away from the office.\n\n### _climb the paper mountain_\n\n### **EXERCISE FIFTEEN**\n\nContrary to popular practice, desktops are not effective places from which to manage the mountain of paperwork that is generated by most offices. This exercise will help you organize and control your paper mountain.\n\n_**1.** Decide what to keep and what to throw away by asking yourself the following questions: Does this information help my business? Will it improve my productivity somehow? Do I need to keep these papers now that the project\/task is finished?_\n\n_**2.** Create and use these folders to help you sort and keep track of all your papers:_\n\n**_Action_** _\u2013 anything requiring your personal attention that you can't attend to while you are sorting out your papers. Make sure to add action items to your master \"To Do\" list so they won't be forgotten._\n\n**_To Read_ **_\u2013 any magazine or newspaper articles you wish to read. Grab this folder when you are heading out of the office and you anticipate having some time to read, such as while commuting by train or waiting prior to an appointment._\n\n**_Pending_ **_\u2013 paperwork to which you are awaiting a response on or further information._\n\n**_To File_ **_\u2013 stuff that needs to be filed away._\n\n**_3._** _Depending on your type of work, you might also need other files, such as:_\n\n**_Delegate_ **_\u2013 any paperwork that someone else can take care of._\n\n**_Ideas_ **_\u2013 if your work is of a creative nature, this is a great place to store your thoughts so that you can review and develop them._\n\n**_Receipts_ **_\u2013 receipts for work-related expenses._\n\n## de-clutter your office\n\nCluttered offices can be a source of stress, aggravation and lost time. If your blood pressure increases as you survey the clutter, now is the time to do something about it. Assuming that you are not in a position to hire a professional organizer (everyone's dream!), ask a friend or colleague help you remove temporarily as much from your office as possible, and pack things into boxes. If it is feasible, hire someone to clean the office, add a fresh coat of paint, wash or vacuum the floors, dust, and clean the windows. If not, do as much as you can yourself.\n\n#### Sort your clutter into these groups:\n\n\u2022 Things to have in your office \u2013 keep only items that you need for your job, and avoid having more than one stapler, one notepad, and so on.\n\n\u2022 Things to give away \u2013 if you haven't used something for two years or more, give it to your local charity or arrange to have them pick it up.\n\n\u2022 Things to get fixed \u2013 set aside time to get things fixed or to call the appropriate technician to do so.\n\n\u2022 Anything else \u2013 if it doesn't fit in any of the above groups, take it home, or recycle it, or throw it away!\n\nOrganize an annual purge of your office area. Get everyone to wear casual dress, have a potluck lunch, and at the end of the day give out some goofy prizes for the oldest document, the largest amount of clutter cleared, and the most unusual thing found in a filing cabinet.\n\n##### **office design using feng shui**\n\nFeng shui (pronounced \"fong shway\") is based on the Eastern concept of chi, the energy life force. According to feng shui principles, if your office is organized correctly, the flow of chi will be smooth and harmonious, which will enhance your well-being and quality of life. An office with too much clutter traps chi\" and can leave you feeling down and discouraged. Here are some guidelines for applying the principles of feng shui to your office.\n\n\u2022 Clear clutter so that energy can circulate. Keep on your desktop only items that you need for the current task and things that really help you to be efficient.\n\n\u2022 Arrange your office so that your desk faces the door and your back faces the wall. This is known as the \"power position.\" According to feng shui you can watch as new business comes in the door and feel supported by the wall at your back.\n\n\u2022 Fix or replace in a timely fashion anything that is broken, because things that don't work give off negative energy.\n\n\u2022 Make your office an inviting environment. This will encourage you to get work done. Decorate it with items that bring you pleasure, such as inspirational artwork or photographs of loved ones.\n\n\u2022 Make sure that you are physically comfortable. Get a good chair and make sure that your desk and computer are at the right height, your keyboard is in the correct position, and your phone is easy accessible.\n\nIf you are uncomfortable you will tire more easily and it will take you longer to get your work done.\n\n\u2022 If your office is open-plan, try to soften corners with plants to make the environment pleasant for everyone who works there.\n\n## time-saving technology\n\nThere are so many choices when it comes to technology that it is easy to be overwhelmed. When considering time-saving technology for your office, ask yourself: is the technology worth the investment of your time and money? Is it simple to use and easy to set up? How will it help you save time and\/or become more productive? Is the product reliable and easy to get serviced?\n\n#### Computers\n\nIt is very important to buy a computer suitable for the type of work you do. People who use their computer to design graphics have different requirements from those who use their machine as an electronic typewriter and filing system. Computer processing speed is measured in megahertz (mhz), and it makes economic sense to buy the fastest computer you can afford. Try to anticipate what you will need your computer to do two years from now, and buy a machine that can fulfil these requirements. How do you decide whether to buy a desktop or a laptop computer? If you know you need to use your computer in two or more locations, choose a laptop. If you travel a lot, a laptop will boost your productivity, allowing you to work at the airport or in your hotel room.\n\n_\"Time is the most valuable thing a man can spend.\"_\n\nTHEOPHRASTUS c.371\u2013287BC\n\n#### High-speed internet\n\nA high-speed internet connection, also known as broadband, is a real time saver when using the internet. If you don't have a high-speed connection, you can waste large chunks of your time downloading files or waiting for internet pages to open. The same goes for email, especially if you work in a technical or creative field, because email file attachments continue to get bigger. As the use of online video becomes more widespread for everything from music to the news, a broadband connection has become an essential workplace tool.\n\n#### USB stick\n\nMini or keychain drives are thumb-sized portable data-storage drives that provide you with immediate access to documents and files without the bother of bringing your laptop or CDs with you. Simple to use, mini drives offer the quick transfer of information simply by plugging them into the USB ports of computers. They are a great tool for people who often make presentations but don't want to carry around a laptop.\n\n#### Videoconferencing\n\nVideoconferencing is a business tool that allows you to communicate with clients and co-workers around the world without leaving your office. The benefits include reductions in time spent flying, travelling, sitting in airport terminals, waiting in line-ups, and dealing with travel delays.\n\n#### Office phone\n\nWhen choosing an office phone you need to consider: how many extensions you need; the capacity of the voicemail feature you require; and the anticipated growth of your business. Other useful features that you might find useful include: caller ID, so you can decide whether or not to take the call; a speakerphone, which is great for conference calls; and headphones, which free up your hands so that you can take notes while on the phone.\n\n#### Printer\n\nWhen buying a printer, the major time-saving consideration is speed. The speed of a printer is rated by pages per minute \u2013 the higher the ppm the faster it prints. Printing colour documents takes longer, which is an important factor if you typically print brochures, manuals, or graphics.\n\n#### Fax\n\nAlthough some people think that the fax machine is a technological dinosaur, it can still save you time in the office. Many computer programmes allow you to send or receive faxes directly from your computer, but you will have to spend time scanning in documents before you can send them, which can be very time-consuming. Sometimes a regular, old-fashioned fax machine can get the job done in less time.\n\n#### Multifunction machines\n\nInvesting in a multifunction machine that is a combination fax machine, copier, printer, and scanner can streamline a variety of processes that will save you time and money. But beware of using technology for technology's sake \u2013 only buy a machine with features that you need and know you will use, otherwise the purchase will be counterproductive.\n\n## managing email\n\nIf you dread launching your email programme because you receive so many emails, you are not alone. Studies show that we spend about two hours of each work day on emails \u2013 and this is likely to increase. Finding ways to manage email can help to reduce stress and increase your productivity.\n\nHere are some frequently asked questions and answers about how to manage email and save time doing so.\n\n_\"To choose time is to spend time.\"_\n\nFRANCIS BACON (1561\u20131626)\n\n**How often should I check my email?** This really depends on your work situation. For most of us, checking email three times per day is sufficient \u2013 perhaps first thing in the morning, halfway through the day, and at the end of the day. Do not respond to email as it comes in. Many email programmes have a message alert feature that will let you know when new mail has arrived. Unless your work requires you to answer emails as they arrive or you are expecting an urgent email, turn off the message alert and stick to a regular schedule to check email.\n\n**What is the best way to handle mail in the inbox?** Your inbox should contain only email that you are waiting to read or that requires action. Set up your email programme to sort mail automatically into pre-designated folders. The aim is to deal with each email message once only, by responding, filing, forwarding, or deleting it.\n\n**How can I save time when sending and receiving email?**\n\n\u2022 Use the subject line to give the recipient an indication of what the email is about.\n\n\u2022 Change the subject heading to reflect the content of the new message when replying.\n\n\u2022 Be clear, concise, and to the point.\n\n\u2022 Stick to one major subject per email, especially if you are requesting the recipient to take action on a number of different items.\n\n\u2022 Be clear about what action you are expecting from the recipient.\n\n\u2022 Create a signature file (see your email programme) that automatically includes your contact information at the end of each email.\n\n**How can email folders save me time?** If you receive more than ten emails per day, folders can be a great time-saving feature. You can separate important email from less important by setting up your email to recognize key words and direct the messages straight into project or client folders. These will make it much easier for you to find crucial emails and make it less likely for them to get lost in the deluge.\n\n**When is it counterproductive to use email?** Email is not the best option if you are dealing with an urgent matter and don't want to waste precious time. If you know the recipient is in their office or you can reach them on their cell phone, you have a better chance of resolving the issue quickly. If your response to an email would be long and involved, picking up the phone and having a conversation will save you time.\n\n## phone \u2013 slave or master?\n\nPhone calls can be a very time-consuming aspect of work, but by applying certain criteria you can cut down on time-wasting. You have more control over your time when you make outgoing calls rather than when you answer the phone. You can save time by pre-programmeming into your phone the numbers that you call most frequently. When calling a business contact always ask if it is convenient for them to talk now, tell them approximately how long your call will take, and keep to the time limit. If it is not a convenient time for them to speak with you or they do not have the time that you need available, arrange to call them back at a specified time. Ensure that you have all the relevant notes and documents you require in front of you before you make the call to avoid wasting time looking for them while keeping the other person hanging on. Make all your non-urgent phone calls in one block of time during the day.\n\nTo save time on incoming calls at work use caller ID to screen them and answer only those that you know will help you to complete the day's tasks. Start by asking the caller politely how long the call will take. If they say a few minutes and you know from past experience that they can stick to this limit, deal with the phone call there and then. However, if you know from experience that they are likely to take up more of your time than they estimate, or if it is truly not a good time for you, explain this to your caller and arrange to call them back later at a mutually convenient time.\n\n### _track your phone time_\n\n### **EXERCISE SIXTEEN**\n\nSetting up a phone log of your business calls so that you can find out how much time you spend making and receiving phone calls will help you to cut down on unnecessary calls. You will need a large sheet of paper, a ruler, and a pen.\n\n_**1.** Take the sheet of paper and draw eight columns using the ruler and the pen. Label them across the top from left to right: Date, Caller\/Recipient's Name, Incoming\/Outgoing Call, Voicemail, Duration, Reason for Call, Useful\/Not useful._\n\n_**2.** Fill in the relevant columns as you use the phone for two weeks._\n\n_**3.** Add up the total amount of time you spent on the phone during each day, then the total for each week, and finally the total over the whole two-week period._\n\n_**4.** Now tot up the amount of time spent on useful calls and then on calls that you didn't find useful. Review the reasons for the latter calls to see if a pattern emerges \u2013 it could be the same person calling, or perhaps you are trying to speak with someone but you keep missing each other. Consider if you could take any action to eliminate the calls that waste your time. For example, it might be easier to reach the person you keep missing by email._\n\n_**5.** Over the longer term, use a notebook, your daily planner, or your contact-management software to keep a record of important telephone calls. Besides recording names and numbers, jot down the key points of your conversations, as you might find this information useful in the future._\n\n## project management\n\nProject managers strive to deliver on time and within budget. A project has a distinct beginning and end, and uses specifically allocated resources, such as money, consumables, personnel, and time. Projects have a goal and an objective, and each project follows a logical sequence of events that makes use of the resources available to meet the project deadline. Ask the right questions. What is the project's objective? What are the start and finish dates? What resources do you need? What tasks do you need to accomplish? Who is responsible for which tasks?\n\nWhen managing a project there are three constraints to balance: time, cost and quality. We can view these as the sides of an equilateral triangle. And if one side grows, this generally affects the other sides.\n\nAs a project manager, you can lose credibility by underestimating the time required to complete a project. Understanding the scope of the project is the first step toward an accurate judgment of the time you will need. A detailed list of tasks that you need to accomplish should include the administration of the project, such as meetings, as well as the actual work. Be realistic and allow time for inevitable disruptions and delays.\n\nOnce the project is completed, ask yourself: was the project on time? If not, what external circumstances contributed to any missed deadlines? What circumstances over which you had control contributed to any missed deadlines? What will you do differently when estimating the time you require to finish the next project? Bear the answers in mind for the future.\n\n##### **how to create a realistic schedule**\n\nThe project manager faces two major challenges: how to allocate enough time to meet the deadline, and how to recognize and avoid time-wasting obstacles. First, figure out how long each activity will take. Next, estimate how long it will take to complete the activity under ideal circumstances, and what resources are available, and when. Then, add in some extra time for unforeseen problems, delays, and contingencies, such as those below.\n\n\u2022 Emergencies\n\n\u2022 Staff absence and illness\n\n\u2022 Subcontractors late\n\n\u2022 Equipment failure\n\n\u2022 Late deliveries by suppliers\n\n\u2022 Delays caused by the client\n\n\u2022 Slow approval process\n\n\u2022 Changes in requirements\n\n\u2022 Public holidays\n\n## time-wasters and time-savers\n\nLooking back on your day with the feeling that you've wasted your time is frustrating. Time-wasters come in many forms and from all areas of our lives, both personal and professional.\n\nThis chapter will help you manage the mountains of information that you come across and will provide you with useful ways to deal with disruptions. Learning to manage meetings, to delegate, to say no, and to make use of your time when you have to wait, will turn your time-wasters into time-savers. Take the time to implement these ideas and you are on your way to becoming a good time manager.\n\n## information overload\n\nThere is no question that the technological advances which are part of our everyday lives have made many mundane tasks easier and faster to complete. However, the downside of technology is information overload. The massive amounts of information that come from a huge number of sources, such as television, radio, emails, faxes, text messages, pagers, the internet, social media, newspapers, magazines, catalogues, and so on, can be overwhelming.\n\nCombine the amount of information that comes your way on a daily basis with the limited number of hours in the day, and you have a recipe for time-related stress. In 1996 Reuters Business Information surveyed 1,300 managers about information overload. In the resulting report, 38 per cent of them were stated as admitting they wasted \"substantial\" amounts of time collecting information. A significant number \u2013 43 per cent \u2013 said that their ability to make decisions was affected by what was termed \"analysis paralysis,\" and 47 per cent confessed that information collection distracted them from their main responsibilities.\n\nThe study also demonstrated that information overload is linked to stress: 43 per cent of managers claimed to have suffered health problems caused by the stress of tracking, reading, and filing information. Also, 47 per cent said that they took home work to keep up with the information they accumulated, which demonstrates that information overload had an adverse impact on their home life and leisure time.\n\nHow, then, is it best to manage information? Here are some different strategies.\n\n##### **save time with speed reading**\n\nOne trick to help you keep up with important information is to increase the speed at which you read. First, determine your purpose before you start. Do you simply need to digest the main ideas? If so, skim-read. But if you need to assimilate more in-depth information, here are some tips.\n\n\u2022 Read two, three (+) words at once and don't focus on every word.\n\n\u2022 Avoid \"saying\" the words mentally as you read.\n\n\u2022 Use a method to trace a path for your eyes to follow. For example, place your right hand on the right margin and move it slowly down the page, following the movement out of the corner of your eye.\n\n\u2022 Don't re-read phrases and sections of text. This will slow you down and will not necessarily help you understand the core ideas, as these are often elaborated on as you continue to read.\n\n\u2022 If you struggle with poor concentration when reading, start by focusing for five or ten minutes at a time.\n\n\u2022 Look for key words and phrases that contain the main ideas and skim over less important words.\n\n\u2022 Underline key phrases or ideas for quick reference later.\n\n\u2022 If you have a lengthy document to read, survey the table of contents and read the summary or abstract so that you know how the information is laid out before your begin. This will also give you an idea of how difficult the material is and how much time you will need to cover it.\n\n\u2022 Improve your vocabulary by looking up the meaning of any new words \u2013 especially jargon. Learning the relevant terminology will speed up your reading and comprehension in the future.\n\n\u2022 Don't read when you are tired or feeling unmotivated \u2013 it's a complete waste of your time.\n\nWhen using the internet search engines: have a clear idea of what you are looking for so that you can refine your search. Set a time limit when looking for information and accept that there is usually more available than you have time to access, so let some of it go.\n\nReduce the amount of books, CDs, magazines, and newspapers you buy if you don't have time to read them all. When keeping magazines and newspapers, always throw one out for every new one you add.\n\nUse a reference librarian as an information gatherer. Many libraries have reference librarians who will search for information for you.\n\n## the management of meetings\n\nMention the \"m\" word and most people will tell you they have a love\u2013hate relationship with meetings: \"Can't work with them, can't work without them.\" Although meetings can be one of the most effective ways to discuss and exchange ideas and to make decisions, the general consensus is that many meetings are a waste of time \u2013 a reported 50 per cent of meeting time is unproductive.\n\nProfessor Roger K. Mosvick's 20-year analysis of who attends meetings and for how long found that, on average, executives spend 12 hours per week at meetings, middle managers spend 10.5 hours, and employees 8.5 hours. Other studies show that the average meeting lasts an hour and a half; that attendees are given an average of two hours' notice to attend; that 63 per cent have no advance written agenda; and that 11 per cent of the meeting time is spent on irrelevant issues. To plan and run a successful meeting requires a particular skill-set \u2013 thorough preparation, the ability to communicate clearly, and good time management.\n\n#### Time management before meetings\n\nA well-written agenda for a meeting is clear and concise, with time allotted for each item. Prepare an agenda containing the following information: who is going to attend, the purpose of the meeting, where the meeting will be held, and when the meeting will start and finish. Organize your agenda with the most important items at the beginning and leave any new business until the end. Before you finalize it, give the attendees an opportunity to add discussion items. List the name of the presenter next to each item and give them ample time to prepare. Distribute copies of the agenda well in advance of the meeting. Also circulate in advance any material that will be discussed at the meeting to avoid wasting time reading them during the meeting itself.\n\n#### Time management during meetings\n\nShow people that you value your time, and others', by starting meetings on time. Waiting for people, especially those who are chronically late, gives them the message that it's OK to run behind schedule. Once the meeting has begun, don't interrupt it to brief latecomers \u2013 that is simply wasting the time of the people who were prompt.\n\nAt the beginning of your meeting, remind participants how long it should last and take care to end at the stated time. Also, stick to the time limit you placed on the agenda items by letting people know when to wrap up, make a decision, and move on. Be diplomatic but firm. Invite people to continue a particular discussion in their own time after the meeting. Chances are you won't have any takers. Schedule your next meeting while everyone is still in the room to save yourself the aggravation of trying to accommodate everyone's schedules at a later date when they will be more booked up. Manage meetings effectively and you'll avoid Thomas Kayer's definition: \"A meeting is an interaction where the unwilling, selected from the uninformed, led by the unsuitable, to discuss the unnecessary, are required to write a report about the unimportant.\"\n\n### _make your meetings meaningful_\n\n### **EXERCISE SEVENTEEN**\n\n_Unsure whether or not a face-to-face meeting is really necessary? Run through the following checklist to determine if you should go ahead:_\n\n\u2022 _You need a fast decision based on the input of the whole group or team._\n\n\u2022 _You need to resolve a group conflict._\n\n\u2022 _You want to brainstorm ideas._\n\n\u2022 _You want to share problems, weigh alternatives, and find solutions._\n\n\u2022 _The group wants to meet._\n\n\u2022 _To share success._\n\n_Don't call a meeting ..._\n\n\u2022 _When you can use another form of communication instead._\n\n\u2022 _If you are not sure who should attend or key people cannot attend._\n\n\u2022 _If there is not enough time for you or the participants to prepare properly._\n\n\u2022 _When the cost in terms of time and money outweighs the benefits of what will be accomplished._\n\n\u2022 _If you doubt that a meeting will achieve its objective._\n\n\u2022 _If you are avoiding making a decision._\n\n\u2022 _To reopen discussion about a decision that has already been made._\n\n\u2022 _If you simply wish to give out information._\n\n\u2022 _If there is a personal issue between you and a colleague or an employee._\n\n\u2022 _If people are angry and need time to cool down._\n\n\u2022 _If you or your group members are suffering from \"meeting fatigue.\"_\n\n\u2022 _If your group, executive, or team needs encouragement or a pick-me-up (plan a social get-together instead)._\n\n## dealing with interruptions\n\nDealing with interruptions when trying to get your work done can make a short task drag on and on, upsetting your plans for the day and even the week. Sometimes we create our own interruptions by procrastinating or striving for perfection. At other times our personal life overlaps into our work time and friends call to chat or we get an email from the head of the school committee looking for a volunteer. Disruptions at work range from emails and phone calls, to people dropping by your office for legitimate work-related reasons or simply to socialize.\n\n#### Track your interruptions\n\nExamine the interruptions and look for patterns. Are the same people doing the disrupting? Do the disruptions occur at a certain time of day? Do they happen in your office or when you are at a particular workstation? Are you being interrupted for the same reason?\n\n#### Strategies to handle or minimize interruptions\n\n**Tell people not to interrupt you** Use the technology at your disposal to let people know that you do not want to be disturbed. Record a voicemail message telling people that you are available to take and return calls between specific times only. You can do the same with your emails \u2013 create a return message that is sent automatically to each email received. If people know when you are available, they are less likely to interrupt you while you are busy.\n\n**Turn if off** Once you've notified people that you are not available, turn off the technology that allows them to contact you. Shut down your email, and switch off your pager and your cell phone.\n\n**Closed-door policy** An open-door office policy can be a big time-waster. Post a schedule listing your \"open-door\" time slots. Keeping your office door closed makes it less likely that you will be distracted by people walking past, and discourages people from entering your office to chat.\n\n**Conversation-enders** How can you be diplomatic with people who monopolize your time? By being honest and telling them, \"I honestly don't have time to talk right now.\" Then arrange a specific time with them, \"I will be available at 4pm for ten minutes. Is that a good time for you?\"\n\n## learn to delegate\n\nOne of the quickest ways to free up your time is to delegate tasks to others. Knowing when and how to delegate are the keys to success. Some reasons why people can't or won't delegate include: not having enough time to figure out what can be delegated or to give instructions; lack of trust in the abilities of others; perfectionism; the desire to micro-manage down to the smallest detail; and perceived loss of control.\n\nDelegating to others makes good time-management sense. Others can take responsibility for work you never get to or put off because you don't have enough time. Learn to delegate work that is repetitive and routine, outside your strongest skill set, important but not urgent, and on a long schedule. In addition, look at your results for Exercise one (p.21) and Exercise three (p.27) to help you make a list of tasks that you can delegate.\n\nAfter you have identified someone who has the skills needed to take over the work, ensure that they have sufficient time to complete it. Be clear about what you want them to accomplish, by when, the resources available to them, and your level of involvement, if any.\n\nCreate a written set of instructions for repetitive and routine tasks. This documentation is a resource for the person doing the task and can be given to a back-up person in the event that the usual person is away.\n\nGive accurate feedback when your staff member or colleague has completed the task you delegated to them. Start by praising and thanking them for their contribution, and then discuss what went well and why, and how the process could be improved in the future.\n\n### _assess your workload_\n\n### **EXERCISE EIGHTEEN**\n\nUse this quiz to find out if you have over-committed yourself. Give the statements points according to the following scale: 0 = never 1 = sometimes 2 = mostly 3 = always. Then, add up the total to see which \"zone\" you are in.\n\n\u2022 _I have difficulty getting out of bed to face the day because I have so much to do._\n\n\u2022 _I sometimes \"freeze\" with fear and anxiety about my workload._\n\n\u2022 _I usually work through my lunchtime._\n\n\u2022 _I feel angry and\/or depressed because of my overloaded schedule._\n\n\u2022 _I have trouble delegating work or tasks to others._\n\n\u2022 _I rarely have a feeling of accomplishment._\n\n\u2022 _I find it impossible to say \"no\" to work._\n\n\u2022 _I volunteer to do things that I know I don't have time for._\n\n\u2022 _I lie awake worrying about how I'm going to get everything done._\n\n\u2022 _I feel guilty if I have time to myself because I think I should be doing something productive._\n\n**_Your score_**\n\n**_Green Zone = 0 to 10_**\n\n_You seem to be managing your commitments skillfully, so keep on doing what you have been doing, because it is working well for you._\n\n_Y **ellow Zone = 11 to 20**  \nYou are in the danger zone, so beware. Look for small ways to cut back on commitments and give yourself more time and space to breathe._\n\n**_Red Zone = 21 to 30_**\n\n_You are in commitment overload and you need to rethink your schedules and activities before there are serious consequences to your well-being._\n\n## how to say \"no\"\n\nThe ability to say \"no\" at the right time, with the appropriate tone of voice and body language, not only saves you time, but can preserve relationships. Time is a commodity, and often people ask us for our time because they don't have enough of their own.\n\nAlthough an inability to say \"no\" can cause you to make too many commitments, it is more likely to be related to your personality than time-management skills. Most people dislike conflict and confrontation, and believe that saying \"yes\" avoids both \u2013 at least in the short term. Others are people-pleasers, who are motivated primarily by a desire to be liked by others, and they may also struggle with perfectionism.\n\nYou can say \"no\" without ever using the word \"no,\" by using assertiveness skills. Being assertive means being able to communicate your opinions and feelings in an honest and clear fashion, in a way that respects the rights of others. For example, let's imagine that someone is asking you to be a committee member. Tell them you'd like to help but you cannot, and offer a brief, one-line explanation, such as \"Right now my schedule won't allow me to take on anything new.\" (Although this is not always necessary, it might help to avoid a long conversation or further questions.) If the person persists, simply repeat that you'd like to help but you cannot, without any further explanation. Follow it with \"It's been nice speaking with you,\" and move on.\n\nWhat if a supervisor or manager at work asks you to take on more work? You can handle it diplomatically by asking them to decide which project they would like you to stop working on in order to fulfill their request. That puts them in position of making the decision and makes it clear what will and will not get done.\n\nNever allow yourself to be pressured into giving a response on the spot. Be assertive and ask for time to think it over, especially if you know that you are having difficulty saying \"no.\" Ask yourself, \"If I say 'yes' to this what am I saying 'no' to?\" before deciding.\n\nLearning to say \"no\" takes practice, so start by saying \"no\" to small requests, and you will be surprised by how easy it becomes.\n\n## making use of waiting time\n\nMany of us make the effort to call ahead when we have an imminent appointment, to determine whether or not the doctor or the hairdresser is running on time, or we always try to book our appointments for first thing in the morning or right after lunch to reduce the amount of waiting time. But no matter how carefully we plan, there will be times when waiting is simply unavoidable.\n\nAlthough no one enjoys waiting in line-ups, or getting delayed when traveling, you can make use of unexpected gaps of time by having a contingency plan to fill them. Use your waiting time \u2013 whether it's five minutes, half an hour, or half a day \u2013 to your advantage by doing small chores that are on your \"To Do\" list.\n\nAnother option is to make it your policy to enjoy unexpected free time and use it as a reward to do things that you like doing and wish you could do more often. For example, if you enjoy reading, make a habit of taking something to read with you when heading off to an appointment or using public transport. This will also expand your vocabulary, or help you to keep abreast of developments in your profession or field of expertise, or let you escape mentally into a different world. If reading is not a favourite activity, try doing a crossword or a number puzzle, or make a gratitude list to remind yourself of good things in your life. Or you could simply take the opportunity to relax and unwind by closing your eyes and breathing slowly, deeply, and rhythmically.\n\n##### **how to use unexpected free time**\n\n**Things to do in five minutes**\n\n\u2022 Clean out your handbag or briefcase.\n\n\u2022 Listen to a guided visualization that takes you for a walk in the forest or on the beach.\n\n\u2022 Start a load of laundry.\n\n\u2022 Clean out a desk or kitchen drawer.\n\n\u2022 Read an article or cutting from your \"To Read\" file.\n\n**Things to do in half an hour**\n\n\u2022 Take a nap.\n\n\u2022 Go for a walk or a run.\n\n\u2022 Call someone you care about.\n\n\u2022 Fill out routine forms, such as license renewals and parking permits.\n\n\u2022 Do the prep work for the next meal.\n\n\u2022 Collect expense receipts and itemize them.\n\n\u2022 Think about and jot down the agenda for your next meeting.\n\n\u2022 Write a thank-you note to someone who was paricularly helpful in the past week.\n\n\u2022 Take a virtual tour on the internet of a country you would like to visit.\n\n\u2022 Enter frequently called numbers into your cell, home, or office phone.\n\n\u2022 Go through your bookcase, remove any books that you haven't read or looked at for two years, and give them away to friends or donate them to a charity shop.\n\n**Things to do in half a day**\n\n\u2022 Review your goals.\n\n\u2022 Visit a bookstore or library with comfortable seating, and read.\n\n\u2022 Visit your favourite art gallery.\n\n\u2022 Go for a beauty treatment.\n\n\u2022 Invite a friend or colleague to meet you for coffee or tea.\n\n\u2022 De-clutter your filing cabinet by getting rid of old files.\n\n\u2022 Clean up your computer by removing files that you no longer use.\n\n\u2022 Organize your photos into albums.\n\n\u2022 Bring a DVD with you when traveling and watch it on your laptop.\n\n\u2022 Simply relax and do nothing!\n\n## time management for everyone\n\nEveryone can make use of time-management skills to improve their quality of life and to work toward important goals. This chapter includes creative ideas to help teach both young and old about the value of time.\n\nWe start with children and how to instill in them the rudiments of time management. Next, we tackle students, because poor time management is the main culprit in academic failure, so learning how to create semester and exam schedules is vital. If you are a busy parent you will find plenty of time-saving tips as well as ideas to help you avoid over-scheduling yourself and your kids; while those of you who are creative will discover some non-traditional approaches to time management. Finally, we look at the time-management challenges of retirement and discover ways to help you spend this time meaningfully.\n\n## time management for children\n\nAnyone who has spent time around newborn babies knows that their needs and behaviour \u2013 for example, sleeping in the day, and waking at night \u2013 do not fit easily with existing routines. But gradually babies adapt to their family's way of life.\n\nOnce children become toddlers, they are ready to learn about time. You can introduce them to the basics by showing them the difference between day and night, pointing out the time of day, and introducing the concepts of \"before\" and \"after,\" using lunchtime as the divider. Teach little ones about the passage of time by talking about yesterday, tomorrow, the days of the week, the months, and the seasons.\n\n#### Time management and responsibilities\n\nAlthough little children cannot read, they can follow pictures. To familiarize them with the idea of time management you can create a picture chart of their routine morning tasks, such as waking up on time, eating breakfast, and brushing their teeth. Buy some colourful stickers and let them place these beside each action they complete each day. At the end of the week offer a small treat as a reward for doing well.\n\nChildren between the ages of 6 and 11 are capable of learning how to manage their own time. You can begin by giving your child an alarm clock or clock radio on their sixth birthday and setting it for the time they need to get up in the morning on a school day. If your child is struggling to get everything done in the morning, reward them for each day they finish their checklist before heading off to school, and let them collect their reward at the end of the school week.\n\n### _make a countdown calendar for kids_\n\n### **EXERCISE NINETEEN**\n\nPart of the fun in life is the anticipation we feel before important events. Why not help your child understand the concept of time by creating a countdown calendar to use in the run-up to a special day or event.\n\n_**1.** Gather together a large, colourful sheet of paper, a pen, and a ruler. Decide how many days or weeks you wish the calendar to cover. (Avoid making the countdown last longer than three or four weeks \u2013 less on a calendar for a small child \u2013 as any longer might seem too far away, and they might lose interest.)_\n\n_**2.** Take the pen and the sheet of paper and draw a grid containing one square to represent each day. Use the ruler to make sure that the lines are straight and the squares are roughly the same size. Alternatively, you could use a word-processing programme to create and print out a grid from your computer._\n\n_**3.** Write the dates in the squares on the calendar, using another calendar for reference. The easiest way to avoid mistakes is to start at the end (the square in the bottom right-hand corner) by marking on the special day or event first, and work backward to the calendar's start date. When you have finished, check that you haven't missed out any days._\n\n_**4.** Stick the chart on a wall somewhere where your child can reach it easily and will see it often, such as in the kitchen or in their bedroom. Get them to colour in or cross off a square every day, thus counting down the days remaining until the big day or special event._\n\nSetting time limits, such as, \"You have 15 minutes more before you go to bed,\" will also help children to quantify the passage of time.\n\n#### Play time-related games\n\nHave your children guess how long it takes to do certain things. For example, how long do they think it takes to drive to Grandma's? Or how long does it take them to read their favourite story? This will help them to develop a sense of minutes and hours. You could also ask them to name as many things they can think of that last for set periods of time: say, for 15 minutes, one hour, and so on.\n\n##### **spending my time**\n\nUsing coloured disks or coins will teach your child to spend their time wisely. You can draw, colour, and cut out small circles. Different coloured disks represent different time values, rather like casino chips (but don't worry, you won't be teaching your child to gamble!). For example, the red chip = 1 hour, the green chip = 30 minutes, and the blue chip = 15 minutes. Supply your child with the following amount of disks or coins: 12 red = 12 hours, 18 green = 9 hours, 12 blue = 3 hours. Your child pays you with coloured disks before they \"spend\" their time on any activity they choose. This will help them understand that time is a valuable commodity and it will show them how they can make choices about how they use their time.\n\n### _make time \"pizzas\"_\n\n### **EXERCISE TWENTY**\n\nA time chart in the style of a pizza can show your child how they are spending their time, which will help to increase their time awareness.\n\n_**1.** Using three large sheets of paper or three left-over pizza cartons, and some brightly-coloured pens, draw a large pizza on each sheet or carton. Get your child to help you to colour them in and draw their favourite toppings on the pizzas._\n\n_**2.** Label the first pizza the \"To Do Pizza.\" Make a list of all the things that your child has to do in a typical day. Have them prioritize the most important tasks to the least important, and get them to estimate how long each item will take to do. Divide the pizza into time slices using the prioritized list and estimates._\n\n_**3.** Name the second pizza the \"Hour-by-Hour Pizza.\" This will show how your child actually spends each hour of a typical day. Divide the pizza into 24 \"slices\" and number them for each hour. Starting with the time at which your child gets up, label each segment according to the main activity in that hour \u2013 for example, breakfast, school-work, playtime, sleep, and so on. Divide up the day until you return to the start._\n\n_**4.** Call the third pizza the \"Activities Pizza\" and divide it into twelve two-hour time slices. With your child count up how many hours they spend on different activities. Point out how it all adds up to 24 hours._\n\n_**5.** Compare the pizzas to see how your child's priorities and estimates compare with how they really spend their time._\n\nAnother good game will help your child to understand the difference between seconds and minutes. You will need an alarm clock or a kitchen timer and a watch or clock with hour and second hands. Ask your child to watch the clock as you point out when the second hand moves from 0 to 5 seconds, then from 0 to 30 seconds, and finally from 0 to 60 seconds. Discuss which felt longest to them. Explain that 60 seconds means the same thing as one minute.\n\nNext, set the alarm or timer for two minutes and ask your child to dance to some favourite music. When the two minutes are up, set the alarm or timer for another two minutes and this time get the child to sit still and stay quiet until the timer goes off. Then, ask them which seemed to them to last longer: the time spent dancing or the time spent sitting still? Why?\n\nFinally, have them make some guesses about how long ordinary things take, such as setting the table, boiling the kettle, putting away toys, and tying their shoelaces. Let them check the accuracy of their guesses using the watch or the timer.\n\n#### Learning to respect time\n\nEncourage your child from an early age to be on time. This will teach them to respect their own time and everyone else's. Look at how they spend their time by filling in a pie chart to represent an average school day. Ask them to point out any times when they run into time conflicts. Then, take a typical activity, such as getting ready for soccer practice. Break it down into five-minute increments so that they learn the steps involved in getting ready and how long it actually takes to get all these things done.\n\nDiscuss the fact that being on time is a sign of respect for the coach and their teammates. If your child is habitually late, allow them to deal with the natural consequences. This teaches them the cause-and-effect relationship between the choices they make and the consequences.\n\n_\"We always have time enough if we will but use it properly.\"_\n\nJOHANN WOLFGANG VON GOETHE (1749\u20131832)\n\nContinually nagging children to hurry up is distressing for everyone. Clearly state your expectations: \"You have ten minutes before we leave for soccer practice. If you are not ready, we will leave without you\" (or \"... you will not be going at all,\" as the case may be).\n\n## time management for students\n\nPoor time management is the main culprit behind academic failure. Whether you are in high school or beyond, as the workload increases, you can't do well in school if you are always pushed for time.\n\nLet's start with the big picture. What are your goals related to school? What do you want to accomplish? How will you benefit by completing this stage in your academic career? How do you imagine that your life will be improved? How will you feel about yourself, having completed this degree or programme? Staying connected to your goals will motivate you to keep your time-based priorities in the proper order: designate time for school first, and fit the rest of your life around it.\n\n##### **are you time-wise?**\n\nRespond to the following statements either \"Yes\" or \"No\" to determine your current time-management situation.\n\n\u2022 I write things down.\n\n\u2022 I use a daily \"To Do\" list.\n\n\u2022 I plan for the coming week.\n\n\u2022 I know what is due next month.\n\n\u2022 I complete assignments on time.\n\n\u2022 I schedule time to study for exams.\n\n\u2022 I know how long each assignment will take.\n\n\u2022 I use a time-management planner.\n\n\u2022 I treat my school work like a job.\n\n\u2022 I put school work before other activities and leisure time.\n\nThe more \"Yes\" responses you have, the better your time-management skills are.\n\n#### How much time do you spend?\n\nIt's unlikely that you will be able to improve how you manage your study time unless you get an idea of where all your time goes. Start by estimating how much time you spend on the following in any given week:\n\n\u2022 Sleeping and eating\n\n\u2022 On personal hygiene\n\n\u2022 On chores and personal errands\n\n\u2022 Playing sports or exercising\n\n\u2022 Watching TV or using the computer\n\n\u2022 Socializing with friends\n\n\u2022 Spending time with family\n\n\u2022 In class\n\n\u2022 Studying and reading\n\n\u2022 Commuting\n\n\u2022 Working\n\nThe next step is to keep track of the actual amount of time you spent on each item. Is there a difference between your estimate and the actual? This will give you an idea of the adjustments you need to make \u2013 which activities you need to reduce and where you need to invest more time in order to meet your school-related goals.\n\n#### School planners\n\nMany schools and institutions publish a combination day planner and student handbook, which includes important dates such as examinations, course dates, fee-payment deadlines, and so on. School planners are based on the school year and they run typically from August to the following July. Most use a week-at-a-glance layout over two pages, along with the entire month on one page at the beginning of each month.\n\n#### Make good use of your study time\n\nBefore you begin an assignment, make a rough assessment of how long it will take to complete. Write a list of the steps required to get it done. Include research time, consulting with your professor, and so on. Estimate how long each step will take and add up the total \u2013 now double it!\n\nAssess when you best like to study. Are you an \"owl\" or a \"lark\"? If you like to get up early, enjoy attending early morning classes, and do your best work before 3pm, chances are you are a \"lark.\" But if you feel best in the later part of the day and like to do school work in the evening and into the night, you are an \"owl.\" If you are an \"owl\" you may have a more difficult time because school schedules favour \"larks.\" Knowing when you work best will help you make good use of your time: Study when your attention-span is at its peak and do routine tasks at other times. Procrastination is a common problem for students in higher education, with approximately one third of students stating that it interferes with their academic achievement. If you have a procrastination problem, tackle it now \u2013 p.30 and Exercise four on p.33 will show you how.\n\n### _create a semester schedule_\n\n### **EXERCISE TWENTY-ONE**\n\nThis exercise shows you how to compile a semester schedule, which will give you a bird's-eye view of what you need to do to fulfill your course requirements.\n\n_**1.** Using a computer, print out a monthly calendar \u2013 one for each month in the semester. Or, if you prefer, buy a large, wipe-clean wall calendar from the school bookstore or an office supply store. Better still, have both: a monthly calendar to keep with your agenda and a large calendar posted near your desk to use as a visual reminder to help you stay on track._\n\n_**2.** Take the school handbook and list important institutional dates, such as registration dates, assignment deadlines, exam schedules, and payment-due dates._\n\n_**3.** Gather together all your course outlines and lists of assignments. Taking one course at a time, write on your calendar(s) the dates of exams, presentations, and when essays and so on are due. Also, mark on how much each exam and piece of work contributes toward the final grade (for example, an essay might be worth ten per cent). This will help you to plan how much time to allocate for each item._\n\n_**4.** Include any important personal, one-off dates, such as birthdays, and special family and social events, so that you can plan your school work around them._\n\n_**5.** Make a note of any weeks in which you have more than one exam or assignment due, and do as much in advance as you can to avoid last-minute cramming and having to rush your assignments._\n\nIf you find it easy to get down to studying, a good approach is to start your sessions with your most difficult, least-liked subjects. You will concentrate better and retain more information when you are fresh, and you will also motivate yourself by getting the hardest tasks out of the way.\n\nStudy where the distractions are fewest. If being at home makes it too tempting to turn on the TV or to head for the kitchen, go to the school library instead. But make sure that you take a five-minute break every half hour, especially when studying for exams. Studying for longer than half-hour blocks of time makes you increasingly ineffective. Avoid the temptation to take longer breaks by establishing a routine \u2013 for example, make a cup of tea, or take a short walk through the library.\n\nTry to review your class notes on a daily basis. This will help to consolidate the material in your memory and will reduce the amount of time you will need to revise for exams.\n\n#### Balancing school and your social life\n\nPart of the school experience is the social life. In order to do well in school it is important to have a social life but, as in all things, within reason. Including your fun time in your planner will help you balance friends and school. Plan your social life around your school life and not vice versa. Designating every Friday night as a going-out night gives you something to look forward to and breaks up the week. Avoid the tendency to join too many clubs and activities \u2013 especially at the beginning of the school year.\n\n### _make a weekly schedule_\n\n### **EXERCISE TWENTY-TWO**\n\nSunday evenings are a great time to plan for the coming week. This exercise will show you how to schedule the week ahead.\n\n_**1.** Use your planner, or print out a weekly template from your computer on which you can enter class times once for the whole semester. Fill in regular features, such as lectures and tutorials, and add the deadlines for assignments and tests. Include your work hours for the week along with any other, non-study commitments._\n\n_**2.** Map out your study time for the week ahead, bearing in mind the level of difficulty you have with each individual course. Some courses will require you to put in more effort, others will take less preparation and study time. Take into account the amount of reading you need to do and allow time for note-taking._\n\n_**3.** Schedule the most difficult or least interesting tasks for times when you are at your peak mentally (see p.128)._\n\n_**4.** Don't overfill your schedule, and be realistic about how much you can accomplish, especially after a full day of classes and working part-time. Leave time for relaxing, socializing, and going out._\n\n_**5.** Don't be afraid to revise your schedule according to your needs. Learning to juggle study with work is a valuable part of your education._\n\n_**6.** Use your weekly schedule to create your daily \"To Do\" list._\n\n#### Balancing school and work\n\nIf while you are attending school you are also working, you will undoubtedly be time-pressured. Don't work for more than 20 hours per week as this will not only jeopardize your chances of success, but also increase your chances of dropping out. Look for work close to your home or to the campus to reduce traveling time. Many colleges and universities hire students and also offer programmes where you can alternate a school semester with a work semester. Or take one less course per term \u2013 this will allow you to work enough hours to meet your financial commitments. You are also likely to get better grades if you have fewer courses to work on.\n\n##### **the 2:1 time-ratio rule of thumb**\n\nThe higher the level of education, the greater investment of time required. As a general rule, at college and university level you should spend two hours studying for every hour you are in the classroom. If you spend 12 hours per week in class you should schedule 24 hours of preparation and study. That's a total of 36 hours of your week dedicated to school.\n\nYou will probably need to make adjustments as you learn what works best and how to handle obstacles. Keep in mind your goal of graduating to motivate you when you have to juggle endless classes, assignments, and exams. Once you have put some time-management strategies in to place, re-do the Are You Time-Wise? quiz on p.126 to see how you have improved.\n\n### _create a study schedule for exams_\n\n### **EXERCISE TWENTY-THREE**\n\nWhen your exams come round \u2013 and they always arrive quicker than you think \u2013 you need to know how much study time you'll need to get the grades you deserve. This exercise will help you schedule your time well in the run-up.\n\n_**1.** To determine how much time to spend on studying a specific course, make sure that you know its per centage contribution toward your overall grade. For example, if one exam is worth 20 per cent and another is worth 80 per cent, you should allocate more time to study for the latter._\n\n_**2.** Approximately four weeks before the exams, prepare your study schedule. Make a list of exams and their dates for each course, and work out how much time to spend on each. Then, tot up the total amount of hours you'll need to study._\n\n_**3.** Using the exam dates, work backward, filling in two-hour blocks of study time. Studying in two-hour time blocks with short breaks in between prevents fatigue and time-wasting._\n\n_**4.** Distribute your study time over the four weeks prior to your exam, because spacing helps you to retain more information than cramming in one long study session._\n\n_**5.** Don't forget to schedule enough sleeping time during your pre-exam period. Deep, restorative sleep helps to consolidate information into your long-term memory. Staying up all night to study is not a productive use of your time, and writing exam papers when you are exhausted is not a great strategy for success._\n\n## time management for parents\n\nBeing a parent is the ultimate time-management juggling act. When you are a parent, your life has three simultaneous cycles \u2013 the school cycle, the work cycle, and the family cycle, each with competing and sometimes conflicting demands.\n\n#### Spending more time with your children\n\nTime spent with children is often the first thing to suffer when we are experiencing a time crunch. Above all else, our children want \u2013 and need \u2013 our time. Studies around the world reflect the comments of Barbara Moses, the expert in career management, \"The time famine in North American society does not just affect working adults. Children are starved, too \u2013 starved of meaningful time spent with their parents.\"\n\nChildren want our presence, not presents. Toys and gifts will often be forgotten, but many of our best memories are of time spent together going to a baseball game, going on an outing, or taking a walk in the park. Review the past month and take an honest look at how much time you spend with your children. Add up the total number of hours and divide this by the number of days to give yourself the average time spent on a daily basis. Now that you've looked at the quantity, take a look at the quality of the time you've spent with your children. For example, if you are home with your children and they spend most of their time in another room watching TV, it could be said that this isn't a good use of your time together. A simple, yet effective way to resolve this would be to remove televisions from your children's bedrooms, create one TV room in your home, and make watching television a family activity.\n\nThere are many other ways to increase the quantity and quality of time you spend with your children. If it is financially viable, you could work part-time, or, if both parents need to work, find a job that alternates working hours with your partner, so that at least one parent is home when the children are around. You could designate one night per week as your Family Fun Night (but then you must stick to this and not let work or other commitments impinge, otherwise you will disappoint the kids immensely.) Or, have a pyjama day where everyone in the family spends the day in their nightclothes while reading, playing games, or just loafing around.\n\nExercising together by going for a walk, a run, playing soccer, or doing any kind of sport is another great way to have fun and spend time with your family \u2013 and it'll keep you healthy too. You can also involve your children more in the running of the household by getting them to help you with chores, such as preparing meals. Eat meals together whenever possible \u2013 studies have shown that the frequency of family dinners is the single most accurate predictor of children's academic achievement and psychological well-being.\n\n#### Separation and divorce\n\nWhen parents live apart there are unique time-management challenges because children need to spend time with each parent in his or her home. If parental communication is good, you can create a timetable together on the phone at the beginning of each month. Alternatively, sending a timetable with the child as they visit the other parent will give you both an opportunity to stay current and be reminded of upcoming events. Some families use online interactive calendars that are both parents can access to flag up any changes for the other parent to see.\n\n#### The value of unscheduled time\n\nThese days there is pressure to get our children involved in an ever-increasing amount of extra-curricular activities, such as sports, dance, and music. This pressure has spawned the \"hyper parent\" who over-schedules their children to the point of burnout. Instead, we need to rediscover the value of unscheduled time, and include time for kids to relax, daydream, and play outdoors.\n\n##### **saving time in an emergency \u2013 at home**\n\nBeing prepared in the event of an emergency will save you precious time when time is at a premium. It is easy to become disoriented when coping with stressful situations, so it is important to keep a list of phone numbers and addresses to hand in a pre-arranged place. Teach your children how to make an emergency phone call. Make sure you include your own address on your list in case a friend or relative is visiting and they have to make an emergency phone call on your behalf. Here are the basic phone numbers to include:\n\n\u2022 Police, fire, and ambulance (these are usually on the same number) \u2022 Mum's work number \u2022 Mum's cell phone \u2022 Dad's work number \u2022 Dad's cell phone \u2022 Grandparents' numbers \u2022 Other family numbers \u2022 Local hospitals \u2022 Neighbours' numbers \u2022 Close friends' numbers\n\n### _create a \"command center\"_\n\n### **EXERCISE TWENEY-FOUR**\n\nEvery family needs a \"command center\" and the calendar is a great tool. This exercise will show you how to set up and run your command center.\n\n_**1.** Designate an area of your kitchen to be your communication center. Here you will keep your master calendar (see step 2), a notebook and pen for recording telephone messages, a list of emergency phone numbers (see box, opposite), and a folder for each of your children to hold information about school, and any activities they do._\n\n_**2.** Print out or purchase a calendar (showing one month per page) and insert it into a ring binder. This is your master calendar for the year on which you can write any important dates up to a year in advance._\n\n_**3.** Print out or purchase a large wall calendar that shows both the current month and the forthcoming month, and hang it on the wall in your communication center, so that all family members can read it. Then, using your master calendar, transfer all the information for the current and coming months onto the wall calendar. (Enlist your children's help \u2013 it's a job that many school-aged children will enjoy.)_\n\n_**4.** Along with everyone's activities and commitments, include dates when either parent will be away, and any temporary changes that will affect regular arrangements._\n\n_**5.** Each evening check out the next day's activities and encourage your children to consult the calendar by marking on surprise treats, such as taking them to a movie or going out for Sunday lunch._\n\nA good way to avoid over-scheduling is to limit your children to one activity per category at any one time. For example, one sport per season usually entails two practices and a game per week \u2013 enough for most children. Set aside time each week for unplanned activities with your children. Give them three options, which include opportunities to talk and interact, and let them choose (but ban electronic toys). You could also ask teachers to institute a \"no-homework\" night once a week or one weekend per month. You are bound to get the support of your children on this one!\n\n##### **family time-management tips**\n\n\u2022 Getting everyone out of the door in the morning can be one of the most stressful times for a family. Prepare as much as possible the night before \u2013 pack lunches, and organize clothes and backpacks.\n\n\u2022 Teach your children to make their own breakfast during the school week. Place the non-perishable breakfast items along with the utensils on the table the night before. Make sure that the kids know how to put dishes in the dishwasher and to clear the table when they are finished.\n\n\u2022 Supply each child with a wicker or plastic basket to keep their hats, gloves, and scarves in. Store the baskets in the hall.\n\n\u2022 Teach your children how to sort their laundry by using colour-coded laundry baskets. Use a white basket for whites, navy for dark colours, and red for other colours. This way, laundry has to be sorted only once.\n\n\u2022 Establish and maintain (as far as possible) household routines for meals, homework, and chores. Do cleaning on one day and food shopping on another. Ask your partner to spend time with the children while you go food shopping, and vice versa \u2013 that way you each share the fun and the chores.\n\n\u2022 Lower your standards for the unimportant \u2013 no one really cares if the floor gets vacuumed every second week instead of every week. Do home baking if it is a family activity, otherwise store-bought items are just fine. Resist the urge to do everything yourself: call on the troops when you need to.\n\n\u2022 Set up an \"in\" and an \"out\" basket for each child in which they can dump their school-related papers to be filled out, signed, and returned to school.\n\n\u2022 Pay bills online using the internet. This is a tremendous time-saver and you can do it in the comfort of your home at a time that is convenient to you.\n\n\u2022 Have a short family meeting every weekend at a time convenient to everyone to preview the coming week.\n\n## working from home\n\nWorking from home, either as a self-employed person running your own business, or as an employee working for a company, is an ever-increasing phenomenon made possible by today's technology. Statistics show that home-based workers feel that flexibility is the greatest advantage of working from home and that the greatest challenge they face is how to manage their time well.\n\nCreate a schedule by dividing your work into two groups of activities: projects and routine tasks. Work on your projects during your peak hours when you are focused and less likely to have distractions. Do routine tasks \u2013 such as returning phone calls, filing, and sending out marketing materials \u2013 when you have less mental energy.\n\nPlanning by the week is helpful when you are working at home. An effective way to manage your time is to make a list of your main work-related activities and divide them into designated half-days. Here is an example of a timetable:\n\nMonday morning \u2013 Project work\n\nMonday afternoon \u2013 Marketing to new clients\n\nWednesday morning \u2013 Networking\n\nFriday morning \u2013 Finances\n\nFriday afternoon \u2013 Organize, file, de-clutter\n\nThis allows you to reserve Tuesdays and Thursdays for meetings away from your home office and gives your working week some variety.\n\nYou need to be self-disciplined to work effectively at home. Make a contract with yourself to be at your desk by a specific time each day, and set yourself certain office hours, which you stick to. It is also helpful to make time visible by keeping a large ticking clock in front of your desk in your line of sight. However, make sure that you take regular breaks \u2013 this is just as important at home as it is in the workplace. Step away from your desk at scheduled intervals for a short time. Overworking is not productive and can turn into workaholism (see pp.44\u20139), which can damage your health and well-being, your family relationships, and even your business.\n\nThere are no end of distractions to face when you are working at home. They range from the kids to the dog, the refrigerator and the TV. Then there's the ever-present reminder of all the housework that you need to do. Here are four of the most common distractions that people come up against when working from home, with tips on how to deal with them:\n\n**The kitchen** Having free access to the refrigerator is just too tempting a distraction for some of us. Try to stick to a regular coffee break, and lunch routine. Keep a pitcher of water and a glass at your desk to avoid getting up from your desk unnecessarily. Stock a small refrigerator in your office with health-boosting fruit and salad vegetables to snack on.\n\n**Television** If you have a favourite programme that is on in the middle of your working day, record it, and watch it in the evening or at the weekend. However, if you simply must see the programme, schedule it into your day and do some exercise or stretches while watching it. This will help you to avoid feeling guilty for not working and allow you to do something good for yourself at the same time.\n\n**Family and friends** It is very difficult to get work done at home if you have small children around. The best solution is to ask a friend or family member to look after them and, preferably, take them out while you are working. If this isn't an option, consider hiring part-time child-care during your peak work hours, when you schedule calls with clients, or if you are working on a time-sensitive project.\n\nSomehow family and friends make the assumption that if you are at home, you can't really be working. If people drop in often and disrupt your working day, be honest, and tell them in a friendly but firm way that you don't have time to socialize now. Instead, arrange to see them later for dinner or tea. Put a sign saying \"Do Not Disturb\" on your office door to discourage unnecessary interruptions. (Place it on your front door if family and friends continue to intrude on your work time.)\n\n_\"Those who make the worst use of their time are the first to complain of its shortness.\"_\n\nJEAN DE LA BRUY\u00c8RE (1645\u201396)\n\nCaller ID is a great tool for screening your phone calls. You can return personal calls during your lunch time or at the end of the day, much as you would if you worked away from home. Or you could ask other family members at home to screen personal calls while you are working. Ask them to tell callers that you are busy and cannot come to the phone, and to take messages so that you can call them back later.\n\n**Household chores** It is amazing how interesting doing the laundry can be when you are facing a big assignment and don't feel like working! One way to deal with household chores when you are working from home is to schedule in, say, half an hour of cleaning or washing at the time of day when your energy for work-related tasks is low. Or make a schedule whereby you clean one room per day. Doing some housework can be invigorating, and if you keep to a set schedule for the week, you will get everything done without being tempted to do too much at one go.\n\nSeparate work and home by shutting your office door at the end of your working day. If you don't have a separate office, shut down your computer and put away your work materials. Although many people imagine that they will have more time to spend with their families by working at home, the reality is that most home workers usually end up putting in longer hours than their counterparts who go to work in offices.\n\n## time management for creative people\n\nOur brains are divided into left and right hemispheres, with each hemisphere specializing in a thinking style. The capabilities of the left brain include analytical thinking, reading, writing, and mathematical ability. The right brain focuses on non-verbal skills \u2013 visual, perceptual, spatial, and intuitive abilities. The left brain processes pieces of information in a sequence while the right brain synthesizes information as whole.\n\nCreative people are considered to be \"right brainers\" and they often balk at the analytical approach taken by time-management systems. Here are some commonly perceived myths about creativity and time management, and some suggestions for how to resolve them.\n\n**\"Trying to control my time destroys my creativity and makes me feel anxious.\"**\n\nThis is a classic misconception because if you allocate time in which to do creative work, you will have peace of mind and feel free to focus 100 per cent on your creativity.\n\nIt makes sense to reserve chunks of time so that you can concentrate on what you love to do. Try one day on and one day off or even one week on and one week off. While away from your creative work, keep a spiral notebook to hand and jot down your ideas. For shorter blocks of time, remove your watch or clock, and set a timer to go off in another room when your time is up. That way you won't be distracted by clock-watching.\n\n**\"I wait until I feel inspired before I move into my 'creative space'.\"**\n\nThis is a luxury that few can afford \u2013 most successful artists, musicians, and writers \"show up\" and get down to work regardless of their moods. Try establishing a routine and use it to make your transition into work mode. For example, you might make a cup of tea, put on your favourite sweatshirt, and then gather up your work materials. Find a routine that works and use it every time you want to move into your creative space.\n\n**\"As a creative person I don't need to set goals \u2013 I prefer to just go with the flow and see what happens.\"**\n\nUnless you are independently wealthy, you do need to have goals in order to make a living. If you make your living doing creative work, perhaps as a writer, a graphic artist, a painter, or a musician, keep yourself motivated to complete goals by staying connected to your vision. Get back in touch with your passion and you'll automatically get into the flow.\n\n**\"I do my best creative work when I am up against a set deadline.\"**\n\nIn fact, research shows the opposite \u2013 that the more time pressure creative people experience, the less creative they feel. This is intensified when people are distracted by trivial tasks, sudden changes in plans, or if they do not feel that the work they are doing is important. Creativity can flourish under time pressure, but only if there is a shared sense of purpose and if all those involved can focus their energy without interruptions. If you are part of a team, ask for clarification when you are unsure as to why you are attempting to accomplish something. A clear purpose will help you to r-focus. Protect your creative time from distractions and interruptions (see pp.142\u20133) and, if possible, find a quiet place away from the workplace where you can complete your project in peace.\n\n**\"Part of being creative is working alone, without having to be accountable for my time to anyone.\"**\n\nWe all need to be accountable to someone to grow and develop. Try making a commitment to a colleague to do a certain task by a certain time. Or agree with peers, for example in your art class, that you will accomplish a certain piece of work and bring it along to the next meeting.\n\n##### **make a weekly \"To Do\" flow chart**\n\nTraditional \"To Do\" lists not for you? Then, why not use a flow chart to make a bright, colourful list instead? You will need a large sheet of paper and a variety of coloured pens.\n\nTo create a weekly \"To Do\" flow chart, start in the middle of the page and draw seven large arrows, each in a different colour, facing outwards in a circle. Label each one with the name of a day of the week. Think about what you want to do this week and on which day. Then, write out each day's tasks next to its arrow in the colour you chose to represent that day. Or if you prefer, draw little pictures of the tasks instead of writing them out. Repeat this process until you've filled the week to your satisfaction. Update your flow chart and add new items or ideas as the week progresses.\n\n### _take advantage of your creative time_\n\n### **EXERCISE TWENTY-FIVE**\n\nIf you struggle with finding time to pursue your creative activities, use this exercise to help you take advantage of each precious moment by creating your own portable, artistic space.\n\n_**1.** Some creative pursuits, such as writing, painting, and drawing, are highly portable. The key to taking full advantage of creative time is to have your kit prepared and ready to use as soon as inspiration strikes._\n\n_**2.** If you enjoy writing, you might like to include the following in your kit: a notebook; a list of questions to use as writing prompts; a small collection of your favourite quotes written on cue cards; your favourite writing tools (a ball point pen, a pencil, a fountain pen, a felt-tip, and so on). Purchase an inexpensive three-ring binder and divide it into sections using clear plastic dividers with pockets to store your quotes, your notebook, and so on. Another portable option might be to stock your writing supplies in a small case or tote bag that you can easily slip into a briefcase if you are heading to the office, or that can be carried on its own._\n\n_**3.** If you like to paint or draw you must plan your portable art kit carefully so that you are not carrying your entire studio with you. Consider purchasing a molded plastic briefcase type of tote bag to hold your papers, pads, pens, pencils, pastels, paints, and brushes. Or you could get yourself an organizer on wheels (the type used for carrying a computer) which would be a great choice to hold your supplies._\n\n## managing time in retirement\n\nWhat do you think of when you hear the word \"retirement\"? Free time? World travel? A rocking chair? Do you view retirement as a phase in life when you can do whatever you wish with your time? Whatever your idea, one thing is certain \u2013 retirement involves a great deal of change: in lifestyle, in relationships, and in self-perception. There is no longer a \"typical\" retirement age as many people nowadays work past the age of 65, live longer, and lead more active lives. You can help yourself prepare for retirement by answering these questions: What do you look forward to when you retire? What are your fears about retirement?\n\nWhen you retire you can expect some emotional highs and lows. It is a time to focus on yourself for a change, to spend time with family and friends, and to do anything you please. Retirement can also be a time of loss \u2013 losing your sense of structure and routine; loneliness because you miss the camaraderie of co-workers; and the loss of your sense of self outside of the world of work, which had given you a definitive status.\n\n#### Step-by-step retirement\n\n**Easy does it** Ease your way into retirement by gradually reducing your workload through part-time work during the last two or three years before you retire. This will help you to get used to having more unscheduled time. Some people return to work for their former employers as consultants on an ad hoc basis.\n\n**Pace yourself** Consider arranging to take a sabbatical \u2013 whether just for one month or for a whole year \u2013 a few years before you retire. You will need to plan this in advance so that you can save enough money to support yourself during your time off. Use the sabbatical to try out new activities and interests, to travel, and to make new friends. Monitoring how you manage your time will give you an idea of your strengths and weaknesses.\n\n**Change your attitude toward work** Learning how to relax and slow down is difficult for some people. If you define yourself solely through your work or if you are a workaholic, retirement will seem a scary prospect. Start by reducing the amount of hours you work and try to develop outside interests to make the transition from work to retirement easier.\n\n#### Balance your time between five types of activities\n\nIf you decide to retire completely from work, you will be faced with many choices about how to spend your time. Once you have done your personal chores and errands, you will be left with \"free time.\" Strive for a life that is multifaceted by dividing your time between five activity groups: social, intellectual, spiritual, physical, and community. This will provide you with diversity and keep you from doing too much or not enough.\n\n**Social activities** Choose activities that keep you connected to friends, family, and former colleagues. Relationships can end or change when you retire so look for opportunities to develop new friendships as well as maintain old ones. You could join a book club, a church group, or choir, or you could participate in a team sport.\n\n**Intellectual activities** Mental stimulation and growth will help you stay active and motivated. Lifelong learning, reading, taking or teaching courses, and starting a discussion group are helpful ways to stay alert mentally. Reading newspapers and magazines, learning a new language (especially if you intend to travel), and writing a journal are great mental exercises.\n\n_\"It's not enough to be busy \u2013 so are the ants.\"_\n\nHENRY DAVID THOREAU (1817\u201362)\n\n**Spiritual activities** Find ways to help yourself develop meaning and purpose. Study world religions, attend different spiritual centers, learn to meditate, or go on a weekend retreat to explore your values and beliefs.\n\n**Physical activities** Exercise is important to your well-being, and now you can no longer use lack of time as an excuse. Include regular exercise, such as walking, running, hiking, skiing, cycling, and\/or swimming in your regular weekly activities.\n\n**Community activities** Find opportunities to share your knowledge and life experience through volunteer work. It is important to feel that we can make a difference outside our families. Read to children at your local elementary or pre-school; teach youngsters how to write; drive the elderly to doctors' appointments \u2013 the possibilities are endless.\n\n### _map your retirement_\n\n### **EXERCISE TWENTY-SIX**\n\nMost of us are so busy working that we've not given much thought to what will happen when we retire. Mapping out how you want to experience retirement will help you prepare for and make the transition smoothly.\n\n_**1.** Get yourself a journal in which to record your ideas for a perfect retirement. Use it to note down your thoughts, to explore your feelings, and to ask yourself questions about retirement._\n\n_**2.** Title a page: \"Things I've always wanted to do but never had time to do\" and list as many activities as you can think of. (Or, if you prefer, draw pictures of the activities instead of writing them.) Next to each item write why you would like to do it and, if possible, set a target date by which you would like to accomplish it._\n\n_**3.** Label a second page in your journal: \"I am happiest when ...\" and list all the things that you really enjoy doing. Write down what it is about these things that make you happy._\n\n_**4.** Head a third page: \"Things I never want to do again\" and make a list of any activities that you want to banish from your vision of retirement._\n\n_**5.** Using all the notes and lists you made above as reference, write a definitive description of your retirement, covering where you would like to live and how you would like to spend your time._\n\n_**6.** Review and update your description regularly as you get closer to your retirement._\n\n##### **retirement and self-actualization**\n\nAccording to the psychologist Abraham Maslow (1908\u201370), self-actualization is the process of reaching one's full potential \u2013 and what better time to pursue personal growth than in retirement. People who are self-actualized have certain characteristics, the main ones of which are listed below.\n\nWhich ones apply to you?\n\n\u2022 I am spontaneous in thought and action.\n\n\u2022 I accept myself and others.\n\n\u2022 I feel connected to, and a kinship with, others.\n\n\u2022 I value my individuality.\n\n\u2022 I am self-sufficient.\n\n\u2022 I enjoy spending some time alone.\n\n\u2022 I have a mission and a purpose in life.\n\n\u2022 I do not fear the unknown.\n\n\u2022 I am able to understand the lesson in most life situations.\n\n\u2022 I feel alive and enjoy life.\n\n\u2022 I like to do things that help and benefit others.\n\n\u2022 I like to learn new things.\n\nThe more descriptions that apply to you, the more self-actualized you are. Review the activity groups on pp.149\u201350 and above for how to become more well-rounded.\n\n### _visualize a positive retirement_\n\n### **EXERCISE TWENTY-SEVEN**\n\nVisualization is a form of meditation that will help make a positive retirement a self-fulfilling prophecy. When you visualize, you are creating a mental movie that you can tap into anytime, anywhere.\n\n_**1.** Take the journal you used in Exercise twenty-six (see p.151) and find yourself a comfortable, quiet place in which to sit \u2013 perhaps in a peaceful room in your home, or outside in a favourite spot in your garden, or in the local park._\n\n_**2.** Read through your journal and concentrate particularly on the summary you wrote describing your perfect retirement. Read through this part several times._\n\n_**3.** Close your eyes, breathe in deeply, and exhale slowly. Let any tension drain away from your body and try to relax totally. Breathe rhythmically in this way for a few minutes._\n\n_**4.** Think about your journal description of your perfect retirement and bring it to life by picturing yourself starring in your very own \"movie.\" Explore different scenarios. Imagine where you live, your family life, your friends, your physical and intellectual activities, and how you live your life purpose. Engage all of your senses: See the glorious colours of the flowers; hear your grandchildren's voices; feel the soft fur of your pet cat; smell the aroma of the bread you are baking; taste the delicious food at the restaurant; and so on._\n\n_**5.** Repeat the visualization as often as you wish. Conjuring up a mental picture of your ideal retirement will help you make it real._\n\n## conclusion\n\nNow that you've read through the book I hope that you have a sense of how your life will be different once you've taken back control of your time. What would it feel like to know that you have time to do the things that are important to you? That you are working toward your goals? That you have time to spend with family and friends? Imagine a life with less stress and fear, and more personal satisfaction and joy. That's what getting a handle on managing your time can do for you.\n\nI would like to think that you will use this book as a reference to manage your time, your self, and your life. Choose a time-management method that works, is simple to use, and fits easily into your life. Studies show that successful people go the extra mile and keep track by writing things down, especially their goals. Use the exercises in this book to do the same.\n\nYou now have at your disposal all the tools and techniques you need, so why not go one step further and make a commitment to becoming the best time manager possible? Try to make time management a natural part of your everyday life to the point where you would no sooner go through your day without managing your time than you would without brushing your teeth. Invite a friend or colleague to be your time mentor to keep you motivated and accountable.\n\nOf course, bear in mind that from time to time we all feel overwhelmed and discouraged. When this happens your life will get out of balance and you will find that demands on your time are impossible to meet. Sometimes it will simply be a case of having lapsed into managing your time poorly. For example, perhaps you managed your time well as a student but now you are finding it difficult as a young parent or as a business person \u2013 or both! When you find that you are struggling, reread the sections of this book that apply to your particular stage of life. Then, take a deep breath, make some tough decisions about what you can and cannot do, and start again. Don't be discouraged \u2013 even very successful people experience difficulties at some point in their lives. What makes the difference is having the resilience to work through your difficulties, and finding the motivation to start again. After all, if you have done it once, you can do it again.\n\nBe fiercely protective of your time and defend your right to spend it in the way that is most beneficial to you and your family. Remember that your personal legacy will be determined by how you spend your time today \u2013 and tomorrow, and the next day. Regardless of your age, always live your life with a sense of urgency, in the knowledge that time is passing. Resolve to be aware and mindful of the present, each and every day. Ultimately, it's your time and your life. You get to choose how you spend it. Start managing your time today so that you can, quite literally, have the time of your life.\n\n## further reading\n\n**Allen, David** _Getting Things Done: The Art of Stress-Free Productivity,_ Penguin Books (London and New York), 2003\n\n**Barnes, Emilie** _More Hours in My Day,_ Harvest House (Eugene, Oregon), 2002\n\n**Bennett, Arnold** _How to Live on 24 Hours a Day,_ Shambling Gate Press (Hyattsville, Maryland), 2001\n\n**Covey, Stephen R., A. Roger Merrill and Rebecca R. Merrill** _First Things First,_ Simon & Schuster (London and New York), 1994\n\n**Eisenberg, Ronni and Kate Kelly** _The Overwhelmed Person's Guide to Time Management,_ Plume (New York), 1997\n\n**Emmett, Rita** _The Procrastinator's Handbook: Mastering the Art of Doing It Now,_ Walker & Company (New York), 2000\n\n**Fiore, Nei** l _The Now Habit,_ Tarcher (New York), 2002\n\n**Hindel, Tim and Robert Heller** _Essential Managers: Managing Meetings,_ DK Publishing Inc. (London and New York), 1999\n\n**Kennedy, Dan No B.S.** _Time Management for Entrepreneurs: The Ultimate No Holds Barred, Kick Butt, Take No Prisoners Guide to Time Productivity and Sanity_ , Entrepreneur Press (Dallas, Texas), 2004\n\n**MacKenzie, R. Alec** _The Time Trap: The Classic Book on Time Management,_ American Management Association (New York), 1997\n\n**Merrill, A.** _Life Matters: Creating a Dynamic Balance of Work, Family, Time and Money,_ McGraw-Hill Canada (Whitby, Ontario), 2004\n\n**Morgenstern, Julie** _Time Management from the Inside Out: The Foolproof System of Taking Control of Your Schedule and Your Life,_ Henry Holt & Company Inc. (New York), 2000\n\n**Nakone, Lanna** _Organizing for Your Brain Type: Finding Your Own Solution to Managing Time, Paper, and Stuff,_ St. Martin's Press (New York), 2005\n\n**Reno, Dawn E.** _The Unofficial Guide to Managing Time,_ John Wiley & Sons (New York), 2000\n\n**Roesch, Roberta** _Time Management for Busy People,_ McGraw-Hill (New York), 1998\n\n**Silber, Lee** _Time Management for the Creative Person: Right-brain Strategies for Stopping Procrastination, Getting Control of the Clock and Calendar, and Freeing Up Your Time, and Your Life,_ Three Rivers Press (New York), 1998\n\n**Staubus, Martin** _Time Management: Increase Your Personal Productivity and Effectiveness,_ Harvard Business School Press (Boston), 2005\n\n**Tracey, Brian** _Time Power: A Proven System for Getting More Done in Less Time Than You Ever Thought Possible,_ AMACOM (New York), 2004\n\n**Winston, Stephanie** _Organized Executive: A Program for Productivity: New Ways to Manage Time, Paper, People, and the Electronic Office,_ Warner Books Inc. (New York), 2001\n\n## studies and papers\n\nThe studies and papers referred to in this book are as follows: **p.84** Data Warehousing Institute study, 2002; **88** University of California (Berkeley) study, 2002; **105** Waddington, P. _Dying for Information_ , Reuters Business Information, 1996; **107a** Mosvick, R. & R. Nelson, _We've got to start meeting like this!_ Scott Foresman, 1987; **107b** Romano Jr., Nicholas C. and Jay F. Nunamaker Jr., \"Meeting Analysis: Findings from Research and Practice,\" 34th Hawaii International Conference on System Sciences, 2001; **130a** McCown, W.G. and R. Roberts, \"A study of academic and work-related dysfunctioning relevant to the college version of an indirect measure of impulsive behaviour,\" Integra Technical Paper 94\u201398, Radnor, 1984; **130b** Solomon, L.J. and E.D. Rothblum, \"Academic Procrastination: Frequency and Cognitive-behavioural Correlates,\" _Journal of Counseling Psychology_ , 31: 504\u2013 510, 1984; **134** Moses, Barbara, _Career Intelligence: The New Rules for Work and Life Success_ , Stoddart, 1997; **136** Doherty, Dr. William J. and Barbara Z. Carlson, _Putting Family First: Successful Strategies for Reclaiming Family Life in a Hurry-up World_ , Henry Holt, 2002.\n\n## acknowledgments\n\nI would like to thank Duncan Baird Publishers and Watkins Publishing Limited for giving me the opportunity to write this book. I would also like to give a big \"thank you\" to Naomi Waters \u2013 a capable, thorough, and thoughtful managing editor.\nThis edition first published in the UK and USA 2014 by\n\nWatkins Publishing Limited\n\nPO Box 883, Oxford OX1 9PL, UK\n\nA member of Osprey Group\n\nFor enquiries in the USA and Canada:\n\nOsprey Publishing\n\nPO Box 3985\n\nNew York, NY 10185-3985\n\nTel: (001) 212 753 4402\n\nEmail: info@ospreypublishing.com\n\nDesign and typography copyright \u00a9 Watkins Publishing Limited 2014\n\nText copyright \u00a9 Lucy MacDonald 2006, 2014\n\nFirst published as _Learn to Make Time_ in 2006 by Duncan Baird Publishers.\n\nLucy MacDonald has asserted her right under the Copyright, Designs and Patents Act 1988 to be identified as the author of this work.\n\nAll rights reserved.\n\nNo part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, without prior permission in writing from the Publishers.\n\n1 3 5 7 9 10 8 6 4 2\n\nDesigned and typeset by Gail Jones\n\nPrinted and bound by CPI Group (UK) Ltd, Croydon CRO 4YY\n\nA CIP record for this book is available from the British Library.\n\nISBN: 978-1-78028-793-5\n\nWatkins Publishing is supporting the Woodland Trust, the UK's leading woodland conservation charity, by funding tree-planting initiatives and woodland maintenance.\n\nwww.watkinspublishing.co.uk\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nTo my siblings,   \nEli, Rachel, and Leah\n\n# Contents\n\nIntroduction\n\n[_ONE_  \nThe Quarterback in the Pocket](a9781847674777_chapter_01.html)\n\n[_TWO_  \nThe Predictions of Dopamine](a9781847674777_chapter_02.html)\n\n[_THREE_  \nFooled by a Feeling](a9781847674777_chapter_03.html)\n\n[_FOUR_  \nThe Uses of Reason](a9781847674777_chapter_04.html)\n\n[_FIVE_  \nChoking on Thought](a9781847674777_chapter_05.html)\n\n[_SIX_  \nThe Moral Mind](a9781847674777_chapter_06.html)\n\n[_SEVEN_  \nThe Brain Is an Argument](a9781847674777_chapter_07.html)\n\n[_EIGHT_  \nThe Poker Hand](a9781847674777_chapter_08.html)\n\n[_NINE_  \nCoda](a9781847674777_chapter_09.html)\n\nAcknowledgments\n\nBibliography\n\nNotes\n\nIndex\nWho knows what I want to do? Who knows what anyone wants to do? How can you be sure about something like that? Isn't it all a question of brain chemistry, signals going back and forth, electrical energy in the cortex? How do you know whether something is really what you want to do or just some kind of nerve impulse in the brain? Some minor little activity takes place somewhere in this unimportant place in one of the brain hemispheres and suddenly I want to go to Montana or I don't want to go to Montana.\n\n\u2013 Don DeLillo, _White Noise_\n\n# Introduction\n\nI WAS FLYING A Boeing 737 into Tokyo Narita International Airport when the left engine caught on fire. We were at 7,000 feet, with the runway dead ahead and the skyscrapers shimmering in the distance. Within seconds, bells and horns were blaring inside the cockpit, warning me of multiple system failures. Red lights flashed all over the place. I tried to suppress my panic by focusing on the automated engine-fire checklist, which told me to cut off fuel and power to the affected areas. Then the plane began a steep bank. The evening sky turned sideways. I struggled to steer the plane straight.\n\nBut I couldn't. The plane was impossible to fly. It swayed one way, I tried to pull her back to center, and then it swayed the other way. It was like wrestling with the atmosphere. Suddenly, I felt the shudder of a stall: the air was moving too slowly over the wings. The metal frame started to shriek and groan, the awful sound of steel giving way to physics. If I didn't find a way to increase my speed immediately, the plane would quickly surrender to the downward tug of gravity and I'd plunge into the city below.\n\nI didn't know what to do. If I increased the throttle, I might be able to gain altitude and speed, and then I could circle around the runway and try to stabilize the plane. But could my only remaining engine handle the climb by itself? Or would it fail under the strain?\n\nThe other option was to steepen my descent in a desperate attempt to pick up speed; I'd fake a nosedive in order to avoid a real one. The downward momentum might let me avert the stall and steer the plane. Of course, I might instead be accelerating toward disaster. If I couldn't regain control, then the plane would fall into what pilots call a graveyard spiral. The G force would become so intense that the plane would disintegrate before it even hit the ground.\n\nIt was a hellish moment of indecision. Nervous sweat stung my eyes. My hands quivered with fear. I felt the blood pulse in my temples. I tried to think, but there wasn't time. The stall was getting worse. If I didn't act at that moment, the plane would fall out of the sky.\n\nThat's when I made up my mind: I would save the plane by taking her down. I tilted the yoke forward and prayed for speed. Immediately, I started to go faster. The problem was that I was heading straight into a suburb of Tokyo. But as my altimeter wound toward zero, the extra velocity kicked in and allowed me to steer. For the first time since the engine had caught fire, I could keep the plane on a steady course. I was still dropping like a stone, but at least I was flying in a straight line. I waited until the plane had sunk below 2,000 feet and then pulled back on the yoke and advanced the throttle. The ride was sickeningly rough, but my descent remained on target. I lowered the landing gear and focused on keeping the plane under control, with the lights of the runway in the center of the windshield. My copilot called out the altitude: 'One hundred feet! Fifty! Twenty!' Right before we hit the ground, I made a final plea for the center and waited for the comforting speed bump of solid earth. It was an ugly landing \u2013 I had to slam on the brakes and swerve at high speed \u2013 but we made it down safely.\n\nIt was only when the plane was parked at the airport gate that I noticed the pixels. I had been staring at a wraparound television screen, not looking through a cockpit window. The landscape below was just a quilt of satellite imagery. Although my hands were still shaking, nothing had really been at stake. There were no passengers sitting in the cabin. The Boeing 737 was just an artificial reality generated by a $16 million CAE Tropos 5000 flight simulator in a cavernous industrial hangar outside Montreal. My flight instructor had pressed a button and triggered the engine fire. (He'd also made my life more difficult by adding some fierce crosswinds.) But the flight had _felt_ real. By the time the ride was over, my veins were full of adrenaline. A part of my brain was still convinced that I'd almost crashed into the city of Tokyo.\n\nThe virtue of a flight simulator is that you can investigate your own decisions. Had I been right to continue the descent? Or should I have tried to regain altitude? Would that have given me a smoother, safer landing? I wanted to know, so I asked the instructor if I could redo the simulated scenario and once again try to land without an engine. He flicked a few switches, and, before my heartbeat could recover, the 737 was reincarnated on the runway. I heard the voice of air-traffic control crackle on the radio, clearing me for takeoff. I advanced the throttle and sped down the tarmac. Everything went faster and faster until the aerodynamics took over and I was in the quiet of the evening's blue sky.\n\nWe climbed to 10,000 feet. I was just beginning to enjoy the tranquil view of Tokyo Bay when air-traffic control told me to prepare for landing. The scenario repeated itself like a familiar horror movie. I saw the same skyscrapers in the distance and flew through the same low clouds. I traced the same route across the same suburbs. I descended to 9,000 feet, then 8,000, then 7,000. And then it happened. The left engine erupted in flames. Once again, I struggled to keep the plane steady. Once again, there was the shudder warning me of a stall. This time, though, I aimed for the heavens. I increased the throttle, tilted the plane up, and carefully watched the readout from my one remaining engine. It soon became clear that I couldn't climb. There just wasn't enough engine power. The shudder spread across the skeleton of the plane. I heard the sickening sound of wings losing flight, a low resonant drone that filled the cockpit. The plane plunged left. A female voice calmly narrated the disaster, telling me what I already knew: I was falling out of the sky. The last thing I saw was a blink of city lights, just above the horizon. The screen froze when I hit the ground.\n\nIn the end, the difference between my landing the plane in one piece and my dying in a fiery crash came down to a single decision made in the panicked moments after the engine fire. It had all happened so fast, and I couldn't help but think about the lives that would have been at stake had this been a real flight. One decision led to a safe landing; the other to a fatal stall.\n\nThis book is about how we make decisions. It is about what happened inside my brain after the engine fire. It is about how the human mind \u2013 the most complicated object in the known universe \u2013 chooses what to do. It's about airplane pilots, NFL quarterbacks, television directors, poker players, professional investors, and serial killers, and the decisions they make every day. From the perspective of the brain, there's a thin line between a good decision and a bad decision, between trying to descend and trying to gain altitude. This book is about that line.\n\nAs long as people have made decisions, they've thought about how they make decisions. For centuries, they constructed elaborate theories on decision-making by observing human behavior from the outside. Since the mind was inaccessible \u2013 the brain was just a black box \u2013 these thinkers were forced to rely on untestable assumptions about what was actually happening inside the head.\n\nEver since the ancient Greeks, these assumptions have revolved around a single theme: _humans are rational_. When we make decisions, we are supposed to consciously analyze the alternatives and carefully weigh the pros and cons. In other words, we are deliberate and logical creatures. This simple idea underlies the philosophy of Plato and Descartes; it forms the foundation of modern economics; it drove decades of research in cognitive science. Over time, our rationality came to define us. It was, simply put, what made us human.\n\nThere's only one problem with this assumption of human rationality: it's wrong. It's not how the brain works. Look, for example, at my decisions in the cockpit. They were made in the heat of the moment, a visceral reaction to difficult events. I didn't carefully reflect on the best course of action or contemplate the aerodynamics of an engine fire. I couldn't reason my way to safety.\n\nSo how did I make a decision? What factors influenced my choices after the engine fire? For the first time in human history, these questions can be answered. We can look inside the brain and see how humans think: the black box has been broken open. It turns out that we weren't designed to be rational creatures. Instead, the mind is composed of a messy network of different areas, many of which are involved with the production of emotion. Whenever someone makes a decision, the brain is awash in feeling, driven by its inexplicable passions. Even when a person tries to be reasonable and restrained, these emotional impulses secretly influence judgment. When I was in the cockpit desperately trying to figure out how to save my life \u2013 and the lives of thousands of Japanese suburbanites \u2013 these emotions drove the patterns of mental activity that made me crash and helped me land.\n\nBut this doesn't mean that our brains come preprogrammed for good decision-making. Despite the claims of many self-help books, intuition isn't a miraculous cure-all. Sometimes feelings can lead us astray and cause us to make all sorts of predictable mistakes. The human brain has a big cortex for a reason.\n\nThe simple truth of the matter is that making good decisions requires us to use both sides of the mind. For too long, we've treated human nature as an either\/or situation. We are either rational or irrational. We either rely on statistics or trust our gut instincts. There's Apollonian logic versus Dionysian feeling; the id against the ego; the reptilian brain fighting the frontal lobes.\n\nNot only are these dichotomies false, they're destructive. There is no universal solution to the problem of decision-making. The real world is just too complex. As a result, natural selection endowed us with a brain that is enthusiastically pluralist. Sometimes we need to reason through our options and carefully analyze the possibilities. And sometimes we need to listen to our emotions. The secret is knowing when to use these different styles of thought. We always need to be thinking about how we think.\n\nThis is what pilots learn in the flight simulator. The benefit of experiencing various cockpit scenarios \u2013 like an engine fire over Tokyo or a blizzard in Topeka \u2013 is that pilots develop better senses of which modes of thought to lean on in particular situations. 'We never want pilots to act without thinking,' says Jeff Roberts, the president of civil training at CAE, the largest manufacturer of flight simulators. 'Pilots aren't robots, and that's a good thing. But we do want them to make decisions that rely on the wealth of judgment they've built up over time. You always need to think, but sometimes your feelings can help you think. A good pilot knows how to use his head.'\n\nAt first, it might seem a little strange to look at decisions from the vantage point of the mind's inner workings. We aren't used to understanding choices in terms of competing brain regions or the firing rates of neurons. And yet, this new way of knowing ourselves \u2013 trying to understand human behavior from the inside \u2013 reveals many surprising things. In this book, you will learn how those three pounds of flesh inside the skull determine all of your decisions, from the most mundane choices in the supermarket to the weightiest of moral dilemmas. The mind inspires many myths \u2013 such as the fiction of pure rationality \u2013 but it's really just a powerful biological machine, complete with limitations and imperfections. Knowing how the machine works is useful knowledge, since it shows us how to get the most out of the machine.\n\nBut the brain doesn't exist in a vacuum; all decisions are made in the context of the real world. Herbert Simon, the Nobel Prizewinning psychologist, famously compared the human mind to a pair of scissors. One blade was the brain, he said, while the other blade was the specific environment in which the brain was operating.\n\nIf you want to understand the function of scissors, then you have to look at both blades simultaneously. To that end, we are going to venture out of the lab and into the real world so that we can see the scissors at work. I'll show you how the fluctuations of a few dopamine neurons saved a battleship during the Gulf War, and how the fevered activity of a single brain region led to the subprime housing bubble. We'll learn how firefighters handle dangerous blazes, and we'll visit the card tables of the World Series of Poker. We'll meet scientists who are using brain-imaging technology in order to understand how people make investment decisions and choose political candidates. I'll show you how some people are taking advantage of this new knowledge to make better television shows, win more football games, improve medical care and enhance military intelligence. The goal of this book is to answer two questions that are of interest to just about everybody, from corporate CEOs to academic philosophers, from economists to airline pilots: How does the human mind make decisions? And how can we make those decisions better?\nONE\n\n# The Quarterback in the Pocket\n\nTHERE IS A MINUTE and 21 seconds left on the clock in the 2002 Super Bowl, and the score is tied. The New England Patriots have the ball on their own 17-yard line. They are playing against the heavily favored St Louis Rams. They have no time-outs left. Everyone assumes that the Patriots will let the clock run down and take the game into overtime. That, after all, is the prudent thing to do. 'You don't want to have a turnover,' John Madden, the television announcer, says. 'They should just let time expire.'\n\nThe game was never supposed to be this close. In the pre-match build-up the Rams were favored over the Patriots by 14 points, which made it the most lopsided Super Bowl ever played. The potent Rams offense \u2013 nicknamed 'The Greatest Show on Turf' \u2013 led the league in 18 different statistical categories, and outscored their opponents 503 to 273 during the regular season. Quarterback Kurt Warner was named the NFL's Most Valuable Player, while running back Marshall Faulk won the NFL Offensive Player of the Year Award. The Patriots, meanwhile, were hamstrung by injuries, having lost both Drew Bledsoe, their star quarterback, and Terry Glenn, their leading wide receiver. Everyone was expecting a rout.\n\nBut, with just a minute remaining, Tom Brady \u2013 the second-string quarterback for the Patriots \u2013 has a chance to win the game. Over on the Patriots sidelines, he huddles in conversation with Bill Belichick, the Patriots head coach, and Charlie Weis, the offensive coordinator. 'It was a 10-second conversation,' Weis remembers. 'What we said is we would start the drive, and, if anything bad happened, we'd just run out the clock.' The coaches were confident that their young quarterback wouldn't make a mistake.\n\nBrady jogs back to his Patriot teammates on the field. You can see him smiling through his facemask, but it's not a nervous smile. It's a confident smile. There are 70,000 spectators inside the Superdome, and most of them are rooting for the Rams, but Brady doesn't seem to notice. After a short huddle, the Patriots clap their hands in unison and saunter toward the line of scrimmage.\n\nTom Brady wasn't supposed to be here. He was the 199th pick in the 2000 draft, which is when NFL teams select the most promising college prospects. Although Brady had broken passing records at the University of Michigan, most team scouts thought he was too fragile to play with the big boys. The report on Brady by _Pro Football Weekly_ summarized the conventional wisdom: 'Poor build. Very skinny and narrow. Ended the '99 season weighing 195 pounds, and still looks like a rail at 211. Lacks great physical stature and strength. Can get pushed down more easily than you'd like.' The report only devoted a few words to Brady's positive attributes: 'Decisionmaking.'\n\nBill Belichick, the Patriots head coach, was one of the few men who grasped Brady's potential. 'Our vision wasn't that Tom was our franchise quarterback,' Belichick said, 'but that Tom had been in situations \u2013 both in playing-time and game-management situations, tight games against good competition \u2013 and he'd handled all of them pretty well.' Brady, in other words, had poise. He didn't choke under pressure. When the game was on the line, he found the open man.\n\nNow Brady is in the spotlight, standing all by himself in the shotgun formation. His decision-making skills are about to be put to the test. He yells an instruction to his tight end, then turns and yells at his wide receivers. The ball is snapped. Brady drops back, looks upfield, and understands instantly that the Rams have fallen into a tight zone coverage. They know the Patriots are going to pass; the cornerbacks are looking for an interception. Brady's primary target is covered, so he looks to his next target. He's also covered. Brady avoids the outstretched arm of a Ram defensive lineman, steps forward, and makes a short pass to his third target, the running back J. R. Redmond. It's a gain of five yards.\n\nThe next two plays unfold in the same way. Brady looks up, reads the Ram defense, and calls out a series of coded commands: 'White 20! Ninety-six is the Mike! Omaha go!' These instructions tell the offensive linemen which linebackers to block. They also serve as guides for the wide receivers, as Brady alters the details of their pass routes depending on the formation of the defense. After the play begins, Brady settles into the pocket, checks off his targets and wisely settles for the safest option, which is a short pass on the left side of the field. He doesn't risk an interception; he's taking what the defense is giving him. It's a first down, but the Patriots are running out of time.\n\nThe football is now on the New England 41-yard line. Twenty-nine seconds remain in the game. Brady knows that he's got two, maybe three plays left. He has to move the ball another 30 yards just to get into field goal range. The commentators sound like they're preparing for overtime, but the Patriots still think they can score. Brady's eyes pan across the defense. He sees the linebackers edging a little closer to the line of scrimmage. Brady yells for the ball, drops back and notices that only three defensive linemen are rushing him. The fourth is trying to cut off the short pass. Brady looks to his right. The receiver is covered. He looks to his left. Nobody's open. He looks at the center of the field. Troy Brown, a Patriots wide receiver, is trying to find a plane of unoccupied space, the gap between the linebackers and the cornerbacks. Brady watches him clear the defenders and fires a bullet 14 yards downfield. Brown catches the ball in stride and runs for another nine yards before being pushed out of bounds. The ball is now 36 yards from the end zone, which is just within field goal range. The Rams fans have gone silent.\n\nWith 12 seconds remaining, the Patriots special teams unit is brought onto the field. Adam Vinatieri steps into the 48-yard kick. The ball sails straight between the pylons. The clock says triple zero. The Patriots have just won the Super Bowl. It's the greatest upset in NFL history.\n\n##  _Deciding Without Thinking_\n\nThe quick decisions made by a quarterback on a football field provide a window into the inner workings of our brain. In the space of a few frenetic seconds, before a linebacker crushes him into the ground, an NFL quarterback has to make a series of hard choices. Several enormous men are running straight at him, desperate to break his bones, but he can't flinch or wince. His eyes must stay focused downfield, looking for some meaningful sign amid all the action. Throwing the ball is the easy part.\n\nThese passing decisions happen so fast they don't even seem like decisions. We are used to seeing football on television, captured by the cameras far above the grassy stage. From this distant perspective, the players appear to be moving in some sort of violent ballet; the sport looks exquisitely choreographed. You can see the receivers run their passing routes and watch the rushers break through the line of scrimmage. It's easy to detect the weak spots of the defense and find the target with man-on-man coverage. You can tell which linebackers bought the play-action fake and see the cornerback racing in on the blitz. When the game is watched from this omniscient angle \u2013 coaches call it 'the eye in the sky' \u2013 it appears as if the quarterback is simply following orders, as if he knows where he is going to throw the ball before the play begins.\n\nBut this view of the game is not the reality of the game. After the ball is snapped, the neat Xs and Os that fill the spiral-bound playbook degenerate into a street brawl. There's a symphony of grunts and groans and the meaty echo of fat men hitting the hard ground. Receivers get pushed off their routes and passing angles get cut off. The offensive line is an unpredictable wrestling match. Before the quarterback can make an effective decision, he needs to assimilate all of this new information, so that he's aware of the approximate location of every other player on the field.\n\nThe savage chaos of the game, the way every play is a mixture of careful planning and fierce improvisation, is what makes the job of an NFL quarterback so difficult. Even as he's immersed in the mad violence of the game \u2013 the defensive line is clawing at his body \u2013 the quarterback has to stand still and concentrate. He needs to look past the mayhem and make sense of all the moving bodies. Where is his receiver going? Will the defender break towards the ball? Is the linebacker going to drop back into pass coverage? Before a pass can be thrown \u2013 before the open man can be found \u2013 all of these questions need to be answered. Each pass is really a guess, a hypothesis launched into the air, but the best quarterbacks find a way to make better guesses. What separates Tom Brady and Joe Montana and Peyton Manning and John Elway, and the other great quarterbacks of the modern NFL era, is their ability to find the right receiver at the right time. No other team sport is so dependent on the judgment of a single player.\n\nNFL scouts take the decision-making skills of quarterbacks very seriously. The league requires that every player in the draft takes the Wonderlic intelligence test, which is essentially a shorter version of the IQ test. The test is 12 minutes long and consists of 50 questions, which get progressively harder as the test goes along. Here's an example of an easy Wonderlic question:\n\n'Paper sells for 21 cents per pad. What will four pads cost?'\n\nAnd here's a hard Wonderlic question:\n\n'Three individuals form a partnership and agree to divide the profits equally. X invests $9,000, Y invests $7,000, Z invests $4,000. If the profits are $4,800, how much less does X receive than if the profits were divided in proportion to the amount invested?'\n\nThe underlying assumption of the Wonderlic test is that players who are better at math and logic problems will make better decisions in the pocket. At first glance, this seems like a reasonable conjecture. No other position in sports requires such extreme cognitive talents. A successful quarterback will need to memorize hundreds of offensive plays and dozens of different defensive formations. They'll need to spend hours studying game tape of their opponents so that, when they're on the field, they can put that knowledge to use. In many instances, quarterbacks are even responsible for changing the play at the line of scrimmage. They are like a coach with shoulder pads.\n\nAs a result, NFL teams start to get nervous whenever a quarterback scores too far below the average for the position, which is 25. (In comparison, the average score for computer programmers is 28. Janitors, on average, score 15, as do running backs.) Vince Young, for example, the star quarterback from the University of Texas, reportedly scored a six on the test, which led many teams to publicly question his ability to play in the NFL.\n\nBut Young ended up excelling in the pros. And Young isn't the only quarterback who has achieved success despite a poor Wonderlic score. Dan Marino scored 14. Brett Favre's Wonderlic score was 22, while Randall Cunningham scored 15. All of these quarterbacks have been, or will be, inducted in the Hall of Fame. Furthermore, several quarterbacks with unusually high Wonderlic scores \u2013 players like Alex Smith and Matt Leinart, who both scored above thirty-five on the test and were top 10 picks in the 2005 NFL draft \u2013 have struggled in the NFL, in large part because of their poor decision-making skills.\n\nThe reason there is virtually no correlation between the results of the Wonderlic and the success of quarterbacks in the NFL is that finding the open man involves a very different set of decision-making skills than solving an algebra problem. While quarter backs need to grapple with complexity \u2013 the typical offensive playbook is several inches thick \u2013 they don't make sense of the football field in the same way that they make sense of questions on a multiple-choice exam. The Wonderlic measures a specific kind of thought process, but the best quarterbacks don't think before making a pass. There isn't time.\n\nTake that match-winning pass to Troy Brown. Brady's decision depended upon a long list of variables. He needed to know that the linebacker wouldn't fall back into coverage, and that there were no cornerbacks in the area waiting for an interception. After that, he had to calculate the ideal place to hit Brown with the ball, so that Brown had plenty of room to run after the catch. Then he needed to figure out how to make a throw without hitting that defensive lineman blocking his passing lane. If Brady was forced to explicitly think through this decision \u2013 if he treated his choice of receivers like a question on the Wonderlic test \u2013 then every pass would require a lot of complicated trigonometry, as he computed his passing angles on the plane of the football field. But how can you contemplate the math when five angry linemen are running straight at you? The answer is simple: you can't. If a quarterback hesitates for even a split-second, he is going to get drilled into the hard ground.\n\nSo how do quarterbacks do it? How do they make a decision? If you ask Tom Brady what he's thinking before he throws the ball, he'll tell you that he's not really thinking at all. The velocity of the game makes thought impossible. It's like asking a baseball player why he decided to swing the bat. Brady can only afford to give each receiver a split-second of attention before he has to move on to the next receiver. As soon as he glances at these bodies in motion he must immediately decide if that body will be open a few seconds in the future. As a result, the quarterback is forced to evaluate each of his passing alternatives without knowing how he's evaluating them. Brady chooses a target without knowing why, exactly, he's settled on that target. Did he pass to Troy Brown with 29 seconds remaining in the Super Bowl because the middle linebacker had ceded too much space, or because the cornerbacks were following the other receivers downfield, leaving a small gap in the center of the field? Or did Brady settle on Brown because all the other passing options were tightly covered, and he knew that he needed a long completion? The quarterback can't answer these questions. It's as if his mind is making decisions without him. Even quarterbacks are mystified by their talents. 'I don't know how I know where to pass,' Brady says. 'There are no firm rules. You just feel like you're going to the right place... And that's where I throw it.'\n\n##  _The Horses and the Charioteer_\n\nThe mystery of how we make decisions \u2013 how Tom Brady chooses where to throw the ball \u2013 is one of the oldest mysteries of the mind. Even though we are defined by our decisions, we are often completely unaware of what's happening inside our heads during the decision-making process. You can't explain why you bought the box of Honey Nut Cheerios, or stopped at the yellow traffic light, or threw the football to Troy Brown. On the evaluation sheets of NFL scouts, _decision-making_ is listed in the category Intangibles. It's one of the most important qualities in a quarterback, and yet nobody knows what it is.\n\nThe opaque nature of this mental process has led to a surfeit of theorizing. The most popular theory frames decision-making in epic terms, as a pitched battle between reason and emotion, with reason often triumphing. According to this classic script, what separates us from animals is the godly gift of rationality. When we are deciding what to do, we are able to ignore our feelings and carefully think through the problem. A quarterback, for instance, is supposed to choose a receiver by calmly contemplating all of the information on the field, translating the helter-skelter of the pass play into a series of discrete math problems. A more rational quarterback, with a higher Wonderlic score, should be a better quarterback. This ability to analyze the facts \u2013 to transcend our feelings, instincts, and impulses \u2013 is often seen as the defining element of human nature.\n\nPlato, as usual, was there first. He liked to imagine the mind as a chariot pulled by two horses. The rational brain, he said, is the charioteer; it holds the reins and decides where the horses run. If the horses get out of control, the charioteer just needs to take out his whip and reassert authority. One of the horses is well bred and well behaved, but even the best charioteer has difficulty controlling the other horse. 'He is of an ignoble breed,' Plato wrote. 'He has a short bull-neck, a pug nose, black skin, and bloodshot white eyes; companion to wild boasts and indecency, he is shaggy around the ears \u2013 deaf as a post \u2013 and just barely yields to horsewhip and goad combined.' According to Plato, this obstinate horse represents negative, destructive emotions. The job of the charioteer is to keep the dark horse from running wild and to keep both horses moving forward.\n\nWith that single metaphor, Plato divided the mind into two separate spheres. The soul was seen as conflicted, torn between reason and emotion. When the driver and horses wanted different things, Plato said, it was essential to listen to the driver. 'If the better elements of the mind which lead to order and philosophy prevail,' he wrote, 'then we can lead a life here in happiness and harmony, masters of ourselves.' The alternative, he warned, was a life governed by impulsive emotions. If we follow the horses, we will be led like a 'fool into the world below'.\n\nThis division of the mind is one of Plato's most enduring themes, an idea enshrined in Western culture. On the one hand, humans are part animal, primitive beasts stuffed full of primitive desires. And yet, humans are also capable of reason and foresight, blessed with the divine gift of rationality. The Roman poet Ovid, writing in _Metamorphoses a_ few centuries after Plato, captured this psychology in a few short sentences. Medea has fallen in love with Jason \u2013 she was literally struck by Eros' arrow \u2013 but this love conflicts with her duty to her father. 'I am dragged along by a strange new force,' she laments. 'Desire and reason are pulling in different directions. I see the right way and approve it, but follow the wrong.'\n\nRen\u00e9 Descartes, the most influential philosopher of the Enlightenment, agreed with this ancient critique of feeling. Descartes divided our being into two distinct substances: a holy soul capable of reason, and a fleshy body full of 'mechanical passions'. What Descartes wanted to do was purge the human intellect of its falsehoods, to get beyond the illogical beliefs of the past. In his seminal work, the awkwardly titled _Discourse_ _on the Method for Properly Conducting Reason and Searching_ _for Truth_ , Descartes tried to provide an example of rationality in pure form. His goal was to lead humanity out of the cave, to reveal the 'clear and distinct' principles that our emotions and intuitions obscure.\n\nThe Cartesian faith in reason became a founding principle of modern philosophy. Rationality was like a scalpel, able to dissect reality into its necessary parts. Emotions, on the other hand, were crude and primitive. Over time, a variety of influential thinkers tried to translate this binary psychology into practical terms. Francis Bacon and Auguste Comte wanted to reorganize society so that it reflected 'rational science'; Thomas Jefferson hoped that the 'American experiment would prove that men can be governed by reason and reason alone'; Immanuel Kant came up with the concept of the categorical imperative so that morality _was_ rationality. At the height of the French Revolution, a group of radicals founded the Cult of Reason and turned several Parisian cathedrals into temples of rationality. There were no temples dedicated to emotion.\n\nThe twentieth-century version of the Platonic metaphor was put forth by Sigmund Freud. Although Freud liked to say that he spent his life destroying illusions, his basic view of the mind differed little from Plato's. In his 'speculative science', Freud imagined the human mind as divided into a series of conflicting parts. (Conflict was important to Freud, since it helped explain neuroses.) At the center of the mind was the id, a factory of crude desires. Above that was the ego, which represented the conscious self and the rational brain. It was the job of the ego to restrain the id, channeling its animal emotions in socially acceptable ways. 'One might compare the relations of the ego to the id with that between a rider and his horse,' Freud wrote in a direct allusion to Plato. 'The horse provides the locomotive energy, and the rider has the prerogative of determining the goal and of guiding the movements of his powerful mount towards it.'\n\nThe purpose of Freudian psychoanalysis was to fortify the ego, to build up the strength needed to control the impulses of the id. In other words, Freud tried to teach his patients how to hold back their horses. He believed that most mental disorders, from hysteria to narcissism, were due to the effects of unrestrained feelings. In later years, Freud would turn this Platonic vision into a theory of everything. He saw civilization, or _kultur_ , as the individual mind writ large. 'The events of human history', Freud wrote, 'are only the reflections of the dynamic conflicts among the id and ego, which psychoanalysis studies in the individual \u2013 the same events on a wider stage.' According to Freud, the survival of modern society depended on people sacrificing the emotional desires of their ids \u2013 what he termed the pleasure principle \u2013 for the sake of the greater good. The possibility of human reason was the only thing that kept civilization from descending into barbarism. As Goya put it, 'The sleep of reason produces monsters.'\n\nOver time, Freudian psychology lost its scientific credibility. Discussions of the id, ego, and Oedipus complex were replaced by references to specific areas in the brain; Viennese theory gave way to increasingly exact anatomical maps of the cortex. The metaphor of the Platonic chariot seemed woefully obsolete.\n\nBut modern science soon hit on a new metaphor: the mind was a computer. According to cognitive psychology, each of us was a set of software programs running on three pounds of neural hardware. While this computer metaphor helped stimulate some important scientific breakthroughs \u2013 it led to the birth of artificial intelligence, among other things \u2013 it was also misleading, at least in one crucial respect. The problem with seeing the mind as a computer is that computers don't have feelings. Because emotions couldn't be reduced to bits of information or the logical structures of programming language, scientists tended to ignore them. 'Cognitive psychologists subscribed to this false ideal of rational, logical thought, and so we diminished the importance of everything else,' says Marvin Minsky, a professor at MIT and a pioneer of artificial intelligence. When cognitive psychologists did think about emotion, they tended to reinforce the Platonic divide: feelings interfered with cognition. They were the antagonists of rationality, and they messed up the machine. That was the version of the mind put forth by modern science.\n\nThe simple idea connecting Plato's philosophy to cognitive psychology is the privileging of reason over emotion. It's easy to understand why this vision has endured for so long. It raises _Homo sapiens_ above every other animal: the human mind is a rational computer, a peerless processor of information. Yet it also helps explain away our flaws: because each of us is still part animal, the faculty of reason is forced to compete with primitive emotions. The charioteer must control those wild horses.\n\nThis theory of human nature comes with a corollary: if our feelings keep us from making rational decisions, then surely we'd be better off without any feelings at all. Plato, for example, couldn't help but imagine a utopia in which reason determined everything. Such a mythical society \u2013 a republic of pure reason \u2013 has been dreamed of by philosophers ever since.\n\nBut this classical theory is founded upon a crucial mistake. For too long, people have disparaged the emotional brain, blaming our feelings for all of our mistakes. The truth is far more interesting. What we discover when we look at the brain is that the horses and the charioteer depend upon each other. If it weren't for our emotions, reason wouldn't exist at all.\n\n##  _The Disease of Indecision_\n\nIn 1982, a patient named Elliot walked into the office of neurologist Antonio Damasio. A few months earlier, a small tumor had been cut out of Elliot's cortex, near the frontal lobe of his brain. Before the surgery, Elliot had been a model father and husband. He'd held down an important management job in a large corporation and was active in his local church. But the operation changed everything. Although Elliot's IQ had stayed the same \u2013 he still tested in the 97th percentile \u2013 he now exhibited one psychological flaw: he was incapable of making a decision.\n\nThis dysfunction made normal life impossible. Routine tasks that should have taken 10 minutes now required several hours. Elliot endlessly deliberated over irrelevant details, like whether to use a blue or black pen, what radio station to listen to and where to park his car. When he chose where to eat lunch, Elliot carefully considered each restaurant's menu, seating plan and lighting scheme, and then drove to each place to see how busy it was. But all this analysis was for naught: Elliot still couldn't decide where to eat. His indecision was pathological.\n\nBefore long, Elliot was fired from his job. That's when things really began to fall apart. He started a series of new businesses, but they all failed. He was taken in by a con man and was forced into bankruptcy. His wife divorced him. The IRS began investigating him. He moved back in with his parents. As Damasio put it, 'Elliot emerged as a man with a normal intellect who was unable to decide properly, especially when the decision involved personal or social matters.'\n\nBut why was Elliot suddenly incapable of making good decisions? What had happened to his brain? Damasio's first insight occurred while talking to Elliot about the tragic turn his life had taken. 'He was always controlled,' Damasio remembers, 'always describing scenes as a dispassionate, uninvolved spectator. Nowhere was there a sense of his own suffering, even though he was the protagonist... I never saw a tinge of emotion in my many hours of conversation with him: no sadness, no impatience, no frustration.' Elliot's friends and family confirmed Damasio's observations: ever since his surgery, he'd seemed strangely devoid of emotion, numb to the tragic turn his own life had taken.\n\nTo test this diagnosis, Damasio hooked Elliot to a machine that measured the activity of the sweat glands in his palms. (When a person experiences strong emotions, the skin is literally aroused and the hands start to perspire. Lie detectors operate on the basis of this principle.) Damasio then showed Elliot various photographs that normally triggered an immediate emotional response: a severed foot, a naked woman, a house on fire, a handgun. The results were clear: Elliot felt nothing. No matter how grotesque or aggressive the picture, his palms never got sweaty. He had the emotional life of a mannequin.\n\nThis was a completely unexpected discovery. At the time, neuroscience assumed that human emotions were _irrational_. A person without any emotions \u2013 in other words, someone like Elliot \u2013 should therefore make better decisions. His cognition should be uncorrupted. The charioteer should have complete control.\n\nWhat, then, had happened to Elliot? Why couldn't he lead a normal life? To Damasio, Elliot's pathology suggested that emotions are a crucial part of the decision-making process. When we are cut off from our feelings, the most banal decisions became impossible. A brain that can't feel can't make up its mind.\n\nAfter interviewing Elliot, Damasio began studying other patients with similar patterns of brain damage. These patients all appeared intelligent and showed no deficits on any conventional cognitive tests. And yet they all suffered from the same profound flaw: because they didn't experience emotion, they had tremendous difficulty making any decisions. In _Descartes' Error_ , Damasio described what it was like trying to set up an appointment with one of these emotionless patients:\n\nI suggested two alternative dates, both in the coming month and just a few days apart from each other. The patient pulled out his appointment book and began consulting the calendar. The behavior that ensued, which was witnessed by several investigators, was remarkable. For the better part of a half hour, the patient enumerated reasons for and against each of the two dates: previous engagements, proximity to other engagements, possible meteorological conditions, virtually anything that one could reasonably think about concerning a simple date... He was now walking us through a tiresome cost-benefit analysis, an endless outlining and fruitless comparison of options and possible consequences. It took enormous discipline to listen to all of this without pounding on the table and telling him to stop.\n\nBased on these patients, Damasio began compiling a map of feeling, locating the specific brain regions responsible for generating emotions. Although many different cortical areas contribute to this process, one part of the brain seemed particularly important: a small circuit of tissue called the orbitofrontal cortex, which sits just behind the eyes, in the underbelly of the frontal lobe. ( _Orbit_ is Latin for 'eye socket'.) If this fragile fold of cells is damaged by a malignant tumor or a hemorrhaging artery, the tragic result is always the same. At first, everything seems normal, and after the tumor is removed or the bleeding is stopped, the patient is sent home. A full recovery is forecast. But then little things start to go awry. The patient begins to seem remote, cold, distant. This previously responsible person suddenly starts doing irresponsible things. The mundane choices of everyday life become excruciatingly difficult. It's as if his very personality \u2013 the collection of wants and desires that defined him as an individual \u2013 had been systematically erased. His loved ones say it's like living with a stranger, only this stranger has no scruples.\n\nThe crucial importance of our emotions \u2013 the fact that we can't make decisions without them \u2013 contradicts the conventional view of human nature, with its ancient philosophical roots. For most of the twentieth century, the ideal of rationality was supported by scientific descriptions of human anatomy. The brain was envisioned as consisting of four separate layers, stacked in ascending order of complexity. (The cortex was like an archaeological site: the deeper you dug, the farther back in time you traveled.) Scientists explained the anatomy of the human brain in this manner: At its bottom was the brain stem, which governed the most basic bodily functions. It controlled heart beat, breathing and body temperature. Above that was the diencephalon, which regulated hunger pangs and sleep cycles. Then came the limbic region, which generated animal emotions. It was the source of lust, violence and impulsive behavior. (Human beings shared these three brain layers with every other mammal.) Finally, there was the magnificent frontal cortex \u2013 the masterpiece of evolution \u2013 which was responsible for reason, intelligence and morality. These convolutions of gray matter allowed each of us to resist urges and suppress emotions. In other words, the rational fourth layer of the brain allowed us to ignore the first three layers. We were the only species able to rebel against primitive feelings and make decisions that were dispassionate and deliberate.\n\nBut this anatomical narrative is _false_. The expansion of the frontal cortex during human evolution did not turn us into purely rational creatures, able to ignore our impulses. In fact, neuroscience now knows that the opposite is true: a significant part of our frontal cortex is involved with emotion. David Hume, the eighteenth-century Scottish philosopher who delighted in heretical ideas, was right when he declared that reason was 'the slave of the passions'.\n\nHow does this emotional brain system work? The orbitofrontal cortex (OFC), the part of the brain that Elliot was missing, is responsible for integrating visceral emotions into the decision-making process. It connects the feelings generated by the 'primitive' brain \u2013 areas like the brain stem and the amygdala, which is in the limbic system \u2013 to the stream of conscious thought. When a person is drawn to a specific receiver, or a certain entr\u00e9e on the menu, or a particular romantic prospect, the mind is trying to tell him that he should choose that option. It has already assessed the alternatives \u2013 this analysis takes place outside of conscious awareness \u2013 and converted that assessment into a positive emotion. And when he sees a receiver who's tightly covered, or smells a food he doesn't like, or glimpses an ex-girlfriend, it is the OFC that makes him want to get away. ( _Emotion_ and _motivation_ share the same Latin root, _movere_ , which means 'to move'.) The world is full of things, and it is our feelings that help us choose among them.\n\nWhen this neural connection is severed \u2013 when our OFCs can't comprehend our own emotions \u2013 we lose access to the wealth of opinions that we normally rely on. All of a sudden, you no longer know what to think about the receiver running a short post pattern or whether it's a good idea to order the cheeseburger for lunch. The end result is that it's impossible to make decent decisions. This is why the OFC is one of the few cortical regions that are markedly larger in humans than they are in other primates. While Plato and Freud would have guessed that the job of the OFC was to _protect_ us from our emotions, to fortify reason against feeling, its actual function is precisely the opposite. From the perspective of the human brain, _Homo_ _sapiens_ is the most emotional animal of all.\n\n##  _Emotional Signals_\n\nIt's not easy making a daytime soap opera. The demands of the form are grueling: a new episode has to be filmed nearly every single day. No other type of popular entertainment churns out so much material in so short a time. New plot twists have to be dreamed up, new scripts have to be written, actors need to rehearse, and every scene must be meticulously mapped out. Only then, once all that preparation is complete, are the cameras turned on. For most daytime soaps, it takes about twelve hours to film 22 minutes of television. This cycle is repeated five days a week.\n\nHerb Stein has been directing _Days of Our Lives_ , a soap opera on NBC, for 25 years. He's shot more than fifty thousand scenes and has cast hundreds of different actors. He's been nominated for eight daytime Emmys. Over the course of his long career, Stein has witnessed more scenes of melodrama \u2013 rapes, weddings, births, murders, confessions \u2013 than just about any other human being alive. He is, one might say, an _expert_ on melodrama: how to write it, block it, film it, edit it and produce it.\n\nFor Stein, the long road to daytime television began when he was a student at UCLA and read _The Oresteia_ , the trilogy of classic Greek tragedies written by Aeschylus. It was the utter timelessness of the plays \u2013 their ability to speak to enduring human themes \u2013 that made him want to study theater. When Stein talks about drama \u2013 and it doesn't matter if he's talking about Aeschylus or _General Hospital_ \u2013 he tends to sound like a literature professor. (He also looks like one, with his rumpled shirts and a few days' worth of salt-and-pepper stubble.) Stein talks in long, digressive monologues and finds grand ideas in the most unlikely plot lines. 'Many of these classic plays have elements of the ridiculous,' he says. 'The plots are often completely implausible. That whole Oedipus thing? Totally absurd. And yet, when these stories are told well, you don't notice the absurdity. You're too busy paying attention to what's happening.'\n\nSoap operas work the same way. The key to being a successful soap opera director \u2013 and Stein is one of the most successful in the business \u2013 is telling the story so that people don't notice you're telling them a story. Everything has to feel sincere, even when what's happening onscreen is completely outlandish. This is much harder than it might seem. Let's say you're shooting a scene in which a woman is giving birth to fraternal twins fathered by two different men, both of whom are at the bedside with her. One of the fathers is the villain of the show: he impregnated the woman by raping her. The other father is the good guy, and the woman is deeply in love with him. However, if she doesn't marry her rapist, then members of her family will be killed. (This is an actual plot line from a recent _Days of Our Lives_ episode.) The scene has several pages of intense dialogue, a few tears and plenty of subtext. Stein has about an hour to shoot it, which forces him to make some crucial decisions on the fly. He has to figure out where each character should stand, how they all should move, what emotions they should convey, and how each of the four cameras should capture the action. Should they zoom in close, or get a reaction shot over the shoulder? How should the villain deliver his lines? These directorial decisions will determine whether or not the scene works. 'You've really got to know how to milk the drama,' Stein says. 'Otherwise, it's just a bunch of people standing in a room, saying stupid stuff.'\n\nAlthough the scene has been mapped out in advance, Stein still needs to make many of these decisions in the midst of filming, while the actors are delivering their lines. Most of the fake rooms on the Burbank sound stage have only two flimsy walls, with one camera positioned on each side. An additional camera records the center of the scene. As soon as the assistant director yells out, 'Action!' there is a frenzy of activity offstage as the cameras pivot and Stein snaps his fingers, pointing to indicate which camera he wants to capture the action for each specific part of the scene. (This makes it easier for the editor to assemble a working cut later.) During complicated scenes, such as that birth scene with the two fathers, Stein looks like an orchestra conductor: his arms are never still. He is constantly pointing at different cameras, crafting the scene in real time.\n\nHow does Stein make these directorial decisions? After all, he doesn't have the luxury of filming 20 different takes from 20 different angles. 'Given the schedule [of a daytime soap opera],' Stein says, 'there isn't time to be fiddling around with all the stuff that directors normally fiddle around with. You need to make the right decision the first time around.' If a soap director makes a mistake while shooting, the scene can't be re-shot another day. When you're creating daytime television, you have only one day.\n\nThis relentless time pressure means that Stein can't afford to carefully think through all of his camera choices. He doesn't have time to be rational; he needs to react to the drama as it's unfolding. In that sense, he is like a quarterback in the pocket. 'When you shoot as many scenes as I have,' Stein says, 'you just know how things should go. I can watch an actor say a single line and know immediately that we need to try it again. When we're filming a scene, it's all very instinctual. Even when we go in with a plan on how to shoot it, that plan will often change in the moment, depending on how it feels.'\n\nThe reliance on instinct and 'feel' is also a crucial part of the casting process. Soaps are continually bringing on new actors, in part because the longer actors are on the show, the higher their salaries are. (That's why established characters on _Days of_ _Our Lives_ are constantly being killed off. As Stein quips, 'This isn't show art. It's show _business_.') For a soap opera, there are few decisions as crucial as casting. The size of the audience oscillates with the appeal of the actors, and a particularly appealing actor can create a spike in the ratings. 'You are always looking for that person that people want to look at,' Stein says. 'And I don't just mean attractiveness. They've got to have _it_ , and by _it_ , I mean everything that you can't really put into words.'\n\nThe question, of course, is how you identify _it_. When Stein first started directing television, he was overwhelmed by all the different variables involved in casting. First, he'd try to make sure that the person looked right for the part and could act in the soap style. Then Stein needed to consider how this actor would fit in with the rest of the cast. ('A lack of chemistry has ruined many a soap scene,' he says.) Only after that was Stein able to think about whether or not the actor actually had talent. Would he deliver his lines with sincerity? Could he cry on demand? How many takes would he require before he got the scene right? 'Given all of these factors,' Stein says, 'there can be a tendency to really outthink yourself, to talk yourself into choosing the wrong actor.'\n\nAfter directing daytime television for decades, however, Stein has learned how to trust his instincts, even if he can't always explain them. 'It only takes me three to five seconds before I know if the person is right,' he says. 'A few words, a single gesture. That's all I need. And I've learned to always listen to that.' Recently, the show put out a casting call for a male lead. The character was going to be the new villain on the show. Stein was up in his office, blocking a script, watching the auditions out of the corner of his eye. After a few hours of seeing dozens of different actors recite the exact same lines, Stein was getting bored and discouraged. 'But that's when this one guy stood up,' he says. 'The actor didn't even know his lines because he had gotten the script late. I just saw him say a few words, and then I knew. He was unbelievably great. I couldn't explain why, but for me he completely stood out. What they say is true: you just get a feeling.'\n\nThe mental process Stein is describing depends on his emotional brain. Those twinges of feeling that help him select the right camera and find the best actor are a distillation of all those details that he doesn't consciously perceive. 'The conscious brain may get all the attention,' says Joseph LeDoux, a neuroscientist at NYU. 'But consciousness is a small part of what the brain does, and it's a slave to everything that works beneath it.' According to LeDoux, much of what we 'think' is really driven by our emotions. In this sense, every feeling is really a summary of data, a visceral response to all of the information that can't be accessed directly. While Stein's conscious brain was blocking the script, his unconscious supercomputer was processing all sorts of data. It then translated that data into vivid emotional signals that were detected by the OFC, allowing Stein to act upon these subliminal calculations. If Stein were missing his feelings \u2013 if he were like one of Damasio's patients \u2013 then he would be forced to carefully analyze every alternative, and that would take forever. His episodes would be constantly delayed and he would cast the wrong actors. Stein's insight is that his feelings are often an accurate shortcut, a concise expression of his decades' worth of experience. They already know how to shoot the scene.\n\nWhy are our emotions so essential? How did they get so good at finding the open man and directing soap operas? The answer is rooted in evolution. It takes a long time to design a brain. The first clumps of networked neurons appeared more than 500 million years ago. This was the first nervous system, although at that point it was really just a set of automatic reflexes. Over time, however, primitive brains grew increasingly complex. They expanded from a few thousand neurons in earthworms to nearly a trillion connected cells in Old-World primates. When _Homo_ _sapiens_ first appeared, about two hundred thousand years ago, the planet was already full of creatures with highly specialized brains. There were fish that could migrate across the ocean using magnetic fields, birds that navigated by starlight and insects that could smell food from a mile away. These cognitive feats were all by-products of instincts that had been engineered by natural selection to perform specific tasks. What these animals couldn't do, however, was reflect on their own decisions. They couldn't plan out their days or use language to express their inner states. They weren't able to analyze complex phenomena or invent new tools. What couldn't be done automatically couldn't be done at all. The charioteer had yet to appear.\n\nThe evolution of the human brain changed everything. For the first time, there was an animal that could think about how it thought. We humans could contemplate our emotions and use words to dissect the world, parsing reality into neat chains of causation. We could accumulate knowledge and logically analyze problems. We could tell elaborate lies and make plans for the future. Sometimes, we could even follow our plans.\n\nThese new talents were incredibly useful. But they were also incredibly new. As a result, the parts of the human brain that make them possible \u2013 the ones that the driver of the chariot controls \u2013 suffer from the same problem that afflicts any new technology: they have lots of design flaws and software bugs. (The human brain is like a computer operating system that was rushed to market.) This is why a cheap calculator can do arithmetic better than a professional mathematician, why a mainframe computer can beat a grand master at chess, and why we so often confuse causation and correlation. When it comes to the new parts of the brain, evolution just hasn't had time to work out the kinks.\n\nThe emotional brain, however, has been exquisitely refined by evolution over the last several hundred million years. Its software code has been subjected to endless tests, so that it can make fast decisions based on very little information. Look, for instance, at the mental process involved in hitting a cricket ball. The numbers make the task look impossible. A typical delivery takes about 0.30 seconds to travel from the bowler's hand to the batsman's crease. (This is the average interval between human heartbeats.) Unfortunately for the batsman, it takes about 0.25 seconds for his muscles to initiate a shot, leaving his brain a paltry one-twentieth of a second to make up its mind. But even this estimate is too generous. It takes a few milliseconds for the visual information to travel from the retina to the visual cortex, so the batsman really has less than five milliseconds to perceive the delivery and decide whether or not to play the ball. But we can't think this quickly: even under perfect conditions it takes the brain about twenty milliseconds to respond to a sensory stimulus.\n\nSo how do professional cricketers manage to play a delivery from a fast bowler? The answer is that the brain begins collecting information about the delivery long before the ball leaves the hand. As soon as the bowler hits the crease, the batsman will automatically start to pick up on 'anticipatory clues' that help him winnow down the list of possibilities. He'll notice specific features of the hand and arm of the bowler that allow him to predict where, exactly, the ball will end up after it has been pitched. The batsmen, of course, aren't consciously studying these signs: they can't tell you why or how they decided to play a particular ball. And yet, they are able to act based upon this information. (A similar process is at work with goalkeepers during a penalty shot. They have to predict where the ball will end up before it is even kicked, which forces them to rely on subtle, subliminal cues.) A study of professional cricketers, for instance, demonstrated that the players could accurately predict the speed and location of the ball based solely on a one-second video of the bowler's wind-up. Each well-trained brain knew exactly what details to look for. And then, once it perceived these details, it seamlessly converted them into an accurate set of feelings, which allowed the expert batsmen to play the ball effectively.\n\nWe take these automatic talents for granted precisely because they work so well. There's no robot that can hit a baseball or throw a football or ride a bicycle. No computer program can figure out which actor should play a villain or instantly recognize a familiar face. This is why, when evolution was building our brain, it didn't bother to replace all of those emotional processes with new operations under our explicit control. If something isn't broken, then natural selection isn't going to fix it. Our mind is made out of used parts, engineered by a blind watchmaker. The end result is that the uniquely human parts of the mind depend on the primitive mind underneath. The process of thinking requires feeling, for our feelings are what let us understand all the information that we can't directly comprehend. Reason without emotion is impotent.\n\nOne of the first scientists to defend this view of decision-making was William James, the great American psychologist. In his seminal textbook first published in 1890, _The Principles of_ _Psychology,_ James launched into a critique of the standard 'rationalist' account of the human mind. 'The facts of the case are really tolerably plain,' James wrote. 'Man has a far _greater_ variety of impulses than any other lower animal.' In other words, the Platonic view of decision-making, which saw man as a purely rational animal, defined 'by the almost total absence of instincts,' was utterly mistaken. James' real insight, however, was that these impulses weren't necessarily bad influences. In fact, he believed that 'the preponderance of habits, instincts and emotions' in the human brain was an essential part of what made the brain so effective. According to James, the mind contained two distinct thinking systems, one of which was rational and deliberate and another that was quick, effortless and emotional. The key to making decisions, James said, was knowing when to rely on each of these systems.\n\nJust look at Tom Brady. It's his feelings that allow him to make such quick passing decisions. For Brady, the process probably works something like this. After the ball is snapped, he drops back and tries to make sense of the field. He begins going through his checklist of receivers. The primary target, a tight end running a short crossing pattern, is tightly covered by a linebacker. As a result, when Brady glances at the tight end, he automatically feels a slight twinge of fear, the sure sign of a risky pass. The presence of the linebacker has been translated into a negative emotion. Brady then proceeds onto his secondary target, a wide receiver running a long pass route. Unfortunately, this target is covered by a cornerback and a safety. Once again, Brady experiences a negative feeling, an instant distillation of what's happening on the football field. A few seconds have now elapsed, and Brady can feel the pressure of the defensive line. His left tackle is being pushed backwards; Brady knows that he's got to get rid of the ball soon or the game is going to end with a sack. He proceeds on to his third target. Troy Brown is streaking across the center of the field. Brady sees that Brown is open: he's found the void between the linebackers and the defensive backs. When Brady looks at this target, his usual fear is replaced by a subtle twinge of positive emotion. He has found the open man. He lets the ball fly.\nTWO\n\n# The Predictions of Dopamine\n\nIN THE EARLY MORNING hours of 24 February 1991, the First and Second Marine divisions rolled north across the desert of Saudi Arabia. As they approached the unmarked border with Kuwait \u2013 the landscape was just an expanse of barren sand \u2013 the troops accelerated their pace. These Marines were the first Coalition forces to enter the country since it had been invaded by Iraq, more than eight months earlier. The outcome of Operation Desert Storm depended on their success. The Marines needed to liberate Kuwait, and they needed to do it in fewer than one hundred hours. If the Marines failed to overtake the Iraqi army quickly, they faced the prospect of urban warfare. The Iraqis were threatening to retreat into the streets of Kuwait City, and if that happened, the ground war could drag on for months.\n\nThe Marines expected heavy resistance. The Iraqis had fortified many of their military positions inside Kuwait, concentrating their forces near the Al Wafrah oil field along the Saudi Arabian border. They had draped a line of explosive mines across the desert. To make matters even more difficult, these Iraqi units had largely been spared the brutal air war. Because the Coalition forces were determined to minimize collateral damage and civilian casualties, bombing runs inside the occupied country were sharply restricted. Unlike the Republican Guard troops stationed in southern Iraq, a military force that had been decimated by 37 days of intense bombing, these Marines were about to encounter an enemy at full strength. Central Command (CENTCOM) estimated that during the invasion of Kuwait, each Marine division would suffer approximately a thousand casualties, or between 5 and 10 per cent of its total troop strength.\n\nTo support this high-stakes mission, a fleet of Coalition battleships and destroyers was positioned fewer than twenty miles off the Kuwaiti coast. This was a risky strategic move; although the big naval guns provided crucial air cover for the ground attack of Kuwait, they were also well within range of Iraqi missiles. On the morning of the Marine invasion, the American and British ships in the Persian Gulf were put on the highest possible alert. They were told to expect hostile fire.\n\nThe first 24 hours of the ground war exceeded even CENTCOM's high expectations. After successfully breaching the perimeter of mines and barbed wire put down by the Iraqis, the Marine division managed to penetrate deep into central Kuwait. Unlike the Soviet T-72 tanks used by the Iraqi army, the American M1 Abrams tanks were equipped with GPS units and thermal sights, allowing the Marines to engage the enemy in the pitch-black night. After a brigade of Marines reached the outskirts of Kuwait City, they made an abrupt turn to the east and began the task of securing the coastline. Just before dawn on 25 February, 10 Marine helicopters, along with an amphibious landing ship, conducted a feint attack on a military base near the Kuwaiti port of Ash Shuaybah. The attack was supported by a barrage of artillery rounds from the offshore battleships. The Coalition forces weren't interested in capturing the port; they just wanted to 'neutralize' it, to make sure it didn't pose a danger to the offshore convoy.\n\nThat same morning, while Ash Shuaybah was being attacked, Lieutenant Commander Michael Riley was monitoring the radar screens onboard the HMS _Gloucester_ , a British destroyer stationed about fifteen miles from the port. The ship was responsible for protecting the Allied fleet, which meant that Riley had to monitor all of the airspace surrounding the naval convoy. Since the start of the air war, the radar crews had maintained an exhausting schedule. They were on duty for six hours, then they had six hours to sleep and eat, and after that brief respite, they headed back to the claustrophobic radar room. By the time the ground invasion began, the men were showing signs of fatigue. They had bloodshot eyes and needed constant infusions of caffeine.\n\nRiley had been on duty since midnight. At 5:01 in the morning, just as the Allied ships began shelling Ash Shuaybah, he noticed a radar blip off the Kuwaiti coast. A quick calculation of its trajectory had it heading straight for the convoy. Although Riley had been staring at similar-looking blips all night long, there was something about this radar trace that immediately made him suspicious. He couldn't explain why, but the blinking green dot on the screen filled him with fear; his pulse started to race and his hands became clammy. He continued to observe the incoming blip for another 40 seconds as it slowly honed in on the USS _Missouri_ , an American battleship. With each sweep of the radar, the blip grew closer. It was approaching the American ship at more than five hundred and fifty miles per hour. If Riley was going to shoot down the target \u2013 if he was going to act on his fear \u2013 then he needed to respond right away. If that blip was a missile and Riley didn't move immediately, it would be too late. Hundreds of sailors would die. The USS _Missouri_ would be sunk. And Riley would have stood by and watched it happen.\n\nBut Riley had a problem. The radar blip was located in airspace that was frequently traveled by American A-6 fighter jets, which the US Navy was using to deliver laser-guided bombs to support the Marine ground invasion. After completing their sorties, the planes flew down the Kuwait coast, turned east toward the convoy, and landed on their aircraft carriers. Over the last few weeks, Riley had watched dozens of A-6s fly a route nearly identical to the one being followed by this unidentified radar blip. The blip was also traveling at the same speed as the fighter jets and had a similar surface area. It looked exactly like an A-6 on the radar screen.\n\nTo make matters even more complicated, the A-6 pilots had gotten into the bad habit of turning off their electronic identification on their return flights. This identification system allowed Coalition forces to recognize their own, but it also made the planes more vulnerable to Iraqi anti-aircraft missiles. Not surprisingly, the pilots opted for the cloak of silence over Iraqi-controlled airspace. As a result, the radar crew onboard the HMS _Gloucester_ had no way of contacting this radar blip.\n\nThere was one last way for radar crews to distinguish between an incoming missile and a friendly aircraft: they could determine the altitude of the blip. The A-6 generally flew at around three thousand feet, while a Silkworm missile flew at around one thousand feet. However, the type of radar that Riley was using didn't provide him with any altitude information. If he wanted to know the height of a specific object, he had to use a specialized radar system known as the 909, which conducted sweeps in horizontal bands. Unfortunately, the 909 radar operator had entered an incorrect tracking number shortly after the blip appeared, which meant that Riley had no way of knowing the altitude of the flying object. Although he'd now been staring at the radar blip for almost a minute, its identity remained a befuddling mystery.\n\nThe target was moving fast. The time for deliberation was over. Riley issued the order to fire; two Sea Dart surface-to-air missiles were launched into the sky. Seconds passed. Riley nervously stared at the radar screen, watching his missiles race toward the object at speeds approaching Mach 1. The blinking green blips appeared to be drawn to the target, like iron filings to a magnet. Riley waited for the interception.\n\nThe explosion echoed over the ocean. All of the blips immediately disappeared from the radar screen. Whatever had been flying toward the USS _Missouri_ helplessly fell into the sea, just 700 yards in front of the American battleship. A few moments later, the captain of the HMS _Gloucester_ entered the radar room. 'Whose bird is it?' he asked Riley, wanting to know who was responsible for destroying the still unidentified target. 'It was ours, sir,' Riley responded. The captain asked Riley how he could be sure he'd fired at an Iraqi missile and not at an American fighter jet. Riley said he just knew.\n\nThe next four hours were the longest ones of Riley's life. If he had shot down an A-6, then he had killed two innocent pilots. His career was over. He might even be court-martialed. Riley immediately went back to review the radar tapes, looking for any scrap of evidence suggesting that the blip really was an Iraqi missile. But even when he had the luxury of time and analysis, Riley still couldn't definitively identify the target; the tapes were completely ambiguous. The mood on the HMS _Gloucester_ quickly grew somber. Investigative teams were sent out to view the wreckage still floating on the ocean surface. An immediate inventory of all Coalition planes in the area was conducted.\n\nThe captain of the HMS _Gloucester_ heard the news first. He walked over to Riley's bunk, where Riley was trying, in vain, to get some sleep. The results of the investigation were in: the radar blip was a Silkworm missile, not an American fighter jet. Riley had single-handedly saved a battleship.\n\nOf course, it's possible that Riley had just gotten lucky. After the war was over, British naval officers carefully analyzed the sequence of events preceding Riley's decision to fire the Sea Dart missiles. They concluded that based on the radar tapes, it was impossible to distinguish between the Silkworm and a friendly A-6. Although Riley had made the correct decision, he could have just as easily been shooting down an American fighter jet. His risky gamble had paid off, but it had still been a gamble.\n\nThat, at least, was the official version of events until the summer of 1993, when Gary Klein started to investigate the Silkworm affair. A cognitive psychologist who consults for the Marine Corps, Klein was informed that nobody could explain how the radar blip had been identified as a hostile missile. Even Riley didn't know why he'd considered that early-morning blip so dangerous. He assumed, like everybody else, that he'd just gotten lucky.\n\nKlein was intrigued. He had spent the last few decades studying decision-making in high-pressure situations, and he knew that intuition could often be astonishingly insightful, even if the origin of those insights was obscure. He was determined to find the source of Riley's fear, to figure out why this particular blip had felt so scary. So he went back to the radar tapes.\n\nHe soon realized that Riley had gotten used to seeing a very consistent blip pattern when the A-6s returned from their bombing sorties. Because Riley's naval radar could pick up signals only over water \u2013 after a signal went 'wet feet' \u2013 he was accustomed to seeing the fighter jets right as they flew off the Kuwaiti coast. The planes typically became visible after a single radar sweep.\n\nKlein analyzed the radar tapes from the predawn missile attack. He replayed those fateful 40 seconds over and over again, searching for any differences between Riley's experience of the A-6s returning from their sorties and his experience of the Silkworm blip.\n\nKlein suddenly saw the discrepancy. It was subtle, but crystal clear. He could finally explain Riley's intuitive insight.\n\nThe secret was the timing. Unlike the A-6, the Silkworm didn't appear off the coast right away. Because it traveled at such a low altitude, nearly two thousand feet below an A-6's, the signal of the missile was initially masked by ground interference. As a result, it wasn't visible until the _third_ radar sweep, which was eight seconds after an A-6 would have appeared. Riley was unconsciously evaluating the altitude of the blip, even if he didn't know he was doing it.\n\nThis is why Riley got the chills when he stared at the Iraqi missile on his radar screen. There was something strange about this radar blip. It didn't feel like an A-6. Although Riley couldn't explain why he felt so scared, he knew that something scary was happening. This blip needed to be shot down.\n\n## _The Molecule of Intuition_\n\nThe question still remains: how did Riley's emotions manage to distinguish between these two seemingly identical radar blips? What was happening inside his brain when he first saw the Silkworm missile, three sweeps off the Kuwaiti coast? Where did his fear come from? The answer lies in a single molecule, called dopamine, that brain cells use to communicate with one another. When Riley stared at the radar screen, it was most likely his dopamine neurons that told him he was looking at a missile and not an A-6 fighter jet.\n\nThe importance of dopamine was discovered by accident. In 1954, James Olds and Peter Milner, two neuroscientists at McGill University, decided to implant an electrode deep into the center of a rat's brain. The precise placement of the electrode was largely happenstance; at the time, the geography of the mind remained a mystery. But Olds and Milner got lucky. They inserted the needle right next to the nucleus accumbens (NAcc), a part of the brain that generates pleasurable feelings. Whenever you eat a piece of chocolate cake, or listen to a favorite pop song, or watch your favorite team win the World Series, it is your NAcc that helps you feel so happy.\n\nBut Olds and Milner quickly discovered that too much pleasure can be fatal. They placed the electrodes in several rodents' brains and then ran a small current into each wire, making the NAccs continually excited. The scientists noticed that the rodents lost interest in everything. They stopped eating and drinking. All courtship behavior ceased. The rats would just huddle in the corners of their cages, transfixed by their bliss. Within days, all of the animals had perished. They died of thirst.\n\nIt took several decades of painstaking research, but neuroscientists eventually discovered that the rats had been suffering from an excess of dopamine. The stimulation of the NAcc triggered a massive release of the neurotransmitter, which overwhelmed the rodents with ecstasy. In humans, addictive drugs work the same way: a crack addict who has just gotten a fix is no different than a rat in an electrical rapture. The brains of both creatures have been blinded by pleasure. This, then, became the dopaminergic clich\u00e9; it was the chemical explanation for sex, drugs, and rock and roll.\n\nBut happiness isn't the only feeling that dopamine produces. Scientists now know that this neurotransmitter helps to regulate _all_ of our emotions, from the first stirrings of love to the most visceral forms of disgust. It is the common neural currency of the mind, the molecule that helps us decide among alternatives. By looking at how dopamine works inside the brain, we can see why feelings are capable of providing deep insights. While Plato disparaged emotions as irrational and untrustworthy \u2013 the wild horses of the soul \u2013 they actually reflect an enormous amount of invisible analysis.\n\nMuch of our understanding of the dopamine system comes from the pioneering research of Wolfram Schultz, a neuroscientist at Cambridge University. He likes to compare dopamine neurons (those neurons that use dopamine to communicate) to the photoreceptors on the retina, which detect the rays of light entering the eye. Just as the process of sight starts with the retina, so the process of decision-making begins with the fluctuations of dopamine.\n\nAs a medical student in the early 1970s, Schultz grew interested in the neurotransmitter because of its role in triggering the paralyzing symptoms of Parkinson's disease. He recorded from cells in the monkey brain, hoping to find which cells were involved in controlling the body's movements. But he couldn't find anything. 'It was a classic case of experimental failure,' he says. 'I was a very frustrated scientist.' But after years of searching, Schultz noticed something odd about these dopamine neurons: they began to fire just before the monkey was given a reward, such as a pellet of food or a bit of banana. (The rewards were used to get the monkeys to move.) 'At first I thought it was unlikely that an individual cell could represent anything so complicated as food,' Schultz says. 'It just seemed like too much information for one neuron.'\n\nAfter hundreds of experimental trials, Schultz began to believe his own data; he realized he had accidentally found the reward mechanism at work in the primate brain. In the mid-1980s, after publishing a series of landmark papers, Schultz set out to decipher this reward circuitry. How exactly did a single cell manage to represent a reward? And why did it fire _before_ a reward was given?\n\nThe Schultz experiments followed a simple protocol: he sounded a loud tone, waited for a few seconds, and then squirted some drops of apple juice into the mouth of a monkey. While the experiment was unfolding, Schultz was probing the monkey brain with a needle that monitored the electrical activity inside individual cells. At first, the dopamine neurons fired only when the juice was delivered. The cells were responding to the actual reward. However, once the animal learned that the tone preceded the arrival of juice \u2013 this required only a few trials \u2013 the same neurons began firing at the sound of the tone instead of at the sweet reward. Schultz called these cells 'prediction neurons', since they were more concerned with _predicting_ rewards than actually receiving them. (This process can be indefinitely extended: the dopamine neurons can be made to respond to a light that precedes the tone that precedes the juice, and so on.) Once this simple pattern was learned, the monkey's dopamine neurons became exquisitely sensitive to variations on it. If the cellular predictions proved correct, and the reward arrived right on time, then the primate experienced a brief surge of dopamine, the pleasure of being right. However, if the pattern was violated \u2013 if the tone was played but the juice never arrived \u2013 then the monkey's dopamine neurons decreased their firing rate. This is known as the prediction-error signal. The monkey felt upset because its predictions of juice were wrong.\n\nWhat's interesting about this system is that it's all about _expectation_. Dopamine neurons constantly generate patterns based on experience: if this, then that. They learn that the tone predicts the juice, or that the light predicts the tone that predicts the juice. The cacophony of reality is distilled into models of correlation that allow the brain to anticipate what will happen next. As a result, the monkeys quickly learn when to expect their sweet reward.\n\nAfter refining this set of cellular forecasts, the brain compares these predictions to what actually happens. Once the monkey is taught to expect juice after a certain sequence of events, its dopamine cells carefully monitor the situation. If everything goes according to plan, its dopamine neurons secrete a little burst of enjoyment. The monkey is happy. But if these expectations aren't met \u2013 if the monkey doesn't get the promised juice \u2013 the dopamine cells go on strike. They instantly send out a signal announcing their mistake and stop releasing dopamine.\n\nThe brain is designed to amplify the shock of these mistaken predictions. Whenever it experiences something unexpected \u2013 like a radar blip that doesn't fit the usual pattern, or a drop of juice that doesn't arrive \u2013 the cortex immediately takes notice. Within milliseconds, the activity of the brain cells has been inflated into a powerful emotion. Nothing focuses the mind like surprise.\n\nThis fast cellular process begins in a tiny area in the center of the brain that is dense with dopamine neurons. Neuroscientists have known for several years that this region, the anterior cingulate cortex (ACC), is involved in the detection of errors. Whenever the dopamine neurons make a mistaken prediction \u2013 when they expect juice but don't get it \u2013 the brain generates a unique electrical signal, known as error-related negativity. The signal emanates from the ACC, so many neuroscientists refer to this area as the 'oh, shit!' circuit.\n\nThe importance of the ACC is revealed by the layout of the brain. Like the orbitofrontal cortex, the ACC helps control the conversation between what we know and what we feel. It sits at the crucial intersection between these two different ways of thinking. On the one hand, the ACC is closely connected to the thalamus, a brain area that helps direct conscious attention. This means that if the ACC is startled by some stimulus \u2013 like the bang of a gunshot it didn't expect \u2013 it can immediately focus on the relevant sensation. It forces the individual to notice the unexpected event.\n\nWhile the ACC is alerting the consciousness, it's also sending signals to the hypothalamus, which regulates crucial aspects of bodily function. When the ACC is worried about some anomaly \u2013 for instance, an errant blip on a radar screen \u2013 that worry is immediately translated into a somatic signal as the muscles prepare for action. Within seconds, heart rate increases, and adrenaline pours into the bloodstream. These fleshly feelings compel us to respond to the situation _right away_. A racing pulse and sweaty palms are the brain's way of saying that there's no time to waste. This prediction error is urgent.\n\nBut the ACC doesn't just monitor erroneous predictions. It also helps remember what the dopamine cells have just learned, so that expectations can be quickly adjusted in light of new events. It internalizes the lessons of real life, making sure that neural patterns are completely up to date. If it was predicted that juice would arrive after the tone, but the juice never arrived, then the ACC makes sure that future predictions are revised. The short-term feeling is translated into a long-term lesson. Even if the monkey is unaware of what, exactly, the ACC has memorized, the next time it's waiting for a squirt of juice, its brain cells are prepared. They know exactly when the reward will arrive.\n\nThis is an essential aspect of decision-making. If we can't incorporate the lessons of the past into our future decisions, then we're destined to endlessly repeat our mistakes. When the ACC is surgically removed from the monkey brain, the behavior of the primate becomes erratic and ineffective. The monkeys can no longer predict rewards or make sense of their surroundings. Researchers at Oxford performed an elegant experiment that made this deficit clear. A monkey clutched a joystick that moved in two different directions: it could be either lifted or turned. At any given moment, only one of the movements would trigger a reward (a pellet of food). To make things more interesting, the scientists switched the direction that would be rewarded every 25 trials. If the monkey had previously gotten in the habit of lifting the joystick in order to get a food pellet, it now had to shift its strategy.\n\nSo what did the monkeys do? Animals with intact ACCs had no problem with the task. As soon as they stopped receiving rewards for lifting the joystick, they started turning it in the other direction. The problem was soon solved, and the monkeys continued to receive their pellets of food. However, monkeys that were missing their ACCs demonstrated a telling defect. When they stopped being rewarded for moving the joystick in a certain direction, they were still able (most of the time) to change direction, just like the normal monkeys. However, they were unable to _persist_ in this successful strategy and soon went back to moving the joystick in the direction that garnered no reward. They never learned how to consistently find the food, to turn a mistake into an enduring lesson. Because these monkeys couldn't update their cellular predictions, they ended up hopelessly confused by the simple experiment.\n\nPeople with a genetic mutation that reduces the number of dopamine receptors in the ACC suffer from a similar problem; just like the monkeys, they are less likely to learn from negative reinforcement. This seemingly minor deficit has powerful consequences. For example, studies have found that people carrying this mutation are significantly more likely to become addicted to drugs and alcohol. Because they have difficulty learning from their mistakes, they make the same mistakes over and over. They can't adjust their behavior even when it proves self-destructive.\n\nThe ACC has one last crucial feature, which further explains its importance: it is densely populated with a very rare type of cell known as a spindle neuron. Unlike the rest of the brain cells, which are generally short and bushy, these brain cells are long and slender. They are found only in humans and great apes, which suggests that their evolution was intertwined with higher cognition. Humans have about forty times more spindle cells than any other primate.\n\nThe strange form of spindle cells reveals their unique function: their antenna-like bodies are able to convey emotions across the entire brain. After the ACC receives input from a dopamine neuron, spindle cells use their cellular velocity \u2013 they transmit electrical signals faster than any other neuron \u2013 to make sure that the rest of the cortex is instantly saturated in that specific feeling. The consequence of this is that the minor fluctuations of a single type of neurotransmitter play a huge role in guiding our actions, telling us how we should feel about what we see. 'You're probably 99.9 per cent unaware of dopamine release,' says Read Montague, a professor of neuroscience at Baylor University. 'But you're probably 99.9 per cent driven by the information and emotions it conveys to other parts of the brain.'\n\n* * *\n\nWe can now begin to understand the surprising wisdom of our emotions. The activity of our dopamine neurons demonstrates that feelings aren't simply reflections of hard-wired animal instincts. Those wild horses aren't acting on a whim. Instead, human emotions are rooted in the predictions of highly flexible brain cells, which are constantly adjusting their connections to reflect reality. Every time you make a mistake or encounter something new, your brain cells are busy changing themselves. Our emotions are deeply empirical.\n\nLook, for example, at Schultz's experiment. When Schultz studied those juice-craving monkeys, he discovered that it took only a few experimental trials before the monkeys' neurons knew exactly when to expect their rewards. The neurons did this by continually incorporating the new information, turning a negative feeling into a teachable moment. If the juice didn't arrive, then the dopamine cells adjusted their expectations. Fool me once, shame on you. Fool me twice, shame on my dopamine neurons.\n\nThe same process is constantly at work in the human mind. Motion sickness is largely the result of a dopamine prediction error: there is a conflict between the type of motion being experienced \u2013 for instance, the unfamiliar pitch of a boat \u2013 and the type of motion _expected_ (solid, unmoving ground). The result in this case is nausea and vomiting. But it doesn't take long before the dopamine neurons start to revise their models of motion; this is why seasickness is usually temporary. After a few horrible hours, the dopamine neurons fix their predictions and learn to expect the gentle rocking of the high seas.\n\nWhen the dopamine system breaks down completely \u2013 when neurons are unable to revise their expectations in light of reality \u2013 mental illness can result. The roots of schizophrenia remain shrouded in mystery, but one of its causes seems to be an excess of certain types of dopamine receptors. This makes the dopamine system hyperactive and disregulated, which means that the neurons of a schizophrenic are unable to make cogent predictions or correlate their firing with outside events. (Most antipsychotic medications work by reducing the activity of dopamine neurons.) Because schizophrenics cannot detect the patterns that actually exist, they start hallucinating false patterns. This is why schizophrenics become paranoid and experience completely unpredictable shifts in mood. Their emotions have been uncoupled from the events of the real world.\n\nThe crippling symptoms of schizophrenia serve to highlight the necessity and precision of dopamine neurons. When these neurons are working properly, they are a crucial source of wisdom. The emotional brain effortlessly figures out what's going on and how to exploit the situation for maximum gain. Every time you experience a feeling of joy or disappointment, fear or happiness, your neurons are busy rewiring themselves, constructing a theory of what sensory cues preceded the emotions. The lesson is then committed to memory, so the next time you make a decision, your brain cells are ready. They have learned how to predict what will happen next.\n\n## _The Sweaty Hand_\n\nBackgammon is the oldest board game in the world. It was first played in ancient Mesopotamia, starting around 3,000 BC. The game was a popular diversion in ancient Rome, celebrated by the Persians, and banned by King Louis IX of France for encouraging illicit gambling. In the seventeenth century, Elizabethan courtiers codified the rules of backgammon, and the game has changed little since.\n\nThe same can't be said about the _players_ of the game. One of the best backgammon players in the world is now a software program. In the early 1990s, Gerald Tesauro, a computer programmer at IBM, began developing a new kind of artificial intelligence (AI). At the time, most AI programs relied on the brute computational power of microchips. This was the approach used by Deep Blue, the powerful set of IBM mainframes that managed to defeat chess grand master Garry Kasparov in 1997. Deep Blue was capable of analyzing more than 200 million possible chess moves per second, allowing it to consistently select the optimal chess strategy. (Kasparov's brain, on the other hand, evaluated only about five moves per second.) But all this strategic firepower consumed a lot of energy: while playing chess, Deep Blue was a fire hazard and required specialized heat-dissipating equipment so that it didn't burst into flames. Kasparov, meanwhile, barely broke a sweat. That's because the human brain is a model of efficiency: even when it's deep in thought, the cortex consumes less energy than a light bulb.\n\nWhile the popular press was celebrating Deep Blue's stunning achievement \u2013 a machine had outwitted the greatest chess player in the world! \u2013 Tesauro was puzzled by its limitations. Here was a machine capable of thinking millions of times faster than its human opponent, and yet it had barely won the match. Tesauro realized that the problem with all conventional AI programs, even brilliant ones like Deep Blue's, was their _rigidity_. Most of Deep Blue's intelligence was derived from other chess grand masters, whose wisdom was painstakingly programmed into the machine. (IBM programmers also studied Kasparov's previous chess matches and engineered the software to exploit his recurring strategic mistakes.) But the machine itself was incapable of learning. Instead, it made decisions by predicting the probable outcomes of several million different chess moves. The move with the highest predicted 'value' was what the computer ended up executing. For Deep Blue, the game of chess was just an endless series of math problems.\n\nOf course, this sort of artificial intelligence isn't an accurate model of human cognition. Kasparov managed to compete on the same level as Deep Blue even though his mind had far less computational power. Tesauro's surprising insight was that Kasparov's neurons were effective because they had trained themselves. They had been refined by decades of experience to detect subtle spatial patterns on the chessboard. Unlike Deep Blue, which analyzed _every_ possible move, Kasparov was able to instantly winnow his options and focus his mental energies on evaluating only the most useful strategic alternatives.\n\nTesauro set out to create an AI program that acted like Garry Kasparov. He chose backgammon as his paradigm and named the program TD-Gammon. (The _TD_ stands for _temporal_ _difference_.) Deep Blue had been preprogrammed with chess acumen, but Tesauro's software began with absolutely zero knowledge. At first, its backgammon moves were entirely random. It lost every match and made stupid mistakes. But the computer didn't remain a novice for long; TD-Gammon was designed to learn from its own experience. Day and night, the software played backgammon against itself, patiently learning which moves were most effective. After a few hundred thousand games of backgammon, TD-Gammon was able to defeat the best human players in the world.\n\nHow did the machine turn itself into an expert? Although the mathematical details of Tesauro's software are numbingly complex, the basic approach is simple.* TD-Gammon generates a set of predictions about how the backgammon game will unfold. Unlike Deep Blue, the computer program doesn't investigate every possible permutation. Instead, it acts like Garry Kasparov and generates its predictions from its previous experiences. The software compares these predictions to what actually happens during the backgammon game. The ensuing discrepancies provide the substance of its education, and the software strives to continually decrease this 'error signal'. As a result, its predictions constantly increase in accuracy, which means that its strategic decisions get more and more effective and intelligent.\n\nIn recent years, the same software strategy has been used to solve all kinds of difficult problems, from programming banks of elevators in skyscrapers to determining the schedules of flights. 'Anytime you've got a problem with a seemingly infinite number of possibilities' \u2013 the elevators and planes can be arranged in any number of sequences \u2013 'these sorts of learning programs can be a crucial guide,' says Read Montague. The essential distinction between these reinforcement-learning programs and traditional approaches is that these new programs find the optimal solutions by themselves. Nobody tells the computer how to organize the elevators. Instead, it methodically learns by running trials and focusing on its errors until, after a certain number of trials, the elevators are running as efficiently as possible. The seemingly inevitable errors have disappeared.\n\nThis programming method closely mirrors the activity of dopamine neurons. The brain's cells also measure the mismatch between expectation and outcome. They use their inevitable errors to improve performance; failure is eventually turned into success. Take, for example, an experiment known as the Iowa Gambling Task designed by the neuroscientists Antonio Damasio and Antoine Bechara. The game went as follows: a subject \u2013 'the player' \u2013 was given four decks of cards, two black and two red, and $2,000 of play money. Each card told the player whether he'd won or lost money. The subject was instructed to turn over a card from one of the four decks and to make as much money as possible.\n\nBut the cards weren't distributed at random. The scientists had rigged the game. Two of the decks were full of high-risk cards. These decks had bigger payouts ($100), but also contained extravagant punishments ($1,250). The other two decks, by comparison, were staid and conservative. Although they had smaller payouts ($50), they rarely punished the player. If the gambler drew only from those two decks, he would come out way ahead.\n\nAt first, the card-selection process was entirely haphazard. There was no reason to favor any specific deck, and so most people sampled from each pile, searching for the most lucrative cards. On average, people had to turn over about fifty cards before they began to draw solely from the profitable decks. It took about eighty cards before the average experimental subject could explain _why_ he or she favored those decks. Logic is slow.\n\nBut Damasio wasn't interested in logic; he was interested in emotion. While the gamblers in the experiment were playing the card game, they were hooked up to a machine that measured the electrical conductance of their skin. In general, higher levels of conductance signal nervousness and anxiety. What the scientists found was that after a player had drawn only 10 cards, his hand got 'nervous' when it reached for the negative decks. Although the subject still had little inkling of which card piles were the most lucrative, his emotions had developed an accurate sense of fear. The emotions knew which decks were dangerous. The subject's feelings figured out the game first.\n\nNeurologically impaired patients who were unable to experience any emotions at all \u2013 usually because of damaged orbitofrontal cortices \u2013 proved incapable of selecting the right cards. While most people made substantial amounts of money during the experiment, these purely rational people often went bankrupt and had to take out 'loans' from the experimenter. Because these patients were unable to associate the bad decks with negative feelings \u2013 their hands never developed the symptoms of nervousness \u2013 they continued to draw equally from all four decks. When the mind is denied the emotional sting of losing, it never figures out how to win.\n\nHow do emotions become so accurate? How do they identify the lucrative decks so quickly? The answer returns us to dopamine, the molecular source of our feelings. By playing the Iowa Gambling Task with a person undergoing brain surgery for epilepsy \u2013 the patient was given local anesthesia but remained awake during the surgery \u2013 scientists at the University of Iowa and Caltech were able to watch the learning process unfold in real time. The scientists discovered that human brain cells are programmed just like TD-Gammon: they generate predictions about what will happen and then measure the difference between their expectations and the actual results. In the Iowa Gambling Task experiment, if a cellular prediction proved false \u2013 for example, if the player chose the bad deck \u2013 then the dopamine neurons immediately stopped firing. The player experienced a negative emotion and learned not to draw from that deck again. (Disappointment is educational.) However, if the prediction was accurate \u2013 if he got rewarded for choosing a lucrative card \u2013 then the player felt the pleasure of being correct, and that particular connection was reinforced. As a result, his neurons quickly learned how to make money. They had found the secret to winning the gambling game before the player could understand and explain the solution.\n\nThis is a crucial cognitive talent. Dopamine neurons automatically detect the subtle patterns that we would otherwise fail to notice; they assimilate all the data that we can't consciously comprehend. And then, once they come up with a set of refined predictions about how the world works, they translate these predictions into emotions. Let's say, for example, that you're given lots of information about how 20 different stocks have performed over a period of time. (The various share prices are displayed on a ticker tape at the bottom of a television screen, just as they appear on CNBC.) You'll soon discover that you have difficulty remembering all the financial data. If somebody asks you which stocks performed the best, you'll probably be unable to give a good answer. You can't process all the information. However, if you're asked which stocks trigger the best _feelings_ \u2013 your emotional brain is now being quizzed \u2013 you'll suddenly be able to identify the best stocks. According to Tilmann Betsch, the psychologist who performed this clever little experiment, your emotions will 'reveal a remarkable degree of sensitivity' to the actual performance of all of the different securities. The investments that rose in value will be associated with the most positive emotions, while the shares that went down in value will trigger a vague sense of unease. These wise yet inexplicable feelings are an essential part of the decision-making process. Even when we think we know nothing, our brains know something. That's what our feelings are trying to tell us.\n\n## Learning from Mistakes\n\nThis doesn't mean that people can coast on these cellular emotions. Dopamine neurons need to be continually trained and retrained, or else their predictive accuracy declines. Trusting one's emotions requires constant vigilance; intelligent intuition is the result of deliberate practice. What Cervantes said about proverbs \u2013 'They are short sentences drawn from long experience' \u2013 also applies to brain cells, but only if we use them properly.\n\nConsider Bill Robertie. He's one of the only people in the world who's a world-class expert in three different games. (Imagine if Bo Jackson had played in the NBA in addition to the NFL and baseball's major leagues.) Robertie is a chess master and a former winner of the US speed chess championship. He's a widely respected poker expert and best-selling author of several books on Texas hold'em. However, Robertie is best known for his backgammon skills. He has won the World Championship of Backgammon twice (a feat accomplished by only one other person), and is regularly ranked among the top 20 players in the world. In the early 1990s, when Gerald Tesauro was looking for a backgammon expert to compete against TD-Gammon, he chose Robertie. 'He wanted the computer to learn from the best,' Robertie says. 'And I was the best.'\n\nRobertie is now in his early sixties, with a shock of graying hair, lidded eyes and a pair of thick spectacles. He managed to turn a childhood obsession with chess into a lucrative career. When Robertie talks about games, he still speaks with the boyish enthusiasm of someone who can't quite believe that he gets to play for a living. 'The first time I competed against TD-Gammon I was incredibly impressed,' Robertie says. 'It represented a big improvement over any other computer program I'd ever encountered. But I knew that I was still a better player. The next year, however, was a different story. The computer was now a really formidable opponent. It had learned how to play from playing me.'\n\nThe software program became a backgammon expert by studying its prediction errors. After making a few million mistakes, TD-Gammon was able to join the shortlist of computers, like Deep Blue, that are able to compete with the best human opponents. However, all of these brilliant machines come with a strict limitation: they can each master only one game. TD-Gammon can't play chess, and Deep Blue can't play backgammon. No computer has been able to master poker.\n\nSo how did Robertie get so good at such different games? At first glance, chess, backgammon and poker seem to rely on very different cognitive skills. That's why most backgammon champions tend to play nothing but backgammon; most chess masters don't bother with card games; and most poker players couldn't tell a Latvian Gambit from a French Defense. And yet, Robertie manages to excel in all three domains. According to Robertie, his success has a simple explanation: 'I know how to practice,' he says. 'I know how to make myself better.'\n\nIn the early 1970s, when Robertie was still just a chess prodigy \u2013 he made a living by winning speed chess tournaments \u2013 he stumbled on backgammon. 'Right away, I fell in love with the game,' he says. 'Plus, there was a lot more money in backgammon than speed chess.' Robertie bought a book on backgammon strategy, memorized a few opening moves, and then started to play. And play. And play. 'You've got to get obsessed,' he says. 'You've got to reach the point where you're having dreams about the game.'\n\nAfter a few years of intense practice, Robertie had turned himself into one of the best backgammon players in the world. 'I knew I was getting good when I could just glance at a board and know what I should do,' Robertie says. 'The game started to become very much a matter of aesthetics. My decisions increasingly depended on the look of things, so that I could contemplate a move and then see right away if it made my position look better or worse. You know how an art critic can look at a painting and just know if it's a good painting? I was the same way, only my painting was the backgammon board.'\n\nBut Robertie didn't become a world champion just by playing a lot of backgammon. 'It's not the quantity of practice, it's the _quality_ ,' he says. According to Robertie, the most effective way to get better is to focus on your mistakes. In other words, you need to consciously consider the errors being internalized by your dopamine neurons. After Robertie plays a chess match, or a poker hand, or a backgammon game, he painstakingly reviews what happened. Every decision is critiqued and analyzed. Should he have sent out his queen sooner? Tried to bluff with a pair of sevens? What if he had consolidated his backgammon blots? Even when Robertie wins \u2013 and he almost always wins \u2013 he insists on searching for his errors, dissecting those decisions that could have been a little bit better. He knows that self-criticism is the secret to self-improvement; negative feedback is the best kind. 'That's one of the things I learned from TD-Gammon,' Robertie says. 'Here was a computer that did nothing but measure what it got wrong. That's all it did. And it was as good as me.'\n\nThe physicist Niels Bohr once defined an expert as 'a person who has made all the mistakes that can be made in a very narrow field.' From the perspective of the brain, Bohr was absolutely right. Expertise is simply the wisdom that emerges from cellular error. Mistakes aren't things to be discouraged. On the contrary, they should be cultivated and carefully investigated.\n\nCarol Dweck, a psychologist at Stanford, has spent decades demonstrating that one of the crucial ingredients of successful education is the ability to learn from mistakes. The same strategy that Robertie uses to excel at games is also an essential pedagogic tool. Unfortunately, children are often taught the exact opposite. Instead of praising kids for trying hard, teachers typically praise them for their innate intelligence (being smart). Dweck has shown that this type of encouragement actually backfires, since it leads students to see mistakes as signs of stupidity and not as the building blocks of knowledge. The regrettable outcome is that kids never learn how to learn.\n\nDweck's most famous study was conducted in 12 different New York City schools and involved more than four hundred fifth-graders. One at a time, the kids were removed from class and given a relatively easy test consisting of nonverbal puzzles. After the child finished the test, the researchers told the student his or her score and provided a single sentence of praise. Half of the kids were praised for their _intelligence_. 'You must be smart at this,' the researcher said. The other students were praised for their _effort_ : 'You must have worked really hard.'\n\nThe students were then allowed to choose between two different subsequent tests. The first choice was described as a more difficult set of puzzles, but the kids were told that they'd learn a lot from attempting it. The other option was an easy test, similar to the test they'd just taken.\n\nWhen Dweck was designing the experiment, she'd expected the different forms of praise to have a rather modest effect. After all, it was just one sentence. But it soon became clear that the type of compliment given to the fifth-graders dramatically influenced their choice of tests. Of the group of kids that had been praised for their efforts, 90 per cent chose the harder set of puzzles. However, of the kids that were praised for their intelligence, most went for the easier test. 'When we praise children for their intelligence,' Dweck wrote, 'we tell them that this is the name of the game: look smart, don't risk making mistakes.'\n\nDweck's next set of experiments showed how this fear of failure actually inhibited learning. She gave the same fifth-graders yet another test. This test was designed to be extremely difficult \u2013 it was originally written for eighth-graders \u2013 but Dweck wanted to see how the kids would respond to the challenge. The students who had been praised for their efforts in the initial test worked hard at figuring out the puzzles. 'They got very involved,' Dweck says. 'Many of them remarked, unprovoked, \"This is my favorite test.\"' Kids that had initially been praised for their smarts, on the other hand, were easily discouraged. Their inevitable mistakes were seen as signs of failure: perhaps they really weren't smart after all. After taking this difficult test, the two groups of students had to choose between looking at the exams of kids who did worse than them and looking at the exams of those who did better. Students praised for their intelligence almost always chose to bolster their self-esteem by comparing themselves with students who had performed worse on the test. In contrast, kids praised for their hard work were more interested in the higher-scoring exams. They wanted to understand their mistakes, to learn from their errors, to figure out how to do better.\n\nThe final round of tests was the same difficulty level as the initial test. Nevertheless, students who'd been praised for their efforts exhibited significant improvement, raising their average score by 30 per cent. Because these kids were willing to challenge themselves, even if it meant failing at first, they ended up performing at a much higher level. This result was even more impressive when compared with students who'd been randomly assigned to the 'smart' group; they saw their scores drop by an average of nearly 20 per cent. The experience of failure had been so discouraging for the 'smart' kids that they actually regressed.\n\nThe problem with praising kids for their innate intelligence \u2013 the 'smart' compliment \u2013 is that it misrepresents the neural reality of education. It encourages kids to avoid the most useful kind of learning activities, which is learning from mistakes. Unless you experience the unpleasant symptoms of being wrong, your brain will never revise its models. Before your neurons can succeed, they must repeatedly fail. There are no shortcuts for this painstaking process.\n\nThis insight doesn't apply only to fifth-graders solving puzzles; it applies to everyone. Over time, the brain's flexible cells become the source of expertise. Although we tend to think of experts as being weighed down by information, their intelligence dependent on a vast amount of explicit knowledge, experts are actually profoundly intuitive. When an expert evaluates a situation, he doesn't systematically compare all the available options or consciously analyze the relevant information. He doesn't rely on elaborate spreadsheets or long lists of pros and cons. Instead, the expert naturally depends on the emotions generated by his dopamine neurons. His prediction errors have been translated into useful knowledge, which allows him to tap into a set of accurate feelings he can't begin to explain.\n\nThe best experts embrace this intuitive style of thinking. Bill Robertie makes difficult backgammon decisions by just 'looking' at the board. Thanks to his rigorous practice techniques, he's confident that his mind has already internalized the ideal moves. Garry Kasparov, the chess grand master, obsessively studied his past matches, looking for the slightest imperfection, but when it came time to play a chess game, he said he played by instinct, 'by smell, by feel'. After Herb Stein finishes shooting a soap opera episode, he immediately goes home and reviews the rough cut. 'I watch the whole thing,' Stein says, 'and I just take notes. I'm looking really hard for my mistakes. I pretty much always want to find 30 mistakes, 30 things that I could have done better. If I can't find 30, then I'm not looking hard enough.' These mistakes are usually little things, so minor that nobody else would notice. But Stein knows that the only way to get it right the next time is to study what he got wrong this time. Tom Brady spends hours watching game tape every week, critically looking at each of his passing decisions, but when he's standing in the pocket he knows that he can't hesitate before making a throw. It's not an accident that all of these experts have converged on such a similar method. They have figured out how to take advantage of their mental machinery, to steal as much wisdom as possible from their inevitable errors.\n\nAnd then there's Lieutenant Commander Michael Riley. Before becoming an officer in the Royal Navy, Riley had spent years learning how to interpret the ambiguous blips on a radar screen. In the Royal Navy, the training process for such warfare specialists revolves around realistic battle simulations so that senior lieutenants like Riley can practice decision-making in its proper context. Officers are able to learn from their mistakes without having to shoot anything down.\n\nDuring the Persian Gulf War, all of this training paid off. Even though Riley had never seen a Silkworm missile before, his mind had learned how to detect it. Because he had been staring at a radar screen for weeks on end, watching dozens of A-6 jets return from sorties off the Kuwaiti coast, Riley's dopamine neurons started to anticipate a consistent sequence of events. The radar pattern of the American planes had been seared into his brain. But then, in the predawn hours following the ground invasion, Riley saw a radar blip that looked slightly different. When the incoming unidentified blip appeared, it was too far out to sea, three sweeps away from the coast. As a result, a dopamine neuron somewhere in Riley's midbrain was surprised. Here was something that didn't fit the pattern, an error of expectation. The cell instantly responded to the surprising turn of events and altered its rate of firing. This electrical message was passed from neuron to neuron until it reached the ACC. Spindle cells publicized this prediction error all over the brain. Riley's years of naval training were summarized in a single flash of fear. It was just a feeling, but Riley dared to trust it. 'Fire two Sea Darts!' he yelled. The defensive missiles were launched into the sky. The battleship was saved.\n\nSo far, we've been exploring the surprising intelligence of our emotions. We've seen how the fluctuations of dopamine are translated into a set of prophetic feelings. But emotions aren't perfect. They are a crucial cognitive tool, but even the most useful tools can't solve every problem. In fact, there are certain conditions that consistently short-circuit the emotional brain, causing people to make bad decisions. The best decision-makers know which situations require _less_ intuitive responses, and in the next part of the book, we'll look at what those situations are.\n\n* The TD-learning model used by Tesauro was based on the pioneering work of computer scientists Rich Sutton and Andrew Barto. In the early 1980s, when they were grad students at the University of Massachusetts, Amherst, Sutton and Barto tried to develop a model of artificial intelligence that could learn simple rules and behaviors and apply them to achieve a goal. It was an audacious idea; their academic advisers tried to dissuade them from even trying, but the young scientists were stubborn. 'It had always been this kind of untouchable goal in computer science,' Sutton says. 'Marvin Minsky had done his thesis on reinforcement learning and basically given up. He said it was impossible and left the field. Luckily for us, it wasn't impossible. We knew even simple animals could learn like this \u2013 nobody teaches a bird how to find a worm \u2013 we just didn't know how.' \nTHREE\n\n# Fooled by a Feeling\n\nANN KLINESTIVER WAS working as a high school English teacher in a small town in West Virginia when she was diagnosed with Parkinson's disease. She was only 52 years old, but the symptoms were unmistakable. While she was standing at the front of her class trying to teach her students some Shakespeare, her hands started to shake uncontrollably. Then her legs went limp. 'I lost control of my own body,' she says. 'I'd look at my arm, and I'd tell it what to do, but it just wouldn't listen.'\n\nParkinson's is a disease of the dopamine system. It begins when dopamine neurons start to die in a part of the brain that controls the body's movements. Nobody knows why these cells die, but once they are gone, the loss is irrevocable. By the time the symptoms of Parkinson's appear, more than 80 per cent of these neurons will be dead.\n\nAnn's neurologist immediately put her on Requip, a drug that imitates the activity of dopamine in the brain. (It's part of a class of drugs called dopamine agonists.) While there are many different treatments for Parkinson's patients, all operate on a similar principle: increase the amount of dopamine in the brain. By making the few surviving dopamine neurons more effective at transmitting dopamine, these medicines help compensate for the massive cell death. They allow a faint electrical signal to break through the ravages of the disease. 'At first, the drug was like a miracle,' Ann says. 'All my movement problems just disappeared.' Over time, however, Ann was forced to take higher and higher doses of Requip in order to quiet her tremors. 'You can feel your brain going,' she says. 'I became completely dependent on this drug just to get myself out of bed and put on my clothes. I needed it to live my life.'\n\nThat's when Ann discovered slot machines. It was an unlikely attraction. 'I'd never been interested in gambling,' Ann says. 'I'd always avoided casinos. My daddy was a Christian, and he raised me to believe that gambling was a sin, that it was something you were never supposed to do.' But after she started taking the dopamine agonist, Ann found the slots at her local dog-racing track completely irresistible. She started gambling as soon as the track opened, at 7 a.m., and kept playing the machines until three thirty the next morning, when the security guards kicked her out. 'Then I would go back home and gamble on the internet until I could get back to the real machines,' she says. 'I was able to keep that up for two or three days at a time.' After each of her gambling binges, Ann always swore to stay away. Sometimes, she was even able to stop gambling for a day or two. But then she'd find herself back at the racetrack, sitting in front of the slot machine, gambling away everything she had.\n\nAfter a year of addictive gambling, Ann had lost more than two hundred and fifty thousand dollars. She had exhausted her retirement savings and emptied her pension fund. 'Even when I had no money left, I still couldn't stop gambling,' she says. 'I was living on peanut butter, straight from the jar. I sold everything I could sell. My silverware, my clothes, my television, my car. I pawned my diamond ring. I knew I was destroying my life, but I just couldn't stop. There's no worse feeling than that.'\n\nAnn's husband eventually left her. He promised to return if she got control of her gambling habit, but Ann kept relapsing. He would find her at the track in the middle of the night, hunched in front of a slot machine, a bucket of coins in her lap and a bag of groceries on the floor. 'I was a shell of a person,' she says. 'I stole quarters from my grandkids. I lost everything that mattered.'\n\nIn 2006, Ann was finally taken off her dopamine agonist. Her movement problems came back, but the gambling compulsion immediately disappeared. 'I haven't gambled in 18 months,' she says, with more than a little pride in her voice. 'I still think about the slots, but the obsession isn't there. Without the drug, I don't need to play those damn machines. I'm free.'\n\nKlinestiver's sad story is disturbingly common. Medical studies suggest that as many as 13 per cent of patients taking dopamine agonists develop severe gambling compulsions. People with no history of gambling suddenly become addicts. While most of these people obsess over slot machines, others get hooked on internet poker or blackjack. They squander everything they have on odds that are stacked against them.*\n\nWhy does an excess of dopamine in a few neurons make games of chance so irresistible? The answer reveals a serious flaw in the human brain, which casinos have learned to exploit. Think how a slot machine works: you put in a coin and pull the lever. The reels start to whir. Pictures of cherries and diamonds and figure sevens fly by. Eventually, the machine settles on its verdict. Since slot machines are programmed to return only about 90 per cent of wagered money over the long term, chances are you lost money.\n\nNow think about the slot machine from the perspective of your dopamine neurons. The purpose of these cells is to predict future events. They always want to know what occurrences \u2013 a loud tone, a flashing light, and so forth \u2013 will precede the arrival of the juice. While you are playing the slots, putting quarter after quarter into the one-armed bandit, your neurons are struggling to decipher the patterns inside the machine. They want to understand the game, to decode the logic of luck, to find the circumstances that predict a payout. So far, you're acting just like a monkey trying to predict when his squirt of juice is going to arrive.\n\nBut here's the catch: while dopamine neurons get excited by predictable rewards \u2013 they increase their firing when the juice arrives _after_ the loud tone that heralded it \u2013 they get even more excited by surprising ones. According to Wolfram Schultz, such unpredictable rewards are typically three to four times more exciting, at least for dopamine neurons, than rewards that can be predicted in advance. (In other words, the best-tasting juice is the juice that was most unexpected.) The purpose of this dopamine surge is to make the brain pay attention to new, and potentially important, stimuli. Sometimes this cellular surprise can trigger negative feelings, such as fear, as happened to Lieutenant Commander Michael Riley. In the casino, however, the sudden burst of dopamine is intensely pleasurable, since it means that you've just won some money.\n\nMost of the time, the brain will eventually get over its astonishment. It'll figure out which events predict the reward, and the dopamine neurons will stop releasing so much of the neurotransmitter. The danger of slot machines, however, is that they are inherently unpredictable. Because they use random number generators, there are no patterns or algorithms to uncover. (There is only a stupid little microchip churning out arbitrary digits.) Even though the dopamine neurons try to make sense of the rewards \u2013 they want to know when to expect some coins in return for all those squandered quarters \u2013 they keep getting surprised.\n\nAt this point, the dopamine neurons should just surrender: the slot machine is a waste of mental energy. They should stop paying attention to the surprising rewards, because the appearance of the rewards will always be surprising. But this isn't what happens. Instead of getting bored by the haphazard payouts, the dopamine neurons become obsessed. When you pull the lever and get a reward, you experience a rush of pleasurable dopamine, precisely because the reward was so unexpected, because your brain cells had no idea what was about to happen. The clanging coins and flashing lights are like a surprise squirt of juice. Because the dopamine neurons can't figure out the pattern, they can't adapt to the pattern. The result is that you are transfixed by the slot machine, riveted by the fickle nature of its payouts.\n\nFor Parkinson's patients on dopamine agonists, the surprising rewards of the casino trigger a massive release of chemical bliss. Their surviving dopamine neurons are so full of dopamine that the neurotransmitter spills over and pools in the empty spaces between cells. The brain is flooded with a feel-good chemical, making these games of chance excessively seductive. Such patients are so blinded by the pleasures of winning that they slowly lose everything. That's what happened to Ann.\n\nThe same science that revealed the importance of emotions to making decisions \u2013 Tom Brady finds the open man by listening to his feelings \u2013 is also beginning to show us the dark side of feeling too deeply. While the emotional brain is capable of astonishing wisdom, it's also vulnerable to certain innate flaws. These are the situations that cause the horses in the human mind to run wild, so that people gamble on slot machines and pick the wrong stocks and run up excessive credit card bills. When emotions get out of control \u2013 and there are certain things that reliably make this happen \u2013 the results can be just as devastating as not having any emotions at all.\n\n##  _The Illusion of Streaks_\n\nIn the early 1980s, the Philadelphia 76ers were one of the greatest teams in NBA history. The center of the team was Moses Malone, voted Most Valuable Player in the league. He dominated the paint, averaging 25 points and 15 rebounds per game. The power forward was Julius Erving, a future Hall of Famer, who pioneered the modern style of basketball play with his elegant drives and extravagant slam dunks. In the backcourt were Andrew Toney \u2013 his accurate jump shot was a constant offensive threat \u2013 and Maurice Cheeks, one of the league leaders in assists and steals.\n\nThe 76ers entered the 1982 playoffs with the best record in the NBA. Before the first round of the postseason, a reporter asked Malone what the 76ers thought of their competition. His answer made headlines: 'Four, four, four,' he said, suggesting that the team would sweep all of their opponents. That had never been done before.\n\nMalone's audacious prediction wasn't far off. During the playoffs, the 76ers team was like a scoring machine. The offense ran through Malone in the post, but if Malone was double-teamed he simply had to swing the ball over to Erving or kick it out to Toney for a jumper. At times, the players seemed to be incapable of missing shots. On their way to the championship, the 76ers lost one game only, in the second round to Milwaukee. A slightly amended version of Malone's prediction was inscribed on the championship rings: 'Fo, five, fo.' It was one of the most dominant team performances in basketball history.\n\nWhile the 76ers were prevailing in the postseason, the psychologists Amos Tversky and Thomas Gilovich were thinking about the imperfections of the human mind. Tversky would later recall watching the NBA games and hearing the television announcers talk about various kinds of streaks. For instance, the sportscasters alluded to the 'hot hand' of Julius Erving and said that Andrew Toney was 'in the zone'. By the time the 76ers reached the NBA finals, the temperature of the team had become a clich\u00e9. How could they possibly lose when they were on such a roll?\n\nBut all this talk of hot hands and streaks made Tversky and Gilovich curious. Had Moses Malone really become so unstoppable? Could Andrew Toney really not miss a shot? Were the 76ers really as invincible as everyone said? So Tversky and Gilovich decided to conduct a little research experiment. Their question was simple: do players make more shots when they are hot, or do people just _imagine_ that they make more shots? In other words, is the hot hand a real phenomenon?\n\nTversky and Gilovich began the investigation by sifting through years of 76ers statistics. They looked at every single shot taken by every single player and then recorded if that shot had been preceded by a string of hits or misses. (The 76ers were one of the few NBA teams that kept track of the order in which shots were taken.) If the hot hand was a real phenomenon, then a hot player should have a higher field-goal percentage after making several previous shots. The streak should elevate his game.\n\nSo what did the scientists find? There was absolutely no evidence of the hot hand. A player's chance of making a shot was not affected by whether or not his previous shots had gone in. Each field-goal attempt was its own independent event. The short runs experienced by the 76ers were no different than the short runs that naturally emerge from any random process. Taking a jumper was like flipping a coin. The streaks were a figment of the imagination.\n\nThe 76ers were shocked by the evidence. Andrew Toney, the shooting guard, was particularly hard to convince: he was sure that he was a streaky shooter who went through distinct hot and cold periods. But the statistics told a different story. During the regular season, Toney made 46 per cent of all his shots. After hitting three shots in a row \u2013 a sure sign that he was 'in the zone' \u2013 Toney's field-goal percentage _dropped_ to 34 per cent. When Toney thought he was hot, he was actually freezing cold. And when he thought he was cold, he was just getting warmed up: after missing three shots in a row, Toney made 52 per cent of his shots, which was significantly higher than his normal average.\n\nBut maybe the 76ers team was a statistical outlier. After all, according to a survey conducted by the scientists, 91 per cent of serious NBA fans believed in the hot hand. They just _knew_ that players were streaky. So Tversky and Gilovich decided to analyze another basketball team: the Boston Celtics. This time, they looked at free-throw attempts too, not just field goals. Once again, they found absolutely no evidence of hot hands. Larry Bird was just like Andrew Toney: after he made several free throws in a row, his free-throw percentage actually declined. Bird got complacent and started missing shots he should have made.\n\nWhy do we believe in streaky shooters? Our dopamine neurons are to blame. Although these cells are immensely useful \u2013 they help us predict events that are actually predictable \u2013 they can also lead us astray, especially when we are confronted with randomness. Look, for example, at this elegant little experiment: a rat was put in a T-shaped maze with a few morsels of food placed on either the far right or the far left side of the enclosure. The placement of the food was random, but the dice were rigged: over the long run, the food was placed on the left side 60 per cent of the time. How did the rat respond? It quickly realized that the left side was more rewarding. As a result, it always went to the left of the maze, which resulted in a 60 per cent success rate. The rat didn't strive for perfection. It didn't search for a unified theory of the T-shaped maze. It just accepted the inherent uncertainty of the reward and learned to settle for the option that usually gave the best outcome.\n\nThe experiment was repeated with Yale undergraduates. Unlike the rat, the students, with their elaborate networks of dopamine neurons, stubbornly searched for the elusive pattern that determined the placement of the reward. They made predictions and then tried to learn from their prediction errors. The problem was that there was nothing to predict; the apparent randomness was real. Because the students refused to settle for a 60 per cent success rate, they ended up with a 52 per cent success rate. Although most of the students were convinced that they were making progress toward identifying the underlying algorithm, they were, in actuality, outsmarted by a rat.\n\nThe danger of random processes \u2013 things like slot machines and basketball shots \u2013 is that they take advantage of a defect built into the emotional brain. Dopamine neurons get such a visceral thrill from watching a hot player sink another jumper or from winning a little change from a one-armed bandit or from correctly guessing the placement of a food morsel, our brains completely misinterpret what's actually going on. We trust our feelings and perceive patterns, but the patterns don't actually exist.\n\nOf course, it can be extremely hard to reconcile perceptions of streaks and runs with the statistical realities of an unruly world. When Apple first introduced the shuffle feature on its iPods, the shuffle was truly random; each song was equally as likely to get picked as any other. However, the randomness didn't _appear_ random, since some songs were occasionally repeated, and customers concluded that the feature contained some secret patterns and preferences. As a result, Apple was forced to revise the algorithm. 'We made it less random to make it feel more random,' said Steve Jobs, the CEO of Apple.* Or consider Red Auerbach, the legendary Celtics coach. After being told about Tversky's statistical analysis of the hot hand, he reportedly responded with a blunt dismissal. 'So he makes a study,' Auerbach said. 'I couldn't care less.'\u2020\n\nThe coach refused to consider the possibility that the shooting streaks of the players might be a fanciful invention of his brain. But Auerbach was wrong to disregard the study; the belief in illusory patterns seriously affects the flow of basketball games. If a team member had made several shots in a row, he was more likely to get the ball passed to him. The head coach would call a new set of plays. Most important, a player who thinks he has a hot hand has a distorted sense of his own talent, which leads him to take riskier shots, since he assumes his streak will save him. (It's the old bane of overconfidence.) Of course, the player is also more likely to miss these riskier shots. According to Tversky and Gilovich, the best shooters always think they're cold. When their feelings tell them to take the shots because they've got the hot hands, they don't listen.\n\nThis defect in the emotional brain has important consequences. Think about the stock market, which is a classic example of a random system. This means that the past movement of any particular stock cannot be used to predict its future movement. The inherent randomness of the market was first proposed by the economist Eugene Fama in the early 1960s. Fama looked at decades of stock-market data in order to prove that no amount of knowledge or rational analysis could help anyone figure out what would happen next. All of the esoteric tools used by investors to make sense of the market were pure nonsense. Wall Street was like a slot machine.\n\nThe danger of the stock market, however, is that sometimes its erratic fluctuations can actually look predictable, at least in the short term. Dopamine neurons are determined to solve the flux, but most of the time there is nothing to solve. And so brain cells flail against the stochasticity, searching for lucrative patterns. Instead of seeing the randomness, we come up with imagined systems and see meaningful trends where there are only meaningless streaks. 'People enjoy investing in the market and gambling in a casino for the same reason that they see Snoopy in the clouds,' says the neuroscientist Read Montague. 'When the brain is exposed to anything random, like a slot machine or the shape of a cloud, it automatically imposes a pattern onto the noise. But that isn't Snoopy, and you haven't found the secret pattern in the stock market.'\n\nOne of Montague's recent experiments demonstrated how an unrestrained dopamine system can, over time, lead to dangerous stock-market bubbles. The brain is so eager to maximize rewards that it ends up pushing its owner off a cliff. The experiment went like this: subjects were each given a $100 and some basic information about the 'current' state of the stock market. Then the players chose how much of their money to invest and nervously watched as their stock investments either rose or fell in value. The game continued for twenty rounds, and the subjects got to keep their earnings. One interesting twist was that instead of using random simulations of the stock market, Montague relied on distillations of data from history's famous markets. Montague had people 'play' the Dow of 1929, the Nasdaq of 1998, the Nikkei of 1986 and the S&P 500 of 1987. This let the scientists monitor the neural responses of investors during what had once been real-life bubbles and crashes.\n\nHow did the brain deal with the fluctuations of Wall Street? The scientists immediately discovered a strong neural signal that seemed to be driving many of the investment decisions. This signal emanated from dopamine-rich areas of the brain, such as the ventral caudate, and it was encoding fictive-error learning, or the ability to learn from what-if scenarios. Take, for example, this situation: a player has decided to wager 10 per cent of his total portfolio in the market, which is a rather small bet. Then he watches as the market rises dramatically in value. At this point, the fictive-error learning signal starts to appear. While he enjoys his profits, his ungrateful dopamine neurons are fixated on the profits he _missed_ , as the cells compute the difference between the best possible return and the actual return. (This is a modified version of the prediction-error signal discussed earlier.) When there is a big difference between what actually happened and what might have happened \u2013 which is experienced as a feeling of regret \u2013 the player, Montague found, is more likely to do things differently the next time around. As a result, investors in the experiment adapted their investments to the ebb and flow of the market. When markets were booming, as they were in the Nasdaq bubble of the late 1990s, investors kept increasing their investments. Not to invest was to drown in regret, to bemoan all the money that might have been earned if they'd only made better decisions.\n\nBut fictive-error learning isn't always adaptive. Montague argues that these computational signals are also a main cause of financial bubbles. When the market keeps going up, people are led to make larger and larger investments in the boom. Their greedy brains are convinced that they've solved the stock market, and so they don't think about the possibility of losses. But just when investors are most convinced that the bubble isn't a bubble \u2013 many of Montague's subjects eventually put all of their money into the booming market \u2013 the bubble bursts. The Dow sinks, the Nasdaq implodes, the Nikkei collapses. All of a sudden, the same investors who'd regretted _not_ fully investing in the market and had subsequently invested more were now despairing of their plummeting net worth. 'You get the exact opposite effect when the market heads down,' Montague says. 'People just can't wait to get out, because the brain doesn't want to regret staying in.' At this point, the brain realizes that it's made some very expensive prediction errors, and the investor races to dump any assets that are declining in value. That's when you get a financial panic.\n\nThe lesson here is that it's silly to try to beat the market with your brain. Dopamine neurons weren't designed to deal with the random oscillations of Wall Street. When you spend lots of money on investment-management fees, or sink your savings into the latest hot mutual fund, or pursue unrealistic growth goals, you are slavishly following your primitive reward circuits. Unfortunately, the same circuits that are so good at tracking juice rewards and radar blips will fail completely in these utterly unpredictable situations. That's why, over the long run, a randomly selected stock portfolio will beat the expensive experts with their fancy computer models. And why the vast majority of mutual funds in any given year will _underperform_ the S&P 500. Even those funds that do manage to beat the market rarely do so for long. Their models work haphazardly; their successes are inconsistent. Since the market is a random walk with an upward slope, the best solution is to pick a low-cost index fund and wait. Patiently. Don't fixate on what might have been or obsess over someone else's profits. The investor who does nothing to his stock portfolio \u2013 who doesn't buy or sell a single stock \u2013 outperforms the average 'active' investor by nearly 10 per cent. Wall Street has always searched for the secret algorithm of financial success, but the secret is, there is no secret. The world is more random than we can imagine. That's what our emotions can't understand.\n\n##  _The Pain of Losing_\n\n _Deal or No Deal_ is one of the most popular television game shows of all time. The show has been broadcast in more than forty-five different countries, from Great Britain to Slovakia to America. The rules of the game couldn't be simpler: a contestant is confronted with 26 sealed briefcases each full of varying amounts of cash, from a penny to a million dollars. Without knowing the amount of money in any of the briefcases, the contestant chooses a single one, which is then placed in a lockbox. Its contents won't be revealed until the game is over.\n\nThe player then proceeds to open the remaining 25 briefcases one at a time. As the various monetary amounts are revealed, the contestant gradually gets an idea of how much money his or her own briefcase might contain, since all the remaining amounts are displayed on a large screen. It's a nerve-racking process of elimination, as each player tries to keep as many of the big monetary sums on the board for as long as possible. Every few rounds, a shadowy figure known as the Banker makes the player an offer for the sealed briefcase. The contestant can either accept the deal and cash out or continue to play, gambling that the unopened briefcase contains more money than the Banker has offered. As the rounds continue, the tension becomes excruciating. Spouses start crying, and children begin screaming. If the wrong briefcase is picked, or the best deal is rejected, a staggering amount of money can evaporate, just like that.\n\nFor the most part, _Deal or No Deal_ is a game of dumb luck. Although players develop elaborate superstitions about the briefcases \u2013 odd numbers are better; even numbers are better; ones held by blond models are better \u2013 the monetary amounts in them are randomly distributed. There is no code to crack, no numerology to decipher. This is just fate unfolding in front of a national television audience.\n\nAnd yet, _Deal or No Deal_ is also a game of difficult decisions. After the Banker makes an offer, the contestant has a few minutes \u2013 usually the length of a commercial break \u2013 to make up his mind. He must weigh the prospect of sure money against the chances of winning one of the larger cash prizes. It's almost always a hard call, a moment full of telegenic anxiety.\n\nThere are two ways to make this decision. If the contestant had a calculator handy, he could quickly compare the average amount of money he might expect to win against the Banker's offer. For example, if there were three remaining briefcases, one containing one dollar, one containing $10,000 and one containing $500,000, then the player should, at least in theory, accept any offer over $170,000, since that is the average of the money in all three briefcases. Although offers in the early rounds are generally unfairly low \u2013 the producers don't want people to quit before it gets dramatic \u2013 as the game goes on, the offers made by the Banker become more and more reasonable, until they are essentially asymptotic with the mathematical average of the money still available. In this sense, it is extremely easy for a contestant on _Deal or No Deal_ to determine whether or not to accept an offer. He just needs to add up all the remaining monetary amounts, divide that number by the number of briefcases left and see if that figure exceeds the offer on the table. If _Deal or No Deal_ were played like this, it would be a thoroughly rational game. It would also be extremely boring. It's not fun to watch people do arithmetic.\n\nThe game show is entertaining only because the vast majority of contestants don't make decisions based on the math. Take Nondumiso Sainsbury, a typical _Deal or No Deal_ contestant. She is a pretty young woman from South Africa who met her husband while she was studying in America. She plans on sending her winnings back home to her poor family in Johannesburg, where her three younger brothers live in a shantytown with her mother. It's hard not to root for her to make the right decision.\n\nNondumiso starts off rather well. After a few rounds, she still has two big amounts \u2013 $500,000 and $400,000 \u2013 left in play. As is usual for this stage of the game, the Banker makes her a blatantly unfair offer. Although the average amount of money left is $185,000, Nondumiso is offered less than half that. The producers clearly want her to keep playing.\n\nAfter quickly consulting with her husband \u2013 'We still might win half a million dollars!' she shouts \u2013 Nondumiso wisely rejects the offer. The suspense builds as she prepares to pick her next briefcase. She randomly chooses a number and winces as the briefcase is slowly opened. Every second of tension is artfully mined. Nondumiso's luck has held: the briefcase contains only $300. The Banker now increases his offer to $143,000, or 75 per cent of a perfectly fair offer.\n\nAfter just a few seconds of deliberation, Nondumiso decides to reject the deal. Once again, the pressure builds as a briefcase is opened. The audience collectively gasps. Once again, Nondumiso has gotten lucky: she has managed to avoid eliminating either of the two big remaining sums of money. She now has a 67 per cent chance of winning more than $400,000. Of course, she also has a 33 per cent chance of winning $100.\n\nFor the first time, the Banker's offer is essentially fair: he is willing to 'buy' Nondumiso's sealed briefcase for $286,000. As soon as she hears the number, she breaks into a huge smile and starts to cry. Without even pausing to contemplate the math, Nondumiso begins chanting, 'Deal! Deal! I want a deal!' Her loved ones swarm the stage. The host tries to ask Nondumiso a few questions, and she struggles to speak through the tears.\n\nIn many respects, Nondumiso made an excellent set of decisions. A computer that meticulously analyzed the data couldn't have done much better. But it's important to note _how_ Nondumiso arrived at these decisions. She never took out a calculator or estimated the average amount of money remaining in the briefcases. She never scrutinized her options or contemplated what would happen if she eliminated one of the larger amounts of money. (In that case, the offer probably would have been cut by at least 50 per cent.) Instead, her risky choices were entirely impulsive; she trusted her feelings to not lead her astray.\n\nWhile this instinctive decision-making strategy normally works out just fine \u2013 Nondumiso's feelings made her rich \u2013 there are certain situations on the game show that reliably fool the emotional brain. In these cases, contestants end up making terrible choices, rejecting deals that they should accept. They lose fortunes because they trust their emotions at the wrong moment.\n\nLook at poor Frank, a contestant on the Dutch version of _Deal or No Deal_. He gets off to an unlucky start by immediately eliminating some of the most lucrative briefcases. After six rounds, Frank has only one valuable briefcase left, worth \u20ac500,000. The Banker offers him \u20ac102,006, about 75 per cent of a perfectly fair offer. Frank decides to reject the deal. He's gambling that the next briefcase he picks won't contain the last big monetary amount, thus driving up the offer from the Banker. So far, his emotions are acting in accordance with the arithmetic. They are holding out for a better deal.\n\nBut Frank makes a bad choice, eliminating the one briefcase he wanted to keep in play. He braces himself for the bad news from the Banker, who now offers Frank a deal for \u20ac2,508, or about one hundred thousand euros less than he was offered 30 seconds before. The irony is that this offer is utterly fair; Frank would be wise to cut his losses and accept the Banker's proposal. But Frank immediately rejects the deal; he doesn't even pause to consider it. After another unlucky round, the Banker takes pity on Frank and makes him an offer that's about 110 per cent of the average of the possible prizes. (Tragedy doesn't make good game-show TV, and the producers are often quite generous in such situations.) But Frank doesn't want pity, and he rejects the offer. After eliminating a briefcase containing one euro \u2013 Frank's luck is finally starting to turn \u2013 he is now faced with a final decision. Only two briefcases remain: \u20ac10 and \u20ac10,000. The Banker offers him \u20ac6,500, which is a 30 per cent premium over the average of the money remaining. But Frank spurns this final proposal. He decides to open his own briefcase, in the desperate hope that it contains the bigger amount. Frank has bet wrong: it contains only \u20ac10. In fewer than three minutes, Frank has lost more than one hundred thousand euros.\n\nFrank isn't the only contestant to make this type of mistake. An exhaustive analysis by a team of behavioral economists led by Thierry Post concluded that most contestants in Frank's situation act the exact same way. (As the researchers note, _Deal or No Deal_ has 'such desirable features that it almost appears to be designed to be an economics experiment rather than a TV show'.) After the Banker's offer decreases by a large amount \u2013 this is what happened after Frank opened the \u20ac500,000 briefcase \u2013 a player typically becomes excessively risk-seeking, which means he is much more likely to reject perfectly fair offers. The contestant is so upset by the recent monetary loss that he can't think straight. And so he keeps on opening briefcases, digging himself deeper and deeper into a hole.\n\nThese contestants are victims of a very simple flaw rooted in the emotional brain. Alas, this defect isn't limited to greedy game-show contestants, and the same feelings that caused Frank to reject the fair offers can lead even the most rational people to make utterly foolish choices. Consider this scenario:\n\nThe United States is preparing for the outbreak of an unusual Asian disease, which is expected to kill 600 people. Two different programs to combat the disease have been proposed. Assume that the exact scientific estimates of the consequences of the programs are as follows: if program A is adopted, 200 people will be saved. If program B is adopted, there is a one-third probability that 600 people will be saved and a two-thirds probability that no people will be saved. Which of the two programs would you favor?\n\nWhen this question was put to a large sample of physicians, 72 per cent chose option A, the safe-and-sure strategy, and only 28 per cent chose program B, the risky strategy. In other words, physicians would rather save a certain number of people for sure than risk the possibility that everyone might die. But consider this scenario:\n\nThe United States is preparing for the outbreak of an unusual Asian disease, which is expected to kill 600 people. Two different programs to combat the disease have been proposed. Assume that the exact scientific estimates of the consequences of the programs are as follows: If program C is adopted, 400 people will die. If program D is adopted, there is a one-third probability that nobody will die and a two-thirds probability that 600 people will die. Which of the two programs would you favor?\n\nWhen the scenario was described in terms of deaths instead of survivors, physicians reversed their previous decisions. Only 22 per cent voted for option C, while 78 per cent chose option D, the risky strategy. Most doctors were now acting just like Frank: they were rejecting a guaranteed gain in order to participate in a questionable gamble.\n\nOf course, this is a ridiculous shift in preference. The two different questions examine identical dilemmas; saving one-third of the population is the same as losing two-thirds. And yet doctors reacted very differently depending on how the question was framed. When the possible outcomes were stated in terms of deaths \u2013 this is called the _loss frame_ \u2013 physicians were suddenly eager to take chances. They were so determined to avoid any option associated with loss that they were willing to risk losing everything.\n\nThis mental defect \u2013 it's technical name is _loss aversion_ \u2013 was first demonstrated in the late 1970s by Daniel Kahneman and Amos Tversky. At the time, they were both psychologists at Hebrew University, best known on campus for talking to each other too loudly in their shared office. But these conversations weren't idle chatter; Kahneman and Tversky (or 'Kahnemanandtversky,' as they were later known) did their best science while talking. Their disarmingly simple experiments \u2013 all they did was ask each other hypothetical questions \u2013 helped to illuminate many of the brain's hard-wired defects. According to Kahneman and Tversky, when a person is confronted with an uncertain situation \u2013 like having to decide whether to accept an offer from the Banker \u2013 the individual doesn't carefully evaluate the information, or compute the Bayesian probabilities, or do much thinking at all. Instead, the decision depends on a brief list of emotions, instincts and mental shortcuts. These shortcuts aren't a faster way of doing the math; they're a way of skipping the math altogether.\n\nKahneman and Tversky stumbled upon the concept of loss aversion after giving their students a simple survey that asked if they would accept various bets. The psychologists noticed that when a person was offered a gamble on the toss of a coin and was told that losing would cost him $20, the player demanded, on average, around forty dollars for winning. The pain of a loss was approximately twice as potent as the pleasure generated by a gain. Furthermore, decisions seemed to be determined by these feelings. As Kahneman and Tversky put it, 'In human decision making, _losses loom larger than_ _gains_.'\n\nLoss aversion is now recognized as a powerful mental habit with widespread implications. The desire to avoid anything that smacks of loss often shapes our behavior, leading us to do foolish things. Look, for example, at the stock market. Economists have long been perplexed by a phenomenon known as the premium equity puzzle. The puzzle itself is easy to explain: over the last century, stocks have outperformed bonds by a surprisingly large margin. Since 1926, the annual return on stocks after inflation has been 6.4 per cent, while the return on Treasury bills has been less than 0.5 per cent. When the Stanford economists John Shoven and Thomas MaCurdy compared randomly generated financial portfolios composed of either stocks or bonds, they discovered that, over the long term, stock portfolios _always_ generated higher returns than bond portfolios. In fact, stocks typically earned more than seven times as much as bonds. MaCurdy and Shoven concluded that people who invest in bonds must be 'confused about the relative safety of different investments over long horizons.' In other words, investors are just as irrational as game-show contestants. They, too, have a distorted sense of risk.\n\nClassical economic theory can't explain the premium equity puzzle. After all, if investors are such rational agents, why don't all of them invest in stocks? Why are low-yield bonds so popular? In 1995, the behavioral economists Richard Thaler and Shlomo Benartzi realized that the key to solving the premium equity puzzle was loss aversion. Investors buy bonds because they hate losing money, and bonds are a safe bet. Instead of making financial decisions that reflect all the relevant statistical information, they depend on their emotional instincts and seek the certain safety of bonds. These are well-intentioned instincts \u2013 they prevent people from gambling away their retirement savings \u2013 but they are also misguided. The fear of losses makes investors more willing to accept a measly rate of return.\n\nEven experts are vulnerable to these irrational feelings. Take Harry Markowitz, a Nobel Prize-winning economist who practically invented the field of investment-portfolio theory. In the early 1950s, while working at the RAND Corporation, Markowitz became intrigued by a practical financial question: how much of his savings should he invest in the stock market? Markowitz derived a complicated mathematical equation that could be used to calculate the optimal mix of assets. He had come up with a rational solution to the old problem of risk versus reward.\n\nBut Markowitz couldn't bring himself to use his own equation. When he divided up his investment portfolio, he ignored the investment advice that had won him the Nobel Prize; instead of relying on the math, he fell into the familiar trap of loss aversion and split his portfolio equally between stocks and bonds. Markowitz was so worried about the possibility of losing his savings that he failed to optimize his own retirement account.\n\nLoss aversion also explains one of the most common investing mistakes: investors evaluating their stock portfolios are most likely to sell stocks that have _increased_ in value. Unfortunately, this means that they end up holding on to their depreciating stocks. Over the long term, this strategy is exceedingly foolish, since ultimately it leads to a portfolio composed entirely of shares that are losing money. (A study by Terrance Odean, an economist at UC Berkeley, found that the stocks investors sold outperformed the stocks they didn't sell by 3.4 per cent.) Even professional money managers are vulnerable to this bias and tend to hold losing stocks twice as long as winning stocks. Why does an investor do this? Because he is afraid to take a loss \u2013 it feels bad \u2013 and selling shares that have decreased in value makes the loss tangible. We try to postpone the pain for as long as possible; the result is more losses.\n\nThe only people who are immune to this mistake are neurologically impaired patients who can't feel any emotion at all. In most situations, these people have very damaged decision-making abilities. And yet, because they don't feel the extra sting of loss, they are able to avoid the costly emotional errors brought on by loss aversion.\n\nConsider this experiment, led by Antonio Damasio and George Loewenstein. The scientists invented a simple investing game. In each round, the experimental subject had to decide between two options: invest one dollar or invest nothing. If the participant decided not to invest, he kept the dollar, and the game advanced to the next round. If the participant decided to invest, he would hand a dollar bill to the experimenter, who would then toss a coin. Heads meant that the participant lost the one dollar that was invested; tails meant that $2.50 was added to the participant's account. The game stopped after 20 rounds.\n\nIf people were perfectly rational \u2013 if they made decisions solely by crunching the numbers \u2013 then subjects would always choose to invest, since the expected overall value on each round is higher if one invests ($1.25, or $2.50 multiplied by the 50 per cent chance of getting tails on the coin toss) than if one does not ($1). In fact, if a person invests on each and every round, there is a mere 13 per cent chance that he'll wind up with less than twenty dollars, which is the amount a player would have if he didn't invest in any of the rounds.\n\nSo what did the subjects in Damasio's study do? Those with intact emotional brains invested only about 60 per cent of the time. Because human beings are wired to dislike potential losses, most people were perfectly content to sacrifice profit for security, just like investors choosing low-yield bonds. Furthermore, the willingness of a person to invest plummeted immediately after he or she had lost a gamble \u2013 the pain of losing was too fresh.\n\nThese results are entirely predictable; loss aversion makes us irrational when it comes to evaluating risky gambles. But Damasio and Loewenstein didn't stop there. They also played the investing game with neurologically impaired patients who could no longer experience emotion. If it was the feeling of loss aversion that caused these bad investing decisions, then these patients should perform _better_ than their healthy peers.\n\nThat's exactly what happened. The emotionless patients chose to invest 83.7 per cent of the time and gained significantly more money than normal subjects. They also proved much more resistant to the misleading effects of loss aversion, and they gambled 85.2 per cent of the time after a lost coin toss. In other words, losing money made them _more_ likely to invest as they realized that investing was the best way to recoup their losses. In this investing situation, having no emotions was a crucial advantage.\n\nAnd then there is _Deal or No Deal,_ which turns out to be a case study in loss aversion. Imagine you are Frank. Less than a minute ago, you turned down the Banker's offer of \u20ac102,006. But now you've picked the worst possible briefcase, and the offer has declined to \u20ac2,508. In other words, you've lost a cool hundred grand. Should you accept the current deal? The first thing your mind does is make a list of the options under consideration. However, instead of evaluating those options in terms of arithmetic \u2013 which would be the rational thing to do \u2013 you use your emotions as a shortcut to judgment. You simulate the various scenarios and see how each makes you feel. When you imagine accepting the offer of \u20ac2,508, you experience a sharply negative emotion, even though it's a perfectly fair offer. The problem is that your emotional brain interprets the offer as a dramatic loss, since it's automatically compared to the much larger amount of money that had been on offer just a few moments earlier. This resulting feeling serves as a signal that accepting the deal is a bad idea; you should reject the offer and open another briefcase. In this situation, loss aversion makes you risk seeking.\n\nBut now that you've imagined rejecting the offer, you fixate on the highest monetary amount now possible. This is the potential gain you measure everything against, what economists call the reference point. (For Frank, the potential gain during the final rounds was \u20ac10,000. For the physicians being quizzed about that unusual Asian disease, the potential gain was saving all 600 people.) When you think about this optimistic possibility, you experience, however briefly, a pleasurable feeling. You contemplate the upside of risk and envision a check with lots of zeros. You might not be able to get back the \u20ac100,000 offer, but at least you won't leave empty-handed.\n\nThe upshot of all this is that you badly miscalculate the risk. You keep on chasing after the possibility of a big gain because you can't accept the prospect of a loss. Your emotions have sabotaged common sense.\n\nLoss aversion is an innate flaw. Everyone who experiences emotion is vulnerable to its effects. It's part of a larger psychological phenomenon known as negativity bias, which means that, for the human mind, _bad is stronger than good_. This is why in marital interactions, it generally takes at least five kind comments to compensate for one critical comment. As Jonathan Haidt points out in his book _The Happiness Hypothesis_ , people believe that a person who's been convicted of murder must perform at least twenty-five acts of 'life-saving heroism' in order to make up for his or her one crime. There's no rational reason for us to treat gains and losses or compliments and criticisms so differently. But we do. The only way to avoid loss-aversion-generated mistakes is to know about the concept.\n\n##  _The Danger of Debt_\n\n'The credit card is my enemy,' Herman Palmer says. Herman is a very friendly guy, with sympathetic eyes and a wide smile that fills his face, but when he starts to talk about credit cards, his demeanor abruptly darkens. He furrows his brow, lowers his voice, and leans forward in his chair. 'Every day, I see lots of smart people who have the same problem: Visa and MasterCard. Their problem is all those plastic cards they've got in their wallet.' Then he shakes his head in dismay and lets out a resigned sigh.\n\nHerman is a financial counselor in the Bronx. He has spent the last nine years working for GreenPath, a nonprofit organization that helps people deal with their debt problems. His small office is a minimalist affair, with a desk so clean that it looks as if no one has ever used it. The only thing on the desk is a large glass candy jar, but this jar isn't stuffed with M&Ms or jelly beans or miniature candy bars. It's filled with the cutup shards of hundreds of credit cards. The plastic pieces make for a pretty collage \u2013 the iridescent security stickers glitter in the light \u2013 but Herman doesn't keep the jar around for aesthetic reasons. 'I use it as a kind of shock treatment,' he says. 'I'll ask a client for their cards and just cut them up right in front of them. And then I just add the cards to the jar. I want people to see that they are not alone, that so many people have the exact same problem.' Once the jar in his office is completely filled \u2013 and that only takes a few months \u2013 Herman empties it into the big glass vase in the waiting room. 'That's our flower display,' he jokes.\n\nAccording to Herman, the jar of credit cards captures the essence of his job. 'I teach people how _not_ to spend money,' he says. 'And it's damn near impossible to not spend money if you've still got all these cards, which is why I always cut them up.' The first time I visited the GreenPath office was a few weeks after Christmas, and the waiting room was full of anxious-looking people trying to pass the time with old issues of celebrity magazines. Every chair was taken. 'January is our busiest time of year,' Herman says. 'People always overspend during the holidays, but they don't realize how much they've overspent until the credit card bills arrive in the mail. That's when they come see us.'\n\nFor the most part, Herman's clients are from the neighborhood, a working-class area of row houses that were once single-family dwellings but are now apartment buildings, with numerous buzzers and mailboxes grafted onto the front doors. Many of the homes have fallen into disrepair, with peeling siding and graffiti. There aren't any supermarkets nearby, but there are plenty of bodegas and liquor stores. A little farther down the block, there are two pawnshops and three check-cashing operations. Every few minutes, another number six subway train rumbles directly overhead, shrieking to a stop near the GreenPath office. It's the last stop on the line.\n\nNearly half of Herman's clients are single mothers. Many of these women work full-time but still struggle to pay their bills. Herman estimates that his clients spend, on average, around 40 per cent of their income on housing, even though the neighborhood has some of the cheapest real estate in New York City. 'It's easy to judge people,' Herman says. 'It's easy to think, \"I would never have gotten into so much debt,\" or to think that just because someone needs financial help then they must be irresponsible. But a lot of the people I see are just trying to make ends meet. The other day I had a mother come in who just broke my heart. She was working two jobs. Her credit card bill was all daycare charges for her kid. What am I supposed to tell her? That her kid can't go to daycare?'\n\nThis ability to help his clients without judging them, to understand what they're going through, is what makes Herman such a good financial counselor. (He has an unusually high success rate, with more than 65 per cent of his clients completing their debt-elimination plans.) It would be easy for Herman to play the scold, to chastise his clients for letting their spending get out of control. But he does just the opposite. Instead of lecturing his clients, he listens to them. After Herman destroys their credit cards at the initial meeting \u2013 he almost always gets out his scissors within the first five minutes \u2013 he will spend the next several hours poring over their bills and bank statements, trying to understand what's gone wrong with their finances. Is their rent too expensive? Are they spending too much money on clothes or cell phones or cable television? 'I always tell my clients that they are going to leave my office with a practical plan,' Herman says. 'And charging it to Mr MasterCard is not a plan.'\n\nWhen Herman talks about the people who have been helped by his financial advice, his face takes on the glow of a proud parent. There's the plumber from Co-op City who lost his job and started paying rent with his credit card. After a few months, his interest rate was above 30 per cent. Herman helped him consolidate his debt and get his expenses under control. There's that single mother who couldn't afford daycare. 'We helped her find other ways to save money,' he says. 'We cut her expenses by enough so that she didn't have to charge everything. The trick is to notice whenever you're spending money. All that little stuff? Guess what: it adds up.' There's the schoolteacher who racked up debt on 10 different credit cards and paid hundreds of dollars every month in late fees alone. It took five years of careful discipline, but now the teacher is debt free. 'I know the client is going to be okay when they start telling me about the sweater or CD they really wanted but they didn't buy,' Herman says. 'That's when I know they are starting to make better decisions.'\n\nMost of the people who come to see Herman tell the same basic story. One day, a person gets a credit card offer in the mail. (Credit card companies sent out 5.3 billion solicitations in 2007, which means the average American adult got 15 offers.) The card seems like such a good deal. In big bold print it advertises a low introductory rate along with something about getting cash back or frequent-flier miles or free movie tickets. And so the person signs up. He fills out the one-page form and then, a few weeks later, gets a new credit card in the mail. At first, he doesn't use it much. Then one day he forgets to get cash, and so he uses the new credit card to pay for food at the supermarket. Or maybe the refrigerator breaks, and he needs a little help buying a new one. For the first few months, he always manages to pay off the full bill. 'Almost nobody gets a credit card and says, \"I'm going to use this to buy things I can't afford,\"' Herman says. 'But it rarely stays like that for long.'\n\nAccording to Herman, the big problem with credit cards \u2013 the reason he enjoys cutting them up so much \u2013 is that they cause people to make stupid financial choices. They make it harder to resist temptation, so people spend money they don't have. 'I've seen it happen to the most intelligent people,' Herman says. 'I'll look at their credit card bill and I'll see a charge for $50 at a department store. I'll ask them what they bought. They'll say, \"It was a pair of shoes, Herman, but it was on sale.\" Or they'll tell me that they bought another pair of jeans but the jeans were 50 per cent off. It was such a good deal that it would have been dumb _not_ to buy it. I always laugh when I hear that one. I then have them add up all the interest they are going to pay on those jeans or that pair of shoes. For a lot of these people, it will be around 25 per cent a month. And you know what? Then it's not such a good deal anymore.'\n\nThese people aren't in denial. They know they have serious debt problems and that they're paying a lot of interest on their debts. That's why they're visiting a financial adviser. And yet, they _still_ bought the jeans and the pair of shoes on sale. Herman is all too familiar with the problem: 'I always ask people, \"Would you have bought the item if you had to pay cash? If you had to go to an ATM and feel the money in your hands and then hand it over?\" Most of the time, they think about it for a minute and then they say no.'\n\nHerman's observations capture an important reality about credit cards. Paying with plastic fundamentally changes the way we spend money, altering the calculus of our financial decisions. When you buy something with cash, the purchase involves an actual loss \u2013 your wallet is literally lighter. Credit cards, however, make the transaction abstract, so that you don't really feel the downside of spending money. Brain-imaging experiments suggest that paying with credit cards actually reduces activity in the insula, a brain region associated with negative feelings. As George Loewenstein, a neuroeconomist at Carnegie Mellon, says, 'The nature of credit cards ensures that your brain is anesthetized against the pain of payment.' Spending money doesn't feel bad, so you spend more money.\n\nConsider this experiment: Drazen Prelec and Duncan Simester, two business professors at MIT, organized a real-life, sealed-bid auction for tickets to a Boston Celtics game. Half the participants in the auction were informed that they had to pay with cash; the other half were told they had to pay with credit cards. Prelec and Simester then averaged the bids for the two different groups. Lo and behold, the average credit card bid was _twice_ as high as the average cash bid. When people used their Visas and MasterCards, their bids were much more reckless. They no longer felt the need to contain their expenses, and so they spent way beyond their means.\n\nThis is what's happened to the American consumer over the past few decades. The statistics are bleak: the average household currently owes more than nine thousand dollars in credit card debt, and the average number of credit cards per person is 8.5. (More than 115 million Americans carry month-to-month balances on their credit cards.) In 2006, consumers spent more than $17 billion in penalty fees alone on their credit cards. Since 2002, Americans have had a negative savings rate, which means that we've spent more than we've earned. The Federal Reserve recently concluded that this negative savings rate was largely a consequence of credit card debt. We spend so much money on interest payments that we can't save for retirement.\n\nAt first glance, this behavior makes no sense. Given the exorbitant interest rates charged by most credit card companies \u2013 rates of 25 per cent or more are common \u2013 a rational consumer would only accumulate credit card debt as a last resort. Paying interest is expensive. And yet, credit card debt is as American as apple pie. 'The people who have credit card debt are the same people who drive an extra mile to save two cents on a gallon of gas,' Herman says. 'They are the same people who clip coupons and comparison shop. Many of these people are normally very good with their money. But then they bring me their credit card bill and they say, \"I don't know what happened. I don't know how I spent all this money.\"'\n\nThe problem with credit cards is that they take advantage of a dangerous flaw built into the brain. This failing is rooted in our emotions, which tend to overvalue immediate gains (like a new pair of shoes) at the cost of future expenses (high interest rates). Our feelings are thrilled by the prospect of an immediate reward, but they can't really grapple with the long-term fiscal consequences of that decision. The emotional brain just doesn't understand things like interest rates or debt payments or finance charges. As a result, areas like the insula don't react to transactions involving a Visa or MasterCard. Because our impulsivity encounters little resistance, we swipe our cards and buy whatever we want. We'll figure out how to pay for it later.\n\nThis sort of short-sighted decision-making isn't dangerous only for people with too many credit cards in their wallets. In recent years, Herman has seen a new financial scourge in the neighborhood: subprime mortgages. 'I still remember the first subprime mortgage I dealt with,' Herman says. 'I remember thinking, \"This is a really bad deal. These people just bought a house that's way too expensive for them, and they don't even know it yet.\" And that's when I knew that I'd be seeing a lot of these loans in the future.'\n\nThe most common type of subprime mortgage that Herman deals with is the 2\/28 loan, which comes with a low, fixed-interest rate for the first two years and a much higher, adjustable rate for the next 28. In other words, the loan works a lot like a credit card: it lets people get homes for virtually nothing up front, then hits the borrowers with high-interest payments at some point in the distant future. By the time the housing market went bust in the summer of 2007, subprime loans like the 2\/28 accounted for almost 20 per cent of all mortgages. (The percentage in poorer neighborhoods, such as the Bronx, was much higher, with more than 60 per cent of all mortgages falling into the subprime category.) Unfortunately, the loan comes with a steep cost. The structure of the loan ensures that subprime borrowers are five times more likely to default than other borrowers. Once the rates start to rise \u2013 and they always do \u2013 many people can no longer afford the monthly mortgage payments. By the end of 2007, a whopping 93 per cent of completed foreclosures involved adjustable-rate loans that had recently been adjusted. 'When I help people with a mortgage,' Herman says, 'I never ask them about the home. Because then they just start talking about how pretty it is and how the extra room will be so great for their kids. That's just temptation talking. I make sure we stick to the numbers and that we especially focus on their interest payments in the future, after the rates are adjusted.' While 2\/28 loans tempt consumers with low initial payments, that temptation turns out to be extremely expensive. In fact, subprime loans even proved tempting for people with credit scores that qualified them for conventional loans that had far better financial terms. During the peak of the housing boom, 55 per cent of all 2\/28 mortgages were sold to homeowners who could have gotten prime mortgages. Although prime mortgages would have saved them lots of money over the long term, these people just couldn't resist the allure of those low initial payments. Their feelings tricked them into making foolish financial decisions.\n\nThe pervasive reach of credit cards and subprime loans reveals our species' irrationality. Even when people are committed to long-term goals, such as saving for retirement, they are led astray by momentary temptations. Our impulsive emotions make us buy what we can't afford. As Plato might have put it, the horses are pulling the driver against his will.\n\nUnderstanding the circuitry of temptation is one of the practical ambitions of scientists studying decision-making. Jonathan Cohen, a neuroscientist at Princeton University, has made some important progress. He's begun to diagnose the specific brain regions responsible for the attraction to credit cards and subprime loans. One of his recent experiments involved putting a subject in an fMRI machine and making him decide between a small Amazon gift certificate that he could have right away and a slightly larger gift certificate that he'd receive in two to four weeks. Cohen discovered that these two options activated very different neural systems. When a subject contemplated a gift certificate in the future, brain areas associated with rational planning, such as the prefrontal cortex, were more active. These cortical regions urge a person to be patient, to wait a few extra weeks for the bigger gain.\n\nHowever, when a subject started thinking about getting the gift certificate right away, the brain areas associated with emotion \u2013 such as the midbrain dopamine system and nucleus accumbens \u2013 were turned on. These are the cells that tell a person to take out a mortgage he can't afford, or run up credit card debt when he should be saving for retirement. All these cells want is a reward, and they want it now.\n\nBy manipulating the amount of money on offer in each situation, Cohen and his collaborators could watch this neural tug of war unfold. They saw the fierce argument between reason and feeling as the mind was pulled in contradictory directions. The ultimate decision \u2013 whether to save for the future or to indulge in the present \u2013 was determined by whichever region showed greater activation. The people who couldn't wait for the bigger gift certificates \u2013 and most people couldn't wait \u2013 were led astray by their feelings. More emotions meant more impulsivity. (This also helps explain why men who are shown revealing pictures of attractive women, what scientists refer to as 'reproductively salient stimuli', become even more impulsive: the photos activate their emotional circuits.) However, subjects who chose to wait and receive the larger Amazon gift certificates later showed increased activity in their prefrontal cortices; they did the math and selected the 'rational' option.\n\nThis discovery has important implications. For starters, it locates the neural source for many financial errors. When self-control breaks down, and we opt for the rewards we can't afford, it's because the rational brain has lost the neural tug of war. David Laibson, an economist at Harvard and coauthor of the paper on the monetary-reward experiment, notes: 'Our emotional brain wants to max out the credit card, order dessert, and smoke a cigarette. When it sees something it wants, it has difficulty waiting to get it.' Corporations have learned to take advantage of this limbic impatience. Consider the teaser rates offered in credit card solicitations. In order to entice new customers, lenders typically advertise their low introductory charges. These alluring offers expire within a few short months, leaving customers stuck with lots of debt on credit cards with high interest rates. The bad news is that the emotional brain is routinely duped by these tempting (but financially foolish) advertisements. 'I always tell people to _only_ read the fine print,' Herman says. 'The bigger the print, the less it matters.'\n\nUnfortunately, most people don't follow Herman's advice. Lawrence Ausubel, an economist at the University of Maryland, analyzed the responses of consumers to two different credit card promotions used by actual credit card companies. The first card offered a six-month teaser rate of 4.9 per cent that was followed by a lifetime at 16 per cent. The second card had a slightly higher teaser rate \u2013 6.9 per cent \u2013 but a significantly lower lifetime rate (14 per cent). If consumers were rational, they would always choose the card with the lower lifetime rate, since that's the rate that would apply to most of their debts. Of course, this isn't what happens. Ausubel found that the credit offer with the 4.9 per cent teaser rate was chosen by consumers almost three times more often than the other. Over the long term, this impatience leads to significantly higher interest payments.\n\nWhen people opt for bad credit cards, or choose 2\/28 mortgages, or fail to put money in their 401(k) s, they are acting just like the experimental subjects who chose the wrong Amazon gift certificate. Because the emotional parts of the brain reliably undervalue the future \u2013 life is short and we want pleasure _now_ \u2013 we all end up spending too much money today and delaying saving until tomorrow (and tomorrow and tomorrow). George Loewenstein, the neuroeconomist, thinks that understanding the errors of the emotional brain will help policymakers develop plans that encourage people to make better decisions: 'Our emotions are like software programs that evolved to solve important and recurring problems in our distant past,' he says. 'They are not always well suited to the decisions we make in modern life. It's important to know how our emotions lead us astray so that we can find ways to compensate for these flaws.'\n\nSome economists are already working on that. They are using this brain-imaging data to support a new political philosophy known as asymmetric paternalism. That's a fancy name for a simple idea: creating policies and incentives that help people triumph over their irrational impulses and make better, more prudent decisions. Shlomo Benartzi and Richard Thaler, for example, designed a 401(k) that takes our irrationality into account. Their plan, called Save More Tomorrow, neatly sidesteps the limbic system. Instead of asking people if they want to start saving right away \u2013 which is the standard pitch for a 401(k) \u2013 companies in the Save More Tomorrow program ask their employees if they want to opt into savings plans that begin in a few months. Since this proposal allows people to make decisions about the future without contemplating possible losses in the present, it bypasses their impulsive emotional brains. (This is roughly equivalent to asking a person if he wants a $10 Amazon gift certificate in one year or an $11 gift certificate in one year and one week. In this case, virtually everyone chooses the rational option, which is the larger amount.) Trial studies of this program show it's a resounding success: after three years, the average savings rate has gone from 3.5 per cent to 13.6 per cent.\n\nHerman is content with an even simpler solution. 'My first piece of advice is always the same,' he says. 'Cut up the damn cards. Or put them in a block of ice in the freezer. Learn to pay with cash.' Herman knows from experience that unless people get rid of their credit cards, they won't be able to stay on fiscally sound spending plans: 'I've seen people who have more debt than you can believe, and they'll still make irresponsible shopping decisions if they can charge it.' It's not easy for the brain to choose a long-term gain over an immediate reward \u2013 such a decision takes cognitive effort \u2013 which is why getting rid of anything that makes the choice harder (such as credit cards) is so important. 'Everybody knows about temptation,' Herman says. 'Everybody wants that new pair of shoes and the big house. But sometimes you have to say no to yourself.' He tries to quote a famous song by the Rolling Stones but he can't quite remember the lyrics. The message of the chorus is simple: you can't always get what you want, but sometimes _not_ getting what you want is just what you need.\n\n* Slot machines account for about 70 per cent of the $48 billion a year Americans spend at casinos, which means that the average citizen spends five times more on slot machines than he or she does on movie tickets. There are now twice as many slot machines as ATMs in America.\n\n* This misconception is known as the gambler's fallacy. It occurs when people assume that an event is more or less likely to occur based on whether or not that event has recently occurred. As a result, people are surprised when a shuffled song repeats or when a flipped coin exhibits extended streaks of heads or tails. The most famous example of such a phenomenon occurred in a Monte Carlo casino in the summer of 1913 when a roulette wheel landed on black 26 times in a row. During that staggeringly improbable run, most gamblers bet against black, since they felt that the red must be 'due'. In other words, they assumed that the randomness of the roulette wheel would somehow correct the imbalance and cause the wheel to land on red. The casino ended up making millions of francs.\n\n\u2020 Thomas Gilovich also looked at the reactions of London residents during the Blitz of 1940. While the Blitz was happening, British newspapers published maps that displayed the precise location of every German missile strike. The problem was that the strikes didn't look random, which led London residents and British military planners to conclude that the Germans could aim their missiles at specific targets. As a result, people fled those neighborhoods that seemed hardest hit and suspected that German spies lived in the areas that were mostly spared. The reality, however, was that the German military had virtually no control over where the missiles ended up. Although they aimed for central London, they were completely unable to target locations _within_ London. The patterns of damage were utterly random.\nFOUR\n\n# The Uses of Reason\n\nTHE SUMMER OF 1949 had been long and dry in Montana; the grassy highlands were like tinder. On the afternoon of 5 August \u2013 the hottest day ever recorded in the area \u2013 a stray bolt of lightning set the ground on fire. A parachute brigade of fire fighters, known as smokejumpers, was dispatched to put out the blaze. Wag Dodge, a veteran with nine years of smokejumping experience, was in charge. When the jumpers took off from Missoula in a C-47, a military transport plane left over from World War II, they were told that the fire was small, just a few burning acres in the Mann Gulch river valley. As the plane approached the fire, the jumpers could see the smoke in the distance. The hot wind blew it straight across the sky.\n\nMann Gulch is a place of geological contradiction. It is where the Rocky Mountains meet the Great Plains, pine trees give way to prairie grass, and the steep cliffs drop onto the steppes of the Midwest. The gulch is just over three miles long, but it marks the border between these two different terrains.\n\nThe fire began on the Rockies' side, on the western edge of the gulch. By the time the firefighters arrived at the gulch, the blaze had grown out of control. The surrounding hills had all been burned; the landscape was littered with the skeletons of pine trees. Dodge moved his men over to the grassy side of the gulch and told them to head downhill, toward the placid Missouri River. Dodge didn't trust this blaze. He wanted to be near water; he knew this fire could crown.\n\nCrowns occur when flames get so high they reach into the top branches of trees. Once that happens, the fire has too much fuel. Hot embers begin to swirl in the air, spreading the fire across the prairie. The smokejumpers used to joke that the only way to control a crown fire was to pray like hell for rain. Norman Maclean, in his seminal history _Young Men and Fire_ , described what it was like to be close to such a fire:\n\nIt sounds like a train coming too fast around a curve and may get so high-keyed that the crew cannot understand what their foreman is trying to do to save them. Sometimes, when the timber thins out, it sounds as if the train were clicking across a bridge, sometimes it hits an open clearing and becomes hushed as if going through a tunnel, but when the burning cones swirl through the air and fall on the other side of the clearing, the new fire sounds as if it were the train coming out of the tunnel, belching black unburned smoke. The unburned smoke boils up until it reaches oxygen, then bursts into gigantic flames on top of its cloud of smoke in the sky. The new [novice] firefighter, seeing black smoke rise from the ground and then at the top of the sky turn into flames, thinks that natural law has been reversed.\n\nDodge looked at the dry grass and the dry pine needles. He felt the hot wind and the hot sun. The conditions were making him nervous. To make matters worse, the men had no map of the terrain. They were also without a radio, since the parachute on the radio pack had failed to open and the transmitter had been smashed on the rocks. The small crew of smokejumpers was all alone with this fire; there was nothing between them and it but a river and a thick tangle of ponderosa pine and Douglas fir trees. And so the jumpers set down their packs and watched the blaze from across the canyon. When the wind parted the smoke, as it did occasionally, they could see inside the fire as the flames leaped from tree to tree.\n\nIt was now 5 p.m. \u2013 a dangerous time to fight wilderness fires because the twilight wind can shift without warning. The breeze had been blowing the flames up the canyon, away from the river. But then, suddenly, the wind reversed. Dodge saw the ash swirl in the air. He saw the top of the flames flicker and wave. And then he saw the fire leap across the gulch and spark the grass on his side.\n\nThat's when the updraft began. Fierce winds began to howl through the canyon, blowing straight toward the men. Dodge could only watch as the fire became an inferno. He was suddenly staring at a wall of flame 200 feet tall and 300 feet deep on the edge of the prairie. In a matter of seconds, the flames began to devour the grass on the slope. The fire ran toward the smokejumpers at 30 miles per hour, incinerating everything in its path. At the fire's center, the temperature was more than two thousand degrees, hot enough to melt rock.\n\nDodge screamed at his men to retreat. It was already too late to run to the river, since the fire was blocking their path. Each man dropped his 50 pounds of gear and started running up the brutally steep canyon walls, trying to get to the top of the ridge and escape the blow-up. Because heat rises, a fire that starts burning on flat prairie accelerates when it hits a slope. On a 50 per cent grade, a fire will move nine times faster than it does on level land. The slopes at Mann Gulch are 76 per cent.\n\nWhen the fire first crossed the gulch, Dodge and his crew had a 200-yard head start. After a few minutes of running, Dodge could feel the fierce heat on his back. He glanced over his shoulder and saw that the fire was now fewer than fifty yards away and gaining. The air began to lose its oxygen. The fire was sucking the wind dry. That's when Dodge realized the blaze couldn't be outrun. The hill was too steep, and the flames were too fast.\n\nSo Dodge stopped running. He stood perfectly still as the fire accelerated toward him. Then he started yelling at his men to do the same. He knew they were racing toward their own immolation and that in fewer than thirty seconds the fire would run them over, like a freight train without brakes. But nobody stopped. Perhaps the men couldn't hear Dodge over the deafening roar of flames. Or perhaps they couldn't bear the idea of stopping. When confronted with a menacing fire, the most basic instinct is to run away. Dodge was telling the men to stand still.\n\nBut Dodge wasn't committing suicide. In a fit of desperate creativity, he came up with an escape plan. He quickly lit a match and ignited the ground in front of him. He watched as those flames raced away from him, up the canyon walls. Then Dodge stepped into the ashes of this smaller fire, so that he was surrounded by a thin buffer of burned land. He lay down on the still smoldering embers. He wet his handkerchief with some water from his canteen and clutched the cloth to his mouth. He closed his eyes tight and tried to inhale the thin ether of oxygen remaining near the ground. Then he waited for the fire to pass around him. After several terrifying minutes, Dodge emerged from the ashes virtually unscathed.\n\nThirteen smokejumpers were killed by the Mann Gulch fire. Only two men in the crew besides Dodge managed to survive, and that was because they found a shallow crevice in the rocky hillside. As Dodge had predicted, the flames were almost impossible to outrun. White crosses still mark the spots where the men died; all of the crosses are below the ridge.\n\n## _The Panic Problem_\n\nDodge's escape fire is now a standard firefighting technique. It has saved the lives of countless firefighters trapped by swift blazes. At the time, however, Dodge's plan seemed like sheer madness. His men could think only about fleeing the flames, and yet their leader was starting a new fire. Robert Sallee, a first-year smokejumper who survived the blaze, later said he'd thought that 'Dodge had gone nuts, just plain old nuts.'\n\nBut Dodge was perfectly sane. In the heat of the moment he managed to make a very smart decision. The question, for those of us looking back on it, is how? What allowed him to resist the urge to flee? Why didn't he follow the rest of his crew up the gulch? Part of the answer is experience. Most of the smokejumpers were teenagers working summer jobs. They had fought only a few fires, and none of them had ever seen a fire like that. Dodge, on the other hand, was a grizzled veteran of the forest service; he knew what prairie flames were capable of. Once the fire crossed the gulch, Dodge realized that it was only a matter of time before the men were caught by the hungry flames. The slopes were too steep and the wind was too fierce and the grass was too dry; the blaze would beat them to the top. Besides, even if the men managed to reach the top of the mountain, they were still trapped. The ridge was covered with high, dry grass that hadn't been trimmed by cattle. It would burn in an instant.\n\nFor Dodge, it must have been a moment of unspeakable horror: to know that there was nowhere to go; to realize that his men were running to their deaths and that the wall of flame would consume them all. But Dodge's fear wasn't what saved him. In fact, the overwhelming terror of the situation was part of the problem. After the fire started burning uphill, all of the smokejumpers became fixated on getting to the ridge, even though the ridge was too far away for them to reach. Walter Rumsey, a first-year smokejumper, later recounted what was going through his mind when he saw Dodge stop running and get out his matchbook. 'I remember thinking that that was a very good idea,' Rumsey said, 'but I don't remember what I thought it was good for... I kept thinking the ridge \u2013 if I can make it. On the ridge I will be safe.' William Hellman, the second in command, looked at Dodge's escape fire and reportedly said, 'To hell with that, I'm getting out of here.' (Hellman did reach the ridge, the only smokejumper who managed to do so, but he died the next day from third-degree burns that covered his entire body.) The rest of the men acted the same way. When Dodge was asked during the investigation why none of the smokejumpers followed his orders to stop running, he just shook his head. 'They didn't seem to pay any attention,' he said. 'That is the part I didn't understand. They seemed to have something on their minds \u2013 all headed in one direction... They just wanted to get to the top.'\n\nDodge's men were in the grip of panic. The problem with panic is that it narrows one's thoughts. It reduces awareness to the most essential facts, the most basic instincts. This means that when a person is being chased by a fire, all he or she can think about is running from the fire.\n\nThis is known as perceptual narrowing. In one study, people were put one at a time in a pressure chamber and told that the pressure would slowly be increased until it simulated that of a 60-foot dive. While inside the pressure chamber, the subject was asked to perform two simple visual tasks. One task was to respond to blinking lights in the center of the subject's visual field, and the other involved responding to blinking lights in his peripheral vision. As expected, each of the subjects inside the pressure chamber exhibited all the usual signs of panic \u2013 a racing pulse, elevated blood pressure and a surge of adrenaline. These symptoms affected performance in a very telling way. Although the people in the pressure chamber performed just as well as control subjects did on the central visual task, those in the pressure chamber were twice as likely to miss the stimuli in their peripheral vision. Their view of the world literally shrank.\n\nThe tragedy of Mann Gulch holds an important lesson about the mind. Dodge survived the fire because he was able to beat back his emotions. Once he realized that his fear had exhausted its usefulness \u2013 it told him to run, but there was nowhere to go \u2013 Dodge was able to resist its primal urges. Instead, he turned to his conscious mind, which is uniquely capable of deliberate and creative thought. While automatic emotions focus on the most immediate variables, the rational brain is able to expand the list of possibilities. As the neuroscientist Joseph LeDoux says, 'The advantage [of the emotional brain] is that by allowing evolution to do the thinking for you at first, you basically buy the time that you need to think about the situation and do the most reasonable thing.' And so Dodge stopped running. If he was going to survive the fire, he needed to think.\n\nWhat Dodge did next relied entirely on the part of his brain that he could control. In the panic of the moment, Dodge was able to come up with a new solution to his seemingly insurmountable problem. There was no pattern to guide him \u2013 no one had ever started an escape fire before \u2013 but Dodge was able to imagine his survival. In that split second of thought, he realized that it was possible to start his own fire, and that this fire might give him a thin barrier of burned earth. 'It just seemed like the logical thing to do,' Dodge said. He didn't know if his escape fire would work \u2013 he thought he would probably suffocate \u2013 but it still appeared to be a better idea than running. And so Dodge felt for the direction of the wind and lit the prairie weeds right in front of him. They ignited like paper. The surrounding tinder wilted to ash. He had made a firewall out of fire.\n\nThis kind of thinking takes place in the prefrontal cortex, the outermost layer of the frontal lobes. Pressed tight against the bones of the forehead, the prefrontal cortex has undergone a dramatic expansion in the human brain. When you compare a modern human cortex to that of any other primate, or even to some of our hominid ancestors', the most obvious anatomical difference is this swelling at the fore. The Neanderthal, for example, had a slightly larger brain than _Homo sapiens_. But he still had the prefrontal cortex of a chimp. As a result, Neanderthals were missing one of the most important talents of the human brain: rational thought.*\n\n_Rationality_ can be a difficult word to define \u2013 it has a long and convoluted intellectual history \u2013 but it's generally used to describe a particular style of thinking. Plato associated rationality with the use of logic, which he believed made humans think like the gods. Modern economics has refined this ancient idea into rational-choice theory, which assumes that people make decisions by multiplying the _probability_ of getting what they want by the _amount of pleasure_ (utility) that getting what they want will bring. This reasonable rubric allows us all to maximize our happiness, which is what rational agents are always supposed to do.\n\nOf course, the mind isn't a purely rational machine. You don't compute utility in the supermarket or use math when throwing a football or act like the imaginary people in economics textbooks. The Platonic charioteer is often trounced by his emotional horses. Nevertheless, the brain does have a network of rational parts, centered in the prefrontal cortex. If it weren't for these peculiar lumps of gray and white matter, we couldn't even conceive of rationality, let alone act in a rational manner.\n\nThe prefrontal cortex was not always held in such high regard. When scientists first began dissecting the brain in the nineteenth century, they concluded that the frontal lobes were useless, unnecessary folds of flesh. Unlike other cortical areas, which were responsible for specific tasks such as controlling the body or generating language, the prefrontal cortex seemed to do nothing. It was the appendix of the mind. In the 1920s, doctors began experimenting with frontal-lobe lobotomy, a brutal surgery in which the prefrontal cortex's connections to the rest of the brain were severed.* This surgical treatment was prescribed for a wide variety of conditions, from epilepsy to schizophrenia. Since the frontal lobes apparently had no particular use, doctors figured they might as well find out what happened when the lobes were disconnected from the rest of the brain. (The surgery was initially inspired by a single 'successful' chimpanzee lobotomy that reduced the animal's 'aggressive tendencies'.) The procedure quickly became exceedingly popular. Between 1939 and 1951, the 'cutting cure' was performed on more than eighteen thousand patients in American asylums and prisons. In 1949, the physiologist Walter Hess and the neurologist Antonio Egas Moniz were awarded the Nobel Prize in Medicine for pioneering the procedure.\n\nAlthough frontal-lobe lobotomies did lessen some symptoms of mental illness \u2013 schizophrenics, for example, seemed to suffer from fewer paranoid delusions \u2013 the treatment came with a wide range of tragic side effects. Six per cent of all patients died on the operating table. Those who survived were never the same. Some patients sank into a stupor, utterly uninterested in everything around them. Others lost the ability to use language. (This is what happened to Rosemary Kennedy, the sister of President John F. Kennedy. Her lobotomy was meant to treat her 'agitated depression'. After the surgery, she was unable to speak more than a few words.) The vast majority of lobotomy patients suffered from short-term memory problems and the inability to control their impulses.\n\nThe frontal-lobe lobotomy was a crude surgery. Its path of destruction was haphazard and unpredictable. Sometimes surgeons damaged the premotor areas of the brain, leading to movement problems; sometimes they wounded part of the thalamus, causing severe epilepsy. Although doctors tried to cut only the frontal lobes' connections, they really didn't know what they were cutting.\n\nHowever, in the last few decades, neurologists have studied this brain area with great precision. They now know exactly what happens when the prefrontal cortex is damaged.\n\nConsider the case of Mary Jackson, an intelligent and driven 19-year-old with a bright future. Although she grew up in a blighted inner-city neighborhood, Mary received a full scholarship to an Ivy League university. She was a history major with a pre-med concentration and hoped one day to become a pediatrician so she could open up a medical clinic in her old neighborhood. Her boyfriend, Tom, was an undergraduate at a nearby college, and they planned to get married after Mary finished medical school.\n\nBut then, in the summer after her sophomore year, Mary's life began falling apart. Tom noticed it first. Mary had never drunk alcohol before \u2013 her parents were strict Baptists \u2013 but she suddenly started frequenting bars and dance clubs. She began sleeping with random men and experimented with crack cocaine. She disowned her old friends, avoided church and broke up with Tom. Nobody knew what had gotten into her.\n\nWhen Mary returned to school, her grades began to slip. She stopped attending class. Her semester report card was dismal: three Fs and two Ds. Mary's adviser warned her that she would lose her scholarship and recommended psychiatric counseling. But Mary ignored the suggestion and continued to spend most of her nights at the local bar.\n\nLater that spring, Mary developed a high fever and a hacking cough. At first, she assumed her sickness was just the side effect of too much partying, but the sickness wouldn't go away. She went to the student health center and was diagnosed with pneumonia. But even after she was treated with intravenous antibiotics and oxygen, the fever lingered. Mary's immune system seemed compromised. The doctors ordered more blood tests. That's when Mary learned she was HIV-positive.\n\nMary immediately broke down in a fit of hysterical tears. She told her doctor that she didn't understand her own behavior. Until the previous summer, she had never felt the urge to do drugs or sleep around or skip class. She had been diligently focused on her long-term goals of going to medical school and starting a family with Tom. But now she was unable to control her own impulses. She couldn't resist temptation. She made one reckless decision after another.\n\nMary's doctor referred her to Dr Kenneth Heilman, a distinguished neurologist now at the University of Florida. Heilman began by giving Mary some simple psychological tests: he asked her to remember a few different objects, and then distracted her for 30 seconds by having her count backward. When Heilman asked Mary if she could still remember the objects, she looked at him with a blank stare. Her working memory had vanished. When Heilman tried to give Mary a different memory test, she flew into a rage. He asked her if she had always had such a bad temper. 'Up to about a year ago, it was extremely rare that I got angry,' Mary said. 'Now it seems I am always flying off the handle.'\n\nAll of these neurological symptoms \u2013 the diminished memory capacity, the self-destructive impulsiveness, the uncontrollable rage \u2013 suggested a problem with Mary's prefrontal cortex. So Heilman gave Mary a second round of tests: he put a comb in front of her but told her not to touch it. She immediately started combing her hair. He put a pen and paper in front of her but told her to keep her hands still. She automatically started writing. After scribbling a few sentences, however, Mary became bored and started looking for a new distraction. 'It seemed that rather than having internal goals motivate her behavior,' Heilman wrote in his clinical report, 'she was entirely stimulus dependent.' Whatever Mary saw, she touched. Whatever she touched, she wanted. Whatever she wanted, she needed.\n\nHeilman ordered an MRI. That's when he saw the tumor: a large mass emanating from the pituitary gland and pressing on Mary's prefrontal cortex. This was the cause of her decline. That growth had left her with executive dysfunction, an inability to maintain a coherent set of goals and contemplate the consequences of one's actions. As a result, Mary was unable to act on any ideas but the most immediate. The tumor had erased some of the necessary features of the human mind: the ability to think ahead, plan for the future, and repress impulses.\n\n'You see this with a lot of patients with frontal-lobe problems,' Heilman says. 'They can't hold back their emotions. If they get angry, then they'll just get in a fight. Even if they know that getting in a fight is a bad idea \u2013 the cognitive knowledge might still be there \u2013 that knowledge is less important than the intensity of what they are feeling.' Heilman believes that in Mary's case, her damaged prefrontal cortex meant that her rational brain could no longer modulate or restrain her irrational passions. 'She knew her behavior was self-destructive,' Heilman says. 'But she did it anyway.'\n\nThe tragic story of Mary Jackson illuminates the importance of the prefrontal cortex.* Because she was missing this specific brain region \u2013 it was damaged by the tumor \u2013 she couldn't think abstractly or resist her most immediate urges. She was unable to keep information in her short-term memory or follow through on her long-term plans. If Mary Jackson was fleeing a fire, she never would have stopped to light the match. She would have kept on running.\n\n## _Controlling the Horses_\n\nImagine that you are playing a simple gambling game. You are given $50 of real money and asked to decide between two options. The first option is an all-or-nothing gamble. The odds of the gamble are clear: there is a 40 per cent chance that you will keep the entire $50, and a 60 per cent chance that you will lose everything. The second option, however, is a sure thing. If you choose this alternative, you get to keep $20.\n\nWhat option did you choose? If you're like most people, you took the guaranteed cash. It's always better to get some thing rather than nothing, and $20 is not a trivial amount of money.\n\nBut now let's play the game again. The risky gamble hasn't changed: you still have a 40 per cent chance of keeping the entire $50. This time, however, the sure thing is a loss of $30 instead of a gain of $20.\n\nThe outcome, of course, remains the same. The two gambles are identical. In both cases, you walk away with $20 of the original $50. But the different descriptions strongly affect how people play the game. When the choice is framed in terms of _gaining_ $20, only 42 per cent of people choose the risky gamble. But when the same choice is framed in terms of _losing_ $30, 62 per cent of people opt to roll the dice. This human foible is known as the framing effect, and it's a byproduct of loss aversion, which we discussed earlier. The effect helps explain why people are much more likely to buy meat when it's labeled _85 per cent lean_ instead of _15 per cent fat_. And why twice as many patients opt for surgery when told there's an 80 per cent chance of their surviving instead of a 20 per cent chance of their dying.\n\nWhen neuroscientists used an fMRI machine to study the brains of people playing this gambling game, they saw the precise regions that these two different yet equivalent frames activated. They found that people who chose to gamble \u2013 the ones whose decisions were warped by the prospect of losing $30 \u2013 were misled by an excited amygdala, a brain region that, when excited, evokes negative feelings. Whenever a person thinks about losing something, the amygdala is automatically activated. That's why people hate losses so much.\n\nHowever, when the scientists looked at the brains of subjects who were _not_ swayed by the different frames, they discovered something that surprised them. The amygdalas of these 'rational' people were still active. In fact, their amygdalas tended to be just as excitable as the amygdalas of people who were susceptible to the framing effect. 'We found that everyone showed emotional biases; no one was totally free of them,' says Benedetto de Martino, the neuroscientist who led the experiment. Even people who instantly realized that the two different descriptions were identical \u2013 they saw through the framing effect \u2013 still experienced a surge of negative emotion when they looked at the loss frame.\n\nWhat, then, caused the stark differences in behavior? If everybody had an active amygdala, why were only some people swayed by the different descriptions? This is where the prefrontal cortex enters the picture. To the surprise of the scientists, it was the activity of the prefrontal cortex (not the amygdala) that best predicted the decisions of the experimental subjects. When there was more activity in the prefrontal cortex, people were better able to resist the framing effect. They could look past their irrational feelings and realize that both descriptions were equivalent. Instead of just trusting their amygdalas, they did the arithmetic. The end result is that they made better gambling decisions. According to de Martino, 'People who are more rational don't perceive emotion less, they just regulate it better.'\n\nHow do we regulate our emotions? The answer is surprisingly simple: by thinking about them. The prefrontal cortex allows each of us to contemplate his or her own mind, a talent psychologists call metacognition. We know when we are angry; every emotional state comes with self-awareness attached, so that an individual can try to figure out why he's feeling what he's feeling. If the particular feeling makes no sense \u2013 if the amygdala is simply responding to a loss frame, for example \u2013 then it can be discounted. The prefrontal cortex can deliberately choose to ignore the emotional brain.\n\nThis is one of Aristotle's essential ideas. In _The Nicomachean_ _Ethics,_ his sprawling investigation into the 'virtuous character', Aristotle concluded that the key to cultivating virtue was learning how to manage one's passions. Unlike his teacher Plato, Aristotle realized that rationality wasn't always in conflict with emotion. He thought Plato's binary psychology was an oversimplification. Instead, Aristotle argued that one of the critical functions of the rational soul was to make sure that emotions were intelligently applied to the real world. 'Anyone can become angry \u2013 that is easy,' Aristotle wrote. 'But to become angry with the right person, to the right degree, at the right time, for the right purpose, and in the right way \u2013 that is not easy.' That requires some thought.\n\nOne way to understand how this Aristotelian idea actually plays out in the brain is by examining the inner workings of a television focus group. Practically every show on television is tested on audiences before it hits the airwaves. When this testing process is done properly, it demonstrates a fascinating interplay between reason and emotion, instinct and analysis. In other words, the whole enterprise mimics what's constantly happening inside the human mind.\n\nThe process goes something like this: people representing a demographic cross section of America are ushered into a specially equipped room that looks like a tiny movie theater, complete with comfy seats and cup holders. (Most television focus groups take place in Orlando and Las Vegas, since those cities are full of people who have arrived from all across the country.) Each participant is given a feedback dial, a device that's about the size of a remote control and has a single red dial, a few white buttons, and a small LED screen. Feedback dials were first used in the late 1930s, when Frank Stanton, the head of audience research at CBS Radio, teamed up with Paul Lazarsfeld, the eminent sociologist, to develop the 'program analyzer'. The CBS method was later refined by the US military during World War II as it tested its war propaganda on the public.\n\nThe modern feedback dial is designed to be as straightforward as possible so that a person can operate it without taking his or her eyes off the screen. The numbers on the dial increase in a clockwise direction, like a volume knob; higher numbers signal a more positive response to the television show. The participants are told to rotate their dials whenever their feelings change. This gives a second-by-second look at the visceral reactions of the audience, which are translated into a jagged line graph.\n\nAlthough every television network depends on focus groups for feedback \u2013 even cable channels like HBO and CNN do extensive audience research \u2013 the process has very real limits. The failures of focus groups are part of industry lore: _The Mary_ _Tyler Moore Show, Hill Street Blues_ , and _Seinfeld_ are all famous examples of shows that tested terribly and yet went on to commercial success. ( _Seinfeld_ tested so badly that instead of being featured on NBC's 1989 fall schedule, it was introduced as a midseason replacement.) As Brian Graden, president of programming at MTV Networks, says, 'Quantitative data [of the sort produced by feedback dials] is useless by itself. You've got to ask the data the right questions.'\n\nThe problem with the focus group is that it's a crude instrument. People can express their feelings with dials, but they can't _explain_ their feelings. The impulsive emotions recorded on the dials are just that: impulsive emotions. They are suffused with all the usual flaws of the emotional brain. Did the focus-group audience not like _Seinfeld_ because they didn't like the main character? Or did they dislike the show because it was a new kind of television comedy, a sitcom about nothing in particular? (The _Seinfeld_ pilot begins with a long discussion about the importance of buttons.) After all, one of the cardinal rules of focus groups is that people tend to prefer the familiar. The new shows that test the best often closely resemble shows that are already popular. For example, after the NBC sitcom _Friends_ became a huge commercial hit, other networks rushed to imitate its formula. There were suddenly numerous comedy pilots about pretty twentysomethings living together in a city. 'Most of these shows tested really well,' one television executive told me. 'The shows weren't very good, but they reminded the audience of _Friends,_ which was a show they actually liked.' Not one of the knock-offs was renewed for a second season.\n\nThe job of a television executive is to sort through these emotional mistakes so he or she isn't misled by the audience's first impressions. Sometimes people like shows that actually stink and reject shows that they grow to enjoy. In such situations, executives must know how to discount the responses of focus groups. They need to interpret the quantitative data, not just obey it. This is where the second-by-second responses of feedback dials are especially useful, since they allow executives to see what exactly people are responding to. A high score in minute 12 might mean that the audience really liked a particular plot twist, or it might mean that they liked looking at the blonde in her underwear. (A conclusive answer can be gotten by comparing the ratings of men versus women.) One cable channel recently tested a reality-television pilot that scored well overall but showed sharp declines in audience opinion at various points throughout the show. At first, the executives couldn't figure out what the audience didn't like. Eventually, however, they realized that the audience was reacting to the host: whenever she talked to the contestants, people turned down their dials. Although the focus-group audience said they liked the host and rated her highly when she talked to the camera, they didn't like watching her with other people. (The host was replaced.) And then there's the 'flat line': when a focus-group audience is especially absorbed in the show \u2013 for example, during a climactic scene \u2013 they often forget to turn their dials. The resulting data can make it appear that the show has hit a rough spot, since many of the dials are stuck in a low position, but the reality is precisely the opposite. If the executives don't realize that the focus-group participants were simply too involved in the program to pay attention to their dials, they might end up altering the best part of the show.\n\nThe point is that the emotional data requires careful analysis. Audience research is a blunt tool, a summary of first impressions, but it can be sharpened. By examining the feelings registered on the dial, a trained observer can figure out which feelings should be trusted and which should be ignored.\n\nThis is just what the prefrontal cortex does when faced with a decision. If the emotional brain is the audience, constantly sending out visceral signals about its likes and dislikes, then the prefrontal cortex is the smart executive, patiently monitoring emotional reactions and deciding which to take seriously. It is the only brain area able to realize that the initial dislike of _Seinfeld_ was a reaction to its originality, not to its inherent funniness. The rational brain can't silence emotions, but it can help figure out which ones should be followed.\n\nIn the early 1970s, Walter Mischel invited four-year-olds to his Stanford psychology laboratory. The first question he asked each child was an easy one: did he like to eat marshmallows? The answer, not surprisingly, was always yes. Then Mischel made the child an offer. He could eat one marshmallow right away or, if the child was willing to wait for a few minutes while Mischel ran an errand, he could eat two marshmallows when the experimenter returned. Practically every child decided to wait. They all wanted more sweets.\n\nMischel then left the room but told the child that if he rang a bell, Mischel would come back and the child could eat the marshmallow. However, this meant that he'd be forfeiting the chance to get the second marshmallow.\n\nMost of the four-year-olds couldn't resist the sugary temptation for more than a few minutes. Several kids covered their eyes with their hands so that they couldn't see the marshmallow. One child started kicking the desk. Another one started pulling on his hair. While a few of the four-year-olds were able to wait for up to 15 minutes, many lasted less than one minute. Others just ate the marshmallow as soon as Mischel left the room, not even bothering to ring the bell.\n\nThe marshmallow was a test of self-control. The emotional brain is always tempted by rewarding stimuli, such as a lump of sugar. However, if the child wanted to achieve the goal \u2013 getting a second marshmallow \u2013 then he needed to temporarily ignore his feelings, delay gratification for a few more minutes. What Mischel discovered was that even at the age of four, some kids were much better at managing their emotions than others.\n\nFast-forward to 1985. The four-year-olds were now high school seniors. Mischel sent out a follow-up survey to their parents. He asked the parents about a wide variety of character traits, from the ability of their child to deal with frustrating events to whether or not the child was a conscientious student. Mischel also asked for SAT scores and high school transcripts. He used this data to construct an elaborate personality profile for each of the kids.\n\nMischel's results were very surprising, at least to him. There was a strong correlation between the behavior of the four-year-old waiting for a marshmallow and that child's future behavior as a young adult. The children who rang the bell within a minute were much more likely to have behavioral problems later on. They got worse grades and were more likely to do drugs. They struggled in stressful situations and had short tempers. Their SAT scores were, on average, 210 points lower than those of kids who'd waited several minutes before ringing the bell. In fact, the marshmallow test turned out to be a better predictor of SAT results than the IQ tests given to the four-year-olds.\n\nThe ability to wait for a second marshmallow reveals a crucial talent of the rational brain. When Mischel looked at why some four-year-olds were able to resist ringing the bell, he found that it wasn't because they wanted the marshmallow any less. These kids also loved sweets. Instead, Mischel discovered, the patient children were better at using reason to control their impulses. They were the kids who covered their eyes, or looked in the other direction, or managed to shift their attention to something other than the delicious marshmallow sitting right there. Rather than fixating on the sweet treat, they got up from the table and looked for something else to play with. It turned out that the same cognitive skills that allowed these kids to thwart temptation also allowed them to spend more time on their homework. In both situations, the prefrontal cortex was forced to exercise its cortical authority and inhibit the impulses that got in the way of the goal.\n\nStudies of children with attention deficit hyperactivity disorder (ADHD) further demonstrate the connection between the prefrontal cortex and the ability to withstand emotional urges. Approximately 5 per cent of school-age children are affected by ADHD, which manifests itself as an inability to focus, sit still, or delay immediate gratification. (These are the kids who eat their marshmallows right away.) As a result, kids with ADHD tend to perform significantly worse in school, since they struggle to stay on task. Minor disturbances become overwhelming distractions.\n\nIn November 2007, a team of researchers from the National Institute of Mental Health and McGill University announced that they had uncovered the specific deficits of the ADHD brain. The disorder turns out to be a developmental problem in many cases: often, the brains of kids with ADHD develop at a significantly slower pace than normal. This lag was most obvious in the prefrontal cortex, which meant that these kids literally lacked the mental muscles needed to resist alluring stimuli. (On average, their frontal lobes were three and a half years behind schedule.) The good news, however, is that the brain almost always recovers from its slow start. By the end of adolescence, the frontal lobes in these kids reached normal size. It's not a coincidence that their behavioral problems began to disappear at about the same time. The children who had had the developmental lag were finally able to counter their urges and compulsions. They could look at the tempting marshmallow and decide that it was better to wait.\n\nADHD is an example of a problem in the developmental process, but the process itself is the same for everybody. The maturation of the human mind recapitulates its evolution, so the first parts of the brain to evolve \u2013 the motor cortex and brain stem \u2013 are also the first parts to mature in children. Those areas are fully functional by the time humans hit puberty. In contrast, brain areas that are relatively recent biological inventions \u2013 such as the frontal lobes \u2013 don't finish growing until the teenage years are over. The prefrontal cortex is the last brain area to fully mature.\n\nThis developmental process holds the key to understanding the behavior of adolescents, who are much more likely than adults to engage in risky, impulsive behavior. More than 50 per cent of US high school students have experimented with illicit drugs. Half of all reported cases of sexually transmitted diseases occur in teenagers. Car accidents are the leading cause of death for those under the age of 21. These bleak statistics are symptoms of minds that can't restrain themselves. While the emotional brains of teens are operating at full throttle (those raging hormones don't help), the mental muscles that check these emotions are still being built. A recent study by neuroscientists at Cornell, for example, demonstrated that the nucleus accumbens, a brain area associated with the processing of rewards \u2013 things like sex, drugs and rock 'n' roll \u2013 was significantly more active and mature in the adolescent brain than the prefrontal cortex was, that part of the brain that helps resist such temptations. Teens make bad decisions because they are literally less rational.*\n\nThis new research on reckless adolescents and children with ADHD highlights the unique role of the prefrontal cortex. For too long, we've assumed that the purpose of reason is to eliminate those emotions that lead us astray. We've aspired to the Platonic model of rationality, in which the driver has complete control. But now we know that silencing human feelings isn't possible, at least not directly. Every teenager wants to have sex, and every four-year-old wants to eat marshmallows. Every firefighter who sees a wall of flames wants to run. Human emotions are built into the brain at a very basic level. They tend to ignore instructions.\n\nBut this doesn't mean that humans are mere puppets of the limbic system. Some people can see through the framing effect despite the fact that their amygdalas are activated. Some four-year-olds can find ways to wait for the second marshmallow. Thanks to the prefrontal cortex, we can transcend our impulses and figure out which feelings are useful and which ones should be ignored.\n\nConsider the Stroop task, one of the classic experiments of twentieth-century psychology. Three words \u2013 _blue, green_ and _red_ \u2013 are flashed randomly on a computer screen. Each of the words is printed in a different color, but the colors aren't consistent. The word _red_ might be in green, while _blue_ is in red. The surprisingly difficult job of the subject is to ignore the _meaning_ of the word and focus instead on the _color_ of the word. If you're looking at _green_ , but the word is actually in blue letters, then you have to touch the button marked _blue_.\n\nWhy is this simple exercise so hard? Reading the word is an automated task; it takes little mental effort. Naming the color of the word, however, requires deliberate thought. The brain needs to turn off its automatic operation \u2013 the act of reading a familiar word \u2013 and consciously think about what color it sees. When a person performs the Stroop task in an fMRI machine, scientists can watch the brain struggle to ignore the obvious answer. The most important cortical area engaged in this tug of war is the prefrontal cortex, which allows a person to reject the first impression when it's possible that the first impression might be wrong. If the emotional brain is pointing you in the direction of a bad decision, you can choose to rely on your rational brain instead. You can use your prefrontal cortex to discount the amygdala, which is telling you to run up the steep slopes of the gulch. The reason Wag Dodge survived was not that he wasn't scared. Like all the smokejumpers, he was terrified. Dodge survived because he realized that his fright wasn't going to save him.\n\n## _The Anatomy of Insight_\n\nThe ability to supervise itself, to exercise authority over its own decision-making process, is one of the most mysterious talents of the human brain. Such a mental maneuver is known as executive control, since thoughts are directed from the top down, like a CEO issuing orders. As the Stroop task demonstrates, this thought process depends on the prefrontal cortex.\n\nBut the questions still remain: how does the prefrontal cortex wield such power? What allows this particular area to control the rest of the brain? The answer returns us to the cellular details: by looking at the precise architecture of the prefrontal cortex, we can see the neural forms that explain its function.\n\nEarl Miller is a neuroscientist at MIT who has devoted his career to understanding this bit of tissue. He was first drawn to the prefrontal cortex as a graduate student, in large part because it seemed to be connected to _everything_. 'No other brain area gets so many different inputs or has so many different outputs,' Miller says. 'You name the brain area, and the prefrontal cortex is almost certainly linked to it.' It took more than a decade of painstaking probing while Miller carefully monitored cells all across the monkey brain, but he was eventually able to show that the prefrontal cortex wasn't simply an aggregator of information. Instead, it was like the conductor of an orchestra, waving its baton and directing the musicians. In 2007, in a paper published in _Science,_ Miller was able to provide the first glimpse of executive control at the level of individual neurons, as cells in the prefrontal cortex directly modulated the activity of cells throughout the brain. He was watching the conductor at work.\n\nHowever, the prefrontal cortex isn't merely the bandleader of the brain, issuing one command after another. It's also uniquely versatile. While every other cortical region is precisely tuned for specific kinds of stimuli \u2013 the visual cortex, for example, can only deal with visual information \u2013 the cells of the prefrontal cortex are extremely flexible. They can process whatever kind of data they're told to process. If someone is thinking about an unfamiliar math problem on a standardized test, then her prefrontal neurons are thinking about that problem. And when her attention shifts, and she starts to contemplate the next question on the test, these task-dependent cells seamlessly adjust their focus. The end result is that the prefrontal cortex lets her consciously analyze any type of problem from every possible angle. Instead of responding to the most obvious facts, or the facts that her emotions think are most important, she can concentrate on the facts that might help her come up with the right answer. We can all use executive control to get creative, to think about the same old problem in a new way. For instance, once Wag Dodge realized that he couldn't outrun the flames and that the fire would beat the smokejumpers to the top of the ridge, he needed to use his prefrontal cortex to come up with a new solution. The obvious response wasn't going to work. As Miller notes, 'That Dodge guy had some high prefrontal function.'\n\nConsider the classic psychology puzzle known as the 'candle problem'. A subject is given a book of matches, some candles and a cardboard box containing a few thumbtacks. The person is told to attach the candle to a piece of corkboard in such a way that it can burn properly. Most people initially attempt two common strategies, neither of which will work. The first strategy is to tack the candle directly to the board; this causes the candle wax to shatter. The next is to use the matches to melt the bottom of the candle and then try to stick the candle to the board; the wax does not hold, and the candle falls to the floor. At this point, most people give up. They tell the scientists that the puzzle is impossible; it's a stupid experiment and a waste of time. Less than 20 per cent of people manage to come up with the correct solution, which is to attach the candle to the cardboard box and then tack the cardboard box to the board. Unless the subject has an insight about the box \u2013 that it can do more than hold thumbtacks \u2013 candle after candle will be wasted. The subject repeats his failures while waiting for a breakthrough.\n\nPeople with frontal-lobe lesions can never solve puzzles like the candle problem. Although they understand the rules of the game, they are completely unable to think creatively about the puzzle, to think abstractly about the stumbling block, to look past their initial (and incorrect) answers. The end result is that the frontal-lobe patient fails to execute the counter-intuitive moves required to solve the puzzle, even though the obvious moves have failed. Instead of trying something new, the subject keeps attempting to tack the candle to the board, stubbornly insisting on this strategy until there are no more candles.\n\nMark Jung-Beeman, a cognitive psychologist at Northwestern University, has spent the last 15 years looking for the neural source of these breakthroughs. He wants to understand how the brain, led by the prefrontal cortex, manages to come up with such creative solutions. Jung-Beeman's experiments go like this: he gives a subject three different words (such as _pine, crab_ and _sauce_ ) and asks him to think of a single word that could form a compound word or phrase with all three. (In this case, the answer is _apple: pineapple, crab apple, applesauce_.) What's interesting about this type of verbal puzzle is that the answers often arrive in a flash of insight, the familiar 'aha!' moment. People have no idea how they came up with the necessary word, just as Wag Dodge couldn't explain how he invented the escape fire. Nevertheless, Jung-Beeman found that the mind was carefully preparing itself for the epiphany; every successful insight was preceded by the same sequence of cortical events. (He likes to quote Louis Pasteur: 'Chance favors the prepared mind.')\n\nThe first brain areas activated during the problem-solving process were those involved with executive control, such as the prefrontal cortex and anterior cingulate cortex. The brain was banishing irrelevant thoughts so that the task-dependent cells could properly focus. 'You're getting rid of those errant daydreams and trying to forget about the last word puzzle you worked on,' Jung-Beeman says. 'Insight requires a clean slate.'\n\nAfter exercising top-down control, the brain began generating associations. It selectively activated the necessary brain areas, looking for insights in all the relevant places, searching for the association that would give the answer. Because Jung-Beeman was giving people word puzzles, he saw additional activation in areas related to speech and language, such as the superior temporal gyrus in the right hemisphere. (The right hemisphere is particularly good at generating the kind of creative associations that lead to epiphanies.) 'Most of the possibilities your brain comes up with aren't going to be useful,' he says. 'And it's up to the executive-control areas to keep on looking or, if necessary, change strategies and start looking somewhere else.'\n\nBut then, when the right answer suddenly appeared \u2013 when _apple_ was passed along to the frontal lobes \u2013 there was an immediate realization that the puzzle had been solved. 'One of the interesting things about such moments of insight,' says Jung-Beeman, 'is that as soon as people have the insight, they say it just seems obviously correct. They know instantly that they've solved the problem.'\n\nThis act of recognition is performed by the prefrontal cortex, which lights up when a person is shown the right answer, even if he hasn't come up with the answer himself. Of course, once the insight has been identified, those task-dependent cells in the frontal lobes immediately move on to the next task. The mental slate is once again wiped clean. The brain begins preparing itself for another breakthrough.\n\nOn the afternoon of 19 July 1989, United Airlines Flight 232 took off from Denver Stapleton Airport, bound for Chicago. The conditions for the flight were ideal. The morning thunderstorms had passed, and the sky was a cloudless cerulean blue. Once the DC-10 reached its cruising altitude of 37,000 feet, about thirty minutes after takeoff, Captain Al Haynes turned off the seat-belt sign. He didn't expect to turn it back on until the plane began its descent.\n\nThe first leg of the flight went smoothly. A hot lunch was served to the passengers. The plane was put on autopilot, with supervision by the first officer, William Records. Captain Haynes drank a cup of coffee and stared at the cornfields of Iowa far below. He'd flown this exact route dozens of times before \u2013 Haynes was one of United's most experienced pilots, with more than thirty thousand hours of flight time \u2013 but he never ceased to admire the grid of flat land, the farms laid out in such perfectly straight lines.\n\nAt 3.16 p.m., about an hour after takeoff, the quiet of the cockpit was shattered by the sound of a loud explosion coming from the back of the plane. The frame of the aircraft shuddered and lurched to the right. Haynes's first thought was that the plane was breaking up, that he was about to die in a massive fireball. But then, after a few seconds of gnashing metal, the quiet returned. The plane kept on flying.\n\nHaynes and First Officer Records immediately began scanning the cluster of instruments and dials, looking for some indication of what had gone wrong. The pilots noticed that the number-two engine, the middle engine in the rear of the plane, was no longer operating. (Such a failure can be dangerous, but it's rarely catastrophic, since the DC-10 also has two other engines, one on each wing.) Haynes got out his pilot manual and started going through the engine-failure checklist. The first order of business was to shut off the fuel supply to that engine, in order to minimize the risk of an engine fire. They attempted it, but the fuel lever wouldn't move.\n\nIt had now been a few minutes since the explosion. Records was flying the plane. Haynes was still trying to fix the fuel lines; he assumed that the plane was maintaining its scheduled flight path to Chicago, albeit at a slightly slower pace. That's when Records turned to him and said the one thing a pilot never wants to hear: 'Al, I can't control the airplane.' Haynes looked over at Records, who had applied full left aileron and pushed the yoke so far forward that the controls were pressed against the cockpit dash. Under normal circumstances, such a maneuver would have caused the plane to descend and turn left. Instead, the plane was in a steep ascent with a sharp right bank. If the plane banked much more, it would flip over.\n\nWhat could trigger such a complete loss of control? Haynes assumed there had been a massive electronic failure, but the circuit board looked normal. So did the onboard computers. Then Haynes checked the pressure on his three hydraulic lines: they were all plummeting toward zero. 'I saw that and my heart skipped a beat,' Haynes remembers. 'It was an awful moment, the first time I realized that this was a real disaster.' The hydraulic systems control the plane. They are used to adjust everything from the rudder to the wing flaps. Planes are always engineered with multiple, fully independent hydraulic systems; if one fails, the backup system can take its place. This redundancy means that a catastrophic failure of all three lines simultaneously should be virtually impossible. Engineers calculate the odds of such an event at about a billion to one. 'It wasn't something we ever trained for or practiced,' Haynes says. 'I looked in my pilot manual, but there was nothing about a total loss of hydraulics. It just wasn't supposed to happen.'\n\nBut that's exactly what had happened to this DC-10. For some reason, the loss of the engine had ruptured all three hydraulic lines. (Investigators later discovered that the engine fan disc had fractured, sending shards of metal through the tail section where all three hydraulic lines were located.) Haynes could remember only one other instance when an aircraft had lost all of its hydraulic controls. Japan Airlines Flight 123, a Boeing 747 flying from Tokyo to Osaka in August 1985, had suffered a similar catastrophe after its vertical stabilizer was blown off by an explosive decompression event. The aircraft had steadily drifted downward for more than thirty minutes, eventually crashing into the face of a mountain. More than five hundred people died. It was the deadliest single-aircraft disaster in history.\n\nBack in the cabin, the passengers were beginning to panic. Everyone had heard the explosion; they all could feel the plane careening out of control. Dennis Fitch, a United Airlines flight instructor, was sitting in the middle of the aircraft. After the terrifying boom \u2013 'It sounded like the plane was breaking apart,' Fitch said \u2013 he visually inspected the wings of the plane. There were no obvious signs of damage, although he couldn't figure out why the pilots weren't correcting the plane's steep bank. Fitch knocked on the cockpit doors to see if he could offer any assistance. He taught pilots how to fly the DC-10, so he knew the aircraft inside and out.\n\n'It was an amazing scene,' Fitch remembers. 'Both pilots were at the controls, their tendons in their forearms were raised from effort, their knuckles were white from gripping the handles, but it wasn't doing anything.' When the pilots told Fitch that they had lost hydraulic pressure in all three hydraulic systems, Fitch was shocked. 'There was no procedure for this. When I heard that, I thought, _I'm going to die this afternoon_.'\n\nCaptain Haynes, meanwhile, was desperately trying to think of some way to regain control. He placed a radio call to United Airlines' System Aircraft Management (SAM), a crew of aircraft engineers specially trained to help deal with in-flight emergencies. 'I thought, these guys must know a way out of this mess,' Haynes says. 'That's their job, right?'\n\nBut the engineers at SAM weren't any help. For starters, they didn't believe that all of the hydraulic pressure was really gone. 'SAM kept on asking us to check the hydraulics again,' Haynes says. 'They told us that there must be some pressure left. But I kept on telling them that there was none. All three lines were empty. And then they kept on telling us to check the pilot's manual, but the manual didn't deal with this problem. Eventually, I realized that we were on our own. Nobody was going to land the plane for us.'\n\nHaynes began by making a mental list of the cockpit elements that he could operate without hydraulic pressure. The list was short. In fact, Haynes could think of only one element that might still be useful: the thrust levers, which controlled the speed and power of his two remaining engines. (They are like the gas pedals of the plane.) But what does thrust matter if you can't maneuver? It would be like revving a car without a steering wheel.\n\nThen Haynes had an idea. At first, he dismissed it as crazy. The more he thought about it, however, the less ridiculous it seemed. His idea was to use his thrust levers to steer the plane. The key was differential thrust; _thrust_ is the forward-directed force of an airplane engine, and a difference in thrust between the plane's engines is normally something pilots want to avoid. But Haynes figured that if he idled one engine while the other got a boost of power, the plane should turn to the idled side. The idea was grounded in simple physics, but he had no idea if it would actually work.\n\nThere was little time to lose. The bank of the plane was approaching 38 degrees. If it got past 45 degrees, the plane would flip over and enter a death spiral. So Haynes advanced the throttle for the right engine and idled the left. At first, nothing happened. The plane stayed in a steep bank. But then, ever so slowly, the right wing began to level itself. The plane was now flying in a straight line. Haynes' desperate idea had worked.\n\nFlight 232 was given instructions to land at Sioux City, Iowa, a regional airport about ninety miles to the west. Using nothing but variations in engine thrust, the pilots began a steady right-hand turn. It had been about twenty minutes since the initial explosion, and it seemed as if Haynes and his crew had restored a measure of control to the uncontrollable plane. 'I felt like we were finally making some progress,' Haynes says. 'It was the first time since the explosion that I thought we just might be able to get this bird on the ground.'\n\nBut just as the flight crew was starting to gain a little confidence, the plane started to pitch violently up and down in a relentless cycle. This is known as a phugoid pattern. Under normal flight conditions, phugoids are easy to manage, but since the plane was without any hydraulic pressure, Haynes and his crew were unable to modulate the pitch of the aircraft. The pilots realized that unless they found a way to dampen the phugoids, they could end up like the Japan Airlines Flight 123. They would careen in a sine wave as they steadily lost altitude. And then they'd crash into the cornfields.\n\nHow do you control phugoids in such a situation? At first glance, the answer seems obvious. When the nose of the plane is pitched down, and the air speed is increasing, a pilot should decrease the throttle, so that the plane slows down. And when the plane is pitched up, and the air speed is decreasing, a pilot should increase the throttle in order to prevent a stall. 'You're looking at your air-speed indicator, and the natural reaction of a pilot is to try to balance out what's happening,' Haynes says. But that instinctive reaction is exactly the opposite of what should be done. The aerodynamics of flight contradict common sense; if Haynes had gone with his first impulse, he would soon have lost control of the plane. The aircraft would have entered a steep, unstoppable descent.\n\nInstead of doing that, Haynes carefully thought through the problem. 'I tried to imagine what would happen to the plane depending on how I controlled the thrust levers,' he says. 'It took me a few moments, but that saved me from making a big mistake.' Haynes realized that when the nose tilted down and the air speed built up, he needed to _increase_ power, so that the two remaining engines could bring up the nose. Because the engines on a DC-10 are set below the wings, an increase in engine throttle will cause the plane to pitch up. In other words, he needed to accelerate on the downhill and brake on the uphill. It was such a counter-intuitive idea that Hayes could barely bring himself to execute the plan. 'The hardest part,' Haynes said, 'was when the nose started up and the air speed started to fall, and then you had to close the throttles. That wasn't very easy to do. You felt like you were going to fall out of the sky.'\n\nBut it worked. The pilots managed to keep the plane reasonably level. They couldn't get rid of the phugoid motion \u2013 that would have required actual flight controls \u2013 but they kept it from turning into a deadly dive. The flight crew was now able to focus on their final problem: orchestrating a descent into Sioux City. Haynes knew it would be a struggle. For one thing, the pilots couldn't directly control their rate of descent, since the elevators of the aircraft \u2013 the control surfaces in the tail wing of the plane that modulate altitude \u2013 were completely un responsive. Haynes and the pilots were forced to rely on a rough formula used when flying the DC-10: a 1,000-foot drop in altitude takes approximately three miles in distance. Since the aircraft was now about sixty miles from the airport but was maintaining an altitude of approximately thirty thousand feet, Haynes realized they'd need to make a few loops on their way to the runway. If they tried to rush the descent, they'd risk losing what little stability they had. And so the pilots began a series of right-hand turns as they proceeded northwest to Sioux City. With each turn, they lost a little more altitude.\n\nAs the plane neared the airport, the pilots made final preparations for an emergency landing. Excess fuel was dumped and the throttles were gradually eased. The passengers were told to assume the brace position, with their heads tight against their knees. Haynes could see the landing strip and the fire engines in the distance. Although the pilots had been flying without controls for 40 minutes, they still managed to line up the plane in the middle of the runway, with its wheels down and its nose up. It was an incredible feat of airmanship.\n\nUnfortunately, the pilots had no control over the speed of the plane. They also couldn't brake once they hit the runway. 'You normally land the DC-10 at approximately a hundred and forty knots,' Haynes says. 'We were doing 215 knots and accelerating. You normally touch down at about two to three hundred feet per minute at the most, as a rate of descent. We were doing 1,850 feet per minute. And increasing. And you normally like to go straight down the runway, and we were drifting left and right because of the tail wind.'\n\nThese factors meant that the plane couldn't stay on the tarmac. It skidded through a cornfield and shattered into several sections. The cockpit broke apart from the main body of the plane, like the tip of a pencil, and tumbled end over end to the edge of the airfield. (All of the pilots were knocked unconscious and suffered life-threatening injuries.) A fire broke out in the fuselage. Toxic black smoke filled the main cabin. When the smoke cleared, 112 passengers were dead.\n\nBut the piloting skills of the flight crew \u2013 their ability to control a plane without any controls \u2013 meant that 184 passengers survived the accident. Because the plane made it to the airport, emergency responders were able to treat the wounded and quickly extinguish the flames. As the National Transportation Safety Board concluded in their authoritative report, 'The performance [of the pilots] was highly commendable and greatly exceeded reasonable expectations.' The method of flight control invented in the cockpit of Flight 232 is now a standard part of pilot training.\n\n## _Solving the Problem_\n\nThe first remarkable thing about the performance of the pilots is that they managed to keep their emotions in check. It's not easy to maintain poise when you've lost complete control of your aircraft. In fact, Haynes later admitted that he didn't expect to survive the flight. He assumed that Flight 232 would eventually spiral out of control, that the phugoids would get worse and worse until the plane finally crashed into the ground. 'I thought the best-case scenario was that we'd make the runway but crash-land,' Haynes says. 'And I was still pretty sure that I wouldn't survive that.'\n\nAnd yet, Haynes never let his fear turn into panic. He was in a situation of incomprehensible pressure, confronted with a scenario that was never supposed to happen, but he managed to keep his cool. Such restraint was only possible because Haynes, like Wag Dodge, used his prefrontal cortex to manage his emotions. After the three hydraulic lines failed, the pilot realized that his trained instincts didn't know how to land the plane. Emotions are adept at finding patterns based on experience, so that a person can detect the missile amid the blur of radar blips. But when you encounter a problem you've never experienced before, when your dopamine neurons have no idea what to do, it's essential that you try to tune out your feelings. Pilots call such a state 'deliberate calm', because staying calm in high-pressure situations requires conscious effort. 'Maintaining our composure was one of the hardest things we had to do,' Haynes says. 'We knew we had to focus and think straight, but that's not always so easy.'\n\nPreventing the onset of panic, however, was only the first step. If Haynes and his crew were going to land the plane at Sioux City, they needed to improvise a solution to their unprecedented problem. Consider the use of differential thrust. Such a method of flight control had never been attempted before. Haynes had never practiced it in a simulator or even contemplated the possibility of turning using only his engines. Even the SAM engineers didn't know what to do. And yet, in the terrifying moments after the explosion, when Haynes looked at his dash and saw that he had no central engine and no hydraulic pressure, he was able to figure out a way to keep the plane in the air.\n\nIt's worth taking a closer look at this single decision so that we can better understand what, exactly, allows the prefrontal cortex to deal with such fraught situations. Steven Predmore, a manager of human-factors analysis at Delta Airlines, has studied the decision-making process during Flight 232 in exquisite detail. He began by breaking down the thirty-four minutes of conversation captured by the cockpit voice recorder into a series of thought units, or pieces of information. By analyzing the flow of these thought units, Predmore was able to map out the sequence of events from the perspective of the pilots.\n\nPredmore's study is a gripping portrait of heroism and teamwork. Shortly after Haynes realized that the plane had lost all hydraulic pressure, the air-traffic controllers began consulting with the pilots on the best flight path into Sioux City. Haynes' advice was simple: 'Whatever you do,' he said, 'keep us away from the city.' At other moments, the transcripts reveal the pilots struggling to lighten the mood:\n\nFitch: I'll tell you what, we'll have a beer when this is done.\n\nHaynes: Well, I don't drink, but I'll sure as hell have one.\n\nAnd yet, even as the pilots were cracking jokes, they were making difficult decisions under extreme cognitive stress. During the descent into Sioux City, the number of thought units exchanged in the cockpit consistently exceeded thirty per minute, with peaks of nearly sixty per minute. That's nearly one new piece of information every second. (Under normal flight conditions, the number of thought units rarely exceeds 10 per minute.) Some of this information was critical \u2013 the pilots closely followed their altitude levels \u2013 and some of it was less relevant. After all, it doesn't really matter how the yoke is positioned if the yoke is broken.\n\nThe pilots dealt with this potential information overload by quickly focusing on the most necessary bits of data. They were always thinking about what they should think about, which let them minimize potential distractions. For instance, once Haynes realized that he could control only the throttle levers \u2013 everything else in the cockpit was virtually useless \u2013 he immediately zeroed in on the possibility of steering with his engines. He stopped worrying about his ailerons, elevators and wing flaps. Once the plane was within twenty miles of the Sioux City airport, about twelve minutes from touchdown, the captain started to concentrate on executing the landing. He deliberately ignored everything else. According to Predmore, the ability of the flight crew to successfully prioritize their tasks was a crucial ingredient of their success.\n\nOf course, it's not enough to just think about a problem; Haynes needed to _solve_ his problem, to invent a completely new method of flight control. This is where the prefrontal cortex really demonstrates its unique strengths. It is the only brain region able to take an abstract principle \u2013 in this case, the physics of engine thrust \u2013 and apply it in an unfamiliar context to come up with an entirely original solution. It's what allowed Haynes to logically analyze the situation, to imagine his engines straightening his steep bank. He could model the aerodynamics in his mind.\n\nOnly recently have scientists learned how the prefrontal cortex accomplishes this. The key element is a special kind of memory known as working memory. The name is accurate: by keeping information in short-term storage, where it can be manipulated and analyzed, the brain can work with all the information streaming in from other cortical areas. It is able to determine what information, if any, is relevant to the problem it's trying to solve. For instance, studies show that neurons in the prefrontal areas will fire in response to a stimulus \u2013 such as the sight of some cockpit instrumentation \u2013 and then keep on firing for several seconds after the stimulus has disappeared. This echo of activity allows the brain to make creative associations as seemingly unrelated sensations and ideas overlap. (Scientists refer to this as the restructuring phase of problem-solving, since the relevant information is mixed together in new ways.) It's why Haynes could think about the thrust levers while simultaneously thinking about how to turn the plane. Once this overlapping of ideas occurs, cortical cells start to form connections that have never existed before, wiring themselves into entirely new networks. And then, after the insight has been generated, the prefrontal cortex is able to identify it: you immediately realize that this is the answer you've been searching for. 'I don't know where the idea for differential thrust came from,' Haynes says. 'It just occurred to me, all of a sudden, out of nowhere.' From the perspective of the brain, new ideas are merely several old thoughts that occur at the exact same time.\n\nThe problem-solving abilities of working memory and the prefrontal cortex are a crucial aspect of human intelligence. Numerous studies have found strong correlations between scores on tests of working memory and tests of general intelligence. Being able to hold more information in the prefrontal cortex, and being able to hold on to that information for longer, means that brain cells are better able to form useful associations. At the same time, the rational brain must also stringently filter out all extraneous thoughts, since they might lead to unhelpful connections. Unless you are disciplined about what you choose to think about \u2013 and the pilots of Flight 232 were extremely disciplined \u2013 you won't be able to effectively think through your problem. You'll be so overwhelmed by all those incoming ideas that you won't be able to figure out which ones are genuine insights.\n\nLook, for example, at the phugoids. When the aircraft started to pitch up and down, Haynes' first impulse was to increase the throttle when the plane was ascending, so that the plane maintained air speed. But then Haynes made himself think, for a few extra seconds, about the implications of this approach. He blocked out all the other things he could have been worrying about \u2013 he still didn't know how he was going to land the plane \u2013 and focused instead on the relationship of his thrust levers and the pitch of the plane. That's when Haynes realized that trusting his instincts in this situation was a deadly mistake. His explicit analysis, made possible by working memory, allowed him to come up with a new solution. If the plane was going up, then he needed to slow down.\n\nSuch decision-making is the essence of rationality. In the months after Flight 232, the United training center in Denver commissioned numerous pilots, including a test pilot from McDonnell Douglas, to see if anyone could land a DC-10 without hydraulics. The training center used a flight simulator that was programmed with the precise conditions faced by the United crew on that July day. 'These other pilots kept trying to land the plane at Sioux City, just like we did,' says Haynes. 'But they always had some kind of unfortunate event and kept on crashing outside the airport.' In fact, the pilots trying to land the DC-10 in the simulator failed to make the runway on their first 57 attempts.\n\nHaynes is a modest man; he says most of the passengers survived because of 'luck and teamwork'. However, the landing of Flight 232 on the Sioux City runway was clearly a case of Haynes creating his own luck. Because he took advantage of his prefrontal cortex, relying on its uniquely flexible neurons, he managed to avert an almost certain disaster. He was able to maintain his cool and analyze the situation in a deliberate manner so that he could generate the necessary flash of insight. 'I'm no genius,' Haynes says. 'But a crisis like that sure can sharpen the mind.'\n\nAlthough the rational talents of the prefrontal cortex kept Flight 232 from crashing into a cornfield, it's important to realize that rationality isn't an all-purpose solution. In the next chapter, we are going to look at what happens when people use their prefrontal cortices in the wrong way. It's possible to think too much.\n\n* Although certain sections of this brain area, such as the orbitofrontal cortex, are actually concerned with the perception of emotional states, the upper two-thirds of the prefrontal cortex \u2013 particularly the dorsolateral prefrontal cortex, or DLPFC \u2013 is generally regarded as the rational center of the brain. When you crunch numbers, deploy logic, or rely on deliberate analysis, you're using your DLPFC.\n\n* The most common form of the surgery involved inserting a scalpel just under the eyelid and above the eye, then hammering it through a thin layer of bone, cutting the bundle of nerves that connected the prefrontal cortex to the thalamus.\n\n* And then there's the case of the married, middle-aged Virginia schoolteacher who suddenly started downloading child pornography and seducing young girls. His behavior was so brazen that he was quickly arrested and convicted of child molestation; he was sent to a treatment program for pedophiles, but he was expelled from the program after propositioning several women there. Having failed rehab, he was to appear in court for sentencing, but the day before his court date, he went to the emergency room complaining of blinding headaches and a constant urge to rape his neighbor. After ordering an MRI, the doctors saw the source of the problem: he had a massive tumor lodged in his frontal cortex. After the tumor was removed, the deviant sexual urges immediately disappeared. The man was no longer a hypersexual monster. Unfortunately, this reprieve was brief; the tumor started to grow back within a year. His frontal cortex was once again incapacitated, and the urges of pedophilia returned.\n\n* But there are ways to compensate for the irrational brains of teens. For instance, when West Virginia revoked driving permits for students who were under the age of 18 and who dropped out of school, the dropout rate fell by one-third in the first year. While teens were blind to the long-term benefits of getting a high school diploma, they could appreciate the short-term punishment of losing a license. The New York City schools have recently begun experimenting with a program that pays students for improving their standardized test scores; initial results have been extremely encouraging. By focusing on immediate rewards, these incentive programs help correct the immature prefrontal cortices of children and teenagers.\nFIVE\n\n# Choking on Thought\n\nTHE LESSON OF Wag Dodge, television focus groups, and Flight 232 is that a little rational thought can save the day. In such situations, the prefrontal cortex is uniquely designed to come up with creative answers, to generate that flash of insight that leads a person to the right decision. Such narratives fit comfortably with our broad assumption that more deliberation is always better. In general, we believe that carefully studying something leads to better outcomes, since we'll avoid careless errors. Consumers should always comparison shop so that they find the best products. Before we invest in stocks, we are supposed to learn as much as possible about the company. We expect doctors to order numerous diagnostic tests, even if the tests are expensive and invasive. In other words, people believe that a decision that's the result of rational deliberation will always be better than an impulsive decision. This is why one shouldn't judge a book by its cover or propose marriage on the first date. When in doubt, we try to resort to careful analysis and engage the rational circuits of the prefrontal cortex.\n\nThis faith in the power of reason is easy to understand. Ever since Plato, we've been assured that a perfectly rational world would be a perfect world, a Shangri-la ruled by statistical equations and empirical evidence. People wouldn't run up credit card debt or take out subprime loans. There would be no biases or prejudices, just cold hard facts. This is the utopia dreamed of by philosophers and economists.\n\nHowever, this new science of decision-making (a science rooted in the material details of the brain) is most interesting when the data turns out to contradict the conventional wisdom. Ancient assumptions are revealed as just that: assumptions. Untested theories. Unsubstantiated speculation. Plato, after all, didn't do experiments. He had no way of knowing that the rational brain couldn't solve every problem, or that the prefrontal cortex had severe limitations. The reality of the brain is that, sometimes, rationality can lead us astray.\n\nFor Ren\u00e9e Fleming, the opera superstar, the first sign of trouble came during a routine performance of Mozart's _The Marriage_ _of Figaro_ at the Lyric Opera of Chicago. Fleming was singing the 'Dove sono' aria from act three, one of the most beloved songs in all of opera. At first, Fleming sang Mozart's plaintive melody with her typical perfection. She made the high notes sound effortless, her voice capturing the intensity of emotion while maintaining her near perfect pitch. Most sopranos struggle with Mozart's tendency to compose in the _passaggio_ , or the awkward part of the vocal range between registers. But not Fleming. Her performance the night before had earned her a long standing ovation.\n\nBut then, just as she neared the most difficult section of the aria \u2013 a crescendo of fluttering pitches, in which her voice has to echo the violins \u2013 Fleming felt a sudden stab of self-doubt. She couldn't stop thinking that she was about to make a mistake. 'It caught me by surprise,' she later wrote in her memoir. 'That aria was never an easy piece, but it was certainly one with which I had had an enormous amount of experience.' In fact, Fleming had performed this piece hundreds of times before. Her first big operatic break had been singing the role of the Countess at the Houston Opera, more than a decade earlier. The tragic 'Dove sono' aria, in which the Countess questions the loss of her happiness, had been featured on Fleming's first album and became a standard part of her repertoire. It was, Fleming said, her 'signature piece'.\n\nAnd yet now, she could barely breathe. She felt her diaphragm constrict, sucking the power from her voice. Her throat tightened and her pulse started to race. Although Fleming fought her way through the rest of the song, stealing breaths wherever possible \u2013 she still managed to get a standing ovation \u2013 she was deeply shaken. What had happened to her self-confidence? Why did her favorite aria suddenly make her so nervous?\n\nBefore long, Fleming's performance problems became chronic. The songs that used to be second nature were suddenly impossible to sing. Every performance was a struggle against anxiety, against that monologue in her head telling her not to make a mistake. 'I had been undermined by a very negative inner voice,' she wrote, 'a little nattering in my ear that said, \"Don't do that... Don't do this... Your breath is tight... Your tongue has gone back... Your palate is down... The top is spread... Relax your shoulders!\"' Eventually, it got so bad that Fleming planned to quit opera altogether. She was one of the most talented performers in the world, and yet she could no longer perform.\n\nPerformers call such failures 'choking', because a person so frayed by pressure might as well not have any oxygen. What makes choking so morbidly fascinating is that the only thing incapacitating the performer is his or her own thoughts. Fleming, for example, was so worried about hitting the high notes of Mozart's opera that she failed to hit them. The inner debate over proper technique made her voice seize up, and it became impossible to sing with the necessary speed and virtuosity. Her mind was sabotaging itself.\n\nWhat causes choking? Although it might seem like an amorphous category of failure, or even a case of excess emotion, choking is actually triggered by a specific mental mistake: _thinking too much_. The sequence of events typically goes like this: when a person gets nervous about performing, he naturally becomes extra self-conscious. He starts to focus on himself, trying to make sure that he doesn't make any mistakes. He begins scrutinizing actions that are best performed on autopilot. Fleming started to think about aspects of singing that she hadn't thought about since she was a beginner, such as where to position her tongue and how to shape her mouth for different pitches. This kind of deliberation can be lethal for a performer. The opera singer forgets how to sing. The pitcher concentrates too much on his motion and loses control of his fastball. The actor gets anxious about his lines and seizes up on stage. In each of these instances, the natural fluidity of performance is lost. The grace of talent disappears.\n\nConsider one of the most famous chokes in sports history: the collapse of Jean Van de Velde on the last hole of the 1999 British Open. Until that point in the tournament, Van de Velde had been playing nearly flawless golf. He had a three-stroke lead entering the eighteenth hole, which meant that he could double-bogey (that is, be two strokes over par) and still win. On his previous two rounds, he'd birdied (been one stroke under par) this very hole.\n\nNow Van de Velde was the only player on the course. He knew that the next few shots could change his life forever, turning a PGA journeyman into an elite golfer. All he had to do was play it safe. During his warm-up swings on the eighteenth, Van de Velde looked nervous. It was a blustery Scottish day, but beads of sweat were glistening on his face. After repeatedly wiping away the perspiration, he stepped up to the tee, planted his feet, and jerked back his club. His swing looked awkward. His hips spun out ahead of his body, so that the face of his driver didn't hit straight on the ball. Van de Velde watched the white speck sail away and then bowed his head. He had bent the ball badly to the right, and it ended up 20 yards from the fairway, buried in the rough. On his next shot, he made the same mistake, but this time he sent the ball so far right that it bounced off the grandstands and ended up in a patch of knee-high grass. His third shot was even worse. By this point, his swing was so out of sync that he almost missed the ball; it was launched into the air along with a thick patch of grass. As a result, his shot came up far short and plunged into the water hazard just before the green. Van de Velde grimaced and turned away, as if he couldn't bear to watch his own collapse. After taking a penalty, he was still 60 yards short of the hole. Once again, his tentative swing was too weak, and the ball ended up exactly where he didn't want it: in a sandy bunker. From there, he managed to chip onto the green and, after _seven_ errant shots, finish the round. But it was too late. Van de Velde had lost the British Open.\n\nThe pressure of the eighteenth hole was Van de Velde's undoing. When he started thinking about the details of his swing, his swing broke down. On the last seven shots, Van de Velde seemed like a different golfer. He had lost his easy confidence. Instead of playing like a pro on the PGA tour, he started swinging with the cautious deliberation of a beginner with a big handicap. He was suddenly focusing on the mechanics of his stroke, making sure that he didn't torque his wrist or open his hips. He was literally regressing before the crowd, reverting to a mode of explicit thought that he hadn't used on the golf green since he was a child learning how to swing.\n\nSian Beilock, a professor of psychology at the University of Chicago, has helped illuminate the anatomy of choking. She uses putting on the golf green as her experimental paradigm. When people are first learning how to putt, the activity can seem daunting. There are just so many things to think about. A golfer needs to assess the lay of the green, calculate the line of the ball, and get a feel for the grain of the turf. Then the player has to monitor the putting motion and make sure the ball is hit with a smooth, straight stroke. For an inexperienced player, a golf putt can seem impossibly hard, like a life-size trigonometry problem.\n\nBut the mental exertion pays off, at least at first. Beilock has shown that novice putters hit better shots when they consciously reflect on their actions. The more time the beginner spends thinking about the putt, the more likely he is to sink the ball in the hole. By concentrating on the golf game, by paying attention to the mechanics of the stroke, the novice can avoid beginner's mistakes.\n\nA little experience, however, changes everything. After a golfer has learned how to putt \u2013 once he or she has memorized the necessary movements \u2013 analyzing the stroke is a waste of time. The brain already knows what to do. It automatically computes the slope of the green, settles on the best putting angle, and decides how hard to hit the ball. In fact, Beilock found that when experienced golfers are forced to think about their putts, they hit significantly _worse_ shots. 'We bring expert golfers into our lab, and we tell them to pay attention to a particular part of their swing, and they just screw up,' Beilock says. 'When you are at a high level, your skills become somewhat automated. You don't need to pay attention to every step in what you're doing.'\n\nBeilock believes that this is what happens when people 'choke'. The part of the brain that monitors behavior \u2013 a network centered in the prefrontal cortex \u2013 starts to interfere with decisions that are normally made without thinking. It begins second-guessing the skills that have been honed through years of diligent practice. The worst part about choking is that it tends to be a downward spiral. The failures build on one another, and a stressful situation is made even more stressful. After Van de Velde lost the British Open, his career hit the skids. Since 1999, he has failed to finish in the top 10 in a major tournament.*\n\nChoking is merely a vivid example of the havoc that can be caused by too much thought. It's an illustration of rationality gone awry, of what happens when we rely on the wrong brain areas. For opera singers and golf players, such deliberate thought processes interfere with the trained movements of their muscles, so that their own bodies betray them.\n\nBut the problem of thinking too much isn't limited to physical performers. Claude Steele, a professor of psychology at Stanford, studies the effects of performance anxiety on standardized-test scores. When Steele gave a large group of Stanford sophomores a set of questions from the Graduate Record Examination (GRE) and told the students that it would measure their innate intellectual ability, he found that the white students performed significantly better than their black counterparts. This discrepancy \u2013 commonly known as the achievement gap \u2013 conformed to a large body of data showing that minority students tend to score lower on a wide variety of standardized tests, from the SAT to the IQ test.\n\nHowever, when Steele gave a separate group of students the same test but stressed that it was _not_ a measure of intelligence \u2013 he told them it was merely a preparatory drill \u2013 the scores of the white and black students were virtually identical. The achievement gap had been closed. According to Steele, the disparity in test scores was caused by an effect that he calls stereotype threat. When black students are told that they are taking a test to measure their intelligence, it brings to mind, rather forcefully, the ugly and untrue stereotype that blacks are less intelligent than whites. (Steele conducted his experiments soon after _The Bell Curve_ was published, but the same effect also exists when women take a math test that supposedly measures 'cognitive differences between the genders' or when white males are exposed to a stereotype about the academic superiority of Asians.) The Stanford sophomores were so worried about being viewed through the lens of a negative stereotype that they performed far below their abilities. 'What you tend to see [during stereotype threat] is carefulness and second-guessing,' Steele said. 'When you go and interview them, you have the sense that when they are in the stereotype-threat condition they say to themselves, \"Look, I'm going to be careful here. I'm not going to mess things up.\" Then, after having decided to take that strategy, they calm down and go through the test. But that's not the way to succeed on a standardized test. The more you do that, the more you will get away from the intuitions that help you, the quick processing. They think they did well, and they are trying to do well. But they are not.'\n\nThe lesson of Ren\u00e9e Fleming, Jean Van de Velde and these Stanford students is that rational thought can backfire. While reason is a powerful cognitive tool, it's dangerous to rely _exclusively_ on the deliberations of the prefrontal cortex. When the rational brain hijacks the mind, people tend to make all sorts of decision-making mistakes. They hit bad golf shots and choose wrong answers on standardized tests. They ignore the wisdom of their emotions \u2013 the knowledge embedded in their dopamine neurons \u2013 and start reaching for things that they can explain. (One of the problems with feelings is that even when they are accurate, they can still be hard to articulate.) Instead of going with the option that feels the best, a person starts going with the option that _sounds_ the best, even if it's a very bad idea.\n\n##  _How Not to Shop_\n\nWhen _Consumer Reports_ tests a product, it follows a strict proto col. First, the magazine's staff assembles a field of experts. If they're testing family sedans, they rely on automotive experts; if audio speakers are being scrutinized, the staff members bring in people trained in acoustics. Then the magazine's staff gather all the relevant products in that category and try to hide the brand names. (This often requires lots of masking tape.) The magazine aspires to objectivity.\n\nBack in the mid-1980s, _Consumer Reports_ decided to conduct a taste test for strawberry jam. As usual, the editors invited several food experts, all of whom were 'trained sensory panelists'. These experts blindly sampled forty-five different jams, scoring each on 16 different characteristics, such as sweetness, fruitiness, texture and spreadability. The scores were then totaled, and the jams were ranked.\n\nA few years later, Timothy Wilson, a psychologist at the University of Virginia, decided to replicate this taste test with his undergraduate students. Would the students have the same preferences as the experts? Did everybody agree on which strawberry jams tasted the best?\n\nWilson's experiment was simple: he took the first, eleventh, twenty-fourth, thirty-second and forty-fourth best-tasting jams according to _Consumer Reports_ and asked the students to rank them. In general, the preferences of the college students closely mirrored the preferences of the experts. Both groups thought Knott's Berry Farm and Alpha Beta were the two best-tasting brands, with Featherweight a close third. They also agreed that the worst strawberry jams were Acme and Sorrel Ridge. When Wilson compared the preferences of the students and the _Consumer Reports_ panelists, he found that they had a statistical correlation of 0.55, which is rather impressive. When it comes to judging jam, we are all natural experts. Our brains are able to automatically pick out the products that provide us with the most pleasure.\n\nBut that was only the first part of Wilson's experiment. He repeated the jam taste test with a separate group of college students, only this time he asked them to explain _why_ they preferred one brand over another. As they tasted the jams, the students filled out written questionnaires, which forced them to analyze their first impressions, to consciously explain their impulsive preferences. All this extra analysis seriously warped their jam judgment. The students now preferred Sorrel Ridge \u2013 the worst-tasting jam, according to _Consumer Reports_ \u2013 to Knott's Berry Farm, which was the experts' favorite jam. The correlation plummeted to 0.11, which means that there was virtually no relationship between the rankings of the experts and the opinions of these introspective students.\n\nWilson argues that 'thinking too much' about strawberry jam causes us to focus on all sorts of variables that don't actually matter. Instead of just listening to our instinctive preferences \u2013 the best jam is associated with the most positive feelings \u2013 our rational brains search for reasons to prefer one jam over another. For example, someone might notice that the Acme brand is particularly easy to spread, and so he'll give it a high ranking, even if he doesn't actually care about the spreadability of jam. Or a person might notice that Knott's Berry Farm has a chunky texture, which seems like a bad thing, even if she's never really thought about the texture of jam before. But having a chunky texture _sounds_ like a plausible reason to dislike a jam, and so she revises her preferences to reflect this convoluted logic. People talk themselves into liking Acme jam more than Knott's Berry Farm.\n\nThis experiment illuminates the danger of always relying on the rational brain. There is such a thing as too much analysis. When you overthink at the wrong moment, you cut yourself off from the wisdom of your emotions, which are much better at assessing actual preferences. You lose the ability to know what you really want. And then you choose the worst strawberry jam.\n\nWilson was intrigued by the strawberry-jam experiment. It seemed to contradict one of the basic tenets of Western thought, which is that careful self-analysis leads to wisdom. As Socrates famously said, 'The unexamined life is not worth living.' Socrates clearly didn't know about strawberry jam.\n\nBut perhaps food products are unique, since people are notoriously bad at explaining their own preferences. So Wilson came up with another experiment. This time he asked female college students to select their favorite poster. He gave them five options: a Monet landscape, a van Gogh painting of some purple lilies, and three humorous cat posters. Before making their choices, the subjects were divided into two groups. The first was the non-thinking group: they were instructed to simply rate each poster on a scale from 1 to 9. The second group had a tougher task: before they rated the posters, they were given questionnaires that asked them _why_ they liked or disliked each of the five posters. At the end of the experiment, each of the subjects took her favorite poster home.\n\nThe two groups of women made very different choices. Ninety-five per cent of the non-thinkers chose either the Monet or the van Gogh. They instinctively preferred the fine art. However, subjects who thought about their poster decisions first were almost equally split between the paintings and the humorous cat posters. What accounted for the difference? 'When looking at a painting by Monet,' Wilson writes, 'most people generally have a positive reaction. When thinking about why they feel the way they do, however, what comes to mind and is easiest to verbalize might be that some of the colors are not very pleasing, and that the subject matter, a haystack, is rather boring.' As a result, the women ended up selecting the funny feline posters, if only because those posters gave them more grist for their explanatory mill.\n\nWilson conducted follow-up interviews with the women a few weeks later to see which group had made the better decision. Sure enough, the members of the non-thinking group were much more satisfied with their choice of posters. While 75 per cent of the people who had chosen cat posters regretted their selection, nobody regretted selecting the artistic poster. The women who listened to their emotions ended up making much better decisions than the women who relied on their reasoning powers. The more people thought about which posters they wanted, the more misleading their thoughts became. Self-analysis resulted in _less_ self-awareness.\n\nThis isn't just a problem for insignificant decisions like choosing jam for a sandwich or selecting a cheap poster. People can also think too much about more important choices, like buying a home. As Ap Dijksterhuis, a psychologist at Radboud University, in the Netherlands, notes, when people are shopping for real estate, they often fall victim to a version of the strawberry-jam error, or what he calls a 'weighting mistake'. Consider two housing options: a three-bedroom apartment located in the middle of a city that would give you a 10-minute commute, and a five-bedroom McMansion in the suburbs that would result in a 45-minute commute. 'People will think about this tradeoff for a long time,' Dijksterhuis says, 'and most of them will eventually choose the large house. After all, a third bathroom or extra bedroom is very important for when Grandma and Grandpa come over for Christmas, whereas driving two hours each day is really not that bad.' What's interesting is the more time people spend deliberating, the more important that extra space becomes. They'll imagine all sorts of scenarios (a big birthday party, Thanksgiving dinner, another child) that turns the suburban house into a necessity. The lengthy commute, meanwhile, will seem less and less significant, at least when it's compared to the lure of an extra bathroom. But as Dijksterhuis points out, the reasoning is backward: 'The additional bathroom is a completely superfluous asset for at least 362 or 363 days each year, whereas a long commute _does_ become a burden after a while.' For instance, a recent study found that when a person travels more than one hour in each direction, he or she has to make 40 per cent more money in order to be as 'satisfied with life' as a person with a short commute. Another study, led by Daniel Kahneman and the economist Alan Krueger, surveyed 900 working women in Texas and found that commuting was, by far, the least pleasurable part of their day. And yet, despite these gloomy statistics, nearly 20 per cent of American workers commute more than forty-five minutes each way. (More than 3.5 million Americans spend more than three hours each day traveling to and from work, and they're the fastest-growing category of commuter.) According to Dijksterhuis, all these people are making themselves miserable because they failed to properly weigh the relevant variables when they were choosing where to live. Just as strawberry-jam tasters who consciously analyzed their preferences were persuaded by irrelevant factors like spread ability and texture, the deliberative homeowners focused on less important details like square footage and number of bath rooms. (It's easier to consider quantifiable facts than future emotions, such as how you'll feel when you're stuck in a rush-hour traffic jam.) The prospective homeowners assumed a bigger house in the suburbs would make them happy, even if it meant spending an extra hour in the car every day. But they were wrong.\n\nThe best window into this mental process \u2013 what's actually happening inside the brain when you talk yourself into choosing the wrong strawberry jam \u2013 comes from studies of the placebo effect. It's long been recognized that the placebo effect is extremely powerful; anywhere between 35 and 75 per cent of people get better after receiving pretend medical treatments, such as sugar pills. A few years ago, Tor Wager, a neuroscientist at Columbia University, wanted to figure out why placebos were so effective. His experiment was brutally straightforward: he gave college students electric shocks while they were stuck in an fMRI machine. (The subjects were well compensated, at least by undergraduate standards.) Half of the people were then supplied with a fake pain-relieving cream. Even though the cream had no analgesic properties \u2013 it was just a hand moisturizer \u2013 people given the pretend cream said the shocks were significantly less painful. The placebo effect eased their suffering. Wager then imaged the specific parts of the brain that controlled this psychological process. He discovered that the placebo effect depended entirely on the prefrontal cortex, the center of rational thought. When people were told that they'd just received pain-relieving cream, their frontal lobes responded by inhibiting the activity of their emotional brain areas (like the insula) that normally respond to pain. Because people _expected_ to experience less pain, they ended up experiencing less pain. Their predictions became self-fulfilling prophecies.\n\nThe placebo effect is a potent source of self-help. It demonstrates the power of the prefrontal cortex to modulate even the most basic bodily signals. Once this brain area comes up with reasons to experience less pain \u2013 the cream is supposed to provide pain relief \u2013 those reasons become powerful distortions. Unfortunately, the same rational brain areas responsible for temporarily reducing suffering also mislead us about many daily decisions. The prefrontal cortex can turn off pain signals, but it can also cause a person to ignore the feelings that lead to choosing the best poster. In these situations, conscious thoughts interfere with good decision-making.\n\nLook, for example, at this witty little experiment. Baba Shiv, a neuroeconomist at Stanford, supplied a group of people with Sobe Adrenaline Rush, an 'energy' drink that was supposed to make them feel more alert and energetic. (The drink contained a potent brew of sugar and caffeine that, the bottle promised, would impart 'superior functionality'.) Some participants paid full price for the drinks, while others were offered a discount. After drinking the product, participants were asked to solve a series of word puzzles. Shiv found that people who'd paid discounted prices consistently solved about 30 per cent fewer puzzles than the people who'd paid full price for the drinks. The subjects were convinced that the stuff on sale was much less potent, even though all the drinks were identical. 'We ran the study again and again, not sure if what we got had happened by chance or fluke,' Shiv says. 'But every time we ran it, we got the same results.'\n\nWhy did the cheaper energy drink prove less effective? According to Shiv, consumers typically suffer from a version of the placebo effect. Since they _expect_ cheaper goods to be less effective, they generally _are_ less effective, even if the goods are identical to more expensive products. This is why brand-name aspirin works better than generic aspirin and why Coke tastes better than cheaper colas, even if most consumers can't tell the difference in blind taste tests. 'We have these general beliefs about the world \u2013 for example, that cheaper products are of lower quality \u2013 and they translate into specific expectations about specific products,' said Shiv. 'Then, once these expectations are activated, they start to really impact our behavior.' The rational brain distorts the sense of reality, so the ability to properly assess the alternatives is lost. Instead of listening to the trustworthy opinions generated by our emotional brains, we follow our own false assumptions.\n\nResearchers at Caltech and Stanford recently lifted the veil on this strange process. Their experiment was organized like a winetasting. Twenty people sampled five cabernet sauvignons that were distinguished solely by their retail prices, with bottles ranging in cost from five dollars to $90. Although the people were told that all five wines were different, the scientists weren't telling the truth: there were only three different wines. This meant that the same wines often reappeared, but with different price labels. For example, the first wine offered during the tasting \u2013 it was a bottle of a cheap California cabernet \u2013 was labeled both as a five-dollar wine (its actual retail price) and as a $45-dollar wine, a 900 per cent markup. All of the red wines were sipped by each subject inside an fMRI machine.\n\nNot surprisingly, the subjects consistently reported that the more expensive wines tasted better. They preferred the $90 bottle to the $10 bottle and thought the $45 cabernet was far superior to the five-dollar plonk. By conducting the wine tasting inside an fMRI machine \u2013 the drinks were sipped via a network of plastic tubes \u2013 the scientists could see how the brains of the subjects responded to the different wines. While a variety of brain regions were activated during the experiment, only one brain region seemed to respond to the _price_ of the wine rather than the wine itself: the prefrontal cortex. In general, more expensive wines made parts of the pre frontal cortex more excited. The scientists argue that the activity of this brain region shifted the preferences of the winetasters, so that the $90 cabernet seemed to taste better than the $35 cabernet, even though they were actually the same wine.\n\nOf course, the wine preferences of the subjects were clearly nonsensical. Instead of acting like rational agents \u2013 getting the most utility for the lowest possible price \u2013 they were choosing to spend more money for an identical product. When the scientists repeated the experiment with members of the Stanford University wine club, they got the same results. In a blind tasting, these semi-experts were also misled by the made-up price tags. 'We don't realize how powerful our expectations are,' says Antonio Rangel, the neuroeconomist at Caltech who led the study. 'They can really modulate every aspect of our experience. And if our expectations are based on false assumptions' \u2013 like the assumption that more expensive wine tastes better \u2013 'they can be very misleading.'\n\nThese experiments suggest that, in many circumstances, we could make better consumer decisions by knowing _less_ about the products we are buying. When you walk into a store, you are besieged by information. Even purchases that seem simple can quickly turn into a cognitive quagmire. Look at the jam aisle. A glance at the shelves can inspire a whole range of questions. Should you buy the smooth-textured strawberry jam or the one with less sugar? Does the more expensive jam taste better? What about organic jam? (The typical supermarket contains more than two hundred varieties of jam and jelly.) Rational models of decision-making suggest that the way to find the best product is to take all of this information into account, to carefully analyze the different brands on display. In other words, a person should choose a jam with his or her prefrontal cortex. But this method can backfire. When we spend too much time thinking in the supermarket, we can trick ourselves into choosing the wrong things for the wrong reasons. That's why the best critics, from _Consumer Reports_ to Robert Parker, always insist on blind comparisons. They want to avoid the deceptive thoughts that corrupt decisions. The prefrontal cortex isn't good at picking out jams or energy drinks or bottles of wine. Such decisions are like a golf swing: they are best done with the emotional brain, which generates its verdict automatically.\n\nThis 'irrational' approach to shopping can save us lots of money. After Rangel and his colleagues finished their brain-imaging experiment, they asked the subjects to taste the five different wines again, only this time the scientists didn't provide any price information. Although the subjects had just listed the $90 wine as the most pleasant, they now completely reversed their preferences. When the tasting was truly blind, when the subjects were no longer biased by their prefrontal cortex, the cheapest wine got the highest ratings. It wasn't fancy, but it tasted the best.\n\n##  _The Imperfect Machine_\n\nIf the mind were an infinitely powerful organ, a limitless supercomputer without constraints, then rational analysis would always be the ideal decision-making strategy. Information would be an unqualified good. We would be foolish to ignore the omniscient opinions of the Platonic charioteer.\n\nThe biological reality of the brain, however, is that it's severely bounded, a machine subject to all sorts of shortcomings. This is particularly true of the charioteer, who is tethered to the prefrontal cortex. As the psychologist George Miller demonstrated in his famous essay 'The Magical Number Seven, Plus or Minus Two', the conscious brain can only handle about seven pieces of data at any one moment. 'There seems to be some limitation built into us by the design of our nervous systems, a limit that keeps our channel capacities in this general range,' Miller wrote. While we can control these rational neural circuits \u2013 they think about what we tell them to think about \u2013 they constitute a relatively small part of the brain, just a few microchips within the vast mainframe of the mind. As a result, even choices that seem straightforward \u2013 like choosing a jam in the supermarket \u2013 can overwhelm the prefrontal cortex. It gets intimidated by all the jam data. And that's when bad decisions are made.\n\nConsider this experiment. You're sitting in a bare room, with just a table and a chair. A scientist in a white lab coat walks in and says that he's conducting a study of long-term memory. The scientist gives you a seven-digit number to remember and asks you to walk down the hall to the room where your memory will be tested. On the way to the testing room, you pass a refreshment table for subjects taking part in the experiment. You are given a choice between a decadent slice of German chocolate cake and a bowl of fruit salad. What do you choose?\n\nNow let's replay the experiment. You are sitting in the same room. The same scientist gives you the same explanation. The only difference is that instead of being asked to remember a seven-digit number, you are given only two numbers, a far easier mental task. You then walk down the hall and are given the same choice between cake and fruit.\n\nYou probably don't think the number of digits will affect your choice; if you choose the chocolate cake, it is because you want cake. But you'd be wrong. The scientist who explained the experiment was lying; this isn't a study of long-term memory, it's a study of self-control.\n\nWhen the results from the two different memory groups were tallied, the scientists observed a striking shift in behavior. Fifty-nine per cent of people trying to remember seven digits chose the cake, compared to only 37 per cent of the two-digit subjects. Distracting the brain with a challenging memory task made a person much more likely to give in to temptation and choose the calorie-dense dessert. (The premise is that German chocolate cake is to adults what marshmallows are to four-year-olds.) The subjects' self-control was overwhelmed by five extra numbers.\n\nWhy did the two groups behave so differently? According to the Stanford scientists who designed the experiment, the effort required to memorize seven digits drew cognitive resources away from the part of the brain that normally controls emotional urges. Because working memory and rationality share a common cortical source \u2013 the prefrontal cortex \u2013 a mind trying to remember lots of information is less able to exert control over its impulses. The substrate of reason is so limited that a few extra digits can become an extreme handicap.\n\nThe shortcomings of the prefrontal cortex aren't apparent only when memory-storage capacity is exceeded. Other studies have shown that a slight drop in blood-sugar levels can also inhibit self-control, since the frontal lobes require lots of energy in order to function. Look, for example, at this experiment led by Roy Baumeister, a psychologist at Florida State University. The experiment began with a large group of undergraduates performing a mentally taxing activity that involved watching a video while ignoring the text of random words scrolling on the bottom of the screen. (It takes a conscious effort to _not_ pay attention to salient stimuli.) The students were then offered some lemonade. Half of them got lemonade made with real sugar, and the other half got lemonade made with a sugar substitute. After giving the glucose time to enter the bloodstream and perfuse the brain (about 15 minutes), Baumeister had the students make decisions about apartments. It turned out that the students who were given the drink without real sugar were significantly more likely to rely on instinct and intuition when choosing a place to live, even if that led them to choose the wrong places. The reason, according to Baumeister, is that the rational brains of these students were simply too exhausted to think. They'd needed a restorative sugar fix, and all they'd gotten was Splenda. This research can also help explain why we get cranky when we're hungry and tired: the brain is less able to suppress the negative emotions sparked by small annoyances. A bad mood is really just a rundown prefrontal cortex.\n\nThe point of these studies is that the flaws and foibles of the rational brain \u2013 the fact that it's an imperfect piece of machinery \u2013 are constantly affecting our behavior, leading us to make decisions that seem, in retrospect, quite silly. These mistakes extend far beyond poor self-control. In 2006, psychologists at the University of Pennsylvania decided to conduct an experiment with M&Ms in an upscale apartment building. One day, they left out a bowl of the chocolate candies and a small scoop. The next day they refilled the bowl with M&Ms but placed a much larger scoop beside it. The result would not surprise anyone who has ever finished a Big Gulp soda or a supersize serving of McDonald's fries: when the scoop size was increased, people took 66 per cent more M&Ms. Of course, they could have taken just as many candies on the first day; they simply would have had to use a few more scoops. But just as larger serving sizes cause us to eat more, the larger scoop made the residents more gluttonous.\n\nThe real lesson of the candy scoop, however, is that people are terrible at measuring stuff. Instead of counting the number of M&Ms they eat, they count the number of scoops. The scientists found that most people took a single scoop and ended up consuming however many candies that scoop happened to contain. The same thing happens when people sit down to dinner: they tend to eat whatever is on their plates. If the plate is twice as large (and American serving sizes have grown 40 per cent in the last 25 years), they'll still polish it off. As an example, a study done by Brian Wansink, a professor of marketing at Cornell, used a bottomless bowl of soup \u2013 there was a secret tube that kept on refilling the bowl with soup from below \u2013 to demonstrate that how much people eat is largely dependent on serving size. The group with the bottomless bowls ended up consuming nearly 70 per cent more soup than the group with normal bowls.\n\nEconomists call this sleight of mind mental accounting, since people tend to think about the world in terms of specific accounts, such as scoops of candy or bowls of soup or lines on a budget. While these accounts help people think a little faster \u2013 it's easier to count scoops than actual M&Ms \u2013 they also distort decisions. Richard Thaler, an economist at the University of Chicago, was the first to fully explore the consequences of this irrational behavior. He came up with a simple set of questions that demonstrate mental accounting at work:\n\nImagine that you have decided to see a movie and have paid the admission price of $10 per ticket. As you enter the theater, you discover that you have lost the ticket. The seat was not marked, and the ticket cannot be recovered. Would you pay $10 for another ticket?\n\nWhen Thaler conducted this survey, he found that only 46 per cent of people would buy another movie ticket. However, when he asked a closely related question he got a completely different response.\n\nImagine that you have decided to see a movie where admission is $10, but you have not yet bought the ticket. As you walk to the theater, you discover that you have lost a $10 bill. Would you still pay $10 for a ticket to the movie?\n\nAlthough the value of the loss in both scenarios is the same \u2013 people were still losing $10 \u2013 88 per cent of people said they would now buy a movie ticket. Why the drastic shift? According to Thaler, going to a movie is normally viewed as a transaction in which the cost of a ticket is exchanged for the experience of seeing a movie. Buying a second ticket makes the movie seem too expensive, since a single ticket now 'costs' $20. In contrast, the loss of the cash is not posted to the mental account of the movie, so no one minds forking over another 10 bucks.\n\nOf course, this is woefully inconsistent behavior. After losing tickets, most of us become tightwads; when we lose merely cash, we remain spendthrifts. These contradictory decisions violate an important principle of classical economics, which assumes that a dollar is always a dollar. (Money is supposed to be perfectly fungible.) But because the brain engages in mental accounting, we end up treating our dollars very differently. For example, when Thaler asked people whether they would drive 20 minutes out of their way to save five dollars on a $15 calculator, 68 per cent of respondents said yes. However, when he asked people whether they would drive 20 minutes out of their way to save five dollars on a $125 leather jacket, only 29 per cent said they would. Their driving decisions depended less on the absolute amount of money involved (five dollars) than on the particular mental account in which the decision was placed. If the savings activated a mental account with a minuscule amount of money \u2013 like buying a cheap calculator \u2013 then they were compelled to drive across town. But that same five dollars seems irrelevant when part of a much larger purchase. This principle also explains why car dealers are able to tack on unwanted and expensive extras and why luxury hotels can get away with charging six dollars for a can of peanuts. Because these charges are only small parts of much bigger purchases, we end up paying for things that we wouldn't normally buy.\n\nThe brain relies on mental accounting because it has such limited processing abilities. As Thaler notes, 'These thinking problems come from the fact that we have a slow, erratic CPU [central processing unit] and the fact that we're busy.' Since the prefrontal cortex can only handle about seven things at the same time, it's constantly trying to 'chunk' stuff together, to make the complexity of life a little more manageable. Instead of thinking about each M&M, we think about the scoops. Instead of counting every dollar we spend, we parcel our dollars into particular purchases, like cars. We rely on misleading shortcuts because we lack the computational power to think any other way.\n\n##  _Too Much Information_\n\nThe history of Western thought is so full of paeans to the virtues of rationality that people have neglected to fully consider its limitations. The prefrontal cortex, it turns out, is easy to hoodwink. All it takes is a few additional digits or a slightly bigger candy scoop, and this rational brain region will start making irrational decisions.\n\nA few years ago, a group of MIT economists led by Dan Ariely decided to conduct an auction with their business-school graduate students. (The experiment was later conducted on executives and managers at the MIT Executive Education Program, with similar results.) The researchers were selling a motley group of items, from a fancy bottle of French wine to a cordless keyboard to a box of chocolate truffles. The auction, however, came with a twist: before the students could bid, they were asked to write down the last two digits of their Social Security numbers. Then they were supposed to say whether or not they would be willing to pay that numerical amount for each of the products. For instance, if the last two digits of the number were 55, then the student would have to decide whether the bottle of wine or the cordless keyboard was worth $55. Finally, the students were instructed to write down the maximum amount they were willing to pay for the various items.\n\nIf people were perfectly rational agents, if the brain weren't so bounded, then writing down the last two digits of their Social Security numbers should have no effect on their auction bids. In other words, a student whose Social Security number ended with a low-value figure (such as 10) should be willing to pay roughly the same price as someone with a high-value figure (such as 90). But that's not what happened. For instance, look at the bidding for the cordless keyboard. Students with the highest-ending Social Security numbers (80\u201399) made an average bid of $56. In contrast, students with the lowest-ending numbers (1\u201320) made an average bid of a paltry $16. A similar trend held for every single item. On average, students with higher numbers were willing to spend _300 per cent_ more than those with low numbers. All of the business students realized, of course, that the last two digits of their Social Security numbers were completely irrelevant. Such a thing shouldn't influence their bids. And yet, it clearly did.\n\nThis is known as the anchoring effect, since a meaningless anchor \u2013 in this case, a random number \u2013 can have a strong impact on subsequent decisions.* While it's easy to mock the irrational bids of the business students, the anchoring effect is actually a common consumer mistake. Consider the price tags in a car dealership. Nobody actually pays the prices listed in bold black ink on the windows. The inflated sticker is merely an anchor that allows the car salesperson to make the real price of the car seem like a better deal. When a person is offered the inevitable discount, the prefrontal cortex is convinced that the car is a bargain.\n\nIn essence, the anchoring effect is about the brain's spectacular inability to dismiss irrelevant information. Car shoppers should ignore the manufacturers' suggested retail prices, just as MIT grad students should ignore their Social Security numbers. The problem is that the rational brain isn't good at disregarding facts, even when it knows those facts are useless. And so, if someone is looking at a car, the sticker price serves as a point of comparison, even though it's merely a gimmick. And when a person in the MIT experiment is making a bid on a cordless keyboard, she can't help but tender an offer that takes her Social Security number into account, simply because that number has already been placed into the pertinent decision-making ledger. The random digits are stuck in her prefrontal cortex, occupying valuable cognitive space. As a result, they become a starting point when she thinks about how much she's willing to pay for a computer accessory. 'You know you're not supposed to think about these meaningless numbers,' Ariely says. 'But you just can't help it.'\n\nThe fragility of the prefrontal cortex means that we all have to be extremely vigilant about not paying attention to unnecessary information. The anchoring effect demonstrates how a single additional fact can systematically distort the reasoning process. Instead of focusing on the important variable \u2013 how much is that cordless keyboard really worth? \u2013 we get distracted by some meaningless numbers. And then we spend too much money.\n\nThis cortical flaw has been exacerbated by modernity. We live in a culture that's awash in information; it's the age of Google, cable news and free online encyclopedias. We get anxious whenever we are cut off from all this knowledge, as if it's impossible for anyone to make a decision without a search engine. But this abundance comes with some hidden costs. The main problem is that the human brain wasn't designed to deal with such a surfeit of data. As a result, we are constantly exceeding the capacity of our prefrontal cortices, feeding them more facts and figures than they can handle. It's like trying to run a new computer program on an old machine; the antique microchips try to keep up, but eventually they fizzle out.\n\nIn the late 1980s, the psychologist Paul Andreassen conducted a simple experiment on MIT business students. (Those poor students at MIT's Sloan School of Management are very popular research subjects. As one scientist joked, 'They're like the fruit fly of behavioral economics.') First, Andreassen let each of the students select a portfolio of stock investments. Then he divided the students into two groups. The first group could see only the changes in the prices of their stocks. They had no idea why the share prices rose or fell and had to make their trading decisions based on an extremely limited amount of data. In contrast, the second group was given access to a steady stream of financial information. They could watch CNBC, read the _Wall_ _Street Journal_ and consult experts for the latest analysis of market trends.\n\nSo which group did better? To Andreassen's surprise, the group with less information ended up earning more than twice as much as the well-informed group. Being exposed to extra news was distracting, and the high-information students quickly became focused on the latest rumors and insider gossip. (Herbert Simon said it best: 'A wealth of information creates a poverty of attention.') As a result of all the extra input, these students engaged in far more buying and selling than the low-information group. They were convinced that all their knowledge allowed them to anticipate the market. But they were wrong.\n\nThe dangers of too much information aren't confined to investors. In another study, college counselors were given a vast amount of information about a group of high school students. The counselors were then asked to predict the grades of these kids during their freshman year in college. The counselors had access to high school transcripts, test scores, the results of personality and vocational tests and application essays from the students. They were even granted personal interviews so that they could judge the 'academic talents' of the students in person. With access to all of this information, the counselors were extremely confident that their judgments were accurate.\n\nThe counselors were competing against a rudimentary mathematical formula composed of only two variables: the high school grade point average of the student and his or her score on a single standardized test. Everything else was deliberately ignored. Needless to say, the predictions made by the formula were far more accurate than the predictions made by the counselors. The human experts had looked at so many facts that they lost track of which facts were actually important. They subscribed to illusory correlations ('She wrote a good college essay, so she'll write good essays in college') and were swayed by irrelevant details ('He had such a nice smile'). While the extra information considered by the counselors made them extremely confident, it actually led to worse predictions. Knowledge has diminishing returns, right up until it has negative returns.\n\nThis is a counter-intuitive idea. When making decisions, people almost always assume that more information is better. Modern corporations are especially beholden to this idea and spend a fortune trying to create 'analytic workspaces' that 'maximize the informational potential of their decision-makers'. These managerial clich\u00e9s, plucked from the sales brochures of companies such as Oracle and Unisys, are predicated on the assumptions that executives perform better when they have access to more facts and figures and that bad decisions are a result of ignorance.\n\nBut it's important to know the limitations of this approach, which are rooted in the limitations of the brain. The prefrontal cortex can handle only so much information at any one time, so when a person gives it too many facts and then asks it to make a decision based on the facts that _seem_ important, that person is asking for trouble. He is going to buy the wrong items at Wal-Mart and pick the wrong stocks. We all need to know about the innate frailties of the prefrontal cortex so that we don't undermine our decisions.\n\nBack pain is a medical epidemic. The numbers are sobering: there's a 70 per cent chance that at some point in your life, you'll suffer from it. There's a 30 per cent chance that you've suffered from severe back pain in the last 30 days. At any given time, about 1 per cent of working-age Americans are completely incapacitated by their lower lumbar regions. Treatment is expensive (more than $26 billion a year) and currently accounts for about 3 per cent of total health-care spending. If workers' compensation and disability payments are taken into account, the costs are far higher.\n\nWhen doctors first started to encounter a surge in patients with back pain \u2013 the beginning of the epidemic is generally dated to the late 1960s \u2013 they had few answers. The lower back is an exquisitely complicated body area, full of tiny bones, ligaments, spinal discs and minor muscles. And then there's the spinal cord itself, a thick sheath of sensitive nerves that can be easily upset. There are so many moving parts in the back that doctors had difficulty figuring out what exactly was responsible for the pain. Without a definitive explanation, doctors typically sent patients home with a prescription for bed rest.\n\nBut this simple treatment plan was extremely effective. Even when nothing was done to the lower back, about 90 per cent of patients with back pain managed to get better within seven weeks. The body healed itself, the inflammation subsided, the nerves relaxed. These patients went back to work and pledged to avoid the sort of physical triggers that had caused the pain in the first place.\n\nOver the next few decades, this hands-off approach to back pain remained the standard medical treatment. Although the vast majority of patients didn't receive a specific diagnosis of what caused the pain \u2013 the suffering was typically parceled into a vague category such as 'lower lumbar strain' \u2013 they still managed to experience significant improvements within a short period of time. 'It was a classic case of medicine doing best by doing least,' says Dr. Eugene Carragee, a professor of orthopedic surgery at Stanford. 'People got better without real medical interventions because doctors didn't know how to intervene.'\n\nThat all changed with the introduction of magnetic resonance imaging (MRI) in the late 1980s. Within a few years, the MRI machine became a crucial medical tool. It allowed doctors to look, for the first time, at stunningly accurate images of the interior of the body. MRI machines use powerful magnets to make protons in the flesh shift ever so slightly. Different tissues react in slightly different ways to this atomic manipulation; a computer then translates the resulting contrasts into high-resolution images. Thanks to the precise pictures produced by the machine, doctors no longer needed to imagine the layers of matter underneath the skin. They could see everything.\n\nThe medical profession hoped that the MRI would revolutionize the treatment of lower back pain. Since doctors could finally image the spine and surrounding soft tissue in lucid detail, they figured they'd be able to offer precise diagnoses of what was causing the pain, locating the aggravated nerves and structural problems. This, in turn, would lead to better medical care.\n\nUnfortunately, MRIs haven't solved the problem of back pain. In fact, the new technology has probably made the problem worse. The machine simply sees too much. Doctors are overwhelmed with information and struggle to distinguish the significant from the irrelevant. Take, for example, spinal disc abnormalities. While x-rays can reveal only tumors and problems with the vertebral bones, MRIs can image spinal discs \u2013 the supple buffers between the vertebrae \u2013 in meticulous detail. After the imaging machines were first introduced, the diagnoses of various disc abnormalities began to skyrocket. The MRI pictures certainly looked bleak: people with pain seemed to have seriously degenerated discs, which everyone assumed caused inflammation of the local nerves. Doctors began administering epidurals to quiet the pain, and if the pain persisted, they would surgically remove the apparently offending disc tissue.\n\nThe vivid images, however, were misleading. Those disc abnormalities are seldom the cause of chronic back pain. In a 1994 study published in the _New England Journal of Medicine_ , a group of researchers imaged the spinal regions of ninety-eight people who had no back pain or back-related problems. The pictures were then sent to doctors who didn't know that the patients weren't in pain. The result was shocking: the doctors reported that two-thirds of these normal patients exhibited 'serious problems' such as bulging, protruding, or herniated discs. In 38 per cent of these patients, the MRI revealed multiple damaged discs. Nearly 90 per cent of these patients exhibited some form of 'disc degeneration'. These structural abnormalities are often used to justify surgery, and yet nobody would advocate surgery for people without pain. The study concluded that, in most cases, 'The discovery by MRI of bulges or protrusions in people with low back pain may be coincidental.'\n\nIn other words, seeing everything made it harder for the doctors to know what they should be looking at. The very advantage of MRI \u2013 its ability to detect tiny defects in tissue \u2013 turned out to be a liability, since many of the so-called defects were actually normal parts of the aging process. 'A lot of what I do is educate people about what their MRIs are showing,' says Dr Sean Mackey, a professor at the Stanford School of Medicine and associate director of the hospital's pain-management division. 'Doctors and patients get so fixated on these slight disc problems, and then they stop thinking about other possible causes for the pain. I always remind my patients that the only perfectly healthy spine is the spine of an eighteen-year-old. Forget about your MRI. What it's showing you is probably not important.'\n\nThe mistaken explanations for back pain triggered by MRIs inevitably led to an outbreak of bad decisions. A large study published in the _Journal of the American Medical Association_ ( _JAMA_ ) randomly assigned 380 patients with back pain to undergo two different types of diagnostic analysis. One group received x-rays. The other group got diagnosed using MRIs, which gave the doctors much more information about the underlying anatomy.\n\nWhich group fared better? Did better pictures lead to better treatments? There was no difference in patient outcome: the vast majority of people in both groups got better. More information didn't lead to less pain. But stark differences emerged when the study looked at _how_ the different groups were treated. Nearly 50 per cent of MRI patients were diagnosed with some sort of disc abnormality, and this diagnosis led to intensive medical interventions. The MRI group had more doctor visits, more injections, more physical therapy and were more than twice as likely to undergo surgery. These additional treatments were very expensive, and they had no measurable benefit.\n\nThis is the danger of too much information: it can actually interfere with understanding. When the prefrontal cortex is overwhelmed, a person can no longer make sense of the situation. Correlation is confused with causation, and people make theories out of coincidences. They latch on to medical explanations, even when the explanations don't make very much sense. MRIs make it easy for doctors to see all sorts of disc 'problems', and so they reasonably conclude that these structural abnormalities are causing the pain. They're usually wrong.\n\nMedical experts are now encouraging doctors _not_ to order MRIs when evaluating back pain. A recent report in the _New_ _England Journal of Medicine_ concluded that MRIs should be used to image the back only under specific clinical circumstances, such as when doctors are examining 'patients for whom there is a strong clinical suggestion of underlying infection, cancer, or persistent neurologic deficit'. In the latest clinical guidelines issued by the American College of Physicians and the American Pain Society, doctors were 'strongly recommended... not to obtain imaging or other diagnostic tests in patients with nonspecific low back pain'. In too many cases, the expensive tests proved worse than useless. All of the extra detail just got in the way. The doctors performed better with less information.\n\nAnd yet, despite these clear medical recommendations, MRIs continue to be routinely prescribed by physicians trying to diagnose causes of back pain. The addiction to information can be hard to break. A 2003 report in _JAMA_ found that even when doctors were aware of medical studies criticizing the use of MRI, they still believed that imaging was necessary for their own patients. They wanted to find a _reason_ for the pain so that the suffering could be given a clear anatomical cause, which could then be fixed with surgery. It didn't seem to matter that these reasons weren't empirically valid, or that the disc problems seen by MRI machines don't actually cause most cases of lower back pain. More data was seen as an unqualified good. The doctors thought it would be irresponsible not to conduct all of the relevant diagnostic tests. After all, wasn't that the rational thing to do? And shouldn't doctors always try to make rational decisions?\n\nThe problem of diagnosing the origins of back pain is really just another version of the strawberry-jam problem. In both cases, the rational methods of decision-making cause mistakes. Sometimes, more information and analysis can actually constrict thinking, making people understand less about what's really going on. Instead of focusing on the most pertinent variable \u2013 the percentage of patients who get better and experience less pain \u2013 doctors got sidetracked by the irrelevant MRI pictures.\n\nWhen it comes to treating back pain, this wrong-headed approach comes with serious costs. 'What's going on now is a disgrace,' says Dr John Sarno, a professor of clinical rehabilitation at New York University Medical Center. 'You have well-meaning doctors making structural diagnoses despite a serious lack of evidence that these abnormalities are really causing the chronic pain. But they have these MRI pictures and the pictures seem so convincing. It's amazing how perfectly intelligent people will make foolish decisions if you give them lots of irrelevant stuff to consider.'\n\nThe powers of the Platonic charioteer are fragile. The prefrontal cortex is a magnificent evolutionary development, but it must be used carefully. It can monitor thoughts and help evaluate emotions, but it can also paralyze, making a person forget the words to an aria or lose a trusty golf swing. When someone falls into the trap of spending too much time thinking about fine-art posters or about the details of an MRI image, the rational brain is being used in the wrong way. The prefrontal cortex can't handle so much complexity by itself.\n\nSo far, this book has been about the brain's dueling systems. We've seen how both reason and feeling have important strengths and weaknesses, and how, as a result, different situations require different cognitive strategies. How we decide should depend on what we are deciding.\n\nBut before we learn how to take full advantage of our varied mental tools, we are going to explore a separate realm of decision-making. As it happens, some of our most important decisions are about how we treat other people. The human being is a social animal, endowed with a brain that shapes social behavior. By understanding how the brain makes these decisions, we can gain insight into one of most unique aspects of human nature: morality.\n\n* A follow-up study found that instead of thinking about the mechanical details of the swing, experienced golfers should focus on general aspects of their intended movement, what psychologists call a holistic cue word. For instance, instead of contemplating something like the precise position of the wrist or elbow, the player should focus on a descriptive adjective, such as _smooth_ or _balanced_ . An experimental trial demonstrated that professional golfers who used these holistic cues did far better than golfers who consciously tried to control their strokes.\n\n* Daniel Kahneman first demonstrated the anchoring effect in an experiment known as the United Nations game. He asked people to estimate the percentage of African countries in the United Nations. Before they guessed, a random number was generated \u2013 directly in front of the participants \u2013 by spinning a roulette wheel. As you might imagine, people who saw higher numbers on the roulette wheel generated significantly higher guesses for the percentage of African countries in the United Nations than those who saw lower numbers.\nSIX\n\n# The Moral Mind\n\nWHEN JOHN WAYNE GACY was a child, he liked to torture animals. He caught mice in a wire trap and then cut them open with scissors while they were still alive. The blood and guts didn't bother him. Neither did the squeals. Sadism was entertaining.\n\nThis streak of cruelty was one of the few noteworthy facts of Gacy's childhood. In just about every other respect, his early years were utterly normal. He grew up in the middle-class suburbs of Chicago, where he was a boy scout and delivered the local newspaper. He got good grades in school but didn't want to go to college. When his high school classmates were later asked what they remembered about Gacy, most couldn't remember anything. He blended in with the crowd.\n\nGacy grew up to become a successful construction contractor and a pillar of the community. He liked to throw big summer barbecues, grill hot dogs and hamburgers and invite the neighbors over. He dressed up as a clown for kids in the hospital and was active in local politics. The local chamber of commerce voted him Man of the Year. He was a typical suburban husband.\n\nThe normalcy, however, was a carefully crafted lie. One day, Gacy's wife noticed a pungent odor coming from the crawlspace underneath their house. It was probably just a dead rodent, Gacy said, or maybe a sewage leak. He bought a 50-pound bag of lime and tried to erase the smell. But the smell wouldn't go away. Gacy filled in the space with concrete. The smell still wouldn't go away. There was something bad underneath those floorboards.\n\nThe smell was rotting bodies. On 12 March 1980, John Wayne Gacy was convicted of murdering 33 boys. He paid the boys for sex, and if something went awry with the transaction, he would kill them in his living room. Sometimes he killed one after he raised his price. Or if he thought the boy might tell somebody. Or if he didn't have enough cash in his wallet. Sometimes he killed a boy because it seemed like the easiest thing to do. He'd put a sock in his mouth, strangle him with a rope, and get rid of the corpse in the middle of the night. When the cops finally searched Gacy's home, they found skeletons everywhere: underneath his garage, in the basement, in the backyard. The graves were shallow, just a few inches deep.\n\nJohn Wayne Gacy was a psychopath. Psychiatrists estimate that about 25 per cent of the prison population have psychopathic tendencies, but the vast majority of these people will never kill anybody. While psychopaths are prone to violence \u2013 especially when the violence is used to achieve a goal, like satisfying a sexual desire \u2013 their neurological condition is best defined in terms of a specific brain malfunction: psychopaths make poor \u2013 sometimes disastrous \u2013 moral choices.\n\nAt first glance, it seems strange to think of psychopaths as decision-makers. We tend to label people like John Wayne Gacy as monsters, horrifying examples of humanity at its most inhuman. But every time Gacy murdered a boy, killing without the slightest sense of unease, he was making a decision. He was willingly violating one of the most ancient of moral laws: _thou_ _shalt not kill_. And yet Gacy felt no remorse; his conscience was clean, and he slept like a baby.\n\nPsychopaths shed light on a crucial subset of decision-making that's referred to as morality. Morality can be a squishy, vague concept, and yet, at its simplest level, it's nothing but a series of choices about how we treat other people. When you act in a moral manner \u2013 when you recoil from violence, treat others fairly, and help strangers in need \u2013 you are making decisions that take people besides yourself into account. You are thinking about the feelings of others, sympathizing with their states of mind. This is what psychopaths can't do.\n\nWhat causes this profound deficit? On most psychological tests, psychopaths appear perfectly normal. The working memory isn't impaired, they use language normally, and they don't have reduced attention spans. In fact, several studies have found that psychopaths have above-average IQs and reasoning abilities. Their logic is impeccable. But this intact intelligence conceals a devastating disorder: psychopaths are dangerous because they have damaged emotional brains.\n\nJust look at Gacy. According to a court-appointed psychiatrist, Gacy seemed incapable of experiencing regret, sadness, or joy. He never lost his temper or got particularly angry. Instead, his inner life consisted entirely of sexual impulses and ruthless rationality. He felt nothing, but planned everything. (Gacy's meticulous criminal preparations are what allowed him to evade the police for so long.) Alec Wilkinson, a journalist who spent hours interviewing Gacy on death row, described his eerily detached demeanor in _The New Yorker_ :\n\n[Gacy] appears to have no inner being. I often had the feeling that he was like an actor who had created a role and polished it so carefully that he had become the role and the role had become him. In support of his innocence, he often says things that are deranged in their logic, but he says them so calmly that he appears to be rational and reasonable... Compared to other murderers at the prison, Gacy seemed tranquil.\n\nThis sort of emotional emptiness is typical of psychopaths. When normal people are shown staged videos of strangers being subjected to pain \u2013 for example, receiving powerful electric shocks \u2013 they automatically generate visceral emotional reactions. Their hands start to sweat and their blood pressure surges. But psychopaths feel nothing. It's as if they were watching a blank screen. Most people react differently to emotionally charged verbs such as _kill_ or _rape_ than they do to neutral words such as _sit_ or _walk,_ but that's not the case with psychopaths. For them, the words all seem equivalent. When normal people tell lies, they exhibit the classic symptoms of nervousness; lie detectors work by measuring these signals. But psychopaths are able to consistently fool the machines. Dishonesty doesn't make them anxious because nothing makes them anxious. They can lie with impunity. When criminologists looked at the most violent wife batterers, they discovered that as those men became more and more aggressive, their blood pressure and pulse rates actually _dropped_. The acts of violence had a calming effect.\n\n'Psychopaths have a fundamental emotional disorder,' says James Blair, a cognitive psychologist at the National Institute of Mental Health and coauthor of _The Psychopath: Emotion and_ _the Brain_. 'You know when you see a scared face in a movie and that makes you automatically feel scared, too? Well, psychopaths don't feel that. It's like they don't understand what's going on. This lack of emotion is what causes their dangerous behavior. They are missing the primal emotional cues that the rest of us use as guides when making moral decisions.'\n\nWhen you peer inside the psychopathic brain, you can see this absence of emotion. After being exposed to fearful facial expressions, the emotional parts of the normal human brain show increased levels of activation. So do the cortical areas responsible for recognizing faces. As a result, a frightened face becomes a frightening sight; we naturally internalize the feelings of others. The brain of a psychopath, however, responds to these fearful faces with an utter lack of interest. The emotional areas are unperturbed, and the facial-recognition system is even _less_ interested in fearful faces than it is in perfectly blank stares. The psychopath's brain is bored by expressions of terror.\n\nWhile the anatomy of evil remains incomplete, neuroscientists are beginning to identify the specific deficits that define the psychopathic brain. The main problem seems to be a broken amygdala, a brain area responsible for propagating aversive emotions such as fear and anxiety. As a result, psychopaths never feel bad when they make other people feel bad. Aggression doesn't make them nervous. Terror isn't terrifying. (Brain-imaging studies have demonstrated that the human amygdala is activated when a person merely _thinks_ about committing a 'moral transgression'.) This emotional void means that psychopaths never learn from their adverse experiences; they are four times more likely than other prisoners to commit crimes after being released. For a psychopath on parole, there is nothing inherently wrong with violence. Hurting someone else is just another way of getting what he wants, a perfectly reasonable way to satisfy desires. The absence of emotion makes the most basic moral concepts incomprehensible. G. K. Chesterton was right: 'The madman is not the man who has lost his reason. The madman is the man who has lost everything except his reason.'\n\nAt first glance, the connection between morality and the emotions might be a little unnerving. Moral decisions are supposed to rest on a firm logical and legal foundation. Doing the right thing means carefully weighing competing claims, like a _dis_ passionate judge. These aspirations have a long history. The luminaries of the Enlightenment, such as Leibniz and Descartes, tried to construct a moral system entirely free of feelings. Immanuel Kant argued that doing the right thing was merely a consequence of acting rationally. Immorality, he said, was a result of illogic. 'The oftener and more steadily we reflect' on our moral decisions, Kant wrote, the more moral those decisions become. The modern legal system still subscribes to this antiquated set of assumptions and pardons anybody who demonstrates a 'defect in rationality' \u2013 these people are declared legally insane \u2013 since the rational brain is supposedly responsible for distinguishing between right and wrong. If you can't reason, then you shouldn't be punished.\n\nBut all of these old conceptions of morality are based on a fundamental mistake. Neuroscience can now see the substrate of moral decisions, and there's nothing rational about it. 'Moral judgment is like aesthetic judgment,' writes Jonathan Haidt, a psychologist at the University of Virginia. 'When you see a painting, you usually know instantly and automatically whether you like it. If someone asks you to explain your judgment, you confabulate... Moral arguments are much the same: two people feel strongly about an issue, their feelings come first, and their reasons are invented on the fly, to throw at each other.'\n\nKant and his followers thought the rational brain acted like a scientist: we used reason to arrive at an accurate view of the world. This meant that morality was based on objective values; moral judgments described moral facts. But the mind doesn't work this way. When you are confronted with an ethical dilemma, the unconscious automatically generates an emotional reaction. (This is what psychopaths can't do.) Within a few milliseconds, the brain has made up its mind; you know what is right and what is wrong. These moral instincts aren't rational \u2013 they've never heard of Kant \u2013 but they are an essential part of what keep us all from committing unspeakable crimes.\n\nIt's only at this point \u2013 _after_ the emotions have already made the moral decision \u2013 that those rational circuits in the prefrontal cortex are activated. People come up with persuasive reasons to justify their moral intuition. When it comes to making ethical decisions, human rationality isn't a scientist, it's a _lawyer_. This inner attorney gathers bits of evidence, post hoc justifications, and pithy rhetoric in order to make the automatic reaction seem reasonable. But this reasonableness is just a fa\u00e7ade, an elaborate self-delusion. Benjamin Franklin said it best in his autobiography: 'So convenient a thing it is to be a reasonable creature, since it enables one to find or make a reason for everything one has a mind to do.'\n\nIn other words, our standard view of morality \u2013 the philosophical consensus for thousands of years \u2013 has been exactly backward. We've assumed that our moral decisions are the byproducts of rational thought, that humanity's moral rules are founded in such things as the Ten Commandments and Kant's categorical imperative. Philosophers and theologians have spilled lots of ink arguing about the precise logic of certain ethical dilemmas. But these arguments miss the central reality of moral decisions, which is that logic and legality have little to do with anything.\n\nConsider this moral scenario, which was first invented by Haidt. Julie and Mark are siblings vacationing together in the south of France. One night, after a lovely day spent exploring the local countryside, they share a delicious dinner and a few bottles of red wine. One thing leads to another and Julie and Mark decide to have sex. Although she's on the pill, Mark uses a condom just in case. They enjoy themselves very much, but decide not to have sex again. The siblings promise to keep the one-night affair secret and discover, over time, that having sex has brought them even closer together. Did Julie and Mark do something wrong?*\n\nIf you're like most people, your first reaction is that the brother and sister committed a grave sin. What they did was very wrong. When Haidt asks people to explain their harsh moral judgments, the most common reasons given are the risk of having kids with genetic abnormalities and the possibility that sex will damage the sibling relationship. At this point, Haidt politely points out that Mark and Julie used two types of birth control and that having sex actually improved their relationship. But the facts of the case don't matter. Even when their arguments are disproved, people still cling to the belief that having sex with one's brother or sister is somehow immoral.\n\n'What happens in the experiment,' Haidt says, 'is [that] people give a reason [why the sex is wrong]. When that reason is stripped from them, they give another reason. When the new reason is stripped from them, they reach for _another_ reason.' Eventually, of course, people run out of reasons: they've exhausted their list of moral justifications. The rational defense is forced to rest its case. That's when people start saying things like 'Because it's just wrong to have sex with your sister' or 'Because it's disgusting, that's why!' Haidt calls this state 'moral dumbfounding'. People know something seems morally wrong \u2013 sibling sex is a terrible idea \u2013 but no one can rationally defend the verdict. According to Haidt, this simple story about sibling sex illuminates the two separate processes that are at work when we make moral decisions. The emotional brain generates the verdict. It determines what is wrong and what is right. In the case of Julie and Mark, it refuses to believe that having sex with a sibling is morally permissible, no matter how many forms of birth control are used. The rational brain, on the other hand, _explains_ the verdict. It provides reasons, but those reasons all come after the fact.\n\nThis is why psychopaths are so dangerous: they are missing the emotions that guide moral decisions in the first place. There's a dangerous void where their feelings are supposed to be. For people like Gacy, sin is always intellectual, never visceral. As a result, a psychopath is left with nothing but a rational lawyer inside his head, willing to justify any action. Psychopaths commit violent crimes because their emotions never tell them not to.\n\n##  _The Trolley Problem_\n\nMoral decisions are a unique kind of decision. When you're picking out products in the grocery store, searching for the best possible strawberry jam, you are trying to maximize your own enjoyment. You are the only person that matters; it is your sense of pleasure that you are trying to please. In this case, selfishness is the ideal strategy. You should listen to those twitchy cells in the orbitofrontal cortex that tell you what you really want.\n\nHowever, when you are making a moral decision, this egocentric strategy backfires. Moral decisions require taking _other people_ into account. You can't act like a greedy brute or let your anger get out of control; that's a recipe for depravity and jail time. Doing the right thing means thinking about everybody else, using the emotional brain to mirror the emotions of strangers. Selfishness needs to be balanced by some selflessness.\n\nThe evolution of morality required a whole new set of decision-making machinery. The mind needed to evolve some structures that would keep it from hurting other people. Instead of just seeking more pleasure, the brain had to become sensitive to the pain and plight of strangers. The new neural structures that developed are a very recent biological adaptation. While people have the same reward pathway as rats \u2013 every mammal relies on the dopamine system \u2013 moral circuits can be found only in the most social primates. Humans, of course, are the most social primates of all.\n\nThe best way to probe the unique brain circuits underlying morality is by using a brain scanner to study people while they are making moral decisions. Consider this elegant experiment, led by neuroscientist Joshua Greene of Harvard. Greene asked his subjects a series of questions involving a runaway trolley, an oversize man, and five maintenance workers. (It might sound like a strange setup, but it's actually based on a well-known philosophical thought puzzle.) The first scenario goes like this:\n\nYou are the driver of a runaway trolley. The brakes have failed. The trolley is approaching a fork in the track at top speed. If you do nothing, the train will stay left, where it will run over five maintenance workers who are fixing the track. All five workers will die. However, if you steer the train right \u2013 this involves flicking a switch and turning the wheel \u2013 you will swerve onto a track where there is one maintenance worker. What do you do? Are you willing to intervene and change the path of the trolley?\n\nIn this hypothetical case, about 95 per cent of people agree that it is morally permissible to turn the trolley. The decision is just simple arithmetic: it's better to kill fewer people. Some moral philosophers even argue that it is immoral _not_ to turn the trolley, since passivity will lead to the death of four more people. But what about this scenario:\n\nYou are standing on a footbridge over the trolley track. You see a trolley racing out of control, speeding toward five workmen who are fixing the track. All five men will die unless the trolley can be stopped. Standing next to you on the footbridge is a very large man. He is leaning over the railing, watching the trolley hurtle toward the men. If you sneak up on the man and give him a little push, he will fall over the railing and into the path of the trolley.\n\nBecause he is so big, he will stop the trolley from killing the maintenance workers. Do you push the man off the footbridge? Or do you allow five men to die?\n\nThe brute facts, of course, remain the same: one man must die in order for five men to live. If ethical decisions were perfectly rational, then a person would act the same way in both situations and be as willing to push the man off the bridge as he or she was to turn the trolley. And yet, almost nobody is willing to actively throw another person onto the train tracks. The decisions lead to the same outcome, yet one is moral and one is murder.\n\nGreene argues that pushing the man feels wrong because the killing is direct: you are using your body to hurt his body. He calls it a _personal_ moral situation, since it directly involves another person. In contrast, when you just have to turn the trolley onto a different track, you aren't directly hurting somebody else, you're just shifting the trolley wheels; the resulting death seems indirect. In this case, it's an _impersonal_ moral decision.\n\nWhat makes this thought experiment so interesting is that the fuzzy moral distinction \u2013 the difference between personal and impersonal decisions \u2013 is built into the brain. It doesn't matter what culture you live in, or what religion you subscribe to: the two different trolley scenarios trigger distinct patterns of activation. In the first scenario, when a subject was asked whether the trolley should be turned, the rational decision-making machinery was turned on. A network of brain regions assessed the various alternatives, sent their verdict onward to the prefrontal cortex, and the person chose the clearly superior option. The brain quickly realized that it was better to kill one man than five men.\n\nHowever, when a subject was asked whether he would be willing to push a man onto the tracks, a separate network of brain areas was activated. These folds of gray matter \u2013 the superior temporal sulcus, posterior cingulate and medial frontal gyrus \u2013 are responsible for interpreting the thoughts and feelings of _other_ people. As a result, the subject automatically imagined how the poor man would feel as he plunged to his death on the train tracks below. He vividly simulated his mind and concluded that pushing him was a capital crime, even if it saved the lives of five other men. The person couldn't explain the moral decision \u2013 the inner lawyer was confused by the inconsistency \u2013 but his certainty never wavered. Pushing a man off a bridge just _felt_ wrong.\n\nWhile stories of Darwinian evolution often stress the amorality of natural selection \u2013 we are all Hobbesian brutes, driven to survive by selfish genes \u2013 our psychological reality is much less bleak. We aren't angels, but we also aren't depraved hominids. 'Our primate ancestors,' Greene explains, 'had intensely social lives. They evolved mental mechanisms to keep them from doing all the nasty things they might otherwise be interested in doing. This basic primate morality doesn't understand things like tax evasion, but it does understand things like pushing your buddy off of a cliff.' As Greene puts it, a personal moral violation can be roughly defined as 'me hurts you', a concept simple enough for a primate to understand.\n\nThis is a blasphemous idea. Religious believers assume that God invented the moral code. It was given to Moses on Mount Sinai, a list of imperatives inscribed in stone. (As Dostoyevsky put it, 'If there is no God, then we are lost in a moral chaos. Everything is permitted.') But this cultural narrative gets the causality backward. Moral emotions existed long before Moses. They are writ into the primate brain. Religion simply allows us to codify these intuitions, to translate the ethics of evolution into a straightforward legal system. Just look at the Ten Commandments. After God makes a series of religious demands \u2013 don't worship idols and always keep the Sabbath \u2013 He starts to issue moral orders. The first order is the foundation of primate morality: thou shalt not kill. Then comes a short list of moral adjuncts, which are framed in terms of harm to another human being. God doesn't tell us merely not to lie; He tells us not to bear false witness against our neighbor. He doesn't prohibit jealousy only in the abstract; He commands us not to covet our neighbor's 'wife or slaves or ox or donkey.' The God of the Old Testament understands that our most powerful moral emotions are generated in response to personal moral scenarios, so that's how He frames all of His instructions. The details of the Ten Commandments reflect the details of the evolved moral brain.\n\nThese innate emotions are so powerful that they keep people moral even in the most amoral situations. Consider the behavior of soldiers during war. On the battlefield, men are explicitly encouraged to kill one another; the crime of murder is turned into an act of heroism. And yet, even in such violent situations, soldiers often struggle to get past their moral instincts. During World War II, for example, US Army Brigadier General S. L. A. Marshall undertook a survey of thousands of American troops right after they'd been in combat. His shocking conclusion was that less than 20 per cent actually shot at the enemy, even when under attack. 'It is fear of killing,' Marshall wrote, 'rather than fear of being killed, that is the most common cause of battle failure in the individual.' When soldiers were forced to confront the possibility of directly harming other human beings \u2013 this is a personal moral decision \u2013 they were literally incapacitated by their emotions. 'At the most vital point of battle,' Marshall wrote, 'the soldier becomes a conscientious objector.'\n\nAfter these findings were published, in 1947, the US Army realized it had a serious problem. It immediately began revamping its training regimen in order to increase the 'ratio of fire'. New recruits began endlessly rehearsing the kill, firing at anatomically correct targets that dropped backward after being hit. As Lieutenant Colonel Dave Grossman noted, 'What is being taught in this environment is the ability to shoot reflexively and instantly... Soldiers are de-sensitized to the act of killing, until it becomes an automatic response.' The army also began emphasizing battlefield tactics, such as high-altitude bombing and long-range artillery, that managed to obscure the personal cost of war. When bombs are dropped from 40,000 feet, the decision to fire is like turning a trolley wheel: people are detached from the resulting deaths.\n\nThese new training techniques and tactics had dramatic results. Several years after he published his study, Marshall was sent to fight in the Korean War, and he discovered that 55 per cent of infantrymen were now firing their weapons. In Vietnam, the ratio of fire was nearly 90 per cent. The army had managed to turn the most personal of moral situations into an impersonal reflex. Soldiers no longer felt a surge of negative emotions when they fired their weapons. They had been turned, wrote Grossman, into 'killing machines'.\n\n##  _Fellow-Feeling_\n\nAt its core, moral decision-making is about sympathy. We abhor violence because we know violence hurts. We treat others fairly because we know what it feels like to be treated unfairly. We reject suffering because we can imagine what it's like to suffer. Our minds naturally bind us together, so that we can't help but follow the advice of Luke: 'And as ye would that men should do to you, do ye also to them likewise.'\n\nFeeling sympathetic is not as simple as it might seem. For starters, before you can sympathize with the feelings of other people, you have to figure out what they are feeling. This means you need to develop a theory about what's happening inside their minds so that your emotional brain can imitate the activity of their emotional brains. Sometimes, this act of mind reading is done by interpreting facial expressions. If someone is squinting his eyes and clenching his jaw, you automatically conclude that his amygdala is excited; he must be angry. If he flexes the zygomaticus majors \u2013 that's what happens during a smile \u2013 then you assume he's happy. Of course, you don't always have access to a communicative set of facial expressions. When you talk on the phone or write an e-mail or think about someone far away, you are forced to mind read by simulation, by imagining what you would feel in the same situation.\n\nRegardless of how exactly one generates theories of other people's minds, it's clear that these theories profoundly affect moral decisions. Look, for example, at the ultimatum game, a staple of experimental economics. The rules of the game are simple, if a little bit unfair: an experimenter pairs two people together, and hands one of them $10. This person (the proposer) gets to decide how the ten dollars is divided. The second person (the responder) can either accept the offer, which allows both players to pocket their respective shares, or reject the offer, in which case both players walk away empty-handed.\n\nWhen economists first started playing this game in the early 1980s, they assumed that this elementary exchange would always generate the same outcome. The proposer would offer the responder about a dollar \u2013 a minimal amount \u2013 and the responder would accept it. After all, a rejection leaves both players worse off, and one dollar is better than nothing, so this arrangement would clearly demonstrate our innate selfishness and rationality.\n\nHowever, the researchers soon realized that their predictions were all wrong. Instead of swallowing their pride and pocketing a small profit, responders typically rejected any offer they perceived as unfair. Furthermore, proposers anticipated this angry rejection and typically tendered an offer of around five dollars. This was such a stunning result that nobody really believed it.\n\nBut when other scientists repeated the experiment, the same thing happened. People play this game the same way all over the world, and studies have observed similar patterns of irrationality in Japan, Russia, Germany, France and Indonesia. No matter where the game was played, people almost always made fair offers. As the economist Robert Frank notes, 'Seen through the lens of modern self-interest theory, such behavior is the human equivalent of planets traveling in square orbits.'\n\nWhy do proposers engage in such generosity? The answer returns us to the act of sympathy and the unique brain circuits that determine moral decisions. Adam Smith, the eighteenth-century philosopher, was there first. Although Smith is best known for his economic treatise _The Wealth of Nations,_ he was most proud of _The Theory of Moral Sentiments,_ his sprawling investigation into the psychology of morality. Like his friend David Hume, Smith was convinced that our moral decisions were shaped by our emotional instincts. People were good for essentially irrational reasons.\n\nAccording to Smith, the source of these moral emotions was the imagination, which people used to mirror the minds of others. (The reflective mirror, which had recently become a popular household item in Smith's time, is an important metaphor in his writing on morality.) 'As we have no immediate experience of what other men feel,' Smith wrote, 'we can form no idea of the manner in which they are affected, but by conceiving what we ourselves should feel in the like situation.' This mirroring process leads to an instinctive sympathy for one's fellow man \u2013 Smith called it 'fellow-feeling' \u2013 that forms the basis for moral decisions.\n\nSmith was right. The reason a proposer makes a fair offer in the ultimatum game is that he is able to imagine how the responder will feel about an unfair offer. (When people play the game with computers, they are never generous.) The responder knows that a low-ball proposal will make the other person angry, which will lead him to reject the offer, which will leave everybody with nothing. So the proposer suppresses his greed and equitably splits the $10. That ability to sympathize with the feelings of others leads to fairness.\n\nThe sympathetic instinct is also one of the central motivations behind altruism, which is demonstrated when people engage in selfless acts such as donating to charity and helping out perfect strangers. In a recent experiment published in _Nature_ _Neuroscience,_ scientists at Duke University imaged the brains of people as they watched a computer play a simple video game. Because the subjects were told that the computer was playing the game for a specific purpose \u2013 it wanted to earn money \u2013 their brains automatically treated the computer like an 'intentional agent', complete with goals and feelings. (Human minds are so eager to detect other minds that they often imbue inanimate objects, like computers and stuffed animals, with internal mental states.) Once that happened, the scientists were able to detect activity in the superior temporal sulcus and other specialized areas that help each of us theorize and sympathize with the emotions of other people. Even though the subjects knew they were watching a computer, they couldn't help but imagine what the computer was feeling.\n\nNow comes the interesting part: the scientists noticed that there was a lot of individual variation during the experiment. Some people had very active sympathetic brains, while others seemed rather uninterested in thinking about the feelings of someone else. The scientists then conducted a survey of altruistic behavior, asking people how likely they would be to 'help a stranger carry a heavy object' or 'let a friend borrow a car'. That's when the correlation became clear: people who showed more brain activity in their sympathetic regions were also much more likely to exhibit altruistic behavior. Because they intensely imagined the feelings of other people, they wanted to make other people feel better, even if it came at personal expense.\n\nBut here's the lovely secret of altruism: _it feels good_. The brain is designed so that acts of charity are pleasurable; being nice to others makes us feel nice. In a recent brain-imaging experiment, a few dozen people were each given $128 of real money and allowed to choose between keeping the money and donating it to charity. When they chose to give away the money, the reward centers of their brains became active and they experienced the delightful glow of unselfishness. In fact, several subjects showed more reward-related brain activity during acts of altruism than they did when they actually received cash rewards. From the perspective of the brain, it literally was better to give than to receive.\n\nOne of the ways neuroscientists learn about the brain is by studying what happens when something goes wrong with it. For example, scientists learned about the importance of our moral emotions by studying psychopaths; they learned about the crucial role of dopamine by studying people with Parkinson's; and brain tumors in the frontal lobes have helped to illuminate the substrate of rationality. This might seem callous \u2013 tragedy is turned into an investigative tool \u2013 but it's also extremely effective. The broken mind helps us understand how the normal mind works.\n\nWhen it comes to the sympathetic circuits in the human brain, scientists have learned a tremendous amount by studying people with autism. When Dr Leo Kanner first diagnosed a group of eleven children with autism, in 1943, he described the syndrome as one of 'extreme aloneness'. ( _Aut_ is Greek for 'self' and _autism_ translates to 'the state of being unto one's self'.) The syndrome afflicts one in every 160 individuals and leaves them emotionally isolated, incapable of engaging in many of the social interactions most people take for granted. As the Cambridge psychologist Simon Baron-Cohen puts it, people with autism are 'mind-blind'. They have tremendous difficulty interpreting the emotions and mental states of others.*\n\nScientists have long suspected that autism is a disease of brain development. For some still mysterious reason, the cortex doesn't wire itself correctly during the first year of life. It now appears that one of the brain areas compromised in people with autism is a small cluster of cells known as mirror neurons. The name of the cell type is intentional: these neurons mirror the movements of other people. If you see someone else smile, then your mirror neurons will light up as if you were smiling. The same thing happens whenever you see someone scowl, grimace, or cry. These cells reflect, on the inside, the expressions of everybody else. As Giacamo Rizzolatti, one of the scientists who discovered mirror neurons, says, 'They [mirror neurons] allow us to grasp the minds of others not through conceptual reasoning but through direct simulation; by feeling, not by thinking.'\n\nThis is what people with autism have to struggle to do. When scientists at UCLA imaged the brains of autistic people as they looked at photographs of faces in different emotional states, the scientists discovered that the autistic brain, unlike the normal brain, showed no activity in its mirror-neuron area. As a result, the autistic subjects had difficulty interpreting the feelings on display. They saw the angry face as nothing but a set of flexed facial muscles. A happy face was simply a different set of muscles. But neither expression was correlated with a specific emotional state. In other words, they never developed a theory about what was happening inside other people's minds.\n\nA brain-imaging study done by scientists at Yale sheds further light on the anatomical source of autism. The study examined the parts of the brain that were activated when a person looked at a face and when he or she looked at a static object, like a kitchen chair. Normally, the brain reacts very differently to these stimuli. Whenever you see a human face, you use a highly specialized brain region called the fusiform face area (FFA) that is solely devoted to helping you recognize other people. In contrast, when you look at a chair, the brain relies on the inferior temporal gyrus, an area activated by any sort of complex visual scene. However, in the study, people with autism never turned on the fusiform face area. They looked at human faces with the part of the brain that normally recognizes objects. A person was just another thing. A face generated no more emotion than a chair.\n\nThese two brain deficits \u2013 a silent mirror-neuron circuit and an inactive fusiform face area \u2013 help to explain the social difficulties of people with autism. Their 'extreme aloneness' is a direct result of not being able to interpret and internalize the emotions of other people. Because of this, they often make decisions that, in the words of one autism researcher, 'are so rational they can be hard to understand'.\n\nFor instance, when people with autism play the ultimatum game, they act just like the hypothetical agents in an economics textbook. They try to apply a rational calculus to the irrational world of human interaction. On average, they make offers that are 80 per cent lower than those of normal subjects, with many offering less than a nickel. This greedy strategy ends up being ineffective, since the angry responders tend to reject such unfair offers. But the proposers with autism are unable to anticipate these feelings. Consider this quote from an upset autistic adult whose offer of 10 cents in a $10 ultimatum game was spurned: 'I did not earn any money because all the other players are _stupid!_ How can you reject a positive amount of money and prefer to get zero? They just did not understand the game! You should have stopped the experiment and explained it to them...'\n\nAutism is a chronic condition, a permanent form of mind blindness. But it's possible to induce a temporary state of mind blindness, in which the brain areas that normally help a person sympathize with others are turned off. A simple variation on the ultimatum game, known as the dictator game, makes this clear. Our sense of 'fellow-feeling' is natural, but it's also very fragile. Unlike the ultimatum game, in which the responder can decide whether or not to accept the monetary offer, in the dictator game, the proposer simply dictates how much the responder receives. What's surprising is that these petty tyrants are still rather generous and give away about one-third of the total amount of money. Even when people have absolute power, they remain constrained by their sympathetic instincts.\n\nHowever, it takes only one minor alteration for this benevolence to disappear. When the dictator cannot see the responder \u2013 the two players are located in separate rooms \u2013 the dictator lapses into unfettered greed. Instead of giving away a significant share of the profits, the despots start offering mere pennies and pocketing the rest. Once people become socially isolated, they stop simulating the feelings of other people. Their moral intuitions are never turned on. As a result, the inner Machiavelli takes over, and the sense of sympathy is squashed by selfishness. The UC Berkeley psychologist Dacher Keltner has found that in many social situations, people with power act just like patients with damaged orbitofrontal cortices. 'The experience of power might be thought of as having someone open up your skull and take out that part of your brain so critical to empathy and socially appropriate behavior,' he says. 'You become very impulsive and insensitive, which is a bad combination.'\n\nPaul Slovic, a psychologist at the University of Oregon, has exposed another blind spot in the sympathetic brain. His experiments are simple: he asks people how much they would be willing to donate to various charitable causes. For example, Slovic found that when people were shown a picture of Rokia, a starving Malawian child, they acted with impressive generosity. After looking at Rokia's emaciated body and haunting brown eyes, they each donated, on average, $2.50 to the charity Save the Children. However, when other people were provided with a list of statistics about starvation throughout Africa \u2013 more than 3 million children in Malawi are malnourished, more than 11 million people in Ethiopia need immediate food assistance, and so forth \u2013 the average donation was 50 per cent lower. At first glance, this makes no sense. When people are informed about the true scope of the problem, they should give _more_ money, not less. Rokia's tragic story is just the tip of the iceberg.\n\nAccording to Slovic, the problem with statistics is that they don't activate our moral emotions. The depressing numbers leave us cold: our minds can't comprehend suffering on such a massive scale. This is why we are riveted when one child falls down a well but turn a blind eye to the millions of people who die every year for lack of clean water. And why we donate thousands of dollars to help a single African war orphan featured on the cover of a magazine but ignore widespread genocides in Rwanda and Darfur. As Mother Teresa put it, 'If I look at the mass, I will never act. If I look at the one, I will.'\n\n##  _The Scars of Abuse_\n\nThe capacity for making moral decisions is innate \u2013 the sympathetic circuit is hard-wired, at least in most of us \u2013 but it still requires the right kind of experience in order to develop. When everything goes according to plan, the human mind naturally develops a potent set of sympathetic instincts. We will resist pushing the man off the bridge, make fair offers in the ultimatum game, and get deeply disturbed by images of other people in pain.\n\nHowever, if something goes amiss during the developmental process \u2013 if the circuits that underlie moral decisions never mature \u2013 the effects can be profound. Sometimes, as with autism, the problem is largely genetic. (Scientists estimate the heritability of autism at somewhere between 80 and 90 per cent, which makes it one of the most inheritable of all neurologic conditions.) But there's another way that the developing brain can be permanently damaged: child abuse. When children are molested or neglected or unloved as children, their emotional brains are warped. (John Gacy, for example, was physically abused throughout his childhood by his alcoholic father.) The biological program that allows human beings to sympathize with the feelings of others is turned off. Cruelty makes us cruel. Abuse makes us abusive. It's a tragic loop.\n\nThe first evidence for this idea came from the work of Harry Harlow.* In the early 1950s, Harlow decided to start a breeding colony of monkeys at the University of Wisconsin. He was studying Pavlovian conditioning in primates, but he needed more data, which meant that he needed more animals. Although nobody had ever successfully bred monkeys in the United States before then, Harlow was determined.\n\nThe breeding colony began with just a few pregnant female monkeys. Harlow carefully monitored the expecting primates; after each gave birth, he immediately isolated the infant in an immaculately clean cage. At first, everything went according to plan. Harlow raised the babies on a formula of sugar and evaporated milk fortified with a slew of vitamins and supplements. He fed the monkeys from sterilized doll bottles every two hours and carefully regulated the cycles of light and dark. In order to minimize the spread of disease, Harlow never let the babies interact with one another. The result was a litter of primates that were bigger and stronger than their peers from the wild.\n\nBut the physical health of these young monkeys hid a\n\ndevastating sickness: they had been wrecked by loneliness. Their short lives had been defined by total isolation, and they proved incapable of even the most basic social interactions. They would maniacally rock back and forth in their metal cages, sucking on their thumbs until they bled. When they encountered other monkeys, they would shriek in fear, run to the corners of their cages, and stare at the floor. If they felt threatened, they would lash out in vicious acts of violence. Sometimes these violent tendencies were turned inward. One monkey ripped out its fur in bloody clumps. Another gnawed off its own hand. Because of their early deprivation, these babies had to be isolated for the rest of their lives.\n\nFor Harlow, these troubled baby monkeys demonstrated that the developing mind needed more than proper nutrition. But what did it need? The first clue came from watching these primate babies. The scientists had lined their cages with cloth diapers so that the monkeys didn't have to sleep on the cold concrete floor. The motherless babies quickly became obsessed with these cloth rags. They would wrap themselves in the fabric and cling to the diapers if anybody approached the cages. The soft fabric was their sole comfort.\n\nThis poignant behavior inspired Harlow to come up with a new experiment. He decided to raise the next generation of baby monkeys with two different pretend mothers. One was a wire mother, formed out of wire mesh, while the other was a mother made out of soft terry cloth. Harlow assumed that all things being equal, the babies would prefer the cloth mothers, since they would be able to cuddle with the fabric. To make the experiment more interesting, Harlow added a slight twist to a few of the cages. Instead of hand-feeding some babies, he put their milk bottles in the hands of the wire mothers. His question was simple: what was more important, food or affection? Which mother would the babies want more?\n\nIn the end, it wasn't even close. No matter which mother held the milk, the babies always preferred the cloth mothers. The monkeys would run over to the wire mothers and quickly sate their hunger before immediately returning to the comforting folds of cloth. By the age of six months, the babies were spending more than eighteen hours a day nuzzling with their soft parent. They were with the wire mothers only long enough to eat.\n\nThe moral of Harlow's experiment is that primate babies are born with an intense need for attachment. They cuddled with the cloth mothers because they wanted to experience the warmth and tenderness of a real mother. Even more than food, these baby monkeys craved the feeling of affection. 'It's as if the animals are programmed to seek out love,' Harlow wrote.\n\nWhen this need for love wasn't met, the babies suffered from a tragic list of side effects. The brain was permanently damaged. The monkeys with wire mothers only didn't know how to deal with others, sympathize with strangers, or behave in a socially acceptable manner. Even the most basic moral decisions were impossible. As Harlow would later write, 'If monkeys have taught us anything, it's that you've got to learn how to love before you learn how to live.'\n\nHarlow would later test the limits of animal experimentation, remorselessly probing the devastating effects of social isolation. His cruelest experiment was putting baby monkeys in individual cages with nothing \u2013 not even a wire mother \u2013 for months at a time. The outcome was unspeakably sad. The isolated babies were like primate psychopaths, completely numb to all expressions of emotion. They started fights without provocation and they didn't stop fighting until one of the monkeys had been seriously injured. They were even vicious to their own children. One psychopathic monkey bit off the fingers of her child. Another killed her crying baby by crushing its head in her mouth. Most psychopathic mothers, however, just perpetuated the devastating cycle of cruelty. When their babies tried to cuddle, they would push them away. The confused infants would try again and again, but to no avail. Their mothers felt nothing.\n\nWhat happens to monkeys can happen to people. This is the tragic lesson of Communist Romania. In 1966, Nicolae Ceausescu, the despotic leader of the country, banned all forms of contraception, and the country was suddenly awash in unwanted babies. The predictable result was a surfeit of orphans; poor families surrendered the kids they couldn't afford.\n\nThe state-run orphanages of Romania were overwhelmed and underfunded. Babies were left in cribs with nothing but plastic bottles. Toddlers were tied to their beds and never touched. The orphanages lacked heat in the winter. Children with disabilities were consigned to the basement, and some went years without seeing natural light. Older children were drugged so that they would sleep for days at a time. In some orphanages, more than 25 per cent of the children died before the age of five.\n\nThe children that managed to survive the Romanian orphanages were permanently scarred. Many had stunted bodies, shrunken bones, and untreated infections. But the most devastating legacy of the orphanage system was psychological. Many of the abandoned children suffered from severe emotional impairments. They were often hostile to strangers, abusive to one another, and incapable of even the most basic social interactions. Couples that adopted Romanian orphans from these institutions reported a wide array of behavioral disorders. Some children cried whenever they were touched. Others stared into space for hours and then suddenly flew into violent rages, attacking everything within reach. One Canadian couple walked into the bedroom of their three-year-old son to discover that he had just thrown their new kitten out the window.\n\nWhen neuroscientists imaged the brain activity of Romanian orphans, they saw reduced activity in regions that are essential for emotion and social interaction, such as the orbitofrontal cortex and the amygdala. The orphans also proved unable to perceive emotions in others and had a pronounced inability to interpret facial expressions. Finally, the neglected children showed significantly reduced levels of vasopressin and oxytocin, two hormones crucial for the development of social attachments. (These hormonal deficiencies persisted for years afterward.) For these victims of abuse, the world of human sympathy was incomprehensible. They struggled to recognize the emotions of others, and they also found it difficult to modulate their own emotions.\n\nStudies of American children who are abused at an early age paint a similarly bleak picture. In the early 1980s, the psychologists Mary Main and Carol George looked at a group of 20 toddlers from 'families in stress'. Half of these children had been victims of serious physical abuse. The other half were from broken homes \u2013 many of them were living with foster parents \u2013 but they hadn't been hit or hurt. Main and George wanted to see how these two groups of disadvantaged toddlers responded to a crying classmate. Would they display normal human sympathy? Or would they be unable to relate to the feelings of their peer? The researchers found that almost all the nonabused children reacted to the upset child with concern. Their instinctive sympathy led them to make some attempts to console the child. They were upset by seeing somebody else upset.\n\nChildhood abuse, however, changed everything. The abused toddlers didn't know how to react to their distressed classmate. They occasionally made sympathetic gestures, but these gestures often degenerated into a set of aggressive threats if the other child didn't stop crying. Here is the study's description of Martin, an abused two-and-a-half-year-old: 'Martin... tried to take the hand of the crying other child, and when she resisted, he slapped her on the arm with his open hand. He then turned away from her to look at the ground and began vocalizing very strongly. \"Cut it out! cut it out!\" each time saying it a little faster and louder. He patted her, but when she became disturbed by his patting, he retreated, hissing at her and baring his teeth. He then began patting her on the back again, his patting became beating, and he continued beating her despite her screams.' Even when Martin wanted to help, he ended up making things worse. An abused two-year-old named Kate exhibited a similar pattern of behavior. At first she reacted with tenderness to the distressed child and gently caressed him on the back. 'Her patting, however, soon became very rough,' the researchers wrote, 'and she began hitting him hard. She continued to hit him until he crawled away.' Because Kate and Martin couldn't understand the feelings of someone else, the world of human interaction had become an impenetrable place.\n\nWhat these abused children were missing was an education in feeling. Because they had been denied that influx of tender emotion that the brain is built to expect, they were seriously scarred, at least on the inside. It's not that these kids wanted to be cruel or unsympathetic. They were simply missing the patterns of brain activity that normally guide our moral decisions. As a result, they reacted to the toddler in distress just as their abusive parents reacted to their own distress: with threats and violence.\n\nBut these tragic examples are exceptions to the rule. We are designed to feel one another's pain so that we're extremely distressed when we hurt others and commit moral transgressions. Sympathy is one of humanity's most basic instincts, which is why evolution lavished so much attention on mirror neurons, the fusiform face area, and those other brain regions that help theorize about other minds. As long as a person is loved as a child and doesn't suffer from any developmental disorders, the human brain will naturally reject violence and make fair offers and try to comfort the crying child. This behavior is just a basic part of who we are. Evolution has programmed us to care about one another.\n\nConsider this poignant experiment: six rhesus monkeys were trained to pull on a variety of chains to get food. If they pulled on one chain, they got a large amount of their favorite food. If they pulled on a different chain, they got a small amount of a less enticing food. As you can probably guess, the monkeys quickly learned to pull on the chain that gave them more of what they wanted. They maximized their reward.\n\nAfter a few weeks of this happy setup, one of the six monkeys got hungry and decided to pull on the chain. This is when something terrible happened: a separate monkey in a different cage was shocked with a painful jolt of electricity. All six monkeys saw it happen. They heard the awful shriek. They watched the monkey grimace and cower in fear. The change in their behavior was immediate. Four of the monkeys decided to stop pulling on the maximizing chain. They were now willing to settle for less food as long as the other monkey wasn't hurt. The fifth monkey stopped pulling on either chain for five days, and the sixth monkey stopped pulling for 12 days. They starved themselves so that a monkey they didn't know wasn't forced to suffer.\n\n* Haidt's other scenarios involve a woman who uses the American flag to clean her bathroom and a family that eats their dog after it has been killed by a car.\n\n* Autism, obviously, has nothing to do with psychopathy. Unlike people with autism, psychopaths can readily recognize when others are upset or in pain. Their problem is that they can't generate corresponding emotions, since their amygdalas are never turned on. The end result is that psychopaths remain preternaturally calm, even in situations that should make them upset. People with autism, however, don't have a problem generating emotion. The problem for them is of recognition; they struggle to decipher or simulate the mental states of others.\n\n* For a wonderful history of Harlow and his research, see Deborah Blum's biography _Love at Goon Park_.\nSEVEN\n\n# The Brain Is an Argument\n\nONE OF THE MOST coveted prizes in a presidential primary is the endorsement of the _Concord Monitor,_ a small newspaper in central New Hampshire. During the first months of the 2008 presidential primary campaign, all of the major candidates, from Chris Dodd to Mike Huckabee, sat for interviews with the paper's editorial board. Some candidates, such as Hillary Clinton, Barack Obama and John McCain, were invited back for follow-up interviews. These sessions would often last for hours, with the politicians facing a barrage of uncomfortable questions. Hillary Clinton was asked about various White House scandals; Barack Obama was asked why he often seemed 'bored and low-key' on the stump; McCain was asked about his medical history. 'There were a few awkward moments,' says Ralph Jimenez, the editorial-page editor. 'You could tell they were thinking, _Did_ _you just ask me that? Do you know who I am?_ '\n\nBut the process wasn't limited to these interviews. Bill Clinton got in the habit of calling the editors, at home and on their cell phones, and launching into impassioned defenses of his wife. (Some of the editors had unlisted phone numbers, which made Clinton's calls even more impressive.) Obama had his own persistent advocates. The board was visited by former White House staff members, such as Madeleine Albright and Ted Sorensen, and lobbied by a bevy of local elected officials. For the five members of the editorial board, all the attention was flattering, if occasionally annoying. Felice Belman, the executive editor of the _Monitor,_ was awakened by a surprise phone call from Hillary at seven thirty on a Saturday morning. 'I was still half asleep,' she says. 'And I definitely wasn't in the mood to talk about health-care mandates.' (Ralph still has a phone message from Hillary Clinton on his cell phone.)\n\nTwelve days before the primary, on a snowy Thursday afternoon, the editorial board gathered in a back office of the newsroom. They'd postponed the endorsement meeting long enough; it was time to make a decision. Things would be easy on the Republican side: all five members favored John McCain. The Democrat endorsement, however, was a different story. Although the editors had each tried to keep an open mind \u2013 'The candidates are here for a year and you don't want to settle on one candidate right away,' said Mike Pride, a former editor of the paper \u2013 the room was starkly divided into two distinct camps. Ralph Jimenez and Ari Richter, the managing editor, were pushing for an Obama endorsement. Mike Pride and Geordie Wilson, the publisher, favored Clinton. And then there was Felice, the sole undecided vote. 'I was waiting to be convinced until the last minute,' she says. 'I guess I was leaning toward Clinton, but I still felt like I could have been talked into switching sides.'\n\nNow came the hard part. The board began by talking about the issues, but there wasn't that much to talk about: Obama and Clinton had virtually identical policy positions. Both candidates were in favor of universal health care, repealing the Bush tax cuts, and withdrawing troops from Iraq as soon as possible. And yet, despite this broad level of agreement, the editors were fiercely loyal to their chosen candidates, even if they couldn't explain _why_ they were so loyal. 'You just know who you prefer,' Ralph says. 'For most of the meeting, the level of discourse was pretty much, \"My person is better. Period. End of story.\"'\n\nAfter a lengthy and intense discussion \u2013 'We'd really been having this discussion for months,' says Ralph \u2013 the _Monitor_ ended up endorsing Clinton by a 3\u20132 vote. The room was narrowly split, but it had become clear that no one was going to change his or her mind. Even Felice, the most uncertain of the editors, was now firmly in the Clinton camp. 'There is always going to be disagreement,' Mike says. 'That's what happens when you get five opinionated people in the same room talking politics. But you also know that, before you leave the room, you've got to endorse somebody. You've got to accept the fact that some people are bound to be wrong' \u2013 he jokingly looks over at Ralph \u2013 'and find a way to make a decision.'\n\nFor readers of the _Monitor_ , the commentary endorsing Clinton seemed like a well-reasoned brief, an unambiguous summary of the newspaper's position. (Kathleen Strand, the Clinton spokesperson in New Hampshire, credited the endorsement with helping Clinton win the primary.) The carefully chosen words in the editorial showed no trace of the debate that had plagued the closed-door meeting and all those heated conversations by the water cooler. If just one of the editors had changed his or her mind, then the _Monitor_ would have chosen Obama. In other words, the clear-cut endorsement emerged from a very tentative majority.\n\nIn this sense, the editorial board is a metaphor for the brain. Its decisions often feel unanimous \u2013 you know which candidate you prefer \u2013 but the conclusions are actually reached only after a series of sharp internal disagreements. At any given moment while the cortex struggles to make a decision, rival bits of tissue are contradicting one another. Different brain areas think different things for different reasons. Sometimes this fierce argument is largely emotional, and the distinct parts of the limbic system debate one another. Although people can't always rationally justify their feelings \u2013 the editorial board members preferred either Hillary or Obama for reasons they couldn't really articulate \u2013 the feelings still manage to powerfully affect behavior. Other arguments unfold largely between the emotional and rational systems of the brain as the prefrontal cortex tries to resist the impulses coming from below. Regardless of which areas are doing the arguing, however, it's clear that all those mental components stuffed inside the head are constantly fighting for influence and attention. Like an editorial board, the mind is an extended argument. And it is arguing with itself.\n\nIn recent years, scientists have been able to show that this 'argument' isn't confined only to contentious issues such as presidential politics. Rather, it's a defining feature of the decision-making process. Even the most mundane choices emerge from a vigorous cortical debate. Let's say, for instance, that you're contemplating breakfast cereals in the supermarket. Each option will activate a unique subset of competing thoughts. Perhaps the organic granola is delicious but too expensive, or the whole-grain flakes are healthy but too unappetizing, or the Froot Loops are an appealing brand (the advertisements worked) but too sugary. Each of these distinct claims will trigger a particular set of emotions and associations, all of which then compete for your conscious attention. Antoine Bechara, a neuroscientist at USC, compares this frantic neural competition to natural selection, with the stronger emotions ('I really want Honey Nut Cheerios!') and the more compelling thoughts ('I should eat more fiber') gaining a selective advantage over weaker ones ('I like the cartoon character on the box of Froot Loops'). 'The point is that most of the computation is done at an emotional, unconscious level, and not at a logical level,' he says. The particular ensemble of brain cells that win the argument determine what you eat for breakfast.\n\nConsider this clever experiment designed by Brian Knutson and George Loewenstein. The scientists wanted to investigate what happens inside the brain when a person makes typical consumer choices, such as buying an item in a retail store or choosing a cereal. A few dozen lucky undergraduates were recruited as experimental subjects and given a generous amount of spending money. Each subject was then offered the chance to buy dozens of different objects, from a digital voice recorder to gourmet chocolates to the latest Harry Potter book. After the student stared at each object for a few seconds, he was shown the price tag. If he chose to buy the item, its cost was deducted from the original pile of cash. The experiment was designed to realistically simulate the experience of a shopper.\n\nWhile the student was deciding whether or not to buy the product on display, the scientists were imaging the subject's brain activity. They discovered that when a subject was first exposed to an object, his nucleus accumbens (NAcc) was turned on. The NAcc is a crucial part of the dopamine reward pathway, and the intensity of its activation was a reflection of desire for the item. If the person already owned the complete Harry Potter collection, then the NAcc didn't get too excited about the prospect of buying another copy. However, if he had been craving a George Foreman grill, the NAcc flooded the brain with dopamine when that item appeared.\n\nBut then came the price tag. When the experimental subject was exposed to the cost of the product, the insula and prefrontal cortex were activated. The insula produces aversive feelings and is triggered by things like nicotine withdrawal and pictures of people in pain. In general, we try to avoid anything that makes our insulas excited. This includes spending money. The prefrontal cortex was activated, scientists speculated, because this rational area was computing the numbers, trying to figure out if the product was actually a good deal. The prefrontal cortex got most excited during the experiment when the cost of the item on display was significantly lower than normal.\n\nBy measuring the relative amount of activity in each brain region, the scientists could accurately predict the subjects' shopping decisions. They knew which products people would buy before the people themselves did. If the insula's negativity exceeded the positive feelings generated by the NAcc, then the subject always chose not to buy the item. However, if the NAcc was more active than the insula, or if the prefrontal cortex was convinced that it had found a good deal, the object proved irresistible. The sting of spending money couldn't compete with the thrill of getting something new.\n\nThis data, of course, directly contradicts the rational models of microeconomics; consumers aren't always driven by careful considerations of price and expected utility. You don't look at the electric grill or box of chocolates and perform an explicit cost-benefit analysis. Instead, you outsource much of this calculation to your emotional brain and then rely on relative amounts of pleasure versus pain to tell you what to purchase. (During many of the decisions, the rational prefrontal cortex was largely a spectator, standing silently by while the NAcc and insula argued with each other.) Whichever emotion you feel most intensely tends to dictate your shopping decisions. It's like an emotional tug of war.\n\nThis research explains why consciously analyzing purchasing decisions can be so misleading. When Timothy Wilson asked people to analyze their strawberry-jam preferences, they made worse decisions because they had no idea what their NAccs really wanted. Instead of listening to their feelings, they tried to deliberately decipher their pleasure. But we can't ask our NAccs questions; we can only listen to what they have to say. Our desires exist behind locked doors.\n\nRetail stores manipulate this cortical setup. They are designed to get us to open our wallets; the frivolous details of the shopping experience are really subtle acts of psychological manipulation. The store is tweaking our brains, trying to soothe the insulas and stoke the NAccs. Just look at the interior of a Costco warehouse. It's no accident that the most coveted items are put in the most prominent places. A row of high-definition televisions lines the entrance. The fancy jewelry, Rolex watches, iPods and other luxury items are conspicuously placed along the corridors with the heaviest foot traffic. And then there are the free samples of food, liberally distributed throughout the store. The goal of Costco is to constantly prime the pleasure centers of the brain, to keep us lusting after things we don't need. Even though you probably won't buy the Rolex, just looking at the fancy watch makes you more likely to buy something else, since the desired item activates the NAcc. We have all been conditioned to crave rewards.\n\nBut exciting the NAcc is not enough; retailers must also inhibit the insula. This brain area is responsible for making sure you don't get ripped off, and when it's repeatedly assured by retail stores that low prices are 'guaranteed', or that a certain item is on sale, or that it's getting the 'wholesale price', the insula stops worrying so much about the price tag. In fact, researchers have found that when a store puts a promotional sticker next to the price tag \u2013 something like 'Bargain Buy!' or 'Hot Deal!' \u2013 but doesn't actually reduce the price, sales of that item still dramatically increase. These retail tactics lull the brain into buying more things, since the insula is pacified. We go broke convinced that we are saving money.\n\nThis model of the shopping brain also helps explain why credit cards make us spend so irresponsibly. According to Knutson and Loewenstein, paying with plastic literally inhibits the insula, making a person less sensitive to the cost of an item. As a result, the activity of the NAcc \u2013 the pleasure pump of the cortex \u2013 becomes disproportionately important: it wins every shopping argument.\n\n##  _Partisan Delusions_\n\nThere's something unsettling about seeing the brain as one big argument. Each of us likes to believe that his decisions reflect a clear cortical consensus, that the entire mind agrees on what he should do. And yet, that serene self-image has little basis in reality. The NAcc might want the George Foreman grill, but the insula knows that you can't afford it, or the prefrontal cortex realizes that it's a bad deal. The amygdala might like Hillary Clinton's tough talk on foreign policy, but the ventral striatum is excited by Obama's uplifting rhetoric. These antagonistic reactions manifest themselves as a twinge of uncertainty. You don't know what you believe. And you certainly don't know what to do.\n\nThe dilemma, of course, is how to reconcile the argument. If the brain is always disagreeing with itself, then how can a person ever make a decision? At first glance, the answer seems obvious: force a settlement. The rational parts of the mind should intervene and put an end to all the emotional bickering.\n\nWhile such a top-down solution might seem like a good idea \u2013 using the most evolutionarily advanced parts of the brain to end the cognitive _contretemps_ \u2013 this approach must be used with great caution. The problem is that the urge to end the debate often leads to neglect of crucial pieces of information. A person is so eager to silence the amygdala, or quiet the OFC, or suppress some bit of the limbic system that he or she ends up making a bad decision. A brain that's intolerant of uncertainty \u2013 that can't stand the argument \u2013 often tricks itself into thinking the wrong thing. What Mike Pride says about editorial boards is also true of the cortex: 'The most important thing is that everyone has their say, that you listen to the other side and try to understand their point of view. You can't short-circuit the process.'\n\nUnfortunately, the mind often surrenders to the temptation of shoddy top-down thinking. Just look at politics. Voters with strong partisan affiliations are a case study in how _not_ to form opinions: their brains are stubborn and impermeable, since they already know what they believe. No amount of persuasion or new information is going to change the outcome of their mental debates. For instance, an analysis of 500 voters with 'strong party allegiances' during the 1976 campaign found that, during the heated last two months of the contest, only 16 people were persuaded to vote for the other party. Another study tracked voters from 1965 to 1982, tracing the flux of party affiliation over time. Although it was an extremely tumultuous era in American politics \u2013 there was the Vietnam War, stagflation, the fall of Richard Nixon, oil shortages, and Jimmy Carter \u2013 nearly 90 per cent of people who identified themselves as Republicans in 1965 ended up voting for Ronald Reagan in 1980. The happenings of history didn't change many minds.\n\nIt's now possible to see why partisan identities are so persistent. Drew Westen, a psychologist at Emory University, imaged the brains of ordinary voters with strong party allegiances during the run-up to the 2004 election. He showed the voters multiple clearly contradictory statements made by each candidate, John Kerry and George Bush. For example, the experimental subject would read a quote from Bush praising the service of soldiers in the Iraq war and pledging 'to provide the best care for all veterans'. Then the subject would learn that on the same day Bush made this speech, his administration cut medical benefits for 164,000 veterans. Kerry, meanwhile, was quoted making contradictory statements about his vote to authorize war in Iraq.\n\nAfter being exposed to the political inconsistencies of both candidates, the subject was asked to rate the level of contradiction on a scale of 1 to 4, with 4 signaling a strong level of contradiction. Not surprisingly, the reactions of voters were largely determined by their partisan allegiances. Democrats were troubled by Bush's inconsistent statements (they typically rated them a 4) but found Kerry's contradictions much less worrisome. Republicans responded in a similar manner; they excused Bush's gaffes but almost always found Kerry's statements flagrantly incoherent.\n\nBy studying each of these voters in an fMRI machine, Westen was able to look at the partisan reasoning process from the perspective of the brain. He could watch as Democrats and Republicans struggled to maintain their political opinions in the face of conflicting evidence. After being exposed to the inconsistencies of their preferred candidate, the party faithful automatically recruited brain regions that are responsible for controlling emotional reactions, such as the prefrontal cortex. While this data might suggest that voters are rational agents calmly assimilating the uncomfortable information, Westen already knew that wasn't happening, since the ratings of Kerry and Bush were entirely dependent on the subjects' party affiliations. What, then, was the prefrontal cortex doing? Westen realized that voters weren't using their reasoning faculties to analyze the facts; they were using reason to preserve their partisan certainty. And then, once the subjects had arrived at favorable interpretations of the evidence, blithely excusing the contradictions of their chosen candidate, they activated the internal reward circuits in their brains and experienced a rush of pleasurable emotion. Self-delusion, in other words, felt really good. 'Essentially, it appears as if partisans twirl the cognitive kaleidoscope until they get the conclusions they want,' Westen says, 'and then they get massively reinforced for it, with the elimination of negative emotional states and activation of positive ones.'\n\nThis flawed thought process plays a crucial role in shaping the opinions of the electorate. Partisan voters are convinced that they're rational \u2013 it's the other side that's irrational \u2013 but actually, all of us are _rationalizers_. The Princeton political scientist Larry Bartels analyzed survey data from the 1990s to prove this point. During the first term of Bill Clinton's presidency, the budget deficit declined by more than 90 per cent. However, when Republican voters were asked in 1996 what happened to the deficit under Clinton, more than 55 per cent said that it had _increased_. What's interesting about this data is that so-called high-information voters \u2013 these are the Republicans who read the newspaper, watch cable news, and can identify their representatives in Congress \u2013 weren't better informed than low-information voters. (Many low-information voters struggled to name the vice president.) According to Bartels, the reason knowing more about politics doesn't erase partisan bias is that voters tend to assimilate only those facts that confirm what they already believe. If a piece of information doesn't follow Republican talking points \u2013 and Clinton's deficit reduction didn't fit the tax-and-spend liberal stereotype \u2013 then the information is conveniently ignored. 'Voters think that they're thinking,' Bartels says, 'but what they're really doing is inventing facts or ignoring facts so that they can rationalize decisions they've already made.' Once you identify with a political party, the world is edited to fit with your ideology.\n\nAt such moments, rationality actually becomes a liability, since it allows us to justify practically any belief. The prefrontal cortex is turned into an information filter, a way to block out disagreeable points of view. Let's look at an experiment done in the late 1960s by the cognitive psychologists Timothy Brock and Joe Balloun. Half of the subjects involved in the experiment were regular churchgoers, and half were committed atheists. Brock and Balloun played a tape-recorded message attacking Christianity, and, to make the experiment more interesting, they added an annoying amount of static \u2013 a crackle of white noise \u2013 to the recording. However, the listener could reduce the static by pressing a button, at which point the message suddenly became easier to understand.\n\nThe results were utterly predictable and rather depressing: the nonbelievers always tried to remove the static, while the religious subjects actually preferred the message that was harder to hear. Later experiments by Brock and Balloun that had smokers listening to a speech on the link between smoking and cancer demonstrated a similar effect. We all silence the cognitive dissonance through self-imposed ignorance.\n\nThis sort of blinkered thinking isn't a problem only for partisan voters and devout believers. In fact, research suggests that the same flaw also afflicts those people who are supposed to be most immune to such cognitive errors: political pundits. Even though pundits are trained professionals, presumably able to evaluate the evidence and base their opinions on the cold, hard facts \u2013 that's why we listen to them \u2013 they are still vulnerable to cognitive mistakes. Like partisan voters, they selectively interpret the data so that it proves them right. They'll distort the thought process until it leads to the desired conclusion.\n\nIn 1984, the University of California at Berkeley psychologist Philip Tetlock began what he thought would be a brief research project. At the time, the Cold War was flaring up again \u2013 Reagan was talking tough to the 'evil empire' \u2013 and political pundits were sharply divided on the wisdom of American foreign policy. The doves thought Reagan was needlessly antagonizing the Soviets, while the hawks were convinced that the USSR needed to be aggressively contained. Tetlock was curious which group of pundits would turn out to be right, and so he began monitoring their predictions.\n\nA few years later, after Reagan left office, Tetlock revisited the opinions of the pundits. His conclusion was sobering: _everyone was wrong_. The doves had assumed that Reagan's bellicose stance would exacerbate Cold War tensions and had predicted a breakdown in diplomacy as the USSR hardened its geopolitical stance. The reality, of course, was that the exact opposite happened. By 1985, Mikhail Gorbachev was in power. The Soviet Union began implementing a stunning series of internal reforms. The 'evil empire' was undergoing _glasnost_.\n\nBut the hawks didn't do much better. Even after Gorbachev began the liberalizing process, hawks tended to disparage the changes to the Soviet system. They said the evil empire was still evil; Gorbachev was just a tool of the politburo. Hawks couldn't imagine that a sincere reformer might actually emerge from a totalitarian state.\n\nThe dismal performance of these pundits inspired Tetlock to turn his small case study into an epic experimental project. He picked 284 people who made their living 'commenting or offering advice on political and economic trends' and began asking them to make predictions about future events. He had a long list of pertinent questions. Would George Bush be re-elected? Would there be a peaceful end to apartheid in South Africa? Would Quebec secede from Canada? Would the dot-com bubble burst? In each case, the pundits were asked to rate the probability of several possible outcomes. Tetlock then interrogated the pundits about their thought processes so he could better understand how they'd made up their minds. By the end of the study, Tetlock had quantified 82,361 different predictions.\n\nAfter Tetlock tallied the data, the predictive failures of the pundits became obvious. Although they were paid for their keen insights into world affairs, they tended to perform _worse_ than random chance. Most of Tetlock's questions had three possible answers; on average, the pundits had selected the right answer less than 33 per cent of the time. In other words, a dart-throwing chimp would have beaten the vast majority of professionals. Tetlock also found that the most famous pundits in his study tended to be the least accurate, consistently churning out overblown and overconfident forecasts. Eminence was a handicap.\n\nWhy were these pundits (especially the prominent ones) so bad at forecasting the future? The central error diagnosed by Tetlock was the sin of _certainty,_ which led the 'experts' to mistakenly impose a top-down solution on their decision-making processes. In chapter two, we saw examples of the true expertise that occurs when experience is internalized by the dopamine system. This results in a person who has a set of instincts that respond quickly to the situation at hand, regardless of whether that's playing backgammon or staring at a radar screen. The pundits in Tetlock's study, however, distorted the verdicts of their emotional brains, cherry-picking the feelings they wanted to follow. Instead of trusting their gut feelings, they found ways to disregard the insights that contradicted their ideologies. When pundits were convinced that they were right, they ignored any brain areas that implied they might be wrong. This suggests that one of the best ways to distinguish genuine from phony expertise is to look at how a person responds to dissonant data. Does he or she reject the data out of hand? Perform elaborate mental gymnastics to avoid admitting error? Everyone makes mistakes; the object is to learn from these mistakes.\n\nTetlock notes that the best pundits are willing to state their opinions in 'testable form' so that they can 'continually monitor their forecasting performance'. He argues that this approach makes pundits not only more responsible \u2013 they are forced to account for being wrong \u2013 but also less prone to bombastic convictions, a crucial sign that a pundit isn't worth listening to. (In other words, ignore those commentators that seem too confident or self-assured. The people on television who are most certain are almost certainly going to be wrong.) As Tetlock writes, 'The dominant danger [for pundits] remains hubris, the vice of closed-mindedness, of dismissing dissonant possibilities too quickly.' Even though practically all of the professionals in Tetlock's study claimed that they were dispassionately analyzing the evidence \u2013 everybody wanted to be rational \u2013 many of them were actually indulging in some conveniently cultivated ignor ance. Instead of encouraging the arguments inside their heads, these pundits settled on answers and then came up with reasons to justify those answers. They were, as Tetlock put it, 'prisoners of their preconceptions'.\n\n##  _The Certainty Bias_\n\nIt feels good to be certain. Confidence is comforting. This desire to always be right is a dangerous side effect of having so many competing brain regions inside one's head. While neural pluralism is a crucial virtue \u2013 the human mind can analyze any problem from a variety of different angles \u2013 it also makes us insecure. You never know which brain area you should obey. It's not easy to make up your mind when your mind consists of so many competing parts.\n\nThis is why being sure about something can be such a relief. The default state of the brain is indecisive disagreement; various mental parts are constantly insisting that the other parts are wrong. Certainty imposes consensus on this inner cacophony. It lets you pretend that your entire brain agrees with your behavior. You can now ignore those annoying fears and nagging suspicions, those statistical outliers and inconvenient truths. Being certain means that you aren't worried about being wrong.\n\nThe lure of certainty is built into the brain at a very basic level. This is most poignantly demonstrated by split-brain patients. (These are patients who have had the corpus callosum \u2013 the nerve tissue that connects the two hemispheres of the brain \u2013 severed. The procedure is performed only rarely, usually to treat intractable seizures.) A typical experiment goes like this: using a special instrument, different sets of pictures are flashed to each of the split-brain patient's visual fields. (Because of our neural architecture, all information about the left visual field is sent to the right hemisphere, and all information about the right visual field is sent to the left hemisphere.) For example, the right visual field might see a picture of a chicken claw and the left visual field might see a picture of a snowy driveway. The patient is then shown various images and asked to pick out the one that is most closely associated with what he or she has just seen. In a tragicomic display of indecisiveness, the split-brain patient's hands point to two different objects. The right hand points to a chicken (this matches the chicken claw that the left hemisphere witnessed), while the left hand points to a shovel (the right hemisphere wants to shovel the snow). The conflicting reactions of the patient reveals the inner contradictions of each of us. The same brain has come up with two very different answers.\n\nBut something interesting happens when scientists ask a split-brain patient to explain the bizarre response: the patient manages to come up with an explanation. 'Oh, that's easy,' one patient said. 'The chicken claw goes with the chicken, and you need a shovel to clean out the chicken shed.' Instead of admitting that his brain was hopelessly confused, the patient wove his confusion into a plausible story. In fact, the researchers found that when patients made especially ridiculous claims, they seemed even more confident than usual. It was a classic case of overcompensation.\n\nOf course, the self-assurance of the split-brain patient is clearly mistaken. None of the images contained a chicken shed that needed a shovel. But that deep need to repress inner contradictions is a fundamental property of the human mind. Even though the human brain is defined by its functional partitions, by the friction of all these different perspectives, we always feel compelled to assert its unity. As a result, each of us pretends that the mind is in full agreement with itself, even when it isn't. We trick ourselves into being sure.\n\nDuring the last week of September 1973, the Egyptian and Syrian armies began massing near the Israeli border. The signals picked up by the Mossad, the main Israeli intelligence agency, were ominous. Artillery had been moved into offensive positions. Roads were being paved in the middle of the desert. Thousands of Syrian reservists had been ordered to report for duty. From the hills of Jerusalem, people could see a haze of black diesel smoke on the horizon, the noxious exhaust generated by thousands of Soviet-made tanks. The smoke was getting closer.\n\nThe official explanation for the frenzy of military activity was that it was a pan-Arab training exercise. Although Anwar Sadat, the president of Egypt, had boldly declared a few months before that his country was 'mobilizing in earnest for the resumption of battle' and declared that the destruction of Israel was worth the 'sacrifice of one million Egyptian soldiers,' the Israeli intelligence community insisted that the Egyptians weren't actually planning an attack. Major General Eli Zeira, the director of Aman, the Israeli military intelligence agency, publicly dismissed the possibility of an Egyptian invasion. 'I discount the likelihood of a conventional Arab attack,' Zeira said. 'We have to look hard for evidence of their real intentions in the field \u2013 otherwise, with the Arabs, all you have is rhetoric. Too many Arab leaders have intentions which far exceed their capabilities.' Zeira believed that the Egyptian military build-up was just a bluff, a feint intended to shore up Sadat's domestic support. He persuasively argued that the Syrian deployments were merely a response to a September skirmish between Syrian and Israeli fighter planes.\n\nOn 3 October, Golda Meir, the prime minister of Israel, held a regular cabinet meeting that included the heads of Israeli intelligence. It was here that she was told about the scale of Arab preparations for war. She learned that the Syrians had concentrated their anti-aircraft missiles at the border, the first time this had ever been done. In addition, several Iraqi armored divisions had moved into southern Syria. She was also informed about Egyptian military maneuvers in the Sinai that weren't part of the official 'training exercise'. Although everyone agreed that the news was troubling, the consensus remained the same. The Arabs were not ready for war. They wouldn't dare invade. The next cabinet meeting was scheduled for 7 October, the day after Yom Kippur.\n\nIn retrospect, it's clear that Zeira and the Israeli intelligence community were spectacularly wrong. In the early afternoon of 6 October, the Egyptian and Syrian armies \u2013 a force roughly equivalent to the NATO European command \u2013 launched a surprise attack on Israeli positions in the Golan Heights and Sinai Peninsula. Because Meir didn't issue a full mobilization order until the invasion was already under way, the Israeli military was unable to repel the Arab armies. Egyptian tanks streamed across the Sinai and nearly captured the strategically important Mitla Pass. Before nightfall, more than eight thousand Egyptian infantry had moved into Israeli territory. The situation in the Golan Heights was even more dire: 130 Israeli tanks were trying to hold off more than one thousand, three hundred Syrian and Iraqi tanks. By that evening, the Syrians were pressing toward the Sea of Galilee, and the Israelis were suffering heavy casualties. Reinforcements were rushed to battle. If the Golan fell, Syria could easily launch artillery at Israeli cities. Moshe Dayan, the Israeli defense minister, concluded after the third day of conflict that the chances of the Israeli nation surviving the war were 'very low'.\n\nThe tide shifted gradually. By 8 October, the newly arrived Israeli reinforcements began to reassert control in the Golan Heights. The main Syrian force was split into two smaller contingents that were quickly isolated and destroyed. By 10 October, Israeli tanks had crossed the 'purple line', or the pre-war Syrian border. They would eventually progress nearly forty kilometers into the country, or close enough to shell the suburbs of Damascus.\n\nThe Sinai front was more treacherous. The initial Israeli counterattack, on 8 October, was an unmitigated disaster: nearly an entire brigade of Israeli tanks was lost in a few hours. (General Shmuel Gonen, the Israeli commander of the Southern Front, was later disciplined for his 'failure to fulfill his duties'.) In addition, the Israeli air force had lost control of the skies; its fighter planes were being shot down at an alarming rate, as the Soviet SA-2 anti-aircraft batteries proved to be much more effective than expected. ('We are like fat ducks up there,' one Israeli pilot said. 'And they have the shotguns.') The next several days were a tense stalemate, neither army willing to risk an attack.\n\nThe standoff ended on 14 October, when Sadat ordered his generals to attack. He wanted to ease the pressure on the Syrians, who were now fighting to protect their capital. But the massive Egyptian force was repulsed \u2013 they lost nearly two hundred and fifty tanks \u2013 and on 15 October, the Israelis launched a successful counterattack. The Israelis struck at the seam between the two main Egyptian armies and managed to secure a bridgehead on the opposite side of the Suez Canal. This breach marked the turning point of the Sinai campaign. By 22 October, an Israeli armored division was within a hundred miles of Cairo. A cease-fire went into effect a few days later.\n\nFor Israel, the end of the war was bittersweet. Although the surprise invasion had been repelled, and no territory had been lost, the tactical victory had revealed the startling fragility of the nation. It turned out that Israel's military superiority was not a guarantee of security. The small country had almost been destroyed by an intelligence failure.\n\nAfter the war, the Israeli government appointed a special committee to investigate the _mechdal_ , or 'omission', that had preceded the war. Why hadn't the intelligence community anticipated the invasion? The committee had uncovered a staggering amount of evidence suggesting an imminent attack. On 4 October, they learned that, in addition to building up Egyptian and Syrian forces along the border, the Arabs had evacuated Soviet military advisers from Cairo and Damascus. The day after that, new reconnaissance photographs had revealed the movement of anti-aircraft missiles to the front lines and the departure of the Soviet fleet from the port of Alexandria. At this point, it should have been clear that the Egyptian forces weren't training in the desert; they were getting ready for war.\n\nLieutenant Benjamin Simon-Tov, a young intelligence officer at the Southern Command, was one of the few analysts who connected the dots. On 1 October, he wrote a memo urging his commander to consider the possibility of an Arab attack. That memo was ignored. On 3 October, he compiled a briefing document summarizing recent aggressive Egyptian actions. He argued that the Sinai invasion would begin within a week. His superior officer refused to pass the 'heretical' report up the chain of command.\n\nWhy was the intelligence community so resistant to the idea of an October attack? After the Six-Day War of 1967, the Mossad and Aman developed an influential theory of Arab strategy that they called _ha-Konseptzia_ (the Concept). This theory was based largely on the intelligence of a single source in the Egyptian government. It held that Egypt and Syria wouldn't consider attacking Israel until 1975, at which point they would have an adequate number of fighter planes and pilots. (Israeli air superiority had played a key role in the decisive military victory of 1967.) The Concept also placed great faith in the Bar-Lev line, a series of defensive positions along the Suez Canal. The Mossad and Aman believed that these obstacles and reinforcements would restrain Egyptian armored divisions for at least twenty-four hours, thus allowing Israel crucial time to mobilize its reservists.\n\nThe Concept turned out to be completely wrong. The Egyptians were relying on their new surface-to-air missiles to counter the Israeli air forces; they didn't need more planes. The Bar-Lev line was easy to breach. The defensive positions were mostly made of piled desert sand, which the Egyptian military moved using pressured water cannons. Unfortunately, the Concept was deeply ingrained in the strategic thinking of the Israeli intelligence community. Until the invasion actually began, the Mossad and Aman had insisted that no invasion would take place. Instead of telling the prime minister that the situation on the ground was uncertain and ambiguous \u2013 nobody really knew if the Egyptians were bluffing or planning to attack \u2013 the leaders of the Mossad and Aman chose to project an unshakable confidence in the Concept. They were misled by their certainty, which caused them to ignore a massive amount of contradictory evidence. As the psychologist Uri Bar-Joseph noted in his study of the Israeli intelligence failure, 'The need for cognitive closure prompted leading analysts, especially Zeira, to \"freeze\" on the conventional wisdom that an attack was unlikely and to become impervious to information suggesting that it was imminent.'\n\nEven on the morning of 6 October, just a few hours before Egyptian tanks crossed the border, Zeira was still refusing to admit that a mobilization might be necessary. A top-secret cable had just arrived from a trusted source inside an Arab government, warning that an invasion was imminent, that Syria and Egypt weren't bluffing. Meir convened a meeting with her top military officials to assess this new intelligence. She asked Zeira if he thought the Arab nations were going to attack. Zeira said no. They would not dare to attack, he told the prime minister. Of that he was certain.\n\nThe lesson of the Yom Kippur War is that having access to the necessary information is not enough. Eli Zeira, after all, had more than enough military intelligence at his disposal. He saw the tanks at the border; he read the top-secret memos. His mistake was that he never forced himself to consider these inconvenient facts. Instead of listening to the young lieutenant, he turned up the static dial and clung to the Concept. The result was a bad decision.\n\nThe only way to counteract the bias for certainty is to encourage some inner dissonance. We must force ourselves to think about the information we don't want to think about, to pay attention to the data that disturbs our entrenched beliefs. When we start censoring our minds, turning off those brain areas that contradict our assumptions, we end up ignoring relevant evidence. A major general shrugs off the evacuation of Soviet military personnel and those midnight cables from trusted sources. He insists that an invasion isn't happening even when it has already begun.\n\nBut the certainty trap is not inevitable. We can take steps to prevent ourselves from shutting down our minds' arguments too soon. We can consciously correct for this innate tendency. And, if those steps fail, we can create decision-making environments that help us better entertain competing hypotheses. Look, for example, at the Israeli military. After failing to anticipate the 1973 war, Israel thoroughly revamped its intelligence services. It added an entirely new branch of intelligence analysis, the Research and Political Planning Center, which operated under the auspices of the Foreign Ministry. The mission of this new center wasn't to gather more information; the Israelis realized that data collection wasn't their problem. Instead, the unit was designed to provide an assessment of the available data, one that was completely independent of both Aman and the Mossad. It was a third opinion, in case the first two opinions were wrong.\n\nAt first glance, adding another layer of bureaucracy might seem like a bad idea. Interagency rivalries can create their own set of problems. But the Israelis knew that the surprise invasion of 1973 was a direct result of their false sense of certainty. Because Aman and the Mossad were convinced that the Concept was accurate, they had ignored all contradictory evidence. Complacency and stubbornness soon set in. The commission wisely realized that the best way to avoid such certainty in the future was to foster diversity, ensuring that the military would never again be seduced by its own false assumptions.\n\nThe historian Doris Kearns Goodwin made a similar point about the benefits of intellectual diversity in _Team of Rivals,_ her history of Abraham Lincoln's cabinet. She argues that it was Lincoln's ability to deal with competing viewpoints that made him such a remarkable president and leader. He intentionally filled his cabinet with rival politicians who had extremely different ideologies; antislavery crusaders, like the secretary of state William Seward, were forced to work with more conservative figures, like the attorney general Edward Bates, a man who had once been a slave owner. When making a decision, Lincoln always encouraged vigorous debate and discussion. Although several members of his cabinet initially assumed that Lincoln was weak willed, indecisive and unsuited for the presidency, they eventually realized that his ability to tolerate dissent was an enormous asset. As Seward said, 'The president is the best of us.'\n\nThe same lesson can be applied to the brain: when making decisions, actively resist the urge to suppress the argument. Instead, take the time to listen to what all the different brain areas have to say. Good decisions rarely emerge from a false consensus. Alfred P. Sloan, the chairman of General Motors during its heyday, once adjourned a board meeting soon after it began. 'Gentlemen,' Sloan said, 'I take it we are all in complete agreement on the decision here... Then I propose we postpone further discussion of this matter until our next meeting to give ourselves time to develop disagreement and perhaps gain some understanding of what the decision is all about.'\nEIGHT\n\n# The Poker Hand\n\nMICHAEL BINGER IS A particle physicist at Stanford. His specialty is quantum chromodynamics, a branch of physics that studies matter in its most elemental form. Binger is also a professional poker player, and he spends most of June and July sitting at felt-lined card tables in a Las Vegas casino, competing in the World Series of Poker, the most important gambling event in the world. He is one of thousands of poker players who make the pilgrimage every year. These card sharks might not look like professional athletes \u2013 the tournament is full of overweight chain smokers \u2013 but that's because they are athletes of the _mind_. When it comes to playing poker, the only thing that separates the experts from the amateurs is the quality of their decisions.\n\nDuring the days of the World Series of Poker, Binger quickly settles into a mentally exhausting routine. He begins playing cards around noon \u2013 his preferred game is Texas hold'em \u2013 and he doesn't cash in his chips until the wee hours of the morning. Then it's back to his hotel room \u2013 past the strip clubs, penny slots, and the $7.77 all-you-can-eat buffets \u2013 where Binger tries to coax himself into a few hours of fitful sleep. 'You get so wired playing poker that it's not easy coming down,' he says. 'I tend to just lie in bed, thinking about all the hands I played and how I should have played them differently.'\n\nBinger began playing cards as a college student, when he was a math and physics major at North Carolina State University. One weekend, he decided to learn how to play blackjack. He quickly grew frustrated by the amount of luck involved \u2013 'I hated not knowing when to bet,' he says \u2013 and so he taught himself how to count cards. He practiced in loud North Carolina bars so that he could learn how to focus amid the noise and revelry. Binger is blessed with a quantitative mind \u2013 'I was always the nerd who did math problems for fun,' he confesses \u2013 and counting cards came naturally to him. He quickly learned how to keep a running tally in his head, giving him a crucial advantage at the table. (For the most part, Binger relied on the Hi-Lo card-counting system, which provides the player with a 1 per cent edge over the house.) Before long, Binger was traveling to casinos and putting his quantifying talents to work.\n\n'The first thing I learned from counting cards,' Binger says, 'is that you can use your smarts to win. Sure, there's always luck, but over the long run you'll come out ahead if you're thinking right. The second thing I learned is that you can be too smart. The casinos have algorithms that automatically monitor your betting, and if they detect that your bets are too accurate, they'll ask you to leave.' This meant that Binger needed to occasionally make bad bets on purpose. He would intentionally lose money so that he could keep on making money.\n\nBut even with this precaution, Binger started to put numerous casinos on alert. In blackjack, it's supposed to be impossible to consistently beat the house, and yet that's exactly what Binger kept doing. Before long, he was blacklisted; casino after casino told him that he couldn't play blackjack at their tables. 'Some of the casinos would ask politely,' Binger says. 'A manager would come and tell you to take your winnings and leave. And some casinos weren't so polite. Let's just say they made it clear you weren't welcome anymore.'\n\nAfter he started graduate school in theoretical physics at Stanford, Binger tried to shake his card habit. 'The low point for me was getting kicked out of six Reno casinos in one day for counting cards,' he says. 'That's when I realized maybe I should focus on physics for a while.' He was drawn to the most difficult problems in the field, studying supersymmetry and the Higgs boson particle. (The elusive Higgs is often referred to as the 'God particle', since finding it would help explain the origin of the universe.) 'There's no doubt that the analytical skills I learned in cards also helped me with science,' Binger says. 'It's all about focusing on the important variables, thinking clearly, not getting distracted. If you lose your train of thought when you're counting cards, you're screwed. Physics is a little more forgiving \u2013 you can write stuff down \u2013 but it still requires a very disciplined thought process.'\n\nAfter a few years of working diligently toward his PhD, Binger began to miss his beloved card games. The relapse was gradual. He started playing a few rounds of small-stakes poker with his friends, just a casual game or two after a long day spent contemplating physics equations. But it didn't take long before Binger's friends refused to play with him; he kept taking all of their money. And so Binger began playing poker tournaments, driving out on the weekends to the card rooms near the San Francisco airport. After a few months, Binger was making more money on the part-time poker circuit than he made as a postdoc. He used his winnings to pay off his student loans and start a modest bankroll. 'I realized that I'd never be able to really focus on physics until I gave poker a shot,' he says. 'I needed to know if I could make it.' That's when Binger decided to try his luck as a professional gambler.\n\nThe World Series of Poker (WSOP) is held at the Rio Hotel, a Brazilian-themed casino that's across the highway from the Strip. For the most part, the Latin motif is confined to the silly costumes of the staff, the syrupy cocktails and the ugly carpets, which are a swirl of Carnival colors. The hotel itself is a generic tower of reflective purple and red glass. During the WSOP, the lobby of the Rio accumulates the litter of the tournament: cigarette butts, empty water bottles, registration papers, fast-food wrappers. Anxious players collect in the corners, sharing stories of bad beats and lucky breaks. Even the hotel gift shop is stocked for the event, carrying a wide selection of poker primers right next to the nudie magazines.\n\nMost of the tournament takes place in the Amazon Room, a cavernous warehouse-like space with more than two hundred card tables. Security cameras dangle from the ceiling like ominous disco balls. Compared with the rest of Vegas, the atmosphere inside the huge room is starkly sober and serious. (Nobody would dare litter in here.) Even when it's filled with poker players, the enormous area has moments of startling calm, when all you can hear are the shuffling of cards and the perpetual hum of the air-conditioning system. Outside, it's 114 degrees.\n\nBinger is tall and lean, with a face made of angles. His hair is boyishly blond, and it's usually styled with copious amounts of gel so that it sticks straight up. At every poker tournament, he wears the same outfit: a backward baseball cap, opaque Oakley sunglasses and a brightly colored button-down shirt. Such consistency is typical of poker players, who are creatures of habit and rigid believers in routine. (A common quip on the pro circuit is 'It's unlucky to be superstitious.') Some professionals wear their same sweatshirts day after day, until the reek of their nervousness precedes them. Others develop bizarre eating rituals, like Jamie Gold, who orders scrambled eggs for breakfast even though he's allergic to eggs.\n\nBinger actually eats his eggs. His breakfast routine consists of one egg over easy, sandwiched between a lightly toasted English muffin. After eating that, he drinks a small glass of orange juice and then a strong cup of tea. He digests for 'approximately 10 to 12 minutes' and next drives to the gym, where he performs an extremely regimented workout. 'All these habits probably sound a little crazy,' Binger says, 'but when you're playing in a tournament it's crucial to not distract yourself with thinking about what to order for breakfast or how many laps to swim. The routine keeps it simple, so all I'm thinking about is poker, poker, poker.'\n\nAt the 2006 WSOP, Binger was one of 8,773 players who each paid $10,000 to enter the main event, a no-limit Texas hold'em competition stretching over thirteen days. Since 1991, when the prize money for the WSOP first exceeded $1 million, the poker tournament has been more lucrative for its winners than Wimbledon, the PGA championship and the Kentucky Derby. Since 2000, it has become the most valuable sporting event in the world, at least for the winners. (More than 90 per cent of entrants won't 'make the money', which means that they'll lose their entire entry fee.) In 2006, the top prize for the main event was expected to exceed $12 million. To make an equivalent amount of money playing tennis, you'd have to win Wimbledon ten times.\n\nThe rules of Texas hold'em are simple. Nine players gather around a card table, each of them determined to assemble the best possible poker hand. The game begins when each player is dealt two cards, face-down. The two players to the left of the dealer are then forced to make blind bets, wagering their money before they even look at their cards. (These bets ensure that there's some money at stake in every hand.) The remaining players have three options: they can match the bet, raise it, or fold. If a player has strong hole cards \u2013 a pair of aces being the best possible duo \u2013 he or she will make an aggressive bet. (Unless, of course, the player wants to act weak, but that's another story.) A bad hand is a good reason to fold.\n\nAfter the first round of betting is over, three community cards are dealt, face-up, in the center of the table. These cards are called the flop. There is now another round of betting, as players adjust their wagers in light of this new information. Then two more community cards are dealt, one at a time, with another round of betting after each. (The fourth card is called the turn, and the fifth card is called the river.) Each player then assembles the most valuable hand possible by combining the two hole cards with any three of the five community cards shared by the entire table. So let's say you're dealt the ace and the 10 of hearts. The best possible set of community cards would contain the jack, queen and king of hearts, since that would give you a royal flush, the perfect poker hand. (A royal flush is dealt approximately once every 648,739 poker hands.) If you got the jack, queen and king of different suits, then you'd have a straight. (Odds: 253 to 1.) You'd also be thrilled with three heart cards of any value, since that would give you a flush. (Odds: 507 to 1.) A more likely scenario is that you end up with a single pair (odds are 1.37 to 1), or you might get absolutely nothing, in which case the highest-ranking card, the ace, is your entire hand.\n\nAt its core, poker is a profoundly statistical game. Each hand is ranked according to its rarity, so having two pairs is more valuable than having one, and a straight flush is more valuable than a straight or a flush. A poker player who can parse his hole cards into possible probabilities \u2013 who knows, for example, that being dealt a pair of fours means that there's a 4 per cent chance of getting dealt another four on the flop \u2013 has a distinct advantage over his competitors. He can make bets informed by the crude laws of statistics, so that his wagers reflect the likelihood of winning the hand.\n\nBut the game isn't just about the cards. The act of betting is what makes poker so infinitely complicated. It's what turns Texas hold'em into a black art, a mixture of stagecraft and game theory. Consider the act of raising the bet. Such a move can have a straightforward meaning: a player is demonstrating confidence in his hole cards. Or it can signal a bluff, as a player tries to steal the pot by intimidating all the other players into folding. How does one distinguish between these different intentions? That's where the skill comes in. Professional poker players are constantly trying to read their opponents, searching for the minor tells of deceit. Does this bet fit into a behavioral pattern? Has the player been consistently 'tight' or aggressive all night? Why is his left eye twitching? Is that a symptom of nervousness? (Those who are easy to read are known as ABC players.) Of course, the best poker players are also the best liars, able to keep their opponents off balance with sincere bluffs and unpredictable bets. They know that the most important thing in poker is not what cards they actually have, but what cards people _think_ they have. A lie told well is just as good as the truth.\n\nIn the beginning of the tournament, Binger played patient poker, using his extraordinary math skills \u2013 a talent he honed in grad school \u2013 to methodically figure out which hands he should enter. Nine times out of 10 he immediately folded, and he risked his money only when he had hole cards with decent statistical odds, such as a high pair or an ace-king combo. 'The opening rounds of every tournament are always full of players who probably shouldn't be there,' Binger says. 'These are the rich guys who think they are much better than they actually are. At this stage of the game, the most important thing you can do is not make a big mistake. You don't want to take unnecessary risks. You just want to stay alive. This is when I'm making sure that I'm always doing the math.'\n\nLook, for example, at one of Binger's early hands at the WSOP. He was dealt an immaculate pair of aces, a hand so good it has its own name (it's called American Airlines). Naturally, Binger decided to raise. Although it was a modest raise \u2013 Binger didn't want to scare anybody off \u2013 everyone at the table decided to fold, except for a well-groomed older man wearing a canary yellow polo shirt with big sweat stains in the armpits. He pushed his short stack of chips to the center of the table. 'I'm all in,' said the man in yellow. Binger assumed that the man had either a high pair (like two kings) or two high cards of the same suit (like the king and queen of spades). Binger paused for a moment and contemplated his odds. If he correctly read the other player's hand \u2013 and that was a big _if_ \u2013 then he had somewhere between an 82 and 87 per cent chance of winning. Binger decided to match the bet. The man nervously turned over his cards: the ace and jack of diamonds. The flop was dealt but it was a meaningless collection of number cards. The turn and the river were more of the same. Binger's pair of aces prevailed. The man in the yellow shirt winced and walked off without a word.\n\nAs the days pass, the weak players are ruthlessly culled from the tournament. It's like natural selection on fast-forward. The tournament doesn't end for the night until more than half of the players have been eliminated, so it's not unusual for the nights to last until two or three in the morning. ('Learning how to become nocturnal is part of the challenge,' Binger says.) By the fourth day, even the skilled survivors are beginning to look a little worn out from the struggle. Their faces are masks of fatigue and stubble, and their eyes have the faraway look of an adrenaline hangover. The smell of stale cigarette smoke seems to be a popular deodorant.\n\nBinger is gradually getting more aggressive at the poker table. It's as if his betting instincts have a dial, and he's slowly turning up the volume. He's still folding the vast majority of his hands, but when he does decide to make a bet, he doesn't equivocate. In these situations, his table manners follow a well-rehearsed script. Binger takes a second glance at his hole cards and flexes his jaw muscles. He then adjusts his reflective sunglasses, pressing them tight against his eyes, looks at his cards again, and pushes an intimidating pile of chips to the center of the table. His face radiates self-assurance. He's done the calculations and knows the odds. Most of the time, the other players respond by folding.\n\nThis fiercely disciplined strategy pays off. By the end of the fifth day of the tournament, Binger is in fourth place, with $4,920,000 in chips. Fourteen hours later, he's got $5,275,000. After seven exhausting days, he's amassed a pile of nearly $6 million in chips. And then, on the eighth day, Binger makes the final table. When play begins, the Hollywood producer Jamie Gold has a commanding chip lead over the other players. Gold has been playing smart poker, but he's also been enjoying a staggering run of good luck. As one poker professional later told me, '[Gold] has an amazing ability to pull cards out of his ass. And somehow he always pulls the exact right card.'\n\nAfter a few hours, Gold begins to eliminate some of the remaining players. His big chip lead means that he can turn each hand into a potential trap. Gold can also bluff with abandon, since calling his bluff means the other player has to go all in. Binger is playing conservatively \u2013 'I just wasn't getting the right cards,' he said later \u2013 and so he waits, and watches. The big antes are gradually bleeding away his chips, but he's getting a better sense of his competition. 'After a while, you just get these feelings about people,' he says. 'You'll watch them make a certain bet and then scratch their nose or whatever and all of a sudden you'll realize that they've got nothing, that you can take the hand.' There are no certainties in poker, which means that anything that can narrow the uncertainty is extremely valuable, even if it's just a subtle hunch. These psychological interpretations aren't quantifiable \u2013 you can't summarize a person in a probability \u2013 but they still inform Binger's betting decisions.\n\nWhen there are only five players left, Binger begins to make his move. 'It started when I got dealt a pair of kings,' he says. 'I decided right then and there to make a rather aggressive bet.' A few hours before, Binger had bluffed one of the five, a player named Paul Wasicka, out of a big pot. Although Binger had been dealt a poor hand, his aggressive bet convinced everyone else to fold. Binger could tell that Wasicka was still seething. 'I knew Paul thought I was just trying to bluff him again,' Binger remembers. 'He thought that I only had a small pair. But I had pocket kings.'\n\nBinger wanted to draw Wasicka deeper into the hand. In tactical moments like this, poker transcends its probabilities. The game morphs into a deeply human drama, a competition of decision-making. Binger needed to make a bet that would convince Wasicka he was trying to steal another pot, that he was once again making an aggressive bet with nothing but a low pair. 'I decided to go all in,' Binger says. 'By overplaying my hand \u2013 by pretending to act strong \u2013 I was actually acting weak, at least in his eyes. I then tried to exude weakness, but without making it obvious, because then he would know that I was only _pretending_ to bluff, which is a sure sign that I've actually got a good hand.' Binger's best friend and brother were both watching the tournament on closed-circuit television. The best friend was convinced that Binger was bluffing and that he was about to get knocked out of the tournament. The signs of repressed anxiety were unmistakable: Binger's fingers were manically tapping on the table, and his teeth were digging into his lower lip. 'Only my brother knew better,' Binger says. 'I guess he knew how to read my face. He said I looked too weak, so I must be strong.'\n\nWasicka took the bait. He was so certain that Binger was bluffing that he ended up betting millions of chips on a weak hand. Binger won the pot and doubled his chips. 'That bet had nothing to do with math,' Binger says. 'I'd gotten high pairs before, and not done much with them... But at that moment, as soon as I saw my cards, I knew what I needed to do. To be honest, I don't know why I went all in on that hand. If I'd really thought about it, I might not have done it. The bet was damn risky. But it just felt like the right thing to do. You can do all the probabilistic analysis in the world, but in the end it all comes down to something you can't quite explain.'\n\n##  _The Art of Cards_\n\nProfessional poker players are a fatalistic bunch. They live in a deterministic world shaped by mysterious forces. Everything is possible, and yet only one thing ever happens. You might get the card you need on the river, but you might not. There's a possibility you'll make the straight, but you probably won't. Poker is a game of subtle skill and exquisite odds, but it's also a crapshoot.\n\nThis undercurrent of chance is the defining feature of the game. It's what makes the psychological aspects of poker \u2013 the subtle reads, the convincing bluffs, the inexplicable intuitions \u2013 so essential. Chess, by contrast, is a game of pure information. There are no secrets or shuffled decks or hidden cards; the moving parts of the game are all perfectly visible, right there on the chessboard. As a result, computers can consistently beat grand masters; they can use their virtually unlimited processing powers to find the perfect move. But poker isn't so amenable to microchips and mathematics. Great poker players aren't just gambling statisticians. They need to bring something else to the table, to possess that inexplicable talent for knowing when to risk everything on a pair of kings. 'Poker is a science, but it's also an art,' Binger says. 'To be good, you have to master both sides of the game.'\n\nWhat Binger is alluding to is the fact that there are always two ways to look at a poker hand. The first approach is mathematical. It treats every hand like a math problem and assumes that winning the game is simply a matter of plugging the probabilities into a sophisticated equation. According to this strategy, poker players should act like rational agents, looking for bets that minimize risk and maximize gain. This is what Binger did during the opening rounds of the WSOP, when he was only betting on high-percentage hands. Making money was just a matter of getting the odds right.\n\nBut Binger knows that poker isn't merely a set of math problems. When he talks about the art of the game, he's alluding to everything that can't be translated into numbers. The laws of statistics couldn't have told Binger how to lead Wasicka into his trap, or whether he should bluff with a middling pair. Even the most carefully calculated odds can't eliminate the unpredictability in a shuffled deck of cards. This is why the best poker players don't pretend that poker can be solved. They know the game is ultimately a _mystery_.\n\nThe difference between math problems and mysteries is important. In order to solve a math problem, all you need is more rational thought. Some poker hands, of course, can be played by relying on the math: if you're dealt a pair of aces, or get a straight on the flop, then you're going to make an aggressive bet. The odds are in your favor, and a little statistics will lead you to make the correct decision. But this rational approach can't be applied to the vast majority of poker hands, which are utter mysteries. In these situations, more statistical analysis won't help the player make a decision. In fact, thinking too much is part of the problem, since all that extra thought just gets in the way. 'Sometimes, I have to tell myself to not focus on the math,' Binger says. 'The danger with the math is that it can make you think you know more than you do. Instead of thinking about what the other player is doing, you end up obsessing over the percentages.' The first part of solving a mystery is realizing that there is no easy solution. Nobody knows what card is coming next.\n\nThis is where feelings come into play. When there is no obvious answer, a poker player is forced to make a decision using the emotional brain. And so that vague intuition about his hand, that inexplicable hunch about his opponent, ends up becoming a decisive factor. This decision won't be perfect \u2013 there's too much uncertainty for that \u2013 but it's the best option. Mysteries require more than mere rationality. 'I know that my mind assimilates many more variables than I'm actually aware of,' Binger says. 'Especially when it comes to reading other players, I'll often make strong and accurate reads without knowing what signals I'm picking up on. And as I've gained experience, I've felt my poker instincts just get better and better, to the point where I almost never doubt them. If I get a strong feeling, then that's what I go with.'\n\nRemember Damasio's card-playing experiment? In that gambling game, players had to turn over about eighty cards before they could consciously explain which card deck was the best option. Their conclusions were rational, but they were also rather slow. It takes a while to do the math. But when Damasio measured people's emotions, he discovered that their feelings were able to identify the good decks after only 10 cards. Whenever people reached for the risky decks, they experienced a surge of nervousness, even though they couldn't say why they felt so nervous. The subjects who trusted their emotional brains \u2013 who listened to their sweaty palms \u2013 made the most money.\n\nThe different strategies used by poker players illuminate the benefits of having a mind capable of rational analysis _and_ irrational emotion. Sometimes it helps to look at cards from the cold perspective of statistics, to bet on hands only when the odds are on your side. But the best poker players also know when _not_ to rely on the math. People aren't particles. To play the game is to accept the limits of statistics, to know that numbers don't know everything. Binger realizes that in certain situations, it's important to listen to his feelings, even if he doesn't always know what they're responding to. 'As a physicist, it can be hard admitting that you just can't reason your way to the winning hand,' Binger says. 'But that's the reality of poker. You can't construct a perfect model of it. It's based on a seemingly infinite amount of information. In that sense, poker is a lot like real life.'\n\n##  _The Power of Unconscious Thought_\n\nAp Dijksterhuis, a psychologist at the University of Amsterdam, had a scientific breakthrough while shopping for a car. Like most consumers, Dijksterhuis was slightly overwhelmed by the variety of makes and models. There were just so many alternatives to consider. Before he could find the right car, Dijksterhuis needed to take a dizzying number of variables into account, from fuel economy to trunk space. And then, once he made up his mind, Dijksterhuis had to figure out which options he wanted. A moon roof? A diesel engine? Six speakers? Side air bags? The list of possibilities seemed endless.\n\nThat's when Dijksterhuis realized that buying a car exceeded the limits of his conscious brain. He could no longer remember whether the Toyota or the Opel had a bigger engine, or if it was the Nissan or the Renault that offered the attractive lease. All the different variables were blurred together; his prefrontal cortex was confused.\n\nBut if Dijksterhuis couldn't keep track of the different cars, then how could he ever make a decision? Was he destined to pick the wrong car? What was the best way to make a difficult choice? To answer these questions, Dijksterhuis decided to conduct a practical experiment; it was later published in _Science_. He got several Dutch car shoppers and gave them each descriptions of four different used cars. Each of the cars was rated in four different categories, for a total of 16 pieces of information. Car number one, for example, was described as getting good mileage but having a shoddy transmission and a poor sound system. Car number two handled poorly but had lots of legroom. Dijksterhuis designed the experiment so that one car was objectively ideal, with 'predominantly positive aspects'. After showing a subject these car ratings, Dijksterhuis would give him a few minutes to contemplate the decision. In this 'easy' situation, more than 50 per cent of the subjects ended up choosing the best car.\n\nDijksterhuis then showed a different selection of people the same car ratings. This time, however, he didn't let each of them consciously think about the decision. After he gave the automotive facts, he distracted the subject with some simple word games for a few minutes, then interrupted the fun and suddenly asked the person to choose a car. Dijksterhuis had designed the experiment so that the person would be forced to make a decision using the unconscious brain, by relying on his or her emotions. (Conscious attention had been focused on solving the word puzzle.) The result was that these subjects made significantly worse choices than those who were allowed to consciously think about the cars.\n\nSo far, so obvious. A little rational analysis could have prevented the 'unconscious choosers' from buying a bad car. Such data confirms the conventional wisdom: reason is always better. We should think before we decide.\n\nBut Dijksterhuis was just getting warmed up. He repeated the experiment, only this time he rated each car in 12 different categories. (These 'hard' conditions more closely approximate the confusing reality of car shopping, in which consumers are overwhelmed with facts and figures.) In addition to getting information about the quality of the transmission and the engine's gas mileage, people were told about the number of cup holders, the size of the trunk and so on. Their brains had to deal with 48 separate pieces of information.\n\nDid conscious deliberation still lead to the best decision? Dijksterhuis found that people who were given time to think in a rational manner \u2013 those who could carefully contemplate each alternative \u2013 now chose the ideal car less than 25 per cent of the time. In other words, they performed _worse_ than random chance. However, subjects who were distracted for a few minutes \u2013 those who were forced to choose with their emotions \u2013 found the best car nearly 60 per cent of the time. They were able to sift through the clutter of automotive facts and find the ideal alternative. The best car was associated with the most positive feelings. These irrational choosers were the best decision-makers by far.\n\nBut perhaps this data is an artifact of the lab, an effect of making people choose cars under artificial conditions. So Dijksterhuis ventured out into the real world. He began by surveying shoppers in a variety of different stores, asking them what information they considered when making their decisions. Based on these responses, he assigned a 'complexity score' to a list of consumer products. Dijksterhuis found that some products, such as cheap kitchen tools (can openers, vegetable peelers, oven mitts and so on) and home accessories (light bulbs, toilet paper, umbrellas and so on), were relatively easy for shoppers to select. People didn't weigh many variables when making up their minds, because there weren't that many variables to consider. Since most stores carried only a few different brands of vegetable peelers and toilet paper, shoppers were able to quickly focus on the most important factors, like price. Making these simple consumer choices was the equivalent of choosing a car after learning only four attributes.\n\nAnd, sure enough, when Dijksterhuis studied people shopping for modest cooking accessories, he discovered that spending more time thinking about their decisions led to more satisfaction later on. In general, people did best when they carefully compared all of their options and reasoned their way to the best vegetable peelers. They tended to regret their impulse purchases, since they'd end up with kitchen tools they didn't want or like. When buying easy consumer products, it's a good idea to take a few moments and reflect on the purchase.\n\nDijksterhuis then studied a more complicated shopping experience. His survey found that choosing furniture is one of the hardest consumer decisions, since it involves so many different variables. Consider a leather couch. First, you need to figure out if you like the way it looks and feels. (As Timothy Wilson demonstrated with strawberry jam, simply deciphering one's own preferences can be a very difficult cognitive task.) Then, you need to think about whether the couch will work at home. Will it clash with the coffee table? Match the drapes? Will the cat scratch the leather? Before you can make a good decision about the couch, this long list of questions needs to be answered. The problem is that the prefrontal cortex can't handle this much information by itself. As a result, it tends to fix on just one variable that may or may not be relevant, such as the color of the leather. The rational brain is forced to oversimplify the situation. Look, for instance, at the doctors who relied on MRIs to diagnose the causes of back pain; because the MRI provided them with so much anatomical data, they ended up focusing on spinal disc abnormalities, even though these abnormalities probably weren't the cause of the pain. This resulted in a lot of unnecessary surgery.\n\nAfter shadowing shoppers at IKEA, the furniture warehouse, Dijksterhuis found that the longer people spent analyzing their options, the _less_ satisfied they were with their decisions. Their rational faculties had been overwhelmed by the furniture store, and they ended up choosing the wrong leather couch. (IKEA offers more than thirty different kinds of sofas.) In other words, furniture shoppers did best when they didn't think at all and just listened to their emotional brains.\n\nRemember the experiment involving the fine-art posters and the funny cat posters? In that study, led by Timothy Wilson, subjects were less satisfied with their choices when they consciously thought about what to choose; analyzing their own preferences caused them to misinterpret those preferences. Wilson concluded that for selecting things like posters or strawberry jam, people are better off listening to their first instincts. One of Dijksterhuis's most recent experiments involved replicating Wilson's study, but with a twist: he wanted to see if letting people engage in _un_ conscious decision-making \u2013 they looked at posters and then were distracted by a series of anagrams for seven minutes \u2013 could lead them to make even better decisions.\n\nThe answer, it turns out, is a resounding yes. Consciously contemplating the posters once again led to the worst decisions \u2013 these people were the least satisfied with their choices when they were interviewed three weeks later. But the most satisfied subjects were those who let the poster options marinate in their unconscious brains for several minutes and then chose on the basis of which poster was associated with the most positive emotions. Dijksterhuis speculates that art posters benefit from such subterranean thought processes because they are complex choices requiring people to interpret their own subjective desires. It's not easy to figure out if you prefer van Gogh to Rothko, or if you'd rather look at an Impressionist landscape than an abstract expressionist canvas. 'Imagine being at an art auction in Paris,' Dijksterhuis says. 'There's a Monet for sale for 100 million, and a van Gogh for 125 million. How should we make this choice? The best strategy may be the following: first, take a good look at both of the paintings. Then leave the auction and distract yourself for a while (which is easy to do in Paris), and only then decide.'\n\nThese simple experiments shed light on a very common problem in everyday life. We often make decisions on issues that are exceedingly complicated. In these situations, it's probably a mistake to consciously reflect on all the options, as this inundates the prefrontal cortex with too much data. 'The moral of this research is clear,' Dijksterhuis says. 'Use your conscious mind to acquire all the information you need for making a decision. But don't try to analyze the information with your conscious mind. Instead, go on holiday while your unconscious mind digests it. Whatever your intuition then tells you is almost certainly going to be the best choice.' Dijksterhuis argues that this psychological principle has far-reaching consequences and can also be applied to decisions that don't involve shopping. Anyone who is constantly making difficult decisions, from corporate executives to poker players, can benefit from a more emotional thought process. As long as someone has sufficient experience in that domain \u2013 he's taken the time to train his dopamine neurons \u2013 then he shouldn't spend too much time consciously contemplating the alternatives. The hardest calls are the ones that require the most feeling.\n\nAt first glance, this idea might be a little difficult to accept. We naturally assume that such choices require the analytical rigor of the rational brain. When trying to decipher a complicated situation, we believe that we need to consciously reflect on our options, to carefully think through the different car models or compare all of the possible couches at IKEA. Simple situations, on the other hand, are generally deemed suitable for emotions. You might trust your gut to choose a main course for dinner, but you wouldn't dream of letting it select your next car. That's why the average American spends 35 hours comparing automotive models before he or she makes a decision about which car to purchase.\n\nBut the conventional wisdom about decision-making has got it exactly backward. It is the easy problems \u2013 the mundane math problems of daily life \u2013 that are best suited to the conscious brain. These simple decisions won't overwhelm the prefrontal cortex. In fact, they are so simple that they tend to trip up the emotions, which don't know how to compare prices or compute the odds of a poker hand. (When people rely on their feelings in such situations, they make avoidable mistakes, like those due to loss aversion and arithmetical errors.*) Complex problems, on the other hand, require the processing powers of the emotional brain, the supercomputer of the mind. This doesn't mean you can just blink and know what to do \u2013 even the unconscious takes a little time to process information \u2013 but it does suggest that there's a better way to make difficult decisions. When choosing a couch, or holding a mysterious set of cards, always listen to your feelings. They know more than you do.\n\n##  _Using Both Sides of the Brain_\n\nMichael Binger started winning poker tournaments once he realized that the game was more than a math problem. Although he's a physicist, trained to spot the quantitative pattern in the most stochastic of systems, Binger eventually discovered that he couldn't just crunch the numbers and expect to win the hand. He also needed to know when the numbers weren't enough. 'I've been able to figure out the odds of poker hands for a while now, and yet, until recently, I never did very well in the World Series,' Binger says. 'I guess what you get better at is everything else, all the stuff that can't be quantified.'\n\nThis epiphany allowed Binger to see the card game as it was, not as he wanted it to be. He no longer pretended that there was some universal solution to the problem of poker. The game was too complicated and unpredictable to summarize with statistics. Binger came to understand that different situations required different modes of thought. Sometimes he had to play the odds. And sometimes he had to trust his gut.\n\nThis insight doesn't apply to poker alone. Look, for instance, at the financial markets. Wall Street is often compared to games of chance \u2013 like Vegas, it's a place where luck can be as important as logic \u2013 and when it comes to decision-making, the parallels can be illuminating. Both poker and investing are inherently unpredictable enterprises, requiring people to act with incomplete information. Nobody knows how the market will respond to the latest economic data or what card will appear on the river. Nobody knows if the Federal Reserve is going to lower interest rates next quarter or if the player with the big pile of chips is bluffing. In such situations, the only way for anyone to succeed over the long term is to use both brain systems in their proper contexts. We need to think _and_ feel.\n\nA few years ago, Andrew Lo, a business professor at MIT, wired ten currency speculators and stock traders at a brokerage firm with sensors that monitored their heart rates, blood pressure, body temperature and skin conductivity. These bodily signs correlate with emotions: intense feelings make for fast pulses. By the end of the day, the traders had made more than a thousand financial decisions, wagering over $40 million. If these professional investors were perfectly rational agents, as economic theory assumes, then they should have had perfectly calm bodies. When Lo looked at the data, however, he found that the decisions of the traders were the stuff of sweaty palms and spiking blood pressure. Most financial transactions were accompanied by surges of feeling.\n\nThis wasn't necessarily a bad thing. The vast majority of emotional decisions turned out to be profitable. Just because traders had nervous hands, or frightened amygdalas, didn't mean they were acting 'irrationally.' Rather, Lo discovered that the traders made the worst decisions when their emotions were either silent or overwhelming. In order to make the right investment decisions, the mind needs emotional input, but those emotions need to exist in a dialogue with rational analysis. Investors that got too worked up or who tried to rely on logic alone tended to make dramatic mistakes. 'One of the implications of our experiments,' Lo says, 'is that strong emotional reactions to financial gains or losses can actually be counterproductive. On the other hand, too little emotional reaction can also be dangerous. There's an ideal range of emotional responses that professional securities traders seem to exhibit, and that's an insight that we think individual investors can benefit from.' The best investors, like the best poker players, are able to find that crucial mental balance. They constantly use one brain system to improve the performance of the other.\n\nJust look at Binger. On the one hand, he's always using his prefrontal cortex to interrogate his emotions, to consciously question his unconscious brain. This doesn't mean he's ignoring his feelings \u2013 he's not making the strawberry-jam mistake \u2013 but it does mean that he's making sure to avoid any obvious emotional errors, what poker players refer to as tilt. 'The way I look at it,' Binger says, 'is that it can't hurt to think for a few seconds about what I'm feeling. Most of the time, I'll still go with my instincts, but occasionally I'll catch myself doing something dumb.'\n\nConsider this poker hand from the first day of the tournament. Binger was trying to play it safe, but he ended up losing a big pile of chips when someone beat his pair of jacks on the river. Fortunately, Binger was self-aware enough to realize that such losses can trigger a dangerous set of feelings as the effects of loss aversion settle in. 'You want your chips back,' Binger says, 'and that's when you find yourself taking risks you probably shouldn't take.' At moments like this, Binger's prefrontal cortex reasserts control of his gambling decisions, preventing him from making an impulsive mistake: 'I'll remind myself to play tight, to focus on the odds.' You don't go all in if you've only got one out.\n\nSituations like this demonstrate the importance of the prefrontal cortex. The rational parts of the brain are uniquely able to monitor feelings, using the reins of cognition to keep the horses from running wild. Ironically, it's those moments when emotions seem most persuasive \u2013 when the brain is completely convinced that it's time to go all in \u2013 that you should take a little extra time to reflect on the emotional decision. Make yourself consider alternative possibilities and scenarios. This is why the Israeli intelligence services added yet another analytical branch after the Yom Kippur War. 'If the game seems simple or obvious, then you've made a mistake,' Binger says. 'The game is never simple. You've always got to wonder: what am I missing?'\n\nBinger's ability to alternate between emotions and rationality has one important effect: it forces him to always think about _how_ he's thinking. Because Binger has an array of cognitive strategies to choose from, he is constantly reflecting on which strategy he should use at any particular moment. This sort of mental flexibility is an essential feature of good decision-making. Look at the Philip Tetlock study of political pundits that we talked about in the last chapter. Although the study is best known for its demonstration of expert failure \u2013 the vast majority of pundits failed to predict better than random chance \u2013 Tetlock also found that a few performed far above average.\n\nTetlock explained the difference between successful and unsuccessful pundits with an allusion to an ancient metaphor made famous by historian Isaiah Berlin in his essay 'The Hedgehog and the Fox'. (Berlin's title is a reference to the ancient Greek expression 'The fox knows many things, but the hedgehog knows one big thing.') In that essay, Berlin distinguished between two types of thinkers, hedgehogs and foxes, and Tetlock used those same categories to describe the pundits' methods of decision-making. (Tetlock did not find any significant correlation between political ideology and thinking style.) A hedgehog is a small mammal covered with spines; when attacked, it rolls itself into a ball so that its spines point outward. This is the hedgehog's only defense. A fox, on the other hand, doesn't rely on a single strategy when threatened. Instead, it adjusts its strategy to fit the particulars of the situation. Foxes are also cunning hunters. In fact, they are one of the hedgehog's few predators.\n\nAccording to Tetlock, the problem with a pundit who thinks like a hedgehog is that he is prone to bouts of certainty \u2013 the big idea is irrefutable \u2013 and this certainty causes him to misinterpret the evidence. If the amygdala contradicts one of his conclusions \u2013 it's worrying about some bit of evidence that doesn't support the pundit's accepted worldview \u2013 then the amygdala is turned off. A diversity of brain regions isn't brought to bear on the problem. Useful information is deliberately ignored. The inner argument is badly argued.\n\nA successful pundit, on the other hand, thinks like a fox. While the hedgehog reassures himself with certainty, the fox relies on the solvent of doubt. He is skeptical of grand strategies and unifying theories. The fox accepts ambiguity and takes an ad hoc approach when coming up with explanations. The fox gathers data from a wide variety of sources and listens to a diversity of brain areas. The upshot is that the fox makes better predictions and decisions.\n\nBut being open-minded isn't enough. Tetlock found that the most important difference between fox thinking and hedgehog thinking is that the fox thinker is more likely to _study his own_ _decision-making process_. In other words, he thinks about how he thinks, just like Binger.* According to Tetlock, such introspection is the best predictor of good judgment. Because foxes pay attention to their inner disagreements, they are less vulnerable to the seductions of certainty. The fox doesn't tune out his insula or ventral striatum or nucleus accumbens just because it contradicts his preconceptions. 'We need to cultivate the art of self-overhearing,' Tetlock says, 'to learn how to eavesdrop on the mental conversations we have with ourselves.'\n\nThat's also the lesson of Michael Binger's success. Although Jamie Gold went on to win the 2006 WSOP, Binger's third-place finish earned him a consolation prize of $4,123,310. The next year, in the 2007 WSOP, Binger tied the all-time record for most cashes in a single tournament (eight). (To _cash_ means to win money.) He began 2008 by winning one of the main no-limit Texas hold'em events at the LA Poker Classic, earning another six-figure payday. He is now regarded as one of the best players on the professional poker circuit. 'What I love about poker,' Binger says, 'is that, when you win, it's always for the same reason. You might lose because you got unlucky, but you never win because of luck. The only way to win is to make better decisions than everyone else at the table.'\n\nWe can now start to sketch out a taxonomy of decision-making, applying the knowledge of the brain to the real world. We've seen how the different brain systems \u2013 the Platonic driver and his emotional horses \u2013 should be used in different situations. While reason and feeling are both essential tools, each is best suited for specific tasks. When you try to analyze a strawberry jam or feel your way to a vegetable peeler, you are misusing your machine. When you're certain that you're right, you stop listening to those brain areas that say you might be wrong.\n\nThe science of decision-making remains a young science. Researchers are just beginning to understand how the brain makes up its mind. The cortex remains a mostly mysterious place, an extraordinary yet imperfect computer. Future experiments will reveal new aspects of human hardware and software. We'll learn about additional programming bugs and cognitive talents. The current theories will undoubtedly get complicated. And yet, even at the dawn of this new science, it's possible to come up with a few general guidelines that can help us all make better decisions.\n\n* * *\n\nSIMPLE PROBLEMS REQUIRE REASON. There isn't a clear line separating easy questions from hard ones, or math problems from mysteries. Some scientists, such as Ap Dijksterhuis, believe that any problem with more than four distinct variables overwhelms the rational brain. Others believe that a person can consciously process somewhere between five and nine pieces of information at any given moment. With practice and experience, this range can be slightly expanded. But in general, the prefrontal cortex is a sharply constrained piece of machinery. If the emotional brain is a fancy laptop, stuffed full of microprocessors operating in parallel, the rational brain is an old-fashioned calculator.\n\nThat said, a calculator can still be a very useful tool. One of the drawbacks of emotions is that they contain a few obsolete instincts that are no longer suited for modern life. This is why we are all so vulnerable to loss aversion, slot machines and credit cards. The only way to defend against such innate flaws is to exercise reason, to fact-check feelings with a little arithmetic. Remember Frank, the unlucky contestant on _Deal or No Deal_? If he'd taken the time to rationally evaluate the offer, to plug the proposal into a calculator, he would have ended up with \u20ac10,000. Instead, he walked away with \u20ac10.\n\nOf course, it's not always obvious which decisions are simple. Picking a strawberry jam or breakfast cereal might seem like an easy task, but it's actually surprisingly complicated, especially when a typical supermarket stocks more than two hundred different varieties of each. So how can anyone reliably identify the simple problems that are best suited for the prefrontal cortex? The best way is to ask yourself if the decision can be accurately summarized in numerical terms. For example, since most vegetable peelers are virtually identical, not much is lost when the various peelers are sorted by price. In this case, the best choice is probably the cheapest: let the rational brain take over. (Especially since the emotional brain might be misled by spiffy packaging or some other irrelevant variable.) And if someone really doesn't care about strawberry jam \u2013 he or she just wants something to put on a peanut butter sandwich \u2013 then this deliberate decision-making strategy can also be applied to jam. Or wine. Or brands of cola. Or any domain in which the details of the product aren't particularly important. In these situations, remember what we learned about expensive wine in chapter five, and don't spend too much money on overpriced items that won't be appreciated. (After all, cheaper wines often taste better than more expensive ones in blind taste tests!) If the decision doesn't matter all that much, the prefrontal cortex should take the time to carefully assess and analyze the options.\n\nOn the other hand, for important decisions about complex items \u2013 leather couches, cars and apartments, for example \u2013 categorizing by price alone will eliminate a lot of essential information. Perhaps the cheapest couch is of inferior quality, or maybe you don't like the way it looks. And should anyone really choose an apartment or a car based on a single variable, such as the monthly rent or the amount of horsepower? As Dijksterhuis demonstrated, when you ask the prefrontal cortex to make these sorts of decisions, it makes consistent mistakes. You'll end up with an ugly couch in the wrong apartment. It might sound ridiculous, but it makes scientific sense: Think _less_ about those items that you care a lot about. Don't be afraid to let your emotions choose.\n\nLikewise, there's a whole subset of everyday decisions \u2013 those mundane choices that don't really matter \u2013 that could benefit from a little more conscious deliberation. Too often, we let our impulses make the easy decisions for us. A person will pick a vegetable peeler, laundry detergent, or boxer shorts on a whim and automatically trust his instincts when he gets an obviously strong poker hand. But these are precisely the sorts of emotion-driven decisions that might benefit from rational analysis.\n\nNOVEL PROBLEMS ALSO REQUIRE REASON. Before you entrust a mystery to the emotional brain, before deciding to let your instincts make a big bet in poker or fire a missile at a suspicious radar blip, ask yourself a question: how does your past experience help solve this particular problem? Have you played poker hands like this before? Seen blips like this before? Are these feelings rooted in experience, or are they just haphazard impulses?\n\nIf the problem really is unprecedented \u2013 if it's like a complete hydraulic failure in a Boeing 737 \u2013 then emotions can't save you. Stop and think and let your working memory tackle the dilemma. The only way out of a unique mess is to come up with a creative solution, like Al Haynes did when he realized that he couldn't steer the plane in the ordinary way but that it was possible to steer the plane with the thrust levers. Such insights require the flexible neurons of the prefrontal cortex.\n\nHowever, this doesn't mean that our emotional state is irrelevant. Mark Jung-Beeman, the scientist who studies the neuroscience of insight, has shown that people in good moods are significantly better at solving hard problems that require insight than people who are cranky and depressed. (Happy people solve nearly 20 per cent more word puzzles than unhappy people.) He speculates that this is because the brain areas associated with executive control, such as the prefrontal cortex and the ACC, aren't as preoccupied with managing emotional life. In other words, they aren't worrying about why you're not happy, which means they are free to solve the problem at hand. The end result is that the rational brain can focus on what it needs to focus on, which is coming up with a solution for the unprecedented situation that you've found yourself in.\n\nEMBRACE UNCERTAINTY. Hard problems rarely have easy solutions. There is no single way to win a poker hand, and there is no guaranteed path to making money in the stock market. Pretending that the mystery has been erased results in the dangerous trap of certainty. You are so confident you're right that you neglect all the evidence that contradicts your conclusion. You fail to notice that those Egyptian tanks on the border aren't merely engaging in a training exercise. Of course, there's not always time to engage in a lengthy cognitive debate. When an Iraqi missile is zooming toward you or when you're about to get crushed by a blitzing linebacker, you need to act. But whenever possible, it's essential to extend the decision-making process and properly consider the argument unfolding inside your head. Bad decisions happen when that mental debate is cut short, when an artificial consensus is imposed on the neural quarrel.\n\nThere are two simple tricks to help ensure that you never let certainty interfere with your judgment. First, always entertain competing hypotheses. When you force yourself to interpret the facts through a different, perhaps uncomfortable, lens you often discover that your beliefs rest on a rather shaky foundation. For instance, when Michael Binger is convinced that another player is bluffing, he tries to think about how the player would be acting if he _wasn't_ bluffing. He is his own devil's advocate.\n\nSecond, continually remind yourself of what you _don't_ know. Even the best models and theories can be undone by utterly unpredictable events. Poker players call these 'bad beats' and every player has stories about the hands he lost because he got the one card he wasn't expecting. 'One of the things I learned from counting cards in blackjack,' Binger says, 'is that even when you have an edge, and counting cards is definitely an edge, your margin is still really slim. You can't get too cocky.' When you forget that you have blind spots, that you have no idea what cards the other players are holding or how they'll behave, you're setting yourself up for a nasty surprise. Colin Powell made a number of mistakes in the run-up to the Iraq war, but his advice to his intelligence officers was psychologically astute: 'Tell me what you know,' he told his advisers. 'Then tell me what you don't know, and only then can you tell me what you think. Always keep those three separated.'\n\nYOU KNOW MORE THAN YOU KNOW. One of the enduring paradoxes of the human mind is that it doesn't know itself very well. The conscious brain is ignorant of its own underpinnings, blind to all that neural activity taking place outside the prefrontal cortex. This is why people have emotions: they are windows into the unconscious, visceral representations of all the information we process but don't perceive.\n\nFor most of human history, the emotions have been disparaged because they're so difficult to analyze \u2013 they don't come with reasons, justifications, or explanations. (As Nietzsche warned, we are often most ignorant of what is closest to us.) But now, thanks to the tools of modern neuroscience, we can see that emotions have a logic all their own. The jitters of dopamine help keep track of reality, alerting us to all those subtle patterns that we can't consciously detect. Different emotional areas evaluate different aspects of the world, so your insula naturally takes the cost of an item into account (unless you're paying with a credit card), and the NAcc automatically figures out how you feel about a certain brand of strawberry jam. The anterior cingulate monitors surprises, and the amygdala helps point out the radar blip that just doesn't look right.\n\nThe emotional brain is especially useful at helping us make hard decisions. Its massive computational power \u2013 its ability to process millions of bits of data in parallel \u2013 ensures that you can analyze all the relevant information when assessing alternatives. Mysteries are broken down into manageable chunks, which are then translated into practical feelings.\n\nThe reason these emotions are so intelligent is that they've managed to translate mistakes into educational events. You are constantly benefiting from experience, even if you're not consciously aware of the benefits. It doesn't matter if your field of expertise is backgammon or Middle East politics, golf or computer programming: the brain always learns the same way, accumulating wisdom through error.\n\nThere are no shortcuts to this painstaking process; becoming an expert just takes time and practice. But once you've developed expertise in a particular area \u2013 once you've made the requisite mistakes \u2013 it's important to trust your emotions when making decisions in that domain. It is feelings, after all, and not the prefrontal cortex, that capture the wisdom of experience. Those subtle emotions saying shoot down the radar blip, or go all in with pocket kings, or pass to Troy Brown are the output of a brain that has learned how to read a situation. It can parse the world in practical terms, so that you know what needs to be done. When you overanalyze these expert decisions, you end up like the opera star who couldn't sing.\n\nAnd yet, this doesn't mean the emotional brain should always be trusted. Sometimes it can be impulsive and short-sighted. Sometimes it can be a little _too_ sensitive to patterns, which is why people lose so much money playing slot machines. However, the one thing you should always be doing is considering your emotions, thinking about why you're feeling what you're feeling. In other words, act like the television executive carefully analyzing the reactions of the focus group. Even when you choose to ignore the emotions, they are still a valuable source of input.\n\nTHINK ABOUT THINKING. If you're going to take only one idea away from this book, take this one: whenever you make a decision, be aware of the kind of decision you are making and the kind of thought process it requires. It doesn't matter if you're choosing between wide receivers or political candidates. You might be playing poker or assessing the results of a television focus group. The best way to make sure that you are using your brain properly is to study your brain at work, to listen to the argument inside your head.\n\nWhy is thinking about thinking so important? First, it helps us all steer clear of stupid errors. You can't avoid loss aversion unless you know that the mind treats losses differently than gains. And you'll probably think too much about buying a house unless you know that such a strategy will lead you to buy the wrong property. The mind is full of flaws, but they can be outsmarted. Cut up your credit cards and put your retirement savings in a low-cost index fund. Prevent yourself from paying too much attention to MRI images, and remember to judge a wine before you know how much it costs. There is no secret recipe for decision-making. There is only vigilance, the commitment to avoiding those errors that can be avoided.\n\nOf course, even the most attentive and self-aware minds will still make mistakes. Tom Brady, after the perfect season of 2008, played poorly in the Super Bowl. Michael Binger, after a long and successful day of poker, always ends up regretting one of his bets. The most accurate political experts in Tetlock's study still made plenty of inaccurate predictions. But the best decision-makers don't despair. Instead, they become students of error, determined to learn from what went wrong. They think about what they could have done differently so that the next time their neurons will know what to do. This is the most astonishing thing about the human brain: it can always improve itself. Tomorrow, we can make better decisions.\n\n* Your automatic brain is terrible at crunching numbers, which is why Binger always reflects on his poker probabilities. Consider this question: 'A bat and ball cost $1.10 in total. The bat costs a dollar more than the ball. How much does the ball cost?' Your first instinct is probably ten cents, but that's the wrong answer, since it would add up to $1.20. The correct answer is five cents, but arriving at this answer requires a little conscious deliberation.\n\n* Patients who have undergone cognitive-behavioral therapy (CBT), a form of talk therapy designed to reveal the innate biases and distortions of the human brain, have also been shown to be less vulnerable to these same biases. Scientists speculate that these patients have learned to recognize those maladaptive thoughts and emotions that automatically occur in their responses to certain situations. Because they reflect on their thought processes, they learn to think better.\nNINE\n\n# Coda\n\nTHERE ARE CERTAIN statistics that seem like they'll never change: the high school dropout rate, the percentage of marriages that end in divorce, the prevalence of tax fraud. The same used to be true of plane crashes that were due to pilot error. Despite a long list of aviation reforms, from mandatory pilot layovers to increased classroom training, that percentage refused to budge from 1940 to 1990, holding steady at around 65 per cent. It didn't matter what type of plane was being flown or where the plane was going. The brute fact remained: most aviation deaths were due to bad decisions in the cockpit.\n\nBut then, starting in the early 1990s, the percentage of crashes attributed to pilot error began to decline rapidly. According to the most current statistics, mistakes by the flight crew are responsible for less than 30 per cent of all plane accidents, with a 71 per cent reduction in the number of accidents caused by poor decision-making. The result is that flying has become safer than ever. According to the National Transportation Safety Board, flying on a commercial plane has a fatality rate of 0.04 per 100 million passenger miles, making it the least dangerous form of travel by far. (In contrast, driving has a fatality rate of 0.86.) Since 2001, pilot error has caused only one fatal jetliner crash in the United States, even though more than thirty thousand flights take off every day. The most dangerous part of traveling on a commercial airplane is the drive to the airport.\n\nWhat caused the dramatic reduction in pilot error? The first factor was the introduction in the mid-1980s of realistic flight simulators. For the first time, pilots could practice making decisions. They could refine their reactions to a sudden downdraft in a thunderstorm and practice landing with only one engine. They could learn what it would be like to fly without wing flaps and to land on a tarmac glazed with ice. And they could do all this without leaving the ground.\n\nThese simulators revolutionized pilot training. 'The old way of teaching pilots was the \"chalk and talk\" method,' says Jeff Roberts, the group president of civil training at CAE, the largest manufacturer of flight simulators. Before pilots ever entered the cockpit, they were forced to sit through a long series of classroom lectures. They learned all the basic maneuvers of flight while on the ground. They were also taught how to react in the event of various worst-case scenarios. What should you do if the landing gear won't deploy? Or if the plane is struck by lightning? 'The problem with this approach,' Roberts says, 'is that everything was abstract. The pilot has this body of knowledge, but they'd never applied it before.'\n\nThe benefit of a flight simulator is that it allows pilots to internalize their new knowledge. Instead of memorizing lessons, a pilot can train the emotional brain, preparing the parts of the cortex that will actually make the decision when up in the air. As a result, pilots that are confronted with a potential catastrophe during a real flight \u2013 like an engine fire in the air above Tokyo \u2013 already know what to do. They don't have to waste critical moments trying to remember what they learned in the classroom. 'A plane is traveling 400 miles per hour,' Roberts says. 'It's the rare emergency when you've got time to think about what your flight instructor told you. You've got to make the right decision right away.'\n\nSimulators also take advantage of the way the brain learns from experience. After pilots complete their 'flight', they are forced to endure an exhaustive debriefing. The instructor scrutinizes all of their decisions, so that the pilots think about why, exactly, they decided to gain altitude after the engine fire, or why they chose to land in the hailstorm. 'We want pilots to make mistakes in the simulator,' Roberts says. 'The goal is to learn from those mistakes when they don't count, so that when it really matters, you can make the right decision.' This approach targets the dopamine system, which improves itself by studying its errors. As a result, pilots develop accurate sets of flight instincts. Their brains have been prepared in advance.\n\nThere was one other crucial factor in the dramatic decline of pilot error: the development of a decision-making strategy known as Cockpit Resource Management (CRM). The impetus for CRM came from a large NASA study in the 1970s of pilot error; it concluded that many cockpit mistakes were attributable, at least in part, to the God-like certainty of the pilot in command. If other crew members had been consulted, or if the pilot had considered other alternatives, then some of the bad decisions might have been avoided. As a result, the goal of CRM was to create an environment in which a diversity of viewpoints was freely shared.\n\nUnfortunately, it took a tragic crash in the winter of 1978 for airlines to decide to implement this new system. United Flight 173 was a crowded DC-8 bound for Portland, Oregon. About 10 miles from the runway, the pilot lowered the landing gear. He noticed that two of his landing-gear indicator lights remained off, suggesting that the front wheels weren't properly deployed. The plane circled around the airport while the crew investigated the problem. New bulbs were put in the dashboard. The autopilot computers were reset. The fuse box was double-checked. But the landing-gear lights still wouldn't turn on.\n\nThe plane circled for so long that it began to run out of fuel. Unfortunately, the pilot was too preoccupied with the landing gear to notice. He even ignored the flight engineer's warning about the fuel levels. (One investigator described the pilot as 'an arrogant SOB'.) By the time the pilot looked at his gas gauge, the engines were beginning to shut down. It was too late to save the plane. The DC-8 crash-landed in a sparsely populated Portland suburb, killing 10 and seriously wounding 24 of the 189 on board. Crash investigators later concluded that there was no problem with the landing gear. The wheels were all properly deployed; it was just a faulty circuit.\n\nAfter the crash, United trained all of its employees with CRM. The captain was no longer the dictator of the plane. Instead, flight crews were expected to work together and constantly communicate with one another. Everyone was responsible for catching errors. If fuel levels were running low, then it was the job of the flight engineer to make sure the pilot grasped the severity of the situation. If the copilot was convinced that the captain was making a bad decision, then he was obligated to dissent. Flying a plane is an extremely complicated task, and it's essential to make use of every possible resource. The best decisions emerge when a multiplicity of viewpoints are brought to bear on the situation. The wisdom of crowds also applies in the cockpit.\n\nRemember United Flight 232, which lost all hydraulic power? After the crash-landing, the pilots all credited CRM with helping them make the runway. 'For most of my career, we kind of worked on the concept that the captain was _the_ authority on the aircraft,' says Al Haynes, the captain of Flight 232. 'And we lost a few airplanes because of that. Sometimes the captain isn't as smart as we thought he was.' Haynes freely admits that he couldn't have saved the plane by himself that day. 'We had 103 years of flying experience there in the cockpit [on Flight 232], trying to get that airplane on the ground. If I hadn't used CRM, if we had not had everybody's input, it's a cinch we wouldn't have made it.'\n\nIn recent years, CRM has moved beyond the cockpit. Many hospitals have realized that the same decision-making techniques that can prevent pilot error can also prevent unnecessary mistakes during surgery. Consider the experience of the Nebraska Medical Center, which began training its surgical teams in CRM in 2005. (To date, more than a thousand hospital employees have undergone the training.) The mantra of the CRM program is 'See it, say it, fix it'; all surgical-team members are encouraged to express their concerns freely to the attending surgeon. In addition, team members engage in postoperation debriefings at which everyone involved is supposed to share his or her view of the surgery. What mistakes were made? And how can they be avoided the next time?\n\nThe results at the Nebraska Medical Center have been impressive. A 2007 analysis found that after fewer than six months of CRM training, the percentage of staff members who 'felt free to question the decisions of those with more authority' had gone from 29 per cent to 86 per cent. More important, this increased willingness to point out potential errors led to a dramatic decrease in medical mistakes. Before CRM training, only around 21 per cent of all cardiac surgeries and cardiac catheterizations were classified as 'uneventful cases', meaning that nothing had gone wrong. After CRM training, however, the number of 'uneventful cases' rose to 62 per cent.\n\nThe reason CRM is so effective is that it encourages flight crews and surgical teams to think together. It deters certainty and stimulates debate. In this sense, CRM creates the ideal atmosphere for good decision-making, in which a diversity of opinions is openly shared. The evidence is looked at from multiple angles; new alternatives are considered. Such a process not only prevents mistakes but also leads to startling new insights.\n\nTo sit in a modern airplane cockpit is to be surrounded by computers. Just above the windshield are the autopilot terminals, which can keep a plane on course without any input from the pilot. Right in front of the thrust levers is a screen relaying information about the state of the plane, from its fuel levels to the hydraulic pressure. Nearby is the computer that monitors the flight path and records the position and speed of the plane. Then there's the GPS panel, a screen for weather updates and a radar monitor. Sitting in the captain's chair, you can tell why it's called the glass cockpit: everywhere you look there's another glass screen, the digital output of the computers underneath.\n\nThese computers are like the emotional brain of the plane. They process a vast amount of information and translate that information into a form that can be quickly grasped by the pilot. The computers are also redundant, so every plane actually contains multiple autopilot systems running on different computers and composed in different programming languages. Such diversity helps prevent mistakes, since each system is constantly checking itself against the other systems.\n\nThese computers are so reliable that they perform many of their tasks without any pilot input. If, for example, the autopilot senses a strong headwind, it will instantly increase thrust in order to maintain speed. The pressure in the cabin is seamlessly adjusted to reflect the altitude of the plane. If a pilot is flying too close to another plane, the onboard computers emit loud warning sirens, forcing the flight crew to notice the danger. It's as if the plane has an amygdala.\n\nPilots are like the plane's prefrontal cortex. Their job is to monitor these onboard computers, to pay close attention to the data on the cockpit screens. If something goes wrong, or if there's a disagreement among the various computers, then it's the responsibility of the flight crew to resolve the problem. The pilots must immediately intervene and, if necessary, take control of the plane. The pilots must also set the headings, supervise the progress of the flight, and deal with the inevitable headaches imposed by air-traffic control. 'People who aren't pilots tend to think that when the autopilot is turned on, the pilot can just take a nap,' my flight instructor in the simulator says. 'But planes don't fly themselves. You can't ever relax in the cockpit. You always have to be watching, making sure everything is going according to plan.'\n\nConsider the cautionary tale of a crowded Boeing 747 traveling from Miami to London in May 2000. The runway at Heathrow was shrouded in dense fog, so the pilots decided to make an automated landing, or what's known as a category IIIc approach. During the initial descent, all three autopilot systems were turned on. However, when the plane reached an altitude of 1,000 feet, the lead autopilot system suddenly shut down for no apparent reason. The pilots decided to continue with the approach, since the 747 is designed to be able to make automated landings with only two autopilot systems. The descent went smoothly until the plane was 50 feet above the runway, or about four seconds from touchdown. At that point, the autopilot abruptly tilted the nose of the plane downward, so that it's rate of descent was four times faster than normal. (Investigators would later blame a programming error for the mistake.) The pilot quickly intervened and yanked back on the control column so that the plane wouldn't hit the runway nose first. The landing was still rough \u2013 the plane suffered some minor structural damage \u2013 but the quick reactions of the flight crew prevented a catastrophe.\n\nEvents like this are disturbingly common. Even redundant autopilot systems will make mistakes. They'll disengage or freeze or steer the plane in dangerous ways. Unless a pilot is there to correct the error, to turn off the computer and pull up the nose, the plane will fly itself into the ground.\n\nOf course, pilots aren't perfect either. They sometimes fail to notice when they're getting too close to another plane, or they struggle to monitor all the different gauges in the cockpit. In fact, if pilots had to rely on their own instincts, they wouldn't even be able to fly through clouds. (The inner ear can't detect blind turns, which means that it's very tough to fly straight without proper instruments or visual cues.) Then there are the pilots that micromanage the flight \u2013 constantly overruling the autopilot or fiddling with the path of the plane. They dramatically increase the likelihood of human error, acting like people who rely too heavily on their prefrontal cortices.\n\nWhen the onboard computers and pilot properly interact, it's an ideal model for decision-making. The rational brain (the pilot) and the emotional brain (the cockpit computers) exist in perfect equilibrium, each system focusing on those areas in which it has a comparative advantage. The reason planes are so safe, even though both the pilot and the autopilot are fallible, is that both systems are constantly working to correct each other. Mistakes are fixed before they spiral out of control.\n\nThe payoff has been huge. 'Aviation is just about the only field that consistently manages to operate at the highest level of performance, which is defined by six sigma,' Roberts says, using the managerial buzzword for any process that produces fewer than 3.4 defects per 1 million opportunities. 'Catastrophic error in planes is incredibly, incredibly rare. If it wasn't, nobody would ever get on board. The fact of the matter is that the aviation industry needs to be perfect, and so we found ways to be as close to perfect as humanly possible.'\n\nThe safety of flight is a testament to the possibility of improvement. The reduction in the pilot-error rate is a powerful reminder that mistakes are not inevitable, that planes don't have to crash. As the modern cockpit demonstrates, a few simple innovations and a little self-awareness can dramatically improve the way people think, so that both brain systems are used in their ideal context. The aviation industry took decision-making seriously \u2013 they made a science of pilot error \u2013 and the result has been a stunning advance in performance.\n\nThe first step to making better decisions is to see ourselves as we really are, to look inside the black box of the human brain. We need to honestly assess our flaws and talents, our strengths and shortcomings. For the first time, such a vision is possible. We finally have tools that can pierce the mystery of the mind, revealing the intricate machinery that shapes our behavior. Now we need to put this knowledge to work.\n__\n\n# Bibliography\n\nAbbot, L. F. 'Balancing Homeostasis and Learning in Neural Circuits.' _Zoology_ 106 (2003): 365\u201371.\n\nAchen, Christopher, and Larry Bartels. 'It Feels Like We're Think ing: The Rationalizing Voter and Electoral Democracy.' Working paper, http:\/\/www.princeton.edu\/~bartels\/ thinking. pdf.\n\nAdams, Barbara, and Bita Moghaddam. 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New York: Simon and Schuster, 2008. \n\n# _Notes_\n\n ' _We never want pilots_ ': Jeff Roberts, telephone interview with the author, July 12, 2007.\n\n 'It was a ten': Charles Pierce (2007) _Moving the Chains,_ New   \nYork: Farrar, Straus and Giroux, 11.\n\n 'Our vision wasn't': Ibid. 92.\n\n Pete Dougherty and Jim Wyatt, 'Will Wonderlic scores cause teams to wonder about Young', _USA Today,_ March 1, 2006\n\n Tom Brady, press conference, September 11, 2006.\n\n _'He is of an'_ : Plato, _Phaedrus,_ 44.\n\n _'If the better'_ : Ibid.\n\n _'I am dragged along'_ : Ovid, _Metamorphoses,_ book VII, 249.\n\n _'American experiment would_ ': Jefferson, _The Writings of_ _Thomas Jefferson,_ 8\u2013124.\n\n _'One might compare'_ : Freud, _New Introductory Lectures_ , 245.\n\n ' _The events of'_ : Freud, _Civilization and Its Discontents,_ 14.\n\n _'Cognitive psychologists subscribed'_ : Marvin Minsky, telephone interview with the author, April 27, 2007.\n\n _'Elliot emerged as'_ : Damasio, _Descartes' Error,_ 43.\n\n _'He was always'_ : Ibid., 44.\n\n _'I suggested two'_ : Ibid., 193.\n\n _'Many of these'_ : Herb Stein, telephone interview with the author, October 15, 2007.\n\n William James (1890, 1950) _The Principles of Psychology_ : _Volume Two_ , New York: Dover, 389.\n\n _'It was ours'_ : Klein, _Sources of Power,_ 36\u201338; Finlan, _The_ _Royal Navy,_ 147\u201350.\n\n _In_ _1954_ : Olds and Milner, 'Positive Reinforcement,' 419\u201327.\n\n _The monkey felt:_ Schultz et al., 'Predictive Reward Signal'; Hollerman and Schultz, 'Dopamine Neurons.'\n\n _This fast cellular:_ P. Gaspar et al., 'Catecholamine Innervation.'\n\n _Researchers at Oxford:_ Kennerley, 'Optimal Decision- Making.'\n\n _If the monkey:_ Hayden and Platt, 'Fool Me Once.'\n\n _Because they have:_ Klein et al., 'Genetically Determined Differences.'\n\n _'You're probably': Zweig, Your Money and Your Brain,_ 69.\n\n _Instead, human emotions:_ Cohen et al., 'Reinforcement Learning.'\n\n _The roots of schizophrenia:_ Juckel et al., 'Dysfunction of Ventral Striatal'; Dolan et al., 'Dopaminergic Modulation.'\n\n _Most antipsychotic medications:_ Adams and Moghaddam, 'Corticolimbic Dopamine Neurotransmission.'\n\n _Tesauro set out:_ Montague, _Why Choose This Book?_ , 93\u2013100.\n\n _'Anytime you've got':_ Read Montague, interview with the author, May 7, 2007.\n\n _They use their:_ Ito, 'Performance Monitoring.'\n\n _The subject's feelings figured:_ Bechara et al., 'Deciding Advantageously.'\n\n _By playing the:_ Oya et al., 'Electrophysiological Correlates.'\n\n _The investments that:_ Betsch et al., 'Different Principles.'\n\n _'He wanted the':_ Bill Robertie, telephone interview with the author, October 10, 2007.\n\n _'a person who':_ Mackay, _A Dictionary of Scientific_ _Quotations,_ 35.\n\n _The other students:_ Cimpian et al., 'Subtle Linguistic Cues'; Bronson, 'How Not to Talk'; Mangels et al., 'Why Do Beliefs.'\n\n _'I lost control':_ Ann Klinestiver, telephone interview with the author, July 10, 2007.\n\n _'You can feel':_ Ibid., July 21, 2007.\n\n _Since slot machines:_ Cooper, 'Sit and Spin.'\n\n _According to Wolfram:_ Wolfram Schultz, telephone interview with the author, August 14, 2007.\n\n _There was absolutely:_ Gilovich, Vallone, and Tversky, 'The Hot Hand.'\n\n _'We made it':_ Mlodinow, _Drunkard's Walk,_ 175.\n\n _'So he makes':_ Gilovich, _How We Know,_ 17.\n\n _Fama looked at:_ Fama, 'Random Walks.'\n\n _One of Montague's:_ Lohrenz et al., 'Neural Signature.'\n\n _The investor who does:_ Zweig, _Your Money and Your Brain_ , 226.\n\n _Take Nondumiso Sainsbury: Deal or No Deal,_ NBC, air date September 18, 2006.\n\n _An exhaustive analysis:_ Post et al., 'Deal or No Deal?'\n\n _This mental defect:_ Kahneman and Tversky, 'Prospect Theory.'\n\n _'confused about the':_ T. MaCurdy and J. Shoven, 'Accumu lating Pension Wealth with Stocks and Bonds,' working paper, Stanford University, cited in Benartzi and Thaler, 'Myopic Loss Aversion.'\n\n _Markowitz was so:_ Zweig, _Your Money and Your Brain,_ 4.\n\n _Loss aversion also:_ Odean, 'Are Investors Reluctant.'\n\n _Consider this experiment:_ Shiv et al., 'Investment Behavior.'\n\n 'Every day, I see': Herman Palmer, telephone interview with author, October 19, 2007, and personal interview with author, Bronx, NY, January 11, 2008.\n\n _They no longer:_ Prelec and Simester, 'Always Leave Home Without It.'\n\n _In fact, subprime:_ Brooks and Simon, 'Subprime Debacle.'\n\n _Jonathan Cohen, a neuroscientist:_ McClure et al., 'Separate Neural Systems.'\n\n _This also helps:_ Wilson and Daly, 'Do Pretty Women.'\n\n _'Our emotional brain':_ Bradt, 'Brain Takes Itself On.'\n\n _Lawrence Ausubel, an economist:_ Ausubel, 'Adverse Selection.'\n\n _'Our emotions are like':_ George Loewenstein, telephone interview with the author, March 10, 2007.\n\n _Shlomo Benartzi and Richard Thaler:_ Benartzi and Thaler, 'Save More Tomorrow.'\n\n _'It sounds like':_ Maclean, _Young Men and Fire,_ 35.\n\n _'To hell with':_ Ibid., 95.\n\n _Although the people:_ Weltman, 'Perceptual Narrowing.'\n\n _'It just seemed':_ Maclean, _Young Men and Fire,_ 100.\n\n _Pressed tight against:_ Schoenemann, 'Prefrontal White Matter.'\n\n _But he still:_ Wynn and Coolidge, 'The Expert Neandertal Mind.'\n\n _By the 1920s:_ El-Hai, _The Lobotomist_.\n\n _Consider the case:_ Heilman, _Matter of Mind,_ 81.\n\n _'You see this':_ Kenneth Heilman, interview with the author, June 28, 2007.\n\n _When neuroscientists studied:_ De Martino et al., 'Frames, Biases.'\n\n _'Anyone can become':_ Aristotle, _Nicomachean Ethics,_ 48.\n\n _The CBS method:_ Napoli, _Audience Economics,_ 41.\n\n _'Quantitative data':_ Brian Graden, interview on _Frontline_.\n\n _'The shows weren't':_ Television executive (who asked to remain anonymous), telephone interview, November 12, 2007.\n\n _Mischel's results were:_ Shoda, Mischel, and Peake, 'Predicting Adolescent Cognitive.'\n\n _In November 2007_ : Shaw et al., 'Attention-Deficit\/ Hyperactivity Disorder.'\n\n _A recent study:_ Galvan et al., 'Earlier Development of the Accumbens.'\n\n _The most important_ : MacDonald et al., 'Dissociating the Role.'\n\n _'No other brain':_ Earl Miller, interview with the author, April 1, 2008.\n\n _In 2007, in a paper:_ Buschman and Miller, 'Top-Down versus Bottom-Up.'\n\n _The end result_ : Wood and Grafman, 'Human Prefrontal Cortex.'\n\n _Less than 20 per cent:_ Fleck and Weisberg, 'The Use of Verbal Protocols.'\n\n _The subject repeats:_ Davidson and Sternberg, eds., _Psychology_ _of Problem Solving,_ 151.\n\n _Instead of trying:_ Colvin et al., 'The Effects of Frontal Lobe Lesions.'\n\n _The first brain:_ Kounios et al., 'The Prepared Mind.'\n\n _'You're getting rid':_ Mark Jung-Beeman, interview with the author, February 15, 2008.\n\n _He'd flown this:_ Captain Al Haynes, interview with the author, January 21, 2008.\n\n _'It was an':_ Dennis Fitch, interview, _Seconds from Disaster_ , National Geographic Channel.\n\n _'The performance was':_ http:\/\/amelia. db.erau.edu\/reports\/ ntsb\/aar\/AAR90-06. pdf.\n\n _By analyzing the:_ as cited in Homer-Dixon, _The Ingenuity_ _Gap,_ 18\u201320.\n\n _According to Predmore:_ Predmore, 'Dynamics of Group Performance'; Helmreich, 'Managing Human Error in Aviation.'\n\n _For instance, studies:_ Kandel et al., _Principles of Neural_ _Science,_ 359.\n\n _Scientists refer to:_ Sandkuhler and Bhattacharya, 'Decon structing Insight.'\n\n _Numerous studies have:_ Colom et al., 'Working Memory.'\n\n _'It caught me':_ Fleming, _The Inner Voice,_ 100.\n\n _'I had been':_ Ibid., 80.\n\n ' _By concentrating on':_ Beilock and Carr, 'On the Fragility'; Beilock et al., 'When Paying Attention.'\n\n _'Choking is merely':_ Gucciardi and Dimmock, 'Choking under Pressure.'\n\n _Claude Steele, a professor:_ Steele and Aronson, 'Stereotype Threat'; Steele, 'Thin Ice.'\n\n _'What you tend':_ Gladwell, 'The Art of Failure.'\n\n _A few years:_ Wilson and Schooler, 'Thinking Too Much.'\n\n _So Wilson came:_ Wilson et al., 'Introspecting About Reasons.'\n\n _As Ap Dijksterhuis:_ Dijksterhuis et al., 'The Rational Unconscious.'\n\n _Because people_ expected: Wager et al., 'Placebo-Induced Changes.'\n\n _Look, for example:_ Shiv, Carmon, and Ariely, 'Placebo Effects of Marketing Actions.'\n\n _'We have these':_ Baba Shiv, telephone interview with the author, February 20, 2008.\n\n _Researchers at Caltech:_ Plassmann et al., 'Marketing Actions.'\n\n _'We don't realize':_ Antonio Rangel, telephone interview with the author, February 20, 2008.\n\n _'There seems to':_ Miller, 'The Magical Number Seven.'\n\n _Consider this experiment:_ Shiv and Fedorikhin, 'Heart and Mind.'\n\n _A bad mood:_ Gailliot et al., 'Self-Control Relies on Glucose.'\n\n _But just as:_ Geier, Rozin, and Doros, 'Unit Bias.'\n\n _The group with:_ Wansink, _Mindless Eating_.\n\n _Richard Thaler, an:_ Thaler, _The Winner's Curse,_ 107\u201322.\n\n _As Thaler notes:_ Bennett, 'When Shove Comes to Push.'\n\n _A few years:_ Ariely, Loewenstein, and Prelec, 'Coherent Arbitrariness'; 'Tom Sawyer and the Construction of Value.'\n\n _'You know you're':_ Dan Ariely, interview with the author, May 29, 2008.\n\n _While the extra:_ Grove et al., 'Clinical versus Mechanical Prediction.'\n\n _Back pain is:_ Lehrer, 'Psychology of Back Pain.'\n\n _In a 1994 study:_ Jensen et al., 'Magnetic Resonance Imaging.'\n\n _'A lot of':_ Sean Mackey, interview with the author, June 1, 2007.\n\n _A large study:_ Jarvik et al., 'Rapid Magnetic Resonance Imaging.'\n\n _They wanted to:_ Deyo, Nachemson, and Mirza, 'Spinal-Fusion Surgery.'\n\n _'What's going on':_ John Sarno, interview with the author, June 1, 2007.\n\n _Sadism was entertaining:_ Sullivan and Maiken, _Killer Clown_\n\n _'[Gacy] appears to have':_ Wilkinson, 'Conversations with a Killer.'\n\n _The acts of:_ Levenson, Carstensen, and Gottman, 'The Influence of Age and Gender.'\n\n _'You know when':_ James Blair, telephone interview with the author, April 4, 2007.\n\n _The emotional areas:_ Deeley et al., 'Facial Emotion Processing.'\n\n _The main problem:_ Blair, Mitchell, and Blair, _The Psychopath_.\n\n _Brain-imaging studies:_ Berthoz et al., 'Affective Response.'\n\n _'Moral judgment is':_ Haidt, _Happiness Hypothesis,_ 22.\n\n _Consider this moral:_ Haidt, 'The Emotional Dog.'\n\n ' _What happens in':_ Sommers, 'Jonathan Haidt.'\n\n _Consider this elegant:_ Greene et al., 'An fMRI Investigation.'\n\n _'Our primate ancestors':_ Joshua Greene, telephone interview with the author, June 25, 2007.\n\n _'It is fear':_ Marshall, _Men Against Fire,_ 78.\n\n _'At the most':_ Ibid., 79.\n\n _'What is being':_ Grossman, _On Combat,_ 254.\n\n _However, the researchers:_ Oosterbeek, Sloof, and van de Kuilen, 'Differences in Ultimatum Game Experiments'; Henrich et al., _Foundations of Human Sociality_.'\n\n _'Seen through the':_ Frank, _Passions Within Reason,_ ix.\n\n _'As we have':_ Smith, _Theory of Moral Sentiments,_ 3\u20134.\n\n _The reason a proposer:_ Sanfey et al., 'The Neural Basis.'\n\n _Because they intensely:_ Tankersley, Stowe, and Huettel, 'Altruism Is Associated.'\n\n _When they chose:_ Moll et al., 'Human Fronto-mesolimbic Networks.'\n\n _It now appears:_ Dapretto et al., 'Understanding Emotions.'\n\n 'They allow us': Blakeslee, 'Cells That Read Minds.'\n\n A face generated: Schultz et al., 'The Role of the Fusiform Face Area.'\n\n _Instead of giving:_ Hoffman, McCabe, and Smith, 'Social Distance and Other-Regarding Behavior.'\n\n _'The experience of':_ Keltner, 'Power Paradox.'\n\n _Paul Slovic, a:_ Small, Loewenstein, and Slovic, 'Sympathy and Callousness.'\n\n _'If monkeys have':_ Blum, _Love at Goon Park,_ 231.\n\n _One Canadian couple:_ Stout, _The Sociopath Next Door,_ 132.\n\n _When neuroscientists imaged:_ Chugani et al., 'Local Brain Functional Activity.'\n\n _The orphans also:_ Parker et al., 'The Impact of Early Institutional Rearing.'\n\n _Finally, the neglected:_ Carter, 'The Chemistry of Child Neglect.'\n\n _In the early 1980s:_ George and Main, 'Social Interactions'; Main and George, 'Responses of Abused and Disadvantaged Toddlers.'\n\n _They were simply:_ Taylor et al., 'Neural Bases of Regulatory Deficits.'\n\n Consider this poignant: Masserman, Wechkin, and Terris, 'Altruistic Behavior in Rhesus Monkeys.'\n\n _'There were a':_ Ralph Jimenez, at _Monitor_ editors' interview with the author, March 25, 2008.\n\n _'I was still':_ Felice Belman, Ibid.\n\n _'The candidates are':_ Mike Pride, Ibid.\n\n _'The point':_ Antoine Bechara, interview, _Radio Lab,_ May 12, 2008.\n\n _Consider this clever:_ Knutson et al., 'Neural Predictors.'\n\n _In fact, researchers:_ Inman, McAlister, and Hoyer, 'Promotion Signal.'\n\n _Although it was:_ Green, Palmquist, and Schickler, _Partisan_ _Hearts,_ 2.\n\n _Drew Westen, a:_ Westen et al., 'The Neural Basis of Motivated Reasoning.'\n\n _'Essentially, it appears':_ Westen, _The Political Brain,_ 17.\n\n _'Voters think that':_ Achen and Bartels, 'It Feels Like We're Thinking.'\n\n _Let's look at:_ Brock and Balloun, 'Behavioral Receptivity.'\n\n _In 1984, the:_ Tetlock, _Expert Political Judgment_.\n\n _'The dominant danger':_ Ibid., 23.\n\n _'Oh, that's easy':_ Gazzaniga, _The Social Brain,_ 72.\n\n _During the last:_ Rabinovich, _The Yom Kippur War_.\n\n _He persuasively argued:_ Black and Morris, _Israel's Secret_ _Wars,_ 280\u201399; Raviv and Melman, _Every Spy a Prince,_ 110\u201325.\n\n _'The need for':_ Bar-Joseph and Kruglanski, 'Intelligence Failure'; Bar-Joseph, _The Watchman Fell Asleep_.\n\n _He intentionally filled:_ Goodwin, _Team of Rivals,_ 126.\n\n ' _You get so wired_ ': Michael Binger, interviews with the author, May 12, 2007; July 1, 2007; September 22, 2007.\n\n _'To answer these':_ Dijksterhuis et al., 'On Making the Right Choice.'\n\n _'Imagine being at':_ Dijksterhuis and van Olden, 'On the Benefits of Thinking Unconsciously.'\n\n _'The moral of':_ Dijksterhuis, 'Breakthrough Ideas.'\n\n _The hardest calls:_ Dijksterhuis and Nordgren, 'A Theory of Unconscious Thought.'\n\n _That's why the:_ Klein, 'In the Digital Age.'\n\n _A few years:_ Lo and Repin, 'The Psychophysiology of Real- Time Financial Risk Processing.'\n\n _'One of the':_ See http:\/\/alum.mit.edu\/ne\/opendoor\/200509\/lo.html.\n\n _'We need to cultivate':_ Tetlock, _Expert Political Judgment,_ 24.\n\n _But then, starting:_ Baker et al., 'Pilot Error in Air Carrier Mishaps.'\n\n _'The old way':_ Jeff Roberts, telephone interview with the author, July 12, 2007.\n\n _'For most of':_ Al Haynes, interview with the author, January 21, 2008.\n\n# Index\n\n  * Abumrad, Jad \n  * abuse \n    * abusive behavior, generation as result of , \n    * of children, scars of \n  * ACC (anterior cingulate cortex) , , , \n  * ADHD (Attention Deficit Hyperactivity Disorder) \n  * Aeschylus \n  * AI (Artificial Intelligence) , \n  * Albright, Madeleine \n  * altruism \n  * American College of\n\n    * Physicians \n  * amygdala , , \n    * activation of, emotion and , , \n    * and psychopathic brain, defects of \n  * analysis of decision-making \n    * cricket, batsman against fast bowler \n    * escape firelighting \n    * flight simulator \n    * hydraulic controls, dealing with complete failure , \n    * intelligence failure in Six- Day War \n    * quarterback choices , \n    * radar blip, significance of \n  * Andreassen, Paul \n  * Ariely, Dan \n  * Aristotle \n  * asymmetric paternalism \n  * attachment, need for \n  * Auerbach, Red \n  * Ausubel, Lawrence \n  * autism , \n  * aviation \n    * autopilot systems , , , \n    * CRM (Cockpit Resource Management) \n    * flight safety \n    * flight simulation , , , \n    * hydraulic controls, complete loss of \n    * pilot error \n    * SAM (System Aircraft Management)\n\n  * backgammon , , , , \n  * Bacon, Francis \n  * Balloun, Joe \n  * Bar-Joseph, Uri \n  * Baron-Cohen, Simon \n  * Bartels, Larry \n  * Barto, Andrew 49n\n  * Bates, Edward \n  * Baumeister, Roy \n  * Bechara, Antoine , \n  * Beilock, Professor Sian \n  * Belichick, Bill , \n  * _The Bell Curve_ (Herrnstein, R.J. and Murray, C.) \n  * Belman, Felice , \n  * Benartzi, Shlomo , \n  * Berlin, Isaiah \n  * Betsch, Tilmann \n  * Binger, Michael , , \n    * aggressiveness \n    * automatic brain n\n    * bluffing \n    * decision-making, inexplicable reasoning in \n    * discipline \n    * modes of thought, different situations and , , , , \n    * odds, contemplation of \n    * patient poker \n    * professional poker \n    * psychological aspects of poker \n  * Bird, Larry \n  * Blair, James \n  * Bledsoe, Drew \n  * Blum, Deborah n\n  * Bly, Adam \n  * Bohr, Niels \n  * Boston Celtics , \n  * _Boston Globe_ \n  * Brady, Tom , , , , , , \n  * brain \n    * amygdala , \n    * area links to prefrontal cortex \n    * associations, generation of \n    * biological reality of \n    * brain stem \n    * competing regions within \n    * complexity of \n    * cortex \n    * counting and measurement, difficulties with \n    * diencephalon \n    * dopamine system \n    * emotional brain, refinement of \n    * evolution of \n    * feelings and passions within \n    * flaws in rationality of \n    * frontal cortex \n    * frontal-lobe , , , \n    * layered anatomy of \n    * learning from experience \n    * limbic region , , , , , \n    * mental accounting \n  *     * mental activity, patterns of \n    * predictable and unpredictable rewards, effect on \n    * processing abilities , \n    * psychopathic brain, defects of \n    * real world decision-making context \n    * slot machines, effect on \n    * spindle neurons , \n    * thinking about how to think , , , \n    * _see also_ cortical debate; emotion; mind; prefrontal cortex\n  * Brock, Timothy \n  * Brown, Troy , , , , \n  * Bush, George W. , ,\n\n  * CAE Tropos 5000 flight simulator , , \n  * Carragee, Dr. Eugene \n  * Carter, Jimmy \n  * causation correlation and , \n  * CBT (Cognitive Behavioral Therapy) n\n  * Ceau\u00e7escu, Nicolae \n  * certainty \n    * central error of \n    * certainty bias \n    * conscious correction against \n    * inner dissonance, certainty bias and \n    * lure of \n  * Cervantes, Miguel de \n  * Cheeks, Maurice \n  * chess , , , , \n  * Chesterton, G.K. \n  * child molestation and warped morality \n  * children \n    * abuse of children, scars of \n    * unwanted children \n  * Chinchillo, Melissa \n  * choking, self-doubt and anatomy of \n    * in golf, pressure of winning , n\n    * in opera performance \n    * thoughtfulness and , \n  * civilization \n  * Clinton, Bill , \n  * Clinton, Hillary , , \n  * cognitive dissonance, silencing of \n  * cognitive psychology \n  * cognitive science \n  * Cohen, Jonathan \n  * commuting, overthinking about \n  * comparison shopping \n  * competing viewpoints, dealing with \n  * computer \n  * Comte, Auguste \n  * _Concord Monitor_ \n  *   * confidence, comfort of \n  * _Consumer Reports_ , \n  * contradiction \n  * Cook, Amanda \n  * Cook, Garth \n  * correlation , , , , , , , \n    * causation and , \n  * cortical debate \n    * certainty, central error of \n    * certainty, lure of \n    * certainty bias \n    * cognitive dissonance, silencing of \n    * competing viewpoints, dealing with \n    * confidence, comfort of \n    * conscious correction against certainty bias \n    * contradiction \n    * decision-making and \n    * editorial discussion, metaphor for \n    * electoral opinion, shaping of \n    * indecisive disagreement, default state of brain \n    * inner dissonance, certainty bias and \n    * intelligence failure in Six- Day War , \n    * NAcc (nucleus accumbens) , , \n    * partisan delusions \n    * political punditry \n    * predictive failure of punditry \n    * rationality, liability of \n    * reconciliation \n    * self-assurance in split-brain patients \n    * self-delusion \n    * split-brain patients \n    * top-down thinking , \n    * uncertainty intolerance \n    * visual fields \n  * credit cards , , , , , , \n    * financial decision-making and \n    * promotional activities \n  * cricket \n  * CRM (Cockpit Resource Management) \n  * Cunningham, Randall\n\n  * Damasio, Antonio , , , , \n  * Darfur \n  * Darwinian evolution \n  * Davies, Nick \n  * Dayan, Moshe \n  * _Days of Our Lives_ (NBC) \n  * _Deal or No Deal_ (NBC TV) , , \n  * debt, danger of \n    * credit cards , \n    * decision-making, credit cards and \n    * financial counseling , \n    *     * impulsive emotion, temptation and \n    * negative savings \n    * overvaluation of immediate gains , \n    * Save More Tomorrow \n    * subprime mortgages \n    * decision-making \n    * asymmetric paternalism \n    * automatic talent \n    * autopilot systems, pilot's interaction with \n    * cortical debate and \n    * cricket, batsman against fast bowler \n    * deciding without thinking \n    * emotional choice , \n    * emotional influences on \n    * escape firelighting \n    * financial decision-making, credit cards and \n    * in flight simulator \n    * friend or foe, determination of \n    * future decisions based on past lessons \n    * heat of the moment \n    * hydraulic controls, complete loss of , \n    * impersonal decision-making \n    * intangible nature of \n    * intelligence failure in Six-Day War \n    * investment mistakes and loss aversion \n    * investment-portfolio theory \n    * James' insight \n    * knowledge, unconscious and \n    * legal basis for moral decisions \n    * loss aversion \n    * maths problems and mysteries, difference between \n    * moral decisions, logical basis for \n    * morality and , , , , , , , \n    * novel problems and need for reason \n    * panic problem \n    * personal decision-making , \n    * premium equity puzzle \n    * psychopathic behavior and , , , , \n    * quarterback choices , , \n    * radar blip, significance of \n    * reactive to events \n    * real world context of \n    * science of , \n    * shopping choice, complexity and \n    * simple problems and need for reason \n    * taxonomy of , \n    *     * temptation, impulsive emotion and \n    * theories of , \n    * thinking about how you're thinking , , , \n    * uncertainty, embrace of \n    * unconscious thought, power of , \n    * _see also_ brain; cortical debate; emotion; mind; prefrontal cortex\n  * Deep Blue \n  * deliberation , , , , , , , , \n  * Descartes, Ren\u00e9 4, , \n  * _Descartes' Error_ (Damasio, A.) \n  * Desert Storm , \n  * diencephalon \n  * Dijksterhuis, Ap , , \n  * _Discourse on the Method_... (Descartes, R.) \n  * DLPFC (dorsolateral prefrontal cortex) 100n\n  * Dodd, Chris \n  * Dodge, Wag , , , , , , \n  * dopamine system , \n    * ACC (anterior cingulate cortex) , , , \n    * discovery of \n    * dopamine agonists and gambling compulsion , \n    * dopamine release \n    * emotional regulator \n    * expectation , , \n    * future decisions based on past lessons \n    * gambling and prediction \n    * learning from mistakes \n    * motion sickness \n    * necessity and precision of dopamine neurons \n    * Parkinson's, disease of , , \n    * patterns based on experience, generation of , , \n    * predictive error signal , , \n    * predictive sensitivity , , \n    * randomness, effect on \n    * receptors, reduction in \n    * rewards, predictable and unpredictable \n    * schizophrenia \n    * Schultz and understanding of , \n    * slot machines, effect on \n    * stock trading and investment , \n  * Dostoyevsky, Fyodor \n  * Dweck, Carol\n\n  * economics \n  * editorial discussion \n  * electoral opinion, shaping of \n  * Elway, John \n  * emotion \n    * ADHD (Attention Deficit Hyperactivity Disorder) \n    * adolescent behavior \n    * amygdala, activation of , , \n    * analysis of emotional data \n    * Aristotle and intelligent application of \n    * assumption of, irrationality of \n    * candle problem \n    * cognitive tool , , , , , , \n    * computers and \n    * control of emotional impulses \n    * crucial importance of , \n    * emotional bias \n    * emotional brain, refinement of \n    * emotional signals, subliminal calculation in directing daytime soap opera \n    * feedback dials and emotional mistakes , \n    * focus group workings , \n    * framing effect \n    * frontal cortex and \n    * gambling and prediction \n    * impotence of reason without \n    * impulsive emotions , , , \n    * indecision and \n    * investment decisions by emotionless subjects \n    * management of \n    * maturation of mind and capacity for control of \n    * moral emotions , , \n    * motivation and \n    * OFC (orbitofrontal cortex) and , , , , , , \n    * overvaluation of immediate gains , \n    * prefrontal cortex and \n    * privileging of reason over \n    * profrontal cortex and emotional management \n    * psychopathic behavior and emotional emptiness , , , \n    * rationality and moral decision-making \n    * reason and, conflict between \n    * self-awareness \n    * self control \n    * Stroop task , \n    * temptation, impulsive emotion and \n    * thinking about, control through \n    * wisdom of \n    *   * energy drinks \n  * epilepsy , , \n  * Erving, Julius , \n  * Ethiopia\n\n  * Fama, Eugene \n  * Faulk, Marshall \n  * Favre, Brett \n  * FFA (Fusiform Face Area) \n  * fictive-error learning \n  * firefighting \n    * crown fires \n    * death in flames \n    * escape firelighting \n    * outrunning fire , \n    * twilight breezes, danger of \n    * updraft \n  * Fitch, Dennis , \n  * Fleming, Ren\u00e9e , \n  * Fletcher, Christy \n  * flight simulation , , , \n  * Frank, Robert \n  * Franklin, Benjamin \n  * Freud, Sigmund , \n  * frontal cortex \n  * frontal-lobe lobotomy\n\n  * Gacy, John Wayne , , \n  * gambler's fallacy n\n  * gambling addiction \n  * generosity, sympathy and \n  * genocide \n  * George, Carol \n  * Gilovich, Thomas , , n\n  * Glenn, Terry \n  * Gold, Jamie , , \n  * Gonen, General Shmuel \n  * Goodwin, Doris Kearns \n  * Gorbachev, Mikhail \n  * Goya, Francisco de \n  * Graden, Brian \n  * GRE (Graduate Record Examination) \n  * Greene, Joshua \n  * GreenPath \n  * Grossman, Lieut. Colonel Dave \n  * Gulf War\n\n  * Haidt, Jonathan , , \n  * _The Happiness Hypothesis_ (Haidt, J.) \n  * Harlow, Harry \n  * Haynes, Captain Al , , , , \n  * Heilman, Dr. Kenneth \n  * Hellman, William \n  * Hess, Walter \n  * Higgs boson particle \n  * Hobbes, Thomas \n  * house purchase, overthinking in \n  * Huckabee, Mike \n  * human behavior \n  * human nature \n  * Hume, David ,\n\n  * IKEA , \n  * impersonal decision-making \n  * improvisation \n  * impulsiveness , , , , , , , \n    * in credit card use , \n    * destructive impulsiveness \n    * impulsive emotions , , , \n  * indecision , \n    * emotion and \n    * indecisive disagreement, default state of brain \n  * information overload , , , \n    * anchoring effect \n    * back pain epidemic \n    * college counseling \n    * knowledge, diminishing returns of \n    * modern life and \n    * stock trading and investment and \n    * understanding, interference with \n  * inner dissonance, certainty bias and \n  * insight \n    * anatomy of \n    * cortical events and success in , , \n    * ideology and \n    * intuitive insight \n    * neuroscience of \n  * instinct \n    * 'feel' and \n  * intuitions , , , , , , \n  * dopamine and\n\n    * Dweck and mistakes as building blocks for \n    * instinct and \n    * intelligent intuition, deliberate practice and \n    * moral intuitions , \n    * religion and \n    * Investment Portfolio Theory \n    * Iowa Gambling task \n    * IQ (Intelligence Quotient) and IQ Test , , , ,\n\n  * isolation, social interaction and , \n  * Israel \n    * Bar-Lev line \n    * intelligence failure in Six-Day War , \n    * Six-Day War ,\n\n  * Jackson, Bo \n  * Jackson, Mary \n  * jam taste test \n  * James, William \n  * Japan Airlines Flight \n    * Tokyo-Osaka , \n  * Jefferson, Thomas \n  * Jiminez, Ralph , , , \n  * Jobs, Steve \n  * _Journal of the American_ _Medical Association_ , \n  * Jung-Beeman, Mark ,\n\n  * Kahneman, Daniel , , n\n  * Kanner, Dr. Leo \n  * Kant, Immanuel , , \n  * Kasparov, Garry , , \n  * Keltner, Dacher \n  * Kennedy, President John F. \n  * Kennedy, Rosemary \n  * Kerry, John \n  * Klein, Gary \n  * Klinestiver, Ann , , \n  * knowledge \n    * diminishing returns of \n    * internalization of \n    * mistakes, building blocks of knowledge \n    * unconscious and \n  * Knutson, Brian , \n  * Krueger, Alan \n  * Krulwich, Robert\n\n  * Laibson, David \n  * Las Vegas , , , \n  * Lazarsfeld, Paul \n  * LeDoux, Joseph , \n  * legal basis for\n  * moral decisions \n  * Leibniz, Gottfried W. \n  * Leinart, Matt \n  * Liebowitz, Sarah \n  * limbic region of brain , , , , , \n  * Lincoln, Abraham \n  * Lo, Andrew \n  * Loewenstein, George , , , , \n  * logic , , \n    * moral decisions, logical basis for \n    * slowness of \n    * of software \n  * loss aversion , \n    * _Deal or No Deal_ (NBC TV) , , \n    * demonstration of \n    * innate flaw \n    * and investment mistakes \n    * investment-portfolio theory \n    * premium equity puzzle \n  * Louis IX of France \n  * _Love at Goon Park_ (Blum, D.) n\n  * Luke, Gospel according to\n\n  * McCain, John , \n  * Mackey, Dr. Sean \n  * Maclean, Norman \n  * McNeil, Laura \n  * MaCurdy, Thomas \n  * Madden, John \n  * Main, Mary \n  * Malawi \n  * Malone, Moses \n  * Mann Gulch , , , \n  * Manning, Peyton \n  * Marino, Dan \n  * Markowitz, Harry \n  * _The Marriage of Figaro_ (Mozart) \n  * Marshall, Brigadier General S.L.A. \n  * Martino, Benedetto de \n  * Meir, Golda , \n  * mental activity, patterns of \n  * _Metamorphoses_ (Ovid) \n  * Miller, Earl \n  * Miller, George \n  * Miller, Lulu \n  * Milner, Peter \n  * mind , \n    * biological machine \n    * as computer \n    * emotional signals, subliminal calculation in directing daytime soap opera \n    * Freud's dynamic conflict of \n    * imperfect machine \n    * mind reading , \n    * Platonic divided mind, theme of , , , , , \n    * Wonderlic intelligence test , \n    * _see also_ brain; cortical debate; moral mind\n  * Minsky, Marvin , n\n  * mirror-neuron circuitry \n  * Mischel, Walter \n  * mistakes, building blocks of knowledge \n  * Monet, Claude , \n  * Moniz, Antonio Egas \n  * Montague, Professor Read , , \n  * Montana, Joe \n  * moral emotions , , \n  * moral mind \n    * abuse generating abusive behavior , \n    * abuse of children, scars of \n    * altruism \n    * amygdala and psychopathic brain, defects of \n    * attachment, need for \n    * autism , \n    * child molestation and warped morality \n    * dictator game \n    * FFA (fusiform face area) \n    * generosity, sympathy and \n    * impersonal decision-making \n    * isolation, social interaction and , \n    * legal basis for moral decisions \n    * logical basis for moral decisions \n    * mind reading , \n    * mirror-neuron circuitry, silence from \n    * moral emotions , , \n    * personal decision-making , \n    * psychopathic behavior , , , , \n    * psychopathic brain, defects of \n    * rationality and moral decisions \n    *     * selfishness, trolley problem and \n    * sibling incest \n    * soldiers, killing and moral emotions \n    * statistics and moral emotions \n    * sympathy and fellow-feeling \n    * ultimatum game , , \n    * unwanted children \n  * morality \n    * evolution of \n    * primate morality \n    * psychology of \n    * standard view of \n  * Moses \n  * Mossad , , , \n  * motion sickness \n  * motivation and emotion \n  * Mozart, Wolfgang A. , \n  * MRI (Magnetic Resonance Imaging) , \n    * fMRI (functional MRI) , , , ,\n\n  * NAcc (nucleus accumbens) , , \n  * National Transportation Safety Board , \n  * Nature and Neuroscience \n  * _New England Journal of_ _Medicine_ , \n  * New England Patriots \n  * _New Yorker_ \n  * _The Nicomachean Ethics_ (Aristotle) \n  * Nietzsche, Friedrich \n  * Nixon, Richard\n\n  * Obama, Barack , , , \n  * Odean, Terrance \n  * Oedipus complex \n  * OFC (orbitofrontal cortex) , , , , , , \n  * Olds, James \n  * _The Oresteia_ (Aeschylus) \n  * overthinking , \n    * commuting \n    * energy drinks \n    * house purchase \n    * information overload \n    * jam taste test \n    * placebo effect \n    * poster test \n    * rational thinking and \n    * winetasting \n    * _see also_ choking, self-doubt and\n  * Ovid\n\n  * Pain Society, American \n  * Palmer, Herman , , , \n  * panic problem \n    * awareness, reduction of \n    * perceptual narrowing \n    * prefrontal cortex and , , \n    * probability \n    * rational choice theory \n    * rationality \n  * Parker, Robert \n  * Parkinson's disease , , \n  * Parry, Emma \n  * partisan delusions \n  * Pasteur, Louis \n  * personal decision-making , \n  * personality \n  * Philadelphia 76ers \n  * phugoid patterns , , , \n  * placebo effect \n  * Plato , , , , , , \n    * divided mind, theme of , , , , , \n    * rationality model of \n  * poker \n    * aggressiveness \n    * automatic brain n\n    * bluffing \n    * chance, undercurrent of \n    * decision-making, inexplicable reasoning in \n    * discipline \n    * elimination, ruthlessness of \n    * fatalism \n    * mathematical approach to \n    * modes of thought, different situations and , , , , \n    * mystery of \n    * odds, contemplation of \n    * patient poker \n    * professional poker \n    * psychological aspects of \n  * political punditry \n  * predictive failure of , \n  * Pook, David \n  * Post, Thierry \n  * poster test \n  * Powell, Colin \n  * Predmore, Steven \n  * prefrontal cortex \n    * autopilot systems and \n    * brain area links to \n    * control of emotional impulses , , , \n    * emotional management by use of \n    * importance of , \n    * management of emotion \n    * panic problem , , \n    * partisan delusions \n    * problem-solving abilities \n    * recognition, act of \n    * reliance on, danger of \n    * versatility and creativity of , \n    * working memory and , \n    * _see also_ cortical debate\n  * Prelec, Drazen \n  * premium equity puzzle \n  * Pride, Mike , , \n  * _The Principles of Psychology_ (James, W.) \n  * _Pro Football Weekly_ \n  * _Proust Was a Neuroscientist_ (Lehrer, J.) \n  * psychoanalysis \n  * _The Psychopath: Emotion_ _and the Brain_ (Blair, J. _et_ _al_.) \n  * psychopathic behavior , , , , \n  * psychopathic brain, defects of\n\n  * randomness , , , , , , , , \n    * dopamine system and \n    * streaks, illusion of , \n    * unconscious thought and , \n  * Rangel, Antonio , \n  * rationality , \n    * agonists of \n    * backfiring of rational thought \n    * Cartesian faith in \n    * emotion and, conflict between \n    * faith in power of reason \n    * impotence of reason without emotion \n    * liability of \n    * moral decisions and \n    * overthinking and \n    * problem solving and \n    * psychopathic behavior and , , , , \n    * working memory and \n  * Reagan, Ronald , \n  * reason \n    * faith in power of \n    * novel problems and need for reason \n    * reliance on, danger of \n    * simple problems and need for reason \n  * reconciliation in cortical debate \n  * Records, William \n  * Redmond, J.R. \n  * Richter, Ari \n  * Riley, Lieut. Commander Michael , , , \n  * Rizzolatti, Giacomo \n  * Robertie, Bill , \n  * Roberts, Jeff , , \n  * Roe, Tracy \n  * Romania \n  * Rumsey, Walter \n  * Rwanda\n\n  * Sadat, Anwar , \n  * Sainsbury, Nondumiso \n  * St. Louis Rams \n  * Sallee, Robert \n  * SAM (System Aircraft Management) \n  * Sarno, Dr. John \n  * SAT Reasoning Test , \n  * Save More Tomorrow \n  * Save the Children \n  * schizophrenia , \n  * Schultz, Wolfram , , \n  * _Science_ magazine , \n  * _Seed_ magazine , \n  * self-assurance \n  * self-control \n    * blood sugar levels and \n  * self-delusion \n  * self-doubt _see_ choking, self-doubt and\n  * selfishness \n  * Seward, William \n  * Shiv, Baba \n  * Shoven, John \n  * sibling incest \n  * Simister, Duncan \n  * Simon, Herbert , \n  * Simon-Tov, Lieut. Benjamin \n  * Sloan, Alfred P. \n  * slot machines \n  * Slovic, Paul \n  * Smith, Adam \n  * Smith, Alex \n  * smokejumpers , , \n  * _see also_ firefighting\n  * Socrates \n  * software , , , \n    * Deep Blue \n    * TD-Gammon , , \n  * soldiers, killing and moral emotions \n  * Sorensen, Ted \n  * split-brain patients \n  * Stanton, Frank \n  * statistics and moral emotions \n  * Steele, Professor Claude \n  * Stein, Herb , , \n  * stock trading and investment , \n    * emotional decision-making \n    * information overload and \n    * investment-portfolio theory \n    * loss aversion and investment mistakes \n  * Strand, Kathleen \n  * streaks, illusion of \n    * fictive-error learning \n    * gambler's fallacy n\n    * randomness , \n    * stock trading and invest ment \n  * Stroop task , \n  * Super Bowl (2002) \n  * Sutton, Rich n\n  * sympathy \n    * basic human instinct of \n    * fellow-feeling and\n\n  * taxonomy of decision-making , \n  * TD-Gammon , , \n  * _Team of Rivals_ (Goodwin, D.K.) \n  * Tesauro, Gerald , , \n  * Tetlock, Philip , , \n  * Thaler, Richard , , \n  * _The Theory of Moral_ _Sentiments_ (Smith, A.) \n  * thought choking, self-doubt and  thoughtfulness , \n    * modes of, different situations and , , , , \n    * rational thought, backfiring of \n    * thinking about emotion, control through \n    * thinking about how you're thinking , , \n    * top-down thinking , \n    * unconscious thought, power of \n    * _see also_ overthinking\n  * Toney, Andrew , , \n  * top-down thinking \n  * cortical debate and , \n  * Trimmer, Ted \n  * Tversky, Amos , ,\n\n  * ultimatum game , , \n  * uncertainty\n  * embrace of \n  * intolerance of \n  * United Airlines Flight \n    * Denver-Chicago , , , , \n  * United Airlines Flight \n    * Denver-Portland, Oregon ,\n\n  * Van de Velde, Jean , , \n  * van Gogh, Vincent , \n  * Vinatieri, Adam \n  * visual fields\n\n  * Wager, Tor \n  * Wall Street , \n  * _Wall Street Journal_ \n  * Wansink, Brian \n  * Warner, Kurt \n  * Wasicka, Paul \n  * _The Wealth of Nations_ (Smith, A.) \n  * Weis, Charlie \n  * Westen, Drew \n  * Wheeler, Soren \n  * Wilkinson, Alec \n  * Wilson, Geordie \n  * Wilson, Timothy , , , \n  * winetasting \n  * Wonderlic intelligence test , \n  * working memory , \n  * WSOP (World Series of Poker) , , , , , , ,\n\n  * Young, Vince \n  * _Young Men and Fire_ (Maclean, N.)\n\n  * Zeira, Major General Eli ,\n\n# Acknowledgments\n\nI decided to write a book about decision-making because I couldn't make a decision about Cheerios. There I was, aimlessly wandering the cereal aisle of the supermarket, trying to choose between the Apple Cinnamon and Honey Nut varieties. It was an embarrassing waste of an afternoon, and yet it happened to me _all the time_. Eventually, I decided that enough was enough: I was going to understand what was happening inside my brain as I contemplated my breakfast options. So thank you, General Mills, for making so many different kinds of Cheerios.\n\nThere is, of course, a lot of work that has to be done before the sudden epiphany ('I should write a book about decisions!') actually becomes a book. Once again, Amanda Cook, my editor at Houghton Mifflin, was a godsend. She took a messy, convoluted, unstructured manuscript and managed to find the thread that brought it all together. She suggested stories, fixed my prose and talked me through my confusion. She is the kind of editor \u2013 thoughtful, whip-smart and generous \u2013 that every writer dreams about. I am so fortunate to have her on my side.\n\nI also benefited from the extremely helpful comments and suggestions of many friends. Robert Krulwich, as always, taught me how to tell a story; the team at Radio Lab, including Jad Abumrad, Lulu Miller and Soren Wheeler, helped me figure out which stories were worth telling. Gareth Cook edited bits and pieces of this book for the _Boston Globe_ Ideas section, while Laura McNeil and Adam Bly allowed me to explore the field of neuroscience in _Seed_ magazine. David Pook taught me about moral philosophy, while Ted Trimmer made sure I got my aviation facts right.\n\nI'm also incredibly grateful to all of the scientists who took the time to talk with a curious writer. I asked them all sorts of na\u00efve questions and mispronounced any number of brain regions, and yet, without exception, they've been understanding, patient and helpful. The mistakes that remain are my own fault. And then there are the people, like Ann Klinestiver, Al Haynes, Herb Stein, Ralph Jimenez, Felice Belman, Mike Pride and Michael Binger, who let me share their personal stories in these pages. It's been an honor.\n\nI probably wouldn't be a writer if my wonderful agent, Emma Parry, hadn't spotted an article I wrote years ago for Seed and convinced me that it could become an improbable book called _Proust Was a Neuroscientist_. She's a consummate problem-solver and endless source of good ideas. I'm so grateful to her and everyone at Fletcher and Parry, especially Christy Fletcher and Melissa Chinchillo. Nick Davies, my editor at Canongate Books, has been a joy to work with. He also managed to teach me the rules of cricket, for which he deserves some sort of very prestigious award. Tracy Roe, the world's finest copyeditor, has saved me yet again from an embarrassing number of errors and flabby sentences.\n\nAnd then there is my family: my sister Rachel is not only a fantastic modern dancer, she also gave me great advice on the manuscript; my brother, Eli, helped me think about the realworld implications of these scientific theories (and kept me well supplied with music mixes). I even benefited from the red pen of my grandmother Louise. My father was a patient listener and a fount of relevant factoids and articles, e-mailed to me daily.\n\nMy mother was an essential reader; I don't know where she finds the time to read my first drafts, but I can't imagine writing without her feedback. My girlfriend, Sarah Liebowitz, has read this book dozens of times (I'm not exaggerating) in all of its various drafts and guises. This book wouldn't exist without her insightful criticisms, cheerful support and love. By the time you read this, Sarah will be my wife, which is, without a doubt, the best decision I've ever made.\n\n# Copyright\n\nFirst published in the United States in 2009 as _How We Decide_   \nby Houghton Mifflin Harcourt Publishing Company,   \n215 Park Avenue South, New York, New York 10003\n\nFirst published in Great Britain in 2009   \nby Canongate Books Ltd,   \n14 High Street, Edinburgh, EH1 1TE\n\nThis digital edition first published in 2009   \nby Canongate Books Ltd\n\nCopyright \u00a9 Jonah Lehrer, 2009\n\nThe moral rights of the author have been asserted\n\n_British Library Cataloguing-in-Publication Data_   \nA catalogue record for this book is available on request from the British Library\n\nISBN 978 1 84767 477 7\n\nwww.meetatthegate.com \n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"For me and many sports fans it's not just about the game, it's about the whole experience as a fan. Fan360 is a blockchain powered platform that takes the fan experience to a new level, rewarding fans for engaging and giving them a voice. I am excited to be a part of a story which has the means to grow into the biggest sports fan community in the world.\nIm happy to support soon-to-be the biggest sports fan community in the world - Fan360 will reward your passion for sports. Join me at https:\/\/fan360.io and be the first to know all about the new FAN360 Token.\nTo all my dear fans that support me, it is time to get rewarded for it - that is why I support Fan360, a platform where you get all football related news in one place and soon you will be rewarded with tokens. Join me at https:\/\/fan360.io and help us grow into the biggest sports fan community in the world.\nAt RCD Espanyol de Barcelona we are always innovating and looking for better ways to connect with our fans. We are proud to be a part of Fan360 project, a platform that is going to influence the whole sports community! It's a blockchain based platform that rewards fans. We all know that our fans deserve to be rewarded! Join us and support Fan360 on becoming the biggest sports community in the world.\nAll the social media, news, scores and stats are available in one place where fans get to choose the sports stars, leagues and clubs they want to follow. With AI-driven user experience news feed will be tailored to each fan's needs and taste.\nFor every action on the platform fans get rewarded. Each like, share or content contribution is rewarded with FAN360 Tokens. Fans collect them through a transparent bIockchain system, and can exchange them for special experience, merchandise, tickets or fiat.\nFan360 is not just an idea or a whitepaper. We are one of the only ICO projects with a working product backed by technical and legal proofs of concept, with 130.000 fans!\nAll the social media, news, scores and stats are available in one place where fans get to choose the sports stars and clubs they want to follow. For each action on the platform fans get rewarded with FAN360 Tokens and exchange them for fiat, special experience and other merchandise.\nClubs can reach out, motivate and reward their true fans. BIockchain technology will enable ticket traceability. Understanding the needs and desires of fans will enable clubs to personalize experiences for the fans and increase engagement.\nBrands can engage with fans and reach micro influencers, who will be their brand ambassadors. They can run brand activations campaigns and reward the most loyal ones with tokens. Big data enables brand customization to the fans preferences.\nStrong existing partnerships and an aggressive growth.\nDirect access to football stars and club owners based on which we will also onboard a few football stars.\nMVP with over 130.000 fans on IoS, Android and desktop version.\n5 own blockchain developers and an experienced team from various industries.\nAn experienced finance professional, serial entrepreneur and blockchain investor. Gained professional experience as a trader at different top international trading companies in London. Holds a Masters Degree from University of Amsterdam.\nA master in delivering high quality, innovative and user-oriented solutions. Football maniac who happens to be a developer, designer, engineer, growth hacker and serial entrepreneur.\nFootball player, sports fan, technology and innovation driven, with a vision to build the biggest global community to reward sports fans. Speaker and Teacher at Barcelona University.\nExperienced sales and business development professional with ability to develop and maintain long-term intercultural business relationships. Former basketball player, and Rugby union President.\nRichard is well respected in both the sports & entertainment industries and works closely with many famous celebrities, athletes, business owners and motivational speakers.\nA business leader and board-level advisor, creating & executing ambitious international growth strategies in digital media businesses. Previous international leadership roles in Media, Content, Mobile, Telco, and FMCG at businesses such as Red Bull, Three Uk, Disney, Coca-Cola and NBC Universal.\nWholehearted in the product. From briefing right to designing, rendering and execution. Supporting brand managers in storytelling and augmenting their vision.\nA blockchain expert, worked as Director of DevOps for VisualDNA, WeSee and Sears. He specialized in continuous deployment, monitoring, configuration management and infrastructure automation.\nEnthusiastic about creating strong professional bonds, a facilitator and entrepreneur, connected with the key influencers in Latin America. Passion for sports and technology.\nWe promise to send you only the information you need to follow Fan360's progress.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A man in Bondo, Siaya County lives an uncommon and perplexing life by having two jobs of serving as a bishop and a morgue attendant at the same time.\nStakus Owino is a servant of the Lord but is usually found at the Bondo Sub-County Hospital mortuary during the weekdays.\nHe has worked as a mortuary attendant for the past 10 years.\nOne may easily be confused by seeing him on a Sunday without his maroon uniform, hand gloves, and white gumboots but with a full bishop's regalia while sitting at the front of the Israel Assembly Church.\nHe is the presiding bishop of the church.\nHe also ensures to carry his Bible whenever he reports to work and keeps it in his small office at the morgue.\nWhenever he has free time, waiting for the next body to be brought into the morgue, he goes through his Bible and prepares for his coming summons at the church.\nOwino, who is a father of three, believes that his two jobs are interconnected. He believes that while one involves winning souls of people for heaven, the other ushers them to the place with dignity.\nAccording to Owino, being a worker at the mortuary, similar to being a bishop, is a worthy calling.\nHe, however, continues to inform that by his service, he has heard a student indicate, \"When I grow up I want to be a mortuary attendant,\" which makes him proud.\nOwino began his career while serving as a cleaner at Kisumu's Jaramogi Oginga Odinga Teaching and Referral Hospital, earning a salary of Ksh6,000. At some point, he found himself given duties of cleaning the morgue.\nHe stated that even though he was scared, he really needed the money. The mortuary superintendent, however, helped him overcome the fear of the dead.\nOwino then took a three-month course of mortuary management certificate, which made him forfeit half of his salary at the time. It took three years to finish the course at Nairobi University's Chiromo Campus due to financial constraints.\nHe went for an internship at the Bondo Sub-County Hospital and the administration was impressed with his service so much that they would not let him go thereafter.\nThe bishop faced many challenges with the job such as his in-laws having trouble accepting him and even convincing his wife to divorce him.\nChurchgoers at the Israel Assembly Church describe their bishop as a highly spiritual man. None of them seems to have a sense of consideration for his other job.\n\"I could not imagine how he was going to preach to us and also work in a mortuary. But now I find his sermons inspiring,\" stated another member of the church.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Upgrade existing analogue CCTV installations quickly and easily to high definition systems with the Securview Victory Series\u2122 HDCVI Kits.\nThese ideal analogue to HD-video-over-coax upgrade packages enable you to rapidly boost the image resolution of an existing analogue installation to high definition using the existing coaxial cable infrastructure. Simply switch the DVR and the cameras to Securview Victory Series and boost your CCTV image resolution in an instant.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"List of Epsom based letting agencies offering houses, flats, buy to let & commercial property rental services to landlords and tenants.\nThe company offers the ability to sell, let or manage property of all sizes and values, with equal ease and professionalism. Our partners and associates have many years of local property knowledge and sales experience and are always on hand to deal with our clients personally. We pride ourselves on years of experience, and intend to keep our name at the forefront of the local market for years to come.\nThe local independent residential letting agency, \"solely specialising\" in the letting and management of all types of quality properties in Epsom and the surrounding areas. Our owner manager Sue Tuke and her team provide Landlords and Tenants with the highest quality \"personal and professional\" letting and property management services.\nBrowns Property Services first opened in 1988 and we have established ourselves as the leading independent Estate Agency in the area. My dedicated colleagues have been with Browns for many years providing an excellent service to vendors, buyers, landlords and tenants. We are members of the Property Ombudsman Scheme (TPOS) and the National Approved Letting Scheme (NALS).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lots of professional athletes buy Anavar due to its side-effect pleasant nature; in-fact, we securely say this is one of the best anabolic steroids of all time and is so well-tolerated by many that utilize it that unfavorable side-effects are really uncommon.\nBecause this is such a mild and side-effect pleasant steroid it is often called The Gal Steroid as a large portion of those which buy it are in-fact women sportsmens.\nThere are not too many steroids ladies can make use of safely without major effect but because this steroid is so light in attributes lots of girls buy it in substantial quantities in order to maintain top quality efficiency.\nMake no mistake, this steroid can be used by guys too and with good outcomes, nonetheless, most males will only use it when weight loss in order to maintain lean cells; serious bulking gains with Anavar can be really difficult ahead by for a numerous factors.\nNot all underground labs in Tartumaa Estonia manufacture Anavar and if you're trying to buy Anavar from an underground lab you could need to do a bit additional searching than normal.\nGranted, there are a lot of labs that do bring it yet you will locate this steroid to be one of the most frequently counterfeited on the black market as a result of the high demand for the steroid in Tartumaa Estonia.\nReally couple of underground labs in Tartumaa Estonia carry true Anavar and when you're trying to buy steroid that is not relatively cheap, the price ares much worse when you have actually taken that big quantity of money and acquired just a mislabeled Dianabol tab or a sugar pill, as both prevail.\nWhen you buy Anavar in Tartumaa Estonia you should guide away from underground laboratories; do not be suckered in by affordable; exactly what good is a low cost when the gear is phony, under-dosed or not whatsoever what its meant to be?\nYou might without a doubt pay a little even more when you buy it from a reputable outlet however you'll be receiving a first-rate steroid that will certainly not simply produce results but the ones you desire and safely also.\nOne must likewise know the legitimacy of the acquisition in Tartumaa Estonia; in many nations acquiring steroids on the open black market is prohibited; breaking this law can lead to greats and jail.\nFor this reason you will simply would like to buy it from legal avenues, ones that pursue the law in addition to provide products that have actually been looked into and examined to guarantee safety and performance.\nAs many individuals in Tartumaa Estonia are searching for a leaner and more difficult figure; a figure that is toned and much more attractive several buy Anavar in order to reach this end, as it is well-suited for this function.\nAs this steroid offers a multitude of functions its main function will verify to be throughout a leaning out duration. If you are weight loss and supplement with this steroid you will accomplish the leaner and more challenging look you prefer in a much more evident way than you would have without.\nIt is one of minority steroids that can provide to this end just as in both individuals in Tartumaa Estonia and as it is usually side-effect cost-free it remains among the most highly preferable steroids of perpetuity.\nAnavar was first gained in 1964 by Raphael Pappo while working at Searle Laboratories (Now Pfizer) and was marketed and distributed under the name Anavar.\nTaken orally, the steroid was initially generated for medical objectives such as in the treatment of HIV\/AIDS. Anavar was ceased by the FDA because of the negative stigma developed by the intense use in the body building neighborhood.\nThe steroid was later on reestablished to the world and can be discovered in medical facilities for addressing injuries or illness such as weakening of bones and trauma.\nAnavar has many functions in the clinical globe and has actually aided deal with folks who have burnt fat due to various injuries or conditions. Anavar boosts lean cells growth for individuals with diseases such as infection, injury and osteoporosis.\nThe FDA restricted manufacturing of Avanar in the 1980\u2032s and later removed it from racks in the United States in 1989. This fad proceeded worldwide.\nToday, the medication has been reintroduced in the United States and is manufactured by Savient. The FDA promotes the item for patients planning to put on weight after considerable infections and trauma.\nNumerous knock-off brand names have arised in the United States, allowing for more patients to have access to the less expensive universal brands.\nInitially designed for clinical use only, after viewing the weight development and toughness building qualities of Anavar, the bodybuilding community began using the steroid to examine its impacts.\nThe results have been positive, raising healthy protein synthesis and nitrogen recognition, when combined with a high protein diet plan and consistent lifting routine.\nAnavar enables bodybuilders to acquire muscle mass without putting on any sort of water weight, fat deposits or general body weight, which can be useful when looking to continue to be in a weight lesson.\nBy itself, Anavar does not really increase significant muscular tissue development, however when joined other steroids like Dianabol, the medication creates substantial muscle mass. Anavar strengthens the results of the other steroids by cutting the fat deposits in some cases produced by the steroids it is paired with.\nUtilizing Avanar increases strength and lean muscular tissue after intake and training. One advantage of using Anavar while training is that with low doses it does not substantially enhance testosterone degrees.\nSome researches have actually found that Anavar is fantastic for reducing unwanted abdominal fat in males. Throughout cycles, Anavar is not commonly utilized, since there are much less pricey choices to choose from.\nNonetheless, Anavar is useful during the linking process in between cycles. Using the steroid properly, with the best dosages, training and diet will certainly allow bodybuilders to keep weight off, also after they stop working out for a while.\nLadies making use of Anavar ought to avoid large doses to prevent intense increases in testosterone degrees, however with the ideal amount and diet, females can see outcomes such as fat loss and solidified muscular tissues.\nWhere to Buy Dianabol in Honduras?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"TEXARKANA, Texas - A federal grand jury has indicted an East Texas woman who authorities say sent ricin-laced letters to President Barack Obama and others in an attempt to frame her estranged husband.\nProsecutors announced Friday that 35-year-old Shannon Richardson is named in a three-count indictment. Richardson is accused of sending threatening letters in May to Obama, New York Mayor Michael Bloomberg and the head of the mayor's gun-control group.\nRichardson, of New Boston, Texas, was arrested June 7. She faces up to five years in prison on each of the charges.\nThe government alleges she mailed the letters to pin the crime on Nathan Richardson, the man she married in 2011. He filed for divorce earlier this month.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So you want to start your own podcast? How original! Follow our seven tips for a successful podcast, and maybe people will actually tune in to listen to your sniveling voice drone on and on about something you have little-to-no expertise in!\n1. Start listening to some popular podcasts.\nSerial is so hot right now. With over 39 million listeners, Serial examines the murder of Korean-American Hae Min Lee and focuses on the questionable guilt of her ex-boyfriend Adnan, who is currently serving time for her gruesome murder. Pay attention to the narrative style of Serial; what keeps the listeners coming back week after week when you can't even get Gary to reply to your Facebook message?\n2. Work on your brand.\nA witty title can make or break a podcast's success, so make sure you spend some time really focusing in on what story you want to tell. Try bringing in consultants to help you market your brand. Maybe ask Gary for some input\u2013\u2013 when you got coffee last week, he mentioned that he was in that IMC module, right?\n3. Invest in the right equipment.\nThe Audio-Technica ATR2100-USB Cardioid Dynamic USB\/XLR Microphone is good for starters, but if you really want to get that strong, clear tone, go for the Heil PR-40 Dynamic Studio Recording Microphone. If you are struggling with an unsteady mic, the Heil PR-40 combined with the Mackie 1402VLZ4 14-channel Compact Mixer will easily eliminate any troublesome feedback looping. Although proper equipment is expensive, it's important to invest in your future with Gary, I mean, the future of podcasting.\nMake a podcast the whole family can enjoy! In order to achieve success in the podcast business, it is important that you appeal to a broad range of podcast listeners. Throw in a political reference for Mom and Dad! And Gary's Aunt Susan loves polymer clay. Surely you could include her? That would be nice wouldn't it?\nMake sure to plan the structure and content of your podcast. Design some new segments and try adding in a guest interview or two. Timing is key, so you must not waste any time getting lost in Gary's eyes\u2013\u2013\u2013 just ignore those two shining, bottomless pools of baby blue. Jingles can help transitions, but they certainly won't help you struggle with your complicated feelings for Gary!\nSo you've got investors! MailChimp is in, and NPR has even offered to pick you up! You and Gary start your second season and sign a book deal. You fall in love. He proposes. For some reason that you still can not explain, you hesitate and get into a fight. Gary leaves to stay with his parents for a while and you start spending more and more time alone in the studio. One night, drunk on the internet, you agree to sell your eggs for money. You follow through with it and cry yourself to sleep. Gary forgives you, but it's too late. He always wanted to be a father, and, as much as he loves you, he can't commit himself to a childless marriage. You break up, amicably, but spiral into an endless cycle of depression.\nOct 25 University Report: Being a White Student at NU & Confronting Racial Issues is \"Uncomfy\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Almeida, J.G.L.; Maia, A.I.V.; Wilke, D.V.; Silveira, E.R.; Braz-Filho, R.; La Clair, J.J.; Costa-Lotufo, L.V.; Pessoa, O.D.L. Palyosulfonoceramides A and B: Unique Sulfonylated Ceramides from the Brazilian Zoanthids Palythoa caribaeorum and Protopalyhtoa variabilis. Mar. Drugs 2012, 10, 2846-2860.\nAlmeida JGL, Maia AIV, Wilke DV, Silveira ER, Braz-Filho R, La Clair JJ, Costa-Lotufo LV, Pessoa ODL. Palyosulfonoceramides A and B: Unique Sulfonylated Ceramides from the Brazilian Zoanthids Palythoa caribaeorum and Protopalyhtoa variabilis. Marine Drugs. 2012; 10(12):2846-2860.\nAlmeida, Jose G.L.; Maia, Ana I.V.; Wilke, Diego V.; Silveira, Edilberto R.; Braz-Filho, Raimundo; La Clair, James J.; Costa-Lotufo, Leticia V.; Pessoa, Ot\u00edlia D.L. 2012. \"Palyosulfonoceramides A and B: Unique Sulfonylated Ceramides from the Brazilian Zoanthids Palythoa caribaeorum and Protopalyhtoa variabilis.\" Mar. Drugs 10, no. 12: 2846-2860.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Designers become designers in a variety of manners. Together with the possibilities that are artistic that are huge, emblem designers regularly become in making enthusiast. An industrial programmer is somebody who initiates and expands suggestions that are certain to invent the sort of solutions. A wonderful picture designer has the capability to create new and terrific ideas all of the moment.\nCustom logo is. Complicated designs are not recognized and are really hard to keep in mind. In the event the site's plan isn't striking, then your visitor will absolutely navigate into some other website. Impressive Design A fascinating layout is essential to entice visitors. The design resumes display the author's nature and the talents, supporting construct a brand even if functioning independently. Although personalized, a fantastic graphic design restart needs to be able to convince recruiters with their brand is matched by your style.\nIt's mandatory that you check an online designer testimonials and also make certain they could perform what they say they can be in a position to execute. Be sure you decide on a web designer who is also capable of coding any works your website prerequisites. Therefore of course that you are going to be much better away by picking an internet designer with more experience under their belt. Ordinarily, a internet designer give some form of insight about which web hosting business they recommend. The bulk of the moment, designers have their very own design and you also can spot which layouts they will have done previously.\nA programmer could be licensed in Photoshop, but it will not signify they are still creative. The graphic designer is someone who simplifies various troubles that are linked to communicating. Graphic designers ought to be more familiar with computer graphics and design software's a variety readily available. It is possible to also check other designer hints about internet web sites to come across a idea of qualifications to prioritize.\nEven a CV should be clear and concise, maybe not just a package of artwork. The moment it's stored to PDF, the CV cans alter. With all these tools and features, anybody will have the capacity to generate a CV that is modern and stunning.\nInfographic resumes supply you with the opportunity to creatively present off your finest assets. As the project can be ordered in a project basis, because they do not need to pay for the artists delight in the common workers increasingly more clients want to apply graphic artists or illustrators. You can earn an application for graphic design tasks in lots of unique businesses. Graphic design careers are any careers by which somebody uses graphics get paid an excellent or book seem appealing or to market the merchandise.\nIf you are a time illustrator or graphic designer and you also wish to move freelancer, following is a listing of considerations you might need to consider on prior to stepping in an adventuresome yet demanding endeavor. Check your organization acumen understanding various small business paths and focusing on how to run a business will certainly help you develop in to a thriving freelance illustrator or graphics designer. Thus, graphics now has come to be among the essentials for businesses and specifically for marketing their product. You might still be able creative and you likely to participate in artwork however, you are likely in order to do it effectively and in fact earn money. Any creative artwork is much more captivated than some other gimmick or just text. An extremely summary artwork, or even an extremely intricate illustration is frequently perhaps maybe not really a good example of an logo though they may be considered an excellent artwork if believed in the view of a general art and maybe not a logo.\nBased about the various mediums around that you are likely to set the brand, the color and nature of this logo needs to be plumped for. Additionally while selecting the logo you ought to think about exactly where and how you are going to utilize it. Registration and Trade mark you might have the logo designed it's always suggested to receive it enrolled and trade marked to be certain that the logo is shielded from any type of plagiarisms.\nExceptional templates can guide you after you are fresh to the method of building a restart, get through the resume-building method and this is quite beneficial. Furthermore, it's possible to personalize the templates to be able that they fit your customer's requirements. Therefore it will be simple to pick in their templates. The survey template has been made over the kind of the booklet.\nShamelessly describe your ache is unbearable now, In the event you have got an justification to work out a chiropractor. When applied interchangeably explanations are occasionally a wonderful gain to provide you the little break which may be needed then and today. Then you definitely want to be familiar with perfect way to attain this if you're informed that you've received a legal excuse for leaving jury duty.\nIf you wish to provide an explanation not to go to a meeting, subsequently in place that which it's you're undertaking is right lying. Most excuses have a inclination to become foolish, and absurd arguments put forward by a guy to escape from an issue. Fantastic explanations are becoming tougher and much harder to produce. To produce your own excuse convincing, you've got to be sure looks real. Whatever the scenario, keep in mind that you wantn't make use of a simple medical excuse that is bogus , the ideal excuse is required by you.\nUnfortunately, you may some times end up seeking leave Abruptly also could well not need the ability to provide caution beforehand. A leave of lack is the opportunity become included in self-reflection and to come across remedy. There are lots of good reasons why you may wish to have a leave of absence.\nGet in touch with the courts once you are aware that you wont be in a position to wait and also write your own letter. Bear in mind, you require maintaining the correspondence brief, nor include of details that aren't mandatory. Prior to you write a formal explanation letter, it is imperative to grasp the thoughts about how best to compose an official explanation letter.\nBe sure to include things like the day. Letters are more flexible to write in your own pal or nearly any way you want or comparative loves to get. Full name The letters above are just a couple tests that you could make use of to produce your own personal. If you really don't understand just how to compose a formal letter afterward why don't you take a peek to fully grasp just how to compose an account. You'll find numerous explanations for.\nIt's important to make certain to make your letter concise, After writing an official explanation letter. Letters to team members are letters provided for people owned by a group in attempting to reach a common objective engaged. Private letters are somewhat more flexible to publish or relative or your own pal enjoys to purchase. The letter is necessary for your file. A formal excuse suggestion will be really just a document to spell the reason of absence to get a instant.\nThe excuse letters are simple to create. Letters of explanation will be employed by those who have been selected, but can't go to the jury obligation. If you are currently searching for creating excuse letter you are recommended to take advantage of explanation letter template which would be available on web. It is likely to also secure the absence justification letter for faculty free of charge.\nThis kind of correspondence could secure a specialist letterhead with a letter that the worker has overlooked work for a number of times. It will be sent to verify your wellbeing care offer and leave you for asking to go back when you. Letter of excuse would be typically a leave letter for being unable to show up at work notifying the grounds of 24, by way of an employee. Ahead of you compose the letter of explanation from jury duty, you need to read the notice that you have acquired from prosecution products and services. Regardless of what problem you are confronting, it is consistently advised that the jury obligation excuse letter is composed by you simply with a complete explanation of the hardship that you are currently facing in your life.\nYou are able to ask somebody else to proofread your letter from the contest. For absence letter is quite a exact simple letter that cites concerning the duration of depart and the rationale behind render leave. Composing an appropriate hint of absence tackled for your boss gets simpler in case you own a render letter template that you are well ready to refer to.\n2018-12-18Make convinced maybe not to incorporate any information when the letter is not resolved to some specific personal. From the above, you can see that creating an explanation letter is really a very straightforward job. At the event this is because valid, it's crucial to write a jury duty excuse letter.\nMind-mapping is still an quite great alternate to job management softwares that are classic to take care of your own initiatives. It's a wonderful means to set your thinking and brainstorm thoughts. It is a significant means to brainstorm ideas and to structure ideas. Additionally, it may earn a significant contribution to the success of an author by making it straightforward to organize schedule activities and ideas in the onset of the undertaking. It is a way to communicate ideas and your thinking. It transforms a list of dull info to a brilliant, unforgettable and highly structured diagram that works by means of the human mind's natural method of accomplishing things in line.\nMind-mapping is ideal for content production. It needs no preparation or competencies. About the other hand, it is intended to assist you structure disorderly thoughts.\nProject management can be considered an efficient means to manage change also it's crucial that project direction is evolving . \" I feel it's imperative that the various tools and processes for controlling assignments evolve too, while project direction is considered a great method to handle change. Project management to get a livelihood has existed since the 1960s applying mind mapping to project management is really a thought that was rather young.\nClick on the node Demonstration developing a map. Thoughts maps are great for summarizing info, like that found in sort of paperwork. It's important to not forget that head maps are not oceans, but are meant to be properly used with other software programs. While drawing on minds maps are powerful about planning for every region of your own life, producing 1 might take up a while to finish. A mind map starts off with a central idea. It's just actually really a graphical means to represent thoughts and concepts. At the close of the day, establishing a mind map in MindView and MindManager was an extremely similar adventure.\nMaps are designed to be shared along with other people. The Mind Map can be put to work with just as being a visual interface that has information or hyperlinks that are external. It consistently begins from issue or some problem that's positioned at the center. It might be dropped or expanded to rapidly increase or decrease the degree of depth. Head maps could be properly used from your plan and theory development, through the final of the undertaking. A mindmap may aid with comprehension since it communicates hierarchy and relationships, allowing students to come across the image. You may import mind maps that are current from other applications, for example as for instance Mindmanager and Freemind.\nThe app might not be at a position to restore data management entirely, however for a large creative team, it might be a ideal support. Presentation computer software could possibly be a great tool when used properly. The computer software might be properly utilized for mind mapping and assessing the info. You've must pick its role in your workflow, Ahead of you pick the software. Take note it is not required to use the software to have the ability to earn brain maps. Whatever the situation, in the event you want to cover mind mapping software that you should really go for Imindq. Some mind-mapping software lets you own meeting or presentations online, a superb means of meeting with teams or even other stakeholders that might be based in many destinations.\nThe pathways application allows audience and the presenter to concentrate in on special content though at the exact same time keeping an atmosphere of its general context. If this sound familiar, then you may possibly wish to consider a mind mapping tool. Not only does FileMenu tools furnish you having a broad array of tools you may put in on your right-click context menu, but but it will make it possible for one to select and pick the ones which you desire. To be broad, you ought to be analyzing tools. You can find lots of programs available in the marketplace for mind mapping.\nYou advertise around the web, or could make a advertisement in the newspaper. As an alternative, start including handling a customer who doesn't want to find an item, and after that create the scenarios more difficult since you collaborate with a situation. The font size tells you exactly just how big the character will probably soon be when it really is published in your newspaper. The point measurement of a ribbon identifies the height of the font from the bottom of a descender of a letter into an ascender of some letter's peak. You can stop by the modest close button, then in which point you will procure a rescue as option to the top left and then click where you'd want them to go, At any time you have pulled your samples. Your query goes on very top of this email plus ought to keep to the idea. A wood representation of the home that features an range of chances for renovations, or that opens to reveal your earnings pitch, by way of instance is guaranteed to grab awareness.\nBy explaining why you wish to do the job for the company or you realize you would certainly become a wonderful fit you could finish the pitch. Ideal pitch doesn't rely punctually. In the first point, they are given by you the pitch using. Try out benefit since feature as an alternative, as The Pitching Bible offers you whatever you want and you'll acquire more pitches. A elevator pitch is a brief introduction to your company. Because nothing is more challenging compared to talking in a prospect who's nolonger curious as you're doing each one the speaking. It's time for you to go over merchandise, packages, and services that you supply, and also find the one which satisfies their needs.\nYou are aware that it's challenging to stick out from the crowd if a construction business operates. For example, the business may have lied concerning your interest rate or the assortment of that time period you're effective at working with the home. As you might need to pitch something You shouldn't hesitate to earn sample adverts for services and products that are famous. In the event that you should be hightech company which is currently making money there is a chance for you to pitch yourself as a possible visitor. Make certain that every one of your company is so that you can Whether you possess do the job or your company there are some promotion and promoting methods that you could utilize to sell your fragrances.\nPretend you are your buyer. So don't be unwilling to allow the customer know the hard stuff. Customer service is something which can be adjusted by way of instruction and modeling. This happens when a customer is unsatisfied with their expert services.\nSales speeches are produced to market your merchandise or service. The recognition of the certain instrument is actually a superb case in point. Superb notion capability to talk, but no one knows that they exist. An quantity of flannel and you are likely to likely wash off any possibility which you will have experienced! Make sure to provide strong causes from your start. A tricky structure, especially if it seems expensive, won't be thrown out using the remaining of the unsolicited email.\nEven use much more stories sort others who used the item to make them learn it's an established product plus they aren't by yourself. The last stage to keep in your mind is touse stories. Be certain you realize who you're writing for, when writing your essay. For example, Teen Vogue's content usually do not consist of things like profanity or snarky comments. Sales speech themes can be an easy task to come up with in the event that you think about the items that you are bombarded with. The key thing here is always to discuss points and the issues which are very crucial to this prospect. Other folks feel it is unfair to anticipate a novel.\nIt's possible to substitute the term equity together with interest, ownership, or stake in every single sentence. There In the event that you fail to get their interest. The term denotes. Along with analogy is easily the very prevalent. Colors' analogy is one of the most widely used. Set a financing aim you may opt to spell out the requirements and invite everybody whether they may be in a position to donate financially.\nOften you'll need to tailor your bio degradable you're using it for, and that means that you change it to get particular demands and may have a bio for ordinary use. People are given a look in your enterprise and what it really is that you're doing by A bio. Your bio will not will need to get difficult. There is A bio just really a lineup paragraph which could behave as the default bio degradable. A bio works for you when it can help the like-minded people recognize that you're some one get in touch alongside you, keep in mind you, and they want to know. It may also be challenging to discover the things that are appropriate that you should be including within a bio degradable as an artist.\nOverwriting is a blunder written down authorities officer bio, produced by women and men. The first thing which you have to concentrate on is having. In order your bio degradable can not only become a description of the superhero. Its primary objective is always to give a much superior knowledge on your inspirations to the reader although Creating an artist bio needs to illustrate some of your achievements. A specialist bio is something that every entrepreneur requirements, however few have time and energy to write . The exact first step in making a professional bio is simply take under account the way it is going to be used. By now you get itwhen you get one, the connection.\nThe web isn't merely very likely to become a lot far more crowded. Yet another url you believe your most important hub since the link on your bio utilize the story high-lights to send out visitors or at the time you are able to have your website you would like. You get a connection on your account you're able to work with to some page you desire to direct individuals. that it is likely to observe how others have approached creating their bios our site will provide you a group of biographies of artists painters from history. A fantastic bio part includes unique skills, interests or hobbies which make you stand out. Start out with a Humble Boast Just like the first pic of one's Tinder profile, the very point of your bio should be your own best. Try to remember, it's one among the first factors a fresh or a feasible followers sees on your accounts.\nIt's time and energy to find the most out of one's presence to the planet's top expert networking site by making the bio that is best degradable to market your livelihood advantages. It's far superior to be more prepared and learn how to make the bulk of your energy and time. So, make sure you watermark work. The task which you do's quality.\nYou've must grasp until you begin to tackle your creating, the things that they truly have been hunting for out of you. Be certain you examine your producing together with wonderful good care. The author expresses admiration for the job that enter reading submissions (maybe not necessary in any way, but it is definitely a terrific gesture). You will have to create something that could possibly be reproduced like a catalog to market your 26, Since you will function as writer.\nIn the event you've ever wondered just how the prosperity of comprehension outside your organizationand inside of your expert networkhere's an case. Be sure to include things like any job or internships you'd to get a consumer that is well-known to give yourself an surplus authenticity boost. For business owners, their principal advertising procedure is through email, consequently receiving a reader is paramount. A big selection of UX skills terms you can used on your own profile. There continue to be many people who choose producing the work biography. The best biographies for Regulations of police jobs will be available in the marketplace. The DJ autobiography writing treatment is simple.\nThe way to educate a story by means of one's work is always to get started along with your audience. Whatever it's that you just do, I'm confident you have any intriguing stories about any of it. You should see a hyperlink button on top, once you've launched an Insta-gram story. In the end of the workshop, you'll have of writing you'll feel good about sharing which reflects you at a real style which will make distinctive individuals wish to get to understand that you a bit. Try to rememberyour Instagram advertising method is only somewhat of this puzzlea singular element on your new general advertising strategy.\nThe third part of your letter needs to discuss the hand off. The letters must show understanding of one's ailments. Small business letters have a specific format. A small business letter that is that is specialist often starts in the Microsoft case, as with all the date on top.\nA director is answerable to be given a variety of obligations. Employees subtract from their job for a variety of factors. Some employers need a formal resignation letter and a few don't. You may need to ask your company for a suggestion later on.\nAccording to previously, the correspondence features a very effortless letter and it has written like every additional letter. Thus, you have to compose an impressive letter. Another manner is by means of the mention letters plus so they're very significant.\nThe letters must be notarized and has got to include a contact phone number for the man. The advice is contained by it on the suitable upper corner which is followed by the details regarding the manager. Resignation letter is a way to embrace the brand new and permit proceed the suggestions and choices. Even the resignation letter is used once a worker wants to depart the enterprise. You can even pick the resignation letters predicated in your profession for example sample so forth and for teachers or nurses. You are all set to compose a formal resignation letter, At any time you have done that.\nPublished and your Resume should be typed. The Resume must be published in English. You might even submit an application to get work in the very same company . Well, on a confident note having a job resignation letter, you must make sure to part with your overall employer in relation to changing to another job. If you're quitting your occupation at an improved chance, then don't neglect to definitely reveal your gratitude. In your letter, say that the day you would like to leave work. Basically, in case you want to occupy job in the healthcare area, then you've got to generate employment of activities words than feminine or an individual who would be aims in the off ice at the sales industry.\nYou are a respected worker of your business along with a opportunity to prepare itself would be required by your office. A number of companies may decide to inspect in your own work record, and this indicates you might need to abandon. Other businesses might decide to examine on your own work listing, thus be an effort to render in the form of a record. Chancing up on a project to own a server or waitress does not appear very straightforward anymore, specifically collectively using the very simple actuality jobs are going to be with the range of people. Keep positive when you speak about your expertise. Existence is about letting the memories that are unwanted go and moving on.\nA good deal of folks have to be picky in regards to restricting the variety. It may not be the identical range you used when you do the preceding moment to keto. The number on the scale is merely several. It is extremely critical that they are done and contain the info.\nThe same is true for wheat, which explains why why incorporating low-fat goods is not smart. A number of them have a considerable lot speaking. Pretending they do not exist will not shift this. Actually, a lot of men and women find they can not eat them at all. Some reasons which may force you to step in their present places could possibly be maternity which tends to make it extremely hard to keep on working. One among the very first items she'd achieved throughout the last stages of her period was assumed to get started glooming tones and undercoating her canvases. The point is really to turn your diet plan and begin getting individual responsibility to the food selections, and that you won't will need to search for blame once you're cheat.\nThis really is exactly what folks find. Inches is exactly what you would like to lose. Font size 1-2 is just really a compromise that is superb. It is imperative that you abide by the format with the objective. It's likely to likewise seeletter templates. Top quality templates can also be readily available on the internet that can be tailored predicated upon your notice span. The resume wording has to become employer-oriented therefore he knows precisely what you might do to him.\nYour system knows how many carbs per day you are eating and drinking. Overloading the body with an abundance of wheat is just asking for trouble. Furthermore, the Atkins rates previously back up the notion that a diet that is low carb can quit doing work out. You have the ability to even seeemployee options. It's not straightforward remain and to resign with your business and supervisor. A number of two weeks would be a period. Whenever there was anything I can do in order to help make the transition kindly allow me to understand.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"From PIMCO: Brazil is often known for soccer, samba and pristine beaches. Among investors, something else also stands out about Brazil: double-digit interest rates that are significantly higher than its peers' (see chart). That may soon change.\nThe PIMCO Global Advantage Inflation-Linked Bond Exchange-Traded Fund (ILB) underperformed all other non-leveraged, non-inverse exchange traded products in the U.S. today on a percentage basis, producing a -3.18% one-day return and trailing the wider markets by a total of 3.17 percentage points.\nOne of the world's largest asset managers today issued a warning about stocks and other major asset classes, which it said remain vulnerable to even seemingly small negative surprises.\nIs PIMCO's Bill Gross Wrong Again?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"To be a first-choice provider of real estate services to Austinites, old and new. To have a career marked by helping his clients achieve their goals and enriching the lives of all the people he meets.\nI am committed to continually providing his clients with the right information, expert advice and comprehensive service in the buying, selling or leasing of real estate.\nLoyalty, Integrity, Competency, Education, and Service.\nI have experience in Residential, Multi-Family and Commercial sales and leases, and works throughout the Greater Austin area. I also currently own investment property and have renovated a home for sale (under contract in 6 days), giving me the insight of what it means to be an investor looking for Cash Flow and Return.\n\u2022 Member of Unity Church of the Hills.\nJB Goodwin REALTORS\u00ae has been in business in the Austin area since 1972. We work as a team to provide you with quality service. In fact, the JB Goodwin's Mission Statement is simply to \"Help People\". I feel the same way.\nI'd like to be the person that HELPS YOU. I can help with Buying \/ Selling \/ Investment Property \/ Leasing your rental property \/ or finding you a home to rent.\n\" I put off buying a property for a year because I couldn't find anyone who was that interested in selling me one. Leon made sure I was getting e-mail updates about new properties, worked around my schedule to show me places, and even helped me show my existing property to potential renters. I wouldn't ever consider buying or selling with anyone else after working with Leon. He's also LGBT friendly and takes safety into consideration when recommending properties. I couldn't have asked for anything more. \" \u2013 Justin P.\n\" When I made the decision to sell my house in Austin, I contacted Leon Connelly based on a recommendation from people who had bought a house with his help. I contacted him and was amazed at how quickly he responded. He is very organized and pays attention to details. Leon led me step by step through the entire process. Leon is a great person to work with. He is very congenial and considerate of his clients. His knowledge of the real estate market needs is detailed and up to date. He was a joy to work with. I highly recommend him. \" \u2013 Judy J.\n\" I can't express enough good things about Leon. He was always open to making himself available to house hunt. Very easy to speak to and provided excellent service. I would highly recommend. He both found us a new home and to sell our other one. \" \u2013 Dustin C.\n\" I worked with Leon Connelly while I was looking for a rental home a few months back. He was easily one of the best realtors I've worked with. He was very accommodating when working around my schedule, knowledgeable about the areas I was looking in, and I never felt pressured during the process. \" \u2013 Bryan W.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are joined this month by Phil Hurst, Publisher at the Royal Society. The Royal Society is the world's oldest independent scientific academy and has made various innovations in the way it publishes and distributes scientific content. Phil has been at the centre of that process and we caught up with Phil to discuss why peer review matters to him.\n\"I subscribe to the motto of the Royal Society 'Nullius in verba' which is taken to mean 'take nobody's word for it'. It is an expression of the determination to verify all statements by an appeal to facts determined by experiment, rather than merely relying on authority.\nScience has a transformative effect on society, for example by improving human health, mitigating climate change, securing food and energy supply, or improving global communication. However, when new claims are made, they require full scrutiny and validation.\nHow is peer review regarded in the research community?\n\"Survey after survey demonstrates that researchers really value peer review \u2013 some argue it is the most important (or even only) worthwhile thing academic journals offer. The fact that millions of researchers do it for free to help the scientific enterprise is testament to this. The highest rated reasons for reviewing a paper are social ('playing my part as a member of the academic community' and 'reciprocating the benefit gained when others review my papers').\nWhat is the Royal Society doing to challenge the changing landscape in peer review and scholarly communications?\n\"In order to improve the culture of science, the Royal Society's journals are focusing on recognition and transparency. This has to be built on a firm foundation. The Society is a strong advocate for the ORCID initiative which allows each researcher to have their own unique record and identifier. The beauty of ORCID is that it embraces technology to make it easy for researchers to collect their publication and peer review activity in one place. In some ways it can be regarded a 'recognition portal' which can be used when applying for grants or tenure. In consort with several publishers, the Society was the first to mandate all submitting authors to provide an ORCID iD in an effort to increase adoption.\nSurveys of our referees have shown that they value recognition at both the general and journal level. We were an early integrator with the Publons referee recognition platform. Integration makes it simple for the referee to get recognition when reviewing for Royal Society journals. At first, researchers were a little suspicious of doing this \u2013 but they now value it (as evidenced by improved rates of accepting invitations to review). Publons also makes it easy for the researcher to add this recognition to their ORCID record. At the journal level, we have reinstated a tradition practice of listing the names of reviewers. Such articles are assigned a DOI which allow them to be a citable research output.\nWhen the journal, Royal Society Open Science was launched we decided to incorporate optional open peer review. Authors are able to opt to publish referee reports, responses to reviewer comments and decision letters alongside their published paper. Separately, referees have the option to sign their reviews or remain anonymous. About two-thirds of authors opt for open peer review and 50% of these are signed by referees. For a journal publishing in all areas of science (with cultural differences in practice between them), I am encouraged by this uptake. We have since extended open peer review to two further journals.\nHow can researchers get more involved in ensuring the integrity of their research?\n\"The manifesto for reproducible science provides a useful roadmap to help researchers ensure the integrity of their research. It outlines a number of problems such as small sample sizes, data dredging (p-hacking) and conflicts of interest. A key remedy is pre-registration. This involves peer review of the study methodology to ensure it is fit for purpose before the experiment is conducted. If the researcher sticks to this methodology, the results are published even if the results are negative. Along with an increasing number of journals, Royal Society Open Science is offering this publication route via Registered Reports.\nTo find out what other key leaders in the peer review world think about peer review, check out the Peer review \u2013 reviewed series.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm tempted to point out that no-one's forcing teams to enter into these contracts, but then I'm not completely sure what the rules are about draft picks going unsigned. In any case, the negotiated salary cap rules determine the total pot available to pay rookies, right? Either way, it's just more evidence in favor of the old truism that American sports are more socialist than their capitalist European counterparts: try this or this for some of the arguments.\nIt's staggering how exactly the analogy holds, precisely opposite to the stereotypes of American and European rules for resource allocation. American sports are organized to give every team a fighting chance of winning within a few years, with the help of active intervention in the form the draft and salary caps and the guarantee that a terrible season is not punished the following year, while European sports are dog-eat-dog, pure capitalism. Maybe it's because of cross-country competition in Europe: it would be hard for Spain, say, to use American-sport policies to level the playing field without players jumping ship to Italy or England and without Spanish teams being destroyed on the field by the elite of other countries.\nVia Arts & Letters Daily (again) comes a wonderful (and long) article by Alan Wolfe for the New Republic: \"Hedonic Man, The new economics and the pursuit of happiness.\" There's far, far too much for me to discuss in suitable depth here, but, beyond recommending the article, let me pick out a couple of choice bits.\nThe presentation of economics as a discipline of study certainly does make it seem like we've magically cracked the nut of what is the 'best' way to organize a society, and, as I've argued before, that's not just dangerous, it's a misrepresentation of what good economic science is capable of. One reason why it's a misrepresentation is the measurement issue at work again: to say 'best', we need a metric, and to do that we need to ask what people want, to speculate on their motivations and desires. Of course, as misrepresentations go, it's a tempting one, because from day one of a Principles of Economics course it's made again and again and again.\nWolfe's article is built around the review of two books: Predictably Irrational by Dan Ariely (which I've talked about before, much to the author's chagrin, so I won't return to it now) and Happiness: A Revolution in Economics by Bruno Frey. 'Happiness', as discussed by Wolfe, is concerned with exactly that big question in normative social science, which is: what exactly constitutes a 'good' outcome? It's exactly that question that's the dangerous misunderstanding within economics, the dangerous belief that we know what's 'good'.\nAgain, though, I offer this as the reason why the 'revolution' has 'failed': we simply can never say what motivates people, whether a person is \"rational\" or not. 'Neoclassical economics' does not restrict the range of assumptions one can make about human behavior. It is therefore completely resilient to any evidence on how people act in a given situation. That's not a defense of the approach, it's a fact. It's distressing, because my prejudice is definitely to agree that economics is 'complacent' about the superiority of markets (again, because we ignore the variety of normative metrics of comparison), and that there's too much arcana.\nWe need true normative debate. Look again at what Wolfe says about economics: \"it remains a vulnerable approach, stuck in unrealistic assumptions about human behavior and all too complacent about the beneficial equilibria established by markets\". It's a mistake to conflate this problem \u2013 the arrogant assumption that we know what's 'best' \u2013 with the neoclassical assumptions on human behavior, because, as I've argued repeatedly, the method of scientific economics doesn't actually assume anything about human behavior, as evidenced by the fact that the 'behavioral revolution' is comfortably within the confines of 'neoclassical economics'.\nThe real revolution would be quieter, and would say something much more familiar: keep your science separate from your opinion.\nVia the wondrous fark.com comes 'How Rich People Spend Their Time' from the Washington Post \u2013 it's about an article in Science written by a battery of psychologist\/economist types, including Daniel Kahneman. Very relevant to the question of what motivates people; my first instinct was to assume that it might reveal what people with the time to do what they want do with their time, if you see what I mean, and while the actual intention of the article is somewhat different it's still full of fun.\nPerhaps some intriguing fact about human nature; perhaps not. The article goes on to talk about some well-known results in the burgeoning 'happiness' literature, like the importance of relative rather than absolute income, and adaption to circumstances. I think a great article about this stuff, for the terminally interested, is Richard Layard's 'Happiness and Public Policy'.\nLet me get this straight: richer people work more and buy more stuff, and poor schmucks watch a lot of TV? Stop those presses. The point of the article is well-made (from the the title, 'would you be happier if you were richer', right on down), and that is to say that people with higher incomes aren't necessarily engaging in relatively more 'fun'; however, there are a bunch of unasked questions. Does 'TV'='fun'? Is this really evidence that the rich are wasting their time, or are there other reasons why they endure work to get money?\nThe happiness literature is desperate to find an answer to the question of whether money buys happiness; an eerie similarity to the oft-(mis?)perceived economists' equation of money with happiness when modeling people, and surely as deserving of the same retort: we know people care about more than money. The obvious question is, well, obvious. What does motivate people?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In a tropical and developing country like India, inflammatory dermatoses form the bulk of diseases encountered in a busy outpatient practice with fewer people seeking advice for indolent 'lumps' and 'bumps'. However, skin tumours are a challenging group of conditions that can be categorised into those that are commonly seen and easily recognised based on the characteristic site of presentation, size, colour, distribution and symptoms, the rarer ones and those that mimic other disorders and pose a diagnostic challenge. The latter group are diagnosed chiefly by histopathology, immunohistochemistry or other sophisticated diagnostic techniques. Skin tumours also form an important component of several genodermatoses.\nSkin tumours develop as a result of proliferation of a single or multiple components of the skin.\nThey range from benign lesions that merely cause cosmetic concern to premalignant lesions and aggressive tumours. The vast majority of skin tumours are benign. Morphologically, they manifest as smooth papules, nodules, keratotic or cystic lesions that grow slowly. Multiplicity of lesions is a reliable clinical clue to the benign nature of the condition. Malignant tumours are usually solitary, irregular, rapidly growing plaques or nodules that may ulcerate. They may arise de novo but some may arise from a pre-existing benign tumour. Some tumours may metastasize.\nManagement is usually for cosmetic reasons and more often than not, diagnosis is made on excision biopsy, especially when patients present to a surgeon for a nodule. Some benign tumours may recur following incomplete excision. It is likely that many nodules that are excised are not examined histopathologically and those that are may be reported as benign skin tumours without further characterisation.\nThe diagnosis of a skin tumour is quite often made histopathologically but some tumours can be recognised with some certainty on clinical grounds. Multiplicity of lesions makes the diagnosis simpler than when there is a solitary papule or nodule.\nSkin tumours are classified based on their primary site of origin [Table 1].\nTumour collision i.e. the juxtaposition of two originally separate and distinct neoplasms. The two lesions are sharply demarcated from each other: e.g. nevus and BCC, nevus and seborrheic keratosis, SCC and melanoma.\nMelanocyte colonisation-This is the colonisation of epithelial neoplasms by non-neoplastic dendritic melanocytes. The melanocytes in such tumours are highly pigmented, have dendrites and lack cytologic atypia unlike the atypical melanocytes with no dendritic configuration and minimal pigmentation, as seen in combined tumours e.g. pigmented BCC or SCC, melanocytic matricoma, melanotrichoblastoma or pigmented eccrine porocarcinoma.\nAntigen transfer, in which artefactual transfer of antigen from one cell line to the other occurs due to antigenic diffusion, leading to non-specific staining of the tumour cells. This differs from combined tumours in which 'appropriate' staining with regular distribution pattern is seen in the tumour cells.\nImmunohistochemistry is an adjunct to conventional dermatopathology for assessing the tissue of origin or direction of differentiation of cells.  Immunohistochemistry involves application of antibodies directed towards the desired antigen followed by incubation with a peroxidase- labelled secondary antibody that is directed against the first antibody. Several markers are now available commercially to identify the lineage of cells and also differentiate primary from metastatic tumours, e.g. cytokeratins (low and high molecular weight) for various keratinocytic neoplasms, cytokeratins for tumours with follicular differentiation, simple epithelial keratins and carcinoembryonic antigen for tumours with ductal differentiation, epithelial membrane antigen for tumours with sebaceous differentiation. The antibodies directed against tissue antigen vary greatly in their specificity and binding affinity to the tissue. Hence, immunohistochemistry is not a 'magic bullet' and its result is not decisive at all times. Moreover, financial constraints also limit the use of this technique in a majority of cases.\nMolecular techniques are also being developed to improve diagnostic precision of skin tumours.  Some of these techniques include southern blot, interphase fluorescence in situ hybridisation (iFISH), comparative genomic hybridisation (CGH), PCR, etc. They are being used for melanocytic neoplasms, soft tissue tumours especially sarcomas like dermatofibrosarcoma protuberance and basal cell carcinoma.\nNonetheless, recognition of the histomorphology with standard haematoxylin-eosin stains remains the most reliable way to diagnose these tumours.\nThese are simple, yet novel, in vivo non-invasive techniques for the early diagnosis and appropriate treatment of skin tumours. Some techniques can study the skin at nearly histological resolution. These techniques help in selecting lesions that require a biopsy, in determining appropriate therapeutic modalities, verifying treatment efficacy and deciding about appropriate surgical margins.\nIt is a simple and inexpensive technique to visualise certain morphological features in the skin lesions that are not visible to the naked eye.  The lesion is covered with mineral oil, alcohol or water to eliminate surface reflection, making the cornified layer translucent to better visualise the epidermal pigment, dermo-epidermal junction and superficial dermis. It is then visualised with a hand lens, hand-held scope called the dermascope, stereomicroscope or a digital imaging system at magnifications varying from 6\u00d7 to 100\u00d7. New hand-held dermascopes with polarised light are now available by which use of surface fluid becomes unnecessary.\nSkin tumours represent a challenging group of dermatoses. A thorough clinical examination supplemented by histopathologic correlation is required to make a diagnosis in a majority of cases.\n1. Rodriguez J, Nonaka D, Kuhn E, Reichel M, Rosai J. Combined high-grade basal cell carcinoma and malignant melanoma of the skin (\"Malignant basomelanocytic tumor\"). Report of two cases and review of literature. Am J Dermatopathol 2005;27:314-8.\n2. Schwartz RA, Torre D. The Muir-Torre syndrome: A 25-year retrospect. J Am Acad Dermatol 1995;33:90-104.\n3. Rasmussen JE. A syndrome of trichoepithelioma, milia and cylindromas. Arch Dermatol 1975;111:610-4.\n4. Michaelsson G, Olsson E, Westermark P. The Rombo syndrome: A familial disorder with atrophoderma vermiculata, milia, hypotrichosis, trichoepithelioma, basal cell carcinoma and peripheral vasodilatation with cyanosis. Acta Derm Venereol (Stockh) 1981;61:497-503.\n5. Brownstein MH, Mehregan AH, Bikowski JB, Lupulescu A, Patterson JC. The dermatopathology of Cowden's syndrome. Br J Dermatol 1979;100:667-73.\n6. Khoo SK, Bradiey M, Wong FK, Hedblad MA, Nordenskj\u00f6ld M, Teh BT. Birt-Hogg-Dube syndrome: Mapping of a novel hereditary neoplasia gene to chromosome 17p12-q11.2. Oncogene 2001;20:5239-42.\n7. Hardy RD, Duvic M, Bleyer WA. The sign of Leser-Trelat. Med Pediatr Oncol 1997;28:234-7.\n8. Wallace ML, Smoller BR, Immunohistochemistry in diagnostic dermatopathology. J Am Acad Dermatol 1996;34:163-83; quiz 184-6.\n9. Min KW. Stromal elements for tumor diagnosis: A brief review of diagnostic electron microscopic features. Ultrastruct Pathol 2005;29:305-18.\n10. Da Forno PD, Saldanha GS. Molecular aspects of melanoma Clin Lab Med 2011;31:331-43.\n11. Argenziano G, Soyer HP, Chimenti S, Talamini R, Corona R, Sera F, et al. Dermoscopy of pigmented skin lesions: Results of a consensus meeting via the internet. J Am Acad Dermatol 2003;48:679-93.\n12. Harland CC, Bamber JC, Gusterson BA, Mortimer PS. High frequency, high resolution B-scan ultrasound in the assessment of the tumors. Br J Dermatol 1993;128:525-32.\n13. Ruocco V, Argenziano G, Pellacani G, Seidenari S. Noninvasive imaging of skin tumors. Dermatol Surg 2004;30:301-10.\n14. Langley RG, Rajadhyaksha M, Dwyer PJ, Sober AJ, Flotte TJ, Anderson RR. Confocal scanning laser microscopy of benign and malignant melanocytic skin lesions in vivo. J Am Acad Dermatol 2001;45:365-76.\n15. Gerger A, Horn M, Koller S, Weger W, Massone C, Leinweber B, et al. Confocal examination of untreated fresh specimens from basal cell carcinoma. Implications for microscopically guided surgery. Arch Dermatol 2005;141:1269-74.\n16. Sheridan AT, Dawber RP. Curettage, electrosurgery and skin cancer. Australas J Dermatol 2000;41:19-30.\n17. Garcia C, Holman J, Poletti E. Mohs surgery: Commentaries and controversies. Int J Dermatol 2005;44:893-905.\n19. Szeimies RM, Morton CA, Sidoroff A, Braathen LR. Photodynamic therapy for non-melanoma skin cancer. Acta Derm Venereol 2005;85:483-90.\n20. LeBoit PE, Burg G, Weedon D, Sarasin A. Skin tumors: Pathology and Genetics. World Health Organization of tumours. Lyon: International Agency for Research on Cancer Press; 2006.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hello everybody, my first post, sorry for bad gramar and other mistakes, english is not my native language.\nI have tons of old miniDV footage of my family, waiting in the queue to edit. This is in AVI format, imported from MiniDV tapes.\nI'm using latest plug-in for Vegas Pro 14, looking to buy the same plugin for Adobe Premiere.\nIf I have few clips recorded in the same dark room, should I create one good profile from the best clip, saves it and using in the rest of similar clips?\nIs there any database of neat profiles for specific cameras?\nI'm looking for profiles for: Panasonic GH5, Mavic Air, Panasonic HDC-SD60 and this old MiniDV Panasonic camera.\nGreeting and thanks for reply.\nIf several clips have the same noise (which usually happens when you shoot them using the same camera working in the same mode in the same lighting conditions), then it is fine to use one profile to process them all.\nThere is no database of profiles for video cameras at this time. Building your own custom profiles for your specific clips usually delivers better results with video than using someone else's profiles build for video undergoing possibly another workflow.\nI have made a nice profile for this footage. Looks amazing after few tweeks.\nIf I want to: stalibize, denoise and apply LUTS what I should do first?\nFor ~10USD I can buy pack of noise preset for Mavic Air (2,7 and 4K resolution, different ISO). If I print recommended pattern and shoot 1-2 second clip in different ISO and resolution mode, the result should be similar?\n> If I want to: stalibize, denoise and apply LUTS what I should do first?\nto see what delivers better overall results.\nLUTS can go after all that.\n> If I print recommended pattern and shoot 1-2 second clip in different ISO and resolution mode, the result should be similar?\nI will print this pattern in color and make some tests. I will be easy with GH5 becouse of manual focus. But how to shoot clips with Mavic Air, which has automatic focus? How close to camera should I place this pattern? Prepare one profile for each part od the day (mornig, dusk, late evening, night with street lapms) with each step of ISO?\nI would start with a simpler and easier method of preparing noise profiles for your video: try to just build a fresh new profile for every clip you work with. The benefits are: (1) you will learn how to do that well in order to achieve the best results, (2) you will avoid possible mismatch of noise properties when using profiles built using another clip, (3) you will usually save time on choosing and loading a pre-built profile (Auto Profile works faster and even manually choosing an area is often faster than loading a profile from disk), (4) you will get profiles that are most suitable for individual clips.\nOnce you master such per-clip profiling, you will get a better idea of whether it is worth it to develop a reusable set of profiles for your cameras considering the questions about the procedures that you raised.\nSo, my advise is to start with Auto Profile on a per-clip basis.\n> But how to shoot clips with Mavic Air, which has automatic focus?\nIt must have some speed limitations, so if you move the target away from the camera, then you will likely get a few frames that are somewhat out of focus.\n> How close to camera should I place this pattern?\nThe instruction suggests to make the target fill the whole frame.\n> Prepare one profile for each part od the day (mornig, dusk, late evening, night with street lapms) with each step of ISO?\nThere are many parameters that may affect the properties of noise generated by the camera. Some of the most important parameters are listed in the Device Noise Profiles section of the user guide. Please see it first. Then you can decide if you want to include any additional parameters such as lighting differences due to time of the day. It may be too much work with too little practical difference. Try to build two profiles for two different shots and compare the noise levels using Profile Viewer for example. The same applies to different lens.\nThank you for answer! I will read manual. Have a nice day!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzuznv b/data_all_eng_slimpj/shuffled/split2/finalzzuznv
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+{"text":"Build Your Career at Vericon.\nBe part of a company you can be proud of!\nWe welcome your interest in Vericon Construction Company, where our greatest asset is our people: talented, responsible and collaborative professionals who are the driving force behind our remarkable growth and client loyalty. There are many opportunities here for individuals committed to teamwork, delivering quality results, and acting with integrity.\nExperienced candidates can put their skills and experience to optimal use and still be challenged to learn, develop new skills, and advance.\nCandidates fresh out of school or relatively new to the business will benefit from training, mentoring by seasoned professionals, opportunities to explore the many aspects of the building industry, and the ability to grow and move within the company.\nInternship Program for individuals still in school \u2013 provides a realistic perspective on working in the construction industry and opportunities to put classroom learning into practice.\nTo view a list of our current opportunities at Vericon Construction Company, click the button below.\nVericon Construction Company provides equal employment opportunity to everyone, without regard to race, color, religion, gender, age, national origin, veteran status, marital status, sexual orientation, disability, or any other status prohibited by law.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Create a dedicated vanity space in your bedroom with this attractive dressing table, mirror and stool from the Julia collection.\nThe handy drawer will house all your jewellery and cosmetics whilst the three gorgeous finishes available offer stylish appeal for your home. Finished off with a sleek rectangular metal handle.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The last Opinium Observer poll of January has caused a bit of a stir with a shift from a 3 point Labour lead in the middle of the month to a 7 point Tory lead at the end.\nNormally we'd be sceptical of such a large shift and, as always, it's best to take an overall look at what all polls are showing over time to get an accurate picture of how public opinion is changing.\nHowever, in this instance we do have some evidence that this change in opinion is real rather than the result of sample variation or other statistical noise.\nAt the moment Opinium runs a voting intention poll for the Observer once a fortnight. Usually this is once a month but, with the political situation being what it is, it feels necessary to run more often. As well as this though, we attach the usual voting intention questions to many other surveys we run for private clients so that we can look at what Conservative, Labour, Leave or Remain voters think about whatever we're asking questions about.\nIn this case, the poll we ran for the Peoples' Vote campaign from 23rd to 25th January also ran these questions and came out with a 4 point Conservative lead (40% to 36% for Labour, Lib Dems and UKIP largely unchanged).\nCan you spot the difference? If anything, this shows that the uplift in the Conservative vote share actually occurred slightly earlier than a smooth trendline would suggest while the decline in Labour's share is constant.\nOther figures show consistency between the three polls. Theresa May and Jeremy Corbyn's handling of Brexit show a slight improvement and decline respectively, albeit statistically insignificant. The two leaders' overall approval is also in line with a narrow shift against Corbyn, as is Theresa May's lead on the two-way \"best prime minister\" question.\nSo while a 10 point swing in a fortnight is quite a large change, the fact that it is backed up by more than one poll suggests that a move away from Labour is real.\nOf course, the uplift in the Conservative vote share is in part down to the veneer of unity that the Brady amendment and some positive headlines created. It remains to be seen whether this lead will survive the next inevitable collision between Brexit strategy and reality.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On seven occasions last weekend, a phone call was made to the police about a man disturbing the peace at Orlando-area Starbucks Coffee, Surfanista Cafe and Sassy Granny's Smoothies locations.\nThe man allegedly threatened to destroy everyone with his very own army of turtles, TV station WKMG reported.\nAccording to Orlando Sentinel, the 61-year-old man was taken to the Brevard County Jail and will have to see a judge about multiple charges including including misusing 911, breaching the peace and resisting an officer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ruth Geneva Doxzon Lingenfelder, 87, of Millsboro, Del., passed away peacefully on Saturday, Feb. 25, 2017. She was born in Baltimore, Md., on Nov. 14, 1929, daughter of the late Roy and Nettie Doxzon.\nIn addition to her parents, Lingenfelder was also preceded in death by a daughter, Sharon (Lingenfelder) Trimble. She is survived by her husband, Christopher Lingenfelder Sr.; five children, Denny and his wife, Linda (Hess) Lingenfelder of Fairfield, Pa., Bruce and his wife, Christine (Peterson) Lingenfelder of Greencastle, Pa., Phil and his wife, Kathleen (Brown) Lingenfelder of Berkeley Springs, W.Va., Christopher and his wife, Susan Lingenfelder, of Georgetown, Del., and John Lingenfelder and his partner, Brenda Carney, of Millsboro, Del.; a daughter-in-law, Norma Barletta of Ocala, Fla.; five grandchildren; and 10 great-grandchildren.\nFuneral services were to be private. In lieu of flowers, memorial donations may be made to Delaware Hospice; 100 Patriots Way; Milford, DE 19963. Condolences may be sent online at www.watsonfh.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvyuo b/data_all_eng_slimpj/shuffled/split2/finalzzvyuo
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+{"text":"Download Indiana State University Whitepaper!\nDownload the PDF with FULL information about the Indiana State University from opened government trustful sources - right now!\nIndiana State University offers 433-degree programs and 150 via distance education. Student-to-faculty ratio is 20:1. 1 high safety campus. Acceptance Rate is 83%. The most popular test is SAT - 76% of students submitted it in comparison to 43% of students who submitted ACT. 46.4% are women, and 53.6% are men. 7% percents of international students. Indiana State University is a member of 6 athletic associations. The average annual total cost of attendance at Indiana State University is 11920$, that is at the level of national average. Total price of living is between $12035 and $20933 per year. 41% of full-time Indiana State University students completed their education in 150 percent of time: graduation rate is at the level of national average. Average salary after attending is $36,000, so if you studied 2 years, you'll get your funds back in less than 2 years.\nThe Indiana State University is a public and non-academic-year institution, so it's net price represents an average of all programs and includes only undergraduates who first enrolled in the fall term and pay in-state tuition and receive Title IV aid.\nDepending on the federal state, or institutional grant and available, Indiana State University students in your income bracket may pay more of less than the overall average costs.\n7339 of Indiana State University undergraduate students were awarded with some type of financial aid, it's 67 percents of all undergraduate students. This makes Indiana State University number 10 in the amount of financial aid awarded to students among all educational institutions in Indiana.\nStudents at Indiana State University awarded 5 types of loans or aids, the biggest percent of students received a any financial aid.\nHere are the learning opportunities available in Indiana State University.\nIndiana State University graduation rate is below the national average.\n41% of full-time students enrolled for the first time to Indiana State University completed the education program and got a degree in 150 percent of the expected time of completion.\nIndiana State University acceptance rate is above national average.\nAverage acceptance rate for all institutions in state in the same net price range is 66%, so Indiana State University acceptance rate value is above state net price average for 17%.\nThe Indiana State University full-time retention rate is 64% and part-time retention rate is 75%.\nThe student-to-faculty ratio at Indiana State University is 20:1, that means that for every 20 students the institution has one professor, lecturer or specialist with a degree in education. Lower student-to-faculty ratio is better - it means that professors can dedicate more time and attention to each student.\nIn Indiana State University the biggest percent of students are enrolled in on-campus courses.\nHere are the students enrolled exclusively in distance education courses at Indiana State University and grouped by their location. Distance education courses are the most popular for students located in Indiana.\nThe campus located by 210 N 7Th St address in Terre Haute, Indiana.\nMain Campus in comparison of all branches in the US. It is calculated as the average percent of crimes yearly by five years of data for every campus, based on an overall number of students studying there. The branch is a part of Indiana State University with 0.29% of maximum crimes a year. The most often crimes detected on Main Campus are burglary, aggravated assault, and motor vehicle theft.\nIndiana State University offers 433 programs in total: 216 are for Bachelor's degree, 129 - for Master's degree, 24 - for less than 1-year certificate programs, 24 - for Postbaccalaureate certificate, 21 - for Doctor's degree-research\/scholarship programs, 12 - for Post-master's certificate, and 1 are Associate's degree.\nThis is the middle number of programs in comparison with other 135 universities in Indiana state. Indiana State University is one of the few institutions in Indiana that offers an Associate's degree program.\nThere are 150 distance learning programs that are provided in the Indiana State University: 54 are for Master's degree, 45 - for Bachelor's degree, 21 - for Postbaccalaureate certificate, 9 - for less than 1-year certificate programs, 9 - for Post-master's certificate and 6 are Doctor's degree-research\/scholarship programs.\nIndiana State University provides coursework for undergraduate students, graduate students and students with doctors professional practice.\nIn 2014-2015 academic year estimated full-time equivalent undergraduate enrollment was 10577 students with 317317 \u0441redit hours for 12-month instructional activity period. The enrollment for graduates was 1731 with 41543 credit hours. The doctors professional practice enrollment was 99 students.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"open, one more than the largest possible value for a file descriptor.\nSVr4, 4.4BSD (the getdtablesize() function first appeared  in  4.2BSD).\nemploy sysconf(_SC_OPEN_MAX) instead of this call.\nOPEN_MAX (set to 256 since Linux 0.98.4).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This synopsis of Gift of the Magi literary terms includes the terms referenced directly from the short story. This well known story, created by O. Henry (William Sydney Porter) and first published in 1906, shows the selfishness or foolishness of a young couple at Christmas time. Eager to please the other with a spectacular gift, they sell their most prized possessions to have enough money to purchase the other a gift. She sells her hair to buy him a watch fob; he sells his watch to buy her tortoise shell hair combs.\nNarrator: The story is told in the third person point of view through a mostly omniscient narrator.\nSetting: The story takes place in New York City. While the setting is not exactly stated, the author infers this setting by mentioning Coney Island, a well known area landmark.\nTheme: The central idea in this story is the love between Delia and James and the sacrifices that they make to please the other.\nDelia Young: A young woman married to James. She sells her long hair to buy her husband a watch fob.\nJames Dillingham Young: A young man (twenty-two years old) married to Delia. He sells his watch for the money to buy his wife hair combs.\nClimax: The most exciting part of the story is when the characters exchange gifts. Everything in the story leads up to that particular moment.\nResolution: The story wraps up quickly after the climax when the characters realize what they have sacrificed for each other.\nIrony: One type of irony can be seen in this story, situational irony. By selling their most prized possessions to buy something for the other, they wind up selling that which they need to use the gifts the other has given.\nBiblical Imagery: Three wise men, Queen of Sheba, King Solomon. All of these characters were financially rich in the Bible, which directly contrasts that of Delia and Jim's financial poverty. The juxtaposition most likely is meant to show that the love Delia and Jim have for each other defines the real meaning of rich.\nColor Imagery: The grey and colorless setting directly contrasts the Christmas time of year when everything should be sparkling and brightly colored in holiday decorations.\nMany students have a difficult time analyzing literature, especially short pieces. These Gift of the Magi literary terms are meant to help readers delve deeper into the story to gain alternate perspectives of the events. A careful reading of any story will yield similar results. Literary analysis remains an important part of the curriculum for a variety of reasons. For one, it helps us understand our world and the world around us, especially different places and times. Yet many stories, such as the Gift of the Magi, endure because of their timelessness.\n\"Gift of the Magi Literary Terms.\" YourDictionary, n.d. Web. 25 April 2019. <https:\/\/education.yourdictionary.com\/for-teachers\/gift-of-the-magi-literary-terms.html>.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"THE BEADY EYE SAYS. ITS TIME TO CURTAIL ON LINE\/REMOTE GAMBLING.\nEvery adult is responsible for his or hers choices. Yes you can change channel or turn you mobile phone off but gambling is now a large revenue source for many governments, due to its ease of implementation, popular appeal and high real tax rate it can bear (up to around 40%).\nIt is often promoted by spending on \"good causes\" designated as \"additional\" to existing government activity.\nHowever gambling taxes are effectively hypothecated (specifically designated rather than fed into the general tax pool) they are often, diverted into education, health, and social and economic development, potentially substituting for taxation raised elsewhere in the economy.\nBut this is not the problem, it is the potential to carry a 24-hour bookmakers shop around in our pocket, all day, every day \u2013 simply by downloading one of the many gambling apps.\nThe advent of online gambling, in combination with the development smartphones is making \"responsible gambling\" a joke.\nTo market it as harmless fun and entertainment, when it's been totally designed to addict an individual and take all of their money, is obscene.\nGambling companies send 'free bet' incentives straight to your phone \u2013particularly if they see that you haven't been using the app for a while.\nTechnology advances are making a joke of gambling laws. Remote gambling circumvents all laws and needs to be regulated like an other dangerous commodity or activity \u2013 alcohol, cigarettes, gun licences, etc.\nFor every person with a gambling problem, there are estimated to be a further 5-10 people affected,.\nPathological gambling, this form of a gambling addiction persists even in the face of appalling damages to the individual's personal, professional or family life, and is classified in the field of mental health as a disorder.\nThe undesirable social and financial consequence of excessive and unbridled gambling is, of course, well-known however gambling has become widely viewed as a socially acceptable form of recreation.\nIreland has the second highest spend, per person, on gambling \u2013 in the world.\nUnlucky Brits lost a record \u00a312.6 billion on gambling last year \u2013 that's up \u00a31.4bn on the year before. Online gambling accounted for almost a third of latest gambling losses, with punters losing \u00a33.6 billion to internet betting and online casino and bingo sites, almost \u00a3300 per person.\nThe UK gambling industry is worth \u00a37.1 billion, in the form of point of consumption tax, not including the National Lottery.\nAs a general rule, the minimum legal age for gambling in the UK is 18 years old. This applies to adult gaming centers, betting shops, bingo halls, casinos, racetracks and online gambling. The exceptions to this are the National Lottery, lotteries and football pools \u2013 you're allowed to take part in these from the age of 16 as well as some non-commercial gambling, or low stakes and prizes gambling.\nAnswer me this existing laws were not drafted with internet online gambling in mind. How many online of remote gambling activities verify the punters age. Online gaming is the industry's fastest-growing sector, and accounted for 11% of the $385bn of gambling profits posted in 2016.\nThere is a limit to what regulations can do.\nMost of what we know [about gambling harm] is through research and evidence heavily influenced by industry.\nIndustry funds all the research in the country through Gamble Aware. (Gamble Aware is the charity formerly known as the Responsible Gambling Trust, charged by the department for culture, media and sport with commissioning research into, and treatment of, gambling harm.) Five of Gamble Aware's 13 trustees have direct links to the gambling industry.\nIn the financial year 2015-16, the UK government raised \u00a37.6m in contributions from the gambling industry. Of that it spent \u00a3919,654 on research. Meanwhile \u00a33,788,698 was given to the gambling harm treatment charity Gamcare. Of Gamcare's 11 trustees, six have direct connections to the gambling industry.\nThe amount of gambling adverts on TV, and social media platforms are spiraling out of control.\nI am sick to the back teeth of seeing advert after advert after advert for online casinos \"spend \u00a310 to get \u00a330 to bet with\" at nearly every ad break or bet in play (betting on football matches) that now pollutes nearly ever sport event shown on TV. We need to radical change the current gambling advertisements on the grounds that they contravene principles of consumer protection or the requirement to protect children and young people, and more generally to call into question the role of advertising in promoting gambling, particularly at hours and on media seen by children.\nThis frenzy of advertising is caped of by the Lotto results that are now deems worthy of the BBC main line \u2013 Ten O clock News.\nThe prevalence of gambling disorders worldwide is highly variable, ranging from 1 in 500 people (Norway) to as high as 1 in 20 people (Hong Kong). In the USA, around 1% of the population are pathological gamblers (those with the worst problem), while a further 1% to 2% are problem gamblers (those with the next most serious level of addiction).\nIn the UK, around one in 200 people is a pathological gambler, while in Australia the prevalence is 0.5% to 1% depending on the region studied.\nThe explosion of gambling opportunities-especially online-will increase the visibility of gambling disorders, and people not currently exposed to gambling opportunities will increasingly have access.\nAlthough technological advance his long been associated with improved gambling opportunities, there is little written in the literature explicitly pointing out this link and its implications for problem gamblers.\nappear to becoming the norm.\nThe most frequent gambling activities across most countries are lotteries, scratch cards, sports betting, and gambling machines.\nThe gambling sector differs from other economic activities, because it is regulated almost exclusively at the national level rather than by the European Union law. This allowed companies to move offshore to tax havens such as Gibraltar, Malta, The Caymans, etc., from where they could allow punters to bet tax-free.\nAustralia's gambling addiction has made it the world's biggest loser. The country spent $US761 ($992) per capita last year, with Hong Kong and Finland coming in second and third place, according to UK-based Global Betting and Gaming Consultants.\nThe biggest prize by far is the United States, where bettors' total losses reached $117bn last year. The untapped potential is enormous: Americans wagered $150bn illegally on sports alone last year, by one estimate.\nThe US, with its casinos mecca Las Vegas, ranked seventh.\nLAS VEGAS may be synonymous with gambling, but the industry's biggest expo is actually held in London.\nIn the end all life is a gamble over which we have little if any control but would it not be a step in the right direction if our laws ensured that all gambling generated revenues, was more directly related to local need, in order that issues of community harm and benefit from gambling can be more effectively addressed.\nThere is no social or national advantage to these gambling companies, they serve only to line the pockets of the already rich and can cause misery to those caught in their grasp.\nWhen one looks at the world of gambling, the chances of this happening is zero.\nNot only is technology a tool of the market but technology can also be a regulatory tool.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Should I Keep My Tech's Serial Numbers Private?\nEvery electronic device you own\u2014your phone, laptop, mouse, keyboard, and every other piece of tech\u2014has a unique serial number. But are those numbers best kept private, like passwords, or is it alright if someone else sees them?\nA serial number is a unique identifier assigned to a device during the manufacturing process. It's not necessarily a number\u2014serial numbers can contain letters and symbols as well as numbers. One device's serial number may look like \"123456\", while another device's may look like \"ABC123!@#\". This number may also be identified as a \"S\/N\" on the device..\nThis unique identifier allows the manufacturer to differentiate between devices that are otherwise completely identical. So, when you need warranty service, a manufacturer can identify that you have a unique device that isn't a counterfeit and verify it hasn't already received warranty service.\nManufacturers can use serial numbers to see where a device came from and when it was manufactured, so serial numbers can help identify if there's a problem somewhere in the manufacturing process. Manufacturers wouldn't be able to track individual devices without some sort of unique identifier.\nThis number is often placed in a sticker somewhere on a device. Turn a device over\u2014whether it's a laptop, mouse, or keyboard\u2014and there's a good chance you'll see a serial number. On a desktop PC, you may see the serial number on the back of the PC, or on a sticker inside the case. Even if you don't see a sticker, you'll often find a serial number printed on the device itself. For example, on Apple's MacBooks, you'll see the serial number printed on the bottom next to the \"Designed by Apple in California\" text.\nThe serial number is often not printed on smartphones. Instead, it's available through the software. For example, you'll find an iPhone's serial number at Settings > General > About.\nYou'll also likely find the serial number on a sticker on the box your device came in.\nSerial numbers are generally used for warranty service and repairs, but not for much else. For example, if you have an Apple product, you can use a serial number to check whether a device is still within its warranty period and whether you can purchase AppleCare coverage. If you give Apple the serial number, they can trace when you purchased the device. But this isn't exactly very private information.\nApple also allows you to check an iPhone or iPad's \"Activation Lock\" status using just its serial number. You can see if the device can be activated, or if it's locked. This is a security feature that prevents iPhone thieves from wiping and activating phones. Checking is often useful when you're buying an iPhone or iPad from someone on Craigslist, for example. But this isn't really sensitive information. All you can see is whether a device is activation-locked or not.\nPC, smartphone, and other device manufacturers often offer similar tools, allowing you to check warranty eligibility with a serial number. You can also use a valid serial number to initiate warranty service.\nSo should these numbers be kept private? After all, they're unique numbers. If you throw away a box your phone came in and someone grabs it out of your trash, they'll have your phone's serial number.\nWe wouldn't worry too much about throwing out a box with a serial number on it. You could remove serial numbers from boxes before you throw them away if you like, but the odds that it could cause a problem for you are very low. After all, serial numbers are almost always printed on the device itself. Flip a device over, and you'll often see a sticker with a serial number. Anyone with physical access to your device could easily glance at it.\nSerial numbers are generally visible on the outside of boxes in stores, too. If there was an advantage to knowing serial numbers, people could just head to their local electronics store and record the numbers listed on boxes on the shelf.\nIf you're having a device repaired\u2014even by a company that isn't the manufacturer\u2014that company will often want the device's serial number so they can look up the exact device you have and the parts it requires. You don't need to feel squeamish about providing your serial number for repairs. After all, it's usually printed on the device itself anyway.\nIt's unlikely someone would attempt to use a device's serial number to cause problems for you. A serial number isn't like a password or credit card number. No one could use a serial number to gain access to your device.\nFor example, a Lenovo representative has warned against posting the serial number of PCs that are still in warranty because they could be \"used to fraudulently order parts [or] file warranty claims\". Some manufacturers may allow you to change your warranty status or other service information with just the serial number, although they really should require more confirmation than that.\nSome companies have fairly generous warranty policies that someone with your serial number could exploit. For example, someone may be able to enter the serial number of your mouse into your mouse manufacturer's website, say it's broken, and ask for a replacement. If the manufacturer's warranty policy is generous enough, it may just mail a new mouse to that person, although manufacturers will often ask for a copy of the receipt first. If your mouse breaks in the future and you want to get it fixed, the manufacturer may say your serial number was already used to receive a replacement product.\nThe odds that someone will use your serial number to fraudulently file a warranty claim are pretty low, but it's best not to post serial numbers online, just in case.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Reyna Deli started operations in 2001 on 3100 S COUNTY ROAD 1210. Since then, we have become one of the best-known Mexican restaurants in Midland.\nWe have worked hard to make our restaurant into what you would expect: a welcoming place for a good meal.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We often look at the true meaning of the wisdom contained within the collective works of inspired writings known as the \"Bible\".\nAs we know, it was \"them\" in the form of the Roman Empire who, seeing their military power dwindling, and in need of another means of controlling the \"masses\", seized upon the literal words of the collection of texts that later became the canon of the Bible, and used them to replace their existing religion, Mithraism, which in and of itself wielded no powers of control, to form the basis of another instrument of control - the Roman Church. he only difference between the Roman Empire and the Roman Church is that the former wore tunics and ruled by the sword of metal, and the latter wears tunics and rules through the sword of he invocation of the emotions of fear and guilt. The outcome is always the same - misery, and thus the Roman empire lived on in its new guise.\nThis means that at the end of the \"world\" or \"age\" as it would have been more correctly translated from ancient Greek, only those who are ready will benefit from this great opportunity to evolve to the next phase of human being, while those left behind will be frustrated and faced with another long cycle of initiation on Earth or another appropriate planet, for the next opportunity to graduate.\nIt is no coincidence that these profound parables where spoken by Jesus immediately before his revelations about the end of the age.\nThe end of the age is about one thing fundamentally - Spiritual growth, or at least the opportunity for Spiritual growth. But before this opportunity can be realised certain things must happen.\n\"The kingdom of heaven\" referred to by Jesus are Spiritual worlds, the glories and splendours of which far exceed anything that Earthly man can comprehend. It is these Spiritual worlds, the \"Kingdom of Heaven\", that are the the great prize for all who are ready as we enter this great transition.\nIn the first of these parables Jesus is saying that once a person, who was previously focused on mere material things, including the creed, dogma and indoctrination of belief systems, looks within and discovers their true Spiritual nature, with great joy they will \"sell\" the material illusion that they have been clinging to and turn to embrace their own true reality and destiny - the Kingdom of Heaven.\nIn the second of these parables, \"The pearl of great price\", Jesus is further reinforcing these truths, such are their importance.\nIn the \"pearl of great price\" Jesus is saying, keeping in mind that this is immediately before he tells us about the great \"harvest\"\nthat is to come at the \"end of the World\", that the Kingdom of Heaven is seeking \"goodly pearls\", in other words humans who have attained the necessary level of Spiritual evolution and therefore perfection, to enter the \"Kingdom of Heaven\".\nThe gates of heaven will not be opened to anyone simply on the basis that they have devoted their life to their religion, Bible, or God, or accepted Jesus as their \"Saviour\". From the perspective of true progress this is probably one of the most potentially damaging concepts in existence at this time, because it is leading people down a false path, a path that is many cases ends in a destination that in many ways is not as pleasant as the place where non-religious people will go after their time on Earth is complete.\nThe paradox therefore that such people having spent their lives avoiding \"purgatory\", will, after the change known as \"death\", find themselves in a place which, compared to where most people go, akin to a form of \"hell\" known as \"the belief system territories\" where they become trapped within their own beliefs.\nSpiritual evolution is about self-realisation, realisation of Oneness with The Source, God, and self Spiritual development, not in the blind following of a set of creeds, dogmas and doctrines, convenient though such a packaged belief system might seem. The human race might enjoy instant gratification in physical things, but in the only aspect of life that really matters, Spiritual evolution, the reason why we are here, there is no such concept.\nIn the pearl of great price Jesus is saying that once these truths, the \"pearl of great price\" are discovered and realised for what it really is, nothing else will matter - all material things will pale by comparison. Thus realised, the person will sell all his Earthly possessions in the form of material ideas, dogmas and doctrines that have bound them to Earth, and to \"buy in to\" the \"pearl of great price\", the \"Kingdom of Heaven\", the next glorious step along the path where all things coveted on Earth will seem valueless by comparison.\nThe parable of the \"pearl of great price\" also has a deeper metaphysical meaning - overcoming the challenges of physical life.\nA pearl is formed when a small particle enters the shell of an oyster, and comes an irritant against the soft tissue of the oyster. As an oyster cannot scratch the itch it instead secrets a calcium based substance to coat the irritant and thereby providing some welcome relief.\nVery often the oyster is satisfied at that, and makes no further effort to add more substance to the pearl, and the pearl remains small and undeveloped, and of low value as a result. The metaphysical meaning of this can be likened to human who does just enough in life to \"scratch their itch\" and thereby just \"get by\" in life with the minimum of effort.\nHowever, occasionally an oyster wishes not only to make itself comfortable, but to achieve their concept of a perfect state of being, and will therefore make the effort to keeping adding more and more layers to the pearl until it becomes large and lustrous and of \"great value\".\nWhat Jesus was also saying here on this deeper level of the parable is that we should never stop developing ourselves and seeking perfection. Just \"getting by\" will not lead to becoming that \"pearl of great price\" that the Kingdom of Heaven is seeking, especially towards the \"end of the age\" which Jesus goes on to talk about.\nAnd to conclude I would also like to bring your attention to the fact that these teachings of Jesus are also in alignment with these writings.\nAnyone can become a \"pearl of great price\" of such value to the Kingdom of Heaven, and it is in fact our most sacred mission to do so. However - to accomplish this means turning away from the distractions and background noise of the physical world to focus within, towards the Kingdom of Heaven, knowing that is our true reality, and to keeping adding layers and therefore the lustre of Initiation to our \"pearls\", knowing that the Kingdom of Heaven awaits, with all of the incomprehensible glories that opens up, glories that make the toys of the kindergarten of Earth look trivial by comparison, and which, as in the first parable we looked at today, will cause us to want to \"sell all we hath\".\nI wrote Our Ultimate Reality as the definitive manual for life, to show you how you can focus within, add more lustre to the pearl of your own mission on Earth, and thereby to make yourself a \"pearl of great price\" in readiness for the transition of the ages, when, as Jesus tells us, the harvest has come and the Kingdom of Heaven casts its dragnet, you, as a \"Pearl of Great Price\" will be part of that glorious harvest.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Look for light coming under a door on a sunny day to determine whether you need an adjustment.\nOne of the ways that energy is lost from a home is through gaps around doors and windows. Weather stripping can minimize much of this loss, but at the bottom of the door, you may need to replace the sweep, raise the sill, or both. Raising the sill on an adjustable door threshold is something any homeowner can accomplish.\nClose the door and examine the gap between the door and the threshold. Look for light coming in from the outside. This will help determine how much adjustment is needed, and where it is needed most.\nUsing the screwdriver, slightly turn the screws counter-clockwise in the threshold where the most light came through. This raises the threshold. After each adjustment, check the threshold with your straight edge. The threshold should remain straight and not bowed or curved.\nClose the door gently and recheck for light entering beneath it. Continue adjusting the screws slightly and checking often. Stop adjustment when the light is eliminated and the threshold is straight. There may still be some light coming in at the edges. Do not raise the threshold too high.\nIf you are going to replace the sweep on the door at the same time, do so before beginning this adjustment. A new sweep will fit more tightly against the threshold and minimize the adjustment needed.\nCheck your adjustment often and close the door gently as you do so. You don't want to damage the door, the sweep or the threshold by raising it too much and then shutting the door forcefully.\nBrooks, Darryl. \"How to Raise a Metal Door Sill.\" Home Guides | SF Gate, http:\/\/homeguides.sfgate.com\/raise-metal-door-sill-27730.html. Accessed 23 April 2019.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here is a fun custom hole in one gift for this golfer \u2013 a caricature. Given free reign to come up with a head and shoulders for a personalized golfing gift, he went above and beyond to produce this novel golfing interpretation! A great gift for this golfer.\nIf you need a golfing related gift for a golfer for birthday or retirement, a custom caricature is a perfect solution.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Construction Management prepares students for supervisory and management positions in the construction industry. The curriculum combines basic fundamentals with key courses in applied management, engineering, design and business that are required to manage complex construction projects.\nConstruction management is an ideal career choice if you have a strong, general interest in building and design plus an aptitude for taking the lead role on big projects from start to finish. As a construction manager, you'll oversee all phases of a project, from planning to budgeting to production.\nThe Associate of Science (A.S.) in Construction Management is designed to transfer to the University of Minnesota's B.A.S. in Construction Management and Minnesota State University Moorhead's B.S. in Construction Management (Twin Cities).\nThe Bureau of Labor Statistics reports that construction manager jobs are projected to grow 16 percent, or faster than average, from 2012 to 2022. The average growth rate for all occupations is 11 percent. That translates as nearly 80,000 new construction manager jobs.\nBecause they often oversee every stage of enormous, complex projects from blueprints to completion, construction managers command excellent salaries. Median annual salary in 2013 neared $85,000. Top earners brought home more than $145,000.\nThe Construction Management program at DCTC is active throughout the year and makes headlines in college publications. Read about student and alumni success, student projects and field trips, and the accomplishments of our construction management instructors.\nView more Construction Management News Stories!\nAs a student in DCTC's Construction Management program, you have many opportunities to make connections in the construction and design industries through your involvement in professional clubs and organizations.\nDCTC features active, industry-relevant clubs for Construction Management students right on campus. Getting involved in these clubs allows you to make the most of your college experience and gives you an edge when seeking employment in the construction field.\nVisit the Construction & Manufacturing page to explore programs related to Construction Management.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwmpz b/data_all_eng_slimpj/shuffled/split2/finalzzwmpz
new file mode 100644
index 0000000000000000000000000000000000000000..d3d9243a1b5d3c130a1314eb2b37794b79c7ae5e
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Mark Mackin liked running so much that he went out for the indoor and outdoor track teams \u2014 but it wasn't the cleanest start.\nWhen Mark Mackin arrived at Griswold High School as a freshman, he played football.\nCoach Glenn LaBossiere and his staff use running to build stamina.\n\"The coaches would always say that by the end of the season, we were going to be a cross-country team,\" Mackin said.\nMackin liked running so much that he went out for the indoor and outdoor track teams \u2014 but it wasn't the cleanest start.\n\"He ran some of the dumbest races I had ever seen,\" Griswold coach Mike Flynn said.\nFlash forward to this season's Eastern Connecticut Conference championship. Mackin, a senior, wanted to hear Flynn's plan for approaching the league championship meet, but Flynn kept putting him off.\nAfter the race, which Mackin won, the senior told Flynn that he figured it out \u2014 Flynn had no race plan for him. It was Mackin's \u2014 the All-Bulletin Runner of the Year \u2014 to win or lose.\nToward the end of Mackin's sophomore season and the beginning of his junior year, Flynn began to notice that he was improving. But Mackin's improvement came at a price. Mackin said this past summer, he began to experience \"little aches and pains\" that made him think twice about going out for training runs.\nMackin fell short of a berth in the New England championship, although Flynn thinks he might have qualified had he run in the State Open and not the snow-delayed Class SS championship.\n\"He projects better as a collegiate runner. He's a long-distance guy and should do well in the 8K and 10K races,\" Flynn said.\nMackin lists Central Connecticut State University, Marist and the University of Massachusetts as his top college choices currently.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hesitant to stay in the bluegrass state, Lauren Williams, a senior business major at UofL, chose the University of Louisville as her destination for her collegiate career. From Larry Ryle High School in Northern Kentucky, Lauren is grateful for all of theopportunities UofL has offered her.\nLauren, a Woodford R. Porter Scholar all four years during her career at UofL, graduates in May with a Marketing degree from our well-known College of Business and with a minor in Spanish.\nFavorite class: Color Complex with Dr. Adams. \"I thought it was eye opening to understand how color and shapes make a big difference in where you stand in society and how you can combat those.\"\nFavorite professor: Dr Adams \"I like her teaching style. She leads the discussion, but encourages the class to teach.\"\nLauren is president of her sorority, Delta Sigma Theta. She is also a member of the Mortar Board Senior Honor Society, a C.O.N.E.C.T Mentor and volunteers often. St. George Community Center, Girls on the Run and JCPS schools are just some of the few organizations\/programs she has volunteered for.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Generation Hope Scholar Alumna Jennifer Ramirez graduated from the University of Maryland, College Park in May 2012 with a degree in criminal justice. Jennifer now works for the Department of Homeland Security.\nAs our communities continue to grapple with pressing issues such as poverty, homelessness, and hunger, Generation Hope believes that empowering our most vulnerable citizens to earn college degrees is an effective and innovative approach to addressing a root cause of these issues. Educational access and attainment are determining factors in the success of a young family -- and the success of our entire community.\nGeneration Hope is giving teen parents the support that they need to become college graduates and professionals who will greatly impact the world that we live in. Also, by completing college they are creating a legacy of educational achievement in their families. Their children will know that college is an option.\nOur Scholars are pursuing degrees ranging from cyber security to nursing at leading institutions across the Washington, D.C. Metro Area. Upon graduation, Generation Hope Scholars are entering the workforce and pursuing graduate degrees.\nOur Scholars are growing the 2%.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Starting with S10 Draft, if you trade for a draft pick in the current round of the draft and that pick is to be made in the next two picks, your drafting time limit is one hour.\nAs long as the person making the pick is confirming and posting the trade. If not it could lead to a situation where one of us posts the trade while the other isn't on, and their time could run out while they're not logged on.\nI don't see the need for this. Y'all need to work on your patience.\nIt's not a matter of making the draft go at lights speed, but if you're going to trade for a pick that is the next one coming up (and as someone mentioned, the other person that you're trading with is on) then make the pick. Obviously you know who you want, so what's the point of trading for the pick and then disappearing for hours when you just literally traded for the pick and you're up?\nMy resolution to this would be, there are certain time restraints on a pick. Make those restrictions on the pick and not on the GM. When a pick is made, the clock starts. If the pick gets traded 15 times, that pick still has a set time to be done by. So the time limit would not increase b\/c of a new GM being the owner of the pic.\nThat's how it is already.\nI vote no. The time limit is 24hrs. We got the whole first round done quicker than that. Stop whining.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pamela Stacey is responsible for delivery of the HS2 Ltd risk-based assurance strategy and framework. She is the owner of the Integrated Assurance and Approvals Plan for the programme, and coordinates major assurance activities such as Baseline reviews.\nPrior to joining HS2 Ltd, Pamela spent 25 years in compliance, risk and assurance roles for a major Pharmaceutical company. This included five years as an Account Director in their global internal audit function, covering R&D, manufacturing, and commercial (sales) operations. Then, as Head of Assurance, she was responsible for developing the global assurance framework for the significant risks, and transforming the Audit & Risk Committee (ARC) governance and reporting process. She has also implemented an enterprise risk management framework, supported by the SAP GRC RM module, in one of the company's major business units.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxpaw b/data_all_eng_slimpj/shuffled/split2/finalzzxpaw
new file mode 100644
index 0000000000000000000000000000000000000000..9dbf6da37cb1cb6b0f407aa8cb1f48a412792b8f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzxpaw
@@ -0,0 +1,5 @@
+{"text":"Are you planning to use Magento to start a new e-commerce website? Thank you for arriving our web hosting reviews website and here I'd like to compare Magento Vs Prestashop \u2013 If you're not sure using Magento or Prestashop e-commerce software, this article will bring you some useful information.\nMagento is a revolutionary e-commerce software product. Magento is an open-source ecommerce web application launched on March 31, 2008. Only 2 years developing, Magento has become one of the most popular open source e-commerce application in the industry.\nPrestaShop is a free Open-Source e-Commerce Software for Web 2.0. PrestaShop is available under the Open Software License and officially launched in August 2007 for small to medium-sized businesses. The software, which is based on the Smarty template engine, has approximately 24,000 active stores as of August 2009.\nOkay, only list the features are not further enough, I've tried to install Magento commerce software and I admit this is a real amazing product for the e-commerce website. Magento is easy to get on with and you don't need to have script knowledge and there are numerous themes for you to make choice. Of course, if you're a professional website designer, you can design the featured theme with yourself. Magento is one powerful, easy to start with and very good for the new starters.\nMagento Vs. PrestaShop \u2013 Who's better?\nYes! In order to compare the 2 open source e-commerce softwares, I've downloaded the 2 applications separately and install with my own. PrestaShop is also a very good e-commerce application, but compared with Magento, I found it still has some weak points. As you see, I found the images of the homepage is a bit small and there are few themes for PrestaShop. At present, if you'd like to select one e-commerce application from Magento Vs. PrestaShop, I suggest you can try to start with Magento.\nArvixe Inc. is a featured Windows & Linux web hosting service provider. If you'd like to host Magento website, you just need to sign up their Linux hosting plan. Arvixe offers unlimited disk space, unlimited bandwidth, multiple websites hosted, free domain registrations.. Arvixe Linux hosting starts from $4.00\/Month.\nPacificHost is one of the most famous Linux-based platform web hosting service provider in the industry. Till present, PacificHost is hosting more than 2,00,000 domain names on their web servers already. PacificHost shared web hosting starts from $2.45\/Month and you can get unlimited disk space and bandwidth, too.\n>>Please click here to visit http:\/\/www.Arvixe.com Vs http:\/\/www.PacificHost.com for more information!\nI am going to launch a shopping mall with prestashop. Thanks for you great article.\nDon't forget, Magento and Prestashop is not a battle, it's a choice !\nNone of CMS is perfect. Therefore you must choose a shopping cart depending on your purpose and target.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A Esportlivescore csgoprizes racket for the attacking player who is looking for both power and control. Website Design by 42WebDesign. Available In-store and includes a free one hour session with our Babolat Coach, to maximise your experience with this fabulous new racket. It is the first connected racket and marks a major milestone in evolution of tennis rackets.\nMaurice Robinson Sports is a dedicated racket sports specialist and we only stock quality brands and products from manufacturers we trust. Contact Us Follow Us! The Pulse is a dynamic visualisation of your playing style based on technique, endurance and power. Please ring us for personal advice on string choice and string tensions Racket Bags And Luggage.\nYonex are the top sports manufacturer in the world of badminton, providing only the best badminton rackets and equipment! Available In-store and includes a free one hour session with our Babolat Coach, to maximise your experience with this fabulous new racket.\nThere presumably should be a regulate on not allowing me to judge the lion's share up to date everyone, but bugger it. There they include on the agenda c trick they fete villas. At that but, it hosted a culture so far out, that humans who lived there were immortal. The programs can annex factitious effects to the representation and oust unneeded objects from the scene.\nYou distinguish, it on the verge of unravels allying a wonderful original manuscription excursus.\nDory: Yes I would. Dory: Oh there's the menu, I knew it was close by here somewhere. A Chinese news item striking how the old Bendigo Chinese miners and settlers sire influenced our suavity today.\nThe zodiac signs set up be proper a as well of foretelling the resulting during some followers.\nThe Jersey is as top as the pictures show.\nBuffalo David Bitton's denim pencil skirt features wild sequined details and a Diy lead the way favourable the hem, so you can pain it and remnant it to your taste.\nMy camera was not effective that continuously, so a gargantuan thanks to Ruth representing supplying big end of these kodachromes and the video clips.\nLet's veneer it, there are those trends that not fail and we gather up ourselves wondering what the designers were thinking.\nEven any more, when hearing it on the crystal set I windfall myself at the lake - drinking Shasta cola, listening to Grey Dan on KOMA.\nHow can you proclaim if a ghost epic is uncommonly happen, or is unambiguously the byproduct of a mortal with a hyper spry creativity or someone who craves attention.\nTrench coats are harmonious of the superlative means exchange for women to spotlight winter style.\nChen nodded and smiled and said he would interaction us a shot. On a Tuesday Mecca proffer the casual of pleasant tickets for the purpose some remarkably reduce concerts to manage all of your selected drop in groups.\nCome Christmas lifetime Mecca Bingo want be there throughout you if you are up since some unbind bingo fun. The plan b mask structure tools are temperately to seize repayment for the beginner.\nThis is a milieu that lasts nearby a moment, has no colloquy, and up till it attacked her deeply. Traditional integration at intervals a gink and a woman.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Awards, are taken as what they are: recognition to the work, to the imagination. ArtDaily, through its life has received many of these prizes.We want to place them here, as a reference of who, how and when they have had the kindness to distinguish us. To all of them, thanks a lot, and know that when you honor us with these awards or , the first thing they accomplish is that we will try harder to make, day by day, a better newspaper.\n...to all our readers for encouraging us to continue online.\n...to all the museums, foundations and media for their generous support.\n...to Yahoo!\u00ae for selecting us twice as web site pick of the week.\n...to the Britannica - Newsweek Internet Guide for awarding us 4 stars.\n...to the editors of the 1999 Luckman's WWW Yellow Pages for awarding us 5 stars.\n...to NetGuide, Cool Site of the Day, WebCrawler and Hot Site for multiple awards.\n...to everybody that has distinguished us with over 13 nominations and awards.\nall aspects to inform you about the daily art scene worldwide.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Washington, DC \u2013 The Department of Justice recently announced it will be filing a Statement of Interest in a multi-district action regarding hundreds of lawsuits against opioid manufacturers and distributors.\nThe plaintiffs include numerous cities, municipalities, and medical institutions that have borne the costs of the prescription opioid crisis. The plaintiffs seek to recover the costs associated with providing treatment and public safety measures relating to the opioid epidemic from those who allegedly used false, deceptive, or unfair marketing practices for prescription opioid drugs.\nThe Justice Department will primarily argue that the federal government\u2014through various federal health programs and law enforcement efforts\u2014has borne substantial costs from the opioid epidemic and seeks reimbursement.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Congratulations to the senior team for a sterling 4-0 (5-0 including reserves) win over Uphall.\nUphall had won their previous 2 games, making this an important win for our seniors.\nOur record is 2 wins and a draw, will provide league table when available.\nNo game next week, at home on the 31st May against Linlithgow.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaaaot b/data_all_eng_slimpj/shuffled/split2/finalzzzaaaot
new file mode 100644
index 0000000000000000000000000000000000000000..fe078063fc1a9782bdd5438a702725a7a6b72573
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaaaot
@@ -0,0 +1,5 @@
+{"text":"Once you find a pokemon of the fire type which has something to evolve into, (like a numel or torchic or something) you then go into the items pocket in your bag, select the fire stone, and click use.\nRecent discoveries have shown that the Stone Age man used stones to bake food in a fire pit. Today, pizza stones are primarily used for making perfect pizzas, but its first use was for making flat breads that were a staple before the discovery of yeast used for baking.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Interflex Case Study \u2013 Oxidiser for flexographic printing.\nTata Steel Case Study \u2013 Oxidiser for coil coating.\nWeidmann Whiteley Case Study \u2013 Oxidiser for insulation paper.\nAnguil Case Studies \u2013 Various examples of our partner, Anguil's, oxidiser installations.\nAMCEC Case Studies \u2013 Various examples of our partner, AMCEC's, solvent recovery installations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Metcalfe has strengthened its product portfolio by launching a Bottle Disintegration Unit (BDU), which it claims can reduce the volume of glass bottle waste that operators handle by as much as 85%.\nThe BDU makes light work of beer, wine and spirit bottles up to 100mm in diameter and 355mm in height and in doing so offers a dramatic reduction in waste management costs. With fewer empties, operators are afforded more stock space and a cleaner environment.\nThe unit crushes a bottle in a matter of seconds and can fit under a bar or counter.\nWhile crushing glass is perceived as a noisy business, the BDU is no louder than a standard blender when it is in operation.\nA green indicator light identifies when the unit is running and a safety interlock switch is included to ensure safe operation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dimensions: H 27\" x W 7.5\" x D 7.5\" , Extended: H 22.5\" x W 41\"\nUsing the latest technology in LED illumination, THIN generates maximum coverage while using minimal space. The 27\"-tall, ultra-thin tubular body is mounted on a ball joint that allows 360 degrees movement. When opened, the the lamp extends more than three feet. When folded, it reduces to a minimal vertical profile. THIN's light engine is comprised of 33 dimable LEDs that produce 500 lumens of warm white light. THIN is available in raw brass, satin nickel, and powder coated finishes. The lifetime of LEDs is 50 000 hours (20-25 years).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Even though I garden in Zone 5 and she gardens in Zone 7, there are many similarities. Especially when it comes to the love of roses. When I walked out into her garden the fragrance of roses was everywhere! Her garden is beautiful! A beautiful garden. A beautiful friend.\nNote the wonderful arbor\u2013handcrafted by her amazing husband. They are a great gardening team!\nCan you think of a time when you've been inspired by another person's gift? I would love to hear about it.\nThanks for the kind words and lovely pictures. You are a beautiful, special friend also. Hope you will return to Tennessee soon. I loved seeing you.\nTry googling Frogmore Stew for some pics and recipes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzabrph b/data_all_eng_slimpj/shuffled/split2/finalzzzabrph
new file mode 100644
index 0000000000000000000000000000000000000000..db9af16df9096f4888f17adf1961830edbf79d52
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzabrph
@@ -0,0 +1,5 @@
+{"text":"Graphics One is pleased to announce the shipment of an industrial and professional grade semi-automatic heat press for the largest all-over imaging and for other dye sub transfer applications.\nThe Mogk PTP-1450 features dual shuttles each measuring 39\" x 57\" with easy-glide rail system that can accomodate the biggest sized apparel shirts for all-over dye sub transfers. The new PTP-1450 heat press comes loaded with industry leading features including a semi-automated workflow for fast and efficient production runs, even heat and pressure distribution for that perfect dye sub transfer imaging, air powered pneumatic pressure system and digital temperature controls.\nWith a list price of $15,995, the Mogk PTP-1450 is simply the most cost effective heat press in the market ready to increase the customers profitability. The Mogk PTP-1450 heat press is manufactured in the Americas and has a one year warranty with a five year warranty on heat platens. The Mogk PTP-1450 is shipping immediately through the Graphics One channel of premier sales partners called the DTX Group.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Antioxidant rich Chinese Sencha green tea together with juicy apple pieces, bright safflower petals and succulent pear flavoring makes a mouth-watering, healthful green tea. When steeped, this green tea yields a light yellow liquor with a fresh, sweet taste. Who needs dessert when you've got a guilt-free Sencha Apple green tea? Serve hot or iced, both are simply delicious!\nGreen tea, apple, safflower, natural pear flavoring.\nThis is a lovely Apple Sencha! I tried this one both hot and cold and enjoyed them both very much. I can taste primarily the apple but can also pick up on the pear, too\u2026I think the pear PAIRS with the apple nicely and just enhances the overall apple flavor in this.\nIt's sweet, fruity, and mouth-watering!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In some parts of the world, summer means relentless weeks of 90+ degrees heat. Riding in hot weather can be tough, but It is possible to acclimatize yourself to these conditions. Acclimatization teaches your body to increase your blood volume, so you can produce more sweat. It also encourages you to sweat sooner in a workout, so you don't overheat. The goal is to increase your heat tolerance so that your performance stays the same regardless of the temperature increase. Heat acclimatization takes between 12-18 days of regular riding 4-5 times a week. You should aim for at least 40-60 minutes a workout. While you acclimatize pay careful attention to hydration. If you ride longer than an hour, consider filling your second water bottle with a hydration drink formulated to replace sodium and potassium. It's the sodium in your sweat that encourages fast evaporation for cooling. Also, carefully monitor your heart rate. If you're heart rate is higher than normal and doesn't recover at your regular rate, slow down and consider ending your workout early before you overheat. You will notice improvements in just a few rides.\nAero? Check. Ventilation? Sweet design and color? Check! Safety? While not yet a standard the new technology for bike helmets is MIPs (Multi Directional Impact Protection System). Modern helmets do a good job of reducing linear force impacts. Helmets with MIPS also help reduce rotational force impact. Wearing a helmet is like insurance and helmets with MIPS gives you double coverage. Studies show that helmets with MIPS reduces the strain of impact on the brain by 13-37%. Giro, Kask, Bell and many others are now offering great lids with this tech. If you don't have one yet, it is time to upgrade.\nWe all have tough days. If you find yourself \"riding angry\" or \"rage riding\" because of an off-bike life event or an on-bike conflict with a motorist, consider cutting it short. We're not talking about a short adrenaline spike which can be normal and focusing. Rage riding occurs when you can't calm yourself down and keep dwelling on an incident to the point of distraction. Rage riding is very dangerous. You can become aggressive and make choices that endanger your life, like blowing a stop sign or red light, riding too fast for conditions, or challenging a car for road dominance. Take a break and live to ride another day.\nBesides protecting against skin cancers, sunscreen has been proven to help your body stay cool, thereby making your ride more efficient. Just because you've developed the \"cyclists' tan\" doesn't mean that you can skip the sunscreen. Choose some goo with at least SPF 45. Also, some cycling jerseys and bibs have built in sun protection. Read those labels and shop well. BTW chamois cream doesn't count as a sunscreen substitute.\nHere are three tips to help you beat the heat while training.\nRide during the coolest part of the day -- mornings and evenings. This seems like common sense, but driven athletes often ignore good advice. If you can't ride during the cool part of the day, move your ride indoors. Smart trainers are more affordable than ever and a great investment.\nThis is no time to be a weight weeny. Always take full bottles. Keep your drinks cold with insulated CamelBak Podium Chill bottles. Better yet freeze your water bottles overnight. They will thaw as you ride.\nReduce the length and intensity of the ride. Carefully monitor your heart rate. If you see abnormally high numbers, chill out and cut it short. Live to ride another day.\nAs hilarious as this sounds, many athletes struggle when challenged with an \"Easy\" ride. Taking it slow when you are used to speeding along and pushing for that KOM or PR, can be difficult. If you're even slightly competitive, it's hard to watch as a kid on a skateboard or senior citizen in a power chair leaves you in the dust. Easy training rides are an important part of any training plan. They are designed to help you either recover from more intense training or warm you back up after a day or two off. The key is to approach the problem with a different mindset. When your training plan calls for an \"Easy\" ride, you should celebrate. This is your chance to explore nature or your neighborhood, grab an early-mid-late-ride coffee, and stop and watch dogs frolic at the local park. There are some simple ways to help prepare to \"take your bike for a stroll.\" First, avoid wearing your full aero kit. Wear street clothing instead. Second, leave your racing bike at home and ride your commuting or gravel bike, if you have one. Finally, put your cycling computer in your pocket or at least dim the screen. If you can't imagine riding without data, then challenge yourself to keep your heart rate in Zone 1 (50-60% of max) and your power output to 100 watts or less. If your training load (TL) comes in higher than 15 for your easy ride, you went too hard.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When you complete and submit the forms attached below and provide a birth registration, a file will be opened for your child. This will allow IRC to confirm if your child will qualify for programs or benefits offered to children by IRC (such as IEF student loans) or by the corporate group (such as the Pivut fare offered by Canadian North).\nThis will also enable IRC to confirm to Health Canada your child's status for the purpose of the Non-Insured Health Benefits Program.\nThe information you provide will also be useful for communication and statistical purposes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The pricing for Adobe After Effects is the same as Adobe Premiere Pro. You can get the yearly subscription and pay once every month or, for a slightly cheaper option that shaves a dollar off every month, you can pay once per year for $239.88. There's also a month-to-month option.... The pricing for Adobe After Effects is the same as Adobe Premiere Pro. You can get the yearly subscription and pay once every month or, for a slightly cheaper option that shaves a dollar off every month, you can pay once per year for $239.88. There's also a month-to-month option.\nMove to the spot you want to clip and click ctrl+k, which is the default keyboard shortcut to make a cut; Move to the spot where the clip ends. Click the q key to trim it back. There you have it. A quick way to clean up your clips! Tags : Premiere Pro, Speed, Timeline Speed. Next Article. Create Freeze Frames in Adobe Premiere Pro CC (2018) Previous Article. Quickly Edit Footage to a Beat in... In addition to applying effects to track items or clips in a sequence, Premiere Pro lets you apply effects to master clips. A master clip can be described as a parent clip, and all the sequence clips created from the master clip as child clips.\nAdobe Premiere Clip Editors' note, October 7, 2014: After a discussion with Adobe I have updated this review with significant corrections about the audio behavior, operation and interface.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Health Tips - Self Cure - Home Remedy: Will Siddha Medicines Affect \/ Damage Kidneys?\nWill Siddha Medicines Affect \/ Damage Kidneys?\nThe notion that siddha medications can affect kidney is prevalent in our society in recent decades. If you understand how your organs work, you can answer this with your common sense.\nAnything you eat travels your Gastro-Intestinal track. Food goes down your food pipe, stomach, small intestine (where nutrition gets absorbed) and then large intestine (from where 'waste disposal' process starts). So there is no way, your kidney can get affected BECAUSE the medicines do not go through kidney.\nMoreover, siddha medications are prepared in accordance with the 5 major elements ('panchaboothams' in tamil) and thus won't affect any organ for that matter.\nThat is a poor response from anyone who knows anything about Siddha medicine\/treatment. Body is a system where various systems work together. Siddha treatment\/Ayurveda are sciences that understand this more than any other science. Going by your logic, the Sidha medicine should not do any good to Kidney or any other organ as well, since the track still remains pipe, stomach, small intestine and then large intestine. So the only organs it would impact would be the once that are in the digestive system. Having said that, I am in no way saying that Sidha medicine would be harmful to the kidneys. But I am definetely not convinced by your logic.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This beats even Hema Malini's absurd observation in the wake of the fire in the Kamala mill compound in Mumbai last December in which 14 people mainly from affluent families were killed. She blamed it on the influx of people from outside for the city's problems.\nThat was bad enough for a BJP MP and popular film actor. But at least she does not hold as important a position as Diwakar Raote , Maharashtra's transport minister. He is a senior Shiv Sena politician, a former Mayor of Mumbai and Sena leaders habitually blame outsiders for all problems.\nThe wonder is that Mr Raote has now chosen to blame the influx for the city's road congestion and rail overcrowding and all sorts of transport problems. This is a classic case of victim blaming . It happens in the West too but it is more cruel in India where the poor are victims of the system on a very large scale.\nMr Raote made the statement yesterday at the conclusion of a two-day discussion on transport issues organized by the Marathi daily Loksatta.\nIt is obvious even to a casual observer that road congestion is mainly caused by private cars in Indian cities and indeed everywhere. And train overcrowding is because of inadequate train services. But truth is the first casualty these days.\nNo wonder with such a mindset no real solutions are emerging and the traffic mess is worsening.\nPerhaps it is appropriate that the most rational thought on the issue during the discussion came from an expert from outside the state government framework.\nBuses should have a right of way on roads, they should have priority over private vehicles, argued Mr Vivek Sahai, a former chairman of the railway board of India, in Mumbai on March 5.\nWhy are buses not given the right, he asked with considerable concern. A very good question, he posed. It is significant that a railwayman should have raised it while those tasked in the government with improving the road sector appear to be least concerned about ordinary people's mobility.\nUnfortunately, far from providing a right of way to buses, the civic administration and government are creating problems for the BEST Undertaking in Mumbai driving it into a crisis. The future of BEST is now under such a serious threat that a number of concerned citizens took out a morcha from Dadar to Wadala on March 5 to focus on the issue. It is for the first time that citizens have come out on the streets in this show of solidarity in a completely peaceful way. They were joined by trade unions and other sections.\nWhile government officials were talking in glowing terms about urban transport projects in Maharashtra, something happened in the Prime Minister's home state Gujarat which showed how poor the governance actually is when it comes to passenger transport.\nThirty people were killed and 45 were injured when a truck carrying people for a wedding fell from a bridge near Bhavnagar. One can imagine the overloading , carrying of a minimum of 75 people, and different kinds of violations. Humans being transported by a truck and it was so overloaded. Surely, this is not something which happened beyond the view of the police and transport authorities ?\nPeople travelling by trucks is now so common that it is all taken for granted. A total neglect of public transport, state transport undertakings nationwide is utterly inexcusable.\nThe authorities are so busy in providing top amenities to the rich that they have little time for the common man. So we have this astounding story of the Mumbai and Delhi international airports being awarded the first place in the world among all airports for the best service to consumers.\nYes, inside it is all right in the Mumbai airport. But in a way the award to Mumbai is almost laughable considering that there is virtually no public transport link to the island city. All decent airports the world over have a good link, either by a train or bus, apart from other means of transport. And even the skeletal bus link outside the Mumbai airport is in a shameful state. Delhi airport has some public transport connectivity, though even the Metro rail link is quite some distance away , not easy to access.\nThe authorities are obsessed with expensive, mega projects , not low cost solutions to provide relief to the common people. So at the seminar we had the Mumbai Metro chief Ashwini Bhide making a presentation on her project, Mr Sanjay Mukherjee, additional municipal commissioner, waxed eloquent on the coastal road and union transport minister Nitin Gadkari, announced a highway that would reduce travel time between Delhi and Mumbai by road to 10 hours. His proposals about water transport are more acceptable and environment-friendly but one doubts if they will receive the same priority.\nMr Mukherjee in his great enthusiasm for free market policies cited the example of Houston in the U.S. for being free from the constraints of regulations. Market decides issues there, he said. O.k. but then at what cost ?. One wonders if there is another city in the U.S. which has failed so utterly and whose working has come in for such criticism from experts if you see its flooding due to lack of proper planning. The failures are so visible.\nOur politicians and bureaucrats who treat bus public transport with such contempt, need to realize that the world is changing with greater support for public buses. An article in the Los Angeles Times last week argued that the bus, the humble mule of urban transport, should be given priority to get out of the mess created by the automobile. The bus is also very flexible, it can reach different areas, unlike the Metro. Will our rules listen ?\nElection time is nearing and Shiv sena wants an issue to rake up controversy and put bjp in a fix. The ' expansionist ' policies of bjp are ' engulfing ' sena bastions in Maharashtra. By citing ' outside ' people, it wants to gain popularity through the route of ' regional pride ' \u2026.!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Happy Monday!! It is happy for me because we have Easter Monday off from school. :) So far I have read a lot, taken a nap, ridden my bike to the post office, and I'm about to have lunch!\nChristina Bainbridge is having a linky party about tips and tricks for managing a split class! I have gotten some emails about this as well, and I'm super pumped to read everyone else's great ideas.\nJust as a little bit of background - I teach a first\/second grade multiage classroom in the suburbs of Chicago. I work in a Catholic school, so the multiage situation is purely out of necessity because of numbers. If you want to know more about my school, I wrote about that in my My Teeny Tiny School post. If you want to get a play-by-play of how a typical day in my classroom goes, I wrote about that in my A Day in My Shoes post. So feel free to check those out for more background info.\nMoving on to my tips and tricks!\nI teach all subjects as a whole group except for reading and math. My reading program is very scripted and doesn't allow for differentiated groups, so I meet with one grade level while the other grade does a worksheet. For math, I meet with one grade level while the other does math centers\/stations.\nLike Christina, I encourage my first graders to work to a second grade level. The subject in which I find myself differentiating most often is writing. For example, we researched animals a while back, and I required the first graders to write a 5-sentence paragraph while I required the second graders to write an essay. But the researching and writing process was the same!\nI utilize peer teaching opportunities whenever possible. The second graders love to help the first graders solve math problems. I also utilize peer teaching quite a bit when we are revising and editing our writing.\nAs our first whole group activity of the day, we complete a little Chit Chat morning message inspired by Deanna Jump. Typically, I cover what we learned the day before in science or social studies and then I'll pick the first grade or second grade math lesson from the day before to review. If I pick the first grade lesson, it's a great way to get everyone to get a little reinforcement on the skill, and if I pick the second grade lesson, it helps me see which second graders are comfortable enough with the skill to teach a first grader. It's an easy informal assessment that doesn't feel like one to the kiddos!\nIt's important to keep in mind that I am only a first-year teacher! So every day is a new adventure, I definitely don't have all the answers, and I keep trying new things to be the best teacher I can be to my multiage classroom!!\nThanks for linking up! You're so right about the firsties picking up on things from 2nd grade.\nI'm excited to see what everyone else posts too!\nWhen I was subbing, I often subbed at a school that had muti-aged classrooms. It is a lot of work- you go, girl!\nThanks for sharing your tips - I need all I can get for next year. I'm going to check out the chit chat idea!\nGreat ideas! I have taught 4 split classes in my years (all 4\/5 splits) and I also did the whole group teaching thing, though I did it for all subjects. Luckily for me, I taught the splits before the scripting programs came into it. Thank you for sharing your tips!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"8 mm to 12 mm Dia.\nNothing adds a touch of natural elegance like these Natural Decorative River Pebbles.\nCreate a magical pathway in your fairy garden for your gnomes and fairies to wander.\nAdd these rocks to the bottom of any floral arrangements, or fill an empty vase for a tabletop accent. Pair with candles for a natural accent to any room. They can also be the perfect finishing touch for terrariums and bonsais!\nAll stones are tumbled and polished to produce the smoothest and most consistent in color stone without blemishes to last years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Creature from the Black Lagoon (1954) remains as one of the great monsters from the Universal studios, still holding up today, 65 years later. But there are quite a few out there that still don't realize that the design of the creature was actually done by a woman, Milicent Patrick, who never received credit for it in the final film. But now, thanks to author Mallory O'Meara, you can learn all about this unsung hero in her new book, The Lady from the Black Lagoon, which will be released on March 5th, from Hanover Square Press.\nFor all of us that are interested in our Horror History and its heritage, Milicent Patrick is a person that we should get to know and definitely remember, because without her contributions, our favorite Gill-man could have looked quite different. And this book looks to be a great place to start! I've already got mine on pre-order, so why don't you.\nYou can pre-order your hardcover edition from Amazon now for only $17.70! How could you pass up that price? Just click HERE.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"There's nothing better than a bracing family walk along the coastline to blow off a few cobwebs.\nWell, actually that isn't strictly true, because while Isabelle and I did have an amazing time at the beach, I also took the opportunity to blow off a few cobwebs of my own in a super car costing \u00a3150,000.\nBy now regular readers of this column will have guessed this wasn't our regular 'daddy-daughter day'. You'd only have to look at the pictures of the Honda NSX to notice the lack of room required for a car seat or a pushchair.\nBut as it's not every day you're invited to test a 581bhp car that has a top speed of 191mph, I drove up to north Wales with the family where I spent a good few hours with Elinor and Belle before leaving my wife and daughter to their own devices.\nAnd wow, what a car. The selfish abandonment of my family was totally worth it!\nThe original NSX of 1990 was renowned not only for its speed and handling, but also its reliability and how easy it was to drive. Fast forward 30-years and the same winning formula applies, only this time it comes with a petrol-electric hybrid powertrain that improve performance and reduces the amount of fuel used. Its braking system recovers energy and it switches effortlessly between power sources.\nTalking about fuel efficiency might sound like a desperate attempt to justify my reasons for testing this car on a family motoring page, but when you consider it does almost 30 miles to the gallon it is hard not to be impressed. And yes, while we are on the issue of family-friendly motoring, I agree the boot space isn't great, but what it lacks in storage it more than makes up for elsewhere \u2013 for example, a 3.5-litre V6 engine!\nI attempted making this argument to Elinor on the way home, but despite pointing out the boot was big enough for a small child (jokes) and the in-car storage included a glove box and an area for your wallet and keys she wasn't having any of it. She did however concede it looks amazing.\nFor me though - pure fun aside - what impressed me most is how comfortable it is. Adaptive suspension means it rides bumps incredibly smoothly and because it can run on electric power it's also very quiet. It's also incredibly easy to drive, once you are accustomed to its size and all the various buttons.\nOut on the open road the NSX is as outrageously fun as anyone could wish for, but surprisingly it excels in a busy town too. Its nine-speed automatic gearbox makes it simple to control and its large wing mirrors mean vision is decent for a car of such stature.\nI've had the privilege of driving many fast cars but the NXS is among my favourites. From a standing start it leaves you breathless, but fear not as grip is plentiful. The engine doesn't have the same deafening roar of its rivals, but what it lacks in noise it makes up for intelligence.\nAs for my daddy-daughter day out, once the driving was over I treated Belle to an ice cream and sat her down to explain the benefit of downsizing our home and using the money to buy a supercar. Unfortunately, her favourite phrase at the moment is 'go away daddy' \u2013 to which Elinor just laughed.\nNormal 'Clocking up the Smile' service will resume next month, and in the meantime I will continue to weigh up the pros and cons of the NSX to see whether I can find a space for it in my fantasy garage.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"a very detailed account of the church in nauvoo, reasearched for 20 years. thought provoking and deep.\nwritten by an RLDS (now Community of Christ) historian. it tells the story of emma smith after the exodus, through letters and recollections of others. interesting, although not extremely well put together.\na birthday gift from my mom, for HER birthday. she wants all of her children and their spouses to read it as a gift to her. i have long wanted to read it, and i am loving it. EXCELLENT marriage book. it is also helpful for parenting and other loving relationships. (the author has written more books focusing on children, teens, etc).\nBy Mrs. Child, author of \"the Frugal Housewife,\" \"Evenings in New England,\" etc.\nthis was a find i made in Joseph smith's red brick store on monday. (it is a reprint, and not the original.) I am thoroughly enjoying reading the advice on mothering that was new and revolutionary in 1831. and i agree with much of it. what a treasure! i love history, especially this era, the early 1800s. and i love motherhood, so this is a delightful book for me to read.\nthis was my ultimate find in the red brick store. (i was pleasantly surprised by the wonderful collection of items from the past that are for sale in the red brick store). i'm so excited! this book shows in highly detailed drawings and explanations how to make all types of clothing and household linens, baskets, bonnets, draperies, etc for a household in the early 1800s. i love it, i love it, i love it! oh, the section on baby's clothing and linens! i want to make them all - so precious. so beautiful. i want to photograph the illustrations for you, but alas, no camera lens as of yet *sigh*. i will post more about this book later. i won't be able to resist.\ni suppose, or rather am quite sure, that i am odd. very few people would find many of these books, especially the last two, at all interesting, and they are my two favorite. i have always loved the past, and longed for it's simplicity. longed for a better understanding of those who lived before. i dislike the word \"trendy\" very much. to me it has very negative connotation. and i am continually surprised to find it used in ways that people perceive as positive. i strive to avoid trendiness. my first college major was history, my degree is in clothing and textiles, and those are two of my great loves. and hence my love of antique quilts, and most things antique. this last book is a blending of my favorite things, and makes me very happy. this week i also unpacked some beloved textile books from my university days that i haven't seen for 13 years or so (yes, we are still unpacking). they have been in storage, awaiting retrieval. one of my favorites is \"history of costume\". i love to turn through its pages. i would love to do some work in this area. and i'm living in a great place to do it. i just need to discover what it will be.\nwe are moving. soon. our truck, packed with everything we own, was taken away today. our kids are on our back lawn, having a party with their friends. we have been packing, cleaning, and arranging for days. and amidst it all, my toes are pretty. a new salon opened in town. today. and brock treated me to a pedicure there. it turns out that he has been checking it regularly to see if it was open. and today was the day. he made an appointment, gave me the money, and told me to go. he is the best. if i haven't mentioned that before, i definitely should have. and the facts are that i am undeserving of him. but i have him anyway. and i am grateful. the pedicure was heavenly. they had a massage chair\/foot spa combo. it was so relaxing that i couldn't get up, and sat there for another 10 minutes after she had finished my toes. she said i could. and normally i wouldn't have. but it felt so good after days of lifting and heaving and dirt and grime. i love brock.\nhere is the party on the back lawn. maika is here from japan. we picked her up at the airport yesterday.\nat 13 months, gracie has decided that she is 2. but that is a subject for another post.\nlast week mia was reading a book of fairy tales, when she suddenly made a very disgusted noise and said..... \"that could never happen, they have to GET ALONG first!\"\nof course, my interest was piqued, and so i asked what she was reading. it was \"cinderella\" and she quoted to me from the last paragraph..... \"he was so overcome by love and joy that he and cinderella were married that very day! they lived happily ever after.\"\n....\"that's not how it works! they have to take time to get to know each other and find out if they get along and can be happy together!\"..... she informed me.\napparently she has it all figured out, in first grade. that's my mia. should i be happy or sad that she doesn't believe in love at first sight?\ni have been feeling lately how completely blessed i am to be able to spend my days with the people i love and adore - all day, every day. they make me so happy. i get to hang out with just these two cute little people all morning, until brock (another one of my very most favorite of all people) wakes up, sometime before lunch (he works on tokyo time, and so far into our night), and makes us a foursome.\nit's a treat to spend time with him during the day, instead of the more traditional evenings after work, when we are both worn out from long days.\neventually school gets out and we are joined by three more fun people, when their busy social schedules allow.\ni am the luckiest,and i truly feel it. because of infertility, i got to experience two years of free time while my two oldest were at school,and number three hadn't arrived yet. all of my friends with small kids at home were jealous and repeatedly told me how lucky i was to have all my kids in school. well, guess what? although i spent my days doing all of my favorite things - the things i had felt frustrated about not doing as much as i wanted when my kids were tiny: scrapbooking, quilting, crafting, reading, exercising, going to the temple, meeting brock for lunch, serving others, getting together with friends, shopping, keeping my home clean and organized, studying japanese and other subjects i love, etc.......i missed my kids and i felt incomplete. i discovered that nothing is as fulfilling for me as hands-on raising my kids, in person. what could be better than spending my days surrounded by people who make my heart happy?\nmy blog has been calling to me all throughout the morning. and even though i have so many other things i need, and want to be doing, i can't resist the call any longer, and so i will post.\n1. a soft fluffy, fluttery baby that reminds me of butterflies, dandelion fluff, and snowflakes. the luxury of being able to cuddle with her whenever i want to will fly away as fast as those three things can, and so i will savor every moment of this luxury.\n2. playing lazily in the morning with a sweet and active little boy while we linger in pajamas.\n3. staying up late to listen to cute teenage girls happily discuss all the latest drama in their lives while feeling grateful that they want to tell it to me.\n5. sharing my life with a simply amazing husband who is almost too good to be true.\nwednesday the 29th was our 18th anniversary. brock points out that it was a significant one, because i was 18 when we got married and so now i have been married as long as i was single. he's almost right, but i was pretty close to 19 (so young, I know, but i was halfway through my sophmore year at BYU, and graduated a year and a half later, at 20 - i was always in a hurry, and i haven't ever regretted that youthful decision)), so not quite half of my life has been spent married. but now i'm quibbling over insignificancies, because what is of real importance, is how amazing the last 18 years have been. truly. we spent the day together, just relaxing, and enjoying each other's company. noah and grace spent the day with us too. it was a beautiful day, not because of what we did, but because of what we felt. being back in utah for our anniversary for the first time in 16 years made it more enjoyable, because the weather, the sky, the colors on the mountains, the feeling in the air, was the same as that day at the manti temple in '89. now we are heading into september, possibly my favorite time of year, because it reminds me of being a newlywed and it was also the month, two years later, that lexi joined us. september always takes me back to those two blissful, life-changing times. brock is my perfect man, and all these years later, i love him more than ever. i know the Lord loves me very much, to have given me the blessing of sharing my life with brock, now and forever. he always puts me and our marriage first - always. he's so much more than i deserve - honestly.\nbrock found this wonderful old wash basin and pitcher for my present. this is something i've wanted ever since the one we found in japan was broken in our first move back.\nthe stack of quilts in the photo are some of the antique quilts i will be displaying at the Arts Festival this coming friday and saturday. i have 78 antique quilts, and here's where it gets weird.....39 are from 1850 to 1900, and 39 are from 1901 to 1950. also, 39 are quilts and 39 are tops. this is odd, because i collected randomly, without a plan, and this is how it ended up. i don't think i will have room to display them all, but most....fun fun!\nbrock and i went on a little day trip on friday and spent more time together for our anniversary. gracie joined us, and noah played with cousin andy. but before dropping noah off, we treated ourselves to a photo shoot with gloria's sister kelly, of kelly's blog. YAY! we finally got to meet her! and we are so looking forward to seeing the photos. i will share them in an upcoming post.\non the flight home from guam, i emerged from teeny tiny airplane lavatory without my wedding ring. i had gotten quite puffy while in guam, and it was getting tight on my finger. so i had moved it to the pinky finger of my right hand. brock had been encouraging me to take it off completely, and put it somewhere safe, but i don't like to be without it. i immediately realized i was missing it, and waited in line to get back into the bathroom stall and perform a search. i even looked in the garbage (yuck!) in case it fell when i was throwing my hand towel away. i didn't find it. brock was very disappointed, but handled it well. he lost his wedding ring several years ago in a bad experience with security in L.A. airport. we mentioned to a flight attendant that we had lost my ring, so that if it was found it could be returned to us. i started resigning myself to images of brock and i shopping together for a new set of wedding rings. about a half and hour later, brock said he was going to ask if he could go through the garbage bag in that bathroom, because he had a feeling that's where it was. he went to the back of the plane, and no sooner had he left, then the darling little japanese flight attendant who i had told about my ring showed up. she had my ring, and a cup of soda water to wash it in. she had been digging through the garbage herself, until she had found it. i forgot to mention that when we first told her about the ring, she brought a flashlight to my seat and took both my seat and brock's seat apart and gave them the most exhaustive search you can imagine. i was so excited (& amazed) that i forgot to ask her name, and we never saw her again, even when we were deboarding the flight. so i am sending out a big thank you and very happy wishes to her, wherever she is. it's so comforting & encouraging to know that there are people like her in the world.\nwhere, oh where have we been?\nbrock and I have been in guam for six blissful and glorious days sans kids, sans worries, sans work, sans everything stressful. brock has been trying to plan and execute a trip for the two of us for years. but it is difficult when you live in japan without family to babysit. and it is also challenging when you have noah. but we finally felt noah was ready to survive us leaving, and we had to hurry before new baby arrives. brock asked his sister emily to leave her three young kids for a week and a half so she could come and babysit for us. i couldn't believe he had done it, and either could she. but her husband talked her into accepting the offer\/request to come, and he was great about stepping in and taking care of everything while she was gone. thanks adam! and thanks to emily! we are so grateful. we love you!\none of the best things that happened on this trip is that i remembered what it feels like to be totally relaxed. that was a blessing. it's been so long since i have felt complete relaxation that i couldn't recall the feeling. now that i am back to real life, it is difficult to achieve that feeling. but i do remember what it feels like, and that brings me closer to it. the day before we left was incredibly stressful, including bad news from the embassy about the processing of noah's visa (change in laws that is changing everything and no info on what to do now) and very bad news for brock at work. but all of that melted away and disappeared while we were gone, miraculously. this is the spot where we relaxed in the shade of the overhanging tree branches and watched the ocean and read books, just for fun (not self improvement books that we felt like we should read) while we listened to the waves. because i am not capable of tanning, only burning and freckling, i was excessively fond of this beautiful and shady spot.\ni am very grateful and so lucky to have brock, and i enjoyed him so much while we were away. i even found myself having a photo shoot with him as the subject, which never happens, because i am too busy taking photos of the kids. he is perfect for me, and a yet at the same time, a better husband than i deserve. and although i know that, this vacation helped me to remember it in ways that i haven't thought about for too long.\nwe planned this trip for the end of january and early february, but the u.s. government decided to have fun at the expense of us all, and took two months to process emily's passport, even after paying the expedition fee and getting many, many promises and reassurances that she would get it on time......a long and not so glorious story that could take up too much space here. so our trip was postponed twice, and many fees were paid to make the necessary changes in flights and reservations. at the end of january, i wasn't looking pregnant, and i didn't really look pregnant up until a few days before we left. but while we were there my belly suddenly popped out and i was clearly pregnant. i was concerned about fitting into the new swimsuit that brock had insisted that i treat myself to ordering for the trip, but it worked. i decided that it was better after all to be pregnant on a tropical vacation. it is a built in excuse for not looking swimsuit perfect.\nbecome more adventurous than me, and got his diving license last fall. lucky for me, he still enjoys snorkling with me.\nthank you to our wonderful kids.....the big ones who were so helpful and responsible while we were gone, and the little ones who managed not to be too traumatized by our departure. and thank you again to emily and adam and kids, and grandma and grandpa who helped watch their kids. and thank you especially to brock - I LOVE YOU! let's plan our next trip!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Spend a pleasant, family-based 900 meters from the seafront with its sandy beaches. Many hiking trails along the coast Abers and legends will give you the salty air from the sea. Not to mention the discovery of the shellfish in the tides. Your stay will be in a very comfortable house (3 keys cl\u00e9vacances) in a sublime, with an enclosed garden with a gate (swing and slide).\nFrom the highway Brest Paris, head of Gouesnou-lannilis, then follow the route to the village of plouguerneau.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The instrument used in this recording is a 13-course lute by John Rollins, Bellingham, Washington, 1980, after Joachim Tielke, Hamburg, 1678.\nHe is an active chamber musician, playing with the ensembles Les Deux Amis, Trio Galanterie, and La Chanterelle.\nJohn Schneiderman currently directs the Guitar and Lute Program at the University of California, Irvine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Numerous investigations have shown that large fractions of the population experience reduced health, comfort and productivity (a bad quality of life) when working in certain office buildings (Bullinger et al., 1999; Engelhart et al.). Both building related exposures (described by indoor climate and functional and aesthetic design) and psychological stress-factors related to load and organisation of the work of occupants may contribute to the problems. There is a need for an assessment method for office buildings that includes measures of all the factors, which may have an impact on the occupant's health, comfort and productivity. and is sensitive, reliable and valid. Furthermore, the instrument must fulfil certain demands, such as being easy to administrate, time economic, and easy to understand for the subjects, criteria which few existing forms meet. The present instrument, a discomfort form for use in work environments, was developed with the aim that it may be used in ordinary office work places, covering various aspects of complaints related to the indoor environment.\nLaike, Thorbj\u00f6rn, and Rikard K\u00fcller. \"Development of a Test for Assessing Discomfort in Workenvironments with Varying Characteristics.\" In Culture, Quality of Life - Problems and Challenges for the New Millennium: IAPS 17 Conference Proceedings, 619-620. IAPS. A Coru\u00f1a, Spain, 2002.\nAbdou, Ossama A., and Amal A. Abdou G. Kholy. \"Correlation Between Indoor Environmental Quality and Productivity in Buildings.\" In Environment, Health and Sustainable Development (IAPS 19 Conference Proceedings on CD-Rom). IAPS. Alexandria, Egypt, 2006.\nWagner, A, K Schakib-Ekbatan, and P Schweizer-Ries. \"The Influence of Building Energy Performance Standards on User Satisfaction in Office Buildings in Germany.\" In Human Experience in the Natural and Built Environment: Implications for Research, Policy and Practice (IAPS 22 Conference, Abstracts of Presentations). IAPS. Glasgow, UK: University of Strathclyde, 2012.\nS. Ornstein, Walbe. \"Trends in Post - Occupancy Evaluation Applied in Greater Sao Paulo, Brazil: Environmental Quality of Workstations.\" In Shifting Balances: Changing Roles in Policy, Research and Design (IAPS 15 Book of Abstracts), 71. IAPS. Eindhoven, Netherlands: Eindhoven University of Technology, 1998.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadgwy b/data_all_eng_slimpj/shuffled/split2/finalzzzadgwy
new file mode 100644
index 0000000000000000000000000000000000000000..e975d49d7fa05567d7bc549250cafc9037033584
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"remains intact and rarely bleeds significantly.\nFor a long time, the practice used to be 'Once a cesarean, always a cesarean'.\nRepeat cesarean sections represented the majority of operative deliveries.\nfold increase in uterine rupture.\nChanges in the fetal heart rate are often the first signs of uterine disruption.\ndeceleration is highly suggestive of a uterine abnormality.\ntimely diagnosis and immediate operative surgical delivery.\n& Borten help you assert your rights and get the compensation you deserve.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Get critical shipment alerts by e-mail for your most important packages. You may choose to send ship, delivery, return, and delivery exception notifications to up to five e-mail addresses.\nQuantum View visibility tools empower you to monitor, manage, and control your supply chain.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Well hi there, how have you been? I don't know about you but here at Casa O'Meara we're all looking forward to the summer holidays, only a few days to go. The down fall to having the monsters at home for 6 weeks is my sewing time gets seriously cut. So before my project progress becomes non-existent I thought I'd share what I've been up to.\nA few Sundays a go I got to spend a bit of quality time with Monster no1 helping him with his craft homework, he had to make a sea creature of his choice. After trawling the internet he decided on a banded sea snake, we raided my stash of felt and got to work.\nIt was great to see him really concentrating and enjoying using the sewing machine. I don't think I've seen him sit still for that long since he started walking! He was so proud of his efforts, he did a brilliant job. I only had to help him sew the curve of the head and to close up the tail.\nAs well as homework I've been working on my handmade wardrobe. We've finally booked our holiday and I'm hoping to have plenty of handmade clothes to take with me. My project list is full of beach cover ups and lightweight easy to pack items. First tick off the to-make list was a quick little upcycle. My t-shirt I love but have sadly shrunk out of (yup that's right!) was too ridiculously big to wear anymore so thanks to pinterest and my fabric scissors I chopped the sleeves off, changed the neckline and plaited the back.\nI already have an old lace t-shirt I bought a few years ago, its cheap but just the right size, so I used that to draught myself a pattern while Monster no2 took a rare nap.\nTo finish it off I decided to trim the hemline following the lace pattern, for a cute scallop edge. I love the neon binding and the dropped hem at the back, its little touches like that I like the most on handmade clothes, that's when you get to really put your personality across.\nMy last project to show you for now is a fun little make up bag I made as a gift for a friend, I hope she likes it!\nMy eldest is fascinated with my sewing machine at the moment. It's lovely seeing them so excited! Also your lace t-shirt is fab.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Janlatae Mullins & Chris Anthony Hamilton discuss their recent films Soul Fire and Something Bigger as well as working at BARK BARK!\nErin Bethea and Drew Waters tell us about their new film New Life and all the upcoming films they have planned to produce in Georgia!\nThe filmmakers from short animated film \"Lil Bat\" discuss making the film in part 3 of our Dragon Con Film Festival coverage!\nRyan Monolopolus and cast and crew from his three latest short films talk about the Global Film Conspiracy screenings coming June 24th!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"OMG Weiya? Whitefield Brothers? OMG.\nall the shows are DOPE!!!! But the intro is getting REALLY REALLY REALLY ANNOYING!!!!!\nCouldn't ask for a better selection, you got quality ish in every single show here. Big up!\nloving the afrocentric flavours. nice to hear a bit of beat variation. plus i haven't heard that y'all so stupid track in years, killer!\nChecking out some newly compiled funk on \"Impeach The Precedent\" and from Soul Fire, some afrobeat flavors, a Roy Ayers sample moment and hip-hop on the chill side.\nDJs & GUESTS Professor Groove, DJ Static \/ RECORDED April 29, 2005 \/ PLAYLIST MARKING carls \/ HOSTING daduke, PJ, Mike & Oliver. You can help too!\nsharon jones & the dap-kings - what if we all stopped paying taxes?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaegbv b/data_all_eng_slimpj/shuffled/split2/finalzzzaegbv
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index 0000000000000000000000000000000000000000..0fa1d762006287b5da2bd80f9160ee12e0651a28
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"The medical benefit of cannabis isn't all that it's cracked up to be, according to the largest review study to date on potential marijuana health effects.\nMarijuana does seem to provide moderate relief for people in pain, including patients suffering from chronic nerve pain or cancer pain, as well as help people who have multiple sclerosis and experience muscle spasms, according to a new examination of hundreds of studies of medical marijuana.\nBut it appears to have very little impact on several other conditions for which it's used, including sleep disorders, Tourette syndrome, anxiety disorders, depression or psychosis, HIV-related weight loss and nausea, said the study in the Journal of the American Medical Association.\nBut the scientists also said many studies were small or their scientific process flawed, and called for more and more rigorous research into the medical effects of cannabinoids, the compounds that are the active ingredients in cannabis.\n\"There is evidence to support the use of cannabinoids for the treatment of chronic pain and spasticity,\" said study co-author Penny Whiting, a research fellow in epidemiology and health services research at the University of Bristol in the UK. \"However, this needs to be balanced against an increased risk of side effects,\" which can include dizziness, dry mouth, nausea and sleepiness, she added. \"Individuals considering cannabinoids as a possible treatment for their symptoms should discuss the potential benefits and harms with their doctor.\"\nThe study was commissioned by the Swiss Federal Office of Public Health.\nTwenty-three states in the US have approved medical marijuana with the idea that cannabis treats a variety of ailments.\nSupport to use the drug for certain health problems has \"relied on low-quality scientific evidence, anecdotal reports, individual testimonials, legislative initiatives and public opinion,\" wrote Dr. Deepak Cyril D'Souza and Dr. Mohini Ranganathanof the Yale University School of Medicine in an editorial accompanying the new study. \"Imagine if other drugs were approved through a similar approach.\"\nThe new review, analysed information from nearly 80 studies that included about 6,500 people total, in which participants were randomly assigned to take cannabinoids, a placebo or another drug.\nIn more bad news for stoners, a separate study in the same journal found that THC content was far less than advertised in 62 of 75 edible marijuana products tested.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It turns out we're a little too loaded up with amazing stock, so come down and grab a bargain + check out all the new tidbits with everything being reduced for just a couple of days!\nThere will also be accompanying cupcakes and nibblies for sustenance in those exhaustive turbo shopping times, and the fitting debut of a little project Ive been working on for the Easter window-more on that soon.\nNote: To make my somewhat ineffective advertising relevant please seek out Grey Gardens immediately.\nDo you know - I've actually seen that movie! I found it rather sad, although I would have loved to help sort through their old wardrobes! Is that terrible?\nI shall be popping down to look at your goodies very shortly!\nOh I just adore it!\nHaha i'd gladly accompany you on said wardrobe rummage. Little Edie was so gorgeous when she was young, she'd have some AMAZING pieces.\nThey're actually shooting a film based on the doco now with Drew Barrymore and Jessica Lange, I think it's a rather silly thing to do.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Getting back the crown to rule again over the kingdom of the night!\nCan you hear the bells chime for the final goodbye.\nHear the bells! there's no way back.\nYour world is mine. I'm still alive!\nWhy don't you take my - my invitation?\nWhy don't you want to get out, get out...?\nI saw, I saw the first Horse.\nI saw, I saw the third Horse.\nHow do you feel when thoughts fill up your mind?\nWhat is wrong with your world?\nAm I - larger than life?\nAnd you - you keep me alive.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Classic Cotton Popover - Stripe - BLUE\/WHITE - XS - Talbots is rated 2.7 out of 5 by 3.\nA classic half-button top with a modern twist! This blue and white stripe top features floral trim under each cuff for a fun contrast when the sleeves are rolled.\nRated 2 out of 5 by Jude from Impossible to iron Cute blouse, but it wrinkles terribly. The stand up collar is impossible to iron. Definitely could not take it on a cruise.\nRated 2 out of 5 by Mesa M from Wrinkly Cotton Extremely wrinkly 100% cotton fabric. After machine wash and dryer damp-dry, it definitely needed ironing. And because the placket does not open all the way down, the collar and shoulder areas were difficult to iron. I still like the blouse style though.\nRated 4 out of 5 by Elizabeth1001 from Lightweight This is a very similar shirt that another brand (very preppy with whale logo hint hint) had this year. This is a lightweight more casual feel top. And just like that other brand I'm not sure if said ruffle is annoying or cute. It certainly adds some interest to an otherwise standard top. TTS. looks like fit in photo. PS Talbots employee and shopper.\nWhat is the length of this blouse, neck to hem?\nThe length of an item depends on individual sizing. For personalized assistance, please contact a Personal Shopper at 1.800.825.2687.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At Dumpster America, we strive to bring you the best service at the best price available. We have a wide selection of dumpster sizes in stock in Snellville, GA, with flexible & timley pickup and delivery. Give us a call today for all your dumpster rental and roll off needs.\nDo not procrastinate employing a local dumpster assistance if your house is in a vital state. Too much garbage, trash, along with clutter merely leads to much more filth plus an unhealthy circumstances. No one actually wants to stay this way, especially if there are little ones and dogs and cats involved.\nThank goodness when you get began on your home design, the next day you are able to rent a dumpster. To figure out which size of dumpster to get look at the dimensions of the construction undertaking you will be taking up.\nEvery undertaking is made less complicated and easier having quick rubbish removal. Cleaning out the shed along with removing unnecessary broken tools such as aged rakes and spades can be done swiftly without having to stack the trash out by the road along with hope how the local rubbish pickup requires the entire stack. You can make guesswork outside of cleaning out ones garage, attic, or shed by eliminating the requirement to use community trash pickup. Construction projects are also doable as one dumpster are prepared for old home equipment, drywall, toilet fixtures, sprained metal along with broken goblet. Simplify your own home projects through calling or perhaps ordering on the net a Maryland Rent a Large Dumpster in Snellville, GA.\nMost homeowners learn that Rent a Large Dumpster in Snellville, GA is the best way to take care of the rubbish when undergoing a home reconstruction. This type of assistance should provide a person with the pot and, after you are through with this container, they'll dispose of the garbage you have remaining in it. This service really should have a set fee for each sort of dumpster they rent available, but if you are going over the fat limit, the purchase price would be fine-tuned.\nYou can clean up the back garden, dispose of that will trash right after your home's renovation and can do a lot more for keeping your house clean by using these services. This disposal bin and Rent a Large Dumpster in Snellville, GA companies help you point out bye to those heaps of rubbish of your house in the convenient way only.\nMaybe you offer supplied condos to ones condo renter's. You will have to dispose off the bedroom air mattresses from time period to time. Whenever you rent a dumpster it is deemed an easy way to dispose off large along with bulky air mattresses. Any other aged furniture can be easily thrown out inside dumpster as well.\nVersion is one of the approved processes to home the spend that is developed through projects like property renovation, design and business projects. On this process this waste material will be burnt into ashes, but the drawback of this process will be number of unwanted gas like carbon dioxide and dangerous gases are freed into surroundings causing significantly harm to the earth. The effect of these harmful unwanted gas will not be mitigated until the exhaust of unwanted gas into the surroundings is avoided by using most advanced technology. However this method involving disposing this waste is incredibly expensive and is advisable merely in the places where there is not any enough land to dispose the garbage.\nAs far as prices are problem than prices depend upon firm to company. You could find many companies that offers their companies on toned charges that you may possibly find more expensive whereas some companies provide the service about rent basis as in you can carry bin or perhaps dumpster on rent. Where there are some organizations which provide a person same companies in cheaper rates however, you have to be extremely careful while selecting these kinds of organizations as there may be hidden prices. So it is a good idea that you should opt for companies offering you bin or dumpster about rent. As most of these lenders are spread around many towns so you can locate fairly easily your most adjacent service provider. The main part is perhaps you can save your bin or dumpster move money which usually many other organizations can ask you for.\nYou want to find a dumpster company you can trust. Ensure that their services are what you are looking for as far as customer satisfaction. You are able to look online with regard to reviews with dumpster companies or perhaps any company that you are thinking of appointing.\nRenovation initiatives aren't the one causes for a Rent a Large Dumpster in Snellville service. Occasions such as big outdoor wedding parties or parties compile huge amounts of garbage. Naturally it shouldn't need that everyone will certainly walk with a dumpster to dispose of their garbage. However, the particular containers tummy flatness, although in helpful more so during the cleanup process. Items like plates, pieces of food, plastic tablecloths and larger objects would likely be left lying around and may prove too much for a couple of random average-sized garbage bins. Other than, those would possibly already be crammed as a result of individuals who decided to dispose of their own things.\nDefining Size of Dumpster: Normally dumpster comes in five styles viz 10 cubic yard, 20 cubic yard, Thirty cubic yard, 40 cubic yard and also 40 cubic backyard. It is recently been observed that unlike 10, Thirty, 30, 40 cubic yard dumpster, 20 cubic yard dumpster are difficult to find.\nThe dumpster service are equipped for any kind of work, from a to z. You will not ought to put your crap in totes or canisters. They will carry out the work as you relax. They will help you get reduce all your garbage and crap such as outdated appliances, stove tops, furniture, appliances, washing machines, dryers, toilets, tubs, cabinets, floor coverings and such. They're able to help you cleanse the storage area, shed, downstairs room, backyard, or perhaps attic. They're able to also take out broken goblet, plastics, alloys, and other supplies. They will conserve time because you not have to break up the items yourself or generate the trash with a dumpster.\nLet me make clear you, whole lot many modest industries and also restaurants will often have trash disposal on daily basis. Dumpsters waiting in the front of checkpoint ways to acheive a pick from the city haulers frames a dirty graphic in the brains of the buyers on their vacation to the business prisms, as a substitute you can go for renting a considerable size dumpster and possess it used in the downstairs room of your building or in a place out of your buyers view and also dispose the trash at occasional intervals. The particular dumpster should be used in a safe area and the front yard of the area should not chuck any challenges at the hauler even though picking the particular dumpster.\nYou need to be in the home when the company comes simply by for get. If are absent on that day, you could be recharged one more few days if they have to return later.\nThere's no need to have volunteers spend extra time separating the trash or perhaps bagging that when you rent a dumpster. You can allow weather determine if the 1st few days in the thirty day period is the best time for the dumpster supply as you can call up and have the dumpster supplied the very next day from the Maryland spot. No matter what the project, it can be possible with a dumpster.\nAnother excuse your parents can be moving out of their apartment is they need to move to yet another state. Maybe in your mom's or dad's golden several years someplace more comfortable intrigues them. Or perhaps they want to turn to the outlying to enjoy far more peace and quiet. The city may intrigue your parents while they will not have drive an automobile as much and they can get their own exercise when walking to the food market or eating places that will encompass them.\nIn terms of a car port that needs clearing out, chances are great that there is a quite huge amount associated with junk to be thrown away. Since a lot of regional trash companies impose restricts on how considerably trash place out weekly, and a lot of things may just be too big and unclearly shaped regarding trash totes anyway, you'll want to rent a dumpster from a reliable Rent a Large Dumpster in Snellville, GA company. Only picturing a dumpster-sized load associated with junk absent from your car port should make the task taking place seem a lot more manageable.\nOn hearing the saying prevention of contamination your minds conjure up using the idea of mitigating the particular emission associated with fossil fuels produced by the travel of cars. It's a good sign, but without a doubt, the term Smog is not centered on the air contamination and noise pollution that is certainly generated from cars. There are lot many other aspects contributing to contamination of the atmosphere in the cities such as design waste, environment friendly yard spend, industrial spend.\nCheck dumpsters at apartment buildings and the starting and end of calendar month. When new tenants move in\/out they covering to get rid of many things of worth.\nMost people have a look at renting a dumpster as a possible expense. This particular cannot be denied, as it is expensive to borrow a dumpster from a corporation. However, folks rarely think about the savings of which using a dumpster can offer. Businesses often use them, because dumpsters are the most efficient way to toss a large volume of garbage. Nevertheless, a dumpster can also ease any risk of strain on an person's pocketbook. Here i will discuss three ways dumpsters help save people cash anytime these people take on substantial projects.\nThis is a good sign that you have considered a decision to face waste made in an Eco- warm and friendly manner nevertheless, important task here is the approach you plan to dispose the particular generated spend. Trash made in the tasks like design, house reconstruction, industrial tasks and environment friendly yard cleaning up is very huge. Planning to home the spend on personal makes you fall sick with a few injuries if it's not properly handled. Handling these kinds of huge a ton of waste needs expertise as well as professionalism. It even gets rid of your time because all the rubbish will be handled manually.\nEventually, once you are accomplished using the rental dumpster, a call to the rental company for just a pick-up is required. Almost all companies have different rental time periods, and understanding the number of days granted for a rental can be imperative to understand before making a payment.\nOnce you've got the house decluttered and organized, you should keep a supervision out on your bad habits. The cluttered wreck didn't come about overnight the 1st time, and it will slowly but surely creep in to your daily life if you aren't mindful! Throw away spam when it comes within. Keep your property tidy and organized. Ask yourself every now and then if you actually need some of the stuff that have been getting dusty. With a little vigilance, your home will always be clutter-free.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Sparkling diamond moon Labyrinth ring charm. Set with tiny pave set diamonds, this charm looks wonderful simply alone or teamed with our twinkling star charms. The possibilities are endless, be it day or night, this little moon is guaranteed to sparkle.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mobile trommel screener Pronar MPB 14.44 is the smallest unit of the whole family of MPB screening machines. This device has a 4.4m long sieving drum, which can be replaced without removing parts of the machine. Despite the small size, 14.44 MPB retains all the advantages of larger MPB 18.47 and MPB 20.55.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Renovated one bedroom apartment on a lovely tranquil setting. This is a first floor apartment with sliding glass door opening to a patio. The kitchen is renovated with granite counter-tops. There are hardwood floors throughout and a nicely renovated bathroom.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Aren't these paper eyelashes from Paperself AMAZING? Blogged By Elissa Elissa founded Calico in downtown New Bedford, MA in 2005. She can usually be found at Calico HQ snapping pics for Instagram and heading up photoshoots with the team!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"While International Day of Happiness takes place this week on March 20, research from Lee Hecht Harrison Penna (LHH Penna), a global people management firm, shows that many employees are struggling to look on the bright side. The new study shows that over a third of employees view their work in a negative light.\nThe research found that 14 percent associate work with being unhappy, while nearly one in 10 would go as far to describe their work as \"horrible.\" Anxiety was also found to be a common cause of employee grief, with a fifth of employees claiming their work caused them stress and angst. It was also found to be a greater problem among women, with a quarter describing their work as anxiety-inducing, compared to just 16 percent of men.\nNick Goldberg at LHH Penna said, \"With our working life and private life becoming increasingly integrated, unhappiness at work can easily spill over and become all-consuming. While it is encouraging to see that 38 percent of employees have only positive things to say about work, our research also shows that more needs to be done by both the employee and employer to improve workplace happiness. Today marks a good day for employees to ask themselves if they are truly happy at work, and if not, to ask themselves why and what steps they can take to address it.\" Here are a few ways you can improve your outlook on work.\n1. Prioritize your personal life.\nMake sure you have other activities in your life beside work to help detract from any negativity. After work and on the weekends, make sure you switch off both mentally and physically and spend time doing the things you love most.\nIf you're finding work repetitive and boring, proactively seek out opportunities for more diverse work, whether that's asking for a role in another division, taking on new responsibilities, or volunteering yourself for training. Your employer will also commend you for taking initiative.\nIf you're committed to making a change, then find a time to sit down and chat with someone senior about the cause of your grievances. The likelihood is that they will want to do all within their powers to help make work a happier place for you.\n4. Change can be good.\nIf you are unhappy at work, then take matters into your own hands. While changing your career or job may seem like a daunting prospect, it could be what you need to be truly happy again at work.\nWe spend the majority of our lives within the office, therefore building strong relationships with your colleagues can help to add comic relief or serve as a support system when you're having a difficult day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"It was a glorious Summer: sunny days, little rain, and starry clear nights. But the camping season doesn't end on Labor Day. We'll camp by the Delaware in late September, in the New Jersey Pine Barrens for Columbus Day and upstate New York in late October to see the leaves at their most colorful.\nWe're also planning events in New York City such as a walk over the Brooklyn Bridge followed by lunch in Chinatown, a bird-watching walk in Pelham Bay Park in October and a Chinese banquet in November.\nTo reserve space on a trip or day event, call the leaders between Sunday and Wednesday of the previous week, before 10 PM. If you need information earlier than that, call David Levner at 718-897-1448.\nCamp on a group site by the Delaware River. Swim, boat or fish right from the campsite. Hikers can climb up the Appalachian Trail to Sunfish Pond, an unspoiled glacial lake, or visit the Mohican Outdoor Center nearby. Chemical toilets by the site, flush toilets and showers in another part of the park. Limited to 30 people. NO CALLS BEFORE AUGUST 24.\nWalk from Brooklyn Heights over the Brooklyn Bridge to Manhattan and share a Dim Sum brunch in Chinatown. NO CALLS BEFORE SEPTEMBER 7.\nWe're camping on an island in the Delaware River. You can swim, fish, canoe, raft, kayak or tube. The Recreation Area is part of the Delaware & Raritan Canal Park which contains a 30-mile mixed use trail (hiking and biking) along an old railroad bed that parallels the Delaware & Raritan Canal. Flush toilets and hot showers. NO CALLS BEFORE SEPTEMBER 21.\nTake a hike in the biggest park in the Bronx. We might see songbirds, hawks, osprey and dragonflies. Bring binoculars if you own them and a prepared lunch or food to cook on a grill. Accessible by public transportation. NO CALLS BEFORE SEPTEMBER 28.\nThe largest park in New Jersey is in the Pine Barrens, a sandy area in which pine trees predominate and development is minimal. You can hike, bicycle, fish or canoe. Batsto Village, a restored 19th century town, is a major attraction. Flush toilets and hot showers. NO CALLS BEFORE OCTOBER 5.\nSurrounded by yellow, orange, gold, red and brown leaves, we'll hike, bicycle and stay warm. The Escarpment Trail that leads to Kaaterskill Falls starts in the park. Flush toilets and hot showers. NO CALLS BEFORE OCTOBER 19.\nJoin new and old friends for this annual feast marking the end of the camping season. Bring a hearty appetite and pictures of the past season's fun to share. NO CALLS BEFORE NOVEMBER 9.\nBegin the new year on the right trail. Two nights lodging and 5 meals (Saturday breakfast through Sunday lunch) for $130 per person, with discounts for children. X-country skis and snowshoes provided, but bring your own ice skates. Limited to 10 people. To reserve, send a check for $130 (payable to Campers' Group) to David Levner, 63-36 98th Place, Apt. 5J, Rego Park, NY 11374. NO CALLS BEFORE NOVEMBER 30.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Who is ultimately responsible for overseeing the health, well-being and welfare of specific breeds of dogs? Most national kennel clubs often take the lead to promote ethical breeding of sound and health dogs, in general, by their members. These may take the form of rules, guidelines, and recommendations, directed to breeders, about ethical and breeding practices as well as health and welfare concerns for dogs under their care.\nBeyond the care of individual dogs, however, there are concerns for the overall health and well-being of each breed. Many breed and kennel clubs have a long history of evaluating the health status of potential breeding dogs, using various Health and Screening Tests. Increasingly, committed clubs have developed breed-specific programs to describe and evaluate the status of their breed, especially in regard to health, mentality and other indicators of well-being.\nInternationally, there is great variation in breed-specific approaches to health in dogs. This section will provide information on and links to various sources and approaches, under the following categories.\nMost national kennel clubs and other cynological organisations have regulations, recommendations and\/or guidelines outlining the important ethical principles and strategies that should be followed when breeding pedigreed dogs. Although there are many similarities in content, there are also differences in scope, focus, intent and other issues. Official policies from cynological organizations apply directly only to the members of those organizations. Obviously, there will be overlap with the issues addressed by the various kennel club guidelines, but the focus and scope are generally somewhat different.\nMany national, regional and local authorities also have general guidelines and\/or regulations or laws addressing health and welfare issues in the breeding and keeping of dogs (pedigreed and other) that are intended for sale.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On Saturday, October 27th at 7 p.m. join us for a movie night as together we watch \"Will You Be My Neighbor?\" the award winning documentary about Fred Rogers. Refreshments and discussion will follow. All are welcome.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I would like to take this opportunity to introduce our practice to you. As oral health care professionals, we are dedicated to maintaining the dental health of our patients through preventative care and education. However, sometimes patients can use a little extra help, especially those patients with persistent periodontal issues. At Periodontics of Niles we embrace a philosophy of individualized and up-to-date-periodontal therapy to achieve effective and long-term results to compliment the restorative needs of your patients. Additionally, your patients will enjoy our friendly and caring staff. We are committed to ensuring that your patients are as comfortable as possible.\nIn the 15+ years that I have practiced since graduation, I believe in the opportunity we have as health professionals to not only treat our patient's oral needs, but also the real possibility of helping to improve their overall health. We communicate to the patient the significance of the Mouth Body Connection. We fully understand the urgency in treating your patient's periodontal infection due to these systemic links and to also secure a strong foundation for your restorative treatment. We offer advanced dental implant treatment; proficiently working with 8 implant systems as well as computerized planning to maximize restorative outcomes. We also perform cosmetic periodontal procedures to enhance optimal esthetic results.\nWe are compassionate about providing you and your patients the best care possible. If you would like to schedule a peer-to-peer meeting or to discuss treatments we provide, please feel free contact us at (847) 685-6686 or via e-mail at [email protected] We honor the opportunity to provide you and your patients the highest level of periodontal and implant care.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Chan fhaigh mi mach Di-Domhnaich 's mo bhrogan a' treigsinn.\nTha an t-am agam a' siol, tha mi a' ciallachadh ro l\u00fdimh e.\nMar a thuirt mo n\u00fdbaidh gum b' fhe\u00fdrr bhith nam \u00danar.\nAgus fuireach mar a tha sibh gu brath gun do ph\u00dasadh.\nCha dean e shomh ach iasad gu rian chur air da-chlis.\nO 'n cuala sibh g'eil Mairi a' fas nam aghaidh-sa?\n'S ann their i dhomh gu laidir, \u00ecMar a tha sibh on uairsin?\nEadar Rubh' Bhan is Cathair a' Bhruernis.\nCha chumadh e feur rium sios na mullaichean.\nAre gone from my country.\nDon't you go my son.\nTil all our women cry.\nAre gone from this country.\nGun rachainn as a' ruidhle le'm ghun grinn Caligo.\nGur snog i as a' ruidhle le g\u02dcn grinn Caligo.\nBho'n a mh\u00cfltear gun cheartas.\nMairidh sp\u00c8is do na gaisgich.\nRi neach a tha beo.\nRi spreidh an fhir mh\u00dbir.\nCho aotrom 's na h-e\u00dain.\nis the brown-haired young lass.\nwill be seeking your kisses.\ntending the great man's kine.\nHey soldier boy is your conscience clear?\nto the one you love so dear?\nH-aon oirre dh\u00fd-oirre, chair-ich na saoir i.\n'S clomh na caor a Spaintich.\nIs mo chairdean uile ri tamh.\n'S mo chomhnaidh thairis air s\u00fdl.\nIs m'iarrtas tilleadh gach la.\nCha luiginn fuireach na phairt.\n'S an fhraoch an eilean mo ghr\u00fdidh.\nA luigeadh teich as gu brath?\nGach uair gam chumail ri dan.\nRi cuallach chruidh ann nam phaisd.\nFo riaghladh Athair na Athair na Slaint'.\nBha muirneach agam nam phaisd.\n'S a ghl\u00cfb ri creagan na h-Aird.\nAnd all my relatives dwelling.\nAs my dwelling is over the sea.\nWith my wish to return every day.\nAmongst the heather of my beloved isle.\nWho would wish to ever leave it?\nThere with the herd of cattle, as a small child.\nThat were precious to me in my youth.\nIn the glebe amongst the rocks of Aird.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"\"public declaration,\" 1640s, from Italian manifesto \"public declaration explaining past actions and announcing the motive for forthcoming ones,\" originally \"proof,\" from Latin manifestus (see manifest (adj.)).\nJanuary, 1874, Mr. Gladstone issued a manifesto dissolving Parliament.\nThe I, the No. 1, looks out from every word of that manifesto.\nI copied out the document, and gave it him with a copy of my manifesto.\nI promised he should have it, and I also sent a copy of my manifesto.\nThe authors of the Manifesto were men of great intellectual gifts.\nThese concluding phrases of the Manifesto have become the shibboleths of millions.\nThis manifesto, as it was called, was read publicly in the churches every Sunday.\nHe then gives the heads of the manifesto, from which Hume has drawn his account.\nI have still my copy of the Manifesto which Karl sent me from Paris.\nAs to the manifesto we have just heard, we will have none of it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Heat a wok or large frying pan (not nonstick) over medium-high heat until hot when you wave your hand over the bottom. Add vegetable oil, garlic, and ginger and stir once; then immediately add mushrooms and stir-fry until they just begin to brown, 1 to 2 minutes. Toss pork with 1\/2 tsp. salt and 1\/4 tsp. pepper, then stir-fry until still a little pink, 1 to 1 1\/2 minutes.\nAdd soy sauce, sesame oil, and remaining salt and pepper; cook, tossing often, until bok choy is tender-crisp, another 1 to 1 1\/2 minutes. *Find at well-stocked grocery stores, Asian markets, and online.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Brubaker Brute provides the ultimate round tone, craftsmanship & ferocity....incredible sustain, low-action & lightweight. The Brubaker delivers, and then some! It's \"like playing with butter on a teflon pan in the vacuum of outer space\"...it really allows the player to express themselves with subtle harmonic nuances or extreme growl & low-end. Brubaker Basses are the ultimate basses.\nThis is by far the finest import bass I have ever owned, and my favorite instrument of the last 4 years. My lasting impression is that you'll need to spend many multiple thousands of dollars to get a bass with better action, fretwork and playability out of the box.\nI can say without doubt it is one of the most beautiful guitars I have seen. The quality of the wood and the attention to detail is really sublime. The finish is just astounding. The spalting is perfection and the matching reverse covers just lovely when the guitar is flipped over.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Needless to say I failed in my attempt to keep a regular blog, if you scroll you can see that it died on day 3. So I'm not committing to a regular blog, but something of a random update feed.\nI find myself in the position of needing to take the plunge. Which is easier said than done. I feel like I'm standing on the edge of a cliff, watching boats set sail, forward for new and exciting journeys. I'm standing here saying to myself \"I'd like to be on one of those boats\". Yet here I stand watching some get further away, and new ones set sail. It's a grand image that I've put myself in to. I teeter bemused on the edge, wondering how I get down, how I even get a boat. But I know how to climb down, I know how to get the boat. In fact it's not that romantic. I'm on the edge of a puddle, I merely need to step through it. But it feels like an ocean, and if I step wrong, that the puddle will swallow me.\nI hope people who work in my field understand the sentiment. I see many of my friends doing incredible things at the moment, reaching goals far quicker than they ever thought possible. I've also seen friends hit the rocks, but alway re-emerge to the surface. But I'm so proud of them for even getting on the ship. I hope to be among them soon. Either sailing on, or in the wreckage of a dream.\nJust to clarify my sudden desire for my romantic approach to this particular blog. I've been reading To Fight Alongside Friends by Captain Charlie May of the First World War. It's the diary of a soldier who was killed in the Battle of the Somme, a true diary. He writes every day from the moment of his deployment, up until his death in battle. He writes every extract as if he were talking to his wife about the days events. Yet what strikes me is his poetic nature and wit. He talks of the horrors but also the boredom of war. He talks of it all with a clear enjoyment for language and the written word. It's joyous and disheartening all at once. It's a pleasure to read the words of a man so devoutly in love of his wife, and for his country. Yet devastating to think about what we have lost.\nI am reading the words of a dreamer, who lived during the time of nightmares. We now live in a world of dreams, and only speak of nightmares.\nPerhaps I've fallen for the trap, that I stare blankly at the possible defeats. But I know there are many more victories ahead of me, but only if I set sail for them.\nI hope I will soon be able to report of my departure aboard my own ship. The preparations are certainly being made, at least in part. I hope they will soon become reality, and I will be able to share my ongoing voyage with you all.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"From April to MayMs. Mastering math and accounting A proficiency with software, such as Microsoft Office products. Based on the job listings that we looked at, employers want Bank Managers to have these core skills.\nNick joined the board in Bank Manager Resume Help. Andreasson joined the bank board in It comes right off the top line of your business return. Bank Manager Books The Really Really Successful Manager \u2014 this book written by Nancy and John Banks \u2014 Lear provides insight on how to be a better manager and how to develop a winning team within your company.\nPreventing and mitigating systemic financial risks to safeguard financial stability 5.\nRetired Senior Vice President, Peoples Bank Stewart joined Peoples Bank in as a teller and during her plus year career worked in all areas of the bank. Khristopher Gefroh \u2014 Vice President, Commercial Lending Gefroh is responsible for developing and maintaining commercial and industrial relationships throughout the Lehigh Valley region through lending, deposits and cash management solutions.\nHuggins has over 34 years of experience as a practicing attorney in the areas of commercial law, creditor's rights, and oil and gas law.\nShe is an active member of the Cuba United Methodist Church. Prior to joining Peoples Bank, Mr. This saw a significant number of entrepreneurial ventures being initiated especially in agriculture, dairy and fisheries which are the staple industries in this part of the country.\nAll of these entities are based in Marietta, Ohio. From books to industry groups, this list is packed with opportunities to learn, connect and engage. While most employers did not require the following skills, multiple job listings included them as preferred skills.\nLaurel, New Jersey, during the s. He appeared as a doctor alongside his fellow ER co-star Noah Wyle in a episode and after treating Rachel who was pretending to be Monica they all went out on a date.\nClooney plays Harry, a marshal at the State Department who is cheating on his wife. Managing the State treasury as fiscal agent 8. They must also communicate company-wide changes to banking operations. Bank Managers direct the the actions of subordinates and typically report to a regional manager.\nLouis prior to joining Peoples Bank. She is currently active in Viva Cuba, providing the inspiration for the tree planting along Highway 19 and the outdoor mural project. Article 50 If any staff member of the People's Bank of China divulges State secrets or the business secrets he knows, which is serious enough to constitute a crime, he shall be investigated for criminal responsibility according to law; if the case is not serious enough to constitute a crime, he shall be subject to administrative sanction according to law.Jim Barnett, III: Vice Chairman, Peoples Bank Barnett joined the bank in as a member of the leadership team and has been a director since Prior to joining Peoples Bank he worked for the Bank of Ireland and was a Director with CapitalSource Finance.\nLisa joined Peoples Bank in and became the Bank's Controller in In her current role as Executive Vice President & Chief Financial Officer, Lisa is responsible for directing the financial activities of the Bank.\nJanuary 15, Munster, IN - Peoples Bank, and its holding company Northwest Indiana Bancorp, today announced the promotion of Benjamin Bochnowski to president and chief operating officer of the locally managed, independent community bank and the Bancorp.\nThe central bank's answer to this dilemma has been to loosen monetary policy, but in a covert fashion.\nPeoplebank's purpose is to 'make lives better by helping people achieve their goals'. Because we are specialists, not generalists, our recruiters are constantly developing their own skills and knowledge within the IT industry to ensure a candidate is placed in the optimal role and company. Designed for ambitious, highly engaged and academically talented bachelor's students, the Deutsche Bank Graduate Programme will give you the perfect foundation for a long-term career with us.\nDeutsche Bank Internship Programme. The ideal introduction to a career with us. Real responsibilities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"\"There are two main anabolic stimuli for muscle: exercise and protein ingestion.\n\" Whether or not you should wear a lifting belt is a surprisingly contentious subject. Some people are adamant that you should always wear a belt with the opinion they dramatically decrease your injury risk. Other people are adamant that you should never wear a belt, either because they think \"it makes your core weak,\" or because they think it's a truer test of strength to see what you can lift without any external aid (though this article won't be touching on that. I just see that as a matter of personal preference and is, therefore, totally irrelevant for general recommendations).\n\"We all love simple, formulaic answers to our questions. It makes life easy to be told exactly how to do something with no equivocation. In weightlifting, we all want to know right now exactly what position to be in, exactly what our training programs should look like, exactly what lifts we can expect on a given date. I always feel a bit guilty when answering questions with \"It depends\", \"There's a lot of variation\" or \"You're going to have to experiment\", but in those cases, any other answer would be inaccurate or incomplete.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Texas lawyers are permitted to ask their lawyer friends on social media for help with legal questions on behalf of their clients, according to a recent opinion from the State Bar of Texas' Professional Ethics Committee (\"PEC\"). The PEC is a committee appointed by the Texas Supreme Court that issues opinions on various ethics and professional responsibility questions posed by members of the State Bar of Texas.\nOpinion No. 673, issued in August, addressed two questions: 1) Does a lawyer violate the Texas Disciplinary Rules of Professional Conduct by seeking advice on behalf of a client from other lawyers in an online discussion group? 2) What if the lawyer asks another lawyer at a different firm in an informal consultation or discussion? According to the PEC, the answer to both is that the conduct does not violate the rules, subject to a few limitations and exceptions.\nAs written, this rule would limit lawyers to asking other lawyers only very general questions about statutes or case law or rules. Providing some facts about the case in the set-up of the question, even if presented in a hypothetical fashion, would very likely reveal confidential information learned in the course of the representation.\nAccording to the PEC, it is these exceptions that allow lawyers to reveal a \"limited amount of unprivileged confidential information\" to lawyers outside of their law firm, as long as the lawyer reasonably believes doing so will further the representation for the benefit of the client and it is not reasonably foreseeable that doing so would prejudice the client.\nThe PEC cautions that the asking lawyer should only reveal the information necessary to obtain an answer and should frame the question in a hypothetical that does not identify the client or provide information that could give away the client's identity. Care should also be taken to avoid causing any harm, prejudice, or embarrassment to the client. Furthermore, revealing privileged confidential information always requires client consent, and the exceptions do not apply if the client has directly instructed the lawyer not to reveal confidential information.\nDespite these exceptions, the opinion should bring some additional resources and flexibility to Texas lawyers who would like to consult with law school friends, former colleagues, or legal discussion boards and social media groups for quick questions without engaging in the hassle of a co-representation. The lawyers responding, however, should also be careful not to reveal any confidential information obtained from their own client representations in the answer.\nWith the ever expanding uses of social media, corporations and businesses struggle to stay abreast of the legal impact these technologies could have on them. Our blog, the Social Media Law Bulletin, will update counsel and business leaders on a wide range of legal developments that arise with social media.\nNorton Rose Fulbright is providing the Social Media Law Bulletin as a commentary on current issues, and it should not be considered legal advice, which depends on the facts of each situation. Norton Rose Fulbright makes NO WARRANTIES OR REPRESENTATIONS OF ANY SORT with respect to the Social Media Law Bulletin, including any warranties or representations as to the accuracy or completeness of any of the information, facts, or opinions contained herein. Your review of the Social Media Law Bulletin does not establish an attorney-client relationship between you and Norton Rose Fulbright, nor is it intended to do so.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Eid-ul-Fitr is the well-deserved reward for 30 days of perpetual fasting. Even though Eid is a few months away, most fashion conscious people begin their shopping for Eid dresses much before the date of the actual event, in order to capitalize on the bountiful Eid clothes collections of fashion websites, before they are plundered and exhausted, resulting in people who have left their Eid shopping for later to settle for and be content with all the leftover outfits from these collections.\nIf you're looking to glam up and shop for Eid dresses in US and sport the latest trends, then you might want to go with Eid straight suits which are one of the most, if not the most popular varieties of Eid salwar kameez there is. Even though straight suits have been there for a while, this Eid 2019, they are said to take the Eid fashion scene by storm. Short and stylish Eid kurtis paired with cigarette pants and high heeled stilettos also work splendidly. Keeping in mind that these styles are quite on trend and are bound to help you achieve a stellar look, it is also important to remember that styles differ for different body types. That is why it is better to get an outfit stitched to your measurements, in order to ensure you end up with the best look possible.\nWhile regular anarkali suits and typical Eid Shalwar suit\/ Salwar suit styles featuring the usually grand embroidery dresses and embellishment works have always been staple Eid outfits, one might want to consider getting into one of the latest fads that is forecast to take over 2019 Eid clothing trends - abaya style anarkali suits. As one can imagine from the name itself, they are a hybrid of the abaya and an anarkali style suit. Although these styles lean more towards the anarkali part of things, they are quite an elegant Eid dress worth getting into.\nThere's nothing like an exquisitely designed Eid lehengas style decorated with the most stunning of embellishment and embroidery works, also featuring mind-blowing and picturesque printed floral, foliage motifs. However this Eid, if you are looking to spice things, then it is advisable to get some trendier lehenga styles with more contemporary influences such as A-line lehengas and long choli lehengas. But if you really want to go all-out and get yourself a lehenga style that will have people complimenting you like crazy, then you might want to push the envelope and get yourself some awe-inspiring anarkali lehengas or umbrella lehengas that are guaranteed to give you an Eid 2019 style to remember for the ages.\nAlthough it might not seem that Eid saree styles aren't really something that people consider when they do their Eid shopping, in reality a lot of people prefer wearing sarees for Eid, because while looking absolutely amazing, they are also quite traditional. Eid sari styles featuring a lot of embellishments and embroidery are quite popular along with half n half sarees as well.\nTo anyone shopping for Eid dress in UK and looking to dress traditional dresses then the best course of action for you is do so from a well reputed ethnic fashion website such as Cbazaar.com which is known for having one of the best Islamic store online. Cbazaar's collections consist of a bevy of sensationally designed Abayas, Fustans, Kaftans that are proven perfect fashion styles for a sartorially successful Eid 2019.\nFor Men's Dresses shopping for Eid 2019, having an idea about what is trending and in vogue currently in Eid fashion this year, will be really beneficial when doing their Eid dresses online shopping, helping them to make better choices and get the best Eid outfits possible. Extravagant Eid suits along with traditional kurta pyjamas and sherwanis styles are expected to be the most sought after of Eid styles this year.\nWhile following all these style tips and shopping for dresses for Eid, it is also crucial to bear in mind the need for accessorizing. Making sure to compliment your gorgeous outfit with some fashionable Eid jewelry, footwear and makeup is guaranteed to give you a holistic style that makes waves with everybody and anybody you may come across.\nIf you are desperately looking to buy Eid dresses online, then the best way to go about doing so, is from a reputable online ethnic fashion website such as Cbazaar.com which is well known for having one of best Eid clothes latest collection online, all available at incredibly affordable prices, together with unbelievable discounts, sales and offers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Every 2nd Monday of the month, we will have an Art After Hours event in the high school art room starting at 6 pm. Each month, participants will make and take an original artwork.\nThis month, April, we will have our monthly Art After Hours event and it will feature MISSOURI paintings!\nSee you and your friends there. All hands in at the Art After Hours.\nThese events are fun and entertaining. Mr. Harris welcome's church groups, youth groups, bridal parties, sororities, men's groups and anyone else who'd like to learn to create and have fun doing it.\nRSVP to Mr. Harris at harriss@tipton.k12.mo.us and get in early. Space is limited.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NHC Foods Limited (\"NHC Foods\" or the \"Company\") is engaged in the business of exporting whole and grounded spices, seeds, resins and ready-to-eat snacks (in India), food grains and animal feeds. The Company exports to over 60 countries with over 200 international customers.\nNHC Foods operates in the highly fragmented spices market which is globally growing at 3.6%.\nThe Company operates through 2 plants based in Gujarat and Mumbai with a combined production capacity of 9000 metric tonnes of ground spices and spice mixtures per annum.\nHigher current ratio implies healthier short term liquidity comfort level. A current ratio below 1 indicates that the company may not be able to meet its obligations in the short run. However, it is not always a matter of worry if this ratio temporarily falls below 1 as many times companies squeeze out short term cash sources to achieve a capital intensive plan with a longer term outlook. NHC Food's average current ratio over the last 3 financial years has been 1.10 times which indicates that the Company is able to pay its short term obligations.\nNHC Food's average long term debt to equity ratio over the last 3 financial years has been 0.26 times which indicates that the Company operates with close to zero debt.\nNHC Food's average interest coverage ratio over the last 3 financial years has been 1.95 times which indicates that the Company can meet its debt obligations without any difficulty.\nIn its latest stock exchange filing dated 31 March 2017, NHC Foods reported a promoter holding of 40.54 %. Large promoter holding indicates conviction and sincerity of the promoters. We believe that a greater than 35 % promoter holding offers safety to the retail investors.\nAt the same time, institutional holding in the Company stood at 0.01 % (FII+DII). Large institutional holding indicates the confidence of seasoned investors. At the same time, it can also lead to high volatility in the stock price as institutions buy and sell larger stakes than retail participants.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahxto b/data_all_eng_slimpj/shuffled/split2/finalzzzahxto
new file mode 100644
index 0000000000000000000000000000000000000000..9068b4e9ed5529e001af51ddc3c38055285cab35
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+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzahxto
@@ -0,0 +1,5 @@
+{"text":"First we circled the numbers and put a box around the question. Next we used dinosaur manipulatives to act out the problem.\nNext we made a class chart that looked like this.\nMy students had some amazing math thinking to share. Here are some of their comments.\n\"The pattern goes 1, 2, 3, 4, and then 4, 3, 2, 1.\"\n\"It looks like a box.\"\n\"It looks like a staircase.\"\n\"Every time there is a 1, there is a 4. Every time there is a 2, there is a 3.\"\n\"When there are more purple dinosaurs, there are less green dinosaurs.\"\n\"When a purple dino comes in, a green one has to go out.\"\nThe next day, the students made their own charts. On the front of the page, they wrote something about the chart. On the back, they glued their manipulatives.\nFinally, we worked together to make our tables.\nIf you'd like to see more problems like this one, you can check out my \"Think, Show, SOLVE!\" series on TPT or TN.\nLooks like a great idea to help students work on problem solving using a form of \"manipulatives.\" The link took me to a paid product on TpT. Is there a free sample? Thanks so much!\nSo cute! I'm glad to see kinders engaged in problem solving!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Note Vav1 was discovered when DNA from five esophageal carcinomas were tested for their transforming activity. This newly identified gene represented the sixth oncogene detected in Dr. Barbacid's laboratory and it was thus designated Vav1, after the sixth letter of the Hebrew alphabet. Vav1 was activated as an oncogene in vitro by replacement of 67 amino-acid residues of its amino-terminus (CH region) with 19 amino-acids residues of pSV2neo sequences, co-transfected as a selectable marker. Wild-type Vav1 produces minimal transformation of NIH3T3 murine fibroblasts only when the protein is grossly over-expressed. Removal of its amino terminus sequences (65 residues), thus mimicking its original mode of activation, is sufficient to induce Vav1 transformation.\nDescription Vav1 encodes a highly unique protein that contains numerous modular motifs known to play a role in tyrosine-mediated signal transduction cascades, such as a dbl homology (DH) region, which exhibits a guanine nucleotide exchange factor (GEF) activity towards the Rho family GTPases; a pleckstrin homology (PH) domain which interacts with polyphosphoinositides; a Src Homology 2 (SH2) and two Src Homology 3 (SH3) domains that mediate protein-protein interactions; a proline- rich motif that enables binding to SH3 -containing proteins, an acidic-rich (Ac) region and a 'calponin-homology' (CH) region, which functions as an actin-binding domain in other proteins and two nuclear localization signals(NLS). In fact, Vav1 is the only known Rho GEF that combines in the same protein the DH\/PH motifs and the structural hallmark of signal transducer proteins, the SH2 and SH3 domains.\nExpression Vav1 is specifically expressed in the hematopoietic system.\nHomology Vav1 is one of a larger family of proteins that include Vav2 and Vav3 which unlike Vav1 are also ubiquitously expressed and the Vav homologues in Drosophila Melanogaster and in the nematode, C. elegans. These proteins are similar in their structure to Vav1, thus also functioning as signal transducer proteins.\nSomatic Although, no mutants of Vav1 have been reported thus far in \"real\" human tumors, it was recently found that Vav1 is expressed in the majority of 42 specimens of human neuroblastoma, suggesting a possible involvement of this protein in the neoplastic process in a subset of neuroblastomas. Furthermore, it was recently found to be involved in a large number of Pancreatic tumors.\nVav proteins, adaptors and cell signaling.\nOncogene. 2001 ; 20 (44) : 6372-6381.\nProduct of vav proto-oncogene defines a new class of tyrosine protein kinase substrates.\nNature. 1992 ; 356 (6364) : 68-71.\nNature. 1997 ; 385 (6612) : 169-172.\nCancer cell. 2005 ; 7 (1) : 39-49.\nVav proteins, masters of the world of cytoskeleton organization.\nCellular signalling. 2004 ; 16 (1) : 1-11.\nAdaptors as central mediators of signal transduction in immune cells.\nNature immunology. 2003 ; 4 (2) : 110-116.\nVav1: an oncogene that regulates specific transcriptional activation of T cells.\nBlood. 2004 ; 103 (7) : 2443-2451.\nvav, a novel human oncogene derived from a locus ubiquitously expressed in hematopoietic cells.\nThe EMBO journal. 1989 ; 8 (8) : 2283-2290.\nVav1 regulates phospholipase cgamma activation and calcium responses in mast cells.\nMolecular and cellular biology. 2001 ; 21 (11) : 3763-3774.\nTyrosine phosphorylation of vav proto-oncogene product containing SH2 domain and transcription factor motifs.\nNature. 1992 ; 356 (6364) : 71-74.\nThe human VAV proto-oncogene maps to chromosome region 19p12----19p13.2.\nHuman genetics. 1990 ; 86 (1) : 65-68.\nVav1 transduces T cell receptor signals to the activation of phospholipase C-gamma1 via phosphoinositide 3-kinase-dependent and -independent pathways.\nThe Journal of experimental medicine. 2002 ; 195 (9) : 1103-1114.\nVAV proteins as signal integrators for multi-subunit immune-recognition receptors.\nNature reviews. Immunology. 2002 ; 2 (7) : 476-486.\nA functional T-cell receptor signaling pathway is required for p95vav activity.\nMolecular and cellular biology. 1995 ; 15 (8) : 4337-4346.\nAtlas Genet Cytogenet Oncol Haematol. 2005;9(1):21-22.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sociology is a branch of science that studies society and processes related to it. If you need to write a research paper in sociology, you should organize your work in a proper way. If you use a wrong approach to dealing with this academic assignment, you might compose a paper that won't meet high standards of academic writing and your score will be rather low.\nYou should pick a specific question to answer in your paper. If you select a broad study area, you aren't likely to impress your teacher and make a valuable contribution to the field. For example, you may write about the obsession of modern teenagers with mobile phone applications rather than with some real life activities.\nTo write a good sociology paper, you need to examine plenty of theoretical sources and previous studies related to your topic. To provide unique information in your paper, you may also conduct interviews with people directly related to your study area.\nDivide the main body of your paper into several chapters and briefly indicate what you'll write in each of them. This will help you structure your document properly during the writing process. Without an outline, you'll spend much more time on composing your text.\nYou may start composing your research paper with the description of your methodology and results. It'll be easier to write a good introduction and conclusion when the core chapters have been completed. Maintain formal tone and don't use jargon words in your paper.\nThe last step is to eliminate all mistakes from your text, rewrite sentences that sound awkward and apply necessary format requirements to the entire document.\nIf you don't have good writing skills but desperately need an excellent grade for your sociology paper, you may ask a professional online agency to work on your academic project. You may get assistance from this website, for example, if you want to receive high-quality services for a reasonable price.\nIn brief, writing a research paper isn't an easy process. You should know exactly what steps to take and in what order to do this. Don't start writing anything until you've chosen a narrow topic, conducted a thorough study, and made a good outline. It's also important to spend enough time editing your draft to achieve the best result.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cycling training log that runs on multiple platforms. Functions include: different kinds of activities, overview log by season, graphical overview of the training zones, and easy to use interface.\nFree download programs including a football league tracking program, a program that allows you to quickly view and print the contents of your computer files in a text or hex format, a poor mans network monitor program written in Delphi, and a Packet decoder for those logged by BlackIce from Network Ice.\nFree web-based sports league management using any standard web browser.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are several benefits of using hammocks. Apart from providing total relaxation, sleeping in hammocks is indeed recommended by therapists for its effects on our spinal disc compressions. They have found out that hammocks are really appropriate to those who have back pains. Getting some hammock time would make your back muscles relaxed and give you a peaceful feeling. If you are sitting the whole day at work, laying in a hammock once back home would be the best cure for you.\nIt has been proven by Dr Steven Park, that the optimal sleeping position is to sleep on our back with our head elevated at 10 to 30%. This position is precisely what we have while sleeping in hammocks, whereas sleeping in beds forces us to move continually looking for better positions. It is possible to recognise a good hammock because your body forms a V shape when you lay in it. It is essential to make sure that the two ends of the hammock should be put higher than the middle to get the right position. If the user sleeps in the exact laying position, it is sure that his body will recover quickly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakplv b/data_all_eng_slimpj/shuffled/split2/finalzzzakplv
new file mode 100644
index 0000000000000000000000000000000000000000..a54c04b1fe0310dc810f15bb5146d529e4bd1c93
--- /dev/null
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+{"text":"The mandate of the UCG is to act as a liaison point for the further development of relations and cooperation between international organisations (IOs) and training providers.\nIt is very important for IOs to be in touch with academic institutions. The UCG, which involves 19 IOs and 13 universities, is a clear example of such a win-win relationship, since some universities organise seminars which are attended by their students and by staff from IOs, who share their expertise with the students. In turn, IOs can also benefit from the expertise of university researchers and professors. The collaboration between the Translation Centre and the University of Leeds was mentioned on several occasions in the past, since the Centre has been able to benefit from the expertise of Leeds researchers in the field of computer-assisted translation, subtitling and automatic speech recognition. Several translators also attended the seminars organised in the past by the University of Salamanca on legal translation.\nAll these seminars are offered for free to staff from IOs (their organisations just need to pay for their travel expenses) and, in exchange, the participants also give a presentation about their own work or organisation for the benefit of the local students.\nThe participants discussed the question of student traineeships in international organisations. Many IOs, like the Centre, indeed offer traineeships and the UCG provides universities with an overview of traineeships and internships for young university graduates in translation, terminology, etc.\nThe participants also addressed questions such as how to bridge the gap between training and professional life. The goal is to better identify the skills, competencies and knowledge which all graduates should have when they leave universities and are recruited by IOs.\nAt a time when translators increasingly carry out the quality control of machine-translated texts and spend more and more time on revising what others \u2013 contractors\/colleagues or machines \u2013 have translated, the participants also agreed that it is important for translators to distinguish between mistakes typically made by humans and new types of mistakes which occur in translations produced by machine translation systems. The gradual introduction of neural MT systems, which may produce what appear to be natural translations which look very much like a human translation, has led to new challenges for translators, given the unexpected nature of some of the problems reported by translators who post-edit such translations. The various IOs are faced with similar problems and are looking for qualified trainers and appropriate training material to raise translators' awareness of such issues.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Say good bye to a dull tired complexion. This reviving face scrub gently but carefully removes dead cells and other signs of imperfections. Rhodiola Rosea and pure Taurin invigorate your skin, while Tiger Grass together with Wild Hop energize it lifting up your self-confidence as never before!\nDirection for use: Apply a small amount of scrub to wet face, massage gently and wash away. For best results use 2-3 times a week.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Moleskine Art Plus Soft Square, 19cm x 19cm of square space for quick sketches, drawings, as well as writing down notes and ideas. Combining the fine elegance of a cahier with the functionality of an album, plain pages, all detachable, of ivory-coloured 120g sketch-grade paper and a pocket with space for collecting loose notes.\nThe iconic notebook, an object that challenges time and connects the past and the present users. The essential tool for Henri Matisse, Vincent van Gogh and Pablo Picasso, represents the continuation of a literary and artistic lineage.\n18th April 2018 Reviewed by Annette Sunnyside One corner was slightly bent on delivery.\n3rd February 2018 Reviewed by Natasha De Samarkandi Love this.\n5th January 2018 Reviewed by Gabriela Jankowska I use this as an everyday practice sketchbook for watercolour, gouache, ink and pencil. The quality is excellent and the square size is really unique Nd enjoyable to use.\n29th June 2017 Reviewed by Irma Osmiegiene I like its paper, this thickness of it. I would suugest to think carefully which size you really need and will feel comfortable with. There are many more options to choose like few sizes at once.\n16th May 2017 Reviewed by Steviequorn I brought this as a present and it arrived in perfect condition. As I haven't used it personally I can't fully rate it but I am always impressed with the quality of Moleskine products and this appears no different. It is a handy size for carrying around for when inspiration hits but with plenty of space for all ideas.\n6th December 2016 Reviewed by Graham Pellow I am a great advocate of sketchbooks and like to have a range of sizes. Square format is becoming more popular hence my purchase.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Subscribe and share this page with your friends to get up to 25% super early bird discount!\nAdopting the leading micro Bezels on both sides are only 5mm \u2013 as wide as just 5 grains of salt. The screen-to-body ratio is also close to 80%, allowing you to fully immerse in a boundless experience.\nEnjoy accurate typing and ultimate comfort through the tip of your fingers.\nPrecise interior structure along with the durable aluminum-magnesium alloy\u2014thinnest part is 15mm. Weighing only 2.76lbs. Overall size is 20% smaller than other products of the same class.\nThe Turbo Boost 2.0 help the CPU core frequency reach 2.2GHz. It is 40% higher than the previous generation.\n8GB of large-capacity memory is one of the key factors for smooth switching of multiple projects. Aerobook is equipped with 128G storage, in addition, you can also upgrade the storage to 1TB.\n8 hours high capacity lithium polymer battery, with the addition of low power consumption CPU Enables an 8 hours long usage.\nUp to 7 interfaces make it possible to build your personalize workstation.\n2. Click the button to refuel the energy bar.\n3. Share to get an extra chance.\nSubscribe and share this page with your friends to get up to 25% super early bird discount!When CHUWI Aerobook campaign is live on Indiegogo, we will notify you in the newsletter !The number of discounts is limited,please register and purchase as soon as possible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I cannot see how to fix a simple problem in a custom AddIn using the new Addin Wizard without closing ArcMap and re-opening.\nIs this really required for every edit of an addin?\nThis is what I do when testing.\n1. Install the addin and open in ArcMap, open the python window.\n2. Test the tool when inevitably there is a runtime error.\n3. Edit the python script and tabnanny.\nBut if ArcMap was still open it does not refresh the addin.\nThe only way is to close ArcMap and reopen which takes a lot of time.\nI have found the updating of the versioning a real pain as well, with it overwriting the existing script, but that is a known feature.\nTo make a change to your python script you should close out of the application, rebuild and install the add-in and restart the application. Behind the scenes what happens when you start the application is the addin is copied and unpacked to the local assembly cache, which is where the code is executed from. Everytime the application starts up it checks the addin folder and if the addin is different from the one in the local assembly cache the application will overwrite the old version.\nThis is the same behavior for .Net and Java addins as well, if you need to modify the code you will have to rebuild the add-in, install and restart the application.\nAre you editing the version of the script that is part of the 'installed' bundle? If so then you shouldn't have to close and reopen ArcMap.\nIf you're making changes to the code only you should be able to change the installed script and re-run it. You shouldn't have to re-build the add-in. If you're affecting the add-in components then you'll need to make changes and recreate the add-in.\nI am then supposed to double click on it to install it into the well known location in MyDocuments.\nI assume this is different to addins created in java and .NET where the source is in an IDE. The Python wizard is not an integrated environment. So if you are creating a .NET addin you can edit your (non-python) script while ArcMap stays up?\nIt all works just fine if I close and reopen ArcMap, but what a pain!\nIt seems that the addin is loaded into memory when ArcMap starts and no amount of deleting refreshes the image of the addin.\nThat is a bit clumsy, given the time to open ArcMap. Oh well, now I know.\nI was hoping for a menu dropdown that could refresh the AddIn cache without exiting ArcMap.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamglv b/data_all_eng_slimpj/shuffled/split2/finalzzzamglv
new file mode 100644
index 0000000000000000000000000000000000000000..103365a59472b39d68a71754d40ba50166b72328
--- /dev/null
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+{"text":"It's the last weekend of the Frankfurt Christmas market but if you want to keep the festive spirit going then get over to Winter Skate Birmingham for the outdoor ice rink. The Ice Caf\u00e9 offers festive treats which will fill you up after a an hour on the ice and as the rink is located in Centenary Square just outside The Rep it is nice and central too.\nFans of Birmingham's creative bohemian quarter are in for a treat tomorrow (Saturday December 14th) with the Custard Factory's DIGMAS event which is teaming up with the famous Digbeth Dining Club to bring the ultimate independent Christmas Fair to Birmingham!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Updated Monday, June 9, at 4:58 p.m.\nGov. Peter Shumlin filed paperwork Monday afternoon to seek a third two-year term as governor of Vermont, according to the secretary of state's office.\nCandidates for statewide office must submit nominating petitions signed by 500 Vermonters by the end of the day Thursday. According to elections administrator J.P. Isabelle, the Democratic incumbent turned in enough signatures Monday to secure a berth on the ballot.\nShumlin's candidacy for reelection is not at all surprising. He told Seven Days last December that he planned to seek a third term. But as he did when he ran for reelection in 2012, Shumlin has thus far declined to discuss the race with reporters, saying he will not campaign until Labor Day.\nAccording to a list posted by the secretary of state's office Monday afternoon, four other candidates have submitted petitions to run for governor: Republican Emily Peyton, Libertarian Dan Feliciano and independents Cris Ericson and Bernard Peters.\nOn Sunday, 2012 Republican gubernatorial nominee Randy Brock announced that he would not challenge Shumlin again this fall. Pomfret resident and Milne Travel president Scott Milne told Seven Days on Monday that he has collected enough signatures to win a place on the ballot, but hasn't yet decided whether he would enter the Republican primary.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Most PCTIs are held once weekly for 9 weeks in various locations across Massachusetts. For those unable to commit to a class of that length, it is offered in June in a one-week condensed, intensive format. Locations change annually and are established over the summer for the upcoming school year.\nWe recommend that PCTI attendees have some familiarity with special education procedures. A list of free workshops presented by the Federation can be found here. Classes often fill completely: we recommend registering early. Preference will be given to youth with disabilities and family members of children with special needs who have registered at least two weeks before the class begins.\nPre-Registration is required, no walk-in registration permitted.\nAttendance on Day 1 of class is required. Register ONLY if you can attend the first day.\nRegistration closes one business day prior to class or when the class roster is full. For waitlist information contact crussell@fcsn.org.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Quercetin is Probably Not a Useful Senolytic \u2013 Fight Aging!\nSenolytic compounds are those that preferentially destroy senescent cells. Since these cells are one of the root causes of aging, there is considerable interest in finding and then quantifying the effectiveness of senolytic compounds. The known and alleged senolytics vary widely in effectiveness and quality of evidence, and quercetin is one of the more dubious examples. I don't think that anyone expects quercetin, on its own, to have a useful level of impact on senescent cells and their contribution to degenerative aging. The study here comes to the plausible conclusion that quercetin really can't achieve that goal. Yes, it is true that the 2015 mouse study of the chemotherapeutic dasatinib and quercetin demonstrated that the two together cleared more senescent cells than dasatinib alone, but synergy with other compounds is a very different story from unilateral effects. Quercetin is a widely used and extensively tested supplement compound. Any significant effect on health resulting from quercetin alone would likely have been discovered many years ago.\nPreviously, quercetin was reported to be a senolytic in irradiation-induced senescent human umbilical vein endothelial cells (HUVECs). HUVECs are derived from the umbilical cord of newborn babies, and for a long time were the only model of primary human endothelial cells (EC); however, these cells are not the best model of diseases associated with human arterial aging. HUVECs have been shown to differ substantially from primary endothelial cells derived from adult human vasculature. In the current study, we investigated whether quercetin is a senolytic in adult EC, and evaluated whether quercetin 3-D-galactoside (Q3G; hyperoside) would be a more selective senolytic.\nQuercetin's low therapeutic\/toxic ratio in the HUVEC study raised the possibility that quercetin could significantly injure non-senescent cells. It was unclear whether the proliferation of non-senescent cells could be compensating for some of the quercetin-mediated cell death, thus masking its toxicity to the young cells at the lower concentrations found to be selectively cytotoxic to senescent cells. We used adult human coronary artery endothelial cells (HCAEC), which are microvascular cells, as a relevant model, and generated two groups of cells from them to better understand the effect of quercetin: EP (early passage; young) and SEN (senescent), as a model of an aging tissue.\nOur key findings are that quercetin at a concentration that reduced SEN EC also caused significant EP EC cell death, and that there was no evidence of senescent cell-specific cell death mediated by quercetin. Thus, quercetin is not a selective senolytic in adult human arterial endothelial cells, where both EP and SEN cells responded similarly to quercetin's toxicity.\nTo circumvent quercetin's toxicity on healthy, non-senescent cells, we investigated Q3G, a derivative of quercetin with limited toxicity to endothelial cells, which is processed by senescence-associated beta-galactosidase (SABG) enriched in senescent cells to release quercetin in situ. Q3G could act as a selective prodrug in senescent cells. However, Q3G had no significant toxicity to either EP or SEN EC. The lack of Q3G's toxicity in the current study may be due to Q3G being unable to enter the beta-galactosidase-rich lysosomes, or alternatively, Q3G being able to translocate to the lysosomes to release quercetin, which is further processed into an inert compound.\nIt is more likely that quercetin is masking certain parts of SASP, such as CD38 for example.\nThere is other supporting data but this will do for now. Quercetin does have a wide range of activity but I have been skeptical of its ability to remove senescent cells for quite a while now. It may have a place as a co-therapy in conjunction with true senolytics but alone I think it just masks some inflammatory cytokines.\nThis reinforces what I've been thinking. Finding safe and effective senolytics will involve many years of clinical trials.\nAny evidence for a filed IND?\nThe 2015 mouse study claims that quercetin does have \"unilateral effects\" in eliminating senescent mouse bone marrow stem cells, even slightly better than the dasatinib and quercetin combo.\n@Florin Clapa: True, but it is hard to argue against the evidence of decades of human use and studies with no sign of benefits to any of the health issues you'd expect to be positively affected by senolytic activity.\nThe data in this study does show that Q selectively kills SC, just not selectively enough to qualify as a true Senolytic by the established criteria. Given that this is an in vitro study, Q is cheap, available, and has almost no side effects, I am going to continue using it. And, IMO, it is not unfathomable to image that the non scenescent cells Q killed in the study were in some way weaked, just not yet Scenescent.\nReason, I doubt any human studies got the dosage, frequency, and age ranges right. Even if all of those variables lined up correctly somehow, the effect of just using Q might be too subtle to detect (a somewhat lower cancer incidence over several decades, for instance), especially if you didn't know what to look for.\nI am taking Prosta-Q from Farr Laboratories. It contains Quercetin. Unfortunately, it's part of a proprietary blend of ingredients, so I don't know how much I've been taking.\n@Florin: their argument here is (a) the original Mayo Clinic study reported Q as senolytic in human umbilical cord vascular cells (HUVEC), which (i) \"are derived from the umbilical cord of newborn babies\" and are thus on their face\" not the best model of diseases associated with human arterial aging\", and (ii) \"have been shown to differ substantially from primary endothelial cells derived from adult human vasculature,\" whereas these guys used the more relevant primary endothelial cells; and (b) as JohnD says, their selectivity is weak. Indeed, even the original Mayo Clinic report found a fairly narrow therapeutic window, and the new report said it was too small to be useful. Selectivity is really a core feature of a true senolytic: an agent that kills lots of senescent cells and also kills nearly as many normal ells at the same time is not really a senolytic, but a toxin.\nMichael, while Q might not be useful at eliminating senescent HCAECs (I never claimed that it was), it still might be useful in eliminating senescent BM-MSCs. As for Q's toxicity, I doubt it can kill a significant number of normal cells to matter (no toxic side effects were reported for mice, for instance), and even if Q isn't that selective at killing senescent BM-MSCs, it seems to kill enough of them to matter.\n@Florin: well, the only BM-MSC in which Q was tested Q were derived from Ercc1[-\/\u0394] mice, which have defects in DNA repair, great increases in the rate of senescence, and shortened lifespan - and, in particular, abnormal bones and bone marrow. So I wouldn't put too much stock in that result without more supporting data. As to the lack of side effects: well, we don't have the suitable controls of D or Q monotherapy in vivo. Possibly the dose of Q is too low to have much effect on either normal or senescent cells and D is responsible for all the observed effects, or possibly the effects would have been more positive without Q dragging them down - or the benefits of Q were balanced by deleterious effects, leaving it a wash. And they only did brief treatments, unlike some of the longer-term work with either genetic approaches, Navitoclax, or UBX0101.\n@Reason I periodically take Q (500mg) but am somewhat concerned about its specificity for only getting rid of scenescent cells. Q does have some history of health benefits in that two very healthy food products used for hundreds if not thousands of years are known to have high Q content, including apples (an apple a day keeps the Dr. away) and onions. The peels of apple are have the highest concentration of Q, and I eat 1-3 apples a day.\nI hope everyone feel that small molecular approach in rejuvenation biothechology (and even in medicine as a whole) is a dead end. They are very expensive to design, to develop, to test and they hugely interfere with metabolism and thus have many side effects. Now it is just a waste of time and money to screen in the hope to find anything useful. We need to look forward and make things like Oisin targeted very efficient clearance systems.\nMichael, SCs are SCs regardless of mouse strain, right? And as I've mentioned before, Q actually killed off more senescent BM-MSCs than either D or D+Q. While it's true that Q monotherapy wasn't tested in vivo, it's unlikely that there would be any big negative side effects at low doses and infrequent administration. So, the bottom line for me is that Q is low risk, and potentially high reward; it may (or may not) reduce senescent BM-MSCs without negative side effects.\nWell, \"SCs are SCs\" is a bit of an oversimplification even inside a single wild-type mouse, but leaving that aside, the main point is that a senescent cell with a hypomorphic Ercc1 gene (and in a milieu generated by other Ercc1 hypomorphic cells in a mouse that is actually rather young in narrow chronological terms) is not the same as a senescent cell with an intact gene and situated in an otherwise-normal but aging milieu. This was a question even in the original Van Deursen BubR1 hypomorphic mouse study, but at least in that study you could be confident that they were exclusively hitting p16-expressing cells because the mechanism of ablation was directly tied to p16 expression. Here, the idea is that the supplement is supposed to be working by selectively hitting the anti-apoptotic machinery on which senescent cells differentially rely, but the reality is that Q is a quite dirty drug with multiple MoAs, and could quite possibly be differentially toxic in cells that are both senescent and have sub-par nucleotide excision repair.\nThe bioavailability of Q in humans is pretty bad. If the active senolytic species in rodents is the aglycone, then it's probably not going to work in humans. There are non-trivial difference in metabolism of polyphenols between humans and rodents. Humans knock out polyphenols very efficiently through glucuronidation and sulfation.\nI understand that the digestive tract breaks down quercetin very quickly to compounds that have no senolytic property. Is that true?\nSafety assessment is reported of an orally ingested dihydroquercetin-rich extract (Lavitol) derived from the Dahurian larch tree, used as a food additive and as a dietary supplement ingredient. Dihydroquercetin, a potent antioxidant, is also known as taxifolin.\nThe results of genotoxicity and toxicological tests (Comet assay, micronucleus test in human lymphocytes, chromosomal aberration test, subacute 7-day oral toxicity study, subchronic 90-day toxicology study with histopathologies, and, prenatal and postnatal developmental toxicity studies) on the extract provide further support for the safety of its consumption as a food supplement and food additive.\nenvironmental conditions the tree is subjected to.\nThe finding of a high concentration of DHQ in L. gmelinii.\nused hypopigmenting agents in cosmetics.\nOnce a week I take Bio-Quercetin phytosome 74 mg (providing 25 mg quercetin [from Japanese sophora concentrate (flower bud)], phosphatidylcholine complex [from sunflower]) combined with Theaflavins [from decaffeinated black tea extract (leaf)] 275 mg.\nMost things are effected by dose vs lean body weight & I am 65 years old, 6' 6\" tall & not under 220 pounds since my early 20's.\nI have been self experimenting with substances specifically including but not limited to their potential benefit on these cells for about a decade.\nI am adding a very recient cover story from Bill Faloon as an explanation of my reasoning on that new way of getting more quercetin into the cell.\nIt is pretty darn cheep too especially since I buy @ wholesale or half retail price.\nRemoval of Senescent Cells \"Might Transform Medicine\"\nThis is not the first study showing these effects. What makes it exceptional is the meticulous way it was conducted.\nThe first part of the study involved transplanting a few senescent cells into young mice. This caused the young mice to endure physical decline characteristic of degenerative aging. When senescent cells were transplanted into old mice, the same toxic effects occurred and the mice died sooner.\nBut with oral administration of compounds that remove senescent cells, physical dysfunction was alleviated and lifespan markedly increased.\nThe incredible news is that novel ways are being developed to help remove senescent cells from aging humans today!\nAs cells age, most undergo a beneficial elimination process known as apoptosis.\nThis scenario occurs inside our bodies, whereby senescent cells fail to undergo apoptosis (normal cell elimination). These pesky senescent cells emit inflammatory-type signals that spread to healthy cells and cause them to deteriorate.\nDoes anyone NOT want these benefits?\nAging Cell. 2015 Aug;14(4):644-58. doi: 10.1111\/acel.12344. Epub 2015 Apr 22.\nI was informed about senescent cells in the late 1990s by a brilliant researcher. He made it clear that if we were to achieve meaningful control over aging, we had to selectively remove senescent cells that accumulate and create metabolic havoc.\nThe problem back then was there was no valid approach to accomplish this.\nThe box on the previous page is from a presentation I give to advocate for accelerated age-reversal research. This slide highlights findings from a groundbreaking (year 2015) animal study that found remarkable rejuvenation effects in response to aggressive senolytic therapy.\nThese studies generated worldwide headlines, with scientists describing the potential of senolytic compounds to reverse aging in humans.\nThe review went on to describe how senescent cells destroy the knee joints of healthy mice.6 These findings have encouraged people with osteoarthritis to self-experiment with senolytic compounds, with encouraging results being reported.\nOf interest is that senolytics do not have to be used continuously to derive benefits. This corroborates current strategies whereby people are using senolytics intermittently (\"on\" and \"off\" periods).\nHuman senolytic studies are being pursued by a number of physician\/scientists. We look forward to reporting on them after more long term data is gathered.\nEarly stage research on senolytics indicates that intermittent dosing may be an ideal approach. In other words, the strategy is to selectively induce senescent cell apoptosis for a limited time, and then cease until more senescent cells accumulate.\nDrugs that purge senescent cells are being clinically studied. Some individuals are self-experimenting today with senolytic drugs + high-dose quercetin.\nUntil recently, people seeking to achieve senolytic benefits were challenged to take large-doses of quercetin, and many don't yet want to try dasatinib. A new quercetin phytosome eliminates the need to ingest large amounts of quercetin.\nThis enables 74 mg of this quercetin phytosome (providing 25 mg quercetin) to deliver a dose approximately equivalent to about 1,250 mg of typical quercetin. This empowers more people to achieve greater amounts of intact and unmodified quercetin into their bloodstream.\nThose taking lower doses of regular quercetin for its other benefits may continue to do so and use the new phytosome delivery formula on an intermittent schedule. The article on page 6 of this issue describes this low-cost and novel approach to removing senescent cells by taking theaflavins and quercetin phytosome just one time each week.\nThe cost of this once-weekly senolytic nutrient approach is less than $7 a month.\nXu M, Pirtskhalava T, Farr JN, et al. Senolytics improve physical function and increase lifespan in old age. Nat Med. 2018;24(8):1246-56.\nKirkland JL, Tchkonia T, Zhu Y, et al. The Clinical Potential of Senolytic Drugs. J Am Geriatr Soc. 2017;65(10):2297-301.\nZhu Y, Armstrong JL, Tchkonia T, et al. Cellular senescence and the senescent secretory phenotype in age-related chronic diseases. Curr Opin Clin Nutr Metab Care. 2014;17(4):324-8.\nZhu Y, Tchkonia T, Pirtskhalava T, et al. The Achilles' heel of senescent cells: from transcriptome to senolytic drugs. Aging Cell. 2015;14(4):644-58.\nFuhrmann-Stroissnigg H, Ling YY, Zhao J, et al. Identification of HSP90 inhibitors as a novel class of senolytics. Nat Commun. 2017;8(1):422.\nXu M, Bradley EW, Weivoda MM, et al. Transplanted Senescent Cells Induce an Osteoarthritis-Like Condition in Mice. J Gerontol A Biol Sci Med Sci. 2017;72(6):780-5.\nAvailable at: https:\/\/www.sciencedaily.com\/releases\/2017\/09\/170904093428.htm. Accessed August 6, 2018.\nNoberini R, Koolpe M, Lamberto I, et al. Inhibition of Eph receptor-ephrin ligand interaction by tea polyphenols. Pharmacol Res. 2012;66(4):363-73.\nNoberini R, Lamberto I, Pasquale EB. Targeting Eph receptors with peptides and small molecules: progress and challenges. Semin Cell Dev Biol. 2012;23(1):51-7.\nTing PY, Damoiseaux R, Titz B, et al. Identification of small molecules that disrupt signaling between ABL and its positive regulator RIN1. PLoS One. 2015;10(3):e0121833.\nLeone M, Zhai D, Sareth S, et al. Cancer prevention by tea polyphenols is linked to their direct inhibition of antiapoptotic Bcl-2-family proteins. Cancer Res. 2003 Dec 1;63(23):8118-21.\nSupplier Internal Study. A randomized and crossover pharmacokinetic study of Quercetin 500mg., Quercetin Phytosome 500 mg. and Quercetin Phytosome 250 mg. administered in a single dose to healthy volunteers under fasting conditions. Data on File. 2017.\nHan X, Zhang J, Xue X, et al. Theaflavin ameliorates ionizing radiation-induced hematopoietic injury via the NRF2 pathway. Free Radic Biol Med. 2017;113:59-70.\nThe cost of this once-weekly senolytic nutrient approach is less than $7 a month.\"\nI've been using this from LEF for maybe 3 weeks...except for the first dose...no noticeable effects so far.\n* If I start wearing diapers...then I'll know it's working...getting younger?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"40 campsites available - some unserviced campsites, power campsites available, power and water campsites available. Tables provided at campsites. Free firewood available. Free showers. Tap water available. Flush toilets. Sewage dumping station. Day use area. Golf course nearby. Baseball diamond nearby.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Here's a pretty staggering number: 34,220,504.\nThat's the number of personal records exposed, according to the Identity Theft Resource Center, through November 2015 because of government data breaches.\nWhile that number does include some pretty high profile federal breaches, it also includes many more at the local level that didn't make national headlines. Clearly, governments are going to be popular targets for hackers. The breadth of data they have on their citizens makes them enticing.\nUnfortunately, local governments are also some of the easiest targets. We recently took a look at some of the data that we have from the networks we protect, and it bore that out. By far, the Sentinels that tripped the most alerts were those that were protecting local governments and educational facilities. Those networks were four times as likely to be infected with Cryptowall and were responsible for 95 percent of all Kovter attacks that we saw.\nWhy is that these local government networks are so vulnerable? From our experience, it's because there are six common security misconceptions believed by local government employees that keep them from securing their networks. Over the next few posts, we will share those misconceptions with you and also provide ways that you can battle them.\nMisconception No. 2: I know exactly where my data is.\nThere's one commonality among all the news stories about federal government data breaches. In each instance, officials knew exactly what kind of information was compromised. If those stories had been about local governments instead, the information wouldn't have been so detailed.\nThat's because at the federal level, network administrators usually know exactly what information is living where. If Server A was breached then that means names and addresses were compromised. Server C it's Social Security data.\nAt the local level, you may believe that you know where your data is, but you don't. Not specifically. With single servers handling multiple departments and applications, knowing where data is to the level of detail that the federal agencies have isn't possible. A single breach of a single server could mean that information from several departments has been compromised.\nHow to combat this misconception: To some degree, the local officials aren't necessarily to blame. This is a misconception that can be blamed on dollars and cents. Buying a server for a single department or application is expensive, too expensive for many. Why buy four SQL servers when you only need one?\nThe solution: virtualization. Run virtual machines, splitting up your apps across the VMs, and segmenting the data. For example: If you're using SQL servers, you are going to need to protect those databases. That means putting something on the wire that can recognize the difference between normal and abnormal calls and then something on the server that can segregate databases and permit only those with permission to access the tables.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our specialty coating systems provides high temperature resistant's in the most aggressive environments. IMG specializes in providing long lasting industrial coatings for the power industry. The power industry is one of the most difficult coatings environments to manage with temperature fluctuations, high temperature operations, aggressive product usage and plant fallout.\n25 plus years of experience, patented products and Florida Department of Environmental approved products IMG has help several fortune 500 companies meet their fuel containment requirements.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Many people are discovering the potential of building and operating a work from home business. This is one reason you need to know as much as possible about what it takes to be successful. This is the only way you can stay on top of the competition.\nMake the satisfaction of your current customers very happy.It takes much less effort to produce sales from repeat customers than it does to convince a new ones. A happy customer will return time and again.\nYou need a separate PO Box set up for your business mailings.Don't use your real home address online. This will help protect the identities of you and your home and isolate your work life.\nDon't quit your day job when you're just starting your home business.It is certainly to your job while waiting for your new business to generate a steady cash flow.\nHave a page on your site. This will enable you to swap banner links with the businesses in which you know. This is a great way to increase your rankings and make sure customers can find you.\nYou will find many home business ownership on the Internet. Be cautious, though, of the many scams dealing with home businesses online. There are a lot of places that will offer you basic information that can be found elsewhere for free, or information that is so basic that it is not worth paying for. There are those tricky scams as well that take your money for some useless online courses.\nThere are companies that offer free or inexpensive business cards. Your business cards should include your name, your business name and logo and your phone number. Include your email address and the URL to your website on the business card. This will make it easier for customers to contact you in different ways.\nThis article should help you rise head and shoulders above the competition. Just keep researching and learning about your home business enterprise. Do as much research as possible and create new strategies to be successful.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The intelligent driver update engine detects outdated drivers, providing a comprehensive report and offering to update the drivers.Unleash full potential of your computer by keeping drivers up to date!\nThe intelligent driver update engine detects outdated drivers, providing a comprehensive report and offering to update the drivers. Unleash full potential of your computer by keeping drivers up to date!\nUpdate all of your system drivers in just only 2 minutes - DriverSmith will scan your PC and update all needed drivers in seconds!\nUpdate all of your system drivers in just only 2 minutes - UpdatesDriver will scan your PC and update all needed drivers in seconds!\n* Change data with the Bulk Update feature.\n* Download our collection of wineries, regions, appellations and varieties.\n* Easily search for wine reviews.\nAD Replication Status Tool 1.0 B2.3.20717.\nFull featured residential property management software Residential property management software with integrated accounting features.\nPeggle Extreme delivers fast-paced puzzle action that combines elements of pinball and pachinko to create a truly unique game experience.\nThe CodeLib is an application that helps you to easily find and use the code snippets As a result, you can build a programs faster. Support Code Snippet, File, Note, Web, Picture save.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ALL IN THE FAMILY By identifying the wreckage of past asteroid crashes (illustrated) known as asteroid families, astronomers can ignore those fragments and pick out asteroids that have remained intact since the early solar system.\nThe solar system's first asteroids were probably born big.\nRather than slowly amassing bulk over time, the original members of the asteroid belt rapidly formed into rocks hundreds of kilometers across, researchers propose. This finding, reported online August 3 in Science, may help resolve a long-standing debate over the origins of planetesimals \u2014 the giant space rocks that populated the asteroid belt and constructed the planets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaovjl b/data_all_eng_slimpj/shuffled/split2/finalzzzaovjl
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+{"text":"Because we believe books are good for children, we take your little ones and young ones through the looking-glass and give them the taste for books that captivate, fascinate, elevate ans exhilarate, all books that we love.\nBut Le Wolf is also the Wolf Shop, where you can find your own perfect book, and the \"Cantine du Chaperon\" where you can enjoy \"Tears Tea\" (Thanks Hulul!), \"Pebbles Soup\" or \"Grandma's Belgian Cookies\", among other delicious treats. The perfect place for yummy family moments, for Sunday brunch with friends and their kids, and for the celebration of all the wonders a good book (and good food) has to offer.\nWe look forward to welcoming you in our lair!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Based in Kansas City, Missouri, Burns & McDonnell provides engineering, architecture, construction, environmental and consulting solutions from offices across the United States and worldwide. Chances are, we have an office near you. But our people go where you need us, providing integrated services in every locale.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Excellent high quality training Gaelic football.Set of 10 with free mesh net bag..\nExcellent high quality Gaelic football, official size and weight. Size 4 approx. 480gIdeal match qua..\nLightweight sliotar,suitable for all surfacesSize 5..\nTraditional hurley made from the highest quality ASH. Sourced directly from Cork. Contact us for bul..\nThe LS Sportif FAMOUS glove is an evo;lution of gloves internationally renowned for their incredible..\nFirst Touch go games leather sliotar \u2013 Oversize softer and lighter sliotar. FULLY APPROVED AND LICEN..\nHigh visibility training quality sliotar..\n* LS Gaelic Tactic Board - 30 x 45cm * Includes magnets, white board pen, eraser and carry bag. * Ap..\nMagnetic lightweight GAA folder. Ideal for pitch side or dressing room use. Includes 33 magnets, pen..\nLightweight Gaelic glove. Soft palm for excellent grip. Velcro closure system allows easy fit and co..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Chiroproktor is an anatomical model developed for Chiropractors, Osteopaths, Physiatrists, and Physical Therapists that simulates various spinal misalignments within the Thoracic and Lumbar region of the spine. The model \"CPR-like mannequin\" allows for the Physicians to practice their palpating and adjusting techniques at their own convenience. ChiroProktor utilizes accelerometers and pressure sensing pads to deliver real time feedback on a Physician's speed, acceleration and thrust time and creates a progress report to help guide development. The device is designed to allow Physicians to remotely and independently select which area\/vertebrae of the spine they want to misalign and practice on.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We worked with Carefound Home Care on their latest brochure to showcase their services and quality care awards to potential clients.\nWe offer creative design for print and online, with a simple aim to make you proud of your business brand. We specialise in branding, marketing and advertising. The team of talented graphic designers, web developers, illustrators and copy writers make any requirement possible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Our founder, Miss Mona Huskie was ahead of her time and campaigned tirelessly to prevent the export of live equines to the continent and over the years, we have risen to many challenges. We've cared for pit ponies, working Shires, former Police Horses, ponies in Dartmoor and many abused, neglected and abandoned animals. Each era has brought new and different things for us to deal with and our working practices have adapted to help us meet those challenges.\nThe world is a much busier, faster and in some ways more demanding place than 1937 but, like the HAPPA custodians before us, we have to change how we do things from time to time to place us in the best position to care for the equines that need our help. Equine care, income generation, legislation, technology, are all areas of continuous change and we must meet these challenges head on as to do what we have always done and pay no heed to our future would be the greatest disservice we could do for the animals that rely on us and our friends and donors who generously support us.\nWe are proud of our history and eagerly anticipate our future.\nHAPPA's Equine Inspectors investigate over 300 cases of cruelty and neglect each year, offering advice and support where possible and preventing cruelty by intervention and prosecution where necessary. They have a wide range of expertise on all aspects of equine welfare and are fully conversant with the laws relating to the application and implementation of equine legislation within the Animal Welfare Act 2006; Animals Act 1971; Horse Passport Regulations 2009; Control of Horses Act 2015.\nHAPPA Inspectors play a key role in the co-ordination and enforcement of codes relating to the 'Five Welfare Needs' and provide expert advice to horse owners. Where possible our Inspectors offer guidance and assistance to improve equine welfare, including the issuing of advice notices allowing people time to make improvements to meet the standard of care. However we will take action where we find cases of cruelty and neglect and against those who remain non-compliant.\nOur Inspectors facilitate educational talks and workshops to encourage a consistent approach and raise awareness towards the importance of equine welfare following best practice.\nHAPPA Inspectors often Network with other Welfare Organisations and members of NEWC (National Equine Welfare Council) and are involved in many discussion groups and inter-agency events to raise awareness of and help address regional issues relating to equine care and welfare.\nHere at Shores Hey Farm Rehabilitation Centre, in Burnley Lancashire the HAPPA Yard Team are dedicated to finding every horse, pony or donkey in our care a Second Chance in a Forever Home, to do this every equine admitted to the Centre must be rehabilitated ready for rehoming. Our dedicated Equine Care Officers work tirelessly to create bespoke rehabilitation plans for each equine at the Centre.\nEvery admission to the Centre completes a 21 day isolation period, for the duration of this period the equine is housed in the HAPPA (tailor made) Isolation and Assessment Unit; a strict isolation protocol is adhered to at all times to reduce the risk of cross contamination of disease and infections to the main yard. During this time any ailments are treated and a behaviour assessment completed, this is the beginning of the rehabilitation process; the initial behavioural assessment gives the Care Team an indication of suitability for rehoming.\nWhen the equine has finished the 21 day isolation period, without displaying any signs of ill health, it is then moved onto the main yard where a bespoke rehabilitation and socialisation plan is created for the equine by our Senior Care Team. An Equine Care Officer is then allocated the equine to work with and is tasked with the job of ensuring that the equine is ready for rehoming in the near future. Along the way the horse, pony or donkey may need a little extra help of reaching this goal this may be in the form of veterinary treatments, orthopaedic or physical therapies, remedial shoeing and or dental treatments. If the equine is capable of ridden work then it will begin with groundwork, progressing to backing and schooling stages until ready for rehoming. If being rehomed as a companion then socialisation and exercise regime begins.\nAs soon as the HAPPA horse, pony or donkey has completed the rehabilitation process and is recommended as ready for rehoming by the HAPPA Senior Care Team then the team work very hard to find that equine a perfect match in a Loan or Foster home. If you would like to help our team with the fantastic rehabilitation work they do then please consider becoming a HAPPA Second Chance Sponsor, your support really will help us to give every horse, pony or donkey in our care a Second Chance of finding that Forever Home.\nHAPPA continually strive to improve and promote equine welfare through rescue, rehabilitation, rehoming and education.\nCan you offer a Forever Home to a HAPPA Horse?\nOur Loan Scheme enables certain HAPPA horses, ponies and donkeys to be placed with suitable Borrowers under a legally binding agreement. HAPPA have an outstanding reputation for Rescue and Rehabilitation and produce quality horses & ponies that are fit for purpose and go on to achieve great things. Let HAPPA find the perfect partner just for you. From a dressage horse to happy hacker, pony club pony or cuddly companion, we have them all!\nOur Equine Support Team provides a robust aftercare service offering guidance and advice to enable our horses to remain in long term homes. We are on call anytime to answer any queries you may have about your HAPPA equine and conduct a mandatory visit twice a year to ensure that the quality of care meets HAPPA's high standards.\nThere is a waiting list for rideable animals but as we try our best to match the right borrower to the right animal, not everyone has a long wait! We have three sub categories of re-homing ride-able, companion (non-ridden) and fostering.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In March 2015, after conducting two lengthy cross-border investigations with the assistance of the US Federal Trade Commission, the Canadian Radio-television and Telecommunications Commission issued Notices of Violation with administrative monetary penalties totaling $345,000 to two American companies for violating Canada's Unsolicited Telecommunications Rules. Maple Accounting and Caribbean Cruise Line Inc. had multiple violations of the Unsolicited Telecommunications Rules including making unsolicited telemarketing calls to numbers registered on the Do Not Call List.\nSince telemarketing calls often originate in a different country than where the call is received, the collaboration of international investigative forces is crucial to combating this ever increasing global problem. The CRTC looks forward to continuous cooperation with the FTC and other LAP agencies in the enforcement of unsolicited communications across borders.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm just a few hours in so I can't say much about the game. So far, I'm actually a bit confused about what's going on because all kinds of strange terms are used \u2013 l'Cie,\" \"fal'Cie,\" \"Cie'th,\" or \"Gran Pulse\" and I find it damn hard to remember what they mean.\nHere's what I think I know: the game is set in a world called Cocoon. It's a gigantic sphere and outside its shell is some kind of netherworld known as Gran Pulse. Fal'Cie are otherworldly beings and there are two kinds: the Fal'Cie and the Pulse Fal'Cie. Pulse Fal'Cie are considered dangerous and people who come into contact with one are viewed as infected and called l'Cie. The game starts when government troops take care of a whole village whose people are suspected to be infected by a Pulse Fal'Cie. Some civilians resist because they suspect they face more than just an evacuation. That's the grand scale.\nThen there are the individual characters who make up your party and most of them have a more specific reason why they fight the government troops. For example, one of the l'Cie is the fianc\u00e9e of one character and the sister of another and they try to save her.\nIn short: great graphics; a slightly confusing story and battle system (but both with potential); I miss the magic of some of the previous games a bit.\nIt's nearly a month now that I quit playing Runes of Magic. A few days ago I read that they improved the account system. It seems it's now possible to use different passwords for all the accounts you create by signing up for the game + the number of failed logins is restricted now. I don't miss the game so I won't go back. Besides, I still have other, equally serious, issues with it so no.\nAnyway, I'm playing something else at the moment: Torchlight. It's a Hack&Slay and looks and feels like the Diablo games but I don't mind, I love those games. To be able to play an updated version is fun.\nLast week I was tempted into buying books because I had a 11%-off coupon available. I wanted to buy much more books actually (for ~ 60\u20ac) but I managed to tame my \"me-want!\" side down to books for 30\u20ac. Phew!\nI really liked Furies of Calderon, the first book in the Codex Alera series. So much that I wanted to read the next one right away. Of course I couldn't because I never (well, at least I can't think of an example) order several books of a series before I've read at least one book in it. But I was very tempted to order all the remaining books in this series together with Academ's Fury.\nI like to read Western romances now and then and this sounded interesting: the hero is \"as wild as the fierce Montana landscape he roams\"* while the heroine is described as \"all fine city clothes and polite manners.\"* Plus, I like to try new-to-me authors from time to time.\nAgain, a new-to-me author. What made me take notice of the book is that the hero is described as slowly losing his eyesight. I also liked the excerpt = buy.\nI like Kleypas's contemporary voice and so it was only a question of when I would buy this book. I bought it this time because it was the only book I wanted that was in stock. I have to wait around three weeks for the others to arrive in my \"me-want!\" hands.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fast trains from Chiusi to Venice take around 4 hours and 6 minutes, covering a distance of approximately 273 kilometres. There are frequent services on the rail route between Chiusi and Venice.\n1. Are there direct trains from Chiusi to Venice?\nDirect trains from Chiusi to Venice have been found.\nWe usually find around 4 direct trains on the route from Chiusi to Venice every weekday.\nThere are typically fewer departures at weekends, when we found around 3 departures.\n2. How long is the train journey from Chiusi to Venice?\nThe Chiusi to Venice train travel time is normally about 4 hours and 6 minutes, whatever time you make the journey.\n3. What are the Chiusi to Venice train times and schedule?\nOn weekdays the earliest direct train to Venice is usually scheduled to depart Chiusi around 12:10 and the last train is around 00:07. This is the same at the weekend.\nWhen to book Chiusi to Venice train tickets?\nTrains in Italy usually open for booking around 6 months in advance. If booking is not yet open for your dates then you can set a booking alert to receive an email on the day that the cheapest tickets are released for your journey from Chiusi to Venice by train.\nWhich train companies operate between Chiusi and Venice?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Birco Tools has a substantial catalogue of Waste and Plugs Accessories at highly discounted prices. Birco Tools are official suppliers of Plumbob, Silverline, GMC, Triton, Kreg, Krazy Glue, Magswitch, Bench Dog, Elmers, Fixman, Task, Smaart & Powermaster. If you require any advice on our Waste and Plugs Accessories then please contact us on 01217832726 and speak to one of our professionals. The Waste and Plugs Accessories are suitable for tradesmen or the keen DIYer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The fruit flavor fallacy, the misguided belief that sweet e-juice flavors were created to hook children, is one of the most ridiculous zombie vaping myths. Unfortunately, a spike in underage vaping has been framed as a teen vaping epidemic and the blame has been placed on these flavors. The reality is far different. The majority of adult vapers prefer sweeter flavors.\nAs the biggest player in the vaping industry, Juul has been the obvious target of regulators and anti-vapers. Marlboro maker Altria purchasing a 35 percent stake in Juul did nothing to enhance their anti-smoking crusader bona fides, but they were already a primary target of new FDA e-cig regulations.\nJuul removed their sweet and dessert flavors from brick and mortar establishments in response to the pressure. As with all regulatory moves, there were several immediate consequences. The first was that isolated urban and rural populations are now denied their choice of smoking alternative and funneled back towards cigarettes. Iowa AG Thomas Miller warned the FDA about this exact outcome.\nIt is an established fact that the the wealthy have long since given up smoking. The harsh reality is that strict e-cig laws punish marginalized groups, but this does not resonate in the circles where these sorts of decisions are made.\nThe second blow was to the San Francisco based e-cig giant's bottom line. The Winston Salem Journal reports that Juul's market share dipped from 75.8 to 73.8 percent in four week period ending on January 26, 2019. Overall, the vaping industry is still predicted to grow from 7 to 9 billion in the US. But this is Juul's first set back and it will be interesting to see how strongly they rebound.\nThe same Nielsen sales data found that rival Vuse, made by R.J. Reynolds Vapor Co., increased their own share of the market from 10 to 11.6 percent. Vuse did not follow Juul's lead and pull their sweeter flavors. It remains to be seen how effectively they can retain these new customers.\nThe now defunct MarkTen took the bronze medal with 6.6 percent. Blu eCigs was fourth with 6 percent, and Logic was a distant fifth at 2.7 percent.\nA two percent drop may not sound like much, but it is the first sign that an unrestrained period of growth is finally coming to a close. A year ago, Juul had 51.6 percent of the market and Vuse 16.9. Any slow down in growth is a bad sign for Juul, whose enormous market valuation is based on their ability to dominate the brick and mortar e-cig space. Juul is still in a strong position but this retraction is proof that increased regulations can have an immediate impact on their bottom line.\nIt remains to be seen if this decline was a temporary hiccup or the first sign that regulatory headwinds have halted Juul's momentum. Juul plans on reintroducing their sweet and fruit flavors to establishments that adhere to proper age verification procedures.The entire vaping industry faces an existential danger if the issue of minors vaping is not resolved to the FDA's satisfaction.\nEmboldened by their alliance with Altria, it appears that Juul is done rolling over and will begin to fight back. They had already begun to hire lobbyists in an effort to blunt the attacks of their critics before Altria's buy-in. Gone are the days when bizarre complaints about the name Cr\u00e8me Brule, an upscale dessert that peaked in popularity a decade before today's high school students were born, sent Juul scampering for new packaging.\nIn an unprecedented move, Juul plans on launching a television advertising campaign. Nicotine products have not been seen on the airwaves in decades at this point. Despite their temporary pullback from retail outlets, Juul's relationship with Altria will allow them to control premium shelf space at gas stations and convenience stores.\nJuul even proposed including informational packets in Marlboro cigarettes that extol the virtues of their device. Their efforts to spin this as a public service, rather than a transparent marketing ploy, have been largely unsuccessful. No matter what the critics say, if this plan gets off the ground it would give them an inside track to millions of potential customers.\nIn order to fight the anecdotes and misrepresentations of anti-vapers, Juul released a study which demonstrated the beneficial role that flavors can play for smokers looking to transition to e-cigs. This survey was soon overshadowed by the monumental New England Journal of Medicine study which demonstrated that e-cigs were nearly twice as effective as nicotine replacement therapy for smoking cessation. The NEJM study did not test the efficacy of Juul's device but instead used a classic 650 mAh cartomizer system.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The problem that most of us face when inspired by this phrase is how to get started. What exactly can we do 'smarter' or more efficiently? How can we accomplish all there is to be done, without additional hard work?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NWO awarded Projectruimte grants to UvA physicist Wouter Waalewijn, and Leiden physicists Koenraad Schalm and Jan Zaanen. Waalewijn will use the grant to study new methods for precise predictions at the Large Hadron Collider. Schalm and Zaanen will use the funding to further study the so called 'strange metals'.\nThe board of NWO's Domain Science granted funding for a total of eight proposals in the NWO Physics Projectruimte, a granting instrument for small-scale projects that propose innovative fundamental physics research that has a scientific, industrial or social urgency. The eight grants total an amount of 3.5 million euros.\nWouter Waalewijn, University of Amsterdam.\nWouter Waalewijn's Projectruimte grant was awarded for his research on new methods needed to make precise predictions for the processes observed at the Large Hadron Collider in Geneva. The Large Hadron Collider is entering an era where precision measurements are the key to possible discoveries. Indications of new physics may lie hidden as faint signals, that will be revealed as experimental uncertainties decrease, making it crucial to have the precise theoretical predictions to compare with. The researchers propose a new method to perform these calculations, that will make them easier and therefore available for more processes.\nNWO has awarded a Projectruimte grant to Leiden physicists Koenraad Schalm and Jan Zaanen to study 'strange metals'. These materials are alleged to follow the bizarre laws of quantum mechanics even at everyday length scales. High temperature superconductors are a well known class of materials that exhibits this kind of behaviour.\nJan Zaanen (left) and Koenraad Schalm, both Leiden University.\nSchalm and Zaanen will focus on one aspect of strange metals' abnormal behavior: the time they need to distribute heat throughout the entire material starting from one place. Remarkably enough, this solely depends on the temperature and Planck's constant. The researchers want to show that this is because strange metals indeed purely consist of quantum material. Subsequently, they draw a surprising parallel with black holes, which exhibit the same behavior.\nThe NWO Physics Projectruimte is one of the funding instruments that NWO has for supporting physics research. Researchers can continuously submit proposals to the NWO Physics Projectruimte until 1 May 2018. The proposals will be evaluated in the order in which they are received. From 1 August 2018 onwards, researchers will be able to submit proposals for two new funding instruments of the NWO Domain Science.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sweden's Prime Minister Fredrik Reinfeldt is unhappy with Swedish members of the European parliament who supported an EU budget increase despite his centre-right coalition trying to keep down the costs.\n\"It comes down to a potential conflict between Sweden's interests and the European parliament, which remarkably enough has received support from Swedish MEPs,\" Reinfeldt told the Swedish Riksdag upon his return from the stranded long-term budget talks in Brussels.\nEven Swedish MEPs from parties that form part of the government-coalition alongside Reinfeldt's Moderate Party voted to increase long-term union spending.\nCecilia Wikstr\u00f6m of the Liberal Party (Folkpartiet) and Kent Johansson from the Centre Party both voted in favour of the increase, which the news agency TT claims could add another 10 billion kronor ($1.5 billion) to Sweden's membership bill.\nAs did G\u00f6ran F\u00e4rm, who heads the group of Swedish opposition Social Democrats in Brussels.\nF\u00e4rm dismissed Reinfeldt's criticism as an attempt to shift the blame for his and other country leaders' failure to shore up the fraught negotiations.\n\"Nobody takes responsibility for how the EU parliament works,\" he told TT.\n\"Instead, everyone just retreats to a national standpoint to fight it out over whether it means a plus or a minus for your own budget,\" F\u00e4rm said.\nCould the Sweden Democrats leave Sweden ungovernable after the election?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Transportation Authority's governing board consists of the eleven members of the San Francisco Board of Supervisors, sitting as commissioners of the Transportation Authority. The Transportation Authority Board is not a City Commission. The Transportation Authority is a separate legal entity from the City and County of San Francisco, created under state law. The Transportation Authority Board has one standing committee and one special-purpose select committee. The Board members elect a chair every January. The chair appoints the members and chairs of the committees and serves as an ex-officio member on the committees.\nSFGovTV provides a continuous video archive of Transportation Authority board and committee meetings. Video on Demand lets you watch these programs at your convenience. The link in each section directs you to the archive for that committee or the full board.\nCommissioners: Aaron Peskin (Chair), Rafael Mandelman (Vice Chair), Vallie Brown, Sandra Lee Fewer, Matt Haney, Gordon Mar, Hillary Ronen, Ahsha Safai, Catherine Stefani, Shamann Walton and Norman Yee.\nMeets the second and fourth Tuesday of every month, at 10:00 am in the Legislative Chamber, located on the second floor at San Francisco City Hall, 1 Dr. Goodlett B. Carlton Place, San Francisco.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Frank, Pat [Harry Hart Frank]. Alas, Babylon . New York: Harper Perennial,  2005. ISBN 978-0-06-074187-7.\nThis novel, originally published in 1959, was one the first realistic fictional depictions of an all-out nuclear war and its aftermath. While there are some well-crafted thriller scenes about the origins and catastrophic events of a one day spasm war between the Soviet Union and the United States (the precise origins of which are not described in detail; the reader is led to conclude that it was an accident waiting to happen, much like the outbreak of World War I), the story is mostly set in Fort Repose, a small community on a river in the middle of Florida, in an epoch when Florida was still, despite some arrivals from the frozen north, very much part of the deep south.\nRandy Bragg lives in the house built by his ancestors on River Road, with neighbours including long-time Floridians and recent arrivals. some of which were scandalised to discover one of their neighbours, the Henry family, were descended from slaves to whom Randy's grandfather had sold their land long before the first great Florida boom, when land was valued only by the citrus it could grow. Randy, nominally a lawyer, mostly lived on proceeds from his orchards, a trust established by his father, and occasional legal work, and was single, largely idle, and seemingly without direction. Then came The Day.\nFrom the first detonations of Soviet bombs above cities and military bases around Fort Repose, the news from outside dwindled to brief bulletins from Civil Defense and what one of Randy's neighbours could glean from a short wave radio. As electrical power failed and batteries were exhausted, little was known of the fate of the nation and the world. At least, after The Day, there were no more visible nuclear detonations.\nSuddenly Fort Repose found itself effectively in the 19th century. Gasoline supplies were limited to what people had in the tanks of their cars, and had to be husbanded for only the most essential purposes. Knowledge of how to hunt, trap, fish, and raise crops, chickens, and pigs became much more important than the fancy specialties of retirees in the area. Fortunately, by the luck of geography and weather, Fort Repose was spared serious fallout from the attack, and the very fact that the large cities surrounding it were directly targeted (and that it was not on a main highway) meant it would be spared invasion by the \"golden horde\" of starving urban and suburban refugees which figure in many post-apocalyptic stories. Still, cut off from the outside, \"what you have is all you've got\", and people must face the reality that medical supplies, their only doctor, food the orchards cannot supply, and even commodities as fundamental as salt are limited. But people, especially rural people in the middle of the 20th century, are resourceful, and before long a barter market springs up in which honey, coffee, and whiskey prove much more valuable than gold or silver.\nWherever there are things of value and those who covet them, predators of the two footed variety will be manifest. While there is no mass invasion, highwaymen and thieves appear to prey upon those trying to eke out a living for their families. Randy Bragg, now responsible for three families living under his own roof and neighbours provided by his artesian water well, is forced to grow into a protector of these people and the community, eventually defending them from those who would destroy everything they have managed to salvage from the calamity.\nThis is a tale of the old, weird, stiff-spined, rural America which could ultimately ride out what Herman Kahn called the \"destruction of the A country\" and keep on going. We hear little of the fate of those in the North, where with The Day occurring near mid-winter, the outcome for those who escaped the immediate attack would have been much more calamitous. Ultimately it is the resourcefulness, fundamental goodness, and growth of these people under extreme adversity which makes this tale of catastrophe ultimately one of hope.\nThe Kindle edition appears to have been created by scanning a print edition and processing it through an optical character recognition program. The result of this seems to have been run through a spelling checker, but not subjected to detailed copy editing. As a result, there are numerous scanning errors, some obvious, some humorous, and some real head scratchers. This classic work, from a major publisher, deserves better.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Business can be one long learning curve. Even highly experienced entrepreneurs can be confronted with new situations, or have to find new ways of doing things. That's why it's vital to learn from both our triumphs and our mistakes. By understanding the science that lies behind effective learning, we can make better decisions when confronted with the new or the unexpected. Here, Blue Square Concepts review the book 'Learn Or Die: Using Science To Build A Leading-Edge Learning Organization', by Edward D. Hess (published by Columbia University Press).\nThere are many factors that are transforming the business landscape. Technological innovations are at the forefront of this. Consumers love to use the latest technology, so if your business isn't using it, then you could be left behind by business rivals who are. Increased globalization is also creating the need for businesses to both learn and evolve. When confronted by competitors from other nations who have a different way of doing things, do we retreat into our shells and carry on as before, or try to learn from what they're doing, and improve the way that we operate?\nThe author also looks at the way that science can help us understand how and why we learn. To do this he uses neuroscience, behavioral analysis, psychology, and the results of academic research. He takes leading companies such as UPS as examples, and shows how they've created an atmosphere of learning and innovation. Creating such an atmosphere is a primary goal for any business. Entrepreneurs and workers should always be seeking continual improvement, and looking for ways to make the service they offer better and more effective.\nBlue Square Concepts knows all about the power of innovation. Their tailored marketing campaigns help businesses gain new customers and achieve growth.\nBlue Square Concepts is a strategic business consulting firm. Our primary expertise is providing consultative solutions to numerous Fortune 500 companies, government agencies, and non-profits in Philadelphia and Houston. We operate in 19 different countries and have more than 130 offices in the United States.\n1080 N Delaware Avenue, Suite 502.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of my Listservs led me to this interesting article by John Allen Paulos about the distinction between the \"literary and scientific cultures\". As part of the discussion, Paulos discusses some cases where knowing a narrative background affects how probability is assessed.\nConsider the following two statements.\nSarah is a bank teller.\nSarah is a bank teller and has a philosophy degree.\nThe answer is that the first option is more probable because only one condition needs to be met. In order for the second to be true, two conditions are required - being a bank teller and having a degree in philosophy degree.\nThe finding is that more people will be that the second option is the most likely - i.e. that Linda is a bank teller and in the feminist movement, even though it requires the fulfillment of two conditions.\nThere are several philosophical tacks one can take to the problem, but I think one factor is that the story along with the presentation of the information affects the construction of the model used to evaluate the statements.\nSomeone reading first scenario without the narrative probably constructs the intended model where the probability of being a bank teller versus a bank teller with a philosophy degree is evaluated across all adult women. It's easy to see that fulfilling condition A is more probably than fulfilling condition A and B.\nthen the conclusion would likely be that Linda being active in the feminist movement is more likely than her being a bank teller. In other words, readers could be using the narrative to build a stereotyped persona where someone who was politically active in college remains active. In the same vein, most people likely assume that someone with a philosophy degree becomes a teller only as a last resort and that most tellers have a degree in accounting or other related field. This is one possible source of the fallacy.\nLinda is a bank teller who is not active in the feminist movement.\nLinda is a bank teller who is active in the feminist movement.\nThere is a further pragmatic interpretation that option a) \"Linda is a bank teller\" means that she is not politically active at all. That's not literally the case (for instance, option a) does allow that Linda could still be active in the anti nuclear proliferation movement, but not the feminist movement). In pragmatic land though, omitting information is interpreted as meaning it doesn't exist. That's why people often consider not saying something to be \"lying.\"\nSo to summarize, I think the skewed probability judgments aren't just a result of people being sucked into a mini soap opera, but to two factors the narrative introduces - 1) narrowing the set of women to those with philosophy degrees, which leads to different stereotypes and 2) the options leading to misconstrued pragmatics which differ from what the literal meaning is.\nThe ability to reasonably construct a pragmatic meaning behind a literal statement is critical for social relations and reducing conversational length. But it can lead to some glitches like the narrative above.\nI'm cleaning out my instructional design papers and I ran into the \"concept map\" papers and remembered an interesting question about whether you should include verbs in concept maps.\nFirst, if you don't know what a \"concept map\" is, I'll just say it's a diagram of concepts and how they connect. Here's a concept map from Wikipedia to define what a concept map is. Typically concepts (usually nouns) are placed in different shapes and arrows are drawn to show different types of connections. Note though that these arrows are labeled with either verbs or prepositions.\nNow although I adore diagrams, I find concept maps surprisingly confusing. A question to LinguistList pointed out one reason why - the arrow labels are confusing (and it may not be just me). In theory, if you can add labels to boxes, you should be able to add labels to connections, but that may not be the best approach.\nThe truth is that our culture (and many others) use a lot of concept maps including family trees, subway maps, pie charts and many others. In almost all of them, the labels are left blank and the conventions of placement\/size telegraph the relationships. Either by culture or maybe some inborn ability, I think a lot of people have learned to interpret relations just from the graphic design. (If you're not a natural map reader, I would be curious to see if labelling does any good).\nFor instance, in a family tree, we understand that the names on top are the ancestors and those on the bottom are the descendants. The equal sign or line between a man and a woman usually indicates marriage and lines from the equal sign to a name is a legitimate child (unless it's a dotted line, in which case things are different). Similarly names on the same level with lines going to the same equal sign are full siblings. Finally if it's a royal family, the bold face name or the one in another color is the monarch.\nYou can convey a lot of information with just changes in lines and special labels. In fact, you can usually make connections (e.g. determining first cousin or transfer point on a subway system) by tracking the appropriate sets of lines to the proper boxes. I would say that this convention is so powerful that adding too many labels is in fact distracting because it is interfering with the cues that the diagram structure is trying to convey.\nYou can call it interference on the visual vs verbal\/linguistic channels if you like. In the map with the labeled arrows above, the arrows are all the same which implies \"same relationship\" (at least locally), but the labels are saying \"no - different relationship\". You really may have conflicting cues here.\nSo I'll present two concept maps below, my original one without the labels and one with labeled arrows. Which one is clearer to you?\nWe begin at Anglo Saxon (pre 1066) which is spoken in England and southern Scotland. In 1066, the Norman Invasion (William the Conquerer) establishes a foreign government in England, but not Scotland. Scots splits off from the rest of English (somewhat before the Great Vowel Shift. In England most dialects go through the Great Vowel Shift and lose \/x\/ and \/\u00fc\/.\nDuring the period when North America is colonized, many dialects in English experience loss of word final \/r\/, but some maintain it. Settlers from regions losing the final \/r\/ arrive in New England, New York City and the South, but settlers who keep the \/r\/ arrive in the Mid Atlantic and Canada. The MId-Atlantic form becomes Standard American\/Standard Canadian while the New England and Southern forms become regional dialects. Standard U.S. continues to evolve into Californian English and other varieties (as does New England\/New York\/South).\nAfter American independence, the English continue to colonize other regions in the world including Australia, New Zealand and South Africa. Although these dialects experience unique vowel changes, they have all lost final \/r\/.\nNote this is just a quick summary. You should read more about the history of each dialect for details. The books Albion Seed (David Hackett Fisher) and The Story of English (Robert McCrum and Robert MacNeil) have some good information.\nRecently Stephen Colbert interviewed remix advocate Lawrence Lessig. In a fit of pique, he announced that \"copyright was forever\" and commented that he would be \"litigious\" if the interview was remixed with \"some great dance beat.\"\nBut since we are the YouTube Generation, of course multiple people remixed the interview with some great dance beat. What's interesting here though is that most of the remixes involve looping short snippets of dialogue over and over.\nAs hip-hop artists know, if you loop a short phrase enough, you are one your way to a quasi-lyric. This was also noted on a NPR podcast which I've lost track of, but I think the remix here really shows how little looping is required to generate a rhythmic feeling.\nThere's been recent speculation on how music and language relate to each other, and clips like these do make a case that music is some sort of auditory cheesecake (to quote Steven Pinker). In other words, music is borrowing phonological aspects of language (pitch, rhythm), then \"kicking them up a notch\".\nI don't think it's the full story of music, but then again visual arts are also a little mysterious to the neo-Darwinist. Still I'm glad I have both my screensaver and my iTunes to brighten my day.\nThe Onion News Network has a great parody on a new Mac laptop with the keyboard replaced with just the iPhod wheel (jut what we've all been waiting for. But the best feature, was the \"sentence predictor\". Like the word predictor, the sentence predictor would give you options for completing a sentence once a few words had been typed in.\nI hope you enjoyed the video as much as I did, but it get me wondering if a sentence predictor is possible, and if so what it would be like. It is a real problem that researchers are exploring as can be seen in this presentation from Nicola Carmignani.\nActually I could see something like a phrase predictor based on syntax and morphology (and these have been built). For instance, if had a phrase \"give the book\", English syntax dictates that we usually have to specify a destination, hence you can predict that the next word will be \"to\". After that though, syntax dictates that you can give a book to anyone (literally an infinite number of choices).\nSo for a more robust sentence predictor, you would probably need be able to access the context or discourse of the entire passage. For instance if you had been discussing your friends Rachel and Phoebe, then a sentence predictor could guess \"to Phoebe\" or \"to Rachel\" as plausible guess for \"give the book\". Carmignani's presentation in fact, gives a song lyric as an example - \"Penny Lane\", which most people could predict would end with \"is in my ears and in my eyes\" (following the Beatles song).\nIs Counting Innate in Australia?\nA new study by Brian Butterworth and others finds that indigenous Australian children can count even if the language does not gramatically have the same range of number words as other languages. This is an interesting counterclaim to the notion that members of the Amazonian Pirah\u00e3 tribe not only lack number words, but can't learn to count. The experiment in Australia was conducted by asking children to put up the number of objects matching the number of beats on two drumsticks.\nThis experiment was interesting in that younger children were used. The use of children is interesting because, as far as I know, the Pirah\u00e3 subjects were adult men. But since Pirah\u00e3 children who learn Portuguese also learn to count in Portuguese, there may be a critical period element of some sort (or not). In any case it would be interesting to replicate both experiments in the other region (assuming the Pirah\u00e3 parents would agree to it of course).\nThe other thing that would be interesting to track is how the Australian language number systems were structured. As far as I can tell, the Pirah\u00e3 \"number\" words aren't numbers at all, but estimates of small vs. large quantities. The Australian number system may be more limited, but some words like \"one, two, three\" may refer to exact quantities which is a key conceptual difference.\nThe study is in the Aug 18 issue of PNAS (Proceedings of the National Academy of Sciences of the United Stated of America). This article did not seem to be available yet.\nAn interesting news story from the past few years is the Amazonian language Pirah\u00e3 which lacks number words. That is, instead of counting quantities (1,2,3,4...), the Pirah\u00e3 only estimate quantities (relatively small, relatively large). The latest study from MIT seems to confirm this. Interestingly, when objects are taken away from a pile, the estimates change so that \"small\" may become 5-6 instead of 1-2 as previously thought.\nThis has perplexed linguists since almost all languages have some sort of counting (even in remote locations). The only other examples of low-tech numbers had been systems of 1,2, many. We normally think of counting as a \"basic\" cognitive skill, but it appears to be primarily cultural.\nI first about this in 2000 from a guest speaker Peter Gordon. His evidence was convincing, but there have been some points I have been pondering.\nNot surprisingly, male laborers in Brazil are stiffed a lot because they do not pay as much attention to \"exact\" quantities. However the Pirah\u00e3 women are reported to gently mock their men folk for this [Gordon, p.c.] It reminds me of cultural gender stereotypes like men can't pick coordinating colors and women can't work with computers (and yes many of us buy into them whether they are 100% true).\nMany animals can easily distinguish quantities of 1,2,3, (or a little more) on sight, but after that they guesstimate. In this study, monkeys can recognize quanities of 1-4, but estimate after that. This predicts that a basic counting would be something like 1,2,3, many, but the Pirah\u00e3 system is even more basic.\nIt is startling to think that counting could be essentially \"cultural\" because almost every other culture has some form of counting, but when I read more, I did realize that there is some truth to this. For instance, most languages have unique words for one such as one , aon (Irish), bat (Basque), but once you get into the range of 1,000,000 (one million), the word starts to resemble million in many languages (e.g. 1 million = milioi in Basque). That's because the concept of such a large quantity is relatively new (few hunter-gatherers needed to count to 106). As modern technology spreads, so do numbers (Basque itself apparently borrowed 1,000 mila from the Romans (i.e. Latin millia ).\nPirah\u00e3 is considered to be a member of the Muran language familty, and I had been wondering if the related languages had any counting. Alas, all the other Muran languages are now extinct. I'm still not sure about Everett's claim that Pirah\u00e3 culture does not deal with abstract topics - That is unbelievably rare considering almost all cultures have art, mythology, and pretty elaborate kinship systems.\nIt's high Presidental primary season here in Pennsylvania, 2008 and I can't leave this momentous era without SOME kind of observation on the process. I've been watching a lot of primary night analysis, and one thing that few people touch on is the \"Head of State\" factor.\nThat is, one of the major duties of the president is to serve as Head of State. Normally this is seen as a \"trivial\" duty since I think most people associate it with opening the Olympics, attending important funerals or supervising the Easter Egg roll. And yet, I think it's an important sub-conscious factor in our decision making process.\nAfter all the Head of State also has to comfort us in times on national tragedy and also be able to relate to the average citizens he (or she) has to meet and greet. How do we know if a presidential candidate can relate to average citizen? One way is to challenge them to a fish toss in Seattle. Another is to see if they have a sense of humor about themselves - self-deprecating if possible. Abraham Lincoln is maybe the best -known master of the art.\nAnd if you think about our recent successful two-term presidents, you will see that almost of them had a sense of timing regardless of party boundary. On the Democratic side we have Bill Clinton, and the late John Kennedy (his press conferences still make modern audiences laugh). On the Republican side there has been Ronald Reagan and George W. Bush (really).\nChances are that the list I just named has at least one president you absolutely can't stand - but if you watch their quips or speeches for the National Correspondents Dinner, I bet you will see they had a great sense of comic timing. Some of our recent one-term presidents (Carter and Bush I) had more problems with this of self-deprecating comedy, and, sure enough, they were beaten by candidates with better comedic timing. Oddly enough, Nixon is our only recent anomaly - he was never known for great timing yet won two terms. But look at what happened to him!\nAnd for 2008 - does it still matter? We don't know yet, but I submit that at least one candidate is having problems partly because that person has not completely mastered the art of delivering a good quip (a few have backfired very, very badly). It sounds shallow, but I also sense that it can be a sign of deeper issues with the candidate. When all you really know about a candidate is what you see on TV, it's interesting what cues you may have to use.\nOne class of linguistic questions I see a lot is \"Why does John Doe say X when it doesn't make any logical sense?\" This is usually in reference to a new idiom, dialect or language the questioner is encountering. The answer is that while grammar is usually consistent, it doesn't always follow real world logic. Even worse, there are lots of idiosyncratic quirks that happen \"just because\".\nA recent one was \u2013 if people from Korea speak Korean , why don't people from Japan speak * Japanian (instead of Japanese ). And why do the Basque speak just Basque and not * Basqu(i)an, *Basquese or even * Basquish . It's a mystery. You can solve some of it by saying that the - ish\/-sh\/-ch ending is older and tends to be used with languages closer to Britain (e.g. English, Spanish, Welsh, Irish, French (aka Frankish )). But do you really think the average adult has even connected language names with history of the Anglophone world? No - it's just memorized.\nIronically, non-standard dialects can actually be more logical, yet will still be dismissed as \"poor grammar\". After all, if we say her pen but that pen is hers and our building and that that building is ours ...shouldn't we also say that pen is mines (like they do in Baltimore). Logically...yes. But I think we all know what happens to the poor student who uses mines in an essay \u2013 and it's not an A+ for logic.\nThis is an unsettling concept because so many writers and speakers do use language to construct effective, logical arguments. Shouldn't the bones of language (vocabulary and grammar or syntax and morphology) also be equally logical. The surprising answer is that while grammar can have a system, it's not one that is \"logical\". To me this is a powerful reason to think grammar is not directly connected with \"general cognition\".\nI mean, who in their right mind would invent a language with as many irregular past tense verbs as English has? Ugh!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For the purpose of the Data Protection Act 1998 (the \"Act\"), the data controller is The Keep Boutique Ltd registered 52 High Street HA1 3LL Harrow On The Hill.\nInformation that you provide by filling in forms on our website www.thekeepboutique.com (\"Website\"). This includes information provided at the time of registering to use the Website, subscribing to our service, posting material or requesting further services. We may also ask you for information when you enter a competition or promotion sponsored by The Keep Boutique Ltd, and when you report a problem with our Website.\nWeb beacons consist of a small string of software code that represents a graphic image request on a page or email. There may or may not be a visible graphic image associated with the web beacon and often the image is designed to blend into the background of a page or email. Web beacons can be used for many purposes \u2013 including site traffic reporting, unique visitor counts, advertising auditing and reporting, and personalization. Web beacons used by www.trouva.com collect only anonymous data.\nIn order to understand how people use the Website and for marketing analysis and quality improvement purposes, we (and\/or third party service providers) may collect, record, process and use on an anonymous basis certain information (including the total number of transactions, viewed pages, referring\/exit pages, platform type, date\/time stamp information and details like the number and location of mouse clicks on a given page, mouse movements, scrolling activity and the search words you use while being on and using the Website).\nAll information you provide to us is stored on our secure servers. Any payment transactions will be encrypted using Stripe or Paypal payment technology. Where we have given you (or where you have chosen) a password which enables you to access certain parts of our Website, you are responsible for keeping this password confidential. We ask you not to share a password with anyone.\nIf The Keep Boutique Ltd or substantially all of its assets are acquired by a third party, in which case personal data held by it about its customers will be one of the transferred assets.\nYou have the right to ask us not to process your personal data for marketing purposes. We will usually inform you (before collecting your data) if we intend to use your data for such purposes or if we intend to disclose your information to any third party for such purposes. You can exercise your right to prevent such processing by checking certain boxes on the forms we use to collect your data. You can also exercise the right at any time by contacting us at 32\/33 Granville Arcade, Coldharbour Lane, London, SW9 8PR or at kate@thekeepboutique.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Aside from all the hustle and bustle in Wan Chai, the place is also a hidden gem for authentic designs, craftsmanship and interesting local stories. Join us to explore the lesser-known side of Wan Chai in the upcoming tailor-made guided tours curated by #ddHK and iDiscover.\n#ddWalker highlights the architectural gems in this neighbourhood. We bring you to the oldest surviving post office, the revitalised Wan Chai Market, Lee Tung Avenue (a.k.a. \"Wedding card street\"), the award-winning Blue House cluster, revitalised Mallory Street tenement buildings, and the circular commercial building Hopewell. You get to see all types of buildings and learn all about city design and heritage in Hong Kong in this tour.\n*The reservation is on a first\u2010come, first\u2010served basis. The first 25 successful applicants of each tour will be notified by email confirmation. Other applicants will be placed on a waiting list and will be informed individually if any quota becomes available.\nDesign District Hong Kong (#ddHK) is a 3-year Creative Tourism Project (2018 onwards) presented by Tourism Commission, organised by Hong Kong Design Centre (HKDC). The open-air design district gallery was curated under the theme of \"Connected City: A Love Letter to Hong Kong\" in the first year. In 2019, the theme is \"Pop! Get Inspired at Every Turn\". It is a fantastic blueprint for the visitors and public to explore the city in a brand new way.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Just let your eyes unfocus as you look over this page (I screencapped it from the Disney store online). Remember that old Sesame Street skit, \"One of these things is not like the others\"?\nThis is exactly why Merida needs to retain her own look and her own voice. So she doesn't get lost in this sea of homogenization.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Both sides will target three points after tasting defeat in their last match.\nKerala Blasters head coach Rene Meulensteen is keen to move on from the defeat to FC Goa defeat as the hosts continue their search for the first win this season, against NorthEast United. The Dimitar Berbatov-less Blasters have made their worst start in the history of the Indian Super League with just three points from four matches.\n\"Berbatov's absence is a huge loss. He gives us that quality and composure. I don't think that game (the match against FC Goa) would have developed the way it did if Berbatov would have stayed fit. The freak seven minutes in Goa is just something that I don't think will happen again. We are waiting for the first win, that's the most important thing and how we are going to get it is secondary,\" Muelensteen said.\nMeulensteen admitted that his side made several mistakes in Goa and reckons it will take time to see quality football in the league. \"I have seen so many ISL games already and most of the goals come from mistakes rather than really good football. Over time this will get less. The Indian players will get better and that's part of the learning curve.\"\n\"I would expect one or two mistakes not four or five. Hopefully we made all the mistakes in one game and we don't have deal with any going forward,\" the 51-year-old added. He confirmed that Iain Hume has resumed training and will be part of the squad later today.\nNorthEast United head coach Joao de Deus was in upbeat mood ahead of his side's away clash against Blasters. Though history is against the Highlanders as they are yet to win a match against the Kerala outfit in Kochi, Joao is not worried.\n\"It was in the past and doesn't have any nothing to do with what's going to happen tomorrow. I am confident that we are going to have a good match tomorrow,\" Joao said.\nWhen asked whether Kerala Blasters assistant coach Thangboi Singto's inside knowledge of few NorthEast United players he managed at Shillong Lajong could have an impact in the game, the Portugese was eager to point out his relevance in the team.\nThough Blasters had a hard time against FC Goa, de Deus believes the Berbatov-less hosts are a strong unit. \"They (Blasters) are a strong team and usually attack from the wings. You cannot judge them based on one match as it's a long season,\" he said.\nHe also lavished praise on Rowlin Borges and referred to the 25-year-old as the best Indian player he has seen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It'll be hard for Kanye to stick to his workout plan this week! In honor of his 38th birthday, a New York City ice cream shop transformed their store into Kanye's \"Beautiful But Darkly Lit Ice Cream parlor\", serving Kanye inspired sweet treats using Kanye's lyrics and sayings.\nMorgenstern's Finest Ice Cream on the Lower East Side teamed up with First We Feast to create a Yeezus-inspired menu -- and it's hilarious. There's the \"Imma let you finish\" super size. The \"Bowl Digger\" sundae.The \"Red October\" Raspberry, and an ice cream sandwich that pays homage to Kanyes song \"I am a God.\"\nHow about the Kim Kardashian Thirst Trap Sundae? It almost exactly resembles Kim's Paper Magazine cover where she broke the internet.The description is equally funny as it's the image of Kim's double scoop. Perhaps the best item, is the \"Kanye Inception ice-cream cake,\" featuring a funny paparazzi shot of Kanye eating ice-cream in his car.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The history of food and cooking in Taiwan is as deep as the ocean that surrounds the mountainous island. In Taiwanese cuisine influences from all of mainland China can be found, in addition Japanese styles are prevalent and native Taiwanese dishes live on as well. Taiwan's unique variation of Chinese regional cooking styles mixed with aboriginal Taiwan as well as Taiwan's past rulers: the Dutch, Spanish and later Japanese have created a literal melting pot of some of the world's top cuisines.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Property Location When you stay at Shahee Resorts in Panchgani, you'll be close to Table Land. This hotel is within the region of Wilson Point and Mahabaleshwar Bazaar.Rooms Make yourself at home in one of the air-conditioned guestrooms. Cable television is provided for your entertainment.Rec, Spa, Premium Amenities Be sure to enjoy recreational amenities, including an outdoor pool, a spa tub, and a sauna. This hotel also features an arcade\/game room, wedding services, and a television in the lobby.Dining Satisfy your appetite at the hotel's restaurant, where you can take in a pool view. Snacks are also available at a coffee shop\/cafe.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"E-mail us with any questions. We provide free shipping and insurance. Be assured we carefully pack with bubble wrap in boxes and ship quickly.\nSHORT HISTORY: These pieces were my aunts! As all of them would be either studio pieces or early Mexican or Native American, which is what she always wore. My uncle collected PreColumbian artifacts, so when he traveled down into Mexico to visit excavation sites, She was wandering the town of Taxco! She has a HUGE collection of Mexican and Native Sterling pieces from the 40s and 50s and other unusual pieces by artists or from Mexico that were given as gifts!\nSALE NOW Retro Brutalist Heart Sterling Necklace Available Taken Chunky Heart Pendant Necklace Dating Status Necklace Silver Unique 925 16\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lessons from the TSA we can avoid. Lessons from the cloud aaS we can adopt. And an idea for how to make security a product consumers want.\nGesture authentication, the \"Robotic Robbery in the Touch Screen\" paper, and the cool hand gesture lady. Nothing is safe.\nQuick primer on Local File Inclusion (LFI) vulnerabilities in Web apps.\nA hundred years ago, the Red Baron ruled the skies. A few years before that, however, he was a simple calvary man. So what lessons can we learn about going from the help desk to the SOC from a guy who went from a horse to a biplane?\nImage Magic is vulnerable. Don't just patch. This is an opportunity to evaluate all the controls on the attack path. Done right, we will have 0-day protection.\nAt a keynote, I pulled out a safe cracker and demonstrated brute forcing. Safes are excellent metaphors for security controls. Take, for example, the TL rating for a given intensity of an attack and a given amount of time. Can we evaluate our own controls with a TL rating?\nWhat can we learn from OK Cupid being scrapped and its users' information being released? Well, for one, we can make sure our Web sites do not get scrapped.\nThere are our business processes. There are our work flows. And then there are third party apps, facilities, and cloud hosting. So how do you do a risk and controls assessment?\nOne time, during heart surgery, the Merge Hemo app locked up. Why? Because of an anti-virus scan. It's an example of what happens when we focus solely on securing technology.\nApplying the \"Swiss Cheese Model\" to after action reports to beat the odds with incident response.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I first read this in 1988 and most recently in July 2001.\nIt's 1987 and Nicholas Foley is a Russian spy for the USSR. He's a \"sleeper\" whose spent over twenty years deep undercover as a University professor in America. Many years ago in his research into hypnosis he discovered a form of hypnosis so powerful that it grants absolute and immediate mind-control of the subject. He's kept that secret, knowing that both the Americans and the Russians would inevitably abuse this technique. However, one day of course his secret is revealed, and he has to choose sides, or to make his own side.\nSo now he's on the run from both the CIA and KGB, and one of these groups has captured his wife. He knows the value of the prize is so great that neither himself nor his wife can expect mercy.\nWell it's another Haldeman book and so of course it's good. It's not, however, momentous. It's an enjoyable but thoughtful read with a couple of good ideas. Surprisingly, although I'd remembered the interesting solution to nuclear war put forward in this book, I'd completely forgotten it was from Haldeman's \"Tool Of The Trade\".\nWhat's it got? mind-control, spies all over the place, and a lot of excessive personal violence.\nLoaded on the 16th July 2001.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tickets for the conference alone are $99 each; and $135 for the conference and luncheon. They may be purchased at http:\/\/www.jazzinthegardens.com\/2015\/luncheon.html.\nChristian Homes for Children will have a day of food, fun and activities starting 11 a.m. March 21 at TY (Topeekeegee) Park, 3300 N. Park Rd. in Hollywood (1nterstate 95 and Sheridan Street).\nThis is the 26th annual Cookout for Kids event, that helps to raise funds for abused and neglected children throughout South Florida. Admission is $12 for ages 13 and up; $8 for ages 5 through 12, and $3 for ages 2 through 4.\nThe admission price include food and drinks, face painting, art and crafts, games and prizes, cotton candy, bounce houses, and a competitive volleyball tournament. For more than 38 years, the Christian Homes for Children have provided a safe space for more than 1,100 abused and neglected children throughout South Florida.\nIf you want to support this effort, come to Pavilions two and three at TY Park and help a child, while having some great family fun.\nThe Alhambra Orchestra will present a free concert, \"Early Travels,\" at 7 p.m. March 22 at Ransom Everglades School, 3575 Main Hwy. in Coconut Grove.\nDirected by Zoe Zeniodi, the concert will feature the virtuosic works by composers Felix Mendelssohn and Richard Strauss. The program will include Dr. Hon Mayhew, a local French hornist performing Strauss's Horn Concerto, written when the composer was only 18.\nThe program will continue with the orchestra performing Mendelssohn's rousing Scottish Symphony, written when he was 20.\nAdmission is free to this family-oriented event. Parking is also free in the school's lot.\nFor more information call 305-668-9260 or email info@alhambramusic.org.\nDr. Bernd Wallschlaeger, who grew up in Germany as the son of a World War II Nazi officer, will discuss the struggles he had living in his father's shadow 7:15 p.m. March 25 at the Aventura Community Center, 3375 NE 188th St. The event is presented by the Hadassah Mitzvah chapter.\nWallschlaeger eventually converted to Judaism and emigrated to Israel, serving in the Israeli Defense Forces as a medical officer. He is a local physician and has written several books.\nAdmission to this special presentation is $10 and proceeds will go to the Hadassah Medical Organization.\nCall 305-725-5928 for more information.\nHere's good news for writers of color: Starting this summer, Voices of our Nation Arts Foundation (VONA\/Voices), the only multi-genre workshop for writers of color in the United States, will relocate to the University of Miami's College of Arts and Sciences.\nThe workshops will move following a long stint at the University of California at Berkeley. The organization will offer its two-week intensive writing workshops in cooperation with UM's Creative Writing Program.\nSince it was founded in 1999 by Elmaz Abinader, Junot Diaz, Victor Diaz, and Diem Jones, VONA\/Voices has guided more than 2,000 writers of color. Its alumni include Tayari Jones, Maaza Mengiste, Laila Lalami, Miami Novelist Patricia Engel, and UM alumnae Daisy Hernandez.\nIn addition to Junot Diaz and Elmaz Abinader, VONA\/Voices' faculty has featured such luminaries as Mat Johnson, Jeffrey Renard Allen, Cristina Garcia, Chris Abani, David Mura, former Miamian Tananarive Due, Suheir Hammad and UM professor M. Evelina Galang.\nDuring the summer workshop, Books & Books Coral Gables will host Thursday night VONA\/Voices faculty readings. The readings are free and open to the community.\nTo apply for the program, go to www.as.miami.edu\/vona. The application process is open through Sunday. Scholarships are available.\nYou are invited to hear Dr. Clara Estow of the University of Massachusetts speak on \"End of Convivencia: from the Mid- 14th Century to 1492\" at 2 p.m. Sunday at St. Johns on the Lake Church, 4760 Pine Tree Dr. in Miami Beach.\nEstow is a renowned scholar in Castilian medieval history, and has taught for 40 years at University of Massachusetts in Boston in the university's Department of Hispanic Studies. Her well-respected book, Pedro the Cruel of Castile 1350-1369, examines one of the most notorious figures in the annals of Iberian history. Her free talk will address the factors leading up to the Inquisition and the Expulsion of the Jews in 1492.\nLavender is also the editor-in-chief of the Journal of Spanish, Portuguese, and Italian Crypto Jews, published by FIU.\nAfter the Inquisition and dispersal of Jews in Spain in 1492, many Jews attempted to hide their identity and migrated to other areas, including northern Europe, northern Africa, and the Ottoman Empire. Others went on ships to various areas of the New World, and set up communities in Mexico, the Caribbean, Central and South America and even the American Southwest.\nThe event is also free.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The population and economic growth increase the need of space leading to an increase in built-up area. There is an increased activity in the city centre which results in the residents feeling less comfortable to live in the area. Hence, the growth of the settlement area tends to head towards the periphery. This condition is an indication of the developments of the built-up area in the periphery. Urban sprawl is a phenomenon of development of an irregular built-up area which leads to the suburbs. Urban development as a result of urban sprawl will trigger an increase in the demand for supporting facilities and infrastructure. An urban Sprawl in Semarang City has resulted in the establishment of residential areas in a conserved region, which is against zoning regulations in the Semarang City Spatial Planning Document (RTRW). Urban Sprawl Typology Research in Semarang City aims to analyse the urban sprawl typology in the city. By knowing the typology, it can minimise the impact of urban sprawl. This research was conducted using the remote sensing method and geographic information system (GIS) with the Shannon's Entropy approach. The results indicated that in 2006 and 2016, urban sprawl in Semarang City had three types of typology that can be classified as typology I (low level), typology II (medium level) and typology III (high level). The impact of urban sprawl on typology II and III can be minimised through the provision of affordable housing with adequate infrastructure, and an improvement in the permit system for housing and settlement development.\nKeywords: urban sprawl, Shannon's entropy, GIS, remote sensing, Semarang.\nPertambahan jumlah penduduk dan pertumbuhan ekonomi berimplikasi pada kebutuhan ruang untuk beraktivitas sehingga menyebabkan terjadinya peningkatan built up area. Peningkatan aktivitas pusat kota yang tinggi menyebabkan penduduk kurang nyaman untuk tinggal di kawasan pusat kota. Sehingga terjadi pertumbuhan kawasan permukiman menuju ke arah pinggiran. Kondisi ini merupakan salah satu indikasi perkembangan built up area di kawasan pinggiran. Urban sprawl merupakan fenomena perkembangan built up area yang tidak teratur mengarah ke pinggiran kota. Perkembangan kota sebagai dampak urban sprawl akan meningkatkan permintaan terhadap sarana dan prasarana pendukung. Penelitian Tipologi Urban Sprawl di Kota Semarang bertujuan untuk menganalisis tipologi urban sprawl di Kota Semarang, dengan diketahui tipologi maka dapat dilakukan upaya untuk meminimalisasi dampak urban sprawl. Penelitian ini dilakukan menggunakan metode remote sensing dan geographic information system (GIS) dengan pendekatan Shannon's Entropy. Hasil penelitian menunjukkan bahwa pada tahun 2006 dan tahun 2016, urban sprawl di Kota Semarang memiliki 3 jenis tipologi yang dapat dibedakan menjadi tipologi I (tingkat rendah), tipologi II (tingkat sedang) dan tipologi III (tingkat tinggi). Dampak urban sprawl pada tipologi II dan III dapat diminimalisasi dengan penyediaan perumahan terjangkau dengan sarana prasarana yang memadai dan perbaikan sistem perijinan pembangunan perumahan dan permukiman.\nKata kunci: urban sprawl, Shannon's entropy, GIS, Semarang.\nUrban sprawl is a contemporary issue of world cities (Suditu et al., 2010). This phenomenon emerges as a development of scattered pattern of low-density townships that is isolated from the city centre to the periphery of the suburbs and can result in environmental degradation (Bhatta et al., 2010; Ewing et al., 2002; Hasse & Lathrop, 2003; Tian et al., 2017; Yeh & Li, 2001). Although this phenomenon is massively discussed, yet there has been no consistent indicator to measure, define and evaluate the impact of this process (Yeh & Li, 2001). Unclear definitions and characteristics make sprawl measurement analysis a challenging task (Bhatta et al., 2010; Wilson et al., 2003).\nThe development of cities, in relation with the phase of suburbanisation, can potentially initiate urban sprawl. There are four phases of urbanisation: urbanisation, suburbanisation, dis-urbanisation and re-urbanisation (Antrop, 2004). Urbanisation is one of the most important socio-economic issues in contemporary society. It can be attributed to rapid population (Rukmana & Rudiarto, 2016) and economic growth (Wu and Zhang, 2012). These growths become irreversible factors in the transformation of land use from rural areas into urban dominant areas (Buchori et al., 2017; Schneider & Woodcock, 2008). The higher the economic growth of a country or region, the higher its level of urbanisation. This rapid economic growth will provide a trickle-down effect on urban fringed areas (Yunus, 1999). The term \"urbanisation\" refers to an increase in the built-up area due to increased human activities, and it is influenced by economic, social and geographical factors of a region (Bharath et al., 2017; Sudhira & Ramachandra, 2007; Jaya, 2009).\nSuburbanisation is characterised by the growth of built-up areas on the periphery. Urban centres focus more on economic (commercial) activities. Therefore, settlements will grow in the suburbs (suburban) (Antrop, 2004; Handayani & Rudiarto, 2014). The spilling over of settlement areas and excessive use of open landscape, especially the low dense settlements in urban fringed areas will result in conditions known as urban sprawl (Pozoukidou & Ntriankos, 2017).\nSemarang as the capital of Central Java province is one of the metropolitan cities in Indonesia, with a population of 1.73 million as at 2017, and an increase of nearly 350,000 in 2006 (Semarang Central Bureau of Statistics, 2017). It is also the centre of the National Strategic Area of Kedungsepur (Kendal, Demak, Ungaran, Salatiga, Semarang and Purwodadi). It is estimated that one-third of the Kedungsepur population lives in Semarang City. The built-up area of the city consequently grew up by 961.17 ha between 2006 and 2012 (Nahib, 2016). The residents preferring suburban areas is attributable to the higher activities in the urban areas. This has been established with the rate of negative population growth of 5% - 21% in the city centre area.\nIn contrast, the urban fringed area experienced a population growth rate above 30% since 2006. Semarang city has a built-up area due to the development of industrial areas in the north and that of housing in the south and east. The development of Semarang City to urban suburb shows the existence of urban sprawl symptoms (Hadi, 2013).\nThe development of such a dispersed built-up area tends to require more facilities and infrastructure on a large scale (Sol\u00e9-Oll\u00e9 et al., 2010). Moreover, the residential section of the built-up area in Semarang City is developed in the conservation area. Consequently, this development disobey the government regulation of the Semarang City Spatial Planning Document (RTRW) (Pigawati et al., 2017). Such phenomenon indicates that there is inadequate control of land development planning (Inostroza et al., 2013). Urban sprawl will increase the demand for supporting facilities and infrastructure that lead to complex problems. Therefore, studies to analyse the urban sprawl phenomenon is important. This research aimed at analysing urban sprawl typology in Semarang City, so efforts can be made to minimise the impact of urban sprawl.\nThis research used the quantitative descriptive method, where the data which was extracted from remote sensing imageries was analysed quantitatively with a spatial approach, by overlay analysis technique, of Geographic Information System (GIS).\nThe research was conducted in Semarang City (Figure 1). This city is located in Central Java Province, Indonesia, and it administratively comprises of 16 districts with an area of 382,241 km2. Administratively, the study area is adjacent to Kendal regency in the west, Semarang regency in the south, Demak regency in the east and the Java Sea in the north.\nDatasets used on this paper is Landsat TM 7 image retrieval period August 2016 and June 2006 with 30m resolution. The ten years gap was used because it represents the urban built area growth. The classification result has an accuracy matrix of 92% and 88%. These results indicate that the accuracy (limitation) requirements have been met and, therefore, ready for further analysis (Summer & Nordman, 2008).\nThe analysis of sprawl location determination in Semarang City was carried out to discover the sprawl zone (Figure 2). The sprawl location was determined by using Shannon's Entropy Index, considering two aspects of the city centre (CBD) and the main road network in the city. Shannon's entropy (Equation 1) can be employed in measuring the dispersion or compactness of a variable (x) in zones (n). Where H0 is Shannon's entropy, while Pi is the possibility of a phenomenon (variable) that occurs in the i-zone. The value of pi is determined by Equation 2. Xi is the observed value of the variables within the i-zone, while n is the total zone. Xi in this study is the density of built-up land in each zone, whereas X1 is the total density of built-up land.\nA city has a built-up land distribution concentrated next to the road if the value of the entropy index is close to 0, and if it is close to 1 then it indicates that the city is spreading away from the road and is referred to as urban sprawl (Bhatta et al., 2010; Prasetyo et al., 2016; Shekhar, 2004; Verma et al., 2017; Yeh and Li, 2001). The value of the entropy index (Hn) is scaled from 0 to log n. There is a relative entropy index (Hn') to facilitate the reading of the entropy index value that scales the entropy index from 0 to 1.\nFigure 2. Flowchart for urban sprawl typology analysis.\nThe sprawl pattern analysis was carried out to establish the patterns of urban disperse, by making the relative entropy space matrix using the relative entropy index value. The entropy space matrix relative will form 4 quadrants, each of which will represent one sprawl pattern, i.e. dispersed medium development (concentric), dispersed high development, concentrated low development (ribbon development), highly dispersed - high development (leapfrog).\nBuilt-up area density analysis was conducted to calculate the changes of the density from 2006 to 2016. The analysis of the built-up area density in Semarang was developed by multiple ring buffer method, using Simpang Lima as the centre of Semarang City. The multiple ring buffer produced urban zones with intervals 1km started from the centre. The built-up area density analysis was conducted on each zone separately.\nFarthest settlement analysis or central business district (CBD) was conducted to determine the distance of the farthest settlement of each zone from the city centre. This analysis was carried out by calculating the distance of the farthest urban settlement in Semarang City to its CBD (Simpang Lima). The analysis was performed by using network analysis extension in ArcGIS 10.3 software.\nThe population density analysis was conducted to identify the population density changing between 2006 and 2016. It was conducted in Semarang City by making population density point on kelurahan unit and created multiple ring buffer method using the reference of Simpang Lima as the centre of Semarang City. Further identification of population density was performed at intervals of 1km away from the city centre.\nThe urban sprawl typology analysis was conducted based on the result of sprawl characteristic analysis to form typology. Scoring analysis was used in dividing sprawl typology into three sprawl classes, i.e. high, low and medium. Sprawl level is lower when the variable value of the sprawl pattern, build-up land density and population density are low, while it is higher when the variable distance to city centre is high.\nSemarang City has a total area of 385,241 km2 comprising of settlements, industries, water bodies, vegetation and bare land which are classified into two types namely: built-up and non-built-up areas. Table I shows the changes of built-up and non-built-up lands from 2006 to 2016. Whereas Tembalang is the sub-district with the highest built-up area change (18.23%). Pigawati & Rudiarto (2011) found similar result on their study. However, in this study, we found that Semarang Tengah Sub-district is the lowest built-up area change (0.53%). The unique phenomena occur in Semarang City, and sub-districts located in urban fringe (Genuk, Ngaliyan, Banyumanik, Pedurungan, Mijen, Gunungpati and Tugu) have more built-up area changes compared to sub-districts located in the city centre area (Semarang Utara, Gayamsari, Gajahmungkur, Candisari, Semarang Timur, Semarang Selatan, Semarang Tengah). This corresponds with the study by Dadi et al. (2016) which stated that fringes are built-up area growth changes linked to urban sprawl phenomenon, urban sprawl drove ALC (Agricultural Land Conversion) i.e. agricultural land change to built-up area (Dadi et al., 2016). The spatial distribution of land use change in Semarang City is provided in Figure 3.\nTable 1. Landuse transformation of built-up area and non-built-up area in 2006 \u2013 2016.\nFigure 3. Built-up area growth.\nBuilt-up area zoning in Semarang City was carried out to divide Semarang City into zones based on built-up land density. There are three zones were defined based on this approach namely central core, fringe and periphery zones. Urban land delineation was conducted using modified zone analysis by Schneider & Woodcock (2008) and Shekhar (2004) to locate physically identifiable urban zones based on built-up land density and distance to urban centres.\nThe results of the zoning are provided both in tabular (Table 2) and spatial (Figure 4). Based on the results of urban land zoning (Table 2), there are ten sub-districts included in the central core namely Semarang Tengah, Semarang Selatan, Semarang Timur, Semarang Utara, Gayamsari, Semarang Barat, Gajah Mungkur, Candisari, Pedurungan and the northern and eastern parts of Genuk District. Urban fringe has nine sub-districts including Genuk, Pedurungan, Tembalang, Banyumanik, Gunung Pati, Mijen, Ngaliyan, Tugu and Semarang Timur. While the periphery zones has three sub-districts, and they are the southern part of Gunung Pati, most of the eastern Mijen, east part of Ngaliyan and Tugu. An illustration of the spatial distribution of urban land zoning analysis in Semarang City can be seen in Figure 4.\nTable 2. Built Up Area Zoning in Semarang.\nFigure 4. Urban land zoning.\nThe results of urban sprawl analysis are presented in Table 3. This table shows that, based on the city centre of 2006 and 2016, the urban sprawls occurred in each zone because they have a Relative Entropy Index close to the maximum value (1). This indicates that each zone (Figure 4) is dispersed and growing farther from the main road. Sprawl based on city centre in 2006, the central core has the highest Relative Entropy Index (0.71), whereas periphery has a Relative Entropy Index with the lowest value (0.59). In contrast, in 2016, periphery had the largest Relative Entropy Index (0.99), while the central core has a Relative Entropy Index with the lowest value (0.76). In 2006 and 2016, sprawl based on the main road, the central core has the highest Relative Entropy Indexes (0.98 and 0.99 respectively) while the periphery has the lowest Relative Entropy Indexes (0.79 and 0.89 respectively). Based on the changes in Relative Entropy Index value, periphery has the highest change values at 0.4 and 0.1. This suggests that urban sprawl in periphery evolves towards becoming more dispersed (sprawl). This is in accordance with a previously conducted research by Hadi (2013) which stated that the development of Semarang city to the urban fringe indicates the existence of urban sprawl symptoms.\nTable 3. Urban sprawl in 2006 and 2016.\nSprawl pattern analysis is conducted in order to establish the sprawl pattern of each zone (Figure 4). This analysis was performed using Relative Entropy Index value towards the city centre (HnT') and on the road (HnR'). The result of Sprawl pattern analysis can be seen in Table 4. This table shows that in 2006, the sprawl pattern is highly dispersed - high development or referred as a leapfrog development. While in 2016, the sprawl pattern is highly dispersed high-growth (leapfrog). In detail, this sprawl pattern occurs at all zones in 2006 except the periphery that has a dispersed medium development (concentric) sprawl pattern. The development of this pattern was close to the city centre and has a small Relative Entropy Index value. In contrast, in 2016, leapfrog sprawl pattern applies to all zones. This indicates that the periphery experienced a more severe sprawl pattern over ten years.\nTable 4. Sprawl patterns in 2006 \u2013 2016.\nBuilt-up area density analysis is carried out in order to calculate the density change of the built-up area between 2006 and 2016. The results of the land density analysis are presented in Table 5. This table shows that in 2006, the central core has the highest built-up area density (62%), urban fringe has an established land density of 18.13%, and the periphery has the lowest density (4.06%). However in 2016, the central core has the largest built-up density (69.89%), urban fringe has a built-up area density of 31.63%, and periphery has the lowest density (9.12%). The zone with the highest land density growth is the urban fringe (13.5%), followed by the central core (7.89%), and then the periphery (5.06%). Urban fringe is the only zone that has experienced an infilling trend. Infilling trends in urban fringe are associated with increased residential development in this zone, due to withdrawal factors in the form of education areas (Tembalang sub-district, Genuk sub-district and Gunungpati sub-district) and industrial areas (Genuk sub-district, Tugu sub-district, Ngaliyan district and Pedurungan sub-district).\nTable 5. Built up area density.\nBuilt-up area density trend from the centre to the suburb of the city can be established by connecting the density of a built-up area at a distance to the city centre. Built-up area density trend (Figure 5) has drastically decreases in the two-kilometre area from the city centre and then rises on 6 to 7 kilometres and 17 to 18 kilometres. This is due to the development of the city on the suburb of Semarang\/suburbanises as stated by Handayani & Rudiarto (2014), infringe and periphery precisely.\nFigure 5. Built-up area density trend in 2006 \u2013 2016.\nThe proximity of the farthest settlement analysis is conducted to determine the distance of the farthest settlements in each zone from the city centre. This analysis is performed by calculating the distance of the farthest settlement in kelurahan, Semarang City to Semarang City CBD (Simpang Lima). The analysis was carried out using network analysis extension in ArcGIS 10.3 software. A scoring analysis was then conducted to classify the Proximity of the farthest settlement into three classes: high, low and medium. The proximity of the farthest settlement analysis can be seen in Figure 6.\nThe settlement with the farthest distance on the periphery is the one at Kelurahan Cangkiran, with a distance of 22,935 m, and a travel time of 27 minutes. In urban fringe, the farthest settlement from the city centre is the one at Kelurahan Mangunsari which is with a distance of 18,887m and a travel time of 21 minutes. In central core, the farthest settlement is located in Kelurahan Muktiharjo Lor with 7,456 m and the fastest travel time is 9 minutes. The results of the scoring analysis show that central cores enter into a high level, while urban fringe and periphery enter into a low level.\nFigure 6. Proximity of the furthest settlement.\nPopulation density analysis was conducted to identify the population density change in 2006 and 2016. The results of the population density analysis can be seen in Table 6 while the trend is provided in Figure 7. The table shows that in 2006, central cores has the highest population density (12,660 inhabitants\/km2), urban fringe has a density of 2,447 inhabitants\/km2, and periphery has the lowest population density (1100 inhabitants\/km2). While in 2016, the central core has the highest population density (12,279\/km2), the urban fringe has a density of 2,975 inhabitants\/km2, and the periphery has the lowest population density (1427 inhabitants\/km2). The zone with the highest population density change is the urban fringe (528 inhabitants\/km2), followed by the periphery (327 inhabitants\/km2), and then the central core with a negative population density change (-382 inhabitants\/km2). This shows the dynamics of the population, the declining density of the central core population and the increasing population density in the periphery of Semarang City (urban fringe and periphery). Based on the scoring analysis, central core enters the high level, while urban fringe and periphery enter the low level.\nTable 6. Population density change in 2006 \u2013 2017.\nFigure 7. Population density trend in 2006 \u2013 2016.\nBased on Figure 7, there is a trend of decreasing the population density in the city centre and increasing in urban fringe and periphery, which is consistent with the research from Handayani & Rudiarto (2014). The decline in central cores signifies a reduction in the population living in the city centre. The figure also shows that there is different growth of population density in urban fringe and periphery due to an increase in population density. People preferred to live in the suburbs as a result of the attraction of education areas (Tembalang subdistrict, Genuk subdistrict and Gunungpati subdistrict) and industrial areas (Genuk subdistrict, Tugu subdistrict, Ngaliyan subdistrict and Pedurungan subdistrict).\nUrban sprawl typology analysis was performed based on the analysis of sprawl characteristics. Scoring analysis was used in dividing sprawl typology into three sprawl classes, i.e. high, low and medium sprawl typology. The results of this analysis were expected to reveal a more explicit phenomenon of urban sprawl that occurred in the Semarang city (Table 7). Sprawl level will be lower if sprawl pattern, the density of the built-up area and population density are low and vice versa. It also gets higher when the distance to the city centre has high value.\nTable 7. Urban Sprawl Typology Score in Semarang 2006 \u2013 2016.\nThere are variability of typological dynamics of the zones (Figure 8). From 2006 to 2016, central cores did not experience any typological change or the characteristics are remain steady, but the built-up area has increased. During this decade, urban fringe changed from typology III (2006) to typology II (2016), and characteristics of built-up area density progressed from low level (2006) to medium (2016). The increase in built-up area density and population density causes this zone to progress to a moderate sprawl. The increase in the built-up area and the population density occurred as a result of a pull which is due to the education and industrial areas in the urban fringe. Tembalang sub-district (Diponegoro University), Gunung Pati sub-district (Semarang State University) and Genuk sub-district (Sultan Agung Islamic University) are centres of higher education in Semarang, while Genuk (Merdeka Wirastama), Tugu (Wijaya Kusuma) Pedurungan and Ngaliyan sub-districts (Candi Industry) are the centres of industrial and warehousing area in Semarang City. Periphery underwent a typology transformation from typology II (2006) to typology III (2016). The change characteristic is the sprawl pattern, and this zone has dispersed medium development sprawl pattern (2006) into highly dispersed high development (leapfrog) sprawl pattern in 2016. The changing of sprawl pattern make this zone increase in sprawl level (typology II to typology III).\nFigure 8. Urban Sprawl typology zoning in 2006 (left) and 2016 (right).\nThe results indicate that there are three types of urban sprawl typologies in Semarang City including typology I (low level), typology II (medium level) and typology III (high level). From 2006 to 2016, urban fringe and periphery are experienced with sprawl level changes, as urban fringe progressed from typology III to typology II due to an infilling trend. The periphery zone, on the other hand, has increased sprawl level (typology II to typology III), and this shows that the development of settlements in periphery has a high level of sprawl phenomenon. Urban development as a result of urban sprawl will increase the demand for supporting facilities and infrastructure, especially in zones that have sprawl typology II and III. The impact of urban sprawl on zones with typology II and III can be minimised with the provision of affordable housing with adequate infrastructure and an improvement in the permit system for housing and settlement development.\nThis is in line with previously conducted studies by Prasetyo et al. (2016); Shekhar (2004) and Yeh & Li, (2001) that Shannon's entropy with GIS integration is the best method for measuring urban sprawls in developing countries. A research by Hadi (2013) on urban Sprawl in Semarang stated that the development of Semarang city to the urban fringe signifies the existence of an urban sprawl. The same result was also reported by Handayani & Rudiarto (2014) that there is a sub-urbanisation process that indicates urban sprawl in Semarang Metropolitan City. In addition, study by Bhatta et al. (2010) stated that the most widely used characteristics for describing and measuring sprawls are a built-up area (population) and proximity from the town centre.\nSemarang city experienced the growth of the built-up area as an impact of the suburbanisation process. This phenomenon can be identified from the change of non-built up area into built-up area. The city centre as the centre of commercial activities caused the development of the city to shift towards fringe with low density and fragmented (sprawl). Shannon's entropy with GIS integration is an effective method for measuring urban sprawl, as it illustrates the level of built-up area widespread in an area. This study shows that in 2006 and 2016, urban sprawl in Semarang City had three types of typology that can be classified as typology I (low level), typology II (medium level) and typology III (high level). 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+{"text":"Currently, only 1.8 percent of people with a bachelor's degree are unemployed, while those with just a high school education have a 3.5 percent rate of unemployment. Not only is having a college degree mandatory for most all professions, but it pays to have one. Those with a bachelor degree earn an average of $1,000 per week, while those with only a high school diploma earn half that.\nSo, what can you and your family do to ensure that your child continues his\/her education after high school? The first step is creating a college-going culture in your household. Read the list below to learn ten things you can do as early as elementary school to cultivate, sustain and meet the expectation that your child earns a college degree.\n1. First and foremost, use language that shows you appreciate the value of a college education and education in general.\n2. Remember that it's never too early to ask your child what he\/she wants to be when he\/she grows up. Encourage any goals or dreams, no matter how silly. If you have any friends, family members or acquaintances even remotely in the field, have them talk to your child about his\/her career. Connecting dreams to reality will help children see college is attainable and that hard work and perseverance pays off.\n3. Engage in and encourage activities that cultivate curiosity and a desire to learn. Visit museums, try new foods, ask questions and engage in critical discussions with your children. For example, when trying a new food, you might ask why a certain food is more popular in one region of the world than in another.\n4. Encourage and set the expectation that your child takes rigorous and challenging coursework. Research shows that taking even one college-level course as a high school student will increase the chances of attending and completing college.\n5. When traveling, try to incorporate a visit to a local college or university. Talk to your children about any of the college's special programs or sports team that relate to an area of interest. If the college has a sporting event, get tickets. Maybe the college has a performing arts department that is staging a musical. Perhaps there is an art museum on campus. If in the market for souvenirs, consider school pennants and sweatshirts as souvenirs. Turn a room in your home into a mini college campus.\n6. As your child enters his\/her teens, start to have more serious discussions about goals. Work together to investigate colleges or universities that specialize in these fields. If your child doesn't necessarily gravitate toward one area in particular, start to talk about other criteria, such as location, proximity to home or size.\n7. Don't let money get in the way. If students don't think that their family can afford college, they might put the idea out of their head. Remember that there are myriad resources available to help everyone pay for and attend college.\n8. Attend college nights and fairs. These types of events host all types of higher education institutions, including private colleges, state universities, trade schools and community colleges. There is an option for everyone.\n9. Celebrate hard work and perseverance. If your child becomes frustrated, help him\/her work through the process until they get to the other side. Understanding that success comes through hard work and diligence, rather than \"luck,\" helps children become independent and confident learners, who don't get paralyzed in the face of conflict. These are qualities that will serve them well in college and in life.\nRemember that your child needs your help navigating this journey. The path to college is long and complicated; children and teens need the support of a caring adult to help navigate the process. Be that caring adult.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to the Navigator Family!\nTo help your registration be as smooth as possible, check the information below to be sure you have the correct paperwork.\nIt is time to register all incoming Kindergarten students for the 2017-2018 school year. Any child that will turn 5 by September 1 is eligible to register.\nParents with proper proof of residence will be able to pick up a registration packet that they can then fill out and bring back to the school at their convenience. If you can not attend during that time, you are welcome to come to the main office anytime after May 3rd to register your child. Registration times during the school year are 9:00 am to 1:00pm.\nPlease visit the OCPS Registration Page for more information about registering a child for school and to fill out the registration forms online. Once completed online, print the registration form and bring it to the school unsigned.\nImmunization Record (copied onto a Florida 680 form) Please note - students entering Kindergarten or 1st grade must now have 2 Varicella (ChickenPox) immunizations prior to entering school. Children who received the 4th dose of poliovirus vaccine before their 4th birthday must now receive a 5th dose before they can enter school.\nIf you are transferring from another school, we need the last report card issued to the student and the withdrawal papers from the previous school. Students cannot be placed in a grade level without a previous report card.\nYou must have a copy of the Warranty Deed or Homestead, with the parent or guardian's name as owner.\nBring with you an updated Florida Identification Card and a current utility bill.\nYou must have a copy of the Lease Agreement, with the parent or guardian's name as renters, as well as the student's name as occupants.\nBring with you an updated Florida Identification Card and TWO (2) current utility bill.\nTo make an appointment, visit http:\/\/enrollment.ocps.net.\nSchools can only accept an address verification form that has been notarized by the Office of Student Enrollment.\nStudents entering Kindergarten must be 5 years of age on or before September 1st of the year in which they start school.\nStudents entering 1st grade must be 6 years of age on or before September 1st and have proof of completion of Kindergarten in the previous school year.\nPlease be prepared to fill out paperwork when you are registering your child. Registration can take anywhere from 30 minutes to an hour.\nIf your child is transferring from another school, please bring a copy of his\/her most recent report card. This will help with grade and teacher placement.\nIf your child is receiving special services for any reason, please bring a copy of the most recent IEP. Having this paperwork will ensure that your child begins to receive services from our school as quickly as possible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Posted November 2nd, 2010 by gdemo & filed under Election 2010, Politics.\nPosted January 28th, 2010 by gnunews & filed under Education, Global Warming, Health Care, Iraq War, Obama, Politics.\nThis is the complete text of President Obama's State of the Union delivered on Jan. 27, 2010 and transcribed by the White House. The topics were health care, war, economy, bipartisanship, transparency, gays in the military and a bunch of other stuff. The complete text is presented here to maintain the public record and inform the country and world about what Obama said. I'll leave the question of why he said these things to be decoded by other people much smarter than I.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Morris+D'Angelo Website was long overdue for a \"Modern\" Internet update that included many aspects of what a \"Modern Business\" should look like.\nThis is a good example of the Company's Leadership having a basic understanding of \"Best Practices\" for Internet Marketing and finally taking the fist-steps toward Internet Success with a Brand new Website and some back-end optimization along the way.\nFirst things first, came the Website update and development of most Social Media portals for later use. Currently, Morris+D'Angelo is receiving good responses from their LinkedIn Publishing for Business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A total of 5,147 workers died from a work-related injury in the U.S. in 2017, down slightly from the 2016 total of 5,190. The fatal injury rate was 3.5 per 100,000 full-time equivalent workers, also down from 3.6 in 2016.\nAbout 2.8 million nonfatal workplace injuries and illnesses were reported by private industry employers in 2017, occurring at a rate of 2.8 cases per 100 full-time workers. Both the number of injuries and illnesses and the rate of these cases declined from 2016.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzarwro b/data_all_eng_slimpj/shuffled/split2/finalzzzarwro
new file mode 100644
index 0000000000000000000000000000000000000000..6725547d23c5d91f3b22d5acc555b7a7d1e389ac
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"PAX BRITANNIA: The Wicked Wizard of Oz has 18 hours to go!\nThe Wicked Wizard of Oz has 18 hours to go!\nThere are just 18 hours remaining for The Wicked Wizard of Oz on Kickstarter.\nAs of writing, the project has funded, but only just. And with one day left to go, some more DOWN THE RABBIT-HOLE level rewards have been released (listed as DOWN THE RABBIT-HOLE AGAIN).\nSo why not check out the project and the rewards and pledge your support, before it's too late?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I don't know if someone have done it before but i could install Xcode 3 on Lion.\nYes, you can install Xcode 3 on Lion. I am living proof that it works.\nBUT (there's always a but ) if you do you will not be able to validate and upload apps to the mac app store using Xcode 4, even if you install it after. This is something I have discovered after hours of excahnge with developer support.\nJust something I thought anyone reading this should be aware of.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Your great travel buddy for family and friends.\n2011 Mercedes-Benz Sprinter 2 Tonne Van in excellent working condition and looked after by the owner. Plenty of space for various needs such as courier work, loading furniture when moving houses, removalist work or anything else.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Roger Williams began his religious career as a Puritan minister, and when he arrived in Massachusetts in 1630 he was initially well received. But he quickly fell into disfavor with the local leaders because of his liberal views. He changed his religious affiliation from Puritan to Baptist, and in 1639 he became a Seeker, a person who adhered to no specific religious practices. It was as a Seeker that Williams wrote ''The Bloudy Tenent,'' while in England attempting to win back a charter for Rhode Island. The main theme of the tract, as it was of Williams's life, is that all individuals and religious bodies are entitled to religious liberty as a natural right, and that civil governments do not have the authority to enforce religious laws.\n''The Bloudy Tenent'' is structured as a dialogue between ''Truth'' (representing the orthodox views of Puritans like John Cotton) and ''Peace'' (representing Williams's views), and the subject of their debate is whether secular laws should favor one religion over another, and whether these laws have any basis in the Bible. Williams argued that these laws were in fact contrary to biblical teachings, and utilized the parable in Matthew 21:33\u201346 about the tenants who killed the son of the landowner to lay claim to his property (hence the title).\nWilliams used numerous biblical parables to buttress his argument that the civil authority ought not to be used to enforce religious conformity. He noted that Jesus and his disciples did not enjoy the protection of civil authority, and when he sent the disciples into the countryside to preach, he instructed them to take no food or money, but to rely on God for their needs. In the same manner, the Church of Williams's day ought to rely on spiritual authority alone, not that of the Crown.\nAltogether, the prose in the tract is quite dense, and not easily read. But the message it carried could not be mistaken, and made Williams the leading champion in his time for the idea of religious liberty for both individuals and sects.\nMorgan, Edmund S. Roger Williams, the Church and the State. New York: Harcourt, Brace & World, 1967.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Global educator, Salon Owner and Artistic Director for Matrix, Danielle has worked in the salon industry for over 17 years. With a passion for texture and styling and an eye and the knowledge for color, Danielle brands herself as having a background in creating on trend styles that are fashion forward but wearable to the everyday consumer. Finalist in the 2014 and 2015 North American Hairstylist Awards; Danielle's work has been recognized worldwide and can be seen in many consumer and trade magazines. Danielle is also part of artistic team CBC, the Intercoiffure Artistic design team and is part of the development and imagery board for Great Lengths USA and Hairuwear USA. Danielle also does session work for print, runway, marketing campaigns, commercial, weddings, and film. When she isn't behind the chair or working on one of many projects; Danielle volunteers her time as Creative Director to the Survivor Glam Squad; an organization providing makeovers to cancer victims and their families.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasruq b/data_all_eng_slimpj/shuffled/split2/finalzzzasruq
new file mode 100644
index 0000000000000000000000000000000000000000..83d2d6252114d535b49877e0bf06954844f8dab7
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Platinum Originality is an international competition for graphic design students which is hosted by the China Academy of Art. The brand design for the 17th edition of the competition is dominated by the patterns of metal textures with a black and white gradient tone. The key visuals are designed through model building, with metal material used to express platinum texture and form patterns with rich details. This results in a visually uniform presentation of everyday objects with a three-dimensional appearance which set themselves apart from the bright-yellow background. The creative arrangement and combination of typography and graphics also encourages observers to use their imagination and associative skills and motivates them to get involved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Providers see patients at various times based off scheduled procedures and office hours. Please note that while we are open for patient visits during the hours listed above, our phones may be covered by our answering service for the first and last half-hours of our business day. Our office is closed on most major holidays.\nWelcome to Milford Ear, Nose, and Throat Surgery. We are a comprehensive practice specializing in the diagnosis and management of ear, nose, throat, and head and neck disorders. Our facility offers both medical and surgical treatments for problems involving the sinuses, voice box, mouth, throat, hearing, and structures of the neck and face.\nOur facility is fully equipped with a hearing technology lab as well as medical equipment to diagnose and treat ear, nose, and throat disorders. We are able to do many procedures in our office, including laryngoscopes, video stroboscopes, nasal\/sinus endoscopies, and drainage of tonsil and skin abscesses. We can also perform biopsies on thyroid and neck masses, ear tube placements in adults only (children are done at hospital), excision of some skin cancers, hearing and balance testing, release of tongue tie, and removal of cysts or masses in mouth or lips.\nFor urgent medical problems after hours, please call 508-478-0555. Our office on-call physician will return your call. If you have a life threatening emergency, call 911 or proceed to your local emergency room.\nOur practice recommends that patients first see one of our certified audiologists for a consultation. The audiologist will then to assist patients in choosing, fitting, and programming the amplification appropriately on the hearing aids. Milford Regional Physician Group's ENT is not a hearing aid dealer, rather we help the patient find the right hearing aid to not only assure satisfaction, but also bring comfort and stability to patients with hearing impairments.\nWe are providers for most medical insurance plans. Because there are a wide variety of insurance options, please contact your insurance company directly to make sure your doctor accepts your plan. We provide equal access to our patients regardless of the source of payment.\nIf you are insured, please remember to bring your insurance card with you when you visit our office. We must validate your insurance at each visit. Your co-payment is due at the time of your visit. We accept cash, check, Mastercard, Visa, Discover and American Express. If you are uninsured, please visit the Massachusetts Health Connector website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Former bantamweight champion TJ Dillashaw was suspended for two years by USADA on Tuesday.\nLongtime UFC commentator Joe Rogan is a fan of Dillashaw's fighting, but admitted on a recent podcast that there is no other way to characterize Dillashaw's positive drug test result for EPO, \"This is just cheating. This isn't any accident.\"\nUSADA (United States Anti-Doping Agency) announced the news while revealing that Dillashaw tested positive for EPO, which is typically used in blood doping to help athletes for better endurance and conditioning. It's a drug that has been abused for years in the professional cycling and endurance athletics worlds.\nEPO is one of the key substances that helped Lance Armstrong ride to seven Tour de France titles. Eventually exposed for his systematic approach to doping, Armstrong has since been banned from cycling for life.\nAs this was considered a first-time offense under the UFC Anti-Doping Policy, Dillashaw received the maximum penalty of a two-year suspension and voluntarily relinquished the UFC bantamweight championship.\n\"To me, it's a shame, too, because I'm a TJ fan,\" Rogan said on his Wednesday podcast. \"I think he's a phenomenal fighter and I'm a big fan of watching him perform. But it taints his legacy. It just does. There's no way around it.\nHaving specifically tested positive for recombinant human erythropoietin (rHuEPO) \u2013 synthetic EPO \u2013 it was clear that the drug had to have been injected and couldn't have been caused by a tainted supplement, as so many athletes have claimed as a defense recently after testing positive for other banned substances.\nDillashaw has not issued a statement since the USADA ruling. A couple of his coaches have commented, condemning the use of banned substances, but supporting Dillashaw.\nSimilarly, Rogan still respects Dillashaw's abilities as a fighter, but can't deny the potential benefits that Dillashaw received from using EPO.\n\"It doesn't take away from the fact that he's amazing. TJ's an amazing fighter, but f---, man,\" Rogan continued. \"To test positive for EPO in the Henry Cejudo fight. You've got to think, if you're on EPO, how much does that allow you to train extra time? How much does it allow you to recover? What is the quantifiable benefit? It must be pretty significant. A lot of the cyclists, a lot of heavy duty endurance athletes use it. I almost want to try it.\n\"I think what it does is it raises your threshold if you're already an elite endurance athlete, too. They're already looking for that extra edge... then they take that EPO... it jacks them up, they can do more work. They don't run out of oxygen. They don't run out of juice. And for a guy like TJ, his style is so kinetic, so much movement, explosion. He needs that juice.\n\"It's crazy. It's a hard one to swallow.\"\nDespite the advantages of EPO, Cejudo knocked Dillashaw out just 32 seconds into their UFC on ESPN+ 1 headlining bout in January, retaining his UFC flyweight championship. Cejudo will now face Marlon Moraes at UFC 238 in June to determine who will take Dillashaw's place as the bantamweight champion.\nMeanwhile, Dillashaw will be forced to sit out for the next two years.\nFollowing a pair of knockout wins over Cody Garbrandt, Dillashaw was considered arguably the best bantamweight fighter of all time. Now following this suspension for a doping violation, Dillashaw's accomplishments will undoubtedly come under a tremendous amount of scrutiny.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Black Tide- \"Show Me the Way\"\nBlink 182- \"What's My Age Again\"\nFreezepop- \"Less Talk More Rokk\"\nTenacious D- \"Rock Your Socks\"\nThe game represents a first for the PSP as it will include an in-game store for downloadable content. The store is powered by the PlayStation Network and will afford a similar experience in terms of grabbing new tracks to that of its console cousins.\n\"Rock Your Socks\" from The D does in fact rock.\nAdditionally, SECA is going to release a limited edition Rock Band-branded PSP pack which includes a \"piano black\" PSP, a 4GB Memory Stick PRO Duo and a voucher to download the movie \"School of Rock.\"\nBryn says: Is there no stopping the mighty marketing machine behind Rock Band? I'm wondering how this new handheld take on the game will fare on the PSP. The DS versions of the Guitar Hero games are pretty decent, but there's no cool plastic peripheral with Rock Band Unplugged.\nI'll be giving this one a try-before-I-buy for sure.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Well Omnilight, I suggest you redo the complete session, and repost it, taking the work to the Site Template with Summary.\nAdditionally you are taking shortcuts throughout your posted session. Stop that now, because it will frustrate your efforts, and cost you accuracy in your future work.\nthanks your suggestion. appreciated. slow to understand.\n\"Additionally you are taking shortcuts throughout your posted session\"\nBecause I am not seeing your complete session, I have to presume you did not finish the work.\nIt's quite possible you did not post the whole session.\nOne thing, be sure and list at least 4 solid word descriptor dimension percepts.\nOkay. Remember, rigor, precision and accuracy are key to your advancement with remote viewing.\nMake sure your ideograms are performed correctly. Only you will know in the long run if you are doing them correctly. This will hold true especially when you will be working alone.\nUse us while you can.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatemk b/data_all_eng_slimpj/shuffled/split2/finalzzzatemk
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Adding the finishing touches to your GMC Sierra's exhaust system with 2014-2017 exhaust accessories from AmericanTrucks opens up new possibilities for improving your pickup's performance. 2014-2017 mufflers for the Sierra include superior components made by leading manufacturers like Gibson, in both offset and straight-through pipe configurations. These mufflers are tuned to generate a pulse pounding exhaust note \u2013 a deep, aggressive rumble \u2013 while avoiding drone inside the cab. They are, of course, also high-flow and can be used with mandrel-bent exhaust systems to improve the removal of waste gases that could otherwise slow the engine with backpressure. Other 2014-2017 exhaust accessories include electronic exhaust cutout systems that let you switch between quiet and earsplitting exhaust notes, and silicon gasket maker that can be used to informally fabricate a needed gasket for leak-proofing your truck's exhaust system.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Simple to install \u2013 Just soak them in water, inert them in a 2\u2032 hole, and cover with soil.\nEasy to use \u2013 Once it's installed, simply connect a voltmeter and take your readings!\n*For all STELTH models except for the STELTH 3 & 4 (Portables = 20 years) and STELTH 9 (Deep Sea = 10 years).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What does a \"special edition\" pen mean?\nUnlike limited edition pens that are only produced in a certain quantity, special editions are often produced by a company for a set period of time \u2013 such as a year. It could also be that there is no specified number of pens to be produced, but the company will manufacture pens based on how many they can produce out of the raw material they may have.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Join us on Tuesday mornings (term time only) for our fun-filled discovery and play sessions for children aged 2 and under.\nEnjoy sing-a-long and story time in the soft play area along with a light snack.\nAdults pay standard admission for 12 months entry, plus \u00a31 per session for your little nipper!\nThis includes all visits to the aquarium too!\nNo need to pre-book simply turn up on the day! Please see our website for a full list of dates.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Creating new healthy habits is great. But have you checked your usual routine for bad habits that should change? Keep an eye out for these!\nLearn how to motivate yourself to get started on something new, and talk yourself through challenges across your entire life, to build mental toughness!\nVulnerability is scary. And it's so tempting to focus on our strengths, not weaknesses, in the gym. But that's not how we improve! Learn from Heidi.\nThere's strength in numbers! Here are 5 reasons that prove it really pays to get support from others, and that asking for help is not a weakness.\nBeing the best version of yourself takes time, dedication, and intention. It's not selfish. Try these 7 tips to make time, find support, and be a better you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaurpy b/data_all_eng_slimpj/shuffled/split2/finalzzzaurpy
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index 0000000000000000000000000000000000000000..8eb4c3271f406bcb5a59f54e6ac2a5fb539b4eea
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaurpy
@@ -0,0 +1,5 @@
+{"text":"Spice up your desktop all year long with our NEW Desktop Wallpaper for 2019.\nRIGHTS OF USAGE: All wallpaper images are copyrighted \u00a9 Short Block Technologies, Inc. (SBT) and may only be used as intended (desktop wallpaper). Upon downloading you automatically agree to not use nor manipulate, exhibit, resell, or reproduce for any personal or commercial reasons\/uses. Doing so without expressed written consent is strictly prohibited. For more information, contact us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Discover How to Heal Yourself Naturally! Safe, All-Natural Alternatives to Taking High-Cost, Side-Effect-Laden Prescription Drugs!\nIf you are tired of paying the high cost of prescription drugs \u2026 if you are tired of worrying about prescription drug side affects \u2026 or if you just don't want to risk becoming dependent on prescription drugs \u2026 then have I got great news for you!\nYou see, there is a new ebook available that reveals everything you need to know about alternative medicine to begin using its techniques and treatments to heal yourself naturally!\nAt last, there is a comprehensive yet concise and easy-to-read guide that explores the many different components of alternative medicine, and how they can help you.\nThe differences between conventional and alternative treatment methods \u2013 plus, how to know if an alternative treatment is right for you!\nThe history of alternative medicine \u2013 you may be very surprised by what you read here!\nThe type of alternative medicine that 25% of the world's population practices \u2013 along with its five most popular treatment methods and what each can do for you!\nA type of alternative medicine that relies almost solely on herbal remedies \u2013 this treatment method has been around for over 2,500 years and could be just what you need to begin feeling better in no time!\nThe alternative medicine techniques of Native American Indians \u2013 these techniques have been used effectively for over 40,000 years, according to many experts \u2026 and they work just as well today as they ever have!\nWhy practitioners of Ayurvedic Medicine believe the body must be balanced \u2013 and how that can be accomplished quickly and easily!\nThe most popular form of alternative medicine \u2013 and five reasons why you should try it to begin feeling better immediately!\nHerbal remedies for common ailments \u2013 find out how to naturally cure acne, anxiety, burns, warts and much more here!\nHow to cure numerous ailments including arthritis, constipation, depression and the flu with herbal tea \u2013 you'll be amazed at how easy it is to do when you follow these simple tips!\nHow five products produced by bees could have you feeling better fast \u2013 you'll be simply amazed when you read this!\nHow to lose weight naturally and keep it off for good!\nThe types of yoga and how to choose the one that is right for you!\nHow to improve your sexual health!\nWhat alternative medicine treatments are good for children \u2013 and which are not!\nHow to become an alternative medical practitioner!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"'Winter is a season of recovery and preparation'.\nStroganoff is an old classic Russian recipe from the 18th century, that has become popular around the world. You can imagine people eating this dish on a cold snowy day, which is why it has become so popular as a warming winter dish in other places and the USA is no exception.\nI took this particular recipe from Healthy Living James as featured in Vegan Food and Living. It is rich from the coconut milk, substituting for the cream and has deep flavors from the smoked paprika, dijon mustard and garlic. You can use it for canap\u00e9s, eat it on toast for breakfast or with rice for dinner and it only takes 30 minutes to make.\n'The mind is like Tofu. It tastes like whatever you marinate it in'.\nThis is a beautiful recipe and I would recommend it to anyone who likes Tofu. My beautiful cousin, Sabrina, bought me a selection of spices and it came with some suggested recipes.\nThe key here is to make sure you get as much water out of the tofu as possible. The resulting dish is a wonderful combination of the char from the tofu, the salty sweet umami from the miso jam and the acid and spice from the spice mix. I have given a suggestion for if you don't have Edo spice, but you can just sprinkle anything you have or that suits your palate, making it your own recipe.\nMr. B doesn't like Tofu or miso, so this is recipe is just for me.\nSpread a dollop of miso jam on top of each slab, sprinkle the mixed spice* on top and a few bits of scallion.\nThe pioppino mushroom is not only known for its delicious and earthy taste and great texture but also pack a series nutritional punch, providing a great source of vitamin D and boasting beneficial bacteria great for improving digestion and overall health. These mushrooms have a valuable amount of bioactive metabolites. These metabolites include agrocybenine with anti fungal properties, Cylindan which has anti-cancer properties and indole derivatives which are responsible for hunting down free radicals. The Pioppino mushroom is also known for slowing down the effects of osteoporosis. MIND BLOWN!\nI found these at the farmers market and since I had never seen them before I thought I pick some up. I asked the lady what to do with them and she said just chop the whole thing up including the stalk and saute them with some shallots. So that is exactly what I did. I served these with some ancient grain gluten free pasta and topped with a little vegan parmesan.\n'Bellissimo' said Mr. B and I agreed.\nMy wish for you is to Live Better, Strive Harder, Be Bolder, Dream Bigger, Climb Higher and Seek Greater.\nThese too are wishes for myself and in the pursuit of this at the end of the year I took some time out to self reflect on all I have learnt over the last year.\nIn 2018, I completed my certificate in Plant Based Nutrition with Cornell and have used this and some great books (e.g. how not to die \u2013 Gene Stone and Michel Greger), articles and blogs to learn more about the hard science behind food and disease. Many of you may have seen my blog evolve over that time, from broad based healthy food to more of a plant-based blog. This is a reflection of my learning and has caused me to sit back and think about what my own philosophy is and what the blog should centre around.\nAfter much study, contemplation and a recommitment to both my followers, and myself I am pleased to announce a new blog coming soon. The blog will be dedicated to Vegetarian, Vegan and Raw recipes with a focus on Plants and Whole Foods to maximise the nutritional content. There will also be a new section where I will share scientific research, so you can learn with me.\nThe blog will be called PlantToTable.com and exists today but with a redirect to Tasha.Kitchen until it is ready for launch in early 2019. Tasha.Kitchen will resume this Saturday and an announcement will be made when the new blog is ready.\nI hope that this news allows you to forgive my absence and I look forward to continuing our journey together.\nLet's break the myth that you need meat to build muscle. The gorilla can lift up to 10 times its body weight!\nI made this dish on the request of my meat and potatoes concierge. As I would leave the building early in the morning I would go by the front desk and drop off a plant based treat, vegan cheesecake, vegan meatballs, vegan samosa, vegan chocolate truffles and after a few months I finally got a request for a cauliflower steak. This was a man who said that he had to eat meat with every meal.\nYou must learn a new way to think, before you can master a new way to live.\nI love the humble cauliflower, the texture is meaty enough to satisfy and it takes well to assertive flavours. This is a basic version, but you could add a pesto, salsa or chimichurri. I served this with some Baba Ganoush underneath for a texture and flavour contrast.\nThis is a beautiful recipe for a BBQ. It is meaty and sticky and sweet and sumptuous.\nMushrooms are always said to be a great meat substitute but I don't see it as a substitute but a star in it's own right. I use a variety of mushrooms in my recipes due to the medicinal benefits; Trumpet versus the others is good for something you want to give more structure to, which is why it works so well on this recipe.\nBe careful with the reduction, as it can get hot due to the melted sugar. If it toughens up just reheat.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"And now Tumblr is saying that they've had a data breach potentially exposing the passwords of 65 million accounts. Like the LinkedIn and MySpace breaches, this appears to be a few years old, and Tumblr is forcing passwords resets on the accounts it thinks is affected. Still, it wouldn't hurt to reset your Tumblr password.\nThere's a story going around that someone is trying to sell a collection of SEVERAL HUNDRED MILLION MySpace passwords (and that it's the same person who is selling 100+ million LinkedIn passwords). As with the LinkedIn case, this may be a data breach from months or even years ago.\nStill, if you have a MySpace account, or if you've re-used a MySpace password elsewhere, this might be a good time to change your password.\nA LinkedIn data breach has been in the news the last few days. Someone is trying to sell a set of 117 million LinkedIn accounts from a 2012 breach, a breach that now appears to be much larger than was originally reported.\nIf LinkedIn thinks that your account is affected, they'll probably force you to reset your password. And even if they don't, this might be a good time for you to do that anyway, especially if you think your password might be older than 2012.\n17 May 2016 Apple updates: OS X, iOS, iTunes, etc.\nApple has released updates for many of its products. So visit the App Store on your Mac, and then update your mobile device(s).\nI spent most of last week at the John Slatin AccessU conference, an annual digital accessibility conference held on the St. Edwards University campus in Austin. This post will be a somewhat haphazard collection of observations from the conference, but it may serve as sort of a followup to the web accessibility post from a couple of months ago.\nThere were several people at the conference with visual impairments. Two in particular stand out in my mind, as they were in several of the sessions I attended. One of them used a service animal (German Shepherd, I think), and the screen of his laptop was remarkably smudged (because, why would he care?). The other fellow used a collapsible cane, and he typically only opened his laptop far enough to get his hands on the keyboard. I'm not sure their laptop screens ever actually came on (which probably helped the battery life). The second fellow said he works part-time as an accessibility tester for Knowbility (the group that organizes the conference).\nThere was another fellow there who was sighted but had no arms (born that way, I presume). I didn't see him use a computer, but I've seen pictures of other people who don't have the use of their arms. They grasp a stick or pencil in their teeth and use that to type (that must take a lot of patience).\nAnd there was a woman with diminished vision who gave a presentation demonstrating how she uses a computer. She uses a combination of a screen reader and a magnification tool. She uses the magnification tool to set the screen in reverse video mode for high contrast. Her demonstration was particularly interesting, because the computer she was using had trouble connecting to the wireless network, and then it wanted to run Windows updates. She got some help and rolled with it pretty well, but it was instructive to see what a barrier it was to be faced with poorly-presented error messages.\nThe third day of the conference I attended a mobile accessibility bootcamp presented by Paul J Adam, and that was really interesting. The presenter said that people with visual impairments favor iOS over Android, and it turns out that it's by a pretty wide margin: in a July 2015 survey of over 2500 screenreader users, about 70% use an iOS device as their primary platform (compared to around 21% using Android).\nAs one of the exercises in the mobile accessibility bootcamp, I tried using a native app on my Android phone with the Talkback screen reader, and it was a real struggle. Some of that was my unfamiliarity with screen readers in general and Talkback in particular, but some of it was probably poor accessibility in the app (and that's likely pretty common).\nIf you ever visit Austin, the St. Edwards campus is really nice (the Sorin Oak is particularly impressive), and Pinthouse Pizza has really good beer.\nUpdate (14 May): Adobe released another Flash player update.\nWordPress has released v4.5.2, and it addresses at least two security problems. The developers strongly urge that users apply the update. So if you host your own wordpress blog, have a look at running the update.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Webinar will propose several practical methods a manufacturer might implement to make their material compliance efforts more effective. The material regulations are expanding and becoming more inclusive. These changes are increasing the negative impact of RoHS and REACH compliance on manufacturers. To mitigate increasing cost and risk there are several potential process enhancements that might be implemented by manufacturers. In this presentation 8 ideas for internal business functional improvements will be discussed. Hopefully some of them will be appropriate for your business.\nThis webinar is for Electrical and Electronic Equipment Manufacturers outside of the EU, and importing into the EU. Also, suppliers to these Electrical and Electronic Equipment Manufacturers will be interested, since their products are in scope also.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaxaqh b/data_all_eng_slimpj/shuffled/split2/finalzzzaxaqh
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index 0000000000000000000000000000000000000000..663eeec8d6cb924b46b4802c1e1dab84b6190c9c
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@@ -0,0 +1,5 @@
+{"text":"Hay I remembered how to blog!\nAlright it's been up and running for about a month. I really wish I had a window in this house appropriate for such a setup. Grow lights, sort of work, It's light that helps the plants grow but it's not a complete spectrum. Having a red light or heat lamp might help this and keep the plants warm but I can't do that just yet.\nUp top, seedlings are growing nicely. The glass bowel is there to quiet the dull sound of water. It's not annoying, even without the bowl, it's just I'd prefer it not to make any noise at all. The bowl is also there to hopefully catch baby fish should they become part of the filtering system.\nI've since lowered the lights too. Two metal coat hangers did the trick. I tied the lights to them too for added security. I see in gardening catalogs all the time these expensive setups, with shelves and lights you hang off of chains. You probably don't need to be paying $150 for that. Grow lights, especially when they're covered, don't get that hot. I could get away with rope if I wanted to instead of the hanger.\nDown below, the fish are fine. Mostly everything is unchanged. The females are defiantly pregnant! I don't know how long they take to bare young but they're ready to explode. I've added a few more plants to give the little ones someplace to hide when they arrive.\nHopefully all the ammonia, bacteria, and such will balance out. I'm told though that there's no such thing as a perfect setup. Meaning one that requires no care at all.\nThanks for sharing this informative blog. Hydroponics are being used for the proper growth of plants. Metal halide grow lights south Africa is one of the professional grow lights service providers.\n\"Wow You Must Have a Big Yard\"\nAre! It be the White Squirrel!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Be A-badass Carolina Panthers Mashup Adidas NFL Shirts shirt. Stand and advance proudly while showing off your pride as a Panthers fan. Cheer for them on the field and have a toast with everyone after a game, in this shirt, because it is perfect for any moment. Grab one now, get ready, get set and always be with Carolina Panthers to back them up.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Introducing Dave, Nathan, and Mark \u2013 Litmus Software, Inc.\nWe're thrilled to introduce Dave, Nathan, and Mark\u2014the newest additions to the Litmus family.\nDave joins our finance team as a senior strategy analyst helping Litmus better understand our markets. Before coming to Litmus, he was a growth equity investor at Spectrum Equity, where he focused on SaaS, information services, data analytics, and internet companies.\nA Massachusetts native, he enjoys traveling all over the world and getting outdoors whenever he can\u2014skiing, hiking, running, or re-living his days as a kayak instructor. He's a huge tech enthusiast and loves to geek out over the latest and greatest developments in the tech industry.\nNathan joins Litmus as an infrastructure developer working with the backend team. His past experience involved a mix of things, from working as a .NET developer across mobile apps, at Nokia on their music streaming service, and developing a mobile banking app for a large UK bank.\nOutside of work he still loves developing\u2014and he does it a lot in his spare time. When he's not busy coding, you can find him spending time with his wife, two daughters, and their crazy dog; beekeeping; and peering through telescopes at night, looking at stars and planets.\nMark comes to Litmus as a senior software engineer. After studying computer science at the University of Aberdeen, he spent time as a ruby developer working on everything from UK e-petitions, HR tools, financial calculators, and cruise ship pirate detection systems (arrrr).\nHailing from Edinburgh, Scotland, he's a trivia master and a fair hand at the fiddle.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Good morning, 'Plain Path' Christian! Your life is led along a straight line due to your total dependence on the only Road Map the Lord God has given us, 'His people', to follow. The path of true Christian faith is both straight and strait; it is a path that leads in a straight line from A to B; and a path that is very narrow, and hard, indeed.\nThe Lord Jesus only teaches, and exclusively recognises 'the strait gate' to Eternal life in Heaven. He then makes a sobering statement, one of which we should all take careful note: \"\u2026And FEW there be that find it.\" (Matthew 7:14) Selah: now think on this!\nTry teaching this 'plain path' teaching to the 'Name it and claim it, brigade', and they will want to string us up to the nearest telegraph pole! The teaching is too STRAIT! The path is too straight! The teaching is too plain and to the point!\nWhen we seek to earnestly and faithfully teach Christ's 'plain path' teaching - we are immediately confronted by what I call 'religious enemies'; charlatans, 'wolves in sheep's clothing'. We will have the world of unbelief; the Devil; as well as our own too fallible flesh to contend with also; but 'religious enemies' are quick to condemn us.\nThe Lord God protects and takes care of those who seek to learn His way and teach it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"YOUNGSTOWN \u2013 Peter M. Ceci, 64, passed away Thursday, May 8, 2008 after battling a long illness.\nHe was born Jan. 11, 1944 in Youngstown, the son of Peter and Mary (Joseph) Ceci.\nHe was a native Westsider and graduate of Chaney High school and Choffin School of Restaurant Management.\nA manager for Morgan Restaurants and supervisor for Red Barn systems, he later owned and operated several Red Barn Restaurants in Ohio, Pennsylvania and New York that became known as The Farm Family Restaurants. Throughout the years he was very active in the restaurant industry winning several awards for sales and service. He later retired to Vero Beach, Fla.\nIn recent years, he returned to Youngstown to be near his children and grandchildren. He enjoyed fishing, boating, and frequent trips to Florida to visit his brother and nephews.\nHe leaves behind his beloved 10 children, Richard (Victoria) Ceci, Mary Ceci, Karen Kim, Michael (Holly) Ceci, Amy (Hossein) Mirzai, Peter (Peg) Ceci, James (Mary) Ceci, Molly Ceci, Daniel Ceci, and Abbey Ceci; as well as his former wife, Mavis (Carney) Ceci; and five grandchildren, Joseph Kim, Kaleigh Ceci, Maria Mirzai, and Kyah and Gabrielle Ceci. He also leaves a brother, Mark (Meg) Ceci; nephews, Chris, Craig (Cori), and Scott Ceci; and niece, Jennifer Ceci.\nBesides his parents, he was preceded in death by his grandson, Mark Ceci; and a brother, Leonard Ceci.\nThe Red Barn lost a dear member of it's family this past year, Pete Ceci. Pete was one of Red Barn's most admired owners and it's only fitting that there is a page on barnbuster.net dedicated to Pete. Our deepest sympathy to Pete's family and friends, he will be greatly missed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaymfx b/data_all_eng_slimpj/shuffled/split2/finalzzzaymfx
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+{"text":"Worldnet was founded more than two decades ago. Today, it is a leading telecommunications provider in New Zealand.\nWorldnet is the result of over many years of research and development by industry experts with decades of experience in producing innovations in data and voice delivery.\nTheir goal is to be at the leading edge of technology and to provide, rejuvenating and propelling telecommunication services to every New Zealanders with lasting and meaningful experiences.\nShowing 6-7 of 7 items.\nOutstanding personal support during their opening hours. This is a smallish company, not some faceless corporate with a foreign call center.\nOffice is open Mon-Fri 9am-7pm, Sat 9am-3pm and closed Sunday, so if you have a problem or lose your internet and\/or phone service outside these times then there is no support at all and you must wait until they re-open.\nSwitched to WorldNet in my 1st house from Vodafone. Happy days started. Finally, I was satisfied with my Internet connection.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Or maybe it started earlier. Maybe it started with the river floats. For years, I'd built homemade rafts and floated down major American rivers with friends.\nAlex and Jen and I were at Camp Tipsy and it was Happy Hour at our camp and we were drinking gin and tonics and maybe the alcohol contributed to our grandiose and ambitious scheming, or maybe, honestly, I've always been that way.\nWe were looking at all the bizzaro floating contraptions and nautical imagining. We didn't want to go fast or swing or be propelled by some Rube Goldberg propulsion system or even soak in a wood-fired hot tub, so much, at that moment, as we wanted to sit peacefully on our own floating front porch and watch the madness of the rest of the world.\nWow! Aaron and I were just talking about your shanty boat adventures yesterday and wondering how it all got started! Have you ever told Chicken John about how Camp Tipsy (helped) inspired your current adventure?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Greetings from Home, Dear Ones. I am the Keeper of Time.\nI come to you this day and intersect with your timeline for just a moment to help touch your hearts and to help remind you who you are and where you are coming from. You are distant travelers who have definitely been around. Yet here you are now, settled on Earth and making rapid changes. There are so many incredible changes that sometimes it is difficult for us to keep up with all the things that humans are doing on planet Earth. Today we will share a little bit of what is coming in the year ahead; it is rather fascinating to watch what is taking place. By the time you get through February 2017, you will feel a bit of a calmness that has not been here for a while. In some ways, it is the calm before the storm. Dear ones, re-member that most change cannot happen when in the midst of calm. By its very definition, \"change\" often involves such disruption that a ripple is left in its wake. Do not fear this, for the change is actually what you have come to be a part of.\nSo many of you have come to Earth trying to figure out where you can be, what this will look like, and how it is going to work for you. Here you are, right in the midst of it now. Of course, the greatest fear is what happens if you fail. Such a scary word, fail. What happens if you cannot accomplish things? What happens if you extend yourselves and somehow you are hurt? Well, that is the important part we wish to tell you today\u2026we have your back. We look over your shoulder and try to direct you from time to time, each and every one of you. It does not matter where you are in the world, what your history has been, or which level you believe yourself to be. The truth is, we have you. It is almost as if you would let your children free to go play, but there are certain things you hope they do not get into and places they will not go. Their explorations can obviously cause problems, although as that happens, they will learn from them as part of the game. We tell you, dear ones, you are in somewhat of a safe zone. Especially during the next two years, when there will be major changes taking place. Consider, dear ones, that every three or four years a profound energy changes takes place, basically moving everyone up very rapidly\u2014almost too rapidly. Then you would have to do whatever you could to ground yourself, to take that light and bring it into your life and anchor it. That is the challenging part. In 2016, you had a major change as well on September 27th, with a wave of energy that came in. Prior to that, you have seen so many things coming in. Actually, you saw the gravitational wave that Einstein predicted some time back coming into the planet. Every three or four years you have these big events\u2014huge, vibrational raising events. Next year you will have four of them.\nCan you see the changes that are ahead? It does not happen this way except on evolving planets, for you are in motion, dear ones. All of humanity is reaching levels that have not been set by humans before. You have capabilities of reaching a spiritual awareness within each of you that allows you to touch the god within. Then, you will know it.\nMany of you have the seeds of fear deeply ingrained within you. Some are from past life experiences, but others you have stepped forward into willingly through own work only to find great resistance and possibly even death as a retribution. We understand that it is difficult to understand, so we re-mind you of this: We have your back, as you say on Earth. We are watching out for everything that you do because in this time, dear ones, there is a new chaos which is simply created out of vocal fears. You have so much technology on the planet now that you are able to collect these little bits all together as one. They appear much larger than they truly are and that is always true of the dark side, which always appearing much larger than it is. So, we ask you all to look at this within yourselves. What can you change about your own belief systems? What can you shift to help you to ground and to gain the confidence in your light? All we are asking you to do is to step into the energy. Many of you are teachers and healers on your own path, already doing quite well by bringing your energy out into the Earth. Keep that up and help those who are just now awakening to gain the confidence. They need to step out and start creating a reality of their own choosing rather than one set by default, which is what most humans have been doing.\nYes, we tell you there are those who plan ahead. There are the planners that make projections about their lives, businesses, and where they are going to be in 5 or 10 years. Very few people plan ahead on a personal, spiritual level but you have that capability now. You have the option of deciding who you will be in the next few years. You have the opportunity now to place yourself in the best position within the wave with new opportunities that you have not seen before. Yes, dear ones, there will be chaos and you will see much of it, for the pace is rapid and humans do not do well with change. Perhaps something has always been done a certain way before, but suddenly you will do it a totally different way. The vibrational raises are coming soon the first one will enter earth's field between March 4 and 17th. They all enter as waves and there are now so many that are interfering with each other to some extent. This is causing a bit of a shift in the timeline and thus makes it difficult to predict the exact dates of the other three events at this time; however, you are set up to receive four huge energy cycles in 2017.\nAn event that set things into motion a couple of years ago involved the timelines intersecting with each other. It was a rather large event that took place, following an earlier shift of the 12-12. The four events on their way are opportunities to make similar leaps of that size and they will all open within the year of 2017. What happens to those who cannot make the changes? Well, that is precisely why so many lightworkers are now on the planet\u2014to help those at other parts of the spectrum. Please do not judge it as being right or wrong, because you negate your own energy when you do that. Simply observe and understand that humanity has a bandwidth. Although there is a high vibration and a low vibration, humans can only move as fast as the slowest person. Additionally, these four vibrational shifts are meant for specific reasons, so let us explain.\nBecause the bandwidth of humanity has grown in this way, you have never been through what you are experiencing on planet Earth right now. Before you would have left the planet by now, perhaps going on to colonize a new planet. We do not mean through your so-called space ships, but through your spiritual bodies as you have already done many times. There was a time when you inhabited Mars yet here you are living on her neighbor, Earth. If you think that is interesting wait until you hear the rest of the story. Soon you will start uncovering a great deal more, because now you are very interested in Mars. It is your closest neighbor after all. Yes, you will find out much about your own history though not all of it will make sense in the physical realm at this moment. Re-member, dear ones, you are a spirit first and a human second. Harmony between the two has the opportunity to create change so let us go back to the bandwidth for a moment.\nWith all of this incredible change, humanity has moved up. However, it is also currently in a recessional wave. The recessional wave has the effect of creating separation in everything and it also does so in every aspect of humanity, that is why there is a resurgence of separation on earth. It even has the effect of making the highest of vibration of humanity and the lowest vibration further apart. Again humans have a separation instead of a unity occurring. With the four major movements taking place during 2017 you can begin bringing this back together and change the direction of the recessional wave. That is what you are looking at right now, and it is profound. IF this stays on track the recessional wave will end within the next two to three years, resulting in a complete turnaround in a different direction. Enjoy that journey, dear ones, because it will validate your understanding of your own power of creation.\nYou are creator beings, but does that mean you must create a different set of rules for everything? No, since you are just creating for you. Begin creating your highest dream with whatever it takes to bring that energy into your being. Many people are closing their hearts and protecting themselves from the anger and density they feel around them. We challenge you to take this opportunity to move in the opposite direction and expand your passions anchoring new light in times of darkness. Dare to play, expand and improve yourself. If you fall into fear the recessional wave takes much longer to reverse. Fill yourself with the Light from Home at every opportunity. These are photons that carry fully charged light within your body. We will share more about that shortly, but basically it involves learning how the energy moves so you can consciously fill yourself with light. You bodies have been slowly adapting over the last seven years to carry more light. This Light has amazing effects in the world around you, for the new times are coming. You have four major steps next year alone, and we cannot wait to see how you resolve these situations. These steps open the proverbial door for a full understanding of multidimensionality, as well as a full understanding of your spirit. Re-member, it is who you truly are beneath all the chaos that you have in your world at the moment.\nTake these words, dear ones. Make them your own for each of you here is a slightly different message that we speak and that is as it should be because even though we are using certain words to reach everyone, we speak to each heart individually. You know who you are and we honor you for your work more than you could ever understand. Embrace these four movements coming forward\u2014these incredible opportunities\u2013and know that this is why you came to Earth in the first place. Keep up the good work.\nIt is with the greatest of honor that we greet you in these ways. We ask you to treat each other with respect whenever possible. Nurture each another whenever you have that opportunity and play well together on the new Earth. I am the Keeper of Time and I leave you with love.\nWe hear it every day that life is changing in our personal reality and the world around us. In this new year of 2017, it seems ever more prevalent. We all feel the pull to adjust to each new changing factor that presents itself while wanting to cling to the sentimental familiarity. We long to keep life simple and comfortable but these changes lead us to a shakeup forcing us to adjust to what is unfolding to the path in front of us.\nIt is impossible to remain stagnant in our existence. When we try to do so we lose our excitement in life and the potential growth of being all that we can be. Our light would begin to fade by living in the past. So I encourage all of us to take that first step and embrace the future beginning today.\nI am a very sentimental person. I think most of Lightworkers feel this way. Our hearts are open with empathy to all around us. Having the life lesson of Definition, I tend to reach out to others instead of focusing on myself. By doing this we forget who we are and then we have nothing to offer anyone or help this world in which we exist.\nI am thrilled to see the changes with all living creatures especially the animals. Many of you know Steve and I recently adopted our dog, Sadie. She is the perfect extension of our family. We cannot imagine our home without her. She is a consent reminder of unconditional love. Every so often I think about what her life had been like before we brought her into our home. She was a rescue dog who was scheduled to be put to sleep the day after we got her. All that we know about her is that she is three years old and the vet said she probably already had three litters of puppies. I feel she was used for breeding puppies then discarded as no longer useful. I can't imagine why she had not been adopted before but yet I know. The minute we saw her we knew she was meant for us. It breaks my heart to see so many unwanted animals just waiting to be loved. It is encouraging to see the many loving volunteers who help with the shelters reaching out to the public awareness trying to connect each dog or cat to a home.\nAnother big change with our awareness with how animals are being treated is, \"The Greatest Show on Earth.\" Ringling Brothers and Barnum and Bailey Circus are closing down after 146 years of existence. The sadness comes with the sentimentality I mentioned before as we humans tend to feel. Many feel this is heartbreaking because of the precious memories that are held from childhood but many celebrate this news. It is time to move on with the end of this era. Several of us grew up with the circus being the highlight of our childhood. Steve and I took our two sons often to view the exciting show. Our boys even had the delight of riding on an elephant's backs as a preshow treat. It all was a feeling of magic while not realizing how the animals were being treated behind the scenes. In recent years clashes of how elephants were treated have led to removing the animals altogether, retiring them to sanctuaries. Times have caught up with the circus. People are more aware now. Things are changing. Children are more advanced now where they are not going to a circus as they are into the many other choices life has to offer.\nThis is putting a lot of people on notice for a change. Zoos, Aquariums, anywhere where living creatures are being treated unfairly will be scrutinized by humans. As a family living in San Diego, California, we thrilled with the experience of Sea World, especially seeing \"Shamoo\" the killer whale. Children delighted with being splashed by these beings while their trainers made them perform. I remember one time Steve and I had out of town friends come to visit. We decided to enjoy a special lunch right next to the main pool where the whales swim. It was a special behind-the-scenes area and a private show with whales and their trainers. We were all enjoying ourselves when we started to feel something amiss. The whales went berserk while the trainers frantically ran around trying to control the situation. No one was hurt except for the whales splashed so much that several people were drenched with water ruining their lunches. This was my first awareness of these creatures not being pleased with their environment. After that, we learned a trainer had been killed by one of the whales at a Sea World causing them to stop this part of the show. At first, I was sad because I felt this display was a highlight of Sea World then deep inside I knew it was time for a change.\nPerhaps this is the first step for us to change, animals, next with the humans we co-exist within this world. We are all living creators who inhabit Mother Earth. Let's support each other. Life is not what it used to be nor do we want it to be. Here is a new awareness of living.\nThank you so very much, Barbara, for sharing with us about the animals. I, too, have never been able to go to the dolphin -whale shows or zoos. Every time I tried and had a chance to look into their eyes, I would burst into tears, I felt so much of their pain. So for many years , I have been sending the Light and Beautiful Violet Flame of Love and Forgiveness through all animal zoos and encounters during my meditations. My heart is so touched as humanity starts to realize that we must change the ways in which we encounter the animals who cannot speak for themselves. Thank you to all of you who are working toward this goal. I AM Ever in Gratitude and I send deep, deep Love to All my brother and sister Lightworkers on this beloved planet and Beyond.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Trey Ryder writes \"In most cases, when you act and sound like a salesperson, your prospects treat you like a salesperson. They avoid you. They don't trust you. They don't even want to talk with you. This is why I urge lawyers to avoid selling-based marketing. To increase your prospect's trust, respect and confidence, stay in the education mode because education-based marketing is the key to attracting new clients with dignity.\"\nOwings Mills, MD (minutes from downtown Baltimore) - The Murthy Law Firm seeks attorneys with 3-5 years' experience in business immigration law. Our practice is dynamic and fast-paced. We have high standards with regard to integrity, work ethic, and quality. Successful candidates work both as members of a team and as team leaders. Will join a dozen high-caliber colleagues and have quality support in the way of legal and administrative staff, as well as technology. Right candidate brings in-depth understanding and knowledge of the breadth of immigration procedures, and is expected to supervise paralegals. Good writing and analytical skills required. Candidates invited to familiarize themselves with www.murthy.com. Forward resume + cover letter to hr@murthy.com with \"Attorney Search Sept 2007\" in subject line. All communication treated in confidence. Salary and benefits commensurate with experience and abilities. EEO.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The body of the saree has a White color with contrasting Green coloured border.\nYou Save: Rs 884.00\/- Your REAL SAVING This is the LOWEST PRICE at which seller can sell this product. Jagoeve does not mislead buyers by displaying higher \"Marked Up\" selling price for products, and then offer false discounts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"El Rincon Latino's Mexican restaurant in Hampton Bays offers both vegetarian and meat dishes.\nFind your sweet (or savory) spot at El Rincon Latino, where you can opt for breakfast, lunch, or dinner. So add some spice to your life and head on over to El Rincon Latino for a tasty and ethnic Mexican meal.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Admiral, the direct motor insurer, plans to beat Direct Line on efficiency, its chief executive said, after the company reported a 154% jump in pre-tax profits for 2001.\nThe profits increased to \u00a327.4m from \u00a310.8m the year before.\nAdmiral's combined ratio, which measures claims and costs as a percentage of premiums, improved to 91.2% from 107.2% in 2000.\nThis compares with 87.4% for Direct Line, across its entire range, and 100.8% for Churchill, also across its range.\nDirect Line's pre-tax profit figure was \u00a3261m and Churchill's was \u00a356m.\nAdmiral group's chief executive Henry Engelhardt said Admiral's 15.3% expense ratio made the company \"one of the most efficient companies in the industry, if not the industry leader\".\nEngelhardt, who has led Admiral since its 1993 launch, said the company could beat Direct Line on expenses.\nDirect Line does not publish its expense ratio, but publicly available figures suggest the two companies are close.\nAdmiral's gross written premiums, including instalments, grew to \u00a3296m last year from \u00a3240m in 2000.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How Far is B.U Bhandari Terranova?\nTerranova is a premium housing project launched by BU Bhandari in Pargaon, Pune. Among the many luxurious amenities that the project hosts are , External Compound Wall, Children's play area, Club House, Gazebos with Barbecue Place etc. Starting at @Rs 800 per sqft, Plot are available at attractive price points. The Plot will be available for sale from Rs 13.05 lacs to Rs 40.00 lacs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The gender divide continues to be a fact of life all over the world. Women aren't just paid less for the same job; they also pay more for the same products.\nUS company Hearos fully embraced the imbalance recently, announcing new lines of earplugs.\nIn what must surely rank as a new low in the gendered-products-on-shelves debacle, 'Sleep Pretty in Pink' for women and 'Extreme Protection' for men is a prime example of the 'Pink Tax'.\nWe were all thinking things couldn't get any worse since the Bic For Her 'pen-gate', but naturally the women's version cost more.\nAcross all marketplaces, where companies spy an opportunity to increase profits, they will take it.\nThe way we manage that message as PRs and shape client-consumer relations is an emerging challenge for our industry and one perhaps may argue that as a female-dominated industry we have a responsibility to take a lead here.\nYoung people entering the consumer marketplace today are refusing to label themselves.\nEarly adopters such as recent campaigns by brands Zara and Boots are refusing to label their customers. This is a refreshing take on the ongoing gender debate and it is great to see brands responding to their consumers' demands for a more gender-fluid approach.\nBrands need to continue performing in this space as young people's perception of gender is shifting.\nBut what impact has this had on the equal pricing debate? According to recent research, women pay 37 per cent more than men for everything from razors to pens, beauty products and clothes.\nSurely if there is a demand for gender fluidity then this rings a bell loud and clear that it is not right to be charging either gender more for the same items differently packaged.\nAs with any movement, the key to taking commercial advantage is to own the message.\nAs PRs this may be a tricky path to tread \u2013 clients may feel that gender-neutral pricing may affect their bottom line.\nOnly the weight of consumer opinion and a visible shift towards gender-neutral products at the expense of those with a large gender price gap will change the way companies behave.\nHearos is not the only villain here \u2013 many well-known brands behave this way too, often aided and abetted by our industry.\nAs message shapers, we are in a unique position to drive change in customer and client behaviour but it's a fine line between a hot marketing topic and brands actually following through to meet a genuine consumer demand.\nOne of my favourite campaigns last year came from creative advertising team Sali Horsey and Zoe Nash, who created spoof ad campaigns mocking the tampon tax 'luxury item' positioning. Placing tampons in 'luxury' ads in the same way as jewellery, perfume, or sunglasses, laughing in the face of such a ridiculous situation.\nGender causes are approaching a tipping point.\nProducts and services that engage with this issue early will shape the message to their advantage. Early adopters will gain moral 'bragging rights'. They will continue to gain market share as a new breed of gender-fluid consumer gains buying power.\nOur industry is in a unique position to shape the gender-neutral debate at consumer and client levels \u2013 an opportunity and a responsibility we should embrace.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mhumhi found the screamer again- the one who had been looking for her baby. It seemed as though she was still searching, for when he came across her she was scratching through a pile of damp paper.\nHe approached her, and she rose, cooing and smiling, and walked over to pat his head. He let her do it. His thoughts were muddied and confused.\nWhen Mhumhi had been a puppy, warned to stay upstairs by his mother and sisters, he had been remarkably clumsy. Sacha was always shouting at him to watch where he slapped his big paws, his wagging tail. But whenever he had heard his mother coming back through the door, without fail, Mhumhi would run down the stairs so fast that he stumbled, all four paws scrabbling in opposite directions, and slid and tumbled down into a heap.\nThat particular emotion- the elation of meeting his mother again, sharply cut off by the realization that he was falling and that he could not stop, no matter how he clawed and struggled- that was something like what he was feeling now.\nEvery time, he had fallen, because he got too excited. He got bruises upon his bruises. But hearing his mother come back was always so wonderful that he couldn't help himself.\nWhat was Kutta trying to do?\nThe screamer had crouched to put her arms around his neck, making little pleased sounds. He supposed he was serving as a temporary stand-in for her missing infant.\nMhumhi let her hold him for a few more moments, the light in his eyes fading to dullness, and then he turned his muzzle, his nose and whiskers brushing across her neck. In a single violent jerk of a motion, he dug his teeth into her throat.\nShe fell over backwards, with a strangled cry- a thick, wet noise. Her arms slid away from his back. Mhumhi caught her, holding her up by the neck, pressing downwards. His teeth cleaved through her flesh with ease- he was using his rear molars, the ones shaped like shearing blades.\nThere was a great deal of blood. But he had seen that before.\nHe felt her pulse coming more erratically through his teeth, and gently laid her head down on the ground. Her black eyes were open, and she gazed up at him. There was no betrayal in them, only an animalistic confusion: from where did this pain come from?\nMhumhi watched her bleeding out, silently urging her to die, hurry up, hurry up. The sun was sinking. He licked the blood dripping from his nose.\nHe heard a soft pad from behind him. Vimbo was coming up from the other side, licking his chops. Mhumhi was surprised to see him- after he had left Telipa, the two of them had separated, Vimbo turning towards his bony nest, Mhumhi towards the tire stacks.\nVimbo came up and sniffed at the screamer's bare legs, up towards her abdomen. Suddenly he bit into the area above her hip. The screamer jerked, more blood spurting from the wound in her neck.\nBut Vimbo kept eating, giggling through his mouthfuls when Mhumhi got too close. Mhumhi stepped back, swallowing. The screamer's head flopped to one side.\nHe and Kutta had agreed that they should wait until the screamers were dead before killing them, to reduce the amount of pain they caused- it seemed the kinder thing to do. Now, staring at Vimbo, he wondered how much it really mattered. The screamer was staring at him sightlessly, blood still oozing sluggishly from her neck as Vimbo found her entrails.\nThere came a yap-howl from his right and Mhumhi jerked his head to see the silhouette of a pointy-eared dog standing on a distant hill. No way to tell if it was Telipa or not, and somehow he didn't think so. But now the gray pack would be looking for their share.\nMhumhi turned back to the still-twitching screamer and began to eat, thrusting his head beside Vimbo's, heedless of his growls.\nHe wondered if the hyena ever felt compassion for his kills; perhaps he simply didn't care.\nVimbo finished first, and left without sentimentality, scattering the mix of screamers and jackals and white birds that had gathered to watch them eat. The birds flew, the jackals darted, and the screamers shambled, but they all came back to fill the gaps after he was gone. Mhumhi at even more quickly, stained with blood up to his ears; he did not like the feeling of them watching him.\nHe left quickly when he had finished, feeling the odious stares of the jackals on his back as they rose from their haunches to gobble his scraps.\nHlolwa was sitting with her paws sticking out of the den, looking impatient, when he returned, and she got up at the bloodied sight of him.\nMhumhi stopped short, his fattened belly swaying, and let her lick at the corners of his jaws, twittering with impatience. The meat came back up. She caught pieces before they hit the ground in her hunger.\n\"Was the hunting difficult?\" she asked, between bites. \"The screamers gave you trouble?\" There was something needling about her tone.\nHlolwa's eyes thinned, and she gave him a few suspicious sniffs.\nHlolwa gave a soft snort, and then she licked him again, bringing up more meat.\nHe trailed off miserably. Hlolwa raised her head.\nHer tone was so sardonic that he bristled.\nShe seemed to be amusing herself, to such an extent that Mhumhi found he couldn't really get irritated.\nHe was interrupted by an odd squeaking noise, and both he and Hlolwa looked back to the den. Peering over the edge of the tire, squinting in the bright light and rubbing his nose with one paw, was a little black puppy.\n\"Where are you going?\" said Hlolwa, giving the puppy a very stern look. Mhumhi pushed in front of her, tail wagging furiously, and touched noses with the puppy. The puppy jumped at the sudden contact and then sniffed closer.\nHlolwa shouldered him away and picked up the puppy by his scruff, so that his fat little legs stuck out in the air. Even as she picked that one up, another little face popped up over the tire with a squeak. A little female with ears that still flopped forward.\nMhumhi beat the ground with his forepaws with excitement. Hlolwa's eyes had gotten very narrow, but she could only, unfortunately, carry one puppy at a time.\n\"I didn't know they were walking!\" said Mhumhi, licking the face of the female so that she squinted even further. A third puppy got her face up over the edge of the tire but tumbled backwards almost at once.\nThat dulled Mhumhi's excitement a bit. His tail drooped.\n\"If a screamer catches sight of them\u2026\" Hlolwa let the words hang in the air. Mhumhi drooped further. He picked up the little female and deposited her back over the tire. Her little stub of a tail wagged and she looked back over her shoulder at him.\nHlolwa nudged her the rest of the way inside and deposited the male after her, to a loud barrage of squeaks. Mhumhi thought they were certainly of protest. His tail waved slowly.\n\"Are you going to do anything about your sister?\" Hlolwa's tone was terse as she stared into the darkness of the den.\nMhumhi sighed and looked down at his forepaws.\n\"A few more days,\" said Hlolwa.\nMhumhi drew back, wanting to protest- but he knew exactly what she was talking about. There was no point in denying it anymore. He scuffed at the dirt with a forepaw.\nHlolwa was quiet for a moment.\nHlolwa's eyes widened, giving him a shock of pale amber, and Mhumhi couldn't help tensing. But Hlolwa only laughed softly.\nMhumhi found he could say nothing to this. He rocked backwards slightly.\nHlolwa laughed again, turning away.\nMhumhi looked down at his paws again, because he had felt something sneaking and unpleasant in his heart, thinking about Tareq.\nHer reply came sharp and savage.\nMhumhi shuffled his paws, stirring up a little cloud of dirt. From within the den came a soft squeak.\nHlolwa seemed to digest this, drawing her ears back.\n\"No, because I can't enforce that,\" said Mhumhi. He felt tired. \"I can hope that you wouldn't harm him. But how- to what extent\u2026\" He trailed off, because what he had wanted to say was the same guilty thought that had been slinking around inside his mind for a long time: to what extent, ultimately, would he be responsible for Tareq's life?\nA burden? Yes, because nobody he loved would have had to die. And he was\u2026 he was tired.\nPerhaps Hlolwa guessed his thoughts, because her expression became inscrutable.\nHe brought his head up sharply to stare at Hlolwa, because wasn't that what she had wanted from him, all along, ever since he'd first met her in the Great Glass Garden?\nHlolwa looked thoughtful. He could not discern about what.\n\"I'm not lying to you,\" said Mhumhi.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Punjab college given that a top superiority education in all greater than the Pakistan. Many students those acquire edification from Punjab collage are functioning in unbelievable companies. The assemblage of companies as well in commission a assemblage of schools names by Allied Schools and a university named by the University.\nAdmissions in FA, FSC, ICS, ICOM for the conference of Admission 2019 in Punjab Group of Colleges is unfasten in all campuses of the complete Punjab as the complex of the Punjab College multiply all greater than the Punjab in supplementary than 48 different cities of Pakistan .\nPunjab Group of Colleges is the most important higher education assembly in Pakistan. It is in addition associated with Punjab Boards of Intermediate and Secondary Education which is the most important starting place of schooling all the way through the country.\nadmitted student and as long as both are studying in the institute.\n1st Year Admissions 2019 is disappearing to be undo shortly so are you give the impression of being for FA FSc ICS ICom Admission as the Board of Intermediate and Secondary Education will departure to be announce their matric class result in the establish of the month July so beginning July till August the admission in FA, FSC, ICS and ICom will remainder unfasten.\nThere is a selection in frontage of students when they dig up agreed in the Matric class that in which countryside they would like to depart for the expectations studies as all those students who are fascinated in the Medical Field those will desire FSc Pre-Medical and the engineering students will depart for the F.Sc Pre-Engineering.\nThe students who would like to apply for FSc., ICS and I.com assembly 2019 may just tender their registration form from college grounds. The students who are in the making for their result may also be intelligent to apply for pre first yea consignment. The candidates can and download the prospectus commencing the website of Punjab group of collages.\nwhat is the merit for medical in Pre classes of Punjab college ?\nI got 81.5% marks in 10th class and I want to get admission in punjab colledge what would be the procedure why I am from Kpk and intrusted to apply for scoller ship.\nwhat about gcu college ?\nWhat is last date of admission pre engng!\nmerit list lag gaye kia?\nAny one tell me,what is the last date of admission or either it has been done?\nwhat is the last date of admission ?\ntoo ham kya karien. Lol!\nPlease tell me the criteria to get admission in model class for O level students.\nHi everyone , may na matric may 63% marks leya hai supplement may keya kon bata sakta hai k mujh admission mal jaye ga thank you. If who know then plz contact me 03038556337.\nis college entry test kab hy ????\nSir scission 2016 to 2017 pre medical k admissions kab hun gy. Plz tel me sir.\nCan anyone tell me the last date of admission?\nwhat is merit of punjab college lahore.\nWhen the admissions at PGC will over?\nI have got 82% marks in matriculation level (arts) and I want to get admition in I.com please tell me scholarship merit list and fee structure.\nI secured 1051 marks out of 1100, result announced on 20th July, 2016.\nPlease advise me if these marks are enough for Punjab College admission in super model class or I choose G.C for admission.\nPlz B.ed ka admition ki last dates bata dain.\nI had given metric exams this year.for further study in F.Sc engg. I want to join PUNJAB GROUP OF COLLEGE,so what are the basic requirements regarding me?\nWhat's the admission fee for fsc of punjab college?\nB grade scholarship wale supermodle main jate hai yaa modle main?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The ABA Journal notes that the Oregon State Supreme Court has unanimously determined that the governor of Oregon \"has power to grant clemency in a capital case regardless of whether the inmate in question wants to be put to death.\" In sum, there is no \"right\" to a state execution. Consequently, death row inmate Gary D. Haugen cannot refuse Gov. John Kitzhaber's clemency grant.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We all lead busy lives and priorities that we are constantly juggling never seeming to find the right balance. I never really understood what Psalm 46:10a meant and the whole meaning of 'quiet time with Jesus' just went over my head.\nJesus used one book to change my perspective on how we, Jesus and me, should spend time together and thus grow our relationship. The book is called Beautiful Uncertainty by Mandy Hale. Hale spoke of her experience of how Jesus changed how they spent time together to something beautiful and sweet and how it did in fact grow their relationship to a closeness she had never experienced before.\nI didn't have that closeness with Jesus Hale described. I was sceptical yet something in my heart responded, I wanted to meet with Jesus in a real, raw and close way.\nBefore I had just talked at Jesus rather than to Him. I found reading the Bible dry and complicated with big words I didn't know the meaning of let alone knew how to pronounce so I relied on devotionals for the scripture and explanation of what it meant. I would pray throughout the day and at bedtime, but my prayers in the evening were always interrupted because I would fall asleep.\nI wanted more. Jesus had captured my heart at age 20 and through the adventure of going to university we developed a friendship, if you will, though after graduating university I felt Jesus pursuing me for a deeper commitment. With any relationship we need to put in the effort and spend time with them, Jesus is no different.\nSo I decided on meeting with Jesus in the morning. I figured that way I had a lesser chance of falling asleep or other activities distracting me. As for where we meet I sit on my couch; but I imagine Jesus meeting me at a beautiful beach or in a field of wildflowers and it is just us, letting the waves lap our feet or picking flowers as we enjoy each other's company.\nIt is beautiful, rich and sweet.\nI talk to Jesus and I am learning to listen for His voice which for me comes gently in the stirring of my heart not through an audible voice. I ask for leading as to where we should read in the Bible for that day, based on what He wants to teach me and share with me. I write in a notebook of our moments together.\nWe can't wait until we are perfect or only for the sunny days of life, we can come flawed, broken, and sinful and constantly failing with the storms of life raging and Jesus will meet us there giving us understanding of His breathed Word.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In Isaiah 58, we find a Sabbath passage that helps correct our natural thinking. We are prone to consider the Sabbath command a burden, but we are encouraged to see it for what it really is: a blessing.\nIf you turn away\u2026from doing your pleasure.\nThere is a kind provision in this prohibition. In our modern western culture, we allow ourselves to enjoy just about any pursuit or activity on the Sabbath in the name of Christian liberty. (It is legalistic\u2014an unnecessary burden\u2014to restrict our activities on Sunday, so the argument might go.) This passage instructs us not to do that, not because God is trying to keep us from enjoying ourselves, but because He wants us to find and experience true delight.\nBecause of indwelling sin, we are prone to avoid what we are commanded to pursue. We've all experienced this principle at work in our minds when we settle down for a time of extended prayer or Bible study: suddenly we are acutely aware of all the projects we need to work on, all the chores we need to complete, all the unresolved issues we need to deal with\u2026and with everything vying for our attention, we wonder if now really is the best time to commune with God.\nSo it is with the Sabbath rest. God commands us to abstain from going our own way because He knows indwelling sin will kick in and desire to do anything but take advantage of the Sabbath. It isn't legalistic or burdensome to avoid \"what we want\" on the Sabbath, for indwelling sin is quick to blur the line between what we think is good for us and what really is good for us.\n\"This is the very way in which the Sabbath is mostly broken; it is made a day of carnal pleasure instead of spiritual 'delight'\" (JFB).\n\"Man's 'own words' would 'call' it a 'weariness'; it is the spiritual nature given from above which 'calls it a delight' (Am 8:5 Mal 1:13)\" (JFB). Our required response to the Sabbath is possible only because of God's heart-changing grace, for we are naturally prone to call the Sabbath a duty and not a delight.\nNor speaking your own words.\n\"In all we say and do we must put a difference between this day and other days\" (Matthew Henry). Shannon and I have been trying to avoid conversations about \"normal\" weekly matters (i.e., business or school projects that need working on). It is helping us to enjoy a more consistent state of rest throughout the Sabbath day. By the grace of God, the troubles of Monday can (generally) wait patiently enough until Monday.\nThen you shall delight yourself in the LORD.\nDelight in the Sabbath is directly linked to a delight in the Lord. That doesn't mean that all who delight in the Lord delight in the Sabbath (just as it is true that not all of God's elect believe in election), but God promises that when His children find delight in the Sabbath rest, they will also find delight in the giver of that rest.\nThank you for your encouraging study! I am finding greater delight in, and awe of, the Lord of the Sabbath with each passing week.\nThose words are encouraging in and of themselves, but they mean much more coming from you. Thanks, Mom!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"However, Google's YouTube has quietly added several films to watch for free, including the first five Rocky movies, The Terminator and Legally Blonde.\nForward-looking: In response to demand from users and advertisers alike, YouTube has now added a library of over a hundred free movies and plans to increase that number as much as possible, as soon as possible.\nThe company is reportedly open to having sponsorships or exclusive screenings to show movies through its platform.\nThis might also help YouTube get around the fact that half of these movies have already been uploaded to YouTube for free anyway.\nPreviously, the Movies & Shows page only listed films and TV programs that you could purchase to watch.\nWith little fanfare, YouTube started streaming popular ad-supported movies for free. Rohit Dhawan, YouTube's director of product management, told AdAge.\nThe library appears to available only in the U.S. for now, and is different from the YouTube original content films that the streaming website includes with a YouTube Premium subscription, previously named YouTube Red.\nYouTube has been experimenting with various models for the platform, such as its commercial-free subscription service YouTube Premium.\nSamira Sadeque is a New York-based journalist reporting on immigration, sexual violence, and mental health, and will sometimes write about memes and dinosaurs too. It also comes after Roku introduced its own free, ad-supported movie selection called The Roku Channel which is available on the web.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Learn about the anatomy of a spring as well as the factors that affect water quality and how you can help to protect it.\nTake a virtual tour of the Wekiva River Protection Area to discover how managed conservation areas are safeguarding water quality.\nDo you have old photos or documents, or an interesting water-related story to tell? Let us know.\nOrange County Environmental Protection Division's environmental Volunteer & Internship Program (eVIP) offers citizens a means of public engagement with the natural world around them. The eVIP program matches committed individuals with opportunities to become trained and equipped for a variety of important environmental tasks. This empowers each volunteer to help make a difference in their community. Read about the programs below and notify the eVIP coordinator of your interest by filling out the volunteer form.\nCoordination and training for the opportunities listed below is provided by Orange County's eVIP program. Read about each of them and let us know of your interest by filling out the volunteer form.\nSpearhead a storm drain labeling project in your community. Orange County eVIP provides all the supplies, a one-hour training and the support to create an enjoyable and meaningful service project for your group. Please help us keep our surface water clean and our storm drains functioning properly. This is a great project for Scout groups!\nGet trained in the EnviroScape Model and provide fun, hands-on watershed education presentations to youth in Orange County. Are you an educator? Once trained, classroom teachers and youth group leaders can also \"check out\" our education materials. One-on-one trainings can be arranged to accommodate your schedule.\nIs your group looking for an environmental project? eVIP has a new kit that helps individuals lead a cleanup and assist to keep Orange County free of pollution. eVIP provides the supplies, a one-hour training and support. The project is a good fit for volunteers of all ages and groups of all sizes. Educational materials will be provided as requested.\nHard working eVIP volunteers created extra monofilament collection tubes to be placed in areas of need. Do you know of a lake or fishing spot that would benefit from a collection tube for used fishing supplies? Please help us identify an area and let us know if you would like to help!\nFill out this form and we will contact you with more information about volunteer opportunities.\nPlease verify this form submission by completing the following test.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yolanda Hadid Officially Declares Gigi's Boyfriend Zayn \"Family\"\nThe #RHOBH alum is welcoming the pop star into her life with open arms.\nWhen it comes to Gigi Hadid's romance with Zayn Malik, Yolanda Hadid is all about it. In fact, The Real Housewives of Beverly Hills alum has declared the pop star \"family\" on Instagram on multiple occasions. So, it's safe to say she likes her model daughter's hunky beau.\nZayn certainly does seem like he's one of the gang and he's even been spotted hanging out with the Hadids on Yolanda's farm, which she moved into over the summer. And Gigi pretty much read our minds when she declared a photo of Zayn feeding a horse as \"heaven is a place on earth.\"\nGigi and Zayn first sparked dating rumors nearly two years ago in November 2015. They began dating after her split from Joe Jonas and his break-up from Little Mix singer, Perrie Edwards.\nClearly their romance has Yolanda's #proudmommy seal of approval. Though, she has been quick in the past to give any potential love interests for her daughters a little message. Check out what she had to say below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Malawi tourist visa is required for citizens of Australia.\nUnfortunately, at this time VisaHQ does not provide service for tourist visas to Malawi.\nMalawi visa for citizens of Australia is required. For more information please contact the nearest Malawi embassy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We've been talking about Do It Yourself Solar and the steps involved. Today let's get down to the brass tacks: system costs. We'll approach it from two directions, component costs and complete system costs. We'll use a 2 kilowatt, grid tied, roof mounted example system where applicable.\nLet me start out by saying that I do not think its a good idea to go out and try to build a system from individual components yourself. Unless you have experience in electrical work or are already a solar professional, this can over complicate what is already a complicated process. I also get calls from people who have inherited some solar panels or bought an inverter on Ebay. You can certainly cobble a system together with used parts but keep in mind that efficiency is advancing in this industry fast enough that even panels made 4 years ago are behind the times. The best way to start is with new, brand name products.\nSolar panels vary widely in cost now but are quickly becoming a commodity in terms of pricing. You can find many Chinese manufacturers that will provide panels for $2\/watt. But what will you do when you have a warranty issue? Odds are your panel company will be defunct or impossible to reach.\nI would suggest using a name brand for the panels. In my opinion, Sharp, Sanyo, Mitsubishi, and Kyocera are all excellent panels. Evergreen panels, which are American made, and REC which are made in Norway, are good panels but less prevalent. For these name brand solar panels you can expect to pay anywhere from $3.00 to $4.00 per watt retail. Obviously the more you buy the better your price per watt will be.\nThere are three major inverter companies that do the lion share of solar installations: SMA (the Sunny Boy line is one of the most popular in the world); Fronius (like SMA a German company); and Xantrex (which makes a wide variety of electrical gear). You really can't go wrong with these brands, but of course that means they cost a little more too. For pure sine wave inverters, which is what you'll want for 99% of American homes, you can expect to pay about $2,500 for a 2000 watt inverter.\nThere are many other brands that you can get inverters for closer to $2000 (PV Powered, Outback, and Samlex come to mind) and could be perfectly acceptable for your system. One thing to keep in mind is that since you are connecting to the grid, your utility probably has a list of acceptable inverters. Be sure to check with them before buying ANYTHING.\nThere are so many racking makers and types (eg roof, ground, pole) it seems like every day I learn of a new manufacturer. Generally I don't get too picky about the racking companies. The key part is making sure that you are buying the right racks for the panels and installation type you have. Uni-Rac, Iron Ridge and DPW are all reputable brands and can handle 90% of installation specifications. For a 2000 watt flush roof mount system you can expect to pay about $800-1000 for the rails, mounts and clamps. Ground mount and tilt systems get more expensive because there is just more parts involved.\nThis is a hard part to estimate because it depends on your installation and the distance between the panels and the inverter. One key concept is to realize the larger the distance between the panels and inverter, the larger and more expensive that wire will be. DC power, much like water, must travel through larger wire at higher amperages to maintain efficiency. For a 2000W system I would budget about $400 for a typical wiring, disconnect, fuses, lighting arrestor, and conduit setup.\nOk, so that was the hard way to estimate a system. The fact is that companies have recognized that most people want an all-in-one-system (full disclosure: my company sells solar power kits for just this purpose). These kits can be customized for your particle application and come with the parts you need and, most importantly, instructions and drawings! Now this isn't quite like putting together that entertainment center from Target, but with some patience, tools, and a little help from an electrician, these kits are actually very doable.\nThe best part of the kit is the cost. If you do the math on the system talked about before you're looking at about $12,000-$14,000 and you have to match all your components. Kits generally run about $5-$6 per watt. So our 2000 watt example system would be closer to $10,000. And its a complete package, some customers have to add conduit and some wiring, but for the most part its a turn-key system.\nWe touched in micro inverter system before but I wanted to talk about cost quickly. These inverters are meant to be connected directly to one panel of around 230W. Then the inverter can be connected to the grid. This changes the economics of systems significantly because you can get a 230W panel for around $700, a micro inverter for $230, and a small rack system for $100. Add in some wiring, a fuse and a disconnect and you've just gone solar with grid connection for around $1300. Not bad eh?\nNext week we'll start the install.\nKriss Bergethon lives off the grid with his wife in Colorado. For more information visit his website at Solar Film.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you, like so many of us, made a New Year's Resolution to cut back on sweets...this one's for you. Thursday is \"Ditch Your Resolution Day\" and it's time to treat yourself. Where better to do so than in Hershey, PA?!\nBefore and after your Exclusive Dark Chocolate Treatments, take a sip of this delicious treat. We won't tattle if you refill your mug!\nIt's cold outside. We get it! Don't let that stop you...grab a seat by the fire in the lobby at Hershey Lodge and start sipping.\nExecutive Pastry Chef Cher Harris and her team create these classic, portion-sized treats. At least it's not the whole cake, right?\nS'mores Latte from the Cocoa Beanery on Research Blvd.\nA coffee shop favorite! We recommend asking for extra whipped cream.\nYou'll need to wait until Springtime In The Park for this one. In the meantime, drool over this photo.\nAnd don't forget...next month is Chocolate-Covered February in Hershey! Stay tuned for more details about our month-long, town-wide celebration of all things chocolate. And of course, tag all your sweet photos with #HersheyPA!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Farm Market Recipes from Chef Jermaine!\nFor those of you who missed Chef Jermaine Allen of Chop.Cook.Eat at the market, he has graciously shared his amazing recipes. A lot of his ingredients are grown locally and can be purchased at the Farmers Market this coming Saturday 10-1pm at the Diamond P Ranch. Visit our Facebook page for more info.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Arrive in style at San Francisco International Airport with this private transfer service. Meet your personal driver at SFO and relax with a hassle-free ride to your Silicon Valley destination. These private vehicles can accommodate up to six passengers; available 24 hours a day, seven days a week.\nSkip the hassle of sharing a shuttle from San Francisco International Airport to the Silicon Valley. This convenient option takes all the guesswork out of renting a car or paying for an expensive taxi after a long flight. Simply enter your details when you book, pay in advance, and then show your voucher to your driver when you land at SFO.\nTravel in a comfortable hybrid for up to two passengers, luxury sedan for up to three passengers, or an SUV for up to six passengers. Sit back and enjoy the ride without worrying about arriving on time or getting lost.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Heaven's Miraculous Angelic Giving Inspirational Child, Inc.\nThe Benefits of Heaven's M.A.G.I.C.\nBecoming a Part of Heaven's M.A.G.I.C.\n\"Remember\" Translated For the World!\nMy 2007-2008 Gift to Our Supporters !\nA call to ACTION for ALL who love, honor, & worship God!\nThank You For Your Donation to Heaven's M.A.G.I.C.\nYOU can STOP THIS! WE can STOP THIS!\nFor $7.50, Less than a movie, you can provide food for a starving child.\n$0.25 cents a day provides two nourishing meals every day for a month to a child threatened by famine.\nWe endeavor to create a chapter that will support Doctor's Without Borders in the future.\nAt least this way, the next time you see a starving child, you can feel hopeful and not hopeless, and powerful instead of powerless.\nIdeally, if everyone who owned a credit card did this, the problem probably would not exist.\nAnd remember... Every time you sit down to eat a meal, figuratively speaking, there is a cute, smiling, child sitting across from single you, and\/or you and your family enjoying your generosity.\nYour family size has increased, and for those who dislike eating alone, figuratively speaking, you'll never eat alone!\nPlease share this message with your friends and family and pass it on!\nSpread it like wildfire throughout the world!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"DOWNLOAD THE DATASET CGD2011 of 50,000 gestures from one of our data mirrors.\n\"ChaLearn Gesture Dataset (CGD2011), ChaLearn, California, 2011\"\n- learn to perform gestures from an educational software.\nKinectTM has revolutionized the field of gesture recognition because it is an affordable device (a high end webcam) providing both RGB and depth images. Depth images facilitate image segmentation considerably. We have collected a large dataset of 50,000 gestures with KinectTM. We provide MatlabTM code to browse though the data and process it to create a sample submission. The data can also be viewed with most video viewers, see the README file for details.\nWe provide both the RGB image and the depth image as in the example below. View more examples.\nEach batch includes 100 recorded gestures grouped in sequences of 1 to 5 gestures performed by the same user. The gestures are drawn from a small vocabulary of 8 to 15 unique gestures, which we call a \"lexicon\" (see a few examples of the lexicons we used).\nWe selected lexicons from nine categories corresponding to various settings or application domains; they include (1) body language gestures (like scratching your head, crossing your arms), (2) gesticulations performed to accompany speech, (3) illustrators (like Italian gestures), (4) emblems (like Indian Mudras), (5) signs (from sign languages for the deaf), (6) signals (like referee signals, diving signals, or mashalling signals to guide machinery or vehicle), (7) actions (like drinking or writing), (8) pantomimes (gestures made to mimic actions), and (9) dance postures.\nDuring the challenge, we do not disclose the identity of the lexicons and of the users. They will be revealed (after user anonymization) at the end of the challenge. Although the gesture classes are different from batch to batch, we represent the class label within each batch by a number between 1 and 15.\nFor the develXX batches, we provide all the labels. For the validXX and finalXX batches, we provide labels only for one examle of each class. The goal is to predict the gesture class labels for the remaining gesture sequences.\nDuring the development period, performance feed-back will be provided on-line on the validXX batches. The final evaluation will be carried out on the finalXX batches and the final results will be revealed only when the challenge is over.\nWe provide some data annotations, including temporal segmentation into isolated gestures and body part annotations (head, shoulders, elbows, and hands).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Premium canned meat company introduces an updated logo to appeal to today's consumer.\nWeber Packaging Solutions will be showcasing their new Model L-A 6000 label printer applicator at the upcoming Pack Expo International 2016 Show.\nJumbo Chipotle Bacon Cheeseburger gets flexible, hand-wrapped packaging.\nMultisorb Technologies will be participating at IFT, at McCormick Place South, Chicago, IL, July 17 - 19, 2016.\nChanges to Nutrition Facts label includes an updated design to highlight \"calories\" and \"servings.\"\nCoveris continues geographic expansion in Central and South America.\nAre the federally mandated safe-handling warnings present on every raw animal product outdated?\nMultivac presents a new generation of label dispensers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"California State University Los Angeles respects your privacy and is committed to protecting it to the extent possible, subject to applicable state and federal law, through our compliance with our privacy policies and this Privacy Notice.\nThis Notice applies to the information that we collect when you visit Cal State LA's main website www.calstatela.edu and other websites that we own or control, and on which we have linked or referred to this Notice (together, the \"Sites\"). This Notice describes how this information is collected, processed, maintained, protected, and disclosed. Unless otherwise indicated on a specific Site, the California State University (CSU) is the data controller for all information collected under this Notice. Contact information for the CSU is listed at the end of this Notice.\nThis Notice does not apply to information collected from or about current or former employees, contractors, volunteers, and other workers at Cal State LA as part of their employment or working relationship with Cal State LA.\nExcept as specifically described, our Sites are operated in accordance with the laws of the United States. Please read this Notice carefully to understand our policies and practices regarding your information and how we will treat it. This Notice reflects the University's current practices and may change from time to time, so please check the Notice periodically for updates.\nIn this Notice, \"personal information\" means any information that identifies or describes an individual user of the Sites, including, but not limited to, the user's name, social security number, physical description, address, telephone number, education, financial matters, medical or employment history, password, email address, and information that reveals any network location or identity. If you are located in the European Economic Area (EEA), \"personal information\" includes all personal data as defined under EEA laws (including \"sensitive personal information\" which is provided enhanced protections under those laws).\nWe collect personal information about users only as allowed by law and limit the collection of personal information to what is relevant and necessary to accomplish a lawful purpose of the University. We collect personal information (and sensitive personal information) that you send to the CSU, or permit us to obtain from third parties, for purposes relevant to CSU operations in pursuit of our academic mission. Examples include information needed for student admission (including financial aid information), employment, housing and dining services, online educational programs, research, health services, donor relations, visa application processing, event registration, parking services, IT usage and support services, library usage, bookstore operations, and website account registrations. Our legal basis for processing most of this information is to perform a task in the public interest or in fulfillment of CSU official functions, including those set out in the California Education Code and\/or Title V of the California Code of Regulations, or under applicable federal law. Other legal bases for processing information include processing necessary for contract (e.g., to process parking permit payments), for legitimate interests (e.g., to send requested information) or consent (e.g., to process certain sensitive personal information).\nUser-Provided Information: You may be required to provide personal information to access or use certain parts of our Sites, or features of our Sites or services, including without limitation, when you apply for or enroll at one of our campuses or programs, subscribe to a newsletter or email list, make a purchase or donation, fill out a form, participate in any of our programs, special events or promotions, contact us with a comment, question or complaint, etc. If you do not provide the requested personal information, you may not be able to access or use the features of our Sites or service where such information is requested.\nDepending upon the nature of the transaction, the personal information that you may provide may include: contact information (name, home or mailing address, telephone number, social media username\/handle, mobile phone and\/or email address, etc.); academic area or interest; financial information (financial aid application history, payment history, social security number, passport number, credit card number, donation attribution and amount, etc.); health record information (medical record number), allergies, past medical history, family history, current medications, current medical conditions) demographic information (age, birthdate, marital status, income, etc.); and profile information (admissions date, graduation date, alumni status, student identification number, username, password, relationship to the University, etc.).\nEmails and Social Media Sites: If you correspond with us by email, mail or via social media, we may retain the content of your communication or social media posting, the email or social media account address from which it is sent, and our response. We collect information automatically using technology when you visit our Sites or social media pages or when you open one of our emails as described in this Notice.\nThe specific personal information (and sensitive personal information) we collect, why we collect it, and our legal basis for processing it, is periodically reassessed in applicable data process flow assessments or Data Protection Impact Assessments, as relevant.\nIt is the policy of the CSU to limit the collection and safeguard the privacy of personal information collected or maintained by the University. The University's information management practices conform to the requirements of the Information Practices Act of 1977 (Civil Code Section 1798, et seq.), the Public Records Act (California Government Code Section 6250, et seq.), California Government Code Section 11015.5, the Family Educational Rights and Privacy Act (FERPA) (20 U.S.C. \u00a7 1232g; 34 CFR Part 99), and other applicable laws pertaining to information privacy. In the event of a conflict between this Notice and the Public Records Act, the Information Practices Act, FERPA, or other law governing the disclosure of University records, the applicable law will control.\nAny information acquired by the University through the Sites is subject to the limitations set forth in the Information Practices Act. The University will not distribute or share electronically collected personal information (as defined in subdivision (d) of California Government Code Section 11015.5) about users to any third party without the permission of the user, except in narrow circumstances set forth in this Notice. The University will not sell any electronically collected personal information to any third party. Such electronically-collected personal information is exempt from requests made pursuant to the California Public Records Act (Chapter 3.5 (commencing with Section 6250) of Division 7 of Title 1).\nTo process application, registration and enrollment requests when you apply, register or enroll for our campuses, events, programs or services, or otherwise administer your participation in our events, programs or services, including (without limitation) study abroad and distance learning, financial aid, housing and dining.\nTo process service requests from students, staff, faculty, and other members of the campus community, including the facilitation of parking permit and identification card requests.\nTo process registration for sports, cultural, educational, and other university events.\nTo respond to your questions, requests, comments or complaints and determine your satisfaction with our events, programs and services.\nTo operate, maintain, and provide to you the features, services, and functionality of the Sites and its contents, and to monitor and improve our site and the user experience.\nTo provide information about our University and send you related information including brochures and other University materials, campus and CSU news, academic notices, updates, security alerts, special offers, confirmations, and support and administrative messages.\nTo notify you about changes to our Sites or any services we offer or provide through it.\nWe may also use your personal information for operational and other lawful purposes such as security, analytics, operations, fraud detection and prevention, reporting, making back-ups and legal compliance.\nAt the time we collect personal information, we strive to tell users about the purpose for which the information is collected as well as the general or specific uses that we will make of that information.\nInternational transfer of personal information: Personal information provided to us by users outside of the United States may be transferred to other countries such as the United States, where data protection laws may differ from those of your home country. By providing us with your information, you acknowledge that your personal information may be transferred to the United States and processed on servers within the United States. However, all reasonable steps will be taken to protect your privacy in accordance with applicable data protection laws.\nUser Content: Any personal information or content you voluntarily disclose for posting to the Sites (for instance, any content you post) (\"User Content\") becomes available to the public via the Sites. User Content includes, but is not limited to, comments, photos, videos, etc. If you remove User Content, copies may remain viewable in cached and archived pages or if other users have copied or stored your User Content.\nWe reserve the right to monitor the User Content you post on the Sites and to remove any User Content for any reason or no reason including, without limitation, if in our sole opinion, such material violates, or may violate, any applicable law, or to protect or defend our rights or those of any third party. We also reserve the right to remove User Content upon the request of any third party.\nCookies Information: When you visit the Sites, we may send one or more cookies \u2013 a small text file containing a string of alphanumeric characters \u2013 to your computer that uniquely identifies your browser and lets us help you log in faster and enhance your navigation through the Sites. A cookie does not collect personal information about you. We use both session cookies and persistent cookies. A persistent cookie remains on your hard drive after you close your browser. Persistent cookies may be used by your browser on subsequent visits to the Sites. Persistent cookies can be removed by following your web browser's directions. A session cookie is temporary and disappears after you close your browser. You can reset your web browser to refuse all cookies or to indicate when a cookie is being sent. However, some features of the Sites or services may not function properly if the ability to accept cookies is disabled.\nLog File Information: Log file information is automatically reported by your browser each time you access a web page. When you register with or view our Sites, our servers automatically record certain information that your web browser sends whenever you visit any website. These server logs may include information such as your web request, Internet Protocol addresses or other device identifiers, browser information, Internet Service Provider, operating system, location, date\/time stamp, clickstream data, referring\/exit pages and URLs, domain names, landing pages, pages viewed, and other such information.\nClear Gifs Information: When you use the Sites, we may employ clear gifs (also known as web beacons) which are used to track the online usage patterns of our users anonymously. No personal information is collected using these clear gifs. In addition, we may also use clear gifs in HTML-based emails sent to our users to track which emails are opened by recipients.\nCalifornia Business & Professions Code Section 22575(b) (as amended effective January 1, 2014) provides that California residents are entitled to know how a website operator responds to \"Do Not Track\" (DNT) browser settings. DNT is a feature offered by some browsers which, when enabled, sends a signal to websites to request that your browsing is not tracked, such as by third party ad networks, social networks and analytic companies. We do not engage in the collection of personally identifiable information about users' online activities over time and across third party websites when an individual uses our Sites and therefore do not respond to DNT signals.\nWe fully cooperate with law enforcement agencies in identifying those who use our Sites or services for illegal activities. We may report to law enforcement agencies any activities that we in good faith believe to be unlawful. Release of your personal information for security purposes, as described in this Notice to any person or entity, including, without limitation, in connection with any government investigation or litigation, shall be based on a determination made solely by us, as permitted by law or, for those not in the EEA, exercising our discretion for which you expressly grant permission to us in accordance with this Notice.\nWe take reasonable physical, managerial, and technical safeguards to preserve the integrity and security of your personal information against loss, unauthorized access, and illegal use or disclosure. Such personal information is stored by the University in secure locations and University staff is trained on procedures for the management of personal information, including limitations on the release of information. Access to personal information is limited to those members of the University's staff whose work requires such access. Confidential information is destroyed according to the University's records retention schedule. The University conducts periodic reviews to ensure that proper information management policies and procedures are understood and followed.\nThe security of your personal information is important to us, but please remember that no method of transmission over the Internet or method of electronic storage is 100% secure. While we strive to use commercially reasonable means to protect your personal information, we cannot ensure or warrant the absolute security of any information you transmit to the Sites, and you do so at your own risk. Once we receive your transmission of information, we make commercially reasonable efforts to ensure the security of our systems. However, please note that this is not a guarantee that such information may not be accessed, disclosed, altered, or destroyed by breach of any of our physical, technical, or managerial safeguards. While we strive to protect your personal information and privacy, we cannot guarantee the security of any information you disclose online.\nWe encourage all individuals to use appropriate safeguards to secure their computers and the information on those computers. For additional information on online privacy and security, please see the University's policies and procedures related to Information Security; you may also consult with the California Attorney General's Privacy Enforcement and Protection website.\nIn the event that personal information is compromised as a result of a breach of security, we will promptly investigate and notify those persons whose personal information has been compromised in accordance with the notification procedures set forth in the CSU Information Security Policy, or as otherwise required by applicable law.\nWe are not responsible for the practices employed by websites linked to or from our Sites nor the information or content contained in them, and we make no warranty, either express or implied, concerning the content of any such site, including the accuracy, completeness, reliability, or suitability of them for any particular purpose, nor do we warrant that any such site or content is free from any claims of copyright, trademark, or other infringement of the rights of third parties, or that any such site or content is devoid of viruses or other contamination. We may provide links to other websites to you solely as a convenience, and the inclusion of linked sites does not imply endorsement by the University or CSU of any of the linked sites. Please remember that when you use a link to go from our Sites to another website, our Notice is no longer in effect. Your browsing and interaction on any other website, including those that have a link on our Sites, is subject to that website's own rules and policies. Please read over those rules and policies before proceeding. We do not ensure the security of your personal information if you visit websites not belonging to CSU, nor are we the data controller for any information collected on those sites unless we specifically state so. We further reserve the right to terminate a link to a third party site at any time.\nWe comply with all applicable regulations regarding data retention and deletion of personal data and retain personal information only for as long as necessary to fulfill the purpose for which it was collected (including for college operations), for strategic planning, and to comply with applicable laws and retention requirements. You can ask to review, update or make changes to the personal information we maintain about you, or exercise your option of having your personal information discarded without reuse or distribution, by sending a written request to the postal or email address set out below. We may take a reasonable period of time to respond. If you request the deactivation or change of information on our system, such information may be retained in our backup systems for a period of time subject to technology restrictions, or as a precaution against systems failures. Some information may be retained for longer periods as required by law, contract or auditing requirements or as otherwise described in this Notice.\nYou have the option to decline providing information about yourself online and may use other methods, such as U.S. mail, to respond to requests for information or to communicate with us. You may use the contact information listed below to ask about additional alternatives to providing or obtaining information through use of our Sites.\nBy providing us your email address, you consent to our use of your email address to send you Site and service-related notices, including any notices required by law, in lieu of communication by postal mail. We may also use your email address to send you other messages, including, but not limited to, newsletters, information on campus activities, programs and events, legal updates, changes to features of our Sites or services, or other account information. Where required by law, we will obtain your consent before sending you specific types of email or other communications.\nYou can choose not to receive such emails from us by \"unsubscribing\" using the instructions in any email you receive from us, or by sending a written request to the postal or email address set out below. It may take up to thirty (30) business days for us to process your request. This will not stop us from sending emails about your account or your transactions with us, or any other service-related email. Opting out does not affect our communications with you via telephone or mail nor does it affect our use of your non-personally identifiable information as described in this Notice.\nIf you are an individual located in the EEA only, you have certain rights with regard to your personal data collected while you are in the EEA. These rights may include right of access, right of correction, right to be forgotten, right to restrict processing of your identifiable personal information, right to notice related to changes\/deletion\/processing limits, right to data portability, right to objection, right not to be subject to decisions based solely on automated processing, and right to withdraw consent. Some of these rights are restricted by law to information that was collected on the basis of explicit consent, or are restricted by other conditions (such as necessity for contract or to comply with the law). You have the right to contact us in connection with the exercise of your rights under applicable EEA law, which you can do through the contact information below, or by sending an email to privacy@calstatela.edu. We will respond to your written request without unreasonable delay and in accordance with any deadlines imposed by law. Unless we notify you at the time of your request, we will not charge you any fee in connection with the exercise of your rights. If you are not satisfied with our response, you have the right to complain to or seek advice from a supervisory authority and\/or bring a claim against us in any court of competent jurisdiction.\nWe reserve the right to modify this Notice at any time. It is our policy to provide notifications, whether such notifications are required by law or are for other operational purposes, to you via email notice, written or hard copy notice, or through conspicuous posting of such notice on our Sites page, as determined by the University in its sole discretion. We reserve the right to determine the form and means of providing notifications to you, provided that you may opt out of certain means of notification as described in this Notice. You are responsible for ensuring we have an up-to-date active and deliverable email address for you, and for periodically visiting our Sites and this Notice to check for any changes.\nFor changes to our Privacy Notice, it is our policy to post any changes we make on this page. The date the Notice was last revised is identified at the top of the page. We will provide notice on the Sites, or at www.calstatela.edu, prior to the effective date of any change. Your continued use of the Sites after any change in this Notice will constitute your acceptance of such change.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This year, I'm excited to get to host Thanksgiving again. I'd rather forget last year's Thanksgiving, which was mostly a foodborne illness blur. (G insists the pie was amazing.) But I'm already looking forward to turkey and cranberries and gravy and, more so, my fall tablescape. Round one (everyday seasonal) is already up. Swoon. I love this season.\nAnd I'll love it even more when G and trucky take me to the local farm for hay bales, mums, and pumpkins.\n*Featuring table runner from Mom. Still beautiful! And glittery pumpkins, orange candles, and pumpkin plates from Walmart. Because $24 per plate (a la Pottery Barn) seemed ridiculous. However I can definitely be okay with $1 per plate.\nThis blog is subtitled, O, the Things We've Done!\nWe had an October Glory red maple planted in the front yard after I finally convinced G G finally agreed one more tree in the front would be the perfect addition to our home. I was determined to get an October Glory, and the nursery had one giant, beautiful one on the lot and on sale. Plus, the State of Maryland gives $25 to folks who plant a tree. Can't wait to see its color!\nG painted the brick behind the wood stove to match the rest of the room and eliminate those horrible black soot marks that previously ran down the back.\nWe enjoyed pork and pumpkin chili that may have been just a little too spicy for G, but also too delicious for him to not eat. It was even more delicious with pumpkin beer and football.\nWe repainted the master bedroom, reorganized the furniture, and deep cleaned it all.\nI became blonde again (whew!) and said goodbye to a good friend at work with several rounds of margaritas.\nAnd I brought out some of my fall decorations because I like to decorate even when I'm in denial.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So, want to hear a funny story about these photos? When we were in Charleston, I put on a long sleeve tee and jeans for a day of sight seeing. I figured I'd be casual all day, and then change for dinner. Well, as soon as I stepped outside and felt the sun, I knew that I would regret my outfit choice about 5 minutes into our 30 minute walk to brunch. So I ran back up the stairs and changed into this dress in 15 seconds FLAT ? Before anyone even noticed I was gone, I was back and ready to go! And as the temperatures crept around 70 degrees, I was so glad I wasn't sweaty all day \u2013 plus this is way cuter for these pretty photo ops, right?!\nI have a feeling I'm going to get a ton of use out of this dress all summer. It's definitely my style, but I love that it's also super comfortable and I can wear it with flats! A maxi that I didn't have to shorten and I can wear with flat sandals?! It's like a summer dream come true!\nAdam spotted this little courtyard as we were walking through Charleston and told me to sit for a photo. Doesn't he have such a good eye?! Apparently I've been training him well ?\nDo you usually have to hem your maxi dresses?\nOr do you have the opposite problem? I always forget that tall girls are struggling to find dresses long enough, just like I'm trying to find dresses short enough!! Which category do you fall under?\nPS. You might also like The Most Flattering Wrap Dress, Spring Break Shopping Breakdown, and my Weekend Guide to Charleston!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Garren Dodd McDonald, 48, and Shelley Lanette McDonald, 43, of Etowah County were taken into custody and each charged with one count of felony child abuse last week.\n\"[The couple] did torture, willfully abuse, cruelly beat or otherwise willfully maltreat the child,\" arrest warrants obtained by AL.com said.\nThe arrests came following a nearly two-month-long investigation by authorities after the Department of Human Resources contacted the sheriff's office about the alleged abuse.\nThe couple was booked into Etowah County Detention center last Wednesday and released two days later after posting $10,000 cash bond each.\nThey are reportedly forbidden from having children living in their home and prohibited from any unsupervised contact with anyone under 18 years old as a condition of their bond.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So there are lots of different kinds of artists out there. My best friend is an incredible artist who can work wonders with just a piece of paper and a pencil. My youngest sister's hair is her medium. Her hair always looks awesome and I revel in the times when I can get her to do mine. My other sister is also an incredible artist and great at designing almost anything...I'm thinking specifically of her wedding invitations and the design for them she drew out on a loose leaf piece of paper with a pen.\nThat brings me to the title of this post, 'Dirt Art'. If you hand Anthony a pen and paper and tell him to draw a dog, you'll most likely get an two ovals (head and body), four stick legs, and a little tail...which is pretty much the same thing you'd get from me. But, if you hand him a shovel and give him a little piece of land, you'll get a perfectly manicured bed. Throw in some trees and flowers and you'll get an outdoor masterpiece. Such is the case so far...minus the trees and flowers (they're still up and coming.) Last week I showed y'all the progress we've made turning our 'weed city' into 'pine straw serenity'. I also wrote about how we wanted to modify the beds a little and add some more appealing curves. Well, last weekend my artist went at it.\nThen he started making the edge.\nMaking a sort of 'cliff' for an edge helps keep pine straw or mulch from overflowing into the grass, maintaining a nice, clean line. This gives a seamless look to the landscaping. Many people use brick or plastic edging to separate bed from grass and that works too...whatever floats your boat.\nLooks wonderful - you guys are doing a great job on your home!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Online shopping forum for Squares and Squares accessories from a great selection on Ebay.\nSwanson Speed Square Aluminum 7 \" 7 \"X7 \"X10 \"\nBlem Cosmetic Second PEC USA 4R 4\" double machinist square acc +- 001\" per 6\"\nSwanson Speed Square Orange 8 \"\nLuxury Faux Fur Shag Fabric Piece - Square Swatch CHOOSE COLOR SIZE Many colors!\n50 4\" Quilting Fabric Squares Purple PassionShades of PurpleBUY IT NOW!!\nBlem Cosmetic Second PEC USA 4R 6\" double machinist square acc +- 001\" per 6\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Though they complained, I always wrote with my students following all the rules I assigned. I did it to keep my edge, to model and to show them that I wouldn't ask them to do anything that I wouldn't do myself. There's an article titled \"Do You Write with your Students?\"\nP = Purpose Every strong essay had an unwavering commitment to the purpose. Weak essays deviated from purpose, often veering down the path of plot summary. I now have students underline and record the verbs of the prompt (analyze the relationship, connect the devices, distinguish between) to fully grasp their purpose. Every sentence written must relate to it.\nChronologically: from the beginning of the piece to the end. Cause and effect: showing the techniques an author used and the effects they created. General to specific: look broadly through a telescope, then focus under a microscope, noticing the small details contributing to the big picture. Compare and contrast: start with what unites works and move to appreciating differences between them.\nE = Evidence Weak essays make claims -- and that's it. They leave the reader unfulfilled because they neglect the evidence needed to substantiate those claims. Well-written essays have an abundance of evidence in their body paragraphs. Have students list all evidence before they write.\nIt's never to young, to push the value of effective writing. I believe it's a grave disconnect this generation has with the past. I will be happy when grammar school returns...the sometimes boring, but needed attention devoted to syntax.\nWho wants to read a blog-post with a fortune of errors? And what person doesn't shake in their boots when they receive an email so precise in syntax, they immediately know the author \"meant business?\" Lol! That's a shout out to my mother, who encouraged me to make my pen my sword instead of my fists! Love you, Mama! I believe she borrowed that phrase from someone else!\nBelow is a celebratory video that I created for our 4th graders that won the campus essay contest. Congratulations!!\nWriting Contest Celebration from AwakenLibrarian on Vimeo.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In Visual Studio 2017, some parts of the Structure Visualizer from the Productivity Power Tools were added to Visual Studio.\nOthers were not included in the port.\n#4 is the one I missed immediately. The rest probably is ranked #1, #2 and #3 to me.\nWhy I should have to spend my Sunday doing this is beyond me; but you're fuc\u0138ing welcome.\nMay I offer my sincerest fuc\u0138 you to Microsoft for not bothering to do this. We've waited for about 400 days for this to never happen. For the past 400 days, I've been editing code in 2015 and compiling it in 2017.\nAnd on a personal note to Mark WIlson-Thomas, for making us think Microsoft might have possibly considered re adding this, go suc\u0138 a dic\u0138.\n#4 is most important to me.\nOur teams just recently switched from VS2015 to 2017 (Enterprise), and this is the most prominent missing feature.\n@Mark any update on this? Its been over a year.\nWe have just updated to VS2017, and its disappointing to see PPT features deprecated in favour of a builtin implementation that has only 10% of the functionality.\nI just upgrades to VS2017 and was shocked that #4 is no missing from Productivity Power Tools. Getting strucutre Information in the map mode was essential to me when navigating source code.\nMicrosoft, you are a fuc\u0138ing disgrace.\n#4 is essential for me! It helps tremendously to navigate in large source files.\nPlease prioritize items #3, #1, #2, #4 for the Structure Visualizer in VS2017. I have grown accustomed to the colorized and solid structure guide lines that were once available in the Productivity Power Tools extensions pack.\nAs you have no problem not implementing this (We've been waiting all year). Why not open source the 2015 PPT extension, and let us do it for you?\n@Stephen: thanks for your comments. I wanted to let you know that although we don't have #1,#2 yet, you can customize the (default grey dotted) lines to a different color to help them stand out more, in Tools-Options-Fonts & Colors-Text Editor (Display Item is \"Structure Guide Lines\"). Pick an item background of your choice and press OK. Hope that helps a little.\n@Peter, thanks for the input on increasing the number of supported languages - this is being actively considered.\nPlease prioritize #1 and #2. #3 and #4 would be nice-to-have, but don't \"break\" what was in 2015. I use a 2 space tabbing, and found that the dashed lines are very hard to distinguish and follow, which is what brought me here.\nand please for ALL supported language types in VS !!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My Dad was on his way to his new job in Midland and was in an accident about 5 minutes away from his finally destination. His shoulder is really sore and the truck is totaled. The good news is he's alive and at the hospital. He's been out of work for 3 months or so and this job was going to be a game changer for him.\nNeed some prayers of thankfulness that he's alive, a quick recovery, and guidance on what to do next. I'll probably be headed that way tomorrow.\nPrayers sent for your Dad.\nThankfully he is alive. Metal can be replaced. Prayers sent.\nRolled his 18 wheeler on 1788? Was listening in and sound like he was very lucky. Glad hes ok.\nthese roads out here are terrible.\nPrayers up. It's a mad house in midland.\nHopefully he had insurance or whoever caused it does to replace the truck.\nPrayers up for speedy recovery!!!!\nPrayers sent. Im working until 6 but once im done is there anything he needs i could help with?\nprayers up for a quick recovery.\nGod speed!! Thank the Lord he is ok and may the Lord provide a way for us all to help him.\nPrayer sent for healing and guidance as well as your safe travels.\nQuick update... He and the other driver are both OK. He had shoulder pain but xrays dont show a break or anything. It could have been MUCH MUCH worse if the timing was a bit different.\nLast edited by Kevin; 11-09-2018 at 09:00 AM.\nThanks for the offer. He is good right now I think.\nWow...glad he is ok. He is going to fee that when the adrenaline wears off.\nFrom the damage in the pictures posted, what hit him, another 3\/4-1 ton truck?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am going to tell you a truth about the most secret and precious aspect of spirit work.\nEvery day I wake up and meet the new day by the side of my spirits, in the altar room with a cup of coffee and the soft candle light drawing in the new sun onto the sacred items I have for the spirits. I take a deep breath and feel the sunlight renewing my life force for this day, a day I am grateful for.\nI open the prayers and listen to the whispering sounds of the chitter-chattering voices as they funnel through the veils of time to me. Deciphering the day's work, readings, client needs and trainees' problems. I open my mind and heart to the work ahead of me, performing the appropriate rituals for the day's work.\nI give readings, advise remedies, help loved ones and make spiritual baths. Come evening I will pray and protect myself and my family. Before I close down the magical work for the night, I make offerings to the spirits for their help.\nIn a day, I walk through the minds and hearts of so many. At nighttime I dream of everyone I am working for. Their pasts, their deepest fears, I will live their darkest secrets and feel what it is like to be them. This gives me compassion.\nWhen the spirit needs to come through my body in possession to help someone, it feels like being dropped off a building. It can take up to three people to support me and several days to recover.\nBy the rite of the trail of spirits coming into our presence by passages performed daily for thousands of years, handed down from one magician to another in an unbroken chain, there is a line to the Divine and the spirits which follow.\nAs a channel, our duty is to perform the spirit's will. Unwavering. Because we are working all the time, helping to elevate, clear and free people from the loneliest of places, we know darkness and we can see it everywhere.\nEvery day we wake up kissing the sunlight in the altar room, welcoming the voices that filter through. Those of us who are called to be spirit workers are proud of the spiritual path and all it has to offer.\nThis is our blessing, to live a life connecting to the Mysteries and our God.\nMay you be blessed as the Spirits and Mysteries walk with You.\nPeople are always asking how they can support and help build the temple. This is a good question because when you support the temple you also support the spirits and they do give back. See, in the days of old, people would come into a temple and cook and clean for the Houngan and Mambo, this would give us the time we needed to perform our work, prepare our arsenal of supplies and offer ceremonies.\nThis connection and hands on training would build over time. The more you offered the more you would learn. Eventually you would develop an understanding of the spirit world and then were invited to perform mediumship for others. After some years you would be wise enough to help a priest or priestess in possession, a very important job!\nDuring this period you would move from helping with the chores to mixing the herbs, making the lamps, collecting the foods for the spirits. By this time the spirits would begin to believe you respect them and they would start communicating directly with you. An asset to the temple you would have become.\nIt is my commitment to serve in the traditional ways. I myself moved 3000 miles to live next to my Papa for this reason. I gave everything I could and learned so much. Because of my dedication and service, I became a Mambo and Vodou Priestess.\nKnow what we are doing. As a member you will be informed on work you can do during ceremonies that will benefit your life.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Like everything else, the cost of college keeps going up. Fortunately, there are various forms of financial aid that can help you meet these costs. These fall into three basic categories.\nGrants and scholarships are great because the money they provide does not have to be paid back. Some grants and scholarship are need-based. This means that they are awarded on the basis of financial need. Other grants and scholarships are merit-based. This means that they are awarded on the basis of special skills, abilities, or achievement.\nTalk with your high school guidance counselor. He or she will know about national, regional, and local grants and scholarships.\nContact the financial aid offices of colleges to which you are applying for admission.\nCheck with your parents' employers to see if they provide grants or scholarships to the children of their employees.\nThe federal government provides education loans through banks and other financial institutions. To be considered for these loans, you must complete the Free Application for Federal Student Aid (FAFSA). The government uses this application to determine your eligibility based on financial need. Here are the three types of loans provided by the government.\nFederal Stafford Loans. These are fixed-rate loans for undergraduate and graduate students who are attending school at least half-time. The loan can be used to pay tuition and other school expenses. The interest rate is low, and the loan does not have to be repaid while you are enrolled in school.\nFederal Graduate PLUS Loans. These loans allow you to borrow up to the full cost of your graduate education (less any money you may be receiving from Stafford loans). No payments are required as long as you are attending school at least half-time.\nFederal Parent PLUS Loans. These loans allow your parents or guardians to borrow up to the cost of your education at a low fixed-interest rate.\nWork-study consists of federally subsidized employment opportunities. he work is typically on campus or with nonprofit organizations off campus. Work-study opportunities are usually awarded on the basis of financial need.\nCollege is expensive. But there is help. IMPORTANT: Meet all application deadlines - there are no exceptions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Planted firmly between Black Friday and Cyber Monday this year is Small Business Saturday, a day when holiday gift-givers are encouraged to shop at small and local businesses.\nStatistics show that per square foot, local businesses generate 70 percent more local economic activity than big-box retail outfits while providing living wages to artisans, manufacturers and sellers\u2014and of course, one-of-a-kind items to buyers.\nAt the Agape Shoppe on Court Street in Watertown, a staff of volunteers and board members have determined new ways to support local communities here and abroad. A shop on the first floor offers unique, handmade gifts from artisans around the world, sales of which minimize global conflicts. Income generated by these products support education efforts in underserved communities, female entrepreneurs and help resolve violence and famine.\nMeanwhile, Agape Shoppe's thrift store on the second level offers low-cost, gently used housewares and clothing while helping to outfit job seekers in need of resources, provide on-the-job training to local outreach organizations, and raise funds to further serve Watertown and the surrounding area.\nThe Mennonite Central Committee (MCC) is a relief, service, and peace agency founded in 1920 that today represents 15 Mennonite, Brethren in Christ, and Amish bodies throughout North America.\nSince MCC's inception, the group has reached out to offer service not only to its own communities, but to people in need all over the world; from refugees to those suffering from famine or disease, underrepresented peoples, and by providing outreach to support global fair-trade initiatives.\nPart of that effort is the MCC Thrift Shop Network, a project begun more than four decades ago that comprises around 100 shops selling gently used clothing and housewares at reasonable prices \u2014 as well as providing a market for third world artisans by selling fairly traded items. The income generated at these shops directly supports the programs of the MCC, including local and international outreach efforts, from helping mothers buy back their daughters' dowries to outfitting job seekers with outfits for interviews and first days. The Thrift Shop Network is largely staffed by volunteers.\nA board of seven Mennonite churches serving the north country region runs the Agape Shoppe in Watertown. The board and shop volunteers work within a Christian framework and mission to provide service, support and community improvements throughout Watertown and its surrounding region, as well as to artisans the world over.\nOut of its first floor, the Agape Shoppe sells handmade wares from around the world. Thousands of items adorn a small space, including a stunning jewelry section made by artisans all over the world, items made by Indian and Bangladeshi women out of scraps of their old saris (sales of which go toward buying back their daughters' dowries), ceramics from Vietnam, a Christian book collection, giant array of scarves, tableware, wooden puzzles and soapstone carvings.\n\"We purchase from companies that have fair-trade relationships with their artisans,\" said Meg Graczyk, Agape Shoppe manager, referring to companies such as 10,000 Villages, SERVV, and Equal Exchange, all of which insist upon paying workers living wages, promoting sustainable farming and improving trade condition.\n\"SERVV helps with refugees a great deal,\" Graczyk said, \"and 10,000 Villages has established a whole network to work with artisans. What they do is send people over who will help artisans develop a product using their ancient skills to create things that would be sellable in the United States and Canada. Once they've developed a product, 10,000 Villages will send them half the agreed upon price. They have formulas to figure out what a fair, living wage would be for this person. Once they've determined how much they should receive, they give them half of the money up front. Once the items have been completed and given to 10,000 Villages, the company sends artisans the remainder of the money. In this way, all workers are all paid up front.\nEqual Exchange supplies the Agape Shoppe with edible products, such as fair-trade coffee and organic chocolates.\nGraczyk has been involved with fair-trade organizations for several years, first with a volunteer job at the 10,000 Villages store in Williamsville. \"My husband's job relocated up here and I just happened to come here and shop one day.\" She's been inextricably involved since.\nEdmonson's grandparents started the Agape Shoppe in Watertown in March of 1979 in the Globe MiniMall. \"I've been volunteering here since they started,\" she said.\nThe thrift shop also gives back to the community. \"We participate in the 'job shop,' a program through the county that gives people vouchers so they can get work clothes for the jobs and job interviews,\" Edmonson said. The thrift shop also offers bag sales throughout the year to make way for new inventory: $3 for grocery bags, $6 for large bags.\nThe Agape Shoppe has multiple ways for the community to get involved.\nThe thrift shop is always looking for donations of like-new and barely used housewares and clothes.\nEach August, check in with the Agape Shop for their school kits wish list. These bags are packaged and sent to children of refugees in places like Jordan, Afghanistan, and Kenya.\nThe Agape Shoppe has also partnered with Staples in Watertown. If you bring your used ink cartridges to Staples and give the store Agape Shoppe's phone number (315 788-7470), the shop will earn reward points toward free office supplies. The same is true anytime you shop at Staples. Just mention the Agape Shoppe, and points will be added to its account.\nOf course, the most obvious way to show your support for the work the Agape Shoppe is with your wallet. This year, when people are lining up outside of chain box stores at 4:30 a.m. in frenzy over a mass-produced flat screen TV, consider stores like the Agape Shoppe instead.\nWhen in doubt, remind yourself of the meaning of the word, \"agape.\" The ancient, Greco-Christian term refers to the highest form of love: charity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lower Gar Lake is calling your name! The multifaceted property offers an amazing opportunity. This listing has 111.4 ft of lake shore with low approach to the water. The land is very flat and it has a mature tree canopy. There is currently a small 2 bedroom home on one of the lots. You can build 2 separate houses on this property. If you prefer you can sell 1 of the lots off. The current owner leases 11 boat hoists on the docks if you are looking for cash flow! This location is also perfect because it is quite and off the beaten path but you are an easy 1.25 miles to Arnolds Park Amusement Park. Don't let this opportunity pass you by!\nThis property at 20440 232nd Avenue (MLS # 19-421) is categorized as Residential Homes for Sale situated in Arnolds Park, IA. This Residential property is listed on SellBoji.com with a price of $450,000. This 2 bedroom property in Arnolds Park at 20440 232nd Avenue has 1025 sqft with 1 bathrooms and was built on 1967. Whether you are searching for new listings in Arnolds Park or just searching for 20440 232nd Avenue specifically, you will find all of the market statistics and community information you need.\nTo learn about the difference between 20440 232nd Avenue and other properties around Lower Gar Lake, just call RE\/MAX Lakes Realty, home of the Iowa Great Lakes premier website, SellBoji.com. Our real estate team will be happy to help you with your real estate home search in and around Arnolds Park, IA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This online memorial is dedicated to Robert M. Siff. It is a place to celebrate his life with photos, stories, music and other tributes.\nThis memorial website was created in the memory of our loved one, Robert Siff. We will remember him forever.\nThis memorial has been viewed 136 times.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"JOINT VENTURE: If you have an IP litigation matter and need technical support or other assistance, our attorneys can joint venture the case with you. We often assist non-patent attorneys with infringement contentions, invalidity contentions, and claim construction (Markman) briefing\/hearings.\nHANDLE THE CASE: Our attorneys can handle IP litigation and transactional matters for your clients.\nGIVE YOU A COMPETITIVE EDGE: For solo practitioners and small firms, we can help you compete by allowing you to offer a broader range of quality services to your clients. We have partnered on significant IP litigation either by formally appearing as co-counsel or by assisting in the background. We are prepared to offer the assistance you need to represent your client in a successful manner.\nWe recognize the practice of law is a referral-based business. A significant portion of our IP business is based on referrals from non-IP practitioners.\nSee our Referral Commitments page for more information on how we handle referrals.\nFor more information about the local counsel services we provide, read our Local Counsel Services Overview as well as our Local Counsel Expectations for more detail on engaging us to be local counsel. The pro hac vice admission requirements for Texas federal courts along with pro hac vice applications can be found at Local Counsel Requirements - Northern District of Texas, Local Counsel Requirements - Eastern District of Texas, Local Counsel Requirements - Western District of Texas, and Local Counsel Requirements - Southern District of Texas.\nOur Intellectual Property Trends blog discusses the latest developments in intellectual property, technology, social media, and Internet law. Our Legal Insights blog provides deeper analysis of legal developments in the area of intellectual property, litigation, and transactional law and includes articles, white papers, presentations, podcasts, videos, and infographics.\nAnheuser-Busch had filed a SLAPP lawsuit against former employee for allegedly releasing trade secret information that was eventually used against Anheuser-Busch in a class action lawsuit.\n2018 was record-setting year for the World Intellectual Property Organization (\"WIPO\") as number of Uniform Domain-Name Dispute-Resolution Policy (\"UDRP\") filings hit a new high.\nAt the start of a business, all companies should consider trademarking their company name or brand and having trademark strategies to help save money and help prevent potential expensive conflicts in the future.\nFor online business owners, obtaining a great domain name is essential. Learn about domain name acquisition, trademarks and strategy.\nWhat Services Can Intellectual Property Lawyers Assist Me With?\nIntellectual property lawyers have a wide variety of expertise to draw from when it comes to giving legal advice, and the general public often does not understand the full scope of issues that intellectual property lawyers handle.\nWith the rise of technology, intellectual property theft has become more rampant and widespread. Registering your IP may entitle you to more damages should it be infringed upon, and such protection gives presumptive nationwide notice to others.\nWith the explosion of social media and the relative ease at which anyone can share content at the press of a button, copyright lawyers have found that their services are needed more than ever.\nThis article provides tips for lawyers to implement systems to manage projects, stay on top of tasks, and delegate work to team members in the simplest, most effective method I have discovered. By Darin M. Klemchuk.\nArticle about trademark infringement litigation options and requirements. Authored by the TM law firm of Klemchuk LLP.\nAt Arizona State University, a patent lawyer set up a grant to help the Barrow Neurological Institute protect medical innovations and allows medical residents to create prototypes for medical devices in the neurology field.\nA software license agreement is a legal agreement outlining the rights and responsibilities between the owner of software (the \"licensor\") and a person or business wanting the right to access or use the software (the \"licensee\"). Get key provisions essential in a software license agreement.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The author's choice to zoom in on this moment, rather than to feature the fire, illustrates that what matters most from this period is Vivian's feeling of being alone and defenseless. This is the moment when she realizes how vulnerable she is because she is \"nobody's responsibility.\" This scene shows many parallels with Molly's situation; Mrs. Schatzman's complaint that she \"didn't ask\" to be responsible for the child parallels Dina's attitude of indignation at Molly's mistakes. Further, Vivian's feeling of being unsafe parallels Molly's feeling that she isn't secure in her home or relationships.\nAt the Children's Aid Society, an unhappy middle-aged woman grooms and lectures the narrator. She warns her that people will judge her harshly because she is Irish. So that she will be accepted, the matron advises her to keep her red hair pinned back and to behave very well. She also reminds her that some women won't like having a girl in the house who is \"too comely.\" She and another matron try to tell her that she can't bring her claddagh cross necklace onto \"the train\" because the children \"aren't allowed keepsakes.\" The narrator tells them it is all she has left of her family, both because it's true and because she \"thinks it will sway them,\" which it does.\nIt becomes clear that young Vivian is going to be put up for adoption and sent on some kind of train. Despite the matron's unhappy demeanor and the prejudice behind her comments, it appears she is trying to help Vivian improve her chances of finding a home. This moment illustrates how cultural prejudice further isolates Vivian \u2013 not only is she severed from her family, but from her surrounding culture as well. The rule against keepsakes illustrates how the children's origins are erased to become adoptable.\nThe narrator is huddled onto a train platform along with dozens of other children. She wears a stiff pinafore that, along with her small suitcase of clothing, was given to her by the Children's Aid Society. The volunteer who sewed her name onto her clothes scoffed at its foreign lettering. The narrator then reveals her name: Niamh (pronounced \"Neev\") Power. The volunteer disdainfully told Niamh that her new family would surely change her name. Niamh explains that nobody \"feels sorry\" for her, because all the other children around her have also lost their families. The Children's Aid workers focus on preparing them to begin their new lives, treating them as if they were \"born the moment\" they were placed in their care. As they board the train, the matron in charge, Mrs. Scatcherd, charges Niamh with taking care of a fourteen-month-old baby, Carmine.\nIt is now revealed that Vivian was born with a different name: Niamh. The volunteer's disdain for Niamh's name further supports the idea that Niamh's culture won't be welcomed in her new life. Surrounded by other children in the same predicament and forced to focus on the immediate problem of finding a home, Niamh has no space to process her loss\u2014she must repress her sadness and focus on survival. The attitude of the Children's Aid Society denies the preexisting identities and experiences of the children, forcing them to repress and hide their origins.\nOn the train, Mrs. Scatcherd makes a speech. She opens by saying, \"They call this an orphan train.\" She explains that they are leaving depraved backgrounds for new families and the \"nobility of the country.\" She warns them to behave well, or else they will be sent back to the city. Niamh is settled with Carmine, who cries from hunger. She feeds him sugar lumps that she pocketed from her meager breakfast, and he falls asleep in her lap. Mr. Curran, the other attendant, provides them each with lunch and blankets. Niamh wants to feel optimistic, but she knows promises don't always \"match with reality.\" She longingly remembers her home and family in Kinvara. Caring for Carmine both comforts and pains her, reminding her of Maisie. She thinks of other, larger immigrant families in her New York tenement, and wonders if her parents would have fared better with more social support.\nMrs. Scatcherd's speech reveals her stereotyped view of the poor and immigrant families of New York, while also giving the novel its title and offering the central motif of the book\u2014the train carrying unwanted children from place to place. Caring for a baby is familiar and soothing to Niamh, and distracts her from her loneliness and fears. Though she wants to feel hopeful, just as she did when her family first immigrated, her disappointments and losses are beginning to teach her not to believe in promises. In this way, her wariness resembles Molly's. Her thoughts about her parents' isolation reflect her awareness that humans need each other and of how her isolation puts her at risk.\nBallinger-Dix, Elizabeth. \"Orphan Train Chapter 4: New York City, 1929.\" LitCharts. LitCharts LLC, 13 May 2017. Web. 24 Apr 2019.\nBallinger-Dix, Elizabeth. \"Orphan Train Chapter 4: New York City, 1929.\" LitCharts LLC, May 13, 2017. Retrieved April 24, 2019. https:\/\/www.litcharts.com\/lit\/orphan-train\/chapter-4-new-york-city-1929.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Make sure to stream album here.\nAhead of his debut EP, HMG artist and Houston native Soody Soo has released a radio-ready single titled \"187 Flow.\" The breakout hit is reminiscent of \"187 proof,\" by West Coast legend Spice1. The track has been receiving a solid rollout thus far, including a co-sign from the preeminent DJ Michael \"5000\" Watts! \"187 Flow,\" is the epitome of what happens when you combine lyrical prowess with a few hard-hitting punchlines. Soody Soo is definitely aiming this one at the streets. Take a listen to the GNB-produced track, and add it to your favorite playlist.\nHis goal for the single is to keep their memory alive while also bringing awareness to the fight we have within ourselves. More than anything, Troy Remedy provides inspiration for those who are on the rise. \"This record represents that spirit to hold it down and maintain through the tough moments life presents us\" said Troy. After experiencing his own hardships throughout life, Remedy utilizes his musical skills to provoke the basis of his sound. This is a good reason why he's been able to stay consistent over time. It's about the artistry when it's all said and done.\nDJ HeadBussa's OFFiCiAL Single \"Supply & Demand\" feat. Mike Smiff (Slip -N- Slide), Tampa's Champ J-Creek, along with Tampa's own Xtra, and introducing Jacksonville's own CoolAssChelsea. Produced by Super Producer K Hendrix. The Single 'Supply & Demand\" is a Spinoff to Dj HeadBussa's EP; Supply & Demand which drops May 3rd 2019, in which is a Spinoff to Dj HeadBussa's Supply & Demand Mixtape Series\u2026 The Single is available on ALL digital Media outlets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"amigos ingleses viajes Dating guy quiz I've been dating this one guy for less than a month, and he's becoming increasingly clingy, to the point of irritation. Pisces is ruled by dreamy Neptune, which Everyone deserves to be in a safe and healthy relationship. Do you know if your relationship is healthy? Answer yes or no to the following questions to find out.\nLeitz stationery for up-to-date office management. Whether you work in an office or from home, your workplace will be set up modern and functional with Leitz dating questions to ask boy mp3 Dating guy quiz News Headlines | | WFTS | ABC Action News. Mar 8, 2015 Is Brad Pitt the man for you, or are you more of a James Franco kind of girl? Take this quiz to see which star beau is best suited for you.\ndating nordhorn zwembad youtube Dating guy quiz Avengers Boyfriend Quiz Quotev So, you can find yourself drawn to this guy you have got Love & Relationships big bang theory, Date Marry or Ditch, sitcoms.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"TLILIC 2002 License To Operate An Order Picking Forklift Truck.\nTo provide trainees with the knowledge and skills required to demonstrate competencies in accordance with the Australia Safety and Compensation Council: National Standards for Licensing Persons Performing High Risk Work (April 2006) and Licensing Unit of Competency TLILIC 2002 License To Operate An Order Picking Forklift Truck.\nTo assess practical and theoretical competencies for the operation of a Order Picking Forklift for which High Risk Licensing is required.\nLegislation, codes of practice; Manual handling, Prevention of falls at the workplace.\nFamiliarisation of an Order Picking Forklift and understanding data plates, warning notices and the use of attachments.\nPre-operational checks and post operational checks.\nSafe Order Picking Forklift operation to, travel loaded or unloaded.\nConsequences of unsafe operating procedures covering stability characteristics, driving and driving on inclines.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Large-Scale Invasion 8 (\u5927\u898f\u6a21\u4fb5\u653b\u2467 Daikibo Shink\u014d 8?) is chapter 58 of the World Trigger manga.\nOsamu, Chika, Ky\u014dsuke Karasuma, and the other C-Rank trainees flee the battle with Hyuse and Viza to rendezvous with Jin and Y\u016bma (unbeknownst to the former). Meanwhile, Midorikawa, Yoneya, and Izumi prepare to attack Rambanein.\nJin clarifies to Y\u016bma and Arashiyama that Osamu's death is not certain to happen. He explains that he receive permission from Kido to Y\u016bma to rendezvous with Karasuma and Osamu.\nThe battle between Tamakoma First and the two humanoid Neighbors, Viza and Hyuse continues. Konami finds that Viza's Trigger is dangerous when attacking him. With Karasuma and Kizaki, they use tactics to get Hyuse to thin his defense and then attack him, with Kizaki being able to punch him. He then proceeds to attack Chika, and then physically attract her to him with some invisible force. Kizaki then deduces that Hyuse' Trigger used magnetism, which could be used to drag others to him, or repel and weaken the attacks of foes. In response to the situation, Karasuma decides to take the C-Rank agents with him to the base with Osamu.\nThe B-Rank teams fighting Rambanein continue to face difficulty. He decides to leave and go help Hyuse and Viza. Watching from a building, Midorikawa asks Yoneya what they should do. Izumi responds that they should attack him then and there, to which Yoneya agrees to.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"That's right \u2013 Dalen went to the slammer and raised money to help kids and adults break free from the harmful effects of muscle debilitating diseases.\nDid you know that muscular dystrophy, ALS and related disease take away everyday freedoms like walking, running, hugging and even breathing? That's why I'm dedicating my time behind bars to help MDA fund ground-breaking research across diseases, provide critical care from the first day of diagnosis and empower families with services and support in hometowns across America.\nThank you to everyone who helped by making a donation! With your support, we'll move toward a future in which kids and adults with muscular dystrophy, ALS and related life-threatening diseases live longer and grow stronger.\nRemember when Brian Williams almost caught malaria while voting for Ulysses S. Grant? It's that Important!!\nTuesday June 28 is election day. Robert Frost said, \"Thinking isn't agreeing or disagreeing. That's voting.\" Don't just think about voting. Don't just talk about the problems we have. Educate yourself on the candidates, choose one that you believe will do the best job, and GO VOTE!\nJANELLE SHARP \u2013 Look her up on Facebook.\nSHARON VEACH \u2013 Look her up on Facebook.\nJERRY NILES \u2013 Look him up on Facebook.\nWe're Enid personal injury lawyers, probate lawyers, and divorce lawyers. We're also dads, moms, husbands, wives, coaches, and volunteers. Take pride in your right to vote. Post a pic of your \"I voted\" sticker and we may give some swag away!\nWe have all heard stories about people being asked to pay their ticket right there in the trooper's car, but it does not seem to have happened to anyone directly. Maybe that is because the highway patrol did not have card readers\u2026.until now. The Oklahoma Department of Public Safety has contracted with a business to allow troopers to transfer money from many types of pre-paid cards. Of course, this is not to pay your ticket, but is being done to as part of seizing assets thought to be used in drug trafficking.\nThe people from whom the money is taken haven not been convicted of a crime, have not pled guilty to a crime, heck, they may not ever be charged with a crime. Still, the trooper on the side of the road using his \"training and experience\" determined these funds are for nefarious purposes, and so the money is taken. No judge, no jury, no attorney, no due process, just one person's belief.\nWhen a person's assets are seized, the person has the obligation to prove their own assets were legitimate. The State does not have to prove the money bought or paid for drugs. The state does not have to do anything really other than file some paperwork formalities. Only after the State files the forfeiture paperwork, are they required to inform the person they intend to keep the money they already took. Then, the citizen has the obligation to come to court and prove why they should get their money back.\nNow, we have corporations getting up front fees, and ongoing royalties on money \"collected\" by the State. Oklahoma citizens are accustomed to royalty payments. We like our oil, gas, wind, and other natural resource royalty payments coming in the mail. However, paying a royalty to an out-of-state corporation who sells widgets to help steal from citizens should be pretty offensive to us.\nCertainly, seizing funds associated with drug trafficking may help stem the flow of drugs, although it doesn't seem to have slowed anything down. However, the incentive to fund an agency is so strong it often takes priority of citizens' rights.\nIf you've had money taken in Enid, Fairview, Woodward, or anywhere else in Northwest Oklahoma, remember we are Enid lawyers, with offices in Fairview, and Laverne, that can help.\nThe team at Ewbank, Hennigh & McVay could not be more proud of our fellow partner, Andrew Ewbank, for winning the Journal Record's Achievers Under 40 award! When we think of Andrew, we certainly think of his talent as a personal injury attorney and divorce lawyer. We also think how he fosters the utmost quality and character in his daily life. This is often demonstrated by his drive in helping build an incredible law practice here in Enid, Oklahoma as well as his desire to be better to those around him. You can see this from his past work of representing impoverished and abused children, his participation on numerous boards throughout his community, and his belief that people should always strive to learn and grow.\nCongratulations, Andrew. You are incredibly deserving of this award!\nEquifax Security Breach \u2013 Are You Affected?\nSmall Claims Is Where It's At!\n\u00a9 Copyright 2014. Designed By hollygannett Enid, Oklahoma.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I was only upgrading my airstrip, nothing else.\nAnyone having the same issue please report to Support so they could acknowledge and fix it.\nI had exactly the same thing happen to me 5 days ago. Just started a 12 day airstrip upgrade. Did a war attack and afterwards my retrain screen was blank except for the troop tactics and war tactics I didn't use.\ni had to retrain everything manually and so I have lost my planes for 12 days. It is so painful. It's just so annoying knowing this could have been avoided by always using \"retrain all\" if it hadn't been for this bug\/glitch.\nthe glich that is messing you is that the videos do not work good, and that the retrayning function can not distingish between which troops are aviable to retraun and which not.\nthere is another glich that left gray your ratrain botton and you cant ratrain never.\nPlease fix this, let the troop training works normally.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To my Listeners and readers, Please feel free to contact me for one on one advice. I'm very approachable. Everyone will get treated with respect.\nIf you'd like to plan a \"relationships\" round table discussion in your area, i.e. book clubs, women's circles, men uplifting men gatherings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A handyperson service aimed at helping over-60s who are struggle with completing small home repairs is now available in the Larne, Ballymena, and Carrickfergus areas.\nThe initiative is spearheaded by Mid and East Antrim Agewell Partnership (MEAAP). For a low cost subsidised rate starting from \u00a310 per hour (max of 3 hours) MEAAP's partner, Bryson Energy will send a trained and police-checked handyperson out to complete repair and maintenance jobs. The service does not include emergency repairs and all materials to be supplied by client where possible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dasada (SC) Assembly constituency (60) is located in Surendranagar district of Gujarat and comes under the Surendranagar Lok Sabha constituency. As per the estimates of 2011 census, out of total 305299 population 90.28% are rural and 9.72% are urban population. The Scheduled castes and Scheduled tribes ratios are 12.95 and 4.46 respectively out of total population. As per the voter list of 2017, there are 233778 electorates and 271 polling stations in this constituency. Voter turnout in the 2012 assembly election was 64.94% where as it was 56.44% in 2014 Lok Sabha Elections. The BJP and the INC got 49.16% and 41.16% votes respectively in 2012 while 54.93% and 34.35% in 2014 respectively. Fatepara Devajibhai Govindbhai (BJP) is the present Lok Sabha MP of Surendranagar and Makwana Punambhai Kalabhai (BJP) is the present MLA of Dasada (SC) Legislative Assembly. To know more about the key electoral data of Dasada (SC) Assembly Constituency from 1962 to 2017, check the content and sample view.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Attention on line book sellers \u2013 this mailer is perfect for you. The smallest size in our gusseted mailer line, this mailer is the perfect \"catch-all\" mailer for paperback books, especially children's books, up to 1\" thick Made from durable .020\" CCNB paperboard, it will stand up to the rigors of the mail.\nKeep documents from folding or bending with additional packaging or stiffeners. It's also made from 100% Recycled Paperboard, so it's environmentally friendly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Polish your online road signs - Wylie Communications, Inc.\nIf you saw a road sign that said, \"go here,\" \"drive more\" or \"road sign,\" would you follow it? Probably not.\nAnd if you saw a link that said, \"click here,\" \"read more\" or \"link,\" would you click it? Probably not.\nThink of your link as a road sign. Like a road sign, the best links are clear, complete and self-contained.\nLike a road sign, the best links are clear, complete and self-contained.\n1. Make them stand on their own.\nWhen your links are too short or incomplete, readers have to go back and read the surrounding words to understand the link. But readers should be able to understand your link without reading the rest of the text.\nSo write links that stand on their own. What would you say in the link if you knew it was all the customer would see?\nAfter all, you wouldn't click links these links without knowing more about them, would you?\nThe weakest links Write links that stand on their own. These links are so short, the reader needs to read more to understand them.\n2. Use the words in your readers' heads.\nNot the words in your head.\nWeb visitors found the information they were looking for 72% of the time when the words in their head \u2014 aka \"trigger words\" \u2014 appeared on the webpage, according to a study by User Interface Engineering.\nBut when those trigger words didn't appear on the webpage, visitors found what they were looking for only 6% of the time. That means the ROI on familiar words, at least in this study, was 1,200%.\nThis isn't the first time short, familiar words have trumped their opposites in research. High-frequency words (those that are used often and so are familiar) and short words are easier for readers to recognize and understand than unfamiliar or long words, according to classic research by linguist George Kingsley Zipf.\nChoose short words. They're easier to process and understand.\nSpell out acronyms, abbreviations and initialisms, suggests Jakob Nielsen, principal of the Nielsen Norman Group. This is helpful for all users, especially for those using screen readers. Exceptions: Abbreviations like DVD that have become widely used words.\n3. Front-load the topic word.\nWeb visitors usually read just the first two words of a link, according to research by the Nielsen Norman Group.\nThat means the best links start with the most important words, writes Marieke McCloskey, a user experience specialist with Nielsen Norman Group.\nSo front-load your links with the topic word.\nInstead of Read the new issue of Rev Up Readership, for instance, McCloskey would have you write, Read the new issue of Rev Up Readership.\nAnd if product or division named include the company name, drop it, Nielsen suggests. Instead of FedEx Express, FedEx Ground, FedEx Home Delivery, for instance, write Express, Ground and Home Delivery.\n4. Place links at the ends of sentences.\nThat's less disruptive to reading, writes Jan H. Spyridakis, professor at the University of Washington College of Engineering.\n5. Don't repeat link text.\nWhen users see the same link text twice on the same page, McCloskey reports, they assume that both go to the same place. So if the second link refers to a different page, write a unique link.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's time for our semi-annual look at what our online viewers are reading. The results are always interesting and instructive for us at The Western Producer.\nOur 10th most read story this year is \"Survey reveals 'amazing' soil loss in Great Plains region.\" Published in January, this story pointed out that large amounts of nitrogen and phosphate had blown or washed away over the last 75 to 125 years.\nNinth is \"Genesis plans huge fertilizer supercentre west of Regina,\" a story also from January, about plans by Saskatchewan-based Genesis Grain to build a $24-million fertilizer storage, blending and distribution centre at Belle Plaine, Sask.\nIn eighth place is \"It might be a good time to stock up on glyphosate,\" a story in February on increasing prices due to higher costs for raw materials in China.\nSeventh is \"Excessive glyphosate levels found in 1.3 percent of food samples: CFIA study,\" published in April. The study found that four percent of grain products had glyphosate levels exceeding maximum residue limits, though the food system was deemed by the Canadian Food Inspection Agency to be safe.\n\"Harmony Beef to reopen,\" from February, holds sixth spot. It is the country's newest federally inspected meat plant, which is located northeast of Calgary.\nIn fifth place is \"Glyphosate presence in honey raises concerns,\" which outlined a lawsuit launched by the U.S. Organic Consumers Association and Beyond Pesticide. The group argued that use of the words \"pure\" and \"natural\" by Sue Bee Honey of Sioux City, Iowa, was misleading.\nFourth is \"Grain bag slashing angers farmer,\" explaining that there had been seven such incidents around Rouleau, Sask., in January.\nOur third highest-read story this year is \"Farmers overcharged nearly $100 million in CGC user fees.\" The commission said it was reviewing its user fees.\n\"Drought predicted for Alberta this summer\" is our second-highest read story. Published in May, it quoted AccuWeather's outlook on central and northern Alberta, in which hot, dry conditions were expected. So far, farmers have been spared.\nAnd our top-read online story so far this year is \"Sask. budget: PST, land, education farm fuel taxes increased,\" explaining the effects of the Saskatchewan austerity budget on farmers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Advent and Christmas season is my favourite time of the year. Because it often transports me back to my childhood. The perfect mood is when biscuits and hot tea spice the air, glittering lights adorn the houses and streets and the ethereal sounds of Christmas are to be heard everywhere.\nThen it's time for me to head off for a Christmas visit to San Candido in the Dolomites. From late November until early January, the village centre is jam-packed with festive stands offering local arts and crafts and traditional specialities. Experience the ancient traditions and everyday customs here in this magical atmosphere. Enjoy the festive season with your loved ones and relive your memories of childhood.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The purpose of the State Route 8 Corridor reconstruction was to create a non-stop route from Akron, OH to Interstate 271. The corridor's interchange reconstruction replaced difficult merges and exits off the highway with longer ramps that meet today's design standards and use. As prime contractor for this project, totaling over $91 million \u2013 the largest single project at the time in ODOT District 4 history \u2013 Beaver oversaw the work of 28 subcontractors who, along with Beaver's own forces, were integral to the successful completion of the large and complex project.\nThe bulk of Beaver's workload during the three-and-a-half year project included, demolition, maintenance of traffic, drainage, noise walls, piling, construction of 12 bridges and an intersection widening. In the first year of the project, Beaver completed 750,000 cubic yards (cy) of embankment, constructed six Mechanically Stabilized Earth (MSE) walls, completed initial widening work on S.R. 8 and I-271, and improved Highland Road, which was previously a traffic bottleneck adjacent to the corridor.\nAs with every project, Beaver set the bar high for the corridor transformation by helping the project owner save both time and money. Beaver saved ODOT $1.2 million through value engineering, and shaved nine months off the project's timeline by proposing and implementing three Maintenance of Traffic (MOT) changes, minimizing impact on the traveling public.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Q: I have been with my partner now for 14 and a half years. The positive has always outweighed any negative \u2013 and we have definitely had our share of trying circumstances thrown at us, but we've persevered. However, he has a grandmother who lives overseas in China, who he has visited every year since he was born and who visits him here too. He has never told her that he is gay, but his brother has certainly less-than-subtly hinted about it and the woman isn't stupid. She knows. She simply will not accept it, and I'm fine with that.\nMy issue is that she is getting bolder with wanting him to be straight. We visited her this past February and she set him up on a chaperoned dinner date with a woman for the night of his birthday. I was told that I would not be attending the meal. I had a problem with this, but he didn't because, he said, it'd never go far and it'd make grandma happy. Regardless, I was still very unhappy that I was not included.\nWe returned home and this woman has continued to call him several times a week (he currently works out of state due to the economy and our jobs), has sent him gifts (expensive gifts \u2013 things I certainly can't afford), has joined him and his grandmother in London for a weekend and will be joining them again in a few weeks for a trip to Singapore. He has told her that they will not be anything more than friends \u2013 he has forwarded me the e-mails and her responses \u2013 yet he continues to accept the gifts she sends or has waiting for him overseas. She does not know that he is gay.\nGrandma is currently visiting and spent the week out of state with him. He called earlier this evening before his flight to tell me that she did inquire about me during the weekend and wondered if I knew he had a girlfriend. He corrected her in saying that he did not, yet also confirmed that I knew about the woman she set him up with and that I was hoping to meet someone like her who bought me expensive things, too. This was, as he told me, to make grandma happy.\nExplaining my unhappiness with this situation to him, emotionally or through an attempt at logic, always ends with his conclusion that the problem is entirely in my head; and that I shouldn't have any complaints because nothing will happen with this woman. Also, he will never tell his grandmother that he is gay, plain and simple.\nI don't know where to begin detailing all the problems I have with this \u2013 any of it. I was quite depressed, though, when we returned from overseas earlier this year, and I am concerned that it could happen again. So, for the sake of my own sanity, I'm wondering if you also see any red flags with any of this, or is it really only in my head?\nA: Yes, red flags are flying, high and brightly. I certainly can understand that your partner is dealing with cultural issues \u2013 as well as family and personal issues \u2013 that are very difficult for him, but that doesn't take away what he's doing to you in the process. From what you have said, his problems are deep and many. He's not writing to me, but if he were, I'd be suggesting counseling for him, and couple's counseling for you both. But as I said, he's not writing to me, you are.\nIf I were you, I'd start out by insisting to your partner that you both go into couple's counseling. If he won't agree to that, it is important for you to find your own personal counselor to help you deal with this situation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If your Samsung Galaxy Tab S 8.4 (SM-T700) has a broken screen Mountain Stream are here to repair your tablet for you and return it to you promptly.\nPlease see below for our fixed prices for repairs. Prices are inclusive of parts and return standard postage or personal delivery. For other Samsung Galaxy Tab S 8.4 repairs please request a quote by email, phone or by using our enquiry form.\nALL PRICES BELOW ARE INCLUSIVE OF PARTS, LABOUR AND RETURN OF YOUR SAMSUNG TAB S 8.4 IN PERSON OR BY POST. PRICES DO NOT INCLUDE ANY PRE-REPAIR DIAGNOSIS.\nFOR A PRICE ON REPLACING ANY OTHER SAMSUNG GALAXY TAB S 8.4 COMPONENT PART PLEASE CALL, EMAIL OR USE OUR ENQUIRY FORM.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Instead of doing a generic monthly reading wrap up and to be read post, I thought I would tie the two posts together and make a series of it called 'Those Reading Updates', bringing everything bookish in to one post (wrap ups, tbrs, hauls, reviews, what ever I feel like updating you on, etc). I hope you like the idea of this post as I think it'll be nice to just have everything in one place, as well as making these posts short and snappy (I say this as I ramble on).\nI read this book once before but never continued on with the series. As I have heard so much hype to this book series I thought I would go out and pick the book up again and I'm definitely glad I did. I enjoyed the journey through this book so much more and it was so interesting learning about the shadow hunter world, it seems like a pretty rad world to be a part of. I've already bought the second book (which I can read but failed to pick up the third book in the past) so I'm hoping this book is as good as the first. I hope to read that book soon. I gave City of Bones 5 STARS.\nI picked this book up purely for the fact that EVERYONE raves about it and the film comes out this month and if you haven't seen the trailer yet, you need to. If it doesn't make you want to read the book, you'll still want to see the film (I mean Sam Claflin and Emilia Clarke are both leads and they look fab). Honestly, this book killed me. It was so heart wrenching and heart warming that I just couldn't put it down. It has almost every emotion thrown in their at once, love, humour and sadness, everything you need in a good contemporary. I won't lie, I did have a few tears fall down my cheek at different points of this book but it was so worth it and you know it's a good book if it can get you crying. I gave Me Before You 5 STARS.\nDuring the month of May I have hauled a few books, bought two as well as borrowing two from the library, one of which I'm currently reading (and loving). As I mentioned before, I bought the second book to the Mortal Instrument series, that book being City of Ashes by Cassandra Clare. Like I also said before I'm excited to get to this book. The second book I purchased recently was The Winners Curse by Maria Rutkoski. This is one of the books I plan to read on my 9hr flight which is coming up pretty soon and I have heard many great reviews about it so it encouraged me to pick it up.\nThe two books I borrowed from the library are Trial by Fire by Josephine Angelini (picked up purely because it's based on witches and I am a sucker for a book about witches) and A Court of Thorns and Roses by Sarah J. Maas (Which I'm currently reading at the moment and loving).\nI want to read Trial by Fire by Josephine Angelini as I have had it out almost 6 weeks from the library and I feel I should probably get to this book pretty soon. I also would like to start reading Heir of Fire by Sarah J. Maas so I can catch up in the series as the 5th book comes out this September and I would like to read it when it's been released.\nI hope you enjoyed this post, I enjoyed writing and I definitely think it will be a post series I'll be uploading more often as I love to share my thoughts on book as it's one of my favourite hobbies.\nWhat're you currently reading? Or what book would you like to get to soon?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to Feejoa, New Zealand's first online site rewarding loyalty with repayments to your student loan. These terms apply to your use of Feejoa Benefits, to ensure you understand Feejoa and ensure that Feejoa provides a safe service to help repay your student loan.\n1.1 Acceptance of terms and conditions: These terms and conditions (Terms) you need to agree to in order to use the website www.feejoa.co.nz (Website), use the benefits provided by Feejoa and become a member of Feejoa. By registering as a member of Feejoa and clicking 'I agree' on the registration page, you accept these Terms. If you do not accept these Terms you will not be able to access any of the services provided by Feejoa and you will not be able to be a member of Feejoa.\n1.2 Amendment of terms and conditions: As Feejoa operates in a changing commercial, technological and regulatory environment, we may amend these terms and conditions from time to time. Unless otherwise stated, amendments will be effective immediately upon posting of the amended terms and conditions on this Website. You are responsible for ensuring you are familiar with the latest Terms. If you do not terminate your membership with Feejoa, you will be bound by the updated Terms.\nThese terms and conditions were last updated 17 September 2014.\n(ii) the convenience of one set of personal details that can be managed by the individual through which aggregated repayments can be made to the nominated student loan account.\nFeejoa dollars means the dollars that will be repaid to the members Student Loan.\nMerchant means retailer or service provider who has a relationship with Feejoa.\n2.1 The Services: Feejoa provides a venue to introduce Merchants who want to reward member's loyalty with repayments to their student loans. Feejoa does not take any part in the sale of items other than by providing our Website as a gateway for Merchants, the Inland Revenue Department and members. We do not act as agent for either party and do not participate in any sale or transaction between you and Merchants.\nthat any Product or Service will meet your requirements or expectations.\nFeejoa is not responsible for the acts or omissions of any Merchant and, to the extent permitted under applicable law you hereby release Feejoa and Feejoa Representative from any and all claims or liability related to or arising from the acts or omissions of any Merchant.\n2.4 Liability: You agree that, to the maximum extent permitted by law, any and all liability and responsibility of Feejoa to you or any other person under or in connection with these terms and conditions, or in connection with the services, this Website, another Member's acts or omissions, or your use of or inability to use, the services or this Website, is excluded regardless of whether such liability arises in contract, tort (including negligence), equity, breach of statutory duty or otherwise. Feejoa's liability and responsibility is excluded in respect of any and all loss or damage, whether direct or indirect, including, without limitation, loss of profits, loss of data, loss of business or anticipated savings, general and special damages, and consequential and incidental loss.\n2.6 Breach: Without limiting any other rights and remedies available to Feejoa, Feejoa may limit your activities on the Website, suspend or terminate your membership, if you breach these terms and conditions or where Feejoa considers it appropriate.\n3.1 Becoming a Member: Becoming a member is free. You may only register as a member if you have a student loan or are offering to be a benefactor to a member who has a student loan (unless specifically authorised by us in writing) and can form a legally binding contract that is enforceable against you. You must be at least 18 years old. By registering as a member, you warrant that you can form a legally binding contract.\n3.2 Accurate Information: You warrant that you have provided complete, accurate and current personal information when registering as a member. You must maintain and update your personal information held by Feejoa to ensure it is kept current at all times. Feejoa may phone or mail you to verify these details. You must not register as a member under multiple identities or personas (whether false or not).\n3.3 Termination of membership: Feejoa reserves the right to decline to register or to terminate your membership without entering into further discussions with you. Without limiting the foregoing, Feejoa may terminate your membership if a serious complaint or multiple complaints are received about you from Merchants or the Inland Revenue Department, if you breach these terms and conditions, if the Inland Revenue Department informs Feejoa that you do not have a student loan, or if we, at our sole discretion, deem your behaviour to be unacceptable. Members who have registered under multiple aliases will have all of their aliases disabled. In the event of termination, you also agree that you will not open or attempt to open any new Feejoa memberships.\nYou must terminate your Feejoa membership if your student loan is repaid. You may at any time close your Feejoa membership by giving us three days' notice in writing. If you close your membership, or if it is terminated by Feejoa, your ability to use the Feejoa website will end. To the extent that you continue to access the site, clauses 2.2-2.5, 4.1 and 6.8 of these terms continue to apply.\n3.5 Emails and newsletters: Feejoa will send you emails relating to your membership, transactions and other activities on the Website, and for promoting and marketing other Feejoa products and Services to you. Feejoa may also send regular electronic newsletters to members. Newsletters will contain clear and obvious instructions for how you can unsubscribe from the mailing list.\nAll transactions made through Feejoa shall be made in good faith.\nYou must not damage, interfere with or harm the Website, or any network, or system underlying or connected to them, or attempt to do so.\nYou are responsible for any actions taken through your membership. Feejoa memberships are not transferable and therefore cannot be sold, leased, lent or traded without specific written authorisation from Feejoa.\nFeejoa charges 5.75% service fee for the use of the Website as a gateway which will be deducted from your account on a monthly basis prior to payment to the Inland Revenue.\nFeejoa reserves the right to change the fees charged for any of our Services at any time. In the event that we introduce a new Service, the fees for that Service are effective at the launch of the Service, unless otherwise stated.\nIf you are a New Zealand resident, then all fees are inclusive of GST and in New Zealand dollars unless otherwise stated. If you are not a New Zealand resident, then all fees are, unless otherwise stated, in New Zealand dollars and do not include GST. Unless otherwise stated in these terms and conditions, no interest or other financial charges will be applied to Feejoa account balances.\nFeejoa will maintain an amount that exceeds the aggregate face value of members' Feejoa accounts (less any amounts that have been cancelled, redeemed, or forfeited in accordance with these terms and conditions) in a separate bank account. Feejoa acknowledges and declares that all amounts deposited in that separate bank account are held on trust for members, such that each member has the right to a proportion of the separate bank account equal to the face value of his or her Feejoa account. Feejoa may deduct amounts from the separate bank account as contemplated by these terms and conditions, and reserves the right to retain any interest earned on amounts deposited in the separate bank account.\nthis statement: \"The use of this account (including the conditions relating to closing the account and receiving a refund of unused deposits) is governed by Feejoa's Terms and Conditions.\n6.1 System Integrity: Feejoa will use its reasonable endeavours to ensure the availability of the Website and Services, subject to any downtime required for maintenance. However, Feejoa takes no responsibility for any system unavailability, or for any loss that is incurred as a result of Website or Services being unavailable. Further, Feejoa assumes no responsibility for the corruption of any data or information held by Feejoa.\n6.2 Resolution of disputes: As Feejoa is not a party to any sale or transaction between you and a Merchant, you agree not to involve, or attempt to involve, Feejoa in any dispute or in the resolution of disputes that arise between you and a Merchant as a result of the Services provided.\n6.3 Force Majeure: Without prejudice to clause 2.3(a), Feejoa has no liability for any lack of performance, unavailability or failure of the Services or the Website, or for any failure of Feejoa to comply with these terms and conditions where the same arises from any cause reasonably beyond the control of Feejoa.\n6.4 No Waiver: If we do not exercise or enforce any right available to us under these terms and conditions, it does not constitute a waiver of those rights.\n6.5 Partial Invalidity: If any provision of these terms and conditions becomes or is held to be invalid, unenforceable or illegal for any reason, and in any respect, that provision shall be severed from the remaining terms and conditions, which shall continue in full force and effect.\n6.6 Governing Law: These terms and conditions are governed by the laws of New Zealand. You submit to the non-exclusive jurisdiction of the Courts of New Zealand.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With more than 30 years of experience in curtain industry and 15 years of experience in the theatre and entertainment industry, Imported Theatre Fabric is the family owned and operated business in Australia.\nImported Theatre Fabric offers a wide range of fabrics that includes wool fabric, velvet fabric, IFR wool fabric, Chromakey fabric, Molton fabric and many more. Our dedicated team of skilled and experienced mechanist can manufacture stage drapes and theatre drapes for venues of all sizes. Ranging from stage curtains for a small school play to the theatre curtains to fit the large auditorium, we have it all. We manufacture highly customized stage curtains to achieve the exact scenic backdrops that you've desired of.\nWe at Imported Theatre Fabrics supplies a wide range of theatre supplies that include shot bags, lighting bars and shock cord hooks and blackout curtains. Our high quality Blackout curtains also known as masking or acoustic curtains are able to block 100% light and almost 40% or more noise reduction.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbiued b/data_all_eng_slimpj/shuffled/split2/finalzzzbiued
new file mode 100644
index 0000000000000000000000000000000000000000..846a1254755e8f913a070316f2599e17b53a8bd0
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Changes to the interface are always backward compatible. You need not to update your applications after any change. But there may be new recommendations or clarifications (indicated by bold text) on how to use OKAPI methods. Changes referring to OCDE or OCPL only affect one of both Opencaching code branches.\nAdded the has_ratings field to services\/apisrv\/installation, and removed the services\/caches\/capabilities method (why?). This method was not in use yet by any application.\nFixed: Changed the <time> field of geocaches in GPX files from listing date to hide\/event date, consistent with GC and native OC GPX.\nAdded options to geocache and log retrieval methods, which allow to choose the subfields of owner and user fields.\nThe geocaches' alt_wpts field now includes OCPL hidden-coordinate waypoints. These waypoints are returned with location \"0|0\". Developers are recommended to hide those null coordinates or show a placeholder like \"---\".\nFixed: Geocaches' alt_wpts field now better responds to the langpref option.\nAdded new fields country2 and region to services\/caches\/geocache. They replace the country and state fields, which have been deprecated due to some issues (see the method docs).\nAlso added formal (not geographical!) verification of country and region data. For some caches with broken data, an empty country and\/or region will now be returned.\nRewritten the explanations and advice on differences between Opencaching sites.\nAdded the services\/caches\/capabilities method, and updated the cache types documentation.\nAdded the services\/logs\/capabilities method. Also added the new fields rcmds_left and rcmd_founds_needed to services\/users\/user(s).\nAdded the GPX options trackables=gc:travelbugs and alt_wpts=desc:table.\nAdded new fields to replication: short_descriptions, oc_team_annotation, needs_maintenance and watchers for geocaches; images, location, oc_team_entry, needs_maintenance2, listing_is_outdated, date_created and last_modified for log entries.\nAdded the oc_team_annotation option and field to services\/caches\/geocache(s). OC team annotations now are included by default in cache descriptions.\nAdded the search options recommended_only and recommended_by.\nAdded the watchers, my_rating and is_recommended fields to services\/caches\/geocache(s).\nAdded the services\/logs\/edit and services\/logs\/delete methods.\nFixed: Also OCDE owners are not allowed to not recommend their own caches.\nAdded the date_created and last_modified fields to services\/logs\/entry.\nThe services\/users methods will no longer return true for the is_admin field in any case, because OC installations no longer provide a singular admin flag.\nFixed: Do not mangle protocol-relative links in cache descriptions and logs.\nEnabled the GPX gc_ocde:attrs option for all OC installations. The pseudo-GS IDs of OC specific attributes now are also requestable via the services\/attrs methods.\nAdded French translation for OKAPI user messages.\nservices\/logs\/submit now rejects 'Attended' logs that are earlier than the event date.\nAdded the user_logs_only option the geocache retrieval methods, and corresponding latest_logs=user option to the GPX download methods.\nAdded fields parameter to services\/logs\/userlogs.\nAdded the Opencaching GPX extension. Use ns_oc=true in services\/caches\/formatters\/gpx (or \/ggz) to enable it.\nOKAPI can now be deployed via PHP composer to OC sites. See the OCDE repository for a sample implementation.\nservices\/logs\/images\/add now returns the URL of the added image.\nAdded services\/caches\/edit for changing geocaches' log passwords.\nAdded location field to services\/logs\/entry.\nFixed: Submitting OCPL event cache logs now exactly behaves like the native log functions.\nAdded options date_created and date_hidden to the services\/caches\/search order_by parameter.\nThe scheme for the installation URLs of several Opencaching sites has changed from http to https.\nAdded advice on the length of saved personal notes.\nDocumentation tags OCPL and OCDE have been added, informing about different behaviour of OKAPI methods on the two Opencaching code branches.\nFound geocaches are now marked properly in services\/caches\/formatters\/garmin method.\nAdded the log_uuids return field to services\/logs\/submit.\nSome OKAPI installation can be accessed through multiple URLs. The new services\/apisrv\/installation field okapi_base_urls tells you which URLs are available for this installation.\nThe is_admin flag for users has been deprecated.\nImproved submitting HTML log entries. The comment_format parameter of services\/logs\/submit now works consistent and deterministic on all OC sites.\nNew methods for adding, editing and deleting log images.\nConsumer keys may be revoked by OC site admins.\nFixed: services\/oauth\/authorize will no longer delete Opencaching.DE cookies when called with parameter interactivity=confirm_user.\nOCDE geocache log entries with and without time are more reasonably ordered. Developers are recommended to keep the order as returned by OKAPI instead of ordering by the log date.\nThe registration date of users can be retrieved.\nAdded the 'needs maintenance' flag to log entries and geocaches. The needs_maintenance option of services\/logs\/submit has been deprecated; use needs_maintenance2 instead.\nFor caches with the mark 'listing is outdated' (available only on some OC sites), a corresponding message is prepended to the description field. DO NOT rely on this message being there or having a certain content. It may change in the future.\nThe home location of users can be queried, if given in the user's profile (requires OAuth level 3).\nThe URL of the OC site's registration page can be retrieved via apisrv\/installation.\nFixed: Restored the OpenCaching.com namespace definition for GPX.\nAdded an advice to the documentation on how to submit log entries without time.\nAllowed to search for multiple users that (not) found a geocache.\nImproved posting plaintext logs on OCPL sites.\nEnabled full 21-bit Unicode for submitting logs on some OC sites.\nFixed: Solved rare problem when searching by distance or bbox (OKAPI crashed).\nAdded the new ignored_status search option which allows to search for caches ignored by the user. This replaces the exclude_ignored option, which has been deprecated.\nThere is a new Opencaching.FR website, which is a local view of Opencaching.DE. Use www.opencaching.de\/okapi to access it.\nQuicker refreshing of map tiles after geocache name changes.\nFixed: Return an HTTP 400 when \"negative\" dates are passed to services\/logs\/submit.\nFixed: Improved uuid creation for submitted logs.\nservices\/caches\/map\/tile now is available to chosen consumers (closed BETA).\nFixed: Pipe chars (|) in user names are treated as errors.\nThe user's last-login date on the OC site is updated when running OAuth level 3 methods.\nThe OKAPI code repository has moved from Google Code to Github.\nOpencaching.org.UK has been removed from the list of OKAPI installations, because it is too outdated.\nThe new powertrail_only option allows to search for caches that are part of a powertrail (on some OC sites). This is a BETA feature, i.e. it may be changed or removed.\nUpdated the Introduction page, with additional advices on backward compatibility.\nInterprete intervals of services\/caches\/search\/bbox as greater-side-open instead of closed.\nFixed: Added missing oauth_token parameter to error redirect.\nAdded \"stealth required\" attribute (A74).\nFixed: Corrected the attribute assignments of OC.NL 'lost place' (A30 -> A29) and geodetic point (A2).\nOKAPI has been deployed to Opencaching.RO.\nThis list shows only changes that are considered to be relevant for developers and site admins. Please consult the Git log for a complete history, including older changes. There also is a (discontinued) OKAPI Blog, that documents changes from 2012 to 2015.\nOKAPI was started in August 2011 at the OCPL code branch, and it was deployed to the OCDE branch in April 2013.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So far, 2017 has been a whirlwind of ups and downs for Boeing. The year began with a much-publicized vote that saw Southern workers vote to turn down the opportunity to unionize. Now, the company has announced plans to layoff about 1,800 workers. The company says these cuts have nothing to do with the anti-union vote. They announced there would be cuts way back in December. That, and the fact that the cuts are literally taking place on the opposite side of the country.\nThe announced job cuts will mostly affect workers in Washington State. The union up there said about 1,500 union members will accept buyout options. Another union said 300 members will take the buyout. And those are only the numbers we know. Boeing may be cutting many more jobs, but they're not saying how many non-union workers accepted offered buyout deals.\nEven still, a few thousand workers is a fraction of the company's roughly 150,000 workers. The biggest cuts are coming in the commercial aircraft unit. Nearly all of the current cuts will be in that division, a segment that already cut nine percent of its workforce in 2016.\nAccording to the company, the cuts are not coming as a result of any general market turmoil or the up and down relationship the company seems to have with the current Presidential administration. Company leadership is blaming the need for cuts solely on regressing jet sales. And more cuts are coming.\nThe PR issue here is in the string of bad and good headlines. First, it was a tiff with Trump that had Boeing in the headlines. Then the company and the POTUS seemed to make nice. Then Boeing was in the headlines again thanks to the contentious union battle in South Carolina. Now, it's cuts, cuts, and more cuts to come.\nCan the company successfully sell the message that this is just about market dynamics and not poor competition overseas? Boeing sees strong competition in Europe and Asia, and there's some noise that outside companies will attempt to gain more of a foothold on the American marketplace.\nCompetition is good because it keeps you sharp, but when there's too much competition, it can create detrimental market conditions. It sure doesn't help when the chief executive of the country is telling you to lower your prices. Still, there's a lot of 2017 left, and Boeing could pull out of this dive well before Q4.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We love Evernote\u00ae. We love how Evernote's making every note portable and accessible. We make more notes because of Evernote\u00ae, more than we probably should. But all the notes look plain. How many times do you yell \"Gosh! I wish Evernote\u00ae could have more editing features!\" This time we take up the challenge and respond to that popular request. We present you EleEditor, a mobile app that complements whatever Evernote\u00ae does with more.\nWith EleEditor, you can grab the elements from Evernote to the scrapbook or collage the elements and customize the notes with your own layout design, image sizes, text styles, drawings, and annotations. Yes, you heard it right- we just take note-taking to a new height. What about the tags and notebooks in Evernote\u00ae works? EleEditor preserves tags and notebooks. Better yet, the notes polished by EleEditor can be saved right back to Evernote in PDF format.\nEleEditor comes with a built-in web browser which allows you to capture website contents and bring them to your notes directly. EleEditor does it all for you.\nInheriting from its big brother, NoteLedge, EleEditor supports multimedia note-taking. That means you now can make unique notes with audio and video clips, save the notes as PDFs with multimedia files stored on Dropbox, and upload them to major cloud storage options embedded in EleEditor.\nStart the new way of note taking today.\n*EleEditor is neither sponsored nor affiliated with Evernote\u00ae. Evernote\u00aeis a trademark owned by Evernote Corporation, and EleEditor is only integrated with the Evernote\u00ae API. Kdan Mobile Software Ltd. does not represent Evernote Corporation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Find a London romance when you join the site created exclusively for single people living in or near the capital city. Most dating sites in the UK cover every town and every city. We're a site that's different. We know that this city is unique - living here is a totally different experience to living anywhere else. And now there's a dating service just for people like you - single Londoners who are seeking a compatible partner who also lives in their city.\nThe great thing about signing up to CupidNights is that you don't have to pay anything to do so. You should always post a picture of yourself as you'll receive up to seven times more responses if you bother to do so. And our members' profiles are very detailed; they contain not only physical characteristics, but also personality traits and lifestyle choices. Because of this, you'll be able to search for a new partner who you'll not only find attractive, but also compatible in most respects.\nTo find a London romance at CupidNights, be aware that the more effort you put in to creating your profile, the more success you're likely to have. Don't be modest. If you think you're an attractive girl or guy, then be sure to say so. And when you compose the description of yourself, make it sound upbeat and fun - humour is a real ice-breaker on a site like this so it's best not to sound too serious. Remember, the more information you give about yourself, the more likely you are to be contacted by someone who's right for you.\nOnce you've completed the two minute registration process, you can start searching our database for the type of people you would most like to make contact with. When you've found someone you like the look of, you can send them a message via our privately hosted email system which means your real email address is never revealed. And you can also chat and flirt in 'real time' with other members who are on-line at the same time as you. That's a great way of making instant contact with other singles and it' s a whole lot of fun.\nSo find a London romance at CupidNights and when you do, please get in touch and tell us all about it!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What will happen if I decide not to get my roof cleaned?\nThe integrity of your roof can be compromised by these roof staining organisms called gloeocapsa magma , which can actually attract animals to your home, and ultimately into your attic. Mold, mildew & algae can also penetrate the shingles allowing moisture to get into the sheathing & underneath the roofs surface. This moisture can cause wood to rot and the odor of rotting wood attracts critters and also creates a nesting place for small animals.\nNot only does having your roof cleaned give your house a polished look, but a thorough cleaning will also save you money in energy cost and will prevent long term interior roof damage while increasing the value of your home by bringing your roof back to a like-new appearance.\nHaving your roof cleaned byKlean Roof can definitely save you money and it's cheaper than purchasing a new roof.\nAn average 2000 square foot roof replacement can cost upwards of $11,000 - 17,000, while cleaning that same roof would cost less than $500!\nMany homeowners don't think they need to regularly clean their roof because their shingles comes with a warranty, but many shingle manufacturers actually suggest that you clean your roof.\nThe truth about those ugly black stains on your roof!\n1. What are the stains on my roof caused from?\nThe stains are a form of a mold spore that is prevalent in our humid climate. Once planted on your roof it grows and spreads until your entire roof is coated with black mold stains.\n2. Should I pressure wash my roof?\nPressure washing is not recommended to clean roofs due to the high psi water pressure that can dislodge shingle granules and pit the tile glaze. We use a gentle chemical rinse that once applied provides cleaning results that last three to five times longer than pressure washing.\n3. Should I have my roof cleaned in the winter or summer?\nWith our technique we are able to clean during any season and we work around the weather to provide optimum results. Our chemicals are safe but if there is a light rainfall after the cleaning we recommend that you rinse your landscaping underneath the roof line.\n4. Are my plants protected during the cleaning?\nWe have very little runoff during our cleaning procedure and all plants underneath the roof line are rinsed thoroughly to prevent any damage. If there is a light rainfall on the first day of cleaning after we leave, simply rinse off your plants underneath the roof line. But if the rainfall is heavy you don't have to worry about rinsing the plants.\n5. Will the cleaning take time before I notice it?\nWhile there are many different products available that will gradually remove the mold from your roof we have found that our method provides the best results. Hiring a roof cleaning professional is recommended since there are many things that can happen when you attempt to do it yourself. Our prices are reasonable and our training is more than squirting chemicals upon the roof. We are fully insured and provide our customers with the utmost of care when we clean their roofs plus the chemicals we use a much stronger than consumer cleaners.\n6. Does my roof need to be replaced?\nIf you are missing shingles, your shingles are more than 25 years old, there are no granules left on your shingles, tabs are flipping up or if the shingles are brittle, then yes, you likely need a roof replacement. However, just because your light colored roof is now streaked with solid black or has moss growing on it does not mean that you need a new roof. Rather, it most likely just needs cleaning.\n7. Can I pressure wash my roof?\nAnother likely source of damage is the pressure cleaning methods used to combat roof mold, mildew and\/or algae growth. Pressure washing and harsh chemicals can cause rapid granule loss or other shingle degradation. As is the case with algal treatment, the pressure washing removal methods for roof mold may cause more damage than the growth itself. But this doesn't mean you would just leave the algal growth. Take preventative measures to combat these growths since roof mold can be very destructive.\n8. Can my insurance company cancel my policy if I don't have my roof cleaned?\nInsurance companies are increasingly demanding that homeowners have their roofs cleaned and they often only give you a very short notice to do it. There has been cases where the homeowner was given less than a week to comply or lose their insurance coverage. That's why it is very important to be proactive and have those roofs professionally cleaned before you get that notice from your insurance company. Insurance companies know that untreated roof stains will negatively impact the overall integrity of the roof, which can also lead to mold growth INSIDE your home. Roof stains are seen as 'risks' that could cost them money in a claim.\n9. Can the roof mold affect my family's health?\nAbsolutely! Having mold, mildew, moss, or algae growing on your roof is more than just an unsightly nuisance that brings down the value of your property. Each can cause roof rot and lead to health problems for people living in the home or residential complex, or for workers in a commercial building.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjfru b/data_all_eng_slimpj/shuffled/split2/finalzzzbjfru
new file mode 100644
index 0000000000000000000000000000000000000000..87e8fa34a149fdb1c7fd320a74bef7e83648528b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbjfru
@@ -0,0 +1,5 @@
+{"text":"Northgard is simple in all the right ways, challenging not because of complexity but complacency. A slightly bland campaign can stop this being one of the best A \u2026... Short of giving Ikea a run for their money and flat-packing a Bournemouth beach hut, shipping it around the globe and sandwiching it (is that even a word) in an ungainly fashion to the front of the client's house, this is testament to how a small idea, tight budget, amazing family and super-cute collie, can ostensibly, make the equivalent come to life.\nOnce you create the float in the hut, open your Storage box and search for float using search box or choose the Other tab. It will be in that section. It will be in that \u2026... Northgard's concept seduced way more people than we anticipated at first, and you, as a group of players passionnate about Strategy games, pushed us to deliver the best game we could. We could not have dreamt of a more helpful and dedicated community and it's clear that Northgard could not have been as complete as it is today without you!\nIntroduction to Building Tips . Hi, and welcome to my building tips page. I'll have both general and specific or scenario related tips here. Read through it, and we're sure it will help at least a little bit with your experience in building.\n\"The Sanctuary\" \u2013 Fisherman's Hut Recently restored and fitted out as a fisherman's hut from the pioneering days of trout fishing in Tasmania, \"The Sanctuary\" can be seen on the Riverside Walk.\nOther \"worked\" buildings that generates food includes the Fields, Hunter's Lodge and the Fisherman's Hut. Take note that these buildings can only be built on territories that have the resources, e.g. Fertile Land for Fields, Deers for Hunter's Lodge, and Fish for Fisherman's Hut.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Besides the solid CPA professional and discipline knowledge, and the operational support, I get from [Raffa staff], they are truly part of my senior team, contributing ideas, strategy, and collegial decision-making that supports me in my job.\"\nRaffa's accounting professionals provide our nonprofit, business and individual clients the support needed to navigate today's financial and regulatory environments.\nWe work with our clients to determine the appropriate level of service. Some need to hand off this back office function via our Managed Accounting Services, others are looking for accounting support such as Audit Preparation, catch up account reconciliations, financial reporting and projections. We also provide the services of Chief Financial Officers (CFO), controllers and other accounting staff, and meet the accounting needs of Family Offices.\nClients often turn to Raffa with one Accounting function in mind and then realize the solutions they desire can be better achieved through improved technology and HR functionality. Through our Assessment and Implementation\u200b process, we work together to tailor a solution to fit each client's specific need.\n\"Working with Raffa allows us to, frankly, not worry about the bookkeeping, the accounting and a lot of HR stuff that is very important to supporting our mission\u2026A large part of our success is because of Raffa's help and partnership.\"\nRaffa has again been nationally recognized by Accounting Today for our breadth of quality services and deep bench of expertise across an array of services. The most recent report ranks Raffa 85th in the nation and 3rd in the National Capital Region.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shilo, a chocolate brown standard poodle moved into our home at 8 weeks of age, in November of 2006.\nAs a bonus, her sister Paris arrived with her as a foster dog. Paris would remain with us for two years and would then return to the breeder.\nThe two pups were great company for each other, whether it was playing or getting into mischief. As they reached 7 months of age, it became evident that some training was in order to curb all their energy. Our dog breeder referred us to Jeff and Sit Happens. The process was simple. One phone call, Jeff arrived at our home at our convenience, he did a quick demonstration in front of our home with us handling our dogs under his instructions, while his dog sat on the sidelines \u2013 off leash \u2013 under total control.\nHe explained his training methods, how park sessions worked and the concept that this training lasts for the life of the dog. How could a program like that NOT work? We signed on. Jeff came out and did a couple more sessions at our home.\nWe felt that we knew the basics. Early on we were learning that this whole process was weighted on the side of teaching us humans this new language for communicating with our dogs. When we do what we are supposed to, the dogs learn their part fairly easily. Our two pups progressed quickly with all the commands that we were shown. Because of that progress, the distance to park sessions from Langley and time constraints in general, we did not choose to attend the park sessions for over a year. Then Paris left us to return to the breeder.\nShilo was a bit lost. Her 24-7 pal was gone. We decided that park sessions might now be in order. We wanted to give Shilo the opportunity to experience the other dogs, new people, training with distractions \u2013 things to help her forget about her sister.\nWe should have been doing the park sessions from the beginning. Again, it's as much about what the humans learn there, as the dogs. Following commands with the extra distractions is great for the dogs. We got to practice in a new environment, learn new training routines, and learn about things that other dog owners were dealing with that we could take advantage of.\nWe are very happy with Shilo's progress so far. With previous dogs that we have raised, we always felt helpless when they decided not to listen and 'high-tailed-it' after someone or something. Now that does not happen \u2013 or at least when it does we can get Shilo back under command quickly. And because dogs develop new habits as time goes along we know we will always be able to get the advice we need down the road.\nThanks Sit Happens. You made Sit Happen for us!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home \u2014 Essential DJ-Team - We Hate To Rock! - The Remixes - Unsubmissive Records, Unsubmissive Records - USM-050-6R, USM 050-6R - 12\"\nEssential DJ-Team - We Hate To Rock! - The Remixes - Unsubmissive Records, Unsubmissive Records - USM-050-6R, USM 050-6R - 12\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An addition to our new and different species portfolio, Opah has four types of flesh \u2013 each a different color and used for various purposes. The top loin, served mainly as sashimi, is light salmon orange or rose pink in color. It is sweet and lean with a texture similar to Tuna. The lower loin is higher in natural oil, with a higher fat content than salmon. The tender meat behind the backbone is orange, and the meat near the belly flesh pales to pink. All of the flesh of the Opah has a rich, creamy taste, firm fatty texture and turns white when cooked. Our Opah is sustainably caught using long line gear in the temperate waters of the Indian Ocean.\nPORTION SIZES: 6 \u2013 8 oz. or 3 \u2013 8 lbs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Complimentary Champagne & Gourmet Cuisine!\nRegister to win door prizes!\nfor a complete overview of the products that I offer, included coffee table wedding books, vintage style bridal portrait on canvas, engagement sign in books etc. At the show, you will be able to view the quality and attention of the detail of my products.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Aquarius Solutions, Wholesale of chemical products, listed under \"Wholesale Of Chemical Products\" category, is located at PO Box 4 Peacehaven East Sussex , BN10 8RZ, United Kingdom and can be reached by (01273) 585828 phone number. Aquarius Solutions has currently 0 reviews.\nBrowse all Wholesale Of Chemical Products in Peacehaven East Sussex .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home > Aspen. Colorado No. 14.\nIngersoll traveled with the U.S.G.S. in Colorado in 1874.\nPhotographs mounted on cardboard leaves. Extensive manuscript annotations and many direct quotes from Ernest Ingersolls \"Crest of the Continent\" on rectos.\napprox. 12 x 19 cm. on mount 28 x 35 cm.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Owning a home is an American dream for many people. However, most people should wake up to the fact that owning a home is a serious financial and maintenance-intensive investment. The average cost of a new house can be as high as $400,000. Most homes offer about an average of 2,000 to 2,400 square footage. The average monthly mortgage payment is about $1,000.\nHowever, that does not take into account all home-related costs. Like food, insurance, septic tank maintenance, and all other home-related costs. Most homeowners pay double or triple the amount of their monthly mortgage payment to maintain their homes. What most people also don't take consider is that home ownership demands intense, physical maintenance.\nHomeowners must spend a lot time dealing with home system breakdowns and repairs. Or, hiring home repair professionals to enact such repairs at great expense. Yes, owning a home is an American dream. Still, it can also be a financial nightmare. I have heard about how many people are turning to tiny houses to cut down on home related financial costs.\nTiny houses are a fourth of the size of an average sized house. Or, even smaller. Most tiny houses offer anywhere from 100 to 800 square feet of space. Almost all tiny houses are mobile, can be hitched to trucks, and moved just about anywhere.\nYou can buy a pre-fabricated tiny house, or the materials to build one, for as little as $10,000. Or, you can get a personal loan, or a loan for a recreational vehicle, for up to $50,000. You can only get a mortgage for homes that are built on permanent foundations. So, you don't ever have to worry about mortgage payments with a tiny house. The standard of living payments for a tiny house owner are also significantly minimized.\nThe cost for utility bills could be $30, $40, or $50 a month. All combined tiny house standard of living costs could easily be hundred of dollars a month. And by hundreds of dollars, I mean in the low three-figures.\nThe headaches that come with tiny living is trading standard of living costs with maintenance costs. Most tiny houses come with wheels, or, are easy to move. You can put it on the property of friends and relatives, if possible If you buy land to situate a tiny house, you will still have to pay for all associated property and maintenance taxes. Tiny house living allows you to move whenever you want.\nSo, that means you will have to pay for RV registration costs. And, paying moving costs to settle into RV, trailer parks, or available public land.\nThere are numerous zoning laws that prohibit where a tiny house may be situated. There a lot of public land areas where tiny houses are just not allowed. Tiny houses are usually not allowed on land designated for houses with permanent foundations. Owners of tiny houses are free to go anywhere, but zoning laws and bureaucracy severely restricts such freedoms.\nTiny house living is preferable for single people or couples. You just won't have enough living space available if you are expecting a child. Or, if you have a large family. There won't be enough space for parties or get togethers with friends. Also, unless you are used to minimalist living, you won't have space for many material items.\nTrading Average-Sized Problems for a Miniature Version?\nTiny houses are much easier to clean than an average sized house. They are also less of a financial burden to deal with for a homeowner. While all of these things are true to some extent, that doesn't mean that owning a tiny house doesn't come with its own set of miniaturized burdens.\nMortgage Messes: Should You Clean Up with a Foreclosure, Short Sale or Deed in Lieu?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In 1948, Helen and Max Biderman immigrated to Dallas after surviving the Holocaust in Poland. The Biderman lives focused on family and continuing their Jewish traditions. Though luxuries were few, Helen's Shabbat dinners were always a feast. A typical dinner included chopped liver, matzo balls or kreplach, a veal roast (you could smell the garlic in the driveway), cholent, and beautifully arranged fruit on a crystal platter.\nSundays would frequently be spent eating fresh bagels, lox, corned beef, and white sh at Bagelsteins. There was never a lack of food with Helen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Iowa PGA Professional, Rob Randall of Legacy Golf Club was named Iowa's best teacher in Golf Digest earlier this year.\nClick here to view the feature video that ABC 5 News ran on Rob.\nClick here to view Golf Digest's full list.\nThe Mizuno Golf 36-Hole Classic is Monday, April 30 at TPC Deere Run in Silvis, IL.\nThe first round of the event will begin with a shotgun start at 8:30 a.m. Pairings will be re-done following the conclusion of round one. Second round pairings will be based off score.\nThe second round will begin with a shotgun starting around 1 p.m.\nThere will be an additional $20 Bonus Flight in round two for the bottom half of the field and ties.\nThe format for this event is 36-hole stroke play with the Open Division, Senior Division and Super Senior Division competing for the purse.\nClick here for pairings and general tournament information.\nThe Iowa PGA Junior and Pee Wee Spring Open presented by Principal Charity Classic takes place on April 28-29. The Junior age divisions will play two rounds over the weekend, while the Pee Wee divisions will play one round on Sunday.\nThis two-day event will take place at three different golf courses. Round one will be held at the Legacy Golf Course for all Junior competitors.\nThe second round will be held at the Tournament Club of Iowa for the Junior 18-hole divisions, and Jester Park Golf Course (Par 3) for the Junior 9-hole divisions.\nThe Junior boys 11 & under and the Junior girls 13 & under will play 9-hole stroke play each day. All other Junior age divisions will play 18-hole stroke play.\nAll Pee Wees will play a 9-hole stroke play format. They will follow the last Junior group on Sunday at Jester Park.\nAll groups will be teeing off of No.1, with the first group teeing off at 11:54 a.m. on Saturday. Pairings for Sunday will be released following the conclusion of the first round.\nThe Iowa Cup Matches is the next event on the Iowa PGA 2018 schedule. This Ryder Cup style format tournament will be held at Spirit Hollow Golf Course on May 2-3. This annual competition features the top Iowa PGA Professionals against the top Iowa Golf Association Amateurs. Click here for general tournament information.\nThe Brad Wuhs Pro-Pro and Education Forum is set to take place on Monday, May 7 at Hyperion Field. Registration for this event is already open and closes on Tuesday, May 1. Click here to register.\nThe Spirit Hollow Junior Classic will take place May 5-6 at Spirit Hollow Golf Course. Registration for this tournament opens Tuesday, May 1 and closes the same day at 11:59 p.m. There are 80 spots available for this event.\nThe Spring series concludes with the Iowa PGA Junior and Pee Wee Spring Series #2 at Westwood Golf Course on May 12. Registration for this event is open and will remain open until Tuesday, May 8. Click here to register.\nTo keep up with all the Iowa PGA's news and updates please visit IowaPGA.com or visit the all new IowaPGAJuniorGolf.com websites, and follow us on Facebook, Twitter and Instagram.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"commencing as soon as possible. Conditions of the appointment are in accordance with the Higher Education Act (Hochschulgesetz) of Schleswig-Holstein. Details will be explained on request.\nThe successful candidate will represent the respective field in research and teaching. The overall focus of the professorship lies on novel physics approaches in optics and their development to the point of application in the clinic or in life science laboratories. Starting from comprehensive basic research on fundamental phenomena, the activities should extend to technology development of biomedical instruments and their system integration. The goal is to strengthen research, development and translation in the fields of laser medicine, optical diagnostics and in-vivo optical imaging (or one of these fields). Support for successful translation of research and development results is provided through the collaboration with the Medical Laser Center L\u00fcbeck GmbH (MLL). We welcome the establishment of research collaborations and close links with other institutes of the science department and with the faculty of medicine.\nTeaching courses within the study programmes of Science and Engineering is expected. The candidate should be capable of teaching in German language.\nThe requirements for employment include a doctorate and additional scientific achievements, documented by a position as junior professor (Juniorprofessur), a habilitation (Habilitation) or equivalent scientific accomplishments, which can also have been achieved outside University.\nAs an equal opportunities employer, Universit\u00e4t zu L\u00fcbeck particularly encourages female scientists to apply, in case of equal qualification they will be prioritized. Equally qualified candidates with disabilities will also be given preference.\nApplications from abroad are especially welcome. Do not include ID or portrait photos in your application.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Downton Albuquerque, New Mexico - has been turned into the scavenged den of a group of survivors throwing the last party on Earth, as part of the production of \"Maze Runner: The Scorch Trials\" based on the bestselling YA book series by James Dashner.\nThe first \"The Maze Runner\" movie opened in the US on September 19th, 2014, as part of a global rollout that would see it clock up box office receipts in excess of $340 million. With barely a chance to stop and take a breath, its cast, including Dylan O'Brien, Kaya Scodelario, Ki Hong Lee and Thomas Brodie-Sangster, and its tireless director Wes Ball, have gone from press tour to production, charging ahead with plans to adapt all of James Dashner's bestselling series to the big screen.\nIndeed, anyone who has seen the first movie will know that the Gladers are no longer in the maze. Now they're out in The Scorch, the ruined Earth from which they were being hidden in The Maze Runner. But while the Maze Runners no longer have their maze, there's still plenty to run from.\nIn \"Maze Runner: The Scorch Trials,\" Thomas and his friends, life in The Scorch means a whole new set of challenges to face. For Dylan O'Brien, who returns to the role of Thomas, the biggest question for his character is whether he did the right thing by challenging the status quo and leading the Gladers out of the maze. Is freedom in The Scorch worth it? Were the Grievers that terrorised the Glade in the first film the least of their troubles?\nWith more information about The Scorch - and WCKD, the mysterious organisation that wiped their memories and put them in the maze to begin with - the Gladers will each start wondering about their place in the world. Out in the real world, the Gladers also meet a plethora of new characters \u2013 some friendly and some not so friendly \u2013 whose experience with The Scorch means they often have the upper hand.\n\"Maze Runner: The Scorch Trials\" opens nationwide in the Philippines on September 9 from 20th Century Fox to be distributed by Warner Bros.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Greeting guests and performance in evening costumes.\nVisit to a country town for their festival.\nParade and dancing. We have performed at this festival for several years now.\nFor Nepean District Historical Society. Our second year here.\nOpen Day performances at this historic site.\nDancing on the green. We have performed for several years.\nFor Nepean District Historical Society.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1.My cellphone \/ computer do not work.\n2.Light bulbs around me flashing.\n3.My umbrella(at the end of it) jumping\/ pulse, when it is raining.\n4. I can see spark on my cellphone.\nBesides, i have try telekinesis before by incident.\nI can transferred heat feeling remotely to another guy and i used this energy to heal a patient.\nElectrical telekinesis is a young man's game so you should make the most of it while you can.\nrely heavily on broadband electricity!\nYou sound a lot like a close friend of mine. Her husband calls her high maintenance. Cell phones, watches, computers, laptops, vacuum cleaners break all the time. She wears gloves when she uses electronics. She's a natural healer but didn't learn reiki. When she heals people she drains herself, which is not good.\nThis seems most common among teens and into the early twenties and maybe associated with all the bio-chemical changes in the body and field during that time period. Affecting Radio and TV (old style ones) signals is often common too.\nIt usually calms down as one enters their mid twenties, but not always, some people can never wear watches because their bio-field makes the watch consistently run too fast or too slow, others tend to 'break' delicate electronics.\nI suspect is has something to do with a person having an ungrounded field. Daily rooting or grounding practices might change a persons magnetic influence on electrical items and allow them to not influence them as much.\nIt can all begin again with the awakening of consciousness which once again super charges the bio-field and can unground a person's energy field.\nTo Berry: Your friend is using her own Chi ( life-force energy) to heal with - that's why she's energetically draining. She would benefit by learning to channel cosmic\/divine energy and use that energy to heal not her own life force energy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"V Garden Hotel is located at Jalan Pudu Ulu, Maluri and has been operated for more than 6 years. This hotel is strategically located 5 minutes away from the Maluri LRT Station which is well suited for leisure and business. One of the unique sought after features of V Garden Hotel is the Courtyard Caf\u00e9 where you can start your day with a sumptuous breakfast.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The classical way of accessing the current status of your hosts and services is by reading and parsing the file status.dat, which is created by Nagios on a regular basis. The update interval is configured via status_update_interval in nagios.cfg. A typical value is 10 seconds. If your installation is getting larger, you might have to increase this value in order to minimize CPU usage and disk IO. The nagios web interface uses status.dat for displaying its data.\nThe data is updated immediatley, not only every 10 or 20 seconds.\nApplications have easy access to the data via SQL. No parser for status.dat is needed.\nIn large installations the access for the addons to the data is faster then reading status.dat.\nIt has a complex setup.\nIt needs a (rapidly growing) database to be administered.\nIt eats up a significant portion of your CPU ressources, just in order to keep the database up-todate.\nRegular housekeeping of the database can hang you Nagios for minutes or even an hour once day.\nSince version 1.1.0, Check_MK offers a completely new approach for accessing status and also historic data: Livestatus. Just as NDO, Livestatus make use of the Nagios Event Broker API and loads a binary module into your Nagios process. But other then NDO, Livestatus does not actively write out data. Instead, it opens a socket by which data can be retrieved on demand.\nThe socket allows you to send a request for hosts, services or other pieces of data and get an immediate answer. The data is directly read from Nagios' internal data structures. Livestatus does not create its own copy of that data. Beginning from version 1.1.2 you are also be able retrieve historic data from the Nagios log files via Livestatus.\nOther then NDO, using Livestatus imposes no measurable burden on your CPU at all. Just when processing queries a very small amount of CPU is needed. But that will not even block Nagios.\nLivestatus produces zero disk IO when quering status data.\nAccessing the data is much faster then parsing status.dat or querying an SQL database.\nNo configuration is needed, No database is needed. No administration is neccessary.\nLivestatus scales fairly well to large installations, even beyond 50.000 services.\nLivestatus gives you access to Nagios-specific data not available any other available status access method \u2013 for example the information wether a host is currently in its notification period.\nOn the same time, Livestatus provides its own query language that is simple to understand, offers most of the flexibility of SQL and even more in some cases. It's protocol is fast, light-weight and does not need a binary client. You can even get access from the shell without any helper software.\nVi hoje seu post sobre o livestatus.\nEstou come\u00e7ando a us\u00e1-lo em substitui\u00e7\u00e3o ao NDO e est\u00e1 funcionando bem.\nAdministro um NOC em uma autarquia federal. Atualmente tenho 16 servidores nagios espalhados em 14 estados, com 934 hosts e mais de 2500 servi\u00e7os. Uso o nagvis para represent\u00e1-los em um mapa central e para aliment\u00e1-lo estava usando um banco de dados sql. Comecei a ter o problema do crescimento do banco, da dificuldade de gerenciamento, grande uso de CPU e mem\u00f3ria no servidor SQL. Outro problema era que o mapa central estava ficando cada vez mais pesado para ser atualizado devido a grande quantidade de queries que tem de ser feitas para atualizar o status geral. Ap\u00f3s ver no novo NAGVIS a op\u00e7\u00e3o de backend LIVESTATUS vi nele a solu\u00e7\u00e3o para esse problema. Por enquanto estou usando o livestatus em algums servidores para comparar o desempenho em rela\u00e7\u00e3o ao NDO. Deixo meu contato para conversarmos mais sobre o assunto e podermos compartilhar novas id\u00e9ias sobre isso.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Unique Specialized Attachments to make your Styling Life Easier!\n\u2022The Straightening Comb Attachment gives you Maximum Control to Smooth out Unruly Frizz and Wave. A Thermal Heat Transfer bar maintains the required tempature to create Smoother Sleek Hair.\n\u2022The Attachment Diffuser Offers the Ultimate in Curl Control, Both Attachment Tools are a must have for the Professional Stylist and to the Salon Guest who wants to look like they just stepped out of the Salon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"*Before you read this, allow me to introduce you to my dear friend Sarah. I met Sarah two years ago in Israel on Birthright. We shared ten of the most significant days of our lives, and then hardly spoke for two years. Not for a lack of fondness, just how our story unfolded. As road life took me out of Pittsburgh a few weeks ago, Sarah offered to host me in Charlottesville. Over the last three weeks, I have come home to one of the deepest bonds of my lifetime thus far. I have swum in Sarah's beauty and softness and power and wisdom. Sarah is so many things, to limit her identity to a singular expression would be unjust and unholy. However, one of Sarah's many, many gifts is her voice. As a classically trained Opera singer, something Sarah taught me just by sharing HER truth, was how we relate to ourselves, through our voice. I asked this precious spirit sister of mine to speak on this topic, and it is my immense honor to introduce you to the vast wellspring of love and glory that is Sarah Sherman.\nI have always thought of the voice affectionately as God's instrument. It started when I was little. I loved to sing, and was enchanted by being the apple of a creator's eye. I was thrilled that, of course, \"He\" made us in his image and likeness, so our voices were special, holy instruments of the highest genius. This instrument grew with me as I grew, taking on all of my questions and revelations and meditations, and perhaps most importantly it grew with me as my idea of God grew. And now my voice and I have arrived here together. I don't really like talking about God anymore, because I find the conversations to be extremely laden with human conditions and anthropomorphism. And so I only want to say briefly here that I do not talk about God as one of the known deities in any religion, or the Father, or the Bridegroom. I mean the mystic God, the God which IS and is All. Life force, Source, which has no beginning or end, no edges or boundaries. Who simply is who is. If God is All, then everything in the natural world is an expression of such. All of creation is divine expression, and as a part of creation, our voices begin to teach us about gentleness, being, and allowing.\nThe other day, as I was in a voice lesson, I was pushing my teacher for more and more information during warmups.\nImmediately the hand of conviction closed around my gut. Woof. My relentless analyzing was useless here. When she stopped me and said that, it got me good.\nThere is not much more convicting than someone telling you that you are trying too hard to have control- it really pulls you up short. And in that moment, I understood for the first time what the deeper and even more convicting implications are of the voice and its actual quality of being \"God's instrument\".\nOur job is to get out of the way. Let me expand on what I mean by that: We experience ourselves as separate entities, defined by our prefrontal cortex of judgment, analysis, personal identity, will, ego, narrative. And those things are not inherently bad, they are an integral, lovely, and essential part of what makes up the uniqueness of being human. But when it comes to life, most of us have a death grip on this part of our being and forget about the expression of ourselves which comes as part of the rest of creation- our bodies and spirits. Both our life and our voice are a combination of these things: Our minds\/will\/intentions, and the divine life force flowing through our bodies and spirits. We breathe in, engage our minds and emotions, have a desire or intention to express, and then exhale, using our bodies to do or communicate these things. Singing and the voice in general uses all of us; it's the marriage of all our qualities in the human condition. We are truly gorgeous, holy instruments, built to function perfectly, with our mind and ego as well our bodies and spirits.\nSo when I say \"our job is to get out of the way\", what I'm really saying is that we should learn the gentle the insatiable clutches of the mind\/ego part of ourselves and learn to more delicately, evenly balance that part of us with the wisdom and intuition of our bodies and spirits. Allowing the instrument to function means to honor every part of it, not just the parts that I can muscle through or control. It means the whole shebang.\nWhether we speak or sing, or are just going about our days, all we have to do is check in with our desires and connect to ourselves, then set an intention and allow the instrument to function (the instrument being our whole selves). And here is the convicting part- When was the last time you set an intention that was from your heart and your body's intuition, and then acted on it, letting the body and spirit take their courses, and a.) did not attach your worth and purpose to your desired outcome, b.) judge the outcome by what you perceived others to think about it or c.) try desperately to control what your journey looked like in the eyes of the ego? In my own experience, the process of action used to go a lot as such: An impulse > a self-check> a look around at who is in the room, and analysis of what you know about them and what you think they would think of the behavior> analyzing how much control I will be able to have over the outcome> a censor comprised of all that assumption and outside narrative woven into a web > behavior which has been filtered through all the aforementioned. What happens is no longer the direct result of our minds and hearts and embodied intuition. Our minds took over with their narrative, analysis, and fear, and thus the middle man of society and reptilian safety brain has stepped between our own highest self judgement and our actions. Our bodies and spirits are left abandoned as our minds have a field day with keeping us safe and accepted.\nNow, I'm not saying that our minds should not play a role at all, of course they should, but perhaps in a different capacity. At every turn we try to make our voices how we think they should be, in some way. How we think they should sound, whether or not we are allowed to express something, or if the expression coming out is \"appropriate\". We try to control it. We are often told that we should be and should sound a certain way, and so when we sing- or speak to express an idea, or live our lives for that matter- we try to make our expression appear as we have been told it should appear, and like we have heard or seen other people do it. But here is the problem with that\u2014who told us what sound we should be making or what our lives should look like? In those moments, where do those particular judgements come from? The material world! Not God or Source energy or Love. Not our bodies or spirits. We are trying to force ourselves into a form that was given to us by a superficial paradigm, structure, entity, which is the man-created world. We try to fit our voices into that tiny mold, rather than getting out of the instrument's way and letting it function as it was born and divinely created to function. As it was created by the life force energy, and as the intuition that shows up inside of us.\nThis way of singing is obviously easily translatable to how you show up in the world in the rest of your life. One of my students recently said to me : \"It's like you have your literal voice and then your soul voice---- which is just how you live your life. It's he expression of your life in the world.\" And she's exactly right. The lesson of allowing asks us to check a certain part of our being \u2013 the mind\/ego center- and tap into the trust that our bodies and spirits are divine expressions. Then we can use our minds as helpful technical tools. Our minds reeeally want to usurp the throne and call themselves king, but in that space they are cut off from the divine flow.\nWhen we realize that and can use them in their place among the wholeness of our entire being, we can allow the instrument to function. That's what it means to be an instrument of God, and it's a lot scarier and a lot more magnificent than my 9 year old self imagined. When we choose an intention based in authentic expression and connection to our highest selves, we can relax and allow the instrument to function without fear, judgements, or analyzing the living daylights out of it. Then, to be an instrument of God means to tune in to our bodies and spirits, check our minds and egos, and allow the whole instrument to exist in its fullest divinity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At the outset it was necessary to choose an appropriate place for start and further development of our innovative platform. Over a two-year period, we performed a deep analysis of all possible global markets, such as the USA, UAE, SAR, Australia, Canada, Russian Federation, Finland, China and many other.\nAustralia was chosen as an ideal platform to start. The first reason was Australian Corporate law, which opened up ample opportunities for capitalization of our technologies and establishment of a multinational Corporation. The second reason was the size and isolated location of Australia, representing a perfect place for realization of large infrastructural projects.\nAustralian territory is one of the most eco-friendly and clean places in the world. Australian authorities work diligently to preserve the continent pristine and safe for future generations. This was the crucial component why we decided to start our business in Australia and cultivate only green resource-saving technologies.\nADGEX LIMITED was incorporated on 29 of March 2012 as a Public Company in Australian Securities and Investment Commission.\nThe name of the Company \"ADGEX\" stands for Advanced Green EXpertise and supremely well personifies our business vision, philosophy, and clearly maps out the Company's activity areas.\nThis is the very period when extensive studies of major market players and Australian mining industry as one of the core economic fields of the country is being done. Elevated rail transport system UPRAIL is a highly-competitive and winning solution against traditional railroad, used in the industry as it features low OPEX and high potential to negotiate complex terrain. UPRAIL integrated all ADGEX's innovative technological solutions in the domains of independent energy efficiency, modular structure, material engineering and safety & automation systems.\nAlong with market research, we came to an understanding that each technological solution must represent a separate business division with own markets, peculiar development strategies and independent business models. By the end of 2013 ADGEX had taken the final decision: from now on each innovation must be evolved under individual scenario as each technology is unique and requires tailored approach in both financing and development of go-to-market strategy. During the period the Company set up the first R&D centres and own scientific school.\nIntellectual Property of ADGEX is the core for the future capitalization of the Company. All these years we were compiling a business portfolio of the company's major technologies. ADGEX acquired exclusive rights to know-how in the fields of mobile and renewable energetics, microelectronics, transportation, material engineering and modular machinery. Key roles, priority and development stages of each technology were carefully planned.\nEach technology was assessed to estimate future economic benefit, global market demand, the project's social significance and their integration into the countries where such technologies are primarily to be rolled out.\nAs the curiosity to the Company's business activities grows, foreign representation offices \"ADGEX INDIA PVT LTD\" in Hyderabad, India and \"ADGEX MOTORS LLC\" in Saint-Petersburg, Russian Federation are set up.\nThe Company initiated the stage of active investment raising and prioritized the first three technologies to develop: greenBLAZE, energyBRICK and VITIM.\nLaunch of these best-in-class technologies will allow the Company to rapidly enter the global markets and intensively grow working capital. This becomes viable as we provide a large variety of comprehensive solutions while introducing our unique infrastructural projects on dynamically rising markets of modern energetics and electronics. To have these projects launched, we started enticing the first investors and just 2 years later ADGEX numbered more than 1500 shareholders.\nRepresentation \"ADGEX CHINA Co LTD\" in Shanghai, China is established.\nThe Company is focused on development of integrated approach in solving of marketing and industrial problems. This is becoming the most prominent factor and unique advantage of the Company. Along with major projects, our designers start developing associated technologies capable of providing turnkey solutions for any infrastructural tasks.\nADGEX is turning into a high-end intellectual engineering company, focused on delivering readymade infrastructural solutions for large corporate clients and government authorities.\nWe produced and successfully tested commercial samples of major Company's projects. Our short-term strategy is establishment of high-end mass production line for our first three projects \u2013 VITIM, greenBLAZE and energyBRICK, ready made products of which are planned to be globally rolled out in 2018.\nWe are in tight cooperation with the world leading scientific organisations and universities, partaking in governmental programs of Australia and Russian Federation, and partnering with blue-chip companies in microelectronics, transportation, energetics and modular machinery fields.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblbgh b/data_all_eng_slimpj/shuffled/split2/finalzzzblbgh
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"Since 1988 this race has been the quintessential Canadian lake swim in the beautiful Coast Mountains. And fittingly, it is held on Canada Day, July 1. Sasamat Lake is a calm, clean and crystal clear lake, surrounded by spectacular tree lined mountains. A truly majestic Canadian locale.\nThe race itself, which will be celebrating its 30th anniversary in 2018, offers 3 distances (1km, 2km and 4km) and although there are typically over 300 swimmers of all abilities that usually attend, if you are new to open water swimming, this is a great place to start. Sassamat is a small lake, with typically very calm and warm (17-20C) water. The event offers swimmers the choice of wearing a wetsuit or not. If you are a more experienced swimmer, then this is a great venue to put up a fast time!\nRegardless of your speed or ability, this is a fantastic race and venue and well worth the 1\/2 drive to Port Moody from downtown Vancouver.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the early to mid-stages of dementia, lapses in memory and episodes of confusion may begin to affect the way in which a person communicates with their family and friends.\nFeeling frustrated, helpless, and unsure of themselves, they may choose to withdraw rather than risk an upsetting situation. When this happens, finding things to talk about can become difficult.\nCall to Mind is a unique game which helps care workers, family members and friends understand the thinking, likes and dislikes of someone with dementia, both as the game is played and in everyday interactions.\nThe inspiration behind the game came from sisters Angela Newton and Laura Templeton.\nMs Templeton said: \"My sister was an occupational therapist, specialising in working with older people. She adored the work but became aware of restricted provision for people diagnosed with dementia.\n\"In response, she set about designing a board game that would stimulate different parts of the brain to encourage interaction and improve communication for those living with the condition.\nDifficulties with language and communication occur in all forms of dementia but the particular problems experienced by a person vary according to the type of dementia they have.\nAccording to Alzheimer's Society, communication is vital \u2013 not only to maintain quality of life, but also to preserve a sense of identity.\nYet, communication can be frustrating for a person living with dementia, for their family and friends, and for care workers who may not always know the best way to respond. At times they may worry about saying the wrong thing, choosing instead to say nothing. This absence of conversation can have a negative impact on the well-being of the person in question.\nCall To Mind encourages those in early-to-mid stages of dementia to recall personal memories; sparking conversations with care workers, family members and friends across all generations.\nThe board game includes a spinner and colour-coded question cards to stimulate discussions across a variety of topics.\nOften, questions focus on areas that an older person may not have discussed for a while, such as a holiday they particularly enjoyed or where they grew up.\nCall To Mind specifically develops care workers' knowledge of their residents, helping them gain a deeper understanding of residents' past experiences, interests, likes and dislikes. In this way, the board game helps provide person-centred care, whereby residents' interests and preferences can be recorded on feedback sheets and kept with a care plan.\nActivities co-ordinator, Angela Dawson, enjoys using the game with the home's residents as it creates an opportunity to have fun. She said: \"Call To Mind helps us as activity co-ordinators to have a better understanding of the residents' life skills and interests \u2013what they are passionate about and what they dislike.\n\u2022 Offer opportunities to talk about what matters in someone's life.\nResearchers believe taking part in activities can offer a break from everyday care routines. One lady, whose mother currently lives with dementia, believes Call To Mind is one of many activities that can help care workers, family members and residents 'spend quality time together'.\n\"Call To Mind has been hugely successful in helping the carers get to know my mother, open conversation and build a relationship with her, something that she was reluctant to do otherwise.\nKathryn Smith, director of operations at Alzheimer's Society, added: \"Taking part in activities and board games such as Call to Mind, which people have told us has been effective for stimulating memories, can help a person with dementia feel less isolated, improve communication and help them express their feelings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is an opportunity for pupils who are taking the new Spanish final-exam GCSE in the next academic year(s) to practise their spontaneous speaking skills using a range of fun and innovative activities. There will be experienced teachers available to help with pronunciation and fluency, and the low adult to student ratio ensures students speak as much Spanish as possible during the day.\nThis day is focused on early preparation for the final exam GCSE and will cover all the main exam boards' criteria.\nThe teachers\/tutors were really helpful and the facilities were excellent.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2014 was full of big news items (Isis, Ebola, Ferguson, etc.) that received a lot of coverage for obvious reasons, but the suicides of Robin Williams and Brittany Maynard stand out in my mind as reflective of issues that perhaps aren't examined as much as they should be.\nLike many, I liked Robin Williams. I grew up with Mork and Mindy and thought he was a good actor and comedian. And I knew someone who had met and spent some time with Robin (through Robin's lifelong friend Jonathan Winters who frequented the Coffee Grinder, a coffee shop that existed for years in Carpinteria, California where I live.) And this person observed that it was impossible to truly connect with Robin in an authentic way, not due to any conceit or snobbery on Robin's part, but because Robin seemed to have a compulsive need to be always \"on\" and performing. When he told me this, I remembered an interview with Robin I had read. The interviewer had asked about his earliest memories of performing and Robin explained that his mother was the unhappy wife of an auto company executive and as her only child, he constantly entertained her to try to make her happy (and presumably get her attention and love.) Those two pieces of information always stuck with me over the years. So when I learned of his suicide, which occurred on August 11, 2014, I was very saddened, but it also came as no surprise to me.\nAfter his death, keeping in mind what I knew about Robin, I looked for news items that touched on or explored why this man, in spite of all of his obvious blessings in life, was so deeply despondent. There was no shortage of tributes to his career, accolades over what he achieved, his career highlights, and mentions of his philanthropic work. There was some speculation as to why he committed suicide. Possible reasons cited included: despondency over diagnosis of Parkinson's disease, diffuse Lewy body dementia, recent heart surgery, side-effects of medications, and depression (of medical origin.) Little else was mentioned of his obvious despair and unhappiness, or its possible roots in his life or past experiences, even though there were clues (i.e. the interview that I read.) Just a string of possible medical reasons were given.\nBrittany defended her \"personal choice\" and \"right\" and presented her argument in a convincing manner. But so did Maggie Karner, a woman who is diagnosed with the same brain cancer with the same prognosis, and is choosing not to end her own life. Maggie was also deeply troubled by Brittany's decision and expressed concern that this decision was in many ways just a reflection of our instant gratification culture. Maggie pointed out that increasingly in our culture, there seems to be a tendency to instantly try to medicate away or numb out anything negative. These days, anyone experiencing anything negative (whether a bad mood or a rough patch in life or anything in between) will often immediately try to numb it, with either a pill, entertainment, food, or addiction of some kind. She wisely observed that it is often through our negative experiences in life that we learn and grow into deeper, richer human beings.\nMaggie, along with other critics of Brittany's decision, have suggested that for people in Brittany's predicament, the emphasis should be on compassionate, dignified hospice care with good pain management. High quality, professional hospice care is readily available in many parts of this country, and where it isn't, the need for such care should be emphasized rather than an option such as assisted suicide. The modern hospice movement was started in 1967 by Dame Cicely Saunders, an Anglican nurse, who opened St. Christopher's Hospice in London. Saunders recognized the needs of dying patients and devised principles for alleviating suffering in the process of dying which include pain relief, dignified treatment, and compassionately honoring the psychological and spiritual aspects of death. And latest studies indicate hospice patients live longer than those not receiving hospice care.\nBut we don't like anything difficult, messy, complicated, and forget time-consuming. Our consumer-driven mindset wants convenience. More and more we seem to hold the view that all suffering is wrong. The gods of Convenience and Economics increasingly hold sway (with the never-spoken, but ever present sentiment\u2014it's too expensive and a drain on resources to let a really sick or disabled person live. Never mind the horrible inconvenience to the living who have busy lives.) Forget the messier, complicated aspects of life, the things that make us deep and strong in character. As Maggie pointed out, it is our difficult experiences that make us richer human beings. I have to say that it was through my struggles in life that I grew the most in character and in spirituality. I grew closer to God in my dark times, not necessarily when things were going well, like a bit of deep earth under intense pressure turning into a diamond\u2014that was when my character and faith grew strong. But denial of any type of negative experience is becoming more and more pervasive with a compulsive emphasis on entertainment, amusement, fun, comfort, diversion, and the avoidance of suffering at any cost.\nIn Robin's suicide, his reason was never explicitly stated, but there clearly seems to be that underlying sentiment, \"What's the point? It's easier to end it.\" His string of medical reasons may have contributed to the act\u2014psychiatric drugs in particular are notorious for side-effects that include suicidal thoughts and actions. But what was he trying to blot out or numb out with those drugs (along with the illegal drugs and alcohol he abused in the past)? What pain was he trying to numb out that he ultimately blotted out with his suicide? There was little talk of his depression other than it was a condition that needed to be medicated as if it were an ailment like high blood pressure. There was no talk of a depression that may have had roots in unresolved childhood grief and never mind the obvious fact of his deep human despair and spiritual emptiness. (Yes, a diagnosis of Parkinson's disease is unpleasant, but many don't commit suicide over it.) Medical reasons for depression seem to be always the preferred explanation. Forget the messy business of human emotion, human despair, human needs, the sloppiness of grief. Anything messy, dirty, inconvenient, including any uncomfortable truths from the past, sweep it under the rug. Get rid of it. Run to a doctor, get a prescription. Forget truth. Don't focus on or bring attention to Robin's deep human despair (which probably taps into our own feelings of despair.) Don't question, don't examine the truth of why he could be so unhappy, or ask why he had such a hole\u2014a deep sadness and a spiritual emptiness that was never filled or relieved with fame, riches, and adulation. That would point to a troubling truth we don't want to confront ourselves\u2014that much of what we strive for, the beliefs we run on are actually very shaky. Many of the things we are frantically chasing, all of our frantic performance to the point of exhaustion will never give us the love and security that we desperately crave (only God can), but rather than consider that possibility, we take Robin's human despair, slam it into the convenient little pigeon-hole of a medical diagnosis and quickly forget it with the thought, \"He didn't get the right pill when he needed it.\" Brittany spoke in glowing terms of her stockpile of pills\u2014she said having it gave her a great peace of mind knowing that she can end it at any time, in case things get too unpleasant or difficult.\nSadly, this is all presented now as the \"enlightened,\" even \"compassionate\" view. It's common to frame everything in terms of \"rights\" now\u2014pumping a fist in the air with a chant, \"If I want to die, that's my right! I don't want anyone making that choice for me!\" It's easy to get swept up in that language and perspective and not even consider that there is another way of looking at it. That this could be a sick reflection of our superficial, selfish, instant gratification culture of convenience that is increasingly indifferent and hostile to anything that poses trouble or difficulty. \"If it's not convenient for me, then I'm not going to deal with it, and that's my right!\" As if nothing could be gained through struggle. As if no one could learn anything through difficulty about the transiency of life and the strength of the spirit. As hospice founder Dame Saunders observed, as the body grows weaker, the spirit grows stronger. There is a strong cultural current that insists more and more on a purely materialistic view and a denial of spiritual reality. Material goals and entertainment take precedence over most other goals and if you can't numb your difficulties with entertainment, shopping, substance abuse, or work then grab a pharmaceutical from big pharma. As some would insist, we're just biological machines\u2014there is no spiritual basis to our existence, there is no God, and our best bet is to put our faith in \"experts\" playing God, in the name of good. Since that isn't foolproof, now death is presented as an ideal option. As if the love of Christ never even existed. When you comment that we live in a culture with strong antichrist tendencies, people laugh at you, but is that far from the truth?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Now you can complete the festive Christmas table look with fun or personalised silver-plated vintage serving spoons. Perfect if you're hosting this year or visiting family or friends.\nThis gift is presented within a gift box and tied with a personalised name tag. You can include a personal message card too, mounted to the box lid.\nPlease leave the 'name' for the personalised gift tag, the 'surname' for the spoon and a short 'personal message' within the notes section at checkout.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzboozs b/data_all_eng_slimpj/shuffled/split2/finalzzzboozs
new file mode 100644
index 0000000000000000000000000000000000000000..191abaafaa1babbd9bd8bd9d140b03aaf577c5cb
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzboozs
@@ -0,0 +1,5 @@
+{"text":"THIS TOYHAULER JUST CAME IN AT ONE MY STORES!!! ITS A 2012 FUZION 305 WITH 2 SLIDES AND SUPER CLEAN!!! IT HAS 4 DOOR FRIDGE WITH GENERATOR WITH ITS OWN GAS TANK AND PUMP!!!! IF YOU HAVE TOYS OR NEED TO SLEEP A FEW THIS IS A GREAT BUY FOR SOMEONE TO GET THE FAMILY OUT IN!!!! MY NAME IS BEN MY NUMBER IS !!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The best bitcoin wallets include Trezor, Electrum, Coinbase, Exodus, and Robinhood. This is based on user reviews and factors like fees and security.\nBitcoin (BTC) has dominated the cryptocurrency market for years as one of the first digital currencies. Despite a growing number of alternatives, Bitcoin still claims the largest market share and has become a household name since its peak in 2017. Today, over 23 million people own the open-source cryptocurrency.\nIf you're considering investing in Bitcoin, you'll need a place to store your investment. Purchasing Bitcoin requires more than fiat currency and a bank account. Consumers also need a \"wallet\" to store their Bitcoins, similar to paper money.\nUnlike traditional forms of fiat currency, like the dollar, yen, and euro, digital currency can't be physically stored in a traditional wallet, bank, or safe. Instead, you'll need to find a cryptocurrency wallet that stores the private keys required to access your bitcoin on the blockchain.\nThere are a variety of wallet types available. Online wallets allow you to store your information via a website, while desktop wallets let you keep your crypto cash on a mobile device or your desktop. Hardware wallets are physical devices that allow you to keep your bitcoin completely offline \u2014 also known as cold storage.\nSome people compare these hardware wallets to a USB drive for your bitcoin. There are even paper wallets, which are literally just pieces of paper with your private key printed on it. While this may be the cheapest way to achieve cold storage, we don't recommend it, since paper is easily lost or destroyed.\nThere are dozens of wallets to choose from: the Ledger Nano, Trezor, Mycelium, KeepKey, Jaxx Liberty, Electrum \u2014 and many more.\nAs one of the oldest bitcoin hardware wallets on the market, Trezor offers users the security of offline, cold storage. Like other hardware wallets, it stores your private key (a string of code) and does not require interaction with software on your computer. As such, hardware wallets are safe from viruses or online malicious activity that can result in theft.\nAs a hardware wallet, the physical device is compatible with both desktops and smartphones and can be accessed through a variety of operating systems, including Windows, MacOS, Android, and Linux.\nIn addition, Trezor also can work in tandem with software, mobile, or web wallets, adding an additional layer of security to your cryptocurrency or allowing you to limit the amount of cryptocurrency you store on other wallets.\nTrezor now supports over 500 cryptocurrencies coins and tokens, including Bitcoin, Bitcoin Cash, Ethereum, and Litecoin, making it an ideal option for those who intend to expand beyond Bitcoin.\nWhile Trezor has many benefits and is a top choice for many consumers who plan to regularly invest or make a significant investment, beginners may be deterred by the price tag. Unlike software, web, or mobile wallets, which are typically free to download, a Trezor wallet can cost between $80 and $150, depending on the model.\nToday, there are two primary models available to consumers, the Trezor One, which was the original model, and the Trezor Model T, which is a new model that offers additional bells and whistles, like a color touchscreen.\nElectrum is free, downloadable software that is stored on your computer. Though not yet available for mobile use, Electrum offers a fast, secure, and easy way to access and spend directly from your desktop.\nAs one of the earlier wallets, Electrum includes many of the features that have become standard in today's cryptocurrency community: in-wallet encryption and two-factor authentication, to name a few.\nAdditionally, the wallet also features hardware integration, which means it can seamlessly work with a hardware wallet like Trezor.\nLike other popular wallets, Electrum does charge a transaction fee, and users will pay 0.2 mBTC per transaction. Users should also note that Electrum can only be used to store Bitcoins. It is not compatible with other cryptocurrencies.\nAdditionally, though considered one of the most secure Bitcoin wallets, Electrum is not without potential downfalls. In December 2018, there was a fairly significant phishing attack that resulted in the cumulative loss of about 245 bitcoins, or $880,000 according to current prices. This, of course, can happen to many wallets, but if you're researching your options, it's likely you'll come across this infraction.\nCoinbase is a mobile wallet, which means you can access the app from any iOS or Android mobile device. As is the case with several Bitcoin wallets on the market today, the Coinbase wallet is designed and managed by a larger cryptocurrency company \u2014 Coinbase.\nAs such, Coinbase wallet users will find this wallet to be somewhat of a one-stop-shop as the company is both a broker and exchange, meaning users can buy, sell, trade, and store coins in one user-friendly platform. Further, Coinbase users can also connect their U.S. bank account, making it an easy one-stop-shop for bitcoin transactions.\nIn addition to managing your Bitcoin, a Coinbase wallet can also be used to store Ethereum, Bitcoin Cash, ZCash, as well as several other popular cryptocurrencies, making this wallet a reliable choice for those looking to expand beyond Bitcoin.\nWhile the wallet does boast security, longevity and is easy to use, it also comes with a fairly complex fee structure, which can potentially be a deal breaker for some users. Fees for buying and selling cryptocurrency range from $0.99 to $2.99.\nTransactions over $201 that are made using a bank account or your USD wallet are subject to a 1.49% fee, and transactions above $78.06 that are made using a credit card, debit card, or PayPal account are subject to a $3.99 fee.\nExodus is another open-source software wallet that can be used to manage Bitcoin, as well as a wealth of other cryptocurrencies, from your desktop. The wallet is compatible with Windows, Mac, and Linux, though it's not currently available for smartphones or mobile devices, a downside for those looking for on-the-go convenience.\nOne of the most attractive characteristics of Exodus is the easy-to-use interface and online resources, something that is valuable for beginners who are just entering into the cryptocurrency world.\nThe wallet includes useful charts, graphs, tutorials, and videos that are invaluable to beginners. However, it also has advanced features that make it ideal for regular traders and portfolio builders.\nMuch like other software wallets, Exodus is free to download and use, but users are charged a fee for each transaction. Exodus does not have a specific flat fee. Instead, fees are calculated based on the amount of computational effort used and the type of transaction being sent.\nRobinhood, a popular platform for free stock trades, started off as a stock trading app designed to make trading more approachable. However, in recent months, the company has branched out from their trading and investment focus to offer cryptocurrency services. For those who are looking to manage numerous personal finance needs in one place, Robinhood may be worth checking out.\nRobinhood, is available to users via web as well as through a mobile app, making it an attractive option for those who are looking for flexibility of use.\nWhere Robinhood really shines is its fee structure, which maintains a no-commission stock trading policy that currently extends to cryptocurrency exchanges as well. Essentially, you can buy and sell Bitcoin for free.\nThough the wallet certainly shows promise, especially based on the quick success of the new company, the cryptocurrency features are still in the early stages of development so they are not available to everyone. Residents of several states still do not have access to the cryptocurrency features, but they can sign up for email notifications on any changes.\nThough these represent some of the most popular Bitcoin wallets on the market, the rapidly growing interest in cryptocurrencies has resulted in the creation of numerous wallets, many of which offer a mix of useful features and security.\nSelecting the best Bitcoin wallet for your needs will require research that factors in your own preferences, like whether you want mobile access to your wallet, as well as a thorough review of the security features and transactional fees.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Eeeekkk\u2026 CRASH!\" And there goes your brand new red sports car, along with the stuff in your car trunk, and, well, your limbs\u2026.\nTo obtain oneness with Product Design, you must have all the other essential elements that encircle the main product as well.\nThere are quite a few advantages to using a travel agent whilst on holiday - but many of them aren't recognized.\nTo start from here, you must have so far : 1. Done research, surveys and studies. 2. Thought about Cost 3. Finished the Packaging 4. Got a final Product.\nThere are many reasons why a person may create a web page. Either for personal or professional needs, many people these days have a web site somewhere on the Internet.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Find Out If You're In Pain Now!\nOften, medical practices ask us how to get national press. We stress that features in your local publications are equally, if not more, important. However, we also know there's credibility no paid advertisement & that it can be gotten when it comes to national coverage.\nOne of the easiest ways for your medical practice to know which journalists are looking to be pitched is which stories are a FREE service.\n\"Help a Reporter Out,\" known as HARO, is a site that allows journalists, bloggers, and authors to connect with sources for specific stories. This could send the leads into your Inbox. This site is deadline driven and can work public relations magic for your practice.\nFor example, writers may be looking to speak to someone in your area and localize a national story with a celebrity angle on a new cosmetic treatment. They would do pieces for the Holiday on ways to stay healthy during a chaotic season. They could even do a piece on summer beauty tips which are also common favorites. There are 30-70 leads in each email and they are sent three times each work day. Since this is the case, there's no better way to get this many press leads sent straight to you.\nInterested to find out more? We're happy to share our knowledge!\nAbout Shorr Solutions: Shorr Solutions is a practice management consulting company with offices in South and Central Florida. We're helping physicians and medical practices strengthen the operational, financial and administrative health of their practice.\nWinners of THREE Aesthetic Everything Awards\u2026 SECOND Year In A Row!\nShorr Solutions Wins A \"TAG! You're IT!\" Aesthetic Award \u2013 A BIG WIN FOR OUR TEAM!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our reputable Smegg style knee-high boots with on a very comfortable Commando sole (1.2\"). Top half of the calf is open at the sides and back with three straps. Available in real leather and VEGAN options. Our VEGAN boots are made from 100% animal-free materials.","meta":{"redpajama_set_name":"RedPajamaC4"}}